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\subsection{Normality of total spaces}
\begin{lemma}
\label{lem:reducing_reduced_fibers}
In the situation of \autoref{notation:CM_semi_positivity}, there exists a finite morphism from a smooth projective curve $\tau :S \to T$ such that if $g : Y \to S$ is the normalized pullback of $f$ (so the normalization of the pullback) and $\pi : Y \to X$ the induced morphism, then $g$ has reduced fibers, and there is an effective $\mathbb{Q}$-divisor $\Gamma$ on $Y$ such that
\begin{enumerate}
\item \label{itm:reducing_reduced_fibers:canonical_bundle_formula} $\pi^* (K_{X/T} + \Delta) = K_{Y/S} + \Gamma$,
\item \label{itm:reducing_reduced_fibers:CM} $\lambda_g = \sigma^* \lambda$, where $\lambda_g$ is the CM line bundle for $g$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\tau$ be any finite cover such that at the closed points $t \in T$ over which the fiber $X_t$ is non reduced, the ramification order of $\tau$ is divisible by all the multiplicities of all the components of $\tau$. Then, $g$ will have reduced fibers, and \autoref{sec:base_change_relative_canonical_smooth_base}
implies the existence of $\Gamma$ (denoted by $\Delta_Z$ there). Finally, \autoref{prop:relative_canonical_base_change_normal}.\autoref{itm:relative_canonical_base_change:base_change} yields point \autoref{itm:reducing_reduced_fibers:canonical_bundle_formula}, and \autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:smooth_base} yields point \autoref{itm:reducing_reduced_fibers:CM}.
\end{proof}
\begin{lemma}
\label{lem:normal_stable_pullback_fiber_product}
If $f : X \to T$ is a surjective morphism from a normal variety to a smooth projective curves with reduced fibers, $m>0$ is an integer and $ \tau : S \to T$ is a finite morphism from another smooth curve, then
\begin{enumerate}
\item \label{itm:normal_stable_pullback_fiber_product:pullback} $X \times_T S$ is normal, and
\item \label{itm:normal_stable_pullback_fiber_product:fiber_product} $X^{(m)}$ is normal (see \autoref{sec:product_notation} for the product notation).
\end{enumerate}
\end{lemma}
\begin{proof}
First we note that $f$ is flat and hence so is $f^{(m)} : X^{(m)} \to T$ by induction on $m$ and the stability of flatness under base-change.
We know that a variety $Z$ is normal if and only if it is $S_2$ and $R_1$. In the particular case, when $Z$ maps to a smooth curve $U$ via a flat morphism $g$, then $Z$ is $S_2$ if and only if the general fibers of $g$ are $S_2$ and the special ones are $S_1$ (so without embedded points) \cite[6.3.1]{Grothendieck_Elements_de_geometrie_algebrique_IV_II} \cite[12.2.4.i]{Grothendieck_Elements_de_geometrie_algebrique_IV_III}, and it is $R_1$ if the general fibers are $R_1$ and the special ones are $R_0$ (so reduced) \cite[12.2.4.ii]{Grothendieck_Elements_de_geometrie_algebrique_IV_III}. It is immediate then that this characterization of $S_2$ and $R_1$ propagates both to fiber powers and to base-changes.
\end{proof}
\subsection{Semi-positivity engine}
\begin{proposition}
\label{prop:semi_positivity_engine_downstairs}
Let $f : (X, \Delta) \to T$ be a surjective morphism from a normal, projective pair to a smooth curve such that $(X_t,\Delta_t)$ is klt for general $t \in T$ (as $X_t$ is normal for $t \in T$ general, $\Delta$ is $\mathbb{Q}$-Cartier at the codimension $1$ points of $X_t$, and hence $\Delta_t$ makes sense), and let $L$ be a Cartier divisor on $X$ such that $L- K_{X/T} - \Delta$ is an $f$-ample and nef $\mathbb{Q}$-Cartier divisor. Then, $f_* \scr{O}_X(L)$ is a nef vector bundle.
\end{proposition}
\begin{proof}
According to \autoref{lem:reducing_reduced_fibers} we may assume that the fibers of $f$ are reduced. According to \cite[Lem 3.4]{Patakfalvi_Semi_positivity_in_positive_characteristics}, it is enough to prove that for all integers $m>0$, the following vector bundle is generated at a general $t \in T$ by global sections:
\begin{equation*}
\omega_T (2t) \otimes \bigotimes_{i=1}^m f_* \scr{O}_X(L) \cong
\underbrace{f^{(m)}_* \scr{O}_{X^{(m)}} \left( L^{(m)} + \left( f^{(m)} \right)^* K_T + 2 X^{(m)}_t \right)}_{\textrm{\cite[Lem 3.6]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}, and see \autoref{sec:product_notation} for the fiber product notation}}
.
\end{equation*}
For that it is enough to prove that the natural restriction homomorphism $
H^0\left( X^{(m)}, N \right) \to H^0\left( X_t^{(m)}, N_t \right)$
is surjective, where
\begin{equation*}
N:=L^{(m)} + \left( f^{(m)} \right)^* K_T + 2 X^{(m)}_t = K_{X^{(m)}} + \Delta^{(m)} + (L - K_{X/T} - \Delta)^{(m)} + 2 X_t^{(m)}.
\end{equation*}
We note here that according to \autoref{lem:normal_stable_pullback_fiber_product}, $X^{(m)}$ is normal. Furthermore, $K_{X^{(m)}} + \Delta^{(m)} = (K_{X/T} + \Delta)^{(m)} + \left( f^{(m)} \right)^* K_T$ is $\mathbb{Q}$-Cartier. We also note that the only generality property of $t$ that we use below is that $X_t$ is normal, $X_t \subsetneq \Supp \Delta_t$ and $(X_t, \Delta_t)$ is klt. Hence, at this point, we fix a $t$ with such properties.
Set $\scr{I}:=\scr{J}_{\left( X^{(m)}, \Delta^{(m)}\right)}$, where $\scr{J}$ denotes the multiplier ideal of the corresponding pair. Then for the above surjectivity the next diagram, the top row of which is exact, shows that it is enough to prove the vanishing of $H^1\left( X^{(m)}, \scr{I} \otimes \scr{O}_{X^{(m)}} \left( N - X_t^{(m)}\right)\right)$.
\begin{equation*}
\xymatrix@C=10pt@R=10pt{
H^0\left( X^{(m)}, \scr{I} \otimes \scr{O}_{X^{(m)}}\left( N \right)\right) \ar[r] \ar@{^(->}[d]
&
H^0\left( X^{(m)}_t, N|_{X^{(m)}_t}\right) \ar[r]
& H^1\left( X^{(m)}, \scr{I} \otimes \scr{O}_{X^{(m)}}\left( N - X^{(m)}_t\right)\right) \\
H^0\left( X^{(m)}, N \right) \ar[ur]
}
\end{equation*}
We note that here we used that $\left(X_t^{(m)}, \Delta_t^{(m)} \right)$ is klt by \autoref{lem:product}, and hence by inversion of adjunction \cite[Thm 5.50]{Kollar_Mori_Birational_geometry_of_algebraic_varieties} so does $\left(X^{(m)}, \Delta^{(m)} \right)$ in a neighborhood of $X_t^{(m)}$. This then implies that $\scr{I}$ is trivial in a neighborhood of $X_t^{(m)}$.
We conclude by noting that the above cohomology vanishing is given by Nadel-vanishing as
\begin{equation*}
N - X_t^{(m)} = K_{X^{(m)}} + \Delta^{(m)} + \underbrace{(L - K_{X/T} - \Delta)^{(m)} + X_t^{(m)}}_{\textrm{ample}}.
\end{equation*}
\end{proof}
\begin{corollary}
\label{cor:semi_positivity_engine_upstairs}
Let $f : (X, \Delta) \to T$ be a surjective morphism from a normal, projective pair to a smooth curve such that $(X_t,\Delta_t)$ is klt for some (or equivalently general) $t \in T$, and let $L$ be a $\mathbb{Q}$-Cartier divisor on $X$ such that
\begin{enumerate}
\item \label{itm:semi_positivity_engine_upstairs:sections_non_zero} $L_t$ is globally generated for $t \in T$ general,
\item $L - K_{X/T} - \Delta$ is an $f$-ample and nef $\mathbb{Q}$-Cartier divisor, and
\item there is a $\mathbb{Q}$-Cartier divisor $N$ on $T$ such that $L + f^* N $ is Cartier.
\end{enumerate}
Then $L $ is nef.
\end{corollary}
\begin{proof}
According to \autoref{lem:reducing_reduced_fibers} we may assume that the fibers of $f$ are reduced, and by further pullback (using \autoref{lem:normal_stable_pullback_fiber_product}.\autoref{itm:normal_stable_pullback_fiber_product:fiber_product}) we may also assume that $N$ is Cartier, whence $L$ is also Cartier. Then, we may apply \autoref{prop:semi_positivity_engine_downstairs} yielding that $f_* \scr{O}_X(L)$ is nef.
Then, our proof is concluded by the natural homomorphism $f^* f_* \scr{O}_X(L) \to \scr{O}_X(L)$, which is surjective over a non-empty open set of $T$ according to assumption \autoref{itm:semi_positivity_engine_upstairs:sections_non_zero} and the fact that cohomology and base change always holds over an opens set.
\end{proof}
\subsection{Moduli applications}
\begin{definition}
\label{def:functor}
Let $\mathfrak{X}$ be a class of $K$-semi-stable Fano varieties over $k$. A pseudo-functor (or equivalently category fibered in groupoid) $\scr{M}_{\mathfrak{X}}$ is called \emph{a moduli functor for $\mathfrak{X}$}, if for normal test scheme $T$ of finite type over $k$:
\begin{equation}
\label{eq:functor}
\scr{M}_{\scr{X}} (T) =
\left\{ \raisebox{16pt}{\xymatrix@R=15pt{ X \ar[d]_f \\ T }} \left| \parbox{26em}{
\begin{enumerate}
\item $f$ is a flat morphism,
\item $K_{X/T}$ is $\mathbb{Q}$-Cartier,
\item for each geometric point $\overline{t} \in T$, $X_{\overline{t}} \in \mathfrak{X}$.
\end{enumerate}} \right. \right\},
\end{equation}
with arrows being given by Cartesian diagrams.
\end{definition}
\begin{remark}
We did not require anything in \autoref{def:functor} for the value of $\scr{M}_{\mathfrak{X}}$ on non-normal test schemes to keep the definition open to future developments of the theory
\end{remark}
\begin{remark}
In higher dimensional moduli theory generally it is required also Koll\'ar's condition, that is, for all integer $m$, $\omega_{X/T}^{[m]}$ is supposed to be flat and compatible with base-change. Here, the latter compatibility precisely means that $\left( \omega_{X/T}^{[m]}\right)_S \cong \omega_{X_S/S}^{[m]}$ for every base change $S \to T$.
We did not require this condition in \autoref{def:functor} as it is known that over reduced bases in characteristic zero, this condition follows from $K_{X/T}$ being $\mathbb{Q}$-Cartier \cite[Thm 3.68]{Kollar_Families_of_varieties_of_general_type}
\end{remark}
\begin{definition}
\label{def:moduli}
Let $\mathfrak{X}$ be a class of $K$-semistable Fano varieties over $k$ and $\scr{M}_{\mathfrak{X}}$ a moduli functor for $\mathfrak{X}$ (as in \autoref{def:functor}). We say that a proper algebraic space $M$ is a \emph{moduli space for $\scr{X}$} with the uniform $K$-stable locus $M^u \subseteq M$ being open, if
\begin{enumerate}
\item $M^u \subseteq M$ is an open sub-algebraic space,
\item $\mu : \scr{M}_{\mathfrak{X}} \to M$ is universal among maps to algebraic spaces (that is, for any algebraic space $N$, composition with $\mu$ induces a bijection $\Hom\left(\scr{M}_{\mathfrak{X}},N\right)\cong \Hom(M,N)$),
\item $\mu(k)$ takes exactly the uniformly $K$-stable varieties from $\mathfrak{X}(k)$ to $M^u(k)$ ,
\item $\mu(k)$ is surjective and it is a bijection when restricted to the uniform $K$-stable varieties,
\item there is a generically finite, proper cover $\pi : Z \to M$ by a scheme, given by a family $f : X \to Z$ in $\scr{M}_{\scr{X}}(Z)$, such that
\begin{enumerate}
\item $\pi$ is finite over $M^u$, and that
\item some positive multiple of the CM line bundle $\lambda_{f}$ on $Z$ is numerically equivalent to a line bundle that descends to $M$.
\end{enumerate}
\end{enumerate}
\end{definition}
\begin{proof}[Proof of \autoref{cor:moduli}]
Consider the generically finite cover $\pi : Z \to M$ given by \autoref{def:moduli}. Also by the assumptions of \autoref{def:moduli}, there is an integer $r>0$ and a line bundle $\scr{L}$ on $M$ such that $\pi^* \scr{L} \equiv r \lambda_f$. Let $\rho : M' \to M$ be the normalization of $M$ and let $\tau : Z \to M'$ be the induced morphism. Set $\scr{N}:= \rho^* \scr{L}$. We note that $\rho^{-1}(M^u)$ is the normalization of $M^u$.
According to \cite[Thm 6.1]{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds} to prove that $\rho^{-1}(M^u)$ is quasi-projective, we have to show that $\scr{N}$ is nef and for all closed, irreducible subspaces $V \subseteq \rho^{-1}(M^u)$, $c_1(\scr{N})^{\dim V} >0$. However, these are immediate, as:
\begin{enumerate}
\item $\tau^* \scr{N} \equiv r \lambda_f$ is nef according to \autoref{thm:semi_positive_no_boundary}.\autoref{itm:semi_positive_no_boundary:nef}, hence so is $\scr{N}$.
\item $\tau^{-1}V$ is a closed subspace of $Z$ intersecting non-trivially $\pi^{-1}(M^u)$. Let $\iota : W \to Z$ be the resolution of a component of $\tau^{-1}V$ dominating $V$. By Nagata's theorem we may assume that $W$ is projective (as opposed to just proper). Then $\iota(W)$ has to intersect $\pi^{-1}(M^u)$, and as $\tau|_{\pi^{-1}(M^u)}$ is finite, $\dim W = \dim V$. By the assumption $\tau^* \scr{N} \equiv r \lambda_f$, it is enough to prove that $\lambda_f|_{\tau^{-1}V}^{\dim W}>0$. As $\lambda_f$ is nef, for this in turn it is enough to show that $\lambda_f|_{W}=\lambda_{f_W}$ is big, where the latter equality is given by \autoref{prop:CM_base_change}. However, $f_W$ is of maximal variation by the finiteness of $\tau|_{\pi^{-1}(M^u)}$. So, we are done by \autoref{thm:semi_positive_no_boundary}.\autoref{itm:semi_positive_no_boundary:big} applied to $f_W$.
\end{enumerate}
\end{proof}
\begin{proof}[Proof of \autoref{cor:bigger_open_set}]
The proof is a verbatim copy of the above proof of \autoref{cor:moduli} with two differences:
\begin{itemize}
\item One does not have to assume the existence of the moduli space $M_{\scr{X}}$, as its semi-normalization is known to exist by \cite[Thm 1.3]{Li_Wang_Xu_Degeneration} and \cite[Thm 1.1]{Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}. As we work on the normalization this is enough for our purposes because the normalization maps to the semi-normalization.
\item The argument eventually boils down to showing that if $g : Y \to V$ is a family of $K$-semi-stable klt Fanos of maximal variation such that the very general fiber is either smooth or uniformly $K$-stable, then $\lambda_g$ is big. In the uniformly $K$-stable case this is \autoref{cor:proper_base}, and in the smooth case this is \cite[Thm 1.1]{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds}.
\end{itemize}
\end{proof}
\subsection{Applications to anticanonical volumes}
\begin{proof}[Proof of \autoref{cor:bounding_volume}]
We have
\begin{multline*}
\vol(-K_X-\Delta)
=
\underbrace{(-K_X - \Delta)^{\dim X}}_{\textrm{$-K_X - \Delta$ is ample}}
= \left((-K_{X/\mathbb{P}^1} - \Delta) - f^* K_{\mathbb{P}^1}\right)^{\dim X}
\\ = (-K_{X/\mathbb{P}^1} - \Delta)^{\dim X} + (\dim X) 2 \vol (-K_F-\Delta_F)
= - \deg \lambda_{f, \Delta} + (\dim X) 2 \vol (-K_F-\Delta_F)
\\ \leq
\underbrace{(\dim X) 2 \vol (-K_F-\Delta_F)}_{\textrm{\autoref{thm:semi_positive_boundary}}}
\end{multline*}
For the second claim, if $F$ is smooth, we apply the bound on the volume of K-semi-stable Fano varieties obtained in \cite[Thm 1.1]{Fujita_Optimal_bounds_for_the_volumes_of_Kahler-Einstein_Fano_manifolds} to $F$; if $F$ is singular, we apply \cite[Thm 3]{Liu_The_volume_of_singular_Kahler-Einstein_Fano_varieties}.
\end{proof}
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\title{Positivity of the CM line bundle for families of K-stable klt Fanos}
\author{Giulio Codogni}
\address{
EPFL\\
SB MATHGEOM CAG \\
MA B3 615 (B\^atiment MA) \\
Station 8 \\
CH-1015 Lausanne}
\email{[email protected]}
\author{Zsolt Patakfalvi}
\address{
EPFL\\
SB MATHGEOM CAG \\
MA B3 635 (B\^atiment MA) \\
Station 8 \\
CH-1015 Lausanne}
\email{[email protected]}
\date{\today}
\DeclareMathOperator{\Td}{Td}
\begin{document}
\maketitle
\begin{abstract}
The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of polarized varieties, in particular on the base of families of klt Fano varieties (also called sometimes $\mathbb{Q}$-Fano varieties). It is conjectured that it yields a polarization on the conjectured moduli space of K-semi-stable klt Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with $K$-semi-stable/$K$-polystable fibers. We prove the necessary semi-positivity statements in the $K$-semi-stable situation,
and the necessary positivity statements in the uniform $K$-stable situation, including in both cases variants assuming $K$-stability only for very general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory.
We also present an application to the classification of Fano varieties. Furthermore, in the semi-positivity case we may allow log-Fano pairs.
\end{abstract}
\section{Introduction}
\input{introduction.tex}
\subsection{Outline of the proof}
\label{sec:outline}
\input{outline_proof.tex}
\subsection{Organization of the paper}
\input{organization.tex}
\subsection{Acknowledgements}
\input{acknowledgement.tex}
\section{Notation}
\label{sec:notation}
\input{notation.tex}
\section{The definition of the CM line bundle}
\label{sec:CM_definition}
\input{CM_definition.tex}
\section{The delta invariant and $K$-stability}
\label{sec:delta}
\input{delta_plus.tex}
\section{Growth of sections of vector bundles over curves}
\label{sec:growth}
\input{growth.tex}
\section{Ancillary statements}
\input{ancillary.tex}
\section{Semi-positivity }
\label{sec:semi_positivity}
\input{proof.tex}
\section{Bounding the nef threshold}
\label{sec:nefness_threshold}
\input{nef_threshold.tex}
\section{Positivity}
\label{sec:positivity}
\input{ positivity.tex}
\section{Applications}
\label{sec:applications}
\input{applications.tex}
\section{Examples}
\label{sec:examples}
\input{example.tex}
\subsection{Definitions}
Basis-type divisors and the delta invariant have been introduced by K. Fujita and Y. Odaka in \cite{FO}, see also \cite{BJ}; in this section we recall their definitions.
\begin{definition}\label{def:basis}
Assume we are in the following situation:
\begin{itemize}
\item $Z$ is a variety over $k$,
\item $L$ is a $\mathbb{Q}$-Cartier divisor on $Z$, and
\item $q>0$ is an integer for which $qL$ is Cartier
\end{itemize}
A divisor $D \in |L|_{\mathbb{Q}}$ is of \emph{$q$-basis type} if there are $D_i \in |qL| \quad (1 \leq i \leq h^0(X,qL))$, for which the corresponding $s_i \in H^0(Z,qL)$ form a $k$-basis of $H^0(Z, qL)$, and $D$ can be expressed as
$$
D=
\frac{1}{qh^0(Z,qL)}\sum_{i=1}^{h^0(Z,qL)} D_i.
$$
$D$ is of \emph{basis type} if it is of $q$-basis type for some integer $q>0$.
\end{definition}
Let $\Delta$ be a fixed effective $\mathbb{Q}$-divisor on $Z$ such that $(Z,\Delta)$ is a klt pair. Given a $\mathbb{Q}$-Cartier effective divisor $D$ on $Z$, we define its log canonical threeshold as
\begin{equation*}
\lct(Z,\Delta; D):=\sup \{t | (Z,\Delta+tD) \textrm{ is klt } \}.
\end{equation*}
Remark that since $(Z,\Delta)$ is klt, the above threshold is a positive number. Let us recall the definition of the $\alpha$ invariant.
\begin{definition}\label{def:alpha}
Let $(Z,\Delta)$ be a klt pair and let $L$ be an effective $\mathbb{Q}$-Cartier divisor on $Z$. The alpha invariant of $(Z,\Delta;L)$ is
\begin{equation*}
\alpha(Z,\Delta; L):=\inf_{D\in |L|_{\mathbb{Q}}}\lct(Z,\Delta;D).
\end{equation*}
We write $\alpha(Z,\Delta)$ for $\alpha(Z,\Delta;-K_Z-\Delta)$.
\end{definition}
The $\alpha$ invariant has been introduced by Tian in relation with the existence problem for K\"{a}hler-Einstein metrics. The delta invariant is a variation on the alpha invariant. The main difference is that in the case of $\alpha$ invariant one considers the log canonical threshold of all divisors in the $\mathbb{Q}$-linear system, while in the $\delta$ invariant is defined using only basis type divisors. In particular, while $\alpha (X) \geq \frac{\dim X}{\dim X + 1}$ only implies $K$-semi-stability \cite{Tian_alpha,Odaka_Sano}, $\delta(X) \geq 1$ happens to be equivalent to it \cite[Theorem B]{BJ}, see also \autoref{def:K_stable}. The delta invariant was introduced in \cite[Definition 0.2]{FO}. In \cite{BJ}, although it was also denoted by $\delta$, it is called the \emph{stability threshold}.
\begin{definition}\label{def:delta}
Let $(Z,\Delta)$ be a klt pair and let $L$ be a $\mathbb{Q}$-Cartier divisor on $Z$.
\begin{enumerate}
\item \label{itm:delta:delta_q} For every positive integer $q$ for which that $qL$ is Cartier and $h^0(Z,qL)>0$, the \emph{$q$-th delta invariant} of $L$ with respect to the pair $(Z,\Delta)$ is
$$
\delta_q(Z,\Delta;L):=\inf_{D \in |L|_{\mathbb{Q}} \textrm{ is of $q$-basis type}} \lct(Z,\Delta;D).
$$
According to \cite[Lem 8.8]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}, this infimum is in fact a minimum.
\item \label{itm:delta:delta} Assume that $L$ is big, and fix an integer $s>0$ such that $sL$ is Cartier and $h^0(Z, sL)>0$, which conditions then also hold for every positive multiple of $s$.
The \emph{delta invariant} of $L$ with respect to $(Z,\Delta)$ i
$$
\delta(Z,\Delta;L):=\limsup_{q\to \infty}\delta_{sq}(Z,\Delta;L).
$$
According to \autoref{lem:scaling_delta_invariant}, the above definition does not depend on the choice of $s$, and the limsup is in fact a limit.
\item
If $(Z,\Delta)$ is a klt Fano pair, we let $\delta_q(Z,\Delta):=\delta_q(Z,\Delta;-K_Z-\Delta)$ and
$\delta(Z,\Delta):=\delta(Z,\Delta;-K_Z-\Delta)$.
\end{enumerate}
\end{definition}
\subsection{Relation to K-stability}
In this section we follow closely \cite{BJ}, as we want to adapt some of their result from Fano varieties over $\mathbb{C}$ to Fano pairs over $k$. Similar adaptation was done also in \cite{Blum_Thesis}. Consider the situation:
\begin{notation}
$(Z,\Delta)$ is a klt pair, $L$ is a $\mathbb{Q}$-Cartier divisor on $Z$, and $s>0$ is an integer such that $sL$ is Cartier and $h^0(Z,sL)\neq 0$.
\end{notation}
Let $v$ be a non-trivial divisorial valuation on $Z$ associated to a prime divisor $E$ over $Z$, we consider the filtration
\begin{equation*}
F_iH^0(Z,qsL):=\{ t \in H^0(Z,qsL) |\textrm{ such that } v(t)\geq i\}=\underbrace{H^0(V,qs\pi^*L-iE)}_{\parbox{80pt}{\tiny $\pi : V \to Z$ is a normal model where $E$ lives}},
\end{equation*}
and the invariant
\begin{multline*}
S_{q}(v):=\frac{1}{qsh^0(Z,qsL)}\sum_i i \dim_k\left(F_iH^0(Z,qsL)/F_{i+1}H^0(Z,qsL)\right)
\\ =\frac{1}{qsh^0(Z,qsL)}\sum_{i \geq 1} \dim_k F_iH^0(Z,qsL).
\end{multline*}
Denote by $B_q$ the set of $qs$-basis type divisors with respect to $qsL$. As observed for instance in \cite[proof of Lemma 2.2]{FO},
\begin{equation}
\label{com_basis}
S_q(v)=\max_{D\in B_q}v(D),
\end{equation}
and the maximum is attained exactly for bases adapted to the filtration $F_i$. When $L$ is big, the asymptotic of $S_q$ is well-understood, see for instance \cite[proof of Theorem 1.3]{FO}, \cite[Corollary 2.12]{BJ} and \cite[Corollary 3.2]{Boucksom_Uniform}:
\begin{equation}
\label{asymp}
S(v):=\lim_{q\to \infty}S_q(v)=\frac{1}{\Vol(L)}\int_0^{+\infty}\Vol(\pi^* L-xE)dx
\end{equation}
The next statement is a logarithmic version of \cite[Theorem 4.4]{BJ}, following very closely the arguments given there.
\begin{theorem}
\label{thm:BJ}
\begin{enumerate}
\item \label{itm:BJ_limit}
If $L$ is a big $\mathbb{Q}$-Cartier divisor, such that $sL$ is a Cartier divisor and $h^0(Z, sL) \neq 0$, then
the sequence $\delta_{qs}(Z,\Delta;L)$ converges to $\delta(Z,\Delta;L)$, i.e. the delta invariant is a limit and not only a limsup; moreover
\begin{equation*}
\delta(Z,\Delta;L)=\inf_v\frac{A(v)}{S(v)},
\end{equation*}
where $A(v)$ is the log-discrepancy of $v$ with respect to the klt pair $(Z,\Delta)$, and the inf is taken over all non-trivial divisorial valuations. In particular, $\delta(Z, \Delta;L)$ is independent of the choice of $s$.
\item \label{itm:BJ_alpha}
Assuming furthermore that $L$ is ample, the following bounds hold
\begin{equation*}
\frac{\dim Z+1}{\dim Z }\alpha(Z,\Delta;L)\leq \delta(Z,\Delta;L)\leq \left(\dim Z+1\right)\alpha(Z,\Delta;L).
\end{equation*}
\end{enumerate}
\end{theorem}
\begin{proof}
{\scshape Point \autoref{itm:BJ_limit}.}
Set $\delta_q:=\delta_{qs}(Z,\Delta;L)$ and $\delta:=\delta(Z,\Delta;L)$. We first prove the inequality
\begin{equation}
\label{sup}
\limsup_{q\to \infty} \delta_q\leq \inf_v\frac{A(v)}{S(v)}
\end{equation}
Thanks to Equations \autoref{com_basis} and \autoref{asymp}, we can write
\begin{equation*}
\inf_v\frac{A(v)}{S(v)}=
\underbrace{\inf_v\lim_{q\to \infty}\inf_{D\in B_q}\frac{A(v)}{v(D)}\geq \limsup_{q\to \infty}\left(\inf_{D\in B_q}\inf_v\frac{A(v)}{v(D)}\right)}_{\parbox{200pt}{\tiny $\forall v' : \displaystyle\inf_{D\in B_q} \frac{A(v')}{v'(D)} \geq \displaystyle\inf_{D\in B_q}\displaystyle\inf_v\frac{A(v)}{v(D)}$ \\ $\Rightarrow \forall v' \displaystyle\lim_{q\to \infty}\displaystyle\inf_{D\in B_q}\frac{A(v')}{v'(D)}\geq \displaystyle\limsup_{q\to \infty}\left(\displaystyle\inf_{D\in B_q}\displaystyle\inf_v\frac{A(v)}{v(D)}\right)$, and then take $\displaystyle\inf_{v'}( \_ )$ on the left side}}
=\underbrace{\limsup_{q\to \infty} \delta_q.}_{\inf_v\frac{A(v)}{v(D)}=\lct(Z,\Delta; D)}
\end{equation*}
We now prove the inequality
\begin{equation}
\label{inf}
\liminf_{q\to \infty} \delta_q\geq \inf_v\frac{A(v)}{S(v)}
\end{equation}
This follows from the key uniform convergence result \cite[Corollary 3.6]{BJ}: for every $\varepsilon>0$ there exists a $q_0=q_0(\varepsilon)$ such that for all $q>q_0$ and all divisorial valuations $v$ we have
\begin{equation*}
(1+\varepsilon)S(v)\geq S_q(v)
\end{equation*}
(The quoted result is above the complex numbers, however its proof works also on $k$.) We thus have for $q$ big enough
\begin{equation*}
\frac{1}{1+ \varepsilon}\inf_v\frac{A(v)}{S(v)}\leq \inf_v\frac{A(v)}{S_q(v)}=\inf_v\inf_{D\in B_q}\frac{A(v)}{v(D)}\underbrace{=\delta_q}_{\inf_v\frac{A(v)}{v(D)}=\lct(Z,\Delta; D)}
\end{equation*}
taking the liminf on $q$ on the right hand side, and then letting $\varepsilon$ go to zero, we get the requested inequality. We obtain point \autoref{itm:BJ_limit} combining Equations \ref{sup} and \ref{inf}.
{\scshape Point \autoref{itm:BJ_alpha}.}
Given a divisorial valuation $v$, we define its $q$-th pseudo-effective threshold as
\begin{equation*}
T_q(v):=\max \left\{ \left. \frac{v(D)}{qs} \; \right| \; D\in |qsL|\right\}
\end{equation*}
and we have
\begin{equation*}
\alpha(Z,\Delta;L)
\inf_q \inf_v \frac{A(v)}{T_q(v)}.
\end{equation*}
When $L$ is ample, \cite[Prop. 3.11]{BJ} gives the following bounds
\begin{equation*}
\frac{\dim(Z)}{\dim(Z)+1}\inf_q T_q(v)\geq S(v)\geq \left(\frac{1}{\dim(Z)+1}\right)\inf_{q}T_q(v),
\end{equation*}
which imply point \autoref{itm:BJ_alpha} (again, the proof in \cite{BJ} is over the complex numbers, but it works also over $k$).
\end{proof}
\begin{corollary}[{\scshape Invariance of the delta invariant by scaling}]
\label{lem:scaling_delta_invariant} In the situation of \autoref{def:delta}.\autoref{itm:delta:delta}, for every positive integer $r>0$, $ \delta(Z,\Delta;L)=r \delta(Z,\Delta;rL)$. Equivalently,
\begin{equation}
\label{eq:delta_invariant_scaling:statement}
\limsup_{q\to \infty}\delta_{rsq}(Z,\Delta;L)= \limsup_{q\to \infty}\delta_{sq}(Z,\Delta;L).
\end{equation}\end{corollary}
\begin{proof}
By \autoref{thm:BJ}, the limsup appearing in Equation \autoref{eq:delta_invariant_scaling:statement} is a limit, so the claim.
\end{proof}
We give the following definition of K-stability, which is equivalent to the more classical one by \cite[Theorem 6.1 (ii)]{Odaka_Sun} and \cite[Theorem 1.5]{Fujita_Uniform_K-stability_and_plt_blowups_of_log_Fano_pairs}.
\begin{definition}
\label{def:K_stability}
A normal Fano pair $(Z,\Delta)$ is
\begin{enumerate}
\item \emph{$K$-semi-stable} if it is klt and for every divisorial valuation $v$, one has $A(v)\geq S(v)$;
\item \emph{uniformly $K$-stable} if it is klt and there exists a positive constant $\varepsilon$ such that for every divisorial valuation $v$, one has $A(v)\geq(1 +\varepsilon) S(v)$.
\end{enumerate}
Here $A(v)$ denotes the log-discrepancy with respect to the pair $(Z, \Delta)$.
\end{definition}
The following corollary is now an immediate consequence of the above definition and \autoref{thm:BJ}
\begin{corollary}[{\scshape Characterization of K-stability}]
\label{def:K_stable}
Let $(Z,\Delta)$ be a normal Fano pair.
Then, $(Z,\Delta)$ is
\begin{enumerate}
\item $K$-semi-stable if and only if $(Z,\Delta)$ is klt and $\delta(Z,\Delta) \geq 1$,
\item uniformly $K$-stable if and only if $(Z,\Delta)$ is klt and $\delta(Z,\Delta) > 1$.
\end{enumerate}
Moreover, if $(Z,\Delta)$ is klt and $\alpha(Z,\Delta)\geq\frac{\dim(Z)}{\dim(Z)+1} $ (resp. $>\frac{\dim(Z)}{\dim(Z)+1}$), then $(Z,\Delta)$ is K-semi-stable (resp. uniformly K-stable); if $(Z,\Delta)$ is klt and $\alpha(Z,\Delta)\leq \frac{1}{\dim(Z)+1}$ (resp. $< \frac{1}{\dim(Z)+1}$), then $(Z,\Delta)$ is not uniformly K-stable (resp. not K-semi-stable).
\end{corollary}
\subsection{Products}
The following conjecture is motivated by the equivalence between K-stability and K\"{a}hler-Einstein metrics in the Fano setting, it has been already proposed in \cite[Conjecture 1.11]{PW}.
\begin{conjecture}\label{conj:delta}
Given two klt Fano pairs $(W,\Delta_W)$ and $(Z,\Delta_Z)$, one has
$$
\delta(W\times Z,\Delta_W\boxtimes \Delta_Z)=\min\{\delta(W,\Delta_W) \, , \, \delta(Z,\Delta_Z)\}
$$
\end{conjecture}
The analogue result for the alpha invariant and any polarization was written down for example in \cite[Proposition 8.11]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}, but used to be present much earlier in a smaller generality for example in Viehweg's works. See also \cite[Thm. 1.10]{PW} and \cite[Lemma 2.29]{Chelstov_log} for the Fano case. We can prove a weaker result for the delta invariant in \autoref{prop:prod_basis}, before which we need a definition and a lemma.
\begin{definition}[Product basis type divisor]\label{def:prod_basis}
Let $(W,\Delta_W)$ and $(Z,\Delta_Z)$ be two klt pairs, let $L_W$ and $L_Z$ $\mathbb{Q}$-Cartier divisors on $W$ and $Z$, respectively, and let $q>0$ be an integer such that both $qL_W$ and $q L_Z$ are Cartier. A divisor $D$ on $W\times Z$ is of $q$-\emph{product basis type} if there exist $q$-basis type divisors $D_W$ on $W$ and $D_Z$ on $Z$ such that
$$
D=p_W^*D_W+p_Z^*D_Z
$$
where $p_W$ and $p_Z$ are the projections.
\end{definition}
\begin{remark}
\label{rem:prod_basis}
In \autoref{def:prod_basis}, if $D_W$ is associated to a basis $s_i$ and $D_Z$ to a basis $t_i$, then $D$ is associated to the basis $s_i\boxtimes t_j$.
\end{remark}
\begin{lemma}
\label{lem:product}
Let $(W,\Delta_W)$ and $(Z,\Delta_Z)$ be two klt (resp. lc) pairs, then also $(W\times Z,\Delta_W \boxtimes \Delta_Z)$ is klt (resp. lc).
\end{lemma}
\begin{proof}
As we work in characteristic zero, we may take the product of a log resolution of $(W,\Delta_W )$ and of $(Z, \Delta_Z )$. This will be a log-resolution for $(W\times Z,\Delta_W\boxtimes \Delta_Z)$, with the union of the discrepancies of the original two log-resolutions, so the claim.
\end{proof}
\begin{proposition}\label{prop:prod_basis}
With the notations of \autoref{def:prod_basis}, let $D$ be a $q$-product basis type divisor. Then,
$$
\lct(W\times Z, \Delta_W \boxtimes \Delta_Z,D)\geq\min\{ \delta_q(W,\Delta_W;L_W)\; , \; \delta_q(Z,\Delta_Z; L_Z)\}
$$
\end{proposition}
\begin{proof}
Take $t<\min\{ \delta_q(W,\Delta_W; L_W),\delta_q(Z,\Delta_Z;L_Z)\}$. We have to show that $(W\times Z,\Delta_W\boxtimes \Delta_Z+tD)$ is log canonical. Recall that
\begin{equation*}
(W\times Z,\Delta_W\boxtimes \Delta_Z+tD)=(W\times Z,\left(\Delta_W+tD_W\right)\boxtimes\left( \Delta_Z+tD_Z\right))
\end{equation*}
and both $(W,\Delta_W+tD_W)$ and $(Z,\Delta_Z+tD_Z)$ are log canonical because of the hypothesis on $t$, so the claim follows from \autoref{lem:product}
\end{proof}
\subsection{Behavior in families}
Here we prove that the $\delta$-invariant is constant on very general geometric points.
Recall that a \emph{geometric point} of $T$ is a map from the spectrum of an algebraically closed field to $T$. Key examples are the closed points and the geometric generic point (i.e. the algebraic closure of the function fields) of $T$.
\begin{proposition}
\label{prop:delta_general_fiber}
If $f : (X,\Delta) \to T$ is a flat, projective family of normal pairs over a normal variety (that is $\Supp \Delta$ does not contain any fiber and $K_{X/T}$ is $\mathbb{Q}$-Cartier), and $L$ is an $f$-ample $\mathbb{Q}$-Cartier divisor on $X$, then there is a very general value of $\delta\left(X_{\overline{t}}, \Delta_{\overline{t}} ; L_{\overline{t}}\right)$. More precisely, there is a real number $d\geq 0$ and there are countably many Zariski closed subsets $T_i \subseteq T$ such that for any geometric point $\overline{t} \in T \setminus \left( \bigcup_i T_i \right)$, $\delta\left(X_{\overline{t}}, \Delta_{\overline{t}}; L_{\overline{t}} \right)=d$.
\end{proposition}
\begin{proof}
We may fix an integer $s>0$ such that $sL$ is Cartier and $f_* \scr{O}_X(qsL)$ is non-empty and commutes with base-change for any integer $q>0$. In particular, then for all $t \in T$, $sL_t$ is Cartier and $h^0(X_t, qsL_t)$ is positive and independent of $t$ for any integer $q>0$.
We prove the statement that \emph{for each integer $q>0$ there is a real number $d >0$ and a non-empty Zariski open set $U \subseteq T$ such that for each geometric point $\overline{t} \in U$, $\delta_{qs}\left(X_{\overline{t}},\Delta_{\overline{t}}; L_{\overline{t}} \right)=d$.}
Setting $T_q:= T \setminus U$ implies then the statement of the proposition. So, we fix an integer $q>0$, and in the rest of the proof we show the above statement in italics. We also set $r:=h^0(X_t, qsL_t)$ and $l:=qsr$, where the former is independent of $t \in T$ by the above choice of $s$.
Set $W:=\mathbb{P}((f_* \scr{O}_X(qsL))^*)$. Then, for any geometric point $\overline{t} \in T$ we have natural bijections:
\begin{equation}
\label{eq:delta_general_fiber:one_div}
\fbox{\parbox{125pt}{$k\left(\overline{t}\right)$-rational points of $
W_{\overline{t}}$}}
\leftrightarrow
\fbox{\parbox{130pt}{lines through the origin in $H^0\left(X_K,\scr{O}_X(qsL)|_{X_{\overline{t}}} \right)$}}
\leftrightarrow
\fbox{\parbox{100pt}{$D \in \left|qsL|_{X_{\overline{t}}}\right|$ } }
\end{equation}
We consider the open subset
\begin{equation*}
Y \subseteq \underbrace{ W \times_T W \times_T \cdots \times_T W}_{r \textrm{ times}}
\end{equation*}
corresponding to linearly independent lines. That is, for any geometric point $\overline{t} \in T$, using \autoref{eq:delta_general_fiber:one_div}, we have a natural bijection
\begin{equation}
\label{eq:delta_general_fiber:basis_type_div}
\fbox{\parbox{120pt}{$k\left(\overline{t}\right)$-rational points of $
Y_{\overline{t}}$}}
\leftrightarrow
\fbox{\parbox{230pt}{$(D_i)=(D_1,\dots,D_r)$ is a basis of $\left|qsL|_{X_{\overline{t}}}\right|$ } }
\end{equation}
Denote by $\overline{y}_{(D_i)}$ the geometric point of $Y$ corresponding to $(D_i)$ via the correspondence \autoref{eq:delta_general_fiber:basis_type_div}, where $D_i\in \left|\left.qsL\right|_{X_{k\left(\overline{y}\right)}} \right|$.
Consider the universal family of $q$-basis type divisors
\begin{equation*}
g : (Z := X \times_T Y, \Delta':=\Delta_Y ; \Gamma ) \to Y
\end{equation*}
such that for any geometric point $\overline{y}:=\overline{y}_{(D_i)} \in Y$,
$\Gamma_{\overline{y}} = \sum_{i=1}^{r} \frac{D_i}{l}$.
Denote by $\pi : Y \to T$ the natural projection.
According to \cite[Lem 8.8]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} , the log canonical threshold functin $\overline{y} \mapsto \lct \left( \Gamma_{\overline{y}};Z_{\overline{y}}, \Delta'_{\overline{y}} \right)$ is lower semi-continuous. Furthermore, the second paragraph of \cite[Lem 8.8]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} shows that there is a dense open set $Y_0 \subseteq Y$ such that $\lct \left( \Gamma_{\overline{y}};Z_{\overline{y}}, \Delta'_{\overline{y}} \right)$ is the same for every $\overline{y} \in Y_0$. Applying this iteratedly to the complement of $Y_0$, we obtain that $\overline{y} \mapsto \lct \left( \Gamma_{\overline{y}};Z_{\overline{y}}, \Delta'_{\overline{y}} \right)$ takes only finitely many values on $Y$, say $r_1>r_2>\dots>r_l$, and the level sets are constructible subsets of $Y$. Hence,
\begin{equation*}
L_i:=\left\{\overline{y} \in Y \left| \lct \left( \Gamma_{\overline{y}};Z_{\overline{y}}, \Delta'_{\overline{y}} \right) \geq r_i \right. \right\}
\end{equation*}
are open sets, and
for any geometric point $\overline{y}:=\overline{y}_{(D_j)}$ of $Y$,
\begin{equation*}
\lct \left(X_{\overline{y}},\Delta_{\overline{y}}; \Gamma_{\overline{y}} \right) = \lct \left(X_{\overline{y}},\Delta_{\overline{y}}; \sum_{j=1}^r \frac{D_j}{l} \right) = \max \{r_i | (D_j) \in L_i \}.
\end{equation*}
It follows that for any geometric point $\overline{t} \in T$,
\begin{equation}
\label{eq:delta_general_fiber:delt_inv}
\delta_{qs}\left(X_{\overline{t}}, \Delta_{\overline{t}}\right)= \max\{ r_i | Y_{\overline{t}} \subseteq (L_i)_{\overline{t}} \}.
\end{equation}
After the above discussion, our claim follows immediately. Indeed, we just need to choose $a$ to be the smallest integer such that $L_a$ contains the generic fiber of $\pi$. Then there is a non-empty open set $U \subseteq T$ contained in
\begin{equation*}
\left( T \setminus \pi(Y \setminus L_a) \right) \cap \pi(L_a \setminus L_{a-1}).
\end{equation*}
In particular, for any geometric point $\overline{t} \in U$:
\begin{enumerate}
\item $Y_{\overline{t}} \subseteq (L_a)_{\overline{t}}$, and
\item $ (L_a \setminus L_{a-1})_{\overline{t}} \neq \emptyset$ and hence $Y_{\overline{t}} \not\subseteq (L_{a-1})_{\overline{t}}$.
\end{enumerate}
Therefore, by setting $d:=r_a$, \autoref{eq:delta_general_fiber:delt_inv} implies that $\delta_{qs}\left( X_{\overline{t}}, \Delta_{\overline{t}}\right) = d$ for all geometric points $\overline{t} \in U$.
\end{proof}
\begin{remark}
\label{rem:delta_invariant_non_albraically_closed_field}
We note that one could define the $\delta$-invariant also over over non algebraically closed base fields, with verbatim the same definition as \autoref{def:delta}. If $(Y_K,\Delta_K)$ is a projective klt pair and $N_K$ is a $\mathbb{Q}$-Cartier divisor defined over a non-closed field $K$, and furthermore we choose a basis type divisor $D= \sum_{i=1}^{h^0(Y_K, qN_K)} \frac{D_i}{q}$ (that is, $D_i$ form a $K$-basis of $H^0(Y_K, qN_K)$), then $\lct (Y_K,\Delta_K, D) = \lct\left(Y_{\overline{K}}, \Delta_{\overline{K}}, D_{\overline{K}} \right)$, where $D_{\overline{K}}$ is a basis type divisor for $N_{\overline{K}}$. Hence, $\delta_q (Y_K,\Delta_K, N_K) \geq \delta_q \left(Y_{\overline{K}}, \Delta_{\overline{K}}, N_{\overline{K}} \right)$. However, $\delta_q (Y_K,\Delta_K, N_K) > \delta_q \left(Y_{\overline{K}}, \Delta_{\overline{K}}, N_{\overline{K}} \right)$ could happen as not all basis type divisors of $N_{\overline{K}}$ come from basis type divisors of $N_K$. A simple example is if $Y_K$ is a conic not isomorphic to $\mathbb{P}^1_K$, $\Delta_K=0$, and $N_K = K_{Y_K}^{-1} $. Then, $\delta_q(Y_K,\Delta_K;N_K)=3$, but $\delta_q\left(Y_{\overline{K}},\Delta_{\overline{K}};N_{\overline{K}}\right)=1$.
In particular, if one takes a conic bundle $f : X \to T$ without a section, and $\eta$ is the generic point of $T$, then for the generic fiber we have $\delta\left(X_{\eta}\right)=2$, but for all geometric fiber (including the geometric generic fiber) outside of the discriminant locus we have $\delta\left(X_{\overline{t}}\right)=1$. So, the $\delta$-invariant is not the same for a general and for the generic point (in general). In particular, one cannot replace "any geometric point $\overline{t} \in T$" in \autoref{prop:delta_general_fiber} with just "any point $t \in T$".
\end{remark}
\begin{remark}
The special case of \autoref{prop:delta_general_fiber} when $d=1$ and $\Delta=0$ (so for $K$-semi-stability via \cite{BJ}) was shown in \cite[Thm 3]{Liu_On_the_semi-continuity_problem_of_normalized_volumes_of_singularities} with other methods.
\end{remark}
\begin{remark}
\autoref{prop:delta_general_fiber} is very weak version of what is expected. It is conjectured (c.f., \cite{Blum_Liu_The_normalized_volume_of_a_singularity_is_lower_semicontinuous}
) that $\delta$ is lower semi-continuous, and furthermore the $\delta \geq 1$ set is also open. Some of this has been proven in \cite[Thm 1.1(i)]{Li_Wang_Xu_Degeneration} and \cite{Blum_Liu_The_normalized_volume_of_a_singularity_is_lower_semicontinuous}.
\end{remark}
\subsection{Boundary free statements}
\label{sec:intro_boundary_free}
For a flat family of $n$-dimensional Fano family $f : X \to T$ (that is, $X$ and $T$ are normal and projective, and $-K_{X/T}$ is an $f$-ample $\mathbb{Q}$-Cartier divisor) the Chow-Mumford (CM) line bundle is the pushforward cycle
\begin{equation}
\label{eq:CM_definition_intro}
\lambda_f := - f_* \left( c_1(-K_{X/T})^{n+1}\right).
\end{equation}
This cycle, up to multiplying with a positive rational number, is the first Chern class of the
a functorial line bundle on $T$ defined in \cite{Paul_tian1,Paul_tian2} (see also \cite{Schumacher,Fine_Ross_A_note_on_positivity_of_the_CM_line_bundle,PRS}).
Our main motivation to consider the CM line bundle originates from the classification theory of algebraic varieties: the \emph{birational part} of classification theory, also called the Minimal Model Program, predicts that up to specific birational equivalences, each projective variety decomposes into iterated fibrations with general fibers of 3 basic types: Fano, weak Calabi-Yau, and general type (to be precise one here needs to allow pairs, see \autoref{sec:intro_logarithmic}, but the boundary free case is a good first approximation). These 3 types are defined by having a specific class of mild singularities and negative/numerically trivial/positive canonical bundles. Then the \emph{moduli part} of the classification theory is supposed to construct compactified moduli spaces for the above 3 basic types of varieties. According to our current understanding the moduli part seems to be doable only in the presence of a (singular) K\"ahler-Einstein metric, which is predicted to be equivalent to the algebraic notion of $K$-polystability.
We note that the above predictions are proven in large cases, e.g., MMP: \cite{Birkar_Cascini_Hacon_McKernan_Existence_of_minimal_models,Hacon_McKernan_Existence_of_minimal_models_for_varieties_of_log_general_type_II,
Hacon_Xu_Existence_of_log_canonical_closures,
Birkar_Existence_of_log_canonical_flips_and_a_special_LMMP,
Fujino_Introduction_to_the_log_minimal_model_program_for_log_canonical_pairs,
flips_and_abundance_for_algebraic_threefolds,Keel_Matsuki_McKernan_Log_abundance_theorem_for_threefolds,
Alexeev_Hacon_Kawamata_Termination_of_many_4-dimensional_log_flips,
Birkar_On_termination_of_log_flips_in_dimension_four,Birkar_Ascending_chain_condition_for_log_canonical_thresholds_and_termination_of_log_flips}; connections between $K$-stability and K\"ahler-Einstein metrics: \cite{Berman_Guenancia_Kahler-Einstein_metrics_on_stable_varieties_and_log_canonical_pairs,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_I_Approximation_of_metrics_with_cone_singularities,
Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_II_Limits_with_cone_angle_less_than_2pi,
Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_III_Limits_as_cone_angle_approaches_2pi_and_completion_of_the_main_proof,
Tian_K-stability_and_Kahler-Einstein_metrics,Odaka_The_GIT_stability_of_polarized_varieties_via_discrepancy,Odaka_The_Calabi_conjecture_and_K-stability,Odaka_Xu_Log-canonical_models_of_singular_pairs_and_its_applications, Recent_Tian}.
In particular, on the Fano side for the moduli part one should construct algebraically a moduli space conjectured below. Furthermore, the polarization on this moduli space should be given by the CM line bundle, as the connection between the K\"ahler-Einstein theory and $K$-stability originates from Tian's idea to view the metrics as polystable objects in an infinite dimensional GIT type setting with respect to the CM line bundle. (See \autoref{def:K_stability} and \autoref{def:K_stable} for a precise definition and characterization used in the present article for $K$-semistability and see \autoref{sec:K_stability_versions} for an explanation on $K$-polystability). The precise conjecture is as follows:
\begin{conjecture}{\rm (e.g., a combination of \cite[Conj 3.1]{Odaka_On_the_moduli_of_Kahler-Einstein_Fano_manifolds} and \cite[Qtn 3, page 15]{Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}, but also implicit to different degrees for example in \cite{Tian2,Paul_tian2,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_III_Limits_as_cone_angle_approaches_2pi_and_completion_of_the_main_proof,Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds,Li_Wang_Xu_Degeneration,SpottiYTD})}
\label{conj:K_ss_moduli}
\begin{enumerate}
\item \label{itm:K_ss_moduli:stack} The stack $\scr{M}_{n,v}^{\Kss}$ of $K$-semistable klt Fano varieties of fixed dimension $n>0$ and anti-canonical volume $v>0$ is an Artin stack of finite type over $k$.
\item \label{itm:K_ss_moduli:algebraic_space} The stack $\scr{M}_{n,v}^{\Kss}$ admits a good moduli space $M^{\Kps}_{n,v}$ (in the sense of \cite{Alper_Good_moduli_spaces_for_Artin_stacks}), which is proper over $k$, and which parametrizes $K$-polystable klt Fano varieties of dimension $n$ and volume $v$. Furthermore, the CM line bundle descends onto $M^{\Kps}_{n,v}$.
\item \label{itm:K_ss_moduli:polarization} $M^{\Kps}_{n,v}$ is a projective scheme via the polarization given by the CM line bundle. In particular, the CM line bundle is
\begin{enumerate}
\item \label{itm:K_ss_moduli:semi_positive} nef in families of $K$-semi-stable klt Fanos, and
\item \label{itm:K_ss_moduli:positive} nef and big in maximally varying families of $K$-polystable klt Fanos.
\end{enumerate}
\end{enumerate}
\end{conjecture}
Our main result concerns point \autoref{itm:K_ss_moduli:polarization} of \autoref{conj:K_ss_moduli}. We completely solve point \autoref{itm:K_ss_moduli:semi_positive} and we solve \autoref{itm:K_ss_moduli:positive} on the uniformly $K$-stable locus (Notes: $K$-polystability usually occurs at the boundary of the uniformly $K$-stable locus, see \autoref{sec:K_stability_versions} for the definitions; see also \autoref{rem:why_only_unif_K_stable} for the reasons of the specific generality in point \autoref{itm:K_ss_moduli:positive}, and see \autoref{sec:intro_logarithmic} for the results for pairs).
\begin{theorem}
\label{thm:semi_positive_no_boundary}
Let $f : X \to T$ be a flat morphism with connected fibers between normal projective varieties such that $-K_{X/T}$ is $\mathbb{Q}$-Cartier and $f$-ample, and let $\lambda_f$ be the CM line bundle defined in equation \autoref{eq:CM_definition_intro}.
\begin{enumerate}
\item \label{itm:semi_positive_no_boundary:pseff} {\scshape Pseudo-effectivity:} If $T$ is smooth and the very general geometric fibers of $f$ are $K$-semistable, then {$\lambda_f$ is pseudo-effective}.
\item \label{itm:semi_positive_no_boundary:nef} {\scshape Nefness:} If all the geometric fibers of $f$ are $K$-semi-stable then {$\lambda_f$ is nef}.
\item \label{itm:semi_positive_no_boundary:big} {\scshape Bigness:} If $T$ is smooth, the very general geometric fibers of $f$ are uniformly $K$-stable, the variation of $f$ is maximal (i.e., there is a non-empty open set of $T$ over which the isomorphism equivalence classes of the fibers are finite), and either $\dim T=1$ or the fibers of $f$ are reduced, then {$\lambda_f$ is big}.
\item \label{itm:semi_positive_no_boundary:ample} {\scshape Ampleness:} If all the geometric fibers of $f$ are uniformly $K$-stable and the isomorphism equivalence classes of the fibers are finite, then {$\lambda_f$ is ample}.
\item \label{itm:semi_positive_no_boundary:q_proj} {\scshape Quasi-projectivity:} If $T$ is only assumed to be a proper normal algebraic space, all the geometric fibers are $K$-semi-stable and there is an open set $U \subseteq T$ over which the geometric fibers are uniformly $K$-stable and the isomorphism classes of the fibers are finite, then $U$ is a quasi-projective variety.
\end{enumerate}
\end{theorem}
\begin{remark}
Notably, \autoref{thm:semi_positive_no_boundary} deals with non-smoothable singular Fanos too, about which we remark that:
\begin{enumerate}
\item This is the first result about (semi-)positivity of the CM line bundle dealing with non-smoothable singular Fanos, as explained in \autoref{rem:partial_results}.
\item Non-smoothable singular Fanos is the general case, for example in the sense that smoothable $K$-semi-stable Fanos are bounded, even without fixing the anti-canonical volume (one can put this together from \cite{Kollar_Miyaoka_Mori_Rational_connectedness_and_boundedness_of_Fano_manifolds,Jiang_Boundedness_of_Q-Fano_varieties_with_degrees_and___alpha-invariants_bounded_from_below}). On the other hand, non-smoothable $K$-semi-stable Fanos are unbounded if one does not fix the volume, as can be seen by considering quasi-\'etale quotients by bigger and bigger finite groups of $\mathbb{P}^2$, which are $K$-semi-stable according to \cite[Cor. 1.7]{Fujita_Uniform_K-stability_and_plt_blowups_of_log_Fano_pairs}.
\end{enumerate}
\end{remark}
\begin{remark}
The proof of \autoref{thm:semi_positive_no_boundary} uses the Central Limit Theorem of probability theory. See \autoref{sec:outline_semi_positivity} for an outline of our argument or \autoref{prop:vectP2} for the precise place where the Central Limit Theorem is used.
\end{remark}
\begin{remark}
Either all or no very general geometric fibers is $K$-semi-stable (resp. uniformly $K$-stable), as shown by the constancy of the $\delta$-invariant on very general generic fibers (\autoref{prop:delta_general_fiber}). In particular, if $k$ is uncountable, say $\mathbb{C}$, the assumptions of \autoref{thm:semi_positive_no_boundary} can be checked on closed fibers. Furthermore, in the $K$-semistable cases, so for points \autoref{itm:semi_positive_no_boundary:pseff} and \autoref{itm:semi_positive_no_boundary:nef} of \autoref{thm:semi_positive_no_boundary}, it is enough to find a single $K$-semistable closed fiber, according to \cite[Thm 3]{Blum_Liu_The_normalized_volume_of_a_singularity_is_lower_semicontinuous}. The uniformly $K$-stable version of \cite[Thm 3]{Blum_Liu_The_normalized_volume_of_a_singularity_is_lower_semicontinuous} is not known, but it is expected too.
We also remark that in \autoref{thm:semi_positive_no_boundary} we carefully said ``geometric fiber'' instead of just ``fiber''. The reason is that we use the $\delta$-invariant description of $K$-stability, and the $\delta$-invariant of a variety is not invariant under base extension to the algebraic closure (see \autoref{rem:delta_invariant_non_albraically_closed_field}). So, for scheme theoretic fibers over non algebraically closed fields the $\delta$-invariant can have non semi-continuous behavior.
\end{remark}
\begin{remark}
We chose the actual generality for \autoref{thm:semi_positive_no_boundary}, as it is the generality in which the relative canonical divisor exists and admits reasonable base-change properties (see \autoref{sec:relative_canonical} for details) on very general curves in moving families of curves on the base.
Nevertheless, in situations where this base-change is automatic, \autoref{thm:semi_positive_no_boundary} directly implies statements over non-normal, non-projective, and even non-scheme bases. This is made precise in the following statement:
\end{remark}
\begin{corollary}
\label{cor:proper_base}
Let $f : X \to T$ be a flat, projective morphism with connected fibers to a proper algebraic space, such that there is an integer $m > 0$ for which $\omega_{X/T}^{[m]}$ is a line bundle and all the geometric fibers are $K$-semi-stable klt Fano varieties. Let $N$ be the CM-line bundle associated to the polarization $\omega_{X/T}^{[-m]}$ as defined over general bases in \cite{Paul_tian2} (see \autoref{notation:Paul_Tian}).
Then, $N$ is nef, and if the variation of $f$ is maximal and the very general geometric fiber is uniformly $K$-stable, then $N$ is big.
\end{corollary}
\begin{remark}
Note that over $\mathbb{C}$ the positivity properties of \autoref{thm:semi_positive_no_boundary} (nefness, pseudo-effectivity, bigness, ampleness) can be characterized equivalently analytically, e.g., \cite[Prop 4.2]{Demailly_Singular_Hermitian_metrics_on_positive_line_bundles}
\end{remark}
\begin{remark}
{\scshape Negativity of $-K_{X/T}$ point of view.}
Unwinding definition \autoref{eq:CM_definition_intro},
we obtain that \autoref{thm:semi_positive_no_boundary} in the case of one dimensional base states that $(-K_{X/T})^{n+1}$ is at most zero/smaller than 0. Using this in conjunction with the base-change property with the CM line bundle (\autoref{prop:CM_base_change}) we obtain that \autoref{thm:semi_positive_no_boundary}, especially the last 3 points, prove strong negativity property of $-K_{X/T}$ for klt Fano families.
There does exist birational geometry statements claiming that $-K_{X/T}$ is not nef, e.g., \cite[Prop 1]{Zhang_On_projective_manifolds_with_nef_anticanonical_bundles}. Our negativity statements point in this direction but go further. However, it is not a coincidence that strong negativity statements on $-K_{X/T}$ did not show up earlier, as in fact \autoref{thm:semi_positive_no_boundary} is not true for every family of klt Fano varieties. Indeed, \autoref{ex:negative_degree} shows that in \autoref{thm:semi_positive_no_boundary} one cannot relax the $K$-semi-stable Fano assumption to just assuming klt Fano.
The development of the notions of $K$-stability in the past decade was essential for creating the chance of proving negativity statements for $-K_{X/T}$ of the above type.
We also note that as $-K_{X/T}$ is not nef usually in the above situation (c.f., \autoref{thm:nef_threshold} and \autoref{ex:not_nef}), the negativity of $(-K_{X/T})^{n+1}$ is independent of the negativity of $\kappa(-K_{X/T})$. In fact, assuming the former, $\kappa(-K_{X/T})$ can be $- \infty$ (\autoref{ex:anti_canonical_no_section}), 0 (\autoref{ex:not_nef}), $\dim X$ (\autoref{ex:positive_and_big}), and also something in between the latter two values (\autoref{ex:anti_canonical_no_section}).
\end{remark}
\begin{remark}
\label{rem:partial_results}
The following are the already known partial results on \autoref{conj:K_ss_moduli}.
\begin{enumerate}
\item On the algebraic side, aiming for all klt singularities, there were no results on point \autoref{itm:K_ss_moduli:polarization} of \autoref{conj:K_ss_moduli} earlier (although the second author with Xu proved the canonically polarized version in \cite{Patakfalvi_Xu_Ampleness_of_the_CM_line_bundle_on_the_moduli_space_of_canonically_polarized_varieties}). Speaking about points \autoref{itm:K_ss_moduli:stack} and \autoref{itm:K_ss_moduli:algebraic_space} of \autoref{conj:K_ss_moduli}, they decompose into statements about different properties of the moduli functor: boundedness, separatedness, properness, openness of K-semi-stability, and contrary to the canonically polarized case, an analysis of the action of automorphisms on the CM-line bundle is also necessary to have it descend to the coarse moduli space. Out of these, only boundedness is known according to \cite[Cor 1.7]{Jiang_Boundedness_of_Q-Fano_varieties_with_degrees_and___alpha-invariants_bounded_from_below}, which uses also crucially the seminal papers of Birkar \cite{Birkar_Anti-pluricanonical_systems_on_Fano_varieties,Birkar_Singularities_of_linear_systems_and_boundedness_of_Fano_varieties}.
\item On the other hand, using analytic methods (sometimes in conjunction with algebraic ones), there are plenty of results about \autoref{conj:K_ss_moduli} on the closure of the locus of smooth Fanos:
\cite{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds,Li_Wang_Xu_Degeneration,SpottiYTD,Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}.
The only significant piece missing from the above results is that the positivity of the CM line bundle is not known on closed subspaces $V$ lying in the boundary of the closure of the locus of smooth Fanos. Our theorem in particular remedies this if the very general Fano parametrized by $V$ is uniformly $K$-stable (and necessarily singular), see \autoref{cor:bigger_open_set}.
\end{enumerate}
\end{remark}
\begin{remark}
\label{rem:why_only_unif_K_stable}
There are two main reasons why our positivity statements (points \autoref{itm:semi_positive_no_boundary:big}, \autoref{itm:semi_positive_no_boundary:ample} and \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary}) work in the uniformly $K$-stable case, but not in the $K$-polystable case:
\begin{enumerate}
\item We rely on the characterization of $K$-semistability and uniform $K$-stability via the $\delta$ invariant given by \cite{Fujita_Odaka_On_the_K-stability_of_Fano_varieties_and_anticanonical_divisors,BJ}. Such characterization is not available for the $K$-polystable case.
\item Our theorem on the nef threshold (\autoref{thm:nef_threshold} below, on which the above 3 points of \autoref{thm:semi_positive_no_boundary} depend) fails in the $K$-polystable case according to \autoref{ex:not_nef}. Hence, one would need a significantly different approach to extend points \autoref{itm:semi_positive_no_boundary:big}, \autoref{itm:semi_positive_no_boundary:ample} and \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary} to the $K$-polystable case.
\end{enumerate}
\end{remark}
\begin{remark}
One could make definition \autoref{eq:CM_definition_intro} also without requiring flatness. We do not know if \autoref{thm:semi_positive_no_boundary} holds in this situation. Nevertheless, we note that it would be interesting to pursue this direction for example for applications to Mori-fiber spaces with higher dimensional bases (see \autoref{cor:bounding_volume}).
Also we expect that the reduced fiber assumption of point \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary} can be removed, as we needed it for technical reasons (certain base changes over movable curves are nice), and also because the conjectured $K$-semi-stable reduction should eliminate it.
\end{remark}
\subsection{Logarithmic statements}
\label{sec:intro_logarithmic}
We prove points \autoref{itm:semi_positive_no_boundary:pseff} and \autoref{itm:semi_positive_no_boundary:nef} of \autoref{thm:semi_positive_no_boundary} in the generality of pairs. We state this separately, in the present subsection, as the statements are more cumbersome (e.g., one needs to guarantee that the boundary can restrict to fibers, etc.).
If $f : (X, \Delta) \to T$ is a flat morphism of relative dimension $n$ from a projective normal pair to a normal projective variety such that $-(K_{X/T} + \Delta)$ is $\mathbb{Q}$-Cartier and $f$-ample. Then we define the CM line bundle by
\begin{equation}
\label{eq:CM_line_bundle_log}
\lambda_{f, \Delta}:= -f_* (-(K_{X/T} + \Delta)^{n+1} ).
\end{equation}
\begin{theorem}
\label{thm:semi_positive_boundary}
Let $f : X \to T$ be a flat morphism of relative dimension $n$ with connected fibers between normal projective varieties and let $\Delta$ be an effective $\mathbb{Q}$-divisor on $X$ such that $-(K_{X/T} + \Delta)$ is $\mathbb{Q}$-Cartier and $f$-ample. Let $\lambda_{f,\Delta}$ be the CM line bundle on $T$ as defined in \autoref{eq:CM_line_bundle_log}.
\begin{enumerate}
\item {\scshape Pseudo-effectivity:} \label{itm:semi_positive_boundary:pseff} If $T$ is smooth and $(X_t, \Delta_t)$ is K-semi-stable for very general geometric fibers $X_t$, then $\lambda_{f, \Delta}$ is pseudo-effective.
\item {\scshape Nefness:} \label{itm:semi_positive_boundary:nef} If all fibers $X_t$ are normal, $\Delta$ does not contain any fibers (so that we may restrict $\Delta$ on the fibers), and $(X_t, \Delta_t)$ is $K$-semi-stable for all geometric fibers $X_t$, then $\lambda_{f, \Delta}$ is nef.
\end{enumerate}
\end{theorem}
\begin{remark}
\label{rem:no_log_pos}
We note that positivity statements, that is, points \autoref{itm:semi_positive_no_boundary:ample}, \autoref{itm:semi_positive_no_boundary:big} and \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary}, are much trickier to prove in the logarithmic case than in the boundary free case (c.f. \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}). The main issue is that a family of pairs can have maximal variations in a way that the underlying family of varieties has no variation, but at the same time the underlying family of varieties is not trivializable on a finite cover (e.g., take the family of \autoref{ex:negative_degree} and put a pair structure on it using a small multiple of varying anti-pluricanonical divisors). The same issue in the canonically polarized case involved a decent amount of work \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension,Patakfalvi_Xu_Ampleness_of_the_CM_line_bundle_on_the_moduli_space_of_canonically_polarized_varieties}.
\end{remark}
\subsection{Applications}
\label{sec:intro_applications}
Our applications of the theorems above are of two type:
\begin{enumerate}
\item moduli theoretic, as we have already suggested in \autoref{sec:intro_boundary_free}, and
\item birational geometric, aiming to understand Mori-fiber spaces with $K$-semistable fibers.
\end{enumerate}
We start with the precise statements of the moduli theoretic ones (\autoref{cor:moduli} and \autoref{cor:bigger_open_set}):
\begin{corollary}
\label{cor:moduli}
If $M$ is a proper algebraic space which is the moduli space for some class $\mathfrak{X}$ of $K$-semi-stable Fanos with the uniform $K$-stable locus $M^u \subseteq M$ being open (see \autoref{def:moduli} for the precise definitions), then the normalization of $M^u$ is a quasi-projective scheme over $k$.
\end{corollary}
\begin{remark}
The space $M^u$ of \autoref{cor:moduli} is many times smooth already, in which case the normalization can be certainly dropped from the statement. In fact, we know that it is smooth at the points corresponding to smooth Fanos
\cite{Ran_Deformations_of_manifolds_with_torsion_or_negative_canonical_bundle,Kawamata_Unobstructed_deformations_A_remark_on_a_paper_of_Z_Ran}, and to terminal Fano $3$-folds
\cite[Thm 1.7]{Sano_On_deformations_of_Q-Fano_3-folds}. Unfortunately, these unobstructedness statements do not hold for all Fanos, as \cite[Rem 2.13]{Sano_On_deformations_of_Q-Fano_3-folds} gives a counterexample. However, the counterexample is a cone over a Del-Pezzo surface of degree 6. Hence, it has infinite automorphism group, and in particular it is not uniformly $K$-stable. This leads to the following question.
\end{remark}
\begin{question}
Is the deformation space of uniformly $K$-stable Fanos (including general klt ones) unobstructed?
\end{question}
In the next corollary, we extend the locus of the moduli space of $K$-semistable Fanos that is known to be quasi-projective from the smooth locus to the union of the smooth locus and the largest open set in the uniformly $K$-stable locus.
\begin{corollary}
\label{cor:bigger_open_set}
If $M$ is the moduli space of smoothable $K$-semistable Fanos (which is known to exist as an algebraic space according to \cite{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds,Li_Wang_Xu_Degeneration,SpottiYTD,Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}), and $M^0$ is an open set parametrizing Fanos that are either smooth or uniformly $K$-stable, then the normalization of $M^0$ is quasi-projective.
\end{corollary}
Fujita showed in \cite[Thm 1.1]{Fujita_Optimal_bounds_for_the_volumes_of_Kahler-Einstein_Fano_manifolds} that $\vol(-K_X) \leq (n+1)^n$ for every $K$-semistable Fano variety $X$ of dimension $n$ (see \cite[Thm 3]{Liu_The_volume_of_singular_Kahler-Einstein_Fano_varieties} for better bounds in the presence of quotient singularities). Using \autoref{thm:semi_positive_boundary} we can show similar bounds for (non-necessarily klt) Fano $X$ admitting a Fano fibration structure with $K$-semi-stable general fiber.
\begin{corollary}
\label{cor:bounding_volume}
If $(X,\Delta)$ is a normal Fano pair, and $f : X \to \mathbb{P}^1$ is a fibration with $K$-semi-stable very general geometric fibers $F$, then
\begin{equation*}
\vol(-(K_X+\Delta)) \leq 2 \dim \left( X\right) \vol (-(K_F+\Delta_F)).
\end{equation*}
If furthermore $\Delta=0$, then
\begin{equation*}
\vol(-K_X) \leq 2\dim \left(X\right)^{\dim \left(X\right)}.
\end{equation*}
\end{corollary}
\begin{remark}
\autoref{cor:bounding_volume} is sharp for surfaces and threefolds. Indeed, a del Pezzo surface of degree $8$ and the blow-up of $\mathbb{P}^3$ at a line (whose anti-canonical volume is $54$) can be fibred over $\mathbb{P}^1$ with $K$-semis-table fibres.
\end{remark}
\begin{remark}
\label{rem:K_stability_classification}
{ \scshape Classification of (uniform) $K$-(semi/poly)-stable Fanos}: to understand the power of \autoref{cor:bounding_volume}, it is useful to know which Fanos are $K$-semi-stable and which are not. In fact, then one want to figure this out for all the four $K$-stability properties (see \autoref{sec:K_stability_versions}), which has been an active area of research recently. To start with, let us recall that $K$-semi-stable Fano varieties are always klt.
A Del-Pezzo surface is K-polystable if and only if it is not of degree $8$ or $7$ \cite{Tian_Yau_Kahler-Einstein_metrics_on_complex_surfaces_with_C_1_at_least_zero,Tian_On_Calabis_conjecture_for_complex_surfaces_with_positive_first_Chern_class}.
Smooth Fano surfaces with discrete automorphism groups are even uniformly K-stable, and their delta invariant (see \autoref{sec:delta}) is bounded away from $1$ in an effective way \cite{PW}. Smoothable singular K-stable Del-Pezzo surfaces are classified in \cite{Odaka_Spotti_Sun}.
K-stable proper intersection of two quadrics in an odd dimensional projective space are classified in \cite{Spotti_Sun} (also \cite{Arezzo_Ghigi_Pirola_Symmetries_quotients_and_Kahler-Einstein_metrics}); in particular, smooth varieties of these types are always K-stable. Cubic $3$-folds are studied in \cite{Li_Xu}, where again smooth ones are $K$-stable, and so are the ones containing only $A_k$ singularities for $k \leq 4$. Under adequate hypotheses, in \cite{Dervan_Finite}, it is shown that Galois covers of K-semistable Fano varieties are K-stable. This can be applied for instance to double solids. Furthermore, birational superigid Fano varieties are K-stable under some addition mild hypothesis \cite{OO13,SZ,Z}. However, according to the best knowledge of the authors, there is not a complete classification of K-stable smooth Fano threefolds.
If one wants to study klt Fano varieties from the point of view of the MMP, it is particularly relevant to see if one can apply \autoref{cor:bounding_volume} to the case of Mori Fibre Spaces. In \cite[Corollary 1.11]{CFST}, it is shown that if a smooth Fano surface or a smooth toric variety can appear as a fibre of MFS, then it is K-semistable. We do not know if the analogous result holds in dimension $3$. However, there are examples of smooth Fano fourfolds with Picard number one (which then can be general fibers of MFS's) that are not K-semistable \cite{Fujita_Examples}, see also \cite{Codogni2018}.
\end{remark}
\subsection{Byproduct statements}
As a byproduct of our method we obtain the following bound on the nef threshold of $-(K_{X/T} + \Delta)$ with respect to $\lambda$ in the uniformly $K$-stable case.
\begin{theorem}
\label{thm:bounding_nef_threshold}
\label{thm:nef_threshold}
Let $f : X \to T$ be a flat morphism with connected fibers from a normal projective variety of dimension $n+1$ to a smooth curve and let $\Delta$ be an effective $\mathbb{Q}$-divisor on $X$ such that
\begin{itemize}
\item $-(K_{X/T} + \Delta)$ is $\mathbb{Q}$-Cartier and $f$-ample, and
\item $\left(X_{\overline{t}},\Delta_{\overline{t}}\right)$ is uniformly $K$-stable for very general geometric fibers $X_{\overline{t}}$.
\end{itemize}
Set
\begin{itemize}
\item \label{itm:thm:bounding_nef_threshold_Delta} set $\delta:=\delta\left(X_{\overline{t}}, \Delta_{\overline{t}}\right)$ for $\overline{t}$ very general geometric point, and
\item let $v:=\left( (-K_{X/T} - \Delta)_t \right)^n$ for any $t \in T$.
\end{itemize}
Then, $- K_{X/T} - \Delta + \frac{\delta}{(\delta -1) v (n+1)} f^* \lambda_{f,\Delta} $ is nef.
\end{theorem}
\begin{remark}
One cannot have a nef threshold statement as in \autoref{thm:nef_threshold} for uniformly $K$-stable replaced with $K$-polystable. Indeed, take the family $f : X \to T$ given by \autoref{ex:not_nef}. It has $K$-polystable fibers, $\deg \lambda_f=0$, but $K_{X/T}$ is not nef. In particular, for any $a \in \mathbb{Q}$, $-K_{X/T}+ a f^*\lambda_f \equiv K_{X/T}$, and hence for any $a \in \mathbb{Q}$, $-K_{X/T}+ a f^*\lambda_f$ is not nef.
\end{remark}
And we have a structure theorem when $\lambda$ is not positive:
\begin{theorem}
\label{thm:lambda_non_big}
Let $f : X \to T$ be a flat morphism of relative dimension $n$ with connected fibers between normal projective varieties and let $\Delta$ be an effective $\mathbb{Q}$-divisor on $X$ such that $-(K_{X/T} + \Delta)$ is $\mathbb{Q}$-Cartier and $f$-ample. Assume that $\left(X_{\overline{t}}, \Delta_{\overline{t}}\right)$ is uniformly $K$-stable for very general geometric fibers $X_{\overline{t}}$. If $H$ is an ample divisor on $T$, such that $\lambda_{f,\Delta} \cdot H^{\dim T -1} =0$, then for every integer $q>0$ divisible enough, $f_* \scr{O}_X(q (-K_{X/T} - \Delta))$ is an $H$-semi-stable vector bundle of slope $0$.
\end{theorem}
\subsection{Similar results in other contexts}
Roughly, there are three types of statements above: (semi-)positivity results, moduli applications, inequality of volumes of fibrations. Although in the realm of $K$-stability ours are the first general algebraic results, statements of these types were abundant in other, somewhat related, contexts: KSBA stability, GIT stability, and just general algebraic geometry. Our setup and our methods are different from these results, still we briefly list some of them for completeness of background. We note that KSBA stability is related to our framework as it is shown to be exactly the canonically polarized $K$-stable situation \cite{Odaka_The_GIT_stability_of_polarized_varieties_via_discrepancy,Odaka_The_Calabi_conjecture_and_K-stability,Odaka_Xu_Log-canonical_models_of_singular_pairs_and_its_applications}. Also, GIT stability is related, as $K$-stability originates from an infinite dimensional GIT, although it is shown that it cannot be reproduced using GIT (e.g., \cite{Wang_Xu_Nonexistence_of_asymptotic_GIT_compactification}).
\noindent
\begin{tabular}{|l|p{110pt}|p{110pt}|p{77pt}|}
\hline
& general algebraic geometry & KSBA stability & GIT stability \\
\hline
(semi-)positivity & \cite{Griffiths_Periods_of_integrals,Fujita_On_Kahler_fiber_spaces,Kawamata_Characterization_of_abelian_varieties,Viehweg_Weak_positivity,Kollar_Subadditivity_of_the_Kodaira_dimension} & \cite{Kollar_Projectivity_of_complete_moduli,Fujino_Semi_positivity_theorems_for_moduli_problems,Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension,Patakfalvi_Xu_Ampleness_of_the_CM_line_bundle_on_the_moduli_space_of_canonically_polarized_varieties} & \cite{CH} \\
\hline
moduli applications & \cite{Viehweg_Quasi_projective_moduli} & \cite{Kollar_Projectivity_of_complete_moduli,Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension,Ascher_Bejleri_Moduli_of_weighted_stable_elliptic_surfaces_and_invariance_of_log___plurigenera} & \cite{CH} \\
\hline
volume (slope) inequalities & \cite{Xiao_Fibered_algebraic_surfaces_with_low_slope} & & \cite{Pardini,Stoppino}\\
\hline
\end{tabular}
\subsection{Overview of K-stability for Fano varieties}
\label{sec:K_stability_versions}
In the present article we define $K$-semi-stability and uniform $K$-stability using valuations (\autoref{def:K_stability}), which is equivalent then to the $\delta$-invariant definition (\autoref{def:K_stable}). These definitions were shown to be equivalent in \cite[Theorem B]{BJ} to the more traditional ones that use test configurations. Also, they have a serious disadvantage: there is no known delta invariant type definition of $K$-stability and $K$-polystability, which although we do not use in any of the statements or in the proofs, they are important notions for the big picture. Hence, for completeness we mention below their definitions using test configurations. We refer the reader to \cite{Tian_Test,Don_Calabi} or more recent papers such as \cite{Dervan_Uniform,Boucksom_Uniform} for more details:
\begin{description}
\item[K-semi-stability] For every normal test configuration, the Donaldson-Futaki invariant is non-negative.
\item[K-stability] For every normal test configuration, the Donaldson-Futaki invariant is non-negative, and it is equal to zero if and only if the test configuration is a trivial test configuration. In particular, there is no $1$-parameter subgroup of $\Aut(X)$.
\item[K-poly-stability] For every normal test configuration the Donaldson-Futaki invariant is non-negative, and it is equal to zero if and only if the test configuration is a product test configuration, i.e. it comes from a one parameter subgroup of the automorphism group of $X$.
\item[Uniform K-stability] There exists a positive real constant $\delta$ such that for every normal test configuration the Donaldson-Futaki invariant is at least $\delta$ times the $L^1$ norm (or, equivalently, the minimum norm) of the test configuration. This notion implies K-stability, and in the case of smooth complex Fanos that the automorphism group of $X$ is finite \cite[Cor E]{Boucksom_Hisamoto_Jonsson_Uniform_K-stability_and_asymptotics_of_energy_functionals_in_Kahler___geometry}.
\end{description}
We also note that the Yau-Tian-Donaldson conjecture asserts that a klt Fano variety admits a singular K\"{a}hler-Einstein metric if and only if it is K-polystable. This is known for smooth \cite{Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_I_Approximation_of_metrics_with_cone_singularities,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_II_Limits_with_cone_angle_less_than_2pi,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_III_Limits_as_cone_angle_approaches_2pi_and_completion_of_the_main_proof,Tian} and smoothable Fano varieties \cite{Li_Wang_Xu_Degeneration} (and independently \cite{SpottiYTD} in the finite automorphism case), and for singular ones admitting a crepant resolution \cite{Recent_Tian}. In the literature, there are also many proposed strenghtening of the notion of K-stability; they should be crucial to extend the YTD conjecture to the case of constant scalar curvature K\"{a}hler metrics. In this paper we are interested in uniform K-stability \cite{Dervan_Uniform,Boucksom_Uniform,Bouck_variational}, which at least for smooth Fano manifold is known to be equivalent to K-stability (we should stress that the proof is via the equivalence with the existence of a K\"{a}hler-Einstein metric). One can also strenghten the notion of K-stability by looking at possibly non-finitely generated filtration of the coordinate ring, see \cite{Nystrom_filtrations,Gabor,Tits}.
\subsection{Fiber product notation}
\label{sec:product_notation}
The most important particular notation used in the article is that of fiber products. That is, for a family $f : X \to T$ of varieties we denote the $m$-times fiber product of $X$ with itself over $T$ by $X^{(m)}$. As in our situation the base is always clear, we omit it from the notation. So, for $X^{(m)}$ means $m$-times fiber product over $T$, and $X_t^{(m)}$ means $m$-times fiber product over $t$. In this situation $p_i : X^{(m)} \to X$ denotes the projection onto the $i$-th factor, and we set for any divisor $D$ or line bundle $\scr{L}$:
\begin{equation*}
D^{(m)}:= \sum_{i=1}^m p_i^* D \textrm{, and } \scr{L}^{(m)}:= \bigotimes_{i=1}^m p_i^* \scr{L}.
\end{equation*}
\subsection{General further notation}
\label{sec:general_notation}
A \emph{variety} is an integral, separated scheme of finite type over $k$. A $(X, \Delta)$ is a \emph{pair} if $X$ is a normal variety, and $\Delta$ is an effective $\mathbb{Q}$-divisor, called the \emph{boundary}.
A projective pair $(X, \Delta)$ over $k$ is \emph{Fano} if $(X, \Delta)$ has klt singularities, and $-(K_X + \Delta)$ is an ample $\mathbb{Q}$-Cartier divisor. To avoid confusion, many times we say \emph{klt Fano} instead of \emph{Fano}, nevertheless we mean the same by the two. If there is not boundary, we mean taking the empty boundary. A projective pair $(X,\Delta)$ is a \emph{normal Fano} pair, if the klt condition is not assumed, that is, we only assume that $-(K_X + \Delta)$ is an ample $\mathbb{Q}$-Cartier divisor.
A \emph{big open set} $U$ of a variety $ X$ is an open set for which $\codim_X (X \setminus U) \geq 2$.
A \emph{vector bundle} is a locally free sheaf of finite rank.
The \emph{$\mathbb{Q}$-linear system} of a $\mathbb{Q}$-divisor $D$ on a normal variety is $|D|_{\mathbb{Q}}:=\{L \textrm{ is a $\mathbb{Q}$-divisor} | \exists m \in \mathbb{Z}, m>0 : m L \sim m D\}$.
A \emph{geometric fiber} of a morphism $f : X \to T$ is a fiber over a geometric point, that is over a morphism $\Spec K \to T$, where $K$ is an algebraically closed field extension of $k$. We say that a condition holds for a \emph{very general geometric point/fiber}, if there are countably many proper closed sets, outside of which it holds for all geometric points/fibers. \emph{General point/fiber} is defined the same way but with excluding only finitely many proper closed subsets. The \emph{(geometric) generic point/fiber} on the other hand denotes the scheme theoretic (geometric) generic point/generic fiber.
\subsection{Relative canonical divisor}
\label{sec:relative_canonical}
For a flat family $f : X \to T$ the relative dualizing complex is defined by $\omega_{X/T}^\bullet:=f^! \scr{O}_T$, where $f^!$ is Grothendieck upper shriek functor as defined in \cite{Hartshorne_Residues_and_duality}. If $f$ is also a family of pure dimension $n$, then the relative canonical sheaf is the lowest non-zero cohomology sheaf $\omega_{X/T}:= h^{-n}(\omega_{X/T}^\bullet)$ of the relative dualizing complex. To obtain the absolute versions of these notions one uses the above definition for $T = \Spec k$. The important facts for this article are:
\begin{enumerate}
\item The sheaf $\omega_{X/T}$ is reflexive if the fibers are normal \cite[Prop A.10]{Patakfalvi_Schwede_Zhang_F_singularities_in_families}.
\item If $T$ is Gorenstein, then $\omega_{X/T} \cong \omega_X \otimes f^* \omega_T^{-1}$ \cite[Lemma 2.4]{Patakfalvi_Semi_negativity_of_Hodge_bundles_associated_to_Du_Bois_families}, and then as $\omega_X$ is $S_2$ \cite[Cor 5.69]{Kollar_Mori_Birational_geometry_of_algebraic_varieties}, it is also reflexive \cite{Hartshorne_Stable_reflexive_sheaves}.
\item By the previous two points, if $f$ is flat and either $T$ is smooth or the fibers are normal, then $\omega_{X/T}$ is reflexive, and then if $X$ is normal, it corresponds to a divisor linear equivalence class which we denote by $K_{X/T}$.
\item \label{itm:Cohen_Macaulay_base_change} On the relative Cohen-Macaulay locus $U \subseteq X$ (that is, on the open set where the fibers are Cohen-Macaulay), $\omega_{U/T} \cong \omega_{X/T}|_U$ is compatible with base-change \cite[Thm 3.6.1]{Conrad_Grothendieck_duality_and_base_change}.
\end{enumerate}
In particular, by the above we always have the following assumptions on our families: $f : X \to T$ is flat with fibers being of pure dimension $n$, and either $T$ is smooth, or the fibers of $f$ are normal. In both cases we discuss base-change properties of the relative canonical divisor below.
\subsubsection{Base-change of the relative log-canonical divisor when the fibers are normal}
\label{sec:base_change_relative_canonical_normal_fibers}
Let us assume that $f : X \to T$ is a projective, flat morphism to normal projective variety with normal, connected fibers (in particular $X$ is also normal), and $\Delta$ is an effective $\mathbb{Q}$-divisor on $X$, such that $\Delta$ does not contain any fiber, and $K_{X/T} + \Delta$ is a $\mathbb{Q}$-Cartier divisor. Let $U \subseteq X$ be the smooth locus of $f$, which is an open set, and by the normality assumption on the fibers, $U \cap X_t$ is a big open set on each fiber $X_t$ (see \autoref{sec:general_notation} for the definition of a big open set).
Let $S \to T$ be a morphism from another normal projective variety. Then, we may define a pullback $\Delta_S$ as the unique extension of the pullback of $\Delta|_U$ to $U_S$ (the key here is that $\Delta|_U$ is $\mathbb{Q}$-Cartier). In particular, $f_S : X_S \to S$ and $\Delta_S$ satisfies all the assumptions we had for $f : X \to T$ and $\Delta$. Moreover, if $\sigma: X_S \to X$ is the induced morphism, then as $\mathbb{Q}$-Cartier divisors
\begin{equation}
\label{eq:pullback_log_canonical_divisor_normal_fibers}
K_{X_S/S} + \Delta_S \sim_{\mathbb{Q}} \sigma^* (K_{X/T} + \Delta).
\end{equation}
Indeed, it is enough to verify this isomorphism on $U$ (as $U$ is big in $X$, and $U_S$ is big in $X_S$). However, over $U$ the linear equivalence \autoref{eq:pullback_log_canonical_divisor_normal_fibers} holds by the definition of $\Delta_S$ and by the base-change property of point \autoref{itm:Cohen_Macaulay_base_change} above.
\subsubsection{Base-change of the relative log-canonical divisor when the base is smooth}
\label{sec:base_change_relative_canonical_smooth_base}
Let $f : X \to T$ be a flat morphism from a normal projective variety to a smooth, projective variety with connected fibers. Let $\Delta$ be an effective $\mathbb{Q}$-divisor on $X$ such $K_{X/T} + \Delta$ is $\mathbb{Q}$-Cartier. Let $T_{\norm} \subseteq T$ be the open set over which the fibers of $X$ are normal.
Note that by the smoothness assumption on $T$, at a point $x \in X$, the fiber $X_{f(x)}$ is Gorenstein if and only if $X$ is relatively Gorenstein if and only if $X$ is Gorenstein.
Let $U \subseteq X$ be the open set of relatively Gorenstein points over $T$. Let $\iota : C \to T$ be a finite morphism from a smooth, projective curve such that $\iota(C) \cap T_{\norm} \neq \emptyset$, and denote by $\sigma: X_C \to X$ the natural morphism.
\emph{We claim that $\sigma^{-1} U$ is big in $X_C$.} This is equivalent to show that for each $c \in C$, $X_c$ is Gorenstein at some point, and that for general $c \in C$, there is a big open set of $X_c$ where $X_c$ is Gorenstein. The former is true for all schemes of finite type over $k$ (hence also for $X_c$), and the latter is true by the $\iota(C) \cap T_{\norm} \neq \emptyset$ assumption. This concludes our claim.
Now, let $\pi : Z \to X_C$ be the normalization of $X_C$, $\rho: Z \to X$ and $g : Z \to C$ the induced morphisms and set $W:= \rho^{-1} U$. The notations are summarized in the following diagram:
\begin{equation*}
\xymatrix{
& W \ar@{^(->}[ld] \ar[r] & \sigma^{-1} U \ar@{^(->}[ld] \ar[r] & U \ar@{^(->}[ld] & \\
Z \ar[dr]_g
\ar@/^1.3pc/[rr]|!{[rru];[r]}\hole^\rho
\ar[r]_{\pi} & X_C \ar[r]_{\sigma} \ar[d]^{f_C} & X \ar[d]^f \\
& C \ar[r]_{\iota} & T & T_{\norm} \ar@{_(->}[l]
}
\end{equation*}
Then, \cite[Lem 9.13]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} tells us that there is a natural injection $\omega_{W/C} \to \left( \pi|_W \right)^* \omega_{\sigma^{-1} U/C}$. To be percise, \cite[Lem 9.13]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} assumes $\sigma^{-1} U$ to be normal, but as the proof does not use it, this is an unnecessary assumption. Combining this injection with the isomorphism
$\left( \sigma|_{\sigma^{-1}U} \right)^* \omega_{U/T} \cong \omega_{\sigma^{-1} U/C}$ given by point \autoref{itm:Cohen_Macaulay_base_change} above we obtain
\begin{equation}
\label{eq:relative_canonical_base_change}
\omega_{W/C} \hookrightarrow \left( \pi|_W \right)^* \omega_{\sigma^{-1} U/C} \cong \left( \pi|_W \right)^* \left( \sigma|_{\sigma^{-1}U} \right)^* \omega_{U/T} \cong \left( \rho|_W \right)^* \omega_{U/T} ,
\end{equation}
which is an isomorphism over the locus $T_{\red}$ over which the fibers of $f$ are reduced. Indeed, over $T_{\red}$ the fibers of $X_C \to C$ are all reduced, and by the $\iota(C) \cap T_{\norm} \neq \emptyset$ assumption the general fiber of $X_C \to C$ is normal. In particular, over $T_{\red}$, $X_C$ is $R_1$ and $S_2$, and hence normal. So, $\pi$ is the identity over $T_{\red}$.
Let $m>0$ be then an integer such that $m(K_{X/T}+ \Delta)$ is Cartier. That is, $\scr{L}:=\scr{O}_X(m(K_{X/T}+ \Delta))$ is a line bundle, and furthermore, $m\Delta$ yields an embedding $\omega_{U/T}^{\otimes m} \hookrightarrow \scr{L}|_U$. Composing this with the $m$-th power of the homomorphism of \autoref{eq:relative_canonical_base_change} we obtain:
\begin{equation}
\label{eq:relative_canonical_base_change_final}
\omega_{W/C}^{\otimes m} \to (\rho|_W)^* \scr{L} \cong \scr{O}_W(m \rho^*(K_{X/T} + \Delta)|_W),
\end{equation}
which map over $T_{\red}$ is given by ``multiplying with $\left(\rho|_{g^{-1}\iota^{-1} T_{\red}}\right)^* m\Delta$''. Indeed, for the latter remark, the main thing to note is that the regular locus of $X$, over which $m \Delta$ is necessarily Cartier, pulls back to a big open set of $g^{-1}\iota^{-1}T_{\red}$ (as general fiber of $f_C$ is normal and special fiber of $f_C$ over $T_{\red}$ are reduced). Hence $\pi$ is an isomorphism over $g^{-1}\iota^{-1}T_{\red}$ and also the pullback $\left(\rho|_{g^{-1}\iota^{-1} T_{\red}}\right)^* m\Delta$ is sensible the usual way: restricting to the regular locus, performing the pullback there, and then taking divisorial extension using bigness of the open set.
Lastly, the map \autoref{eq:relative_canonical_base_change_final} is given by an effective divisor $D$. If we set $\Delta_Z:= \frac{D}{m}$, using that $W$ is big in $Z$, we obtain:
\begin{proposition}
\label{rem:reduced_fibers_no_boundary}
\label{prop:relative_canonical_base_change_normal}
In the above situation, there is an effective $\mathbb{Q}$-divisor $\Delta_Z$ on $Z$ such that:
\begin{enumerate}
\item \label{itm:relative_canonical_base_change:base_change} $K_{Z/C} + \Delta_Z \sim_{\mathbb{Q}} \rho^*( K_{X/T} + \Delta)$,
\item \label{itm:relative_canonical_base_change:red_fibers} $X_C$ is normal over $T_{\red}$ and $\Delta_Z|_{g^{-1}\iota^{-1}T_{\red}} = \left(\rho|_{g^{-1}\iota^{-1} T_{\red}}\right)^* \Delta$, and
\item $\Delta_Z|_{g^{-1}\iota^{-1}T_{\norm}}$ agrees with the pullback of $\Delta|_{f^{-1}T_{\norm}}$ in the sense of \autoref{sec:base_change_relative_canonical_normal_fibers}.
\end{enumerate}
\end{proposition}
\subsubsection{ Semi-positivity statements.}
\label{sec:outline_semi_positivity}
As nefness and pseudo-effectivity can be checked via non-negative intersection with effective or moving 1-cycles, respectively, points \autoref{itm:semi_positive_boundary:pseff} and \autoref{itm:semi_positive_boundary:nef} of \autoref{thm:semi_positive_boundary} can be reduced to the case of a curve base. Hence, we assume that the base of our fibration $f: X \to T$ is a curve, in which case pseudo-effectivity and nefness are both equal to the degree being at least zero. So, we are supposed to prove that $\deg \lambda_f \geq 0$ or equivalently that $(-K_{X/T})^{n+1} \leq 0$ (see \autoref{eq:CM_line_bundle_log}).
We argue by contradiction, so we assume that $(-K_{X/T})^{n+1}>0$. If we fix a $\mathbb{Q}$-divisor $H$ on $T$ of small enough positive degree, then by the continuity of the intersection product $(-K_{X/T} - f^*H )^{n+1}>0$ also holds. As $X$ is normal and fibered over the curve $T$ over which $-K_{X/T}$ is ample, this implies via a Riemann-Roch computation that the $\mathbb{Q}$-linear system $|-K_{X/T} - f^* H|_{\mathbb{Q}}$ is non-empty (see \autoref{rem:big}). Our initial idea is to obtain a contradiction from this, as \autoref{prop:cont} tells us that there can exist no $\Gamma \in |-K_{X/T} - f^* H|_{\mathbb{Q}}$ such that $(X_t,\Gamma_t)$ is klt for general $t \in T$. The only problem is that there are examples where $|-K_{X/T} - f^* H|_{\mathbb{Q}}$ is non-empty but the above klt condition fails. Indeed, every family with negative CM line bundle has to be an example like that according to \autoref{prop:cont}. An explicit example is given in \autoref{ex:negative_degree}.
Our second idea is that maybe the $K$-stable assumption leads us to a $\Gamma$ as above that also satisfies the klt condition. According to the delta invariant description of $K$-semi-stability (\autoref{def:K_stable}), if $X_t$ is $K$-semi-stable, then up to a little perturbation one can obtain klt divisors the following way: for $q \gg 0$, let $D_1,\dots,D_l$ be divisors corresponding to any basis of $H^0\left(X_t, -qK_{X_t} \right)$; then the divisor $D := \displaystyle\sum_{i=1}^l \frac{D_i}{ql} \in |-K_{X_t}|_{\mathbb{Q}}$
is such that $(X_t, D)$ is klt.
Now, we would like to lift such a divisor to $|-K_{X/T} - f^* H|_{\mathbb{Q}}$. To this end, it is enough to lift for $q \gg 0$, every element of a basis of $H^0\left(X_t, -qK_{X_t} \right)$ to elements of $H^0(X,q(-K_{X/T} - f^* H))$. Using some perturbation argument, one can get by by finding linearly independent sections $s_1,\dots,s_l \in H^0\left(X_t, -qK_{X_t} \right)$ such that $s_i$ lifts, and $\frac{l}{h^0\left(-qK_{X_t} \right)}$ is close enough to $1$.
This in turn would be implied by the following: let $\scr{E}_q$ be the subsheaf of $f_* \scr{O}_X(-qK_{X/T})$ spanned by the global sections. Then we would need to show that
\begin{equation}
\label{eq:outline_goal}
\displaystyle\lim_{q \to \infty} \frac{\rk \scr{E}_q}{\rk f_* \scr{O}_X(q(-K_{X/T}-f^*H))}=1.
\end{equation}
(For the readers more familiar with the language of volumes and restricted volumes, we note that \autoref{eq:outline_goal} is equivalent to showing that the restricted volume of $-K_{X/T}$ over a general fiber is equal to the anti-canonical volume of the fibers.)
Unfortunately, \autoref{eq:outline_goal} is still not doable. For example, if one takes the isotrivial family
\begin{equation*}
X:=\mathbb{P}_{T} (\scr{O}_{T}(-n)\oplus \underbrace{\scr{O}_T(1)\oplus \dots \oplus \scr{O}_T(1)}_{\textrm{$n$ times}})
\end{equation*}
of $\mathbb{P}^n$'s over $T:= \mathbb{P}^1$ (as in \autoref{ex:negative_degree} for $n=2$), then
\begin{equation*}
f_* \scr{O}_X(-qK_{X/T}) \cong S^{(n+1)q}(\scr{O}_{T}(-n)\oplus \scr{O}_T(1)\oplus \dots \oplus \scr{O}_T(1)).
\end{equation*}
In this situation $\scr{E}_q$ is the direct sum of the factors with degree greater than $q \deg H \sim q \varepsilon$ (here $1 \gg \varepsilon>0$). Then one can compute that \autoref{eq:outline_goal} does not hold. For example, in the case of $n=1$,
\begin{equation*}
S^{2q}(\scr{O}_T(-1) \oplus \scr{O}_T(1)) = \scr{O}_T(-q) \oplus \scr{O}_T(-q+1) \oplus \dots \oplus \scr{O}_T(q).
\end{equation*}
So, we see that the limit of \autoref{eq:outline_goal} is $\frac{1}{2} - \varepsilon$.
The idea that saves the day at this point is the \emph{product trick}, which was pioneered in the case of semi-posivity questions by Viewheg \cite{Viehweg_Weak_positivity}. The precise idea is to replace $X$ by an $m$-times self fiber product $X^{(m)}$ over $T$. Let $f^{(m)} : X^{(m)} \to T$ be the induced morphism (\autoref{sec:product_notation}). Then, one can replace the initial goal with showing that there exists $\Gamma \in \left| - K_{X^{(m)}/T} - \left( f^{(m)}\right)^* m H \right|_{\mathbb{Q}}$ such that $\left(X^{(m)}_t, \Gamma_t \right)$ is klt for $t \in T$ general. Running through the previous arguments for $X^{(m)}$ instead of $X$, this would boil down to showing that
\begin{equation}
\label{eq:outline_goal_2}
\displaystyle\lim_{m \to \infty} \frac{\rk \scr{E}_{q,m}}{\rk f_*^{(m)} \scr{O}_{X^{(m)}} \left(q\left(-K_{X^{(m)}/T}- \left(f^{(m)} \right)^* m H\right)\right) }=1,
\end{equation}
where $\scr{E}_{q,m}$ is a subsheaf given by certain condition specified below of the subsheaf generated by global sections of
\begin{equation}
\label{eq:tensor_product_bundle_outline}
f_*^{(m)} \scr{O}_{X^{(m)}} \left(q\left(-K_{X^{(m)}/T}- \left(f^{(m)} \right)^* m H\right)\right) \cong \bigotimes_{\textrm{$m$ times}}f_* \scr{O}_X(q(-K_{X/T}-f^*H)).
\end{equation}
The extra condition in the definition of $\scr{E}_{q,m}$ is due to the need that $\Gamma$ has to be klt on a general fiber. This would be automatic if the conjecture that products of $K$-semi-stable klt Fanos are $K$-semi-stable was known. Unfortunately this is a surprisingly hard unsolved conjecture in the theory of $K$-stability. Hence, we elude it by considering only bases of $H^0\left( X^{(m)}_t, -q K_{X^{(m)}_t} \right) \cong \displaystyle\bigotimes_{m \textrm{ times}} H^0 \left( X_t, -q K_{X_t}\right)$ that are induced from bases of $H^0 \left( X_t, -q K_{X_t}\right)$. As log canonical thresholds are known to behave well under taking products (\autoref{prop:prod_basis}),
if the restriction $\Gamma|_{X_t^{(m)}}$ to a general fiber is a divisor corresponding to such basis, the $K$-stability of $X_t$ implies that $\left(X_t^{(m)}, \Gamma|_{X_t^{(m)}} \right)$ is klt. Hence, the additional condition in the definition of $\scr{E}_{q,m}$ is that it is the biggest subsheaf as above such that $\left( \scr{E}_{q,m} \right)_t$ is spanned by simple tensors for a basis $t_1,\dots,t_l$ of $\left( f_* \scr{O}_X(q(-K_{X/T}-f^*H)) \right)_t$ to be specified soon.
So, we are left to specify a basis of $\left( f_* \scr{O}_X(q(-K_{X/T}-f^*H)) \right)_t \cong H^0(X_t, -qK_{X_t})$ for which \autoref{eq:outline_goal_2} holds. For that we use the Harder-Narasimhan filtration $0=\scr{F}^0 \subseteq \dots \subseteq \scr{F}^r$ of $f_* \scr{O}_X(q(-K_{X/T}-f^*H))$. Let the basis $v_1,\dots, v_l$ be any basis adapted to the restriction of this filtration over $t$, that is, to $0=\scr{F}^0_t \subseteq \dots \subseteq \scr{F}^r_t$. The lower part of the filtration, until the graded pieces reach slope $2g$ (where $g$ is the genus of $T$), is globally generated. Furthermore, there is an induced Harder-Narasimhan filtration on \autoref{eq:tensor_product_bundle_outline}. The slope at least $2g$ part of the latter filtration is globally generated such that its restriction over $t \in T$ is generated by simple tensors in $v_i$ (\autoref{prop:HN_Tensor_product}). Hence, if $\scr{E}_{q,m}'$ is this part of the Harder-Narasimhan filtraton, then it is enough to prove that
\begin{equation}
\label{eq:outline_goal_3}
\displaystyle\lim_{m \to \infty} \frac{\rk \scr{E}_{q,m}'}{\rk f_*^{(m)} \scr{O}_{X^{(m)}} \left(q\left(-K_{X^{(m)}/T}- \left(f^{(m)} \right)^* m H\right)\right) }=1,
\end{equation}
The final trick of the semi-positivity part is then that \autoref{eq:outline_goal_3} can be translated to a probability limit, which then is implied by the central limit theorem of probability theory (\autoref{prop:vectP2}).
We explain here the probability theory argument via the example of
\begin{equation*}
\scr{F}_m:=\displaystyle\bigotimes_{m \textrm{ times}} (\scr{O}_{\mathbb{P}^1}(-1) \oplus \scr{O}_{\mathbb{P}^1}(2)).
\end{equation*}
The claim then is that as $m$ goes to infinity the rank of the non-negative degree part of $\scr{F}_m$ over the rank of $\scr{F}_m$ converges to $1$. It is easy to see that this is the following limit:
\begin{equation*}
\sum_{0 \leq i \leq m, 2i - (m-i) \geq 0 } {m \choose i} \left(\frac{1}{2} \right)^n =
\underbrace{\sum_{0 \leq i \leq m, i \geq \frac{m}{3} } {m \choose i} \left(\frac{1}{2} \right)^n \geq \sum_{0 \leq i \leq m, i \geq \frac{m}{2}- A\frac{\sqrt{m}}{4} } {m \choose i} \left(\frac{1}{2} \right)^n }_{\textrm{for $m$ big enough, where $A>0$ is an arbitrary fixed real number}}
\end{equation*}
The latter is the probability when flipping a coin $m$ times one gets at least $\frac{m}{2} - A\frac{\sqrt{m}}{4}$ heads. Note that for this $m$-times flipping the expected value is $\frac{m}{2}$ and $\sqrt{m}$-times the square deviation is $\frac{\sqrt{m}}{4}$. Hence, the above probability converges to $\int_A^\infty \frac{1}{\sqrt{2 \pi}} e^{\frac{-x^2}{2}} dx$ by the classical De Moivre-Laplace theorem, a special case of the central limit theorem. We obtain \autoref{eq:outline_goal_3} by taking $A \to \infty$ limit, and using that the above integral integrates the density function of the standard Gaussian normal distribution.
\subsubsection{ Nefness threshold.}
\label{sec:outline_nefness_threshold}
This part uses the same ideas as the above semi-positivity part, but in a different logical framework. That is, the argument is not a proof by contradiction. Instead, the starting point is that $\left(-K_{X/T} + \left( f^{(m)} \right)^* \left(\frac{\lambda_f}{v (n+1)} + H \right) \right)^{n+1}>0$. Hence, again up to a little perturbation and by using the ideas of the previous point, there is an integer $m>0$ such that there exists a $\Gamma \in \left|-\delta K_{X^{(m)}/T} + \left( f^{(m)} \right)^* m \left(\frac{\delta \lambda_f}{v (n+1)} + H \right) \right|_{\mathbb{Q}}$ for which $\left( X^{(m)}_t,\Gamma_t \right)$ is klt for $t \in T$ general. Then standard semi-positivity argument (\autoref{prop:semi_positivity_engine_downstairs}) shows that
\begin{equation*}
K_{X^{(m)}/T} -\delta K_{X^{(m)}/T} + \left( f^{(m)} \right)^* m \left(\frac{\delta \lambda_f}{v (n+1)} + H \right) = (1- \delta) K_{X^{(m)}/T} + \left( f^{(m)} \right)^* m \left(\frac{\delta \lambda_f}{v (n+1)} + H \right)
\end{equation*}
is nef. Lastly, one divides by $\delta -1$, converges to $0$ with $H$, and lastly by a standard lemma (\autoref{lem:product_nef}) removes the $(\_)^{(m)}$.
\subsubsection{ Positivity.}
\label{sec:outline_positivity}
The rough idea here is to use a twisted version of the ampleness lemma (\cite[3.9 Ampleness Lemma]{Kollar_Projectivity_of_complete_moduli}, with little modifications in \cite[Thm 5.1]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}). We need a twisted version of the ampleness lemma as the techniques developed until this point in the article do not work directly over higher dimensional bases. The main idea here is that to get bigness of $\lambda_f$ it is enough to show positivity of $\lambda_f$ over a very general element $C$ of each moving family of curves of $T$, in a bounded way. Below we explain how we do this.
The main benefit of proving the nefness threshold result above
is the following: one can prove, again using standard semi-positivity arguments (\autoref{prop:semi_positivity_engine_downstairs}), that $\scr{Q}:=f_* \scr{O}_X(-r K_{X/T} + \alpha f^* \lambda_f)$ is nef, for some constants $r$ and $\alpha$. Furthermore, these constants $r$ and $\alpha$ can be chosen to be uniform, as $f$ runs through all families obtained by base-changing on a very general element $C$ of a moving family of curves on $T$. Then, the ampleness lemma (\autoref{thm:ampleness}) gives an ample line bundle $B$ on $T$ such that for all curves $C$ as above, $C \cdot B \leq C \cdot \det \scr{Q}$. Then one can use another trick from (semi-)positivity theory, which we also learned from Viehweg's works. That is, for $q := \rk \scr{Q}$, there is an embedding
\begin{equation*}
\det \scr{Q} \to \bigotimes_{q \textrm{ times}} f_* \scr{O}_X(-r K_{X/T} + \alpha f^* \lambda_f) \cong f_*^{(q)} \scr{O}_X\left(-r K_{X^{(q)}/T} + q \alpha \left( f^{(q)} \right)^* \lambda_f \right),
\end{equation*}
Using the adjunction of $f^{(q)}_*$ and $\left( f^{(q)} \right)^*$ this implies the inequality of divisors
\begin{equation*}
\left(f^{(q)}\right)^* B \leq \left(f^{(q)}\right)^* \det \scr{Q} \leq - r K_{X^{(q)}/T} + q \alpha \left( f^{(q)} \right)^* \lambda_f,
\end{equation*}
which survives the restriction over $C$ by the genericity assumption in the definition of the latter.
From here, a simple intersection computation shows that $C \cdot B$ bounds $\deg \lambda_f|_C$ from below up to some uniform constants, not depending on the choice of $C$ (see the end of the proof of point \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary}).
\subsection{Variation}
\label{sec:variation}
\begin{definition}
\label{def:var}
Let $f : X \to T$ be a flat morphism between normal projective varieties, with $-K_{X/T}$ $\mathbb{Q}$-Cartier and $f$-ample. Let $q_0$ be an integer such that $q_0K_{X/T}$ is Cartier, and for all positive integers $q_0|q$, set $\scr{L}_q:= \scr{O}_X(-qK_{X/T})$. As $\scr{L}_q$ provides a relatively ample polarization, the $\Isom$ scheme $I:=\Isom_{T \times T} ( p_1^* f , p_2^* f)$ exists together with the two natural projections $q_i : I \to T$. Let $I'$ be the image of $(q_1, q_2) : I \to T \times T$. Then, there is a non-empty open set $U \subseteq T$ where the fibers of $p_1|_{I'}: I' \to T$ have the same dimension, say $d$. This dimension is the dimension of a general isomorphism equivalence class of the fibers of $f$. As these isomorphism equivalence classes (at least general ones) would be exactly the fibers of any reasonable moduli map, one defines the \emph{variation} of $f$ as
\begin{equation}
\label{eq:var}
\var (f):= \dim T - d.
\end{equation}
$f$ has \emph{maximal variation}, if $\var (f)=\dim T$.
\end{definition}
\subsection{Curve base}
\begin{notation}
\label{notation:CM_positivity}
In the situation of \autoref{notation:CM_semi_positivity}, assume that
\begin{enumerate}
\item $\delta>1$, where $\delta= \delta\left(X_{\overline{t}}, \Delta_{\overline{t}}\right)$ for very general geometric points $\overline{t} \in T$, and
\item $\deg \lambda_{f, \Delta} =0$.
\end{enumerate}
\end{notation}
\begin{theorem}
\label{thm:no_divisor}
In the situation of \autoref{notation:CM_positivity}, for each ample $\mathbb{Q}$-divisor $L$ on $T$, $|-K_{X/T} -\Delta -f^*L|_{\mathbb{Q}}=\emptyset$.
\end{theorem}
\begin{proof}
Assume that $\Gamma \in |-K_{X/T} -\Delta -f^*L|$. Using \cite[Thm 1.2]{Fujita_Openness_results_for_uniform_K-stability}, \autoref{def:K_stable} and \autoref{prop:delta_general_fiber}, choose a small rational number $\varepsilon>0$ such that for very general geometric points $\overline{t} \in T$ we have $\delta\left(X_{\overline{t}}, \Delta_{\overline{t}} + \varepsilon \Gamma_{\overline{t}}\right)>1$. Then,
\begin{multline*}
0 \geq
\underbrace{ (-K_{X/T} - \Delta - \varepsilon \Gamma)^{n+1} }_{\textrm{by \autoref{thm:semi_positivity_curve}}}
= (-K_{X/T} - \Delta + \varepsilon (K_{X/T} + \Delta + f^*L))^{n+1}
%
=
%
(-(1-\varepsilon) (K_{X/T} + \Delta) + \varepsilon f^* L )^{n+1}
%
\\ =
%
(1-\varepsilon)^n \left( (1-\varepsilon)(- K_{X/T} - \Delta)^{n+1} + (n+1)\varepsilon(-K_{X/T} - \Delta)^n f^* L \right)
%
\\ =
%
\underbrace{ (n+1)\varepsilon(1-\varepsilon)^n (-K_{X_t} - \Delta_t)^n \deg L }_{(- K_{X/T} - \Delta)^{n+1} =0} >0.
\end{multline*}
This is a contradiction.
\end{proof}
\begin{notation}
\label{notation:CM_positivity_HN}
In the situation of \autoref{notation:CM_positivity},
\begin{enumerate}
\item let $q_0>0$ be an integer such that $q_0 (-K_{X/T} - \Delta)$ is Cartier,
\item for each integer $q_0 |q$, define $\scr{E}_q:= f_* \scr{O}_X(q(-K_{X/T} - \Delta))$, and set $0=\scr{F}^0_q \subseteq \scr{F}^1_q \subseteq \dots \subseteq \scr{F}^{s_q-1}_q \subseteq \scr{F}^{s_q}_q$ be the Harder-Narasimhan filtration of $\scr{E}_q$. Set $\scr{G}^i_q:= \scr{F}^i_q/ \scr{F}^{i-1}_q$,
\item let $g$ be the genus of $T$.
\end{enumerate}
\end{notation}
\begin{lemma}
\label{lem:2g_slope_bound}
In the situation of \autoref{notation:CM_positivity_HN}, for every positive integer $q_0 | q$, $\mu(\scr{F}_q^1) \leq 2g$.
\end{lemma}
\begin{proof}
Assume the contrary, that is, $\mu(\scr{F}_q^1)>2g$, and let $t \in T$ be an arbitrary closed point. According to \autoref{prop:semi_stable_globally_generated}, $\scr{F}_q^1(-t)$ is globally generated. So, there is a $\Gamma'\in |q (-K_{X/T} - \Delta) - f^* L'|$, where $L'$ is the divisor determined by $t$ on $T$. Hence, for $\Gamma := \frac{\Gamma}{q}$ and $L:= \frac{L'}{q}$ we have $\Gamma \in |-K_{X/T} - \Delta - f^* L |_{\mathbb{Q}}$. This contradicts \autoref{thm:no_divisor}.
\end{proof}
\begin{proposition}
\label{prop:no_positive_slopes}
In the situation of \autoref{notation:CM_positivity_HN}, for every positive integer $q_0 | q$, $\mu(\scr{F}_q^1) \leq 0$.
\end{proposition}
\begin{proof}
Assume that $\mu\left( \scr{F}_q^1\right) >0$, and let $\scr{H}$ be the image of
\begin{equation*}
\xi: \left( \scr{F}_q^1 \right)^{\otimes m } \to \scr{E}_{qm}
\end{equation*}
for some $m \gg 0$. \emph{We claim that $\scr{H}$ is not zero} because of the following: Let $\eta$ be the generic point of $T$. Then any $x \in \left( \scr{F}_q^1\right)_\eta$ can be identified with some $\tilde{x} \in H^0\left( X_\eta,q \left( - K_{X_\eta} - \Delta_\eta \right) \right)$, in which case $\xi \left( x^{\otimes m}\right)$ gets identified with $\tilde{x}^m \in H^0\left( X_\eta, mq \left(- K_{X_\eta} - \Delta_\eta \right) \right)$. In particular, the following implications conclude our claim:
$
x \neq 0 \Rightarrow \tilde{x} \neq 0 \Rightarrow \tilde{x}^m \neq 0 \Rightarrow \xi\left( x^{\otimes m} \right) \neq 0$.
Let then $j$ be the smallest integer such that $\scr{F}_{qm}^j$ contains $\scr{H}$, and let $\scr{H}'$ be the image of $\scr{H}$ in $\scr{G}_{qm}^j$. By the choice of $j$, $\scr{H}' \neq 0$, and as $\scr{H}'$ is a surjective image of $ \left( \scr{F}_q^1 \right)^{\otimes m }$:
\begin{equation*}
\mu\left(\scr{F}_{mq}^1\right)
\geq
\underbrace{\mu \left(\scr{G}_{mq}^j\right)}_{\parbox{57pt}{\tiny by the definition of the Harder-Narasimhan filtration}} \geq
\underbrace{\mu(\scr{H}') }_{\parbox{45pt}{\tiny $\scr{G}_{mq}^j$ is semi-stable}}
>
\underbrace{\mu \left( \left( \scr{F}_q^1 \right)^{\otimes m } \right)}_{\parbox{65pt}{\tiny $ \left( \scr{F}_q^1 \right)^{\otimes m }$ is semi-stable according to \autoref{prop:tensor_product_semi_stable}}}
=
\underbrace{m \mu \left(\scr{F}_q^1\right)}_{\textrm{\autoref{prop:tensor_product_semi_stable}}}
>
\underbrace{ 2g}_{\parbox{53pt}{\tiny $m \gg 0$, and we assumed that $\mu\left( \scr{F}_q^1 \right)>0$}} .
\end{equation*}
This contradicts \autoref{lem:2g_slope_bound}.
\end{proof}
\begin{theorem}
\label{thm:positivity_curve}
In the situation of \autoref{notation:CM_positivity}, if $q>0$ is an integer such that $-q(K_{X/T} + \Delta)$ is Cartier, then
$f_* \scr{O}_X(-q (K_{X/T} + \Delta))$ is a semi-stable vector bundle of slope $0$.
\end{theorem}
\begin{proof}
First,
\autoref{thm:bounding_nef_threshold} yields that $-K_{X/T} - \Delta$ is nef. Then, $f_* \scr{O}_X( q(-K_{X/T}- \Delta))$ is also nef, by \autoref{prop:semi_positivity_engine_downstairs} taking into account the $\mathbb{Q}$-linear equivalence
\begin{equation*}
q(-K_{X/T}- \Delta) \sim_{\mathbb{Q}} K_{X/T} + \Delta + (q+1)(-K_{X/T}- \Delta).
\end{equation*}
Finally, \autoref{prop:no_positive_slopes}, concludes our proof.
\end{proof}
\subsection{Ampleness lemma}
\autoref{thm:ampleness} is an extract of the argument of the ampleness lemma of \cite{Kollar_Projectivity_of_complete_moduli} (one assumption removed in \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}). It will be one of the main technical ingredients for the proof of items \autoref{itm:semi_positive_no_boundary:big} and \autoref{itm:semi_positive_no_boundary:ample} of \autoref{thm:semi_positive_no_boundary} given in \autoref{sec:arbitrary_base_pos}.
\begin{theorem}
\label{thm:ampleness}
Let $V$ be a vector bundle of rank $v$ on a normal projective variety $T$ over $k$, and let $\phi : W:=\Sym^d (V) \twoheadrightarrow Q$ be a surjective homomorphism onto another vector bundle, where the ranks are $w$ and $q$, respectively. Assume that there is an open set, where the map of sets $T(k) \to \Gr(w,k)/\GL(v, k)$ induced by $\phi$ is finite to one. Then, for each ample Cartier divisor $B$ on $T$ there is an integer $m>0$ and a non-zero homomorphism
\begin{equation*}
\Sym^{qm}\left( \bigoplus_{i=1}^w W \right) \to \scr{O}_T(-B) \otimes ( \det Q)^m.
\end{equation*}
\end{theorem}
\begin{proof}
The proof is contained in \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}, but not presented there ideally for our purposes, so we give a recipe of how to turn \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} into the above statement. First, specialize \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} to the case of projective base and the special choice of $W = \Sym^d (V)$ and $G=\GL(v,k)$, where the latter two choices are fine according to \cite[Rem 5.3]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}. At this point the assumptions of \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} become identical to ours, except that in \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} was assumed to be weakly positive. However, this assumption is not used until the last three lines of the proof. In particular, the existence of the non-zero homomorphism of the displayed equation \cite[(5.5.5)]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} exists even without the weakly positive assumption. This is exactly our statement, taking into account the isomorphism of the displayed equation of the proof of \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} after \cite[(5.5.5)]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}.
\end{proof}
\subsection{Arbitrary base}
\label{sec:arbitrary_base_pos}
\begin{proof}[Proof of point \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary}]
As in the proofs of \autoref{thm:semi_positivity_curve} and \autoref{thm:nef_threshold}, we may assume that $k$ is uncountable.
Let $\eta$ be the generic point of $T$.
\begin{enumerate}
\item Set $n:= \dim X - \dim T$, $v:=K_{X_\eta}^n$, $\delta:= \delta\left(X_{\overline{\eta}}\right)$.
\item Fix a rational number $\alpha$ such that $\alpha > \max\left\{ 1, \frac{\delta}{(\delta -1)v(n+1)} \right\}$.
\end{enumerate}
Throughout the proof $\iota : C \to T$ denotes the normalization of a very general member of an arbitrary covering curve family of $T$. Very general here means that it is not contained in countably many divisors $S_i$, which we will specify during the proof explicitly. Set:
\begin{itemize}
\item $\eta_C$ to be the generic point of $C$,
\item $Z:=X_C$ (note that as the fibers of $f$ are reduced, and the general ones are normal, $Z$ is normal),
\item $\sigma : Z \to X$ and $g : Z \to C$ be the induced morphisms,
\item $\lambda:=\lambda_g$.
\end{itemize}
Then the following holds:
\begin{itemize}
\item $\sigma^* K_{X/T} \cong K_{Z/C}$ by \autoref{rem:reduced_fibers_no_boundary}.\autoref{itm:relative_canonical_base_change:base_change}, and
$\lambda=\lambda_f|_C$ by \autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:smooth_base}.
\item a $\mathbb{Q}$-Cartier divisor $L$ is pseudo-effective if and only if $L \cdot C \geq 0$ (for any such $C$),
\item according to \autoref{prop:delta_general_fiber}, $\delta\left(X_{\overline{\eta_C}} \right) = \delta$ (assuming we add the countably many divisors to $S_i$, over which $\delta(X_t)<\delta$, which are given by \autoref{prop:delta_general_fiber}). In particular, as $\delta>1$ the very general fibers of $g$ are uniformly $K$-stable, and hence klt, see \cite[Theorem 1.3]{Odaka_The_GIT_stability_of_polarized_varieties_via_discrepancy}.
\item in particular, by \autoref{thm:semi_positivity_curve}, $\deg \lambda \geq 0$,
\item by \autoref{thm:bounding_nef_threshold}, $-K_{Z/C} + \alpha g^* \lambda$ is nef and $g$-ample.
\end{itemize}
It is important that throughout the proof all constants (so all rational numbers) will be fixed independently of the particular choice of $C$ (for which there are two choices, first one choses a covering family, and then a very general member of that). For this reason, whenever such a constant is fixed, we do it in a numbered list item (see points above and below).
Choose integers $r \geq 2$ and $d>0$ such that
\begin{enumerate}[resume]
\item $rK_{X/T}$ and $r\alpha \lambda_f$ are Cartier,
\item $h^i(X_t, -rK_{X_t})=0$ for all $i>0$ and all $t \in T$,
\item $-rK_{X_t}$ is very ample for all $t \in T$,
\item the multiplication maps $W:=\Sym^d f_* \scr{O}_X (-r K_{X/T}) \to f_* \scr{O}_X (-dr K_{X/T})=:Q$ are surjective, and
\item \label{itm:positivity:classifying_map} for all $t \in T$, $K_{t}:=\Ker \left( \Sym^d H^0\left(X_t, -r K_{X_t}\right) \to H^0\left(X_t,-d r K_{X_t}\right) \right)$ generates $\scr{I}(d)$, where $\scr{I}$ is the ideal of $X_t$ via the embedding $\varphi_{\left|-rK_{X_t} \right|} : X_t \to \mathbb{P}^{v-1}$, where $v:= \rk f_* \scr{O}_X (-r K_{X/T})$ and $\varphi_{\left|-rK_{X_t} \right|}$ is defined only up to the action of $\GL(v,k)$ on the target. Note that this is achievable because $\scr{I}$ form a flat family as $t$ varies.
In particular, if we set $w:= \rk W$ and $q:= \rk Q$, then for every $t \in T(k)$, $K_{t} \subseteq W_t$ determines $X_t \hookrightarrow \mathbb{P}^{v-1}$ up to the action of $\GL(v, k)$, which then means that the orbit of $K_{t}$ in $\Gr(w,q)/G(v,k)$ determines $X_t$ up to isomorphism. Therefore if we apply \autoref{thm:ampleness} for $W \to Q$, then the fibers of the classifying map $T(k) \to \Gr(w,q)/G(v,k)$ are contained in the isomorphism classes of the fibers of $f$ and hence, by the maximal variation assumption, there is an open set where these fibers are finite.
\end{enumerate}
As,
\begin{equation*}
r (- K_{Z/C} + 2\alpha g^* \lambda) = K_{Z/C} + \underbrace{ (r+1)( -K_{Z/C} + \alpha g^* \lambda) + \underbrace{(r-1) \alpha}_{>1} g^* \lambda}_{\textrm{nef and $g$-ample}},
\end{equation*}
by \autoref{prop:semi_positivity_engine_downstairs}, $g_* \scr{O}_Z(r (- K_{Z/C} + 2\alpha g^* \lambda)) $ is a nef vector bundle. Set
\begin{equation*}
M:= r( -K_{X/T} + 2 \alpha f^* \lambda_f).
\end{equation*}
Note that the conclusions of point \autoref{itm:positivity:classifying_map} about the finiteness of the classifying map hold also for $-rK_X$ replaced by $M$, as $f_* \scr{O}_X(M)$ and $f_* \scr{O}_X(dM)$ differs from $f_* \scr{O}_X(-rK_{X/T})$ and $f_* \scr{O}_X(-rdK_{X/T})$ only by a twist with $r2 \alpha \lambda_f$ and $dr2 \alpha \lambda_f$, respectively. So,
\autoref{thm:ampleness} yields an ample divisor $B$ on $T$, an integer $m>0$ and a non-zero homomorphism as follows (see point \autoref{itm:positivity:classifying_map} above for the definition of $w$ and $q$):
\begin{equation*}
\xi : \Sym^{qm}\left( \bigoplus_{i=1}^w \Sym^d (f_* \scr{O}_X(M)) \right) \to \scr{O}_X(-B) \otimes \left(\det f_* \scr{O}_X(dM) \right)^m.
\end{equation*}
As the target of $\xi$ is a line bundle, there exists a divisor, on the complement of which $\xi$ is surjective. Let us add this divisor to $S_i$. Then $\xi|_C$ is a non-zero homomorphism as follows:
\begin{equation*}
\xi_C : \Sym^{qm}\left( \bigoplus_{i=1}^w \Sym^d( g_* \scr{O}_Z(M_C)) \right) \to \scr{O}_C(-B_C) \otimes \left( \det g_* \scr{O}_Z(dM_C) \right)^m.
\end{equation*}
Define
\begin{equation*}
\scr{A}:=\det f_* \scr{O}_X(dr ( -K_{X/T} + 2 \alpha f^* \lambda_f))= \det f_* \scr{O}_X(dM),
\end{equation*}
and let $A$ be a divisor corresponding to $\scr{A}$.
As $g_* \scr{O}_Z(M)$ is nef and hence so is every bundle that admits a generically surjective map from the left side of $\xi_C$, we obtain that
\begin{equation}
\label{eq:positivity:B_A}
\deg \scr{A}|_C = \deg \det g_* \scr{O}_Z(dM_C)) \geq \frac{B \cdot C}{m} .
\end{equation}
Consider now, the natural embedding:
\begin{equation*}
\alpha : \det f_* \scr{O}_X(dM) \hookrightarrow \bigotimes_{i=1}^q f_* \scr{O}_X(dM) \cong f^{(q)}_* \scr{O}_{X^{(q)}}\left(dM^{(q)}\right),
\end{equation*}
given by the embedding of representations $\det \to \bigotimes_{i=1}^q $ of $\GL(q,k)$.
Hence, by adjunction of $f_*^{(q)}(\_)$ and $\left(f^{(q)}\right)^*(\_)$ one can write $\left(f^{(q)} \right)^* A + D = dM^{(q)}$, where $D$ is an effective divisior on $X^{(q)}$. Furthermore, as $\alpha$ is a morphism of vector bundles, and the formation of $f_* \scr{O}_X(dM)$ is compatible with base-change, $D$ does not contain any fiber. By the continuity of log canonical threshold, there is a $0<\varepsilon < \frac{1}{rd}$ such that $\left(X_t^{(q)}, \varepsilon D_t\right)$ is klt for general closed points $t \in T$. In particular by the genericity of $C$ the same holds also for general $t \in C$. Then, if we define $N:= d r( -K_{X/T} + 3 \alpha f^* \lambda_f)$,
according to \autoref{cor:semi_positivity_engine_upstairs}, the following divisor is nef ($Z^{(q)}$ is normal by \autoref{lem:normal_stable_pullback_fiber_product}.\autoref{itm:normal_stable_pullback_fiber_product:fiber_product}).
\begin{multline*}
\underbrace{K_{Z^{(q)}/C} + \varepsilon D_C }_{\parbox{80pt}{\tiny $\left(Z^{(q)}_t, \varepsilon \left(D_C\right)_t \right)$ is klt for $t \in C$ general}}
+
\underbrace{\left(dr + 1 - \varepsilon rd \right)\left( -K_{Z^{(q)}/C} + 2 \alpha q \left(g^{(q)}\right)^* \lambda \right) + (dr-2) \alpha q \left(g^{(q)}\right)^* \lambda}_{\textrm{nef and $f$-ample $(r \geq 2, d>0)$}}
%
\\ \sim \left( N^{(q)}_C - \varepsilon \left(g^{(q)} \right)^* A_C \right) = \left(N_C - g^* \frac{\varepsilon }{q} A_C \right)^{(q)}
\end{multline*}
Set $\varepsilon':= \frac{\varepsilon}{qr d}$. Then we have that $\frac{N_C}{dr} - \varepsilon' g^*A_C$ is nef according to \autoref{lem:product_nef}.
So,
\begin{multline*}
0 \leq (-K_{Z/C} + 3\alpha g^* \lambda - \varepsilon' g^* A_C)^{n+1}
=
\left(\underbrace{-K_{Z/C} + \frac{g^*\lambda}{v(n+1)}}_{\parbox{90pt}{\tiny top self-intersection is $0$ by the definition of $\lambda$}} + \left(3\alpha - \frac{1}{v(n+1)} \right) g^* \lambda - \varepsilon' g^*A_C \right)^{n+1}
\\ =
(n+1) v \deg \left( \left(3\alpha - \frac{1}{v(n+1)} \right) \lambda - \varepsilon' A_C \right)
\leq
\underbrace{(n+1) v \deg \left( \left(3\alpha - \frac{1}{v(n+1)} \right) \lambda - \varepsilon' \frac{B_C}{m} \right)}_{\textrm{equation \autoref{eq:positivity:B_A}}}
\end{multline*}
Hence, $ \left(3\alpha - \frac{1}{v(n+1)} \right) \lambda_f - \varepsilon' \frac{ B}{m}$ is pseudo-effective (as it dots to at least zero with each movable class). Therefore, $\lambda_f$ is the sum of an ample and a pseudo-effective $\mathbb{Q}$-divisor, so $\lambda_f$ is big.
\end{proof}
\begin{proof}[Proof of point \autoref{itm:semi_positive_no_boundary:ample} of \autoref{thm:semi_positive_no_boundary}]
By Nakai-Moishezon it is enough to prove that for all normal varieties $V$ mapping finitely to $X$, $\left( \lambda_f|_V \right)^{\dim V} >0$. However, using \autoref{prop:CM_base_change}, this we may obtain by replacing $f : X \to T$ with $f_V : X \times_T V \to V$, and applying point \autoref{itm:semi_positive_no_boundary:big} to $f_V$.
\end{proof}
\begin{proof}[Proof of point \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary}]
Let $m>0$ be the integer such that $L:=-mK_{X/T}$ is an $f$-very ample divisor, and let $N$ be the line bundle, which is the inverse of the leading term of the Knudsen-Mumford expansion for $L$ (\autoref{notation:Paul_Tian}). According to \cite[Thm 6.1]{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds} it is enough to prove that $N$ is nef and that for all closed subvarieties $V$ of $T$ intersecting $U$, $\left(N|_V\right)^{\dim V}>0$. In fact, also nefness is a similar intersection question, that is, $\deg N|_C \geq 0$ for all curves $C$ of $T$.
Note that the normalization of $C$ is automatically a scheme, hence we may assume that $C$ is a smooth scheme.
Furthermore, $V$ has a finite cover by a scheme \cite[Tag 04V1]{stacks-project} and by Nagata's theorem and resolution of singularities we may also assume that $V$ is projective and smooth. Therefore, by replacing $V$ by this generically finite cover we may assume that $V$ is also a smooth scheme.
As $N$ is compatible with base-change (\autoref{lem:Knudsen_mumford_base_change}), by relaxing the isomorphism class assumption, we can replace $T$ by $C$ or $V$. Then, the base is smooth and projective, and we have to prove that without any variation assumption $N$ is nef, and if furthermore the variation is maximal, then $N$ is even big. By \autoref{prop:2_defs_CM_same}.\autoref{itm:2_defs_CM_same:leading_term}, we can prove this for the CM line bundle, instead of $N$, which is then shown in points \autoref{itm:semi_positive_no_boundary:nef} and \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary}.
\end{proof}
\begin{proof}[Proof of \autoref{thm:lambda_non_big}]
Choose $q$ big enough such that $-q(K_{X/T} + \Delta)$ is Cartier and without higher cohomology on the fibers.
Let $H_i \in |H|$ be general for $i=1,\dots, \dim T -1$, and set $C:= \bigcap_{i=1}^{\dim T -1} H_i$. By the above generic choices, $Z:=X_C$ is normal. Furthermore, $C$ lies in the smooth locus of $T$, hence for base-change properties along $C \to T$ we may assume that $T$ is smooth. In particular, there is an induced boundary $\Delta_Z$ on $Z$ (\autoref{sec:base_change_relative_canonical_smooth_base}), for which $K_{X/T} + \Delta|_Z = K_{Z/C} + \Delta_Z$ (\autoref{prop:relative_canonical_base_change_normal}), and consequently
\begin{equation}
\label{eq:lambda_non_big:base_change}
f_* \scr{O}_X(-q(K_{X/T} + \Delta))|_C \cong \left(f_C\right)_* \scr{O}_{Z}\left(-q\left(K_{Z/C} + \Delta_Z \right) \right).
\end{equation}
Furthermore,
\begin{equation*}
0 =
\underbrace{\lambda_{f,\Delta} \cdot H^{\dim T -1}}_{\textrm{assumption}}
= \deg \lambda_{f,\Delta}|_C
=
\underbrace{\deg \lambda_{f_C, \Delta_Z} .}_{\textrm{\autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:smooth_base}}}
\end{equation*}
Therefore, according to \autoref{thm:positivity_curve}, $\left(f_C\right)_* \scr{O}_{Z}\left(-q\left(K_{Z/C} + \Delta_Z \right) \right)$ is a semi-stable vector bundle of slope $0$. However, then the isomorphism \autoref{eq:lambda_non_big:base_change} implies that $f_* \scr{O}_X (- q(K_{X/T} + \Delta))$ is $H$-semi-stable of slope $0$: if it had a subsheaf $\scr{F}$ of $H$-slope bigger than $0$, then for the saturation $\scr{F}'$ of $\scr{F}$, $\scr{F}'|_C$ would be a subbundle of positive degree of $\left(f_C\right)_* \scr{O}_{Z}\left(-q\left(K_{Z/C} + \Delta_Z \right) \right)$, which is a contradiction.
\end{proof}
\begin{proof}[Proof of \autoref{cor:proper_base}]
The proof is very similar to that of point \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary} above.
As in the above proof, $T$ has a generically finite cover by a smooth, projective scheme. By base-changing over this cover one may assume that the base is smooth and projective. By \autoref{prop:2_defs_CM_same}, we may replace $N$ by the CM-line bundle notion used in the present article (\autoref{def:CM}), and then points \autoref{itm:semi_positive_no_boundary:nef} and \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary} concludes the proof.
\end{proof}
\subsection{Framework and results}
In this and the next sections we work in the following setup:
\begin{notation}
\label{notation:CM_semi_positivity}
Let $f \colon (X,\Delta) \to T$ satisfy the following assumptions:
\begin{enumerate}
\item $T$ is a smooth, projective curve,
\item $X$ is a normal, projective variety of dimension $n+1$,
\item $f$ is a projective and surjective morphism with connected fibers,
\item $\Delta$ is an effective $\mathbb{Q}$-divisor on $X$,
\item $-(K_X + \Delta)$ is an $f$-ample $\mathbb{Q}$-Cartier divisor.
\item $(X_t,\Delta_t)$ is klt for general $t \in T$.
\end{enumerate}
\end{notation}
The main result of the section is the following, from which the statements of the introduction will follow in a quite straightforward manner.
\begin{theorem}
\label{thm:semi_positivity_curve}
In the situation of \autoref{notation:CM_semi_positivity}, if $\delta\left(X_{\overline{t}},\Delta_{\overline{t}}\right) \geq 1$ for very general geometric points $t \in T$, then $\deg \lambda_{f,\Delta} \geq 0$.
\end{theorem}
\subsection{Proofs}
\label{sec:semi_pos_proof}
The proof of \autoref{thm:semi_positivity_curve} will be by contradiction with the next proposition.
\begin{proposition}\label{prop:cont}
In the situation of \autoref{notation:CM_semi_positivity}, let $H$ be an ample $\mathbb{Q}$-divisor on $T$. Then, there do not exist $\mathbb{Q}$-Cartier divisors $\Gamma$ and $\widetilde{\Gamma}$ on $X$ such that:
\begin{enumerate}
\item\label{class} $\Gamma + \widetilde{\Gamma} \sim_{\mathbb{Q}} -K_{X/T}-\Delta-f^* H$,
\item \label{ample} $\widetilde{\Gamma}$ is nef, and
\item \label{klt} $(X_t,\Delta_t +\Gamma_t)$ is klt for $t \in T$ general.
\end{enumerate}
\end{proposition}
\begin{proof}
Assume that there exist $\Gamma$ and $\widetilde{\Gamma}$ as above.
Let $a>0$ be a rational number such that $-K_{X/T} - \Delta + af^* H$ is ample. Fix a rational number $\varepsilon > 0$ such that $\varepsilon a - (1-\varepsilon)< 0$. The following computation the yields a contradiction, as \autoref{cor:semi_positivity_engine_upstairs} yields that the right side is nef.
\begin{equation*}
\underbrace{ (\varepsilon a - (1-\varepsilon)) f^*H}_{\varepsilon a - (1-\varepsilon)<0 \Rightarrow \textrm{ this is not nef}}
\sim_{\mathbb{Q}}
\underbrace{K_{X/T} + \Delta + (1-\varepsilon) \Gamma }_{(X_t,\Delta_t +(1-\varepsilon) \Gamma_t ) \textrm{ is klt }}
+ \underbrace{(1-\varepsilon)\widetilde{\Gamma} + \varepsilon\left(- K_{X/T} -\Delta + af^* H \right)}_{\textrm{ample}}
\end{equation*}
\end{proof}
\begin{proof}[Proof of \autoref{thm:semi_positivity_curve}]
As both the consequences and the conditions of the theorem are invariant under base-extension to another algebraically closed field, we may assume that $k$ is uncountable. In particular, the very general geometric fibers show up also amongst closed fibers.
First, according to \autoref{lem:reducing_reduced_fibers} we may assume that all fibers of $f$ are reduced. This is to guarantee that the $m$-times iterated fiber product $X^{(m)}$ is normal for any integer $m>0$, according to \autoref{lem:normal_stable_pullback_fiber_product}.
We argue by contradiction, so assume that $\deg \lambda_{f,\Delta}<0$. For $m$ big enough, we are going to produce divisors $\Gamma$ and $\widetilde{\Gamma}$ on $\left( X^{(m)},\Delta^{(m)}\right) $ whose existence contradicts \autoref{prop:cont}.
Fix a closed point $t$ in $T$ such that $X_t$ is normal, $X_t \not\subseteq \Supp \Delta_t$, $(X_t,\Delta_t)$ is klt and $\delta(X_t, \Delta_t) \geq 1$ (using \autoref{prop:delta_general_fiber}). Let $H$ be an ample line bundle on $T$. Fix rational numbers $a,\varepsilon>0$ and $0<c<1$ and an integer $q>0$, such that:
\begin{enumerate}[itemsep=4pt]
\item the intersection product inequality $(-K_{X/T} -\Delta- \varepsilon f^* H)^{n+1}>0$ holds. This is possible because \autoref{def:CM_log}
and the assumption $\deg \lambda_{f,\Delta}<0$ implies that $ (-K_{X/T} - \Delta)^{n+1}>0$. Set $M:= -K_{X/T} - \Delta - \varepsilon f^* H$.
\item $D:=-K_{X/T}-\Delta+af^*H$ is ample.
\item \label{negative}
$c<\frac{\varepsilon}{a+\varepsilon}$.
\item \label{Cartier} $qM$ is Cartier, which is possible, as $M$ is $\mathbb{Q}$-Cartier.
\item \label{higher_cohomology} $R^if_*\scr{O}_X(qM)=0$ for all $i>0$, which is possible, as $M$ is $f$-ample.
\item \label{degree} $\deg\left(f_*\scr{O}_X(qM)\right)>0$, using \autoref{lem:Mumford_line_bundles_over_curve}.
\item \label{delta} $\delta_q(X_t,\Delta_t)>1-c$, using \autoref{thm:BJ}.
\end{enumerate}
From now on, let $\scr{E}:=f_*\scr{O}_X(qM)$. Remark that according to \cite[Lemma 3.6]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} for every integer $m>0$,
$$
\scr{E}^{\otimes m}=f_*^{(m)} \scr{O}_{X^{(m)}} \left( q M^{(m)} \right) \cong f_*^{(m)} \scr{O}_{X^{(m)}}\left( q\left( - K_{X^{(m)}/T} -\Delta^{(m)}- m\varepsilon \left(f^{(m)}\right)^* H\right)\right),
$$
and by item \autoref{higher_cohomology}, the following base change holds
$$
\scr{E}^{\otimes m}_t=H^0\left(X_t^{(m)}, q\left( - K_{X_t^{(m)}} -\Delta^{(m)}_t \right) \right).
$$
In general, it is not possible to lift a basis of $\scr{E}_t$ to sections of $\scr{E}$. However, thanks to \autoref{prop:vectP2}, we can choose a basis $e_i$ of $\scr{E}_t$, an integer $m>0$, and $\ell$ global sections $s_i$ of $\scr{E}^{\otimes m}$ so that the sections $s_i$, when restricted over $t$, are linearly independent pure tensor in the $e_i$, and furthermore
\begin{equation}\label{bound}
\frac{\ell}{h^0\left(X_t^{(m)},-q\left(K_{X_t^{(m)}}+\Delta_t^{(m)}\right)\right)}
>
\underbrace{\frac{1-c}{\delta_q(X_t,\Delta_t)}}_{\parbox{57pt}{\tiny $<1$ according to assumption \autoref{delta}}} \,.
\end{equation}
We are now ready to construct $\Gamma$ and $\widetilde{\Gamma}$ on $X^{(m)}$ as in \autoref{prop:cont}. We let
$$ \Gamma:=(1-c)\frac{1}{q\ell}\sum_{i=1}^{\ell}\{s_i=0\}\,,$$
and
$$ \widetilde{\Gamma}:=cD^{(m)} \,.$$
To complete the proof of \autoref{thm:semi_positivity_curve}, we have to prove that $\Gamma$ and $\widetilde{\Gamma}$ are as in \autoref{prop:cont}, with $f$ replaced by $f^{(m)}$. To check item \autoref{class}, remark that
$$\Gamma+\widetilde{\Gamma} \sim_{\mathbb{Q}} -K_{X^{(m)}/T}-\Delta^{(m)}+ m\left( c a - (1 - c) \varepsilon \right) \left(f^{(m)}\right)^*H.$$
Furthermore, because of assumption \autoref{negative}, $c a - (1 - c) \varepsilon <0$ holds; so, we may apply \autoref{prop:cont} replacing $H$ by $-m\left( c a - (1 - c) \varepsilon \right) H$. Item \autoref{ample} of \autoref{prop:cont} follows from the ampleness of $D$.
To prove of item \autoref{klt} of \autoref{prop:cont}, we compute the log canonical threshold. We first remark that, since the sections $s_i$ restricted to $X_t^{(m)}$ are linearly independent pure tensors in the $e_i$, we have that
$$
\frac{\ell}{h^0\left(X_t^{(m)},-q\left(K_{X_t^{(m)}}+\Delta^{(m)}_t\right)\right)} \Gamma_t \leq (1-c)P
$$
for the $q$-product basis type divisor $P$ on $X_t^{(m)}$ associated to $\{e_i\}$, as in \autoref{def:prod_basis} and \autoref{rem:prod_basis}. Using \autoref{prop:prod_basis}, we obtain that $\lct\left(X^{(m)}_t,\Delta^{(m)}_t;P_t\right)\geq \delta_q(X_t,\Delta_t)$. This yields
$$
\lct\left(X_t^{(m)},\Delta_t^{(m)};\Gamma_t\right)
\geq
\underbrace{\frac{\delta_q(X_t,\Delta_t)\ell}{(1-c)h^0\left(X_t^{(m)},-q\left(K_{X_t^{(m)}}+\Delta^{(m)}_t\right)\right)} >1}_{\textrm{by assumption \autoref{bound}}}.
$$
Hence, all assumptions of \autoref{prop:cont} are verified, implying that $\Gamma$ and $\widetilde{\Gamma}$ cannot exist. Therefore, we obtained a contradiction with our initial assumption that $\deg \lambda_{f,\Delta} <0$.
\end{proof}
\begin{proof}[Proof of \autoref{thm:semi_positive_boundary}]
\emph{The proof of point \autoref{itm:semi_positive_boundary:pseff}:}
As at the beginning of the proof of \autoref{thm:semi_positivity_curve}, we may assume that $k$ is uncountable.
According to \cite[Thm 0.2]{Boucksom_Demailly_Paun_Peternell_The_Pseudo_effective_cone_of_a_compact_Kahler_manifold_and_varieties_of_negative_Kodaira_dimension}, it is enough to show that $\lambda_{f,\Delta} \cdot C \geq 0$ for every morphism $\iota: C\to X$ from a smooth projective curve such that $C \to \iota(C)$ is the normalization and $\iota(C)$ is a very general curve in a family covering $T$. In particular, for a very general closed point $t \in \iota(C)$, $X_t$ is normal, $(X_t, \Delta_t)$ is klt and $\delta\left(X_t, \Delta_t\right) \geq 1$. Let $Z \to X_C$ be the normalization, $g : Z \to C$ the induced morphism and $\Delta_Z$ the boundary induced by $\Delta$ on $Z$ as explained in \autoref{sec:base_change_relative_canonical_smooth_base}. According to \autoref{prop:relative_canonical_base_change_normal}.\autoref{itm:relative_canonical_base_change:base_change}, $g : (Z, \Delta_Z) \to C$ satisfies the assumptions of \autoref{thm:semi_positivity_curve}. Hence the following computation concludes the proof of point \autoref{itm:semi_positive_boundary:pseff}:
\begin{equation*}
0 \leq
\underbrace{\deg \lambda_{g, \Delta_Z}}_{\textrm{\autoref{thm:semi_positivity_curve}}}
=
\underbrace{C \cdot \lambda_{f, \Delta}}_{\textrm{\autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:smooth_base}}}.
\end{equation*}
\emph{The proof of point \autoref{itm:semi_positive_boundary:nef}:}
In this case for each finite morphism $C \to T$ from a smooth projective curve, according to \autoref{sec:base_change_relative_canonical_normal_fibers}, $f_C : (X_C, \Delta_C) \to C$ satisfy the assumptions of \autoref{thm:semi_positivity_curve}. So:
\begin{equation*}
0 \leq
\underbrace{\deg \lambda_{f_C, \Delta_C}}_{\textrm{\autoref{thm:semi_positivity_curve}}}
=
\underbrace{C \cdot \lambda_{f, \Delta}}_{\textrm{\autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:normal_fiber}}}.
\end{equation*}
\end{proof}
\begin{proof}[Proof of points \autoref{itm:semi_positive_no_boundary:pseff} and \autoref{itm:semi_positive_no_boundary:nef} of \autoref{thm:semi_positive_no_boundary}]
These are special cases of \autoref{thm:semi_positive_boundary}.
\end{proof}
| {'timestamp': '2018-06-20T02:10:09', 'yymm': '1806', 'arxiv_id': '1806.07180', 'language': 'en', 'url': 'https://arxiv.org/abs/1806.07180'} |
\section{Conclusion\label{section:conclusion}}
We have implemented a simulation with the following characteristics:
\begin{itemize}
\item The simulation is composed of a cellular automaton embedded in a reactive environment providing global instants and broadcast events.
\item The basic components are threads that are executed in synchronous parallelism. Each cell of the cellular automaton is implemented by a thread. After detection, each particle is animated by several threads of the reactive environment.
\item The cells living at instant $t$ code the superposition state of the particle at instant $t$. Before detection, the evolution of the particle is defined by the cellular automaton rule ({{\bf U}} procedure).
\item Detection of a particle generates a broadcast event, instantaneously received by all the corresponding living cells of the cellular automaton.
\item After detection, living cells are instantaneously reset, and a particle animated by the reactive environment is created. The state of the created particle is randomly chosen among the basic states composing the superposition ({{\bf R}} procedure). This is the only source of nondeterminism in the simulation.
\item Entanglement of particles basically means that they share the same procedure {{\bf R}}. The shared event corresponding to {{\bf R}} is instantaneously broadcast to all entangled particles by the reactive environment.
\end{itemize}
There are at least two regards in which the simulation could certainly be improved:
\begin{itemize}
\item In the current simulation, superposition states are segments; they could be replaced by circular shapes centered on the source. Hexagonal CA's could be useful for that purpose.
\item One may wonder how to introduce gravity, for particles that have a mass. With real particles, this is not a problem: gravity can be naturally implemented as a field. But it is not clear how to introduce gravity when the particle is virtual (the cell rule of the cellular automaton should implement it).
\end{itemize}
\section{Introduction}
The work described here is an exercise in parallel programming inspired
by Physics (more precisely, by R. Penrose's books
\cite{Penrose-NewMind,Penrose-Road}). Actually, it presents a simulation
intended to reflect three rather intriguing aspects of Quantum Mechanics (QM):
\begin{enumerate}
\item {\it Self-interference}: this is basically Young's
experiment where a particle interferes with itself when passing through
two slits.
\item {\it Superposition}: the state of a particle is a superposition of
several basic states which disappears when a measure occurs.
\item {\it Entanglement}: a measure performed on one element of a pair
of entangled particles has an instantaneous effect on the other.
\end{enumerate}
The main objective of the simulation is to build global behaviours as
{\it parallel compositions} of elementary ones. The variant of
parallelism used is the {\it synchronous} one
\cite{Halbwachs-Synchrone}. The goal is not to exactly model reality
(actually, one gets something which is is {\it certainly not} a good
model of reality as it does not compute over probabilities) but to
mimic some aspects of it, linked to QM.
\input{mc}
We make the assumption of a discrete world built over a cellular
automaton (CA) \cite{ToffoliMargolus-CA}, implementing the {{\bf U}}
procedure. We do not model fields, which are non-discrete objects, but
CA's which are discrete ones. Actually, in CA's, both space and time
are discrete. The rate at which a CA is simulated is the limit
speed\footnote{it can be considered as the ``light speed'' of the system.}
of the system. CA's are embedded in a synchronous world which is
actually the one of the simulation. There are thus {\it global
instants} during which actions are considered as simultaneous. The
basic hypothesis is that the {{\bf U}} procedure is deterministic and local
(thus, not instantaneous, as it can take several instants for a change
to propagate in space). On the opposite, the {{\bf R}} procedure is
nondeterministic, global, and instantaneous (in a sense to be made
precise later). The main basic notions are: synchronous parallelism,
determinism of {{\bf U}} and nondeterminism of {{\bf R}}, discrete space (CA)
and discrete time (synchrony hypothesis).
In the simulation, a (virtual) particle at instant $t$ is represented by a
set of cells living at instant $t$. Each cell is in a basic state of the
particle, and the particle global state is the superposition of all the
basic states of the associated cells. Basic states are identified by
specific colors and cells are drawn according to the basic state they
hold. When superposition disappears, after execution of procedure {{\bf R}}, one
cell is chosen randomly and the created (real) particle falls in the
basic state of the chosen cell (the particle then receives the color
of the cell). Thus, the probability for a particle to fall in a given
basic state $\ket a$ depends on the number of cells holding this state: the
more there are cells holding $\ket a$, the more the probability to
choose $\ket a$ is high.
A measure is implemented by the emission of a broadcast event that is
instantaneously received by all the cells implementing the measured
object. This is basically the {{\bf R}} procedure. In reaction to {{\bf R}}, one cell
is nondeterministically chosen and the real particle is produced from
that cell (with its basic state).
A particle is virtual when it is implemented by the CA. It turns to
real after application of the {{\bf R}} procedure. A real particle is animated
by the synchronous simulation; it is a data structure with coordinates
and speed, animated by several elementary parallel behaviours (for
example, an inertia behaviour, which at each instant sets the
coordinates according to the speed, and a bouncing behaviour which
makes the particle bounce on the simulation borders\footnote{borders
do not have any physical significance; they are there just to contain
particles in a limited area.}.
The simulation basically shows the following aspects:
\begin{enumerate}
\item Self-interference is illustrated by Young's
experiment where a particle is emitted against a
wall with two slits. The particle passes through both slits, which
produces interferences. The simulation shows the production of
interferences, and their disappearance when one slit is obstructed.
\item The simulation shows the production of a real particle when a
detector reacts to the presence of a living cell; in this case, the {{\bf R}} procedure
entails the reduction to a basic state.
\item Pairs of entangled particles are considered. The measure of one
element of the pair triggers the {{\bf R}} procedure for it, which produces a
real particle. The measure also triggers the {{\bf R}} procedure for the
second particle. Moreover, it forces the choice of the basic state of the real
particle produced from the second virtual particle.
\end{enumerate}
Section \ref{section:simulation} describes the simulation. Section
\ref{section:implementation} describes the implementation and gives the
most important pieces of code. Some related work is described in
Section \ref{section:related-work}. Finally, Section
\ref{section:conclusion} concludes the text.
\subsection{Young's Experiment}
A new wall is introduced in Figure \ref{slits} which separates the
simulation in two parts. In the left image, two slits are present; in
the right image, the right slit is obstructed. This basically
corresponds to Young's experiment. In the right image, the structure
under the separation wall is reproduced over the
separation wall. In the left part, the two structures produced from the
two slits interfere when they overlap.
\begin{figure}[!htbf]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=220pt]{slit2.eps} &
\includegraphics[width=220pt]{slit1.eps}
\end{tabular}
\end{center}
\caption{Young's Experiment}
\label{slits}
\end{figure}
We now consider the detection of particles in Young's
experiment. Figure \ref{slits-2} shows the situation after
detection of 100 particles (in presence of 2 slits).
\begin{figure}[!htbf]
\begin{center}
\includegraphics[width=300pt]{prob2.eps}
\end{center}
\caption{Young's Experiment - 2}
\label{slits-2}
\end{figure}
Once the number of slits is fixed, all particles reach the detector in
the same superposition of basic states (because {{\bf U}} is deterministic and
local). When the {{\bf R}} procedure is executed, a cell is randomly chosen
from the superposition and the particle gets the cell basic state. The
probability for a particle to fall on the basic state $\ket a$ only depends
on the percentage of cells with state $\ket a$ appearing in the
superposition. The superposition state thus directly codes the
probabilities associated with the basic states. The probability of
a basic state $\ket a$ in a superposition $S$ is:
\begin{equation}
Prob (\ket a, S) = \frac {\#a \in S} {\#S}
\end{equation}
These probabilities depend on the number of slits, on the localisation
of the detector and on the localisation of the slits. Here is for
example a typical result (with 1000 particles):
\begin{center}
\begin{tabular}{|c|cccccc|}
\hline
basic state &0 &1 &2 &3 &4 &5 \\ \hline
1 slit & 0.327 & 0.023 & 0.281 & 0.153 & 0.017 & 0.199 \\ \hline
2 slits & 0.226 & 0.039 & 0.298 & 0.171 & 0.116 & 0.150 \\ \hline
\end{tabular}
\end{center}
\noindent
In the configuration from which this table is produced, a particle
has for example probability 0.327 to fall in state 0 with 1 slit,
while it has probability 0.226 to fall in the same state with 2 slits.
| {'timestamp': '2011-01-13T18:32:47', 'yymm': '1101', 'arxiv_id': '1101.2133', 'language': 'en', 'url': 'https://arxiv.org/abs/1101.2133'} |
\section{Introduction}
Many computerized methods have been developed in the past 30 years to build
numerical models of sedimentary terrains from seismic and well data, where
geological layers are often both folded and faulted. Estimations and forecasts
based on such models may impact economic decisions, so numerical
representations of available data must be as accurate and consistent as
possible.
\ \\
One way of checking whether any sub-surface model is consistent is to bring it
back in time, to a state prior to faulting and folding for a given geological
horizon \citep{Moretti2008,Maerten2015}. If such a process fails, the
incriminated areas may point out inconsistencies in the present-day structural
model. If it is successful, the geologist can use this simpler restored model
to refine his interpretations and build a geological history of the study area
\citep{DurandRiard2013}.
\ \\
Most 3D geological restoration techniques based on mechanics of continuous media
assume that geological layers deform in a linear elastic manner
\citep{Maerten2015}. However, the faulted subsurface is a discontinuous medium
in which large, non-linear plastic deformations occur. Large deformations are
taken into account by some restoration methods
(e.g.~\citet{Muron2005,Moretti2006}) but induce a time-consuming, heavy
computation load for each restored stage. Moreover, mechanical restoration
methods may result in restored models with gaps and overlaps close to
fault-induced discontinuities, which are then minimized through debatable
numerical post-processes.
\ \\
\citet{Lovely2018} present a simple, purely geometrical restoration method based
on the commercially available implementation of the GeoChron theory
\citep{Mallet2014} provided by Emerson Paradigm\textsuperscript{\textregistered} in the
SKUA\textsuperscript{\texttrademark}
software package.
In this paper, we derive a full geometrical restoration theory from the fundamental
equations of the GeoChron model and show complete implementation results.
For input GeoChron models of any degree of geometrical and topological
complexity, our method handles both small and large deformations,
does not assume elastic behavior and does not require any prior knowledge of
geo-mechanical properties. Finally, the fundamental equations of this method intrinsically integrate
minimization of gaps and overlaps along faults without any need for
post-processing.
\section{Overview of the mathematical GeoChron framework}
This section briefly describes the main elements of the mathematical GeoChron
framework used in this article. The complete theory may be found in~\citet{Mallet2014}.
\ \\
As illustrated by Figure~\ref{GEOCHRONO_F2}, consider a
geo-stationary satellite pointing a camera
vertically towards a region of interest on the surface of the Earth.
This camera delimits a right-handed frame of three
orthogonal unit vectors $\{\bar\r_u, \bar\r_v, \bar\r_t\}$ where
$\bar\r_t$ is orthogonal to the surface of the Earth and oriented
upward. These three vectors define the edges of a box where
images shot by the camera are stacked in
chronological order throughout geological-time. For coherency with the
geological processes, the camera is geo-stationary in the sense
that its origin and $\{\bar\r_u,\bar\r_v,\bar\r_t\}$ vectors are ``attached''
to the tectonic plate which contains the domain of interest.
\begin{figure}
\centerline{\psfig{figure=./figures/GEOCHRONO_F2,width=116mm}}
\caption{
$\overline{G}$-space viewed as a continuous chronological stack of images $\{\overline{H}_t\}$ of
the surface of the Earth shot from a geo-stationary camera throughout
geological-time (in \citet{Mallet2014}, courtesy of EAGE).
}
\label{GEOCHRONO_F2}
\end{figure}\noindent
\ \\
Let $\overline{H}_\tau$ be a horizontal plane orthogonal to the
vector $\bar\r_t$ and corresponding to the one-to-one map of the sea floor at geological-time $\tau$.
$\overline{H}_\tau$ is identical to a picture of the sea floor taken at geological-time
$\tau$ and is therefore parallel to the pair of orthogonal unit
vectors $\{\bar\r_u, \bar\r_v\}$, which can thus be used as a 2D frame
for $\overline{H}_\tau$. As a consequence, for any given
origin $\bar{\bf p}_0(t)$ belonging to $\overline{H}_\tau$,
the pair of vectors $\{\bar\r_u, \bar\r_v\}$ induces a rectilinear coordinate
system $(u,v)$ on $\overline{H}_\tau$ such that:
\begin{equation} \label{eqn:geochron:3}
\bar{\bf p}\in \overline{H}_\tau \quad \Longleftrightarrow \quad \exists\
(u,v) \in {I\!\!R}^2 \ : \
\bar{\bf p} \ = \ \bar{\bf p}_0(\tau) + u\cdot\bar\r_u + v\cdot\bar\r_v
\end{equation}\noindent
At geological-time $\tau$, the $(u,v)$ rectilinear coordinate system
so defined can thus be used to locate on map
$\overline{H}_\tau$ any particle of sediment being deposited at that geological-time.
Therefore, the $(u,v)$ pair is called
``paleo-geographic coordinate'' system. On sea floor $H_\tau$, the reverse
image of the rectilinear coordinate axes $(u)$ and $(v)$ consists in a
curvilinear coordinate system.
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-UVT-transform,width=116mm}}
\caption{
Graphical characterization of direct and inverse $uvt$-transforms (in
\citet{Mallet2014}, courtesy of EAGE).
}
\label{GBR-UVT-transform}
\end{figure}\noindent
\ \\
Let $G$, also called $G$-space, be the domain of interest in stratified
sedimentary terrains. Two distinct coordinate systems characterize any particle
of sediment observed today at location $\r\in G$ in the subsurface:
\begin{itemize}
\item First, present-day horizontal geographic
coordinates $\{x(\r),y(\r)\}$ and altitude $z(\r)$ with respect to a given
direct 3D frame consisting of three orthogonal unit vectors $\{\r_x,\r_y,\r_z\}$
associated with the $G$-space, where $\r_z$ is vertical and oriented upward:
\begin{equation} \label{eqn:GBR-.a1}
\r \ = \ x(\r)\cdot\r_x + y(\r)\cdot\r_y + z(\r)\cdot\r_z
\quad \in G
\end{equation}\noindent
\item Second, paleo-geographic coordinates
$\{u(\r),v(\r)\}$ as they could have been observed at geological-time $t(\r)$
when the particle was deposited. These paleo-coordinates
$\{u(\r),v(\r),t(\r)\}$ define the location $\overline{\r}$ of the particle in a
``depositional'' space $\overline{G}$ also called $\overline{G}$-space:
\begin{equation} \label{eqn:GBR-1.b}
\overline{\r} \ = \ u(\r)\cdot\overline{\r}_u + v(\r)\cdot\overline{\r}_v + t(\r)\cdot\overline{\r}_t
\quad \in \overline{G}
\end{equation}\noindent
In this definition, the $\overline{G}$-space is associated with a given, direct
3D frame of three unit orthogonal vectors
$\{\overline{\r}_u,\overline{\r}_v,\overline{\r}_t\}$ where $\overline{\r}_t$ is
vertical and oriented upward. From now on, $\overline{G}$ is identified with the
box associated with the camera shown in Figure~\ref{GEOCHRONO_F2}.
\end{itemize}\noindent
As Figure~\ref{GBR-UVT-transform} shows, the equations above can be viewed as
a ``direct'' $uvt$-transform of point $\r\in G$ into a point
$\overline{\r}=\overline{\r}(\r)\in\overline{G}$ and, conversely, a ``reverse''
$uvt$-transform of point $\overline{\r}\in \overline{G}$ into a point
$\r=\r(\overline{\r})\in G$. Furthermore, the $uvt$-transform also applies as
follows to any function $\varphi$ defined in $G$:
\begin{equation}\label{eqn:uvt-transform-phi}
\overline{\varphi}(\overline{\r}) = \varphi(\r)\quad \forall\:\r \in G
\end{equation}\noindent
This concept of $uvt$-transform both of points and functions plays a central
role in the GeoChron-based restoration method presented in this article. When
referring to a function $\varphi$ defined in $G$, the following
notations may be used interchangeably for clarity:
\begin{equation} \label{eqn:geochron:dual-func-3}
\begin{array}{llllllllllllllll}
\quad &
\varphi(x,y,z)
&\equiv&
\varphi({\r})
&\equiv&
\varphi\big(\ {\r}(\bar{\r}) \ \big)
\\ \\
\equiv&
\bar{\varphi}(u,v,t)
&\equiv&
\bar{\varphi}(\bar{\r})
&\equiv&
\bar{\varphi}\big(\ \bar{\r}({\r}) \ \big)
&\equiv&
\overline{\varphi(\r)}
\end{array}
\end{equation}\noindent
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-Tectonic-Style,width=116mm}}
\caption{
Vertical cross section of a cylindrical, constant thickness layer. If
the tectonic style is flexural slip, arc lengths $A^fB^f$ and $C^fD^f$
are equal. If the tectonic style is minimal deformation, arc length
$A^mB^m$ is smaller than arc length $C^mD^m$. As an example, one can
picture flexural slip occurring when the material being bent is a stack
of paper sheets, but minimal deformation would rather occur with a piece
of rubber.
}
\label{GBR-Tectomic-Style}
\end{figure}\noindent
\ \\
According to geological context, a geologist can
choose one of two different tectonic styles to characterize the behavior of
geological layers subject to tectonic forces.
Figure~\ref{GBR-Tectomic-Style} illustrates both these options, referred to as
``flexural slip'' and ``minimal
deformation'' (see definitions on pages 53 and 54 of
\citet{Mallet2014}). In the GeoChron framework, these tectonic styles each
translate as a different set of equations which constrain the behavior of
paleo-geographic coordinates $\{u,v\}$.
\ \\
Let $\partial_\alpha\varphi$ denote $\partial\varphi/\partial \alpha \: \forall
\, \alpha \in \{x,y,z\}$ and $\grad\, \varphi$ denote the gradient
$\partial_x\varphi\, \r_x + \partial_y\varphi\, \r_y + \partial_z\varphi\, \r_z$
of a function $\varphi$ over frame $\{\r_x,\r_y,\r_z\}$.
The mathematical GeoChron theory presented in \citep{Mallet2014} states that,
depending on tectonic style, functions\footnote{ From now on,
we use the following concise notation: $\{f,g,\cdots\}_x\equiv\{f(x),g(x),\cdots\}$.}
$\{u,v\}_\r$
are assumed to honor, in a least squares sense, the following differential
equations (see pages~70 to 74 in \citet{Mallet2014}):
\begin{equation} \label{eqn:geochron:43E3}
\begin{array}{cc}
\mbox{Minimal deformation style:}
&
\left\{
\begin{array}{cc}
1 \& 2)&
||\grad\, u|| \simeq 1
\quad ; \quad
||\grad\, v|| \simeq 1
\\ \\
3)&
\grad\, u \cdot \grad\, v \ \simeq \ 0
\\
4)&
\grad\, t \cdot \grad\, u \ \simeq \ 0
\\
5)&
\grad\, t \cdot \grad\, v \ \simeq \ 0
\end{array}
\right.
\end{array}
\end{equation}
\noindent
\begin{equation} \label{eqn:geochron:49e}
\begin{array}{cc}
\mbox{Flexural slip style:}
&
\left\{
\begin{array}{cccccccc}
1 \& 2)&
||\grad_H\,u|| \simeq 1
\quad ; \quad
||\grad_H\,v|| \simeq 1
\\ \\
3)&
\grad_H\,u \cdot \grad_H\,v \ \simeq \ 0
\end{array}
\right.
\end{array}
\end{equation}\noindent
where $\grad\,\varphi(\r)$ is the gradient of scalar function $\varphi$ at
location $\r$,
$H\equiv H_{t(\r)}$ is the horizon passing through point
${\r\in G}$, defined as the set of particles of sediment which were deposited at
geological-time $t(\r)$,
and $\grad_H\,\varphi({\r})$ is the projection of gradient
$\grad\, \varphi({\r})$ onto said horizon $H$.
\ \\
Equations \ref{eqn:geochron:43E3} or \ref{eqn:geochron:49e} can be honored
exactly only in the particular case where horizons are perfectly planar and
parallel. In all other cases, local deformations of terrains entail that these
equations can only be approximated in a least squares sense.
As Figure~\ref{GBR-UVT-transform} shows, functions $\{u,v,t\}_\r$ are
continuous and smooth everywhere in $G$ except
across faults.
\ \\
For any equivalent system of GeoChron functions $\{u,v,t\}_\r$ and any tectonic style, it may be
shown\footnote{ See Equation~2.25 on page~64 of \citet{Mallet2014}.} that the
component ${\cal E}_{\alpha\beta}(\r)$ of the strain (deformation) tensor
$\Bmath{\cal E}(\r)$ at any point ${\r\in G}$ on the global frame
$\{\r_x,\r_y,\r_z\}$ honors the following equation:
\begin{equation} \label{eqn:Strain-6.4}
2\cdot {\cal E}_{\alpha\beta}(\r) \ = \
\delta_{\alpha\beta} \ - \
\biggl\{
\partial_\alpha u\cdot\partial_\beta u + \partial_\alpha v\cdot\partial_\beta v
+ \frac{N^\alpha\cdot N^\beta}{ (1 - \phi)^2 }
\biggr\}_{\r}
\qquad \forall\ (\alpha,\beta)\in\{x,y,z\}^2
\end{equation}\noindent
where $\phi({\r})$ denotes the compaction coefficient at point $\r$ defined on
page~38 of \citet{Mallet2014}
whilst $\{N^x,N^y,N^z\}_{\r}$ denote the components on $\{\r_x,\r_y,\r_z\}$ of
the unit vector $\B{N}(\r)$ orthogonal to horizon $H_{t(\r)}$ passing through
$\r$ and oriented in the direction of younger terrains:
\begin{equation} \label{GBR-N}
\B{N}(\r) \ = \ \frac{\grad\,t(\r)}{||\grad\,t(\r)||}
\end{equation}\noindent
\section{GeoChron framework for 3D restoration}
The restoration method presented in this paper uses as input an initial GeoChron
model of the studied domain $G$, which provides (see
Figure~\ref{GBR-Tetrahedral_Mesh-1} and \citet{Mallet2014}):
\begin{itemize}
\item Fault network topology and geometry;
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-Tetrahedral_Mesh-2,width=116mm}}
\caption{
Exploded view of a faulted, 3D geological domain $G$.
During restoration, twin faces $F^-$ and $F^+$ of fault $F$
must slide on one another. Points $(\r_{\msc{f}}^+,\r_{\msc{f}}^-)$, which
were collocated on horizon $H_t$ at deposition time $t$ prior to
faulting, are denoted a pair of ``twin-points''.
}
\label{GBR-Tetrahedral_Mesh-1}
\end{figure}\noindent
\item For each geological fault $F$, two disconnected surfaces $F^+$ and
$F^-$ bordering $F$ on either side. As observed today, $F^+$ and $F^-$ are collocated; however,
during the restoration process, $F^+$ and $F^-$ should slide on one another,
without generating gaps or overlaps between adjacent fault blocks;
\item For each fault $F$, a set of pairs of points $(\r_{\msc{f}}^+,\r_{\msc{f}}^-)$
called ``twin-points'' and such that:
\begin{enumerate}
\item $\r_{\msc{f}}^+\in F^+$ and $\r_{\msc{f}}^-\in F^-$;
\item Before $F$ induced any movement in the subsurface, the particles
of sediment which are observed today at locations $\r_{\msc{f}}^+$ and
$\r_{\msc{f}}^-$ were collocated.
\end{enumerate}
\noindent
During the activation of fault $F$, particles of sediment initially located
on $F$ are assumed to slide along apparent fault-striae defined as the
shortest path, on $F$, between pairs of twin points
$(\r_{\msc{f}}^+,\r_{\msc{f}}^-)$ (see example of fault network with
fault-striae in Figure~\ref{GBR-fault-striae}). From now on and for concision's sake,
``apparent'' fault-striae will simply be called ``fault-striae'';
\item A tectonic style which may be either ``minimal deformation'' or
``flexural slip'';
\item A triplet $\{u,v,t\}_\r$ of piecewise continuous functions defined on
the $G$-space such
that, for a particle of sediment observed today at location $\r\in G$, the numerical
values $\{u(\r),v(\r)\}$ represent the paleo-geographic coordinates of the particle
at geological-time $t(\r)$ when it was deposited.
\end{itemize}\noindent
Moreover, inherent to the GeoChron model, the following points are of relevance
to the restoration method presented in this paper:
\begin{itemize}
\item Each geological horizon $H_{\tau}$ is the set of particles of
sediment deposited at a given geological-time $\tau$:
\begin{equation} \label{eqn:Restoration-00}
\r\in H_{\tau}
\quad \Longleftrightarrow \quad
t(\r) = \tau
\end{equation}
\noindent
In other words, $H_{\tau}$ is defined as a level-set of the
geological-time function $t(\r)$;
\item Paleo-geographic coordinates functions $\{u,v\}_\r$ and twin-points are
linked by the following equations:
\begin{equation} \label{GBR:TP-1}
\begin{array}{c}
\{\ (\r_{\msc{f}}^+,\r_{\msc{f}}^-) \mbox{ is a pair of twin-points } \}
\quad \Longleftrightarrow \quad
\left\{
\begin{array}{ccccccc}
1)& \r_{\msc{f}}^+ \in F^+ \ \mbox{ \B{\&} } \ \r_{\msc{f}}^- \in F^- \\ \\
2)& u(\r_{\msc{f}}^-) = u(\r_{\msc{f}}^+) \\
3)& v(\r_{\msc{f}}^-) = v(\r_{\msc{f}}^+) \\
4)& t(\r_{\msc{f}}^-) = t(\r_{\msc{f}}^+)
\end{array}
\right.
\end{array}
\end{equation}\noindent
As shown in Figure~\ref{GBR-fault-striae}, each pair of twin-points $(\r_{\msc{f}}^+,\r_{\msc{f}}^-)$
is the intersection of a level set of function $t(\r)$ with a
fault-stria. As a consequence of constraints 2-3-4 above, fault-striae
characterize the paleo-geographic coordinates $\{u,v\}_\r$, and vice versa;
\item Each point $\r\in G$ is characterized by its coordinates
$\{x(\r),y(\r),z(\r)\}$ with respect to a direct frame $\{\r_x,\r_y,\r_z\}$ consisting
of three mutually orthogonal unit vectors with $\r_z$ oriented upward;
\item At any location $\r$ in geological domain $G$, the
equation $\x(s|\r)$ of curve ${\cal N}(\r)$ passing through $\r$, denoted ``normal-line'',
where $s$ is the arc length abscissa along ${\cal N}(\r)$, obeys the following
differential relationship:
\begin{equation} \label{GBR:N}
\frac{d \x(s|\r)}{ds} \ = \ \B{N}(\x(s|\r))
\end{equation}\noindent
where $\B{N}(\r)$ is the unit vector defined by equation \ref{GBR-N}.
\end{itemize}
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-fault-striae,width=116mm}}
\caption{
Example of fault-striae drawn on a fault network and deduced from
functions $\{u,v,t\}_\r$ of a GeoChron model. Data courtesy
of Total.
}
\label{GBR-fault-striae}
\end{figure}
\subsection*{Problem to address}
\label{Restoration-PB}
Let us assume that, at given geological-time $\tau$, horizon $H_{\tau}$ to be
restored coincided with a given, smooth surface $\widetilde{H}_\tau$ considered
as the sea floor, whose altitude at geological-time $\tau$ is a given
function\footnote{ In practice, $z_\tau^o(u,v)$ should be negative everywhere in the studied domain.}
$z_\tau^o(u,v)$ of GeoChron paleo-geographic coordinates $(u,v)$:
\begin{equation} \label{eqn:Restoration-So}
\{ \ \r_\tau^o\in{H}_\tau \ \}
\quad \Longleftrightarrow \quad
\{ \ z_\tau^o(\r_\tau^o) = z_\tau^o(u(\r_\tau^o),v(\r_\tau^o)) \ \}
\end{equation}\noindent
Let $G_\tau$ be the part of the $G$-space observed today and geologically
deposited up to geological-time $\tau$:
\begin{equation} \label{eqn:Restoration-Gtau}
\begin{array}{|c|} \hline \\
\quad
\r_\tau\in G_\tau \quad \Longleftrightarrow \quad t(\r_\tau)\leq \tau
\quad
\\ \\ \hline \end{array}
\end{equation}\noindent
The problem then consists in:
\begin{enumerate}
\item Restoring horizon $H_{\tau}$ to its initial, unfolded and unfaulted state
$\widetilde{H}_\tau$ corresponding to sea floor $\widetilde{\cal S}_\tau(0)$ at geological-time $\tau$;
\item Reshaping the terrains in such a way that, for each point $\r_\tau\in G_\tau$ stratigraphically located below $H_\tau$:
\begin{enumerate}
\item the particle of sediment currently located at point $\r_\tau$ moves to its
former, restored location $\{\bar\r_\tau=\bar\r_\tau(\r_\tau)\}$, where it was at geological-time
$\tau$;
\item no overlaps or voids are created in the subsurface;
\item volume variations are minimized whilst also taking compaction into account.
\end{enumerate}
\end{enumerate}
\begin{figure}
\centerline{\psfig{figure=./figures/GBRestoration-XX,width=116mm}}
\caption{
Folding / unfolding a book. Modified after \citet{Mallet2014}.
}
\label{GBRestoration-XX}
\end{figure}
\subsection*{Comment}
Figure~\ref{GBRestoration-XX} shows a book, considered as the analogue of a
stack of geological layers, being folded, then laid flat again. Two distinct
types of differential equations drive the book's geometrical transformations:
\begin{itemize}
\item \B{Folding:}
\begin{enumerate}
\item Initial state:
\begin{itemize}
\item Book pages are parallel and flat, which implies that the
book's initial geometry is known;
\item Mechanical laws which control the behavior of the pages
(e.g., elasticity) are known;
\item Physical properties (e.g., Lam\'e coefficients) of the
pages are known;
\item External forces applied to the book are known.
\end{itemize}\noindent
\item Final state:
\begin{itemize}
\item The book folds under the action of given external forces;
\item Its final geometry may be deduced from the aforementioned
mechanical laws and physical properties.
\end{itemize}\noindent
\end{enumerate}\noindent
In this first case, the geometry of the final state is dictated by a set of
differential equations controlled by initial geometry and mechanical
properties. It seems quite obvious that, with the application of similar
external forces, the book's final geometry will differ considerably
according to whether its pages are made of paper, plastic or steel.
\item \B{Unfolding:}
\begin{enumerate}
\item Initial state:
\begin{itemize}
\item Book pages are folded and their geometry is given;
\end{itemize}\noindent
\item Final state:
\begin{itemize}
\item The top page of the book is flat and its geometry is given,
\item All book pages remain parallel, without any void or intersection.
\end{itemize}\noindent
\end{enumerate}\noindent
In this second case, the geometry of the final state is dictated by a set of
differential equations controlled by initial and final conditions only and
does not depend on the pages' mechanical properties.
\end{itemize}
\noindent
This analysis shows that differential equations which rule the folding and
unfolding cases differ and do not require the same input and boundary
conditions. In particular, unfolding does not require the mechanical properties
of the medium (pages of the book) to be known. This is why we state that
geologic restoration can be purely geometrical, without relying on
geo-mechanical laws and physical properties of geologic layers.
\subsection*{Prior art}
Since seminal article \citet{Dahlstrom1969} was published half a century ago,
dozens of methods have been proposed to restore sedimentary terrains as they
were at a given geologic time $\tau$ (e.g. \citet{Gibbs1983, Suppe1985, Muron2005,
Moretti2006, Moretti2008, Maerten2015}), including some that represent
horizons as level-sets of a geological-time function
\citep{DurandRiard2010}. So far, only the one developed by \citet{Lovely2018} is
based on the GeoChron model paradigm.
\ \\
Figure~\ref{GBR-UVT-transform} illustrates that the $uvt$-transform of the
subsurface has the general look of restored stratified terrains. However, this
is not true restoration because, in the ``unfolded'' $\overline{G}$-space, all
horizons are transformed into parallel, horizontal planes, so lateral variations
in layer thicknesses are generally not preserved.
\ \\
Note that, barring compaction, in the very particular case where all layers have
a constant thickness and the following equation holds
\begin{equation} \label{GBR:THGrad}
||\grad\,t_\tau(\r_\tau)|| = 1
\qquad \forall\ \r_\tau\in G_\tau
\end{equation}\noindent
then, the $uvt_\tau$-transform $\overline{G}_\tau$ of $G_\tau$ preserves the
thickness of each layer, which implies that $\overline{G}_\tau$ so obtained
could be considered as a restored version of $G_\tau$. This observation led to
the following comment on page~91 in \citet{Mallet2014}, recalled here:
\[
\left.
\begin{array}{c}
\mbox{
\begin{minipage}[t]{115mm}
\begin{small}\sl
$\ll$
[\ldots] using a GeoChron model as an \B{input}, \\ it is possible to develop new breeds of unfolding algorithms.
$\gg$
\end{small}
\end{minipage}\noindent
}
\end{array}
\right.
\]
\noindent
By adding minimal functionalities to commercial
SKUA\textsuperscript{\tiny\textregistered} software designed to implement the
GeoChron model, \citet{Lovely2018} proposed a first, easy to implement
restoration algorithm which consists in using classical GeoChron equations to
compute restoration functions.
These ``native'' GeoChron equations were not devised with restoration of
sedimentary terrains as a goal. Despite this, by clever use of the software,
\citeauthor{Lovely2018} achieved remarkable first restoration results. As a
remedy to some weaknesses their study pointed out, in this paper we adapt the
GeoChron model theory, rather than its implementation.
The new set of differential equations and boundary conditions
obtained as a result are specifically designed to solve geometric restoration
problems and provide geologically and geometrically consistent
restored models. As we develop each step in our method, we will point out the
differences with \citeauthor{Lovely2018}'s work.
\section{GeoChron-Based Restoration (GBR) }
\label{GBR-GBR}
This section describes a purely geometrical method directly derived from the
GeoChron mathematical framework and aimed at restoring terrains at a given
geological-time $\tau$, whatever the structural complexity of horizons and
faults in studied domain $G$. This GeoChron-Based Restoration (GBR) method can be
intuitively introduced by the ``jelly block'' analogy depicted in
Figure~\ref{GBRestoration-1}.
\begin{figure}
\centerline{\psfig{figure=./figures/GBRestoration-1,width=140mm}}
\caption{
Jelly principle (minimal deformation style):
The $u_\tau v_\tau t_\tau$-transform of jelly block $G_\tau$
consists of a jelly block $\overline{G}_\tau$ identical to the restored
version of $G_\tau$ as it was at geological-time $\tau$.
In \citet{Tertois2019}.
}
\label{GBRestoration-1}
\end{figure}
\subsection*{Jelly block $\Bmath{\overline{G}_\tau}$}
Figure~\ref{GBRestoration-1}-A shows an arbitrarily-shaped ``jelly block'',
denoted $\overline{G}_\tau$, which contains a direct frame of orthogonal unit
vectors $\{\bar\r_{u_\tau},\bar\r_{v_\tau},\bar\r_{t_\tau}\}$
and a family of smooth, continuous horizontal surfaces $\{\overline{\cal S}_\tau(d):d\geq 0\}$
intersected only once by any vertical straight line parallel to $\bar\r_{t_\tau}$:
\begin{enumerate}
\item For each point ${\bar\r_\tau}\in \overline{G}_\tau$,
$\{u_\tau({\bar\r_\tau}),v_\tau({\bar\r_\tau})\}$ represent the horizontal
coordinates of ${\bar\r_\tau}$ with respect to horizontal unit frame vectors $\{\bar\r_{u_\tau},
\bar\r_{v_\tau}\}$ whilst $t_\tau(\bar\r_\tau)$ represents its altitude with respect
to $\bar\r_{t_\tau}$, oriented upward;
\item Any point $\bar\r_\tau^o\in\overline{\cal S}_\tau(0)$ is located at altitude
$(t_\tau(\bar\r_\tau^o)=0)$ with respect to the vertical unit vector $\bar\r_{t_\tau}$
oriented upward;
\item $\overline{\cal S}_\tau(d)$ is located at algebraic vertical
distance $(d)$ from $\overline{\cal S}_\tau(0)$ in such a way that:
\begin{equation} \label{GBR:X0001}
\left|
\begin{array}{cccccccc}
d<0 &\Longleftrightarrow& \overline{\cal S}_\tau(d) & \mbox{is located above } \overline{\cal S}_\tau(0)
\\
d>0 &\Longleftrightarrow& \overline{\cal S}_\tau(d) &
\mbox{is located below } \overline{\cal S}_\tau(0)
\end{array}
\right.
\end{equation}\noindent
\end{enumerate}
\noindent
Contrarily to classical Free Form Deformation methods which, following the
principles formulated by \citet{Sederberg1986}, introduce the
concept of jelly block, $\overline{G}_\tau$ may be of arbitrary shape and may be
discontinuous across surfaces dividing it, either partially or totally.
\subsection*{Jelly block $\Bmath{G_\tau}$}
Figure~\ref{GBRestoration-1}-B shows the
folded jelly block $G_\tau$ resulting from the deformation of jelly block
$\overline{G}_\tau$ under tectonic forces induced either by minimal deformation
or flexural slip tectonic forces:
\begin{equation} \label{GBR:X2}
\begin{array}{|c|} \hline \\
\quad
\begin{array}{c}
\overline{G}_\tau \quad \longrightarrow\mbox{Tectonic
forces}\longrightarrow \quad G_\tau
\end{array}
\quad
\\ \\ \hline \end{array}
\end{equation}\noindent
In spaces $\overline{G}_\tau$ and $G_\tau$:
\begin{enumerate}
\item Using reverse and direct $u_\tau v_\tau t_\tau$-transforms,
each point ${\bar\r_\tau}\in\overline{G}_\tau$ is transformed into point
$\r_\tau\in G_\tau$, and conversely:
\begin{equation} \label{GBR:X1a}
{\bar\r_\tau}\in\overline{G}_\tau \quad \longleftrightarrow
\quad \r_\tau\in G_\tau
\end{equation}\noindent
\item For each point $\r_\tau\in G_\tau$:
\begin{itemize}
\item $\{x(\r_\tau),y(\r_\tau)\}$ represent the horizontal geographic coordinates
of $\r_\tau$ with respect to $\{\r_x,\r_y\}$, whilst $z(\r_\tau)$ represents its
altitude with respect to the vertical unit frame vector $\r_z$ oriented
upward;
\item $\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ are functions defined as follows in $G_\tau$:
\begin{equation} \label{GBR-XXX-1}
\begin{array}{|c|} \hline \\
u_\tau(\r_\tau) = u_\tau(\bar{\r}_\tau) \quad ; \quad
v_\tau(\r_\tau) = v_\tau(\bar{\r}_\tau) \quad ; \quad
t_\tau(\r_\tau) = t_\tau(\bar{\r}_\tau)
\\ \\ \hline
\end{array}
\end{equation}
\end{itemize}\noindent
\item Each horizontal surface $\overline{\cal S}_\tau(d)\in \overline{G}_\tau$ is
transformed into a curved surface ${\cal S}_\tau(d)\in G_\tau$ ``parallel\footnote{ The
notion of ``parallelism'' is linked to eikonal Equation~\ref{GBR:eikonal-equation}.}'' to ${\cal S}_\tau(0)$ and each surface ${\cal S}_\tau(d)$ is a level-set of function $t_\tau(\r_\tau)$;
\item The images of rectilinear coordinate axes $(u_\tau)$, $(v_\tau)$ and
$(t_\tau)$ contained in jelly block $\overline{G}_\tau$ consist of curved lines
in folded jelly block $G_\tau$.
\end{enumerate}
\noindent
From now on, without loss of generality and for the sake of simplicity,
$\overline{G}_\tau$-space frame $\{\bar\r_{u_\tau},\bar\r_{v_\tau},\bar\r_{t_\tau}\}$
and its origin
$\overline{O}_{u_\tau v_\tau t_\tau}$ are identified with $G$-space frame
$\{\r_x,\r_y,\r_z\}$ and its origin $O_{xyz}$:
\begin{equation} \label{GBR:X00}
\bar\r_{u_\tau} \equiv \r_x \quad;\quad
\bar\r_{v_\tau} \equiv \r_y \quad;\quad
\bar\r_{t_\tau} \equiv \r_z \quad;\quad
\overline{O}_{u_\tau v_\tau t_\tau} \equiv O_{xyz}
\end{equation}\noindent
Equivalently to Equations~\ref{GBR:X00}, we can state that the jelly particle
observed at location $\r_\tau\in G_\tau$ may be moved (i.e. restored) to its
former, initial location $\bar\r_\tau=\bar\r_\tau(\r_\tau)$
defined as follows, where $\B{R}_\tau(\r_\tau)$ is called ``restoration vector
field'':
\begin{equation} \label{GBR:X00.a}
\begin{array}{|c|} \hline \\
\begin{array}{c}
\bar\r_\tau(\r_\tau) \ = \ \r_\tau \ + \ \B{R}_\tau(\r_\tau)
\qquad \forall\ \r_\tau\in G_\tau
\\ \\
\mbox{with : }\quad
\B{R}_\tau(\r_\tau) = [\r_x, \r_y, \r_z] \cdot
\left[ \hspace{-2mm}
\begin{array}{c}
u_\tau(\r_\tau)-x(\r_\tau) \\
v_\tau(\r_\tau)-y(\r_\tau) \\
t_\tau(\r_\tau)-z(\r_\tau)
\end{array}
\hspace{-2mm} \right]
\end{array}
\\ \\ \hline \end{array}
\end{equation}\noindent
\subsection*{Fundamental GeoChron-Based Restoration principle}
We can conclude from the statements above that jelly block $G_\tau$ may be
considered as a pseudo-subsurface whose geometry at time of deposition $\tau$
was identical to $\overline{G}_\tau$ and where all pseudo-horizons $\{{\cal S}_\tau(d): d\geq 0\}$ are assumed
to be parallel. Therefore, for any point $\r$ within jelly
block $G_\tau$, restoration functions $u_\tau(\r_\tau)$ and $v_\tau(\r_\tau)$ may be identified with pseudo
paleo-geographic coordinates and $t_\tau(\r_\tau)$ may be identified with a pseudo
geological-time of deposition, which leads us to derive the following ``fundamental GeoChron-based restoration principle'':
\begin{equation}\label{GBR-fundamental-principle}
\begin{array}{|c|} \hline
\mbox{
\begin{minipage}[t]{130mm}
\ \\
{\sc Fundamental GBR principle: } \\ \sl
Barring the effects of compaction, equations established for the
GeoChron functions $\{u,v,t\}_\r$ also apply to functions $\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$
\end{minipage}\noindent
}
\\ \\ \hline \end{array}
\end{equation}
\subsection*{GeoChron-Based Restoration (GBR) algorithm }
Assume that a numerical GeoChron model characterized by functions $\{u,v,t\}_\r$
defined on a possibly faulted geological domain $G$ is given. To restore the terrains to their
state at given restoration geological-time $\tau$, as Figure~\ref{GBRestoration-1} shows, the following GeoChron-Based Restoration algorithm is proposed:
\begin{enumerate}
\item Identify the part of the subsurface stratigraphically located below
horizon $H_\tau$ with jelly block $G_\tau$.
\item Identify our restoration problem with a jelly block restoration problem.
For that purpose, make the following assumptions:
\begin{enumerate}
\item reverse $u_\tau v_\tau t_\tau$-transform ${S}_\tau(0)$ of
sea floor $\overline{S}_\tau(0)$ is identified with horizon $H_\tau$:
\begin{equation} \label{GBR-H-tau}
{\cal S}_\tau(0) \ \equiv \ H_\tau
\end{equation}
\item at depositional time $\tau$, $\overline{S}_\tau(0)$ is
temporarily assumed to be flat and horizontal and is identified with sea
level with an altitude of zero; in other words, the following temporary
assumption is made:
\begin{equation} \label{GBR-G-tau-A}
t_\tau(\r_\tau^o) \ = \ 0
\qquad \forall\ \r_\tau^o \in H_\tau
\end{equation}
\noindent
\item in $G_\tau$, terrain compaction is temporarily ignored;
\end{enumerate}\noindent
\item Using the jelly block paradigm, to restore subsurface geometry to
geological-time $\tau$:
\begin{enumerate}
\item using equations and numerical techniques described in sections
\ref{BGR:Characterizing-t-tau} to \ref{GBR-taking-faults-into-account},
compute numerical approximations of functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ on $G_\tau$;
\item compute restoration vector field $\B{R}_\tau(\r_\tau)$ defined by
Equation~\ref{GBR:X00.a} and generate restored jelly block
$\overline{G}_\tau$ as the $u_\tau v_\tau t_\tau$-transform of $G_\tau$;
\begin{equation} \label{GBR-R-tau}
\bar\r_\tau(\r_\tau)\in \overline{G}_\tau \quad
\Longleftrightarrow \quad \bar\r_\tau(\r_\tau) = \r_\tau + \B{R}_\tau(\r_\tau)
\end{equation}
\item using a specific algorithm described in section
\ref{GBR:::compaction}, reverse compaction assumption \#2.c;
\item to reverse the flat $\overline{S}_\tau(0)$ assumption \#2.b, move each point $\overline{\r}_\tau\in
\overline{G}_\tau$ downward\footnote{ The sea floor is located below the
sea level which implies that $z_\tau^o$ is constantly negative.} as follows
\begin{equation} \label{GBR-T-tau}
\overline{\r}_\tau \quad \longleftarrow \quad
\overline{\r}_\tau \ + \
\{
t_\tau(\overline{\r}_\tau) + z_\tau^o( u_\tau(\overline{\r}_\tau),v_\tau(\overline{\r}_\tau))
\} \cdot \overline{\r}_{t_\tau}
\qquad \forall\ \overline{\r}_\tau\in \overline{G}_\tau
\end{equation}
where $z_\tau^o(u,v)$ is assumed to be a given function of GeoChron
paleo-geographic coordinates; In practice, $z_\tau^o(u,v)$ may be
defined on $H_\tau$.
\end{enumerate}\noindent
\end{enumerate}
\noindent
At first glance, replacing $(u,v)$ by $(u_\tau,v_\tau)$ in
Equation~\ref{GBR-T-tau} may seem dubious. To justify this, in
$\overline{G}_\tau$, consider the vertical straight line
$\overline{\Delta}(u_\tau,v_\tau)$ with constant paleo-geographic coordinates
$(u_\tau,v_\tau)$. The straight line $\overline{\Delta}(u_\tau,v_\tau)$ so
defined cuts the horizontal plane $\overline{\cal S}_\tau(0)$ at a point with
paleo-geographic coordinates $(u_\tau,v_\tau,t_\tau=0)$. The crux point of our
argument is that, if the restoration process is coherent with the input GeoChron
model then, on $\{\overline{S}_\tau(0)\equiv\overline{H}_\tau\}$,
paleo-geographic coordinates $(u_\tau,v_\tau)$ are exactly the
same\footnote{ See Equations~\ref{GBR:Indet}.} as the GeoChron paleo-geographic
coordinates $(u,v)$. Therefore, Equation~\ref{GBR-T-tau} simply states that, in
$\overline{G}_\tau$, the entire column of sediments located on line
$\overline{\Delta}(u_\tau,v_\tau)$ is rigidly moved downward in such a way that
the particle of sediment at the top of this column, which was at altitude zero of
sea level, is moved to the correct, given altitude of the sea floor at
geological-time $\tau$.
\subsection*{Preservation of GeoChron functions}
In the proposed GBR process, the ``true'' GeoChron paleo-geographic coordinate
functions $\{u,v\}_\r$ and ``true'' geological-time function $t(\r)$ of the
GeoChron model provided as input are transformed passively. In other words,
after restoration, paleo-geographic coordinates
$\{u(\r_\tau),v(\r_\tau),t(\r_\tau)\}$ attached to the particle of sediment
observed today at a point $\r_\tau\in G_\tau$ remain preserved:
\begin{equation} \label{GBR:ModelPreservation}
u(\overline{\r}_\tau) = u({\r}_\tau) \quad ; \quad
v(\overline{\r}_\tau) = v({\r}_\tau) \quad ; \quad
t(\overline{\r}_\tau) = t({\r}_\tau)
\end{equation}
As a consequence:
\begin{equation}\label{GBR-is-a-GeoChronModel}
\begin{array}{|c|} \hline
\mbox{
\begin{minipage}[t]{130mm}
\ \\
Given a present-day GeoChron model of the subsurface characterized
by GeoChron functions $\{u,v,t\}_\r$, its GBR
restoration at a given geological-time $\tau$ in the past is also a GeoChron model characterized by the same GeoChron functions.
\end{minipage}
}
\\ \\ \hline \end{array}
\end{equation}
\noindent
Therefore, at any restoration time $\tau$, any tool or application developed for
a GeoChron model may be applied as is on the GBR-restored version $\overline{G}_\tau$ of this model.
\ \\
The most important application of geological restoration is to validate the
geometry of the input GeoChron model. At any geological-time $\tau$, this
restored geometry is simpler, which makes validation and editing easier. This
validation process is
robust only if the restoration method is both precise and consistent with the
initial GeoChron model provided as input and we will show how to implement a solution which
addresses these concerns.
\section{Characterizing function $\protect\Bmath{t_\tau(\r_\tau)}$}
\label{BGR:Characterizing-t-tau}
In this section, assume that, at geological-time $\tau$, the effect of
compaction is omitted in $G_\tau$ and that sea floor $\overline{\cal S}_\tau(0)$
coincides with sea level at altitude zero.
\ \\
For any tectonic style, after applying tectonic forces to
jelly block $\overline{G}_\tau$, the images $\{{\cal S}_\tau(d): d\geq 0\}$ of
horizontal surfaces $\{\overline{\cal S}_\tau(d): d\geq 0\}$ remain parallel. For
any $d\geq 0$ and any infinitely small increment $\varepsilon>0$, parallel
surfaces ${\cal S}_\tau(d)$ and ${\cal S}_\tau(d+\varepsilon)$ may be considered
as the top and base of a jelly layer with constant thickness $\varepsilon$. In
other words, for any point $\r_\tau\in{\cal S}_\tau(d+\varepsilon)$, the shortest
path to ${\cal S}_\tau(d)$ measures $\varepsilon$ and is orthogonal to both ${\cal S}_\tau(d)$ and ${\cal S}_\tau(d+\varepsilon)$.
\ \\
As a consequence:
\begin{itemize}
\item Starting from any arbitrary point $\r_\tau\in G_\tau$ there is, recursively defined, a
curvilinear ``normal-line\footnote{ See Equation~\ref{GBR:N}.}''
${\cal N}_\tau(\r_\tau)$ constantly orthogonal to the
family of parallel surfaces $\{{\cal S}_\tau(d):d\geq 0\}$ and linking $\r_\tau$ to the nearest point on $\{{\cal S}_\tau(0)\equiv H_\tau\}$;
\item The value of $t_\tau(\r_\tau)$ is defined as the negative distance along
${\cal N}_\tau(\r_\tau)$ from point $\r_\tau$ to surface $\{{\cal S}_\tau(0)\equiv H_\tau\}$:
\begin{equation} \label{GBR:normal-line}
\begin{array}{|c|} \hline \\
t_\tau(\r_\tau) \ = \ \Bmath{-}\,
\biggl\{\ \mbox{ arc length of normal-line between $\r_\tau$ and ${\cal S}_\tau(0)$}\ \biggr\}
\quad \forall\ \r_\tau\in G_\tau
\\ \\ \hline \end{array}
\end{equation}
\end{itemize}
\noindent
Moreover, $t_\tau(\r_\tau)$ is also equal to the vertical coordinate $t_\tau(\bar\r_\tau)$ of $\bar\r_\tau$ in $\overline{G}_\tau$.
\ \\
For any derivable function $\varphi(\r)$ and unit vector $\B{u}$, the following
equation holds\footnote{ E.g., see Equation~13.43 on page 316 of
\citet{Mallet2014}.}:
\begin{equation} \label{GBR-JRP-1.0}
\frac{d\varphi(\r+s\cdot\B{u})}{ds}\bigg|_{s=0}
\ = \
\grad\,\varphi(\r)\cdot \B{u}
\end{equation}\noindent
Therefore, if we denote $\B{N}_\tau(\r_\tau)$ the unit vector at location $\r_\tau\in
G_\tau$ which is orthogonal to surface ${\cal S}_\tau(d(\r_\tau))$ passing through
$\r_\tau$ and oriented in the direction of younger terrains, then:
\begin{equation} \label{GBR-JRP-1}
\frac{d t_\tau(\r_\tau+s\cdot\B{N}_\tau(\r_\tau))}{ds}\bigg|_{s=0}
\ = \
\grad\,t_\tau(\r_\tau)\cdot \B{N}_\tau(\r_\tau)
\ = \
\grad\,t_\tau(\r_\tau)\cdot \frac{\grad\,t_\tau(\r_\tau)}{||\grad\,t_\tau(\r_\tau)||}
\ = \ ||\grad\,t_\tau(\r_\tau)||
\end{equation}\noindent
According to Equation~\ref{GBR:normal-line}, $d
t_\tau(\r_\tau+s\cdot\B{N}_\tau(\r_\tau))$ represents the thickness $ds>0$ of
the micro layer between ${\cal S}_\tau(d(\r_\tau))$ and ${\cal
S}_\tau(d(\r_\tau)-ds)$, from which we can write:
\begin{equation} \label{GBR-JRP-2}
\{\ dt_\tau(\r_\tau+s\cdot\B{N}_\tau(\r)) \ = \ ds \}
\quad \Longleftrightarrow \quad
\biggl\{
||\grad\,t_\tau(\r_\tau)||
\ = \
\frac{d t_\tau(\r+s\cdot\B{N}_\tau(\r_\tau))}{ds}
\ = \ 1
\biggr\}
\end{equation}\noindent
Moreover, on horizon $H_\tau$, we have
\begin{equation} \label{GBR-JRP-2-a}
\forall\ \r_\tau^o\in \{{\cal S}_\tau(0)\equiv H_\tau\} \ : \quad
\left|
\begin{array}{lll}
1)& t_\tau(\r_\tau^o) = 0
\\ \\
2)& \displaystyle
\B{N}_\tau(\r_\tau^o) = \B{N}(\r_\tau^o)
\end{array}
\right.
\end{equation}\noindent
where $\B{N}(\r_\tau^o)$, defined by equation \ref{GBR:N}, is given.
\subsection*{The eikonal equation}
In a jelly block $G_\tau$ of any geometrical and topological complexity, we may
conclude from the equations above that $t_\tau(\r_\tau)$ must honor the following
fundamental differential equation, called the ``eikonal equation'',
characterizing the parallelism of surfaces $\{{\cal S}_\tau(d):d\geq 0\}$,
subject to specific boundary conditions:
\begin{equation} \label{GBR:eikonal-equation}
\begin{array}{|c|} \hline \\
\begin{array}{cccc}
1)& ||\grad\,t_\tau(\r_\tau)|| = 1 \qquad \forall\ \r_\tau\in G_\tau
\\ \\
2)& \mbox{subject to : \ }
\left\{
\begin{array}{cc}
a)& t_\tau(\r_\tau^o) = 0
\\ \\
b)& \displaystyle
\grad\,t_\tau(\r_\tau^o) = \B{N}(\r_\tau^o)
\end{array}
\right\} \quad \forall\ \r_\tau^o\in \{{\cal S}_\tau(0)\equiv H_\tau\}
\end{array}\noindent
\\ \\ \hline \end{array}
\end{equation}\noindent
Physicists would use the well-known eikonal Equation~\ref{GBR:eikonal-equation} to
describe the time of first arrival at point $\r_\tau\in G_\tau$ of a light wave-front
emitted by $\{{\cal S}_\tau(0)\equiv H_\tau\}$ and propagating at constant, unit
speed. To go further with this analogy, faults would be considered as opaque barriers which induce
discontinuities in functions $\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$. As
Figure~\ref{GBR-dark-fault-block} shows, fault blocks which are not illuminated by
$H_\tau$ are called ``$\tau$-dark fault blocks''. More precisely, a point
$\r_\tau\in G_\tau$ belongs to a $\tau$-dark fault block if and only if, within the
studied domain, no continuous path (i.e. uncut by faults) exists between $\r_\tau$ and $\{{\cal S}_\tau(0)\equiv H_\tau\}$.
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-dark-fault-block,width=116mm}}
\caption{
Vertical cross section in a structural model depicting a ``$\tau$-dark''
fault block (darker yellow), which cannot be illuminated by light emitted by horizon $H_\tau$.
}
\label{GBR-dark-fault-block}
\end{figure}\noindent
\section{Characterizing functions $\protect\Bmath{\{u_\tau, v_\tau\}_{\r_\tau}}$}
\label{BGR:Characterizing-uv-tau}
Solving eikonal Equation~\ref{GBR:eikonal-equation} provides us with the values
of $t_\tau(\r_\tau)$ over space $G_\tau$. Assuming that $t_\tau(\r_\tau)$ is now
known, this section shows how differential
equations characterizing functions $\{u_\tau, v_\tau\}_{\r_\tau}$ can be derived from
the jelly paradigm and fundamental GBR principle~\ref{GBR-fundamental-principle}.
\subsection*{First type boundary conditions for $\protect\Bmath{\{u_\tau,v_\tau\}_{\r_\tau}}$ on $\protect\Bmath{H_\tau}$}
By definition, for any point $\r_\tau^o\in H_\tau$:
\begin{itemize}
\item In the $G_\tau$ space, $\{u(\r_\tau^o),v(\r_\tau^o)\}$ are the given GeoChron paleo-geographic coordinates of the particle of sediment observed today at location $\r_\tau^o$;
\item In the $\overline{G}_\tau$ restored space,
$\{u_\tau(\bar\r_\tau^o),v_\tau(\bar\r_\tau^o)\}$ are unknown geographic coordinates
of the particle of sediment which would have been observed at location $\bar\r_\tau^o$ on sea floor $\overline{\cal S}_\tau(0)$.
\end{itemize}
\noindent
Obviously, taking Equations~\ref{GBR-XXX-1} into account and for any point
$\r_\tau^o\in H_\tau$, coordinates
$\{u(\r_\tau^o),v(\r_\tau^o)\}$ and $\{u_\tau(\r_\tau^o),v_\tau(\r_\tau^o)\}$
should be identical. As a consequence, the following first type boundary
conditions, where $u(\r_\tau^o)$ and $v(\r_\tau^o)$ are known, must be honored:
\begin{equation} \label{GBR:Indet}
\begin{array}{|c|} \hline \\
\forall\ \r_\tau^o\in \{{\cal S}_\tau(0)\equiv H_\tau\} \ : \quad
\left|
\begin{array}{cccl}
1)& u_\tau(\r_\tau^o) &=& u(\r_\tau^o)
\\ \\
2)& v_\tau(\r_\tau^o)&=& v(\r_\tau^o)
\end{array}
\right.
\\ \\ \hline \end{array}
\end{equation}\noindent
\citet{Lovely2018} do not set the constraints specified above, which
implies that functions $\{u_\tau,v_\tau\}_{\r_\tau}$ are not synchronized with known
paleo-geographic functions $\{u,v\}_\r$. As a consequence, the $uvt$-transform
and $u_\tau v_\tau t_\tau$-transform of $H_\tau$ are not constrained to be
identical, implying that erroneous deformations may appear on
$\overline{H}_\tau$ obtained as $u_\tau v_\tau t_\tau$-transform of $H_\tau$. As
an example, consider the (generally curvilinear) patch $H_\tau(u_o,v_o,\Delta)$
defined on $H_\tau$ as follows:
\begin{equation} \label{eqn:Restoration-Rescal-4}
\r\in H_\tau(u_o,v_o,\Delta)
\quad \Longleftrightarrow \quad
\left\{
\begin{array}{ccccc}
t(\r)=\tau
\\
u_o\leq u(\r) \leq u_o+\Delta
\\
v_o\leq v(\r) \leq v_o+\Delta
\end{array}
\right.
\end{equation}\noindent
and consider also $\overline{H}_\tau^\oplus(u_o,v_o,\Delta)$ and
$\overline{H}_\tau^\ominus(u_o,v_o,\Delta)$ as the restored images of
$H_\tau(u_o,v_o,\Delta)$ on $\overline{H}_\tau$ with and without constraints~\ref{GBR:Indet},
respectively. If
constraints~\ref{GBR:Indet} are omitted, then
$\overline{H}_\tau^\oplus(u_o,v_o,\Delta)$ and $\overline{H}_\tau^\ominus(u_o,v_o,\Delta)$
may have different areas and/or shapes, implying that the restoration
process may induce deformations which are incoherent with
those described by the $\{u,v,t\}_\r$ GeoChron functions provided as input.
\subsection*{Second type boundary conditions for $\protect\Bmath{\{u_\tau,v_\tau\}_{\r_\tau}}$ on $\protect\Bmath{H_\tau}$}
In the restored space $\overline{G}_\tau$, terrains older than $\tau$ are
generally still folded and, similarly to terrains in the $G$-space, their
deformation may be characterized by the ``partial'' strain tensor at geological-time $\tau$ denoted $\Bmath{\cal E}(\r|\tau)$.
For coherency's sake, on $H_\tau$ the ``total'' strain tensor $\Bmath{\cal
E}(\r)$ characterized by Equation~2.20 on page~63 of \citet{Mallet2014} and the ``partial'' strain tensor $\Bmath{\cal E}(\r|\tau)$ should be equal:
\begin{equation} \label{GBR:BCUV-2T.X1}
\Bmath{\cal E}(\r_\tau^o) \ = \ \Bmath{\cal E}(\r_\tau^o|\tau)
\qquad\forall\ \r_\tau^o\in H_\tau
\end{equation}\noindent
On horizon $H_\tau$, as a consequence of boundary conditions
\ref{GBR:eikonal-equation}.2b set on $t_\tau(\r_\tau)$, we have:
\begin{equation} \label{GBR:BCUV-2T.0}
\B{N}(\r_\tau^o) \ = \ \frac{\grad\,t(\r_\tau^o)}{||\grad\,t(\r_\tau^o)||} \ = \ \frac{\grad\,t_\tau(\r_\tau^o)}{||\grad\,t_\tau(\r_\tau^o)||} \ = \ \B{N}_\tau(\r_\tau^o)
\qquad\forall\ \r_\tau^o\in H_\tau
\end{equation}\noindent
Therefore, barring the effects of compaction,
according to GBR principle~\ref{GBR-fundamental-principle} and
Equation~\ref{GBR:BCUV-2T.X1}, for all indexes $(\alpha,\beta)\in\{x,y,z\}^2$, Equation~\ref{eqn:Strain-6.4} implies:
\begin{equation} \label{GBR:BCUV-2T.X2}
\forall\ \r_\tau^o\in H_\tau \ : \quad
\left|
\begin{array}{llllllllll}
&&
\{
\partial_\alpha u\cdot\partial_\beta u + \partial_\alpha v\cdot\partial_\beta v + {N^\alpha\cdot N^\beta }
\}_{\r_\tau^o}
\\ \\
&=&
\{
\partial_\alpha u_\tau\cdot\partial_\beta u_\tau + \partial_\alpha v_\tau\cdot\partial_\beta v_\tau + {N^\alpha\cdot N^\beta }
\}_{\r_\tau^o}
\end{array}
\right.
\end{equation}\noindent
\begin{equation} \label{GBR:BCUV-2T.4}
\Longleftrightarrow\quad
\forall\ \r_\tau^o\in H_\tau \ : \quad
\left|
\begin{array}{cllllll}
&&
\{
\partial_\alpha u\cdot\partial_\beta u + \partial_\alpha v\cdot\partial_\beta v
\}_{\r_\tau^o}
\\ \\
&=&
\{
\partial_\alpha u_\tau\cdot\partial_\beta u_\tau + \partial_\alpha v_\tau\cdot\partial_\beta v_\tau
\}_{\r_\tau^o}
\end{array}
\right.
\end{equation}\noindent
A straightforward solution to these equations consists in constraining
restoration functions $\{u_\tau,v_\tau\}_{\r_\tau}$ as follows, where
$\{u,v\}_\r$ are known:
\begin{equation} \label{GBR:Indet-1}
\begin{array}{|c|} \hline \\
\forall\ \r_\tau^o\in \{{\cal S}_\tau(0)\equiv H_\tau\} \ : \quad
\left|
\begin{array}{cccl}
1)& \grad\,u_\tau(\r_\tau^o) &=& \grad\,u(\r_\tau^o)
\\ \\
2)& \grad\,v_\tau(\r_\tau^o)&=& \grad\,v(\r_\tau^o)
\end{array}
\right.
\\ \\ \hline \end{array}
\end{equation}\noindent
This second type of boundary conditions are not implemented in
\citet{Lovely2018}'s method, which may jeopardize the consistency of
restored models with respect to the initial GeoChron model.
\subsection*{Comment}
In order to maintain consistency of restoration functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ with input GeoChron functions
$\{u,v,t\}_{\r_\tau}$, boundary conditions~\ref{GBR:Indet} and~\ref{GBR:Indet-1}
must be honored as strictly as possible. If they do not conflict with these
boundary conditions, other, not so strict constraints may be added and honored
in a least squares sense, as for example the following ``$\tau$-twin-pins''
constraint.
\ \\
By definition, we suggest denoting ``$\tau$-twin-pins'' a pair of points
$(\r_\tau^1,\r_\tau^2)$ in $G_\tau$ such that the structural geologist has
reason to believe that their restored images
$(\overline{\r}_\tau^1,\overline{\r}_\tau^2)$ at restoration time $\tau$ must be
located on a single vertical line in $\overline{G}_\tau$. In order to take that
constraint into account, restoration functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ may be computed such that, in a least
squares sense:
\begin{equation} \label{GBR:Pin-Point}
\left|
\begin{array}{ccccccccc}
u_\tau(\r_\tau^1) &\simeq& u_\tau(\r_\tau^2)
\\ \\
v_\tau(\r_\tau^1) &\simeq& v_\tau(\r_\tau^2)
\end{array}
\right.
\end{equation}\noindent
By setting this type of constraint repeatedly on pairs of points located, for
instance, on a line in $G_\tau$, it is possible to make the restored version of
this line vertical in $\overline{G}_\tau$.
\subsection*{Characterizing functions $\protect\Bmath{\{u_\tau,v_\tau\}_{\r_\tau}}$ in $G_\tau$}
According to GeoChron theory\footnote{ See equation \ref{eqn:Strain-6.4}.}, everywhere in studied domain $G$, terrain
deformation is characterized by the gradients of geological-time function
$t(\r)$ and paleo-geographic functions $\{u,v\}_\r$, which may be considered as
deformation ``records'' taken into account by boundary
conditions~\ref{GBR:Indet} and~\ref{GBR:Indet-1}. As explained below, to
propagate these boundary conditions over the entire $G_\tau$-space, the
gradients of $\{u_\tau,v_\tau\}_{\r_\tau}$ have to honor specific differential
equations.
\ \\
Referring to fundamental GBR principle~\ref{GBR-fundamental-principle}, to
characterize the restoration functions $\{u_\tau,v_\tau,t_\tau\}_\r$, we should
simply substitute these functions to $\{u,v,t\}_\r$ in
Equation~\ref{eqn:geochron:43E3} or Equation~\ref{eqn:geochron:49e}. However:
\begin{enumerate}
\item In $G_\tau$ and in accordance with GeoChron theory:
\begin{enumerate}
\item ``soft'' constraints~\ref{eqn:geochron:43E3}-1\&2
or~\ref{eqn:geochron:49e}-1\&2 may only be honored in a least squares
sense;
\item due to local deformations of horizons and layers induced by
tectonic forces, left hand sides of
constraints~\ref{eqn:geochron:43E3}-1\&2 or~\ref{eqn:geochron:49e}-1\&2
may slightly differ from the specified value ``1''.
\end{enumerate}\noindent
\item On $H_\tau$, ``hard'' constraints~\ref{GBR:Indet} and~\ref{GBR:Indet-1}
strictly specify the values of restoration functions $\{u_\tau,v_\tau\}$ and
their 3D gradients which, according to item 1.b above, may conflict with
constraints~\ref{eqn:geochron:43E3}-1\&2 or~\ref{eqn:geochron:49e}-1\&2.
\end{enumerate}\noindent
Therefore, to resolve such a conflict, we suggest removing
constraints~\ref{eqn:geochron:43E3}-1\&2 or~\ref{eqn:geochron:49e}-1\&2
and constraining $\{u_\tau,v_\tau\}_{\r_\tau}$ in a least squares sense as follows:
\begin{itemize}
\item In a minimal deformation tectonic style context:
\begin{equation} \label{GBR:X11}
\forall\ \r_\tau\in G_\tau \ : \quad
\begin{array}{|c|} \hline \\
\begin{array}{cc}
1)&
\{\grad\,u_\tau \cdot \grad\,v_\tau\}_{\r_\tau} \ \simeq \ 0
\\
2)&
\{\grad\,t_\tau \cdot \grad\,u_\tau\}_{\r_\tau} \ \simeq \ 0
\\
3)&
\{\grad\,t_\tau \cdot \grad\,v_\tau\}_{\r_\tau} \ \simeq \ 0
\end{array}
\\ \\ \hline \end{array}
\end{equation}
\item In a flexural slip tectonic style context, denoting
$\grad_{\msc{s}}\varphi(\r_\tau)$ the orthogonal projection of $\grad\,\varphi(\r_\tau)$ onto
the plane tangent to the level-set ${\cal S}_\tau(-t_\tau(\r_\tau))$ of function
$t_\tau(\r_\tau)$ at location $\r_\tau$:
\begin{equation} \label{GBR:S1}
\forall\ \r_\tau\in G_\tau\ : \quad
\begin{array}{|c|} \hline \\
\{\grad_{\msc{s}}\,u_\tau \cdot \grad_{\msc{s}}\,v_\tau\}_{\r_\tau} \ \simeq \ 0
\\ \\ \hline \end{array}
\end{equation}\noindent
\end{itemize}
It must be noted, however, that boundary conditions~\ref{GBR:Indet}
and~\ref{GBR:Indet-1} fully specify $\{u_\tau,v_\tau\}_{\r_\tau}$ only on
horizon $H_\tau$. To propagate these conditions downward throughout the
whole domain $G_\tau$ whilst honoring constraints \ref{GBR:X11} or \ref{GBR:S1}, we propose to specify that the gradients of these
functions must vary as smoothly as possible in $G_\tau$. In practice, this may
be achieved in a least squares sense thanks to the following, additional constraint:
\begin{equation} \label{GBR-UtauVtau-smoothness}
\sum_{\alpha\in\{x,y,z\}}
\int_{G_\tau}
\biggl[
|| \partial_\alpha \grad\,u_\tau(\r_\tau) ||^2 +
|| \partial_\alpha \grad\,v_\tau(\r_\tau)||^2
\biggr] \cdot d\r_\tau \qquad \mbox{minimum}
\end{equation}\noindent
\section{Accounting for faults}
\label{GBR-taking-faults-into-account}
As shown on Figure~\ref{GBR-Tetrahedral_Mesh-1}, 3D geological domain $G$ may be
cut by faults and GeoChron functions $\{u,v,t\}_\r$ are discontinuous
across these faults. Similarly, faults induce discontinuities in GeoChron
functions $\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ defined on $G_\tau$. However, based on
geological arguments presented below, values and gradients of functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ on either side of a fault should generally honor geometric constraints which are
specific to particular types of faults.
\subsection*{$\Bmath{\tau}$-active \& $\Bmath{\tau}$-inactive faults}
With respect to a given restoration time $\tau$, we classify faults according to the
two following categories:
\begin{itemize}
\item A fault which intersects horizon $H_\tau$ is a
``$\tau$-active'' fault (e.g $F$ in Figure~\ref{GBR-DyingFaults});
\item A fault which belongs to $G_\tau$ and does not intersect
horizon $H_\tau$ is a ``$\tau$-inactive'' fault (e.g.
$F^{\prime}$ and $F^{\prime\prime}$ in Figure~\ref{GBR-DyingFaults}).
\end{itemize}
\noindent
``$\tau$-active'' or ``$\tau$-inactive'' status is defined relatively to
restoration time $\tau$: At an older restoration time
$\tau^\prime<\tau$, a $\tau$-inactive fault which intersects $H_{\tau^\prime}$
may become a $\tau^\prime$-active fault.
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-DyingFaults,width=116mm}}
\caption{
Vertical cross section where multicolored bold lines represent horizons
$\{H_t: t\leq\tau\}$ and green dashed lines depict surfaces $\{{\cal S}_\tau(d):d\geq 0\}$ parallel to $\{H_\tau\equiv{\cal S}_\tau(0)\}$. For each pair of
twin-points $(\r_{\msc{f}}^+,\r_{\msc{f}}^-)$ on $H_\tau$, we have both
$\{t_\tau(\r_{\msc{f}}^+)=t_\tau(\r_{\msc{f}}^-)=z_\tau^o\}$ and
$\{t(\r_{\msc{f}}^+)=t(\r_{\msc{f}}^-)\}$. Level sets $\{{\cal S}_\tau(d):d\geq 0\}$ of
function $t_\tau(\r_\tau)$ are continuous across $\tau$-inactive faults (in gray).
}
\label{GBR-DyingFaults}
\end{figure}\noindent
\ \\
When horizon $H_\tau$ is restored, the jelly block $G_\tau$
underneath $H_\tau$ must behave as if it were only impacted by $\tau$-active
faults. All other geologic objects such as horizons and $\tau$-inactive faults
embedded in the jelly block are passively deformed by this restoration
process. In other words, restoration functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ must be continuous across $\tau$-inactive faults.
\ \\
However, geological domain $G_\tau$ is topologically
discontinuous across faults of any type.
In order to ensure functions $\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ are ${\cal C}^1$-continuous across
$\tau$-inactive faults, the following constraints may be set on pairs of
``$\tau$-mate-points'' $(\r_{\msc{f}}^\oplus,\r_{\msc{f}}^\ominus)_\tau$
defined as collocated points lying on $F^+$
and $F^-$, respectively:
\begin{equation} \label{GBR-Secondary-faults}
\begin{array}{|c|} \hline \\
\left|
\begin{array}{cccccccccccccccccc}
1)& u_\tau(\r_{\msc{f}}^\oplus) &=& u_\tau(\r_{\msc{f}}^\ominus)
\\
2)& v_\tau(\r_{\msc{f}}^\oplus) &=& v_\tau(\r_{\msc{f}}^\ominus)
\\
3)& t_\tau(\r_{\msc{f}}^\oplus) &=& t_\tau(\r_{\msc{f}}^\ominus)
\end{array}
\right.
\quad \Bmath{\&} \quad
\left|
\begin{array}{cccccccccccccccccc}
4)& \grad\,u_\tau(\r_{\msc{f}}^\oplus) &=& \grad\,u_\tau(\r_{\msc{f}}^\ominus)
\\
5)& \grad\,v_\tau(\r_{\msc{f}}^\oplus) &=& \grad\,v_\tau(\r_{\msc{f}}^\ominus)
\\
6)& \grad\,t_\tau(\r_{\msc{f}}^\oplus) &=& \grad\,t_\tau(\r_{\msc{f}}^\ominus)
\end{array}
\right.
\\ \\
\forall\ (\r_{\msc{f}}^\oplus,\r_{\msc{f}}^\ominus)_\tau \in \ \mbox{$\tau$-inactive fault $F$}
\\ \\ \hline \end{array}
\end{equation}
\noindent
\subsection*{Boundary conditions for $\protect\Bmath{\{u_\tau,v_\tau\}_{\r_\tau}}$ on $\protect\Bmath{\tau}$-active faults}
When horizon $H_\tau$ is restored, terrains located on either side of
$\tau$-active faults must slide along lines called $\tau$-fault-striae, tangent
to these faults. For geological consistency, $\tau$-fault-striae must be
identical to fault-striae (see Figure~\ref{GBR-fault-striae}) associated with the $\{u,v\}_\tau$
paleo-geographic coordinates of the GeoChron model provided as input to the
proposed restoration method:
\begin{equation} \label{GBR-fault-striae-identity}
\begin{array}{|c|} \hline \\
\Bmath{\tau}\mbox{-fault-striae}\ \equiv \ \mbox{fault-striae}
\qquad \forall\ \tau
\\ \\ \hline \end{array}
\end{equation}\noindent
\noindent
As \citet{Lovely2018} use the standard
SKUA\textsuperscript{\tiny\textregistered} algorithm to reestablish continuity
of functions $\{u_\tau,v_\tau\}_{\r_\tau}$ through faults, $\tau$-twin-points are
recomputed from the geometry and
topology of level-sets $\{{\cal S}(d):d\geq 0\}$ of $t_\tau(\r_\tau)$ in $G_\tau$. As
a consequence, these $\tau$-twin-points may not be located on the same
fault-striae as those induced by known $\{u,v,t\}_\r$ functions of the input
GeoChron model (e.g., see Figure~\ref{GBR-fault-striae}), which may break the
consistency between input and restored model close to faults.
\ \\
Similarly to GeoChron twin-points $(\r_{\msc{f}}^+,\r_{\msc{f}}^-)$\footnote{ See
definition~\ref{GBR:TP-1}.}, $\tau$-twin-points
$(\r_{\msc{f}}^+,\r_{\msc{f}}^-)_\tau$ are characterized by the following
equations:
\begin{equation} \label{GBR:TP-1.tau}
\{\ (\r_{\msc{f}}^+,\r_{\msc{f}}^-)_\tau \mbox{ is a pair of $\tau$-twin-points } \}
\quad \Longleftrightarrow \quad
\left\{
\begin{array}{ccccccc}
1)& \mbox{$F$ is a $\tau$-active fault} \\
2)& \r_{\msc{f}}^+ \in F^+ \ \mbox{ \B{\&} } \ \r_{\msc{f}}^- \in F^- \\ \\
3)& u_\tau(\r_{\msc{f}}^-) = u_\tau(\r_{\msc{f}}^+) \\
4)& v_\tau(\r_{\msc{f}}^-) = v_\tau(\r_{\msc{f}}^+) \\
5)& t_\tau(\r_{\msc{f}}^-) = t_\tau(\r_{\msc{f}}^+)
\end{array}
\right.
\end{equation}\noindent
\ \\
In order to restore terrains along $\tau$-active faults without creating voids
or overlaps,
constraints~\ref{GBR:TP-1.tau}-3 to \ref{GBR:TP-1.tau}-5 must be honored
all along $\tau$-active faults.
Function $t_\tau(\r_\tau)$ is independent from $\{u_\tau,v_\tau\}_{\r_\tau}$ and
is assumed to be already known. As a consequence, $\{u_\tau,v_\tau\}_{\r_\tau}$ simply have to honor the
following constraints:
\begin{equation} \label{GBR-Sigma-6}
\begin{array}{|c|} \hline \\
\left.
\begin{array}{ccccccccccccc}
1)& u_\tau(\r_{\msc{f}}^+) &=& u_\tau(\r_{\msc{f}}^-)
\\ \\
2)& v_\tau(\r_{\msc{f}}^+) &=& v_\tau(\r_{\msc{f}}^-)
\end{array}
\right\}
\quad \forall\ (\r_{\msc{f}}^+,\r_{\msc{f}}^-)_\tau \in \mbox{ $\tau$-active fault}
\\ \\ \hline \end{array}
\end{equation}\noindent
\subsection*{Why distinguish $\Bmath{\tau}$-active from $\Bmath{\tau}$-inactive faults?}
At restoration geological-time $\tau$, any fault that isolates a fault block in
$G_\tau$ such that pseudo light emitted by $H_\tau$ cannot reach it, must be
considered as $\tau$-inactive in order for this fault block to be restored.
\ \\
Next, considering faults that do not intersect $H_\tau$ as active may result in
erroneous distortion of restored terrains. The top left corner of
Figure~\ref{GBR-DyingFaults-1} shows the same cross section as
Figure~\ref{GBR-DyingFaults} but the restoration for horizon
$H_\tau$ is computed while considering fault $F^{\prime\prime}$, which does not
intersect $H_\tau$, as a $\tau$-active fault.
\ \\
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-DyingFaults-1,width=120mm}}
\caption{
Vertical cross section in which fault $F^{\prime\prime}$ is
erroneously considered as a $\tau$-active fault. In the restoration process
induced by a $u_\tau v_\tau t_\tau$-transform, the distance between points
$\B{A}$ and $\B{B}$ cannot be correctly preserved.
}
\label{GBR-DyingFaults-1}
\end{figure}\noindent
If fault $F^{\prime\prime}$ is considered as a $\tau$-active fault, pairs of
points $(\B{a},\B{A})$ and $(\B{b},\B{B})$ shown in the top left part of
Figure~\ref{GBR-DyingFaults-1} are considered as $\tau$-twin-points.
After restoration, $\tau$-twin-points are collocated, implying that
$(\overline{\B{a}}=\overline{\B{A}})$ and $(\overline{\B{b}}=\overline{\B{B}})$
(right hand side of Figure~\ref{GBR-DyingFaults-1}).
\ \\
As distance $(\B{AB}=\overline{\B{AB}^\star})$ differs from
$(\overline{\B{AB}}=\overline{\B{ab}})$, in the neighborhood
of faults $F$ and $F^{\prime\prime}$, a restoration
performed via $u_\tau v_\tau t_\tau$-transform would generate incorrect
variations in lengths and volumes.
\ \\
To avoid these inconsistencies,
distinguishing $\tau$-active and $\tau$-inactive faults is a key component of
the GBR method and an improvement over the first implementation of a GeoChron
restoration method by \citet{Lovely2018}.
\subsection*{Manually activating faults}
According to geological context, structural geologists may prefer some
$\tau$-inactive faults to be considered as $\tau$-active, for example when a
thrust fault is known to be active at a particular time even though it did not
break through to the sea floor. Technically, this is possible for nearly any
fault, which is then constrained with Equations~\ref{GBR-Sigma-6} as any other
active fault.
\ \\
However, any fault bordering a $\tau$-dark fault block (e.g.
Figure~\ref{GBR-dark-fault-block}) must be handled as $\tau$-inactive, otherwise
functions $\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ would be undetermined inside this
$\tau$-dark fault block which, as a consequence, could not be restored.
\section{Taking compaction into account}
\label{GBR:::compaction}
Compaction is defined as pore space reduction in sediments due to increased load
during deposition. As this process changes the geometry of geological
layers as their depths increase, restoration workflows frequently handle
compaction as an option.
\ \\
As we have done so far, assume that an initial version of restoration functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ has been obtained without taking compaction
into account. In other words, the compacted thicknesses of layers in studied domain
$G_\tau$, as observed today, have been approximately preserved in restored domain
$\overline{G}_\tau$. In effect, restoration uplifts and unloads terrains, which
should induce decompaction resulting in increased layer thicknesses in the restored domain.
\ \\
In this section we show how $\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ can be replaced
by new restoration functions in such a way that
the new $u_\tau v_\tau t_\tau$-transform
$\overline{G}_\tau$ of $G_\tau$ so obtained restores the terrains and induces
thickness variations as a consequence of decompaction, which should be the
exact inverse of the compaction that occurred between geological-time $\tau$ and
the present geological-time.
\subsection*{Athy's law}
As the concept of decompaction may be easier to grasp in the restored space, we
refer to Figure~\ref{GBRestoration-1}-D which shows the subsurface restored at geological-time
$\tau$. Let $\overline{V}(\overline{\r}_\tau)$ be an infinitely small
volume of sediment centered on point $\overline{\r}_\tau\in\overline{G}_\tau$
underneath the sea floor $\{\overline{\cal S}_\tau(0)\equiv\overline{H}_\tau\}$.
\ \\
Laboratory experiments on rock samples show that, during burial when sediments
contained in $\overline{V}(\overline{\r}_\tau)$
compact under their own weight, their porosity $\overline\Psi(\overline{\r}_\tau)$
exponentially decreases according to Athy's law \citep{Athy1930}:
\begin{equation} \label{GBR::Compaction-0}
\overline\Psi(\overline{\r}_\tau) \ \simeq \ \overline\Psi_o(\overline{\r}_\tau) \cdot exp\biggl\{ -\overline\kappa(\overline{\r}_\tau)\cdot\delta(\overline{\r}_\tau) \biggr\}
\qquad \forall\ \overline{\r}_\tau\in \overline{G}_\tau
\end{equation}\noindent
In this equation, $\delta(\overline{\r}_\tau)$ is
the absolute distance, or depth, from point
$\overline{\r}_\tau\in\overline{G}_\tau$ to sea floor $\overline{\cal
S}_\tau(0)$ measured at geological-time $\tau$ whilst
$\overline\Psi_o(\overline{\r}_\tau)<1$ and
$\overline\kappa(\overline{\r}_\tau)>0$ are known non-negative coefficients
which only depend on rock type at location $\overline{\r}_\tau$.
As an example, assuming that $\delta(\overline{\r}_\tau)$ is expressed in
meters, the following average coefficients for sedimentary terrains observed
in southern Morocco have been reported \citep{Labbassi1999}:
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
$Rock\ type$ & $\overline\Psi_o$ & $\overline\kappa$ \\
\hline
Siltstone & 0.62 & 0.57$\times 10^{-3}$\\
Clay & 0.71& 0.77$\times 10^{-3}$\\
Sandstone & 0.35& 0.60$\times 10^{-3}$\\
Carbonates & 0.46& 0.23$\times 10^{-3}$\\
Dolomites & 0.21& 0.61$\times 10^{-3}$\\
\hline
\end{tabular}
\end{center}
\ \\
Keeping in mind that, in the restored $\overline{G}_\tau$-space,
$\{-{t}_\tau(\overline{\r}_\tau)\}$ measures the vertical distance from
$\overline{\r}_\tau$ to sea floor $\overline{S}_\tau(0)$, in
Equation~\ref{GBR::Compaction-0}, depth function
$\delta(\overline{\r}_\tau)$ can be expressed as follows:
\begin{equation} \label{GBR:decompaction-D-1}
\delta(\overline{\r}_\tau) \ = \ - {t}_\tau(\overline{\r}_\tau)
\qquad \forall\ \overline{\r}_\tau \in \overline{G}_\tau
\end{equation} \noindent
As a consequence, in the context of our GBR method, Athy's law may straightforwardly be reformulated as:
\begin{equation} \label{GBR::Compaction-AthyGBR}
\overline\Psi(\overline{\r}_\tau) \ \simeq \ \overline\Psi_o(\overline{\r}_\tau) \cdot exp\biggl\{ \overline\kappa(\overline{\r}_\tau)\cdot{t}_\tau(\overline{\r}_\tau) \biggr\}
\qquad \forall\ \overline{\r}_\tau\in \overline{G}_\tau
\end{equation}\noindent
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-Compaction,width=116mm}}
\caption{
Porosity versus compaction: Infinitely small column of
sediment where, under vertical compaction, the
vertical dimension of pores represented in yellow is halved. As a
result, $(d\bar{h}^\oplus=3)$ becomes $(d\bar{h}=2.5)$. Note that
yellow and gray cells may be randomly swapped in the vertical
direction without altering these results.
}
\label{GBR-Compaction}
\end{figure}\noindent
\subsection*{Decompaction in $\Bmath{\overline{G}_\tau}$}
Elasto-plastic mechanical frameworks developed to model compaction rely on a
number of input parameters which may be difficult for a geologist or geomodeler
to assess and require solving a complex system of equations
\citep{Schneider1996}. Isostasic approaches are simpler to parameterize and
still provide useful information on basin evolution \citep{DurandRiard2011}.
Therefore, we will consider compaction as a mainly one-dimensional, vertical
process induced by gravity which mainly occurs in the early stages of sediment burial
when horizons are still roughly horizontal surfaces close to the sea floor. At
any point $\overline{\r}_\tau\in\overline{G}_\tau$ within a layer, the decompacted thickness
$d\bar{h}^\oplus(\overline{\r}_\tau)$ of a vertical probe consisting of an infinitely short column of sediment roughly
orthogonal to the restored horizon passing through
$\overline{\r}_\tau$ is linked to the thickness
$d\bar{h}(\overline{\r}_\tau)$ of the shorter, compacted vertical column by the following
relationship:
\begin{equation} \label{GBR:decompaction-1.u.1}
\forall\ \overline{\r}_\tau\in\overline{G}_\tau \ : \quad
\left|
\begin{array}{c}
d\bar{h}^\oplus(\overline{\r}_\tau)
\ = \ \displaystyle
\frac{1}{1-\overline{\phi}_\tau(\overline{\r}_\tau)}\cdot
d\bar{h}(\overline{\r}_\tau)
\\ \\
\mbox{with : } \overline{\phi}_\tau(\overline{\r}_\tau) = \overline\Psi_o(\overline{\r}_\tau)-\overline\Psi(\overline{\r}_\tau) \in [0,1[
\end{array}
\right.
\end{equation}
\noindent
In this equation, $\overline{\phi}(\overline{\r}_\tau)$ denotes the ``compaction coefficient''
which characterizes vertical shortening of the probe at restored location
$\overline{\r}_\tau\in\overline{G}_\tau$. As an example,
Figure~\ref{GBR-Compaction} shows the same infinitely short vertical column of
sediment where average porosity is equal to $(\overline\Psi_o=1/3)$ before
compaction and $(\overline\Psi=1/6)$ after compaction. The compaction
coefficient $(\overline\Psi_o-\overline\Psi)$ is then equal to
$(\overline{\phi}=1/6)$ and column shortening $(1-\overline{\phi})$ is
$(5/6)$.
\subsection*{Taking present day compaction in $\Bmath{\overline{G}_\tau}$ into account}
So far, in the context of our GBR method, the restored $\overline{G}_\tau$-space
has been built assuming that there is no compaction. As a consequence,
$\overline{G}_\tau$ we obtained so far is incorrect because it has undergone
compaction characterized by present day porosity
$\overline{\Psi}_p(\overline{\r}_\tau)$.
\ \\
Let $\bar{\phi}_\tau^\ominus(\overline{\r}_\tau)$ and
$\bar{\phi}_\tau^\oplus(\overline{\r}_\tau)$ be the pair of given functions
defined by:
\begin{equation} \label{GBR:decompaction-Step-1}
\forall\ \overline{\r}_\tau\in\overline{G}_\tau \ : \quad
\left|
\begin{array}{llllllllllll}
\bar{\phi}_\tau^\ominus(\overline{\r}_\tau)
&=& \bar\Psi_o(\overline{\r}_\tau)-\overline\Psi_p(\overline{\r}_\tau)
\\ \\
\bar{\phi}_\tau^\oplus(\overline{\r}_\tau)
&=& \bar\Psi_o(\overline{\r}_\tau)-\overline\Psi(\overline{\r}_\tau)
\end{array}
\right.
\end{equation}\noindent
where, for coherency with Athy's law, present day porosity $\overline\Psi_p(\overline{\r}_\tau)$ is assumed to honor the following constraint:
\begin{equation} \label{GBR:decompaction-Step-1.C}
\overline\Psi_\p(\overline{\r}_\tau) \ \leq \ \overline\Psi(\overline{\r}_\tau)
\qquad
\forall\ \overline{\r}_\tau\in\overline{G}_\tau
\end{equation}\noindent
Note that such a constraint implies that $\phi_\tau^\oplus({\r}_\tau)\leq \phi_\tau^\ominus({\r}_\tau)$.
\ \\
Considering once again the vertical probe introduced above in restored space
$\overline{G}_\tau$ and using equation \ref{GBR:decompaction-1.u.1} twice in a
forward then backward way, to take compaction into account, we propose the
following two steps:
\begin{enumerate}
\item First, to cancel out the compaction characterized by given, present day
porosity $\overline\Psi_p(\overline{\r}_\tau)$, a ``total'', vertical decompaction
is applied by updating $d\bar{h}(\overline{\r}_\tau)$ as follows:
\begin{equation} \label{GBR:decompaction-Step-1z}
d\bar{h}_o(\overline{\r}_\tau)
\ = \ \displaystyle
\frac{1}{1-\bar{\phi}_\tau^\ominus(\overline{\r}_\tau)}\cdot
d\bar{h}(\overline{\r}_\tau)
\end{equation}\noindent
After this first operation, probe porosity is equal to $\overline{\Psi}_o(\overline{\r}_\tau)$.
\item Next, a ``partial'' recompaction is applied as a function of the actual porosity $\overline\Psi(\overline{\r}_\tau)$
approximated by Athy's law \ref{GBR::Compaction-AthyGBR} at geological-time $\tau$:
\begin{equation} \label{GBR:decompaction-Step-1zz}
d\bar{h}^\oplus(\overline{\r}_\tau)
\ = \ \displaystyle
\{1-\bar{\phi}_\tau^\oplus(\overline{\r}_\tau)\}\cdot
d\bar{h}_o(\overline{\r}_\tau)
\end{equation}\noindent
After this second operation, probe porosity is equal to $\overline{\Psi}(\overline{\r}_\tau)$.
\end{enumerate}\noindent
Therefore, to take present-day compaction into account, Equation~\ref{GBR:decompaction-1.u.1} must be replaced by:
\begin{equation} \label{GBR:decompaction-Step-3}
d\bar{h}^\oplus(\overline{\r}_\tau)
\ = \ \displaystyle
\frac{1-\bar{\phi}_\tau^\oplus(\overline{\r}_\tau)}{1-\bar{\phi}_\tau^\ominus(\overline{\r}_\tau)}\cdot
d\bar{h}(\overline{\r}_\tau)
\qquad \forall\ \overline{\r}_\tau\in\overline{G}_\tau
\end{equation}
\noindent
\subsection*{GBR approach to decompaction in $\Bmath{\overline{G}_\tau}$}
In the restored $\overline{G}_\tau$-space, ${t}_\tau(\overline{\r}_\tau)$ may be
interpreted as an arc-length abscissa ${s}(\overline{\r}_\tau)$ along the
vertical straight line passing through $\overline{\r}_\tau$ oriented in the same
direction as the vertical unit frame vector\footnote{ See equation \ref{GBR:X00}.}
$\{\overline{\r}_{t_\tau}=\r_z\}$. Therefore, in the $\overline{G}_\tau$-space,
\begin{equation} \label{GBR::Compaction-dt}
d{t}_\tau(\overline{\r}_\tau)
\ = \
ds(\overline{\r}_\tau)
\ = \
d\overline{h}(\overline{\r}_\tau)
\end{equation}\noindent
is the height of an infinitely short vertical column of restored sediment
located at point $\overline{\r}_\tau\in\overline{G}_\tau$, subject to present-day compaction.
As a consequence, to take compaction into account in the restored
$\overline{G}_\tau$-space, according to Equations~\ref{GBR:decompaction-Step-3}
and \ref{GBR::Compaction-dt}, function ${t}_\tau(\overline{\r}_\tau)$ must be
replaced by a ``decompacted'' function ${t}_\tau^\oplus(\overline{\r}_\tau)$ such that:
\begin{equation} \label{GBR:decompaction-A}
\frac{d{t}_\tau^\oplus}{d{t}_\tau}\bigg|_{\overline{\r}_\tau}
\ = \ \displaystyle
\frac{d\bar{h}^\oplus(\overline{\r}_\tau)}{d\bar{h}(\overline{\r}_\tau)}
\ = \ \displaystyle
\frac{1-\bar{\phi}_\tau^\oplus(\overline{\r}_\tau)}{1-\bar{\phi}_\tau^\ominus(\overline{\r}_\tau)}
\end{equation} \noindent
Assuming that $\{\overline{\r}_{t_\tau}=\r_z\}$ is the unit vertical frame vector of the
$\overline{G}_\tau$-space, it is well known that
\begin{equation} \label{GBR:decompaction-B}
\grad\,{t}_\tau^\oplus(\overline{\r}_\tau)\cdot \ \overline{\r}_{t_\tau}
\ = \
\frac{d{t}_\tau^\oplus(\overline{\r}_\tau + {s}\cdot\overline{\r}_{t_\tau})}{d{s}}\bigg|_{s=0}
\ = \
\frac{d{t}_\tau^\oplus}{d{t}_\tau}\bigg|_{\overline{\r}_\tau}
\end{equation} \noindent
from which we can conclude that the current altitude ${t}_\tau(\overline{\r}_\tau)$
of point $\overline{\r}_\tau\in\overline{G}_\tau$ should be transformed into a
decompacted altitude ${t}_\tau^\oplus(\overline{\r}_\tau)$ honoring the following differential equation:
\begin{equation} \label{GBR:decompaction-C}
\begin{array}{|c|} \hline \\
\quad
\displaystyle
\grad\,{t}_\tau^\oplus(\overline{\r}_\tau)\cdot \overline{\r}_{{t}_\tau}
\ = \ \frac{1-{\phi}_\tau^\oplus(\overline{\r}_\tau)}{1-{\phi}_\tau^\ominus(\overline{\r}_\tau)}
\qquad \forall\ \overline{\r}_\tau\in \overline{G}_\tau
\\ \\
\mbox{with : }\quad
\bar{\phi}_\tau^\oplus(\overline{\r}_\tau) = \overline{\Psi}_o(\overline{\r}_\tau)-\overline{\Psi}(\overline{\r}_\tau)
\quad \& \quad
\bar{\phi}_\tau^\ominus(\overline{\r}_\tau) = \overline{\Psi}_o(\overline{\r}_\tau)-\overline{\Psi}_p(\overline{\r}_\tau)
\quad
\\ \\ \hline \end{array}
\end{equation}
\noindent
Due to the vertical nature of compaction, on $\{\overline{\cal
S}(0)\equiv\overline{H}_\tau\}$,
function ${t}_\tau^\oplus(\overline{\r}_\tau)$ should vanish and its gradient
should be vertical. In other words, in addition to constraint
\ref{GBR:decompaction-C}, function ${t}_\tau^\oplus(\overline{\r}_\tau)$ must
also honor the following boundary conditions where $\overline{\r}_{u_\tau}$ and
$\overline{\r}_{v_\tau}$ are the unit horizontal frame vectors of the $\overline{G}_\tau$-space:
\begin{equation} \label{GBR:decompaction-D}
\begin{array}{|c|} \hline \\
\quad
\forall\ \overline{\r}_\tau^o\in \{\overline{\cal S}_\tau(0)\equiv \overline{H}_\tau\}\ : \quad
\left|
\begin{array}{cc}
1)& t_\tau^\oplus(\bar\r_\tau^o) = \ 0
\\ \\
2)& \grad\,{t}_\tau^\oplus(\overline{\r}_\tau^o)\cdot \overline{\r}_{\bar{u}_\tau} \ = \ 0
\\
3)& \grad\,{t}_\tau^\oplus(\overline{\r}_\tau^o)\cdot
\overline{\r}_{\bar{v}_\tau} \ = \ 0
\end{array}
\right.
\quad
\\ \\ \hline \end{array}
\end{equation}
\noindent
As compaction is a continuous process, ${t}_\tau^\oplus(\overline{\r}_\tau)$
must be ${\cal C}^o$-continuous across all faults affecting $\overline{G}_\tau$.
As a consequence, in addition to constraints \ref{GBR:decompaction-C} and
\ref{GBR:decompaction-D}, for any fault $\overline{F}$ in $\overline{G}_\tau$, function
${t}_\tau^\oplus(\overline{\r}_\tau)$ must also honor the following boundary
conditions where $(\overline{\r}_{\msc{f}}^\oplus,\overline{\r}_{\msc{f}}^\ominus)_\tau$
are pairs of ``$\tau$-mate-points''defined as collocated points lying on the
positive face $\overline{F}^+$
and negative face $\overline{F}^-$ of $\overline{F}$ at geological-time $\tau$:
\begin{equation} \label{GBR:decompaction-E}
\begin{array}{|c|} \hline \\
\begin{array}{c}
{t}_\tau(\overline{\r}_{\msc{f}}^\oplus) \ = \ {t}_\tau(\overline{\r}_{\msc{f}}^\ominus)
\\ \\
\forall\ \overline{F}\in \overline{G}_\tau \quad \&\quad
\forall\
(\overline{\r}_{\msc{f}}^\oplus,\overline{\r}_{\msc{f}}^\ominus)_\tau \in \ \overline{F}
\end{array}
\\ \\ \hline \end{array}
\end{equation}
\noindent
Using an appropriate numerical method, ${t}_\tau^\oplus(\r_\tau)$ must be
computed in $\overline{G}_\tau$ whilst ensuring that differential
equation~\ref{GBR:decompaction-C} and boundary conditions
\ref{GBR:decompaction-D} and \ref{GBR:decompaction-E} are honored. To ensure
smoothness and uniqueness of
${t}_\tau^\oplus(\r_\tau)$, the following constraint may also be added:
\begin{equation} \label{GBR:decompaction-F}
\sum_{(a,b)\in \{u_\tau,v_\tau, t_\tau\}^2}
\int_{\overline{G}_\tau}
\biggl\{\partial_a\partial_b\, {t}_\tau^\oplus(\overline{\r}_\tau) \biggr\}^2
\cdot d\overline{\r}_\tau
\qquad \mbox{minimum}
\end{equation}\noindent
\ \\
As a conclusion, to take compaction into account, the following GBR approach may be used:
\begin{enumerate}
\item Compute a numerical approximation of ${t}_\tau^\oplus(\overline{\r}_\tau)$ in $\overline{G}_\tau$ and use the reverse $u_\tau v_\tau t_\tau$-transform to update ${t}_\tau({\r}_\tau)$ in ${G}_\tau$:
\begin{equation} \label{GBR:decompaction-G}
{t}_\tau({\r}_\tau) \longleftarrow \ {t}_\tau^\oplus(\overline{\r}_\tau)
\qquad \forall\ {\r}_\tau\in G_\tau;
\end{equation}\noindent
\item Recompute numerical approximations of restoration functions $u_\tau(\r_\tau)$ and $v_\tau(\r_\tau)$ in $G_\tau$ to prevent voids
and overlaps in the restored space, as,
according to Equations~\ref{GBR:X11} and \ref{GBR:S1}, $u_\tau(\r_\tau)$ and $v_\tau(\r_\tau)$
depend on $t_\tau(\r_\tau)$;
\item Build the ``decompacted'' restored space $\overline{G}_\tau$ as the new, direct $u_\tau v_\tau t_\tau$-transform of geological space $G_\tau$ observed today.
\end{enumerate}\noindent
\ \\
This approach to decompaction is fully derived from the GBR framework described
in this paper and differs from the sequential decompaction following
Athy's law along IPG-lines applied by~\citet{Lovely2018}.
\section{Constraints summary}
Among all the equations presented so far,
Equations~\ref{GBR:eikonal-equation}-1, \ref{GBR:Indet}, \ref{GBR:Indet-1} and \ref{GBR-Sigma-6}
are the most critical.
\ \\
First and above all, honoring constraint~\ref{GBR:eikonal-equation}-1 as
closely as possible is the very heart of the proposed {GBR} method. Due to local
deformations of horizons, this equation may generally be honored only in a
least squares sense.
However, if $||\grad\,t_\tau||_\r$ deviates too much from 1, then, during the
restoration process, layer thicknesses will not be preserved, which may induce
undesirable volume variations; and due to constraints~\ref{GBR:X11} or
\ref{GBR:S1} based on $t_\tau(\r_\tau)$, restoration functions $\{u_\tau,v_\tau\}_{\r_\tau}$ will be
incorrect.
\ \\
Next, constraints~\ref{GBR-Sigma-6} are of paramount importance because,
during restoration of horizon $H_\tau$, they prevent gaps and overlaps from
appearing in $\overline{G}_\tau$ along faults.
\ \\
Finally, constraints~\ref{GBR:Indet} and \ref{GBR:Indet-1} are also extremely important because
they preserve coherency of restored surface $\overline{H}_\tau$ viewed either as
the $u_\tau v_\tau t_\tau$-transform or the regular GeoChron $uvt$-transform of
$H_\tau$. Without constraints~\ref{GBR:Indet}, the GBR method would not be
consistent with the input GeoChron model.
\subsection*{Comment: Volume preservation}
Barring the effects of compaction, let us consider, in the $G_\tau$-space, a
pseudo-layer $L(d,\varepsilon)$ with infinitely small thickness $\varepsilon$
bounded by pseudo-horizons ${\cal S}(d)$ and ${\cal S}(d-\varepsilon)$. Because
of eikonal constraint \ref{GBR:eikonal-equation}, in the
$\overline{G}_\tau$-space, restored layer $\overline{L}(d,\varepsilon)$ holds
as closely as possible the same thickness $\varepsilon$ as $L(d,\varepsilon)$.
\ \\
Consider now, in the $G_\tau$-space, an infinitely small compact patch
$\Delta{\cal S}(d)$ drawn on ${\cal S}(d)$ and let $\Delta{\cal
S}(d-\varepsilon)$ be the projection of this patch onto ${\cal
S}(d-\varepsilon)$ along lines with constant
$\{u_\tau,v_\tau\}$ coordinates\footnote{ In GeoChron theory, these lines are
called ``Iso-Paleo-Geographic'' lines and abbreviated IPG-lines.}
passing through ${\cal S}(d)$. Let $\Delta V(d,\varepsilon)$ be the infinitely
small volume bounded by $\Delta{\cal S}(d)$, $\Delta{\cal S}(d-\varepsilon)$ and
the field of lines defined above. During restoration, depending on the
structural style, two cases have to be considered:
\begin{itemize}
\item if the structural style is flexural slip, by definition\footnote{ See
\citet{Mallet2014}, page 72.}, areas and angles
on surfaces ${\cal S}(d)$ and ${\cal S}(d-\varepsilon)$ are preserved;
\item if the structural style is minimal deformation, by definition\footnote{ See
\citet{Mallet2014}, page 71.}, deformations
of areas and angles on surfaces ${\cal S}(d)$ and ${\cal S}(d-\varepsilon)$ are
minimized, as much as possible.
\end{itemize}\noindent
Therefore, omitting compaction, as in both cases thickness $\varepsilon$ is
preserved as much as possible, volumes of $\Delta V(d,\varepsilon)$ and its restored version
$\overline{\Delta} V(d,\varepsilon)$ are as identical as possible.
\section{Numerically approximating $\protect\Bmath{\{u_\tau,v_\tau,t_\tau\}}$}
\label{GBR-Improving-t-tau}
From a theoretical standpoint, restoration functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ are solutions to a wide system of partial
differential equations presented so far in this paper. However, from a practical
perspective, these equations are often non linear and coupled, which makes them
difficult to solve. Many general numerical techniques known in the art
could be employed but, as we show in the following, the geological nature of our
problem makes it possible for us to replace these complex
differential equations by surrogates which are easier to solve.
\subsection*{About the eikonal equation}
As pointed out in the previous section, computing a function $t_\tau(\r_\tau)$ which honors eikonal
Equation~\ref{GBR:eikonal-equation} is the corner-stone of our
proposed {GBR} method but Equation~\ref{GBR:eikonal-equation}-1, recalled below,
is not linear:
\begin{equation} \label{GBR:eikonal-equation-1}
||\grad\,t_\tau(\r_\tau)|| = 1 \qquad \forall\ \r_\tau\in G_\tau
\end{equation}\noindent
Through Equations~\ref{GBR:X11} or \ref{GBR:S1}, any excessive violation of this
constraint also impacts functions $\{u_\tau,v_\tau\}_{\r_\tau}$ and the resulting
restoration is then inevitably incorrect.
\ \\
Based on the test example shown in Figure~\ref{GBR-Ramp-test}, where
horizon $H_\tau$ to restore is the central sigmoid surface, results obtained
with two different numerical techniques are compared and shown on
Figure~\ref{GBR-Ramp-results}.
This seemingly simple test is actually highly significant because it shows
local variations in curvature which make eikonal Equation~\ref{GBR:eikonal-equation-1} difficult to approximate numerically.
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-Ramp-test,width=116mm}}
\caption{
Test example: {GBR} of a ``ramp'' structure. Restored horizon $H_\tau$ is the central, sigmoid surface. Note that main curvature of $H_\tau$ locally varies and the associated curvature center moves from one side to the opposite side of $H_\tau$.
}
\label{GBR-Ramp-test}
\end{figure}\noindent
\subsection*{Computing $\Bmath{t_\tau(\r_\tau)}$: Surrogate (weak) eikonal equations}
We have stated before that:
\begin{equation} \label{GBR:X8}
\grad\,t_\tau(\r_\tau^o) \ = \ \B{N}(\r_\tau^o) \ = \ \frac{\grad\,t(\r_\tau^o)}{||\grad\,t(\r_\tau^o)||}
\qquad \forall\ \r_\tau^o\in H_\tau
\end{equation}\noindent
which means that eikonal Equation~\ref{GBR:eikonal-equation} is approximately
equivalent to the following system called ``surrogate-eikonal'' equation:
\begin{equation} \label{GBR:X9}
\left|
\begin{array}{cccc}
1)& \displaystyle
\sum_{\alpha\in\{x,y,z\}}
\int_{G_\tau} ||\partial_\alpha\, \grad\, t_\tau(\r_\tau) ||^2 \cdot d\r_\tau \qquad \mbox{minimum}
\\ \\
2)& \mbox{subject to : \ }
\left\{
\begin{array}{cclllll}
a)& t_\tau(\r_\tau^o) = 0
\\ \\
b)& \displaystyle \grad\,t_\tau(\r_\tau^o) = \B{N}(\r_\tau^o)
\end{array}
\right\} \quad \forall\ \r_\tau^o\in H_\tau
\end{array}\noindent
\right.
\end{equation}\noindent
Eikonal Equations~\ref{GBR:eikonal-equation}-2 are strictly honored
on $H_\tau$ and Equation~\ref{GBR:X9}-1 is assumed to smoothly
propagate $\grad\,t_\tau(\r)$ in such a way that, everywhere inside $G_\tau$ and
similarly to Equation~\ref{GBR:eikonal-equation}-1, $\grad\,t_\tau(\r_\tau)$
roughly remains a unit vector field.
In practice, according to techniques known in the art, Equation~\ref{GBR:X9}-1 may be
linearly approximated so that each Equation~\ref{GBR:X9} is linear and,
therefore, easier to solve than ``true'' eikonal equation ~\ref{GBR:eikonal-equation}.
\ \\
As mentioned above, at any point $\r_\tau\in G_\tau$, Equation~\ref{GBR:X9}-1 should
ensure that $||\grad\,t_\tau(\r_\tau)||$ is equal to its unit starting value on
$H_\tau$. Unfortunately, away from $H_\tau$, numerical drift usually makes
$||\grad\,t_\tau(\r_\tau)||$ deviate from target value 1. As a consequence, eikonal
constraint~\ref{GBR:eikonal-equation}-1 is generally not perfectly honored away from
$\{{\cal S}_\tau(0)\equiv H_\tau\}$, which implies that, after restoration,
distortions inevitably appear in the vertical direction of the
$\overline{G}_\tau$-space.
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-Ramp-results,width=100mm}}
\caption{
Test example showing histograms of $||\grad\,t_\tau(\r_\tau)||$ in the
studied domain corresponding to Figure~\ref{GBR-Ramp-test}. Depending on the
method used to compute $t_\tau(\r_\tau)$, the resulting magnitude of
$||\grad\,t_\tau(\r_\tau)||$ may severely deviate from value ``1'' required by
eikonal Equation~\ref{GBR:eikonal-equation}-1. Note that, for clarity's sake,
the vertical axis of histogram (B) has been shrunk by a factor of 3.
}
\label{GBR-Ramp-results}
\end{figure}\noindent
\ \\
On Figure~\ref{GBR-Ramp-results}-A, the histogram of $||\grad\,t_\tau(\r_\tau)||$
so obtained in $G_\tau$ with surrogate eikonal Equations~\ref{GBR:X9} applied to our test example clearly shows that
eikonal Equation~\ref{GBR:eikonal-equation}-1 is not honored correctly.
First, $||\grad\,t_\tau(\r_\tau)||$ is never equal to 1. Next, the median value is about
0.89, which represents an error of 11~\%. Finally, standard deviation is 0.032
and the spread between 25$^{th}$ and 75$^{th}$ percentiles is 0.051.
\ \\
We can conclude from these figures that approximating eikonal
Equation~\ref{GBR:eikonal-equation}-1 by Equations~\ref{GBR:X9} does not give
precise enough results.
\subsection*{Computing $\Bmath{t_\tau(\r_\tau)}$: A precise incremental solution}
Generally, even though eikonal Equation~\ref{GBR:eikonal-equation}-1 is not
perfectly honored, function $t_\tau(\r_\tau)$ generated by
Equations~\ref{GBR:X9} may be considered as an approximation of
the actual solution. In other words, assuming that $t_\tau^\star(\r)$ is a first approximation of $t_\tau(\r_\tau)$, there is an unknown function
$\varepsilon_\tau(\r_\tau)$ which may be used as follows to compute, in a post-processing step, an improved version of $t_\tau(\r_\tau)$:
\begin{equation} \label{GBR-ImprovT-10}
t_\tau(\r_\tau) \ = \ t_\tau^\star(\r_\tau) +
\varepsilon_\tau(\r_\tau)
\end{equation}
\noindent
with
\begin{equation} \label{GBR-ImprovT-11}
\varepsilon_\tau(\r_\tau^o) \ = \ - t_\tau^\star(\r_\tau^o)
\qquad\ \forall\ \r_\tau^o\in H_\tau
\end{equation}
\noindent
and where $t_\tau^\star(\r_\tau)$ is assumed to be precise enough to honor:
\begin{equation} \label{GBR-ImprovT-11-XX}
||\grad\, \varepsilon_\tau(\r_\tau)|| \ \ll \ ||\grad\, t_\tau^\star(\r_\tau)|| \ \simeq \ 1 \qquad\ \forall\ \r_\tau\in G_\tau
\end{equation}
\noindent
\ \\
Through faults, $\varepsilon_\tau(\r_\tau)$ is assumed to behave in a
similar way to function $t_\tau(\r_\tau)$. In other words,
referring to constraints~\ref{GBR-Secondary-faults}, for any pair of
$\tau$-mate-points $(\r_{\msc{f}}^\oplus,\r_{\msc{f}}^\ominus)$ located on a $\tau$-inactive fault $F$,
function $\varepsilon_\tau(\r_\tau)$ and its gradient must honor the following
equations:
\begin{equation} \label{GBR-Secondary-faults-XXX}
\left.
\begin{array}{cccccccccccccccccc}
1)& \varepsilon_\tau(\r_{\msc{f}}^\oplus) - \varepsilon_\tau(\r_{\msc{f}}^\ominus)
&=& t_\tau^\star(\r_{\msc{f}}^\ominus) - t_\tau^\star(\r_{\msc{f}}^\oplus)
\\ \\
2)& \grad\,\varepsilon_\tau(\r_{\msc{f}}^\oplus) - \grad\,\varepsilon_\tau(\r_{\msc{f}}^\ominus)
&=& \grad\,t_\tau^\star(\r_{\msc{f}}^\ominus) - \grad\,t_\tau^\star(\r_{\msc{f}}^\oplus)
\end{array}
\right\}
\quad \forall\ (\r_{\msc{f}}^\oplus,\r_{\msc{f}}^\ominus) \in \mbox{$\tau$-inactive fault}
\end{equation}
\noindent
\ \\
In addition to constraints~\ref{GBR-ImprovT-11} and
\ref{GBR-Secondary-faults-XXX}, to better fit eikonal
Equation~\ref{GBR:eikonal-equation}-1, the unknown function $\varepsilon_\tau(\r_\tau)$
should also honor the following non linear constraint:
\begin{equation} \label{GBR-ImprovT-12}
1 \ = \
||\grad\,\{t_\tau^\star(\r_\tau)+\varepsilon_\tau(\r_\tau)\}||^2 \ = \
||\grad\,t_\tau^\star(\r_\tau)||^2 \ + \
||\grad\,\varepsilon_\tau(\r_\tau)||^2 \ + \
2\cdot \grad\,t_\tau^\star(\r_\tau) \cdot \grad\,\varepsilon_\tau(\r_\tau)
\end{equation}
\noindent
According to Equation~\ref{GBR-ImprovT-11-XX}, second order term
$||\grad\,\varepsilon_\tau(\r_\tau)||^2$ may be neglected in order to linearize the
equation above:
\begin{equation} \label{GBR-ImprovT-13}
\quad
\grad\,\varepsilon_\tau(\r_\tau) \cdot \grad\,t_\tau^\star(\r_\tau) \ \simeq \
\frac{1}{2}\cdot
\{
1 - ||\grad\,t_\tau^\star(\r_\tau)||^2
\}
\qquad \forall\ \r_\tau\in G_\tau
\quad
\end{equation}
\noindent
This linear constraint to be honored in a least squares sense, in addition to
constraints~\ref{GBR-ImprovT-11} and \ref{GBR-Secondary-faults-XXX}, fully
characterizes function $\varepsilon_\tau(\r_\tau)$ in a unique way. Similarly to
Equation~\ref{GBR:X9}-1, the following constraint may be added to ensure
$\varepsilon_\tau(\r_\tau)$ is smooth:
\begin{equation} \label{GBR:Epsilon-smooth}
\sum_{\alpha\in\{x,y,z\}}
\int_{G_\tau} ||\partial_\alpha\, \varepsilon_\tau({\r}_\tau)||^2 \cdot d\r_\tau
\qquad \mbox{minimum}
\end{equation}\noindent
\ \\
On Figure~\ref{GBR-Ramp-results}-B, the histogram of $||\grad\,t_\tau(\r_\tau)||$
obtained on our test example with the above incremental approach shows
that eikonal Equation~\ref{GBR:eikonal-equation}-1 is now correctly honored:
$||\grad\,t_\tau(\r_\tau)||$ is, in average, very close to 1. The median value stands
at 1.0, standard deviation is 0.012 and the spread between 25$^{th}$ and
75$^{th}$ percentiles is reduced to 0.0092.
\ \\
From these observations, we can conclude that the above incremental
approximation of eikonal Equation~\ref{GBR:eikonal-equation}-1 is well suited
to computing function $t_\tau(\r_\tau)$.
Similar results may be observed on other test examples of varying complexity.
\subsection*{Computing $\protect\Bmath{\{u_\tau, v_\tau\}_{\r_\tau}}$}
Assuming that $t_\tau(\r_\tau)$ has already been numerically approximated, to
compute an approximation of $\{u_\tau,v_\tau\}_{\r_\tau}$, our approach derives from a technique
suggested on page 123 of \citep{Mallet2014}:
\begin{enumerate}
\item assuming that $\B{N}_\tau(\r)$ is defined as follows in $G_\tau$:
\begin{equation} \label{GBR-AxeCoaxe-2}
{\bf N}_\tau(\r_\tau) \ = \
\frac{ \grad\,t_\tau(\r_\tau) }
{||\grad\,t_\tau(\r_\tau)||}
\qquad \forall\ \r_\tau\in G_\tau
\end{equation}\noindent
we compute global structural axis $\B{A}_\tau$ defined as a unit vector
averagely orthogonal to vector field $\B{N}_\tau(\r_\tau)$;
\item for any point $\r_\tau\in G_\tau$, we compute local structural axis
$\Bmath{a}_\tau(\r_\tau)$ and co-axis $\Bmath{b}_\tau(\r_\tau)$ as follows:
\begin{equation} \label{GBR-AxeCoaxe-1}
\Bmath{a}_\tau(\r_\tau) \ = \
\frac{ {\bf N}_\tau(\r_\tau)\times{\bf A}_\tau\times{\bf N}_\tau(\r_\tau) }
{||{\bf N}_\tau(\r_\tau)\times{\bf A}_\tau\times{\bf N}_\tau(\r_\tau)||}
\, ; \qquad
\Bmath{b}_\tau(\r_\tau) \ = \ {\bf
N}_\tau(\r_\tau)\times\Bmath{a}_\tau(\r_\tau) \, ;
\end{equation}\noindent
\item depending on tectonic style, for any point $\r_\tau\in G_\tau$, restoration functions
$\{u_\tau,v_\tau\}_{\r_\tau}$ are set to honor the following surrogate equations in a
least squares sense:
\begin{itemize}
\item in a minimal deformation context, Equation~\ref{GBR:X11} may be
approximated by:
\begin{equation} \label{GBR-MinDefStyle}
\left|
\begin{array}{ccccccccccccc}
\grad\,u_\tau(\r_\tau) &\times& \Bmath{a}_\tau(\r_\tau) &\simeq& \B{0}\\
\grad\,v_\tau(\r_\tau) &\times& \Bmath{b}_\tau(\r_\tau)
&\simeq& \B{0}
\end{array}
\right.
\end{equation}\noindent
\item in a flexural slip context, Equation~\ref{GBR:S1} may be
approximated by:
\begin{equation} \label{GBR-FlexSlipStyle}
\left|
\begin{array}{cccccccccccc}
\grad_{\textsc{s}}\,u_\tau(\r_\tau) &\times& \Bmath{a}_\tau(\r_\tau) &\simeq& \B{0} \\
\grad_{\textsc{s}}\,v_\tau(\r_\tau) &\times&
\Bmath{b}_\tau(\r_\tau) &\simeq& \B{0}
\end{array}
\right.
\end{equation}\noindent
\end{itemize}\noindent
In the particular case where $H_\tau$ is a perfect cylindrical surface, it can be shown that these approximations are exact.
Compared to similar Equations~3.124 and 3.125 on page 123 of \citep{Mallet2014},
the surrogate equations above have been slightly adapted not to conflict
with Equation~\ref{GBR:Indet-1} on $H_\tau$;
\item finally, to ensure smoothness and uniqueness of functions $\{u_\tau,v_\tau\}_{\r_\tau}$,
constraint~\ref{GBR-UtauVtau-smoothness} is added.
\end{enumerate}\noindent
In practice, numerical results so obtained generally yield sufficiently precise
approximations for restoration functions $\{u_\tau,v_\tau\}_{\r_\tau}$. If more
precision is required, these approximations could be improved with an incremental technique
similar to the one proposed above for restoration function $t_\tau(\r_\tau)$.
\section{Examples of 3D restoration}
\label{GBR-Validating-1}
Figure~\ref{rainbow} shows the restoration of a synthetic model with four
horizons, modeled on a grid with about 84,000 cells. Restoration functions
$\{u_\tau,v_\tau,t_\tau\}_{\r_\tau}$ and the associated restoration vector field
$\B{R}_\tau(\r_\tau)$ are computed on the grid for each
restoration time $\tau$ from the initial GeoChron functions $\{u,v,t\}_{\r}$
using the framework and algorithms described in this paper. The full structural
model is then updated on demand to reflect the restored state specified by the
user.
This synthetic example was
designed to illustrate the correct behavior of the GBR method on layers with
varying thickness and horizons with extreme deformation as their extremities on
either side are vertical.
\ \\
Total computation time on an average workstation is 2.25~s per horizon to
restore. Switching between two restored states then takes 0.07~s. Using the
flexural slip tectonic style, variations in area for horizons from
present-day state (A) to restored state (B) are -3.91~\% for the top horizon and
+0.225~\%
for the bottom horizon. As expected, areal variations are higher if the minimal
deformation tectonic style (C) is used (-14.9~\% and +9.10~\% for top and bottom
horizons, respectively). The neutral axis in this model when the minimal
deformation regime is applied is located close to the third, blue horizon at the
top of the blue layer for which areal variation at this restoration stage
is 0.243~\%. As this model is essentially a two-dimensional example
with no variation in geometry in the third dimension, volume variation figures
are similar, with a -1.67~\% global volume variation between initial and restored
states for top horizon in the flexural slip case and -4.46~\% in the minimal
deformation case.
\ \\
This extreme example illustrates that restoration results depend on the initial
GeoChron paleo-coordinates $\{u,v\}_\r$ from which restoration functions
$\{u_\tau,v_\tau\}_{\r_\tau}$ are computed. When the tectonic style is minimal
deformation, specifying the location of the horizon with the minimal amount of
deformation in the initial GeoChron model would help compute $\{u_\tau,v_\tau\}_{\r_\tau}$
such that in restored states, deformations on that specific horizon are
minimized.
\ \\
Figure~\ref{clyde_sections} shows a
full structural volume model restored to deposition time of various horizons.
On an average workstation, computation time in this grid with 845,150
cells was 29.8~s per horizon to restore. Switching from one
restored state to the next then takes 1.15~s.
\ \\
The top-left block diagram shows the model restored to present-day sea floor
geometry, used as an approximation of paleo-topography. The horizon being
restored is an erosive surface and the volume below shows the geological-time
function for the eroded terrains. The image to the right shows the location of a
seismic cross section rendered at different restored times $\{\tau_3, \tau_2,
\tau_1\}$. The top cross section is the present-day geometry of horizons and
faults painted over the seismic image. The cross sections below show horizons,
faults and seismic image restored at times $\tau_3$ when the blue horizon was
deposited, $\tau_2$ when the green, erosive horizon was deposited and $\tau_1$
when the yellow, first horizon modeled in the eroded sequence was deposited.
\ \\
Each restored model is consistent: Despite the complexity of the fault network,
there are no gaps between faults and horizons and no overlaps between fault
blocks. Interval times between horizons, highlighted by identical black arrows
on each cross section, are a constant 360~ms for \textbf{a}, 500 ms for
\textbf{b} and 395~ms for \textbf{c}.
\section{Conclusions}
In this paper, we propose a new restoration method based on the GeoChron model.
Contrary to classical, mechanical methods based on elasticity theory,
this new method is purely geometrical and, therefore, does
not require prior knowledge of geo-mechanical properties of the terrains. This
method works equally well for small and large deformations and for any possible
mechanical behavior (elastic, plastic, \ldots) of the terrains. Moreover, the
restoration process in itself handles consistency around faults and with the
tectonic style chosen by the geomodeler. Finally, a new technique aimed at
taking compaction into account is also proposed.
\ \\
This restoration method also requires less
computation and fewer user inputs than classical geo-mechanical methods. As a
consequence, it is fast and simple to use, which allows
geologists to routinely check and validate structural model
consistency. At any given geological-time $\tau$, if inconsistencies are
spotted, the geological-time function $t(\r)$ ruling the geometry of the
horizons of a restored GeoChron model may be locally interactively edited. Such
changes of $t(\r)$ can automatically and instantly
be back-propagated to the initial GeoChron model corresponding to the
present-day subsurface, without any additional computations.
\section{Acknowledgments}
The authors would like to thank Emerson for their support and for permission to
publish this paper.
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-Rainbow,width=100mm}}
\caption{
Vertical cross-section of a three-dimensional synthetic model with four horizons
showing both variations in layer thickness and extreme deformation as towards
the extremities of the model, horizons become vertical (A). Restoration using
the flexural slip tectonic style (B) results in better conservation of horizon
area than using the minimal deformation style (C).
}
\label{rainbow}
\end{figure}\noindent
\begin{figure}
\centerline{\psfig{figure=./figures/GBR-Clyde,width=170mm}}
\caption{
Restoration of several horizons in a three-dimensional model with an erosive
stratigraphic sequence. Location of seismic image section is shown in top right
corner. Sections below are painted with seismic image, faults and horizons in the
present-day model and at three restoration times. Interval times between two
horizons highlighted by thick, black arrows are preserved.
}
\label{clyde_sections}
\end{figure}\noindent
\bibliographystyle{aapg}
| {'timestamp': '2021-05-17T02:11:58', 'yymm': '2105', 'arxiv_id': '2105.06137', 'language': 'en', 'url': 'https://arxiv.org/abs/2105.06137'} |
\section{Introduction}
The Glauber model~\cite{glauber} was the first theoretical approach which calculated the effects of shadowing in hadron-nucleus interactions. The model however, was essentially non-relativistic. In the pioneering papers by V.N.~Gribov~\cite{gribov69,gribov70}
it was realised that the length scales of interaction rising with energy significantly
change the pattern of hadron-nucleus interaction at high energies.
In particular, particles created in an inelastic collision with one nucleon, can be subsequently absorbed by another bound nucleons. Such corrections make the nuclear medium more transparent for hadrons, and consequently lead to a reduction of the total hadron-nucleus cross section. These corrections, called Gribov inelastic shadowing,
improve the Glauber model, making it well-founded.
\section{From the Glauber model to the Gribov-Glauber theory}
The amplitude of probability for a hadron to interact with the nucleus is one minus the probability amplitude of no interaction with any of the bound nucleons. So the $hA$ elastic amplitude at impact parameter $b$ has the eikonal
form,
\beq
\Gamma^{hA}(\vec b;\{\vec s_j,z_j\}) =
1 - \prod_{k=1}^A\left[1-
\Gamma^{hN}(\vec b-\vec s_k)\right]\ ,
\label{1.40}
\eeq
where $\{\vec s_j,z_j\}$ denote the coordinates of the target nucleon
$N_j$. $i\Gamma^{hN}$ is the elastic scattering amplitude on a nucleon
normalized as,
\beqn
\sigma_{tot}^{hN} &=& 2\int d^2b\,\mbox{Re}\,\Gamma^{hN}(b);\nonumber\\
\sigma_{el}^{hN}&=& \int d^2b\, |\Gamma^{hN}(b)|^2\ .
\label{1.60}
\eeqn
In the approximation of single particle nuclear density one can calculate
a matrix element between the nuclear ground states.
\beqn
\left\la0\Bigl|\Gamma^{hA}(\vec b;\{\vec s_j,z_j\})
\Bigr|0\right\ra =
1-\left[1-{1\over A}\int d^2s\,
\Gamma^{hN}(s)\int\limits_{-\infty}^\infty dz\,
\rho_A(\vec b-\vec s,z)\right]^A\ ,
\label{1.80}
\eeqn
where
\beq
\rho_A(\vec b_1,z_1) = \int\prod_{i=2}^A
d^3r_i\,
|\Psi_A(\{\vec r_j\})|^2\ ,
\label{1.100}
\eeq
is the nuclear single particle density.
Eq.~(\ref{1.80}) is related via unitarity to the total $hA$
cross section,
\beqn
\sigma_{tot}^{hA}&=&2\mbox{Re}\,\int d^2b\,\left\{1 -
\left[1-{1\over A}\int d^2s\,
\Gamma^{hN}(s)\,T_A(\vec b-\vec s)\right]^A\right\}
\nonumber\\ &\approx&
2\int d^2b\, \left\{1-
\exp\left[-{1\over2}\,\sigma_{tot}^{hN}\,(1-i\rho_{pp})\,
T^h_A(b)\right]\right\}\ ,
\label{1.120}
\eeqn
where $\rho_{pp}$ is the ratio of the real to imaginary parts
of the forward $pp$ elastic amplitude;
\beq
T^h_A(b)= \frac{2}{\sigma_{tot}^{hN}}\int d^2s\,
\mbox{Re}\,\Gamma^{hN}(s)\,T_A(\vec b-\vec s)\ ;
\label{1.140}
\eeq
and
\beq
T_A(b) = \int_{-\infty}^\infty dz\,\rho_A(b,z)\ ,
\label{1.160}
\eeq
is the nuclear thickness function.
We use Gaussian form of $\Gamma^{hN}(s)$ in what follows,
\beq
\mbox{Re}\, \Gamma^{hN}(s) =
\frac{\sigma_{tot}^{hN}}{4\pi B_{hN}}\,
\exp\left(\frac{-s^2}{2B_{hN}}\right)\ ,
\label{1.180}
\eeq
where $B_{hN}$ is the slope of the differential $hN$ elastic cross
section. Notice that the accuracy of the optical approximation (the second line in
(\ref{1.120})) is quite high for heavy nuclei. For the sake of simplicity, we use the optical form
throughout the paper although for numerical evaluations always rely on the accurate expression (the first line in (\ref{1.120})).
The effective nuclear thickness, Eq.~(\ref{1.140}) implicitly contains energy dependence, which is extremely weak.
In what follows we also neglect the real part of the elastic amplitude, unless specified, since it gives a vanishing correction $\sim
\rho_{pp}^2/A^{2/3}$.
Besides the total cross section Eq.~(\ref{1.120}) one can calculate within the Glauber model also elastic, quasielastic (break-up of the nucleus without particle production) and inelastic cross section. One can find details of such calculations, as well as numerical results, in Ref.\cite{mine,xsect,kps-ciofi}, and below in Sect.~\ref{hA}.
The Glauber model is intensively used nowadays as a theoretical tool to study heavy ion collisions. However, this model is subject to significant Gribov corrections \cite{gribov69}.
\subsubsection{Intermediate state diffractive excitations}\label{sect-kk}
\label{sec:intermediate}
The Glauber model is a single-channel approximation, therefore it misses the
possibility of diffractive excitation of the projectile in the intermediate
state as is illustrated in Fig.~\ref{fig:diff}.
\begin{figure}[htb]
\centerline{\includegraphics[width= 8.5 cm]{diff.pdf}}
\caption{Diagonal and off-diagonal diffractive multiple excitations of the
projectile hadron in intermediate state}
\label{fig:diff}
\end{figure}
Inclusion of multiple diffractive transitions between different excitations, like depicted in Fig.~\ref{fig:diff}, obviously is a challenge, because diffractive transitions between different excited states cannot be measured in diffractive processes.
There is, however, one case free of these problems,
shadowing in hadron-deuteron interactions. In this case no interaction in the intermediate
state is possible, and knowledge of diffractive cross section $hN\to XN$
is sufficient for calculations of the inelastic correction with no further
assumptions. In this case the Gribov correction to the Glauber model for the total cross section has the simple form
\cite{gribov69},
\beq
\Delta\sigma^{hd}_{tot} = -
2\int dM^2\int dp_T^2\,
\frac{d\sigma_{sd}^{hN}}{dM^2dp_T^2}\,
F_d(4t).
\label{1.360}
\eeq
Here $F(t)$ is the deuteron electromagnetic formfactor;
$\sigma_{sd}^{hN}$ is the cross section of single diffractive
dissociation $hN\to XN$ with longitudinal momentum transfer
\beq
q_L=\frac{M^2-m_h^2}{2E_h};
\label{1.340}
\eeq
and $t=-p_T^2-q_L^2$.
The formula for the lowest order inelastic
corrections to the total hadron-nucleus cross section was suggested in
\cite{kk},
\beqn
\Delta\sigma^{hA}_{tot} &=&
- 8\pi\int d^2b\,
e^{-{1\over2}\sigma^{hN}_{tot}T_A(b)}\!\!\!
\int\limits_{M_{min}^2}\!\! dM^2
\left.\frac{d\sigma_{sd}^{hN}}
{dM^2\,dp_T^2}\right|_{p_T=0}
\nonumber\\ &\times&
\int\limits_{-\infty}^{\infty}dz_1\,
\rho_A(b,z_1)
\int\limits_{z_1}^{\infty}dz_2\,
\rho_A(b,z_1)\,e^{iq_L(z_2-z_1)}\ ,
\label{1.320}
\eeqn
This correction takes care of the onset of
inelastic shadowing via phase shifts controlled by $q_L$ and does a good job describing data at low energies \cite{murthy,gsponer}, as one can also see
in Fig.~\ref{fig:murthy}.
\begin{figure}[htb]
\centerline{\includegraphics[width=7 cm]{murthy.pdf}}
\caption{Data and calculations \cite{murthy} for the total neutron-lead
cross section. The dashed and solid curves correspond to the
Glauber model and corrected for Gribov shadowing respectively.\label{fig:murthy}}
\end{figure}
In a similar way Gribov corrections can be calculated for neutrino-nucleus cross section
at low $Q^2$, when PCAC hypothesis is at work.
Then one can employ the Adler relation, which connects neutrino and pion induced
cross sections at $Q^2=0$, and write the nucleus-to-proton ration of total neutrino-nucleus cross sections as \cite{nu-tot},
\beqn
R^\nu_{A/N}&=&1 - \frac{8\pi}{A\sigma^{\nu N}_{tot}}
\left.\frac{d\sigma^{\nu\to\pi}_{diff}}{dp_T^2}\right|_{p_T=0}
\int d^2b\int\limits_{-\infty}^\infty dz_1\,\rho_A(b,z_1)
\nonumber\\&\times&
\int\limits_{-\infty}^{z_1} dz_2\,\rho_A(b,z_2)
e^{-iq_L^\pi(z_2-z_1)}\,e^{-{1\over2}\sigma^{\pi N}_{tot}T_A(b,z_2,z_1)},
\label{2.120}
\eeqn
where $q_L^\pi=(Q^2+m_{\pi}^2)/2\nu$.
At very low energy where $q_L^\pi$ is large, the second term in (\ref{2.120}) is suppressed, and the first one, corresponding to the cross section proportional to $A$, dominates. At high energies $q_L^\pi\ll R_A$ can be neglected and the integrations over $z_{1,2}$ can be performed analytically \cite{nu-tot}. Due to the Adler relation the first term in (\ref{2.120}) and the volume part of the second term cancel, and the rest is the "surface" term $\propto A^{2/3}$,
\beq \left.\frac{d^2\sigma(\nu A\to
l\,X)}{dQ^2\,d\nu}\right|_{q_L\ll1/R_A} =
\frac{G^2}{2\pi^2}\,f_\pi^2\,\frac{E-\nu}{E\nu}
\,\sigma(\pi A\to X),
\label{2.40}
\eeq
as could be anticipated in accordance with the Adler relation.
Notice that the high-energy regime actually starts at rather low energies if $Q^2\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}
m_\pi^2$.
The results of numerical evaluation of the shadowing effect Eq.~(\ref{2.120})
for neon are plotted in Fig.~\ref{fig:shad-1}.
\begin{figure}[htb]
\parbox{.471\textwidth}{
\centerline{\includegraphics[width=5 cm]{shad-1.pdf}}
\caption{The neon to nucleon ratio of total neutrino cross sections
at different $Q^2$ \cite{nu-tot}. Dashed and solid curves
correspond to the Glauber and Gribov corrected calculations. }
\label{fig:shad-1}}
\hfill
\parbox{.471\textwidth}{
\centerline{\includegraphics[width=5 cm]{shad-2.pdf}}
\caption{ The neon to proton ratio of the total neutrino cross sections,
calculated in Ref.\cite{nu-tot} for $x<0.2$ and $Q^2<0.2\,\mbox{GeV}^2$. The data present
the BEBC results \cite{wa59}.}\label{fig:shad-2}}
\end{figure}
The calculations \cite{nu-tot} done within
the Glauber approximation, Eq.~(\ref{2.120}), and also including the Gribov's inelastic
corrections (important at high energies $\nu>m_{a_1}^2R_A$) are plotted in Fig.~\ref{fig:shad-1} by solid curves as function of energy, for different $Q^2$.
The calculated shadowing effects are compared with BEBC data \cite{wa59} in Fig.~\ref{fig:shad-2}.
As was anticipated, the shadowing exposes an early onset, and a significant suppression occurs at small $Q^2$ in
the low energy range of hundreds MeV. This is an outstanding feature of
the axial current. This seems to be supported by data, although
with a rather poor statistics.
Concluding this section, we notice that although the approximation of lowest order Grobov corrections does in some cases good job, the
higher order off-diagonal transitions neglected in (\ref{1.320}) and (\ref{2.120}), might be important, but unknown. Indeed, the
intermediate state $X$ has definite mass $M$, but no definite
cross section. It was fixed in (\ref{1.320}) at $\sigma_{tot}^{hN}$ with no justification.
Thus, the lowest order Gribov correction Eq.~(\ref{1.320}) has a nice feature
of including phase shifts, which allows to describe the onset of Gribov shadowing,
like is presented in Fig.~\ref{fig:murthy}. However, the uncertainties related to the missed higher order corrections and the unknown absorption in (\ref{1.320}) seem to be incurable, while one works in the hadronic (eigenstates of the mass matrix) representation.
\subsection{Interaction eigenstate representation}\label{eigen}
If a hadron were an eigenstate of interaction, i.e. could undergo only
elastic scattering (as a shadow of inelastic channels) and no diffractive
excitation was possible, the Glauber formula would be exact and no
inelastic shadowing corrections were needed. This simple observation
gives a hint that one should switch from the hadronic basis to a complete set of mutually orthogonal eigenstates of the scattering amplitude operator. This
was the driving idea of the description of diffraction in terms of elastic
amplitudes \cite{pom,gw}, and becomes a powerful tool for calculation of
inelastic shadowing corrections in all orders of multiple interactions
\cite{kl,zkl}. Notice that this idea also was used by Gribov \cite{gribov69} to explain
the role of the coherence length increasing with energy, on the example of an incoming proton, fluctuating to a nucleon-pion pair.
Physical states (including leptons, photons) can be expanded
over the complete set of states $|k\ra$,
\beq
|h\ra=\sum\limits_{k}\,\Psi^h_k\,|k\ra\ ,
\label{b.1}
\eeq
which are the eigenstates of the scattering amplitude operator, $\hat f\,|k\ra=
f_{el}^{kN}|k\ra$. For the sake of simplicity we neglect the real part of the amplitude, i.e. assume that $f_{el}^{kN}=i\,\sigma_{tot}^{kN}/2$.
$\Psi^h_k$ in (\ref{b.1}) are weight factors (amplitudes) of the Fock state
decomposition. They obey the orthogonality conditions,
\beqn
\sum\limits_{k}\,\left(\Psi^{h'}_k\right)^{\dagger}\,\Psi^h_k
&=&\delta_{h\,h'}\ ;
\nonumber\\
\sum\limits_{h}\,\left(\Psi^{h}_l\right)^{\dagger}\,\Psi^h_k
&=&\delta_{lk}\ .
\label{b.2}
\eeqn
We also assume
that the amplitude is integrated over impact parameter, {\it i.e.} that
the forward elastic amplitude is normalized as
$|f_{el}^{kN}|^2=4\,\pi\,d\sigma_{el}^{kN}/dt|_{t=0}$.
The amplitudes of elastic $f_{el}(hh)$ and off diagonal diffractive
$f_{sd}(hh')$ transitions can be expressed as,
\beq
f_{el}^{hN}=2i\,\sum\limits_k\,\left|
\Psi^h_k\right|^2\,\sigma_{tot}^{kN}
\equiv 2i\,\la\sigma\ra\ ;
\label{b.3}
\eeq
\beq
f_{sd}^{hN}(h\to h')=
2i\,\sum\limits_k\, (\Psi^{h'}_k)^{\dagger}\,
\Psi^h_k\,\sigma_{tot}^{kN}\ .
\label{b.4}
\eeq
Note that if all the eigenamplitudes were equal, the diffractive
amplitude (\ref{b.4}) would vanish due to the orthogonality relation,
(\ref{b.2}). The physical reason is obvious. If all the $f_{el}^{kN}$ are
equal, the interaction does not affect the coherence between
different eigen components $|k\ra$ of the projectile hadron $|h\ra$.
Therefore, the off-diagonal transitions are possible only due to differences
between the eigenamplitudes.
By summing up in the diffractive cross section
using completeness Eq.~(\ref{b.2}), and
excluding the elastic channels, one gets \cite{kl,mp,zkl},
\beqn
16\pi\,\frac{d\sigma^{hN}_{sd}}{dt}\biggr|_{t=0}&=&
\sum\limits_k \left|\Psi^h_k\right|^2
\left(\sigma^{kN}_{tot}\right)^2
- \biggl(\sum\limits_i
\left|\Psi^h_k\right|^2\sigma^{kN}_{tot}\biggr)^2
\nonumber\\
&\equiv& \left\la\left(\sigma^{kN}_{tot}\right)^2\right\ra -
\left\la\sigma^{kN}_{tot}\right\ra^2\ .
\label{b.6}
\eeqn
Each of the eigenstates propagating through the nucleus can experience only elastic
scatterings, so the Glauber eikonal approximation becomes
exact for such a state. Then, the cross sections for
hadron-nucleus collisions should be averaged over the Foch states, chosen to be interaction eigenstates \cite{kl,zkl},
\beq
\sigma^{hA}_{tot} = 2\int d^2b\,\left\{1 -
\left\la\exp\left[-{1\over2}\,\sigma^{kN}_{tot}\,T^h_A(b)\right]
\right\ra\right\}.
\label{b.10}
\eeq
This is to be compared with the Glauber approximation. The difference is
obvious, in Eq.~(\ref{b.10}) the exponential is averaged,
while in the Glauber approximation the exponent is averaged,
\beq
\sigma^{hA}_{tot}\Bigr|_{Gl} = 2\int d^2b\,\left\{1 -
\exp\left[-{1\over2}\,\left\la\sigma_{tot}^{kN}\right\ra\, T^h_A(b)\right]
\right\},
\label{b.30}
\eeq
where $\la\sigma^{kN}_{tot}\ra = \sigma_{tot}^{hN}$. The difference between
Eqs.~(\ref{b.10}) and (\ref{b.30}), is the Gribov inelastic correction
calculated in all orders of opacity expansion, which was impossible to do within hadronic representation (see above). This result can be compared with the expression (\ref{1.320})
for the lowest order correction expanding the exponentials in (\ref{b.10}) and
(\ref{b.30}) in number of collisions up to the lowest order.
Applying (\ref{b.6}) we find,
\beqn
\sigma^{hA}_{tot} - \sigma^{hA}_{tot}\Bigr|_{Gl} &=&
\int d^2b\, {1\over4}\,\Bigl[\left\la\sigma^{kN}_{tot}\right\ra^2 -
\left\la\left(\sigma^{kN}_{tot}\right)^2\right\ra\Bigr]\,T^h_A(b)^2
\nonumber\\ &=&
- 4\pi\int d^2b\,T^h_A(b)^2\int dM^2\,
\frac{d\sigma^h_{sd}}{dM^2dt}\biggr|_{t=0}.
\eeqn
This result is indeed identical to Eq.~(\ref{1.320}), if to neglect the
phase shift vanishing at high energies, and also to expand the
exponential.
\section{Color dipoles as eigenstates of interactions}
The dipole representation in QCD, first proposed in Ref.\cite{zkl},
allows to calculated Gribov corrections in all orders, because a high energy dipole
is an eigenstate of interaction. Indeed, the transverse dipole separation, which controls the scattering amplitude, is preserved during propagation and remains intact after multiple soft interactions within
a nuclear target.
To perform proper averaging in (\ref{b.10}) one should expend the beam particle
(hadrons, vector or axial currents) over Fock states. Each Fock component is a colorless dipole, consisted of two or more partons. After averaging over intrinsic dipole distances, one should sum up different Fock states with proper weights.
\subsection{Deep-inelastic scattering}
The parton model interpretation of the space-time development of reactions is not Lorentz invariant. While Deep-inelastic scattering (DIS) at small $x\ll1$ is interpreted in the Bjorken reference frame as absorption of the virtual photon by a target parton carrying corresponding fractional momentum $x$ of the target, in the target rest frame it looks differently.
The high energy virtual photon fluctuates into a $\bar qq$ pair, which interacts with the targets and is produced on mass shell.
The two main contributions to the diffractive cross section in
(\ref{1.320}) at high energies come from the triple-Regge terms $\mathbb{P}\Pom\mathbb{R}$
and $\mathbb{P}\Pom\mathbb{P}$, which generate $1/M^3$ and $1/M^2$ mass dependences respectively. The quark-gluon structure of these contributions in diffraction is illustrated in Fig.~\ref{fig:pom-reg}.
\begin{figure}
\centerline{\includegraphics[width=8 cm]{pom-reg.pdf}}
\caption{Gribov corrections with $\bar qq$ and $\bar qq+g$ intermediate states. }
\label{fig:pom-reg}
\end{figure}
Apparently, the $\bar qq$ and $\bar qqg$ states, propagating through the nucleus,
attenuate with an absorptive cross section, which cannot be trivially fixed, like was done
in (\ref{1.320}), but should be properly treated within the dipole approach.
\subsubsection{"Frozen" dipoles}
If the lifetime of partonic fluctuations of a photon significantly exceeds the nuclear size, the dipole approach allows to calculate easily the shadowing effects in DIS. Indeed, in this case one can rely on the ``frozen" approximation Eq.~(\ref{b.10}), which for interaction of a virtual photon has the form,
\beqn
\sigma_{T,L}^{\gamma^*A}(x,Q^2)&=&
2\int d^2b\int\limits_0^1d\alpha\int d^2r_T
\left|\Psi_{q\bar q}^{T,L}(\alpha,r_T,Q^2)\right|^2
\nonumber\\ &\times&
\left[1-e^{-{1\over2}\sigma_{q\bar q}(r_T,x)T_A(b)}\right],
\label{4.200}
\eeqn
where $r_T$ is the dipole transverse separation; $\alpha$ is the fractional light-cone momentum of the quark;
the distribution functions $\Psi_{q\bar q}^{T,L}(\alpha,r_T)$ in the quadratic form read,
\beqn
\left|\Psi_{q\bar q}^{T}(\alpha,r_T)\right|^2&=&
\frac{2N_c\alpha_{em}}{(2\pi)^2}\sum\limits_{f=1}^{N_f}Z_f^2
\left\{\left[1-2\alpha(1-\alpha)\right]\varepsilon^2{\rm K}^2_1(\varepsilon r_T)
\right.\nonumber\\ &+&\left.
m_f^2{\rm K}^2_0(\varepsilon r_T)\right\};
\label{4.210}\\
\label{psil}
\left|\Psi_{q\bar q}^{L}(\alpha,r_T)\right|^2&=&
\frac{8N_c\alpha_{em}}{(2\pi)^2}\sum\limits_{f=1}^{N_f}Z_f^2
Q^2\alpha^2(1-\alpha)^2{\rm K}^2_0(\varepsilon r_T).
\label{4.220}
\eeqn
The advantage of the dipole description is pretty obvious,
Eq.~(\ref{4.200}) includes Gribov inelastic shadowing corrections to all orders of multiple interactions~\cite{zkl},
what is hardly possible within hadronic representation.
On the other hand, the dipoles having a definite size, do not have any definite mass, therefore the phase shifts between amplitudes on different nucleons cannot be calculated as simple as in Eq.~(\ref{1.320}). A solution for this problem was proposed in Ref.\cite{krt1,zakharov}.
\subsubsection{Path integral technique}
The lifetime of the "frozen" dipole, or coherence length (or time), is given by,
\beq
l_c=\frac{2\nu}{Q^2+M_{\bar qq}^2},
\label{bh100}
\eeq
where $\nu$ is the dipole energy in the target rest frame; $M_{\bar qq}^2=(m_q^2+k_T^2)/\alpha(1-\alpha)$. At very small Bjorken
$x=Q^2/2m_N\nu\ll1$ the coherence length can significantly exceed the nuclear size, $l_c\gg R_A$, and one can safely rely on the "frozen" approximation.
However, if $t_c\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} R_A$ such an approximation is not appropriate and one should correct for the dipole size fluctuations during propagation through the nucleus,
which corresponds to inclusion of the phase shifts between DIS amplitudes on different bound nucleons in Eq.~(\ref{1.320}). Within the dipole description this can be done employing the path integral technique \cite{feynman}, which sums up different propagation paths of the partons.
For a $\bar qq$ component of a transversely or longitudinally polarized virtual photon Eq.~(\ref{4.200}) should be replaced by,
\beqn
\Bigl(\sigma^{\gamma^*A}_{tot}\Bigr)^{T,L}\! &=&
A\Bigl(\sigma^{\gamma^*N}_{tot}\Bigr)^{T,L}
\nonumber\\
&-& \frac{1}{2} Re\int d^2b
\!\int\limits_0^1 d\alpha
\!\int\limits_{-\infty}^{\infty} dz_1 \!\int\limits_{z_1}^{\infty} dz_2
\!\int d^2r_1\int d^2r_2
\nonumber\\ & \times &
\Psi^{T,L^\dagger}_{\bar qq}\!\!\left(\varepsilon,\vec r_2\right)
\rho_A\left(b,z_2\right)\sigma_{q\bar{q}}^N\left(s,\vec r_2\right)
G\left(\vec r_2,z_2\,|\,\vec r_1,z_1\right)
\nonumber\\ & \times &
\rho_A\left(b,z_1\right)\sigma_{q\bar{q}}^N\left(s,\vec r_1\right)
\Psi^{T,L}_{\bar qq}\!\left(\varepsilon,\vec r_1\right).
\label{4.240}
\eeqn
The Green's function
$G\left(\vec r_2,z_2\,|\,\vec r_1,z_1\right)$ describes propagation
of a $\bar qq$ pair in an
absorptive medium, having initial separation $\vec r_1$ at the initial position $z_1$, up to the point $z_2$, where it gets separation $\vec r_2$, as is illustrated in Fig.~\ref{fig:GF}.
\begin{figure}[htb]
\centerline{\includegraphics[width=6 cm]{GF.pdf}}
\caption{Propagation of a $q\bar q$-pair through a nucleus between points with longitudinal coordinates $z_1$ and $z_2$. The evolution of the $\bar qq$ separation from the initial, $\vec r_1$, up to the final, $\vec r_2$, due to the transverse motion of the quarks, is described by the
Green's function $G\left(\vec\rho_2,z_2;\vec\rho_1,z_1\right)$, a solution of Eq.~(\ref{4.260}). }
\label{fig:GF}
\end{figure}
It satisfies the evolution equation \cite{kz91,kst1,zakharov},
\beq
\left[i\frac{\partial}{\partial z_2}
+\frac{\Delta_\perp\left(r_2\right)-\varepsilon^2}
{2\nu\alpha\left(1-\alpha\right)}
+U(r_2,z_2)
\right]
G\left(\vec r_2,z_2\,|\,\vec r_1,z_1\right)
= 0
\label{4.260}
\eeq
The light-cone potential in the left-hand side of
this equation describes nonperturbative interactions within the dipole, and its absorption in the medium.
The real part the potential responsible for nonperturbative quark interactions was modelled and fitted to data of $F_2^p$ in Ref.\cite{kst2}. Here we fix $\mbox{Re}\, U(r_2,z_2)=0$, and treat quarks as free particles for the sake of simplicity.
The imaginary part of the potential describes the attenuation of the dipole in the medium,
\beq
\mbox{Im}\, U(r,z)=-{1\over2}\,\sigma_{\bar qq}(r)\,\rho_A(b,z).
\label{4.270}
\eeq
The numerical results of the calculations, which are performed either disregarding or including the real part of the potential, modelled in Ref.\cite{kst2},
are plotted in Fig.~\ref{fig:nmcc} by dashed and solid curves respectively. One can see that inclusion of the nonperturbative effects does not lead to a significant change of the magnitude of shadowing. Comparison with NMC data \cite{nmc,nmc951} shows pretty good agreement.
\begin{figure}[htb]
\centerline{\includegraphics[width=8cm]{nmcc.pdf}}
\caption{Comparison between calculations for shadowing in DIS and experimental
data from NMC \cite{nmc,nmc951}
for the structure functions of
different nuclei relative to carbon.
$Q^2$ ranges within $3\le Q^2\le 17\,\mbox{GeV}^2$.
The solid and dashed curves are calculated including or
excluding the real part of the potential respectively.
}
\label{fig:nmcc}
\end{figure}
We remind that this is a parameter-free description. More example of comparison with data can be found in Ref.\cite{krt2}. Notice also that we
did not include any mechanism of nuclear enhancement at $x>0.1$, the effect
called antishadowing, which affect the small-$x$ region as well.
\subsubsection{Gluon shadowing}
As was mentioned above, the contribution of all Fock components in the cross section should be summed up
\beq
\sigma_{tot}^{\gamma^*A} = A\,\sigma_{tot}^{\gamma^*N}\,
-\, \Delta\sigma_{tot}(\bar qq)\,
-\, \Delta\sigma_{tot}(\bar qqg)\,
-\, \Delta\sigma_{tot}(\bar qq2g)\, -\,...
\label{4.1}
\eeq
The next after the $\bar qq$ Fock component is $\bar qq+g$.
It has a considerably shorter coherence time, compared with $\bar qq$, because
of specific nonperturbative effects increasing the mean transverse momentum of gluons
\cite{kst2,spots}. Correspondingly, the next component $|\bar qq2g\ra$ has even a much shorter coherence time $t_c$, which makes the 4th term in (\ref{4.1}) negligibly small within the currently achieved kinematic range. Thus, we keep only first two low Fock components.
Notice that a successful attempt to sum up all Fock components was done in the form of Balitsky-Kovchegov equation \cite{b,k}
The 3rd term in the total cross section Eq.~(\ref{4.1}) is illustrated in Fig.~\ref{fig:GF-g}.
\begin{figure}
\centerline{\includegraphics[width=6 cm]{GF-g.pdf}}
\caption{Propagation through a nucleus of the $q\bar q-g$ fluctuation of a longitudinally polarized photon.
Neglecting the small, $\sim1/Q^2$ size of the color-octet $\bar qq$ pair, the effective octet-octet dipole propagation is described by the Green's function $G_{gg}(\vec r_{2g},z_2;\vec r_{1g},z_1)$.
}
\label{fig:GF-g}
\end{figure}
Differently from the case of a $|\bar qq\ra$ Fock state,
where we found that at high $Q^2$ perturbative QCD can be safely used for shadowing calculations,
the nonperturbative effects remain important for the
$|\bar q\,q\,g\ra$ component even for highly virtual photons.
High $Q^2$ squeezes the $\bar qq$ pair down to a size $\sim 1/Q$,
while the mean quark-gluon separation at $\alpha_g \ll 1$
depends on the strength of nonperturbative gluon interaction which
is characterised in this limit by a small separation $r_0\approx 0.3\,\mbox{fm}$ \cite{kst2}.
which considerably smaller than the confinement radius $1/\Lambda_{QCD}$. This is confirmed
by various experimental observations \cite{spots}, in particular by the observed strong suppression of the
diffractive gluon radiation \cite{kst2}. In nonperturbative QCD models this scale is related to the instanton size \cite{shuryak,zahed}.
The nonperturbative quark-gluon wave function
was found in Ref.\cite{kst2} to have the form,
\beq
\Psi_{qg}\left(\vec r_{g},\alpha_g\right)\Bigr|_{\alpha_g\ll1}=
- \frac{2i}{\pi}\,
\sqrt\frac{\alpha_s}{3}\
\frac{\vec e\cdot\vec r_{g}}{r^2_{g}}\,
\exp\left(-\frac{r^2_{g}}{2r_0^2}\right)\ ,
\label{4.460}
\eeq
where $\vec e$ is the gluon polarization vector.
For $Q^2\gg 1/r_0^2$ the $\bar qq$ is small,
$r^2_{\bar qq} \ll r_g^2$, and one can treat the $\bar qqg$ system
as a color octet-octet dipole, as is illustrated in Fig.~\ref{fig:GF-g}.
Then the three-body Green's function factorizes,
\beq
G_{q\bar qg}\left(\vec r_{2g},
\vec r_{2},z_2;\vec r_{1g},\vec r_{1},z_1
\right)\Rightarrow
G_{q\bar q}\left(\vec r_{2},z_2;
\vec r_{1},z_1\right)
G_{gg}\left(\vec r_{2g},z_2;\vec r_{1g},z_1\right).
\label{4.400}
\eeq
The color octet-octet Green function $G_{gg}$, describing the propagation of a glue-glue dipole with $\alpha_g\ll1$ through the medium, satisfies the simplified evolution equation \cite{kst2},
\beq
\left[i\frac{\partial}{\partial z_2}-\frac{Q^2}{2\nu}
-V(\vec r_{2g},z_2)
\right]
G_{gg}\left(\vec r_{2g},z_2;\vec r_{1g},z_1\right)
=0
\label{4.420}
\eeq
Here
\beq
\mbox{Im}\, V(\vec r_{2g},z)=-{1\over2}\sigma_{gg}(r,x)\rho_A(z),
\label{bh200}
\eeq
with the color-octet dipole cross section, which reads,
\beq
\sigma_{gg}\left(r,x\right)
=\frac{9}{4}\,
\sigma_{q\bar q}\left(r,x\right).
\label{4.480}
\eeq
The real part of the potential must correctly reproduce the wave function Eq.~(\ref{4.460}).
\beq
\mbox{Re}\, V(\vec r_{2g},z)=
\frac{r_{2g}^2}
{2\nu \alpha_gr_0^4}
\label{bh300}
\eeq
Longitudinal photons can be used to disentangle between the effects of higher twist quark shadowing (2d term in (\ref{4.1})), and leading twist gluon shadowing (3rd term in (\ref{4.1})). The contribution, which mixes up these two types of Gribov corrections, come from so called aligned jet configurations \cite{bjorken} of the $\bar qq$ pair.
Namely the mean $\bar qq$ separation $r_{\bar qq}^2\sim Q^2/\alpha_q(1-\alpha_q)$
is small, unless the large value of $Q^2$ is compensated by smallness of $\alpha_q$ or $(1-\alpha_q)$. In the wave function of longitudinal photons, Eq.~(\ref{4.220}) such aligned-jet configurations are suppressed, so shadowing of longitudinal photons
should represent the net effect of gluon shadowing.
While the distance between the $q$ and the $\bar q$ is small, of order $1/Q^2$, the
gluon can propagate relatively far at a distance $r_g\sim r_0$ from the $q\bar q$-pair, which after the emission of the gluon is in a color-octet state. Therefore, the
entire $|q\bar qg\ra$-system appears as a octet-octet $gg$-dipole, and the shadowing
correction to the longitudinal cross section directly gives the magnitude of gluon
shadowing, which we
want to calculate.
Thus, the cross section of longitudinal photons is proportional to the gluon distribution function, therefore,
\beq
\frac{g_A(x,Q^2)}{g_N(x,Q^2)}\approx
\frac{\sigma^L_A(x,Q^2)}{\sigma^L_N(x,Q^2)}
\label{4.520}
\eeq
The shadowing
correction to $\sigma^L_A(x,Q^2)$ has the form (compare with (\ref{4.240})),
\beqn
\Delta\sigma^L_A(x,Q^2) &=&- \mbox{Re}\,\int d^2b
\int\limits_{-\infty}^{\infty} dz_1
\int\limits_{z_1}^{\infty} dz_2\,
\rho_A(b,z_1)\rho_A(b,z_2)
\nonumber\\ &\times&
\int d^2r_{2g}\,d^2r_{2\bar qq}\,d^2r_{1g}\,d^2r_{1\bar qq}
\int d\alpha_q\,d{\rm ln}(\alpha_g)
F^{\dagger}_{\gamma^*\to\bar qqg}
(\vec r_{2g},\vec r_{2\bar qq},\alpha_q,\alpha_g)
\nonumber\\ &\times&
G_{\bar qqg}(\vec r_{2g},\vec r_{2\bar qq},z_2;\vec r_{1g},\vec r_{1\bar qq},z_1)
F_{\gamma^*\to\bar qqg}(\vec r_{1g},\vec r_{1\bar qq},\alpha_q,\alpha_g)
\label{4.540}
\eeqn
Here
\beq
F^{\dagger}_{\gamma^*\to\bar qqg}
(\vec r_{g},\vec r_{\bar qq},\alpha_q,\alpha_g)
=
-\,\Psi^L_{\bar qq}\left(\vec r_{\bar qq},
\alpha_q\right)\ \vec r_{g}\cdot\vec\nabla\,
\Psi_{qg}\left(\vec r_{g}\right)\
\sigma_{gg}^N\left(x,r_{g}\right)\ ,
\label{4.440}
\eeq
Assuming $Q^2\gg 1/r_0^2$ we can neglect $r_{\bar qq}\ll r_g$. The net diffractive amplitude $F_{\gamma^*\to\bar qqg}(\vec r_{1g},\vec r_{1\bar qq},\alpha_q,\alpha_g)$ takes the form of Eq.~(\ref{4.440}), and we can rely on the factorized relation (\ref{4.400}) for the 3-body Green's function, with equation (\ref{4.420}) for the evolution of the gluonic dipole.
The results of numerical calculation of (\ref{4.540}) for the ratio
\beq
R_g(x,Q^2)=\frac{g_A(x,Q^2)}{A\,g_N(x,Q^2)},
\label{4.600}
\eeq
are depicted in Fig.~\ref{fig:glue} as function of Bjorken $x$ for $Q^2=4$ and $40\,\mbox{GeV}^2$.
\begin{figure}[htb]
\centerline{\includegraphics[width=7 cm]{glue.pdf}}
\caption{Ratio (\ref{4.600}) for carbon, copper and lead
at small Bjorken $x$ and $Q^2 = 4\,\mbox{GeV}^2$ (solid curves) and $40\,\mbox{GeV}^2$ (dashed curves).
}
\label{fig:glue}
\end{figure}
The predicted small magnitude of gluon shadowing was confirmed by next-to-leading
(NLO) global analyses of DIS data \cite{florian,kumano}. It also goes along with the
well known smallness of the triple-Pomeron coupling, measured at low virtuality
\cite{kklp}. The latter controls large mass diffraction,
which proceeds via gluon radiation, and its smallness leads to suppression of gluon radiation and gluon shadowing related to corresponding Gribov corrections.
The observed weakness of these effects is interpreted in the dipole approach a smallness
of the parameter $r_0\approx0.3\,\mbox{fm}$ in Eq.~(\ref{4.460}) \cite{kst2,spots}.
\subsection{Hadron-nucleus cross sections}\label{hA}
Applications of the dipole approach to calculation of Gribov corrections to the hadron-nucleus total cross sections contains more uncertainties and modelling compared with
hard reactions, like DIS. Nevertheless, it allows to make a progress compared with the hadronic representation, which involves ad hoc assumptions about the interaction cross section, of an excited hadronic state, and the unknown higher terms in the opacity
expansion.
\subsubsection{Excitation of the valence quark skeleton of the
proton}
First of all one should rely on a parametrization of the dipole cross section,
which allows an extension to the soft, large separation region. Following
\cite{kst2,xsect} we chose the saturated shape of the cross section, which rises as $r_T^2$
at small $r_T$, but levels off at large $r_T$,
\beq
\sigma_{\bar qq}(r_T,s)=\sigma_0(s)\,\left[
1-{\rm exp}\left(-\frac{r_T^2}
{R_0^2(s)}\right)\right]\ ,
\label{180}
\eeq
where $R_0(s)=0.88\,fm\,(s_0/s)^{0.14}$ and $s_0=1000\,GeV^2$
\cite{kst2}. The energy dependent factor $\sigma_0(s)$ is defined as,
\beq
\sigma_0(s)=\sigma^{\pi p}_{tot}(s)\,
\left(1 + \frac{3\,R^2_0(s)}{8\,\la r^2_{ch}\ra_{\pi}}
\right)\ ,
\label{190}
\eeq
where $\la r^2_{ch}\ra_{\pi}=0.44\pm 0.01\,fm^2$ \cite{pion} is the mean
square of the pion charge radius.
This dipole cross section is normalized to reproduce the pion-proton total
cross section, $\la\sigma_{\bar qq}\ra_\pi=\sigma_{tot}^{\pi p}(s)$. The
saturated shape of the dipole cross section is inspired by the popular
parametrization given in Ref.~\cite{gbw1,gbw2}, which is fitted to the low-$x$ and
high $Q^2$ data for $F^p_2(x,Q^2)$ from HERA. However, that should not be used
for our purpose, since is unable to provide the correct energy dependence of
hadronic cross sections. Namely, the pion-proton cross section cannot exceed
$23\,\mbox{mb}$. Besides,
Bjorken $x$ is not a proper variable for soft reactions, since at small $Q^2$
the value of $x$ is large even at low energies. The $s$-dependent dipole cross
section Eq.~(\ref{180}) was fitted \cite{kst2} to data for hadronic cross
sections, real photoproduction and low-$Q^2$ HERA data for the proton
structure function. The cross section (\ref{190}) averaged with the pion wave
function squared (see below) automatically reproduces the pion-proton cross
section.
In the case of a proton beam one needs a cross section for a three-quark dipole,
$\sigma_{3q}(\vec r_1,\vec r_2,\vec r_3)$, where $\vec r_i$ are the transverse
quark separation with a condition $\vec r_1+\vec r_2+\vec r_3=0$. In order to
avoid the introduction of a new unknown phenomenological quantity, we
express the
three-body dipole cross section via the conventional dipole cross section $\sigma_{\bar qq}$
\cite{mine},
\beq
\sigma_{3q}(\vec r_1,\vec r_2,\vec r_3) =
{1\over2}\,\Bigl[\sigma_{\bar qq}(r_1)+
\sigma_{\bar qq}(r_2)+
\sigma_{\bar qq}(r_3)\Bigr]\ .
\label{195}
\eeq
This form satisfies the limiting conditions, namely, turns into
$\sigma_{\bar qq}(r)$ if one of three separations is zero. Since all these
cross sections involve nonperturbative effects, this relation hardly can
be proven, but should be treated as a plausible assumption.
The 3-quark valence wave function is modelled assuming that
the dipole cross section is independent of the sharing
of the light-cone momentum among the quarks, so the wave function squared of
the valence Fock component of the proton, $\left|\Phi(\vec
r_i,\alpha_j\right|^2$ should be integrated over fractions $\alpha_i$. The
result depends only on transverse separations $\vec r_i$. The form of the
nonperturbative valence quark distribution is unknown, therefore for the
sake of simplicity we assume the Gaussian form,
\beqn
&&\left|\Psi_N(\vec r_1,\vec r_2,\vec r_3)\right|^2 =
\int\limits_0^1 \prod\limits_{i=1}^3 d\alpha_i\,
\left|\Phi(\vec r_i,\alpha_j)\right|^2\,
\delta\left(1-\sum\limits_{j=1}^3 \alpha_j\right)
\nonumber\\ &=&
\frac{2+r_p^2/R_p^2}{(\pi\,r_p\,R_p)^2}
\exp\left(-\frac{r_1^2}{r_p^2}-\frac{r_2^2+r_3^2}{R_p^2}\right)\,
\delta(\vec r_1+\vec r_2+\vec r_3)\ ,
\label{200}
\eeqn
where $\vec r_i$ are the interquark transverse distances. The two scales
$r_p$ and $R_p$ characterizing the mean transverse size of a diquark and
the mean distances to the third quark.
For the sake of simplicity here we assume
that the forces binding the valence quarks are of an iso-scalar nature,
therefore the quark distribution is symmetric, i.e. $r_p=R_p$ in
(\ref{200}). In this case the mean interquark separation squared is
$\la \vec r_i^{\,2}\ra={2\over3}R_p^2 = 2\la r_{ch}^{2}\ra_p$.
See other possibilities of an asymmetric valence structure in Ref.\cite{xsect}.
Apparently, any model for the dipole cross section and valence quark distribution in the proton, must reproduce correctly data for diffractive excitation of the proton,
otherwise the Gribov corrections will come out wrong. It was demonstrated
that with the above choice of the dipole cross section and proton wave function
one reproduces quite well the results of the global analysis \cite{kklp} of single diffraction data, namely the triple-Reggeon term $\mathbb{P}\Pom\mathbb{R}$, as well as
the triple-Pomeron one $\mathbb{P}\Pom\mathbb{P}$, controlling diffractive gluon radiation.
Now we are in a position to calculate the Gribov corrections.
\subsubsection{Excitation of the valence quark skeleton of the
proton}
The total cross sections reads,
\beqn
\sigma_{tot}^{pA} &=&
2\int d^2b\,\left[1-
\left\la e^{-{1\over2}\,
\sigma_{3q}(r_i)\,T_A(b)}\right\ra\right]
\nonumber \\ &\equiv&
2\int d^2b
\left[1-
\int \prod\limits_{i=1}^3 d^2r_i\,
\left|\Psi_N(r_j)\right|^2\,
e^{-{1\over2}\,
\sigma_{3q}(\vec r_k)\,T_A(b)}\right].
\label{312}
\eeqn
Using the wave function Eq.~(\ref{200}) with $r_p=R_p$ and the cross
section (\ref{180}) we get the following forward elastic cross section,
\beq
\left.\frac{d\sigma^{pp}_{el}}
{dp_T^2}\right|_{p_T=0} =
\frac{\gamma^2}{(1+{2\over3}\gamma)^2}\
\frac{\sigma_0^2(s)}{16\pi},
\label{265}
\eeq
where
$\gamma = 3\la r_{ch}^2\ra_p/R_0^2(s)$.
\section{Gluon shadowing}\label{gluons}
Eikonalization of the lowest Fock state $|3q\ra$ of the proton done in
(\ref{312}) corresponds to the Bethe-Heitler regime of gluon radiation.
Indeed, gluon bremsstrahlung is responsible for the rising energy dependence
of the cross section, and in the eikonal form
(\ref{312}) one assumes that the whole spectrum of gluons is radiated in each of multiple
interactions.
However, the Landau-Pomeranchuk-Migdal effect \cite{lp,m} is known to
suppress radiation in multiple interactions. Since a substantial part of the
inelastic cross section at high energies is related to gluon radiation, the
LPM effect suppresses the cross section. This is a
quantum-mechanical interference phenomenon and it is a part of the suppression
called Gribov inelastic shadowing. In
the QCD dipole picture it come from inclusion of higher Fock states, $|3qg\ra$,
etc. Each of these states represents a colorless dipole and its elastic
amplitude on a nucleon is subject to eikonalization.
As we already mentioned, the eikonalization procedure requires the
fluctuation lifetime to be much longer than the nuclear size. Otherwise,
one has to take into account the "breathing" of the fluctuation during
propagation through a nucleus, which can be done by applying the
light-cone Green function technique \cite{kz91,kst1,kst2}. In hadronic
representation this is equivalent to saying that all the longitudinal
momenta transfers must be much smaller than the inverse mean free path of the
hadron in the nucleus. Otherwise, one should employ the path-integral technique, described above.
The c.m. energies of HERA-B, RHIC and LHC are sufficiently high to treat
the lowest Fock state containing only valence quarks as "frozen" by the
Lorentz time dilation during propagation through the nucleus. Indeed, for
the excitations with the typical nucleon resonance masses, the coherence
length is sufficiently long compared to the nuclear size. This is why we
applied eikonalization without hesitation so far. Such an approximation,
however, never works for the higher Fock states containing gluons. Indeed,
since the gluon is a vector particle, the integration over effective mass of the
fluctuation is divergent, $dM^2/M^2$, which is the standard triple-Pomeron
behaviour. Therefore, the energy of collisions can never be sufficiently high
to neglect the large-mass tail. For this reason the inelastic shadowing
corrections, related to excitation gluonic degrees of freedom never saturates,
and keeps rising logarithmically with energy.
There are, however non-linear effects which are expected to stop the rise
of inelastic corrections at high energies. This is related to the
phenomenon of gluon saturation \cite{glr,al} or color glass condensate
\cite{mv}. The strength of these nonlinear effect is expected to be rather mild
due to smallness of the gluonic spots in the nucleus \cite{spots}.
The reason is simple, in spite of a sufficient
longitudinal overlap of gluon clouds originated from different nucleons,
there is insufficient overlap in the transverse plane. This fact leads to
a delay of the onset of saturation up to very high energies, since the
transverse radius squared of the gluonic clouds rise with energy very
slowly, logarithmically, with a small coefficient of the order of
$0.1\,\mbox{GeV}^{-2}$.
The details of the calculation of inelastic corrections related to excitation
of gluonic degrees of freedom can be found in Ref.\cite{kst2,mine}. The
numerical results
for nuclear cross sections at the energies of RHIC and LHC can be found in Ref.\cite{xsect,kps-ciofi}. As an example, the total proton-lead cross section,
calculated at $\sqrt{s}=5\,\mbox{TeV}$ in the Glauber approximation, and corrected to Gribov shadowing related to excitation of valence quarks and gluons, results in
$\sigma^{pPb}_{tot}=4242.5\,\mbox{mb};\ 4235.2\,\mbox{mb};$ and $4207.1\,\mbox{mb}$ respectively.
\section{Summary}
The dipole phenomenology in QCD has been intensively developing over the past three decades, due to both theoretical efforts and precise experimental data, in particular from HERA. This theoretical tool allows to calculate the effects of Gribov inelastic shadowing
on a more solid ground and in all orders of opacity expansion. In this note we presented several explicit examples.
\vspace*{25px}
\noi
{\bf Acknowledgements:} Work was partially
supported by Fondecyt (Chile) grant 1130543.
\vspace{2ex}
\input{BH-ref.tex}
\end{document}
| {'timestamp': '2016-02-02T02:10:16', 'yymm': '1602', 'arxiv_id': '1602.00298', 'language': 'en', 'url': 'https://arxiv.org/abs/1602.00298'} |
\section*{Acknowledgement}
This work has been supported in part by the European Research
Council grant n.~226455, \textit{``SUPERSYMMETRY, QUANTUM GRAVITY
AND GAUGE FIELDS (SUPERFIELDS)"}.
This work was conducted during the period B.N.T. served as a
postdoctoral research fellow at the \textit{``INFN-Laboratori
Nazionali di Frascati, Roma, Italy''}.
\section*{Appendix}
In this appendix, we provide explicit forms of the higher
principle minors of the state-space metric tensor of the eight
charged nonextremal nonlarge black holes. Our analysis illustrates
that the stability properties of the specific state-space
hypersurface may exactly be exploited in general. The definite
behavior of state-space properties, as accounted in the concerned
main section suggests that the various intriguing hypersurfaces of
the state-space configuration include the nice feature that they
do have definite stability properties, except for some specific
values of the charges and anticharges. As mentioned in the main
sections, these configurations are generically well-defined and
indicate an interacting statistical basis. Herewith, we discover
that the state-space geometry of the general black brane
configurations in string theory indicate the possible nature of
the underlying statistical fluctuations. Significantly, we notice
from the very definition of the intrinsic metric tensor that the
related factors of the principle minors take the following
expressions
\begin{eqnarray}
\tilde{p}_5&:=& n_2 \sqrt{m_1}+2 \sqrt{n_2} \sqrt{m_2}
\sqrt{m_1}+m_2 \sqrt{m_1}+\sqrt{n_1} n_2+2 \sqrt{n_1} \sqrt{n_2}
\sqrt{m_2}+\sqrt{n_1} m_2,\nonumber\\
\tilde{p}_6&:=& n_2 \sqrt{m_1} \sqrt{n_3}+n_2 \sqrt{m_1}
\sqrt{m_3}+2 \sqrt{n_2} \sqrt{m_1} \sqrt{m_2} \sqrt{n_3}+2
\sqrt{n_2} \sqrt{m_1} \sqrt{m_2} \sqrt{m_3} \nonumber\\ && +\sqrt{m_1} m_2
\sqrt{n_3}+\sqrt{m_1} m_2 \sqrt{m_3}+\sqrt{n_1} n_2
\sqrt{n_3}+\sqrt{n_1} n_2 \sqrt{m_3} \nonumber\\ && +2 \sqrt{n_1}
\sqrt{n_2} \sqrt{m_2} \sqrt{n_3}+2 \sqrt{n_1} \sqrt{n_2}
\sqrt{m_2} \sqrt{m_3}+\sqrt{n_1} m_2 \sqrt{n_3}+\sqrt{n_1} m_2
\sqrt{m_3},\nonumber\\
\tilde{p}_7&:=& 2 \sqrt{n_1} n_2 \sqrt{n_3} \sqrt{m_3}
\sqrt{m_4}+2 m_2 \sqrt{n_3} \sqrt{m_3} \sqrt{m_4} \sqrt{m_1}+8
\sqrt{n_2} \sqrt{m_2} n_3 \sqrt{n_4} \sqrt{m_1} \nonumber\\ && +8
\sqrt{n_1} \sqrt{n_2} \sqrt{m_2} n_3 \sqrt{n_4}+8 m_2 \sqrt{n_3}
\sqrt{m_3} \sqrt{n_4} \sqrt{m_1}+16 \sqrt{n_2} \sqrt{m_2}
\sqrt{n_3} \sqrt{m_3} \sqrt{n_4} \sqrt{m_1} \nonumber\\ && +2 \sqrt{n_2}
\sqrt{m_2} n_3 \sqrt{m_4} \sqrt{m_1}+2 \sqrt{n_1} \sqrt{n_2}
\sqrt{m_2} m_3 \sqrt{m_4}+4 \sqrt{n_1} \sqrt{n_2} \sqrt{m_2}
\sqrt{n_3} \sqrt{m_3} \sqrt{m_4} \nonumber\\ && +8 n_2 \sqrt{n_3}
\sqrt{m_3} \sqrt{n_4} \sqrt{m_1}+8 \sqrt{n_1} n_2 \sqrt{n_3}
\sqrt{m_3} \sqrt{n_4}+8 \sqrt{n_1} m_2 \sqrt{n_3} \sqrt{m_3}
\sqrt{n_4} \nonumber\\ && +8 \sqrt{n_1} \sqrt{n_2} \sqrt{m_2} m_3
\sqrt{n_4}+8 \sqrt{n_2} \sqrt{m_2} m_3 \sqrt{n_4} \sqrt{m_1}+2
\sqrt{n_2} \sqrt{m_2} m_3 \sqrt{m_4} \sqrt{m_1} \nonumber\\ && +2
\sqrt{n_1} \sqrt{n_2} \sqrt{m_2} n_3 \sqrt{m_4}+16 \sqrt{n_1}
\sqrt{n_2} \sqrt{m_2} \sqrt{n_3} \sqrt{m_3} \sqrt{n_4}+2
\sqrt{n_1} m_2 \sqrt{n_3} \sqrt{m_3} \sqrt{m_4} \nonumber\\ && +\sqrt{n_1}
n_2 n_3 \sqrt{m_4}+4 \sqrt{n_1} m_2 m_3 \sqrt{n_4}+n_2 m_3
\sqrt{m_4} \sqrt{m_1}+4 m_2 n_3 \sqrt{n_4} \sqrt{m_1} \nonumber\\ && +4
\sqrt{n_1} m_2 n_3 \sqrt{n_4}+\sqrt{n_1} m_2 m_3 \sqrt{m_4}+4 m_2
m_3 \sqrt{n_4} \sqrt{m_1}+4 n_2 m_3 \sqrt{n_4} \sqrt{m_1} \nonumber\\ &&
+m_2 n_3 \sqrt{m_4} \sqrt{m_1}+m_2 m_3 \sqrt{m_4}
\sqrt{m_1}+\sqrt{n_1} n_2 m_3 \sqrt{m_4}+4 \sqrt{n_1} n_2 m_3
\sqrt{n_4} \nonumber\\ && +4 \sqrt{n_1} n_2 n_3 \sqrt{n_4}+\sqrt{n_1} m_2
n_3 \sqrt{m_4}+4 n_2 n_3 \sqrt{n_4} \sqrt{m_1}+n_2 n_3 \sqrt{m_4}
\sqrt{m_1} \nonumber\\ && +2 n_2 \sqrt{n_3} \sqrt{m_3} \sqrt{m_4}
\sqrt{m_1}+4 \sqrt{n_2} \sqrt{m_2} \sqrt{n_3} \sqrt{m_3}
\sqrt{m_4} \sqrt{m_1},\nonumber\\
\tilde{p}_8&:=& 2 \sqrt{n_1} n_2 \sqrt{n_3} \sqrt{m_3}
\sqrt{m_4}+2 m_2 \sqrt{n_3} \sqrt{m_3} \sqrt{m_4} \sqrt{m_1}+2
\sqrt{n_2} \sqrt{m_2} n_3 \sqrt{n_4} \sqrt{m_1} \nonumber\\ && +2
\sqrt{n_1} \sqrt{n_2} \sqrt{m_2} n_3 \sqrt{n_4}+2 m_2 \sqrt{n_3}
\sqrt{m_3} \sqrt{n_4} \sqrt{m_1} +4 \sqrt{n_2} \sqrt{m_2}
\sqrt{n_3} \sqrt{m_3} \sqrt{n_4} \sqrt{m_1} \nonumber\\ && +2 \sqrt{n_2}
\sqrt{m_2} n_3 \sqrt{m_4} \sqrt{m_1}+2 \sqrt{n_1} \sqrt{n_2}
\sqrt{m_2} m_3 \sqrt{m_4}+4 \sqrt{n_1} \sqrt{n_2} \sqrt{m_2}
\sqrt{n_3} \sqrt{m_3} \sqrt{m_4} \nonumber\\ && +2 n_2 \sqrt{n_3}
\sqrt{m_3} \sqrt{n_4} \sqrt{m_1}+2 \sqrt{n_1} n_2 \sqrt{n_3}
\sqrt{m_3} \sqrt{n_4}+2 \sqrt{n_1} m_2 \sqrt{n_3} \sqrt{m_3}
\sqrt{n_4} \nonumber\\ && +2 \sqrt{n_1} \sqrt{n_2} \sqrt{m_2} m_3
\sqrt{n_4}+2 \sqrt{n_2} \sqrt{m_2} m_3 \sqrt{n_4} \sqrt{m_1}+2
\sqrt{n_2} \sqrt{m_2} m_3 \sqrt{m_4} \sqrt{m_1} \nonumber\\ && +2
\sqrt{n_1} \sqrt{n_2} \sqrt{m_2} n_3 \sqrt{m_4}+4 \sqrt{n_1}
\sqrt{n_2} \sqrt{m_2} \sqrt{n_3} \sqrt{m_3} \sqrt{n_4}+2
\sqrt{n_1} m_2 \sqrt{n_3} \sqrt{m_3} \sqrt{m_4} \nonumber\\ && +\sqrt{n_1}
n_2 n_3 \sqrt{m_4}+\sqrt{n_1} m_2 m_3 \sqrt{n_4}+n_2 m_3
\sqrt{m_4} \sqrt{m_1} +m_2 n_3 \sqrt{n_4} \sqrt{m_1} \nonumber\\ &&
+\sqrt{n_1} m_2 n_3 \sqrt{n_4} +\sqrt{n_1} m_2 m_3 \sqrt{m_4}+m_2
m_3 \sqrt{n_4} \sqrt{m_1}+n_2 m_3 \sqrt{n_4} \sqrt{m_1} \nonumber\\ &&
+m_2 n_3 \sqrt{m_4} \sqrt{m_1} +m_2 m_3 \sqrt{m_4}
\sqrt{m_1}+\sqrt{n_1} n_2 m_3 \sqrt{m_4}+\sqrt{n_1} n_2 m_3
\sqrt{n_4} \nonumber\\ && +\sqrt{n_1} n_2 n_3 \sqrt{n_4}+\sqrt{n_1} m_2
n_3 \sqrt{m_4}+n_2 n_3 \sqrt{n_4} \sqrt{m_1}+n_2 n_3 \sqrt{m_4}
\sqrt{m_1} \nonumber\\ && +2 n_2 \sqrt{n_3} \sqrt{m_3} \sqrt{m_4}
\sqrt{m_1}+4 \sqrt{n_2} \sqrt{m_2} \sqrt{n_3} \sqrt{m_3}
\sqrt{m_4} \sqrt{m_1}.
\end{eqnarray}
| {'timestamp': '2011-02-14T02:02:02', 'yymm': '1102', 'arxiv_id': '1102.2391', 'language': 'en', 'url': 'https://arxiv.org/abs/1102.2391'} |
\section*{Acknowledgments}
I thank C.M. Ko for asking the questions which lead to this study.
This work was supported by the National Science Foundation under grant
number PHY-9403666.
| {'timestamp': '1996-04-24T03:17:50', 'yymm': '9604', 'arxiv_id': 'nucl-th/9604036', 'language': 'en', 'url': 'https://arxiv.org/abs/nucl-th/9604036'} |
\section{Introduction}
Orthonormal transformations play a key role in most matrix decomposition techniques and spectral methods \cite{Belabbas369}. As such, manipulating them in an efficient manner is essential to many practical applications. While general matrix-vector multiplications with orthogonal matrices take $O(d^2)$ space and time, it is natural to ask whether faster approximate computations (say $O(d \log d)$) can be achieved while retaining enough accuracy.\\
Approximating an orthonormal matrix with just a few building blocks is hard in general. The standard decomposition technique meant to reduce complexity is a low-rank approximation. Unfortunately, for an orthonormal matrix, which is perfectly conditioned, this approach is meaningless.\\
In this work, we are inspired by the fact that several orthonormal/unitary transformations that exhibit low numerical complexity are known. The typical example is the discrete Fourier transform with its efficient implementation as the fast Fourier transform \cite{doi:10.1137/1.9781611970999} together with other Fourier-related algorithms: fast Walsh-Hadamard transforms \cite{1674569}, fast cosine transforms \cite{1163351}, and fast Hartley transforms \cite{1457236}. Other approaches include fast wavelet transforms \cite{doi:10.1002/cpa.3160440202}, banded orthonormal matrices \cite{SIMON2007120, Strang12413} and fast Slepian transforms \cite{KARNIK2019624}. Decomposition of orthogonal matrices into $O(d)$ Householder reflectors or $O(d^2)$ Givens rotations \cite{Golub1996}[Chapter 5.1] are known already. These basic building blocks have been further extended, for example we have fast Givens rotations \cite{Rath1982} and two generalizations of the Givens rotations \cite{BILOTI201356} and \cite{GGR}. In theoretical physics, unitary decompositions parametrize symmetry groups \cite{Tilma:2002ke}, and they are compactly parametrized using $\sigma$-matrices \cite{Spengler_2010} or symmetric positive definite matrices \cite{Barvinok}. To the best of our knowledge, none of these factorizations focus on reducing the computational complexity of using the orthogonal/unitary transformations but rather they model properties of physical systems.\\
Our idea is to approximately factor any orthonormal matrix into a product of a fixed number of sparse matrices such that their application (to a vector)
has linearithmic complexity. In this paper, we derive structured approximations to orthonormal matrices that can be found efficiently and applied remarkably fast. We pose the search for an efficient approximation as an optimization problem over a product of structured matrices, the extended orthogonal Givens transformations. These structures extend Givens rotations to also include reflectors with no computational drawback and suggest a decomposition of the main optimization problem into sub-problems that are easy to understand and solved via a greedy approach. The theoretical properties of the obtained solution are characterized in terms of approximation bounds while the empirical properties are studied extensively.\\
We illustrate our approach considering dimensionality reduction with principal component analysis (PCA).
Here the goal is not to propose a new fast algorithm to compute the principal directions, plenty of efficient algorithms for this task \cite{FastMCSVD}\cite{Golub1996}[Chapter~10] \cite{FindingStructureWithRandomness}. Rather, we aim at constructing fast dimensionality reduction operators.
While the calculation of the principal components is a one-off computation, a numerically efficient projection operation is critical since it is required multiple times in downstream applications. The problem of deriving fast projections has also been previously studied. Possible approaches include: fast wavelet transforms \cite{WaveletsForPCA}, sparse PCA \cite{MINIMAXOPTIMALSPCA, SparsePCA}, structured transformations such as circulant matrices \cite{CirculantDRoperator}, Kronecker products \cite{KroneckerPCA}, Givens rotations \cite{DCTandKLT, Treelets, LeCunFastApproximations} or structured random projections \cite{FastJL, ToeplitzJL}. Compared to these works, we propose a new way to factorize any orthogonal matrix, including PCA directions, into simple orthogonal structures that we call extended orthogonal Givens transformations and which naturally lead to optimization problems that have closed-form solutions and are therefore efficiently computed.\\
We note that our approach provides new perspectives on the structure of the orthogonal group and how to coarsely approximate it, which might have an impact on other open research questions.\\% where orthogonal matrices are involved.\\
The paper is organized as follows: Section 2 describes the basic building blocks and algorithm that we propose, Section 3 gives the theoretical guarantees for our contributions, Section 4 details the application of our method to PCA projections and Section 5 shows the numerical experiments.
\section{The proposed algorithm}
Given a $d\times d$ orthonormal $\mathbf{U}$, the matrix-vector multiplication $\mathbf{Ux}$ takes $O(d^2)$ operations. We want to build $\mathbf{\bar{U}}$ such that $\mathbf{U} \approx \mathbf{\bar{U}}$ and $\mathbf{\bar{U}x}$ takes $O(d \log d)$ operations. Parametrizations of orthonormal matrices \cite{doi:10.1137/0908055} are known, but to be best of our knowledge, the problem of accurately approximating $\mathbf U$ as product of only a few $O(d \log d)$ transformations is open. Given a $d \times p$ diagonal $\mathbf{\Sigma}_p$, in the spirit of previous work minimizing Frobenius norm approximations \cite{Kondor2014MMF, lemagoarou:hal-01416110}, we consider the problem
\begin{equation}
\underset{\mathbf{\bar{U}}, \ \mathbf{\bar{\Sigma}}_p}{\text{minimize}}\ \| \mathbf{U} \mathbf{\Sigma}_p - \mathbf{\bar{U}}\mathbf{\bar{\Sigma}}_p \|_F^2 \text{ subject to } \mathbf{\bar{U}} \in \mathcal{F}_g,
\label{eq:theoptimizationproblem_only_orthonormal}
\end{equation}
where $\mathbf{\bar{U}}$ is $d \times d$ and $\mathbf{\bar{\Sigma}}_p$ is a $d \times p$ diagonal matrix. Choosing $p = d$ while $\mathbf{\Sigma}_p$ and $\mathbf{\bar{\Sigma}}_p$ to be the identity, we simply approximate $\mathbf U$. We will also use $\mathbf{U}_p$ do denote the first $p \leq d$ columns of $\mathbf{U}$.
The above general formulation allows to also consider cases where different directions might have different importance.
Then, $\mathcal{F}_g$ is a set of orthogonal matrices -- defined next -- that can be applied fast and allow to efficiently but approximately solve
\eqref{eq:theoptimizationproblem_only_orthonormal}.
\subsection{The basic building blocks}
Classic matrix building blocks that are numerically efficiency include circulant/Toeplitz matrices or Kronecker products. These choices are inefficient as they depend on $O(d)$ free parameters but their matrix-vector product cost is $O(d \log d)$ or even $O(d \sqrt{d})$, i.e., they do not scale linearly with the number of parameters they have. Consider the sparse orthogonal matrices
\begin{equation}
\mathbf{G}_{ij} = \begin{bmatrix} \mathbf{I}_{i-1} & & &\\
& * & & * \\
& & \mathbf{I}_{j-i-1} & \\
& * & & * \\
& & & & \mathbf{I}_{d-j} \\
\end{bmatrix}, \mathbf{\tilde{G}}_{i j} \in \left\{ \! \begin{bmatrix} c & -s \\ s & c \end{bmatrix}\!,\! \begin{bmatrix} c & s \\ s & -c \end{bmatrix} \! \right\},\text{ such that } c^2 + s^2 = 1,
\label{eq:theG}
\end{equation}
where the non-zero part (denoted by $*$ and $\mathbf{\tilde{G}}_{i j}$) on rows and columns $i$ and $j$. The transformation in \eqref{eq:theG}, with the first option in $\mathbf{\tilde{G}}_{i j}$, is a Givens (or Jacobi) rotation. With the second option, we have a very sparse Householder reflector. These transformations were first used by \citet{FastSparsifyingTransforms} to learn numerically efficient sparsifying dictionaries for sparse coding. The $\mathbf{G}_{ij}$s have the following advantages: i) they are orthogonal; ii) they are sparse and therefore fast to manipulate: matrix-vector multiplications $\mathbf{G}_{ij}\mathbf{x}$ take only $6$ operations; iii) there are two degrees of freedom to learn: $c$ (or $s$) and the binary choice; and iv) allowing both sub-matrices in $\mathbf{\tilde{G}}_{i j}$ enriches the structure and as we will see, leads to an easier (closed-form solutions) optimization problem.\\
We propose to consider matrices $\mathbf{\bar{U}} \in \mathcal{F}_g$ that are products of $g$ transformations from \eqref{eq:theG}, that is
\begin{equation}
\mathbf{\bar{U}} =
\prod_{k=1}^{g} \mathbf{G}_{i_k j_k} = \mathbf{G}_{i_1 j_1} \dots \mathbf{G}_{i_g j_g}.
\label{eq:approxu}
\end{equation}
Matrix-vector multiplication with $\mathbf{\bar{U}}$ takes $6g$ operations -- when $g$ is $O(d \log d)$ this is significantly better than $O(d^2)$, while the coding complexity of each $\mathbf{G}_{i j}$ is approximately $2\log_2 d + C$: $2\log_2 d -1$ bits to encode the choice of the two indices, a constant factor $C$ for the pair $(c,s)$ and 1 bit for the choice between the rotation and reflector. The coding complexity of $\mathbf{\bar{U}}$ scales linearly with $g$.\\
We note that Givens rotations have been used extensively to build numerically efficient transformations \cite{5560826, EffectiveGivens, Kondor2014MMF, lemagoarou:hal-01416110, Treelets}. However, $2 \times 2$ reflector was not used before. This may be because in linear algebra (e.g. in QR factorization) and in optimization \cite{Shalit14} considering also the reflector has no additional benefit: the rotation alone introduces each zero in the QR factorization and the reflector does not have an exponential mapping on the orthogonal manifold, respectively. As we will show, considering both the rotation and the reflector has the advantage of providing a closed-form solution to our problem.
\subsection{The proposed greedy algorithm}
We propose to solve the optimization problem in \eqref{eq:theoptimizationproblem_only_orthonormal} with a greedy approach: we keep $\mathbf{\bar{\Sigma}}_p$ and all variables fixed except for a single $\mathbf{G}_{i_k j_k}$ from $\mathbf{\bar{U}}$ and minimize the objective function. When optimizing w.r.t. $\mathbf{G}_{i_k j_k}$ it is convenient to write
\begin{equation}
\! \| \mathbf{U}\mathbf{\Sigma}_p \! \! - \! \mathbf{\bar{U} \bar{\Sigma}}_p \|_F^2
\! = \! \|
\underbrace{ \prod_{t=1}^{k-1} \mathbf{G}_{i_t j_t}^T
\mathbf{U} \mathbf{\Sigma}_p \! \! }_{{\mathbf L}^{(k)}}
- \mathbf{G}_{i_k j_k}
\underbrace{
\prod_{t=k+1}^{g} \mathbf{G}_{i_t j_t}
\mathbf{\bar{\Sigma}}_p}_{{\mathbf N}^{(k)}} \|_F^2 \! = \! \| \mathbf{L}^{(k)} \! - \! \mathbf{G}_{i_k j_k} \mathbf{N}^{(k)} \|_F^2.
\label{eq:thestart}
\end{equation}
The next result characterizes the Givens transformation $\mathbf{G}_{i_k j_k}$ minimizing the above norm. We drop the dependence on $k$ for ease of notation.
\begin{theorem}
[Locally optimal $\mathbf{G}_{ij}$]
Let $\mathbf{L}$ and $\mathbf{N}$ be two $d \times p$ matrices. Further, let $\mathbf{Z} = \mathbf{L} \mathbf{N}^T$ and $\mathbf{Z}_{\{i,j\}} = \begin{bmatrix} Z_{ii} & Z_{ij} \\ Z_{ji} & Z_{jj} \end{bmatrix}$ then we have
\begin{equation}
\ C_{ij} = \| \mathbf{Z}_{ \{i,j\} } \|_* - \text{tr}(\mathbf{Z}_{ \{ i,j \} }) = \left\{ \begin{array}{lr} \sqrt{ (Z_{ii} + Z_{jj})^2 + (Z_{ij} - Z_{ji})^2} -Z_{ii} -Z_{jj} , \text{if } \det( \mathbf{Z}_{ \{ i,j \} } ) \geq 0 \\ \sqrt{ (Z_{ii} - Z_{jj})^2 + (Z_{ij} + Z_{ji})^2} -Z_{ii} -Z_{jj}, \text{if } \det( \mathbf{Z}_{ \{ i,j \} } ) < 0 \end{array}\right.
\label{eq:theCijcomputed}
\end{equation}
Let $\mathbf{Z}_{\{i^\star,j^\star\}} \! = \! \mathbf{V}_1 \mathbf{S V}
_2^T$ be the SVD of $\mathbf{Z}_{\{i^\star,j^\star\}}$ , with
\begin{equation}
(i^\star, j^\star) = \underset{(i, j),\ j > i}{\arg \max}\ \ C_{ij}.
\label{eq:theCij}
\end{equation}
Then, the extended orthogonal Givens transformation that minimizes $\| \mathbf{L} - \mathbf{G}_{i j} \mathbf{N} \|_F^2$ is given by
$\mathbf{\tilde{G}}_{i^\star j^\star}^\star = \mathbf{V}_1 \mathbf{V}_2^T.$
\end{theorem}
The above theorem derives a locally optimal way to construct an approximation $\mathbf{\bar{U}}$. We iteratively apply the result to find, for each component $k$ in \eqref{eq:approxu}, the extended orthogonal Givens transformation that best minimizes the objective function \eqref{eq:theoptimizationproblem_only_orthonormal}. The full procedure is in Algorithm 1 and can be viewed in two different ways: i) a coordinate minimization algorithm; or ii) a hierarchical decomposition where each stage is extremely sparse. The proposed algorithm is guaranteed to converge, in the sense of the objective function \eqref{eq:theoptimizationproblem_only_orthonormal}, to a stationary point. Indeed, no step in the algorithm can increase the objective function, since the sub-problems are minimized exactly: we choose the best indices and then perform the best $2 \times 2$ transformation. We note three remarks on the properties of Algorithm 1.
\begin{rem}[Complexity of Algorithm 1]
The computational complexity of the iterative part of Algorithm 1 is $O(d g)$ and the initialization is dominated by the computation of all the scores $C_{ij}$ which takes $O(d^2)$. Note that, the $C_{ij}$s are computed from scratch only once in the initialization phase. After that, at each step $k$ we need to recompute the $C_{ij}$ (redo the $2\times 2$ singular value decompositions) only for the indices $(i_k, j_k)$ currently used (the Givens transformations act on two coordinates at a time). All other scores are update by the same quantity: in \eqref{eq:theCij}, the $C_{ij}$ are the same except when $i$ or $j$ belong to the set $(i_k, j_k)$. This observation substantially reduces the running time.
\end{rem}
\begin{rem}[Complexity of applying $\mathbf{\bar{U} \mathbf{\bar{\Sigma}}}_p$]
When $p < d$ the computational complexity of $6g$ operations is an upper bound. Since we keep only $p$ components, we need be careful not to perform operations whose result is thrown away by the mask $\mathbf{\bar{\Sigma}}_p$. Consider for example a transformation $\mathbf{G}_{1d}$ applied to a vector of size $d$ projected to a $p < d$ dimensional space. The three operations that take place on the $d^\text{th}$ component are unnecessary. Then, after computing $\mathbf{\bar{U}}$, a pass is made through each of the $g$ transforms to decide which of two coordinates the computations are necessary for the final result. As we will show, this further improves the numerical efficiency of our method.
\end{rem}
\begin{rem}[On the choice of indices] Algorithm 1 greedily chooses at each step $k$ the indices according to \eqref{eq:theCij}. Other factors might be considered: i) choosing indices based on previous choices so that only a select group of indices are used throughout the algorithm, or ii) make multiple choices at each step in order to speed up the algorithm.
\end{rem}
\begin{algorithm}[tb]
\caption{Approximate orthonormal matrix factorization with extended Givens transformations}
\label{alg:fpca}
\begin{algorithmic}
\State {\bfseries Input:} The $p$ orthogonal components $\mathbf{U}_p$ and their weights (singular values) $\mathbf{\Sigma}_p$, the size $g$ of the approximation \eqref{eq:approxu}, the update rule for $\mathbf{\bar{\Sigma}}_p$ in \{ `identity', `original', `update' \} and the stopping criterion $\epsilon$ (default taken to be $\epsilon = 10^{-2}$).
\State {\bfseries Output:} The linear transformation $\mathbf{\bar{U}} \mathbf{\mathbf{\bar{\Sigma}}}_p$, the approximate solution to \eqref{eq:theoptimizationproblem_only_orthonormal}.
\State {\bfseries Initialize:} $\mathbf{G}_{i_k j_k} \! = \! \mathbf{I}_{d \times d},k\! =\! 1,\dots,g$ and compute all scores $C_{ij}$ according to \eqref{eq:theCijcomputed} with $\mathbf{Z} = \mathbf{U}_p \mathbf{\bar{\Sigma}}_p^T$, where $\mathbf{\bar{\Sigma}}_p = \begin{bmatrix} \mathbf{I}_{p \times p}; & \mathbf{0}_{(d-p) \times p} \end{bmatrix}$ if the update rule is `identity' and $\mathbf{\bar{\Sigma}}_p = \mathbf{\Sigma}_p$ otherwise.
\Repeat
\State Set $\mathbf{L}^{(0)} = \mathbf{U}_p \mathbf{\Sigma}_p$ and set $\mathbf{N}^{(0)} = \mathbf{G}_{i_1 j_1} \dots \mathbf{G}_{i_g j_g} \mathbf{\bar{\Sigma}}_p$.
\For{$k = 1$ {\bfseries to} $g$ }
\State Update $\mathbf{N}^{(k)} = \mathbf{G}_{i_k j_k}^T \mathbf{N}^{(k-1)}$ and find best score according to \eqref{eq:theCij}.
\State Compute the best $k^\text{th}$ transformation by Theorem 1.
\State Update $\mathbf{L}^{(k)} = \mathbf{G}_{i_k j_k}^T \mathbf{L}^{(k-1)}$ and then update all scores $C_{ij}$ in \eqref{eq:theCijcomputed} but only for indices $i, j \in \{i_k, j_k\}$ with $\mathbf{Z}\! = \! \mathbf{L}^{(k)} (\mathbf{N}^{(k)})^T$ -- all other scores are unchanged.
\EndFor
\State Set $\mathbf{\bar{\Sigma}}_p = \begin{bmatrix}
\text{diag}(\mathbf{L}^{(g)}); & \mathbf{0}_{(d-p) \times p}
\end{bmatrix}$ if rule is `update', $i \leftarrow i+1$ and $\epsilon_i = \| \mathbf{L}^{(g)} - \mathbf{\bar{\Sigma}}_p \|_F^2 $.
\Until $ |\epsilon_{i-1} - \epsilon_i| < \epsilon$, if $i > 1$.
\end{algorithmic}
\end{algorithm}
\section{Analysis of the proposed algorithm}
We consider $p = d$, i.e., $\mathbf{\bar{\Sigma}}_p = \mathbf{I}_{d \times d}$ and therefore $\mathbf{Z} = \mathbf{U}$. We model the $\mathbf{U}$ as a random orthonormal matrix with Haar measure \cite{HowToRandomUnitary} updated so that the diagonal is positive. We perform this update because multiplication by a diagonal matrix with $\pm 1$ entries has no computational cost but it brings $\mathbf{U}$ closer to $\mathbf{I}_{d \times d}$.
The goal of this section is to establish upper bounds for the distance between $\mathbf{U}$ and $\mathbf{\bar{U}}$, as a function of $d$ and $g$. We first comment on the inherent difficulty of the problem.
\begin{rem}[The approximation gap] Since the orthogonal group has size $O(d^2)$, by the pigeonhole principle a random orthogonal matrix as \eqref{eq:approxu} and only $g \ll d^2$ degrees of freedom cannot be exactly approximated with less than $O(d^2)$ operations.
For our purposes, think $g$ either $O(d)$ or $O(d \log d)$.
Our goal is to show that the fast structures we propose can perform well in practice and have theoretical bounds that guarantee worse case or average accuracy.
\end{rem}
\noindent Next, we show two approximations bounds depending on the number of Givens transformations \eqref{eq:theG}.
\begin{theorem}[A special bound] Given a random $d \times d$ orthonormal $\mathbf{U}$, for large $d$, its approximation $\mathbf{\bar{U}}$ from \eqref{eq:theoptimizationproblem_only_orthonormal} with $g = d/2$ transformations from \eqref{eq:theG} obeys
\begin{equation}
\mathbb{E}[ \| \mathbf{U} - \mathbf{\bar{U}} \|_F^2] \leq 2d - \sqrt{2\pi d}.
\end{equation}
\end{theorem}
\begin{theorem}[A general bound] Given a random $d \times d$ orthonormal $\mathbf{U}$, for large $d$, its approximation $\mathbf{\bar{U}}$ from \eqref{eq:theoptimizationproblem_only_orthonormal} with $g \leq d(d-1)/2$ transformations from \eqref{eq:theG} is bounded by
\begin{equation}
\mathbb{E} \left[ \| \mathbf{U} - \mathbf{\bar{U}} \|_F^2 \right] \leq 2(d - \lfloor r \rfloor) - \frac{2\sqrt{2}}{\sqrt{\pi}} \sqrt{d - \lfloor r \rfloor},\text{ where } r = d - \frac{1 + \sqrt{(2d-1)^2 - 8g}}{2}.
\label{eq:theresultsofu3}
\end{equation}
\end{theorem}
Theorem 2 shows that, on average, the performance might degrade with increasing $d$. As stated in Remark 4, this is not surprising since the orthonormal group is much larger than the structure we are trying to approximate it with. The next result provides a bound for other values of $g$. In Theorem 3, taking $g = c_1 d \log d$ for some positive constant $c_1$ we have that $r \approx c_1 \log d$. This means that whenever $p \ll d$ we will roughly need $O(d)$ Givens transformations from \eqref{eq:theG} to improve the $\lfloor r \rfloor$ term. Since the proof of the theorem uses only rotations (and furthermore, in a particular order of indices $(i_k,j_k)$) we expect our algorithm to perform much better than the bound indicates as it allows for a richer structure \eqref{eq:theG} and uses greedy steps that maximally improve the accuracy at each step.\\
The previous theorems consider the Frobenius norm. In the Jacobi iterative process for diagonalizing a symmetric matrix with Givens rotations \cite{Golub1996}[Chapter~8.4] the progress of the procedure (convergence) is measured using the off-diagonal ``norm'' $\text{off}(\mathbf{U}) = \sqrt{\sum_{t}^d \sum_{q \neq t}^d U_{tq}^2}$.\\
\begin{theorem}[Convergence in the off-diagonal norm] Given a $d \times d$ orthonormal $\mathbf{U}$ and a single Givens transformation $\mathbf{G}_{ij}$, assuming $\text{det}(\mathbf{U}_{ \{i,j\} }) \geq 0$ we have
\begin{equation}
\text{off}(\mathbf{UG}_{ij}^T )^2 \leq \text{off}(\mathbf{U})^2 \! + \frac{1}{2}((U_{ii} - U_{jj})^2- (U_{ij} - U_{ji})^2).
\end{equation}
\end{theorem}
This result shows that, unlike with the Jacobi iterations, monotonic convergence in this quantity is not guaranteed and depends on the relative differences between the diagonal and the off-diagonal entries of $\mathbf{U}_{ \{i,j\} }$. Our method convergence monotonically to a stationary point when we measure the progress in the Frobenius norm.
\begin{rem}[The effect of a single $\mathbf{G}_{ij}$] Given a $d \times d$ orthonormal $\mathbf{U}$ and a Givens transformation $\mathbf{G}_{ij}$ from \eqref{eq:theG} we have that: i) $\mathbf{U}\mathbf{G}_{i j}^T$ is closer to the identity matrix in the sense that $\mathbf{\tilde{G}}_{ij}$ makes a positive contribution to the diagonal elements, i.e., $\text{tr}(\mathbf{U}\mathbf{G}_{i j}^T) = \text{tr}(\mathbf{U}) + C_{i j}$; and ii) $\mathbb{E}[C_{ij}] \approx 0.6956 d^{-1/2}$ if $\mathbf{U}$ is random with Haar measure and positive diagonal for large $d$.
\end{rem}
The above remark suggests a metric to study the convergence of the proposed method: each Givens transformation adds the score $C_{ij}$ to the diagonal entries of the current approximation (and therefore ensures that $\mathbf{U \bar{U}}^T $ converges to the identity -- the only diagonal orthonormal matrix). By choosing the maximum $C_{ij}$ we are taking the largest step in this direction.
\begin{rem}[Evolution of $C_{i_kj_k}$ with $k$] Given a fixed $0 < u < 1$, consider the toy construction $\mathbf{U}_{ \{i,j\} } = \begin{bmatrix} u & z_2 \\ z_1 & u \end{bmatrix}$, i.e., a $2 \times 2$ sub-matrix of a $d \times d$ orthonormal matrix where diagonal elements are equal and off-diagonals are two independent truncated standard normal random variables in the interval $[-\sqrt{1-u^2}, \sqrt{1-u^2}]$ (since rows and columns of $\mathbf{U}$ are $\ell_2$ normalized). Then, for large $d$, by direct calculation we have that the expected score $\mathbb{E}[ C_{ij}(u) ]$, i.e., $C_{ij}$ as a function of $u$, obeys
\begin{equation}
\begin{aligned}
\mathbb{E}[C_{ij}(u)] \propto (1-u)^2,
\end{aligned}
\end{equation}
i.e., the expected $C_{ij}$ decreases on average quadratically with the increase in the diagonal elements.\\
The remark is intuitive: as $k$ increases $\mathbf{\bar{U}}$ is more accurate and $\mathbf{U\bar{U}}^T$ becomes diagonally dominant, i.e., $\mathbf{U\bar{U}}^T \to \mathbf{I}_d$ as $k \to O(d^2)$, and we do expect to reach lower scores $C_{i_k j_k}$, i.e., few Givens transformations from \eqref{eq:theG} provide a rough estimation while very good approximations require $k \approx d^2$.
\end{rem}
\subsection{Other ways to measure the approximation error}
Throughout this paper we use the Frobenius norm to measure and study the approximation error we propose. In this section we discuss this choice and explore a few alternatives.\\
In the linear algebra literature, a natural way to measure approximation error is through the operator norm. This is especially true in the randomized linear algebra field (where matrix concentration inequalities which bound the operator norm play a central role). Moreover, the review manuscript \citep{10.1561/2200000048} deals explicitly with some potential issues that might arise from using the Frobenius, instead of the operator, norm in matrix approximations: the discussion in Chapter 6 of \citep{10.1561/2200000048} entitled ``Warning: Frobenius--Norm Bounds''. The text highlights situations where the matrix to be approximated is low rank and corrupted by noise and/or scaling issues are present. While that discussion holds true, in our case we deal with a given perfectly conditioned matrix $\mathbf{U}$ and its approximation displays the same scaling ($\mathbf{\bar{U}}$ has normalized columns just as $\mathbf{U}$). Consider the following remark.
\noindent \textbf{Remark 7. [Simplifying the Frobenius norm objective function]} \textit{Consider for simplicity $p = d$ and $\mathbf{\bar{\Sigma}}_d = \mathbf{\Sigma}_d$ in \eqref{eq:theoptimizationproblem_only_orthonormal} and see that $\| (\mathbf{U} - \mathbf{\bar{U}})\mathbf{\Sigma}_d \|_F^2 = 2\sum_{i=1}^d \sigma_i (1 - \mathbf{u}_i^T \mathbf{\bar{u}}_i) = 2\sum_{i=1}^d \sigma_i (1 - \cos (\theta_i))$,
where $\theta_i$ is the angle between column vectors $\mathbf{u}_i$ and $\mathbf{\bar{u}}_i$ (the columns of $\mathbf{U}$ and $\mathbf{\bar{U}}$, respectively), and $\sigma_i > 0$ are the diagonal elements of $\mathbf{\Sigma}_d$.}\hfill$\square$
Therefore, the proposed optimization objective function minimizes the weighted sum of the cosines of the angles between the original columns and their approximations. As such, low-rank or scaling issues cannot arise. The only potential problem is that different columns might have very different approximation errors (i.e., there could exist $i$ such that $\cos(\theta_i) \approx 1$ while there could be some $j$ for which $\cos(\theta_j) \ll 1$). This issue can be mitigated by increasing $g$ in \eqref{eq:approxu} or choosing carefully the indices $(i_k, j_k)$ where the proposed transformations operate.
Assume that we consider the operator norm as the approximation error, i.e., we want to minimize $\|\mathbf{U} - \mathbf{\bar{U}}\|_2$ in \eqref{eq:theoptimizationproblem_only_orthonormal}. We have the following two results.
\noindent \textbf{Remark 8. [The spectrum of the error matrix]} \textit{Consider $p = d$ and $\mathbf{\bar{\Sigma}}_d = \mathbf{\Sigma}_d = \mathbf{I}_d$ in \eqref{eq:theoptimizationproblem_only_orthonormal} then for any orthonormal $\mathbf{U}$ and $\mathbf{\bar{U}}$ the error matrix $\mathbf{U} - \mathbf{\bar{U}}$ is normal and has all its eigenvalues on a circle of radius one centered at $(1,0)$ in the complex plane. As such, we have that $\| \mathbf{U} - \mathbf{\bar{U}} \|_2 \leq 2$.}\hfill$\square$
\noindent \textbf{Theorem 5. [A bound on the operator norm of the error matrix]} \textit{Consider $p = d$ and $\mathbf{\bar{\Sigma}}_d = \mathbf{\Sigma}_d = \mathbf{I}_d$ in \eqref{eq:theoptimizationproblem_only_orthonormal} and that $\mathbf{u}^T_i \mathbf{\bar{u}}_i \geq 0$ for all $i=1,\dots,d$, then the operator norm of the error matrix obeys $\| \mathbf{U} - \mathbf{\bar{U}} \|_2 \leq 1-\epsilon_\text{min} + \sqrt{(d-1) (1-\epsilon_\text{min}^2) }$ where $\epsilon_\text{min} = \underset{i}{\min} \ \mathbf{u}^T_i \mathbf{\bar{u}}_i = \underset{i}{\min} \ \cos(\theta_i)$. The bound in Remark 8 is met when $\epsilon_{\min} \geq (d-2)/d$.}\hfill$\blacksquare$
Remark 8 describes the full spectrum of the error matrix. It is interesting to notice that the upper bound $\| \mathbf{U} - \mathbf{\bar{U}} \|_2 \leq 2$ coincides with the expectation result from \cite{CollinsMale2011}: as $d \to \infty$ if both $\mathbf{U}$ and $\mathbf{\bar{U}}$ are chosen uniformly at random with Haar measure then almost surely $\|\mathbf{U} + \mathbf{\bar{U}} \| \to 2$. Theorem 5 describes an upper bound on the operator norm of the error matrix. The bound is tight only for very high values of $\epsilon_{\min}$ and therefore is meant to give a qualitative measure of the approximation. As in Remark 7, the key quantity is $\cos(\theta_i)$ but now the bound depends on the worst approximation: while the Frobenius norm objective function minimizes the sum of the pairwise distances between the columns of $\mathbf{U}$ and $\mathbf{\bar{U}}$, when using the operator norm the approximation accuracy depends on the largest distance between the same pairwise columns. The assumption that $\mathbf{u}^T_i \mathbf{\bar{u}}_i \geq 0$ is not restrictive at all as we expect $\mathbf{U}^T \mathbf{\bar{U}}$ to be diagonally dominant with positive elements on the diagonal (see Remark 5 and the discussion after Remark 6). We note that in this context the Frobenius norm approach could be used to minimize a proxy for the operator norm. As in Remark 7 we have the choice of $\sigma_i$ we could update these values with each iteration of the proposed algorithm such that $\sigma_i^{\text{(new)}} \leftarrow \frac{\mathbf{u}_{i_{\max}}^T \mathbf{\bar{u}}_{i_{\max}} }{\mathbf{u}_i^T \mathbf{\bar{u}}_i}$ where $i_{\max} = \underset{i}{\arg \max}\ \mathbf{u}_i^T \mathbf{\bar{u}}_i$. This choice will encourage the algorithm to improve upon the worst pairwise column approximation (i.e., the highest $\mathbf{u}_i^T \mathbf{\bar{u}}_i$). This approach would be in the spirit of an iteratively reweighted least squares algorithm such as \cite{doi:10.1002/cpa.20303}.
Finally, the last measure of accuracy we consider is the angle between two subspaces as described by \cite{10.2307/2005662}. This value can be non-zero only when $p<d$, a case which is interesting for PCA projections and which we detail in the next section. Following \cite{doi:10.1137/S1064827500377332}, given $\mathbf{U}_p$ and $\mathbf{\bar{U}}_p$, whose columns span two $d$-dimensional subspaces of size $p$, we compute the angles between the two subspaces as
\begin{equation}
\beta_i = \arccos (\tau_i),\ i=1,\dots,p,
\end{equation}
where $0 \leq \tau_i \leq 1$ are the singular values of $\mathbf{U}_p^T \mathbf{\bar{U}}_p$ and where $0\leq \beta_1 \leq \dots \leq \beta_p \leq \pi/2$. We take the principal angle to be the largest angle above, i.e., $\beta = \beta_p = \arccos(\tau_p)$.
Finally, we would like to note that all the approximation errors we have previously discussed measure (in different ways) how well $\mathbf{U}_p^T \mathbf{\bar{U}}_p$ approaches the identity matrix.
\section{Application: fast PCA projections}
Consider a training set of $d$-dimensional points $\{\mathbf{x}_i \}_{i=1}^N$ and the $d \times N$ matrix $\mathbf{X} = \begin{bmatrix} \mathbf{x}_1 & \dots & \mathbf{x}_N \end{bmatrix}$. Given $1 \leq p < d$, PCA provides the optimal $p$-dimensional projection that minimally distorts, on average, the data points. The projection is given by the eigenvectors of the $p$ largest eigenvalues of $\mathbf{X} \mathbf{X}^T$ or, equivalently, the left singular vectors of the $p$ largest singular values of $\mathbf{X}$, that is
\begin{equation}
\mathbf{XX}^T \approx \mathbf{U}_p (\mathbf{\Sigma}_p \mathbf{\Sigma}_p^T ) \mathbf{U}_p^T \text{ and } \mathbf{X} \approx \mathbf{U}_p \mathbf{\Sigma}_p \mathbf{V}_p^T.
\label{eq:eigandsvd}
\end{equation}
Given the above decompositions we can approximate $\mathbf{X} $ by $\mathbf{\bar{X}} = \mathbf{\bar{U}} \mathbf{\bar{\Sigma}}_p \mathbf{V}_p^T$, i.e., we keep $\mathbf{V}_p$ but we modify the principal components and their singular values, such that we minimize the error given by
\begin{equation}
\| \mathbf{U}_p \mathbf{\Sigma}_p \mathbf{V}_p^T - \mathbf{\bar{U}} \mathbf{\bar{\Sigma}}_p \mathbf{V}_p^T \|_F^2 = \| \mathbf{U}_p \mathbf{\Sigma}_p - \mathbf{\bar{U}} \mathbf{\bar{\Sigma}}_p \|_F^2,
\label{eq:Visnotneeded}
\end{equation}
where $\mathbf{U}_p$ is $d \times p$, the diagonal matrix $\mathbf{\Sigma}_p$ is $p \times p$, $\mathbf{\bar{U}}$ is of size $d \times d$, $\mathbf{\bar{\Sigma}}_p$ is $d \times p$ and is zero except for its main $p \times p$ diagonal. In this paper, we work with $\mathbf{X}$, as opposed to $\mathbf{XX}^T$, to keep the relationship with $\mathbf{U}_p$ linear, rather than quadratic. In the context of applying our approach to PCA, we use our decomposition on the principal components $\mathbf{U}_p$ which we assume are already calculated together with the associated singular values $\mathbf{\Sigma}_p$ which we may use as weights in \eqref{eq:theoptimizationproblem_only_orthonormal}.\\
Note that in \eqref{eq:Visnotneeded}, $\mathbf{V}_p$, which has size $N$, is not necessary and that the two-step procedure is equivalent to computing the projections $\mathbf{\bar{U}}$ directly from $\mathbf{X}$. Also note that based on \eqref{eq:eigandsvd}, we could factor $\mathbf{X} \approx \mathbf{\bar{U}} \mathbf{\Sigma} \mathbf{\bar{V}}^T$ where $\mathbf{\bar{U}}$ and $\mathbf{\bar{V}}^T$ are approximations in $\mathcal{F}_{O(d \log d)}$ and $\mathcal{F}_{O(N \log N)}$, respectively. The difficulty here is the dependency of $\mathbf{\bar{V}}$ on $N \gg d$ which would require a large running time.\\% can be very large and therefore finding $\mathbf{\bar{V}}$ would require a large running time. It is for this reason that we choose to find $\mathbf{U}$ or $\mathbf{U}_p$ first by some well-established algorithms and then perform our approximation $\mathbf{\bar{U}}$ or $\mathbf{\bar{U}}_p$ depending only on dimensions $d$ and $p$.
With $\mathbf{\bar{U}}$ fixed, for $\mathbf{\bar{\Sigma}}_p$ we have several strategies: i) set it to the identity, i.e., flatten the spectrum; ii) keep it to the original singular values $\mathbf{\Sigma}_p$; or iii) continuously update it to minimize the Frobenius norm, i.e., get the new ``singular values'' that are optimal with the approximation $\mathbf{\bar{U}}$. The first approach favors the accurate reconstruction of all components while the other approach favors mostly the few leading components only (depending on the decay rate of the corresponding singular values).\\%The good choice depends on the application at hand.
\subsection{Comparison with the symmetric diagonalization by Givens rotations approach} Because of the locally optimal way the Givens transformations are chosen (see Theorem 1), our proposed factorization algorithm is computationally slower than the Jacobi diagonalization process which chooses the Givens rotations on indices $(i,j)$ corresponding to the largest off-diagonal entry of the covariance matrix. Furthermore, the Jacobi decomposition uses each rotation to zero the largest absolute value off-diagonal entry and because of this sub-optimal choice needs $O(d^2 \log d)$ Givens rotations \cite{ManySweeps} to complete de diagonalization (more than the $\frac{d(d-1)}{2}$ needed to fully represent the orthonormal group).\\
In all other aspects, our approach provides advantages over the Jacobi approach: i) we define a clear objective function that we locally optimize exactly; ii) it is known that the Jacobi process converges slowly when the number of rotations is low \cite{JacobiIsLInearEarlyOn}, which is exactly the practically relevant scenario we have, i.e., $g \ll d^2$; iii) with the same computational complexity, i.e., $g$ terms in the factorization, our proposed approach is always more accurate since we include as a special case the Givens rotations.
\subsection{Comparison with structured matrix factorization} Our approach requires the explicit availability of the orthonormal principal directions $\mathbf{U}_p$. Previous methods that factor using only Givens rotations are not applied directly on an orthogonal matrix. These methods rely on receiving as input a symmetric object (e.g., $\mathbf{XX}^T$) and then using a variant of Jacobi iterations for matrix diagonalization \cite{lemagoarou:hal-01416110} or multiresolution factorizations \cite{Kondor2014MMF} to find the good rotations that approximate the orthonormal eigenspace. Applying Givens transformations directly to $\mathbf{X}$ on the left, i.e., $\mathbf{G}_{ij} \mathbf{X}$, cannot lead to the computation of the PCA projections $\mathbf{U}$ but only to the polar decomposition. On the other hand, when applying Givens rotations on both sides of the covariance matrix, i.e., $\mathbf{G}_{ij} \mathbf{XX}^T \mathbf{G}_{ij}^T$, then the right eigenspace $\mathbf{V}$ cancels out in the product \eqref{eq:eigandsvd} and we are able to directly recover $\mathbf{U}$ (but we need $\mathbf{XX}^T$ explicitly). Finally, note that the diagonalization process approximates the full eigenspace $\mathbf{U}$ and cannot separate from the start the $p$ principal components $\mathbf{U}_p$ because they are solving the following problem
\begin{equation}
\underset{\mathbf{\bar{U}},\ \mathbf{\bar{\Lambda}}}{\text{minimize}}\ \| \mathbf{XX}^T - \mathbf{\bar{U}}\mathbf{\bar{\Lambda}} \mathbf{\bar{U}} \|_F^2 \text{ subject to } \mathbf{\bar{U}} \in \mathcal{F}_g.
\end{equation}
This formulation is useful to approximate the whole symmetric matrix $\mathbf{XX}^T$ (or $\mathbf{U}$), but not necessarily the $p$ principal eigenspace $\mathbf{U}_p$. To get these we would need to complete the diagonalization process, find the $p$ largest entries on the diagonal of $\mathbf{\bar{\Lambda}}$ and then work backward to identify the rotations that contributed diagonalizing those largest elements. This procedure would be prohibitively expensive. Previous work, e.g. \cite{Kondor2014MMF, Treelets}, deals with approximating $\mathbf{XX}^T$ rather than computing PCA. In the same line of work, the one-sided Jacobi algorithm for SVD \cite{doi:10.1137/05063920X} can also be applied.
\section{Experimental results}
Given the rather pessimistic guarantees, we tackle problems: how well does Algorithm 1 recover random orthogonal matrices and principal components such that we benefit from the computational gains but do not significantly impact the approximation/classification accuracy. Source code available
\footnote{https://github.com/cristian-rusu-research/fast-orthonormal-approximation}
\subsection{Synthetic experiments}
For fixed $d$ we generate random orthonormal matrices from the Haar measure \cite{HowToRandomUnitary}.
Figure \ref{fig:random} shows the representation error $(2d)^{-1} \| \mathbf{U} - \mathbf{\bar{U}} \|_F^2$ for the proposed method. The plot shows that allowing for the Givens transformations $\mathbf{\tilde{G}}_{i j}$ in \eqref{eq:theG} brings a 17\% relative benefit as compared to using only the Givens rotations while, for the same $g$, the computational complexity is the same. The circulant approximation performs worst because it has the lowest number of degrees of freedom, only $d$ (computationally, it is comparable with the Givens and proposed approaches for $g = 100$). Lastly, we can observe that the bound is very pessimistic, especially for these values of $g$. In Figure \ref{fig:random_evo_stages} (left) we show for fixed number of transformations $g$ the progress that the proposed algorithm makes with each iteration. It is interesting to observe that the initialization steps (first $g$ steps) significantly decrease the approximation error while the other step make only moderate improvements. This indicates that convergence is slow and might take a large number of iterations with little progress made by the latter steps. Also in Figure \ref{fig:random_evo_stages} (right) we show the number of stages in each transformation. A stage is a set of extended orthogonal Givens transformations that can be applied in parallel, i.e., they do not share any indices $(i_k, j_k)$ among them. This is important from an implementation perspective as parallel processing can be exploited to speedup the transformations.
\subsection{MNIST digits and fashion}
\begin{figure}[!tbp]
\centering
\begin{minipage}[t]{0.495\textwidth}
\includegraphics[trim = 18 5 30 22, clip, width=0.49\textwidth]{random_orthonormal_50-eps-converted-to.pdf}
\includegraphics[trim = 18 5 28 22, clip, width=0.49\textwidth]{random_orthonormal-eps-converted-to.pdf}
\caption{Average approximation errors and standard deviations over 100 realizations of random orthonormal matrices of size $d = 50$ (left) and $d = 100$ (right). For reference we show the bound developed in Theorem 3, the approximation accuracy of the circulant \cite{CirculantDRoperator} (Toeplitz performed just marginally better) and that of the using the factorization \eqref{eq:approxu} but allowing only Givens rotations. As expected, performance degrades with large $d$.}
\label{fig:random}
\end{minipage}
\hfill
\begin{minipage}[t]{0.495\textwidth}
\includegraphics[trim = 18 5 28 20, clip, width=0.49\textwidth]{random_orthogonal_with_iterations-eps-converted-to.pdf}
\includegraphics[trim = 18 5 30 20, clip, width=0.49\textwidth]{random_orthogonal_number_of_stages-eps-converted-to.pdf}
\caption{Left: for $d=100$ and $g \in \{ 332, 664, 1328 \}$ we show the evolution (mean and standard deviation) of the objective function with the number of iterations. In each case, the first $g$ iterations are the initialization process. Right: for $d \in \{50, 100\}$ we show the number of stages in each transformation we learrn with the proposed algorithm as a function of $g$. For both plots results are averaged over 100 realizations.}
\label{fig:random_evo_stages}
\end{minipage}
\end{figure}
\begin{figure}[!tbp]
\centering
\begin{minipage}[t]{0.495\textwidth}
\includegraphics[trim = 18 5 26 20, clip, width=0.49\textwidth]{mnist_digits_new-eps-converted-to.pdf}
\includegraphics[trim = 18 5 23 20, clip, width=0.49\textwidth]{mnist_fashion_new-eps-converted-to.pdf}
\caption{Classification accuracy obtained by the k-NN algorithm for the MNIST digits (left) and fashion (right) datasets as a function of the complexity of the proposed projections. Dimensionality reduction was done with $p = 15$ principal components. The bold text represents the speedup (FLOPS) compared to the cost of projecting with the unstructured, optimal, PCA components which takes $2pd$ operations. For the sparser JL the variable $s$ is the number of non-zeros in each column.}
\label{fig:mnistdetails}
\end{minipage}
\hfill
\begin{minipage}[t]{0.495\textwidth}
\includegraphics[trim = 18 5 28 20, clip, width=0.49\textwidth]{usps_with_iterations-eps-converted-to.pdf}
\includegraphics[trim = 18 5 30 20, clip, width=0.49\textwidth]{digits_number_of_stages-eps-converted-to.pdf}
\caption{For the USPS dataset ($d = 256$) we have, left: and $g \in \{ 60, 120, 240 \}$ we show the evolution (mean and standard deviation) of the objective function with the number of iterations -- again, in each case, the first $g$ iterrations are the initialization steps; right: the number of stages in each transformation created by the proposed method as a function of the number of extended orthogonal Givens transformations $g$.}
\label{fig:usps_evo_stages}
\end{minipage}
\end{figure}
We now turn to a classification problem. We use the MNIST digits
and fashion
datasets. The points have size $d = 400$ (we trimmed the bordering whitespace) and we have $N = 6\times 10^4$ training and $N_\text{test} = 10^4$ test points. In all cases, we use the k-nearest neighbors (k-NN) algorithm with $k = 10$, and we are looking to correctly classify the test points. Before k-NN we apply PCA and our proposed method. Results are shown in Figure \ref{fig:mnistdetails}. We deploy two variants of the proposed method: approximate the principal components as if they had equal importance and approximate the principal components while simultaneously also updating estimates of the singular values.\\
For comparison, we also show the sparse JL \cite{SparserJL}. In this case, the target dimension is $p \in \{15, 30\}$ while the random transformation of size $p \times d$ only has three non-zero entries per column. More non-zeros did not have any significant effect on the classification accuracy while increasing (doubling, in this case) the target dimension $p$ increases the classification accuracy by 10\%. The results reported in the plots are averages for 100 realizations and the standard deviation is below 1\%. Of course, the significant advantage of the JL approach over PCA is that no training is needed. The disadvantage is that if we choose greedily the target projection dimension, i.e., low $p$, the accuracy degrades significantly for JL. Results are identical for PCA when $p$ is $15$ of $30$.\\
Since we are dealing with image data, we also project using the discrete cosine transform (DCT). For the digits dataset, the performance is poor but, surprisingly, for the fashion dataset, this approach is competitive given the large speedup it reports (we used the sparse fast Fourier transform \cite{Indyk:2014:SSF:2634074.2634110} as we want only the largest $p$ components). We have also performed the projection by fast wavelet transforms with `haar' and several of the Daubechies `dbx' filters but the results were always similar to that of the DCT. For clarity of exposition, we did not add these results to the figures.\\
Our proposed methods report a clear trade-off between the classification accuracy and the numerical complexity of the projections. If we insist on an accuracy level close (within 1--2\%) of the full PCA then the speedup is only about x3. Reasonable accuracy is obtained for a speedup of x4--x5 after which the results degrade quickly. For $p=15$ better performance seems impossible via randomization. In this figure, the speedup is measured in terms of the number of operations (FLOPS).\\
Finally, in Figure \ref{fig:gt_vs_spca} (left) we compare our proposed methods against the sparse PCA on MNIST digits. sPCA performs exceptionally well in terms of the classification accuracy given the computational budget (a similar result is replicated for MNIST fashion). On other datasets where the principal components capture some global features (not local like in our example) we expect this performance to degrade. The training time of sPCA exceeds by 60\% the running time of PCA plus that of our method. We used the implementation of \citet{MINIMAXOPTIMALSPCA}.
\begin{table}[t]
\caption{Average classification accuracies for k-NN when using PCA projections and our approximations without spectrum update, for various datasets. We also show the speedup (FLOPS and actual running time) and the number of features selected in the calculations as a proportion out of the total $d$ (see also Remark 2). Results are averaged over 100 random realizations (train/test splits).}
\label{tb:sample-table}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccccr}
\toprule
DATASET & FULL PCA & \multicolumn{4}{c}{PROPOSED ALGORITHM} \\
& accuracy & accuracy & speedup(FLOPS) & speedup(TIME) & selection \\
\midrule
PENDIGITS & 95 $\pm$ 0.7& 91 $\pm$ 2.1 & $\times$1.6 & $\times$1.1 & 1 \\
ISOLET & 92 $\pm$ 0.4 & 90 $\pm$ 1.0 & $\times$12 & $\times$10.1 & 1 \\
USPS & 95 $\pm$ 1.2& 94 $\pm$ 0.8 & $\times$7.7 & $\times$4.7 & 0.64 \\
UCI & 90 $\pm$ 1.9& 87 $\pm$ 1.5 & $\times$2.5 & $\times$1.6 & 0.72 \\
20NEWS & 80 $\pm$ 3.1 & 77 $\pm$ 2.1 & $\times$3.1 & $\times$2.5 & 0.3 \\
EMNIST digits & 97 $\pm$ 2.5& 95 $\pm$ 1.8 & $\times$13 & $\times$11.3 & 0.37 \\
MNIST 8m & 96 $\pm$ 2.0& 94 $\pm$ 0.9 & $\times$15 & $\times$13.7 & 0.28 \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
Because of ideas in Remark 1, we perform only the finally useful calculations and further reduce the computational cost on average by one third (these are accounted for already in the numbers in the plots).
\subsection{Experiments on other datasets}
The 20-newsgroups
dataset consists of $18827$ articles from 20 newsgroups (approximately 1000 per class). The data set was tokenized using the rainbow package (www.cs.cmu.edu/~mccallum/bow/rainbow).
Each article is represented by a word-count vector for the $d = 2\times10^4$ common words in the vocabulary. For this dataset we have $N_\text{test} = 5648$, $p = 200$, and as shown in Figure \ref{fig:gt_vs_spca} (right) in this case we outperform sparse PCA.\\
We also apply our algorithm to several other popular dataset from the literature: PENDIGITS (www.ics.uci.edu/$\sim$mlearn/MLRepository.html)
with 10 classes and $d = 16$, $N = 7494$, $N_\text{test} = 3498$, $p = 4$; ISOLET (archive.ics.uci.edu/ml/datasets/isolet)
with 26 classes and $d = 617$, $N = 6238$, $N_\text{test} = 1559$, $p = 150$; USPS (github.com/darshanbagul/USPS\_Digit\_Classification)
with 10 classes and $d = 256$, $N = 7291$, $N_\text{test} = 2007$, $p = 12$; UCI (ftp.ics.uci.edu/pub/machine-learning-databases/optdigits)
with 10 classes and $d = 64$, $N = 3823$, $N_\text{test} = 1797$, $p = 6$; EMNIST digits (nist.gov/itl/iad/image-group/emnist-dataset)
with 10 classes and $d = 784$, $N = 24\times10^4$, $N_\text{test} = 4\times10^4$, $p = 15$; MNIST 8m (leon.bottou.org/papers/loosli-canu-bottou-2006)
with 10 classes and $d = 784$, $N = 6.4 \times 10^6$, $N_\text{test} = 1.7 \times 10^6$, $p = 15$. All the results are shown in Table \ref{tb:sample-table}. Here we provide two measures for the speedup: the FLOPS (number of arithmetic operations, additions and multiplications) and the actual running time (in seconds). For the time speedup, we first computed $\mathbf{\bar{U}}$ and then implemented the matrix-vector multiplication and compared it to the generic matrix-vector multiplication with $\mathbf{U}$, both compiled in C using the gcc compiler and --O3 flag. Once computed by the proposed algorithm, the transformation could also be hardcoded and a further speedup improvement could be achieved. The running time speedups are slightly below the FLOPS speedups due to overhead in modern CPUs (fetching data, register loading etc.). Aside the computational benefits of the proposed transformations we note that while complex instruction set computing machines are closing the speedup gap in terms of running time the price is higher energy consumption and more complex execution pipelines/circuitry.\\
\begin{figure}[!tbp]
\centering
\includegraphics[trim = 110 0 132 12, clip, width=\textwidth]{fig_all_other_norms_new-eps-converted-to}
\caption{We show for three datasets used in Section 5.3, as a function of the number of Givens transformations $g$ in $\mathbf{\bar{U}}_p$ that goes like $g = \alpha p \log_2 d$, the Frobenius (left most) and operator norms errors (second from the left), angle between subspaces distance measured in radians (third from the left) and correlations (right most shows the average correlations but also minimum and maximum values). The results are averaged over 10 realizations (random training/testing data samples) but the variance is always below 0.05.}
\label{fig:all_other_norms}
\end{figure}
In Figure \ref{fig:usps_evo_stages}, for the USPS dataset, we provide insights into the behavior of the proposed algorithm with each iteration and the number of stages in each transformation that we construct. Results are similar to the ones shown for the random orthogonal approximation. We observe again that the initialization step is very efficient in reducing the approximation error (especially when compared to the iterative process that follows) and that the number of stages that have to be applied sequentially can further improve the running time when implementing the proposed transformations.\\
Finally, one of the most famous applications of PCA is in the field of computer vision for the problem of human face recognition. The eigenfaces \cite{Sirovich:87} approach was used successfully for face recognition and classification tasks. Here, we want to reproduce the famous eigenfaces by using the proposed methods. The original eigenfaces and their approximations (sparse eigenfaces \cite{SparseFaces}), with different $g$ and therefore different levels of detail, are shown in Figure \ref{fig:eigenfaces}.
\begin{figure}[!tbp]
\centering
\begin{minipage}[t]{0.495\textwidth}
\includegraphics[trim = 18 5 28 20, clip, width=0.49\textwidth]{gt_vs_spca_new-eps-converted-to.pdf}
\includegraphics[trim = 18 5 30 20, clip, width=0.49\textwidth]{gt_vs_spca_20news_new-eps-converted-to.pdf}
\caption{Classification accuracy versus number of operations on the MNIST digits (left) and 20NEWS (right) datasets for our proposed methods and the sparse PCA method \cite{MINIMAXOPTIMALSPCA, SparsePCA}.}
\label{fig:gt_vs_spca}
\end{minipage}
\hfill
\begin{minipage}[t]{0.495\textwidth}
\centering
\includegraphics[trim = 18 15 30 20, clip, width=0.75\textwidth]{eigenfaces_new-eps-converted-to.pdf}
\caption{A few eigenfaces obtained by the optimal PCA (top) and by our proposed method with $g = 3059$ and $g = 1020$ (middle and bottom, respectively). The projection speedup (FLOPS) is x4.1 and x13.9, respectively.}
\label{fig:eigenfaces}
\end{minipage}
\end{figure}
\subsection{Results on other approximation errors}
In Section 3.1 we have explored several other ways to measure the approximation accuracy of the proposed algorithm. We note that in the proposed method we keep the Frobenius norm objective function but we also measure the operator norm, the angle between subspaces distance and the average/minimum/maximum correlations. All results for three datasets are shown in Figure \ref{fig:all_other_norms}, the choices of parameters are the same as in Section 5.3. The first observation is that the approximation errors increase with larger $d$ (as already clear from Remark 4 and Theorem 3), i.e, best results are obtained for UCI ($d = 64$) and the worst for MNIST ($d = 400$). Second, note that improving the Frobenius norm error we also improve all the other error measures. Thirdly, note the similar behavior between the operator norm and subspace distance errors. It is interesting to observe that the two plots seem to suggest experimentally that $\arccos(\sigma_{\min}( \mathbf{U}_p^T \mathbf{\bar{U}}_p )) \approx 1-\sigma_{\max}( \mathbf{I}_p - \mathbf{U}_p^T \mathbf{\bar{U}}_p )$ which hold over a large array of number of Givens transformations $g$. Lastly, the two right-most plots highlight the connection between the operator norm optimization and the maximization of the lowest coherence between columns and their approximations (as per Theorem 5).\\
Depending on the application at hand we can choose how to measure the approximation error.
\subsection{Details of the implementation}
\begin{table}[t]
\caption{Speed-up achieved with the proposed approximations $\mathbf{\bar{U}}_p$ against a vanilla C implementation of dense matrix-vector multiplication and BLAS Level 2 functions, in both cases with no parallelism.}
\label{}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccccr}
\toprule
DATASET: & ISOLET & USPS & EMNIST digits & MNIST 8m \\
\midrule
C language (TIME) & $\times$10.1 & $\times$4.7 & $\times$11.3 & $\times$13.7 & \\
BLAS (TIME) & $\times$4.8 & $\times$2.5 & $\times$5.7 & $\times$6.8 & \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
The source code attached to this paper is written for the Matlab environment. Besides the implementation of Algorithm 1, we also provide the code to efficiently perform the matrix-vector multiplication with the proposed $\mathbf{\bar{U}}$ and $\mathbf{\bar{U}}_p$, respectively (also taking into account Remark 3). Still, due to the characteristics of Matlab, our implementation is not faster than the dense matrix-vector multiplication ``*''. To provide an appropriate comparison, we implement the matrix-vector multiplication with our structures in the compiled lower-level programming language C. We provide comparisons of this implementation against two scenarios: a vanilla matrix-vector multiplication (using the appropriate ordering of the loops to exploit locality) and the Basic Linear Algebra Subprograms (BLAS) Level 2 routines for matrix-vector multiplication (SGEMV). We use the single thread variant of BLAS as for the dimensions $d$ we consider the overhead of the parallel implementation is significant.\\
Applying the proposed transformations on batch features, i.e., matrix-matrix multiplications, would require careful application of the proposed transformations \eqref{eq:theG} to fully use the computing architecture and maximize performance.
\section{Conclusions}
This paper proposes a new matrix factorization algorithm for orthogonal matrices based on a class of structured matrices called extended orthogonal Givens transformations. We show that there is a trade-off between the computational complexity and accuracy of the approximations created by our approach. We apply our method to the approximation of a fixed number of principal components and show that, with a minor decrease in performance, we can reach significant computational benefits.\\
Future research directions include strengthening the theoretical guarantees since they are way above what we observe experimentally. Further, it would be of interest to improve the complexity of the proposed algorithm either by a parallel implementation or using randomization (e.g. computing a random subset of the $O(d^2)$ scores). As an immediate application, it would be interesting to apply our decomposition to the recently proposed unitary recurrent neural networks \cite{URNN}.
\section*{Acknowledgment}
This material is based upon work supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216, and the Italian Institute of Technology. Part of this work has been carried out at the Machine Learning Genoa (MaLGa) center, Universita di Genova (IT). Lorenzo Rosasco acknowledges the financial support of the European Research Council (grant SLING 819789), the AFOSR projects FA9550-17-1-0390 and BAA-AFRL-AFOSR-2016-0007 (European Office of Aerospace Research and Development). Cristian Rusu acknowledges support by the Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-III-P1-1.1-TE-2019-1843, within PNCDI III.
{\small
\bibliographystyle{plainnat}
| {'timestamp': '2021-03-24T01:29:13', 'yymm': '1907', 'arxiv_id': '1907.08697', 'language': 'en', 'url': 'https://arxiv.org/abs/1907.08697'} |
\section{Introduction} \label{sec:intro}
Herbig Ae (HAe) stars are a class of intermediate mass (\SIrange[range-units = single, range-phrase = --]{1.5}{5}{\solarmass}) pre-main-sequence stellar objects characterized by strong excess emission in the near-infrared (NIR) and millimeter wavelengths, typically peaking around \SI{3}{\micro\meter}. Due to their young age and high luminosity, HAe stars are the ideal targets for observing accretion and planet formation processes {\em in situ}. The bulk of the NIR excess originates from within the inner few astronomical units surrounding these stars. Since even the closest HAe systems are located at distances exceeding \SI{100}{\parsec}, the milliarcsecond resolution required to observe features on \si{\au} scales is far beyond the capabilities of even the largest solitary optical telescopes. However, advancements made in long-baseline optical interferometry over the course of the last two decades have made it possible to probe these spatial scales.
Early HAe investigations of \cite{Hillenbrand_1992}, limited only to spectral energy distribution (SED) measurements, were able to reproduce the photometric NIR excess measurements successfully by assuming a flat, optically-thick accretion disk that extends down to a few stellar radii. This theoretical picture, however, was \emph{not confirmed} by the early infrared interferometry observations by \citet{MillanGabet_1999} of \object{AB~Aurigae} with the Infrared Optical Telescope Array, where they found NIR disk sizes many times larger than predicted by optically-thick, geometrically thin disk models. \citet{Natta_2001} and \citet{Dullemond_2001} put forward a new inner disk model where the star is surrounded by an optically-thin cavity with a ``puffed-up'' inner rim wall located at the radius where the equilibrium temperature is equal to the dust's characteristic sublimation temperature. Additional interferometer measurements by \citet{MillanGabet_2001} along with Keck aperture masking observations of \object{LkH$\alpha$~101} \citep{Tuthill_2001} supported the idea that the bulk of the NIR disk emission comes from a ring located at the dust sublimation radius. With these results, \citet{Monnier_2002} published the first ``size-luminosity diagram'' finding dust sublimation temperatures between \SIrange[range-units = single, range-phrase = --]{1500}{2000}{\kelvin}, broadly consistent across the sample. The similarity between this characteristic temperature at $\sim$\SI{1800}{\kelvin} and known silicate sublimation temperatures indicated that the NIR emission is tracing a silicate dust sublimation rim. \citet{Dullemond_2010} \revtwo{presents} a comprehensive overview of the theoretical and observational picture for the inner disks of young stellar objects (YSOs) that is still largely up-to-date.
The first sub-milliarcsecond interferometric observations (\citealt{Tannirkulam_2008}) of HAe stars \object{HD~163296} and \object{AB Aurigae} unexpectedly discovered that a significant fraction of the flux responsible for the NIR excess originated from well-within the supposed dust sublimation radius. Moreover, they found it was unlikely that the distribution had a sharp edge
with an illuminated inner rim. Subsequent observations (for example, see \citealt{Benisty_2010} and \citealt{Lazareff_2017}) have confirmed this basic result across a large number of HAe stars, in contradiction to the prevailing theory of HAe inner disk structure. Several phenomena have been proposed to explain this large amount of inner emission, including emission from refractory dust grains (\citealt{Benisty_2010}) and optically thick free-free/bound-free emission from a hot accreting gas (\citealt{Kraus_2008}; \citealt{Tannirkulam_2008}). Further multi-wavelength measurements of more HAe objects are required to determine the relevant mechanisms dominating the emission.
In this paper, we conduct a multi-wavelength interferometric study in H and K bands for two HAe objects: \object{HD~163296} (MWC~275) and \object{HD~190073} (V1295~Aql). A list of basic stellar properties and NIR photometry from the literature for these two objects is reproduced in Table \ref{tab:basicprop}\rev{, where we adopt photosphere temperatures based solely on the measured spectral type, as literature temperature estimates range on the order of a few \revtwo{hundred} Kelvin for both objects}. Our \rev{interferometric} data are collected on longer baselines with more complete $(u, v)$ coverage than is currently available for HAe stars in the literature. The higher angular resolution better constrain the orientation and radial distribution of the material producing the emission than previous work while the multi-wavelength data probe the mechanism producing the mysterious interior emission. We analyze the measurements by fitting simple geometrical models to the brightness distribution; our goal in this paper is to characterize the sizes and general profiles of these stars rather than try to model small scale details to which our data is not adequately sensitive. We then validate our interferometric model fitting by comparing to the object SEDs and compute new luminosity, mass, and age estimates of these YSOs. Finally, we speculate on the physical origin of the interior NIR excess emission.
\begin{deluxetable}{lcc}
\tablecaption{Literature Stellar Properties and Photometry of Target Sources}
\tablehead{\colhead{Property} & \colhead{\object{HD~163296}} & \colhead{\object{HD~190073}}}
\startdata
$\alpha$ (J2000) & \SI{17}{\hourangle}\,\SI{56}{\minuteangle}\,\SI{21.29}{\secondangle}
& \SI{20}{\hourangle}\,\SI{03}{\minuteangle}\,\SI{02.51}{\secondangle} \\
$\delta$ (J2000) & \ang[angle-symbol-over-decimal]{-21; 57; 21.87}
& \ang[retain-explicit-plus,angle-symbol-over-decimal]{+05; 44; 16.66} \\[4pt]
Spectral Type\tablenotemark{a} & A1 Vepv & A2 IVev \\
$T_\text{eff}$\tablenotemark{b} & \rev{\SI{9230}{\kelvin}} & \rev{\SI{8970}{\kelvin}} \\
Distance\tablenotemark{c} & \SI{101.5 \pm 1.2}{\parsec} & \SI{891 \pm 53}{\parsec} \\
Luminosity\tablenotemark{d} & \SI{28}{\solarluminosity}
& \SI{780}{\solarluminosity} \\[4pt]
$V$ mag\tablenotemark{e} & \num{6.84 \pm 0.06} & \num{7.79 \pm 0.06} \\
$H$ mag\tablenotemark{e} & \num{5.48 \pm 0.07} & \num{6.61 \pm 0.07} \\
$K$ mag\tablenotemark{e} & \num{4.59 \pm 0.08} & \num{5.75 \pm 0.08} \\
\enddata
\tablerefs{\textsuperscript{a}\cite{Mora_2001};\rev{ \textsuperscript{b}\cite{Kenyon_1995}, based on reported spectral type;} \textsuperscript{c}\cite{Gaia_DR2}; \textsuperscript{d}\cite{Monnier_2006}, rescaled to Gaia DR2 distances; \textsuperscript{e}\cite{Tannirkulam_2008}}
\label{tab:basicprop}
\end{deluxetable}
\section{Observations and Data Reduction} \label{sec:obsdata}
In this section, we describe and present the most complete published sample of broadband H and K band long-baseline interferometric data collected of the HAe stars \object{HD~163296} and \object{HD~190073} by combining interferometer observations conducted at various facilities. We will provide a copy of the calibrated data used in our modeling using the OI-FITS format \citep{Pauls_2005} and uploaded to the Optical interferometry Database (OiDb) \citep{Haubois_2014} developed by the Jean-Marie Mariotti Center (Grenoble, France).
\subsection{CHARA Interferometric Data} \label{sec:chara}
New observations were conducted at the Center for High Angular Resolution Astronomy (CHARA) interferometer \citep{ten_Brummelaar_2005}, located on Mt.\@ Wilson, California, with baselines of up to \num{330} meters and at various orientations. These data together yield a maximum nominal angular resolution of $\lambda/2B = \num{0.51}$ milliarcseconds (\si{\milliarcsecond}) in H band and \SI{0.67}{\milliarcsecond} in K band, where $\lambda$ is the filter central wavelength and $B$ is the longest baseline length measured.
We used the CHARA 2--telescope ``Classic'' beam-combiner instrument (\citealt{ten_Brummelaar_2013}) to collect broad H-band ($\lambda_{\text{eff}} = \SI{1.673}{\micro\meter}$, $\Delta\lambda = \SI{0.304}{\micro\meter}$) and K-band ($\lambda_{\text{eff}} = \SI{2.133}{\micro\meter}$, $\Delta\lambda = \SI{0.350}{\micro\meter}$) squared visibility ($\mathcal{V}^2$) measurements of our target stars between July 2004 and July 2010. We supplement the eight nights of HD~163296 Classic observations previously published in \cite{Tannirkulam_2008} with 8 additional nights of data. We also present sixteen new, formerly unpublished nights of observations of HD~190073. A summary of the Classic measurements is given in Table~\ref{tab:classic}.
\begin{deluxetable}{rrlcclh}
\tablecaption{CHARA/Classic Observations}
\tablehead{\colhead{Target} & \colhead{Date} & \colhead{Configuration} & \colhead{Band} & \colhead{n.\,$V^2$} & \colhead{Calibrator(s)} & \nocolhead{Notes}}
\startdata
HD~163296 & 2004-07-09 & S1-W1 & K & 2 & \object{HD 164031}, \object{HD 163955} & \nodata \\
& 2005-07-20 & W1-W2 & K & 2 & \object{HD 164031}, \object{HD 163955} & obslog says 3 sets \\
& 2005-07-22 & W1-W2 & K & 3 & \object{HD 164031}, \object{HD 163955} & \nodata \\
& 2005-07-26 & W1-W2 & K & 4 & \object{HD 164031}, \object{HD 163955} & \nodata \\
& 2006-06-22 & S2-W2 & K & 5 & \object{HD 164031}, \object{HD 163955} & obslog says 3 sets \\
& 2006-06-23 & W1-E1 & K & 2 & \object{HD 163955}, \object{HD 164031} & obslog says 1 set \\
& & S2-W2 & K & 1 & \object{HD 163955}, \object{HD 164031} & \nodata \\
& 2006-08-22 & S2-E2 & K & 1 & \object{HD 166295} & \nodata \\
& 2006-08-23 & S2-E2 & K & 3 & \object{HD 166295}, \object{HD 164031}, \object{HD 163955} & \nodata \\
& 2007-06-15 & S2-W1 & K & 5 & \object{HD 161023}, \object{HD 162255} & \nodata \\
& 2007-06-16 & S2-W1 & K & 1 & \object{HD 164031} & \nodata \\
& 2007-06-17 & S2-W1 & K & 5 & \object{HD 156365}, \object{HD 164031} & \nodata \\
& 2008-06-12 & W1-W2 & K & 5 & \object{HD 164031}, \object{HD 156365} & \nodata \\
& 2008-06-15 & W1-W2 & H & 3 & \object{HD 161023}, \object{HD 162255} & \nodata \\
& & S2-W1 & H & 2 & \object{HD 161023}, \object{HD 162255} & \nodata \\
& 2008-06-16 & S2-W1 & H & 1 & \object{HD 161023}, \object{HD 162255} & \nodata \\
& 2009-06-24 & S2-E2 & H & 3 & \object{HD 156365}, \object{HD 164031} & \nodata \\
& 2009-06-25 & S2-E2 & H & 2 & \object{HD 156365}, \object{HD 164031} & obslog says 3 sets \\ \hline
HD~190073 & 2007-06-08 & W1-W2 & K & 6 & \object{HD 187923}, \object{HD 193556} & \nodata \\
& 2007-06-09 & W1-W2 & K & 6 & \object{HD 187923}, \object{HD 193556} & \nodata \\
& 2008-06-14 & W1-W2 & H & 3 & \object{HD 187923}, \object{HD 193556} & \nodata \\
& 2008-06-17 & W1-S2 & K & 1 & \object{HD 187923}, \object{HD 193556} & clouds, large count fluctuations \\
& & W1-E2 & K & 1 & \object{HD 187923} & obslog says not reduced \\
& 2008-06-18 & W1-S2 & K & 2 & \object{HD 187923}, \object{HD 193556} & \nodata\\
& 2009-06-19 & W1-S2 & K & 2 & \object{HD 193556} & \nodata \\
& 2009-06-20 & W1-S2 & K & 2 & \object{HD 187923}, \object{HD 193556} & obslog says 1 set \\
& 2009-06-22 & W2-E2 & K & 5 & \object{HD 183303} & obslog says 4 sets \\
& & E1-S1 & K & 1 & \object{HD 183303} & not in obslog \\
& 2009-06-24 & E2-S2 & H & 3 & \object{HD 183303}, \object{HD 183936} & obslog says 2 sets \\
& 2009-06-25 & E2-S2 & K & 5 & \object{HD 183303} & \nodata \\
& 2009-10-31 & W2-S2 & K & 5 & \object{HD 183936} & \nodata \\
& 2010-06-14 & W2-S1 & K & 2 & \object{HD 150366}, \object{HD 188385} & \nodata \\
& & E2-S1 & K & 3 & \object{HD 187897}, \object{HD 189509} & \nodata \\
& 2010-06-15 & W1-E2 & K & 1 & \object{HD 189509} & \nodata \\
& 2010-06-16 & E1-S1 & K & 5 & \object{HD 177305}, \object{HD 177332}, \object{HD 188385}, \object{HD 189509} & obslog says 4 sets \\
& 2010-06-17 & W1-S1 & K & 5 & \object{HD 141597}, \object{HD 188385}, \object{HD 206660} & \nodata \\
& 2010-07-22 & S1-E1 & K & 3 & \object{HD 188385}, \object{HD 189509} & \nodata \\
\enddata
\tablenotetext{}{The ``n.\@ $V^2$'' column indicates the number of squared visibility measurements collected per the particular observation.}
\label{tab:classic}
\end{deluxetable}
Squared visibility and closure phase measurements were also obtained with the 3--telescope CLIMB instrument (\citealt{ten_Brummelaar_2013}) between July 2010 and June 2014. Broad H-band ($\lambda_{\text{eff}} = \SI{1.673}{\micro\meter}$, $\Delta\lambda = \SI{0.274}{\micro\meter}$) and K-band ($\lambda_{\text{eff}} = \SI{2.133}{\micro\meter}$, $\Delta\lambda = \SI{0.350}{\micro\meter}$) observations were collected over the span of 22 nights for HD~163296 and 29 nights for HD~190073; a summary of these measurements is given in Table~\ref{tab:climb}.
\begin{deluxetable}{rrlccclh}
\tablecaption{CHARA/CLIMB Observations}
\tablehead{\colhead{Target} & \colhead{Date} & \colhead{Configuration} & \colhead{Band} & \colhead{n.\,$V^2$} & \colhead{n.\,$T^3$} & \colhead{Calibrator(s)} & \nocolhead{Notes}}
\startdata
HD~163296 & 2010-07-11 & S1-S2-W2 & K & 11 & 1 & \object{HD 164031} & \nodata \\
& 2010-07-12 & W1-E1-E2 & K & 7 & 1 & \object{HD 164031} & \nodata \\
& 2010-07-13 & E1-E2 & K & 4 & 0 & \object{HD 156365} & \nodata \\
& 2010-07-14 & W1-E1-E2 & K & 7 & 1 & \object{HD 159743}, \object{HD 164104} & \nodata \\
& 2010-07-15 & W1-E1-E2 & K & 14 & 2 & \object{HD 159743}, \object{HD 170657} & \nodata \\
& 2011-06-15 & W1-W2-E1 & K & 14 & 2 & \object{HD 162255}, \object{HD 170680} & \nodata \\
& 2011-06-20 & W1-W2-E1 & K & 14 & 2 & \object{HD 162255}, \object{HD 163955} & \nodata \\
& 2011-06-23 & S1-W1-W2 & K & 28 & 4 & \object{HD 162255}, \object{HD 163955} & \nodata \\
& 2011-06-24 & S1-W2-E2 & K & 35 & 5 & \object{HD 162255}, \object{HD 163955} & \nodata \\
& 2011-06-27 & S1-W1-E2 & K & 21 & 3 & \object{HD 163955} & \nodata \\
& 2011-06-30 & W2-E2 & K & 3 & 0 & \object{HD 163955} & obslog says this is S2-W2 \\
& 2011-07-01 & S2-W2-E2 & K & 43 & 5 & \object{HD 165920}, \object{HD 163955} & \nodata \\
& 2012-06-27 & W1-W2-E2 & K & 28 & 4 & \object{HD 163955} & \nodata \\
& 2012-06-28 & S1-E1-E2 & K & 35 & 5 & \object{HD 159743}, \object{HD 170622}, \object{HD 163955} & \nodata \\
& 2012-06-29 & W1-E1-E2 & K & 42 & 6 & \object{HD 159743}, \object{HD 163955} & \nodata \\
& 2012-06-30 & S1-S2-E1 & K & 14 & 2 & \object{HD 163955} & \nodata \\
& 2012-07-01 & S1-S2-W1 & K & 14 & 2 & \object{HD 159743}, \object{HD 163955} & \nodata \\
& 2013-06-12 & S1-W1-W2 & K & 21 & 3 & \object{HD 163955}, \object{HD 170622} & \nodata \\
& 2013-06-16 & W1-W2-E2 & K & 14 & 2 & \object{HD 170657} & \nodata \\
& 2014-06-09 & W1-E1-E2 & H & 23 & 3 & \object{HD 163955}, \object{HD 164031} & \nodata \\
& 2014-06-11 & S2-E2 & H & 2 & 0 & \object{HD 150366} & \nodata \\
& 2014-06-12 & S2-W1-E2 & H & 7 & 1 & \object{HD 164031} & \nodata \\ \hline
HD~190073 & 2010-09-29 & S2-W2-E2 & K & 17 & 2 & \object{HD 183936} & \nodata \\
& 2011-06-15 & W1-W2-E1 & K & 21 & 3 & \object{HD 188385}, \object{HD 189509} & \nodata \\
& 2011-06-18 & W1-W2-E1 & K & 23 & 2 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2011-06-20 & W1-W2-E1 & K & 27 & 3 & \object{HD 188385}, \object{HD 189509} & \nodata \\
& 2011-06-23 & S1-W1-W2 & K & 35 & 5 & \object{HD 189509}, \object{HD 188385}, \object{HD 190498} & \nodata \\
& 2011-06-24 & S1-W2-E2 & K & 49 & 7 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2011-06-25 & S1-W2-E2 & K & 7 & 1 & \object{HD 189509} & \nodata \\
& 2011-06-26 & S1-W1-W2 & K & 28 & 4 & \object{HD 188385}, \object{HD 189509}, \object{HD 190498} & \nodata \\
& 2011-06-27 & S1-W1-E2 & K & 21 & 3 & \object{HD 189509}, \object{HD 188385}, \object{HD 190498} & \nodata \\
& 2011-06-28 & S1-W2-E1 & K & 21 & 3 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2011-07-01 & S2-W2 & K & 2 & 0 & \object{HD 188385}, \object{HD 189509} & \nodata \\
& 2011-08-03 & S1-W1-E2 & K & 9 & 1 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2012-06-27 & W1-W2-E2 & K & 35 & 5 & \object{HD 188385} & \nodata \\
& 2012-06-28 & S1-E1-E2 & K & 42 & 6 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2012-06-29 & W1-E1-E2 & K & 35 & 5 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2012-06-30 & S1-S2-E1 & K & 49 & 7 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2012-07-01 & S1-S2-W1 & K & 21 & 3 & \object{HD 189509}, \object{HD 188385} & \nodata \\
& 2013-06-09 & W1-W2 & K & 3 & 0 & \object{HD 188385} & \nodata \\
& 2013-06-10 & W1-W2 & K & 21 & 0 & \object{HD 188385}, \object{HD 190498} & \nodata \\
& 2013-06-12 & S1-W1-W2 & K & 7 & 1 & \object{HD 188385} & \nodata \\
& 2013-06-16 & W1-W2-E2 & K & 14 & 2 & \object{HD 188385} & \nodata \\
& 2014-05-29 & W2-E2 & H & 12 & 0 & \object{HD 188385}, \object{HD 189712} & \nodata \\
& 2014-05-30 & S2-W2-E2 & H & 24 & 12 & \object{HD 188385}, \object{HD 189712} & \nodata \\
& 2014-06-02 & S2-W1-E1 & H & 11 & 1 & \object{HD 188385}, \object{HD 189712} & \nodata \\
& 2014-06-03 & S2-W1-E1 & H & 16 & 2 & \object{HD 188385}, \object{HD 189712} & \nodata \\
& 2014-06-04 & S2-W1-E2 & H & 23 & 12 & \object{HD 188385}, \object{HD 189712} & \nodata \\
& 2014-06-05 & S2-W1-W2 & H & 7 & 4 & \object{HD 188385}, \object{HD 189712} & \nodata \\
& 2014-06-07 & S2-W1-W2 & H & 18 & 8 & \object{HD 188385} & \nodata \\
& 2014-06-11 & E1-E2 & H & 4 & 0 & \object{HD 188385} & \nodata \\
\enddata
\tablenotetext{}{The ``n.\@ $V^2$'' column indicates the number of squared visibility measurements collected per the particular observation and the ``n.\@ $T^3$'' column indicates the number of closure phase measurements collected.}
\label{tab:climb}
\end{deluxetable}
The data from both Classic and CLIMB instruments were reduced using a pipeline developed in-house at the University of Michigan (available from the authors upon request). The pipeline is designed specifically for robust selection of fringes of faint/low-visibility objects such as YSOs. Squared visibilities are estimated using the power spectrum method and spectral windows are user-selectable based on the changing seeing conditions. For CLIMB, the closure phases were averaged using short, coherent time blocks across the fringe envelopes weighted by the magnitude of the triple-amplitude. The transfer function was monitored with regular observations of calibrators stars chosen to be single and nearly unresolved by the interferometer, selected with the JMMC \texttt{SearchCal} tool \rev{(\citealt{Chelli_2016})}. The pipeline was validated end-to-end using the known binary \object{WR~140} (\citealt{Monnier_2011}), including the sign of the closure phase.
Due to occasional sudden changes in sky or instrument conditions, the estimated calibration transfer function can be incorrect. While we are working on objective, automated procedures based on signal-blind diagnostics, we currently rely on detecting outliers based on sudden, unphysical changes in our data as a function of time or baseline coverage. While subjective, our outlier removal eliminated only \SI{3.6}{\percent} of the CHARA/Classic data and \SI{2.1}{\percent} of the CHARA/CLIMB data, and should not strongly affect our fitting results. Lastly, a minimum error floor is applied to the data from the pipeline to account for expected errors in our transfer function estimation mentioned above. Based on earlier studies \citep{Tannirkulam_2008,Monnier_2011} and a new study from CLIMB using WR~140, we apply a minimum squared visibility errors of \SI{10}{\percent} relative error or additive \num{0.02} error, whichever is largest. Generally for high visibilities, the relative error floor dominates the error budget while the additive error comes into play for long-baseline low-visibility points.
\subsection{VLTI Interferometric Data}
We augment our CHARA observations with squared visibility and closure phase measurements obtained at the Very Large Telescope Interferometer (VLTI) at Cerro Paranal, Chile. These data greatly increased the $(u, v)$ coverage of our data set at intermediate-length baselines of up to \num{128} meters.
We made use of archival data of HD~163296 and HD~190073 collected with \rev{the} AMBER instrument (\citealt{Petrov_2007}) spanning multiple nights in 2007-08 and in 2012, respectively (see Table \ref{tab:amber}). These
measurements \rev{were recorded} in low spectral resolution mode ($R = 30$) \rev{spanning H- and K-bands}. The data were reduced using \rev{\texttt{amdlib} (v3.0.9; http://www.jmmc.fr/data\_processing\_amber.htm)} software where we derived
K-band visibilities
and closure phases. During the data reduction, only data with the \SI{20}{\percent} best fringe \rev{signal-to-noise ratio (}SNR\rev{)} were taken into account \citep{Tatulli_2007}. Because the observations were conducted without a fringe tracker, the data showed a typical spread of the optical path difference (OPD) between \SIrange[range-units = single, range-phrase = --]{20}{60}{\micro\meter}. To minimize the effect of the OPD spread and the resulting influence of atmospheric degradation on the calibrated visibilities, we used the histogram equalization method introduced by \cite{Kreplin_2012}.
Even with these corrections, the majority of the \rev{AMBER data spanning} H-band
\rev{remained of low quality due to poor calibration.}
Since we had access to
\rev{superior H-band} PIONIER data for both targets \rev{(see next paragraph)}, we excluded the AMBER H-band data \rev{in the present analysis},
\rev{and kept} only the K-band data
\rev{here}. To maintain congruity with the CHARA observations, AMBER data with spectral measurements in the range of \SIrange[range-units = single, range-phrase = --]{1.995}{2.385}{\micro\meter} were collapsed to a single point by averaging the individual squared visibility values to simulate a broadband measurement (adopting the median wavelength of each spectral measurement set as the effective wavelength). The squared visibility errors on the collapsed measurements were taken to be
\begin{equation}\label{eq:error}
\sigma_{\mathcal{V}^2} = \frac{\sum_i^N \Delta \mathcal{V}_i^2}{N\sqrt{N-1}},
\end{equation}
\noindent with $N$ referring to the total number of measurements in each collapsed data point (typically 7) and $\Delta \mathcal{V}_i^2$ the pipeline estimated errors on the individual measurements. We then applied a minimum error floor of \SI{10}{\percent} relative error or additive \num{0.02} error, whichever is largest, adopted to be consistent with the errors chosen for the CHARA experiments. Finally, we applied our outlier removal, in accord with the CHARA data procedure, which resulted in fewer than \SI{3}{\percent} of the data rejected.
\begin{deluxetable}{lcrlccl}
\tablecaption{VLTI/AMBER Observations}
\tablehead{\colhead{Target} & \colhead{Ref.} & \colhead{Date} & \colhead{Configuration} & \colhead{Band} & \colhead{n.\,sets} & \colhead{Calibrator(s)}}
\startdata
\object{HD~163296} & a & 2007-08-31 & H0-E0-G0 & K & 1 & \object{HD 172051} \\
& a & 2007-09-03 & H0-E0-G0 & K & 2 & \object{HD 139663} \\
& a & 2007-09-06 & D0-H0-G1 & K & 1 & \object{HD 172051} \\
& b & 2008-05-25 & D0-H0-A0 & K & 8 & \object{HD 156897}, \object{HD 164031} \\
& b & 2008-05-27 & D0-H0-G1 & K & 1 & \object{HD 156897} \\
& b & 2008-06-04 & H0-E0-G0 & K & 2 & \object{HD 106248}, \object{HD 156897} \\
& b & 2008-06-05 & H0-E0-G0 & K & 5 & \object{HD 156897}, \object{HD 163955} \\
& c & 2008-06-24 & UT1-UT2-UT4 & K & 1 & \object{HD 156897} \\
& d & 2008-07-07 & D0-H0-G1 & K & 5 & \object{HD 160915} \\
& d & 2008-07-09 & K0-A0-G1 & K & 4 & \object{HD 160915} \\
& e & 2008-07-19 & K0-A0-G1 & K & 1 & \object{HD 156897} \\ \hline
\object{HD~190073} & f & 2012-06-07 & UT2-UT3-UT4 & K & 4 & \object{HD 179758}, \object{HD 188385} \\
\enddata
\tablerefs{\textsuperscript{a}Dataset 60.A-9054; \textsuperscript{b}Dataset 081.C-0794; \textsuperscript{c}Dataset 081.C-0851; \textsuperscript{d}Dataset 081.C-0124; \textsuperscript{e}Dataset 081.C-0098; \textsuperscript{f}Dataset 089.C-0130.}
\label{tab:amber}
\end{deluxetable}
Additionally, we include in our data analysis recently published squared visibility and closure phase measurements from the PIONIER instrument (\citealt{Le_Bouquin_2011}) in H band, conducted by \cite{Lazareff_2017}; see observations and reduction summary therein. Since these data were conducted in a low spectral resolution mode ($R = 15$), we collapsed the measurements to a single value in H band by averaging $V^2$ measurements (in like manner as we did for the AMBER instrument data in K band). The PIONIER data consisted of 3 data points in the range \SIrange[range-units = single, range-phrase = --]{1.55}{1.80}{\micro\meter}.
\rev{We estimated the squared visibility errors by applying Equation \ref{eq:error} and then imposed a relative error floor of \SI{5}{\percent} to each measurement to account for visibility variations within H-band in each measurement. No minimum additive error was applyed. While these error floors are smaller than those applied to the Classic, CLIMB, and AMBER datasets, the overall data quality produced by the more recent PIONIER instrument is higher, as it employs fringe scanning mode that is better able to correct for seeing effects than AMBER and the Classic/CLIMB instruments.}
\subsection{Keck Aperture Masking Observations}
Finally, we include in our analysis archival short-baseline measurements of HD~163296 from the Keck aperture masking experiment in both H and K bands, collected on September 29, 1998. These short baseline data constrain the fraction of the total light distributed in a large-scale ``halo'' and also could detect stellar companions, if present. Detailed descriptions of the experiment and the data analysis method were published by \cite{Tuthill_2000}. \rev{No additional percentage or \revtwo{additive} error floors were applied to the Keck data, which is dominated by systematic errors; all estimated $\mathcal{V}^2$ errors exceed \SI{12}{\percent} which is larger than any estimated seeing variations.} Unfortunately, \rev{Keck aperture masking} observations of HD~190073 were not conducted.
Note that the Keck masking experiment has a roughly 5 times larger field-of-view (FOV FWHM $\sim$\ang{;;1}/\ang{;;1.3} for H/K bands) for ``incoherent flux'' than the single-mode fiber combiner PIONIER (FOV FWHM $\sim$\ang{;;0.18} for H band) and similar field of view to the CHARA combiners (seeing limited FOV FWHM $\sim$\ang{;;1}--\ang{;;2}) -- light beyond this FOV does not get included in the calibration of the visibility.
\section{Data}
\subsection{Squared Visibilities}
The combined squared visibility measurements and full $(u,v)$ coverage from all instruments are shown in Figure \ref{fig:mwc275raw} for HD~163296 and in Figure \ref{fig:v1295aqlraw} for HD~190073.
The $(u,v)$ coverages for both objects are rather well sampled in K band, providing a sufficient number of points at all covered baselines to infer a good picture of the radial brightness distribution at sub-\si{\au} scales. Nonetheless, there are notable gaps in the data. HD~163296 does not have excellent coverage at long baselines along the north-south direction, but is well sampled otherwise. HD~190073 is missing intermediate baseline information in the north-south direction as well as intermediate/long baseline measurements along the east-west direction. Moreover, due to the lack of Keck aperture masking measurements for HD~190073, there are no K band $(u,v)$ points at baselines shorter than \SI{29.9}{\meter} which are necessary to constrain the emission contribution due to a large scale, over-resolved, halo component in the system. The $(u,v)$ coverage for both objects in H band is not as well sampled, particularly at long baselines.
We can learn important qualitative features about both objects by inspecting their visibility curves. For HD~190073, there are sufficient short baseline measurements from PIONIER to show that any large scale halo component present in the system contributes negligibly to the total flux received from the system, as the short baseline points indicate that the squared visibility is consistent with unity at zero baseline, within instrumental uncertainties.
For both targets, at baseline separations exceeding \SI{120}{\meter}, the squared visibilities remain approximately constant, especially in the K-band data; the lack of discernible oscillations about the asymptotic value at long baselines strongly suggests that the inner emission lacks sharp boundaries where the emission intensity changes quickly over a small radius range, a result first hinted at by \citet{Tannirkulam_2008} for HD 163296 with a mere eight squared visibility measurements at long baselines. In the H band, at first glance it appears that there may be some oscillation about the asymptotic value; however, this is coupled with larger uncertainties than in K-band due to poorer seeing as well as a sparser $(u, v)$ coverage, indicating that the disk profile in H-band may nonetheless be consistent with a smooth distribution as it is in the K band. Some high H-band visibilities for HD~190073 between \SIrange[range-units = single, range-phrase = --]{200}{300}{\meter} may be due to mis-calibration. Additionally, although our observations do not appear to coincide with periods of flux variability, we can not rule out their occurrence within our dataset \citep{Ellerbroek_2014}.
\begin{figure}[htbp]
\begin{center}
\includegraphics{fig1.pdf}
\caption{H and K band squared visibilities measurements of HD~163296 shown on the left and corresponding $(u, v)$ coverage on the right. New data presented here from Classic/CLIMB data sets (Tables \ref{tab:classic} and \ref{tab:climb}) and from the AMBER experiment (Table \ref{tab:amber}). For the $(u,v)$ plots, $u$ increases to the left (east), $v$ increases to the top (north) and the concentric axis circles are drawn with increasing radii in units of \SI{100}{\meter}.}
\label{fig:mwc275raw}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics{fig2.pdf}
\caption{H and K band squared visibility measurements HD~190073. New data presented here from Classic/CLIMB data sets (Tables \ref{tab:classic} and \ref{tab:climb}) and from the AMBER experiment (Table \ref{tab:amber}).}
\label{fig:v1295aqlraw}
\end{center}
\end{figure}
\subsection{Closure Phases}\label{sec:closurephase}
The closure phase measurements for both objects in H and K bands are presented in Figure \ref{fig:closurephase}. Per convention, each closure phase measurement is plotted with respect to the longest baseline in the bispectrum closing triangle with the standard caveat that this is not necessarily the baseline along which the true Fourier phase is the strongest. Note that the AMBER and PIONIER measurements plotted in the figure have not been ``collapsed'' as they have been the squared visibility case, instead the full spectral range of measurements are plotted individually within their respective bands. To distinguish individual wavelength measurements collected at the same baseline, we plot spatial frequency rather than baseline distance along the abscissa.
We note that, for the most part, closure phase measurements with closing triangles on short to intermediate baselines are consistent with a value of zero, indicating that the brightness distribution is roughly point symmetric on larger spatial scales. At longer baselines, we see a departure in the presumed point symmetry, indicating that fine scale structure is present, perhaps due to orbiting clumps in the inner disk. It is intriguing that we see no clear evidence of this fine-scale structure in the visibility curve although the high data uncertainties\rev{, especially at long baselines where the SNR is on the order of 1-2,} likely mask the expected variations.
Finally, we note that the strongest closure phases, measured with the CHARA/CLIMB instrument, were conducted over a span of 4 years, during which time variability in the disk structure may have occurred. Thus, the presented closure phase measurements cannot necessarily be considered as a representative snapshot of the disk asymmetries at any given time.
\begin{figure}[htbp]
\begin{center}
\includegraphics{fig3.pdf}
\caption{Top: measured closure phases of HD~163296. Bottom: measured closure phases of HD~190073.}
\label{fig:closurephase}
\end{center}
\end{figure}
\section{Modeling}
\begin{deluxetable}{lCp{1.5in}}
\tablecaption{Geometric Primitive Models}
\tablehead{\colhead{Name} & \colhead{Visibility $V_{ext}(b)$} & \colhead{Comment}}
\startdata
Gaussian & \displaystyle \exp\left[\frac{-(\pi\theta b)^2}{4\ln 2}\right] & $\theta$ is the FWHM in \si{\milliarcsecond}. \\[16pt]
Disk & \displaystyle \frac{2 J_1(\pi\theta b)}{\pi\theta b} & $\theta$ is the diameter in \si{\milliarcsecond}. \\[16pt]
Thin Ring & \displaystyle J_0(2\pi\theta b) & $\theta$ is the diameter in \si{\milliarcsecond}. \\[16pt]
\enddata
\tablecomments{These models assume point symmetry where $b = \sqrt{u^2 + v^2}$ is the baseline in dimensionless units $B/\lambda_\text{eff}$. The building blocks were taken from \cite{Berger_2007}}
\label{tab:primitive}
\end{deluxetable}
We fit simple geometrical models to characterize the global properties of the extent and configuration of the inner-\si{\au} emission between our two targets. Our focus is, primarily, to compare the spatial sizes of the emission seen in the H and K bands without trying to impose a particular interpretation on the data. Therefore, we opt not to perform the sort of detailed radiative transfer modeling typically found in the current literature (eg.\@ \citealt{Aarnio_2017}), based on many as-of-yet unobservable chemical and geometrical properties of these systems.
While the new data presented in this paper provide the critical long baseline information necessary to constrain the sharpness of features in the inner disk to sub-\si{\au} resolution, the instrumental sensitivities of the squared visibility measurements are, unfortunately, not quite high enough to uniquely determine the true emission distribution. We are however able to paint a crude general picture of the surface brightness distribution in the inner few \si{\au} of HD~163296 and HD~190073, which we hope will be able to provide a starting point for future radiative transfer modeling efforts.
\subsection{Model Building}\label{sec:modelbuild}
We tested three simple geometric emission models to fit to the squared visibility measurements, each composed of three components: a point source representing the unresolved star, an extended circumstellar emission component, and an over-resolved halo component. These components provide fractional flux contributions $f_{ps}$, $f_{ext}$, and $f_{halo}$, respectively, such that they sum to unity.
A crucial initial step to our model analysis was to de-project the data into a face-on orientation. To do this, our data were rotated and stretched onto an ``effective baseline'
by transforming the $(u, v)$ coordinates of each measurement into a $(u', v')$ frame via
\rev{
\begin{equation}\label{uvprime}
\left(\begin{array}{c} u' \\ v' \end{array}\right) =
\left(\begin{array}{cc}\sin P\!A & \cos P\!A \\ -\cos P\!A \cos i & \sin P\!A \cos i \end{array}\right)
\left(\begin{array}{c} u \\ v \end{array}\right)
\end{equation}
}
\noindent where $i$ and $P\!A$ are the inclination and position angles of the disk, respectively. These quantities were left as free parameters for out fitting routine.
With the data correctly de-rotated, the functional form of the total squared visibility measured at an effective baseline length $b$ is given by
\begin{equation}
V^2(b) = \left[f_{ps} + f_{ext} V_{ext}(b)\right]^2,
\end{equation}
\noindent where $V_{ext}(b)$ is a ``primitive'' function describing an extended emission component from Table \ref{tab:primitive} and the visibilities of the star itself provide a constant offset contribution since the diameters of the stars themselves are unresolved at the baselines we probe. Note that the halo flux contribution is not explicitly expressed in the equation; the addition of this component is to allow the model fits to have sub-unity visibility values at zero baseline. The halo, being over-resolved, is dominant at baseline separations smaller than the shortest baseline measurements available and in reality raises the visibility to unity at the origin while making a negligible contribution at the baselines measured.
Our extended geometrical models were chosen to distinguish between two general scenarios of the inner disk emission: either the emission is largely constrained to a thin ring, representing the illuminated inner rim of a truncated dust disk, or the emission emanates primarily from a region within this supposed sublimation radius.
In the latter case, we fit two possible models: one in which the emission has a uniform disk surface brightness and another in which the brightness distribution is Gaussian in form. \rev{In all cases, we allow the H- and K-band sizes to fit independently in order to best minimize their residuals; we note that this is the case even in our ``inner rim'' ring models where the H- and K-band sizes would physically be expected to be the same.}
We constructed all our models with point-symmetric brightness distributions. Strictly speaking, such an assumption is only valid if the closure phase measurements are all consistent with \SI{0}{\degree} or \SI{180}{\degree}, which is not the case in the present data, especially for HD~163296 at longer baselines. However, as we argued in \S\ref{sec:closurephase}, the closure phases likely indicate fine scale asymmetries in the overall brightness distribution. Since our purpose in this paper is only to characterize the general large-scale emission profile's size and sharpness, choosing a point symmetric brightness distribution is justified considering that the closure phases measured along the primary lobe in visibility in both targets are consistent with zero.
\subsection{Model Fitting}
Each model was fit simultaneously to the H- and K-band data for each object. Because the emission component was assumed to be coplanar, the inclination and position angles of the extended emission component were fixed together. This is generally not desirable, but was a necessary constraint due to the relative overall lack of H-band squared visibility data. Component flux contributions and size parameters were allowed to vary, but the sum of the point source and extended emission fluxes (or in other words, the expected halo contribution) were fixed together between the two bands, due to the lack of reliable data at short baselines between the different instruments, which is partially a consequence of their differing fields of view. In the case of HD~190073, the data do not suggest the presence of a halo component. In fact, since \rev{nominal} fits of the data for HD~190073 tended to slightly overshoot \rev{a value of 1.0} at zero baseline \rev{(a nonphysical result), we added an additional constraint that the overall summed total fractional flux contributions of the point source and extended emission equaled unity there for this object, effectively eliminating any halo contribution.}
Our fits employed the Levenberg-Marquadt gradient descent algorithm to minimize the $\ell$1-norm statistic. To ensure we found the best fit, we initialized each model with \num{200} random sets of starting parameter values, and selected the fit with the overall lowest $\ell$1-norm value. This statistic was selected over the more typical $\ell$2-norm since our squared visibility error estimates are non-Gaussian due to seeing and calibration effects and the underlying covariance is generally poorly understood. Moreover, there are several obvious outliers in the data set, such as squared visibility measurements with negative values due to bias corrections, to which the $\ell$1-norm statistic is more robust. That said, the final fit parameters were not found to be significantly different when the fits were conducted with the $\ell$2-norm.
The resulting parameter values and reduced $\ell$1-norms of our model fits for HD~163296 and HD~190073 are presented in Tables \ref{tab:mwcres} and \ref{tab:v12res}, respectively. We estimated the uncertainties on the fit parameters by performing \num{1000} sets of bootstraps on the individual data points with \num{10} random sets of starting values in the neighborhood (of a few $\sigma$) of each of the best fit values using the same fitting procedure described above. Reported uncertainties in the fit parameters were found by locating the most compact \SI{68}{\percent} in the bootstraps and computing two sided errors around the best fit value of all the data. For all parameters except the inclination and position angles, the two sided errors were nearly symmetric. It should be noted however that the stated errors are likely under-estimating the true uncertainties due to our neglect of possible correlated errors in our datasets expected due to seeing variations. Another concern is that incomplete (and in some locations, especially in H-band, very sparse) $(u,v)$ coverage causes certain data points to have an artificially large weight on the final fit result, amplifying possible systematic uncertainties.
In Figures \ref{fig:mwcres} and \ref{fig:v12res}, we overplot the best fit models on the de-projected interferometric measurements data with the modulus of the visibility along the ordinate and the effective baselines along the abscissa. While the fits themselves were conducted against the actual squared visibility measurements, visibility provides a better sense of the contribution importance of the various model components. As per convention, measurements recorded with negative squared visibilities retain that sign in the absolute visibility plots as well. We also inset stacked spatial image reconstructions for each of the \num{1000} bootstrap fit results in the aforementioned figures to give a visual sense of the presumed spatial distributions and range of uncertainties in our model fits.
\section{Results and Discussion}\label{sec:results}
We found that, of the simple models we tested, the Gaussian surface brightness distributions of the inner disk systematically outperformed the other models, visually and statistically, in describing the squared visibility measurements for both targets. For HD~163296, where the visibility at zero baseline was allowed to fit as a free parameter, only the Gaussian model provided good agreement with the data at short baselines; the uniform disk and thin ring models significantly underestimated the visibility measurements. A key feature of the Gaussian model is that the visibility quickly and monotonically asymptotes to the flux value of the star, without any ``ringing'' seen at longer baselines. This description is consistent with the observed data, to within the degree of uncertainty assumed given the systematics of the instruments used. We note that the apparent ringing seen in the H-band CLIMB long-baseline data may be due to poor calibration and sparse $(u, v)$ plane sampling rather than a real effect, as systematic errors are more prominent in H-band and the long-baseline K-band data are consistent with a flat visibility profile. While monotonically decreasing behavior at long baselines is not unique to Gaussian models and also describe power-law-like models such as the Pseudo-Lorentzian profile described in \cite{Lazareff_2017}, such models disagree with observations at short baselines where the observed data indicate a concave down
visibility profile whereas Pseudo-Lorentzian models are concave up.
The uniform disk models we tried fit fairly well to the data based solely on the $\ell$1-norm values, though were overall not as successful at describing the data as the Gaussian models. For instance, in the case of HD~163296 where the fit visibility at zero baseline was allowed to vary as a free parameter, the disk models do noticeably underestimate data at short baselines. In K-band for both targets, the ringing at intermediate baselines predicted by the uniform disk models do not visually match the observations. At H-band, there appears to be better convergence between the model and the data at long baselines, but as mentioned earlier, this may well be due to \rev{sparse} (u,v) coverage and instrumental systematic effects in the H-band calibration of the CLIMB data.
Finally, the thin ring models, which prior to the work of \cite{Tannirkulam_2008} were the favored picture of the inner emission, performed the worst during the fitting process. In the case of HD~163296, the ring models dramatically underestimate the observed visibilities at short baselines. For HD~190073, the chosen inclination and position angles of the ``best'' fit appear to merely exploit our data's under-sampled $(u, v)$ coverage, and even still produce $\ell$1-norm values in H and K band which are larger than for the other models tested, respectively. \rev{It is interesting to note that the fits in \cite{Lazareff_2017} favored a more ring-like emission profile for HD~190073 to that of the disk-like geometry we find in this work. It is difficult to assess the root of this discrepancy, which may point to deficiencies in our models since the true distributions are likely not exact Gaussians or rings. We do note that the discrepancies in fit position angles may play a role in the presence or \revtwo{absence} of \rev{oscillations} in $\mathcal{V}^2$ characteristic of a ring-like geometry. Furthermore, HD~190073 is more \revtwo{susceptible} to errors in fitting the position angle as it appears to be more face-on than HD~163296. We point out that the longer baseline CHARA data we examine in this work is better able to constrain the general geometry and orientation better than PIONIER data alone. However, the overall $(u,v)$ sampling of the square visibility data presented in this work and especially in \cite{Lazareff_2017} contain large gaps in coverage, and are likely the most important factor in the discrepancies noted. We point out that our results for HD~163296 are in good agreement with \cite{Lazareff_2017} indicating robustness between the methodologies employed in both works.}
\begin{figure}[htbp]
\begin{center}
\includegraphics{fig4.pdf}
\caption{Model fit results of HD~163296, with H band fits on the left and K band fits on the right. Comparison of the different models tested is shown vertically. Inset in each plot is a stacked image of the emission distribution generated by the individual bootstrap fit parameters.}
\label{fig:mwcres}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics{fig5.pdf}
\caption{Model fit results of HD~190073. See the caption in Figure \ref{fig:mwcres} for details of the layout.}
\label{fig:v12res}
\end{center}
\end{figure}
\begin{deluxetable}{lcccccccl}
\tablecaption{HD~163296 Results}
\tablehead{\colhead{Model} & \colhead{Band} & \colhead{$i$ [\si{\degree}]} & \colhead{$P\!A$ [\si{\degree}]} & \colhead{$\Theta$ [\si{\milliarcsecond}]$^\tablenotemark{a}$} & \colhead{$f_{ext}$} & \colhead{$f_{ps}$} & \colhead{$\ell$1-norm/N} & \colhead{comment}}
\startdata
\multirow{2}{*}{Gaussian} & H & \multirow{2}{*}{$45.0^{+2.8}_{-1.1}$} & \multirow{2}{*}{$131.3^{+1.9}_{-2.3}$} & \num{ 4.08 \pm 0.10} & \num{ 0.64 \pm 0.02} & \num{ 0.26 \pm 0.01} & 0.85 & \multirow{2}{1in}{Best fit} \\
& K & & & \num{ 3.98 \pm 0.08} & \num{ 0.78 \pm 0.02} & \num{ 0.12 \pm 0.01} & 0.81& \\
\multirow{2}{*}{Uniform Disk} & H & \multirow{2}{*}{$43.7^{+2.3}_{-5.9}$} & \multirow{2}{*}{$124.5^{+4.7}_{-3.2}$} & \num{ 5.85 \pm 0.28} & \num{ 0.51 \pm 0.05} & \num{ 0.30 \pm 0.02} & 1.26 & \multirow{2}{1in}{Underestimates short baseline data} \\
& K & & & \num{ 4.97 \pm 0.16} & \num{ 0.70 \pm 0.04} & \num{ 0.10 \pm 0.01} & 1.14& \\
\multirow{2}{*}{Thin Ring} & H & \multirow{2}{*}{$46.9^{+1.1}_{-6.6}$} & \multirow{2}{*}{$128.6^{+5.2}_{-8.0}$} & \num{ 3.12 \pm 0.12} & \num{ 0.54 \pm 0.07} & \num{ 0.18 \pm 0.05} & 1.98 & \multirow{2}{1in}{Underestimates short baseline data} \\
& K & & & \num{ 3.20 \pm 0.28} & \num{ 0.60 \pm 0.06} & \num{ 0.11 \pm 0.02} & 1.25& \\
\enddata
\tablenotetext{a}{$\Theta$ indicates FWHM for the Gaussian model and the diameter for the uniform disk and ring models. See \S\ref{sec:modelbuild} for definitions of the remainder of the parameters.}
\label{tab:mwcres}
\end{deluxetable}
\begin{deluxetable}{lcccccccl}
\tablecaption{HD~190073 Results}
\tablehead{\colhead{Model} & \colhead{Band} & \colhead{$i$ [\si{\degree}]} & \colhead{$P\!A$ [\si{\degree}]} & \colhead{$\Theta$ [\si{\milliarcsecond}]\tablenotemark{a}} & \colhead{$f_{ext}$} & \colhead{$f_{ps}$} & \colhead{$\ell$1-norm/N} & \colhead{comment}}
\startdata
\multirow{2}{*}{Gaussian} & H & \multirow{2}{*}{$32.2^{+5.0}_{-4.08}$} & \multirow{2}{*}{$81.6^{+5.6}_{-8.20}$} & \num{ 3.44 \pm 0.37} & \num{ 0.63 \pm 0.01} & \num{ 0.37 \pm 0.01} & 1.15 & \multirow{2}{1in}{Best fit} \\
& K & & & \num{ 3.67 \pm 0.04} & \num{ 0.82 \pm 0.01} & \num{ 0.18 \pm 0.01} & 0.67& \\
\multirow{2}{*}{Uniform Disk} & H & \multirow{2}{*}{$36.8^{+1.9}_{-2.7}$} & \multirow{2}{*}{$22.6^{+5.9}_{-2.3}$} & \num{ 5.53 \pm 0.40} & \num{ 0.64 \pm 0.01} & \num{ 0.36 \pm 0.01} & 1.26 & \multirow{2}{1in}{Decent fit} \\
& K & & & \num{ 6.07 \pm 0.11} & \num{ 0.82 \pm 0.01} & \num{ 0.18 \pm 0.01} & 0.90& \\
\multirow{2}{*}{Thin Ring} & H & \multirow{2}{*}{$50.9^{+1.5}_{-2.1}$} & \multirow{2}{*}{$48.0^{+6.3}_{-3.9}$} & \num{ 4.28 \pm 0.16} & \num{ 0.59 \pm 0.02} & \num{ 0.41 \pm 0.02} & 2.13 & \multirow{2}{1in}{Over-fits to poor $(u,v)$ coverage} \\
& K & & & \num{ 3.63 \pm 0.14} & \num{ 0.88 \pm 0.01} & \num{ 0.12 \pm 0.01} & 1.51& \\
\enddata
\tablenotetext{a}{$\Theta$ indicates FWHM for the Gaussian model and the diameter for the uniform disk and ring models. See \S\ref{sec:modelbuild} for definitions of the remainder of the parameters.}
\label{tab:v12res}
\end{deluxetable}
\subsection{Comparison to Photometry}
In Figure \ref{fig:sed}, we use photometry collected by \cite{Tannirkulam_2008} in June, 2006, and by \cite{Lazareff_2017} in 2014, in conjunction with near-mid infrared SED measurements collected in June/July, 2007 for HD~190073 and in March, 2011 for HD~163296, first presented in \cite{MillanGabet_2016}, to construct crude SED profiles of the two objects spanning from near-ultraviolet to near-infrared wavelengths. \rev{The long baseline interferometry allows us to estimate the flux contribution of the unresolved point source stellar photosphere in H and K bands by investigating the asymptotic value of the visibility oscillations.}
We note that any interstellar dust reddening should not significantly affect the H--K color at the distances to the stars, so fitting model photospheres to the interferometry data has the two-fold effect of testing the extended flux models and also determining the reddening to the source by extrapolating the NIR interferometry to measured visible fluxes.
\rev{We begin by \revtwo{modeling} the photosphere of HD~163296. Due to it's proximity to earth (\SI{101.5}{\parsec}), the amount of interstellar dust extinction to the star is negligible (see \texttt{Bayestar17}; \citealt{Green_2018}). Using measured B, V, and R band photometric points (\citealt{Lazareff_2017}), we fit tabulated model photospheres of \cite{Castelli_2004} with the closest available temperature, \SI{9250}{\kelvin}, and all $\log g$ values available (ranging from 1.5 to 5.0 in increments of 0.5). We do not use the U band photometric point in the fit\revtwo{,} as it is expected to be amplified by accretion shocks\revtwo{, nor} points redder than R band\revtwo{,} as the disk contribution begins to dominate in the infrared. We can report that unpublished visible light interferometry using the CHARA-VEGA instrument confirms that \SI{94 \pm 6}{\percent} of the R-band flux is coming from an unresolved source (private communication from Karine Perraut and Denis Mourard).}
\rev{The H- and K-band flux contributions to the resulting model stellar photosphere SED match the slope and has values within errors of the points estimated by the Gaussian disk model (see Figure~\ref{fig:sed}), further indicating this is a good approximation of the true emission profile. From the photosphere fits, we infer the stellar luminosity and radius, and note that all model photospheres (with different $\log g$ values) obtain radii within \SI{3}{\percent}. We compare these results to model isochrones generated by the \texttt{PARSEC} code (\citealt{Bressan_2012}) to predict a luminosity of \SI{16.1}{\solarluminosity}, a mass of \SI{1.9}{\solarmass} with $\log g = 4.3$, and an age of \SI{10.4}{\mega\year}.}
\rev{We repeat this process for HD 190073, using model photospheres with temperatures at \SI{9000}{\kelvin}. Due to its greater distance (\SI{891}{\parsec}), we apply a reddening correction according to \cite{Clayton_1989}, assuming a typical $R_V = 3.1$. We find the Gaussian fit interferometric points at H and K band are consistent with $A_V \sim 0.19$, which is in excellent agreement with the \texttt{Bayestar17} value of $A_V$ = \num{0.186 \pm 0.062}. Primarily as a result of the new \textit{Gaia} DR2 distance estimate, we find that this star is incredibly young, at \SI{320}{\kilo\year}, and bright, with a luminosity of \SI{560}{\solarluminosity}. Our fits also suggest this star is twice as massive as previously assumed, with $M = \SI{5.6}{\solarmass}$ and $\log g = 3.2$ (formerly \SI{2.84}{\solarmass}; \citealt{Catala_2007}).}
\begin{deluxetable}{lcc}
\tablecaption{\rev{Adopted} Stellar Properties of Target Sources}
\tablehead{\colhead{Property} & \colhead{\object{HD~163296}} & \colhead{\object{HD~190073}}}
\startdata
$A_V$ & 0 & \rev{0.19} \\
Mass & \rev{\SI{1.9}{\solarmass}} & \rev{\SI{5.6}{\solarmass}} \\
Radius & \rev{\SI{1.6}{\solarradius}} & \rev{\SI{9.8}{\solarradius}} \\
$T_\text{eff}$ & \rev{\SI{9250}{\kelvin}} & \rev{\SI{9000}{\kelvin}} \\
Luminosity & \rev{\SI{16}{\solarluminosity}} & \rev{\SI{560}{\solarluminosity}} \\
Age & \rev{\SI{10.4}{\mega\year}} & \rev{\SI{320}{\kilo\year}}
\enddata
\tablenotetext{}{}
\label{tab:fitprop}
\end{deluxetable}
\begin{figure}[htbp]
\begin{center}
\includegraphics{fig6.pdf}
\caption{SEDs of our target objects and point source contributions at H and K bands as inferred by the interferometric models tested in this paper. Note the interferometric points are slightly staggered for clarity. Over-plotted on the data are model photospheres of \cite{Castelli_2004} \rev{at a range of extinctions fit to B, V, and R photometric observations by \cite{Lazareff_2017}.} Left (HD~163296): \rev{Model photospheres with $T_{e\!f\!f} = \SI{9250}{\kelvin}$ and $\log g = 4.5$.} Right (HD~190073): \rev{Model photospheres with $T_{e\!f\!f} = \SI{9000}{\kelvin}$ and $\log g = 3.0$.}}
\label{fig:sed}
\end{center}
\end{figure}
\subsection{Disk Orientation}
We note that the values we \revtwo{fit for} the disk inclination \revtwo{and} position angle \revtwo{of} HD~163296 were consistent with one another for all three models tested. In particular, the values obtained with our Gaussian model fits ($i = \SI{45.0}{\degree}^{+\SI{2.8}{\degree}}_{-\SI{1.1}{\degree}}$, $P\!A = \SI{131.3}{\degree}^{+\SI{1.9}{\degree}}_{-\SI{2.3}{\degree}}$) agree well with values quoted in the literature of gas measured via millimeter interferometry (\citealt{Flaherty_2015}: $i=\SI{48.4}{\degree}$, $P\!A = \SI{132}{\degree}$) and dust from scattered light studies on large scales (\citealt{Monnier_2017}: $i=\SI{48}{\degree}$, $P\!A = \SI{136}{\degree}$).
The situation is not as clear for HD~190073. While the Gaussian and uniform disk models found similar inclination angles, their fit position angles were incompatibly different. The thin ring model fit also produced radically differing values in inclination and position angle to the other two models. Unfortunately, there isn't much available in the literature to compare our derived values to. \cite{Lazareff_2017} found values of $i = \SI{30 \pm 4}{\degree}$ and $\theta = \SI{159 \pm 3}{\degree}$ in their best fit model, however, these values are highly suspect due to a poorly sampled $(u, v)$ coverage in their \revtwo{dataset}. Moreover, these data were used as part of the present analysis, and different results for the position angle were deduced. Since large-scale scattered light images or mm-wave ALMA images of HD~190073 have not yet been obtained, there is not yet an independent measure with which to test consistency. \cite{Fukagawa_2010} included HD~190073 in their survey sample of scattered light targets with the Subaru telescope, but obtained a null result in the poor seeing conditions where observation of the target was attempted.
We remind the reader that we tied the H and K band disk inclination and position angles together due to our limited (u,v) coverage. While the similar H band K band sizes for these objects suggest a common physical origin, there are mechanisms that might produce orthogonal position angles, such as dust caught in a disk wind. Future higher-quality data should investigate this possibility.
\subsection{Disk Sizes and Possible Emission Sources}\label{sec:emission}
Our Gaussian fit model for HD~163296 indicates that the inner disk is the same size in H and K bands, within uncertainty, with an H/K size ratio of \num{1.03 \pm 0.03}. At a distance of \SI{101.5}{\parsec}, the 3.98~mas FWHM from our Gaussian model has a physical size of approximately \SI{0.41}{\au}. For conventional large grains assuming our revised luminosity of \rev{\SI{16}{\solarluminosity}}, we expect dust at this distance $R\sim\SI{0.20}{\au}$ at an equilibrium temperature of \rev{$T\sim\SI{1750}{\kelvin}$} \citep[assuming dust wall backwarming,][]{Monnier_2005}.
In the case of HD~190073, the Gaussian fits indicate an H/K size ratio of \num{0.94 \pm 0.10}. At a distance of \SI{891}{\parsec}, the 3.67~mas FWHM of our best-fitting Gaussian brightness profile in K band has a physical size of \SI{3.27}{\au}. Using a revised luminosity of \rev{\SI{560}{\solarluminosity}}, we find dust at the distance $R\sim\SI{1.64}{\au}$ has an equilibrium temperature of \rev{$T\sim\SI{1500}{\kelvin}$}.
For both sources, the dust temperatures expected at the Gaussian FWHM location are close to the sublimation temperature for most dust species expected in the inner disks of YSOs.
However, a significant fraction of the inner NIR emission appears to originate within the sublimation radius.
The current dataset analyzed here does not yield any definitive conclusions pertaining to the dominant source of the inner emission observed, however, some insights may be gleaned via the fit results.
We examine briefly a few possibilities:
\begin{enumerate}
\item{Conventional dust species might exist inside the expected dust evaporation radius due to shielding by optically-thick inner gas in the midplane. Alternatively, some unidentified refractory dust species might exist that survive at $T>\SI{2000}{\kelvin}$. Testing these hypotheses requires detailed radiative transfer \revtwo{modeling} of the gas distribution in the inner sub-AU and is beyond the scope of this paper, although the similar sizes for H and K band qualitatively support this scenario since we would not expect strong differences in the emission between H and K bands for these temperatures.}
\item{Transparent dust grains could exist close to the star without evaporating. For a giant star, \citet{Norris_2012} inferred the presence of iron-free silicates close to the stellar surface within the expected dust evaporation radius.
\rev{They} suggested that species such as forsterite (\chem{Mg_2SiO_4}) and enstatite (\chem{MgSiO_3}) are almost transparent at wavelengths of \SI{1}{\micro\meter} and we speculate that these grains might be found in YSO disks as well. Observations of YSOs in polarized light might identify the same scattering signature as was done by \citet{Norris_2012} for mass-losing evolved stars.
}
\item{Hot ionized gas within \SI{0.1}{\au} could be optically-thick due to free-free/bound-free opacity, as seen for Be star disks (eg. \citealt{Sigut_2007}). Since A stars do not emit enough ionizing radiation to maintain a sufficient reservoir of ionized gas, there would need to be a local heating source in the midplane for this mechanism to be viable, possibly due to viscous or magnetic heating. This mechanism produces a sharply rising spectrum into the infrared and would have the largest impact on the differential H and K band sizes. If such an optically-thick gas were located in a thin disk that extended all the way to the surface of the star, with an inclination of \SI{45}{\degree} in conjunction with the rest of the inner-disk, this could in principle block up to \SI{14}{\percent} of the central stars' NIR flux.
\end{enumerate}
\section{Summary}
We combined broad-band infrared interferometric observations of \object{HD~163296} and \object{HD~190073} at H and K bands collected at CHARA, VLTI, and Keck to present the most complete $(u,v)$ sampled set of observations at the longest baselines available to date for these two objects. These observations allow for us to examine the inner disk structures on milliarcsecond (sub-\si{\au}) scales.
We characterize these observations with simple point-symmetric geometric models to estimate the orientation, sharpness, and spatial emission distribution of the inner disk, fixing the inclination and position angle between H and K bands. We focus on extracting general, qualitative features concerning the multi-wavelength emission geometry without a detailed radiative transfer analysis at this time. Our models however are able to constrain the basic size and profile of the surface brightness distribution of in the few \si{\au} immediately surrounding the central pre-main-sequence star.
We find that a 2D Gaussian disk profile is best able to reproduce the squared visibility measurements collected for both targets in both H and K bands, producing superior fits than both uniform disk and ring models. If the inner disk is optically-thin, we confirm earlier indications that the bulk of the inner emission originates close in to the host star, well within the supposed dust sublimation radius inferred by SED modeling. In conjunction with AB~Aurigae and HD~163296 previously studied by \cite{Tannirkulam_2008}, along with \object{MWC~614} studied by \citet{Kluska_2018}, \rev{and \object{HD~142666} studied by \cite{Davies_2018}}, HD~190073 is now the \rev{fifth} HAe object observed by CHARA with sufficient angular resolution to
rule-out a \rev{sharp, thin} ring-like geometry for the bulk of the NIR disk emission.
For both HD~163296 and HD~190073, we find small, near-zero closure phases for all baselines probing the main lobe of the disk emission (baselines $<$\SI{130}{\meter}) suggesting the large scale emission is point-symmetric. That said, some triangles with longer baselines show significant non-zero closure phases that indicate asymmetries on the $<$\SI{2}{\milliarcsecond} scale.
We speculate that asymmetries or clumpy emission in the inner disk could explain this and should motivate future monitoring campaigns to see if these clumps exist and if they show orbital motion.
We speculate on the origin of the innermost disk emission that is closer to the star than expected.
One major new result here is that the H- and K-band sizes are nearly the same for both objects, within \SI{3 \pm 3}{\percent} for HD~163296 and within \SI{6 \pm 10}{\percent} for HD~190073. This points towards a single emission mechanism throughout the inner \si{\au} such as thermal dust emission, as opposed to a combination free-free gas emission close to star and thermal dust emission farther out as has been previously suggested. We also highlight the possibility that the inner-\si{\au} could be filled, at least partially, with glassy grains nearly transparent at UV and visible wavelengths, but scattering/emitting in the near-infrared. With the advent of next generation polarimetric modes on upcoming instruments at CHARA, we will be able to test this hypothesis by measuring the inner emission in scattered light.
\rev{Finally, we use the \revtwo{extracted} point source contribution \revtwo{of} our Gaussian fit results along with \revtwo{measured} photometry to \revtwo{model} the photospheric contribution of our \revtwo{targets'} SEDs\revtwo{, allowing} us to estimate the age, mass, luminosity and radius of the central stars. For HD~163296, we find that a photosphere with a temperature of \SI{9250}{\kelvin} provides a fit to our interferometric data and measured visible photometry assuming zero interstellar reddening. For HD~190073, \revtwo{we find that} a photosphere \revtwo{at} \SI{9000}{\kelvin} observed through 0.19 magnitudes of visible extinction to \rev{matches} the interferometric and photometric data. \revtwo{Coupled with the new} \textit{Gaia} distance of \SI{891}{\parsec}, \revtwo{we find HD~190073 to be} a very luminous and massive young star with age $<$\SI{0.4}{\mega\year}.}
\acknowledgments
JDM and BRS acknowledge support from NSF-AST 1506540 and AA acknowledges support from NSF-AST 1311698. CLD, AK, and SK acknowledge support from the ERC Starting Grant ``ImagePlanetFormDiscs'' (Grant Agreement No. 639889), STFC Rutherford fellowship/grant (ST/J004030/1, ST/K003445/1) and Philip Leverhulme Prize (PLP-2013-110).
FB acknowledges support from NSF-AST 1210972 and 1445935.
MS acknowledges support by the NASA Origins of Solar Systems grant NAG5-9475, and NASA Astrophysics Data Program contract NNH05CD30C.
We thank William Danchi and Peter Tuthill for use of the previously unpublished Keck aperture masking data and also Paul Boley for his Python OIFITS module (available at \url{https://github.com/pboley/oifits}).
The CHARA Array is supported by the National Science Foundation under Grant No. AST-1211929, AST-1636624, and AST-1715788. Institutional support has been provided from the GSU College of Arts and Sciences and the GSU Office of the Vice President for Research and Economic Development.
This research has made use of the Jean-Marie Mariotti Center \texttt{SearchCal} (\citealt{Chelli_2016}) service (available at \url{http://www.jmmc.fr/searchcal}) co-developed by LAGRANGE and IPAG, and of CDS Astronomical Databases SIMBAD and VIZIER (available at \url{http://cdsweb.u-strasbg.fr/}).
This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} \citep{Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the {\it Gaia} Multilateral Agreement.
This research has also made use of the SIMBAD database (\citealt{simbad}), operated at CDS, Strasbourg, France, and
the NASA's Astrophysics Data System Bibliographic Services.
\vspace{5mm}
\facilities{CHARA, VLTI, Keck}
\software{SearchCal, amdlib, OIFITS.py, Bayestar17, PARSEC}
\bibliographystyle{aasjournal}
| {'timestamp': '2018-11-12T02:06:49', 'yymm': '1811', 'arxiv_id': '1811.03778', 'language': 'en', 'url': 'https://arxiv.org/abs/1811.03778'} |
\section{Introduction}
Our manipulations rely heavily on plethystic notation and the terminology used in \cite{GHRY}. In \cite{HRW15}, the reader can find detailed explanations for all of the notations used in this paper.
Recall that Dyck paths in the $n\times n$ lattice square $R_n$ are paths from $(0,0)$ to $(n,n)$ proceeding by north and east unit steps, always remaining weakly above the main diagonal of $R_n$. These paths are usually represented by their area sequence
$(a_1,a_2,\ldots, a_n)$, where $a_i$ counts the number of complete cells
between the north step in the $i^{th}$ row and the diagonal. Notice that the $x$-coordinate of the north step in the $i^{th}$ row is simply the difference $u_i=i-1-a_i$.
A parking function $PF$ supported by the Dyck path $D\in R_n$ is obtained by labeling the north steps of $D$ with $1,2,\ldots ,n$ (usually referred as ``cars''), where the labels increase along the north segments of $D$. Parking functions can be represented as two line arrays
$$
PF=\begin{pmatrix}
c_1 & c_2 & \cdots & c_n\\
a_1 & a_2 & \cdots & a_n\\
\end{pmatrix}
$$
with cars $c_i$ and area numbers $a_i$ listed from bottom to top.
We also set
$$
area(PF)= \sum_{i=1}^n a_i,
\enskip\ess\enskip
dinv(PF)= \hskip -.08in \sum_{1\le i<j\le n}\hskip -.08in \Big(
\chi(c_i<c_j \enskip\&\enskip a_i=a_j)\, + \,
\chi(c_i>c_j \enskip\&\enskip a_i=a_j+1)
\Big).
$$
Moreover, the word $w(PF)$ is the permutation obtained by reading the cars
in the two line array by decreasing area numbers and from right to left.
This given, the Haglund factor of a Dyck path $D$ is obtained by setting
$$
H_D(z;t)\enskip = \enskip \prod_{i=2}^{n}\big(1+{z\over t^{a_i}}\big)^{\chi(u_{i-1}= u_i)}.
$$
The $LLT$ polynomial constructed
from the Dyck path $D$ is obtained by setting
$$
LLT_D(X;q,t)\enskip = \enskip \sum_{D(PF)=D}t^{area(PF)}q^{dinv(PF)}s_{p\big(ides(w(PF)\big)}[X]
$$
where the sum is over parking functions supported by $D$ and the last factor is the Schur function indexed by the composition giving the descent set of the inverse of $w(PF)$.
The special version of the Delta Conjecture of \cite{HRW15} we refer to here is the equality
\begin{equation}\label{eq:theDC}
\Delta'_{e_{k-1}} e_n\enskip = \enskip
\sum_{D\in R_n}
LLT_D(X;q,t)\enskip
H_D(z;t)\, \Big|_{z^{n-k}}
\end{equation}
where $\Delta'_{F}$ is the eigen-operator of the modified Macdonald polynomial
defined by setting for any symmetric function $F$
$$
\Delta'_{F}{\widetilde H}_\mu[X;q,t]\enskip = \enskip
F\big[B_\mu(q,t)-1\big]\, {\widetilde H}_\mu[X;q,t]
\enskip\ess\enskip
(\hbox{for all $\mu$}).
$$
As mentioned previously, it was proved in \cite{GHRY} that
the equality in (\ref{eq:theDC}) is valid when both sides are evaluated at $q=0$. Since the left hand side is easily shown to be symmetric in $q$ and $t$, then it must also remain valid when both sides are evaluated at $t=0$.
The main result in \cite{GHRY} is the equality of the symmetric functions on the right hand sides of
the following two equations
\begin{equation}\label{eq:24fromGHRY}
\sum_{\lambda\vdash n}LHS_{k,\lambda}s_{\lambda'}[x(1-q)]=
\sum_{\mu\vdash n}q^{- n(\mu)}
P_\mu[X;1/q]
\Big[{ \ell(\mu)-1\atop k-1 }\Big]_q
(q;q)_{\ell(\mu)},
\end{equation}
and
\begin{equation}\label{eq:25fromGHRY}
\sum_{\lambda\vdash n}RHS_{k,\lambda}s_{\lambda'}[X(1-q)]\enskip = \enskip
q^{-k(k-1)}
(q;q)_k\sum_{\substack{\mu\vdash n\\ \ell(\mu)=k}}
q^{n(\mu)}
P_\mu[X;q],
\end{equation}
(these are labeled (24) and (25) in that paper).
Where
\[
LHS_{k,\lambda}=q^{-\binom{k}{2}}\langle\Delta_{e_{k-1}} ' e_n, s_\lambda\rangle\Big|_{\substack{t=0}}
,
\enskip\ess\enskip
RHS_{k,\lambda}= \sum_{D\in R_n}
\big\langle LLT_D(X;q,t) ,s_\lambda\big\rangle
H_D(z;t) \Big|_{z^{n-k}}\Big|_{t=0},
\]
and $ P_\mu[X;q]$, $ Q_\mu[X;q]$ are the Hall-Littlewood polynomials with Cauchy Kernel
\[
\sum_{\mu\vdash n}P_\mu[X;q]\, Q_\mu[Y;q]
\,=\, h_n\big[XY(1-q)\big]
\]
The present work was started by Jeff Remmel who sadly passed away before its completion. Remmel proposed the possibility of extending the Delta Conjecture when
the symmetric function side
``$\Delta'_{e_k}e_n$'' is replaced by
``$\Delta'_{s_\nu}e_n$'' , with $\nu$ an arbitrary partition. Remmel asked
the first author to obtain computer data to see if there was any similarity to the data that was obtained in the classical case. One of the most surprising features of the classical case
is the discovery that the polynomial in (\ref{eq:25fromGHRY}) contains only hook Schur functions in its Schur expansion.
It is precisely this experimental discovery that made the proof of the equality of the polynomials in (\ref{eq:24fromGHRY}) and (\ref{eq:25fromGHRY}) substantially less challenging.
This given, we began an exploration of the Schur expansion of the polynomial
\begin{equation}\label{eq:SymSide}
LHS_{\nu,n}[X;q]\enskip = \enskip\sum_{\lambda\vdash n}\big\langle\Delta'_{s_\nu}e_n\hskip .25em , \hskip .25em
s_\lambda\big\rangle\Big|_{t=0} s_{\lambda'}[X(1-q)].
\end{equation}
To our surprise, this polynomial also yielded Schur expansions containing only hook Schur functions.
\vskip .06truein
A crucial feature of \cite{GHRY} was the discovery of a new method for proving the equality of two symmetric functions.
More precisely, the equality of the functions in (\ref{eq:24fromGHRY}) and (\ref{eq:25fromGHRY}) as well as their hook Schur function expansion
was obtained simply by showing that both could be expressed as linear combinations of the following shifted Cauchy kernel, using the same coefficients $c_i(q)$
\[
\sum_{\mu\vdash n}P_\mu[X;q]\, Q_\mu[\tttt{ 1-q^i \over 1-q };q]
\,=\,h_n\big[X(1-q^i)\big]
\hskip .5truein (\hbox{for $1\le i\le n$})
\]
The data obtained, in the present case, suggested that all these desired features are present only when
$\nu$ is restricted to be a hook partition $(m-k,1^k)$ with $m<n$.
This discovery prompted us to study the symmetric function
\begin{equation}\label{eq:polyToStudy}
{\it LHS}_{k,m,n}[X,q]\enskip = \enskip \omega\Big(\Delta_{s_{m-k,1^k}}'e_n\Big|_{t=0}
\Big) [X(1-q)].
\end{equation}
Following the basic steps carried out in \cite{GHRY} we prove here that (\ref{eq:polyToStudy}) is equivalent to the identity
\begin{equation}\label{eq:equivalentPoly}
{\it LHS}_{k,m,n}[X,q]= q^{m+{k+1\choose2}}
\sum_{\mu\vdash n} q^{-n(\mu)}(q;q)_{\ell (\mu)}
\Big[{ m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q P_\mu [X;q^{-1}].
\end{equation}
To mimic the methods used in the classical case, we now need to produce a ``combinatorial side''. A simple comparison of the right hand sides of (\ref{eq:24fromGHRY}) and (\ref{eq:25fromGHRY}) shows that, in the case of the Delta Conjecture,
the symmetric function produced by the ``combinatorial side'' could be obtained by expanding the symmetric function side in terms of the basis
$\{P_\mu[X;q]\}_\mu$.
\vskip .06truein
This led to the decision to declare the symmetric function obtained by expanding
the polynomial in (\ref{eq:SymSide}) in terms of the basis $\{P_\mu[X;q]\}_\mu$ as the ``combinatorial side'' of (\ref{eq:SymSide}). This decision led us to conjecture the following ``combinatorial side'' of
(\ref{eq:polyToStudy}).
\begin{equation}\label{eq:combSide}
RHS_{k,m,n}[X;q]
=q^m\sum_{j=2+k}^{m+1}
q^{{k+2\choose 2}-(k+2)j+1}
\Big[{j-2\atop k}\Big]_q\Big[{m-1\atop j-2}\Big]_q
(q;q)_j\hskip -.15in\sum_{\mu\vdash n;\ell(\mu)=j}
\hskip -.15in q^{n(\mu)}P_\mu[X;q].
\end{equation}
In this paper, we first prove that
\begin{equation}\label{eq:symSideEqualsCombSide}
LHS_{k,m,n}[X;q]\enskip = \enskip RHS_{k,m,n}[X;q].
\end{equation}
\vskip .06truein
Jeff Remmel succeeded in formulating many of the conjectures needed to prove (\ref{eq:symSideEqualsCombSide}) by precisely following the methods developed in \cite{GHRY}.
In the first section we will outline the proof of (\ref{eq:symSideEqualsCombSide}) and walk through the steps
used by Jeff Remmel to formulate his conjectures needed to complete this proof. In the second section, we present the technical details carried out by the remaining authors to prove Remmel's conjectures and ultimately prove (\ref{eq:symSideEqualsCombSide}).
\vskip .06truein
After this project was completed, we learned that Brendon Rhoades and Mark Shimozono had already constructed, for any partition $\nu$, a symmetric function to be viewed as the ``combinatorial side'' and conjectured it to be equal to the
polynomial
\begin{equation}\label{eq:RSSymSide}
{\it LHS}_{\nu,n}[X,q]\enskip = \enskip \omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}
\Big)[X].
\end{equation}
Even more importantly, Jim Haglund communicated to us that he was able to prove the Rhoades-Shimozono conjectures using solely the results in \cite{GHRY}. We show here that an appropriate modification of Haglund's argument proves that the polynomial in (\ref{eq:RSSymSide}) plethystically evaluated at $X(1-q)$ expands only in terms of hook Schur functions for all $\nu$. This confirms our original experimental findings about the polynomial in (\ref{eq:SymSide}).
\vskip .06truein
These truly surprising circumstances demanded at least two additional investigations.
The first was to determine whether or not there was any relation between our method of predicting a ``combinatorial side'' and
the Rhoades-Shimozono conjectures. The second was to find a symmetric function reason explaining Haglund's result. In the final section of the paper, we present our comments about these two problems. Here we will add a few words.
\vskip .06truein
For the first problem the evidence we gathered confirms that in this case
our combinatorial side predicts the Rhoades-Shimozono combinatorial side.
\vskip .06truein
To be precise, we show that the symmetric function
\[
LHS_{\nu,n}[X,q]\enskip = \enskip \omega\Big(\Delta'_{s_\nu}\, e_n\Big|_{t=0}\Big)[X(1-q)]
\]
expands in terms of the $\{P_\mu[X, q^{-1}]\}_\mu$ basis as
\begin{equation}\label{eq:expSymSidePmu}
LHS_{\nu,n}[X,q]
\enskip = \enskip
q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over 1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
P_\mu(X,q^{-1}).
\end{equation}
Expanding the polynomial in (\ref{eq:expSymSidePmu}) in terms of the basis $\{P_\mu[X,q]\}_\mu$ yielded our conjectured ``combinatorial side'' to be the symmetric function
\begin{align*}
RHS_{\nu,n}[X,q]
=&
q^{|\nu| }
\sum_{k={\ell}(\nu)}^{|\nu|}(q;q)_k
\sum_{|\rho|=|\nu|, , \,
{\ell}(\rho)=k}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}\,\times
\\
&\hskip .5truein\bigsp\hskip .5truein
\times \,
q^{-k(k+1)}
(q;q)_{k+1}\hskip -.08in\uu\sum_{\multi{\mu\vdash n\, ;\, {\ell}(\mu)=k+1}}\hskip -.08in\uu
q^{n(\mu)}
P_\mu[X;q]
\end{align*}
It turns out that this is precisely the Rhoades-Shimozono ``combinatorial side'' plethystically evaluated at $X(1-q)$.
\section{Jeff Remmel's conjectures in the hook case}
In this section, we will outline the steps followed by Remmel to formulate the conjectures necessary to establish the equality in (\ref{eq:symSideEqualsCombSide}), that is,
\[
LHS_{k,m,n}[X;q]\enskip = \enskip RHS_{k,m,n}[X;q]
\]
with the polynomial in (\ref{eq:equivalentPoly}) :
\[
{\it LHS}_{k,m,n}[X,q]= q^{m+{k+1\choose2}}
\sum_{\mu\vdash n} q^{-n(\mu)}(q;q)_{\ell (\mu)}
\Big[{ m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q P_\mu [X;q^{-1}]
\]
as the ``symmetric function side'', and the polynomial in (\ref{eq:combSide}):
\[
RHS_{k,m,n}[X;q]
=q^m\sum_{j=2+k}^{m+1}
q^{{k+2\choose 2}-(k+2)j+1}
\Big[{j-2\atop k}\Big]_q\Big[{m-1\atop j-2}\Big]_q
(q;q)_j\hskip -.15in\sum_{\mu\vdash n;\ell(\mu)=j}
\hskip -.15in q^{n(\mu)}P_\mu[X;q]
\]
as the ``combinatorial side''.
To follow the classical case, Remmel used the identity
\begin{equation}\label{eq:CauchyKernelIdentity}
{h_n[X(1-q^i)]\over 1-q^i}\enskip = \enskip\sum_{\mu\vdash n}
q^{n(\mu)} P_\mu[X;q]
\prod_{j=2}^{{\ell}(\mu)}
(1-q^{i- j+1 } )
\end{equation}
and then tried to solve for the $c_i^{k,m}(q)$ in the equations
\[
RHS_{k,m,n}[X;q]\enskip = \enskip
\sum_{i=1}^n c_i^{k,m}(q)\,
\sum_ {\mu\vdash n }q^{n(\mu)} P_\mu[X;q]
\prod_{r=2}^{\ell(\mu)}
(1-q^{i- r+1 } ),
\]
which may be best rewritten as
\begin{equation}\label{eq:RHSexp}
RHS_{k,m,n}[X;q]=\sum_ {\mu\vdash n }q^{n(\mu)} P_\mu[X;q]
\sum_{i=1}^n c_i^{k,m}(q)\,
\prod_{r=2}^{\ell(\mu)}
(1-q^{i- r+1 } ).
\end{equation}
Likewise (\ref{eq:combSide}) may also be rewritten as
\begin{equation}\label{eq:RHSaltExp}
RHS_{k,m,n}[X;q] =
\sum_ {\mu\vdash n }
q^{n(\mu)} P_\mu[X;q] q^m q^{{k+2\choose 2}-(k+2)\ell(\mu)+1}\Big[{\ell(\mu)-2\atop k}\Big]_q\Big[{m-1\atop \ell(\mu)-2}\Big]_q
(q;q)_j.
\end{equation}
Since $\{P_\mu[X;q]\}_\mu$ is a basis, the equality of (\ref{eq:RHSexp}) and (\ref{eq:RHSaltExp}) can be true if and only if we have
\begin{equation}\label{eq:RemmelConj1}
\sum_{i=1}^n c_i^{k,m}(q)\,
\prod_{r=2}^{j}
(1-q^{i- r+1 })
=
q^{m+{k+2\choose 2}-(k+2)j+1}\Big[{j-2\atop k}\Big]_q\Big[{m-1\atop j-2}\Big]_q
(q;q)_j.
\end{equation}
A careful examination of computer data led Jeff Remmel to conjecture that the solution of the equations in (\ref{eq:RemmelConj1}) are the
coefficients
\begin{equation}\label{eq:coeffs}
c_{s}^{k,m}(q)
\enskip = \enskip
(-1)^{m+1-s}q^{{ m+1-s \choose 2}-(k+1)m+{k+1\choose 2}}
\Big[
{ m-1 \atop k
}
\Big]_q
\Big[
{ k+2 \atop m+1-s
}
\Big]_q (1-q^s)
\end{equation}
It turns out that the proof of the Remmel conjecture is an easy consequence of the nature of the equations in (\ref{eq:RemmelConj1}). This gives the validity of (\ref{eq:RemmelConj1}) with the $c_i^{k,m}(q)$ given by (\ref{eq:coeffs}). This also proves the identity
\[
\sum_{i=1}^n c_{i}^{k,m}(q)
{h_n\big[X(1-q^i)\big]\over 1-q^i
}\enskip = \enskip RHS_{k,m,n}[X;q]
\enskip\ess\enskip
(\hbox{for all $1\le k\le m-1 $ and
$m<n$})
\]
This given, to prove (\ref{eq:symSideEqualsCombSide}) we only
need to show that we also have
\[
\sum_{i=1}^n c_{i}^{k,m}(q)
{h_n\big[X(1-q^i)\big]\over 1-q^i
}\enskip = \enskip LHS_{k,m,n}[X;q]
\enskip\ess\enskip
(\hbox{for all $1\le k\le m-1 $ and
$m<n$}).
\]
However here, as in the classical case, rather than the expression in (\ref{eq:CauchyKernelIdentity}) Remmel was forced to use the equivalent expression
\[
{h_n[X(1-q^i)]\over 1-q^i}\enskip = \enskip
\sum_{\mu\vdash n}
q^{-n(\mu)} P_\mu[X;1/q]
\prod_{j=2}^{{\ell}(\mu)}
(1- q^{i+ j-1 } ).
\]
This given, his next goal was to
prove the identity
\begin{align*}
&
\sum_{\mu\vdash n}
q^{-n(\mu)} P_\mu[X;1/q]
\sum_{i=1}^n c_{i}^{k,m}(q)
\prod_{j=2}^{{\ell}(\mu)}
(1- q^{i+ j-1 } )\enskip = \enskip
\\
\enskip = \enskip
&
\sum_{\mu\vdash n}
q^{-n(\mu)}P_\mu [X;q^{-1}]
q^{m-k-1+{k+2\choose2}}
\Big[{ m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q(q;q)_{\ell (\mu)}.
\end{align*}
Since $\{P_\mu [X;q^{-1}]\}_\mu $ is a symmetric function basis, equating the coefficients of $P_\mu [X;q^{-1}]$ on both sides reduced us to verifying the following $q$-identity for all $1\le {\ell}\le n$
\begin{equation}\label{eq:RemmelConj2}
\sum_{i=1}^n c_{i}^{k,m}(q)
\prod_{j=2}^{{\ell}}
(1- q^{i+ j-1 } )=
q^{m+{k+1\choose2}}
\Big[{ m -1\atop k}\Big]_q
\Big[{m +{\ell} -(k+2)\atop m}\Big]_q(q;q)_{{\ell}}.
\end{equation}
Actually, in order to prove (\ref{eq:symSideEqualsCombSide}), we need only show that by means of the Remmel's coefficients defined in (\ref{eq:coeffs}), both of his conjectures (\ref{eq:RemmelConj1}) and (\ref{eq:RemmelConj2}) hold. The following section contains all the details needed to carry this out.
\section{Technical details}
In this section, we provide the technical details that are needed to prove the Remmel conjectures. We begin with a particular $q$-binomial identity.
\begin{prop}\label{prop:mainqidentity}
Given nonnegative integers $m,k,\ell$ with $k+2\leq \ell\leq m+1$,
\[ \sum_{i=0}^{\min(k+2,m+1-\ell)}(-1)^i q^{\binom{i}{2}} \qbin{k+2}{i}{q} \qbin{m+1-i}{\ell}{q}=q^{(k+2)(m+1-\ell)} \qbin{m-k-1}{\ell-2-k}{q}.\]
\end{prop}
\begin{proof}
We will show that the proposition is a consequence of a well known hypergeometric series identity. First, we put it in standard form. Let $$t_j=(-1)^j q^{\binom{j}{2}} \qbin{k+2}{j}{q} \qbin{m+1-j}{\ell}{q}.$$ Then, the ratio of consecutive terms in the summation is $\frac{t_{j+1}}{t_j}$ which after some simplification can be shown to be equal to $\frac{-q^{k-\ell+2}(1-q^{-2-k}q^j)(1-q^{\ell-m-1}q^j)}{(1-q^{j+1})(1-q^{-m-1}q^j)}.$ Thus we can write the summation appearing on the left hand side of the proposition as a hypergeometric series, \begin{equation}\label{eq:hgs} \setlength\arraycolsep{1pt}
\qbin{m+1}{\ell}{q}{}_2 \Phi_1\left(\begin{matrix}q^{-2-k}, &q^{\ell-m-1}\\& q^{-m-1}\end{matrix}\bigg\rvert q;q^{k-\ell+2}\right).\end{equation}
The $q$-Vandermonde hypergeometric series identity asserts that
$${}_2 \Phi_1\left(\begin{matrix}A, &q^{-n}\\& C\end{matrix}\bigg\rvert q;\frac{C}{Aq^{-n}}\right)=\frac{(\frac{C}{A};q)_n}{(C,q)_n}.$$ Applying this to (\ref{eq:hgs}) yields \begin{equation}\label{eq:evalhgs} \qbin{m+1}{\ell}{q}\frac{(q^{k-m+1};q)_{m-\ell+1}}{(q^{-m-1};q)_{m-\ell+1}}.\end{equation}
Using the identity
\begin{equation}(q^{-n};q)_m=q^{m(m-2n-1)/2}(-1)^m(q^{n-m+1};q)_m, \end{equation}
equation (\ref{eq:evalhgs}) can be simplified to
\[ q^{(k+2)(m+1-\ell)} \qbin{m+1}{\ell}{q}\frac{(q^{\ell-k-1};q)_{m-\ell+1}}{(q^{\ell+1};q)_{m-\ell+1}}, \] which can easily be manipulated to become the right hand side of the proposition.
\end{proof}
The identity given in Proposition (\ref{prop:mainqidentity}) gives rise to the following corollary under the substitution $m\rightarrow m-1+\ell$.
\begin{cor}\label{cor:mainqidentity}
\[ \sum_{i=0}^{\min(k+2,m)} (-1)^iq^{\binom{i}{2}}\qbin{k+2}{i}{q}\qbin{m+\ell-i}{\ell}{q}=q^{(k+2)m}\qbin{m+\ell-(k+2)}{\ell-(k+2)}{q}. \]
\end{cor}
What follows next is a proposition which completely verifies Remmel's conjectures. Namely, that given the coefficients defined in (\ref{eq:coeffs}), both (\ref{eq:RemmelConj1}) and (\ref{eq:RemmelConj2}) hold.
\begin{prop}\label{prop:simplifyingproduct}Given nonnegative integers $k,m,n,\ell$ with $k+2\leq \ell \leq m+1\leq n$,
\begin{enumerate}
\item $\displaystyle \sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i-j+1})=q^{m+\binom{k+2}{2}-(k+2)\ell+1}\qbin{\ell-2}{k}{q}\qbin{m-1}{\ell-2}{q}(q;q)_\ell$ \label{prop:simpa}
\item $\displaystyle \sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i+j-1})=q^{m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{m+\ell-(k+2)}{m}{q}(q;q)_\ell$ \label{prop:simpb}
\end{enumerate}
\end{prop}
\begin{proof}
First, it is worth noting that by our definitions $c_i^{k,m}=0$ when either $i>m+1$ or $i<m-k-1$.
To prove part~\ref{prop:simpa}, notice that when $i<\ell$ the product contains a 0 term. Thus,
\begin{eqnarray*}
&&\sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i-j+1})\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i-j+1})\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} c_i^{k,m}\frac{(1-q)\cdots(1-q^{i-1})}{(1-q)\cdots(1-q^{i-\ell})}\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} c_i^{k,m}\qbin{i}{\ell}{q}\frac{(q;q)_\ell}{1-q^i}\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} (-1)^{m+1-i}q^{\binom{m+1-i}{2}-(k+1)m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{k+2}{m+1-i}{q}\qbin{i}{\ell}{q}(q;q)_\ell\\
&=&\sum_{i=0}^{\min(k+2,m+1-\ell)} (-1)^{i}q^{\binom{i}{2}-(k+1)m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{k+2}{i}{q}\qbin{m+1-i}{\ell}{q}(q;q)_\ell\\
&=&q^{\binom{k+1}{2}-m(k+1)}\qbin{m-1}{k}{q}\sum_{i=0}^{\min(k+2,m+1-\ell)} (-1)^{i}q^{\binom{i}{2}}\qbin{k+2}{i}{q}\qbin{m+1-i}{\ell}{q}(q;q)_\ell.\\
\end{eqnarray*}
Then using Proposition \ref{prop:mainqidentity},
\begin{eqnarray*}
&=&q^{\binom{k+1}{2}-m(k+1)}\qbin{m-1}{k}{q} q^{(k+2)(m+1-\ell)} \qbin{m-k-1}{\ell-2-k}{q}(q;q)_\ell\\
&=&q^{m+\binom{k+2}{2}-(k+2)\ell+1} \qbin{\ell-2}{k}{q}\qbin{m-1}{\ell-2}{q}(q;q)_\ell.\\
\end{eqnarray*}
This completes the proof of part~\ref{prop:simpa}. To prove part~\ref{prop:simpb},
\begin{eqnarray*}
&&\sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i+j-1})\\
&=&\sum_{i=\max(m-k-1,1)}^{m+1} c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i+j-1})\\
&=&\sum_{i=0}^{\min(k+2,m)} c_{m+1-i}^{k,m}\prod_{j=2}^{\ell}(1-q^{m+j-i})\\
&=&q^{\binom{k+1}{2}-(k+1)m}\qbin{m-1}{k}{q}\sum_{i=0}^{\min(k+2,m)} (-1)^iq^{\binom{i}{2}}\qbin{k+2}{i}{q}(1-q^{m+1-i})\prod_{j=2}^{\ell}(1-q^{m+j-i})\\
&=&q^{\binom{k+1}{2}-(k+1)m}\qbin{m-1}{k}{q}\sum_{i=0}^{\min(k+2,m)} (-1)^iq^{\binom{i}{2}}\qbin{k+2}{i}{q}\qbin{m+\ell-i}{\ell}{q}(q;q)_\ell\\
&=&q^{\binom{k+1}{2}-(k+1)m}\qbin{m-1}{k}{q}q^{(k+2)m}\qbin{m+\ell-(k+2)}{\ell-(k+2)}{q}(q;q)_\ell\\
&=&q^{m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{m+\ell-(k+2)}{m}{q}(q;q)_\ell
\end{eqnarray*}
The next to last step is justified by Corollary \ref{cor:mainqidentity}.
\end{proof}
\section{Additional investigations}
To begin the investigation of whether our ``combinatorial side'' was related to that of Rhodes-Shimozono, we first expanded the symmetric function $\omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}\Big)[X(1-q)]$ in terms of the basis $\{P_\mu(X,q^{-1})\}_\mu$. This is done in the following theorem.
\begin{thm}\label{thm:genSymSideExp}
\begin{equation}\label{eq:thmGenSymSideExp}
\omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}
\Big)[X(1-q)]
\enskip = \enskip q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over 1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
P_\mu(X,q^{-1})
\end{equation}
\end{thm}
\begin{proof}
We begin with the following expansion of $e_n$ (Lemma 2.1 in \cite{GHRY}),
\[ e_n (X) = \sum_{\mu\vdash n}\frac{(1-q)(1-t)\widetilde{H}_\mu (X;q,t) \Pi_\mu ^{\prime}(q,t)B_\mu (q,t) }{w_\mu (q,t)}.
\]
Recognizing that the left hand side does not contain the indeterminates $q$ and $t$, we can interchange them and obtain
\[ e_n (X) = \sum_{\mu\vdash n}\frac{(1-q)(1-t)\widetilde{H}_\mu (X;t,q) \Pi_\mu ^{\prime}(t,q)B_\mu (t,q) }{w_\mu (t,q)}.
\]
Then using the definition of $\Delta'$, and setting $t=0$, we have
\begin{equation}\label{eq:expOfEn}\Delta_{s_{\nu}}'e_n\Big|_{t=0} = \sum_{\mu\vdash n}\frac{(1-q)s_{\nu}\big[B_\mu(0,q) -1 \big]\widetilde{H}_\mu (X;0,q) \Pi_\mu ^{\prime}(0,q)B_\mu (0,q) }{w_\mu (0,q)}.
\end{equation}
In \cite{GHRY}, it was noted that
\begin{align*}
B_\mu (0,q) &= 1+q+\dotsb +q^{\ell (\mu)-1}=\frac{1-q^{\ell(\mu)}}{1-q},\\
\Pi_\mu ' (0,q) &= (q;q)_{\ell (\mu)-1},\\
w_\mu (0,q) &= \prod_{c\in \mu}q^{l(c)} \cdot \prod_{\substack{c\in \mu\\ a(c)=0}}(1-q^{l(c)+1})\cdot \prod_{\substack{c\in \mu \\ a(c)>0}}(-q^{l(c)+1})\\
&= (-1)^{n-\ell (\mu)}q^{2n(\mu)+n-\sum_i \binom{m_i(\mu) +1}{2}}\prod_{i}(q;q)_{m_i(\mu)},
\end{align*}
where $(q;q)_m = (1-q)\cdots (1-q^m)$.
Substituting these into (\ref{eq:expOfEn}) and simplifying gives
\[
\Delta_{s_{\nu}}'e_n\Big|_{t=0} = \sum_{\mu\vdash n}(-1)^{n-\ell (\mu)}s_{\nu}\big[q+q^2+\dotsb +q^{\ell (\mu)-1} \big]q^{-2n(\mu)-n+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \widetilde{H}_\mu (X;0,q).
\]
Replacing $X$ by $X(1-q)$ and factoring a $q$ out of the plethystic evaluation, the right hand side becomes
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{n-\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-2n(\mu)-n+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \widetilde{H}_\mu (X(1-q);0,q),
\]
and then expanding $\widetilde{H}_\mu (X(1-q);0,q)$ yields
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{n-\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-2n(\mu)-n+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \sum_{\lambda\vdash n}s_\lambda\left[X(1-q)\right]\widetilde{K}_{\lambda , \mu}(q).
\]
But, since $s_\lambda\left[X(1-q)\right]=(-q)^ns_{\lambda'}\left[X(1-1/q)\right]$ and $\widetilde{K}_{\lambda , \mu}(q) = q^{n(\mu)}K_{\lambda,\mu}(q^{-1})$, we can now apply $\omega$ and eventually arrive at
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-n(\mu)+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \sum_{\lambda\vdash n}s_{\lambda}\left[X(1-q^{-1})\right]K_{\lambda , \mu}(q^{-1}).
\]
We will next need two facts stated in \cite{GHRY}:
\begin{equation}\label{eq:fact1fromGHRY}
Q_\mu(X,q) = \sum_{\lambda \vdash n}s_\lambda\left[X(1-q)\right]K_{\lambda,\mu}(q)
\end{equation}
and
\begin{equation}\label{eq:fact2fromGHRY}
P_\mu(X,q^{-1}) = \frac{(-1)^{\ell(\mu)} q^{\sum_{i}{m_i(\mu)+1\choose 2}}}{\prod (q;q)_{m_i(\mu)}}Q_\mu(X,q^{-1}),
\end{equation}
where $\mu$ is a partition of $n$.
Applying (\ref{eq:fact1fromGHRY}) at $q^{-1}$ gives
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-n(\mu)+\sum_i \binom{m_i(\mu) +1}{2}}
{(q;q)_{{\ell}(\mu)}
\over
\prod_{i=1}^{{\ell}(\mu)}(q;q)_{m_i(\mu)}
}\,\, Q_\mu(X,q^{-1}),
\]
and then applying (\ref{eq:fact2fromGHRY}) we prove the theorem, namely,
$$
\omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}
\Big)[X(1-q)]
\enskip = \enskip q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over 1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
P_\mu(X,q^{-1}).
$$
\end{proof}
\begin{cor}
The identity (\ref{eq:equivalentPoly}), namely,
\[
{\it LHS}_{k,m,n}[X,q]=
q^{m +{k+1\choose2}}
\sum_ {\mu\vdash n}
\Big[{ m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q q^{-n(\mu)}(q;q)_{\ell (\mu)} P_\mu [X;q^{-1}],
\]
is none other but a specialization of Theorem \ref{thm:genSymSideExp}, at $\nu=(m-k,1^k)$.
\end{cor}
\vskip -.12in
\begin{flushright}
\begin{tikzpicture}[scale=0.46, line width=1pt]
\draw (0,0) grid (1,-5);
\draw (1,-4) grid (4,-5);
\node at (.5,-.5) {-4};
\node at (.5,-1.5) {-3};
\node at (.5,-2.5) {-2};
\node at (.5,-3.5) {-1};
\node at (.5,-4.5) {0};
\node at (1.5,-4.5) {1};
\node at (2.5,-4.5) {2};
\node at (3.5,-4.5) {3};
\node[right] at (2,-.5) {$m$-$k$-$1$};
\draw[->](2,-.5) -- (1,-.5);
\node[left] at (2.75,-3) {$k$};
\draw[->](2.5,-3.25) -- (3.5,-4);
\node at (2,-5.5) {$c(x)$};
\draw (6,0) grid (7,-5);
\draw (6,-4) grid (10,-5);
\node at (6.5,-.5) {1};
\node at (6.5,-1.5) {2};
\node at (6.5,-2.5) {3};
\node at (6.5,-3.5) {4};
\node at (6.5,-4.5) {8};
\node at (7.5,-4.5) {3};
\node at (8.5,-4.5) {2};
\node at (9.5,-4.5) {1};
\node[right] at (7.75,-2.25) {$m$-$k$-$1$};
\draw[->](8,-2.5) -- (6.75,-3.5);
\node[right] at (7.85,-3.5) {$m$};
\draw[->](8,-3.5) -- (6.75,-4.5);
\node at (8,-5.5) {$h(x)$};
\end{tikzpicture}
\end{flushright}
\vskip -1.6in
\vskip .3in
\begin{proof}
Recall that the definition of the left hand side of (\ref{eq:equivalentPoly}) is
\vskip .1in
\enskip\ess\enskip$
{\it LHS}_{k,m,n}[X,q]= \omega\Big(\Delta_{s_{m-k,1^k}}'e_n\Big|_{t=0} \Big)[X(1-q)]
$
\vskip .1in
\noindent
Now the Macdonald formula for the plethystic evaluation of $s_\lambda$
\vskip .1in
\noindent
at $ 1+q+\cdots+q^{n-1}$ is
\vskip -.12in
$$
s_\lambda[1+q+\cdots+q^{n-1}]
= q^{n(\lambda)}\Big[{n \atop \lambda'} \Big]_q
$$
\vskip -.16in
\noindent
where
$$
\Big[{n \atop \lambda } \Big]_q
\enskip = \enskip \prod_{x\in \lambda}{ 1-q^{n-c(x)} \over 1-q^{h(x)}}
$$
With $c(x)$ and $h(x)$ the {\it content} and the {\it hook} of cell $x\in \lambda$.
\vskip .06truein
Now for $\lambda=(m-k,1^k)$ we have $n(\lambda)={k+1\choose 2}$ and
$\lambda'=(k+1,1^{m-k-1})$. We thus obtain
\begin{align*}
s_{m-k,1^k}[1+q+\cdots+q^{\ell-2}]&=
q^ {k+1\choose 2}
{
(1-q^{\ell-1+0 })\cdots (1-q^{\ell-1 +m-k-1})(1-q^{\ell-1-1})\cdots (1-q^{\ell-1-k})
\over
(q;q)_k(1-q^m)(q;q)_{m-k-1}
}\\
&=
q^ {k+1\choose 2}
{ (q^{{\ell}-k-1},q)_m
\over
(q;q)_k(1-q^m)(q;q)_{m-k-1}
}
\end{align*}
See the illustration above where the statistics $c(x)$ and $h(x)$ are computed for the hook partition $(m-k,1^k)$
{\vskip .125truein}
\noindent
Notice next that we have
\begin{align*}
\Big[{ m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q &\enskip = \enskip
{1
\over
(q;q)_{k}(q;q)_{m-1-k}
}
\enskip
{(q;q)_{m +\ell(\mu) -(k+2)}
\over
(1-q^m)(q;q)_{\ell(\mu) -(k+2)}}
\\
&\enskip = \enskip
{(q^{{\ell}-k-1},q)_m
\over
(q;q)_{k}(1-q^m)(q;q)_{m-1-k}
}
\end{align*}
\noindent
To prove that for $\nu=(m-k,1^k)$
(\ref{eq:thmGenSymSideExp}) reduces to (\ref{eq:equivalentPoly}), we need only verify the equality
\[
q^{m }
s_{m-k,1^k}\big[\tttt{1-q^{{\ell} -1}\over 1-q }\big]\enskip = \enskip q^{m +{k+1\choose2}}
\Big[{ m -1\atop k}\Big]_q
\Big[{m +\ell -(k+2)\atop m}\Big]_q
\]
However, the above calculations show exactly that.
\end{proof}
Theorem \ref{thm:genSymSideExp} provides an expansion of the symmetric function side in terms of the\\ $\{P_\mu(X,q^{-1})\}$ basis. We now seek an appropriate ``combinatorial side'' by expanding the same symmetric function in terms of the $\{P_\mu(X,q)\}$ basis. In order to do this, we will use a special evaluation given in the following theorem.
\begin{thm}\label{thm:specEval}
\begin{equation}\label{eq:thmSpecEval}
s_\nu\Big[\tttt{1-q^{j-1}\over 1-q }\Big]
\enskip = \enskip
\sum_{k={\ell}(\nu)}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{j-1}
\over (q;q)_{j-1-k}}
\end{equation}
\end{thm}
\begin{proof}
Recall that from \cite{Mac}, we get the identity
\[
s_\nu[X]\enskip = \enskip \sum_{\rho\vdash |\nu|}K_{\nu,\rho}(q)P_\rho[X,q],
\]
which can be also written as
\[
s_\nu[X]\enskip = \enskip \sum_{\rho\vdash |\nu|}K_{\nu,\rho}(q){Q_\rho[X,q]\over \prod_{i=1}^m
(q;q)_{m_i(\rho)}}
\]
and $X{ \rightarrow } X/(1-q)$ gives
\[
s_\nu\big[\tttt{X\over 1-q}\big]\enskip = \enskip \sum_{\rho\vdash|\nu|}H_\rho[X ;q] { K_{\nu,\rho}(q)\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}.
\]
Now the replacement $X{ \rightarrow } 1-q^{j-1}$
yields
\[
s_\nu\big[\tttt{ 1-q^{j-1}\over 1-q}\big]\enskip = \enskip \sum_{\rho\vdash|\nu|}H_\rho[ 1-q^{j-1} ;q] { K_{\nu,\rho}(q)\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
\]
This can be rewritten in the form
\begin{equation}\label{eq:thmSpecEvalRew}
s_\nu\big[\tttt{1-q^{j-1}\over 1-q }\big]
\enskip = \enskip
\sum_{k=1}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,H_\rho[1-q^{j-1};q ]
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}.
\end{equation}
Now, the Macdonald reciprocity in the Hall-Littlehood case yields
\[
H_\rho[1-u;q ]\enskip = \enskip q^{n(\rho)}
\prod_{s=1}^{{\ell}(\rho)}
(1-u/q^{ s-1 } ).
\]
In particular, the replacement $u{ \rightarrow } q^{j-1}$ gives (for ${\ell}(\rho)=k$)
\[
H_\rho[1-q^{j-1};q ]\enskip = \enskip q^{n(\rho)}
\prod_{s=1}^{k}
(1- q^{j -s } )\enskip = \enskip q^{n(\rho)}(1-q^{j-k})\cdots (1-q^{j-1})
\]
Thus (\ref{eq:thmSpecEvalRew}) becomes
\[
s_\nu\big[\tttt{1-q^{j-1}\over 1-q }\big]
\enskip = \enskip
\sum_{k=1}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{j-1}
\over (q;q)_{j-1-k}}
\]
Since the coefficient $K_{\nu,\rho}(q)$ fails to vanish only when $\nu\ge\rho$ in dominance,
the hypothesis ${\ell}(\rho)=k$ forces ${\ell}(\nu)\le k$. This given we can write
\[
s_\nu\big[\tttt{1-q^{j-1}\over 1-q }\big]
\enskip = \enskip
\sum_{k={\ell}(\nu)}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{j-1}
\over (q;q)_{j-1-k}}
\]
\end{proof}
\vskip -.35in
Now Theorem \ref{thm:genSymSideExp} gives that our symmetric function side has the expansion
\[
LHS_{\nu,n}[X,q]
\enskip = \enskip
q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over 1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
P_\mu(X,q^{-1}).
\]
Using Theorem \ref{thm:specEval}, this can be rewritten as
\[
q^{|\nu| } \sum_{\mu\vdash n}
\sum_{k={\ell}(\nu)}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{{\ell}(\mu)-1}
\over (q;q)_{{\ell}(\mu)-1-k}}
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
P_\mu(X,q^{-1}),
\]
or better,
\[
q^{|\nu| }
\sum_{k={\ell}(\nu)}^{|\nu|}(q;q)_k
\sum_{\multi{|\rho|=|\nu\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}\,
\sum_{\mu\vdash n}\Big[{{\ell}(\mu)-1\atop k}\Big]_q
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
P_\mu(X,q^{-1}).
\]
Recall that in \cite{GHRY}, for the classical case of the Delta conjecture at $t=0$, we proved the identity
\[
\sum_{\mu\vdash n}q^{- n(\mu)}
P_\mu[X;q^{-1}]
\Big[{ {\ell}(\mu)-1\atop k }\Big]_q
(q;q)_{{\ell}(\mu)}
\enskip = \enskip q^{-k(k+1)}
(q;q)_{k+1}\hskip -.08in\uu\sum_{\multi{\mu\vdash n\, ;\, {\ell}(\mu)=k+1}}\hskip -.08in\uu
q^{n(\mu)}
P_\mu[X;q]
\]
This permits us to obtain
the expansion of the symmetric function side in terms of the basis
$\big\{P_\mu[X;q]\big\}_\mu$ and use our recipe to obtain what we would label as the ``combinatorial side''. Namely,
\begin{align*}
RHS_{\nu,n}[X,q]
\enskip = \enskip&
q^{|\nu| }
\sum_{k={\ell}(\nu)}^{|\nu|}(q;q)_k
\sum_{\multi{|\rho|=|\nu\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}\,\times
\\
&\hskip .5truein\bigsp
\times \,
q^{-k(k+1)}
(q;q)_{k+1}\hskip -.08in\uu\sum_{\multi{\mu\vdash n\, ;\, {\ell}(\mu)=k+1}}\hskip -.08in\uu
q^{n(\mu)}
P_\mu[X;q]
\end{align*}
Additionally, the last sum appearing in ${\it RHS}_{\nu,n}[X,q]$ was proved to have a hook Schur function expansion in \cite{GHRY}.
We have thus proved the following generalization of (\ref{eq:symSideEqualsCombSide}).
\begin{thm}It is not only true that
\begin{equation}\label{eq:thmGeneralSymSideEqualsCombSide}
{\it LHS}_{\nu,n}[X,q]\enskip = \enskip {\it RHS}_{\nu,n}[X,q],
\end{equation}
but also that the Schur expansion of both sides contains only hook
Schur functions.
\end{thm}
\begin{remark}
The right hand side of this identity is none one other than the Rhodes-Shimozono ``combinatorial side'' transformed to our set up, (see the righthand side of Theorem 1.2 in \cite{HagRhoShi}).
\end{remark}
\begin{remark}
In \cite{GHRY} (see Lemma 4.2) it is shown that
$$
h_n[X(1-u)] \enskip = \enskip (1-u)\sum_{s=0}^{n-1}(-u)^s
s_{(n-s,1^s)}[X]
$$
It follows from this identity that any symmetric polynomial whose Schur functions expansion contains only hook Schur functions may be expanded as linear combination of the shifted Cauchy kernel $h_n(X(1-q^i)]$. What forced Remmel to restrict himself to $\Delta_{s_\nu} e_n$ in the hook case of $\nu$ is that in the hook case the needed coefficients are products of $q$-analogues of integers. This facilitated conjecturing their exact nature. With the wisdom of hindsight we can now explain Haglund result as due to the fact that Schur function expansions of the appropriately modified polynomials
$\Delta_{s_\nu}e_n$ contain only hook Schur functions in full generality. However, this
circumstance is only an artifact of the specialization at $t=0$. In fact, without this specialization, computer data reveals the dimension of the space spanned by the polynomials $\Delta_{s_\nu}e_n$ to be much larger than $n$.
The data suggests that, more likely, this dimension is the number of partitions of $n$.
\end{remark}
\section{Acknowledgements}
The authors want to express their gratitude to Jim Haglund for making his result and his argument available to us, (unpublished manuscript). We are also grateful to Dennis Stanton for providing us with the tools we needed to be able to prove (in section 3) Remmel's $q$-binomial conjectures.
| {'timestamp': '2018-01-24T02:04:17', 'yymm': '1801', 'arxiv_id': '1801.07385', 'language': 'en', 'url': 'https://arxiv.org/abs/1801.07385'} |
\section{Time-evolution in Quantum Mechanics with
A certain climax of our present letter comes when we address the
problem of the time-evolution in Quantum Mechanics with $\Theta
\neq I$ and with a manifest time-dependence in all our operators.
Once we prepare an initial state as a normalized vector
$|\varphi(t) \!\succ \in {\cal H}_{phys}^{(stand)}$ at $t=0$, we can
rely only on our understanding of the evolution caused by the
auxiliary self-adjoint Hamiltonians $h(t)$. In particular, we may
immediately solve any time-dependent Schr\"{o}dinger equation
\begin{equation}
{\rm i}\,\partial_t |\varphi(t)\!\!\succ \ \ =\ h(t)\,
|\varphi(t)\!\!\succ\,,\ \ \ \ \ \
|\varphi(t)\!\!\succ \ \ = \ u(t)\,|\varphi(0)\!\!\succ
\label{timeq}
\end{equation}
and we are sure that the related evolution operator given by
equation
\begin{equation}
{\rm i}\partial_t u(t)=h(t)\,u(t)\,
\label{seh}
\end{equation}
is {\em certainly} unitary in ${\cal H}_{phys}^{(stand)}$,
\[
\prec\! \varphi(t) \,|\,
\varphi(t)\!\succ=
\prec\! \varphi(0) \,|\,
\varphi(0)\!\succ\,.
\]
In the next step of our considerations we recollect the pull-backs
$|\Phi(t)\rangle=\Omega^{-1}(t)\, |\varphi(t)\!\!\succ$ and $\langle\!\langle
\Phi(t)\,|=\prec\!\!\varphi(t)\,|\,\Omega(t)$ carrying, by
assumption, their own time dependence. It is represented,
formally, by the ``right-action" evolution rule
\begin{equation}
|\Phi(t)\rangle=U_R(t)\, |\Phi(0)\rangle\,,\ \ \ \ \ \
U_R(t)=\Omega^{-1}(t)\,u(t)\,\Omega(0)
\end{equation}
accompanied by its ``left-action" parallel
\begin{equation}
|\Phi(t)\rangle\!\rangle=U_L^\dagger(t)\, |\Phi(0)\rangle\!\rangle\,,\ \ \ \ \ \
U_L^\dagger(t)=\Omega^\dagger(t)\,u(t)\,
\left [\Omega^{-1}(0)\right ]^\dagger\,.
\end{equation}
The respective non-Hermitian analogues of the Hermitian evolution
rule~(\ref{seh}) are now obtained by the elementary
differentiation and insertions yielding the two separate
differential operator equations
\begin{equation}
{\rm i}\partial_t U_R(t)=
-\Omega^{-1}(t)
\left [{\rm i}\partial_t\Omega(t)
\right ]\, U_R(t)+H(t)\, U_R(t)\,
\end{equation}
and
\begin{equation}
{\rm i}\partial_t U_L^\dagger(t)=
H^\dagger(t)\, U_L^\dagger(t)
+
\left [{\rm i}\partial_t\Omega^\dagger(t)
\right ]\,
\left [
\Omega^{-1}(t)
\right ]^\dagger\, U_L^\dagger(t)\,.
\end{equation}
We are prepared to verify what happens with the norm $ \langle\!\langle
\Phi(t)\,|\,\Phi(t)\rangle $ of states which evolve with time in the
physical space ${\cal H}^{(\Theta)}$. The elementary
differentiation gives
\begin{eqnarray}
{\rm i}\partial_t\langle\!\langle \Phi(t)\,|\,\Phi(t)\rangle
={\rm i}\partial_t
\langle\!\langle \Phi(0)\,|\,U_L(t)\,U_R(t) \,|\,\Phi(0)\rangle=
\\
=\langle\!\langle \Phi(0)\,|\,
\left [ {\rm i}\partial_t
U_L(t)
\right ]
\,U_R(t) \,|\,\Phi(0)\rangle
+\langle\!\langle \Phi(0)\,|\,U_L(t)\,
\left [ {\rm i}\partial_t
U_R(t)
\right ]
\,|\,\Phi(0)\rangle=
\\
= \langle\!\langle \Phi(0)\,|\, U_L(t)\,
\left [ -H(t)+\Omega^{-1}(t)
\left [{\rm i}\partial_t\Omega(t)
\right ]
\right ]
\,U_R(t) \,|\,\Phi(0)\rangle+\\
+\langle\!\langle \Phi(0)\,|\,U_L(t)\,
\left [ H(t)
-\Omega^{-1}(t)
\left [{\rm i}\partial_t\Omega(t)
\right ]
\right ]\,U_R(t)
\,|\,\Phi(0)\rangle=0\,.
\end{eqnarray}
We see that the norm remains constant also in ${\cal
H}^{(\Theta)}$. In the other words, the time-evolution of the
system specified by the quasi-Hermitian and time-dependent
Hamiltonian $H(t)$ is unitary.
Once we abbreviate $\dot{\Omega}(t)\equiv \partial_t\Omega(t)$,
our latter observation can be rephrased as an explicit
specification of {\em the same} time-evolution generator
\begin{equation}
H_{(gen)}(t)=H(t) -{\rm i}\Omega^{-1}(t)
\dot{\Omega}(t)
\end{equation}
entering {\em both} the present quasi-Hermitian updates
\begin{eqnarray}
{\rm i}\partial_t|\Phi(t)\rangle
=H_{(gen)}(t)\,|\Phi(t)\rangle\,,
\label{SEA}\\
{\rm i}\partial_t|\Phi(t)\rangle\!\rangle
=H_{(gen)}^\dagger(t)\,|\Phi(t)\rangle\!\rangle\,
\label{SEbe}
\end{eqnarray}
of the current textbook time-dependent Schr\"{o}dinger equation
for wave functions. Such a confirmation of the preservation of the
overall unitarity of the evolution (at the cost of a not too
difficult and explicit redefinition $H \to H_{(gen)}$ of the
generator of time evolution which remains the same for both the
left and right action) may be read as really good news for the
theory.
We arrived at the answer to the question given in the title. In a
concise discussion we should emphasize that such an answer is
quite surprising because the standard connection between the
Hamiltonian and the time evolution of the system is usually
interpreted as a certain ``first principle" of the quantum
dynamics. In such a context our present constructive argument
against the current postulate $H_{(gen)}=H$ should be perceived as
a formal foundation of an apparently counterintuitive innovative
idea that once we admit some dynamically motivated time dependence
of the Hamiltonian itself, we might also contemplate a parallel
assumption of some equally arbitrary, phenomenologically motivated
time dependence of some other observable quantities.
The key reason for the possible mathematical consistency as well
as for a formal acceptability of such a fairly unusual scenario
should be seen in the deep formal ambiguity of the metric
operator. In principle, this observation (made also in refs.
\cite{Geyer} and \cite{ali}) gives us a huge space and freedom for
the {\em time-dependent} variability of the metric
$\Theta=\Theta(t)$. Let us emphasize that in the generic case with
$\dot{H}(t)\neq 0$ the $t-$dependence of $H\neq H^\dagger$ becomes
immediately transferred to its left and right eigenvectors and,
subsequently (i.e., via formula (\ref{identify})), to the metric
$\Theta=\Theta(t)$ itself. Moreover, in parallel to the freedom of
our choice of the $t-$dependence of the Hamiltonian $H(t)$, there
exists, at least in principle, no mathematical obstruction to an
unrestricted time-dependence freedom introduced directly in all
the complex coefficients $\mu_n=\mu_n(t)$ (cf., once more,
eq.~(\ref{identify})).
We have to admit that we did not really expect that the ``old"
idea of a mapping $\Omega$ between spaces \cite{Geyer} (and, of
course, between Hamiltonian $H$ and its ``suitable" Hermitian
partner $h$) will survive so easily its transfer to the
time-dependent case with $\Omega=\Omega(t)$. At the same time,
having arrived at our final pair of the generalized quantum
time-evolution differential equations (\ref{SEA}) and (\ref{SEbe})
we find them quite natural and consistent. Summarizing our present
results, the Hamiltonian (i.e., energy-operator $H=H(t)$) was
assumed quasi-Hermitian (i.e., $H^\dagger(t) =
\Theta(t)\,H(t)\,\Theta^{-1}(t)$ at some nontrivial metric
$\Theta(t)=\Theta^\dagger(t)>0$). In parallel, some other
quasi-Hermitian operators $A_j$ of observables were assumed to
carry an independent time-dependence (i.e., we were allowed to
demand that $A_j^\dagger(t) = \Theta(t)\,A_j(t)\,\Theta^{-1}(t)$).
Under these assumptions we demonstrated that the evolution of the
system in question still remains unitary.
The latter observation throws new light upon some applications of
the models with $\Theta \neq I$ (cf., e.g., the ones reviewed in
\cite{Carl}) where the values of all the coefficients
$\mu_n=\mu_n(t)$ are fixed using a suitable phenomenological
postulate. In the other applications reviewed in \cite{Geyer} the
variability of $\mu_n$s is also being removed, step by step, via
an appropriate selection of some other operators decided to play
the role of observables. We may conclude that there is nothing
counterintuitive in the manifest ``decoupling" of $H_{(gen)}(t)$
from $H(t)$. It is fairly easily accepted immediately after one
imagines that the information about the quantum evolution {\em
can} be carried {\em not only} by the manifestly time-dependent
Hamiltonian $H(t)$ {\em but also} by some other manifestly
time-dependent observables $A_j(t)$, $j=1,2, \ldots$. Naturally,
the latter information must also influence the overall evolution
of the system.
In each of the eligible scenarios and approaches, the constraints
imposed upon the variability of the coefficients in $\Omega(t)$
may be considered controllable and at our disposal. After an
acceptance of such a concept of the time-dependence in quantum
dynamics, our present main {\em technical} contribution may be
seen in an almost elementary check of the mutual compatibility and
independence of the two ``input-information" hypotheses involving
$H=H(t)$ {\em and} $\Theta=\Theta(t)$.
\section*{Acknowledgement}
Work supported by GA\v{C}R, grant Nr. 202/07/1307, Institutional
Research Plan AV0Z10480505 and by the M\v{S}MT ``Doppler
Institute" project Nr. LC06002.
| {'timestamp': '2007-11-05T10:29:43', 'yymm': '0711', 'arxiv_id': '0711.0535', 'language': 'en', 'url': 'https://arxiv.org/abs/0711.0535'} |
\section{Introduction}
The fifth generation (5G) of the cellular systems, which will be introduced in the early 2020s, outperforms the previous generations in terms of data rates and capacity, in addition to supporting new communication protocols to deal with demanding requirements such as latency, reliability and efficiency in MTC~\cite{tullberg2016metis}. In MTC or machine-to-machine communications (M2M), data is automatically exchanged between two MTC nodes or an MTC node and a server where the human cooperation is minimized~\cite{shariatmadari2015machine}.
5G system outperforms current networks in terms of spectrum efficiency via decreasing the latency and guaranteeing reliability up to $99.999 \%$. Authors in~\cite{andrews2014will} discuss about the 5G technology in more details in terms of requirements and challenges such as data rate, latency, energy and cost issues.
\subsection{Machine-Type Communication}
In recent years, MTC has received much attention due to the vast applications in wireless communications via supporting transmission/reception of short data packets in comparison with the current communication systems which carry long data packets, and also being a cost-effective and energy efficient technology~\cite{lee2016packet}.
MTC technology facilitates wide range of applications and is characterized as massive MTC (mMTC) and ultra-reliable MTC (uMTC)~\cite{bockelmann2016massive}. In mMTC, where the blocklength is short, a huge number of devices in a certain domain are covered with low-rate and low-power connectivity, and also considerable reliability in order to cover critical situations, e.g. in sensor networks, smart meters, actuators, etc,~\cite{7041045},~\cite{singh2016selective}. In addition, timing constraints from few seconds to even extremely low end-to-end deadlines in particular applications, are critical concerns in MTC~\cite{biral2016impact}. In uMTC, the connection is supported by transferring the short data packets with ultra-high reliability and low latency, in the scope of less than a millisecond which can be a requirement in several applications such as cloud connectivity, road safety, industrial control and safe interconnection between vehicles~\cite{bockelmann2016massive},~\cite{7529226}. Therefore, owing to the vast applicability of MTC with short packets in cellular network infrastructure, covering a novel wireless mode, namely ultra-high reliable communication (URC) is a critical concern in 5G~\cite{schotten2014availability}.
Supporting the ultra-high reliability and low latency are crucial requirements in the upcoming networks in order to improve the security, functionality and the ability of the interaction between different types of communication units such as human-to-human, human-to-machine or machine-to-machine which creates novel business models and applications~\cite{tullberg2016metis}.
In MTC with short packet transmissions, since the length of metadata and information bits are similar, an unsuccessful encoding of the metadata increases the performance loss in the wireless communication systems, particularly under the FB regime~\cite{7529226}. There are several works which have studied different aspects of FB coding. For instance, authors in~\cite{5452208}, provide a tight approximation of achievable coding rate for a specified outage probability under the FB transmissions since majority of the theoretical results assume infinite blocklength (IFB) codes.
In~\cite{iscan2016comparison}, authors provide an overview of some coding schemes for FB which may be used in 5G. They indicate that the performance of wireless communications with short data packets considerably enhances via novel coding schemes with better minimum distance between the codewords; albeit, the decoders become computationally more sophisticated. Furthermore, authors in~\cite{park2012new}, study the impact of FB on the attainable coding rate. They propose a new power allocation strategy, called modified water-filling, over the AWGN channels.
They succeed to maximize the lower bound of the achievable coding rate in comparison to the conventional water-filling technique in FB regime
\subsection{Cooperative Relaying with Finite Blocklength}
In recent years, relaying has become an interesting and challenging research topic. The transmission from the source to the destination is improved with help of intermediate auxiliary nodes. Moreover, relaying is able to combat the wireless fading generated due to the multipath propagation through taking advantage of spatial diversity~\cite{zimmermann2005performance}. The most usual relaying protocol is decode-and-forward (DF), where the relay decodes, encodes and retransmits the message~\cite{zimmermann2005performance}.
Authors in~\cite{hu2015performance} investigate the performance of a two-phase relaying model under the FB and IFB regimes. They consider perfect CSI at the destination where the channel gains of the relaying link and the direct link are combined. They propose different schemes based on the different error scenarios where relaying has a performance advantage over the direct transmission (DT) under the FB regime, particularly when the blocklength is small. Authors in~\cite{swamy2015cooperative} propose cooperative relaying as a way to meet the high reliability and latency requirements.
In~\cite{7569613}, authors examine the performance of a multi-relay DF protocol with FB under the perfect CSI and partial CSI.
They indicate that with perfect CSI, the throughput of FB coding is higher than the throughput of IFB coding.
Furthermore, in our previous work~\cite{parisa2017}, we consider cooperative relaying scenarios with perfect CSI for Rayleigh fading channels. We illustrate how the reliability improves via relaying and how we can meet the URC requirements. We examine the probability of successful transmission as a function of the number of information bits and coded blocklength. Moreover, we provide an approximation to the outage probability in closed form. We show that relaying consumes less transmit power compared to DT in order to enable URC with FB coding.
\subsection{Our Contribution}
In this paper, we further investigate the reliability of relaying under the FB regime. We study protocols, namely selection combining (SC) and maximum ratio combining (MRC), where relaying outperforms DT. Furthermore, we illustrate the superiority of MRC over SC in terms of power consumption, latency and reliability under the FB regime and, different from~\cite{hu2015performance}, we present closed form expressions for the outage probability.
The following are considered the contributions of this paper.
\begin{itemize}
\item We provide the general expression of the outage probability for each relaying protocol considered in this work and incremental decode-and-forward relaying under general Nakagami-$m$ fading. Therefore, we generalize our previous work in~\cite{parisa2017}.
\item We extend the work in~\cite{hu2016blocklength} by comparing the MRC scenario with SC relaying. Also, a generalized closed-form expression for the outage probability is attained, thus extending \cite{6888474}.
\item We show the trade-off between information payload and blocklength under the UR region as a function of the quality of the link, which is reflected by the $m$ parameter of the Nakagami-$m$ distribution. We examine the minimum delay in relaying schemes required to perform in the UR region.
\end{itemize}
{\textbf{Notation}:} Throughout this paper, $f_{W}(\cdot)$ and $F_{W}(\cdot)$ are the probability density function (PDF) and cumulative distribution function (CDF) of a random variable (RV) W, respectively. $\operatorname{Q}^{-1}(\cdot)$ represents the inverse of the $\operatorname{Q}$-function which is defined as $\operatorname{Q(w)} = \int_{w}^{\infty} \tfrac{1}{\sqrt{2\pi}}\operatorname{e}^{-t^2/2}dt$~\cite[\S F.2]{miller2012probability}. The outage probability is denoted by $\epsilon$ and $\operatorname{E[\cdot]}$ is the expectation. $\Gamma(\cdot)$, $\Gamma(\cdot,\cdot)$ and $\operatorname {\cal P}(\cdot,\cdot)$ are the Gamma function~\cite[\S 6.1.1]{abramowitz1964handbook}, incomplete Gamma function~\cite[\S 6.5.2]{abramowitz1964handbook} and Gamma regularized function~\cite[\S 6.5.1]{abramowitz1964handbook}, respectively. $_1F_{1}(a;b;z)$ denotes the regularized hypergeometric function~\cite[\S 15.1.1]{abramowitz1964handbook}.
\section{System Model} \label{sc:system_model}
Consider a source $S$, a destination $D$ and a relay $R$ as illustrated in Fig.~\ref{fig:System Model}. We consider a normalized distance between $S$ and $D$ as $d=1$m, and that $R$ can move in a straight line between $S$ and $D$, while the distance between $D$ and $R$ is denoted by $\beta$. The links denoted as $X$, $Y$ and $Z$ represent the $S$-$R$, $R$-$D$, and $S$-$D$ links respectively, and each transmission takes $n_i$ channel uses where $i \in \{S, R\}$. This means that $n_S$ channel uses for $S$ and $n_R$ channel uses for $R$, respectively. In this scenario, first $S$ sends data to $D$ and $R$ which is known as broadcasting phase. Thereafter, in the relaying phase, $R$ forwards to $D$ only if it decoded the message correctly~\cite{hu2015performance},~\cite{7569613}.
The received signals are written as $y_{1}\!=\!
h_{1}x + w_{1}$ and $y_{2}\!=\! h_{2}x + w_{2}$ at $D$ and $R$ respectively, and if $R$ participates, the received signal at $D$ is $y_{3}\!=\! h_{3}x + w_{3}$, where $x$ is the transmitted signal with power $P$ and $w_i$ is the AWGN noise vector with power $N_0\!=\!1$ where $i\in \{X,Y,Z\}$. Nakagami-$m$ quasi static fading channels in the $S$-$D$, $S$-$R$ and $R$-$D$ links are denoted as $h_1$, $h_2$, and $h_3$, respectively.
\begin{figure}[b!]
\centering
\includegraphics[width=\columnwidth]{SystemModel2.pdf}
\caption{System model for the relaying scenario with a source ($S$), destination ($D$) and a relay ($R$). The links between $S$ to $D$, $S$ to $R$ and $R$ to $D$ are referred as the direct link $h_{1}$, the backhaul link $h_{2}$ and the relaying link $h_{3}$ each of which with $n_{i \in \{S, R\}}$ channel uses respectively.}
\label{fig:System Model}
\end{figure}
In a DF-based cooperative transmission, the instantaneous SNR for each link depends on the total power constraint $P=P_S+P_R=\eta P+ (1-\eta)P$ which is given by $\Omega_Z \!=\!\eta P|h_{1}|^2/N_{0}$, $\Omega_X \!=\!\eta P|h_{2}|^2/N_{0}$ and $\Omega_Y \!=\! (1-\eta) P|h_{3}|^2/N_{0}$, where $0 <\eta\leqslant1$ and the average SNR in each link is $\gamma_Z =\eta P / N_{0}$, $\gamma_X =\eta P / N_{0}$ and $\gamma_Y = (1-\eta) P / N_{0}$. We consider $\eta$ as the power allocation factor in order to provide a fair comparison between DT and cooperative schemes. Moreover, note that the PDF of Nakagami-$m$ fading is a function of two parameters: the fading parameter $m$ and the scale parameter $\Omega> 0$~\cite{simon2005digital}. The squared envelop is then Gamma distributed as $X \thicksim \mathsf{Gamma}(m_{i} , \Omega_{i} /m_{i} ) $ where $i\in \{X,Y,Z\}$.
\subsection{Coding Rate and Error Probability at Finite Blocklength } \label{sc:Finite Block length}
In this section, we revisit the concept of coding rate under the FB regime. In a communication system, first $k$ information bits are mapped to a sequence known as the codeword with a blocklength of $n$ symbols. Thereupon, the generated codeword is sent through the wireless channel. Subsequently, the decoder maps the channel outputs into an estimate of the information bits. Therefore, we can define the coding rate ${\cal R}$ as the ratio of the information bits $k$ to the number of channel symbols $n$ as ${\cal R} \!=\! k/n$~\cite{7529226}. The maximum coding rate ${\cal R}^*(n,\epsilon)$ in bits per channel use (bpcu) is~\cite{7529226}
\begin{equation} \label{eq:maximum rate}
{\cal R}^*(n,\epsilon) = C(\rho) - \sqrt{\frac {V(\rho)}{n}}\operatorname{Q}^{-1}(\epsilon)\log_{2}\operatorname{e},
\end{equation}
where $C$ and $V$ are the positive channel capacity and the channel dispersion, respectively, defined as $C(\rho) = \log_{2}(1+\rho)$ and $V(\rho) = \rho (2+\rho) \big/(1+\rho)^2$, where $\rho$ is the average SNR as $P/N_0$.
For the AWGN channel, $h_{i}=1$ and $\frac{1}{n}\sum_{i}^{n}|x_{i}|^2\leq\rho$ holds~\cite{7529226}.
\begin{figure*}[!t]
\begin{equation}\label{eq:outage_naka}
\epsilon\!=\! \dfrac{\mu\bigg[\theta\left(\Gamma\left(m,\dfrac{m\varrho}{\Omega}\right)\!-\!\Gamma\left(m,\dfrac{m\vartheta}{\Omega}\right)\right)\!+\!\Omega \bigg(\Gamma\left(1\!+\!m,\dfrac{m\vartheta}{\Omega}\right)
\!-\!\Gamma\bigg(1\!+\!m,\dfrac{m\varrho}{\Omega}\bigg)\bigg)\bigg]}{\sqrt{2\pi}}\!+\!\dfrac{\operatorname{\cal P}\left(m,\dfrac{m\vartheta}{\Omega}\right)\!+\!\operatorname{\cal P}\left(m,\dfrac{m\varrho}{\Omega}\right)}{2}.
\end{equation}
\hrule
\end{figure*}
The outage probability is then the expectation over the instantaneous SNR distribution as follows\footnote{This approximation is accurate for $n$ > 100, as proved for AWGN channels \cite[Figs. 12 and 13]{polyanskiy2010channel}, as well as for fading channels~\cite{yang2014quasi}.}~\cite{hu2016blocklength}
\begin{equation} \label{outage_fading}
\epsilon\approx \operatorname{E}\Bigg[\operatorname{Q}\Bigg(\sqrt{n}\frac{C(\rho|h|^2)-{\cal R}^{*}(n,\epsilon) }{\sqrt{V(\rho|h|^2)}}\Bigg)\Bigg].
\end{equation}
\subsection{Closed-form Expression of the Outage Probability} \label{Q-function}
The outage probability in (\ref{outage_fading}) does not have a closed-form expression, but it can be tightly approximated as we shall see next.
Let us first define $g(x)\!=\!\sqrt{n}\tfrac{C(\rho)-\cal R}{\sqrt{V(\rho)}}$. Then, we resort to a linearization of the Q-function~\cite{7106474},~\cite{6888474}.
\begin{equation}\label{eq:W(t)}
K(x) \approx \operatorname{Q}(g(x)) =
\left\{
\begin{array}{@{}ll@{}}
\ 1 & x\leqslant\varrho\\
\ \dfrac{1}{2}-\dfrac{\mu}{\sqrt{2\pi}}(x-\theta) & \varrho<x< \vartheta \;\;\;\;,\\
\ 0 & x\geq \vartheta
\end{array}\right.
\end{equation}
where, $\theta = \tfrac{2^{\cal R}-1}{P}$, $\vartheta=\theta+\sqrt{\tfrac{\pi}{2}\mu^{-2}}$, $\varrho=\theta-\sqrt{\tfrac{\pi}{2}\mu^{-2}}$ and $\mu\!=\!\sqrt{\frac{n}{2\pi}}(e^{2\cal R}-1)^{-\tfrac{1}{2}}$.
Therefore, the outage probability in (\ref{outage_fading}) is reformulated as
\begin{equation}\label{outage_fading_linear}
\epsilon= \operatorname{E}[\operatorname{Q}\left(g(x)\right)] = \int_{0}^{\infty}K(x)f_{X}(x)dx,
\end{equation}
where $f_{X}(x)$ represents the PDF of the SNR of link $X$.
\begin{proposition}\label{propose1}
The approximated outage probability of a communication link following Nakagami-$m$ fading is given in closed-form in (\ref{eq:outage_naka}) on top of the next page, where $\varrho$, $\vartheta$, $\mu$ and $\theta$ are defined in (\ref{eq:W(t)}).
\end{proposition}
\begin{proof}
See Appendix A.
\end{proof}
\begin{corollary}
Notice that for $m=1$, (\ref{eq:outage_naka}) reduces to the approximated outage probability for the Rayleigh fading as ~\cite[\S 5]{parisa2017}.
\end{corollary}
\noindent Thus, our proposed approximation in (\ref{eq:outage_naka}) includes~\cite[\S 5]{parisa2017}, which was first reported in~\cite{6888474}, as a special case.
\section{Performance Analysis of Relaying}\label{sec:performance}
In this section, we examine the outage probability in all relaying schemes considered in this work, as well as for the case of DT which is used for comparison purposes.
\subsection{Cooperative Transmission} \label{sec:Cooperative schemes}
\subsubsection{Selection Combining (SC)}
In this protocol $D$ tries to decode the messages sent by both $S$ and $R$, separately.
The overall outage probability becomes~\cite{5956530},~\cite{alves2012throughput}
\begin{equation}
\epsilon_{SC}=\epsilon_{Z}\left(\epsilon_{X}+\left(1-\epsilon_{X}\right)\epsilon_{Y}\right),
\end{equation}
where $\epsilon_{Z}$ , $\epsilon_{X}$ and $\epsilon_{Y}$ are the outage probabilities of the $S$-$D$, $S$-$R$ and $R$-$D$ links, respectively. The outages $\epsilon_{Z}$, $\epsilon_{X}$ and $\epsilon_{Y}$ are equal to (\ref{eq:outage_naka}) with $m$ and $\Omega$ replaced by $m_{Z}$ and $\Omega_{Z}$, $m_{X}$ and $\Omega_{X}$, and $m_{Y}$ and $\Omega_{Y}$ for each link, respectively.
\subsubsection{Maximum Ratio Combining (MRC)}\label{sec:MRC}
In this protocol, the transmissions from the source and from the relay are coherently combined at the receiver. Hence, the instantaneous SNR after the source and relay transmissions is $\Omega_{W} = \Omega_{Z} + \Omega_{Y}$~\cite{5956530},~\cite{alves2012throughput}. The outage probability is~\cite{alves2012throughput}
\begin{align}\label{eq:MRC_outage}
\epsilon_{MRC} =\epsilon_{Z}\left(\epsilon_{X}\!+\!\left(1\!-\!\epsilon_{X}\right)\frac{\epsilon_{SRD}}{\epsilon_{Z}}\right),
\end{align}
where $\epsilon_{SRD}$ is the outage probability after MRC of the transmissions from the source to the destination. Moreover, the ratio $\tfrac{\epsilon_{SRD}}{\epsilon_{Z}}$ comes from the conditioning of $\epsilon_{SRD}$ on the fact that the transmission from the source to the destination failed.
In order to calculate (\ref{eq:MRC_outage}), first we need to define the PDF of $W$, a Gamma random variable equal to $Z + Y$ , the sum of the instantaneous SNRs of the $S$-$D$ and $R$-$D$ links, as follows.
PDF of random variable $W$ is
\begin{equation}\label{pdf_mrc}
\begin{split}
f_{W}(w)&=\operatorname{exp}\left(-\dfrac{w}{\Omega_{y}}\right)w^{-1+m_{z}+m_{y}}\Omega_{z}^{-m_{z}}\Omega_{y}^{-m_{y}}\\ &\times _1F_{1}\bigg(m_{z},m_{z}+m_{y},w\left(\dfrac{1}{\Omega_{y}}-\dfrac{1}{\Omega_{z}}\right)\bigg),
\end{split}
\end{equation}
{\noindent where $\Omega_{Z}$, $\Omega_{Y}$ and $m_{Z}$, $m_{Y}$ are the instantaneous SNRs and the fading severity of the $S$-$D$ and $R$-$D$ links, respectively.}
\begin{proposition}
The outage probability after combining the source and relay transmissions, $\epsilon_{SRD}$, is
\begin{equation}\label{mrc_SRD}
\begin{split} \epsilon_{SRD}&=\int_{0}^{\varrho}f_W{\left(w|m,\Omega\right)}dw+\int_{\varrho}^{\vartheta}\left(\tfrac{1}{2}\!-\!\tfrac{\mu}{\sqrt{2\pi}}(w\!-\!\theta)\right)\\
&\times f_W{\left(w|m,\Omega\right)}dw.
\end{split}
\end{equation}
where, $\vartheta$ and $\varrho$ are specified in (\ref{eq:W(t)})
\begin{proof}
By plugging (\ref{pdf_mrc}) into (\ref{outage_fading_linear}) and multiplying with $K(x)$, we attain (\ref{mrc_SRD}). Solving (\ref{mrc_SRD}) is difficult and it does not have a closed form expression. Hence, we only work with its numerical integration whenever $m>1$.
\end{proof}
\end{proposition}
Since (\ref{mrc_SRD}) does not have a closed form expression for the outage probability under general Nakagami-$m$, we analyze the particular case when $m=1$, which corresponds to Rayleigh fading. Therefore, the PDF of $W$ becomes~\cite[\S 10]{parisa2017}.
\begin{lemma}\label{proposition2}
For $m=1$, the outage probability, $\epsilon_{SRD}$, is equal to~\cite[\S 6]{parisa2017}.
\end{lemma}
\begin{proof}
The proof can be found in~\cite{parisa2017}.
\end{proof}
\subsection{Direct Transmission (DT)}
In DT, the source sends the data packets directly to the destination. Since the channel coefficients are Nakagami-$m$ distributed, the instantaneous SNR of DT scheme is $\Gamma_{Z} \thicksim \mathsf{Gamma}(m_{Z}, \Omega_{Z} /m_{Z})$. Then, the outage probability of DT is calculated according to (\ref{eq:outage_naka}), where $m = m_{Z}$, $\Omega_{Z} = P_S/ N_0 $, and is denoted as $\epsilon_{DT}$.
\section{Numerical Results}\label{sc:result}
In this paper we study the performance of incremental relaying protocols under the FB regime for the Nakagami-$m$ fading. First, we compare the cooperative schemes with DT as a function blocklength and transmit power. We show that the cooperative relaying protocols outperform the DT in terms of transmit power and the blocklength under the UR region.
The accuracy of our analytical model is proved via the numerical results. Unless stated otherwise, we assume $n=500$ where $n_{S}=n_{R}=n$, $k=250$, average SNR as $10$ dB and that $R$ is exactly midway between $S$ and $D$, with $\beta=\tfrac{1}{2}$.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{1-c.pdf}
\vspace{-2mm}
\caption{Outage probabilities for DT, SC, MRC with $k=250$ and average SNR as 10 dB.}\label{fig:outageVSblocklength}
\end{figure}
\subsection{On the impact of reliability improvement}
Fig.\ref{fig:outageVSblocklength} compares two distinct relaying protocols with DT in terms of the overall outage probability and fading severity. It can be clearly seen that MRC and SC outperform DT and perform closely in the entire range. In addition, higher values of $m$ results in lower outage
probability due to the improvement of LOS. Therefore, the availability of LOS should be taken into account when designing and deploying networks with stringent reliability requirements under the FB regime.
\begin{figure}[!htp]
\centering
\includegraphics[width=\columnwidth]{2-c.pdf}
\vspace{-2mm}
\caption{Outage probabilities for DT, SC, MRC with $k=250$ and $n=500$. }\label{fig:outageVSpower}
\end{figure}
Fig.\ref{fig:outageVSpower} shows that the cooperative protocols outperform DT in terms of transmit power consumption. The diversity gain achieved by the relaying schemes is more evident at high SNR and MRC performs better than SC. Hence, URC becomes feasible in case of less severe fading (larger $m$) and/or by
utilizing cooperative schemes. Note also that the analytical and numerical results match very well.
\begin{figure*}[!t]
\centering
\includegraphics[width=2\columnwidth,height=1.9in]{3-c.pdf}
\vspace{0mm}
\caption{Probability of successful transmission in SC and MRC protocols as a function of the blocklength $n$ and payload $k$ with average SNR as 10 dB.}\label{fig:contour}
\end{figure*}
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{4-c.pdf}
\vspace{-2mm}
\caption{Impact of power allocation factor $\eta$ on the outage probability of SC and MRC with different blocklengths $n$. We consider $m=2$, $k=250$ and average SNR as $10$ dB. The points marked in red are the minimum outage probabilities of SC and MRC.}\label{fig:powerallocation}
\end{figure}
\subsection{URC requirements}
In Fig.\ref{fig:contour}, we examine the impact of coding rate on the performance of SC and MRC protocols with different values of $m$ under the UR region. We illustrate that MRC protocol outperforms SC under th FB regime. For instance, with $m=1$ and $n=300$, MRC provides $99.99 \%$ reliability with $k=45$, while SC supports equal reliability with $k=40$. Hence, SC is more affected by the coding rate growth compared to the MRC protocol. Moreover, we can clearly see that improving the LOS (larger $m$), considerably improves the reliability in wireless communications; thus, the size of the payload is not a critical concern anymore.
In Fig.\ref{fig:powerallocation}, we show the effect of power allocation factor on the outage probability. The optimal value of $\eta$, where the outage probability is minimized with different blocklengths, is approximately $0.62$ and $0.9$ for SC and MRC, respectively. From the figure, we see that in good LOS conditions more power is allocated to $S$, though $R$ is needed to provide the diversity order with respect to DT and thus achieve the required reliability target.
In the following figure, we compare the delay in relaying schemes with equal power allocation $\eta =0.5$ and optimal power allocation $\eta=0.62$ and $\eta=0.9$ for SC and MRC, respectively. The choice of the minimum delay $\delta$ is in such a way that minimizes the outage probability constrained to a specific interval of interest which gives the optimal values of $n_S$ and $n_R$, and is a nonlinear optimization problem. The optimization problem numerically is solved via the Matlab function $fmincon$\footnote{Interior point algorithm is used to solve the nonlinear optimization problem~\cite{waltz2006interior}.}. Delay $\delta$ is equal to the symbol time $T_s$ multiplied with blocklength $n$ plus the number of retransmissions. For example, in Fig.\ref{fig:totallatency}, where $\eta=0.5$, future releases of LTE foresee a minimum symbol period of $T_s = 8.33\mu$s~\cite{yilmazultra} and regardless of the retransmissions, $99.99 \%$ reliability is feasible via MRC protocol with $\delta = 3.5 $ms while latency reduces to $\delta = 1.66$ms with $99.9 \%$ reliability. On the other hand, SC protocol provides $99.99 \%$ reliability with $\delta = 3.9$ms when $n_S = n_R$. Note that DT would not be able to cope with such stringent latency requirements. Therefore, we should consider a good trade-off between the reliability and latency in cooperative communication protocols.
Moreover, we indicate that applying the optimal values of $\eta$ shown in Fig.\ref{fig:powerallocation}, slightly reduce the latency in relaying schemes compared to the equal power allocation to $S$ and $R$ under the same assumptions; however, behavior of the protocols alters under the optimal $\eta$. Moreover, in SC with $n_S \neq n_R$ and $\eta = 0.5$, latency reduces in comparison to the case when $n_S = n_R$. For instance, when $n_s=227$ and $n_R=219$, $99.99 \%$ reliability is feasible with $\delta=3.7$ms compared to the case when $n_S = n_R$. Hence, different parameters such as LOS, transmit power, coding rate of each link, delay and reliability should be all considered in designing the networks to support the URC requirements.
\section{Conclusions and Final Remarks} \label{sc:conc}
Herein we assess the relay communication under the finite blocklength regime. The performance of two relaying schemes, SC and MRC, are compared to the DT in terms of the outage probability under the Nakagami-$m$ fading distribution. We indicate that the relaying improves the communication efficiency which is more obvious at high SNR regime through providing higher order of diversity and consuming less transmit power. Moreover, we show the optimal number of channel uses for each link for all relaying schemes considered in this work in order to meet the latency constraint under the UR region. We show that lower coding rates, improve the reliability up to $99.999 \%$.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth, height =2in]{7-c.pdf}
\vspace{-4mm}
\caption{Total delay in terms of channel uses with equal power allocation ($\eta=0.5$) and $\eta=0.62$ and $\eta=0.9$ for SC and MRC, respectively with $m=2$, $k=250$ and average SNR as $10$ dB.}\label{fig:totallatency}
\end{figure}
Furthermore, we indicate that increasing the LOS, increases the reliability while the transmit power is kept low. In addition, we show that if $S$ and $R$ transmit with different coding rates in SC scenario, the latency reduces in comparison to the case of having equal coding rates for both direct and relay links.
We also prove the correctness of our analytical model via the numerical results.
All in all, we conclude that the relaying is a desirable technique in order to improve the system quality, particularly in M2M communications where low latency is a concern with short packet transmissions. In addition, relaying requires less transmit power to work under the UR region and increasing the LOS improves the reliability. More
over, MRC outperforms SC and DT in terms of power consumption and latency requirements. Finally, in our future work, we will work on the cooperative relaying under the imperfect CSI with FB coding.
\begin{figure*}[!t]
\begin{equation}\label{eq:Outage_overall}
\begin{split}
\epsilon&\!=\!\dfrac{F_X{\left(\vartheta,m,\Omega\right)}\!+\!F_X{\left(\varrho,m,\Omega\right)}}{2}\!+\!\dfrac{\mu\theta\bigg(F_X{\left(\vartheta,m,\Omega\right)}\!-\!F_X{\left(\varrho,m,\Omega\right)}\bigg)}{\sqrt{2\pi}}\!-\!\dfrac{\mu}{\sqrt{2\pi}}\bigg(\bigg(\bigg(\dfrac{1}{\Omega}\bigg)^{-m} \Omega^{1\!-\!m}\bigg(\Gamma\left(1\!+\!m,\dfrac{m\varrho}{\Omega}\right)\\
&\!-\!\Gamma\left(1\!+\!m, \dfrac{m\vartheta}{\Omega}\right)\bigg)\bigg)\bigg/ \Gamma\left(1\!+\!m\right)\bigg)\!=\!\dfrac{1}{2}\bigg(\operatorname{\cal P}\left(m,\dfrac{m\vartheta}{\Omega}\right)\!+\!\operatorname{\cal P}\left(m,\dfrac{m\varrho}{\Omega}\right)\bigg)\!+\!\dfrac{\mu m \theta}{\sqrt{2\pi}\Gamma\left(1\!+\!m\right)}\bigg(\Gamma\left(m,\dfrac{m\varrho}{\Omega}\right)\\
&\!-\!\Gamma\left(m,\dfrac{m\vartheta}{\Omega}\right)\bigg)\!+\!\dfrac{\mu\Omega}{\sqrt{2\pi}\Gamma(1\!+\!m)}\bigg(\Gamma\left(1\!+\!m,\dfrac{m\vartheta}{\Omega}\right)\!-\!\Gamma\left(1\!+\!m,\dfrac{m\varrho}{\Omega}\right)\bigg)\!=\!\dfrac{\mu}{\sqrt{2\pi}}\bigg[\theta\bigg(\Gamma\left(m,\dfrac{m\varrho}{\Omega}\right)\!-\!\Gamma\left(m,\dfrac{m\vartheta}{\Omega}\right)\bigg)\\
&\!+\!\Omega \bigg(\Gamma\left(1\!+\!m,\dfrac{m\vartheta}{\Omega}\right)\!-\!\Gamma\left(1\!+\!m,\dfrac{m\varrho}{\Omega}\right)\bigg)\bigg]\!+\!\dfrac{1}{2}\bigg(\operatorname{\cal P}\left(m,\dfrac{m\vartheta}{\Omega}\right)\!+\!\operatorname{\cal P}\left(m,\dfrac{m\varrho}{\Omega}\right)\bigg).\\
\end{split}
\end{equation}
\hrule
\end{figure*}
\section*{Appendix A}
\section*{Proof of the Proposition~\ref{propose1}}
\begin{proof}
Let the PDF of the instantaneous SNR of the link denoted by the random variable $X \thicksim \mathsf{Gamma}(m,\Omega/m)$ be
%
\begin{align}\label{eq:pdf_gamma}
f_X{(x|m,\tfrac{\Omega}{m})} =(\frac{\Omega}{m})^{-m}\frac{\exp(-\frac{mx}{\Omega})x^{m-1}}{\Gamma(m)} &\quad x>0
\end{align}
%
then, plugging (\ref{eq:W(t)}) and (\ref{eq:pdf_gamma}) into (\ref{outage_fading_linear}), we attain the outage probability equal to
\begin{equation}
\epsilon\!=\! \int_{0}^{\varrho}f_X{\left(x|m,\Omega\right)}dx\!+\!\int_{\varrho}^{\vartheta}\left(\tfrac{1}{2}\!-\!\tfrac{\mu}{\sqrt{2\pi}}(x\!-\!\theta)\right)f_X{\left(x|m,\Omega\right)}dx.
\end{equation}
After some algebraic manipulations we attain (\ref{eq:Outage_overall}) on top of the this page, where $F_X(x;m,\Omega)\!=\!\operatorname{\cal P}(m,\tfrac{mx}{\Omega})$ and $\tfrac{m\mu\theta\Gamma(m,\tfrac{mx}{\Omega})}{\sqrt{2\pi}\Gamma\left(1\!+\!m\right)}$ can be simplified as $\tfrac{\mu\theta\Gamma\left(m,\tfrac{mx}{\Omega}\right)}{\sqrt{2\pi}\Gamma(m)}$.
\end{proof}
\section*{Acknowledgments}
This work is partially supported by Aka Project SAFE (Grant no. 303532), and by Finnish Funding Agency for Technology and Innovation (Tekes), Bittium Wireless, Keysight Technologies Finland, Kyynel, MediaTek Wireless, Nokia Solutions and Networks, and CNPq (Brazil).
| {'timestamp': '2018-01-24T02:05:18', 'yymm': '1801', 'arxiv_id': '1801.07423', 'language': 'en', 'url': 'https://arxiv.org/abs/1801.07423'} |
\section{Introduction}
Metal nanoparticles (NPs) are used in a multitude of applications, but perhaps the oldest is as a colourizing agent when dispersed in a host dielectric, as in the (dichroic) Lycurgus cup\cite{Freestone2007,Barch2015}. Exposed to optical radiation, metal NPs exhibit scattering properties due to excited plasmons, which depend on their shape, size, composition and the host medium\cite{Mie,Doyle1989,Murray2007,Liz2006}.\\
Producing colours by exploiting plasmonic effects on metal nanostructures is of interest because the colours can last a long time (e.g., the Lycurgus cup), can be rendered down to the diffraction limit\cite{Kumar2012}, and can be used in any metal colouring or marking application where inks, paints or pigments should be avoided for environmental, health, cost or other reasons.
\\
Fabrication techniques for plasmonic colouring of metal surfaces include laser interference lithography (LIL)\cite{Gallinet2015}, electron beam lithography (EBL)\cite{Roberts2014,Tan2014}, ion beam lithography (IBL) or milling (IBM)\cite{Cheng2015}, and hot embossing or nanoimprint lithography (NIL) \cite{Gallinet2015,Clausen2014}. Kumar et al.\cite{Kumar2012} fabricated coloured images via EBL having a resolution as high as ≈ 100 000 pixels/inch$^{2}$. However, the production of large coloured surfaces is incompatible with the demands of low-cost mass-manufacturing. Furthermore, such processes generally require flat surfaces, and with the exception of periodic structures below the diffraction limit of visible light \cite{Clausen2014,Wu2013}, colours produced are generally angle-dependent.
Femtosecond (fs) lasers have been considered ideal for metal colourization due to their ablation characteristics, e.g., the tendency to preferentially release NPs, compared to the large clusters and chunks produced by nanosecond pulses \cite{Tillack2004,Perriere2007,Balling2013}. Guo et al. \cite{Vorobyev2013a} showed that exposing metals to fs pulses could yield highly absorptive surfaces (so called ‘black metals’), or surfaces producing a specific colour. However, only one set of laser parameters was reported for each colour \cite{Fan2014,Vorobyev2008,Fan2013b}, the colour palette was limited and, due to the underlying regular structure, in the case of femtosecond lasers, were angle-dependent \cite{Vorobyev2008c}. In addition, the low pulse energy of fs lasers restricts their use to low repetition rates, making the colouring process very time-consuming.
Picosecond (ps) lasers have lower costs and higher pulse energies. Fan et al. showed a limited colour palette on copper and each colour was associated to one set of laser parameters \cite{Fan2014}. Thermal effects for ps pulses are likely to play a role in the creation of colours, unlike femtosecond lasers, whose pulses are shorter than the thermal expansion time \cite{Nolte1997}.
Here, we report on a universal, high-throughput and deterministic process for producing a complete angle-independent colour palette composed of thousands of colours using a picosecond laser. This is achieved on unpolished noble metal surfaces, including surfaces with frosting and millimetre-scale topographical features. We demonstrate the process by colouring silver coins produced at the Royal Canadian Mint, set to be released to the general public at the end of 2016. Scanning electron microscopy (SEM) reveals that a full colour palette can be obtained by controlling the particle density on the surface.
Our experiments demonstrate that a very large set of laser parameter combinations can produce a given colour, as long as the total accumulated fluence remains the same, making the process scalable and time-efficient.
\begin{figure}[H]
\centering
\begin{tabular}{@{}p{1\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{white}{}}}{Figure1abcd.png}\\
\subfigimg[width=\linewidth]{\textbf{\textcolor{white}{}}}{Figure1ef.png}
\end{tabular}
\caption{25 mm$^2$ coloured squares obtained by 1 $\mu$m changing $L_s$ (marked above) for (a) a laser fluence of $\phi=1.67$ J/cm$^{2}$ with a laser marking speed of $v=50$ mm/s, and (b) $\phi=29.75$ J/cm$^{2}$ at $v=1000$ mm/s. (c) Photograph of blue angle-independent colouring of a flat silver coin observed at various angles. (d) Hue versus total accumulated fluence for ($\begingroup\color{blue}\bullet\endgroup$) $\phi=1.12$ J/cm$^{2}$ at $v=11$ mm/s, ($\begingroup\color{red}\star\endgroup$) $\phi=1.67$ J/cm$^{2}$ at $v=50$ mm/s, ($\begingroup\color{green}\filledtriangledown\endgroup$) $\phi=2.59$ J/cm$^{2}$ at $v=100$ mm/s, ($\begingroup\color{black}\filledtriangleleft\endgroup$) $\phi=6.26$ J/cm$^{2}$ at $v=250$ mm/s, ($\begingroup\color{mycolor1}\pentagram\endgroup$) $\phi=13.23$ J/cm$^{2}$ at $v=500$ mm/s, ($\begingroup\color{cyan}\filledsquare\endgroup$) $\phi=20.58$ J/cm$^2$ at $v=750$ mm/s, and ($\begingroup\color{mycolor2}\filledtriangleright\endgroup$) $\phi=29.75$ J/cm$^2$ at $v=1000$ mm/s; Hue can be seen to make a full 360$^{\circ}$ rotation (dashed line). (e) Polar plot representation of (d) with Hue plotted azimuthally and the logarithm of the total accumulated fluence plotted radially; a full rotation in Hue can be observed with increasing total accumulated fluence. (f) CIE diagram of all the colors obtained using the different laser parameters from (d). }
\label{fig1}
\end{figure}
To theoretically understand the colour formation, we used large-scale computational electrodynamics to simulate the scattering from periodic distributions of different sized nanoparticles, with geometrical parameters based on statistical analysis of SEM images. Simulations show that several NP arrangements can produce the same colour, similar to what is observed experimentally, and that narrow band absorption due to plasmonic resonances in heterogeneous nanoclusters are critical for colour production.
\section{Results and Discussion}
\subsection{Angle-independent laser colouring of silver}
The exposure of pure silver to different laser parameters is observed to produce vivid angle-independent colours, Fig. 1 (a,b,c). The same process was used to produce similar colour palettes on copper and gold. These colours are highly reproducible and found to depend on the density of NPs covering the metal surface.
\\
During laser ablation by raster scanning the sample surface, the number of particles re-deposited and accumulated from one line to the next is dictated by the inter-line spacing $L_s$, i.e., the spacing between two consecutive laser lines. The change in particle density with increasing distance from the laser ablated line can be seen in the supplementary material, Fig. S. 1.\\
Figs. 1 (a,b) show the colours produced on silver by raster scanning the surface, where different colours are produced by simply increasing Ls. In particular, increasing Ls from 1 to 12 $\mu$m with a laser marking speed of v = 50 mm/s results in a production rate of $\eta$ =0.05 to 0.8 mm$^2$/s (Fig. 1 (a)), while augmenting the laser marking speed to v = 1000 mm/s with Ls ranging from 1 to 25 $\mu$m (Fig. 1 (b)) results in production rates of $\eta$ =1 to 25 mm$^2$/s. The production rate is defined as $\eta$ = L$_s$v. The colour palettes in Figs. 1 (a,b) can be extended by simply reducing the line spacing between each successive line (i.e., increasing total accumulated fluence). The angle-independent colours produced on silver, Fig. 1 (c), cover the spectral and the nonspectral regions (e.g., Magenta) through a 360$^{\circ}$ rotation in Hue, Fig. 1 (d-f).
\\
The total accumulated fluence is defined as
\begin{equation}
\Phi = \phi N_{eff} =\frac{a^{2}Ef}{vL_s},
\end{equation}
where the laser fluence is given by
\begin{equation}
\phi = \frac{2E}{\pi w_o^2},
\end{equation}
the number of effective laser shots is
\begin{equation}
N_{eff}= \underbrace{\sqrt{\frac{\pi}{2}}\frac{aw_of}{v}}_\text{intra-line}\underbrace{\sqrt{\frac{\pi}{2}}\frac{aw_o}{L_s}}_\text{inter-line},
\end{equation}
and $E$ is the laser pulse energy, ωo the beam waist radius, $f$ the laser repetition rate, and $a$ a correction factor due to the larger modified region when using a higher pulse energy. This correction factor was determined from semi-logarithmic plots used to determine the laser spot size, following the procedure detailed in \cite{Jandeleit1996}. The intra-line component of $N_{eff}$ encompasses the shot overlap within a local region, where $f/v$ is the distance traveled between successive laser pulses in a single laser line. The inter-line component, in comparison, considers the geometrical overlap, $a\omega_o/L_s$, between successive laser lines. At large spacings (\textit{i.e.}, lower total accumulated fluence and $a\omega_o/L_s$ $\rightarrow$ 1) the colours are observed to converge to yellow due to the absence of overlap between consecutive laser lines. Further evidence of this accumulation process can be observed from the absence of colours, other than yellow, in the last line of each of the coloured squares. The last line does not undergo the process of particle accumulation as there are no subsequent lasered lines. In addition, in the absence of overlap and for large distances between successive lines, LIPSS structures can be observed to form next to the laser ablated lines causing the yellow colors to be angle-dependent. In the overlap case no LIPSS structures are created or they are destroyed and the surface remains relatively smooth, Fig. S. 2 in supplementary materials.
\begin{figure}[H]
\centering
\begin{tabular}{@{}p{1\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{white}{}}}{Figure2.png}
\end{tabular}
\caption{Photographs of silver topographical butterfly coins (a) before and (b) after laser colouring; the topography has an overall height of $\sim$ 2 mm (surfaces as high as 4 mm were also coloured) without adjusting focus. The distinctive white areas on the butterfly are frosted ($\it{i.e}$., melted) finish; (c) optical microscope close up of the frosted regions, height of $\sim$ 1 to 2 $\mu$m. (d) Photograph of a silver eagle coin. The eagle is frosted and its surface is similar to that shown in (c). The difference in elevation is observed to create a colour gradient on the eagle wings (e) and tail (f). The eagles in (d-f) originate from different coins coloured using different laser parameters. (g) Blue Jay colouring on a flat silver coin, with permission from National Geographic. (h) Colouring of a butterfly, $\sim$ 10 mm wide, on a flat silver coin}
\label{fig2}
\end{figure}
Previous work on picosecond and femtosecond laser colouring of metals \cite{Vorobyev2013a, Fan2014} only report a single set of laser parameters to obtain a given colour (less than 10 angle-independent colours). We show that there are in fact a very large number of laser parameter combinations that could be used to obtain a given Hue, as long as the total accumulated fluence remains the same -- as highlighted by the trend (master curve) in Fig. 1 (d) -- under the condition that the repetition rate remains fixed. Fig. 1 (e) shows a polar plot of Fig. 1 (d) with Hue plotted azimuthally and the logarithm of the total accumulated fluence plotted radially; a full 360$^{\circ}$ rotation in Hue is observed. The strength of the process and the master curve can also be seen in the aesthetic control of a single colour. For example, not only are we able to produce the colour blue, but we can also control the type of blue (\textit{e.g.}, navy blue, sky blue, etc...), as the Lightness of the colours, for the same Hue, scales up with laser fluence, $\phi$, as seen in Figs. 1 (b). The process allows us to generate thousands of colours, well-beyond what has been previously reported, enabling this technique to rival paint based processes currently used at the Royal Canadian Mint. Chroma, however, is observed to remain unaffected for each Hue when using different laser fluences, $\phi$. Figure 1 (f) is a CIE xy chromaticity diagram of the colours obtained via the different laser parameters in Fig. 1 (d). The rotation and overlapping of the data points on the diagram shows explicitly that the xy values of CIE XYZ colour space can be recovered for a fixed laser fluence, $\phi$, by simply adjusting the marking speed or line spacing. Deviations from the master curve in Fig. 1 (d), are attributed to the increasingly chaotic nature of the surface with increasing total accumulated fluence. While we are able to render reds reproducibly, there is less variety in the reds than for other colours due to the high slope in the red region on the master curve (Hue: 345 to 15). From equation (1), $v$, $L_s$ and $E$ can be changed independently to obtain a specific total accumulated fluence corresponding to a desired colour, Fig. 1 (d). \\
However, changing the repetition rate, $f$, with either $v$, $L_s$ and $E$ to obtain a desired total accumulated fluence resulted in colours other than what was expected based on the master curve of Fig. 1(d). To study the effect of the laser repetition rate on the colours, $f$ was changed proportionally with speed, $v$, to maintain a fixed total accumulated fluence with $E$ and $L_s$ kept constant. The colour palette obtained was similar to that shown in Fig. 1(a) but with fewer colours. Increasing $f$ and keeping it fixed while changing $v$, $L_s$ and $E$ would only create reduced colour palettes. The baseline in Fig. 1(d) increased with increasing repetition rate gradually cutting off the lower colours (\textit{i.e.}, blues, purples and reds). No colours other than yellow were observed at repetition rates above 400 kHz suggesting a local thermal accumulation effect. \\
Figs. 2 (a) and (b) show before and after images of the colouring process applied to silver coins of varying topography (up to 2 mm in height). The different colours were selected prior to laser colouring using the corresponding total accumulated fluence values from the master curve (Fig. 1 (d)). Precise colouring of the coin features was done using vision alignment software with pattern recognition. The vectors were obtained from an artist at the Royal Canadian Mint. The process is capable of uniform and viewing angle-independent colouring of elevated and complex surfaces that could not be coloured via traditional paint-based manufacturing processes. Moreover, compared to top-down fabrication techniques, this colouring process is efficient regardless of surface quality, allowing for the uniform colouring of frosted (\textit{i.e.}, melted) surfaces, Figs. 2 (a-f).\\
In addition, due to the dependence of the colours on the total accumulated fluence, colour gradients can also be produced by focusing the laser beam slightly above the surface, outside the confocal volume, making the colouring process sensitive to topography - see the eagle wings (e) and tail (f) in Fig. 2. The extent of the number of colours in the colour gradient is governed by the distance between the surface of the coin and the laser focus. At small distances (while still out of the confocal region) the colour gradient is observed to have a smooth transition in colours over a large area. Flat Figs. 2(g,h) and topographical (Fig. 2 (b)) surfaces situated within the confocal volume are, however, coloured uniformly. White and perfect black were also achieved using the appropriate laser parameter combinations, as shown in Fig. 2 (g).
\\
The long term stability of the colours was observed to depend on the colour. However, passivation coatings formed via atomic layer deposition on coloured silver surfaces protected the colours during aggressive humidity and tarnish tests carried out at the Royal Canadian Mint. The colours were, however, slightly red-shifted due to the passivation layer, an effect that can be pre-compensated by altering the write laser parameters.
\subsection{Surface Analysis}
\begin{figure}[H]
\centering
\begin{tabular}{@{}p{1\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{black}{}}}{Figure3.png}
\end{tabular}
\caption{(a,b,c) Low-magnification optical microscope images of coloured surfaces; (d,e,f) low- and (g,h,i) high-magnification SEM images of corresponding surfaces. Surfaces processed using $\phi=1.12$ J/cm$^2$ at $v=11$ mm/s for (a,d,g) $L_s=5$ $\mu$m (Hue = 216.5, Cyan), (b,e,h) $L_s=10$ $\mu$m (Hue = 269.3, Blue), and (c,f,i) $L_s=30$ $\mu$m (Hue = 17.2, Red). The number of medium NPs is observed to decrease with increasing $L_s$ (d,e,f -- scale 1 $\mu$m), and the number of small NPs to increase with $L_s$ (g,h,i -- scale 0.2 $\mu$m).}
\label{fig3}
\end{figure}
Extensive SEM analyses of regions exhibiting different colours on silver reveal 3 distinct classes of particles differentiated by size: large ($r\geq75$ nm), medium ($10.7\leq r$ $\textless$ 75 nm) and small ($r$ $\textless$ 10.7 nm) where $r$ is the radius of a particle.
These particle classes were obtained from statistical analysis of SEM images which produced histograms with well-defined ranges of particle sizes (see Fig. S. 4(a,b,c), supplementary material).
Figs. \ref{fig3}(a-c) show three coloured surfaces with corresponding low- and high-magnification SEM images. The particle density is found to change significantly with line spacing, as shown in Figs. \ref{fig3}(d,g).
In Figs. \ref{fig3}(d-i), the small and medium particles are seen to form random networks with the medium-sized particles sparsely covering the surface and the small ones more densely distributed across the irradiated region. The formation of NPs is believed to come from the combination of thermal effects\cite{Che2008, Galhenage2013, Roque2005, Fazio2014} and the re-deposition of particles following laser ablation\cite{Perriere2007, Tillack2004a, Tillack2004}.
Upon close examination of the SEM images, it appears that the small particles are in reality approximately hemispherical, an observation supported by a cross-section of a coloured silver sample obtained through focused ion beam milling and imaging. The small particles can be viewed as spheres partially embedded into the silver surface.
\\
SEM analyses of coloured regions reveal that the number density of the small particles, produced using different laser parameters, follows its own clear distinctive trend with total accumulated fluence, (Fig. S. 3 (a), supplementary materials), similar to that of Fig. 1 (d), whereas the number density of the medium particles does not (Fig. S. 3 (b), supplementary materials). This observation suggests that the small particles play a major role in the colours perceived, even though they have not been considered in previous works. The mean radius of small and medium particles was found to remain approximately constant as a function of line spacing (Figs. S. 3 (c,d), determined from the analysis of three SEM images per line spacing or colour). However, the mean inter-particle distance (wall-to-wall) changes with line spacing, Figs. S. 3 (e,f), suggesting that the colours are affected by near-field interactions between nanoparticles in close proximity \cite{Rechberger2003,Romero2006,Jain2010,Liz2006}, particularly the associated surface plasmon resonance frequency \cite{Liz2006}.
\\
Wavelength-dispersive spectroscopy (WDS) analysis of the different colours in Figure 1, showed no difference in the amount of oxidation measuring 2.8 $\pm$ 0.4$\%$ oxygen content for all Hue values tested. Monte Carlo simulations (WinXray) of silver oxide layers, under the same WDS operating conditions, gave an equivalent oxide thickness of $\sim$ 2 nm. The same conclusion was gathered from the invariability of the oxygen content in separate energy dispersive spectroscopy (EDS) and X-ray photoelectron spectroscopy (XPS) measurements of the different coloured surfaces. The XPS analyses of the coloured surfaces also showed no sulphur content.
\subsection{FDTD Simulations}
Our detailed numerical simulations show that colour formation can be viewed as a selective absorption process occurring due to plasmonic resonances. White light incident on a nanostructured metallic surface is not fully reflected, as it would be for a smooth surface. Rather, narrow-band spectral components (colours) are subtracted due resonant and absorptive processes in nanoclusters (homogeneous and/or heterogeneous), single particles, lattice resonant modes, and absorption in the medium, consistent with another study \cite{Ng2015}. What survives is reflected (there is no transmittance).
\begin{figure}[H]
\centering
\begin{tabular}{@{}p{.8\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{black}{a)}}}{Figure4a.pdf}\\
\subfigimg[width=\linewidth]{\textbf{\textcolor{black}{b)}}}{Figure4b.pdf}
\end{tabular}
\begin{tabular}{@{}p{.7\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{black}{c)}}}{Fig4c.png}
\end{tabular}
\caption{(a) Computed reflectance spectra considering only small NPs embedded by $R_s$/2, only medium NPs embedded by $R_m$/2, and medium and small NPs embedded by $R_m$/2 and $R_s$/2, respectively. (b) Reflectance spectra computed by varying the embedding of the small NPs in steps of 0.5 nm (the medium NPs are embedded $R_m$/2). (c) Graph of Hue values vs. total accumulated fluence comparing the measured and computed values for small NPs embedding of 1.5-2.5 nm, and medium NPs not embedded.}
\label{fig5}
\end{figure}
The analysis of SEM images produced statistics of the distributions, \textit{i.e.}, average radii and average inter-particle distances for small, medium and large NPs. We carried out simulations considering only small and medium Ag NPs uniformly distributed on a Ag substrate. This is supported by the two discernible bumps (bimodal distribution) noted in the histograms (Fig. S. 4 (a,b,c), supporting information). The large particles are neglected due to their low density. The radii of small and medium sized NPs are $R_s$ and $R_m$, and the inter-particle distances (centre-to-centre) are $D_s$ and $D_m$, respectively.
In each case the NPs were periodically arranged in a hexagonal configuration, which produces a hexamer unit cell. The hexamer has $D_{6h}$ symmetry, and based on group theory, exhibits polarization independence (isotropy), as verified by simulations \cite{Hent2010}. We chose $D_m$ as an integer multiple of $D_s$ to allow the application of periodic boundary conditions (PBCs). The periodicity of the unit cell is small enough to preclude coupling by diffraction into surface plasmon waves on the Ag surface.
\\
By changing the surface density of small and medium sized NPs, their radii, and their level of embedding into the substrate, several types of heterogeneous nanoclusters can be formed. Embedding transforms the NPs gradually from spherical to hemispherical. A nanocluster has a central nanoparticle (cNP) with nearby nanoparticles arranged in a ring (rNPs). Plasmonic nanoclusters embedded in a homogeneous medium or on a dielectric have been extensively studied (dimer, trimer, quadrumer, tetramer, hexamer, heptamer)\cite{Luk2010,Mirin2009,Hent2010}, but not on a metal substrate, and not arranged and embedded as inspired by SEM images of laser-processed metal surfaces. Plasmonic nanoclusters exhibit resonances which can lead to selective absorption\cite{Mirin2009}.
We have observed different types of resonant modes in our simulations; we will refer to them as cluster (or collective) resonances.
In the simulations that follow, we consider the case of line spacing $L_s=5$ $\mu$m marked at $v=50$ mm/s with a laser fluence $\phi=1.67$ J/cm$^2$, having statistical average parameters $R_s=4$ nm, $R_m=34.3$ nm, $D_s=13.5$ nm and $D_m=108$ nm.
In Fig. 4 (a) we show the computed reflectance spectrum considering small NPs only, medium NPs only, and the combination of both NP sizes; the NPs are half-embedded, \textit{i.e.}, embedded by half their radius. The medium size NPs are observed to produce colours in the absence of the small ones, but changing $D_m$ and the level of embedding does not reproduce the range of colours observed experimentally. Alternatively, considering only small NPs produced colours that are too light to be discernible. However, the interaction between small and medium NPs drastically alters the computed reflection spectrum, as noted by the appearance of a deep dip at $\lambda$ = 650 nm in Fig. 4 (a), which is due to a new cluster resonance. This will produce a discernible final colour. In our simulations we observed a very high sensitivity to embedding, which changes dramatically the geometry of the nanoclusters, and consequently, their resonance characteristics. In particular, the embedding of the small NPs increases the size of the nano-gaps between the cNP and the rNPs. In Fig. 4 (b) we show the reflectance by embedding the small NPs in steps of 0.5 nm. A blue-shift in the cluster resonance is observed when the embedding of small NPs is increased. A similar blue-shift due to cluster expansion was reported in \cite{Hent2010}.
In Fig. 4 (b) we also see other smaller dips in the reflectance curves. These are due to nanocluster resonances arising from aggregates of small NPs. In Fig. 4 (c) we compare the experimental results with the FDTD simulations. A qualitative agreement was found for medium NPs not embedded and small NPs embedded into the substrate by 2.5 nm. We were able to reproduce by simulations the full colour palette, supporting the role of plasmons in the colour rendition. We found that a large set of geometries can produce the same reflectance, in the same way as a large set of laser parameters can produce the same colour.
In Figs. \ref{fig6}(a-d) we show the electric field distribution at the free-space optical wavelengths of $390$ nm (a,c) and $650$ nm (b,d), which are the wavelengths at which we observe absorption dips in Fig. \ref{fig5}(a).
We show $xz$ planes cut 2 nm above the silver surface for medium NPs only (a,b), and medium and small NPs (c,d).
In Fig. \ref{fig6}(a) we observe the resonance of the medium sized NP at $390$ nm, as predicted by Mie theory for the same sphere in air.
In our case, the resonance produces a dip in the reflectance due to the presence of the substrate.
Fig. \ref{fig6}(b) shows that the medium NPs only do not produce any absorption at $650$ nm, and the reflectance is very high.
In Fig. \ref{fig6}(c) we observe that the near-field intensity of the medium NPs at $390$ nm is reduced by the presence of the small NPs, which act as plasmonic chain waveguides \cite{Maier2002}, coupling the medium size NPs.
\\
In Figs. 5 (a-d) we show the electric field distribution at the free-space optical wavelengths of $\lambda$ = 390 nm (a,c) and $\lambda$ = 650 nm, which are the wavelengths at which we observe absorption dips in Fig. 4 (a). We show xz planes cut 2 nm above the silver surface for medium NPs only (Figs. 5 (a,b)), and medium and small NPs (Figs. 5 (c,d)). In Fig. 5 (a) we observe the resonance of the medium sized NPs at $\lambda$ = 390 nm, as predicted by Mie theory for the same sphere in air. In our case, the resonance produces a dip in the reflectance due to the presence of the substrate. Fig. 5 (b) shows that medium NPs only do not produce any absorption at $\lambda$ = 650 nm, and the reflectance is very high. In Fig. 5 (c) we observe that the near-field intensity of the medium NPs at $\lambda$ = 390 nm is reduced by the presence of the small NPs, which act as plasmonic waveguides \cite{Maier2002}, coupling the medium size NPs. When small and medium NPs are illuminated at $\lambda$ = 650 nm (Fig. 5(d)), a heterogeneous cluster resonance is excited, producing field enhancement in the nano-gaps, and ultimately the strong absorption observed in Fig. 5 (a). In Fig. 5 (e) we show a snapshot of the time-domain simulation in a yz plane cut through the centre of the medium NPs. The plane wave excitation pulse has just hit the nanostructures inducing localized surface plasmons in all NPs which are clearly visible. The time-evolution of the excitation in the xz and yz planes is shown in Movies S. 1 and S. 2 (supporting information), respectively.
In Fig. 5 (f) we show an SEM image for the case of $\phi$ = 1.67 J/cm2 at $v$ = 50 mm/s with $L_s$ = 8 $\mu$m, which further justifies our simulation approach. We observe the presence of two sets of particles (small and medium sizes) having a random distribution. In our simulations we considered perfect periodicity of the NPs, which gives isotropy with respect to the incident polarization, narrowband resonances, and ultimately colour selectivity. This can be considered a good qualitative approximation. In \cite{Nishijima2012} random clusters were investigated, and the authors reported a broadening of the resonances. This matches our experiments which show broader spectra with respect to simulations.
\begin{figure}[H]
\centering
\begin{tabular}{@{}p{.8\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{white}{}}}{Figure5abcd.pdf}
\end{tabular}
\begin{tabular}{@{}p{.8\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{black}{e)}}}{Figure5e.pdf}
\end{tabular}
\begin{tabular}{@{}p{.7\linewidth}@{}}
\subfigimg[width=\linewidth]{\textbf{\textcolor{white}{f)}}}{Figure5f.png}
\end{tabular}
\caption{FDTD simulations showing the electric field distribution for only medium size NPs at (a) $\lambda$ = 390 nm and (b) $\lambda$ = 650 nm, and for medium and small NPs at (c) $\lambda$ = 390 nm and (d) $\lambda$ = 650 nm. (e) Time-domain snapshot of medium and small nanoparticles embedded by $R_{m}$/2 and $R_{s}$/2, respectively. (f) SEM image for $\phi$=1.67 J/cm$^2$ at $v=50$ mm/s with $L_s=8$ $\mu$m.}
\label{fig6}
\end{figure}
\section{Conclusion}
We described a universal and deterministic process for colouring noble metals with and without topography. Each individual colour can be linked to a total accumulated fluence. The colours originate from random distributions of small and medium nanoparticles embedded into the surface, induced and controlled by laser exposure. The randomness of the nanoparticle networks is modelled effectively by assuming a periodic structure defined by statistical averages of nanoparticle size and separation. We have demonstrated that plasmonic effects arising in heterogeneous nanoclusters explain the full palette of experimental colours. The small nanoparticles in particular, which have always been neglected in post-surface analyses following laser exposure, play a fundamental role in the colour formation. In fact, they change the geometry of the nanoclusters, the cluster resonance, and ultimately the perceived colour. The new proposed method decreases the colouring time of large metal surfaces, and opens the door to large-scale industrial applications for anti-counterfeiting, bio-sensing, bio-compatibility, and the decoration of consumer products such as jewels, art, architectural elements and fashion items.
\section{Methods}
In our experiments, 1064 nm light from a 15 W Duetto (Nd:YVO$_{4}$, Time-Bandwidth Product) mode-locked MOPA laser, operating at a repetition rate of 50 kHz and producing 10 ps pulses, was focused on the metal surface using an F-theta lens (f=163 mm, Rodenstock). The pulse energy for the presented results ranged between 3.4 and 91.4 $\mu$J. The laser was fully electronically integrated and enclosed by a third party for industrial applications (GPC-PSL, FOBA). For accurate focusing, the surface of the samples was located using a touch probe system. The silver samples were of 99.99$\%$ purity and not polished prior to machining to meet requirements of reproducibility in industrial applications. For machining, the samples were placed on a 3-axis translation stage with a resolution of 1 $\mu$m in both the lateral and axial directions. The samples were raster scanned using galvanometric XY mirrors (Turboscan 10, Raylase) displacing the beam in a top to bottom fashion with a mechanical shutter blocking the beam between successive lines. The laser polarization was parallel to the marking direction. The laser power was computer controlled via a laser interface and calibrated using a power meter (3A-P-QUAD, OPHIR). A spot size of 14 $\mu$m was obtained from semi-logarithmic plot of the square diameter of the modified region, measured with a scanning electron microscope (SEM), as a function of energy, following the procedure described in \cite{Jandeleit1996}. High resolution SEM (JSM-7500F FESEM, JEOL) images were obtained using secondary electron imaging (SEI) mode. Colours were quantified using a Chroma meter (CR-241, Konica Minolta) with the CIELCH colour space, 2 observer and illuminant C (North sky daylight); where L is colour Lightness, C is Chroma (colour saturation) and H is Hue (colour value associated with a 360 polar scale). For analysis of the SEM images, a Matlab program was written to locate the position of each particle and record its diameter and the wall-to-wall inter-distance spacing to its nearest-neighbours.
Three dimensional finite-difference time-domain (FDTD) simulations\cite{Taflove2005, Taflove2013} have been performed to determine the origin of the colour formation process. We used in-house 3D-FDTD parallel code \cite{Lesina2015,Vaccari2014} on an IBM BlueGene/Q supercomputer (64k cores) part of the Southern Ontario Smart Computing Innovation Platform (SOSCIP).
The nanoparticles are arranged on the $xz$-plane. The system is excited by a $z$-polarized plane wave. This is a broadband electromagnetic pulse propagating along the $y$-direction from air and impinging on the nanostructured surface.
The analysis was performed over the wavelength range 350-750 nm in a single run of the code by in-line discrete Fourier transform (DFT). A space-step of $0.25$ nm was used for the simulations, and $0.125$ nm for visualization quality (Figs. \ref{fig6}(a-e) and Movies).
The dispersion of silver was introduced by the Drude+2CP model \cite{Vial2011}. This model was implemented in FDTD by the auxiliary differential equation (ADE) technique \cite{Prokopidis2013}. The simulation domain in the direction of the plane wave propagation is truncated by convolutional perfectly matched layers (CPML) absorbing boundary conditions \cite{Roden2000}. The simulations required up to 16k cores.
The theoretical reflectance spectrum is calculated by integration of the Poynting vector in the backward far-field region.
The experimental reflectance spectra (colours) were reconstructed using an in-house Matlab code, weighting each frequency composing the spectra to the spectral sensitivity of the eye.
| {'timestamp': '2016-09-12T02:06:03', 'yymm': '1609', 'arxiv_id': '1609.02874', 'language': 'en', 'url': 'https://arxiv.org/abs/1609.02874'} |
\section{Introduction}
\label{sec:intro}
\vspace{-0.25cm}
The freely available multispectral satellite imagery and advancement in modern machine learning have paved the way for a wide variety of applications. These include disaster assessment~\cite{dao2015object}, crop classification~\cite{csillik2019object, kobayashi2020crop}, urbanization~\cite{deepCountISPRS19} and environment monitoring~\cite{donlon2012global}.
The increasing quality and resolution of available remote sensing imagery have made it possible to perform robust crop monitoring and yield estimation over large areas~\cite{battude2016estimating}. For instance, the Sentinel-2 satellite imagery with a spatial resolution of 10m and Landsat-8 imagery with a spatial resolution of 30m is available every five days and 16 days, respectively. Thus, enabling us to perform detailed spatio-temporal analysis~\cite{vuolo2018much}.
\par
Supervised learning is considered to be the state-of-the-art approach to produce crop maps. Some traditional machine learning algorithms like Support Vector Machines (SVM) \cite{saini2018crop}, K-nearest neighbors (KNN) \cite{thanh2018comparison}, Cart \cite{sonobe2017experimental}, and Random Forest \cite{rodriguez2012assessment} have been widely applied for crop classification. Among these methods, those using a series of images at different time stamps~\cite{gomez2016optical} have shown better results than the one using a single image \cite{saini2018crop}.
\par
Due to the popularity of deep learning, recently, the problem of crop classification has also been attempted using different deep learning architectures. In earlier work, a 1D convolutional neural network model has been proposed~\cite{zhou2018crops} which stacked features of different time stamps as in the case of random forest. To use the temporal information more efficiently, ~\cite{pelletier2019temporal} has proposed a new 1D CNN model. This method feeds multivariate time series to 1D CNN instead of stacked values of features. Similarly, a new 1D CNN architecture based on the idea of the Inception network has been introduced~\cite{zhong2019deep}. It has shown improved performance on traditional machine learning algorithms, including Random Forest and XGboost~\cite{abdi2020land}.
\par
The 2D convolutional neural networks show outstanding image classification performance, and many architectures, including Alexnet~\cite{krizhevsky2012imagenet}, Inception, Resnets~\cite{he2016deep}, etc., have been introduced. A new dataset comprising Sentinel-2 images of different landcover types from around the world has been developed~\cite{helber2019eurosat} and different 2D CNN architectures, including Resnet50 and GoogleNet~\cite{szegedy2015going}, have been compared to classify the multispectral images. Recently, an ensemble of 2D CNN has been used for crop classification using multi-temporal Sentinel-1 and Landsat-8 imagery~\cite{kussul2017deep}. This study shows the significance of using spatial information in addition to temporal one.
\par
The recurrent neural networks are very popular for applications where time-series data is used. The idea of using LSTM \cite{zhou2019long} has also been applied to landcover classification using multi-temporal SAR imagery~\cite{zhong2019deep}. This work has claimed that LSTM performs worse than Random Forest classifier for crop classification using multispectral imagery. The recurrent convolutional neural network (R-CNN)~\cite{rcnn} uses LSTM followed by 2D CNN, but unlike~\cite{kussul2017deep}, this architecture uses the information of only one pixel. The idea of combining CNN and LSTM has also been introduced for landcover classification~\cite{kwak2019combining}\cite{russwurm2018multi}.
\par
The 3D convolutional neural networks are widely considered for applications related to video, medical imaging, and remote sensing~\cite{BhimraICASSP2019}. In \cite{ji20183d}, a 3D CNN architecture has been presented showing better performance than 2D CNN as it uses the temporal information better than the 2D CNN where timestamps of all features are stacked on one axis.
\begin{figure*}[t]
\centering
\includegraphics[width = 18cm]{images/model.png}
\caption{Proposed model architectures showing combination of spatio-temporal and temporal convolution blocks. Spatio-temporal block via 3D CNN are shown in blue whereas the temporal only block via 1D convolution are shown in green.}
\vspace{-0.4cm}
\label{fig:model}
\end{figure*}
Existing literature suggests various ways to exploit available spatio-temporal data~\cite{BhimraICASSP2019,ji20183d} however, these methods are either good at utilizing spatial information or temporal information but not both. We instead propose a novel DCNN based architecture that combines both spatial as well as temporal analysis. In the first stage of our architecture, we propose to use 3D convolutions that perform spatio-temporal analysis without collapsing the temporal dimension. Once the spatio-temporal features are extracted via multiple 3D CNN layers, we introduce temporal only analysis to further extract important information from the temporal dimension only. This spatio-temporal followed by temporal only analysis helps eliminate noise, usually present in the temporal only analysis. Thus the proposed approach outperforms classical as well as state-of-art-methods on benchmark datasets.
\section{Methodology}
\label{sec:method}
\vspace{-0.25cm}
\subsection{Data Pre-processing}
\vspace{-0.2cm}
We have divided the pre-processing of data into five steps. The first step is to select all the images of a multispectral satellite for the region of interest in a suitable period, covering every crop's cropping season under consideration. For this purpose, the NDVI time series for different crops have been examined. The second step is to select the least cloudy images in that period. In our case, the images with cloud cover less than $10\%$ have been selected. The third step is the cloud masking of the chosen images. The fourth step is to take the median of images chosen for each month. This step is for larger areas, as in our case, because the region can't be covered in a single image of a satellite. In the fifth and final step, the missing pixels due to the cloud or other reasons have been filled, which can be done using simple interpolation or linear regression. All these pre-processing steps have been performed on the Google Earth Engine Platform.
%
\vspace{-0.2cm}
\subsection{Feature Selection}
\vspace{-0.2cm}
For each image in our dataset, $13$ features have been selected. The selected features have six multi-spectral satellite imagery bands, including Blue, Green, Red, Near Infrared (NIR), and two Short Wave Infrared (SWIR) bands. Table~\ref{tab:bands} shows the selected Sentinel-2 and Landsat-8 bands. The other seven features are indices derived from these bands, including Normalized Difference Vegetation Index (NDVI), Enhanced Vegetation Index, Green Normalized Difference Vegetation Index (GNDVI), Soil Adjusted Vegetation Index (SAVI), Bare Soil Index (BSI), Normalized Difference Water Index (NDWI) and Normalized Difference Buildup Index (NDBI). These indices~\cite{pal2017comparison,vermote2016preliminary} are calculated as follows:
%
\begin{table}[b]
\centering
\caption{Selected Bands of Landsat-8 and Sentinel-2\Bstrut}
\begin{tabular}{>{\centering}p{1.6cm} >{\centering} p{0.5cm}>{\centering}p{0.8cm}>{\centering}p{0.6cm}>{\centering}p{0.6cm}p{0.8cm}p{0.8cm}}
\hline
\textbf{Bands}&\textbf{Blue}&\textbf{Green}&\textbf{Red}&\textbf{NIR}&\textbf{SWIR1}& \textbf{SWIR2}\Tstrut\Bstrut\\
\hline\hline
\textbf{Sentinel-2}&B2&B3&B4&B8&B11&B12\Tstrut\\
\textbf{Landsat-8}&B2&B3&B4&B5&B6&B7\\
\hline
\end{tabular}
\label{tab:bands}
\end{table}
%
\begin{center}
$\text{NDVI} = \frac{NIR - RED}{NIR + RED}~~~~,~~\text{GNDVI} = \frac{NIR - GREEN}{NIR + GREEN}$
\end{center}
\begin{center}
$\text{EVI} = 2.5\times\frac{NIR - RED}{NIR + 6\times RED-7.5\times BLUE + 1}$
\end{center}
\begin{center}
$\text{SAVI} = 1.5\times\frac{NIR - RED}{NIR + RED + 0.5}$
\end{center}
\begin{center}
$\text{BSI} = \frac{(SWIR + RED) - (NIR + BLUE)}{(SWIR + RED) + (NIR + BLUE)}$
\end{center}
\begin{center}
$\text{NDBI} = \frac{SWIR1 - NIR}{SWIR1 + NIR}~~,~~~\text{NDWI}= \frac{GREEN - NIR}{GREEN + NIR}$
\end{center}
%
\subsection{Model Architecture}
\vspace{-0.2cm}
The model architecture is shown in Fig.~\ref{fig:model}. The model consists of three parts concatenated to each other. The first part shown in blue performs spatio-temporal convolution using 3D CNN. The second part shown in green performs the temporal convolution. The output of 3D CNN is squeezed before feeding to 1D CNN. These two parts extract the features from input. The third part consists of a fully connected neural network, which predicts the label from multi-temporal input images.
\par
The model input shape is $(r, r, t, c)$ where $r$ is the spatial dimension of input, $t$ is the number of timestamps, and $c$ is the number of channels. In our implementation, spatial dimension $r$ is $7$, timestamps $t$ are $9$ and channels $c$ are $13$. Each convolutional block has a batch normalization layer before activation. It has been observed that with batch-normalization, the model converges faster as well as shows better accuracy.
\section{RESULTS AND EVALUATION}
\label{sec:results}
\vspace{-0.2cm}
\subsection{Experimental Setup}
\vspace{-0.2cm}
The experiments have been performed on two different California counties, including Imperial County and Yolo County. For both regions, the crop survey data provided by the California Department of Water Resources (CDWR) \cite{dataset} for the year 2016 have been used as ground truth information.
%
\begin{figure}[t]
\centering
\includegraphics[width=8.5cm]{images/mapsamples.png}
\caption{The regions of interest showing training (red) and test (blue) data distribution for Yolo (left) and Imperial Counties (right) distribution.}
\vspace{-0.2cm}
\label{fig:mapsamples}
\end{figure}
%
\par
For Imperial County, $10$ major classes have been considered, while $14$ classes have been taken into account for Yolo County. The "Others" class contains either minor crop types, fallow land, or build up. There are about $7256$ and $7340$ total parcels for Imperial County and Yolo County, respectively, as mentioned in Table~\ref{tab:yolo_samples}. These parcels have been split into $60\%$ training data and $40\%$ test data. The fields have been divided into training and test sets before sampling to ensure that no sample from the training region is included in the test set. The Fig.~\ref{fig:mapsamples} shows the distribution of training and testing parcels. The imagery of Sentinel-2 and Landsat-8 has been used for Imperial County and Yolo County, respectively. In both cases, nine scenes from March to November 2016 have been used. The same dataset has been used to train all the classifiers, including proposed and state-of-the-art methods, in both pixel and parcel-based analysis.
\begin{table}[ht]
\centering
\small
\caption{Summary of survey data from Imperial and Yolo counties provided by California Department of Water Resources (CDWR)}
\begin{tabular}{p{1.6cm} p{0.8cm} p{0.8cm} | p{1.6cm} p{0.8cm} p{0.8cm}}
\hline
\multicolumn{3}{c}{\textbf{Imperial County}} & \multicolumn{3}{c}{\textbf{Yolo County}}\Tstrut\Bstrut\\
\hline
\multirow{2}{*}{\textbf{Class}} &\textbf{Parcels} & \textbf{\%age} & \multirow{2}{*}{\textbf{Class}} &\textbf{Parcels} & \textbf{\%age}\Tstrut\\
& \textbf{Count} & \textbf{area} & & \textbf{Count} & \textbf{area}\Bstrut\\
\hline \hline
Alfalfa & 2156 & 31.3\% & Rice & 350 & 11.2\%\Tstrut\\
Pastures & 1150 & 15.4\%& Safflower & 161 & 2.4\%\\
Lettuce& 619 & 8.2\%& Corn & 150 & 2.1\%\\
Wheat & 315 & 4.3\%& Field crops & 407 & 6.6\%\\
Onions & 229& 3.0\% & Alfalfa & 575 & 8.7\%\\
Truck Crops & 1026 & 11.6\% & Pastures & 440 & 4.3\%\\
Corn & 232& 3.5\% & Cucurbits & 168 & 1.8\%\\
Field Crops & 192 & 2.9\% & Tomatoes & 496 & 9.8\%\\
Subtropical &271 & 1.6\% & Truck Crops & 289 & 1.3\%\\
Others & 1066 & 18.2\%& Almonds & 799 & 8.9\%\\
& & & Deciduous & 867 & 5.3\%\\
& & & Subtropical & 176 & 1.1\%\\
& & & Vineyard & 688 & 5.6\%\\
& & & Others & 1874 & 30.9\%\\\hline
\end{tabular}
\vspace{-0.4cm}
\label{tab:yolo_samples}
\end{table}
\subsection{Comparative Analysis}
\vspace{-0.2cm}
We compared our proposed hybrid model with four state-of-the-art methods, namely Random Forest, 1D CNN \cite{zhong2019deep}, 2D CNN\cite{kussul2017deep} and 3D CNN\cite{ji20183d}
\begin{table*}[t]
\centering
\small
\caption{Table showing comparison of our proposed 3D$\rightarrow$1D CNN with state-of-the-art methods on Imperial and Yolo counties. The comparison is performed using pixel-wise and parcel-wise strategies (provides single classification within parcel boundary)}
\resizebox{0.9\textwidth}{!}{
\begin{tabular}
{c|cc|cc|cc|cc}
\hline
\multirow{3}{*}{\textbf{Classifier}}&\multicolumn{4}{c}{\textbf{Imperial County}} & \multicolumn{4}{c}{\textbf{Yolo County}}\Tstrut\Bstrut\\
\cline{2-9}
&\multicolumn{2}{c|}{\textbf{Pixel-wise}} & \multicolumn{2}{c|}{\textbf{Parcel-wise}}&\multicolumn{2}{c|}{\textbf{Pixel-wise}} & \multicolumn{2}{c}{\textbf{Parcel-wise}}\Tstrut\Bstrut\\
\cline{2-9}
& \textbf{Accuracy}&\textbf{F1-Score}&\textbf{Accuracy}&\textbf{F1-Score}& \textbf{Accuracy}&\textbf{F1-Score}&\textbf{Accuracy}&\textbf{F1-Score}\Tstrut\Bstrut\\
\hline \hline
Random Forest & 78.96\% & 78.60\% & 80.61\% & 80.39\% & 86.68\% & 86.25\%& 82.23\% & 81.21\%\Tstrut\\
1D CNN \cite{zhong2019deep} & 79.46\% & 79.05\%& 80.33\% & 80.24\%& 87.94\% & 87.80\%& 84.45\% & 83.90\%\\
2D CNN \cite{kussul2017deep} & 79.71\% & 79.48\%& 80.54\% & 80.40\% & 88.50\% & 88.36\%& 84.55\% & 84.18\%\\
3D CNN \cite{ji20183d} & 79.95\% & 79.70\%& 80.80\% & 80.72\%&88.01\% & 87.82\%& 83.17\% & 82.74\%\\
3D$\rightarrow$1D CNN (Ours)& \textbf{81.23\%} & \textbf{81.16\%}& \textbf{81.76\%} & \textbf{81.76\%} & \textbf{90.23\%} & \textbf{90.08\%} & \textbf{85.87\%} & \textbf{85.51\%}\\
\hline
\end{tabular}
}
\label{tab:results}
\end{table*}
\begin{figure*}[ht]
\centering
\begin{minipage}[b]{0.3\linewidth}
\centering
\centerline{\includegraphics[width=5cm]{images/originalimperialmap3.png}}
\centerline{}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.3\linewidth}
\centering
\centerline{\includegraphics[width=5cm]{images/imperialpapermap3.png}}
\centerline{}\medskip
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\centerline{\includegraphics[width=5cm]{images/imperial3d1dmap3.png}}
\centerline{}\medskip
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\centerline{\scalebox{-1}[1]{\includegraphics[width=5cm]{images/originalyolomap3.png}}}
\centerline{(a) Ground Truth}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.3\linewidth}
\centering
\centerline{\scalebox{-1}[1]{\includegraphics[width=5cm]{images/yolopapermap3.png}}}
\centerline{(b) 3D CNN\cite{ji20183d}}\medskip
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\centerline{\scalebox{-1}[1]{\includegraphics[width=5cm]{images/yolo3d1dmap3.png}}}
\centerline{(c) 3D$\rightarrow$1D CNN (Ours)}\medskip
\end{minipage}\vspace{-0.2cm}
\caption{Ground truth and maps produced for Imperial County (Top) and Yolo County (Bottom) by 3D CNN and our network.\label{fig:maps}}
\vspace{-0.2cm}
\end{figure*}
For random forest, the number of trees has been set to $100$, the minimum number of samples required to split a node is two, and the maximum number of features is set to the square root of the number of features. For the 1D Convolutional Neural Network, the architecture proposed in \cite{zhong2019deep} has been used except that more features have been used instead of just EVI. The idea given by \cite{pelletier2019temporal} has been applied to consider more features in 1D CNN. For 2D Convolutional Neural Network, five ensembles of 2D CNN have been used as proposed in \cite{kussul2017deep}. The number of filters in the network ensembles is 160, 170, 180, 190, and 200. Due to the larger number of input features, more filters have to be used. For 3D Convolutional Neural Network, the model proposed in \cite{ji20183d} using the concept of VGG has been used, but the number of filters has been increased to 64 and 128 in both layers. Our proposed method has the same architecture as explained in Sec.~\ref{sec:method}.
\par
All the deep learning networks have been trained for about $50$ epochs using Adam optimizer with a default learning rate of $0.001$. The batch size of $128$ has been used, and each network's best results have been recorded. The accuracy and weighted F1-score for each classifier are mentioned in the Tables~\ref{tab:results}. The table shows the pixel-wise accuracy and F1-score as well as the parcel-wise results. It is evident from both tables that our proposed model has outperformed all the other classifiers in both experiments. Also, other deep learning models don't show much better accuracy than the random forest, and parcel-wise accuracy for the Imperial county of random forest is greater than other deep learning models except for our proposed one. On the other hand, our 3D$\rightarrow$1D CNN model improves the accuracy and F1-score by $2\%$. The parcel-wise results have been calculated by assigning each parcel a label based on the majority voting of pixels in it.
\par
Contrary to our expectation, the parcel-wise accuracy is $4-5\%$ less than pixel-wise accuracy for each classifier in the case of Yolo County. The Yolo County survey data analysis revealed that $3.5\%$ parcels in test data were tagged as multi-use, i.e., more than one crop was sown in those fields. In contrast, Imperial County data have no field labeled as multi-use. Therefore, accuracy increased in the case of Imperial County while decreased in the case of Yolo County. Another reason is the presence of larger polygons in the test set of Yolo County.
\par
The \textbf{Qualitative Analysis} is shown in Fig.~\ref{fig:maps} which shows the original labels from CDWR data and results produced by 3D CNN \cite{ji20183d} and our proposed 3D$\rightarrow$1D CNN model. In the figure, there are some blank spaces where ground truth was not available, and it can be observed that those blank spaces have been predicted as "Others" represented by black color. The zoomed portion of the results in Fig.~\ref{fig:maps} further illustrates that the maps produced by our hybrid model have lesser noise compared to the ones produced by vanilla 3D CNN.
\begin{table}[tbh]
\centering
\caption{Performance Comparison of State of the art models}
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{c c c}
\hline
\textbf{Classifier} & \textbf{Parameters} & \textbf{Inference time (ms)}\Tstrut\Bstrut\\
\hline \hline
1D CNN \cite{zhong2019deep} & 1,083,970 & 6.39\Tstrut\\
2D CNN \cite{kussul2017deep} & 4,278,282 & 7.11\\
3D CNN \cite{ji20183d} & 394,050 & 4.72\\
3D$\rightarrow$1D CNN (Ours) & 361,406 & 4.58\\
\hline
\end{tabular}
}
\label{tab:performance}
\end{table}
\vspace{-0.4cm}
\subsection{Computational Cost}
\par
Table~\ref{tab:performance} shows the comparison of the number of parameters in different deep learning models and their inference time. Our model has a lesser number of parameters as well as the fastest inference time. This is because, instead of flattening the output of 3D CNN, we fed it to 1D CNN, which reduced the number of parameters and improved the performance.
\vspace{-0.2cm}
\section{Conclusion}
\vspace{-0.4cm}
In this paper, a new architecture and a method for large-area crop classification have been presented. Our hybrid model combines the spatio-temporal representation via 3D CNN with temporal only representation via 1D CNN. Different neural network architectures proposed in the past for crop mapping have been implemented and compared with our designed model. Experiments have been performed using multispectral imagery of different satellites to show the generality of our method. The experimentation has revealed that the proposed hybrid network performs better on benchmark CDWR datasets.
In the future, we aim to extend our approach to fields that have been marked as multi-use (having more than one crop in a small parcel) as multi-use is a more commonly observed pattern in agriculture-based economies, particularly among developing countries.
\small
\bibliographystyle{IEEEbib}
| {'timestamp': '2021-03-19T01:13:11', 'yymm': '2103', 'arxiv_id': '2103.10050', 'language': 'en', 'url': 'https://arxiv.org/abs/2103.10050'} |
\section{Context}
\label{sec:context}
Maritime traffic surveillance has played an important role, not only in navigation safety, but also in other Maritime Situational Awareness (MSA) aspects. Among others, the early detection of abnormal behaviours helps identify suspicious activities, enforce law, perform efficiently search and rescue, etc.
In recent years, the world has experienced an unprecedented development of Big Data and Artificial Intelligence (AI), of which Deep Learning~\cite{lecun_deep_2015} is the core. People have tried to leverage Big Data and AI in many domains, and maritime surveillance is not an exception. The rich information provided by the Automatic Identification System (AIS) makes it a very appealing research topic. In~\cite{mantecon_deep_2019}, Airbus Defence and Space used a ResNet-based architecture to re-identify the navigation status of vessels. Similarly, convolutional neural networks were used to discover mobility modes in~\cite{chen_use_2018}. Some work went into detecting fishing pattern using a data mining approach~\cite{souza_improving_2016}. NATO STO Centre for Maritime Research and Experimentation used recurrent neural networks to predict vessel position in~\cite{forti_prediction_nodate}. Our previous work~\cite{nguyen_multi-task_2018,nguyen_geotracknet-maritime_2019} takes advantage of the high modeling capacity of deep neural network to create a probabilistic presentation of AIS tracks for anomaly detection.
Although those models have given impressive experimental results, applying them to operational systems is still called into question: (i) the black (or gray) box nature of Deep Learning raises concern about their reliability, (ii) the massive volume of AIS data requires a suitable deployment strategy. Besides that, the absence of reference groudtruthed dataset also questions the ability to evaluate the operational usefulness of such learning-based strategies.
In this paper, we report our progress in making a research prototype, namely the \textit{GeoTrackNet } model presented in \cite{nguyen_multi-task_2018} and \cite{nguyen_geotracknet-maritime_2019}, reach operational needs.
\revise{Specifically, we analyse the types of anomalies detected by the model, and evaluate scalability as well as the possibility of deploy it in real-time a big data platform.}
The paper is organised as follows: in Section~\ref{sec:geoTrackNet}, we give a brief introduction of \textit{GeoTrackNet}. The report on the validation of the output of the model is presented in Section~\ref{sec:validation}. Section~\ref{sec:onlineDetection} demonstrates the ability to deploy \textit{GeoTrackNet } in a real-time distributed system. We end the paper by the conclusions and perspectives in Section~\ref{sec:conclusions}.
\section{GeoTrackNet}
\label{sec:geoTrackNet}
In this section, we summary the principles of \textit{GeoTrackNet}. For more details, readers are encouraged to refer to~\cite{nguyen_geotracknet-maritime_2019}.
\textit{GeoTrackNet } is a probabilistic model to detect abnormal behaviours in maritime traffic. The model is based on the hypothesis that most of vessels' behaviours in the training set are normal and the model can learn the dynamics of those tracks. In the training phase, \textit{GeoTrackNet } learns a distribution representing the data (see Fig.~\ref{fig:GeoTrackNet}). This distribution is then used in the test phase to state how likely a new AIS track is.
Any track that does not follow this distribution will be flagged as abnormal. To capture the high complexity of AIS data, \textit{GeoTrackNet } uses a Variational Recurrent Neural Network (VRNN)~\cite{chung_recurrent_2015}, which is the state of the art in text and speech modeling.
Because vessel density and vessels' behaviours are not geographically homogeneous, \textit{GeoTrackNet } also exploits a geographically-dependent \textit{a contrario} detection rule. The Region Of Interest (ROI) is divided into small cells, then a local threshold is applied to state the normality of each AIS message in this cell.
\begin{figure}
\centering
\includegraphics[width=85mm]{figures/geoTrackNet_principle.png}
\caption{\textit{GeoTrackNet } principle. In the training phase, the model learns a distribution representing AIS tracks in the ROI. Any track that does not follow this distribution will be associated with a very low likelihood and will be considered as abnormal in the test phase.}
\label{fig:GeoTrackNet}
\end{figure}
\section{Analysis of the performance of GeoTrackNet}
\label{sec:validation}
Bringing such research prototype to an operational system requires, \textit{inter alia}, to address its validation in terms of relevance of the unusual trajectories that are detected, the explainability of the results and the scalability of the model.
\subsection{Relevance of the detected anomalies}
\label{sec:analysis_relevance}
Since \textit{GeoTrackNet } is a probabilistic model, it detects what is probabilistically unusual. These anomalies may not correspond to suspicious activities. As mentioned above, maritime anomaly detection is an intermediate step in MSA. The final purpose is to detect whether actions are needed when an anomaly happens, e.g. enforcing law when a smuggling activity is committed. It is thus operationally important to better understand the types of unusual trajectories raised by the system together with its limitations\footnote{\revise{The aim of this paper is to analyse the types of vessels' behaviours flagged as abnormal by \textit{GeoTrackNet}. For the comparison of the performance of of \textit{GeoTrackNet} and other state-of-the-art models', please refer to \cite{nguyen_geotracknet-maritime_2019}}}.
\revise{It is also important to note that maritime traffic anomaly detection is an ill-defined problems. No universal definitions of abnormal behaviours or groundtruthed datasets are available for this problem, hence the validation of the outcomes of the detector is subjective. We focus on understanding the types of vessels' behaviours will be flagged by \textit{GeoTrackNet}, rather than calculating quantitative measure.}
This analysis involved a manual inspection of the results of \textit{GeoTrackNet } over AIS data acquired around the Ushant Traffic Separation Scheme (TSS) by maritime domain experts
from CLS Group (Collecte Localisation Satellites).
Those experts are in charge of the operational monitoring of maritime traffic using multiple sources of data (AIS, Earth Observation, Open Source Intelligence, ELINT, etc). For this analysis they took into consideration their knowledge of the maritime activities in the area, the AIS tracks, the log of the vessels, the weather conditions, etc. This analysis allows to highlight the types of trajectories that are raised (or not raised) as unusual.
\subsubsection{Unusual trip}
\label{sec:unusual_trip}
\begin{figure}
\centering
\includegraphics[width=80mm]{figures/running_in.png}
\caption{Running-in. This vessel made a test voyage after being repaired at the port of Brest.}
\label{fig:running-in}
\end{figure}
Fig.~\ref{fig:running-in} shows the trajectory of a vessel of type cargo raised by \textit{GeoTrackNet }, steaming to sea and then turning back to the departing port. The visual inspection of the vessels tracks in the dataset shows that this behaviour is indeed unusual. One could consider that this voyage might involve a transshipment. However, after checking other sources of information, we figured out that this vessel had been repaired in the port of Brest (it was berthed in the shipyard), and most probably performed a test trip before going back to the port.
Even if finally not suspicious, we consider relevant to report such tracks as statistically unusual in this location.
\subsubsection{Effect of weather conditions}
\label{sec:weather_conditions}
\begin{figure}
\centering
\includegraphics[width=80mm]{figures/wind.png}
\caption{Anomaly due to extreme weather condition. Because of a strong wind blew from the opposite direction, this vessel slowed down (in the red zone). Some similar cases were detected at the same night.}
\label{fig:wind}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm]{figures/wind20170325.png}
\caption{Wind speed and wind direction on 2017-03-25 at 21:00 UTC from NCEP GFS weather forecast. From this model the wind was blowing from North East in the opposite direction of vessels entering the English Channel with a speed of around 30 knots.}
\label{fig:wind_speed}
\end{figure}
\textit{GeoTrackNet } does not take into account the environmental conditions.
In extreme weather/ocean conditions, vessels' behaviours change.
By nature, extreme conditions are rare events. Hence, the model did not see enough of these types of behaviours in the training set to include them into the normalcy model.
The changes of behaviour due to specific weather condition may then be flagged as abnormal behaviours.
An example of such cases is shown in Fig.~\ref{fig:wind}. On the night of March 25, 2017, there was a strong wind blowing from the North East. Some vessels in the area were highly affected by this wind at the entrance of the English Channel just after turning right after the Ushant TSS, while the wind was blowing in the opposite direction with a speed of around of 30 knots.
Again, this behaviour is not suspicious, but raised as statistically unusual. We consider relevant to report such tracks as in such specific weather conditions. Some vessels may be impacted differently by wind drag.
\subsubsection{Weak changes of trajectories}
\label{sec:weak}
\begin{figure}
\centering
\includegraphics[width=80mm]{figures/stop1.png}
\caption{Example of a weak change in trajectory detected by \textit{GeoTrackNet }.}
\label{fig:stop}
\end{figure}
As expected \textit{GeoTrackNet } succeeds to raise anomalies on small changes of trajectories mixing both unusual changes of speed and location. Such weak change of trajectory is illustrated in Fig.~\ref{fig:stop}. A vessel left the English channel, slowed down slightly off the usual route, and then started again. Similar behaviours were observed near the TSS in the opposite track (vessel stopping briefly before entering the Channel).
We associated this type of behaviours to the \textit{MARPOL NOx Tier III Regulation before entering/leaving Sulphur Emission Control Areas (SECA)}~\cite{international_maritime_organization_imo_international_nodate}.
Such SECA has been in place since 2015 for the North Sea including the English Channel with a limit at the 5°W meridian. The reporting of such behaviours is relevant as it is likely that the vessels stop their engine before changing their fuel. This may relate to higher risks of engine failure (not restart after stop) with vessels potentially drifting in the traffic lane or to the coast.
\subsubsection{Unreported anomalies}
\label{sec:unreported}
As \textit{GeoTrackNet } is designed to report statistically unusual tracks, it does not report those that are not sufficiently represented in the training dataset. Without surprise, it thus fails to report vessels crossing the traffic lanes and entering the separation area. This TSS is heavily monitored by the local authorities and the Maritime Rescue Co-ordination Centre (MRCC) and the separation areas can easily be seen for instance in Fig. \ref{fig:new_results}. However, such trajectories can easily be reported by simpler techniques, by just checking the position of AIS messages with respect to those forbidden areas.
\subsection{Scalability of the model}
\label{sec:scalability}
Former results were demonstrated on an area of interest with a limited geographic coverage based on training over terrestrial AIS data~\cite{nguyen_geotracknet-maritime_2019}. Using \textit{GeoTrackNet } on different scenarios raises the following questions: Does it provide similar results when trained and applied over a larger geographic area? Does it provide similar performance using a combination of both terrestrial and satellite AIS data? Does it provide relevant results on areas where maritime traffic is less structured? The evaluation of these questions is ongoing. However, we have preliminary results related to the scalability of the model.
\begin{figure}
\centering
\includegraphics[width=80mm]{figures/new_ROI.png}
\caption{New dataset. The ROI of the new dataset is 3.4 times bigger than the original. It contains both S-AIS and T-AIS, while the original training set contains T-AIS only.}
\label{fig:new_ROI}
\end{figure}
We tested the model on a new dataset, which contains both satellite AIS messages (S-AIS) and terrestrial AIS messages (T-AIS)\footnote{The original dataset contains T-AIS only.}, over the same period (January to March 2017). The ROI of the new dataset is 3.4 times bigger than the original one. We used the same hyper-parameters as those used in the previous setting, except the dimensions of the ROI.
\begin{figure}
\centering
\includegraphics[width=80mm]{figures/new_results.png}
\caption{AIS tracks detected as abnormal by \textit{GeoTrackNet } trained on the new dataset. Blue: AIS tracks in the training set; other colours: detected tracks.}
\label{fig:new_results}
\end{figure}
The result is shown in Fig.~\ref{fig:new_results}. Almost all the tracks flagged as abnormal by the previous model are detected again by the new model. There are new abnormal tracks because they do not exist in the original dataset (S-AIS tracks or outside of the previous ROI).
The combination of S-AIS and T-AIS seems to improve the performance of \textit{GeoTrackNet}. As shown in Fig.~\ref{fig:s-ais}, a vessel which was falsely detected by the original model is correctly flagged as normal by the new model thanks to the improved training dataset.
\begin{figure}
\centering
\includegraphics[width=80mm]{figures/trainingdata_issue.png}
\caption{An AIS track flagged as abnormal by the original model but considered as normal by the new model. This track is actually normal. However, in the original training set, there are not many tracks of this type (because they are outside of the coverage zone of the terrestrial AIS station), hence it was flagged as abnormal. The coverage area of the new dataset is better, \textit{GeoTrackNet } then succeeded to tag this track as normal.}
\label{fig:s-ais}
\end{figure}
\section{Online Detection}
\label{sec:onlineDetection}
Within an operational surveillance system of a given area of interest, it is mandatory
to detect suspicious behaviours as early as possible in order to be able to
react accordingly if necessary. For this application, \textit{GeoTrackNet } could be a
tool helping the analysts to sort through the ever increasing flow of AIS data
they have to monitor in the area of interest. In this context, the ability of \textit{GeoTrackNet}, possibly implemented within a distributed framework, to process real AIS data streams and raise alerts in real time is a key issue.
In this section, we first present the proposed framework for the implementation of \textit{GeoTrackNet } in a streaming context. Then, we report numerical experiments to evaluate scale-up performance.
\subsection{High level principles}
\label{subsec:highLevelPrinciples}
In a streaming context, \textit{GeoTrackNet } is wrapped in a streaming operator: the
\emph{GeoTrackNet Operator}\xspace. This operator is responsible for (a) preprocessing the stream of AIS
messages and (b) triggering the calls to the anomaly detection
function. The preprocessing phase consists in building incrementally the
tracks from the stream of AIS messages. A track is composed of AIS messages
belonging to the same ship (same MMSI) where erroneous positions ({e.g.,}\xspace not in
the ROI) or erroneous speed messages ({e.g.,}\xspace greater than 30 knots) have been
removed. The time difference between consecutive messages in a track is
assumed to be less than a threshold (here 4 hours). Otherwise, they will be
part of two different tracks. Note that tracks are re-sampled to a resolution
of 10 minutes using a linear interpolation.
The \emph{GeoTrackNet Operator}\xspace must then keep hold of the currently active tracks until a minimum
track duration is reached before triggering the anomaly detection. To cope with a
potentially high velocity of the AIS messages stream, the \emph{GeoTrackNet Operator}\xspace can be
replicated. In this situation the stream of input messages needs to be
partitioned such that two messages belonging to the same ship are sent to the
same instance of the \emph{GeoTrackNet Operator}\xspace. We choose to partition the stream of messages
according to the value of the MMSI field to achieve this requirement.
The current implementation of the \emph{GeoTrackNet Operator}\xspace is based on the
Faust\footnote{https://faust.readthedocs.io} library. This library implements
the Kafka\footnote{https://kafka.apache.org} protocol. The Kafka protocol
allows for partitioning the stream of AIS messages regardless of the number of
\emph{GeoTrackNet Operators}\xspace actually deployed. Consequently, the infrastructure can get scaled up and
down when necessary to keep up with the message rate.
\subsection{Synthetic evaluation}
The performance of \textit{GeoTrackNet } is critical as it allows both early anomaly
detection and low resource usage. In this section, a quantitative
evaluation of the computational complexity of \textit{GeoTrackNet } is proposed. For this purpose, our benchmarking experiments rely on
a synthetic set of tracks built from the original dataset~\cite{nguyen_geotracknet-maritime_2019} but covering the
period from July 2011 to January 2018 \footnote{The code to reproduce the experiment in this Section is available at: https://github.com/msimonin/MultitaskAIS/tree/online\_detection/bench}.
In total, $237\,863$ tracks were built, $147\,786$ did not pass the preprocessing
step while $90\,077$ were actually tested for normality. For each tested track the execution
time of \textit{GeoTrackNet } was recorded. Note that this time includes the preprocessing
time. The evaluation has been carried on the Grid'5000 testbed~\cite{balouek_adding_2013}, specifically
on a Dell PowerEdge C6220 of the \textit{Paravance} cluster.
\textit{GeoTrackNet } was given one CPU core and the execution time
statistics are depicted in Fig.~\ref{fig:synthetic}.
\begin{figure}
\begin{center}
\begin{tabular}{lr}
mean & $2.07$ \\
\hline
std & $0.22$ \\
\hline
min & $1.49$ \\
\hline
$q_1$ & $1.93$ \\
\hline
median& $2.04$ \\
\hline
$q_3$ & $2.19$ \\
\hline
max & $3.74$ \\
\hline
\end{tabular}
\caption{Statistics of the execution time of \textit{GeoTrackNet } on $90\,077$
tracks. On average, it takes $2,07$s to test the normality of a track.
Only the tracks that passed the preprocessing phase are accounted here.}
\label{fig:synthetic}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{figures/uniq_mmsi.pdf}
\caption{CDF of the number of unique MMSIs received in a 10 minutes time window for the
original dataset (2011-2018). Reading: 80\% of the time less than 80 unique
MMSIs are received on a 10 minutes time window.}
\label{fig:synthetic:cdf}
\end{center}
\end{figure}
In our setup, \textit{GeoTrackNet } can process on average 0.5 tracks per second using one
CPU core. Additionally, due to the preprocessing phase, points are added to a
track every 10 minutes if the rate of messages received for this MMSI is at some point lower than that. Fig.~\ref{fig:synthetic:cdf} depicts the amount of unique MMSI for which at least one message is received in a fixed window. More specifically, according to the graph, a maximum of 400 unique MMSIs are received in a time window of 10 minutes. In other words, for this dataset and ROI, the peak rate at which the \emph{GeoTrackNet Operator}\xspace has to be called is 400 times per 10 minutes, but most of the time, it is significantly lower that that (as can be seen on the picture, 90\% of the time, less than 100 unique MMSI are received in a time window of 10 minutes). This leads us to the conclusion that 2 CPU cores are sufficient to treat all the tracks of the ROI.
\section{Conclusions and Perspectives}
\label{sec:conclusions}
\textit{GeoTrackNet } is a computer-aided system designed to report statistically unusual vessels' tracks. We analysed the validity of this system in terms of relevance of the reported anomalies, scalability of the model, computing performance and possibility to process data streams.
We showed that the model is able to report unusual behaviour that may be difficult to detect with simple rule-based geofencing and limits on course over ground (COG) or speed over ground (SOG). As expected, it fails to detect rare events in area of low traffic. However, most of them can be easily detected by other techniques such as geofencing. It is thus a promising complement in an operational system.
The current version of \textit{GeoTrackNet } does not take into account environmental information. A post-processing step where the ocean current, the wind speed, etc. are used to detect unusual vessels' behaviours due to extreme weather can be added in the future.
\section{Acknowledgment}
\label{sec:Acknowledgment}
This work was supported by public funds (Minist\`ere de l'Education Nationale, de l'Enseignement Sup\'erieur et de la Recherche, FEDER, R\'egion Bretagne, Conseil G\'en\'eral du Finist\`ere, Brest M\'etropole) and by Institut Mines T\'el\'ecom, received in the framework of the VIGISAT program managed by ``Groupement Bretagne T\'el\'ed\'etection'' (BreTel). The authors acknowledge the support of DGA (Direction G\'en\'erale de l'Armement) and ANR (French Agence Nationale de la Recherche) under reference ANR-16-ASTR-0026 (SESAME initiative) and AI Chair OceaniX, the labex Cominlabs, the Brittany Council and the GIS BRETEL (CPER/FEDER framework).
The dataset used in this paper is provided by CLS and Erwan Guegueniat.
The analysis of abnormal tracks in this paper was carried out using the Maritime Awareness System (MAS) of CLS.
Experiments in Section \ref{sec:onlineDetection} were carried out using the Grid'5000
testbed, supported by a scientific interest group hosted by Inria and including
CNRS, RENATER and several Universities as well as other organizations.
| {'timestamp': '2020-08-13T02:22:25', 'yymm': '2008', 'arxiv_id': '2008.05443', 'language': 'en', 'url': 'https://arxiv.org/abs/2008.05443'} |
\section{Introduction}
Cognitive deficit of older adults is one of the biggest global public health challenges in elderly care. Approximately 5.2 million people of 65 and older are suffered with any form of cognitive impairments in United States in 2012 \cite{stat12}. Dementia is one of the major causes of the cognitive impairments which is more acute among 85 and older population (50\%) \cite{stat12}. However, the costs (financial and time) of health care and long-term care for individuals with Alzheimer's (special form of dementia) or other dementias are substantial. For example, during 2016, about 15.9 million family and friends in United States provided 18.2 billion hours of unpaid assistance to those with cognitive impairments which is a contribution to the nation valued at \$230.1 billion. One the other hand, total payments for all individuals with all form of cognitive impairments are estimated at \$259 billion. Total annual payments for health care, long-term care and hospice care for people with Alzheimer's or other dementias are projected to increase from \$259 billion in 2017 to more than \$1.1 trillion in 2050. Among the above costs, a significant amount are relevant to clinical and diagnostic tests \cite{stat17}. Although clinical and diagnostic tests have become more precise in identifying dementia, studies have shown that there is a high degree of underrecognition especially in early detection. However, there are many advantages to obtaining an early and accurate diagnosis when cognitive symptoms are first noticed as the root cause findings of impairment always lessen the progress of impairment status and sometimes symptoms can be reversible and cured.
With the proliferation of emerging ubiquitous computing technologies, many mobile and wearable devices have been available to capture continuous functional and physiological behavior of older adults. Wearable sensors are now capable of estimating number of steps being taken, physical activity levels, sleep patterns and physiological outcomes (heart rate, skin conductance) of older adults \cite{sano15}. Ambient sensors also help capture the movement patterns of objects and humans for activity and behavior recognition \cite{dawadi14,dawadi15}. Researchers also proved the existence of correlations between cognitive impairment and everyday task performance \cite{dawadi14, akl15,alam16} as well as physiological symptoms \cite{alam16,sano15}. Although current studies showed some successes in IoT-assisted cognitive health assessment in different domains individually, there are several existing challenges in developing and validating a fully automated multi-modal assessment model.
\begin{enumerate}
\item \emph{Real-time IoT System}: A real-time IoT system must include a continuous and fault tolerant data streaming capability among central hub, wearable sensors and ambient sensors regardless of network communication protocol (WiFi, Ethernet, Bluetooth etc.) which are not available in existing researches.
\item \emph{Multi-modal Context Fusion}: Though several offline clinically validated cognitive health assessment tools exist \cite{wai03, starling99, krapp07, yesavage82, zung71}, there is no universally accepted method for IoT-assisted automatic cognitive health assessment in smart home environment that can fuse multi-modal sensor contexts altogether. For example, some researchers showed ambient sensors based Activities of Daily Livigin (ADLs) sequence pattern can signify the cognitive health status of older adults \cite{akl15, dawadi15}. Researchers also showed wearable Electrodermal Activity pattern analysis may carry the significance of cognitive status \cite{sano15}. However, for validation of IoT based cognitive health assessment, self-reported surveys, clinical diagnosis and observation based tools are used individually by prior researchers \cite{akl15, dawadi15, sano15, alam16}.
\end{enumerate}
Regarding aforementioned challenges for the automation of cognitive health assessment, \emph{AutoCogniSys} considers (i) reproducibility of our model in any smart home system consists of ambient motion sensors, wearable accelerometer (ACC) sensors, wearable Electrodermal Activity (EDA) and Photoplethysmography (PPG) sensors individually or combined streams; (ii) context awareness based on ambient motion sensors and wearable ACC sensors in any types of activities such as hand gestural, postural and complex ADLs; and (iii) high accuracy, i.e., a recall rate of over 90\% with less than 5\% false positive rate. More specifically, \emph{AutoCogniSys} extends our existing work \cite{alam16} in three dimensions,
\emph{(1) True Automation:} We first investigate the correlations of cognitive impairment with human activities and stress where we manually labeled activities, extract the corresponding physiological sensor (EDA and PPG) features of each activity, and use statistical method to find correlations. Then, we propose automatic complex activity recognition based on a Hierarchical Dynamic Bayesian Network (HDBN) model, fine-grained extraction of physiological sensor features and finally machine learning classification of cognitive impairment.
\emph{(2) Noises Elimination:} We define different types of noises on ACC, EDA and PPG sensors, propose extensive signal processing techniques to remove noises and show significant improvement can be achieved in cognitive impairment classification.
\emph{(3) Implementation and Evaluation:} Finally, we design and implement IoT system and analytic methods and minimize the human involvement to automate our proposed cognitive health assessment approach by considering effective smart home sensor customization and deployment, data collection, screening, cleaning and filtering, feature computation, normalization and classification, and activity model training.
\textbf{Research Questions:} \emph{AutoCogniSys} consequently tackles the following key research questions.
$\bullet$ Can we detect simultaneously the periodic rhythms of both hand gestures and postural activities from wrist-worn ACC sensor signal for diverse population (population with same activity but diverse ways such as walking with walker, stretcher or normally)? If so, how can we incorporate the hand gesture, posture and ambient sensor data streams to help improve the ADLs recognition models?
$\bullet$ How can we exploit and relate the micro-activity features into noise free physiological sensor signals processing to automate cognitive health assessment process? What are the critical roles of clinical survey and technology guided assessment methodologies and their inter-relationships for automating the different intermediate steps of cognitive health assessment process?
To tackle these, we make the following \textbf{key contributions}:
$\bullet$ We employ an extensive signal deconvolution technique that in conjunction with machine learning technique helps facilitate a wrist-worn ACC-based multi-label (hand gestural and postural) activity recognition for diverse population. We then leverage multi-label context sets with ambient and object sensor signals for complex activity recognition based on HDBN model.
$\bullet$ We propose a novel collaborative filter for EDA signal processing by postulating signal as a mixture of three components: \emph{tonic phase, phasic phase} and \emph{motion artifacts}, and employ convex optimization technique for filtering out the motion artifacts. We also propose a novel PPG signal processing technique to filter out the inherent motion artifacts and noises using improved Periodic Moving Average Filtering (PMAF) technique.
$\bullet$ We design and prototype an IoT system consisting of multiple devices (wearable wrist band, IP camera, object and ambient sensors) connected with central hub via WiFi, Ethernet and Bluetooth communication protocols. We collected data from 22 older adults living in a continuing care retirement community center in a very natural setting (IRB \#HP-00064387).
$\bullet$ Finally, we employ statistical and machine learning techniques to jointly correlate the activity performance metrics and stress (EDA and PPG) features that helps achieve max. 93\% of cognitive impairment status detection accuracy. We evaluate \emph{AutoCogniSys} on 5 clinically validated offline assessment tools as ground truth.
\section{Related Works}
\emph{AutoCogniSys} builds on previous works on wearable devices based low-level (postural and hand gestural) activity recognition and their integration with ambient sensors to recognize complex ADLs, the underlying signal processing and applications on cognitive health assessment automation.
\subsection{Wearable Sensor Signal Processing}
Wearable sensors can be two types: physical and physiological. Physical sensors (accelerometer, gyroscope etc.) signal values change over the movements of the sensor devices. Physiological sensors change over physiological condition of body such as EDA changes over stress and PPG changes over heart rate. However, physical movements also impose noises on physiological sensor signals which is called \emph{motion artifacts}.
\subsubsection{Physiological Signal Processing}
A continuous and descrete decomposition of EDA, and time and frequency domain analytics of PPG signal have been investigated before to extract relevant physiological features which were contaminated with noises and motion artifacts \cite{alam16}. \cite{setz10} denoised and classified EDA from cognitive load and stress with accuracy higher than 80\%. Though motion artifacts removal techniques such as exponential smoothing \cite{hern11} and low-pass filters \cite{poh10, hernandez14} provide significant improvement in filtering EDA signals, wavelet transforms offer more sophisticated refinement for any kind of physiological sensors such as electroencephalogram \cite{krish06, zikov02}, electrocardiogram \cite{erc06,alfa08}, and PPG \cite{lee03}. \cite{chen15} proposed a stationary wavelet transform (SWT) based motion artifacts removal technique. `cvxEDA' proposed a convex optimization technique considering EDA as a mixture of white gaussian noise, tonic and phasic components where white gaussian noise includes motion artifacts and external noises \cite{greco16}. \emph{AutoCogniSys} intelligently combines SWT and `cvxEDA' together to remove noises and motion artifacts from EDA signal. On the other hand, it is more difficult to remove motion artifacts from PPG signal due to its periodicity of nature \cite{wang13}. Researchers proposed different methods such as frequency analytics \cite{garde13,wang13}, statistical analytics \cite{peng14} and digital filter \cite{lee10} to reduce noises and motion artifacts from PPG. \emph{AutoCogniSys} used Periodic Moving Average Filter (PMAF) in this regard \cite{lee07}.
\subsubsection{Physical Sensor Signal Processing}
ACC based hand gesture recognition has been explored by several researchers in past such as discrete hidden markov model \cite{liu10}, artificial neural network \cite{arce11}, weighted naive bayes and dynamic time warping \cite{mace13}. Akl et. al. proposed 18 gesture dictionary based Support Vector Machine (SVM) classifier \cite{akl11}. Wrist-worn ACC based postural activity recognition approach has been proposed using Decision Tree, Random Forest, Support Vector Machines, K-Nearest Neighbors, Naive Bayes and deep neural networks \cite{gj14, wang16}, the accuracy stagnates at 85\% using SVM method \cite{martin16}. However, neither of past works proposed any technique that can provide single body worn ACC sensor-based multiple body contexts recognition nor works efficiently for diverse posture say walking normally, with walker, with double walker or wheel chair. Our proposed 8-hand gesture recognition technique assisted sparse-deconvolution method improves classification performances on both normal and diverse postures. However, we incorporated hand gestures and postures in conjunction with ambient sensors into single-inhabitant HDBN model \cite{alam16b} that provides significant improvement in complex activity recognition.
\subsection{Cognitive Health Assessment}
Smart home environment has been used for providing automated health monitoring and assessment in the ageing population before \cite{dawadi14, gong15, akl15, dawadi15}. `SmartFABER' proposed a non-intrusive sensor network based continuous smart home environmental sensor data acquisition and a novel hybrid statistical and knowledge-based technique to analyz the data to estimate behavioral anomalies for early detection of mild-cognitively impairment \cite{riboni16}. \cite{skubic15} presented an example of unobtrusive, continuous monitoring system for the purpose of assessing early health changes to alert caregivers about the potential signs of health hazards. Though, prior researches proposed a sequence of ambient motion sensor streams as complex activity components in activity based health assessment \cite{dawadi14, gong15, akl15, dawadi15}, we consider inclusion of an wearable wrist-band with in-built ACC sensor to detect hand gesture and posture, augmenting with the ambient sensor readings to help recognize complex activities as well as cognitive health assessment of older adults. Additionally, we propose intelligent use of physiological features of skin through different physiological sensor signals (EDA, PPG) processing in daily activity tasks and incorporate context-awareness for automation of cognitive health assessment that have not been explored before.
\begin{figure}[!htb]
\begin{center}
\epsfig{file=flowchart.pdf,height=1.6in, width=3.5in}
\caption{Overall flow of \emph{AutoCogniSys} pipeline.}
\label{fig:overview}
\end{center}
\end{figure}
\section{Overall Architecture}
We first investigate existing IoT-based cognitive health care frameworks that covers every aspects of wearable (physical, physiological) and ambient (passive infrared and object) sensor signals computing. \emph{AutoCogniSys} is comprised of three component modules: (i)~sensing, (ii)~processing, and (iii)~analysis. The `sensing' module consists of clinical assessment tools (surveys, observation and clinical backgrounds) and sensing signals (ambient and wearable sensors). `Sensor processing' module is comprised of three sub-modules: a)~clinical assessment feature extraction from assessment tools; b)~ambient sensor feature extraction; and c)~wearable sensor processing (noise removal, segmentation, feature extraction, classification etc.). `Analysis' module is comprised of machine learning and statistical analytics-based score prediction of cognitive impairment. Automation of each module's functionality and inter-intra modular transactions without human interference can be called {\it true automation} of cognitive health assessment. Fig.~\ref{fig:overview} shows the overall flow of \emph{AutoCogniSys} which is discussed in details in the following sections.
\subsection{Demographic Ground Truth Data Collection}
Currently in standard clinical practice and research, the most accurate evaluations of cognitive health assessment are one-to-one observation and supervision tasks/questionnaires for monitoring an individual's functional abilities and behavior \cite{resnick15}. In the first stage of this pilot study, we have investigated current literatures and carefully chosen the clinically proven functional and behavioral health assessment survey tools \cite{resnick15}. On the other hand, to cross check with the survey based evaluations, we have also chosen clinically justified observation based behavioral assessment methods. First, following the resident consent, our clinical research evaluator collects demographic and descriptive data (age, gender, race, ethnicity, marital status, education and medical commodities). She has performed two types of clinical assessments: (1) \emph{Observation based} where the resident's cognition is assessed using the Saint Louis University Mental Status (SLUMS) scale \cite{wai03}. (2) \emph{Survey based} where five widely used and clinically well validated surveys are taken into account: (a) \emph{Yale Physical Activity Survey} \cite{starling99}; (b) \emph{Lawton Instrumental Activities of Daily Living}; (c) \emph{Barthel Index of Activities of Daily Living} \cite{krapp07}; (d) \emph{Geriatric Depression Rating scale} \cite{yesavage82}; and (e) \emph{Zung Self-Rating Anxiety scale} \cite{zung71}.
\subsection{Smart Environment Creation}
For an ideal IoT-based system, instrumenting and deploying it at each participant's natural living environment warrants for assembling a flexible set of hardware and software interfaces to ease the system configuration, setup, and network discovery processes. The sensor system placed in the residences of volunteers needs to meet several specific physiological signals and activity monitoring needs. However, we must confirm that the devices are reliable with potential for re-deployment as well as appear unintimidating to the participants. Inspired by the above requirements, we developed a real testbed IoT system, {\it SenseBox}, by customizing Cloud Engine PogoPlug Mobile base station firmware to integrate with WiFi (connect ambient and object sensors) and Bluetooth (connect wristband) protocol. The smart home components are as follows: (i) PogoPlug base server with a continuous power supply, (ii) 3 binary Passive Infrared sensors in three different rooms (kitchen, livingroom and bedroom) to capture room level occupancy, (iii) 7 binary object sensors attached with closet door, entry door, telephone, broom, laundry basket, trash can and trash box, (iv) three IP cameras in the appropriate positions to collect the ground truth data and (v) an Empatica E4 \cite{empatica} wrist-band (integrated sensors: PPG at 64 Hz, EDA at 4 Hz, Body temperature at 1 Hz and a triaxial ACC at 32 Hz) on the participant's dominating hand.
\section{Activity Recognition}
We aim to detect single wrist-worn ACC sensor based hand gesture and postural activities and insert these into an HDBN graphical model in conjunction with ambient and object sensor values for complex activity recognition. We consider the recognition problem asan activity tupple of $\langle gesture,posture,ambient,object \rangle$. Though, Alam et. al. provides significant performance improvement for single wrist-worn ACC sensor aided 18-hand gesture based postural activity recognition in lab environment \cite{alam17}, it faces some practical challenges in real-time smart environment with older adults due to the diversity of their postures. For example, some older adults use walker, double walking sticks or wheel chair for walking in which cases collecting 18 hand gestures and corresponding postural activities for training requires endless efforts and carefulness. To reduce the complexity of ground truth labeling and later state space explosion for graphical model (HDBN), we propose to use rotational normalization method that can merge some hand-gestures subject to directional differences and forms an 8-hand gesture model. However, our proposed Feature Weight Naive Bayes (FWNB) classifier adds significant improvement on Alam et. al. proposed sparse-deconvolution method as well as recognition in diverse postural environment.
\begin{figure}[!htb]
\begin{center}
\epsfig{file=hand_gestures.pdf,height=0.5in, width=3in}
\vspace{-.2in}
\caption{8 hand gesture dictionary with direction}
\label{fig:hand_gestures}
\vspace{-.2in}
\end{center}
\end{figure}
\subsection{Hand Gesture Recognition}
\label{sec:hand_gesture}
\emph{AutoCogniSys} proposes an 8-gesture dictionary (as shown in Fig. \ref{fig:hand_gestures}) and a Feature Weighted Naive Bayesian (FWNB) framework for building, modeling and recognizing hand gestures. The method comprises of the following steps: (i) \emph{Preprocessing:} wrist-worn ACC sensor provided 3-axis data are passed through 0.4Hz low-pass filter to remove the data drift. (ii) \emph{Rotation normalization:} Normalizing the rotation of hand gestures provides greater accuracy and allows for more realistic, orientation-independent motion. At first, we find the best fit plane of the acceleration vectors thus if the motion lies in a single plane, then the acceleration vectors of a closed shape should on average lie in that main plane. Then, we take all acceleration segments between points of inflection to form one single vector called reference vector that provides us the general direction of user's motion. After that, each vector is normalized relative to the reference vector. This normalization helps remove a lot of hand gestures from prior considered 18 hand gestures resulting a reduced dictionary of 8 gestures. (iii) \emph{Feature Weighted Naive Bayesian model:} Naive Bayes classifier is light-weight and efficient technique for hand gesture recognition. We extract 12 ACC features \cite{alam17} and calculate weight for each feature type based on the similarity of feature measures of the trained gestures (0$<$weight$<$1). While recognizing gestures, the proximity of each feature measure to the average trained feature measure of each gesture type is calculated by a normal distribution. Then, the proximity value is multiplied by the feature weight that was calculated in the training phase. All of these multiplied values are added together and the system predicts the gesture type with the greatest value as the user gesture. In the learning data points, there should be static postural activities (such as sitting, lying etc.) to avoid unexpected noises over wrist-worn ACC sensors. In the final hand gesture dictionary, we save the reference vector as our signal dictionary.
\subsection{Postural Activity Recognition}
In normal lab environment, wrist-worn ACC sensor signal is a mixture (convolution) of actual hand gesture and postural activity relevant signals \cite{alam17}. \emph{AutoCogniSys} improves the idea by reducing the number of hand gestures and postural activities to 8 (as shown in Fig.\ref{fig:hand_gestures}) using rotation normalization and 4 (walking, sitting, standing and lying). Then, we use sparse-deconvolution method (with 31\% signal reconstruction error) to get Approximately Sparse Factor. The summary of the entire process is stated bellow:
{\it Building Deconvolution Method:} We first consider the wrist-worn ACC sensor signals (3-axis values) as a convolution of hand gesture and postural activity effects and build a deconvolution framework. The deconvolution framework takes a known signal (hand gesture effects) and a equalizer parameter ($\lambda$) as input and provides an Approximately Sparse Factor signal (postural activity effects) as output. For 3-axis ACC signals, we need to learn associated 3 equalizer parameters for each hand gesture. Moreover, each equalizer parameter is involved with 4 postural activities that results a total 96 ($8\times 3\times 4$) equalizer parameters to learn.
{\it Learning Classification Model:} We use the Approximately Sparse Factor signal to extract 12 statistical features and SVM with sequential machine optimization (SMO) \cite{cao06} for postural activity recognition.
{\it Prediction Model:} After recognizing the hand gestures following the method explained in Sec.~\ref{sec:hand_gesture}, we take the corresponding reference vector as known signal and extract the Approximately Sparse Factor signals incorporating corresponding 3 equalizer parameters ($\lambda$) for the sparse-deconvolution method. Then, we apply feature extraction and prior learned SMO based SVM classifier \cite{cao06} to classify final postural activity. Fig.~\ref{fig:deconvolution} illustrates a single axis example of the deconvolution.
\begin{figure}[!htb]
\begin{center}
\epsfig{file=deconvolution.pdf,height=1.6in, width=3in}
\vspace{-.15in}
\caption{Sample deconvolution example of X-axis. The raw x-axis of accelerometer signal, reference vector of the sample gesture and the extracted corresponding ASF signal of walking.}
\label{fig:deconvolution}
\end{center}
\vspace{-.15in}
\end{figure}
\subsection{Complex Activity Recognition}
We build a HDBN based complex activity recognition framework for single inhabitant scenario smart home environment \cite{alam16b} taking the advantage of detected hand gestural and postural activities along with the ambient and object sensor streams. At first, we obtain instant hand gestural and postural activities from our above proposed models, and additionally motion sensor and object sensor readings from our IoT-system for every time instant generating a 4-hierarchy of HDBN model. Considering the context set $\langle gestural, postural, ambient,object\rangle$ as a hierarchical activity structure (extending two 2-hierarchical HDBN \cite{alam16b}), we build complex activity recognition model for single inhabitant scenario. Finally, we infer the most-likely sequence of complex activities (and their time boundaries), utilizing the well-known Expectation Maximization (EM) algorithm \cite{dempster77} for training and the Viterbi algorithm \cite{forney73} for run-time inference.
\section{Automatic Activity Features Estimation}
The effects of cognitive ability on daily activity performance have been studied before \cite{dawadi14,akl15}. They experimentally and clinically validated that cognitive impairment highly reduces the daily activity performances and this activity performance can be computed as an indicator of cognitive ability status of older adults. The standard activity features refer to completeness of task (TC), sequential task ability (SEQ), interruption avoidance capabilities (INT) etc. In current behavioral science literature, the above activity features carry specific definition based on the sub-tasks involved with a complex activity \cite{dawadi14,akl15}. Completeness of task refers to how many sub-tasks are missed by the participants. Sequential task ability refers to how many sequences of sub-tasks are missed referring the gerontologist defined standard sequences of the sub-task for the particular complex activity. Interruption avoidance capability refers to how many times the participants stop or interleave while doing any sub-task. The final goal of activity features estimation is to provide overall task score. The task score is proportional to the functional ability of participants in performance daily activities. Our behavioral scientist team, comprises with Nursing professor, gerontologist and retirement community caregivers, carefully discus, optimize and choose 87 sub-tasks in total for 13 complex activities.
Each of the sub-task comprises with sequential occurrences of hand gesture and postural activities. However, no researchers ever considered hand gesture for activity features estimation due to complexity of multi-modal wearable and ambient sensors synchronization and multi-label activity classification \cite{dawadi14,akl15}. \emph{AutoCogniSys} exploited single wrist-worn sensor based hand gesture and postural activity recognition, and proposed an activity features (TC, SEQ and INT) estimation method including these two parameters in conjunction with object and ambient sensor features that provide significant improvement of cognitive health assessment of older adults.
\subsection{Machine Learning Based Complex Activity Features Estimation}
In current cognitive health assessment literature, complex activity features can be defined as $\langle TC,SEQ,INT,TS\rangle$. We used supervised method to estimate TC, SEQ and INT, and unsupervised method to estimate TS. We first, formulate the automated scoring as a supervised machine learning problem in which machine learning algorithms learn a function that maps $\langle${\it hand gesture, posture, object, ambient sensor}$\rangle$ feature set to the direct observation scores. We use bagging ensemble method to learn the mapping function and SMO based SVM \cite{cao06} as base classifier. The learner averages by boostrapping individual numeric predictions to combine the base classifier predictions and generates an output for each data point that corresponds to the highest-probability label. We train three classifiers considering observation as ground truth for TC, SEQ and INT scores and test on the testing dataset. We derive unsupervised scores using dimensionality reduction technique for each feature set. First, we take all features of each activity, apply optimal discriminant analysis technique as a dimensionality reduction process \cite{zhang09} and reduce the feature sets into single dimensional value which represents the automated task completeness scores of the particular user activity. A min-max normalization is applied that provides us a uniform range of the variables using $
z_i=\frac{x_i-min(x)}{max(x)-min(x)}$ equation where $x=\{x1,\ldots,x_n\}$ and $z_i$ is $i^{th}$ normalized data. The final single dimensional score represents machine learning based TS score.
\section{Physiological Sensor Signals Processing}
The autonomic nervous system (ANS) restrains the body's physiological activities including the heart rate, skin gland secretion, blood pressure, and respiration. The ANS is divided into sympathetic (SNS) and parasympathetic (PNS) branches. While SNS actuates the body's resources for action under arousal conditions, PNS attenuates the body to help regain the steady state. Mental arousal (say stress, anxiety etc.) activates the sweat gland causing the increment and reduction of Skin Conductance on SNS and PNS physiological conditions respectively. However, Instant Heart Rate also has similar effect on SNS and PNS physiological condtions i.e., a higher value of heart rate is the effect of SNS and lower value is the outcome of PNS. EDA and PPG sensors are widely used to estimate the instant value of skin conductance and heart rate respectively \cite{alam16}.
\subsection{EDA Sensor Signal Processing}
EDA is the property of the human body that causes continuous variation in the electrical characteristics of the skin which varies with the state of sweat glands in the skin. There are three types of arousal: \emph{cognitive, affective and physical}. \emph{Cognitive} arousal occurs when a person tries to solve any problem using her cognitive ability. \emph{Affective} arousal occurs when a person is worried, frightened or angry either doing daily activities or in resting position. On the other hand, \emph{physical} arousal is related to the brain command to move bodily parts which is imposed on the total arousal as an artifact, called \emph{motion artifact}. However, there are always some noises due to the weather conditions (temperature, humidity etc.) and device motion. This \emph{motion artifact} can be the prime cause of signal contamination of physiological outcomes while performing daily activities which must be removed. \emph{AutoCogniSys} proposes an EDA sensor signal processing method consists of three steps: (i) noise and motion artifacts removal, (ii) separation of tonic component and phasic component (explained later) from contamination free EDA signal and (iii) feature extraction on the response window.
\subsubsection{Motion Artifacts Removal}
There are many types of motion artifacts but the unsual steep rise is the mostly occured ones associated with EDA signal while performing daily activities \cite{edel67}. We use well-known steep rising noises reduction technique, SWT \cite{chen15}. We first consider EDA signal as a mixture of a slow variant tonic and fast variant phasic component, i.e., SWT coefficient is modeled as a mixture of two Gaussian components, phasic (close to zero valued signal) and tonic (high rising signal). After expanding EDA signal into multiple levels of scaling and wavelet coefficients, we choose adaptively a threshold limit at each level based on the statistical estimation of the wavelet coefficients' distribution, and employ that on the wavelet coefficients of all levels. Finally, an inverse wavelet transform technique is applied to the thresholded wavelet coefficients to obtain the artifacts free EDA signal. Fig~.\ref{fig:eda_artifact_removal} shows a sample of raw and motion artifacts free EDA signal.
\begin{figure}[!htb]
\begin{center}
\vspace{-.1in}
\epsfig{file=eda_signal_artifact.pdf,height=1.6in, width=3.5in}
\caption{Dashed line represents noisy EDA signal and solid red line represents \emph{AutoCogniSys} proposed motion artifact free EDA signal}
\label{fig:eda_artifact_removal}
\end{center}
\end{figure}
\subsubsection{Convex Optimization Technique to EDA Deconvolution}
After the motion artifact removal, we consider EDA as the sum of three components for $N$ sample: a slow tonic driver ($t$), fast (compact, bursty) non-negative sparse phasic driver ($r$) and a reminder error term ($\epsilon_r$).
\begin{equation}
\label{eq:eda_signal}
y = t + r + \epsilon_r
\end{equation}
This additive error $\epsilon_r$ is a White Gaussian Noise. The central problem associated with the deconvolution method is to get tonic $t$ component from the above equation. \cite{greco16} showed that EDA signal deconvolution (separation of tonic, phasic and noise terms from EDA signal) is a quadratic optimization problem and defined tonic component as follows:
\begin{equation}
\label{eq:tonic}
t = Bl + Cd,
\end{equation}
where $B$ is a tall matrix whose columns are cubic $B$-spline basis functions, $l$ is the vector of spline coefficients, $C$ is a $N\times 2$ matrix, $d$ is a $2\times 1$ vector with the offset and slope coefficients for the linear trend. The above equation is subject to the following optimization problem,
\begin{eqnarray}
minimize \frac{1}{2} {||Mq + Bl + Cd- y||}^2_2 +\alpha {||Aq||}_1 + \frac{\lambda}{2} {||l||}^2_2\\
subject\;to\; Aq \geq 0\nonumber
\end{eqnarray}
where $M$ and $A$ are tridiagonal matrices and $q$ is an auxiliary variable. After solving the above equation, we can get the optimal values for $\{q,l,d\}$ that can be used to obtain tonic component from the equation~\ref{eq:tonic}. The reminder of the equation~\ref{eq:eda_signal} ($r+\epsilon_r$) is considered as a mixture of White Gaussian Noise ($\epsilon_r$) and a fast variant phasic component ($r$). We employ butterworth low-pass filter (5Hz) and hanning smoothing with window size 4 (optimal) to remove $\epsilon_r$ from phasic component ($r$).
\subsection{PPG Signal Processing}
PPG is used mainly for measuring the oxygen saturation in the blood and blood volume changes in skin. An ideal PPG signal processing must contain the following steps: noise and motion artifacts removal, heart rate detection, heart rate variability estimation and feature extraction.
\subsubsection{PPG Signal Noise and Motion Artifacts Removal}
Similar to EDA signal, PPG signal is also contaminated with motion artifacts and noises. However, unlike EDA signal, PPG produce quasiperiodicity in a time series spectrum \cite{mete30}. We use Periodic Moving Average Filter (PMAF) to remove motion artifacts and noises \cite{lee07}. We first segment the PPG signal on periodic boundaries and then average the $m^{th}$ samples of each period. After filtering the input PPG signal with a 5-Hz $8^{th}$-order Butterworth low-pass filter, we estimate the maximum and minimum value of each period. The mean of each period are obtained from the maximum and minimum values applying the zero crossing method. These points of the means help determine the boundaries of each period. Then, interpolation or decimation is performed to ensure that each period had the same number of samples \cite{lee07}.
\subsubsection{Heart Rate and Heart Rate Variability Estimation}
We first apply PMAF on PPG signal to remove noises and motion artifacts, refine PPG by smoothing the signal using 1-dimensional Gaussian Filter and Convolution, calculate first derivative of the convoluted signal and finally find the differences between two consecutive peak values which is called HRV \cite{sel08}. The occurrences of total peak values (R-peak or beat) in each minute is called Heart Rate (HR) with an unit of Beat Per Minute. The signal value property of HRV and HR are inversely proportional which means the mental arousal that increases HR should decrease HRV in the time segment window. Fig~\ref{fig:ppg_artifact_removal} shows a sample of the noisy and filtered PPG signal and their corresponding Instant Heart Rate.
\begin{figure}[!htb]
\vspace{-.1in}
\begin{center}
\epsfig{file=ppg_artifact_removal.pdf,height=1.4in, width=3.5in}
\vspace{-.15in}
\caption{Top figure illustrates the noisy signal (dotted line) and filtered signal from PPG sensor based on our filtering method. Bottom figure illustrates instant heart rate calculated from noisy signal (dotted line) and filtered signal}
\label{fig:ppg_artifact_removal}
\end{center}
\vspace{-.15in}
\end{figure}
\subsection{Physiological Sensor Signal Feature Extraction}
Using the above mentioned methods, we removed the noises and motion artifacts from EDA and PPG signals and generated two time series signal from EDA (tonic and phasic components) and one time series signal from PPG (HRV). Then, we segment each of the time series signal based on our prior detected complex activities such that each response window starts and ends with the starting and ending points of each complex activity. We extract 7 statistical time-series features for EDA (as shown in Table~\ref{tab:eda_features}) and 8 features for HRV (Table~\ref{tab:hrv_features}) within the response window).
\begin{table}[!t]
\begin{center}
\renewcommand{\arraystretch}{1}
\caption{EDA Features Within The Response Window}
\begin{scriptsize}
\label{tab:eda_features}
\begin{tabular}{|c|l|}
\hline
\bfseries Features& \bfseries Description\\
\hline
nSCR & Number of SCRs within response window (wrw)\\
\hline
Latency & Response latency of first significant SCR wrw\\
\hline
AmpSum & Sum of SCR-amplitudes of significant SCRs wrw\\
\hline
SCR & Average phasic driver wrw\\
\hline
ISCR & Area (i.e. time integral) of phasic driver wrw\\
\hline
PhasicMax & Maximum value of phasic activity wrw\\
\hline
Tonic & Mean tonic activity wrw\\
\hline
\end{tabular}
\end{scriptsize}
\end{center}
\end{table}
\begin{table}[!t]
\begin{center}
\renewcommand{\arraystretch}{1}
\vspace{-.3in}
\caption{Heart Rate Variability features}
\label{tab:hrv_features}
\begin{scriptsize}
\begin{tabular}{|c|l|}
\hline
\bfseries Feature& \bfseries Description\\
\hline
$\overline{RR}$&Mean RR intervals\\
\hline
SDNN&Standard deviation of RR intervals\\
\hline
SDSD&Std of successive RR interval differences\\
\hline
RMSSD&Root mean square of successive differences\\
\hline
NN50&\#successive intervals differing more than 50 ms\\
\hline
pNN50&relative amount of NN50\\
\hline
HRVTI&Total number of RR intervals/height of the histogram\\
\hline
TINN&Width of RR histogram through triangular interpolation\\
\hline
\end{tabular}
\end{scriptsize}
\end{center}
\end{table}
\section{Experimental Evaluation}
In this section, we explain our data collection, available benchmark dataset, baseline methods and evaluation.
\subsection{Datasets and Baseline Methods}
We validate and compare \emph{AutoCogniSys} with baseline methods on both publicly available and our collected datasets.
\subsubsection{RCC Dataset: Collection and Ground Truth Annotation}
For collecting Retirement Community Center Dataset (RCC Dataset), we recruited 22 participants (19 females and 3 males) with age range from 77-93 (mean 85.5, std 3.92) in a continuing care retirement community with the appropriate institutional IRB approval and signed consent. The gender diversity in the recruited participants reflects the gender distribution (85\% female and 15\% male) in the retirement community facility. A trained gerontology graduate student evaluator completes surveys with participants to fill out the surveys. Participants are given a wrist band to wear on their dominant hand, and concurrently another trained IT graduate student have the IoT system setup in participants' own living environment (setup time 15-30 minutes). The participants are instructed to perform 13 \emph{complex ADLs}. Another project member remotely monitors the sensor readings, videos and system failure status. The entire session lasts from 2-4 hours of time depending on participants' physical and cognitive ability.
We follow the standard protocol to annotate demographics and activities mentioned in the IRB. Two graduate students are engaged to annotate activities (postural, gestural and complex activity) whereas the observed activity performances are computed by the evaluator. Two more graduate students are engaged to validate the annotations on the videos. In overall, we are able to annotate 13 complex activities (total 291 samples) labeling for each participant; 8 hand gestures (total 43561 samples) and 4 postural activities (total 43561 samples) labeling. Annotation of postural and complex activities outcomes no difficulties from recorded videos. However, annotation of hand-gestures is extremely difficult in our scenario. We used video based hand tracker that can track and sketch wrist movements from a video episode \cite{hugo14}. This sketching can help us significantly to identify which particular hand gesture is being performed in the time segment.
\subsubsection{EES Datasets: EDA and PPG Sensor Datasets}
We used Eight-Emotion Sentics (EES) dataset to validate \emph{AutoCogniSys} proposed physiological signal processing approaches \cite{picard01}. The dataset consists of measurements of four physiological signals (PPG/Blood Volume Pulse, electromyogram, respiration and Skin Conductance/EDA) and eight affective states (neutral, anger, hate, grief, love, romantic love, joy, and reverence). The study was taken once a day in a session lasting around 25 minutes for 20 days of recordings from an individual participant. We consider only PPG and EDA for all of the affective states in our study.
\subsubsection{Baseline Methods}
Though no frameworks ever combined all modalities together into real-time automated cognitive health assessment, we evaluate \emph{AutoCogniSys} performance by comparing the performances of its components individually with upto date relevant works. For hand gesture and postural activity recognition, we consider \cite{alam17} proposed method as baseline. For complex activity recognition, we compare our hand gesture and postural activity classifiers aided HDBN model with three-level Dynamic Bayesian Network \cite{zhu12} framework. For activity performance estimation, activity performance based cognitive health assessment; and EDA and PPG based cognitive health assessment, we have considered \cite{alam16} proposed method as baseline.
\subsection{Activity Recognition Evaluation}
The standard definition for \emph{accuracy} in any classification problem is $\frac{TP+TN}{TP+TN+FP+FN}$ where $TP,TN,FP$ and $FN$ are defined as true positive, true negative, false positive and false negative. For complex activity recognition evaluation, we additionally consider \emph{start/end duration error} as performance metric that can be explained as follows: consider that the true duration of ``cooking'' is 30 minutes (10:05 AM - 10:35 AM) and our algorithm predicts 29 minutes (10.10 - to 10.39 AM). Then, the start/end duration error is 9 minutes ($|$5 minutes delayed start$|$ + $|$4 minutes hastened end$|$), in an overall error of e.g., 30\% (9/30=0.3). We measure cross-participant accuracy using leave-two-participants-out method for performance metrics, i.e., we take out two of the participants' data points from the entire dataset, train our proposed classification models, test the model accuracy on the two left-out participants relevant data points, and continue the process for entire dataset.
\begin{figure*}[!htb]
\begin{minipage}{0.45\textwidth}
\begin{center}
\epsfig{file=hand_gesture_accuracy.pdf,height=1.6in, width=3in}
\caption{Feature Weighted Naive Bayes (FWNB) classification accuracy comparisons with baseline approaches (graphical signatures of all hand gestures are shown).}
\label{fig:hand_gesture_accuracy}
\end{center}
\end{minipage}
\begin{minipage}{0.29\textwidth}
\begin{center}
\vspace{-.12in}
\epsfig{file=posture_accuracy_normal.pdf,height=1.6in, width=2.1in}
\caption{4-class postural level activity recognition performance and comparisons with baseline method}
\label{fig:posture_accuracy_normal}
\end{center}
\end{minipage}
\begin{minipage}{0.25\textwidth}
\begin{center}
\vspace{-.12in}
\epsfig{file=posture_accuracy_extended.pdf,height=1.6in, width=2.1in}
\caption{6-class diverse postural activity recognition framework accuracy comparisons with the baseline approach.}
\label{fig:posture_accuracy_extended}
\end{center}
\end{minipage}
\end{figure*}
Fig~\ref{fig:hand_gesture_accuracy} displays Feature Weighted Naive Bayes (FWNB) based the 8-hand gestural activity recognition accuracies comparisons with the baseline methods which clearly depicts the outperformance of our method (5\% improvement) with an overall accuracy of 92\% (FP rate 6.7\%) in RCC dataset. For postural activity recognition, dataset achieving 91\% postural activity recognition accuracy (FP rate 9.5\%) which outperforms the baseline approach significantly (8\% improvement). Now, we expand the postural activities for RCC datasets into 3 diverse `walking' postures: `normal walking', `walking with walker', `walking with single stick' and the accuracy goes down to 88\% (FP 7.9\%). Fig.~\ref{fig:posture_accuracy_normal} and Fig.~\ref{fig:posture_accuracy_extended} illustrate 4-class postural and extended 6-class postural classifier accuracies respectively which clearly posit that \emph{AutoCogniSys} outperforms in each case of postural activities as well as overall performances (8\% and 7\% improvement respectively).
For complex activity classification, we choose RCC dataset to train our HDBN model. Our leave-two-participants out method results an accuracy of 85\% (FP Rate 3.6\%, precision 84.2\%, recall 84.5\%, ROC Area 98.2\%) with a start/end duration error of 9.7\%. We run the entire evaluation for baseline complex activity recognition algorithm too achieving an overall accuracy of 78\% (FP Rate 5.2\%, precision 79.6\%, recall 78.5\%, ROC Area 82.7\%) which is clearly lower performed method than our approach. Fig. \ref{fig:complex_activity_roc} and Fig~\ref{fig:complex_activity_accuracy} illustrate the ROC curve and each complex activity recognition accuracy comparisons with baseline method which depict the outperformance of our framework over baseline methods (7\% improvement). Fig~\ref{fig:complex_activity_accuracy} also shows that inclusion of postural activity improves the final complex activity recognition (4\% improvement).
\begin{figure} [!htb]
\begin{minipage}{0.15\textwidth}
\begin{center}
\epsfig{file=complex_activity_roc.pdf,height=1.4in, width=1.1in}
\caption{ROC curve for complex activity recognition}
\label{fig:complex_activity_roc}
\end{center}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\begin{center}
\epsfig{file=complex_activity_accuracy.pdf,height=1.4in, width=2.3in}
\caption{Complex ADLs recognition accuracy improvement and comparison with baseline \cite{zhu12} and HMM based method}
\label{fig:complex_activity_accuracy}
\end{center}
\end{minipage}
\end{figure}
\subsection{Quantification of Performance Score}
To characterize both the qualitative and quantitative health assessment performance scores, we start with four different feature groups ranging from both functional and physiological health measures: (i) observation based activity features, (ii) automatic activity performance features, (iii) EDA features and (iv) PPG features.
In \emph{observation based activity features}, we design a complex activity set comprised of multiple subtasks which are involved with task {\it interruption, completion and sequencing}. Participants are instructed to perform the complex activities while the trained evaluator observed the aforementioned functional activity performance measures. Each incorrect attempt of performance measure will be assigned one point thus higher score reflects lower performance of functional activities \cite{dawadi14}. We first detect hand gesture and postural activities. Then, we feed the low-level activity contexts (gestural and postural) combined with ambient contexts (object and ambient motion sensor readings) into HDBN for single inhabitant model \cite{alam16b} to recognize complex activities. The complex activity recognition framework provides both activity labels and activity window (start-end points). Then, we extract features of object sensor, ambient sensor, gestural activity and postural activity events for each activity window. The features are number of occurrences, mean number of occurrences, consecutive 1, 2, 3, $\ldots$ 20 occurrences, top 10, 20, 30, $\ldots$, 90 percentile etc (29 features in total). In \emph{physiological features} we first detect 13 complex activities using HDBN algorithm which provides activity labels and activity window (start-end points), apply noise reduction, motion artifacts removal, extract 7 EDA features and 8 HRV features for each activity and take the mean of them over time (minutes) to get 15 (7+8) complex activity physiological features set for each participant. In summary, we extract 3 observation based activity features, 29 automatic activity performance features, 7 EDA features and 8 HRV features.
\subsection{Physiological Signal Processing Performance Evaluation}
Standard evaluation technique should use both experimental and publicly available datasets to confirm the outperformance of the novel approaches. We first evaluate our physiological signal processing techniques using a publicly available dataset (EES Dataset \cite{picard01}) to detect 8 human emotions. Then, in next section, we evaluate our methods in assessing cognitive health status of older adults using RCC dataset.
For EDA, we first apply SWT method to remove motion artifacts and noises. Then, we use cvxEDA method to separate tonic and phasic components of EDA. Then, we extract 7 EDA features on a sliding window of 4 seconds. Finally, we feed the 7 EDA features into a SMO based SVM algorithm \cite{cao06}. We use 10-fold cross validation to classify eight emotions achieving 87\% of overall accuracy (FP rate 6\%). For PPG, we first apply our proposed PMAF based noises and motion artifacts removal technique. Then, we calculate HRV and perform time-domain feature extraction to extract 8 HRV features on a sliding window of 4 seconds. We feed these features into a SMO based SVM algorithm \cite{cao06}. Our 10-fold cross validation shows accuracy of 79\% (FP rate 11.5\%) of detecting 8 emotions on EES Dataset. Fig. \ref{fig:ees_eda} and Fig. \ref{fig:ees_ppg} clearly depict that \emph{AutoCogniSys} proposed EDA and PPG signal processing techniques significantly improve the accuracy over the baseline \cite{alam16} method (10\% and 12\% improvement).
\begin{figure}[!htb]
\begin{minipage}{0.24\textwidth}
\begin{center}
\epsfig{file=ees_eda.pdf,height=1.2in, width=1.8in}
\caption{(EES Databaset) EDA features based Eight Emotion classification accuracy comparisons with baseline method}
\label{fig:ees_eda}
\end{center}
\end{minipage}
\begin{minipage}{0.23\textwidth}
\begin{center}
\epsfig{file=ees_ppg.pdf,height=1.2in, width=1.7in}
\caption{(EES Dataset) PPG features based 8-Emotion classification accuracy comparisons with baseline method}
\label{fig:ees_ppg}
\end{center}
\end{minipage}
\end{figure}
\subsection{Evaluation of Performance Scores}
The feature subsets used in the experimentation for observation and survey based clinical assessments and technology guided physiological and activity initiated health assessments are depicted in Table~\ref{tab:feature_subset}. From our 6 demographics surveys, we find significant distributions in terms of cognition only for SLUMS Score (S-Score). Based on that, we divide our participants pool into three groups: \emph{Not Cognitively Impaired (NCI), Mild Cognitively Impaired (MCI) and Cognitively Impaired (CI)} where the number of participants are $5$, $7$ and $10$ respectively.
\begin{table}[!t]
\begin{scriptsize}
{\centering
\renewcommand{\arraystretch}{.6}
\caption{Feature Subsets}
\label{tab:feature_subset}
\begin{tabular}{|l|L{5.5cm}|}
\hline
\bfseries Feature& \bfseries Description\\
\hline
Observation & Task Completeness (TC), Sequencing (SEQ), Interruptions (INT)\\
\hline
Survey & SLUMS Score (S-Score), ZUNG Score (Z-Score), IADL Score (I-Score), Yale Score (YPAS), Barthel Score (B-Score), GDS Score (G-Score)\\
\hline
EDA and HRV & 7 and 8 Features\\
\hline
Activity Performance& Supervised (TC, SEQ, INT), Unsupervised\\
\hline
Arousal& EDA and HRV features of each complex activity window\\
\hline
\end{tabular}
}
\end{scriptsize}
\end{table}
\begin{figure}[!htb]
\begin{center}
\epsfig{file=group_correlation.pdf,height=1in, width=3.3in}
\caption{\emph{AutoCogniSys} Proposed Method Based Group Correlation analysis ( $r-value$) NCI, MCI and CI represent not cognitive, mild cognitive and cognitively impaired group of population. TC, INT, SEQ, EDA and HRV represent task completeness, interruption scores, sequencing scores, electrodermal activity features and heart rate variability features}
\label{fig:group_correlation}
\end{center}
\vspace{-.2in}
\end{figure}
\begin{figure}[!htb]
\begin{center}
\epsfig{file=group_correlation_baseline.pdf,height=1in, width=3.3in}
\caption{Baseline \cite{alam16} method based Group Correlation analysis ( $r-value$)}
\label{fig:group_correlation_baseline}
\vspace{-.25in}
\end{center}
\end{figure}
\subsection{Statistical Correlation Analysis of Cognitive Health}
We used Pearson correlation coefficients with significance on $p<0.05$* for individual feature and partial correlation coefficients with significance on $p<0.005$** for group of features correlation analysis. Fig. \ref{fig:group_correlation} and Fig. \ref{fig:group_correlation_baseline} show the group correlation analysis results based on \emph{AutoCogniSys} proposed framework and baseline \cite{alam16} framework respectively. It can be clearly depicted that our proposed framework improves the correlation with the ground truths.
\subsection{Machine Learning Classification of Cognitive Health}
We evaluate using machine learning classifiers to predict cognitive status of older adults using both individual modalities and combined features. We use leave-two-participants out method to train and test classification accuracy.
We first choose the individual activity features (machine learning method based interruption scores, sequencing scores, unsupervised scores) and their combined features to train and test cognitive impairment status classification for SMO based SVM algorithm \cite{cao06}. The classification accuracies are 72\%, 69\%, 76\% and 83\% respectively. Then we consider 7 EDA-activity features and 8 HRV-activity features individually in training and testing phase of SMO based SVM algorithm \cite{cao06} resulting 85\% and 80\% accuracy respectively.
\begin{figure}[!htb]
\begin{minipage}{0.24\textwidth}
\begin{center}
\epsfig{file=combined_classification.pdf,height=1.2in, width=1.7in}
\vspace{-.15in}
\caption{Individual and combined classification accuracies comparison with baseline method for cognitive impairment status detection}
\label{fig:combined_classification}
\end{center}
\end{minipage}
\begin{minipage}{0.23\textwidth}
\begin{center}
\epsfig{file=each_activity_cognitive_assessment.pdf,height=1.2in, width=1.7in}
\caption{Machine learning based cognitive health assessment accuracy for each complex activity in terms of activity, EDA and HRV features.}
\label{fig:each_activity_cognitive_assessment}
\end{center}
\end{minipage}
\end{figure}
For combined classifier, we first applied sequential forward feature selection to find the best combinations of 1- 3 features for cognitive impairment classification group MCI, NCI and CI in terms of combined activity features (29 features), EDA-activity features (7 features) and HRV-activity features (8) features. Our final combined classifier (SMO based SVM algorithm \cite{cao06}) provides an accuracy of {\bf 93\%} in detecting the cognitive impairment status of older adults. Fig. \ref{fig:combined_classification} shows our proposed individual and combined methods outperform the baseline \cite{alam16} significantly (13\% improvement). Fig. \ref{fig:each_activity_cognitive_assessment} shows the cognitive impairment status prediction accuracy for each modality (activity feature, EDA and HRV) per individual complex activity.
\subsection{Discussion}
If we exclude the postural activities from automated activity performance scoring, we find reduced statistical correlation with original task completeness performance for \{NCI, MCI, CI\} participant group (INT 0.53*, SEQ 0.21' and unsupervised 0.49'). However, if we skip our proposed motion artifact removal stage, we find reduced statistical correlation with \{NCI, MCI\} and \{MCI, CI\} groups of participants (EDA and HRV correlations respectively \{0.51*, -0.51*\} and \{-0.53*,0.46\}). To test our proposed motion artifacts removal impact on EDA signals more rigorously, we choose 5 random participants, engage one expert motion artifact annotator to annotate motion artifacts segment on each participant's first 30 minutes of complex dataset using recorded video and apply both baseline and our methods to detect motion artifact segments. While baseline method achieves 75.5\% (FP rate 20.3\%) accuracy in detecting motion artifact segments, \emph{AutoCogniSys} outperforms achieving 89.9\% (FP rate 8.9\%) accuracy. In terms of experience, we have seen 100\% acceptance of wearing wrist-band, 71\% of acceptance for signing consent on using cameras and 0\% failure rate of collecting continuous data.
\section{Conclusion}
We propose, \emph{AutoCogniSys}, an IoT inspired design approach combining wearable and ambient sensors embedded smart home design, extensive signal processing, machine learning algorithms and statistical analytics to automate cognitive health assessment in terms of complex activity performances and physiological responses of daily events. Additionally, our postural activity detection approach in diverse population cum improved activity performance measurement and fundamental physiological sensor artifacts removal from physiological sensors help facilitate the automated cross-sectional cognitive health assessment of the older adults. Our efficient evaluation on each modality (physical, physiological and ambient) and each activity mode proves that any of the mode (say single activity and single sensor) also can provide significant improved cognitive health assessment measure.
| {'timestamp': '2020-03-18T01:05:03', 'yymm': '2003', 'arxiv_id': '2003.07492', 'language': 'en', 'url': 'https://arxiv.org/abs/2003.07492'} |
\section{Introduction}
\label{section:introduction}
The temperature anisotropies in the cosmic microwave background (CMB) anisotropy spectrum measured by the Wilkinson Microwave Anisotropy Probe (WMAP) \citep{WMAP9} and PLANCK \citep{Planck_13}, and observations of the large-scale ($k \lower0.6ex\vbox{\hbox{$ \buildrel{\textstyle <}\over{\sim}\ $}} 0.1$ Mpc $h^{-1}$) galaxy clustering spectrum measured by the 2dF Galaxy Redshift Survey \citep{Cole_etal05} and Sloan Digital Sky Survey \citep{Tegmark_etal06} have shown that the large-scale structure formation is consistent with the $\Lambda$ cold dark matter (CDM) model~\citep{Frenk:2012ph}. The observation of the CDM component implies physics beyond the standard model, and many dark matter candidates exist within extensions to the standard model of particle physics that behave as CDM~\citep{jungman_etal96b}. Though CDM is theoretically well-motivated, there are both theoretical (e.g.~\cite{Abazajian:2012ys,Zurek:2013wia}) and observational~\citep{Weinberg:2013aya,Bulbul2014,Boyarsky2014,Boyarsky_etal14b} interests in considering alternatives. In fact, a broad exploration of particle dark matter candidates finds that many viable models behave differently than CDM, particularly on small scales, implying that observations of dark matter structure on small scales may provide a unique test of different particle dark matter candidates.
Aside from the Magellanic Clouds, the eight largest satellite galaxies of the Milky Way are dwarf spheroidal (dSph) galaxies and the internal kinematics of these dSphs offer one of the best prospects for understanding the properties of particle dark matter from small scale astronomical observations (for a recent review see~\cite{Walker:2012td}). Stellar kinematics unambiguously indicate that the dSphs are dark matter-dominated~\citep{Walker:2007ju}, and their measured potentials have been used to determine whether their dark matter profiles are consistent with an NFW density profile long predicted by CDM~\citep{Navarro_etal96,navarro_etal97}. However, in spite of the high quality data sets, at present it is unclear whether the data indicate that the dark matter distributions in dSphs are in conflict with the NFW model~\citep{Walker_etal11, Amorisco_etal13} or are consistent with it~\citep{Jardel:2012am,Breddels:2013qqh,Strigari_etal14}.
Improvements in N-body simulations and in hydrodynamic simulations of Milky Way-like dark matter halos and their corresponding populations of subhalos now provide even more detailed predictions for the dark matter distributions of the dSphs. Detailed fitting of the stellar kinematics and photometry to subhalos in CDM N-body simulations indicate that the dSphs reside in dark matter halos with maximum circular velocity of approximately 20-25 km/s~\citep{Strigari_etal10}. More general fits to the dSph stellar kinematics in alternative dark matter model simulations such as warm dark matter (WDM)~\citep{Lovell_etal14}, decaying dark matter (DDM) ~\citep{Wang_etal14}, and self-interacting dark matter (SIDM)~\citep{Vogelsberger_etal12} indicate that the maximum circular velocities are larger than in CDM~\citep{Boylan-Kolchin_etal11}. Recent CDM hydrodynamic simulations find that these dark matter-only simulations neglect the important effect of baryons, which modify the $z=0$ maximum circular velocities of dSphs by about 15$\%$~\citep{Zolotov_etal12,Brooks_etal13,Brooks_etal14,Sawala_etal14,Sawala:2014xka}.
Identifying subhalos that are consistent with hosting dSphs in CDM as well in alternative dark matter scenarios can shed light on the cosmological evolution of those subhalos and, perhaps, the galaxies that they contain. Such identifications may pave the way for the development of specific predictions of both CDM and alternative models that can serve as true tests of the models. Exploiting stellar kinematics for this purpose is complementary to and significantly more robust than using predictions for the luminosities of galaxies within subhalos, because these predicted luminosities are extremely uncertain and rely on extrapolating phenomenological scaling relations outside of their established domain~\citep{Garrison-Kimmel:2014vqa}. It is additionally complementary to methods that utilize measurements of the orbital motions of Milky Way satellites~\citep{Rocha_etal12b,Sohn_etal13,Kallivayalil:2013xb}.
In this paper we identify dark matter subhalo host candidates of the Fornax dSph in CDM, WDM, and DDM N-body simulations by matching to the observed kinematics and photometry. The non-CDM based models that we study exhibit DM free-streaming effects that are not present in CDM. For WDM models, previous studies suggest that an equivalent thermal relic mass $\sim$ keV generates a truncation of the matter power spectrum on scales $\sim$ a few Mpc~\citep{Bode_etal01}. This effect suppresses the formation of small structure below the WDM free-streaming scale, resulting in delayed formation of halos.~\cite{Lovell_etal14} use galactic zoom-in simulations to show that typical galactic subhalos are less concentrated than their CDM counterparts because they form at later times. Sterile neutrinos are a canonical WDM candidate, and the decays of sterile neutrinos provide a possible origin for the detection of an unexplained X-ray line observed at 3.55 keV in the Galactic Centre, M31, and galaxy clusters ~\citep{Bulbul2014,Boyarsky2014,Boyarsky_etal14b}. In DDM models, the DM free-streaming is delayed since the excess velocity imparted from the DM decay is introduced with a lifetime comparable or greater to Hubble time. This can avoid the tight limits placed by high-redshift phenomena like Lyman-$\alpha$ forest \citep{Wang_etal13} and can impact galactic substructure~\citep{Wang_etal14}. In DDM, at high redshift the structure formation is similar to CDM until the age of the Universe is comparable to decay lifetime~\citep{Wang_etal12}.
We focus on the stellar kinematics of Fornax because it has a high-quality kinematic data sample and the dark matter potentials of subhalos in all of our simulations are well-resolved on the scale of the Fornax half-light radius. With the candidate host subhalos of Fornax identified, we determine the dynamical properties of the host subhalos, such as the present day mass, the peak mass, and the maximum circular velocity. With candidate host subhalos identified in each simulation, we examine the assembly histories of the Fornax host candidates. To assign luminosity to our Fornax candidates we use a simple relationship between the stellar mass and peak halo mass, and from this we identify Fornax subhalo candidates in CDM, WDM, and DDM that are consistent with both its kinematics and luminosity. These Fornax candidates allow us to predict the infall times and degree of tidal stripping of these subhalos, and we use these quantities to connect to models for the Fornax star formation history in each simulation.
\begin{figure}
\includegraphics[height=11.8cm]{mr_combine.eps}
\caption{
\textit{Top}: DM distribution of subhalos selected from three simulations with different DM models. The navy line represents a subhalo drawn from the Aquarius simulation (CDM), the aqua line is from the WDM simulation with WDM thermal relic mass = 2.3 keV, and the orange line is from the DDM simulations. These three subhalos provide good-fits to the Fornax stellar kinematic profile and photometry data. Mass estimations for Fornax from previous studies are also shown. The squared data points with 1 $\sigma$ error bars show the mass estimation at the 3D half-light radius from \citet{Wolf_etal10}, and the diamond points with 1 $\sigma$ error bars from \citet{Walker_etal09}. \textit{Middle}: Slopes of logarithmic mass profiles from these three subhalos. The horizontal dash line marks where slope =2, which is the theoretical predicted slope for NFW profiles as R $\to$ 0. The non-CDM subhalos have slope $>$ 2.0 in the inner region. \textit{Bottom}: The best-fit line-of-sight velocity dispersion derived using the jeans equation formalism with $\beta$ = constant model.
}
\label{fig:mr}
\end{figure}
The outline of the paper is as follows. In \S~\ref{section:simulations}, we briefly describe the properties of simulations used in our analysis. In~\S~\ref{section:fitting} we review our procedures for identifying Fornax subhalo candidates using its stellar kinematic data and photometry data. In~\S~\ref{section:results} we present our results for the subhalo assembly histories, and discuss how these quantities can be used to determine the Fornax infall time, star formation history, and quenching mechanisms. Lastly we draw our conclusions in \S~\ref{section:conclusion}.
\section{Simulations}
\label{section:simulations}
In this section we briefly describe the cosmological simulations that we utilize. For more details we refer to the original simulation papers (CDM : \cite{Springel_etal08}, WDM : \cite{Lovell_etal14}, and DDM : \cite{Wang_etal14}).
\begin{figure*}
\includegraphics[height=4.6cm]{cor_con.eps}
\caption{ The 68$\%$ and 95$\%$ contour region for the stellar density slope at r=600 pc versus velocity anisotropy $\beta$. From left to right, the contours are drawn from the MCMC results of the Jeans equation fitting for the subhalos shown in Figure~\ref{fig:mr} (left: CDM, middle: WDM, right: DDM).
}
\label{fig:contour}
\end{figure*}
For the CDM model, we utilize the Aquarius simulations, which are six realizations (Aquarius A to F) of galactic zoom-in simulations~\citep{Springel_etal08}. These simulations are generated using the Gadget code~\citep{Springel_etal08} and use cosmological parameters consistent with the one-year and five-year~\textit{Wilkinson Microwave Anisotropy Probe} (WMAP) data: $H_0$ = 73 km/s/Mpc, $\Omega_m$ = 0.25, $\Omega_{\Lambda}$ = 0.75, $\sigma_8$ = 0.9, and $n_s$ = 1. We adopt the level-2 resolution simulations as our main sample (with Plummer gravitational softening length $\epsilon$ = 65.8 pc and particle mass $m_p$ = 1.399$\times10^4M_{\odot}$$-$6.447$\times10^3M_{\odot}$), and utilize the highest level-1 resolution for Aquarius A (with $\epsilon$ = 20.5 pc and particle mass $m_p$ = 1.712$\times10^3M_{\odot}$) to perform resolution tests to understand the effects of the force softening scale on our analysis. The details of these tests are described in the Appendix~\S~\ref{Resolution}. The properties of the Aquarius simulations are shown in Table~\ref{tb:simulation}.
For the WDM models, we use the simulations described in~\cite{Lovell_etal14}. To account for WDM physics,~\cite{Lovell_etal14} re-simulate the Aquarius A halo using initial condition wave amplitudes that are rescaled with thermal relic WDM power spectra from~\cite{Viel_etal05}. We adopt their ``high resolution" suite that corresponds to level-2 in the original Aquarius notation with $m_p$ = 1.55 $\times10^4 M_{\odot}$ and $\epsilon$ = 68.1 pc. This suite includes two WDM simulations with equivalent thermal relic masses of $2.3$ keV and $1.6$ keV, and their CDM counterpart simulations. The $2.3$ keV thermal relic is a good approximation to the matter power spectrum of a 7 keV sterile neutrino that is resonantly produced in a lepton asymmetry $L_6\sim$20, which translates to a transfer function warmer than that inferred from the 3.55 keV line ~\citep{Venumadhav15, Lovell_etal15}. We take $1.6$ keV model as a case that is ruled out by current Lyman-$\alpha$ forest limits (e.g. ~\citep{Viel_etal13}) and $2.3$ keV as the model likely allowed by such limits for comparison. The cosmological parameters are derived from WMAP7: $H_0$ = 70.4 km/s/Mpc, $\Omega_m$ = 0.272, $\Omega_{\Lambda}$ = 0.728, $\sigma_8$ = 0.81, and $n_s$ = 0.967. Note that this is different than the original Aquarius simulations for which WMAP1 cosmology was implemented. We describe the effects of the different cosmology on our results in Appendix~\S~\ref{w1w7}. The self-bound halos in Aquarius simulations and WDM simulations were identified using the \textsc{SUBFIND} algorithm~\citep{Springel_etal01}.
For our DDM model, we use the set of simulations that implement late-time DDM physics from~\cite{Wang_etal14}. In these DDM models, a dark matter particle of mass $M$ decays into a less massive daughter particle of mass $m = (1-f)M$ with $f \ll 1$ and a significantly lighter, relativistic particle, with a lifetime $\Gamma^{-1}$, where $\Gamma$ is the decay rate. The stable daughter particle acquires a recoil kick velocity, $V_{k}$, the magnitude of which depends upon the mass splitting between the decaying particle and the daughter particle. The DDM simulations are generated using a modified version of ~\textsc{GADGET-3}~\citep{peter_etal10}. Here we consider the case with decay lifetime $\Gamma^{-1}$= 10 Gyr and kick velocity $V_{k}$ = 20 km/s. This model has been show to have interesting implications for dark matter small-scale structure~\citep{Wang_etal14} and is allowed by current Lyman-$\alpha$ forest limits~\citep{Wang_etal13}. The cosmology used is based on WMAP7 results with $H_0$=71 km/s/Mpc, $\Omega_m$=0.266, $\Omega_{\Lambda}$=0.734, $\sigma_8$=0.801, and $n_s$=0.963. We use \textsc{Amiga Halo Finder (AHF)} \citep{Knollmann_etal09} for halo finding and the merger tree is constructed using~\textsc{Consistent Trees}~\citep{Behroozi_etal13b}.
We define the virial mass $(M_{200b})$ of the galactic halo as the mass enclosed within the region of 200 times the background for all simulations, which corresponds to the $M_{50}$ in~\citet{Springel_etal08}. We adopt this definition for the same reason quoted in~\citet{Springel_etal08}, namely that it yields the largest radius among other conventional halo definition and hence the largest number of substructures.
\begin{table*}
{\renewcommand{\arraystretch}{1.3}
\renewcommand{\tabcolsep}{0.2cm}
\begin{tabular}{l c c c c c c c}
\hline
\hline
Simulations & Particle mass $m_p$& Force Softening $\epsilon$ & $M_{200b}$ & $r_{200b}$ &Dark Matter Properties\\
& [$M_{\odot}$] & [pc] & [$M_{\odot}$] & [kpc] & \\
\hline
Aq-A1 &1.712$\times 10^3$&20.5 &2.52$\times 10^{12}$ & 433.5 &CDM\\
Aq-A2 &1.370$\times 10^4$&65.8 &2.52$\times 10^{12}$& 433.5 &CDM\\
Aq-B2 &6.447$\times 10^3$&65.8 &1.05$\times 10^{12}$& 323.1 &CDM\\
Aq-C2 &1.399$\times 10^4$&65.8 &2.25$\times 10^{12}$& 417.1 &CDM\\
Aq-D2 &1.397$\times 10^4$&65.8 &2.52$\times 10^{12}$& 433.2 &CDM\\
Aq-E2 &9.593$\times 10^3$&65.8 &1.55$\times 10^{12}$& 368.3 &CDM\\
Aq-F2 &6.776$\times 10^3$&65.8 &1.52$\times 10^{12}$& 365.9 &CDM\\
\hline
Aq-A2 w7 &1.545$\times 10^4$&68.2 &2.53$\times 10^{12}$ & 432.1&CDM\\
Aq-A2-$m_{1.6}$ &1.545$\times 10^4$&68.2 &2.49$\times 10^{12}$& 429.9 &WDM ($m_{\mathrm{\small{WDM}}}$=1.6keV)\\
Aq-A2-$m_{2.3}$ &1.545$\times 10^4$&68.2 &2.52$\times 10^{12}$& 431.4 &WDM ($m_{\mathrm{\small{WDM}}}$=2.3keV)\\
\hline
Z13-CDM &2.40$\times 10^4$ &72.0 &1.31$\times 10^{12}$ & 335.2 &CDM\\
Z13-t10-v20 & 2.40$\times 10^4$ &72.0 &1.16$\times 10^{12} $ & 336.2 &DDM $(\Gamma^{-1}$=10 Gyr, $V_k$=20.0 km/s)\\
\end{tabular}
\medskip
\caption{Parameters of simulations. The mass of the galactic halo $M_{200b}$ is defined as the mass enclosed within the region of 200 times of the background density. }
}
\label{tb:simulation}
\end{table*}
\section{Fitting subhalos to stellar kinematics}
\label{section:fitting}
In this section we discuss our method for fitting subhalos in simulations to the stellar kinematic and photometric data of Fornax. For our theoretical analysis we use the spherical jeans equation, and allow for a constant but non-zero anisotropic stellar velocity dispersion. We utilize this theory in order to efficiently identify a large sample of Fornax candidates. For comparison studies with similar motivations have been undertaken~\citep{Boylan-Kolchin_etal11,Garrison-Kimmel:2014vqa} that have used the mass estimator of~\citet{Walker_etal09} and \citet{Wolf_etal10} to estimate maximum circular velocities for the dSphs. As we show below our results are in good agreement with these previous results. Since we start at the level of the jeans equation, we are able to identify preferred orbital structure of the stars given the underlying subhalo potentials.
We assume that the potential is spherically symmetric, dispersion-supported, and in dynamical equilibrium, so that we can derive the stellar line-of-sight velocity dispersion profile as a function of projected radius $R$~\citep{Binney_etal82}:
\begin{equation}
\sigma^2_{los}(R)={2 \over I_{\ast}(R)} \int^{\infty}_{R}[1-\beta(r){R^2 \over r^2}] {\rho_{\ast}(r)\sigma_r^2r \over \sqrt{r^2-R^2}} dr
\label{eq:vd_los}
\end{equation}
with
\begin{equation}
I_{\ast}(R)=2 \int^{\infty}_{R} {\rho_{\ast}(r)r \over \sqrt{r^2-R^2}} dr.
\label{eq:Abel}
\end{equation}
Here $\rho_{\ast}(r)$ is the 3D stellar density profile and $ I_{\ast}(R)$ is its 2D projection.
The velocity anisotropy parameter is $\beta(r)\equiv$ 1-$\sigma^2_t(r)/\sigma^2_r(r)$, where $\sigma_r(r)$ is the radial velocity dispersion of stars and $\sigma_t(r)$ is the tangential velocity dispersion. These quantities satisfy the Jeans equation \citep{Binney_etal08}:
\begin{equation}
r{d(\rho_*\sigma^2_r) \over dr}+2\beta(r)\rho_*\sigma^2_r=-\rho_*(r){GM(<r)\over r}.
\label{eq:Jeans}
\end{equation}
For the case of constant non-zero $\beta$, the solution of Eq.~\ref{eq:Jeans} has a simple form:
\begin{equation}
\rho_{\ast}\sigma_r^2(r)={1 \over r^{2\beta}}\int^{\infty}_{r} {\rho_{\ast}(r')GM(<r') r'^{2\beta} \over r'^2} dr'.
\label{eq:rho_sigma}
\end{equation}
To model the 3D stellar density profile $\rho_{*}(r)$ as a function of radius we use this general form:
\begin{equation}
\rho_{\ast}(r) \propto {1 \over x^{a}(1+x^b)^{(c-a)/b}},
\label{eq:sp}
\end{equation}
where x=$r$/$r_0$ and $a, b, c$ are free parameters that capture the stellar distribution slopes over different radii. A density profile of this form has been found to adequately describe the photometry of the classical dSphs~\citep{Strigari_etal10}. Though the conversion of 3D to 2D profile is not a one-to-one relation, implying that different choice of parameters for the 3D profiles can provide similar 2D projected profiles, the present photometry data do give good constraints on the Fornax stellar distributions with profiles of the form Eq.~\ref{eq:sp}. We fix the normalization of stellar mass by assuming mass-to-light ration $M_*$/L $= 1$ and adopt the Fornax V-band luminosity value of $L_v$=1.7$\times 10^{7} M_{\odot}$~\citep{McConnachie2012}.
With the above equations we utilize the following algorithm to model the Fornax line-of-sight velocity dispersion given the subhalos in each of our simulations. We begin by finding subhalos in our galactic zoom-in simulations using halo finders, and for each subhalo determine the dark matter distribution and thus the potential as a function of radius. Given the potential of each subhalo, the stellar distribution described by the parameters $a,b,c,r_0$, and velocity anisotropy by $\beta$, we solve the jeans equation to determine the line-of-sight velocity dispersion. We then fit to the photometric and kinematic data by marginalizing over these parameters $a,b,c,r_0$, and $\beta$ via a Markov Chain Monte Carlo (MCMC) method to determine the best fitting value for the model parameters.
For our fits to the Fornax stellar kinematics we define
\begin{equation}
\chi^2_{\sigma} = \sum^{N_{bins}}_{i=1} {[\hat{\sigma}_i-\sigma_{los}(R_i)]^2 \over \epsilon^2_i},
\label{eq:chi_vd}
\end{equation}
where $N_{bins}$ is the number of annuli, $R_i$ is the mean value of the projected radius of stars in the $i^{th}$ annulus, and $\sigma_{los}(R_{i}) $ is the derived velocity dispersion for a given subhalo in the $i^{th}$ annulus. The line-of-sight velocity dispersion from the binned data is $\hat{\sigma_i}$ and the corresponding error is $\epsilon_i$ in each annulus.
The kinematic datasets that we use consist of line-of-sight stellar velocities from the samples of~\cite{Walker_etal09b}. We consider only stars with $>$ 90$\%$ probability of membership, which gives us a sample of 2409 Fornax member stars. We bin the velocity data in circular annuli and derive the mean line-of-sight velocity in each annulus as function of $R_i$. We calculate the line-of-sight velocity dispersion $\hat{\sigma}$ and their error $\epsilon$ in each annulus following the method described in~\cite{Strigari_etal10}.
For our fits to the Fornax photometry we define
\begin{equation}
\chi^2_{IR} = \sum^{N_{IR}}_{i=1} {[\hat{I_*}(R_i)-I_*(R_i)]^2 \over \epsilon^2_{IR, i}},
\label{eq:chi_IR}
\end{equation}
where $N_{IR}$ is the number of radial bins, $R_i$ is the radius of the $i^{th}$ data point, and $\hat{I_*}(R_i) $ is the derived 2D stellar surface density in the $i^{th}$ radius bin and $ \epsilon^2_{IR, i}$ is the uncertainty. For our photometric data we use the measurements from~\cite{Coleman_etal05}. Note that in our analysis we assume that Fornax contains a single population of stars.
For our MCMC, we adopt uniform priors over the following range: 0 $\leq$ a $\leq$ 2, 0.5 $\leq$ b $\leq$ 5, 4 $\leq$ c $\leq$ 8, 0.5 $\leq$ $r_0$ $\leq$ 2, -1.0 $\leq$ $\beta$ $\leq$ 1.0. The MCMC calculation is performed using a modified version of the publicly available code \textsc{CosmoMC} \citep{Lewis_etal02} as an MCMC engine with our own likelihood functions. We adopt a simple form of the likelihood function as the summation of the $\chi^2$ from both the stellar kinematic data and photometry over their degree of freedom: $\chi^2_{tot}/d.o.f.$ = $\chi^2_{\sigma}/d.o.f.$ + $\chi^2_{IR}/d.o.f.$, where $\chi_{\sigma}^2$ and $\chi_{IR}^2$ are calculated using Eq.~\ref{eq:chi_vd} and Eq.~\ref{eq:chi_IR} with degree of freedom (\textit{d.o.f.}) equal to number of data bins minus the number of free parameters plus one. We have 15 radial bins in the velocity dispersion data and 19 radial bins in the photometry light profile data. We then select the subhalos that satisfy the condition of $\chi^2_{tot}/d.o.f. \leq 3.0$ as good-fits to the Fornax kinematic and photometry data. We choose these criteria in order to obtain a conservatively large subhalo sample of candidates that can host Fornax.
\begin{figure*}
\includegraphics[height=5.8cm]{Fornax_vcir_beta.eps}
\caption{ Circular velocity curves of subhalos that are good fits to the Fornax stellar kinematic and photometry data. Notice that for each curve the stellar mass contribution has been included. The dark gray shaded areas include 68$\%$ of the subhalo curves from Aquarius simulation A to F, and the light gray areas are for 95 $\%$ of the subhalos. The solid black lines are the average of all the Aquarius CDM fits. From left to right panel, the colored lines show the subhalos from the WDM simulation with 2.3 keV thermal relic mass (left panel, aqua), the WDM simulation with 1.6 keV thermal relic mass (middle panel, green), and the DDM simulation with lifetime =10 Gyr and kick velocity = 20 km/s (right panel, orange)
}
\label{fig:vcir}
\end{figure*}
\section{Results}
\label{section:results}
We now present the results of our analysis. We begin by presenting the properties of the dark matter subhalos in the different simulations that are candidate Fornax hosts using stellar kinematics alone. We then combine with a simple model for subhalo luminosities to identify subhalos that match both the luminosity and kinematics of Fornax, and present results for the assembly history of these subhalos.
\subsection{Properties of Fornax Candidate Host Subhalos at z=0}
In this subsection we discuss the properties of Fornax candidate host subhalos at $z=0$. We begin by discussing the density profiles. for which we compare the central densities of the subhalos from the different simulations. We then discuss the total subhalo masses and maximum circular velocities of the host subhalo candidates.
\label{subsection: Best-fit subhalos}
\subsubsection{Subhalo density profiles and correlations}
We begin by examining the density profiles of the subhalos that provide good fits to the photometry and line-of-sight velocity dispersion profiles. Figure~\ref{fig:mr} shows the line-of-sight velocity dispersions for three different representative subhalos, one from each of the CDM, WDM, and DDM simulation. For CDM, here we choose the subhalo that has the minimum value of $\chi^2_{tot}/d.o.f.$, while for DDM and WDM we choose the subhalos that have $\chi^2_{tot}/d.o.f.$ $<$ 3 and have the shallowest central density profiles. For WDM (DDM), the $\chi^2_{tot}/d.o.f.$ is specifically 1.67 (0.85), and the p-value for the velocity dispersion fit is 0.4 (0.84). For the CDM subhalo, the $\chi^2_{tot}/d.o.f.$ is 0.93 while the p-value for the velocity dispersion fit is 0.8. Thus in all three cases the fits are good from a statistical standpoint .
Figure~\ref{fig:mr} also shows both the mass distribution and the log-slopes as a function of radius of the selected subhalos. The CDM subhalo is the most centrally-concentrated at radii $\lower0.6ex\vbox{\hbox{$ \buildrel{\textstyle <}\over{\sim}\ $}}$ 500 pc, generating a central slope of $\Delta log_{10}M/\Delta log_{10} R$ $\sim$ 2, which is consistent with the predicted value from an NFW profile. The DDM subhalo has a central slope of $\sim$ 2.6, while the WDM subhalo has a central slope of $\sim$ 2.5. Thus the WDM subhalos have ``soft" cores, which are expected since WDM subhalos are less concentrated than their CDM counterparts because they form at a later time~ \citep{Lovell_etal12}. The DDM subhalo has a slightly shallower core-like feature than in WDM because the effects of dark matter decay are scale-dependent~\citep{Wang_etal14}.
As Figure~\ref{fig:mr} indicates, the predicted line-of-sight velocity dispersions are very similar, even though the best fitting values of $a,b,c,r_0$ and $\beta$ are different in each case. This highlights an explicit degeneracy between the dark matter potential, the velocity anisotropy, and the stellar density profile. At the level of a jeans-based analysis, the degeneracy between the dark matter potential and the velocity anisotropy has been well-studied and has been known to preclude determination of the dark matter profile shape~\citep{Strigari:2007vn}.
The degeneracy between the stellar density profile and both the dark matter potential and the velocity anisotropy is less well-understood (see however~\cite{Strigari_etal10,Strigari_etal14}), so it is interesting to examine this further within the context of a Jeans-based analysis. Figure~\ref{fig:contour} highlights this degeneracy for the case of a single subhalo by showing the correlation between the stellar density slope at $r = 600$ pc versus anisotropy parameter. For all models, within 68$\%$ contour region $\beta$ spans a wide range and is degenerate with the stellar density concentration. The specific allowed regions change in each of the models, because they predict different DM density slopes.
\begin{figure*}
\includegraphics[height=8.3cm]{Fornax_mcmc_param_check_hist.eps}
\centering
\caption{Histograms of subhalo properties for the Fornax good-fits. Here we show two subhalo present-time properties: maximum circular velocity ($V_{max}$, left panels), and subhalo mass ($M_{sub}$, right panels). In the upper two panels the navy histograms show the Aquarius A results, the aqua ones are for the WDM 2.3 keV simulation, and the green ones are for the WDM 1.6 keV simulation. In the lower two panels the dark red histograms show the Z13 CDM results, and the orange ones are for the DDM simulation. We note that for the same galactic halo realization, the alternative DM scenarios presented here (WDM, DDM) generate much less substrucure than their CDM couterparts (see text in \S~\ref{subsubsection: Subhalo masses}).
}
\label{fig:param}
\end{figure*}
\begin{table}
\caption{Average subhalo properties of Fornax kinematic good-fits }
{\renewcommand{\arraystretch}{1.3}
\renewcommand{\tabcolsep}{0.2cm}
\begin{tabular}{l c c c c c c c}
\hline
\hline
Simulations & $M_{sub}$ & $M_{peak}$ & $V_{max}$ \\
& [$M_{\odot}$] & [$M_{\odot}$] & [km/s] \\
\hline
Aq-A2-w7 & 3.72$\pm$1.91$\times 10^8$& 1.55$\pm$2.01$\times 10^9$&17.70$\pm$1.02\\
Aq-A2-$m_{2.3}$&1.30$\pm$1.32$\times 10^9$&4.23$\pm$2.09$\times 10^9$&20.31$\pm$2.26\\
Aq-A2-$m_{1.6}$&3.52$\pm$3.15$\times 10^9$&5.05$\pm$2.67$\times 10^9$&23.22$\pm$4.57\\
\hline
\hline
Z13-CDM &4.50$\pm$2.00$\times 10^8$& 8.77$\pm$2.94$\times 10^8$&18.26$\pm$0.71\\
Z13-t10-v20&2.11$\pm$1.78$\times 10^9$& 4.87$\pm$5.10$\times 10^9$ &25.03$\pm$3.18\\
\hline
Aq-A2 &3.46$\pm$1.27$\times 10^8$&1.60$\pm$2.14$\times 10^9$&17.69$\pm$0.80\\
Aq-B2 &2.34$\pm$1.06$\times 10^8$&1.04$\pm$1.76$\times 10^9$&17.78$\pm$0.98\\
Aq-C2 &2.60$\pm$1.10$\times 10^8$&0.91$\pm$1.25$\times 10^9$&17.61$\pm$0.44\\
Aq-D2 &3.40$\pm$2.30$\times 10^8$&8.28$\pm$4.23$\times 10^8$&17.81$\pm$1.13\\
Aq-E2 &2.84$\pm$1.23$\times 10^8$&6.62$\pm$3.79$\times 10^8$&17.60$\pm$0.97\\
Aq-F2 &3.58$\pm$2.43$\times 10^8$&1.00$\pm$0.59$\times 10^9$&17.98$\pm$1.04\\
\hline
Aq Average &3.07$\pm$1.77$\times 10^8$&1.02$\pm$1.29$\times 10^9$&17.76$\pm$0.94\\
Aq Max &1.18$\times 10^9$&8.83$\times 10^9$&21.71\\
Aq Min &1.09$\times 10^8$&3.08$\times 10^8$&15.79\\
\hline
\end{tabular}
\medskip
\\
}
\label{tb:summary}
\end{table}
\subsubsection{Subhalo masses and circular velocities}
\label{subsubsection: Subhalo masses}
Though the density profiles of the CDM, WDM, and DDM subhalos are indistinguishable from the photometric and the kinematic data, the subhalos are distinct when considering more global properties. In Figure~\ref{fig:vcir} we show the circular velocity curves, $V_{circ}=\sqrt{G M(<r)/r}$, of subhalos that provide good fits to the Fornax kinematic data and photometry data. Note that for each curve the stellar mass contribution has been included, so that at a radius of 0.9 kpc, the enclosed mass contribution from stars is about 10$\%$ of the DM mass. The circular velocity curves in the WDM and DDM models are shallower in the center than the CDM subhalos, which is consistent with the discussion above on the individual best-fitting density profiles. Further, the WDM and DDM subhalos have larger maximum value, $V_{max}$, which is consistent with previous analyses~\citep{Lovell_etal12,Wang_etal14}.
\begin{figure*}
\includegraphics[height=5.8cm]{Fornax_mcmc_mpeak_hist.eps}
\caption{
The $M_{peak}$ distribution of the Fornax good-fit subhalos. The color notation in the left panel is the same as previous figures, and the right panel shows the distribution from all six Aquarius level-2 simulations (A-F). We note that there is only one galactic halo realization in the left and middle panel, while there are six in the left panel. The yellow shaded areas indicate the region where the luminosity of these objects are consistent with the Fornax luminosity using abundance matching methods from \citet{Behroozi_etal13}.
}
\label{fig:mpeak}
\end{figure*}
Figure~\ref{fig:param} shows the distribution of $V_{max}$ and the total mass of the subhalo, $M_{sub}$, for the CDM subhalo host candidates that are good fits to Fornax. Here we show only results from the CDM simulations Aq-A2-w7 and Z13, because from these simulations we are able to identify the subhalo counterparts from the WDM and DDM simulations, respectively. From Figure~\ref{fig:param} and also Table~\ref{tb:summary}, we show that CDM host subhalo candidates have $10^{8} M_{\odot} \le M_{sub}\le 10^{9} M_{\odot}$, $16 \, {\rm km/s} \, \le V_{max}\le 22 $ km/s. Table~\ref{tb:summary} shows the average $M_{sub}$ and $V_{max}$ of Fornax candidates from all our simulations. As is indicated, Aq-A2-w7 and Z13 provide a good representation of our entire sample of CDM simulations. The $V_{max}$ range of candidates from six Aquarius simulations (see Table~\ref{tb:summary}) have an average $\sim$17 km/s and an associated small variance, which agree well with results from previous study~\citep{Boylan-Kolchin_etal12}.
For comparison Figure~\ref{fig:param} shows the WDM and DDM subhalos that match the Fornax kinematics and photometry. In the WDM simulation with 2.3 keV WDM mass we identify seven host subhalo candidates with $M_{sub} \sim 0.3-3\times10^9 M_{\odot}$ and $V_{max} \sim$ 18-23 km/s. In the WDM simulation with 1.6 keV particle mass, we identify six host subhalo candidates with $M_{sub} \sim 0.2-8\times10^9 M_{\odot}$ and $V_{max} \sim$ 17-28 km/s. In the DDM simulation we identify three host subhalo candidates with $M_{sub} \sim 0.8-4\times10^9 M_{\odot}$ and $V_{max} \sim$ 22-28 km/s. These results indicate that from our Jeans-based modeling WDM and DDM predict a population of massive Fornax host subhalos that have shallower central dark matter density profiles. We note that alternative DM scenarios always predict much fewer Fornax candidates than their CDM counterpart simulations, simply because the number of subhalos are reduced significantly due to different DM properties. For example for the WDM case, Fig. 11 in~\cite{Lovell_etal14} shows that the numbers of subhalos with $V_{max} \ge$ 15 km/s are 14 (1.6 keV), 28 (2.3 keV) and 120 (CDM). For the DDM case the subhalo number with $V_{max} \ge$ 15 km/s is 16 and 56 for the CDM counterpart. The subhalo numbers in different CDM halos reflect mainly the differences in halo mass and merger history.
\subsection{Predictions for Luminosity and Effects of Reionization}
\label{subsection: luminosity}
To this point we have not used the luminosity of Fornax as a constraint, other than to consider its impact on the stellar kinematics and photometry. In doing so we have implicitly assumed that all of the subhalos are suitable hosts of a galaxy with the present day luminosity of Fornax. We have made this assumption in order to analyze a large statistical sample of well-resolved Fornax host candidates in N-body simulations that do not account for the effects of baryons on the subhalo evolution. High resolution Local Group simulations with baryons have been recently undertaken~\citep{Sawala_etal14,Sawala:2014xka}, though at present the statistical sample of subhalos is smaller in these simulations than the corresponding sample from N-body simulations. For this reason, we use semi-analytic models for the luminosities of our Fornax subhalo host candidates, and leave to a future study the analysis of the simulations that include the baryons.
Our subhalo luminosities are motivated by the results of several studies. First, we consider the results of~\cite{Sawala_etal14}, who utilize hydrodynamic simulations to predict the luminosity of subhalos before they fall into the Milky Way halo and at $z=0$. For subhalos with the present day luminosity of Fornax, Sawala et al. find that subhalos with the luminosity of Fornax have a total mass in the range $\sim 2-10 \times 10^9$ M$_\odot$, and a total peak halo mass in the range $\sim 7-20 \times 10^9$ M$_\odot$. The high end of this mass range is consistent with previous semi-analytic models~\citep{Cooper:2009kx} and with the predictions from the abundance matching method~\citep{Behroozi_etal13}. For dSphs fainter than Fornax, the predicted range of subhalo masses for a fixed luminosity is larger, and also there are stronger deviations from the predictions of abundance matching and that of~\cite{Sawala_etal14}. A well-known caveat, however, is that the abundance matching technique is not calibrated at both sub-galactic scales and in non-CDM cosmologies. Therefore results in these regimes depend on extrapolating the existing models and assuming the average galaxy formation history holds for non-CDM based models.
In order to determine the relevant peak subhalo mass range for Fornax, we first take the lower bound on the stellar mass of Fornax to be $M_{*}\lower0.6ex\vbox{\hbox{$ \buildrel{\textstyle >}\over{\sim}\ $}}$1$\times10^{7}M_{\odot}$. The lower bound is derived from the 1 $\sigma$ lower bound on the Fornax V-band luminosity of $L_V =$ 1.7 $\pm ^{0.5}_{0.4}\times 10^7 L_{\odot}$, and assuming a stellar mass-to-light ratio $M_{*}$/L = 0.8 ($M_{*}\sim1.3\times 10^7\times 0.8 M_{\odot}$ for the lower bound). For this adopted stellar mass, we then select Fornax candidates by their peak mass that predict the same stellar mass range using the abundance matching description in~\citet{Behroozi_etal13}. In Figure~\ref{fig:mpeak} we show the distribution of peak subhalo masses for these Fornax candidates. These results indicate that while subhalos that are consistent with the Fornax kinematics have subhalo peak masses in the range $M_{sub} \sim 10^8$-$10^{10} M_{\odot}$, only a handful at the very massive end predict the right luminosity. For example from the right panel in Figure~\ref{fig:mpeak} we find that 2 out of 124 candidates from the six Aquarius simulations are consistent with the Fornax luminosity. Interestingly, these two candidates come from the Aquarius A and B simulations, which are, respectively, the most massive and the least massive halos among all six Aquarius simulations. This would indicate that the likelihood of generating a Fornax candidate may be only weakly correlated with the galactic halo mass.
\begin{figure*}
\includegraphics[height=6.5cm]{Fornax_mcmc_mprog_hist.eps}
\caption{
The distribution of the progenitor mass at z=6 for the Fornax candidates. The color scheme in the left panel is the same as Figure~\ref{fig:param} showing DDM and Z13 simulation candidates, and right panel shows all the candidates from all six Aquarius halos. We note that there is only one galactic halo realization in the left panel, while there are six in the left panel. The filled histograms indicate the subhalos with luminosity that match Fornax at z=0. The shaded areas indicate the progenitor mass range with baryon content subject to UV background suppression~\citep{Okamoto_etal08}.
}
\label{fig:mz6}
\end{figure*}
For comparison, in Figure~\ref{fig:mpeak} we show the Fornax candidates from the WDM and DDM simulations, as well as their CDM counterparts. Though the sample of host subhalos is smaller than in the case of CDM, there is a higher probability for WDM and DDM Fornax candidates to reside in more massive subhalos. Therefore, the WDM and DDM candidates are more likely to match the bright luminosity of Fornax than the CDM candidates. In each case, we find 1-2 candidates that match the kinematics and luminosity, out of 3-7 candidates that just match the kinematics. Note again the caveat that the relationship between the stellar mass and subhalo progenitor mass may differ from this relation which is derived from CDM simulations.
With candidate host subhalos identified we are now in position to study their evolution and the masses of their progenitors near the redshift of reionization. The effect of reionization on suppressing star formation in small halos shows that halos below a few $10^{8} M_{\odot}$ likely have no stars due to the UV-background suppression. This implies that any subhalo with a progenitor less massive would be unlikely to host a visible galaxy today~\citep{Okamoto_etal08, Sawala_etal14}.
In Figure~\ref{fig:mz6} we show the subhalo progenitor mass at z=6 for the Fornax candidates. According to the criteria of~\citet{Okamoto_etal08}, more than half of the Fornax candidate in Aquarius would not host any stars. The subhalos that match both the Fornax kinematic and photometry data as well as the luminosity are on the massive end of the distribution.
In the left panel of Figure~\ref{fig:mz6} we show the case for the DDM simulation and its CDM counterpart. We do not include WDM simulations here because their structure formation and galaxy formation at high redshift are expected to be significantly different than in CDM. Also for the WDM models that we considered here, majority of the subhalo progenitors are not yet formed or identified by halo finders at z=6. However, for the DDM model with a decay lifetime of $10$ Gyr, it is safe to assume a similar reionization history as CDM. Since at $z=6$ less than about 10$\%$ of the dark matter has decayed with a small recoil kick velocity of $20$ km/s, the structure growth of both the dark matter and baryon components should be similar to CDM. Again the majority of the CDM subhalos should be UV-suppressed, and all three DDM subhalos are above the threshold, with one object that matches the Fornax luminosity.
\subsection{Mass Assembly Histories}
\label{subsection: assembly}
We now move on to discuss the mass assembly histories of our Fornax candidates. The left panel of Figure~\ref{fig:mz} shows the subhalo mass as a function of redshift for all 124 of our Fornax candidates from all six Aquarius halos. The middle panel shows the corresponding candidates from the WDM model with 2.3 keV mass and the right panel shows the candidates from DDM simulations. From the discussion above, we identify the subhalos that match the Fornax luminosity, and also determine an approximate infall time for those subhalos that match both the Fornax luminosity and kinematics. From Aquarius simulations, we can see that the candidates that match the kinematics and luminosity, which are shown in blue lines rather than the gray lines for candidates only fit stellar kinematics, also have the largest ratio of M($z_{peak})/M(z=0)$, with typical peak values for this ratio of $\sim$ 30$-$60 at $z\sim$2. For WDM, subhalos that match the kinematics and luminosity also show a large peak value at $z \sim 1.5$. On the other hand the corresponding DDM subhalo shows a relatively low peak value of M($z_{peak})/M(z=0) \sim$ 3 at $z~\lower0.6ex\vbox{\hbox{$ \buildrel{\textstyle <}\over{\sim}\ $}} 1$.
From the results presented in Figure~\ref{fig:mz}, we can identify three important trends and predictions regarding tidal stripping, infall times, and star formation histories.
\subsubsection{Tidal Stripping}
\label{Tidal stripping}
Using the above results in CDM we have a clear prediction that Fornax has lost a significant amount of dark matter mass after it falls into the Milky Way. Does this translate into a more specific observational signature? One possibility is that this tidal stripping of dark matter is manifest in the tidal stripping of the stars. However, since our models only directly include dark matter, it is not clear whether we should expect stellar tidal tails to be observable. It is possible that the stellar components are embedded deeply in the center of the subhalo potential, so that the dark matter is mostly stripped and the stars left unstripped. For example, ~\cite{Watson_etal12} shows that if subhalos experience substantial dark matter mass loss before mass is lost within the galaxy, this explains how satellite galaxies lose stellar mass and contribute to ``intrahalo light" (IHL).
\begin{figure*}
\includegraphics[height=7.0cm]{Fornax_mz.eps}
\caption{
Mass assembly histories for the Fornax good-fit subhalos. The gray lines show the fits with luminosity dimmer than Fornax using abundance matching technique from \citet{Behroozi_etal13}, and the blue lines show those with luminosity that matches the Fornax luminosity. The vertical dotted lines show the infall time of the blue curves. In the left panel the dashed black lines include 68$\%$ of the Aquarius simulation fits, and the black solid lines show the mean value.
}
\label{fig:mz}
\end{figure*}
For comparison the Fornax candidates in WDM, shown in the middle panel of Figure~\ref{fig:mz}, are similar to CDM in that both require large $M_{peak}/M(z=0)$. In DDM, as shown in the right panel of Figure~\ref{fig:mz}, the ratio is much smaller, $M_{peak}/M(z=0)~\sim 3$. These results hint at the interesting possibility that hosts of Fornax have undergone different degrees of tidal stripping in different dark matter models. Thus with future larger statistical samples of non-CDM models, as well as the inclusion of baryons in the simulation, the degree of tidal stripping of Milky Way-satellites may provide a new test of dark matter models.
\subsubsection{Infall Time}
\label{Infall}
From the peaks in the evolution of the subhalo masses in Figure~\ref{fig:mz}, we can identify the infall times of the subhalos into the Milky Way halo. In CDM, the subhalos that match the Fornax kinematics and luminosity have infall redshifts of $z=1.25$ and $z=1.37$, corresponding to lookback times of $\sim 9$ Gyr. For the 2.3 keV WDM model the infall redshift is $z=1.02$, corresponding to a lookback time of $\sim$ 8 Gyr. For DDM model the infall redshift is at $z=0.28$, corresponding to a lookback time of $\sim$ 3 Gyr. So the DDM host subhalo is predicted to fall into the Milky Way more recently than the CDM and WDM candidates.
It is informative to compare the lookback times that we deduce to previous estimates of the infall times of Milky Way satellites.~\cite{Rocha_etal12b} estimate infall times using subhalo Galactocentric positions and orbital motions from Via Lactea II simulation. Using these criteria Fornax is most likely to have fallen into the Milky Way halo $\sim 5-9$ Gyr ago. So even though our criteria for identifying Fornax subhalo candidates in CDM simulations are different from those in~\cite{Rocha_etal12b}, the infall times are in good agreement. We note that the Aquarius simulations that we utilize and our stricter matching criteria using luminosity and stellar kinematic data allows for a larger range of merger histories than considered in~\cite{Rocha_etal12b}, who utilize a single Milky Way halo.
\subsubsection{Star-Formation Quenching Mechanism }
\label{Quenching}
The quenching of star formation in Milky Way dSphs may be related to their infall times. For example ram-pressure stripping can remove the cold gas at the center of the satellites as a result of the high-speed interaction with the hot gas halo of the host halo \citep{Gunn_etal1972}. Ram pressure stripping has been invoked as the quenching mechanism for Milky Way and M31 dwarf satellites with $M_\star~\lower0.6ex\vbox{\hbox{$ \buildrel{\textstyle <}\over{\sim}\ $}}~10^8 M_{\odot}$ with an extremely short quenching timescale $\sim$ 2 Gyr~\citep{Fillingham_etal15}.
As shown in Figure~\ref{fig:sf}, Fornax is observed to have an enhanced star formation activity $\sim$ 3-4 Gyr ago and quenching $\sim$ 2 Gyr ago \citep{Coleman_etal08}. If we match these timescales with the infall times that we determined above, the enhanced star formation event is most consistent with the infall time of our DDM subhalo candidate. The grey shaded area shows the possible Fornax infall time range from ~\citet{Rocha_etal12b}. This covers a large infall time range including those predicted by the CDM and WDM simulations. However, it is interesting to note that other recent Fornax star formation history studies either indicate similar peak at $\sim$ 4 Gyr~\citep{deBoer_etal12} or draw different conclusions with a star formation peak at $\sim$ 8 Gyr~\citep{del_Pino_etal13}. Deep photometric studies of Fornax will help resolve this discrepancy and provide more insight on the connection of the Fornax star formation history with its infall time.
To explain the Fornax star formation history in the context of the earlier subhalo infall times predicted in our CDM and WDM models, we can consider a scenario in which the Fornax star formation was quenched when it merged into Milky Way halo $\sim$ 9 Gyr ago. A close encounter with the Milky Way at the last perigalactic passage would then trigger star formation. This mechanism has been invoked to explain the star formation history of Leo I~\citep{Mateo_etal08}, which had a burst in its star formation $\sim 3-4$ Gyr ago. Indeed proper motion and line-of-sight velocity measurements indicate that Leo I is likely on a fairly eccentric, nearly unbound orbit~\citep{Sohn_etal13}. In contrast with Leo I, Fornax likely has a much less eccentric orbit~\citep{Lux_etal10,Piatek_etal07}, so a much closer encounter with the Milky Way may be more difficult to invoke in the case of Fornax. More precise measurement of orbital velocity and detailed orbit reconstruction will shed more light on this issue.
\section{Conclusions and Discussion}
\label{section:conclusion}
We have fit the kinematics and photometry of the Fornax dSph to subhalo potentials predicted by numerical simulations of different dark matter models.
CDM subhalo candidates are typically in the mass range of $10^8-10^9 M_{\odot}$. WDM and DDM models predict on average larger mass subhalo hosts of Fornax.
The diversities of subhalo candidate properties in different dark matter models and the different time evolution of dark matter free-streaming effects lead to diversities in subhalo formation history, and also likely their stellar formation history. With this in mind we utilize simulation merger trees to investigate the formation history of the subhalos that match the Fornax kinematics. We implement simple models to match the Fornax luminosity to dark matter halo mass in order to understand possible star formation histories of these Fornax subhalo candidates. Under the assumption that the mass of the most massive progenitor is correlated with the current stellar content in each galaxy, we derive current stellar masses for each subhalo.
For CDM subhalos, our best candidates require the ratio between the peak mass and current mass to be $\lower0.6ex\vbox{\hbox{$ \buildrel{\textstyle >}\over{\sim}\ $}}$ 10. This suggests that these systems experience significant tidal stripping after they fall into Galactic halo and are subject to substantial dark matter mass loss. It also implies that their infall times are $\lower0.6ex\vbox{\hbox{$ \buildrel{\textstyle >}\over{\sim}\ $}}$ 9 Gyr ago, so that they fall well inside the galactic halo where the tidal stripping is efficient. Interestingly, similar findings also have been pointed out by~\cite{Cooper:2009kx} using Aquarius simulations with a ``particle tagging" technique to simulate Galactic stellar halos. Using Fornax and Carina as two examples, these authors find that they can only match their observed dwarf galaxy surface brightness and velocity dispersion profiles simultaneously by choosing model satellites that have suffered substantial tidal stripping.
Our WDM subhalo candidates show only mild deviation from their CDM counterparts, in that they have similar peak mass to current mass ratio and infall time. A caveat to this analysis is the real stellar content might be different, and full hydrodynamic simulations are needed to calibrate galaxy formation process in WDM. For the DDM candidates, using similar assumptions our subhalo candidates have relatively small peak mass/current mass ratio ($\sim$ 3) and they are more massive at $z=0$ than the CDM candidates. Therefore the DDM candidate only experiences mild mass loss since merging into the Galactic halo approximately 3 Gyr ago.
We note that our results are not yet able predict the existence of visible stellar tidal tails. Since galaxies are embedded deep in the center of the halo potential, most of the tidal stripping effects are expected to be on the DM component. Determining whether or not stars are stripped out requires N-body simulations with star particles.~\cite{Cooper:2009kx} have shown that 20$\%$ of the stellar mass and 2$\%$ of DM mass of their Fornax candidate remains at present time. Also recent work from \cite{Battaglia_etal15} have examined the tidal effects on Fornax along its possible orbits. Current and forthcoming wide field deep imaging surveys may be sensitive to faint stellar tidal tails and can provide an observational test of our models. ~\citet{Bate_etal15} use data from VLT Survey Telescope (VST) ATLAS Survey to study a region of 25 square degrees centered on Fornax. They have excluded a shell structure outside of Fornax's tidal radius that was reported from a previous study \citep{Coleman_etal05}. In the near future the Dark Energy Survey (DES) will cover a wide area around Fornax with much improved imaging depth, and possibly provide a test of the tidal stripping hypothesis.
We find that most of the subhalo candidates are subject to the cosmic reionization UV-background so that their star formation is highly suppressed. In our sample more than half of the CDM candidates are likely not visible at all due to reionization. For the WDM model the reionization history is more uncertain because high-redshift structure formation is significantly delayed. However, for DDM the early structure formation follows CDM until late times when the decay process becomes significant, so a similar reionization history can be applied. Most of our DDM candidates are not affected by the UV-background since they are already massive at z=6. This is largely because for the same Fornax kinematic data, DDM candidates tend to be more massive because they have less concentrated DM profiles than their CDM counterparts.
The difference in Fornax infall times in the different models may imply a different star formation quenching mechanism if environmental effects play an important role in dSph formation processes. From the observed Fornax star formation history, there is an enhanced star formation event at 3-4 Gyr ago and it is quenched $\sim$ 2 Gyr ago~\citep{Coleman_etal08}. For those subhalo candidates that have infall times well before the star formation peak, star formation may be triggered by a close passage to the Milky Way. For those that have infall times after the star formation peak, the infall into Galactic halo may cause rapid gas loss due to ram-pressure stripping. More precise predictions can be made by detailed Fornax orbital motion reconstruction from its proper motion and line-of-sight velocity measurement, which may be possible with the GAIA satellite. Larger sets of simulations that sample more possible halo formation histories in different dark matter scenarios will also help to confirm our results.
\begin{figure}
\includegraphics[height=5.8cm]{Fornax_SF.eps}
\caption{
Qualitative Fornax star-formation histories (black solid lines, from \citet{Coleman_etal08}) compared to estimated infall times from \citet{Rocha_etal12b} using Galactocentric positions and orbital motions (grey filled area) and our results (vertical dashed lines). The two navy dashed lines indicate the infall time for the two CDM candidates from the Aquarius simulations that match both the Fornax luminosity and stellar kinematics. The aqua line is for the WDM 2.3 keV candidate, and the orange line is for the DDM candidate. Notice the DDM candidate infall time is significantly lower than the \citet{Rocha_etal12b} prediction and our CDM and WDM candidates, and also it happens at the time when the Fornax star formation rate starts to drop after a burst 3-4 Gyr ago. All others predict infall times before the enhanced star formation event.
}
\label{fig:sf}
\end{figure}
\section*{Acknowledgments}
We would like to thank Mike Boylan-Kolchin, Andrew Cooper, Yao-Yuan Mao, Julio Navarro, and Till Sawala for useful discussions. M.-Y. W. and L.E.S. acknowledge support from NSF grant PHY-1522717. This work is part of the D-ITP consortium, a programme of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). This work was supported by the Science and Technology Facilities Council (grant number ST/F001166/1) and the European Research Council (grant numbers GA 267291 "Cosmiway"). It used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. The work of ARZ was funded in part by the Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (Pitt PACC) at the University of Pittsburgh.
| {'timestamp': '2015-09-23T02:12:53', 'yymm': '1509', 'arxiv_id': '1509.04308', 'language': 'en', 'url': 'https://arxiv.org/abs/1509.04308'} |
\section{Introduction}
\label{sec:introduction}
\PARstart{I}{n} the new era of Open Science \cite{Woelfe2011}, data sharing has become increasingly crucial for efficient and fair development of science and industry. Especially in the field of medical image science, various datasets have been released and used for the development of new methods and benchmarks. There have been attempts to create publicly open databases consisting of medical images, demographic data, and clinical information, such as ADNI, AIBL, PPMI, 4RTN, PING, ABCD and UK BioBank. In the near future, clinical images acquired with medical indications will become available for research use.
Big data, consisting of large amounts of brain magnetic resonance (MR) images and corresponding medical records, could provide new evidence for the diagnosis and treatment of various diseases.
Clearly, search technology is essential for the practical and effective use of such big data.
Currently, text-based searching is widely used for the retrieval of brain MR images. However, since this approach requires skills and experience during retrieval and data registration, there is a strong demand from the field to realize content-based image retrieval (CBIR) \cite{Kumar2013}.
To build a CBIR system that is feasible for brain MR imaging (MRI) databases, obtaining an appropriate and robust low-dimensional representation of the original MR images that reflects the characteristics of the disease in focus is extremely important.
Various methods have been proposed, including those based on classical feature description
\cite{Tu2010,Huang2012,Murala2014},
anatomical phenotypes \cite{Faria2015}, and deep learning techniques
\cite{Kruthika2019,Swati2019,Onga2019}.
The latter two techniques
\cite{Swati2019,Onga2019} acquire similar low-dimensional representations for similar disease data by introducing the idea of distance metric learning \cite{Alipanahi2008}\cite{Hoffer2016}.
Their low-dimensional representations adequately capture disease characteristics rather than individual variations seen on gyrification patterns in the brain. However, the application of these methods to a heterogeneous database containing MRIs from various scanners and scan protocols is hampered by the scanner or protocol bias, which is not negligible.
In brain MRI, such non-biological experimental variations (i.e., magnetic field strength, scanner manufacturer, reconstruction method) resulting from differences in scanner characteristics and protocols can affect the images in various ways and have a significant impact on the subsequent process
\cite{Clark2006,Han2006,Yu2018,Oishi2019,Onga2019,Wachinger2021}.
Wachinger et al. \cite{Wachinger2021} analyzed 35,320 MR images from 17 open datasets and performed the 'Name That Dataset' test, that is guessing which dataset it is based on the images alone. They reported a prediction accuracy of 71.5\% based only on volume and thickness information from 70\% of the training data. This is evidence that there are clear features left among datasets.
Removing those variabilities is essential in multi-site and long-term studies and for building a robust CBIR system. There has been an increase in recent research on data harmonization, i.e., eliminating or reducing variation that is not intrinsically related to the brain's biological features.
Perhaps the most straightforward image harmonization approach is to reduce the variations in the intensity profile \cite{Gao2019,Um2019}. In the methods in both \cite{Gao2019} and \cite{Um2019}, correction of the luminance distribution for each sub-region reduces the variability of the underlying statistics between images, whereas histogram equalization reduces the variability of neuroradiological features. However, these methods are limited to approximating rudimentary statistics that can be calculated from images, and they are based on the assumption that the intensity histogram is similar among images. This assumption is invalid when images that contain pathological findings that affect intensity profile are included. While some improvement in unintended image variability can be expected, the effect on practical tests that utilize data from multiple sites is unknown.
In the field of genomics, Johnson et al. \cite{Johnson2007} proposed an empirical Bayes-based correction method to reduce batch effects, which are non-biological differences originating from each batch of micro-array experiments obtained from multiple tests. This effective statistical bias reduction method is now called ComBat, and it has recently been published as a tool for MRI harmonization \cite{Fortin2017-COMBAT}. This tool has been applied to several studies \cite{Fortin2017a,Fortin2017b,Yu2018,Wachinger2021}.
The ComBat-based methods standardize each cortical region based on an additive and use multiplicative linear transform to compensate for variability. Some limitations of these models have been pointed out, such as the following: (i) they might be insufficient for complex multi-site and area-level mapping, (ii) the assumption of certain prior probabilities (Gaussian or inverse gamma) is not always appropriate, and (iii) they are susceptible to outliers \cite{Zhao2019}
Recently, advancements in machine learning techniques \cite{Zhao2019,Dewey2019,Moyer2020,Dinsdale2021}
have provided practical solutions for MR image harmonization.
DeepHarmony \cite{Dewey2019} uses a fully convolutional U-net to perform the harmonization of scanners.
The researchers used an MRI dataset of multiple sclerosis patients in a longitudinal clinical setting to evaluate the effect of protocol changes on atrophy measures in a clinical study. As a result, DeepHarmony confirmed a significant improvement in the consistency of volume quantification across scanning protocols. This study was practical in that it aimed to directly standardize MR images using deep learning to achieve long-term, multi-institutional quantitative diagnosis.
However, this model requires ''traveling head'' (participants are scanned using multiple MRI scanners) to train the model.
Zhao et al. \cite{Zhao2019} attempted to standardize a group of MR images of infants taken at multiple sites into a reference group using CycleGAN \cite{Zhu2017}, which has a U-net structure in the generator. The experiment validated the evaluation of cortical thickness with several indices (i.e., ROI (region-of-interest)-base, distribution of low-dimensional representations).
They argued that the retention of the patient's age group was superior to ComBat in evaluating group difference.
Moyer et al. \cite{Moyer2020} proposed a sophisticated training technique to reconstruct bias-free MR images by acquiring a low-dimensional representation independent of the scanner and condition. Their method is an hourglass-type unsupervised learning model based on variational autoencoders (VAE) with an encoder--decoder configuration.
The input $\vt{x}$ and output $\vt{x'}$ are the same MR images, and their low-dimensional representation is $\vt{z}$ (i.e., $\vt{x}\rightarrow \vt{z} \rightarrow \vt{x'}$).
The model is trained with the constraint that $\vt{z}$ and site- and scanner-specific information $\vt{s}$ are orthogonal (actually relaxed), such that the $\vt{s}$ in $\vt{z}$ is eliminated. They demonstrated the advantages of their method on diffusion MRI, but their technological framework is applicable to other modalities.
Dinsdale et al. \cite{Dinsdale2021} also proposed a data harmonization method based on the idea of domain adaptation \cite{Ganin2016}.
Their model uses adversarial learning, where the feature extractor consisting of convolutional neural networks (CNN) following the input is branched into a fully connected net for the original task (e.g., segmentation and classification) and other fully connected nets for domain discriminators (e.g., scanner type or site prediction) to make the domain unknown while improving the accuracy of the original task. They have confirmed its effectiveness in age estimation and segmentation tasks.
The methods developed by Moyer et al. and Dinsdale et al. aim to generate a low-dimensional representation with ''no site information'', and they are highly practical and generalizable techniques for data harmonization. Nevertheless, for CBIR, a method that is applicable for a large number of legacy images is necessary. Here, it is not realistic to collect images from each site and train the model to harmonize them. Practically, a method that can convert heterogeneous images in terms of variations in scanners and scan parameters into images scanned by a given pseudo-''standard'' environment by applying a learned model is highly desired.
In this paper, we propose a novel framework called disease-oriented image embedding with pseudo-scanner standardization (DI-PSS) to obtain a low-dimensional representation of MR images for practical CBIR implementation. The PSS, the key element of the proposal, corrects the bias caused by different scanning environments and converts the images so that it is as if the same equipment had scanned them. Our experiments on ADNI and PPMI datasets consisting of MR images captured by three manufacturers' MRI systems confirmed that the proposed DI-PSS plays an important role in realizing CBIR.
The highlights of this paper's contribution are as follows:
\begin{itemize}
\item To the best of the authors' knowledge, this is the first study of the acquisition and quantitative evaluation of an effective low-dimensional representation of brain MR images for CBIR, including scanner harmonization.
\item Our DI-PSS framework reduces undesirable differences caused by differences in scanning environments (e.g., scanner, protocol, dataset) by converting MR images to images taken on a predefined pseudo-standard scanner, and a deep network using a metric learning acquires a low-dimensional representation that better represents the characteristics of the disease.
\item DI-PSS provides appropriately good low-dimensional representations for images from other vendors’ scanners, diseases, and datasets that are not used for learning image harmonization. This is an important feature for the practical and robust CBIR, which applies to a large amount of legacy MRIs scanned at heterogeneous environments.
\end{itemize}
\section{Clarification of the issues addressed in this paper}
\label{sec:sec2}
\subsection{Overlooking the problem}
We begin by presenting the issues to be solved in this paper.
As mentioned above, to realize CBIR for brain MRI, Onga et al. proposed a new technique called disease-oriented data concentration with metric learning (DDCML), which acquires low-dimensional representations of 3D brain MR images that are focused on disease features rather than the features of the subject's brain shape \cite{Onga2019}.
DDCML is composed of 3D convolutional autoencoders (3D-CAE) effectively combined with deep metric learning. Thanks to its metric learning, DDCML could acquire reasonable low-dimensional representations for unlearned diseases according to their severity, demonstrating the feasibility of CBIR for brain MR images. However, we found that such representations are highly sensitive to differences in datasets (i.e., differences in imaging environments, scanners, protocols, etc.), which is a serious challenge for CBIR.
Figure \ref{fig:fig1} shows the low-dimensional distribution obtained by DDCML and visualized by t-SNE \cite{Laurens2008}. Here, DDCML was trained on Alzheimer's disease (AD) and healthy cases (clinically normal; CN) in the ADNI2 dataset and evaluated ADNI2 cases not used for training and healthy control (Control -- equivalent to CN) and Parkinson's disease (PD) cases in the untrained PPMI dataset. From the perspective of CBIR, it is desirable to obtain similar low-dimensional representations for CN and Control. However, it can be confirmed that the obtained low-dimensional representations are more affected by the differences in the environment (dataset) than by the disease. As mentioned above, differences in imaging environments, including scanners, are a major problem in multi-center and time series analysis, and inconsistent low-dimensional representations because of such differences in datasets are a fatal problem in CBIR implementation. The purpose of this paper is reducing these differences and to obtain a low-dimensional representation that better captures the characteristics of the disease and is suitable for appropriate CBIR.
\begin{figure}[t]
\includegraphics[width=0.9\linewidth]{fig/fig1_old.png}
\caption{Plots of low-dimensional representations of 3D MRI obtained from different datasets.}
\small{The impact of different scanners (CN$\Leftrightarrow$Control; they are medically equivalent) is greater than the impact of the disease (AD$\Leftrightarrow$CN).}
\label{fig:fig1}
\end{figure}
\subsection{Our data harmonization strategy for realizing CBIR}
In studies dealing with multi-site and long-term data, it is undoubtedly important to reduce non-biological bias originating from differences among sites and datasets.
Since the methods of Moyer et. al.\cite{Moyer2020} and Dinsdale et al. \cite{Dinsdale2021} are theoretical and straightforward learning method that utilizes images of the target site to achieve data harmonization, their robustness to unexpected input (i.e. from another site or dataset) is questionable. Therefore, in principle, the images of all target sites (scanners, protocols) need to be learned in advance.
Since CBIR requires more consideration of the use of images taken in the past, the number of environments that need to be addressed can be larger than for general data harmonization.
It will be more difficult to implement a harmonization method that learns all the data of multiple environments in advance.
Therefore, in contrast to their approaches, we aim to achieve data harmonization by converting images taken in each environment into images that can be regarded as having been taken in one predetermined ''standard'' environment (e.g., the scanner currently used primarily at each site). However, in addition to the problems described above, it is practically impossible to build an image converter for each environment.
With this background, we have developed a framework that combines CycleGAN, which realizes robust image transformation, with deep metric learning to achieve a certain degree of harmonization even for images in untrained environments. In this paper, we validate the feasibility of our framework, which converts MR images captured in various environments into pseudo standard environment images using only one type of image converter.
\section{Disease-oriented image embedding with pseudo-scanner standardization (DI-PSS)}
\label{sec:sec3}
The aim of this study is to obtain a low-dimensional embedding of brain MRI that is independent of the MRI scanner and individual characteristics but dependent on the pathological features of the brain, to realize a practical CBIR system for brain MRI.
To accomplish this, we propose a DI-PSS framework, which is composed of the three following components: (1) pre-process, (2) PSS, and (3) embedding acquisition.
\subsection{The Pre-processing component (skull stripping with geometry and intensity normalization)}
The pre-processing component performs the necessary pre-processing for future image scanner standardization processing and low-dimensional embedding acquisition processing.
Specifically, for all 3D brain MR image data, skull stripping was performed using a multi-atlas label-fusion algorithm implemented in the MRICloud \cite{Mori2016}.
The skull-stripped images were linearly aligned to the JHU-MNI space using a 12-parameters affine transformation function implemented in the MRICloud, resulting in aligned brain images.
This feature makes a significant contribution to the realization of the proposed PSS in the next stage. It is important to note here that since brain volume information is the feature that contributes most to the prediction of the dataset \cite{Wachinger2021}, the alignment to a standard brain with this skull stripping technique should also contribute to the harmonization of the data.
In addition, because the intensity and contrast of brain MR images are arbitrarily determined, there is a large inter-image variation. In brain MR image processing using machine learning, the variation in the average intensity confounds the results. Therefore, we standardized the intensity so that the average intensity value of each case was within mean $\mu=18$ and margin $\epsilon = 1.0$ by performing an iterative gamma correction process, as in previous studies \cite{Arai2018}\cite{Onga2019}.
\subsection{The PSS component}
\subsubsection{The concept of PSS}
The proposed PSS is an image conversion scheme that converts a given raw MR image into a synthesized image that looks like an MR image scanned by a standard scanner and a protocol. Since there are numerous combinations of scanners and scan parameters, building scanner- and parameter-specific converters is not practical.
Therefore, in our PSS scheme, we only construct a 1:1 image conversion model (i.e., PSS network) that converts images from a particular scanner $Y$ to a standard scanner $X$.
That is, a particular PSS network is used to convert images captured by other scanners $(Z_1, Z_2,\cdots)$ as well. This strategy is in anticipation of the generalizability of the PSS network, backed by advanced deep learning techniques.
In this paper, we evaluate the robustness of our image transformations provided by PSS on MR images taken by other vendors' scanners and on images in different datasets.
Figure \ref{fig:fig2} gives an overview of our PSS network that realizes the PSS. The PSS network makes effective use of CycleGAN \cite{Zhu2017}, which has achieved excellent results in 1:1 image transformation.
Here, training of CycleGAN generally requires a lot of training data, especially in the case of 3D data, because the degree of freedom of the model parameters is large. However, it is difficult to collect such a large amount of supervised labeled 3D MRI data to keep up with the increase. Since the position of any given slice is almost the same in our setting thanks to MRICloud in the skull stripping process, a 3D image can be treated as a set of 2D images containing position information. With these advantages, our PSS suppressed the problems an overwhelmingly insufficient amount of training data and the high degree of freedom of the transformation network. In sum, arbitrary slices are cut out from the input 3D image and converted to slices corresponding to the same position in the 3D image as the target domain using the PSS network based on common (2D) CycleGAN. Note that the PSS process is performed using the trained generator $G_X$.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\linewidth]{fig/fig2.png}
\caption{Overview of pseudo-scanner standardization (PSS) network.}
\small{Our PSS network is based on CycleGAN, and PSS is performed with trained generator $G_X$.}
\label{fig:fig2}
\end{figure*}
\subsubsection{Implementation of the PSS network}
The structure of the PSS network that realizes the proposed PSS is explained according to the CycleGAN syntax, with images captured by a standard scanner as domain $X$ and images captured by a certain different scanner as domain $Y$.
Generator $G_Y$ transforms (generates) an image $\vt{y’}=G_Y(\vt{x})$ with the features of domain $Y$ from an image $\vt{x}$ of the original domain $X$.
Discriminator $D_Y$ determines the authenticity of the real image $\vt{y}$ belonging to domain $Y$ or the generated $\vt{y'}=G_Y(\vt{x})$.
Similarly, the conversion from domain $Y$ to domain $X$ is performed by generator $G_Y$, and discriminator $D_X$ judges the authenticity of the image. The goal of this model is to learn maps of two domains $X$ and $Y$ given as training data. Note here again that we use the trained module $G_X$ (maps $Y$ to $X$) as an image converter.
The training of the model proceeds by repeating the transformation of the training data sample $\vt{x_i} \in X$ and the training data sample $\vt{y_j} \in Y$.
The overall objective function of the PSS network, $L_{PSS}$ to be minimized, consists of the three following loss components: adversarial loss ($L_{GAN}$), cycle consistency loss ($L_{eye}$), and identity mapping loss ($L_{identity}$). This is expressed as follows:
\begin{equation}
\begin{split}
L_{PSS} (G_Y,&G_X,D_Y,D_X) = \\
&L_{GAN} (G_Y,D_Y) + L_{GAN} (G_X,D_X) \\
\hspace*{3em}&+\lambda_1 L_{eye}+ \lambda_2 L_{identity}.
\end{split}
\end{equation}
The adversarial loss ($L_{GAN}$) is defined based on the competition between the generator, which tries to produce the desired other domain image, and the discriminator, which sees through the fake generated image; this minimization implies a refinement of both. From the point of view of image transformation, the minimization of this loss means that the probability distribution generated by the generator is closer to the probability distribution of the counterpart domain, which means that a higher quality image can be obtained. This loss is defined in both directions, $X\rightarrow Y$ and $Y\rightarrow X$, and these are expressed in order as follows:
\begin{equation}
\begin{split}
L_{GAN}(G_Y,D_Y) =
&E_{\vt{y}\sim p_{data}(\vt{y})}[(D_Y(\vt{y})-1)^2] \\
+&E_{\vt{x}\sim p_{data}(\vt{x})}[(D_Y(G_Y(\vt{x}))^2],
\end{split}
\end{equation}
\begin{equation}
\begin{split}
L_{GAN}(G_X,D_X) =
&E_{\vt{x}\sim p_{data}(\vt{x})}[(D_X(\vt{x})-1)^2] \\
+&E_{\vt{y}\sim p_{data}(\vt{y})}[(D_X(G_X(\vt{y}))^2].
\end{split}
\end{equation}
The cycle consistency loss ($L_{eye}$) is a constraint to guarantee that mutual transformation is possible by cycling two generators:
\begin{equation}
\begin{split}
L_{eye}(G_X,G_Y) =
&E_{\vt{x} \sim p_{data} (\vt{x})} || G_X (G_Y (\vt{x}))-\vt{x}||_1 \\
+ &E_{\vt{y} \sim p_{data} (\vt{y})} ||G_Y (G_X (\vt{y}))-\vt{y}||_1
\end{split}
\end{equation}
Finally, the identity mapping loss ($L_{identity}$) is a constraint to maintain the original image features without performing any transformation when the image of the destination domain is input:
\begin{equation}
\begin{split}
L_{identity}(G_X,G_Y)=
&E_{\vt{x} \sim p_{data} (\vt{x})} || G_X(\vt{x})-\vt{x}||_1 \\
+&E_{\vt{y} \sim p_{data} (\vt{y})} ||G_Y(\vt{y})-\vt{y}||_1
\end{split}
\end{equation}
It has been confirmed that the introduction of this constraint can suppress the learning of features that are not important in either domain, such as unneeded tints. Here, $\lambda_1$ and $\lambda_2$ are hyper-parameters and we set $\lambda_1 =10.0$ and $\lambda_2=0.5$ as in the original setting.
\subsubsection{The Embedding acquisition component}
In the embedding acquisition component, the low-dimensional embedding of 3D brain MRI images is obtained by our embedding network after the PSS process.
Our embedding network is a 3D-CAE model consisting of encoders and decoders with distance metric learning, referring to Onga et al.'s DDCML\cite{Onga2019}.
Distance metric learning is a learning technique that reduces the Euclidean distance between feature representations of the same label and increases the distance between feature representations of different labels.
Thanks to the introduction of metric learning, 3D-CAE has been found to yield embedding that is more focused on disease features.
According to Hoffer’s criteria \cite{Hoffer2016}, the distance distribution in the low-dimensional embedding space for input $\vt{x}$ for class $i$ ($i \in 1, \cdots c$; where $c$ is the number of types of disease labels in the dataset) is calculated by
\begin{equation}
P(\vt{x};\vt{x_1},\cdots, \vt{x_c} )_i = \frac{ \exp(-{||f(\vt{x})-f(\vt{x_i})||}^2) }{ \sum_j^c \exp(-{||f(\vt{x})-f(\vt{x_j})||}^2)}.
\end{equation}
Here, $\vt{x_i}$ ($i \in 1, \cdots, c$) is randomly sampled data from each class $i$, and $f$ denotes the operation of the encoder (i.e., encoder part of the 3D-CAE in our implementation). This probability can be thought of as the probability that the data $\vt{x}$ belong to each class $i$.
The loss function $L_{dist}$ is calculated by the cross-entropy between the $c$-dimensional vector $\vt{P}$ described above and the $c$-dimensional one-hot vector $\vt{I}(\vt{x})$ with bits of the class to which $\vt{x}$ belongs as
\begin{equation}
L_{dist} (\vt{x},\vt{x_1},\cdots,\vt{x_c}) = H(\vt{I}(\vt{x}),\vt{P}(\vt{x};\vt{x_1},\cdots,\vt{x_c}))
\end{equation}
Here, $H(\vt{I}(\vt{x}),\vt{P}(\vt{x};\vt{x_1},\cdots,\vt{x_c}))$ takes a small value when the probability that the element firing in $\vt{I}(\vt{x})$ belongs to the class it represents is high, whereas it takes a large value when the probability is low. Thus, $L_{dist}$ aims at the distribution of the sampled data at locations closer to the same class and farther from the different classes on the low-dimensional feature space.
Finally, the objective function $L_{CAE}$ of our low-dimensional embedding acquisition network consisting of 3D-CAE and metric learning is finally expressed by the following equation:
\begin{equation}
L_{CAE}=L_{RMSE} + \alpha L_{dist} (\vt{x},\vt{x_1},\cdots,\vt{x_c} )
\end{equation}
Here, $L_{RMSE}$ is the pixel-wise root mean square error normalized by image size in CAE image reconstruction. Furthermore, $\alpha$ is a hyper-parameter set to 1/3 based on the results of preliminary experiments.
\section{Experiments}
\label{sec:sec4}
In CBIR, cases of the same disease should be able to acquire similar low-dimensional representations, regardless of the individual, scanner, or protocol. We investigated the effectiveness of the proposed DI-PSS by quantitatively evaluating how PSS changes the distribution of embeddings within and between data groups (i.e., combination of scanner type and disease). In addition, we compared the clustering performance of the obtained embeddings against diseases with and without PSS.
\subsection{Dataset}
In this experiment, we used the ADNI2 and PPMI datasets, in which the vendor information of the scanners (Siemens [SI], GE Medical Systems [GE], Philips Medical Systems [PH]) was recorded along with the disease information. Statistics of those datasets used in the experiment are shown in Table \ref{table:table1}.
We used Alzheimer's disease (ADNI-AD or AD) and clinically normal cases (ADNI-CN) from ADNI2 dataset with vendor information.
From the PPMI dataset, we used two types of labeled images, Parkinson's disease (PD) and Control. We did not utilize the scanner information for this dataset in evaluating the versatility of the proposed method.
Note that ADNI-CN and Control can be considered medically equivalent. Furthermore, PD is known to show little or no difference in MRI from healthy cases \cite{Postuma2015}\cite{Meijer2017}.
The ADNI and PPMI are longitudinal studies that include multiple time points, and the datasets contain multiple scans for each participant. To avoid duplication, one MRI was randomly selected from each participant.
The MRICloud (\url{https://mricloud.org/}) was used to skull strip the T1-weighted MRIs and affine transform to the JHU-MNI atlas \cite{Oishi2009}.
A neurologist with more than 20 years of experience in brain MRI research performed the quality control of the MRIs and removed MRIs that the MRICloud did not appropriately pre-process.
Due to the neural network model used in the experiments, the skull-stripped and affine-transformed brain MR images were converted to 160$\times$160$\times$192 pixels after cropping the background area.
Training and evaluation of the PSS network and embedding network were performed using five-fold cross validation. In the evaluation experiments described below,, the evaluation data of each fold is not included in the training data for either the PSS network or the embedding network.
Note that even skilled and experienced neuroradiologists cannot separate PD from CN or Control by visual inspection of the T1-weighted images.
Therefore, we did not expect these two conditions to be separable by unsupervised clustering methods even after applying the DI-PSS.
\begin{table}[t]
\caption{Dataset used in our study}
\label{table:table1}
\begin{tabular}{ccllccc}
\toprule
\multicolumn{2}{c}{dataset} & vendor & label & \#used & \#patients & \#total \\ \hline
\multirow{6}{*}{ADNI} & \multirow{3}{*}{CN} & Siemens & CN\_SI & 92 & 103 & 439 \\
& & GE & CN\_GE & 93 & 101 & 494 \\
& & Philips & CN\_PH & 27 & 27 & 119 \\ \cline{2-7}
& \multirow{3}{*}{AD} & Siemens & AD\_SI & 80 & 84 & 254 \\
& & GE & AD\_GE & 80 & 92 & 302 \\
& & Philips & AD\_PH & 20 & 24 & 73 \\ \hline
\multirow{2}{*}{PPMI} & & \multirow{2}{*}{n/a} & Control & 75 & 75 & 114 \\
& & & PD & 149 & 149 & 338 \\
\bottomrule
\end{tabular}
\footnotesize{ADNI-CN and Control can be considered medically equivalent. \\
There are no PD-related anatomical features observable on T1-weighted MRI.}
\end{table}
\begin{figure*}[t]
\begin{minipage}[t]{0.57\hsize}
\centering
\includegraphics[width=0.95\linewidth]{fig/fig3a.png}
\subcaption{Generator}
\label{fig3a}
\end{minipage}
\begin{minipage}[t]{0.38\hsize}
\centering
\includegraphics[width=0.95\linewidth]{fig/fig3b.png}
\subcaption{Discriminator}
\label{fig3b}
\end{minipage}
\caption{Architecture of (a) Generators ($G_X, G_Y$) and (b) Discriminators ($D_X, D_Y$) in the PSS network.}
\footnotesize{(a) kernel size (f$\times$f), stride size (s), padding size (p), $\times$ \# of kernel + instance norm + ReLU*\\
(b) convA : kernel size (f$\times$f), stride size (s), padding size (p), $\times$ \# of kernel + LeakyReLU\\
\hspace*{1em} convB : kernel size (f$\times$f), stride size (s), padding size (p), $\times$ \# of kernel + instanceNorm + LeakyReLU\\
\hspace*{1em} convC : kernel size (f$\times$f), stride size (s), padding size (p), $\times$ \# of kernel \\
FC : fully connected layer (400$\rightarrow$1) }
\label{fig:fig3}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.95\linewidth]{fig/fig4.png}
\caption{Architecture of embedding network.}
\footnotesize{conv : kernel 3$\times$3, stride size=1, padding size=1, $\times$ (\# of kernel) + ReLU\\
deconv : kernel 3$\times$3, stride size=1, padding size=1, $\times$ (\# of kernel) + ReLU\\
average pooling, up-sampling (bi-linear interpolation): 2$\times$2$\times$2:1.}
\label{fig:fig4}
\end{figure*}
\subsection{Detail of the PSS network and its training}
Figures \ref{fig:fig3}a and \ref{fig:fig3}b show the architecture of the generator ($G_X, G_Y$) and the discriminator ($D_X, D_Y$), respectively of the PSS network.
They are basically the same as the original CycleGAN for 2D images.
Since PSS is to reduce the bias caused by variations in scanners and scan parameters, the disease-related anatomical variations should be minimized in the training images.
Therefore, we used only ADNI-CN cases, in which disease features do not appear in the brain structure, to train the PSS network.
In this experiment, we chose the Siemens scanner as the standard scanner because it has largest market share, and we chose the GE scanner as the specific vendor of image conversion source.
In other words, our PSS network is designed to convert CN images taken by GE scanners from the ADNI2-dataset (CN\_GE) to synthetic images similar to those scanned by the Siemens scanners (CN\_SI). We evaluated the applicability of the PSS to the diseased brain MRIs (AD and PD), as well as the generalizability to the non-GE scanners (see Section IV.D). In PSS network, we used coronal images for the training. The number of training images of each fold in the PSS network is (93+92)$\times$4/5 (5-fold CV)$\times$192 (slices).
\subsection{Detail of the embedding network and its training}
Figure \ref{fig:fig4} shows the architecture of our 3D-CAE-based embedding network.
Our embedding network embeds each 3D brain MR image into 150-dimensional vectors. The size of the MRIs handled by the embedding network is halved at each side, as in DDCML \cite{Onga2019}, to improve the learning efficiency. Note that the compression ratio of our embedding network is (80$\times$80$\times$96):150 = 4,096:1. The embedding network was trained and evaluated using ADNI2 and PPMI datasets with the five-fold cross-validation strategy.
As mentioned above, PD and CN cannot even be diagnosed from images by skilled neuroradiologists, so for training 3D-CAE to obtain low-dimensional representations, two classes of metric learning are used so that the representations of AD and (CN + Control) are separated. The low-dimensional representations of brain MR images are acquired by five-fold cross validation of 3D-CAE. In addition to AD, CN, and Control in each test fold, the low-dimensional representation of PD, which was not included in the training, is analyzed to quantitatively verify the effectiveness of the proposed DI-PSS evaluation.
\subsection{Evaluation of the PSS}
To evaluate the effectiveness of the proposed DI-PSS framework, we evaluate the three following elements:
\begin{enumerate}
\item Changes in MR images
\item Distribution of the embedding.
\item Clustering performance of the embedding.
\end{enumerate}
In (1), we assess how the images are changed by our scanner standardization.
We quantitatively evaluate the difference between the original (raw) image and the synthetic image with peak signal-to-noise ratio (PSNR), root mean squared error (RMSE), and structured similarity (SSIM).
To ensure that the evaluation is not affected by differences in brain size, these evaluations were performed on brain regions only.
Although MRICloud, which is used in skull stripping in this experiment, standardizes the brain size to the standard brain size, reducing the differences in brain size between cases, this method was adopted for a more rigorous evaluation.
In (2), we quantitatively examine the effect of PSS by analyzing the distribution of the obtained low-dimensional representations.
Specifically, for each category (e.g., CN\_SI, AD\_GE) we investigate the following: (i) variation (i.e., standard deviation) of the embedding and (ii) the mean and standard deviation of the distance from each embedding to the centroid of a different category, where the distance between the centroids of ADNI-CN\_Siemens (CN\_SI) and ADNI-AD\_Siemens (AD\_SI) are normalized to 1.
In addition, we visualize those distributions in 2D space using t-SNE \cite{Laurens2008} as supplemental results for intuitive understanding.
In (3), we evaluate the separability of the resulting embeddings.
In this study, we performed spectral clustering \cite{Ng2001} to assess its potential quality for CBIR.
In the spectral clustering, we used a normalized graph Laplacian based on 10-nearest neighbor graphs with a general Gaussian-type similarity measure.
We set the number of clusters to be two (AD vs. CN + Control + PD), which is the number of disease categories to be classified. Here, the consistency of the distance between the embedded data because of the difference in folds is solved by standardizing the distance between CN\_SI and AD\_SI per fold to be 1, as mentioned above.
The clustering performance was evaluated using two methodologies.
The first was evaluation with six commonly used criteria (i.e., silhouette score, homogeneity, completeness, V-measure, adjusted Rand-index [ARI], and adjusted mutual information [AMI]) implemented on the scikit-learn machine learning library (\url{https://scikit-learn.org/}).
The other is a diagnostic capability based on clustering results.
Here, as with other clustering evaluations in the literature, we swap the columns so that each fold results in the optimal clustering result and then sum them.
\section{Results}
\label{sec:sec5}
\subsection{Changes in MR images by PSS}
Figure \ref{fig:fig5} shows an example of each MR image converted to an image taken on a pseudo-standard (= Siemens) scanner with PSS and the difference visualized.
Table \ref{table:table2} summarizes the statistics of the degree of change in the images in the brain regions. Here, the background region was excluded from the calculation to eliminate the effect of differences in brain size.
For the ADNI dataset, the differences obtained by the PSS image transformation were not significant between CN, AD, and scanner vendors, although the Philips scanners showed less variation on average.
For the PPMI dataset that was not used for training, the change in the image because of PSS is clearly larger compared with ADNI (approx. $\times$ 1.5 in RMSE).
In all categories, the amount of change because of PSS varied from case to case, but the PSS treatment did not cause any visually unnatural changes in the images.
Figure \ref{fig:fig6} shows the cumulative intensity changes of images by PSS in each category. This time, the background areas other than the brain are also included in the evaluation. The number of pixels where the intensity has not changed because of PSS exceeds 80\% for all categories, indicating that no undesired intensity changes have occurred for the background (as also seen in Figures. \ref{fig:fig5} and \ref{fig:fig6}).
There is no significant difference in the distribution of intensity change by vendor, and the PPMI dataset has a larger amount of intensity change overall.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\linewidth]{fig/fig5.png}
\caption{Example of image change by PSS (in coronal plane).}
\footnotesize{In each category, from left to right, the original image, the PSS processed image, and the difference between them.}
\label{fig:fig5}
\end{figure*}
\begin{figure*}[t]
\centering
\begin{minipage}[t]{0.45\hsize}
\centering
\includegraphics[width=0.95\linewidth]{fig/fig6a.png}
\subcaption{overall view}
\label{fig6a}
\end{minipage}
\begin{minipage}[t]{0.45\hsize}
\centering
\includegraphics[width=0.95\linewidth]{fig/fig6b.png}
\subcaption{enlarged view}
\label{fig6b}
\end{minipage}
\caption{Cumulative intensity changes of MR images by PSS.}
\label{fig:fig6}
\end{figure*}
\begin{table}[t]
\centering
\caption{Summary of image changes by PSS}
\label{table:table2}
\begin{tabular}{clccc}
\toprule
dataset & label & PSNR (db) & RMSE & SSIM \\ \hline
\multirow{6}{*}{ADNI}
& CN\_SI & 31.52 ± 2.85 & 7.17 ± 2.57 & 0.9743 ± 0.0048\\
& CN\_GE & 31.67 ± 2.45 & 6.94 ± 2.13 & 0.9748 ± 0.0041 \\
& CN\_PH & 32.18 ± 2.65 & 6.58 ± 2.02 & 0.9747 ± 0.0038 \\
& AD\_SI & 31.64 ± 3.04 & 7.13 ± 2.77 & 0.9746 ± 0.0043 \\
& AD\_GE & 31.65 ± 2.52 & 6.98 ± 2.32 & 0.9750 ± 0.0044 \\
& AD\_PH & 32.33 ± 2.13 & 6.36 ± 1.63 & 0.9751 ± 0.0031 \\ \hline
\multirow{2}{*}{PPMI} & Control & 30.16 ± 5.47 & 9.81 ± 8.32 & 0.9596 ± 0.0346 \\
& PD & 29.40 ± 5.94 & 11.60 ± 10.89 & 0.9539 ± 0.0473 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Distribution of low-dimensional embedded data}
\subsubsection{Distance between centers of the data distribution by category}
Table \ref{table:table3} shows the variation (standard deviation; SD) of the 150-dimensional embedded representation in each category. Again, it should be noted here that CN\_SI and AD\_SI were normalized to 1.
The average reduction in SD for all data by PSS was 8.27\%.
Tables \ref{table:table4} shows the statistics of distances from each embedding to the centroid of a different category. This shows the distribution of the data, considering the direction of variation, which is more practical for CBIR application.
With PSS, the average distance between centroids across categories is almost unchanged, but the variability is greatly reduced for all categories.
\begin{table}[t]
\caption{Variation (SD) of the embedding in category $\dagger$}
\label{table:table3}
\begin{tabular}{clcccc}
\toprule
dataset & label & \#data & baseline & with PSS & $-$ SD (\%)\\ \hline
\multirow{6}{*}{ADNI}
& CN\_SI & 92 & 0.697 & 0.648 & 7.12 \\
& CN\_GE & 93 & 0.784 & 0.716 & 8.66 \\
& CN\_PH & 27 & 0.622 & 0.619 & 0.48 \\
& ADNI-CN &212 & 0.753 & 0.701 & 6.91 \\ \cline{2-6}
& AD\_SI & 80 & 0.863 & 0.783 & 9.24 \\
& AD\_GE & 80 & 0.849 & 0.806 & 5.09 \\
& AD\_PH & 20 & 0.771 & 0.706 & 8.46 \\
& ADNI-AD &180 & 0.876 & 0.823 & 6.09 \\ \hline
\multirow{2}{*}{PPMI}
&Control & 75 & 0.607 & 0.554 & 8.74 \\
&PD & 149& 0.603 & 0.515 & 14.71 \\ \hline
both & CN & 92 & 0.759 & 0.704 & 7.20 \\ \hline
\multicolumn{2}{c}{all} &616& 0.755& 0.693 & 8.27 \\
\bottomrule
\end{tabular}
\small{$\dagger$:Distance between CN\_SI and AD\_SI were normalized to 1.}
\end{table}
\begin{table}[t]
\caption{Mean and variability of embedding across categories of data $\dagger$}
\label{table:table4}
\begin{tabular}{llrrrrr}
\toprule
\multicolumn{2}{c}{}&\multicolumn{2}{c}{baseline} & \multicolumn{2}{c}{with PSS} & -SD (\%) \\
from & to & mean & SD & mean & SD & \\ \hline
ADNI-CN & AD & \multirow{2}{*}{0.879} & 0.669 & \multirow{2}{*}{0.890}& 0.541 & 19.1 \\
AD & ADNI-CN & & 0.875 & & 0.729 & 16.6 \\
Control & AD & \multirow{2}{*}{1.354} & 0.745 & \multirow{2}{*}{1.329}& 0.537 & 28.0 \\
AD & Control & & 0.907 & & 0.702 & 22.6 \\
PD & Control & \multirow{2}{*}{0.256} & 0.469 & \multirow{2}{*}{0.297}& 0.312 & 33.5 \\
Control & PD & & 0.414 & & 0.368 & 11.2 \\
ADNI-CN & PD & \multirow{2}{*}{0.364} & 0.609 & \multirow{2}{*}{0.362}& 0.474 & 22.2 \\
PD & ADNI-CN & & 0.373 & & 0.255 & 31.4 \\
AD & PD & \multirow{2}{*}{1.164} & 0.939 & \multirow{2}{*}{1.091}& 0.770 & 18.0 \\
PD & AD & & 0.620 & & 0.434 & 29.9 \\
CN & AD & \multirow{2}{*}{0.996} & 0.753 & \multirow{2}{*}{0.997}& 0.583 & 22.6 \\
AD & CN & & 0.917 & & 0.773 & 15.8 \\
CN & PD & \multirow{2}{*}{0.249} & 0.593 & \multirow{2}{*}{0.269}& 0.466 & 21.4 \\
PD & CN & & 0.349 & & 0.264 & 24.2 \\
\bottomrule
\end{tabular}
\small{$\dagger$:Distance between CN\_SI and AD\_SI were normalized to 1.}
\end{table}
\subsubsection{Visualization of the distribution of the embedding}
Figures \ref{fig:fig7}a and \ref{fig:fig7}b show scatter plots of the embedding of test data with and without PSS, respectively in an arbitrary fold by t-SNE.
Specifically, this is a scatter plot of the AD, CN, and Control test cases (data excluded from the training in the five-fold cross-validation) along with the untrained PD cases on the model.
Here, PD has been randomly reduced to 1/5 for better visualization. Without PSS (baseline; 3D CAE + metric learning), AD and CN are properly separated, but the distribution of Control + PD (i.e., the difference in datasets) is separated from that of CN to a discernible degree (left). It can be confirmed that by performing PSS, the distribution of Control + PD becomes closer to that of CN, and the separation between AD and other categories becomes better (right).
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\linewidth]{fig/fig7.png}
\caption{Distribution of embedding visualized with t-SNE \cite{Laurens2008}:}
\small{(left) baseline (3D-CAE+metric learning), (right) baseline with PSS.}
\label{fig:fig7}
\end{figure*}
\subsection{Clustering performance of the embedding}
In this section, we compare the separation ability of the obtained low-dimensional embedding of MR images with and without PSS (baseline).
Tables \ref{table:table5} summarizes the clustering performance evaluated with six commonly used criteria.
These are the silhouette score (silh), homogeneity score (homo), completeness score (comp), V-measure (harmonic mean of homogeneity and completeness; V) , ARI, and AMI implemented on the scikit-learn library.
In each category, 1 is the best score and 0 is a score based on random clustering.
It can be confirmed that PSS improved the clustering ability in all evaluation items.
Table \ref{table:table6} is a summary of the clustering performance evaluated with the diagnostic ability. Table \ref{table:table6} (a) is a confusion matrix. Here, the numbers of CN, Control and AD cases are the sum of each fold in the cross-validation.
In each fold, we tested all PD cases (not included in the training), and the number was divided by five and rounded to the nearest whole number.
Tables \ref{table:table6}b and \ref{table:table6}c summarize the diagnostic performance calculated from Table \ref{table:table6} (a) without and with PD cases, respectively. It can be confirmed that PSS enhances the separation of AD and other categories (i.e., CN, Control and PD) in the low-dimensional representation.
\begin{table}[t]
\begin{center}
\caption{Clustering performance evaluated with common criteria $\dagger$}
\label{table:table5}
\begin{tabular}{crrrrrr}
\toprule
& \multicolumn{1}{c}{silh} & \multicolumn{1}{c}{homo} & \multicolumn{1}{c}{comp} & \multicolumn{1}{c}{V} & \multicolumn{1}{c}{ARI} & \multicolumn{1}{c}{AMI} \\ \hline
baseline & 0.236 & 0.220 & 0.301 & 0.250 & 0.251 & 0.241 \\
+PSS & 0.246 & 0.301 & 0.351 & 0.324 & 0.387 & 0.317 \\
\bottomrule
\end{tabular}
\end{center}
\small{$\dagger$: Score 1 is the best in each category. 0 is the score for random clustering.}
\end{table}
\begin{table*}[t]
\caption{Evaluation of clustering ability by diagnostic ability.}
\label{table:table6}
\begin{subtable}[t]{0.9\textwidth}
\begin{center}
\caption{Confusion matrix}
\begin{tabular}{l|cc|cc}
\toprule
\multirow{2}{*}{}&\multicolumn{2}{c|}{baseline} & \multicolumn{2}{c}{with PSS} \\
& CN+Control (+PD)& AD & CN+Control (+PD)& AD \\ \hline
CN+Control &284& 3& 274& 13\\
(+PD) &(+104)&(+45)&(+113)&(+36)\\ \hline
AD &114 & 66& 75&105 \\
\bottomrule
\end{tabular}
\end{center}
\end{subtable}
\vspace*{0.5em}
\begin{subtable}[t]{0.9\textwidth}
\begin{center}
\caption{Clustering performance (excluded PD cases)}
\begin{tabular}{l|rrr|rrr|rr}
\toprule
\multirow{2}{*}{}&\multicolumn{3}{c|}{CN+Control} & \multicolumn{3}{c|}{AD}& \multirow{2}{*}{accuracy}&\multirow{2}{*}{ macro-F1} \\
& precision & recall & \multicolumn{1}{c|}{F1} & precision & recall & \multicolumn{1}{c|}{F1} & & \\ \hline
baseline & 71.36 & 98.95 & 82.92 & 95.65 & 36.67 & 53.01 & 74.9 & 68.0 \\
+PSS & 78.51 & 95.47 & 86.16 & 88.98 & 58.33 & 70.47 & 81.1 & 78.3 \\
\bottomrule
\end{tabular}
\end{center}
\end{subtable}
\vspace*{0.5em}
\begin{subtable}[t]{0.9\textwidth}
\begin{center}
\caption{Clustering performance (included PD cases)}
\begin{tabular}{l|rrr|rrr|rrr}
\toprule
\multirow{2}{*}{}&\multicolumn{3}{c|}{CN+Control} & \multicolumn{3}{c|}{AD}& \multirow{2}{*}{accuracy}&\multirow{2}{*}{ macro-F1} & Specificity\\
& precision & recall & F1 & precision & recall & F1 & & & of PD\\ \hline
baseline & 77.29 & 88.99 & 82.73 & 57.89 & 36.67 & 44.90 & 73.7 & 63.8 & 69.7 \\
+PSS & 83.77 & 88.76 & 86.19 & 68.18 & 58.33 & 62.87 & 79.9 & 74.5 & 75.8 \\
\bottomrule
\end{tabular}
\end{center}
\end{subtable}
\end{table*}
PSS improved the diagnostic performance by about 6.2\% (from 73.7 to 79.9\%) for micro-accuracy and about 10.7\% (from 63.8 to 74.5\%) for macro-F1. The specificity for PD was also improved by 6.1\% (from 69.7\% to 75.8\%).
\section{Discussion}
\label{sec:sec6}
\subsection{Changes on MR images by PSS}
Our PSS network transforms healthy cases taken with GE scanners to those taken with Siemens scanners.
As can be seen from Figure \ref{fig:fig6} and Table \ref{table:table2}, the amount of change in the images because of PSS was almost the same for both AD and CN images in the ADNI dataset, including the Philips case.
The amount of conversion of the image for the PPMI dataset was larger than that for the ADNI dataset. This is thought to be due to the process of absorbing the differences in the datasets that exist in the image but are invisible to the eye.
However, in all cases, the converted images have a natural appearance without destroying the brain structure.
This can be objectively confirmed in SSIM, which evaluates the structural similarity on the image, maintains a high value. As discussed in detail below, PSS can reduce disease-specific variation in the resulting low-dimensional embedding, absorb differences among datasets and scanner vendors, and improve the separability of diseases.
Given these factors, we can conclude that this PSS transformation was done properly.
\subsection{Contributions of DI-PSS for CBIR}
This section discusses the effects of our DI-PSS framework from the perspective of CBIR implementation.
\subsubsection{Distribution of embedding}
Based on the results in Tables \ref{table:table3} and \ref{table:table4}, we first discuss the effectiveness of the proposed DI-PSS.
From Table \ref{table:table3}, PSS reduces the inter-cluster variability for all data categories.
In particular, the SD of ADNI-CN and ADNI-AD, which are taken by scanners from three different companies in the same dataset, are reduced by 6.9\% and 6.1\%, respectively.
This indicates that the PSS reduces the difference caused by different scanners.
In addition, the SD of ALL\_CN, which is a combination of ADNI-CN and Control from a different PPMI dataset, is also reduced by 7.2\%, which clearly shows that the proposed PSS can absorb differences in datasets.
This benefit can also be seen in Figure \ref{fig:fig7}.
The reduction of PD variability by PSS is more pronounced ($-$14.7\%) than the others, and it is ultimately the category with the lowest variability. This is mentioned later in this section.
From Table \ref{table:table4}, PSS also succeeds in reducing the variability from each piece of data to all the different cluster centers (inter-cluster variability). What is noteworthy here is the degree of decrease in the standard deviation, which reached an average of 22.6\%.
This ability to reduce not only the variability of data in the same category, but also the directional variability up to different data categories is an important feature in CBIR.
In this experiment, we only built an image transformer (i.e. PSSnetwork) that converts CN\_GE to CN\_SI cases, but we could confirm that the harmonization is desirable for categories that are not included in the training in this way.
This strongly suggests that the strategy we have adopted -- that is, not having to build image harmonizers for all scanner types -- may have sufficient harmonization effects for many types of scanners.
Incidentally, the distances between PD and CN (ADNI-CN vs. PD and ALL-CN vs. PD) are closer than the distances between other categories. This supports the validity of the assumption we made in our experiment that PD and CN are outwardly indistinguishable, and therefore, they can be treated as the same class.
In contrast, if we look closely, we can see that the distances of the gravity centers between PD and CN (0.249$\rightarrow$0.269) and PD and Control (0.256$\rightarrow$0.297) are slightly increased by PSS, and Table \ref{table:table3} shows that the variation of PD is greatly reduced by PSS.
From this, we can say that the PSS is moving the PDs into smaller groups away from CN and Control.
This can be taken as an indication that the model trained by DI-PSS tends to consider PD as a different class that is potentially separated from the CN category.
Since the size of the dataset for this experiment was limited, we would like to run tests with a larger dataset in the future.
\subsubsection{Separability of the embedding for CBIR}
Thanks to the harmonization of scanners by PSS, the proposed DI-PSS not only reduces the variability of low-dimensional representations of each disease category, which could not be reduced by deep metric learning learning alone as adopted in DCMML \cite{Onga2019}, but also reduces the differences among datasets, resulting in a significant performance improvement in the clustering ability of low-point representations.
The PD data are different from the ADNI data used for training, and thus, it is an unknown dataset from our model. The improvement of clustering performance by the proposed DI-PSS for PD as well is an important and noteworthy result for the realization of CBIR.
\subsection{Validity of the model architecture}
The recently proposed data harmonization methods for brain MR images by Moyer et al \cite{Moyer2020} and Dinsdale et al. \cite{Dinsdale2021} have been reported to be not only logically justified but also very effective.
However, as mentioned above, these methods are difficult to apply to CBIR applications because images from all scanners are theoretically needed to train the model. Our DI-PSS is a new proposal to address these problems.
Although DI-PSS only learned the transformation from CN\_GE to CN\_Siemens, the improvement of the properties of the obtained embeddings was confirmed even for combinations that included other companies' scanners, such as the Philips scanner, and different disease categories (AD) that were not included in the training.
The results are evidence of proper data harmonization. We think this is due to the combination of MRICloud, an advanced skull stripping algorithm that performs geometric and volumetric positioning, and CycleGAN's generic style transformation capabilities and distance metric learning, which make up the PSS network.
Experiments with large-scale data from more diverse disease classes are needed, but in this experiment, we could confirm the possibility of obtaining effective scanner standardization by building one model that translates into a standard scanner.
\section*{Limitations of this study}
The number of data and diversity of their conditions used in these experiments are limited. There is also a limit to the number of diseases we considered. In the future, verification using more data is essential.
\section{Conclusion}
In this paper, we proposed a novel and effective MR image embedding method, DI-PSS, which is intended for application to CBIR. DI-PSS achieves data harmonization by transforming MR images to look like those captured with a predefined standard scanner, reducing the bias caused by variations in scanners and scan protocols, and obtaining a low-dimensional representation preserving disease-related anatomical features. The DI-PSS did not require training data that contained MRIs from all scanners and protocols; One set of image converters (i.e., CN\_GE to CN\_Siemens) was sufficient to train the model. In the future, we will continue the validation with more extensive and diverse data.
\section*{Acknowledgment}
This research was supported in part by the Ministry of Education, Science, Sports and Culture of Japan (JSPS KAKENHI), Grant-in-Aid for Scientific Research (C), 21K12656, 2021–2023.
\begin{footnotesize}
The MRI data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12–2-0012).
ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research \& Development, LLC.; Johnson \& Johnson Pharmaceutical Research \& Development LLC.; Lumosity; Lundbeck; Merck \& Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics.
The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California.
ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California.
An additional MRI data used in the preparation of this article were obtained from the Parkinson’s Progression Markers Initiative (PPMI) database (\url{www.ppmi-info.org/data}).
For up-to-date information on the study, visit www.ppmi-info.org. PPMI – a public-private partnership – is funded by the Michael J. Fox Foundation for Parkinson’s Research and funding partners, including AbbVie, Allergan, Avid Radiopharmaceuticals, Biogen, Biolegend, Bristol-Myers Squibb, Celgene, Denali, GE Healthcare, Genentech, GlaxoSmithKline, Lilly, Lundbeck, Merck, Meso Scale Discovery, Pfizer, Piramal, Prevail Therapeutics, Roche, Sanofi Genzyme, Servier, Takeda, Teva, UCB, Verily, Voyager Therapeutics, and Golub Capital.
\end{footnotesize}
\bibliographystyle{IEEEtran}
| {'timestamp': '2021-08-17T02:09:46', 'yymm': '2108', 'arxiv_id': '2108.06518', 'language': 'en', 'url': 'https://arxiv.org/abs/2108.06518'} |
\section{Method: additional details}
\subsection{Scale ambiguity in SFM}\label{s:sfm}
In Sec. 3.2 in the paper, we explain that the scale ambiguity of structure from motion (SFM) causes each reconstruction of a sequence $S^i$ to be known only up to
a global sequence specific scaling factor $\lambda^i$.
Since $\lambda^i$ is not required to learn \ensuremath{\Phi_\text{vp}}, but it is important for depth prediction (as discussed in Sec. 3.3 from the paper), we estimate it as well.
To do so, we note that, given a pair of frames $(t,t^\prime)$ from sequence $S^i$, one can estimate the sequence scale as
$
\lambda^i_{t,t^\prime}
=
\frac
{\| T^i_{t^\prime} - R^i_{t^\prime t} T^i_{t} \|}
{\| \hat T^i_{t^\prime} - R^i_{t^\prime t} \hat T^i_{t} \|}.
$
This expression allows us to conveniently estimate $\lambda^i$ on the fly as a moving average during the SGD iterations used to learn \ensuremath{\Phi_\text{vp}}, as samples $\lambda^i_{t,t^\prime}$ can be computed essentially for free during this process.
\subsection{The VpDR-Net\xspace architecture: further details}\label{s:arch}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figures/network-hc.pdf}
\caption{\textbf{The core architecture of VpDR-Net\xspace.} This figure extends the Viewpoint \& Depth estimation block from Figure 2 in the paper and describes the architecture of the hypercolumn (HC) module. \label{fig:overview-hc}}
\end{figure*}
This section contains additional details about the layers that compose the VpDR-Net\xspace architecture.
\myparagraph{The core architecture}
The architecture of the VpDR-Net\xspace (introduced in Sec. 3.2 from the paper) is a variant of the ResNet-50 architecture~\cite{he16resnet} with some modifications to improve its performance as a viewpoint and depth predictor that we detail below.
In order to decrease the degree of geometrical invariance of the network, we first replace all $1\times 1$ downsampling filters with full $2\times 2$ convolutions. We then attach bilinear upsampling layers that first resize features from 3 different layers of the architecture (res2d, res3d, res4d) into fixed-size tensors and then sum them in order to create a multiscale intermediate image representation which resembles hypercolumns (HC) \cite{hariharan2015hypercolumns}. An extension of Fig. 2 from the paper that contains the diagram of this HC module can be found in \Cref{fig:overview-hc}.
\myparagraph{Architecture of the viewpoint factorization network \ensuremath{\Phi_\text{vp}}}
HC is followed by 3 modified $3\times 3$ downsampling residual layers that produce the final viewpoint prediction. While the standard downsampling residual layers do not contain the residual skip connection due to different sizes of the input and output tensors, here we retain the skip connection by performing $3\times 3$ average pooling over the input tensor and summing the result with the result of the second $3\times 3$ downsampling convolution branch. We further remove the ReLU after the final residual summation layer. \Cref{fig:overview-vp} contains an overview of the viewpoint estimation module together with a detailed illustration of the modified downsampling residual blocks.
\myparagraph{Architecture of the depth prediction \ensuremath{\Phi_\text{depth}}}
The depth prediction network (introduced in Sec. 3.3 from the paper) shares the early HC layers with the viewpoint factorization network \ensuremath{\Phi_\text{vp}}. The remainder of the pipeline is based on the state-of-the-art depth estimation method of \cite{laina2016deeper}. More precisely, after attaching 2 standard residual blocks to the HC layers, the network also contains
two 2x2 up-projection layers from \cite{laina2016deeper} leading to a 64-dimensional representation of the same size as the input image. This is followed by 1x1 convolutional filters that predict the depth and confidence maps $\hat D_t$ and $\hat \sigma_{d_j}$ respectively.
\Cref{fig:overview-depth} contains an illustration of \ensuremath{\Phi_\text{depth}}.
\myparagraph{Architecture of the point cloud completion network \ensuremath{\Phi_\text{pcl}}}
Differently from the two previous networks, the point cloud completion network $\ensuremath{\Phi_\text{pcl}}$ (introduced in Sec. 3.4 from the paper) is not convolutional but uses a residual multi-layer perceptron (MLP), \ie a sequence of residual fully connected layers.
In more details, the network starts by appending to each 3D point $\hat p_i \in \hat P^G_f \subset \mathbb{R}^3$ an appearance descriptor $a_i$ and processes
this input with an MLP with an intermediate pooling operator:
$$
(\hat S, \hat \delta) =
\ensuremath{\Phi_\text{pcl}}(\hat P^G_f) =
\operatorname{MLP}_2
\left(
\operatornamewithlimits{pool}_{1\leq i \leq |\hat P^G_f|}
\operatorname{MLP}_1(\hat p_i, a_i)
\right).
$$
The intermediate pooling operator, which is permutation invariant, removes the dependency on the number and order of input points $\hat P_f^G$. In practice, the pooling operator uses both max and sum pooling, stacking the results of the two.
For the appearance descriptors, recall that each point $\hat p_i$ is the back-projection of a certain pixel $(u_i,v_i)$ in image $f$. To obtain the appearance descriptor $a_i$ we reuse the HC features from the core architecture and sample a column of feature channels at location $(u_i,v_i)$ using differentiable bilinear sampling. Note that, following \cite{tatarchenko16multi}, the fully connected residual blocks contain leaky-ReLUs with the leak factor set to 0.2. A diagram depicting \ensuremath{\Phi_\text{pcl}}{} can be found in \Cref{fig:overview-pcl}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/figureB_supp.pdf}
\caption{\textbf{The architecture of \ensuremath{\Phi_\text{depth}}.} \label{fig:overview-depth}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/figureC_supp.pdf}
\caption{\textbf{The architecture of \ensuremath{\Phi_\text{vp}}.} Top: the layers of \ensuremath{\Phi_\text{vp}}, bottom: A detail of the 3x3 downsampling residual block.\label{fig:overview-vp}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{figures/figureD_supp.pdf}
\caption{\textbf{The architecture of \ensuremath{\Phi_\text{pcl}}.} Top: The overview of the point cloud completion network, bottom: A detail of the fully connected residual block.
Orange boxes denote the sizes of the layer outputs. \label{fig:overview-pcl}}
\end{figure}
\section{Experimental evaluation}
In this section we provide additional details about the learning procedures of the baseline networks
and about the experimental evaluation.
\subsection{Learning details of BerHu-Net\xspace and VPNet}
In this section we provide learning details for the BerHu-Net and VPNet baselines.
The learning rates and batch sizes were in all cases adjusted empirically such that the convergence is achieved on the
respective training sets.
\textbf{BerHu-Net\xspace} is trained with stochastic gradient descent with a momentum of 0.0005, initial learning rate $10^{-3}$ and a batch size of 16.
The learning rate was lowered tenfold when no further improvement
in the training losses was observed. The BerHu loss
uses the adaptive adjustment of the loss cut-off threshold as explained in \cite{laina2016deeper}.
For the 2x2 up-projection layers we used the implementation of \cite{laina2016deeper}.
For each test image, we repeat the depth map extraction 70 times\footnote{We empirically verified that 70 repetitions are enough for convergence of the variance estimates.}
with the dropout layer turned on and compute the variance of the predictions in order to obtain the per-pixel depth confidence values.
The final feed-forward pass turns off the dropout layer and produces the actual depth predictions.
\textbf{VPNet} is trained with stochastic gradient descent with a momentum of 0.0005, initial learning rate $10^{-2}$ and a batch size of 128.
The learning rate was lowered tenfold when no further improvement
in the training losses was observed.
For VPNet trained on aligned FrC, we adjusted the produced bounding box and viewpoint annotations in the same fashion as
done for adjusting the Pascal3D annotations in sec. 5.1. in the paper,
ensuring that the aligned FrC dataset is as compatible as possible with the target Pascal3D dataset.
For LDOS, the produced dataset was adjusted in the same way except that we did not use the bounding boxes predicted by \cite{sedaghat15unsupervised}
because the input video frames already focus on full/truncated views of the object category.
\subsection{Additional results}
In sec. 5.1. in the paper we compared VpDR-Net\xspace to
\cite{sedaghat15unsupervised} on an adjusted version of the Pascal3D dataset. In this section, we additionally report the standard
AVP measure \cite{xiang2014beyond} on the original Pascal3D dataset in order to present a better comparison
with fully supervised state-of-the-art on this dataset. Because the AVP measure requires an object detector,
we extract viewpoints from the same set of RCNN detections as in \cite{tulsiani2015viewpoints}.
Due to the fact that the AVP measure, as well as most other measures from sec. 5.1. in the paper, depends on the dataset-specific
global alignment transformation $\mathcal{T}_G$, we estimate it from the ground truth annotations
of the training set of \cite{xiang2014beyond} using the same method as described in sec. 5.1. in the paper.
Due to the additional measurement noise brought by the estimation of $\mathcal{T}_G$, we report results only
for the coarsest resolution of 4 azimuth bins. Our VpDR-Net\xspace obtained 33.4 and 14.7 AVP for the car and chair classes
vs. 29.4 and 14.3 AVP obtained by \cite{sedaghat15unsupervised} using the same detections from \cite{tulsiani2015viewpoints}.
Our approach performs on par with some fully supervised approaches such as 3D DPM \cite{pepik2012teaching},
while being inferior to the fully supervised state-of-the-art by the same margin as for the other metrics reported in table 1 in the paper.
\subsection{Absolute pose evaluation protocol}
As noted in the paper, the absolute pose error metrics $e_R$ and $e_C$
can be computed only after aligning the implicit global coordinate frames of the benchmarked network
and of the ground truth annotations. This procedure is explained in detail below.
Given a set of ground truth camera poses $g_i^\ast\ = (R_i^\ast,T_i^\ast)$
and the corresponding predictions $\hat g_i = ( \hat R_i,\hat T_i)$,
we want to estimate a global similarity transform $\mathcal{T}_G = (R_G,T_G,s_G)$, parametrized by a
scale $s_G \in \mathbb{R}$, translation $T_G \in \mathbb{R}^3$ and rotation $R_G \in SO(3)$, such that
the coordinate frames of $g_i^\ast$ and $\hat g_i$ become aligned.
In more detail, the desired global similarity transform
satisfies the following equation:
\begin{equation}
\hat R_i ( R_G X + T_G ) + s_G \hat T_i = R_i^\ast X + T_i^\ast ~ ; ~ \forall X
\label{eq:globadjust}
\end{equation}
\ie given an arbitrary world-coordinate point $X \in \mathbb{R}^3$, its projection into the coordinate frame of $g_i^\ast$ (the right part of \cref{eq:globadjust})
should be equal to the projection of $X$ into the coordinate frame of $\hat g_i$ after transforming $X$ with $R_G$, $T_G$ and
scaling the corresponding camera translation vector $\hat T_i$ with $s_G$ (the left side of \cref{eq:globadjust}).
Note that for LDOS data $\mathcal{T}_G$ corresponds to a rigid motion and $s_G=1$.
Given $\mathcal{T}_G$, the adjusted camera matrices $\hat g_i^{ADJUST}$ for which $\hat g_i^{ADJUST} \approx g_i^\ast$
are then computed with
$$
\hat g_i^{ADJUST} = ( ~~ \hat R_i R_G ~~ , ~~\hat R_i T_G + s_G \hat T_i ~~ )
$$
In order to estimate $\mathcal{T}_G$, $X$ is substituted in \cref{eq:globadjust} with $X = C_i^\ast = -{R_i^\ast}^T T_i^\ast$,
\ie $X$ is set to be the center of the ground truth camera $g_i^\ast$ which is a valid point of the world coordinate frame.
After performing some additional manipulations, we end up with the following constraint:
\begin{equation}
\frac{1}{s_G} R_G C_i^\ast + \frac{1}{s_G} T_G = \hat C_i ~ ; ~ \forall i
\label{eq:umeyama}
\end{equation}
where $\hat C_i = - \hat R_i^T \hat T_i$ is the center of the predicted camera $\hat g_i$.
Given the corresponding camera pairs $ \{ (g_i^\ast,\hat g_i) \}_{i=1}^N $
the constraint in \cref{eq:umeyama} is converted to a least squares minimization problem:
\begin{equation}
\arg\min_{R_G,T_G,s_G} \sum_{i=1}^N || \frac{1}{s_G} R_G C_i + \frac{1}{s_G} T_G - \hat C_i ||^2
\label{eq:umeyama2}
\end{equation}
and solved using the UMEYAMA algorithm \cite{umeyama1991least}.
For Pascal3D we estimate $\mathcal{T}_G$ from the held-out training set
and later use it for evaluation on the test set.
For LDOS, due to the absence of a held-out annotated training set,
we estimate $\mathcal{T}_G$ on the test set.
\subsection{Point cloud prediction}
The normalized point cloud distance of~\cite{rock2015completing} is computed as
$
D_\text{pcl}(C,\hat C) =
\frac{1}{|C|}
\sum_{c \in C}
\min_{\hat c \in \hat C}
\| \hat c - c \|
+
\frac{1}{|\hat C|}
\sum_{\hat c \in \hat C}
\min_{c \in C} \|\hat c - c\|.
$
For the VIoU measure, a voxel grid is setup around each ground truth point-cloud $C$ by uniformly subdividing $C$'s bounding volume into $30^3$ voxels.
The point clouds are compared within the local coordinate frames of each frame's camera (whose focal length is assumed to be known). Furthermore, since the
SFM reconstructions are known only up to a global scaling factor, we adjust each point cloud prediction $\hat C$ from the FrC dataset by multiplying it with
a scaling factor $\zeta$ that aligns the means of $\hat C$ and $C$. Note that $\zeta$ can be computed analytically with:
$$
\zeta = \frac{\mu_{C}^T \mu_{\hat C} } { \mu_{\hat C}^T \mu_{\hat C} },
$$
where $\mu_{C} = \frac{1}{|C|} \sum_{c_m \in C} c_m $ is the centroid of the point cloud $C$.
\myparagraph{Ablative study} In table 2 in the paper, we have presented a comparison of VpDR-Net\xspace to the baseline approach from \cite{aubry14seeing}.
Here we provide an additional ablative study that evaluates the contribution of the components of $\ensuremath{\Phi_\text{pcl}}$.
More exactly, \cref{tab:completion} extends table 2 from the paper with the following flavours of VpDR-Net\xspace:
(1) VpDR-Net\xspace-$\hat P_f$ which only predicts the partial point cloud $P_f$,
(2) VpDR-Net\xspace-Chamfer which removes the density predictions $\hat \delta$ and replaces $l_{pcl}(\hat S)$ with a Chamfer distance loss and
(3) VpDR-Net\xspace-$\hat S$ that predicts the raw unfiltered and untruncated point cloud $\hat S$.
The drops in performance by predicting solely the raw and partial point clouds $\hat P_f$ and $\hat S$ emphasize the importance of the point cloud completion and density prediction components respectively. The Chamfer distance loss brings marginal improvements in $D_{pcl}$ but a significant decrease of VIoU due to the inability of the network to represent and discard outliers.
\begin{table}
\centering
\footnotesize
\setlength\tabcolsep{2pt}
\begin{tabular}{lrrrr}
\toprule
Test set & \multicolumn{2}{c}{\textbf{LDOS}} & \multicolumn{2}{c}{\textbf{FrC}} \\
\midrule
Metric & $\uparrow$ mVIoU & $\downarrow$ m$D_{pcl}$ & $\uparrow$ mVIoU & $\downarrow$ m$D_{pcl}$ \\
\midrule
Aubry \cite{aubry14seeing} & 0.06 & 1.30 & 0.21 & 0.41 \\
VpDR-Net\xspace-$\hat P_f$ & 0.10 & 0.37 & 0.11 & 0.56 \\
VpDR-Net\xspace-Chamfer & 0.09 & \textbf{0.18} & 0.20 &\textbf{0.24} \\
VpDR-Net\xspace-$\hat S$ & 0.12 & 0.27 & 0.18 & 0.50 \\
\textbf{VpDR-Net\xspace (ours) } & \textbf{0.13} & 0.20 & 0.24 & 0.28 \\
\textbf{VpDR-Net\xspace-Fuse (ours)} & \textbf{0.13} & 0.19 & \textbf{0.26} & 0.26 \\
\bottomrule
\end{tabular}
\caption{ \textbf{Point cloud prediction ablative study}.
Comparison between VpDR-Net\xspace and the method of Aubry \etal \cite{aubry14seeing} and an additional ablative study.}
\label{tab:completion}
\vspace{-2em}
\end{table}
\myparagraph{Related methods} Note that apart from \cite{aubry14seeing}, there exist newer works that tackle the problem of single-view
3D reconstruction \cite{huang2015single,massa2016deep}, however these were not considered due to their requirement of renderable
mesh models which are not available in our supervision setting.
\section{Conclusion}\label{s:conclusions}
We have demonstrated the power of motion cues in replacing manual annotations and synthetic data in learning 3D object categories. We have done so by proposing a single neural network that simultaneously performs monocular viewpoint estimation, depth estimation, and shape reconstruction. This network is based on two innovations, a new image-based viewpoint factorization method and a new probabilistic shape representation. The contribution of each component was assessed against suitable baselines.
\section{Experiments}\label{s:exp}
We assess viewpoint estimation in~\cref{s:exp-pose}, depth prediction in~\cref{s:exp-depth}, and point cloud prediction in~\cref{s:exp-pcc}.
\myparagraph{Datasets} Throughout the experimental section, we consider three datasets for training and benchmarking our network: (1) \textbf{FreiburgCars (FrC)}~\cite{sedaghat15unsupervised} which consists of RGB video sequences with the camera circling around various types of cars; (2) the \textbf{Large Dataset of Object Scans (LDOS)}~\cite{choi2016large} containing RGBD sequences of man-made objects; and (3) \textbf{Pascal3D}~\cite{xiang2014beyond}, a standard benchmark for pose estimation~\cite{tulsiani2015viewpoints,su2015render}.
For viewpoint estimation, Pascal3D already contains viewpoint annotations. For LDOS, experiments focus on the \textit{chair} class. In order to generate ground truth pose annotations for evaluation, we manually aligned 3D reconstructions of 10 randomly-selected chair videos and used 50 randomly-selected frames for each video as a test set.
For depth estimation, we evaluate on LDOS as it provides high quality depth maps one can use as ground truth.
For point cloud reconstruction, we use FrC and LDOS. Ground truth point clouds for evaluation are obtained by merging the SFM or RGBD depth maps from all frames of a given test video sequence, sampling $3\cdot10^4$ points and post-processing those using a 3D Laplacian filter. For FrC, five videos were randomly selected and removed from the train set, picking 60 random frames per video for evaluation. For LDOS the pose estimation test frames are used.
\myparagraph{Learning details} VpDR-Net\xspace is trained with stochastic gradient descent with a momentum of 0.0005 and an initial learning rate of $10^{-2}$. The weights of the losses were empirically set to achieve convergence on the training set.
Better convergence was observed by training VpDR-Net\xspace in two stages. First, $\ensuremath{\Phi_\text{depth}}$ and $\ensuremath{\Phi_\text{vp}}$ were optimized jointly, lowering the learning rate tenfold when no further improvement in the training losses was observed. Then, $\ensuremath{\Phi_\text{pcl}}$ is optimized after initializing the bias of its last layer, which corresponds to an average point cloud of the object category, by randomly sampling points from the ground truth models.
\subsection{Pose estimation}\label{s:exp-pose}
\myparagraph{Pascal3D} First, we evaluate the VpDR-Net\xspace viewpoint predictor on the Pascal3D benchmark \cite{xiang2014beyond}. Unlike previous works \cite{su2015render,tulsiani2015viewpoints} that focus on estimating the object/camera viewpoint represented by a 3 DoF rotation matrix, we evaluate the full 6 DoF camera pose represented by the rotation matrix $R$ together with the translation vector $T$.
In Pascal3D, the camera poses are expressed relatively to the whole scenes instead of the objects themselves, so we adjust the dataset annotations.
We crop every object using bounding box annotations after reshaping the box to a fixed aspect ratio, and resize the crop to $240\times 320$ pixels. The camera pose is adjusted to the cropped object using the P3P algorithm to minimize the reprojection error between the camera-projected vertices of the ground truth CAD model and the original projection after cropping and resizing.
\myparagraph{Absolute pose evaluation} We first evaluate absolute camera pose estimation using two standard measures: the angular error
$
e_R = 2^{-\frac{1}{2}}
\|\ln R^\ast \hat R^\top \|_F
$
between the ground truth camera pose $R^\ast$ and the prediction $\hat R$ \cite{tulsiani2015viewpoints,su2015render}, as well as the camera-center distance
$
e_C = \|\hat C - C^\ast\|_2
$
between the predicted camera center $\hat C$ and the ground truth $C^\ast$. Following the common practice \cite{tulsiani2015viewpoints,su2015render} we report median $e_R$ and $e_C$ over all pose predictions on each test set.
Note that, while object viewpoints in Pascal3D and our method are internally consistent for a whole category, they may still differ between them by an arbitrary global 3D similarity transformation. Thus, as detailed in the supplementary material, the two sets of annotations are aligned by a single global similarity $\mathcal{T}_G$ before assessment.
\myparagraph{Relative pose evaluation} To assess methods with measures independent of $\mathcal{T}_G$ we also evaluate: (1) the relative rotation error between pairs of ground truth relative camera motions $R_{t t^\prime}^\ast$ and the corresponding predicted relative motions $\hat R_{t t^\prime}$ given by
$
e_R^\text{rel} =
2^{-\frac{1}{2}}
\|\ln
R_{t t^\prime}^\ast
\hat R_{t t^\prime}^\top
\|_F
$
and (2) the normalized relative translation error
$
e_T^\text{rel}
= \|
\hat T_{t t^\prime} - T_{t t^\prime}^\ast
\|_2
$%
, where both $\hat T_{t t^\prime}$ and $T_{ t t^\prime}^\ast$ are $\ell_2$-normalized so the measure is invariant to the scaling component of $\mathcal{T}_G$. We report the median errors over all possible image pairs in each test set.
\begin{figure}[t]
\centering
\pbox{\textwidth}{
\includegraphics [width=0.47\linewidth] {figures/depthRMSDepthSortchart.pdf}
\includegraphics [width=0.48\linewidth] {figures/depthRMSConfSortchart.pdf}\\
\includegraphics [width=1\linewidth] {figures/depthLegend.pdf}
}
\caption{\textbf{Monocular depth prediction.} Cumulative RMS depth reconstruction error for the LDOS data, when pixels are ranked by ground truth depth (left) and by confidence (right).}
\label{fig:depthest}
\end{figure}
\begin{figure}[t]
\newcommand{0.84cm}{0.84cm}
\centering
\includegraphics[height=0.84cm]{figures/depth/car/00002_image_small.png}
\includegraphics[height=0.84cm]{figures/depth/car/00001_image_small.png}
\includegraphics[height=0.84cm]{figures/depth/car/00006_image_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_image_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_image_2_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_image_3_small.png}\\
\includegraphics[height=0.84cm]{figures/depth/car/00002_depth_pred_small.png}
\includegraphics[height=0.84cm]{figures/depth/car/00001_depth_pred_small.png}
\includegraphics[height=0.84cm]{figures/depth/car/00006_depth_pred_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_depth_pred_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_depth_pred_2_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_depth_pred_3_small.png}\\
\includegraphics[height=0.84cm]{figures/depth/car/00002_depth_confidence_small.png}
\includegraphics[height=0.84cm]{figures/depth/car/00001_depth_confidence_small.png}
\includegraphics[height=0.84cm]{figures/depth/car/00006_depth_confidence_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_depth_confidence_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_depth_confidence_2_small.png}
\includegraphics[height=0.84cm]{figures/depth/chair/00001_depth_confidence_3_small.png}\\
\caption{\textbf{Monocular depth prediction.} Top: input image; middle: predicted depth; bottom: predicted depth confidence.
Depth maps are filtered by removing low confidence pixels.
\label{fig:depth_qual}}
\vspace{-1em}
\end{figure}
\myparagraph{Pose prediction confidence evaluation} A feature of our model is to produce confidence scores with its viewpoint estimates. We evaluate the reliability of these scores by correlating them with viewpoint prediction accuracy. In order to do so, predictions are divided into ``accurate'' and ``inaccurate'' by comparing their errors $e_R$ and $e_C$ to thresholds (set to $e_R=\frac{\pi}{6}$ following~\cite{su2015render,tulsiani2015viewpoints} and $e_C=15$ and $0.5$ for Pascal3D or LDOS respectively). Predictions are then ranked by decreasing confidence scores and the average precisions $AP_{e_R}$ and $AP_{e_C}$ of the two ranked lists are computed.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figures/pcls.pdf}
\caption{\textbf{Point cloud prediction.} From a single input image of an unseen object instance (top row), VpDR-Net\xspace predicts the 3D geometry of that instance in the form of a 3D point cloud (seen from two different angles, middle and bottom rows).\label{fig:pcl_qual}}
\vspace{-1em}
\end{figure*}
\myparagraph{Baselines} We compare our viewpoint predictor to a strong baseline, called~\textbf{VPNet}, trained using absolute viewpoint labels. VPNet is a ResNet50 architecture \cite{he16resnet} with the final softmax classifier replaced by a viewpoint estimation layer that predicts the 6 DoF pose $\hat g_t^i$. Following \cite{tulsiani2015viewpoints}, rotation matrices are decomposed in Euler angles, each discretized in 24 equal bins.
This network is trained to predict a softmax distribution over the angular bins and to regress a 3D vector corresponding to the camera translation $T$. The average softmax value across the three max-scoring Euler angles is used as a prediction confidence score.
We test both an unsupervised and a fully-supervised variant of VPNet. VPNet-unsupervised is comparable to our setting and is trained on the output of the global camera poses estimated from the videos by the state-of-the-art sequence-alignment method of \cite{sedaghat15unsupervised}. In the fully-supervised setting, VPNet is trained instead by using ground-truth global camera poses provided by the Pascal3D training set.
\myparagraph{Results} \Cref{tab:posest} compares VpDR-Net\xspace to the VPNet baselines. First, we observe that our baseline VPNet-unsupervised is very strong, as we report $e_R=49.6$ error for the full rotation matrix, while the original method of \cite{sedaghat15unsupervised} reports an error of 61.5 just for the azimuth component. Nevertheless, VpDR-Net\xspace outperforms VPNet in all performance metrics except for a single case ($e_R$ for LDOS chairs). Furthermore, the advantage is generally substantial, and the unsupervised VpDR-Net\xspace reduces the gap with fully-supervised VPNet by 20 \% or better in the vast majority of the cases. This shows the advantage of the proposed viewpoint factorization method compared to aligning 3D shapes as in~\cite{sedaghat15unsupervised}. Second, we observe that the confidence scores estimated by VpDR-Net\xspace are significantly more correlated with the accuracy of the predictions than the softmax scores in VPNet, providing a reliable self-assessment mechanism. The most confident viewpoint predictions of VpDR-Net\xspace are shown in \cref{fig:vp_qual}.
\myparagraph{Ablation study} We evaluate the importance of the different components of VpDR-Net\xspace by turning them off and measuring performance on the \textit{chair} class. In \cref{tab:ablation}, \textbf{VpDR-Net\xspace{}-NoProb} replaces the robust probabilistic losses $\mathcal{L}_R$ and $\mathcal{L}_T$ with their non-probabilistic counterparts $\ell_R$ and $\ell_T$, and confidence predictions are replaced with random scores for AP evaluation. \textbf{VpDR-Net\xspace{}-NoDepth} removes the depth prediction and point cloud prediction branches during training, retaining only the $\ensuremath{\Phi_\text{vp}}$ subnetwork. \textbf{VpDR-Net\xspace{}-NoAug} does not use the data augmentation mechanism of \cref{s:augmentation}.
We observe a significant performance drop when each of the components is removed. This confirms the importance of all contributions in the network design.
Interestingly, we observe that the depth prediction branch $\ensuremath{\Phi_\text{depth}}$ is crucial for pose estimation (\eg~-34.27 $e_R$ on LDOS).
\input{tab_completion}
\subsection{Depth prediction}\label{s:exp-depth}
The monocular depth prediction module of VpDR-Net\xspace is compared against three baselines: \textbf{VpDR-Net\xspace-Rand} uses VpDR-Net\xspace to estimate depth but predicts random confidence scores. \textbf{BerHu-Net\xspace} is a variant of the state-of-the-art depth prediction network from \cite{laina2016deeper} based on the same $\ensuremath{\Phi_\text{depth}}$ subnetwork as VpDR-Net\xspace (but dropping $\ensuremath{\Phi_\text{pcl}}$ and $\ensuremath{\Phi_\text{vp}}$). Following \cite{laina2016deeper}, for training it uses the BerHu depth loss and a dropout layer, which allows it to produce a confidence score of the depth measurements at test time using the sampling technique of~\cite{kendall2015bayesian,gal2016Bayesian}. Finally, \textbf{BerHu-Net\xspace-Rand} is the same network, but predicting random confidence scores.
\myparagraph{Results} Fig.~\ref{fig:depthest} (right) shows the cumulative root-mean-squared (RMS) depth reconstruction error for LDOS after sorting pixels by their confidence as estimated by the network. By fitting better to inlier pixels and giving up on outliers, VpDR-Net\xspace produces a much better estimate than alternatives for the vast majority of pixels. Furthermore, accuracy is well predicted by the confidence scores. Fig.~\ref{fig:depthest} (left) shows the cumulative RMS by depth, demonstrating that accuracy is better for pixels closer to the camera, which are more likely to be labeled with correct depth. Qualitative results are shown in \cref{fig:depth_qual}.
\subsection{Point cloud prediction}\label{s:exp-pcc}
We evaluate the point cloud completion module of VpDR-Net\xspace by comparing ground truth point clouds $C$ to the point clouds $\hat C$ predicted by $\ensuremath{\Phi_\text{pcl}}$ using: (1) the voxel intersection-over-union (VIoU) measure that computes the Jaccard similarity between the volumetric representations of $\hat C$ and $C$, and (2) the normalized point cloud distance of~\cite{rock2015completing}. We average these measures over the test set leading to mVIoU and m$D_{pcl}$ (see supp. material for details).
VpDR-Net\xspace is compared against the approach of Aubry~\etal~\cite{aubry14seeing} using their code. \cite{aubry14seeing}
is a 3D CAD model retrieval method which first trains a large number of exemplar models which, in our case, are represented by individual video frames with their corresponding ground truth 3D point clouds.
Then, given a testing image, \cite{aubry14seeing} detects the object instance
and retrieves the best matching model from the database. We align the retrieved point cloud to the object location in the testing image using the P3P algorithm.
For VpDR-Net\xspace, we evaluate two flavors. The original VpDR-Net\xspace that predicts the point cloud $\hat C$ and VpDR-Net\xspace-Fuse which further merges $\hat C$ with the predicted partial depth map point cloud $\hat P$.
\Cref{tab:completion} shows that our reconstructions are significantly better on both metrics for both LDOS chairs and FrC cars. Fusing the results with the original depth map produces a denser point cloud estimate and marginally improves the results.
Qualitative results are shown in \cref{fig:pcl_qual}.
\section{Introduction}
Despite their tremendous effectiveness in tasks such as object category detection, most deep neural networks do not understand the 3D nature of object categories. Reasoning about objects in 3D is necessary in many applications, for physical reasoning, or to understand the geometric relationships between different objects or scene elements.
The typical approach to learn 3D objects is to make use of large collections of high quality CAD models such as \cite{shapenet2015} or \cite{xiang2016objectnet3d}, which can be used to fully supervise models to recognize the objects' viewpoint and 3D shape. Alternatively, one can start from standard image datasets such as PASCAL VOC
\cite{Everingham10}, augmented with other types of supervision, such as object segmentations and keypoint annotations~\cite{carreira16lifting}. Whether synthetically generated or manually collected, annotations have so far been required in order to overcome the significant challenges of learning 3D object categories, where both viewpoint and geometry are variable.
In this paper, we develop an alternative approach that can learn 3D object categories in an \emph{unsupervised manner} (\cref{fig:front}), replacing synthetic or manual supervision with \emph{motion}. Humans learn about the visual word by experiencing it continuously, through a variable viewpoint, which provides very strong cues on its 3D structure. Our goal is to build on such cues in order to learn the 3D geometry of object categories, using videos rather than images of objects. We are motivated by the fact that videos are almost as cheap as images to capture, and do not require annotations
\begin{figure}[t]
\vspace{-1em}
\includegraphics[width=\linewidth,trim=0 1em 0 1em]{figures/splash/splashv2.pdf}
\caption{We propose a convolutional neural network architecture to learn the 3D geometry of object categories from videos only, without manual supervision. Once learned, the network can predict i)~viewpoint, ii) depth, and iii) a point cloud, all from a single image of a new object instance.\label{fig:front}}
\end{figure}
\begin{figure*}
\includegraphics[width=\linewidth]{figures/network.png}
\caption{\textbf{Overview of our architecture.} As a preprocessing, structure from motion (SFM) extracts egomotion and a depth map for every frame. For training, our architecture takes pairs of frames $f_t$, $f_{t'}$ and produces a viewpoint estimate, a depth estimate, and a 3D geometry estimate. At test time, viewpoint, depth, and 3D geometry are predicted from single images.\label{fig:overview}}
\end{figure*}
We build on mature structure-from-motion (SFM) technology to extract 3D information from individual video sequences. However, these cues are specific to each object instance as contained in different videos. The challenge is to integrate this information in a global 3D model of the object category, as well as to work with noisy and incomplete reconstructions from SFM.
We propose a new deep architecture composed of three modules (\cref{fig:overview}). The first module estimates the \emph{absolute viewpoint} of objects in all video sequences (\cref{s:viewpoint}). This aligns different object instances to a common reference frame where geometric relationships can be modeled more easily. The second estimates the 3D shape of an object from a given viewpoint, producing a \emph{depth map} (\cref{s:depth}). The third \emph{completes the depth map to a full 3D reconstruction} in the globally-aligned reference frame (\cref{s:recon}).
Combined and trained end-to-end without supervision, from videos alone, these components constitute VpDR-Net\xspace,
a network for \textbf{v}iew\textbf{p}oint, \textbf{d}epth and \textbf{r}econstruction,
capable of extracting viewpoint and shape of a new object instance from a single image.
One of our main contributions is thus to demonstrate the utility of using motion cues in learning 3D categories. We also introduce two significant technical innovations in the viewpoint and shape estimation modules as well as design guidelines and training strategies for 3D estimation tasks.
The first innovation (\cref{s:viewpoint}) is a new approach to align video sequences of different 3D objects based on a \emph{Siamese viewpoint factorization network}. While existing methods~\cite{sun09multi,sedaghat15unsupervised} align shapes by looking at 3D features, we propose to train VpDR-Net\xspace to directly estimate the absolute viewpoint of an object. We train our network to reconstruct \emph{relative camera motions} and we show that this implicitly aligns different objects instances together. By avoiding explicit shape comparisons in 3D space, this method is simpler and more robust than alternatives.
The second innovation (\cref{s:recon}) is a new network architecture that can generate a complete point cloud for the object from a partial reconstruction obtained from monocular depth estimation. This is based on a shape representation that predicts the support of a point probability distribution in 3D space, akin to a flexible voxelization, and a corresponding space occupancy map.
As a general design guideline, we demonstrate throughout the paper the utility of allowing deep networks to \emph{express uncertainty} in their estimate by predicting probability distributions over outputs (\cref{s:method}), yielding more robust training and useful cues (such as separating foreground and background in a depth map). We also demonstrate the significant power of \emph{geometry-aware data augmentation}, where a deep network is used to predict the geometry of an image and the latter is used to generate new realistic views to train other components of the system (\cref{s:augmentation}). Each component and design choice is thoroughly evaluated in~\cref{s:exp}, with significant improvements over the state-of-the-art.
\section{Geometry-aware data augmentation}\label{s:augmentation}
As viewpoint prediction with deep networks benefits significantly from large training sets~\cite{su2015render}, we increase the effective size of the training videos by \emph{data augmentation}. This is trivial for tasks such as classification, where one can translate or scale an image without changing its identity. The same is true for viewpoint recognition if the task is to only estimate the viewpoint orientation as in~\cite{su2015render,tulsiani2015viewpoints}, as images can be scaled and translated without changing the equivalent viewpoint orientation. However, this assumption is not satisfied if, as in our case, the goal is to estimate all 6 DoF of the camera pose.
Inspired by the approach of~\cite{gupta2016synthetic}, we propose to solve this problem by using the estimated scene geometry to \emph{generate new realistic viewpoints} (\cref{f:aug}). Given a sample $(f^i_t, g^i_t, D^i_t)$, we apply a random perturbation to the viewpoint (with a forward bias to avoid unoccluding too many pixels) and use depth-image-based rendering (DIBR)~\cite{morvan2009acquisition} to generate a new sample $(f^i_*, g^i_*, D^i_*)$, warping both the image and the depth map.
Sometimes the depth map $D^i_t$ from KF contains too many holes to yield satisfactory DIBR results (\cref{f:aug}, bottom); we found preferable to use the depth $\hat D^i_t = \ensuremath{\Phi_\text{depth}}(f_t)$ estimated by the network which is less accurate but more robust, containing almost no missing pixels (\cref{f:aug}, top).
\section{Method}\label{s:method}
We propose a single Convolutional Neural Network (CNN), VpDR-Net\xspace, that learns a \emph{3D object category} by observing it from a~\emph{variable viewpoint} in videos and no supervision (\cref{fig:overview}). Videos do not solve the problem of modeling intra-class shape variations, but they provide powerful yet noisy cues about the 3D shape of individual objects.
VpDR-Net\xspace takes as an input a set of $K$ video sequences $S^1, ..., S^K$ of an object category (such as cars or chairs), where a video $S^i = (f_1^i, ... , f_{N^i}^i)$ contains $N^i$ RGB or RGBD frames $f_t^i \in \mathbb{R}^{H \times W \times \mathcal{C} }$ (where $\mathcal{C}=3$ for RGB and $\mathcal{C}=4$ for RGBD data)
and learns a model of the 3D category. This model has three components: i) a predictor $\ensuremath{\Phi_\text{vp}}(f_i^t)$ of the \emph{absolute viewpoint} of the object (implicitly aligning the different object instances to a common reference frame; \cref{s:viewpoint}), ii) a \textit{monocular depth} predictor $\ensuremath{\Phi_\text{depth}}(f_i^t)$
(\cref{s:depth}) and iii) and a \textit{shape} predictor $\ensuremath{\Phi_\text{pcl}}(f_i^t)$ that extends the depth map to a point cloud capturing the complete shape of the object (\cref{s:recon}). Learning starts by preprocessing videos to extract instance-specific egomotion and shape information (\cref{s:sfm}).
\subsection{Sequence-specific structure and pose}\label{s:sfm}
Video sequences are pre-processed to extract from each frame $f_t^i$ a tuple $(K_t^i,g_t^i,D_t^i)$ consisting of: (i) the camera calibration parameters $K_t^i$, (ii) its pose $g_t^i\in SE(3)$, and (iii) a depth map $D_t^i\in \mathbb{R}^{H \times W}$ associating a depth value to each pixel of $f_t^i$. The camera pose $g_t^i = (R_t^i,T_t^i)$ consists of a rotation matrix $R_t^i \in SO(3)$ and a translation vector $T_t^i \in \mathbb{R}^3$.\footnote{We use the convention that $g_t^i$ transforms world-relative coordinates $p_\text{world}$ to camera-relative coordinates $p_\text{camera} = g_t^i p_\text{world}$.}
We extract this information using off-the-shelf methods: the structure-from-motion (SFM) algorithm COLMAP for RGB sequences \cite{schoenberger2016sfm,schoenberger2016mvs}, and an open-source implementation~\cite{rusu2011pcl} of KinectFusion (KF) \cite{newcombe2011kinectfusion} for RGBD sequences. The information extracted from RGB or RGBD data is qualitatively similar, except that the scale of SFM reconstructions is arbitrary.
\subsection{Intra-sequence alignment}\label{s:viewpoint}
Methods such as SFM or KF can reliably estimate camera pose and depth information for single objects and individual video sequences, but are not applicable to~\emph{different instances and sequences}. In fact, their underlying assumption is that geometry is fixed, which is true for single (rigid) objects, but false when the geometry and appearance differ due to intra-class variations.
Learning 3D object categories requires to relate their variable 3D shapes by identifying and putting in correspondence analogous geometric features, such as the object front and rear. For rigid objects, such correspondences can be expressed by rigid transformations that \emph{align} occurrences of analogous geometric features.
The most common approach for aligning 3D shapes, also adopted by~\cite{sedaghat15unsupervised} for video sequences, is to extract and match 3D feature descriptors. Once objects in images or videos are aligned, the data can be used to supervise other tasks, such as learning a monocular predictor of the absolute viewpoint of an object~\cite{sedaghat15unsupervised}.
One of our main contributions, described below, is to reverse this process by learning a viewpoint predictor \emph{without} explicitly matching 3D shapes. Empirically (\cref{s:exp}), we show that, by skipping the intermediate 3D analysis, our method is often more effective and robust than alternatives.
\myparagraph{Siamese network for viewpoint factorization} Geometric analogies between 3D shapes can often be detected in image space directly, based on visual similarity. Thus, we propose to train a CNN $\ensuremath{\Phi_\text{vp}}$ that maps a single frame $f_t^i$ to its \emph{absolute viewpoint} $\hat g_t^i = \ensuremath{\Phi_\text{vp}}(f_t^i)$ in the globally-aligned reference frame. We wish to learn this CNN from the viewpoints estimated by the algorithms of~\cref{s:sfm} for each video sequence. However, these estimated viewpoints are \emph{not} absolute, but valid only within each sequence; formally, there are unknown sequence-specific motions $h^i = (R^i,T^i) \in SE(3)$ that map the sequence-specific camera poses $g^i_t$ to global poses $\hat g_t^i = g_t^i h^i$.\footnote{$h^i$ composes to the right: it transforms the world reference frame and then moves it to the camera reference frame.}
To address this issue, we propose to supervise the network using \emph{relative pose changes within each sequence}, which are invariant to the alignment transformation $h^i$. Formally, the transformation $h^i$ is eliminated by computing the relative pose change of the camera from frame $t$ to frame $t^\prime$:
\begin{equation}\label{e:fund}
\hat g^i_{t^\prime} (\hat g^i_{t})^{-1}
=
g^i_{t^\prime} h^i (h^i)^{-1} (g^i_{t})^{-1}
=
g^i_{t^\prime} (g^i_{t})^{-1}.
\end{equation}
Expanding the expression with $\hat g_t^i = (\hat R_t^i,\hat T_t^i)$, we find equations
expressing the relative rotation and translation
\begin{align}
\hat R_{t^\prime}^i (\hat R_t^i)^\top
&=
R_{t^\prime}^i (R_t^i)^\top,
\label{e:fund1}
\\
\hat T^i_{t^\prime} - \hat R_{t^\prime}^i (\hat R_t^i)^\top \hat T^i_{t}
&=
T^i_{t^\prime} - R_{t^\prime}^i (R_t^i)^\top T^i_{t}.
\label{e:fund2}
\end{align}
Eqs.~\eqref{e:fund1} and \eqref{e:fund2} are used to constrain the training of a \emph{Siamese architecture}, which, given two frames $t$ and $t'$, evaluates the CNN twice to obtain estimates $(\hat R^i_t, \hat T^i_t) = \ensuremath{\Phi_\text{vp}}(f^i_t)$ and $(\hat R^i_{t^\prime}, \hat T^i_{t^\prime}) = \ensuremath{\Phi_\text{vp}}(f^i_{t^\prime})$. The estimated poses are then compared to the ground truth ones, $(R^i_t,T^i_t)$ and $(R^i_{t^\prime},T^i_{t^\prime})$, in a relative manner by using losses that enforce the estimated poses to satisfy \cref{e:fund1,e:fund2}:
\begin{align}
\ell_R
(\hat R^i_t, \hat T^i_t, \hat R^i_{t^\prime}, \hat T^i_{t^\prime})
&\ensuremath{\overset{\cdot}{=}}
\|
\ln \hat R_{ t t^\prime}^i (R_{t t^\prime}^i)^\top
\|_F
\label{e:loss1}
\\
\ell_T (\hat R^i_t, \hat T^i_t, \hat R^i_{t^\prime}, \hat T^i_{t^\prime})
&\ensuremath{\overset{\cdot}{=}}
\|
\hat T_{t t^\prime}^i - T_{t t^\prime}^i
\|_2
\label{e:loss2}
\end{align}
where $\ln$ is the principal matrix logarithm and
\begin{align*}
R^i_{t^\prime t}
&\ensuremath{\overset{\cdot}{=}}
R_{t^\prime}^i (R_t^i)^\top,
&
\hat R^i_{t^\prime t}
&\ensuremath{\overset{\cdot}{=}}
\hat R_{t^\prime}^i (\hat R_t^i)^\top,
\\
T^i_{t^\prime t}
&\ensuremath{\overset{\cdot}{=}}
T^i_{t^\prime} - R^i_{t^\prime t} T^i_{t},
&
\hat T^i_{t^\prime t}
&\ensuremath{\overset{\cdot}{=}}
\hat T^i_{t^\prime} - \hat R^i_{t^\prime t} \hat T^i_{t}.
\end{align*}
While this CNN is only required to correctly predict relative viewpoint changes \emph{within each sequence}, since the \emph{same CNN} is used for all videos, the most plausible/regular solution for the network is to assign similar viewpoint predictions $(\hat R_t^i$, $\hat T_t^i)$ to images viewed from the same viewpoint, leading to a globally consistent alignment of the input sequences. Furthermore, in a large family of 3D objects, different ones (e.g. SUVs and sedans) tend to be mediated by intermediate cases.
This is shown empirically in \cref{s:exp}.
\myparagraph{Scale ambiguity in SFM} \label{s:sfmambiguity} For methods such as SFM, there is an additional ambiguity: reconstructions are known only up to sequence-specific scaling factors $\lambda^i > 0$, so that the camera pose is parametrized as
$
g^i_t(\lambda^i) = (R_t^i, \lambda^i T_t^i).
$
This ambiguity leaves~\cref{e:fund1} unchanged, but~\cref{e:fund2} becomes:
$$
\hat T^i_{t^\prime} - \hat R^i_{t^\prime t} \hat T^i_{t}
=
\lambda^i (T^i_{t^\prime} - R^i_{t^\prime t} T^i_{t})
\quad\Rightarrow\quad
\hat T^i_{t^\prime t} = \lambda^i T^i_{t^\prime t}
$$
During training, the ambiguity can be removed from loss~\eqref{e:loss2} by dividing vectors $T^i_{t^\prime t}$ and $\hat T^i_{t^\prime t}$ by their Euclidean norm. Note that for KF sequences $\lambda^i = 1$.
As the viewpoints are learned, an estimate of $\hat \lambda^i$ is computed using a moving average over training iterations for the other network modules to use (see supplementary material for details).
\myparagraph{Probabilistic predictions} Due to intrinsic ambiguities in the images or to errors in the SFM supervision (caused for example by reflective or textureless surfaces), $\ensuremath{\Phi_\text{vp}}$ is occasionally unable to predict the ground truth viewpoint accurately. We found beneficial to allow the network to explicitly learn these cases and express uncertainty as an additional input-dependent prediction. For the translation component, we modify the network to predict the absolute pose $\hat T^i_t$ as well as its confidence score $\sigma_{\hat T^i_t}$ (predicted as the output of a soft ReLU units to ensure positivity). We then model the relative translation as a Gaussian distribution with standard deviation $\sigma_T = \sigma_{\hat T_{t^\prime}^i} + \sigma_{\hat T_{t}^i}$ and our model is now learned by minimizing the negative log-likelihood $\mathcal{L}_T$ which replaces the loss $\ell_T$:
\begin{equation}\label{e:robust1}
\mathcal{L}_T =
-\ln \frac{1}{(2\pi\sigma^2_T)^\frac{3}{2}}
\exp\left(
-\frac{1}{2} \frac{\ell_T^2}{\sigma_T^2}
\right).
\end{equation}
The rotation component is more complex due to the non-Euclidean geometry of $SO(3)$, but it was found sufficient to assume that the error term~\eqref{e:loss1} has Laplace distribution and optimize
$
\mathcal{L}_R = - \ln \frac{1}{C_R}
\exp\left(
-\frac{\sqrt{2} \ell_R}{\sigma_R}
\right),$
$\sigma_R =
\sigma_{\hat R_{t^\prime}^i}
+
\sigma_{\hat R_{t}^i},
$ where $C_R$ is a normalization term ensuring that the probability distribution integrates to one.
During training, by optimizing the losses $\mathcal{L}_R$ and $\mathcal{L}_T$ instead of $\ell_R$ and $\ell_T$, the network can discount gross errors by dividing the losses by a large predicted variance.
\myparagraph{Architecture} The architecture of $\ensuremath{\Phi_\text{vp}}$ is a variant of ResNet-50~\cite{he16resnet} with some modifications to improve its performance as viewpoint predictor. The lower layers of $\ensuremath{\Phi_\text{vp}}$ are used to extract a multiscale intermediate representation (denoted HC for hypercolumn \cite{hariharan2015hypercolumns} in \cref{fig:overview}). The upper layers consist of $2\times 2$ downsampling residual blocks that predict the viewpoint (see supp. material for details).
\input{fig_aug}
\subsection{Depth prediction}\label{s:depth}
The depth predictor module $\ensuremath{\Phi_\text{depth}}$ of VpDR-Net\xspace takes individual frames $f^i_t$ and outputs a corresponding depth map $\hat D_t = \ensuremath{\Phi_\text{depth}}(f^i_t)$, performing monocular depth estimation.
Estimating depth from a single image is inherently ambiguous and requires comparing the image to internal priors of the object shape. Similar to pose, we allow the network to explicitly \emph{learn and express uncertainty} about depth estimates by predicting a posterior distribution over possible pixel depths.
For robustness to outliers from COLMAP and KF, we assume a Laplace distribution
with negative log-likelihood loss
\begin{equation}
\mathcal{L}_D = \sum_{j=1}^{WH}
-\ln \frac{\sqrt{2}}{2\hat\sigma_{d_j}} ~
\exp\left(
-\frac{\sqrt{2} ~ |d_j - \hat {\lambda^i}^{-1} \hat d_j|}{\hat \sigma_{d_j} }
\right),
\end{equation}
where $d_j$ is the noisy ground truth depth output by SFM or KF for a given pixel $j$, and $\hat d_j$ and $\hat \sigma_{d_j}$ are respectively the corresponding predicted depth mean and standard deviation. The relative scale $\hat \lambda^i$ is 1 for KF and is estimated as explained in~\cref{s:sfmambiguity} for SFM.
\input{viewpoint_qualitative.tex}
\subsection{Point-cloud completion}\label{s:recon}
Given any image $f$ of an object instance, its \textit{aligned 3D shape} can be reconstructed by estimating and aligning its depth map using the output of the viewpoint and depth predictors of~\cref{s:depth,s:viewpoint}. However, since a depth map cannot represent the occluded portions of the object, such a reconstruction can only be partial. In this section, we describe the third and last component of~VpDR-Net\xspace, whose goal is to generate a full reconstruction of the object, beyond what is visible in the given view.
\myparagraph{Partial point cloud} The first step is to convert the predicted depth map $\hat D_f = \ensuremath{\Phi_\text{depth}}(f)$ into a partial point cloud
$
\hat P_f
\ensuremath{\overset{\cdot}{=}}
\{ \hat p_j : j=1,\dots,HW \},
$
$
\hat p_j
\ensuremath{\overset{\cdot}{=}}
K^{-1}
\begin{bmatrix} u_j & v_j & \hat d_i \end{bmatrix}^\top,
$
where $(u_j,v_j)$ are the coordinates of a pixel $j$ in the depth map $\hat D_f$ and $K$ is the camera calibration matrix. Empirically, we have found that the reconstruction problem is much easier if the data is aligned in the global reference frame established by VpDR-Net\xspace. Thus, we transform $\hat P_f$ into a globally-aligned point cloud as $\hat P^G_f = \hat g^{-1} \hat P_f$, where $\hat g=\ensuremath{\Phi_\text{vp}}(f)$ is the camera pose estimated by the viewpoint-prediction network.
\myparagraph{Point cloud completion network} Next, our goal is to learn the point cloud completion part of our network $\ensuremath{\Phi_\text{pcl}}$ that takes the aligned but incomplete point could $\hat P^G_f$ and produces a complete object reconstruction $\hat C$. We do so by predicting a 3D occupancy probability field. However, rather than using a volumetric method that may require a discrete and fixed voxelization of space, we propose a simple and efficient alternative. First, the network $\ensuremath{\Phi_\text{pcl}}$ predicts a set of $M$ 3D points $\hat S = (\hat s_1,\dots, \hat s_M) \in \mathbb{R}^{3\times M}$ that, during training, closely fit the ground truth 3D point cloud $C$. This step minimizes the fitting error:
\begin{equation}
\ensuremath{\ell_\text{pcl}}(\hat S) = \frac{1}{|C|}\sum_{c \in C}
\min_{m=1,\dots,M} \left\| c - \hat s_m \right\|_2.
\end{equation}
The 3D point cloud $\hat S$ provides a good coverage of the ground truth object shape. However, this point cloud is conservative and distributed \emph{in the vicinity} of the ground truth object. Thus, while this is not a precise representation of the object shape, it works well as a support of a probability distribution of space occupancy. In order to estimate the occupancy probability values, the network $\ensuremath{\Phi_\text{pcl}}(\hat P^G_f)$ predicts additional scalar outputs
$$
\delta_m = | \{ c \in C : \forall m^\prime: \| \hat s_m - c \|_2 \leq \| \hat s_{m^\prime} - c \|_2 \} | / |C|
$$
proportional to the number of ground truth surface points $c \in C$ for which the support point $\hat s_m$ is the nearest neighbor. The network is trained to compute a prediction $\hat\delta_m$ of the occupancy masses $\delta_m$ by minimizing the squared error loss
$
\ell_\delta(\hat \delta, \delta) = \sum_{m=1}^M (\hat \delta_m - \delta_m)^2.
$
Given the network prediction $(\hat S, \hat \delta) = \ensuremath{\Phi_\text{pcl}}(\hat P^G_f)$, the completed point cloud is then defined as the subset of points $\hat C$ that have sufficiently high occupancy, defined as:
$
\hat C_\tau = \{ \hat s_m \in \hat S : \delta_m \geq \tau \}
$
where $\tau$ is a confidence parameter. The set $\hat C_\tau$ can be further refined by using e.g.\ a 3D Laplacian filter to smooth out noise.
\myparagraph{Architecture} The point cloud completion network $\ensuremath{\Phi_\text{pcl}}$ is modeled after PointNet \cite{qi16pointnet}, originally proposed to semantically \emph{segment} a point clouds. Here we adapt it to perform a completely different task, namely 3D shape reconstruction. This is made possible by our model where shape is represented as a cloud of 3D support points $\hat S$ and their occupancy masses $\hat \delta$.
Differently from $\ensuremath{\Phi_\text{vp}}$ and $\ensuremath{\Phi_\text{depth}}$, $\ensuremath{\Phi_\text{pcl}}$ is \emph{not} convolutional but uses a sequence of fully connected layers to process the 3D points in $\hat P_f^G$, after appending an appearance descriptor to each of them. A key step is to add an intermediate orderless pooling operator to remove the dependency on the order and number of input points (see the supplementary material for details). The architecture is configured to predict $M=10^4$ points $\hat S$.
\input{tab_posest}
\myparagraph{Leave out} During training the incomplete point cloud $\hat P_f^G$ is downsampled by randomly selecting between $M = 10^3$ and $10^4$ points based on their depth prediction confidence as estimated by $\ensuremath{\Phi_\text{depth}}$. Similar to dropout, dropping points allows the network to overfit less, to become less sensitive to the size of the input point cloud, and to implicitly discard background points (as these are assigned low confidence by depth prediction). For the latter reason, leave out is maintained at test time too with $M=10^4$.
\section{Related work}
\myparagraph{Viewpoint estimation} The vast majority of methods for learning the viewpoint of object categories use manual supervision~\cite{savarese073d,Ozuysal09pose,glasner11viewpoint,pepik14multi,xiang2014beyond,mottaghi2015coarse,tulsiani2015viewpoints} or synthetic~\cite{su2015render} data. In \cite{ummenhofer16demon}, a deep architecture predicts a relative camera pose and depth for a pair of images. Only a few works have used videos~\cite{sun09multi,sedaghat15unsupervised}. \cite{sedaghat15unsupervised} solves the shape alignment problem using a global search strategy based on the pairwise alignment of point clouds, a step we avoid by means of our Siamese viewpoint factorization network.
\myparagraph{3D shape prediction}
A traditional approach to 3D reconstruction is to use handcrafted 3D models~\cite{lawrence63machine, lowe87three}, and more recently 3D CAD models~\cite{shapenet2015,xiang2014beyond}. Often the idea is to search for the 3D model in a CAD library that best fits the image~\cite{lim13parsing,aubry14seeing,gupta15aligning,bansal16marr}. Alternatively, CAD models can be used to train a network to directly predict the 3D shape of an object~\cite{girdhar16learning,wu16learning,tatarchenko16multi,choy163d}. Morphable models have sometimes been used ~\cite{zia13detailed,kar15category}, particularly for modeling faces~\cite{blanz03face,liu16joint}. All these methods require 3D models at train time.
\myparagraph{Data-driven approaches for geometry} Structure from motion (SFM) generally assumes fixed geometry between views and is difficult to apply directly to object categories due to intra-class variations. Starting from datasets of unordered images, methods such as~\cite{zhu10model} and \cite{prasad10finding} use SFM and manual annotations, such as keypoints in \cite{carreira16lifting,kar15category}, to estimate a rough 3D geometry of objects. Here, we leverage motion cues and do not need extra annotations.
| {'timestamp': '2017-08-28T02:01:11', 'yymm': '1705', 'arxiv_id': '1705.03951', 'language': 'en', 'url': 'https://arxiv.org/abs/1705.03951'} |
\section{Introduction} \label{sec:intro}
The total solar flux\footnote{We have observed the total solar 'flux density' practically. It is commonly, and perhaps inaccurately, referred to as 'flux' in the literature. In the paper, we will use the common term 'flux' rather than 'flux density'.} in microwaves, especially the F10.7 index (total solar flux at 2.8 GHz), is widely used as an indicator of solar activity in the fields of heliophysics and geophysics, because the F10.7 index indicates the variation of solar UV emission in comparison to sunspot number. The time variation in microwaves is traditionally classified into three components based on the timescale of enhancements. The components are a background component from the quiet Sun, a slowly varying component, and a sporadic (burst) component \citep{1965sra..book.....K}. The background component is considered to originate from optically thick thermal bremsstrahlung emission from the atmosphere above the upper chromosphere, but the emission mechanism of the background component is not so simple at lower frequencies (1$\sim$10 GHz). Actually, a simple calculation of thermal bremsstrahlung emission from optically thick layers cannot explain the quiet Sun spectrum measured by \cite{1991ApJ...370..779Z}. The reason for this discrepancy is that the transition region, which is an important contributor in this frequency range, is optically thin \citep[e.g.][]{2011SoPh..273..309S}. The contribution of coronal emission is also a reason for the discrepancy \citep{1991ApJ...370..779Z, 1996ApJ...473..539B}. To reproduce the spectrum of the background component at low frequencies, we need to consider the vertical thermal structure of the quiet Sun from the chromosphere to the corona. Therefore, modeling of the background component is needed, such as those based on recent Radiative-MHD simulations \citep{2004A&A...419..747L}.
The emission sources of a slowly varying component were investigated vigorously using the Westerbork Synthesis Radio Telescope and Very Large Array \citep{1959AnAp...22....1K, 1977A&A....61...79C, 1980A&A....82...30A, 1982ApJ...258..384L, 1982SoPh...80...71C, 1983ApJ...269..698M, 1983A&A...124..103D, 1984ApJ...283..413S, 1987ApJ...315..716W, 1991ApJ...379..366G, 1992ApJS...78..599W, 1994ApJ...426..434A}. From these studies, it is widely accepted that the slowly varying component originates in the thermal bremsstrahlung emission of coronal loops above active regions and thermal gyro-resonance emission from strong magnetic field regions, such as sunspots. A sporadic component is related to flares and coronal mass ejections, and is emitted from the non-thermal electrons accelerated in these phenomena.
The total solar flux at 2.8 GHz has been observed from 1947 \citep{1697669}. Nevertheless, the long-term variation of solar microwave emission itself has not been well investigated except in a few studies \citep[e.g.][]{1969JRASC..63..125C}, because it is hard to study the properties of microwave emission from the single-frequency data. In Japan, the total solar fluxes at 1, 2, 3.75, and 9.4 GHz have been observed continuously from 1957 \citep{1953PRIAN..1..71T, 1957PRIAN..4..60T}, and the dataset is calibrated well with the established method \citep{1973SoPh...29..243T}. \cite{1994SoPh..152..167S, 1995JGR...10019851S} investigated the microwave spectrum of 1980s data obtained in Japan. Because the main purpose of their investigation is the comparison between the solar irradiance measured with the Active Cavity Radiometer Irradiance Monitor (ACRIM) aboard the Solar Maximum Mission (SMM) and microwave spectrum, the period of their investigation is just only 10 years. Hence, there is no studies of microwave spectrum variation in the period that is longer than one solar cycle yet.
In the paper, we examine the long-term variation of solar microwave emission using the dataset of the last half century, and report the properties of the solar cycle variation in microwaves.
\section{Observation \& Dataset} \label{sec:obs}
In November 1951, monitoring of the total solar flux density at 3.75 GHz started at Toyokawa Observatory, the Research Institute of Atmospherics, Nagoya University \citep{1953PRIAN..1..71T}. Observations at 9.4 GHz started in May 1956, and the 1 and 2 GHz observations started in May and June 1957, respectively at the same site \citep{1957PRIAN..4..60T}. The solar radio telescopes for the 1, 2, and 9.4 GHz observations were prepared for the coordinated observations in the International Geophysical Year lasting from July 1957 to December 1958 \citep{1958BRIA....67T}. For the analysis described in the paper, we need data at all four frequencies, 1, 2, 3.75, and 9.4 GHz. Therefore, we analyzed the data obtained from June 1957.
In March 1994, the telescopes for the 1, 2, and 9.4 GHz observations were moved to Nobeyama Solar Radio Observatory (NSRO), a branch of the National Astronomical Observatory of Japan (NAOJ), and their operation resumed late in May 1994. At the same time, a new telescope for a 3.75 GHz observation was constructed in Nobeyama. Before the transfer and construction of the telescopes, the 17, 35, and 80 GHz monitoring observations were carried out by NSRO \citep{1985PASJ...37..163N}. Thus, the monitoring observations of the total solar fluxes at 1, 2, 3.75, 9.4, 17, 35, and 80 GHz started at the same site from 1994 May. The monitoring system was named Nobeyama Radio Polarimeters (NoRP). Due to the change of the observing site, the total solar fluxes at 1, 2, and 9.4 GHz were not measured from March to May 1994. Although there is no data loss at 3.75 GHz, we did not analyze the data of this period because no spectrum can be obtained for this study.
NSRO closed on 31 March 2015. Since 1 April 2015, Nobeyama Radio Observatory (NRO), a branch of NAOJ, has continued the operation of NoRP. The dataset used in the paper covers until December 2016.
NSRO released calibrated daily flux densities at each observing frequency in tabulated form every month at its FTP site\footnote{ftp://solar-pub.nao.ac.jp/pub/nsro/norp/data/daily/}. The calibration method is described in \cite{1973SoPh...29..243T}. In cooperation with the consortium for NoRP scientific operations\footnote{http://solar.nro.nao.ac.jp/norp/html/policy\_new.html}, NRO continues to release the tabulated data using the same FTP site. We used the daily fluxes in the tables for our analysis. The value in the table is not the same as the value calculated using the analysis package of NoRP that is included in the SolarSoftWare \citep[SSW:][]{1998SoPh..182..497F}. The value in the tables is derived after manually removing the effects of radio bursts, bad weather conditions, and instrumental problems. In contrast, when the SSW package is used, the daily flux is automatically calculated from the raw data without such treatment. The difference can be neglected for a study of flares, but the values in the tables are well calibrated and much more reliable for a study of the long-term variation. The total solar flux used in the paper is corrected based on the seasonal variation of the distance between the Sun and the Earth. To compare the solar cycle variation in microwaves with that of the sunspot number, the monthly mean total sunspot number and 13-month smoothed monthly total sunspot number provided from WDC-SILSO, Royal Observatory of Belgium are used \citep{sidc}.
\section{Solar Cycle Variation in Microwave} \label{sec:flux}
Panels (a) and (b) in Figure \ref{fig:fig1} show the long-term variations of the total solar fluxes in microwaves from June 1957 to December 2016. As mentioned in the previous section, the effect of radio bursts is carefully removed. Therefore, the total solar fluxes in panels (a) and (b) in Figure \ref{fig:fig1} are composed of a background component and a slowly varying component, and we can consider that the enhancement around the solar maxima (vertical dashed lines) is caused by the increasing of coronal plasma, and the rising of the number, size, and magnetic field strength of sunspots.
\begin{figure}[h]
\epsscale{.70}
\plotone{fig1.eps}
\caption{The variation of the total solar flux in microwaves. (a) The daily total solar fluxes in microwaves and the monthly mean total sunspot number. (b) The monthly mean microwave fluxes, the monthly mean total sunspot number (solid lines) and the monthly standard deviations (MSD: asterisk). In the two panels, Blue, Red, Orange, Green, and Black indicate 1 GHz, 2 GHz, 3.75 GHz, 9.4 GHz and sunspot number, respectively. (c) The black asterisks indicate the monthly mean total sunspot number, and the red line is the 13-month smoothed monthly total sunspot number. The green asterisks indicate the AMSD, and the blue line is the 13-month smoothed AMSD. (d) The parameters of the model function. The red indicates a: intercept, and the black indicates b: slope. The vertical dashed lines and dotted-dashed lines indicate the solar maxima and solar minima defined from the monthly mean total sunspot number, as shown in Table 1. \label{fig:fig1}}
\end{figure}
Comparing the daily and monthly total solar fluxes in microwaves and the monthly mean total sunspot number, their variations are very similar at the solar maxima. We can find the counterparts of almost all microwave peaks in the time profile of sunspot number. On the other hand, there are large gaps between them at the solar minima. The total solar fluxes in microwaves do not decrease as significantly as the sunspot number does, because there is a contribution of the emission from the quiet Sun even at solar minima.
To characterize the microwave spectrum in each month, we calculated not only the monthly mean flux but also the monthly standard deviation (MSD) from the daily total solar flux in each frequency (panel (b) in Figure \ref{fig:fig1}). We found that the MSD also shows the solar cycle variation. The MSD around the solar maxima becomes large because the flux from active regions can change up to 40 \% from one day to the next \citep{1978SoPh...56..335F, 1982SoPh...80...71C, 1983A&A...124..103D, 1991ApJ...379..366G}. The emission that causes the large MSD is a slowly varying component, and the variation of the emission strongly depends on the coronal activity and magnetic field strength. At solar minima, no magnetic field is strong enough for gyro-resonance emission, and the coronal activity that enhances the fluctuation of coronal plasma is very low. Consequently, the MSD at the solar minima becomes very small.
Although the four MSDs can be calculated from the total solar fluxes at the four observing frequencies (1, 2, 3.75, and 9.4 GHz), their variations are quite similar. To simplify our discussion, the averaged value of the four MSDs is used hereafter, and we call the averaged value ``Averaged Monthly Standard Deviation" (AMSD). Panel (c) in Figure \ref{fig:fig1} shows the long-term variation of the AMSD, 13-month smoothed AMSD, monthly mean total sunspot number, and 13-month smoothed monthly total sunspot number. The variation of the AMSD is very similar to that of the monthly mean total sunspot number, and the correlation coefficient between them is 0.71. When we calculate the correlation coefficient between the 13-month smoothed values of the monthly mean total sunspot number and AMSD, the coefficient increases to 0.91. Considering the emission mechanisms at the observing frequencies reviewed at Section \ref{sec:intro}, the AMSD directly reveals the variation of the strong magnetic field regions and the activities in the atmosphere higher than the upper chromosphere.
Table \ref{tab:tab1} presents the month when the monthly mean total sunspot number or AMSD becomes the maximum/minimum value in each solar cycle. In the table, we can see the difference between the months derived from the monthly mean total sunspot number and that from the AMSD. The difference no doubt results from the differing atmospheric layers that dominate the two indexes, as mentioned above.
\begin{table}[h]
\begin{center}
\begin{tabular}{c c c c || c c c c}
\hline \hline
Cycle & Max/Min & SN & AMSD & Cycle & Max/Min & SN & AMSD\\
\hline
19 & Min. & & & 22 & Min. & Jun. 1986 & \bf{Sep. 1986} \\
& Max. & Oct. 1957 & \bf{Jan. 1959} & & Max. & Jun. 1989 & \bf{Mar. 1989} \\
\hline
20 & Min. & Jul. 1964 & \bf{Sep. 1964} & 23 & Min. & Oct. 1996 & \bf{Jan. 1997} \\
& Max. & Mar. 1969 & \bf{Jun. 1969} & & Max. & Jul. 2000 & \bf{Oct. 2003} \\
\hline
21 & Min. & Jul. 1976 & \bf{Jul. 1976} & 24 & Min. & Aug. 2009 & \bf{Jul. 2008} \\
& Max. & May 1980 & \bf{Jun. 1982} & & Max. & Feb. 2014 & \bf{Jul. 2014} \\
\hline \hline
\end{tabular}
\end{center}
\caption{The months when the monthly mean total sunspot number (SN) or averaged monthly standard deviation (AMSD) becomes the maximum/minimum value in each solar cycle. \label{tab:tab1}}
\end{table}
\section{Microwave Spectra at Solar Maxima and Minima} \label{sec:spec}
To investigate the solar cycle variation of the microwave spectrum, we carried out the fitting of monthly mean microwave spectra using several simple functions. Finally, we found that an exponential function can give a satisfactory fit to most of the spectra: log$_{10}$(Flux) = a + b $\times ~\nu$, $\nu$ is frequency in Hz. Panel (d) in Figure \ref{fig:fig1} shows the time variation of the fitting model parameters (a: the intercept, b: the slope of the spectrum). The intercept correlates weakly with the solar cycle variation. On the other hand, the slope of the spectrum indicates the anti-correlation. Gyro-resonance emission in lower frequency is stronger than in higher frequency when the emission regions of both frequencies are the same. Because the property of gyro-resonance emission tends to flatten the spectrum at solar maxima, the anti-correlation is a matter of course. We mention that the maximum values of the spectral slopes at the solar minima are about 9.3$\pm$0.3$\times$10$^{-8}$ (3 \% variation) and that the difference is so small though the minimum value of the spectral slope at the solar maximum significantly differs from that in the other solar maxima (5.3$\pm$1.0$\times$10$^{-8}$, $\sim$19 \% variation).
To compare the microwave spectra in the solar cycles, we plot the monthly mean solar microwave spectra at the solar maxima and solar minima (Figure \ref{fig:fig2}). To investigate the states where the activity in the upper chromosphere -- corona is highest/lowest, from here, we define ``solar maximum" and ``solar minimum" as the months when the AMSD becomes the maximum/minimum value in each solar cycle. The months are written in bold, Table \ref{tab:tab1}. As shown in the time variation of the fitting model parameter, the spectrum at a solar maximum significantly differs among the cycles. The absolute values of the total solar fluxes at the solar maxima also vary by over 100 \% between the cycles. The difference in the spectra tends to be related to the strength of the solar cycle variation.
\begin{figure}[h]
\plotone{fig2.eps}
\caption{ The monthly mean microwave spectra in the months when the AMSD becomes the maximum value (Left panel) or the minimum value (Right panel) in each solar cycle. Color indicates the number of the solar cycle. The months of the solar maxima and minima are shown in bold, Table \ref{tab:tab1}. \label{fig:fig2}}
\end{figure}
A solar minimum is a suitable moment for investigating the spectrum of a background component, because there is no object that emits a slowly varying component and a sporadic component. To obtain the pure spectrum of a background component, we need to select the moment when the coronal activity is extremely low. For this reason, the monthly mean spectrum when the AMSD is lowest in its cycle is suitable for this study. The right panel in Figure \ref{fig:fig2} displays the five monthly mean spectra at the solar minima defined from the AMSD, and the months of the spectra are written in bold, Table \ref{tab:tab1}. The slopes of these spectra are very similar because the variance of the total solar fluxes at the solar minima is only 5.6 SFU (12 \%) at 1 GHz, 4.72 SFU (9 \%) at 2 GHz, 2.23 SFU (3 \%) at 3.75 GHz, and 11.2 SFU (4 \%) at 9 GHz. The values are similar to the precisions of the instruments.
\section{Summary \& Discussion} \label{sec:sum}
We investigated the total solar fluxes at 1, 2, 3.75, and 9.4 GHz obtained from June 1957 to December 2016, and found that the averaged monthly standard deviation (AMSD) of microwave solar fluxes can reveal the long-term variation of solar activity well. Because of the high sensitivity to the activities in the transition region and corona, AMSD can be very useful for investigating the solar-terrestrial environment.
Although an exponential function can give a satisfactory fit to most of the spectra, there is no physical background of the model function. When we execute the fitting of the daily spectra, the chi-square value of the fitting becomes very large around the solar maxima. The tendency can be also seen in the spectra at the solar maxima in Figure \ref{fig:fig2}. The result is consistent with \cite{1994SoPh..152..167S}, and it can be understood as the effect of gyro-resonance emission from sunspots, as mentioned by them, because the spectrum of the emission has a power-law distribution \citep[e.g.][]{1985ARA&A..23..169D}.
To investigate the spectrum of a background component in each cycle, we compared the monthly mean spectra at the solar minima in Cycle 20$\sim$24 defined from the AMSD. Based on the absolute values of the fluxes and the slopes of the spectra, we found that the difference in the spectra is very small. As mentioned in Section \ref{sec:intro}, a background component should reveal the average atmospheric structure of the transition region and lower corona in the quiet Sun. Therefore, the results indicate that the average atmospheric structure above the upper chromosphere in the quiet Sun at solar minima, which may be related to the energy input for atmospheric heating from the sub-photosphere to the corona, has not varied for half a century. The peaks of sunspot numbers at the solar maxima varied during the period. In other words, the Sun’s magnetic activity has been changing significantly \citep[e.g.][]{2012ApJ...757L...8L, 2012SoPh..281..577P, 2012ApJ...750L..42G, 2012ApJ...753..157S, 2013ApJ...763...23S, 2013PASJ...65S..16S}. Nevertheless, our observing results suggest that the energy input for atmospheric heating from the sub-photosphere to the corona has not changed in the quiet Sun. It is commonly supposed that the energy input in the quiet Sun is caused by convection motion and the local dynamo mechanism in the sub-photosphere. Therefore, our results may indicate that the properties of convection motion and the local dynamo mechanism have not changed for half a century, and the solar cycle differences in strong magnetic field generated by the global dynamo mechanism.
\acknowledgments
The authors would like to express thank to the people who have participated in the operation of the solar radio telescopes at Toyokawa and Nobeyama Radio Polarimeters for over half a century. The Nobeyama Radio Polarimeters (NoRP) are operated by Nobeyama Radio Observatory, a branch of National Astronomical Observatory of Japan, and their observing data are verified scientifically by the consortium for NoRP scientific operations. This work was carried out on the Solar Data Analysis System operated by the Astronomy Data Center of the National Astronomical Observatory of Japan. M.S. was supported by JSPS KAKENHI Grant Number JP17K05397. A.A. was supported by JSPS KAKENHI Grant Numbers JP15H05816 and JP15K17772.
\facility{NoRP}
| {'timestamp': '2017-09-13T02:06:04', 'yymm': '1709', 'arxiv_id': '1709.03695', 'language': 'en', 'url': 'https://arxiv.org/abs/1709.03695'} |
\section{Hypergraphs and quantum states}
Given a hypergraph $G=(V,E)$ and non-negative edge weights $w\colon E\to\mathbb R_+$, the \emph{cut function} $\delta\colon 2^V\to\mathbb R_{\geq0}$ is defined as $c(S) = \sum_{e\in\delta(S)} w(e)$, where $\delta(S)$ is the set of hyperedges that contain vertices both in~$S$ and~$V\setminus S$.
Fixing a subset of terminals~$T\subseteq V$, the \emph{min-cut function} $m\colon 2^T \to \mathbb R_{\geq0}$ is then given by $m(A) = \min_{S : S \cap T = A} c(S)$.
It is well-known that hypergraph cut functions are symmetric and submodular.
This property extends directly to min-cut functions.
We now turn to quantum states.
Given a quantum state~$\rho$ on a finite-dimensional tensor product Hilbert space $\mathcal H = \bigotimes_{t\in T} \mathcal H_t$, the \emph{entropy function} $S\colon 2^T\to\mathbb R_{\geq0}$ assigns to each subset $A\subseteq T$ the von Neumann entropy $S(A) = -\tr[\rho_A \log \rho_A]$ of the reduced state~$\rho_A = \tr_{T\setminus A}[\rho]$.
By a celebrated theorem of Lieb-Ruskai, the entropy function is submodular, and it is also symmetric if we restrict to pure states.
This coincidence begs the question if hypergraph min-cuts can always be realized by quantum entropies.
Such a result has been proved for graphs~\cite{hayden2016holographic,nezami2016multipartite}, motivated by research on the holographic principle in theoretical high-energy physics~\cite{bao2015holographic}.
To state the question more precisely, define the \emph{hypergraph cone}~$\Chyper n \subseteq \mathbb R_{\geq0}^{2^n}$ as the set of min-cut functions obtained from arbitrary weighted hypergraphs with terminals~$T=[n]$~\cite{bao2020quantum}.
Similarly, define the \emph{quantum entropy cone}~$\Cquantum n \subseteq \mathbb R_{\geq0}^{2^n}$ as the closure of the set of entropy functions obtained by varying over all pure quantum states on finite-dimensional tensor product Hilbert spaces as above~\cite{pippenger2003inequalities}.
Both sets are convex cones.
It is an well-known open problem in quantum information theory to determine the cones~$\Cquantum n$ for~$n \geq 4$~\cite{pippenger2003inequalities,linden2005new}.
Due to its import, this problem has also been studied for the class of stabilizer states, which are a versatile family of quantum states that have many applications in quantum information theory~\cite{gottesman1997stabilizer,gross2017schur}.
Thus, let $\Cstab n$ denote the closed convex cone generated by the set of entropy functions of pure stabilizer states (over any fixed prime).
In general, $\Cstab n \subseteq \Cquantum n$ is a proper subcone~\cite{linden2013quantum,gross2013stabilizer}.
Our main result is as follows:
\begin{thm}\label{thm:main}
For any $n\in\mathbb N$, we have that $\Chyper n \subseteq \Cstab n \subseteq \Cquantum n$.
\end{thm}
\noindent
\Cref{thm:main} proves a conjecture made in \cite{bao2020quantum}, where the authors verified the inclusion for $n \leq 4$.
However, not even the inclusion $\Chyper n \subseteq \Cquantum n$ was known before this work.
To prove \cref{thm:main}, we start by constructing for any weighted hypergraph a quantum state whose entropies realize the cut-function.
We then apply a general construction for transforming entropies into minimized entropies, and show that it succeeds with high probability.
\section{Tensor network states for hypergraph cuts}
Fix a hypergraph $G=(V,E)$ with integral edge weights~$w\colon E \to \mathbb N$.
For each vertex $x\in V$, define the Hilbert space $\mathcal H_x = \bigotimes_{e \in E : x \in e} \mathcal H_{x,e}^{\otimes w(e)}$, where $\mathcal H_{x,e} = \mathbb C^D$.
The dimension $D$ will later be taken to be large.
Now let $\ket\Omega$ be the state given by
\begin{align}\label{eq:omega}
\ket\Omega = \bigotimes_{e\in E} \ket{\GHZ(e)}^{\otimes w(e)} \in \bigotimes_{x\in V} \mathcal H_x.
\end{align}
Here, $\ket{\GHZ(e)} = \frac1{\sqrt D} \sum_{i=1}^D \ket i^{\otimes \abs e} \in \bigotimes_{x \in e} \mathcal H_{x,e}$ denotes an $\abs e$-partite GHZ state of local dimension~$D$.
In general, one can also develop the following theory for other multipartite entangled states than the GHZ state, but for our purposes this construction will suffice.
The GHZ state has the property that it is a pure state whose reduced states have~$D$ nonzero eigenvalues which are all equal to~$1/D$.
This implies that not only the von Neumann entropy of any subsystem $S \subseteq V$, but in fact any R\'enyi-$\alpha$ entropy can be calculated by
\begin{align}\label{eq:cut entropies}
S_\alpha(\Omega_S)
= \sum_{e\in E} w(e) \, S_\alpha\bigl(\GHZ(e)_{S \cap e}\bigr)
= \log(D) \sum_{e\in \delta(S)} w(e) = \log(D) \, c(S).
\end{align}
We have thus found a family of quantum states, parameterized by $D$, whose entropy function is exactly proportional to the cut function~$c\colon 2^V\to\mathbb R_+$ of the given hypergraph.
\section{Random tensor network states for hypergraphs min-cuts}
We will now explain how to turn the state~\eqref{eq:omega} into one that approximates the min-cut function~$c\colon 2^T\to\mathbb R_+$ of the hypergraph $G=(V,E)$ for any fixed choice of terminals~$T\subseteq V$.
We may assume that any connected component of $G$ touches $T$ (otherwise we can remove this component without impacting the min-cut function).
Our main tool is the following construction, which is implicit in~\cite{hayden2016holographic} (cf.~\cite{horodecki2007quantum,dutil2010one}).
Define the (not necessarily normalized) pure state
\begin{align}\label{eq:projected state}
\ket{\Psi} = \Bigl( \bigotimes_{x \in V \setminus T} \bra{\phi_x} \Bigr) \ket\Omega \in \mathcal H = \bigotimes_{x\in T} \mathcal H_x,
\end{align}
obtained by projecting the tensor factors for each non-terminal vertex~$x\not\in T$ onto pure states~$\ket{\phi_x}$ (which we will below choose at random).
We note that~$\Psi$ can be understood as a tensor network state of bond dimension~$D$.
We now relate the entropies of~$\Psi$ to those of the state~$\Omega$.
For this, recall that for any quantum state~$\rho$,
\begin{align}\label{eq:renyi mono}
S_2(\rho) \leq S(\rho) \leq S_0(\rho),
\end{align}
where $S_2(\rho) = -\log\tr[\rho^2]$ the \emph{R\'enyi-2 entropy}, $S(\rho) = -\tr[\rho\log\rho]$ the \emph{von Neumann entropy}, and $S_0(\rho) = \log\rk[\rho]$ the \emph{log-rank}.
If $\rho$ is not normalized then we define $S_\alpha(\rho)$ in terms of the normalization (this makes no difference for the log-rank).
It is easy to see that for any state of the form~\eqref{eq:projected state} and any $A \subseteq T$,
\begin{align}\label{eq:rank upper}
S_0(\Psi_A) \leq \min_{S : S \cap T = A} S_0(\Omega_S) = \log(D) m(A).
\end{align}
Indeed, any cut $S$ for $A$ gives us an upper bound on the rank of $\Psi_A$ in terms of the rank of $\Omega_S$, which can be calculated using~\eqref{eq:cut entropies} for $\alpha=0$.
In view of~\eqref{eq:renyi mono}, we would like to complement this upper bound by a similar lower bound on the R\'eniy-2 entropy.
For this, we choose each $\phi_x$ independently at random from a \emph{projective 2-design}.
This means that the first two moments agree with the unitarily invariant probability measure on pure states, i.e., $\mathbb E[\phi_x] = \tfrac I {D_x}$ and $\mathbb E[\phi_x^{\otimes 2}] = \tfrac{I + F}{D_x(D_x+1)}$, where~$D_x = \dim \mathcal H_x$ and~$F$ denotes the swap operator on $\mathcal H_x^{\otimes 2}$.
For example, we may choose~$\phi_x$ to be a uniformly random stabilizer state~\cite{klappenecker2005mutually,gross2007evenly}, in which case~$\Psi$ is again a stabilizer state~\cite{hayden2016holographic}.
We now compute the expected trace and purity of any subsystem.
\begin{lem}\label{lem:moments}
Let the $\phi_x$ be chosen independently at random from a 2-design.
Then the state~$\Psi$ in~\eqref{eq:projected state} satisfies
\begin{align*}
\mathbb E[\tr[\Psi]] = \frac1{D_b}
\quad\text{and}\quad
\mathbb E[\tr[\Psi_A^2]] = \frac1{D_b^2} D^{-m(A)} \left( k_A + O(D^{-1}) \right)
\end{align*}
for all $A \subseteq T$, where $D_b = \prod_{x\in V\setminus T} D_x$ and $k_A$ denotes the number of minimal cuts for~$A$.
\end{lem}
\begin{proof}
From the formula for the first moment of a 2-design,
\begin{align*}
\mathbb E[\tr[\Psi]]
= \tr\Bigl[\Omega \bigl( \bigotimes_{x \in V \setminus T} \mathbb E[\phi_x] \bigr) \Bigr]
= \frac 1 {D_b},
\end{align*}
(as is common we suppress tensor products with identity operators).
For $\tr[\Psi_A^2]$, we first use the swap trick,
\begin{align*}
\tr[\Psi_A^2]
= \tr[\Psi^{\otimes 2} F_A]
= \tr\Bigl[\bigl( \bigotimes_{x \in V \setminus T} \bra{\phi_x}^{\otimes 2} \bigr) \Omega^{\otimes 2} \bigl( \bigotimes_{x \in V \setminus T} \ket{\phi_x}^{\otimes 2} \bigr) F_A\Bigr]
= \tr\Bigl[\Omega^{\otimes 2} \bigl( \bigotimes_{x \in V \setminus T} \phi_x^{\otimes 2} \bigr) F_A\Bigr].
\end{align*}
Here, $F_A$ denotes the product of the swap operators on~$\mathcal H_x^{\ot2}$ for $x\in A$.
In the last step, we used the cyclicity of the trace and the fact that the swap operator $F_A$ `commutes' with $\bra{\phi_x}^{\otimes 2}$ for~$x\in V\setminus T$.
We can then use the compute the average by using the formula for the second moment of a 2-design:
\begin{align*}
\mathbb E[\tr[\Psi_A^2]]
&= \tr\Bigl[\Omega^{\otimes 2} \bigl( \bigotimes_{x \in V \setminus T} \mathbb E[\phi_x^{\otimes 2}] \bigr) F_A\Bigr]
= \tr\Bigl[\Omega^{\otimes 2} \bigl( \prod_{x \in V \setminus T} \frac {I_x + F_x} {D_x(D_x+1)} \bigr) F_A\Bigr] \\
&= \prod_{x\in V\setminus T} \frac1{D_x(D_x+1)} \sum_{S : S \cap T = A} \tr\bigl[\Omega^{\otimes 2} F_S \bigr]
\end{align*}
where the last step is again by the swap trick.
The prefactor is $D_b^{-2} (1 + O(D^{-1}))$, while the sum can be estimated using~\eqref{eq:cut entropies},
\begin{align*}
\sum_{S : S \cap T = A} \tr\bigl[\Omega^{\otimes 2} F_S \bigr]
= \sum_{S : S \cap T = A} 2^{-S_2(\Omega_S)}
= \sum_{S : S \cap T = A} D^{-c(S)}
= D^{-m(A)} \left( k_A + O(D^{-1}) \right),
\end{align*}
where we recall that~$k_A$ denotes the number of minimal cuts for $A$.
Together we obtain the desired bound.
\end{proof}
For $A=\emptyset$, the min-cut is empty and nondegenerate by our assumption that any connected component of~$G$ touches~$T$, so \cref{lem:moments} states that $\mathbb E[\tr[\Psi]^2] = D_b^{-2} (1 + O(D^{-1}))$.
Thus, $\tr[\Psi]$ is concentrated around its mean, suggesting that, with high probability, $\Psi\neq0$ and $S_2(\Psi_A) / \! \log(D) \approx m(A)$ for large~$D$.
The following lemma which follows the proof strategy of~\cite{hayden2016holographic} makes this intuition precise.
\begin{lem}\label{lem:concentration}
Let~$\Psi$ be defined as in~\eqref{eq:projected state}, with each $\phi_x$ chosen independently at random from a 2-design. Then the following properties hold for large~$D$:
\begin{enumerate}[label={(\alph*)}]
\item\label{it:nonzero}
$\mathbb P(\Psi\neq0) = 1 - O(D^{-1})$.
\item\label{it:expected renyi}
$\mathbb E[S_2(\Psi_A) | \Psi\neq0] \geq \log(D) \, m(A) - \log(k_A) - O(D^{-1/4})$, with $k_A$ the number of minimal cuts for~$A$.
\item\label{it:high probability entropy}
For any~$\delta>0$, it holds that
\begin{align*}
\mathbb P\Bigl(\Psi\neq0 \text{ and } \abs[\Big]{ \frac{S_2(\Psi_A)}{\log(D)} - m(A) } \leq \delta \text{ for all } A \subseteq T\Bigr) = 1 - O\Bigl(\frac1{\delta\log(D)}\Bigr)
\end{align*}
The same statement holds for the the von Neumann entropy~$S(\Psi_A)$ instead of the R\'enyi entropy~$S_2(\Psi_A)$.
\end{enumerate}
\end{lem}
\begin{proof}
As just noted, $\mathbb E[\tr[\Psi]^2] = \mathbb E[\tr[\Psi]]^2 (1 + O(D^{-1}))$, so Chebyshev's inequality shows that, for any~$\varepsilon>0$,
\begin{align}\label{eq:cheby}
\mathbb P\bigl(\lvert D_b\tr[\Psi] - 1 \rvert \leq \varepsilon \bigr)
\geq 1 - O\Bigl(\frac1{\varepsilon^2 D}\Bigr).
\end{align}
This establishes~\ref{it:nonzero}.
Next we prove~\ref{it:expected renyi}.
Let $E$ denote the event that $\lvert D_b\tr[\Psi] - 1 \rvert \leq D^{-1/4}$.
By~\eqref{eq:cheby},
\begin{align}\label{eq:p_E}
p_E := \mathbb P(E) = 1 - O(D^{-1/2}).
\end{align}
Since~$E$ implies that~$\Psi\neq0$, we have
\begin{align}\label{eq:E[S_2|nonzero]}
\mathbb E[S_2(\Psi_A) | \Psi\neq0]
\geq \mathbb P(\Psi\neq0) \, \mathbb E[S_2(\Psi_A) | \Psi\neq0]
\geq p_E \, \mathbb E[S_2(\Psi_A) | E]
\end{align}
We now bound
\begin{equation}\label{eq:E[S_2|E]}
\begin{aligned}
\mathbb E[S_2(\Psi_A) | E]
&= -\mathbb E\bigl[\log(D_b^2 \tr[\Psi_A^2]) \big| E \bigr] + 2 \mathbb E\bigl[ \log(D_b\tr[\Psi]) \big| E \bigr] \\
&\geq -\log\mathbb E\bigl[D_b^2 \tr[\Psi_A^2] \big| E \bigr] + 2 \log\bigl(1-D^{-1/4}\bigr) \\
&= -\log\mathbb E\bigl[D_b^2 \tr[\Psi_A^2] \big| E \bigr] - O(D^{-1/4})
\end{aligned}
\end{equation}
where we used Jensen's inequality to lower-bound the first term.
Using~$p_E \, \mathbb E[\tr[\Psi_A^2] | E] \leq \mathbb E[\tr[\Psi_A^2]]$, we obtain
\begin{align*}
-\log\mathbb E\bigl[D_b^2 \tr[\Psi_A^2] \big| E \bigr]
&\geq -\log \mathbb E[D_b^2 \tr[\Psi_A^2]] + \log(p_E) \\
&= \log(D) \, m(A) - \log \bigl( k_A + O(D^{-1}) \bigr) + \log\bigl( 1 - O(D^{-1/2}) \bigr) \\
&= \log(D) \, m(A) - \log(k_A) - O(D^{-1/2})
\end{align*}
by \cref{lem:moments} and~\eqref{eq:p_E}.
Together with~\eqref{eq:E[S_2|nonzero]}, \eqref{eq:E[S_2|E]}, and~$S_2(\Psi_A) \leq \log(D)m(A)$, which holds by~\eqref{eq:renyi mono} and \eqref{eq:rank upper}, we find
\begin{align*}
\mathbb E[S_2(\Psi_A) | \Psi\neq0]
&\geq \log(D) \, m(A) - \log(k_A) - O(D^{-1/4}),
\end{align*}
proving~\ref{it:expected renyi}.
To prove \ref{it:high probability entropy}, we note that $\log(D)m(A) - S_2(\Psi_A)$ is a nonnegative random variable. Thus, for any fixed~$A\subset T$,
\begin{align*}
\mathbb P\Bigl(m(A) - \frac{S_2(\Psi_A)}{\log(D)} > \delta \Big| \Psi\neq0 \Bigr)
\leq \frac{\log(D) m(A) - \mathbb E[S_2(\Psi_A) | \Psi\neq0 ]}{\delta \log(D)}
\leq \frac{\log(k_A) + O(D^{-1/4})}{\delta \log(D)} = O\left(\frac1{\delta\log(D)}\right)
\end{align*}
where we first used the Markov inequality and then part~\ref{it:expected renyi}.
By taking the union bound over all the finitely many subsets~$A \subseteq T$ and using part~\ref{it:nonzero}, we obtain
\begin{align*}
\mathbb P\Bigl(\Psi\neq0 \text{ and } m(A) - \frac{S_2(\Psi_A)}{\log(D)} \leq \delta \text{ for all } A \subseteq T\Bigr) = 1 - O\Bigl(\frac1{\delta\log(D)}\Bigr).
\end{align*}
In view of $S_2(\Psi_A) \leq S(\Psi_A) \leq \log(D)m(A)$, this proves part~\ref{it:high probability entropy}.
\end{proof}
\noindent
\Cref{lem:concentration} readily implies our main result.
\begin{proof}[Proof of \cref{thm:main}]
We only need to show that $\Chyper n \subseteq \Cstab n$.
Since $\Cstab n$ is a closed cone, it suffices to show that for any $\delta>0$ and any hypergraph with integral edge weights, terminal set~$T=[n]$, and min-cut function~$m$, there exists a number~$c>0$ and a stabilizer state $\Psi$ on an $n$-partite Hilbert space such that
\begin{align*}
\abs[\Big]{ \frac{S(\Psi_A)}{c} - m(A) } \leq \delta
\end{align*}
for all~$A \subseteq T$.
This follows from \cref{lem:concentration} if we use the set of stabilizer states as the 2-design and choose~$D$ to be sufficiently large.
Indeed, if each~$\phi_x$ is a stabilizer state then so is $\Psi$.
\end{proof}
\small
\bibliographystyle{alpha}
| {'timestamp': '2020-03-02T02:01:17', 'yymm': '2002', 'arxiv_id': '2002.12397', 'language': 'en', 'url': 'https://arxiv.org/abs/2002.12397'} |
\section{Introduction}
The statistical evolution of the morphology of galaxies is one of the
most fundamental observational clues to understand the establishing
process of the Hubble sequence of galaxies seen in the local universe.
Especially rest-frame optical morphology is important because it
reflects stellar mass distribution of galaxies. High-resolution imaging
observations with Hubble Space Telescope (HST), covering rest-frame
optical wavelength up to $z\sim1$, have shown that the Hubble
sequence is already established at $z\sim1$. Morphological studies
on galaxies at higher redshifts started following the discovery of
a large number of high redshift galaxies using Lyman Break selection (Steidel et al. 2003).
HST NICMOS observations of the Hubble Deep Field North (HDFN) show that
the morphology of the LBGs is essentially independent of the
wavelength up to the rest-frame 4000{\AA} (Giavalisco 2002).
However, the NICMOS observations are not sufficient to
conclude the distribution of the stellar mass
in $z\sim3$ LBGs. The NICMOS observations covers up to the $H$-band,
which is just at 4000{\AA} in the rest-frame of $z\sim3$ galaxies.
In addition, the NICMOS observations are mostly limited to the
$z\sim3$ LBGs in the small HDFN area and only include object as
bright as $M_{B}\sim-22$mag, which is still 0.5mag fainter than
the characteristic absolute magnitude of the $z\sim3$ LBGs ($M_{V}^{*}=-24.0$ mag).
In order to examine the rest-frame $V$-band morphology of
the $z\sim3$ LBGs covering a wide luminosity range ($M_{V}^{*}-0.5$ mag $-$
$M_{V}^{*}+3.0$ mag), we conducted AO-assisted imaging
observations of 36 of them in the $K$-band, which corresponds to the rest-frame $V$-band
in $z\sim3$ galaxies (Akiyama et al. 2007). Thirty one of the LBGs are detected in the deep
imaging observations with typical effective integration time of 5 hours.
\section{Results of AO imaging observations of the $z\sim3$ LBGs}
The AO-assisted $K$-band images of the 4 brightest $z\sim3$ LBGs
are shown in the upper left panel of Figure~\ref{akiyama_f01} as examples.
The AO observations clearly resolve most of the 31 $z\sim3$ LBGs at
the resolution of FWHM$\sim0.\!^{\prime\prime}2$.
We examined their total $J-K$ colors, which corresponds to the rest-frame
$U-V$ colors, using seeing-limited $J$-band data of
thirty of the LBGs.
The bright LBGs show red rest-frame $U-V$ colors
(average of $0.2$ mag), while
most of the fainter LBGs show blue rest-frame $U-V$ color (average of $-0.4$ mag).
The color distribution of the LBGs show that the LBG sample covers
not only blue less massive galaxies but also red massive galaxies at among $2<z<3$
galaxies selected with photometric redshifts.
\begin{figure}[t]
\begin{center}
\includegraphics[width=110mm]{akiyama_f01.eps}
\end{center}
\caption{
Left) $K$-band images (upper) and optical images with $K$-band contour (lower)
of the 4 brightest $z\sim3$ LBGs
in the sample. Each panel has a $3.\!^{\prime\prime}5 \times 3.\!^{\prime\prime}5$
(27kpc$\times$27kpc at $z=3$) FoV.
North is to the top and east is to the left.
We do not match the PSF size of the $K$-band
contour to that of the optical images, thus only the positions of the peaks should
be compared.
Right) Distance between $K$-band and optical peaks as a function of
$K$-band magnitudes for the $z\sim3$ LBGs (filled squares) and
field object (crosses).
}\label{akiyama_f01}
\end{figure}
In the lower left panels of Figure~\ref{akiyama_f01}, the
$K$-band contours of the LBGs are compared with the
optical images with the seeing-limited resolution.
The peaks in the $K$-band images of the LBGs show
significant offset from those in the optical images. In the right
panel of Figure~\ref{akiyama_f01}, the shifts of the LBGs are shown as a
function of $K$-band magnitude with filled squares for the whole sample. In order to
evaluate the uncertainties of the shift measurements, we also plot the differences
of $K$-band and optical positions of compact field objects with crosses.
The peaks in the $K$-band images of 7 of the LBGs with $K<22$ mag
show significant or marginal shifts from those in the optical images.
The presence of the shifts among red luminous LBGs implies that the UV-bright star-forming regions
are not necessarily centered at their main body observed in the $K$-band.
In order to examine whether their light profiles are similar
to those of local galaxies or not, we apply one component S\'ersic
profile fitting to the AO-assisted $K$-band images of the LBGs with $K<21.5$mag.
An example of the S\'ersic profile fitting is shown in the left panel
of Figure~\ref{akiyama_f02} (DSF2237b-MD81 with $K=20.2$ mag at $z=2.82$).
As can be seen in the panel, the exponential profile (S\'ersic profile with
$n=1$) fit the profile better than the $r^{1/4}$ profile (with $n=4$), and
the best-fit $n$ is 0.9 with free $n$ fitting.
The results of the S\'ersic profile fitting of the LBGs
with $K<21.5$ mag are shown in the right panel of Figure~\ref{akiyama_f02}
with filled and open squares, respectively. We also examined the profiles
of serendipitously observed Distant Red Galaxies in the FoVs.
The images of all but one of the LBGs and DRGs with $K<21.5$ mag are fitted
well with S\'ersic profile with $n$ index less than 2, similar to
disk galaxies in the local universe.
In order to directly compare the best-fit parameters with those
of galaxies at intermediate redshifts, we make simulated images of
$z\sim3$ galaxies by "cloning" HST/ACS images of galaxies at
$z=0.4-0.7$. Their distribution is shown with blue crosses in the
panel.
The LBGs and DRGs locate similar part of the $R_{e}$ vs. $n$ plane
to the disk galaxies at intermediate redshifts.
\begin{figure}[t]
\begin{center}
\includegraphics[width=110mm]{akiyama_f02.eps}
\end{center}
\caption{
Left) $K$-band profiles of an LBG along the semi-major axis.
The red-solid, blue-dashed, and green-dotted lines show the best
fit S\'ersic (with free $n$), $r^{1/4}$ (S\'ersic with $n=4$), and
exponential (S\'ersic with $n=1$) profiles, respectively. Long dashed
line show the estimated profiles of the PSFs at the object positions.
Right) $R_{e}$ vs. $n$ for the $z\sim3$ LBGs (filled squares) and DRGs (open squares)
with $K<21.5$ mag. Simulated $z=3$ galaxies that are brighter than $K=21.5$ mag
in the 2 mag PLE model are plotted with blue small crosses.}\label{akiyama_f02}
\end{figure}
Assuming that the $z\sim3$ LBGs and DRGs have a disk shape, we
compare their size-luminosity and size-stellar mass relations with
those of disk galaxies at low/intermediate redshifts in
the left and right panels of Figure~\ref{akiyama_f03}, respectively.
It should be noted that still there is a possibility that the $z\sim3$ LBGs
and DRGs have a spheroidal shape with $n=1$ profile instead of the disk shape.
The $z\sim3$ LBGs are brighter than $z=0$ and $1$ disk
galaxies at the same $R_{e}$. The surface brightness of the LBGs,
which are estimated from $M_{V}$ and $R_{e}$, are 2.2-2.9 mag
and 1.2-1.9 mag brighter than those of the disk galaxies at $z=0$ and $1$,
respectively.
The size-stellar mass relation indicates that
the $z\sim3$ LBGs brighter than $M_{V}^{*}$ have
the average
surface stellar mass density 3-6 times larger than those
of the $z=0$ and $1$ disk galaxies.
On the contrary, for less-luminous $z\sim3$ LBGs,
their size-stellar mass relation is similar to
those of $z=0$ and $1$ disk galaxies.
\begin{figure}[t]
\begin{center}
\includegraphics[width=110mm]{akiyama_f03.eps}
\end{center}
\caption{Left) $M_{V}$ vs. $R_e$ for the LBG sample (filled squares). Large
and small symbols indicate $R_{e}$ obtained with the S\'ersic profile fitting
with free $n$ and fixed $n=1$, respectively. Open squares show the DRG sample.
Contours show the distributions
of $z=0$ (dashed blue contour) and $z=1$ (solid red contour) disk-galaxies
from Barden et al. (2005). The blue and red dashed lines show $<\mu_{V}>$
of disk galaxies at $z=0$ ($20.84$ mag arcsec$^{-2}$) and $z=1$ (19.84 mag arcsec$^{-2}$),
respectively.
Right) $M_{*}$ vs. $R_{e}$ of the LBG (filled squares) and DRG (open squares)
samples. Contours show the distributions of $z=0$ and $1$ disk-galaxies
from Barden et al. (2005). The solid line represents the relation $\log \Sigma_{M}
(M_{\odot}$kpc$^{-2})=8.50$ derived from the disk galaxies at $z=0-1$
(Barden et al. 2005).}\label{akiyama_f03}
\end{figure}
\section{Speculation: LBGs before $z\sim3$ and after}
Because in the local universe, such
disk with high surface stellar mass density as the massive $z\sim3$ LBGs is rare, we expect that the disks of the
massive $z\sim3$ LBGs are destroyed between $z\sim3$ to $1$. The
strong spatial clustering of the LBGs implies that they reside in massive
dark halos and that they evolve into local spheroids (Adelberger et al. 2005).
Therefore, the disks with high surface stellar mass density would evolve into
local spheroids through "dry" merging events.
Considering the difference between the mass of the dark matter
halos the $z\sim3$ LBGs reside ($2-6\times10^{11} M_{\odot}$;
Adelberger et al. 2005) and the Jeans mass at the time of
reionization ($\sim10^{10} M_{\odot}$), we naively expect that the
$z\sim3$ LBGs form from builing blocks through several major merges.
The high fraction of disk-like galaxies in the high-redshift universe
would be explained with a hypothesis that they have been going through
only gas-rich "wet" merges at that time (e.g., Springel \& Hernquist 2005).
We start an AO observing program with laser guide stars in order to
extend the sample of bright $z\sim3$ galaxies, and to
establish the morphological evolution from $z=3$ to $0$.
The high-resolution $K$-band imaging observations of the galaxies at $z\ge3$
should be one of the unique fields that can be explored only with LGS AO on
ground-based 8-10m class telescopes until the launch of the James Webb Space Telescope.
| {'timestamp': '2007-09-21T22:09:09', 'yymm': '0709', 'arxiv_id': '0709.3523', 'language': 'en', 'url': 'https://arxiv.org/abs/0709.3523'} |
\section{INTRODUCTION}
One of the major achievements of observational cosmology in the 20th
century has been the detailed reconstruction of the large-scale
structure of what is now called the `local Universe' ($z\le 0.2$).
Large redshift surveys such as the 2dFGRS \citep{colless01,colless03}
and SDSS \citep{york00,sdss_dr7} have assembled samples
of over a million objects, precisely characterizing large-scale
structure in the nearby Universe on scales ranging from 0.1 to
$100\,h^{-1}{\rm Mpc}$. The SDSS in particular is
still extending its reach, using Luminous Red Galaxies
as highly effective dilute tracers of large volumes
\citep{eisenstein11,sdss_dr9}.
In addition to changing our view of the galaxy distribution around us,
the quantitative analysis of galaxy redshift surveys has
consistently yielded important advances in our knowledge of the
cosmological model.
Galaxy clustering on large scales is
one of the most important relics of the initial conditions that
shaped our Universe, and the observed shape of the power spectrum $P(k)$ of
density fluctuations [or of its Fourier transform, the correlation
function $\xi(r)$] indicates that we live in a low-density Universe
in which only $25-30\%$ of the mass-energy density is provided by
(mostly dark) matter. Combined with other observations, particularly
anisotropies in the Cosmic Microwave Background (CMB), this observation
has long argued for the rejection of open models in favour of
a flat universe dominated by a negative-pressure cosmological constant
\citep{efstathiou1990}. This conclusion predated the more direct
demonstration via the Hubble diagram of distant Type Ia Supernovae
\citep{riess98, perlmutter99}
that the Universe is currently in a phase of accelerated expansion.
Subsequent LSS and CMB data
\citep[e.g.][]{cole05,komatsu09,hinshaw2012} have only reinforced
the conclusion that the Universe is dominated by a repulsive `dark
energy'. Current observations are consistent with the latter
being in the simplest form already suggested by Einstein with his
Cosmological Constant, i.e. a fluid with non-evolving equation of
state $w=-1$.
Theoretical difficulties with the cosmological constant,
specifically the smallness and fine-tuning problems
\citep[e.g.][]{weinberg89} make scenarios with evolving
dark energy an appealing alternative. This is the motivation for
projects aiming at detecting a
possible evolution of $w(z)$. Redshift surveys are playing a crucial
role in this endeavour, in particular after the discovery of the
signature of Baryonic Acoustic Oscillations (BAO) from the pre-recombination
plasma into large-scale structure. This `standard rod' on a comoving scale of
$\sim 110\,h^{-1}{\rm Mpc}$ \citep{percival01,cole05,eisenstein05} provides us
with a powerful mean to measure the expansion history $H(z)$ via the
angular diameter distance (e.g. \citejap{percival10}, \citejap{blake11},
\citejap{anderson12}).
An even more radical explanation of the observed accelerated
expansion could be a breakdown of General Relativity (GR) on
cosmological scales \citep[see e.g.][]{carroll04,jain2010}. Such a scenario is
fully degenerate with dark energy in terms of $H(z)$, a degeneracy
that in principle can be lifted by measuring the growth rate of
structure, which depends on the specific theory describing gravity.
There are in principle several experimental ways to measure the growth
of structure. Galaxy peculiar motions, in particular, directly
reflect such growth. When the redshift is used as a distance
proxy, they produce a measurable effect on
clustering measurements, what we call {\it Redshift Space
Distortions\/} (RSD: \citejap{kaiser87}). The
anisotropy of statistical measurements like the two-point correlation
function is proportional to the growth rate of cosmic structure
$f(z)$, which is a trademark of the gravity theory: if GR holds, we
expect to measure a growth rate $f(z)=[\Omega_M(z)]^{0.55}$
\citep{peebles80, lahav1991}. If gravity is modified on large scales, different
forms are predicted \citep[e.g.][]{dvali00, linder2007}. In fact, although the
RSD effect has been well
known since the late 1980s (\citejap{kaiser87}), its potential in the
context of dark energy and modified gravity has become clear only
recently \citep{zhang07, guzzo08}. The RSD method is now considered to be one of
the most promising probes for future dark energy experiments,
as testified by the exponential growth in
the number of works on both measurements \citep[e.g.][]{beutler10,
blake11, reid12}, and theoretical modelling \citep[e.g.][]{song09,
percival09, white09, scoccimarro04, taruya10, kwan12, reid11,
delatorre12}. Redshift surveys are thus expected to be as
important for cosmology in the present Century as they
were in the previous one, as suggested
by their central role in several planned experiments -- especially
the ESA dark-energy mission, Euclid \citep{laureijs11}.
The scientific yield of a redshift survey, however, extends
well beyond fundamental cosmological aspects. It is equally
important to achieve an understanding of the relationship between
the observed baryonic components in galaxies and the dark-matter
haloes that host them. For this purpose, we need to build
statistically complete samples of
galaxies with measured positions, luminosity, spectral properties and
(typically) colours and stellar masses; in providing
such data, redshift surveys are thus a vital
probe of galaxy formation and evolution.
Significant statistical progress has been made in relating the galaxy
distribution to the underlying dark matter, via
Halo Occupation Distribution modelling of accurate estimates of the galaxy
two-point correlation function, for samples selected in luminosity, colour and
stellar mass \citep[e.g.][]{seljak00, peacock00, cooray02, zheng04}.
At the same time, important global galaxy population trends
involving properties such as luminosities, stellar masses, colours and
structural parameters can be precisely measured when these parameters are available
for $\sim10^6$ objects, as in the case of the SDSS
\citep[e.g.][]{kauffmann03}.
In more recent years, deeper redshift surveys over areas of 1-2
deg$^2$ have focused on exploring how this detailed picture
emerged from the distant past. This was the direct consequence of
the development during the 1990s of multi-object spectrographs on 8-m class
telescopes. The most notable
projects of this kind have been the VIMOS VLT Deep Survey
(VVDS; \citejap{lefevre05}), the DEEP2 survey \citep{coil08} and the zCOSMOS
survey \citep{lilly09}, which adopted various strategies aimed at
covering an extended redshift range, up to $z\sim 4.5$. Such depths
inevitably limit the angular size and thus the volume explored in a
given redshift interval, reflecting the desire of these projects to
trace galaxy evolution back to its earliest phases, while
understanding its relationship with environment over a limited range
of scales. Evolutionary trends in the dark-matter/galaxy connection
were explored using these surveys \citep{zheng07, abbas10}, but
none of these samples had sufficient volume to produce
stable and reliable comparisons of e.g. the amplitude and shape
of the correlation function. Only the Wide extension of VVDS
\citep{garilli08}, started to have sufficient volume as to attempt
cosmologically meaningful computations at $z\sim 1$
\citep{guzzo08}, albeit with large error bars. In general, clustering
measurements at $z\sim 1$ from these samples remained dominated by
field-to-field fluctuations (cosmic variance), as dramatically shown by
the discrepancy observed between the VVDS and zCOSMOS correlation
function estimates at $z\simeq 0.8$ \citep{delatorre10}.
At the end of the past decade it was therefore clear that a new step in
deep redshift surveys was needed, if these were to produce
statistical results that could be
compared on an equal footing with those derived from surveys of the local
Universe, such as 2dFGRS and SDSS.
Following those efforts, new generations of cosmological surveys have focused on
covering the largest possible volumes at intermediate depths, utilizing
relatively low-density tracers, with the main goal of measuring the BAO signal
at redshifts 0.4-0.8. This is the case with the SDSS-3 BOSS project
\citep{eisenstein11}, which extended the concept pioneered by the SDSS
selection of Luminous Red Galaxies
\citep[e.g.][]{anderson12, reid12}. Similarly, the WiggleZ survey
further exploited the long-lived 2dF positioner on the AAT 4-m telescope, to
target emission-line galaxies selected from UV observations of the
GALEX satellite \citep{drinkwater10, blake11, blake11b}. Both these
surveys are characterized by a very large volume
($1-2\, h^{-3}{\rm Gpc}^3$), and a relatively sparse galaxy population ($\sim
10^{-4}\,h^{3}{\rm Mpc}^{-3}$). This is typical of surveys performed
with fibre positioning spectrograph, which normally can observe
500-1000 galaxies over areas of 1-2 square degrees. Higher galaxy densities
can be achieved with such systems via multiple visits, although this then
limits the redshift and/or volume surveyed. This approach has been taken by the GAMA survey
\citep{driver2011}, which aims to achieve similar numbers of redshifts
to the 2dFGRS ($\sim 200,000$), but working to $r<19.8$ and out to $z\simeq 0.5$.
Indeed, the high sampling density of GAMA makes it an important intermediate
step between the local surveys and the higher redshifts probed by
the survey we are presenting in this paper, i.e. VIPERS.
\begin{figure}
\centering
\includegraphics[width=\hsize]{tilesW1.pdf}
\includegraphics[width=\hsize]{tilesW4.pdf}
\caption{The areas covered by VIPERS within the CFHTLS-Wide W1
(top) and W4 (bottom)
fields. The internal numbering reported on each tile is
linked to the CFHTLS naming convention in
Table~\ref{tab:W1_tiles} and \ref{tab:W4_tiles}
in the Appendix. Also
shown are the positions of the VVDS-Deep \citep{lefevre05} and
VVDS-Wide \citep{garilli08} survey fields.
}
\label{fig:tilesW1W4}
\end{figure}
VIPERS stands for VIMOS Public Extragalactic Redshift Survey and has been
designed to measure redshifts for approximately
100,000 galaxies at a median redshift $z\simeq 0.8$. The central goal
of this strategy is to build a data set capable of achieving an
order of magnitude improvement on the key
statistical descriptions of the galaxy distribution and internal
properties, at an epoch when the Universe was about
half its current age. Such a data set would allow combination
with local samples on a comparable statistical footing.
Despite being centred at $\bar z\sim 0.7$, in terms of volume and number
density VIPERS is similar to local surveys like
2dFGRS and SDSS. All these surveys are characterized by
a high sampling density, compared to the sparser samples of the
recent generation of BAO-oriented surveys.
In this paper we provide an overview of the VIPERS survey design and
strategy, discussing in some detail the construction of the target
sample. The layout of the paper is as follows: in
\S~\ref{sec:design}, we discuss the survey design;
in \S~\ref{sec:photcat} we describe the properties of the VIPERS
parent photometric data and the build-up of a homogeneous sample over
24~deg$^2$; in \S~\ref{sec:mag-col} we discuss how from these
data the specific VIPERS target sample at $z>0.5$ has been selected,
using galaxy colours; in \S~\ref{sec:obs} the details of the VIMOS
observations and the general properties of the spectroscopic sample
are presented; in \S~\ref{sec:selfun} we discuss the various selection
effects and how they have been accounted for; finally, in
\S~\ref{sec:results} we present the redshift and large-scale spatial
distribution of the current sample, summarizing the scientific
investigations that are part of separate papers currently submitted or
in preparation.
As a public survey, we hope and expect that the range of science that
will emerge from VIPERS will greatly exceed the core analyses
from the VIPERS Team. This paper is therefore also to introduce the
new VIPERS data, in view of the first Public Data Release (PDR-1),
which will be available at {\tt http://vipers.inaf.it} in September
2013 and that will be described in more detail in a specific
accompanying paper.
\section{SURVEY DESIGN}
\label{sec:design}
VIPERS was conceived in 2007 with a focus on clustering and RSD at
$z\simeq 0.5-1$,
but with a desire to enable broader goals involving large-scale
structure and galaxy evolution, similarly to the achievements of 2dFGRS
and SDSS at $z\simeq 0.1$. The survey design was also strongly
driven by the specific features of the
VIMOS spectrograph, which has a relatively small field of view
compared to fibre positioners ($\simeq 18 \times 16\, {\rm arcmin}^2$;
see \S~\ref{sec:selfun}), but a
larger yield in terms of redshifts per unit area.
Given the luminosity function of galaxies and results
from previous VIMOS surveys as VVDS \citep{lefevre05,garilli08} and
zCOSMOS \citep{lilly09}, we knew that
a magnitude-limited sample with $i_{AB}<22.5-23.0$ would cover
the redshift range out to $z\sim1.2$, and could be assembled with
fairly short VIMOS exposure times
($<1$ hour). Also, taking 2dFGRS as a local
reference, a survey volume around $5 \times 10^{7}\,h^{-3}{\rm Mpc}^3$ could be
explored by observing an area of $\sim 25$~deg$^2$. The first attempt
towards this kind of survey was VVDS-Wide, which covered $\sim 8$~deg$^2$
down to a magnitude $i_{AB}=22.5$, but observing all objects (stars and
galaxies), with low sampling ($\simeq20\%$).
Building upon this experience,
VIPERS was designed to maximize the number of galaxies observed in
the range of interest, i.e. at $z>0.5$, while at the same time
attempting to select against stars, which represented a contamination up to 30\% in some of the
VVDS-Wide fields. The latter criterion requires multi-band photometric information
and excellent seeing quality, but these qualities also benefit the
galaxy sample, where a wider range of ancillary science is enabled
if the galaxy surface-brightness profiles can be well resolved. The
outstanding imaging dataset that was available for these purposes was
the Canada-France-Hawaii Telescope Legacy Survey
(CFHTLS) Wide photometric catalogue, as described below in \S~\ref{sec:photcat}.
The desired redshift range was isolated through a simple
and robust colour-colour selection on the $(r-i)$ vs $(u-g)$ plane
(as shown in Fig. \ref{fig:col-col}). This is one of many ways in which
we have been able to benefit from the experience of previous VIMOS
spectroscopic surveys: we could be confident in advance that this
selection method would efficiently remove galaxies at $z<0.5$, while
yielding $>98\%$ completeness for $z>0.6$, as verified in the results
shown below. A precise calibration of this separation method
was made possible by the location of the VVDS-Wide ($i_{AB}<22.5$) and VVDS-Deep
($i_{AB}<24$) samples within the W4 and W1 fields of CFHTLS,
respectively. This was an important reason for locating the VIPERS
survey areas within these two CFHTLS
fields while partly overlapping the original VVDS areas, as shown in
Fig.~\ref{fig:tilesW1W4}. The magnitude
limit was set as in VVDS-Wide, i.e. $17.5\le i_{AB} \le 22.5$ (after
correction for Galactic extinction).
The details of the star-galaxy
separation are discussed in Appendix~\ref{app:stars},
while the colour-colour selection is described in \S\ref{sec:mag-col}.
\section{PHOTOMETRIC SOURCE CATALOGUE}
\label{sec:photcat}
The VIPERS target selection is derived from the `T0005' release of
the CFHTLS Wide which was available for the first observing season
2007/2008. This object selection was completed and improved using the
subsequent T0006 release, as we will now describe.
The mean limiting AB magnitudes of CFHTLS Wide (corresponding to the
50\% completeness for point sources) are $\sim
25.3,25.5,24.8,24.48,23.60$ in $u^\ast,g',r',i',z'$, respectively. To
construct the CFHTLS catalogues used here, objects in each tile were
detected on a $gri$-$\chi^2$ image \citep{szalay99} and
galaxies were selected using \texttt{SEXtractor}'s
`\texttt{mag\_auto}' magnitudes \citep{bertin96}, in the
AB
system\footnote{http://terapix.iap.fr/rubrique.php?id\_rubrique=252}. These
are the magnitudes used throughout this work, after they have been
corrected for foreground Galactic extinction using the following
prescription:
\begin{eqnarray}
u &=& u^\ast_{raw} - 4.716 * E(B-V) \\
g &=& g'_{raw} - 3.654 * E(B-V) \\
r &=& r'_{raw} - 2.691 * E(B-V) \\
i &=& i'_{raw} - 1.998 * E(B-V) \\
z &=& z'_{raw} - 1.530 * E(B-V) \;\;\; ,
\end{eqnarray}
%
where the extinction factor $E(B-V)$ is derived at
each galaxy's position from the Schlegel dust maps \citep{schlegel98}.
When the first target catalogues were generated, the CFHTLS survey
included some photometrically incomplete areas (`holes'
hereafter). In these areas one or more bands was either corrupted or
missing. In particular, all of the VIPERS W1 field at right
ascensions less than $RA\simeq 02^h \; 09^\prime$ were missing one
band as CFHTLS Wide observations had not been completed. Smaller
survey holes were mostly due to the partial failure of amplifier
electronics (since all CCDs have two outputs, some images are missing
only half-detector areas).
In general, these missing bands meant that we were not able to select
VIPERS targets in the affected areas and they were therefore excluded
from our first two observing seasons (2008 and 2009). The majority of
these problems were fixed in Summer 2010 using the CFHTLS-T0006, which
was carefully merged with the existing VIPERS target list. The T0005
and T0006 catalogs, limited to $i_{AB} < 23.0$, were positionally
matched over the area of each hole, using a search radius of 0.6
arcsec. All matches with a compatible $i$-band magnitude (defined as
having a difference less than 0.2 mag) were considered as good
identifications and used to verify the consistency between the two
releases.
For objects near the VIPERS faint limit, i.e. $i_{AB} \sim 22.5$, the
{\it rms} magnitude offset between the two catalogues was found to
range between 0.02 to 0.04 mag (larger in the $u$-band), and
smaller than this for brighter objects. Given this result, we
concluded that the T0006 version of galaxy magnitudes could be used
directly to replace the bad or missing magnitudes for the original
T0005 objects in the holes. This solution was definitely preferable to
replacing \textit{all} magnitudes with their T0006 values, an operation that
would have modified the target sample at the faint limit simply due to
statistical scatter.
Only a few of the T0005 holes arising from CCD failures were not
filled by the T0006 release. To complete these remaining areas,
Director's Discretionary Time (DDT) was awarded at CFHT with MegaCam
in summer 2009 (S. Arnouts \& L. Guzzo, private communication). At the
end of 2010, the combination of new T0006 observations and the DDT
data resulted in a virtually complete coverage in all five bands of
the two VIPERS areas in W1 and W4. The last problem to be resolved
was re-calibrating a few small areas which were observed in T0006 with
a new $i$-band filter, called `$y$', as the original $i$-band filter
broke in 2007. This procedure is described in Appendix~\ref{app:col}.
\begin{figure}
\centering
\includegraphics[width=\hsize]{tile9tile11uggr.pdf}
\caption{One of largest tile-to-tile magnitude zero-point
variations in the T0005 data. The position of the
stellar sequence in the $(g-r)$ vs. $(u-g)$ plane is compared
for tile \#9 and tile \#11 in the W4 VIPERS area (see \S~\ref{tile-numbers}),
showing an offset of $\sim 0.15$ magnitudes in $(g-r)$ and
$\sim 0.06$ in $(u-g)$ between
the two tiles.
}
\label{fig:zero-point-shift-example}
\end{figure}
\subsection{Tile-to-Tile Zero-Point Homogenisation}
\label{color-corrections}
The CFHTLS data are provided in single tiles of $\sim 1$ deg side,
overlapping each other by $\sim 2$ arcmin to allow for
cross-calibration. These are shown in Fig.~\ref{fig:tilesW1W4} for the W1 and W4
fields, together with the
position of the two VIPERS areas. To build the VIPERS global catalogue we merged
adjacent tiles, eliminating duplicated objects. In these cases, the
object in a pair having the best Terapix flag was chosen; if the flags
were identical, the object at the greater distance from the tile
border was chosen. Tiles were merged proceeding first in right
ascension rows and then merging the rows into a single catalogue.
For any galaxy survey planning to measure large-scale clustering it is
crucial that the photometric or colour selection is as homogeneous as
possible over the full survey area in order to avoid creating spurious
object density fluctuations that could be mistaken as real
inhomogeneities. Given the way the CFHTLS-Wide catalogue has been
assembled, verifying and correcting any tile-to-tile variation of this
kind is therefore of utmost importance. In fact, it was known and
directly verified that each tile in T0005 still had a small but
non-negligible zero-point offset in some of the photometric
bands. These offsets are a consequence of non-photometric images being
used as photometric anchor fields in the global photometric solution.
These tile-to-tile colour variations are evident when stars are
plotted in a colour-colour diagram, as in
Fig.~\ref{fig:zero-point-shift-example}. In this figure we show the
$(u-g)$ vs. $(r-i)$ colours for stellar objects in two particularly
discrepant tiles (see Appendix~\ref{app:stars} for details on how stars and galaxies are
separated). Such offsets can produce two kinds of systematic effects
in a survey like VIPERS. First, a tile-to-tile difference in the
selection magnitude ($i$ band) would introduce a varying survey depth
over the sky and thus a variation in the expected number counts and redshift
distribution. Secondly, the colours would be affected, and thus any
colour-colour selection (as the one applied to select galaxies at
$z>0.5$ for the VIPERS target catalogue -- see next section), would vary
from one tile to another.
The well-defined location of stars in colour-colour space, as shown in
Fig.~\ref{fig:zero-point-shift-example}, suggests a technique for a
possible correction of the colour variations, i.e. using the observed
stellar sequence as a colour calibrator \citep[see][for a similar more
recent application of this regression technique]{high09}. An
important assumption of this correction procedure is that stars and
galaxies are affected by similar zero-point shifts, and thus that
stellar sequences can also be used to improve the photometric
calibration of extended objects. This assumption is quite reasonable
and it is the same adopted at Terapix in the past to check internal
calibration until the second-last release, i.e. T0006. With the latest
release, T0007, there are indications that a contribution to these
zero-point discrepancies could be also due to a dependence on seeing
of \texttt{mag\_auto} when applied to stellar objects. This effect is
not fully understood yet and its amplitude is smaller than the
corrections we originally applied to the T0005 data. The potential
systematic impact of this uncertainty, in particular on clustering
analyses of the PDR-1 sample, is explicitly addressed in the
corresponding papers \citep[see e.g.][]{delatorre13}.
The colour corrections were carried out assuming (a) that the $i$-band
magnitude had a negligible variation from tile to tile, and (b) taking
the colours measured in tile W1-25 (see Fig.~\ref{fig:tilesW1W4}) as the
reference ones. W1-25 is the tile overlapping the VVDS-Deep survey,
which was used to calibrate the colour selection criteria as discussed
in \S~\ref{sec:mag-col}. By referring all colours to that tile, we
assured (at least) that the colour-redshift correlation we calibrated
was applied self-consistently to all tiles. For all tiles covered by
VIPERS we measured therefore the $(u-g)$ value of the blue-end cut-off
in the stellar sequence, clearly visible in
Fig.~\ref{fig:zero-point-shift-example}, together with the zero points
derived from a linear regression to the $(g-r)\, vs\, (u-g)$ and
$(r-i)\, vs\, (u-g)$ relationships for stars. These two regressions
give a consistent slope of 0.50 and 0.23, respectively, over all
tiles. This allowed us to compute three colour offsets $\delta_{ug}$ ,
$\delta_{gr}$ and $\delta_{ri}$ for each tile, corresponding to the
values required to match the same measurements in W1-25.
All following steps in the selection of VIPERS target galaxies were then operated
on colours corrected using these offsets, i.e.
\begin{eqnarray}
(u-g) & = & (u-g)_{uncorr} - \delta_{ug} \\
(g-r) & = & (g-r)_{uncorr} - \delta_{gr} \\
(r-i) & = & (r-i)_{uncorr} - \delta_{ri} \,\,\,\,\, .
\end{eqnarray}
%
\section{SELECTION OF VIPERS GALAXY TARGETS}
\label{sec:mag-col}
\begin{figure}
\centering
\includegraphics[width=\hsize]{color_color_sel.pdf}
\caption{Distribution in the $(r-i)$ vs $(u-g)$ plane of
$i_{AB}<22.5$ galaxies with known redshift from the VVDS-Deep survey, showing the
kind of selection applied to construct the VIPERS
target sample. The colour selection of eq.~\ref{eq:col-col}
is described by the continuous line, which empirically splits the sample into $z>0.5$ (red filled
circles) and $z<0.5$ (blue stars) by optimizing the completeness
and contamination of the high-redshift sample.
}
\label{fig:col-col}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\hsize]{check_hist.pdf}
\caption{Test of the colour-colour redshift selection, using
galaxies with known redshift from the VVDS-Deep survey. The
colour locus in Fig.~\ref{fig:col-col} is used to separate
{\it a priori} galaxies lying below (blue-dashed line) and above
(red solid line) $z\simeq 0.5$. The dotted black line shows
the global $dN/dz$ of the sample. The VVDS-Deep sample has been
limited to objects belonging to tile \# 25 (where the bulk of
the sample is concentrated), given that this has been used
as the reference for the global colour calibration discussed in
the text.
}
\label{fig:col-hist}
\end{figure}
Around half of the galaxies in a magnitude-limited sample with
$i_{AB}<22.5$ are at $z<0.5$. At the same time, the average number
of slits that can be accommodated within one of the four VIMOS
quadrants (see below) is approximately fixed, for a parent sample
with a given clustering. This means that in a pure magnitude
limited survey at this depth, around half of the slits would fall
on $z<0.5$ galaxies. Given the original goal of VIPERS to build a
sample complementary to local surveys, a strategy was devised as to
select \textit{a priori} only galaxies at higher redshifts,
doubling in this way the sampling over the high-redshift range.
Using available magnitude-limited VVDS data, a simple yet effective
and robust colour selection criterion was devised through a series
of experiments. The most effective criteria are shown in
Fig.~\ref{fig:col-col} applied to the VVDS data. Galaxies are
retained in the source list if their colours obey the
following relationship:
\begin{equation}
(r-i) > 0.5 (u-g) \;\;\;\ {\rm OR} \;\;\; (r-i)>0.7 \,\,\,\, .
\label{eq:col-col}
\end{equation}
The resulting true redshift distribution of the selected samples is
shown in Fig.~\ref{fig:col-hist}, with the corresponding level of
completeness as a function of redshift explicitly computed in
Fig.~\ref{fig:CSR}. In the latter figure, we used the VVDS data (both Deep
and Wide), and plot the ratio of the numbers of objects in a
VIPERS-like selected sample, to the total sample. We call this
quantity the Colour Sampling Ratio (CSR). As this figure
shows, the VIPERS selection does not introduce any colour bias
(i.e. it selects virtually all galaxies) above $z\sim 0.6$, with an
acceptable contamination ($\sim 5 \%$) of low-redshift interlopers.
An alternative technique to select a high-redshift sample could have
been to use photometric redshifts computed using all five bands. We
verified that this method provides comparable performance in terms of
completeness and contamination to the colour-colour selection. However
we preferred a simple colour-colour criterion, as it can be reproduced
precisely at any time, while photometric redshifts depend inevitably
also on the features of the specific codes and template selection
used, which will evolve with time.
\begin{figure}
\centering
\includegraphics[width=\hsize]{csr.pdf}
\caption{A direct verification of the completeness of the VIPERS
colour selection as a function of redshift,
using both VVDS-Deep and VVDS-Wide data, in W1 and W4
respectively. Note that the original colour criteria were
defined based only on the VVDS-Deep data. The curves and points give the
Colour Sampling Rate (CSR), i.e. the ratio of the number of
galaxies satisfying the VIPERS criteria
within a redshift bin and the total number of galaxies in that
same bin. Both fields provide consistent
selection functions, indicating that the colour-colour
selection function is basically unity above $z=0.6$ and can be
consistently modelled in the transition region $0.4<z<0.6$.
}
\label{fig:CSR}
\end{figure}
Finally, to further broaden the scientific yield of VIPERS,
the galaxy target catalogue was supplemented with two small additional samples
of AGN candidates. These include a sample of X-ray selected AGNs from
the XMM-LSS survey in the W1 field \citep{pierre07}, and a sample of
colour-defined AGN
candidates selected among objects classified as stars in the previous
phase. These two
catalogues contributed on average 1-3 objects per quadrant (against
about 90 galaxy targets) with negligible impact on the galaxy
selection function. These AGN candidates are excluded from the current PDR-1
sample. All the details on the selection criteria and the properties
of the resulting objects will be discussed in a future paper.
\section{VIMOS OBSERVATIONS}
\label{sec:obs}
\subsection{ The VIMOS Spectrograph}
The VIPERS project is designed around VIMOS (VIsible
Multi-Object Spectrograph), on `Melipal', the ESO Very Large
Telescope (VLT) Unit 3 \citep{lefevre03}.
VIMOS is a 4-channel imaging spectrograph; each channel
(a `quadrant') covers $ \sim 7 \times 8 $ arcmin$^2$ for
a total field of view (a `pointing') of $\sim218$ arcmin$^2$.
Each channel is a complete spectrograph with the
possibility to insert $\sim30\times30$ cm$^2$ slit masks
at the entrance focal plane, as well as broad-band filters or
grisms. The standard lay-out of the four quadrants on the sky is
reproduced in Fig.~\ref{fig:VIMOS-layout}. The figure shows the slit
positions and the resulting location of the spectra, overlaid on the
direct pre-image of pointing P082 in the W1 field.
\begin{figure*}
\centering
\includegraphics[width=14truecm]{W1P082_slits_bw.jpg}
\caption{Example of the detailed footprint and disposition of the
four quadrants in a full VIMOS pointing (W1P082 in this
case). Note the reconstructed boundaries (solid red lines), which
have been traced pointing-by-pointing through an automatic
detection algorithm that follows the borders of the
illuminated area. These can vary in general among different
pointings in the database, in particular due to the CCD
refurbishment of 2010 and sometimes to
vignetting by the telescope guide probe arm.
}
\label{fig:VIMOS-layout}
\end{figure*}
The pixel scale on the CCD detectors is 0.205 arcsec/pixel,
providing excellent sampling of the Paranal mean image quality and
Nyquist sampling for a slit 0.5 arcsecond in width. For the VIPERS
survey, we use slits of 1 arcsecond, together with the
`Low-Resolution Red' (LR-Red) grism, which provides a spectral
resolution $R\simeq 250$. The instrument has no atmospheric
dispersion compensator, given the large size of its field-of-view
at the VLT Nasmyth focus ($\simeq1$m). For this reason,
observations have to be limited to airmasses below 1.7. For VIPERS
observations we rarely went above an airmass of 1.5.
To prepare the MOS masks, direct exposures (`pre-images') need to be
observed beforehand under the same instrumental conditions. Object
positions in these images are then cross-correlated with the target
catalogue in order match its astrometric coordinates to the actual
instrument coordinate system. This operation is performed during the
mask preparation using VMMPS, the standard package for automatic
optimisation of the positions and total number of slits
\citep{bottini05}.
In summer 2010, VIMOS was upgraded with new red-sensitive CCDs in each of
the 4 channels, as well as with a new active flexure compensation
system. The reliability of the mask exchange system was also improved
\citep{hammersley10}. The original thinned E2V detectors
were replaced by twice-thicker E2V devices, considerably lowering
the fringing and increasing the global instrument efficiency by up to
a factor 2.5 (one magnitude) in the redder part of the
wavelength range. This upgrade significantly improved the average
quality of VIPERS spectra, resulting in a significantly higher
redshift measurement success rate.
\subsection{Data Reduction, Redshift Measurement and Validation}
\label{SpectralDataRed}
VIPERS is the first VIMOS redshift survey for which the data reduction
is performed with a fully automated pipeline, starting from the raw
data and down to the calibrated spectra and redshift measurements. The pipeline
includes and updates algorithms from the original VIPGI system
\citep[][]{scodeggio05} within a complete purpose-built environment.
Within it, the standard CCD data reduction, spectral extraction and
calibration follow the usual recipes discussed in previous VIMOS
papers \citep{lefevre05, lilly09}. The difference in the case of
VIPERS is that the only operation for which we still require human
intervention is the verification and validation of the measured
redshift. All data reduction has been centralised in our data
reduction and management centre at INAF - IASF Milano. When ready, the fully
reduced data are made available to the team within a dedicated
database. The full management of these operations within the
`EasyLife' environment is described in \citet{garilli08}.
Fig.~\ref{fig:spectra} shows a few examples of VIPERS spectra, for
galaxies with varying redshift and quality flag. In common with
previous VIMOS surveys \citep[e.g.][]{lefevre05, lilly09}, all
redshifts have been validated independently by two scientists but with
some simplification to increase efficiency given the very large number
of spectra. Nevertheless, this required a very strong team effort.
Two team members are assigned the same VIMOS field to review, with one
of the two being the primary person responsible for that pointing. At
the end of the process discrepant redshifts resulting from the two
reviewers are discussed and reconciled.
\begin{figure*}
\centering
\vskip -3truecm
\includegraphics[width=\hsize]{vipers_8spectra.pdf}
\vskip -5truecm
\caption{Examples of representative VIPERS spectra of early- and late-type
galaxies, chosen among the different quality classes
(i.e. quality flags)
and at different redshifts. The typical absorption and
emission features are marked.
}
\label{fig:spectra}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\hsize]{vel_error.pdf}
\caption{The distribution of the differences between two
independent redshift measurements of the same object, obtained
from a set of 1215 VIPERS galaxies with quality flag $\ge 2$.
In the bottom panel, the darker dots correspond to
top-quality redshifts (i.e. flags 3 and 4), which show a
dispersion substantially similar to the complete sample (see
text). Catastrophic failures (defined as being discrepant by
more than $\Delta z = 6.6\times 10^{-3} (1+z)$) have obviously been excluded. Top: distribution of the
corresponding differences $\Delta v = c\Delta z /(1+z)$. The
best-fitting Gaussian has a dispersion of $\sigma_2=200$ km s$^{-1}$,
corresponding to a single-object {\it rms} error
$\sigma_v=\sigma_2/\sqrt{2}=141$ km s$^{-1}$. In terms of
redshift, this translates into a standard deviation of
$\sigma_z=0.00047(1+z)$ for a single galaxy measurement.
}
\label{fig:double-obs}
\end{figure}
The quality of the measured redshifts is quantified at the time of
validation through a similar grading scheme to that described in
\citet{lefevre05, lilly09}. The corresponding confidence levels are
estimated from repeated observations, as explained in
\S~\ref{sec:error} and \S~\ref{sec:conf}):
\begin{itemize}
\item
Flag 4.X: a high-confidence, highly secure redshift, based on a high SNR spectrum and
supported by obvious and consistent spectral features. The
combined confidence level of Flag 4 + Flag 3 measurements is estimated to be $>99\%$
\item
Flag 3.X: also a very secure redshift, comparable in confidence
with Flag 4, supported by clear spectral features in the spectrum, but not necessarily with high SNR.
\item
Flag 2.X: a fairly secure, $\sim 95\%$ confidence redshift measurement,
with sufficient spectral features in support of the measurement.
\item
Flag 1.X: a tentative redshift measurement, based on weak spectral features
and/or continuum shape, for which there is $\sim 50\%$ chance that
the redshift is actually wrong.
\item
Flag 0.X: no reliable spectroscopic redshift measurement was possible.
\item
Flag 9.X: redshift is based on only one single clear spectral emission feature, usually
identified (in the VIPERS range) with [OII]3727 \AA.
\item
Flag -10: spectrum with clear problems in the observation or data
processing phases. In most cases this is a failed extraction by VIPGI
\citep[][]{scodeggio05}, or a bad sky subtraction because the object is too
close to the edge of the slit.
\end{itemize}
Serendipitous objects appearing by chance within the slit of the main
target are identified by adding a `2' in front of the main flag.
A decimal part of the flag `.X' is then added to the main flag
defined in this way after the final human review of the
redshifts. This is performed by an automatic algorithm, which
cross-correlates the spectroscopic measurement ($z_{spec}$) with the
corresponding photometric redshift ($z_{phot}$), estimated from the
five-band CFHTLS photometry using the {\it Le Phare} code
\citep{ilbert06,arnouts11}. The 68\% confidence interval $[z_{ph-min},
z_{ph-max}]$ (in general not symmetric), is provided by the code based
on the PDF of the estimated $z_{phot}$, allowing us to verify the
statistical agreement between the two values. If $z_{ph-min} <
z_{spec} < z_{ph-max}$, then they are considered in agreement and a
flag 0.5 is added to the primary flag. Thus, a flag `*.5' is an
indication, whatever the primary integer flag is, supporting the
correctness of the redshift. This is particularly useful in the case
of highly uncertain, flag=1 objects, for which confidence can be
increased. Flag 0.4, for which the redshifts were only in marginal
agreement, was assigned for cases in which the two redshifts are
compatible only at the $2\sigma$ level, where $\sigma$ is now the
global (median) symmetric scatter of the photometric redshifts,
$2\sigma_z(z_{phot}) = 0.05(1+z_{phot})$. These cases are considered
only if this $2\sigma$ interval is larger than the primary 68\%
confidence interval (if not, they go back to the first category). This
allows us to signal cases in which the PDF of the single measurement
is rather narrow, but still the spectroscopic redshift is close. We
finally have the cases `0.2', when neither of the two criteria is
satisfied, and `0.1', when no $z_{phot}$ estimate is available.
In all VIPERS papers redshifts with flags ranging between 2.X and
9.X
are referred to as secure redshifts and are the only ones normally
used in the science analyses.
\subsection{Error on Redshift Measurements}
\label{sec:error}
For 783 galaxies in the VIPERS PDR-1 sample a repeated, reliable
redshift measurement exists. These are objects lying at the border of
the quadrants, where two quadrants overlap, and were therefore
observed by two independent pointings. In addition, during the
re-commissioning of VIMOS after the CCD refurbishment in summer 2010,
a few pointings were re-observed to verify the performances with the
new set-up \citep{hammersley10}, targeting another 1357
galaxies.
In total, this gives a
sample of 1941 galaxies with double observations. 1215 of these
yield a reliable redshift (i.e. with a flag $\ge 2$) in both
measurements and can be conveniently used to obtain an estimate of the
internal {\it rms} value of the redshift error of VIPERS galaxies.
The bottom panel of Figure~\ref{fig:double-obs} shows the distribution
of the differences between these double measurements. The sign of
these differences is clearly arbitrary. These have been computed as
$z_2-z_1$, where `1' and `2' are chronologically ordered in terms of
observation date. Once normalised to the corresponding redshift
expansion factor $1+z$, the overall distribution of these measurements
is very well described by a Gaussian with a dispersion of
$\sigma_2=200$ km s$^{-1}$, corresponding to a single-object $1\sigma$
error $\sigma_v=\sigma_2/\sqrt{2}=141$ km s$^{-1}$. In terms of
redshift, this yields a standard deviation on the redshift
measurements of $0.00047(1+z)$. If we restrict ourselves to the
highest quality spectra (i.e. flags 3 and 4), we are left with 655
double measurements; the resulting rest-frame 2-object dispersion
changes very little, decreasing to $\sigma_2=193$ km s$^{-1}$. This
indicates that flags 2, 3 and 4 are substantially equivalent in terms
of redshift precision.
\subsection{Confidence Level of Quality Flags}
\label{sec:conf}
\begin{table}
\caption{Redshift confidence levels corresponding to the VIPERS quality
flags, estimated from pairs of measurements of the same galaxy.}
\label{tab:flags}
\centering
\begin{tabular}{l c}
\hline\hline
Flag Class & $z$ confidence level\\
\hline
{\bf 3+4} & 99.6\%\\
{\bf 2} & 95.1\%\\
{\bf 1} & 57.5\%\\
\hline\hline
\end{tabular}
\end{table}
Repeated observations allow us to quantify in an objective way the
statistical meaning of our quality flags, which are by nature
subjective; they are assigned by individuals in a large, geographically
dispersed team. Remarkably, the grading system turns out to be quite
stable and well-defined as we will now see.
Let us define two redshifts as `in agreement' when $\Delta
z/(1+z) < 3 \sigma_z \simeq 0.0025$. We compare the redshifts of
double measurements from the VIPERS sample only, considering the flag
assigned to both measurements. Flags 3 and 4 are considered together,
as they should not be different in practice in terms of strict
redshift reliability. We therefore consider pairs of measurements, in
the following cases:
\begin{enumerate}
\item both measurements have flag=3 or 4: out of 655 pairs, 5 have discrepant redshift.
\item one measurement has flag=2 and the other 3/4: In this case we assume the
measurement with flag 3/4 to be the correct one. We have 10 flag=2
redshifts that are discrepant, out of 345.
\item both measurements have flag=2: 22 out of 148 pairs have discrepant redshift
\item one measurement has flag=1 and the other has 2, 3 or 4: 121 out of 301 are discrepant
\item both measurements have flag=1: 56 out of 74 are discrepant
\end{enumerate}
With the reasonable assumption that when two redshifts are in
agreement they are both correct from these data we can derive
a confidence level of the redshift measurements for each flag class,
which we report in Table~\ref{tab:flags}.
\begin{figure*}
\centering
\includegraphics[width=16cm]{vipersmask_1deg.jpg}
\caption{The masks developed for VIPERS, within a 1 deg$^2$
region of the survey. The new bright-star mask
is marked by the magenta circles and
cross patterns, while the original mask distributed by Terapix, based
on the four-point star template, is shown in green; orange
polygons are drawn around selected extended sources. The quadrants that make up the
VIPERS pointings are plotted in red. In the background is the
CFHTLS T0006 $\chi^2$ image of the field 020631-050800
produced by Terapix. Note the significant gain in usable sky
obtained with the new VIPERS-specific mask.
}
\label{fig:photo_mask}
\end{figure*}
\section{SURVEY SELECTION FUNCTION}
\label{sec:selfun}
The VIPERS angular selection function is the result of the
combination of several different angular completeness functions. Two
of these are binary masks, the first related to defects in the parent photometric
sample (mostly areas masked by bright stars) and the other to the
specific footprint of VIMOS and how the
different pointings are tailored together to mosaic the VIPERS area.
Moreover, within each of the four VIMOS quadrants only an average
40\% of the available targets satisfying the selection criteria are
actually placed behind a slit and observed, defining what we call
the Target Sampling Rate (TSR). This fraction varies
with location on the sky due to fluctuations in the surface density
of objects. This is a significant issue when working with
VIPERS data, since it is hard to make the distribution of slit centres
strongly clustered; rather, the slit assignment algorithm attempts
to maximize the number of spectra in a given quadrant. Thus, the observed
sky distribution is near to uniform, reflecting a TSR that is
inversely proportional to the surface density. In practice, we choose
to evaluate the TSR on a per-quadrant basis, as shown in Fig.~\ref{fig:tsr},
using the ratio of assigned targets to potential targets.
Finally, varying observing conditions and technical
issues determine a variation from quadrant to quadrant of the Spectroscopic
Success Rate (SSR), i.e. of the actual number of redshifts measured
with respect to the number of targeted galaxies; again, this can be
measured empirically and is shown in Fig.~\ref{fig:ssr}. Both these
quantities are discussed in more detail in the following.
Detailed knowledge
of all these contributions is a crucial ingredient for any
quantitative measurement of galaxy clustering.
In principle,
there will also be variations of the TSR and SSR within a single quadrant, owing
to the details of the response of slit assignment to small-scale clustering, and
to internal distortions that may cause the slits to be slightly misplaced
on the sky. These effects are hard to represent simply, since they cannot
be viewed purely as a position-dependent probability of obtaining a
redshift. This is because the finite size of the slits mean that close
pairs of galaxies cannot be sampled, and there will always be some complex
structure in the statistics of pair separations owing to the survey
selection. Once the main quadrant-based corrections are made, the only
practical way of dealing with these is to use the known statistics of
angular clustering in the initial photometric catalogue in order to make
a final small correction to the estimated clustering statistics
\citep{delatorre13}.
\subsection{Revised CFHTLS Photometric Mask}
The photometric quality across the CFHTLS images is tracked with a set
of masks accounting for imaging artefacts and non-uniform coverage.
We use the masks to exclude regions from the survey area with
corrupted source extraction or degraded photometric quality. The
masks consist primarily of patches around bright stars ($B_{\rm
Vega}<17.5$) owing to the broad diffraction pattern and internal
reflections in the telescope optics. At the core of a saturated
stellar halo there are no reliable detections, leaving a hole in the
source catalogue, while in the halo and diffraction spikes spurious
sources may appear in the catalogue due to false detections. We
also add to the mask extended extragalactic sources that may be
fragmented into multiple detections or that may obscure potential
VIPERS sources. The masks are stored in DS9 region file format
using the \texttt{polygon} data structure.
Terapix included a bright star mask as part of the T0006 data release
consisting of star-shaped polygons centred on the stellar halos. We
found this mask to be too restrictive for VIPERS; in
particular, we found that the area lost was excessive near diffraction
spikes and within stellar halos. We follow the same strategy in
constructing the VIPERS mask, but instead use a circular template with a cross
pattern. The angular size of the template is scaled based upon the
magnitude of the star.
Our starting point for the bright star mask was the USNO-B 1.0
catalogue \citep{monet03}, from which we selected a sample of
stars with $B_{\rm Vega}<17.5$. Using the full CFHTLS area (130 deg$^2$),
we measured the mean source density in the photometric catalogue
as a function of distance from a bright star. We used the density
profile to calibrate a size-magnitude relation for the stellar halo.
We derived the following relations for the star magnitude $B$
and the halo radius $R$ in arcminutes:
\begin{eqnarray}
B < 15.19: & \log_{10}(R) = -2.60 \log_{10}(B) + 2.33 \\
B \ge 15.19: & \log_{10}(R) = -6.55\log_{10}(B) + 6.99.
\end{eqnarray}
For stars brighter than $B=17$ we include a cross pattern to cover the
diffraction spikes. For the brightest $\sim$200 stars with $B<11$, we
inspected the $\chi^2$ image \citep[see][]{szalay99} and adjusted the masks
individually. The
USNO B catalogue includes a number of extended sources that in many
cases have multiple entries. We cross-checked the catalogue against
the 2MASS Extended Source Catalogue to remove duplicates. A zoom into
the W1 field, showing the various masks, is displayed in Fig.~\ref{fig:photo_mask}.
Although significant attention was given to constructing a homogeneous
imaging survey in five bands, a handful of patches exist within the W1
field that have degraded photometric quality in one band. These
regions were identified based upon high values of the photometric
redshift $\chi^2$. We include these regions as rectangular patches in
the photometric mask, visible in Fig.~\ref{fig:photo_mask}. No such
regions were identified in the W4 field.
\subsection{Spectroscopic mask and weights}
Although the general lay-out of VIMOS is well known, the precise
geometry of each quadrant's observations need to be specified
carefully, in order to perform precise clustering measurements with the
VIPERS data. Although it happens rarely, a quadrant may be partly
vignetted by the guide probe arm, in those cases in which no
better located guide star could be found. In addition, the
accurate size and geometry of each quadrant was changed between the pre- and
the post-refurbishment data (i.e. from mid-2010 on), due to the
dismounting of the instrument and the technical features of the new
CCDs. We had therefore to build our own extra mask of the spectroscopic
data, accounting for all these aspects at any given point on the sky
covered by the survey.
The masks for the W1 and W4 data were constructed from the pre-imaging
observations by running an
image analysis routine that identifies `good' regions. First, a
polygon is defined that traces the edge of the image. The mean and
variance of the pixels are computed in small patches at the vertices
of the polygon. These measurements are compared to the statistics at
the centre of the image. The vertices of the polygon are then
iteratively moved inward toward the center until the statistics along
the boundary are within an acceptable range. The boundary that
results from this algorithm is used as the basis for
the field geometry. The polygon is next simplified to reduce the
vertex count: short segments that are nearly co-linear are replaced by
long segments. The WCS information in the fits header is used to
convert from pixel coordinates to sky coordinates. Each mask was then
examined by eye. Features due to
stars at the edge of an image were removed, wiggly segments were
straightened and artefacts due to moon reflections were corrected.
The red lines in Fig.~\ref{fig:photo_mask} show the detailed borders
of the VIMOS quadrants, describing the spectroscopic mask.
Before scientific analyses can be performed on the observed
data, knowledge of two more selection functions (angular masks) is
needed, as discussed briefly above.
First, we need to know how many potential targets in each VIMOS
quadrants have been actually observed: this is what we call the Target Sampling
Rate (TSR). As shown in Fig.~\ref{fig:tsr}, this varies
on a quadrant-by-quadrant basis due to the intrinsic fluctuations in
the number density of galaxies as a
function of position on the sky. Thanks to the adopted strategy
(i.e. having discarded through the colour selection almost half of the
magnitude-limited sample lying at $z<0.5$), the average TSR of VIPERS
is $>40\%$, a fairly high value that represents one of the specific
important features of VIPERS. This can also be appreciated in
Fig.~\ref{fig:mag_tsr_ssr} (bottom histogram), where we plot the TSR
integrated over the whole survey, as a function of galaxy magnitude.
Note how the TSR is completely independent of the target magnitude.
The second incompleteness that varies from quadrant to quadrant
is the fraction of succesfully measured redshifts, out of the total
number of targeted galaxies. This defines what we call the
Spectroscopic Success Rate (SSR), which can also be defined for each
VIMOS quadrant. This is shown in
Fig.~\ref{fig:ssr}, where one can appreciate how for the majority of
the survey area we have SSR$>80\%$. A few observations under
problematic conditions (either technical or atmospheric) are clearly
marked out by the brown and purple rectangles. Also the SSR can be
plotted, integrated over the whole survey, as a function of the target
magnitude. This is also shown in Fig.~\ref{fig:mag_tsr_ssr} as the top
red histogram.
More discussion on the details of the TSR and SSR will be presented in
the paper accompanying the PDR-1 catalogue.
\begin{figure*}
\centering
\includegraphics[width=\hsize]{tsr_v2.pdf}
\caption{Lay-out on the sky of all pointings included in the PDR-1
catalogue, for the two fields W1 and W4. Each of the four quadrants composing
the pointings is shown and colour-coded according to the
specific Target Sampling Rate (TSR) over its area. The TSR is
simply the ratio of the number of targeted galaxies over the
number of potential targets. As shown, the average TSR is
around 40\%. Black
quadrants correspond to a failure in the insertion of the mask
for that specific quadrant and the consequent loss of all data.
}
\label{fig:tsr}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\hsize]{ssr_v2.pdf}
\caption{Same as Fig.~\ref{fig:tsr}, but now with the colour coding
measuring the Spectroscopic
Success Rate (SSR), i.e. the ratio of the number of reliably
measured redshifts (i.e. quality flag $\ge 2$)
to the number of targeted galaxies. Also in this figure a few
problematic areas emerge: purple and brown quadrants correspond
to regions in which the fraction of successful measurements is,
respectively, below 50\% and 70\%. As can be seen, for the
majority of quadrants the success rate is larger than 80\%
}
\label{fig:ssr}
\end{figure*}
\section{RESULTS AND PERSPECTIVES}
\label{sec:results}
Experience with the first half of the VIPERS dataset fully confirms the expected general
performance and science potential of the survey. As shown here, the average quality
of the redshifts is as expected, with typical redshift measurement
errors that are even better than in previous similar surveys with VIMOS.
Fig.~\ref{fig:nz} shows the redshift distribution of the data
collected so far in the two fields. The combination of the two fields
provides an impressively smooth distribution, averaging over local
structure. As discussed earlier, the survey is
complete beyond $z=0.6$, with a transition region at
$0.4<z<0.6$ produced by the colour-colour selection. A substantial
tail of galaxies out to $z=1.4$ is also apparent. This redshift range benefits
particularly strongly from the increased sensitivity and lack of substantial
fringing with the refurbished VIMOS CCDs, allowing a clearer detection of the
[OII]3727 line or the 4000\,\AA\ break beyond 8000\,\AA.
The most striking result from this first significant set of VIPERS observations is
provided by the new maps of the 3D galaxy
distribution in the range $0.5<z<1.2$, which we show in the cone
diagrams of Fig.~\ref{fig:coneW1W4}. As
demonstrated by these plots, VIPERS provides an unpredecented combination
of overall size and detailed sampling, yielding
a representative picture of the overall galaxy population and
large-scale structure when the Universe was
about half its current age. A direct comparison of VIPERS with
local surveys, in terms of size and redshift, is shown in
Fig.~\ref{fig:ben_cones}. Here the VIPERS redshift data are
plotted together with those from the SDSS-Main and SDSS-LRG
surveys. The fidelity with which structure can be seen in VIPERS
(covering linear scales $\sim$ Gpc) is comparable, at high
redshifts, to that of SDSS-Main at $z<0.1$, while the lower
density of the LRG sample conveys little visual impression of
significant structure.
\begin{figure}
\centering
\includegraphics[width=\hsize]{mag_tsr_ssr.pdf}
\caption{Plots of the Target Sampling Rate (TSR, lower darker
histogram) and the
Spectroscopic Success Rate (SSR, two upper lighter histograms), as a function of galaxy
magnitudes. The TSR is shown to be independent of
galaxy magnitudes, indicating that there is no bias in terms
of apparent luminosity in the process of assigning galaxy
targets to slits. As for the efficiency in measuring
redshifts, the two top histograms
correspond to the SSR when all measured redshifts (flag $\ge
1$) are considered and to when reliable redshifts (flag $\ge
2$) are used, as in the case of Fig.~\ref{fig:tsr}
SSR in measuring redshifts is however obviously dependent
on magnitude.
}
\label{fig:mag_tsr_ssr}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\hsize]{nz.pdf}
\caption{The redshift distribution of galaxies with a
measured redshift from the full VIPERS PDR-1 catalogue (black solid line),
and within the W1 and W4 fields (red and blue solid lines,
respectively). All measured redshifts (flag=1 and above) have
been plotted here. The redshift histogram restricted to only the most
reliable redshifts (flag$ >1$) does not show significant differences.
}
\label{fig:nz}
\end{figure}
\begin{figure*}
\centering
\includegraphics[angle=90, height=22truecm]{cone_W1.jpg}
\includegraphics[angle=90, height=22truecm]{cone_W4.jpg}
\caption{The large-scale galaxy distribution unveiled by the VIPERS PDR-1
catalogue in the CFHTLS W1 and W4 fields (left and right respectively), currently including
$\sim 55,000$ redshifts. Galaxy positions are projected
along the declination direction, where the width is $\simeq 1^\circ$ for W1 and
$\simeq 1.5^\circ$ for W4. Note the high-resolution sampling of
large-scale structure in VIPERS, comparable to that of SDSS
Main and 2dFGRS at $z<0.2$.
}
\label{fig:coneW1W4}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\hsize]{SDSS_vipers_conediagram_bw.png}
\caption{Putting VIPERS in perspective. This plot shows the
complementarity of the $0.5<z<1.5$ regions probed by the two
VIPERS deep fields, and the SDSS main and LRG samples at lower
redshift (for which a 4-degree-think slice is shown). The LRG samples are excellent statistical probes on the
largest scales, but (by design) they fail to register the details
of the underlying nonlinear structure, which is clearly exposed by VIPERS.
}
\label{fig:ben_cones}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\hsize]{cone_W1_color_darkergreen.png}
\caption{A zoom into the cone diagram of the W1 field, where now
the additional dimension represented by galaxy rest-frame colours has been
added. Galaxies are here marked in blue, green or reddish,
depending on whether their $U-B$ rest-frame colour is
respectively $< 0.9$, between 0.9 and 1.2 or $> 1.2$. Also in this case the
size of the dots has been set proportionally to the B-band
luminosity of the corresponding galaxy. The plot shows
clearly that the colour-density relation for galaxies is already
in place at these redshifts \citep{cucciati06}, with red early-type galaxies
tracing the backbone of structure and blue/green star-forming
objects filling the more peripheral lower-density
regions. This picture gives an example of the potential of
VIPERS for studying the clustering of galaxies as a function
of galaxy properties, over scales ranging from less
than a Mpc to well above 100 Mpc.
}
\label{fig:cone_colours}
\end{figure*}
New statistical measurements of clustering are being obtained with
these results. Moreover, the rich and high-quality set of ancillary
photometric data, combined with the distance information, is
allowing us to compute the key metadata (SED, luminosities, stellar
masses) for quantifying the connection between galaxy properties and
the surrounding structure at these early epochs. An example of
the power of correlating galaxy properties with the surrounding
large-scale structure is provided by Fig.~\ref{fig:cone_colours},
which represents a zoom into part of the W1 VIPERS volume. Here
galaxies have been coloured according to their rest-frame $U-B$
colour, providing in this way obvious evidence that the present-day
colour-density relation had already been established at these redshifts.
Scientific activities using this rich dataset within
the VIPERS Team are concentrating on a series of specific
aims, which we summarize briefly here:
\begin{itemize}
\item To measure in detail the clustering of galaxies on
small/intermediate scales at $0.5<z<1$, quantifying its dependence
on luminosity and stellar mass \citep{marulli13}. The final goal here is
to describe the relation between baryons and Dark Matter,
measuring the evolution of the Halo Occupation
Distribution (HOD) of galaxies.
\item To measure the power spectrum of the galaxy distribution $P(k)$
on both small and large scales at $z\simeq 0.8$, constraining
the overall matter density parameter
\citep{bel13},
and the neutrino mass and number of species \citep{granett12, xia12}.
\item To measure the growth of structure
between $z=1.2$ and 0.5, by modelling the anisotropy
of clustering \citep{delatorre13}. The initial application is to
the galaxy population treated as a whole, but the high sampling
and good spectroscopic completeness means that we will be able to
exploit the use of multiple populations to reduce statistical and
systematic errors in this measurement.
\item To measure the luminosity and stellar mass functions to high
statistical accuracy at $0.5<z<1$, in particular at the
bright/massive end \citep{davidzon13}.
\item More generally, to make a full characterization of the evolution of galaxies over this
important range of redshifts, in terms of the distributions of other
fundamental properties like colours, spectral types and
star-formation rates \citep{fritz13}.
\item To measure higher-order clustering statistics at this early
epoch, where mass fluctuations are closer to the linear regime, measuring the moments of the galaxy
distribution (Cappi et al., in preparation) and the evolution and nonlinearity of
galaxy biasing (Di Porto et al., in preparation).
\item To construct a large and well-defined sample of
optically-selected groups and clusters at at $0.5<z<1$, to
investigate the properties of these systems and in particular the
evolution of galaxies in different environments (Iovino et al., in preparation).
\item To reconstruct the density field over a large volume
and dynamic range at $0.5<z<1$, to produce an order-of-magnitude
improvement in our knowledge of crucial relationships between
galaxies and their environment, as the colour-density relation
(Cucciati et al., in preparation).
\item To construct a massive spectroscopic and multi-band photometric
database, with automatic spectral classifications through
SED-fitting, Principal Component Analysis \citep{marchetti13} and other
techniques, such as supervised learning algorithm methods \citep{malek13}.
\item To cross-correlate the detailed 3D maps of the galaxy
distribution with the dark-matter maps reconstructed using weak
lensing from the CFHTLS high-quality images.
\item To measure the faint end of the AGN luminosity function and
their correlation with large-scale structure, through a dedicated
sub-sample.
\end{itemize}
This is a substantial list of what should prove to be exciting developments,
representing a major advance in our knowledge of the structure in the
Universe around redshift unity. But all these applications should benefit
from more detailed investigation, and there are many fruitful topics
beyond those listed above. We hope, and expect, that VIPERS will follow
in the path of the major low-redshift surveys in generating many more
important papers from open use of the public data. We therefore encourage
readers to stay tuned for the forthcoming PDR-1 data release, which
will become available in September 2013 at
{\tt http://vipers.inaf.it/}. This should serve to increase anticipation
for what may be achieved with the final VIPERS dataset, which will be
roughly double the present size.
\begin{acknowledgements}
We acknowledge the crucial contribution of the ESO staff for the
management of service observations. In particular, we are deeply
grateful to M. Hilker for his constant help and support of this
program. Italian participation to VIPERS has been funded by INAF
through PRIN 2008 and 2010 programs. LG and BRG acknowledge
support of the European Research Council through the Darklight
ERC Advanced Research Grant (\# 291521). OLF acknowledges
support of the European Research Council through the EARLY ERC
Advanced Research Grant (\# 268107). Polish participants have
been supported by the Polish Ministry of Science (grant N N203
51 29 38), the Polish-Swiss Astro Project (co-financed by a
grant from Switzerland, through the Swiss Contribution to the
enlarged European Union), the European Associated Laboratory
Astrophysics Poland-France HECOLS and a Japan Society for the
Promotion of Science (JSPS) Postdoctoral Fellowship for Foreign
Researchers (P11802). GDL acknowledges financial support from
the European Research Council under the European Community's
Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement
n. 202781. WJP and RT acknowledge financial support from the
European Research Council under the European Community's Seventh
Framework Programme (FP7/2007-2013)/ERC grant agreement
n. 202686. WJP is also grateful for support from the UK Science
and Technology Facilities Council through the grant
ST/I001204/1. EB, FM and LM acknowledge support from grants
ASI-INAF I/023/12/0 and PRIN MIUR 2010-2011. YM acknowledges
support from CNRS/INSU (Institut National des Sciences de
l’Univers) and the Programme National Galaxies et Cosmologie
(PNCG). CM is grateful for support from specific project funding
of the {\it Institut Universitaire de France} and the LABEX
OCEVU.
\end{acknowledgements}
\bibliographystyle{aa}
| {'timestamp': '2013-03-14T01:01:35', 'yymm': '1303', 'arxiv_id': '1303.2623', 'language': 'en', 'url': 'https://arxiv.org/abs/1303.2623'} |
\section{Introduction}
Fourth order parabolic equations and corresponding initial-boundary value problems appear in the modelling in areas as diverse as biology, phase-field modelling and image processing to name a few. In most cases of practical interest one has to resort to numerical methods for their solution, due to complex geometry and/or the presence of non-linearities.
During the last five decades, finite element methods (FEMs) have been widely used to numerically solve fourth order elliptic or parabolic problems; see, e.g., \cite{baker, ciarlet, brezzi.fortin:mixed,destuynder, hughes, brenner,moz.suli:nipg} and the references therein for earlier works. There are, generally speaking, three families of FEMs developed for fourth order problems: conforming, mixed and non-conforming. The classical conforming methods (see, e.g., \cite{ciarlet} and the references therein) require the construction of complicated elements with a number of degrees of freedom devoted to ensuring $C^1$-continuity across the element interfaces. This results into limitations in the applicability of conforming methods on general, possibly irregular, meshes \cite{stogner:06} and their non-trivial extensions to dimensions three (or higher). Mixed methods (see, e.g., \cite{brezzi.fortin:mixed,destuynder} and the references therein), whereby the fourth order operator is first transformed into a system of second order operators are widely used in practice, but they require very careful treatment in the imposition of essential and natural boundary conditions. Non-conforming methods for fourth order problems were first presented by \cite{baker} and then further developed in \cite{hughes, brenner,moz.suli:nipg,MR2142199} and other works. The key idea in non-conforming methods is the use of penalties to ensure convergence into the natural energy space of the variational problem, despite finite element basis functions being either just continuous ($C^0$-interior penalty procedures; see, e.g., \cite{hughes, brenner}) or completely discontinuous (discontinuous Galerkin interior penalty procedures; see, e.g., \cite{baker,moz.suli:nipg,MR2142199,feng_kar}).
Adaptive FEMs based on a posteriori error estimates has been an active field of research in recent years, especially for second order elliptic and parabolic problems. For the case of fourth order elliptic problems a posteriori error estimators and indicators have been developed, e.g., in \cite{MR1426319,verfurth,MR1902705,MR1836871,MR2291934,MR2373172,MR2670114, MR2752872,georgoulis_houston_virtanen}. A posteriori bounds and adaptive algorithms for parabolic fourth order problems are far less developed in the literature. For instance, the development of adaptive algorithms based on various types of a posteriori indicators for the Cahn-Hilliard fourth order parabolic problem can be found in \cite{larsson-mesforush,feng-wu:08,banas-nuernberg:09}. Error control for variational methods for fourth order parabolic equations has been predominantly focused to space-discrete mixed or conforming formulations. The recent work \cite{larsson-mesforush} deals with goal-oriented error estimation for the fully discrete Cahn-Hilliard problem. Therefore, the development of adaptive algorithms based on a posteriori estimators for fully discrete methods for fourth order parabolic problems is still largely an unexplored area.
Advances in a posteriori error analysis of fully discrete schemes with non-conforming spatial discretizations of second order parabolic problems have been recently presented \cite{ern-vohralik:10, georgoulis_lakkis_virtanen}. In \cite{georgoulis_lakkis_virtanen}, an adaptive algorithm based on the derived a posteriori estimates is also considered.
Local residual a posteriori error bounds for semi-discrete conforming and mixed spatial discretizations ffor the Cahn-Hilliard problem and the Hele-Shaw flow are presented in \cite{feng-wu:08}. Finally, a posteriori error estimates in an $L^2(H^2)$-type norm and adaptive algorithms for fully discrete schemes with discontinuous Galerkin methods for fourth order problems are proposed in \cite{thesis_juha}. The derivation of reliability bounds in \cite{thesis_juha} is based on the elliptic reconstruction framework of Makridakis and Nochetto \cite{MR2034895}; we also refer to \cite{lakkis-makridakis:06,georgoulis_lakkis_virtanen} for some relevant extensions.
This work is concerned with the derivation of a posteriori error estimates in weaker than $L^2(H^2)$-norms and their use within an adaptive algorithm for a class of discontinuous Galerkin interior penalty methods for a fully discrete approximation of the problem:
\begin{eqnarray}
u_{t} + \Delta^2 u &=& f \quad \; \text{ in } \; \Omega \times (0,T], \label{modelpde1}\\
u = \nabn{u} &= &0 \quad \; \, \text{ in } \; \partial\Omega \times (0,T] \quad \text{ and } \label{modelpde2}\\
u &= &u_0 \quad \text{ in } \; \Omega \times \{0\} \quad \text{} \label{modelpde3}
\end{eqnarray}
with $\Omega \subset \mathbb{R}^d$, $d=2,3$ a convex polygonal domain with boundary $\partial\Omega$. More specifically, we derive a posteriori error estimators for the error measured in $L^\infty(L^2)$ and $L^2(L^2)$ norms for a numerical scheme consisting of discontinuous Galerkin method in space and simple backward-Euler time-stepping for the problem (\ref{modelpde1}) - (\ref{modelpde3}). The a posteriori analysis is performed for convex domains (as is usual for these norms) in two and three space dimensions for local spatial polynomial degrees $r \ge 2$. To enable the optimality of the a posteriori estimators in the $L^\infty(L^2)$ and $L^2(L^2)$ norms, the elliptic reconstruction framework is employed. Moreover, the $L^2(L^2)$-norm analysis employs a special test function construction inspired from the a priori analysis of FEMs for wave problems in \cite{baker_wave}. Somewhat surprisingly, the use of this special testing, in conjunction with the elliptic reconstruction, results into the derivation of $L^2(L^2)$-norm a posteriori estimators via a standard energy argument. The efficiency of the a posteriori estimators is assessed numerically. The reliability bounds are used within two variants of a space-time adaptive algorithm. The adaptive algorithm is able to achieve the same error reduction with far fewer degrees of freedom compared to uniform meshes, thereby highlighting the relevance of the derived a posteriori estimates in practical computations. The simple model problem (\ref{modelpde1}) - (\ref{modelpde3}) appears to be sufficient in highlighting some of the challenges in the error estimation and adaptivity of finite element methods for more complex fourth order parabolic problems. It appears that the derived a posteriori bounds and the respective adaptive algorithms can be modified in a straightforward fashion to include the original dG method of Baker \cite{baker} and $C^0$-interior penalty methods \cite{hughes, brenner}.
The remaining of this work is organised as follows. In Section \ref{notaandprem}, notation is introduced and some standard results needed in the subsequent analysis are recalled. The discontinuous Galerkin (dG) method for the biharmonic problem, along with the derivation of posteriori error bounds for the dG approximation of the biharmonic problem in $L^2$-norm are derived in Section \ref{section:DGmethod}. The respective fully discrete scheme for the parabolic model problem (\ref{modelpde1}) - (\ref{modelpde3}) is given in Section \ref{section:nummethod}, while Section \ref{apost_parabolic} contains the derivation of residual type a posteriori estimates of errors in $L^\infty(L^2)$ and $L^2(L^2)$ norms for the fully discrete scheme. The efficiency and reliability of the a posteriori estimators is tested on a range of uniform meshes in Section \ref{numerics}. The adaptive algorithm utilizing the a posteriori estimates in a series of numerical experiments are also presented in Section \ref{numerics}. Some concluding remarks regarding the results and possible extensions are given in Section \ref{finalremark}.
\section{Notation and preliminaries}\label{notaandprem}
The standard Hilbertian Lebesgue space is denoted by $L^2(\omega)$, for a domain $\omega\subset\mathbb{R}^d$, $(d=2,3)$,
with corresponding inner product $\langle\cdot,\cdot,\rangle_{\omega}$ and norm $\|\cdot\|_{\omega}$;
when $\omega=\Omega$, we shall drop the subscript writing $\langle\cdot,\cdot,\rangle$ and $\|\cdot\|$, respectively.
We also denote by $H^s(\omega)$, the standard Hilbertian Sobolev
space of index $s\ge 0$ of real-valued functions defined on
$\omega\subset\mathbb{R}^d$, along with the corresponding norm and seminorm
$\|\cdot\|_{s,\omega}$ and $|\cdot|_{s,\omega}$, respectively. For $1\le p\le +\infty$, we also define the spaces $L^p(0,T,H^s(\omega))$, consisting of all measurable functions $v: [0,T]\to H^s(\omega)$,
for which
\begin{equation}
\begin{gathered}
\|v\|_{L^p(0,T;H^s(\omega))}
:=\Big(\int_0^T \|v(t)\|_{s,\omega}^p\mathrm{d} t\Big)^{1/p}<+\infty,
\quad \text{for}\quad 1\le p< +\infty, \\
\|v\|_{L^{\infty}(0,T;X)}:={\ess \sup}_{0\le t\le
T}\|v(t)\|_{s,\omega}<+\infty,\quad \text{for}\quad p = +\infty.
\end{gathered}
\end{equation}
Let $\wavy{T}$ be a subdivision of $\Omega$ into disjoint elements $ \kappa \in \mathcal{T} $.
The subdivision $\wavy{T}$ is assumed to be shape-regular (see, e.g.,
p.124 in \cite{ciarlet}) and is constructed via smooth mappings
$F_{\kappa}:\hat{\kappa}\to\kappa$ with uniformly bounded Jacobian throughout the mesh family considered, where $\hat{\kappa}$ is the reference element.
The above mappings are assumed to be constructed so as to ensure $\bar{\Omega}=\cup_{ \kappa \in \mathcal{T} }\kappa$ and that
the elemental edges are straight segments (i.e., lines or planes). Note that we also use the expression edge to mean side when $d=3$.
The broken Laplacian, $\bdel{u}$, is defined element-wise by
$(\bdel{u})|_{\kappa}:=\del({u}|_{\kappa})$ for all $\kappa \in \wavy{T}$.
For a nonnegative integer $r$, we denote by $\mathcal{P}_r(\hat{\kappa})$, the set of all polynomials of
total degree at most $r$, if $\hat{\kappa}$ is the reference simplex, or of degree at most $r$ in each variable, if $\hat{\kappa}$ is the reference hypercube.
We consider the finite element space
\begin{equation}\label{eq:FEM-spc}
S^r :=\{v\in L^2(\Omega):v|_{\kappa}\circ F_{\kappa}
\in\mathcal{P}_{r}(\hat{\kappa}),\, \kappa \in \mathcal{T} \}.
\end{equation}
By $\Gamma$ we denote the union of all $(d-1)$-dimensional element
edges associated with the subdivision $\wavy{T}$, including the boundary. Further, we decompose $\Gamma$ into
two disjoint subsets $\Gamma=\partial\Omega\cup\Gamma_{\text{\rm int}}$,
where $\Gamma_{\text{\rm int}}:=\Gamma\backslash\partial\Omega$.
For two (generic) elements $\kappa^+,\kappa^-\in\wavy{T}$ sharing an edge $e=\kappa^+\cap\kappa^-$, we define the outward normal unit vectors $\mbf{n}^+$ and $\mbf{n}^-$ on $e$ corresponding
to $\partial\kappa^+$ and $\partial\kappa^-$, respectively. For functions $v:\Omega\to\mathbb{R}$ and
$\mbf{q}:\Omega\to\mathbb{R}^d$, that may be discontinuous across $\Gamma$,
we define the following quantities.
For $v^+:=v|_{e\subset\partial\kappa^+}$, $v^-:=v|_{e\subset\partial\kappa^-}$,
$\mbf{q}^+:=\mbf{q}|_{e\subset\partial\kappa^+}$, and $\mbf{q}^-:=\mbf{q}|_{e\subset\partial\kappa^-}$, we set
\[ \mean{v}:=\frac{1}{2}(v^+ + v^-),\ \mean{\mbf{q}}:=\frac{1}{2}(\mbf{q}^+ + \mbf{q}^-),
\qquad \jumptwo{v}:=q^+\mbf{n}^++q^-\mbf{n}^-,\ \jump{\mbf{q}}:=\mbf{q}^+\cdot \mbf{n}^++\mbf{q}^-\cdot \mbf{n}^-;\]
if $e\in \partial\kappa\cap\partial\Omega$, these definitions are modified to $\mean{v}:=v^+$, $\mean{\mbf{q}}:=\mbf{q}^+$,
$\jumptwo{v}:=v^+\mbf{n}$, $\jump{\mbf{q}}:=\mbf{\mbf{q}}^+\cdot \mbf{n}$.
With the above definitions, it is easy to verify the identity
\begin{equation}\label{scalarformula}
\elsum{\ints{\partial \kappa}{v\,\mbf{q} \cdot \vect{n} } } =
\ints{\Gamma}{ \jumptwo{v}\cdot\mean{\mbf{q}}} + \ints{\Gamma_{\text{\rm int}}}{ \mean{v}\jump{\mbf{q}}},
\end{equation}
with $\mbf{n}$ denoting the outward normal unit vector on $\partial\kappa$, corresponding to $\kappa$.
We define the element size $h_{\kappa}:=(\mu_{d}(\kappa))^{1/d}$, where $\mu_{d}$ is the $d$-dimensional Lebesgue measure; we collect the element sizes into the
into the element-wise constant function ${\bf h}:\Omega\to\mathbb{R}$, with
${\bf h}|_{\kappa}=h_{\kappa}$, $ \kappa \in \mathcal{T} $ and $\mbf{h}=\mean{{\mbf h}}$ on $\Gamma$. Also, for two (generic) elements $\kappa^+$, $\kappa^-$ sharing an edge $e:=\partial\kappa^+\cap\partial\kappa^-\subset\Gamma_{\text{\rm int}}$, we
define $h_{e}:=\mu_{d-1}(e)$.
As we shall be dealing with mesh adaptive algorithms below, we assume that all sequences of meshes considered in this work are locally quasi-uniform, i.e.,
there exists constant $c\ge 1$,
independent of ${\bf h}$, such that, for any pair of
elements $\kappa^+$ and $\kappa^-$ in $\wavy{T}$ which share an edge,
\begin{equation}\label{eq:bdd-var}
c^{-1}\le h_{\kappa^+}/h_{\kappa^-}\le c.
\end{equation}
Finally we recall a series of some (standard) results used throughout this work; their proofs can be found, e.g., in \cite{ciarlet,dupont-scott:80,braess:01,brenner-scott:02,brenner-wang-zhao:02}.
\begin{lemma}[approximation property] \label{lemma:approximationproperty}
Let $0 \le m \le r + 1$ and $\wavy{T}$ be a subdivision of $\Omega$, $\Omega \subset \mathbb{R}^d$. Then there exists a constant $C_{\text{app}}$, independent of $h_{\kappa}$, such that for any $u \in H^m(\Omega)$ and $\kappa \in \wavy{T}$, there exists $p:C(\kappa)\to\mathbb{R}$, with $p\circ F_{\kappa} \in \mathcal{P}_r(\kappa)$ and
\begin{equation}\label{approximationproperty}
\snorm{u-p}{j,\kappa} \le C_{\text{app}} \ h^{m-j}_{\kappa} \ \snorm{u}{m,\kappa} \ , 0 \le j \le m .
\end{equation}
\end{lemma}
\begin{lemma}[inverse estimate] \label{lemma:inverseestimate}
There exists a constant $C_{\text{inv}}$, independent of $h_{\kappa}$, such that
\begin{equation}\label{inverseestimate}
\snorm{p}{j,\kappa} \le C_{\text{inv}} \ h^{i-j}_{\kappa} \ \snorm{p}{i,\kappa} \ , 0 \le i \le j \le 2,
\end{equation}
for all $p:C(\kappa)\to\mathbb{R}$, with $p\circ F_{\kappa} \in \mathcal{P}_r(\kappa)$.
\end{lemma}
\begin{lemma}[trace inequality] \label{lemma:traceinequality}
For every $u \in H^1(\kappa)$, with $\kappa \in\wavy{T}$, there exists a constant $C_{\text{tr}} > 0$ independent of $h_{\kappa}$ such that
\begin{equation} \label{traceinequality}
\norm{u}{0,\partial \kappa}^2 \le C_{\text{tr}} ( h^{-1}_{\kappa} \norm{u}{0,\kappa}^2 + h_{\kappa} \snorm{u}{1,\kappa}^2 ).
\end{equation}
\end{lemma}
\begin{lemma}[Poincar\'e-Friedrichs inequality \cite{brenner-wang-zhao:02}] \label{lemma:pfinequality}
There exists a constant $C_{\text{pf}}$, independent of $h_{\kappa}$, such that for any $u\in L^2(\Omega)$, with $u|_{\kappa} \in H^2(\kappa)$ for all $\kappa\in\wavy{T}$, we have
\begin{equation} \label{pfinequality}
\norm{u}{0,\Omega}^2 + \snorm{u}{1,\Omega}^2 \le C_{\text{pf}} \left( \snorm{u}{2,\Omega}^2 +\norm{\mbf{h}^{-3/2}\jumptwo{u}}{0,\Gamma}^2 + \norm{\mbf{h}^{-1/2}\jump{\nabla u}}{0,\Gamma}^2 \right).
\end{equation}
\end{lemma}
\section{Discontinuous Galerkin method for the biharmonic problem} \label{section:DGmethod}
We consider the biharmonic equation
\begin{equation}\label{pde}
\Delta^2 \tilde{u}= \phi \quad\text{in } \Omega,
\end{equation}
with homogeneous essential boundary conditions
\begin{equation}\label{modelbc}
\tilde{u}=0\quad,\qquad
\nabla \tilde{u} \cdot\mbox{\boldmath$\rm{n}$}=0\quad\text{on }\partial\Omega ,
\end{equation}
where $\mbox{\boldmath$\rm{n}$}$ denotes the unit outward normal vector to $\partial\Omega$ and $\phi \in L^2(\Omega)$. Then the regularity of the problem implies that $\tilde{u} \in H^4(\Omega) \cap H^2_0(\Omega)$ \cite{grisvard:elliptic}.
Upon defining the lifting operator $\mathcal{L}:\mathcal{S}:=S^r+H_0^2(\Omega)\to S^r$ by
\begin{equation}
\int_{\Omega} \mathcal{L}(\nu) \psi \,\mathrm{d} x = \int_{\Gamma}
\Big(\jumptwo{\nu}\cdot\mean{\nabla \psi} - \mean{\psi}\jump{\nabla \nu}\Big)\,\mathrm{d} s\quad\forall \psi\in S^r,
\ee{lifting}
the \emph{(symmetric) interior penalty discontinuous Galerkin (dG) method} for (\ref{pde}), (\ref{modelbc}) is given by:
\begin{equation}
\text{find}\ \tilde{u}_h \in S^r\ \text{such that}\quad B(\tilde{u}_h,v_h)=l(v_h)\quad\forall v_h\in S^r,
\ee{dgfem}
where the bilinear form $B:\mathcal{S}\times\mathcal{S}\to\mathbb{R}$ and the linear form $l:\mathcal{S}\to\mathbb{R}$ are given by
\begin{equation}\label{bilinear}
\begin{aligned}
B(w,v) :=& \int_{\Omega} \Big(\Delta_h w \Delta_h v
+ \lift{w}\Delta_h v+\Delta_h w\lift{v}\Big)\,\mathrm{d} x+ B_p(w,v)
\end{aligned}
\end{equation}
with
\[
B_p(w,v):=\int_{\Gamma}\Big( \sigma\jumptwo{w}\cdot\jumptwo{v}
+ \xi\jump{\nabla w}\jump{\nabla v} \Big)\,\mathrm{d} s,
\]
and
\begin{equation}
l(v): = \int_{\Omega} \phi v \,\mathrm{d} x,
\ee{linear}
respectively, for $w,v\in\mathcal{S}$. The piecewise constant discontinuity penalization parameters $\sigma,\xi: \Gamma \rightarrow \mathbb{R}$ are given by
\begin{equation}
\sigma|_{e} = \sigma_0 (\mbf{h}|_{e})^{-3}, \quad \xi|_{e} = \xi_0 (\mbf{h}|_{e})^{-1},
\ee{penaltyparametersdef}
respectively, where $\sigma_0 > 0$ and $\xi_0 > 0$. To guarantee the stability of the IPDG method defined
in (\ref{dgfem}), $\sigma_0$ and $\xi_0$ must be selected sufficiently large.
Note that this formulation is inconsistent for trial and test functions belonging to the solution space $\mathcal{S}$.
However, when $w,v\inS^r$, in view of (\ref{lifting}), (\ref{bilinear}) gives
\begin{equation}\label{bilinear_original}
\begin{aligned}
B(w,v) =& \int_{\Omega} \Delta_h w \Delta_h v\,\mathrm{d} x+
\int_{\Gamma}\Big( \mean{\nabla\Delta w}\cdot\jumptwo{v}+\mean{\nabla\Delta v}\cdot\jumptwo{w}\\
&-\mean{\Delta w}\jump{\nabla v}-\mean{\Delta v}\jump{\nabla w}
+\sigma\jumptwo{w}\cdot\jumptwo{v}
+ \xi\jump{\nabla w}\jump{\nabla v} \Big)\,\mathrm{d} s;
\end{aligned}
\end{equation}
therefore, (\ref{dgfem}) coincides with the symmetric version interior penalty method presented in \cite{suli.moz:sipg}.
For the bilinear form $B(\cdot,\cdot)$ in (\ref{bilinear}) we have the continuity and coercivity with respect to the energy norm on $\mathcal{S}$ defined by
\bale{.}{rcl}
\enorm{w}&=&\roots{ \norm{\Delta_h w}{\Omega}^2 + \norm{\sqrt{\sigma} \jumptwo{w}}{\Gamma}^2 + \norm{\sqrt{\xi}\jump{\nab{w}}}{\Gamma}^2 }{1}{2}.
\eale{.}{energynorm}
\begin{lemma}[\cite{georgoulis_houston}] \label{contcoerlemma}
For sufficiently large $\sigma_0>0$ and $\xi_0 > 0$ there exist positive constants $C_{\text{\it{cont}}}$ and $C_{\text{\it{coer}}}$, depending only on the mesh parameters such that
\begin{equation}
|B(u,v)| \leq C_{\text{\it{cont}}} \enorm{u} \hspace{.05in} \enorm{v} \hspace{.05in} \forall u,v \in \mathcal{S} \quad \text{and}
\ee{continuitythm}
\begin{equation}
B(u,u) \geq C_{\text{\it{coer}}} \enorm{u}^2 \hspace{.05in} \forall u \in \mathcal{S} \hspace{.1cm} \text{.}
\ee{coercivitythm}
\end{lemma}
An a posteriori bound for the energy norm error of the dG method (\ref{dgfem}) for (\ref{pde}) -- (\ref{modelbc}) has been considered in \cite{georgoulis_houston_virtanen}. Now, we shall present an a posteriori bound for the $L^2$-norm error (cf. \cite{riviere-wheeler:03} for a corresponding result for the second order problem).
\begin{theorem}[$L^2$-a posteriori bounds for the elliptic problem] \label{theorem:ltwo_apost_biharmonic}
Let $\tilde{u} \in H^4(\Omega) \cap H^2_0(\Omega)$ be the solution of (\ref{pde})--(\ref{modelbc}), $\tilde{u}_h \in S^r$
be dG approximation (\ref{dgfem}) associated with the mesh $\wavy{T}$.
Then, there exists a positive constant $\const{(\ref{theorem:ltwo_apost_biharmonic})}$, independent of $\wavy{T}$, $\mbf{h}$, $\tilde{u}$ and $\tilde{u}_h$, such that
\begin{equation}\label{ltwo_apost_bound_bound}
\ltwo{\tilde{u}-\tilde{u}_h}{} \le \mathcal{E} \left(\wavy{T}, \tilde{u}_h, \phi \right),
\end{equation}
where
\begin{equation}\label{ltwo_apost_estimator}
\begin{aligned}
\mathcal{E} \left(\wavy{T}, \tilde{u}_h, \phi \right) & := \const{(\ref{theorem:ltwo_apost_biharmonic})}\Big( \ltwo{\mbf{h}^{4 - \lambda/2}(\phi - \Delta_h^2 \tilde{u}_h)}{}^2
+ \ltwo{\mbf{h}^{(7 - \lambda)/2}\jump{ \nabla \Delta \tilde{u}_h}}{\Gamma_{\text{\rm int}}}^2 + \ltwo{\mbf{h}^{(5 - \lambda)/2}\jumptwo{ \Delta \tilde{u}_h}}{\Gamma_{\text{\rm int}}}^2 \\
& \phantom{ := } + \sum_{e \in \Gamma} \left( h_{e}^{3 - \lambda} \left( 1 + \xi_0^2 \right) {\ltwo{\jump{ \nabla \tilde{u}_h}}{}}^2 + h_{e}^{1 - \lambda} \left( 1 + \sigma_0^2 \right) {\ltwo{\jumptwo{ \tilde{u}_h}}{}}^2 \right)
\end{aligned}
\end{equation}
and $\lambda := 2 \, (2 - \min \{2, r - 1\})$.
\end{theorem}
\begin{proof}
The dual problem
\begin{equation}\label{dualpde}
\Delta^2 z= \tilde{u} - \tilde{u}_h=:\tilde{e} \quad\text{in } \Omega,
\end{equation}
with homogeneous essential boundary conditions $z=\nabla \tilde{u} \cdot\mbox{\boldmath$\rm{n}$}=0 \;\text{on }\partial\Omega $ clearly satisfies $\tilde{e} \in L^2(\Omega)$ and, therefore, the following regularity estimate holds
\begin{equation}\label{dualpdereg}
\ltwo{z}{4,\Omega} \le \const{\text{reg}} \ltwo{\tilde{e}}{}.
\end{equation}
Using (\ref{dualpde}), integrating by parts twice, applying (\ref{scalarformula}) and (\ref{lifting}) as well as the regularity of the dual solution, $z \in H^4(\Omega)$, we have
\begin{equation}\label{ltwo_apost_estimator_identity1}
\begin{aligned}
\ltwo{\tilde{e}}{}^2 =& \sum_{\kappa \in \wavy{T}} \int_{\kappa} \, \Delta^2 z \; \tilde{e} \, \mathrm{d} x
= \int_{\Omega} \Delta_h z \Delta_h\tilde{e} \mathrm{d} x
- \int_{\Gamma} \jump{\nabla\tilde{e}} \mean{\Delta z } \mathrm{d} s + \int_{\Gamma} \jumptwo{\tilde{e}} \cdot \mean{\nabla \Delta z} \mathrm{d} s.
\end{aligned}
\end{equation}
By using the fact that $\tilde{u}$ is a weak solution and integrating the term involving $\Delta_h z \Delta_h \tilde{u}_h$ by parts, we arrive at,
\begin{equation}\label{ltwo_apost_estimator_intbypar2}
\begin{aligned}
\ltwo{\tilde{e}}{}^2
= &\; B(\tilde{u}, z) - \int_{\Omega} \Delta_h z \Delta_h \tilde{u}_h \mathrm{d} x
+ \int_{\Gamma} \jump{\nabla \tilde{u}_h} \mean{\Delta z } \mathrm{d} s - \int_{\Gamma} \jumptwo{\tilde{u}_h} \cdot \mean{\nabla \Delta z} \mathrm{d} s \\
= &\; l(z) - \int_{\Omega} z \Delta^2 \tilde{u}_h \mathrm{d} x
+ \ints{\Gamma_{\text{\rm int}}}{\mean{z}\jump{\nabla\Delta_h \tilde{u}_h}}
- \ints{\Gamma_{\text{\rm int}}}{\mean{\nabla z}\cdot\jumptwo{\Delta_h{\tilde{u}_h}}} \\
& \;+ \int_{\Gamma} \jump{\nabla \tilde{u}_h} \mean{\Delta z } \mathrm{d} s - \int_{\Gamma} \jumptwo{\tilde{u}_h} \cdot \mean{\nabla \Delta z} \mathrm{d} s.
\end{aligned}
\end{equation}
\noindent Using the standard orthogonal $L^2$-projection, $\Pi : \mathcal{S} \to S^r$, of $z$ , we can derive the following identity by integrating by parts and using (\ref{scalarformula}) and (\ref{lifting}) as follows,
\begin{equation}\label{ltwo_apost_estimator_conferror}
\begin{aligned}
& 0 = l( - \Pi z) - B(\tilde{u}_h, - \Pi z) \\
=&\sum_{\kappa \in \wavy{T}} \intx{\kappa}{\big((\phi-\dels{\tilde{u}_h})(- \Pi z) -\wavy{L}(\tilde{u}_h)\Delta_h{(- \Pi z)}\big)}
+ \ints{\Gamma_{\text{\rm int}}}{\mean{- \Pi z}\jump{\nabla\Delta_h \tilde{u}_h}} \\
&- \ints{\Gamma_{\text{\rm int}}}{\mean{\nabla(- \Pi z)}\cdot\jumptwo{\Delta_h{\tilde{u}_h}}}
- \ints{\Gamma}{\big(\sigma\jumptwo{\tilde{u}_h}\cdot\jumptwo{- \Pi z}+\xi\jump{\nabla\tilde{u}_h}\jump{\nabla(- \Pi z)}\big)} .
\end{aligned}
\end{equation}
Using (\ref{lifting}) in (\ref{ltwo_apost_estimator_conferror}) and combining (\ref{ltwo_apost_estimator_intbypar2}) and (\ref{ltwo_apost_estimator_conferror}), we get
\begin{equation}\label{ltwo_apost_estimator_identity2}
\begin{aligned}
\ltwo{\tilde{e}}{}^2 \!= & \ltwo{\tilde{e}}{}^2 + l( - \Pi z) - B(\tilde{u}_h, - \Pi z) \\
\!= & \! \int_{\Omega} (\phi - \Delta_h^2\tilde{u}_h) {(z - \Pi z)} \mathrm{d} x
+ \! \int_{ \Gamma_{\text{\rm int}}}\!\! \Big(\mean{z - \Pi z} \jump{\nabla \Delta \tilde{u}_h} \mathrm{d} s
- \jumptwo{\Delta \tilde{u}_h} \cdot \mean{\nabla (z - \Pi z)} \Big) \mathrm{d} s \\
& -\int_{\Gamma} \jumptwo{\tilde{u}_h} \cdot \Big( \mean{\nabla \Delta (z - \Pi z) } +\sigma_0 \mbf{h}^{-3} \jumptwo{z - \Pi z} \Big) \mathrm{d} s\\
& +\int_{\Gamma} \jump{\nabla(\tilde{u}_h)} \Big( \mean{ \Delta (z - \Pi z) } +\xi_0 \mbf{h}^{-1} \jump{\nabla(z - \Pi z)} \Big) \mathrm{d} s .
\end{aligned}
\end{equation}
The assertion then follows by applying Young's inequality, the trace inequality (\ref{traceinequality}) where appropriate, the approximation property (\ref{approximationproperty}) and the regularity of the dual problem on each of the terms on the right hand side of (\ref{ltwo_apost_estimator_identity2}).
\end{proof}
\begin{remark} \label{remark:smoothsubspace}
If a smooth $C^1$ subspace of the finite element space exists, such as Argyris elements in two dimensions, or corresponding constructions in three dimensions, it is possible to establish an a posteriori $L^2$ bound without dependence on penalty parameters; indeed, these terms would vanish from (\ref{ltwo_apost_estimator_identity2}) if the projection, $\Pi$, could be defined onto the smooth subspace of $S^r$.
\end{remark}
\begin{remark} \label{remark:suboptimality}
It is interesting to note that the a posteriori error bound of (\ref{ltwo_apost_bound_bound}) reflects the suboptimal $L^2$-norm error convergence of the dG method when quadratic polynomials are applied. Similar behaviour is observed theoretically and numerically in \cite{georgoulis_houston} and in \cite{suli.moz:sipg} in the context of the a priori error analysis of the same method.
\end{remark}
\section{DG method for the parabolic problem} \label{section:nummethod}
Throughout the remaining of this work, we shall denote by $u$ the weak solution of the problem (\ref{modelpde1})--(\ref{modelpde3}) in variational form:
find $u \in H^1(0,T; H^4(\Omega) \cap H^2_0(\Omega) )$ such that
\begin{equation} \label{varmodelpde}
\begin{aligned}
\langle u_t, \phi \rangle + B(u,\phi)&=\langle f,\phi\rangle \quad \forall \phi \in H^2_0(\Omega), \\
u &= u_0 \in L^2(\Omega) \quad \text{ in } \; \Omega \times \{0\}.\quad\quad
\end{aligned}
\end{equation}
We consider a subdivision of the time interval $(0,T]$ to be the family of intervals $\{(t^{n-1},t^{n}]$ ; $n=1,\dots, N$, with $t^0=0$, $t^{n-1} \le t^{n}$ and $t^N=T\}$ , with local time-step $\lambda_n:=t^{n}-t^{n-1}$. Associated with this time-subdivision, let $\wavy{T}_{n}$, $n=0,\dots, N$, be a sequence of meshes which are
assumed to be \emph{compatible}, in the sense that for any two
consecutive meshes $\wavy{T}_{n-1}$ and $\wavy{T}_{n}$, $\wavy{T}_{n}$ can be obtained
from $\wavy{T}_{n-1}$ by locally coarsening some of its elements and then locally refining some (possibly other) elements.
The finite element space corresponding to $\wavy{T}_{n}$ will be denoted by $\femspacen{n}$ and the respective dG bilinear form by $B^n(\cdot,\cdot)$. The backward Euler-dG method for approximating (\ref{varmodelpde}) is then given by: for each $n=1,\dots,N$, find
\begin{equation}\label{eqn:fully-discrete-Euler-pure}
\begin{split}
\ U^{n}\in \femspacen{n}\text{ such that }
\langle \frac{U^{n}- U^{n-1}}{\lambda_{n}},V\rangle+B^n(U^{n},V)=\langle \tilde{f}^{n},V\rangle
\quad \forall V\in \femspacen{n},
\end{split}
\end{equation}
where $\tilde{f}^0(\cdot):=f(\cdot,0)$ and $\tilde{f}^n(\cdot)$ for $n=1,\dots,N$ is a piecewise polynomial of degree $p$ in time $L^2$-projection in time of the source function $f$. In practice, it suffices to take $p=0$ to achieve a first-order-in-time convergent method. We also set $U^0:=\Pi^0 u_0$, with $\Pi^0: L^2(\Omega) \to \femspacen{0}$ is the orthogonal $L^2$-projection operator onto the finite element space $\femspacen{0}$.
\section{A posteriori bounds for the parabolic problem}\label{apost_parabolic}
We shall derive a posteriori error bounds for the backward Euler-dG method (\ref{eqn:fully-discrete-Euler-pure}) measured in $L^\infty(L^2)$- and $L^2(L^2)$-norms. To this end, we shall employ an energy argument (with carefully defined test functions) in conjunction with the elliptic reconstruction technique \cite{MR2034895,lakkis-makridakis:06,georgoulis_lakkis_virtanen}.
We begin by extending the sequence ${\{U^n\}}_{n=1,\dots, N}$ of numerical solutions into a continuous piecewise linear
function of time
\begin{equation} \label{eqn:fully-discrete-solution-extended}
U (0)=\Pi^0 u_0 \quad \text{ and } \quad U(t):= \sfrac{t - t^{n-1}}{\lambda_n} U^n + \sfrac{t^{n} - t}{\lambda_n} U^{n-1}
\end{equation}
for $t \in (t_{n-1},t_{n}]$ and $n=1,\dots, N$. Further, the \emph{discrete elliptic operator} $A^n:\femspacen{n} \to \femspacen{n}$ is defined by
\begin{equation}\label{ell_rec2}
\text{for}\ \phi \in \femspacen{n},\quad \langle A^n \phi, \chi \rangle = B^n(\phi,\chi)\quad \forall \chi\in \femspacen{n}.
\end{equation}
We now give definitions of the estimators involved in the estimation of the parabolic part of the error. Estimators at time step $n$ are denoted by ${\tiny \infty,n}$ subscript and ${\tiny 2,n}$ subscript will be used for the cases of $L^\infty(L^2)$- and $L^2(L^2)$-bounds presented below, respectively.
\begin{definition}[estimators for the parabolic error] \label{def_estimators}
We define the \emph{coarsening} or \emph{mesh-change estimators} by
\begin{equation}\label{apost_parabolic_coarse_estimators}
\gamma_{\infty,n} := \frac{1}{\lambda_n} \ltwo{(I-\Pi^n)U^{n-1}}{}^2,\quad
\gamma_{2,n} := {\ltwo{(I-\Pi^n)U^{n-1}}{}}^2 + \sum_{i=1}^{n-1} {\ltwo{ (\Pi^i - \Pi^{i-1}) U^{i-1} }{}}^2,
\end{equation}
the \emph{time-error evolution estimators} by
\begin{equation}\label{apost_parabolic_time_estimators}
\eta_{\infty,n} := \ltwo{g^n - g^{n-1}}{}^2 {\lambda_n}, \quad
\eta_{2,n} := \ltwo{g^n - g^{n-1}}{}^2 \lambda_n^{2} + \sum_{i=1}^{n-1} \lambda_{i}^{2} \ltwo{ g^i - g^{i-1} }{}^2,
\end{equation}
$g^n := A^n U^n-\Pi^n \tilde{f}^n +\tilde{f}^n$; the \emph{data approximation error in time estimators} by
\begin{equation}\label{apost_parabolic_data_estimators}
\beta_{\infty,n} := \int_{t^{n-1}}^{t^{n}} \ltwo{ \tilde{f}^n - f}{}^2\mathrm{d} t, \quad
\beta_{2,n} := \lambda_n \int_{t^{n-1}}^{t^{n}} \ltwo{ \tilde{f}^n - f}{}^2\mathrm{d} t,
\end{equation}
and an additional space estimator given by
\begin{equation}\label{apost_elliptic_extra_linfty_estimator}
\tilde{\eta}_{\infty,n} := \mathcal{E}\Big(\hat{\wavy{T}}_n, U^n-U^{n-1},g^n - g^{n-1} \Big)^2,
\end{equation}
where $\hat{\wavy{T}}_n := \wavy{T}_{n} \cap \wavy{T}_{n-1}$ {\it finest common coarsening} of $\wavy{T}_{n}$ and $\wavy{T}_{n-1}$ for each $n=1,\dots, N$.
\end{definition}
Using the notation above, we are ready to state the main result.
\begin{theorem}[a posteriori bound] \label{theorem:apost_parabolic}
Let $u \in L^2(0,T;H^4(\Omega) \cap H^2_0(\Omega))$ be the solution of (\ref{varmodelpde}), $ U$
be the approximation obtained by the dG method (\ref{eqn:fully-discrete-Euler-pure}) and defined by (\ref{eqn:fully-discrete-solution-extended}).
Then there exist positive constants $\const{\infty}$ and $\const{2}$, independent of $\wavy{T}_n$, $\mbf{h}$, $u$ and $U$, for any $n=1,\dots, N$ such that
\begin{eqnarray}
\ltwo{e}{L^\infty(0,T;L^2(\Omega))} &\le & \const{\infty} \; \Bigg( \ltwo{e(0)}{} + {\left( \sum_{n=1}^{N} \left( \gamma_{\infty,n} +\eta_{\infty,n}+\beta_{\infty,n} \right) \lambda_n \right)}^{\frac{1}{2}} \nonumber \\
\phantom{\ltwo{e}{L^\infty(0,T;L^2(\Omega))}} &\phantom{\le} & \phantom{\const{\infty} \; \Bigg( }
+ {\left( \sum_{n=1}^{N} \tilde{\eta}_{\infty,n} \right)}^{\frac{1}{2}}
+ \max_{0 \le n \le N} \{{\mathcal{E} \left(\wavy{T}_n, U^n, g^n \right)}\} \Bigg) \label{apost_parabolic_boundLinfty} \\
\ltwo{e}{L^2(0,T;L^2(\Omega))} &\le & \const{2} \; \Bigg( \ltwo{e(0)}{} + {\left( \sum_{n=1}^{N} \left( \gamma_{2,n} +\eta_{2,n}+\beta_{2,n} \right) \; \lambda_n \right)}^{\frac{1}{2}} \nonumber \\
\phantom{\ltwo{e}{L^2(0,T;L^2(\Omega))} }&\phantom{\le }&
\phantom{\const{2} \; \Bigg( \ltwo{e(0)}{} + \sum_{n=1}^{N} \Bigg( \eta_{1,2,n} } + {\left( \sum_{n=1}^{N} { \; {\mathcal{E} \left(\wavy{T}_n, U^n, g^n \right)}^2 \; \lambda_n }\right)}^{\frac{1}{2}} \Bigg) . \label{apost_parabolic_boundL2}
\end{eqnarray}
\end{theorem}
The proof of this theorem will be the content of the remaining of this section, split into a number of intermediate results.
We begin by defining the \emph{elliptic reconstruction} $\omega^n \in H^2_0(\Omega)$, of $U^n$ to be
the solution of the elliptic problem
\begin{equation}\label{ell_rec}
B^n(\omega^n, v) = \langle g^n, v \rangle\quad \forall v \in H^2_0(\Omega)
\end{equation}
where, as above, $g^n := A^n U^n-\Pi^n \tilde{f}^n +\tilde{f}^n$.
We note that under the assumptions on the domain $\Omega$, we also have $\omega^n \in H^4(\Omega)$.
We also extend the elliptic reconstruction into a continuous piecewise linear-in-time function
\begin{equation} \label{eqn:fully-discrete-ellrec-extended}
\omega(t):= \sfrac{t - t^{n-1}}{\lambda_n} \omega^n + \sfrac{t^{n} - t}{\lambda_n} \omega^{n-1}
\end{equation}
for $t \in (t_{n-1},t_{n}]$ and $n=1,\dots, N$.
Finally, we introduce the error decomposition
\begin{equation}\label{errorsplitting}
e:=U-u=\rho-\epsilon,\ \text{where}\ \epsilon:=\omega-U,\ \text{and}\ \rho:=\omega-u,
\end{equation}
where $\rho$ and $\epsilon$ are understood as the \emph{parabolic} and \emph{elliptic} error, respectively, and we set $\epsilon_n:=\epsilon(t_n)$.
\begin{lemma}[Error identity] \label{lemma:parerridentity}
For all $t \in (t_{n-1},t_{n}]$, $n=1,2,\dots,N$, we have
\begin{equation} \label{eqn:parerridentity}
\langle \rho_t,v\rangle + B^n(\rho,v) =\langle \epsilon_t,v\rangle + \langle (I - \Pi^n) U_t ,v\rangle + \sfrac{t- t^n}{\lambda_n}\langle g^n - g^{n-1},v\rangle +\langle \tilde{f}^n - f ,v\rangle,
\end{equation}
for any $v\in H^2_0(\Omega)$, with $I$ denoting the identity mapping.
\end{lemma}
\begin{proof}
Firstly, from (\ref{ell_rec}) and (\ref{ell_rec2}) we have
\begin{equation} \label{eqn:parerridentity1}
B^n(\omega^n,v) - \langle \tilde{f}^n ,v\rangle = B^n(U^n,\Pi^n v) - \langle \Pi^n \tilde{f}^n , \Pi^n v\rangle.
\end{equation}
Also, using the method (\ref{eqn:fully-discrete-Euler-pure}) and the definition of the $L^2$-projection we deduce
\begin{equation}\label{eqn:parerridentity2}
\langle U_t ,v\rangle
= \langle (I - \Pi^n) U_t ,v\rangle + \langle U_t ,\Pi^n v\rangle
= \langle (I - \Pi^n) U_t ,v\rangle - ( B^n(U_n,\Pi^n v) - \langle \Pi^n\tilde{f}^n ,\Pi^n v\rangle) .
\end{equation}
For the elliptic reconstruction error we also have,
\begin{equation}\label{eqn:parerridentity3}
B^n(\omega - \omega^n,v)
= \sfrac{t- t^n}{\lambda_n}\langle g^n - g^{n-1},v\rangle .
\end{equation}
Lastly, for the terms on the left hand side of (\ref{eqn:parerridentity}), we compute
\begin{equation}\label{eqn:parerridentity4}
\langle e_t,v\rangle + B^n(\rho,v)
= \langle U_t,v\rangle + B^n(\omega ,v) - \left( \langle u_t,v\rangle + B^n(u,v) \right)
= \langle U_t,v\rangle + B^n(\omega ,v) - \langle f ,v\rangle.
\end{equation}
Using (\ref{eqn:parerridentity1}), (\ref{eqn:parerridentity1}), and (\ref{eqn:parerridentity3}) in (\ref{eqn:parerridentity4}), along with the identity $e=\rho-\epsilon$ completes the proof.
\end{proof}
The a posteriori bounds (\ref{apost_parabolic_boundLinfty}) and (\ref{apost_parabolic_boundL2}) will be derived by selecting special test functions $v$ in the energy identity (\ref{eqn:parerridentity}) above, along with estimation of the terms on the right-hand side of (\ref{eqn:parerridentity}).
More specifically, we consider the following two test functions, $\tilde{v}:=\rho$ for the $L^\infty(L^2)$ case, and
\begin{equation} \label{def:testfuncLtwo}
\bar{v} (t,\cdot) := \int_t^{T} \rho(s,\cdot) \mathrm{d} s,\quad t\in[0,T],
\end{equation}
for the $L^2(L^2)$ case; this choice is motivated by Baker \cite{baker:76}, who used a similar construction for the proof of a priori bounds for the second order wave problem. The latter test function has most notably the following properties:
\begin{equation}\label{prop:testfuncLtwo}
\begin{aligned}
&\bar{v} \in H^4(\Omega) \cap H^2_0(\Omega) \quad \text{ as } \rho \in H^4(\Omega) \cap H^2_0(\Omega)\; \text{a.e. in}\ [0,T] ,\\
&\bar{v} (T,\cdot)=0=\Delta \bar{v} (T,\cdot),\quad \nabla \bar{v}(T,\cdot)=0,\quad\text{and}\quad \\
&\bar{v}_t(t,\cdot)=-\rho(t,\cdot),\quad \text{a.e. in}\ [0,T].
\end{aligned}
\end{equation}
Next, we consider two auxiliary functions which are needed in the consequent proofs. More specifically, on each interval $t \in (t_{n-1},t_{n}]$, for $n=1,\dots, N$, we define
\begin{equation} \label{def:auxiliaryfunc}
\tilde{G}(t) := (I-\Pi^n)U + \psi^{n}, \quad\text{ with }\quad \psi^{n}:= -(I-\Pi^n)U^{n-1}+\psi^{n-1},\quad \psi^0 :=0,
\end{equation}
and
\begin{equation} \label{def:auxiliaryfunc2}
G(t) := \sfrac{\lambda_{n}}{2} \left( \sfrac{t-t^{n}}{\lambda_{n}} \right)^2 (g^n - g^{n-1}) + \theta^{n}
, \quad\text{ with }\quad\theta^{n}:= - \sfrac{\lambda_{n}}{2} (g^n - g^{n-1}) + \theta^{n-1},\quad \theta^0=0.
\end{equation}
We note that, for each $n=1,\dots, N$, we then have $\tilde{G}(t^{n})=\psi^{n}$, ${{G}}(t^{n})=\theta^{n}$,
\begin{equation} \label{prop:auxiliaryfunc}
\tilde{G}_t(t) := (I-\Pi^n)U_t, \quad \text{ and } \quad {G}_t(t) := \sfrac{t- t^n}{\lambda_n} \left(g^n - g^{n-1} \right) .
\end{equation}
The following estimates will be used in the proof of Theorem \ref{theorem:apost_parabolic}.
\begin{lemma} \label{lemma:linftyextratermsestimate}
Let $\tau \in (0,T]$. Then, we have
\begin{eqnarray}
\int_{0}^{\tau} \langle {\tilde{G}}_t ,\rho\rangle \; \mathrm{d} t &\le& \sum_{n=1}^{N} \ltwo{(\Pi^n - I) U^{n-1}}{} \max_{0 \le t \le T} \ltwo{\rho}{} \label{eqn:linftyextratermsestimate1} \\
\int_{0}^{\tau} \langle G_t ,\rho\rangle \; \mathrm{d} t &\le& \sum_{n=1}^{N} \lambda_{n} \ltwo{g^{n} - g^{n-1}}{}
\max_{0 \le t \le T} \ltwo{\rho}{} \label{eqn:linftyextratermsestimate2} \\
\int_{0}^{\tau} \langle \tilde{f}^n - f ,\rho\rangle \; \mathrm{d} t &\le& \int_{0}^{\tau} \ltwo{\tilde{f}^n - f }{} \mathrm{d} t \max_{0 \le t \le T} \ltwo{\rho}{} \label{eqn:linftyextratermsestimate3} .
\end{eqnarray}
\end{lemma}
\begin{proof}
The proofs of these estimates are immediate via Cauchy-Schwarz-in-space and H\"older-in-time inequalities.
\end{proof}
In the following three lemmata, we prove bounds for the corresponding terms to the ones in Lemma \ref{lemma:linftyextratermsestimate} when testing with $\bar{v}$ given in (\ref{def:testfuncLtwo}).
\begin{lemma} \label{lemma:ltewoextratermsestimate1}
With the above notation, we have
\begin{equation}
\label{eqn:ltwoextratermsestimate1}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}}\!\! \langle (I - \Pi^n) U_t , \bar{v} \rangle \; \mathrm{d} t \le \sum_{n=1}^{N} \Big( \lambda_n {\ltwo{(I - \Pi^n) U^{n-1}}{}}^2
+ \lambda_n \ltwo{ \sum_{i=1}^{n-1} (\Pi^i - \Pi^{i-1}) U^{i-1} }{}^2 \Big)^{\frac{1}{2}} \Big(\int_{t^{n-1}}^{t^{n}} \ltwo{\rho}{}^2\mathrm{d} t \Big)^{\frac{1}{2}} .
\end{equation}
\end{lemma}
\begin{proof}
Recalling the definition of $\tilde{G}$, an integration by parts with respect to time gives
\begin{equation}
\label{eqn:ltwoextratermsestimateproof1}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle (I - \Pi^n) U_t , \bar{v} \rangle \; \mathrm{d} t = \sum_{n=1}^{N} \left[ \langle \tilde{G}(t) , \bar{v}(t) \rangle \right]_{t^{n-1}}^{t^{n}} + \sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle \tilde{G} , - \bar{v}_t \rangle \; \mathrm{d} t
= \sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle \tilde{G} , - \bar{v}_t \rangle \; \mathrm{d} t.
\end{equation}
We recall the properties of $\bar{v}$ in (\ref{prop:testfuncLtwo}) and we estimate theright-hand term further:
\begin{equation}
\label{eqn:ltwoextratermsestimateproof2}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle \tilde{G} , - \bar{v}_t \rangle \; \mathrm{d} t
\le \sum_{n=1}^{N} \Big( \int_{t^{n-1}}^{t^{n}} \ltwo{ \tilde{G}}{}^2 \; \mathrm{d} t \Big)^{\frac{1}{2}} \Big( \int_{t^{n-1}}^{t^{n}} \ltwo{\rho}{}^2 \; \mathrm{d} t \Big)^{\frac{1}{2}} .
\end{equation}
The assertion then follows by estimation of the time integral of $\ltwo{ \tilde{G}}{}^2$:
\begin{equation}
\label{eqn:ltwoextratermsestimateproof3}
\int_{t^{n-1}}^{t^{n}} \ltwo{ \tilde{G}}{}^2\mathrm{d} t \le \lambda_n \ltwo{ (I-\Pi^n) U^{n-1} }{}^2 + \lambda_n \ltwo{ \psi_{n-1} }{}^2,
\end{equation}
and noting that
$
\psi_{n-1} = \sum_{i=1}^{n-1} (\Pi^i - \Pi^{i-1}) U^{i-1}
$.
\end{proof}
\begin{lemma} \label{lemma:ltewoextratermsestimate2}
With the above notation, we have
\begin{equation}
\label{eqn:ltwoextratermsestimate2}
\sum_{n=1}^{N}\!\! \int_{t^{n-1}}^{t^{n}} \sfrac{t- t^n}{\lambda_n}\langleg^n - g^{n-1} , \bar{v} \rangle \; \mathrm{d} t \le \sum_{n=1}^{N} \Big( \lambda_n^3 \ltwo{g^n - g^{n-1} }{}^2 {+ \lambda_n \ltwo{ \sum_{i=1}^{n-1} \sfrac{\lambda_{i}}{2} (g^i - g^{i-1}) }{}^2 \Big)}^{\frac{1}{2}} \Big( \int_{t^{n-1}}^{t^{n}} \ltwo{\rho}{}^2\mathrm{d} t \Big)^{\frac{1}{2}} .
\end{equation}
\end{lemma}
\begin{proof}
Recalling the definition of $G$, an integration by parts with respect to time gives
\begin{equation}
\label{eqn:ltwoextratermsestimateproof21}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \sfrac{t- t^n}{\lambda_n}\langle g^n - g^{n-1} , \bar{v} \rangle \mathrm{d} t
= \sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle G , - \bar{v}_t \rangle \; \mathrm{d} t.
\end{equation}
We recall the properties of $\bar{v}$ in (\ref{prop:testfuncLtwo}) and estimate the right-hand side further:
\begin{equation}
\label{eqn:ltwoextratermsestimateproof22}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle G , - \bar{v}_t \rangle \; \mathrm{d} t
\le \sum_{n=1}^{N} {\Big( \int_{t^{n-1}}^{t^{n}} \ltwo{ G}{}^2 \; \mathrm{d} t \Big)}^{\frac{1}{2}} {\Big( \int_{t^{n-1}}^{t^{n}} \ltwo{\rho}{}^2 \; \mathrm{d} t \Big)}^{\frac{1}{2}} .
\end{equation}
The assertion then follows by estimation of the integral of $\ltwo{ G}{}^2$:
\begin{equation}
\label{eqn:ltwoextratermsestimateproof23}
\int_{t^{n-1}}^{t^{n}} \ltwo{G}{}^2\mathrm{d} t \le \lambda_n^3 \ltwo{g^n - g^{n-1} }{}^2 + \lambda_n \ltwo{ \theta_{n-1} }{}^2
\end{equation}
and noting that $
\theta_{n-1} = \sum_{i=1}^{n-1} - \sfrac{\lambda_{i}}{2} (g^i - g^{i-1})$.
\end{proof}
\begin{lemma} \label{lemma:ltewoextratermsestimate3}
With the above notation, we have
\begin{equation}
\label{eqn:ltwoextratermsestimate3}
\begin{aligned}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle \tilde{f}^n - f , \bar{v} \rangle \; \mathrm{d} t &\le C_{\text{app}} \sum_{n=1}^{N} \Big(\int_{t^{n-1}}^{t^{n}} \lambda_n^2 \ltwo{ \tilde{f}^n - f}{}^2\mathrm{d} t\Big)^{1/2} \Big(\int_{t^{n-1}}^{t^n} \ltwo{\rho}{}^2\mathrm{d} t\Big)^{1/2}.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
As $\tilde{f}^n$ is the $L^2$-projection of $f$ in time, we have
\begin{equation}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle \tilde{f}^n - f,\bar{v} \rangle\mathrm{d} t
=
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle \tilde{f}^n - f,\bar{v} - \zeta^n\rangle \mathrm{d} t,
\end{equation}
for the lowest order time approximation $\zeta^n(\cdot):=\lambda_n^{-1}\int_{t^{n-1}}^{t^n}\bar{v}(t,\cdot) \mathrm{d} t$ of $\bar{v}$.
With the approximation property of $\zeta^n$ in time, we deduce
\begin{equation}
\int_{t^{n-1}}^{t^n}\ltwo{\bar{v} - \zeta^n}{}^2\mathrm{d} t \le C_{\text{app}}^2 \lambda_n^2 \int_{t^{n-1}}^{t^n}\ltwo{\bar{v}_t}{}^2\mathrm{d} t,
\end{equation}
and recalling that $\bar{v}_t=-\rho$, the Cauchy-Schwarz inequality implies
\begin{equation}
\sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \langle \tilde{f} - f,\bar{v} \rangle\mathrm{d} t
\le C_{\text{app}}
\sum_{n=1}^{N} \Big(\int_{t^{n-1}}^{t^{n}} \lambda_n^2 \ltwo{ \tilde{f}^n - f}{}^2\mathrm{d} t\Big)^{1/2} \Big(\int_{t^{n-1}}^{t^n} \ltwo{\rho}{}^2\mathrm{d} t\Big)^{1/2}.
\end{equation}
\end{proof}
To complete a posteriori error bounds in Theorem \ref{theorem:apost_parabolic}, we also need the following two lemmata in which the elliptic error terms $\epsilon$ and $\epsilon_t$ are estimated by fully computable residuals.
\begin{lemma} \label{lemma:ellipticerrorsestimate1}
Let $\epsilon$ be as in (\ref{errorsplitting}). Then, we have
\begin{equation}
\label{eqn:ellipticerrorsestimate1}
\int_{0}^{T} \ltwo{\epsilon}{}^2 \; \mathrm{d} t \le \sfrac{2\lambda_N}{3} \mathcal{E}(\wavy{T}_N, U^N,g^N) + \sum_{n=1}^{N-1} \sfrac{4\lambda_n}{3} \mathcal{E} \left(\wavy{T}_n, U^n, g^n \right)^2
\end{equation}
\end{lemma}
\begin{proof}
Noting that $((t- t^{n-1})/\lambda_n)^2\le \frac{1}{3}$ and $((t^{n}-t)\lambda_n)^2\le \frac{1}{3}$, we have
\begin{equation}
\int_{0}^{T} \ltwo{\epsilon}{}^2 \mathrm{d} t
\le \sum_{n=1}^{N} \sfrac{2\lambda_n}{3} \Big( \ltwo{\epsilon_{n}}{}^2 + \ltwo{\epsilon_{n-1}}{}^2 \Big).
\end{equation}
The assertion then follows by Theorem \ref{theorem:ltwo_apost_biharmonic}.
\end{proof}
\begin{lemma} \label{lemma:ellipticerrorsestimate2}
Let $\epsilon$ be as in (\ref{errorsplitting}) and $\tau \in [0,T]$; then, we have
\begin{equation}
\label{eqn:ellipticerrorsestimate2}
\int_{0}^{\tau} \langle \epsilon_t , \rho \rangle \mathrm{d} t \le \sum_{n=1}^{N}
\mathcal{E} (\hat{\wavy{T}}_n, U^n-U^{n-1},g^n - g^{n-1} )\max_{0 \le t \le T} \ltwo{\rho}{} ,
\end{equation}
where
$\hat{\wavy{T}}_n := \wavy{T}_{n} \cap \wavy{T}_{n-1}$ denotes the finest common coarsening of $\wavy{T}_{n}$ and $\wavy{T}_{n-1}$, $n=1,\dots, N$.
\end{lemma}
\begin{proof}
We have
$ \epsilon_t(t) = (\epsilon_n-\epsilon_{n-1})/\lambda_n$, for $ t \in (t_{n-1},t_{n}]$ and $n=1,\dots, N$.
Denoting $\tau := t_{r+1/2}$ and $r:=\max \{k : t_k \le \tau, k=1,\ldots, N \}$, we then have
\begin{equation} \label{eqn:ellipticerrorsestimate_proof1}
\begin{aligned}
\int_{0}^{\tau} \langle \epsilon_t , \rho \rangle \mathrm{d} t &= \sum_{n=1}^{r+1/2} \int_{t^{n-1}}^{t^{n}} \sfrac{1}{\lambda_n} \langle \epsilon_n-\epsilon_{n-1} , \rho \rangle \mathrm{d} t
\le \max_{0 \le t \le T} \ltwo{\rho}{}\sum_{n=1}^{r+1/2} \ltwo{\epsilon_n-\epsilon_{n-1}}{}.
\end{aligned}
\end{equation}
We now observe that the finite element function $\tilde{z}$ in the proof of Theorem \ref{theorem:ltwo_apost_biharmonic} can be selected from a subspace of $S^r$: in particular, we can select the finite element subspace corresponding to the finest common coarsening mesh $\hat{\wavy{T}}_n $, for $n=1,\dots,N$. Then, following completely analogous argument as in the proof of of Theorem \ref{theorem:ltwo_apost_biharmonic}, we can arrive to the bound
\[
\ltwo{\epsilon_n-\epsilon_{n-1}}{} \le \mathcal{E} \Big(\hat{\wavy{T}}_n, U^n-U^{n-1},g^n - g^{n-1} \Big),
\]
which already yields the result.
\end{proof}
\begin{remark} \label{remark:ellertimederivativebound}
Note that the following simpler alternative bound for the term in Lemma \ref{lemma:ellipticerrorsestimate2} is also possible,
\begin{equation}
\label{eqn:ellipticerrorsestimate2alt}
\begin{aligned}
\int_{0}^{\tau} \langle \epsilon_t , \rho \rangle \mathrm{d} t & \le \; \sum_{n=0}^{N} \mathcal{E} \left(\wavy{T}_n, U^n, g^n \right) \; \max_{0 \le t \le T} \ltwo{\rho}{} .
\end{aligned}
\end{equation}
This bound shifts the emphases from the finest common coarsening mesh, $\hat{\wavy{T}}_n$, in Lemma \ref{lemma:ellipticerrorsestimate2} to the elliptic estimators acting on meshes at each time step only which can be of practical importance when implementing adaptive algorithms based on the estimators.
\end{remark}
\begin{proof} {\bf of Theorem \ref{theorem:apost_parabolic}} To conclude the proof,
we estimate the left-hand side of (\ref{eqn:parerridentity}) in each case of the test functions: $\bar{v}$ to derive $L^2(L^2)$-norm a posteriori bound and $\rho$ for the $L^\infty(L^2)$-norm bound.
First we deal with the $L^2(L^2)$ case; we start by integrating (\ref{eqn:parerridentity}) by parts in time,
\begin{equation}\label{eqn:apost_parabolic_ltwo_parerridentity}
\begin{aligned}
\int_{0}^{T} \langle e_t , \bar{v} \rangle + B(\rho,\bar{v}) \mathrm{d} t& = \int_{0}^{T} \langle e , -\bar{v}_t \rangle \mathrm{d} t + \left[ \langle e ,\bar{v} \rangle \right]_{0}^{T} - \int_{0}^{T} B(\bar{v}_t,\bar{v}) \mathrm{d} t \\
& = \int_{0}^{T} \langle \rho , \rho \rangle \mathrm{d} t - \int_{0}^{T} \langle \epsilon , \rho \rangle \mathrm{d} t - \langle e(0) ,\bar{v}(0) \rangle - \int_{0}^{T} \sfrac{1}{2} \sfrac{d}{dt} B(\bar{v},\bar{v}) \mathrm{d} t \\
& = \int_{0}^{T} \ltwo{\rho}{}^2 \mathrm{d} t - \int_{0}^{T} \langle \epsilon , \rho \rangle \mathrm{d} t - \langle e(0) ,\bar{v}(0) \rangle + \sfrac{1}{2} B(\bar{v}(0),\bar{v}(0)) .
\end{aligned}
\end{equation}
We also have
\begin{equation}\label{eqn:apost_parabolic_ltwo_energyterm}
\langle e(0) ,\bar{v}(0) \rangle \le \ltwo{e(0)}{} \ltwo{\bar{v}(0)}{} \le \ltwo{e(0)}{} C_{\text{pf}} B(\bar{v}(0),\bar{v}(0)).
\end{equation}
Using (\ref{eqn:apost_parabolic_ltwo_parerridentity}) and (\ref{eqn:apost_parabolic_ltwo_energyterm})
in (\ref{eqn:parerridentity}) after integration over each interval $(t^{n-1},t^{n}]$ and summation with respect to $n$, we get,
\begin{equation}
\ltwo{\rho}{L^2(0,T,L^2(\Omega))}^2 \le \ltwo{e(0)}{}^2 + \sum_{n=1}^{N} \int_{t^{n-1}}^{t^{n}} \Big( \langle \epsilon , \rho \rangle + \langle (I - \Pi^n) U_t ,\bar{v}\rangle + \sfrac{t- t^n}{\lambda_n }\langle g^n - g^{n-1} ,\bar{v}\rangle +\langle \tilde{f}^n - f ,\bar{v}\rangle \Big)\mathrm{d} t .
\end{equation}
The bound (\ref{apost_parabolic_boundL2}) now follows upon using the triangle inequality
\[
\ltwo{e}{L^2(0,T,L^2(\Omega))} \le \ltwo{\rho}{L^2(0,T,L^2(\Omega))} + \ltwo{\epsilon}{L^2(0,T,L^2(\Omega))},
\]
Young's inequality and Lemmata \ref{lemma:ltewoextratermsestimate1}, \ref{lemma:ltewoextratermsestimate2}, \ref{lemma:ltewoextratermsestimate3} and \ref{lemma:ellipticerrorsestimate1}.
For the $L^\infty(L^2)$-norm case, upon testing with $v=\rho$, we deduce for the left-hand side of (\ref{eqn:parerridentity}) by integrating by parts to some $\tau \in [0,T]$,
\begin{equation}\label{eqn:apost_parabolic_linfty_parerridentity}
\int_{0}^{\tau} \langle e_t , \rho \rangle + B(\rho,\rho) \mathrm{d} t
= \ltwo{ \rho(\tau)}{}^2 - \ltwo{ \rho(0)}{}^2 - \int_{0}^{\tau} \langle \epsilon_t , \rho \rangle \mathrm{d} t+ \int_{0}^{\tau} B(\rho,\rho) \mathrm{d} t .
\end{equation}
Choosing $\tau$ such that $\ltwo{ \rho(\tau)}{}=\max_{0 \le t \le T} \ltwo{\rho}{} $, using the triangle inequality,
$\ltwo{\rho(0)}{} \le \ltwo{e(0)}{} + \ltwo{\epsilon(0)}{}$,
and (\ref{eqn:apost_parabolic_linfty_parerridentity}) in (\ref{eqn:parerridentity}), we get,
\begin{equation}
\ltwo{\rho}{L^\infty(0,T,L^2(\Omega))}^2 +\int_{0}^{\tau} B(\rho,\rho) \mathrm{d} t \le \ltwo{e(0)}{}^2+\ltwo{\epsilon(0)}{}^2 + \int_{0}^{\tau} \Big( \langle \epsilon_t , \rho \rangle + \langle \tilde{G}_t ,\rho\rangle
+ \langle {G}_t,\rho\rangle +\langle \tilde{f}^n - f ,\rho\rangle \Big)\mathrm{d} t
\end{equation}
where $G$ and $\tilde{G}$ are given by (\ref{def:auxiliaryfunc2}) and (\ref{def:auxiliaryfunc}).
The bound (\ref{apost_parabolic_boundLinfty}) follows again using triangle inequality
\begin{equation}
\ltwo{e}{L^\infty(0,T,L^2(\Omega))} \le \ltwo{\rho}{L^\infty(0,T,L^2(\Omega))} + \ltwo{\epsilon}{L^\infty(0,T,L^2(\Omega))},
\end{equation}
Lemmata \ref{lemma:linftyextratermsestimate} and \ref{lemma:ellipticerrorsestimate2} as well as
$\max_{0 \le t \le T} \ltwo{\epsilon}{} \le \max_{0 \le t \le T} \mathcal{E} \left(\wavy{T}_n, U^n, g^n \right)$.
\end{proof}
We note that a posteriori bounds in the $L^2(H^2)$-norm of the error have been already considered in \cite{thesis_juha}, along with their application within an adaptive algorithm. The $L^2(H^2)$-norm theoretical and numerical results appear to be of the expected order of convergence; they are omitted here for brevity.
\section{Numerical Experiments}\label{numerics}
For $t\in [0,1]$ and $\Omega := (0,1)^2$, we consider two benchmark problems for which $u_0$ and $f$ are chosen so that the exact solution $u$ of problem (\ref{varmodelpde}) coincides with one of the following solutions:
\begin{gather}
\label{eqn:numerics:example1}
u_1(x,y,t)
= \sin(\pi t) \; 10^2 \; \sin^2(\pi x) \; \sin^2(\pi y) e^{-10(x^2+y^2)},
\\
\label{eqn:numerics:example2}
u_2(x,y,t)
= \sin(20 \pi t) \sin^2(\pi x)\sin^2(\pi y) e^{-10(x^2+y^2)}.
\end{gather}
Solutions $u_1$ and $u_2$ are both smooth but $u_2$ oscillates much faster where as $u_1$ exhibits greater space dependency of the error. They are defined so as to emphasize different aspects of the estimators at hand. Similar examples have been studied elsewhere, for example in
\cite{thesis_juha} in the context of $L^2(H^2)$-norm a posteriori estimators; see also \cite{lakkis-makridakis:06,LakkisPryer:10,georgoulis_lakkis_virtanen} for similar examples in the context of second order problems.
For the numerical experiments, the library FEniCS ({\tt http://fenicsproject.org/}) was used. For each of the examples, we compute the solution of (\ref{eqn:fully-discrete-Euler-pure}) using quadratic simplicial finite element spaces and with interior penalty parameters $\sigma_0=\xi_0=20$ in (\ref{penaltyparametersdef}), which is sufficient to guarantee stability of the numerical scheme. The interior penalty parameters have a known effect on the effectivity indices, cf., \cite{thesis_juha,karakashian-pascal:07}.
We study the asymptotic behavior of the indicators by setting all constants appearing in Theorem
\ref{theorem:apost_parabolic} equal to one. We monitor the evolution of the values and the experimental order of
convergence of the estimators and the error as well as of the effectivity
index over time on a sequence of uniformly refined meshes with
$
h_{\kappa,i}:=2^{-i/2-1}$, $i=1,\dots,5$, $\kappa \in \wavy{T}
$
with fixed time steps $\lambda \approx \max_\kappa h_\kappa ^3 $ and $\lambda \approx \max_\kappa h_\kappa^2 $. To this end, we define \emph{experimental order of convergence} ($EOC$) of
a given sequence of positive quantities $a(i)$ defined on a sequence
of meshes of size $h(i)$ by
\begin{equation}
EOC( a,i ) = \frac{\log(a(i+1)/a(i))}{\log(h(i+1)/h(i))},
\end{equation}
the accumulated coarsening or mesh change estimators by
\begin{equation}\label{acc_apost_parabolic_coarse_estimators}
\mathbb{E}_{\text{coarsen},\infty,m} :=\Big( \sum_{n=1}^{m} \gamma_{\infty,n} \lambda_n \Big)^{\frac{1}{2}}\quad \text{ and }\quad
\mathbb{E}_{\text{coarsen},2,m} := \Big(\sum_{n=1}^{m} \gamma_{2,n} \lambda_n \Big)^{\frac{1}{2}},
\end{equation}
accumulated time error evolution estimators by
\begin{equation}\label{acc_apost_parabolic_time_estimators}
\mathbb{E}_{\text{time},\infty,m} := \Big(\sum_{n=1}^{m} ( \eta_{\infty,n} + \beta_{\infty,n}) \lambda_n + \sum_{n=1}^{m} \tilde{\eta}_{\infty,n}\Big)^{\frac{1}{2}} \quad \text{ and } \quad
\mathbb{E}_{\text{time},2,m} := \Big(\sum_{n=1}^{m} ( \eta_{2,n} + \beta_{2,n}) \lambda_n\Big)^{\frac{1}{2}} ,
\end{equation}
accumulated space error estimators by
\begin{equation}\label{acc_apost_elliptic_space_estimators}
\mathbb{E}_{\text{space},\infty,m} := \max_{0 \le n \le N} \{{\mathcal{E} \left(\wavy{T}_n, U^n, g^n \right)}\} \quad \text{ and } \quad
\mathbb{E}_{\text{space},2,m} := \Big(\sum_{n=1}^{m} {\mathcal{E} \left(\wavy{T}_n, U^n, g^n \right)}^2 \lambda_n\Big)^{\frac{1}{2}} ,
\end{equation}
and the \emph{inverse effectivity index}
\begin{equation}
IEI_m = \frac{\ltwo{e}{L^\infty(0,t_m;L^2(\Omega))}}{\mathbb{E}_{\text{time},\infty,m} + \mathbb{E}_{\text{space},\infty,m}} \quad \text{or} \quad IEI_m = \frac{\ltwo{e}{L^2(0,t_m;L^2(\Omega))}}{\mathbb{E}_{\text{time},2,m} + \mathbb{E}_{\text{space},2,m}},
\end{equation}
for the case $L^\infty(L^2)$ and $L^2(L^2)$, respectively. The IEI conveys
the same information as the (standard) effectivity index and has the advantage of relating directly
to the constants appearing in Theorem \ref{theorem:apost_parabolic}.
The results of numerical experiments on uniform meshes, depicted in Figures 1 - 4, indicate that the error estimators
are reliable and also efficient which can be seen from the effectivity index behaviour and the EOC of the error and the
time and space estimators for both $L^2(L^2)$- and $L^\infty(L^2)$-norm a posteriori bounds.
To further evaluate practical aspects of the derived a posteriori estimators, they are incorporated within in two adaptive algorithms; these are outlined in pseudocode as follows
\hspace{-1.5cm}\\
\begin{tabular}{p{8cm} p{8cm}}
\vspace{0pt}
{\footnotesize
\begin{tabular}{llllll}
\multicolumn{6}{l}{{\bf ImplicitTimeStepControl}}\\
\multicolumn{6}{l}{{\bf Input:} $U_{0}, f, \text{TOL}_{\text{time,min}}, \text{TOL}_{\text{time}}, \text{TOL}_{\text{space}}, \ldots $ }\\
& \multicolumn{5}{l}{ $ \text{TOL}_{\text{coarse}}, \lambda_0, t_0, T, \ldots $} \\
& \multicolumn{5}{l}{ $ \mathcal{T}_{0}, \xi_{refine}, \text{{\bf SpaceAdaptivity}}, \ldots $} \\
& \multicolumn{5}{l}{ $ \text{{\bf InitialSpaceAdaptivity}}$} \\
\multicolumn{6}{l}{\{ Initial condition interpolation and mesh refinement \}} \\
\multicolumn{6}{l}{$(U_0,\mathcal{T}_{0})$:={\bf InitialSpaceAdaptivity}($U_{0}, f, \mathcal{T}_{0}, \xi_{refine}$).} \\
\multicolumn{6}{l}{\{Initialize.\}} \\
\multicolumn{6}{l}{{\bf Set:} $n=1$, $\lambda_n=\lambda_{n-1}$.} \\
\multicolumn{6}{l}{{\bf While} $(t_n \le T)$ }\\
& \multicolumn{5}{l}{{\bf Set:} $\mathbb{E}_{\text{time}} := \text{TOL}_{\text{time}} + 1$}\\
& \multicolumn{5}{l}{{\bf While} $(\mathbb{E}_{\text{time}} > \text{TOL}_{\text{time}})$}\\
& & \multicolumn{4}{l}{$t_{n}:=t_{n-1}+\lambda_n$}\\
& & \multicolumn{4}{l}{{\bf Set:} $\mathcal{T}_{t} := \mathcal{T}_{n}$}\\
& & \multicolumn{4}{l}{ $(U_n,\mathcal{T}_{n})$ := {\bf SpaceAdaptivity}($U_{n-1}, \ldots$ }\\
& & & \multicolumn{3}{l}{ $\phantom{(U_n,\mathcal{T}_{n}):=}$ $ f, \text{TOL}_{\text{space}}, \text{TOL}_{\text{coarse}}, \ldots $}\\
& & & \multicolumn{3}{l}{ $\phantom{(U_n,\mathcal{T}_{n}):=}$ $ \lambda_n, t_n, T, \mathcal{T}_{n-1}, \xi_{refine}$) }\\
& & \multicolumn{4}{l}{ compute $\mathbb{E}_{\text{time}}$ .}\\
& & \multicolumn{4}{l}{{\bf if } $(\mathbb{E}_{\text{time}} > \text{TOL}_{\text{time}})$ {\bf then }}\\
& & & \multicolumn{3}{l}{ \{Shorten timestep.\} }\\
& & & \multicolumn{3}{l}{ $\lambda_{n} := \lambda_n / 2 $ }\\
& & & \multicolumn{3}{l}{{\bf Set:} $\mathcal{T}_{n} := \mathcal{T}_{t}$}\\
& & \multicolumn{4}{l}{{\bf endif}}\\
& \multicolumn{5}{l}{{\bf End While}}\\
& \multicolumn{5}{l}{ $\lambda_{n+1} := \lambda_n * 2$ }\\
& \multicolumn{5}{l}{$n:=n+1$}\\
\multicolumn{6}{l}{{\bf End While}}\\
\multicolumn{6}{l}{{\bf Output:} $U_n$} \\
\end{tabular}
\label{thetimealgorithm2}
}
& \vspace{0pt}
{\footnotesize
\begin{tabular}{llllll}
\multicolumn{6}{l}{{\bf ExplicitTimeStepControl}}\\
\multicolumn{6}{l}{{\bf Input:} $U_{0}, f, \text{TOL}_{\text{time,min}}, \text{TOL}_{\text{time}}, \text{TOL}_{\text{space}}, \ldots $ }\\
& \multicolumn{5}{l}{ $ \text{TOL}_{\text{coarse}}, \lambda_0, t_0, T, \ldots $} \\
& \multicolumn{5}{l}{ $ \mathcal{T}_{0}, \xi_{refine}, \text{{\bf SpaceAdaptivity}}, \ldots $} \\
& \multicolumn{5}{l}{ $ \text{{\bf InitialSpaceAdaptivity}}$} \\
\multicolumn{6}{l}{\{ Initial condition interpolation and mesh refinement \}} \\
\multicolumn{6}{l}{$(U_0,\mathcal{T}_{0})$:={\bf InitialSpaceAdaptivity}($U_{0}, f, \mathcal{T}_{0}, \xi_{refine}$).} \\
\multicolumn{6}{l}{\{Initialize.\}} \\
\multicolumn{6}{l}{{\bf Set:} $n=1$, $\lambda_n=\lambda_{n-1}$ and $t_n=t_{n-1} + \lambda_n$.} \\
\multicolumn{6}{l}{{\bf While} $(t_n \le T)$ }\\
& \multicolumn{5}{l}{ $(U_n,\mathcal{T}_{n})$ := {\bf SpaceAdaptivity}($U_{n-1}, \ldots$ }\\
& & \multicolumn{4}{l}{ $\phantom{(U_n,\mathcal{T}_{n}):=}$ $ f, \text{TOL}_{\text{space}}, \text{TOL}_{\text{coarse}}, \ldots $}\\
& & \multicolumn{4}{l}{ $\phantom{(U_n,\mathcal{T}_{n}):=}$ $ \lambda_n, t_n, T, \mathcal{T}_{n-1}, \xi_{refine}$) }\\
& \multicolumn{5}{l}{ compute $\mathbb{E}_{\text{time}}$ .}\\
& \multicolumn{5}{l}{{\bf if } $(\mathbb{E}_{\text{time}} > \text{TOL}_{\text{time}})$ {\bf then }}\\
& & \multicolumn{4}{l}{ $\lambda_{n+1} := \lambda_n / \sqrt{2} $ }\\
& \multicolumn{5}{l}{{\bf elseif } $(\mathbb{E}_{\text{time}} < \text{TOL}_{\text{time,min}})$ {\bf then }}\\
& & \multicolumn{4}{l}{ $\lambda_{n+1} := \lambda_n * \sqrt{2}$ }\\
& \multicolumn{5}{l}{{\bf endif}}\\
& \multicolumn{5}{l}{$t_{n+1}:=t_n+\lambda_n$}\\
& \multicolumn{5}{l}{$n:=n+1$}\\
\multicolumn{6}{l}{ {\bf End While} }\\
\multicolumn{6}{l}{{\bf Output:} $U_n$} \\
\end{tabular}
\label{thetimealgorithm1}
}
\\
\end{tabular}
\noindent where {\bf SpaceAdaptivity} (and {\bf InitialSpaceAdaptivity}) are performed using a standard {\em D\"orfler marking} strategy expressed in pseudocode as follows
\vspace{4pt}
{\footnotesize
\begin{tabular}{llllll}
\multicolumn{6}{l}{{\bf SpaceAdaptivity}}\\
\multicolumn{6}{l}{{\bf Input:} $U_{n-1}, f, \text{TOL}_{\text{space}}, \text{TOL}_{\text{coarse}}, \tau_n, t_n, T, \mathcal{T}_{n-1}, \xi_{refine}$} \\
\multicolumn{6}{l}{{\bf Set:} $\mathcal{T}_{n} := \mathcal{T}_{n-1}$.} \\
\multicolumn{6}{l}{$\mathcal{T}_{n} := $ \bf{SpaceCoarsening}($U_{n-1},\text{TOL}_{\text{coarse}}, \tau_n, \mathcal{T}_{n}$)} \\
\multicolumn{6}{l}{\{Refinement\}} \\
\multicolumn{6}{l}{compute local elliptic estimators, $(\text{LocalEst}_{n,\kappa})_{\kappa \in \mathcal{T}_{n}}$.}\\
\multicolumn{6}{l}{sum up local estimators and set $\text{Sum}_{\text{total}} := \sum_{\kappa \in \mathcal{T}_{n}} \text{LocalEst}_{n,\kappa}$, and compute $\mathbb{E}_{\text{space}}$.}\\
\multicolumn{6}{l}{{\bf While} $(\mathbb{E}_{\text{space}} > \text{TOL}_{\text{space}})$ }\\
& \multicolumn{5}{l}{sort $(\text{LocalEst}_{n,\kappa})_{\kappa \in \mathcal{T}_{n}}$ in descending order, set $Q := \emptyset$.}\\
& \multicolumn{5}{l}{{\bf Set:} $\text{Sum} = 0$.} \\
& \multicolumn{5}{l}{{\bf While} $(( \text{Sum} < \xi_{refine} * \text{Sum}_{\text{total}} )$ and $(\kappa \in \mathcal{T}_{n}))$ }\\
& \multicolumn{5}{l}{\{D\"orfler marking \} }\\
& & \multicolumn{4}{l}{ $\text{Sum} := \text{Sum} + \text{LocalEst}_{n,\kappa}$}\\
& & \multicolumn{4}{l}{{\bf if} $(Sum < \xi_{refine} * \text{Sum}_{\text{total}})$ }\\
& & & \multicolumn{3}{l}{Mark $\kappa$ for refinement; $Q := \{ \kappa \} \cup Q$. }\\
& \multicolumn{5}{l}{{\bf End While} }\\
& \multicolumn{5}{l}{Refine all elements in $Q$ to obtain new mesh $\mathcal{T}_{n}$. }\\
& \multicolumn{5}{l}{Solve $I^n U_{n-1}$. }\\
& \multicolumn{5}{l}{Solve (\ref{eqn:fully-discrete-Euler-pure}) for $U_n$ with $\Pi^n U_{n-1}, \Pi^n f^n, \tau_n$ and $t_n$ on $\mathcal{T}_n$. }\\
& \multicolumn{5}{l}{compute local elliptic estimators, $(\text{LocalEst}_{n,\kappa})_{\kappa \in \mathcal{T}_{n}}$.}\\
& \multicolumn{5}{l}{sum up local estimators and set $\text{Sum}_{\text{total}} := \sum_{\kappa \in \mathcal{T}_{n}} \text{LocalEst}_{n,\kappa}$, and compute $\mathbb{E}_{\text{space}}$.}\\
\multicolumn{6}{l}{ {\bf End While} }\\
\multicolumn{6}{l}{{\bf Output:} $U_n, \mathcal{T}_n$}
\end{tabular}
}
\noindent The refinement ratio $0 < \xi_{refine} \le 1$ and the tolerances $\text{TOL}_{\text{space}}>0,\text{TOL}_{\text{space}}>0$ and $ \text{TOL}_{\text{coarse}}>0$ are predefined quantities. The value of $\xi_{refine}:=0.75$ was used throughout the experiments in adaptive algorithms. Note that the coarsening tolerance, $\text{TOL}_{\text{coarse}}$, (as well as the tolerance for the alternative space estimator in $L^\infty$ case of Remark \ref{remark:ellertimederivativebound}) had to be determined experimentally for given space and time tolerances and depending on an example.
The results of experiments with adaptive algorithms as well as a comparison between the two algorithms, are detailed in Figures 6-8 where we monitor time step size, accumulated degrees of freedom and error evolution in comparison to the uniform approach leading to the desired tolerance. The results of these test cases imply substantial reduction in degrees of freedom by both adaptive algorithms in order to reach the same error tolerance as compared with the uniform approach. This implies a potential efficiency gain in solving PDE problems addressed in this work.
The estimators presented here are found to be suitable for both adaptive time stepping algorithms due to their good separated scaling properties in time and in space. The numerical results appear to be less sensitive to mesh change, compared to the same adaptive algorithms based on the $L^2(H^2)$-norm a posteriori error estimators presented in \cite{thesis_juha}.
For instance, terms involving $({g}^{n} - {g}^{n-1})$ which is sensitive to mesh change (coarsening as well as refinement) scale down sufficiently fast with the a posteriori estimators presented in this work, resulting to robust error reduction in an adaptive algorithm.
Finally, we note that the considerably more computationally efficient {\bf ExplicitTimeStepControl} algorithm (due to absence of time step searching step) was found to reach desired error tolerances (even though this is not guaranteed in general) in these numerical experiments.
\begin{figure}[ht]
\includegraphics[clip,width=\textwidth]{figures1}
\end{figure}
\begin{figure}[ht]
\includegraphics[clip,width=\textwidth]{figures2}
\end{figure}
\begin{figure}[ht]
\includegraphics[clip,width=\textwidth]{figures3}
\end{figure}
\begin{figure}[ht]
\includegraphics[clip,width=\textwidth]{figures4}
\end{figure}
\begin{figure}[ht]
\includegraphics[clip,width=\textwidth]{figures5}
\end{figure}
\begin{figure}[ht]
\includegraphics[clip,width=\textwidth]{figures6}
\end{figure}
\section{Concluding remarks}\label{finalremark}
Residual type a posteriori estimates of errors measured in $L^\infty(L^2)$- and $L^2(L^2)$-norms for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method of local polynomial degrees $r \ge 2$ in space for linear parabolic fourth order problems in space dimensions $2$ and $3$ are presented. Numerical experiments confirming the practical efficiency and reliability of the a posteriori estimators are also presented, along with the use of these a posteriori estimator within adaptive algorithms. It appears that the derived a posteriori bounds and the respective adaptive algorithms can be modified in a straightforward fashion to the original dG method of Baker \cite{baker} and to the $C^0$-interior penalty methods of \cite{hughes, brenner}. Moreover, second order operators can be included in the present analysis, as was done in \cite{thesis_juha}. An extension of these results to nonlinear fourth order parabolic equations remains a future challenge.
\def$'${$'$}
| {'timestamp': '2013-03-12T01:04:04', 'yymm': '1303', 'arxiv_id': '1303.2524', 'language': 'en', 'url': 'https://arxiv.org/abs/1303.2524'} |
\section{Introduction}\label{sec:intro}
\begin{figure}[t]
\begin{center}
{\small
\tikzstyle{block} = [rectangle, draw, fill=blue!20,
text width=9em, text centered, rounded corners, minimum height=3em]
\tikzstyle{line} = [draw, -latex']
\tikzstyle{container} = [draw, rectangle, rounded corners, inner sep=.4cm, fill=gray!20,minimum height=2.2cm]
\begin{tikzpicture}[node distance = 1.7cm and -1.1cm, auto]
\node [block] (mka) {modal Kleene algebras};
\node[block, below right = of mka] (sta) {state transformers};
\node [block, above right = of sta] (back) {predicate transformer quantales};
\node [block, below right = of back] (rel) {binary relations};
\node [block, above right = of rel] (mon) {predicate transformer quantaloids};
\node [block, below left = of sta] (dyn) {dynamical systems};
\node [block, below right = of sta] (pl) {Lipschitz continuous vector fields};
\node [block, below right = of rel] (gen) {continuous vector fields};
\begin{scope}[on background layer]
\node [container,fit = (dyn) (pl) (gen)] (container) {};
\end{scope}
\node [below of = pl, node distance = .8cm] (hyb) {hybrid store dynamics};
\path [line] (mka) -- (sta);
\path [line] (mka) -- (rel);
\path [line] (back) -- (sta);
\path [line] (back) -- (rel);
\path [line] (mon) -- (sta);
\path [line] (mon) -- (rel);
\path [line] (sta) -- (dyn);
\path [line] (sta) -- (pl);
\path [line] (sta) -- (gen);
\path [line] (rel) -- (dyn);
\path [line] (rel) -- (pl);
\path [line] (rel) -- (gen);
\end{tikzpicture}
}
\caption{Isabelle framework for hybrid systems verification}
\label{fig:framework}
\end{center}
\end{figure}
Hybrid systems combine continuous dynamics with discrete control.
Their verification is increasingly important, as the number of
computing systems controlling real-world physical systems is rising.
Mathematically, hybrid system verification requires integrating
continuous system dynamics, often modelled by differential
equations, and discrete control components into hybrid automata,
hybrid programs or similar domain-specific modelling formalisms, and
into analysis techniques for these. Such techniques include state
space exploration, reachability or safety analysis by model checking
and deductive verification with domain-specific
logics~\cite{DoyenFPP18}.
A prominent deductive approach is differential dynamic logic
$\mathsf{d}\mathcal{L}$~\cite{Platzer10}, an extension of dynamic logic~\cite{HarelKT00}
to hybrid programs for reasoning with autonomous systems of
differential equations, their solutions and invariant sets. It is
supported by the KeYmaera X tool~\cite{FultonMQVP15} and has proved
its worth in several case
studies~\cite{JeanninGKSGMP17,LoosPN11,Platzer10}. KeYmaera X
verifies Hoare-style correctness specifications for hybrid programs
using a domain-specific sequent calculus, which itself is based on an
intricate substitution calculus. For pragmatic reasons, its language
has been restricted to differential terms of real
arithmetic~\cite{FultonMQVP15} (that of hybrid automata is usually
restricted to polynomial or linear constraints~\cite{DoyenFPP18}).
Our initial motivation has been to formalise a $\mathsf{d}\mathcal{L}$-style approach to
hybrid system verification in the Isabelle/HOL proof
assistant~\cite{MuniveS18} by combining Isabelle's mathematical
components for analysis and ordinary differential
equations~\cite{HolzlIH13,Immler12,ImmlerH12a,ImmlerT19} with
verification components for modal Kleene algebras~\cite{GomesS16}. We
are using a shallow embedding that, in general, encodes semantic
representations of domain-specific formalisms within a host-language
(deep embeddings start from syntactic representations using data types
to program abstract syntax trees). This benefits not only from the
well known advantages of shallowness: more rapid developments and
simpler, more adaptable components. It has also shifted our focus from
encoding $\mathsf{d}\mathcal{L}$'s complex syntactic proof system to developing
denotational semantics for hybrid systems and supporting the natural
style in which mathematicians, physicists and engineers reason about
them---without any proof-theoretic baggage. After all, we get
Isabelle's proof system and simplifiers for free, and our expressive
power is only limited by its type theory and higher-order logic.
Our main contribution is the first semantic framework for the
deductive verification of hybrid systems in a general purpose proof
assistant. Using a shallow embedding, we currently support abstract
predicate transformer algebras using modal Kleene
algebras~\cite{DesharnaisS11}, quantales of lattice endofunctions or
quantaloids of functions between lattices~\cite{BackW98}. They are
instantiated first to intermediate relational or state transformer
semantics for $\mathsf{d}\mathcal{L}$-style hybrid programs, and then to concrete
semantics over hybrid stores: for dynamical systems with global flows,
Lipschitz continuous vector fields with local flows and continuous
vector fields allowing multiple solutions. A fourth verification
component is based directly on flows. This demonstrates
compositionality of our approach. Figure~\ref{fig:framework} shows its
basic anatomy. The instantiations are seamless with Isabelle's type
polymorphism.
The approach benefits from compositionality and algebra in various
ways. They allow us to localise the development of novel concrete
state transformer semantics for evolutions commands of hybrid programs
that declare a continuous vector field and a guard. These commands are
interpreted as unions of all orbits of solutions of the vector field
at some initial value, subject to the guard constraining the durations
of evolutions. This concrete semantics covers situations beyond the
remits of the Picard-Lindel{\"o}f theorem~\cite{Hirsch09,Teschl12};
the other two form instances. We can then plug the predicate
transformers for evolution commands into the generic algebras for
while program and their rules for verification condition generation.
Verification condition generation for evolution commands is supported
by three procedures that are inspired by $\mathsf{d}\mathcal{L}$, but work in more
general situations. The first one requires users to supply a flow and
a Lipschitz constant for the vector field specified by the evolution
command. After certifying the flow conditions and checking Lipschitz
continuity of the vector field, as dictated by the Picard-Lindel\"of
theorem, the orbit for the flow can be used to compute the weakest
liberal preconditions for the evolution command. The second procedure
requires users to supply an invariant set for the vector field in the
sense of dynamical systems theory~\cite{Hirsch09,Teschl12}. After
certifying the properties for invariant sets---which does not presume
any knowledge of solutions---a correctness specification for the
evolution command and the invariant set can be used in place of a
weakest liberal precondition. The third one uses flows ab initio in
evolution commands and thus circumvents checking any continuity,
existence, uniqueness or invariant conditions of vector fields mentioned.
Hybrid program verification is then performed within the concrete
semantics, but verification condition generation eliminates all
structural conditions automatically so that proof obligations are
entirely about the dynamics of the hybrid program store. They can be
calculated in mathematical textbook style by equational reasoning, and
ultimately by external solvers. We have already created simple tactics
that help automating the computation of derivatives in multivariate
Banach spaces and that of polynomials and transcendental functions.
The entire approach, and the entire mathematical development in this
article has been formalised with Isabelle. All Isabelle components can
be found in the Archive of Formal
Proofs~\cite{afp:kad,afp:vericomp,afp:transem,afp:hybrid}. We are
currently using them to verify hybrid programs post hoc in the
standard weakest liberal precondition style outlined above. Yet the
approach is flexible enough to support Hoare-style reasoning, symbolic
execution with strongest postconditions, program refinement in the
style of Back and von Wright~\cite{BackW98} and reasoning about
program equivalences \`a la Kleene algebra with tests~\cite{Kozen97}.
The remainder of this article is organised as follow:
Section~\ref{sec:KA}-\ref{sec:pt-monad} introduce the algebras of
relations, state and predicate transformers needed.
Section~\ref{sec:discrete-store} explains the shallow embedding used
to formalise verification components for while programs. After
recalling the basics of differential equations in
Section~\ref{sec:ODE}, we introduce our semantics for evolution
commands in
Section~\ref{sec:hybrid-store}-\ref{sec:differential-invariants} and
explain our procedures for computing weakest liberal preconditions and
reasoning with differential invariants for
them. Section~\ref{sec:isa-pt}-\ref{sec:isa-wlp} summarise the
corresponding Isabelle components. Section~\ref{sec:dL} and
\ref{sec:isa-dL} briefly list the derivation and formalisation of
semantic variants of $\mathsf{d}\mathcal{L}$ inference rules.
Section~\ref{sec:examples} presents four verification examples in our
framework. Section~\ref{sec:flow-component} outlines the verification
component based directly on flows. Section~\ref{sec:conclusion}
concludes the article. A glossary of cross-references between theorems in
the text and the Isabelle theories is presented in
Appendix~\ref{sec:crossref}.
\section{Kleene Algebra}\label{sec:KA}
This section presents the mathematical foundations for our simplest
and most developed predicate transformer algebra---modal Kleene
algebra. It introduces the basics of Kleene algebras, and in
particular the state transformer model and relational model used
across this article. The relational model is standard for Kleene
algebra, yet the state transformer model has so far received little
attention and is therefore explained in detail.
A \emph{dioid} $(S,+,\cdot,0,1)$ is a semiring in which addition is
idempotent, $\alpha+\alpha=\alpha$ holds for all $\alpha\in S$. The
underlying abelian monoid $(S,+,0)$ is therefore a semilattice with
order defined
by $\alpha\le \beta\leftrightarrow \alpha+\beta=\beta$. The order is
preserved by $\cdot$ and $+$ in both arguments and $0$ is its least
element.
A \emph{Kleene algebra} $(K,+,\cdot,0,1,^\ast)$ is a dioid expanded by
the Kleene star $(-)^\ast:K\to K$ that satisfies the left and right
unfold and induction axioms
\begin{align*}
1+\alpha\cdot\alpha^\ast \le \alpha^\ast, \qquad \gamma+\alpha\cdot
\beta\le \beta\rightarrow \alpha^\ast \cdot \gamma\le \beta,\qquad
1+\alpha^\ast\cdot\alpha \le \alpha^\ast, \qquad \gamma+\beta\cdot
\alpha\le \beta\rightarrow \gamma\cdot \alpha^\ast \le \beta.
\end{align*}
By these axioms, $\alpha^\ast\cdot \gamma$ is the least fixpoint of
the function $\gamma+\alpha\cdot (-)$ and $\gamma\cdot \alpha^\ast$
that of $\gamma+(-)\cdot \alpha$. The fixpoint $\alpha^\ast$ arises
as a special case. The more general induction axioms combine its
definition with sup-preservation or continuity of left and right
multiplication.
Opposition is an important duality of Kleene algebras: swapping the
order of multiplication in any Kleene algebra yields another one. The
class of Kleene algebras is therefore closed under opposition.
Kleene algebras were conceived as algebras of regular expressions. Yet
we interpret their elements as programs. Addition models their
nondeterministic choice, multiplication their sequential composition
and the Kleene star their unbounded finite iteration. The element $0$
models abort; $1$ models the ineffective program. These intuitions
are grounded in concrete program semantics.
With the relational composition of $R\subseteq X\times Y$ and
$S\subseteq Y\times Z$ defined as
$(R;S)\, x \,z$ if $R\, x\, y$ and $S\, y\, z$ for some $y\in Y$, with
$\mathit{Id}_X\, x\, y$ if $x=y$, and
the reflexive-transitive closure of $R\subseteq X\times X$ defined
as $R^\ast=\bigcup_{i\in\mathbb{N}} R^i$, where $R^0=\mathit{Id}_X$ and
$R^{i+1}=R;R^i$, where we write $R\, x\, y$ instead of $(x,y)\in R$, the following holds.
\begin{proposition}\label{P:rel-ka}
Let $X$ be a set. Then
$\mathsf{Rel}\, X = (\mathcal{P}\, (X\times
X),\cup,;,\emptyset,\mathit{Id}_X,^\ast)$
forms a Kleene algebra---the \emph{full relation Kleene algebra}
over $X$.
\end{proposition}
A \emph{relation Kleene
algebra} over $X$ is thus any subalgebra of $\mathsf{Rel}\, X$.
Opposition can be expressed in $\mathsf{Rel}\, X$ by conversion, where the
\emph{converse} of relation $R$ is defined by
$R^\smallsmile\, x\, y \leftrightarrow R\, y\, x$. It satisfies in particular
$(R;S)^\smallsmile = S^\smallsmile ; R^\smallsmile$.
The isomorphism $\mathcal{P}\, (X\times Y) \cong (\mathcal{P}\, Y)^X$ between
categories of relations and non-deterministic functions---so-called
\emph{state transformers}---yields an alternative representation. It
is given by the bijections
\begin{equation*}
\mathcal{F}:\mathcal{P}\, (X\times Y) \to (\mathcal{P}\, Y)^X,\qquad
\mathcal{R}:(\mathcal{P}\, Y)^X\to \mathcal{P} (X\times Y)
\end{equation*} defined by
$\mathcal{F}\, R\, x = \{y\in Y\mid R\, x\, y\}$ and by
$\mathcal{R}\, f\, x\, y \Leftrightarrow y \in f\, x$. State
transformers $f:X\to \mathcal{P}\, Y$ and $g:Y\to \mathcal{P}\, Z$ are composed by
the Kleisli composition of the powerset monad,
\begin{equation*}
(f\circ_K g)\, x = \bigcup\{g\, y\mid y \in f\ x \},
\end{equation*}
$\eta_X = \{-\}$ is a unit of this monad. The functors $\mathcal{F}$
and $\mathcal{R}$ preserve arbitrary sups and infs, extended pointwise
to state tranformers, and stars
$f^{\ast_K}\, x = \bigcup_{i\in\mathbb{N}} f^{i_K}\, x$, which are
defined with respect to Kleisli composition.
\begin{proposition}\label{P:kleisli-ka}
Let $X$ be a set. Then
$\mathsf{Sta}\, X = ((\mathcal{P}\, X)^X,\cup,\circ_K,\lambda x.\
\emptyset, \eta_X,^{\ast_K})$
forms a Kleene algebra---the \emph{full state transformer Kleene algebra} over $X$.
\end{proposition}
A \emph{state transformer Kleene algebra} over $X$ is any subalgebra of
$\mathsf{Sta}\, X$. Opposition is now expressed using the
(contravariant) functor
$(-)^{\mathit{op}} = \mathcal{F}\circ (-)^\smallsmile\circ
\mathcal{R}$
that associates $f^{\mathit{op}}:Y\to \mathcal{P}\, X$ with every
$f:X\to \mathcal{P}\, Y$.
The category $\mathbf{Rel}$, via relations or
state transformers, is beyond mono-typed Kleene algebra.
For a more refined hierarchy of variants of Kleene algebras, their
calculational properties and the most important computational models,
see our formalisation in the Archive of Formal
Proofs~\cite{afp:ka}. The state transformer model has been formalised
with Isabelle
for this article.
\section{Modal Kleene Algebra}\label{sec:MKA}
Kleene algebras must be extended to express conditionals or while loop
more specifically. This requires tests, which are not prima facie
actions, but propositions. Assertions and correctness
specifications cannot be expressed directly either.
Two standard extensions bring Kleene algebra closer to program
semantics. Kleene algebra with tests~\cite{Kozen97} yields a simple
algebraic semantics for while-programs and a partial correctness
semantics for these in terms of an algebraic propositional Hoare
logic---ignoring assignments. Predicate transformer
semantics, however, cannot be expressed~\cite{Struth16}.
Alternatively, Kleene algebras can be enriched by modal box and
diamond operators in the style of propositional dynamic logic
($\mathsf{PDL}$), which yields test and assertions as well as predicate
transformers. Yet once again, assignments cannot be expressed within
the algebra. We outline the second approach.
An \emph{antidomain semiring} \cite{DesharnaisS11} is a semiring $S$
expanded by an \emph{antidomain operation} $\mathit{ad}:S\to S$ axiomatised by
\begin{align*}
\mathit{ad}\, \alpha \cdot \alpha = 0,\qquad
\mathit{ad}\, \alpha + \mathit{ad}^2\, \alpha = 1,\qquad \mathit{ad}\, (\alpha\cdot \beta) \le \mathit{ad}\, (\alpha \cdot \mathit{ad}^2\, \beta).
\end{align*}
By opposition, an \emph{antirange semiring} \cite{DesharnaisS11} is a
semiring $S$ expanded by an \emph{antirange operation} $\mathit{ar}:S\to S$
axiomatised by
\begin{equation*}
\alpha\cdot \mathit{ar}\, \alpha = 0,\qquad
\mathit{ar}\, \alpha + \mathit{ar}^2\, \alpha = 1,\qquad \mathit{ar}\, (\alpha\cdot \beta) \le
\mathit{ar}\, (\mathit{ar}^2\, \alpha \cdot \beta).
\end{equation*}
Antidomain and antirange semirings
are a fortiori dioids.
The antidomain $\mathit{ad}\, \alpha$ of program $\alpha$ models the set of
those states from which $\alpha$ cannot be executed. The operation
$d=\mathit{ad}^2$ thus defines the \emph{domain} of a program: the set of
those states from which it can be executed. Dually, the antirange
$\mathit{ar}\, \alpha$ of $\alpha$ yields those states into which $\alpha$
cannot be executed and $r=\mathit{ar}^2$ defines the \emph{range} of $\alpha$:
those states into which it can be executed.
A \emph{modal Kleene algebra} ($\mathsf{MKA}$) \cite{DesharnaisS11} is a
Kleene algebra that is both an antidomain and an antirange Kleene
algebra in which $d\circ r=r$ and $r\circ d = d$.
In a $\mathsf{MKA}$ $K$, the set $\mathcal{P}\, \mathit{ad}\, K$---the image of $K$ under
$\mathit{ad}$---models the set of all tests or propositions. We henceforth
often write $p,q,\dots$ for its elements. Moreover,
$\mathcal{P}\, \mathit{ad}\, K=\mathcal{P}\, d\, K=\mathcal{P}\, r\, K =\mathcal{P}\, ar\, K = K_d=K_r$,
where $K_f=\{\alpha\in S\mid f\, \alpha = \alpha\}$ for $f\in\{d,r\}$.
Hence $p\in \mathcal{P}\, \mathit{ad}\, K \leftrightarrow d\, p = p$. It follows that
the class $\mathsf{MKA}$ is closed under opposition. In addition, $K_d$ forms a
boolean algebra with least element $0$, greatest element $1$, join
$+$, meet $\cdot$ and complementation $\mathit{ad}$---the algebra of
propositions, assertions or tests.
Axiomatising $\mathsf{MKA}$ based on domain and range would lack the power to
express complementation: $K_d$ would only be a distributive lattice.
The programming intuitions for $\mathsf{MKA}$ are once again grounded in
concrete semantics.
\begin{proposition}\label{P:rel-mka}
If $X$ is a set, then $\mathsf{Rel}\, X$ is the full relation $\mathsf{MKA}$
over $X$ with
\begin{align*}
\mathit{ad}\, R\, x\, x \leftrightarrow \neg \exists y \in X.\
R\, x\, y \qquad\text{ and }\qquad
\mathit{ar}\, R = \mathit{ad}\, R^\smallsmile.
\end{align*}
\end{proposition}
Every subalgebra of a full relation $\mathsf{MKA}$ is a \emph{relation}
$\mathsf{MKA}$.
Similarly, $\mathit{ar} = \mathit{ad} \circ (-)^\smallsmile$,
$d = r \circ (-)^\smallsmile$ and $r = d\circ (-)^\smallsmile$.
Furthermore,
\begin{equation*}
(\mathcal{P}\, (X\times X))_d= \{P\mid P\subseteq \mathit{Id}_X\}.
\end{equation*}
We henceforth often identify such relational subidentities, sets and
predicates and their types via the isomorphisms
$(\mathcal{P}\, (X\times X))_d\, \cong\, X\to \mathbb{B}\, \cong\, \mathcal{P}\, X$.
\begin{proposition}\label{P:kleisli-mka}
Let $X$ be a set. Then $\mathsf{Sta}\, X$ is the full state
transformer $\mathsf{MKA}$ over $X$ with
\begin{align*}
\mathit{ad}\, f\, x =
\begin{cases}
\eta_X\, x, & \text{ if } f\, x = \emptyset,\\
\emptyset, & \text{ otherwise},
\end{cases}
\qquad\text{ and }\qquad
\mathit{ar}\, f = \mathit{ad}\, f^{\mathit{op}}.
\end{align*}
\end{proposition}
Every subalgebra of a full relation $\mathsf{MKA}$ is a \emph{state
transformer} $\mathsf{MKA}$. Similarly,
\begin{align*}
d\, f\, x =
\begin{cases}
\emptyset, & \text{ if } f\, x = \emptyset,\\
\eta_X\, x, & \text{ otherwise},
\end{cases}
\qquad\text{ and }\qquad
r\, f = d\, f^{\mathit{op}}.
\end{align*}
These propositions generalise again beyond mono-types, but algebras of
such typed relations and state transformers cannot be captured by
$\mathsf{MKA}$.
In every $\mathsf{MKA}$, $p\cdot \alpha$ and $\alpha\cdot p$
model the domain and range restriction of $\alpha$ to states
satisfying $p$. Conditionals and while loops can thus be
expressed:
\begin{align*}
\IF{p}{\alpha}{\beta} = p\cdot \alpha + \bar p \cdot
\beta\qquad\text{ and }\qquad
\WHILE{p}{\alpha} = (p\cdot \alpha)^\ast \cdot \bar p,
\end{align*}
where we write $\bar p = \mathit{ad}\, p = \mathit{ar}\, p$. Together with sequential
composition $\alpha ; \beta= \alpha\cdot \beta$ this yields an
algebraic semantics of while programs without assignments. It
is grounded in the relational and the state transformer semantics.
A more refined hierarchy of variants of $\mathsf{MKA}$s, starting from domain
and antidomain semigroups, their calculational properties and the most
important computational models, can be found in the Archive of Formal
Proofs~\cite{afp:kad}. The state transformer model of $\mathsf{MKA}$ has been
formalised with Isabelle for this article.
\section{Modal Kleene Algebra, Predicate Transformers and Invariants}\label{sec:mka-pt}
$\mathsf{MKA}$ can express the modal operators of $\mathsf{PDL}$, both with a
relational Kripke semantics and a coalgebraic state transformer semantics.
\begin{equation*}
|\alpha\rangle p = d\, (\alpha\cdot p), \qquad |\alpha ] p = \mathit{ad}\,
(\alpha\cdot \mathit{ad}\, p),\qquad
\langle \alpha | p = r\, (p\cdot \alpha), \qquad [\alpha |p = \mathit{ar}\, (\mathit{ar}
\, p\cdot \alpha).
\end{equation*}
This is consistent with J\'onsson and Tarski's boolean algebras with
operators~\cite{JonssonT51}: Each of $|\alpha\rangle$,
$\langle\alpha |$, $|\alpha ]$ and $[\alpha |$ is an endofunction
$K_d\to K_d$ on the boolean algebra $K_d$. Yet another view of modal
operators is that of predicate transformers. The function $|-]-$
yields the \emph{weakest liberal precondition} operator $\mathit{wlp}$;
$\langle -|-$ the \emph{strongest postcondition} operator.
The boxes and diamonds of $\mathsf{MKA}$ are related by De Morgan duality:
\begin{equation*}
|\alpha\rangle p = \overline{ |\alpha ]\bar p}, \qquad |\alpha ] p =
\overline{ |\alpha \rangle \bar p},\qquad
\langle\alpha| p = \overline{ [\alpha |\bar p}, \qquad [\alpha | p =
\overline{ \langle \alpha | \bar p}\, ;
\end{equation*}
their dualities are captured by the adjunctions and conjugations
\begin{alignat*}{4}
|\alpha\rangle p \le q &\leftrightarrow p \le [\alpha|q, &\qquad
\langle \alpha |p \le q &\leftrightarrow p \le |\alpha]q,\\
|\alpha\rangle p\cdot q = 0 &\leftrightarrow p\cdot \langle
\alpha|q=0,&\qquad |\alpha]p+ q = 1&\leftrightarrow
p+[\alpha|q=1.
\end{alignat*}
In $\mathsf{Rel}\, X$, as in standard Kripke semantics,
\begin{equation*}
|R\rangle P = \{x\mid \exists y\in X.\ R\, x\, y \wedge
P\, y\}\quad\text{ and }\quad
|R]P = \{x\mid \forall y\in X.\ R\, x\, y \rightarrow P\, y\},
\end{equation*}
where we identify predicates and subidentity relations. Moreover,
$\langle -|=|-\rangle\circ (-)^\smallsmile$ and
$ [-|=|-]\circ (-)^\smallsmile$. Hence $|R\rangle P$ is the preimage
of $P$ under $R$ and $\langle R|P$ the image of $P$ under $R$. The
isomorphism between subidenties, predicates and sets also allows us to
see $|R\rangle$, $\langle R|$, $|R]$ and $[R|$ as operators on the
complete atomic boolean algebra $\mathcal{P}\, X$, which carries algebraic
structure beyond $K_d$.
In $\mathsf{Sta}\, X$,
\begin{equation*}
\langle f | P =\{y\mid \exists x.\ y \in
f\, x \wedge P\, x\}\quad \text{ and }\quad
|f]P = \{x\mid f\, x \subseteq P\}.
\end{equation*}
Moreover, $|-\rangle = \langle-|\circ (-)^{\mathit{op}}$ and
$[-| = |-]\circ (-)^{\mathit{op}}$. Here, $\langle f|$ is the Kleisli
extension of $f$ for the powerset monad and $|f\rangle$ that of the
opposite function (see Section~\ref{sec:pt-monad}).
The isomorphism $\mathcal{P}\, (X\times X) \cong (\mathcal{P}\, X)^X$ makes the
approaches coherent:
\begin{align*}
|f\rangle = |\mathcal{R}\, f\rangle,\qquad |R\rangle =
|\mathcal{F}\,
R\rangle, \qquad
| f] = |\mathcal{R}\, f],\qquad |R] = |\mathcal{F}\, R],
\end{align*}
and, dually, $\langle f| = \langle \mathcal{R}\, f|$, $\langle R| =
\langle \mathcal{F}\,
R|$, $[f| = [\mathcal{R}\, f|$ and $[R| = [\mathcal{F}\, R|$.
Predicate transformers are useful for specifying program correctness
conditions and for verification condition generation. The identity
\begin{equation*}
p\le |\alpha]q
\end{equation*}
captures the standard partial correctness specification for programs:
if $\alpha$ is executed from states where precondition $p$ holds, and
if it terminates, then postcondition $q$ holds in the states where it
does. Verifying it amounts to computing $|\alpha]q$ recursively over
the program structure from $q$ and checking that the result is greater
or equal to
$p$. Intuitively, $|\alpha]q$ represents the largest set of states
from which one must end up in set $q$ when executing $\alpha$, or
alternatively the weakest precondition from which postcondition $q$
must hold when executing $\alpha$.
Calculating $|\alpha]q$ for straight-line programs is completely
equational, but loops require invariants. To this end one usually
adds annotations to loops,
\begin{equation*}
\WHILEI{p}{i}{\alpha} = \WHILE{p}{\alpha},
\end{equation*}
where $i$ is the \emph{loop invariant} for $\alpha$, and calculates
$\mathit{wlp}$s as follows~\cite{GomesS16,afp:vericomp}. For all
$p,q,i,t\in K_d$ and $\alpha,\beta\in K$,
\begin{gather}
|\alpha \cdot \beta] q = |\alpha] |\beta] q,\label{eq:wlp-seq}\tag{wlp-seq}\\
|\IF{p}{\alpha}{\beta}] q = (\bar p + |\alpha] q)(p + |\beta] q) = p
\cdot |\alpha] q+ \bar p \cdot |\beta] q,\label{eq:wlp-cond}\tag{wlp-cond}\\
i\le |\alpha]i \rightarrow i\le |\alpha^\ast]i,\label{eq:wlp-star}\tag{wlp-star}\\
p \le i \wedge i\cdot t\le |\alpha] i \wedge i\cdot \bar t\le q\, \rightarrow \, p \le
|\WHILEI{t}{i}{\alpha}] q.\label{eq:wlp-loop}\tag{wlp-while}
\end{gather}
In the rule (\ref{eq:wlp-star}), $i$ is a an invariant for the star as
well.
More generally, an element $i\in K_d$ is an \emph{invariant} for
$\alpha$ if it is a postfixpoint of $|\alpha]$ in $K_d$:
\begin{equation*}
i \le |\alpha]i.
\end{equation*}
By the adjunction between boxes and diamonds, this is the case if and
only if $\langle\alpha | i \le i$, that is, $i$ is a prefixpoint of
$\langle \alpha|$ in $K_d$. We return to this equivalence in the
context of differential invariants and invariant sets of vector fields
in Section~\ref{sec:differential-invariants}. We write
$\mathsf{Inv}\, \alpha $ for the set of invariants of $\alpha$.
\begin{lemma}\label{P:inv-lemma}
In every $\mathsf{MKA}$, if $i,j\in \mathsf{Inv}\, \alpha$, then $i+j,i\cdot j\in \mathsf{Inv}\, \alpha$.
\end{lemma}
As a generalisation of the rule (\ref{eq:wlp-loop}) for annotated
while-loops we can derive a rule for commands annotated with
tentative invariants $\alpha\, \mathbf{inv}\ i = \alpha$. For all
$i,p,q\in K_d$ and $\alpha\in K$,
\begin{equation}
\label{eq:wlp-cmd}\tag{wlp-cmd}
p\le i \land i \le |\alpha] i \land i\le q\rightarrow p\le
|\alpha\, \mathbf{inv}\, i] q.
\end{equation}
Combining (\ref{eq:wlp-cmd}) with (\ref{eq:wlp-star}) then yields, for
$\mathbf{loop}\, \alpha\, \mathbf{inv}\, i = \alpha^\ast$,
\begin{equation}
\label{eq:wlp-star-inv}\tag{wlp-loop}
p\le i \land i\le |\alpha]i \land i\le q\rightarrow p\le |\mathbf{loop}\, \alpha\,
\mathbf{inv}\, i]q.
\end{equation}
We use such annotated commands for reasoning about differential
invariants and loops of hybrid programs below.
The modal operators of $\mathsf{MKA}$ have, of course, a much richer calculus
beyond verification condition generation. For a comprehensive list see
the Archive of Formal Proofs~\cite{afp:kad}. We have already derived
the rules of propositional Hoare logic, which ignores assignments, and
those for verification condition generation for symbolic execution
with strongest postconditions in this setting~\cite{afp:vericomp}. A
component for total correctness is also available, and it supports
refinement proofs in the style of Back and von
Wright~\cite{BackW98}. The other two abstract predicate transformer
algebras from Figure~\ref{fig:framework} are surveyed in the following
two sections.
\section{Predicate Transformers \`a la Back and von
Wright}\label{sec:pt-backvwright}
While $\mathsf{MKA}$ is so far our preferred and most developed setting for verifying
hybrid programs, our framework is compositional and supports other predicate
transformer algebras as well. Two of them are outlined in this and the
following section. Their Isabelle formalisation~\cite{afp:transem} is
discussed in Section~\ref{sec:pt-monad}.
The first approach follows Back and von Wright~\cite{BackW98} in
modelling predicate transformers, or simply \emph{transformers}, as
functions between complete lattices. To obtain useful laws for
program construction or verification, conditions are imposed.
A function $f:L_1\to L_2$ between two complete lattices $(L_1,\le_1)$
and $(L_2,\le_2)$ is \emph{order preserving} if
$x\le_1 y \rightarrow f\, x \le_2 f\, y$, \emph{sup-preserving} if
$f \circ \bigsqcup = \bigsqcup \circ \mathcal{P}\, f$ and
\emph{inf-preserving} if
$f \circ \bigsqcap = \bigsqcap \circ \mathcal{P}\, f$. All sup- or
inf-preserving functions are order preserving.
We write $\mathcal{T}(L)$ for the set of transformers over the
complete lattice $L$, and $\mathcal{T}_\le(L)$,
$\mathcal{T}_{\sqcup}(L)$, $\mathcal{T}_{\sqcap}(L)$ for the subsets
of order-, sup- and inf-preserving transformers. Obviously,
$\mathcal{T}_{\sqcap}(L)=\mathcal{T}_{\sqcup}(L^\mathit{op})$. The
following fact is well known~\cite{BackW98,GierzHKLMS80}.
\begin{proposition}\label{P:pt-lattice}
Let $X$ be a set and $L$ a complete lattice. Then $L^X$ forms a
complete lattice with order and sups extended pointwise.
\end{proposition}
Infs, least and greatest elements can then be defined from sups on
$L^X$ as usual. Function spaces $L^L$, in particular, form monoids
with respect to function composition $\circ$ and $\mathit{id}_L$. In
addition, $\circ$ preserves sups and infs in its first argument, but
not necessarily in its second one. Algebraically, this is captured as
follows.
A \emph{near-quantale} $(Q,\le,\cdot)$ is a complete lattice $(Q,\le)$
with an associative composition $\cdot$ that preserves sups in its
first argument. It is \emph{unital} if composition has a unit $1$. A
\emph{prequantale} is a near-quantale in which composition is order
preserving in its second argument. A \emph{quantale} is a near
quantale in which composition preserves sups in its second argument.
\begin{proposition}\label{P:trafo-quantale}
Let $L$ be a complete lattice. Then
\begin{enumerate}
\item $\mathcal{T}(L)$ and
$\mathcal{T}(L^\mathit{op})$ form unital near-quantales;
\item $\mathcal{T}_\le(L)$ ($\mathcal{T}_\le(L^\mathit{op}))$ forms
a unital sub-prequantale of $\mathcal{T}(L)$ ($\mathcal{T}(L^\mathit{op})$);
\item $\mathcal{T}_\sqcup(L)$ ($\mathcal{T}_\sqcap(L)$) forms a
unital sub-quantale of $\mathcal{T}_\le(L)$ ($\mathcal{T}_\le(L^\mathit{op})$).
\end{enumerate}
\end{proposition}
Transformers for while-loops are obtained by connecting quantales with
Kleene algebras. This requires fixpoints of
$\varphi_{\alpha\gamma}= \gamma\sqcup \alpha\cdot (-)$ and
$\varphi_\alpha=1\sqcup \alpha\cdot (-)$ as well as the Kleene star
$\alpha^\ast=\bigsqcup_{i\in \mathbb{N}}\alpha^i$. A
\emph{left Kleene algebra} is a dioid in which $\varphi$ has a least
fixpoint that satisfies
$\mathit{lfp}\, \varphi_{\alpha\gamma} = \mathit{lfp}\, \varphi_\alpha
\cdot \gamma$.
Hence $\varphi_\alpha$ satisfies the left unfold and left induction
axioms $1\sqcup \alpha\cdot \varphi_\alpha\le \varphi_\alpha$ and
$\gamma\sqcup \alpha\cdot \beta\le \beta\rightarrow
\varphi_\alpha\cdot \gamma\le \beta$.
Dually, a \emph{right Kleene algebra} is a dioid in which the least
fixpoint of a dual function $1\sqcup (-)\cdot \alpha$ satisfies the right
unfold and right induction axioms.
\begin{proposition}\label{P:quantale-ka}~
\begin{enumerate}
\item Every near-quantale is a right Kleene algebra with
$\mathit{lfp}\, \varphi_\alpha = \alpha^\ast$.
\item Every prequantale is also a left Kleene algebra.
\item Every quantale is a Kleene algebra with
$\mathit{lfp}\, \varphi_\alpha = \alpha^\ast$.
\end{enumerate}
\end{proposition}
The proofs of (1) and (3) use sup-preservation and Kleene's fixpoint
theorem. That of (2) uses Knaster-Tarski's fixpoint theorem to show
that $\varphi_{\alpha\gamma}$ has a least fixpoint, and fixpoint
fusion~\cite{MeijerFP91} to derive
$\mathit{lfp}\, \varphi_{\alpha\gamma} = \mathit{lfp}\, \varphi_\alpha
\cdot \gamma$,
which yields the left Kleene algebra axioms. In prequantales,
$\mathit{lfp}\, \varphi_\alpha \cdot \gamma\le \alpha^\ast \cdot
\gamma$;
equality generally requires sup-preservation in the first argument of
composition.
The fixpoint and iteration laws on functions spaces, which follow from
Proposition~\ref{P:quantale-ka} and~\ref{P:trafo-quantale}, still need
to be translated into laws for transformers operating on the
underlying lattice. This is achieved again by fixpoint
fusion~\cite{BackW98}. In $\mathcal{T}_\le(L)$,
\begin{equation*}
\mathit{lfp}\, (\lambda g.\ \mathit{id}_L\sqcup f \circ g)\, x =
\mathit{lfp}\, (\lambda y.\ x
\sqcup f\, y),
\end{equation*}
and $\mathit{lfp}$ preserves isotonicity. In $\mathcal{T}_\sqcup(L)$,
moreover,
\begin{equation*}
f\, x \le x\rightarrow f^\ast \, x \le x,
\end{equation*}
$\mathit{id}_L\sqcup f\circ f^\ast = f^\ast=f^\ast \circ f \sqcup \mathit{id}_L$ and
$(-)^\ast$ preserves sups. All results dualise to inf-preserving
transformers.
Relative to $\mathsf{MKA}$, backward diamonds correspond to sup-preserving
forward transformers and forward boxes to inf-preserving backward
transformers in the opposite quantale, where the lattice has been
dualised and the order of composition been swapped. An analogous
correspondence holds for forward diamonds and backward boxes. Sup- and
inf-preserving transformers over complete lattices are less
general than $\mathsf{MKA}$ in that preservation of arbitrary sups or infs is
required, whereas that of $\mathsf{MKA}$ is restricted to finite sups and
infs. Isotone transformers, however, are more general, as not even
finite sups or infs need to be preserved, and finite sup- or
inf-preservation implies order preservation.
We are mainly using the $\mathit{wlp}$ operator for verification condition
generation and hence briefly outline $\mathit{wlp}$s for conditionals and
loops in this setting. We assume that the underlying lattice $L$ is a
complete boolean algebra. We can then lift elements of $L$ to $\mathit{wlp}$s
as $|p]q= p \to q$ and define, in $\mathcal{T}_\le(L^\mathit{op})$,
\begin{align*}
\mathbf{if}\, p\, \mathbf{then}\, f\, \mathbf{else}\, g = |p]\circ f
\sqcap |q]\circ g\qquad \text{ and } \qquad
\mathbf{while}\, p\, \mathbf{do}\, f =\mathit{lfp}\, \varphi_{|p]f}
\circ |\bar p].
\end{align*}
In $\mathcal{T}_\sqcap(L)$, we even obtain
\begin{equation*}
\mathbf{while}\, p\, \mathbf{do}\, f = (|p]\circ f)^\ast \circ |\bar p].
\end{equation*}
These equations allow generating verification conditions as with
(\ref{eq:wlp-cond}) and (\ref{eq:wlp-loop}) from
Section~\ref{sec:mka-pt}. Overall, our Isabelle components for
lattice-based predicate transformers in the Archive of Formal
Proofs~\cite{afp:transem} contain essentially the same equations and
rules for verification condition generation as those for $\mathsf{MKA}$.
We have so far restricted the approach to endofunctions on a complete
lattice to relate it to $\mathsf{MKA}$. Yet it generalises to functions in
$L_2^{L_1}$ and hence to categories~\cite{BackW98}. The corresponding
typed generalisations of quantales are known as
\emph{quantaloids}~\cite{Rosenthal96}. In particular, composition is
then a partial operation.
\section{Predicate Transformers from the Powerset Monad}\label{sec:pt-monad}
A second, more coalgebraic approach to predicate transformers starts
from monads~\cite{Maclane71}. In addition, it details the relational
and state transformer semantics of $\mathsf{MKA}$ beyond the scope of the
algebraic approach.
Recall that $(\mathcal{P},\eta_X,\mu_X)$, for $\mathcal{P}:\mathbf{Set}\to \mathbf{Set}$,
$\eta_X:X\to \mathcal{P}\, X$ defined by $\eta_X= \{-\}$ and
$\mu_X:\mathcal{P}^2\, X\to \mathcal{P}\, X$ defined by $\mu_X = \bigcup$ is the
monad of the powerset functor in the category $\mathbf{Set}$ of sets and
functions. The morphisms $\eta$ and $\mu$ are natural transformations.
They satisfy, for every $f:X\to Y$,
\begin{equation*}
\eta_Y\circ \mathit{id}\, f = \mathcal{P}\, f \circ \eta_X\qquad\text{ and }\qquad
\mu_Y \circ \mathcal{P}^2\, f = \mathcal{P}\, f \circ \mu_X.
\end{equation*}
From the monadic point of view, state transformers $X\to \mathcal{P}\, Y$ are
arrows $X\to Y$ in the Kleisli category $\mathbf{Set}_\mathcal{P}$ of $\mathcal{P}$ over
$\mathbf{Set}$. They are composed by Kleisli composition
$f\circ_K g = \mu\circ \mathcal{P}\, g\circ f$ as explained before
Proposition~\ref{P:kleisli-ka} in Section~\ref{sec:KA}. The category
$\mathbf{Set}_\mathcal{P}$ is isomorphic to $\mathbf{Rel}$, the category of sets and binary
relations.
The isomorphism between state and forward predicate transformers is
based on the contravariant functor
$(-)^\dagger:\mathbf{Set}_\mathcal{P}(X,\mathcal{P}\, Y)\to \mathbf{Set}_\mathcal{P}(\mathcal{P}\, X,\mathcal{P}\,
Y)$---the
Kleisli extension. Its definition $f^\dagger = \mu \circ \mathcal{P}\, f$
implies that $(-)^\dagger =\langle - |$ on morphisms, which is the
strongest postcondition operator.
The structure of state spaces---boolean algebras for $\mathsf{MKA}$, complete
lattices in Back and von Wright's approach---is captured by the
Eilenberg-Moore algebras of the powerset monad. It is well known that
$(-)^\dagger$ embeds $\mathbf{Set}_\mathcal{P}$ into their category. Its objects are
complete (sup-semi)lattices; its morphisms sup-preserving functions,
hence transformers. More precisely, $(-)^\dagger$ embeds into powerset
algebras, complete atomic boolean algebras that are the free objects
in this category.
The isomorphism
$\mathbf{Set}_\mathcal{P}(X, \mathcal{P}\, Y)\cong
\mathbf{Set}^\sqcup(\mathcal{P}\, X,\mathcal{P}\, Y)$
between state transformers and sup-preserving predicate transformers
then arises as follows. The embedding $\langle -|$ has an injective
inverse $\langle-|^{-1}$ on the subcategory of sup-preserving
transformers. It is defined by $\langle -|^{-1} = (-)\circ \eta$,
which can be spelled out as
$\langle \varphi|^{-1}\, x = \{y\mid y \in \varphi\, \{x\}\}$. The
isomorphism preserves the quantaloid structures of state and predicate
transformers that is, compositions (contravariantly), units and sups,
hence least elements, but not necessarily infs and greatest
elements. These results extend to
$\mathbf{Set}^\sqcup(\mathcal{P}\, X,\mathcal{P}\, Y)
\cong\mathbf{Rel}(X, Y)$
via $\mathbf{Set}_\mathcal{P}\cong \mathbf{Rel}$. In addition, predicate
transformers $\langle f|: \mathcal{P}\, X\to \mathcal{P}\, Y$ preserve of course
sups in powerset lattices, hence least elements, but not necessarily
infs and greatest elements.
Forward boxes or $\mathit{wlp}$s can be obtained from state transformers via a
(covariant) functor
$|-]$ of type $\mathbf{Set}_\mathcal{P}(X,\mathcal{P}\, Y)\to
\mathbf{Set}(\mathcal{P}\, Y,\mathcal{P}\, X)$,
embedding Kleisli arrows into the opposite of the category of
Eilenberg-Moore algebras formed by complete (inf-semi)lattices and
inf-preserving functions. It is defined on morphisms as
$|-]=\partial_F\circ \langle- |\circ (-)^{\mathit{op}}$, where
$\partial_F\, f = \partial\circ f \circ \partial$ and $\partial$
dualises the lattice. Unfolding definitions, once again
$|f]\, P=\{x\mid f\, x\subseteq P\}$.
Its inverse $|-]^{-1}$ on the subcategory of inf-preserving
transformers is
$|\varphi]^{-1}\, x = \bigcap\{P\mid x \in \varphi\, P\}$. The duality
$\mathbf{Set}_\mathcal{P}(X,\mathcal{P}\, Y)\cong \mathbf{Set}^\sqcap(\mathcal{P}\, Y,\mathcal{P}\, X)$ reverses
Kleisli arrows and preserves the quantaloid structures up-to lattice
duality, mapping sups to infs and vice versa. It extends to relations
as before. In addition, predicate transformers $|f]$ preserve of
course infs of powerset lattices, hence greatest elements, but not
necessarily sups and least elements.
The remaining transformers $|-\rangle$ and $[-|$ and their inverses
arise from $\langle -|$ and $|-]$ by opposition:
$|-\rangle= \langle -| \circ (-)^\mathit{op}$,
$|-\rangle^{-1} = (-)^\mathit{op}\circ \langle -|^{-1}$,
$[-|= |-] \circ (-)^\mathit{op}$ and
$[-|^{-1} = (-)^\mathit{op}\circ |-]^{-1}$. Taken together, the four
modal operators satisfy the laws of the $\mathsf{MKA}$ modalities outlined in
Section~\ref{sec:mka-pt} and those of the abstract sup/inf-preserving
transformers discussed in Section~\ref{sec:pt-backvwright}, they give
in fact semantics to the algebraic developments, when restricted to
mono-types, and once again yield the same rules for verification
condition in the state transformer and the relational semantics,
albeit in a more general categorical setting.
The categorical approach to predicate transformers outlined is not
new, apart perhaps from the emphasis on quantales and quantaloids. The
emphasis on monads is due at least to Manes~\cite{Manes92}. More recently,
Jacob's work on state-and-effect triangles~\cite{Jacobs17}
has explored similar connections and their generalisation far beyond
sequential programs. A formalisation with a proof assistant like
Isabelle, which is further discussed in Section~\ref{sec:isa-pt}, is
one of the contributions of this article.
\section{Assignments}\label{sec:discrete-store}
Two important ingredients for concrete program semantics and
verification condition generation are still missing: a mathematical
model of the program store and program assignments, and rules for
calculating $\mathit{wlp}$s for these basic commands. To prepare for hybrid
programs (see Section~\ref{sec:hybrid-store} for a syntax) we model
stores and assignments as discrete dynamical systems over
state spaces.
Formally, a \emph{dynamical system}~\cite{Arnold,Teschl12} is an
\emph{action} of a monoid $(M,\star,e)$ on a set or state space $S$,
that is, a monoid morphism $\varphi :M\to S\to S$ into the
\emph{transformation monoid} $(S^S,\circ, \mathit{id}_S)$ on $S^S$. Thus, by definition,
\begin{equation*}
\varphi\, (m\star n) = (\varphi\, m) \circ (\varphi\, n)\qquad\text{ and }\qquad
\varphi\, e = \mathit{id}_S.
\end{equation*} The first action axiom captures the inherent
determinism of dynamical systems. Conversely, each transformation
monoid $(S^S,\circ,\mathit{id}_S)$ determines a monoid action in which the
action $\varphi:S^S\to S\to S$ is function application.
States of simple while programs are functions $s:V\to E$ from program
variables in $V$ to values in $E$. State spaces for such discrete
dynamical systems are function spaces $S=E^V$. An update function
$f_a:V\to (S \to E) \to S\to S$ for assignment commands can
be defined as
\begin{equation*}
f_a\, v\, e\, s = s[v\mapsto e\, s],
\end{equation*}
where $f[a\mapsto b]$ updates $f:A\to B$ by associating $a\in A$ with
$b$ and every $y\neq a$ with $f\, y$. The ``expression'' ${e:S\to E}$
is evaluated in state $s$ to $e\, s$. The maps $f_a\, v\, e$ generate
a transformation monoid, hence a monoid action $S^S\to S \to S$ on
$S^S$. They also connect the concrete program store semantics with the
$\mathit{wlp}$ semantics used for verification condition generation.
We lift $f_a\, v\, e:S\to S$ to a state transformer
$v:=_\mathcal{F} e:S \to \mathcal{P}\, S$ as
\begin{equation*}
(v:=_\mathcal{F} e) = \eta_{E^V} \circ (f_a\, v\, e) = \lambda s.\ \{
f_a\, v\, e\, s\},
\end{equation*}
thus creating a semantic illusion for syntactic assignment commands in
the $\mathsf{MKA}$ $\mathsf{Sta}\, S$. Form $\mathsf{Rel}\, S$, the isomorphism between $\mathbf{Set}_\mathcal{P}$ and $\mathbf{Rel}$
yields
\begin{equation*}
(v :=_\mathcal{R} e) = \mathcal{R}\, (v:=_\mathcal{F} e),
\end{equation*}
hence
$(v :=_\mathcal{R} e) = \{(s,f_a\, v\, e\, s)\mid s\in
E^V\}=\{(s,s[v\mapsto e\, s]\mid s\in E^V\}$.
Alternatively, we could have defined the state transformer semantics
from the relational one via
$(v :=_\mathcal{F} e) = \mathcal{F}\, (v:=_\mathcal{R} e)$.
The $\mathit{wlp}$s for assignment commands in $\mathsf{Rel}\, S$ and $\mathsf{Sta}\, S$ are
of course the same, hence we drop the indices $\mathcal{F}$ and
$\mathcal{R}$ and write
\begin{equation}
|v:= e] Q = \lambda s.\ Q\,
(s[v\mapsto e\, s]) = \lambda s.\ Q\, (f_a\, v\, e\,
s).\label{eq:wlp-asgn}\tag{wlp-asgn}
\end{equation}
Adding the $\mathit{wlp}$ law for assignments in either semantics to the
algebraic ones for the program structure suffices to generate
data-level verification conditions for while programs.
The approach outlined so far is ideally suited for building
verification components via shallow embeddings with proof assistants
such as Isabelle. The predicate transformer algebras of the previous
sections, as shown in the first row of Figure~\ref{fig:framework}, can
all be instantiated to intermediate state transformer and relational
semantics, as shown in Proposition~\ref{P:rel-mka} and
\ref{P:kleisli-mka} for $\mathsf{MKA}$. These form the second row in
Figure~\ref{fig:framework}. Each of these can be instantiated further
in a compositional way to the concrete semantics with predicate
transformers for assignments described in this section.
In Isabelle, these instantiations benefit from type polymorphism. If
modal Kleene algebras have type ${\isacharprime}a$, then the
intermediate semantics have the type of relations or state
transformers over ${\isacharprime}a$, and Proposition~\ref{P:rel-mka}
and \ref{P:kleisli-mka} can be formalised, so that all facts known for
$\mathsf{MKA}$ are available in the intermediate semantics. The concrete
semantics then require another simple instantiation of the types of
relations or state transformers to the types of program stores. All
facts known for $\mathsf{MKA}$ and the two intermediate semantics are then
available in the concrete predicate transformer semantics for while
programs. A particularity of the semantic approach and the shallow
embedding is that assignment semantics are based on function updates
instead of substitutions---see the rule (\ref{eq:wlp-asgn}). This
greatly simplifies the construction of verification
components~\cite{afp:vericomp}. The overall approach discussed has
been developed---for Hoare logics---in~\cite{ArmstrongGS16} and
simplified in~\cite{Struth18}; it has been adapted to predicate
transformer semantics based on $\mathsf{MKA}$ in~\cite{GomesS16}.
\section{Ordinary Differential Equations}\label{sec:ODE}
Before developing relational and state transformer models for the
basic evolution commands of hybrid programs in the next section, we
briefly review some basic facts about continuous dynamical systems and
ordinary differential equations that are needed for our approach.
Continuous dynamical systems $\varphi:T\to S\to S$ are \emph{flows},
which often represent solutions of systems of ordinary differential
equations (ODEs)~\cite{Arnold,Hirsch09,Teschl12}. They are called
continuous because $T$, which models time, is assumed to form a
submonoid of $(\mathbb{R},+,0)$, and the state space or phase space $S$ is
usually a manifold. By definition, flows are monoid actions. Hence
$\varphi$ satisfies, for all $t_1,t_2\in T$,
\begin{equation*}
\varphi\, (t_1 + t_2) = \varphi\,
t_1 \circ \varphi\, t_2\qquad \text{ and } \qquad \varphi\, 0 = \mathit{id}.
\end{equation*}
We always assume that $T$ is an open interval in $\mathbb{R}$ and $S$ an
open subset of $\mathbb{R}^n$. Beyond that one usually assumes that
actions are compatible with the structure on $S$. As $S$ is a
manifold, we assume that flows are continuously differentiable.
The \emph{trajectory} of $\varphi$ through state $s\in S$ is the
function $\varphi_s:T \to S$ defined by
$\varphi_s = \lambda t.\ \varphi\, t\, s$, that is,
$\varphi_s\ t=\varphi\, t\, s$. It describes the system's evolution in
time passing through state $s$.
The \emph{orbit} of $s$ is the set of all states on the trajectory
passing through it. We model it by the function
$\gamma^\varphi:S\to \mathcal{P}\, S$ defined by
\begin{equation*}
\gamma^\varphi\, s= \mathcal{P}\, \varphi_s\, T,
\end{equation*}
the canonical map sending each $s\in S$ to its equivalence class
$\gamma^\varphi\, s$. Orbit functions are state transformers; they form
our basic semantics for hybrid programs.
Flows arise from ODEs as follows. In a system of ODEs
\begin{equation*}
x_i'\ t = f_i\ (t, (x_1\ t),\dots,(x_n\ t)),\qquad (1\le i\le n),
\end{equation*}
each $f_i$ is a continuous real-valued function and
$t\in T\subseteq\mathbb{R}$. Any such system can be made
time-independent---or \emph{autonomous}---by adding the equation
$x_0'\, t=1$. We henceforth restrict our attention to autonomous
systems and write
\begin{equation*}
X'\, t=
\begin{pmatrix}x_1'\ t\\ x_2'\ t\\ \vdots\\ x_n'\ t \end{pmatrix}=
\begin{pmatrix}f_1\ (x_1\ t) \dots (x_n\ t)\\
f_2\ (x_1\ t)\dots (x_n\ t)\\
\vdots\\ f_n\ (x_1\ t)\dots (x_n\ t)\end{pmatrix}=
f\, (X\,t).
\end{equation*}
The continuous function $f:S\to S$ on $S\subseteq \mathbb{R}^n$ is a
\emph{vector field}. It assigns a vector to each point in $S$.
An autonomous system of ODEs is thus simply a
vector field $f$, and a \emph{solution} a continuously differentiable
function $X:T\to S$ that satisfies $X'\, t=f\, (X\,t)$ for all
$t\in T$, or more briefly $X'=f\circ X$.
An \emph{initial value problem} (IVP) is a pair $(f,s)$ of a vector
field $f$ and an initial value
$(0,s)\in T\times S$~\cite{Hirsch09,Teschl12}, where $t_0=0$ and $s$
represent the initial time and initial state of the system. A
solution to the IVP $(f,s)$ satisfies
\begin{equation*}
X'=f\circ X\qquad \text{ and }\qquad X\, 0 = s.
\end{equation*}
If solutions $X$ to an IVP $(f,s)$ are unique and $T=\mathbb{R}$, then it
is easy to show that $X = \varphi^f_s$ is the trajectory of the flow
$\varphi^f$ through $s$.
Geometrically, $\varphi_s^f$ is the unique curve in $S$
that is parametrised by $t$, passes through $s$ and is tangential to
$f$ at any point. As trajectories arise from integrating both sides of
$(\varphi_s^f)'=f\circ\varphi_s^f$, they are also called \emph{integral
curves}. We henceforth write $\varphi_s$, when the dependency on $f$
is clear.
\begin{example}[Particles in fluid]\label{ex:fluid}
The autonomous system of ODEs
\begin{equation*}
x'\, t = v,\qquad
y'\, t = 0,\qquad
z'\, t = - \sin\, (x\, t),
\end{equation*}
where $v\in \mathbb{R}$ is a constant, models the movement of particles in
a three-dimensional fluid. Its vector field
$f:\mathbb{R}^3\to\mathbb{R}^3$,
\begin{equation*}
f\begin{pmatrix}x\\ y\\ z\end{pmatrix} =
\begin{pmatrix}v\\ 0\\ -\sin x\end{pmatrix},
\end{equation*}
associates a velocity vector with each point of $S=\mathbb{R}^3$
(blue vectors in Figure \ref{fig:velvecs}).
For each point $s=(s_1,s_2,s_3)^T$, the solutions
$\varphi_s:\mathbb{R}\to \mathbb{R}^3$ of the IVP $(f,s)$ are uniquely
defined. They are the trajectories of particles through time passing
through state $s$ (red dot and line in Figure \ref{fig:velvecs}),
given by
\begin{equation*}
\varphi_s\, t=\begin{pmatrix}s_1\\ s_2\\ s_3-\cos
s_1\end{pmatrix}+\begin{pmatrix}vt\\ 0\\ \cos\, (s_1+vt)\end{pmatrix}.
\end{equation*}
Checking that they are indeed solutions to the IVP requires simple calculations:
\begin{align*}
\varphi_s'\, t &=
\begin{pmatrix}v\\ 0\\ -\sin\, (s_1+vt)\end{pmatrix}=
f\, \begin{pmatrix}s_1 + vt\\ s_2\\ s_3-\cos s_1+\cos\, (s_1+vt)\end{pmatrix}
= f\, (\varphi_s\, t),\\
\varphi_s\, 0 &=
\begin{pmatrix}s_1\\ s_2\\ s_3-\cos s_1\end{pmatrix}+\begin{pmatrix}v0\\ 0\\ \cos\, (s_1+v0)\end{pmatrix}
=
\begin{pmatrix}s_1\\ s_2\\ s_3\end{pmatrix} = s.
\end{align*}
Checking that $\varphi:\mathbb{R}\to\mathbb{R}^3\to\mathbb{R}^3$,
$\varphi\, t\, s =\varphi_s\, t$, is a flow is calculational as well:
\begin{align*}
\varphi\, t_1 (\varphi\, t_2\, s)
&=\begin{pmatrix}s_1+vt_2\\ s_2\\ s_3-\cos\, s_1\end{pmatrix}
+
\begin{pmatrix}
vt_1\\
0\\
\cos\, (s_1 + vt_2 + vt_1)
\end{pmatrix}
=\begin{pmatrix}
s_1\\
s_2\\
s_3-\cos\, s_1
\end{pmatrix}
+
\begin{pmatrix}
v(t_1+t_2)\\
0\\
\cos\, (s_1 + v(t_1 + t_2))
\end{pmatrix}\\
&= \varphi\, (t_1 + t_2)\, s.
\end{align*}
The condition $\varphi\, 0\, s = s$ has already been checked. \qed
\end{example}
\begin{figure}
\begin{center}
\includegraphics[scale=0.5, trim=0 70 0 20,clip]{figure1.jpg}
\caption{Vector field and trajectory for a particle in a fluid
(Example~8.1)}
\label{fig:velvecs}
\end{center}
\end{figure}
Not all IVPs admit flows: not all of them have unique solutions, and
in many situations, flows exist locally on a subset of $\mathbb{R}$ that
does not form a submonoid. Conditions for their local existence and
uniqueness are provided by Picard-Lindel\"of's
theorem~\cite{Hirsch09,Teschl12}, which we briefly discuss, as it its
important for our approach.
By the fundamental theorem of calculus, any solution to an IVP must
satisfy
\begin{equation*}
X\, t - X\, 0 = \int_0^t f\, (X\, \tau)d\tau.
\end{equation*}
It can be shown that this equation holds if, for $X\, 0= s$, the
function
\begin{equation*}
h\, x\, t = s + \int_0^t f\, (x\, \tau)d\tau
\end{equation*}
has a fixpoint. This, in turn, is the case if the limit $X$ of the sequence
$(h^n)_{n\in\mathbb{N}}$ defined by $h^0\, x\, t= s$ and
$h^{n+1}= h\circ h^n$, exists. Indeed, with this assumption,
\begin{equation*}
X\, t = \lim_{n\to\infty} \left(s + \int_0^t f\, (h^{n-1}\, \tau)d\tau \right) = s + \int_0^t f\, (X\, \tau)d\tau,
\end{equation*}
using continuity of addition, integration and
$f$ in the second step. Finally, existence of the limit of $(h^n)_{n\in\mathbb{N}}$ is
guaranteed by constraining the domain of the $h^n$, and by Banach's
fixpoint theorem there must be a \emph{Lipschitz constant}
$\ell\geq 0$ such that
\begin{equation*}
\lVert f\, s_1 - f\, s_2 \rVert \le \ell \lVert s_1- s_2\rVert,
\end{equation*}
for any $s_1,s_2\in S$, where $\lVert-\rVert$ is the Euclidean norm on
$\mathbb{R}^n$. Vector fields satisfying this condition are called
\emph{Lipschitz continuous}.
\begin{theorem}[Picard-Lindel\"of]\label{P:picard-lindeloef}
Let $S\subseteq\mathbb{R}^n$ be an open set and $f:S\to S$ a Lipschitz
continuous vector field. The IVP $(f,s)$ has then a unique solution
$X:T_s\to S$ on some open interval $T_s\subseteq\mathbb{R}$.
\end{theorem}
It is thus possible to patch together intervals $T_s$ to a set
$U=\bigcup_{s \in S} T_s \times \{s\}\subseteq \mathbb{R} \times S$, from
which a global interval of existence $T=\bigcup_{s\in S} T_s$ can be
extracted. One can then define a \emph{local flow}
$\varphi:T\to S\to S$ such that $\varphi_s\, t$ is the maximal integral
curve at $s$. The monoid action identities $\varphi\, 0 = \mathit{id}$
and $\varphi\, (t_1+t_2)\, s = \varphi\, t_1 (\varphi\, t_2\, s)$ can thus
be shown for all $t_2,t_1+t_2\in T_s$~\cite{Teschl12}, but $U$ need
not be closed under addition. The Picard-Lindel\"of theorem, in the
form presented, thus provides sufficient conditions for the existence
and uniqueness of local flows for autonomous systems of ODEs. Flows
are global and hence monoid actions if $T$ is equal to $\mathbb{R}$ or its
non-negative or non-positive subset.
Hybrid systems deal mainly with dynamical systems where
$T=T_s= \mathbb{R}$ for any $s\in S$ and $S$ is isomorphic to $\mathbb{R}^n$
for some $n\in\mathbb{N}$. Yet our approach supports local flows with
$T\subset \mathbb{R}$ and $S\subset \mathbb{R}^n$, and even IVPs with multiple
solutions beyond the realm of Picard-Lindel\"of.
\section{Evolution Commands for Lipschitz Continuous Vector Fields}\label{sec:hybrid-store}
Simple hybrid programs of $\mathsf{d}\mathcal{L}$~\cite{Platzer10} are defined by the syntax
\begin{equation*}
\mathcal{C}\ ::= \ x:=e \mid x' = f \, \&\, G \mid ?P\mid \mathcal{C};\mathcal{C}\mid \mathcal{C}+\mathcal{C}\mid \mathcal{C}^*,
\end{equation*}
which adds \emph{evolution commands} $x' = f \ \&\ G$ to the program
syntax of dynamic logic. Intuitively, evolution commands introduce a
vector field $f$ for an autonomous system of ODEs and a \emph{guard}
$G$, which models boundary conditions or similar constraints that
restrict temporal evolutions. Guards are also known as \emph{evolution
domain restrictions} or \emph{invariants} in the hybrid automata
literature~\cite{DoyenFPP18}. The commands for nondeterministic choice
and finite iteration can be adapted for modelling conditionals and
while loops as in $\mathsf{MKA}$.
We are only interested in the semantics of hybrid programs. Hence it
remains to define the $\mathit{wlp}$s for evolution commands, which requires
relational and state transformer semantics for evolution commands over
hybrid program stores. In this section we assume that vector fields
are Lipschitz continuous, such that the Picard-Lindel\"of theorem
guarantees at least local flows. This is slightly more general than needed for
dynamical systems. A further generalisation to continuous vector
fields is presented in the next section.
We begin with hybrid program stores for $\mathsf{d}\mathcal{L}$~\cite{Platzer12}. These
are maps $s:V\to\mathbb{R}$ that assign real numbers to variables in
$V$. Variables may appear both in differential equations and the
discrete control of a hybrid system. One usually assumes that $|V|=n$
for some $n\in\mathbb{N}$, which makes $\mathbb{R}^V$ isomorphic to the
vector space $\mathbb{R}^n$. The results from Section \ref{sec:ODE} then
apply to any state space $S\subseteq\mathbb{R}^V$.
Next we describe a state transformer semantics and a $\mathsf{d}\mathcal{L}$-style
relational semantics of evolution commands with Lipschitz continuous
vector fields. Intuitively, the semantics of $x' = f \, \&\, G$ in
state $s\in S\subseteq \mathbb{R}^V$ is the longest segment of the
trajectory $\varphi^f_s$ at $s$ along which all points satisfy $G$.
For the remainder of this section we fix a Lipschitz continuous vector
field $f:S\to S$ for $S\subseteq \mathbb{R}^V$, its (local) flow
$\varphi:T\to S\to S$ for $T\subseteq \mathbb{R}$ with $0\in T$ and a guard
$G:S\to\mathbb{B}$. We freely consider $G$, and any other function of that
type, as a set or a predicate. Finally, we fix a set $U\subseteq T$ with $0 \in U$, which allows us to compute $wlp$s over subintervals of the interval of
existence $T$, typically $[0,t]$, from the time at
which the system dynamics starts to a maximal time $t$ of interest, or
$\mathbb{R}_+$, the set of non-negative real numbers. For each $t\in U$,
let ${\downarrow} t= \{t'\in U \mid t' \le t\}$.
The $G$-\emph{guarded orbit} on $U$ at $s\in S$ is then defined via
$\gamma^\varphi_{G,U}:S\to \mathcal{P}\, S$ as
\begin{equation*}
\gamma^\varphi_{G,U}\, s = \bigcup\{\mathcal{P}\, \varphi_s\, {\downarrow}t
\mid t\in U \land \mathcal{P}\, \varphi_s\, {\downarrow}t \subseteq G\}.
\end{equation*}
Intuitively, $\gamma^\varphi_{G,U}\, s$ is thus the orbit at $s$ defined
along the longest interval of time in $U$ that satisfies guard
$G$. This intuition is more apparent in the following lemma.
\begin{lemma}\label{P:g-orbit-props}
Let $s\in S$. Then
\begin{enumerate}
\item $ \gamma^\varphi_{G,U}\, s = \bigcup\{\gamma^{\varphi|_{{\downarrow}t}}\, s
\mid t\in U\land \gamma^{\varphi|_{{\downarrow}t}}\, s\subseteq G\}$,
\item $\gamma^\varphi_{G,U}\, s= \{\varphi_s\, t
\mid t\in U \land \forall \tau\in {\downarrow}t.\ G\, (\varphi_s\,
\tau)\}$.
\end{enumerate}
\end{lemma}
We have not formalised (1) with Isabelle due to its limitations in
dealing with partial functions. As a special case, for
$U=T_+$, a subinterval of $\mathbb{R}_+$,
\begin{equation*}
\gamma^\varphi_{G,T_+}\, s = \{\varphi_s\, t \mid t\in T_+\land \forall
\tau\in [0,t].\ G\, (\varphi_s\, \tau)\}.
\end{equation*}
We can now define the state transformer semantics of $x'= f\, \&\, G$
simply as
\begin{equation*}
{(x'=_\mathcal{F} f\, \&\, G)_U} = \gamma^\varphi_{G,U}.
\end{equation*}
Hence the denotation of an evolution command in state $s$ is the
guarded orbit at $s$ in time interval $U$. Alternatively, in $\mathsf{Rel}\,
S$,
\begin{equation*}
{(x'=_\mathcal{R} f\, \&\, G)_U} = \mathcal{R}\, {(x'=_\mathcal{F} f\, \&\,
G)_U} = \{(s,\varphi\, t\, s)\mid t\in U\land \forall \tau\in {\downarrow}t.\ G\, (\varphi_s\,
\tau)\}
\end{equation*}
like in Section \ref{sec:discrete-store}. Restricting this further to
$T=\mathbb{R}$ and $U=\mathbb{R}_+$ yields the standard semantics of
evolution commands of $\mathsf{d}\mathcal{L}$.
It remains to derive the $\mathit{wlp}$s for evolution commands. These are the
same in $\mathsf{Rel}\, S$ and $\mathsf{Sta}\, S$, so we drop $\mathcal{F}$ and
$\mathcal{R}$.
\begin{proposition}\label{P:wlpprop}
Let $Q: S\to\mathbb{B}$. Then
\begin{equation*}
|{(x'=f\, \&\, G)_U}] Q = \lambda s\in S.\ \{s\mid \forall t\in U.\
\mathcal{P}\, \varphi_s\, {\downarrow}t \subseteq G \rightarrow \mathcal{P}\, \varphi_s\, {\downarrow}t
\subseteq Q\}.
\end{equation*}
\end{proposition}
By Lemma~\ref{P:g-orbit-props}, alternatively,
\begin{equation*}
|{(x'=f\, \&\, G)_U}] Q
= \lambda s\in S.\ \{s\mid \forall t\in U.\
\gamma^{\varphi|_{{\downarrow}t}}\, s \subseteq G \rightarrow \gamma^{\varphi|_{{\downarrow}t}}\,
s\subseteq Q\}.
\end{equation*}
For verification condition generation, the following variant is most useful.
\begin{lemma}\label{P:wlpprop-var}
Let $Q: S\to\mathbb{B}$. Then
\begin{equation}
|{(x'=f\, \&\, G)_U}] Q = \lambda s\in S.\forall t\in U.\ (\forall
\tau\in {\downarrow}t.\ G\, (\varphi_s\, \tau)) \rightarrow Q\, (\varphi_s\, t).\label{eq:wlp-evl}\tag{wlp-evl}
\end{equation}
\end{lemma}
In particular, for $T=\mathbb{R}$ and $U=\mathbb{R}_+$,
\begin{equation*}
|{(x'=f\, \&\, G)_{\mathbb{R}_+}}] Q = \lambda s\in S.\forall t\in \mathbb{R}_+.\ (\forall
\tau\in [0,t].\ G\, (\varphi_s\, \tau)) \rightarrow Q\, (\varphi_s\,
t).
\end{equation*}
Accordingly, and consistently with $\mathsf{d}\mathcal{L}$, $Q$ is no longer a
postcondition in the traditional sense: by definition it is supposed
to hold along the trajectory and therefore on any orbit at any
particular initial condition $s$ guarded by $G$.
For a more categorical view on the $\mathit{wlp}$ of evolution commands,
remember from Section~\ref{sec:pt-monad} that
$\langle (x'=f\, \&\, G)_U| = (\gamma^\varphi_{G,U})^\dagger$, where
$(-)^\dagger$ is the Kleisli extension map, and that the $\mathit{wlp}$ of
$(x'=f\, \&\, G)_U$ is its right adjoint. It therefore satisfies
\begin{align*}|(x'=f\,
\&\, G)_U]P
=\bigcup\{Q\mid (\gamma^\varphi_{G,U})^\dagger\, Q \subseteq P\}
= \{s \mid \gamma^\varphi_{G,U}\, s \subseteq P\}.
\end{align*}
The identity in Proposition~\ref{P:wlpprop} can then be calculated
from there.
The $\mathit{wlp}$ laws in Proposition~\ref{P:wlpprop} and
Lemma~\ref{P:wlpprop-var} complete the laws for verification condition
generation for hybrid programs in the relational and state transformer
semantics. In practice, Proposition~\ref{P:wlpprop},
Lemma~\ref{P:wlpprop-var} and the Picard-Lindel\"of theorem support
the following procedure for computing the $\mathit{wlp}$ of an evolution
command $x'=f\, \&\, G$ on a set $U$ for a Lipschitz continuous
vector field:
\begin{enumerate}
\item check that the vector field $f$ is indeed Lipschitz continuous and
$S\subseteq R^V$ open;
\item supply a (local) flow $\varphi$ with interval of existence $T$ around $0$ for $f$;
\item check that $\varphi_s$ is indeed the unique solution for $(f,s)$
for any $s\in S$ and for $T$:
\begin{enumerate}
\item $\varphi_s' = f \circ \varphi_s$ for any $s\in S$,
\item $\varphi_s\, 0 = s$ for any $s\in S$,
\item $U$ is subset of open set $T$ with $0\in U$;
\end{enumerate}
\item if successful, apply the identity in Proposition~\ref{P:wlpprop} or Lemma~\ref{P:wlpprop-var}.
\end{enumerate}
Computer algebra tools can of course be used for finding flows in
practice.
The following classical example illustrates our algebraic approach and
gives a first impression of the mathematics involved. A formal verification
with Isabelle can be found in Example~\ref{ex:bouncing-ball-flow}.
\begin{example}[Bouncing ball]\label{ex:ball}
A ball of mass $m$ is dropped from height $h\geq 0$. Its state space
is $s\in\mathbb{R}^V$ for $V=\{x,v\}$, where $x$ denotes its position
and $v$ its velocity. Its kinematics is given by the vector field
$f:\mathbb{R}^V\to\mathbb{R}^V$ with
\begin{equation*}
f\,
\begin{pmatrix}
s_x\\
s_v
\end{pmatrix}
=
\begin{pmatrix}
s_v\\
-g
\end{pmatrix},
\end{equation*}
where $g$ is the acceleration due to gravity and we abbreviate
$s_x = s\, x$ and $s_v = s\, v$. The ball is assumed to bounce back
from the ground in an elastic collision. This is modelled using a
discrete control that checks whether $s_x=0$ and then flips the
velocity. A guard $G=(\lambda s.\ s_x\geq 0)$ prevents any motion
below the ground. The system is modelled by the hybrid
program~\cite{Platzer10}
\begin{align*}
\mathsf{Cntrl} &= \IF {(\lambda\, s.\ s_x=0)} {v:=(\lambda\, s.\ - s_v)} \mathit{skip},\\
\mathsf{Ball} &= ({x'=f\, \&\, G}\, {;}\, \mathsf{Cntrl})^\ast,
\end{align*}
where $\isa{skip}$ denotes the program that does not alter the state
(represented by $1$ in $\mathsf{MKA}$). Its correctness specification is
\begin{equation*}
P\leq|\mathsf{Ball}]Q\qquad\text{ for }\quad
P= (\lambda s.\ s_x = h\land s_v = 0)\quad\text{ and }\quad Q = (\lambda s.\ 0\leq s_x\leq h).
\end{equation*}
We also need the loop invariant
\begin{equation*}
I = \left(\lambda s.\ 0\le s_x \land \frac{1}{2}s_v^2= g(h - s_x)\right),
\end{equation*}
which uses a variant of energy conservation with $m$ cancelled out.
The first step of our verification proof shows that $P\le I$ and
$I\le Q$. The first inequality holds because
$\frac{1}{2} 0^2 = 0 = h - h$; the second one because $0\le s_x$
appears both in $I$ and in $Q$ and because $s_x \le h$ is guaranteed
by $g(h-s_x)\ge 0$, which holds as $\frac{1}{2}s_v^2 \ge 0$. With
transitivity and isotonicity of boxes, we can thus bring the
correctness specification into the form $I \leq|\mathsf{Ball}]I$.
Applying (\ref{eq:wlp-star}) then yields the proof obligation
$I\le |{x'=f\ \&\ G}\, {;}\, \mathsf{Cntrl}]I$. To discharge it, we
use (\ref{eq:wlp-seq}) to calculate the $\mathit{wlp}$s
\begin{align*}
J &=|\IF {(\lambda\, s.\ s_x=0)} {v:=(\lambda\, s.\ - s_v)} \mathit{skip}]I,\\
K &=|{x'=f\, \&\, G}]J
\end{align*}
incrementally and finally show that $I\le K$.
For the first $\mathit{wlp}$ we calculate, with (\ref{eq:wlp-cond}) and for $T=
(\lambda\, s.\ s_x=0)$,
\begin{align*}
J &= (T \to |v:=(\lambda\, s.\ - s_v)] I) \cdot (\overline{T} \to I)\\
&= \left( T \to |v:=(\lambda\, s.\ - s_v)] \left(\lambda s.\ 0\le s_x
\land \frac{1}{2}s_v^2= g(h - s_x)\right)\right) \cdot (\overline{T} \to I)\\
&= \left(T \to \left(\lambda s.\ 0\le s_x \land \frac{1}{2}(-s_v)^2=
g(h - s_x)\right)\right) \cdot (\overline{T} \to I)\\
&= (T \to I) \cdot (\overline{T} \to I)\\
&= I.
\end{align*}
For the second $\mathit{wlp}$, we wish to apply (\ref{eq:wlp-evl}). This
requires checking that $f$ is Lipschitz continuous---$\ell = 1$
does the job, supplying a flow and checking that it solves the IVP
$(f,s)$ for all $s\in S$ and satisfies the flow conditions for
$T=\mathbb{R}$ and $S=\mathbb{R}^V$. We leave it to the reader to verify that
$\varphi:\mathbb{R}\to\mathbb{R}^V\to\mathbb{R}^V$ defined by
\begin{equation*}
\varphi_s\, t =
\begin{pmatrix}
s_x\\
s_v
\end{pmatrix}
+
\begin{pmatrix}
s_v\\
-g
\end{pmatrix}
t
-
\frac{1}{2}\begin{pmatrix}
g\\
0
\end{pmatrix}
t^2
\end{equation*}
meets the requirements in the procedure outlined above,
cf. Example~\ref{ex:fluid}. Then, expanding definitions and applying
(\ref{eq:wlp-evl}) from Lemma~\ref{P:wlpprop-var},
\begin{align*}
K\, s
&= \left(\forall t\in\mathbb{R}_+.\ (\forall\tau\in
[0,t].\ 0 \le \varphi_s\, \tau\, x)\rightarrow 0\le
\varphi_s\, t\,
x \land \frac{1}{2}\left(\varphi_s\, t\, v\right)^2=
g(h - \varphi_s\, t\, x)\right)\\
\ & = \left(\forall t.\ (\forall\tau\in
[0,t].\ 0 \le \varphi_s\, \tau\, x)\rightarrow \frac{1}{2}(\varphi_s\, t\, v)^2=
g(h - \varphi_s\, t\, x)\right)\\
\ & = \left(\forall t. \left(\forall\tau\in
[0,t].\ 0 \le
s_x + s_vt -\frac{1}{2}g\tau^2\right) \rightarrow \frac{1}{2}(s_v-gt)^2=
g\left(h -s_x - s_v t + \frac{1}{2}gt^2\right)\right).
\end{align*}
Finally, for $I\le K$, suppose that $0\le s_x$,
$\frac{1}{2}s_v^2= g\left(h - s_x\right)$ and
$0 \le s_x+s_v\tau -\frac{1}{2}g\tau^2$ for all $\tau\in [0,t]$.
It remains to show that $\frac{1}{2}\left(s_v-gt\right)^2= g\left(h
-s_x-s_vt +\frac{1}{2}gt^2\right)$. Indeed, using the second assumption in
the second step,
\begin{align*}
\frac{1}{2}(s_v-gt)^2
= \frac{1}{2}s_v^2-g\left(s_v t+\frac{1}{2}gt^2\right)
= g(h-s_x)-g\left(s_v t+\frac{1}{2}gt^2\right)
=g\left(h-s_x +s_vt + \frac{1}{2}gt^2\right).
\end{align*}
\qed
\end{example}
Certifying solutions of systems of ODEs can be tedious and hard to
automate. It is possible to circumvent this obstacle to practical
verification applications in various ways.
A first approach is pursued by $\mathsf{d}\mathcal{L}$. It analyses properties specific
to the vector field $f$, such as phase portraits or invariant sets. If
these are strong enough to relate the precondition $P$ with the
postcondition $Q$, one can prove the partial correctness specification
$P\leq |{x'=f\ \&\ G}] Q$ without computing solutions to $f$. We
investigate this method further in the following sections.
A second approach aims at particular types of vector fields for which
(global) flows always exist and are easy to compute. A classical
example are linear systems of ODEs~\cite{Hirsch09,Teschl12}, for which
we have already developed methods that will be described in a
successor article.
A final approach abandons differential equations and vector fields
altogether and starts from flows---as with hybrid
automata~\cite{DoyenFPP18}. This requires changing the syntax of
hybrid programs. The approach is outlined in
Section~\ref{sec:flow-component}.
\section{Evolution Commands for Continuous Vector Fields}\label{sec:generalisation}
As the semantic approach to evolution commands developed in the
previous section depends mainly on orbits, which are nothing but sets
of states, it can be generalised beyond trajectories and flows. In
this section we drop the requirement that unique solutions to IVPs
exist and hence assume that vector fields are merely continuous. In
fact, if vector fields are non-continuous, the set of solutions
defined below will simply be empty.
Consider the IVP $(f,s)$ for a continuous vector field $f:S\to S$ and initial state
$s\in S\subseteq\mathbb{R}^V$. Let
\begin{equation*}
\mathop{\mathsf{Sols}} f\, T\, s = \{X \mid \forall t\in T.\ X'\, t = f\, (X\, t)\land X\, 0 = s\}
\end{equation*}
denote its set of solutions on $T\subseteq \mathbb{R}$. Here, $T$ is no
longer the maximal interval of existence defined by
Picard-Lindel\"of's theorem; it can be changed like the set
$U$ in the previous section. Then each solution $X$ is still
continuously differentiable and thus $f\circ X$ integrable in
$T$.
For all $X\in \mathop{\mathsf{Sols}}\, f\, T\, s$ and $G:S\to\mathbb{B}$, we define the $G$-\emph{guarded
orbit} of $X$ along $T$ in $s$ via the function $\gamma^X_G:S\to \mathcal{P}\, S$ as
\begin{equation*}
\gamma^X_{G}\, s = \bigcup\{\mathcal{P}\, X\,
{\downarrow}t \mid t\in T\land \mathcal{P}\, X\, {\downarrow}t\subseteq G\},
\end{equation*}
which simplifies to
$\gamma^X_{G}\, s= \{X\, t\mid t\in T\land \forall \tau\in
{\downarrow}t.\ G\, (X\, \tau)\}$.
We define the $G$-\emph{guarded orbital} of $f$ along $T$ in $s$ via
the function $\gamma^f_G:S\to \mathcal{P}\, S$ as
\begin{equation*}
\gamma^f_G\ s = \bigcup\{\gamma^X_G\, s\mid X\in \mathop{\mathsf{Sols}}\, f\, T\, s\}.
\end{equation*}
\begin{lemma}\label{P:gorbital}
Let $f:S\to S$ be continuous and $G:S\to \mathbb{B}$. Then
\begin{equation*}
\gamma^f_G\, s = \{X\, t \mid t\in T \land \mathcal{P}\, X\,
{\downarrow}t\subseteq G \land X\in \mathop{\mathsf{Sols}}\, f\, T\, s\}.
\end{equation*}
\end{lemma}
If $\top$ either
denotes the predicate that holds of all states in $S$ or the set $S$
itself, we simply write $\gamma^f$ instead of $\gamma^f_\top$.
The state transformer semantics of the evolution command for a
continuous vector field $f$ can then be defined as
\begin{equation*}
{(x'=_\mathcal{F} f\, \&\, G)} = \gamma^f_G.
\end{equation*}
The corresponding relational semantics is
\begin{equation*}
{(x'=_\mathcal{R} f\, \&\, G)} = \{(s,X\, t)\mid t\in T\land
\forall \tau\in {\downarrow}t.\ G\, (X\, \tau) \land X\in \mathop{\mathsf{Sols}} f\, T\, s\}.
\end{equation*}
Once again, $\langle x'= f\, \&\, G| = (\gamma^f_G)^\dagger$, and this
leads to a $\mathit{wlp}$ for evolution commands.
\begin{proposition}\label{P:wlpprop-gen}
Let $S\subseteq\mathbb{R}^V$ and $T\subseteq \mathbb{R}$. Let $f: S\to S$ be a
continuous vector field and $G,Q: S\to\mathbb{B}$. Then
\begin{align*}
|x'=f\, \&\, G] Q
= \lambda s\in S.\ \{s\mid \forall X\in \mathop{\mathsf{Sols}}\, f\, T\, s.\forall t\in T.\
\mathcal{P}\, X\, {\downarrow}t \subseteq G \rightarrow \mathcal{P}\, X\, {\downarrow}t
\subseteq Q\}.
\end{align*}
\end{proposition}
This identity can be rewritten, for predicates, as
\begin{equation*}
|{x'=f\, \&\, G}] Q = \lambda s\in S.\forall X\in \mathop{\mathsf{Sols}}\, f\, T\, s.\forall t\in T.\ (\forall
\tau\in {\downarrow}t.\ G\, (X\, \tau)) \rightarrow Q\, (X\, t).
\end{equation*}
Whether this fact is useful for verification applications, as
outlined above, remains to be seen; in many cases,
infinitely many solutions exists. Yet the next section
shows that it is certainly useful for reasoning with differential
invariants. The following corollary is important for
verification proofs with invariants as well.
\begin{corollary}\label{P:wlpprop-gen2}
Let $f:S\to S$, $S\subseteq\mathbb{R}^V$, be a continuous vector field,
$T\subseteq\mathbb{R}$
and $G,Q: S\to\mathbb{B}$. Then
\begin{equation*}
|{x'=f\, \&\, G}] Q = |{x'=f\, \&\, G}](G\cdot Q).
\end{equation*}
\end{corollary}
\section{Invariants for Evolution
Commands}\label{sec:differential-invariants}
In $\mathsf{d}\mathcal{L}$, differential invariants are predicates $I$ that satisfy
$I\leq |{x'=f\ \&\ G}] I$~\cite{Platzer12}. In the terminology of
Section~\ref{sec:mka-pt}, they are simply invariants for evolution
commands. They play a crucial role in $\mathsf{d}\mathcal{L}$ and KeYmaera X because of
the limited support for solving ODEs.
In dynamical systems theory, when all guards are $\top$ and global
flows exist, and in (semi)group theory, \emph{invariant sets} for
actions or flows $\varphi:T\to S\to S$ are sets $I\subseteq S$
satisfying $\gamma^\varphi\, s\subseteq I$ for all
$s\in I$~\cite{Teschl12}. Based on the results from
Section~\ref{sec:generalisation}, we generalise both notions uniformly.
A predicate or set $I:S\to\mathbb{B}$ is an \emph{invariant} of the
continuous vector field $f:S\to S$ and guard $G:S\to\mathbb{B}$
\emph{along} $T\subseteq \mathbb{R}$ if
\begin{equation*}
(\gamma^f_G)^\dagger\, I\subseteq I.
\end{equation*}
Note that the parameter $T$ is hidden in the definition of
$\gamma^f_G$.
For $G=\top$, when $(\gamma^f)^\dagger\, I\subseteq I$, we
call $I$ simply an \emph{invariant} of $f$ along $T$.
The following proposition yields an interesting structural insight in
the relationship between invariant sets of dynamical systems and
differential invariants of $\mathsf{d}\mathcal{L}$ in terms of an adjunction.
\begin{proposition}\label{P:inv-prop}
Let $f:S\to S$ be continuous, $G:S\to\mathbb{B}$ and
$T\subseteq \mathbb{R}$. Then the following are equivalent.
\begin{enumerate}
\item $I$ is an invariant for $f$ and $G$ \emph{along} $T$;
\item $\langle x'= f\, \&\, G | I \subseteq I$;
\item $I \subseteq | x'= f\, \&\, G ] I$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{equation*}
(\gamma^f_G)^\dagger I \subseteq I
\leftrightarrow \langle x'= f\, \&\, G | I \subseteq I
\leftrightarrow I \subseteq | x'= f\, \&\, G ] I.
\end{equation*}
The first step uses the definition of backward diamonds as Kleisli
extensions in Section~\ref{sec:pt-monad} and that of the semantics
of evolution commands in Section~\ref{sec:generalisation}. The final
step uses the adjunction between boxes and diamonds from
Section~\ref{sec:mka-pt}.
\end{proof}
For our $\mathit{wlp}$-calculus, condition (3) is of course most useful. Yet instead of
checking that a flow is a solution to a vector field, as previously,
we now need to check whether a predicate is an invariant---without
having to solve the system of ODEs. The following lemmas lead to a
procedure. We show some proofs although they have been formalised with
Isabelle, as they explain why the approach works.
First, we may ignore guards when checking for invariants and we can use a
simple second-order formula.
\begin{lemma}\label{P:inv-props2}
Let $f:S\to S$ be continuous and $I:S\to\mathbb{B}$. Then
\begin{enumerate}
\item $ I \subseteq |x'= f\, \&\, \top]I \rightarrow I \subseteq |x'= f\,
\&\, G]I$,
\item $I \subseteq |x'= f\, \&\, \top]I \leftrightarrow \left(I\, s \to \forall X \in \mathop{\mathsf{Sols}}\, f\, T\, s.\forall t\in T.\ I\,
(X\, t)\right)$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (1), $\gamma^f_G\subseteq \gamma^f$ for all $G$ and hence
$\langle x'=f\, \&\, G|I \subseteq \langle x'=f\, \&\top|I \subseteq
I$. The proof of (2) is a simple calculation.
\end{proof}
Second, we can recurse over predicates as follows.
\begin{lemma}\label{P:invrules}
Let $f:S\to S$ be a continuous vector field, $\mu,\nu:S\to\mathbb{R}$
differentiable and $T\subseteq \mathbb{R}$ with $0\in T$.
\begin{enumerate}
\item If $(\mu\circ X)' =(\nu\circ X)'$ for all $X\in \mathop{\mathsf{Sols}} f\, T\, s$, then $\mu = \nu$ is an invariant for $f$ along $T$,
\item if $(\mu\circ X)'\, t\leq(\nu\circ X)'\, t$ when $t> 0$, and $(\mu\circ X)'\, t\geq(\nu\circ X)'\, t$ when $t< 0$, for all $X\in \mathop{\mathsf{Sols}} f\, T\, s$,
then $\mu < \nu$ is an invariant for $f$ along $T$,
\item if $\mu < \nu$ and $\nu < \mu$ are invariants for $f$ along $T$, then $\mu\neq \nu$ is too (and conversely if $0$ is the least element in $T$),
\item if $\mu < \nu$ and $\nu < \mu$ are invariants for $f$ along $T$, then $\mu\neq \nu$ is too (and conversely if $0$ is the least element in $T$),
\item $\mu \not\le \nu$ is an invariant for $f$ along $T$ if and only if $\nu < \mu$ is too.
\end{enumerate}
\end{lemma}
\begin{proof}
We only show the proof of (1), as it reveals the main idea of the
procedure outlined below. By definition, $\mu = \nu$ is an
invariant for $f$ along $T$ if and only if $\mu\, s = \nu\, s$
implies $\mu\, (X\, t) = \nu\, (X\, t)$ for all
$X\in\mathop{\mathsf{Sols}}\, f\, T\, s$. It is a well known consequence of the mean
value theorem that two continuously differentiable functions are the
same if and only if they intersect at some point and have the same
derivative. Hence $(\mu\circ X)' =(\nu\circ X)'$ and
$\mu\, s = \nu\, s$ imply $\mu\, (X\, t) = \nu\, (X\, t)$ for all
$X\in\mathop{\mathsf{Sols}}\, f\, T\, s$.
\end{proof}
Proposition~\ref{P:wlpprop-gen2}, the properties in this section---in
particular Lemma~\ref{P:invrules}---and
Lemma~\ref{P:inv-lemma} yield the following procedure for proving a
correctness specification $P\le |x' = f\, \&\, G]Q$ using a
differential invariant.
\begin{enumerate}
\item Check whether a candidate predicate $I$ is a differential
invariant:
\begin{enumerate}
\item transform $I$ into negation normal form;
\item if $I$ is complex, reduce it with
Lemma~\ref{P:inv-lemma}, and Proposition~\ref{P:invrules}(3) and (4);
\item if $I$ is atomic, apply Proposition~\ref{P:invrules}(1) and (2);
\end{enumerate}
(if successful, $I\le |x' = f\, \&\, G]I$ holds by
Proposition~\ref{P:inv-prop}(3) and Lemma~\ref{P:inv-props2});
\item if successful, prove $P\le I$ and $|x' = f\, \&\, G](G\cdot I)
\le |x' = f\, \&\, G]Q$.
\end{enumerate}
For $G=\top$ and Lipschitz continuous vector fields, the notions of invariant can be
strengthened.
\begin{corollary}\label{P:inv-props3}
Let $f:S\to S$ be Lipschitz continuous. Then the following are equivalent.
\begin{enumerate}
\item $I$ is an invariant for $f$ along $T$;
\item $\langle x'= f\, \&\, \top | I = I$;
\item $I = | x'= f\, \&\, \top ] I$.
\end{enumerate}
\end{corollary}
The identities (2) and (3) hold because $0\in T$.
Next we revisit the the bouncing ball example from
Section~\ref{sec:hybrid-store} to show our procedure for reasoning
with differential invariants at work. Once again we give detailed
mathematical calculations to indicate the kind of mathematical
reasoning involved. An Isabelle verification can be found in
Example~\ref{ex:bouncing-ball-inv}.
\begin{example}[Bouncing ball with differential
invariant]\label{ex:ball-inv}
We can avoid solving the system of ODEs in
Example~\ref{ex:ball} using a differential invariant to show that
\begin{equation*}
I\le |{x'=f\, \&\, G}]I
\end{equation*}
for the loop invariant $I$ and vector field
$f\, (s_x,s_v)^T = (s_v,-g)^T$. The most natural candidate for a
differential invariant is of course energy conservation, that is,
$\frac{1}{2}m s_v^2=mg (h-s_x)$. Cancelling the mass, we use
\begin{equation*}
I_d = \left(\lambda s.\ \frac{1}{2} s_v^2=g (h-s_x)\right).
\end{equation*}
We now apply our procedure for reasoning with differential
invariants.
\begin{enumerate}
\item We use Proposition \ref{P:invrules} with
$\mu\, s = \frac{1}{2}s_v^2$ and $\nu\, s = g(h- s_x)$ to check that
$I_d$ is indeed an invariant. We thus need to show that
$(\mu\circ X)' =(\nu\circ X)'$ for all $X\in \mathop{\mathsf{Sols}}\, f\, T\, s$,
which unfolds to
\begin{equation*}
\left(\frac{1}{2}(X\, t\, v)^2\right)' = g (h - X\, t\, x)',
\end{equation*}
because $s= X\, t$ and therefore $s_v = X\, t\, v$ and $s_x = X\, t\, x$.
And indeed,
\begin{align*}
\left(\frac{1}{2}(X\, t\, v)^2\right)' &= (X\, t\, v) (X'\, t\, v) =
(X\, t\, v) (f\, (X\,t)\, v) = -(X\, t\, v)g\\
&= -g(f\, (X\,t)\, x) =
-g(X'\, t\, x) = \left( g (h - X\, t\, x)\right)'.
\end{align*}
By Proposition~\ref{P:invrules}(1), $I_d$ is thus an invariant for $f$
along $\mathbb{R}^V$. Proposition \ref{P:inv-prop}(3) and
Lemma~\ref{P:inv-props2} then
imply that
\begin{equation*}
I_d \le |{x'=f\ \&\ G}] I_d.
\end{equation*}
\item It remains to show that $I\le I_d$ and $|{x'=f\ \&\ G}] I_d \le
|{x'=f\ \&\ G}]I$.
\begin{itemize}
\item The first inequality is trivial.
\item For the second one, we calculate
\begin{equation*}
(G \cdot I_d)\, s = \left(0\le s_x \land \frac{1}{2}s_v^2= g(h - s_x)\right) = I\, s.
\end{equation*}
By Corollary~\ref{P:wlpprop-gen2}, therefore,
\begin{equation*}
|{x'=f\ \&\ G}] I_d = |{x'=f\ \&\ G}](G\cdot I_d) = |{x'=f\ \&\ G}]I.
\end{equation*}
\end{itemize}
\end{enumerate}
This shows that $I\le |{x'=f\, \&\, G}]I$. The remaining proof of
$P\leq|\mathsf{Ball}]Q$ is the same as in Example~\ref{ex:ball}.
\qed
\end{example}
This example shows that one can reason about invariants of evolution
commands in a natural mathematical style as it can be found in
textbooks on differential equations~\cite{Arnold,Hirsch09,Teschl12}.
By contrast, $\mathsf{d}\mathcal{L}$ relies on syntactic substitution-based reasoning in
the term algebra of differential rings~\cite{Platzer12} to check
invariants, and complex domain-specific inference rules to manipulate
them. The following section shows that we can derive semantic
variants of the most important $\mathsf{d}\mathcal{L}$ inference rules for those who
like this style of reasoning.
Finally, we briefly specialise our approach to $\mathsf{d}\mathcal{L}$-invariants, the
invariants sets used in dynamical systems theory and those in
(semi)group theory. We assume a setting where global flows exist and
indices $U$ can be dropped.
\begin{corollary}\label{P:dl-invset}
Let $f:S\to S$ be Lipschitz continuous. Then $I:S\to \mathbb{B}$ is a
$\mathsf{d}\mathcal{L}$-invariant for $x'=f\, \&\, \top$ if and only if $I$ is an
invariant set for $\varphi^f$.
\end{corollary}
\begin{proof}
It is easy to check that
$(\forall s\in I.\ I\, s \rightarrow \gamma^\varphi\, s \subseteq I)
\leftrightarrow (\gamma^\varphi)^\dagger I \subseteq I$.
The claim then follows from Proposition~\ref{P:inv-prop}. In the
Lipschitz continuous case, of course,
$\mathop{\mathsf{Sols}}\, f\, T\, s=\{\varphi^f\}$.
\end{proof}
It remains to point out that the difference between the definition of
invariant sets for dynamical systems and that for (semi)group actions
is merely notational: In group theory, an invariant set $I$ of a
(semi)group action $\varphi:T\to S\to S$ satisfies
$T\cdot I\subseteq I$, where
$T\cdot I= \{\varphi\, t\, s \mid t\in T\land s\in I\}$. In the presence
of a unit, therefore $T\cdot I = I$. Yet of course
$(\gamma^\varphi)^\dagger\, I = \{\varphi\, t\, s \mid t\in T\land s\in
I\}$ as well.
\section{Derivation of $\mathsf{d}\mathcal{L}$ Inference Rules}\label{sec:dL}
As a proof of concept, we derive semantic variants of some axioms and
inference rules of $\mathsf{d}\mathcal{L}$. The first one introduces solutions
of IVPs with constant vector fields~\cite{BohrerRVVP17}. It is a
trivial instance of Proposition~\ref{P:wlpprop} with
$f = \lambda s.\ c$ for some $c\in \mathbb{R}$. Such vector fields are
Lipschitz continuous; their flows are $\varphi\, t\, s = s + ct$. Hence
\begin{equation}
|{x'= (\lambda s.\ c)\ \&\ G}] Q = \lambda s\in S.\ \forall
t\in T.\ (\forall \tau\le t.\ G\, (s + c\tau))\rightarrow Q\, (s +
ct). \tag{DS}\label{eq:DS}
\end{equation}
For a second $\mathsf{d}\mathcal{L}$ inference rule we simply rewrite the $\mathit{wlp}$ in
Proposition~\ref{P:wlpprop} as a Hoare-style inference rule.
\begin{lemma}\label{P:solve} Let $S\subseteq\mathbb{R}^V$ and
$T=\mathbb{R}$. Let $\varphi:T\to S\to S$ be the flow for the Lipschitz
continuous vector field $f:S\to S$, and
$G,Q:S\to\mathbb{B}$. Then
\begin{equation}
\inferrule*
{\forall s\in S.\ P\, s\rightarrow (\forall t\in T.\
(\forall \tau\le t.\ G (\varphi_s\, \tau))\rightarrow Q\, (\varphi_s\, t))}
{P\leq |{x'=f\, \&\, G}] Q}\label{eq:dSolve}\tag{dSolve}
\end{equation}
\end{lemma}
To apply this rule, the procedure in Section~\ref{sec:hybrid-store}
must be followed.
Next we derive generalisations of five $\mathsf{d}\mathcal{L}$ axioms and inference
rules for differential invariants in the setting of
Section~\ref{sec:differential-invariants}.
\begin{lemma}\label{P:dcut} Let $P,G,I,Q:S\to\mathbb{B}$, $T\subseteq \mathbb{R}$ and
$f:S\to S$ be a continuous vector field. Then, with $\eta_S$ the unit of the power set monad,
\begin{gather}
|{x'=f\, \&\, G}] I = \eta_S \to
|{x'= f\, \&\, (\lambda\, s.\, G\, s \land I\, s)}] Q = |{x'=f\, \&\, G}] Q,\tag{DC}\label{eq:DC}\\
\inferrule*
{P\leq |{x'=f\, \&\, G}] I\\
P\leq |{x'= f\, \&\, (\lambda\, s.\ G\, s \land I\, s)}] Q}
{P\leq |{x'=f\, \&\, G}] Q}\tag{dC}\label{eq:dC}\\
|{x'=f\ \&\ G}] (\lambda\, s.\ G\, s\to Q\, s) = |{x'=f\ \&\ G}] Q,\tag{DW}\label{eq:DW}\\
\inferrule*{G\leq Q}{P\leq |{x'=f\ \&\ G}] Q}\tag{dW}\label{eq:dW}
\end{gather}
Finally, if $I$ is a
differential invariant for $f$
along $T$, then
\begin{equation}
\inferrule*
{P\leq I\qquad I\leq Q}
{P\leq |{x'=f\ \&\ G}] Q}\tag{dI}\label{eq:dI}
\end{equation}
\end{lemma}
The \emph{differential cut} axiom and rule (\ref{eq:DC}) and
(\ref{eq:dC}) introduce differential invariants in guards of evolution
commands. \emph{Differential weakening}, (\ref{eq:DW}) and
(\ref{eq:dW}), summarises the obvious fact that if a guard is strong
enough to imply a postcondition, then no invariant or solution needs
to be found. Finally, the \emph{differential induction} rule
(\ref{eq:dI}) follows from Proposition~\ref{P:inv-prop}(3), transitivity
and isotonicity of boxes.
These rules are typically applied backwards as follows: $\mathit{dC}$
introduces an invariant. Its left premise is discharged via
$\mathit{dI}$, Proposition~\ref{P:inv-prop} and logical reasoning,
while itsright premise is discharged via $\mathit{dW}$. Verification
examples using these rules and the $\mathsf{d}\mathcal{L}$ approach can be found in our
Isabelle components.
A differential ghost rule, and sometimes a differential effect axiom
have also been proposed for reasoning with invariants in
$\mathsf{d}\mathcal{L}$~\cite{BohrerRVVP17}, yet our semantics approach has so far no
need for these.
\section{Isabelle Components for $\mathsf{MKA}$ and Predicate Transformers}\label{sec:isa-pt}
The entire mathematical development of $\mathsf{MKA}$ in
Section~\ref{sec:KA}-\ref{sec:mka-pt} has been formalised with
Isabelle~\cite{afp:ka,afp:kad}. Verification components for Isabelle
and the relational store model in Section~\ref{sec:discrete-store}
have been developed, too~\cite{GomesS16,afp:vericomp}, using the
shallow embedding approach discussed in Section~\ref{sec:intro} and
\ref{sec:discrete-store}. Predicate transformers \`a la Back and von
Wright have been formalised previously in Isabelle by
Preoteasa~\cite{Preoteasa11,Preoteasa11a}. Our alternative
formalisation emphasises the quantalic structure of
transformers~\cite{afp:quantales,afp:transem}, as in
Section~\ref{sec:pt-backvwright}, and we have added a third component
based on quantaloids~\cite{afp:transem}. It is based on the powerset
monad~\cite{afp:transem}, as outlined in Section~\ref{sec:pt-monad}.
Our formalisation is compositional in that all three approaches to
predicate transformers can be combined with relational and state
transformer semantics and different models of the (hybrid) program
store, as shown in Figure~\ref{fig:framework}.
This section summarises the Isabelle components for predicate
transformers and the verification component based on $\mathsf{MKA}$. More
detailed information can be found in the proof documents for these
components~\cite{afp:vericomp,afp:transem}.
The $\mathsf{MKA}$ component is integrated into the Kleene algebra hierarchy
that formalises variants of Kleene algebras~\cite{afp:ka} and modal
Kleene algebras~\cite{afp:kad}, as outlined in Section~\ref{sec:KA}
and~\ref{sec:MKA}. In these mathematical components, algebras are
formalised as type classes, their models via instantiation and
interpretation statements. For Kleene algebras, a large number of
computationally interesting models has been developed; for $\mathsf{MKA}$ only
the relational model is present in the Archive of Formal Proofs. The
state transformer model has been formalised for quantales in a
different component~\cite{afp:transem}.
Instantiation and interpretation statements have several
purposes. They make algebraic facts available in all models, establish
soundness of algebraic hierarchies and ultimately make the axiomatic
approaches consistent with respect to Isabelle's small trustworthy
core. Finally, they unify developments of multiple concrete semantics.
In our $\mathsf{MKA}$-based verification components~\cite{afp:vericomp},
program syntax is absent and semantic illusions of program syntax are
provided in the concrete program semantics, as outlined in
Section~\ref{sec:discrete-store}. Consequently, verification
conditions for the control structure of programs are generated within
the algebra; those for assignments in the concrete store semantics.
We currently model stores simply as functions from strings
representing variables to values of arbitrary type. Expressions are
simulated by functions from stores to values, as outlined in
Section~\ref{sec:discrete-store}; stores with poly-typed values are
modelled via sum-types. An extension to verification components for
hybrid programs is described in the following sections.
A second component is based on Back and von Wright style predicate
transformers~\cite{BackW98}, for which we have built special purpose
components with advanced features for orderings and
lattices~\cite{afp:order} and for quantales~\cite{afp:quantales}.
These structures are once again formalised as type classes. Predicate
transformers, however, are modelled as global functions that may have different
source and target types. Isabelle's simple type system can infer most
general types for definitions. These can be associated with
predicate transformers by sort constraints; definitions can often be declared in
the point-free style of functional programming. This makes the
formalisation of quantaloids of transformers with partial compositions
straightforward. Mono-typed transformer algebras are obtained from
these via subtyping. They are linked with quantales and
Kleene algebras by interpretation or instantiation.
Isabelle's type system is too weak for a deep embedding of general
categorical concepts such as monads, but formalising instances such as
the powerset monad, its Kleisli category and Eilenberg-Moore algebras
is possible and relatively straightforward. We have formalised the
isomorphisms and dualities between relations, state transformers and
the four predicate transformers corresponding to backward and forward
boxes and diamonds in this setting, but using these dualities to
transfer theorems automatically seems infeasible without writing
low-level tactics.
We have created a second verification component for hybrid systems
based on Back and von Wright's approach, using the monadic
transformers to obtain a concrete semantics. Finally, we have once again
restricted the categorical approach to the mono-typed case in a third
component. Via subtyping we can then show that the categorical
transformers form quantales, and more specifically $\mathsf{MKA}$s.
Everything Isabelle knows about $\mathsf{MKA}$ is then available in this
instance.
\section{Isabelle Components for ODEs and Orbits}\label{sec:isa-ODE}
This and the following two sections describe the formalisation of the
material in Sections~\ref{sec:ODE}-\ref{sec:dL} in Isabelle, from
mathematical components for ODEs and orbits to verification components
for hybrid programs based on (local) flows, differential invariants
and $\mathsf{d}\mathcal{L}$-style inference rules.
We begin with summarising Immler and H\"olzl's formalisation of the
Picard-Lindel\"of theorem based on the impressive Isabelle hierarchy
for analysis and ordinary differential
equations~\cite{HolzlIH13,Immler12,ImmlerH12a,ImmlerT19}. We have
adapted their results to show that unique solutions to IVPs for
autonomous systems of ODEs guaranteed by this theorem satisfy the
local flow conditions, as presented in previous sections.
H\"olzl and Immler have proved Picard-Lindel\"of's theorem for
time-dependent vector fields. These have type
${\isachardoublequoteopen}\mathit{real}\ {\isasymRightarrow}\
{\isacharparenleft}{\isacharprime}a{\isacharcolon}{\isacharcolon}{\isacharbraceleft}heine{\isacharunderscore}borel{\isacharcomma
}banach{\isacharbraceright}{\isacharparenright}\ {\isasymRightarrow}\
{\isacharprime}a{\isachardoublequoteclose}$~\cite{ImmlerH12a}.
They have called their theorem
$\mathit{unique}{\isacharunderscore}\mathit{solution}$ and have
formalised it within a locale called
$\isa{unique{\isacharunderscore}on{\isacharunderscore
}bounded{\isacharunderscore}closed}$
to bundle the assumptions for the local existence of unique
solutions within a closed interval in $\mathbb{R}$. They have specialised and
hence extended this locale to various cases.
Our approach is based on their extension
$\isa{ll{\isacharunderscore}on{\isacharunderscore
}open{\isacharunderscore}it}$
that bundles more or less the conditions of
Theorem~\ref{P:picard-lindeloef}, yet still for the time-dependent
case. Adding the condition $t_0\in T$ for convenience, we have
generated the following variant.
\begin{isabellebody}
\isanewline
\isacommand{locale}\isamarkupfalse%
\ picard{\isacharunderscore}lindeloef\ {\isacharequal}\isanewline
\ \ \isakeyword{fixes}\
f{\isacharcolon}{\isacharcolon}{\isachardoublequoteopen}real\
{\isasymRightarrow}\
{\isacharparenleft}{\isacharprime}a{\isacharcolon}{\isacharcolon}{\isacharbraceleft}heine{\isacharunderscore}borel{\isacharcomma
}banach{\isacharbraceright}{\isacharparenright}\ {\isasymRightarrow}\
{\isacharprime}a{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\
T{\isacharcolon}{\isacharcolon}{\isachardoublequoteopen}real\
set{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\
S{\isacharcolon}{\isacharcolon}{\isachardoublequoteopen}{\isacharprime}a\
set{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ t\isactrlsub {\isadigit{0}}{\isacharcolon}{\isacharcolon}real\isanewline
\ \ \isakeyword{assumes}\ open{\isacharunderscore}domain{\isacharcolon}\ {\isachardoublequoteopen}open\ T{\isachardoublequoteclose}\ {\isachardoublequoteopen}open\ S{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ interval{\isacharunderscore}time{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}interval\ T{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ init{\isacharunderscore}time{\isacharcolon}\ {\isachardoublequoteopen}t\isactrlsub {\isadigit{0}}\ {\isasymin}\ T{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ cont{\isacharunderscore}vec{\isacharunderscore}field{\isacharcolon}\ {\isachardoublequoteopen}{\isasymforall}s\ {\isasymin}\ S{\isachardot}\ continuous{\isacharunderscore}on\ T\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f\ t\ s{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ lipschitz{\isacharunderscore}vec{\isacharunderscore}field{\isacharcolon}\ {\isachardoublequoteopen}local{\isacharunderscore}lipschitz\ T\ S\ f{\isachardoublequoteclose}\isanewline
\isakeyword{begin}
\isanewline
\isanewline
\isacommand{sublocale}\isamarkupfalse%
\ ll{\isacharunderscore}on{\isacharunderscore}open{\isacharunderscore}it\ T\ f\ S\ t\isactrlsub {\isadigit{0}}
\isanewline
\ \ $\langle \isa{proof}\rangle$
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ unique{\isacharunderscore}solution{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\ xivp{\isacharcolon}\
{\isachardoublequoteopen}D\ X\ {\isacharequal}\
{\isacharparenleft}{\isasymlambda}t{\isachardot}\ f\ t\
{\isacharparenleft}X\ t{\isacharparenright}{\isacharparenright}\ on\
{\isacharbraceleft}t\isactrlsub
{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}{\isachardoublequoteclose}\
{\isachardoublequoteopen}X\ t\isactrlsub {\isadigit{0}}\
{\isacharequal}\ s{\isachardoublequoteclose}\
{\isachardoublequoteopen}X\ {\isasymin}\
{\isacharbraceleft}t\isactrlsub
{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}\
{\isasymrightarrow}\ S{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}t\ {\isasymin}\ T{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ yivp{\isacharcolon}\
{\isachardoublequoteopen}D\ Y\ {\isacharequal}\
{\isacharparenleft}{\isasymlambda}t{\isachardot}\ f\ t\
{\isacharparenleft}Y\ t{\isacharparenright}{\isacharparenright}\ on\
{\isacharbraceleft}t\isactrlsub
{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}{\isachardoublequoteclose}\
{\isachardoublequoteopen}Y\ t\isactrlsub {\isadigit{0}}\
{\isacharequal}\ s{\isachardoublequoteclose}\
{\isachardoublequoteopen}Y\ {\isasymin}\
{\isacharbraceleft}t\isactrlsub
{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}\
{\isasymrightarrow}\ S{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}s\ {\isasymin}\ S{\isachardoublequoteclose}\ \isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}X\ t\ {\isacharequal}\ Y\ t{\isachardoublequoteclose}
\isanewline
\ \ $\langle \isa{proof}\rangle$
\isanewline
\isacommand{end}
\isanewline
\end{isabellebody}
\noindent The locale declaration lists the assumptions of
Picard-Lindel\"of's theorem: the vector field $f$---which is still
time-dependent---is defined on an open time interval $T$ that contains
the initial time $t_0$, and an open subset $S$ of the state space. The
vector field $f$ is continuous in time and, for each
$(t,s)\in T\times S$, Lipschitz continuous on a closed subset of
$T\times S$ around $(t,s)$. The sublocale statement shows that these
assumptions imply those of the locale
$\isa{ll{\isacharunderscore}on{\isacharunderscore
}open{\isacharunderscore}it}$.
Lemma $\mathit{unique}{\isacharunderscore}\mathit{solution}$ ensures
that Picard-Lindel\"of's theorem is derivable within this locale. The
notation $D\, X$ stands for $X'$, and $g\in A\to B$ indicates that
function $g$ maps from the set $A$ into the set $B$, as opposed to the
type of $g$, which can be larger. The notation
${\isacharbraceleft}t\isactrlsub
{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}$
indicates the set of real numbers between
$t\isactrlsub {\isadigit{0}}$ and $t$ (including both), where $t$ may
be above or below $t_0$. The formalisation of Picard-Lindel\"of's
theorem comprises a formal definition of solutions to IVPs of system
of ODEs in Isabelle. As an abbreviation, we have defined the set
$\mathop{\mathsf{Sols}}\, f\, T\, s$ of Section~\ref{sec:generalisation} with the
additional requirement that $X\in T\to S$.
\begin{isabellebody}
\isanewline
\isacommand{definition}\isamarkupfalse%
\ ivp{\isacharunderscore}sols\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}real\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a {\isacharcolon}{\isacharcolon}\ real{\isacharunderscore}normed{\isacharunderscore }vector{\isacharparenright}{\isacharparenright}\ {\isasymRightarrow}\ real\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow} \isanewline
\ \ real\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}real\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ set {\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}Sols{\isachardoublequoteclose}{\isacharparenright}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}Sols\ f\ T\ S\ t\isactrlsub {\isadigit{0}}\ s\ {\isacharequal}\ {\isacharbraceleft}X\ {\isacharbar}X{\isachardot}\ {\isacharparenleft}D\ X\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f\ t\ {\isacharparenleft}X\ t{\isacharparenright}{\isacharparenright}\ on\ T{\isacharparenright}\ {\isasymand}\ X\ t\isactrlsub {\isadigit{0}}\ {\isacharequal}\ s\ {\isasymand}\ X\ {\isasymin}\ T\ {\isasymrightarrow}\ S{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\end{isabellebody}
\noindent We have restricted locale $\isa{picard-lindeloef}$ to
autonomous systems and to $t_0=0$, while introducing the variable
$\varphi$ for the local flow of the vector field.
\begin{isabellebody}
\isanewline\isacommand{locale}\isamarkupfalse%
\ local{\isacharunderscore}flow\ {\isacharequal}\ picard{\isacharunderscore}lindeloef\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymlambda}\ t{\isachardot}\ f{\isacharparenright}{\isachardoublequoteclose}\ T\ S\ {\isadigit{0}}\ \isanewline
\ \ \isakeyword{for}\
f{\isacharcolon}{\isacharcolon}{\isachardoublequoteopen}{\isacharprime}a{\isacharcolon}{\isacharcolon}{\isacharbraceleft}heine{\isacharunderscore}borel{\isacharcomma
}banach{\isacharbraceright}\ {\isasymRightarrow}\
{\isacharprime}a{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ T\ S\ L\ {\isacharplus}\isanewline
\ \ \isakeyword{fixes}\ {\isasymphi}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{assumes}\ ivp{\isacharcolon}\isanewline
\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}\ t\ s{\isachardot}\ t\ {\isasymin}\ T\ {\isasymLongrightarrow}\ s\ {\isasymin}\ S\ {\isasymLongrightarrow}\ D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isasymphi}\ t\ s{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f\ {\isacharparenleft}{\isasymphi}\ t\ s{\isacharparenright}{\isacharparenright}\ on\ {\isacharbraceleft}{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}\ s{\isachardot}\ s\ {\isasymin}\ S\ {\isasymLongrightarrow}\ {\isasymphi}\ {\isadigit{0}}\ s\ {\isacharequal}\ s{\isachardoublequoteclose}\isanewline
\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}\ t\ s{\isachardot}\ t\ {\isasymin}\ T\ {\isasymLongrightarrow}\ s\ {\isasymin}\ S\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isasymphi}\ t\ s{\isacharparenright}\ {\isasymin}\ {\isacharbraceleft}{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}\ {\isasymrightarrow}\ S{\isachardoublequoteclose}\isanewline
\end{isabellebody}
\noindent Within this locale, we have shown that $T$ is the maximal
interval of existence (\isa{ex{\isacharunderscore}ivl}). Thus $\varphi$ is the unique solution on
the whole of $T$---and not only on its subsets
${\isacharbraceleft}{\isadigit{0}}{\isacharminus}{\isacharminus}t{\isacharbraceright}$.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ex{\isacharunderscore}ivl{\isacharunderscore}eq{\isacharcolon}\
{\isachardoublequoteopen}s\ {\isasymin}\ S{\isachardoublequoteclose}
\ {\isasymLongrightarrow}\
{\isachardoublequoteopen}ex{\isacharunderscore}ivl\ s\ {\isacharequal}\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ has{\isacharunderscore}vderiv{\isacharunderscore}on{\isacharunderscore }domain{\isacharcolon }\
{\isachardoublequoteopen}s\ {\isasymin}\ S{\isachardoublequoteclose}
\ {\isasymLongrightarrow}\
{\isachardoublequoteopen}D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isasymphi}\ t\ s{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f\ {\isacharparenleft}{\isasymphi}\ t\ s{\isacharparenright}{\isacharparenright}\ on\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ in{\isacharunderscore}ivp{\isacharunderscore}sols{\isacharcolon}\
{\isachardoublequoteopen}s\ {\isasymin}\ S{\isachardoublequoteclose}
\ {\isasymLongrightarrow}\
{\isachardoublequoteopen}{\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isasymphi}\ t\ s{\isacharparenright}\ {\isasymin}\ Sols\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f{\isacharparenright}\ T\ S\ {\isadigit{0}}\ s{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ eq{\isacharunderscore}solution{\isacharcolon}\
{\isachardoublequoteopen}X\ {\isasymin}\
Sols\
{\isacharparenleft}{\isasymlambda}t{\isachardot}\
f{\isacharparenright}\ T\ S\ {\isadigit{0}}\
s{\isachardoublequoteclose}
{\isasymLongrightarrow}\
{\isachardoublequoteopen}t\ {\isasymin}\
T{\isachardoublequoteclose}
\ {\isasymLongrightarrow}\
{\isachardoublequoteopen}s\ {\isasymin}\ S{\isachardoublequoteclose}
\ {\isasymLongrightarrow}\
{\isachardoublequoteopen}X\ t\ {\isacharequal}\ {\isasymphi}\ t\ s{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent Finally, if the maximal interval of existence $T$ is equal
to $\mathbb{R}$, then the flow $\varphi$ is global and hence a proper monoid
action.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\
ivp{\isacharunderscore}sols{\isacharunderscore}collapse{\isacharcolon}\
{\isachardoublequoteopen}T\ {\isacharequal}\
UNIV{\isachardoublequoteclose}
{\isasymLongrightarrow}\
s\ {\isasymin}\ {\isachardoublequoteopen}S{\isachardoublequoteclose}\
{\isasymLongrightarrow}\
{\isachardoublequoteopen}Sols\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f{\isacharparenright}\ T\ S\ {\isadigit{0}}\ s\ {\isacharequal}\ {\isacharbraceleft}{\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isasymphi}\ t\ s{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ is{\isacharunderscore}monoid{\isacharunderscore }action{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymin}\
S{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}T\ {\isacharequal}\ UNIV{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymphi}\
{\isadigit{0}}\ s\ {\isacharequal}\
s{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isasymphi}\ {\isacharparenleft}t\isactrlsub {\isadigit{1}}\ {\isacharplus}\ t\isactrlsub {\isadigit{2}}{\isacharparenright}\ s\ {\isacharequal}\ {\isasymphi}\ t\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\isasymphi}\ t\isactrlsub {\isadigit{2}}\ s{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent We have not generated a locale for this case, as
the assumptions needed remain unchanged. Locale
$\mathit{picard}{\isacharunderscore}\mathit{lindeloef}$ thus
guarantees the existence of unique solutions for IVPs of
time-dependent systems. Locale
$\mathit{local}{\isacharunderscore}\mathit{flow}$ specialises it to
autonomous systems with Lipschitz continuous vector fields and local
flows. It covers dynamical systems with global flows and thus the
verification of hybrid systems. This provides the basic Isabelle
infrastructure for formalising the concrete semantics for hybrid
systems with Lipschitz continuous vector fields from
Figure~\ref{fig:framework}.
Next we describe our formalisation of the orbits and orbitals from
Section~\ref{sec:generalisation}. These form the basis for our
verification components for continuous vector fields beyond the scope
of Picard-Lindel\"of's theorem, as shown in
Figure~\ref{fig:framework}. Yet we can instantiate all concepts to
settings where (local) flows exist. First, we have formalised the
$G$-guarded orbit $\gamma^X_{G}$ of $X$ along $T$, with
$\isa{down\ T\ t}$ standing for ${\downarrow}t$.
\begin{isabellebody}
\isanewline
\isacommand{definition}\isamarkupfalse%
\ g{\isacharunderscore}orbit\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}real\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ real\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymgamma}{\isachardoublequoteclose}{\isacharparenright}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}{\isasymgamma}\ X\ G\ T\ {\isacharequal}\ {\isasymUnion}{\isacharbraceleft}{\isasymP}\ X\ {\isacharparenleft}down\ T\ t{\isacharparenright}\ {\isacharbar}t{\isachardot}\ {\isasymP}\ X\ {\isacharparenleft}down\ T\ t{\isacharparenright}\ {\isasymsubseteq}\ {\isacharbraceleft}s{\isachardot}\ G\ s{\isacharbraceright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ g{\isacharunderscore}orbit{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}{\isasymgamma}\ X\ G\ T\ {\isacharequal}\ {\isacharbraceleft}X\ t\ {\isacharbar}t{\isachardot}\ t\ {\isasymin}\ T\ {\isasymand}\ {\isacharparenleft}{\isasymforall}{\isasymtau}{\isasymin}down\ T\ t{\isachardot}\ G\ {\isacharparenleft}X\ {\isasymtau}{\isacharparenright}{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent We have also formalised the $G$-guarded orbital of $f$ along
$T$ in $s$ ($\gamma^f_G\, s$) together with Lemma~\ref{P:gorbital}.
\begin{isabellebody}
\isanewline
\isacommand{definition}\isamarkupfalse%
\ g{\isacharunderscore}orbital\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ real\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ real\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcolon}{\isacharcolon}real{\isacharunderscore}normed{\isacharunderscore }vector{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set{\isachardoublequoteclose}\ \isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}g{\isacharunderscore}orbital\ f\ G\ T\ S\ t\isactrlsub {\isadigit{0}}\ s\ {\isacharequal}\ {\isasymUnion}{\isacharbraceleft}{\isasymgamma}\ X\ G\ T\ {\isacharbar}X{\isachardot}\ X\ {\isasymin}\ Sols\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f{\isacharparenright}\ T\ S\ t\isactrlsub {\isadigit{0}}\ s{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ g{\isacharunderscore}orbital{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}g{\isacharunderscore}orbital\ f\ G\ T\ S\ t\isactrlsub {\isadigit{0}}\ s\ {\isacharequal}\ \isanewline
\ \ {\isacharbraceleft}X\ t\ {\isacharbar}t\ X{\isachardot}\ t\ {\isasymin}\ T\ {\isasymand}\ {\isasymP}\ X\ {\isacharparenleft}down\ T\ t{\isacharparenright}\ {\isasymsubseteq}\ {\isacharbraceleft}s{\isachardot}\ G\ s{\isacharbraceright}\ {\isasymand}\ X\ {\isasymin}\ Sols\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f{\isacharparenright}\ T\ S\ t\isactrlsub {\isadigit{0}}\ s\ {\isacharbraceright}{\isachardoublequoteclose}\ \isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent We have shown that these definitions generalise those for
dynamical systems by instantiating them to the parameters
of the locale $\isa{local{\isacharunderscore}flow}$. The $\top$-guarded orbital
of $f$ along $T$ in $s$ then becomes the standard orbit of $s$, and
its $G$-guarded version is the set in Lemma~\ref{P:g-orbit-props}.
\begin{isabellebody}
\isanewline
\isacommand{context}\isamarkupfalse%
\ local{\isacharunderscore}flow\isanewline
\isakeyword{begin}
\isanewline
\isanewline
\isacommand{definition}\isamarkupfalse\ orbit {\isacharcolon}{\isacharcolon} {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set{\isachardoublequoteclose}\
{\isacharparenleft}{\isachardoublequoteopen}{\isasymgamma}\isactrlsup {\isasymphi}{\isachardoublequoteclose}{\isacharparenright}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}{\isasymgamma}\isactrlsup {\isasymphi}\ s\ {\isacharequal}\ g{\isacharunderscore}orbital\ f\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ True{\isacharparenright}\ T\ S\ {\isadigit{0}}\ s{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\
orbit{\isacharunderscore}eq{\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\
{\isachardoublequoteopen}s\ {\isasymin}\ S{\isachardoublequoteclose}
{\isasymLongrightarrow}\
{\isachardoublequoteopen}{\isasymgamma}\isactrlsup {\isasymphi}\ s\
{\isacharequal}\ {\isacharbraceleft}{\isasymphi}\ t\ s {\isacharbar}t{\isachardot}\ t\ {\isasymin}\ T{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ g{\isacharunderscore}orbital{\isacharunderscore}collapses{\isacharcolon}\ {\isachardoublequoteopen}s\ {\isasymin}\
S{\isachardoublequoteclose}
{\isasymLongrightarrow}\
{\isachardoublequoteopen}g{\isacharunderscore}orbital\ f\ G\ T\ S\
{\isadigit{0}}\ s\ {\isacharequal}\ {\isacharbraceleft}{\isasymphi}\
t\ s {\isacharbar}t{\isachardot}\ t\ {\isasymin}\ T\ {\isasymand}\ {\isacharparenleft}{\isasymforall}{\isasymtau}{\isasymin}down\ T\ t{\isachardot}\ G\ {\isacharparenleft}{\isasymphi}\ {\isasymtau}\ s{\isacharparenright}{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isakeyword{end}
\isanewline
\end{isabellebody}
Overall, the set theoretic concepts introduced in
Section~\ref{sec:generalisation} are easily definable in
Isabelle. Similarly, lemmas formalising their properties and relating
them are often proved automatically in one or two lines. Analytical
properties like the existence of derivatives in a region of space or
the uniqueness of solutions for IVPs are harder to prove. Such lemmas
often require long structured proofs with case analyses and explicit
calculations, that is, a considerable amount of user interaction. Yet
most proofs remain at least roughly at the level of textbook
reasoning.
\section{Isabelle Components for Hybrid Programs}\label{sec:isa-wlp}
This section describes the integration of the state transformer and
relational semantics for dynamical systems and Lipschitz-continuous
vector fields from Section~\ref{sec:hybrid-store} and the continuous
vector fields from Section~\ref{sec:generalisation} into the three
verification components for predicate transformers outlined in
Section~\ref{sec:isa-pt} and Figure~\ref{fig:framework}. This
requires formalising hybrid stores and the semantics of evolution
commands for dynamical systems, Lipschitz continuous vector fields
with local flows and continuous vector fields. As explained in
Section~\ref{sec:hybrid-store} and \ref{sec:differential-invariants},
this supports two different work flows using the procedures introduced
in these sections: the first one is for reasoning with (local) flows
and orbits, the second one for reasoning with invariants.
First we explain our formalisation of the hybrid store type
$\mathbb{R}^V$. We use Isabelle's type
${\isacharparenleft}\mathit{real}{\isacharcomma}{\isacharprime}n{\isacharparenright}\
\mathit{vec}$
(abbreviated as $\mathit{real}{\isacharcircum}{\isacharprime}n$) of
real valued vectors of dimension $n$, formalised as the type
${\isacharprime}n\ {\isasymRightarrow}\ \mathit{real}$ of functions
from the finite type ${\isacharprime}n$ into $\mathbb{R}$. This represents
hybrid stores in $\mathbb{R}^V$ with $|V|=n$. Isabelle uses the notation
$s{\isachardollar}i$ for the $i$th coordinate of a vector $s$ and
hence the value of store $s$ at variable $i$. More mathematically,
${\isachardollar}$ is the bijection from
$\mathit{real}{\isacharcircum}{\isacharprime}n$ to
${\isacharprime}n\ {\isasymRightarrow}\ \mathit{real}$. Its inverse is
written using a binder ${\isasymchi}$ that replaces
$\lambda$-abstraction. Thus
${\isasymchi} i{\isachardot}\ s{\isachardollar}i = s$ for any
$s{\isacharcolon}{\isacharcolon}\mathit{real}{\isacharcircum}{\isacharprime}n$
and $({\isasymchi} i{\isachardot}\ x){\isachardollar}i = x$ for
any $x{\isacharcolon}{\isacharcolon}\mathit{real}$.
Our state transformer semantics uses functions of type
$\mathit{real}{\isacharcircum}{\isacharprime}n\ {\isasymRightarrow}\
{\isacharparenleft}\mathit{real}{\isacharcircum}{\isacharprime}n{\isacharparenright}\
\mathit{set}$,
which we abbreviate as
${\isacharparenleft}\mathit{real}{\isacharcircum}{\isacharprime}n{\isacharparenright}\
\mathit{nd}{\isacharunderscore}\mathit{fun}$
(for non-deterministic functions). These are instances of the more
general type
${\isacharprime}a\ \mathit{nd}{\isacharunderscore}\mathit{fun}$ of
nondeterministic endofunctions.
Alternatively, we use relations of type
${\isacharparenleft}\mathit{real}{\isacharcircum}{\isacharprime}n{\isacharparenright}\
\mathit{rel}$,
which are instances of ${\isacharprime}a\ \mathit{rel}$. For both
intermediate semantics we have shown with Isabelle that they form
$\mathsf{MKA}$s, yet we have also integrated them into the two quantalic predicate
transformer semantics in Figure~\ref{fig:framework}.
\begin{isabellebody}
\isanewline
\isacommand{interpretation}\ rel{\isacharunderscore}aka{\isacharcolon}\ antidomain{\isacharunderscore}kleene{\isacharunderscore}algebra
\ Id\ {\isacharbraceleft}{\isacharbraceright}\ {\isacharparenleft}{\isasymunion}{\isacharparenright}\ {\isacharparenleft}{\isacharsemicolon}{\isacharparenright}\ {\isacharparenleft}{\isasymsubseteq}{\isacharparenright}\ {\isacharparenleft}{\isasymsubset}{\isacharparenright}\ rtrancl\ rel{\isacharunderscore}ad\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{instantiation}\isamarkupfalse%
\ nd{\isacharunderscore}fun\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharparenright}\ antidomain{\isacharunderscore}kleene{\isacharunderscore }algebra\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent After these proofs, all statements proved in Isabelle's $\mathsf{MKA}$
components are available for state transformers and relations. We
have formalised $\mathit{wlp}$s for both models, where
${\isasymlceil}-{\isasymrceil}$ ambiguously denotes the isomorphism
between predicates and binary relations or nondeterministic functions.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ wp{\isacharunderscore}rel{\isacharcolon}{\isachardoublequoteopen }\ wp\ R\ {\isasymlceil}P{\isasymrceil}\ {\isacharequal}\ {\isasymlceil}{\isasymlambda}\ x{\isachardot}\ {\isasymforall}\ y{\isachardot}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ R\ {\isasymlongrightarrow}\ P\ y{\isasymrceil}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ wp{\isacharunderscore}nd{\isacharunderscore}fun{\isacharcolon}\ {\isachardoublequoteopen}wp\ F\ {\isasymlceil}P{\isasymrceil}\ {\isacharequal}\ {\isasymlceil}{\isasymlambda}\ x{\isachardot}\ {\isasymforall}\ y{\isachardot}\ y\ {\isasymin}\ {\isacharparenleft}F\ x{\isacharparenright}\ {\isasymlongrightarrow}\ P\ y{\isasymrceil}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
Alternatively, we use the categorical forward diamond operator
$\mathit{fb}\isactrlsub {\isasymF}$ for Kleisli arrows of type
$F{\isacharcolon}{\isacharcolon} {\isacharprime}a\
{\isasymRightarrow}\ {\isacharprime}b\ \mathit{set}$ described in Section~\ref{sec:pt-monad},
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ffb{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}fb\isactrlsub {\isasymF}\ F\ X\ {\isacharequal}\ {\isacharbraceleft}x{\isachardot}\ {\isasymforall}y{\isachardot}\ y\ {\isasymin}\ F\ x\ {\isasymlongrightarrow}\ y\ {\isasymin}\ X{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent or its relational counterpart \isa{fb\isactrlsub {\isasymR}}.
We now switch to the categorical approach to predicate transformers
based on state transformers and the Kleisi monad of the powerset
functor, as a preliminary $\mathsf{MKA}$-based one with relations has already
been described elsewhere~\cite{MuniveS18}. Apart from typing and some
minor syntactic differences, the other approaches---predicate
transformers based on $\mathsf{MKA}$ and quantales, and an intermediate
relational semantics for these---yield analogous results and are
equally suitable for verification.
The state and predicate transformer semantics of assignment commands
is based on store update functions, as described in
Section~\ref{sec:discrete-store}. For hybrid programs, it must be
adapted to type ${\isacharprime}a{\isacharcircum}{\isacharprime}n$.
\begin{isabellebody}
\isanewline
\isacommand{definition}\isamarkupfalse%
\ vec{\isacharunderscore}upd\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isacharcircum}{\isacharprime}n\ {\isasymRightarrow}\ {\isacharprime}n\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharcircum}{\isacharprime}n{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}vec{\isacharunderscore}upd\ s\ i\ a\ {\isacharequal}\ {\isacharparenleft}{\isasymchi}\ j{\isachardot}\ {\isacharparenleft}{\isacharparenleft}{\isacharparenleft}{\isachardollar}{\isacharparenright}\ s{\isacharparenright}{\isacharparenleft}i\ {\isacharcolon}{\isacharequal}\ a{\isacharparenright}{\isacharparenright}\ j{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{definition}\isamarkupfalse%
\ assign\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}n\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcircum}{\isacharprime}n\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isacharcircum}{\isacharprime}n\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcircum}{\isacharprime}n{\isacharparenright}\ set{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}{\isacharparenleft}{\isadigit{2}}{\isacharunderscore}\ {\isacharcolon}{\isacharcolon}{\isacharequal}\ {\isacharunderscore}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{7}}{\isadigit{0}}{\isacharcomma}\ {\isadigit{6}}{\isadigit{5}}{\isacharbrackright}\ {\isadigit{6}}{\isadigit{1}}{\isacharparenright}\ \isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}{\isacharequal}\ e{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isacharbraceleft}vec{\isacharunderscore}upd\ s\ x\ {\isacharparenleft}e\ s{\isacharparenright}{\isacharbraceright}{\isacharparenright}{\isachardoublequoteclose}\ \isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ffb{\isacharunderscore}assign{\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}fb\isactrlsub {\isasymF}\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}{\isacharequal}\ e{\isacharparenright}\ Q\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isacharparenleft}{\isasymchi}\ j{\isachardot}\ {\isacharparenleft}{\isacharparenleft}{\isacharparenleft}{\isachardollar}{\isacharparenright}\ s{\isacharparenright}{\isacharparenleft}x\ {\isacharcolon}{\isacharequal}\ {\isacharparenleft}e\ s{\isacharparenright}{\isacharparenright}{\isacharparenright}\ j{\isacharparenright}\ {\isasymin}\ Q{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent We write
${\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}{\isacharequal}\
e{\isacharparenright}$
for the semantic illusion of assignment commands, as Isabelle uses
$f{\isacharparenleft}i\ {\isacharcolon}{\isacharequal}\
a{\isacharparenright}$
for function update $f[i\mapsto a]$. Lemma
$\mathit{ffb}{\isacharunderscore}\mathit{assign}$ is then a direct
consequence of $\mathit{ffb}{\isacharunderscore}\mathit{eq}$, and it
coincides with~(\ref{eq:wlp-asgn}) in Section~\ref{sec:discrete-store}
up to minor syntactic differences. Similarly, $\mathit{wlp}$s for the control
structure commands of hybrid programs (equations~\ref{eq:wlp-seq},
\ref{eq:wlp-cond} and~\ref{eq:wlp-star}) are easily derivable.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ffb{\isacharunderscore}kcomp{\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}fb\isactrlsub {\isasymF}\ {\isacharparenleft}G\ {\isacharsemicolon}\ F{\isacharparenright}\ P\ {\isacharequal}\ fb\isactrlsub {\isasymF}\ G\ {\isacharparenleft}fb\isactrlsub {\isasymF}\ F\ P{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ffb{\isacharunderscore}if{\isacharunderscore}then{\isacharunderscore }else{\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}fb\isactrlsub {\isasymF}\ {\isacharparenleft}IF\ T\ THEN\ X\ ELSE\ Y{\isacharparenright}\ Q\ {\isacharequal}\isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ T\ s\ {\isasymlongrightarrow}\ s\ {\isasymin}\ fb\isactrlsub {\isasymF}\ X\ Q{\isacharbraceright}\ {\isasyminter}\ {\isacharbraceleft}s{\isachardot}\ {\isasymnot}\ T\ s\ {\isasymlongrightarrow}\ s\ {\isasymin}\ fb\isactrlsub {\isasymF}\ Y\ Q{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ffb{\isacharunderscore}loopI{\isacharcolon}\ {\isachardoublequoteopen}P\ {\isasymle}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ \ {\isasymLongrightarrow}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymle}\ Q\ {\isasymLongrightarrow}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ F\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymLongrightarrow} P\ {\isasymle}\ fb\isactrlsub {\isasymF}\ {\isacharparenleft}LOOP\ F\ INV\ I{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent In these lemmas, ${\isacharsemicolon}$ is syntactic sugar for the
Kleisli composition $\circ_K$, and \isa{LOOP} stands for the Kleene
star for state transformers with its annotated loop-invariant after
the keyword \isa{INV}, along the lines of Section~\ref{sec:mka-pt}.
As in Section~\ref{sec:generalisation}, the general semantics of
evolution commands for continuous vector fields is given by
$G$-guarded orbitals of $f$ along $T$. We have formalised the $\mathit{wlp}$s
in Proposition~\ref{P:wlpprop-gen}, and a specialisation to local
flows in the context of our locale $\isa{local{\isacharunderscore}flow}$
given by Lemma~\ref{P:wlpprop-var} (equation~(\ref{eq:wlp-evl})).
\begin{isabellebody}
\isanewline\isacommand{notation}\isamarkupfalse%
\ g{\isacharunderscore}orbital\ {\isacharparenleft}{\isachardoublequoteopen}{\isacharparenleft}{\isadigit{1}}x{\isasymacute}{\isacharequal}{\isacharunderscore}\ {\isacharampersand}\ {\isacharunderscore}\ on\ {\isacharunderscore}\ {\isacharunderscore}\ {\isacharat}\ {\isacharunderscore}{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\isanewline
\isacommand{lemma}\isamarkupfalse%
\
ffb{\isacharunderscore}g{\isacharunderscore}orbital{\isacharcolon} {\isachardoublequoteopen}fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G\ on\ T\ S\ {\isacharat}\ t\isactrlsub {\isadigit{0}}{\isacharparenright}\ Q\ {\isacharequal}\ \isanewline
\ \ \ \ {\isacharbraceleft}s{\isachardot}\ {\isasymforall}X{\isasymin}Sols\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ f{\isacharparenright}\ T\ S\ t\isactrlsub {\isadigit{0}}\ s{\isachardot}\ {\isasymforall}t{\isasymin}T{\isachardot}\ {\isacharparenleft}{\isasymforall}{\isasymtau}{\isasymin}down\ T\ t{\isachardot}\ G\ {\isacharparenleft}X\ {\isasymtau}{\isacharparenright}{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}X\ t{\isacharparenright}\ {\isasymin}\ Q{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharparenleft}\isacommand{in}\
local{\isacharunderscore}flow{\isacharparenright}\
ffb{\isacharunderscore}g{\isacharunderscore}ode{\isacharcolon} {\isachardoublequoteopen}fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\
{\isacharampersand}\ G\ on\ T\ S\ {\isacharat}\
{\isadigit{0}}{\isacharparenright}\ Q\ {\isacharequal}\isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymin}\ S\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymforall}t{\isasymin}T{\isachardot}\ {\isacharparenleft}{\isasymforall}{\isasymtau}{\isasymin}down\ T\ t{\isachardot}\ G\ {\isacharparenleft}{\isasymphi}\ {\isasymtau}\ s{\isacharparenright}{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymphi}\ t\ s{\isacharparenright}\ {\isasymin}\ Q{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent As Lemma $\isa{ffb-g-ode}$ is defined in locale
$\isa{local-flow}$, users are required to check the conditions of the
Picard-Lindel{\"o}f theorem to access this locale and certify that
$\varphi$ is indeed a solution of the IVP.
Finally, we describe our component for reasoning with differential
invariants in the general setting of continuous vector fields. We start with
their definition and a basic property from
Proposition~\ref{P:inv-prop}.
\begin{isabellebody}
\isanewline
\isacommand{definition}\isamarkupfalse%
\ diff{\isacharunderscore}invariant\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharparenleft}{\isacharprime}a{\isacharcolon}{\isacharcolon}real{\isacharunderscore}normed{\isacharunderscore }vector{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ real\ set\ {\isasymRightarrow}\ \isanewline
\ \ {\isacharprime}a\ set\ {\isasymRightarrow}\ real\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}diff{\isacharunderscore}invariant\ I\ f\ T\ S\ t\isactrlsub {\isadigit{0}}\ G\ {\isacharequal}\ {\isacharparenleft}{\isacharparenleft}{\isacharparenleft}g{\isacharunderscore}orbital\ f\ G\ T\ S\ t\isactrlsub {\isadigit{0}}{\isacharparenright}\isactrlsup {\isasymdagger}{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymsubseteq}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ffb{\isacharunderscore}diff{\isacharunderscore}inv{\isacharcolon}\ {\isachardoublequoteopen}diff{\isacharunderscore}invariant\ I\ f\
T\ S\ t\isactrlsub {\isadigit{0}}\ G\ {\isacharequal}\
{\isacharparenleft}{\isacharbraceleft}s{\isachardot}\ I\
s{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G\ on\ T\ S\ {\isacharat}\ t\isactrlsub {\isadigit{0}}{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}{\isacharparenright}\ {\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
We have formalised the most important rules for reasoning with
differential invariants, including those for the procedure of
Section~\ref{sec:differential-invariants} via
Corollary~\ref{P:wlpprop-gen2} and Lemmas~\ref{P:inv-lemma}
and~\ref{P:invrules}. The formalisation of the first two is
straightforward. We have proved the clauses of~\ref{P:invrules} in
various lemmas, and bundled them under the name
$\mathit{diff}{\isacharunderscore}\mathit{invariant}{\isacharunderscore}\mathit{rules}$. We
show one of these clauses as an example.
\begin{isabellebody}
\isanewline
\isacommand{named{\isacharunderscore}theorems}\isamarkupfalse%
\ diff{\isacharunderscore}invariant{\isacharunderscore}rules\ {\isachardoublequoteopen}compilation\ of\ rules\ for\ differential\ invariants{\isachardot}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}diff{\isacharunderscore}invariant{\isacharunderscore }rules{\isacharbrackright}{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\
{\isachardoublequoteopen}is{\isacharunderscore}interval\
T{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{and}\ {\isachardoublequoteopen}t\isactrlsub {\isadigit{0}}\ {\isasymin}\ T{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isasymforall}X{\isachardot}\ {\isacharparenleft}D\ X\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}{\isasymtau}{\isachardot}\ f\ {\isacharparenleft}X\ {\isasymtau}{\isacharparenright}{\isacharparenright}\ on\ T{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}D\ {\isacharparenleft}{\isasymlambda}{\isasymtau}{\isachardot}\ {\isasymmu}\ {\isacharparenleft}X\ {\isasymtau}{\isacharparenright}\ {\isacharminus}\ {\isasymnu}\ {\isacharparenleft}X\ {\isasymtau}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenleft}{\isacharasterisk}\isactrlsub R{\isacharparenright}\ {\isadigit{0}}{\isacharparenright}\ on\ T{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}diff{\isacharunderscore}invariant\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isasymmu}\ s\ {\isacharequal}\ {\isasymnu}\ s{\isacharparenright}\ f\ T\ S\ t\isactrlsub {\isadigit{0}}\ G{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ ffb{\isacharunderscore}g{\isacharunderscore}odei{\isacharcolon}\ {\isachardoublequoteopen}P\ {\isasymle}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymLongrightarrow}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal}\ f\ {\isacharampersand}\ G\ on\ T\ S\ {\isacharat}\ t\isactrlsub {\isadigit{0}}{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymLongrightarrow}\ \isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ I\ s\ {\isasymand}\ G\ s{\isacharbraceright}\ {\isasymle}\ Q\ {\isasymLongrightarrow}\ P\ {\isasymle}\ fb\isactrlsub {\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal}\ f\ {\isacharampersand}\ G\ on\ T\ S\ {\isacharat}\ t\isactrlsub {\isadigit{0}}\ DINV\ I{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent Lemma
\isa{ffb{\isacharunderscore}g{\isacharunderscore}odei{\isacharunderscore
}inv} completes the procedure of
Section~\ref{sec:differential-invariants} by formalising step 2, which
annotates invariants in evolution commands, following the approach
outlined for loops and general commands in $\mathsf{MKA}$ at the end of
Section~\ref{sec:mka-pt}. With Isabelle, we use the \isa{DINV} keyword.
The two procedures for proving partial correctness specifications with
evolution commands require users to discharge proof obligations for
derivatives. In the case of flows, these must be solutions for vector
fields; in the case of differential invariants, the procedure of
Section~\ref{sec:differential-invariants} requires proving the
assumptions of Lemma~\ref{P:invrules}. To increase proof automation
when reasoning about derivatives, we have bundled several derivative
properties under the name \isa{poly{\isacharunderscore}derivatives} as
a tactic.
\begin{isabellebody}
\isanewline
\isacommand{named{\isacharunderscore}theorems}\isamarkupfalse%
\ poly{\isacharunderscore}derivatives\ {\isachardoublequoteopen}compilation\ of\ optimised\ miscellaneous\ derivative\ rules{\isachardot}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{declare}\isamarkupfalse%
\ has{\isacharunderscore}vderiv{\isacharunderscore}on{\isacharunderscore}const\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright }\isanewline
\ \ \ \ \isakeyword{and}\ has{\isacharunderscore}vderiv{\isacharunderscore}on{\isacharunderscore}id\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright }\isanewline
\ \ \ \ \isakeyword{and}\ derivative{\isacharunderscore}intros{\isacharparenleft}{\isadigit{1}}{\isadigit{9}}{\isadigit{1}}{\isacharparenright}\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright }\isanewline
\ \ \ \ \isakeyword{and}\ derivative{\isacharunderscore}intros{\isacharparenleft}{\isadigit{1}}{\isadigit{9}}{\isadigit{2}}{\isacharparenright}\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright }\isanewline
\ \ \ \ \isakeyword{and}\ derivative{\isacharunderscore}intros{\isacharparenleft}{\isadigit{1}}{\isadigit{9}}{\isadigit{4}}{\isacharparenright}\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright }\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}D\ f\ {\isacharequal}\ f{\isacharprime}\ on\ T\ {\isasymLongrightarrow}\ g\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isacharminus}\ f{\isacharprime}\ t{\isacharparenright}\ {\isasymLongrightarrow}\ D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isacharminus}\ f\ t{\isacharparenright}\ {\isacharequal}\ g\ on\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\
{\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright}{\isacharcolon}\
{\isachardoublequoteopen}{\isacharparenleft}a{\isacharcolon}{\isacharcolon}real{\isacharparenright}\
{\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ D\ f\
{\isacharequal}\ f{\isacharprime}\ on\ T\ {\isasymLongrightarrow}\ g\
{\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\
{\isacharparenleft}f{\isacharprime}\
t{\isacharparenright}{\isacharslash}a{\isacharparenright}\
{\isasymLongrightarrow} D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharslash}a{\isacharparenright}\ {\isacharequal}\ g\ on\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}n\ {\isasymge}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ D\ {\isacharparenleft}f{\isacharcolon}{\isacharcolon}real\ {\isasymRightarrow}\ real{\isacharparenright}\ {\isacharequal}\ f{\isacharprime}\ on\ T\ {\isasymLongrightarrow}\isanewline
\ \ g\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ n\ {\isacharasterisk}\ {\isacharparenleft}f{\isacharprime}\ t{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharcircum}{\isacharparenleft}n{\isacharminus}{\isadigit{1}}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharcircum}n{\isacharparenright}\ {\isacharequal}\ g\ on\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}D\ {\isacharparenleft}f{\isacharcolon}{\isacharcolon}real\ {\isasymRightarrow}\ real{\isacharparenright}\ {\isacharequal}\ f{\isacharprime}\ on\ T\ {\isasymLongrightarrow}\isanewline
\ \ g\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isacharminus}\ {\isacharparenleft}f{\isacharprime}\ t{\isacharparenright}\ {\isacharasterisk}\ sin\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ cos\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ g\ on\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}D\ {\isacharparenleft}f{\isacharcolon}{\isacharcolon}real\ {\isasymRightarrow}\ real{\isacharparenright}\ {\isacharequal}\ f{\isacharprime}\ on\ T\ {\isasymLongrightarrow}\ g\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isacharparenleft}f{\isacharprime}\ t{\isacharparenright}\ {\isacharasterisk}\ cos\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
\ \ D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ sin\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ g\ on\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}poly{\isacharunderscore}derivatives{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}D\ {\isacharparenleft}f{\isacharcolon}{\isacharcolon}real\ {\isasymRightarrow}\ real{\isacharparenright}\ {\isacharequal}\ f{\isacharprime}\ on\ T\ {\isasymLongrightarrow}\ g\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isacharparenleft}f{\isacharprime}\ t{\isacharparenright}\ {\isacharasterisk}\ exp\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
\ \ D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ exp\ {\isacharparenleft}f\ t{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ g\ on\ T{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
With this basic tactic, Isabelle can apply rules iteratively and
determine, for pairs of functions, if one is a derivative of the
other. In many cases this is fully automatic. The following lemma
shows an example that involves a mix of polynomials and transcendental
functions beyond differential fields.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}c\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ D\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ a{\isadigit{5}}\ {\isacharasterisk}\ t{\isacharcircum}{\isadigit{5}}\ {\isacharplus}\ a{\isadigit{3}}\ {\isacharasterisk}\ {\isacharparenleft}t{\isacharcircum}{\isadigit{3}}\ {\isacharslash}\ c{\isacharparenright}\ {\isacharminus}\ a{\isadigit{2}}\ {\isacharasterisk}\ exp\ {\isacharparenleft}t{\isacharcircum}{\isadigit{2}}{\isacharparenright}\ {\isacharplus}\ a{\isadigit{1}}\ {\isacharasterisk}\ cos\ t\ {\isacharplus}\ a{\isadigit{0}}{\isacharparenright}\isanewline
\ \ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isadigit{5}}\ {\isacharasterisk}\ a{\isadigit{5}}\ {\isacharasterisk}\ t{\isacharcircum}{\isadigit{4}}\ {\isacharplus}\ {\isadigit{3}}\ {\isacharasterisk}\ a{\isadigit{3}}\ {\isacharasterisk}\ {\isacharparenleft}t{\isacharcircum}{\isadigit{2}}\ {\isacharslash}\ c{\isacharparenright}\ {\isacharminus}\ {\isadigit{2}}\ {\isacharasterisk}\ a{\isadigit{2}}\ {\isacharasterisk}\ t\ {\isacharasterisk}\ exp\ {\isacharparenleft}t{\isacharcircum}{\isadigit{2}}{\isacharparenright}\ {\isacharminus}\ a{\isadigit{1}}\ {\isacharasterisk}\ sin\ t{\isacharparenright}\ on\ T{\isachardoublequoteclose}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
{\isacharparenleft}auto\ intro{\isacharbang}{\isacharcolon}\ poly{\isacharunderscore}derivatives{\isacharparenright}\isanewline
\end{isabellebody}
The formalisation of more advanced heuristics, or even decision
procedures for such classes of functions, is beyond the scope of this
article.
The complete Isabelle formalisation, including the other two predicate
transformer algebras and the relational semantics, can be found in the
Archive of Formal Proofs~\cite{afp:hybrid}, including extensive proof
documents.
We briefly reflect on our experience with the Isabelle formalisation
of our framework. $\mathsf{MKA}$, its relational model and the concrete
relational semantics for traditional while-programs are so far the
most developed and versatile starting point for our hybrid systems
verification components. The full formalisation of a rudimentary Hoare
logic component for this setting using a generalised Kleene algebra
from Isabelle's main libraries fits on two A4 pages~\cite{Struth18}; a
similar development with a predicate transformer component seems
plausible. Our current standalone $\mathsf{MKA}$-based verification component
for traditional while programs fills about seven A4 pages. For hybrid
programs, in theory, only a concrete semantics for hybrid programs
needs to be plugged in as a replacement of the semantics described in
Section~\ref{sec:discrete-store}. In practice, however, Isabelle's
instantiations often make theory hierarchies non-compositional as each
type can only be instantiated in one way. We faced such a clash of
instances between Isabelle's Kleene algebra and analysis hierarchies
and hence had to customise the former for our purposes.
Replacing the intermediate relational semantics by state transformers
required some background work, simply because the former are well
supported by Isabelle whereas the latter are new. In theory, it should
be possible to propagate theorems automatically along the isomorphisms
between these semantics like for type classes, locales and their
instantiations and interpretation. Yet in practice, Isabelle provides
no comparable mechanism to achieve this outside of locales.
The categorical approach to predicate transformer quantaloids is more
complex---both conceptually and from a formalisation point of
view---than the $\mathsf{MKA}$ based one, in particular when state
transformers are integrated via the powerset monad. At the level of
verification conditions generation, however, there are almost no
differences. Once again a stripped down component can be generated
that just suffices for verification condition generation. Relative to
Isabelle's main libraries it fills merely four
pages~\cite{afp:hybrid}. Working with quantales instead of quantaloids
might seem mathematically simpler, but with Isabelle it is actually
more tedious, as subtypes for endofunctions need to be created.
In sum, for simple verification tasks, the lightweight stripped down
predicate transformer algebras obtained from $\mathsf{MKA}$ or quantaloids
seem preferable; for more complex program transformations or
refinements, the integration into the full $\mathsf{MKA}$ hierarchy or
categorical predicate transformer component is certainly beneficial.
\section{Isabelle Support for $\mathsf{d}\mathcal{L}$-Style Reasoning}\label{sec:isa-dL}
This section lists our formalisation of semantic variants of the
most important axioms and inference rules of $\mathsf{d}\mathcal{L}$ in Isabelle
outlined in Section~\ref{sec:dL}. It covers all
three predicate transformer semantics as well as the relational and
state transformer model. Once again, we only show state
transformers in the categorical approach.
We have formalised a generalised version of the $\mathsf{d}\mathcal{L}$-rules with
parameters $T$, $S$ and $t_0$. We can easily instantiate them to
$\mathbb{R}$, $\mathbb{R}^V$ and $0$, respectively. This enables users to
perform verification proofs in the style of $\mathsf{d}\mathcal{L}$. First we show our
formalisations of (\ref{eq:DS}) and (\ref{eq:dSolve}).
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ DS{\isacharcolon}\ \isanewline
\ \ \isakeyword{fixes}\ c{\isacharcolon}{\isacharcolon}{\isachardoublequoteopen}{\isacharprime}a{\isacharcolon}{\isacharcolon}{\isacharbraceleft}heine{\isacharunderscore}borel{\isacharcomma}\ banach{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fb\isactrlsub
{\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal} {\isacharparenleft}{\isasymlambda}s{\isachardot}\ c{\isacharparenright}\ {\isacharampersand}\ G{\isacharparenright}\ Q\ {\isacharequal}\ {\isacharbraceleft}x{\isachardot}\ {\isasymforall}t{\isachardot}\ {\isacharparenleft}{\isasymforall}{\isasymtau}{\isasymle}t{\isachardot}\ G\ {\isacharparenleft}x{\isacharplus}{\isasymtau}\ {\isacharasterisk}\isactrlsub R\ c{\isacharparenright}{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharplus}t\ {\isacharasterisk}\isactrlsub R\ c{\isacharparenright}\ {\isasymin}\ Q{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ solve{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}local{\isacharunderscore}flow\ f\ UNIV\ UNIV\ {\isasymphi}{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isasymforall}s{\isachardot}\ s\ {\isasymin}\ P\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}{\isasymforall}{\isasymtau}{\isasymle}t{\isachardot}\ G\ {\isacharparenleft}{\isasymphi}\ {\isasymtau}\ s{\isacharparenright}{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymphi}\ t\ s{\isacharparenright}\ {\isasymin}\ Q{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}P\ {\isasymle}\
fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent Next we list semantic variants of the five $\mathsf{d}\mathcal{L}$ axioms and
inference rules for reasoning with differential invariants discussed
in Section~\ref{sec:dL}.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ DW{\isacharcolon}\ {\isachardoublequoteopen}fb\isactrlsub
{\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal}\ f\
{\isacharampersand}\ G{\isacharparenright}\ Q\ {\isacharequal}\
fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ G\ s\ {\isasymlongrightarrow}\ s\ {\isasymin}\ Q{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ dW{\isacharcolon}\
{\isachardoublequoteopen}{\isacharbraceleft}s{\isachardot}\ G\
s{\isacharbraceright}\ {\isasymle}\ Q\ {\isasymLongrightarrow}\ P\
{\isasymle}\ fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ DC{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}fb\isactrlsub
{\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ C\ s{\isacharbraceright}\ {\isacharequal}\ UNIV{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fb\isactrlsub
{\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal}\ f\
{\isacharampersand}\ G{\isacharparenright}\ Q\ {\isacharequal}\
fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ G\ s\ {\isasymand}\ C\ s{\isacharparenright}{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ dC{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}P\ {\isasymle}\
fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ C\ s{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}P\ {\isasymle}\
fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ G\ s\ {\isasymand}\ C\ s{\isacharparenright}{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}P\ {\isasymle}\
fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\isacommand{lemma}\isamarkupfalse%
\ dI{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}P\ {\isasymle}\
{\isacharbraceleft}s{\isachardot}\ I\
s{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\
{\isachardoublequoteopen}diff{\isacharunderscore}invariant\ I\ f\
UNIV\ UNIV\ {\isadigit{0}}\ G{\isachardoublequoteclose}\isanewline
\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharbraceleft}s{\isachardot}\ I\ s{\isacharbraceright}\ {\isasymle}\ Q{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}P\ {\isasymle}\
fb\isactrlsub {\isasymF}\
{\isacharparenleft}x{\isasymacute}{\isacharequal} f\ {\isacharampersand}\ G{\isacharparenright}\ Q{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\section{Verification Examples}\label{sec:examples}
This section explains the formalisation of the bouncing ball examples
from Section~\ref{sec:hybrid-store} and
\ref{sec:differential-invariants} with Isabelle; and we add two
verification examples using a simple circular pendulum. All four
examples use Isabelle's type $\isa{\isadigit{2}}$ of two elements. It
is used to denote the set of variables $V$ of hybrid programs over the
state space $\mathbb{R}^V$ for $|V|=2$. We write
\isa{{\isadigit{0}}{\isacharcolon}{\isacharcolon}{\isadigit{2}}} and
\isa{{\isadigit{1}}{\isacharcolon}{\isacharcolon}{\isadigit{2}}} for
the two variables and their type.
\begin{example}[Bouncing Ball via Flow]\label{ex:bouncing-ball-flow}
First, we formalise Example~\ref{ex:ball} with our verification
components for flows. We write
\isa{{\isadigit{0}}{\isacharcolon}{\isacharcolon}{\isadigit{2}}} for
the ball's position starting from height $h$ and
\isa{{\isadigit{1}}{\isacharcolon}{\isacharcolon}{\isadigit{2}}} for
its velocity, and $s\, {\isachardollar}\, {\isadigit{0}}$
$s\, {\isachardollar}\, {\isadigit{1}}$ for $s_x$ and $s_v$. We
formalise the vector field $f\, (s_x,s_v)^T = (s_v,-g)^T$ from
example~\ref{ex:ball} as follows.
\begin{isabellebody}
\isanewline
\isacommand{abbreviation}\isamarkupfalse%
\ fball\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ real{\isacharcircum}{\isadigit{2}}\ {\isasymRightarrow}\ real{\isacharcircum}{\isadigit{2}}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}f{\isachardoublequoteclose}{\isacharparenright}\ \isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}f\ g\ s\ {\isasymequiv}\ {\isacharparenleft}{\isasymchi}\ i{\isachardot}\ if\ i{\isacharequal}{\isadigit{0}}\ then\ s{\isachardollar}{\isadigit{1}}\ else\ g{\isacharparenright}{\isachardoublequoteclose}\isanewline
\end{isabellebody}
\noindent We can now state the partial correctness specification for
the bouncing ball in Isabelle, where the loop invariant $I$ is
that of Section~\ref{sec:hybrid-store}, but written
slightly differently to enhance proof automation.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ bouncing{\isacharunderscore}ball{\isacharcolon}\ {\isachardoublequoteopen}g\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ h\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ \isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharequal}\ h\ {\isasymand}\ s{\isachardollar}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{0}}{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ \isanewline
\ \ {\isacharparenleft}LOOP\ {\isacharparenleft}\isanewline
\ \ \ \ {\isacharparenleft}x{\isasymacute}{\isacharequal}{\isacharparenleft}f\ g{\isacharparenright}\ {\isacharampersand}\ {\isacharparenleft}{\isasymlambda}\ s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}\ {\isacharsemicolon}\ \isanewline
\ \ \ \ {\isacharparenleft}IF\ {\isacharparenleft}{\isasymlambda}\ s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ THEN\ {\isacharparenleft}{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}{\isacharequal}\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isacharminus}\ s{\isachardollar}{\isadigit{1}}{\isacharparenright}{\isacharparenright}\ ELSE\ skip{\isacharparenright}{\isacharparenright}\isanewline
\ \ INV\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isadigit{0}}\ {\isasymle}\ s{\isachardollar}{\isadigit{0}}\ {\isasymand}{\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharminus}\ {\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ h\ {\isacharminus}\ s{\isachardollar}{\isadigit{1}}\ {\isasymcdot}\ s{\isachardollar}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}\isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ {\isadigit{0}}\ {\isasymle}\
s{\isachardollar}{\isadigit{0}}\ {\isasymand}\
s{\isachardollar}{\isadigit{0}}\ {\isasymle}\
h{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\end{isabellebody}
\noindent The proof of this lemma is shown below. It follows that in
Example~\ref{ex:ball}, but requires some intermediate lemmas. For
example, if we first apply rule \isa{ffb{\isacharunderscore}loopI}
(\ref{eq:wlp-star}), the subgoals $P\leq I$ and $I\leq Q$, for
$P= (\lambda s.\ s_x = h\land s_v = 0)$ and
$Q = (\lambda s.\ 0\leq s_x\leq h)$, need to be proven. They can be
discharged automatically after supplying some lemmas about real
arithmetic, which have been bundled under the name
\isa{bb{\isacharunderscore}real{\isacharunderscore}arith}. We show one
of them below to give an impression.
\begin{isabellebody}
\isanewline
\isacommand{named{\isacharunderscore}theorems}\isamarkupfalse%
\ bb{\isacharunderscore}real{\isacharunderscore}arith\ {\isachardoublequoteopen}real\ arithmetic\ properties\ for\ the\ bouncing\ ball{\isachardot}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}bb{\isacharunderscore}real{\isacharunderscore }arith{\isacharbrackright}{\isacharcolon}\
{\isachardoublequoteopen}{\isadigit{0}}\ {\isachargreater}\
g{\isachardoublequoteclose}
{\isasymLongrightarrow}\ {\isadigit{2}}\
{\isasymcdot}\ g\ {\isasymcdot}\ x\ {\isacharminus}\ {\isadigit{2}}\
{\isasymcdot}\ g\ {\isasymcdot}\ h\ {\isacharequal}\ v\ {\isasymcdot}\
{\isasymLongrightarrow}\
{\isachardoublequoteopen}{\isacharparenleft}x{\isacharcolon}{\isacharcolon}real{\isacharparenright}\ {\isasymle}\ h{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent These properties depend on distributivity and commutativity
properties that Isabelle cannot simplify immediately. As we are not
working within a well defined language, such as differential rings or
fields, we have not attempted to automate them any further, so that
proofs require some user interaction.
The remaining rules, that is, \isa{ffb{\isacharunderscore}kcomp}
(\ref{eq:wlp-seq}),
\isa{ffb{\isacharunderscore}if{\isacharunderscore}then{\isacharunderscore}else}
(\ref{eq:wlp-cond}), and \isa{ffb{\isacharunderscore}assign}
(\ref{eq:wlp-asgn}), have been added to Isabelle's automatic proof
tools. It then remains to compute the $\mathit{wlp}$ for the evolution command
of the bouncing ball. To use
\isa{local{\isacharunderscore}flow{\isachardot
}ffb{\isacharunderscore}g{\isacharunderscore}ode}
(\ref{eq:wlp-evl}), we follow the procedure in
Section~\ref{sec:hybrid-store}. We need to check that the vector field
is Lipschitz continuous, supply the local flow as in
Example~\ref{ex:ball}, and check that it solves the IVP and satisfies
the flow conditions.
\begin{isabellebody}
\isanewline
\isacommand{abbreviation}\isamarkupfalse%
\ ball{\isacharunderscore}flow\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ real\ {\isasymRightarrow}\ real{\isacharcircum}{\isadigit{2}}\ {\isasymRightarrow}\ real{\isacharcircum}{\isadigit{2}}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymphi}{\isachardoublequoteclose}{\isacharparenright}\ \isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}{\isasymphi}\ g\ t\ s\ {\isasymequiv}\ {\isacharparenleft}{\isasymchi}\ i{\isachardot}\ if\ i{\isacharequal}{\isadigit{0}}\ then\ g\ {\isasymcdot}\ t\ {\isacharcircum}\ {\isadigit{2}}{\isacharslash}{\isadigit{2}}\ {\isacharplus}\ s{\isachardollar}{\isadigit{1}}\ {\isasymcdot}\ t\ {\isacharplus}\ s{\isachardollar}{\isadigit{0}}\ else\ g\ {\isasymcdot}\ t\ {\isacharplus}\ s{\isachardollar}{\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ local{\isacharunderscore}flow{\isacharunderscore}ball{\isacharcolon}\ {\isachardoublequoteopen}local{\isacharunderscore}flow\ {\isacharparenleft}f\ g{\isacharparenright}\ UNIV\ UNIV\ {\isacharparenleft}{\isasymphi}\ g{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent The arithmetic computations with real numbers at the end
of Example~\ref{ex:ball} are then discharged automatically by adding
the rules in \isa{bb{\isacharunderscore}real{\isacharunderscore}arith}
to Isabelle's automatic tools. The resulting two-line proof of the
bouncing ball is shown below.
\begin{isabellebody}\isanewline
\isacommand{apply}\isamarkupfalse%
{\isacharparenleft}rule\ wp{\isacharunderscore}loopI{\isacharcomma}\ simp{\isacharunderscore}all\ add{\isacharcolon}\ local{\isacharunderscore}flow{\isachardot}wp{\isacharunderscore }g{\isacharunderscore}ode{\isacharbrackleft}OF\ local{\isacharunderscore}flow{\isacharunderscore}ball{\isacharbrackright}{\isacharparenright}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
\ {\isacharparenleft}auto\ simp{\isacharcolon}\ bb{\isacharunderscore}real{\isacharunderscore }arith{\isacharparenright}\isanewline
\end{isabellebody}
Overall, the verification proof covers less than a page and a half in
the proof document---and this is mainly due to the few arithmetic
calculations that require user interaction. All other proofs make
heavy use of Isabelle's simplifiers and are by and large automatic.
\qed
\end{example}
\begin{example}[Bouncing Ball via Invariant]\label{ex:bouncing-ball-inv}
This example formalises the invariant-based proof from
Example~\ref{ex:ball-inv}. The correctness specification changes in
that we annotate the differential invariant ab initio.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ bouncing{\isacharunderscore}ball{\isacharunderscore }invariants{\isacharcolon}\ {\isachardoublequoteopen}g\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ h\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ \isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharequal}\ h\ {\isasymand}\ s{\isachardollar}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{0}}{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ \isanewline
\ \ {\isacharparenleft}LOOP\ {\isacharparenleft}\isanewline
\ \ \ \ {\isacharparenleft}x{\isasymacute}{\isacharequal}{\isacharparenleft}f\ g{\isacharparenright}\ {\isacharampersand}\ {\isacharparenleft}{\isasymlambda}\ s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}\ DINV\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharminus}\ {\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ h\ {\isacharminus}\ s{\isachardollar}{\isadigit{1}}\ {\isasymcdot}\ s{\isachardollar}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}\ {\isacharsemicolon}\ \isanewline
\ \ \ \ {\isacharparenleft}IF\ {\isacharparenleft}{\isasymlambda}\ s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ THEN\ {\isacharparenleft}{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}{\isacharequal}\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isacharminus}\ s{\isachardollar}{\isadigit{1}}{\isacharparenright}{\isacharparenright}\ ELSE\ skip{\isacharparenright}{\isacharparenright}\isanewline
\ \ INV\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isadigit{0}}\ {\isasymle}\ s{\isachardollar}{\isadigit{0}}\ {\isasymand}{\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharminus}\ {\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ h\ {\isacharminus}\ s{\isachardollar}{\isadigit{1}}\ {\isasymcdot}\ s{\isachardollar}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}\isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ {\isadigit{0}}\ {\isasymle}\ s{\isachardollar}{\isadigit{0}}\ {\isasymand}\ s{\isachardollar}{\isadigit{0}}\ {\isasymle}\ h{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{apply}\isamarkupfalse%
{\isacharparenleft}rule\ ffb{\isacharunderscore}loopI{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}\isanewline
\ \ \ \ \isacommand{apply}\isamarkupfalse%
{\isacharparenleft}force{\isacharcomma}\ force\ simp{\isacharcolon}\ bb{\isacharunderscore}real{\isacharunderscore }arith{\isacharparenright}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
{\isacharparenleft}rule\ ffb{\isacharunderscore}g{\isacharunderscore}odei{\isacharparenright }\ {\isacharparenleft}auto\ intro{\isacharbang}{\isacharcolon}\ diff{\isacharunderscore}invariant{\isacharunderscore}rules\ poly{\isacharunderscore}derivatives{\isacharparenright}\isanewline
\end{isabellebody}
As before, the first line of the proof applies the non-evolution
$\mathit{wlp}$-rules; the second one discharges $P\leq I$ and $I\leq Q$ for
loop invariant $I$. It remains to show that
$I\leq |{x'=f\, \&\, G\ \mathsf{DINV}\ I_d}]I$ for differential
invariant $I_d$.
For this we unfold the annotated invariant rule
\isa{ffb{\isacharunderscore}g{\isacharunderscore}odei}, which performs
step (2) of Example~\ref{ex:ball-inv} and generates the proof
obligation $I_d\leq |{x'=f\, \&\, G}]I_d$. The proof of this fact is
automatic because the rule
\isa{ffb{\isacharunderscore}diff{\isacharunderscore}inv}
(Lemma~\ref{P:inv-props2}) has been added to Isabelle's
simplifiers. Step (1) is checked with our rules for derivatives
\isa{poly{\isacharunderscore}derivatives} and differential invariants
\isa{diff{\isacharunderscore}invariant{\isacharunderscore}rules}
(Proposition~\ref{P:invrules}). The full verification covers less than
a page in the proof document.\qed
\end{example}
\begin{example}[Circular Pendulum via Invariant]\label{ex:pendulum-inv}
The ODEs
\begin{equation*}
x'\, t= y\, t\qquad \text{ and } \qquad y'\, t = -x\, t,
\end{equation*}
which correspond to
the vector field $f:\mathbb{R}^V\to \mathbb{R}^V$,
\begin{equation*}
f\,
\begin{pmatrix}
s_x\\
s_y
\end{pmatrix}
=
\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}
\begin{pmatrix}
s_x\\
s_y
\end{pmatrix},
\end{equation*}
for $V=\{x,y\}$, describe the kinematics of a circular pendulum. All
orbits are ``governed'' by the separable differential equation
\begin{equation*}
\frac{dy}{dx} = \frac{y'}{x'} = -\frac{x}{y},
\end{equation*}
obtained by parametric derivation. Rewriting it as $xdx+ydy= 0$ and
integrating both sides yields $x^2+y^2=r^2$, for some constant $r>0$,
which describes the circular orbits of the ODEs. This leads to the
differential invariant
\begin{equation*}
I = \left(\lambda s.\ s_x^2 + s_y^2 = r^2\right),\qquad (r\ge 0).
\end{equation*}
Once again we apply our procedure from
Section~\ref{sec:differential-invariants} to show that
\begin{equation*}
I = |x'= f\, \&\, \top] I
\end{equation*}
with Lemma \ref{P:inv-props3}, as the guard is trivial.
\begin{enumerate}
\item Using Proposition \ref{P:invrules} with $\mu\, s = s_x^2$ and
$\nu\, s = r^2 - s_y^2$ we check that $I$ is an invariant, showing
that $(\mu\circ X)' =(\nu\circ X)'$ for all $X\in \mathop{\mathsf{Sols}}\, f\, T\,
s$, and hence
\begin{equation*}
\left((X\, t\, x)^2\right)' = \left(r^2 - (X\, t\, y)^2\right)'.
\end{equation*}
We calculate
\begin{equation*}
\left((X\, t\, x)^2\right)' = 2(X\, t\, x)(X'\, t\, x)
= -2(X'\, t\, y)(X\, t\, y) =
\left(r^2 - (X\, t\, y)^2\right)'.
\end{equation*}
It therefore follows from Proposition~\ref{P:invrules}(1) that $I$ is an invariant for $f$
along $\mathbb{R}^V$; $I = |{x'=f\ \&\ \top}] I$ holds by Lemma \ref{P:inv-props3}.
\item As $P=I=Q$, there is nothing to show.
\end{enumerate}
In the Isabelle formalisation, we introduce a name for the vector
field and show that $I$ is an invariant for it---as the invariant is
the pre- and postcondition, an annotation is not needed. This is
straightforward following the work flow of the previous example, and
even simpler because the pre- and postconditions are just the
differential invariant.
\begin{isabellebody}
\isanewline
\isacommand{abbreviation}\isamarkupfalse%
\ fpend\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}real{\isacharcircum}{\isadigit{2}}\ {\isasymRightarrow}\ real{\isacharcircum}{\isadigit{2}}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}f{\isachardoublequoteclose}{\isacharparenright}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}f\ s\ {\isasymequiv}\ {\isacharparenleft}{\isasymchi}\ i{\isachardot}\ if\ i{\isacharequal}{\isadigit{0}}\ then\ s{\isachardollar}{\isadigit{1}}\ else\ {\isacharminus}s{\isachardollar}{\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ pendulum{\isacharcolon}\ {\isachardoublequoteopen}{\isacharbraceleft}s{\isachardot}\ r\isactrlsup {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}s{\isachardollar}{\isadigit{0}}{\isacharparenright}\isactrlsup {\isadigit{2}}\ {\isacharplus}\ {\isacharparenleft}s{\isachardollar}{\isadigit{1}}{\isacharparenright}\isactrlsup {\isadigit{2}}{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal}\ f\ {\isacharampersand}\ G{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ r\isactrlsup {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}s{\isachardollar}{\isadigit{0}}{\isacharparenright}\isactrlsup {\isadigit{2}}\ {\isacharplus}\ {\isacharparenleft}s{\isachardollar}{\isadigit{1}}{\isacharparenright}\isactrlsup {\isadigit{2}}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
\ {\isacharparenleft}auto\ intro{\isacharbang}{\isacharcolon}\ diff{\isacharunderscore}invariant{\isacharunderscore}rules\ poly{\isacharunderscore}derivatives{\isacharparenright}\isanewline
\end{isabellebody}
\item Isabelle performs this proof automatically if we supply the
tactic for derivative rules. \qed
\end{example}
\begin{example}[Circular Pendulum via Flow]\label{ex:pendulum-flow}
Alternatively, the kinematic equations for the circular pendulum from
Example~\ref{ex:pendulum-flow}
can of course be solved by linear
combinations of trigonometric functions.
Yet first we need to show that the vector field $f$ is Lipschitz
continuous with constant 1. Next we supply the flow
\begin{equation*}
\varphi_s\, t =
\begin{pmatrix}
\cos t & \sin t\\
-\sin t &\cos t
\end{pmatrix}
\begin{pmatrix}
s_x\\
s_y
\end{pmatrix}.
\end{equation*}
We need to check that it solves the IVP $(f,s)$ for all
$s\in \mathbb{R}^V$ and that it satisfies the flow conditions for
$T=\mathbb{R}$ and $S=\mathbb{R}^V$. As an example calculation,
\begin{align*}
\varphi_s'\, t =
\begin{pmatrix}
-\sin t & \cos t\\
-\cos t &-\sin t
\end{pmatrix}
\begin{pmatrix}
s_x\\
s_y
\end{pmatrix}
=
\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}
\begin{pmatrix}
\cos t &\sin t\\
-\sin t &\cos t
\end{pmatrix}
\begin{pmatrix}
s_x\\
s_y
\end{pmatrix}
= f\, (\varphi_s\, t).
\end{align*}
The remaining conditions are left to the reader.
To compute $|x'=f\, \&\, \top]I$, we expand (\ref{eq:wlp-evl}). This
yields
\begin{align*}
|x'=f\, \&\, \top]I\, s &= \forall t.\ I\, (\varphi_s\, t)\\
& = \left( \forall t.\ (\varphi_s\, t\, x)^2 + (\varphi_s\, t\, y)^2 =
r^2\right)\\
& = \left( \forall t.\ (s_x\cos t + s_y\sin t)^2 + (s_y\cos t -
s_x\sin t)^2 = r^2\right)\\
& = \left( \forall t.\ s_x^2(\sin^2 t +\cos^2 t) + s_y^2(\sin^2 t +
\cos^2 t) = r^2\right)\\
&= I\, s.
\end{align*}
In the Isabelle proof along these lines, we first prove that the
vector field satisfies the conditions of Picard-Lindel\"of's
theorem. To this end we need to unfold the locale definitions, then
introduce the Lipschitz constant, and call Isabelle's simplifiers.
Next, to prove that the solution supplied is a flow and a
solution to the IVP, we unfold definitions and finish the proof by
checking that the derivative of the flow in each coordinate coincides
with the vector field in that coordinate. The introduction of the flow
and these lemmas are shown below.
\begin{isabellebody}
\isanewline
\isacommand{abbreviation}\isamarkupfalse%
\ pend{\isacharunderscore}flow\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ real{\isacharcircum}{\isadigit{2}}\ {\isasymRightarrow}\ real{\isacharcircum}{\isadigit{2}}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymphi}{\isachardoublequoteclose}{\isacharparenright}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}{\isasymphi}\ t\ s\ {\isasymequiv}\ {\isacharparenleft}{\isasymchi}\ i{\isachardot}\ if\ i\ {\isacharequal}\ {\isadigit{0}}\ then\ s{\isachardollar}{\isadigit{0}}\ {\isasymcdot}\ cos\ t\ {\isacharplus}\ s{\isachardollar}{\isadigit{1}}\ {\isasymcdot}\ sin\ t\ else\ s{\isachardollar}{\isadigit{1}}\ {\isasymcdot}\ cos\ t\ {\isacharminus}\ s{\isachardollar}{\isadigit{0}}\ {\isasymcdot}\ sin\ t{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isacommand{lemma}\isamarkupfalse%
\ local{\isacharunderscore}flow{\isacharunderscore}pend{\isacharcolon}\ {\isachardoublequoteopen}local{\isacharunderscore}flow\ f\ UNIV\ UNIV\ {\isasymphi}{\isachardoublequoteclose}\isanewline
\ \ $\langle \isa{proof}\rangle$\isanewline
\end{isabellebody}
\noindent The proof of the correctness specification requires
only an application of the $\mathit{wlp}$ rule
\isa{local{\isacharunderscore}flow{\isachardot
}ffb{\isacharunderscore}g{\isacharunderscore}ode}
(\ref{eq:wlp-evl}) and Isabelle's simplifier.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ pendulum{\isacharcolon}\ {\isachardoublequoteopen}{\isacharbraceleft}s{\isachardot}\ r\isactrlsup {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}s{\isachardollar}{\isadigit{0}}{\isacharparenright}\isactrlsup {\isadigit{2}}\ {\isacharplus}\ {\isacharparenleft}s{\isachardollar}{\isadigit{1}}{\isacharparenright}\isactrlsup {\isadigit{2}}{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ {\isacharparenleft}x{\isasymacute}{\isacharequal}f\ {\isacharampersand}\ G{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ r\isactrlsup {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}s{\isachardollar}{\isadigit{0}}{\isacharparenright}\isactrlsup {\isadigit{2}}\ {\isacharplus}\ {\isacharparenleft}s{\isachardollar}{\isadigit{1}}{\isacharparenright}\isactrlsup {\isadigit{2}}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
\ {\isacharparenleft}force\ simp{\isacharcolon}\ local{\isacharunderscore}flow{\isachardot}ffb{\isacharunderscore}g{\isacharunderscore}ode{\isacharbrackleft}OF\ local{\isacharunderscore}flow{\isacharunderscore}pend{\isacharbrackright}{\isacharparenright}
\end{isabellebody}
\hfill\qed
\end{example}
All four example have been based on the categorical approach and the
state transformer semantics. Alternative formalisations for the other
predicate transformer algebras and the relational semantics can be
found in other verification components~\cite{afp:hybrid}. In the
$\mathsf{MKA}$-based component, the proofs using the relational and the state
transformer semantics are precisely the same, which underpins the
modularity of our approach. In the other components we could certainly
achieve the same effect by simply rewriting names and adjusting some
types.
Transcendental functions are beyond $\mathsf{d}\mathcal{L}$ and KeYmaera X, yet we can
use them smoothly and easily with Isabelle with the tactic outlined
in Section~\ref{sec:isa-wlp}. Both the differential invariant
approach and the flow-based approach benefit from these rules. In
fact, both approaches are very similar for the pendulum example: both
need a handful of lemmas to prove the partial correctness
specification $I = |{x'=f\ \&\ \top}] I$, and both require a creative
step in the form of introducing a differential invariant or the flow
for the system.
We have presented the pendulum example in matrix notation as this
points to a common feature of many applications: their dynamics can be
described by linear systems of ODEs that are representable by
matrices and have uniform solutions given by a matrix exponential that
can be computed with standard methods from linear algebra.
The development of domain-specific techniques for linear systems with
Isabelle is the subject of a successor article.
\section{Outlook: A Flow-Based Verification Component
}\label{sec:flow-component}
The verification components presented so far adhered very much to the
pessimistic interactive theorem proving mind set that requires the
internal reconstruction of all external results. This section briefly
outlines a fourth more optimistic verification component that deviates
entirely from the vector-field-based approach of $\mathsf{d}\mathcal{L}$ and works
directly with flows or solutions to IVPs. It shifts responsibility for
the correctness of solutions entirely to users---or the computer
algebra system they might have used. This is common practice for
instance when working with hybrid automata~\cite{DoyenFPP18}, and of
course it greatly simplifies proofs.
The topological or differentiable structure of the underlying state
space is then of secondary interest; with Isabelle, such structure and
additional conditions can always be imposed by instantiating types
with sort constraints as they arise.
Hence we start from a setting that covers both discrete and continuous
evolutions and use a general type for time instead of $\isa{real}$,
$\isa{rat}$ or $\isa{int}$. The evolution commands are now
arbitrary guarded $\varphi$-type functions instead of vector fields. The
type of time needs to admit an order relation, which is indicated by
the sort constraint $\isa{ord}$ below, yet specific properties, such
as reflexivity or transitivity, need not be imposed ab initio.
Apart from that, the definition of the guarded-orbit semantics and the
$\mathit{wlp}$ rule is as before, but side conditions on Lipschitz
continuity or the Picard-Lindel\"of theorem are superfluous.
\begin{isabellebody}
\isanewline
\isacommand{definition}\isamarkupfalse%
\ g{\isacharunderscore}evol\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharparenleft}{\isacharprime}a{\isacharcolon}{\isacharcolon}ord{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ pred\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b\ set{\isacharparenright}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}EVOL{\isachardoublequoteclose}{\isacharparenright}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}EVOL\ {\isasymphi}\ G\ T\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ g{\isacharunderscore}orbit\ {\isacharparenleft}{\isasymlambda}t{\isachardot}\ {\isasymphi}\ t\ s{\isacharparenright}\ G\ T{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ fbox{\isacharunderscore}g{\isacharunderscore}evol{\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ \isanewline
\ \ \isakeyword{fixes}\ {\isasymphi}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a{\isacharcolon}{\isacharcolon}preorder{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fb\isactrlsub {\isasymF}\ {\isacharparenleft}EVOL\ {\isasymphi}\ G\ T{\isacharparenright}\ Q\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isacharparenleft}{\isasymforall}t{\isasymin}T{\isachardot}\ {\isacharparenleft}{\isasymforall}{\isasymtau}{\isasymin}down\ T\ t{\isachardot}\ G\ {\isacharparenleft}{\isasymphi}\ {\isasymtau}\ s{\isacharparenright}{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymphi}\ t\ s{\isacharparenright}\ {\isasymin}\ Q{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{unfolding}\isamarkupfalse%
\ g{\isacharunderscore}evol{\isacharunderscore}def\ g{\isacharunderscore}orbit{\isacharunderscore}eq\ ffb{\isacharunderscore}eq\ \isacommand{by}\isamarkupfalse%
\ auto\isanewline
\end{isabellebody}
Using the flows of the bouncing ball and the circular pendulum from
previous examples, verification proofs are now fully automatic.
\begin{isabellebody}
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ pendulum{\isacharunderscore}dyn{\isacharcolon} {\isachardoublequoteopen}{\isacharbraceleft}s{\isachardot}\ r\isactrlsup {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}s{\isachardollar}{\isadigit{0}}{\isacharparenright}\isactrlsup {\isadigit{2}}\ {\isacharplus}\ {\isacharparenleft}s{\isachardollar}{\isadigit{1}}{\isacharparenright}\isactrlsup {\isadigit{2}}{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ {\isacharparenleft}EVOL\ {\isasymphi}\ G\ T{\isacharparenright}\ {\isacharbraceleft}s{\isachardot}\ r\isactrlsup {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}s{\isachardollar}{\isadigit{0}}{\isacharparenright}\isactrlsup {\isadigit{2}}\ {\isacharplus}\ {\isacharparenleft}s{\isachardollar}{\isadigit{1}}{\isacharparenright}\isactrlsup {\isadigit{2}}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
\ force\isanewline
\isacommand{lemma}\isamarkupfalse%
\ bouncing{\isacharunderscore}ball{\isacharunderscore}dyn{\isacharcolon}\ {\isachardoublequoteopen}g\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ h\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ \isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharequal}\ h\ {\isasymand}\ s{\isachardollar}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{0}}{\isacharbraceright}\ {\isasymle}\ fb\isactrlsub {\isasymF}\ \isanewline
\ \ {\isacharparenleft}LOOP\ {\isacharparenleft}\isanewline
\ \ \ \ {\isacharparenleft}EVOL\ {\isacharparenleft}{\isasymphi}\ g{\isacharparenright}\ {\isacharparenleft}{\isasymlambda}\ s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}\ T{\isacharparenright}\ {\isacharsemicolon}\ \isanewline
\ \ \ \ {\isacharparenleft}IF\ {\isacharparenleft}{\isasymlambda}\ s{\isachardot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ THEN\ {\isacharparenleft}{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}{\isacharequal}\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isacharminus}\ s{\isachardollar}{\isadigit{1}}{\isacharparenright}{\isacharparenright}\ ELSE\ skip{\isacharparenright}{\isacharparenright}\isanewline
\ \ INV\ {\isacharparenleft}{\isasymlambda}s{\isachardot}\ {\isadigit{0}}\ {\isasymle}\ s{\isachardollar}{\isadigit{0}}\ {\isasymand}{\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ s{\isachardollar}{\isadigit{0}}\ {\isacharminus}\ {\isadigit{2}}\ {\isasymcdot}\ g\ {\isasymcdot}\ h\ {\isacharminus}\ s{\isachardollar}{\isadigit{1}}\ {\isasymcdot}\ s{\isachardollar}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}\isanewline
\ \ {\isacharbraceleft}s{\isachardot}\ {\isadigit{0}}\ {\isasymle}\ s{\isachardollar}{\isadigit{0}}\ {\isasymand}\ s{\isachardollar}{\isadigit{0}}\ {\isasymle}\ h{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{by}\isamarkupfalse\ %
{\isacharparenleft}rule\ ffb{\isacharunderscore}loopI{\isacharparenright}\ {\isacharparenleft}auto\ simp{\isacharcolon}\ bb{\isacharunderscore}real{\isacharunderscore }arith{\isacharparenright}\isanewline
\end{isabellebody}
In these examples there is no longer a link between the flows and an
initial specification in terms of system of ODEs, from which a user
might have started. Hence there is no longer any formal guarantee from
Isabelle that the function $\varphi$ provided satisfies assumptions as
those of \isa{local{\isacharunderscore}flow}.
Further elaboration of this approach, in particular in the direction
of hybrid automata, is left for future work.
\section{Conclusion}\label{sec:conclusion}
We have presented a new semantic framework for the deductive
verification of hybrid systems with the Isabelle/HOL proof assistant.
The approach is inspired by differential dynamic logic, but the design
of our verification components, the focus of our framework and the
workflow for verifying hybrid systems is different.
First of all, as we use a shallow embedding, the basic verification
components generated are quite minimalist and conceptually
simple. They merely require the integration of a $\mathit{wlp}$ semantics for
basic evolution commands into standard predicate transformer
algebras. Our preferred semantics for such commands are state
transformers, which in most cases simply map states to the guarded
orbits of their temporal evolutions. Beyond that, no domain-specific
inference rules are needed, verification condition generation is
automatic---even our approach to differential invariants is based
entirely on general purpose algebraic invariant laws. Our examples
show that mathematical reasoning about differential equations
follows standard textbook style and hence comes close to the natural
way mathematicians, physicists or engineers reason about such systems.
Secondly, we currently aim at an open experimental platform that is
only limited by Isabelle's ODE components, the expressivity of its
higher-order logic and type system, and the proof support it
provides. We could, for instance, have developed our semantics for
time-dependent vector fields, but the restriction to autonomous
systems, which does not affect generality, seems preferable in
practice. The integration of internal or external solvers for
differential algebras, transcendental functions or computer algebra
systems for computing Lipschitz constants or flows in the style of
Isabelle's Sledgehammer tactic are certainly interesting and very
important avenues for future work, but not a main concern in this
article.
Two specialisations of our framework are the topic of successor
papers. The first one restricts our approach to linear systems of
differential equations, where exponential solutions exist and can be
computed with standard methods from linear algebra. The second one
specialises the predicate transformer semantics to algebraic variants
of Hoare logics and to refinement calculi for hybrid programs along
the lines of previous components for traditional
while-programs~\cite{GomesS16}.
Beyond that we expect that a recent formalisation of Poincar\'e maps
with Isabelle~\cite{ImmlerT19} will allow us to extend our framework
to discrete dynamical systems and more computational approaches to
hybrid systems.
Finally, differential-algebraic systems of equations~\cite{HairerW96},
which mix differential equations and algebraic equations, and partial
differential equations~\cite{John86} are important for many
applications in control engineering and physics. Extending our
approach most probably requires significant background work on
mathematical components with Isabelle. While, in both settings, some
simple cases can be reduced to systems of ODEs, numerical methods, for
instance Runge-Kutta, are usually needed to work with such systems.
Whether the workflow of mathematicians, physicists and engineers with
such more computational approaches can be approximated easily with
Isabelle remains to be seen.
\textbf{Acknowledgements}
This work was funded by a CONACYT scholarship. We are grateful to
Achim Brucker, Simon Foster, Peter H\"ofner, Andr\'e Platzer, Thao
Dang as well as the
participants of the RAMiCS 2018 conference and our Oslo lecture
series on Isabelle/HOL for helpful discussions.
\bigskip
\bibliographystyle{abbrv}
| {'timestamp': '2020-01-15T02:15:24', 'yymm': '1909', 'arxiv_id': '1909.05618', 'language': 'en', 'url': 'https://arxiv.org/abs/1909.05618'} |
\section{Introduction}
The abundance of most stable nuclei above iron in the universe can be
understood as produced by various types of neutron
capture \cite{mat85,bar06}. However, production of about 40 stable isotopes cannot be explained in this
way, but only through similar proton capture processes
\cite{bur57,arn03,rau13,rei14,pal14}. The basic ignition fuel is a
large proton flux arising from a stellar explosion. The sequence of
these reactions are then one proton capture after another until the
proton dripline is reached and further captured protons are
immediately emitted. This dripline nucleus usually must wait to beta-decay to a more stable nucleus which in turn can
capture protons anew. This is the "rp-process" \cite{sch98,bro02}. These p-nuclei are
also believed to be produced by other methods: gamma-proton \cite{arn03,woo78} and
neutrino-proton processes \cite{fro06,fro06b}.
The beta-decay waiting time is large for some of these nuclei along
the dripline, which for that reason are denoted waiting points
\cite{oin00}. However, another path is possible to follow for
borromean proton dripline nuclei where two protons, in contrast to
one, are necessary to produce a bound nucleus. Then two protons can
be captured before beta-decay occurs \cite{gri05,gri01}. The capture time and the
corresponding mechanism are therefore important for the description of
the outcome of these astrophysical processes \cite{gor95,sch06}.
We focus in this letter on one of these two-proton capture reactions
leading from a prominent waiting point nucleus, $^{68}$Se
\cite{sch07,tu11}, to formation of the borromean proton dripline
nucleus, $^{70}$Kr~($^{68}$Se$+p+p$). The specific experimental
reaction information is not available, and theoretical estimates are,
at least at the moment, unavoidable \cite{tho04,erl12,pfu12}.
Traditionally, the reactions have been described as sequential
one-proton capture by tunneling through the combined Coulomb plus
centrifugal barrier. The tunneling capture mechanisms have been
discussed as direct, sequential and virtual sequential decay
\cite{jen10,gar05,che07,rod08}. They are all accounted for in the present formulation.
The capture rate depends on temperature through the assumed
Maxwell-Boltzmann energy distribution. It is then important to know
the energy dependence of the capture cross section for given resonance
energies, and especially in the Gamow window \cite{chr15}.
Clearly the desired detailed description requires a
three-body model which is available and even applied to the present
processes \cite{hov16,gar04}. However, the crucial proton-core potentials
have so-far been chosen phenomenologically to produce the essentially
unknown, but crucial, single-particle energies. A new model
involving both core and valence degrees of freedom was recently
constructed to provide mean-field proton potentials derived from the
effective nucleon-nucleon mean-field interaction \cite{hov17,hov17b,hov17c}. In turn
these potentials produce new and different effective three-body
potentials, which in the present letter is exploited to investigate
the two-proton capture rates.
The techniques are in place all the way from the solution of the
coupled core and valence proton system \cite{hov16,hov17b,hov17c}, over the
self-consistent three-body input and subsequent calculations
\cite{nie01,jen04}, to the capture cross sections and rates
\cite{gar15,die10,die11,die10b}. We shall first sketch the steps in
the procedure used in the calculations. Then we shall discuss in more
details the numerical results of interest for the astrophysical
network computations, which calculates the abundances of the isotopes in the Universe.
\section{Theoretical description}
The basic formulation and the procedure are described in
\cite{hov16,gar15,gar15b}. The framework is the three-body technique but based
on the proton-core potential derived through the self-consistently
solved coupled core-plus-valence-protons equations \cite{hov17,hov17b}. The procedure is
first to select the three-body method, second to formulate how to
calculate the capture rate, and finally to choose the numerical
parameters to be used in the computations.
\subsection{Wave functions}
First, the many-body problem is solved for a mean-field treated core
interacting with two surrounding valence protons \cite{hov16}. The
details of this recent model are very elaborate, but already applied
on two different neutron dripline nuclei \cite{hov16,hov17}. It then
suffices to sketch the corresponding details for the present
application. Briefly, the novel features are to find the mean-field
solution for the core-nucleons in the presence of the external field
from the two valence protons. In turn, folding the basic
nucleon-nucleon interaction and the core wave function provides the
interaction between each valence proton and the core nucleus. The
interactions between core and valence nucleons then depend on their
respective wave functions, which are found self-consistently by
iteration. We emphasize that the same nucleon-nucleon interaction is
used both in the core and for this valence-proton core calculation.
The crucial main ingredient in the three-body solution is this
interaction, which then is provided by the procedure and determined
independent of subsequent applications.
The present application exploits the properties of the derived
three-body solution. The three-body problem is solved by adiabatic
expansion of the Faddeev equations \cite{gar04} in hyperspherical coordinates. When necessary
the continuum is discretized by a large box confinement \cite{gar15a}. The
main coordinate is the hyperradius, $\rho$, defined as the mean radial
coordinate in the three-body system \cite{gar15b,hov14}. More
specifically we have
\begin{align}
(2m_n+m_c) \rho^2
= m_n(\bm{r}_{v_1} - \bm{r}_{v_2})^2
+ m_{c} \sum_{i=1}^2 (\bm{r}_{v_i} - \bm{R}_{c})^2
\label{rhodef}
\end{align}
where $m_n$, $m_{c}$, $\bm{r}_{v_1}$, $\bm{r}_{v_2}$ and
$\bm{R}_{c}$ are neutron mass, core mass, valence-proton
coordinates and core center-of-mass coordinate, respectively. The
three-body wave function is found through this procedure for given
angular momenta as functions of the hyperspherical coordinates for the
required ground state ($\Psi_{J}$). When necessary the continuum is
discretized by a large box confinement and the discretized continuum
states ($\psi_{j}^{(i)}$) are calculated.
\subsection{Reactions}
The two-proton capture reaction $p+p+c \leftrightarrow A + \gamma$ cross section $\sigma_{ppc}$ and the photodissociation cross section $\sigma^{\lambda}_{\gamma}$ of order ${\lambda}$ are related \cite{gar15b}.
The three-body energy, $E$, and the ground state energy, $E_{gr}$,
determine the photon energy by $E_{\gamma} = E + |E_{gr}|$. The
dissociation cross section is then given by
\begin{align}
\sigma^{\lambda}_{\gamma}(E_{\gamma})
=& \frac{(2 \pi)^3 (\lambda +1)}{\lambda ((2\lambda +1)!!)^2} \left( \frac{E_{\gamma}}{\hbar c}\right)^{2\lambda-1} \frac{d}{d E}\mathcal{B}({E}\lambda, 0 \rightarrow \lambda), \label{eq siggam}
\end{align}
where the strength function for the ${E}\lambda$ transition,
\begin{align}
\frac{d}{d E} \mathcal{B}({E}\lambda, 0 \rightarrow \lambda) =
\sum_i \left| \braket{\psi_{\lambda}^{(i)} | | \hat{\Theta}_{\lambda} | |
\Psi_{0}} \right|^2 \delta(E-E_i), \label{eq tran}
\end{align}
is given by the reduced matrix elements,
$\braket{\psi_{\lambda}^{(i)} | | \hat{\Theta}_{\lambda} | | \Psi_{0}}$,
where $\hat{\Theta}_{\lambda}$ is the electric multipole operator,
$\psi_{\lambda}^{(i)}$ is the wave function of energy, $E_i$, for all
bound and (discretized) three-body continuum states in the summation.
The capture reaction rate, $R_{ppc}$, is given by Ref.~\cite{die11}
\begin{align}
R_{ppc}(E) = & \frac{8 \pi}{(\mu_{cp} \mu_{cp,p})^{3/2}} \frac{\hbar^3}{c^2}
\left( \frac{E_{\gamma}}{E} \right)^2 \sigma^{\lambda}_{\gamma}(E_{\gamma}),
\label{eq rate}
\end{align}
where $\mu_{cp}$ and $\mu_{cp,p}$ are reduced masses of proton and
core and proton-plus-core and proton, respectively. For the astrophysical processes in a gas of temperature, $T$, we have to average the rate in Eq.~(\ref{eq rate}) over the
corresponding Maxwell-Boltzmann distribution, $B(E,T) = \frac{1}{2}
E^2 \exp(-E/T)/T^3$,
\begin{align}
\braket{R_{ppc}(E)} = \frac{1}{2T^3} \int E^2 R_{ppc}(E) \exp(-E/T)
\, dE, \label{eq ave rate}
\end{align}
where the temperature is in units of energy.
\subsection{Interactions}
The decisive interaction is first of all related to the mean-field
calculation of the core. We use the Skyrme interaction SLy4 \cite{cha98} with acceptable global average properties. The
application on one specific nuclear system requires some adjustment to
provide the known borromean character, that is unbound
proton-core $f_{5/2}$ resonance at $0.6$~MeV \cite{san14} and two protons bound to the core. With a minimum of changes we achieve this by shifting all energies while leaving the
established structure almost completely unaltered. The simplest
consistent such modification is by scaling all the main Skyrme
strength parameters, $t_i$, by the same factor, $0.9515$.
The density dependence of the Skyrme interaction can be viewed as a
parametrized three-body potential. To simulate that effect we employ
a short-range Gaussian, $S_0\exp(-\rho^2/b^2)$, which depends on the three-body hyperradial coordinate $\rho$. We choose $b=6$~fm and leave
$S_0$ to fine-tune each of the $0^+$ and $2^+$ three-body energies. This
is necessary since the keV-scale of binding is crucial for tunneling
through single MeV height barriers. This level of accuracy is beyond
the present capability of many-body model calculations. To reproduce
the predicted $0^+$ energy of $-1.34$~MeV \cite{wan12} a three-body
strength $S_0 = -17.5$~MeV is needed. The unknown $2^+$ energy is
varied from almost bound, zero energy, to the top of the barrier by
$S_0$ changing from $-35.05$~MeV to $-26.22$~MeV.
\section{Effective three-body potentials \label{sec:3b}}
The elaborate numerical calculations produce the sets of coupled
``one-body'' effective potentials depending on hyperradius as shown in
Fig.~\ref{fig:potsw} for both $0^{+}$ and $2^{+}$. The continuations
beyond the $20$~fm in the figure are almost quantitatively Coulomb
plus centrifugal behavior and as such reveal no surprises. The kinks
and fast bends reflect avoided crossings and related structure
changes. They are especially abundant at small distances and around
the barriers. The diagonal non-adiabatic coupling terms as well as
the diagonal structure-less three-body Gaussian potentials are included in the calculations, but not in the figure. They both leave the structure
essentially unaltered although they may change the energies
of the solutions rather substantially.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{CompEffPots.pdf}
\vspace*{-6mm}
\caption{The effective, adiabatic potentials for the $0^{+}$ (red,
solid), and the $2^{+}$ (light-blue, dashed) configuration in
$^{68}\text{Se}+p+p$ using the SLy4 Skyrme interaction between core
and valence protons, scaled to reproduce the experimental $f_{5/2}$
resonance energy of $0.6$~MeV in $^{68}\text{Se}+p$ \cite{san14}. The dotted horizontal line is the $0^+$ energy
at $-1.34$~MeV from $S_0= -17.5$~MeV.
\label{fig:potsw}}
\end{figure}
The $0^{+}$ ground-state at $-1.34$~MeV, predicted from systematics
\cite{aud12}, is reproduced with the chosen parameters. The structure
corresponds to the configuration of the pronounced minimum in the
lowest $0^{+}$ potential. No $0^{+}$ resonance are produced by the
potentials in Fig.~\ref{fig:potsw}. The ground state is the final
state in the capture process independent of the specific mechanism.
However, the decisive capture process proceeds within the $2^{+}$
continuum from the large to the short-distance attractive region of
the potentials shown in Fig.~\ref{fig:potsw}. This lowest minimum is
rather similar to the $0^{+}$ minimum but the non-adiabatic repulsive
terms increase the energy substantially. Unfortunately nothing is
known about a $2^{+}$ resonance which would strongly influence the
capture rate. Consequently the strength, $S_0$, is used to vary the
position of the $2^{+}$ resonance from almost bound to disappearance
above the barriers. Both the resonance energy, the height, and the
rather broad Coulomb shape of the barrier strongly influence the
capture process.
The structure of these potentials is substantially simpler than those
obtained in \cite{hov16} where low-lying single-proton states
$p_{3/2}$ and $f_{5/2}$ both appeared. The present simplification is
an automatic result of the procedure using the nucleon-nucleon
mean-field effective interaction to calculate the proton-core
potential. This is not an ad hoc assumption, but arises naturally due to
identical interactions for both core and valence particles. As such
it is a novel deduction embedded in the design of our model. The
lack of single-particle states of different parity implies that no
$1^-$ three-body resonance states appear in the low-energy region. The transition is then necessarily an $E2$ transition, which contributes to the longer effective lifetime of the system, and could very well be part of the reason this system is a critical waiting point.
\section{Quantitative results}
The all-important core-valence proton potential is derived naturally and unambiguously by our mean-field core treatment, as discussed in the previous sections. As a result the two-proton capture cross section follows directly, only depending on the three-body resonance level. This is discussed in the following section, after which the resulting temperature averaged reaction rates are presented. This is supplemented by a discussion of the reaction mechanism and its implication for the possible reactions.
\subsection{Cross section}
The incident flux of low-energy protons on the core nucleus may result
in capture. The corresponding cross section is most easily obtained
from calculation of the inverse reaction, that is photodissociation of
the $0^+$ ground state, $\Psi_{0}$, of $^{70}$Kr. The discretized
continuum states, $\psi_{\lambda}^{(i)}$, are computed and the cross
section is obtained from Eqs.~(\ref{eq tran}) and (\ref{eq siggam}) with
$\lambda =2$. The two-proton capture cross section of $^{68}$Se,
obtained from Eq.~(\ref{eq siggam}), is shown in Fig.~\ref{fig:cross} as
function of the three-body energy.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{cross_gam.pdf}
\vspace*{-6mm}
\caption{The electromagnetic $E2$ dissociation cross section,
$\sigma^{(\lambda=2)}_{\gamma}(E_{\gamma})$, for the proces,
$^{70}\text{Kr} + \gamma \rightarrow ^{68}\text{Se}+p+p$, as a
function of photon energy. The $0^+$ final state energy is
$-1.34$~MeV and the $2^+$ resonance energies are $E=0.5, \, 1.0, \,
2.0,$ and $4.0$~MeV, respectively. The discretized continuum states
are obtained using box sizes of $\rho_{max} = 150, 200$~fm.
\label{fig:cross}}
\end{figure}
The peaks in the capture cross section occur at experimentally unknown
resonance energies where the tunneling probability is large. We
therefore vary the energy from $0.5$~MeV to $4.0$~MeV where the widths
of the peaks in the cross section increase with energy as the top of
the barrier is approached. We emphasize that the crucial quantity is
the resonance energy. This can be tested by varying the number of
adiabatic potentials used in the calculation. This results in
somewhat different resonance energy which however can be
compensated for by use of the three-body potential, which in turn
recover the cross section in Fig.~\ref{fig:cross}.
These features are simply understood as enhanced spatial overlaps
between the $2^+$ continuum states in the resonance region and the
ground state wave function, expressed through Eq.~(\ref{eq tran}). Beside the resonance contributions we
also find significant, although several orders of magnitude smaller,
background contributions, which incidentally is
independent of the size of the discretization box, as long as it is sufficiently large \cite{gar15a}.
\subsection{Capture rates}
The capture cross sections are the main ingredient in the calculation
of the two-proton absorption rate appropriate for the temperature
dependent astrophysical network computation. The average rate in
Eq.~(\ref{eq ave rate}) are shown in Fig.~\ref{fig:rate} as function
of temperature. The Boltzmann smearing factor produces very smooth
curves of the same qualitative behavior. They are zero at zero
temperature and energy, because the barrier is infinitely thick. All
rates then increase to a maximum at the Gamow peak where the best
compromise is reached between the decreasing temperature distribution
and the increasing tunneling probability.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{rate.pdf}
\vspace*{-6mm}
\caption{The reaction rate for the radiative capture process
$^{68}\text{Se}+p+p \rightarrow ^{70}\text{Kr} + \gamma$, as
function of temperature for the different $2^+$ resonance energies
in Fig.~\ref{fig:cross}. The black dashed curve is the background
contribution.
\label{fig:rate}}
\end{figure}
The peak contribution moves to higher energy and becomes smoother with
increasing resonance energy. Above temperatures of a few GK the
average rate variation is moderate and the size roughly of order
$\simeq 6 \times 10^{-11}$~cm$^{6}[N_{A} mol]^{-2} s^{-1}$. A low-lying
resonance energy corresponds to low-lying peak position of larger
height. We emphasize that the background without resonance
contribution obviously is smaller but only by roughly a factor of two
as soon as the temperature exceeds about $4$~GK ($\sim0.34$~MeV). In
other words, if temperatures are in the astrophysically interesting
range below about $1$~GK, the size variations are substantial, and
vice versa above a few GK the details from the microscopic origin are
smeared out.
The actual size of the rate may reveal deceivingly little variation at
the relatively high temperatures. However, the barrier height
and width are all-decisive and both may easily be different for other
systems where the single-particle structure at the Fermi energy is
different and perhaps more complicated as studied in \cite{hov16}.
The relatively large $2^+$ background contribution might suggest
significant corresponding $0^+$ continuum contributions. However, the
$0^+$ barriers in Fig.~\ref{fig:potsw} are larger and the $0^+
\rightarrow 0^+$ transition as well require processes involving atomic
electrons. It is again worth emphasizing that a superficially more
complete calculation with for example many coupled potentials would
provide the same rates after adjusting to the same resonance energy.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{AngDist.pdf}
\vspace*{-6mm}
\caption{The probability of the three-body, $^{68}\text{Se}+p+p$, wave
function for the lowest allowed potential, integrated over
directional angles $\left( \sin^2 (\alpha) \cos^2 (\alpha) \int
|\Phi_n(\alpha, \rho, \Omega_x, \Omega_y)|^2 d\Omega_x
d\Omega_y\right)$, as a function of hyperradius, $\rho$, and
hyperangle, $\alpha$, related to the Jacobi coordinate system where
"x" is between core and proton. \label{fig:angDist}}
\end{figure}
\subsection{Reaction mechanism}
The rate depends on the capture mechanism. We are here only concerned
with three-body capture, but a dense environment would enhance
four-body capture processes as discussed in \cite{die10}. The overall
three-body process is tunneling through a barrier of particles in a
temperature distribution of given density. Once inside the relatively
thick barrier they have essentially only the option of emitting
photons to reach the bound ground-state. However, the first of this
two-step process can occur through different mechanisms, where the
most obvious possibility is to be captured in different angular
momentum states. The conservation of angular momentum and parity
quantum numbers are crucial in connection with resonance positions. If
low-lying $1^-$ continuum states are allowed they would be preferred,
and vice versa if prohibited $2^+$ continuum states would be
preferred. Low-lying resonances enhance the contributions
substantially. This selection depends strongly on the nucleus under
investigation.
For a given angular momentum of the three-body continuum states, we
still may encounter several qualitatively different ways of absorbing
two protons from the continuum \cite{gar11}. These mechanisms were discussed in
\cite{jen10} for the inverse process of dissociation, that is direct,
sequential and virtual sequential decay. They are all accounted for
in the present formulation. In \cite{hov16} we concluded that the
direct process is most probable for very low three-body energy when
two-body subsystems are unbound. If the energy is larger than stable
two-body substructures such intermediate vehicles enhance the rates
and the mechanism is sequential.
Even when it is energetically forbidden to populate two-body resonance
states it may be advantageous to exploit these structures virtually
while tunneling through an also energetically forbidden barrier. This
is appropriately named the virtual sequential two-body mechanism. It
may be appropriate to emphasize that a similar three-body virtual
mechanism is forbidden because the three-body energy is conserved in
contrast to the energy of any two-body subsystem.
The mechanism for the present capture process is revealed in
Fig.~\ref{fig:angDist} where the $2^+$ probability integrated over the directional
angles is shown for the lowest potential as function of hyperradius
and one of the Jacobi angles. It is a strikingly simple structure for
hyperradii larger than about $15$~fm, which for these coordinates is
equivalent to one proton at that distance from the center of mass of
the combined proton-core system. Since the Jacobi angle, $\alpha$, is
either close to zero or $\pi/2$, this simply means that one proton is
staying very close to the core for all these hyperrradii. Eventually
also this proton has to move away from the core since no bound state
exist. But the process is sequential through this substructure which
can be determined to be the proton-core $f_{5/2}$ resonance.
The higher-lying configurations corresponding to the three following
potentials also show precisely the same $f_{5/2}$ structure. This is
explained by combining the compact proton-core $f_{5/2}$ resonance
with one non-interacting (apart from Coulomb and centrifugal) distant
proton in any angular momentum configuration consistent with a $2^+$ structure.
The angular momenta capable of combining with $f_{5/2}$ to produce $2^+$ are $p_{1/2}, \, p_{3/2}, \, f_{5/2}, \, f_{7/2},$ and $h_{9/2}$. This also implies that for temperatures much smaller
than the $f_{5/2}$ resonance energy it would be energetically
advantageous to start the capture process in a configuration
corresponding to direct three-body capture. The change of structure,
around avoided level crossings, to two-body resonance configurations
would then greatly reduce the barrier and substantially enhance the
capture rate.
\section{Conclusion}
The new model that treats the core and the two valence particles
self-consistently and simultaneously is applied on the waiting point
nucleus ($^{68}$Se) for the astrophysical rp-process. This is done
essentially without any free parameters or phenomenological fitting,
which makes the results much less arbitrary than usual three-body
calculations. Adding two protons, but not one, produces a bound system,
$^{70}$Kr, which is then a borromean nucleus. A moderate overall
scaling of the Skyrme interaction SLy4 reproduces the scarcely known
properties of these dripline nuclei. Other Skyrme interactions
provide very similar results.
We calculate the radiative two-proton capture rate as function of
temperature for different resonance energies. We investigate the
mechanism and find that sequential capture of one proton after the
other by far is dominating. The first available single-particle
resonance state, $f_{5/2}$, is the vehicle, whereas the other proton
can approach in continuum states of even higher angular momentum. In practice, after tunneling through the barrier into the $2^+$ resonance
state, in practice only $E2$ electric transition to the ground state
is allowed. Background capture through non-resonance continuum states
also contributes significantly to the capture process. The sequential
$2^+$ capture mechanism might for other nuclei be replaced by for
example the normally larger $1^-$ capture.
In conclusion, the two-proton capture rates at a waiting point at the
dripline are successfully calculated with a conceptually relatively
simple, but technically advanced, new model. The same effective
nucleon-nucleon interaction is used for both the nuclear mean-field
and the proton-core calculations. The temperature dependent rate and
the corresponding capture mechanism are calculated with less ambiguity
than in previous calculations. A number of applications are now
feasible.
\section*{Acknowledgements}
This work was funded by the Danish Council for Independent Research DFF Natural Science and the DFF Sapere Aude program. This work has been partially supported by the Spanish Ministerio de Economia y Competitividad under Project FIS2014-51971-P.
\section*{References}
| {'timestamp': '2017-09-06T02:04:39', 'yymm': '1709', 'arxiv_id': '1709.01270', 'language': 'en', 'url': 'https://arxiv.org/abs/1709.01270'} |
\section{Introduction}
\label{sec:intro}
The use of prox-functions in convex programming has become standard in recent decades. Originally, they were introduced in the context of mirror descent methods \cite{nemirovski:1983}.
In order to explain how prox-functions enter into optimization methods, we recall their definition.
\begin{definition}[Prox-function] We say that $d:\mathbb{R}^n \rightarrow \mathbb{R}\cup\{\infty\}$ is a prox-function on a closed convex set $Q \subset \mathbb{R}^n$ if
\begin{itemize}
\item[(1)] $d$ is continuous with the domain containing $Q$, i.\,e. $Q \subset \mbox{dom}\, d$.
\item[(2)] $d$ is strongly convex on $Q$ with respect to a norm $\|\cdot\|$, i.\,e.
there exists a constant $\beta >0$ such that for all $x, y \in Q$ and $\alpha \in [0,1]$ it holds:
\[
d(\alpha x +(1-\alpha)y) \leq \alpha d(x) + (1-\alpha) d(y) - \frac{\beta}{2} \alpha (1-\alpha) \|x-y\|^2.
\]
\item[(3)] The computation of the convex conjugate
\begin{equation}
\tag{A}\label{eq:pc}
d^*(s) = \max_{x \in Q} \, \left\langle s, x \right\rangle - d(x)
\end{equation}
is simple, i.\,e. the unique maximizer $x(s)$ can be easily obtained for any $s \in \mathbb{R}^n$.
\end{itemize}
\end{definition}
Auxiliary optimization problem (\ref{eq:pc})
is known to be a key ingredient for minimizing a convex function $f$ on $Q$ by subgradient schemes. Let us take for $s$ subgradients of $f$ or their weighted aggregates. Then, (\ref{eq:pc}) defines a mapping from the dual space with subgradient information $s$ into the primal space with feasible iterates $x(s) \in Q$. This idea leads in particular to primal-dual subgradient methods for convex problems \cite{nesterov:2013}.
There are at least two advantages of using prox-funcions in (\ref{eq:pc}):
\begin{itemize}
\item[(a)] It is well known that the complexity bounds for optimization methods heavily depend on the size of the feasible set $Q$. This value has been traditionally defined with respect to Euclidean norm. However, the size of $Q$, measured with respect to another norm, can be smaller. Thus, by introducing prox-functions, which are strongly convex with respect to an appropriate norm $\|\cdot\|$, it is possible to take into account a particular geometry of the feasible set $Q$.
\item[(b)] More interestingly, prox-functions often allow natural interpretations of the iteration steps (\ref{eq:pc}) within the convex optimization framework. This feature is important in order to explain agents' behavioral dynamics as being driven by unintentional optimization.
Let us illustrate this for the well known entropic prox-function
\[
d(p)= \sum_{i=1}^{n} p^{(i)} \ln p^{(i)}
\]
on the $(n-1)$-dimensional simplex
\[
\Delta=\left\{p \in \mathbb{R}^n \,\left|\, \sum_{i=1}^{n} p^{(i)} =1, p^{(i)} \geq 0\right.\right\}.
\]
For the feasible set $Q=\Delta$ the auxiliary optimization problem (A) reads:
\[
\max_{p \in \Delta} \, \sum_{i=1}^{n} s^{(i)} p^{(i)} - \sum_{i=1}^{n} p^{(i)} \ln p^{(i)}.
\]
Its unique solution is given by
\[
p^{(i)}(s) = \frac{e^{s^{(i)}}}{\displaystyle \sum_{i=1}^{n} e^{s^{(i)}}}, \quad i=1,\ldots,n.
\]
This formula is in accordance with the logit model of discrete choice: $p^{(i)}(s)$ can be viewed as the choice probability of detecting $s^{(i)}$ to be maximal among $s^{(1)}, \ldots, s^{(n)}$.
\end{itemize}
In this paper we introduce prox-functions on the simplex which allow similar probabilistic interpretation as above. They are derived from the additive random utility models of discrete choice \cite{depalma:1994}. Note that the logit model is just one prominent example within this class. Our main result in Section \ref{sec:prox} states that the convex conjugate of the surplus function, associated with an additive random utility model, is a prox-function on the simplex. For the convex conjugate of the corresponding surplus function we show continuity (Section \ref{ssec:co}), strong convexity (Section \ref{ssec:sc}), and simplicity (Section \ref{ssec:sim}). In particular, we explicitly derive the convexity parameter for the class of
generalized extreme value models \cite{mcfadden:1978}, and specifically of generalized nested logit models \cite{wen:2001}. Section \ref{sec:ea} is devoted to an economic application of discrete choice prox-functions in consumer theory. The discrete choice prox-functions are incorporated into the dual averaging scheme from \cite{nesterov:2013} for consumer's utility maximization.
This ensures that the update of internal prices for goods' qualities is due to an additive random utility model.
The dual averaging scheme corresponds to a natural consumption cycle which successively leads to an optimal consumption of goods (Section \ref{sec:cyc}). We mention that the proposed consumption cycle generalizes \cite{nesterov:2016} where the update of internal prices is due to the logit model.
{\bf Notation.} Our notation is quite standard. We denote by $\mathbb{R}^n$ the space of $n$-dimensional column vectors $x =
\left(x^{(1)}, \dots , x^{(n)}\right)^T$, by $\mathbb{R}^n_+$ the set of all vectors with nonnegative components.
If the components of $x \in \mathbb{R}^n$ are nonnegative (positive), we write $x \geq 0$ ($x > 0$). For $x \in \mathbb{R}^n$ we write $x^{(-i)} \in \mathbb{R}^{n-1}$ meaning that the $i$-th component of $x$ is missing. Analogously, we write $x^{(-i,j)} \in \mathbb{R}^{n-2}$ meaning that both $i$-th and $j$-th components of $x$ are missing.
For $x, y \in \mathbb{R}^n$ we introduce the standard scalar product and
(if additionally $y > 0$) the vector division:
\[
\langle x, y \rangle = \sum\limits_{i=1}^n x^{(i)} y^{(i)}, \quad
\frac{x}{y} = \left(\frac{x^{(1)}}{y^{(1)}}, \ldots, \frac{x^{(n)}}{y^{(n)}}\right)^T.
\]
For $x \in \mathbb{R}^n$ we use the following norms:
\[
\|x\|_1 = \sum_{i=1}^{n} \left| x^{(i)}\right|, \quad \|x\|_\infty= \max_{1 \leq i \leq n} \left| x^{(i)}\right|.
\]
Note that they are dual to each other, i.\,e.
\[
\|x\|_\infty = \sup_{\left\|y \right\|_1 \leq 1} \langle y,x\rangle, \quad
\|x\|_1 = \sup_{\left\|y \right\|_\infty \leq 1} \langle x,y\rangle.
\]
We denote by $e_j \in \mathbb{R}^n$ the $j$-th coordinate vector of $\mathbb{R}^n$. All components of the vector $e \in \mathbb{R}^n$ are equal to one.
The space of $(n \times n)$-matrices with real-valued entries is denoted by $\mathbb{R}^{n \times n}$. We use the induced matrix norm for $A \in \mathbb{R}^{n\times n}$:
\[
\left\| A\right\|_{\infty,1} = \max_{\|z\|_\infty \leq 1} \left\| A z\right\|_1.
\]
Given a twice differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}$, $\nabla f$ denotes the gradient and $\nabla^2 f$ stands for the Hessian matrix.
\section{Discrete choice prox-functions on the simplex}
\label{sec:prox}
We derive discrete choice prox-functions on the simplex from additive random utility models.
\subsection{Additive random utility models}
\label{sec:aru}
The additive random utility framework has been first introduced in economic context \cite{mcfadden:1978}. It aims to model the discrete choice from a finite number of alternatives $\{1, \ldots, n\}$ by a rational decision-maker prone to some random errors. Accordingly, the $i$-th alternative is endowed with the utility
\[
u^{(i)} + \epsilon^{(i)},
\]
where $u^{(i)} \in \mathbb{R}$ is its deterministic part and $\epsilon^{(i)}$ is a random error. We denote by
\[
u = \left(u^{(1)}, \ldots, u^{(n)}\right)^T, \quad \epsilon = \left(\epsilon^{(1)}, \ldots, \epsilon^{(n)}\right)^T
\]
the vectors of deterministic utilities and of random utility shocks, respectively.
The following assumption on the stochastic errors is standard, see e.\,g. \cite{depalma:1994}.
\begin{assumption}
\label{ass:rnd}
The random vector $\epsilon$ follows a joint distribution with finite mean that is absolutely continuous with respect to the Lebesgue measure and fully supported on $\mathbb{R}^n$.
\end{assumption}
Since a rational decision-maker chooses alternatives with the maximal utility, the corresponding surplus is given by the expectation
\[
E(u) = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right).
\]
It is well-known that the surplus function $E$ is convex and differentiable \cite{depalma:1994}. In particular, its partial derivatives can be expressed as choice probabilities:
\begin{equation}
\label{eq:der1}
\frac{\partial E(u)}{\partial u^{(i)}} = \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right), \quad i=1, \ldots, n.
\end{equation}
The latter means that the $i$-th partial derivative of $E$ corresponds to the probability of perceiving the $i$-th alternative as one with the maximal utility among the others. This result is known as the Williams-Daly-Zachary theorem in the discrete choice literature \cite{mcfadden:1978, mcfadden:1981}. The formula (\ref{eq:der1}) is valid due to the fact that, under Assumption \ref{ass:rnd}, the ties between the alternatives occur with zero-probability, i.\,e.
\[
\mathbb{P}\left(\epsilon^{(i)} - \epsilon^{(j)} = c \right) =0 \quad
\mbox{for all } i \not = j \mbox{ and } c \in \mathbb{R}.
\]
\subsection{Convex conjugate of the surplus function}
\label{sec:cc}
We turn our attention to the convex conjugate $E^*:\mathbb{R}^n \rightarrow \mathbb{R}\cup\{\infty\}$ of the surplus function:
\[
E^*(p) = \sup_{ u \in \mathbb{R}^n} \, \langle p, u \rangle - E(u),
\]
where $p=\left(p^{(1)}, \ldots, p^{(n)}\right)^T \in \mathbb{R}^n$ is the vector of dual variables.
\subsubsection{Continuity}
\label{ssec:co}
We discuss the continuity of the convex conjugate $E^*$ on its domain
\[
\mbox{dom}\, E^* = \left\{ p \in \mathbb{R}^n \,|\, E^*(p) < \infty \right\}.
\]
For that, we need some elementary properties of the surplus function $E$ listed below.
\begin{lemma}[Elementary properties of $E$]
\label{lem:el}
For the surplus function $E$ it holds:
\begin{itemize}
\item[(E1)] $E(u+\gamma e) = E(u) + \gamma$ for all $\gamma \in \mathbb{R}, u \in \mathbb{R}^n$.
\item[(E2)] $E(u) \geq E(v)$ for all $u, v \in \mathbb{R}^n$ with $u \geq v$.
\item[(E3)] $E(u) \geq \displaystyle \max_{1 \leq i \leq n} u^{(i)} + \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right)$ for all $u \in \mathbb{R}^n$.
\end{itemize}
\end{lemma}
\bf Proof: \rm \par \noindent
\begin{itemize}
\item[(E1)] The linearity of the expectation provides:
\[
E(u+\gamma e) = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} \left(u^{(i)} + \gamma + \epsilon^{(i)}\right) \right) =
\mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} \left( u^{(i)} + \epsilon^{(i)} \right) + \gamma \right)=
E(u) + \gamma.
\]
\item[(E2)] The monotonicity of the expectation provides:
\[
E(u) = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right) \geq
\mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} v^{(i)} + \epsilon^{(i)} \right) = E(v).
\]
\item[(E3)] Due to the finite mean condition from Assumption \ref{ass:rnd}, we have for every $i \in \{1, \ldots, n\}$:
\[
E(u) = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right) \geq
\mathbb{E}_\epsilon \left(u^{(i)} + \epsilon^{(i)} \right) \geq u^{(i)} + \min_{1 \leq i \leq n}\mathbb{E}_\epsilon \left(\epsilon^{(i)} \right).
\]
\end{itemize}
\hfill $\Box$ \par \noindent \medskip
\begin{theorem}[Continuity of $E^*$]
\label{th:ct}
The convex conjugate $E^*$ is continuous on its domain $\mbox{dom}\, E^*$ which coincides with the simplex $\Delta$.
\end{theorem}
\bf Proof: \rm \par \noindent Let us first show that $\mbox{dom}\, E^* \subseteq \Delta$. For $p \in \mathbb{R}^n$ with $\langle p,e\rangle \not =1$ we have:
\[
E^*(p) \geq \sup_{ \gamma \in \mathbb{R}}\, \langle p,v + \gamma e \rangle - E(v + \gamma e) \overset{\mbox{(E1)}}{=} \langle p,v \rangle - E (v) + \sup_{ \gamma \in \mathbb{R}} \, \gamma (\langle p,e \rangle - 1) = \infty,
\]
where $v \in \mathbb{R}^n$ is fixed. For $p \in \mathbb{R}^n$ with $p^{(i)} < 0$ for an $i \in \{1, \ldots, n\}$ we have:
\[
E^*(p) \geq \sup_{ \gamma \leq 0} \, \langle p, \gamma e_i \rangle - E\left(\gamma e_i\right) \overset{\mbox{(E2)}}{\geq}
\sup_{ \gamma \leq 0} \, \gamma p^{(i)} - E\left(0\right) = \infty,
\]
where $e_i$ denotes the $i$-th coordinate vector.
Secondly, we prove that $\mbox{dom}\, E^* \supseteq \Delta$. For that, it is sufficient to show that $E^*$ is bounded from above on $\Delta$. Due to (E3) from Lemma \ref{lem:el}, it holds:
\begin{equation}
\label{eq:ub}
\begin{array}{rcl}
\displaystyle \sup_{p \in \Delta} \, E^*(p) &=& \displaystyle \sup_{p \in \Delta} \left(\sup_{ u \in \mathbb{R}^n} \, \langle p, u \rangle - E(u) \right) =
\sup_{ u \in \mathbb{R}^n} \left( \sup_{p \in \Delta} \, \langle p, u \rangle - E(u) \right) \\ \\
&=& \displaystyle
\sup_{ u \in \mathbb{R}^n} \left(\max_{1 \leq i \leq n} u^{(i)} - E(u)\right) \leq - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right).
\end{array}
\end{equation}
Further, we discuss the continuity of $E^*$ on the simplex $\Delta$.
Since $E^*$ is convex, it is continuous on the relative interior $\mbox{rint}(\Delta)$ of its domain. The continuity of $E^*$ on the whole domain $\Delta$ can be deduced by an application of the Gale-Klee-Rockafellar theorem.
The Gale-Klee-Rockafellar theorem says that a convex function is upper semi-continuous at every point at
which its domain is polyhedral \cite{gkr:1968}. Note that the domain of $E^*$ -- the $(n-1)$-dimensional simplex $\Delta$ -- is polyhedral. Moreover, the convex conjugate of a function is always lower semi-continuous, so is $E^*$ on $\Delta$. Together, the lower and upper semi-continuity of $E^*$ on $\Delta$ provides the claim. \hfill $\Box$ \par \noindent \medskip
Theorem \ref{th:ct} says that the convex conjugate $E^*$ is finite on the simplex $\Delta$.
The latter can be viewed as the set of probability distributions. Hence, the dual variables $p$ can be interpreted as the probabilities attached to the alternatives $\{1, \ldots, n\}$.
\begin{corollary}[Upper bound for $E^*$]
\label{cor:ub}
The convex conjugate $E^*$ is bounded from above on its domain $\Delta$, namely it holds:
\[
E^*(p) \leq - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right) \quad \mbox{for all } p \in \Delta.
\]
\end{corollary}
\bf Proof: \rm \par \noindent The assertion follows from the derivation in (\ref{eq:ub}). \hfill $\Box$ \par \noindent \medskip
\subsubsection{Strong Convexity}
\label{ssec:sc}
We show that the convex conjugate $E^*$ is strongly convex under suitable assumptions, and estimate its convexity parameter.
\begin{definition}[Strong convexity of $E^*$] The convex conjugate $E^*: \Delta \rightarrow \mathbb{R}$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm if for all $p, q \in \Delta$ and $\alpha \in [0,1]$ we have:
\[
E^*(\alpha p +(1-\alpha)q) \leq \alpha E^*(p) + (1-\alpha) E^*(q) - \frac{\beta}{2} \alpha (1-\alpha) \|p-q\|_1^2.
\]
The positive constant $\beta$ is called the convexity parameter of $E^*$.
\end{definition}
The strong convexity of $E^*$ is closely related to the strong smoothness of $E$.
\begin{definition}[Strong smoothness of $E$] The surplus function $E: \mathbb{R}^n \rightarrow \mathbb{R}$ is $L$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$ if for all $u, v \in \mathbb{R}^n$ we have:
\[
E(u+v) \leq E(u) + \langle \nabla E(u), v \rangle + \frac{L}{2} \|v\|_\infty^2.
\]
The positive constant $L$ is called the smoothness parameter of $E$.
\end{definition}
The following duality result between the strong convexity of $E^*$ and the strong smoothness of $E$ can be easily deduced from \cite[Theorem 6]{kak:2009} shown there in the general setting.
\begin{lemma}[Strong convex/smooth duality]
\label{lem:sd}
The convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm if and only if
the surplus function $E$ is $\frac{1}{\beta}$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$.
\end{lemma}
\bf Proof: \rm \par \noindent We apply \cite[Theorem 6]{kak:2009} which says that a closed and convex function is $\beta$-strongly convex with respect to a norm if and only if its convex conjugate is $\frac{1}{\beta}$-strongly smooth with respect to the dual norm. For that, we note that $E^*$ is proper and lower semi-continuous, hence, closed. Moreover, by the Fenchel-Moreau theorem we have
\[
E^{**} = E,
\]
since $E$ is, in particular, a proper, lower semi-continuous, and convex function. Finally, the dual of the $\|\cdot\|_1$ norm is the maximum norm $\|\cdot\|_\infty$.
\hfill $\Box$ \par \noindent \medskip
In view of Lemma \ref{lem:sd}, we may focus on the strong smoothness of $E$. For the characterization of the latter property, we use the fact that the surplus function $E$ is twice differentiable. Let us compute the second order partial derivatives of $E$. Recall that its $i$-th partial derivative can be written as the choice probability
\[
\begin{array}{rcl}
\displaystyle \frac{\partial E(u)}{\partial u^{(i)}} &=& \displaystyle \mathbb{P} \left(\epsilon^{(-i)} - \epsilon^{(i)} \leq u^{(i)} -u^{(-i)} \right) \\ \\ &=& \displaystyle \int_{-\infty}^{u^{(i)} - u^{(-i)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i)} +x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i)}, \end{array}
\]
where $f_\epsilon$ is the probability density function of the random utility shocks $\epsilon$. Here, the inner integral is the probability density function of the $(n-1)$-dimensional vector of random differences $\epsilon^{(-i)}-\epsilon^{(i)}$ \cite{depalma:1994}. By differentiating this formula with respect to $u^{(j)}$ for $j\not =i$, we obtain mixed partial derivatives of $E$:
\[
\frac{\partial^2 E(u)}{\partial u^{(i)} \partial u^{(j)}} = -
\int_{-\infty}^{u^{(i)} - u^{(-i,j)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}.
\]
This integral can be interpreted as the probability density that $\epsilon^{(j)} - \epsilon^{(i)}=u^{(i)}-u^{(j)}$, and $\epsilon_{-i,j} - \epsilon^{(i)} \leq u^{(i)}-u^{(-i,j)}$, i.\,e. both alternatives $i$ and $j$ yield the maximal utility.
Analogously, we obtain the second order partial derivative of $E$ with respect to $u^{(i)}$:
\[
\frac{\partial^2 E(u)}{\partial u^{{(i)}2}} = \sum_{j \not =i}
\int_{-\infty}^{u^{(i)} - u^{(-i,j)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}.
\]
\begin{lemma}[$C^2$-characterization of strong smoothness of $E$]
\label{lem:c2} The surplus function $E$ is $L$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$ if for all $u \in \mathbb{R}^n$ it holds:
\[
\left\| \nabla^2 E(u)\right\|_{\infty,1} \leq L.
\]
\bf Proof: \rm \par \noindent For any $u, v \in \mathbb{R}^n$ we have:
\[
\nabla E(u) - \nabla E(v) = \int_0^1 \diff \nabla E(v + \tau (u-v)) =
\int_0^1 \nabla^2 E(v + \tau (u-v)) \cdot (u-v) \diff{\tau}.
\]
Hence, the gradient of $E$ is Lipschitz continuous:
\[
\begin{array}{rcl}
\left\|\nabla E(u) - \nabla E(v)\right\|_1 &\leq& \displaystyle
\int_0^1 \left\| \nabla^2 E(v + \tau (u-v))\cdot(u-v) \right\|_1 \diff{\tau} \\ \\
&\leq& \displaystyle
\int_0^1 \left\| \nabla^2 E(v + \tau (u-v))\right\|_{\infty,1} \cdot \left\|u-v\right\|_\infty \diff{\tau} \leq L \cdot \left\|u-v\right\|_\infty.
\end{array}
\]
Further, we have:
\[
E(u+v) - E(u) = \int_0^1 \diff E(u+\tau v) =
\int_0^1 \langle \nabla E(u+\tau v), v\rangle \diff \tau.
\]
Due to the Lipschitz continuity of $\nabla E$, we obtain:
\[
\begin{array}{rcl}
E(u+v) - E(u) - \langle \nabla E(u), v\rangle &=& \displaystyle \int_0^1 \langle \nabla E(u+\tau v) - \nabla E(u), v\rangle \diff \tau \\ \\
&\leq& \displaystyle \int_0^1 \left\| \nabla E(u+\tau v) - \nabla E(u) \right\|_1 \cdot \left\|v\right\|_\infty \diff \tau \\ \\
&\leq& \displaystyle
\int_0^1 L \cdot \left\| u+\tau v - u \right\|_\infty \cdot \left\|v\right\|_\infty \diff \tau = \frac{L}{2} \left\|v\right\|^2_\infty.
\end{array}
\]
\hfill $\Box$ \par \noindent \medskip
\end{lemma}
Now, let us consider the set $\mathcal{A}$ of symmetric matrices $A=\left(a_{ij}\right) \in \mathbb{R}^{n\times n}$ satisfying:
\begin{itemize}
\item[(A1)] $\displaystyle a_{ii} \geq 0$ for all $i = 1, \ldots, n$, and $a_{ij} \leq 0$ for all $i \not = j$,
\item[(A2)] $\displaystyle a_{ii} + \sum_{j\not = i}^{} a_{ij} = 0$ for all $i = 1, \ldots, n$.
\end{itemize}
Note that the set $\mathcal{A}$ is closed under matrix addition and multiplication by nonnengative scalars, i\,e. for any $A,B \in \mathcal{A}$ and $\lambda \geq 0$ it holds:
\[
A+B, \lambda A \in \mathcal{A}.
\]
Moreover, the Hessian matrix $\nabla^2 E(u)$ of the surplus function is an element of $\mathcal{A}$.
\begin{lemma}[Representation of $\|\cdot\|_{\infty,1}$]
\label{lem:nr}
For $A \in \mathcal{A}$ it holds:
\begin{equation}
\|A\|_{\infty,1} = 4 \max \, \left\{ \left.\sum_{i,j \in K} a_{ij} \,\right|\, K \subset \{1, \ldots, n\}, |K| \leq \left\lfloor \frac{n}{2} \right\rfloor \right\}.
\end{equation}
\bf Proof: \rm \par \noindent The maximum in the definition of the matrix norm
\[
\|A\|_{\infty,1} = \max_{\|z\|_\infty \leq 1} \left\| A z\right\|_1
\]
is attained at some vertex of the feasible set $\left\{z \in \mathbb{R}^n \,\left|\, \|z\|_\infty \leq 1 \right. \right\}$. These are the vectors
\[
z_K = e_K - e_{K^c},
\]
where $e_K \in \mathbb{R}^n$ is the indicator vector of the subset $K\subset \{1, \ldots, n\}$, i.\,e. $e_K^{(i)}=1$ if $i \in K$, and $e_K^{(i)}=0$ if $i \not \in K$.
We may restrict the choice of $K$ by the condition $|K| \leq \left\lfloor \frac{n}{2} \right\rfloor$, since
\[
\left\| A z_{K^c}\right\|_1 =
\left\| A \left(-z_{K}\right)\right\|_1 =
\left\| A z_{K}\right\|_1.
\]
For such a fixed subset $K$ we compute
\[
\begin{array}{rcl}
\displaystyle \left\| A z_{K}\right\|_1 &=& \displaystyle \sum_{i \in K} \left| a_{ii} + \sum_{j \in K\backslash \{i\}} a_{ij} - \sum_{j \in K^c} a_{ij} \right|+ \sum_{i \in K^c} \left| \sum_{j \in K} a_{ij} - a_{ii} - \sum_{j \in K^c\backslash \{i\}} a_{ij} \right| \\ \\
&\overset{\mbox{(A2)}}{=}& \displaystyle 2 \sum_{i \in K} \left| a_{ii} + \sum_{j \in K\backslash \{i\}} a_{ij} \right|+ 2 \sum_{i \in K^c} \left| \sum_{j \in K} a_{ij} \right| \\ \\
&\overset{\mbox{(A1)}}{=}& \displaystyle 2 \sum_{i \in K} \left(a_{ii} + \sum_{j \in K\backslash \{i\}} a_{ij} \right)- 2 \sum_{i \in K^c} \sum_{j \in K} a_{ij} \\ \\
&=& \displaystyle
2 \sum_{j \in K} \left(a_{jj} + \sum_{i \in K\backslash \{j\}} a_{ij} - \sum_{i \in K^c} a_{ij} \right) \\ \\
&\overset{\mbox{(A2)}}{=}& \displaystyle
2 \sum_{j \in K} \left(2 a_{jj} + 2 \sum_{i \in K\backslash \{j\}} a_{ij} \right) =
4 \sum_{i,j \in K} a_{ij}.
\end{array}
\]
\hfill $\Box$ \par \noindent \medskip
\end{lemma}
We use Lemma \ref{lem:nr} to estimate the $\|\cdot\|_{\infty,1}$ norm on a particular subset of $A$, which will appear in what follows.
\begin{corollary}
\label{cor:pp}
For $p\in \Delta$ we define the matrix $R={\rm diag \,}{p} - p\cdot p^T$. It holds for the latter:
\[
R \in \mathcal{A}, \quad \|R\|_{\infty,1} \leq 1.
\]
\bf Proof: \rm \par \noindent The symmetric matrix $R=(r_{ij})$ fulfills (A1), since in view of $p \in \Delta$ it holds for all $i = 1, \ldots, n$, and $i \not = j$:
\[
r_{ii}=p^{(i)}\left(1-p^{(i)}\right) \geq 0, \quad r_{ij}=-p^{(i)}p^{(j)} \leq 0.
\]
It also fulfills (A2) due to the following derivation:
\[
R \cdot e = {\rm diag \,}{p}\cdot e - p\cdot p^T \cdot e = p - p= 0.
\]
Let us fix a subset of indices $K \subset \left\{1, \ldots,n \right\}$.
We estimate the following expression from Lemma \ref{lem:nr} uniformly for all $p \in \Delta$:
\[
\sum_{i,j \in K} r_{ij} = \sum_{i \in K} p^{(i)} \left( 1- \sum_{j \in K} p^{(j)} \right).
\]
For that, let us solve the maximization problem
\[
\max \, \sum_{i \in K} p^{(i)} \left( 1- \sum_{i \in K} p^{(i)} \right) \quad \mbox{s. t.} \quad \sum_{i \in K} p^{(i)} \leq 1, \quad p^{(i)} \geq 0 \mbox{ for all } i \in K.
\]
Without loss of generality, we may assume that for its solution holds:
\[
\sum_{i \in K} p^{(i)} < 1, \quad p^{(i)} > 0 \mbox{ for all } i \in K.
\]
Otherwise, the optimal value vanishes or we pass over to a smaller subset of indices. The first order optimality condition reads:
\[
1- \sum_{i \in K} p^{(i)} - \sum_{i \in K} p^{(i)} =0.
\]
Hence, we get $\displaystyle \sum_{i \in K} p^{(i)} = \frac{1}{2}$,
and the optimal value is
\[
\sum_{i \in K} p^{(i)} \left( 1- \sum_{i \in K} p^{(i)} \right) =
\sum_{i \in K} p^{(i)} \left( 1- \frac{1}{2} \right) = \frac{1}{4}.
\]
The application of Lemma \ref{lem:nr} provides the assertion. \hfill $\Box$ \par \noindent \medskip
\end{corollary}
\begin{lemma}[Estimation of $\|\cdot\|_{\infty,1}$]
\label{lem:tr}
For $A \in \mathcal{A}$ it holds:
\[
\|A\|_{\infty,1} \leq 2 \tr{A},
\]
where $\tr{A}$ denotes the trace of the matrix $A$.
\bf Proof: \rm \par \noindent For any $z\in \mathbb{R}^n$ it holds:
\[
\left\| A z\right\|_1 = \sum_{i=1}^{n} \left| \sum_{j=1}^{n} a_{ij} z^{(j)} \right|
\leq \sum_{i=1}^{n} \sum_{j=1}^{n} \left|a_{ij}\right| \cdot \left|z^{(j)}\right|
\leq \sum_{i=1}^{n} \sum_{j=1}^{n} \left|a_{ij}\right| \cdot \left\|z\right\|_\infty.
\]
Additionally, we have:
\[
\sum_{i=1}^{n} \sum_{j=1}^{n} \left|a_{ij}\right|
\overset{\mbox{(A1)}}{=} \sum_{i=1}^{n} \left( a_{ii} - \sum_{j\not = i}^{} a_{ij} \right) \overset{\mbox{(A2)}}{=} \sum_{i=1}^{n} \left( a_{ii} + a_{ii} \right) = 2 \tr{A}.
\]
Overall, we obtain the inequality:
\[
\|A\|_{\infty,1} = \max_{\|z\|_\infty \leq 1} \left\| A z\right\|_1 \leq
\max_{\|z\|_\infty \leq 1} 2 \tr{A} \left\|z\right\|_\infty = 2 \tr{A}.
\]
\hfill $\Box$ \par \noindent \medskip
\end{lemma}
Now, we are ready to state the general result on the strong convexity of $E^*$. It is given in terms of the differences $\epsilon^{(j)} - \epsilon^{(i)}$, $i \not = j$, of random utility shocks. We recall that the density function of $\epsilon^{(j)} - \epsilon^{(i)}$ can be written as
\[
g_{i,j} \left(z\right)= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, z +x^{(i)},x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}.
\]
Any point $\bar z_{i,j} \in \mathbb{R}$ which maximizes the density function $g_{i,j}$ is called a mode of the random variable $\epsilon^{(j)} - \epsilon^{(i)}$.
\begin{theorem}[Strong convexity of $E^*$]
\label{th:sc}
Let the differences $\epsilon^{(j)} - \epsilon^{(i)}$ of random utility shocks have modes $\bar z_{i,j} \in \mathbb{R}$, $i \not = j$.
Then, the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
\beta = \frac{1}{\displaystyle 2\sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right)}.
\]
\bf Proof: \rm \par \noindent
We estimate the second order derivative of the surplus function $E$ with respect to $u^{(i)}$:
\[
\begin{array}{rcl}
\displaystyle \frac{\partial^2 E(u)}{\partial u^{{(i)}2}} &=& \displaystyle \sum_{j \not =i} \int_{-\infty}^{u^{(i)} - u^{(-i,j)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}\\ \\
&\leq& \displaystyle \sum_{j \not =i} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)} \\ \\
&=& \displaystyle \sum_{j \not =i} g_{i,j} \left(u^{(i)}-u^{(j)}\right) \leq
\sum_{j \not =i} g_{i,j} \left(\bar z_{i,j}\right).
\end{array}
\]
Hence, we obtain for the trace of $\nabla^2 E$ the following inequality:
\[
\tr{\nabla^2 E(u)} = \sum_{i=1}^{n} \frac{\partial^2 E(u)}{\partial u^{{(i)}2}} \leq
\sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right).
\]
In view of Lemma \ref{lem:tr}, whose application is justified by the fact that the matrix $\nabla^2 E(u)$ fulfills (A1)--(A2), it holds:
\[
\left\|\nabla^2 E(u)\right\|_{\infty,1} \leq 2 \tr{\nabla^2 E(u)} \leq 2 \sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right) = \frac{1}{\beta}.
\]
The latter provides, due to Lemma \ref{lem:c2}, that the surplus function $E$ is $\frac{1}{\beta}$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$. Finally, we apply Lemma \ref{lem:sd} to conclude that the convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm.\hfill $\Box$ \par \noindent \medskip
\end{theorem}
\begin{remark}[Existence of modes] \textup{We note that the condition on the existence of modes in Theorem \ref{th:sc} cannot be weakened. This can be seen already in case of two alternatives, i.\,e. $n=2$. Then, the second order derivative of the surplus function $E$ is
\[
\nabla^2 E(u) = \left( \begin{array}{rr}
g_{1,2}\left(u^{(1)}-u^{(2)}\right) & -g_{1,2}\left(u^{(1)}-u^{(2)}\right) \\
-g_{2,1}\left(u^{(2)}-u^{(1)}\right) & g_{2,1}\left(u^{(2)}-u^{(1)}\right)
\end{array}\right).
\]
After a moment of reflection we realize that
\[
g_{1,2}\left(u^{(1)}-u^{(2)}\right) = g_{2,1}\left(u^{(2)}-u^{(1)}\right).
\]
Due to Lemma \ref{lem:nr}, it holds:
\[
\left\|\nabla^2 E(u)\right\|_{\infty,1} = 4 g_{1,2}\left(u^{(1)}-u^{(2)}\right).
\]
From now on we assume that $g_{1,2}$ is continuous.
Hence, the reverse implication in Lemma \ref{lem:c2} becomes valid, and $E$ is strongly smooth -- or, equivalently, $E^*$ is strongly convex -- if and only if the density function $g_{1,2}$ is bounded on $\mathbb{R}$. The latter property can be characterized by the existence of a mode $\bar z_{1,2}$ of $\epsilon^{(2)}-\epsilon^{(1)}$. Additionally, the convexity parameter of $E^*$ from Theorem \ref{th:sc} can be expressed as
\[
\beta = \frac{1}{\displaystyle 2\sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right)} = \frac{1}{\displaystyle 4 g_{1,2}\left(\bar z_{1,2}\right)} = \frac{1}{\left\|\nabla^2 E\left(\bar u\right)\right\|_{\infty,1}},
\]
where the utilities $\bar u = \left(\bar u^{(1)},\bar u^{(2)} \right)^T$ are chosen to satisfy $\bar u^{(1)}-\bar u^{(2)}=\bar z_{1,2}$. This formula provides that the convexity parameter of $E^*$ cannot be larger than $\beta$.}\hfill $\Box$ \par \noindent \medskip
\end{remark}
The estimation of the convexity parameter of $E^*$ in Theorem \ref{th:sc} is rather pessimistic when the number of alternatives increases. The dependence of $\beta$ on $n$ becomes explicit in case of independent and identically distributed random utility shocks.
\begin{corollary}[Strong convexity of $E^*$ for IID utility shocks]
\label{cor:iid}
Let the random utility shocks $\epsilon^{(i)}$, $i=1,\ldots,n$, be independent and identically distributed with the common probability density function $f$ having a mode $\bar z \in \mathbb{R}$.
Then, the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
\beta = \frac{1}{\displaystyle 2n(n-1) f\left(\bar z\right)}.
\]
\bf Proof: \rm \par \noindent
We estimate the probability distribution function of $\epsilon^{(j)}-\epsilon^{(i)}$:
\[
\begin{array}{rcl}
\displaystyle g_{i,j} \left(z\right) &=&
\displaystyle \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, z +x^{(i)},x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)} \\ \\
&=&
\displaystyle
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \prod_{k \not = i,j} f\left(y_k +x^{(i)}\right) f\left(z+x^{(i)}\right) f\left(x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)} \\ \\
&=&
\displaystyle
\int_{-\infty}^{\infty} f\left(z+x^{(i)}\right) f\left(x^{(i)}\right) \prod_{k \not = i,j}\int_{-\infty}^{\infty} f\left(y_k +x^{(i)}\right) \diff y_{k} \diff x^{(i)}.
\end{array}
\]
For all $k\not =i,j$, and $x^{(i)} \in \mathbb{R}$ we have:
\[
\int_{-\infty}^{\infty} f\left(y_k +x^{(i)}\right) \diff y_{k} = 1.
\]
Moreover, $f\left(z+x^{(i)}\right) \leq f\left(\bar z \right)$ for all $x^{(i)} \in \mathbb{R}$. Altogether, we get
\[
\displaystyle g_{i,j} \left(z\right) \leq f\left(\bar z \right).
\]
The application of Theorem \ref{th:sc} yields the assertion.
\hfill $\Box$ \par \noindent \medskip
\end{corollary}
In what follows we concentrate on some special distributions of random utility shocks widely used in the discrete choice literature. For them we obtain better estimations of the convexity parameter of $E^*$, in particular, not dependent on the number of alternatives. First, we consider the class of generalized extreme value models \cite{mcfadden:1978}. The vector $\epsilon= \left(\epsilon^{(1)}, \ldots, \epsilon^{(n)}\right)^T$ of random utility shocks defines a generalized extreme value model (GEV) if it follows the joint distribution given by the probability density function
\[
f_\epsilon\left(y^{(1)}, \ldots, y^{(n)}\right) = \frac{\partial^n \exp\left(-G\left(e^{-y^{(1)}},\ldots, e^{-y^{(n)}}\right)\right)}{\partial y^{(1)} \cdots \partial y^{(n)}},
\]
where the generating function $G:\mathbb{R}^n_+ \rightarrow \mathbb{R}_+$ has the following properties:
\begin{itemize}
\item[(G1)] $G$ is homogeneous of degree $\nicefrac{1}{\mu} > 0$.
\item[(G1)] $G\left(x^{(1)}, \ldots, x^{(i)}, \ldots, x^{(n)}\right) \rightarrow \infty$ as $x^{(i)} \rightarrow \infty$, $i=1, \ldots,n$.
\item[(G3)]
For the partial derivatives of $G$ with respect to $k$ distinct variables it holds:
\[
\frac{\partial^k G\left(x^{(1)},\ldots, x^{(n)}\right)}{\partial x^{\left(i_1\right)} \cdots \partial x^{\left(i_k\right)}} \geq 0 \mbox{ if } k \mbox{ is odd}, \quad
\frac{\partial^k G\left(x^{(1)},\ldots, x^{(n)}\right)}{\partial x^{\left(i_1\right)} \cdots \partial x^{\left(i_k\right)}} \leq 0 \mbox{ if } k \mbox{ is even}.
\]
\end{itemize}
It is well known from \cite{mcfadden:1978} that the surplus function for GEV is
\[
E(u) = \mu \ln G\left(e^{u}\right) + \mu \gamma,
\]
where $\gamma$ is Euler's constant and we set $e^u=\left(e^{u^{(1)}}, \ldots, e^{u^{(n)}}\right)^T$ for the sake of brevity.
The choice probability of the $i$-th alternative is given by the $i$-th partial derivative of the GEV surplus function $E$:
\[
\mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)=\frac{\partial E(u)}{\partial u^{(i)}} = \mu \frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}}\cdot \frac{e^{u^{(i)}}}{G\left(e^{u}\right)}.
\]
We state a sufficient condition for the strong convexity of $E^*$ in terms of the generating function $G$.
\begin{theorem}[Strong convexity of $E^*$ for GEV]
\label{th:gev}
Let a generating function $G$ for GEV satisfy the following inequality for all $x=\left(x^{(1)}, \ldots, x^{(n)}\right)^T\in \mathbb{R}^n_+$:
\[
\sum_{i=1}^{n} \frac{\partial^2 G(x)}{\partial x^{{(i)}2}} \cdot x^{{(i)}2} \leq M \cdot G(x)
\]
with some constant $M \in \mathbb{R}$. Then, the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
\beta = \frac{1}{2(\mu M +1)-\nicefrac{1}{\mu}}.
\]
\bf Proof: \rm \par \noindent
For the second order partial derivative of $E$ with respect to $u^{(i)}$ it holds:
\[
\begin{array}{rcl}
\displaystyle \frac{\partial^2 E(u)}{\partial u^{{(i)}2}} &=& \displaystyle \mu \left( \frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}}\cdot \frac{e^{u^{(i)}}}{G\left(e^{u}\right)} \left(1-\frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}} \cdot\frac{e^{u^{(i)}}}{G\left(e^{u}\right)} \right)
+
\frac{\partial G^2\left(e^{u}\right)}{\partial x^{{(i)}2}}\cdot\frac{\left(e^{u^{(i)}}\right)^2}{G\left(e^{u}\right)} \right) \\ \\
&=& \displaystyle \frac{1}{\mu}\frac{\partial E(u)}{\partial u^{(i)}} \left( 1 - \frac{\partial E(u)}{\partial u^{(i)}}\right) + \left(1-\frac{1}{\mu}\right)\frac{\partial E(u)}{\partial u^{(i)}}
+ \mu
\frac{\partial G^2\left(e^{u}\right)}{\partial x^{{(i)}2}}\cdot\frac{\left(e^{u^{(i)}}\right)^2}{G\left(e^{u}\right)}.
\end{array}
\]
Analogously, we obtain the mixed partial derivative of $E$ for $j\not = i$:
\[
\begin{array}{rcl}
\displaystyle
\frac{\partial^2 E(u)}{\partial u^{(i)} \partial u^{(j)}} &=& \displaystyle
\mu \left( -\frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}} \frac{e^{u^{(i)}}}{G\left(e^{u}\right)} \cdot \frac{\partial G\left(e^{u}\right)}{\partial x^{(j)}} \frac{e^{u^{(j)}}}{G\left(e^{u}\right)}
+
\frac{\partial G^2\left(e^{u}\right)}{\partial x^{(i)} \partial x^{(j)}}\cdot \frac{e^{u^{(i)}} e^{u^{(j)}}}{G\left(e^{u}\right)} \right) \\ \\
&=&\displaystyle -\frac{1}{\mu}\frac{\partial E(u)}{\partial u^{(i)}} \cdot \frac{\partial E(u)}{\partial u^{(j)}}
+ \mu
\frac{\partial G^2\left(e^{u}\right)}{\partial x^{(i)} \partial x^{(j)}}\cdot\frac{e^{u^{(i)}} e^{u^{(j)}}}{G\left(e^{u}\right)}.
\end{array}
\]
Equivalently, we have in matrix form:
\[
\nabla^2 E(u) = \frac{1}{\mu} R(u) + S(u),
\]
where
\[
\begin{array}{rcl}
R(u) &= & \displaystyle {\rm diag \,}{\nabla E(u)} - \nabla E(u) \cdot \nabla^T E(u), \\ \\
S(u) &=& \displaystyle \left(1-\frac{1}{\mu}\right) {\rm diag \,}{\nabla E(u)} + \mu \frac{{\rm diag \,}{e^u} \cdot \nabla^2 G\left(e^u\right)\cdot {\rm diag \,}{e^u}}{G\left(e^u\right)}.
\end{array}
\]
Since $\nabla E(u) \in \Delta$, we may apply Corollary \ref{cor:pp} to derive that
\[
R(u) \in \mathcal{A}, \quad \|R(u)\|_{\infty,1} \leq 1.
\]
In view of $\nabla^2E(u), R(u) \in \mathcal{A}$, the matrix $S(u)$ fulfills (A2):
\[
S(u) \cdot e = \underbrace{\nabla^2 E(u)\cdot e}_{=0} - \frac{1}{\mu} \underbrace{R(u)\cdot e}_{=0} = 0.
\]
The off-diagonal entries of $S(u)$ are nonpositive due to (G3), hence, $S(u)$ fulfills also (A1). We apply Lemma \ref{lem:tr} to estimate the $\|\cdot\|_{\infty,1}$ norm of $S(u)\in \mathcal{A}$:
\[
\begin{array}{rcl}
\displaystyle
\|S(u)\|_{\infty,1} &\leq& 2 \tr{S(u)} \\ \\ &=& \displaystyle 2 \left(\left(1-\frac{1}{\mu}\right) \sum_{i=1}^{n} \frac{\partial E(u)}{\partial u^{(i)}} + \mu \sum_{i=1}^{n} \frac{\partial G^2\left(e^{u}\right)}{\partial x^{{(i)}2}}\cdot\frac{\left(e^{u^{(i)}}\right)^2}{G\left(e^{u}\right)}\right) \\ \\
&\leq& \displaystyle 2 \left(\left(1-\frac{1}{\mu}\right) + \mu M \right).
\end{array}
\]
Overall, we have:
\[
\|\nabla^2 E(u)\|_{\infty,1} \leq \frac{1}{\mu} \|R(u)\|_{\infty,1} + \|S(u)\|_{\infty,1} \leq \frac{1}{\mu} + 2 \left(\left(1-\frac{1}{\mu}\right) + \mu M \right)=\frac{1}{\beta}.
\]
\end{theorem}
The latter provides, due to Lemma \ref{lem:c2}, that the surplus function $E$ is $\frac{1}{\beta}$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$. Finally, we apply Lemma \ref{lem:sd} to conclude that the convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm.\hfill $\Box$ \par \noindent \medskip
Now, we consider generalized nested logit models (GNL) introduced in \cite{wen:2001}. GNL is a particular class of GEV models defined by the generating function
\[
G(x)= \sum_{\ell \in L} \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}}.
\]
Here, $L$ is a generic set of nests. The parameters $\sigma_{i\ell} \geq 0$ denote the shares of the $i$-th alternative with which it is attached to the $\ell$-th nest. For any fixed $i \in \{1, \ldots,n\}$ they sum up to one:
\[
\sum_{\ell \in L} \sigma_{i\ell} = 1,
\]
and $\sigma_{i\ell}=0$ means that the $\ell$-th nest does not contain the $i$-th alternative. Hence, the set of alternatives within the $\ell$-th nest is
\[
N_\ell = \left\{i \,|\, \sigma_{i\ell} >0\right\}.
\]
The nest parameters $\mu_\ell > 0$ describe the variance of the random errors while choosing alternatives within the $\ell$-th nest. Analogously, $\mu >0$ describes the variance of the random errors while choosing among the nests. For the function $G$ to fulfill (G1)-(G3) we require:
\[
\mu_\ell \leq \mu \quad \mbox{for all } \ell \in L.
\]
The underlying choice process can be viewed to comprise two stages:
\begin{itemize}
\item[(1)] the probability of choosing the $\ell$-th nest is
\[
q_\ell = \frac{e^{\nicefrac{v_\ell}{\mu}}}{\displaystyle
\sum_{\ell \in L}e^{\nicefrac{v_\ell}{\mu}}},
\]
where
\[
v_\ell = \mu_\ell \ln \left( \sum_{i =1}^{n} \left(\sigma_{i\ell} \cdot e^{u^{(i)}}\right)^{\nicefrac{1}{\mu_\ell}} \right)
\]
stands for the utility attached to the $\ell$-th nest;
\item[(2)] the probability of choosing the $i$-th alternative within the $\ell$-th nest is
\[
p_{i\ell} = \frac{\left(\sigma_{i\ell} \cdot e^{u^{(i)}}\right)^{\nicefrac{1}{\mu_\ell}}}{\displaystyle
\sum_{i=1}^{n} \left(\sigma_{i\ell} \cdot e^{u^{(i)}}\right)^{\nicefrac{1}{\mu_\ell}}}.
\]
\end{itemize}
Overall, the choice probability of the $i$-th alternative according to GNL amounts to
\[
\mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)= \mu \frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}}\cdot\frac{e^{u^{(i)}}}{G\left(e^{u}\right)}= \sum_{\ell \in L} q_\ell \cdot p_{i\ell}.
\]
\begin{corollary}[Strong convexity of $E^*$ for GNL]
\label{cor:gnl}
For GNL the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
\beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-\nicefrac{1}{\mu}}.
\]
\bf Proof: \rm \par \noindent Let us estimate the constant $M$ from Theorem \ref{th:gev}. We have:
\[
\frac{\partial G\left(x\right)}{\partial x^{(i)}}= \frac{1}{\mu}\sum_{\ell \in L} \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1}
\left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-1} \cdot \sigma_{i\ell},
\]
and, further:
\[
\begin{array}{rcl}
\displaystyle \frac{\partial^2 G(x)}{\partial x^{{(i)}2}} &=& \displaystyle \frac{1}{\mu}\sum_{\ell \in L}
\frac{1}{\mu_\ell}\left(\frac{\mu_\ell}{\mu}-1\right)
\left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-2}
\left(\left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-1} \cdot\sigma_{i\ell}\right)^2 \\ \\
&+& \displaystyle \frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1}
\left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-2} \cdot\sigma_{i\ell}^2.
\end{array}
\]
Due to $\mu_\ell \leq \mu$, $\ell \in L$, we get:
\[
\frac{\partial^2 G(x)}{\partial x^{{(i)}2}} \leq \frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1}
\left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-2} \cdot\sigma_{i\ell}^2.
\]
We multiply these terms by $x^{{(i)}2}$ and sum up over $i=1, \ldots, n$:
\[
\begin{array}{rcl}
\displaystyle
\sum_{i=1}^{n} \frac{\partial^2 G(x)}{\partial x^{{(i)}2}} \cdot x^{{(i)}2} &\leq& \displaystyle
\frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1} \\ \\ &&\displaystyle \cdot
\sum_{i=1}^{n}\left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-2} \cdot \sigma_{i\ell}^2 \cdot x^{{(i)}2} \\ \\
&=& \displaystyle \frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}}\\ \\
&\leq& \displaystyle\frac{1}{\mu} \max_{\ell \in L} \left( \frac{1}{\mu_\ell} -1\right) \cdot G(x)= \frac{1}{\mu} \left( \frac{1}{\displaystyle \min_{\ell \in L}\mu_\ell} -1\right) \cdot G(x).
\end{array}
\]
Hence, we may apply Theorem \ref{th:gev} with
\[
M = \frac{1}{\mu} \left( \frac{1}{\displaystyle \min_{\ell \in L}\mu_\ell} -1\right).
\]
\hfill $\Box$ \par \noindent \medskip
\end{corollary}
\begin{example}[Multinomial logit] \textup{Let in GNL there be just one nest, i.\,e. $L = \{1\}$, and $\mu_1 = \mu$. Then, the generating function
\[
G(x)= \sum_{i =1}^{n} \left(x^{(i)}\right)^{\nicefrac{1}{\mu}}
\]
leads to the multinomial logit (ML). The corresponding surplus function is
\[
E(u) = \mu \ln \sum_{i =1}^{n} e^{\nicefrac{u^{(i)}}{\mu}} + \mu \gamma,
\]
and the choice probabilities are
\[
\mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)= \frac{e^{\nicefrac{u^{(i)}}{\mu}}}{\displaystyle
\sum_{i=1}^{n}e^{\nicefrac{u^{(i)}}{\mu}}}, \quad i=1, \ldots, n.
\]
Note that ML can be deduced from the IID random utility shocks $\epsilon^{(i)}$, $i=1, \ldots,n$, each of them following the Gumbel distribution with zero mode and variance $\nicefrac{\mu \pi}{\sqrt{6}}$ \cite{depalma:1994}. For ML the convex conjugate of the surplus function can be explicitly given:
\[
E^*(p) = \mu \sum_{i=1}^{n} p^{(i)} \ln p^{(i)} - \mu \gamma = \mu H(p) - \mu \gamma,
\]
where $H$ is the (negative) entropy. It is well known that $H$ is $1$-strongly convex with respect to the $\|\cdot\|_1$ norm due to the Pinsker inequality. Hence, $E^*$ is $\mu$-strongly convex with respect to the $\|\cdot\|_1$ norm. The same result also follows from Corollary \ref{cor:gnl} with the convexity parameter
\[
\beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-\nicefrac{1}{\mu}}= \frac{1}{\nicefrac{2}{\mu}-\nicefrac{1}{\mu}}= \mu.
\]
} \hfill $\Box$ \par \noindent \medskip
\end{example}
\begin{example}[Nested logit]
\textup{Let in GNL for every alternative $i \in \{1,\ldots,n\}$ there be a unique nest $\ell_i \in L$ with $\sigma_{i \ell_i}=1$, and $\mu=1$. Then,
the nests $N_\ell=\left\{i \,|\, \ell_i=\ell\right\}$ are mutually exclusive, and
the generating function
\[
G(x)=\sum_{\ell \in L} \left( \sum_{i \in N_\ell} x^{{(i)}\nicefrac{1}{\mu_\ell}} \right)^{\mu_\ell}
\]
leads to the nested logit (NL). The corresponding surplus function is
\[
E(u) = \mu \ln \sum_{\ell \in L} \left( \sum_{i \in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}} \right)^{\mu_\ell} + \mu \gamma,
\]
and the choice probabilities for $i \in N_\ell$, $\ell \in L$ are
\[
\mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)=
\frac{e^{\mu_\ell \ln \sum_{i \in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}}}}{\displaystyle
\sum_{\ell \in L} e^{\mu_\ell \ln \sum_{i \in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}}}}\cdot \frac{e^{\nicefrac{u^{(i)}}{\mu_\ell}}}{\displaystyle
\sum_{i\in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}}}.
\]
For NL the convex conjugate of the surplus function is explicitly given in \cite{fosgerau:2017}:
\[
E^*(p) = \sum_{\ell \in L} \mu_\ell \sum_{i\in N_\ell} p^{(i)} \ln p^{(i)}
+ \sum_{\ell \in L} \left(1-\mu_\ell\right) \left(\sum_{i\in N_\ell} p^{(i)} \right) \ln \left(\sum_{i\in N_\ell} p^{(i)} \right)
- \mu\gamma.
\]
By examining the first part of this formula, it can be shown that $E^*$ is $\displaystyle \left(\min_{\ell \in L} \mu_\ell\right)$-strongly convex with respect to the $\|\cdot\|_1$ norm. Corollary \ref{cor:gnl} provides the convexity parameter, which is of the same order:
\[
\beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-\nicefrac{1}{\mu}}= \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-1} > \frac{1}{2} \min_{\ell \in L} \mu_\ell.
\]
} \hfill $\Box$ \par \noindent \medskip
\end{example}
\begin{example}[GNL's from literature] \textup{For other specifications of GNL -- except of ML and NL -- the explicit form of the convex conjugate $E^*$ is not known yet. In this case Corollary \ref{cor:gnl} can be applied in order to estimate the convexity parameter of $E^*$.
\begin{itemize}
\item[(i)] {\bf Ordered GEV} \cite{small:1987} is a GNL model with
\[
L=\{ 1, \ldots, n+m\}, \quad \mu=1,
\]
\[
\sigma_{i\ell}>0 \mbox{ for all } \ell \in \{i, \ldots,i+m\}, \quad
\sigma_{i\ell}=0 \mbox{ for all } \ell \in L \backslash \{i, \ldots,i+m\}.
\]
There are $n+m$ overlapping nests $N_\ell=\left\{i \,|\, \ell-m \leq i \leq \ell\right\}$, and every alternative lies exactly in $m+1$ of them, namely $i \in N_\ell$ for $\ell=i, \ldots, i+m$. Then,
the generating function is
\[
G(x)=\sum_{\ell=1}^{n+m} \left( \sum_{i \in N_\ell} \left(\sigma_{i\ell} x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\mu_\ell}.
\]
\item[(ii)] {\bf Paired combinatorial logit} \cite{koppelman:2000} is a GNL model with
\[
L=\{ (i,j)\in \{1,\ldots, n\} \,|\, i \not = j\}, \quad \mu=1,
\]
\[
\sigma_{i\ell}=
\left\{ \begin{array}{cl}
\displaystyle\frac{1}{2(n-1)} & \mbox{if } \ell = (i,j), (j,i) \mbox{ with } j \not =i, \\
0 & \mbox{else.}
\end{array}
\right.
\]
There are $n^2-n$ nests corresponding to the pairs of alternatives, and every alternative lies in $2(n-1)$ of them. Then,
the generating function is
\[
G(x)=\sum_{\ell=(i,j), i \not =j} \left( \left(\sigma_{i\ell} x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} + \left(\sigma_{j\ell} x^{(j)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\mu_\ell}.
\]
\item[(iii)] {\bf Principles of differentiation GEV} \cite{bresnahan:1997} is a GNL model with
\[
L = \mathop{\dot{\bigcup}}_{d \in D} L_d, \quad \mu=1, \quad
\mu_\ell = \mu_d \mbox{ for all } \ell \in L_d,
\]
\[
\sigma_{i\ell}=
\left\{ \begin{array}{cl}
\displaystyle \sigma_d & \mbox{if } i \in N_{\ell d} \mbox{ and } \ell \in L_d, \\
0 & \mbox{else,}
\end{array}
\right.
\]
where
\[
\{1,\ldots,n\}=\mathop{\dot{\bigcup}}_{\ell \in L_d} N_{\ell d}.
\]
The set $D$ represents the dimensions of alternatives. Along the $d$-th dimension alternatives can be clustered into the disjoint nests $N_{\ell d}$, $\ell \in L_d$. The shares $\sigma_{i\ell}$ of all alternatives within these nests depend only on the dimension $d$. The parameters $\mu_\ell$ also coincide for all $\ell \in L_d$.
Then, the generating function is
\[
G(x)=\sum_{d \in D} \sigma_d \sum_{\ell \in L_d} \left( \sum_{i\in N_{\ell d}} \left(x^{(i)}\right)^{\nicefrac{1}{\mu_d}} \right)^{\mu_d}.
\]
\end{itemize}
Note that the convexity parameter of $E^*$ for GNL specifications (i)-(iii) depends only on the smallest of the nest parameters $\mu_\ell$, $\ell \in L$:
\[
\beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-1}.
\]
\hfill $\Box$ \par \noindent \medskip
}
\end{example}
Let us relate our results on the strong convexity of $E^*$ to those existing in the literature.
\begin{remark}[Cross moment model]
In CMM \cite{mishra:2012} the random vector $\epsilon \sim (0,\Sigma)$
follows a joint distribution with zero mean and a given covariance matrix $\Sigma$. Additionally, it maximizes the surplus function:
\[
Z(u) = \max_{\epsilon \sim (0,\Sigma)} \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right).
\]
Assuming that the covariance matrix $\Sigma$ is positive definite, the following dual representation of $Z$ has been derived in \cite{li:2019}:
\[
Z(u) = \max_{ p \in \Delta} \, \langle p, u \rangle + \mbox{tr} \left( \left(\Sigma^{\nicefrac{1}{2}}\left( \mbox{diag}(p) - p p^T \right) \Sigma^{\nicefrac{1}{2}}\right)^{\nicefrac{1}{2}} \right).
\]
Moreover, the solution $p\in \Delta$ of the latter optimization problem provides the choice probabilities corresponding to the random error $\epsilon(u)\sim (0,\Sigma)$ that maximizes the surplus function above, i.\,e.
\[
p^{(i)} = \mathbb{P} \left( u^{(i)} + \epsilon^{(i)}(u) = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}(u)\right), \quad i=1, \ldots, n.
\]
Hence, the convex conjugate of $Z$ is
\[
Z^*(p) = - \mbox{tr} \left( \left(\Sigma^{\nicefrac{1}{2}}\left( \mbox{diag}(p) - p p^T \right) \Sigma^{\nicefrac{1}{2}}\right)^{\nicefrac{1}{2}} \right).
\]
In \cite{li:2019}, $Z^*$ is shown to be strongly convex on the simplex w.r.t. Euclidean norm (however, its convexity parameter is not given explicitly). We point out that $Z^*$ can be therefore used as a discrete choice prox-function on the simplex as well. \hfill $\Box$ \par \noindent \medskip
\end{remark}
\subsubsection{Simplicity}
\label{ssec:sim}
In view of the discussion in Section \ref{ssec:sc}, we assume that the convex conjugate $E^*$ is strongly convex.
\begin{theorem}[Simplicity]
\label{th:sim}
The unique maximizer of the optimization problem
\begin{equation}
\label{eq:sim}
E(u) = \sup_{ p \in \Delta} \, \langle p, u \rangle - E^*(p)
\end{equation}
is given by the choice probabilities
\[
p^{(i)} = \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right), \quad i=1, \ldots, n.
\]
\bf Proof: \rm \par \noindent Let a vector $u \in \mathbb{R}^n$ of deterministic utilities be given. The vector $p$ of corresponding probabilities lies in the relative interior of the simplex $\Delta$, i.\,e. $p \in \mbox{rint}(\Delta)$. This is due to the fact that the distribution of the random vector $\epsilon$ is fully supported on $\mathbb{R}^n$, see \cite{norets:2013}. Hence, we have:
\[
p = \nabla E(u),
\]
or, equivalently, by the convex duality \cite{rockafellar:1970}:
\[
u \in \partial E^*(p).
\]
The latter is the first order optimality condition for the optimization problem (\ref{eq:sim}). Additionally, note that
the dual representation of the surplus function is valid due to the Fenchel-Moreau theorem.
\hfill $\Box$ \par \noindent \medskip
\end{theorem}
\begin{corollary}[Lower bound for $E^*$]
\label{cor:lb}
The unique minimizer $p_0$ of the convex conjugate $E^*$ consists of the choice probabilities with respect to the zero-utility, i.\,e.
\[
p^{(i)}_0 = \mathbb{P} \left( \epsilon^{(i)} = \max_{1 \leq i \leq n} \epsilon^{(i)}\right), \quad i=1, \ldots, n.
\]
Moreover, it holds:
\[
E^*\left(p\right) \geq
E^*\left(p_0\right) = - \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n}\epsilon^{(i)} \right) \quad \mbox{for all } p \in \Delta.
\]
\bf Proof: \rm \par \noindent
Setting $u=0$ in Theorem \ref{th:sim} yields the assertion.
\hfill $\Box$ \par \noindent \medskip
\end{corollary}
\section{Economic application}
\label{sec:ea}
We apply discrete choice prox-functions for the natural adjustment of consumer's demand.
\subsection{Lancaster's approach to consumer theory}
\label{sec:lan}
We briefly describe the Lancaster's approach to consumer theory as presented in \cite{lancaster:1966}. For that, let $x \in \mathbb{R}^m_+$ denote the consumer's demand vector of $m$ goods. Every such demand vector $x$ generates the vector $z \in \mathbb{R}^n$ of $n$ qualities (sometimes called characteristics):
\[
z = Q x,
\]
where $Q=\left(q_{ij}\right) \in \mathbb{R}^{n\times m}$ is a fixed quality matrix. Its entries $q_{ij}$ denote the amounts of the $i$-th quality while consuming one unit of the $j$-th good. Further, the consumer assigns utility $u(z)$ to the goods' qualities $z$, and tries to maximize it by adjusting the demand $x$. Hereby, the budget constraint need to be satisfied, i.\,e.
\[
\langle \pi, x \rangle \leq w,
\]
where $\pi \in \mathbb{R}^m_{+}$ is a fixed vector of positive goods' prices, i.\,e. $\pi >0$, and $w >0$ is a fixed available budget. Overall, the Lancaster's approach to consumer theory consists in solving the following maximization problem:
\[
\max_{\scriptsize \begin{array}{c} x \geq 0 \end{array}} \, u(z) \quad \mbox{s.t.} \quad z = Q x, \quad \langle \pi, x \rangle \leq w.
\]
In what follows we focus on the Leontieff utility function by setting
\[
u(z)=\min \left\{ \frac{z^{(1)}}{\sigma^{(1)}}, \ldots, \frac{z^{(n)}}{\sigma^{(n)}}\right\},
\]
where $\sigma=\left(\sigma^{(1)}, \ldots, \sigma^{(n)}\right)^T\in \mathbb{R}^n_{+}$ is a fixed vector of positive quality standards, i.\,e. $\sigma >0$.
Additionally,
we assume that there exists a feasible demand vector which delivers positive Leontieff utility. Otherwise, the Lancaster's consumption problem is trivially solved by driving the demand to zero. On the contrary, under the proposed assumption the whole budget $w$ will be spent at any optimal demand. Hence, the budget constraint can be taken tight without loss of generality. The primal Lancaster's consumption problem becomes
\[
\mbox{P:}\quad \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w \end{array}} U(x),
\]
with the concave objective function
\[
U(x)= u(Qx)=\min_{1\leq i \leq n} \frac{(Qx)^{(i)}}{\sigma^{(i)}}.
\]
The optimization problem (P) consists in adjusting demand $x$ by spending the budget $w$ in order to maximize the worst ratio of consumed qualities $(Qx)^{(i)}$ in relation to their desired standards $\sigma^{(i)}$ taking over $i=1, \ldots, n$.
Let us derive a dual optimization problem for (P). For that, we introduce dual variables
\[
\lambda=\left(\lambda^{(1)}, \ldots, \lambda^{(n)}\right)^T \in \mathbb{R}^n,
\]
which can be interpreted as internal prices of qualities. Due to the duality of linear programming, we have:
\[
\begin{array}{rcl}
\displaystyle
\max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w \end{array}} U(x) &=& \displaystyle
\max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w \end{array}} \min_{1\leq i \leq n} \frac{(Qx)^{(i)}}{\sigma^{(i)}} \\ \\ &=& \displaystyle
\max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w \end{array}} \min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1 \end{array}} \langle Qx, \lambda \rangle \\ \\
&=& \displaystyle
\min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1 \end{array}} \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w \end{array}} \left\langle x, Q^T\lambda \right\rangle \\ \\ &=& \displaystyle
\min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1 \end{array}} w \max_{1\leq j \leq m} \frac{\left(Q^T \lambda\right)^{(j)}}{\pi^{(j)}} = \displaystyle
\min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1 \end{array}} \Phi(\lambda).
\end{array}
\]
The dual Lancaster's consumption problem becomes
\[
\mbox{D:}\quad \min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1 \end{array}} \Phi(\lambda),
\]
with the convex objective function
\[
\Phi(\lambda)=w \max_{1\leq j \leq m} \frac{\left(Q^T \lambda\right)^{(j)}}{\pi^{(j)}}.
\]
The optimization problem (D) consists in adjusting internal prices of qualities $\lambda$ in order to minimize the best ratio of quality estimates $\left(Q^T \lambda \right)^{(j)}$ of goods in relation to their market prices $\pi^{(j)}$ taking over $j=1, \ldots, m$.
Note that usually the number $n$ of goods is considerably bigger than the number $m$ of their qualities. From a consumer it will be thus more plausible to expect that (D) is successively solved rather than (P).
\subsection{Dual Averaging Scheme}
\label{sec:da}
We rewrite the dual optimization problem (D) by introducing new variables
\[
p^{(i)} = \sigma^{(i)} \lambda^{(i)}, \quad i =1, \ldots, n.
\]
In terms of the variables $p=\left(p^{(1)},\ldots,p^{(n)} \right)^T$ we equivalently obtain the following auxiliary optimization problem on the simplex:
\[
\mbox{A:}\quad \min_{\scriptsize \begin{array}{c} p \in \Delta \end{array}} \Psi\left(p\right),
\]
where the objective function is
\[
\Psi(p)= \Phi\left(\frac{p}{\sigma}\right).
\]
For solving (A) we apply the dual averaging scheme from \cite{nesterov:2013}. For that, we use the family of prox-functions on the simplex discussed in Section \ref{sec:prox}:
\[
d(p)=E^*(p) - E^*\left(p_0\right).
\]
Note that $d$ is continuous on the simplex $\Delta$ (Theorem \ref{th:ct}), strongly convex with convexity parameter $\beta>0$ (Theorems \ref{th:sc} and \ref{th:gev}, Corollary \ref{cor:gnl}). The computation of its convex conjugate
$
d^*(u) = E(u) - E\left(0\right)
$
is simple (Theorem \ref{th:sim}). Additionally, Corollary \ref{cor:lb} provides us with the prox-center of $\Delta$:
\[
p_0 = \mbox{arg} \min_{p \in \Delta} \, d(p).
\]
In view of $d\left(p_0\right)=0$, the dual averaging scheme can be initialized by $p_0$.
\[
\begin{tabular}{|c|}
\hline \\
\begin{tabular}{c}
{\bf Dual Averaging Scheme for (A)}
\end{tabular}\\ \hline \\
\begin{tabular}{ll}
{\bf 1.} Compute $\nabla \Psi\left(p_k\right)$. \\ \\
{\bf 2.} Set $\displaystyle s_{k+1}=\frac{1}{k+1}\sum_{\ell=0}^{k} \nabla \Psi\left(p_\ell\right)$. \\ \\
{\bf 3.} Update
$\displaystyle p_{k+1} = \mbox{arg} \min_{p \in \Delta} \left\{ \left\langle s_{k+1}, p \right\rangle + \frac{d(p)}{\sqrt{k+1}} \right\}$.\\
\end{tabular}
\\ \\ \hline
\end{tabular}
\]
The convergence properties of the dual averaging scheme follow from \cite[Theorem 1]{nesterov:2013}. For $k\geq 0$ it holds:
\begin{equation}
\label{eq:ineq}
\delta_k\leq \frac{D}{\sqrt{k+1}}+\frac{M^2}{2 \beta} \cdot \frac{1}{k+1}\left(1+\sum_{\ell=1}^{k} \frac{1}{\sqrt{\ell}}\right),
\end{equation}
where
\[
\delta_k = \max_{p \in \Delta} \, \frac{1}{k+1} \sum_{\ell=0}^{k} \left\langle \nabla \Psi\left(p_\ell\right), p_\ell - p \right\rangle, \quad
M = \max_{p \in \Delta} \left\|\nabla \Psi\left(p\right) \right\|_\infty, \quad D\geq \max_{p \in \Delta} \, d(p).
\]
\subsection{Consumption cycle}
\label{sec:cyc}
Let us state the dual averaging scheme from Section \ref{sec:da} in terms of the primal and dual Lancaster consumption problem (P) and (D), respectively.
\begin{itemize}
\item[\bf 1.] We compute
\[
\nabla \Psi\left(p_k\right) = \nabla \Phi\left(\frac{p_k}{\sigma}\right) = \frac{\nabla \Phi \left(\lambda_k\right)}{\sigma},
\]
where $\lambda_k$ denotes the internal prices of qualities at the $k$-th iteration. It holds:
\[
\nabla \Phi \left(\lambda_k\right) =
\nabla \left( w \max_{1\leq j \leq m} \frac{\left(Q^T \lambda_k\right)^{(j)}}{\pi^{(j)}} \right) = w \frac{Q y_k}{\pi},
\]
where the sharing vector $y_k \in \Delta$ fulfills
\[
y^{(j)}_k = 0 \quad \mbox{for } j \not \in J\left(\lambda_k\right)
\]
with the active index set
\[
J\left(\lambda_k\right) = \left\{ j \in \{1, \ldots, m\} \, \left| \, \frac{\left(Q^T \lambda_k\right)^{(j)}}{\pi^{(j)}}=\max_{1\leq j \leq m} \frac{\left(Q^T \lambda_k\right)^{(j)}}{\pi^{(j)}} \right.\right\}.
\]
In other words, $J\left(\lambda_k\right)$ contains goods with the best quality/price ratio estimated by means of internal prices $\lambda_k$.
We set the demand at the $k$-th iteration as
\[
x_k = w\frac{y_k}{\pi}.
\]
Note that $x_k$ is feasible for (P), since
\[
\left\langle \pi,x_k\right\rangle = \left\langle \pi,w\frac{y_k}{\pi}\right\rangle = w \left\langle e,y_k\right\rangle =w.
\]
Moreover, the demand is concentrated on the goods with the best quality/price ratio, i.\,e. $x^{(j)}_k \not = 0$ if and only if $j \in J\left(\lambda_k\right)$. Overall, we obtain:
\[
\nabla \Psi\left(p_k\right) = \frac{Q x_k}{\sigma},
\]
which are the ratios of the consumed qualities $Q x_k$ in relation to their standards $\sigma$.
\item[\bf 2.] We set
\[
s_{k+1}=\frac{1}{k+1}\sum_{\ell=0}^{k} \nabla \Psi\left(p_\ell\right) = \frac{1}{k+1}\sum_{\ell=0}^{k} \frac{Q x_\ell}{\sigma}
= \frac{Q \bar x_k}{\sigma}
\]
with the average demand
\[
\bar x_k = \frac{1}{k+1}\sum_{\ell=0}^{k} x_\ell.
\]
Again, $s_{k+1}$ relates the average consumption $Q \bar x_k$ to the standards $\sigma$.
\item[\bf 3.] We update
\[
p_{k+1} = \mbox{arg} \min_{p \in \Delta} \left\{ \left\langle s_{k+1}, p \right\rangle + \frac{d(p)}{\sqrt{k+1}} \right\}.
\]
Due to Theorem \ref{th:sim}, we equivalently obtain for $i=1, \ldots, n$:
\[
p_{k+1}^{(i)} = \mathbb{P} \left( s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}} = \min_{1 \leq i \leq n} s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}}\right).
\]
For the internal prices we have:
\[
\lambda_{k+1}^{(i)} = \frac{1}{\sigma^{(i)}} \mathbb{P} \left( s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}} = \min_{1 \leq i \leq n} s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}}\right).
\]
Thus, the internal price $\lambda_{k+1}^{(i)}$ of the $i$-th quality is proportional to the probability of detecting its average consumption $\left(Q \bar x_k\right)^{(i)}$ as the lowest one in comparison to the standard $\sigma^{(i)}$. Moreover, this detecting process is in accordance with additive random utility models allowing behavioral interpretations. E.\,g., the additive random errors $\frac{\epsilon^{(i)}}{\sqrt{k+1}}$ are diminishing with respect to the iteration number $k$, which accounts for the learning effect.
We further mention that $\lambda_{k+1}$ is feasible for (D), since
\[
\left\langle \sigma,\lambda_{k+1}\right\rangle = \left\langle e,p_{k+1}\right\rangle =1.
\]
So are the average internal prices
\[
\bar \lambda_k = \frac{1}{k+1}\sum_{\ell=0}^{k} \lambda_\ell.
\]
\end{itemize}
Let us examine the convergence properties of the proposed consumption cycle.
\begin{theorem}[Consumption cycle]
The duality gap between (P) and (D) evaluated at the average demand and the average internal prices is closing at the optimal rate $\mbox{O}\left(\frac{1}{\sqrt{k+1}}\right)$. Namely, it holds for $k \geq 0$:
\[
0 \leq \Phi\left( \bar \lambda_k\right) - U\left(\bar x_k\right) \leq
\left(D+\frac{M^2}{\beta}\right) \frac{1}{ \sqrt{k+1}},
\]
where
\[
M = w \max_{\scriptsize \begin{array}{c} 1 \leq i \leq n\\ 1 \leq j \leq m \end{array}} \frac{\left|q_{i,j}\right|}{\sigma^{(i)} \cdot \pi^{(j)}}, \quad
D = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n}\epsilon^{(i)} \right) - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right).
\]
\end{theorem}
\bf Proof: \rm \par \noindent
We compute the gap bound
\[
\begin{array}{rcl}
\displaystyle \delta_k &=& \displaystyle \max_{p \in \Delta} \, \frac{1}{k+1} \sum_{\ell=0}^{k} \left\langle \nabla \Psi\left(p_\ell\right), p_\ell - p \right\rangle =
\frac{1}{k+1} \sum_{\ell=0}^{k} w \left\langle \frac{Q y_\ell}{\pi}, \lambda_\ell\right\rangle - \min_{p \in \Delta} \left\langle s_{k+1}, p \right\rangle \\ \\
&=& \displaystyle \frac{1}{k+1} \sum_{\ell=0}^{k} w \sum _{j \in J\left(\lambda_\ell \right)} y_\ell^{(j)} \cdot \frac{\left(Q^T \lambda_\ell\right)^{(j)}}{\pi^{(j)}} - \min_{1 \leq i \leq n} s_{k+1}^{(i)} \\ \\ &=&\displaystyle \frac{1}{k+1} \sum_{\ell=0}^{k} w \max_{1 \leq j \leq m} \frac{\left(Q^T \lambda_\ell\right)^{(j)}}{\pi^{(j)}}- \min_{1 \leq i \leq n} \frac{\left(Q \bar x_k\right)^{(i)}}{\sigma^{(i)}} = \frac{1}{k+1} \sum_{\ell=0}^{k} \Phi\left( \lambda_\ell\right) - U\left(\bar x_k\right).
\end{array}
\]
The uniform bound for subgradients of $\Psi$ is given by
\[
M=\max_{p \in \Delta} \left\|\nabla \Psi\left(p\right) \right\|_\infty = \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w \end{array}} \left\| \frac{Q x}{\sigma} \right\|_\infty = w \max_{\scriptsize \begin{array}{c} 1 \leq i \leq n\\ 1 \leq j \leq m \end{array}} \frac{\left|q_{i,j}\right|}{\sigma^{(i)} \cdot \pi^{(j)}}.
\]
By applying Corollaries
\ref{cor:ub} and \ref{cor:lb}, we estimate the constant $D$:
\[
\max_{p \in \Delta} \, d(p) = \max_{p \in \Delta} \,
E^*(p) - E^*\left(p_0\right) \leq - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right) +\mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n}\epsilon^{(i)} \right)= D.
\]
It is straightforward to see that
\[
\frac{1}{k+1}\left(1+\sum_{\ell=1}^{k} \frac{1}{\sqrt{\ell}}\right) \leq \frac{2}{\sqrt{k+1}}.
\]
Finally, by using (\ref{eq:ineq}) we get:
\[
\begin{array}{rcl}
\displaystyle \Phi\left( \bar \lambda_k\right) - U\left(\bar x_k\right) &\leq& \displaystyle \frac{1}{k+1} \sum_{\ell=0}^{k} \Phi\left( \lambda_\ell\right) - U\left(\bar x_k\right) = \delta_k \\ \\ &\leq& \displaystyle
\frac{D}{\sqrt{k+1}}+\frac{M^2}{2 \beta} \cdot \frac{1}{k+1}\left(1+\sum_{\ell=1}^{k} \frac{1}{\sqrt{\ell}}\right)=\left(D+\frac{M^2}{\beta}\right) \frac{1}{ \sqrt{k+1}}.
\end{array}
\]
\hfill $\Box$ \par \noindent \medskip
| {'timestamp': '2019-09-13T02:14:29', 'yymm': '1909', 'arxiv_id': '1909.05591', 'language': 'en', 'url': 'https://arxiv.org/abs/1909.05591'} |
\section{Constructive Upper Bounds}
The multiplicative upper bound inequality for vertex
Folkman numbers stated in Theorem \ref{K-XLSW} below
was proved independently in \cite{Nkolev2008} and \cite{XLSW}.
A related constructive upper bound for $F_v(k,k;k+1)$
was obtained in \cite{XLSW}, which improved
earlier known bounds, however it is still much weaker
than the best known probabilistic upper bound
for these parameters \cite{drr14}.
\medskip
\begin{theorem} \label{K-XLSW}
{\bf \cite{Nkolev2008,XLSW}}
If $\max\{a_1, \cdots , a_r\} \leq a$ and
$\max\{b_1, \cdots , b_r\} \leq b$, then
$$F_v(a_1 b_1, \cdots, a_r b_r; ab+1) \leq
F_v(a_1, \cdots, a_r; a+1) F_v(b_1, \cdots, b_r; b+1).$$
\end{theorem}
For graphs $G$ and $H$, we will use their
{\em lexicographic product graph} $G[H]$ defined
on the set of vertices $V(G) \times V(H)$ with
$\{(u_1,v_1),(u_2,v_2)\} \in E(G[H])$
if and only if
$\{u_1,u_2\} \in E(G)$ or
($u_1=u_2$ and $\{v_1,v_2\} \in E(H)$).
\medskip
\begin{lemma} \label{prodarrow}
For graphs $G, H$ and $H_i$, and integers $a_j \ge 2$,
$1 \le i,j \le r$,
if $G \rightarrow (a_1, \cdots,a_r)^v$
and $H \rightarrow (H_1, \cdots, H_r)^v$, then
$G[H] \rightarrow (K_{a_1}[H_1], \cdots, K_{a_r}[H_r])^v$.
\end{lemma}
\begin{proof}
Let $G$ and $H$ be any graphs as in the assumptions
of the lemma. Let their sets of vertices be
$U=V(G)$ and $V=V(H)$, respectively, and consider
any partition $V(G[H])=\bigcup_{i=1}^{r}X_i$,
i.e. $r$-coloring $C_v$
of the vertices of $G[H]$. We need to show that for
some color $i$, $1 \le i \le r$, the subgraph induced
by $X_i$ contains $K_{a_i}[H_i]$. Note that for each fixed
$u \in U$, the vertices
$V(u)=\{(u,v)\;|\;v \in V\}$ induce a graph
isomorphic to $H$ in $G[H]$. Hence, for each $u \in U$
there exists a color $i(u)$,
$1 \le i(u) \le r$, such that
the subgraph induced by $V(u)$ contains $H_{i(u)}$
in color $i(u)$.
Next, consider the $r$-coloring $C_v^{'}$
of vertices of $G$ defined by $i(u)$.
Since $G \rightarrow (a_1, \cdots,a_r)^v$,
then there exists $j$ such $C_v^{'}$
contains $K_{a_j}$ in color $j$ in $G$, or equivalently,
in the vertex $r$-coloring $C_v$
of $G[H]$ we have $a_j$ isomorphic copies of $H$,
each of them containing $H_j$ in color $j$,
and all of them are interconnected by edges in $G[H]$.
\end{proof}
\medskip
Let $cl(H)$ denote the clique number of graph $H$,
i.e. the largest integer $s$ such that $K_s \subset H$.
The following generalizes Theorem \ref{K-XLSW}.
\medskip
\begin{theorem} \label{FolkmangeneralGH}
If $\max \{a_1, \cdots, a_r\} \leq a$ and
$\max \{cl(H_1), \cdots, cl(H_r)\} \leq b$, then
$$F_v(K_{a_1}[H_1], \cdots, K_{a_r}[H_r];ab+1)
\leq F_v(a_1, \cdots, a_r; a+1) F_v(H_1, \cdots, H_r;b+1).$$
\end{theorem}
\begin{proof}
Consider any graph
$G \in \mathcal{F}_v(a_1,\cdots,a_r;a+1)$
with the set of vertices $V(G)=U=\{u_1,\cdots,u_s\}$,
where $s=F_v(a_1,\cdots,a_r;a+1)$,
and any graph $H$ such that
$H \in \mathcal{F}_v(H_1,\cdots,H_r;b+1)$
and $V(H)=V=\{v_1,\cdots,v_t\}$,
where $t=F_v(H_1,\cdots,H_r;b+1)$.
Note that $st=|V(G[H])|$ is also equal to
the right hand side of the target inequality.
By the construction of $G[H]$
one can easily see that $cl(G[H]) \le ab$. Finally,
Lemma \ref{prodarrow} implies that
$G[H] \rightarrow (K_{a_1}[H_1], \cdots, K_{a_r}[H_r])^v$,
which completes the proof.
\end{proof}
\medskip
We note now, and will also use it later, that
Theorem \ref{FolkmangeneralGH} is specially
interesting in the cases involving graphs $J_k$,
because we can use the fact that $J_{sk+1}$
is a subgraph of $K_s[J_{k+1}]$. For instance,
using Theorem \ref{FolkmangeneralGH} for two
colors with $s=a_1=a_2=2$, $k=b$, $F_v(2,2;3)=5$
and $H_1=H_2=J_{k+1}$, we obtain
$$F_v(K_{2k+2}-2K_2,K_{2k+2}-2K_2;2k+1)
\leq 5F_v(J_{k+1},J_{k+1};k+1).$$
\smallskip
Further, since
$J_{2k+1}$ is a subgraph of $K_{2k+2}-2e$,
it also holds that
$$F_v(J_{2k+1},J_{2k+1};2k+1) \leq
5F_v(J_{k+1},J_{k+1};k+1).$$
\medskip
In fact, we can do a little better on 3 out of 5
blocks of $F_v(J_{k+1},J_{k+1};k+1)$ vertices,
as stated in the next theorem.
\medskip
\begin{theorem} \label{FvK{2k+1}-e}
For every integer $k \geq 2$, we have that
$F_v(J_{2k+1},J_{2k+1};2k+1) \leq
2F_v(k,k;k+1) + 2F_v(J_{k+1},J_{k+1};k+1)
+ F_v(K_k,J_{k+1};k+1).$
\end{theorem}
\begin{proof}
Consider any graphs $H_1, H_2, H_3$ such that
$H_1 \in \mathcal{F}_v(k,k;k+1)$,
$H_2 \in \mathcal{F}_v(K_k,J_{k+1};k+1)$, and
$H_3 \in \mathcal{F}_v(J_{k+1},J_{k+1};k+1)$,
and they have the smallest possible number of vertices,
i.e. $|V(H_1)| = F_v(k,k;k+1)$,
$|V(H_2)| = F_v(K_k,J_{k+1};k+1)$, and
$|V(H_3)| = F_v(J_{k+1},J_{k+1};k+1)$.
Let $H_4$ be an isomorphic copy of $H_1$,
and $H_5$ an isomorphic copy of $H_3$.
The clique number of all graphs $H_i$ is
equal to $k$.
Our goal is to construct graph
$G \in \mathcal{F}_v(J_{2k+1},J_{2k+1};2k+1)$
on the set of vertices
$V(G)=\bigcup_{i=1}^5 V(H_i)$,
which has $cl(G)=2k$.
This will suffice to complete the proof.
The set of edges of graph $G$ is defined by
$E(G) = \bigcup_{i=1} ^5 E(H_i) \cup E(1,3)
\cup E(1,5) \cup E(2,4) \cup E(2,5) \cup E(3,4)$,
where
$E(i,j) = \{ \{ u,v\} \;|\; u \in V(H_i), v \in V(H_j),
i \not= j, 1 \le i,j \le 5\}$.
One can easily check that $G$ is $K_{2k+1}$-free,
since the edges of types $E(i,j)$ do not form
any triangles. It remains to be shown that
$G \rightarrow (J_{2k+1},J_{2k+1})^v$.
For a contradiction, suppose that there exists a partition
$V(G)=R \cup B$, i.e. a red-blue coloring of the vertices
of $G$, which has no monochromatic $J_{2k+1}$.
Without loss of generality we may assume that
$H_1$ contains a red $K_k$. Therefore, there is no red
$J_{k+1}$ in $H_3$ and no red $J_{k+1}$ in $H_5$,
otherwise we would have a red $J_{2k+1}$.
Hence, there are blue $J_{k+1}$'s in both
$H_3$ and $H_5$.
Therefore, there is no blue $K_k$ in $H_2$
and no blue $K_k$ in $H_4$.
Hence, there is a red $J_{k+1}$ in $H_2$
and a red $K_k$ in $H_4$, and together
they form a red $J_{2k+1}$.
Thus $G \rightarrow (J_{2k+1},J_{2k+1})^v$,
which completes the proof.
\end{proof}
\smallskip
As an application of the last theorem
we consider an interesting case of $F_v(J_5,J_5;5)$.
Likely, it is just somewhat larger (and harder to compute)
than the well studied classical case of
$F_v(4,4;5)$, for which the currently best
known bounds are
$17 \le F_v(2,3,4;5) \le F_v(4,4;5) \le 23$ \cite{Fv445}.
\begin{claim}
$F_v(J_5,J_5;5) \le 36$.
\end{claim}
\begin{proof}
The first three graphs $H_i$ in the proof
of Theorem \ref{FvK{2k+1}-e}
for this case are of orders equal to
$F_v(2,2;3)$, $F_v(J_3,J_3;3)$ and
$F_v(K_2,J_3;3)$, respectively. The first of them,
equal to 5, is uniquely witnessed by the cycle
$C_5$. Obtaining upper bounds for the other two
requires some work when analyzing
possible triangle-free graphs arrowing
the corresponding graph parameters
(note that $J_3=P_3$ and thus
any $J_3$-free graph
must be of the form $sK_1 \cup tK_2$).
An example of graph witnessing
$F_v(J_3,J_3;3) \le 9$ can be constructed
by dropping one vertex from the graph
$C_5[2K_1]$, and for $F_v(K_2,J_3;3) \le 8$
by adding four main diagonals
to the cycle $C_8$. Putting it all together,
by Theorem \ref{FvK{2k+1}-e} applied to this case we obtain
$F_v(J_5,J_5;5) \le 2\cdot 5+2 \cdot 9+8=36$.
\end{proof}
We expect that the actual value of $F_v(J_5,J_5;5)$
is still smaller, but probably not much less so.
How to obtain better bound in this and other similar
cases by detailed analysis is an interesting and
challenging problem.
\medskip
We end this section with two more
upper bounds on $F_v(J_k,J_k;k)$.
Let $E_s$ denote the empty graph on $s$ vertices.
Thus, for example, $K_s[E_t]$ is the same as
the standard complete
$s$-partite graph with all parts of order $t$,
or equivalently, the Tur\'{a}n graph $T_{st,s}$.
One can easily show, similarly as in the proof
of Lemma \ref{prodarrow},
that if $G \rightarrow (k-1,k-1)^v$, then
$G[E_3] \rightarrow (K_{k-1}[E_2],K_{k-1}[E_2])^v$.
Moreover, the same assumption also gives
$G[E_3] \rightarrow (J_k,J_k)^v$,
because $J_k$ is a subgraph of $K_{k-1}[E_2]$.
An even stronger result following from similar
considerations is presented in the next theorem.
\medskip
\begin{theorem} \label{FvKk-e}
If $G \in \mathcal{F}_v(k-1,k-1;k)$,
$|V(G)| = F_v(k-1,k-1;k)$ and $f(G)$
is the largest order of any $K_{k-1}$-free induced
subgraph in $G$, then
$$F_v(J_k,J_k;k) \leq 3 F_v(k-1,k-1;k) - f(G).$$
\end{theorem}
\begin{proof}
Let $G$ be any graph in
$\mathcal{F}_v(k-1, k-1; k)$
with the smallest possible number of vertices
$|V(G)|=F_v(k-1, k-1; k)$. Let us denote
$V(G)=U=\{u_1,\cdots,u_s\}$, so that the vertices
$X=\{u_1,\cdots,u_{f(G)}\}$ induce the largest
$K_{k-1}$-free subgraph in $G$. Note that the
vertices of every $K_{k-1}$ in $G$ have
nonempty intersection with
$Y=\{u_{f(G)+1},\cdots,u_s\}$.
We will construct a graph $H$
on $3|V(G)|-f(G)$ vertices such that
$H \in \mathcal{F}_v(J_k,J_k;k)$, which will
complete the proof of the theorem.
Let $V=\{ v_1,v_2,v_3\}$.
First, take the graph $G[E_3]$ with the set
of vertices $U \times V$. Then $H$ is obtained
from it by dropping $f(G)$ vertices forming the set
$\{\{ u_i,v_3\}\;|\;1 \le i \le f(G)\}$ with
all incident edges. Clearly, graph $H$ has the right
number of vertices and $cl(H)=k-1$. It remains to
be shown that $H \rightarrow (J_k,J_k)^v$.
Let $V(H)=R \cup B$ be any partition
of the vertices of $H$ into two parts,
i.e. any red-blue coloring of $V(H)$.
Note that for each fixed
$u \in U$, there are 2 or 3 vertices in the set
$V(u)=\{(u,v)\;|\;(u,v) \in V(H)\}$.
Let $i(u) \in \{R,B\}$ be a color of at least
half of vertices in $V(u)$ (1 or 2).
Next, consider the red-blue coloring of vertices
of $G$ defined by $i(u)$.
Since $G \rightarrow (k-1,k-1)^v$, then
for the coloring $i(u)$
there exists a set of vertices $S \subset V(G)$
containing a monochromatic $K_{k-1}$ in $G$.
Considering the properties of $X$ and $Y$, $S$ must
contain at least one vertex $u \in Y \cap S$,
and consequently at least two vertices
$(u,x), (u,x) \in V(H)$ are of color $i(u)$.
Now, $S$ expanded to vertices of $H$ of
the same color must contain a monochromatic $J_k$.
\end{proof}
\medskip
\begin{cor}
$$F_v(J_k,J_k;k) \leq
\Bigl\lceil {{5 F_v(k-1, k-1; k)}\over 2}\Bigr\rceil.$$
\end{cor}
\medskip
\begin{proof}
Set $m=\lceil{(F_v(k-1, k-1; k)-1)/2}\rceil$, and let
$G$ and $f(G)$ be as in Theorem \ref{FvKk-e}.
Observe that for every vertex $v \in V(G)$,
$G-v \not\rightarrow (k-1,k-1)^v$ and thus there
exists a $K_ {k-1}$-free set $S \subset V(G) - \{v\}$
satisfying $|S| \ge m$.
Therefore, $f(G) \ge m$ and the corollary easily
follows from Theorem \ref{FvKk-e}.
\end{proof}
\medskip
Using arguments similar to those in
Lemma \ref{prodarrow} and Theorem \ref{FvKk-e},
it is easy to see that
$C_5 [E_{2t-1}] \rightarrow (K_{t,t}, K_{t,t})^v,$
and since $|C_5 [E_{2t-1}]| = 10t-5$, we have an
upper bound
$$F_v (K_{t,t}, K_{t,t}; K_3) \leq 10t -5.$$
\noindent
Improving this bound can be difficult, and
obtaining a good lower bound even harder but
interesting. Hence, we propose the following
problem:
\begin{Prob} \label{F_vK_{t,t}}
For $t \ge 2$,
\smallskip
\noindent
{\rm (a)}
obtain tight bounds for $F_v (K_{t,t}, K_{t,t};3)$, and
\noindent
{\rm (b)}
obtain good bounds for $F_v (K_{t,t}, K_{t,t};k)$, for $k \ge 4$.
\end{Prob}
\medskip
In another application of Theorem \ref{FolkmangeneralGH}
for two colors, with $a_1=a_2=2$, $b=r$ and
$H_1=H_2=T_{tr,r}$,
and for all $t,r \ge 2$, we obtain
$$F_v(T_{2tr,2r},T_{2tr,2r};2r+1) \leq
5 F_v(T_{tr,r},T_{tr,r}; r+1).$$
In particular,
note that for $r=2$ we have $T_{tr,r}=K_{t,t}$, and
thus the last inequality implies
$F_v(T_{4t,4},T_{4t,4}; K_5) \leq 50t -25.$
\bigskip
The current authors recently studied
the so-called chromatic Folkman numbers \cite{XLR0}, which have
one additional requirement for their witness Folkman
graphs $G$, namely that their chromatic number
$\chi(G)$ is the smallest possible
($\chi(G)=1+ \sum_{i=1}^r (a_i - 1)$
for vertex colorings, and
$\chi(G)=R(a_1, \cdots, a_r)$ for edge colorings).
Some of the constructions in this section lead to
upper bounds involving larger graphs but with
the same chromatic number, mainly because
$\chi(G[E_s]) = \chi(G)$. Thus, these techniques
potentially could lead to stronger claims about upper bounds,
where the chromatic number is as small as possible.
\bigskip
\section{Two Concrete Upper Bounds}
\medskip
If the graphs we wish to arrow to are not of the form
$K_{a_i}[H_i]$ as in Theorem \ref{FolkmangeneralGH},
then the constructions for upper bounds may become
little more complex. For instance, when we deal with
complete but unbalanced multipartite graphs such as
$K_{1,2,2}$ or $K_{2,2,3}$. The cases we study in this section
also involve edge colorings, and the corresponding
generalized edge Folkman numbers. Good upper bounds
for the edge cases seem to be even harder to obtain
than for vertex colorings.
We will focus mainly on two small but puzzling cases
of $F_e(J_4,J_4;5)$ and $F_e(K_3,J_4;5)$.
By the monotonicity of arrowing we have
$$15 = F_e(3,3;5) \le F_e(K_3,J_4;5)
\le F_e(J_4,J_4;5) \le F_e(J_4,J_4;4) \le 30193.$$
\medskip
The equality $F_e(3,3;5)=15$ was obtained
in \cite{Nenov81,PRU}, where in the latter
it was also shown, with the help of computer algorithms,
that $F_v(3,3;4)=14$. Furthermore, in the same work
the authors obtained all 153 graphs on 14 vertices
in the set $\mathcal{F}_v(3,3;4)$. These two parameter
scenarios are connected, since it is known that
$G+u \in \mathcal{F}_e(3,3;5)$ holds whenever
$G \in \mathcal{F}_v(3,3;4)$ (see Lemma \ref{miscarr}(a)).
Two examples of such graphs, $G_a$ and $G_b$,
are presented in Figures A and B.
They are used in the constructions of the following
theorems, which exploit enhancements of
the implication in Lemma \ref{miscarr}(a).
Consult \cite{XLR1}
for the discussion of similar bounds and for
additional pointers to the literature.
\bigskip
\begin{center}
{\small
0 1 1 0 0\ \ 1 1 0 0 0\ \ 1 0 0\ \ 1\\
1 0 0 1 0\ \ 0 0 0 1 1\ \ 0 0 1\ \ 1\\
1 0 0 0 1\ \ 0 0 1 0 1\ \ 0 1 0\ \ 1\\
0 1 0 0 1\ \ 1 0 1 0 0\ \ 0 1 0\ \ 1\\
0 0 1 1 0\ \ 0 1 0 1 0\ \ 0 0 1\ \ 1\\
\smallskip
1 0 0 1 0\ \ 0 0 0 1 1\ \ 1 1 0\ \ 1\\
1 0 0 0 1\ \ 0 0 1 0 1\ \ 1 0 1\ \ 1\\
0 0 1 1 0\ \ 0 1 0 1 0\ \ 1 1 0\ \ 1\\
0 1 0 0 1\ \ 1 0 1 0 0\ \ 1 0 1\ \ 1\\
0 1 1 0 0\ \ 1 1 0 0 0\ \ 0 1 1\ \ 1\\
\smallskip
1 0 0 0 0\ \ 1 1 1 1 0\ \ 0 1 1\ \ 0\\
0 0 1 1 0\ \ 1 0 1 0 1\ \ 1 0 1\ \ 0\\
0 1 0 0 1\ \ 0 1 0 1 1\ \ 1 1 0\ \ 0\\
\smallskip
1 1 1 1 1\ \ 1 1 1 1 1\ \ 0 0 0\ \ 0\\
}
\end{center}
\medskip
\begingroup\leftskip=15pt\rightskip=15pt
\noindent
{\small
{\bf Figure A.} Adjacency matrix of 14-vertex graph
$G_a \in \mathcal{F}_v(3,3;4)$, with a vertex of
maximum degree equal to 10. It is one of 60 such
graphs, all of them enumerated in \cite{PRU}.
Graph $G_a$ has a specially nice structure:
vertices 1--5 and 6--10 induce $C_5$'s,
vertices 11--13 span $K_3$, 10 neighbors of
vertex 14 induce a graph with 320 automorphisms
(which is necessarily triangle-free),
while the entire $G_a$ has only 2 automorphisms.
$G_a$ has independence number 5, it has 41 triangles,
and $|E(G_a)|=48$.
}
\par\endgroup
\eject
\medskip
\begin{center}
{\small
0 0 0 0 0 0 0\ \ \ 1 1 1 1 0 0 0\\
0 0 0 0 0 0 0\ \ \ 1 1 1 0 1 0 0\\
0 0 0 0 0 0 0\ \ \ 1 1 0 1 0 1 0\\
0 0 0 0 0 0 0\ \ \ 1 0 1 0 1 0 1\\
0 0 0 0 0 0 0\ \ \ 0 1 0 1 0 1 1\\
0 0 0 0 0 0 0\ \ \ 0 0 1 0 1 1 1\\
0 0 0 0 0 0 0\ \ \ 0 0 0 1 1 1 1\\
\smallskip
1 1 1 1 0 0 0\ \ \ 0 1 1 0 0 1 1\\
1 1 1 0 1 0 0\ \ \ 1 0 0 1 1 0 1\\
1 1 0 1 0 1 0\ \ \ 1 0 0 1 1 1 0\\
1 0 1 0 1 0 1\ \ \ 0 1 1 0 1 1 0\\
0 1 0 1 0 1 1\ \ \ 0 1 1 1 0 0 1\\
0 0 1 0 1 1 1\ \ \ 1 0 1 1 0 0 1\\
0 0 0 1 1 1 1\ \ \ 1 1 0 0 1 1 0\\
}
\end{center}
\medskip
\begingroup\leftskip=15pt\rightskip=15pt
\noindent
{\small
{\bf Figure B.} Adjacency matrix of the Nenov
graph $G_b$ \cite{Nenov81},
which is the unique 14-vertex graph
in the set $\mathcal{F}_v(3,3;4)$
with independence number 7.
In graph $G_b$, vertices 1--7 form
the only 7-independent set and vertices 8--14
induce $\overline{C_7}$.
Graph $G_b$ has 14 automorphisms,
35 triangles, and $|E(G_b)|=42$,
which is the smallest number of edges
among all graphs in $\mathcal{F}_v(3,3;4)$.
}
\par\endgroup
\bigskip
We will also need some simple facts about arrowing.
They are collected in the following lemma.
\smallskip
\begin{lemma} \label{miscarr}
All of the following hold:
\smallskip
\noindent
{\rm (a)}
if $G\rightarrow (3,3)^v$ and $u$ is a new vertex,
then $G+u \rightarrow (3,3)^e$,
\smallskip
\noindent
{\rm (b)}
if $G\rightarrow (3,3)^v$, then
$G[E_{2k-1}] \rightarrow (K_{k,k,k},K_{k,k,k})^v$ for $k \ge 1$,
\smallskip
\noindent
{\rm (c)}
$K_{1,2,2} \rightarrow (J_3,K_3)^e$, and
\smallskip
\noindent
{\rm (d)}
$K_{2,2,3} \rightarrow (J_3,J_4)^e$.
\end{lemma}
\begin{proof}
Part (a) is a basic property of arrowing
used by many authors in scenarios similar to ours,
for example, in \cite{Nenov81}.
For part (b) observe that $K_3[E_k]=K_{k,k,k}$
and use Lemma \ref{prodarrow} for two colors with
$r=2$, $a_1=a_2=3$ and $H_1=H_2=E_k$. For parts
(c) and (d), first note that
$K_{1,2,2}=K_1+C_4$ and $K_{2,2,3}=E_3+C_4$,
and then consider possible edges of $C_4$ in the
color avoiding $J_3$; there are at most two such edges.
There are just a few more choices of edges in the
$J_3$-free color since in total there are at most
2 or 3 such edges in $K_{1,2,2}$ and $K_{2,2,3}$,
respectively. A routine check easily shows that
the edges in the other color must contain a $K_3$
and $J_4$, respectively.
\end{proof}
\medskip
\begin{theorem} \label{FvJ4J4K5}
$F_e(J_4,J_4;5) \leq 51$.
\end{theorem}
\begin{proof}
The skeleton of the proof is as follows.
First, we will construct a $K_4$-free graph $H$
on 50 vertices such that
$H \rightarrow (K_{2,2,3},K_{2,2,3})^v$.
Then we will claim that the 51-vertex graph
$G=K_1+H$ is in the set $\mathcal{F}_e(J_4,J_4;5)$,
or equivalently, it is a witness of the upper bound
in the theorem.
We start with the graph $G_a \in \mathcal{F}_v(3,3;4)$
described in Figure A.
Let $u$ be the only vertex of degree 10.
Consider the partition of the set $V(G_a)$ into
$X \cup Y$, where
$X=\{v\;|\;\{u,v\}\in E(G_a)\}$,
so that $|X|=10$ and $|Y|=4$.
Note that every triangle in $G_a$ must have
at least one vertex in $Y$.
Next, define the graph $G_1=G_a[E_5]$ which
has 70 vertices and it is $K_4$-free.
By Lemma \ref{miscarr}(b), it holds that
$G_1 \rightarrow (K_{3,3,3},K_{3,3,3})^v$,
which is too strong for our purpose, so we can
drop some vertices from $G_1$.
We define graph $H$ as an induced subgraph
of $G_1$ following the idea of the proof of
Theorem \ref{FvKk-e}, which now is to
drop $2|X|$ vertices from $G_1$ formed
by two triangle-free parts of $G_a$.
More precisely, if the product graph $G_1$ uses
$V(E_5)=\{v_1,\cdots,v_5\}$, then
$V(H)=V(G_1) \setminus X \times \{v_4,v_5\}$
and $|V(H)|=50$.
By an argument very similar to that in the proof
of Theorem \ref{FvKk-e} we can see that
$H \rightarrow (K_{2,2,3},K_{2,2,3})^v$.
Being a subgraph of $G_1$, the graph $H$ is $K_4$-free.
Finally, define graph $G$ to be $K_1+H$, and
let $u$ be the vertex in $V(G) \setminus V(H)$.
Clearly, $G$ is $K_5$-free and we have
$V(G)=51$. It remains to be shown that
$G \rightarrow (J_4,J_4)^e$. Consider any
red-blue coloring $C_e$ of the edges $E(G)$.
Define a red-blue vertex coloring $C_v$
by $C_v(x)=C_e(\{u,x\})$ for $x \in V(H)$.
Since $H \rightarrow (K_{2,2,3},K_{2,2,3})^v$,
then $H$ contains a monochromatic $K_{2,2,3}$
in $C_v$, say, with the vertex set $U$.
Without loss of generality assume that
all vertices in $U$ are red.
Now, by Lemma \ref{miscarr}(d),
the set of edges induced by $U$ must contain
a red $J_3$ or blue $J_4$ in $C_e$.
If it is blue $J_4$, then we are done, otherwise
red $J_3$ together with vertex
$u$ induces a red $J_4$.
\end{proof}
\medskip
\begin{theorem} \label{FvK3J4K5}
$F_e(K_3,J_4; K_5) \leq 27$.
\end{theorem}
\begin{proof}
The reasoning is very similar to that in the proof of
Theorem \ref{FvJ4J4K5}, just the parameters vary.
We start with the graph $G_b$ described in Figure B,
and let $G_1=G_b[E_3]$.
By Lemma \ref{miscarr}(b), it holds that
$G_1 \rightarrow (K_{2,2,2},K_{2,2,2})^v$,
so we can drop some vertices from $G_1$.
Note that $J_4=K_{1,1,2}$.
We will define an induced subgraph $H$ of $G_1$
on 26 vertices
such that $H \rightarrow (K_{1,1,2},K_{1,2,2})^v$.
Then the graph $K_1+H$ will be a witness
of the upper bound.
Consider a partition of the set $V(G_b)$ into
$X \cup Y \cup Z$, where
$X$ consists of the vertices of 7-independent set
(the first 7 vertices), $Y$ is the pair of vertices
of any nonedge contained in the second block of 7 vertices,
$Z$ is formed by the remaining vertices,
and thus they have orders 7, 2 and 5, respectively.
Note that every triangle in $G_b$ must have
at least one vertex in $Z$, and at least
two vertices in $Y \cup Z$.
Drop $2|X|$ vertices from $V(G_1)$, 2 associated with
each vertex in $X$, and similarly drop $|Y|$ vertices
associated with $Y$. This defines an induced
subgraph $H$ on $|X|+2|Y|+3|Z|=26$ vertices.
Now,
for every red-blue vertex coloring $C_1$ of $V(G_1)$
define the coloring $C_b$ of $V(G_b)$ by
assigning to $u \in V(G_b)$ color
of the majority of vertices in the set
$\bigl\{\{u,v\}\;|\;\{u,v\}\in V(H)\bigr\}$ under $C_1$.
Note that these sets have the cardinality 1, 2 and 3
for $u$ in $X$, $Y$ and $Z$, respectively.
Thus, every monochromatic triangle in $G_b$ can be
expanded in $H$ to at least 1, 1, and 3 vertices, or
at least 1, 2, and 2 vertices, respectively.
Hence, similarly as in the proof of
Theorem \ref{FvKk-e}, we can see that
$H \rightarrow (K_{1,1,2},K_{1,2,2})^v$. Furthermore,
being a subgraph of $G_1$, the graph $H$ is $K_4$-free.
Finally, define the graph $G$ as $K_1+H$,
so that $G$ is $K_5$-free and we have
$V(G)=27$. It remains to be shown that
$G \rightarrow (K_3,J_4)^e$.
Let $u$ be the vertex in $V(G) \setminus V(H)$
and consider any
red-blue coloring $C_e$ of the edges $E(G)$.
Define a red-blue vertex coloring $C_v$
by $C_v(x)=C_e(\{u,x\})$ for $x \in V(H)$.
Since $H \rightarrow (K_{1,1,2},K_{1,2,2})^v$,
then $H$ contains a red $K_{1,1,2}$ or blue
$K_{1,2,2}$ in $C_v$, say with the vertex set $U$.
Recall that $J_4=K_{1,1,2}$.
First suppose that
the vertices in $U$ are forming a red $J_4$ in $C_v$.
If at least one of the edges with both endpoints
in $U$ is red in $C_e$, then we have red $K_3$
spanned by this edge and $u$ in $C_e$. Otherwise,
all edges induced by $U$ are blue in $C_e$, so we
have blue $J_4$. On the other hand, suppose that
the vertices in $U$ form a blue $K_{1,2,2}$ in $C_v$.
By Lemma \ref{miscarr}(c),
the set of edges induced by $U$ must contain
a red $K_3$ or blue $J_3$ in $C_e$.
If it is red $K_3$, then we are done, otherwise
blue $J_3$ together with vertex
$u$ induces a blue $J_4$.
\end{proof}
\medskip
The upper bounds of the last two theorems are
likely larger than the actual values, despite that
the graphs $G_a$ and $G_b$ where chosen
to be the best for our method, namely: none of the graphs on
14 vertices in $\mathcal{F}_v(3,3;4)$ has any induced triangle-free
subgraph on more than 10 vertices and none has two independent
sets whose union has more than 9 vertices, respectively.
On the other hand, improving on either of the bounds
$F_v(K_{2,2,3},K_{2,2,3};4) \le 50$ or
$F_v(K_{1,1,2},K_{1,2,2};4) \le 26$, currently
used in the proofs, would lead to better upper bounds
in Theorems 10 and 11, respectively. Finally, let us
note that Folkman numbers with other parameters
could be studied by the current method, for example,
such as when exploiting the inequality
$F_e(3,4;7) \leq 1+F_v(K_4,K_3+C_5;6)$, which relies on
the well known case of edge arrowing $K_3+C_5 \rightarrow (3,3)^e$.
\bigskip
\section{Folkman numbers} \label{Folkman}
Let $r, s, a_1, \cdots, a_r$ be integers such
that $r \ge 2$,
$s > \max \{a_1, \cdots, a_r \}$ and
$\min \{a_1, \cdots, a_r \} \ge 2$.
We write $G \rightarrow (a_1 ,\cdots ,a_r)^v$
(resp. $G \rightarrow (a_1 ,\cdots ,a_r)^e$)
if for every
$r$-coloring of $V(G)$ (resp. $E(G)$), there exists
a monochromatic $K_{a_i}$ in $G$ for some color
$i \in \{1, \cdots, r\}$.
The Ramsey number $R(a_1, \cdots, a_r)$
is defined as the smallest integer $n$ such that
$K_n \rightarrow (a_1, \cdots, a_r)^e$.
The sets of vertex and edge Folkman graphs are defined as
$$\mathcal{F}_v(a_1, \cdots, a_r; s)=\{G \;|\; G \rightarrow
(a_1, \cdots, a_r)^v \textrm{ and } K_s \not\subseteq G\},\textrm{ and}$$
$$\mathcal{F}_e(a_1, \cdots, a_r;s)=\{G\;|\; G \rightarrow
(a_1, \cdots, a_r)^e \textrm{ and } K_s \not\subseteq G\},$$
respectively, and the vertex and edge Folkman numbers
are defined as the smallest orders of graphs in these sets,
namely
$$F_v(a_1, \cdots, a_r;s)=\min\{|V(G)|\;|\;G\in
\mathcal{F}_v(a_1, \cdots, a_r;s) \},\textrm{ and}$$
$$F_e(a_1, \cdots, a_r;s)=\min\{|V(G)|\;|\;G\in
\mathcal{F}_e(a_1, \cdots, a_r;s) \}.$$
The generalized vertex and edge Folkman numbers,
$F_v(H_1, \cdots, H_r;H)$ and $F_e(H_1, \cdots, H_r;H)$,
are defined analogously by considering arrowing graphs
$H_i$ while avoiding $H$, instead of arrowing complete
graphs $K_{a_i}$ while avoiding $K_s$.
The edge Folkman number
$F_e(a_1, \cdots, a_r; k)$ can be seen as a generalization
of the classical Ramsey number $R(a_1, \cdots, a_r)$,
since for $k > R(a_1, \cdots, a_r)$ we clearly have
$F_e(a_1, \cdots, a_r; k) = R(a_1, \cdots, a_r).$
\medskip
In 1970, Folkman \cite{Folkman} proved that
for positive integers $k$ and $a_1, \cdots, a_r$,
$F_v (a_1, \cdots, a_r; k)$ and $F_e (a_1, a_2; k)$
exist if and only if $k > \max \{a_1, \cdots$, $a_r\}$.
Folkman's method did not work for
edge colorings for more than two colors.
The existence of $F_e(a_1, \cdots, a_r; k)$
was proved by Ne\v{s}et\v{r}il and R\"{o}dl
in 1976 \cite{NesetrilRodl}.
Folkman numbers have been studied by many
other authors, in particular in
\cite{drr14,RT,Nkolev2008,LiLin2017a,
LinLi2012a,RodlRS,XuShao2010a,XLSW}.
The current authors studied chromatic variations
of Folkman numbers \cite{XLR0}, and some
existence questions for
$F_v(H_1, \cdots, H_r;H)$ and $F_e(H_1, \cdots, H_r;H)$
\cite{XLR1}.
Perhaps the most wanted Folkman number is $F_e(3,3;4)$,
for which the currently best known bounds are 20
\cite{BikovNenov2016a} and 785 (an unpublished
improvement from 786 obtained by Kauffman, Wickus and the third
author, for more information about the upper bound see
\cite{XLR1}). Further improvements of the
bounds on $F_e(3,3;4)$ seem very difficult,
but some insights can be made into similar
cases involving almost complete graphs $K_k-e$.
For vertex-disjoint graphs $G$ and $H$, their {\em join}
$G+H$ has the vertices $V(G) \cup V(H)$ and edges
$E(G) \cup E(H) \cup E(G,H)$, where $E(G,H)$ is the set
of all possible edges between $V(G)$ and $V(H)$.
Let us also denote $K_k-e$ by $J_k$.
In \cite{XLR1}, we proved
the existence of
$F_e(K_{k+1},K_{k+1};J_{k+2})$ and
$F_v(K_k,K_k;J_{k+1})$, for all $k \ge 3$.
In the same paper we discussed the existence
of some generalized Folkman numbers, especially
in the cases of the form $F_e(K_3,K_3;H)$ for some
small graphs $H$. The latter
includes proofs of nonexistence of the numbers
$F_e(K_3,K_3;J_4)$, $F_e(K_3,K_3;K_2+3K_1)$ and
$F_e(K_3,K_3;K_1+P_4)$, and poses some open cases,
like that for $F_e(K_3,K_3;K_1+C_4)$.
In Section 2 we overview some of the prior constructions
and related upper bounds, and we present our new
constructions. They lead to some new concrete upper bounds,
presented in Section 3, for some special cases including
$F_e(K_3,J_4; K_5) \leq 27$ and
$F_e(J_4,J_4; K_5) \leq 51$.
\input body1.tex
\input ref1.tex
\end{document}
| {'timestamp': '2017-08-02T02:03:09', 'yymm': '1708', 'arxiv_id': '1708.00125', 'language': 'en', 'url': 'https://arxiv.org/abs/1708.00125'} |
\section{Introduction}
We will discuss the physics of correlated electron systems from an
experimental viewpoint, focussing on optical spectroscopy. The
interaction of light and matter will be discussed first from a
classical point of view, based on the Maxwell equations. This
review will be the basis for a discussion of optical techniques
that are most commonly used. We will then continue with a
discussion of the quantum mechanical description of the
interaction between light and matter, using the Kubo-formalism. We
finally discuss the application of sum rules to correlated systems
and what happens when interactions, like the electron-phonon
interaction, become important. The first part of our review is not
meant to be complete. Readers with interest for further details
are referred to references \citep{Wooten} and \citep{Dressel}. In
the following all fields, currents, charge densities etc. are
implied to be position and time dependent if not written
explicitly. Bold quantities imply vectors or matrices.
\newpage
\section{Electromagnetism and Matter}
\subsection{Maxwell's equations}
We start this review with the microscopic Maxwell equations,
\begin{eqnarray}
\nabla\cdot \mathbf{e} &=& 4\pi\rho_{micro},\label{micromax1} \\
\nabla\times \mathbf{e} &=& -\frac{1}{c}\frac{\partial}{\partial t}\mathbf{b},\label{micromax2} \\
\nabla\cdot \mathbf{b} &=& 0, \label{micromax3}\\
\nabla\times \mathbf{b} &=& \frac{1}{c}\frac{\partial}{\partial
t}\mathbf{e}+\frac{4\pi}{c}\mathbf{j}_{micro}.\label{micromax4}
\end{eqnarray}
Here \textbf{e} and \textbf{b} are the microscopic electric and
magnetic fields respectively. $\rho_{micro}$ is the total
microscopic charge distribution and $j_{micro}$ the total
microscopic current distribution (i.e. due to internal and
external sources). Note that these equations are written in the
C.G.S. system of units. To convert them to S.I. units, simply
replace $4\pi$ by $1/\varepsilon_{0}$. The charge distribution for
a collection of point sources with charge $q_{i}$ can be written
classically,
\begin{equation}
\rho_{micro}=\sum_{i}q_{i}\delta(\mathbf{r}-\mathbf{r_{i}}),
\end{equation}
or quantum mechanically as,
\begin{equation}
\rho_{micro}=-e\Psi^{*}(\mathbf{r})\Psi(\mathbf{r}).
\end{equation}
Equations (\ref{micromax1}-\ref{micromax4}) are not very practical
to work with. As a first step we rewrite them in a more familiar
form. To do this we average the fields, charge and current
distributions over a volume $\Delta V$,
\begin{eqnarray}
\rho_{total}(\mathbf{r})=\frac{1}{\Delta V}\int_{\Delta
V}\rho_{micro}(\mathbf{r}+\mathbf{r'})d^{3}\mathbf{r'},\\
\mathbf{J}_{total}(\mathbf{r})=\frac{1}{\Delta V}\int_{\Delta
V}\mathbf{J}_{micro}(\mathbf{r}+\mathbf{r'})d^{3}\mathbf{r'},
\end{eqnarray}
and similarly for $\mathbf{e}$ and $\mathbf{b}$. This is a
sensible procedure under the condition that $a_{0}\ll\Delta
V\ll(2\pi c/\omega)^{3}$ where $a_{0}$ is the Bohr radius. Using
these averaged distributions we arrive at the standard Maxwell
equations in free space,
\begin{eqnarray}
\nabla\cdot \mathbf{E}_{total}(\mathbf{r},t) &=& 4\pi\rho_{total}(\mathbf{r},t),\label{max1} \\
\nabla\times \mathbf{E}_{total}(\mathbf{r},t) &=& -\frac{1}{c}\frac{\partial}{\partial t}\mathbf{B}_{total}(\mathbf{r},t),\label{max2} \\
\nabla\cdot \mathbf{B}_{total}(\mathbf{r},t) &=& 0, \label{max3}\\
\nabla\times \mathbf{B}_{total}(\mathbf{r},t) &=&
\frac{1}{c}\frac{\partial}{\partial
t}\mathbf{E}_{total}(\mathbf{r},t)+\frac{4\pi}{c}\mathbf{J}_{total}(\mathbf{r},t).\label{max4}
\end{eqnarray}
In order to see how matter interacts with propagating
electromagnetic waves we have to distinguish between induced and
external sources. We write $\mathbf{J}_{total}\equiv
\mathbf{J}_{ext}+\mathbf{J}_{ind}$ and $\rho_{total}\equiv
\rho_{ext}+\rho_{ind}$. Both the induced and external charge and
current distributions have to obey the continuity equations
separately,
\begin{equation}
\nabla\cdot \mathbf{J}_{ind/ext}+\frac{\partial}{\partial
t}\rho_{ind/ext}=0.
\end{equation}
We can distinguish three different types of macroscopic internal
sources,
\begin{equation}\label{Jind}
\mathbf{J}_{ind}=\mathbf{J}_{cond}+\frac{\partial
\mathbf{P}}{\partial t}+c\nabla\times \mathbf{M}.
\end{equation}
The first term on the right hand side, $\mathbf{J}_{cond}$,
corresponds to the response of the free charges. The second term
is the current due to changes in the polarization state of the
system. Finally, we include a term representing a current due to
magnetization. Note that this last term is purely transversal (the
divergence of a rotation is always zero) and so is easy to
distinguish from the other two terms. Since the induced free
charge current due to photons is necessarily transversal,
$\nabla\cdot \mathbf{J}_{cond}=0$, we can use the continuity
equations to show that the induced free charge density has to be
zero and as a consequence that the total induced charge density
\begin{equation}\label{rho_ind}
\rho_{ind}=-\nabla\cdot P.
\end{equation}
It is convenient to introduce new fields
\begin{eqnarray}
\mathbf{D}(\mathbf{r},t)\equiv \mathbf{E}_{ext}(\mathbf{r},t)\equiv\mathbf{E}(\mathbf{r},t)+4\pi\mathbf{P}(\mathbf{r},t),\label{Dfield}\\
\mathbf{H}(\mathbf{r},t)\equiv\mathbf{B}(\mathbf{r},t)-4\pi\mathbf{M}(\mathbf{r},t),\label{Hfield}
\end{eqnarray}
so that using equations (\ref{Jind}-\ref{Hfield}) in equations
(\ref{max1}) and (\ref{max4}) we find,
\begin{eqnarray}
\nabla\cdot \mathbf{D}(\mathbf{r},t) &=& 4\pi\rho_{ext}(\mathbf{r},t),\label{mattermax1} \\
\nabla\times \mathbf{H}(\mathbf{r},t) &=&
\frac{1}{c}\frac{\partial}{\partial
t}\mathbf{D}(\mathbf{r},t)+\frac{4\pi}{c}\mathbf{J}_{ext}(\mathbf{r},t)+\frac{4\pi}{c}\mathbf{J}_{cond}(\mathbf{r},t).\label{mattermax4}
\end{eqnarray}
\subsection{Linear Response Theory}
In the spirit of linear response theory we assume that the
response of polarization, magnetization or current are linear in
the applied fields:
\begin{eqnarray}
\mathbf{P}=\chi_{e}\mathbf{E},\\
\mathbf{M}=\chi_{m}\mathbf{H},\label{Mag}\\
\mathbf{J}=\sigma\mathbf{E}.
\end{eqnarray}
The electric and magnetic susceptibilities can be expressed in
terms of a dielectric function
$\mathbf{\varepsilon'}=1+4\pi\chi_{e}$ and magnetic permittivity
$\mathbf{\mu'}=1+4\pi\chi_{m}$. The dielectric function is a
response function that connects the external field
$\textbf{E}_{ext}$ at position \textbf{r} and time $t$ with the
field $\textbf{E}$ at all other times and positions. So in
general,
\begin{equation}
\mathbf{E}_{ext}(\mathbf{r},t)=\int_{-\infty}^{t}\int\varepsilon'(\mathbf{r},\mathbf{r'},t,t')\mathbf{E}(\mathbf{r'},t')d^{3}r'dt'.
\end{equation}
We will be mainly interested in the Fourier transform of
$\varepsilon'\equiv\varepsilon(\textbf{q},\omega)$ however. It is
an easy exercise to express the Maxwell equations in terms of
$\textbf{q}$ and $\omega$ which we leave to the reader. We can use
these definitions to once again rewrite the Maxwell equations in
the following form,
\begin{eqnarray}
\nabla\cdot(\varepsilon'\mathbf{E}) &=& 4\pi\rho_{ext},\label{linresmax1} \\
\nabla\times \mathbf{E} &=& -\frac{1}{c}\frac{\partial}{\partial t}(\mu')\mathbf{H},\label{linresmax2} \\
\nabla\cdot\mu'\mathbf{H} &=& 0, \label{linresmax3}\\
\nabla\times \mathbf{H} &=& \frac{1}{c}\frac{\partial}{\partial
t}(\varepsilon'\mathbf{E})+\frac{4\pi\sigma}{c}\mathbf{E}.\label{linresmax4}
\end{eqnarray}
We are now in a position to study the response of a solid to an
externally applied field or light wave. For simplicity we assume
that our solid is homogeneous so that $\nabla\varepsilon'=0$ and
$\nabla\mu'=0$. We can describe light waves by plane waves, i.e.
\begin{eqnarray}
\mathbf{E}(\mathbf{r},t)=\mathbf{E_{0}}e^{i(\mathbf{q\centerdot
r}-\omega t)},\label{E}\\
\mathbf{B}(\mathbf{r},t)=\mathbf{B_{0}}e^{i(\mathbf{q\centerdot
r}-\omega t)}\label{B}.
\end{eqnarray}
Using (\ref{B}) on the right hand side of Faraday's equation
(\ref{max2}) and rearranging we find,
\begin{equation}
\mathbf{B}=\frac{c}{i\omega}\nabla\times\mathbf{E}.
\end{equation}
If we now take the curl of this equation and use the fact that we
can express $\textbf{M}$ in terms of $\textbf{B}$ as (see
equations (\ref{Hfield}) and (\ref{Mag})),
\begin{equation}
\mathbf{M}=\frac{\mu'^{-1}-1}{4\pi}\mathbf{B}.
\end{equation}
we find that
\begin{eqnarray}
\nabla\times\mathbf{M}&=&\frac{\mu'^{-1}-1}{4\pi}\nabla\times\mathbf{B}=\frac{c}{i\omega}\frac{\mu'^{-1}-1}{4\pi}\nabla\times\nabla\times\mathbf{E}\nonumber\\
&=&\frac{c}{i\omega}\frac{\mu'^{-1}-1}{4\pi}(\nabla^{2}\mathbf{E}-\nabla(\nabla\cdot\mathbf{E})=\frac{cq^{2}}{i\omega}\frac{\mu'^{-1}-1}{4\pi}\mathbf{E}^{T}.
\end{eqnarray}
Note that in this equation we are left with only the transversal
field since the curl of a curl is transverse. We give two further
identities for completeness,
\begin{equation}
\frac{\partial\mathbf{P}}{\partial
t}=-i\omega\frac{1-\varepsilon'}{4\pi}\mathbf{E},
\end{equation}
and,
\begin{equation}
J_{cond}=\sigma\mathbf{E}.
\end{equation}
Finally, we note that inside the solid $\rho_{ext}=J_{ext}=0$.
With this we have all the ingredients to express equation
(\ref{mattermax4}) in terms of $\textbf{E}$ and $\textbf{J}$. We
split this equation in transversal and longitudinal parts to find,
\begin{eqnarray}
\frac{\mathbf{J}_{ind}^{T}(\mathbf{q},\omega)}{\mathbf{E}^{T}(\mathbf{q},\omega)}&\equiv&\frac{i\omega}{4\pi}\{1-\varepsilon'(\mathbf{q},\omega)-
\frac{i4\pi}{\omega}\sigma(\mathbf{q},\omega)-\frac{c^{2}q^{2}}{\omega^{2}}(1-\frac{1}{\mu
(\mathbf{q},\omega)})\},\\
\frac{\mathbf{J}_{ind}^{L}(\mathbf{q},\omega)}{\mathbf{E}^{L}(\mathbf{q},\omega)}&\equiv&\frac{i\omega}{4\pi}\{1-\varepsilon'(\mathbf{q},\omega)-
\frac{i4\pi}{\omega}\sigma(\mathbf{q},\omega)\}.
\end{eqnarray}
We can define a new dielectric function with longitudinal and
transverse components and write the previous equation in a more
compact form,
\begin{equation}
\frac{\mathbf{J}_{ind}^{L,T}(\mathbf{q},\omega)}{\mathbf{E}^{L,T}(\mathbf{q},\omega)}\equiv\frac{i\omega}{4\pi}\{1-\varepsilon^{L,T}(\mathbf{q},\omega)\}.
\end{equation}
This new dielectric function $\varepsilon$ is now a complex
quantity:
$\varepsilon\equiv\varepsilon'+i\varepsilon''=\varepsilon'+i4\pi\sigma/\omega$.
Using the last relation we can also define a complex conductivity
$\hat{\sigma}\equiv\sigma'+i\sigma''$ and it is related to the
dielectric function by,
\begin{equation}
\hat{\sigma}=\frac{i\omega}{4\pi}(1-\varepsilon).
\end{equation}
The real part of $\varepsilon$ is often called the reactive part
and the imaginary part the dissipative part. The real and
imaginary parts are also indicated with a subscript 1 and 2
instead of (') and ('').
\subsection{Kramers-Kronig relations}\label{KK}
A fundamental principle in physics is the principle of causality:
an effect cannot precede its cause. This principle provides us
with a very useful relation between the real and imaginary parts
of a response function like the optical conductivity as we now
show. First we express the induced current due to an electric
field in terms of a memory function,
\begin{equation}\label{memfunc}
j(t)=\int^{t}_{-\infty}M(t-t')E(t')dt'.
\end{equation}
The memory function has the property that $M(\tau<0)=0$. This is
simply a restatement of the causality principle: we switch on a
driving force at time $\tau=0$ so before that time there can be no
current. We define the Fourier transform of $M$ in equation
(\ref{memfunc}) as,
\begin{equation}\label{FTmem}
\hat{\sigma}(\omega)=\int_{0}^{\infty}M(\tau)e^{i\omega\tau}d\tau.
\end{equation}
To do the integral we change to the complex frequency plane,
$\omega\to z=\omega_{1}+i\omega_{2}$. The exponential in Eq.
(\ref{FTmem}) now splits in a complex and real part,
\begin{equation}\label{FTmem2}
\hat{\sigma}(\omega)=\int_{0}^{\infty}M(\tau)e^{i\omega_{1}\tau}e^{-\omega_{2}\tau}d\tau.
\end{equation}
The second exponent in this integral is bounded in the upper half
plane for $\tau>0$ and in the lower half plane for $\tau<0$, so
that we can evaluate the integral in the upper half plane since
$M(\tau<0)=0$. We use the contour shown in figure \ref{contour}.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{contour.png}
\caption{\label{contour}Contour used to derive the KK-relations.}
\end{figure}
Since all poles occur on the real axis, the complete contour is
zero,
\begin{equation}
\oint dz \frac{\hat{\sigma}(z)}{z-\omega}=0.
\end{equation}
The integral along the large semi circle is also zero. So we are
left with,
\begin{equation}
\int_{-\infty}^{\omega-\varepsilon} dz
\frac{\hat{\sigma}(z)}{z-\omega} +
\int^{\infty}_{\omega+\varepsilon} dz
\frac{\hat{\sigma}(z)}{z-\omega} + \int_{\pi}^{0}d(\omega+\epsilon
e^{i\phi})\frac{\hat{\sigma}(\omega+\epsilon e^{i\phi})}{\epsilon
e^{i\phi}}=0.
\end{equation}
The first two integrals give the principle value of the integral
for $\epsilon\to 0$,
\begin{equation}
\mathcal{P}\int_{-\infty}^{\infty} d\omega'
\frac{\hat{\sigma}(\omega')}{\omega'-\omega}-\pi
i\hat{\sigma}(\omega')=0.
\end{equation}
From which the Kramers-Kronig relations follow,
\begin{eqnarray}
\sigma_{1}(\omega)&=&\frac{1}{\pi}\mathcal{P}\int_{-\infty}^{\infty}\frac{\sigma_{2}(\omega')}{\omega'-\omega}d\omega',\label{KK1}\\
\sigma_{2}(\omega)&=&-\frac{1}{\pi}\mathcal{P}\int_{-\infty}^{\infty}\frac{\sigma_{1}(\omega')}{\omega'-\omega}d\omega'.\label{KK2}
\end{eqnarray}
Using $Im(M(\tau))=0$ we see that
$\hat{\sigma}(-\omega)=\hat{\sigma}^{*}(\omega)$, which implies
that $\sigma_{1}(-\omega)=\sigma_{1}(\omega)$ and
$\sigma_{2}(-\omega)=-\sigma_{2}(\omega)$. These relations can be
used to rewrite equations (\ref{KK1}) and (\ref{KK2}),
\begin{eqnarray}
\sigma_{1}(\omega)&=&\frac{2}{\pi}\mathcal{P}\int_{0}^{\infty}\frac{\omega'\sigma_{2}(\omega')}{\omega'^{2}-\omega^{2}}d\omega',\\
\sigma_{2}(\omega)&=&-\frac{2\omega}{\pi}\mathcal{P}\int_{0}^{\infty}\frac{\sigma_{1}(\omega')}{\omega'^{2}-\omega^{2}}d\omega'.
\end{eqnarray}
The relations (\ref{KK1}) and (\ref{KK2}) between the real and
imaginary parts of the optical conductivity are examples of the
general relations between real and imaginary parts of causal
response functions and they are referred to as Kramers-Kronig (KK)
relations.
\subsection{Polaritons}\label{polariton}
In this section we discuss the properties of electromagnetic waves
propagating through solids. Such a wave is called a polariton. A
polariton is a photon dressed up with the excitations that exist
inside solids. For example one can have phonon-polaritons which
are photons dressed up with phonons. Although the solutions of the
Maxwell equations, i.e. the fields $\textbf{E}$ and $\textbf{B}$,
have the same form as before (Eq. (\ref{E}) and (\ref{B})) they
obey different dispersion relations as we will now see. As before
we assume that $\nabla\varepsilon'=\nabla\mu'=0$ and that
$\rho_{ext}=J_{ext}=0$. Taking the curl of Eq. (\ref{linresmax2})
we obtain for the left-hand side,
\begin{equation}
\nabla\times\nabla\times\mathbf{E}=-\nabla^{2}\mathbf{E}^{T},
\end{equation}
where the T indicates that we are left with a purely transverse
field. We then use Eq. (\ref{linresmax4}) to work out the
right-hand side of Eq. (\ref{linresmax2}) and we obtain the wave
equation,
\begin{equation}
\nabla^{2}\mathbf{E}^{T}=\frac{\varepsilon'\mu}{c^{2}}\frac{\partial^{2}\mathbf{E}^{T}}{\partial
t^{2}}+\frac{4\pi\sigma\mu}{c^{2}}\frac{\partial\mathbf{E}^{T}}{\partial
t}.
\end{equation}
From this wave equation we easily obtain the dispersion relation
for polaritons travelling through a solid by substituting Eq.
(\ref{E}),
\begin{equation}\label{poldisp}
\mu(\mathbf{q},\omega)\{\varepsilon'(\mathbf{q},\omega)+i\frac{4\pi\sigma(\mathbf{q},\omega)}{\omega}\}\omega^{2}=\mu\varepsilon^{L}\omega^{2}=\mathbf{q}^{2}c^{2}.
\end{equation}
The dispersion relation for longitudinal waves can be found by
observing that for longitudinal waves $\nabla\times\textbf{E}=0$
and hence the dispersion relation is simply,
\begin{equation}
\mu(\mathbf{q},\omega)\{\varepsilon'(\mathbf{q},\omega)+i\frac{4\pi\sigma(\mathbf{q},\omega)}{\omega}\}=0.
\end{equation}
The polariton solutions to Eq. (\ref{poldisp}) are of the form
\begin{equation}
\mathbf{E}(\mathbf{r},t)=\mathbf{E_{0}}e^{i(\mathbf{q\centerdot
r}-\omega t)},
\end{equation}
with
\begin{equation}
|q|=\frac{\sqrt{\mu\varepsilon}\omega}{c}.
\end{equation}
We now define the refractive index,
\begin{equation}
\hat{n}(\mathbf{q},\omega)=n+ik\equiv\sqrt{\mu\varepsilon}.
\end{equation}
In all cases considered here $n>0$ and $k>0$. We also note that
$Im(\varepsilon)\geq 0$ but it is possible to have
$Re(\varepsilon)<0$. If $k>0$ the wave travelling through the
solid gets attenuated,
\begin{equation}
\mathbf{E}(\mathbf{r},t)=\mathbf{E_{0}}e^{i\omega(nr/c-t)-r/\delta}.
\end{equation}
The extinction of the wave occurs over a characteristic length
scale $\delta$ called the skin depth,
\begin{equation}\label{skin}
\delta=\frac{c}{\omega k}=\frac{c}{\omega
Im\sqrt{\mu\varepsilon_{1}+i4\pi\mu\sigma_{1}/\omega}}.
\end{equation}
\begin{figure}[hbt]
\begin{minipage}{8cm}
\includegraphics[width=8cm]{Drudecond.png}
\caption{\label{drudecond} Real part of the optical conductivity
for parameter values indicated in the graph. The curve is
calculated using equation (\ref{drude}).}
\end{minipage}\hspace{2pc}%
\begin{minipage}{8cm}
\includegraphics[width=8cm]{Drudediel.png}
\caption{\label{dielfunc} Dielectric function corresponding to
equation (\ref{drudediel}) with the same parameters as in figure
\ref{drudecond}.}
\end{minipage}
\end{figure}
Note that we can have $k>0$ if $Im(\varepsilon)=0$ and
$Re(\varepsilon)<0$ so that the wave gets attenuated even though
there is no absorption. In table \ref{skindepth} we indicate some
limits of the skin depth.
\begin{table}[tbh]
\begin{tabular}{lrlrl}
\hline\hline
Insulator & & $\frac{4\pi\sigma_{1}}{\omega}\ll\varepsilon_{1}$ &
&
$\delta\approx\frac{c}{2\pi\sigma_{1}}\sqrt{\frac{\varepsilon_{1}}{\mu}}$\\
Metal & & $\frac{4\pi\sigma_{1}}{\omega}\gg\varepsilon_{1}$ & &
$\delta\approx\frac{c}{\sqrt{2\pi\mu\sigma_{1}\omega}}$\\
Superconductor & &
$\frac{4\pi\sigma_{1}}{\omega}\ll\varepsilon_{1}=-\frac{c^{2}}{\lambda^{2}\omega^{2}}$
& & $\delta\approx\frac{\lambda}{\sqrt{\mu}}$\\
\hline\hline
\end{tabular}
\caption{Some limiting cases of the general expression Eq.
(\ref{skin}). $\lambda$ in the last line is the London penetration
depth. }\label{skindepth}
\end{table}
To illustrate some of the previous results we now have a look at
the simplest model of a metal: the Drude-model. The optical
conductivity in the Drude model is,
\begin{equation}\label{drude}
\hat{\sigma}=\frac{ne^{2}}{m}\frac{1}{\tau^{-1}-i\omega}.
\end{equation}
Often $1/\tau$, the time in between scattering events, is
written as a scattering rate $\gamma$. The plasma frequency is
defined as $\omega^{2}_{p}\equiv 4\pi ne^{2}/2m$. The
dielectric function can now be written as,
\begin{equation}\label{drudediel}
\varepsilon(\omega)=1+4\pi\chi_{bound}-\frac{4\pi
ne^{2}}{m}\frac{1}{\omega(\gamma-i\omega)}=\varepsilon_{\infty}-\frac{4\pi
ne^{2}}{m}\frac{1}{\omega(\gamma-i\omega)},
\end{equation}
where for completeness we have included the contribution due to
the bound charges, represented by a high energy contribution
$\varepsilon_{\infty}$. Figure \ref{drudecond} shows the optical
conductivity given by equation (\ref{drude}) for parameter values
typical of a metal. Using the same parameters we can calculate the
dielectric function given by equation (\ref{drudediel}). The
results are shown in figure \ref{dielfunc}. We note that the real
part of the dielectric function is negative for
$\omega<\omega_{p}/\sqrt{\varepsilon_{\infty}}$ and positive for
$\omega>\omega_{p}/\sqrt{\varepsilon_{\infty}}$. The point where
it crosses zero is called the screened plasma frequency
$\omega^{*}_{p}$ (screened by interband transitions).
We can also easily calculate the optical constants,
\begin{equation}\label{Druden}
\hat n = \sqrt {\varepsilon _\infty - \frac{{\omega _p^2 }}
{{\omega \left( {\omega + i\tau ^{ - 1} } \right)}}}.
\end{equation}
The real and imaginary part are displayed in figure
\ref{druden}. We see that at the screened plasma frequency both
$n$ and $k$ show a discontinuity.
\begin{figure}[hbt]
\begin{minipage}{8cm}
\includegraphics[width=8cm]{Druden.png}
\caption{\label{druden} Optical constants corresponding to
equation (\ref{drudediel}) with the same parameters as in figure
\ref{drudecond}.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{8cm}
\includegraphics[width=8cm]{poldisp.png}
\caption{\label{figpoldisp} Polariton dispersion calculated with
the same parameters as in figure \ref{drudecond}.}
\end{minipage}
\end{figure}
The polariton dispersion follows from equation (\ref{poldisp}).
Here we assume that $\mu$ = 1 and frequency independent and use
Eq. (\ref{drudediel}) to solve (\ref{poldisp}) for $\omega(q)$.
The polariton dispersion consists of two branches the lowest
one for 0 $\le$ $\omega$ $\le$ $1/\tau$ and one for $\omega$
$\ge$ $\omega_{p}/\sqrt{\varepsilon_{\infty}}$.
Finally we show the skin depth in figure \ref{figskindepth}.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{skindepth.png}
\caption{\label{figskindepth} Skin depth calculated with the same
parameters as in figure \ref{drudecond}.}
\end{figure}
We see that for frequencies smaller than the scattering rate,
$\gamma$, light waves can enter the material. This is called the
classical skin effect. For frequencies larger than the screened
plasma frequency the material becomes transparent again.
\section{Experimental Techniques}
The goal of optical spectroscopy is to determine the complex
dielectric function or equivalently the complex optical
conductivity. Since electromagnetic waves have small momenta
compared to the typical momenta of a solid, i.e. $\textbf{q}\ll
1/a_{0}$, we usually only probe the $q\to 0$ limit of the optical
constants. In this limit,
\begin{eqnarray}
\lim_{q\to
0}(\varepsilon^{T}(\mathbf{q},\omega)-\varepsilon^{L}(\mathbf{q},\omega))=0,\\
\varepsilon(\mathbf{q\to
0},\omega)=\varepsilon_{1}(\omega)+i\frac{4\pi}{\omega}\sigma_{1}(\omega).
\end{eqnarray}
In some cases we can directly obtain information on both real and
imaginary components seperately, but more often we obtain
information where the contributions are mixed. We then make use of
some form the KK-relations to disentangle the two.
\subsection{Reflection and Transmission at an interface}
When we shine light on an interface between vacuum and a material,
part of the light is reflected and another part is transmitted as
in figure \ref{reflection}.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{reflection.png}
\caption{\label{reflection}Electromagnetic waves reflecting from a
material. The reflected wave has a smaller amplitude and is phase
shifted with respect to the incoming wave. The transmitted wave is
continuously attenuated inside the material.}
\end{figure}
At the boundary the electromagnetic waves have to obey the
following boundary conditions,
\begin{eqnarray}
\mathbf{E}_{i}+\mathbf{E}_{r}=\mathbf{E}_{t},\label{boundcond1}\\
\mathbf{E}\times\mathbf{H}\;//\;\mathbf{k}.
\end{eqnarray}
From these two equations it follows that the reflected magnetic
field suffers a phase shift at the boundary,
\begin{equation}\label{boundcond2}
\mathbf{H}_{i}-\mathbf{H}_{r}=\mathbf{H}_{t}.
\end{equation}
Using equation (\ref{E}) in equation (\ref{linresmax2}) we obtain,
\begin{equation}
iqc\mathbf{E}^{T}=i\omega\mu\mathbf{H}.
\end{equation}
so that, using the dispersion relation (\ref{poldisp}),
\begin{equation}
\frac{\mathbf{H}}{\mathbf{E}^{T}}=\sqrt{\frac{\varepsilon}{\mu}}.
\end{equation}
From now on we set $\mu=1$ unless otherwise indicated. In that
case $\mathbf{H}/\mathbf{E}^{T}=\hat{n}$. Combining this result
with Eq. (\ref{boundcond2}) we get,
\begin{equation}
\mathbf{E}_{i}-\mathbf{E}_{r}=\hat{n}.
\end{equation}
Together with Eq. (\ref{boundcond1}) we can now solve for
$\textbf{E}_{r}/\textbf{E}_{i}$ and
$\textbf{E}_{t}/\textbf{E}_{i}$,
\begin{eqnarray}
\hat{r}\equiv\mathbf{E}_{r}/\mathbf{E}_{i}=\frac{1-\hat{n}}{1+\hat{n}},\label{rhat}\\
\hat{t}\equiv\mathbf{E}_{t}/\mathbf{E}_{i}=\frac{2}{1+\hat{n}}.\label{that}
\end{eqnarray}
The two quantities $\hat{r}$ and $\hat{t}$ are the complex
reflectance and transmittance.
\subsection{Reflectivity experiments}
The real reflection coefficient $R(\omega)$ which is measured in a
reflection experiment is related to $\hat{r}$ via
\begin{equation}
R=|\hat{r}|^{2}=|\frac{(n-1)^{2}+k^{2}}{(n+1)^{2}+k^{2}}|.
\end{equation}
Note that in this experiment we obtain no information on the phase
of $\hat{r}$. In these experiments the angle of incidence is as
close to normal incidence as possible. To measure $R(\omega)$ one
first measures the reflected intensity $I_{s}$ from the sample
under study. To normalize this intensity one then has to take a
reference measurement. This can be done by replacing the sample
with a mirror (i.e a piece of aluminum or gold) and again measure
the reflected intensity, $I_{ref}$. The reflection coefficient is
then $R(\omega)\equiv I_{s}(\omega)/I_{ref}(\omega)$. A better way
is to evaporate a layer of gold or aluminum \textit{in-situ} and
measure the reflected intensity as a reference. This way one
automatically corrects for surface imperfections and, if done
properly, there are no errors due to different size and shape of
the reflecting surface. To obtain the optical constants from such
an experiment we have to make use of KK-relations. If we define,
$\hat{r}(\omega)\equiv\sqrt{R(\omega)}e^{i\theta}$, then the
logarithm of $\hat{r}(\omega)$ is
\begin{equation}
\ln\hat{r}(\omega)=\ln\sqrt{R(\omega)}+i\theta.
\end{equation}
The phase $\theta$ in this expression is the unknown we want to
determine. If we interpret $\hat{r}$ as a response function we can
use the same arguments as in the section on KK relations and
calculate $\theta({\omega})$ from,
\begin{equation}
\theta(\omega)=-\frac{\omega}{\pi}P\int_{0}^{\infty}\frac{\ln
R(\omega')}{\omega'^{2}-\omega^{2}}d\omega',
\end{equation}
which is just the same as the KK-relation for $\hat{\epsilon}$.
The complex dielectric function is calculated from $R(\omega)$ and
$\theta(\omega)$ using,
\begin{equation}\label{KKofR}
\hat{\varepsilon}(\omega ) = \left( {\frac{{1 - \sqrt {R\left(
\omega \right)} e^{i\theta \left( \omega \right)} }}{{1 + \sqrt
{R\left( \omega \right)} e^{i\theta \left( \omega \right)} }}}
\right)^{\rm 2} {\rm }.
\end{equation}
Although in principle exact, this technique is in practice only
approximate. The reason is that we cannot measure $R(\omega)$ from
zero to infinite frequency. Most experiments probe a frequency
range between a few meV and a few eV. To do the integral in Eq.
(\ref{KKofR}) one then has to use extrapolations in the frequency
ranges where no data is available. For metals the low frequency
extrapolation which is most often used is the so-called
Hagen-Rubens approximation,
\begin{equation}
R(\omega)=1-\alpha\sqrt{\omega}.
\end{equation}
For frequencies above the interband transitions one often uses an
extrapolation that is proportional to $\omega^{-4}$.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{Druderefl.png}
\caption{\label{druderefl} Reflectivity calculated using
parameters typical for a metal. The inset shows the low energy
reflectivity on an enhanced scale.}
\end{figure}
As an example of a possible experimental result we show in figure
\ref{druderefl} the reflectivity calculated from the Drude model
for the same parameters as in section on polaritons.
The reflectivity is close to one until just below the plasma
frequency. At the zero crossing of $\varepsilon_{1}$ the
reflectivity has a minimum. The inset shows a blow up of the
"flat" region below 50 meV. Here one can clearly see the
Hagen-Rubens behavior mentioned above. If the sample under
investigation is anisotropic one has to use polarized light along
one of the principle crystal axes to perform the experiment.
\subsection{Grazing Incidence Experiments}
A closely related technique is to measure reflectance under a
grazing angle of incidence. Here one has to distinguish between
experiments performed with different incoming polarizations as
shown in figure \ref{grazincid}.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{grazincidence.png}
\caption{\label{grazincid} Grazing incidence experiment. The
result of the experiment is extremely sensitive to the precise
orientation of the crystal axes with respect to the incoming
light.}
\end{figure}
We distinguish between p-polarized light and s-polarized light.
For p-polarization the electric field is parallel to the plane of
incidence, whereas for s-polarization it is perpendicular to it (s
stands for senkrecht). Since in principal the optical constants
along the three crystal axes can be different, we use the labels
$a$, $b$ and $c$ for the optical constants as indicated in figure
\ref{grazincid}. For p-polarized light the complex reflectance is,
\begin{equation}\label{rp}
{\rm }r_{\rm p} = \frac{{{ \hat n}_c { \hat n}_b \cos \theta -
\sqrt {{ \hat n}_c ^2 - \sin ^2 \theta } }}{{{ \hat n}_c { \hat
n}_b \cos \theta + \sqrt {{ \hat n}_c ^2 - \sin ^2 \theta } }}.
\end{equation}
The angle $\theta$ in this equation is the angle relative to the
surface normal under which the experiment is performed. For
s-polarized light the complex reflectance is,
\begin{equation}\label{rs}
{\rm }r_{\rm s} = \frac{{\cos \theta - \sqrt {{ \hat n}_a ^2 -
\sin ^2 \theta } }}{{\cos \theta + \sqrt {{ \hat n}_a ^2 - \sin
^2 \theta } }}.
\end{equation}
An example of such an experiment is shown in figure
\ref{grazincidbi2212}.
\begin{figure}[tbh]
\includegraphics[width = 6 cm]{grazincidbi2212.png}
\caption{\label{grazincidbi2212} Grazing incidence reflectivity of
Bi-2201, Bi-2212, Tl-2201 and Tl-2212. The inset in panel (b)
indicates the measurement geometry. The figure is adapted from
ref. \citep{tsvetkov}.}
\end{figure}
In this example the samples are from the bismuth based family of
cuprates \cite{tsvetkov}. They have a layered structure consisting
of conducting copper-oxygen sheets, interspersed with insulating
bismuth-oxygen layers. Since the bonding between layers is not
very strong it is very difficult to obtain samples that are
sufficiently thick along the insulating c-direction. The grazing
incidence technique is used here to probe the optical constants of
the c-axis without the need of a large ac-face surface area. A
disadvantage in this particular experiment is that it is not
possible to determine accurately the absolute value of the optical
constants. It is possible however to determine the so-called loss
function $Im(-1/\hat{\varepsilon}_{c})$. The experiment is
performed on the ab-plane of the sample using p-polarized light
and we can simplify the expression for $\hat{r}_{p}$ by using the
fact that the $a$ and $b$ direction are almost isotropic. The
resulting expression for $\hat{r}_{p}$ is,
\begin{equation}
\hat{r}_{\rm p} = \frac{{\sqrt {\hat \varepsilon _b } \cos \theta
- \sqrt {1 - {{\sin ^2 \theta } \mathord{\left/
{\vphantom {{\sin ^2 \theta } {\hat \varepsilon _c }}} \right.
\kern-\nulldelimiterspace} {\hat \varepsilon _c }}} }}{{\sqrt {\hat \varepsilon _b } \cos \theta + \sqrt {1 - {{\sin ^2 \theta } \mathord{\left/
{\vphantom {{\sin ^2 \theta } {\hat \varepsilon _c }}} \right.
\kern-\nulldelimiterspace} {\hat \varepsilon _c }}} }}.
\end{equation}
From this equation we can derive the following relation between
the grazing incidence reflectivity and a pseudo loss-function
$L(\omega)$ \cite{dvdm},
\begin{equation}
L(\omega)\equiv\frac{{\left( {1 - R_p } \right)}} {{\left( {1 +
R_p } \right)}} \approx Im \frac{{2e^{i\phi _p } }} {{\left| {n_b
} \right|\cos \theta }}\sqrt {1 - \frac{{\sin ^2 \theta }} {{\hat
\varepsilon _c }}}.
\end{equation}
The function
$\sqrt{1-\frac{\sin^{2}\theta}{\hat{\varepsilon}_{c}}}$ has maxima
at the same position as the true loss-function. In this way
information was gained on the phonon structure of the c-axis of
this material.
\subsection{Spectroscopic Ellipsometry}
The third technique we introduce here is spectroscopic
ellipsometry. This relatively new technique has two major
advantages over the previous techniques. Firstly, the technique is
self-normalizing meaning that no reference measurement has to be
done and secondly, it provides directly both the real and
imaginary parts of the dielectric function.
As with the grazing incidence technique we have to distinguish
between s- and p-polarized light and label the crystal axes.
Instead of measuring $R_{p}$ or $R_{s}$ independently, we now
measure directly the amplitude and phase of the ratio
$\hat{r}_{p}/\hat{r}_{s}=|\hat{r}_{p}/\hat{r}_{s}|e^{i(\eta_{p}-\eta_{s})}$.
To see how this can be done we first describe the experimental
setup. There are a number of different setups one can use and here
we describe the simplest. This setup consists of a source followed
by a polarizer. With this polarizer we can change the orientation
of the polarization impinging on the sample.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{ellipsometry.png}
\caption{\label{ellipsometry} Result of an ellipsometric
measurement. The phase shift $A_{0}$ and amplitude $2\gamma$ are
the two quantities that we are interested in.}
\end{figure}
The light reflected from the sample passes through another
polarizer (called analyzer) and then hits the detector. Depending
on the orientation of the first polarizer we can change the
electric field strength of s- and p-polarized light according to,
\begin{eqnarray}
E_p = \left| {E_i } \right|\cos \left( P \right), \\
E_s = \left| {E_i } \right|\sin \left( P \right). \\
\end{eqnarray}
From the expressions for $\hat{r}_{p}$ and $\hat{r}_{s}$,
(\ref{rp}) and (\ref{rs}), in the previous section it follows
that,
\begin{equation}\label{rp/rs}
{\rm }\hat \rho \equiv {\rm }\frac{{r_{\rm p} }}{{r_{\rm s} }} =
\frac{{\sqrt {{ \hat n}_c ^2 - \sin ^2 \theta } - { \hat n}_c {
\hat n}_b \cos \theta }}{{\sqrt {{ \hat n}_c ^2 - \sin ^2 \theta
} + { \hat n}_c { \hat n}_b \cos \theta }} \cdot \frac{{\sqrt {{
\hat n}_a ^2 - \sin ^2 \theta } + \cos \theta }}{{\sqrt {{ \hat
n}_a ^2 - \sin ^2 \theta } - \cos \theta }}.
\end{equation}
Our task is now to invert this equation and express the optical
constants in terms of measured quantities and instrument
parameters. For an isotropic sample this can be done quite easily.
We define the pseudodielectric function $\hat{\varepsilon}$ such
that:
\begin{equation}
\hat \rho \equiv \frac{{\sin \theta \tan \theta - \sqrt {{
\tilde \varepsilon } - \sin ^2 \theta } }}{{\sin \theta \tan
\theta + \sqrt {{ \tilde \varepsilon } - \sin ^2 \theta } }},
\end{equation}
where we note that
$\hat{\varepsilon}=\varepsilon_{a}=\varepsilon_{b}=\varepsilon_{c}$
in an optically isotropic medium. This equation can be inverted to
obtain $\tilde \varepsilon$,
\begin{equation}\label{ellipseps}
\tilde \varepsilon (\omega ) = \sin ^2 \theta \left[ {1 + \tan ^2
\theta \left( {\frac{{1 - \rho }}{{1 + \rho }}} \right)^2 }
\right].
\end{equation}
So all that is left to do is to express $\hat{\rho}$ in terms of
experimental parameters. The experiment is done in the following
way: we fix the polarizer at some angle $0<P<90$ and then we
record the intensity while rotating the analyzer 360 degrees. The
result is shown in figure \ref{ellipsometry}. We then measure the
amplitude of the resulting sine wave, $\gamma$ and the phase
offset with respect to zero, $A_{0}$ (we assume here that for
$P=0$ the polarizer and analyzer are aligned parallel to each
other). With some goniometry and figure \ref{ellipsometry} we can
show that,
\begin{equation}
\tan A_0 = \frac{{2\tan \left( P \right)}}{{\left| \rho \right|^2
+ \tan ^2 \left( P \right)}}\rho _1,
\end{equation}
and
\begin{equation}
\sqrt {1 - \gamma ^2 } = \frac{{2\tan \left( P \right)}}{{\left|
\rho \right|^2 + \tan ^2 \left( P \right)}}\rho _2.
\end{equation}
Combining these two equations leads to,
\begin{equation}\label{ellipsrho}
\rho = \frac{{1 \pm \sqrt {\gamma ^2 - \tan ^2 A_0 } }}{{\tan
A_0 - i\sqrt {1 - \gamma ^2 } }}\tan \left( P \right).
\end{equation}
\begin{figure}[bth]
\includegraphics[width=8.5 cm]{ellipshg1201.png}
\caption{\label{ellipshg1201}Dielectric function measured
ellipsometrically on a HgBa$_{2}$CuO$_{4}$ sample. The true
dielectric function is shown in solid lines. The pseudo
dielectric function (i.e. actually measured) is shown as a
dashed line. Data taken from ref. \citep{heumen}.}
\end{figure}
The combination of Eq. (\ref{ellipsrho}) with Eq.
(\ref{ellipseps}) is all we need to describe an ellipsometric
experiment on an isotropic sample. For an anisotropic sample
the problem is slightly more difficult. However, there exists a
theorem due to Aspnes which states that the inversion of Eq.
(\ref{rp/rs}) results in Eq. (\ref{ellipseps}) but now the
dielectric function on the left-hand side is a so-called
pseudo-dielectric function. This pseudo-dielectric function is
mainly determined by the component parallel to the intersection
of sample surface and plane of incidence (component along $b$
in figure \ref{grazincid}), but still contains a small
contribution of the two other components. If we perform three
measurements, each along a different crystal axis, we can
correct the pseudo dielectric functions and obtain the true
dielectric functions. If the sample is isotropic along two
directions, as is the case in high temperature superconductors
for example, only two measurements are required. Figure
\ref{ellipshg1201} shows in dashed lines the pseudo dielectric
function of HgBa$_{2}$CuO$_{4}$. In this case the $a$ and $b$
axes have the same optical constants. The c-axis dielectric
function was determined from reflectivity measurements and
subsequently used to correct the pseudo dielectric function
measured by ellipsometry on the ab-plane. The true dielectric
function after this correction is shown as the solid line.
\subsection{Transmission Experiments}
A technique complementary to the reflection techniques is
transmission spectroscopy. This technique is, obviously, most
suitable for transparent samples. In principle the technique can
also be applied to metallic samples but this requires very thin
samples or films. The reflection experiments discussed above are
usually good methods to obtain accurate estimates of the real part
of the optical conductivity. In contrast the transmission
experiments discussed below are more sensitive to weak absorptions
or, in other words to the imaginary part of the optical
conductivity. Note that the simultaneous knowledge of reflection
and transmission spectra allows one to directly determine the full
complex dielectric function without any further approximations.
Examples of weak absorptions which are better probed in a
transmission experiment are multi-phonon or magnon absorptions.
The equations for transmission experiments are slightly more
difficult then those for the reflection experiments. These
equations simplify if we do the experiment on a wedged sample as
shown in figure \ref{wedgedsample}.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{wedgedsample.png}
\caption{\label{wedgedsample}Transmission experiment on a wedged
sample. After the initial ray is partially reflected back from the
front surface all following rays are no longer parallel to the
first transmitted ray.}
\end{figure}
At the boundary between vacuum and the sample, part of the light
is reflected and part transmitted. The part that is transmitted is
given by,
\begin{equation}\label{tcoef1}
{ \hat t}_{v,s} = {\rm }\frac{2}{{1 + { \hat n}}}{\rm }.
\end{equation}
Inside the wave propagates according to $ e^{i\psi}$ where,
\begin{equation}
\psi \equiv \hat nd\omega /c.
\end{equation}
At the next boundary again part of the beam is reflected back into
the sample and part is transmitted. Now we can see the advantage
of the wedged sample: the part of the light that is reflected
propagates away at an angle and after another reflection the
second transmitted ray is no longer parallel to and spatially
separated from the first transmitted ray. This means that we only
have to care about the first transmitted ray. The transmission
coefficient at the boundary from sample to vacuum is given by,
\begin{equation}\label{tcoef2}
{ \hat t}_{s,v} = \frac{{2{ \hat n}}}{{{ \hat n} + 1}}{\rm },
\end{equation}
so that the total transmission coefficient is,
\begin{equation}\label{tcoef3}
{ \hat t}_{v,s} e^{{\rm i}\psi } { \hat t}_{s,v}.
\end{equation}
Putting Eq.'s (\ref{tcoef1})-(\ref{tcoef3}) together and taking
the absolute value to calculate the transmission $T$ gives,
\begin{equation}\label{T}
T = \frac{{\left| {4{ \hat n}} \right|^2 }}{{\left| {1 + { \hat
n}} \right|^4 }}\exp \left\{ { - \frac{{2d}}{\delta }} \right\}.
\end{equation}
In most transmission experiments
$\varepsilon_{1}\gg\varepsilon_{2}$ and the classical skindepth
$\delta$ can be approximated by,
\begin{equation}
\delta \left( \omega \right) \approx \frac{{2\pi \sigma _1 \left(
\omega \right)}}{{{\rm c}\sqrt {\varepsilon _1(\omega)} }}.
\end{equation}
Moreover in these cases $\varepsilon_{1}$ is often dispersion-less
so that we can use the expression for $\delta$ to invert
(\ref{T}),
\begin{equation}
\sigma _1 \left( \omega \right) = \left\{ { - \ln \left( T
\right) + 2\ln \left( {\frac{{4\left| {{ \hat n}}
\right|}}{{\left| {1 + { \hat n}} \right|^2 }}} \right)}
\right\}\frac{{c\sqrt {\varepsilon _1 } }}{{4\pi d}}.
\end{equation}
\begin{figure}[tbh]
\includegraphics[width=6 cm]{grueninger2.png}
\caption{\label{reftrans}Comparison of reflectivity and
transmission measured on the same sample. Note the strong
absorptive features present in the transmission spectrum that are
completely invisible in the reflectance spectra. Figure adapted
from \citep{Grueninger}}
\end{figure}
As an example of this technique we show in figure \ref{reftrans} a
comparison between the reflectivity and transmission spectra of
undoped YBa$_{2}$Cu$_{3}$O$_{7}$ \cite{Grueninger}. This material
is an (Mott) insulator which is clearly visible from the
reflectivity spectrum. The large structure at low energies is an
optical phonon. At higher energy the reflectivity spectrum appears
to be rather featureless. Focussing our attention on the
transmission spectrum we see that it is almost zero in the phonon
range but then above the phonon range a whole series of sharp dips
shows up. The optical conductivity consists of a set of smaller
peaks at energies between 100 meV and 300 meV which are due to
multi-phonon absorptions whereas the larger peak just above 300
meV is due to a two magnon plus one phonon absorption (see also
the section on spin interactions below).
\begin{figure}[tbh]
\includegraphics[width=6 cm]{transmission.png}
\caption{\label{transmission}Pictorial of a transmission
experiment on a plan parallel sample.}
\end{figure}
We can also do the experiment on a sample with two
plan-parallel sides as depicted in figure \ref{transmission}.
We can immediately realize that for a given thickness of the
sample there will be interference effects between different
transmitted rays for certain frequencies. These will cause
oscillations in the transmission spectra which are called
Fabry-Perot resonances. We now analyze the transmission
coefficients for this experiment. The coefficient for the first
ray is off course the same as in Eq. (\ref{tcoef3}). The
coefficients for the higher order rays are formed by
multiplying $\hat{t}_{v,s}e^{i\psi}$ on the right with a factor
$f$,
\begin{equation}
f\equiv\hat{r}_{s,v}e^{i2\psi}\hat{r}_{s,v},
\end{equation}
followed by a factor $\hat{t}_{s,v}$. So the total transmission
coefficient for the second transmitted ray is given by,
\begin{equation}
{\hat t}_{v,s} e^{{\rm i}\psi } { \hat r}_{s,v} e^{{\rm i}\psi }
{\hat r}_{s,v} e^{{\rm i}\psi } { \hat t}_{s,v}={\hat t}_{v,s}
e^{{\rm i}\psi } { \hat t}_{s,v} f.
\end{equation}
The coefficients $\hat{t}_{v,s}$ and $\hat{t}_{s,v}$ are given
by Eq. (\ref{tcoef1}) and (\ref{tcoef2}). The coefficient for
reflection on a boundary from sample to vacuum is given by,
\begin{equation}
{\hat r}_{s,v} = {\rm }\frac{{{\hat n} - 1}}{{{\hat n} + 1}}.
\end{equation}
It is easy to see that if we sum over all transmitted rays the
total transmission coefficient is given by,
\begin{equation}\label{tpl}
\hat t = {\hat t}_{v,s} e^{{\rm i}\psi } {\hat t}_{s,v} \left( {1
+ f + f^2 + ..} \right)= \frac{{2{\hat n}}}{{2{\hat n}\cos \psi -
i(1 + {\hat n}^2 )\sin \psi }}.
\end{equation}
For thin films the phase factor $\psi\ll1$ and we can simplify
this equation to,
\begin{equation}
\hat t \approx \frac{1}{{1 + \frac{{2\pi d}}{c}\sigma _1 -
i\frac{{\omega d}}{{2c}}(1 + \varepsilon ')}},
\end{equation}
and so,
\begin{equation}
T\left( \omega \right) \approx \frac{1}{{1 + 4\pi dc^{ - 1}
\sigma _1 \left( \omega \right)}}.
\end{equation}
\begin{figure}[htb]
\begin{minipage}{8.5 cm}
\includegraphics[width = 8.5cm]{STOtrans.png}
\end{minipage}\hspace{2pc}%
\begin{minipage}{8.5 cm}
\includegraphics[width = 6 cm]{STOdisp.png}
\end{minipage}
\caption{\label{STO}Left: Far infrared transmission spectrum for
SrTiO$_{3}$. The positions of the peaks determine the polariton
dispersion. The dashed line at low frequency is an extrapolation
to zero frequency. Right: Dispersion relation of polaritons in STO
as derived from transmission spectrum in the left panel.}
\end{figure}
More generally from Eq. (\ref{tpl}) we obtain,
\begin{equation}
T_{LR} = \frac{{4\left| \varepsilon \right|}}{{\left|
{4\varepsilon \cos ^2 \psi } \right| + \left| {\left( {1 +
\varepsilon } \right)^2 \sin ^2 \psi } \right| + 2{\mathop{\rm
Im}\nolimits} \left\{ {(1 + \varepsilon )\sin 2\psi } \right\}}}.
\end{equation}
In the case that the sample under investigation has only weak
absorptions, i.e. $Im(\hat{n})\approx0$, this equation simplifies
to,
\begin{equation}\label{TLR}
T_{LR} \approx \frac{{4n^2 }}{{4n^2 + \left( {1 - n^2 }
\right)^2 \sin ^2 \left( {nd\omega /c} \right)}}.
\end{equation}
This equation gives us some insight to the occurrence of
Fabry-Perot resonances: if $\omega=cm\pi/nd$ with m=0,1,2,... the
sinus is equal to zero and the transmission $T=1$. In between
these maxima the transmission has minima and
$T\approx4\hat{n}^{2}/(1+\hat{n}^{2})^{2}$. In reality the
transmission will never reach 1 due to the fact that
$Im(\hat{n})\neq0$ in which case our approximations are no longer
valid. As an example we display in the left panel of figure
\ref{STO} the transmission spectrum of SrTiO$_{3}$
\cite{mechelen}. This material is very close to being
ferroelectric and as a result it has a very large dielectric
constant. The non-sinusoidal shape of the peaks is due to this
large dielectric constant. One can use the Fabry-Perot resonances
to measure the polariton dispersion as we now show. Note that at
each maximum in the transmission spectrum we know precisely the
value of the argument of the sine function in Eq. (\ref{TLR}).
\begin{figure}[htb]
\begin{minipage}{8.5 cm}
\includegraphics[width = 8.5cm]{LSCO.png}
\end{minipage}\hspace{2pc}%
\begin{minipage}{8.5 cm}
\includegraphics[width = 8.5 cm]{LSCOdisp.png}
\end{minipage}
\caption{\label{transLSCO}Left: Transmission spectrum of LSCO at a
temperature just above $T_{c}$ and one far below. Note the shift
in peak positions. Figure adapted from ref. \citep{kuzmenko}.
Right: Dispersion relation of polaritons in LSCO as derived from
left panel. The squares are derived from the spectrum in the
superconducting state whereas the circles are determined at a
temperature slightly above $T_{c}$.}
\end{figure}
We can read off the value of $\omega$ from the graph and using Eq.
(\ref{poldisp}) we can replace the argument in the sine function
by $n\omega d/c=qd$ so the momentum at a given maximum is,
\begin{equation}
q_{m}=\frac{m\pi}{d}\quad m=0,1,2,...
\end{equation}
So given the thickness of the sample we can make a plot of
$\omega(q)$. The result for STO is shown in the right panel of
figure \ref{STO}. We see that the dispersion is linear, indicating
that $n$ is dispersion-less in this range. The slope of the curve
directly gives us $n\approx20.5$. Another interesting application
of this is to superconductors. In figure \ref{transLSCO} we show
the transmission spectrum of LSCO at a temperature slightly above
T$_{c}$ and far below \cite{kuzmenko}. One can see that the
position of the maxima has changed and this shows up in the
polariton dispersion in an interesting way (see right panel figure
\ref{transLSCO}). As in STO we see that in the normal state the
dispersion is linear and extrapolates to zero. In the
superconducting state the dispersion has acquired a $q^{2}$
dependence and no longer extrapolates to zero for $q\to0$. This is
the result of the opening of the superconducting gap and it
implies that the polaritons in the superconducting state have
acquired a mass. This is an example of the Anderson-Higgs
mechanism\cite{anderson}, the same mechanism via which the
Higgs-field gives a mass to the W and Z bosons in elementary
particle physics. In the superconductor the order parameter plays
the role of the Higgs-field and the spontaneously broken symmetry
is that of the U(1) gauge symmetry.
\subsection{TeraHertz time-domain spectroscopy}
This relatively new technique is the last we will discuss here.
\begin{figure}[htb]
\includegraphics[width=8.5 cm]{THz.png}
\caption{\label{THz}Left: recorded signal v.s. delay distance
without sample. Right: recorded signal v.s. delay distance with
sample. Note the extra peaks in the signal on the right due to
multiple reflections in the sample.}
\end{figure}
This technique uses a powerful laser pulse and records the
detector output as a function of time, more often expressed in an
optical delay distance. The result for an experiment in vacuum is
shown in figure \ref{THz} on the left. If we now insert a sample
that is transparent to terahertz radiation in the path of the beam
we expect that due to the different optical path length in the
sample the pulse will arrive at a later time. In fact, if we use a
sample with to plane parallel surfaces we expect a series of peaks
due to multiple reflections in the sample (see right panel of
figure \ref{THz}. These peaks are just a different manifestation
of the Fabry-Perot oscillations observed in transmission
spectroscopy. By Fourier transforming this signal to the frequency
domain and doing the same for the signal without sample we can
again obtain the transmission spectrum. The frequency domain
spectrum corresponding to the time domain spectra of figure
\ref{THz} is shown in figure \ref{STO}.
\section{Quantum theory}
We now move to the quantum theoretical description of the
interaction of light and matter using the Kubo-formalism. So far
we have been using a "geometrical" or macroscopic view of this
interaction, but in this section we will consider the effects of
the absorption of photons by electrons. Consider for simplicity a
metal. The electronic states of the system are described by a set
of bands, some of which are fully occupied, some partially and the
rest empty, figure \ref{bandstruct}.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{bandstruct.png}
\caption{\label{bandstruct}The indicated transition is an
interband transition. Al states below the dashed line indicated by
E$_{F}$ are occupied all states above are empty.}
\end{figure}
When photons interact with these band electrons they can be
absorbed and in this process the electron is excited to a higher
lying band leaving behind a hole. In this way we create
electron-hole pairs and this (dipole) transition from a state
$|\Psi_{\nu}^{N}\rangle$ to a state $|\Psi_{\mu}^{N}\rangle$ is
characterized by an optical matrix element,
\begin{equation}
M_{\mu \nu } (\vec q) = \left\langle {\Psi _\mu ^N } \right.\left|
{{\bf \hat v}_q } \right|\left. {\Psi _\nu ^N } \right\rangle.
\end{equation}
If the transition is from one band to another band we call the
transition an \textit{interband} transition and if the
transition is within a band we call it a \textit{intraband}
transition. In figure \ref{KCL} we show the optical
conductivity of KCl. In this compound a strong onset is seen in
the optical conductivity around $\approx$8.7 eV. This onset is
due to the excitation of electrons from the occupied p-band
related to the Cl$^{-}$ ions to the unoccupied s-band of the
K$^{+}$ ions. Since this particular transition involves moving
charge from the chlorine atoms to the potassium atoms this type
of excitation is called a charge transfer (CT) excitation
\cite{zaanen}.
\begin{figure}[bth]
\includegraphics[width=6 cm]{KCL.png}
\caption{\label{KCL}Optical conductivity of KCl. The series of
strong peaks are due to excitons. The onset in absorption
around 9 eV is the onset of charge transfer excitations.}
\end{figure}
Another important feature in figure \ref{KCL} are the strong
peaks seen around 7.5 eV. Many theories often neglect so-called
vertex corrections because these corrections cancel if the
interactions between electrons are isotropic. However in real
materials interactions are more often than not anisotropic and
this means that these corrections have to be taken into
account. The peaks seen in figure \ref{KCL} are due to
transitions from bound states of electron-hole pairs, called
excitons, which arise due to the vertex corrections. Before we
start our display of the Kubo-formalism we first introduce some
notation. We introduce the field operators,
\begin{equation}
\psi^{\dagger}_{\sigma}(\mathbf{r})=\sum_{k}e^{-i\mathbf{k}\cdot\mathbf{r}}\hat{c}^{\dagger}_{k,\sigma}.
\end{equation}
The density operator is given by,
\begin{equation}
\hat{n}_{\sigma}(\mathbf{r})=\psi^{\dagger}_{\sigma}(\mathbf{r})\psi_{\sigma}(\mathbf{r}).
\end{equation}
The Fourier transform of $\hat{n}_{\sigma}(\mathbf{r})$ is,
\begin{equation}
\hat{n}_{\sigma}(\mathbf{r})=\frac{1}{V}\sum_{q}e^{-i\mathbf{q}\cdot\mathbf{r}}\rho_{q},
\end{equation}
with
\begin{equation}
\rho_{q}=\sum_{k,\sigma}\hat{c}^{\dagger}_{k-q/2,\sigma}\hat{c}_{k+q/2,\sigma}.
\end{equation}
The velocity operator is defined as,
\begin{equation}
\mathbf{\hat{v}}_{q}=\frac{\hbar}{m}\sum_{k,\sigma}\mathbf{k}\hat{c}^{\dagger}_{k-q/2,\sigma}\hat{c}_{k+q/2,\sigma}.
\end{equation}
Finally, we note that the operators $\hat{n}_{\sigma}(\mathbf{r})$
and $\mathbf{\hat{v}}_{q}$ satisfy,
\begin{equation}
\frac{i}{\hbar}\left[\hat{n}_{\sigma}(\mathbf{r}),\hat{H}\right]+\nabla\cdot\mathbf{\hat{v}}_{q}=0.
\end{equation}
\subsection{The Kubo-formalism}
To calculate the optical conductivity from a microscopic starting
point we have to add to the Hamiltonian of the system a term that
describes the interaction with the electromagnetic field described
by,
\begin{equation}\label{Eop}
{\bf E}^T ({\bf r},t) = \frac{{i\omega }}{c}\sum\limits_q {{\bf
A}_q e^{i({\bf q} \cdot {\bf r} - \omega t)} },
\end{equation}
Note that we have chosen the transverse gauge which we will use
throughout the rest of the chapter. The interaction Hamiltonian is
given by,
\begin{equation}
H' = - \frac{{e\hbar }}{c}\sum\limits_q {e^{i({\bf q} \cdot {\bf
r} - \omega t)} {\bf A}_q \cdot {\bf \hat v}_{ - q} },
\end{equation}
and in the presence of an electromagnetic field we use the minimal
coupling,
\begin{equation}
{\bf \hat v}_q \to{\bf \hat v}_q - \frac{{e\hbar }}{{mc}}{\bf
A}_q e^{i({\bf q} \cdot {\bf r} - \omega t)} \hat \rho _q.
\end{equation}
We now start by examining the current operator
$\textbf{J}(\textbf{r},t)=\textbf{J}^{(1)}(\textbf{r},t)+\textbf{J}^{(2)}(\textbf{r},t)$.
It consists of two terms the first of which is called the
diamagnetic term,
\begin{equation}\label{J1}
{\bf J}^{(1)} (\mathbf{r},t) = - \frac{{ne^2 }}{{mc}}{\bf
A}(r,t)=\frac{{ine^2 }}{{m\omega}}{\bf E}^{T}(r,t),
\end{equation}
where in the last equality we have used Eq. (\ref{Eop}). The
second term is more difficult. It is given by,
\begin{equation}
{\bf J}^{(2)} (\mathbf{r},t) = \frac{{e^2 }}{V}\int\limits_{ -
\infty }^{\rm t} {\left\langle {{\rm e}^{{\rm i}H'\tau } {\rm
e}^{{\rm - i}H\tau } {\bf \hat v}(r,t){\rm e}^{{\rm i}H\tau }
{\rm e}^{{\rm - i}H'\tau } } \right\rangle {\rm e}^{{\rm i}\omega
\left( {{\rm t - }\tau } \right)} {\rm d}\tau } {\rm }.
\end{equation}
We make here the approximation of using linear response theory: we
expand the exponentials $e^{iH'\tau}$ to first order in
$\textbf{A}(\textbf{r},t)$ and then stop the series expansion.
After some algebra we arrive at,
\begin{equation}\label{J2}
\frac{J^{(2)}}{\mathbf{E}(\mathbf{r},t)}=\frac{ie^2}{\omega
V}\sum_{n}\mathbf{v}_{-q}^{nm}\mathbf{v}_{q}^{mn}\left[\frac{1}{\omega-E_{n}+E_{m}+i0^{+}}-\frac{1}{\omega+E_{n}-E_{m}+i0^{+}}\right],
\end{equation}
where we have defined,
\begin{equation}
\mathbf{v}_{q}^{mn}\equiv\langle\Psi_{m}|\mathbf{\hat{v}}_{q}|\Psi_{n}\rangle.
\end{equation}
The result we have obtained is for zero temperature but is easily
generalized to finite $T$ if we use the grand canonical ensemble.
Combining Eq. (\ref{J1}) and Eq. (\ref{J2}) we find for the
optical conductivity,
\begin{equation}\label{conduc}
\sigma_{\alpha,\alpha}(\mathbf{q},\omega)=\frac{{iNe^2
}}{{mV\omega}}+\frac{{ie^2 }}{{V\omega}}\sum_{n,m\neq
n}e^{\beta(\Omega-E_{n})}\left[\frac{v_{\alpha,q}^{nm}
v_{\alpha,-q}^{mn}}{\omega-\omega_{mn}+i\eta}-\frac{v_{\alpha,-q}^{nm}
v_{\alpha,q}^{mn}}{\omega+\omega_{mn}+i\eta}\right],
\end{equation}
where we have defined $\omega_{mn}\equiv E_{m}-E_{n}$. The optical
conductivity consists now of three contributions: the diamagnetic
term followed by a contribution to positive frequencies and a
contribution to negative frequencies. We note that in general
$\sigma_{\alpha,\alpha}(\mathbf{q},\omega)$ is a tensor as indicated
by the $\alpha$ subscripts. We further note that the diamagnetic
term does not give a real contribution to the conductivity. This
term gives a $\delta$-function contribution at zero frequency and
this is exactly canceled by a delta function in the second part.
This can be seen by using the fact that for every $n$ we have the
following relationship,
\begin{equation}\label{velsum}
\sum_{n,m\neq n}\frac{v_{\alpha,q}^{nm}
v_{\alpha,-q}^{mn}}{\omega_{mn}}=\frac{N}{2m}.
\end{equation}
So we can rewrite Eq. (\ref{conduc}) as,
\begin{equation}\label{cond}
\sigma_{\alpha,\alpha}(\mathbf{q},\omega)=\frac{{ie^2}}{{V}}\sum_{n,m\neq
n}\frac{e^{\beta(\Omega-E_{n})}}{\omega_{mn}}\left[\frac{v_{\alpha,q}^{nm}
v_{\alpha,-q}^{mn}}{\omega-\omega_{mn}+i\eta}+\frac{v_{\alpha,-q}^{nm}
v_{\alpha,q}^{mn}}{\omega+\omega_{mn}+i\eta}\right],
\end{equation}
From here on we take the limit $q\to0$ and define a generalized
oscillator strength $\Omega_{mn}$ as,
\begin{equation}\label{Omeganm}
\Omega_{mn}^{2}\equiv\frac{8\pi
e^{2}e^{\beta(\Omega-E_{n})}|v^{nm}_{\alpha}|^{2}}{\omega_{mn}V}.
\end{equation}
With this definition we are lead to the Drude-Lorentz expansion of
the optical conductivity,
\begin{equation}\label{DL}
\sigma_{\alpha,\alpha}(\omega)=\frac{i\omega}{4\pi}\sum_{n,m\neq
n}\frac{\Omega_{mn}^{2}}{\omega(\omega+i\gamma_{mn})-\omega^{2}_{mn}}.
\end{equation}
\subsection{Sum Rules}
Sum rules play an important role in optics. Using the equations of
the previous section we derive the Thomas-Reich-Kuhn sum rule also
known as the f-sum rule. The f-sum rule states that, apart from
some constants, the area under $\sigma_{1}$ is proportional to the
number of electrons and inversely proportional to their mass. This
can be shown as follows: integrating Eq. (\ref{DL}) we have,
\begin{equation}
{\mathop{\rm Re}\nolimits} \int\limits_{ - \infty }^\infty {\sigma
\left( \omega \right)d\omega = \frac{1}{4}} \sum\limits_{n,m \ne
n} {\Omega _{_{mn} }^2 }.
\end{equation}
Using the expression for $\Omega_{mn}$, Eq. (\ref{Omeganm}), and
expression (\ref{velsum}) we can rewrite the sum on the right hand
side as,
\begin{equation}
\sum\limits_{n,m \ne n} {\Omega _{_{mn} }^2 } = \frac{{4\pi e^2
N}}{{mV}}\sum\limits_n {e^{\beta \left( {\Omega - E_n } \right)}
= } \frac{{4\pi e^2 N}}{{mV}}.
\end{equation}
So the f-sum rule states that,
\begin{equation}
\int_{-\infty}^{\infty}\sigma_{1}(\omega)d\omega=\frac{{\pi e^2
N}}{{mV}},
\end{equation}
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{Alsumrule.png}
\caption{\label{ALsumrule}Effective number of carriers
$n_{eff}(\Omega_{c})$ as a function of cutoff frequency
$\Omega_{c}$ for Al. Figure adapted from \citep{smith}.}
\end{figure}
as promised. This is the full universal sum rule. It is often
rewritten as an integral over positive frequencies only and
using the definition of the plasma frequency $\omega_{p}$,
\begin{equation}
\omega_{p}^{2}\equiv\frac{{4\pi e^2 N}}{{mV}},
\end{equation}
as,
\begin{equation}
\int_{0}^{\infty}\sigma_{1}(\omega)d\omega=\frac{\omega_{p}^{2}}{8}.
\end{equation}
We can also define \textit{partial} sum rules, i.e. sum rules
where we integrate up to a certain frequency cutoff $\Omega_{c}$.
In such a case the sum rule is not universal (this means for
instance that the value of this sum rule can depend on
temperature) and we can now define a plasma frequency that depends
on the chosen cutoff frequency,
\begin{equation}
\omega_{p,valence}^{2}\equiv\frac{4\pi
e^{2}}{m}n_{eff}(\Omega_{c}).
\end{equation}
A nice example of the application of the partial sum rule is shown
in figure \ref{ALsumrule}. Here the partial sum rule is applied to
the optical conductivity of aluminum \cite{smith}. Here the
effective number of carriers contributing to the sum rule is
plotted as a function of $\Omega_{c}$. $n_{eff}(\Omega_{c})$
slowly increases to a value of roughly three around 50 eV. This
means that as we increase the cutoff from zero to 50 eV we are
slowly integrating over the intraband transitions and when we
reach a value of 50 eV we have integrated over all transitions
involving the three valence electrons. For higher energies the
interband transitions start to contribute with a sharp onset near
80 eV. Finally at 10$^{4}$ eV the sum rule saturates at 13
electrons, the total number of electrons of aluminum.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{sigmaSC.png}
\caption{\label{sigmaSC}Optical conductivity of Bi-2212 at $T_{c}$
and below. The difference in area between the two curves is an
estimate of the superfluid density.}
\end{figure}
Another application of sum rules can be found in superconductors.
In a superconductor the electrons form a superfluid condensate.
This condensate shows up in the optical conductivity as a delta
function at zero frequency (it contributes a diamagnetic term as
in Eq. (\ref{conduc})). At the same time a gap opens up in low
frequency part of the spectrum where the optical conductivity is
(close to) zero, see figure \ref{sigmaSC}. In the normal state the
system is usually metallic and characterized by a Drude peak. In
optical experiments we cannot measure the zero frequency response
and so we cannot directly measure the spectral weight
$\omega_{p,s}^{2}$ of the condensate. However, using sum rules we
can estimate its spectral weight because the total spectral weight
has to remain constant. This is summarized in the
Ferrel-Glover-Tinkham (FGT) sum rule \cite{ferrel}, which states
that the difference in spectral weight between the optical
conductivity in the superconducting and normal state is precisely
the spectral weight of the condensate,
\begin{equation}
\omega _{p,s} (T)^2 = 8\int\limits_{0^ + }^\infty {\left\{
{\sigma (\omega ,T_c ) - \sigma (\omega ,T)} \right\}d\omega }.
\end{equation}
Note that we integrate here from 0$^{+}$.
There also exist sum rules for mixtures of different types of
particles,
\begin{equation}\label{phonsum}
\int\limits_0^\infty {\sigma _1 \left( {\omega '} \right)}
d\omega ' = \sum\limits_j {\frac{{\pi n_j q_j ^2 }}{{2m_j }}},
\end{equation}
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{MgO.png}
\caption{\label{MgO}Optical conductivity due to phonon mode in
MgO. The area under the peak is proportional to the effective
charge of the mode. The inset shows the effective charge
calculated using (\ref{phonsum2}). Data from \citep{damascelli}.}
\end{figure}
here the index j labels the different species. This sum rule can
be applied to measure the charge of ions involved in vibrational
modes. If we can separate the contribution to the optical
conductivity due to the optical modes we can invert Eq.
(\ref{phonsum}) to calculate the effective charge related to the
mode. For example, in MgO (figure \ref{MgO}) both ions contribute
an equal charge $q_{Mg}=-q_{O}$. We define the effective mass
$\mu$ as $\mu^{-1}=m_{Mg}^{-1}+m_{O}^{-1}$ and assume that the
density of the two is equal. In that case we can rewrite Eq.
(\ref{phonsum}) as,
\begin{equation}\label{phonsum2}
Z(\omega )^2 \equiv \left( {\frac{{q_T^* (\omega )}}{e}}
\right)^2 \equiv \frac{2\mu}{{\pi ne^2
}}\int\limits_{\omega_{min}}^{\omega_{max}} {\sigma _{ph} \left(
{\omega '} \right)d\omega '},
\end{equation}
where the integral has to be taken in a frequency range such that
it includes the spectral weight of the optical phonon mode but
nothing else.
We will now derive expressions for the conductivity sum rule from
a more microscopic point of view. To do that we return to the Kubo
expression for the optical conductivity,
\begin{equation}
\sigma _1 \left( \omega \right) = \frac{{\pi e^2
}}{V}Tr\left\langle {\Psi _n } \right|{\bf \hat v}\left\{
{\frac{{\delta \left( {\omega - \hat H + E_n } \right)}}{{\hat H
- E_n }} + \frac{{\delta \left( {\omega + \hat H - E_n }
\right)}}{{\hat H - E_n }}} \right\}{\bf \hat v}\left| {\Psi _n }
\right\rangle.
\end{equation}
The Hamiltonian in this expression is that of the system of
interacting electrons without the interaction of light. It
represents the optical conductivity for the system in an arbitrary
(ground or excited) many-body state $|\Psi\rangle$. A peculiar
point of this expression is that although the velocity operators
create a single electron-hole pair, due to the fact that the
hamiltonian in the denominator of this expression still contains
the interactions between all particles in the system, the optical
conductivity represents the response from the full collective
system of electrons. If we integrate this expression over
frequency we get,
\begin{equation}\label{microsum}
\int\limits_{ - \infty }^\infty {\sigma _1 \left( \omega
\right)d\omega } = \frac{{2\pi e^2 }}{V}Tr\left\langle {\Psi _n }
\right|{\bf \hat v}\frac{1}{{\hat H - E_n }}{\bf \hat v}\left|
{\Psi _n } \right\rangle.
\end{equation}
We now take a closer look at the right-hand side of this
expression. Remember that,
\begin{equation}\label{commutator}
{\bf \hat v} = \frac{i}{\hbar }\left[ {\hat H,{\bf \hat x}}
\right].
\end{equation}
Using the commutator we can rewrite,
\begin{equation}
- 2i\hbar {\bf \hat v}\frac{1}{{\hat H - E_n }}{\bf \hat v} = \left( {\hat H{\bf \hat x} - {\bf \hat x}\hat H} \right)\frac{1}{{\hat H - E_n }}{\bf \hat v} + {\bf \hat v}\frac{1}{{\hat H - E_n }}\left( {\hat H{\bf \hat x} - {\bf \hat x}\hat H}
\right).
\end{equation}
Inserting this back into Eq. (\ref{microsum}) we find after some
rearranging
\begin{equation}\label{sumrule}
\int\limits_{ - \infty }^\infty {\sigma _1 \left( \omega
\right)d\omega } = \frac{{i\pi e^2 }}{{\hbar V}}\left\langle
{[{\bf \hat v},{\bf \hat x}]} \right\rangle,
\end{equation}
where $\langle...\rangle$ stands for the trace over all many-body
states.
Here we have used that,
\begin{equation}
\mathbf{\hat{x}}\hat{H}\frac{1}{\hat{H}-E_{n}}\mathbf{\hat{v}}=\mathbf{\hat{x}}\left(\hat{H}-E_{n}\right)\frac{1}{\hat{H}-E_{n}}\mathbf{\hat{v}}+\mathbf{\hat{x}}E_{n}\frac{1}{\hat{H}-E_{n}}\mathbf{\hat{v}}=\mathbf{\hat{x}}\mathbf{\hat{v}}+\mathbf{\hat{x}}E_{n}\frac{1}{\hat{H}-E_{n}}\mathbf{\hat{v}},
\end{equation}
and the fact that $\hat{H}|\Psi_{n}\rangle=E_{n}|\Psi_{n}\rangle$.
We can now obtain different expressions for the sum rule by
working out the commutator on the right-hand side of Eq.
(\ref{sumrule}) based on different model assumptions. In table
\ref{modelexpr} we summarize some results.
\begin{table}[tbh]
\begin{tabular}{lrl}
\hline\hline Free electrons & \quad\quad & $[{\bf \hat v},{\bf
\hat x}]
=\frac{\hbar }{{im}}\sum\limits_{k\sigma } {\hat n_{k\sigma } }$\\
Band electrons & \quad\quad & $[{\bf \hat v},{\bf \hat x}] =
\frac{\hbar }{{im}}\sum\limits_{k\sigma } {\hat n_{k\sigma } }
[{\bf \hat v},{\bf \hat x}] = \frac{\hbar }{i}\sum\limits_{k\sigma
} {\frac{{\partial ^2 \varepsilon _{k\sigma } }}{{\partial k^2
}}\hat n_{k\sigma } }$\\
N.N. & \quad\quad & $[{\bf \hat v},{\bf \hat x}] = -
\frac{{\hbar a^2 }}{i}\sum\limits_{k\sigma } {\varepsilon
_{k\sigma } \hat n_{k\sigma } }$\\
\hline\hline
\end{tabular}
\caption{Expressions for the commutator in Eq. (\ref{sumrule}) for
three different cases. N.N. stands for Nearest Neighbors tight
binding model} \label{modelexpr}
\end{table}
The sum rule for band electrons is in practice the most useful.
Suppose that we have a system with only a single reasonably well
isolated band around the Fermi level that can be approximated by a
tight binding dispersion $\varepsilon _k = - t\cos \left( {ka}
\right)$. In that case we find an interesting relation,
\begin{equation}
\int\limits_0^{\Omega _c } {\sigma _1 (\omega ,T)} d\omega = -
\frac{{\pi e^2 a^2 }}{{2\hbar ^2 V}}\sum\limits_{k,\sigma }
{\left\langle {\hat n_{k\sigma } \varepsilon _k } \right\rangle _T
} = - \frac{{\pi e^2 a^2 }}{{2\hbar ^2 V}}E_{kin} (T).
\end{equation}
This sum rule states that by measuring the optical spectral weight
we are in fact measuring the kinetic energy of the charge carriers
contributing to the optical conductivity. In real systems this
relation only holds approximately: usually there are other bands
lying nearby and the integral on the left contains contributions
from these as well. Often the bands are described by more
complicated dispersion relations in which case the relation
$\partial^{2}\varepsilon_{k}/\partial k^{2}=-\varepsilon_{k}$ does
not hold. We can make some other observations from the sum rule
for band electrons. Suppose again we have a single empty cosine
like band (it is only necessary that the band is symmetric but it
simplifies the discussion) at $T=0$. Since the band is empty, the
spectral weight is equal to zero. If we start adding electrons the
spectral weight starts to increase until we reach half-filling. If
we add more electrons the spectral weight will start to decrease
again because the second derivative becomes negative for
$k>\pi/2a$. If we completely fill the band the contributions from
$k>\pi/2a$ will precisely cancel the contributions from $k<\pi/2a$
and the spectral weight is again zero. Now consider what happens
if we have a half-filled band and start to increase the
temperature. Due to the smearing of the Fermi-Dirac distribution
higher energy states will get occupied leaving behind lower energy
empty states. The result of this is that the spectral weight
starts to decrease. One can show using the Summerfeld expansion
that the spectral weight follows a $T^{2}$ temperature dependence.
In the extreme limit of $T\to\infty$ something remarkable happens:
the Fermi-Dirac distribution is 1/2 everywhere and the electrons
are equally spread out over the band. The metal has become an
insulator!
\subsection{Applications of sum rules to superconductors}
Before we have a look at some applications of sum rules to
superconductors we first summarize some results from BCS theory.
We want to apply our ideas to cuprate superconductors so we use a
modified version from the original theory to include the
possibility of d-wave superconductivity. In other words we suppose
that there is some attractive interaction between the electrons
that has a momentum dependence. The energy difference between the
normal and superconducting state due to interactions can be
written as \cite{marel},
\begin{equation}
\langle\hat{H}_{s}^{int}\rangle-\langle\hat{H}_{s}^{int}\rangle=\int
d^{3}r g(r)V(r)=\sum_{k}g_{k}V_{k},
\end{equation}
where $g(r)$ and $g_{k}$ are the pair correlation function and its
fourier transform respectively.
\begin{figure}[tbh]
\includegraphics[width=6 cm]{correlation.png}
\caption{\label{correlation} Real and momentum space picture of
the correlation functions $g(r)$ and $g_{k}$. Figure adapted from
\citep{marel}.}
\end{figure}
We can find an expression for $g_{k}$,
\begin{equation}
g_{k}=\sum_{q}\frac{\Delta_{q+k}\Delta^{*}_{q}}{4E_{q+k}E_{q}}.
\end{equation}
As usual,
\begin{equation}
E_{k}=\sqrt{(\varepsilon_{k}-\mu)^{2}+\Delta^{2}},
\end{equation}
and the temperature dependence of $\Delta_{k}$ is given by,
\begin{equation}
\Delta_{k}=\sum_{q}\frac{V_{q}\Delta_{q}}{2E_{q}}\tanh\left(\frac{E_{k}}{2k_{b}T}\right).
\end{equation}
We now use a set of parameters extracted from ARPES measurements
to do some numerical simulations. First of all we calculate
$g_{k}$ and fourier transform it to obtain $g(r)$. The results are
shown in figure \ref{correlation}.
Although $g_{k}$ is not so illuminating $g(r)$ is. This function
is zero at the origin and strongly peaked at the nearest neighbor
sites. This is a manifestation of the d-wave symmetry. We also
note that the correlation function drops off very fast for sites
removed further from the origin.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{correnergy.png}
\caption{\label{correnergy} Correlation energy and kinetic energy
as a function of temperature for a d-wave BCS superconductor.
Figure adapted from \citep{marel}.}
\end{figure}
In figure \ref{correnergy} we show the results for a calculation
of the correlation and kinetic energy using the parameters
extracted from ARPES measurements on Bi-2212. The kinetic energy
is calculated from,
\begin{equation}
\langle\hat{H}_{kin}\rangle=\sum_{k}\varepsilon_{k}\{1-\frac{\varepsilon_{k}-\mu}{E_{k}}\tanh\left(\frac{E_{k}}{2k_{b}T}\right)\}.
\end{equation}
We see that the kinetic energy \textit{increases} in the
superconducting state. This can be easily understood by looking at
what happens to the particle distribution function below $T_{c}$,
as indicated in the left panel of figure \ref{distribution}: when
the system enters the superconducting state the area below the
Fermi energy decreases and the area above the Fermi energy
increases thereby increasing the total kinetic energy of the
system.
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{distribution2.png}
\caption{\label{distribution} Left: Distribution function for the
normal (Fermi liquid like) state and the superconducting state.
Right: Distribution function for a non-Fermi Liquid like state and
the superconducting state.}
\end{figure}
Nevertheless the total internal energy, which is the sum of the
interaction energy and the kinetic energy, decreases and this is
of course why the system becomes superconducting. Now let us take
a look at what happens in the cuprates. In figure \ref{WBi2223} we
display the optical spectral weight $W(\Omega_{c},T)$ as a
function of $T^{2}$ for Bi-2223.
\begin{figure}[tbh]
\includegraphics[width=6 cm]{Wbi2223.png}
\caption{\label{WBi2223} Temperature dependent spectral weight of
Bi-2223. Data taken from ref. \citep{carbone}.}
\end{figure}
To compare this to the BCS kinetic energy we have plotted here
$-W(\Omega_{c},T)$. This result is contrary to the result from our
calculation: the kinetic energy decreases in the superconducting
state. This experimental result, observed first by Molegraaf
\textit{et al.} \cite{molegraaf}, has sparked a lot of interest
both experimentally \cite{heumen,basov,syro,kuz2,carbone} and
theoretically
\cite{hirsch,anderson2,eckl,wrobel,haule,toschi,marsiglio,mayer,norman}.
We note that DMFT calculations with the Hubbard model as starting
point have shown the same effect as observed here \cite{mayer}.
Roughly speaking the effect is believed to be due to the
"strangeness" of the normal state (right panel figure
\ref{distribution}). It is well known that the normal state of the
cuprates shows non Fermi-liquid behavior. So if the distribution
function in the normal state does not show the characteristic step
of the Fermi liquid at the Fermi energy but is rather a broadened
function of momentum it is very well possible that the argument we
made for the increase of the kinetic energy (see above) is
reversed.
\subsection{Applications of sum rules: the Heitler-London model}
Another interesting application of sum rules is that we can use
them in some cases to extract the hopping parameters of a system.
In order to see how this works we express the optical conductivity
at zero temperature,
\begin{equation}
\sigma _1 \left( \omega \right) = \frac{{\pi e^2
}}{V}\left\langle {\Psi _g } \right|{\bf \hat v}\frac{{\delta
\left( {\omega - \hat H + E_g } \right)}}{{\hat H - E_g }}{\bf
\hat v}\left| {\Psi _g } \right\rangle,
\end{equation}
in terms of the dipole operator. Here $|\Psi_{g}\rangle$ is the
groundstate of the system. To do this we make use of the
commutator Eq. (\ref{commutator}) and the insertion of a complete
set of states. After integrating over frequency we get
\begin{equation}\label{dipsumrule}
\int\limits_0^\infty {\sigma _1 \left( \omega \right)d\omega } =
\frac{{\pi e^2 }}{{\hbar ^2 V}}\sum\limits_n {\left( {E_n - E_g }
\right)\left| {\left\langle n \right|{\bf \hat x}\left| g
\right\rangle } \right|^2 }.
\end{equation}
We note that this can be done only for finite system sizes. Now
consider the special case of a diatomic molecule with two energy
levels, one on each atom and a hopping parameter t and distance
$d$ between the two atoms. We also assume that there is a
splitting $\Delta$ between the two levels. The hamiltonian for
such a system is,
\begin{equation}
H = t{\rm }\sum\limits_\sigma {\left( {\psi _{L,\sigma }^t \psi
_{R,\sigma } + \psi _{R,\sigma }^t \psi _{L,\sigma } } \right)} +
\frac{\Delta }{2}\left( {\hat n_R - \hat n_L } \right) + U\left(
{\hat n_{L \uparrow } \hat n_{L \downarrow } + \hat n_{R \uparrow
} \hat n_{R \downarrow } } \right).
\end{equation}
The indices $L$ and $R$ indicate the left and right atom
respectively. If we now put 1 electron in this system we have a
two-level problem that is easily diagonalized. As usual we make
bonding and anti-bonding states,
\begin{eqnarray}
\left| {\psi _{g,\sigma } } \right\rangle = u\left| {\psi _{l,\sigma } } \right\rangle + v\left| {\psi _{R,\sigma } } \right\rangle, \\
\left| {\psi _{e,\sigma } } \right\rangle = v\left| {\psi
_{l,\sigma } } \right\rangle - u\left| {\psi _{R,\sigma } }
\right\rangle.
\end{eqnarray}
The coefficients $u$ and $v$ are given by,
\begin{equation}
u = \frac{1}{{\sqrt 2 }}\sqrt {1 + \frac{\Delta }{{E_{CT}
}}};\quad\quad v = \frac{1}{{\sqrt 2 }}\sqrt {1 - \frac{\Delta
}{{E_{CT} }}}.
\end{equation}
The bonding and anti-bonding states are split by an energy
$E_{CT}$,
\begin{equation}
E_{CT} = \sqrt {\Delta ^2 + 4t^2 }.
\end{equation}
We are now in position to calculate the transition matrix element
appearing in Eq. (\ref{dipsumrule}). The position operator can be
represented by,
\begin{equation}
{\bf \hat x} = \frac{d}{2}{\rm }\left( {\hat n_R - \hat n_L }.
\right).
\end{equation}
So the matrix element is,
\begin{eqnarray}
\left\langle {\psi _{g,\sigma } } \right|{\bf \hat x}\left| {\psi
_{e,\sigma } } \right\rangle =
\left(u\langle\Psi_{L}|+v\langle\Psi_{R}|\right)\frac{d}{2}{\rm
}\left( {\hat n_R - \hat n_L }
\right)\left(u|\Psi_{L}\rangle-v|\Psi_{R}\rangle\right) \nonumber\\
=-\frac{d}{2}(uv)=- \frac{t}{{E_{CT} }}d
\end{eqnarray}
Using this in the sum rule Eq. (\ref{dipsumrule}) finally gives us
the spectral weight of this model,
\begin{equation}\label{tsum}
\int\limits_0^\infty {\sigma _1 \left( \omega \right)d\omega }
=\frac{{e^2 \pi d^2 }}{{\hbar ^2 V}}\frac{{t^2 }}{{\sqrt {\Delta
^2 + 4t^2 } }}.
\end{equation}
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{NaVO.png}
\caption{\label{NaVO} Optical conductivity of
$\alpha$-NaV$_{2}$O$_{5}$ for two polarizations: one with the
field parallel to the chains and one perpendicular. Data taken
from ref. \citep{damascelli2}.}
\end{figure}
We see that there is a very simple relation between the spectral
weight of this model and the hopping parameter. This sumrule has
been applied to $\alpha$-NaV$_{2}$O$_{5}$ \cite{damascelli2}. This
compound is a so-called ladder compound. It consists of double
chains of vanadium atoms forming ladders which are weakly coupled
to each other. Each unit cell contains 4 V atoms and 2 valence
electrons. The vanadiums on the rungs of the ladder are more
strongly coupled than those along the legs, i.e. $t_{\perp}\gg
t_{\parallel}$. The Heitler-London model we have discussed above
can be applied to this system since each rung forms precisely a
two level system with different levels. The only difference that
we have to take into account is that this is a crystal consisting
of $N$ independent two level systems. Figure \ref{NaVO} shows the
optical conductivity of $\alpha$-NaV$_{2}$O$_{5}$.
There are two measurements shown: one with light polarized
parallel to the chains and one with light polarized perpendicular
to the chains. We can immediately read of $E_{CT}\approx 1$ eV.
Integrating the contribution to the optical conductivity of the
peaks we find that the spectral weight perpendicular to the chains
is roughly 4 times larger then the spectral weight parallel to the
chains, so $t_{\perp}\approx 4t_{\parallel}$. Inverting Eq.
(\ref{tsum}) we can calculate $t_{\perp}$ and we find
$t_{\perp}\approx0.3$ eV. The second strong peak at approximately
3.2 eV is a charge transfer peak from vanadium to oxygen. We will
come back to this example in the last section on spin
interactions.
\subsection{Generalized Drude formalism}
We have already encountered the Drude formula for the optical
conductivity of a metal (see the section on polaritons). Even
though this model is based on a classical gas of non-interacting
particles it describes amazingly well the optical properties of a
good metal. This is even more surprising if one realizes that in a
metal electrons reside in bands and that the transitions we are
making with photons are vertical due to the negligible photon
momentum. So from the band picture point of view, when we consider
a single band of electrons interacting with photons we should
expect a single delta function at the origin. The reason that this
is not what is observed is because we have neglected the other
interactions in the system. Electrons in solids interact with the
lattice vibrations, impurities and/or other collective modes. Due
to the electron-phonon interaction for instance we can have
processes where a photon creates an electron-hole pair in which
the electron "shakes off" a phonon. In this process the phonon can
carry away a much larger momentum then originally provided by the
photon. Due to this effect we can have phonon-assisted transitions
which give a width $1/\tau$ to the delta function. This width is
called the scattering rate. If the interactions are inelastic, as
in the interactions with impurities, this scattering rate is just
a constant. Otherwise, this scattering rate can depend on
frequency. However, if we define the scattering rate in Eq.
(\ref{drude}) to be frequency dependent,
$1/\tau\equiv1/\tau(\omega)$, the KK-relations force us to
introduce a frequency dependent effective mass as well. This is
what is done in the generalized Drude formalism \cite{Allen}. The
optical conductivity is written as,
\begin{equation}
\sigma \left( \omega \right) = \frac{{ne^2 /m}}{{\tau ^{ - 1}
(\omega ){\rm - i}\omega m*(\omega )/m}}.
\end{equation}
Having measured a conductivity spectrum we can invert these
equations to calculate 1/$\tau(\omega)$ or $m^{*}(\omega )/m$ via,
\begin{equation}\label{1/tau}
\tau ^{ - 1} (\omega ) \equiv {\mathop{\rm Re}\nolimits}
\frac{{ne^2 /m}}{{\sigma \left( \omega \right)}} = \Sigma
''(\omega ),
\end{equation}
and
\begin{equation}\label{mstar/m}
\frac{{m^{*}(\omega )}}{m} \equiv {\mathop{\rm Im}\nolimits}
\frac{{ - ne^2 /m}}{{\omega \sigma \left( \omega \right)}} = 1 +
\frac{{\Sigma '(\omega )}}{\omega }.
\end{equation}
In the last equality of these equations we have defined an optical
self-energy. Note that this quantity is \textit{not} equivalent to
the self-energy used in the context of Green's functions. We can
rewrite the optical conductivity in terms of $\Sigma(\omega)$ as,
\begin{equation}\label{optcondSigma}
\sigma \left( \omega \right) = \frac{{ne^2 }}{m}\frac{i}{{\omega
+ \Sigma \left( \omega \right)}}.
\end{equation}
In the case of impurity scattering $\Sigma(\omega)$ is simply
given by
\begin{equation}
\Sigma \left( \omega \right) = i/\tau _0,
\end{equation}
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{sigmaCe.png}
\caption{\label{sigmaCe} Optical conductivity of Ce in the
$\alpha$ and $\gamma$ phases. Data taken from ref.
\citep{vandereb}.}
\end{figure}
so that $1/\tau(\omega)=1/\tau_{0}$ and $m*(\omega )/m=1$. We can
also capture the effect of the interaction of the electrons with
the static lattice potential in a self-energy,
\begin{equation}
\Sigma \left( \omega \right) = \lambda \omega,
\end{equation}
which gives $\tau^{-1}(\omega )=0$ and
$m^{*}(\omega)/m=1+\lambda$. This is also called static mass
renormalization. Finally we consider dynamical mass
renormalization where the electrons couple to a spectrum of
bosons,
\begin{equation}\label{SEboson}
\Sigma \left( \omega \right) = \frac{{\lambda \omega }}{{1 -
i\omega /T^* }},
\end{equation}
Here $\lambda$ is a coupling constant and $T^{*}$ is a
characteristic temperature scale related to the bosons. In this
case we find,
\begin{equation}
\tau ^{ - 1} (\omega ) = \lambda T^* \frac{{\omega ^2 }}{{T^{*2} +
\omega ^2 }},
\end{equation}
and
\begin{equation}
\frac{{m^{*}(\omega )}}{m} = 1 + \lambda \frac{{T^{*2} }}{{T^{*2}
+ \omega ^2 }}.
\end{equation}
As an example we will discuss the $\alpha$-phase to $\gamma$-phase
transition in pure Cerium. When Cerium is grown at elevated
temperatures it forms in the so called $\gamma$-phase. At low
temperatures a volume collapse occurs and the resulting phase is
called the $\alpha$-phase. This iso-structural transition is first
order. The reduction in volume can be as much as 20 to 30 $\%$. Ce
has 4 valence electrons and these can be distributed between
\begin{figure}[tbh]
\includegraphics[width=6 cm]{tauCe.png}
\caption{\label{tauCe} Scattering rate and effective mass of Ce in
the $\alpha$ and $\gamma$ phases. Data taken from ref.
\citep{vandereb}.}
\end{figure}
localized $4f$ states and the $5d$ states that form the conduction
band. If occupied, the 4f states will act as paramagnetic
impurities. In the $\gamma$-phase the Kondo temperature
$T_{K}\approx$ 100 K whereas in the $\alpha$-phase $T_{K}\approx$
2000 K. This difference can be understood to be simply due to the
larger lattice spacing in the $\gamma$-phase: the hopping integral
$t$ is smaller and hence $T_{K}$ is smaller. Figure \ref{sigmaCe}
shows the optical conductivity of $\alpha$- and $\gamma$-Cerium.
These measurements were done by depositing Ce films on a substrate
at high and low temperature to form either the $\alpha$ or
$\gamma$ phase. We see that $\gamma$-Cerium is less metallic than
$\alpha$-Cerium. In the $\gamma$-phase there is only a weak Kondo
screening of the impurity magnetic moments and this gives rise to
spin flip scattering, which is the main source of scattering. In
the $\alpha$-phase the moments are screened and form renormalized
band electrons in a very narrow band. This suppresses the
scattering. The difference in scattering rates shows up in the
optical conductivity as a narrower Drude peak for the
$\alpha$-phase (see figure \ref{sigmaCe}). Figure \ref{tauCe}
displays the scattering rate and effective mass extracted from the
optical conductivity in figure \ref{sigmaCe} using Eq.
(\ref{1/tau}) and (\ref{mstar/m}). In the $\gamma$-phase
$1/\tau(\omega)$ extrapolates to a finite value due to local
moment or spin-flip scattering. We can rewrite the real part of
the optical conductivity in Eq. (\ref{optcondSigma}) with the
self-energy of Eq. (\ref{SEboson}) as follows,
\begin{equation}
\frac{{4\pi }} {{\omega _{_p }^{*2} }}\sigma \left( \omega \right)
= \frac{i} {\omega }\frac{1} {{1 + \frac{\lambda } {{1 - i\omega
/T^* }}}}.
\end{equation}
Note that we have defined a renormalized plasma frequency,
$\omega_{p}^{*}$, since the spectral weight is not conserved when
adding $\Sigma(\omega)$ to $\sigma(\omega)$. With some simple
algebra this can be rewritten as,
\begin{equation}
\frac{{4\pi }} {{\omega _{_p }^{*2} }}\sigma \left( \omega \right)
= \frac{i} {\omega } + \lambda \frac{{T^* }} {{\omega ^2 +
i\omega (1 + \lambda )T^* }}.
\end{equation}
It follows that the real part of this expression is then,
\begin{equation}
\frac{{4\pi }} {{\omega _{_p }^{*2} }}Re \sigma \left( \omega
\right) = \frac{\pi } {2}\delta (\omega ) + \lambda \frac{{T^* }}
{{\omega ^2 + (1 + \lambda )^2 T^{*2} }}.
\end{equation}
We see that the optical conductivity is split into two
contributions: a $\delta$-function which represents the
coherent part of the charge response and an incoherent
contribution. The $\delta$-function is usually broadened due to
other scattering channels present in the system. In this case
the $\delta$-function represents the contribution due to the
Kondo-peak whereas the incoherent contribution is due to the
side-bands. This splitting of the conductivity in a coherent
and incoherent contribution is nicely observed in the
$\alpha$-phase of Cerium as indicated in figure \ref{sigmaCe}.
\begin{figure}[thb]
\begin{minipage}{6cm}
\includegraphics[width=8cm]{ZrB12sig.png}
\end{minipage}\hspace{2pc}%
\begin{minipage}{6cm}
\includegraphics[width=6cm]{ZrB12extdrude.png}
\end{minipage}
\caption{\label{ZrB12sig}Left: Optical conductivity of ZrB$_{12}$
at selected temperatures. Right: $1/\tau$ and $m^{*}/m_{b}$ for
several temperatures. Data taken from ref. \citep{teyssier}.}
\end{figure}
This splitting of the optical conductivity
in coherent and incoherent contributions is much more general
however and is frequently observed in correlated electron
systems.
\section{Electron-phonon coupling}
Electron-phonon coupling is most easily described in the framework
of Migdal-Eliashberg theory. The application of the theory to
optics can be found in the papers by Allen \cite{Allen}. In the
so-called Allen approximation the self-energy in Eq.
(\ref{optcondSigma}) is calculated using,
\begin{equation}\label{SEallenaprox}
\Sigma \left( \omega \right) = - 2i\int\limits_0^\infty {d\Omega
\alpha _{tr} ^2 F} (\Omega )K(\frac{\omega }{{2\pi
T}},\frac{\Omega }{{2\pi T}}).
\end{equation}
Here the kernel $K(\frac{\omega }{{2\pi T}},\frac{\Omega }{{2\pi
T}})$ is given by,
\begin{equation}
{\rm K(x}{\rm ,y)} = \frac{{\rm i}}{{\rm y}} + \frac{{y -
x}}{x}\left[ {\Psi (1 - ix + iy) - \Psi (1 + iy)} \right] +
\frac{{y + x}}{x}\left[ {\Psi (1 - ix - iy) + \Psi (1 - iy)}
\right].
\end{equation}
where the $\Psi(x)$ are Digamma functions. The function
$\alpha^{2}_{tr}F(\Omega)$ appearing in Eq. (\ref{SEallenaprox})
is the phonon spectral function. The label "tr" stands for
transport indicating that the spectral function is related to a
transport property. This function is different by a multiplicative
factor from the true $\alpha^{2}F(\Omega)$ as measured by for
instance tunnelling. The electron-phonon coupling strength is
easily calculated from $\alpha^{2}_{tr}F(\Omega)$ by integration,
\begin{equation}\label{lambdatr}
\lambda_{tr}=2\int_{0}^{\infty}\frac{\alpha^{2}_{tr}F(\Omega)}{\Omega}d\Omega.
\end{equation}
This approach was first applied by Timusk and Farnworth in a
comparison of tunnelling and optical measurements on the
superconducting properties of Pb \cite{Timusk}. As an example
we discuss the application of this formalism to the optical
properties of ZrB$_{12}$ \cite{teyssier}. Figure \ref{ZrB12sig}
shows the optical conductivity of ZrB$_{12}$. The spectrum
consists of what appears to be a Drude peak and some interband
contributions. Also shown are the calculated $1/\tau(\omega)$
and $m^{*}(\omega)/m_{b}$. The temperature dependence of
$1/\tau(\omega)$ is what is usually observed for a narrowing of
the Drude peak with decreasing temperature whereas the strong
frequency dependence is suggestive of electron-phonon
interaction. Using the McMillan formula (\ref{lambdatr}) the
coupling strength was estimated to be $\lambda_{tr}\approx0.7$.
In figure \ref{ZrB12ref} the reflectivity of ZrB$_{12}$
together with calculations based on Eq. (\ref{optcondSigma})
and (\ref{SEallenaprox}) is shown. It is clear that a simple
Drude form is not capable of describing the observed
reflectivity. The first fit (fit 1) is a fit where the
$\alpha^{2}_{tr}F(\Omega)$ that was used as input was derived
from specific heat measurements \cite{Lortz}. Although it gives
an improvement over the standard Drude fit there is still some
discrepancy between the data and the fit. To make further
improvements $\alpha^{2}_{tr}F(\Omega)$ was modelled using a
sum of $\delta$-functions. The results of this modelling are
indicated as fit 2 and fit 3. Using Eq. (\ref{lambdatr}) we
find coupling strengths $\lambda_{tr}\approx$ 1 - 1.3. Another
method to roughly estimate $\alpha^{2}_{tr}F(\Omega)$ is due to
Marsiglio \cite{Marsiglio1,Marsiglio2}. It states that a rough
estimate of the shape of $\alpha^{2}_{tr}F(\Omega)$ can be
found by simply differentiating the optical data,
\begin{figure}[tbh]
\includegraphics[width=8.5 cm]{ZrB12ref.png}
\caption{\label{ZrB12ref}Reflectivity of ZrB$_{12}$ around 20 K
together with calculations as explained in the text. Figure
adapted from ref. \citep{teyssier}.}
\end{figure}
\begin{equation}
\alpha^{2}_{tr}F(\Omega)=\frac{1}{2\pi}\frac{\Omega_{p}^{2}}{4\pi}\frac{d^{2}}{d\omega^{2}}Re(\frac{1}{\sigma(\omega)}),
\end{equation}
where $\Omega_{p}$ is the plasma frequency. The obvious problem
with this method is that it requires the double derivative of the
data. Because of the inevitable noise in the data usually some
form of smoothing is required. Applied to ZrB$_{12}$ the extracted
$\alpha^{2}_{tr}F(\Omega)$ shows peaks at the same positions as
the ones extracted before and a coupling strength
$\lambda_{tr}\approx1.1$. These results indicate a medium to
strong electron-phonon coupling for ZrB$_{12}$.
\section{Polarons}
There exist many definitions of what is a polaron. Electrons
coupled to a phonon have been called polaron as have free
electrons moving around in an insulator. Here we will consider the
Landau-Pekar approximation for a polaron \cite{landau,landau2}.
The idea is that when an electron moves about the crystal it
polarizes the surrounding lattice and this in turn leads to an
attractive potential for the electron. If the interaction between
electron and lattice is sufficiently strong this potential is
capable of trapping the electron and it becomes more or less
localized. The new object, electron plus polarization cloud is
called polaron. This self-trapping of electrons can occur in a
number of different situations and different names are used. For
instance, one talks about small polarons in models where only
short range interactions are considered, because this typically
leads to polaron formation with polarons occupying a single
lattice site. From the Landau-Pekar formalism we can get some
feeling of when polarons form and what their
\begin{figure}[thb]
\includegraphics[width=6 cm]{polaroncond.png}
\caption{\label{polcond} Schematic of the optical conductivity of
electrons interacting with a single Einstein mode.}
\end{figure}
properties will be. First of all, the coupling constant $\alpha$
is given by,
\begin{equation}
\alpha^{2}=\frac{Ry}{\hbar\omega_{0}\tilde{\varepsilon}_{\infty}^{2}}\frac{m_{b}}{m_{e}},
\end{equation}
where $Ry$ stands for the unit Rydberg (1 Rydberg =
$m_{e}e^{4}/2\hbar^{2}$ = 13.6 eV), $\omega_{0}$ is the oscillator
frequency of the (Einstein) phonon mode involved $m_{b}$ and
$m_{e}$ are the band and free electron mass respectively and
$\tilde{\varepsilon}$ is given by,
\begin{equation}
\frac{1}{\tilde{\varepsilon}}=\frac{1}{\varepsilon_{\infty,IR}}-\frac{1}{\varepsilon(0)}.
\end{equation}
For strong coupling (small polarons) the polaron mass is expressed
in terms of the coupling constant as,
\begin{equation}
m_{pol}=m_{b}(1+0.02\alpha^{4}),
\end{equation}
\begin{figure}[bht]
\includegraphics[width=6 cm]{LSCOdopdep.png}
\caption{\label{LSCOdopdep} Doping dependence of the room
temperature optical conductivity of
La$_{2-x}$Sr$_{x}$CuO$_{4}$. Figure adapted from Uchida
\textit{et al.}, ref. \citep{uchida}.}
\end{figure}
with the polaron binding energy given by,
\begin{equation}
E_{pol}=0.1\frac{Ry}{\tilde{\varepsilon}_{\infty}^{2}}\frac{m_{b}}{m_{e}}.
\end{equation}
The polaron binding energy typically is of the order of a few
100 meV. The polaron mass is typically of the order of 50-100
times the electron mass. The effect of polaron formation on the
optical conductivity can be described by assuming a gas of
non-interacting polarons (i.e. low polaron density). This
results in a spectrum that can be described by a Drude peak and
a so-called Holstein side-band. If we assume that the electrons
interact with a single Einstein mode the spectrum will look as
in figure \ref{polcond}. The spectrum consists of a
zero-phonon, coherent part (n = 0) with a spectral weight
1/(1+0.02$\alpha^{4}$) followed by a series of peaks that
describe the incoherent movement of polarons assisted by
n=1,2,3.. phonons. In real solids the peaks are smeared out due
to the fact that phonons form bands. The real part of the
optical conductivity can thus be described as,
\begin{equation}
{\mathop{\rm Re}\nolimits} \frac{{4\pi }}{{\omega _{_p }^{*2}
}}\sigma \left( \omega \right) = \frac{1}{{1 + 0.02\alpha ^4
}}\frac{{\pi \delta (\omega )}}{2} + \frac{{0.02\alpha ^4 }}{{1 +
0.02\alpha ^4 }}E_{pol}^{ - 1} \exp \left\{ { - \left(
{\frac{{\omega ^2 - E_{pol}^2 }}{{cE_{pol}^2 }}} \right)^2 }
\right\}.
\end{equation}
The first term in this expression describes the coherent part
of the spectrum, which in real solids will also be smeared out
to finite frequency by other forms of scattering, and an
incoherent term given by the second term which is called the
Holstein band. The shape of the side-band can be qualitatively
understood by imagining how a polaron has to move through the
lattice. In order to move from one site to another the lattice
deformation around the original site has to relax and be
adjusted on the new site. This relaxation process results in
the multi-phonon side-bands of the Drude peak.
\begin{figure}[thb]
\begin{minipage}{6 cm}
\includegraphics[width=4cm]{LaTiOres.png}
\end{minipage}\hspace{2pc}%
\begin{minipage}{6cm}
\includegraphics[width=8 cm]{LaTiOsig.png}
\end{minipage}
\caption{\label{LaTiO} Left panel: temperature dependent
resistivity of LaTiO$_{3.41}$. Right panel: Optical conductivity
for selected temperatures. Figure adapted from ref.
\citep{kuntscher}.}
\end{figure}
The observation of the Holstein side-band is somewhat
complicated because it is not possible to distinguish between
normal interband transitions and the effects due to polaron
formation. There have been some claims that a band observed in
the mid infrared region ($\approx$ 100 - 500 meV) of the
spectrum of high-T$_{c}$ superconductors is due to polaron
formation but many other interpretations exist. Figure
\ref{LSCOdopdep} shows the doping dependence of
La$_{2-x}$Sr$_{x}$CuO$_{4}$. The peak that occurs around 0.5 eV
for the 0.02 doped sample has been interpreted as the Holstein
side-band. Another example where polarons could play a role is
in LaTiO$_{3.41}$ \cite{kuntscher}. In this material the
resistivity (figure \ref{LaTiO}) shows a quasi one dimensional
behavior with an upturn of the resistivity at lower
temperatures. This could be due to polaron formation but it has
also been interpreted as due to a charge density wave. The
optical conductivity at low temperatures shows that a large
part of the spectral weight is contained in a side-band around
300 meV (see figure \ref{LaTiO}). If this peak would be due to
polarons we expect that when we warm up the system to higher
temperatures its spectral weight should be diminished. This is
because the increased temperature unbinds electrons from their
self-trapping potential and therefore shifts spectral weight
from the Holstein band to the Drude peak. This is also what is
observed and at the same time explains the decrease of
resistivity with increasing temperature.
\begin{figure}[bth]
\includegraphics[width=6 cm]{NaV6O15struct.png}
\caption{\label{NaVOstruct}Crystal structure of
NaV$_{6}$O$_{15}$.}
\end{figure}
The last example we will discuss is NaV$_{6}$O$_{15}$. The
structure of this compound is build up out off octahedra and
tetrahedra of vanadium and oxygen atoms where the tetrahedra
form quasi 1-dimensional zig-zag chains (see figure
\ref{NaVOstruct}). There are 3 different types of vanadium
sites in this structure: 2 of them are ionic with a charge 5+
on the vanadium which has then a $3d^{0}$ configuration. The
third site has half an electron more leading to a charge of
4.5+ on the vanadium atom in a $3d^{1/2}$ configuration.
Because of this we expect a quarter filled band and metallic
behavior. Figure \ref{NaVOsigma} shows the optical conductivity
of $\beta$- NaV$_{6}$O$_{15}$. The chains are along the
direction labelled \textit{b}. At energies around 3000
cm$^{-1}$ we observe a broad peak for light polarized along the
$b$-direction which could be due to polarons although these
transitions also correspond well with the energies predicted by
the Hubbard model for d-d transitions. If we compare the
conductivity with polarization parallel and perpendicular to
the b-axis, we see that the conductivity perpendicular to the
b-axis is insulating whereas the one along the b-axis is
conducting. This conducting behavior is due to the quarter
filled bands.
\begin{figure}[thb]
\includegraphics[width=6 cm]{NaV6O15sigma.png}
\caption{\label{NaVOsigma}Optical conductivity of
$\beta$-NaV$_{6}$O$_{15}$ for light polarized along and
perpendicular to the b-axis. Figure adapted from \cite{presura}}
\end{figure}
Are polarons playing an important role in the above examples ? It
is nearly impossible to answer this question experimentally due to
the above mentioned difficulty in separating polaronic behavior
from normal interband transitions. Moreover, in most cases where
polarons are invoked, other theories are also able to reproduce
the experimental results. To close this section we briefly discuss
what happens if the density of polarons becomes larger. Imagine
what happens if we increase the density of polarons such that we
are getting close to a system with one polaron on each site. In
that case the original lattice will almost be completely deformed
and one can wonder wether the electrons are still capable to
self-trap. It seems reasonable that in this limit the polaron
picture no longer applies. Another possibility is the formation of
bipolarons. Since the deformation energy of the lattice is
proportional to the electron charge $E_{pol}\propto -1/2Cq^{2}$,
the binding energy of two polarons is $\propto -Cq^{2}$. The
binding energy of a bipolaron (two electrons trapped by the same
polarization cloud) is twice as large however $E_{bipol}\propto
-1/2C(2q)^{2}$. This binding energy is usually not enough to
overcome the Coulomb repulsion between the electrons.
\section{Spin interactions}\label{spininteraction}
As mentioned in the previous section the signatures for the
presence of polarons can often be interpreted with different
ideas. Most often these models are based on coupling to magnetic
interactions. Consider for example the spectrum of the parent
(undoped) compound YBa$_{2}$Cu$_{3}$O$_{6}$ which is a Mott
insulator (see bottom panel of figure \ref{reftrans}). Below 100
meV we see a series of peaks which are due to phonons. But what
about the structure between 100 meV and 1 eV ? One of the
difficulties in explaining this structure is that light does not
directly couple to spin degrees of freedom. It is however possible
to indirectly make spin flips with photons (see figure
\ref{spinflip}).
\begin{figure}[thb]
\includegraphics[width=10 cm]{spinflip.png}
\caption{\label{spinflip}Interaction diagram for the indirect
interaction of light with spin degrees of freedom.}
\end{figure}
For this process to occur we have to include phonon-magnon
interaction. When a photon enters the material it gets dressed
with phonons forming a polariton which is then coupled to the spin
degrees by the phonon-magnon interaction. This leads to the
possibility of so-called phonon assisted absorption of spin-flip
excitations \cite{Lorenzana}. We see from figure \ref{spinflip}
that the polariton creates a bi-magnon. This is because the
intermediate state has to have spin S = 0. The dashed square
represents all magnon-magnon interactions. The coupling constant
for this process was first calculated by Lorenzana and Sawatzky
and is
\begin{equation}
J_{ph-mag}=\frac{1}{2J}<\frac{d^{2}J}{du^{2}}><u^{2}>,
\end{equation}
\begin{figure}[thb]
\includegraphics[width=8.5 cm]{spinsigma.png}
\caption{\label{spinsigma}Optical conductivity of (a):
La$_{2}$CuO$_{4}$, (b): La$_{2}$NiO$_{4}$ and (c):
Sr$_{2}$CuO$_{4}$. Dashed lines are fits using the
Lorenzana-Sawatzky model.}
\end{figure}
where $J$ is the superexchange constant and $u$ is the atomic
displacement vector. In the process momentum and energy have to be
conserved and this leads to
\begin{equation}
k_{\text{magnon 1}} + k_{\text{magnon 2}} + k_{\text{phonon}} =
k_{\text{photon}} \approx 0.
\end{equation}
and
\begin{equation} \omega _{\text{magnon 1}} + \omega _{\text{magnon 2}} +
\omega _{\text{phonon}} = \omega _{\text{photon}}.
\end{equation}
for the process in figure \ref{spinflip}. This gives constraints
on the possible absorptions. In figure \ref{spinsigma} some
examples are shown of materials in which we believe this process
to play a role. One of the compounds where the predicted optical
conductivity fits the spectrum very well is in the case of
Sr$_{2}$CuO$_{3}$. To make the fit the magnon dispersion as
measured with neutron scattering was used. The reason that this
theory works so well for Sr$_{2}$CuO$_{3}$ is that the conduction
is nearly one dimensional. This gives a good starting point
because the magnon spectrum is completely understood. On the
contrary, the theory is not completely capable of predicting the
spectrum of La$_{2}$CuO$_{4}$. Most likely the peaks around 0.6
and 0.75 eV are due to 4 and 6 magnon absorption. In the case of
YBa$_{2}$Cu$_{3}$O$_{6}$ the situation gets even more complicated
due to the presence of two layers per unit cell. Because of the
doubling of the unit cell, there are now acoustic and optical
magnon branches just as what would happen in the case of phonons.
The effect of this on the optical conductivity was first discussed
by Grueninger \textit{et al.} \cite{Grueninger,Grueninger2}.
Another example of probing of spin excitations occurs in
NaV$_{2}$O$_{5}$. As already discussed in the previous section
this compound has quasi one dimensional chains as shown in figure
\ref{NaVOstruct}. These chains can be seen to form a so-called
ladder structure, with the ladders parallel to the $b$ direction.
\begin{figure}[thb]
\includegraphics[width=3 cm]{NaVOlad.png}
\caption{\label{NaVOlad}Schematic of the ladder structure of
$\alpha$- NaV$_{2}$O$_{5}$. Arrows indicate the position of the
lectrons and their spin orientation.}
\end{figure}
Each adjacent ladder is shifted with respect to the previous
such that the rungs of one ladder fall in between those of the
next (figure \ref{NaVOlad}). The vanadium atoms that form the
ladders have an average charge of +4.5. It has been claimed
\cite{carpy} that the charge distribution is inhomogeneous with
most of the charge on one side of the ladder as indicated in
figure \ref{NaVOlad}.
\begin{figure}[thb]
\begin{minipage}{7cm}
\includegraphics[width=7cm]{NaVOmag.png}
\caption{\label{NaVOmag}Magnetization of $\alpha$-
NaV$_{2}$O$_{5}$. Figure adapted from ref. \citep{Isobe}.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{7cm}
\includegraphics[width=5cm]{NaVOphase.png}
\caption{\label{NaVOphase}Schematic representation of the low
and high temperature phase of $\alpha$- NaV$_{2}$O$_{5}$.}
\end{minipage}
\end{figure}
The temperature dependence of the magnetic susceptibility can
be modelled pretty well using a Bonner-Fischer model for a
spin-1/2 Heisenberg chain \cite{Isobe} for temperatures higher
then 34 K (see figure \ref{NaVOmag}). Below 34 K, X-ray
analysis shows a doubling of the a- and b- axes and a
quadrupling of the c-axis. It indicates that the new unit cell
consists of 64 vanadium atoms and 32 valence electrons. At the
same temperature the susceptibility shows an abrupt drop. An
explanation for this transition is in terms of a spin-Peierls
transition. In the high temperature phase (T $>$ T$_{SP}$) the
left side of the ladder has a uniform spin distribution, as
indicated in the left panel of figure \ref{NaVOphase}, which is
reasonably well described with an anti-ferromagnetic (AF) S =
1/2 Heisenberg spin chain with uniform exchange coupling $J$.
For T $<$ T$_{SP}$ the system dimerises due to a deformation of
the lattice, leading to an alternation of exchange couplings
(see right panel figure \ref{NaVOphase}). Here we focus on the
high temperature phase. If the charge inhomogeneity is present
it would lead to a breaking of the inversion symmetry which in
turn leads to a non-zero optical matrix element for two magnon
absorption \cite{damascelli3}. The idea is similar to the
Lorenzana-Sawatzky model discussed above. In the latter case
the phonon effectively lowers the symmetry making the process
optically allowed. The optical conductivity of $\alpha$-
NaV$_{2}$O$_{5}$ is shown in figure \ref{alphaNaVO}. We can
model $\alpha$- NaV$_{2}$O$_{5}$ with independent ladders where
the hopping probability along a rung ($t_{\perp}$) is much
larger than that along the ladder ($t_{\parallel}$).
Furthermore we assume a large on-site repulsion U. We can then
model a ladder by independent rungs. Assuming a quarter filled
ladder (one electron per rung) we have a simple two level
problem leading to bonding and anti-bonding levels (see also
the discussion in the section on applications of sum rules). If
we also include a potential energy difference $\Delta$ between
the sites the wavefunctions become asymmetric with higher
probability on the low potential site and one can show that
this again leads to bonding and anti-bonding solutions which
are split by an energy \cite{damascelli3},
\begin{figure}[bht]
\includegraphics[width=8.5 cm]{alphaNaVO.png}
\caption{\label{alphaNaVO}Optical conductivity of $\alpha$-
NaV$_{2}$O$_{5}$. Inset (a) shows the low energy continuum
attributed to charged bi-magnon excitations and inset (b) shows
the temperature dependence of the spectral weight of this
continuum. Figure adapted from ref. \citep{damascelli3}.}
\end{figure}
\begin{equation}\label{Ect}
E_{CT}=\sqrt{\Delta^{2}+4t^{2}_{\perp}}.
\end{equation}
Transitions from the bonding to anti-bonding band are optically
active and involve charge transfer (CT) from the left side of the
ladder to the right. The large peak seen in figure \ref{alphaNaVO}
around 1 eV is due to these transitions. The energy position of
the peak indirectly gives evidence for the charge inhomogeneity:
band structure calculations and exact diagonalization of finite
clusters give $t_{\perp}\approx$ 0.35 eV, which would put the
charge transfer peak around 0.7 eV. The observed value of 1 eV
thus indicates $\Delta\neq$ 0. The spectral weight of the peak
allows us to make an estimate for $\Delta$. One can show that,
\begin{equation}
\int_{peak}\sigma_{1}(\omega)=\pi
e^{2}Nd^{2}_{\perp}t^{2}_{\perp}\hbar^{-2}E_{CT}^{-1}.
\end{equation}
Using Eq. (\ref{Ect}) we find $t_{\perp}\approx$ 0.3 eV and
$\Delta\approx$ 0.8 eV. Besides the large CT peak seen in
figure \ref{alphaNaVO} there is also a broad continuum in the
infrared region of the spectrum for $E\parallel a$ (see inset).
This part of the spectrum can be understood if we include the
coupling between rungs of the ladder. For parallel spins on
different rungs this coupling would have no effect since the
Pauli principle would forbid hopping between the sites. For an
anti-parallel spin configuration the system can gain some
kinetic energy from virtual hopping of an electron from one
rung to the next, putting two electrons on one rung. For very
large U this electron would occupy the righthand side of the
rung. Starting from an anti-parallel configuration a spin-flip
transition on one rung thus leads to a net dipole displacement
which leads to optical activity of this transition. We note
that because of spin conservation rules we have to make two
spin-flips. These excitations have been dubbed charged
bi-magnon excitations.
| {'timestamp': '2008-07-21T15:22:50', 'yymm': '0807', 'arxiv_id': '0807.3261', 'language': 'en', 'url': 'https://arxiv.org/abs/0807.3261'} |
\section{Introduction}
\label{sec:intro}
The existing contemporary communications systems can be abstractly characterized by the conceptual seven-layer Open Systems Interconnection model. The lowest (or first) layer, known as the \emph{physical layer}, aims to describe the communication process over an actual physical medium. Due to the increasing demand for flexibility, information exchange nowadays often occurs via antennas at the transmitting and receiving end of a wireless medium, \emph{e.g.}, using mobile phones or tablets for data transmission and reception.
An electromagnetic signal transmitted over a wireless channel is however prone to interference, fading, and environmental effects caused by, \emph{e.g.}, surrounding buildings, trees, and vehicles, making reliable wireless communications a challenging technological problem.
With the advances in communications engineering, it was soon noticed that increasing the number of spatially separated transmit and receive antennas, as well as adding redundancy by repeatedly transmitting the same information encoded over multiple time instances\footnote{'Time instances' are commonly referred to as \emph{channel uses}.}, can dramatically improve the transmission quality. A code representing both diversity over time and space is thus called a \emph{space--time code}. Let us consider a channel with $n_t$ and $n_r$ antennas at its transmitting and receiving end, respectively, and assume that transmission occurs over $T$ consecutive time instances. If $n_t = n_r$, the channel is called \emph{symmetric}, and otherwise asymmetric, which more precisely typically refers to the case $n_r < n_t$. For the time being, a space--time code $\mathcal{X}$ will just be a finite collection of complex matrices in $\Mat(n_t\times T, \mathbb{C})$. The channel equation in this multiple-input multiple-output (MIMO) setting is given by
\begin{equation}
\label{eqn:mimo}
Y_{n_r\times T} = H_{n_r\times n_t}X_{n_t\times T} + N_{n_r\times T},
\end{equation}
where $Y$ is the received matrix, and $X = \left[x_{ij}\right]_{i,j}\in\mathcal{X}$ is the \emph{space--time code} matrix. In the above equation, we adopt the Rayleigh fading channel model, \emph{i.e.}, the entries of the random \emph{channel matrix} $H = \left[h_{ij}\right]_{i,j}$ are complex variables with identically distributed real and imaginary parts,
\begin{align*}
\Re(h_{ij}), \Im(h_{ij}) \sim \mathcal{N}(0,\sigma_h^2),
\end{align*}
yielding a Rayleigh distributed envelope
\begin{align*}
|h_{ij}| = \sqrt{\Re(h_{ij})^2+\Im(h_{ij})^2} \sim \text{Ray}(\sigma_h)
\end{align*}
with scale parameter $\sigma_h$. We assume further that the channel remains static during the entire transmission of the codeword matrix $X$, and then changes independently of its previous state. The additive noise\footnote{The noise is a combination of thermal noise and noise caused by the signal impulse.} is modeled by the \emph{noise matrix} $N$, whose entries are independent, identically distributed complex Gaussian random variables with zero mean.
Let us briefly discuss what constitutes a "good" code. Consider a space--time code $\mathcal{X}$, and let $X, X'$ be code matrices ranging over $\mathcal{X}$.
Two basic design criteria can be derived in order to minimize the probability of error \cite{TSC}.
\begin{itemize}
\item[i)] The \emph{diversity gain} of a code is the asymptotic slope of the error probability curve with respect to the signal-to-noise ratio ($\SNR$) in a $\log-\log$ scale, and relates to the minimum rank $\rk(X-X')$ over all pairs of distinct code matrices $(X,X') \in \mathcal{X}^2$. The minimum rank of $\mathcal{X}$ should satisfy
\begin{align*}
\min_{X \neq X'} \rk(X-X') = \min\{n_t,T\},
\end{align*}
in which case $\mathcal{X}$ is called a \emph{full-diversity} code.
\item[ii)]
The \emph{coding gain} measures the difference in $\SNR$ required for two different codes to achieve the same error probability. For a full-diversity code this is proportional to the determinant
\begin{align*}
\det \left( (X-X')(X-X')^{\dagger}\right).
\end{align*}
\end{itemize}
We define the \emph{minimum determinant} of a code $\mathcal{X}$ as the infimum
\begin{align*}
\Delta_{\min}(\mathcal{X}) := \inf\limits_{X \neq X'}{\det \left( (X-X')(X-X')^{\dagger}\right)}
\end{align*}
as the code size increases, $|\mathcal{X}| \to \infty$. If $\Delta_{\min}(\mathcal{X}) > 0$, the space--time code is said to have the \emph{nonvanishing determinant} property \cite{BR}. In other words, a nonvanishing determinant guarantees that the minimum determinant is bounded from below by a positive constant even in the limit, and hence the error probability will not blow up when increasing the code size.
In 2003, the usefulness of central simple algebras to construct space--time codes meeting both of the above criteria was established in \cite{SRS}; especially of (cyclic) division algebras, for which the property of being division immediately implies full diversity. Thereupon the construction of space--time codes started to rely on cleverly designed algebraic structures, leading to the construction of multiple extraordinary codes, such as the celebrated Golden code \cite{BRV}, or general Perfect codes \cite{ORBV,ESK}.
It was later shown in \cite{BR} that in a cyclic division algebra based code, achieving the nonvanishing determinant property can be ensured by restricting the entries of the codewords to certain subrings of the algebra alongside with a smart choice for the base field, and that ensuring nonvanishing determinants is enough to achieve the optimal trade-off between diversity and multiplexing.
Further investigation carried out in \cite{HLL,VHLR} showed that codes constructed from orders, in particular \emph{maximal orders}, of cyclic division algebras performed exceptionally well. The main observation is that the discriminant of the order is directly related to the offered coding gain, and should be as small as possible in order to maximize the coding gain.
Maximal orders were then the obvious candidates, as they maximize the normalised density of the corresponding lattice and hence also maximize the coding gain. Unfortunately, they are in general very difficult to compute and may result in highly skewed lattices making the bit labeling a delicate problem on its own. Therefore, \emph{natural orders} with a simpler structure have become a more frequent choice as they provide a good compromise between the two common extremes: using maximal orders to optimize coding gain, on the one hand, and restricting to orthogonal lattices to simplify bit labeling, encoding, and decoding, on the other.
However, the current explicit constructions are typically limited to the symmetric case, while the asymmetric case remains largely open. The main goal of this article is to fill this gap,
though our interest is not to analyze the performance of explicit codes. Instead, we focus on the algebraic setup and provide lower bounds for the smallest possible discriminants of natural orders for the considered setups, and give explicit field extensions and corresponding cyclic division algebras meeting the lower bounds.
This article is structured as follows. In Section~\ref{sec:stc} we will shortly introduce MIMO space--time coding and the construction of space--time codes using representations of orders in central simple algebras. Section~\ref{sec:nat_orders} contains the main results of this article. We will consider the most interesting asymmetric MIMO channel setups and fix $F = \mathbb{Q}$ or $F = \mathbb{Q}(i)$ as the base field to guarantee the nonvanishing determinant property\footnote{From a mathematical point of view, any imaginary quadratic number field would give a nonvanishing determinant, but the choice $\mathbb{Q}(i)$ matches with the quadrature amplitude modulation (QAM) commonly used in engineering.}. For each considered setup $(F, n_t, n_r)$, we will find an explicit field extension $F \subset L \subset E$ and an explicit $L$-central cyclic division algebra over $E$, such that the norm of the discriminant of its natural order is minimal. This will translate into the largest possible determinant (see \eqref{eqn:density} and \cite{VHLR,HL} for the proof) and thus provide us with the maximal coding gain one can achieve by using a natural order.
\section{Space--time codes from orders in central simple algebras}
\label{sec:stc}
From now on, and for the sake of simplicity, we set the number $n_t$ of transmit antennas equal to the number $T$ of time slots used for transmission and shortly denote $n:=n_t=T$. Thus, the considered codewords will be square matrices.
\subsection{Space--time lattice codes}
\label{sec:sec2}
Very simplistically defined, a space--time code is a finite set of complex matrices. However, in order to avoid accumulation points at the receiver, in practical implementations it is convenient to impose an additional discrete structure on the code, such as a lattice structure. We define a \emph{space--time code} to be a finite subset of a \emph{lattice}
$$
\Lambda =\left\{\sum_{i=1}^k z_i B_i\,|\, z_i\in\mathbb{Z}\right\}\subset \Mat(n,\mathbb{C}),
$$
where $\{B_1,\ldots,B_k\}\subset \Mat(n,\mathbb{C})$ is a \emph{lattice basis}. We recall that a lattice in $\Mat(n,\mathbb{C})$ is \emph{full} if ${\rm rank}(\Lambda) := k= 2 n^2$. We call a space--time lattice code \emph{symmetric}, if its underlying lattice is full, and \emph{asymmetric}\footnote{This definition relates to the fact that a symmetric code carries the maximum amount of information (\emph{i.e.}, dimensions) that can be transmitted over a symmetric channel without causing accumulation points at the receiving end. In an asymmetric channel, a symmetric code will result in accumulation points, and hence asymmetric codes, \emph{i.e.}, non-full lattices are called for. See \cite{HL} for more details.} otherwise.
Due to linearity, given a lattice $\Lambda \subset \Mat(n,\mathbb{C})$ and $X, X' \in \Lambda$,
\begin{align*}
\Delta_{\min}(\Lambda) := \inf\limits_{X \neq X'} \det \left((X-X')(X-X')^{\dagger} \right) =
\inf\limits_{X \in \Lambda\backslash\left\{0\right\}}{\left|\det(X)\right|^2}.
\end{align*}
This implies that any lattice $\Lambda$ satisfying the nonvanishing determinant property can be scaled so that $\Delta_{\min}(\Lambda) $ achieves any wanted nonzero value.
Consequently, a meaningful comparison of different lattices requires some kind of normalization. To this end,
consider the Gram matrix of $\Lambda$,
\begin{align*}
G_{\Lambda} := \left[\Re\left(\Tr\left(B_i B_j^\dagger\right)\right)\right]_{1 \le i,j \le k},
\end{align*}
where $\Tr$ denotes the matrix trace. The volume $\nu(\Lambda)$ of $\Lambda$ is related to the Gram matrix as $\nu(\Lambda)^2 = \det(G_{\Lambda})$.
\begin{itemize}
\item[i)] The \emph{normalized minimum determinant} \cite{VHLR} of $\Lambda$ is the minimum determinant of $\Lambda$ after scaling it to have a unit size fundamental parallelotope, that is,
\begin{align*}
\delta(\Lambda) = \frac{\Delta_{\min}(\Lambda)}{\nu(\Lambda)^{\frac{n}{k}}}.
\end{align*}
\item[ii)] The \emph{normalized density} \cite{VHLR} of $\Lambda$ is
\begin{align}
\label{eqn:norm_density}
\mu(\Lambda) = \frac{\Delta_{\min}(\Lambda)^{\frac{k}{n}}}{\nu(\Lambda)}.
\end{align}
\end{itemize}
We get the immediate relation $\delta(\Lambda) = \mu(\Lambda)^{{\frac{n}{k}}}$, from which it follows that in order to maximize the coding gain it suffices to maximize the density of the lattice. Maximizing the density, for its part, translates into a certain \emph{discriminant minimization problem} \cite{VHLR,HL}, as we shall see in Section~\ref{subsec:stc_orders} (cf. \eqref{eqn:density}). This observation is crucial and will be the main motivation underlying Section~\ref{sec:nat_orders}.
\subsection{Central simple algebras and orders}
We recall that a finite dimensional algebra over a number field $L$ is an $L$-\emph{central simple algebra}, if its center is precisely $L$ and it has no nontrivial ideals.
An algebra is said to be \emph{division} if all of its nonzero elements have a multiplicative inverse.
By \cite[Prop.~1]{SRS}, as long as the underlying algebraic structure of a space--time code is a division algebra, the full-diversity property of the code will be guaranteed.
It turns out that if $L$ is an algebraic number field, then every $L$-central simple algebra is a \emph{cyclic algebra} \cite[Thm.~32.20]{Re}.
Let $E/L$ be a cyclic extension of number fields of degree $n$ with respective rings of integers $\mathcal{O}_E$ and $\mathcal{O}_L$, and cyclic Galois group ${\rm Gal}(E/L) = \langle \sigma \rangle$. We fix a nonzero element $\gamma\in L^\times$ and consider the right $E$-vector space
\begin{align*}
\mathcal{C} := (E/L,\sigma,\gamma) = \bigoplus_{i = 0}^{n-1}u^i E,
\end{align*}
with left multiplication defined by $xu= u\sigma(x)$ for all $ x\in E$, and $u^n=\gamma$. The triple $\mathcal{C}$ is referred to as a \emph{cyclic algebra} of \emph{index} $n$.
The obvious choice of lattices in $\mathcal{C}$ will be its \emph{orders}. We recall that if $R \subset L$ is a Dedekind ring, an $R$-order in $\mathcal{C}$ is a subring $\mathcal{O} \subset \mathcal{C}$ which shares the same identity as $\mathcal{C}$, is a finitely generated $R$-module, and generates $\mathcal{C}$ as a linear space over $L$. Furthermore, an order is \emph{maximal} if it not properly contained in any other $R$-order of $\mathcal{C}$. Of special interest in this article is the $\mathcal{O}_{L}$-module
\begin{align*}
\mathcal{O}_{\nat} := \bigoplus\limits_{i=0}^{n-1}{u^i\mathcal{O}_{E}},
\end{align*}
which we refer to as the \emph{natural order} of $\mathcal{C}$.
Throughout the paper, we will denote the relative field norm map of the extension $E/L$ by $\Nm_{E/L}$ and the absolute norm map by $\Nm_E=\Nm_{E/\mathbb{Q}}$. The restriction of this map to orders may be specified in the notation as $\Nm_{\mathcal{O}_E/\mathcal{O}_L}$ and $\Nm_{\mathcal{O}_E}=\Nm_{\mathcal{O}_E/\mathbb{Z}}$.
If the element $\gamma$ fails to be an algebraic integer, then $\mathcal{O}_{\nat}$ will not be closed under multiplication. Furthermore, a necessary and sufficient condition for an index-$n$ cyclic algebra $(E/L,\sigma,\gamma)$ to be division is that
\begin{align}
\label{eqn:non_norm}
\gamma^{n/p} \notin \Nm_{E/L}(E^\times)
\end{align}
for all primes $p \mid n$. This is a simple extension of a well-known result due to A. Albert, for more details and a proof see \cite[Prop. 3.6]{VHLR}. In what follows we will refer to such a non-zero element $\gamma \in \mathcal{O}_{L}$ as a \emph{non-norm element} for $E/L$.
\begin{remark}
We recall that given a Dedekind ring $R \subset L$ and an $R$-order $\mathcal{O}$ with basis $\{x_1,\ldots, x_n^2\}$ over $R$, the $R$-discriminant of $\mathcal{O}$ is the ideal
\begin{align*}
\disc(\mathcal{O}/R) = \left(\det\left(\tr_{\mathcal{C}/L}(x_i x_j)_{i,j=1}^{n^2}\right)\right),
\end{align*}
where $\tr(\cdot)$ denotes the reduced trace, which will be defined in \eqref{eqn:lrr}.
While the ring of algebraic integers is the unique maximal order in an algebraic number field, an $L$-central simple division algebra may contain several maximal orders. They all share the same discriminant \cite[Thm.~25.3]{Re}, known as the discriminant $d_{\mathcal{C}}$ of the algebra ${\mathcal{C}}$. Given two $\mathcal{O}_{L}$-orders $\Gamma_1$, $\Gamma_2$, it is clear that if $\Gamma_1\subseteq\Gamma_2$, then $\disc(\Gamma_2/\mathcal{O}_{L}) \mid \disc(\Gamma_1/\mathcal{O}_{L}$). Consequently, $d_{\mathcal{C}}|\disc(\Gamma/\mathcal{O}_{L})$ for every $\mathcal{O}_{L}$-order $\Gamma$ in ${\mathcal{C}}$, and the ideal norm $\Nm_{\mathcal{O}_{L}}(d_{\mathcal{C}})$ is the smallest possible among all $\mathcal{O}_{L}$-orders of $\mathcal{C}$.
\end{remark}
\subsection{Algebraic space--time codes from representations of orders}
\label{subsec:stc_orders}
Let $\mathcal{C} = (E/L,\sigma,\gamma)$ be a cyclic division algebra of index $n$. We fix compatible embeddings of $L$ and $E$ into $\mathbb{C}$, and identify $L$ and $E$ with their images under these embeddings.
The $E$-linear transformation of $\mathcal{C}$ given by left multiplication by an element $c \in \mathcal{C}$ results in an $L$-algebra homomorphism
$\rho: \mathcal{C} \to \Mat(n,E),$
to which we refer to as the \emph{maximal representation}.
An element $c = c_0 + u c_1+ \cdots + u^{n-1} c_{n-1} \in \mathcal{C}$ can be identified via $\rho$
with the matrix
\begin{equation}
\label{eqn:lrr}
\rho(c) = \begin{bmatrix}
c_0& \gamma\sigma(c_{n-1})&\gamma \sigma^2(c_{n-2})& \ldots &\gamma \sigma^{n-1}(c_{1})\\
c_1& \sigma(c_{0})&\gamma\sigma^2(c_{n-1})& \ldots &\gamma \sigma^{n-1}(c_{2})\\
\vdots & \vdots & \vdots & & \vdots \\
c_{n-1} & \sigma(c_{n-2}) & \sigma^2(c_{n-3})& \cdots &\sigma^{n-1}(c_{0})
\end{bmatrix}.
\end{equation}
The determinant $\nr_{\mathcal{C}/L}(c) := \det(\rho(c))$ and trace $\tr_{\mathcal{C}/L}(c) := \Tr(\rho(c))$ define the \emph{reduced norm} and \emph{reduced trace} of $c \in \mathcal{C}$, respectively. We may shortly denote $\nr=\nr_{\mathcal{C}/L}$ and $\tr=\tr_{\mathcal{C}/L}$, when there is no danger of confusion.
Next, given an order $\mathcal{O}$ in $\mathcal{C}$, we may use the maximal representation to define an injective map $\rho: \mathcal{O} \hookrightarrow \Mat(n,E) \subset \Mat(n, \mathbb{C})$. If the center $L$ of the algebra is quadratic imaginary and $\mathcal{O}$ admits an $\mathcal{O}_L$ basis, then $\rho(\mathcal{O})$ is a lattice, and any finite subset $\mathcal{X}$ of $\rho(\mathcal{O})$ will be a space--time lattice code, in the literature often referred to as \emph{algebraic space--time code}.
\begin{remark}
Due to the algebra being division, the above matrices will be invertible, and hence any algebraic space--time code constructed in this way will have full diversity. Moreover, if $c \in \mathcal{O}\backslash \left\{0\right\}$, we have $\nr(c) \in \mathcal{O}_L\backslash\left\{0\right\}$ \cite[Thm. 10.1]{Re}, guaranteeing nonvanishing determinants for $L = \mathbb{Q}$ or quadratic imaginary.
\end{remark}
We now relate the minimum determinant of a code to the density of the underlying lattice $\rho(\mathcal{O})$, which is the main motivation for choosing orders with small discriminant. If the center $L$ of the cyclic algebra is quadratic imaginary and the considered order $\mathcal{O}$ admits an $\mathcal{O}_L$-basis, the volume $\nu(\rho(\mathcal{O}))$ of the lattice relates to the discriminant of the order as \cite{VHLR}
\begin{align}
\label{eqn:density}
\nu(\rho(\mathcal{O})) = c(L,n)\left|\disc(\mathcal{O}/\mathcal{O}_L)\right|,
\end{align}
where $c(L,n)$ is a constant which depends on the center and $[E:L]$. Thus, for fixed minimum determinant, the density of the code (cf. \eqref{eqn:norm_density}) -- and consequently the coding gain -- is maximized by minimizing the discriminant of the order.
\begin{example}
\label{exp:golden}
Let $E/L$ be a quadratic real extension of number fields with Galois group ${\rm Gal}(E/L) = \langle \sigma \rangle$. Let $L$ be of class number 1 and $\mathcal{O}_{L}$, $\mathcal{O}_{E} = \mathcal{O}_{L}[\omega]$ the respective rings of integers.
We choose $\gamma \in \mathcal{O}_L\setminus\left\{0\right\}$ such that $\gamma \notin \Nm_{E/L}(E^\times)$, and define the cyclic division algebra
\begin{align*}
\mathcal{C} = (E/L,\sigma,\gamma) = E \oplus uE,
\end{align*}
where $u^2 = \gamma$. We consider the natural order $\mathcal{O}_\text{nat}$ of $\mathcal{C}$ and construct an algebraic space--time code as a finite subset
\begin{align*}
\mathcal{X} \subset \left\{
\begin{bmatrix} x_1 + x_2\omega & \gamma(x_3 + x_4\sigma(\omega)) \\ x_3 + x_4\omega & x_1 + x_2\sigma(\omega) \end{bmatrix} \ \middle|\ x_i\in \mathcal{O}_{L}
\right\}= \rho(\mathcal{O}_\text{nat}).
\end{align*}
The choice of fields $(E,L) = (\mathbb{Q}(i,\sqrt{5}),\mathbb{Q}(i))$ and element $\gamma = i$ gives rise to the Golden algebra and to the well-known \emph{Golden code} \cite{ORBV}.
\end{example}
The above example relates to the symmetric scenario, \emph{i.e.}, it has an underlying lattice that is full. Full lattices can be efficiently decoded when $n_r = n_t$, and the same lattice codes can also be employed when $n_r > n_t$. There is no simple optimal decoding method for symmetric codes, however, when $n_r < n_t$\footnote{Having too few receive antennas will cause the lattice to collapse resulting in accumulation points, since the received signal now has dimension $2n_rn_t<2n_t^2$. Hence, partial brute-force decoding of high complexity has to be carried out.}.
Building upon symmetric codes, we now briefly introduce \emph{block diagonal asymmetric} space--time codes \cite{HL}, better suited for the asymmetric scenario.
Let $F \subset L \subset E$ be a tower of field extensions with extension degrees $[E:L] = n_r$, $[L:F] = n$, and $[E:F]=n_t=n_rn$, and with Galois groups ${\rm Gal}(E/F) = \langle \tau\rangle $ and ${\rm Gal}(E/L) = \langle \sigma\rangle= \langle \tau^n \rangle$. We fix a non-norm element $\gamma \in \mathcal{O}_{L}\setminus\{0\}$, and consider the cyclic division algebra
\begin{align*}
\mathcal{C} = (E/L,\sigma,\gamma) = \bigoplus\limits_{i=0}^{n_r-1}u^i E.
\end{align*}
Given any order $\mathcal{O} $ in $ \mathcal{C}$, we identify each element $c \in \mathcal{O}$ with its maximal representation $\rho(c)$ and construct the following infinite block-diagonal lattice achieving the nonvanishing determinant property, provided that the base field $F$ is either $\mathbb{Q}$ or quadratic imaginary \cite{HL}:
\begin{align*}
\mathcal{L}(\mathcal{O}) = \left\{\left.\begin{bmatrix} \rho(c) & 0 & \cdots & 0 \\ 0 & \tau\left(\rho(c)\right) & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \cdots & 0 & \tau^{n-1}\left(\rho(c)\right) \end{bmatrix} \in \Mat(n_t,\mathbb{C})\, \right|\, c \in \mathcal{O} \right\}.
\end{align*}
\begin{remark}
The \emph{code rate} \cite{HL} of a space--time code carved out from $\mathcal{L}(\mathcal{O})$ in (complex) symbols per channel use is
\begin{align*}
R = \begin{cases}
nn_r^2/nn_r=n_r &\mbox{if $F$ is quadratic imaginary,} \\
nn_r^2/2nn_r=n_r/2 &\mbox{if $F=\mathbb{Q}$.}
\end{cases}
\end{align*}
We point out that $n_r$ is the maximum code rate that allows for avoiding accumulation points at the receiving end with $n_r$ receive antennas.
\end{remark}
In summary, in order to construct an algebraic space--time code, we first choose a central simple algebra over a suitable base field and then look for a dense lattice in it. This amounts to selecting an adequate order in the algebra. As motivated earlier, we will opt for natural orders as a compromise between simplicity and maximal coding gain.
\section{Natural orders with minimal discriminant}
\label{sec:nat_orders}
As an illustration of the general algebraic setup, consider the tower of extensions depicted in Figure~\ref{fig:tower}.
\begin{figure}[!h]
\centering
\includegraphics[trim={4cm 18.5cm 13cm 4.5cm},clip,scale=.9]{tower}
\caption{Tower of Field Extensions.}
\label{fig:tower}
\end{figure}
In order to get the nonvanishing determinant property, we fix the base field $F \in \left\{\mathbb{Q},\mathbb{Q}(i)\right\}$, as well as the extension degrees $n = [L:F]$ and $n_r = [E:L]$. With these parameters fixed, our goal is to find an explicit field extension $E/L$ with ${\rm Gal}(E/L) = \langle \sigma \rangle$ and a non-norm element $\gamma \in \mathcal{O}_{L}\setminus\{0\}$ such that $E/F$ is a cyclic extension,
$(E/L,\sigma,\gamma)$ is a cyclic division algebra,
and the absolute value $|\Nm_{\mathcal{O}_F}(\disc(\mathcal{O}_{\nat}/\mathcal{O}_{F}))|$ is the minimum possible among all cyclic division algebras satisfying the fixed conditions.
Our constructions rely on some key properties of cyclic division algebras and their orders that we will next present as lemmata.
\begin{lemma}{\cite[Lem.~5.4]{VHLR} and \cite[Prop.~5.3]{HL}}
\label{pro:disc}
Let $(E/L,\sigma,\gamma)$ be a cyclic division algebra of index $n_r$ and $\gamma \in \mathcal{O}_{L}\setminus\{0\}$ a non-norm element. We have
\begin{align*}
\disc(\mathcal{O}_{\nat}/\mathcal{O}_{L}) = \disc(E/L)^{n_r}\cdot\gamma^{n_r(n_r-1)},
\end{align*}
with $\disc(E/L)$ the $\mathcal{O}_L$-discriminant of $\mathcal{O}_E$. Hence, if $F \subset L$, then by the discriminant tower formula
\begin{align}
\label{eqn:disc}
\begin{split}
\disc(\mathcal{O}_{\nat}/\mathcal{O}_{F}) &=
\Nm_{L/F}(\disc(\mathcal{O}_{\nat}/\mathcal{O}_{L}))\cdot \disc(L/F)^{n_r^2} \\
&= \disc(E/F)^{n_r}\cdot \Nm_{L/F}^{n_r(n_r-1)}(\gamma).
\end{split}
\end{align}
\end{lemma}
\begin{lemma} \cite[Thm.~2.4.26]{Veh}
\label{pro:bound}
Let $L$ be a number field and $(\mathfrak{p}_1, \mathfrak{p}_2)$ a pair of norm-wise smallest prime ideals in $\mathcal{O}_{L}$. If we do not allow ramification on infinite primes, then the smallest possible discriminant of all central division algebras over $L$ of index $n_r$ is
\begin{equation}
\label{eqn:bound}
(\mathfrak{p}_1\mathfrak{p}_2)^{n_r(n_r-1)}.
\end{equation}
\end{lemma}
We have arrived at the following \emph{optimization problem:} in order to minimize the discriminant $\disc(\mathcal{O}_{\nat}/\mathcal{O}_{F})$ of a natural order of an index-$n_r$ L-central division algebra over a fixed base field $F\subset L$, we must jointly minimize the relative discriminant of the extension $E/F$ and the relative norm of the non-norm element $\gamma$.
Our findings are summarized in the following table and will be proved, row by row, in the subsequent five theorems. Here, $\alpha$ is a root of the polynomial $X^2+X-i$ and $\beta$ denotes a root of the polynomial $X^3-(1+i)X^2+5iX-(1+4i)$.
\begin{table}[H]
\scalebox{0.94}{
\begin{tabular}{lllllllll}
$F$ & $n$ & $n_r$ & $n_t$ & rate & $\Nm_{\mathcal{O}_{F}}(\disc(\mathcal{O}_{\nat}/\mathcal{O}_{F}))$ & $L$ & $E$ & $\gamma$ \\
\hline
$\mathbb{Q}$ & 1 & 2 & 2 & 1 & $2^2\cdot 3^2$ & $\mathbb{Q}$ & $\mathbb{Q}(i\sqrt{3})$ & 2 \\
$\mathbb{Q}$ & 2 & 2 & 4 & 1 & $2^4\cdot 5^6$ & $\mathbb{Q}(\sqrt{5})$ & $\mathbb{Q}(\zeta_5)$ & -4 \\
$\mathbb{Q}(i)$ & 2 & 2 & 4 & 2 & $2^4\cdot 17^3 $ & $\mathbb{Q}(i,\alpha)$ & $\mathbb{Q}(i, \sqrt{\alpha} )$ &$1+ i$ \\
$\mathbb{Q}(i)$ & 2 & 3 & 6 & 3 & $3^{18}\cdot 13^{12}$ & $\mathbb{Q}(i, i\sqrt{3})$ & $\mathbb{Q}(i, i\sqrt{3},\beta)$ & $\frac{1+i\sqrt{3}}{2}$ \\
$\mathbb{Q}(i)$ & 3 & 2 & 6 & 2 & $2^6\cdot 3^{12}\cdot 13^8$ & $\mathbb{Q}(i, \beta)$ & $\mathbb{Q}(i, i\sqrt{3},\beta)$ & $1+i$
\end{tabular}}
\caption{Main results summarized.}
\label{tab:results}
\end{table}
\begin{remark} It is often preferred that $|\gamma|=1$ for balanced transmission power. However, there are good `remedy' techniques for the case when $|\gamma|>1$, see \emph{e.g.}, \cite{ESK,VHO}.
\end{remark}
\subsection{General Strategy}
We briefly elaborate on the three-step strategy that we will follow to prove each of the theorems. Let $F$, $n$ and $n_r$ be fixed.
\paragraph{\underline{Step 1.}}
We start by finding an explicit cyclic extension $E/L$ of degree $n_r$, $F\subset L$, such that $[L:F] = n$ and $E/F$ is cyclic with $|\Nm_{\mathcal{O}_F}(\disc(E/F))|$ smallest possible. In the cases where $F = \mathbb{Q}$, our extension $E/F$ will either be quadratic or quartic cyclic, and we simply use well-known formulas for computing $\disc(E/\mathbb{Q})$.
For $F = \mathbb{Q}(i)$ we resort to the following results from Class Field Theory (all the details can be found in \cite{CS}).
Let $F\subset E $ be an abelian extension of number fields. For each prime $\mathfrak p$ of $F$ that is unramified in $E$ there is a unique element $\frob_{\mathfrak p}\in {\rm Gal}(E/F)$ that induces the Frobenius automorphism $x\mapsto x^{\sharp k_{\mathfrak p}}$ on the residue field extensions $k_{\mathfrak p}\subset k_{\mathfrak q}$ for the primes $\mathfrak q $ in $E$ extending $\mathfrak p$. The order of $\frob_{\mathfrak p}$ in ${\rm Gal}(E/F)$ equals the residue class degree $[k_{\mathfrak q}:k_{\mathfrak p}]$, and the subgroup $\langle\frob_{\mathfrak p}\rangle$ of ${\rm Gal}(E/F)$ is the decomposition group of $\mathfrak p$. The \textit{Artin map} for $E/F$ is the homomorphism
\begin{align*}
\psi_{E/F}: I_F(\disc(E/F))&\longrightarrow {\rm Gal}(E/F)\\
\mathfrak p &\longmapsto \frob_{\mathfrak p}
\end{align*}
on the group $I_F(\disc(E/F))$ of fractional $\mathcal{O}_F$- ideals generated by the prime ideals $\mathfrak p$ of $F$ that do not divide the discriminant $\disc(E/F)$. These are unramified in $E$. For an ideal $\mathfrak a$ in $I_{F}(\disc(E/F))$ we call $\psi_{E/F}(\mathfrak a)$ the \textit{Artin symbol} of $\mathfrak a$ in ${\rm Gal}(E/F)$.
The \textit{Artin Reciprocity Law} states that if $F\subset E$ is an abelian extension, then there exists a nonzero ideal $\mathfrak m_0\mathcal{O}_L$ such that the kernel of the Artin map $\psi_{E/L}$ contains all principal $\mathcal{O}_L$-ideals $x\mathcal{O}_F$ with $x$ totally positive and $x\equiv 1\zmod {\mathfrak m_0}$.
We define a \emph{modulus} of $L$ to be a formal product $\mathfrak m = \mathfrak m_0 \mathfrak m_\infty$, where $\mathfrak m_0$ is a nonzero $\mathcal{O}_E$-ideal and $\mathfrak m_\infty$ is a subset of the real primes of $L$. We write $x\equiv 1 \zmod{^{\times}\mathfrak m}$ if $\hbox{ord}_{\mathfrak p}(x-1)\ge\hbox{ord}_{\mathfrak p}(\mathfrak m_0)$ at the primes $\mathfrak p$ dividing the finite part $\mathfrak m_0$ and $x$ is positive at the real primes in the infinite part $\mathfrak m_{\infty}$.
In the language of moduli, Artin's Reciprocity Law asserts that there exists a modulus $\mathfrak m$ such that the kernel of the Artin map contains the \textit{ray group} $R_{\mathfrak m}$ of principal $\mathcal{O}_L$-ideals $x\mathcal{O}_L$ generated by elements $x\equiv 1 \zmod{^{\times}\mathfrak m}$. The set of these \textit{admissible} moduli for $L/E$ consists of the multiples of some minimal modulus $\mathfrak f_{E/L}$, the \textit{conductor} of $L/E$. The primes occurring in $\mathfrak f_{E/L}$ are the primes of $L$, both finite and infinite, that ramify in $E$.
If $\mathfrak m=\mathfrak m_0\mathfrak m_{\infty}$ is an admissible modulus for $L/E$, and $I_{\mathfrak m}$ denotes the group of fractional $\mathcal{O}_L$-ideals generated by the primes $\mathfrak p$ coprime to $\mathfrak m_0$, then the Artin map induces a surjective homomorphism
\begin{align}
\label{eqn:artin}
\begin{split}
\psi_{E/L}: \Cl_{\mathfrak m} = I_{\mathfrak m}/R_{\mathfrak m} &\longrightarrow {\rm Gal}(E/L),\\
[\mathfrak p] &\longmapsto \frob_{\mathfrak p},
\end{split}
\end{align}
where $\Cl_{\mathfrak m}$ is the \emph{ray class group}, and $\ker(\psi_{E/F}) = A_{\mathfrak m}/R_{\mathfrak m}$ with
\begin{equation}
\label{eqn:norm}
A_{\mathfrak m} = \Nm_{E/F}(I_{\mathfrak m\mathcal{O}_F})\cdot R_{\mathfrak m}.
\end{equation}
The \textit{existence theorem} from Class Field Theory states that for every modulus $\mathfrak m$ of $F$, there exists an extension $F\subset H_{\mathfrak m}$ for which the map in \eqref{eqn:artin} is an isomorphism. Inside some fixed algebraic closure of $F$, the \textit{ray class field} $H_{\mathfrak m}$ is uniquely determined as the maximal abelian extension of $F$ in which all the primes in the ray group $R_{\mathfrak m}$ split completely. Conversely, if $F\subset E$ is abelian then $E \subset H_{\mathfrak m}$ whenever $\mathfrak m$ is an admissible modulus for $F\subset E$. For $E = H_{\mathfrak m}$, we have $A_{\mathfrak m} = R_{\mathfrak m}$ in \eqref{eqn:norm} and an \textit{Artin isomorphism}
\begin{equation}
\label{eqn:artiniso}
\Cl_{\mathfrak m} \simeq {\rm Gal}(H_{\mathfrak m}/F).
\end{equation}
For all $\mathfrak m$, the ray group $R_{\mathfrak m}$ is contained in the subgroup $P_{\mathfrak m} \subset I_{\mathfrak m}$ of principal ideals in $I_{\mathfrak m}$, with $I_{\mathfrak m}/P_{\mathfrak m} = \Cl_{F}$, the class group of $F$. There is a natural exact sequence
\begin{equation}
\label{eqn:exact}
\mathcal{O}_{F}^{\times} \longrightarrow (\mathcal{O}_F/\mathfrak m)^{\times} \longrightarrow \Cl_{\mathfrak m} \longrightarrow \Cl_{F}\longrightarrow 0,
\end{equation}
and the residue class in $\Cl_{\mathfrak m}$ of $x \in \mathcal{O}_L$ coprime to $\mathfrak m_0$ in the finite group $(\mathcal{O}_F/\mathfrak m)^{\times} = (\mathcal{O}_F/\mathfrak m)^{\times}\times\prod_{\mathfrak p\mid \mathfrak m_{\infty}}\langle -1 \rangle $ consists of its ordinary residue class modulo $\mathfrak m_0$ and the signs of its images under the real primes $\mathfrak p \mid \mathfrak m_{\infty}$.
Finally, we compute the discriminant $\disc(E/F)$ using Hasse's conductor-discriminant formula
\begin{equation}
\label{eqn:condisc}
\disc(E/F)=\prod_{\chi:I_{\mathfrak m}/A_{\mathfrak m}\to\mathbb{C}^{\times}}\mathfrak f(\chi)_0,
\end{equation}
where $\chi$ ranges over the characters of the finite group $I_{\mathfrak m}/A_{\mathfrak m}\simeq {\rm Gal}(E/F)$ and $\mathfrak f(\chi)_0$ denotes the finite part of the conductor $\mathfrak f(\chi)$ of the ideal group $A_{\chi}$ modulo $\mathfrak m$ satisfying $A_{\chi}/A_{\mathfrak m}=\ker \chi$.
\paragraph{\underline{Step 2.}}
Unfortunately, having $|\Nm_{\mathcal{O}_F}(\disc(E/F))|$ smallest possible is not sufficient for $|\Nm_{\mathcal{O}_F}(\disc(\mathcal{O}_{\nat}/\mathcal{O}_{F}))|$ to be smallest possible as well.
Using \eqref{eqn:disc} and \eqref{eqn:bound} we can derive a positive lower bound on the size of a non-norm element $\gamma \in \mathcal{O}_L$ as
\begin{equation}
\label{eqn:constant}
\left|\Nm_{\mathcal{O}_L}(\gamma ^{n_r-1})\right|\ge \frac{\left|\Nm_{\mathcal{O}_L}(\mathfrak{p}_1\mathfrak{p}_2)^{n_r-1}\right|}{\left|\Nm_{\mathcal{O}_L}(\disc(E/L))\right|} =: \lambda_{E,L} \in \mathbb{N},
\end{equation}
where $(\mathfrak{p}_1, \mathfrak{p}_2)$ is a pair of norm-wise smallest prime ideals in $\mathcal{O}_{L}$.
If $\disc(E/L)$ is minimal and $\lambda_{E,L} > 1$, as will be the case in Theorems~\ref{thm:res1} and \ref{thm:res2} below, we need to balance the size of $\disc(E/F)$ and $\gamma$ in order to achieve minimality of $|\Nm_{\mathcal{O}_F}(\disc(\mathcal{O}_{\nat}/\mathcal{O}_{F}))|$.
In the last three theorems, we will then proceed as follows. Given two cyclic algebras $(E/L, \sigma, \gamma)$ and $(E'/L',\sigma', \gamma')$ of index $n_r$, where $\langle \sigma \rangle = {\rm Gal}(E/L)$, $\langle \sigma' \rangle = {\rm Gal}(E'/L')$ and such that $\mathbb{Q}(i)\subset L, L'$, we have by \eqref{eqn:disc}, since norms in $\mathbb{Q}(i)$ are positive,
\begin{align}
\label{eqn:discsize}
\begin{split}
\Nm_{\mathbb{Z}[i]}(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i])) &\le \Nm_{\mathbb{Z}[i]}(\disc(\mathcal{O}'_{\nat}/\mathbb{Z}[i])) \\
&\Leftrightarrow \\
D_{E/L}(\gamma) &\le D_{E'/L'}(\gamma'),
\end{split}
\end{align}
with
\begin{equation}
\label{eqn:balance}
D_{E/L}(\gamma) := \Nm_{\mathbb{Z}[i]}(\disc(E/\mathbb{Q}(i)))\cdot \Nm_{\mathcal{O}_L}(\gamma)^{n_r-1}.
\end{equation}
Our strategy will be to fix a non-norm element $\gamma \in \mathcal{O}_L$ of smallest possible norm, compute $D_{E/L}(\gamma)$ and, along the lines of Step 1, compare $D_{E/L}(\gamma)$ with $D_{E'/L'}(\gamma')$, where $E'/L'$ runs over all degree-$n_r$ cyclic field extensions such that $\mathbb{Q}(i)\subset L'$ and $\gamma' \in \mathcal{O}_L$ is a non-norm element for $E'/L'$ of smallest possible norm.
\paragraph{\underline{Step 3.}}
If $\disc (E/L)$ is smallest possible and $\lambda_{E,L} < 1$, as will be the case in Theorems \ref{thm:res3}, \ref{thm:res4} and \ref{thm:res5}, the optimal situation would be to be able to choose a non-norm element $\gamma$ which is a unit, $\gamma \in \mathcal{O}_L^{\times}$. Hasse's Norm Theorem
will help us decide whether such an element exists and, if it does, how to find it. We use the following strategy from the theory of local fields to compute $\Nm_{K/k}(K^{\times})$ when $K/k$ is an extension of non-Archimedean local fields (all the details can be found in \cite[Chp.~7]{JM}).
Let $k$ be a field, complete under a discrete valuation $v_k$. Let $A_k$ be its valuation ring with maximal ideal $\mathfrak m_k$, generated by a local uniformizing parameter $\pi_k$ with $v_k(\pi_k)=1$, and $\overline{k} = A_k/\mathfrak m_k$ the residue field, where $|\overline k| = q_k$ is a power of a rational prime $p$. Denote by $U_k = A_k-\mathfrak m_k$ the multiplicative group of invertible elements $A_k^{\times}$ of $A_k$, and set $U_k^i = 1+\mathfrak m_k^i$, $i\ge 1$.
Then, $U_k = S_k\times U_k^1$, where $S_{k}$ is a complete set of representatives of $\overline k$, and
\begin{align*}
k^{\times} = \langle \pi_k\rangle U_k = S_k \times \langle \pi_k \rangle \times U_k^1.
\end{align*}
As $\overline k^{\times}$ is cyclic of order $q_k-1$, we may take $S_k =\left\{0\right\} \cup \left\{\zeta_{q_k-1}^i \mid 1 \le i \le q_k-1\right\} $, where $ \zeta_n$ denotes a primitive $n$-th root of unity in $\overline k$.
Let $K$ be finite separable extension of $k$, $A_K$ the integral closure of $A_k$ in $K$, and let $v_K$, $\mathfrak m_K$, $\pi_K$, $\overline K$, $q_K$ and $U_K^i$ be defined as above. As usual, we denote the ramification index and residue degree of $\mathfrak m_K$ in $K/k$ by $e_{K/k}$ and $f_{K/k}$, respectively. We have $e_{K/k}\cdot f_{K/k} = [K:k]$ and $\pi_k = \pi_K^{e_{K/k}}\times u$ with $u$ a unit, so that $\Nm_{K/k}(\pi_K) = \pi_k^{f_{K/k}}$.
The group $\Nm_{K/k} (U_K)$ is a subgroup of $U_k$ with $[U_k: \Nm_{K/k}(U_K)]=e_{K/k} $, and $\Nm_{K/k}(U_K^1) \subset U_k^1$ with $[U_k^1: \Nm_{K/k}(U_K^1)]$ a power of $p$.
Consequently, if the extension is unramified, i.e. $e_{K/k}=1$, then $\Nm_{K/k}(U_K)=U_k$ and every unit is a norm.
Suppose hereinafter that $K/k$ is a totally tamely ramified extension, thus $f_{K/k} = 1$, $(p, e_{K/k}) = 1$, $\overline K = \overline k$, $q_K = q_k$ and the \emph{local conductor}, \emph{i.e.}, the smallest integer $f$ such that $U_ k^f \subset \Nm_{K/k}(K^{\times})$, is 1 (\cite[Aside 1.9]{JM2}). Then $\Nm_{K/k}(\zeta_{q_K-1})=\zeta_{q_k-1}^{[K:k]}$ and $\Nm_{K/k}(U_K^1)=U_k^1$.
Consequently,
\begin{align*}
\Nm_{K/k}(K^{\times}) = \langle \Nm_{K/k}(\pi_K), \zeta_{q_k-1}^{[K:k]}\rangle U_k^1.
\end{align*}
Since $\pi_K$ is not a unit, we conclude that in the totally tamely ramified case,
\begin{equation}
\label{eqn:nonnorm}
U_k \cap \Nm_{K/k}(K^{\times})= \langle \zeta_{q_k-1}^{[K:k]}\rangle U_k^1.
\end{equation}
With the above information at hand, we go back to the extension $E/L$ found in Step 1 with $\lambda_{E,L} < 1$. In order to produce a suitable unit $\gamma \in \mathcal{O}_L^{\times}$ which is not a local norm at some prime ramifying in the extension, we compute the ramified primes as well as $\langle \zeta_{q_k-1}^{[K:k]}\rangle U_k^1$ in the corresponding local extension $K/k$. Considering $\mathcal{O}_L$ as a subset of $A_k$ and using Hensel's lemma, we look for a unit in $\mathcal{O}_L^{\times}$ such that its image in $\overline{k}$ lies outside $\langle \zeta_{q_k-1}^{[K:k]}\rangle$.
Unfortunately, if $\mathcal{O}_L^{\times} \subset \langle \zeta_{q_k-1}^{[K:k]}\rangle U_k^1$, as will be the case in Theorems~\ref{thm:res3} and \ref{thm:res5}, a non-norm unit for $E/L$ will not exist. In those cases the sizes of $\disc(E/F)$ and $\gamma$ in \eqref{eqn:disc} must be balanced using \eqref{eqn:balance} in order to achieve minimality of $|\Nm_{\mathcal{O}_F}(\disc(\mathcal{O}_{\nat}/\mathcal{O}_{F}))|$.
\subsection{Main Results}
We are now ready to state and prove the main results of this article.
\begin{theorem}
\label{thm:res1}
Let $ \mathbb{Q} \subset E = \mathbb{Q}(\sqrt{d})$, $d \in \mathbb{Z}$ square-free, and ${\rm Gal}(E/L)=\langle \sigma \rangle$. Any cyclic division algebra $\left(E/\mathbb{Q}, \sigma, \gamma\right)$ of index 2 satisfies
$|\disc(\mathcal{O}_{\nat}/\mathbb{Z})| \ge 36$, and equality is achieved for $E = \mathbb{Q}(i\sqrt{3})$, $\gamma = 2$.
\end{theorem}
\begin{proof}: The proof follows the strategy described above.
\medskip
\noindent\emph{\underline{Step 1.}}
The smallest possible quadratic discriminant over $\mathbb{Q}$ is $\disc(E/\mathbb{Q}) = 3$, corresponding to the field $E = \mathbb{Q}(i\sqrt{3}) = \mathbb{Q}(\omega)$, $\omega$ a primitive cubic root of unity. Let ${\rm Gal}(E/\mathbb{Q}) = \langle \sigma \rangle$.
\medskip
\noindent\emph{\underline{Step 2.}}
A pair of smallest primes in $\mathbb{Z}$ is $(2,3)$, so that $\lambda_{E, \mathbb{Q}} = \frac{6}{3} > 1$ (cf. \eqref{eqn:constant}). Thus, any non-norm element $\gamma \in \mathbb{Z}$ satisfies $|\gamma | \ge 2$.
To ensure that we can choose $\gamma = 2$, we first show that the equation $x^2+3y^2=2$ has no solution in $\mathbb{Q}$. Consequently, $2\not \in \Nm_{E/\mathbb{Q}}(E^{\times})$ and $(\mathbb{Q}(i\sqrt{3})/\mathbb{Q}, \sigma, 2)$ is a division algebra with $\disc(\mathcal{O}_{\nat}/\mathbb{Z}) = 36$.
Suppose that $\left(\frac{a}{b}\right)^2+3 \left(\frac{c}{d}\right )^2 = 2$, with $a, b, c, d\in \mathbb{Z}$ and such that $(a,b) = (c,d) = 1$. Then,
\begin{equation}
\label{eqn:norm2}
(ad)^2+3(bc)^2=2(bd)^2.
\end{equation}
It is easy to deduce from \eqref{eqn:norm2} that $3^s, s \ge 0$, is the largest power of 3 dividing $b$ if and only if it is the largest power of $3$ dividing $d$. If $(3,bd) = 1$, then equation \eqref{eqn:norm2} has no solution in $\mathbb{Z}$, as 2 is not a square $\mathrm{mod}$ 3. Set $b = 3^sb'$ and $d = 3^sd'$, with $(3,b'd') = 1$, $s\ge 1$. Substituting into \eqref{eqn:norm2} yields
\begin{align*}
(ad')^2+3(b'c)^2=2.3^s(b'd')^2,
\end{align*}
which is absurd, since $(3,ad') = 1$.
Next, we use \eqref{eqn:disc} to see that for $E' = \mathbb{Q}(\sqrt d)$ with $d \ne 1, -3$ a square-free integer and ${\rm Gal}(E'/\mathbb{Q}) = \langle \sigma '\rangle$, the cyclic division algebra $\left(E'/\mathbb{Q}, \sigma', \gamma'\right)$ satisfies $ |\disc(\mathcal{O}_{nat}'/\mathbb{Z})|>36$ for any choice of non-norm element $\gamma\in \mathbb{Z}$.
\begin{itemize}
\item[i)] If $d\equiv 2,3 \zmod 4$, then $|\disc(E'/\mathbb{Q})|=4|d| \ge 8$,
and \eqref{eqn:disc} guarantees that for any $\gamma' \in \mathbb{Z}$, $|\disc (\mathcal{O}_{nat}'/\mathbb{Z})|\ge 8^2\gamma'^2\ge 64$.
\item[ii)] If $d\equiv 1 \zmod 4$ and $|d|\ge 7$, then $\left|\disc(E'/\mathbb{Q})\right| = |d|$ and \eqref{eqn:disc} implies $|\disc (\mathcal{O}_{nat}'/\mathbb{Z})|\ge 7^2\gamma'^2\ge 49$.
\item[iii)] For $d = 5$, we have $\disc(E'/\mathbb{Q}) = 5$ and $\lambda_{E', \mathbb{Q}} = \frac{6}{5} = 2$ (cf. \eqref{eqn:constant}). Using \eqref{eqn:disc} we conclude that for any non-norm element $\gamma' \in \mathbb{Z}$, $\disc(\mathcal{O}_{nat}'/\mathbb{Z})\ge 5^2\cdot 2^2>36$.
\end{itemize} \qed
\end{proof}
\begin{theorem}
\label{thm:res2}
Let $\mathbb{Q} \subset L \subset E$ with $[E:\mathbb{Q}] = 4$, $[E:L] = 2$ and ${\rm Gal}(E/L)=\langle \sigma \rangle$. If $\left( E/L, \sigma, \gamma\right)$ is a cyclic division algebra, then $\Nm_\mathbb{Z}(\disc(\mathcal{O}_{\nat}/\mathbb{Z})) \ge 2^4\cdot 5^6$. Equality is achieved for $L = \mathbb{Q}(\sqrt{5}),$ $E = \mathbb{Q}(\zeta_5)$ and $\gamma = -4$, with $\zeta_5$ a primitive $5^\text{th}$ root of unity.
\end{theorem}
\begin{proof}:
The fields $L$ and $E$ can be uniquely expressed as $L = \mathbb{Q}(\sqrt{D})$ and $E = \mathbb{Q}\left(\sqrt{A(D+B\sqrt D)}\right )$, with $A,B,C,D \in \mathbb{Z}$ such that $A$ is square-free and odd, $D = B^2+C^2$ is square-free, $B,C > 0$ and $(A,D) = 1$ (\cite{HW}, \cite{HSW}).
\medskip
\noindent\emph{\underline{Step 1.}}
\begin{itemize}
\item[i)]If $D \equiv 0 \zmod 2$, then $\disc(E/\mathbb{Q}) = 2^8\cdot A^2\cdot D^3 \ge 2^{11}$. The lower bound is attained for $D = 2$, $B = C = 1$, and $|A| = 1$. Using \eqref{eqn:disc} we deduce that $|\disc(\mathcal{O}_{\nat}/\mathbb{Z})| \ge 2^{22} > 2^4\cdot 5^6$ for any choice of $\gamma \in \mathcal{O}_L$.
\item[ii)] If $D \equiv B\equiv 1 \zmod 2$, then $\disc(E/\mathbb{Q}) = 2^6\cdot A^2 \cdot D^3 \ge 2^6\cdot 5^3$. This expression attains its minimum value for $D = 5$ and $|A| = B = 1$. Hence, $|\disc(\mathcal{O}_{\nat}/\mathbb{Z})| \ge 2^{12}\cdot 5^6 > 2^4\cdot 5^6$ for any choice of $\gamma \in \mathcal{O}_L$.
\item[iii)] If $D\equiv 1 \zmod 2$, $B\equiv 0 \zmod 2$ and $A+B\equiv 3 \zmod 4$, we have $\disc(E/\mathbb{Q})=2^4 \cdot A^2 \cdot D^3 \ge 2^4\cdot 5^3$.
The minimum value is attained for $D = 5$, $B = 2$, and $A = 1$, so that
$|\disc(\mathcal{O}_{\nat}/\mathbb{Z})| \ge 2^8\cdot 5^6> 2^4\cdot 5^6$ for any choice of $\gamma \in \mathcal{O}_L$.
\item[iv)] Finally, if $D\equiv 1 \zmod 2$, $B\equiv 0 \zmod 2$, $A+B\equiv 1 \zmod 4$ and $A\equiv \pm C \zmod 4$, we have $\disc(E/\mathbb{Q}) = A^2 \cdot D^3 \ge 5^3$. The minimum of this expression is attained for $D = 5$, $B = 2$ and $A = -1$, corresponding to the fields $E = \mathbb{Q}(\zeta_5)$ and $L = \mathbb{Q}(\zeta_5+\zeta_5^{-1}) = \mathbb{Q}(\sqrt 5)$.
\end{itemize}
The last case provides us with a field extension $E/L = \mathbb{Q}(\zeta_5)/\mathbb{Q}(\sqrt 5)$ with $\disc(E/L) = 5$ and $\disc(E/\mathbb{Q}) = 5^3$. If we show $|\disc(\mathcal{O}_{\nat}/\mathbb{Z})| = 2^4\cdot 5^6$ for $\gamma = -4$ a non-norm element, the theorem will be proved.
\medskip
\noindent\emph{\underline{Step 2.}}
The two smallest prime ideals in $\mathcal{O}_L$ are $\mathfrak p_2 = 2\mathcal{O}_L$ and $\mathfrak p_5$, a factor of $5\mathcal{O}_L = \mathfrak p_5 \mathfrak p_5'$, of respective norms 4 and 5. Hence, $\lambda_{E,L} = 4$, and any suitable non-norm element for $E/L$ will satisfy $|\Nm_{\mathcal{O}_L}(\gamma)|\ge 4$ (cf. \eqref{eqn:constant}). We show that $-4 \not \in \Nm_{E/L}(E^\times)$. Since the norm is multiplicative and $4 = \Nm_{E/L}(2)$, it suffices to show that $-1 \notin \Nm_{E/L}(E^\times)$. Suppose that $x\in L$ with $\Nm_{E/L}(x) = -1$. Then $\Nm_{E/\mathbb{Q}}(x) = 1$, and we can write $x = \zeta v$ with $\zeta$ a root of unity and $v \in L^\times$ \cite[Prop.~6.7]{JM}. Consequently, $-1 = \Nm_{E/L}(\zeta)\cdot\Nm_{E/L}(v) = v^2$. But $-1$ is not a square in $L$. \qed
\end{proof}
In the remaining cases, our base field will be $F = \mathbb{Q}(i)$. For each rational prime $p$, we write $p\mathbb{Z}[i] = \mathfrak p_p^2$, $\mathfrak p_p\mathfrak p_p'$ or $\mathfrak p_p = (p)$ for the cases where $p$ is ramified, split or inert in $\mathbb{Z}[i]$, respectively.
\begin{theorem}
\label{thm:res3}
Let $\mathbb{Q}(i) \subset L \subset E$, with $[E:\mathbb{Q}(i)] = 4 $ and $[E:L] = 2$. Let ${\rm Gal}(E/L)=\langle \sigma \rangle$. Any cyclic division algebra $\left(E/L, \sigma, \gamma \right)$ under these assumptions satisfies $\Nm_{\mathbb{Z}[i]}(\disc({\cal O}_{\nat}/\mathbb{Z}[i])) \ge 2^4\cdot 17^6$, with quality attained for $L = \mathbb{Q}(i,\alpha),$ $ E = \mathbb{Q}(i,\sqrt{\alpha})$ and {$\gamma=1+i$, with $\alpha$ a root of $f(X)=X^2+X-i$}.
\end{theorem}
\begin{proof}:
\noindent\emph{\underline{Step 1.}}
We start by finding the smallest possible discriminant over $\mathbb{Q}(i)\\
$ for cyclic extensions of degree 4. Let $E$ any be any such extension. By the existence theorem of Class Field Theory, we know that $E$ is contained in the ray class field $ H_{\mathfrak m}$, where $\mathfrak m$ is an admissible modulus for the extension $E/\mathbb{Q}(i)$. The smallest ray class field will have conductor $\mathfrak f = \mathfrak f_{E/\mathbb{Q}(i)}$, which, since $\mathbb{Q}(i)$ is a complex field, will be an ideal of $\mathbb{Z}[i]$.
The restriction of the Artin map \eqref{eqn:artin} to $I_{\mathfrak f}$ gives us a canonical isomorphism ${\rm Gal}(H_{\mathfrak f}/\mathbb{Q}(i)) \simeq I_{\mathfrak f}/P_{\mathfrak f} = C_{\mathfrak f}$ (cf. \eqref{eqn:artiniso}), which implies $[H_{\mathfrak f}:\mathbb{Q}(i)] = |C_{\mathfrak f}|$. Furthermore, since the class group $C_{\mathbb{Q}(i)}$ is trivial and $\mathbb{Z}[i]^{\times}=\langle i \rangle$, by \eqref{eqn:exact} we have the exact sequence
\begin{align*}
0\to \langle i \rangle \to (\mathbb{Z}[i]/{\mathfrak f})^{\times} \to C_{\mathfrak f} \to 0.
\end{align*}
Thus, $C_{\mathfrak f} \simeq (\mathbb{Z}[i]/f)^{\times}/ \hbox{Im} \langle i \rangle$.
The ray class field of conductor 2 is trivial, which is not our case, so $\mathfrak f$ does not divide 2. The map $\langle i \rangle \to(\mathbb{Z}[i]/\mathfrak f)^{\times} $ is injective, so that $[H_{\mathfrak f}:Q(i)] = \frac{1}{4}\left|(\mathbb{Z}[i]/\mathfrak f)^{\times}\right|$. Consequently, $$4 = [E:Q(i)] \mid [H_{\mathfrak f}:Q(i)] \Rightarrow 16 \mid \left | (\mathbb{Z}[i]/\mathfrak f)^{\times}\right | = \Nm(\mathfrak f) \text{ and } \Nm(\mathfrak f)\ge17 .$$
Fortunately we can find ideals of norm 17 with the required properties.
We fix $\mathfrak f = 1+4i$ (or $\mathfrak f=1-4i$),
so that the ray class field of conductor $\mathfrak f$ is precisely $H_{\mathfrak f} = \mathbb{Q}(i, \sqrt[4]{1+4i})$.
Since $\mathfrak f$ is a prime ideal, all non-trivial characters of ${\rm Gal}(H_{\mathfrak f}/\mathbb{Q}(i))$ have conductor $\mathfrak f$, so that by the conductor-discriminant formula \eqref{eqn:condisc}, $\disc(H_{\mathfrak f}/\mathbb{Q}(i)) = \mathfrak f^3$. The absolute discriminant is $\disc(H_{\mathfrak f}/\mathbb{Q}) = 17^3\cdot 4^4$.
We choose $L = \mathbb{Q}(i, \sqrt{1+4i})$ and $E = H_{\mathfrak f}$, and prove that this choice yields the smallest possible discriminant of a cyclic extension of degree 4 over $\mathbb{Q}(i)$. To that end, let $\mathfrak m$ be any ideal of $\mathbb{Z}[i]$ of norm different from 17 for which $16 \mid \left |(\mathbb{Z}[i]/\mathfrak m)^\times\right |$, and let $E' \subset H_{\mathfrak m}$ be a subfield with $E'/\mathbb{Q}(i)$ cyclic of degree 4. Assume that $\disc(E'/\mathbb{Q})\le 17^3 \cdot 4^4$. Since $E'$ has conductor $\mathfrak m$, by the minimality of $\mathfrak m$ the quartic characters of ${\rm Gal}(E'/Q(i))$ have conductor $\mathfrak m$.
The quadratic character could have smaller conductor, but it cannot be smaller than 3, since $\mathbb{Q}(i)$ admits no ray class field of conductor $\mathfrak m$ with $\Nm(\mathfrak m) < 9$.
Using the conductor-discriminant formula and as norms in $\mathbb{Q}(i)$ are positive, we have
\begin{align}
\label{condisc}
\begin{split}
&9 \cdot \Nm(\mathfrak m^2) \le \Nm(\disc(E'/\mathbb{Q}(i))), \\
\Rightarrow\ &9 \cdot \Nm(\mathfrak m)^2\cdot4^4 \le \disc(E'/\mathbb{Q})< 17^3 \cdot 4^4, \\
\Rightarrow\ &\Nm(\mathfrak m) < 23,36\ldots,
\end{split}
\end{align}
and so $\left |(\mathbb{Z}[i]/\mathfrak m)^*\right |<23.$ Since $16 \mid \left|(\mathbb{Z}[i]/\mathfrak m)^\times\right|$, we have $\left|(\mathbb{Z}[i]/\mathfrak m)^\times\right| = 16$. But then necessarily $\Nm(\mathfrak m) = 17$, a contradiction to our assumption.
\medskip
\noindent\emph{\underline{Step 2.}}
Let $\mathcal{O}_L = \mathbb{Z}[i,\alpha]$, with $\alpha$ a root of $f(X)=X^2+X-i$. A pair of norm-wise smallest primes in $\mathcal{O}_L$ is $(\mathfrak p_2, \mathfrak p_5)$, of respective norms 2 and 5. Consequently, $\lambda_{E,L} = \frac{20^2}{17^6} < 1$ (cf. \eqref{eqn:constant}), so that $\Nm_{\mathbb{Z}[i]}(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i])$ will achieve the smallest possible value among all central division algebras satisfying the given conditions for a unit non-norm element $\gamma \in \mathcal{O}_L^{\times}$. Unfortunately, as we will show next, there is no suitable unit in this case, forcing us to consider other non-norm elements.
\medskip
\noindent\emph{\underline{Step 3.}}
By the Hasse Norm Theorem, to show that an element in $L^{\times}$ is not a norm, it suffices to show that it is not a local norm at some prime of $E$. We need to produce a unit $\gamma \in \mathcal{O}_L^\times$ with $\gamma \notin \Nm_{E/L}(E^{\times})$, and since in an unramified local extension all units are norms, we consider the local extension corresponding to the ramifying prime $\mathfrak f = (1+4i)\mathbb{Z}[i]$. It is not difficult to verify that $\mathfrak f\mathcal{O}_L=\mathfrak p_{L}^2$, $\mathfrak p_{L}\mathcal{O}_E=\mathfrak p_{E}^2$ and $\Nm_{L}(\mathfrak p_{L})=\Nm_{E}(\mathfrak p_{E})={17}$.
Let $k=L_{\mathfrak p_L}$ and $K= E_{\mathfrak p_E}$ be the completions of $E$ and $L$ with respect to the discrete valuations associated to the primes $\mathfrak p_L$ of $L$ and $\mathfrak p_E$ of $E$. Then $K/k$ is a totally and tamely ramified cyclic local extension of degree 2, with $\overline{K}=\overline {k} = \mathbb{F}_{17}$. Using \eqref{eqn:nonnorm}, $U_k \cap \Nm_{K/k}(K^{\times})=\langle{ \zeta}_{16}^2~\rangle U_{L}^{(1)}$ and, by Hensel's lemma, an element in $\mathcal{O}_L^{\times}$ will be a non-norm element for $E/L$ if and only if its image in $\mathbb{F}_{17}$ is not a square in $\mathbb{F}_{17}^{\times}$.
Since $\Nm_{L/\mathbb{Q}}(x)=\Nm_{\mathbb{Q}(i)/\mathbb{Q}}(\Nm_{L/\mathbb{Q}(i)}(x))$ and the norm of every unit in $\mathbb{Z}[i]$ is $1$, we have $\Nm_{L/\mathbb{Q}}(\mathcal{O}_L^{\times})=\{1^2\}$. Consequently, the norm of the image in $\mathbb{F}_{17}$ of every element in $\mathcal{O}_L^{\times}$ is a square. Over finite fields, this is the case if and only it such an image is itself a square, which for its part implies that every element in $\mathcal{O}_L^{\times}$ maps to $\langle{\zeta}_{16}^2~\rangle$ and, hence, is in the image of the map $ \Nm_{E/L}$. We conclude that there exists no unit which is a non-norm element for $E/L$, forcing us to use\eqref{eqn:balance} to balance the sizes of $\disc(E/\mathbb{Q}(i))$ and $\gamma$ in \eqref{eqn:disc} in order to achieve minimality of $\Nm(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i]))$.
\medskip
Since the norm of an element $x\in \mathcal{O}_L$ is the product of the norms of the prime ideals dividing $x$, it is easy to verify that the smallest possible norm in $\mathcal{O}_L$ is 4. We set $\gamma = 1+i \in \mathcal{O}_L$ with $\Nm_L(1+i) = 4$, which yields $D_{E/L}(\gamma) = 17^3 \times 2^2$.
In order to study the possible values of $D_{E'/L'}(\gamma')$, let $\mathfrak m$ be any ideal of $\mathbb{Z}[i]$ of norm different from 17, for which $16 \mid \left |(\mathbb{Z}[i]/\mathfrak m)^\times\right |$, and let $E'$ be a subfield of the ray class field $H_{\mathfrak m}$, with $E'/\mathbb{Q}(i)$ cyclic of degree 4.
The smallest possible norms for $\mathfrak m$ are given by
\begin{alignat*}
34 &= \Nm(\mathfrak p_{17}\mathfrak p_2),\quad 49 &&= \Nm(\mathfrak p_7), \quad &&64 = \Nm(\mathfrak p)^6, \\
68 &= \Nm(\mathfrak p_{17}\mathfrak p_2^2),\quad 128 &&= \Nm(1+i)^7, \quad &&\ldots
\end{alignat*}
Using \eqref{condisc}, we see that $9\cdot \Nm(\mathfrak m)^2 \le \Nm(\disc(E'/\mathbb{Q}(i)))$, so that we only need to consider the cases for which $9\cdot \Nm(\mathfrak m)^2 \le 17^3 \times 2^2 \Rightarrow \Nm(\mathfrak m) \le 46,728...$, \emph{i.e.}, only the case $\mathfrak m = (1+4i)(1+i)$.
We compute $\Nm(\disc(E'/F))$. The smallest quadratic discriminant for an extension in which both $(1+i)$ and a prime of norm $17$ ramify is $(4+i)(1+i)^2$, corresponding to the extension $L' = \mathbb{Q}(\sqrt{4+i})$. Thus, $E' = \mathbb{Q}(\sqrt[4]{4+i})$, with $\disc(E'/\mathbb{Q}(i))=(4+i)^3(1+i)^4$ and $\Nm(\disc(E'/\mathbb{Q}(i)) = 17^3 \cdot 2^4$. Consequently, for all $\gamma'\in \mathcal{O}_{L'}$, $D_{E'/L'}(\gamma') \ge 17^3\times 2^4 > 17^3 \times 2^2 = D_{E/L}(\gamma)$. By \eqref{eqn:discsize}, we are done.
\qed
\end{proof}
\begin{theorem}
\label{thm:res4}
Let $\mathbb{Q}(i) \subset L \subset E$ with $[E:\mathbb{Q}(i)]=6$ and
$ [E:L] = 3$. Any cyclic division algebra $\left(E/L, \sigma, \gamma\right)$ satisfies $\Nm(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i]))\ge3^{18}\cdot 13^{12}$. The lower bound is achieved for $L = \mathbb{Q}(\zeta_{12})$, $E = L(\beta)$ and $\gamma=\frac{1+i\sqrt 3}{2}\in \mathcal{O}_L^{\times}$, where $\zeta_{12}$ is a primitive $12^\text{th}$ root of unity and $\beta$ a root of $f(X) = X^3-(1+i)X^2+5iX-(1+4i)$.
\end{theorem}
\begin{proof}:
\noindent\emph{\underline{Step 1.}}
We start by finding the smallest possible discriminant over $\mathbb{Z}[i]$ for cyclic extensions of degree 6. We denote by $L_2$, $L_3$ and $E = L_2L_3$ cyclic extensions of degree 2 and 3 over $F = \mathbb{Q}(i)$ and their compositum, respectively. Using \eqref{eqn:exact}, \eqref{eqn:condisc} and arguments similar to those used in Theorem~\ref{thm:res3}, we deduce that the smallest possible cubic discriminant is $\mathfrak p_{13}^2 = (2-3i)^2$, corresponding to the extension $\mathbb{Q}(i, \beta)/\mathbb{Q}(i)$ \cite[Table p. 883, row 32]{BeMOl}, where $\beta$ is a root of the polynomial $f(X) = X^3-(1+i)X^2+5iX+(-1-4i)$.
Since we want to minimize $\disc(L_2 L_3/\mathbb{Q}(i))$, the use of \eqref{eqn:exact}, \eqref{eqn:condisc} requires that we consider separately the cases where $\mathfrak m$ is and is not relatively prime to $\mathfrak p_{13}$, and check which case yields a smaller value for this discriminant.
\begin{itemize}
\item[i)] $(\mathfrak m, \mathfrak p_{13}) = 1$. The smallest possible discriminant corresponds to the extension $L_2=\mathbb{Q}(i)(\sqrt{-3}) =\mathbb{Q}(\zeta_{12})$ of $\mathbb{Q}(i)$, with $\disc(L_2L_3/\mathbb{Q}(i)) = 3^3\mathfrak p_{13}^4$, of norm $\Nm(\disc(L_2L_3/\mathbb{Q}(i))) = 13^4\cdot 3^6$.
\item[ii)] $(\mathfrak m, \mathfrak p_{13}) \neq 1$. If $\disc(L_2'/\mathbb{Q}(i))=\mathfrak p_{13}\times \mathfrak a$ for some ideal $\mathfrak a \ne(1)$, the best possibility is $\mathfrak p_{13}\mathfrak p_2^2$, corresponding to the extension $L_2'=\mathbb{Q}(i)(\sqrt{3-2i})$ of $\mathbb{Q}(i)$ with discriminant ideal $\mathfrak p_{13}\times \mathfrak p_2^2$. This choice yields\footnote{The factor $\mathfrak p_{13}^5$ comes from the fact that $E/L_3$ is tamely ramified and, thus, has as discriminant the prime $\mathfrak q_{13}$ lying above 13 in $L_3$.} $\disc(L_2'L_3/\mathbb{Q}(i)) = \mathfrak p_{13}^5\times \mathfrak p_2^6$, with $\Nm(\disc(L_2'L_3/\mathbb{Q}(i))) = 13^5\cdot 2^6 > 13^4\cdot 3^6$.
\end{itemize}
We conclude that the smallest possible discriminant over $\mathbb{Z}[i]$ for cyclic extensions of degree 6 is $3^6 \cdot 13^4$, achieved in the extension $E = L_2L_3$. The involved rings of integers are $\mathcal{O}_{2} = \mathbb{Z}\left [i, \frac{i+\sqrt{3}}{2}\right]$
and $ \mathcal{O}_{3} = \mathbb{Z}[i, \beta]$.
The discriminants of the extensions involved are summarized in Table~\ref{tab:disc} below.
\begin{table}[H]
\centering
\scalebox{0.98}{
\begin{tabular}{l|ll|ll|ll|ll}
& $\disc(\cdot/\mathbb{Q}(i))$ & $\Nm_{\mathbb{Z}[i]}$ & $\disc(\cdot/L_2)$ & $\Nm_{\mathcal{O}_2}$ & $\disc(\cdot/L_3)$ & $\Nm_{\mathcal{O}_3}$ \\
\hline
$E$ & $\mathfrak p_3^3\mathfrak p_{13}^4$ & $3^6\cdot 13^4$ & $\mathfrak q_{13}^2\mathfrak q_{13}'^2$ & $13^4$ & $\mathfrak s_3$ & $3^6$ \\
$L_3$ & $\mathfrak p_{13}^2$ & $13^2$ & & & & \\
$L_2$ & $\mathfrak p_{3}$ & $3^2$ & & & &
\end{tabular}}
\caption{Relative discriminants of the field extensions involved.}
\label{tab:disc}
\end{table}
\medskip
\noindent\emph{\underline{Step 2.}}
A pair of smallest primes in $\mathcal{O}_2$ is $(\mathfrak q_2, \mathfrak q_3)$ of norms 4 and 9, respectively, where $\mathfrak q_2$ is any prime above 2, and $\mathfrak q_3$ is any prime above 3. Consequently, $\lambda_{E,L} = \frac{4^29^2}{13^4}<1$ (cf. \eqref{eqn:constant}), and $\Nm_{\mathbb{Z}[i]}(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i])$ will achieve its smallest possible value
for a unit non-norm element $\gamma \in \mathcal{O}_2^{\times}$, $\gamma \notin \Nm_{E/L_2}(E^\times)$.
\medskip
\noindent\emph{\underline{Step 3.}} To simplify notation, we set, $L_2 = L$, and $\mathcal{O}_2 = \mathcal{O}_{L}$. We prove that the unit $\gamma = \frac{1+i\sqrt 3}{2}\in \mathcal{O}_L^{\times}$ satisfies $\gamma \notin \Nm_{E/L}(E^\times)$.
The prime $\mathfrak q_{13}$ ramifies in the extension $E/L$. Let $\mathfrak t_{13}$ be a prime of $E$ extending $\mathfrak q_{13}$, and $ k = L_{\mathfrak q_{13}}$, $K = E_{\mathfrak t_{13}}$ be the completions of $L$ and $E$ with respect to the corresponding valuations, with $|\overline {k}|=|\overline{K}|=13$.
The local extension $K/k$ is a totally and tamely ramified extension of degree 3. Since the image of the unit $\gamma = \frac{1+i\sqrt 3}{2}$ in the residue field $\mathbb{F}_{13}$ is 4, which has multiplicative order 6, we deduce that ${\gamma}\not \in \langle ~ \overline{\zeta}_{12}^3 \rangle U_{L}^{(1)}$. By \eqref{eqn:nonnorm} and Hasse's Norm Theorem, the theorem follows.
\qed
\end{proof}
\begin{theorem}
\label{thm:res5}
Let $\mathbb{Q}(i) \subset L \subset E$ with $[E:\mathbb{Q}(i)]=6$ and $ [E:L]=2$, and let $(E/L,\sigma,\gamma)$ be a cyclic division algebra. Then,
$\Nm(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i]))\ge2^{6}\cdot 3^{12}\cdot 13^{8}$ with equality for $\gamma = 1+i$ and $E = LL_2$, where $L_2 = \mathbb{Q}(i, i\sqrt{3}) = \mathbb{Q}(\zeta_{12})$ and $L = \mathbb{Q}(i, \beta)$ with $\beta$ a root of the polynomial $f(X) = X^3-(1+i)X^2+5iX-(1+4i)$.
\end{theorem}
\begin{proof}:
\noindent\emph{\underline{Step 1, 2.}}
Let $E$ and $L$ be as in the statement of the theorem. By Theorem~\ref{thm:res4}, the same choice of field $E$ ensures the minimality of the discriminant $\disc(E/\mathbb{Q}(i)) = 3^6\cdot 13^4$ among all possible discriminants of cyclic sextic extensions over $\mathbb{Q}(i)$.
Let $L = \mathbb{Q}(i,\beta)$. A pair of smallest primes in $\mathcal{O}_L$ is $(\mathfrak s_{2}, \mathfrak s_{13})$, of norms $2^3$ and 13, respectively. Consequently, $\lambda_{E,L} = \frac{2^3 13}{3^3} < 1$ (cf. \eqref{eqn:constant}), and the norm $\Nm_{\mathbb{Z}[i]}(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i])$ will attain its smallest possible value for a unit non-norm element $\gamma \in \mathcal{O}_L^{\times}$.
\medskip
\noindent\emph{\underline{Step 3.}}
We encounter here the same situation as in Theorem~\ref{thm:res3}. On the one hand, since $\Nm_{L/\mathbb{Q}}(x)=\Nm_{\mathbb{Q}(i)/\mathbb{Q}}(\Nm_{L/\mathbb{Q}(i)}(x))$ and the norm of every unit in $\mathbb{Z}[i]$ is $1$, every element in $\mathcal{O}_L^{\times}$ maps to a square in the residue field. On the other hand, $[E:L]=2$, and \eqref{eqn:nonnorm} tells us that $\mathcal{O}_L^{\times}\cap \Nm_{E/L}(E^{\times})$ consists of those elements in $\mathcal{O}_L^{\times}$ which map to squares in the residue field of any ramifying prime. Consequently, every element in $\mathcal{O}_L^{\times}$ is in the image of the map $ \Nm_{E/L}$, and there exists no non-norm unit element for $E/L$. We thus need to balance the sizes of $\disc(E/\mathbb{Q}(i))$ and $\gamma$ in \eqref{eqn:disc} to achieve minimality of $\Nm(\disc(\mathcal{O}_{\nat}/\mathbb{Z}[i]))$. We observe that this argument holds for any choice of $E$ and $L$, as long as $\mathbb{Q}(i)\subset L$ and $[E:L] = 2$.
We fix $\gamma = 1+i$ of norm $2^3$, corresponding to the smallest possible norm in $\mathcal{O}_L$. Substituting into \eqref{eqn:balance}, we get $D_{E/L}(\gamma) = 3^6\cdot 13^4\cdot 2^3$. We conclude the proof by showing that $D_{E/L}(\gamma) \le D_{E'/L'}(\gamma')$ for any other choice of $E', L'$ and $\gamma'$ under the given assumptions.
Any possible $E' \ne E$ is of the form $E' = L_2'L$ with $L_2'\ne L_2$ or $E' = L_2'L'$ with $L'\ne L$ ($L_2'$ could be equal to $L_2$) and $[L':\mathbb{Q}(i)] = 3$. In the first case, by the minimality arguments in the choice of $L_2$ and $\gamma$, for all choices of $L_2'$ and $\gamma' \in \mathcal{O}_L \setminus \mathcal{O}_L^{\times}$,
\begin{align*}
3^6\cdot 13^4\cdot 2^3\le \Nm(\disc(L_2'L/\mathbb{Q}(i))) \cdot \Nm(\gamma),
\end{align*}
and so $D_{E/L}(\gamma) \le D_{L_2'L/L}(\gamma')$.
Suppose next that $E' = L_2'L'$ with $L'\ne L$ and $[L':\mathbb{Q}(i)] = 3$. As we saw in Step~1 of the proof of Theorem~\ref{thm:res4}, the conductor of a cubic extension $L'\ne L$ of $\mathbb{Q}(i)$ is ideal $\mathfrak m \in \mathbb{Z}[i]$ of norm greater than 13 and such that $|(\mathbb{Z}[i]/\mathfrak m)^\times|$ is a multiple of 12. By the conductor-discriminant formula, the corresponding extension $L'$ will have discriminant $\mathfrak m^2$, and by the minimality in the choice of $L_2$, $\Nm(\disc (L_2L'/\mathbb{Q}(i))=3^6 \Nm(\mathfrak m)^4\le \disc(L_2L_3' /\mathbb{Q}(i))$ for any quadratic extension $L_2'$ of $\mathbb{Q}(i)$. Consequently,
\begin{align*}
3^6 \Nm(\mathfrak m)^4\cdot 2\le \disc(L_3'L_2 /\mathbb{Q}(i))\cdot \Nm(\gamma') = D_{E'/L'}(\gamma')
\end{align*}
for all choices of $\gamma' \in \mathcal{O}_L'\setminus \mathcal{O}_L^{\times}$.
Now,
\begin{align*}
3^6 \Nm(\mathfrak m)^4\cdot 2\ge 3^6\cdot 13^4\cdot 2^3 \Leftrightarrow \Nm (\mathfrak m) \ge 13\sqrt 2> 13.
\end{align*}
We conclude that $D_{E/L}(\gamma) \le D_{E'/L'}(\gamma')$ for all possible choices of $E', L'$ and $\gamma'$, and the theorem follows.
\qed
\end{proof}
\section{Conclusions}
In this article we have introduced the reader to a technique used in multiple-input multiple-output wireless communications known as space--time coding. Within this framework, we have recalled several design criteria which have been derived in order to ensure a good performance of codes constructed from representations of orders in central simple algebras. In particular, we have explained why it is crucial to choose orders with small discriminants as the underlying algebraic structure in order to maximize the coding gain. While maximal orders achieve the minimal discriminant and hence the maximal coding gain among algebraic space--time codes, we have motivated why in practice it may sometimes be favorable to use the so-called natural orders instead. However, one should bare in mind that orthogonal lattices have yet additional benefits such as simple bit labeling and somewhat simpler encoding and decoding, so there is a natural tradeoff between simplicity and coding gain.
For the base fields $F=\mathbb{Q}$ or $F$ imaginary quadratic (corresponding to the most typical signaling alphabets), and pairs of extension degrees $(n_t,n_r)$ in an asymmetric channel setup, we have computed an explicit number field extension $(E/L)$ and a non-norm element $\gamma \in \mathcal{O}_{L}\setminus\left\{0\right\}$ giving rise to a cyclic division algebra whose natural order $\mathcal{O}_{\nat}$ achieves the minimum discriminant among all cyclic division algebras with the same degree and base field assumptions. This way we have produced explicit space--time codes attaining the optimal coding gain among codes arising from natural orders.
\begin{acknowledgements}
A. Barreal and C. Hollanti are financially supported by the Academy of Finland grants \#276031, \#282938, and \#303819, as well as a grant from the Foundation for Aalto University Science and Technology.
\medskip
The authors thank Jean Martinet, Ren\'e Schoof, and Bharath Sethuraman for their useful suggestions, and the anonymous reviewers for their valuable comments to improve the quality of the manuscript.
\end{acknowledgements}
| {'timestamp': '2017-08-04T02:04:14', 'yymm': '1701', 'arxiv_id': '1701.00915', 'language': 'en', 'url': 'https://arxiv.org/abs/1701.00915'} |
\section{Introduction}
Let $H$ be a separable Hilbert space, denoting its inner product by
the symbol $( \ , \ )$ which is conjugate-linear in its first
coordinate and linear in its second. A result of Wigner in
\cite{wigner} shows that every weakly continuous one-parameter group
of $*$-automorphisms $\{\alpha_t\}_{t \in \R}$ of $B(H)$ is
implemented by a strongly continuous unitary group $\{U_t\}_{t \in
\R}$ in that $\alpha_t(A)=U_tAU_t^*$ for all $A \in B(H)$ and $t \in
\R$. This leads us to pursue the more general task of classifying
all suitable semigroups of $*$-endomorphisms of $B(H)$:
\begin{defn}
We say a family $\{\alpha_t\}_{t \geq 0}$ of $*$-endomorphisms of
$B(H)$ is an \emph{$E_0$-semigroup} if:
\begin{enumerate}
\item $\alpha_{s+t}=\alpha_s \circ \alpha_t$ for all $s, t \geq 0$,
and $\alpha_0 (A)=A$ for all $A \in B(H)$.
\item For each $f, g \in H$ and $A \in B(H)$, the inner product
$( f, \alpha_t (A) g)$ is continuous in $t$.
\item $\alpha_t (I)=I$ for all $t \geq 0$ (in other words, $\alpha$ is
\emph{unital}).
\end{enumerate}
\end{defn}
We have two different notions of what it means for two
$E_0$-semigroups to be the same, namely conjugacy and cocycle
conjugacy, the latter of which arises from Alain Connes' definition
of outer conjugacy.
\begin{defn} Let $\alpha$ and $\beta$ be $E_0$-semigroups on $B(H_1)$ and
$B(H_2)$, respectively. We say that $\alpha$ and $\beta$ are
\emph{conjugate} if there is a $*$-isomorphism $\theta$ from
$B(H_1)$ onto $B(H_2)$ such that $\theta \circ \alpha_t = \beta_t
\circ \theta$ for all $t \geq 0$. We say that $\alpha$ and $\beta$
are \emph{cocycle conjugate} if $\alpha$ is conjugate to $\beta'$,
where $\beta'$ is an $E_0$-semigroup on $B(H_2)$ satisfying the
following condition: For some strongly continuous family of
unitaries $U=\{U_t: t \geq 0\}$ acting on $H_2$ and satisfying
$U_{t+s}=U_t \beta_t(U_s)$ for all $s, t \geq 0$, we have $\beta'_t
(A) = U_t \beta_t (A) U_t^*$ for all $A \in B(H_2)$ and $t \geq 0$.
Such a family of unitaries is called a \emph{unitary cocycle} for
$\beta$.
\end{defn}
$E_0$-semigroups are divided into three types based upon the
existence, and structure of, their units. More specifically, let
$\alpha$ be an $E_0$-semigroup on $B(H)$. A \textit{unit} for
$\alpha$ is a strongly continuous semigroup of bounded operators
$U=\{U(t): t \geq 0\}$ such that $\alpha_t(A)U(t) = U(t)A$ for all
$A \in B(H)$. Let $\mathcal{U}_\alpha$ be the set of all units for
$\alpha$. We say $\alpha$ is \textit{spatial} if $\mathcal{U}_\alpha
\neq \emptyset$, while we say that $\alpha$ is \textit{completely
spatial} if, for each $t \geq 0$, the closed linear span of the set
$\{ U_{1}(t_1) \cdots U_n(t_n) f: f \in H, t_i \geq 0 \textrm{ and }
U_i \in \mathcal{U}_\alpha \ \forall \ i, \sum t_i = t \}$ is $H$.
If an $E_0$-semigroup $\alpha$ is completely spatial, we say
it is of type I. If $\alpha$ is spatial but is not completely
spatial, we say $\alpha$ is of type II. If $\alpha$ has no units, we say
it is of type III.
If $\alpha$ is of type I or II, we may further assign an integer $n
\in \mathbb{Z}_{\geq 0} \cup \{ \infty \}$ to $\alpha$, in which
case we say $\alpha$ is of type I$_n$ or II$_n$. We call $n$ the
index of $\alpha$. It was initially defined in different ways in
\cite{powersdiff} and \cite{arvindex}, and the connection between
these definitions was explored in \cite{powersprice}. The index of
$\alpha$ is the dimension of a particular Hilbert space associated
to its units, and it is perhaps the most fundamental cocycle
conjugacy invariant for spatial $E_0$-semigroups. Arveson showed in
\cite{arvindex} that the type I $E_0$-semigroups are entirely
classified (up to cocycle conjugacy) by their index: the type I$_0$
$E_0$-semigroups are semigroups of $*$-automorphisms, while for $n
\in \mathbb{N} \cup \{\infty\}$, every type I$_n$ $E_0$-semigroup is
cocycle conjugate to the CAR flow of rank $n$.
However, at the present time, we do not have such a classification
for those of type II or III. The first type II and type III
examples were constructed by Powers in \cite{typeII} and
\cite{typeIII}. Through Arveson's theory of
product systems, Tsirelson became the first to exhibit
uncountably many mutually non-cocycle conjugate $E_0$-semigroups
of types II and III (see \cite{T2}). A dilation theorem of Bhat in \cite{Bhat} shows
that every unital $CP$-flow $\alpha$ can be dilated to an
$E_0$-semigroup, and that there is a minimal dilation $\alpha^d$ of
$\alpha$ which is unique up to conjugacy. Using Bhat's result,
Powers proved in \cite{hugepaper} that every spatial $E_0$-semigroup
can be obtained from the boundary weight map of a $CP$-flow over a
separable Hilbert space $K$. In \cite{bigpaper}, he constructed
spatial $E_0$-semigroups using boundary weights over $K$ when
$dim(K)=1$ and then began to investigate the case when $dim(K)=2$.
Our goal is to use boundary weight maps to induce unital $CP$-flows
over $K$ for $1<dim(K)<\infty$ and to classify their minimal
dilations to $E_0$-semigroups up to cocycle conjugacy. To do so, we
define a natural boundary weight map $\rho \rightarrow \omega
(\rho)$ using a unital completely positive map $\phi$ and a
normalized boundary weight $\nu$ over $L^2(0, \infty)$. The
necessary and sufficient condition that this map induce a unital
$CP$-flow $\alpha$ is that $\phi$ satisfies a definition of
$q$-positive analogous to that from \cite{hugepaper} (see Definition
\ref{qpos} and Proposition \ref{bdryweight}), in which case we say
that $\alpha$ is the $CP$-flow induced by the boundary weight double
$(\phi, \nu)$. We develop a comparison theory for boundary weight
doubles $(\phi, \nu)$ and $(\psi, \nu)$ ($\phi$ and $\psi$ unital)
in the case that $\nu$ is a normalized unbounded boundary weight
over $L^2(0, \infty)$ of the form $\nu(\sqrt{I - \Lambda(1)} B
\sqrt{I - \Lambda(1)}) = (f,Bf)$, finding that the doubles induce
cocycle conjugate $E_0$-semigroups if and only if there is a
hyper maximal $q$-corner from $\phi$ to $\psi$ (see Definition
\ref{hyp} and Proposition \ref{hypqc}).
The problem of determining hyper maximal $q$-corners from $\phi$ to
$\psi$ becomes much easier if we focus on a particular class of
$q$-positive maps, called the $q$-pure maps, which have the least
possible $q$-subordinates (Definition \ref{qpure}). Given a
$q$-positive map $\phi$ acting on $M_n(\C)$ and a unitary $U \in
M_n(\C)$, we can form a new map $\phi_U$ by
$\phi_U(A)=U^*\phi(UAU^*)U$. We describe the order isomorphism
between the $q$-subordinates of $\phi$ and those of $\phi_U$, which
in turn leads to the existence of a hyper maximal $q$-corner from
$\phi$ to $\phi_U$ if $\phi$ is unital and $q$-pure (Proposition
\ref{basischange}). With this result in mind, we begin the task of
classifying the unital $q$-pure maps. We find that the rank one
unital $q$-pure maps $\phi: M_n(\C) \rightarrow M_n(\C)$ are
precisely the maps $\phi(A)=\rho(A)I$ for faithful states $\rho$ on
$M_n(\C)$ (Proposition \ref{littlestates}). That these maps give us
an enormous class of mutually non-cocycle conjugate $E_0$-semigroups
in one of our main results (Theorem \ref{statesbig}). Furthermore,
for $n>1$, none of the $E_0$-semigroups constructed from boundary
weight doubles satisfying the conditions of Theorem \ref{statesbig} are cocycle
conjugate to any of the $E_0$-semigroups obtained from
one-dimensional boundary weights by Powers in \cite{bigpaper} (Corollary
\ref{thesearenew}).
We turn our attention to the unital $q$-pure maps that are
invertible. These maps are best understood through their
(conditionally negative) inverses. In Theorem \ref{phiu}, we find a
necessary and sufficient condition for an invertible unital map
$\phi$ on $M_n(\C)$ to be $q$-pure. In this case, however, if $\nu$
is a normalized unbounded boundary weight of the form $\nu(\sqrt{I -
\Lambda(1)} B \sqrt{I - \Lambda(1)}) = (f,Bf)$, then the
$E_0$-semigroup induced by the boundary weight double $(\phi, \nu)$
is entirely determined by $\nu$. This $E_0$-semigroup is the one
induced by $\nu$ in the sense of \cite{bigpaper}.
\section{Background}
\subsection{Completely positive maps}
Let $\phi: \mathfrak{U} \rightarrow \mathfrak{B}$ be a linear map
between $C^*$-algebras. For each $n \in \mathbb{N}$, define
$\phi_n: M_n(\mathfrak{U}) \rightarrow M_n(\mathfrak{B})$ by
\begin{displaymath} \phi_n \left(\begin{array}{ccc} A_{11} & \cdots & A_{1n}
\\ \vdots & \ddots & \vdots \\ A_{n1} & \cdots & A_{nn} \end{array}\right)
= \left(\begin{array}{ccc} \phi(A_{11}) & \cdots & \phi(A_{1n})
\\ \vdots & \ddots & \vdots \\ \phi(A_{n1}) & \cdots & \phi(A_{nn})
\end{array} \right). \end{displaymath}
We say that $\phi$ is completely positive if $\phi_n$ is positive
for all $n \in \mathbb{N}$. A linear map $\phi: B(H_1)
\rightarrow B(H_2)$ is completely positive if and only if for all
$A_1, \ldots A_n \in B(H_1)$, $f_1, \ldots, f_n \in H_2$, and $n \in
\mathbb{N}$, we have
$$\sum_{i,j=1}^n (f_i, \phi(A_i^*A_j)f_j) \geq 0.$$
Stinespring's Theorem asserts that if
$\mathfrak{U}$ is a unital $C^*$-algebra and $\phi: \mathfrak{U}
\rightarrow B(H)$ is a unital completely positive map, then $\phi$
dilates to a $*$-homomorphism in that there is a Hilbert space $K$,
a $*$-homomorphism $\pi: \mathfrak{U} \rightarrow B(K)$, and an
isometry $V: H \rightarrow K$ such that
$$\phi(A)=V^*\pi(A)V$$ for all $A \in \mathfrak{U}$.
From the work of Choi (\cite{choi}) and Arveson (\cite{arveson}),
we know that a normal linear map $\phi: B(H_1)
\rightarrow B(H_2)$ is completely positive if and only if it can be
written in the form
$$\phi(A) = \sum_{i=1}^n S_i A S_i^*$$ for
some $n \in \mathbb{N} \cup \{\infty\}$
and maps $S_i: H_1
\rightarrow H_2$ which are linearly independent over
$\ell_2(\mathbb{N})$ in the sense that if $\sum_{i=1}^{r \leq n} z_i S_i = 0$
for a sequence $\{z_i\}_{i=1}^r \in
\ell_2(\mathbb{N})$, then $z_i=0$ for all $i$. With these
hypotheses satisfied, the number $n$ is unique. We will use the
above conditions for complete positivity interchangeably.
\subsection{Conditionally negative maps}
We say a self-adjoint linear map $\psi: B(K) \rightarrow B(K)$ is
\emph{conditionally negative} if, whenever $\sum_{i=1}^m
A_if_i=0$ for $A_1, \ldots, A_m \in B(K)$, $f_1, \ldots, f_m \in K$,
and $m \in \mathbb{N}$, we have $\sum_{i=1}^m (f_i,
\psi(A_i^*A_j)f_j) \leq 0$. If $K=\C^n$, then from the literature
(see, for example, Theorem 3.1 of \cite{pownxn}) we
know that $\psi$ has the form
$$\psi(A) = sA + YA + AY^* - \sum_{i=1}^p \lambda_i S_iAS_i^*,$$ where $s
\in \R$, $tr(Y)=0$, and for all $i$ and $j$ we have $\lambda_i
>0$, $tr(S_i)=0$ and $tr(S_i^*S_j)=n \delta_{ij}$, where
$p \leq n^2$ is independent of the maps $S_i$.
This form for $\psi$ is unique in the sense that if $\psi$ is
written in the form
$$\psi(A)= tA + ZA +AZ^* - \sum_{i=1}^p \mu_i T_i A T_i^*,$$ where $t \in
\R$, $tr(Z)=0$, and for all $i$ and $j$ we have $\mu_i >0$,
$tr(T_i)=0$, and $tr(T_i^*T_j)= n \delta_{ij}$, then $s=t$, $Z=Y$,
and $\sum_{i=1}^p \lambda_i S_iAS_i^*= \sum_{i=1}^p \mu_i T_i A
T_i^*$ for all $A \in M_n(\C)$. Indeed, let $\{v_k\}_{k=1}^n$ be any
orthonormal basis for $\C^n$, let $h_k = v_k / \sqrt{n}$ for each
$k$, let $f \in \C^n$ be arbitrary, and for $k=1, \ldots, n,$ define
$A_k \in M_n(\C)$ by $A_k = f h_k^*$. Using the trace conditions, we
find
\begin{eqnarray*} \sum_{k=1}^n \psi(A_k)h_k & = & \sum_{k=1}^n (h_k, h_k)sf
+ \sum_{k=1}^n (h_k, h_k)Yf + \sum_{k=1}^n (h_k, Y^* h_k)f \\
& \ & - \sum_{k=1}^n \Big(\sum_{i=1}^p \lambda_i(h_k, S_i^*h_k)S_i
f\Big)
\\
&= & sf + Yf + 0 - \sum_{i=1}^p \Big( \sum_{k=1}^n \lambda_i(h_k,
S_i^* h_k)
S_i f \Big) \\
& = & sf + Yf - \sum_{i=1}^p \lambda_i (0) S_i f = sf + Yf.
\end{eqnarray*}
An analogous computation shows that $\sum_{k=1}^n \psi(A_k)h_k = tf
+ Zf$. Since $f \in \C^n$ was arbitrary, we conclude $(t-s)I= Y-Z$.
Therefore, $tr((t-s)I) = tr(Y-Z)=0$, so $t=s$ and $Y=Z$.
Consequently, $\sum_{i=1}^p \lambda_i S_iAS_i^*= \sum_{i=1}^p \mu_i
T_i A T_i^*$ for all $A \in M_n(\C)$.
\subsection{$CP$-flows and Bhat's theorem} Let $K$ be a separable Hilbert
space and let $H= K \otimes L^2(0, \infty)$. We identify $H$ with
$L^2((0, \infty); K)$, the space of $K$-valued measurable functions
on $(0, \infty)$ which are square integrable. Under this
identification, the inner product on $H$ is
$$(f,g) = \int_0 ^\infty (f(x), g(x)) dx.$$ Let
$U=\{U_t\}_{t \geq 0}$ be the right shift semigroup on $H$, so for
all $t \geq 0$ and $f \in H$ we have $(U_t
f)(x) = f(x-t)$ for $x>t$ and $(U_t f)(x)=0$ otherwise. Let
$\Lambda: B(K) \rightarrow B(H)$ be the map defined by
$(\Lambda(A)f)(x)= e^{-x}Af(x)$ for all $A \in B(K), f \in H$.
\begin{defn} Assume the above notation.
A strongly continuous semigroup $\alpha=\{\alpha_t: t \geq 0\}$ of
completely positive contractions of $B(H)$ into itself is a
\textit{CP-flow} if $\alpha_t(A)U_t = U_tA$ for all $A \in B(H)$.
\end{defn}
A theorem of Bhat in \cite{Bhat} allows us to generate
$E_0$-semigroups from unital $CP$-flows, and, more generally, from
strongly continuous completely positive semigroups of unital maps on
$B(H)$, called $CP$-semigroups. We give a reformulation of Bhat's
theorem (see Theorem 2.1 of \cite{bigpaper}):
\begin{thm}\label{dilation} Suppose $\alpha$ is a unital $CP$-semigroup of $B(H_1)$.
Then there is an $E_0$-semigroup $\alpha^d$ of $B(H_2)$ and an
isometry $W: H_1 \rightarrow H_2$ such that
$$\alpha_t(A)=W^*\alpha_t ^d(WAW^*)W$$ and $\alpha_t(WW^*) \geq
WW^*$ for all $t > 0$. If the projection $E=WW^*$ is minimal in
that the closed linear span of the vectors
$$\alpha_{t_1}^d(EA_1E) \cdots \alpha_{t_n}^d(EA_nE)Ef$$ for $f \in
K, A_i \in B(H_1)$ and $t_i \geq 0$ for all $i=1, 2, \ldots, n$ and
$n=1, 2, \ldots$ is $H_2$,
then $\alpha^d$ is unique up to conjugacy.
\end{thm}
In \cite{hugepaper}, Powers showed that every spatial
$E_0$-semigroup acting on $B(\mathfrak{H})$ (for $\mathfrak{H}$ a
separable Hilbert space) is cocycle conjugate to an $E_0$-semigroup
which is a $CP$-flow, and that every $CP$-flow over $K$ arises from
a \textit{boundary weight map} over $H=K \otimes L^2(0, \infty)$.
The boundary weight map $\rho \rightarrow \omega (\rho)$ of a
$CP$-flow $\alpha$ associates to every $\rho \in B(K)_*$ a boundary
weight, that is, a linear functional $\omega(\rho)$ acting on the
null boundary algebra
$$\mathfrak{A}(H)= \sqrt{I_H - \Lambda(I_K)}B(H)\sqrt{I_H-\Lambda(I_K)}
$$ which is normal in the following sense: If we define a linear
functional $\ell(\rho)$ on $B(H)$ by
$$\ell(\rho)(A)=\omega(\rho)\Big(\sqrt{I_H-\Lambda(I_K)}A \sqrt{I_H -
\Lambda(I_K)}\Big),$$ then $\ell(\rho) \in B(H)_*$. If
$\omega(\rho)(I_H-\Lambda(I_K)) = \rho(I_K)$ for all $\rho \in
B(K)_*$, then $\alpha$ is unital. For the sake of neatness, we will
omit the subscripts $H$ and $K$ from the previous sentence when
they are clear. Let $\delta$ be the generator of $\alpha$, and define $\Gamma:
B(H) \rightarrow B(H)$ by $\Gamma(A)= \int_0 ^ \infty e^{-t}U_t AU_t^*$.
The resolvent $R_\alpha:= (I - \delta)^{-1}$ of $\alpha$ satisfies $R_\alpha(A) = \int_0 ^\infty e^{-t}\alpha_t(A) dt$
for all $A \in B(H)$.
Its associated predual map $\hat{R}_\alpha$ is given by
\begin{equation}\label{resolvent}
\hat{R}_\alpha(\eta) = \hat{\Gamma}(\omega(\hat{\Lambda}\eta) + \eta)
\end{equation}
for all $\eta \in B(H)_*$.
A $CP$-flow $\alpha$ over $K$ is entirely determined by a set of normal
completely positive contractions $\pi^{\#}=\{\pi^{\#}_t: t > 0\}$ from
$B(H)$ into $B(K)$,
called the \textit{generalized boundary representation} of $\alpha$.
Its relationship to the boundary weight map is as follows. For
each $t>0$, denote by $\hat{\pi}_t: B(K)_* \rightarrow B(H)_*$
the predual map induced by $\pi_t^\#$. For the
truncated boundary weight maps $\rho \rightarrow \omega_t(\rho) \in
B(H)_*$ defined by
\begin{eqnarray}\label{truncate} \omega_t(\rho)(A)=
\omega(\rho)\Big(U_tU_t^*AU_tU_t^*\Big),\end{eqnarray} we have
$\hat{\pi}_t=\omega_t(I + \hat{\Lambda}\omega_t)^{-1}$ and $\omega_t
= \hat{\pi}_t(I - \hat{\Lambda} \hat{\pi}_t)^{-1}$ for all $t>0$.
The maps $\{\pi_b^\#\}_{b>0}$ have a $\sigma$-strong limit
$\pi_0^\#$ as $b \rightarrow 0$ for each $A \in \bigcup_{t>0} U_t
B(H) U_t^*$, called the \textit{normal spine} of $\alpha$. If
$\alpha$ is unital, then the index of $\alpha^d$ as an
$E_0$-semigroup is equal to the rank of $\pi_0^\#$ as a completely
positive map (Theorem 4.49 of \cite{hugepaper}).
Having seen that every $CP$-flow has an associated boundary weight map, we
would like to approach the situation from the opposite direction.
More specifically, under what conditions is a map
$\rho \rightarrow \omega(\rho)$ from $B(K)_*$ to weights acting on $\mathfrak{A}(H)$ the boundary
weight map of a $CP$-flow over $K$? Powers has found the answer (see Theorem 3.3
of \cite{bigpaper}):
\begin{thm}\label{powersthm}
If $\rho \rightarrow \omega(\rho)$ is a completely positive mapping
from $B(K)_*$ into weights on $B(H)$ satisfying
$\omega(\rho)(I-\Lambda(I_K)) \leq \rho(I_K)$ for all positive
$\rho \in B(K)_*$, and if the maps $\hat{\pi}_t : = \omega_t(I +
\hat{\Lambda}\omega_t)^{-1}$ are completely positive contractions
from $B(K)_*$ into $B(H)_*$ for all $t >0$, then $\rho \rightarrow
\omega(\rho)$ is the boundary weight map of a $CP$-flow over $K$.
The $CP$-flow is unital if and only if $\omega(\rho)(I-\Lambda(I_K))
= \rho(I_K)$ for all $\rho \in B(K)_*$.
\end{thm}
If $dim(K)=1$, the boundary weight map is just $c \in \C \rightarrow
\omega(c) = c \omega(1)$, so we may view our boundary weight map as
a single positive boundary weight $\omega:= \omega(1)$ acting on
$\mathfrak{A}(L^2(0, \infty))$. Since the functional $\ell$ defined
on $B(H)$ by $$\ell(A)=\omega\Big(\sqrt{I - \Lambda(1)} A \sqrt{I -
\Lambda(1)}\Big)$$ is positive and normal, it has the form $\ell(A)
= \sum_{k=1}^n (f_k, A f_k)$ for some mutually orthogonal vectors
$\{f_k\}_{k=1}^{n \in \mathbb{N} \cup \{\infty\}}$, so
$$\omega \Big(\sqrt{I - \Lambda(1)}A\sqrt{I - \Lambda(1)}\Big) =
\sum_{k=1}^n (f_k, A f_k)$$ for all $A \in B(H)$. If $\omega$ is
\textit{normalized} (that is, $\omega(I-\Lambda(1))=1$), then
$\sum_{k=1}^n ||f_k||^2 = 1$. In \cite{bigpaper}, Powers induced
$E_0$-semigroups using normalized boundary weights over $L^2(0,
\infty)$. The \ type \ of \ $E_0$-semigroup \ $\alpha^d$
\ induced \ by \ a \
normalized \ boundary \ weight $\omega(\sqrt{I - \Lambda(1)}A \sqrt{I -
\Lambda(1)}) = \sum_{k=1}^n (f_k, A f_k)$ depends on whether
$\omega$ is bounded in the sense that for some $r>0$ we have
$|\omega(B)| \leq r ||B||$ for all $B \in \mathfrak{A}(H)$. Results
from \cite{hugepaper} imply that $\alpha^d$ is of type I$_n$ if
$\omega$ is bounded and of type II$_0$ if $\omega$ is unbounded. If
$\omega$ is unbounded, then both $\omega_t(I)$ and $\omega_t(\Lambda(1))$ approach infinity
as $t$ approaches zero. We will focus on normalized unbounded boundary weights over $L^2(0,
\infty)$ of the form $\omega(\sqrt{I - \Lambda(1)}A \sqrt{I -
\Lambda(1)})=(f,Af).$ We note that, as discussed in detail in
\cite{markie}, such boundary weights are not normal weights.
If $\alpha$ and $\beta$ are $CP$-flows, we say that $\alpha \geq \beta$ if $\alpha_t
- \beta_t$ is completely positive for all $t \geq 0$. The subordinates of a
$CP$-flow are entirely determined by the
subordinates of its generalized boundary representation (see Theorem 3.4
of \cite{bigpaper}):
\begin{thm} \label{bdryrep} Let $\alpha$ and $\beta$ be $CP$-flows over $K$ with
generalized boundary representations $\pi^\#=\{\pi^{\#}_t\}$ and
$\xi^\# =\{\xi^{\#}_t\}$, respectively. Then $\beta$ is subordinate
to $\alpha$ if and only if
$\pi^{\#}_t - \xi^{\#}_t$ is completely positive for all $t>0$.
\end{thm}
Given
two unital $CP$-flows $\alpha$ and $\beta$, it is natural to ask when their
minimally dilated $E_0$-semigroups are cocycle conjugate. The following definition
from \cite{hugepaper} provides us with a key:
\begin{defn}
Let $\alpha$ and $\beta$ be $CP$-flows over $K_1$ and $K_2$, respectively, where
$H_1 = K_1 \otimes L^2(0, \infty)$ and $H_2= K_2 \otimes L^2(0, \infty)$. We say
that a family of linear maps
$\gamma=\{\gamma_t: t \geq 0\}$ from $B(H_2, H_1)$ into itself is a
flow corner from $\alpha$ to $\beta$ if the family of maps
$\Theta=\{\Theta_t: t \geq 0\}$ defined by
\begin{displaymath} \Theta_t \left( \begin{array}{cc} A_{11} & A_{12} \\
A_{21} & A_{22} \\
\end{array} \right)=
\left( \begin{array}{cc} \alpha_t(A_{11}) & \gamma_t(A_{12}) \\
\gamma_t^*(A_{21}) & \beta_t(A_{22}) \\
\end{array} \right)
\end{displaymath} is a $CP$-flow over $K_1 \oplus K_2$.
If $\gamma$ is a flow corner from $\alpha$ to $\beta$, we consider
subordinates $\Theta'=\{\Theta_t': t \geq 0\}$ of $\Theta$ that are $CP$-flows of the form
\begin{displaymath} \Theta_t' \left( \begin{array}{cc} A_{11} & A_{12} \\
A_{21} & A_{22} \\
\end{array} \right):=
\left( \begin{array}{cc} \alpha'_t(A_{11}) & \gamma_t(A_{12}) \\
\gamma_t^*(A_{21}) & \beta'_t(A_{22}) \\
\end{array} \right).
\end{displaymath} We say that $\gamma$ is a hyper
maximal flow corner from $\alpha$ to $\beta$ if, for every such
subordinate $\Theta'$ of $\Theta$, we have $\alpha=\alpha'$ and
$\beta=\beta'$.
\end{defn}
Our results will involve type II$_0$ $E_0$-semigroups. These are spatial
$E_0$-semigroups which are not semigroups of $*$-automorphisms
and have only one unit $V=\{V_t\}_{t \geq 0}$ up to scaling by $e^{t \lambda}$ for
$\lambda \in \C$.
In the case that unital $CP$-flows $\alpha$ and $\beta$
minimally dilate to type II$_0$ $E_0$-semigroups, we have a necessary and
sufficient condition for $\alpha^d$ and $\beta^d$ to be cocycle
conjugate (Theorem 4.56 of \cite{hugepaper}):
\begin{thm}\label{hyperflowcorn} Suppose $\alpha$ and $\beta$ are unital
$CP$-flows over
$K_1$ and $K_2$ and $\alpha^d$ and $\beta^d$ are their minimal dilations
to $E_0$-semigroups. Suppose $\gamma$ is a hyper maximal flow corner from $\alpha$
to $\beta$. Then $\alpha^d$ and $\beta^d$ are cocycle conjugate. Conversely,
if $\alpha^d$ is a type II$_0$ and $\alpha^d$ and $\beta^d$ are cocycle conjugate,
then there is a hyper maximal flow corner from $\alpha$ to $\beta$.
\end{thm}
We will later use this theorem
to determine a necessary and sufficient condition for some of the
$E_0$-semigroups we construct to be cocycle conjugate (see Definition \ref{hyp} and
Proposition \ref{hypqc}).
\section{Our boundary weight map}
Recall that a completely positive linear map $\phi$ can have
negative eigenvalues. Moreover, even if $I + t \phi$ is invertible
for a given $t$, it does not necessarily follow that $\phi(I + t
\phi)^{-1}$ is completely positive. In our boundary weight
construction, we will require a special kind of completely positive
map:
\begin{defn}\label{qpos} A linear map $\phi: M_n(\C) \rightarrow
M_n(\C)$ is
\emph{$q$-positive} if $\phi$ has no negative eigenvalues and
$\phi(I + t \phi)^{-1}$ is completely
positive for all $t \geq 0$.
\end{defn}
Henceforth, we naturally identify a finite-dimensional Hilbert space
$K$ with $\C^n$ and $B(K \otimes L^2(0, \infty))$ with $M_n(B(L^2(0,
\infty)))$. Under these identifications, the right shift $t$ units
on $K \otimes L^2(0, \infty)$ is the matrix whose $ij$th entry is
$\delta_{ij} V_t$ for $V_t$ the right shift on $L^2(0, \infty)$. The
map $\Lambda_{n \times n}: B(K) \rightarrow B(K \otimes L^2(0,
\infty))$ sends an $n \times n$ matrix $B=(b_{ij}) \in M_n(\C)$ to
the matrix $\Lambda_{n \times n}(B)$ whose $ij$th entry
is $b_{ij} \Lambda(1) \in B(L^2(0, \infty))$. The null boundary
algebra $\mathfrak{A}(H)$ is simply $M_n (\mathfrak{A}(L^2(0,
\infty)))$.
Given a boundary weight $\nu$ over $L^2(0, \infty)$, we write
$\Omega_{\nu, n \times k}$ for the map that sends an $n \times k$
matrix $A=(A_{ij}) \in M_{n \times k}(\mathfrak{A}(L^2(0, \infty)))$
to the matrix $\Omega_{\nu, n \times k} (A) \in M_{n \times k}(\C)$
whose $ij$th entry is $\nu(A_{ij})$. We will suppress the
integers $n$ and $k$ when they are clear, writing the above maps as
$\Omega_\nu$ and $\Lambda$. In the proposition and corollary that
follow, we show how to construct a $CP$-flow using a $q$-positive
map $\phi: M_n(\C) \rightarrow M_n(\C)$, a normalized boundary
weight $\nu$ over $L^2(0, \infty)$, and the map $\Omega_\nu:=
\Omega_{\nu, n \times n}: \mathfrak{A}(H) \rightarrow M_n(\C)$. The
map $\Omega_\nu$ is completely positive since $\nu$ is positive.
\begin{prop} \label{bdryweight} Let $H =\C^n \otimes L^2(0, \infty)$.
Let $\phi: M_n(\C) \rightarrow M_n(\C)$ be a unital completely
positive map with no negative eigenvalues, and let $\nu$ be a normalized unbounded boundary
weight over $L^2(0, \infty)$. Then the map $\rho \rightarrow
\omega(\rho)$ from $M_n(\C)^*$ into boundary weights on
$\mathfrak{A}(H)$ defined by
$$\omega (\rho) (A) = \rho(\phi(\Omega_\nu(A))).$$
is completely positive. Furthermore, the maps $\hat{\pi}_t:= \omega_t (I + \hat{\Lambda}\omega_t)^{-1}$
define normal completely positive contractions $\pi_t ^\#$ of $B(H)$ into $M_n(\C)$
for all $t
> 0$ if and only if $\phi$ is $q$-positive.
\end{prop}
\begin{pf}
The map $\rho \rightarrow \omega(\rho)$ is completely positive since
it is the composition of two completely positive maps. Before
proving either direction, we let $s_t= \nu_t(\Lambda(1))$
for all $t>0$ and prove the equality
\begin{eqnarray} \label{pi} \hat{\pi}_t(\rho) = \rho \Big(\phi(I + s_t \phi)^{-1}
\Omega_{\nu_t}\Big) \end{eqnarray} for all $\rho \in M_n(\C)^*$. Denoting by $U_t$
the right shift on $H$ for every $t>0$, we claim
that $(I+\hat{\Lambda}\omega_t)^{-1} = (I + s_t \hat{\phi})^{-1}$.
Indeed, for arbitrary $t>0$, $B \in M_n(\C)$, and $\rho \in
M_n(\C)^*$, we have
$$\hat{\Lambda}\omega_t(\rho)(B)=
\rho\Big(\phi\Big(\Omega_\nu(U_tU_t^*\Lambda(B)U_tU_t^*)\Big)\Big)=
\rho\Big(\phi\Big(\Omega_{\nu_t}\Big(\Lambda(B)\Big)\Big)\Big)=s_t \rho(\phi(B)),$$
hence $\hat{\Lambda}\omega_t = s_t \hat{\phi}$ and
$(I+\hat{\Lambda}\omega_t)^{-1} = (I + s_t \hat{\phi})^{-1}$.
For any $t>0$ and $A \in B(H)$, we have
\begin{eqnarray*} \hat{\pi}_t(\rho)(A) & = &\omega_t
(I + \hat{\Lambda}\omega_t)^{-1}(\rho)(A) = \Big((I +
\hat{\Lambda}\omega_t)^{-1}(\rho)\Big)(\phi(\Omega_{\nu_t}(A)))
\\ & = & \Big((I + s_t \hat{\phi})^{-1}(\rho)\Big)(\phi(\Omega_{\nu_t}(A)))
= \rho\Big((I + s_t \phi)^{-1} \phi(\Omega_{\nu_t}(A))\Big)\\ & = &
\rho\Big(\phi(I + s_t \phi)^{-1}(\Omega_{\nu_t}(A))\Big),
\end{eqnarray*} establishing \eqref{pi}.
Assume the hypotheses of the backward direction and let $t>0$. By construction,
$\hat{\pi}_t$ maps $M_n(\C)^*$ into $B(H)_*$. It is also a contraction,
since for all $\rho \in M_n(\C)^*$ we have
\begin{eqnarray*} ||\hat{\pi}_t(\rho)|| & = & \Big| \Big|\rho\Big(\phi(I +
s_t\phi)^{-1}\Omega_{\nu_t}\Big)\Big| \Big| \leq ||\rho|| \ ||\phi(I + s_t \phi)^{-1}
\Omega_{\nu_t}|| \\ & = & ||\rho|| \ ||\phi(I + s_t \phi)^{-1}
\Omega_{\nu_t}(I)||
= ||\rho|| \ \Big|\Big| \phi(I + s_t \phi)^{-1}\Big(\nu_t(I)I_{\C^n}\Big)
\Big|\Big| \\ & = & ||\rho|| \Big| \Big|\frac{\nu_t(I)}{1+s_t} I_{\C^n} \Big|\Big| =
||\rho || \frac{\nu_t(I)}{1+\nu_t(\Lambda(1))}\leq ||\rho||,
\end{eqnarray*}
where the last inequality follows from the fact that
$$\nu_t(I-\Lambda(1)) \leq
\nu(I-\Lambda(1)) = 1.$$ Therefore, for every $t>0$, $\hat{\pi}_t$ defines
a normal contraction $\pi_t ^\#$ from $B(H)$ into $M_n(\C)$
satisfying $\hat{\pi}_t(\rho) = \rho \circ \pi_t ^\#$ for all $\rho \in M_n(\C)^*$.
From Eq.\eqref{pi}
we see $\pi_t ^\# = \phi(I + s_t \phi)^{-1} \Omega_{\nu_t}$, so $\pi_t^\#$ is
the composition of completely positive maps and is thus completely positive for all $t>0$.
Now assume the hypotheses of the forward direction. By
unboundedness of $\nu$, the (monotonically decreasing) values
$\{s_t\}_{t>0}$ form a set equal to either $(0, \infty)$ or $[0,
\infty)$. Choose any $t>0$ such that $s_t>0$. Let $T \in B(H)$ be
the matrix with $ij$th entry $(1/\nu_t(I))I$, and let
$\kappa_t: M_n(\C) \rightarrow B(H)$ be the map that sends
$B=(b_{ij}) \in M_n(\C)$ to the matrix $\kappa_t(B) \in B(H)$ whose
$ij$th entry is $(b_{ij}/\nu_t(I)) I$. We note that
$\kappa_t$ is the Schur product $B \rightarrow B \cdot T$, which is
completely positive since $T$ is positive. For all $B \in M_n(\C)$,
we have
$$\phi(I + s_t \phi)^{-1}(B)= \pi^\#_t (\kappa_t(B)),$$
so $\phi(I + s_t \phi)^{-1}$ is the composition of completely
positive maps and is thus completely positive. As noted above, the
values $\{s_t\}_{t>0}$ span $(0, \infty)$, so $\phi$ is
$q$-positive. \qed \end{pf}
\begin{cor}\label{bdryweightcor} The map $\rho \rightarrow
\omega(\rho)$ in Proposition \ref{bdryweight} is the boundary weight
map of a unital $CP$-flow $\alpha$ over $\C^n$, and the Bhat minimal
dilation $\alpha^d$ of $\alpha$ is a type II$_0$ $E_0$-semigroup.
\end{cor}
\begin{pf} The first claim of the corollary follows immediately
from Theorem \ref{powersthm} and Proposition \ref{bdryweight} since
\begin{equation}\label{tech} \omega(\rho)(I-\Lambda(I_{\C^n}))=\rho(\phi(I_{\C^n}))
=\rho(I_{\C^n})
\end{equation} for all $\rho \in M_n(\C)^*$. For the second
assertion, we note that by Theorem 4.49 of \cite{hugepaper}, the
index of $\alpha^d$ is equal to the rank of the normal spine
$\pi_0^{\#}$ of $\alpha$, where $\pi_0^\#$ is the $\sigma$-strong
limit of the maps $\{\pi_b^\#\}_{b>0}$ for each $A \in \bigcup_{t>0}
U_tB(H)U_t^*$. Fix $t>0$, and let $A \in U_tB(H)U_t^*$. From formula
\eqref{pi},
$$\pi_b^{\#}(A) = \phi(I + \nu_b(\Lambda(1)) \phi)^{-1}
(\Omega_{\nu_b}(A)).$$ For all $b<t$ we have
$||\Omega_{\nu_b}(A)||=||\Omega_{\nu_t}(A)||< \infty$. Since
$\nu_b(\Lambda(1)) \rightarrow \infty$ as $b \rightarrow 0$, we
conclude $\lim_{b \rightarrow 0} ||\pi_b^\# (A)|| =0,$ hence
$\pi_0^{\#} = 0$ and the index of $\alpha$ is zero. However,
$\alpha^d$ is not completely spatial since $\alpha$ is
not derived from the zero boundary weight map (see Lemma 4.37 and Theorem
4.52 of \cite{hugepaper}), so $\alpha^d$ is of type II$_0$. \qed \end{pf}
Given a $q$-positive $\phi: M_n(\C) \rightarrow M_n(\C)$ and a
normalized unbounded boundary weight $\nu$ over $L^2(0, \infty)$, we
call $(\phi, \nu)$ a \textit{boundary weight double}. As we have
seen, if $\phi$ is unital then the boundary weight double naturally
defines a boundary weight map through the construction of
Proposition \ref{bdryweight}, inducing a type II$_0$ $E_0$-semigroup
$\alpha^d$ which is unique up to conjugacy by Theorem
\ref{dilation}. We should note that it is not necessary for $\phi$
to be unital in order for the boundary weight double to induce a
$CP$-flow: If $\phi$ is any $q$-positive contraction such that $||\nu_t(I) \phi(I +
\nu_t(\Lambda(1)) \phi)^{-1}|| \leq 1$ for all $t
> 0$, then the arguments given in the proofs of Proposition
\ref{bdryweight} and Corollary \ref{bdryweightcor} show that the
boundary weight double $(\phi, \nu)$ induces a $CP$-flow $\alpha$.
However, if $\phi$ is not unital, then by Eq.\eqref{tech} and
Theorem \ref{powersthm}, neither is $\alpha$.
Motivated by \cite{hugepaper}, we make the following definition:
\begin{defn} Suppose $\alpha: B(H_1) \rightarrow B(K_1)$ and $\beta: B(H_2) \rightarrow
B(K_2)$ are normal and completely positive. Write each $A \in B(H_1 \oplus H_2)
$ as $A=(A_{ij})$, where $A_{ij} \in B(H_j, H_i)$ for each
$i,j=1,2$. We say a linear map $\gamma:
B(H_2, H_1) \rightarrow B(K_2, K_1)$ is a corner from $\alpha$ to
$\beta$ if $\psi: B(H_1 \oplus H_2) \rightarrow B(K_1 \oplus K_2)$
defined by
\begin{displaymath} \psi \left( \begin{array}{cc} A_{11} & A_{12} \\
A_{21} & A_{22} \end{array} \right) = \left(
\begin{array}{cc} \alpha(A_{11}) & \gamma(A_{12}) \\ \gamma^*(A_{21})
& \beta(A_{22}) \end{array} \right)
\end{displaymath} is a normal completely positive map. \end{defn}
We will repeatedly use the following lemma, which gives us the form of any corner between
normal completely positive contractions of finite index. We believe
that this result is already present in the literature, but
we present a proof here for the sake of completeness:
\begin{lem}\label{corners} Let $H_1, H_2, K_1,$ and $K_2$
be separable Hilbert spaces. Let $\alpha: B(H_1) \rightarrow
B(K_1)$ and $\beta: B(H_2) \rightarrow B(K_2)$ be normal completely
positive contractions of the form
$$\alpha(A_{11})= \sum_{i=1}^n S_i A_{11}S_i^*, \beta(A_{22}) =
\sum_{j=1}^p T_j A_{22} T_j^*,$$ where $n,p \in \mathbb{N}$ and the
sets of maps $\{S_i\}_{i=1}^n$ and $\{T_j\}_{j=1}^p$ are both
linearly independent. A linear map $\gamma: B(H_2, H_1) \rightarrow
B(K_2,K_1)$ is a corner from $\alpha$ to $\beta$ if and only if for
all $A_{12} \in B(H_2,H_1)$ we have
$$\gamma(A_{12})= \sum_{i,j} c_{ij} S_i A_{12} T_j^*,$$ where
$C=(c_{ij}) \in M_{n \times p}(\C)$ is any matrix such that $||C||
\leq 1$.
\end{lem}
\begin{pf} For the backward direction, let $C=(c_{ij}) \in M_{n
\times p}(\C)$ be any contraction, and define a linear map $\gamma:
B(H_2, H_1) \rightarrow B(K_2,K_1)$ by $\gamma(A) = \sum_{i,j}
c_{ij} S_iAT_j^*$. We need to show that the map
\begin{displaymath}L \left( \begin{array}{cc} A_{11} & A_{12}
\\ A_{21} & A_{22} \end{array} \right)=
\left( \begin{array}{cc} \alpha(A_{11}) &
\gamma(A_{12}) \\ \gamma^*(A_{21}) & \beta(A_{22}) \end{array} \right)
\end{displaymath} is normal and completely positive. To prove this, we first assume
that $n \geq p$ and note that by Polar Decomposition we may write
$C_{n \times p} = V_{n \times p} T_{p \times p}$, where $V_{n \times
p}$ is a partial isometry of rank $p$ and $T$ is positive. Unitarily
diagonalizing $T$ we see $C_{n \times p} = V_{n \times p} W_{p
\times p}^* D_{p \times p} W_{p \times p}$. We may easily add
columns to $V_{n \times p} W_{p \times p}^*$ to form a unitary
matrix in $M_n(\C)$, which we call $U^*$. Defining $\tilde{D}
= (d_{ij}) \in M_{n \times p}(\C)$
to be the matrix obtained from $D$ by adding $n-p$ rows of zeroes,
we see $U^*\tilde{D} = V_{n \times p} W_{p \times p}^*D$, so $C_{n
\times p} = U^* \tilde{D} W_{p \times p}$ and
$$UC_{n \times p}W_{p \times p}^*=\tilde{D}.$$ In other words, \begin{displaymath}
\sum_{i,j}
c_{ij} u_{ki} \overline{w_{\ell j}} = \left\{
\begin{array}{ll} \delta_{k \ell}
d_{k \ell} & \textrm{ if } k \leq p \\
0 & \textrm{ if } k > p \\ \end{array} \right \}.
\end{displaymath}
Next, define $\{S_i'\}_{i=1}^n: H_1 \rightarrow K_1$ and
$\{T_j\}_{j=1}^p: H_2 \rightarrow K_2$ by
$$S_i ' = \sum_{k=1}^n \overline{u_{ik}} S_{k}, \ \ T_j ' =
\sum_{\ell =1}^p \overline{w_{j \ell}} T_j,$$ so $S_i = \sum_{k=1}^n
{u_{ki}} S_k'$ and $T_j = \sum_{\ell=1}^p w_{\ell j} T_j'$ for all
$i$ and $j$. \\ \\ Since $U$ and $W$ are unitary, it follows that $||D||=||C|| \leq 1$ and that the maps
$\{S_i'\}_{i=1}^n$ are linearly independent, as are the maps
$\{T_j'\}_{j=1}^p$.
We observe that for any $A_{11} \in B(H_1)$ and $A_{22} \in B(H_2)$,
$$\sum_{i=1}^n S_i A_{11} S_i ^* = \sum_{i=1}^n S_i ' A_{11} (S_i')^* \textrm{
and } \sum_{j=1}^p T_j A_{22} T_j ^* = \sum_{j=1}^p T_j' A_{22}
(T_j')^*.$$ Finally, for any $A_{12} \in B(H_2,H_1)$, we use our above computations
to find that
\begin{eqnarray*} \sum_{i,j} c_{ij} S_i A_{12} T_j^* & = & \sum_{i,j,k,
\ell} c_{ij}u_{ki}\overline{w_{\ell j}} S_k' A_{12} (T_\ell')^* =
\sum_{k, \ell} \Big(\sum_{i,j} c_{ij}u_{ki}\overline{w_{\ell j}}
S_k' A_{12} (T_\ell')^*\Big) \\
& = & \sum_{(k \leq p), \ell} \Big(\sum_{i,j}
c_{ij}u_{ki}\overline{w_{\ell j}} S_k' A_{12} (T_\ell')^*\Big) \\ & \ & \
+ \sum_{(k>p) , \ell} \Big(\sum_{i,j}
c_{ij}u_{ki}\overline{w_{\ell j}} S_k' A (T_\ell')^*\Big) \\
& = & \sum_{k \leq p} d_{kk} S_k' A_{12} (T_k')^* + 0 = \sum_{k=1}^p
d_{kk} S_k' A (T_k')^*.
\end{eqnarray*}
We have shown that
\begin{displaymath}
L(A) = \left(\begin{array}{cc} \sum_{i=1}^n S_i' A_{11} (S_i')^*
& \sum_{i=1}^p d_{ii} S_i' A_{12} (T_i') ^* \\
\sum_{i=1}^p \overline{d_{ii}} T_i' A_{21} (S_i')^* & \sum_{i=1}^p
T_i' A_{22} (T_i')^* \end{array} \right) \end{displaymath} for all
\begin{displaymath} A =\left( \begin{array}{cc} A_{11} & A_{12} \\
A_{21} & A_{22} \end{array} \right) \in B(H_1 \oplus H_2).
\end{displaymath}
For each $i=1, \ldots, p$, define $Z_i: H_1 \oplus H_2 \rightarrow
K_1 \oplus K_2$ by
\begin{displaymath} Z_i = \left( \begin{array}{cc}d_{ii} S_i' & 0 \\ 0 &
T_i' \end{array} \right), \end{displaymath} so
\begin{displaymath} L(A) = \sum_{i=1}^p Z_i A Z_i^* + \sum_{i=1}^p \left( \begin{array}{cc}
(1-|d_{ii}|^2) S_i' A_{11} S_i'^* & 0 \\ 0 & 0
\end{array} \right) + \sum_{i=p+1}^n \left( \begin{array}{cc}
S_i' A_{11} S_i'^* & 0 \\ 0 & 0
\end{array} \right).
\end{displaymath} Since $||D|| \leq 1$, the line above shows
that $L$ is the sum of three normal completely positive maps and is
thus normal and completely
positive. Therefore, $\gamma$ is a corner
from $\alpha$ to $\beta$. If, on the other hand, $n<p$, then the
same argument we just used shows that $\gamma^*$ is a corner from $\beta$ to
$\alpha$, which is equivalent to showing that $\gamma$ is a corner
from $\alpha$ to $\beta$.
For the forward direction, suppose that $\gamma$ is a corner from
$\alpha$ to $\beta$, so the map $\Upsilon: B(H_1 \oplus H_2)
\rightarrow B(K_1 \oplus K_2)$ defined by
\begin{displaymath}
\Upsilon\left( \begin{array}{cc} A_{11} & A_{12}
\\ A_{21} & A_{22} \end{array} \right)=
\left( \begin{array}{cc} \sum_{i=1}^n S_iA_{11}S_i^* &
\gamma(A_{12})
\\ \gamma^*(A_{21}) & \sum_{j=1}^p T_jA_{22}T_j^* \end{array} \right)
\end{displaymath}
is normal and completely positive. Therefore, for some $q \in \mathbb{N}
\cup \{\infty\}$ and maps $Y_i: H_1 \oplus H_2 \rightarrow K_1
\oplus K_2$ for $i=1,2,...$, linearly independent over
$\ell_2(\mathbb{N})$, we have
$$\Upsilon(\tilde{A})= \sum_{i=1}^q Y_i\tilde{A}Y_i^*$$ for all $\tilde{A}
\in B(H_1 \oplus H_2)$. For $i=1,2$, let $E_i \in B(H_1 \oplus H_2)$
be projection onto $H_i$, and let $F_i \in B(K_1 \oplus K_2)$ be
projection onto $K_i$. Since $\alpha$ and $\beta$ are contractions
we have $\Upsilon(E_1) \leq F_1$ and $\Upsilon(E_2) \leq F_2$, so
$Y_i E_j Y_i^* \leq F_j$ for each $i$ and $j$. It follows that each
$Y_i$, $i=1, \ldots q$, can be written in the form
\begin{displaymath} Y_i = \left( \begin{array}{cc} \tilde{S}_i & 0
\\ 0 & \tilde{T}_i \end{array} \right)\end{displaymath} for some $\tilde{S}_i \in
B(H_1, K_1)$ and $\tilde{T}_i \in B(H_2, K_2)$.
Note that $\alpha(A_{11})= \sum_{i=1}^n S_i A_{11}S_i^*= \sum_{i=1}^q \tilde{S}_i A_{11} \tilde{S_i}^*$
for all $A_{11}\in B(H_1)$. For each
$\tilde{S}_i$, define a completely positive map $L_i$ by
$L_{i}(A) = \tilde{S}_i A\tilde{S}_i^*$
for $A \in B(H_1)$. Since
$\alpha - L_i$ is completely positive, it follows from the work of Arveson
in \cite{arveson} that $\tilde{S}_i$ can be written as
$$\tilde{S}_i = \sum_{j=1}^n r_{ij} S_j$$ for some complex coefficients $\{r_{ij}\}_{j=1}^n$.
The same argument shows that for each $\tilde{T}_i$ we have
$$\tilde{T}_i = \sum_{j=1}^p b_{ij}
T_j$$ for some coefficients $\{b_{ij}\}_{j=1}^p$. It now follows from
linear independence of the maps $\{Y_i\}_{i=1}^q$ that
$q \leq n+p$. Let $R=(r_{ij}) \in M_{q \times n}(\C)$ and
$B=(b_{ij}) \in M_{q \times p}(\C)$, and let $A \in B(H_1)$. We
calculate
\begin{eqnarray} \sum_{i=1}^n S_iAS_i^* &=& \sum_{i=1}^q \tilde{S}_iA\tilde{S}_i^*
= \sum_{i=1}^q \Big(\sum_{j,k=1}^n r_{ij}\overline{r_{ik}} S_j A
S_k^*
\Big) \nonumber \\
&=& \label{end?} \sum_{j,k=1}^n \Big(\sum_{i=1}^q r_{ij}
\overline{r_{ik}}\Big)S_jAS_k^*.
\end{eqnarray}
Let $M=R^T(R^T)^* \in M_{n}(\C)$, so its $jk$th entry is $ m_{jk}
= \sum_{i=1}^q r_{ij}\overline{r_{ik}}$. Unitarily diagonalizing
$M$ as $UMU^* = D$ for some diagonal $D$ and defining maps $\{S_i'\}_{i=1}^n$ by $S_i' =
\sum_{k=1}^n \overline{u_{ik}} S_k$, we see that Eq.\eqref{end?}
and the same linear algebra technique from the proof of the backward direction
yield $$\sum_{i=1}^n S_i'AS_i'^* =
\sum_{i=1}^n S_iAS_i^* = \sum_{j,k=1}^n m_{jk} S_jAS_k^*
= \sum_{i=1}^n d_{ii} S_i'AS_i'^*.$$ Therefore $D=I$ and consequently
$M=I$, hence $||R|| =1$. An identical argument shows that $||B|| =
1$.
Let \begin{displaymath} \tilde{A} = \left( \begin{array}{cc} A_{11}
& A_{12}
\\ A_{21} & A_{22} \end{array} \right) \in B(H_1 \oplus H_2)
\end{displaymath} be arbitrary. Let $C= (c_{jk}) \in
M_{n \times p}(\C)$ be the matrix $C=(B^*R)^T$, noting that $||C||
\leq 1$. A straightforward computation of
$\Upsilon(\tilde{A})=\sum_{i=1}^q Y_i \tilde{A} Y_i^*$ yields
\begin{eqnarray*} \gamma(A_{12})&=& \sum_{i=1}^q
S_i'A_{12}T_i'^* = \sum_{i=1}^q \Big(\Big(\sum_{j=1}^n a_{ij}
S_j\Big)A_{12}\Big(\sum_{k=1}^p \overline{b_{ik}} T_k^*\Big)\Big) \\
&=& \sum_{j,k} \Big(\sum_{i=1}^q a_{ij} \overline{b_{ik}}\Big) S_j
A_{12} T_k^* = \sum_{j,k} c_{jk} S_jA_{12}T_k^*, \end{eqnarray*}
hence $\gamma$ is of the form claimed. \qed \end{pf}
\section{Comparison theory for $q$-positive maps}
Just as in the general study of various classes of linear operators,
it is natural to impose, and examine, an order structure for
$q$-positive maps. If $\phi$ and $\psi$ are $q$-positive maps
acting on $M_n(\C)$, we say that $\phi$ $q$-dominates $\psi$ (and
write $\phi \geq_q \psi$) if $\phi(I + t \phi)^{-1} - \psi(I + t
\psi)^{-1}$ is completely positive for all $t \geq 0$. We would
like to find the $q$-positive maps with the least complicated
structure of $q$-subordinates. That last statement is not as simple
as it seems. We might think to define a $q$-positive map $\phi$ to
be ``$q$-pure'' if $\phi \geq_q \psi \geq_q 0$ implies $\psi =
\lambda \phi$ for some $\lambda \in [0,1]$, but there exist
$q$-positive maps $\phi$ such that for every $\lambda \in (0,1)$ we
have $\phi \ngeq _q \lambda \phi$. One such example is the Schur
map $\phi$ on $M_2 (\C)$ given by
\begin{displaymath} \phi\left( \begin{array}{cc} a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{array} \right)=
\left( \begin{array}{cc} a_{11} & (\frac{1+i}{2}) a_{12} \\
(\frac{1-i}{2}) a_{21} & a_{22} \\
\end{array} \right).
\end{displaymath}
As it turns out, every $q$-positive map is guaranteed to have a
one-parameter family of $q$-subordinates of a particular form:
\begin{prop}\label{qsubs} Let $\phi \geq_q 0$. For each $s \geq 0$,
let $\phi^{(s)}= \phi(I + s \phi)^{-1}$. Then $\phi^{(s)} \geq_q 0$
for all $s \geq 0$. Furthermore, the set $\{\phi^{(s)}\}_{s \geq
0}$ is a monotonically decreasing family of $q$-subordinates of
$\phi$, in the sense that $\phi^{(s_1)} \geq_q \phi^{(s_2)}$ if $s_1
\leq s_2$.
\end{prop}
\begin{pf} For all $s \geq 0$ and $t \geq 0$, we have
\begin{eqnarray*} \phi^{(s)}(I + t \phi^{(s)})^{-1} & = &
\phi(I+s \phi)^{-1}\Big(I + t \phi(I + s\phi)^{-1}\Big)^{-1} \\ &=&
\phi\Big[\Big(I + t \phi(I + s \phi)\Big)(I + s\phi)\Big]^{-1}\\ & = & \phi(I + (s+t)\phi)^{-1},
\end{eqnarray*} which is completely positive by $q$-positivity of
$\phi$. Therefore, $\phi^{(s)} \geq_q 0$ for all $s \geq 0$.
To prove that $\phi^{(s_1)} \geq_q \phi^{(s_2)}$ if $s_1 \leq s_2$,
we let $t \geq 0$ be arbitrary and examine the map $$\Phi:
=\phi^{(s_1)}(I + t \phi^{(s_1)})^{-1} - \phi^{(s_2)}(I + t
\phi^{(s_2)})^{-1}.$$ Letting $t_1 = s_1+t$ and $t_2=s_2+t$, we
make the following observations:
\begin{equation}\label{darn1} \phi^{(s_j)}(I + t \phi^{(s_j)})^{-1}=
\phi^{(t_j)} \textrm{ for } j=1,2,
\end{equation}
\begin{equation} \label{darn2} \phi^{(t_1)} - \phi^{(t_2)} =
(I + t_2 \phi)^{-1} \Big( (I + t_2 \phi)\phi - \phi (I + t_1 \phi)
\Big) (I + t_1 \phi)^{-1}.\end{equation}
Equations \eqref{darn1} and \eqref{darn2} give us
\begin{eqnarray*} \Phi & = & (I + t_2 \phi)^{-1} \Big(
(I + t_2 \phi)\phi - \phi (I + t_1 \phi) \Big) (I + t_1 \phi)^{-1}\\
& = &
(I + t_2 \phi)^{-1} \Big((t_2-t_1) \phi^2\Big)(I + t_1 \phi)^{-1} \\
& = & (t_2-t_1) \Big(\phi(I + t_2 \phi)^{-1}\Big) \Big( \phi(I + t_1
\phi)^{-1}\Big).
\end{eqnarray*}
The last line is a non-negative multiple of a composition of
completely positive maps and is thus completely positive. We
conclude that $\phi^{(s_1)} \geq_q \phi^{(s_2)}$. \qed
\end{pf}
We now have the correct notion of what it means to be $q$-pure:
\begin{defn}\label{qpure} Let $\phi: M_n(\C) \rightarrow M_n(\C)$
be unital and $q$-positive. We say that $\phi$ is \emph{q-pure} if
its set of $q$-subordinates is precisely $\{0\} \cup
\{\phi^{(s)}\}_{s \geq 0}$.
\end{defn}
\begin{lem}\label{bdryrep2} Let $\nu$ be a normalized unbounded boundary
weight over $L^2(0, \infty)$ of the form $$\nu(\sqrt{I - \Lambda(1)}
B \sqrt{I - \Lambda(1)}) = (f,Bf).$$ Let $\phi: M_n(\C) \rightarrow
M_n(\C)$ be a $q$-positive contraction such that $||\nu_t(I) \phi(I +
\nu_t(\Lambda(1)) \phi)^{-1}|| \leq 1$ for all
$t>0$, and let $\alpha$ be the $CP$-flow derived from the boundary
weight double $(\phi, \nu)$, with boundary generalized representation
$\pi=\{\pi_t^\#\}_{t>0}$.
Let $\beta$ be any $CP$-flow over $\C^n$, with generalized boundary
representation $\xi^\#=\{\xi_t^\#\}_{t>0}$ and boundary weight map
$\rho \rightarrow \eta(\rho)$. Then $\alpha \geq \beta$ if and only
if $\beta$ is induced by the boundary weight double $(\psi, \nu)$,
where $\psi: M_n(\C) \rightarrow M_n(\C)$ is a $q$-positive map satisfying $\phi \geq_q \psi$.
\end{lem}
\begin{pf} As before, for each $t>0$ we let $s_t =
\nu_t(\Lambda(1))$. Assume the hypotheses of the backward
direction. Then $\xi_t^\#= \psi(I + s_t \psi)^{-1} \Omega_{\nu_t}$,
and the direction now follows from Theorem \ref{bdryrep} since the
line below is completely positive for all $t>0$:
$$\pi_t^\# - \xi_t^\# = (\phi(I + s_t \phi)^{-1}-\psi(I + s_t \psi)^{-1})
\Omega_{\nu_t}.$$ Now assume the hypotheses of the forward
direction. Recall that by construction of $\nu$, the set
$\{s_t\}_{t
> 0}$ is decreasing. If $s_t>0$ for all $t>0$ we define $P=\infty$.
Otherwise, we define $P$ to be the smallest positive number such
that $s_P = 0$. Fix any $t_0 \in (0,P)$. Notationally, write each
$g \in H:= \C^n \otimes L^2(0, \infty)$ in its components as $g(x)=(g_1(x), \ldots, g_n(x))$, and
write $f_{t_0}$ for the function $V_{t_0}V_{t_0}^*f \in L^2(0, \infty)$, where
$V_{t_0}$ is the right shift $t_0$ units on $L^2(0, \infty)$. Let $U_{t_0}$
be the right shift $t_0$ units on $H$. Under our
identifications, $U_{t_0}U_{t_0}^*$ is the diagonal matrix in
$M_n(B(L^2(0, \infty)))$ with $ii^{th}$ entries $V_{t_0}V_{t_0}^*$. Define $S: H
\rightarrow \C^n$ by
$$Sg = ((f_{t_0},g_1), \ldots, (f_{t_0}, g_n)),$$ noting
that $\Omega_{\nu_{t_0}}(A) = SAS^*$ for all $A \in B(H)$. Since
$\phi(I + s_{t_0}\phi)^{-1}$ is completely positive, we know it has
the form $\phi(I
+ s_{t_0} \phi)^{-1}(M) = \sum_{i=1}^m R_i M R_i^*$ for some $R_1,
\ldots, R_m \in M_n(\C)$. Therefore,
$$\pi^\#_{t_0}(A) = \Big(\phi(I + s_{t_0} \phi)^{-1}\Big) (\Omega_{\nu_{t_0}}(A))=
\sum_{i=1}^m R_iSAS^*R_i^*.$$ The map $\xi^\#_{t_0}$ is a
subordinate of $\pi^\#_{t_0}$, so from Arveson's work in metric
operator spaces in \cite{arveson}, we know that $\xi^\#_{t_0}$ has the form
$$\xi^\#_{t_0}(A)= \sum_{i,j =1}^m c_{ij} R_iSAS^*R_j^*,$$ for some
complex numbers $\{c_{ij}\}$. Let $L_{t_0}$ be the map $L_{t_0}(M)= \sum_{i,j}
c_{ij} R_i M R_j^*$, noting that $\xi^\#_{t_0}(A)=
L_{t_0}(SAS^*)=L_{t_0}(\Omega_{\nu_{t_0}}(A))$ for all $A \in B(H)$.
Defining $\psi_{t_0}: M_n(\C) \rightarrow M_n(\C)$ by $\psi_{t_0} =
(I - \xi^\#_{t_0} \Lambda)^{-1}L_{t_0},$ we find that for arbitrary $A \in
B(H)$ and $\acute{A} \in M_n(\C)$,
\begin{eqnarray}\label{etaone} \eta_{t_0}(\rho)(A) & = & \Big(\hat{\xi}_{t_0}(I -
\hat{\Lambda} \hat{\xi}_{t_0})^{-1}\Big)(\rho)(A) \nonumber =
\rho\Big((I - \xi^\#_{t_0} \Lambda)^{-1}(\xi^\#_{t_0}(A))\Big) \\ &
= & \rho\Big((I - \xi^\#_{t_0} \Lambda)^{-1}L_{t_0}
(\Omega_{\nu_{t_0}}(A))\Big) =\rho(\psi_{t_0}(\Omega_{\nu_{t_0}}A))
\end{eqnarray} and
\begin{equation} \label{etatwo} \hat{\Lambda} \eta_{t_0} (\rho) (\acute{A})
= \eta_{t_0}(\rho)(\Lambda(\acute{A})) =
\rho\Big(\psi_{t_0}(\Omega_{\nu_{t_0}}(\Lambda(\acute{A})))\Big) =
s_{t_0} \rho(\psi_{t_0}(\acute{A})),\end{equation} so $\hat{\Lambda}
\eta_{t_0}= s_{t_0} \hat{\psi}_{t_0}.$
Using formulas \eqref{etaone} and \eqref{etatwo} and the fact that
$\hat{\xi}_{t_0} = \eta_{t_0}(I + \hat{\Lambda} \eta_{t_0})^{-1}$,
we find
\begin{eqnarray*}\rho(\xi^\#_{t_0}) & = & \hat{\xi}_{t_0}(\rho) = \eta_{t_0}(I +
\hat{\Lambda} \eta_{t_0})^{-1}(\rho) =\Big((I + \hat{\Lambda}
\eta_{t_0})^{-1}(\rho)\Big) (\psi_{t_0} \Omega_{\nu_{t_0}})
\\ & = & \Big((I + s_{t_0} \hat{\psi}_{t_0})^{-1}(\rho)\Big)
(\psi_{t_0}\Omega_{\nu_{t_0}}) = \rho \Big((I + s_{t_0}
\psi_{t_0})^{-1} \psi_{t_0} \Omega_{\nu_{t_0}}\Big) \\ & = & \rho
\Big(\psi_{t_0}(I + s_{t_0} \psi_{t_0})^{-1} \Omega_{\nu_{t_0}}\Big)
\end{eqnarray*} for all $\rho \in M_n(\C)^*$, hence $\xi_{t_0}^\# =
\psi_{t_0}(I + s_{t_0} \psi_{t_0})^{-1}\Omega_{\nu_{t_0}}$.
We now show that the maps $\{\psi_t\}_{t>0}$ are constant on the
interval $(0, P)$. Let $t \in [t_0,P)$ be arbitrary. For each
$\acute{A}=(a_{ij}) \in M_n(\C)$, let $A \in B(H)$ be the matrix
with $ij$th entry $(a_{ij}/\nu_t(I))
V_{t} V_{t}^*$. Let $\rho \in M_n(\C)^*$. Straightforward
computations using formula \eqref{truncate} yield $\Omega_{t_0}(A)=
\Omega_{t}(A) = \acute{A}$ and
$\eta_{t_0}(\rho)(A)=\eta_{t}(\rho)(A)$. Combining these equalities
gives us \begin{eqnarray*} \rho(\psi_{t_0}(\acute{A}))& = &
\rho(\psi_{t_0}\Omega_{\nu_{t_0}}(A))= \eta_{t_0}(\rho)(A)
\\ & = & \eta_{t}(\rho)(A)= \rho(\psi_{t}\Omega_{\nu_{t}}(A)) =
\rho(\psi_{t}(\acute{A})).\end{eqnarray*} Since the above formula
holds for every $\acute{A} \in M_n(\C)$ and $\rho \in M_n(\C)^*$, we
have $\psi_{t_0}=\psi_{t}$. But both $t_0 \in (0,P)$ and $t \in
[t_0, P)$ were chosen arbitrarily, so the previous sentence shows
that $\psi_t = \psi_{t_0}$ for all $t \in (0, P)$.
Letting $\psi = \psi_{t_0}$, we have
\begin{eqnarray}\label{xi} \xi_t^\# = \psi(I + s_t \psi)^{-1}
\Omega_{\nu_t}\end{eqnarray} for all $t \in (0, P)$. Defining
$\kappa_t$ as in the proof of Proposition \ref{bdryweight}, we
observe that $\psi(I + s_t \psi)^{-1}= \xi_t^\# \kappa_t$ for all $t
\in (0,P)$, where the right hand side is completely positive by
hypothesis. Since every $t \in (0, \infty)$ can be written as $t =
s_{t'}$ for some $t' \in (0,P)$, it follows that $\psi(I + t
\psi)^{-1}$ is completely positive for all $t>0$. Furthermore,
$\psi(I + s_t \psi)^{-1} \rightarrow \psi$ in norm as $t\rightarrow
\infty$, hence $\psi \geq_q 0$. Similarly, since $\pi_t^\#-
\xi_t^\#$ is completely positive for all $t>0$ by assumption, it
follows from our formula
$$\phi(I + s_t \phi)^{-1} - \psi(I + s_t \psi)^{-1}= (\pi_t^\# - \xi_t^
\#) \kappa_t$$ that $\phi(I + s_t \phi)^{-1} - \psi(I + s_t
\psi)^{-1}$ is completely positive for all $t>0$, and so its norm
limit (as $t \rightarrow \infty$) $\phi - \psi$ is completely
positive. Therefore, $\phi \geq_q \psi$. Finally, since the
$CP$-flow $\beta$ is entirely determined by its generalized boundary
representation $\xi^\#$, which itself is determined by any
sequence $\{\xi_{t_n}^\#\}$ with $t_n$ tending to $0$ (see the remarks
preceding Theorem 4.29 of \cite{hugepaper}), it follows
from \eqref{xi} that $\beta$ is induced by the boundary weight
double $(\psi, \nu)$. \qed
\end{pf}
In a manner analogous to that used by Powers in \cite{bigpaper} and
\cite{hugepaper}, we define the terms \textit{$q$-corner} and
\textit{hyper maximal $q$-corner}:
\begin{defn}\label{hyp} Let $\phi: M_n(\C) \rightarrow M_n(\C)$ and $\psi:
M_k(\C) \rightarrow M_k(\C)$ be $q$-positive maps. A corner $\gamma:
M_{n \times k}(\C) \rightarrow M_{n \times k}(\C)$ from $\phi$ to
$\psi$ is said to be a $q$-corner from $\phi$ to $\psi$ if the map
\begin{displaymath} \Upsilon \left( \begin{array}{cc} A_{n \times n} &
B_{n \times k} \\ C_{k \times n} & D_{k \times k}
\end{array} \right) = \left( \begin{array}{cc} \phi(A_{n \times n}) &
\gamma (B_{n \times k}) \\
\gamma^* (C_{k \times n}) & \psi(D_{k \times k}) \\
\end{array} \right)
\end{displaymath}
is $q$-positive. A $q$-corner $\gamma$ is called hyper maximal if,
whenever
\begin{displaymath} \Upsilon \geq_q \Upsilon' = \left( \begin{array}{cc} \phi' & \gamma \\
\gamma^* & \psi'\\
\end{array} \right) \geq_q 0,
\end{displaymath}
we have $\Upsilon = \Upsilon'$.
\end{defn}
\begin{prop}\label{basischange} For any $q$-positive $\phi: M_n(\C)
\rightarrow M_n(\C)$ and unitary $U \in M_n(\C)$, define a map
$\phi_U$ by
$$\phi_U (A)= U^* \phi(UAU^*)U.$$
\begin{enumerate} \item The map $\phi_U$ is $q$-positive, and there
is an order isomorphism between $q$-positive maps $\beta$ such that
$\phi \geq_q \beta$ and $q$-positive maps $\beta_U$ such that
$\phi_U \geq \beta_U$. In particular, $\phi$ is $q$-pure if and
only if $\phi_U$ is $q$-pure.
\item If $\phi$ is unital and $q$-pure, then there is a hyper maximal
$q$-corner from $\phi$ to $\phi_U$.
\end{enumerate}
\end{prop}
\begin{pf} To prove the first assertion, we define a completely
positive map $\zeta$ on $M_n(\C)$ by $\zeta(A)= U^*AU$, noting that
$\zeta^{-1}$ is also completely positive. For every $t \geq 0$ and
$A \in M_n(\C)$, we find that $(I + t \phi_U)^{-1}(A) = U^* (I + t
\phi)^{-1}(UAU^*)U$ and
\begin{eqnarray}\label{qposo} \phi_U(I + t \phi_U)^{-1}(A) & = & U^* \phi\Big(U(U^*(I + t
\phi)^{-1}(UAU^*)U)U^* \Big)U \nonumber \\ & = & U^*\phi(I + t
\phi)^{-1}(UAU^*)U \nonumber \\ & = & \zeta \circ \phi(I + t
\phi)^{-1} \circ \zeta^{-1}(A),
\end{eqnarray} so $\phi_U \geq_q 0$. Given any $q$-positive map
$\beta$ such that $\phi \geq_q \beta$, define $\beta_U$ by
$\beta_U(A)= U^* \beta(UAU^*)U$. Then $\beta_U$ is $q$-positive by
\eqref{qposo}, and for each $t \geq 0$ we have
$$\phi_U (I + t \phi_U)^{-1} - \beta_U(I + t \beta_U)^{-1} = \zeta
\circ (\phi(I + t \phi)^{-1} - \beta(I + t \beta)^{-1}) \circ
\zeta^{-1},$$ hence $\phi_U \geq_q \beta_U$. Of course, since
$\phi=(\phi_U)_{U^*}$, the argument just used gives an identical
correspondence between $q$-subordinates $\alpha$ of $\phi_U$ and
$q$-subordinates $\alpha_{U^*}$ of $\phi$. Our first assertion now
follows.
To prove the second statement, we define $\gamma: M_n(\C)
\rightarrow M_n(\C)$ by $\gamma(A)= \phi(AU^*)U$. By Lemma
\ref{corners}, $\gamma$ is a corner from $\phi$ to $\phi_U$, so the
map
\begin{displaymath} \Theta \left(
\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22}
\end{array} \right) = \left( \begin{array}{cc} \phi(A_{11}) &
\gamma(A_{12}) \\ \gamma^*(A_{21}) & \phi_U(A_{22})
\end{array} \right) \end{displaymath} is completely positive. We
calculate $\gamma(I + t \gamma)^{-1}(A) = \phi(I + t
\phi)^{-1}(AU^*)U$, so for each $t \geq 0$ and $\tilde{A}=(A_{ij})
\in M_{2n}(\C)$, we have
\begin{displaymath} \Theta(I + t \Theta)^{-1} (\tilde{A})
= \left( \begin{array}{cc} \phi(I + t \phi)^{-1}(A_{11}) & \phi(I +
t \phi)^{-1}(A_{12}U^*)U \\ U^*\phi(I + t \phi)^{-1}(UA_{21}) &
\phi_U(I + t \phi_U)^{-1}(A_{22})
\end{array} \right). \end{displaymath} This shows that $\gamma(I + t
\gamma)^{-1}$ is a corner from $\phi(I + t\phi)^{-1}$ to $\phi_U (I
+ t\phi_U)^{-1}$ for all $t \geq 0$, so $\gamma$ is a $q$-corner.
Finally, if
\begin{displaymath} \Theta' \left(
\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22}
\end{array} \right) = \left( \begin{array}{cc} \alpha(A_{11}) &
\gamma(A_{12}) \\ \gamma^*(A_{21}) & \beta(A_{22})
\end{array} \right) \end{displaymath} is $q$-positive and $\Theta \geq_q
\Theta'$, then since $\phi$ and $\phi_U$ are $q$-pure
we have $\alpha= \phi(I + t \phi)^{-1}$ for some $t \geq 0$ and
$\beta= \phi_U(I + s \phi_U)^{-1}$ for some $s \geq 0$. Complete
positivity of $\Theta'$ implies that
\begin{displaymath} \Theta' \left(
\begin{array}{cc} I & U \\ U^* & I
\end{array} \right) = \left( \begin{array}{cc} \frac{1}{1+t}I &
U \\ U^* & \frac{1}{1+s}I
\end{array} \right) \geq 0, \end{displaymath} so $s=t=0$ and
$\Theta=\Theta'$, hence $\gamma$ is hyper maximal. \qed \end{pf}
We have arrived at the key result of the section, which
tells us that, under certain conditions, the problem of determining
whether two $E_0$-semigroups induced by boundary weight doubles are
cocycle conjugate can be reduced to the much simpler problem of
finding hyper maximal $q$-corners between $q$-positive maps:
\begin{prop}\label{hypqc}
Let $\nu$ be a normalized unbounded boundary weight over $L^2(0,
\infty)$ which has the form $\nu(\sqrt{I - \Lambda(1)}B\sqrt{I -
\Lambda(1)})=(f, Bf)$. Let $\phi$ and $\psi$ be unital $q$-positive
maps on $M_n(\C)$ and $M_k(\C)$, respectively, and induce $CP$-flows
$\alpha$ and $\beta$ through the boundary weight doubles $(\phi,
\nu)$ and $(\psi, \nu)$.
Then $\alpha^d$ and $\beta^d$ are cocycle conjugate if and only if
there is a hyper maximal $q$-corner from $\phi$ to $\psi$.
\end{prop}
\begin{pf} Let $N=n+k$. For the forward direction, suppose
$\alpha^d$ and $\beta^d$ are cocycle conjugate. Since $\alpha^d$
and $\beta^d$ are of type II$_0$, we know from Theorem
\ref{hyperflowcorn} that there is a
hyper maximal flow corner $\sigma$ from $\alpha$ to $\beta$, with associated
$CP$-flow
\begin{displaymath}\Theta =
\left (\begin{array}{cc} \alpha & \sigma
\\ \sigma^* & \beta \end{array} \right).
\end{displaymath}
Let $\Pi^\#=\{\Pi_t^\#\}$, $\pi^\#=\{\pi_t^\#\}$, and $\xi^\#=\{\xi_t^\#\}$
be the generalized boundary
representations for $\Theta$, $\alpha$, and $\beta$, respectively. Define $s_t= \nu_t(\Lambda(1))$ for
all positive $t$, so for each $t
>0$ there is some $\mathfrak{Z}_t$ such that
\begin{displaymath}\Pi_t^\# =
\left (\begin{array}{cc} \pi_t^\# & \mathfrak{Z}_t
\\ \mathfrak{Z}_t^* & \xi_t^\# \end{array} \right) = \left(
\begin{array}{cc} \phi(I + s_t \phi)^{-1} \circ \Omega_{\nu_t, n \times n}
& \mathfrak{Z}_t \\ \mathfrak{Z}_t^* & \psi(I + s_t \psi)^{-1}
\circ \Omega_{\nu_t, k \times k} \end{array} \right).
\end{displaymath}
\\
Since each $\mathfrak{Z}_t$ is a corner from $\phi(I + s_t
\phi)^{-1} \circ \Omega_{\nu_t, n \times n}$ to $\psi(I + s_t
\phi)^{-1} \circ \Omega_{\nu_t, k \times k}$, we have
$\mathfrak{Z}_t = L_t \circ \Omega_{\nu_t, n \times k}$ for some
$L_t$. Define $B_t$ for each $t>0$ by
\begin{displaymath}B_t =
\left(\begin{array}{cc} \phi(I + s_t \phi)^{-1} & L_t
\\ L_t ^* & \psi(I + s_t \psi)^{-1} \end{array} \right).
\end{displaymath} We observe that $\Pi_t^\# = B_t \circ \Omega_{\nu_t,N \times N}$ for
all $t>0$, whereby the same argument given in the proof of
Lemma \ref{bdryrep2} shows that each $B_t$ has the form $B_t =
W_t (I + s_t W_t)^{-1}$ for some $W_t: M_n(\C)
\rightarrow M_n(\C)$
and that the maps $W_t$ are independent of $t$.
Therefore, for some $\gamma: M_{n \times k}(\C) \rightarrow M_{n \times k}(\C)$,
we have
$$\mathfrak{Z}_t = \gamma(I + s_t \gamma)^{-1} \circ
\Omega_{\nu_t, n \times k}$$ for all $t>0$. Define
$\kappa_{t, N \times N}: M_N(\C) \rightarrow B(H)$ as in Proposition
\ref{bdryweight}. Letting
\begin{displaymath}\vartheta =
\left(\begin{array}{cc} \phi & \gamma
\\ \gamma^* & \psi \end{array} \right),
\end{displaymath}
we observe for each $t$ that $\vartheta(I+s_t \vartheta)^{-1} =
\Pi_{t}^\# \circ \kappa_{t, N \times N}$ is the composition of
completely positive maps and is thus completely positive, hence
$\vartheta \geq_q 0$. Suppose that for some map $\vartheta'$ we
have
\begin{displaymath}\vartheta \geq_q \vartheta' =
\left (\begin{array}{cc} \phi' & \gamma
\\ \gamma^* & \psi' \end{array} \right) \geq_q 0.
\end{displaymath}
As in Proposition \ref{bdryweight}, the boundary weight map $\rho
\in M_{N}(\C)^* \rightarrow L(\rho)$ defined by $L(\rho)(C)=
\rho(\vartheta'(\Omega_{\nu,N \times N} (C))$ induces a $CP$-flow
$\Theta'$ over $\C^N$, where for some $CP$-flows $\alpha'$ over $\C^n$ and $\beta'$
over $\C^k$, we have
\begin{displaymath}\Theta' =
\left (\begin{array}{cc} \alpha' & \sigma
\\ \sigma^* & \beta' \end{array} \right).
\end{displaymath}
By Lemma \ref{bdryrep2}, we have $\Theta \geq \Theta'$
since $\vartheta \geq_q \vartheta'$.
But $\Theta$ is a hyper maximal flow corner, so
$\Theta=\Theta'$. Our formulas for the generalized boundary representations
imply that $\phi(I + t \phi)^{-1}= \phi'(I + t \phi')^{-1}$
and $\psi(I + t \psi)^{-1} = \psi'(I + t \psi')^{-1}$ for all $t>0$,
hence $\phi=\phi'$ and $\psi=\psi'$. We
conclude that $\gamma$ is a hyper maximal $q$-corner.\\ \\
For the backward direction, suppose there is a hyper maximal
$q$-corner $\gamma$ from $\phi$ to $\psi$, so the map $\Upsilon:
M_{N}(\C) \rightarrow M_{N}(\C)$ defined by
\begin{displaymath} \Upsilon \left( \begin{array}{cc} A_{n \times n} &
B_{n \times k} \\ C_{k \times n} & D_{k \times k}
\end{array} \right) = \left( \begin{array}{cc} \phi(A_{n \times n}) &
\gamma (B_{n \times k}) \\
\gamma^* (C_{k \times n}) & \psi(D_{k \times k}) \\
\end{array} \right)
\end{displaymath} is $q$-positive. By Proposition \ref{bdryweight},
the boundary weight map $\rho \in M_N(\C)^* \rightarrow \Xi(\rho)$
defined by
$$\Xi(\rho)(A) = \rho(\Upsilon(\Omega_{\nu, N \times N}(A)))$$ is the boundary weight map of a $CP$-flow
$\theta$ over $\C^{N}$, where for some $\Sigma$ we have
\begin{displaymath}\theta =
\left (\begin{array}{cc} \alpha & \Sigma
\\ \Sigma^* & \beta \end{array} \right).
\end{displaymath}
Let \begin{displaymath}\theta' = \left (\begin{array}{cc} \alpha' &
\Sigma
\\ \Sigma^* & \beta' \end{array} \right)
\end{displaymath}
be any $CP$-flow such that $\theta \geq \theta'$. Letting
$\mathcal{Z}_t= \gamma(I + s_t \gamma)^{-1} \circ \Omega_{\nu_t,n
\times k}$ for all $t>0$, we see the generalized boundary representations
$\Pi^\#=\{\Pi_t ^\#\}$ and $\Pi'=\{\Pi_t'\}$ for $\theta$ and $\theta'$
satisfy
\begin{displaymath}\Pi_t ^\# = \left (\begin{array}{cc} \pi_t^\# &
\mathcal{Z}_t
\\ \mathcal{Z}_t^* & \xi_t ^\# \end{array} \right)
\geq \Pi_t' = \left (\begin{array}{cc} \pi_t' & \mathcal{Z}_t
\\ \mathcal{Z}_t^* & \xi_t' \end{array} \right)
\end{displaymath}
for all $t> 0$. Lemma \ref{bdryrep2} implies that for some $\phi'$
and $\psi'$ with $\phi \geq_q \phi' \geq_q 0$ and $\psi \geq_q \psi'
\geq_q 0$ we have $\pi_t' = \phi'(I + s_t \phi')^{-1}\circ
\Omega_{\nu_t, n \times n}$ and $\xi_t' = \psi'(I + s_t
\psi')^{-1}\circ \Omega_{\nu_t, k \times k}$ for all $t>0$. Defining
$\Upsilon': M_N(\C) \rightarrow M_N(\C)$ by
\begin{displaymath} \Upsilon' \left( \begin{array}{cc} A_{n \times n} &
B_{n \times k} \\ C_{k \times n} & D_{k \times k}
\end{array} \right) = \left( \begin{array}{cc} \phi'(A_{n \times n}) &
\gamma (B_{n \times k}) \\
\gamma^* (C_{k \times n}) & \psi'(D_{k \times k}) \\
\end{array} \right),
\end{displaymath}
we observe that $\Pi_{t}' \circ \kappa_{\nu_t,N \times N}= \Upsilon'(I +
s_t \Upsilon')^{-1}$ for all $s_t>0$, hence $\gamma$ is a $q$-corner from
$\phi'$ to $\psi'$. Hyper maximality of $\gamma$ implies $\phi=\phi'$
and $\psi=\psi'$, thus $\theta=\theta'$. Therefore, $\sigma$ is a
hyper maximal flow corner from $\alpha$ to $\beta$, so $\alpha^d$ and
$\beta^d$ are cocycle conjugate by Theorem \ref{hyperflowcorn}.
\qed \end{pf}
\section{$E_0$-semigroups obtained from rank one unital $q$-pure maps}
Any unital linear map $\phi: M_n(\C) \rightarrow M_n(\C)$ of rank
one is of the form $\phi(A) = \tau(A)I$ for some linear functional
$\tau$. If $\phi$ is positive, then $\tau$ is positive and
$\tau(I)=1$, so $\tau$ is a state. On the other hand, given any
state $\rho$, the map $\phi$ defined by $\phi(A)=\rho(A)I$ is unital
and completely positive. Furthermore, $\phi$ is $q$-positive since
$\phi(I+t \phi)^{-1}= (1/(1+t)) \phi$ for all $t>0$. The rank one unital
$q$-positive maps are therefore precisely the maps $A
\rightarrow \rho(A) I$ for states $\rho$.
The goal of this section is to determine when such maps are
$q$-pure, and then to determine when the
$E_0$-semigroups induced by $(\phi, \nu)$ and $(\psi, \nu)$ are cocycle conjugate,
where $\phi$ and $\psi$ are rank one unital $q$-pure maps and
$\nu$ is a normalized unbounded boundary weight of the form
$\nu(\sqrt{I - \Lambda(1)} B \sqrt{I - \Lambda(1)}) = (f,Bf)$ (Theorem
\ref{statesbig}). We also
obtain a partial result for comparing $E_0$-semigroups induced by $(\phi, \nu)$
and $(\psi, \mu)$ for rank one unital $q$-pure maps $\phi$ and $\psi$ and any
normalized unbounded boundary
weights $\nu$ and $\mu$ over $L^2(0, \infty)$ (Corollary \ref{thesearenew}). \\
\\ We begin with a lemma:
\begin{lem}\label{tauca} Let $\rho$ be a faithful state on $M_n(\C)$, and
define a unital $q$-positive map $\phi:M_n(\C) \rightarrow M_n(\C)$
by $\phi(A)= \rho(A) I$. For any non-zero positive linear functional
$\tau$ on $M_n(\C)$ and non-zero positive operator $C \in M_n(\C)$,
define $\psi_{\tau, C}: M_n(\C) \rightarrow M_n(\C)$ by $\psi_{\tau,
C}(A) = \tau(A)C$.
Then $\psi_{\tau, C}$ is $q$-positive, and $\phi \geq_q \psi_{\tau,
C}$ if and only if $\psi_{\tau, C} = \lambda \phi$ for some $\lambda
\in (0,1]$.
\end{lem}
\begin{pf} Note that for all $A \in M_n(\C)$ and $t
\geq 0$, we have $(I+t \psi_{\tau, C})^{-1}(A)= A - t
\tau(A)/(1+t \tau(C))C$, so \begin{eqnarray}\label{tauc}\psi_{\tau,
C}(I+t \psi_{\tau, C})^{-1}(A) = \frac{\tau(A)}{1 + t \tau(C)}
C,\end{eqnarray} hence $\psi_{\tau, C}$ is $q$-positive. It follows
from \eqref{tauc} that $\phi(I+t\phi)^{-1}(A) = (\rho(A)/(1+t))
I$ for all $A \in M_n(\C)$.
Assume the hypotheses of the forward direction. Since $\phi \geq_q
\psi_{\tau, C}$, we have \begin{equation}\label{first} \frac{\rho(A)
I}{1+t} \geq \frac{\tau(A)C}{1 + t \tau (C)}\end{equation} for all
$t \geq 0$ and $A \geq 0$. This is impossible if $\tau(C)=0$, so we
may assume $\tau(C) \neq 0$. Letting $t \rightarrow \infty$ in
\eqref{first} yields \begin{equation} \label{second} \rho(A)I\geq
\frac{\tau(A) C}{\tau(C)}\end{equation} for all $A \geq 0$. Setting
$A=C$ in \eqref{second}, we see $\rho(C) I - C \geq 0$, yet
$$\rho \Big(\rho(C)I-C\Big)= \rho(C)-\rho(C)= 0,$$ hence
$C=\rho(C)I$ by faithfulness of $\rho$. Rewriting \eqref{second} as $$\rho(A) I \geq
\frac{\tau(A)}{\tau(\rho(C)I)} \rho(C)I = \frac{\tau(A)}{||\tau||}I$$
for all $A\geq 0$, we see that $\rho-\tau / ||\tau||$ is a
positive linear functional. Therefore, $$\Big|\Big|\rho -
\frac{\tau}{||\tau||}\Big|\Big| = \rho(I) - \frac{\tau(I)}{||\tau||}
= 1-1=0,$$ hence $\tau = ||\tau||\rho$. Setting $t=0$ and $A=I$ in
\eqref{first} gives us $||\tau||= \tau(I) = \lambda / \rho(C)$
for some $\lambda \in (0,1]$. Therefore,
$$\psi_{\tau, C} (A) = \tau(A)C =||\tau|| \rho(A) \rho(C) I = \lambda
\rho(A)I = \lambda \phi(A)$$ for all $A \in M_n(\C)$, proving the
forward direction.
The backward direction follows from Proposition \ref{qsubs} since
$\lambda \phi = \phi^{(-1 + 1/\lambda)}$ for every
$\lambda \in (0,1]$.
\qed \end{pf}
{\bf Remark:} Let $\psi: M_n(\C) \rightarrow M_n(\C)$ be a non-zero
$q$-positive
contraction such that the maps $L_{\psi_t} : = t \psi(I + t
\psi)^{-1}$ satisfy $||L_{\psi_t}|| <1$ for all $t
> 0$. By compactness of the unit ball of $B(M_n(\C))$, the maps $L_{\psi_t}$
have some norm limit as $t \rightarrow \infty$. This limit is
unique: Pick any orthonormal basis with respect to the trace inner
product $(A,B)= tr(A^*B)$ of $M_n(\C)$, and let $M_t$ be the $n^2
\times n^2$ matrix of $L_{\psi_t}$ with respect to this basis. From
the cofactor formula for $(I + t \psi)^{-1}$, we know that the
$ij$th entry of $M_t$ is a rational function $r_{ij}(t)$.
Uniqueness of $\lim_{t \rightarrow \infty}L_{\psi_t}$ now follows
from the fact that each $r_{ij}(t)$ has a unique limit as $t
\rightarrow \infty$. We call this limit $L_\psi$. Noting that
$$t \psi
= L_{\psi_t}(I - L_{\psi_t})^{-1} = L_{\psi_t} + L_{\psi_t}^2 +
\ldots$$ for each $t>0$, we claim that $L_{\psi}$ fixes a positive
element $T$ of norm one. To prove this, we first observe for each $k \in \mathbb{N}$
and $t>0$ that
\begin{eqnarray*}t ||\psi|| & =& t ||\psi(I)|| \leq ||L_{\psi_t}(I)|| + \ldots +
||(L_{\psi_t})^{k-1}(I)|| + k \sum_{n=1}^\infty ||(L_{\psi_t})^{kn}(I)|| \\
& < & (k-1) + k \sum_{n=1}^\infty ||(L_{\psi_t})^k(I)||^n
,\end{eqnarray*} hence
$$1= \lim_{t\rightarrow \infty} ||(L_{\psi_t})^k(I)|| =
||(L_{\psi})^k(I)||.$$
Therefore, all elements of the sequence $\{T_k\}_{k \in
\mathbb{N}}$ defined by $T_k = (L_{\psi})^k(I)$ satisfy $T_k \geq 0$
and $||T_k||=1$. Since $T_k-T_{k+1} = (L_{\psi})^k(I-T_1) \geq 0$
for all $k$, the sequence $\{T_k\}_{k \in \mathbb{N}}$ is
monotonically decreasing and therefore has a positive norm limit $T$
with $||T||=1$. Finally, $L_\psi$ fixes $T$ since
$L_{\psi}(T)=\lim_{k \rightarrow \infty} L_{\psi}^{k+1}(I)=T$. The
information at hand suffices in showing that a large class of maps
is $q$-pure:
\begin{prop}\label{littlestates} Let $\rho$ be a state on $M_n(\C)$, and define a
$q$-positive map $\phi$ on $M_n(\C)$ by $\phi(A)= \rho(A) I$. Then
$\phi$ is $q$-pure if and only if $\rho$ is faithful.
\end{prop}
\begin{pf} For the forward direction, we prove the
contrapositive. If $\rho$ is not faithful, then for some $k<n$ and
mutually orthogonal vectors $f_1, \ldots, f_k$ with $\sum_{i=1}^k
||f_i||^2=1$, we have $\rho(A)= \sum_{i=1}^k (f_i, Af_i)$ for all $A
\in M_n(\C)$. Let $P$ be the projection onto the $k$-dimensional
subspace of $\C^n$ spanned by the vectors $f_1, \ldots, f_k$, and
define a $q$-positive map $\psi: M_n(\C) \rightarrow M_n(\C)$ by
$\psi(A) = \rho(A) P$. For each $t \geq 0$ and $A \in M_n(\C)$, we
find
$$(\phi^{(t)}-\psi^{(t)})(A)= \frac{1}{1+t}(\phi(A)-\psi(A)) =
\frac{1}{1+t} \rho(A) (I-P),$$ so $\phi \geq_q \psi$. Obviously,
$\psi
\neq \phi^{(s)}$ for any $s \geq 0$, so $\phi$ is not $q$-pure. \\
\\ To prove the backward direction,
suppose $\phi \geq_q \psi \geq_q 0$ for some $\psi \neq 0$, and form
$L_\psi$ and $L_\phi$. Since $L_{\phi_t} = (t/(1+t))\phi$ for
each $t>0$, we have $L_\phi=\phi$. The map $L_{\phi_t}-L_{\psi_t}$
is completely positive for all $t$, so by taking its limit as $t \rightarrow
\infty$ we see $\phi - L_\psi$ is
completely positive. By the remarks preceding this proposition, we
know that $L_\psi$ fixes a positive $T$ with $||T||=1$. But $(\phi
-L_\psi)(T) = \rho(T)I - T \geq 0$, so $\rho(T)=1$, hence $T=I$ by
faithfulness of $\rho$.
By complete positivity of $\phi-L_\psi$, we have $||\phi - L_\psi||
= ||\phi(I) - L_\psi(I)|| = 0$, so $\phi = L_{\psi}$. Therefore,
\begin{eqnarray} 0 & = & \lim_{t \rightarrow \infty} \Big((\phi -
L_{\psi_t}) \Big(\frac{I}{t} + \psi\Big)\Big) = \lim_{t \rightarrow
\infty} \Big( \phi\Big(\frac{I}{t} + \psi\Big) -
L_{\psi_t}\Big(\frac{I}{t}
+ \psi\Big) \Big) \nonumber \\
& = & \lim_{t \rightarrow \infty} \Big( \frac{\phi}{t} + \phi \psi -
t \psi(I + t \psi)^{-1}\Big(\frac{I}{t} + \psi\Big)\Big) =
\lim_{t\rightarrow \infty} \frac{\phi}{t} + \phi \psi - \psi \nonumber\\
\label{third} & = & \phi \psi - \psi.
\end{eqnarray}
Letting $\tau$ be the positive linear functional $\tau= \rho \circ
\psi$, we conclude from \eqref{third} that $\psi(A)=\rho(\psi(A)) I
= \tau(A)I$ for all $A \in M_n(\C)$. Lemma \ref{tauca} implies that
$\psi = \lambda \phi = \phi^{(-1+1/\lambda)}$ for some
$\lambda \in (0,1]$.
\qed \end{pf}
To prove the main result of the section, we need the following:
\begin{lem}\label{generalnormone} Let $\phi: M_n(\C) \rightarrow
M_n(\C)$ and $\psi: M_k(\C) \rightarrow M_k(\C)$ be rank one unital
$q$-pure maps, and let $\nu$ and $\mu$ be normalized
unbounded boundary weights over
$L^2(0, \infty)$. If the boundary weight doubles $(\phi, \nu)$ and
$(\psi, \mu)$ induce cocycle conjugate $E_0$-semigroups $\alpha^d$
and $\beta^d$, then there is a corner $\gamma$ from $\phi$ to $\psi$
such that $||\gamma||=1$.
\end{lem}
\begin{pf} By construction, $\alpha^d$ and $\beta^d$
are type II$_0$ $E_0$-semigroups. If they are cocycle conjugate, then
by Theorem \ref{hyperflowcorn},
there is a hyper maximal flow corner $\sigma$ from $\alpha$ to
$\beta$ with associated $CP$-flow $\Theta$ over $K_1 \oplus K_2$, where
\begin{displaymath}\Theta =
\left (\begin{array}{cc} \alpha & \sigma
\\ \sigma^* & \beta \end{array} \right).
\end{displaymath}
Let $H_1 = \C^n \otimes L^2(0, \infty)$ and $H_2 = \C^k \otimes L^2(0, \infty)$.
Write the boundary representation $\Pi=\{\Pi_t^\#\}$ for $\Theta$ as
\begin{displaymath} \Pi_t^\# = \left (\begin{array}{cc} \frac{1}{1+
\nu_t(\Lambda(1))} \phi \circ \Omega_{\nu_t, n \times n} & \mathfrak{Z}_t \\
\mathfrak{Z}_t^* & \frac{1}{1+\mu_t(\Lambda(1))}\psi \circ
\Omega_{\mu_t, k \times k}\end{array} \right)
\end{displaymath}
for some maps $\{\mathfrak{Z}_t\}_{t>0}$ from $B(H_2, H_1)$ into $B(K_2, K_1)$. Let $\rho_{11}
\rightarrow \omega(\rho_{11})$ and $\rho_{22} \rightarrow
\eta(\rho_{22})$ denote the boundary weight maps for $\alpha$ and
$\beta$, respectively. Let $\rho \rightarrow \Xi(\rho)$ be the
boundary weight map for $\Theta$, so for some map $\rho_{12}
\rightarrow \ell(\rho_{12})$ from $M_{n \times k}(\C)^*$ to weights
on $B(H_2, H_1)$ we have
\begin{displaymath}
\Xi \left (\begin{array}{cc} \rho_{11} & \rho_{12} \\ \rho_{21} &
\rho_{22}
\end{array} \right)= \left (\begin{array}{cc} \omega(\rho_{11}) &
\ell(\rho_{12})
\\ \ell^*(\rho_{21}) & \eta(\rho_{22})
\end{array} \right).
\end{displaymath}
Denote by $U_t$ the right shift $t$ units on $H$, and let $\pi^\#$
and $\xi^\#$ be the generalized boundary representations for $\alpha$
and $\beta$, respectively. For every
$A=(A_{ij}) \in \mathcal \bigcup_{t>0} U_t B(H) U_t^*$ and bounded
family of functionals $\{\rho(t)=(\rho_{ij}(t))\}_{t > 0}$ in
$M_{n+k}(\C)^*$, we observe that the argument used in
Corollary 3.3 to show that $\pi_0 ^\# = \xi_0^\# = 0$ implies
$$\lim_{t \rightarrow 0}\omega_t(I +
\hat{\Lambda}\omega_t)^{-1}(\rho_{11}(t))(A_{11}) = \lim_{t
\rightarrow 0} \eta_t(I +
\hat{\Lambda}\eta_t)^{-1}(\rho_{22}(t))(A_{22})=0,$$ so by complete
positivity of the generalized boundary representation, we have
\begin{eqnarray}\label{ell} \lim_{t \rightarrow 0} \ell_t(I +
\hat{\Lambda}\ell_t)^{-1}(\rho_{12}(t))(A_{12})=0.\end{eqnarray}
We claim that $\rho_{12} \rightarrow \ell(\rho_{12})$
is unbounded. If $\ell$ is bounded, then for each $\rho_{12} \in M_{n \times
k}(\C)^*$, the family $\rho_{12}(t):=(I +
\hat{\Lambda}\ell_t)(\rho_{12})$ is bounded, and it follows from
\eqref{ell} that
\begin{eqnarray}\label{ell22} \lim_{t \rightarrow 0}
\ell_t(\rho_{12})(A_{12})=0 \end{eqnarray} for each $A_{12} \in
\bigcup_{t>0} W_t B(H_2, H_1) X_t^*$, where $W_t$ and $X_t$ are the
right shift $t$ units on $H_1$ and $H_2$, respectively. Let $A_{12}
\in \bigcup_{t>0} W_t B(H_2, H_1) X_t^*$, so $A_{12} = W_s B X_s^*$
for some $s>0$ and $B \in B(H_2,H_1)$. For all $b<s$, we have
\begin{eqnarray*} \ell_b(\rho_{12})(A_{12}) & = & \ell_b(\rho_{12})(W_sBX_s^*)
= \ell(\rho_{12})(W_bW_b^*W_sBX_s^*X_bX_b^*)\\ & = &
\ell(\rho_{12})(W_b W_{s-b} B X_{s-b}^*X_b^*) = \ell(\rho_{12})(W_s
B X_s^*) \\ & = & \ell(\rho_{12})(A_{12}). \end{eqnarray*}
Therefore, by equation \eqref{ell22} we have
$\ell(\rho_{12})(A_{12})=0.$ Let $A \in B(H_2, H_1)$, $\rho_{12}
\in M_{n \times k}(\C)^*$, and $t>0$ be arbitrary. From above we
have $$\ell_t(\rho_{12})(A)= \ell(\rho_{12})(W_tAX_t^*)=0,$$ hence
$\ell_t \equiv 0$ for all $t>0$. We conclude from uniqueness of the
generalized boundary representation that $\rho_{12} \rightarrow
\ell(\rho_{12})$ is the zero map. The boundary weight map $\rho
\rightarrow \Xi'(\rho)$ defined by
\begin{displaymath}
\Xi' \left (\begin{array}{cc} \rho_{11} & \rho_{12} \\ \rho_{21} &
\rho_{22}
\end{array} \right)= \left (\begin{array}{cc} \omega(\rho_{11}) &
0
\\ 0 & 0
\end{array} \right)
\end{displaymath}
gives rise to the $CP$-flow
\begin{displaymath}\Theta' =
\left (\begin{array}{cc} \alpha & \sigma
\\ \sigma^* & \beta', \end{array} \right)
\end{displaymath}
where $\beta'$ is the non-unital $CP$-flow
$\beta_t'(A_{22})=X_tA_{22}X_t^*$. Trivially, $\Theta \neq \Theta'$
and $\Theta \geq \Theta'$, contradicting hyper maximality of
$\sigma$. Therefore, the map $\rho_{12} \rightarrow \ell(\rho_{12})$
is unbounded.
Since $\Pi_t^\#$ is a contraction for every $t>0$, so is
$\mathfrak{Z}_t$, hence the map $\mathfrak{Z}_t \circ \Lambda: M_{n
\times k}(\C) \rightarrow M_{n \times k} (\C)$ is a contraction for
each $t>0$. A compactness argument shows that $\mathfrak{Z}_{t_n}
\circ \Lambda$ has a norm limit $\gamma$ for some sequence $\{t_n\}$
tending to zero, where $||\gamma|| \leq 1$. From unboundedness of
$\ell$ and the formula $\ell_t=\hat{\mathfrak{Z}}_t(I
-\hat{\Lambda}\hat{\mathfrak{Z}}_t)^{-1}$ for all $t>0$, it follows
that $I-\gamma$ is not invertible, so $||\gamma|| \geq 1$, hence
$||\gamma||=1$. We claim that $\gamma$ is a corner from $\phi$ to
$\psi$. Indeed, for the family of completely positive maps
$\{R_t\}_{t>0}$ defined by $R_t = \Pi_t^\# \circ \Lambda$, we have
\begin{displaymath} \lim_{n \rightarrow \infty} R_{t_n} = \lim_{n
\rightarrow \infty} \left( \begin{array}{cc}
\frac{\nu_{t_n}(\Lambda(1))}{1+\nu_{t_n}(\Lambda(1))} \phi & \mathfrak{Z}_{t_n} \circ \Lambda \\
(\mathfrak{Z}_{t_n} \circ \Lambda)^* &
\frac{\mu_{t_n}(\Lambda(1))}{1+\mu_{t_n}(\Lambda(1))} \psi.
\end{array} \right) = \left( \begin{array}{cc}\phi & \gamma \\
\gamma^* & \psi \end{array} \right).
\end{displaymath}
\qed \end{pf}
If \ $\nu$ \ is \ a normalized \ unbounded \ boundary \ weight \ over \ $L^2(0, \infty)$ \
of \ the \ form $\nu(\sqrt{I - \Lambda(1)}B\sqrt{I - \Lambda(1)})=(f,Bf)$ and if $\phi: M_n(\C) \rightarrow M_n(\C)$
is unital and $q$-pure, we know from Propositions \ref{basischange}
and \ref{hypqc}
that the condition $\psi=\phi_U$ is sufficient for the boundary weight
doubles $(\phi, \nu)$ and
$(\psi, \nu)$ to induce cocycle conjugate $E_0$-semigroups. In the case
that $\phi$ is a rank one unital $q$-pure map, this condition is also necessary:
\begin{thm}\label{statesbig} Let $\phi_1: M_n(\C) \rightarrow M_n(\C)$ and
$\phi_2: M_k(\C) \rightarrow M_k(\C)$ be rank
one unital $q$-pure maps. Let
$\nu$ be a normalized unbounded boundary weight over $L^2(0,
\infty)$ of the form $\nu(\sqrt{I - \Lambda(1)} B \sqrt{I -
\Lambda(1)}) = (f,Bf)$.
Then the boundary weight doubles $(\phi_1, \nu)$ and $(\phi_2, \nu)$
induce cocycle conjugate $E_0$-semigroups if and only if $n=k$ and
$\phi_2=(\phi_1)_U$ for some unitary $U \in M_n(\C)$.
\end{thm}
\begin{pf} The backward direction follows immediately from
Propositions \ref{basischange} and \ref{hypqc}. Assume the
hypotheses of the forward direction. Since $\phi_1$ and $\phi_2$ are
rank one, unital, and $q$-pure, there exist faithful states $\rho_1$
on $M_n(\C)$ and $\rho_2$ on $M_k(\C)$ such that $\phi_1(M)=
\rho_1(M)I_{n \times n}$ and $\phi_2(B) = \rho_2(B)I_{k \times k}$
for all $M \in M_n(\C)$, $B \in M_k(\C)$. By Lemma
\ref{generalnormone}, there is a corner $\gamma$ from $\phi_1$ to
$\phi_2$ such that $||\gamma||=1$. Therefore, for some $A_0 \in M_{n
\times k}(\C)$ of norm one and unit vectors $f_0 \in \C^n$ and $g_0
\in \C^k$, we have $|(f_0, \gamma(A_0)g_0)|=1$. Define $\omega \in
M_{n \times k}(\C)^*$ by $\omega(A) = (f_0, \gamma(A)g_0)$, noting
that $||\omega||=|\omega(A_0)|=1$. We claim that the map
$\tilde{\psi}: M_{n+k}(\C) \rightarrow M_2(\C)$ defined by
\begin{displaymath} \tilde{\psi} \left( \begin{array}{cc} A_{11} & A_{12}
\\ A_{21} & A_{22} \end{array} \right)=
\left( \begin{array}{cc} \rho_1(A_{11}) & \omega(A_{12}) \\
\omega^*(A_{21}) & \rho_2(A_{22}) \end{array} \right)
\end{displaymath} is completely positive. To see this, let
$\{\tilde{F_i}\}_{i=1}^\ell$ be arbitrary vectors in $\C^2$, writing
each $\tilde{F_i}$ as
\begin{displaymath}\tilde{F_i}
= \left( \begin{array}{cc} \lambda_{1i}\\
\lambda_{2i} \end{array} \right) \end{displaymath} for some complex
numbers $\{\lambda_{1i}\}_{i=1}^\ell$ and $\{\lambda_{2i}
\}_{i=1}^\ell$.
Since the map $\psi: M_{n+k}(\C) \rightarrow M_{n+k}(\C)$ defined by
\begin{displaymath} \psi \left( \begin{array}{cc} A_{11} & A_{12}
\\ A_{21} & A_{22} \end{array} \right)=
\left( \begin{array}{cc} \rho_1(A_{11})I & \gamma(A_{12}) \\
\gamma^*(A_{21}) & \rho_2(A_{22})I \end{array} \right)
\end{displaymath} is completely positive by assumption, we know
that for any $A_1, \ldots, A_\ell \in M_{n+k}(\C)$ and the vectors
\begin{displaymath} F_i = \left( \begin{array}{cc} \lambda_{1i} f_0 \\
\lambda_{2i} g_0 \end{array} \right) \in \C^{n+k}, \ \ i=1, \ldots,
k,
\end{displaymath} we have
$$ \sum_{i,j=1}^\ell \Big(F_i, \psi(A_i^*A_j) F_j\Big) \geq 0.$$
However, for each $i$ and $j$ we find that \begin{eqnarray*}
\Big(F_i, \psi(A_i^*A_j)F_j\Big)_{\C^{n+k}}& = &
\overline{\lambda_{1i}}\lambda_{1j} \rho_1((A_i^*A_j)_{11}) +
\overline{\lambda_{1i}}\lambda_{2j} \omega((A_i^*A_j)_{12})\\ & \ &
+ \overline{\lambda_{2i}}\lambda_{j1}
\overline{\omega([(A_i^*A_j)_{21}] ^*)} +
\overline{\lambda_{2i}}\lambda_{2j} \rho_2((A_i^*A_j)_{22}) \\
& = & \Big(\tilde{F_i}, \tilde{\psi}(A_i^*A_j) \tilde{F_j}
\Big)_{\C^2}.
\end{eqnarray*}
Therefore, for all $\ell \in \mathbb{N}$,
$A_1, \dots, A_\ell \in
M_{n+k}(\C)$, and $\tilde{F_1}, \ldots, \tilde{F_\ell} \in \C^{2}$
, we have $\sum_{i,j=1}^\ell \Big(\tilde{F_i}, \tilde{\psi}
(A_i^*A_j) \tilde{F_j}\Big) \geq 0$,
so $\tilde{\psi}: M_{2n}(\C) \rightarrow M_2(\C)$ is
completely positive. Since $\rho_1$ and $\rho_2$ are positive
linear functionals (hence completely positive maps), $\omega$ is a
corner from $\rho_1$ to $\rho_2$.
By faithfulness of $\rho_1$ and $\rho_2$, there exist monotonically
increasing sequences of strictly positive numbers
$\{\lambda_i\}_{i=1}^n$ and $\{\mu_j\}_{j=1}^k$ with $\sum_{i=1}^n
\lambda_i^2 = \sum_{j=1}^k \mu_j^2 =1$, along with orthonormal sets
of vectors $\{f_i\}_{i=1}^n$ and $\{g_j\}_{j=1}^k$, such that
$\rho_1(M)= \sum_{i=1}^n \lambda_i^2 (f_i, M f_i)$ and $\rho_2(B) =
\sum_{j=1}^k \mu_j^2 (g_j, B g_j)$ for all $M \in M_n(\C)$, $B \in
M_k(\C)$. Given $A \in M_{n \times k}(\C)$, let $\tilde{A}$ be the
matrix whose $ji$th entry is $(f_i, A g_j)$, observing that
$||\tilde{A}|| =||A||$. Let $D_\lambda$ and $D_\mu$ be the diagonal
matrices whose $ii$th entries are $\lambda_i$ and $\mu_i$,
respectively, for all $i$, and let $D_{\lambda^2}$ and $D_{\mu^2}$
be the diagonal matrices whose $ii$th entries are $\lambda_i^2$ and
$\mu_i^2$, respectively, observing that $D_{\lambda^2} =
(D_\lambda)^2$ and $D_{\mu^2} = (D_{\mu})^2$.\\ \\
By Proposition \ref{corners}, $\omega$ has the form
$$\omega(A)= \sum_{i,j} c_{ij} \lambda_i \mu_j (f_i, A g_j) =
tr(C D_\mu \tilde{A} D_\lambda)= tr\Big(C D_\mu (D_\lambda
\tilde{A}^*)^* \Big)$$ for some $C=(c_{ij}) \in M_{n\times k}(\C)$
such that $||C|| \leq 1$.
By the Cauchy-Schwartz inequality for the inner product $(B,A)=tr(AB^*)$ on $M_{n
\times k} (\C)$, we have
\begin{eqnarray}1&=& |\omega(A_0)|^2 = |tr(C D_\mu (D_\lambda
\tilde{A_0}^*)^*) |^2 =|(C D_\mu, D_\lambda \tilde{A_0}^*)|^2 \nonumber \\
& \leq & (C D_\mu, C D_\mu) (D_\lambda
\tilde{A_0}^*,D_\lambda \tilde{A_0}^*)= tr(D_\mu C^* C D_\mu)
tr(D_\lambda
\tilde{A_0}^*\tilde{A_0} D_\lambda) \nonumber \\
\label{ii} & \leq & tr(D_{\mu^2}I_k)tr(D_{\lambda^2}I_n) \leq 1*1=1.
\end{eqnarray}
Since equality holds in all the inequalities above, we have $m C D_\mu = D_\lambda
\tilde{A_0}^*$ for some $m \in \C$. It follows from \eqref{ii}
that $|m|=1$ since $||CD_\mu||_{tr}=||D_\lambda \tilde{A_0}^*||_{tr}=1$.
Furthermore, since equality holds in \eqref{ii} and the trace map is faithful, we
have $C^*C=I_k$ and $\tilde{A_0}^*\tilde{A_0}=I_n$. But $C \in M_{n
\times k}(\C)$ and $\tilde{A_0}^* \in M_{n \times k}(\C)$, so $n=k$,
hence $C$ and $\tilde{A_0}$ are unitary.
Writing $D_\lambda = m C D_\mu \tilde{A_0}= (m C
\tilde{A_0})(\tilde{A_0}^* D_\mu \tilde{A_0}),$ we observe that $m C \tilde{A_0}$
is unitary and $\tilde{A_0}^* D_\mu \tilde{A_0}$ is positive. Uniqueness
of the right Polar Decomposition for the invertible matrix
$D_\lambda$ implies
$$D_\lambda = \tilde{A_0}^* D_\mu \tilde{A_0}.$$ Since the diagonal entries in
$D_\lambda$ and $D_\mu$ are listed in increasing order, it follows
that $D_\lambda=D_\mu$, hence $\rho_2$ is of the form $\rho_2(M) =
\sum_{i=1}^n \lambda_i^2 (g_i, M g_i)$. Defining a unitary $U \in
M_n(\C)$ by letting $Ug_i = f_i$ for all $i$ and extending linearly,
we observe that
$$\rho_2(M) = \sum_{i=1}^n \lambda_i^2 (U^*f_i, M U^*f_i) =
\sum_{i=1}^n \lambda_i^2 (f_i, UMU^*f_i)= \rho_1(UMU^*)$$ for all $M
\in M_n(\C)$. In other words, $\phi_2=(\phi_1)_U$. \qed
\end{pf}
In \cite{bigpaper}, Powers constructed $E_0$-semigroups using
boundary weights over $L^2(0, \infty)$. It is routine to check that
in our notation, these are the $E_0$-semigroups arising from the
boundary weight doubles $(\imath_\C, \eta)$, where $\imath_\C$ is
the identity map on $\C$ and $\eta$ is any boundary weight over
$L^2(0, \infty)$.
\begin{cor}\label{thesearenew}
Let $\phi: M_n(\C) \rightarrow M_n(\C)$ and $\psi: M_k(\C)
\rightarrow M_k(\C)$ be unital rank one $q$-pure maps, and let $\nu$
and $\eta$ be normalized unbounded boundary weights over $L^2(0,
\infty)$. Denote by $\alpha^d$ and $\beta^d$ the Bhat minimal
dilations of the $CP$-flows induced by the boundary weight doubles
$(\phi, \nu)$ and $(\psi, \mu)$, respectively.
If $n \neq k$, then $\alpha^d$ and $\beta^d$ are not cocycle
conjugate. In particular, if $n \neq 1$, then $\alpha^d$ is not
cocycle conjugate to the $E_0$-semigroup induced by $(\imath_\C,
\mu)$.
\end{cor}
\begin{pf} From the proof of Theorem \ref{statesbig}, we know
that every corner $\gamma$ from $\phi$ to $\psi$ satisfies
$||\gamma||<1$ since $n \neq k$. The result now follows from Lemma
\ref{generalnormone}. \qed \end{pf}
\section{Invertible unital $q$-pure maps}
Now that we have classified the unital $q$-pure maps on $M_n(\C)$ of
rank one, we explore the unital $q$-pure maps $\phi$ which are
invertible. In a stark contrast to the rank one case, we find that
for a given normalized unbounded boundary weight of the form
$\nu(\sqrt{I - \Lambda(1)} B \sqrt{I - \Lambda(1)})=(f, Bf)$ on
$L^2(0, \infty)$, the doubles $(\phi, \nu)$ and $(\psi, \nu)$
\textit{always} induce cocycle conjugate $E_0$-semigroups if $\phi$
and $\psi$ are unital invertible $q$-pure maps on $M_n(\C)$ and
$M_k(\C)$, respectively.
\newline \newline
The following proposition gives us a bijective correspondence
between invertible unital $q$-positive maps $\phi: M_n(\C)
\rightarrow M_n(\C)$ and unital conditionally negative maps $\Psi:
M_n(\C) \rightarrow M_n(\C)$:
\begin{prop}\label{cnegone} If $\phi: M_n(\C) \rightarrow M_n(\C)$
is an invertible unital $q$-positive map, then $\phi^{-1}$ is
conditionally negative. On the other hand, if $\Psi: M_n(\C)
\rightarrow M_n(\C)$ is a unital conditionally negative map, then
$\Psi$ is invertible and $\Psi^{-1}$ is $q$-positive.
\end{prop}
\begin{pf} Let $\psi=\phi^{-1}$. Since $\phi$ is self-adjoint, so
is $\psi$, and the first statement of the proposition now follows
from the fact that for large positive $t$ we have $$t \phi(I+t
\phi)^{-1} = t \psi^{-1}(I + t \psi^{-1})^{-1} = t (\psi+ tI)^{-1} =
\Big(I + \frac{\psi}{t}\Big)^{-1}=I- \frac{\psi}{t} +
\frac{\psi}{t}^2 - \ldots.$$
To prove the second statement, let $\Psi: M_n(\C) \rightarrow
M_n(\C)$ be any unital conditionally negative map. Since $\Psi$ is
conditionally negative, it follows from a result of Evans and Lewis
in \cite{lewev} that $e^{-s \Psi}$ is completely positive for all $s
\geq 0$. Therefore, $||e^{-s \Psi}||=||e^{-s\Psi}(I)||=
||e^{-s}I||= e^{-s}$ for all $s \geq 0$, and the integral $\int_0 ^
\infty e^{-s \Psi} ds$ converges. Observing that $(d/ds)
(-e^{-s \Psi}) = \Psi e^{-s \Psi}$, we find that
$$\Psi \Big(\int_0 ^ \infty e^{-s \Psi} ds\Big) = \int_0 ^ \infty
\Psi e^{-s \Psi}ds = \lim_{s \rightarrow \infty} (-e^{-s\Psi})\vert_0
^s = I,$$ so $\Psi$ is invertible and
$\Psi^{-1}= \int_0 ^ \infty
e^{-s \psi} ds$. Since $\Psi^{-1}$ is the integral of completely
positive maps, it is completely positive.
Furthermore, we find that $tI + \Psi$ is invertible for every $t>0$ and
that $\Psi^{-1}
\geq_q 0$, since the following holds for all $t>0$:
$$\int_0 ^\infty e^{-st}e^{-s \Psi}ds = \int_0 ^\infty
e^{-s(tI+\Psi)}ds =
(tI + \Psi)^{-1} = \Psi^{-1}(I + t \Psi^{-1})^{-1}.$$ \qed
\end{pf}
Examining the inverse of a unital invertible $q$-positive map $\phi$
is the key to finding the invertible $q$-subordinates of $\phi$, as
we find in the following proposition and corollary:
\begin{prop}\label{invsubs} Let $\phi_1: M_n(\C) \rightarrow M_n(\C)$
be an invertible unital $q$-positive map, and let
$\psi_1=\phi_1^{-1}$. Suppose $\psi_2: M_n(\C) \rightarrow M_n(\C)$
is conditionally negative and $\psi_2-\psi_1$ is completely
positive. Then $\psi_2$ is invertible, and $\phi_2:=(\psi_2)^{-1}$
satisfies $\phi_1 \geq_q \phi_2 \geq_q 0$.
\end{prop}
\begin{pf} Assume the hypotheses of the proposition, and let $s>0$
be arbitrary. Define a function $f$ on $\R$ by $f(t) = e^{-ts
\psi_1}e^{(t-1)s\psi_2}$. The equality below is $f(1)-f(0)=\int_0^1
f'(t) dt$:
$$e^{-s\psi_1}-e^{-s\psi_2} = \int_0^1 s e^{-ts\psi_1}(\psi_2-\psi_1)
e^{(t-1)s\psi_2} dt.$$ The inside of the integral above
is the composition of completely positive maps, so
$e^{-s\psi_1}-e^{-s\psi_2}$ is completely positive. This implies
$e^{-s \psi_1}(I) - e^{-s \psi_2}(I) \geq 0$, so
$$||e^{-s \psi_2}||=||e^{-s \psi_2}(I)|| \leq ||e^{-s \psi_1}(I)|| =
||e^{-s}(I)||=e^{-s}.$$ Now the argument given in the previous
proposition shows that $\int_0 ^\infty e^{-s \psi_2} ds$ converges
and is equal to $\psi_2^{-1}$. Letting $\phi_2 = \psi_2^{-1}$, we
observe that $\phi_1 \geq_q \phi_2$ since the quantity below is
completely positive for every $t \geq 0$:
$$\phi_1(I + t \phi_1)^{-1} - \phi_2(I + t \phi_2)^{-1}
= \int_0 ^\infty e^{-st}(e^{-s\psi_1} - e^{-s \psi_2}) ds.$$ \qed \end{pf}
\begin{cor}\label{invsubsone} Let $\phi_1: M_n(\C) \rightarrow M_n(\C)$
be an invertible unital $q$-positive map, and let $\phi_2: M_n(\C)
\rightarrow M_n(\C)$ be linear and invertible.
Then $\phi_1 \geq_q \phi_2 \geq_q 0$ if and only if $\phi_2^{-1}$ is
conditionally negative and $\phi_2^{-1} - \phi_1^{-1}$ is completely
positive.
\end{cor}
\begin{pf} The backward direction follows from Proposition
\ref{invsubs}. Assume the hypotheses of the forward direction and
let $\psi_1=\phi_1^{-1}$ and $\psi_2= \phi_2^{-1}$. Since
$\phi_2$ is self-adjoint, so is $\psi_2$. For sufficiently large
positive $t$ we have
$$t \phi_2(I + t \phi_2)^{-1} = \Big(I + \frac{\psi_2}{t}\Big)^{-1}=I-
\frac{\psi_2}{t} + \frac{\psi_2^2}{t^2} - \ldots$$ and
$$t^2(\phi_1(I + t \phi_1)^{-1} - \phi_2(I + t \phi_2)^{-1}) = \psi_2 -
\psi_1 + \Big(\frac{\psi_2^2-\psi_1^2}{t} - \frac{\psi_2^3 -
\psi_1^3}{t^2} + \ldots\Big).$$ The first equation shows that
$\phi_2^{-1}$ is conditionally negative, while the second shows that
$\phi_2^{-1} - \phi_1^{-1}$ is completely positive. \qed
\end{pf}
Now that we know how to find all invertible $q$-subordinates of an
invertible unital $q$-positive map $\phi$, we ask if there can be
any other $q$-subordinates of $\phi$. We will find that the answer
is no (see Proposition \ref{invert}). Proving this will require the
use of some machinery (notably Lemma \ref{eps}), which we now build.
\begin{defn}\label{defeps} For every $\phi: M_n(\C) \rightarrow M_n(\C)$ and
$ \epsilon \in [0,1]$, we define a map $\phi_\epsilon$ by
$\phi_\epsilon = \epsilon I + (1- \epsilon)\phi$.
\end{defn} If $\phi$ is $q$-positive, then $\phi_\epsilon$ is
invertible for all $\epsilon \in (0,1]$. In the lemmas that follow,
we make frequent use of the fact that for all $t \geq 0$ we have
\begin{eqnarray}\label{I-} t \phi(I + t \phi)^{-1} = I - (I + t
\phi)^{-1}.\end{eqnarray} We present a quick consequence of
\eqref{I-} for all $a \geq 0$ and $b \geq 0$:
\begin{eqnarray} \label{I-better} a(I + bt \phi)^{-1}=aI - abt\phi(I + bt \phi)^{-1}
\end{eqnarray}
\begin{lem} Let $\phi: M_n(\C) \rightarrow M_n(\C)$ be completely positive.
If $\phi_{\epsilon_k} \geq_q 0$ for some monotonically decreasing
sequence $\{\epsilon_k\}$ of positive real numbers tending to $0$,
then $\phi \geq_q 0$.
\end{lem}
\begin{pf} Assume the hypotheses of the lemma. Let $k$ be
arbitrary. Since $\phi_{\epsilon_k} \geq_q 0$, we know $I- (I
+ t \phi_{\epsilon_k})^{-1}$ is completely positive for all $t \geq
0$. Noting that
$$I - (I + t \phi_\epsilon)^{-1} = I - \Big((1 +
t \epsilon I) + (1-\epsilon)t \phi \Big)^{-1}= I - \frac{1}{1+t
\epsilon} \Big(I + \frac{t(1- \epsilon)}{1+t \epsilon} \phi
\Big)^{-1}$$ and substituting $t'=t(1-\epsilon_k)/(1+t
\epsilon_k)$, we see
$$I - (I + t \phi_\epsilon)^{-1}=
I - \frac{1}{1+ (\frac{\epsilon_k}{1-\epsilon_k + t'\epsilon_k})t'}
(I + t' \phi)^{-1}.$$ Varying $t$ throughout $[0, \infty)$, we find
that the above equation is completely positive for all $t' \in [0,
-1+1/\epsilon_k)$. Of course, for any $t' \in [0,
-1+1/ \epsilon_k )$, we have $t' \in [0, -1+1/\epsilon_\ell)
$ for all $ \ell \geq k$ by
monotonicity of the sequence $\{\epsilon_n\}$. Therefore, we may
repeat the same argument to conclude that for any $t' \in [0,
-1+1/\epsilon_k)$, the map $$I - \frac{1}{1+
(\frac{\epsilon_\ell}{1-\epsilon_\ell + t'\epsilon_\ell})t'} (I + t'
\phi)^{-1}$$ is completely positive for all $\ell \geq k$.
Now fix any $t' > 0$, so $t' \in (0, -1+1/\epsilon_k)$ for some $k \in \mathbb{N}$. A
straightforward computation shows that the sequence $\{c_n\}$
defined by $c_n = \epsilon_n / (1-\epsilon_n + t' \epsilon_n)$
monotonically decreases to $0$. From the previous paragraph, we know
that the map
$$I - \frac{1}{1+ c_\ell t'} (I + t'
\phi)^{-1}$$ is completely positive for all $\ell \geq k$. Since
$c_n \downarrow 0$ it follows that
$$I - (I + t' \phi)^{-1}$$ is completely positive. In other words,
$t' \phi(I + t' \phi)^{-1}$ is completely positive. Since $t'>0$ was
chosen arbitrarily and $\phi$ is completely positive, the lemma
follows. \qed
\end{pf}
\begin{lem} If $\phi: M_n(\C) \rightarrow M_n(\C)$
and $\phi \geq_q 0$, then $\phi_\epsilon \geq _q 0$ for all
$\epsilon \in [0,1)$.
\end{lem}
\begin{pf} Suppose that $\phi \geq_q 0$, and let $\epsilon \in
[0,1)$ be arbitrary. For each $t>0$, we apply formula
\eqref{I-better} to $a=1/(1+t\epsilon)$ and
$b=t(1-\epsilon)/(1+t \epsilon)$ to find
\begin{eqnarray*}I - (I + t \phi_\epsilon)^{-1} & = & I - \frac{1}{1+t
\epsilon} \Big(I + \frac{t(1- \epsilon)}{1+t \epsilon} \phi
\Big)^{-1} \\ & = & \Big(1-\frac{1}{1+t\epsilon}\Big)I +
\frac{t(1-\epsilon)}{(1+t \epsilon)^2} \phi \Big(I +
\frac{t(1-\epsilon)}{1+t \epsilon} \phi\Big)^{-1},
\end{eqnarray*} where both terms on the last line are completely
positive by assumption. Furthermore, $\phi_\epsilon$ is completely
positive, hence $\phi_\epsilon \geq_q 0$. \qed \end{pf}
\begin{cor} Let $\phi: M_n(\C) \rightarrow M_n(\C)$ be a completely positive map.
Then $\phi \geq_q 0$ if and only if $\phi_\epsilon \geq_q 0$ for all
$\epsilon \in (0,1)$.
\end{cor}
\begin{lem}\label{eps} Let $\phi: M_n(\C) \rightarrow M_n(\C)$ and $\psi: M_n(\C)
\rightarrow M_n(\C)$ be $q$-positive maps. Then $\phi \geq_q \psi$
if and only if $\phi_\epsilon \geq_q \psi_\epsilon$ for all
$\epsilon \in (0,1)$.
\end{lem}
\begin{pf} For any $\epsilon \in (0,1)$ we have $\phi_\epsilon -
\psi_\epsilon = \epsilon(\phi-\psi)$, so $\phi- \psi$ is completely
positive if and only if $\phi_\epsilon - \psi_\epsilon$ is
completely positive for all $\epsilon \in (0,1)$. For all $t'>0$ we
have \begin{eqnarray}\label{short} t' \Big( \phi(I + t' \phi)^{-1} -
\psi(I + t' \psi)^{-1} \Big) = (I + t' \psi)^{-1} - (I + t'
\phi)^{-1},\end{eqnarray} and for all $t>0$ we have
\begin{eqnarray} t (\phi_\epsilon(I + t \phi_\epsilon)^{-1} - \psi_\epsilon(I +
t \psi_\epsilon)^{-1}) = \Big(I - (I + t \phi_\epsilon)^{-1}\Big) -
\Big(I - (I + t \psi_\epsilon)^{-1} \Big) \nonumber \\ \label{long}
= \frac{1}{1+t \epsilon} \Big((I + \frac{t(1-\epsilon)}{1+t
\epsilon} \psi)^{-1} - (I + \frac{t(1-\epsilon)}{1+t \epsilon}
\phi)^{-1}\Big).\end{eqnarray}
Assume the hypotheses of the forward direction. Showing that
$\phi_\epsilon \geq_q \psi_\epsilon$ for all $\epsilon \in (0,1)$ is
equivalent to proving that \eqref{long} is completely positive for
every $t \in (0, \infty)$ and $\epsilon \in (0,1)$. But this
follows from complete positivity of $\eqref{short}$ since
$t(1-\epsilon)/(1+t \epsilon) \in (0, \infty)$ for every
$\epsilon \in (0, 1)$ and $t \in (0, \infty)$. Now assume the
hypotheses of the backward direction. Any $t' \in (0, \infty)$ can
be written as $t(1-\epsilon)/(1+t \epsilon)$ for some $\epsilon
\in (0,1)$ and $t \in (0, \infty)$, so complete positivity of
\eqref{long} for all such $\epsilon$ and $t$ implies that
\eqref{short} is completely positive for all $t' >0$, hence $\phi
\geq_q \psi$. \qed \end{pf}
We are now in a position to prove what is perhaps the most striking
result of the section:
\begin{prop}\label{invert}
Let $\xi: M_n(\C) \rightarrow M_n(\C)$ be an invertible unital
$q$-positive map. If $\phi: M_n(\C) \rightarrow M_n(\C)$ is
$q$-positive and $\xi \geq_q \phi$, then $\phi$ is either invertible
or identically zero.
\end{prop}
\begin{pf} For every $\epsilon \in (0,1)$, form $\xi_\epsilon$ and
$\phi_\epsilon$ as in Definition \ref{defeps}, and let
$\psi_\epsilon: = (\phi_\epsilon)^{-1}$. By Lemma \ref{eps}
we have $\xi_{\epsilon} \geq_q \phi_{\epsilon}$ for each $\epsilon$, so
$\psi_\epsilon$
is conditionally negative and $\psi_\epsilon - (\xi_\epsilon)^{-1}$
is completely positive by Corollary \ref{invsubsone}. We first
examine the case when the norms $||\psi_\epsilon||$ remain bounded
as $\epsilon \rightarrow 0$. More precisely, suppose that for all
$\epsilon$ sufficiently small we have $||\psi_\epsilon|| < r$ for
some $r >0$. By compactness of the closed unit ball of radius $r$
in $B(M_n(\C))$, there is a decreasing sequence $\{\epsilon_k\}_{k
\in \mathbb{N}}$ converging to $0$ such that
$\{\psi_{\epsilon_k}\}_{k \in \mathbb{N}}$ has a (bounded) norm
limit $\psi$ as $k \rightarrow \infty$. Noting that
$$I- \phi\psi = \phi_{\epsilon_k}\psi_{\epsilon_k} - \phi \psi =
(\phi_{\epsilon_k} - \phi)(\psi_{\epsilon_k}-\psi) +
\phi(\psi_{\epsilon_k} - \psi) + (\phi_{\epsilon_k} - \phi)\psi$$
and then applying the triangle inequality, we find that
\begin{eqnarray*}
||I - \phi \psi|| &=& ||\phi_{\epsilon_k}\psi_{\epsilon_k} - \phi
\psi|| \\ & \leq & ||\phi_{\epsilon_k} - \phi|| \
||\psi_{\epsilon_k} -\psi|| + ||\phi|| \ ||\psi_{\epsilon_k}-\psi||
+ ||\phi_{\epsilon_k}-\phi|| \ ||\psi||
\end{eqnarray*} for all $k \in \mathbb{N}$. But $\phi$ and $\psi$ are
bounded maps while $\psi_{\epsilon_k}\rightarrow \psi$ in norm and
$\phi_{\epsilon_k} \rightarrow \phi$ in norm, so the above equation
tends to $0$ as $k \rightarrow \infty$. We conclude that $\phi \psi
= I$. Similarly $\psi \phi = I$, hence $\phi$ is invertible and
$\psi=\phi^{-1}$.
If the first case does not hold, then for some decreasing sequence
$\{\epsilon_k\}$ tending to zero, the norms
$\{||\psi_{\epsilon_k}||\}_{k \in \mathbb{N}}$ form an unbounded
sequence. For each $k \in \mathbb{N}$, we write
$$(\xi_{\epsilon_k})^{-1}(A)= s_k A + Y_k A + AY_k^* -
\sum_{i=1}^{m_k} S_{k_i} A S_{k_i}^*$$ $$\textrm{ and }$$
$$\psi_{\epsilon_k}(A)= t_k A + Z_k A + AZ_k^* -
\sum_{i=1}^{\ell_k}T_{k_i} A T_{k_i}^*,$$ where $m_k, \ell_k \leq n^2$,
$s_k \in \R$, $t_k
\in \R$, $tr(Y_k)=tr(Z_k)=0$,
$tr(S_{k_i})=0$ and $tr(S_{k_i}^*S_{k_j})$ is non-zero if and only if $i=j$ ($i,j \leq m_k)$,
and $tr(T_{k_i})=0$ and $tr(T_{k_i}^* T_{k_j})$ is non-zero if and only if $i=j$ ($i,j \leq \ell_k$).
Since $\psi_{\epsilon_k} -
(\xi_{\epsilon_k})^{-1}$ is completely positive for all $k \in
\mathbb{N}$, we know that for each $k$,
there exist $p_k \leq n^2$, complex numbers $\{x_{k_i}\}_{i=1}^{p_k}$,
and maps $\{X_{k_i}\}_{i=1}^{p_k}$ with $tr(X_{k_i})=0$, such
that for all $A \in M_n(\C)$,
\begin{eqnarray} (\psi_{\epsilon_k} - (\xi_{\epsilon_k})^{-1})(A) &
= &
\sum_{i=1}^{p_k} (X_{k_i}+x_{k_i} I) A (X_{k_i} + x_{k_i}I)^* \nonumber \\
& = & \Big(\sum_{i=1}^{p_k} |x_{k_i}|^2\Big)A +
\Big(\sum_{i=1}^{p_k} \overline{x_{k_i}}X_{k_i}\Big)A+
A\Big(\sum_{i=1}^{p_k} \overline{x_{k_i}}X_{k_i}\Big)^* \nonumber \\
\label{l1} & \ & \ + \sum_{i=1}^{p_k} X_{k_i}AX_{k_i}^*.
\end{eqnarray}
Simultaneously, for all $A \in M_n(\C)$ we have
\begin{eqnarray} (\psi_{\epsilon_k} - (\xi_{\epsilon_k})^{-1})(A) &
= & (t_k -s_k)A + (Z_k-Y_k)A +A(Z_k-Y_k)^* \nonumber \\ \label{l2} &
\ & + \Big(\sum_{i=1}^{m_k} S_{k_i} A S_{k_i}^* -
\sum_{i=1}^{\ell_k}T_{k_i} A T_{k_i}^* \Big).
\end{eqnarray}
We claim that \begin{eqnarray}\label{Xis} \Big|\Big|\sum_{i=1}^{p_k}
X_{k_i}AX_{k_i}^*\Big|\Big| \leq \Big|\Big| \sum_{i=1}^{m_k} S_{k_i}
A S_{k_i}^*\Big|\Big|\end{eqnarray} for all $k \in \mathbb{N}$. To
prove this, we let $\{v_j\}_{j=1}^n$ be any orthonormal basis for
$\C^n$, let $h_j=v_j / \sqrt{n}$ for each $i$, let $f \in \C^n$ be
arbitrary, and define maps $A_j$ for $j=1, \ldots, n$ by $A_j = f
h_j^*$. Using the trace conditions on the maps $Y_k$, $Z_k$,
$\{T_{k_i}\}$, $\{S_{k_i}\}$, and $\{X_{k_i}\}$, we find that
\begin{eqnarray*} \sum_{j=1}^n(\psi_{\epsilon_k} - (\xi_{\epsilon_k})^{-1})(A_j)
h_j & = & (t_k -s_k) f + (Z_k - Y_k)f \\ & = & \Big(\sum_{i=1}^{p_k}
|x_{k_i}|^2\Big)f + \Big(\sum_{i=1}^{p_k} \overline{x_{k_i}}
X_{k_i}\Big)f.\end{eqnarray*} Since $f$ was arbitrary, it follows
that $$\Big(t_k - s_k - \sum_{i=1}^{p_k} |x_{k_i}|^2\Big)I =
\Big(\sum_{i=1}^{p_k} \overline{x_{k_i}} X_{k_i} \Big) -
(Z_k-Y_k).$$ Taking the trace of both sides yields
$$0 = tr\Big(\Big(\sum_{i=1}^{p_k} \overline{x_{k_i}} X_{k_i} \Big) - (Z_k-Y_k)\Big) =
tr\Big((t_k - s_k -\sum_{i=1}^{p_k} |x_{k_i}|^2)I\Big),$$ so $t_k -
s_k = \sum_{i=1}^{p_k} |x_{k_i}|^2$ and $Z_k - Y_k =
\sum_{i=1}^{p_k} \overline{x_{k_i}}X_{k_i}$. Formulas \eqref{l1}
and \eqref{l2} now imply that $$\sum_{i=1}^{p_k} X_{k_i}AX_{k_i}^*
=\Big(\sum_{i=1}^{m_k} S_{k_i} A S_{k_i}^* -
\sum_{i=1}^{\ell_k}T_{k_i} A T_{k_i}^* \Big).$$ Therefore, the map
$A \rightarrow \sum_{i=1}^{m_k} S_{k_i} A S_{k_i}^* -
\sum_{i=1}^{\ell_k}T_{k_i} A T_{k_i}^*$ is completely positive, and
$$\Big|\Big|\sum_{i=1}^{p_k} X_{k_i}X_{k_i}^*\Big|\Big| =
\Big|\Big|\sum_{i=1}^{m_k} S_{k_i}S_{k_i}^* -
\sum_{i=1}^{\ell_k}T_{k_i}T_{k_i}^*\Big|\Big| \leq
\Big|\Big|\sum_{i=1}^{m_k} S_{k_i}S_{k_i}^*\Big|\Big|,$$
establishing
\eqref{Xis}.
We now show that there exists some $M \in \mathbb{N}$ such that
\begin{eqnarray}\label{Xibound}||X_{k_i}|| \leq M\end{eqnarray}
for all $k \in \mathbb{N}$ and $i \in \{1, \ldots, p_k\}$. To do
this, we first note that since the sequence of invertible maps
$\{\xi_{\epsilon_k}\}_{k \in \mathbb{N}}$ converges in norm to the
invertible map $\xi$, the sequence $\{(\xi_{\epsilon_k})^{-1}\}_{k
\in \mathbb{N}}$ converges in norm to $\xi^{-1}$. Write $\xi^{-1}$
in the form
$$\xi^{-1}(A) = sA + YA + AY^* - \sum_{i=1}^m S_i A S_i^*,$$ where $m \leq n^2$,
$s \in \R$,
$tr(Y)=0$, and for all $i$ and $j$, $tr(S_i)=0$ and $tr(S_iS_j^*)$ is non-zero if
and only if $i=j$.
Let $f \in \C^n$ be arbitrary, and define vectors $\{h_j\}_{j=1}^n$
and maps $\{A_j\}_{j=1}^n$ exactly as we did earlier in the proof. Then
$\sum_{j=1}^n (\xi_{\epsilon_k})^{-1}(A_j)h_j = s_k f + Y_kf$ for all $k \in \mathbb{N}$
and $\sum_{j=1}^n \xi^{-1}(A_j)h_j = s f + Yf$. Since $(\xi_{\epsilon_k})^{-1}$
converges to $\xi^{-1}$ as $k \rightarrow \infty$, we see that
$(s_k - s) f + (Y_k - Y)f$
converges to $0$ as $k \rightarrow \infty$. But $f$ was arbitrary, so
$$\lim_{k \rightarrow \infty} \Big((s_k - s) I + Y_k - Y \Big)= 0.$$ The
limit of the trace
of the above equation must also be zero, so $s_k$ converges to $s$
and consequently $Y_k$ converges to $Y$.
This implies that not only are the sequences of complex numbers $\{s_k\}_{k=1}^\infty$
and maps $\{Y_k\}_{k=1}^\infty$ both bounded, but that
the sequence of linear maps $\{W_k\}_{k=1}^\infty$ defined by
$W_k(A)= \sum_{i=1}^{m_k} S_{k_i}AS_{k_i}^*$
is bounded and converges to the map $W(A) = \sum_{i=1}^m S_i A S_i^*$.
Choose $M \in \mathbb{N}$ so that $M^2 \geq n^2 \sup_{k
\in \mathbb{N}} \{||W_k||\}$. For every $k \in \mathbb{N}$ and $i
\in \{1, \ldots, m_k\}$, we have $||S_{k_i}||^2 \leq ||W_k|| \leq
M^2/n^2$. Combining this fact with \eqref{Xis}, we find that for
every $k \in \mathbb{N}$ and $i \in \{1, \ldots, p_k\}$,
\begin{eqnarray*}||X_{k_i}||^2 & = & ||X_{k_i}X_{k_i}^*|| \leq
||\sum_{i=1}^{p_k} X_{k_i}X_{k_i}^*|| \leq ||\sum_{i=1}^{m_k}
S_{k_i}S_{k_i}^*|| \leq \sum_{i=1}^{m_k}||S_{k_i}||^2
\\ & \leq & n^2 \max\{||S_{k_i}||^2: i=1, \dots, m_k\} \leq M^2,
\end{eqnarray*} proving \eqref{Xibound}.
Since $||\psi_{\epsilon_k}||
\rightarrow \infty$ as $k \rightarrow \infty$ while
$||(\xi_{\epsilon_k})^{-1}|| \rightarrow ||(\xi)^{-1}|| < \infty$,
there is a sequence of maps $\{A_{\epsilon_k}\}$ of norm one such
that
$||(\psi_{\epsilon_k}-(\xi_{\epsilon_k})^{-1})(A_{\epsilon_k})||
\rightarrow \infty$ as $k \rightarrow \infty$. However, we also
have
\begin{eqnarray}||(\psi_{\epsilon_k}-(\xi_{\epsilon_k})^{-1})(A_{\epsilon_k})||
& = &\Big|\Big| \Big(\sum_{i=1}^{p_k} |x_{k_i}|^2\Big)A_{\epsilon_k}
+ \Big(\sum_{i=1}^{p_k}
\overline{x_{k_i}}X_{k_i}\Big)A_{\epsilon_k} \nonumber \\ & \ & \ +
A_{\epsilon_k}\Big(\sum_{i=1}^{p_k} \overline{x_{k_i}}X_{k_i}\Big)^*
+ \sum_{i=1}^{p_k} X_{k_i}A_{\epsilon_k}X_{k_i}^*\Big|
\Big| \nonumber \\
& \label{above} \leq & \sum_{i=1}^{p_k} |x_{k_i}|^2 + 2 M \sum_{i=1}^{p_k}
|x_{k_i}| + p_k M^2.
\end{eqnarray}
We note that
\begin{eqnarray} \label{prob} \Big(\sum_{i=1}^{p_k}|x_{k_i}|\Big)^2 \geq
\sum_{i=1}^{p_k}|x_{k_i}|^2 \geq
\frac{(\sum_{i=1}^{p_k}|x_{k_i}|)^2}{p_k} \geq
\frac{(\sum_{i=1}^{p_k}|x_{k_i}|)^2}{n^2}\end{eqnarray} for all $k$.
For each $k$, let $\lambda_k = \sum_{i=1}^{p_k} |x_{k_i}|$, noting that
$\lambda_k \rightarrow \infty$ as $k \rightarrow
\infty$ since Eq.\eqref{above} tends to infinity as $k \rightarrow \infty$.
Let $A
\in M_n(\C)$ be any matrix such that $||A||=1$, and let $C= \sup_{k
\in \mathbb{N}} ||(\xi_{\epsilon_k})^{-1}||< \infty$. Using the
reverse triangle inequality and \eqref{prob}, we find that for each
$k \in \mathbb{N}$,
\begin{eqnarray}||\psi_{\epsilon_k}(A)|| & \geq &
||(\psi_{\epsilon_k}-(\xi_{\epsilon_k})^{-1})(A)|| -
||(\xi_{\epsilon_k})^{-1}(A)|| \nonumber \\ \label{last} & \geq &
\frac{\lambda_k^2}{n^2} - 2M \lambda_k - n^2 M^2-C.\end{eqnarray}
Since $\lim_{k \rightarrow \infty} \lambda_k = \infty$,
Eq.\eqref{last} tends to infinity as $k \rightarrow \infty$. For all
$k$ large enough that Eq.\eqref{last} is positive, we have
$$||\phi_{\epsilon_k}|| =\frac{1}{\inf \{||\psi_{\epsilon_k}(A)||: ||A||=1
\}} \leq \frac{1}{\lambda_k^2 / n^2 - 2M \lambda_k - n^2 M^2-C},$$
so $\lim_{k \rightarrow \infty} ||\phi_{\epsilon_k}|| = 0$. But the
sequence $\{\phi_{\epsilon_k}\}_{k=1}^\infty$ converges to $\phi$ in
norm, hence $\phi \equiv 0$. \qed \end{pf}
\begin{prop}\label{cnegpure} An invertible unital linear map $\phi: M_n(\C)
\rightarrow M_n(\C)$ is $q$-pure if and only if $\phi^{-1}$ is of
the form
$$\phi^{-1}(A) = A + YA + AY^*$$ for some $Y=-Y^* \in M_n(\C)$ such
that $tr(Y)=0$.
\end{prop}
\begin{pf} Let $\psi=\phi^{-1}$. Assume the hypotheses of the
forward direction. Write
$$\psi(A) = sA + YA + AY^* - \sum_{i=1}^k \lambda_i X_i A X_i^*,$$
where $s \in \R$, $tr(Y)=0$, and for each $i$ and $j$ we have
$\lambda_i \geq 0$, $tr(X_i)=0$, and $tr(X_i^*X_j)= n \delta_{ij}$.
Defining $\psi': M_n(\C) \rightarrow M_n(\C)$ by
$$\psi'(A) = sA+YA+AY^*,$$
we note that $\psi'$ is conditionally negative, and $\psi'-\psi$ is
completely positive since $(\psi'-\psi)(A)= \sum_{j=1}^k \lambda_j
X_j A X_j^*$ for all $A$. By Lemma \ref{invsubs}, it follows that
$\psi'$ is invertible and that $\phi':=(\psi')^{-1}$ satisfies $\phi
\geq_q \phi' \geq_q 0$.
Since $\phi$ is $q$-pure, there is some $t_0 \geq 0$ such that
$\phi' = \phi^{(t_0)}$, hence
$$\psi'=(\phi')^{-1}=\Big(\phi(I + t_0 \phi)^{-1}\Big)^{-1} =\Big(\psi^{-1}(I + t_0 \psi^{-1})\Big)^{-1}= \Big((t_0
I + \psi)^{-1}\Big)^{-1}= t_0I + \psi.$$ Therefore, for all $A \in
M_n(\C)$ we have
$$\psi'(A) =\psi(A)+ \sum_{j=1}^k
\lambda_j X_j A X_j^* = \psi(A) + t_0A,$$ so the map $L: A
\rightarrow \lambda_j X_j A X_j^*$ satisfies $L=t_0 I$. We repeat a
familiar argument: Let $f \in \C^n$ be arbitrary, choose an
orthonormal basis $\{v_k\}_{k=1}^n$ of $\C^n$, define $h_k = v_k/
\sqrt{n}$ for each $k$, and form $\{A_k\}_{k=1}^n$ by $A_k =
fh_k^*$. The trace conditions for the maps $\{X_j\}$ imply that
$\sum_{k=1}^n L(A_k)h_k=0$. However, since $L=t_0 I$, we must also
have $\sum_{k=1}^n L(A_k)h_k = t_0 f$. From arbitrariness of $f$, we
conclude $t_0=0$. Therefore, $\psi$ has the form $\psi(A)=sA + YA +
AY^*$. Since $\psi(I)=I=sI+Y+Y^*$ and $tr(Y)=0$, we have $s=1$ and
consequently $Y=-Y^*$.
Now assume the hypotheses of the backward direction. Note that
$\psi$ is conditionally negative and unital, hence $\phi$ is
$q$-positive by Proposition \ref{cnegone}. Let $\Phi$ be any non-zero
$q$-positive map such that $\phi \geq_q \Phi$, so by Corollary
\ref{invsubsone} and Proposition \ref{invert}, $\Phi$ is invertible
and $\Psi:=(\Phi)^{-1}$ is a conditionally negative map such that
$\Psi-\psi$ is completely positive. Write $\Psi$ in the form
$$\Psi(A) = s'A+ ZA + AZ^* - \sum_{i=1}^{m} \mu_i T_i A T_i^*,$$ where
$s' \in \R$ and for all $i$ and $j$, $\mu_i > 0$, $tr(T_i)=0$, and
$tr(T_i^*T_j)=n \delta_{ij}$. Writing $C=Z-Y$, we have
$$(\Psi-\psi)(A)= (s'-1)A + CA+AC^* - \sum_{i=1}^m \mu_i T_i A T_i^*.$$
By a familiar argument, complete positivity of $\Psi - \psi$ and the trace conditions for
the above maps imply that $s' \geq 1$, $C=0$, and $T_i=0$ for all
$i$. Therefore $\Psi = \psi + (s'-1)I$, so $\Phi=\Psi^{-1}=
\phi^{(s'-1)}$. We conclude that $\phi$ is $q$-pure. \qed \end{pf}
Let the matrices $\{e_{jk}\}_{j,k=1}^n$ denote the standard basis for $M_n(\C)$,
writing each $A= (a_{jk}) \in M_n(\C)$ as $A= \sum_{j,k} a_{jk}e_{jk}$. The
following theorem classifies all unital invertible $q$-pure maps on $M_n(\C)$:
\begin{thm}\label{phiu}An invertible unital linear map $\phi: M_n(\C)
\rightarrow M_n(\C)$ is $q$-pure if and only if for some unitary $U
\in M_n(\C)$, the map $\phi_U$ is the Schur map
\begin{equation*} \phi_U(a_{jk}e_{jk}) = \left\{
\begin{array}{cc}
\frac{a_{jk}}{1+i(\lambda_j - \lambda_k)}e_{jk} & \textrm{if } j<k \\
a_{jk}e_{jk} & \textrm{if } j=k \\
\frac{a_{jk}}{1-i(\lambda_j - \lambda_k)}e_{jk}& \textrm{if } j>k
\end{array} \right.
\end{equation*}
for all $A= (a_{jk}) \in M_n(\C)$ and $j,k=1, \ldots, n$, where $\lambda_1, \ldots, \lambda_n \in
\R$ and $\lambda_1 + \ldots + \lambda_n = 0$.
\end{thm}
\begin{pf} Assume the hypotheses of the forward direction. By the previous
proposition, $\psi:=\phi^{-1}$ has the form $\psi(A) = A +
\tilde{Y}A + A \tilde{Y}^*$ for some $\tilde{Y} \in M_n(\C)$ with
$\tilde{Y}=-\tilde{Y^*}$ and $tr(\tilde{Y})=0$. Let $B=
-i\tilde{Y}$, so $B=B^*$. Defining $Y:= (1/2)I
+\tilde{Y}=(1/2)I+iB$, we find $\psi(A)= YA+AY^*$ for all
$A\in M_n(\C)$. Since $B$ is self-adjoint, there is some unitary $U
\in M_n(\C)$ such that $U^* B U$ is a diagonal matrix $D$. For each
$k \in \{1, \ldots, n\}$ let $\lambda_k \in \R$ be the $kk$ entry of
$D$. Note that since $tr(B)=0$ we have $\sum_{k=1}^n \lambda_k =
0$, and that $U^*YU$ is the diagonal matrix $M$ whose $kk$ entry is
$1/2 +i \lambda_k$. Defining a map $\psi_U$ by
$\psi_U(A)=U^*\psi(UAU^*)U$ for all $A \in M_n(\C)$, we find that
\begin{eqnarray*}\psi_U(A) &=& U^*(YUAU^* + UAU^*Y^*)U
\\ &=&(U^*YU)A+A(U^*YU)^* = MA+AM^*.\end{eqnarray*}
A quick calculation shows that this is just the Schur map
\begin{equation*} \psi_U(a_{jk}e_{jk}) = \left\{
\begin{array}{cc}
({1+i(\lambda_j - \lambda_k)})a_{jk}e_{jk} & \textrm{if } j<k \\
a_{jk}e_{jk} & \textrm{if } j=k \\
({1-i(\lambda_j - \lambda_k)})a_{jk}e_{jk}& \textrm{if } j>k
\end{array} \right.,
\end{equation*}
and so $(\psi_U)^{-1}$ has the form
\begin{equation*} (\psi_U)^{-1}(a_{jk}e_{jk}) = \left\{
\begin{array}{cc}
\frac{a_{jk}}{1+i(\lambda_j - \lambda_k)}e_{jk} & \textrm{if } j<k \\
a_{jk}e_{jk} & \textrm{if } j=k \\
\frac{a_{jk}}{1-i(\lambda_j - \lambda_k)}e_{jk}& \textrm{if } j>k
\end{array} \right.. \end{equation*} It is straightforward to verify
that $(\psi_U)^{-1}$ is the map $\phi_U(A)= U^*\phi(UAU^*)U$.
Assume the hypotheses of the backward direction. Let $T$ be the
diagonal matrix whose $kk$th entry is $\lambda_k$ for every $k=1,
\ldots, n$. We observe that $tr(T)=0$ and $T=T^*$. Now let $C=iT$,
and let $\tilde{T}= (1/2)I +C$. We routinely verify that
$C=-C^*$ and $tr(C)=0$, and that $(\phi_U)^{-1}$ satisfies
$(\phi_U)^{-1}(A)=\tilde{T}A+A\tilde{T}^* = A + CA + AC^*$ for all
$A \in M_n(\C)$. Proposition \ref{cnegpure} implies that $\phi_U$
is $q$-pure, whereby $\phi$ is $q$-pure by Proposition
\ref{basischange}.
\qed \end{pf}
As it turns out, boundary weight doubles $(\phi, \nu)$ for invertible unital $q$-pure maps
$\phi: M_n(\C) \rightarrow M_n(\C)$ and normalized unbounded boundary weights
$\nu$ over $L^2(0,
\infty)$ of the form $\nu(\sqrt{I - \Lambda(1)} B \sqrt{I -
\Lambda(1)}) = (f,Bf)$ give us
nothing new in terms of $E_0$-semigroups:
\begin{thm} Let $\phi: M_n(\C) \rightarrow M_n(\C)$ be unital, invertible, and
$q$-pure, and let $\nu$ be a normalized unbounded boundary weight
over $L^2(0,\infty)$ of the form $\nu(\sqrt{I - \Lambda(1)} B
\sqrt{I - \Lambda(1)}) = (f,Bf)$. Then $(\phi, \nu)$ and
$(\imath_\C, \nu)$ induce cocycle conjugate $E_0$-semigroups.
\end{thm}
\begin{pf} By Theorem \ref{phiu} and Propositions
\ref{basischange} and \ref{hypqc}, we may assume that $\phi$ is the
Schur map
\begin{equation*} \phi(a_{jk}e_{jk}) = \left\{
\begin{array}{cc}
\frac{a_{jk}}{1+i(\lambda_j - \lambda_k)}e_{jk} & \textrm{if } j<k \\
a_{jk}e_{jk} & \textrm{if } j=k \\
\frac{a_{jk}}{1-i(\lambda_j - \lambda_k)}e_{jk}& \textrm{if } j>k
\end{array} \right. \end{equation*}
for some $\lambda_1, \ldots, \lambda_n \in \R$ with $\sum_{k=1}^n
\lambda_k = 0$.
By Proposition \ref{hypqc}, it suffices to find a hyper maximal
$q$-corner from $\phi$ to $\imath_\C$. For this, define $\gamma:
M_{n \times 1}(\C) \rightarrow M_{n \times 1}(\C)$ by
\begin{displaymath} \gamma \left(
\begin{array}{c}
b_1 \\ b_2 \\ \vdots \\ b_n \end{array} \right) = \left(
\begin{array}{c}
\frac{1}{1+i\lambda_1 } b_1 \\
\frac{1}{1+i\lambda_2} b_2 \\
\vdots \\
\frac{1}{1+i\lambda_n} b_n \end{array} \right).
\end{displaymath}
Now define $\Upsilon: M_{n+1}(\C) \rightarrow M_{n+1}(\C)$ by
\begin{displaymath} \Upsilon \left(
\begin{array}{ccc}
A_{n \times n} & B_{n \times 1} \\
C_{1 \times n} & a \end{array} \right) = \left(
\begin{array}{ccc}
\phi(A_{n \times n}) & \gamma(B_{n \times 1}) \\
\gamma^*(C_{1 \times n}) & a
\end{array} \right).
\end{displaymath}
Letting $\lambda_{n+1}=0$, we observe that $\Upsilon$ is the Schur
map satisfying
\begin{equation*} \Upsilon(a_{jk}e_{jk}) = \left\{
\begin{array}{cc}
\frac{a_{jk}}{1+i(\lambda_j - \lambda_k)}e_{jk} & \textrm{if } j<k \\
a_{jk}e_{jk} & \textrm{if } j=k \\
\frac{a_{jk}}{1-i(\lambda_j - \lambda_k)}e_{jk}& \textrm{if } j>k
\end{array} \right. \end{equation*} for all $j,k=1, \ldots, n+1$ and
$A=(a_{jk}) \in M_n(\C)$. Since $\sum_{i=1}^{n+1} \lambda_k =
\sum_{i=1}^{n} \lambda_k = 0$, it follows from Theorem \ref{phiu}
that $\Upsilon$ is $q$-positive (in fact, $q$-pure), hence $\gamma$
is a $q$-corner from $\phi$ to $\imath_\C$. Now suppose that
$\Upsilon \geq_q \Upsilon' \geq_q 0$ for some $\Upsilon'$ of the
form
\begin{displaymath} \Upsilon' \left(
\begin{array}{ccc}
A_{n \times n} & B_{n \times 1} \\
C_{1 \times n} & a \end{array} \right) = \left(
\begin{array}{ccc}
\phi'(A_{n \times n}) & \gamma(B_{n \times 1}) \\
\gamma^*(C_{1 \times n}) & \imath'(a)
\end{array} \right).
\end{displaymath}
Since $\Upsilon$ is $q$-pure and $\Upsilon'$ is not the zero map, we
know that $\Upsilon' = \Upsilon^{(t)}$ for some $t \geq 0$, and a
quick calculation gives us
\begin{displaymath} \Upsilon' \left(
\begin{array}{ccc}
A_{n \times n} & B_{n \times 1} \\
C_{1 \times n} & a \end{array} \right) = \left( \begin{array}{cc}
\phi^{(t)}(A_{n \times n}) & \gamma^{(t)}(B_{n \times 1}) \\
(\gamma^*)^{(t)}(C_{1 \times n}) & \frac{1}{1+t}(a)
\end{array} \right).
\end{displaymath}
By inspecting the two formulas for $\Upsilon'$ we see
$\gamma=\gamma^{(t)}$. But $\gamma^{(t)}$ has the form
\begin{displaymath} \gamma^{(t)} \left(
\begin{array}{c}
b_1 \\ b_2 \\ \vdots \\ b_n \end{array} \right) = \left(
\begin{array}{c}
\frac{1}{1+t+i\lambda_1 } b_1 \\
\frac{1}{1+t+i\lambda_2} b_2 \\
\vdots \\
\frac{1}{1+t+i\lambda_n} b_n \end{array} \right),
\end{displaymath} hence $t=0$. Therefore, $\Upsilon'=\Upsilon$,
and we conclude the $q$-corner $\gamma$ is hyper maximal. \qed
\end{pf}
In conclusion, we approach the broader question of simply
finding all unital $q$-pure maps $\phi: M_n(\C) \rightarrow M_n(\C)$, as
they provide us with the simplest way to construct and compare
$E_0$-semigroups
through boundary weight doubles. We believe that all $q$-pure maps
are invertible or have rank one. For $n=2$, we find in \cite{Me2}
that this conjecture holds: There is no unital $q$-pure map $\phi:
M_2(\C) \rightarrow M_2(\C)$ of rank 2, and there is no unital
$q$-positive map $\phi: M_2(\C) \rightarrow M_2(\C)$ of rank 3.
It seems that for $n=3$, the key to classifying unital $q$-pure
maps is through investigation of the limits $L_\phi = \lim_{t
\rightarrow \infty} t \phi(I + t \phi)^{-1}$, though the situation
becomes very complicated if $n>3$.
\begin{center} {\bf Acknowledgments} \end{center}
The author very gratefully thanks his thesis advisor, Robert Powers,
for his boundless enthusiasm, constant encouragement, and guidance
in research. His help in the author's thesis work has been
indispensable. The author would also like to thank Geoff Price for
proofreading an earlier draft of the paper and making suggestions.
| {'timestamp': '2010-06-01T02:03:18', 'yymm': '0905', 'arxiv_id': '0905.2708', 'language': 'en', 'url': 'https://arxiv.org/abs/0905.2708'} |
\section{Introduction}
\label{introduction}
Disk galaxies support stellar spiral density waves \citep{bertinetal89a}
that interact with each other at resonances \citep{tagger87} to produce
complex, regenerating structures \citep[see review in][]{sellwood2014}.
These waves also interact with the gas to produce even more complex
structures, including gas clumps in the arms \citep{kim01}, gas feathers
downstream from the main arms \citep{kimostriker02}, and multiple shocks
\citep{lugovskii14}. These and other gas structures have further
connections with star formation, which can produce its own
structure, such as shells, pillars, and local blowout cavities
\citep[see review in][]{elmegreen12}. The quantification of all this
structure for the purpose of comparing different regions or different
galaxies, or comparing theory with observations, is difficult because the
structure is multi-scale, multi-component (gas, old stars, young stars,
etc.), and spatially varying.
Here we describe a new method for quantifying spiral structure and its
spatial variations covering a wide range of scales. We apply this method
to the characterization of spiral arms and feathers for two grand design
galaxies, M81 and M51. The $8\mu$m images from Spitzer are used because
they show essentially all of the diffuse gas structures in PAH emission
with high angular resolution; $0.748^{\prime\prime}$ for M81, which is 13
pc for a distance of 3.6 Mpc, and $1.222^{\prime\prime}$ for M51, which is
63 pc for a distance of 10.6 Mpc (NASA/IPAC Extragalactic Database,
http://ned.ipac.caltech.edu).
\cite{sandage1961} noted the prominence of dust lanes in late-type spiral
galaxies, describing stellar ``branches'' in the main arms and dust
``filaments'' that cut across the arms, especially in the case of M51.
\cite{lynds1970} studied 17 late-type spirals using archival photographic
plates from the Mount Wilson and Palomar observatories. She also noted the
presence of main dust lanes in the arms and thin ``feathers'' of dust with
large pitch angles cutting across the arms. \cite{weaver70} placed the sun
in a stellar ``spur'' or ``offshoot" of the Sagittarius arm in the Milky
Way. \cite{elmegreen80} studied seven spirals to investigate in more detail
the properties of {\it stellar} spurs and determined that they have pitch
angles $\sim50^\circ$ larger than the main arms. She noted that gaseous
feathers and stellar spurs have similar pitch angles and suggested they may
have a common origin \citep[see also][]{piddington1973}. A large survey of
dust feathers in spiral galaxies was conducted by \cite{lavigneetal2006}
using archival data from the Hubble Space Telescope. They found a decrease
in spur separation with increasing gas density, suggesting that
gravitational instabilities are involved. Periodic dust feathers occurred
in 20\% of their Sb and Sc types.
In what follows, we define ``spurs'' as interarm stellar features that jut
out from otherwise continuous stellar arms, as distinct from branches,
where one arm ends and turns into two arms further out. Similarly, we
define ``feathers'' as gas or dust features that jut out from the main
stellar arms. These terms are consistent with the usages referenced above.
The present study is about feathers because we use dust emission to
delineate the structures.
Interarm dust feathers were first seen in emission at $15\mu$m for M51
\citep{block97} with ISOCAM \citep{cesarsky96}. They were first seen at
arcsec resolution in M81 with Spitzer $8\mu$m images
\citep{willneretal2004}. Feathers are bright in molecular line emission as
well, and they often have associated H$\alpha$ \citep[for M51,
see][]{scoville01,corderetal2008,koda09,schinnereretal2013}.
\cite{chandar11} note that two prominent feathers in M51 contain clusters
$\sim10^8$ yrs old and suggest this is consistent with simulations of
global spiral density wave arms in \cite{dobbs10}.
Efforts to explain spurs and feathers began with \cite{balbus88}, who
performed a local gas dynamical stability analysis of a single-fluid
polytropic flow through a spiral arm. He found that gravitational
instabilities in the presence of reverse shear and expansion downstream
from the arm can be reinforced by epicyclic motions and lead to the growth
of features with large pitch angles. Two-dimensional, time-dependant,
magnetohydrodynamic simulations of a self-gravitating, differentially
rotating piece of a gas disk with a steady spiral potential were studied by
\cite{kimostriker02}. They also found that gravitational instabilities in
the compressed arm gas produced feathers downstream as a result of
expansion and reverse shear. Extension to three-dimensions confirmed these
results, although the separation between feathers increased because of the
dilution of the in-plane component of gravity \citep{kim06}.
\cite{chakrabartietal2003} studied the growth of small-scale gas features
in a steadily imposed stellar spiral wave, finding feathers, branches, and
other chaotic features with a link to ultraharmonic resonances and
gravitational instabilities in the arms.
\cite{wadakoda2004} discovered an additional ``wiggle'' instability that
operates even without self-gravity when the reverse shear downstream from
an arm triggers something like a Kelvin-Helmholtz instability. Magnetic
fields \citep{shetty06,dobbsprice08} and three-dimensional effects such as
vertical shear \citep{kim06} may stabilize this, however.
\cite{dwarkadas96} suggested that the radial component of the post-shock
flow stabilizes the Kelvin-Helmholtz instability. \cite{kim14} looked at
the wiggle instability again and showed analytically and with hydrodynamic
non-gravitating simulations that it can arise from accumulated potential
vorticity in gas that flows successively through many irregular shock
fronts. They also explained why \cite{dwarkadas96} did not see the
instability; i.e., the growth rate for their long wavelength perturbations
was too slow. \cite{dobbs06} also got interarm feathers without
self-gravity, from sheared and expanded gas structures that form randomly
in the cold gas of spiral arms and then flow into the interarm region.
\cite{shetty06} extended the \cite{kimostriker02} two-dimensional
self-gravitating simulations to global scales and confirmed that feathers
grow from perturbations in spiral shocks for the inner regions. In the
outer regions, they found that interarm filaments can grow by local
self-gravity even without an imposed spiral. \cite{wada08} included
realistic heating and cooling in a three-dimensional self-gravitating
hydrodynamics simulation with an imposed spiral and found feathers
connected with the main shocks. In contrast, \cite{wada11} got no spurs or
feathers in N-body+SPH models with live stars and gas because the spirals
corotated with the disk while forming and re-forming, and the gas fell into
the spiral potential from both sides. In all of these models, gas
irregularities in the flow through a spiral wave of stars is required for
feather formation downstream.
Recently, \cite{leeshu2012} and \cite{lee2013} present an analytic
formulation of two-dimensional feathering in a magnetic, self-gravitating
gas that circulates in a galaxy with a steady stellar spiral. This
formulation involves perturbing the shocked flow in spiral coordinates. A
similar procedure was followed by \cite{elmegreen91} to determine the
growth rate and effective Toomre $Q$ for shearing gravitational
instabilities in a regular spiral density wave flow. Both studies suggest
that these instabilities drive the formation of giant molecular clouds and
star formation in spiral galaxies \citep[see
also][]{khoperskov13,elmegreenetal2014}.
\cite{kimkim14} studied angular momentum transfer and radial gas drift in
two-dimensional hydrodynamical simulations of gas flow in an imposed spiral
arm potential. They show the density distribution in the ($\ln R,\theta$)
plane, which is similar to our display here. Interarm feathers with large
pitch angles are clearly present in their work, and they are connected with
density irregularities in the spiral arm shocks.
Renaud et al. (2013) performed an N-body/adaptive mesh simulation of a
Milky Way-size galaxy at extremely high resolution and got curled spiral
arm structures that resembled Kelvin-Helmholtz instabilities. They made a
distinction between small-scale structures, which host star formation at
their tips, and extended lower-density interarm structures, which have no
star formation.
To provide a quantitative basis for these studies, we introduce a method to
determine the distribution of pitch angles for spiral arms and their
smaller scale substructures. Using M81 and M51 as examples (Section
\ref{thedata}), we first review the standard Fourier transform techniques
and introduce an improved method using correlations in $2\pi$ azimuthal
windows, which give results in agreement with the Fourier techniques
(Section \ref{1D2DFourier_and_correlation}). Section \ref{new_correlation}
then presents a better method that calculates correlations using small
circular windows in $(\ln R,\theta)$ space. This gives local pitch angles
at different scales, depending on the window size. Finally, in Section
\ref{results}, we present the results of this new method for M81 and M51.
\section{The data}
\label{thedata}
Incorporating large-format infrared detector arrays, with the intrinsic
sensitivity of a cryogenically-cooled mirror and the high observing
efficiency of a heliocentric orbit, the Spitzer Space Telescope gives
unparalleled opportunities to study spiral galaxy morphology and dust grain
emission. The IRAC instrument \citep{fazioetal04} comprises four detectors,
which after launch in August 2003, operated for several years at four
wavelengths, these being 3.6 $\mu$m (channel 1), 4.5 $\mu$m (channel 2),
5.8 $\mu$m (channel 3) and 8.0 $\mu$m (channel 4). The IRAC filter band
center is 7.87 $\mu$m in channel 4 \citep[see][] {gehrzetal07}.
For both M81 and M51, we selected IRAC channel 4, wherein one observes
emission from ultra-small ($\sim 0.01 \mu$m) dust grains as well as from
macromolecules (the 8 micron band contains emission from polycyclic
aromatic hydrocarbons (PAHs) at $\lambda$ = 7.7 microns). Small
carbonaceous grains and PAHs undergo temperature spiking
\citep{greenberg1968,sellgren84,greenberg1996,li2004} and can become hotter
than 1000K - 2000K as they transiently absorb a photon from the
interstellar radiation field. It is primarily the emission from such warm
($\sim$ 60 K) tiny dust grains and PAHs, subject to temperature spiking,
which is detected at 8 microns. As emphasized by \cite{bendoetal2008}, PAH
$8\mu$m emission seems to appear in shell-like features around star-forming
regions, and is thus an excellent tracer of these. In contrast, longer
wavelength $24 \mu$m emission is shown by \cite{bendoetal2008} to peak
within star-forming regions. Channel 4 is thus our preferred choice to
trace spiral arm structure, as well as star formation, in the disks of M81
and M51. Our re-binned channel 4 images of M81 and M51 have scales of
$0.748^{\prime\prime}$ pixel$^{-1}$ and $1.222^{\prime\prime}$
pixel$^{-1}$, respectively.
\section{1D and 2D Fourier techniques \& a correlation method}
\label{1D2DFourier_and_correlation}
The most common way to quantify $m$-armed spiral structure in disk galaxies
is with 1D Fourier transforms of the azimuthal profiles \citep[see
e.g.][]{grosboletal2004}. Essentially, for each azimuthal profile
$I_R(\theta)$, the order-$m$ Fourier coefficients $A_m$ are calculated as
functions of radius $R$ using the equation
\begin{equation}
A_m(R)=\int_{-\pi}^{\pi} I_R(\theta)e^{-im\theta}d\theta .
\label{equation_1dft}
\end{equation}
The amplitude of $A_m(R)$ determines the radial region where a given $m$
structure is important; the phase of $A_m(R)$ can be used to estimate the
pitch angle of the order-$m$ spiral arm.
Bidimensional Fourier techniques have been discussed in a number of papers
\citep[e.g.,][amongst
others]{kalnajs75,consathan82,iyeetal82,krakowetal1982,pueraridottori92,
puerari93,davidivanio99,puerarietal00}. They give global pitch angles,
averaged over a galaxy, or average pitch angles as a function of
galactocentric radius \citep{savchenko2012,davisetal2012}. The
bidimensional Fourier coefficients $A(m,p)$ in a basis of logarithmic
spirals can be calculated as
\begin{equation}
A(m,p)=\int_{u_{min}}^{u_{max}} \int_{-\pi}^{\pi} I(u,\theta)e^{-i(m\theta+pu)}d\theta du
\label{equation_2dft}
\end{equation}
where $u=\ln R$, $m$ is the azimuthal frequency (related to $\theta$), and
$p$ is the frequency related to $\ln R$. The pitch angle $P$ of the
order-$m$ spiral arm is given by $\tan P=-m/p$. $I(u,\theta)$ is the light
distribution of the galaxy in a $(u,\theta)$ plane. Hence, once the radial
annulus to be analyzed is chosen by fixing $u_{min}$ and $u_{max}$, the
amplitude of the complex matrix $A(m,p)$ will show the most probable pitch
angle $P$ of that $m$ structure in the annulus.
We discuss now a new method to extract the same information as the
bidimensional Fourier transform. We construct a family of synthetic
logarithmic spirals inside a $2\pi$ window in $(\ln R, \theta)$
coordinates, i.e., with a given $\Delta \ln R$ and $\Delta \theta=2\pi$ for
radius $R$ and azimuthal angle $\theta$. The synthetic spirals have a pitch
angle $P$ and a number of arms $m$, so their curves of constant phase are
given by $\ln R={m\over p}\theta+\Phi$ for some constant $\Phi$. The pitch
angle $P$ of the spiral in this formalism is given by $\tan P={{\Delta
R\over R\Delta \theta}} = {{\Delta \ln R\over \Delta \theta}} = {-{m\over p}}$.
The phase of the spiral is defined to be zero, which corresponds to the
crest of the synthetic arm, at the center of the window, so that $\Phi=\ln
R_{\rm min}+0.5\Delta \ln R$ for minimum window radius $R_{\rm min}$. Then
the synthetic spiral, $S$, is the cosine function of the phase,
\begin{equation}
S_{m,p}(\ln R,\theta)=\cos(m\theta+p\ln R+p\ln R_{\rm mid})
\end{equation}
With this definition of $\Phi$, the synthetic arm peak is in the middle of the
window for all assumed $m$ and $p$. Note that for $p>0$, the azimuthal angle
increases in the usual sense for cylindrical coordinates, counter-clockwise, as the
radius increases, corresponding to a ``Z''-type morphology, while for $p<0$, the
spiral has an ``S''-type morphology.
The deprojected image $I$ of a galaxy - now sampled in a $(\ln R, \theta)$
plane - is then cross-correlated with the synthetic logarithmic spiral,
giving the correlation
\begin{equation}
C_m(\ln R,\theta,p)=\sum_x \sum_y S_{m,p}(\ln R+y,\theta+x)I(\ln R,\theta)
\label{corr_method_1}
\end{equation}
where the summation is over pixels in the range $x=-\pi$ to $\pi$ and
$y=\ln R-0.5\Delta \ln R$ to $\ln R+0.5\Delta \ln R$. This is the sum over
the local $(x,y)$ coordinates inside the window. There is a different
correlation sum for each image coordinate $(\ln R,\theta)$, and for each
$p$ and $m$, i.e., $C$ is a 4 dimensional matrix. The pitch angle for an
$m$ structure at a given radius $\ln R_{\rm mid}$, is calculated using the
$p$ of the the maximum $C_m(\ln R_{mid},\theta, p)$, maximized in $\theta$.
This corresponds to the best-fit pitch angle for an $m-$arm spiral going
through the image coordinate $(\ln R_{mid},\theta)$.
The results of the application of the techniques discussed above are shown
for M81 in Figures \ref{galaxy_xy_lrt} to \ref{pitch_angle_all_methods}. In
Figure \ref{galaxy_xy_lrt}, we display the deprojected $8\mu$m image of M81
\citep[$PA=157^{\circ}$ and inclination $\omega=58^{\circ}$,][]{rc3} in 2
ways: in the plane of the galaxy, and in $(\ln R,\theta)$ space. Figure
\ref{1dft} displays the results of the application of equation
\ref{equation_1dft}. The amplitudes of the $m$ coefficients show that $m=2$
is the most prominent feature for this grand design galaxy. We can
distinguish two ``sets'' of $m=2$ structures: one from $R=150$ to $R=247$
arcsec and another from $R=247$ to $R=606$ arcsec. Figure
\ref{savchenko_ours} shows results for the 2D Fourier transform as a
function of radius \citep{savchenko2012} and for our correlation method
($2\pi$ window) for $\Delta\ln R=u_{max}-u_{min}=0.35$. Note that our
correlation method using $m$ synthetic spirals in a $2\pi$ window give
results very similar to those coming from a 2D Fourier transform. Finally,
in Figure \ref{pitch_angle_all_methods} we plot the estimation of pitch
angles for the $m=2$ component as a function of radius for three different
techniques: \cite{savchenko2012}, \cite{davisetal2012} and our correlation
method ($2\pi$ window). Note that Savchenko's and our methods agree very
well for all radii. \cite{davisetal2012} only changed $\ln R_{min}$ in
their 2D Fourier transform. Definitely, the inner point calculated with the
Davis et al. method is affected by the outer structures.
\section{A new correlation method: circular windows in the $(\ln R, \theta)$ space}
\label{new_correlation}
Having checked our correlation method in a $2\pi$ window and getting
results in full agreement to those coming from the 2D Fourier transform, we
introduce a new method, now using circular windows in $(\ln R, \theta)$
space. By using these new windows, we no longer have a restriction on $m$
symmetry. Furthermore, by changing the diameter of the window, we can study
the distribution of the pitch angles at various scales. Inside each window,
we define a filter which is a sine function from 0 to $\pi$, i.e., we have
the maximum of the filter at its center, and the values falling to zero at
the borders of the window (Figure \ref{circular_windows}). So, for each
galaxy image, and for each position in $(\ln R, \theta)$ space, we
correlate the galaxy image with the filter function for different pitch
angles and window diameters. Let's describe the filter as $F=F(\ln R_c,
\theta_c, P_w, D_w)$, that is the circular filter centered at $(\ln R_c,
\theta_c)$, with pitch angle $P_w$, in a window of diameter $D_w$. We have
analysed our images for pitch angles $P_w$ from $-1^\circ$ to $-90^\circ$
and from $+90^\circ$ to $+1^\circ$, and for four different $D_w$ with
values 0.1, 0.2, 0.35 and 0.51 in units of $\ln R$. The algorithm first
transforms the deprojected $(x,y)$ image of a galaxy to a $(\ln R,
\theta)$ matrix. Our correlation method can be described as
\begin{equation}
C(\ln R_c, \theta_c, P_w, D_w)={{\sum_{i=1,N} F(\ln R_c, \theta_c, P_w, D_w)
I(\ln R,\theta)}\over{\sum_{i=1,N} I(\ln R,\theta)}}
\end{equation}
where $\sum_{i=1,N}$ represents the summation over the $N=\pi(D_w/2)^2$
pixels on the circular window. $\sum_{i=1,N} I(\ln R,\theta)$ is a
normalization factor. The correlation will be higher when the pitch angle
of some structure centered at $(\ln R_c, \theta_c)$ coincides with the
pitch angle of the filter $F$. By changing the diameter of the window,
different spatial scales can be analyzed.
\section{Results for the locally determined pitch angles at different scales}
\label{results}
We deprojected the Spitzer $8\mu$m image of M81 using the values of
$PA=157^{\circ}$ and inclination $\omega=58^{\circ}$ \citep{rc3}. For M51,
$PA=170^{\circ}$ and inclination $\omega=20^{\circ}$ were taken, following
\cite{shettyetal2007}. The deprojected images were subjected to our cross
correlation method, using sine filters on circular windows in the $(\ln R,
\theta)$ space as described above.
In Figure \ref{regionsdelineated} we present once again M81 in $(x,y)$ and
$(\ln R, \theta)$, but now we delineate the areas in which we add all the
correlations calculated for a given window diameter for each pitch angle.
Four areas were chosen, two over the arms (blue and red), and two placed in
the interarm regions (green and black). The minimum and maximum radii for
the regions are 330 and 525 arcsec, or $\ln R=$5.8 and 6.26. As already
noted, we use four different window diameters $D_w$= 0.1, 0.2, 0.35, and
0.51 in units of $\ln R$. These circular windows are drawn in the bottom
panel of the figure, and represent the sizes of the features we are
measuring.
Figure \ref{main_results} shows the sum of correlations in each area for
the different window diameters, as a function of pitch angle (left panels,
interarm regions; right panels, arm regions). Thicker lines represent
larger window diameters and therefore represent longer spiral structures.
Thinner lines represent the smaller scale structures related to feathers.
For the interarm regions (green and black) and at smaller scales, we have a
broad range of pitch angles. For the black area, the distribution of pitch
angles has a peak at $P=-28^\circ$. For larger scales, the distribution is
affected by the main spiral arms at the azimuthal edge of the analysed
areas; pitch angles $P=90^\circ-P_{spiral}$ therefore get larger
correlations ($P_{spiral}$ is the pitch angle of the main spiral
structure).
For the arm regions in Figure \ref{main_results} (blue and red), the small
scales present a broad distribution of pitch angles and the large scales
show the main $m=2$ structure of this galaxy, which has a pitch angle of
$P\sim-17^\circ$ (see the marked peak for the red region in Figure
\ref{main_results}; see also Section \ref{1D2DFourier_and_correlation}).
The arm inside the blue area has two main components with $P\sim-22^\circ$
and $P\sim-10^\circ$. These two peaks can be recognized in the blue area in
Figure \ref{regionsdelineated}, bottom panel. At smaller radii, there is a
bright structure presenting a more open pitch angle. The structure at large
radius is similar to that in the red area.
In Figure \ref{main_extra}, we present a zoom of the sum of correlations
for one interarm (black) and one arm (red) region, for the two intermediate
scales $D_w$= 0.2 and 0.35. The arm peaks at a pitch angle around
$P=-17^\circ$, but for the interarm region, the sum of the correlations
points to a more open structure, with a broader distribution in pitch
angles and a peak around $P=-28^\circ$.
As discussed in the introduction, observations by \cite{lavigneetal2006}
and others show that feathers are more open compared to main spiral arms
(see their Figures 1 and 2 where they outline the feathers they detect by
eye in M51 and NGC 628). Numerical simulations \citep[e.g.,][]{kimkim14}
also show that sub-structures evolving from the main arms into the interarm
regions have larger pitch angles compared to the main spiral.
In Figure \ref{cartoon}, we show a sketch of the results of Figure
\ref{main_extra}. The main spiral arm in the red region has $P=-17^\circ$
and is designated by a thick red curve. The dashed lines in the black
region have $P=-28^\circ$. These calculated values agree well with the
structures in the figure.
Figures \ref{regionsdelineatedm51} to \ref{main_extra_m51} are similar to
Figures \ref{regionsdelineated} to \ref{main_extra}, but now for M51. The
observed $(\ln R,\theta)$ map resembles the model at slow pattern speed in
\cite{kimkim14}. Figure \ref{regionsdelineatedm51} delineates the areas we
analysed for M51. We use minimum and maximum radii of 44 and 106 arcsec, or
$\ln R=$3.78 and 4.66. Figure \ref{main_results_m51} shows the sum of the
correlations for all regions. As for M81, the interarm regions on small
scales present a broad distribution of pitch angles up to a large value
(see the arrows in the left panels). The interarms on large scales have a
pitch angle distribution that is dominated by the main spiral arms, giving
large correlations for $P=90^\circ-P_{spiral}$. For the arm regions (right
panels), the spiral arm in the red region is more symmetric than the arm in
the blue region. The main peak for the spiral arm in the blue region is
quite asymmetric. Here again we find secondary peaks for pitch angles
larger than the pitch angle of the main spiral arms. In Figure
\ref{main_extra_m51} the pitch angle for the main spiral arm in the red
region is $P=-19^\circ$, with a secondary bump for $P\sim-52^\circ$ (bottom
panel). The structures in the interarm region have larger pitch angles,
$P\le -40^\circ$.
The pitch angles determined for the $m=2$ spiral arms are $P\sim -18^\circ$
for M81 and $P\sim -19^\circ$ for M51. For M81, \cite{kendalletal2008} used
Spitzer IRAC $3.6$ and $4.5\mu$m images to make an ``eye-ball'' fit giving
$P\sim -23^\circ$. Bash \& Kaufman (1986) estimated the pitch angles in
radio continuum maps. They noticed that the two main arms differ in pitch
angles, deriving $P\sim -17^\circ$ for the eastern arm and $P\sim
-23^\circ$ for the western arm. Similar numbers were determined with
bidimensional Fourier transforms by \cite{puerarietal2009} in a
multiwavelength study using GALEX and Spitzer IRAC images. For M51,
\cite{shettyetal2007} used the CO distribution to determine a pitch angle
of $P=-21.1^\circ$. More recently, \cite{huetal2013} used $P=-17.5^\circ$
from their models of M51. \cite{fletcheretal2011} report an average pitch
angle of $P=-20^\circ$. Evidently, many methods give similar results for
the pitch angles of the main spiral arms in these two galaxies. Here we
reproduce those values on large scales, but also find a broad range of
values up to larger pitch angles on small scales. There is a multiplicity
of spiral structures on different scales, as expected from gas flows in
gravitating, turbulent and shearing interstellar media.
\begin{acknowledgements}
I.P. acknowledges support from the Mexican foundation CONACyT and from the
University of the Witwatersrand.
\end{acknowledgements}
| {'timestamp': '2014-08-19T02:01:54', 'yymm': '1408', 'arxiv_id': '1408.3658', 'language': 'en', 'url': 'https://arxiv.org/abs/1408.3658'} |
\section{Introduction}
\subsection{Cusps of modular curves at infinite level}
Let $p$ be a prime and let $K$ be a perfectoid field extension of $\mathbb{Q}_p$. Throughout we shall assume that $K$ contains all $p^n$-th unit roots for all $n\in \mathbb{N}$. Let $N\geq 3$ be coprime to $p$ and let $\mathcal X^{\ast}$ be the compactified modular curve over $K$ of some rigidifying tame level $\Gamma^p$ such that $\Gamma(N)\subseteq \Gamma^p\subseteq \operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$. Here we consider $\mathcal X^{\ast}$ as an analytic adic space.
The first goal of this article is to give a detailed analytic description of the geometry at the cusps in the inverse system of modular curves with varying level structures at $p$, as well as for the modular curves at infinite level introduced by Scholze in \cite{torsion}. In doing so, we aim to complement results on the boundary of infinite level Siegel varieties for $\mathrm{GSp}_{2g}$ for $g\geq 2$ from \cite[\S III.2.5]{torsion}, proved there using machinery like Hartog's extension principle and a perfectoid version of Riemann's Hebbarkeitssatz: Due to assumptions on the codimension of the boundary to be $\geq 2$, these tools do not apply in the elliptic case. Instead, in this case one can get a much more explicit description with more elementary means.
The way we study the boundary in the elliptic case is in terms of adic analytic parameter spaces for Tate curves. Fix a cusp $x$ of $\mathcal X^\ast$, this is in general a point defined over a finite field extension $K\subseteq L_x\subseteq K[\zeta_N]$ depending on $x$, and it corresponds to a $\Gamma^p$-level structure on the Tate curve $\mathrm{T}(q^{e_x})$ over $\O_{L_x}\cc{q}$ for some $1\leq e_x\leq N$. The analytic Tate curve parameter space in this situation is simply the adic open unit disc $\mathcal D_x$ over $L_x$, and there is a canonical open immersion $\mathcal D_x\hookrightarrow \mathcal X^{\ast}$ that sends the origin to $x$.
In order to state our main result, let us recall the tower of anticanonical moduli spaces from \cite[\S3 III]{torsion}: Away from the cusps, the modular curve $\mathcal X^{\ast}$ is the moduli space of elliptic curves $E$ together with a $\Gamma^p$-level structure.
Let $\mathcal X^{\ast}_{\Gamma_0(p)}\to \mathcal X^{\ast}$ be the finite flat cover that relatively represents (away from the cusps) the data of a cyclic subgroup scheme of rank $p$ of $E[p]$.
By the theory of the canonical subgroup, for small enough $\epsilon>0$, the $\epsilon$-overconvergent neighbourhood of the ordinary locus $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)\subseteq \mathcal X^{\ast}_{\Gamma_0(p)}$ decomposes into two disjoint open components: the canonical locus $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_c$ and the anticanonical locus $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a$. In order to understand the cusps of the perfectoid modular curve at infinite level, we first study the tower of compactified modular curves of higher level at $p$ relatively over $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a$. Specifically, for any $n\in \mathbb{N}$, the pullback of $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a\subseteq \mathcal X^{\ast}_{\Gamma_0(p)}$ defines a tower
\[\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a \to \mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a \to \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a \to\mathcal X^{\ast}(\epsilon)\]
of anticanonical loci. Here the rightmost map is finite flat and totally ramified at the cusps, whereas the map $\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a \to \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ is finite Galois with group
\[ \Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z}):=\{\smallmat{\ast}{\ast}{0}{\ast}\in \operatorname{GL}_2(\mathbb{Z}/p^n\mathbb{Z})\}.\]
The cusps of these moduli spaces of higher finite level can be described using analogous parameter spaces for Tate curves: As we shall discuss in detail in \S2, it is essentially an adic analytic version of the classical calculus of cusps of Katz--Mazur \cite{KatzMazur} that for any cusp $x$ of $\mathcal X^{\ast}$, there are Cartesian diagrams of adic spaces of topologically finite type over $K$
\begin{equation*}
\begin{tikzcd}
\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x} \arrow[d, hook] \arrow[r] & \underline{(\mathbb{Z}/p^n\mathbb{Z})^{\times}}\times \mathcal D_{n,x} \arrow[d, hook] \arrow[r] & \mathbb \mathcal D_{n,x} \arrow[r] \arrow[d, hook] & \mathcal D_x \arrow[d, hook] \\
\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}(\epsilon),
\end{tikzcd}
\end{equation*}
where the top left map is $\smallmat{a}{b}{0}{d}\mapsto d$ in the first component and the identity in the second, and where $\mathcal D_{n,x}$ is the open unit disc in the variable $q^{1/p^n}$ over $L_x$,
\[\mathcal D_{n,x}= \text{ open subspace of }\operatorname{Spa}(L_x\langle q^{1/p^n}\rangle,\O_{L_x}\langle q^{1/p^n}\rangle) \text{ defined by }|q|<1.\]
In the limit $n\to \infty$, these open discs become parameter spaces for Tate curves with infinite $\Gamma_0$-level structure at $p$, given by perfectoid open unit discs
\[\mathcal D_{\infty,x}= \text{ open subspace of }\operatorname{Spa}(L_x\langle q^{1/p^\infty}\rangle,\O_{L_x}\langle q^{1/p^\infty}\rangle) \text{ defined by }|q|<1.\]
We then get the following description of the cusps at infinite level: Let
\[\Gamma_0(p^\infty):=\{\smallmat{\ast}{\ast}{0}{\ast}\}\subseteq \operatorname{GL}_2(\mathbb{Z}_p)\]
and let $\underline{\Gamma_0(p^\infty)}$ be the associated profinite perfectoid space.
\begin{Theorem}\label{t:cusps of X_Gamma(p^infty)}
Let $x$ be any cusp of $\mathcal X^{\ast}$.
\begin{enumerate}
\item There is a Cartesian diagram of perfectoid spaces over $K$
\begin{equation*}
\begin{tikzcd}
\underline{\Gamma_0(p^\infty)}\times \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] & \underline{\mathbb{Z}_p^\times}\times \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] & \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] & \mathcal D_x\arrow[d,hook]\\
\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a \arrow[r]&\mathcal X^{\ast}(\epsilon).
\end{tikzcd}
\end{equation*}
\item Define a right action of $\mathbb{Z}_p$ on $\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x}$ by $(\gamma,q^{1/p^n})\cdot h\mapsto (\gamma\smallmat{1}{0}{h}{1},q^{1/p^n}\zeta^{h/e_x}_{p^n})$. Let $\operatorname{GL}_2(\mathbb{Z}_p)$ act on the left via the first factor. Then there is a Cartesian diagram
\begin{equation*}
\begin{tikzcd}
(\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x})/\mathbb{Z}_p \arrow[d,hook] \arrow[r] & \mathcal D_x\arrow[d,hook]\\
\mathcal X^{\ast}_{\Gamma(p^\infty)} \arrow[r] &\mathcal X^{\ast}
\end{tikzcd}
\end{equation*}
for which the left map is a $\operatorname{GL}_2(\mathbb{Z}_p)$-equivariant open immersion.
\item The Hodge--Tate period map $\pi_{\operatorname{HT}}\colon \mathcal X^{\ast}_{\Gamma(p^\infty)} \rightarrow \P^1$ restricts to the locally constant map
\[
(\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x})/\mathbb{Z}_p \to\underline{\P^1(\mathbb{Z}_p)},\quad \big(\!\smallmat{a}{b}{c}{d},q\big)\mapsto (b:d).
\]
\end{enumerate}
\end{Theorem}
We refer to Thm.~\ref{Theorem: Tate parameter spaces at level Gamma(p^infty)} and Thm.~\ref{t:Main-Theorem-parts-2-3} for slightly more precise statements.
In other words, the part of the boundary of $\mathcal X^{\ast}_{\Gamma(p^\infty)}$ lying over $x$ is given by the closed profinite subspace \[\underline{\operatorname{GL}_2(\mathbb{Z}_p)/\smallmat{1}{0}{\mathbb{Z}_p}{1}}\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}\]
and has an open neighbourhood given by a Tate curve parameter space $(\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x})/\mathbb{Z}_p$.
We will prove part (1) of the theorem step by step in \S\ref{s:infinite-level-Gamma_0}-\ref{s:infinite-level-Gamma}.
We then deduce (2) from (1) via $\operatorname{GL}_2(\mathbb{Z}_p)$-translations. For this one needs to describe the action of the larger group
\[\Gamma_0(p):=\{\smallmat{\ast}{\ast}{c}{\ast}|c\in p\mathbb{Z}_p\}\subseteq \operatorname{GL}_2(\mathbb{Z}_p)\]
on the Tate curve parameter space $\underline{\Gamma_0(p^\infty)}\times \mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a $, which also takes into account isomorphisms of Tate curves of the form $q\mapsto \zeta^h_{p^n}q$ for $h\in\mathbb{Z}_p$. We will do so in \S\ref{s:action-of-Gamma_0}.
\begin{Remark}
We note that Pilloni--Stroh \cite{pilloni2016cohomologie} in their construction of perfectoid tilde-limits of toroidal compactifications of Siegel moduli varieties also describe the boundary of $\mathcal X^{\ast}_{\Gamma(p^\infty)}$:
More precisely, the second part of Thm.~\ref{t:cusps of X_Gamma(p^infty)} also follows from \cite[Prop. A.14]{pilloni2016cohomologie}. While their proposition is much more general, the above description is arguably slightly more explicit. We will also identify the canonical and anticanonical subspaces.
\end{Remark}
On the way, we discuss in \S2 some aspects of modular curves as analytic adic space that are not visible in the rigid setting as treated in \cite{conrad2006modular}. For example, in the adic setting there is also a larger analytic Tate curve parameter space $\overline{\mathcal D}_x\to \mathcal X^{\ast}$ of the form
\[\overline{\mathcal D}_x:=\operatorname{Spa}(\O_{L_x}\bb q[\tfrac{1}{p}],\O_{L_x}\bb q)\]
where $\O_{L_x}\bb q$ is endowed with the $p$-adic topology (rather than the $(p,q)$-adic one). This gives rise at infinite level to a map $\overline{\mathcal D}_{\infty,x}\to \mathcal X_{\Gamma_0(p^\infty)}^{\ast}(\epsilon)_a$ where
\[\overline{\mathcal D}_{\infty,x}:=\operatorname{Spa}(\O_{L_x}\bb {q^{1/p^\infty}}_p[\tfrac{1}{p}],\O_{L_x}\bb{q^{1/p^\infty}}_p)\sim \textstyle\varprojlim_{q\mapsto q^p} \overline{\mathcal D}_x.\]
Here $\O_{L_x}\bb{q^{1/p^\infty}}_p$ is the $p$-adic completion of $\varinjlim_n \O_{L_x}\bb{q^{1/p^n}}$.
While these are no longer open immersions, they are sometimes useful, for example because in contrast to $\mathcal D_x$, the spaces $\overline{\mathcal D}_x$ for all $x$ together with the good reduction locus $\mathcal X_{\mathrm{gd}}$ cover the adic space $\mathcal X^{\ast}$. More precisely, we have $\overline{\mathcal D}\setminus \mathcal D= \operatorname{Spa}(\O_{L_x}\lauc{q}[\tfrac{1}{p}],\O_{L_x}\lauc{q}^+)$ where $\O_{L_x}\lauc{q}$ is the $p$-adic completion of $\O_{L_x}\bb{q}[q^{-1}]$, a local ring, and $\O_L\lauc{q}^+$ is a certain valuation subring of rank 2. The image of this rank $2$ point in $\mathcal X^{\ast}$ is a closed point that is neither contained in $\mathcal X_{\mathrm{gd}}$ nor in $\mathcal D_x$. We discuss this situation in more detail in \S\ref{s:points-of-Xast}.
\subsection{cusps of perfectoid modular curves in characteristic $p$}
There are natural analogues of the above descriptions for modular curves in characteristic $p$, which we treat in \S\ref{s:cusps-in-char-p}: We shall work over the perfectoid field $K^{\flat}$ and choose $\varpi^{\flat}\in K^{\flat}$ with $|\varpi^{\flat}|=|p|$. Let $\mathcal X'^{\ast}$ be the compactified modular curve of level $\Gamma^p$ over $K^{\flat}$, considered as an analytic adic space. Then, again, for every cusp $x$ of $\mathcal X'^{\ast}$, there is a Tate parameter space $\mathcal D'_x\hookrightarrow \mathcal X'^{\ast}$ where now $\mathcal D'_x$ is the adic open unit disc over a finite cyclotomic extension $L_x^{\flat}\subseteq K^{\flat}[\zeta_N]$ (here the notation as the tilt of an extension of $K$ corresponding to $x$ will be justified later).
Recall that over any overconvergent neighbourhood $\mathcal X^{\ast}(\epsilon)$ of the locus of ordinary reduction, there is a finite \'etale Igusa curve $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)\to \mathcal X'^{\ast}(\epsilon)$. In the limit over the relative Frobenius morphism, and over $n\to \infty$, these give rise to a pro-\'etale morphism of perfectoid spaces over $\mathcal X'^{\ast}(\epsilon)$
\[\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{{\operatorname{perf}}}\to \mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}.\]
Let now $\mathcal D_{\infty,x}'$ denote the perfectoid open unit disc over $L_x^{\flat}$, which is the perfection of $\mathcal D_x'$. Then we have the following analogue of Thm.~\ref{t:cusps of X_Gamma(p^infty)} in characteristic $p$:
\begin{Theorem}
For every cusp $x$ of $\mathcal X'^{\ast}$, there are Cartesian diagrams
\begin{center}
\begin{tikzcd}
\underline{\mathbb{Z}_p^\times}\times \mathcal D_{\infty,x}' \arrow[d,hook] \arrow[r] & \mathcal D'_{\infty,x} \arrow[d,hook]\arrow[r]&\mathcal D'_{x} \arrow[d,hook] \\
\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{\perf} \arrow[r] & \mathcal X'^{\ast}(\epsilon)^\perf \arrow[r] & \mathcal X'^{\ast}(\epsilon).
\end{tikzcd}
\end{center}
\end{Theorem}
We then compare this diagram to the situation in characteristic $0$ via tilting:
\subsection{A tilting isomorphism at level $\Gamma_1(p^\infty)$}
In \cite[Cor.~III.2.19]{torsion}, Scholze identifies the tilt of $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$ by proving that there is a canonical isomorphism
\begin{equation}\label{eq:tilting the Gamma_0 tower}
\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a^{\flat} \isomarrow\mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}
\end{equation}
of perfectoid spaces over $K^\flat$. In the case of Siegel spaces parametrising abelian varieties of dimension $g\geq 2$, he then extends this tilting isomorphism to level $\Gamma_1(p^\infty)$.
Using Tate curve parameter spaces, we complement this result in \S\ref{s:tilting-isomorphism} by treating the case $g=1$ of elliptic curves. Moreover, we work out the precise situation at the cusps: It follows from \eqref{eq:tilting the Gamma_0 tower} that the cusps of $\mathcal X^{\ast}$ (which can be identified with those of $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$) and the cusps of $\mathcal X'^\ast$ (which can be identified with those of $\mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}$) can be identified via tilting, and the same is true for the field extensions $L_x$ and $L^{\flat}_x$. Using these identifications, we have:
\begin{Theorem}\label{t: tilting the Gamma_1(p^infty)-tower}
\begin{enumerate}
\item There is a canonical isomorphism
\[\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a^{\flat} \isomarrow \mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{{\operatorname{perf}}}\]
which is $\mathbb{Z}_p^\times$-equivariant and makes the following diagram commute:
\begin{equation*}
\begin{tikzcd}[row sep = 0.15cm]
\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a^{\flat} \arrow[d,"\sim"labelrotate,equal] \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a^{\flat}\arrow[d,"\sim"labelrotate,equal] \\ \mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{{\operatorname{perf}}} \arrow[r] & \mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}.
\end{tikzcd}
\end{equation*}
\item For any cusp $x$ of $\mathcal X^{\ast}$ with corresponding cusp $x^{\flat}$ of $\mathcal X'^{\ast}$, the diagram
\begin{equation*}
\begin{tikzcd}[row sep = 0.15cm]
\underline{\mathbb{Z}_p^{\times}}\times \mathcal D_{\infty,x}^{\flat} \arrow[d,"\sim"labelrotate,equal] \arrow[r,hook] &
\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a^{\flat} \arrow[d,"\sim"labelrotate,equal]\\
\underline{\mathbb{Z}_p^{\times}}\times \mathcal D'_{\infty,x^{\flat}}\arrow[r,hook]& \mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{{\operatorname{perf}}}
\end{tikzcd}
\end{equation*}
commutes, where the left map is given by the canonical identification $\mathcal D_{\infty,x}^{\flat}\cong \mathcal D'_{\infty,x^{\flat}}$.
\end{enumerate}
\end{Theorem}
We are interested in this result because of an application to $p$-adic modular forms: In \cite{heuer-thesis}, we use Thm~\ref{t: tilting the Gamma_1(p^infty)-tower} to give a perfectoid perspective on the $t$-adic modular forms at the boundary of weight space of Andreatta--Iovita--Pilloni.
\subsection{\texorpdfstring{$q$}{q}-expansion principles}
As a second application, Tate curve parameter spaces give a way to talk about $q$-expansions of functions on modular curves:
For any $f\in \O(\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a)$, we may define the $q$-expansion of $f$ at a cusp $x\in \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$ to be its restriction to the associated open subspace $\mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$, i.e.\ the image under
\[\O(\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a)\to \O(\mathcal D_{\infty,x})\]
which will automatically lie in $\O_K\bb{q^{1/p^\infty}}[\tfrac{1}{p}]\subseteq \O(\mathcal D_{\infty,x})$. One has analogous definitions for other infinite level modular curves, or open subspaces thereof, as well as for profinite families of cusps.
Such $q$-expansions can be useful when working with modular curves at infinite level, as they often allow one to extend constructions which are a priori defined only away from the cusps (or even just on the good reduction locus), for instance maps defined using moduli functors, to the compactifications. For example:
\begin{Lemma}
A function $f$ on the uncompactified modular curve $\mathcal X_{\Gamma(p^\infty)}(\epsilon)_a$ extends to a function on $\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$ if and only if at every cusp $x$ of $\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$, the $q$-expansion of $f$ is already contained in $\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket[\frac{1}{p}]\subseteq \O_{L_x}\lauc {q^{1/p^\infty}}[\frac{1}{p}]$. Any such extension is unique.
\end{Lemma}
This is what we mean when we say that $q$-expansions can be used as a replacement for Hartog's extension principle in the elliptic case of $g=1$.
An instance of such an application of Tate parameter spaces in the context of $p$-adic modular forms can be found in \cite{ChrisChris-HilbertCHJ}.
Finally in this article, we show in \S\ref{s:q-expansion-principles} that in the spirit of Katz' $q$-expansion principle for modular forms \cite[Thm.~1.6.1]{p-adicMSMF}, one can use Tate parameter spaces to prove various $q$-expansion principles for functions on perfectoid modular curves.
\begin{Proposition}[$q$-expansion principle I]\label{p: q-expansion principle I}
Let $\mathcal C$ be a collection of cusps of $\mathcal X^{\ast}$ such that each connected component of $\mathcal X^{\ast}$ contains at least one $x\in \mathcal C$. Then restriction of functions to the Tate curve parameter spaces associated to $\mathcal C$ defines injective maps
\begin{alignat*}{4}
&\O(\XaGea{0}{\infty})&\hookrightarrow& \textstyle\prod_{x \in \mathcal C}\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket[\tfrac{1}{p}],\\
&\O(\XaGea{1}{\infty})&\hookrightarrow& \textstyle\prod_{x \in \mathcal C}\operatorname{Map}_{{\operatorname{cts}}}(\mathbb{Z}_p^\times,\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket)[\tfrac{1}{p}],\\
&\O(\XaGea{}{\infty})&\hookrightarrow& \textstyle\prod_{x \in \mathcal C}\operatorname{Map}_{{\operatorname{cts}}}(\Gamma_0(p^\infty),\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket)[\tfrac{1}{p}],\\
&\O(\mathcal X'^{\ast}(\epsilon)^\perf)&\hookrightarrow& \textstyle\prod_{x \in \mathcal C}\O_{L_x^{\flat}}\llbracket q^{1/p^\infty}\rrbracket[\tfrac{1}{\varpi^{\flat}}],\\
&\O(\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{\perf})&\hookrightarrow& \textstyle\prod_{x \in \mathcal C}\operatorname{Map}_{{\operatorname{cts}}}(\mathbb{Z}_p^\times,\O_{L_x^{\flat}}\llbracket q^{1/p^\infty}\rrbracket)[\tfrac{1}{\varpi^{\flat}}].
\end{alignat*}
\end{Proposition}
We note that $p$-adic modular forms can be described as functions on $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$ satisfying a certain equivariance property, see \cite{CHJ}, \cite{ChrisChris-HilbertCHJ}. Prop.~\ref{p: q-expansion principle I} may thus be seen as a generalisation of its classical version from modular forms to more general functions.
Similarly, one can detect on $q$-expansions whether a function comes from finite level:
\begin{Proposition}[$q$-expansion principle II]\label{p: q-expansion principle II}
Let $f\in \O(\XaGea{0}{\infty})$. Then for any $n\in \mathbb{Z}_{\geq 0}$, the following are equivalent:
\begin{enumerate}
\item $f$ comes via pullback from $\XaGea{0}{n}$, i.e.\ $f\in\O(\XaGea{0}{n})\subseteq \O(\XaGea{0}{\infty})$.
\item The $q$-expansion of $f$ at every cusp $x$ is contained in $\O_{L_x}\llbracket q^{1/p^n}\rrbracket[\frac{1}{p}]\subseteq \O_{L_x}\llbracket q^{1/p^\infty}\rrbracket[\frac{1}{p}]$.
\item On each connected component of $\XaGea{0}{n}$, there is at least one cusp $x$ at which the $q$-expansion of $f$ is already in ${\O_{L_x}}\llbracket q^{1/p^n}\rrbracket[\frac{1}{p}]\subseteq \O_{L_x}\llbracket q^{1/p^\infty}\rrbracket[\frac{1}{p}]$.
\end{enumerate}
The analogous statements for $\mathcal X'^{\ast}(\epsilon)^\perf$ are also true.
\end{Proposition}
For $\epsilon=0$, i.e.\ on the ordinary locus, one can see on $q$-expansions whether a function is integral, i.e.\ bounded by 1. We note, however, that this fails for $\epsilon>0$.
\begin{Proposition}[$q$-expansion principle III]\label{p: q-expansion principle III}
For $f\in \O(\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a)$ are equivalent:
\begin{enumerate}
\item $f$ is integral, i.e.\ it is contained in $\O^+(\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a)$.
\item The $q$-expansion of $f$ at every cusp $x$ is already in $\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket$.
\item On each connected component of $\XaGea{0}{n}$, there is at least one cusp $x$ at which the $q$-expansion of $f$ is in $\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket$.
\end{enumerate}
The analogous statements for $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(0)_a$, $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a$, $\mathcal X'^{\ast}(0)^{{\operatorname{perf}}}$ and $\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(0)^{\operatorname{perf}}$ are also true when we replace $\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket$ by the respective algebra in Prop.~\ref{p: q-expansion principle I}.
\end{Proposition}
Finally, there is also a version of $q$-expansions for the good reduction locus:
\begin{Proposition}[$q$-expansion principle IV]\label{p: q-expansion principle IV}
Let $\mathcal C$ be a collection of cusps of $\mathcal X^{\ast}$ such that each connected component contains at least one $x\in \mathcal C$. Then a function on the good reduction locus $\mathcal X_{\mathrm{gd}}(\epsilon)$ extends to all of $\mathcal X^{\ast}(\epsilon)$ if and only if its $q$-expansion with respect to $\overline{\mathcal D}(|q|\geq 1)\to \mathcal X_{\mathrm{gd}}(\epsilon)$ at each $x\in \mathcal C$ is already in $\O_{L_x}\llbracket q\rrbracket[\tfrac{1}{p}]\subseteq \O_{L_x}\lauc{q}[\tfrac{1}{p}]$. In this case, the extension is unique. The analogous statements for $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$ $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$, $\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$, $\mathcal X'^{\ast}(\epsilon)$, $\mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}$ and $\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{\operatorname{perf}}$ are also true.
\end{Proposition}
\section*{Acknowledgements}
I would like to thank Johannes Ansch\"utz, Kevin Buzzard, Ana Caraiani, Vincent Pilloni and Peter Scholze for helpful discussions.
This work was carried out while the author was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.
\section{Adic analytic theory of cusps at finite level}\label{s:adic-cusps-finite}
\subsection{Recollections on the classical theory of cusps}\label{s:classical-recollections-on-cusps}
We start by recalling from \cite[\S8.6-8.11]{KatzMazur} some basic facts about cusps of modular curves that we will use freely throughout, and fix some notation and conventions:
Let $N\geq 3$ and let $R$ be an excellent Noetherian regular $\mathbb{Z}[1/N]$-algebra, for instance $R=\mathbb{Z}[1/N]$. Let $X_R$ be the modular curve of some rigidifying level $\Gamma(N)\subseteq \Gamma\subseteq \operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ over $R$.
By definition, the compactification $X_{R}^{\ast}$ of $X_{R}$ is then the normalisation in $\P_R^1$ of the finite flat $j$-invariant $j\colon X_{R}\to \mathbb{A}_R^1$ . The divisor of cusps is defined as the closed subscheme
\[\partial X_{R}^{\ast} := (X_{R}^{\ast}\backslash X_{R})^{\mathrm{red}}=j^{-1}(\infty)^{\mathrm{red}}\hookrightarrow X_{R}^{\ast},\] which is finite \'etale over $\operatorname{Spec}(R)$. When we refer to ``a cusp'' we shall mean by this a (not necessarily geometrically) connected component of $\partial X_{R}^{\ast}$.
We recall from \cite[\S8.11]{KatzMazur} that the divisor of cusps can be computed explicitly using the Tate curve $\mathrm{T}(q)$: This is an elliptic curve over $\mathbb{Z}\cc{q}$ of $j$-invariant $1/q+744+\dots$, which we may base--change to $\mathrm{T}(q)_{R\cc{q}}\to \operatorname{Spec}(R\cc{q})$. Then we have:
\begin{Proposition}[{\cite[{Thm. 8.11.10}]{KatzMazur}}]\label{p:completion-at-cusp}
The completion $\hat{\partial} X_{R}^{\ast}$ of $X_{R}^{\ast}$ along $\partial X_{R}^{\ast}$ is the normalisation of $R\llbracket q\rrbracket $ in the finite flat scheme over $R\cc{q}$ that represents $\Gamma$-level structures of $\mathrm{T}(q)_{R\cc{q}}$. Via the $j$-invariant, $\hat{\partial} X_{R}^{\ast}$ is finite over the completion $R\llbracket q\rrbracket $ of $\P_R^1$ at $\infty$.
\end{Proposition}
To say more concretely what $\partial X_{R}^{\ast}$ looks like, recall that $\mathrm{T}(q)[N]$ is an extension
\[0\to \mu_{N}\to \mathrm{T}(q)[N]\to \underline{\mathbb{Z}/N\mathbb{Z}}\to 0\]
over $\mathbb{Z}\cc{q}$ which becomes split over $\mathbb{Z}\cc{q^{1/N}}$. Consequently, the $\Gamma$-level structures of $\mathrm{T}(q)_{R\cc{q}}$ are defined over various subrings of $R[\zeta_N]\cc{q^{1/N}}$. In particular, each component of $\hat{\partial} X_{R}^{\ast}$ will be of the form $\operatorname{Spf}(R[\zeta_d]\llbracket q^{1/e}\rrbracket )$ for some $d|N$ and $e|N$.
\begin{notation}\label{n:e}
In order to lighten notation, we wish to reduce the amount of $N$-th roots of $q$ throughout. Therefore, we shall by convention renormalise this ring of definition at each cusp to be a subring of the form $R[\zeta_d]\cc{q}$ for some $d|N$, by passing from $\mathrm{T}(q)$ to $\mathrm{T}(q^{e})$.
\end{notation}
This means that depending on the cusp, the completion of the $j$-invariant at the cusp is now given by a map of the form $\operatorname{Spf}(R[\zeta_d]\llbracket q\rrbracket )\to \operatorname{Spf}(R\llbracket q\rrbracket )$ that sends $q\mapsto q^e$.
If $R'$ is another Noetherian excellent regular $\mathbb{Z}[1/N]$-algebra, then $\partial X_{R'}^{\ast}$, $\hat{\partial} X_{R'}^{\ast}$, etc.\ agree with the base-changes via $R\to R'$ \cite[Prop.\ 8.6.6]{KatzMazur}. Therefore, more generally, for any $\mathbb{Z}[1/N]$-algebra $S$ we may simply define $X_S$, $X^{\ast}_S$, $\partial X_S^{\ast}$, $\hat{\partial} X_S^{\ast}$, etc.\ by base-change.
\subsection{The analytic setup}
We now switch to a $p$-adic analytic situation and recall the setup from \cite{torsion}.
Let $p$ be a prime and let $K$ be a perfectoid field extension of $\mathbb{Q}_p$ like in the introduction. We denote by $\mathfrak m$ the maximal ideal of the ring of integers $\O_K$ and by $k$ the residue field. We fix a complete algebraically closed extension $C$ of $K$ and assume that $K$ contains all $p$-power unit roots in $C$. We moreover fix a pseudo-uniformiser $\varpi\in K$ with $|\varpi|=|p|$ such that $\varpi$ contains arbitrary $p$-power roots in $K$. This is possible since $K$ is perfectoid.
Let $N$ be coprime to $p$ and let $\Gamma(N)\subseteq \Gamma^p\subseteq \operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ be a rigidifying tame level. Let $X:=X_K$ be the modular curve of level $\Gamma^p$ over $K$ and let $X^{\ast}:=X^{\ast}_K$ be its compactification.
We denote by $\mathfrak X$ and $\mathfrak X^{\ast}$ the respective $p$-adic completions of $X_{\O_K}$ and $X^{\ast}_{\O_K}$. Let $\mathcal X$ and $\mathcal X^{\ast}$ be the respective adic analytifications of $X$ and $X^{\ast}$. This is the only way in which we deviate from the notation in \cite{torsion}, where $\mathcal X$ denotes the good reduction locus, which we shall instead denote by $\mathcal X_{\mathrm{gd}}\subseteq \mathcal X\subseteq \mathcal X^\ast$.
For any of the classical level structures $\Gamma=\Gamma_0(p^n),\Gamma_1(p^n),\Gamma(p^n)$, $n\in \mathbb{N}$, we denote by $X_\Gamma\to X$ the representing moduli scheme. These all have compactifications $X^{\ast}_{\Gamma}\to X^{\ast}$, and we have associated adic spaces $\mathcal X_{\Gamma}\to \mathcal X$ and $\mathcal X^{\ast}_{\Gamma}\to \mathcal X^{\ast}$.
The uncompactified spaces have a natural moduli interpretation in the category $\mathbf{Adic}$ of adic spaces:
\begin{Lemma}\label{Lemma: adic moduli interpretation of X_Gamma^an}
Let $S$ be an honest adic space over $\operatorname{Spa}(K,\mathcal O_K)$. Then
\[\operatorname{Hom}_{\mathbf{Adic}}(S,\mathcal X_{\Gamma})=X(\mathcal O_S(S)).\]
In particular, the $S$-points of $\mathcal X_{\Gamma}$ are in functorial correspondence with isomorphism classes of elliptic curves over $\mathcal O_S(S)$ with tame level structure $\Gamma^p$ and level structure $\Gamma$ at $p$.
\end{Lemma}
\begin{proof}
The scheme $X_{\Gamma}$ over $K$ is an affine curve \cite[Cor.\ 4.7.2]{KatzMazur}. Let $\mathbf{LRS}$ be the category of locally ringed spaces, then by the universal property of the analytification $\mathcal X_\Gamma = X_{\Gamma}^{\mathrm{an}}$: \[\operatorname{Hom}_{\mathbf{Adic}}(S,\mathcal X_{\Gamma})=\operatorname{Hom}_{\mathbf{LRS}}(S,X_{\Gamma})=X_{\Gamma}(\O_S(S))\]
where the last step is the adjunction of Spec and global sections for locally ringed spaces.
\end{proof}
Let $0\leq \epsilon<\tfrac{1}{2}$ be such that $|p|^{\epsilon}\in |K|$. Using local trivialisations of the Hodge bundle and lifts $\mathrm{Ha}$ of the Hasse invariant one defines an open subspace $\mathcal X^{\ast}(\epsilon)\subseteq \mathcal X^{\ast}$ cut out by the condition that $|\mathrm{Ha}|\geq |p|^{\epsilon}$.
This has a canonical integral model $\mathfrak X^{\ast}(\epsilon)\to \mathfrak X^{\ast}$, for example by \cite[Lemma III.2.13]{torsion}.
In general, for any morphism $S\to \mathcal X^{\ast}$ we shall write
\[S(\epsilon):=S\times_{\mathcal X^{\ast}}\mathcal X^{\ast}(\epsilon).\]
In particular, for any of the classical level structures $\Gamma=\Gamma_0(p^n),\Gamma_1(p^n),\Gamma(p^n)$ the modular curve $\mathcal X^{\ast}_{\Gamma}\to \mathcal X^{\ast}$ restricts to a morphism $\mathcal X^{\ast}_{\Gamma}(\epsilon)\to \mathcal X^{\ast}(\epsilon)$.
We note that the open subspace $\mathcal X^{\ast}(0)$ is precisely the ordinary locus of $\mathcal X^{\ast}$.
\begin{Definition}
We shall say that an elliptic curve $E$ is $\epsilon$-nearly ordinary if $|\mathrm{Ha}(E)|\geq |p|^{\epsilon}$.
\end{Definition}
By the theory of the canonical subgroup, the forgetful morphism $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)\to \mathcal X^{\ast}(\epsilon)$ has a canonical section. We denote by $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_c$ the image of this section, that is the component of $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)$ that parametrises the $\Gamma_0(p)$-structure given by the canonical subgroup. This is called the canonical locus. We denote its complement by $\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a$ and call it the anticanonical locus. For any adic space $S\to \mathcal X^{\ast}_{\Gamma_0(p)}$ we denote by
\[S(\epsilon)_a:=S\times_{\mathcal X^{\ast}_{\Gamma_0(p)}} \mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a\]
the open subspace that lies over the anticanonical locus.
\begin{Definition}
For any adic space $S\to \mathcal X(\epsilon)$ corresponding to an $\epsilon$-nearly ordinary elliptic curve $E$ over $\O_S(S)$, we shall call a $\Gamma$-level structure anticanonical if it corresponds to a point of $\mathcal X_{\Gamma}(\epsilon)_a\subseteq \mathcal X_{\Gamma}$. For instance, a $\Gamma_0(p^n)$-level structure is a locally free subgroup scheme $G_n\subseteq E[p^n]$, \'etale locally cyclic of rank $p^n$, and it is anticanonical if $G_n\cap C_1=0$. Similarly, a $\Gamma(p^n)$-level structure, given by an isomorphism of group schemes $\alpha\colon(\mathbb{Z}/p^n\mathbb{Z})^2 \to E[p^n]$ is anticanonical if the subgroup scheme generated by $\alpha(1,0)$ is anticanonical.
\end{Definition}
For any $n\in\mathbb{N}$, the transformation of moduli functors that sends an elliptic curve $E$ together with an anticanonical $\Gamma_0(p^n)$-structure $G_n$ to $E/G_n$ induces an isomorphism
\begin{equation}\label{eq:Atkin--Lehner}
\mathcal X_{\Gamma_0(p^n)}(\epsilon)_a \xrightarrow{\sim} \mathcal X(p^{-n}\epsilon)
\end{equation}
that is called the Atkin--Lehner isomorphism. The inverse is given by sending $E$ with its canonical subgroup $C_n$ of rank $p^n$ to the data of $E/C_n$ with $\Gamma_0(p^n)_a$-structure $E[p^n]/C_n$. The Atkin--Lehner isomorphism uniquely extends to the cusps for all $n$, and for varying $n$ the resulting isomorphisms fit into a commutative diagram of towers
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^2)}(\epsilon)_a \arrow[d, no head,"\sim" labelrotate] \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a \arrow[d, no head,"\sim" labelrotate] \arrow[r] & \mathcal X^{\ast}(\epsilon) \arrow[d, no head,equal] \\
\cdots \arrow[r] & \mathcal X^{\ast}(p^{-2}\epsilon) \arrow[r,"\phi"] & \mathcal X^{\ast}(p^{-1}\epsilon) \arrow[r,"\phi"] & \mathcal X^{\ast}(\epsilon)
\end{tikzcd}
\end{equation*}
where in the bottom row, the morphism $\phi$ is the ``Frobenius lift'' defined in terms of moduli by sending $E$ to $E/C_1$. The resulting tower is called the ``anticanonical tower''.
It is a crucial intermediate result in \cite{torsion} that the anticanonical tower ``becomes perfectoid'' in the inverse limit. More precisely:
\begin{Theorem}[{\cite[Cor.~III.2.19]{torsion}}]
There is an affinoid perfectoid tilde-limit
\[\XaGea0\infty \sim \textstyle\varprojlim_{n\in\mathbb{N}}\XaGea0n.\]
\end{Theorem}
Since the forgetful morphisms $\XaGea 1 n\to \XaGea 0 n$ are finite \'etale $(\mathbb{Z}/p^n\mathbb{Z})^{\times}$-torsors, even over the cusps, one immediately deduces that in the inverse limit these give rise to an affinoid perfectoid space $\XaGea 1 \infty\sim \varprojlim_n\XaGea 1 n $ together with a forgetful map \[\XaGea 1 \infty \rightarrow \XaGea 0 \infty\]
that is a pro-\'etale $\mathbb{Z}_p^\times$-torsor. Similarly, for full level $\Gamma(p^n)$, one obtains an affinoid perfectoid space $\XaGea{}{\infty}$ together with a forgetful map that is a pro-\'etale $\Gamma_0(p^\infty)$-torsor $\XaGea{}{\infty}\rightarrow \XaGea{0}{\infty}$, where we set:
\begin{Definition}\label{df: Gamma_0(p^m)}
For any $m\in \mathbb{Z}_{\geq 0}\cup\{\infty\}$, let $\Gamma_0(p^m)=\{\smallmat{\ast}{\ast}{c}{\ast}\in \operatorname{GL}_2(\mathbb{Z}_p)\mid c\equiv 0 \bmod p^m\}$.
\end{Definition}
All in all, we have a tower of morphisms
\begin{equation*}
\begin{tikzcd}
\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a
\end{tikzcd}
\end{equation*}
which is a pro-\'etale $\Gamma_0(p)$-torsor away from the boundary, but not globally: One reason is that there is ramification over the cusps in $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a \to \mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a$, but we note that the tower is still no torsor on the quasi-pro-\'etale site, as we will see on $q$-expansions.
\subsection{Analytic Tate curve parameter spaces at tame level}\label{subsection: adic cusps and Tate curves}
In this subsection, we recall the universal analytic Tate curves at the cusps, as developed by \cite{conrad2006modular}. The main technical difference is that we work with analytic adic spaces instead of rigid spaces. In particular, instead of the generalisation of Berthelot's functor constructed in \S3 of \textit{loc. cit.} we may use the adic generic fibre functor.
For now, we shall focus on the adic analytic modular curve $\mathcal X^{\ast}$ over $K$. We remark that everything in this section and the next will also work for $\mathcal X_{\Gamma_0(p^n)}^{\ast}$, except for the additional phenomenon of ramification at the cusps. In order to separate the discussion of these two topics, and to simplify the exposition, we shall therefore focus on $\mathcal X^{\ast}$ for now.
As discussed in \S\ref{s:classical-recollections-on-cusps}, the subscheme $\partial X^{\ast}\subseteq X^{\ast}$ decomposes into a union of points of the form $x\colon\operatorname{Spec}(L)\rightarrow X^{\ast}$ where $K\subseteq L\subseteq K[\zeta_N]$ is a subfield depending on $x$. For example, if $\Gamma^p=\Gamma(N)$, we have $L=K[\zeta_N]$ at every cusp. We now switch to an analytic setup:
\begin{Definition}\label{d:field-of-definition-of-Tate-curve}
By a cusp of $\mathcal X^{\ast}$ we shall mean a connected (but not necessarily geometrically connected) component of $(\mathcal X^{\ast}\backslash \mathcal X)^{\mathrm{red}}$.
Given a fixed cusp $x$, we shall denote by $L=L_x\subseteq K[\zeta_N]$ the coefficient field of definition of the corresponding Tate curve. We have $L=K[\zeta_d]$ for some $d|N$. Let $\mathfrak m_L$ be the maximal ideal of $\O_L$ and let $k_L$ be the residue field.
\end{Definition}
From now on for the rest of this section, let us fix a cusp $x\in \mathcal X^{\ast}$. To simplify notation, we will write $L=L_x$ and $e=e_x$. We note that the cusp $x$ will decompose into $[L:K]$ disjoint $L$-points after base-changing $\mathcal X^{\ast}$ from $K$ to $L$.
By Prop.~\ref{p:completion-at-cusp}, the completion along $x$ is canonically of the form
\begin{equation}\label{eq:completion-at-schematic-cusp}
\operatorname{Spf}(\O_{L}\llbracket q\rrbracket)\rightarrow X_{\O_K}^{\ast}
\end{equation}
where $\O_{L}\llbracket q\rrbracket$ carries the $q$-adic topology. Here we recall from Notation~\ref{n:e} that we have renormalised parameters from $q^{1/e}$ to $q$, so that the universal Tate curve is $\mathrm{T}(q^{e})$. Upon $\pi$-adic completion, this becomes a morphism
\[\operatorname{Spf}(\O_{L}\llbracket q\rrbracket)\rightarrow \mathfrak X^{\ast}\]
where now $\O_{L}\llbracket q\rrbracket$ is endowed with the $(p,q)$-adic topology.
We note that this morphism restricts to $\operatorname{Spf}(\O_{L}\llbracket q\rrbracket)\rightarrow \mathfrak X^{\ast}(0)$ since the supersingular locus is disjoint from the cusps.
On the $p$-adic generic fibre, we obtain a morphism of analytic adic spaces over $K$
\[\mathcal D:=\operatorname{Spf}(\mathcal O_{L}\llbracket q\rrbracket)^{\mathrm{ad}}_\eta \rightarrow \mathcal X^{\ast}.\]
Here $\mathcal D$ is the open unit disc over $K$, a topologically finite type but non-quasicompact, non-affinoid analytic adic space. We emphasize that in general, this depends on $x$, as $L=L_x$ does. If the cusp $x$ is not clear from the context, we shall therefore denote this space by $\mathcal D_x$.
The global functions on $\mathcal D$ are given by $\mathcal O^+(\mathcal D) =\O_{L}\llbracket q\rrbracket$ and
\[\mathcal O(\mathcal D)=\Big \{ \textstyle\sum_{n\geq 0}a_nq^n \in L\llbracket q\rrbracket \text{ such that } |a_n|z^n\to 0\text{ for all }0\leq z<1 \Big\}.\]
More classically, if we regarded $\mathcal D$ as a rigid space, it would be associated to the formal scheme $\operatorname{Spf}(\mathcal O_{L}\llbracket q\rrbracket)$ via Conrad's non-Noetherian generalisation of Berthelot's rigid generic fibre construction, \cite[Thm.\ 3.1.5]{conrad2006modular}.
\begin{Lemma}\label{l: Conrad's theorem on generic fibres of completions around the cusp}
The map $\mathcal D \hookrightarrow \mathcal X^{\ast}$ is an open immersion that sends the origin to the cusp.
\end{Lemma}
\begin{Remark}
This is part of \cite[Thm.~3.2.8]{conrad2006modular} for $\Gamma^p = \Gamma_1(N)$, and in general follows from \cite[Thm.~3.2.6]{conrad2006modular}. These moreover give a moduli interpretation in terms of analytic generalised elliptic curves, as well as a universal analytic generalised Tate curve over $\mathcal D$.
\end{Remark}
\begin{proof}
In an affine open formal neighbourhood $\operatorname{Spf}(A)\subseteq \mathfrak X^{\ast}(0)$ of the cusp, $x$ is cut out by a principal ideal $(f)$ for some non-zero-divisor $f$. The completion along the cusp is then $A\llbracket T\rrbracket/(T-f)$. The adic generic fibre is thus the union over the spaces $\operatorname{Spa}(A\langle f^n/p\rangle[1/p])$, and is thus the open subspace of $\operatorname{Spa}(A[1/p])\subseteq \mathcal X^{\ast}$ defined by the condition $|f|<1$.
\end{proof}
\begin{Lemma}\label{lemma for comparing schematic cusp to adic cusp}
The morphism of locally ringed spaces $\mathcal D\rightarrow \operatorname{Spec} (\mathbb Z_p\llbracket q\rrbracket\otimes_{\mathbb{Z}_p} { \O_{L}})$
induced by the inclusion $\mathbb Z_p\llbracket q\rrbracket \hookrightarrow \mathcal O^+(\mathcal D)$ fits into a commutative diagram of ringed spaces:
\begin{equation*}
\begin{tikzcd}
\operatorname{Spec}(\mathbb Z_p\llbracket q\rrbracket\otimes_{\mathbb{Z}_p} { \O_L}) \arrow[r] & X^{\ast}_{\O_K} \\
\mathcal D \arrow[r,hook] \arrow[u] & \mathcal X^{\ast} \arrow[u].
\end{tikzcd}
\end{equation*}
\end{Lemma}
\begin{proof}
Let $R:=\mathbb Z_p[\zeta_d]$ and $R_0:=\mathbb Z_p[\zeta_{d_0}]$ where $d_0$ is the largest divisor of $d$ such that $K$ contains a primitive $d_0$-th unit root.
The adification of the $p$-adic completion of
$f\colon\operatorname{Spf}(R\llbracket q\rrbracket)\rightarrow \operatorname{Spec}(R\llbracket q\rrbracket)\rightarrow X^{\ast}_{R_0}$ fits into a commutative diagram of ringed spaces
\begin{equation*}
\begin{tikzcd}
{\operatorname{Spec}(R\llbracket q\rrbracket)} \arrow[r,"f"] & X_{R_0}^{\ast}\arrow[r]& \operatorname{Spec}(R_0) \\
{\operatorname{Spa}(R\llbracket q\rrbracket,R\llbracket q\rrbracket)} \arrow[u] \arrow[r,"\hat{f}"] & {\mathfrak X_{R_0}^{\ast \mathrm{ad}}} \arrow[u]\arrow[r]&\operatorname{Spa}(R_0,R_0).\arrow[u]
\end{tikzcd}
\end{equation*}
The lemma follows upon taking the fibre over $\operatorname{Spa}(K,\O_K)\to \operatorname{Spec}(\O_K)$.
\end{proof}
We thus have the following moduli interpretation of $\mathring{\mathcal D}:=\mathcal D(q\neq 0)\subseteq \mathcal D$:
\begin{Lemma}\label{l:moduli interpretation of adic Tate parameter space}
Let $S$ be an honest adic space over $K$ and let $\varphi\colon S\rightarrow \mathcal X$ be a morphism corresponding to an elliptic curve $E$ over $\mathcal O_S(S)$ with $\Gamma^p$-level structure $\alpha_N$.
Then $\varphi$ factors through the punctured open unit disc $\mathring{\mathcal D}\rightarrow \mathcal X$ at the cusp $x$ if and only if $E\cong \mathrm{T}(q_E)$ is a Tate curve for some $q_E\in S$, the $\Gamma^p$-structure $\alpha_N$ corresponds to $x$ under Prop.~\ref{p:completion-at-cusp}, and $q_E$ is locally topologically nilpotent on $S$, i.e.\ $v_z(q_E)$ is cofinal in the value group for all $z\in S$.
\end{Lemma}
\begin{proof}
If $\varphi\colon S\rightarrow \mathcal X^{\ast}\to X^{\ast}$ factors through $\mathring{\mathcal D}\hookrightarrow \mathcal X$, then by Lemma~\ref{lemma for comparing schematic cusp to adic cusp} it factors through the map $\operatorname{Spec}(\mathcal O_{L}\cc q)\to X^{\ast}$. Consequently, $E$ is a Tate curve and we obtain a parameter $q_E\in \mathcal O_S(S)$ as the image of $q\in \mathcal O_{L}\cc q\subseteq \O(\mathcal D)$ on global sections. This is locally topologically nilpotent because $q\in \mathcal O(\mathcal D)$ is locally topologically nilpotent.
Conversely, assume that $E$ is a Tate curve such that $q_E\in \mathcal O(S)$ is topologically nilpotent, with $\Gamma^p$-level structure associated to $x$. The latter condition implies that $\O(S)$ is naturally an $L$-algebra.
It therefore suffices to consider the case that $S=\operatorname{Spa}(B,B^{+})$ is an affinoid adic space over $\operatorname{Spa}(L,\mathcal O_L)$. The condition that $q_E$ is topologically nilpotent implies that for any $x$ there is $n$ such that $|q_E(x)|^n\leq |\varpi|$. Since $S$ is affinoid and thus quasicompact, we can find $n$ that works for all $x\in S$. Similarly, since $E$ is a Tate curve, $q_E\in B$ is a unit and we thus have $0<|q_E(x)|$ for all $x\in S$. Again by compactness, we can find $m$ such that $|\varpi|^m\leq |q_E|$.
But then $q_E^n/\varpi,\varpi^m/q_E\in B^{+}$ and there is a natural morphism of affinoids
\[(L\langle q,q^n/\varpi,\varpi^m/q\rangle,\mathcal O_{L}\langle q,q^n/\varpi,\varpi^m/q\rangle)\,\rightarrow\, (B,B^{+}),\quad q\mapsto q_E\]
through which the map $\mathcal O_{L}\cc{q}\rightarrow B$ defining the Tate curve structure factors. Since the algebra on the left defines an affinoid open of $\mathring{\mathcal D}$, this gives the desired factorisation.
\end{proof}
\begin{Definition}
Consider $\O_L\cc q$ with the $p$-adic topology, that is we suppress the topology coming from $q$.
Let $\O_L\lauc{q}:=\O_L\cc{q}^{\wedge}$ be the $p$-adic completion. Explicitly,
\[\O_L\lauc{q}=\big\{\textstyle\sum_{n\in\mathbb{Z}} a_nq^n\in \O_L\llbracket q^{\pm 1}\rrbracket\mid a_n\to 0 \text{ for }n\to-\infty\big\}.\]
This is a discrete valuation ring with maximal ideal $(p)$ and residue field $k_L\cc q$.
\end{Definition}
\begin{Remark}\label{remark: Tate curve of good reduction}
The adic space $S=\operatorname{Spa}(\O_L\lauc{q}[\tfrac{1}{p}],\O_L\lauc{q})$ consists of a single point. As we have suppressed the $q$-adic topology, it is clear that $q$ is not topologically nilpotent on $S$. We conclude that the map
\[S\rightarrow \operatorname{Spec} \mathcal O_{L}\cc{q}\to X^{\ast}\]
does not factor through $\mathring{\mathcal D}\hookrightarrow \mathcal X^{\ast}$ even though it corresponds to a Tate curve. The point is that this Tate curve has good reduction: Concretely, this means that it is already an elliptic curve over $\mathcal O_{L}\cc{q}$. Its reduction mod $\mathfrak m_L$ is simply the Tate curve $\mathrm{T}(q)$ over $k_L\cc{q}$. It therefore gives rise to a point in the good reduction locus $\mathcal X_{\mathrm{gd}}\subseteq \mathcal X\subseteq \mathcal X^{\ast}$.
\end{Remark}
We can enlarge the Tate parameter space $\mathcal D$ so that it includes the above example:
\begin{Definition}\label{d: OK bb q^1/p^infty _p}
Let $\overline{\mathcal D}=\operatorname{Spa}(\O_L\llbracket q\rrbracket[\tfrac{1}{p}],\O_L\llbracket q\rrbracket)$ where, contrary to our usual convention, $\O_L\llbracket q\rrbracket$ is endowed with the $p$-adic topology. We let $\O_L\llbracket q^{1/p^\infty}\rrbracket _p:=(\varinjlim_m \O_L\llbracket q^{1/p^m}\rrbracket )^{\wedge}$ where, crucially, the completion is the $p$-adic one. This is a perfectoid $\O_L$-algebra. If we use the $(p,q)$-adic topology instead, we obtain a $(p,q)$-adically complete ring $\O_L\llbracket q^{1/p^\infty}\rrbracket $. There is a natural inclusion $\O_L\llbracket q^{1/p^\infty}\rrbracket _p\hookrightarrow\O_L\llbracket q^{1/p^\infty}\rrbracket $, but this is not an isomorphism: For example, $\sum_{n\in\mathbb{N}} q^{n+\frac{1}{p^n}}$ defines an element contained in the codomain but not in the image.
As before, we emphasize that $\overline{\mathcal D}$ depends on our chosen cusp $x$. If this cusp is not clear from the context, we shall also write $\overline{\mathcal D}_x$ for the parameter space associated to $x$.
\end{Definition}
\begin{Lemma}\label{l:overline{D} is sousperfectoid}
\begin{enumerate}
\item The Huber pair $(\O_L \llbracket q\rrbracket[\tfrac{1}{p}],\O_L \llbracket q \rrbracket)$, where $\O_L \llbracket q\rrbracket$ is endowed with the $p$-adic topology, is sous-perfectoid in the sense of \cite[\S6.3]{ScholzeBerkeleyLectureNotes}, and thus sheafy.
\item We have an open immersion $
\mathcal D=\cup_m\overline{\mathcal D}(|q|\leq |p|^{1/p^m})\hookrightarrow \overline{\mathcal D}$.
\item We have an open immersion
$\operatorname{Spa}(\O_L\lauc{q}[\tfrac{1}{p}],\O_L\lauc{q})= \overline{\mathcal D}(|q|\geq 1)\hookrightarrow \overline{\mathcal D}$.
\end{enumerate}
\end{Lemma}
Parts (1)-(2) of the lemma say that we may think of $\overline{\mathcal D}$ as a compactification of $\mathcal D$, albeit a non-standard one. As discussed in Example~\ref{ex: rank-2-point} below, $\mathcal D$ and $\overline{\mathcal D}(|q|\geq 1)$ form a disjoint open cover of $\overline{\mathcal D}$ up to one additional point of rank $2$ which lies between them.
\begin{proof}
The traces $\O_L\llbracket q^{1/p^n}\rrbracket\to \O_L\llbracket q\rrbracket$, $q^{i/p^n}\mapsto 0$ for $0<i<p^n$ in the $p$-adically completed limit give rise to an $\O_L\bb{q}$-linear section $\O_L \llbracket q^{1/p^\infty}\rrbracket_p\to \O_L \llbracket q\rrbracket$. This shows the first part.
Part (3) is clear as $\overline{\mathcal D}(|q|\geq 1)$ is formed by adjoining $q^{-1}$ and completing $p$-adically.
Part (2) follows from $\mathcal D=\cup_m \mathcal D(|q|^m\leq |p|)$ and \[\O(\overline{\mathcal D}(|q|^m\leq |p|))=\O_L\llbracket q\rrbracket\langle \tfrac{q^m}{p}\rangle[\tfrac{1}{p}]=\O_L\langle \tfrac{q^m}{p}\rangle[\tfrac{1}{p}]=\O(\mathcal D(|q|^m\leq |p|)).\qedhere\]
\end{proof}
\begin{Lemma}\label{l:full-Tate-curve-parameter-space}
For every cusp of $\mathcal X^{\ast}$ there is a natural morphism $\overline{\mathcal D}\to \mathcal X^{\ast}(\epsilon)$, extending the morphism $\mathcal D\hookrightarrow \mathcal X^{\ast}(\epsilon)$. The fibre of the good reduction locus is precisely $\overline{\mathcal D}(|q|\geq 1)$.
\end{Lemma}
\begin{proof} For the first part, we take the morphism $\operatorname{Spec}(\O_L\llbracket q\rrbracket )\to X_{\O_K}^{\ast}$ from Lemma~\ref{lemma for comparing schematic cusp to adic cusp}, complete $p$-adically and pass to the generic fibre.
To see the second part, we note that morphisms $\operatorname{Spa}(R,R^+)\to \mathcal X_{\mathrm{gd}}$ correspond to elliptic curves over $R^+$. The locus of $\overline{\mathcal D}$ where the Tate curve is defined over $\O^+$ is precisely that where $q$ is in $\O^{+,\times}$. Equivalently, as we have $|q|\leq 1$ on $\overline{\mathcal D}$, this means that $|q|\geq 1$.
\end{proof}
\begin{Remark}
\begin{enumerate}
\item The map $\overline{\mathcal D}\to \mathcal X^{\ast}(\epsilon)$ is no longer an open immersion. Indeed, $\overline{\mathcal D}$ is not even of locally topologically finite type over $K$. We therefore do not have a rigid analogue of $\overline{\mathcal D}$, and can only capture this map in the setting of adic spaces.
\item Here and in the following, by base-change we could replace $\O_{L}\llbracket q\rrbracket$ by the smaller ring $\mathbb{Z}_p\llbracket q\rrbracket\hat{\otimes}_{\mathbb{Z}_p}\O_{L}$ where the completion is $p$-adic. But we will not need this.
\item If we form the union over all cusps $\mathcal C$, we get two maps \[(i)\: \bigsqcup_{x\in \mathcal C} {\mathcal D}_x\times \mathcal X_{\mathrm{gd}}\to \mathcal X ,\quad\quad (ii) \: \bigsqcup_{x\in \mathcal C} \overline{\mathcal D}_x\times \mathcal X_{\mathrm{gd}}\to \mathcal X\]
of which (ii) is a cover (for example in the $v$-topology), in contrast to (i) whose image misses some points of rank 2. This is what we shall discuss next.
\end{enumerate}
\end{Remark}
\subsection{The points of the adic space \texorpdfstring{$\mathcal X^{\ast}$}{X*}}\label{s:points-of-Xast}
Our next goal is to see how much of the adic space $\mathcal X^{\ast}$ is captured by the Tate curve parameter spaces $\mathcal D$ in conjunction with the good reduction locus. The answer is that they give everything except for a finite set of higher rank points whose moduli interpretation in terms of Tate curves we shall describe.
Indeed, since the loci $\mathcal X_{\mathrm{gd}}$ and $\mathcal D$ in $\mathcal X^{\ast}$ are the preimages of a cover by an open and a closed set of the special fibre $X_{k}$ under the specialisation map $ |\mathcal X^{\ast}|\rightarrow |X_{k}|$, it follows from the adaptation of \cite[Lemma~7.2.5]{de1995crystalline}, to the setting of \cite{conrad2006modular} that if we worked with rigid spaces, then $\mathcal D$ and $\mathcal X$ would cover $\mathcal X^{\ast}$ set-theoretically, but not admissibly so \cite[\S 7.5.1]{de1995crystalline}.
\begin{Proposition}\label{p: classification of points of X^ast}
Let $z\in \mathcal X^{\ast}$ be any point, then we are in either of the following cases:
\begin{enumerate}[label=(\alph*)]
\item $z\in \mathcal X^{\ast}$ is contained in the good reduction locus $\mathcal X_{\mathrm{gd}}$,
\item $z\in \mathcal D_x\hookrightarrow \mathcal X^{\ast}$ is contained in a Tate parameter space around a cusp $x$ of $\mathcal X^{\ast}$,
\item $z\in \mathcal X^{\ast}\backslash\mathcal X_{\mathrm{gd}}$ is of rank $>1$ and its unique height $1$ vertical generisation $z'$ is in $\mathcal X_{\mathrm{gd}}$. \end{enumerate}
When we denote by $j$ the global function on $\mathcal X$ induced by the $j$-invariant $j\colon \mathcal X\rightarrow \mathbb{A}^{1,\mathrm{an}}$, then the above are respectively equivalent to
\begin{enumerate}[label=(\alph*')]
\item $|j(z)|\leq 1$,
\item $|j(z)|> 1$ and its inverse is cofinal in the value group,
\item $|j(z)|> 1$ and its unique rank 1 generisation $z'$ satisfies $|j(z')|=1$.
\end{enumerate}
\end{Proposition}
We note that the analogous description also holds after adding level at $p$.
\begin{proof}
The space $\mathcal X^{\ast}$ is analytic, hence the valuation $v_z$ is always microbial. This implies that $z$ has a unique generisation $z'$ of height 1, so statements ($c$) and ($c$') make sense.
The case of the cusps is clear, so let us without loss of generality assume that $z\in \mathcal X$.
We start by proving that ($a$) and ($a'$) are equivalent. Recall that $X_{\O_K}$ is the preimage of $\mathbb{A}_{\O_K}^{1}$ under the morphism of $\mathcal O_K$-schemes $j\colon X_{\O_K}^{\ast}\rightarrow \P_{\O_K}^{1}$. Upon formal completion and passing to the adic generic fibre, $j$ becomes $j^{\mathrm{an}}$ while $\mathbb{A}^{1}_{\O_K}\subseteq \P^{1}_{\O_K}$ is sent to the open disc $B_{1}(0)\subseteq \mathbb{A}_K^{1,\mathrm{an}}\subseteq \P_K^{1,\mathrm{an}}$ defined by $|j(z)|\leq 1$. Since the adic generic fibre of the completion of $X_{\O_K}\subseteq X_{\O_K}^{\ast}$ is $\mathcal X_{\mathrm{gd}}\subseteq \mathcal X^{\ast}$, this shows that $\mathcal X_{\mathrm{gd}}$ is precisely the preimage of $B_{1}(0)$ under $j^{\mathrm{an}}\colon \mathcal X=X^{\mathrm{an}}\rightarrow \mathbb{A}^{1,\mathrm{an}}$.
Next, let us prove that ($b$) and ($b$') are equivalent. We can always find a morphism
\[r_z\colon \operatorname{Spa}(C,C^{+})\rightarrow \mathcal X\]
where $(C,C^{+})$ is a complete algebraically closed non-archimedean field, such that $z$ is in the image of $r_z$. It thus suffices to show that $r_z$ factors through some $\mathcal D\hookrightarrow \mathcal X^{\ast}$. By Lemma~\ref{l:moduli interpretation of adic Tate parameter space} it suffices to show that ($b$') holds if and only if the elliptic curve $E$ over $C$ that $r_z$ represents is a Tate curve with nilpotent parameter $q_E\in C$.
The image of $j$ in $C$ is precisely the $j$-invariant $j_E$ of $E$. Since in a non-archimedean field the elements with cofinal valuation are precisely the topologically nilpotent ones, condition ($b$') is equivalent to $j_E\ne 0$ and $j_E^{-1}$ being topologically nilpotent.
We can now argue like in the classical case of $p$-adic fields to see that this is equivalent to $E$ being a Tate curve with $q_E$ topologically nilpotent:
Assume the latter, then $j_E=q_E^{-1}+744+196884q+\dots \ne 0$ has valuation $|j_E|=|1/q_E|$ in $C$ and thus $j_E$ satisfies ($b$').
To see the converse, recall that in the formal Laurent series ring $\mathbb{Z}\cc{q}$ the formula $j(q)=q^{-1}+744+\dots$ reverses to
\[q(j^{-1}) = j^{-1}+744j^{-2}+750420j^{-3}+\dots \in \mathbb{Z}\llbracket j^{-1}\rrbracket. \]
If now $j_E^{-1}$ is topologically nilpotent, the above series converges in $C$ and we obtain a topologically nilpotent element $q_E\in C^{\times}$ with $j_E=1/q_E+744+\dots=j(q_E)$. The Tate curve $\mathrm{T}(q_E)_C$ over $C$ thus has the same $j$-invariant as $E$, and since $C$ is algebraically closed we conclude that $E\cong \mathrm{T}(q_E)_C$. This shows that ($b$) and ($b$') are equivalent.
Next, let us show that ($c$') holds if and only if ($a$') and ($b$') don't hold. Recall that we always have a unique height 1 vertical generisation $z'$. Clearly $|j(z)|\neq 0$ if and only if $|j(z')|\neq 0$, and if in this case $|j(z)|^{-1}$ is cofinal then $|j(z')|^{-1}$ is cofinal. This implies that (b') and (c') can't hold at the same time. On the other hand, if $|j(z)|>1$, then either $|j(z')|=1$, or $|j(z')|>1$ in which case $|j(z')|^{-1}<1$ is cofinal because $v_{z'}$ has height 1. This shows that if $|j(z)|>1$ then we are either in case ($b$') or in ($c$').
Since clearly ($c$) implies that ($a$) and ($b$) do not hold, it remains to prove ($c$') implies ($c$). But this follows from applying the equivalence of ($a$) and ($a'$) first to $z$ and then to $z'$.
\end{proof}
\begin{Example}\label{ex: rank-2-point}
Let us work out an example for an elliptic curve corresponding to a point of type $(c)$: Let $x$ be a cusp and $L=L_x$.
Let $\mathbb{R}_{>0}\times \gamma^{\mathbb{Z}}$ be the totally ordered group for which $\gamma$ is such that $z<\gamma^n<1$ for all $n\in \mathbb{Z}_{\geq 1}$ and all $z\in \mathbb{R}_{<1}$. Equip the field $F=\mathcal O_{L}\lauc{q}[\tfrac{1}{p}]$ with the valuation
\[v_{1^{-}}\colon F\rightarrow (\mathbb{R}_{> 0}\times \gamma^{\mathbb{Z}}) \cup \{0\},\quad \sum a_nq^n\mapsto \max_{n\in\mathbb{Z}} |a_n|\gamma^n. \]
Recall that $\mathfrak{m}_{L}\subseteq \mathcal O_{L}$ is the maximal ideal. The valuation subring of $F$ defined by $v_{1^{-}}$ is
\[ F^{+}=\left\{\sum_{n\gg -\infty}^{\infty}a_nq^n \in \mathcal O_{L}\lauc{q} \middle | a_n\in \mathfrak m_L\text{ for all }n<0\right\}. \]
Indeed, we have $v_{1^{-}}(\sum_{n\gg -\infty}^{\infty}a_nq^n)\leq 1$ if and only if $|a_n|\gamma^n\leq 1$ for all $n$. For $n\geq 0$ we have $|a_n|\gamma^n\leq 1$ if and only if $|a_n|\leq 1$, that is $a_n\in \mathcal O_{L}$. For $n<0$, on the other hand, $|\gamma|^n$ is ``infinitesimally'' bigger than $1$, so that $|a_n|\gamma^n\leq 1$ if and only if $|a_n|<1$, that is $a_n\in \mathfrak{m}_L$.
The Tate curve $\mathrm{T}(q^{e})$ over $F$ for $e=e_x$ equipped with the $\Gamma^p$-structure corresponding to $x$, gives rise to a map
\[ \varphi\colon \operatorname{Spa}(F, F^{+})\rightarrow \mathcal X^{\ast}.\]
We claim that $\varphi$ sends $v_{1^{-}}$ neither into $\mathcal X_{\mathrm{gd}}$ nor into any of the Tate parameter spaces $\mathcal D\subseteq \mathcal X^{\ast}$. Indeed, the $j$-invariant of $\mathrm{T}(q^{e})$ is
\begin{equation}\label{equation: j-invariant of Tate curve over rank 2 point}
j=q^{-e}+744+q^e(\dots) \not \in F^{+}
\end{equation}
which is not contained in $F^+$ by the above description. This shows that $\mathrm{T}(q^e)$ does not extend to an elliptic curve over $F^+$. On the other hand, $q$ is not locally topologically nilpotent in $L$. Thus, by Lemma~\ref{l:moduli interpretation of adic Tate parameter space}, $v_{1^{-}}$ does not land in the open subspaces $\mathcal D_x\hookrightarrow \mathcal X^{\ast}$.
Prop.~\ref{p: classification of points of X^ast} explains this as follows: The unique rank 1 vertical generisation of $v_{1^{-}}$ is
\[ v\colon F \rightarrow \mathbb{R}_{\geq 0},\quad \sum a_nq^n\mapsto \max_{n\in\mathbb{Z}} |a_n|\]
with larger valuation ring $\mathcal O_{L}\lauc{q}\supseteq F^{+}$.
We see from equation~\eqref{equation: j-invariant of Tate curve over rank 2 point} that
\[|j(v_{1^{-}})|=\gamma^{-e}<1, \text{ while } |j(v)|=1.\]
This shows that $\varphi$ sends $v_{1^{-}}$ to one of the points of type (c) in Prop.~\ref{p: classification of points of X^ast}, while its generisation $v$ goes to the point of $\mathcal X_{\mathrm{gd}}$ defined in Remark~\ref{remark: Tate curve of good reduction}.
\end{Example}
\subsection{Tate curve parameter spaces at level $\Gamma_0(p^n)$}
Next, we discuss the behaviour of the Tate curve parameter spaces in the anticanonical tower. For this we first recall the situation at the cusps on the level of schemes:
Consider the forgetful morphism $f\colon X^{\ast}_{\Gamma_0(p)}\to X^{\ast}$. Over each cusp of $X^{\ast}$ there are precisely two cusps of $X^{\ast}_{\Gamma_0(p)}$: One is called the \'etale cusp, it corresponds to the $\Gamma_0(p)$-level structure $\mu_{p}\subseteq \mathrm{T}(q)[p]$ on the Tate curve. The other is the ramified cusp, it corresponds to the level structure $\langle q^{1/p}\rangle \subseteq \mathrm{T}(q)[p]$. In particular, this latter level structure is only defined over $\mathbb{Z}\cc{q^{1/p}}$, and by Prop.~\ref{p:completion-at-cusp} the completion at this cusp is given by
\[\operatorname{Spf}(\O_L\llbracket q^{1/p}\rrbracket )\to X_{\Gamma_0(p)}^{\ast}.\]
The names reflect that the map $X^{\ast}_{\Gamma_0(p)}\to X^{\ast}$ is \'etale at the one sort of cusps, but is ramified at the other: Over the \'etale cusp the map induced on completions is the identity
\[\O_L\llbracket q\rrbracket\to \O_L\llbracket q\rrbracket\]
whereas over the ramified cusp it is the inclusion
\[\O_L\llbracket q\rrbracket\to \O_L\llbracket q^{1/p}\rrbracket.\]
For higher level structures $\Gamma_0(p^n)$, the curve $X^{\ast}_{\Gamma_0(p^n)}\to X^{\ast}$ has more cusps of different degrees of ramification $p^i$ with $i\in\{0,\dots,n\}$, and corresponding completions of the form $\operatorname{Spf}(\O_L\llbracket q^{1/p^i}\rrbracket )\to X^{\ast}_{\Gamma_0(p^n)}$. There is, however, exactly one \'etale cusp, corresponding to the $\Gamma_0(p^n)$-level structure $\mu_{p^n}\subseteq\mathrm{T}(q)[p^n]$, and exactly one purely ramified one, corresponding to $\langle q^{1/p^n}\rangle$. Relatively with respect to $X^{\ast}_{\Gamma_0(p^n)}\to X^{\ast}_{\Gamma_0(p)}$, the purely ramified cusps lie over the ramified cusps of $X^{\ast}_{\Gamma_0(p)}$, while all other cusps of $X^{\ast}_{\Gamma_0(p^n)}$ lie over the \'etale cusps of $X^{\ast}_{\Gamma_0(p)}$.
All constructions from the last two sections now go through with the same proofs for $\mathcal X^{\ast}$ replaced by $\mathcal X^{\ast}_{\Gamma_0(p^n)}$, and $\O_L\llbracket q\rrbracket $ replaced by $\O_L\llbracket q^{1/p^i}\rrbracket $ where $i$ depends on the cusp:
\begin{Definition}
For any $n\in \mathbb{Z}_{\geq 0}$, we write $\mathcal D_i:=\operatorname{Spa}(L\langle q^{1/p^i}\rangle,\O_L\langle q^{1/p^i}\rangle)(|q|>1)$ for the open unit disc over $L$ in the parameter $q^{1/p^i}$. We write $\mathring{\mathcal D}_i$ for the open subspace obtained by removing the origin, i.e.\ the point defined by $q^{1/p^i}\mapsto 0$. When we want to emphasize the dependence on field $L=L_x$ determined by a cusp $x\in \mathcal X^{\ast}$, we write these as $\mathcal D_{i,x}$ and $\mathring{\mathcal D}_{i,x}$.
\end{Definition}
Then like in Lemma~\ref{l: Conrad's theorem on generic fibres of completions around the cusp}, we have for any cusp $x$ of $\mathcal X^{\ast}$ and any cusp of $\mathcal X^{\ast}_{\Gamma_0(p^n)}$ over $x$ of ramification degree $p^i$ a canonical open immersion
\[\mathcal D_{i,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p^n)}.\]
From now on until the next chapter, we shall focus exclusively on the anticanonical locus $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$. Here the ramification is very easy to describe, by the following proposition:
\begin{Proposition}\label{Proposition: Tate parameter spaces in the Gamma_0-tower}
Fix a cusp $x\in \mathcal X^{\ast}$.
\begin{enumerate}
\item The cusps of $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ are precisely the purely ramified cusps of $\mathcal X^{\ast}_{\Gamma_0(p^n)}$. In particular, there is a canonical open immersion $\mathcal D_{n,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ over $x$.
\item The forgetful map $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a\to \mathcal X^{\ast}_{\Gamma_0(p^{n-1})}(\epsilon)_a$ gives a bijection between the respective cusps.
The associated Tate curve parameter spaces fit into Cartesian diagrams
\begin{equation*}
\begin{tikzcd}
\mathcal D_{n,x} \arrow[r] \arrow[d,hook] & \mathcal D_{n-1,x} \arrow[d,hook] \\
\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a \arrow[r]
& \mathcal X^{\ast}_{\Gamma_0(p^{n-1})}(\epsilon)_a,
\end{tikzcd}
\end{equation*}
where $\mathcal D_n\to \mathcal D_{n-1}$ is the canonical finite flat map of degree $p$ which sends $q\mapsto q$.
\item For any adic space $S$ over $(K,\mathcal O_K)$, the $S$-points of $\mathring{\mathcal D}_{n,x}\hookrightarrow \mathcal X_{\Gamma_0(p^n)}(\epsilon)_a$ correspond functorially to Tate curves $\mathrm{T}(q)$ over $\mathcal O(S)$ with topologically nilpotent parameter $q\in\mathcal O(S)$, a $\Gamma^p$-structure corresponding to $x$ and a choice of $p^n$-th root $q^{1/p^n}$ of $q$ defining a $\Gamma_0(p^n)$-structure $\langle q^{1/p^n}\rangle \subseteq \mathrm{T}(q)$.
\end{enumerate}
\end{Proposition}
\begin{proof}
Since the canonical subgroup of the Tate curve is given by $\mu_{p}\subseteq \mathrm{T}(q)[p]$, the cusps of $\mathcal X^{\ast}_{\Gamma_0(p^n)}$ contained in $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ are precisely the ones over the ramified cusps in $X^{\ast}_{\Gamma_0(p)}$. But the cusps of $ X^{\ast}_{\Gamma_0(p^n)}$ over the ramified cusps of $ X^{\ast}_{\Gamma_0(p)}$ are precisely the purely ramified ones. This proves (1). Part (3) follows immediately.
The diagram in (2) commutes because by construction it is the generic fibre of a commutative diagram of formal schemes. Since the morphisms are open immersions, it suffices to check that it is Cartesian on the level of points, where it follows from (3) and Lemma~\ref{l:moduli interpretation of adic Tate parameter space}.
\end{proof}
\subsection{Tate parameter spaces of \texorpdfstring{$\mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a$}{X*_G_1(p^n)(e)_a}}
We now pass to higher level at $p$ and describe Tate curve parameter spaces of the form $\mathcal D_n\hookrightarrow \mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a$.
We note that the integral theory of cusps for $\Gamma_1(p^n)$ is slightly complicated in general, see \S4.2 of \cite{conrad2007arithmetic} for a thorough discussion. However, over the anticanonical locus, the story is very simple:
\begin{Lemma}\label{l:Tate parameter spaces of X^{ast}_{Gamma_1(p^n)}}
Let $x$ be a cusp of $\mathcal X^{\ast}$. Then there are $\mathbb{Z}_p^\times$-equivariant Cartesian squares
\begin{equation*}
\begin{tikzcd}
\underline{(\mathbb{Z}/p^{n+1}\mathbb{Z})^{\times}} \times \mathcal D_{n+1,x} \arrow[d] \arrow[r] &\underline{(\mathbb{Z}/p^n\mathbb{Z})^{\times}}\times \mathcal D_{n,x}\arrow[d, hook] \arrow[r] & \mathcal D_{n,x} \arrow[d, hook] \\
\mathcal X^{\ast}_{\Gamma_1(p^{n+1})}(\epsilon)_a \arrow[r]&\mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a,
\end{tikzcd}
\end{equation*}
in which the morphism on the top left is given by the reduction $(a,q)\mapsto (\overline{a},q)$.
\end{Lemma}
\begin{proof}
As the morphisms in the bottom row are finite \'etale Galois torsors for the groups $\mathbb{Z}/p^n\mathbb{Z}$ and $\mathbb{Z}/p^{n+1}\mathbb{Z}$, it suffices to produce a section $\mathcal D_{n,x}\rightarrow
\mathcal X^{\ast}_{\Gamma_1(p^n)}$.
By Prop.~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower}, the cusp of $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ over $x$ corresponds to the choice of $\langle q^{1/p^n}\rangle\subseteq \mathrm{T}(q)$ as a $\Gamma_0(p^n)$-structure. This can be lifted canonically to the $\Gamma_1(p^n)$-structure given by the $p^n$-th root $q^{1/p^n}$ in $\O(\mathcal D_{n,x})$. Upon normalisation, we thus get a canonical lift
\begin{equation*}
\begin{tikzcd}
& \operatorname{Spec}(\mathbb{Z}_p\llbracket q^{1/p^n}\rrbracket\otimes L_x) \arrow[d] \arrow[ld, dashed] \\
X^{\ast}_{\Gamma_1(p^n)} \arrow[r] & X^{\ast}_{\Gamma_0(p^n)}.
\end{tikzcd}
\end{equation*}
The factorisation $\mathcal D_{n,x}\rightarrow \operatorname{Spec}(\mathbb{Z}_p\llbracket q^{1/p^n}\rrbracket\otimes L_x)\rightarrow X^{\ast}_{\Gamma_0(p^n)}$ from the analogue for level $\Gamma_0(p^n)$ of Lemma~\ref{lemma for comparing schematic cusp to adic cusp} together with the universal property of the analytification now give rise to the desired section
\begin{equation*}
\begin{tikzcd}
& \mathcal D_{n,x} \arrow[d] \arrow[ld, dashed] \\
\mathcal X^{\ast}_{\Gamma_1(p^n)} \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^n)}.
\end{tikzcd}
\end{equation*}
This shows that the right square is Cartesian.
That the outer square is Cartesian follows in combination with Prop.~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower}. Consequently, the left square is also Cartesian.
\end{proof}
\subsection{Tate parameter spaces of \texorpdfstring{$\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a$}{X*_G(p^n)(e)_a}}
Next, we look at what happens with the cusps in the transition $\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a\rightarrow \mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a$.
Let us fix notation for the {left} action of $\operatorname{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ on $\mathcal X^{\ast}_{\Gamma(p^n)}$ in terms of moduli: For any $\gamma\in\operatorname{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ it is given by sending a trivialisation $(\mathbb{Z}/p^n\mathbb{Z})^{2}\xrightarrow{\sim} E[p^n] $ to \[(\mathbb{Z}/p^n\mathbb{Z})^{2}\xrightarrow{\gamma^\vee} (\mathbb{Z}/p^n\mathbb{Z})^{2}\xrightarrow{\sim} E[p^n]\]
where $\gamma^{\vee}=\det(g)\gamma^{-1}$. Here the inverse is necessary to indeed obtain a left action, and the twist by $\det(g)$ ensures that the action on the fibres of the map $\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a\rightarrow \mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a$ is given by matrices of the form $\smallmat{\ast}{\ast}{0}{1}$ rather than $\smallmat{1}{\ast}{0}{\ast}$.
\begin{Definition}
For $0\leq m\leq n\in\mathbb{N}$, we denote by $\Gamma_0(p^m,\mathbb{Z}/p^n\mathbb{Z})\subseteq \operatorname{GL}_2(p^m,\mathbb{Z}/p^n\mathbb{Z})$ the subgroup of matrices of the form $\smallmat{\ast}{\ast}{c}{\ast}$ with $c\equiv 0 \bmod p^m$.
\end{Definition}
The forgetful map $X^{\ast}_{\Gamma(p^n)}\to X^{\ast}_{\Gamma_0(p)}$ is given by reducing $(\mathbb{Z}/p^n\mathbb{Z})^{2}\xrightarrow{\sim} E[p^n]$ mod $p$ to $(\mathbb{Z}/p\mathbb{Z})^{2}\xrightarrow{\sim} E[p]$ and sending it to the subgroup generated by $(1,0)$. Consequently, the action of $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$ leaves the forgetful morphism $X^{\ast}_{\Gamma(p^n)}\to X^{\ast}_{\Gamma_0(p)}$ invariant. We see from this that the action of $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$ restricts to an action on $\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a\subseteq \mathcal X^{\ast}_{\Gamma(p^n)}$.
\begin{Lemma}\label{l:Tate parameter spaces of X_Gamma(p^n)(epsilon)_a}
Let $x$ be a cusp of $\mathcal X^{\ast}$.
\leavevmode
\begin{enumerate}
\item Depending on our chosen primitive root $\zeta_{p^n}$, there is a canonical Cartesian diagram
\begin{equation*}
\begin{tikzcd}
\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x} \arrow[d,hook] \arrow[r] & \mathcal D_{n,x} \arrow[d,hook] \\
\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a.
\end{tikzcd}
\end{equation*}
where the left map is $\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})$-equivariant for the action via the first factor.
\item Let $x_{\gamma}$ be the cusp of $\mathcal X^{\ast}_{\Gamma(p^n)}(0)_a$ over $x$ determined by $\gamma=\smallmat{a}{b}{0}{d}\in \Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})$.
Then for any honest adic space $S$ over $K$, the $S$-points of $x_{\gamma}\colon \mathring{\mathcal D}_{n,x}\hookrightarrow \mathcal X_{\Gamma(p^n)}(\epsilon)_a$ correspond functorially to Tate curves $E=\mathrm{T}(q)$ with topologically nilpotent parameter $q_E\in\mathcal O(S)$, a $\Gamma^p$-structure corresponding to $x$, and the basis $(q_E^{d/p^n},q_E^{-b/p^n}\zeta_{p^n}^{a})$ of $E[p^n]$, where $q_E^{1/p^n}$ is the $p^n$-th root of $q_E$ determined by $x$.
\item The reduction map $\Gamma_0(p^{n+1},\mathbb{Z}/p^{n+1}\mathbb{Z})\rightarrow \Gamma_0(p^{n},\mathbb{Z}/p^{n}\mathbb{Z})$ gives a Cartesian diagram
\begin{equation*}
\begin{tikzcd}[column sep = 2cm]
\underline{\Gamma_0(p^{n+1},\mathbb{Z}/p^{n+1}\mathbb{Z})} \times \mathcal D_{n+1,x} \arrow[d,hook] \arrow[r,"{(\gamma,q)\mapsto (\overline{\gamma},q)}"] & \underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x} \arrow[d,hook] \\
\mathcal X^{\ast}_{\Gamma(p^{n+1})}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a.
\end{tikzcd}
\end{equation*}
\end{enumerate}
\end{Lemma}
\begin{proof}
\begin{enumerate}
\item
Arguing as in Lemma~\ref{l:Tate parameter spaces of X^{ast}_{Gamma_1(p^n)}}, it suffices to produce a splitting
\begin{equation*}\label{diagram: lifting Tate curve from Gamma_0 to Gamma}
\begin{tikzcd}
& \operatorname{Spec}(\mathbb{Z}_p\llbracket q^{1/p^n}\rrbracket\otimes L_x) \arrow[d] \arrow[ld, dashed] \\
X^{\ast}_{\Gamma(p^n)} \arrow[r] & X^{\ast}_{\Gamma_1(p^n)}
\end{tikzcd}
\end{equation*}
which we construct as follows: Consider the Tate curve $T=\mathrm{T}(q^e)_R$ over $R=\mathbb{Z}_p\cc{q^{1/p^n}}\otimes \O_L$ with its Weil pairing $e_{p^n}\colon T[p^n]\times T[p^n]\to \mu_{p^n}$. Specialising at $q^{e/p^n}\in T[p^n]$, we obtain an isomorphism \[e(q^{e/p^n},-)\colon C_n\to \mu_{p^n}\]
where $C_n$ is the canonical subgroup of $T$. The preimage of $\zeta_{p^n}$ gives the second basis vector of $T[p^n]$ of an anticanonical $\Gamma(p^n)$-structure on $T[p^n]$, defining the desired lift.
By analytification we then obtain the map $g\colon \mathcal D_{n,x}\rightarrow \mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a$ which gives the Cartesian diagram as $\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a\to\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ is Galois with group $\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}$.
\item We have just seen that the cusp label $1\in \Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})$ corresponds via $q^e\mapsto q_E$ to the isomorphism $\varphi\colon (\mathbb{Z}/p^n\mathbb{Z})^2\to E[p^n]$ defined by the ordered basis $(q_E^{1/p^n},\zeta_{p^n})$. For the general case, we use that the action of $\gamma=\smallmat{a}{b}{0}{d}$ is given by
\[
\begin{tikzcd}[row sep = 0.2cm]
{\smallvector{1}{0}} \arrow[r, "\gamma^{\vee}", maps to] & {\smallvector{d}{0}} \arrow[r, "\varphi", maps to] & q_E^{d/p^n} \\
{\smallvector{0}{1}} \arrow[r, "\gamma^{\vee}", maps to] & {\smallvector{-b}{a}} \arrow[r, "\varphi", maps to]& q_E^{-b/p^n}\zeta_{p^n}^a
\end{tikzcd}
\]
\item This follows from (1)\ and Lemma~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower} as $X^{\ast}_{\Gamma(p^n)}\to X^{\ast}_{\Gamma(p^{n-1})}$ is equivariant for $\pi$.\qedhere
\end{enumerate}
\end{proof}
All in all, we get the following description of Tate curve parameter spaces at finite level:
\begin{Proposition}\label{p: structure of Tate parameter spaces from tame level to Gamma(p^n)}
Let $x$ be any cusp of $\mathcal X^{\ast}$
Then depending on our choice of $\zeta_{p^n}\in K$, there is a canonical tower of Cartesian squares
\begin{equation*}
\begin{tikzcd}[column sep = 1.4cm]
\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x} \arrow[d, hook,"\varphi"] \arrow[r,"\smallmat{a}{b}{0}{d}\mapsto d"] & \underline{(\mathbb{Z}/p^n\mathbb{Z})^{\times}}\times \mathcal D_{n,x} \arrow[d, hook] \arrow[r] & \mathcal D_{n,x} \arrow[r] \arrow[d, hook] & \mathcal D_x \arrow[d, hook] \\
\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}(\epsilon).
\end{tikzcd}
\end{equation*}
\end{Proposition}
\begin{proof}
The square on the right is Prop.~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower}.(2).
The square in the middle is Lemma~\ref{l:Tate parameter spaces of X^{ast}_{Gamma_1(p^n)}}.
The Cartesian diagram on the left exists as a consequence of Lemma~\ref{l:Tate parameter spaces of X_Gamma(p^n)(epsilon)_a}.1 together with the fact that $X^{\ast}_{\Gamma(p^n)}\to X^{\ast}_{\Gamma_1(p^{n})}$ is equivariant with respect to the map $ \smallmat{a}{b}{0}{d}\mapsto d$.
\end{proof}
Lemma~\ref{l:Tate parameter spaces of X_Gamma(p^n)(epsilon)_a} describes the $\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})$-action at the cusps of $\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a$, but there is also an action of the larger group $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$. While the action of $\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})$ just permutes the different copies of $\mathcal D_{n,x}$ over $x$, the action of $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$ in general has a non-trivial effect on each of these Tate curve parameter space, because it also takes into account isomorphisms of Tate curves of the form $q^{1/p^n}\mapsto \zeta_{p^n}q^{1/p^n}$, as we shall now discuss.
\begin{Proposition}\label{Proposition: the action of Gamma_0(p,Z/p^nZ) on cusps of Gamma(p^n)}
Over any cusp $x$ of $\mathcal X^{\ast}$, the $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$-action on $\mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a$ restricts to an action on $\varphi\colon \underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a$ that can be described as follows: Equip $\underline{\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x}$ with a right action by $p\mathbb{Z}/p^n\mathbb{Z}$ given by $(\gamma,q)\mapsto (\gamma\smallmat{1}{0}{h}{1},\zeta^{h/e_x}_{p^n}q)$, then
\[(\underline{\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x})/(p\mathbb{Z}/p^n\mathbb{Z})=\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x}\]
and the left action of $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$ is the natural left action via the first factor.
Explicitly, for any $\gamma_1\in \Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$, the action is given by
\begin{alignat*}{3}
\gamma_1\colon &&\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x}&\;\xrightarrow{\sim}\;&& \underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x}\\
&& \gamma_2,q^{1/p^n} &\; \;\mapsto\; && \smallmat{\det( \gamma_3)/d_3}{b_3}{0}{d_3},\zeta_{p^n}^{-\frac{c_3}{d_3e_x}}q^{1/p^n}
\end{alignat*}
where $\gamma_3=\smallmat{a_3}{b_3}{c_3}{d_3}:=\gamma_1\cdot \gamma_2$.
\end{Proposition}
Here we recall that $e_x$ was introduced in Notation~\ref{n:e}.
\begin{proof}
Recall that the reason for the pullback of $\mathcal D_{n,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p)}$ to $\mathcal X^{\ast}_{\Gamma(p^n)}$ to be of the form $\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x}$ even though $\mathcal X_{\Gamma(p^n)}\rightarrow \mathcal X_{\Gamma_0(p)}$ has larger Galois group $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$ is that in the step from $\mathcal X^{\ast}$ to $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ the isomorphism
\begin{equation*}\label{eq:automorphisms of D_{n,x}}
\psi_h\colon \mathcal D_{n,x}\rightarrow \mathcal D_{n,x}, \quad q^{1/p^n}\mapsto \zeta^{h}_{p^n}q^{1/p^n}
\end{equation*}
for any $h\in \mathbb{Z}/p^n\mathbb{Z}$ induces an isomorphism of moduli for the universal Tate curve $T=\mathrm{T}(q^{e_x})$ over $\mathcal D_{x}$ that sends the anti-canonical $\Gamma_0(p^n)$-level structure $\langle q^{{e_x}/p^n} \rangle$ to $\langle \zeta^{he_x}_{p^n}q^{{e_x}/p^n} \rangle$.
For the action of $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$, this means the following:
Consider the Tate parameter space $\varphi(1)\colon \mathcal D_{n,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^n)}(\epsilon)_a$ at $1\in \Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$, associated to the isomorphism $\alpha\colon (\mathbb{Z}/p^n\mathbb{Z})^2\to T[p^n]$ defined by the ordered basis $(q^{e_x/p^n},\zeta_{p^n})$. Then the action of $\gamma_1=\smallmat{1}{0}{h}{1}$ sends this to the isomorphism $\alpha\circ \gamma^\vee$ defined by $(1,0)\mapsto \zeta_{p^n}^{-h}q^{e_x/p^n}$ and $(0,1)\mapsto \zeta_{p^n}$. The isomorphism $\psi_{-h/e_x}$ identifies this with the basis $(q^{1/p^n},\zeta_{p^n})$:
\begin{equation*}
\begin{tikzcd}
q^{1/p^n}\arrow[d,mapsto]&\langle q^{e_x/p^n}\rangle \arrow[d] \arrow[r] & \mathring \mathcal D_{n,x}\arrow[d,"\psi_{-h/e_x}"'] \arrow[r, hook,"\varphi(1)"] & \mathcal X_{\Gamma(p^n)}(\epsilon)_a \arrow[d, "\gamma_1"] \\
\zeta_{p^n}^{-h/e_x}q^{1/p^n}&\langle \zeta^{-h}_{p^n}q^{e_x/p^n}\rangle \arrow[r] & \mathring{\mathcal D_{n,x}} \arrow[r, hook,"\varphi(1)"] & \mathcal X_{\Gamma(p^n)}(\epsilon)_a.
\end{tikzcd}
\end{equation*}
The action of $\smallmat{1}{0}{h}{1}$ on the component $\{1\}\times \mathcal D_{n,x}$ of $\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{n,x}$ defined by $1\in\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})$ is thus given by $\psi_{-h/e_x}$.
In general, in order to describe the action of $\gamma_1$ on the component $\{\gamma_2 \}\times \mathcal D_{n,x}$, it suffices to compute the action of $\gamma_3=\gamma_1\cdot \gamma_2$ on $\{1\}\times \mathcal D_{n,x}$, since we have a commutative diagram
\[\begin{tikzcd}
{\{1 \}\times \mathcal D_{n,x}} \arrow[d, "\gamma_2"] \arrow[r, dotted] \arrow[rd, "\gamma_3"] & {\{1 \}\times \mathcal D_{n,x}} \arrow[d, dotted] \\
{\{\gamma_2 \}\times \mathcal D_{n,x}} \arrow[r, "\gamma_1"] & {\{\gamma_3 \}\times \mathcal D_{n,x}}.
\end{tikzcd}\]
One can now decompose $\gamma_3$ into the actions which we have already computed, using
\begin{equation}\label{equation: decomposition of Gamma_0(p) into Gamma_0(p^n){1}{0}{ast}{1}}
\smallmat{a}{b}{c}{d}=\smallmat{\det(\gamma)/d}{b}{0}{d}\smallmat{1}{0}{c/d}{1}.
\end{equation}
Applying this to $\gamma_3$, we get the desired action by equivariance of $\varphi$ under $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$.
\end{proof}
\section{Adic theory of cusps at infinite level}\label{s:adic-cusps-infinite}
We now pass to infinite level, starting with $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a\sim\varprojlim \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$. We first note:
\begin{Lemma}\label{l: moduli interpretation of modular curve at infinite level}
Let $(R,R^+)$ be a perfectoid $K$-algebra. Then the set $\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a(R,R^+)$ is in functorial bijection with isomorphism classes of triples $(E,\alpha_N,(G_n)_{n\in \mathbb N})$ of elliptic curves $E$ over $R$ that are $\epsilon$-nearly ordinary, together with a $\Gamma^p$-structure $\alpha_N$ and a collection of anticanonical cyclic subgroups $G_n\subseteq E[p^n]$ of order $p^n$ for all $n$ that are compatible in the sense that $G_n = G_{n+1}[p^n]$. Equivalently, one could view $G=(G_n)_{n\in \mathbb N}$ as a p-divisible subgroup of $E[p^\infty]$ of height 1 such that $D_1$ is anticanonical.
\end{Lemma}
\begin{proof}
Since $(R,R^+)$ is perfectoid, one has
\[\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a(R,R^+)= \varprojlim_{n\in \mathbb{N}} \mathcal X_{\Gamma_0(p^n)}(\epsilon)_a(R,R^+)\]
by \cite[Prop~2.4.5]{ScholzeWeinstein}. The statement thus follows from Lemma~\ref{Lemma: adic moduli interpretation of X_Gamma^an}.
\end{proof}
\begin{Definition}
We shall call the $p$-divisible group $D$ an {anticanonical $\Gamma_0(p^\infty)$-structure}.
\end{Definition}
We wish to study the cusps at infinite level, by which we mean the following:
\begin{Definition}
By the cusps of $\XaGea{0}{\infty}$, $\XaGea{1}{\infty}$ etc.\ we mean the preimage of the divisor of cusps of $\mathcal X^{\ast}$ endowed with its induced reduced structure.
\end{Definition}
We note that at infinite level, the cusps are in general a profinite Zariski-closed subspace of the respective infinite level modular curve.
However, at level $\Gamma_0(p^\infty)$, we will see that the map $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a\rightarrow \mathcal X^{\ast}$ isomorphically identifies the cusps of $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$ and those of $\mathcal X^{\ast}$.
\subsection{The perfectoid Tate parameter space at level \texorpdfstring{$\Gamma_0(p^\infty)$}{G_0(p^infty)}}\label{s:infinite-level-Gamma_0}
We start our discussion by having a closer look at the cusps in the anticanonical tower.
Like before, we fix a cusp $x$ of $\mathcal X^{\ast}$ and let $L$ be the field of definition of the associated Tate curve, like in Def.~\ref{d:field-of-definition-of-Tate-curve}. In particular, if $K$ contains a primitive $N$-th unit root, we simply have $L=K$.
By Lemma~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower}, there is for $x$ a tower of Cartesian squares
\begin{equation}\label{diagram: Tate parameter spaces in anticanonical tower, forgetful morphism version}
\begin{tikzcd}
\dots \arrow[r] & \mathcal D_2 \arrow[r]\arrow[d,hook] & \mathcal D_1 \arrow[d,hook] \arrow[r] & \mathcal D \arrow[d,hook] \\
\dots \arrow[r] &\mathcal X^{\ast}_{\Gamma_0(p^2)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}(\epsilon).
\end{tikzcd}
\end{equation}
\begin{Lemma}\label{the perfectoid parameter space of the Tate curve at infinite level}
\begin{enumerate}
\item The perfectoid open disc $\mathcal D_\infty:=\operatorname{Spa}(L\langle q^{1/p^\infty}\rangle,\O_L\langle q^{1/p^\infty}\rangle)(|q|< 1)$ is a tilde-limit
$\mathcal D_\infty \sim \varprojlim_{n\in\mathbb{N}} \mathcal D_n$.
\item Denote by $\mathcal O_{L}\llbracket q^{1/p^\infty}\rrbracket$ the $(p,q)$-adic completion of $\varinjlim_{n \in \mathbb{N}} \mathcal O_{L}\llbracket q^{1/p^n}\rrbracket$. Then $\mathcal D_\infty$ is the adic generic fibre of the formal scheme $\operatorname{Spf}(\mathcal O_{L}\llbracket q^{1/p^\infty}\rrbracket)$.
\item The global sections of $\mathcal D_\infty$ are given by $\mathcal O^+(\mathcal D_\infty)= \O_L\llbracket q^{1/p^\infty}\rrbracket$ and
\[\mathcal O(\mathcal D_\infty) = \Bigg \{ \sum_{n\in \mathbb{Z}[\frac{1}{p}]_{\geq 0}}a_nq^n \in L\llbracket q^{1/p^\infty}\rrbracket \Bigg |\quad\stackanchor{ |a_n|q^n\to 0\text{ for all }0\leq q<1, }{|a_n|\to 0 \text{ on bounded intervals}} \Bigg\}\]
where the second condition means that for any $\delta> 0$ and for any bounded interval $I\subseteq \mathbb{Z}[\frac{1}{p}]_{\geq 0}$ there are only finitely many $n\in I$ such that $|a_n|> \delta$.
\end{enumerate}
\end{Lemma}
\begin{proof}
It is clear on global sections that
\[\operatorname{Spa}(L\langle q^{1/p^\infty}\rangle,\O_L\langle q^{1/p^\infty}\rangle)\sim\varprojlim_{n\in\mathbb{N}} \operatorname{Spa}(L\langle q^{p^n}\rangle,\O_L\langle q^{1/p^n}\rangle).\]
The first part follows from \cite[Prop.~2.4.3]{ScholzeWeinstein} since $\mathcal D$ is the restriction to the open subspace defined by $|q|<1$.
More explicitly, this means that $\mathcal D_\infty$ is given by glueing the affinoid perfectoid unit discs of increasing radii $<1$ given by
\[\mathcal D_\infty(|q|^{p^m}\leq|\varpi|) = \operatorname{Spa}(L\langle (q/\varpi^{1/p^m})^{1/p^\infty} \rangle,\mathcal O_L\langle (q/\varpi^{1/p^m})^{1/p^\infty} \rangle)\]
for all $m\in \mathbb{N}$. These can be obtained by rescaling the perfectoid unit disc. Computing the intersection of their respective global functions gives $\O(\mathcal D_\infty)$ and $\O^+(\mathcal D_\infty)$.
Part (2) is not just a formal consequence as tilde-limits do not necessarily commute with taking generic fibres. But it follows from the construction: Let
\[S=\operatorname{Spa}(\mathcal O_{L}\llbracket q^{1/p^\infty}\rrbracket,\mathcal O_{L}\llbracket q^{1/p^\infty}\rrbracket)\]
and consider the subspaces $S(|q|^{p^m}\leq |\varpi|\neq 0)$ which are rational because $(q^n,\varpi)$ is open. As usual, one shows that since $\mathcal O_{L}\llbracket q^{1/p^\infty}\rrbracket$ has ideal of definition $(q,\varpi)$, the element $|q(x)|$ must be cofinal in the value group for any $x \in S$. This shows that
\[S^{\mathrm{ad}}_\eta =S(|\varpi|\neq 0) =\bigcup S(|q|^{p^m}\leq |\varpi|\neq 0).\]
Let $B_m^{+}=\O^+(S(|q|^{p^m}\leq |\varpi|\neq 0))$, then as $q^{p^m}/\varpi \in B_m^{+}$, we have $(q,\varpi)^{p^m}\subseteq(\varpi)$ and the ring $B_m^{+}$ thus has the $\varpi$-adic topology. From this one deduces that $B_m^{+} = \mathcal O_L\langle q^{1/p^\infty}/\varpi^{1/mp^\infty} \rangle$ and thus the spaces $S(|q|^{p^m}\leq |\varpi|\neq 0)$ and $\mathcal D_\infty(|q|^{p^m}\leq |\varpi|\neq 0)$ coincide.
\end{proof}
\begin{Remark}\label{Remark: D_infty is not affinoid}
Note that $\mathcal D_\infty$ is not affinoid, even though it is the generic fibre of an affine formal scheme, as we did not equip $\O_L\llbracket q^{1/p^\infty}\rrbracket$ with the $p$-adic topology.
\end{Remark}
\begin{Definition}
The origin in $\mathcal D_\infty$ is the closed point $x\colon \operatorname{Spa}(L,\mathcal O_L)\rightarrow \mathcal D_\infty$ where $q=0$. By removing this point, we obtain a space $\mathring{\mathcal D}_\infty := \mathcal D_\infty\backslash \{x\}$ that satisfies $\mathring{\mathcal D}_\infty \sim \varprojlim \mathring{\mathcal D}_n$.
\end{Definition}
\begin{Definition}
Let $\overline{\mathcal D}_\infty:=\operatorname{Spa}(\O_L\llbracket q^{1/p^\infty}\rrbracket_p[\tfrac{1}{p}],\O_L\llbracket q^{1/p^\infty}\rrbracket_p)$ with the $p$-adic topology on $\O_L\llbracket q^{1/p^\infty}\rrbracket_p$ (see Def.~\ref{d: OK bb q^1/p^infty _p}). Then it is clear from the definition that $\overline{\mathcal D}_\infty\sim \varprojlim_{q\mapsto q^p} \overline{\mathcal D}$.
\end{Definition}
We are now ready to discuss cusps at infinite level and the corresponding Tate curves.
\begin{Proposition}\label{p: the Tate parameter space at infinite level Gamma_0(p^infty)}
Fix a cusp $x$ of $\mathcal X^{\ast}$, with corresponding cusps in the anticanonical tower.
\begin{enumerate}
\item The open immersions $\mathcal D_n\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ over $x$ in the limit $n\to \infty$ give rise to an open immersion
$\mathcal D_\infty\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$ that fits into a Cartesian diagram
\begin{equation*}
\begin{tikzcd}
\mathcal D_\infty \arrow[d] \arrow[r, hook] & \overline{\mathcal D}_\infty \arrow[d] \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a \arrow[d] \\
\mathcal D \arrow[r, hook] &\overline{\mathcal D} \arrow[r, hook] & \mathcal X^{\ast}(\epsilon).
\end{tikzcd}
\end{equation*}
\item Consider the restriction $\mathring{\mathcal D}_\infty\hookrightarrow \mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a$. For any perfectoid $K$-algebra $(R,R^+)$, \[\mathring{\mathcal D}_\infty(R,R^{+})\subseteq \mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a(R,R^{+})\]
is in functorial bijection with isomorphism classes of triples $(E,\alpha_N,(q_E^{1/p^n})_{n\in \mathbb{N}})$ where $E\cong\mathrm{T}(q_E)$ is a Tate curve over $R$ for some topologically nilpotent unit $q_E\in R$, where $\alpha_N$ is a $\Gamma^p$-level structure, and $(q_E^{1/p^n})_{n\in \mathbb{N}}$ is a compatible system of $p^n$-th roots of $q_E$, defining an anticanonical $\Gamma_0(p^\infty)$-structure on $E$.
\end{enumerate}
\end{Proposition}
Exactly as before, we shall also write $\mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$ for the open immersion in the proposition if we wish to emphasize the cusp $x$ we are working over.
\begin{proof}
The map $\mathcal D_\infty\rightarrow \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$ is induced from Prop.~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower} and Prop.~\ref{the perfectoid parameter space of the Tate curve at infinite level} by the universal property of the perfectoid tilde-limit. The outer square in part (1) is now Cartesian using \cite[Prop~2.4.3]{ScholzeWeinstein}, and the fact that the squares in diagram~\eqref{diagram: Tate parameter spaces in anticanonical tower, forgetful morphism version} are Cartesian.
It is clear that the left square is Cartesian.
To show that the right square is as well,
it now suffices to prove this away from the cusps, where it follows from the relative moduli interpretation: Giving an anticanonical $\Gamma_0(p^\infty)$-level structure on $\mathrm{T}(q)$ over some $\O_L\llbracket q\rrbracket$-algebra where $q$ is invertible corresponds to giving a system of $p^n$-th roots of $q$.
The moduli interpretation of $\mathring{\mathcal D}_\infty$ follows from diagram~\eqref{diagram: Tate parameter spaces in anticanonical tower, forgetful morphism version} and Cor.~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower}.
\end{proof}
\subsection{Tate curve parameter spaces of $\XaGea{1}{\infty}$}\label{s:infinite-level-Gamma_1}
Next, we discuss the Tate parameter spaces in the pro-\'etale map $\XaGea{1}{\infty}\to \XaGea{0}{\infty}$. This is just a matter of pulling back the descriptions from finite level: Let
\[\mathcal X^{\ast}_{\Gamma_1(p^n)\cap \Gamma_0(p^\infty)}(\epsilon)_a:=\XaGea{1}{n}\times_{\XaGea{0}{n}}\XaGea{0}{\infty}.\]
\begin{Lemma}\label{l: Tate parameter spaces of X^ast_Gamma_1(p^m)cap Gamma_0(p^infty)}
Let $x$ be any cusp of $\mathcal X^{\ast}$. Let $n\in \mathbb{Z}_{\geq 0}$.
\begin{enumerate}
\item The map $\mathcal X^{\ast}_{\Gamma_1(p^n)\cap\Gamma_0(p^\infty)}(\epsilon)_a\to \XaGea{1}{n}$ restricts to an isomorphism on the cusps.
\item There are canonical Cartesian cubes
\begin{equation*}
\begin{tikzcd}[row sep ={1cm,between origins}, column sep ={2.1cm,between origins}]
& \underline{(\mathbb{Z}/p^{n+1}\mathbb{Z})^{\times}}\times \mathcal D_{n+1} \arrow[rr]\arrow[dd,hook] & & \underline{(\mathbb{Z}/p^n\mathbb{Z})^{\times}}\times \mathcal D_{n,x} \arrow[dd, hook] \arrow[rr] & & \mathcal D_{n,x} \arrow[dd, hook] \\
\underline{(\mathbb{Z}/p^{n+1}\mathbb{Z})^{\times}}\times \mathcal D_{\infty,x} \arrow[dd, hook] \arrow[rr,crossing over] \arrow[ru] & & \underline{(\mathbb{Z}/p^n\mathbb{Z})^{\times}}\times \mathcal D_{\infty,x} \arrow[rr,crossing over] \arrow[ru] & & {\mathcal D_{\infty,x}} \arrow[dd, hook] \arrow[ru] & \\
& \mathcal X^{\ast}_{\Gamma_1(p^{n+1})}(\epsilon)_a \arrow[rr]& & \mathcal X^{\ast}_{\Gamma_1(p^{n})}(\epsilon)_a \arrow[rr] & & \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a \\
\mathcal X^{\ast}_{\Gamma_1(p^{n+1})\cap \Gamma_0(p^\infty)}(\epsilon)_a \arrow[rr] \arrow[ru] & & \mathcal X^{\ast}_{\Gamma_1(p^n)\cap \Gamma_0(p^\infty)}(\epsilon)_a \arrow[rr]\arrow[from =uu, hook,crossing over] \arrow[ru] & & \mathcal X^{\ast}_{\Gamma_0(p^{\infty})}(\epsilon)_a. \arrow[from =uu, hook,crossing over]\arrow[ru] &
\end{tikzcd}
\end{equation*}
\end{enumerate}
\end{Lemma}
\begin{proof}
In part (2), the bottom faces are Cartesian by definition, the back faces are Cartesian by Lemma~\ref{l:Tate parameter spaces of X^{ast}_{Gamma_1(p^n)}}, the rightmost square is Cartesian by Prop.~\ref{p: the Tate parameter space at infinite level Gamma_0(p^infty)}, and the top faces are clearly also Cartesian. Thus all other faces are Cartesian.
Part (1) follows immediately.
\end{proof}
We now take the limit $n\to \infty$ to get Tate parameter spaces for $\XaGea{1}{\infty}$: In doing so, we need to account for the fact that in the inverse limit, the divisor of cusps becomes a profinite set of points, rather than just a disjoint union of closed points.
\begin{Definition}
Let $S$ be a profinite set, and let $(S_i)_{i\in I}$ be a system of finite sets with $S=\varprojlim_{i\in I} S_i$. Then we define $\underline{S}$ to be the unique perfectoid tilde-limit $\underline{S}\sim \varprojlim_{i\in I} \underline{S_i}$. This is independent of the choice of $S_i$:
Explicitly, $\underline{S}$ is the affinoid perfectoid space
\[\underline{S} = \operatorname{Spa}(\operatorname{Map}_{{\operatorname{cts}}}(S,K),\operatorname{Map}_{{\operatorname{cts}}}(S,\O_K)).\]
\end{Definition}
\begin{Proposition}\label{Proposition: Tate parameter spaces at level Gamma_1(p^infty)}
Let $x$ be a cusp of $\mathcal X^{\ast}$ with Tate parameter space $\mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$.
Then in the limit, the open immersions $\underline{(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}}\times \mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_1(p^n)\cap \Gamma_0(p^\infty)}(\epsilon)_a$ give a $\mathbb{Z}_p^\times$-equivariant open immersion $\underline{\mathbb{Z}_p^\times}\times \mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$ that fits into a Cartesian diagram
\begin{equation*}
\begin{tikzcd}
\underline{\mathbb{Z}_p^\times}\times \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] & \mathcal D_{\infty,x} \arrow[d,hook]\\
\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a.
\end{tikzcd}
\end{equation*}
\end{Proposition}
\begin{Remark}
Upon specialisation to the origin $\operatorname{Spa}(L,\O_L)\to \mathcal D_{\infty,x}$, this shows that the subspace of cusps of $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$ over $x$ can be identified with $(\underline{\mathbb{Z}^\times_p})_{L}$, the base-change of the profinite adic space $\underline{\mathbb{Z}_p^\times}$ to $L$. In particular, for any $a \in \mathbb{Z}_p^\times$, specialisation at $a$ gives rise to a locally closed immersion $\mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$ but in contrast to the case of $\Gamma_0(p^\infty)$ this is not going to be an open immersion due to the non-trivial topology on the cusps.
\end{Remark}
\begin{proof}
This follows in the inverse limit over the front of the cubes in Lemma~\ref{l: Tate parameter spaces of X^ast_Gamma_1(p^m)cap Gamma_0(p^infty)}.2, since
\begin{equation}\label{eq:tilde-limit-distributes-over-product}
\underline{\mathbb{Z}_p^\times}\times \mathcal D_{\infty,x}\sim \varprojlim \underline{(\mathbb{Z}/p^n\mathbb{Z})^\times}\times \mathcal D_{\infty,x}.
\end{equation}
That this holds is easy to verify, see for example \cite[Lemma~12.2]{perfectoid-covers-Arizona}.
\end{proof}
\subsection{Tate curve parameter spaces of $\XaGea{}{\infty}$}\label{s:infinite-level-Gamma}
Next, we look at the Tate parameter spaces in the pro-\'etale map $\XaGea{}{\infty}\to \XaGea{1}{\infty}$. As before, we do so by looking at the limit of the finite level morphisms
\[\mathcal X^{\ast}_{\Gamma(p^n)\cap \Gamma_0(p^\infty)}:=\XaGea{}{n}\times_{\XaGea{0}{n}}\XaGea{0}{\infty}\to\XaGea{0}{\infty}. \]
Combining the moduli descriptions of $\mathcal X_{\Gamma(p^n)}$ and Lemma~\ref{l: moduli interpretation of modular curve at infinite level}, we see:
\begin{Lemma}\label{l:moduli interpretation of modular curve of level Gamma(p^n) cap Gamma_0(p^infty)}
Let $(R,R^+)$ be a perfectoid $K$-algebra. Then $\mathcal X_{\Gamma(p^n)\cap\Gamma_0(p^\infty)}(\epsilon)_a(R,R^{+})$ is in functorial bijection with isomorphism classes of tuples $(E,\alpha_N,G,\beta_n)$ with $E,\alpha_N,G$ as in Lemma~\ref{l: moduli interpretation of modular curve at infinite level} and $\beta_n$ an isomorphism $(\mathbb{Z}/p^n\mathbb{Z})^2\rightarrow E[p^n]$ such that $\alpha(1,0)$ generates $G_n$.
\end{Lemma}
We have the following description of the cusps of $\mathcal X^{\ast}_{\Gamma(p^m)\cap\Gamma_0(p^\infty)}(\epsilon)_a$:
\begin{Lemma}\label{l: Tate parameter spaces of X^ast_Gamma(p^m)cap Gamma_0(p^infty)}
Let $x$ be a cusp of $\mathcal X^{\ast}$.
\begin{enumerate}
\item The map $\mathcal X^{\ast}_{\Gamma(p^m)\cap\Gamma_0(p^\infty)}(\epsilon)_a\rightarrow \mathcal X^{\ast}_{\Gamma(p^m)}(\epsilon)_a$ induces an isomorphism on cusps. The cusps of $\mathcal X^{\ast}_{\Gamma(p^m)\cap\Gamma_0(p^\infty)}(\epsilon)_a$ over $x$ are thus parametrised by $\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})$ where $\smallmat{a}{b}{0}{d}$ corresponds to the ordered basis $(q^{d/p^n},\zeta_{p^n}^{a}q^{-b/p^n})$ of $\mathrm{T}(q)[p^n]$.
\item There is the following Cartesian diagram,
where the top map is given by $\smallmat{a}{b}{0}{d}\mapsto d$:
\begin{equation*}
\begin{tikzcd}
\underline{\Gamma_0(p^n,\mathbb{Z}/p^n\mathbb{Z})}\times \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] &\underline{(\mathbb{Z}/p^n\mathbb{Z})^\times}\times \mathcal D_{\infty,x} \arrow[d,hook]\\
\mathcal X^{\ast}_{\Gamma(p^n)\cap\Gamma_0(p^\infty)}(\epsilon)_a\arrow[r] & \mathcal X^{\ast}_{\Gamma_1(p^n)\cap\Gamma_0(p^\infty)}(\epsilon)_a.
\end{tikzcd}
\end{equation*}
\end{enumerate}
\end{Lemma}
\begin{proof}
Part (1) follows from Lemma~\ref{l:Tate parameter spaces of X_Gamma(p^n)(epsilon)_a} and Prop.~\ref{p: the Tate parameter space at infinite level Gamma_0(p^infty)} exactly like in Lemma~\ref{l: Tate parameter spaces of X^ast_Gamma_1(p^m)cap Gamma_0(p^infty)}.
Part (2) follows from a similar Cartesian cube using the left square in Prop.~\ref{p: structure of Tate parameter spaces from tame level to Gamma(p^n)} and Lemma~\ref{l: Tate parameter spaces of X^ast_Gamma_1(p^m)cap Gamma_0(p^infty)}.
\end{proof}
\begin{Definition}
Let $\Gamma_0(p^{\infty})=\smallmat{\mathbb{Z}_p^\times}{\mathbb{Z}_p}{0}{\mathbb{Z}_p^\times}$ be the subgroup of $\operatorname{GL}_2(\mathbb{Z}_p)$ of upper triangular matrices. This is a profinite group with $\Gamma_0(p^{\infty})=\varprojlim_n \Gamma_0(p^{n},\mathbb{Z}/p^n\mathbb{Z})$.
\end{Definition}
\begin{Lemma}\label{l:moduli interpretation of Gamma_1(p^infty)(epsilon)_a}
Let $(R,R^+)$ be a perfectoid $K$-algebra. Then $\mathcal X_{\Gamma(p^\infty)}(\epsilon)_a(R,R^+)$ is in functorial bijection with isomorphism classes of triples $(E,\alpha_N,\beta)$ of an $\epsilon$-nearly ordinary elliptic curve $E$ over $R$, together with a $\Gamma^p$-structure $\alpha_N$, and an isomorphism of $p$-divisible groups $\beta\colon (\mathbb{Q}_p/\mathbb{Z}_p)^2\rightarrow E[p^\infty]$ over $R$ (or equivalently an isomorphism $\mathbb{Z}_p^2\rightarrow T_pE(R)$) such that the restriction of $\beta$ to the first factor is an anti-canonical $\Gamma_1(p^\infty)$-structure.
\end{Lemma}
\begin{proof}
This is an immediate consequence of Lemma~\ref{l:moduli interpretation of modular curve of level Gamma(p^n) cap Gamma_0(p^infty)} and \cite[Prop.~2.4.5]{ScholzeWeinstein}.
\end{proof}
We are now ready to give the main result of this section, which summarises the discussion so far and moreover describes the cusps of $\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$. For the statement, let us recall that for any $n$, the Tate curve $\mathrm{T}(q)$ over $\mathcal D_\infty$ has an anticanonical ordered basis for $\mathrm{T}(q)[p^n]$ given by $(q^{1/p^n},\zeta_{p^n})$. In particular, an anticanonical ordered basis of the Tate module $T_p\mathrm{T}(q)$ is given by the compatible system $(q^{1/p^n})_{n\in\mathbb{N}}$, that we denote by $q^{1/p^\infty}$, and the compatible system $(\zeta_{p^n})_{n\in\mathbb{N}}$, that we denote by $\zeta_{p^\infty}$.
\begin{Theorem}\label{Theorem: Tate parameter spaces at level Gamma(p^infty)}
Let $x$ be a cusp of $\mathcal X^{\ast}$ with corresponding morphism $\mathcal D_x\hookrightarrow \mathcal X^{\ast}$.
\begin{enumerate}
\item We have a tower of Cartesian squares, where the top left map sends $\smallmat{a}{b}{0}{d}\mapsto d$:
\begin{equation*}
\begin{tikzcd}
\underline{\Gamma_0(p^\infty)}\times \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] & \underline{\mathbb{Z}_p^\times}\times \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] & \mathcal D_{\infty,x} \arrow[d,hook] \arrow[r] & \mathcal D_{x}\arrow[d,hook]\\
\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a \arrow[r] & X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a \arrow[r] & \mathcal \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a \arrow[r]&\mathcal X^{\ast}(\epsilon) .
\end{tikzcd}
\end{equation*}
\item For any $\gamma=\smallmat{a}{b}{0}{d}\in \Gamma_0(p^\infty)$, the cusp of $\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$ obtained by specialising $\underline{\Gamma_0(p^\infty)}\times \mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$ at $\gamma$ is the one corresponding to the
isomorphism $\mathbb{Z}_p^2\rightarrow T_p\mathrm{T}(q)$ defined by the ordered basis $(q^{d/p^\infty}, \zeta^{a}_{p^\infty}q^{-b/p^\infty})$ of $T_p\mathrm{T}(q)$.
\end{enumerate}
\end{Theorem}
\begin{proof}
\begin{enumerate}
\item
The only statement we have not yet proved is that the left square is Cartesian. But this follows from Lemma~\ref{l: Tate parameter spaces of X^ast_Gamma(p^m)cap Gamma_0(p^infty)}.(2) in the limit $n\to \infty$. Here we use that
\[ \underline{\Gamma_0(p^\infty)}\times \mathcal D_{\infty,x}\sim \varprojlim_{n\in\mathbb{N}} \underline{\Gamma_0(p^n,\mathbb{Z}/p^n)}\times \mathcal D_{n,x}\]
as well as the analogous statement from \eqref{eq:tilde-limit-distributes-over-product}, which hold by the same argument.
\item This follows from Lemma~\ref{l: Tate parameter spaces of X^ast_Gamma(p^m)cap Gamma_0(p^infty)}.(1) in the limit.\qedhere
\end{enumerate}
\end{proof}
We note the following easy consequence. The analogue of this for Siegel moduli spaces for dimension $g>1$ is shown in the proof of \cite[Lemma III.2.35]{torsion}.
\begin{Corollary}\label{c: over ordinary locus, Gamma to Gamma_1 is split}
For any $n\in \mathbb{N}\cup \{\infty\}$, our choice of $\zeta_{p^n}$ induces a canonical isomorphism
\[\mathcal X^{\ast}_{\Gamma(p^n)}(0)_a = \bigsqcup_{\Gamma(p^n)/\Gamma_1(p^n)}\mathcal X^{\ast}_{\Gamma_1(p^n)}(0)_a.\]
\end{Corollary}
\begin{proof}
For $n=\infty$, there is away from the cusps a canonical section induced by $T_pE=T_pC\times T_pG$ and the canonical isomorphism $T_pC=T_pG^\vee$ induced by the Weil pairing. On Tate parameter spaces, one checks that this splitting is given by the map \[\underline{\mathbb{Z}_p^\times}\times \mathring{\mathcal D}_\infty \to \underline{\Gamma_0(p^\infty)}\times \mathring{\mathcal D}_\infty, \quad(a,q)\mapsto \left(\smallmat{a}{0}{0}{a^{-1}},q\right).\]
This clearly extends over the puncture. Similarly for $n<\infty$.
\end{proof}
\subsection{The action of $\Gamma_0(p)$ on the cusps of $\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$}\label{s:action-of-Gamma_0}
Next, we discuss the full action of $\Gamma_0(p)$ on the Tate parameter spaces at infinite level.
Since the full $\operatorname{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$-action on each $\mathcal X^{\ast}_{\Gamma(p^n)}$ restricts to a $\Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$-action on $\XaGea{}{n}$ as discussed in Prop.~\ref{Proposition: the action of Gamma_0(p,Z/p^nZ) on cusps of Gamma(p^n)}, we see that the $\operatorname{GL}_2(\mathbb{Z}_p)$-action on $\mathcal X^{\ast}_{\Gamma(p^\infty)}$ restricts to an action of $\Gamma_0(p)=\varprojlim_n \Gamma_0(p,\mathbb{Z}/p^n\mathbb{Z})$ on $\XaGea{}{\infty}$.
\begin{Proposition}\label{p: description of the action of Gamma_0(p) on the Tate parameter spaces}
Over any cusp $x$ of $\mathcal X^{\ast}$, the $\Gamma_0(p)$-action on $\mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$ restricts to an action on $\underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$ where it can be described as follows: Equip $\underline{\Gamma_0(p)}\times \mathcal D_\infty$ with a right action by $p\mathbb{Z}_p$ via $(\gamma,q^{1/p^m})\mapsto (\gamma\smallmat{1}{0}{h}{1},\zeta^{h/e_x}_{p^m}q^{1/p^m})$ for $h\in p\mathbb{Z}_p$, then
\[(\underline{\Gamma_0(p)}\times \mathcal D_\infty)/p\mathbb{Z}_p=\underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty\]
as sheaves on $\mathbf{Perf}_K$ and the left action of $\Gamma_0(p)$ is the one induced by letting $\Gamma_0(p)$ act on the first factor of $\underline{\Gamma_0(p)}\times \mathcal D_\infty$.
Explicitly, in terms of any $\gamma_1\in \Gamma_0(p)$, the action is given by
\begin{alignat*}{3}
\gamma_1\colon &&\underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty&\;\xrightarrow{\sim}\;&& \underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty\\
&& \gamma_2,q^{1/p^m} &\; \;\mapsto\; && \smallmat{\det( \gamma_3)/d_3}{b_3}{0}{d_3},\zeta_{p^m}^{-\frac{c_3}{d_3e_x}}q^{1/p^m}.
\end{alignat*}
where $\gamma_3=\smallmat{a_3}{b_3}{c_3}{d_3}:=\gamma_1\cdot \gamma_2$.
\end{Proposition}
\begin{proof}
That the action restricts to an action on $\underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty$ is a consequence of Prop.~\ref{Proposition: the action of Gamma_0(p,Z/p^nZ) on cusps of Gamma(p^n)} in the limit over $n$. The same argument gives the explicit formula.
It remains to verify the isomorphism of sheaves. For this we check that the following diagram commutes, where to ease notation, let $\Gamma_m:=\Gamma_0(p^m,\mathbb{Z}/p^m\mathbb{Z})$ and $\Gamma'_m:=\Gamma_0(p,\mathbb{Z}/p^m\mathbb{Z})$:
\[
\begin{tikzcd}[column sep = 0.4cm,row sep = {1.5cm,between origins},column sep = 1.1cm]
\underline{\Gamma'_{n+1}}\times \mathcal D_{n+1} \arrow[d] \arrow[r] & \underline{\Gamma_{n+1}}\times \mathcal D_{n+1}, \arrow[d] \\
\underline{\Gamma'_n}\times \mathcal D_n \arrow[r] & \underline{\Gamma_n}\times \mathcal D_n,
\end{tikzcd}
\quad
\begin{tikzcd}[column sep = 0.4cm,row sep = {1.5cm,between origins}]
\smallmat{a}{b}{c}{d},q^{1/p^{n+1}} \arrow[d, maps to] \arrow[r, maps to] & \smallmat{\det( \gamma)/d}{b}{0}{d},\zeta_{p^{n+1}}^{-\tfrac{c}{de_x}}q^{1/p^{n+1}} \arrow[d, maps to] \\
\smallmat{a}{b}{c}{d},q^{1/p^n} \arrow[r, maps to] & \smallmat{\det( \gamma)/d}{b}{0}{d},\zeta_{p^{n}}^{-\tfrac{c}{de_x}}q^{1/p^n}
\end{tikzcd}
\]
where $\gamma=\smallmat{a}{b}{c}{d}$ (we emphasize that on the right we describe the maps in terms of \textit{points} rather than \textit{functions}). This diagram is $p\mathbb{Z}_p$-equivariant when we endow the spaces on the left with the $p\mathbb{Z}_p$-actions from Prop.~\ref{Proposition: the action of Gamma_0(p,Z/p^nZ) on cusps of Gamma(p^n)}, and the spaces on the right with the trivial $p\mathbb{Z}_p$-action. The diagram is moreover equivariant for the $\Gamma_0(p)$-action on the left via the natural reduction maps.
In the limit we therefore obtain a $p\mathbb{Z}_p$-invariant morphism
\[\underline{\Gamma_0(p)}\times \mathcal D_\infty\rightarrow \underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty,\]
equivariant for the $\Gamma_0(p)$-action via the first factor on the left, and the action described in the proposition on the right. This induces a morphism of sheaves
\[(\underline{\Gamma_0(p)}\times \mathcal D_\infty)/p\mathbb{Z}_p\rightarrow \underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty.\]
On the other hand, the inclusion $\Gamma_0(p^\infty)\subseteq\Gamma_0(p)$ defines an inverse of this map.
\end{proof}
\subsection{The Hodge--Tate period map on Tate parameter spaces}\label{s:action-of-HT}
Next, we give an explicit description of the Hodge--Tate map on Tate parameter spaces.
Recall that over the ordinary locus, the kernel of the Hodge--Tate map $T_pE\rightarrow \omega_E$ is the Tate module $T_pC$ of the canonical $p$-divisible subgroup, and thus the Hodge--Tate filtration is given by $T_pC\subseteq T_pE$. In particular, this means that $\pi_{\operatorname{HT}}(\mathcal X^{\ast}_{\Gamma(p^\infty)}(0))\subseteq \P^1(\mathbb{Z}_p)$.
When we further restrict to the anticanonical locus, the image lies in the points of the form $(a:1)\in \P^1(\mathbb{Z}_p)$ with $a\in \mathbb{Z}_p$. Denote by $B_1(0)\subseteq \P^1(\mathbb{Z}_p)$ the ball of radius $1$ inside the canonical chart $\mathbb{A}^1\subseteq \P^1$ around $(0:1)$, then the Hodge--Tate period map thus restricts to
\[\pi_{\operatorname{HT}}(\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a)\subseteq B_1(0)\subseteq \P^1(\mathbb{Z}_p).\]
\begin{Proposition}\label{proposition: Hodge--Tate period map on Tate parameter spaces}
Let $x$ be a cusp of $\mathcal X^{\ast}$. Then the Hodge--Tate period map $\pi_{\operatorname{HT}}\colon \mathcal X^{\ast}_{\Gamma(p^\infty)} \rightarrow \P^1$ restricts on $ (\underline{\Gamma_0(p)}\times \mathcal D_{\infty,x})/p\mathbb{Z}_p\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a$ to the locally constant map
\[
(\underline{\Gamma_0(p)}\times \mathcal D_{\infty,x})/p\mathbb{Z}_p \to\underline{\P^1(\mathbb{Z}_p)}\subseteq \P^1,\quad \big(\!\smallmat{a}{b}{c}{d},q\big)\mapsto (b:d).
\]
\end{Proposition}
We deduce this from the following lemma:
\begin{Lemma}\label{l:constant-functions}
Let $f\colon \mathcal D_\infty\rightarrow \mathbb{A}^1_K$ be a function such that $f$ is constant on $(C,\mathcal O_C)$-points with value $a\in L$. Then the corresponding $f\in\mathcal O(\mathcal D_\infty)$ is the constant $a\in L\subseteq \mathcal O(\mathcal D_\infty)$.
\end{Lemma}
\begin{proof}
It suffices to prove this for the spaces $\mathcal D_\infty(|q|\leq \varpi^n)$. After rescaling, we are reduced to showing the lemma for $\mathcal D_\infty$ replaced by $\operatorname{Spa}(L\langle q^{1/p^\infty}\rangle,\mathcal O_L\langle q^{1/p^\infty}\rangle)$.
One can now argue like in the classical proof of the maximum principle: We can regard $f$ as a function
\[f\in L\langle q^{1/p^\infty}\rangle,\quad f=\sum_{m\in \mathbb{Z}[\frac{1}{p}]_{\geq 0}}a_mq^{m}.\]
We need to prove that if $f((x^{1/p^i})_{i\in\mathbb{N}})=a$ for all $(x^{1/p^i})_{i\in\mathbb{N}}\in \mathcal \varprojlim_{x\mapsto x^p} \mathcal O_C$ then $f=a$.
After subtracting by $a=a_0$, we may assume that $f(x)=0$ for all $x\in \mathcal O_C$.
Suppose $f\neq 0$. The convergence condition on coefficients ensures that $\sup_{m\in \mathbb{Z}[\frac{1}{p}]}|a_m|>0$ is attained and after renormalising we may assume that $|f|=\max_{m\in \mathbb{Z}[\frac{1}{p}]}|a_m|=1$.
Consider the reduction
\[r\colon \mathcal O_L\langle q^{1/p^\infty}\rangle \rightarrow k_L[q^{1/p^n}|n\in\mathbb{N}]\]
modulo $\mathfrak m_L\subseteq \mathcal O_L$. After replacing $q\mapsto q^{p^m}$ we may assume that $r(f)\in k_L[q]$. As $\mathcal O_C$ is perfectoid, the projection $\varprojlim \O_C\rightarrow \O_C\rightarrow k_C$ to the residue field is surjective, and the assumption on $f$ now implies that $r(f)$ is a non-zero polynomial in $k_C[q]$ which evaluates to $0$ on all $q\in k_C$, a contradiction as $k_C$ is infinite.
\end{proof}
\begin{proof}[Proof of Prop.~\ref{proposition: Hodge--Tate period map on Tate parameter spaces}]
By Lemma~\ref{l:constant-functions} it suffices to prove that for any $\gamma\in \Gamma_0(p^\infty)$, the map
\[\mathcal D_\infty\xrightarrow{q\mapsto (\gamma,q)} \underline{\Gamma_0(p^\infty)}\times \mathcal D_{\infty,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a\xrightarrow{\pi_{\operatorname{HT}}}\P^1\]
is constant with image $b/d$. To see this, we use the moduli description on $(C,\mathcal O_C)$-points:
On the ordinary locus, $\pi_{\operatorname{HT}}$ sends any isomorphism $\mathbb{Z}_p^2\rightarrow T_pE$ to the point of $\P^1(\mathbb{Z}_p)$ defined by the line $T_pC\subseteq T_pE$ where $C$ is the canonical $p$-divisible subgroup. By Thm.~\ref{Theorem: Tate parameter spaces at level Gamma(p^infty)}.(2), any $(C,\O_C)$-point of $\mathcal D_\infty\xrightarrow{(q\mapsto \gamma,q)} \underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty$ corresponds to a Tate curve $E=\mathrm{T}(q_E)$ with an ordered basis of $T_pE$ given by $(e_1,e_2)=(q_E^{d/p^\infty}, \zeta^{a}_{p^\infty}q_E^{-b/p^\infty})$. Then (using additive notation on $T_pE$) \[be_1+de_2=q_E^{bd/p^\infty}\zeta^{ad}_{p^\infty}q_E^{-db/p^\infty}=\zeta^{ad}_{p^\infty}\]
which spans the line $\langle \zeta_{p^\infty}\rangle = T_pC\subseteq T_pE$. Consequently, the image of $(\gamma,q)$ under $\pi_{\operatorname{HT}}$ is
\[\pi_{\operatorname{HT}}(\gamma,q)=(b:d)=(b/d:1)\in \underline{(\mathbb{Z}_p^\times:1)} \subseteq \P^1(\mathbb{Z}_p).\]
We conclude from this that the function $f\in\operatorname{Map}_{{\operatorname{cts}}}(\Gamma_0(p^\infty),\mathcal O(\mathcal D_\infty))$ defined by the restriction $\pi_{\operatorname{HT}}\colon \underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty \to B(0)$ evaluates at $\gamma$ to $f(\gamma)=b/d$. Since this is true for all $\gamma\in \Gamma_0(p^\infty)$, we see that $f$ is given by a function in
\[\operatorname{Map}_{{\operatorname{cts}}}(\Gamma_0(p^\infty),\mathbb{Z}_p^\times)\subseteq \operatorname{Map}_{{\operatorname{cts}}}(\Gamma_0(p^\infty),\mathcal O(\mathcal D_\infty)).\]
We conclude that $\pi_{\operatorname{HT}}$ has the desired description
\[\underline{\Gamma_0(p^\infty)}\times \mathcal D_\infty \xrightarrow{(\gamma,q)\mapsto b/d} \underline{\mathbb{Z}_p^\times}=\underline{(\mathbb{Z}_p^\times:1)}\subseteq \P^{1}(\mathbb{Z}_p) \hookrightarrow \P^1, \quad (\gamma,q)\mapsto (b/d:1).\qedhere \]
\end{proof}
\subsection{Tate parameter spaces of the modular curve at infinite level}
As an immediate consequence of the above, we can now consider the entire modular curve $\mathcal X^{\ast}_{\Gamma(p^\infty)}$. Recall that by the very construction in \cite{torsion}, this is the space $\operatorname{GL}_2(\mathbb{Q}_p)\XaGea{}{\infty}$ defined by glueing translates of $\XaGea{}{\infty}$. This allows us to deduce the parts (2)-(3) of Thm.\ \ref{t:cusps of X_Gamma(p^infty)}:
\begin{Theorem}\label{t:Main-Theorem-parts-2-3}
Let $x\in \mathcal X^{\ast}$ be any cusp. Define a right action of $\mathbb{Z}_p$ on $\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x}$ by $(\gamma,q^{1/p^n})\cdot h\mapsto (\gamma\smallmat{1}{0}{h}{1},q^{1/p^n}\zeta^{h/e_x}_{p^n})$. Then the quotient $(\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x})/\mathbb{Z}_p$ exists as a perfectoid space, and there is a Cartesian diagram
\begin{equation*}
\begin{tikzcd}
(\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x})/\mathbb{Z}_p \arrow[d,hook] \arrow[r] & \mathcal D_x\arrow[d,hook]\\
\mathcal X^{\ast}_{\Gamma(p^\infty)} \arrow[r] &\mathcal X^{\ast}.
\end{tikzcd}
\end{equation*}
where the top map is induced by the projection from the second factor.
The left map is $\operatorname{GL}_2(\mathbb{Z}_p)$-equivariant for the left action on $(\underline{\operatorname{GL}_2(\mathbb{Z}_p)}\times \mathcal D_{\infty,x})/\mathbb{Z}_p$ via the first factor.
Under this description, the fibres of the canonical and anticanonical locus are precisely
\begin{equation}\label{eq:in-thm-descr-of-Tate-in-antican-and-can}
\begin{tikzcd}[row sep = 0cm]
\Bigg(\underline{\smallmat{\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^{\times}}}\times\mathcal D_{\infty,x}\Bigg)/\mathbb{Z}_p\arrow[r,hook]& \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a\\
\Bigg(\underline{\smallmat{\mathbb{Z}_p}{\mathbb{Z}_p^\times}{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}}\times\mathcal D_{\infty,x}\Bigg)/\mathbb{Z}_p\arrow[r,hook]& \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_c.
\end{tikzcd}
\end{equation}
\end{Theorem}
\begin{proof}
We need to translate the $\Gamma_0(p)$-equivariant open immersion from Prop~\ref{p: description of the action of Gamma_0(p) on the Tate parameter spaces}
\[(\Gamma_0(p)\times \mathcal D_{\infty,x})/p\mathbb{Z}_p\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(\epsilon)_a\]
according to the $\operatorname{GL}_2(\mathbb{Z}_p)$-action on the right hand side.
We first rewrite the left hand side: We have
\[\Gamma_0(p)\smallmat{1}{0}{\mathbb{Z}_p}{1}=\smallmat{\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^\times},\]
and by extending the $p\mathbb{Z}_p$-action to a $\mathbb{Z}_p$-action in the natural way, we get the equivalent description of anticanonical Tate parameter spaces stated in \eqref{eq:in-thm-descr-of-Tate-in-antican-and-can} in the theorem.
Next, we note that we may without loss of generality replace $\mathcal X^{\ast}_{\Gamma(p^\infty)}$ by
\[\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)=\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a\sqcup \mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_c.\]
To simplify the discussion of translates,
we introduce an auxiliary open subspace
\[\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a\subseteq \mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a.\]
parametrising isomorphisms $\alpha\colon \mathbb{Z}_p^2\to T_pE$ such that $\alpha(0,1)\bmod p$ generates the canonical subgroup (``first basis vector anticanonical, second canonical''). More precisely, this subspace can be constructed as follows:
According to Cor.~\ref{c: over ordinary locus, Gamma to Gamma_1 is split}, there is a canonical splitting
\[\mathcal X_{\Gamma_1(p)}(0)_a\to \mathcal X_{\Gamma(p)}(0)_a\]
that identifies the image with a component of $\mathcal X_{\Gamma(p)}(0)_a$. Let $\mathcal X^{\ast}_{\Gamma(p)}(0)^c_a$ be the finite union of the $\smallmat{(\mathbb{Z}/p\mathbb{Z})^{\times}}{0}{0}{1}$-translates of the image. Then $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a$ is defined as the pullback
\begin{center}
\begin{tikzcd}[row sep = 0.15cm]
\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a \arrow[d,"\cap" description] \arrow[r] & \arrow[d,"\cap" description] \mathcal X^{\ast}_{\Gamma(p)}(0)^c_a\\
\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a \arrow[r] & \mathcal X^{\ast}_{\Gamma(p)}(\epsilon)_a.
\end{tikzcd}
\end{center}
It is clear from this definition that $ \mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a$ defines an open and closed subspace of $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a$. By tracing the Tate parameter spaces through the construction, we moreover see that their fibre over $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a$ is
\begin{equation}\label{eq:Tate paramter space for antican-can}
\Bigg(\underline{\smallmat{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^\times}}\times \mathcal D_{\infty,x}\Bigg)/\mathbb{Z}_p \hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a
\end{equation}
We can now identify $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a$ with the finite union of translates
\begin{equation}\label{eq:equivar-antican-case}
\smallmat{1}{\mathbb{Z}_p}{0}{1}\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a=\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a.
\end{equation}
Indeed, away from the cusps this follows on moduli functors using Lemma~\ref{l:moduli interpretation of Gamma_1(p^infty)(epsilon)_a}, whereas over the cusps it follows from the above explicit description using
\[\smallmat{1}{\mathbb{Z}_p}{0}{1} \smallmat{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^\times}=\smallmat{\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^\times}. \]
On the other hand, inside $\mathcal X^{\ast}_{\Gamma(p^\infty)}$ we have an identification
\begin{equation}\label{eq:equivar-can-case}
\smallmat{0}{1}{1}{0}\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)^c_a=\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_c.
\end{equation}
Indeed, one can first check this for $\mathcal X^{\ast}_{\Gamma(p)}(0)_c$ on moduli functors, extend to compactifications, and then pull back to infinite level.
On Tate parameter spaces, the identity \[\smallmat{0}{1}{1}{0}\smallmat{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^\times}=\smallmat{\mathbb{Z}_p}{\mathbb{Z}_p^\times}{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}\]
therefore gives the desired open immersion onto a neighbourhood of the cusps over $x$
\[
\Bigg(\underline{\smallmat{\mathbb{Z}_p}{\mathbb{Z}_p^\times}{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}}\times\mathcal D_{\infty,x}\Bigg)\slash\mathbb{Z}_p\hookrightarrow \mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_c.\]
Taking the disjoint union of the morphisms in~\eqref{eq:in-thm-descr-of-Tate-in-antican-and-can}, we get the desired description.
To check $\operatorname{GL}_2(\mathbb{Z}_p)$-equivariance, we note that every $\gamma\in \operatorname{GL}_2(\mathbb{Z}_p)$ can be decomposed into $\gamma_1\cdot \gamma_2$ where $\gamma_2\in \smallmat{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^\times}$ and either $\gamma_1\in \smallmat{1}{\mathbb{Z}_p}{0}{1}$ or $\gamma_1=\smallmat{0}{1}{1}{0}$. Since the open immersion in \eqref{eq:Tate paramter space for antican-can} is $\smallmat{\mathbb{Z}_p^\times}{p\mathbb{Z}_p}{\mathbb{Z}_p}{\mathbb{Z}_p^\times}$-equivariant, it thus suffices to check this for $\gamma_1$, for which this follows from equivariance in the anticanonical case \eqref{eq:equivar-antican-case}, and glueing in the canonical case \eqref{eq:equivar-can-case}.
\end{proof}
\begin{proof}[Proof of Thm.~\ref{t:cusps of X_Gamma(p^infty)}.(3)]
This follows from Prop.~\ref{proposition: Hodge--Tate period map on Tate parameter spaces} by $\operatorname{GL}_2(\mathbb{Z}_p)$-equivariance of $\pi_{\operatorname{HT}}$.
\end{proof}
\section{Modular curves in characteristic $p$}\label{s:cusps-in-char-p}
We now switch to moduli spaces in characteristic $p$. We start by recalling the general setup:
Let $R$ be any $\mathbb{F}_p$-algebra. Recall from \S\ref{s:adic-cusps-finite} that $X_R$ denotes the modular curve over $R$ of tame level $\Gamma^p$, and $X_R^{\ast}$ denotes its compactification. We write $X_{R,\mathrm{ord}}\subseteq X_R$ for the affine open subscheme where the Hasse invariant $\mathrm{Ha}$ is invertible. Similarly, one defines $X^{\ast}_{R,\mathrm{ord}}\subseteq X^{\ast}_R$ which is also an affine open subspace.
In this section, we consider analytic modular curves over the perfectoid field $K^{\flat}$, the tilt of $K$. We fix a pseudo-uniformiser $\varpi^{\flat}$ such that $\varpi^{\flat\sharp}=\varpi$.
Following the notational conventions in \cite{torsion}, we shall denote modular curves over $K^{\flat}$ with a prime, e.g.\ $X':=X_{K^{\flat}}$ and $X'^{\ast}:=X^{\ast}_{K^{\flat}}$, to distinguish them from the modular curves over $K$.
Let $\mathfrak X'$ be the $\varpi^{\flat}$-adic completion of $X_{\O_{K^{\flat}}}$ and let $\mathcal X'$ be the analytification of $X'$ over $\operatorname{Spa}(K^{\flat},\O_{K^{\flat}})$. We analogously define $\mathfrak X'^{\ast}$ and $\mathcal X'^{\ast}$.
Like in characteristic $0$, for $0\leq \epsilon<1/2$ such that $|\varpi^{\flat}|^\epsilon\in |K^\flat|$ we denote by $\mathcal X'^{\ast}(\epsilon)$ the open subspace of $\mathcal X'^{\ast}$ where $|\mathrm{Ha}|\geq |\varpi^{\flat}|^{\epsilon}$.
Like before, this has a canonical formal model $\mathfrak X'^{\ast}(\epsilon)\to \mathfrak X'^{\ast}$. For any adic space $\mathcal Y\to \mathcal X'^{\ast}$ we write $\mathcal Y(\epsilon):=\mathcal Y\times_{\mathcal X'^{\ast}}\mathcal X'^{\ast}(\epsilon)$. Finally, let $\mathcal X'^{\ast}_{\mathrm{ord}}$ be the analytification of $X_{\mathrm{ord}}'^{\ast}=X_{K^{\flat},\mathrm{ord}}^{\ast}$.
\begin{Remark}
We recall that while the elliptic curves parametrised by $\mathcal X'(\epsilon)$ might have good supersingular \textit{reduction}, the condition on the Hasse invariant ensures that \textit{generically}, these elliptic curves are always ordinary. In other words, $\mathcal X'(\epsilon)\subseteq \mathcal X'_{\mathrm{ord}}$ even for $\epsilon>0$.
\end{Remark}
\subsection{Igusa curves}\label{subsection: adding Igusa structure in adic setting}\label{s:Igusa-curves}
In characteristic $p$, one has the Igusa moduli problem:
\begin{Definition}[\cite{KatzMazur}, Def.~12.3.1]
Let $S$ be a scheme of characteristic $p$ and let $E$ be an elliptic curve over $S$. Consider the Verschiebung morphism $\operatorname{ker} V^n\colon E^{(p^n)}\rightarrow E$.
An Igusa structure on $E$ is a group homomorphism $\phi\colon \mathbb{Z}/p^n\mathbb{Z}\rightarrow E^{(p^n)}(S)$ that is a Drinfeld generator of $\operatorname{ker} V^n$. This means that the Cartier divisor
\[\sum_{a\in \mathbb{Z}/p^n\mathbb{Z}}[\phi(a)]\subseteq E^{(p^n)}\]
coincides with $\operatorname{ker} V^n$.
The Igusa problem $[\operatorname{Ig}(p^n)]$ is the moduli problem defined by the functor sending $E|S$ to the set of Igusa structures on $E$. If $E|S$ is ordinary, the group scheme $\operatorname{ker} V^n$ is \'etale and naturally isomorphic to the Cartier dual $C_n^\vee$ of the canonical subgroup $C_n$. In particular, in this situation, an $\operatorname{Ig}(p^n)$-structure is the same as an isomorphism of group schemes
\[\underline{\mathbb{Z}/p^n\mathbb{Z}}\isomarrow C_n^\vee,\]
or equivalently, an isomorphism of the Cartier duals $\mu_{p^n}\isomarrow C_n$.
\end{Definition}
For any $n\geq 0$, the Igusa problem $[\operatorname{Ig}(p^n)]$ is relatively representable, finite and flat of degree $\varphi(p^n)$ over $\mathbf{Ell}|R$ by \cite{KatzMazur}, Thm.~12.6.1. In particular, the simultaneous moduli problem $[\operatorname{Ig}(p^n),\Gamma^p]$ is representable by a moduli scheme $X_{R,\operatorname{Ig}(p^n)}$ over $R$. The forgetful map
$X_{R,\operatorname{Ig}(p^n)}\rightarrow X_{R}$
is finite and flat, and is an \'etale $(\mathbb{Z}/p^n\mathbb{Z})^{\times}$-torsor over the ordinary locus $X_{R,\mathrm{ord}}\subseteq X_{R}$.
One defines by normalisation a compactification $X^{\ast}_{R,\operatorname{Ig}(p^n)}$. The morphism $X_{R,\operatorname{Ig}(p^n)}\rightarrow X_{R}$ then extends to
\[X^{\ast}_{R,\operatorname{Ig}(p^n)}\rightarrow X^{\ast}_{R}\]
which is still finite Galois with group $(\mathbb{Z}/p^n\mathbb{Z})^{\times}$ over the ordinary locus.
For any map $\operatorname{Spec}(R'\cc{q})\rightarrow X_{R}$ corresponding to a choice of $\Gamma^p$-structure on $\mathrm{T}(q^{e})$ over some cyclotomic extension $R'$ of $R$ and for some $1\leq e\leq N$, the canonical isomorphism \[\mu_{p^n}\isomarrow C_n(\mathrm{T}(q^e))\subseteq \mathrm{T}(q^e)[p^n]\] induces a canonical lifting to a map $\operatorname{Spec}(R'\cc{q})\rightarrow X
_{R,\operatorname{Ig}(p^n)}$.
In particular, over any cusp $x$ of $X^{\ast}_{R}$, the subscheme of cusps of $X^{\ast}_{R,\operatorname{Ig}(p^n)}$ consists of $\varphi(p^n)$ disjoint copies of $x$.
\subsection{Tate parameter spaces in the Igusa tower}
Returning to our analytic setting over $K^{\flat}$, we let $X'^{\ast}_{\operatorname{Ig}(p^n)}:=X^{\ast}_{K^{\flat},\operatorname{Ig}(p^n)}$. We write $\mathfrak X^{\ast}_{\operatorname{Ig}(p^n)}$ for the $\varpi^\flat$-adic completion of $X^{\ast}_{\O_{K^{\flat}},\operatorname{Ig}(p^n)}$ and we write $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}$ for the analytification of $X'^{\ast}_{\operatorname{Ig}(p^n)}$. We then get an open subspace $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)$. Since $\mathcal X'^{\ast}(\epsilon)\subseteq\mathcal X'^{\ast}_{\mathrm{ord}}$, the morphism $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)\to \mathcal X'^{\ast}(\epsilon)$ is a finite \'etale $(\mathbb{Z}/p^n\mathbb{Z})^\times$-torsor.
Like in Lemma~\ref{Lemma: adic moduli interpretation of X_Gamma^an}, one can use that $X'_{\operatorname{Ig}(p^n)}$ is affine to show that these spaces represent the obvious adic moduli functors.
\begin{Definition}
The inverse system of natural forgetful morphisms
\[\dots \to\mathcal X'^{\ast}_{\operatorname{Ig}(p^{n+1})}(\epsilon)\to \mathcal X'^{\ast}_{\operatorname{Ig}(p^{n})}(\epsilon)\to \dots \to \mathcal X'^{\ast}(\epsilon)\]
is called the Igusa tower.
Note that all transition maps in this inverse system are finite \'etale.
\end{Definition}
\begin{question}
For $\epsilon=0$, one can show that this system has a sous-perfectoid (but not perfectoid) tilde limit $\mathcal X'^{\ast}_{\operatorname{Ig}(p^{\infty})}(0)\sim \varprojlim_{n\in\mathbb{N}}\mathcal X'^{\ast}_{\operatorname{Ig}(p^{n})}(0)$. Is this still true for $\epsilon>0$?
\end{question}
\begin{Definition}\label{d:field-of-definition-of-Tate-curve}
As in characteristic $0$, by a cusp we shall mean a (not necessarily geometrically) connected component of the closed subscheme $X'^{\ast}_{\operatorname{Ig}(p^n)}\backslash X'_{\operatorname{Ig}(p^n)}\subseteq X'^{\ast}_{\operatorname{Ig}(p^n)}$ with its induced reduced structure.
\end{Definition}
Given a fixed cusp $x$ of $X'_{\operatorname{Ig}(p^n)}$, we denote by $L_{x}\subseteq K^{\flat}[\zeta_N]$ the field of definition of the associated Tate curve.
Then the completion of $X'^{\ast}_{\O_{K^{\flat}},\operatorname{Ig}(p^n)}$ along the integral extension of $x$ is canonically of the form $\operatorname{Spf}(\mathcal O_{L_{x}}\llbracket q\rrbracket)\rightarrow X'^{\ast}_{\operatorname{Ig}(p^n)}$. Upon $\varpi$-adic completion this becomes
\[\operatorname{Spf}(\mathcal O_{L_x}\llbracket q\rrbracket)\rightarrow \mathfrak X'^{\ast}_{\operatorname{Ig}(p^n)}\]
where $\mathcal O_{L_x}\llbracket q\rrbracket$ carries the $(\varpi^\flat,q)$-adic topology.
Denote by \[\mathcal D' \rightarrow \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}\]
the adic generic fibre, a morphism of adic spaces over $\operatorname{Spa}(K^{\flat},\O_{K^{\flat}})$. Then like before, $\mathcal D'$ is the open unit disc over $L_{x}$ in the variable $q$.
Exactly like in Lemma~\ref{l: Conrad's theorem on generic fibres of completions around the cusp} one sees:
\begin{Lemma}\label{l: Conrad's theorem on generic fibres of completions around the cusp for Ig(p^n)}
The morphism $\mathcal D' \hookrightarrow \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}$ is an open immersion.
\end{Lemma}
If we want to indicate the dependence on the cusp $x$, we shall also call this $\mathcal D'_{x} \hookrightarrow \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}$.
The following lemma explains how the above individual descriptions fit together for different cusps of $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}$ lying over the same cusp of $\mathcal X'^{\ast}$.
\begin{Lemma}\label{l:Cartesian diagrams for Tate parameter spaces in the Igusa tower}
Let $x$ be a cusp of $\mathcal X'^{\ast}$. Then there are Cartesian diagrams
\begin{equation*}
\begin{tikzcd}[column sep=2cm]
\underline{\mathbb{Z}/p^{n+1}\mathbb{Z}}\times \mathcal D'_{x} \arrow[d,hook] \arrow[r]&\underline{\mathbb{Z}/p^n\mathbb{Z}}\times \mathcal D'_{x} \arrow[d,hook] \arrow[r] &
\mathcal D'_x \arrow[d,hook] \\
\mathcal X'^{\ast}_{\operatorname{Ig}(p^{n+1})}(\epsilon) \arrow[r]&\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon) \arrow[r]& \mathcal X'^{\ast}(\epsilon).
\end{tikzcd}
\end{equation*}
\end{Lemma}
\begin{proof}
Using the canonical lift described in \S\ref{s:Igusa-curves}, this can be seen exactly like in Lemma~\ref{l:Tate parameter spaces of X^{ast}_{Gamma_1(p^n)}}, based on the analogue of Lemma~\ref{lemma for comparing schematic cusp to adic cusp} in this setting.
\end{proof}
Like in the $p$-adic case, there is also a larger, quasi-compact Tate curve parameter space:
\begin{Definition}\label{d:overline Dp}
Let $\overline{\mathcal D}'=\overline{\mathcal D}'_x=\operatorname{Spa}(\O_{L_x}\llbracket q\rrbracket[\tfrac{1}{\varpi^{\flat}}],\O_{L_x}\llbracket q\rrbracket)$ where $\O_{L_x}\llbracket q\rrbracket$ is endowed with the $\varpi^{\flat}$-adic topology. Like in Lemma~\ref{l:overline{D} is sousperfectoid}, one sees that this is a sousperfectoid adic space with an open immersion $\mathcal D'=\cup_n\overline{\mathcal D}'(|q|^n\leq |\varpi^{\flat}|)\hookrightarrow \overline{\mathcal D}'$.
\end{Definition}
\begin{Lemma}
For any cusp of $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)$, the map $\mathcal D'\hookrightarrow \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)$ extends uniquely to a natural map $\overline{\mathcal D}'\to \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)$. The fibre of the good reduction locus is $\overline{\mathcal D}'(|q|\geq 1)$.
\end{Lemma}
\begin{proof}
Like in Lemma~\ref{l:full-Tate-curve-parameter-space}.
\end{proof}
\begin{Lemma}\label{Lemma: Frobenius gives Cartesian diagram on Tate parameter spaces of Igusa curves}
Let $n\in\mathbb{Z}_{\geq 0}$. For any cusp $x$ of $\mathcal X'^{\ast}_{\operatorname{Ig}(p^{n})}$, the following squares are Cartesian:
\begin{equation*}
(1)
\begin{tikzcd}
\mathcal D' \arrow[d,"q\mapsto q^p"] \arrow[r, hook] &\overline{\mathcal D}' \arrow[d,"q\mapsto q^p"] \arrow[r] & \mathcal X'^{\ast}_{\operatorname{Ig}(p^{n})}(p^{-1}\epsilon) \arrow[d,"F_{\mathrm{rel}}"] \\
\mathcal D' \arrow[r, hook] & \overline{\mathcal D}' \arrow[r] & \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon).
\end{tikzcd}
\quad (2) \begin{tikzcd}
\mathcal X_{\operatorname{Ig}(p^{n+1})}'^{\ast}(p^{-1}\epsilon)\arrow[d,"F_\mathrm{rel}"] \arrow[r] & \mathcal X_{\operatorname{Ig}(p^n)}'^{\ast}(p^{-1}\epsilon) \arrow[d,"F_\mathrm{rel}"] \\
\mathcal X_{\operatorname{Ig}(p^{n+1})}'^{\ast}(\epsilon) \arrow[r] & \mathcal X_{\operatorname{Ig}(p^n)}'^{\ast}(\epsilon).
\end{tikzcd}
\end{equation*}
\end{Lemma}
\begin{proof}
It is clear that the diagrams commute by functoriality of the relative Frobenius morphism. The second diagram is Cartesian because the bottom map is \'etale.
To see that the first diagram is Cartesian, we first consider the outer square. For this it suffices to check this on $(C^{\flat},C^{\flat+})$-points because the horizontal compositions are open immersions. It is clear that the cusps correspond under $F_{\mathrm{rel}}$, and $q\mapsto q^p$ sends the origin to the origin. Away from the cusps, we can check on moduli interpretations that the square is Cartesian: The desired statement follows as $F_{\mathrm{rel}}$ sends $\mathrm{T}(q)$ to $\mathrm{T}(q^{p})$, and the $\operatorname{Ig}(p^n)$-structure $\langle q\rangle\subseteq \mathrm{T}(q)^{(p^n)}=\mathrm{T}(q^{p^n})$ to $\langle q^p\rangle\subseteq \mathrm{T}(q^p)^{(p^n)}=\mathrm{T}(q^{p^{n+1}})$.
This argument extends to $\overline{\mathcal D}'$, which (away from $x$) we may regard as the moduli space of Tate curves $\mathrm{T}(q)$ with level structure associated to $x$ over adic spaces $S$ over $\O_{L_x}\llbracket q\rrbracket[\tfrac{1}{p}]$. Lifts of maps $S\to \overline{\mathcal D}'$ along the right vertical morphism correspond to Tate curves $\mathrm{T}(q)^{(p^{-1})}=\mathrm{T}(q^{1/p})$ over $S$ whose base change along $F_{\mathrm{rel}}$ is $\mathrm{T}(q)$, and thus to $p$-th roots of $q\in \O^+(S)$.
\end{proof}
\subsection{Perfections of Igusa curves}
In this section, we discuss perfectoid Igusa curves and their Tate curve parameter spaces. We first recall the perfection functor in characteristic $p$:
\begin{Definition}[\cite{torsion}, Def.~III.2.18]
Let $\mathcal Y$ be an analytic adic space over $(K^\flat,\mathcal O_K^\flat)$. Then there is a perfectoid space $\mathcal Y^{{\operatorname{perf}}}$ over $(K^\flat,\mathcal O_K^\flat)$ such that $\mathcal Y^{{\operatorname{perf}}} \sim \varprojlim_{F_{\mathrm{rel}}} \mathcal Y$
where we identify $\mathcal Y^{(p)}$ with $\mathcal Y$ using that $K^\flat$ is perfect. We call $\mathcal Y^{{\operatorname{perf}}}$ the perfection of $\mathcal Y$. The formation $\mathcal Y\mapsto \mathcal Y^{{\operatorname{perf}}}$ is functorial and defines a left-adjoint to the forgetful functor from perfectoid spaces over $K^{\flat}$ to analytic adic spaces over $K^{\flat}$.
\end{Definition}
In the case of $\mathcal Y=\mathcal X'^{\ast}(\epsilon)$, we can first take the inverse limit
${\mathfrak X'^{\ast}}(\epsilon)^{{\operatorname{perf}}}:= \varprojlim_{F_\mathrm{rel}} \mathfrak X'^{\ast}(p^{-n}\epsilon)$
in the category of formal schemes. Its generic fibre is then the tilde limit
\[\mathcal {X'^{\ast}(\epsilon)}^{{\operatorname{perf}}} = \mathfrak {X'^{\ast}(\epsilon)}^{{\operatorname{perf}}}_{\eta} \sim \textstyle\varprojlim_{F_{\mathrm{rel}}} (\mathfrak X'^{\ast}(p^{-n}\epsilon))^{\mathrm{ad}}_{\eta}\]
by \cite[Prop. 2.4.2]{ScholzeWeinstein} and it is clear on any affine open formal subscheme of $\mathfrak X'^{\ast}(\epsilon)$ that this space is perfectoid. The analogous construction also works for $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)^{{\operatorname{perf}}}$.
\begin{Lemma}\label{Lemma moduli interpretation of Ig(p^n)^perf}
Let $(R,R^+)$ be a perfectoid $K^{\flat}$-algebra. Then $\mathcal X'_{\operatorname{Ig}(p^n)}(\epsilon)^{{\operatorname{perf}}}(R,R^+)$ is in functorial bijection with isomorphism classes of triples $(E,\alpha_N,\beta_n)$ of $\epsilon$-nearly ordinary elliptic curves $E$ over $R$ with a $\Gamma^p$-structure $\alpha_N$ and an isomorphism of group schemes \[\beta_n\colon \underline{\mathbb{Z}/p^n\mathbb{Z}}\to \operatorname{ker} (V:E^{(p^n)}\to E)\cong \operatorname{ker} (V:E\to E^{(p^{-n})})\subseteq E[p^n].\]
\begin{proof}
By adjunction, we have $\mathcal X'_{\operatorname{Ig}(p^n)}(\epsilon)^{{\operatorname{perf}}}(R,R^+)=\mathcal X'_{\operatorname{Ig}(p^n)}(\epsilon)(R,R^+)$.
\end{proof}
\end{Lemma}
\begin{Lemma}\label{l: Cartesian diagram Igusa versus perfection}
For any cusp $x$ of $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}$, the perfection of the corresponding Tate parameter space $\mathcal D'\hookrightarrow \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)$ fits into a Cartesian diagram
\begin{equation*}
\begin{tikzcd}
\mathcal D'_\infty \arrow[d]\arrow[r,hook] & \overline{\mathcal D}'_\infty \arrow[d] \arrow[r, hook]& \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)^{\perf} \arrow[d] \\
\mathcal D' \arrow[r, hook] & \overline{\mathcal D}' \arrow[r] & \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon).
\end{tikzcd}
\end{equation*}
Here $\overline{\mathcal D}'_\infty:=\overline{\mathcal D}'^{{\operatorname{perf}}}$, and $\mathcal D'_\infty:=\mathcal D'^{{\operatorname{perf}}}$ can be canonically identified with the open subspace of the perfectoid unit disc $\operatorname{Spa}(K^{\flat}\langle q^{1/p^\infty}\rangle, \O_{K^{\flat}}\langle q^{1/p^\infty}\rangle)$ defined by $|q|<1$.
\end{Lemma}
\begin{proof}
This follows in the limit over the Cartesian diagrams from Lemma~\ref{Lemma: Frobenius gives Cartesian diagram on Tate parameter spaces of Igusa curves}.(1).
\end{proof}
\begin{Lemma}\label{l:perfectoid Igusa tower is proetale}
The following diagram is Cartesian:
\begin{equation*}
\begin{tikzcd}
\mathcal X_{\operatorname{Ig}(p^{n})}'^{\ast}(\epsilon)^{{\operatorname{perf}}} \arrow[d] \arrow[r] &\mathcal X'^{\ast}(\epsilon)^\perf \arrow[d] \\
\mathcal X_{\operatorname{Ig}(p^{n})}'^{\ast}(\epsilon) \arrow[r] & \mathcal X'^{\ast}(\epsilon).
\end{tikzcd}\end{equation*}
\end{Lemma}
\begin{proof}
This follows from Lemma~\ref{Lemma: Frobenius gives Cartesian diagram on Tate parameter spaces of Igusa curves}.(2) in the limit over $F_{\mathrm{rel}}$ because perfectoid tilde-limits commute with fibre products.
\end{proof}
\begin{Definition}
Consider the tower of affinoid perfectoid spaces with finite \'etale maps
\[\dots \to\mathcal X'^{\ast}_{\operatorname{Ig}(p^{n+1})}(\epsilon)^{\operatorname{perf}}\to \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)^{\perf}\to \dots \to \mathcal X'^{\ast}(\epsilon)^\perf. \]
We denote by $\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{\perf}$ the unique affinoid perfectoid tilde-limit of this system.
\end{Definition}
\begin{Proposition}\label{p:Tate parameter spaces in perfectoid Igusa tower}
Let $x$ be any cusp of $\mathcal X'^{\ast}$. Then there are natural Cartesian diagrams
\begin{equation*}
(1) \begin{tikzcd}
\underline{(\mathbb{Z}/p^n\mathbb{Z})^\times}\times \mathcal D_\infty' \arrow[d,hook] \arrow[r] & \mathcal D'_\infty \arrow[d,hook] \\
\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)^{\perf} \arrow[r] & \mathcal X'^{\ast}(\epsilon)^\perf,
\end{tikzcd}
\quad(2)
\begin{tikzcd}
\underline{\mathbb{Z}_p^\times}\times \mathcal D_\infty' \arrow[d,hook] \arrow[r] & \mathcal D'_\infty \arrow[d,hook] \\
\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{\perf} \arrow[r] & \mathcal X'^{\ast}(\epsilon)^\perf.
\end{tikzcd}
\end{equation*}
\end{Proposition}
\begin{proof}
Part (1) follows from Lemma~\ref{l:Cartesian diagrams for Tate parameter spaces in the Igusa tower}, Lemma~\ref{l: Cartesian diagram Igusa versus perfection} and Lemma~\ref{l:perfectoid Igusa tower is proetale} using the Cartesian cube that these three squares span.
Part (2) follows in the inverse limit $n\to \infty$.
\end{proof}
\section{Tilting isomorphisms for modular curves}\label{s:tilting-isomorphism}
\subsection{The tilting isomorphism at level $\Gamma_0(p^\infty)$}
While so far we have studied modular curves in characteristic $0$ and $p$ separately, we now compare the two worlds via tilting. This is possible based on the following result:
\begin{thm}[{\cite[Cor.~III.2.19]{torsion}}]\label{theorem torsion paper: tilting the perfectoid modular curve}
There is a canonical isomorphism
\[\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a^\flat \isomarrow \mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}.\]
\end{thm}
Let us recall how this is proved: Via $\O_K/p\cong\O_{K^{\flat}}/\varpi^\flat$
we have an identification of the reductions $\mathfrak X^{\ast}/p=\mathfrak X'^{\ast}/\varpi^{\flat}$ which by explicit inspection extends to a natural isomorphism
\begin{equation}\label{equation: identification of mXae mod p and mXpae mod t}
\mathfrak X^{\ast}(\epsilon)/p\cong\mathfrak X'^{\ast}(\epsilon)/\varpi^{\flat}.
\end{equation}
The morphism $\mathcal X^{\ast}_{\Gamma_0(p^{n+1})}(\epsilon)\to \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)$ gets identified via the Atkin--Lehner isomorphism \eqref{eq:Atkin--Lehner} with a map $\mathcal X^{\ast}(p^{-(n+1)}\epsilon)\to \mathcal X^{\ast}(p^{-n}\epsilon)$ that has a formal model $\phi\colon \mathfrak X^{\ast}(p^{-(n+1)}\epsilon)\to \mathfrak X^{\ast}(p^{-n}\epsilon)$. One can then prove that mod $p^{1-\delta}$ where $\delta:=\frac{p+1}{p}\epsilon$, the map $\phi$ gets identified with $F_{\mathrm{rel}}$ in the sense that the following diagram commutes:
\[\begin{tikzcd}[row sep=0.2cm]
\mathfrak X^{\ast} (p^{-(n+1)}\epsilon)/p^{1-\delta} \arrow[r,"\phi"] \arrow[d,equal,"\sim"labelrotate] & \mathfrak X^{\ast} (p^{-n}\epsilon)/p^{1-\delta} \arrow[d,equal,"\sim"labelrotate] \\
\mathfrak X'^{\ast}(p^{-(n+1)}\epsilon)/\varpi^{\flat(1-\delta)} \arrow[r,"F_{\mathrm{rel}}"] & \mathfrak X'^{\ast}(p^{-n}\epsilon)/\varpi^{\flat(1-\delta)}.
\end{tikzcd}
\]
In the inverse limit, this gives the result by \cite[Thm~5.2]{perfectoid-spaces}.
The following lemma says that the isomorphism of Thm.~\ref{theorem torsion paper: tilting the perfectoid modular curve} identifies the cusps:
\begin{Lemma}\label{l:tilting cusps}
The cusps of $\mathcal X^{\ast}$ and $\mathcal X'^{\ast}$ correspond via tilting: For each cusp $x$ of $\mathcal X^{\ast}$, considered as a finite \'etale adic space over $K$, its tilt can be canonically identified with a cusp $x^{\flat}$ of $\mathcal X'^{\ast}$. In particular, $(L_{x})^{\flat} =L_{x^{\flat}}$ for the fields of definition.
\end{Lemma}
\begin{proof}
The cusp $x$ can be described as a closed immersion $\operatorname{Spa}(L_x)\hookrightarrow \mathcal X'^{\ast}$. It has a canonical formal model $\operatorname{Spf}(\O_{L_x})\to \mathfrak X'^{\ast}$ that reduces mod $p$ to a morphism $\operatorname{Spec}(\O_{L_x}/p)\to X^{\ast}_{\O_K/p}$
which in turn can be interpreted as cusp of $X^{\ast}_{\O_{K^{\flat}/\varpi^{\flat}}}$. Lifting to $\mathfrak X'^{\ast}$ and taking generic fibre gives a cusp $x^{\flat}$ defined by a closed point
\[\operatorname{Spa}((L_x)^{\flat})\hookrightarrow \mathcal X'^{\ast}.\]
It is clear from this construction that this can be identified with the tilt of $x$ via the equivalence of \'etale sites.
Reversing this argument shows that this defines a bijection on cusps.
\end{proof}
\begin{Proposition}\label{p: description of the tilting isomorphism at level Gamma0 on Tate parameter spaces}
Let $x$ be any cusp of $\mathcal X^{\ast}$. Then the canonical isomorphism of $K^{\flat}$-algebras
\[L_x\llbracket q^{1/p^\infty}\rrbracket^{\flat} =L_{x^{\flat}}\llbracket q^{1/p^\infty}\rrbracket\]
defines isomorphisms $\overline{\mathcal D}_{\infty,x}^\flat\cong\overline{\mathcal D}'_{\infty,x^{\flat}} $ and $\mathcal D_{\infty,x}^\flat\cong\mathcal D'_{\infty,x^{\flat}}$ that fit into a commutative diagram
\[
\begin{tikzcd}[row sep = 0.15cm]
\mathcal D_{\infty,x}^{\flat} \arrow[d,equal,"\sim"labelrotate] \arrow[r, hook] & \overline{\mathcal D}_{\infty,x}^{\flat} \arrow[d,equal,"\sim"labelrotate] \arrow[r, hook] & \XaGea{0}{\infty}^{\flat} \arrow[d,equal,"\sim"labelrotate] \\
\mathcal D'_{\infty,x^{\flat}} \arrow[r, hook] & \overline{\mathcal D}'_{\infty,x^{\flat}}\arrow[r, hook] & \mathcal X'^{\ast}(\epsilon)^\perf.
\end{tikzcd}\]
\end{Proposition}
\begin{proof}
For the proof we use that $\overline{\mathcal D}_\infty$ has a very simple $p$-adic formal model. Here and in the following, let us for simplicity drop the additional $x$ and $x^{\flat}$ in the index.
We can without loss of generality assume $\epsilon=0$. Using the identifications
\[\mathcal D_\infty^{\flat}=\cup_n \overline{\mathcal D}_\infty(|q|^n\leq |\varpi|)^{\flat}= \cup_n \overline{\mathcal D}'_\infty(|q|^n\leq |\varpi^{\flat}|)=\mathcal D'_\infty,\]
it is clear that the left square commutes. It therefore suffices to consider the right square.
Recall that the morphism $\overline{\mathcal D}\to \mathcal X^{\ast}(0)$ arises as the adic generic fibre of the morphism $\overline{\mathfrak D}\to \mathfrak X^{\ast}(0)$ where $\overline{\mathfrak D}:=\operatorname{Spf}(\O_L\llbracket q\rrbracket)$ is endowed with the $p$-adic topology. Similarly, $\overline{\mathcal D}'\to \mathcal X'^{\ast}(0)$ is the adic generic fibre of $\overline{\mathfrak D}'\to \mathfrak X'^{\ast}(0)$ where $\overline{\mathfrak D}':=\operatorname{Spf}(\O_{L^{\flat}}\llbracket q\rrbracket)$. The reductions mod $\varpi$ and $\varpi^{\flat}$ of these formal models can be canonically identified with the map \[\operatorname{Spec}((\O_L/p)\llbracket q\rrbracket )\to \mathfrak X^{\ast}(0)/p=\mathfrak X'^{\ast}(0)/\varpi^{\flat}\]
associated to the Tate curve for the corresponding cusp of $X^{\ast}_{\O_K/p}$.
In the limit over $\varphi$ and $F_{\mathrm{rel}}$, these identifications therefore fit into a commutative diagram
\[\begin{tikzcd}[row sep =0.15cm]
\varprojlim_{q\mapsto q^p} \overline{\mathfrak D}/p\arrow[r]\arrow[d,equal,"\sim"labelrotate]&\varprojlim_{\phi}\mathfrak X^{\ast}(0)/p\arrow[d,equal,"\sim"labelrotate]\\
\varprojlim_{q\mapsto q^p} \overline{\mathfrak D}'/\varpi^{\flat}\arrow[r]& \mathfrak X'^{\ast}(0)^{{\operatorname{perf}}}/\varpi^{\flat}.
\end{tikzcd}
\]
As these are perfectoid schemes over $\O_K^{a}/p$ (or $\O_{K^{\flat}}^{a}/\varpi^{\flat}$) with corresponding perfectoid spaces $\overline{\mathcal D}_\infty\to \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a$ and $\overline{\mathcal D}'_\infty\to \mathcal X'^{\ast}(0)^{{\operatorname{perf}}}$ via \cite[Thm~5.2]{perfectoid-spaces}, this gives the desired identification of the tilts.
\end{proof}
\begin{Remark}
The correspondence of moduli of Tate curves implicit in Prop.~\ref{p: description of the tilting isomorphism at level Gamma0 on Tate parameter spaces} can be made explicit as follows: Let $(R,R^+)$ be a perfectoid $K$-algebra, then Thm~\ref{theorem torsion paper: tilting the perfectoid modular curve} gives a correspondence \[\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a(R,R^{+})=\mathcal X'(\epsilon)^\perf(R^{\flat},R^{\flat+})\]
of elliptic curves with extra data. If now $E_q$ is a Tate curve with parameter $q\in R$, equipped with $\Gamma_0(p^\infty)$-structure $(q^{1/p^n})_{n\in\mathbb{N}}$ and $\Gamma^p$-structure, corresponding to a point in $\mathcal D_\infty(R,R^+)$, then via $\mathcal D_\infty(R,R^+)=\mathcal D'_\infty(R^{\flat},R^{\flat+})$, this corresponds to the Tate curve $E_{q'}$ with parameter
\[q':=(q^{1/p^n})_{n\in\mathbb{N}}\in \varprojlim_{q\mapsto q^p}R^{\times}=R^{\flat\times}.\]
One can moreover identify the $\Gamma^p$-structure of $E_{q'}$ using that $E_{q}[N]^{\flat}=E_{q'}[N]$.
\end{Remark}
\subsection{The tilting isomorphism at level $\Gamma_1(p^\infty)$}
We now extend the tilting isomorphism of Thm.~\ref{theorem torsion paper: tilting the perfectoid modular curve} to level $\Gamma_1(p^\infty)$ by proving the following theorem stated in the introduction:
\begin{Theorem}\label{t: tilting the Gamma_1(p^infty)-tower-2}
\begin{enumerate}
\item There is a canonical isomorphism
\[\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a^{\flat} \isomarrow \mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{{\operatorname{perf}}}\]
which is $\mathbb{Z}_p^\times$-equivariant and makes the following diagram commute:
\begin{equation*}
\begin{tikzcd}[row sep = 0.15cm]
\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a^{\flat} \arrow[d,"\sim"labelrotate,equal] \arrow[r] & \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a^{\flat}\arrow[d,"\sim"labelrotate,equal] \\ \mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{{\operatorname{perf}}} \arrow[r] & \mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}.
\end{tikzcd}
\end{equation*}
\item The cusps of $\XaGea{1}{\infty}$ and $\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)$ correspond via the isomorphism in (1). Moreover, for any cusp $x$ of $\mathcal X^{\ast}$, the following diagram commutes:
\begin{equation*}
\begin{tikzcd}[row sep = 0.15cm]
\underline{\mathbb{Z}_p^{\times}}\times \mathcal D_{\infty,x}^{\flat} \arrow[d,"\sim"labelrotate,equal] \arrow[r,hook] &
\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a^{\flat} \arrow[d,"\sim"labelrotate,equal]\\
\underline{\mathbb{Z}_p^{\times}}\times \mathcal D'_{\infty,x^{\flat}}\arrow[r,hook]& \mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(\epsilon)^{{\operatorname{perf}}},
\end{tikzcd}
\end{equation*}
where the left map is given by the canonical identification $\mathcal D_{\infty,x}^{\flat}\cong \mathcal D'_{\infty,x^{\flat}}$.
\end{enumerate}
\end{Theorem}
For the proof, we use the univeral anticanonical subgroup at infinite level:
\begin{Definition}
For any $n\in \mathbb{Z}_{\geq 1}$,
we denote by $G_n\to \mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a$ the universal anticanonical subgroup of rank $n$. This can be defined via pullback from finite level $\mathcal X_{\Gamma_0(p^n)}(\epsilon)_a$, and is a finite \'etale morphism of perfectoid spaces.
Let $\mathcal E'\to \mathcal X'$ be the analytification of the universal elliptic curve over $X'$, and write $\mathcal E'(\epsilon)\to \mathcal X'(\epsilon)$ for the pullback. We denote by $G'_n\to \mathcal X'(\epsilon)^\perf $ the finite \'etale morphism of perfectoid spaces given by the perfection of $\operatorname{ker} V^n\subseteq \mathcal E'(\epsilon)^{(p^n)}$.
\end{Definition}
\begin{Lemma}\label{Lemma G_n identifies with ker V^n upon tilting}
There is a natural isomorphism making the following diagram commutative:
\begin{equation*}
\begin{tikzcd}[column sep=0.15cm]
G_n^{\flat} \arrow[r,"\sim"] \arrow[d] & G'_n \arrow[d]\\
\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a^{\flat}\arrow[r,"\sim",equal]& \mathcal X'(\epsilon)^{{\operatorname{perf}}},
\end{tikzcd}
\end{equation*}
\end{Lemma}
This lemma is a slight extension of \cite[Lemma III.2.26]{torsion}, from the good reduction locus to the whole uncompactified modular curve (we reiterate that \cite{torsion} writes $\mathcal X$ for the good reduction locus, whereas we use this symbol to denote the whole open modular curve).
\begin{proof}
It suffices to see this locally on $\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a$. The case of good reduction is \cite[Lemma~III.2.26]{torsion}. It therefore suffices to prove the lemma over the ordinary locus $\mathcal X_{\Gamma_0(p^\infty)}(0)_a$.
Over $\mathfrak X^{\ast}(0)$, the universal semi-abelian scheme has a canonical subgroup, a finite flat group scheme $C_n\to \mathfrak X^{\ast}(0)$. Via the Atkin--Lehner isomorphism $\mathcal X^{\ast}(0)\isomarrow\mathcal X^{\ast}_{\Gamma_0(p^n)}(0)_a$, the generic fibre of its dual $(C_n^\vee)^{\mathrm{ad}}_{\eta}$ can be identified over $\mathcal X_{\Gamma_0(p^n)}(0)_a$ with the universal anticanonical subgroup over $\mathcal X_{\Gamma_0(p^n)}(0)_a$.
Similarly, over $\mathfrak X'^{\ast}(0)$, we have a canonical subgroup $C'_n\to\mathfrak X^{\ast}(0)$, and the dual $(C_n'^\vee)^{\mathrm{ad}}_{\eta}$ restricted to $\mathcal X'(0)$ can be identified with the kernel of Verschiebung of $\mathcal E'(0)$ over $\mathcal X'(0)$.
It follows from these descriptions that after pullback we have identifications
\begin{alignat*}{2}
G_n&=&&\big(C_n^{\vee}\times _{\mathfrak X^{\ast}(0)} \varprojlim_{\phi}\mathfrak X^{\ast}(0)\big)^{\mathrm{ad}}_{\eta} \text{ restricted to }\mathcal X_{\Gamma_0(p^\infty)}(0)_a,\\
G_n'&=&&\big(C'^{\vee}_n\times _{\mathfrak X'^{\ast}(0)} \mathfrak X'^{\ast}(0)^{{\operatorname{perf}}}\big)^{\mathrm{ad}}_\eta \text{ restricted to }\mathcal X'(0)^{{\operatorname{perf}}}.
\end{alignat*}
To prove the lemma, it therefore suffices to prove that the formal models on the right hand side can be identified after reduction to $\O_K/p=\O_{K^{\flat}}/\varpi^{\flat}$, for which it suffices to prove that $C_n^{\vee}/p=C'^{\vee}_n/\varpi^{\flat}$ on $\mathfrak X^{\ast}(0)/p=\mathfrak X'^{\ast}(0)/\varpi^{\flat}$. But over the ordinary locus, $C_n/p=C'_n/\varpi^{\flat}$ are both the kernel of Frobenius, and Cartier duals commute with base change.
\end{proof}
We can now complete the proof of Thm.~\ref{t: tilting the Gamma_1(p^infty)-tower-2} stated in the introduction:
\begin{proof}[Proof of Thm.~\ref{t: tilting the Gamma_1(p^infty)-tower-2}]
We start by proving that for any $n\in\mathbb{Z}_{\geq 1}$, there is a natural isomorphism
\begin{equation}\label{dg:proof-of-thm-1.11-case-of-n<infty}
\begin{tikzcd}
\mathcal X^{\ast}_{\Gamma_1(p^n)\cap \Gamma_0(p^\infty)}(\epsilon)_a^{\flat} \arrow[r,"\sim"] \arrow[d] & \mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)^{{\operatorname{perf}}} \arrow[d] \\ \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a^{\flat} \arrow[r,equal] & \mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}
\end{tikzcd}
\end{equation}
making the diagram commute. Part (1) of the theorem then follows in the limit $n\to \infty$.
Away from the cusps, the desired isomorphism is induced by
the natural isomorphism from Lemma~\ref{Lemma G_n identifies with ker V^n upon tilting}, using the moduli interpretations in Lemma~\ref{l:moduli interpretation of modular curve of level Gamma(p^n) cap Gamma_0(p^infty)} and Lemma~\ref{Lemma moduli interpretation of Ig(p^n)^perf}.
We need to extend this over the cusps. One way of doing this is to give the vertical maps in the diagram a relative moduli interpretation that extends to the cusps. More in the spirit of our arguments so far, we shall instead give a more explicit proof using Tate parameter spaces, which also proves part (2).
To this end, fix a cusp $x$ of $\mathcal X^{\ast}$. By Prop.~\ref{p: description of the tilting isomorphism at level Gamma0 on Tate parameter spaces}, the isomorphism $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a^{\flat} \to \mathcal X'^{\ast}(\epsilon)^{{\operatorname{perf}}}$ restricts to the canonical isomorphism $\mathcal D_{\infty}^{\flat}=\mathcal D'_{\infty}$ over $x$.
Using the description of the Tate curve parameter spaces in $\mathcal X^{\ast}_{\Gamma_1(p^m)\cap \Gamma_0(p^\infty)}(\epsilon)_a\to \XaGea{0}{\infty}$ from Lemma~\ref{l: Tate parameter spaces of X^ast_Gamma_1(p^m)cap Gamma_0(p^infty)} and similarly in $\mathcal X'^{\ast}_{\operatorname{Ig}(p^n)}(\epsilon)^{\perf}\to \mathcal X'^{\ast}(\epsilon)^\perf$ from Prop.~\ref{p:Tate parameter spaces in perfectoid Igusa tower}.(1), it now suffices to prove that the isomorphism $G_n^\flat=G_n'$ over the Tate parameter spaces becomes the natural map
\begin{equation}\label{dg:proof-of-thm-1.11-situation-over-cusps}
\begin{tikzcd}
(\underline{\mathbb{Z}/p^n\mathbb{Z}}\times \mathring{\mathcal D}_\infty)^{\flat} \arrow[r,"\sim"] \arrow[d] & \underline{\mathbb{Z}/p^n\mathbb{Z}}\times \mathring \mathcal D'_\infty \arrow[d] \\ \mathring \mathcal D_\infty^{\flat} \arrow[r,"\sim"] & \mathring \mathcal D'_\infty.
\end{tikzcd}
\end{equation}
It is then clear that the diagram extends uniquely over the cusps.
To see this, we note that the restriction of $G_n$ to $\mathring{\mathcal D}_{\infty}$ is indeed canonically isomorphic to $\underline{\mathbb{Z}/p^n\mathbb{Z}}\times \mathring{\mathcal D}_{\infty}$ due to the canonical section given by the element $q^{1/p^n}$ of the anticanonical subgroup $\langle q^{1/p^n}\rangle\subseteq \mathrm{T}(q)[p^n]$. Similarly, $G'_n$ is isomorphic to $\underline{\mathbb{Z}/p^n\mathbb{Z}}\times \mathring{\mathcal D}'_\infty$ on $\mathring{\mathcal D}'_\infty\to \mathcal X'(\epsilon)^\perf$. By considering the dual trivialisations of the respective canonical subgroups, it follows from the construction in the proof of Lemma~\ref{Lemma G_n identifies with ker V^n upon tilting} that these isomorphisms are compatible with tilting and make diagram~\eqref{dg:proof-of-thm-1.11-situation-over-cusps} commute, as desired.
Part (2) now follows from diagram~\eqref{dg:proof-of-thm-1.11-situation-over-cusps} in the limit $n\to \infty$.
\end{proof}
\section{$q$-expansion principles}\label{s:q-expansion-principles}
In this section, we prove various $q$-expansion principles for functions on the infinite level spaces $\XGea{0}{\infty}$, $\XGea{1}{\infty}$, $\XGea{}{\infty}$, etc, based on our discussion of cusps in \S\ref{s:adic-cusps-finite}-\S\ref{s:cusps-in-char-p}.
\subsection{Detecting vanishing}
We begin with the proof of $q$-expansion principle I, Prop.~\ref{p: q-expansion principle I} in the introduction, recalled below. On the way, we also prove principles III and IV. We focus on the case of characteristic $0$, the case of characteristic $p$ is completely analogous.
\begin{Proposition}\label{p: q-expansion principle I,second version}
Let $\mathcal C$ be a collection of cusps of $\mathcal X^{\ast}$ such that each connected component of $\mathcal X^{\ast}$ contains at least one $x\in \mathcal C$. Let $n\in \mathbb{Z}_{\geq 0}\cup\{\infty\}$ and let $\Gamma$ be one of $\Gamma_0(p^n),\Gamma_1(p^n), \Gamma(p^n)$. Define $\mathcal D_{\mathcal C,\Gamma}$ as the pullback
\[
\begin{tikzcd}
\mathcal D_{\mathcal C,\Gamma}\arrow[r,hook]\arrow[d]&\mathcal X^{\ast}_{\Gamma}(\epsilon)_a\arrow[d]\\
\bigsqcup_{x\in \mathcal C}\mathcal D_{x}\arrow[r,hook]& \mathcal X^{\ast}(\epsilon).
\end{tikzcd}
\]
Then the map $\mathcal O(\mathcal X^{\ast}_{\Gamma}(\epsilon)_a)\rightarrow \mathcal O(\mathcal D_{\mathcal C,\Gamma})$
is injective.
\end{Proposition}
This is an analogue of saying that for any affine irreducible integral variety over $K$, completion at any $K$-point gives rise to an injection on function, which is a consequence of Krull's Intersection Theorem. As this requires Noetherianess, we first reduce to the Noetherian situation using that all of the above spaces have natural models over $\mathbb{Z}_p$.
The proof is in two steps: We first consider $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a$ where it is easy to reduce to the Noetherian case. In a second step, we then show that restriction of functions from $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$ to $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(0)_a$ is injective, which is a straight-forward computation on power series.
We start with the case $\epsilon=0$. On the way we will also see Prop.~\ref{p: q-expansion principle III}.
\begin{proof}[Proof of Prop.~\ref{p: q-expansion principle I} for $\epsilon=0$]
The case of $\Gamma=\Gamma(p^n)$ reduces to the one of $\Gamma_1(p^n)$ by Cor.~\ref{c: over ordinary locus, Gamma to Gamma_1 is split}.
We first consider the case of $n<\infty$. Then the case of $\Gamma=\Gamma_0(p^n)$ further reduces to the case of tame level via the Atkin--Lehner isomorphism $\mathcal X^{\ast}(0)\cong\mathcal X^{\ast}_{\Gamma_0(p^n)}(0)_a$. We are therefore left with the case of $\Gamma_1(p^n)$ for $n\in \mathbb{Z}_{\geq 0 }$ (the case of tame level being $n=0$).
The space $\mathcal X^{\ast}_{\Gamma_1(p^n)}(0)_a$ has an affine formal model $\mathfrak X^{\ast}_{\Gamma_1(p^n)}(0)_a=\operatorname{Spf}(R)$ for some complete $\mathbb{Z}_p$-algebra $R$. Let $\mathcal C'$ be the pullback of $\mathcal C$ to $\mathfrak X^{\ast}_{\Gamma_1(p^n)}(0)_a$ and let
\[\sqcup_{x\in \mathcal C'} \operatorname{Spf} \mathcal O_{L_x}\llbracket q\rrbracket\rightarrow \mathfrak X^{\ast}_{\Gamma_1(p^n)}(0)_a\]
be the completion along $\mathcal C'$.
It suffices to show that the map on global sections
\[\varphi:R\to \prod_{x\in \mathcal C'} \mathcal O_{L_x}\llbracket q\rrbracket\] is injective.
As these are flat $\O_K$-algebras, it suffices to see that the reduction
\begin{equation}\label{eq:q-exp-III-for-n<infty}
R/p\to \prod_{c \in \mathcal C'} \mathcal O_{L_x}/p\llbracket q\rrbracket
\end{equation}
is injective.
But this reduction can be interpreted as the completion of $X^{\ast}_{\mathcal O_K/p,\operatorname{Ig}(p^n),\mathrm{ord}}$ at the divisor of cusps $\mathcal C'$.
By base change from $\mathbb{F}_p$ to $\O_K/p$, we can now reduce to showing that for $Y:=X^{\ast}_{\mathbb{F}_p,\operatorname{Ig}(p^n),\mathrm{ord}}$, completion at $\mathcal C'$ defines an injection
\[\O(Y)\to \prod_{x\in \mathcal C'}\mathbb{F}_p(x)\llbracket q\rrbracket \]
where $\mathbb{F}_p(x)\subseteq \mathbb{F}_p[\zeta_N]$ is the coefficient field of definition of the level structure on the Tate curve corresponding to the cusp $x\in \mathcal C'$.
Since $Y$ is a smooth affine curve over $\mathbb{F}_p$, and by considering each connected component separately, the desired injectivity follows as for an integral Noetherian ring $A$, completion at any maximal ideal $\mathfrak m\subseteq A$ gives an injection $A\to \hat{A}_{\mathfrak m}$ by Krull's intersection theorem.
The case of $n=\infty$ can be deduced in the limit:
As the natural restriction map \[\O^+(\overline{\mathcal D}_\infty)\hookrightarrow\O^+({\mathcal D}_\infty)\]
is injective (see Def.~\ref{d: OK bb q^1/p^infty _p}), it suffices to prove the statement for $\overline{\mathcal D}_\infty\sim \varprojlim\overline{\mathcal D}_n$,
while conversely it is clear from $\O^+(\overline{\mathcal D}_n)=\O^+({\mathcal D}_n)$ for $n<\infty$ and the first part that the corresponding result at finite level holds for $\mathcal D_n$ replaced by $\overline{\mathcal D}_n$.
For any $m\in\mathbb{N}$, let $\mathfrak Y_m=\mathfrak X^{\ast}_{\Gamma_0(p^m)}(0)_a$ or $\mathfrak Y_m=\mathfrak X^{\ast}_{\Gamma_1(p^m)}(0)_a$. Then $\mathfrak Y=\varprojlim \mathfrak Y_m$ is a formal model of $\mathcal X^{\ast}_{\Gamma}(0)_a$. To see the result it suffices to prove that the natural maps
\begin{alignat*}{4}
\O(\mathfrak X^{\ast}_{\Gamma_0(p^m)}(0)_a)\to& \prod_{x\in \mathcal C}\O(\mathfrak D_{\infty,x})\\
\O(\mathfrak X^{\ast}_{\Gamma_1(p^m)}(0)_a)\to& \prod_{x\in \mathcal C}\operatorname{Map}_{{\operatorname{cts}}}(\mathbb{Z}_p^\times,\O(\mathfrak D_{\infty,x})).
\end{alignat*}
are injective. By flatness, it suffices to prove this on the reduction mod $\varpi$. But here it follows in the direct limit over $m\to \infty$ from the case of finite level.
\end{proof}
The proof of Prop.~\ref{p: q-expansion principle I} is completed by the following two lemmas:
\begin{Lemma}\label{l:restricting functions from (epsilon) to (0) is injective}
\leavevmode
Let $n\in\mathbb{Z}_{\geq 0}\cup \{\infty\}$ and
let $\mathcal Y\rightarrow \mathcal X^{\ast}$ be one of $\mathcal X^{\ast}_{\Gamma_0(p^n)}$, $\mathcal X^{\ast}_{\Gamma_1(p^n)}$, $\mathcal X^{\ast}_{\Gamma(p^n)}$. Then the open immersion $\mathcal Y(0)\rightarrow \mathcal Y(\epsilon)$ on sections gives an injection $\mathcal O(\mathcal Y(\epsilon))\rightarrow \mathcal O(\mathcal Y(0))$.
\end{Lemma}
\begin{proof}[Proof of Lemma~\ref{l:restricting functions from (epsilon) to (0) is injective}]
It suffices to prove this locally. Let $\mathcal Y_{\mathrm{gd}}(\epsilon):=\mathcal Y(\epsilon)\times_{\mathcal X}\mathcal X_{\mathrm{gd}}$. Then since $\mathcal Y(0)\to\mathcal Y(\epsilon)$ is an open immersion, and $\mathcal Y(0)$ and $\mathcal Y_{\mathrm{gd}}(\epsilon)$ cover all of $\mathcal Y(\epsilon)$, it suffices to prove the statement for $\mathcal Y_{\mathrm{gd}}(\epsilon)$.
Let thus $\mathfrak Y$ be one of $\mathfrak X_{\Gamma_0(p^n)}$ , $\mathfrak X_{\Gamma_1(p^n)}$, $\mathfrak X_{\Gamma(p^n)}$, each for any $n\in\mathbb{Z}_{\geq 0}\cup \{\infty\}$. It suffices to prove that for any affine open $\mathfrak U=\operatorname{Spf}(R)\subseteq \mathfrak Y$ where $\omega$ is trivial, the natural map $\mathfrak Y(0)\rightarrow \mathfrak Y(\epsilon)$ induces an injection $
\mathcal O(\mathfrak Y(\epsilon)|_{\mathfrak U})\rightarrow \mathcal O(\mathfrak Y(0)|_{\mathfrak U})$. We have
$\mathfrak Y(\epsilon)|_{\mathfrak U}=\operatorname{Spf}(S)$ where $S=R\langle X\rangle /(X\mathrm{Ha} -p^{\epsilon} )$, and $\mathfrak Y(0)|_{\mathfrak U}=\operatorname{Spf}(R\langle \mathrm{Ha}^{-1}\rangle)$. Since $\mathrm{Ha}$ is a non-zero-divisor on $R/p^n$, Lemma~\ref{l:restricting functions from (epsilon) to (0) on algebras} below now gives the desired statement.
\end{proof}
\begin{Lemma}\label{l:restricting functions from (epsilon) to (0) on algebras}
Let $A$ be any ring, let $0\neq \varpi \in A$ be a non-zero-divisor and let $H\in A$ be such that its image in $A/\varpi$ is a non-zero-divisor. Endow $A$ with the $\varpi$-adic topology. Then
\[\varphi\colon A\langle X\rangle /(XH-\varpi)\xrightarrow{X\mapsto \varpi X } A\langle X \rangle/(XH-1) \]
is injective.
\end{Lemma}
\begin{proof}
We first note that the assumption on $H\in A$ implies that $H$ is a non-zero-divisor in any $A/\varpi^n$.
Suppose $f=\sum a_nX^n$ is in the kernel of $A\langle X\rangle\rightarrow A\langle X\rangle/(XH-\varpi)\xrightarrow{\varphi} A\langle X\rangle /(XH-1)$. Then there is $g=\sum b_nX^n\in A\langle X\rangle$ such that
\[f(\varpi X)=\sum a_n\varpi^nX^n=(XH-1)g=(XH-1)\sum b_nX^n.\]
Reducing mod $\varpi^m$, we see that
\[a_0+\dots +a_{m-1}\varpi^{m-1} X^{m-1} \equiv (XH-1)\sum b_nX^n \bmod \varpi^m\]
By comparing degrees as polynomials in $A/\varpi^m[X]$, we conclude from $H$ being a non-zero-divisor mod $\varpi^m$ that $\deg(\sum b_nX^n \bmod \varpi^m)<m-1$, thus $b_k\equiv 0 \bmod \varpi^m$ for $k\geq m-1$.
Consequently, there are elements $c_m={b_m}/{\varpi^{m+1}}\in A$ for all $m$ and in $A\llbracket X\rrbracket$ we have
\[f':= (XH-\varpi)\sum \frac{b_m}{\varpi^{m+1}}X^m \stackrel{X\mapsto \varpi X}{\longmapsto} (XH-1)\sum b_mX^m.\]
Thus $f'(\varpi X)=f(\varpi X)$ in $A\llbracket X\rrbracket$ which implies $f'=f$ since $\varpi$ is a non-zero-divisor.
It remains to prove that $\sum c_mX^m$ converges in $A\langle X\rangle$: Since $f\in A\langle X\rangle$, for every $k\in \mathbb{N}$ there is an $N_k$ such that $v(a_{m})\geq k$ for all $m\geq N_k$, where $v$ is the $\varpi$-adic valuation. In particular, we then have $v(\varpi^ma_{m})\geq k+m$ for all $m\geq N_k$. Consequently, for all $m\geq N_k$
\[a_0+\dots +a_{m-1}\varpi^{m-1} X^{m-1} \equiv (XH-1)\sum b_mX^m \bmod \varpi^{m+k}.\]
This shows that $v(b_{m-1})\geq m+k$, and thus $v(c_m)\geq k$ for all $m\geq N_k$. Thus $\sum c_mX^m \in A\langle X\rangle$ as desired.
We conclude that $f$ is already in $(XH-\varpi)A\langle X\rangle$. Thus $\varphi$ is injective.
\end{proof}
We can extract from this argument a proof of $q$-expansion principle III in the introduction:
\begin{Proposition}[$q$-expansion principle III]
For $f\in \O(\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a)$ are equivalent:
\begin{enumerate}
\item $f$ is integral, i.e.\ it is contained in $\O^+(\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a)$.
\item The $q$-expansion of $f$ at every cusp $x$ is already in $\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket$.
\item On each connected component of $\XaGea{0}{n}$, there is at least one cusp $x$ at which the $q$-expansion of $f$ is in $\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket$.
\end{enumerate}
Equivalently, the natural map
$\varphi\colon \O^+(\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a)/p\to \prod_{x}(\O_{L_x}/p)\llbracket q^{1/p^\infty}\rrbracket$
is injective.
The analogous statements for $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(0)_a$, $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a$, $\mathcal X'^{\ast}(0)^{{\operatorname{perf}}}$ and $\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(0)^{\operatorname{perf}}$ are also true when we replace $\O_{L_x}\llbracket q^{1/p^\infty}\rrbracket$ by the respective algebra from Prop.~\ref{p: q-expansion principle I}.
\end{Proposition}
\begin{proof}
It is clear from $\O^+(\mathcal D_{\infty,x})= \O_{L_x}\llbracket q^{1/p^\infty}\rrbracket$ that (1) implies (2) implies (3). To prove that (3) implies (1), it suffices to see that $\O^+(\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a)/p\to \prod_{x}(\O_{L_x}/p)\llbracket q^{1/p^\infty}\rrbracket$ is injective.
We have already seen in \eqref{eq:q-exp-III-for-n<infty} in the proof of Prop.~\ref{p: q-expansion principle I,second version} that
\[ \O(\mathfrak X^{\ast}_{\Gamma_0(p^n)}(0)_a)/p\hookrightarrow \prod_{x\in \mathcal C}(\O_{L_x}/p)\llbracket q^{1/p^n}\rrbracket \]
is injective for any $n\in \mathbb{Z}_{\geq 0}\cup\{\infty\}$. Since by
by \cite[Prop.~4.1.3]{heuer-thesis}, we have \[\O^+(\mathcal X^{\ast}_{\Gamma_0(p^n)}(0)_a)=\O(\mathfrak X^{\ast}_{\Gamma_0(p^n)}(0)_a),\]
this gives the desired statement in the case of $\Gamma_0(p^\infty)$.
By the same argument, the cases of $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(0)_a$, $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a$, $\mathcal X'^{\ast}(0)^{{\operatorname{perf}}}$ and $\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(0)^{\operatorname{perf}}$ also follow from \eqref{eq:q-exp-III-for-n<infty} in the limit $n\to \infty$ using instead \cite[Lemma A.2.2.3]{heuer-thesis}.
\end{proof}
We can also use the lemmas for the proof of $q$-expansion principle IV:
\begin{Proposition}[$q$-expansion principle IV]
Let $\mathcal C$ be a collection of cusps of $\mathcal X^{\ast}$ such that each connected component contains at least one $x\in \mathcal C$. Then a function on the good reduction locus $\mathcal X_{\mathrm{gd}}(\epsilon)$ extends to all of $\mathcal X^{\ast}(\epsilon)$ if and only if its $q$-expansion with respect to $\overline{\mathcal D}(|q|\geq 1)\to \mathcal X_{\mathrm{gd}}(\epsilon)$ at each $x\in \mathcal C$ is already in $\O_{L_x}\llbracket q\rrbracket[\tfrac{1}{p}]\subseteq \O_{L_x}\lauc{q}[\tfrac{1}{p}]$. In this case, the extension is unique. The analogous statements for $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(0)_a$ $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(0)_a$, $\mathcal X^{\ast}_{\Gamma(p^\infty)}(0)_a$, $\mathcal X'^{\ast}(0)$, $\mathcal X'^{\ast}(0)^{{\operatorname{perf}}}$ and $\mathcal X'^{\ast}_{\operatorname{Ig}(p^\infty)}(0)^{\operatorname{perf}}$ are also true.
\end{Proposition}
\begin{proof}
As before, one can reduce to the case of finite level. For simplicity, let us treat $\mathcal X_{\mathrm{gd}}$, the other cases are similar. By Lemma~\ref{l:restricting functions from (epsilon) to (0) is injective} we can reduce to $\epsilon=0$. We then need to prove that the following sequence is left exact:
\[ 0\to \O(\mathfrak X^{\ast}(0)) \to \O(\mathfrak X(0))\times \textstyle\prod_{x\in \mathcal C} \O_{L_x} \bb{q}\xrightarrow{(f,g)\mapsto f-g} \textstyle\prod_{x\in \mathcal C}\O_{L_x}\lauc{q}.\]
It suffices to prove that this is true mod $\varpi^n$ for all $n$. By tensoring with the flat $\mathbb{F}_p$-algebra $\O_L/\varpi^n$, the statement then follows from the following sequence being left-exact:
\[ 0\to \O(X_{\mathbb{F}_p,\mathrm{ord}}^{\ast}) \to \O(X_{\mathbb{F}_p,\mathrm{ord}})\times \textstyle\prod_{x\in \mathcal C} \mathbb{F}_p(x)\bb{q}\xrightarrow{(f,g)\mapsto f-g} \textstyle\prod_{x\in \mathcal C}\mathbb{F}_p(x)(\!(q)\!).\]
This holds as $X_{\mathbb{F}_q}^{\ast}$ is the normalisation of $j\colon X_{\mathbb{F}_q}\to \mathbb{A}_{\mathbb{F}_q}^1$ in $\P_{\mathbb{F}_q}^1$, and thus a function $f$ extends to the cusp $x$ if and only if it is finite over the completion $\mathbb{F}_p\llbracket q\rrbracket $ of $\P_{\mathbb{F}_p}^1$ at $\infty$.
\end{proof}
\subsection{Tate traces and detecting the level}
While the transition from $\Gamma_0(p^\infty)$ to $\Gamma(p^\infty)$ is controlled by the Galois action, the transition from $\Gamma_0(p)$ to $\Gamma_0(p^\infty)$ is controlled by normalised Tate traces, as discussed in \cite[\S III.2.4]{torsion}:
\begin{Proposition}[{\cite[Cor.~III.2.23]{torsion}}]
Let $0\leq n\leq k\in\mathbb{N}$. Then the normalised traces
\[\operatorname{tr}_{k,n}\colon \O_{\mathfrak X^{\ast}_{\Gamma_0(p^k)}(\epsilon)_a}\to \O_{\mathfrak X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a}[\tfrac{1}{p}]\]
of the finite flat forgetful map $\mathfrak X^{\ast}_{\Gamma_0(p^k)}(\epsilon)_a\to \mathfrak X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$ in the limit $k\to \infty$ give rise to compatible continuous $\O_{\mathfrak X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a}$-linear morphisms with bounded image
\[\operatorname{tr}_{n}\colon \O_{\mathfrak X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a}\to \O_{\mathfrak X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a}[\tfrac{1}{p}].\]
\end{Proposition}
\begin{proof}
Via the Atkin--Lehner isomorphism $\mathfrak X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a\cong \mathfrak X^{\ast}(p^{-n}\epsilon)$,
this is the statement of \cite[Cor.~III.2.23]{torsion}, except that we use the compactified $\mathfrak X^{\ast}$ instead of $\mathfrak X$: This is possible since in contrast to the higher dimensional Siegel moduli spaces, the minimal compactification of the modular curve $\mathfrak X^{\ast}$ is a smooth formal scheme, and thus Cor.~III.2.22 applies over all of $\mathfrak X^{\ast}$, not just over $\mathfrak X$, which means that the proof of III.2.23 goes through for $\mathfrak X^{\ast}$.
\end{proof}
\begin{Definition}
Taking global sections and inverting $p$, the trace $\operatorname{tr}_n$ gives a $K$-linear map
\[\operatorname{tr}\colon \O(\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a)\to \O(\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a).\]
\end{Definition}
\begin{Proposition}\label{Proposition: effect of trace on q-expansions}
Let $x$ be any cusp of $\mathcal X^{\ast}$, with corresponding Tate curve parameter space $\mathcal D_{n,x}\hookrightarrow \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$. Then the normalised Tate trace fits into a commutative diagram
\begin{equation*}
\begin{tikzcd}[row sep={0.7cm,between origins}]
\O(\XaGea{0}{\infty}) \arrow[r, "\operatorname{tr}_n"] \arrow[dd, '] & \O(\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a) \arrow[dd, ']&&\\
&& {\displaystyle\sum_{m\in\mathbb{Z}[\frac{1}{p}]_{\geq 0}} a_mq^m} \arrow[r, maps to]& {\displaystyle\sum_{m\in \tfrac{1}{p^n}\mathbb{Z}_{\geq 0}} a_mq^m}
\\
\O(\mathcal D_{\infty,x}) \arrow[r, "\operatorname{tr}_n"] & \O(\mathcal D_{n,x}),&&
\end{tikzcd}
\end{equation*}
where the bottom map is given by forgetting all coefficients $a_m$ for $m\not \in \tfrac{1}{p^n}\mathbb{Z}_{\geq 0}$.
\end{Proposition}
\begin{proof}
Let us treat the case of $n=0$, the other cases are completely analogous.
By continuity, $\operatorname{tr}_n$ is uniquely determined by the normalised traces $\operatorname{tr}_{k,0}$. By Lemma~\ref{Proposition: Tate parameter spaces in the Gamma_0-tower}, this is on $q$-expansions the trace of the inclusion $\O_L\llbracket q\rrbracket\to \O_L\llbracket q^{1/p^k}\rrbracket$. Since after inverting $q$, this map becomes Galois with automorphisms $q^{1/p^k}\mapsto q^{1/p^k}\zeta^d_{p^k}$ for $d\in \mathbb{Z}/p^k\mathbb{Z}$, we compute
\[\operatorname{tr}_{k,0}\left(\sum_{i=0}^\infty a_{\frac{i}{p^k}}q^{\frac{i}{p^k}}\right)= \frac{1}{p^k}\sum_{i=0}^\infty a_{\frac{i}{p^k}}(1+\zeta_{p^k}^i+\dots+\zeta_{p^k}^{(p^{k}-1)i})q^{{\frac{i}{p^k}}}=\sum_{i=0}^\infty a_{i}q^i\]
as $1+\zeta_{p^k}^i+\dots+\zeta_{p^k}^{(p^{k}-1)i}=0$ unless $p^k|i$, when it is $=p^k$, giving the desired description.
\end{proof}
\begin{Proposition}[$q$-expansion principle II]\label{p: q-expansion principle II}
Let $f\in \O(\XaGea{0}{\infty})$. Then for any $n\in \mathbb{Z}_{\geq 0}$, the following are equivalent:
\begin{enumerate}
\item $f$ comes via pullback from $\XaGea{0}{n}$, i.e.\ $f\in\O(\XaGea{0}{n})\subseteq \O(\XaGea{0}{\infty})$.
\item The $q$-expansion of $f$ at every cusp $x$ is contained in $\O_{L_x}\llbracket q^{1/p^n}\rrbracket[\frac{1}{p}]\subseteq \O_{L_x}\llbracket q^{1/p^\infty}\rrbracket[\frac{1}{p}]$.
\item On each connected component of $\XaGea{0}{n}$, there is at least one cusp $x$ at which the $q$-expansion of $f$ is already in ${\O_{L_x}}\llbracket q^{1/p^n}\rrbracket[\frac{1}{p}]\subseteq \O_{L_x}\llbracket q^{1/p^\infty}\rrbracket[\frac{1}{p}]$.
\end{enumerate}
The analogous statements for $\mathcal X'^{\ast}(\epsilon)^\perf\to \mathcal X'^{\ast}(\epsilon)$ are also true.
\end{Proposition}
\begin{proof}
It suffices to prove that (3) implies (1). Clearly $f$ is in $\O(\XaGea{0}{n} )$ if and only if $\operatorname{tr}_n(f)=f$. By Prop.\ \ref{p: q-expansion principle I,second version}, this can be checked on $q$-expansions on each component. By Prop.\ \ref{Proposition: effect of trace on q-expansions}, we have $\operatorname{tr}_n(f)=f$ if and only if the $q$-expansion at each $x$ is in $\O_L\llbracket q^{1/p^n}\rrbracket[\frac{1}{p}]$.
The case of $\mathcal X'^{\ast}(\epsilon)^\perf$ is completely analogous, by replacing the normalised Tate traces of \cite{torsion} with those of \cite[\S 6.3]{AIP}.
\end{proof}
| {'timestamp': '2020-02-10T02:01:00', 'yymm': '2002', 'arxiv_id': '2002.02488', 'language': 'en', 'url': 'https://arxiv.org/abs/2002.02488'} |
\section{Appendices}
\section{Comparing web traffic data with social media engagement signals}
\label{section:comparison}
As we mentioned previously, existing research on news consumption mostly focuses on how news URLs are shared on social media platforms, especially Twitter and Facebook. While social media signals can tell us how people \textit{share} news, they do not answer how many people actually \textit{visit} each URL. Is there any correlation between social media sharing behavior and actual news consumption behavior? If so, how strong is the correlation? To answer those questions, we first collect Facebook and Twitter metrics that measure popularity of~\textit{TGP } links shared on each platforms. We then test the correlational strength of different social media metrics against website visit count, and identify metrics that are good estimations of actual web visit counts.
\subsection{Collecting posts from Facebook and Twitter}
Among 1070 online articles published by~\textit{TGP } during our one-month data collection, 1020 received more than 10,000 unique web visits. To ensure the stability of our experiment, we focus on those 1020 URLs and discard URLs with lower web visit counts. We use Crowdtangle API to collect Facebook posts that contain any one of the 1020 URLs published by~\textit{TGP}. Crowdtangle is a data intelligence service that tracks aggregated engagements and interactions of posts from Facebook pages and groups (both public and private)~\cite{crowdtangle-api}. We use Twitter Academic API~\cite{twitter-api} to collect all original and public tweets that contain any one of the 1020 URLs. For each URL, we calculate seven metrics, shown in Table~\ref{table:metric-source}.
\begin{table}[!h]
\begin{tabular}{p{0.6\linewidth}p{0.35\linewidth}}
\hline
metric & source \\
\hline
number of unique visits (all) & web traffic dataset \\
\# unique visits (from facebook.com) & web traffic dataset \\
\# unique visits (from twitter.com) & web traffic dataset \\
total number of FB reactions & Crowdtangle API \\
total number of FB interactions & Crowdtangle API \\
total number of likes & Twitter API \\
total number of retweets & Twitter API\\
\hline
\end{tabular}
\caption{We calculate seven metrics to quantify the popularity of an article URL. We later correlate web traffic-based metrics with social media-based metrics.}
\label{table:metric-source}
\end{table}
\subsection{Measuring correlations}
We calculate Pearson correlations between each social media metric and (a) the number of visits from all traffic and (b) the number of visits from platform-specific traffic. Pearson correlation is the normalized covariance between two variables, and is used to summarize the strength of the linear relationship between two variables~\cite{pearson-correlation}.
We first observe that Facebook metrics correlate better with traffic that only originated from Facebook than traffic originated from all sites. The same is true for Twitter metrics. For example, Figure~\ref{fig:correlation-scatter-facebook} shows that the Pearson correlation between total Facebook interaction and number of visits from~\url{facebook.com} is 0.894, while the correlation is only 0.595 for number of visits from all sites. Since social media metrics cannot capture URL sharing activities outside of the platform, the correlation significantly decreases when using number of visits from all sites.
We also observe that Facebook metrics correlate better with web visit count than Twitter metrics. For example, when we compare the first row of Figure~\ref{fig:correlation-scatter-facebook} against the first row of Figure~\ref{fig:correlation-scatter-twitter}, we see that Facebook interactions have a higher correlation with web visit counts than Twitter likes. The former metric has a Pearson correlation of 0.595 while the latter has a correlation of 0.435. Why is there a discrepancy? One reason could be that Facebook metrics we collect count both private and public posts, while Twitter metrics only count public posts. Understanding what other factors affect the correlation is a subject for future research.
To summarize, we validate that all social media metrics have a positive correlation with web visit counts. Therefore it is reasonable to use social media engagement signals as a proxy for URL popularity. However, there are limitations when using social media metrics, as they only capture link sharing activities on one platform. Given those insights, researchers should carefully choose from which platform to collect data and which engagement signals to use, as each metric has varying correlational strength. In the future, we plan to test more metrics to further understand correlations between how people share news on social media versus how people actually read news.
\begin{figure}
\centering
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation_fb_reaction.png}
}
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation_fb_interaction.png}
}
\hspace{1mm}
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation__came_from_fb_fb_reaction.png}
}
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation__came_from_fb_fb_interaction.png}
}
\caption{Pearson correlations between Facebook engagement metrics (y-axis) and web visit count (x-axis). Both axes are in log scale. Top row is based on all traffic, and bottom row is based on traffic only from facebook.com. In each scatter plot, a dot represents a unique~\textit{TGP } article URL.}
\label{fig:correlation-scatter-facebook}
\end{figure}
\begin{figure}
\centering
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation_twitter_retweet.png}
}
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation_twitter_like.png}
}
\hspace{1mm}
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation__came_from_twitter_twitter_retweet.png}
}
\subfloat{
\includegraphics[width=0.49\linewidth]{image/correlation__came_from_twitter_twitter_like.png}
}
\caption{Pearson correlations between Twitter engagement metrics (y-axis) and web visit count (x-axis). Both axes are in log scale. Compared with Facebook metric, Twitter metrics have weaker correlation with actual web visit count.}
\label{fig:correlation-scatter-twitter}
\end{figure}
\section{Discussion and Conclusion}
\label{section:conclusion}
In this paper, we collect and analyze a unique website traffic data set that contains more than 68 million visits to\textit{ The Gateway Pundit (TGP)}, a major far-right website known to spread fake news and conspiracy theories. We find that search engines and social media platforms are the main drivers that bring traffic to the site. Our geo-location analysis reveals that~\textit{TGP } is more popular in counties that vote for Donald Trump, and our topic analysis shows that conspiratorial stories are more viral. Finally, we compare engagement signals derived from Twitter and Facebook posts with actual website visit counts, and find varying degrees of correlations. Our population-level behavioral analysis can help researchers design robust intervention methods to counter the spread of misinformation.
One major difficulty encountered during our research was our inability to analyze other comparable web traffic data sets. We reached out to several organizations who could offer such a data set, but did not move forward due to insufficient response. In the future, we plan to collaborate more with industry partners that have direct access to population-level news consumption data. Potential collaborators include web tracking companies and Internet service providers. As misinformation spreads over multiple platforms with an increasing speed, researchers need to be able to access more direct measurement data to quantify and understand the phenomenon of how people access low quality news websites.
\section{Introduction}
\label{section:introduction}
Fake news site is a major threat on today's Internet~\cite{persily_tucker_2020, fakenews-2016-election, spread-of-true-and-false-news}. How to measure the consumption of fake news URLs remains a challenge. Since there is no single metric to quantify the spread of information, the choice of metrics can affect downstream analysis and alter final conclusions. There are two approaches to measuring fake news consumption: indirect and direct.
For indirect measurement, a common method is to collect social media posts containing the URL of interest, calculate engagement signals, and use those metrics as a proxy for URL popularity~\cite{fakenews-2020-election, less-than-you-think, cracking-open-the-news-feed}. Indirect measurements reveal how people \textit{share} news URLs, but not how people actually \textit{visit} those URLs~\cite{fake-news-not-antitrust}.
Only a few studies use direct measurement data. For example,~\cite{rise-and-fall-of-fake-news-sites} collects visit data to fake news sites from third party services such as SimilarWeb and CheckPageRank to assess user engagement. In another study, ~\cite{microsoft-measurement} gathers browsing data from Microsoft Internet Explorer and Edge to analyze visiting patterns to fake news sites before the 2016 US Election. As far as we know, one unexplored data source is web traffic data collected on the server side. This type of web traffic data has rich features that alternative sources do not have. Even though most news websites record their traffic, few make the data publicly available.
During an audit of popular far-right and extreme news websites, we discovered that~\textit{TGP } makes its website traffic available to the general public.~\textit{TGP } is one of the top three right-wing news sites with the largest percentage of traffic surge from December 2019 to December 2020~\cite{surge-in-traffic}. It is also one of ``the top-three most cited domains in tweets spreading false and misleading narratives about voter fraud in 2020''~\cite{fakenews-2020-election}. Even though -- or perhaps because -- the site constantly shares misinformation~\cite{The-Gateway-Pundit-NewsGuard-Nutrition-Label, partisanship-propaganda-disinformation}, it remains highly influential. For example, its articles were cited by Former President Trump’s lawyer and referenced in Trump’s Impeachment Defense Memo~\cite{trail-memorandum}. All of these features make~\textit{TGP } an ideal case study to understand online extreme news consumption behavior.
Given this opportunity, we crawl the entire web traffic from~\textit{TGP } for one month from February 4, 2021 to March 3, 2021. We collect a total of 68 million website visits. Our analysis is two-fold: we first explore available features within the web traffic data to understand how people consume low quality news; we then collect additional social media posts to test correlations between social media engagement signals and actual web visit counts. Our substantive findings include:
\begin{enumerate}[leftmargin=*]
\item Search engines such as Google, Duckduckgo and Bing account for 88.5\% of external referral traffic to~\textit{TGP } home page. Social media platforms including Twitter, Facebook, Telegram and Gab account for more than 42\% of external referral traffic to~\textit{TGP } article pages.
\item At the county level,~\textit{TGP } is more popular in counties that vote for Trump. At the state level,~\textit{TGP } is more popular in ``swing states'' such as Georgia and Arizona.
\item Topic modeling reveals that articles that mention ``2020 US election fraud'' are visited by 29\% more users, and spread 8 hours longer, compared with articles of other topics. Those viral articles usually cover events happening in swing states.
\item Social media engagement signals positively correlate with actual website visit counts. Not all metrics are the same: Facebook metrics achieve a stronger correlation than Twitter metrics.
\end{enumerate}
To the best of our knowledge, our work is the first to analyze server-side web traffic data of a popular low quality news site, and the first to correlate social media engagement signals with actual web traffic counts. In the future, we plan to apply our method to similar server-side web traffic data, although getting access to additional data sets remains challenging.
\section{Method}
\label{section:method}
In this section, we first explain how we collect the entire visit traffic from~\textit{TGP } for one month. We then give an overview of the collected data, and address issues related to data integrity, missing data and data privacy.
\subsection{Data collection}
\textit{TGP } uses StatCounter, a web traffic service, to capture visitor traffic. To access traffic data to~\textit{TGP }, users can either visit a publicly available web portal, or download the data by sending an HTTP GET request to a URL endpoint, which we refer to as the ~\textit{download URL}. Users need to specify two parameters in the URL, which we refer to as \textit{StartTime} and \textit{EndTime}.\footnote{The web portal is available at ~\url{https://statcounter.com/p9449268/summary/?guest=1}. The~\textit{download URL} follows the following pattern: \textit{\url{https://statcounter.com/p9449268/csv/download_log_file?range=StartTime--EndTime}}. \textit{StartTime} and \textit{EndTime} must be in ISO format, such as \textit{2021-04-12T02:18:41}. For a formal definition of ISO format, refer to \url{https://www.w3.org/TR/NOTE-datetime}.}
During our testing phase, we find that no matter what \textit{StartTime} and \textit{EndTime} we set, the downloaded CSV file always contains traffic captured during the most recent 20 minutes. To collect website traffic continuously, we set up a Selenium Chrome Browser to visit the~\textit{download URL} every 15 minutes, from February 3, 2021 to March 3, 2021. We choose a 15 minute interval because it is below the 20-minute interval with a safe margin. One side effect is that our data has duplicates. To remove duplicates, we identify that each website visit is uniquely defined by the combination of five features: \textit{datetime, url, ip, os}, and \textit{browser}. Therefore, we only keep the first record if multiple records have the same five-feature combination.
\begin{table}
\begin{tabular}{p{0.17\linewidth}p{0.4\linewidth}p{0.45\linewidth}}
\toprule
feature & example & description \\
\midrule
datetime & 2021-02-02 18:23:31 & date and time of the visit \\
ip & 10.11.123.12 & IPv4 adddress \\
os & IOS & operating system \\
url & thegatewaypundit.com & url visited \\
isp & Verizon & Internet service provider \\
country & USA & country of IP \\
city & Houston & city of IP \\
region & Texas & region \\
referrer & google.com & previous url \\
page title & Expert claims... & title of the article \\
browser & Safari & browser \\
resolution & 375 $\times$ 667 & resolution \\
\bottomrule
\end{tabular}
\caption{\label{table:overview-datasett}Features in~\textit{TGP } web traffic data set.}
\end{table}
\subsection{Data integrity}
To validate that our collection method captures the entire traffic, we compare the daily number of visits reported by Statcounter against the number calculated from our collection after de-duplication.~\footnote{The official aggregated counts is accessible from the following Statcounter URL:~\url{https://statcounter.com/p9449268/summary/daily-pur-labels-bar-20210204_20210303/}} Figure~\ref{fig:data-integrity} shows that our data set has a completeness ratio of more than 99.8\% on a daily basis. We define the completeness ratio as our number of visits divided by Statcounter's number of visits. The lost entries are possibly caused by parsing errors or corrupted network packages. We believe that this small number of missing entries (less than 0.2\%) will not affect trends we observe.
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/data-integrity.png}
\caption{Total number of visits per day. Blue bar is the official count from~\textit{TGP}, and red bar is the count from our data set. Our data collection has a completeness ratio of more than 99.8\% on a daily basis.}
\label{fig:data-integrity}
\end{figure}
\subsection{Missing data and bot traffic}
Even though we capture the entire web traffic, our data source (StatCounter) has several inherent problems. One potential issue is under-counting. For example, anyone who blocks HTTP and HTTPS request to StatCounter will not have their visits logged by the server. This can happen if people install certain anti-tracking plug-ins. Unfortunately, it is impossible to know exactly how many users install anti-tracking tools, as those tools are designed to hide web visit history.
Another problem is the presence of bot traffic. Bots are programs that automatically visit web pages. According to the documentation, StatCounter does not record most bots or crawlers, because clients have to actually load javascript for their hit to be logged in the system~\cite{statcounter-ignore-crawler}. For more advanced bots that emulate human behavior (load javascript, click buttons), there is no way to distinguish their traffic from real human traffic. To sum up, even though the amount of missing data and advanced bot traffic is unknown and undetectable, we believe that those irregularities will not affect the overall trend during our analysis.
\subsection{Data privacy}
To address concerns regarding data privacy, we first note that our web traffic data does not contain any personal identifiable information such as name, phone number, cookie, session ID, device ID, email address, etc,. Additionally, all of our results presented below are aggregated.
\subsection{Robustness of findings}
Despite the challenges noted above associated with our data collection process, two factors give us confidence in the robustness of our findings. First, our data is about as close to the ground truth as one can hope for in any sort of online data analysis, with a 99.8\% completeness rate. Second, we collect more than 68 million page visits that span a full month. This extended period of time ensures that any daily or hourly data irregularity is smoothed out and the overall trend preserved.
\section{Related Work}
\label{section:related-work}
Measuring who consumes and how people consume fake news is an important but challenging research area. Previous work mostly studies the spread of fake news on social media platforms~\cite{fakenews-2020-election}. For example, ~\cite{spread-of-true-and-false-news} collects tweets containing links to fake news sites, and concludes that fake news spread faster and further than traditional news. In another study using Twitter data,~\cite{fakenews-2016-election} claims that ``fake news accounted for nearly 6\% of all news consumption, but it was heavily concentrated'' on a small percentage of users. Similarly,~\cite{less-than-you-think}, \cite{cracking-open-the-news-feed} collect Facebook posts to understand news consumption behavior, and find that older people are more susceptible and share more fake news.
While social media engagement signals can tell us how people share news on different platforms, they do not necessarily translate into web traffic to the news site~\cite{fake-news-not-antitrust}. One way to bridge this gap is to directly gather data from volunteers via browsing extensions. For example,~\cite{misinformation-in-action} asked participants to install a browser extension to measure their exposure to fake news. However, this approach is usually expensive and the sample size is small.
To understand population-level news consumption behavior, there is an urgent need to collect ``unique datasets with increased validity''~\cite{tackling-misinformation}. Web traffic data is a direct measurement of news consumption. In one study, ~\cite{rise-and-fall-of-fake-news-sites} assesses user engagement by collecting traffic data from tracking services such as SimilarWeb and CheckPageRank.~\cite{microsoft-measurement} gathers browsing data from Microsoft Internet Explorer and Edge, and analyzes visitor patterns to a list of fake news domains before the 2016 US Election.
Different from all previous approaches, we focus on collecting the entire web traffic to a single but important news site (\textit{TGP}). Our data set enables us to validate and extend previous traffic-based analysis. As far as we know, we are the first to test correlations between social media engagement signals and web traffic counts, by combining Twitter and Facebook posts with web traffic data.
\section{Insights from 68 million web visits}
\label{section:result}
In this section, we take a multi-pronged approach to analyze our one-month web visit data along multiple dimensions. To better understand how people consume low quality news, we start by visualizing when people visit the site and from what type of device. We then analyze referrer links to understand which sites bring users to~\textit{TGP }. We also leverage geo-spatial information to validate if people who visit~\textit{TGP } come from areas that voted more favorably for Donald Trump in the 2020 Presidential Election. Finally, we apply topic clustering techniques to quantify what topics are discussed, and what topics are more likely to go viral.
\subsection*{Finding 1: The majority of users visit the site during the day on mobile devices.}
Our data collection contains 68,268,818 unique visits, from February 3, 2021 to March 3, 2021. Figure~\ref{fig:visit-per-hour} plots the number of visits per hour. Since more than 95\% of the visits come from the United States, we see a regular and circadian pattern where the traffic increases during the day, and decreases during the night. The daily peak hourly visit is around 200,000. The only exception is one hour in February 13, 2021, with a recorded visit of nearly 300,000. February 13, 2021 is the day Donald Trump was acquitted on impeachment charges. After checking the data set, we find that the two most visited articles published that day are both about impeachment charges.\footnote{The two articles are \url{https://www.thegatewaypundit.com/2021/02/breaking-senate-votes-57-43-acquit-donald-trump-seven-republicans-voted-convict/} and \url{https://www.thegatewaypundit.com/2021/02/breaking-trump-releases-statement-following-impeachment-acquittal/}}
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/total_visit_per_hour.png}
\caption{Number of visits per hour, from February 4, 2021 to March 3, 2021. There is a peak on February 13, 2021, the day Donald Trump was acquitted on impeachment charges. Our data shows that the two most visited articles during that day both covered this event.}
\label{fig:visit-per-hour}
\end{figure}
To understand how people visit~\textit{TGP}, we look at the operating systems (OS) column as it reveals what device people use. According to Figure~\ref{fig:top-10-os}, more than 80\% of visits come from mobile operating systems including IPhone and Android devices. If this finding holds for other low quality news sources, it suggests that research that mostly focuses on desktop users may miss a large proportion of the population that visits low quality news sources~\cite{misinformation-in-action}.
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/top-10_os.png}
\caption{Most used operating systems among~\textit{TGP } visitors. 80\% of visits come from mobile devices.}
\label{fig:top-10-os}
\end{figure}
\subsection*{Finding 2: Search engines and social media sites are the main drivers of traffic to \textit{TGP}}
Knowing what websites bring people to~\textit{TGP } helps us to identify the source of traffic and to design intervention strategies to slow down the spread of fake news. To reconstruct traffic flows, we use the \textit{referrer} column in our web traffic data. When a browser navigates to URL $B$ from URL $A$, it usually includes a string called referrer in the HTTP request (in our example $A$ is the referrer of $B$). Among 68,268,818 visits, 35,296,042 (52\%) have referrers. For visits that do not have referrers, either users visit a URL directly, or the browser strips the referrer, which can happen when certain privacy-enhancing features are turned on~\cite{mozilla-referrer}. To aggregate referrers that belong to the same site, we normalize each referrer URL to its domain name, removing hostname, path, and other query parameters. We consider two referral behaviors based on the destination URL: sites that bring users to the home page, and sites that bring users to an article page. A \textbf{home page} URL points to domain \textit{thegatewaypundit.com}, while an \textbf{article page} URL has the form \textit{thegatewaypundit.com\textbackslash ARTICLE}. Each type of traffic flow has its own characteristics, which we analyze separately.\\
\noindent
\textbf{Websites that bring users to the home page.}
Figure~\ref{fig:referrer-homapage} shows the top 15 domains that bring visitors to \textit{TGP } home page. Three major search engines (Google, Duckduckgo and Bing) account for 88.5\% of external referral traffic. Among them,~\textit{Google.com} is the top driver of home page traffic (66\%). The anonymous search engine~\textit{duckduckgo.com} is the fourth (13\%), and the Microsoft-developed \textit{bing.com} ranked the fifth (9\%).\footnote{This finding is likely indicative of the role that search engines play in people's quest for political information generally, although it would be interesting to see if these percentages were similar for a mainstream news publication such as \textit{The New York Times}.}
Second to Google are internal~\textit{TGP} article pages. This shows when people browse articles on~\textit{TGP}, they usually navigate back to the home page from different article pages. The third referrer is~\textit{TGP } home page. This is likely caused by people clicking links to the home page when they are already at the home page. Further down the list are far-right and conservative news sites such as \textit{drudgereport.com}, \textit{63red.com} and \textit{protrumpnews.com}.
We also identify referrers from suspected phishing domains. One such domain is \textit{\url{netlix.com}}, ranked number ten. The domain name used to be a center of a lawsuit. According to a legal complaint filed by Netflix in 2009, the video streaming company claims that the domain name ``netlix'' looks too similar to ``netflix,'' and requests \textit{\url{netlix.com}} to be transferred to Netflix.~\footnote{\url{https://www.adrforum.com/domaindecisions/1287043.htm}} The court rejected the order, and~\textit{\url{netlix.com}} still belongs to its original owner. As of September 14, 2021, the website does not host actual content, but automatically redirects users to~\textit{TGP}. We do not know what motivates the owner of \textit{\url{netlix.com}} to redirect visitors to~\textit{TGP}. We will keep monitoring the site as previous research demonstrates that URL redirection is a common technique to distribute unwanted or malicious software~\cite{url-redirection-campaign}.\\
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/top15_referrer_homepage.png}
\caption{Top 15 domains that bring users to the home page.}
\label{fig:referrer-homapage}
\end{figure}
\noindent
\textbf{Websites that bring users to an article page.}
Figure~\ref{fig:referrer-article} shows the top 15 domains that bring visitors to an article page. The top two referrers -- home page and article page -- are both internal traffic. This indicates that (a) most users first land on the home page before clicking an individual article, (b) some users click a new article page while browsing an existing article page, since different articles are interlinked together. When we exclude referrers from~\url{thegatewaypundit.com}, we can classify the rest of sites into two groups:
\begin{enumerate}[leftmargin=*]
\item \textbf{Social media platforms} including Twitter, Facebook, and emerging platforms such as Telegram and Gab. Together they account for 42\% of external referral traffic.
\item \textbf{Conservative news sites} such as \textit{protrumpnews.com}, \textit{thelibertydaily.com}, \textit{populist.press} and \textit{whatfinger.com}. Those sites repost articles from~\textit{TGP } on a regular basis.
\end{enumerate}
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/top15_referrer_article.png}
\caption{Top 15 domains that bring users to an article page. The top two referrers are~\textit{TGP } home page and~\textit{TGP } article pages. Together they account for more than 80\% of referral traffic. We drop those two domains for better visualization.}
\label{fig:referrer-article}
\end{figure}
To further understand how much role each social media platform plays in driving the traffic, we plot the daily number of visits with referrers from four different social media platforms, shown in Figure~\ref{fig:daily-referrer-platform}. The overall trend shows that Twitter and Facebook drive more traffic than Telegram and Gab. Daily traffic volume fluctuates and can be affected by external events. For example, Jim Hoft, founder of~\textit{TGP}, was suspended by Twitter on Februrary 6, 2021.~\footnote{\url{https://www.forbes.com/sites/ajdellinger/2021/02/06/twitter-suspends-gateway-pundit-jim-hoft/?sh=761c0cff3653}} The suspension is likely related to the decline of traffic from Twitter on that day. Other than several peaks in late February, the volume of traffic from Twitter and Facebook has continued to decline. This finding suggests that suspending social media accounts that spread low quality URLs may indeed be an effective way to reduce the spread of misinformation.
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/daily_referrer_by_platform.png}
\caption{Daily referrers by platform. Facebook and Twitter are the top two traffic drivers, although their referral traffic volumes started to decline since February 6, 2021, when Jim Hoft, founder of~\textit{TGP }, was suspended by Twitter.}
\label{fig:daily-referrer-platform}
\end{figure}
\subsection*{Finding 3: Visitors to the site are more likely to be from areas that voted for Donald Trump during the 2020 presidential election}
Our web traffic data records IP and city-level geo-location label for every request. To better understand what types of audiences visit~\textit{TGP}, and the audiences' political preferences, we leverage the geo-location information to answer two key questions: given the fact that articles published on~\textit{TGP } are pro-Trump, pro-Republican Party, and often related to the 2020 US election~\cite{fakenews-2020-election}, (1) is~\textit{TGP } more popular in counties that voted Trump? and (2) is~\textit{TGP } more popular in Republican states, Democratic states, or Swing states?
We assume that each unique IP address is one unique visitor, and each visitor is a voter during the 2020 US Presidential Election. In reality, our assumption might not always be true. For example, multiple people in a household can share the same IP, or one person can visit the site from multiple IP addresses, or the person who visits the site is not eligible to vote. Even though those limitations exist, IP address is the most accurate proxy to real human traffic in our dataset. IP is also commonly used in security research to generate threat intelligence from traffic logs~\cite{microsoft-measurement, cisco-umbrella}. Additionally, we also filter out cities that have fewer than 1,000 unique visitors, because those cities are too small to allow a safe margin of error. For example, an IP address that belongs to a small city might get erroneously assigned to a neighboring city. After the filtering our data set has 596 US cities.\\
\noindent
\textbf{Question 1: Is~\textit{TGP } more popular in counties that voted Trump?}
To answer this question, we first collect county-level 2020 US election results, including the total number of voters, number of voters who voted for Trump, and number of voters who voted for Biden. Then in our web traffic data set, we group number of visits per city into number of visits per county. Finally for each county, we calculate two metrics:
\begin{align*}
\text{\% voters who visited GP} (y) &= \frac{\text{unique number of IP}}{\text{total number of voters}} \\
\text{\% voters who voted for Trump} (x) &= \frac{\text{\# voters who voted Trump}}{\text{total number of voters}}
\end{align*}
Figure~\ref{fig:county-visit-vs-vote} shows the scatter plot of $x$ and $y$. The red line is the expected value of $y$ given $x$, based on a linear regression model. The r-squared value is 0.17 and the slope is 0.037, which indicates a positive correlation between \% voters who visited~\textit{TGP } and \% voters who voted for Trump. Thus we do in fact find that~\textit{TGP } is more popular in counties that voted Trump. This finding is consistent with a 2016 study that shows people from counties that voted for Trump are more likely to visit fake news sites~\cite{microsoft-measurement}.
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/county_visitor_vs_voter_updated.png}
\caption{Scatter plot of city-level \% voters who voted for Trump (x-axis) in the 2020 US Election versus \% voters who visited~\textit{TGP } (y-axis). The red line shows the expected value of $y$ given $x$, based on a linear regression model. The model has a slope of 0.037 and a r-squared value of 0.17, which means that $x$ and $y$ are positively correlated.}
\label{fig:county-visit-vs-vote}
\end{figure}
\noindent\\
\textbf{Question 2: Is~\textit{TGP } more popular in Republican states, Democratic states, or Swing states?}
On a state level, is there any correlation between a state's voting preference and the number of visits to~\textit{TGP}? To answer this question, we first assign each state to one of three categories, based on the voting result during the 2020 US Election:~\footnote{We use the classification from \url{https://swingleft.org/p/super-states}}
\begin{enumerate}[leftmargin=*]
\item Republican. Most voters in this state tend to vote for Republican candidates. 18 states fall into this category.
\item Democratic. Most voters in this state tend to vote for Democratic candidates. 20 states fall into this category.
\item Swing States. 12 states fall into this category: Pennsylvania, Michigan, Arizona, Wisconsin, Georgia, Colorado, Texas, Florida, Ohio, North Carolina, Iowa, and Maine.
\end{enumerate}
We aggregate city-level counts into state-level counts. For each state, we calculate total and per-capita number of visitors. Per-capita number of visitors is defined as the total number of visitors divided by the population of the state. We then aggregate state-level counts based on voting preference, and calculate the mean and median number of visitors in Swing, Republican and Democratic states. Table~\ref{table:state-level-swing-rep-dem} shows the results.
Interestingly, we find that~\textit{TGP } is more popular in Swing states and Republican states than in Democratic states. We perform pairwise t-test and find that the difference is statistically significant when using the per-capita count, but is not significant when using the absolute count. To understand why more people from Swing states visit the site, in the next section we analyze all articles published on the site during our data collection period. We show that most popular articles frequently mention topics related to Swing states, including ``2020 US Election fraud'', ``missing ballot'', or ``voting irregularity'', all of which are unverified or false claims. Those stories have more direct impact on people from Swing states than those from Republican or Democratic states.
\begin{table}
\begin{tabular}{p{0.39\linewidth}p{0.31\linewidth}p{0.22\linewidth}}
\toprule
voting preference & Mean number of visits (total,per-capita) & Median number of visits \\
\midrule
Swing (12 states) & 4171, 0.025 & 2122, 0.020 \\
Republican (18 states)& 3116, 0.022 & 1900, 0.019 \\
Democratic (20 states) & 3096, 0.017 & 1652, 0.015 \\
\bottomrule
\end{tabular}
\caption{We group 50 states into three categories based on a state's voting preference. On average, more visitors come from Swing and Republican states than from Democratic states.}
\label{table:state-level-swing-rep-dem}
\end{table}
\subsubsection*{Visualizing hotspots.}
To better understand where people visit \textit{TGP}, we visualize cities with a high concentration of visitors on two maps. We separate visits into two categories: those coming outside of the United States, and those coming from the United States. Figure~\ref{fig:city-non-us},~\ref{fig:city-us} show top US and non-US cities. The radius of each dot is in proportion to the percentage of city population that visited~\textit{TGP } within a month. Most non-US visits come from cities in Canada, Australia, New Zealand and Israel. For visits within the United States, some come from metropolitan areas such as Denver, Houston and Chicago. Others come from cities within Swing or Republican states such as Florida, Texas and Arizona.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\linewidth]{image/map_city_non_us.png}
\caption{Top non-US cities based on \% visitors that visited \textit{TGP}. The larger a circle, the higher the percentage. Most non-US hotspots are located in Canada, Australia and Israel.}
\label{fig:city-non-us}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\linewidth]{image/map_city_us.png}
\caption{Top US cities based on \% visitors that visited~\textit{TGP }. The larger a circle, the higher the percentage. Some cities belong to large metropolitan areas such as Denver, Houston and Chicago. There are also clusters of cities from Swing or Republican states such as Florida, Texas and Arizona.}
\label{fig:city-us}
\end{figure}
\subsection*{Finding 4: Topics related to ``election fraud'' receive more clicks and remain popular on the site for a longer period of time than other topics}
\label{subsection:topic-analysis}
During the one-month period of our study,~\textit{TGP } published 1070 articles. Some stories go viral, others do not. What topics are discussed? What makes one topic goes viral? Is virality associated with the ``fakeness'' of the story? To better understand those connections, we use topic clustering technique to group~\textit{TGP } articles into ten distinct topics. We then design two metrics to quantify the popularity of an article: number of unique visits (volume based), and number of minutes it takes to receive 50\%/90\%/95\% of all visits (time based). We first show the distribution of those two metrics over all articles, and then aggregate metrics into topics to identify viral content.\\
\noindent
\textbf{What topics are discussed?}
Each article published on~\textit{TGP } comes with a one-sentence title with references to key names and events. For example, one article published on February 18, 2021 is titled ``\textit{Maricopa County Audits Are Proving to Be a Waste of Time and Money, They Were Never Created to Identify the Suspected Election Fraud in the County}.'' Given the rich information from the title, we use non-negative matrix factorization (NMF) to cluster 1070 article titles into different topics. NMF is an unsupervised algorithm to extract topics from text corpus. In our case, the input to NMF is an article-word matrix, where each entry is the tf-idf weight of a word in an article. NMF then factorizes this matrix into a word-topic matrix, and a topic-article matrix. The number of topic is a user-defined parameter. After experimenting with different values, we set the parameter as 10, because the resulting topics are coherent and distinct among each other. Table~\ref{table:topic-cluster} shows keywords associated with each topic.\\
\begin{table}
\begin{tabular}{p{0.1\linewidth}p{0.9\linewidth}}
\toprule
topic & keywords \\
\midrule
1 & \textit{president donald trump acquittal } \\
2 & \textit{joe biden kamala harris} \\
3 & \textit{2020 election fraud voter integrity} \\
4 &\textit{marjorie taylor greene, liz cheney} \\
5 &\textit{capitol riot antifa police fbi} \\
6 &\textit{governor andrew cuomo new york} \\
7 & \textit{democrat impeachment trial} \\
8 & \textit{maricopa arizona county ballot shredded dumpster} \\
9 &\textit{covid 19 vaccine virus cdc} \\
10 &\textit{dominion voting machine} \\
\bottomrule
\end{tabular}
\caption{Keywords associated with each topic. We use non-negative matrix factorization to cluster 1070 articles into 10 topics.}
\label{table:topic-cluster}
\end{table}
\noindent
\textbf{What topics receives more visits?} We first use the number of unique visit to measure article virality. Each unique IP address counts as one unique visit. Figure~\ref{fig:hist-article-visit} shows the histogram of number of unique visits per article. Overall, the average number of visits per article is 32,488; the median number of visits is 22,434. The distribution is skewed to the right, suggesting that some articles have very high number of unique visits.
\begin{figure}[!h]
\centering
\includegraphics[width=0.7\linewidth]{image/hist_page_visit_unique_ip.png}
\caption{Histogram of unique visits per article. We use IP address as a proxy for visitor. The mean number of visits is 32,488, and the median is 22,434. This discrepancy suggests that some articles receive very high number of visits.}
\label{fig:hist-article-visit}
\end{figure}
We then aggregate article-level number of visits into topic-level. Figure~\ref{fig:topic-distribution} shows the mean and median number of visits per topic. The most visited topic is \#3, which according to Table~\ref{table:topic-cluster} is related to ``2020 US election fraud'', an unverified claim pushed by far-right news media. The second most visited topic is \#8, which covers ``voting irregularity and ballot counting in Maricopa county'', another unfounded claim. The popularity of those topics indicate that readers of \textit{TGP} had a huge appetite for articles about electoral fraud stories. The fact that those articles are published and remain popular three month after the US 2020 election shows that this type of misinformation can have a long-lasting effect on readers, and that misinformation does not have to cover real-time topics to remain popular.\\
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/topic-distribution.png}
\caption{Mean and median number of unique visits per topic. The most visited topics are both related to the US 2020 election. Topic \#3 is related to election fraud, and topic \#8 is related to maricopa county ballot. Both claims are unverified conspiracy theories.}
\label{fig:topic-distribution}
\end{figure}
\noindent
\textbf{Do viral topics last longer?} To quantify the popularity of an article in the time dimension, we measure how long it takes an article to receive 50\% ($t_1$), 90\% ($t_2$), and 95\% ($t_3$) of all visits, since the publication of the article. Figure~\ref{fig:hist-article-visit-length} shows that in median values, 50\% of visits come from the first 237 minutes (4 hours); 90\% of visits come from the first 1,177 minutes (20 hours), and 95\% of visits come from the first 1,634 minutes (28 hours). In general, it is rare to have an article that stays viral for more than a day.
\begin{figure}[!h]
\centering
\includegraphics[width=0.8\linewidth]{image/page_visit_hist_50.png}
\includegraphics[width=0.8\linewidth]{image/page_visit_hist_90.png}
\includegraphics[width=0.8\linewidth]{image/page_visit_hist_95.png}
\caption{Histogram of number of minutes to reach 50\%, 90\% and 95\% of total unique visits. On average, an article receives 50\% of all visit traffic within 4 hours (280 minutes) of publication, and receives 90\% of all visit traffic within 20 hours (1,200 minutes) of publication.}
\label{fig:hist-article-visit-length}
\end{figure}
We then aggregate article-level count to topic-level count. Figure~\ref{fig:topic-distribution-length} shows the median $t_2$ and median $t_3$ for each topic. Topics that trend longer include \#3 (``US election fraud''), \#8 (``Arizona county ballot''), and \#10 (``Dominion voting machine''). Topics that trend shorter include \#1 (``President Donald Trump acquittal'') and \#7 (``Democrat Impeachment Trial''). The longest-trending topics are also the most visited topics. This suggests that viral topics are not only read by more people, but also last for a longer period of time. In general, topics related to conspiracy theories are more popular, while topics that state a known fact are less viral.
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{image/topic_read_length_90_95.png}
\caption{Median number of minutes to reach 90\% and 95\% of total visits per topic. Topics that trend longer include \#3 (US election fraud), \#8 (Arizona county ballot), and \#10 (Dominion voting machine). Topics that trend relatively shorter include \#1 (president donald trump acquittal) and \#7 (democrat impeachment trail). In general, conspiratorial topics last longer, while topics that report a known fact do not last as long.}
\label{fig:topic-distribution-length}
\end{figure}
| {'timestamp': '2022-01-13T02:05:36', 'yymm': '2201', 'arxiv_id': '2201.04226', 'language': 'en', 'url': 'https://arxiv.org/abs/2201.04226'} |
\section{Introduction}
Temporal action detection, which aims at locating specified actions from untrimmed videos in the temporal dimension, has been widely applied to various tasks such as video understanding~\cite{sun2015temporal,wang2018rgb}, surveillance~\cite{sultani2018real}, \etc.
Similar to object detection, the pipeline of temporal action detection is usually divided into two stages: temporal action proposal generation (TAPG) and action classification.
The performance of such two-stage temporal action detectors is largely determined by the proposal quality from TAPG.
\begin{figure}[t]
\begin{subfigure}{.5\textwidth}
\includegraphics[width=0.9\linewidth]{figure1.png}
\centering
\caption{BMN+Actionness predictor. }
\label{figure1:a}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\includegraphics[width=0.9\linewidth]{figure2.png}
\centering
\caption{Global context capture. }
\label{figure1:b}
\end{subfigure}
\caption{Benefit of using long-range dependencies.
We apply BMN~\cite{lin2019bmn} and our method to detect the specified action \textit{cricket match} in a video segment, which also includes an audience cheering segment highlighted in red dotted boxes.
We plot the curves of snippet-wise actionness scores for predictions by BMN without using global context in (a) and our method using global context in (b).
It can be seen the one with global context shows higher confidence scores for the frames of audience cheering, demonstrating the benefits brought by global context for temporal action proposal generation.}
\label{figure1}
\end{figure}
Prior works focus on exploiting the local temporal context information~\cite{chao2018rethinking,lin2019bmn,chao2018rethinking,xu2020g} for generating high quality proposals.
For example,~\cite{chao2018rethinking,lin2019bmn} apply a pre-defined ratio to extend the temporal action boundaries, and use temporal convolutional layers or dilated convolutional layers to increase the receptive field;~\cite{xu2020g} leverages a GCN model to aggregate local temporal and semantic context.
Using local context information can effectively build relations among neighbors of current snippets, which works well for locating simple action instances with clean frames and clear boundaries, but is sensitive to action/background representation.
However, the real-world cases are often complicated, with many irrelevant frames and background clutters over the interested actions.
Only exploiting the local temporal context lacks the semantic understanding for the whole video, often leading to failures of locating precise boundaries.
As shown in Figure \ref{figure1:a}, an actionness predictor is added into a recent method BMN~\cite{lin2019bmn} to show its performance on a complicated video.
We can see that, if we only leverage local temporal context for detecting the specified action (i.e. \textit{cricket match}), the long audience cheering segment embedded in frames of the specified action obviously gives low action confidence scores.
But if a longer range is observed (in Figure \ref{figure1:b}), the model is able to understand that the audience cheering is also part of the match, and thus treating this segment and the actions before and after as a whole can avoid wrong boundary prediction.
From this example,
we claim that long-range temporal dependencies should also be exploited for more comprehensive video understanding in complicated situations.
In this work, we propose an augmented transformer with adaptive graph network (ATAG) to take advantage of the two types of contexts for effective temporal proposal generation. It includes an augmented transformer which
mines long-range temporal context for noisy action instance localization, and an adaptive GCN which captures local temporal context.
Specifically,
to locate noisy action instance, one way is to adequately understand the holistic video and regard the noisy frames as a part of action according to the semantics. The transformer is adopted to achieve this purpose, which has shown superior capability of capturing long-term dependencies~\cite{Girdhar_2019_CVPR,han2020mm}.
However, the vanilla transformer has two issues if directly applied in our task. First, the loss functions in the traditional TAPG methods only supervise proposal-level signal and cannot directly guide the transformer to aggregate long-range features to snippet-level features.
Therefore, we apply a snippet actionness loss to binarily classify each snippet into action or background according to the snippet-level features output by the transformer.
The snippets, especially for those noisy snippets, need to extract helpful information from other snippets to be correctly classified, thus the snippet actionness loss explicitly forces the transformer to selectively learn long-range dependencies.
On the other hand, in the vanilla transformer, when generating the queries and keys, the receptive field is constraint to the snippet itself and it is not a robust feature learning style for these noisy snippets. To deal with this problem, we introduce a front block on top of vanilla transformer. The front block is a convolution-based lightweight network, which expands the temporal receptive field and filters out noisy frame features.
For the local temporal context capture, we focus on exploiting the position and gradient/difference information between adjacent features.
We propose an adaptive GCN to build the local temporal context,
where two adjacency matrices are carefully designed. One matrix has all the elements generated during training. It is similar to conventional convolutional layers but different positions correspond to different kernels, which represents the common pattern for all the training data and position information. It can adaptively determine whether the farther snippets should have smaller weights than the central one, or whether the snippets near the edge of the video should be assigned with smaller weights.
The other is a content-based adjacency matrix, which captures the difference between node pairs and represents the unique pattern for each data.
By combining the above two adjacency matrices, our data-driven graph increases the flexibility for graph construction and brings more generality to build local context relationships in various samples.
We demonstrate the effectiveness of ATAG on two challenging datasets, THUMOS14 and ActivityNet13. Our model outperforms well established state-of-the-art methods significantly.
\begin{figure*}
\includegraphics[width=1.\linewidth]{structure.png}
\centering
\caption{Illustration on the architecture of our method. It first employs a two-stream network to extract snippet-level encoded features. Then, it deploys an augmented transformer and an adaptive graph to capture global and local context, respectively. After fusing global and local features, an output module generates the desired predictions.}
\label{figure:structure}
\end{figure*}
\section{Related work}
\paragraph{Attention mechanism for long-range dependencies.}
Long-range attention mechanism is to compute the response at a position in a sequence by accessing all positions and taking their weighted average in the embedding space.
It is broadly leveraged in natural language processing (NLP) and computer vision.
Vaswani et al.~\cite{vaswani2017attention}
introduce a self-attention mechanism, called transformer, capturing long-range dependencies among words in one sentence to address the machine translation task.
Following~\cite{vaswani2017attention}, many transformer-based models are proposed and show great potential to tackle various tasks.
Devlin et al.~\cite{devlin2018bert} apply the bidirectional training of Transformer and successfully deal with a broad set of NLP tasks.
Girdhar et al.~\cite{Girdhar_2019_CVPR} utilize transformer to aggregate features from the spatiotemporal context for recognizing and localizing human actions.
Carion et al.~\cite{carion2020endtoend} employ transformers to build an end-to-end object detection model.
Besides transformers, there are some other manners to utilize the attention mechanism to capture long-range dependencies.
For example, Wang et al.~\cite{Wang_2018_CVPR} embed non-local structure into the action recognition network to capture long-range dependencies.
Wu et al.~\cite{Wu_2019_CVPR} introduce long-term feature banks to analyze videos.
How to effectively exploit temporal context is a pivotal problem in video analysis and long-range dependencies has been applied widely~\cite{Girdhar_2019_CVPR,Wang_2018_CVPR,Wu_2019_CVPR,kozlov2020lightweight}.
However, the long-range dependencies for TAPG has not been well explored, especially in the complicated noisy scenarios.
\paragraph{Temporal action proposal generation (TAPG).}
Early TAPG methods use temporal sliding window~\cite{oneata2014lear,wang2014action} and snippet-wise probability~\cite{shou2016temporal,zhao2017temporal} to generate candidate proposals.
The former fails to handle ground truth action instances with various durations, while the latter is sensitive to noise and can hardly handle actions with long duration.
Recently, most methods adopt multi-scale anchors to generate proposals~\cite{chao2018rethinking,lin2018bsn,Liu_2019_CVPR,lin2019bmn,xu2020g,zhao2020bottom,bai2020boundary}, similar with the idea in anchor-based object detection~\cite{ren2015faster}.
Chao et al.~\cite{chao2018rethinking} adopt a multi-tower architecture to align the receptive field and extend proposal boundaries by a pre-defined ratio which is widely applied by later methods.
In BSN~\cite{lin2018bsn}, Lin et al. directly predict the probability of boundaries and cover flexible temporal duration of action instances.
Based on BSN, Lin et al.~\cite{lin2019bmn} further propose a boundary-matching mechanism to evaluate the completeness of densely distributed proposals.
The ideas of GCN have been adopted in some work,
Xu et al.~\cite{xu2020g} extract temporal context and semantic context by GCN, but applies conventional temporal convolutional layers for actual implementation in temporal context extraction.
Bai et al.~\cite{bai2020boundary} perform a GCN to build the relationship between the boundaries and the content.
Therefore, they do not apply GCN to capture local temporal context.
Above methods have made significant progress in this field. However, they have limited capability of building local dependencies from the local context, leading to degraded performance for complicated cases.
Our proposed ATAG has the ability to build both long-range and local temporal dependencies to achieve better video understanding.
\section{Method}
The overall architecture of the proposed augmented transformer with adaptive graph network (ATAG) is shown in Fig.~\ref{figure:structure}.
In this section, we first describe the problem formulation and backbone feature extraction in Sec.~\ref{ssec:formulation}.
The core augmented transformer with adaptive graph architecture is then introduced in Sec.~\ref{ssec:transformer}, including augmented transformer and adaptive GCN as well, followed by the output module (Sec.~\ref{ssec:output}) used to generate final prediction, and the training~(Sec.~\ref{ssec:training}) and inference~(Sec.~\ref{ssec:inference}) stage.
\subsection{Problem formulation}
\label{ssec:formulation}
We denote an untrimmed video $\mathcal{X}$ as a frame sequence $\mathcal{X} = \{x_n\}^{l}_{n=1}$ with $l$ frames, where $x_n$ is its
$n$-th RGB frame; and the temporal annotations for the $N_g$ action instances
included in $\mathcal{X}$ are denoted as $\psi_g = \{\varphi_n\}^{N_g}_{n=1}$ with $\varphi_n = (t_{s,n}, t_{e,n})$. Here $t_{s,n}$ and $t_{e,n}$ denote the start and end time index of the action instance $\varphi_n$ respectively.
The target of TAPG is to generate temporal proposals each covering a ground truth action instance as accurate as possible. Unlike the temporal action detection task, here the action instance categories need not be considered.
Following previous TAPG methods~\cite{lin2019bmn,xu2020g,bai2020boundary}, we adopt a pre-trained encoding model to extract features of the input video $\mathcal{X}$, which serve as inputs to our developed TAPG model.
In particular,
for any given video $\mathcal{X}$, we split it into $T$ continuous and non-overlapped snippets.
We then adopt a two-stream network~\cite{wang2016temporal} for extracting $T$ snippet features of $C$-dimension, which are then concatenated to form the video features $F \in \mathbb{R}^{T\times C}$ and fed into the our model for proposal generation.
\subsection{Augmented transformer with adaptive graph architecture.}
\label{ssec:transformer}
ATAG introduces a dual-path module, consisting of an augmented transformer and adaptive GCN for global and local temporal context capture, respectively.
The input feature $F$ is split into two parts along the channel dimension, denoted as $F^g$ and $F^l$, which are then fed into the two modules.
\paragraph{Augmented transformer.}
We apply transformers to improve the semantic representation of snippet-level features by capturing long-range dependencies.
A vanilla transformer block is composed of a self-attention layer and position-wise feed-forward layers, performing multi-head attention to compute a weighted sum of input snippet features $F^g\in \mathbb{R}^{T\times C}$, resulting an augmented feature $\tilde{F}^g$ with global context.
In particular, for each attention head, the attention map is computed by matching the transformed input features (a.k.a. the queries) $Q=f(F^g)$ to
another transformation of the input features (a.k.a. the keys) $K=g(F^g)$, with $f$ and $g$ being learnable linear transformation.
Namely,
\begin{equation}
\label{eqn_1}
A = Attention(F) = \mathrm{softmax}(\frac{QK^\top }{\sqrt{d}}),
\end{equation}
where $A\in \mathbb{R}^{T\times T}$ is the generated attention map, and $d$ is the dimension for $Q$ and $K$.
The above attention map computation can find the relationship between features $Q$ and $K$ of global snippets and then aggregate their information into another linear transformation of input features $V=h(F^g)$ (a.k.a. the values).
The multi-head attention is followed by an FFN layer, which contains
two linear layers with ReLU activation and a residual connection after each layer. We also include layer normalization and dropout to facilitate training.
Directly employing the vanilla transformer presents two issues here. First, the main task of TAPG is the boundary regression, and the corresponding loss functions supervise proposal-level signal and cannot guide the transformer to effectively learn long-term semantic relations for each snippet.
Second, according to Eq.~\ref{eqn_1}, the computation of any element $A_{mn}$ in the attention map matrix $A$ depends on only the features of snippet m and n, namely, $F_m^g$ and $F_n^g$, which indicates that the attention map generation does not consider any temporal context, especially in limit transformer layers.
To solve the above issues, we add a snippet actionness loss and a front block to the vanilla transformer and it is dubbed augmented transformer.
The snippet actionness loss explicitly guides the transformer to learn effective long-range dependencies at snippet level.
An actionness predictor is equipped upon the FFN, and it is used to predict the probability of an action instance existing in the input snippet, by minimizing the following binary classification loss $L_{a}$ over action/background categories:
\begin{equation}
L_{a}=\sum^T_{i=1}\alpha^+\cdot g_i^a\cdot \log p^a_i+\alpha^-\cdot(1-g_i^a)\cdot \log(1-p_i^a),
\end{equation}
where $\alpha^+={T}/\sum(g_i^a)$, $\alpha^-={T}/{\sum(1-g_i^a)}$, and $p^a_i$ and $g_i^a$ are the actionness predicted probability and action/background label of snippet $i$.
This loss function supervises the network to obtain snippet-wise classification only depending on each snippet feature from the output of transformer, which is critical to achieve noisy action instances localization. For those noisy snippets that have action labels, the network shall build correct relations among the noisy snippets and others to make correct decisions. Thus, this loss function helps the transformer learn how to selectively capture long-range dependencies and understand the complicated noisy action instances.
The front block is a light-weight network consisting of three parts to expand the temporal receptive field. We first apply a gated linear unit~\cite{dauphin2017language}, followed by a parallel 1 $\times$ 1 convolutional layer and 3 $\times$ 1 average pooling layer with stride 1 to expand the receptive field, and the small 3 $\times$ 1 average pooling layer can also smooth the snippet-level feature to filter out the tiny noisy frames.
The last part is a convolutional layer with a large kernel size, for example, 7 $\times$ 1. To avoid over-fitting from large size kernels, we adopt the depth-wise separable convolution. We apply residual connection for each part and layer normalization behind each part. The structure of front block is depicted in the top-right corner in Fig.~\ref{figure:structure}.
\paragraph{Adaptive graph convolutional layer.}
To capture the local context, we design a new graph convolution layer to construct the local branch. We build a video graph $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, where $\mathcal{V}=\{v_i\}_{i=1}^T$ and $\mathcal{E}$ represent the node and edge sets, respectively. Each node represents a snippet and each edge shows the dependency between two snippets. For local context modeling, edges between two nodes are constructed according to their temporal distance. The edge set is defined as:
\begin{equation}
\mathcal{E}=\{(v_i, v_{i+k})|i \in \{1, ..., T\}; k \in \{0, \pm{1},...,\pm{\delta}\}\},
\end{equation}
where $T$ is the number of snippets, and $\delta$ is defined as the maximum connection distance.
Our graph convolution operation $\mathcal{F}$ is defined as:
\begin{equation}
\label{eq2}
\mathcal{F}(F^l) = W\cdot F^l\cdot (A_a+A_d)\odot{M},
\end{equation}
where $W\in{\mathbb{R}^{C_{\mathrm{out}}\times C_{\mathrm{in}}}}$ includes trainable weights, $M$ is a fixed $T\times T$ binary mask matrix to limit the connection range according to $\delta$, and $\odot$ is the dot product.
Instead of using a pre-defined adjacency matrix,
the matrix in Eq. \ref{eq2} is divided into two parts: $A_a$ and $A_d$ and treated differently.
The first part ($A_a$) is completely parameterized and optimized in the training phase. The elements in $A_a$ can be arbitrary values without any constraints, which means that the edges of the graph are completely learned based on the training data and the positions of nodes in the video.
The second part ($A_d$) is a data-dependent graph which adaptively learns a unique graph for each video. Our GCN is used to capture local temporal context, therefore we pay more attention to the difference of features. To determine whether there is a connection between two nodes and how strong the connection is, the difference of two nodes $m$ and $n$ is computed as:
\begin{equation}
A_{d, mn} = \frac{exp(\sigma({\theta}^T|f_m-f_n|))}{\sum_{i=1}^N exp(\sigma({\theta}^T|f_m-fi|))},
\end{equation}
where $\sigma$ is an activation function, ${\theta}^T$ is the trainable parameter vector to reduce the dimension of $f_m-f_n$ to $1$. The detailed visual description is shown in Figure~\ref{figure:structure}.
\paragraph{Global-local fusion.}
Early fusion and late fusion are widely used in two-stream feature fusion. In our method, we use the early fusion, \ie, the global and local features are fused by concatenation before the output module.
\subsection{Output module}
\label{ssec:output}
After obtaining global and local features,
we feed them into our output module to generate the proposal boundaries and completeness scores, which are applied for proposal generation during inference.
\paragraph{Temporal boundary classification.}
Due to our global-local combination mechanism, the two-branch features
have precise and discriminative action/background representation.
Here, we input them into two same convolutional networks consisting of two temporal convolutional layers and a sigmoid activation function to generate the probability of the start $p^s$ and end $p^e$ for each snippet, respectively.
\paragraph{Completeness regression.} Besides the prediction of boundaries, the network is trained to predict the completeness of proposals to boost the final results. Our completeness regression network follows ~\cite{lin2019bmn} and also introduces the Boundary-Matching mechanism to generate completeness scores for densely distributed proposals. For each proposal, we sample $32$ features from the corresponding two-branch features. Sampled features are then input into the completeness prediction network to generate two completeness score maps $M^{cc}\in \mathbb{R}^{T\times D}$ for completeness classification, and $M^{cr}\in \mathbb{R}^{T\times D}$ for completeness regression, where $D$ is the maximum proposal duration and is set depending on the dataset.
\subsection{Training}
\label{ssec:training}
We apply weighted binary logistic regression loss function $L_{bl}$ for start and end classification losses, denoted as $L_s$ and $L_e$, where $L_{bl}$ is denoted as:
\begin{equation}
\label{bl}
L_{bl}=\sum^T_{i=1}(\alpha^+\cdot g_i\cdot \log p_i+\alpha^-\cdot(1-g_i)\cdot \log(1-p_i)),
\end{equation}
where $\alpha^+={T}/ \sum(g_i) $, $\alpha^-= {T} / {\sum(1-g_i)}$, and $p_i$ and $g_i$ are the predicted probability and ground truth of $i$-th snippet. Following BMN~\cite{lin2019bmn}, our completeness loss $L_{com}$ consists of two different loss functions, binary classification loss and regression loss:
\begin{equation}
L_{com} = L_{c}(G^c, M^{cc}) + \lambda L_r(G^c, M^{cr}),
\end{equation}
where $L_c$ is also a binary logistic regression loss function and its formula is as same as Eq. \ref{bl} and $L_r$ is $l_2$ loss, and $\lambda$ is set as $10$.
The model is trained in the form of a multi-task loss function, with the overall loss function defined as:
\begin{equation}
\label{overall loss}
L = L_{com} + \lambda_1 L_a + \lambda_2 L_s + \lambda_3 L_e,
\end{equation}
where $\lambda_1$, $\lambda_2$, and $\lambda_3$ are three scalars to balance the three terms. The method of label assignment will be described in detail in the supplementary material.
\subsection{Inference}
\label{ssec:inference}
During inference, a proposal set $\psi_p=\{\phi_n=(t_s,t_e,p^s_{t_s},p^e_{t_e},p^{cc}_{t_s,t_e},p^{cr}_{t_s,t_e})\}_{n=1}^N$ is generated, where $p^s_{t_s}$, $p^e_{t_e}$ are start and end probabilities in $t_s$ and $t_e$, and $p^{cc}_{t_s,t_e}$, $p^{cr}_{t_s,t_e}$ are completeness classification score and completeness regression score of proposal from $[t_s,t_e]$. To obtain final results, it is necessary to perform steps of score fusion and redundant proposal suppression.
\paragraph{Score fusion.}
In order to make full use of various predicted scores for each proposal $\phi_n$, we fuse its boundary probabilities and completeness scores by multiplication. The confidence score $p^f$ can be defined as:
\begin{equation}
p^f=p^s_{t_s}\cdot p^e_{t_e}\cdot p^{cc}_{t_s,t_e}\cdot p^{cr}_{t_s,t_e}.
\end{equation}
Hence, the proposals set can be re-written as $\psi_p=\{\phi_n=(t_s,t_e,p^f)\}_{n=1}^N$.
\paragraph{Redundant proposals suppression.}
Because of our dense proposal generation, it is inevitable to output numerous redundant proposals which highly overlap with each other. Thus, based on the completeness score for proposals, we apply the Soft-NMS algorithm to remove redundant proposals. Candidate proposal set $\psi_p$ turns to be $\psi_p^\prime=\{\phi_n=(t_s,t_e,p^{f\prime})\}_{n=1}^{N^\prime}$, where $p^{f\prime}$ and $N^\prime$ is the final completeness score and the number of final proposals, respectively.
\section{Experiments}
\subsection{Datasets and setup}
\paragraph{Datasets.}
The evaluation is performed on two popular TAPG benchmark datasets, THUMOS14 \cite{jiang2014thumos} and ActivityNet1.3 \cite{Heilbron_2015_CVPR}.
We use the subset of THUMOS14 that provides frame-wise action annotations. The model is trained with 200 untrimmed videos from its validation set and evaluated using 213 untrimmed videos from the test set. This dataset is challenging due to the large variations of the frequency and duration of action instances across videos.
ActivityNet1.3 is a large-scale dataset covering 200 complicated human activity classes with 19,994 untrimmed videos, which are divided into training, validation and test set in the ratio 2:1:1. The model is trained on the training set and evaluated with the validation set.
\paragraph{Implementation details.}
For feature encoding, following previous works~\cite{lin2019bmn,bai2020boundary}, we adopt two-stream network~\cite{wang2016temporal} pre-trained on training set of Activity1.3. The frame interval is set to $5$ and $16$ on THUMOS14 and Activity1.3, respectively. For ActivityNet1.3, we resize video feature sequences by linear interpolation to 100, \ie $T=100$, and set the maximum duration length as $100$. For THUMOS14, we slide the window on video feature sequence with $50\%$ overlap and $T=128$ and set the maximum duration length as $64$, which can cover $98\%$ action instances.
Adam~\cite{kingma2014adam} optimizer is adopted and the learning rate is set to $10^{-3}$ and decayed by a factor of 0.1 after every 10 epochs. To preserve the positional information in the transformer, we adopt sine positional encoding and add it to queries and keys. There are 8 heads in the multi-head attention module is $8$. $\delta = 2$ in local branch. In Eq. \ref{overall loss}, the coefficients $\lambda_1$, $\lambda_2$ and $\lambda_3$ are all set to $1$. Early fusion is adopted unless otherwise specified.
\subsection{Temporal action proposal generation}
Temporal action proposal generation aims to produce high quality proposals that have high IoU with ground truth action instances and high recall. To verify proposal quality, Average Recall (AR) under multiple IoU thresholds are calculated. For ActivityNet1.3 and THUMOS14, the IoU thresholds used are $[0.5:0.05:0.95]$ and $[0.5:0.05:1.0]$, respectively. AR under different Average Number of proposals (AN) is defined as AR@AN, and the area under the AR vs. AN curve is named AUC. We use AR@AN and AUC as our metrics to evaluate our model.
\paragraph{Comparison with state-of-the-art methods.}
Our model is compared with state-of-the-art TAPG methods on THUMOS14 and ActivityNet1.3. Table~\ref{Table 1} shows the comparison on THUMOS14. To ensure a fair comparison, both C3D and two-stream features used by previous methods are adopted in our experiments. Following previous methods~\cite{lin2018bsn,lin2019bmn,lin2020fast,bai2020boundary}, we apply NMS and soft-NMS as post-processing methods to evaluate our method, respectively. For both C3D and two-stream features, our results outperform other state-of-the-art methods by a large margin. Additionally, NMS achieves better average recall performance than soft-NMS under small proposal numbers.
The results on ActivityNet1.3 are summarized in Table~\ref{Table 2}. Our method again surpasses all of the leading methods.
These experiments demonstrate the effectiveness of our model by adequately exploiting the global and local context.
\paragraph{Visualization and analysis.}
Fig.~\ref{visualization} visualizes some representative results on ActivityNet1.3 and THUMOS14 featuring different challenging cases. The proposals with the highest $k$ scores are visualized in each video, where $k$ is the number of ground truth. In the top video, background frames and action frames have the similar scene and there are many irrelevant frames which are nearly the same as background frames in the action. However, our top-1 proposal still successfully aligns the position of ground truth action instance.
Our network is trained to regard the noisy snippets in action instance as action. We concern that whether the network will overfit on the noisy frames and many background snippets are mistakenly treated as action instance. In the middle video, a cheering segment exists between the true action and background so it needs to be classified as background, which our model predicts correctly. Note that this is different from the case in Fig.~\ref{figure1}, where the cheering segment is embedded in an action instance. The correct prediction made by our model shows that it correctly understands the semantic information and does not overfit on the noisy frames. The bottom video has multiple ground truth action instances, while our top-5 proposals perfectly cover them in an accurate way, suggesting the high quality of our generated proposals. More examples and comparison with the previous method are shown in the supplement material.
\begin{table}[t]
\begin{center}
\small
\resizebox{!}{5.6cm}{
\begin{tabular}{c|c|p{1.3em}p{1.4em}p{1.4em}p{1.4em}p{1.6em}}
\toprule
\multirow{2}{2.5em}{Feature} & \multirow{2}{3em}{Methods} & \multicolumn{5}{c}{AR@AN} \\
& & @50 & @100 & @200 & @500 & @1000 \\
\midrule
\multirow{14}{2.5em}{C3D}
& SCNN-prop \cite{shou2016temporal} & 17.22 & 26.17 & 37.01 & 51.57 & 58.20 \\
& SST \cite{Buch_2017_CVPR} & 19.90 & 28.36 & 37.90 & 51.58 & 60.27 \\
& TURN \cite{Gao_2017_ICCV} & 19.63 & 27.96 & 38.34 & 53.52 & 60.75 \\
& BSN \cite{lin2018bsn}+NMS & 27.19 & 35.38 & 43.61 & 53.77 & 59.50 \\
& BSN \cite{lin2018bsn}+SNMS & 29.58 & 37.38 & 45.55 & 54.67 & 59.48 \\
& MGG \cite{Liu_2019_CVPR} & 29.11 & 36.31 & 44.32 & 54.95 & 60.98 \\
& BMN \cite{lin2019bmn}+NMS & 29.04 & 37.72 & 46.79 & 56.07 & 60.96 \\
& BMN \cite{lin2019bmn}+SNMS & 32.73 & 40.68 & 47.86 & 56.42 & 60.44 \\
& DBG \cite{lin2020fast}+NMS & 32.55 & 41.07 & 48.83 & 57.58 & 59.55 \\
& DBG \cite{lin2020fast}+SNMS & 30.55 & 38.82 & 46.56 & 56.42 & 62.17 \\
& BC-GNN+NMS \cite{bai2020boundary} & 33.56 & 41.20 & 48.23 & 56.54 & 59.76 \\
& BC-GNN+SNMS \cite{bai2020boundary} & 33.31 & 40.93 & 48.15 & 56.62 & 60.41 \\
\cline{2-7}
& ATAG (Ours)+NMS & \bf{34.91} & \bf{42.34} & 48.64 & 57.96 & 62.03 \\
& ATAG (Ours)+SNMS & 34.47 & 41.92 & \bf{49.60} & \bf{58.49} & \bf{62.24} \\
\midrule
\multirow{14}{2.5em}{2stream}
& TAG \cite{zhao2017temporal} & 18.55 &29.00 & 39.61 & - & - \\
& TURN \cite{Gao_2017_ICCV} & 21.86 & 31.89 & 43.02 & 57.63 & 64.17 \\
& CTAP \cite{Gao_2018_ECCV} & 32.49 & 42.61 & 51.97 & - & - \\
& BSN \cite{lin2018bsn}+NMS & 35.41 & 43.55 & 52.23 & 61.35 & 65.10 \\
& BSN \cite{lin2018bsn}+SNMS & 37.46 & 46.06 & 53.21 & 60.64 & 64.52 \\
& BMN \cite{lin2019bmn}+NMS & 37.15 & 46.75 & 54.84 & 62.19 & 65.22 \\
& BMN \cite{lin2019bmn}+SNMS & 39.36 & 47.72 & 54.70 & 62.07 & 65.49 \\
& MGG \cite{Liu_2019_CVPR} & 39.93 & 47.75 & 54.65 & 61.36 & 64.06 \\
& DBG \cite{lin2020fast}+NMS & 40.89 & 49.24 & 55.76 & 61.43 & 61.95 \\
& DBG \cite{lin2020fast}+SNMS & 37.32 & 46.67 & 54.50 & 62.21 & 66.40 \\
& BC-GNN+NMS \cite{bai2020boundary} & 41.15 & 50.35 & 56.23 & 61.45 & 66.00 \\
& BC-GNN+SNMS \cite{bai2020boundary} & 40.50 & 49.60 & 56.33 & 62.80 & 66.57 \\
\cline{2-7}
& ATAG (Ours)+NMS & \bf{43.60} & \bf{52.21} & \bf{59.67} & 65.98 & 69.24 \\
& ATAG (Ours)+SNMS & 43.52 & 51.86 & 59.48 & \bf{66.04} & \bf{70.28} \\
\bottomrule
\end{tabular}
}
\end{center}
\vspace{-4mm}
\caption{Comparison of our model with state-of-the-art methods on THUMOS14 testing set. All the results are reported in percentage. SNMS stands for Soft-NMS. }
\label{Table 1}
\end{table}
\begin{table} [t]
\small
\begin{center}
\small
\resizebox{!}{0.95cm}{
\begin{tabular}{c|cccccc}
\toprule
Method &\cite{lin2018bsn} & \cite{lin2019bmn} & \cite{Liu_2019_CVPR} & \cite{bai2020boundary} & \cite{zhao2020bottom} & ours \\
\midrule
AR@100(val) & 74.16 & 75.01 & 74.54 & 76.73 & 75.27 & \bf{76.75} \\
AUC(val) & 66.17 & 67.10 & 66.43 & 68.05 & 66.51 & \bf{68.50} \\
AUC(test) & 66.26 & 67.19 & 66.47 & - & - & \bf{68.45} \\
\bottomrule
\end{tabular}
}
\end{center}
\vspace{-5mm}
\caption{Comparison of our model with state-of-the-art methods BSN~\cite{lin2018bsn}, BMN~\cite{lin2019bmn}, MGG~\cite{Liu_2019_CVPR}, BC-GCN~\cite{bai2020boundary}, BUMR~\cite{zhao2020bottom} on ActivityNet1.3. All the results are reported in percentage.}
\label{Table 2}
\end{table}
\begin{figure}[t]
\includegraphics[width=1.0\linewidth]{visualization.png}
\centering
\caption{Visualization examples of generated proposals on ActivityNet1.3 and THUMOS14. The red boxes highlight some frames in the action instances. }
\label{visualization}
\vspace{-4mm}
\end{figure}
\subsection{Ablation study}
To verify the effectiveness of our method, we conduct the following ablation studies on the validation set of ActivityNet1.3 dataset.
\paragraph{Component analysis.}
The augmented transformer and adaptive GCN are the core components in our method. We evaluate several variants of this model by ablating its augmented transformer and GCN in Table~\ref{Ablation1}. \textit{Base} represents the version without augmented transformer and adaptive GCN. We can see that any component can individually improve the performance. The existence of snippet actionness loss and front block can boost the effect of vanilla transformer, which demonstrates that they can assist the transformer to learn long-range dependencies. For local context capture, the result shows that adaptive graph is beneficial for this task and removing any adjacency matrix will degrade the performance. With the global and local fusion, the full model reaches the best result.
\paragraph{Early fusion \vs. late fusion.}
We explore other fusion strategies. Specifically, in the early fusion, besides our early fusion with concatenation, we also evaluate the global and local features are fused by summation; in the late fusion, each path is individually fed into independent prediction networks of the output module and concatenate the dual-branch features till entering classifiers to aggregates predictions of boundaries and completeness scores.
We show the results in Table~\ref{Ablation1}. The Early fusion with concatenation outperforms other fusion strategies.
\paragraph{Number of transformer layers.}
Stacking multi-layer transformers normally brings better performance, which is demonstrated in other tasks, for instance, NLP~\cite{vaswani2017attention,devlin2018bert} and object detection~\cite{carion2020endtoend}. We stack different numbers of transformers and show their results in Table~\ref{Ablation2}. Different from other tasks, more transformers in our model do not improve the performance. We believe the reason is that the scale of datasets in this task is too small to train a deeper transformer network. Although there are about 20k videos in ActivityNet1.3, the density of action instances is low and each video includes only 1.5 action instances on average, which is far less than the datasets in other tasks.
\paragraph{Our GCN \vs. other schemes.}
For the local context capture, we use a GCN with a hybrid adaptive adjacency matrix. Further, we experiment with different structures to replace our GCN, including (i) a general GCN with predefined adjacency matrix, (ii) a self-attention style GCN where the adjacency matrix $A$ is calculated by $A=softmax(X^TW_\theta^TW_\phi X)$ where $X$ is the input feature sequence and $W_\theta$ and $W_\phi$ are trainable weights, and (iii) a convolutional layer. Table~\ref{Ablation3} compares all these variants, with our choice outperforming other two variations.
\begin{table} [t]
\begin{center}
\small
\begin{tabular}{c|cc}
\toprule
Variants of model & AR@100 & AUC \\
\midrule
Base & 75.10 & 66.80 \\
Base + VT & 75.42 & 67.48 \\
Base + VT + Snippet actionness loss & 75.76 & 67.74 \\
Base + VT + Front block & 75.85 & 67.78 \\
Base + Augmented transformer & 76.19 & 68.01 \\
Base + GCN wo/$A_a$ & 75.50 & 67.31\\
Base + GCN wo/$A_d$ & 75.58 & 67.54\\
Base + GCN & 75.79 & 67.81 \\
\midrule
Late fusion & 76.26 & 68.25\\
Early fusion - summation & 75.99 & 67.88 \\
Early fusion - concatenation & 76.75 & 68.50 \\
\bottomrule
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Performance evaluation on different components of our model and comparison of different fusion strategies. VT is short for vanilla transformer.}
\label{Ablation1}
\end{table}
\begin{table} [t]
\begin{center}
\small
\begin{tabular}{c|cc}
\toprule
No. of Transformer & AR@100 & AUC \\
\midrule
1 & 76.75 & 68.50 \\
2 & 76.64 & 68.35 \\
3 & 75.94 & 67.83 \\
\bottomrule
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Performance evaluation on different number of transformer layers.}
\label{Ablation2}
\end{table}
\begin{table} [t]
\small
\begin{center}
\begin{tabular}{c|cc}
\toprule
Structure & AR@100 & AUC \\
\midrule
General GCN & 75.28 & 67.24 \\
Self-attention GCN & 74.76 & 66.50 \\
Convolutional layer & 75.20 & 67.20 \\
Our GCN & 75.79 & 67.81 \\
\bottomrule
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Comparison of our GCN with other schemes.}
\label{Ablation3}
\end{table}
\subsection{Temporal action detection with our proposals}
The important usage of action proposals is for temporal action detection. Thus, evaluating the performance of proposals in the temporal action detection task is another aspect to verify the quality of proposals. We adopt the evaluation metric Mean Average Precision (mAP) from the temporal action detection task. mAP with IoU thresholds $[0.5:0.05:0.95]$ are used on ActivityNet1.3 and mAP with IoU thresholds $\{0.3,0.4,0.5,0.6,0.7\}$ are used on THUMOS14.
We adopt the two-stage ``detection by classifying proposals'' temporal action detection framework. On ActivityNet1.3, the proposals are ranked according to their confidence scores and top 100 proposals per video are selected. Then, for each video, its top-1 video-level classification result will be obtained from ~\cite{zhao2017cuhk} and all the proposals of this video share this classification result as their action classes. On THUMOS14, we adopt top-2 video-level classification results generated by UntrimmedNet~\cite{Wang_2017_CVPR} as labels for top 200 proposals per video.
The results on ActivityNet1.3 and THUMOS14 are shown in Table~\ref{Table Activity} and Table~\ref{Table thumos}, respectively. Our method achieves new state-of-the-art results on both datasets, which validates the quality of our proposals and the effectiveness of our method. The high mAP reflects that our model can predict the good proposals with high scores, and reduce the number of false positives.
\begin{table} [t]
\begin{center}
\begin{tabular}{c|cccc}
\toprule
Method & 0.5 & 0.75 & 0.95 & Average \\
\midrule
CDC~\cite{Shou_2017_CVPR} & 43.83 & 25.88 & 0.21 & 22.77 \\
SSN~\cite{xiong2017pursuit} & 39.12 & 23.48 & 5.49 & 23.98 \\
BSN~\cite{lin2018bsn}+\cite{zhao2017cuhk} & 46.45 & 29.96 & 8.02 & 30.03 \\
BMN~\cite{lin2019bmn}+\cite{zhao2017cuhk} & 50.07 & 34.78 & 8.29 & 33.85 \\
GTAD~\cite{xu2020g}+\cite{zhao2017cuhk} & 50.36 & 34.60 & 9.02 & 34.09 \\
BC-GNN~\cite{bai2020boundary}+\cite{zhao2017cuhk} & 50.56 & 34.75 & 9.37 & 34.26 \\
\midrule
ATAG (Ours)+\cite{zhao2017cuhk} & \bf{50.92} &
\bf{35.35} & \bf{9.71} & \bf{34.68}\\
\bottomrule
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Action detection results on validation set of ActivityNet1.3 dataset in terms of mAP at IoU 0.5, 0.75 and 0.95, and average mAP.}
\label{Table Activity}
\end{table}
\begin{table} [t]
\begin{center}
\begin{tabular}{c|ccccc}
\toprule
Method & 0.7 & 0.6 & 0.5 & 0.4 & 0.3 \\
\midrule
TURN~\cite{Gao_2017_ICCV} & 6.3 & 14.1 & 24.5 & 35.3 & 46.3 \\
BSN~\cite{lin2018bsn} & 20.0 & 28.4 & 36.9 & 45.0 & 53.5 \\
MGG~\cite{Liu_2019_CVPR} & 21.3 & 29.5 & 37.4 & 46.8 & 53.9 \\
BMN~\cite{lin2019bmn} & 20.5 & 29.7 & 38.8 & 47.4 & 56.0 \\
GTAD~\cite{xu2020g} & 23.4 & 30.8 & 40.2 & 47.6 & 54.5 \\
BC-GNN~\cite{bai2020boundary} & 23.1 & 31.2 & 40.4 & 49.1 & 57.1 \\
\midrule
ATAG (Ours) & \bf{28.0} & \bf{38.0} & \bf{47.3} & \bf{53.1} & \bf{62.0}\\
\bottomrule
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Comparison between our approach and other temporal action detection methods on THUMOS-14.}
\label{Table thumos}
\end{table}
\section{Conclusion}
In this paper, we propose an augmented transformer with adaptive graph network (ATAG) for temporal action proposal generation, which presents an augmented transformer and a new adaptive GCN to capture long-range temporal context and local temporal context, respectively. Our augmented transformer can effectively enhance the video understanding and noisy action instance localization. For local context capture, the newly designed adaptive GCN with two types of matrices is leveraged to build local temporal relationships. Extensive experiments show that our model achieves new state-of-the-art performance in temporal action proposal generation and action detection on THUMOS14 and ActivityNet1.3 datasets.
\section*{Acknowledgement}
This work was supported by Alibaba Group through Alibaba Research Intern Program.
{\small
\input{main.bbl}
\bibliographystyle{IEEEtran}
}
\clearpage
\section{Supplementary }
\subsection{Label assignment}
The methods of label assignment in each loss functions are described in detail below.
For boundary classification loss, we need to generate ground truth of boundary label including start $g^s$ and end $g^e$. Given a ground truth action instance $\phi_n=(t_s, t_e)$, we denote its start and end region as $r_s=[t_s-1.5\Delta t, t_s+1.5\Delta t]$ and $r_e=[t_e-1.5\Delta t, t_e+1.5\Delta t]$ respectively, where $\Delta t$ is the temporal interval between two adjacent snippets. Then, we compute overlap ratio IoR of each snippet interval with $r_s$ and $r_e$ separately, where IoR is
defined as the overlap ratio with ground truth proportional. If one snippet interval is overlapped with multiple actions, we take the maximum IoR. Finally, we assign positive labels to the locations with IoR $>0.5$; otherwise negative labels.
For actionness classification loss, we need to generate ground truth of actionness label $g^a$. Similar to boundary label generation, we calculate the IoR of each snippet interval with ground truth of action instances and use the same threshold $0.5$ to assign a positive or negative label.
For proposal completeness loss, we need to generate proposal completeness label map $G^c$. For a proposal $\phi_{i,j}=(t_j, t_j+t_i)$, we compute its Intersection-over-Union (IoU) with all the ground truth action instances, and denote the maximum IoU as $G^c[i,j]$.
\subsection{Additional visualization}
Figure \ref{visualization} illustrates more visualization examples and comparison with classical method BMN [\textcolor{green}{16}]. The proposals of our method and BMN\footnote{The reproduction code we use is from \url{https://github.com/JJBOY/BMN-Boundary-Matching-Network}, whose performance is sightly higher than paper's.} with the highest $k$ scores are visualized in each video, where $k$ is the number of ground truth. All the examples belong to complicated scenarios where interested actions involve irrelevant frames and background clutters. Traditional BMN fails to be robust for noisy frames and outputs incomplete proposal in the first four videos. In the last two videos, although its predictions cover the whole action instances, we speculate the reason is that the network overfits the noisy frames since those proposals also contain massive background frames. By comparison, our method presents precise boundary predictions in all the cases. Those examples adequately demonstrate that our method can understand video semantics and deal with complicated and noisy action instances by modeling long-range and local temporal context, by equipping our novel augmented transformer and adaptive GCN; in contrast, previous methods, such as BMN, focusing on the single local temporal context lack this capacity.
\subsection{Detailed architecture}
Table \ref{architecture} presents the architecture of our actionness predictor in augmented transformer and output module including completeness prediction network and boundary prediction network.
\begin{table} [t]
\begin{center}
\small
\resizebox{!}{3.5cm}{
\begin{tabular}{c|ccp{1.5em}p{1.5em}c}
\toprule
layer & kernel size & stride & group & act & output size \\
\midrule
\multicolumn{6}{c}{Actionness Predictor} \\
\midrule
conv1d & 3 & 1 & 4 & ReLU & $256{\times}T$ \\
\midrule
conv1d & 1 & 1 & 1 & Sigmoid & $1{\times}T$\\
\midrule
\multicolumn{6}{c}{Completeness Prediction Network} \\
\midrule
BM layer & - & - & - & - & $256{\times}32{\times}D{\times}T$\\
conv3d & (32,1,1) & (32,0,0) & 1 & ReLU & $512{\times}1{\times}D{\times}T$ \\
squeeze & - & - & - & - & $512{\times}D{\times}T$ \\
conv2d & (1,1) & (1,1) & 1 & ReLU & $128{\times}D{\times}T$ \\
conv2d & (3,3) & (1,1) & 1 & ReLU & $128{\times}D{\times}T$ \\
conv2d & (3,3) & (1,1) & 1 & ReLU & $128{\times}D{\times}T$ \\
conv2d & (1,1) & (1,1) & 1 & Sigmoid & $2{\times}D{\times}T$\\
\midrule
\multicolumn{6}{c}{Boundary Prediction Network} \\
\midrule
conv1d & 3 & 1 & 4 & ReLU & $256{\times}T$\\
\midrule
conv1d & 1 & 1 & 1 & Sigmoid & $2{\times}T$\\
\bottomrule
\end{tabular}
}
\end{center}
\vspace{-5mm}
\caption{The detailed architectures of some modules in our method.$T$ is the number of snippets and $D$ represents the maximum duratin length.}
\label{architecture}
\end{table}
\begin{figure*}[t]
\includegraphics[width=1.0\linewidth]{supplement_2.png}
\centering
\caption{Visualization examples of generated proposals on ActivityNet1.3 and THUMOS14. The red boxes highlight some frames in the action instances. }
\label{visualization}
\end{figure*}
\end{document}
| {'timestamp': '2021-03-31T02:12:30', 'yymm': '2103', 'arxiv_id': '2103.16024', 'language': 'en', 'url': 'https://arxiv.org/abs/2103.16024'} |
\section{Introduction}
There is an impressive number of experimental verifications, in many different scientific sectors, that coarse-grain properties of systems, with simple laws with respect to the fundamental microscopic algorithms, emerge at different levels of magnification providing important tools for explaining and predicting new phenomena.
For example, the Gompertz law (GL)~\cite{GL}, initially applied to human mortality tables (i.e. aging), describes tumor growth~\cite{steel,norton}, kinetics of enzymatic reactions, oxygenation of hemoglobin, intensity of photosynthesis as a function of CO2 concentration, drug dose-response curve, dynamics of growth, (e.g., bacteria, normal eukaryotic organisms). Analogously, the Logistic Law (LL)~\cite{LL} has been applied in population dynamics, in economics, in material science and in many other sectors.
The ability of macroscopic growth laws in describing the underlying complex dynamics is sometime surprising.
A clear and timely example is given by the infection spreading (Coronavirus \cite{oms1,oms2} in particular), where different Governments impose strong containment efforts on the basis of the mortality rate and of the growth rate of the cumulative total number of detected infected people $N(t)$ at time $t$, which has a strong impact on the national health systems.
However, there is a large number of infected people without any symptoms who contribute to the disease spreading but are not explicitly taken into account in $N(t)$, since not detected.
More precisely, a macroscopic growth law for $N(t)$ is solution of the general differential equation
\be\label{eq:1}
\frac{dN}{dt} = \alpha_{eff}(t) N(t),
\ee
where $\alpha_{eff}(t)$ is the specific growth rate at time $t$. For example, if $\alpha_{eff(t)}=$constant one obtains the exponential behavior. Various growth patterns have been very recently applied to the time evolution of the CoviD-19 infection \cite{np7,np8,np9,np10,np11,np12,np13}. On the other hand, by using the previous equation to describe the epidemic phase, the cumulative number of asymptomatic infected individuals, $A(t)$, seems completely neglected. Indeed, the second term of the equation depends on $N(t)$ only.
For the CoviD-19 infection, since $A(t)$ is unknown, many Governments have correctly applied strong constraints to slow down the spreading: social isolation, information on the localization of infected individuals and the use of a very large number of medical swabs.
However, the question arises: is the information obtained by monitoring $N(t)$ reliable in understanding the epidemic phase?
By a simple model of the interaction between the symptomatic cumulative detected population $N(t)$ and the asymptomatic one, $A(t)$, we discuss how the effective coarse-grain equation (1) takes into account the asymptomatic population in the transient phase of the spreading. Moreover one gets useful indications on the number of asymptomatic individuals and on the effective lethality.
In Sec.1 different macroscopic growth laws are applied to the cumulative number of detected infected individuals in China, South Korea and Italy, where the containment effort
started earlier, to describe the phases of the spreading. A simple dynamical model where the asymptomatic population and the isolation effect play an important role is discussed in Sec.2.
The results are in Sec.3 and Sec.4 is devoted to comments and conclusions.
\section{Infection spreading phases: macroscopic description}
The macroscopic growth laws in eq.~\eqref{eq:1}, can be classified by considering the time derivative of $\alpha_{eff}$, as shown in ref.~\cite{cast1,cast2}. For example the GL and the LL are in the, so called, universality class U2~\cite{cast1}, where $d\alpha_{eff}/dt= a_1 \alpha+a_2 \alpha^2$. Moreover, the Gompertz and the logistic equations can be written in a different way, which clarifies the feedback effect of the increasing population , i.e.
\be\label{eq:G}
\frac{1}{N(t)}\frac{dN(t)}{dt} = - k_g\;\ln \frac{N(t)}{N_\infty^g} \qquad \text{Gompertz}\;,
\ee
\be
\frac{1}{N(t)}\frac{dN(t)}{dt} = k_l \left(1- \frac{N(t)}{N_{\infty}^l}\right)
\qquad
\text{logistic}
\ee
where $k_g$, $N_\infty^g$, $k_l$ and $N_\infty^l$ are constants.
In the GL and in the LL, $k_g \ln(N_\infty^g)$ and $k_l$ are respectively the initial exponential rates and the other terms determine their slowdown.
The comparison of the growth laws, solutions of the previous differential equations, with the data on the cumulative number of infected individuals is reported in fig.~\ref{fig:China} for China,
showing that the coronavirus spreading has three phases: an initial exponential behavior, followed by a Gompertz one and a final logistic phase.
For South Korea, after the three phases, the spreading seems to restart (see fig.~\ref{fig:SK}). Italy is still in the Gompertz growth phase (see
fig. \ref{fig:Italy}).
The previous equations neglect the large (undetected) asymptomatic population and the isolation effects, since they concern the time evolution of $N(t)$ only.
However, a simple model in the next section shows that the GL takes into account a more complex underlying dynamics. The analysis can be repeated for the LL.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{FigChina}
\caption{Comparison of the growth laws with the data of the cumulative number of infected individuals in China: continuous line si the logistic curve, dotted line the Gompertz, the dashed one the exponential. Time zero corresponds to the initial day - 22/01.}
\label{fig:China}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{FigSouthKorea}
\caption{Comparison of the growth laws with the data of the cumulative number of infected individuals in South Korea: continuous line si the logistic curve, dotted line the Gompertz, the dashed one the exponential. Time zero corresponds to the initial day - 20/02.}
\label{fig:SK}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{FigItaly}
\caption{Comparison of the growth laws with the data of the cumulative number of infected individuals in Italy: continuous line si the logistic curve, dotted line the Gompertz, the dashed one the exponential. Time zero corresponds to the initial day - 22/02.}
\label{fig:Italy}
\end{figure}
\section{A simple model: asymptomatic population and containment effects}
In the model, we call $T(t)$ the cumulative total number of infected people, which is the sum of the number of the cumulative detected infected individuals $N(t)$ and of the asymptomatic ones $A(t)$: $T(t) = N(t) + A(t)$.
In the data of $N(t)$ the number of dead and of healed people is included, since they have been previously infected. In the fast growing phase, their total number is much smaller than $N(t)$ and therefore they are not included in the dynamics (the re-infection is considered very unlikely). The previous condition defines the transient phase.
Therefore one has to take into account the rates of the following processes ($n$ = detected infected person, $a$ = asymptomatic infected person):
$ n \rightarrow n +n$, $a \rightarrow n +a$, $n \rightarrow n +a$, $a \rightarrow a + a$ with rates $c_1(t)$, $c_2(t)$, $c_3(t)$, $c_4(t)$ respectively.
Accordingly, the corresponding mean field equations are
\be\label{eq:A}
\frac{dN(t)}{dt} = c_1(t) N(t) + c_2(t) A(t),
\ee
\be\label{eq:B}
\frac{dA(t)}{dt} = c_3(t) N(t) + c_4(t) A(t).
\ee
The functions $c_i(t)$, $i = 1, 4$, describe the damping effects due to the containment effort and we assume they have an exponential decreasing behavior:
\be
c_i(t) \simeq e^{-\lambda t}
\ee
i.e. the rates of the dynamical processes decrease in time due to social isolation and other external constraints. This effect should be independent on the status of the infected individual (n or a). On the other hand, in the processes involving a symptomatic individual, he/she is (or should be) rapidly detected, together with those who belong to his/her chain of infection transmission, independently on their symptomatic or asymptomatic condition. Therefore the rate of the processes involving infected persons should be suppressed with respect to the transmission rate among asymptomatic persons, which is ``invisible''.
In other terms, one expects that, in the transient phase, the rate of the process among asymptomatic individuals only, $a \rightarrow a + a$, decreases slowly than the other ones.
As a first step, one assumes that the functions $c_1(t)$,
$c_2(t)$, $c_3(t)$, $c_4(t)$ are given by
\be\label{eq:C}
c_1(t)=c_1^0 e^{-\lambda_1 t}, \phantom{...} c_2(t)= c_2^0 e^{-\lambda_1 t}, \phantom{...} c_3(t) = c_3^0 e^{-\lambda_1 t}, \phantom{...} c_4(t)=c_4^0 e^{-\lambda_2 t},
\ee
with $\lambda_2 < \lambda_1$ and $c_i^0 > 0$. The coefficients $c_i^0$ are the initial rates and $\lambda_1$, $\lambda_2$ parameterize their reduction due to the isolation conditions.
it is an effective way to take into account the dynamics in eqs.~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}).
To clarify that the solution of eq.(2) takes into account, in an effective way, the dynamics in eqs.~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}), the system is analytically solved (see appendix)
for $\lambda_1=\lambda_2$. For $\lambda_2 < \lambda_1$ there is no analytical solution and the numerical one will be compared with the Italian data and the previous GL fit in the next section.
\section{Results}
The Gompertz fit (from eq.~\eqref{eq:G}) of Italian data is compared with the numerical solution of the dynamical system~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}) with $\lambda_2 < \lambda_1$ in fig.~\ref{fig:B}. The two curves are well compatible, indicating that the interaction between symptomatic and asymptomatic populations is contained in eq.~\eqref{eq:G} in an effective way. Moreover the solution of the system gives an estimate of the asymptomatic population, plotted in fig.~\ref{fig:C}.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{Fig2B}
\caption{Cumulative number of infected individuals in Italy from February 22nd (day 0) to April the 16th. The GL fit (red) and the solution of the system in eq.~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}) (black) are plotted versus the observed data. The parameters are: $N(0)=62$, $A(0)=N(0)$, $c_1^0=0.23$, $c_2^0=0.26$, $c_3^0=0.24$, $c_4^0=0.32$, $\lambda_1=0.0732$, $\lambda_2=0.045$. The two curves overlap.}
\label{fig:B}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{Fig3B}
\caption{Cumulative number of infected individuals in Italy from February 22nd(day 0) to March 22nd. The solution of the system in eqs.~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}) are plotted: detected in black, asymptomatic in blue, total in orange. }
\label{fig:C}
\end{figure}
The previous analysis clearly suggests that:
\begin{itemize}
\item[a)] the curve $N(t)$ takes into account the dynamics of the asymptomatic individuals;
\item[b)] in the fast growing phase there is a large asymptomatic population: to stop the infection spreading the strong social isolation is the best method;
\item[c)] the time evolution of $N(t)$ is an effective macroscopic growth laws solution of eq.~\ref{eq:G} for Italy, completely consistent with the system in eqs.~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}).
\end{itemize}
In this respect the choice of monitoring the time evolution of $N(t)$ to understand the epidemic phase is reliable, since the
underling dynamics is included in the time dependence of $\alpha_{eff}$ in eq.~\eqref{eq:1}. This is probably related to the result
that a general classification of the growth laws depends on the time derivative of $\alpha_{eff}$~\cite{cast1,cast2}.
On the other hand, the system of differential equations give more information than the single equation for $N(t)$. In particular, the ratio, $R(t)$ (see appendix), between symptomatic and asymptomatic populations is plotted as a function of time in fig.~\ref{fig:R}. Moreover, one can evaluate in a more reliable way the lethality of the infection, defined as the ratio of the number of dead individuals and the total number of infected ones. The italian lethality is $13\%$ neglecting the asymptomatic population, but, in our model, its value reduces to $\simeq 3 \%$ considering the total infected population.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{FigRB}
\caption{Ratio between asymptomatic and symptomatic populations.}
\label{fig:R}
\end{figure}
\section{Comments and Conclusions}
The model in eqs.~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}) is a simplified version of a more complex dynamics~\cite{net1,net2} and can be improved including other specific populations~\cite{smaller}.
Different analyses of the CoviD-19 spreading predict a large asymptomatic population. In ref.~\cite{np10} the number of asymptomatic individuals in Italy on March 12th turns out to be more than 100.000 versus a symptomatic detected population of about 12.000. Our analysis suggests an asymptotic population of about 27500 individuals on March 12th, 76000 on March 19th and 129000 on March 24th. In ref.~\cite{lan}, the outbreak in Italy has been estimated, giving a ratio of about 1:3 between symptomatic and asymptomatic individuals on February 29Th. The evaluation of the asymptomatic population is a difficult task and the results are strongly model dependent.
Macroscopic laws in eq.~\eqref{eq:1} include the dynamics of the infection with the advantage of a reduction of the free parameters: the simple system in eqs.~(\ref{eq:A},~\ref{eq:B},~\ref{eq:C}) has in principle 5 parameters plus 2 initial conditions, but the Gompertz curve has 2 parameters and 1 initial condition.
The reliability of $N(t)$ as an index of the spreading has been checked by a devoted analysis, including the crucial effects of the social isolation and the role of the asymptomatic individuals.
A further confirmation of this conclusion comes from the direct comparison between the daily rate $N(t+1)-N(t)$ and the corresponding mortality one, reported in figs.~(\ref{fig:ConfDay},\ref{fig:DDay}), which shows that the time evolution of $N(t)$ anticipates the other by about 8 days.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{ConfirmedXday}
\caption{Confirmed daily rate.}
\label{fig:ConfDay}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{DeathsXday}
\caption{Mortality daily rate.}
\label{fig:DDay}
\end{figure}
It should be clear that the previous analysis implicitly assumes that the protocol concerning the detection of infected individuals, as for example the number of medical swabs (per day), does not change in time. Indeed, a modification in the detecting protocol (as in the Chinese case) or the sudden, strong, variation in the number of swabs could mimic an increase or a decrease in $N(t)$ of not dynamical origin.
\section{Acknowledgements}
P.C. thanks V. Latora for useful comments and suggestions.
\section{Appendix}
Let us consider the system of differential equations
\begin{equation}\label{eq:s}
\begin{cases}
\displaystyle \frac{dN(t)}{dt} = c_1(t) N(t) + c_2(t) A(t)
\\\\
\displaystyle \frac{dA(t)}{dt} = c_3(t) N(t) + c_4(t) A(t)
\end{cases}
\;,
\end{equation}
with
\begin{equation}\label{1}
c_i = c_i^0 \; e^{-\lambda \; t}\;,
\end{equation}.
It can be solved by defining the ratio
\begin{equation}
R(t) \equiv \frac{A(t)}{N(t)}
\;,
\end{equation}
which satisfies the equation
\begin{equation}\label{eq:R}
\frac{dR(t)}{dt}
=
e^{-\lambda\;t} \left[c_3^0-R(t)\;\left[c_1^0-c_4^0+c_2^0\;R(t)\right]\right]
\;.
\end{equation}
The analytical solution of the previous eq.~\eqref{eq:R} is
\begin{equation}
R(t)
=
\frac{R_+ - R_-\;\frac{R_0-R_+}{R_0-R_-}\;e^{-\frac{c_2^0}{\lambda}\;(R_+ - R_-)\;\left(1-e^{-\lambda \; t}\right)}}{1-\frac{R_0-R_+}{R_0-R_-}\;e^{-\frac{c_2^0}{\lambda}\;(R_+ - R_-)\;\left(1-e^{-\lambda \; t}\right)}}
\;,
\end{equation}
where
\begin{equation}
R_\pm
=
\frac{1}{2}\left[-\frac{c_1^0 -c_4^0}{c_2^0} \pm \sqrt{\left(\frac{c_1^0 -c_4^0}{c_2^0}\right)^2 + \frac{4\;c_3^0}{c_2^0} }\;\right]
\end{equation}
and $R_0=R(t=0)=A_0/N_0$, being $N_0$ and $A_0$ the number of detected and asymptomatic infected individuals at the initial time respectively.
Putting $A(t) = R(t)\;N(t)$ in eq.~\eqref{eq:s} it turns out that
\begin{equation}\label{eq:N}
N(t)
=
\frac{N_0\;e^{\frac{c_1^0+c_2^0\;R_+}{\lambda}\left(1-e^{-\lambda t}\right)}}{R_--R_+}\;
\left[(R_0-R_+)\;e^{\frac{c_2^0}{\lambda} (R_--R_+) \left(1-e^{-\lambda t}\right)}-R_0+R_-\right]
\;,
\end{equation}
and
\begin{equation}
A(t)
=
\frac{N_0\;e^{\frac{ c_1^0+c_2^0\;R_+}{\lambda}\left(1-e^{-\lambda t}\right)}}{R_+-R_-}
\Bigg[
R_-\;(R_+-R_0)\;e^{\frac{c_2^0 (R_--R_+)}{\lambda} \left(1 -e^{-\lambda t}\right)}
+
R_+ (R_0-R_-)\Bigg]
\;.
\end{equation}
Finally, for the specific case
$c_3^0=c_1^0 $ and $c_2^0=c_4^0 = k\;c_1^0$, with $k$ some constant,
\begin{equation}\label{eq:a}
N(t)
=
\frac{N_0}{k+1}\;\left((k\;R_0+1) e^{\frac{c_1^0(k+1)}{\lambda} \;\left(1-e^{-\lambda t}\right)}+k (1-R_0)\right)
\;,
\end{equation}
\begin{equation}
A(t)
=
\frac{N_0}{k+1}
\; \left((k\;R_0+1)\;e^{\frac{c_1^0 (k+1)}{\lambda}\;\left(1-e^{-\lambda t}\right)}+R_0-1\right)
\;,
\end{equation}
and
\begin{equation}
R(t)
=
1+
\frac{(R_0-1)(1+k)}{(k\;R_0+1)\;e^{\frac{c_1^0\; (k+1)}{\lambda}\; \left(1-e^{-\lambda t}\right)}+k (1-R_0)}
\end{equation}
The typical time dependence of the previous solutions, $N(t)$ and $A(t)$, is plotted in figure~\ref{fig:App}.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{FigApp}
\caption{Solution of eq.~\eqref{eq:s} with $N(0)=62$, $A(0)=N(0)$, $c_1^0=0.23$, $c_2^0=0.26$, $c_3^0=0.24$, $c_4^0=0.32$. Black curves are for $\lambda=0.0732$, the red ones for $\lambda=0.045$. Continuous curves give the cumulative number of detected infected
individuals ($N(t)$), the dotted one correspond to the asymptomatic ones, $A(t)$, and the dashed line is the total.}
\label{fig:App}
\end{figure}
| {'timestamp': '2020-04-21T02:14:37', 'yymm': '2003', 'arxiv_id': '2003.12457', 'language': 'en', 'url': 'https://arxiv.org/abs/2003.12457'} |
\section{Introduction}
\subsection{Background}
Let $M$ be an oriented 7-manifold with a ${\mathbb G}_2$-structure, defined by a positive 3-form $\phi.$
Endow $M$ with the corresponding Riemannian metric $g = g_\phi$ and Hodge star operator $* = *_g$ (see \S \ref{ss:npg2} below).
A connection $A$ on a vector bundle $E \to M$ is said to be a \emph{${\mathbb G}_2$-instanton} if its curvature $F_A$ satisfies
\begin{equation}\label{instantondef1}
F_A + * \left( \phi \wedge F_A \right) = 0.
\end{equation}
This equation appeared in the physics literature in the early 1980s \cite{corrigandevchand}. In the mid-1990s, not long after Joyce's construction \cite{joyceg2} of compact Riemannian manifolds with $\mathrm{Hol}(g) ={\mathbb G}_2,$ Donaldson and Thomas \cite{donaldsonthomas} proposed to define an invariant of (torsion-free) ${\mathbb G}_2$-structures
by ``counting'' ${\mathbb G}_2$-instantons in a manner analogous to the Casson invariant in dimension three. Although much progress has been made in the intervening two decades, instantons on compact ${\mathbb G}_2$-holonomy manifolds
are still quite difficult to construct,
and even more so to count.
This article investigates a species of ${\mathbb G}_2$-instanton that arises naturally from the geometry of the 7-sphere.
Although $S^7$ is not a ${\mathbb G}_2$-holonomy manifold, it carries a ``standard'' ${\mathbb G}_2$-structure, $\phi_{std},$ coming from the octonionic structure of ${\mathbb R}^8.$
This is an example of a \emph{nearly parallel ${\mathbb G}_2$-structure}, satisfying
\begin{equation}\label{nearlypar}
d \phi = \tau_0 \psi
\end{equation}
where $\psi = *_{g_{\phi}} \phi$ is the dual 4-form and $\tau_0$ is a
constant.
Nearly parallel ${\mathbb G}_2$-structures
have much in common with the torsion-free case
\cite{friedrichkathmoroianusemmelmann}.
Our
motivation for studying ${\mathbb G}_2$-instantons on
$S^7$ is threefold. First, we would ideally like
to determine the complete moduli space of instantons in this model case, as achieved by ADHM \cite{adhm} for the 4-sphere. This is a very difficult problem; we are limited here to studying the deformations of a given ${\mathbb G}_2$-instanton, in the spirit of Atiyah, Hitchin, and Singer \cite{ahs}. As we shall see, instantons on $S^7$ tend to have large spaces of infinitesimal deformations---one of many ``pathological phenomena'' that may also potentially occur in the torsion-free setting.
The second motivation comes from another difficulty in higher-dimensional gauge theory: the appearance of
instantons with essential singularities. Given a ${\mathrm{Spin} }(7)$-instanton with an isolated singularity on a ${\mathrm{Spin} }(7)$-manifold,
the cross-section of the tangent cone at the singular point
is a ${\mathbb G}_2$-instanton on the standard $S^7.$
Hence, these are the ``building blocks'' for the simplest non-removable singularities in dimension eight.
The third motivation is the relationship with gauge theory in fewer than seven dimensions.
Since ${\mathbb G}_2$-holonomy metrics typically collapse at the boundary of the moduli space,
it is important to understand instantons on model ${\mathbb G}_2$-manifolds coming from lower-dimensional geometries. Recently, Y. Wang \cite{yuanqicyproduct} has shown that
any ${\mathbb G}_2$-instanton on $S^1 \times X,$ for $X$ a Calabi-Yau 3-fold, is gauge-equivalent to the pullback of a Hermitian-Yang-Mills connection.
This is the first case where the full moduli space of ${\mathbb G}_2$-instantons on a given manifold with holonomy contained in ${\mathbb G}_2$ has been identified.
In the present case, a direct link with 4-dimensional gauge theory
is provided by the \emph{quaternionic Hopf fibration}
\begin{equation}\label{quathopf}
S^3 \to S^7 \to S^4.
\end{equation}
With the proper conventions, the pullback
of any anti-self-dual (ASD) instanton on $S^4$
is a ${\mathbb G}_2$-instanton on $S^7,$ giving us an abundant source of examples.
The resulting families of ${\mathbb G}_2$-instantons can be further enlarged by the action of the automorphism group ${\mathrm{Spin} }(7),$
which interacts with the Hopf fibration in a nontrivial way.
Our main results
concern the dimension and completeness properties of these families.
\subsection{Summary}
In \S \ref{sec:conventions}, we set our conventions and derive the basic results concerning instantons on nearly parallel ${\mathbb G}_2$-manifolds, while reviewing the literature in this area.
In \S \ref{sec:quatinst}, we take a concrete approach to the special case of the standard instanton. In quaternionic notation on ${\mathbb R}^8 \cong \H^2,$
the pullback by the (right) Hopf fibration of the standard ASD instanton
may be given by
\begin{equation}\label{standardg2instanton}
A_0(x,y) = \frac{1}{|x|^2 + |y|^2} \Im \left[ |y|^2 x^{-1} \, dx + |x|^2 y^{-1} \, dy \right].
\end{equation}
\begin{comment}
As written here, this defines a smooth connection over
${\mathbb R}^8\setminus \{0\}$
with curvature
valued in ${\mathrm{Lie}}({\mathrm{Sp} }(2)) \otimes \Im \H;$ in particular, it is a ${\mathrm{Spin} }(7)$-instanton.
\end{comment}
The restriction of $A_0$ to $S^7$ is dubbed the \emph{standard ${\mathbb G}_2$-instanton}.
Proposition \ref{prop:15deformations} describes a 15-dimensional space of infinitesimal deformations of $A_0,$
which will turn out to be its full space of deformations as a ${\mathbb G}_2$-instanton.
In \S \ref{sec:generaldefos}, we determine the space of infinitesimal deformations of the pullback by the Hopf fibration
of a general irreducible ASD instanton on $S^4.$
The argument relies
on a vanishing result for the vertical component of an infinitesimal deformation in Coulomb gauge, Theorem \ref{thm:vanishing}, which follows from a delicate analysis of the Weitzenbock formula for the deformation operator. Having established that the vertical component vanishes, we use another trick, Lemma \ref{lemma:horiz}, to calculate the horizontal component. This leads directly to our main result, Theorem \ref{thm:generaldefos}, which equates the full space of infinitesimal deformations of the pullback,
as a ${\mathbb G}_2$-instanton,
with three copies of the ASD deformations.
In \S \ref{sec:global}, we briefly discuss the global structure of these families of ${\mathbb G}_2$-instantons.
In the case of charge $\kappa = 1$ and structure group ${\mathrm{SU} }(2),$ which includes the standard instanton,
we have:
\begin{thm}\label{thm:grassmann} The connected component of $A_0$ in the moduli space of ${\mathbb G}_2$-instantons on $S^7$ is diffeomorphic to the tautological 5-plane bundle over the oriented real Grassmannian ${\mathbb G}^{or}(5,7).$
\end{thm}
\noindent
In this description, the base
corresponds to the orbit of the Hopf fibration (\ref{quathopf}) under conjugation by ${\mathrm{Spin} }(7),$ and the fiber corresponds to the pullback of the unit-charge ASD moduli space.
The total space is 15-dimensional, agreeing with
the dimension formula of Theorem \ref{thm:generaldefos}.
By contrast, in the case of higher charge,
we do not expect all of the infinitesimal deformations identified by Theorem \ref{thm:generaldefos}
to be integrable.
Lastly, we state Conjecture \ref{conj:donaldson}, due to Donaldson, which asserts that every ${\mathbb G}_2$-instanton on $S^7$ having integral Chern-Simons value should arise from the pullback construction.
\subsection{Acknowledgements} The author thanks Simon Donaldson
and Thomas Walpuski for discussions, and thanks Aleksander Doan for comments on the manuscript. This research was supported during 2017-18 by the Simons Collaboration on Special Holonomy.
\section{Conventions and basic results}\label{sec:conventions}
\begin{comment}
Let ${\mathbb R}^8 = {\mathbb R}^4 \oplus {\mathbb R}^{4'},$ and choose bases $\omega_i, \omega'_i,$ $i = 1, 2,3,$ for the self-dual 2-forms on ${\mathbb R}^4$ and ${\mathbb R}^{4'},$ respectively. We may take
\begin{equation}
\begin{split}
\omega_1 = dx^{12} + dx^{34}, \qquad \omega'_1 = dx^{56} + dx^{78} \\
\omega_2 = dx^{13} - dx^{24}, \qquad \omega'_2 = dx^{57} - dx^{68} \\
\omega_3 = dx^{14} + dx^{23}, \qquad \omega'_3 = dx^{58} + dx^{67}. \\
\end{split}
\end{equation}
\end{comment}
\subsection{Quaternions, 2-forms, and Fueter maps}\label{ss:quatsandfueter} Let $\H$ denote the 4-dimensional algebra of quaternions, generated by $\mathbf{1}, {\mathbf i}, {\mathbf j},{\mathbf k},$ and subject to the relations
\begin{equation*}
{\mathbf i}^2 = {\mathbf j}^2 = {\mathbf k}^2 = {\mathbf i} {\mathbf j} {\mathbf k} = -\mathbf{1}.
\end{equation*}
The 3-dimensional space of imaginary quaternions $\Im \H,$ with commutator bracket,
is isomorphic to the Lie algebra $\mathfrak{su}(2).$
\begin{comment}
Then the 2-forms (\ref{2formbases}) act on ${\mathbb R}^4,$ in the basis (\ref{quaternionbasis}), by quaternion multiplication on the right:
\begin{equation}\label{quaternionaction}
\cdot \left( - {\mathbf i} \right) = \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_1, \qquad
\cdot \left( - {\mathbf j} \right) = \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_2, \qquad
\cdot \left( - {\mathbf k} \right) = \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_3 .
\end{equation}
We define the three complex structures $I_1, I_2, I_3$ on ${\mathbb R}^4.$
Then, in the basis (\ref{quaternionbasis}), one can check that
\begin{equation}\label{quaternionaction}
I_1 = \cdot \left( - {\mathbf i} \right) , \qquad
I_2 = \cdot \left( - {\mathbf j} \right) , \qquad
I_3 = \cdot \left( - {\mathbf k} \right) .
\end{equation}
\end{comment}
We shall identify $\H$ with ${\mathbb R}^4$
as follows:
\begin{equation}\label{quaternionbasis}
\{ e_0, e_1, e_2, e_3 \} = \{ \mathbf{1}, -{\mathbf i}, -{\mathbf j}, -{\mathbf k}\}.
\end{equation}
Denote by $\H_l$ and $\H_r$ the commuting subalgebras of $\mathrm{End} \, {\mathbb R}^4$ corresponding to left- and right-multiplication by $\H,$ respectively, and let ${\mathrm{Sp} }(1)_l$ and ${\mathrm{Sp} }(1)_r$ be the corresponding subgroups of ${\mathrm{SO} }(4)$ generated by unit quaternions.
With this convention, we have
\begin{equation*}
{\mathrm{Lie}}({\mathrm{Sp} }(1)_l) = \Lambda^{2-}, \qquad {\mathrm{Lie}}({\mathrm{Sp} }(1)_r) = \Lambda^{2+}
\end{equation*}
where $\Lambda^{2 \pm} \subset \Lambda^2 {\mathbb R}^4$ denotes the space of (anti-)self-dual 2-forms with respect to the Euclidean metric.
Choose the following standard basis for the self-dual 2-forms on ${\mathbb R}^4:$
\begin{equation}\label{2formbases}
\mathbf{\omega}_1 = dx^{01} + dx^{23}, \qquad \mathbf{\omega}_2 = dx^{02} - dx^{13}, \qquad \mathbf{\omega}_3 = dx^{03} + dx^{12}.
\end{equation}
Here we abbreviate $dx^{01} = dx^0 \wedge dx^1,$ {\it etc.} Define the complex structures
$I_1, I_2,$ and $I_3,$ by
\begin{equation}\label{quaternionaction}
I_1(v) = v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_1, \qquad
I_2(v) = v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_2, \qquad
I_3(v) = v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_3
\end{equation}
for $v \in T {\mathbb R}^4.$
Then $I_1, I_2,$ and $I_3$ correspond, respectively, to right-multiplication by $e_1, e_2,$ and $e_3,$ in the basis (\ref{quaternionbasis}).
We define a \textit{Fueter map} $L : {\mathbb R}^4 \to {\mathbb R}^4$ to be an endomorphism satisfying
\begin{equation}\label{fueter}
L + I_1 L I_1 + I_2 L I_2 - I_3 L I_3 = 0.
\end{equation}
The 12-dimensional subspace of Fueter maps $\mathfrak{F} \subset {\text{End}} \, {\mathbb R}^4$ is the direct sum:
\begin{equation}\label{fueterfactors}
\mathfrak{F} = \H_l \oplus \H_l I_1 \oplus \H_l I_2.
\end{equation}
In particular, $\mathfrak{F}$ contains the space of linear maps for the standard complex structure, $I_1.$
\subsection{${\mathrm{Spin} }(7)$ and its subgroups}\label{ss:spin7andg2}
The standard 4-form on ${\mathbb R}^8 = {\mathbb R}^4_x \oplus {\mathbb R}^4_y$ is defined by
\begin{equation}\label{psidef}
\Psi_0 = dx^{0123} + dy^{0123} + \omega^x_1 \wedge \omega^y_1 + \omega^x_2 \wedge \omega^y_2 - \omega^x_3 \wedge \omega^y_3.
\end{equation}
\begin{comment}
\begin{equation}\label{omegaidef}
\begin{split}
\omega^x_1 & = dx^{01} + dx^{23}, \qquad \omega^y_1 = dy^{01} + dy^{23} \\
\omega^x_2 & = dx^{02} - dx^{13}, \qquad \omega^y_2 = dy^{02} - dy^{13} \\
\omega^x_3 & = dx^{03} + dx^{12}, \qquad \omega^y_3 = dy^{03} + dy^{12}.
\end{split}
\end{equation}
\end{comment}
The group ${\mathrm{Spin} }(7)$ consists of all linear transformations of ${\mathbb R}^8$ that preserve $\Psi_0$ under pullback. It is a simply-connected, simple, Lie subgroup of ${\mathrm{SO} }(8)$ of dimension 21 (see {\it e.g.} Walpuski and Salamon \cite{walpuskimasters}, \S 9).
The Lie algebra of ${\mathrm{Spin} }(7)$ corresponds to the subspace of 2-forms $\eta \in \Lambda^2 {\mathbb R}^8$
satisfying
\begin{equation}\label{spin7def}
\eta + \ast \left( \Psi_0 \wedge \eta \right) = 0.
\end{equation}
Let $\Lambda^{2+}_{d} \subset \Lambda^2 {\mathbb R}^8$ be the subalgebra spanned by the three elements
$$\omega_1^x - \omega_1^y, \qquad \omega_2^x - \omega_2^y, \qquad \omega_3^x + \omega_3^y.$$
\begin{comment}
\begin{equation*
\left( \begin{array}{cc}
I_1 & 0 \\
0 & - I_1 \end{array} \right), \quad \left( \begin{array}{cc}
I_2 & 0 \\
0 & - I_2 \end{array} \right), \quad \left( \begin{array}{cc}
I_3 & 0 \\
0 & I_3 \end{array} \right).
\end{equation*}
\end{comment}
Also denote the subspace
\begin{equation}
\mathfrak{F}_{x,y} = \left\{L_{ij} dy^i \wedge dx^j \mid L \in \mathfrak{F} \right\} \subset \Lambda^2 {\mathbb R}^8.
\end{equation}
Then we have the following decomposition:
\begin{equation}\label{spin7splitting}
{\mathrm{Lie}}({\mathrm{Spin} }(7)) = \Lambda^{2-}_{x} \oplus \Lambda^{2-}_{y} \oplus \Lambda^{2+}_{d} \oplus \mathfrak{F}_{x,y} .
\end{equation}
One readily checks that each factor of (\ref{spin7splitting}) satisfies (\ref{spin7def}).
\begin{comment}
The standard complex structure on ${\mathbb R}^8 = {\mathbb C}^4 = \H^2$ is given by
\begin{equation*
\left( \begin{array}{cc}
I_1 & 0 \\
0 & I_1 \end{array} \right).
\end{equation*}
\end{comment}
Take complex coordinates
$$z^1 = x^0 + i x^1, \quad z^2 = x^2 + i x^3, \quad z^3 = y^0 + i y^1, \quad z^4 = y^2 + i y^3$$
for ${\mathbb R}^8 \cong {\mathbb C}^4.$
We have the standard holomorphic volume form and K\"ahler form
\begin{equation}\label{stdkahlerforms}
\Omega = dz^1 \wedge dz^2 \wedge dz^3 \wedge dz^4, \qquad {\bf \omega} = \frac{i}{2} \sum_{i = 1}^4 dz^i \wedge d\bar{z}^i = \omega_1^x + \omega_1^y.
\end{equation}
One may verify that
\begin{equation}\label{psi0kahler}
\Psi_0 = \frac{1}{2} \omega \wedge \omega + \Re \Omega.
\end{equation}
The 15-dimensional group ${\mathrm{SU} }(4),$ with its standard presentation, is therefore a subgroup of ${\mathrm{Spin} }(7).$ In particular, the 10-dimensional group ${\mathrm{Sp} }(2)$ of orthogonal quaternionic matrices, linear over $I_1, I_2,$ and $I_3,$ is also a subgroup.
The 14-dimensional Lie group ${\mathbb G}_2$ is the subgroup of ${\mathrm{Spin} }(7)$ stabilizing a point on $S^7.$ Equivalently, ${\mathbb G}_2$ is the subgroup of $\mathrm{GL}(7)$ that preserves the model 3-form
\begin{equation}\label{phi0}
\phi_0 = \frac{\partial}{\partial x^0} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \Psi_0 = dx^{123} + dx^1 \wedge \omega_1^y + dx^2 \wedge \omega_2^y - dx^3 \wedge \omega_3^y.
\end{equation}
Denote the dual 4-form by
\begin{equation}\label{psi0}
\psi_0 = \ast_{{\mathbb R}^7} \phi_0 = dy^{1234} + dx^{23} \wedge \omega_1^y - dx^{13} \wedge \omega_2^y - dx^{12} \wedge \omega_3^y.
\end{equation}
The Lie algebra ${\mathrm{Lie}}({\mathbb G}_2)$ corresponds to the subspace of 2-forms $\xi \in \Lambda^2 {\mathbb R}^7$ satisfying
\begin{equation*}
\xi + * \left( \phi_0 \wedge \xi \right) = 0
\end{equation*}
or equivalently
\begin{equation*}
\psi_0 \wedge \xi = 0.
\end{equation*}
\begin{rmk}\label{rmk:convention}
The (standard) presentation of ${\mathrm{Spin} }(7),$ in which the diagonal ${\mathrm{Sp} }(1)_r$ is a subgroup, can be obtained from (\ref{psidef}) by the coordinate change
\begin{equation*}
\left( \begin{array}{cc}
\mathbf{1} & 0 \\
0 & I_3 \end{array} \right).
\end{equation*}
However, Lemma \ref{lemma:instapullback} below fails in that convention.
In fact, there is no convention in which both ${\mathrm{Sp} }(2)$ (giving a transitive group of automorphisms) and a commuting copy of ${\mathrm{Sp} }(1)$ (giving the Hopf fibration) are subgroups of ${\mathrm{Spin} }(7);$
this is an essential difficulty.
\end{rmk}
\subsection{Nearly parallel ${\mathbb G}_2$-structures}\label{ss:npg2} Let $M$ be an oriented 7-manifold. Recall that a \emph{${\mathbb G}_2$-structure} on $M$ is defined by a global 3-form $\phi$ that is \emph{positive}, in the sense that
\begin{equation}
G_\phi(v) = \left( v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \wedge \left( v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \wedge \phi > 0
\end{equation}
for all $x \in M$ and $v \neq 0 \in T_x M.$ Any such $\phi$ is pointwise equivalent to the model 3-form $\phi_0$ given by (\ref{phi0}) above (see \cite{walpuskimasters}, Theorem 3.2).
A positive 3-form defines a unique Riemannian metric $g_\phi$ on $M$ by the requirement
\begin{equation}\label{g2metric}
6 g_\phi(v,v) Vol_{g_\phi} = G_\phi(v) \,\, \forall \,\, v \in TM.
\end{equation}
To any $\phi,$ we also associate the dual 4-form
\begin{equation}
\psi = \ast_{g_{\phi}} \phi.
\end{equation}
Recall that a ${\mathbb G}_2$-structure is said to be \emph{closed} if $d \phi = 0,$ and \emph{coclosed} if $d \psi = 0.$
We are concerned with ${\mathbb G}_2$-structures satisfying (\ref{nearlypar}), where we assume
\begin{equation}\label{pm4}
\tau_0 = \pm 4.
\end{equation}
The basic reference for nearly parallel structures is Friedrich, Kath, Moroianu, and Semmelmann \cite{friedrichkathmoroianusemmelmann}. With the normalization (\ref{pm4}), nearly parallel ${\mathbb G}_2$-manifolds are Einstein, with
\begin{equation}
\mathrm{Ric}_g = 6 g.
\end{equation}
There are three further equivalent formulations of the nearly parallel condition (\ref{nearlypar}).
The first is that the induced ${\mathrm{Spin} }(7)$-structure on the cone over $M$ be torsion-free. The second and most frequently used condition
is that $M$ possess a nonzero Killing spinor.
The third is as follows:
\begin{lemma}\label{lemma:gradphi} A ${\mathbb G}_2$-structure $\phi$ is nearly parallel if and only if
\begin{equation*}
\nabla \phi = \frac{\tau_0}{4} \psi.
\end{equation*}
Here $\nabla$ is the Levi-Civita connection associated to the metric $g_\phi$ defined by (\ref{g2metric}).
\end{lemma}
\begin{proof} See Karigiannis \cite{karigiannisflowsofg2}, Theorem 2.27.
\end{proof}
\begin{example} Define the \emph{standard ${\mathbb G}_2$-structure on $S^7$} by
\begin{equation}\label{phistdzerothdef}
\phi_{std} = \left. \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \Psi_0 \right|_{S^7}
\end{equation}
where
$$\vec{r} = x^i \frac{\partial}{\partial x^i} + y^i \frac{\partial}{\partial y^i}$$
is the coordinate vector field on ${\mathbb R}^8.$
The group of global automorphisms of $\phi_{std}$ is ${\mathrm{Spin} }(7).$
To obtain a more explicit expression for $\phi_{std},$ we define the 3-form
\begin{equation*}
\nu^x = \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathrm{Vol}_{{\mathbb R}^4_x}= x^0 dx^{123} - x^1 dx^{023} + x^2 dx^{013} - x^3 dx^{012}
\end{equation*}
on ${\mathbb R}^4_x,$ and the 1-forms
\begin{equation}\label{zetaidef}
\begin{split}
\zeta_1^x &= \vec{r}\mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \omega_1^x = x^0 dx^1 - x^1 dx^0 + x^2 dx^3 - x^3 dx^2 \\
\zeta_2^x &= \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \omega_2^x = x^0 dx^2 - x^2 dx^0 - x^1 dx^3 + x^3 dx^1 \\
\zeta_3^x & = \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \omega_3^x = x^0 dx^3 - x^3 dx^0 + x^1 dx^2 - x^2 dx^1.
\end{split}
\end{equation}
Define $\nu^y$ and $\zeta_i^y$ similarly. We then have
\begin{equation}\label{phistdfirstdef}
\begin{split}
\phi_{std}
& = \nu^x + \nu^y + \zeta_1^x \wedge \omega_1^y + \zeta_1^y \wedge \omega_1^x + \zeta_2^x \wedge \omega_2^y + \zeta_2^y \wedge \omega_2^x - \zeta_3^x \wedge \omega_3^y - \zeta_3^y \wedge \omega_3^x.
\end{split}
\end{equation}
It is easy to check, using the ${\mathrm{Spin} }(7)$-invariance, that $\phi_{std}$ defines the round metric
on $S^7.$
Notice that $d \nu^x = 4 {\mathrm{Vol}}_{{\mathbb R}^4_x}$ and $d \zeta_i^x = 2 \omega_i^x,$ and similarly for $y,$ so
\begin{equation}\label{phiPsi0}
d \phi_{std} = \left. 4 \Psi_0 \right|_{S^7}.
\end{equation}
Also note that
\begin{equation}\label{psistd}
\psi_{std} = \ast_{std} \phi_{std} = \left. \ast_{{\mathbb R}^8} \Psi_0 \right|_{S^7} = \left. \Psi_0 \right|_{S^7}.
\end{equation}
It follows from (\ref{phiPsi0}-\ref{psistd}) that $\phi_{std}$ is a nearly parallel ${\mathbb G}_2$-structure.
\begin{comment}
It will be useful to rewrite $\phi_{std}$ using global ${\mathrm{Sp} }(2)$-invariant coframes. Let
\begin{equation*
\begin{split}
\zeta_i & = \zeta_i^x + \zeta_i^y, \qquad \omega_i^\circ = \left. \omega^x_i + \omega^y_i \right|_{S^7} \\
\bar{\omega}_i & = \omega_i^\circ - \frac{1}{2}\epsilon_{ijk} \zeta_j \wedge \zeta_k, \qquad i = 1,2,3 \\
\nu & = \zeta_1 \wedge \zeta_2 \wedge \zeta_3.
\end{split}
\end{equation*}
From (\ref{phistdfirstdef}) and the ${\mathrm{Sp} }(2)$-invariance, we have
\begin{equation
\begin{split}
\phi_{std}
& = \nu + \zeta_1 \wedge \bar{\omega}_1 - \zeta_2 \wedge \bar{\omega}_2 + \zeta_3 \wedge \bar{\omega}_3.
\end{split}
\end{equation}
\end{comment}
\end{example}
\begin{example}\label{ex:squashed} Define the \emph{squashed ${\mathbb G}_2$-structure}
\begin{equation*}
\phi_{sq} = \frac{27}{25} \left( \begin{split} \frac{1}{5} (\nu_x + \nu_y) + \frac{16}{5} \left( \begin{split}\zeta_1^x \wedge \zeta_2^x \wedge \zeta_3^y + \zeta_1^x \wedge \zeta_2^y \wedge \zeta_3^x + \zeta_1^y \wedge \zeta_2^x \wedge \zeta_3^x \\ + \zeta_1^x \wedge \zeta_2^y \wedge \zeta_3^y + \zeta_1^y \wedge \zeta_2^x \wedge \zeta_3^y + \zeta_1^y \wedge \zeta_2^y \wedge \zeta_3^x \end{split} \right) \\
- \zeta^x_1 \wedge \omega^y_1 - \zeta_1^y \wedge \omega_1^x - \zeta^x_2 \wedge \omega^y_2 - \zeta_2^y \wedge \omega_2^x - \zeta^x_3 \wedge \omega^y_3 - \zeta_3^y \wedge \omega_3^x \end{split}\right).
\end{equation*}
The squashed ${\mathbb G}_2$-structure was discovered by Awada, Duff, and Pope \cite{awadaduffpope}, and has automorphism group ${\mathrm{Sp} }(2) {\mathrm{Sp} }(1).$ Appendix \ref{appendix:squashed} includes a proof that $\phi_{sq}$ is nearly parallel. This is a special case of the theorem of Friedrich {\it et al.} \cite{friedrichkathmoroianusemmelmann}, which constructs a strictly-nearly-parallel ${\mathbb G}_2$-structure by shrinking the fibers of any 3-Sasakian 7-manifold.
\end{example}
\begin{rmk}
Alexandrov and Semmelmann \cite{alexandrovsemmelmannnearlyparallel} have shown that both the standard and the squashed structures are rigid among nearly parallel ${\mathbb G}_2$-structures. These remain the only known nearly parallel structures on the 7-sphere, with any of its differentiable structures.
We also note that the (non-)existence of a \emph{closed}
${\mathbb G}_2$-structure on the 7-sphere is a well-known open problem.
\end{rmk}
\subsection{Instantons on nearly parallel ${\mathbb G}_2$-manifolds} Recall that a connection $A$ is called a \emph{${\mathbb G}_2$-instanton} if its curvature satisfies (\ref{instantondef1}),
or equivalently
\begin{equation}\label{instantondef2}
\psi \wedge F_A = 0.
\end{equation}
If the ${\mathbb G}_2$-structure $\phi$ is nearly parallel, then from (\ref{instantondef1}) and (\ref{nearlypar}), we have
\begin{equation*}
\begin{split}
0 & = D_A^*F_A - * D_A \left( \phi \wedge F_A \right) \\
& = D_A^*F_A - * \left( \tau_0 \psi \wedge F_A -\phi \wedge D_A F_A \right) \\
& = D_A^*F_A.
\end{split}
\end{equation*}
We have used (\ref{instantondef2}) and the Bianchi identity in the last line. Hence, in the nearly-parallel case, any ${\mathbb G}_2$-instanton is Yang-Mills.
The linearization of (\ref{instantondef2}) is
$$\psi \wedge D_A \alpha = 0$$
for $\alpha \in \Omega^1\left( {\mathfrak g}_E \right),$ and an infinitesimal gauge transformation $u \in \Omega^0 \left( {\mathfrak g}_E \right)$ acts by $u \mapsto D_A u.$ The infinitesimal deformations of a ${\mathbb G}_2$-instanton $A,$ modulo gauge, therefore correspond to the first cohomology group of the following self-dual elliptic complex:
\begin{equation}\label{deformationcomplex}
\Omega^0 \left( {\mathfrak g}_E \right) \stackrel{D_A}{\longrightarrow} \Omega^1 \left( {\mathfrak g}_E \right) \stackrel{ \psi \wedge D_A }{\longrightarrow} \Omega^6 \left( {\mathfrak g}_E \right) \stackrel{D_A}{\longrightarrow} \Omega^7 \left( {\mathfrak g}_E \right).
\end{equation}
Folding (\ref{deformationcomplex}), and writing $d = D_A,$ we obtain the \emph{deformation operator}
\begin{equation}\label{deformationoperator}
\begin{split}
{\mathscr L}_A & : \Omega^0\left( {\mathfrak g}_E \right) \oplus \Omega^1 \left( {\mathfrak g}_E \right) \to \Omega^0\left( {\mathfrak g}_E \right) \oplus \Omega^1 \left( {\mathfrak g}_E \right) \\
& \begin{pmatrix}
u \\
\alpha
\end{pmatrix}
\mapsto
\begin{pmatrix}
d^*\alpha \\
du + * \left( \psi \wedge d\alpha \right)
\end{pmatrix}.
\end{split}
\end{equation}
This is a first-order, self-adjoint, elliptic operator.
Squaring (\ref{deformationoperator}), we obtain
\begin{equation}\label{lausquared}
\begin{split}
{\mathscr L}_A^2 \begin{pmatrix}
u \\
\alpha
\end{pmatrix} & =
\begin{pmatrix}
d^* d u - *\left( d \psi \wedge d \alpha + \psi \wedge F_A \wedge \alpha \right) \\
d d^* \alpha + * \left( \psi \wedge d \left( * \left( \psi \wedge d \alpha \right) + F_A \wedge u \right) \right)
\end{pmatrix} \\
& = \begin{pmatrix}
d^* d u \\
d d^*\alpha + * \left( \psi \wedge d * \left( \psi \wedge d\alpha \right) \right)
\end{pmatrix}.
\end{split}
\end{equation}
Over a compact manifold, integration by parts implies
\begin{equation*}
\ker \mathscr{L}_A = \ker \mathscr{L}_A^2.
\end{equation*}
\begin{comment}
\begin{equation*}
\mathscr{L}_A \begin{pmatrix} u \\ \alpha \end{pmatrix} = 0 \Leftrightarrow \mathscr{L}_A^2 \begin{pmatrix} u \\ \alpha \end{pmatrix} = 0.
\end{equation*}
\end{comment}
Hence, $du \equiv 0$ for any infinitesimal deformation on a compact nearly parallel ${\mathbb G}_2$-manifold, and $u \equiv 0$ if $A$ is irreducible.
We shall use the following interior product notation:
\begin{equation}\label{2formaction}
\begin{split}
dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \frac{\partial}{\partial x^1} = dx^0, \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \frac{\partial}{\partial x^0} = - dx^1, \quad \text{{\it etc.}}
\end{split}
\end{equation}
\begin{comment}
\begin{equation}\label{2formaction}
\begin{split}
e_0 \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} dx^{01} & = e_1, \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} e_1 = e_0\\
e_1 \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} dx^{01} & = - e_0, \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} e_0 = - e_1, \quad \text{{\it etc.}}
\end{split}
\end{equation}
\end{comment}
We also take interior products between differential forms, {\it e.g.}
$$dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} dx^1 = dx^0, \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} dx^{01} = 1$$
and similarly in general using the metric.
In particular, for a 2-form $b,$ we have
$$* \left( \psi \wedge b \right) = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} b.$$
For a ${\mathfrak g}_E$-valued 1-form $\alpha,$ we shall write
\begin{equation*}
{\mathscr L}_A(\alpha) = {\mathscr L}_A \begin{pmatrix}
0 \\
\alpha
\end{pmatrix} = \begin{pmatrix}
d^*\alpha \\
\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d\alpha
\end{pmatrix}.
\end{equation*}
Then (\ref{lausquared}) becomes
\begin{equation}\label{lasquaredfirst}
{\mathscr L}_A^2(\alpha) = d d^*\alpha + \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \right).\end{equation}
\begin{comment}
\begin{proof} This is an algebraic identity, and can be checked for the standard forms $\phi_0$ and $\psi_0$ above (\ref{phi0}-\ref{psi0}). The result is immediate from (\ref{phi0}-\ref{psi0}).
\end{proof}
\end{comment}
\begin{lemma}[\cite{bryantlaplacianremarks}, (2.7-8)]\label{lemma:phiidents} The following identities hold between any positive 3-form $\phi,$ the associated metric $g = g_\phi,$ and the dual 4-form $\psi = *_{\phi} \phi:$
\begin{equation*}
g^{pq} \phi_{p ij} \phi_{q k \ell} = g_{i k} g_{j\ell} - g_{i\ell} g_{j k} + \psi_{i j k \ell}
\end{equation*}
\begin{equation*}
g^{pq} g^{\ell m} \phi_{p\ell i} \psi_{qm jk} = 2 \phi_{ijk}.
\end{equation*}
\end{lemma}
\begin{proof} Since these are zeroth-order identities, it suffices to check them for the standard 3- and 4-form, given by (\ref{phi0}-\ref{psi0}), and the standard metric. This is easily accomplished using the fact that ${\mathbb G}_2$ acts transitively on orthonormal pairs of vectors.
\end{proof}
\begin{lemma}[{\it Cf.} \cite{yuanqideformations}, Lemma 7.1]\label{lemma:dbstar}
For a nearly parallel $G_2$-structure and any 2-form $b,$ there holds
\begin{equation}\label{dbstar:maineq}
d\left( b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi = d^*b - db \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi + \frac{\tau_0}{2} b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi.
\end{equation}
\end{lemma}
\begin{proof}
Write $b = \frac12 b_{ij} dx^i \wedge dx^j$ in normal coordinates. We then have
\begin{equation*}
\begin{split}
\left( d \left( b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right)_k & = \frac12 \left( \frac12 \left( \nabla_m (b_{ij} \phi_{ij\ell}) - \nabla_\ell (b_{ij} \phi_{ijm}) \right) \right) \phi_{m \ell k} \\
& = \frac14 \left( \nabla_m b_{ij} \phi_{ij \ell} + b_{ij} \nabla_m \phi_{ij\ell} - \nabla_\ell b_{ij} \phi_{ijm} - b_{ij} \nabla_{\ell} \phi_{ijm} \right) \phi_{m \ell k} \\
& = - \frac12 \nabla_m b_{ij} \phi_{\ell ij} \phi_{\ell mk} + \frac{\tau_0}{8} b_{ij} \psi_{m ij \ell} \phi_{m \ell k}
\end{split}
\end{equation*}
where we have used Lemma \ref{lemma:gradphi} in the last line.
By Lemma \ref{lemma:phiidents}, this becomes
\begin{equation*}
\begin{split}
\left( d \left( b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right)_k & = - \frac12 \nabla_m b_{ij} \left( g_{im} g_{jk} - g_{ik} g_{jm} + \psi_{ijmk} \right) + \frac{\tau_0}{4} b_{ij} \phi_{ijk} \\
& = - \nabla_i b_{ik} - \frac12 \nabla_m b_{ij} \psi_{mijk} + \frac{\tau_0}{4} b_{ij} \phi_{ijk}
\end{split}
\end{equation*}
which agrees with the expression (\ref{dbstar:maineq}).
\end{proof}
\begin{prop}\label{prop:weitz} For a ${\mathbb G}_2$-instanton with respect to a nearly parallel ${\mathbb G}_2$-structure $\phi,$ we have
\begin{equation}\label{weitwithsa}
\begin{split}
\mathscr{L}_A^2(\alpha)
& = \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha + \mathscr{S}_A \left( \alpha \right)
\end{split}
\end{equation}
where
$$\mathscr{S}_A\left( \alpha \right) = \nabla_A^*\nabla_A \alpha + {\mathrm{Ric}}(\alpha) - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha \right]$$
is the Yang-Mills stability operator (see Bourguignon-Lawson \cite{bourguignonlawson}).
In the case of the round $7$-sphere, we have
\begin{equation}\label{LAsquared}
\begin{split}
\mathscr{L}_A^2(\alpha ) & = 2 \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha + \nabla_A^*\nabla_A \alpha + 6 \alpha - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha \right] .
\end{split}
\end{equation}
\end{prop}
\begin{proof} From (\ref{lasquaredfirst}) and Lemma \ref{lemma:dbstar}, we have
\begin{equation}\label{weitzmess}
\begin{split}
{\mathscr L}_A^2(\alpha) & = d d^*\alpha + \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d\alpha \right) \\
& = d d^*\alpha + d^* d \alpha - d^2 \alpha \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \psi + \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \\
& = \left( d d^* + d^* d \right) \alpha - \left( F_A \wedge \alpha \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi + \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \\
& = \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha + \left( d d^* + d^* d \right) \alpha + \left( F_A \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi \right) \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha.
\end{split}
\end{equation}
But since $A$ is an instanton, we have
$$F_A \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi = * \left( F_A \wedge \phi \right) = - F_A.$$
Substituting into (\ref{weitzmess}) and applying the Bochner formula yields (\ref{weitwithsa}). Then (\ref{LAsquared}) is obtained by substituting $\tau_0 = 4$ and ${\mathrm{Ric}}_g = 6g$ on the round 7-sphere.
\end{proof}
\begin{rmk} Instantons on compact manifolds with holonomy ${\mathbb G}_2$ (other than the Levi-Civit{\`a} connection) were first constructed by Walpuski \cite{walpuskig2onkummer}, and have subsequently been produced in a few different contexts.
For the state of the art, see \cite{walpuskiarithmeticinstantons}.
Instantons on nearly parallel ${\mathbb G}_2$-manifolds have been studied by Harland and N{\"o}lle \cite{harlandnolle}, and by Ball and Oliveira \cite{balloliveira} in the case of Aloff-Wallach spaces. Ragini Singhal \cite{raginisinghal} also studies instantons on homogeneous nearly parallel ${\mathbb G}_2$-manifolds, using methods similar to those of Charbonneau and Harland \cite{charbharland} in the context of nearly K{\"a}hler 6-manifolds.
\end{rmk}
\section{Hopf fibration and standard instantons}\label{sec:quatinst}
In this section, we give an explicit description of the standard (A)SD instanton and its ${\mathbb G}_2$ relative. We shall use the following variant of Atiyah's quaternionic notation \cite{atiyahgeometry}:
\begin{equation*}
\begin{split}
x & = x^0 \mathbf{1} - x^1 {\mathbf i} - x^2 {\mathbf j} - x^3 {\mathbf k}, \qquad \qquad \,\,\,\, \bar{x} = x^0 \mathbf{1} + x^1 {\mathbf i} + x^2{\mathbf j} + x^3 {\mathbf k} \\
dx & = dx^0 \mathbf{1} - dx^1 {\mathbf i} - dx^2 {\mathbf j} - dx^3 {\mathbf k} , \qquad d\bar{x} = dx^0 \mathbf{1} + dx^1 {\mathbf i} + dx^2 {\mathbf j} + dx^3 {\mathbf k}.
\end{split}
\end{equation*}
Here $x$ and $\bar{x}$ are $\H$-valued functions, and $dx$ and $d\bar{x}$ are $\H$-valued differential forms on ${\mathbb R}^4,$ with signs determined by the convention (\ref{quaternionbasis}). We define $y$ and $\bar{y}$ similarly, and will identify
\begin{equation*}
{\mathbb R}^8 = \H_x \oplus \H_y
\end{equation*}
as above.
\subsection{The quaternionic Hopf fibration(s)}\label{ss:hopf}
The (right) Hopf fibration
\begin{equation}\label{hopf}
\pi: \H^2 \setminus \{ (0,0) \} \rightarrow \H \P^1
\end{equation}
is the quotient projection under right-multiplication by $\H^{\times}.$ The restriction of (\ref{hopf}) to the 7-sphere and to multiplication by ${\mathrm{Sp} }(1)_r$ gives the fibration (\ref{quathopf}).
To see the identification $\H \P^1 \cong S^4$
explicitly, observe that ${\mathrm{Sp} }(2)$ acts on $S^7$ by isometries commuting with ${\mathrm{Sp} }(1)_r,$ where the stabilizer of a fiber is the subgroup ${\mathrm{Sp} }(1)_l \times {\mathrm{Sp} }(1)_l \subset {\mathrm{Sp} }(2).$ The group ${\mathrm{Sp} }(2)$ has a 5-dimensional irreducible representation $W,$ given by the conjugation action on the space of $2\times 2$ traceless self-adjoint quaternionic matrices:
\begin{equation}\label{wdef}
W = \left\{ \left( \begin{array}{cc}
a & \bar{z} \\
z & -a \end{array} \right) \mid a \in {\mathbb R}, z \in \H \right\}.
\end{equation}
This action sets up a map
\begin{equation*}\label{quotientid}
\H \P^1 = {\mathrm{Sp} }(2)/ {\mathrm{Sp} }(1)_l \times {\mathrm{Sp} }(1)_l \,\, \tilde{\longrightarrow} \,\, {\mathrm{SO} }(5) / {\mathrm{SO} }(4) = S^4
\end{equation*}
which is an isometry, up to a factor of $1/2.$
\begin{comment}
We claim that the tautological bundle over $\H \P^1$ is the positive spin bundle $S^+ \to S^4,$ and its orthogonal complement is $S^-.$
To check this, it suffices to write down the Clifford multiplication $\gamma : TS^4 \to \mathrm{Hom} \left( S^+ , S^- \right),$ satisfying
\begin{equation}\label{cliffordrule}
\gamma(v)^* \gamma(v) = |v|^2.
\end{equation}
But a tangent vector $v \in T_x\H \P^1$ is exactly an $\H_r$-linear map from the tautological bundle to its complement, {\it i.e.}, $\gamma(v): S^+_x \to S^-_x.$ Such a map is necessarily given by multiplication by a quaternion on the left, which automatically satisfies (\ref{cliffordrule}).
With the orientation convention on $\H \P^1$ coming from (\ref{quaternionbasis}), one checks that $\gamma$ sets up an isomorphism
\begin{equation*
\Lambda^2_+(T^*S^4) \tilde{\longrightarrow} {\text{End}}_0 \, S^+
\end{equation*}
as required.
\end{comment}
We write $P^+$ for the principal ${\mathrm{Sp} }(1)$-bundle associated to the right Hopf fibration. The \emph{standard self-dual instanton} is defined to be the connection on $P^+$ induced by the round metric on the total space. The corresponding connection form on $P^+$ is the $\mathfrak{su}(2)$-valued 1-form
\begin{equation*
\frac{ \Im \left[ \bar{x} \, d x + \bar{y} \, d y \right] }{ |x|^2 + |y|^2 }.
\end{equation*}
Pulling back to ${\mathbb R}^4$ by the map $x \mapsto (x,1),$
we obtain the well-known connection matrix
\begin{comment} instanton on ${\mathbb R}^4:$
\begin{equation*}\label{standardsdinst}
\frac{\Im \bar{x} dx }{1 + |x|^2}.
\end{equation*}
Its curvature is the $\mathfrak{su}(2)$-valued self-dual 2-form
\begin{equation*}
\frac{ d\bar{x} \wedge d x}{\left( 1 + |x|^2 \right)^2 }.
\end{equation*}
\end{comment}
\begin{equation*
\frac{\Im \bar{x} d x }{1 + |x|^2}
\end{equation*}
whose curvature is the $\mathfrak{su}(2)$-valued self-dual 2-form
\begin{equation*
\frac{ d \bar{x} \wedge d x }{\left( 1 + |x|^2 \right)^2 }.
\end{equation*}
See Atiyah \cite{atiyahgeometry}, \S 1, for these formulae, as well as a generalization giving the complete ADHM construction.
Similarly, we define the \emph{standard anti-self-dual (ASD) instanton} $P^-$ to be the principal bundle associated to the left Hopf fibration, with connection form
\begin{equation}\label{stdasdconnform}
\frac{ \Im \left[ w\, d \bar{w} + z \, d\bar{z} \right] }{ |w|^2 + |z|^2 }.
\end{equation}
In the stereographic chart on ${\mathbb R}^4,$ this has a connection matrix
\begin{equation}\label{stdinstaconnmatrix}
B_0(x) = \frac{\Im x d \bar{x} }{1 + |x|^2}
\end{equation}
and curvature the $\Im \H$-valued self-dual 2-form
\begin{equation}\label{stdinstacurvature}
F_{B_0}(x) = \frac{ d x \wedge d \bar{x} }{\left( 1 + |x|^2 \right)^2 }.
\end{equation}
The basic link between instantons on $S^7$ and $S^4$ is as follows; a more general statement appears in Proposition \ref{prop:hym} below.
\begin{comment}
It follows from (\ref{quotientid}) that the tautological bundle over $\H \P^1_r$ is the positive spin bundle $S^+ \to S^4,$ and its orthogonal complement is $S^-.$
To check this, it suffices to write down the Clifford multiplication $\zeta : TS^4 \to \mathrm{Hom} \left( S^+ , S^- \right),$ satisfying
\begin{equation}\label{cliffordrule}
\zeta(v)^* \zeta(v) = |v|^2
\end{equation}
for all tangent vectors $v.$ But a tangent vector $v$ to $\H \P^1_r$ at a point $x$ is just an $\H_r$-linear map from the tautological bundle to its complement ({\it i.e.} $\zeta(v): S^+_x \to S^-_x$); such a map is necessarily given by multiplication by a quaternion on the left, which satisfies (\ref{cliffordrule}).
\end{comment}
\begin{comment}
Explicitly, given a point $(a,b)^{T} \in S^7$ and a tangent vector $(x,y)^{T},$ both considered as vectors in $\H^2,$ with
$$ \bar{x}a + \bar{y} b = 0$$ the Clifford multiplication on the total spin bundle $S^+ \oplus S^- \simeq \underline{{\mathbb R}^8}$ is given by left-multiplication by
\begin{equation}
\zeta_{\left( \begin{array}{c} a \\ b \end{array} \right) }\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} a \\ b \end{array} \right) \left( \begin{array}{cc} \bar{x} & \bar{y} \end{array} \right) - \left( \begin{array}{c} x \\ y \end{array} \right)\left( \begin{array}{cc} \bar{a} & \bar{b} \end{array} \right) = \left( \begin{array}{cc}
a \bar{x} - x \bar{a} & x \bar{b} - a \bar{y} \\
y \bar{a} - b \bar{x} & y \bar{b} - b \bar{y} \end{array} \right).
\end{equation}
This interchanges $S^+$ and $S^-,$ and squares to $- \left( |x|^2 + |y|^2 \right),$ as required.
\end{comment}
\begin{comment}
With the orientation convention on $\H \P^1_r$ coming from (\ref{quaternionbasis}), one can also check that $\zeta$ sets up an isomorphism
\begin{equation}\label{selfdualS+}
\Lambda^2_+(T^*S^4) \tilde{\longrightarrow} {\text{End}}_0 \, S^+
\end{equation}
as required.
\end{comment}
\begin{lemma}\label{lemma:instapullback} Let $B$ be a connection on a principal bundle over $S^4.$ Then $B$ is an ASD instanton if and only if the pullback $A = \pi^*B$ by the (right) Hopf fibration is a ${\mathbb G}_2$-instanton, for either the standard or the squashed nearly parallel ${\mathbb G}_2$-structure on $S^7.$
\end{lemma}
\begin{proof}
By ${\mathrm{Sp} }(2)$-invariance both of $\phi_{std}$ and of the fibration, it suffices to consider the point $(1,0).$
From (\ref{phistdfirstdef}), we have
\begin{equation}\label{phis30}
\phi (x,0) = \nu^x + \zeta_1^x \wedge \omega_1^y + \zeta_2^x \wedge \omega_2^y - \zeta_3^x \wedge \omega_3^y.
\end{equation}
and
\begin{equation*}
\phi(1,0) = dx^{123} + dx^1 \wedge \omega_1^y + dx^2 \wedge \omega_2^y - dx^3 \wedge \omega_3^y.
\end{equation*}
Then it is clear that $\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} F_A = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \pi^*F_B$ vanishes if and only if $B$ is ASD.
The same argument applies on $\phi_{sq}.$
\end{proof}
\subsection{The standard ${\mathbb G}_2$-instanton}\label{ss:stdg2inst}
Let
\begin{equation}
P = P^-\times_{S^4} P^+ \stackrel{\pi_2}{\longrightarrow} S^7
\end{equation}
be the fiber product over $S^4$ of $P^-$ with $P^+,$ considered as a principal ${\mathrm{Sp} }(1)$-bundle via the projection to $P^+ = S^7.$
According to \S \ref{ss:hopf} and Lemma \ref{lemma:instapullback}, the pullback of the standard ASD instanton on $P^-$ is a ${\mathbb G}_2$-instanton on $P \to S^7,$ which we call the \emph{standard ${\mathbb G}_2$-instanton}, $A_0.$
We may obtain a connection matrix for $A_0$ by pulling back the connection form of the standard ASD instanton (\ref{stdasdconnform})
by the fiber-preserving map $\H^{\times 2}_r \to \H^{\times 2}_l $ given by
\begin{equation*}
\begin{split}
w & = y^{-1} \\
z & = x^{-1}.
\end{split}
\end{equation*}
\begin{comment}
\begin{equation*}
\begin{split}
\begin{pmatrix}
x \\
y
\end{pmatrix} & \mapsto \begin{pmatrix}
y^{-1} \\
x^{-1}
\end{pmatrix}.
\end{split}
\end{equation*}
\end{comment}
This gives
\begin{equation*}
\begin{split}
A_0(x,y) & = \frac{1}{ |x|^{-2} + |y|^{-2} } \Im \left[ x^{-1} d \left( \bar{x}^{-1} \right) + y^{-1} d \left( \bar{y}^{-1} \right) \right] \\
& = \frac{|x|^2|y|^2}{ |x|^{2} + |y|^{2} } \Im \left[ - x^{-1} \bar{x}^{-1} d \bar{x} \bar{x}^{-1} - y^{-1} \bar{y}^{-1} d\bar{y} \bar{y}^{-1} \right] \\
& = \frac{-1}{|x|^2 + |y|^2} \Im \left[ |y|^2 d\bar{x} \, \bar{x}^{-1} + |x|^2 d\bar{y} \, \bar{y}^{-1} \right] \\
& = \frac{1}{|x|^2 + |y|^2} \Im \left[ |y|^2 x^{-1} \, dx + |x|^2 y^{-1} \, dy \right]
\end{split}
\end{equation*}
which is (\ref{standardg2instanton}) above.
The singularity along the $x$-axis can be removed by applying the gauge transformation $g(x,y) = y/|y|:$ noting that
$dg g^{-1} = - \Im y d\bar{y} / |y|^2,$ we obtain
\begin{equation*}
\begin{split}
g(A_0) = gA_0 \bar{g} - dg \bar{g} & = \frac{1}{|x|^2 + |y|^2} \Im \left[ y x^{-1} dx \bar{y} + \frac{|x|^2}{|y|^2} dy \bar{y} + \frac{|x|^2 + |y|^2}{|y|^2} y d\bar{y} \right] \\
& = \frac{1}{|x|^2 + |y|^2} \Im \left[ y x^{-1} dx \bar{y} + \frac{|x|^2}{|y|^2} dy \bar{y} - \frac{|x|^2 }{|y|^2} dy \bar{y} + y d\bar{y} \right] \\
& = \frac{1}{|x|^2 + |y|^2} \Im \left[ y x^{-1} dx \bar{y} + y d\bar{y} \right].
\end{split}
\end{equation*}
The singularity along the $y$-axis can be removed similarly.
However,
they cannot be removed simultaneously, since the bundle $P \to S^7$ is nontrivial;
recall that topological ${\mathrm{SU} }(2)$-bundles on $S^7$ are classified by $\pi_6(S^3) = {\mathbb Z}_{12}.$
\begin{prop}[Crowley and Goette \cite{crowleygoettekreck}, (1.18)]
For an $SU(2)$-bundle on $S^4$ with $c_2(E) = \kappa,$ the pullback bundle
on $S^7$ has homotopy class
\begin{equation*}
\frac{\kappa \left( \kappa + 1 \right)}{2} \in {\mathbb Z}_{12}.
\end{equation*}
\end{prop}
\subsection{Curvature calculation}\label{ss:curvcalc} We check directly that $A_0$ is a ${\mathbb G}_2$-instanton on $S^7.$ By (\ref{phistdzerothdef}), this is equivalent to showing that (\ref{standardg2instanton}) is a ${\mathrm{Spin} }(7)$-instanton on ${\mathbb R}^8.$ We calculate
\begin{equation*}
\begin{split}
d \left( x^{-1} dx \right)
& = - x^{-1} dx \wedge x^{-1} dx
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
d \left( \frac{ |y|^2}{|x|^2 + |y|^2} \right)
& = \frac{ |x|^2|y|^2}{\left( |x|^2 + |y|^2 \right)^2} 2 \Re \left[ y^{-1} dy - x^{-1} dx \right] \\
& = - \, d \left( \frac{|x|^2}{|x|^2 + |y|^2} \right).
\end{split}
\end{equation*}
This gives
\begin{equation}\label{dacalc}
\begin{split}
dA_0 & = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ \begin{split}
& x^{-1} dx \wedge \left(-2 |x|^2|y|^2 \Re \left[ y^{-1} dy - x^{-1} dx \right] - \left( |y|^4 + |x|^2 |y|^2 \right) x^{-1} dx \right) \\
& + y^{-1} dy \wedge \left(-2 |x|^2 |y|^2 \Re \left[ x^{-1} dx - y^{-1} dy \right] - \left( |x|^4 + |x|^2 |y|^2 \right) y^{-1} dy \right) \end{split} \right] \\
& = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ \begin{split}
& x^{-1} dx \wedge \left(-2 |x|^2|y|^2 \Re y^{-1} dy + |y|^2 d\bar{x} x - |y|^4 x^{-1} dx \right) \\
& + y^{-1} dy \wedge \left( - 2 |x|^2 |y|^2 \Re x^{-1} dx + |x|^2 d\bar{y} y - |x|^4 y^{-1} dy \right) \end{split} \right] \\
& = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ \begin{split}
& |y|^2 x^{-1} dx \wedge d\bar{x} x + |x|^2 y^{-1} dy \wedge d\bar{y} y - \bar{x} dx \wedge d \bar{y} y - \bar{y} dy \wedge d\bar{x} x \\
& - 2 |x|^2 |y|^2 y^{-1} dy \wedge x^{-1} dx - |x|^4 y^{-1} dy \wedge y^{-1} dy - |y|^4 x^{-1} dx \wedge x^{-1} dx \end{split} \right].
\end{split}
\end{equation}
On the other hand, we have
\begin{equation}\label{asquaredcalc}
A_0 \wedge A_0 = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ |y|^4 x^{-1} dx \wedge x^{-1} dx + |x|^4 y^{-1} dy \wedge y^{-1} dy + 2 |x|^2 |y|^2 x^{-1} dx \wedge y^{-1} dy \right].
\end{equation}
Adding (\ref{dacalc}) and (\ref{asquaredcalc}), we obtain the curvature form
\begin{equation}\label{stdg2instcurv}
\begin{split}
F_{A_0} (x,y) & = dA_0 + A_0 \wedge A_0 \\
& = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[
|y|^2 x^{-1} dx \wedge d\bar{x} x + |x|^2 y^{-1} dy \wedge d\bar{y} y - 2 \bar{x} dx \wedge d \bar{y} y \right] \\
& = \frac{1}{ \left(|x|^2 + |y|^2 \right)^2 |x|^2 |y|^2 } \Im \left[ \overline{\left( d\bar{x} x |y|^2 - d \bar{y} y |x|^2 \right)} \wedge \left(d\bar{x} x |y|^2 - d \bar{y} y |x|^2 \right) \right].
\end{split}
\end{equation}
The 2-forms $dx \wedge d\bar{x},$ $dy \wedge d\bar{y},$ and $dx \wedge d \bar{y}$ are each invariant under ${\mathrm{Sp} }(1)_r.$ Therefore, the 2-form part of $F_{A_0}$ lies in ${\mathrm{Lie}}({\mathrm{Sp} }(2)) \subset {\mathrm{Lie}}({\mathrm{Spin} }(7)),$ as claimed.
\begin{comment}
\begin{rmk}
If we apply the gauge transformation $g(x,y) = \bar{y}/|y|,$ as before, we get
\begin{equation}\label{stdg2curvatureingauge}
\begin{split}
F_{g(A_0)} = g F_{A_0} \bar{g} (x,y) & = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[
\bar{y} x d\bar{x} \wedge dx x^{-1} y + |x|^2 d\bar{y} \wedge dy - 2 d \bar{y} \wedge dx \bar{x} y \right].
\end{split}
\end{equation}
Notice that in this gauge, the curvature along the $x$-axis is given by
\begin{equation}
\begin{split}
F_{g(A_0)} (x,0) & = \frac{
d\bar{y} \wedge dy }{ |x|^2 }.
\end{split}
\end{equation}
Lemma \ref{lemma:eigenvalues} and the Weitzenbock formula (\ref{LAsquared}) can be used to show that all deformations of the standard instanton are of the form $ F_{A_0} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} X$ for vector fields $X$ on $S^7.$
\end{rmk}
\end{comment}
\subsection{Linear deformations}\label{ss:lineardefos}
Let $A$ be a conical instanton on ${\mathbb R}^8 \setminus \{0\}$
whose curvature $F_A$ takes values in ${\mathrm{Lie}}({\mathrm{Sp} }(2)) \otimes {\mathfrak g}_E.$ Suppose that $F_A(x,y)$ spans ${\mathrm{Lie}}({\mathrm{Sp} }(2))$ as $(x,y)$ varies over ${\mathbb R}^8.$ From (\ref{stdg2instcurv}), this is seen to be the case for $A_0.$
Given any $8\times 8$ matrix $M,$ we associate the vector field
\begin{equation*}
X_M = M \cdot \vec{r} = M^i{}_{j} x^j \frac{\partial}{\partial x^i}
\end{equation*}
on ${\mathbb R}^8,$ as well as the ${\mathfrak g}_E$-valued 1-form
\begin{equation}\label{lineardeformation}
\alpha_M = X_M \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} F_{A} \in \Omega^1 \left( {\mathfrak g}_E \right).
\end{equation}
Notice that (\ref{lineardeformation}) corresponds to pushforward by the diffeomorphism generated by $X_M,$ with its horizontal lift to the bundle. For, working in a local gauge where $A_{X_M} = 0,$ we have
$$\frac{d}{dt} \exp(-tX_M)^* A = X_M \left( A \right) = F(X_M, -) = \alpha_M.$$ The action on the curvature is given by
\begin{equation}\label{deformationexpthing}
\frac{d}{dt} \exp(-tX_M)^* F_A = \frac{d}{dt} F_{\exp^*(-t X_M) A} = D_{A} \alpha_M.
\end{equation}
\begin{lemma}\label{lemma:coulomb} We have
\begin{equation}
\left( D_{A}^{S^7} \right)^* \alpha_M = 0
\end{equation}
if and only if $M \in {\mathrm{Lie}}({\mathrm{Sp} }(2))^\perp \subset{\mathbb R}^{8\times 8}.$
\end{lemma}
\begin{proof} Let $\alpha = \alpha_M$ and $F = F_A.$ In coordinates on ${\mathbb R}^8,$ we have
\begin{equation}\label{R8coulomb}
\left( D_A^{{\mathbb R}^8} \right)^*\alpha = \nabla^i X^j F_{ij} + X^j \nabla^i F_{ij} = M_{ij} F_{ij}.
\end{equation}
Since the curvature spans ${\mathrm{Lie}}({\mathrm{Sp} }(2)),$ the expression (\ref{R8coulomb}) vanishes identically if and only if $M$ lies in the orthogonal complement of ${\mathrm{Lie}}({\mathrm{Sp} }(2)).$
The result then follows from the formula
$$\left( D_A^{{\mathbb R}^8} \right)^*\alpha = \left( D_A^{S^7} \right)^*\alpha - \langle \hat{r}, \nabla_{\hat{r}} \left( \alpha \left( \hat{r} \right) \right) \rangle$$
and the fact that $\alpha(\hat{r}) \equiv 0$ for a conical instanton.
\end{proof}
\begin{prop}\label{prop:15deformations} Let $W$ be the 5-dimensional space (\ref{wdef}).
Then $\ker {\mathscr L}_A$ contains the 15-dimensional space
\begin{equation}\label{15deformations}
\{ \alpha_M \mid M \in W \oplus W I_1 \oplus W I_2 \}
\end{equation}
where $\alpha_M$ is defined by (\ref{lineardeformation}).
\end{prop}
\begin{proof}
According to (\ref{deformationexpthing}), the space (\ref{15deformations}) must be contained in $\ker {\mathscr L}_A.$ For, the subspace $\{\alpha_M \mid M \in W\}$ corresponds to pushforward by elements of ${\mathrm{GL} }(2, \H),$ which preserve the algebra ${\mathrm{Lie}}({\mathrm{Sp} }(2)).$
Meanwhile, per the decomposition (\ref{spin7splitting}), we have
$$W I_1 \oplus W I_2 \subset \Lambda^2_d \oplus \mathfrak{F}_{x,y} \subset {\mathrm{Lie}} \left( {\mathrm{Spin} }(7) \right).$$
Since the curvature $F_A$ spans ${\mathrm{Lie}}({\mathrm{Sp} }(2))$ pointwise, these elements are all nonzero. By Lemma \ref{lemma:coulomb}, the space (\ref{15deformations}) is in Coulomb gauge.
\end{proof}
\begin{comment}
\begin{proof}[Proof of Proposition \ref{prop:15deformations}] By the construction of \S \ref{ss:stdg2inst}, we know that $A_0$ is invariant modulo gauge under ${\mathrm{Sp} }(2).$ Hence, the action of ${\mathrm{Sp} }(2) \subset {\mathrm{Spin} }(7)$ on vector fields gives an action on the space $\{ X \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} F_{A_0} \} \subset \Omega^1\left( {\mathfrak g}_E \right),$ which preserves the kernel of the deformation operator ${\mathscr L}_{A_0}.$
Let $R = {\mathrm{Lie}}({\mathrm{Sp} }(2)) \subset \Lambda^2 {\mathbb R}^8$ be the adjoint representation of ${\mathrm{Sp} }(2),$ and $W$ as above. We may decompose ${\mathbb R}^8 \otimes {\mathbb R}^8 = \Lambda^2 {\mathbb R}^8 \oplus {\mathrm{Sym} }^2{\mathbb R}^8$ into irreducible ${\mathrm{Sp} }(2)$-representations as
\begin{equation}\label{15deformations:so8}
\begin{split}
\Lambda^2 {\mathbb R}^8 = R \oplus W I_1 \oplus W I_2 \oplus W I_3\oplus \langle I_1 \rangle \oplus \langle I_2 \rangle \oplus \langle I_3 \rangle
\end{split}
\end{equation}
and
\begin{equation}\label{15deformations:sym8}
{\mathrm{Sym} }^2{\mathbb R}^8 = W \oplus R I_1 \oplus R I_2 \oplus R I_3 \oplus \langle \mathrm{Id} \rangle.
\end{equation}
We first check the factors of (\ref{15deformations:so8}). According to (\ref{spin7splitting}), $W I_1 \oplus W I_2$ belongs to ${\mathrm{Lie}}({\mathrm{Spin} }(7)),$ and by Lemma \ref{lemma:coulomb}, the subspace $\{ \alpha_M \mid M \in W I_1 \oplus W I_2\}$ is in Coulomb gauge. Hence this subspace is contained in $\ker {\mathscr L}_{A_0}.$ Also by Lemma \ref{lemma:coulomb}, the subspace $\{\alpha_M \mid M \in R \}$ is not in Coulomb gauge (and indeed is trivial modulo gauge).
The three trivial representations in (\ref{15deformations:so8}) yield zero when substituted into $F = F_{A_0}.$
It remains to rule out the factor $W I_3,$ for which it suffices to check that the vector field
\begin{equation}
X = \left( \begin{array}{cc}
I_3 & 0 \\
0 & -I_3 \end{array} \right) \vec{r} = \left( \begin{array}{c}
-x \cdot {\mathbf k} \\
y \cdot {\mathbf k} \end{array} \right)
\end{equation}
does not yield an element of $\ker {\mathscr L}_{A_0},$ as follows.
As remarked above, these correspond to known deformations of $A_0,$ which are in Coulomb gauge by Lemma \ref{lemma:coulomb}. The given space (\ref{15deformations}) therefore lies within $\ker {\mathscr L}_A.$
\end{proof}
\end{comment}
\begin{comment}
\begin{thm}\label{thm:stddefs} The space of infinitesimal deformations of $A_0$ consists of the 15-dimensional space of linear deformations (\ref{15deformations}).
\end{thm}
\begin{proof} This is a special case of Theorem \ref{thm:generaldefos} below, for which an independent proof is given in the Appendix.
\end{proof}
\end{comment}
\section{Infinitesimal deformations}
\label{sec:generaldefos}
In this section, we calculate the infinitesimal deformations, as a ${\mathbb G}_2$-instanton, of the pullback to $S^7$ of a general irreducible ASD instanton on $S^4.$ We write
$$\nabla = \nabla^{S^7} = \pi_{S^7} \circ \nabla^{{\mathbb R}^8}$$
for the Levi-Civita connection on $S^7_{std},$ which we shall couple to the connection on any auxiliary bundle. We shall also write
\begin{equation}\label{nablavh}
\nabla^v = \nabla_{\pi_v}, \qquad \nabla^h = \nabla_{\pi_h}.
\end{equation}
Here $\pi_v$ is the orthogonal projection to the the vertical tangent space of the Hopf fibration, and $\pi_h$ is the complementary projection.
\subsection{Vertical and horizontal components}
Let $\Omega^1_h$ be the annihilator of vertical vector fields along the Hopf fibration, and $\Omega^1_v$ its orthogonal complement. We have
\begin{equation*}
\Omega^1_{S^7} = \Omega^1_v \oplus \Omega^1_h.
\end{equation*}
Letting
$$\Omega^{(p,q)} = \Lambda^p \Omega^1_v \otimes \Lambda^q \Omega^1_h \subset \Omega^{p + q}_{S^7}$$
we have a decomposition
$$\Omega^{k}_{S^7} = \bigoplus_{p + q = k} \Omega^{(p,q)}.$$
An element of $\Omega^{(p,q)}$ will be referred to as a $(p,q)$-form.
Let $\nu$ denote the $(3,0)$ volume form of the Hopf fibration, and let
$$\bar{\nu} = * \nu.$$
The $(0,2)$-forms split as
$$\Omega^{(0,2)} = \Omega^{2+}_h \oplus \Omega^{2-}_h$$
where $\Omega^{2\pm}_h$ are the (anti-)self-dual components
with respect to the $(0,4)$ volume form $\bar{\nu}.$
For a $(0,1)$-form $b,$ we shall write $d^v b$ for the $(1,1)$ part of $db$ and $d^h b$ for the $(0,2)$ part. A similar notation will be used for $(1,0)$-forms (see Lemma \ref{lemma:dalpha} below).
\begin{defn}\label{defn:coframes} Let $\omega_i^x, \omega_i^y, \zeta_i^x,$ and $\zeta_i^y$ be as in (\ref{2formbases}) and (\ref{zetaidef}). For $i = 1,2,3,$ define the global ${\mathrm{Sp} }(2)$-invariant forms on $S^7$:
\begin{equation*}
\begin{split}
\zeta_i & = \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \left( \omega^x_i + \omega^y_i \right) \\
\omega_i^\circ & = \left. \omega^x_i + \omega^y_i \right|_{S^7} \\
\bar{\omega}_i & = \omega_i^\circ - \frac{1}{2}\epsilon_{ijk} \zeta_j \wedge \zeta_k \\
\end{split}
\end{equation*}
Notice that $\{\zeta_i\}$ and $\{ \bar{\omega}_i \}$ are global frames for $\Omega^1_v$ and $\Omega^{2+}_h,$ respectively. The vertical volume form is given by
$$\nu = \zeta_1 \wedge \zeta_2 \wedge \zeta_3.$$
From (\ref{phistdfirstdef}) and the ${\mathrm{Sp} }(2)$-invariance, we may re-express $\phi_{std}$ as follows:
\begin{equation}\label{phistdsp2def}
\begin{split}
\phi_{std}
& = \stackrel{ (3,0) }{\overbrace{ \nu} } + \stackrel{ (1,2) }{\overbrace{ \zeta_1 \wedge \bar{\omega}_1 + \zeta_2 \wedge \bar{\omega}_2 - \zeta_3 \wedge \bar{\omega}_3 }}.
\end{split}
\end{equation}
\end{defn}
\noindent We now derive the basic properties of these frames, and use them to decompose the Laplace operator into horizontal and vertical parts.
\begin{lemma}\label{lemma:gradgammai} The frame $\{\zeta_i\}$ is coclosed, and satisfies
\begin{equation}\label{gradgammai:gradvh}
\nabla^v \zeta_i = \frac{1}{2} \epsilon_{ijk} \zeta_j \wedge \zeta_k, \qquad \nabla^h \zeta_i = \bar{\omega}_i
\end{equation}
\begin{equation}\label{gradgammai:laplace}
\nabla^* \nabla^v \zeta_i = 2 \zeta_i, \qquad \nabla^* \nabla^h \zeta_i = 4 \zeta_i.
\end{equation}
Here $\nabla^v, \nabla^h$ are defined by (\ref{nablavh}) above.
\end{lemma}
\begin{proof} From the Definition \ref{defn:coframes} and (\ref{zetaidef}), for $X, Y \in T S^7,$ we have
\begin{equation}\label{gradgammai:grad}
\left( \nabla_X \zeta_i \right) (Y) = \left( \nabla^{{\mathbb R}^8}_X \zeta_i \right)(Y) = \left( \omega^x_i + \omega^y_i \right)(X,Y) = \omega^\circ_i (X, Y).
\end{equation}
Coclosedness of $\zeta_i$ and (\ref{gradgammai:gradvh}) follow directly from (\ref{gradgammai:grad}).
Since $\zeta_i$ is coclosed and equal to the restriction of a linear form, we have
\begin{equation}\label{gradgammai:spherelaplace}
\nabla^*\nabla_{S^3} \zeta_i = 2 \zeta_i, \qquad \nabla^*\nabla_{S^7} \zeta_i = 6 \zeta_i
\end{equation}
as can be verified directly from (\ref{gradgammai:grad}). Then (\ref{gradgammai:laplace}) follows from (\ref{gradgammai:spherelaplace}) and the fact that $\nabla = \nabla^v + \nabla^h.$
\end{proof}
\begin{lemma}\label{lemma:dalpha}
Let $\alpha = a + b$ be a 1-form on $S^7,$ with $a = f_i \zeta_i \in \Omega^1_v $ and $b \in \Omega^1_h .$ Then
\begin{equation*}
\begin{split}
d \alpha = \,\, \stackrel{ (2,0) }{\overbrace{ d^v a }} & + \stackrel{ (1,1)}{ \overbrace{d^h \! f_i \wedge \zeta_i } } + \stackrel{ (0,2) }{\overbrace{2 f_i \bar{\omega}_i }} \\
& + d^v b \qquad + \, d^h b. \end{split}
\end{equation*}
\end{lemma}
\begin{proof} This is a standard decomposition result, which can be seen directly as follows. By Lemma \ref{lemma:gradgammai}, we have
\begin{equation*}
d \zeta_i = 2 \omega_i^\circ, \quad d^v \zeta_i = \epsilon_{ijk} \zeta_j \wedge \zeta_k, \quad i = 1,2,3.
\end{equation*}
Therefore
\begin{equation}\label{dgammai}
d \zeta_i = \stackrel{ (2,0) }{\overbrace{ \epsilon_{ijk} \zeta_j \wedge \zeta_k }} + \stackrel{ (0,2) }{\overbrace{ 2 \bar{\omega}_i. }}
\end{equation}
The result follows directly from (\ref{dgammai}).
\end{proof}
\begin{lemma}\label{lemma:coclosed} If $a = f_i \zeta_i$ is (co)closed on each $S^3$ fiber, then $\left( \nabla^*\nabla^h f_i \right) \zeta_i$ is again fiberwise (co)closed.
\end{lemma}
\begin{proof} Let $U_j$ be the dual vector field of $\zeta_j,$ for $j = 1,2,3.$ The Killing vector fields $U_j$ commute with the operators $\nabla^*\nabla$ and $\nabla^*\nabla^v,$ hence also with $\nabla^* \nabla^h = \nabla^*\nabla - \nabla^*\nabla^v.$ Since $d^v a$ and $(d^v)^* a = d^*a$ are determined by $U_j (f_i),$ we conclude that $\nabla^* \nabla^h$ preserves (co)closedness on the fibers.
\end{proof}
\begin{comment}
Let $\{e_j\}_{j = 1}^7$ be a basis of vector fields which is orthonormal and satisfies $\nabla_{e_j} e_k (p)= 0$ at a given point $p,$ with
\begin{equation}
e_j(p) \in T_v S^7, \quad j = 1,2,3.
\end{equation}
\end{comment}
\begin{prop}\label{prop:laplacev} Let $\alpha = a + b$ be a 1-form as above, where $a = f_i \zeta_i$ and $b \in \Omega^1_h.$ The vertical component of the Laplacian on $S^7$ is given by
\begin{equation}\label{laplacev:maineq}
\left( \nabla^* \nabla \alpha \right)^v = \nabla^* \nabla^v a + 4a + \left(\nabla^* \nabla^h \! f_i + 2\langle d^h b, \bar{\omega}_i \rangle \right)\zeta_i.
\end{equation}
Here $\nabla^* \nabla^v$ denotes the Laplacian on the $S^3$ fiber.
\end{prop}
\begin{comment}
\begin{rmk}
Here the inner product as 2-tensors is intended:
$$\langle \omega_i , \nabla b \rangle = \langle \omega_i(e_j, e_k), \nabla_{e_j} b (e_k ) \rangle.$$
\end{rmk}
\end{comment}
\begin{proof}
We have
\begin{equation*}
\left( \nabla^*\nabla \alpha \right)^v = \left( \nabla^*\nabla a \right)^v + \left( \nabla^*\nabla b \right)^v.
\end{equation*}
For the first term, we write
\begin{equation}\label{laplacev:avfirst}
\left( \nabla^*\nabla a \right)^v = \nabla^*\nabla^v a + \left( \nabla^* \nabla^h a \right)^v
\end{equation}
and
\begin{equation*}
\begin{split}
\left( \nabla^* \nabla^h a \right)^v & = \left( \nabla^* \left( \nabla^h \! f_i \zeta_i + f_i \nabla^h \zeta_i \right) \right)^v \\
& = \left( \nabla^* \nabla^h f_i \right) \zeta_i + \left( \nabla^h f_i \nabla^*\zeta_i - \nabla f_i \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \bar{\omega}_i \right)^v + f_i \nabla^* \nabla^h \zeta_i \\
& = \left( \nabla^* \nabla^h f_i \right) \zeta_i + 4 f_i \zeta_i
\end{split}
\end{equation*}
where we have used Lemma \ref{lemma:gradgammai}. Then (\ref{laplacev:avfirst}) yields
\begin{equation}\label{laplacev:av}
\left( \nabla^*\nabla a \right)^v = \nabla^*\nabla^v a + 4 a + \nabla^* \nabla^h f_i \zeta_i
\end{equation}
which gives the case $b = 0$ of the formula (\ref{laplacev:maineq}).
Assuming now that $Y$ is horizontal and $b$ is of type $(0,1),$ we compute
\begin{equation*}
\begin{split}
0 \equiv \nabla_Y \langle \zeta_i , b \rangle & = \langle \nabla_Y \zeta_i , b \rangle + \langle \zeta_i, \nabla_X b \rangle \\
\langle \zeta_i, \nabla_Y b \rangle & = - \langle \bar{\omega}_i(Y, -) , b \rangle.
\end{split}
\end{equation*}
For $U$ vertical, we have
\begin{equation*}
\begin{split}
0 \equiv \nabla_U \langle \zeta_i , b \rangle & = \langle \nabla_U \zeta_i , b \rangle + \langle \zeta_i, \nabla_U b \rangle
\end{split}
\end{equation*}
and
\begin{equation*}
\langle \zeta_i, \nabla_U b \rangle = 0.
\end{equation*}
Hence, for $X \in TS^7,$ we have
\begin{equation*}
\left( \nabla_X b \right)^v = - \zeta_i \langle \bar{\omega}_i(X, -) , b \rangle.
\end{equation*}
Next, let $\{e_j\}_{j = 1}^7$ be an orthonormal basis of vector fields that satisfies $\nabla_{e_j} e_k = 0$ at a given point. We compute
\begin{equation*}
\begin{split}
0 & \equiv \nabla_{e_j} \nabla_{e_j} \langle \zeta_i , b \rangle \\
& = \langle \nabla_{e_j} \nabla_{e_j} \zeta_i , b \rangle + 2 \langle \omega_i^\circ \left(e_j, - \right), \nabla_{e_j} b \rangle + \langle \zeta_i, \nabla_{e_j} \nabla_{e_j} b \rangle \\
& = - 6 \langle \zeta_i , b \rangle + 2 \langle \omega_i^\circ \left(e_j, - \right), \nabla_{e_j} b \rangle + \langle \zeta_i, \nabla_{e_j} \nabla_{e_j} b \rangle.
\end{split}
\end{equation*}
Since $\langle \zeta_i, b \rangle \equiv 0,$ we are left with
\begin{equation*}
\langle \zeta_i, \nabla_{e_j} \nabla_{e_j} b \rangle = -2 \langle \omega_i^\circ \left(e_j, - \right), \nabla_{e_j} b \rangle
\end{equation*}
which may be rewritten as
\begin{equation}\label{laplacev:bv}
\left( \nabla^* \nabla b \right)^v = 2 \zeta_i \langle \bar{\omega}_i , d b \rangle.
\end{equation}
Combining (\ref{laplacev:av}) and (\ref{laplacev:bv}) yields (\ref{laplacev:maineq}).
\end{proof}
\begin{rmk} The previous results carry over when $\nabla$ is coupled to a connection that is trivial along the fibers of the Hopf fibration.
\end{rmk}
\subsection{Vanishing of the vertical component} This subsection proves our vanishing theorem for the vertical component of an infinitesimal deformation in Coulomb gauge.
\begin{lemma}\label{lemma:horizproperties} Let $B$ be an ASD instanton on $S^4,$ and put $A = \pi^*B.$ Assume that $f$ is a section of $\pi^* {\mathfrak g}_E \to S^7$ satisfying
\begin{equation}\label{horizproperties:assumption}
\nabla_A^h f = 0.
\end{equation}
Then $f = \pi^*h$ is the pullback of a section of ${\mathfrak g}_E \to S^4,$ with
\begin{equation}\label{horizproperties:conclusion}
\nabla_B h = 0.
\end{equation}
\end{lemma}
\begin{proof} We have
\begin{equation}\label{horizproperties:mixedpartials}
0 = \nabla^v \nabla^h f = \nabla^h \nabla^v f
\end{equation}
since $f$ is a $0$-form and the curvature $F_A$ is purely horizontal. By (\ref{horizproperties:assumption}), we have
$$D_A f = D_A^v f.$$
Applying $D_A$ to both sides, by Proposition \ref{lemma:dalpha}, we obtain
\begin{equation*
\begin{split}
D_A^2 f = \left[ F_A , f \right] & = D_A D_A^v f \\
& = (D_A^v)^2 f + D_A^h D_A^v f + 2 D^v f (U_i) \bar{\omega}_i.
\end{split}
\end{equation*}
By (\ref{horizproperties:mixedpartials}) and the fact that $\left( D_A^v \right)^2 = 0$ for a pullback connection, the first two terms vanish, yielding
\begin{equation}\label{horizproperties:curvaturecomparison}
\left[ F_A , f \right] = 2 \left( \nabla_{U_i} f \right) \bar{\omega}_i.
\end{equation}
Each side of (\ref{horizproperties:curvaturecomparison}) is of type $(0,2);$ however, the LHS is anti-self-dual and the RHS is self-dual. Therefore both sides of (\ref{horizproperties:curvaturecomparison}) must vanish, giving
$$\nabla^v f = 0.$$
Hence, $f$ is constant on the fibers, so $f = \pi^*h$ for a section $h$ of ${\mathfrak g}_E.$ Then (\ref{horizproperties:assumption}) implies (\ref{horizproperties:conclusion}), as desired.
\end{proof}
\begin{thm}\label{thm:vertvanishing}\label{thm:vanishing} Let $B$ be an irreducible ASD instanton on a bundle $E \to S^4,$ and $A = \pi^* B$
its pullback under the Hopf fibration.
If $\alpha \in \Omega^1 \left( \pi^*{\mathfrak g}_E \right)$ satisfies ${\mathscr L}_A \alpha = 0,$ then the vertical part of $\alpha$ vanishes.
\end{thm}
\begin{proof} We write $\nabla = \nabla_A$ and $d = D_A$ throughout the proof.
Decompose $\alpha = a + b$ into vertical and horizontal parts as above. Let $a = f_i \zeta_i,$ and write the self-dual part of $d^h b$ as
\begin{equation*}
\left( d^h b \right)^+ = c_i \bar{\omega}_i .
\end{equation*}
From Proposition \ref{lemma:dalpha} and (\ref{phistdsp2def}), our assumption ${\mathscr L}_A \alpha = 0$ implies
\begin{equation}\label{phidalpha}
\begin{split}
0 = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha
& = \stackrel{(1,0)}{ \overbrace{ \quad \qquad \nu \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d^v a \qquad \quad } } + \stackrel{(0,1)}{ \overbrace{ \phi_{1,2} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \left( d^h a + d^v b \right) }} \\
& \,\, + \phi_{1,2} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \left( 2 f_i \bar{\omega}_i + d^h b \right).
\end{split}
\end{equation}
Noting that $\bar{\omega}_i \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \bar{\omega}_i = 2,$ per (\ref{phistdsp2def}), the $(1,0)$-part of (\ref{phidalpha}) comes out to
\begin{equation*}
\begin{split}
0 & = \zeta_1 \left( *d^v a(U_1) + 4 f_1 + 2 c_1 \right) \\
& + \zeta_2 \left( *d^v a(U_2) + 4 f_2 + 2 c_2 \right) \\
& + \zeta_3 \left( *d^v a(U_3) - 4 f_3 - 2 c_3 \right)
\end{split}
\end{equation*}
where $U_j$ is the dual vector field to $\zeta_j,$ for $j = 1,2,3,$ as above. We conclude that
\begin{equation}\label{ciformulafirst}
\begin{split}
c_1 & = -2 f_1 - \frac12 *d^v a (U_1) \\
c_2 & = -2 f_2 - \frac12 *d^v a (U_2) \\
c_3 & = -2 f_3 + \frac12 *d^v a (U_3).
\end{split}
\end{equation}
We rewrite this as
\begin{equation}\label{ciformula}
c_i \zeta_i = -2a + \frac{1}{2} \mu(a)
\end{equation}
where $\mu: \Omega^1_v \to \Omega^1_v,$ defined by (\ref{ciformulafirst}), obeys
\begin{equation}\label{munorm}
|\mu (a)| = |*d^v a | = |d^v a |.
\end{equation}
Now, the decomposition of the Laplacian (\ref{laplacev:maineq}) reads
\begin{equation}\label{laplacedecompfordef}
\begin{split}
\left( \nabla^* \nabla \alpha \right)^v & = \nabla^* \nabla^v a + 4 a + 4 c_i\zeta_i + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i \\
& = \nabla^* \nabla^v a - 4 a + 2 \mu(a) + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i
\end{split}
\end{equation}
where we have used (\ref{ciformula}). Returning to (\ref{LAsquared}), we have
\begin{equation*}
\begin{split}
0 = \mathscr{L}_A^2 \alpha & = 2 \phi_{std} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha + \nabla^*\nabla \alpha - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha \right] + 6 \alpha \\
& = \nabla^*\nabla \alpha - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} b \right] + 6 \alpha.
\end{split}
\end{equation*}
Taking the vertical part and inserting (\ref{laplacedecompfordef}), since $F_A$ is purely horizontal, we obtain
\begin{equation}\label{decomp0}
\begin{split}
0 = \left( \mathscr{L}_A^2 \alpha \right)^v = \nabla^* \nabla^v a + 2a + 2 \mu(a) + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i.
\end{split}
\end{equation}
Applying the Bochner formula on $S^3$ yields
\begin{equation}\label{decomp1}
0 = \left( d^* d^v + d^v d^* \right) a + 2 \mu(a) + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i.
\end{equation}
We now split $a$ into fiberwise closed and coclosed parts
$$a = a_{cl} + a_{cocl}$$
and write
$$a_{cocl} = g_i \zeta_i.$$
Taking an inner product with $a_{cocl}$ in (\ref{decomp1}), and integrating over $S^7,$ yields
\begin{equation}\label{decomp2}
0 = \| d^v a \|^2 + 2 (a_{cocl}, \mu(a)) + \left( g_i, \nabla^* \nabla^h \! g_i \right).
\end{equation}
Here we have used the orthogonality between fiberwise closed and coclosed vertical 1-forms, the fact that $d^*a_{cocl} = (d^v)^*a_{cocl} = 0,$ and Lemma \ref{lemma:coclosed}.
Note that the first eigenvalue of the Hodge Laplacian on coclosed 1-forms on $S^3$ is 4, so
$$\| d^v a \| \geq 2 \| a_{cocl} \|.$$
Inserting this into (\ref{decomp2}), using Cauchy-Schwartz and (\ref{munorm}), we obtain
\begin{equation*}
\begin{split}
0 &\geq 2 \| a_{cocl} \| \| d^v a \| - 2 \| a_{cocl} \| \| \mu(a) \| + \sum_{i = 1}^3 \| \nabla^h \! g_i \|^2 \\
& \geq 2 \| a_{cocl} \| \| d^v a \| - 2 \| a_{cocl} \| \| d^v a \| +\sum_{i = 1}^3 \| \nabla^h \! g_i \|^2 \\
& \geq \sum_{i = 1}^3 \| \nabla^h \! g_i \|^2.
\end{split}
\end{equation*}
This yields
$$\nabla^h g_i \equiv 0$$
for $i = 1,2,3.$
Since $B$ is assumed irreducible, Lemma \ref{lemma:horizproperties} implies that $g_i \equiv 0,$ and hence $a_{cocl} \equiv 0.$
But then $a = a_{cl}$ is fiberwise closed, so $\mu(a) = 0$ by (\ref{munorm}). Since the RHS in (\ref{decomp0}) is positive,
we conclude that $a \equiv 0,$ as desired.
\end{proof}
\subsection{Space of infinitesimal deformations} This subsection proves our main theorem, which calculates the deformations of a pulled-back instanton on $S^7$ in terms of the ASD deformations on $S^4.$
Consider the operator
$$\varphi = \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d(-) \right)^h : \Omega^1_h \to \Omega^1_h$$
given by the restriction of $\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d(-)$ to the space of horizontal 1-forms. For $b \in \Omega^1_h,$ we have
\begin{equation*
\varphi(b) = \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d b \right)^h = \left( \phi^{(1,2)} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d^v b \right)^h.
\end{equation*}
Let $S^3_0 \subset \{ (x, 0) \} \subset \H^2$ be the fiber of the Hopf fibration contained in the $x$-axis, and let
$$\varphi_0= \left. \varphi \right|_{S^3_0}.$$
Notice that
\begin{equation*}
\begin{split}
dy^j \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \zeta_i & = \left( \zeta_i^y \right)^j = \omega_{ijk} y^k.
\end{split}
\end{equation*}
We may therefore define a local frame for $\Omega^1_h$ near $S^3_0$ by
\begin{equation*}
\bar{e}^j = dy^j - \zeta_i \omega_{ijk} y^k, \qquad j = 0, \ldots, 3.
\end{equation*}
We calculate
\begin{equation*}
\begin{split}
\left. d \bar{e}^j \right|_{S_0^3} & = \omega_{ijk} \zeta_i \wedge \bar{e}^k \\
& = \left. d^v \bar{e}^j \right|_{S_0^3}.
\end{split}
\end{equation*}
Then
\begin{equation*}
\begin{split}
\varphi_0 \left( \bar{e}^j \right) = \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d^v \bar{e}^j \right)^h & = - \left( \omega_{1jk} \omega_{1 k \ell} + \omega_{2jk} \omega_{2 k \ell} - \omega_{3jk} \omega_{3 k \ell} \right) \bar{e}^\ell \\
\end{split}
\end{equation*}
and
\begin{equation}\label{phi0ebar}
\varphi_0 \left( \bar{e}^j \right) = \bar{e}^j.
\end{equation}
\begin{lemma}\label{lemma:horiz} The kernel of $\varphi_0$ consists of sections of the form
\begin{equation*}
L_{ij} x^i \bar{e}^j
\end{equation*}
where $L \in \mathfrak{F}$ is a Fueter map of ${\mathbb R}^4,$ per (\ref{fueter}) above.
\end{lemma}
\begin{proof}
Let $\alpha = \alpha_j \bar{e}^j$ be an arbitrary section of $\Omega^1_h$ near $S^3_0.$ Per (\ref{phi0ebar}) and (\ref{phis30}), we have
\begin{equation}\label{phi0fueter}
\begin{split}
\varphi_0 \left( \alpha \right) & = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \left( U_i \left( \alpha_j \right) \zeta^i \wedge \bar{e}^j \right) + \alpha_j \bar{e}^j \\
& = \left( U_1 \left( \alpha_j \right) \omega_{1 j \ell} + U_2 \left( \alpha_j \right) \omega_{2 j \ell} - U_3 \left( \alpha_j \right) \omega_{3 j \ell} + \alpha_\ell \right) \bar{e}^\ell \\
& =: \beta_\ell \bar{e}^\ell.
\end{split}
\end{equation}
Further, we calculate
\begin{equation*}
\begin{split}
\varphi_0^2 \left( \alpha \right) \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \bar{e}^i & = U_1 \left( \beta_\ell \right) \omega_{1 \ell i} + U_2 \left( \beta_\ell \right) \omega_{2 \ell i} - U_3 \left( \beta_\ell \right) \omega_{3 \ell i} + \beta_i \\
& = \omega_{1 \ell i} U_1 \left(U_1 \alpha_j \omega_{1j\ell} + U_2 \alpha_j \omega_{2j\ell} - U_3 \alpha_j \omega_{3j\ell} + \alpha_\ell \right) \\
& \quad + \omega_{2 \ell i} U_2 \left( U_1 \alpha_j \omega_{1j\ell} + U_2 \alpha_j \omega_{2j\ell} - U_3 \alpha_j \omega_{3j\ell} + \alpha_\ell \right) \\
& \quad - \omega_{3 \ell i} U_3 \left( U_1 \alpha_j \omega_{1j\ell} + U_2 \alpha_j \omega_{2j\ell} - U_3 \alpha_j \omega_{3j\ell} + \alpha_\ell \right) \\
& \quad + \beta_i \\
& = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i \\
& \quad + \omega_{2j\ell} \omega_{1 \ell i} U_1 U_2\left( \alpha_j \right) + \omega_{1 j \ell} \omega_{2 \ell i} U_2 U_1 \left( \alpha_j \right) \\
& \quad - \omega_{3j\ell} \omega_{1 \ell i} U_1 U_3 \left( \alpha_j \right) - \omega_{1 j \ell} \omega_{3 \ell i} U_3 U_1 \left( \alpha_j \right) \\
& \quad - \omega_{3j\ell} \omega_{2 \ell i} U_2 U_3 \left( \alpha_j \right) - \omega_{2 j \ell} \omega_{3 \ell i} U_3 U_2 \left( \alpha_j \right) \\
& \quad + \left( \beta_i - \alpha_i \right) + \beta_i \\
& = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i + \omega_{3 j i} \left[ U_1, U_2 \right] \alpha_j + \omega_{2 j i} \left[ U_1, U_3 \right] \alpha_j - \omega_{1 j i} \left[ U_2, U_3 \right] \alpha_j \\
& \quad + 2\beta_i - \alpha_i \\
& = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i + 2 \left( - \omega_{3 ji } U_3 + \omega_{2ji} U_2 + \omega_{1ji} U_1 \right) \alpha_j \\
& \quad + 2\beta_i - \alpha_i.
\end{split}
\end{equation*}
Again applying (\ref{phi0fueter}), we obtain
\begin{equation}\label{horiz:phi02}
\varphi_0^2 \left( \alpha \right) \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \bar{e}^i = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i + 4\beta_i - 3 \alpha_i.
\end{equation}
Notice from (\ref{gradgammai:gradvh}) that $\nabla_{U_i} U_i = 0.$
\begin{comment} that
\begin{equation*}
\begin{split}
\nabla_{U_1} U_1 & = \pi_{S^3} \left( U_1 \left( x^j \right) \omega_{1\ell j} \frac{\partial }{\partial x^\ell} \right) \\
& = \pi_{S^3} \left( x^k \omega_{1kj} \omega_{1\ell j} \frac{\partial }{\partial x^\ell} \right) \\
& = \pi_{S^3} \left( x^\ell \frac{\partial }{\partial x^\ell} \right) = 0
\end{split}
\end{equation*}
and similarly for $U_2$ and $U_3.$
\end{comment}
Hence the first term on the RHS of (\ref{horiz:phi02}) is the Laplace-Beltrami operator on $S^3_0,$ applied component-wise to $\alpha.$
We conclude that
\begin{equation*}
\varphi_0^2 \left( \alpha \right) = \Delta_{S^3} \alpha + 4 \varphi_0 \left( \alpha \right) - 3 \alpha.
\end{equation*}
In particular, if $\varphi_0(\alpha) = 0,$ we have
\begin{equation*}
\Delta_{S^3} \alpha_j = 3 \alpha_j.
\end{equation*}
Hence, $\alpha_j$ lies in the first nonzero eigenspace, and is the restriction of a linear function on ${\mathbb R}^4:$
$$\alpha_j = x^i L_{ij}.$$
Substituting back into (\ref{phi0fueter}), we have
\begin{equation*}
\varphi_0(\alpha) = x^i \left( L_{ij} + \omega_{1ik} L_{k\ell}\omega_{1 \ell j} + \omega_{2 i k} L_{k\ell}\omega_{2 \ell j} - \omega_{3 i k} L_{k \ell}\omega_{3 \ell j} \right) \bar{e}^j = 0
\end{equation*}
which recovers (\ref{fueter}), as desired.
\end{proof}
\begin{prop}\label{prop:kerphi} The kernel of $\varphi$ is the
subspace of $\Omega^1_h$ given by
\begin{equation}\label{3subsheaves}
\pi^* \Omega^1_{S^4} \oplus I_1 \left( \pi^* \Omega^1_{S^4} \right) \oplus I_2 \left( \pi^* \Omega^1_{S^4} \right).
\end{equation}
Here the complex structures $I_1$ and $I_2$ act pointwise on $\Omega^1_h \subset \Omega^1_{{\mathbb R}^8}.$
\end{prop}
\begin{proof} Over the fiber $S^3_0,$ this follows directly from Lemma \ref{lemma:horiz} and the description (\ref{fueterfactors}) of $\mathfrak{F}.$ Since ${\mathrm{Sp} }(2)$ preserves $\pi^* \Omega^1_{S^4}$ and commutes with $I_1$ and $I_2,$ the subspace (\ref{3subsheaves}) is also invariant under ${\mathrm{Sp} }(2).$ Hence, it agrees with $\ker \varphi$ globally.
\end{proof}
\begin{thm}\label{thm:generaldefos} Let $A = \pi^*B$ be the pullback of an irreducible ASD instanton on $S^4,$ and let
$$V = \ker \left( D_B^+ \oplus D_B^* \right) \subset \Omega^1_{S^4} \left( {\mathfrak g}_E \right)$$
denote the space of infinitesimal deformations of $B.$ The space of infinitesimal deformations of $A,$ as a ${\mathbb G}_2$-instanton, is given by
\begin{equation}\label{generaldefos:maineq}
\ker {\mathscr L}_A = \pi^* V \oplus I_1 \left( \pi^* V \right) \oplus I_2 \left( \pi^* V \right) \subset \Omega^1_{S^7} \left( {\mathfrak g}_{\pi^*E} \right).
\end{equation}
For structure group ${\mathrm{SU} }(2),$ this has dimension $3 \left( 8\kappa - 3 \right).$
\end{thm}
\begin{proof} Let $\alpha \in \ker {\mathscr L}_A.$ According to Theorem \ref{thm:vertvanishing}, the vertical component of $\alpha$ vanishes, so it is a global section of $\Omega^1_h ({\mathfrak g}_E).$ The image $\varphi(\alpha) \in \Omega^1_h$ under the horizontal component of the deformation operator must therefore vanish; by Proposition \ref{prop:kerphi}, we have
$$\alpha = \alpha_{0} + \alpha_{1} + \alpha_{2}$$
according to (\ref{3subsheaves}).
It remains to show that ${\mathscr L}_A(\alpha) = 0$ if and only if $d^*\alpha_{i} = 0$ and $(d^h\alpha_{i})^+ = 0,$ for $i = 0,1,2.$ Clearly $d^*\alpha = 0$ if and only if $d^*\alpha_{i} = 0 \, \forall \, i.$
We have
\begin{equation}\label{dhalphaplus}
(d^h \alpha)^+ = \sum_{i = 1}^3 (d^h \alpha_{i} )^+
\end{equation}
where $(d^h \alpha_{i} )^+$
are linearly independent.
But the map $\phi \, \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} $ is a bundle isomorphism from $\Omega^{2+}_h$ to $\Omega^1_v,$ so
$$\left(\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \right)^v = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} (d^h \alpha)^+ = 0$$
if and only if $(d^h \alpha)^+ = 0.$ By (\ref{dhalphaplus}), this implies that $(d^h \alpha_{i} )^+ = 0$ for $i = 0,1,2,$ as desired.
The dimension formula for structure group ${\mathrm{SU} }(2)$ follows from the Atiyah-Hitchin-Singer Theorem \cite{ahs}.
\end{proof}
\section{Global picture}\label{sec:global}
In this section, we discuss the global structure of the components of the moduli space obtained by pullback. We first prove Theorem \ref{thm:grassmann} on the structure of the $\kappa = 1$ component. For higher charge, the picture necessarily involves Hermitian-Yang-Mills connections on the twistor space ${\mathbb C} \P^3 \to S^4.$
\subsection{Proof of Theorem \ref{thm:grassmann}}
Let $W$ be given by (\ref{wdef}). Taking $A_0$ in the gauge (\ref{standardg2instanton}), we may define the smooth 5-dimensional family of connections
\begin{equation*}
V = \{\exp(w)^* A_0 \mid w \in W \} \subset {\mathscr A}_E.
\end{equation*}
By the construction of \S \ref{ss:hopf}, this family is equal to the pullback by the Hopf fibration of the 5-dimension family of unit-charge ASD instantons on $S^4.$
Further define a smooth map
\begin{equation}\label{firstg2map}
\begin{split}
{\mathrm{Spin} }(7) \times V & \to {\mathscr A}_E \\
\left(\sigma , A \right) & \mapsto \sigma^*A .
\end{split}
\end{equation}
By construction, the image of (\ref{firstg2map}) consists of ${\mathbb G}_2$-instantons.
Denote the principal bundle
$$Q = {\mathrm{Spin} }(7) \to {\mathrm{Spin} }(7) / {\mathrm{Sp} }(2) \times {\mathrm U}(1) .$$
Note that ${\mathrm{Sp} }(2) \times {\mathrm U}(1)$ acts on $V,$ modulo gauge, by the 5-dimensional representation of ${\mathrm{Sp} }(2)$ and the trivial representation of ${\mathrm U}(1).$ Taking the quotient by the gauge group ${\mathscr G}_E,$ the map (\ref{firstg2map}) descends to a smooth map from the associated bundle
$$X = Q \times_{{\mathrm{Sp} }(2) \times {\mathrm U}(1)} V$$
to the space of connections modulo gauge:
\begin{equation}\label{secondg2map}
X
\to {\mathscr A}_E / {\mathscr G}_E.
\end{equation}
Notice that
$${\mathrm{Spin} }(7)/ {\mathrm{Sp} }(2) \times {\mathrm U}(1) \cong {\mathrm{SO} }(7) / {\mathrm{SO} }(5) \times {\mathrm{SO} }(2) = {\mathbb G}^{or}(5,7).$$
Hence, the domain $X$ in (\ref{secondg2map}) is equal to the vector bundle over ${\mathbb G}^{or}(5,7)$ associated to the standard representation of ${\mathrm{SO} }(5),$ {\it i.e.}, the tautological 5-plane bundle.
We claim that the map (\ref{secondg2map}) is a proper embedding.
By Proposition \ref{prop:15deformations}, the image of the differential of (\ref{secondg2map}) has rank at least 15 at each point. But $X$ is 15-dimensional, so the map is an embedding.
The base space ${\mathbb G}^{or}(5,7)$ is compact, and as $x \in X$ tends to infinity in a fiber $V,$ the curvature of the ASD instanton blows up, as does the curvature of the pullback. This implies that the connection tends to infinity in ${\mathscr A}_E / {\mathscr G}_E,$ so (\ref{secondg2map}) is a proper map.
By Theorem \ref{thm:generaldefos}, every element of the image of (\ref{secondg2map}), being equivalent modulo ${\mathrm{Spin} }(7)$ to a pullback from $S^4,$ has a 15-dimensional space of infinitesimal deformations. Since $\dim(X) = 15,$
we conclude that (\ref{secondg2map}) is a diffeomorphism onto a connected component of the ${\mathbb G}_2$-instanton moduli space.
\qed
\begin{rmk} By the same construction, we may compactify $X$ fiberwise to a $\bar{B}^5$-bundle,
where a boundary point records bubbling along an associative great sphere.
\end{rmk}
\subsection{Chern-Simons functional and Hermitian-Yang-Mills connections}
Given two connections $A$ and $A_1$ on a bundle $E,$ let $a = A - A_1$ and define the relative Chern-Simons 3-form:
\begin{equation}
cs(A, A_1) = - Tr \left( a \wedge \left( F_{A_1} + \frac{1}{2} d_{A_1}a + \frac{1}{3} a \wedge a \right) \right).
\end{equation}
This satisfies
$$d cs(A, A_1) = Tr \left( F_A \wedge F_A - F_{A_1} \wedge F_{A_1} \right).$$
On a 7-manifold $M$ with ${\mathbb G}_2$-structure, we may define the global relative \emph{Chern-Simons functional}
\begin{equation}
CS_{\psi}(A, A_1) = \frac{-1}{4 \pi^4}\int_M \psi \wedge cs(A, A_1).
\end{equation}
Let
\begin{equation}\label{bigpi}
\Pi : S_{std}^7 \to {\mathbb C} \P^3
\end{equation}
be the natural projection for the standard complex structure $I_1.$
\begin{prop}\label{prop:hym} Let $B$ be a connection on an ${\mathrm{SU} }(n)$-bundle $E \to {\mathbb C}\P^3.$ Then $A = \Pi^*B$ is a ${\mathbb G}_2$-instanton on $S^7_{std}$ if and only if $B$ is Hermitian-Yang-Mills.
Given any two such connections $(E,B)$ and $(E_1,B_1)$ for which $\Pi^* E \cong \Pi^*E_1,$ we have
\begin{equation*}
CS_\psi \left(A, A_1 \right) = \langle \left[ \omega_{FS} \right] \cup \left( c_2(E) - c_2(E_1) \right), \left[ {\mathbb C}\P^3 \right] \rangle.
\end{equation*}
\end{prop}
\begin{proof} By ${\mathrm{SU} }(4)$-invariance, it suffices to consider the point $p = (1, 0, \ldots, 0) \in S^7.$ We shall write
\begin{equation}
T_p S^7 = {\mathbb R}_{x_1} \oplus {\mathbb C}^3.
\end{equation}
Then, from the expression (\ref{psi0kahler}), we have
\begin{equation}
\phi_{std}(p) = \frac{\partial}{\partial x_0} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \Psi_0 = dx^1 \wedge \omega + \Re dz_2 \wedge dz_3 \wedge dz_4
\end{equation}
where $\omega$ is the standard K{\"a}hler form on ${\mathbb C}^3,$ given by (\ref{stdkahlerforms}). Letting
$$F = F_A(p) = \left. \Pi^*F_B \right|_{T_p S^7} = \left. F_B \right|_{{\mathbb C}^3}$$
we have
\begin{equation}
\phi_0 \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} F = dx^1 \left( \omega . F \right) + \Re \, dz_2 \wedge dz_3 \wedge dz_4 \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} F^{2,0} .
\end{equation}
Therefore, $A$ is a ${\mathbb G}_2$-instanton if and only if
$$\omega .F = 0, \quad F^{2,0} = 0$$
which is to say, $B$ is Hermitian-Yang-Mills on ${\mathbb C} \P^3.$
Next, we have
\begin{equation}\label{thingtointegrate}
\begin{split}
- 4 \pi^4 CS_\psi (A, A_1) & = \int_{S^7} \psi \wedge cs(A, A_1) \\
& = \frac{1}{4} \int_{S^7} d \phi \wedge cs(A, A_1) \\
& = - \frac{1}{4} \int_{S^7} \phi \wedge d cs(A, A_1) \\
& = - \frac{1}{4} \int_{S^7} \phi \wedge Tr \left( F_A \wedge F_A - F_{A_1} \wedge F_{A_1} \right).
\end{split}
\end{equation}
Assume that both $A$ and $A_1$ are pullbacks of Hermitian-Yang-Mills connections on ${\mathbb C} \P^3.$ Computing at $p$ as above, we have $F = F_A^{1,1}$ and are left with
\begin{equation}\label{thisexpression}
\phi_0 \wedge Tr \left( F \wedge F \right) = dx^1 \wedge \omega \wedge Tr \left( F \wedge F \right).
\end{equation}
Since the Fubini-Study metric on ${\mathbb C}\P^3$ satisfies
$$\Pi^*\omega_{FS} = \frac{\omega}{\pi}$$
on $S^7,$ the expression (\ref{thisexpression}) agrees globally with
$$\pi \zeta_1 \wedge \Pi^* \left( \omega_{FS} \wedge Tr \left( F_B \wedge F_B \right) \right).$$
Integrating (\ref{thingtointegrate}) over the fibers of $\Pi,$ we obtain
\begin{equation*}
\begin{split}
-4 \pi^4 CS_\psi (A, A_1) & = - \frac{\pi}{4} \int_{S^7} \zeta_1 \wedge \Pi^* \left( \omega_{FS} \wedge Tr \left( F_B \wedge F_B - F_{B_1} \wedge F_{B_1} \right) \right) \\
& = - \frac{\pi^2}{2} \int_{{\mathbb C}\P^3} \omega_{FS} \wedge Tr \left( F_B \wedge F_B - F_{B_1} \wedge F_{B_1} \right) \\
& = - \frac{\pi^2}{2} \langle \left[ \omega_{FS} \right] \cup 8 \pi^2 \left(c_2(E) - c_2(E_1) \right) , \left[ {\mathbb C} \P^3 \right] \rangle \\
& = -4 \pi^4 \langle \left[ \omega_{FS} \right] \cup \left(c_2(E) - c_2(E_1) \right) , \left[ {\mathbb C} \P^3 \right] \rangle
\end{split}
\end{equation*}
as claimed.
\end{proof}
\subsection{Discussion}\label{ss:discussion}
Theorem \ref{thm:generaldefos} can be explained in light of Proposition \ref{prop:hym}, as follows.
Owing to the Ward correspondence, the space of infinitesimal deformations of the pullback of an ASD instanton to the twistor space ${\mathbb C} \P^3 \to S^4,$ as a Hermitian-Yang-Mills connection, is the complexification of the space of ASD deformations.
In fact, there is an $S^1$ family of complex structures
\begin{equation*
\cos (\theta) I_1 + \sin (\theta) I_2
\end{equation*}
such that the quotient map $\Pi_\theta$
satisfies Proposition \ref{prop:hym} and factorizes the given Hopf fibration:
\[
\xymatrix{
S^7 \ar@{->}[r]^{\Pi_\theta} \ar@{->}[dr]_{\pi} & {\mathbb C}\P^3 \ar@{->}[d] \\
& S^4.}
\]
The span of the pullbacks by $\Pi_\theta$ of the deformation space over ${\mathbb C} \P^3,$ for $\theta \in S^1,$ gives the larger space (\ref{generaldefos:maineq}).
According to this argument, the fibration to ${\mathbb C}\P^3$
accounts for all of the infinitesimal deformations identified by Theorem \ref{thm:generaldefos}.
As such, we expect that only the deformations coming from (\ref{bigpi}), or its 6-dimensional family of ${\mathrm{Spin} }(7)$ conjugates,
are integrable. So for $\kappa > 1,$ the generic dimension of the pulled-back component of the ${\mathbb G}_2$-instanton moduli space should be
\begin{equation}\label{higherchargedimformula}
2\left( 8 \kappa - 3 \right) + 6 = 16 \kappa.
\end{equation}
These components appear to be singular along the subset of instantons coming from $S^4$ ({\it i.e.}, instanton bundles on ${\mathbb C}\P^3$ satisfying a reality condition),
which are themselves manifolds of dimension
$$ 8 \kappa - 3 + 10 = 8 \kappa + 7.$$
More broadly, we would like to know where the instantons obtained via pullback
fit within the full moduli space of ${\mathbb G}_2$-instantons on $S^7.$ While
examples that are unrelated to any fibration
may exist,
the following guess is appropriate based on \cite{yuanqicyproduct} and the present work.
For the statement, fix a reference Hermitian-Yang-Mills connection $B_1$ on an ${\mathrm{SU} }(n)$-bundle $E_1 \to {\mathbb C} \P^3,$ and let $E = \Pi^* E_1 \to S^7$ and $A_1 = \Pi^* B_1.$
\begin{conj}[Donaldson]\label{conj:donaldson} Let $A$ be a ${\mathbb G}_2$-instanton on $E \to S^7_{std},$ for which
$$CS_{\psi}(A, A_1) \in {\mathbb Z}.$$
Then $A$ is equivalent, modulo the action of ${\mathrm{Spin} }(7)$ and ${\mathscr G}_E,$ to the pullback of a Hermitian-Yang-Mills connection on ${\mathbb C} \P^3.$
\end{conj}
| {'timestamp': '2020-06-15T02:03:10', 'yymm': '2002', 'arxiv_id': '2002.02386', 'language': 'en', 'url': 'https://arxiv.org/abs/2002.02386'} |
\section{Introduction}
Among the most important achievements in theoretical numerical analysis during the last decade was
the development of mathematical techniques for analyzing the performance of adaptive finite element methods.
A crucial notion in this theory is that of {\em approximation classes}, which we discuss here in a simple but very paradigmatic setting.
Given a polygonal domain $\Omega\in\mathbb{R}^2$, a conforming triangulation $P_0$ of $\Omega$, and a number $s>0$,
we say that a function $u$ on $\Omega$ belongs to the approximation class $\mathscr{A}^s$
if for each $N$, there is a conforming triangulation $P$ of $\Omega$ with at most $N$ triangles,
such that $P$ is obtained by a sequence of newest vertex bisections from $P_0$,
and that $u$ can be approximated by a continuous piecewise affine function subordinate to $P$
with the error bounded by $cN^{-s}$, where $c=c(u,P_0,s)\geq0$ is a constant independent of $N$.
In a typical setting, the error is measured in the $H^1$-norm, which is the natural energy norm for second order elliptic problems.
To reiterate and to remove any ambiguities, we say that $u\in H^1(\Omega)$ belongs to $\mathscr{A}^s$ if
\begin{equation}\label{e:approx-class-std-intro}
\min_{\{P\in\mathscr{P}:\#P\leq N\}} \inf_{v\in S_P} \|u-v\|_{H^1} \leq c N^{-s},
\end{equation}
for all $N\geq\#P_0$ and for some constant $c$,
where $\mathscr{P}$ is the set of conforming triangulations of $\Omega$ that are obtained by a sequence of newest vertex bisections from $P_0$,
and $S_P$ is the space of continuous piecewise affine functions subordinate to the triangulation $P$.
Approximation classes can be used to reveal a theoretical barrier on any procedure that is designed to approximate $u$
by means of piecewise polynomials and a fixed refinement rule such as the newest vertex bisection.
Suppose that we start with the initial triangulation $P_0$, and generate a sequence of conforming triangulations by using newest vertex bisections.
Suppose also that we are trying to capture the function $u$ by using continuous piecewise linear functions subordinate to the generated triangulations.
Finally, assume that $u\in\mathscr{A}^s$ but $u\not\in\mathscr{A}^\sigma$ for any $\sigma>s$.
Then as far as the exponent $\sigma$ in $cN^{-\sigma}$ is concerned,
it is obvious that the best asymptotic bound on the error we can hope for is $cN^{-s}$, where $N$ is the number of triangles.
Now supposing that $u$ is given as the solution of a boundary value problem,
a natural question is if this convergence rate can be achieved by any practical algorithm,
and it was answered in the seminal works of \cite*{BDD04} and \cite*{Stev07}: These papers established that the convergence rates of certain adaptive finite element methods
are optimal, in the sense that if $u\in\mathscr{A}^s$ for some $s>0$, then the method converges with the rate not slower than $s$.
One must mention the earlier developments \cite*{Dorf96,MNS00,CDD01,GHS07}, which paved the way for the final achievement.
Having established that the smallest approximation class $\mathscr{A}^s$ in which the solution $u$ belongs to essentially determines how fast
adaptive finite element methods converge, the next issue is to determine how large these classes are
and if the solution to a typical boundary value problem would belong to an $\mathscr{A}^s$ with large $s$.
In particular, one wants to compare the performance of adaptive methods with that of non-adaptive ones.
A first step towards addressing this issue is to characterize the approximation classes in terms of classical smoothness spaces,
and the main work in this direction so far appeared is \cite*{BDDP02}, which, upon tailoring to our situation and a slight simplification,
tells that $B^{\alpha}_{p,p}\subset\mathscr{A}^s\subset B^{\sigma}_{p,p}$
for $\frac2p=\sigma<1+\frac1p$ and $\sigma<\alpha<\max\{2,1+\frac1p\}$ with $s=\frac{\alpha-1}2$.
Here $B^\alpha_{p,q}$ are the standard Besov spaces defined on $\Omega$.
This result has recently been generalized to higher order Lagrange finite elements by \cite{GM13}.
In particular, they show that the direct embedding $B^{\alpha}_{p,p}\subset\mathscr{A}^s$ holds in the larger range $\sigma<\alpha<m+\max\{1,\frac1p\}$,
where $m$ is the polynomial degree of the finite element space, see Figure \ref{f:direct-intro}(a).
However, the restriction $\sigma<1+\frac1p$ on the inverse embedding $\mathscr{A}^s\subset B^{\sigma}_{p,p}$ cannot be removed,
since for instance, any finite element function whose derivative is discontinuous cannot be in $B^{\sigma}_{p,p}$ if $\sigma\geq1+\frac1p$ and $p<\infty$.
To get around this problem, Gaspoz and Morin proposed to replace the Besov space $B^{\sigma}_{p,p}$ by the approximation space $A^\sigma_{p,p}$
associated to uniform refinements\footnote{This space is denoted by $\hat B^\sigma_{p,p}$ in \cite{GM13}. In the present paper we are adopting the notation of \cite{Osw94}.}.
We call the spaces $A^\sigma_{p,p}$ {\em multilevel approximation spaces}, and their definition will be given in Subsection \ref{ss:multilevel}.
For the purposes of this introduction, and roughly speaking, the space $A^\sigma_{p,p}$ is the collection of functions $u\in\Leb{p}$ for which
\begin{equation}
\inf_{v\in S_{P_k}} \|u-v\|_{\Leb{p}} \leq c h_k^{\sigma},
\end{equation}
where $\{P_k\}\subset\mathscr{P}$ is a sequence of triangulations such that $P_{k+1}$ is the uniform refinement of $P_k$,
and $h_k$ is the diameter of a typical triangle in $P_k$.
Note for instance that finite element functions are in every $A^\sigma_{p,p}$.
With the multilevel approximation spaces at hand, the inverse embedding $\mathscr{A}^s\subset A^{\sigma}_{p,p}$ is recovered for all $\sigma\leq\frac2p$.
In this paper, we prove the direct embedding $A^{\alpha}_{p,p}\subset\mathscr{A}^s$,
so that the existing situation $B^{\alpha}_{p,p}\subset\mathscr{A}^s\subset A^{\sigma}_{p,p}$ is improved to $A^{\alpha}_{p,p}\subset\mathscr{A}^s\subset A^{\sigma}_{p,p}$.
It is a genuine improvement, since $A^{\alpha}_{p,p}(\Omega)\supsetneq B^{\alpha}_{p,p}(\Omega)$ for $\alpha\geq1+\frac1p$.
Moreover, as one stays entirely within an approximation theory framework, one can argue that the link between $\mathscr{A}^s$ and $A^{\alpha}_{p,p}$ is more natural
than the link between $\mathscr{A}^s$ and $B^{\alpha}_{p,p}$.
Once the link between $\mathscr{A}^s$ and $A^{\alpha}_{p,p}$ has been established, one can then invoke the well known relationships between $A^{\alpha}_{p,p}$ and $B^{\alpha}_{p,p}$.
It seems that this two step process offers more insight into the underlying phenomenon.
We also remark that while the existing results are only for the newest vertex bisection procedure and conforming triangulations,
we deal with possibly nonconforming triangulations, and therefore are able to handle the red refinement procedure,
as well as newest vertex bisections without the conformity requirement.
\begin{figure}[ht]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{diag0}
\subcaption{If the space $B^\alpha_{p,p}$ is located strictly above the solid line and below the dashed line,
then $B^\alpha_{p,p}\subset\mathscr{A}^s$ with $s=\frac{\alpha-1}2$.
The inverse embeddings $\mathscr{A}^{s+\varepsilon}\subset B^\alpha_{p,p}$ hold on the solid line and below the (slanted) dotted line.}
\end{subfigure}
\qquad
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{poisson}
\subcaption{If the space $B^\alpha_{p,p}$ is located above or on the solid line,
and if $u\in\mathscr{A}^s$ and $\Delta u \in B^\alpha_{p,p}$ with $s=\frac{\alpha+1}2$,
then $u\in \mathscr{A}^s_*$.
It is as if the approximation of $\Delta u$ is taking place in $H^{-1}$,
with the proviso that the shaded area is excluded from all considerations.}
\end{subfigure}
\caption{Illustration of various embeddings.
The point $(\frac1p,\alpha)$ represents the space $B^\alpha_{p,p}$.}
\label{f:direct-intro}
\end{figure}
The approximation classes $\mathscr{A}^s$ defined by \eqref{e:approx-class-std-intro} are associated to measuring the error of an approximation in the $H^1$-norm.
Of course, this can be generalized to other function space norms, such as $\Leb{p}$ and $B^\alpha_{p,p}$, which we will consider in Section \ref{s:Lagrange}.
However, we will not stop there, and consider more general approximation classes corresponding to ways of measuring the error between a general function $u$ and a discrete function $v\in S_P$
by a quantity $\rho(u,v,P)$ that may depend on the triangulation $P$ and is required to make sense merely for discrete functions $v\in S_P$.
An example of such an error measure is
\begin{equation}\label{e:total-error-intro}
\rho(u,v,P)
= \left( \|u-v\|_{H^1}^2
+ \sum_{\tau\in P} (\mathrm{diam}\,\tau)^{2}\|f-\Pi_\tau f\|_{\Leb{2}(\tau)}^2 \right)^{\frac12} ,
\end{equation}
where $f=\Delta u$,
and $\Pi_\tau:\Leb{2}(\tau)\to\mathbb{P}_{d}$ is the $\Leb{2}(\tau)$-orthogonal projection onto $\mathbb{P}_{d}$,
the space of polynomials of degree not exceeding $d$.
It has been shown in \cite*{CKNS08} that if the solution $u$ of the boundary value problem
\begin{equation}\label{e:bvp-intro}
\Delta u = f
\quad\textrm{in}\,\,\Omega\qquad\textrm{and}\qquad
u|_\Omega=0 ,
\end{equation}
satisfies
\begin{equation}\label{e:approx-class-gen-intro}
\min_{\{P\in\mathscr{P}:\#P\leq N\}} \inf_{v\in S_P} \rho(u,v,P) \leq c N^{-s},
\end{equation}
for all $N\geq\#P_0$ and for some constants $c$ and $s>0$,
then a typical adaptive finite element method for solving \eqref{e:bvp-intro} converges with the rate not slower than $s$.
Moreover, there are good reasons to consider that the approximation classes $\mathscr{A}^s_*$ defined by the condition \eqref{e:approx-class-gen-intro}
are more attuned to certain practical adaptive finite element methods than the standard approximation classes $\mathscr{A}^s$ defined by \eqref{e:approx-class-std-intro}, see Section \ref{s:2nd-order}.
Obviously, we have $\mathscr{A}^s_*\subset\mathscr{A}^s$ but we cannot expect the inclusion $\mathscr{A}^s\subset\mathscr{A}^s_*$ to hold in general.
In \cite{CKNS08}, an effective characterization of $\mathscr{A}^s_*$ was announced as an important pending issue.
In the present paper, we establish a characterization of $\mathscr{A}^s_*$ in terms of memberships of $u$ and $f=\Delta u$ into
suitable approximation spaces, which in turn are related to Besov spaces.
For instance, we show that if $u\in\mathscr{A}^s$ and $f\in B^\alpha_{p,p}$ with $\frac\alpha2\geq\frac1p-\frac12$ and $s=\frac{\alpha+1}2$, then $u\in\mathscr{A}^s_*$, see Figure \ref{f:direct-intro}(b).
Note that the approximation rate $s=\frac{\alpha+1}2$ is as if we were approximating $f$ in the $H^{-1}$-norm, which is illustrated by the arrow downwards.
However, the parameters must satisfy $\frac\alpha2\geq\frac1p-\frac12$ (above or on the solid line), which is more restrictive compared to $\frac{\alpha+1}2>\frac1p-\frac12$ (above the dashed line),
the latter being the condition we would expect if the approximation was indeed taking place in $H^{-1}$.
This situation cannot be improved in the sense that if $\frac\alpha2<\frac1p-\frac12$ then $B^\alpha_{p,p}\not\subset \Leb{2}$,
hence the quantity \eqref{e:total-error-intro} would be infinite in general for $f\in B^\alpha_{p,p}$.
At this point, the reader might be wondering if we can deduce $f\in B^\alpha_{p,p}$ from $u\in\mathscr{A}^s$.
If so, everything would follow from the single assumption $u\in\mathscr{A}^s$,
which would then render the theory more in line with the traditional results.
However, $u$ can be in a space slightly smaller than $H^1$, and in this case we cannot guarantee $f\in L^2$, and hence the quantity \eqref{e:total-error-intro} would not be defined.
On the other hand, there are standard examples where $f$ is smooth but $u$ is barely in $H^{1+\varepsilon}$ for a small $\varepsilon$.
Furthermore, since we are solving a PDE, and $f$ is ``given'', there would generally be much more information available about $f$ than about $u$,
and so it is not an urgent matter to deduce the regularity of $f$ from that of $u$.
The results of Section \ref{s:2nd-order} and some of the results of Section \ref{s:Lagrange} are proved by invoking abstract theorems that are established in Section \ref{s:general}.
These theorems extend some of the standard results from approximation theory to deal with generalized approximation classes such as $\mathscr{A}^s_*$.
We decided to consider a fairly general setting in the hope that the theorems will be used for establishing characterizations of other approximation classes.
For example, adaptive boundary element methods and adaptive approximation in finite element exterior calculus seem to be amenable to our abstract framework,
although checking the details poses some technical challenges.
This paper is organized as follows.
In Section \ref{s:general}, we introduce an abstract framework that is more general than usually considered in approximation theory of finite element methods,
and collect some theorems that can be used to prove embedding theorems between adaptive approximation classes and other function spaces.
In Section \ref{s:Lagrange}, we recall some standard results on multilevel approximation spaces and their relationships with Besov spaces,
and then prove direct embedding theorems between multilevel approximation spaces and adaptive approximation classes.
The main results of this section are Theorem \ref{t:direct-Lp}, Theorem \ref{t:direct-App}, and Theorem \ref{t:direct-Lp-disc}.
Finally, in Section \ref{s:2nd-order},
we investigate approximation classes associated to certain adaptive finite element methods for variable coefficient second order boundary value problems.
We emphasize that the actual results in Section \ref{s:2nd-order} are in terms of the approximation spaces that are studied in Section \ref{s:Lagrange},
and in order to relate them to Besov spaces, one has to appeal to Section \ref{s:Lagrange}.
\section{General theorems}
\label{s:general}
\subsection{The setup}
\label{ss:setup}
Let $M$ be an $n$-dimensional topological manifold, equipped with a compatible measure, in the sense that all Borel sets are measurable.
What we have in mind here is $M=\mathbb{R}^n$ with the Lebesgue measure on it,
or a piecewise smooth surface $M\subset\mathbb{R}^N$ with its canonical Hausdorff measure.
With $\Omega\subset M$ a bounded domain,
we consider a class of partitions (triangulations) of $\Omega$, and finite element type functions defined over those partitions.
Ultimately, we are interested in characterizing those functions on $\Omega$ that can be well approximated by such finite element type functions.
In order to make these concepts precise, we will use in this section a fairly abstract setting,
which we believe to be a good compromise between generality and readability.
By a {\em partition} of $\Omega$ we understand a collection $P$ of finitely many disjoint open subsets of $\Omega$,
satisfying $\overline\Omega = \bigcup_{\tau\in P}\overline\tau$.
We assume that a set $\mathscr{P}$ of partitions of $\Omega$ is given, which we call the set of {\em admissible partitions}.
For simplicity, we will assume that for any $k\in\mathbb{N}$ the set $\{P\in\mathscr{P}:\#P\leq k\}$ is finite.
In practice, $\mathscr{P}$ would be, for instance, the set of all {\em conforming} triangulations obtained from a fixed initial triangulation $P_0$
by repeated applications of the newest vertex bisection procedure.
Another class of important examples arises when we want to allow partitions with hanging nodes.
In this case, an admissibility criterion on a partition has been discussed in \cite{BN10}.
Here and in the following, we often write triangles and edges et cetera to mean $n$-simplices and $(n-1)$-dimensional faces et cetera,
which seems to improve readability.
Hence note that the use of a two dimensional language does {\em not} mean that the results we discuss are valid only in two dimensions.
We will assume the existence of a {\em refinement procedure} satisfying certain requirements.
Given a partition $P\in\mathscr{P}$ and a set $R\subset P$ of its elements,
the refinement procedure produces $P'\in\mathscr{P}$,
such that $P\setminus P'\supset R$, i.e., the elements in $R$ are refined at least once.
Let us denote it by $P'=\mathrm{refine}(P,R)$.
In practice, this is implemented by a usual naive refinement possibly producing a non-admissible partition,
followed by a so-called {\em completion} procedure.
We assume the existence of a constant $\lambda>1$ such that $|\tau|\leq\lambda^{-n}|\sigma|$ for
all $\tau\in P'$ and $\sigma\in R$ with $\tau\cap\sigma\neq\varnothing$.
Note that we have $\lambda=2$ for red refinements, and $\lambda=\sqrt[n]2$ for the newest vertex bisection.
Moreover, we assume the following on the efficiency of the refinement procedure:
If $\{P_k\}\subset\mathscr{P}$ and $\{R_k\}$ are sequences such that $P_{k+1}=\mathrm{refine}(P_k,R_k)$
and $R_k\subset P_k$ for $k=0,1,\ldots$,
then
\begin{equation}\label{e:complete}
\#P_k-\#P_0\lesssim \sum_{m=0}^{k-1} \#R_m,
\qquad k=1,2,\ldots.
\end{equation}
Here and in what follows, we shall often dispense with giving explicit names to constants, and use the Vinogradov-style notation $X\lesssim Y$,
which means that $X\leq C\cdot Y$ with some constant $C$ that is allowed to
depend only on $\mathscr{P}$ and (the geometry of) the domain $\Omega$.
Assumption \eqref{e:complete} is justified for newest vertex bisection algorithm in
\cite{BDD04,Stev08},
and the red refinement rule is treated in \cite{BN10}.
Next, we shall introduce an abstraction of finite element spaces.
To this end, we assume that there is a quasi-Banach space $X_0$,
and for each $P\in\mathscr{P}$, there is a nontrivial, finite dimensional subspace $S_P\subset X_0$.
The space $X_0$ models the function space over $\Omega$ in which the approximation takes place,
such as $X_0=H^t(\Omega)$ and $X_0=\Leb{p}(\Omega)$.
The spaces $S_P$ are, as the reader might have guessed, models of finite element spaces, from which we approximate
general functions in $X_0$.
Obviously, a natural notion of error between an element $u\in X_0$ and its approximation $v\in S_P$ is the quasi-norm $\|u-v\|_{X_0}$.
However, we need a bit more flexibility in how to measure such errors, and so
we suppose that there is a function $\rho(u,v,P)\in[0,\infty]$ defined for $u\in X_0$, $v\in S_P$, and $P\in\mathscr{P}$.
Note that this error measure, which we call a {\em distance function}, can depend on the partition $P$, and it is only required to make sense for functions $v\in S_P$.
We allow the value $\rho=\infty$ to leave open the possibility that for some $u\in X_0$ we have $\rho(u,\cdot,\cdot)=\infty$.
The most important distance function is still $\rho(u,v,P)=\|u-v\|_{X_0}$, but other examples will appear later in the paper,
see e.g., Example \ref{eg:Poisson}, Subsection \ref{ss:disc} and Section \ref{s:2nd-order}.
Given $u\in X_0$ and $P\in\mathscr{P}$, we let
\begin{equation}\label{e:E-lin}
E(u,S_P)_\rho = \inf_{v\in S_P} \rho(u,v,P),
\end{equation}
which is the error of a best approximation of $u$ from $S_P$.
Furthermore, we introduce
\begin{equation}\label{e:E-nonlin}
E_k(u)_\rho = \inf_{\{P\in\mathscr{P}:\#P\leq2^kN\}} E(u,S_P)_\rho,
\end{equation}
for $u\in X_0$ and $k\in\mathbb{N}$, with the constant $N$ chosen sufficiently large in order to ensure that the set $\{P\in\mathscr{P}:\#P\leq2N\}$ is nonempty.
In a certain sense, $E_k(u)_\rho$ is the best approximation error when one tries to approximate $u$ within the budget of $2^kN$ triangles.
Finally, we define the main object of our study, the {\em (adaptive) approximation class}
\begin{equation}\label{e:A-rho-def}
\mathscr{A}^s_q(\rho) = \mathscr{A}^s_q(\rho,\mathscr{P},\{S_P\}) = \{ u\in X_0 : |u|_{\mathscr{A}^s_q(\rho)}<\infty \} ,
\end{equation}
where $s>0$ and $0<q\leq\infty$ are parameters, and
\begin{equation}\label{e:apps-inf}
|u|_{\mathscr{A}^s_q(\rho)} = \|(2^{ks}E_k(u)_\rho)_{k\in\mathbb{N}}\|_{\leb{q}} ,
\qquad u\in X_0.
\end{equation}
In the following, we will use the abbreviation $\mathscr{A}^s(\rho)=\mathscr{A}^s_\infty(\rho)$.
Note that $u\in\mathscr{A}^s_q(\rho)$ implies $E_k(u)_\rho\leq c2^{-ks}$ for all $k$ and for some constant $c$, and these two conditions are equivalent if $q=\infty$.
We have $\mathscr{A}^s_q(\rho)\subset\mathscr{A}^s_r(\rho)$ for $q\leq r$,
and $\mathscr{A}^s_q(\rho)\subset\mathscr{A}^\alpha_r(\rho)$ for $s>\alpha$ and for any $0<q,r\leq\infty$.
The set $\mathscr{A}^s_q(\rho)$ is not a linear space without further assumptions on $\rho$ and $\mathscr{P}$.
However, in a typical situation, it is indeed a vector space equipped with the quasi-norm $\|\cdot\|_{\mathscr{A}^s_q(\rho)} = \|\cdot\|_{X_0} + |\cdot|_{\mathscr{A}^s_q(\rho)}$.
\begin{remark}\label{r:quasi-norm}
Suppose that $\rho$ satisfies
\begin{itemize}
\item $\rho(\alpha u, \alpha v,P)=|\alpha|\rho(u, v,P)$ for $\alpha\in\mathbb{R}$, and
\item $\rho(u+u',v+v',P)\lesssim \rho(u,v,P) + \rho(u',v',P)$.
\end{itemize}
Then $|\cdot|_{\mathscr{A}^s_q(\rho)}$ is a {\em quasi-seminorm}, in the sense that it is positive homogeneous and satisfies the generalized triangle inequality.
Moreover, $\mathscr{A}^s_q(\rho)$ is a quasi-normed vector space.
If only the second condition holds, then $\mathscr{A}^s_q(\rho)$ would be a quasi-normed abelian group, in the sense of \cite{BL76}.
Even though we will not use this fact, it is worth noting that $\rho$ has the aforementioned properties for all applications we have in mind.
\end{remark}
We call the approximation classes associated to $\rho(u,v,\cdot)=\|u-v\|_{X_0}$
{\em standard approximation classes}, and write $\mathscr{A}^s_q(X_0)\equiv\mathscr{A}^s_q(\rho)$ and $\mathscr{A}^s(X_0)\equiv\mathscr{A}^s(\rho)$.
These standard spaces are within the scope of the general theory of approximation spaces, cf., \cite{Piet81,DL93}.
However, to treat the more general spaces $\mathscr{A}^s(\rho)$, the standard theory needs to be reworked, which is the aim of this section.
We want to characterize $\mathscr{A}^s(\rho)$ in terms of an auxiliary quasi-Banach space $X\hookrightarrow X_0$.
The main examples to keep in mind are $X_0=L^{p}$ and $X=B^{\alpha}_{q,q}$, with $\frac\alpha{n} > \frac1q - \frac1p$.
We assume that the space $X$ has the following local structure:
There exist a constant $0<q<\infty$, and a function $|\cdot|_{X(G)}:X\to\mathbb{R}^+$ associated to each open set $G\subset\Omega$,
such that
\begin{equation
\sum_{k} |u|_{X(\tau_k)}^q \lesssim \|u\|_{X}^q
\qquad (u\in X),
\end{equation}
for any finite sequence $\{\tau_k\}\subset P$ of non-overlapping elements taken from any $P\in\mathscr{P}$.
Finally, for any $\tau\in P$ with $P\in\mathscr{P}$, we let $\hat\tau\subset\Omega$ be a domain containing $\tau$,
which will, in a typical situation, be the union of elements of $P$ surrounding $\tau$.
We express the dependence of $\hat\tau$ on $P$ as $\hat\tau=P(\tau)$.
Then as an extension of the above sub-additivity property, we assume that
\begin{equation}\label{e:loc-X-lower-overlap}
\sum_{k} |u|_{X(P_k(\tau_k))}^q \lesssim \|u\|_{X}^q
\qquad (u\in X),
\end{equation}
for any finite sequences $\{P_k\}\subset\mathscr{P}$ and $\{\tau_k\}$, with $\tau_k\in P_k$ and $\{\tau_k\}$ non-overlapping.
A trivial example of such a structure is $X=\Leb{q}(\Omega)$ with $|\cdot|_{X(G)}=\|\cdot\|_{\Leb{q}(G)}$.
Here the sub-additivity \eqref{e:loc-X-lower-overlap} can be guaranteed if the underlying triangulations satisfy a certain local finiteness property.
\subsection{Direct embeddings for standard approximation classes}
\label{ss:direct-standard}
The following theorem shows that the inclusion $X\subset\mathscr{A}^s(\rho)$ can be proved by exhibiting a direct estimate.
A direct application of this criterion is mainly useful for deriving embeddings of the form $X\subset\mathscr{A}^s(X_0)$.
In the next subsection, it will be generalized to a criterion that is valid in a more complex situations.
\begin{theorem}\label{t:direct-std}
Let $0< p\leq\infty$ and let $\delta>0$.
Assume \eqref{e:complete} on the complexity of completion,
and assume \eqref{e:loc-X-lower-overlap} on the local structure of $X$.
Then for any $k\in\mathbb{N}$ sufficiently large there exists a partition $P\in\mathscr{P}$ with $\#P\leq k$ satisfying
\begin{equation}
\left( \sum_{\tau\in P} |\tau|^{p\delta} |u|_{X(\hat\tau)}^p \right)^{\frac1p} \lesssim k^{-s} \|u\|_{X} ,
\end{equation}
with $s=\delta + \frac1q - \frac1p>0$,
where $\hat\tau=P(\tau)$ is as in \eqref{e:loc-X-lower-overlap},
and the case $p=\infty$ must be interpreted in the usual way (with a maximum replacing the discrete $p$-norm).
In particular, if $u\in X$ satisfies
\begin{equation}\label{e:rho-bound}
E(u,S_P)_\rho \lesssim \left( \sum_{\tau\in P} |\tau|^{p\delta} |u|_{X(\hat\tau)}^p \right)^{\frac1p} ,
\end{equation}
for all $P\in\mathscr{P}$,
then we have $u\in\mathscr{A}^s(\rho)$ with $|u|_{\mathscr{A}^s(\rho)}\lesssim \|u\|_X$.
\end{theorem}
\begin{proof}
What follows is a slight abstraction of the proof of Proposition 5.2 in \cite{BDDP02};
we include it here for completeness.
We first deal with the case $0<p<\infty$.
Let
\begin{equation}\label{e:err-ind-0}
e(\tau,P) = |\tau|^{p\delta} |u|_{X(\hat\tau)}^p,
\end{equation}
for $\tau\in P$ and $P\in\mathscr{P}$.
Then for any given $\varepsilon>0$, and any $P_0\in\mathscr{P}$,
below we will specify a procedure to generate a partition $P\in\mathscr{P}$ satisfying
\begin{equation}\label{e:rho-bnd-pf}
\sum_{\tau\in\mathscr{P}} e(\tau,P) \leq c' (\#P) \varepsilon,
\end{equation}
and
\begin{equation}\label{e:card-P-bnd-pf}
\#P - \#P_0 \leq c \varepsilon^{-1/(1+ps)} \|u\|_{X}^{p/(1+ps)},
\end{equation}
where $c'$ depends only on the implicit constant of \eqref{e:rho-bound},
and $c$ depends only on $|\Omega|$, $\lambda$, and the implicit constants of \eqref{e:complete}, and \eqref{e:loc-X-lower-overlap}.
Then, for any given $k>0$, by choosing
\begin{equation}
\varepsilon = (c/k)^{1+ps} \|u\|_{X}^p,
\end{equation}
we would be able to guarantee a partition $P\in\mathscr{P}$ satisfying
$\#P\leq\#P_0+k$ and
\begin{equation}
\sum_{\tau\in\mathscr{P}} e(\tau,P) \lesssim k^{-sp} \|u\|_{X}^p.
\end{equation}
This would imply the lemma, as $k^{-s}$ can be replaced by $(\#P_0+k)^{-s}$ for, e.g., $k\geq\#P_0$.
Let $\varepsilon>0$ and let $P_0\in\mathscr{P}$.
We then recursively define $R_k=\{\tau\in P_k:e(\tau,P_k)>\varepsilon\}$ and $P_{k+1}=\mathrm{refine}(P_k,R_k)$ for $k=0,1,\ldots$.
For all sufficiently large $k$ we will have $R_k=\varnothing$ since $|u|_{X(\hat\tau)} \lesssim \|u\|_{X}$ by \eqref{e:loc-X-lower-overlap},
and $|\tau|$ is reduced by a constant factor $\mu=\lambda^{-n}<1$ at each refinement.
Let $P=P_k$, where $k$ marks the first occurrence of $R_k=\varnothing$.
Then recalling \eqref{e:err-ind-0}, and taking into account that $e(\tau,P_k)\leq\varepsilon$ for $\tau\in P_k$,
we obtain \eqref{e:rho-bnd-pf}.
In order to get a bound on $\#P$, we estimate the cardinality of $R=R_0\cup R_1\cup\ldots\cup R_{k-1}$, and use \eqref{e:complete}.
Let $\Lambda_j=\{\tau\in R: \mu^{j+1}\leq|\tau|<\mu^{j}\}$ for $j\in\mathbb{Z}$, and let $m_j=\#\Lambda_j$.
Note that the elements of $\Lambda_j$ (for any fixed $j$) are disjoint, since if any two elements intersect,
then they must come from different $R_k$'s as each $R_k$ consists of disjoint elements,
and hence by assumption on the refinement procedure, the ratio between the measures of the two elements must lie outside $(\mu,\mu^{-1})$.
This gives the trivial bound
\begin{equation}
m_j \leq \mu^{-j-1} |\Omega|.
\end{equation}
On the other hand, we have $e(\tau,P_k)>\varepsilon$ for $\tau\in \Lambda_j$ with some $k$, which gives
\begin{equation}\label{e:bnd-epd-pf}
\varepsilon < |\tau|^{p\delta} |u|_{X(\hat\tau)}^p
< \mu^{jp\delta} |u|_{X(\hat\tau)}^p,
\end{equation}
where $\hat\tau$ is defined with respect to $P_k$, and $k$ may depend on $\tau$.
Summing over $\tau\in\Lambda_j$, we get
\begin{equation}
m_j\varepsilon^{q/p}
\leq \mu^{jq\delta} \sum_{\tau\in \Lambda_j} |u|_{X(\hat\tau)}^q
\lesssim \mu^{jq\delta} \|u\|_{X}^q,
\end{equation}
where we have used \eqref{e:loc-X-lower-overlap}.
Finally, summing for $j$, we obtain
\begin{equation}
\#R
\leq \sum_{j=-\infty}^\infty m_j
\lesssim \sum_{j=-\infty}^\infty \min\left\{ \mu^{-j}, \varepsilon^{-q/p} \mu^{jq\delta} \|u\|_{X}^q \right\}
\lesssim \varepsilon^{-q/(p+pq\delta)} \|u\|_{X}^{q/(1+q\delta)},
\end{equation}
which, in view of \eqref{e:complete} and $q/(1+q\delta) = p/(1+ps)$, establishes the bound \eqref{e:card-P-bnd-pf}.
We only sketch the case $p=\infty$ since the proof is essentially the same.
We use
\begin{equation}
e(\tau,P) = |\tau|^{\delta} |u|_{X(\hat\tau)},
\end{equation}
instead of \eqref{e:err-ind-0}, and run the same algorithm.
This guarantees that the resulting partition $P$ satisfies
\begin{equation}
\max_{\tau\in P} e(\tau,P) \leq \varepsilon.
\end{equation}
We bound the cardinality of $P$ in the same way,
which formally amounts to putting $p=1$ into \eqref{e:bnd-epd-pf} and proceeding.
The final result is
\begin{equation}
\#P - \#P_0 \leq c \varepsilon^{-q/(1+q\delta)} \|u\|_{X}^{q/(1+q\delta)} = c \varepsilon^{-1/s} \|u\|_{X}^{1/s},
\end{equation}
where $s = \delta+\frac1q$.
The proof is complete.
\end{proof}
\begin{exampl}\label{eg:bdd-gm}
The main argument of the preceding proof can be traced back to \cite{BirSol67}.
Recently, in \cite{BDDP02}, this argument was applied to obtain an embedding of a Besov space into
$\mathscr{A}^s(X_0)$, i.e., the case where the distance function $\rho$ is given by $\rho(u,v,\cdot)=\|u-v\|_{X_0}$.
We want to include here one such application.
Let $\Omega\subset\mathbb{R}^n$ be a bounded polyhedral domain with Lipschitz boundary,
and take $\mathscr{P}$ to be the set of conforming triangulations of $\Omega$ obtained from a fixed conforming triangulation $P_0$ by
means of newest vertex bisections.
For $P\in\mathscr{P}$, let $S_P$ be the Lagrange finite element space of continuous piecewise polynomials of degree not exceeding $m$.
Thus, for instance, the piecewise linear finite elements would correspond to $m=1$.
Moreover, for $P\in\mathscr{P}$ and $\tau\in P$, let $\hat\tau=P(\tau)$ be the interior of $\bigcup \{\overline\sigma:\sigma\in P,\,\overline\sigma\cap\overline\tau\neq\varnothing\}$.
Finally, let us put $X_0=\Leb{p}(\Omega)$, $X=B^{\alpha}_{q,q}(\Omega)$, and $\rho(u,v,\cdot)=\|u-v\|_{\Leb{p}(\Omega)}$.
Then in this setting, the estimate \eqref{e:rho-bound} holds with the parameters $p$ and $\delta=\frac\alpha{n}+\frac1p-\frac1q$,
as long as $0<\alpha<m+ \max\{1,\frac1q\}$ and $\delta>0$,
cf. \cite{BDDP02,GM13}.
Hence the preceding lemma immediately implies that $B^{\alpha}_{q,q}(\Omega)\hookrightarrow\mathscr{A}^s(\Leb{p}(\Omega))$ with $s=\frac\alpha{n}$.
\end{exampl}
In the rest of this subsection, we want to record some results involving interpolation spaces.
For $u\in X_0$ and $t>0$, the {\em $K$-functional} is
\begin{equation}\label{e:K-functional}
K(u,t;X_0,X) = \inf_{v\in X} \left( \|u-v\|_{X_0} + t \|v\|_X \right),
\end{equation}
and for $0<\theta<1$ and $0<\gamma\leq\infty$, we define the (real) {\em interpolation space}
$(X_0,X)_{\theta,\gamma}$ as the space of functions $u\in X_0$ for which the quantity
\begin{equation}\label{e:interp-norm-disc}
|u|_{(X_0,X)_{\theta,\gamma}} = \left\| [\lambda^{\theta m} K(u,\lambda^{-m};X_0,X)]_{m\geq0} \right\|_{\leb{\gamma}} ,
\end{equation}
is finite.
These are quasi-Banach spaces with the quasi-norms $\|\cdot\|_{X_0} + |\cdot|_{(X_0,X)_{\theta,\gamma}}$.
The parameter $\lambda>1$ can be chosen at one's convenience, because the resulting quasi-norms are all pairwise equivalent.
\begin{corollary}\label{c:direct-std}
Let $0< p\leq\infty$, $\delta>0$ and let $s=\delta + \frac1q - \frac1p>0$.
Assume
\begin{equation}
E(u,S_P)_{X_0} \lesssim \left( \sum_{\tau\in P} |\tau|^{p\delta} |u|_{X(\hat\tau)}^p \right)^{\frac1p} ,
\end{equation}
for all $u\in X$ and $P\in\mathscr{P}$.
Then we have $(X_0,X)_{\alpha/s,\gamma}\subset\mathscr{A}^\alpha_\gamma(X_0)$ for $0<\alpha<s$ and $0<\gamma\leq\infty$.
\end{corollary}
\begin{proof}
For $u\in (X_0,X)_{\alpha/s,\gamma}$ and $v\in X$, we have
\begin{equation}
E_k(u)_{X_0} \lesssim \|u-v\|_{X_0} + E_k(v)_{X_0} ,
\end{equation}
Theorem \ref{t:direct-std} yields $E_k(v)_{X_0}\lesssim 2^{-ks} \|v\|_X$, leading to
\begin{equation}
E_k(u)_{X_0} \lesssim \|u-v\|_{X_0} + 2^{-ks} \|v\|_X.
\end{equation}
After minimizing over $v\in X$,
the right hand side gives the $K$-functional $K(u,2^{-ks};X_0,X)$,
and \eqref{e:interp-norm-disc} implies that $u\in \mathscr{A}^\alpha_\gamma({X_0})$.
\end{proof}
\subsection{Direct embeddings for general approximation classes}
\label{ss:direct}
As mentioned in the introduction, our study is motivated by algorithms for approximating the solution of the operator equation $Tu=f$.
Hence it should not come as a surprise that we assume the existence of a continuous operator $T:X_0\to Y_0$, with $Y_0$ a quasi-Banach space.
An example to keep in mind is the Laplace operator sending $H^1_0$ onto $H^{-1}$.
In this subsection we do not assume linearity, although all examples of $T$ we have in this paper are linear.
We also need an auxiliary quasi-Banach space $Y\hookrightarrow Y_0$,
satisfying the properties analogous to that of $X$, in particular, \eqref{e:loc-X-lower-overlap} with some $0<r<\infty$ replacing $q$ there.
If $Y_0=H^{-1}$, a typical example of $Y$ would be $B^{\sigma-1}_{r,r}$ with $\frac\sigma{n} > \frac1r - \frac12$.
It is obvious that $\mathscr{A}^s(\rho)\subset\mathscr{A}^s(X_0)$, provided we have $\|u-v\|_{X_0}\lesssim\rho(u,v,\cdot)$.
The latter condition is satisfied for all practical applications we have in mind.
We will shortly present a theorem providing a criterion for affirming embeddings such as $\mathscr{A}^s(X_0)\cap T^{-1}(Y)\subset\mathscr{A}^s(\rho)$.
Before stating the theorem, we need to introduce a bit more structure on the set $\mathscr{P}$.
The structure we need is that of \emph{overlay} of partitions:
We assume that there is an operation
$\oplus:\mathscr{P}\times\mathscr{P}\to\mathscr{P}$ satisfying
\begin{equation}\label{e:overlay}
S_{P} + S_{Q} \subset S_{P\oplus Q},
\qquad
\textrm{and}
\qquad
\#(P\oplus Q)\lesssim \#P+\#Q,
\end{equation}
for $P,Q\in\mathscr{P}$.
In addition, we will assume that
\begin{equation}\label{e:overlay-rho}
\rho(u,v,P\oplus Q)\lesssim\rho(u,v,P).
\end{equation}
In the conforming world, $P\oplus Q$ can be taken to be the smallest
and common conforming refinement of $P$ and $Q$, for which
\eqref{e:overlay} is demonstrated in \cite{Stev07}, see also \cite{CKNS08}.
The same argument works for nonconforming partitions satisfying a certain admissibility criterion, cf. \cite{BN10}.
\begin{theorem}\label{t:direct}
Let $0< p\leq\infty$, $\delta>0$, and let $s=\delta + \frac1r - \frac1p>0$.
Assume \eqref{e:complete} on the complexity of completion,
as well as \eqref{e:loc-X-lower-overlap} on the local structure of $Y$, with $r$ replacing $q$ there.
There are no additional assumptions at this point on $\rho$, except \eqref{e:overlay-rho} and the obvious conditions we have imposed in \S\ref{ss:setup}.
Suppose that $u\in\mathscr{A}^s(X_0)\cap T^{-1}(Y)$ satisfies
\begin{equation}\label{e:rho-bound-u-Au}
E(u,S_P)_\rho \lesssim E(u,S_P)_{X_0}
+ \left( \sum_{\tau\in P} |\tau|^{p\delta} |Tu|_{Y(\hat\tau)}^p \right)^{\frac1p} ,
\end{equation}
for all $P\in\mathscr{P}$ (in particular $E(u,\cdot)_\rho$ is always finite).
Suppose also that
\begin{equation}\label{e:Au-red}
\left( \sum_{\tau\in P\oplus Q} |\tau|^{p\delta} |Tu|_{Y(\hat\tau)}^p \right)^{\frac1p} \lesssim \left( \sum_{\tau\in P} |\tau|^{p\delta} |Tu|_{Y(\hat\tau)}^p \right)^{\frac1p} ,
\end{equation}
for any $P,Q\in\mathscr{P}$.
Then we have $u\in\mathscr{A}^s(\rho)$ with $|u|_{\mathscr{A}^s(\rho)}\lesssim |u|_{\mathscr{A}^s(X_0)}+\|Tu\|_Y$.
\end{theorem}
\begin{proof}
Let $k\in\mathbb{N}$ be an arbitrary number.
Then by definition of $\mathscr{A}^s(X_0)$, there exists a partition $P'\in\mathscr{P}$ such that
\begin{equation}
E(u,S_{P'})_{X_0} \leq 2^{-ks} |u|_{\mathscr{A}^s(X_0)},
\qquad\textrm{and}\qquad
\#P'\leq 2^kN .
\end{equation}
Similarly, by applying Theorem \ref{t:direct-std} with $Y$ in place of $X$, and with $Tu$ in place $u$,
we can generate a partition $P''\in\mathscr{P}$ such that
\begin{equation}
\left( \sum_{\tau\in P''} |\tau|^{p\delta} |Tu|_{Y(\hat\tau)}^p \right)^{\frac1p} \lesssim 2^{-ks} \|Tu\|_{Y},
\qquad\textrm{and}\qquad
\#P''\leq 2^kN .
\end{equation}
Then for $P=P'\oplus P''$ we have $\#P\lesssim2^k$ by \eqref{e:overlay}.
Moreover, \eqref{e:Au-red} together with the obvious monotonicity
\begin{equation}
E(u,S_P)_{X_0} \leq E(u,S_{P'})_{X_0},
\end{equation}
guarantee that the right hand side of \eqref{e:rho-bound-u-Au} is bounded by a multiple of $2^{-ks}(|u|_{\mathscr{A}^s(X_0)} + \|Tu\|_{Y})$,
which completes the proof.
\end{proof}
\begin{remark}
Suppose that we replace the condition \eqref{e:rho-bound-u-Au} in the statement of the preceding theorem by the new condition
\begin{equation}
E(u,S_P)_\rho \lesssim E(u,S_P)_{X_0} + E(u,S_P)_{\rho_1}
+ \left( \sum_{\tau\in P} |\tau|^{p\delta} |Tu|_{Y(\hat\tau)}^p \right)^{\frac1p} ,
\end{equation}
where $\rho_1$ is some distance function.
Then by the same argument, we would be able to conclude that
$u\in\mathscr{A}^s(\rho)$ with $|u|_{\mathscr{A}^s(\rho)}\lesssim |u|_{\mathscr{A}^s(X_0)}+|u|_{\mathscr{A}^s(\rho_1)}+\|Tu\|_Y$.
We will use similar strtaightforward extensions of the preceding theorem later in the paper, for instance, in the proof of Theorem \ref{t:elliptic-2nd}.
\end{remark}
\begin{exampl}\label{eg:Poisson}
We would like to illustrate the usefulness of Theorem \ref{t:direct} by sketching a simple application.
For full details, we refer to Section \ref{s:2nd-order}, as the current example is a special case of the results derived there.
We take $\Omega$ and $\mathscr{P}$ as in Example \ref{eg:bdd-gm},
and for $P\in\mathscr{P}$, let $S_P$ be the Lagrange finite element space of continuous piecewise polynomials of degree not exceeding $m$,
with the homogeneous Dirichlet boundary condition.
Moreover, we set $T=\Delta$ the Laplace operator, sending $X_0=H^1_0(\Omega)$ onto $Y_0=H^{-1}(\Omega)$.
Then in this context, it is proved in \cite{CKNS08} that certain adaptive finite element methods converge optimally
with respect to the approximation classes $\mathscr{A}^s(\rho)$, with the distance function $\rho$ given by
\begin{equation}\label{e:Poisson-rho-eg}
\rho(u,v,P)
= \left( \|u-v\|_{H^1}^2
+ \sum_{\tau\in P} |\tau|^{2/n}\|f-\Pi_\tau f\|_{\Leb{2}(\tau)}^2 \right)^{\frac12} ,
\end{equation}
where $f=\Delta u$, and $\Pi_\tau:\Leb{2}(\tau)\to\mathbb{P}_{d}$ is the $\Leb{2}(\tau)$-orthogonal projection onto $\mathbb{P}_{d}$,
with $d\geq m-2$ fixed.
The sum involving $f-\Pi_\tau f$ is known as the {\em oscillation term}.
Let $0< r,\alpha<\infty$ satisfy $\delta = \frac{\alpha}n - \frac1r + \frac12\geq0$ and $\alpha<d+\max\{1,\frac1r\}$.
Then we claim that for each $u\in H^1_0(\Omega)$ with $\Delta u\in B^{\alpha}_{r,r}(\Omega)$,
there exists $u_P\in S_P$ such that
\begin{equation}
\rho(u,u_P,P) \lesssim \inf_{v\in S_P} \|u-v\|_{H^1} + \left( \sum_{\tau\in P} |\tau|^{2(\delta+1/n)} |\Delta u|_{B^{\alpha}_{r,r}(\tau)}^2 \right)^{\frac12},
\end{equation}
for all $P\in\mathscr{P}$.
In light of the preceding theorem, this would imply that each function $u\in\mathscr{A}^s(H^1_0(\Omega))$ with $\Delta u\in B^{\alpha}_{r,r}(\Omega)$,
satisfies $u\in\mathscr{A}^s(\rho)$, cf. Figure \ref{f:direct-intro}(b).
Note that since we can choose $d$ at will, the restriction $\alpha<d+\max\{1,\frac1r\}$ is immaterial.
To prove the claim, we take $u_P$ to be the Scott-Zhang interpolator of $u$, preserving the Dirichlet boundary condition.
Then we have
\begin{equation}
\|u-u_P\|_{H^1} \lesssim \inf_{v\in S_P} \|u-v\|_{H^1},
\end{equation}
for all $P\in\mathscr{P}$.
The oscillation term in \eqref{e:Poisson-rho-eg} can be estimated as
\begin{equation}
\|f-\Pi_\tau f\|_{\Leb{2}(\tau)}
\leq \|f - g\|_{\Leb{2}(\tau)}
\lesssim
|\tau|^{\delta} \|f-g\|_{\Leb{r}(\tau)} + |\tau|^{\delta} |f|_{B^{\alpha}_{r,r}(\tau)},
\end{equation}
for any $g\in\mathbb{P}_{d}$,
where we have used continuity of the embedding $B^{\alpha}_{r,r}(\tau)\subset \Leb{2}(\tau)$
and the fact that $|g|_{B^{\alpha}_{r,r}(\tau)}=0$ when the Besov seminorm is defined using $\omega_{d+1}$.
Furthermore, if $g$ is a best approximation of $f$ in the $\Leb{r}(\tau)$ sense,
the Whitney estimate gives
\begin{equation}
\|f-g\|_{\Leb{r}(\tau)}
\lesssim \omega_{d+1}(f,\tau)_r
\lesssim |f|_{B^{\alpha}_{r,r}(\tau)},
\end{equation}
which yields the desired result.
In closing the example, we note that for this argument to work,
the constants in the Whitney estimates and in the embeddings $B^{\alpha}_{r,r}(\tau)\subset \Leb{2}(\tau)$ must be uniformly bounded independently of $\tau$.
While such investigations on Whitney estimates can be found in \cite{DL04a,GM13}, it seems difficult to locate similar studies on Besov space embeddings.
To remove any doubt, the arguments in the following sections are arranged so that we do not use Besov space embeddings.
Instead, we use embeddings between approximation spaces, and give a self contained proof that the embedding constants are suitably controlled.
\end{exampl}
\section{Lagrange finite elements}
\label{s:Lagrange}
\subsection{Preliminaries}
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain.
Then for $0<p\leq\infty$, we define the $r$-th order {\em $\Leb{p}$-modulus of smoothness}
\begin{equation}
\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{\Leb{p}(\Omega_{rh})}
\end{equation}
where $\Omega_{rh}=\{x\in\Omega:[x,x+rh]\subset\Omega\}$,
and $\Delta_h^r$ is the $r$-th order forward difference operator defined recursively by $[\Delta_h^1u](x)=u(x+h)-u(x)$ and $\Delta_h^ku=\Delta_h^1(\Delta_h^{k-1})u$,
i.e.,
\begin{equation}
\Delta_h^ru (x) = \sum_{k=0}^r (-1)^{r+k} \binom{r}{k} u(x+kh).
\end{equation}
Furthermore, for $0<p,q\leq\infty$, $\alpha\geq0$, and $r\in\mathbb{N}$,
the {\em Besov space} $B^\alpha_{p,q;r}(\Omega)$ consists of those $u\in \Leb{p}(\Omega)$ for which
\begin{equation}
|u|_{B^\alpha_{p,q;r}(\Omega)} = \| t\mapsto t^{-\alpha-1/q}\omega_r(u,t,\Omega)_p \|_{\Leb{q}((0,\infty))},
\end{equation}
is finite.
Since $\Omega$ is bounded, being in a Besov space is a statement about the size of $\omega_r(u,t,\Omega)_p$ only for small $t$.
From this it is easy to derive the useful equivalence
\begin{equation}\label{e:Besov-norm-disc}
|u|_{B^\alpha_{p,q;r}(\Omega)} \eqsim \left\| (\lambda^{j\alpha}\omega_r(u,\lambda^{-j},\Omega)_p)_{j\geq0} \right\|_{\leb{q}},
\end{equation}
for any constant $\lambda>1$.
The mapping $\|\cdot\|_{B^\alpha_{p,q;r}(\Omega)}=\|\cdot\|_{\Leb{p}(\Omega)}+|\cdot|_{B^\alpha_{p,q;r}(\Omega)}$ defines a norm when $p,q\geq1$ and a quasi-norm in general.
If $\alpha>r+\max\{0,\frac1p-1\}$ then the space $B^\alpha_{p,q;r}$ is trivial in the sense that $B^\alpha_{p,q;r}=\mathbb{P}_{r-1}$.
On the other hand, so long as $r>\alpha-\max\{0,\frac1p-1\}$, different choices of $r$ will result in quasi-norms that are equivalent to each other, and in this case we have the classical Besov spaces $B^\alpha_{p,q}(\Omega)=B^\alpha_{p,q;r}(\Omega)$.
In the borderline case, the situation depends on the index $q$.
If $0<q<\infty$ and $\alpha=r+\max\{0,\frac1p-1\}$, then $B^\alpha_{p,q;r}=\mathbb{P}_{r-1}$.
The case $q=\infty$ gives nontrivial spaces: For instance, we have $B^r_{p,\infty;r}(\Omega)=W^{r,p}(\Omega)$ for $p>1$.
A proof can be found in \cite[page 53]{DL93} for the one dimensional case, and the same proof works in multi-dimensions.
The following result, often called the {\em discrete Hardy inequality}, will be used many times in the subsequent sections.
We include the statement here for convenience. A proof can be found in \cite[page 27]{DL93}.
\begin{lemma}\label{e:Hardy-ineq-disc}
Let $(a_j)_{j\in\mathbb{Z}}$ and $(b_k)_{k\in\mathbb{Z}}$ be two sequences satisfying either
\begin{equation}\label{e:Hardy-ineq-disc-1}
|a_j| \leq C \left( \sum_{k=j}^\infty |b_k|^\mu \right)^{1/\mu}, \qquad j\in\mathbb{Z},
\end{equation}
for some $\mu>0$ and $C>0$, or
\begin{equation}\label{e:Hardy-ineq-disc-2}
|a_j| \leq C 2^{-\theta j} \left( \sum_{k=-\infty}^j |2^{\theta k} b_k|^\mu \right)^{1/\mu}, \qquad j\in\mathbb{Z}.
\end{equation}
for some positive $\theta$, $\mu$, and $C$.
Then we have
\begin{equation}\label{e:Hardy-ineq-disc-3}
\|(2^{\alpha j}a_j)_j\|_{\leb{q}} \lesssim C\|(2^{\alpha k}b_k)_k\|_{\leb{q}},
\end{equation}
for all $0<q\leq\infty$ and $0<\alpha<\theta$, with the convention that $\theta=\infty$ if \eqref{e:Hardy-ineq-disc-1} holds,
and with the implicit constant depending only on $q$ and $\alpha$.
\end{lemma}
\subsection{Quasi-interpolation operators}
Let $\Omega\subset\mathbb{R}^n$ be a bounded polyhedral domain with Lipschitz boundary,
and fix a conforming partition $P_0$ of $\Omega$.
We also fix a refinement rule, which is either the newest vertex bisection or the red refinement.
This ensures that all partitions are shape regular.
The set $\mathscr{P}$ can be chosen to be, in case of the newest vertex bisection, the set of all conforming triangulations arising from $P_0$.
More generally, we want to deal with possibly nonconforming partitions,
and we require that $\mathscr{P}$ satisfy the {\em finite support condition} \eqref{e:finite-support} stated below.
We remark that the results in this paper are insensitive to the exact definition of $\mathscr{P}$, so long as the family $\mathscr{P}$ satisfies \eqref{e:finite-support}.
We note that for nonconforming partitions, the degrees of freedom will be so arranged that they give rise to $H^1$-conforming finite element spaces.
In this regard, the terminology ``nonconforming partition'' may be a bit confusing.
We define the Lagrange finite element spaces
\begin{equation}\label{e:Lagrange-fem-space}
{S}_{P}={S}^m_{P}=\left\{u\in C(\Omega) : u|_{\tau}\in \mathbb{P}_{m}\,\forall\tau\in P\right\} ,
\qquad P\in\mathscr{P} ,
\end{equation}
where $\mathbb{P}_m$ is the space of polynomials of degree not exceeding $m$.
Thus, for instance, the piecewise linear finite elements would correspond to $m=1$.
Following \cite{GM13}, we will now construct a quasi-interpolation operator $\tilde Q_{P}:\Leb{p_0}(\Omega)\to S_P$ for $p_0>0$ small.
Their construction works almost verbatim here but we need to be a bit careful since we want to include partitions with hanging nodes into the analysis.
Let $\tau_0=\{x\in\mathbb{R}^n:x_1+\ldots+x_n<m\}\cap(0,m)^n$ be the standard simplex.
Then an $n$-simplex $\tau\subset\mathbb{R}^n$ is the image of $\tau_0$ under an invertible affine mapping.
To each $n$-simplex $\tau$, we associate its nodal set $N_\tau=F(\bar\tau_0\cap\mathbb{Z}^n)$, where $F:\tau_0\to\tau$ is any invertible affine mapping.
The {\em nodal set} $N_P$ of a possibly nonconforming partition $P\in\mathscr{P}$ is defined by the requirement that
$z\in\bigcup_{\tau\in P} N_\tau$ is in $N_P$ if and only if $z\in N_\tau$ for all $\tau$ satisfying $\bar\tau\ni z$, see Figure \ref{f:Lagrange-nodes}.
Furthermore, we define the {\em nodal basis} $\{\phi_z:z\in N_P\}\subset S_P$ of $S_P$ by $\phi_z(z')=\delta_{z,z'}$ for $z,z'\in N_P$.
The aforementioned {\em finite support condition} is as follows.
We require that there is a constant $C>0$ independent of $P\in\mathscr{P}$ and $z\in N_P$, such that
\begin{equation}\label{e:finite-support}
\#\{\tau\in P:\tau\subset\mathrm{supp}\,\phi_z\} \leq C,
\end{equation}
for all $P\in\mathscr{P}$ and $z\in N_P$.
It is obvious that conforming triangulations satisfy this requirement.
For partitions with hanging nodes, we refer to \cite{BN10}.
Given $\tau\in P$ and $\sigma\in P$, let us write $\tau\sim\sigma$ if (and only if) there is $z\in N_P$ such that $\tau\cup\sigma\subset\mathrm{supp}\,\psi_z$.
Then, taking into account the refinement rule and the definition of the nodal basis,
one can show that the finite support condition is equivalent to the {\em strong local finiteness condition}:
\begin{equation}\label{e:local-finite}
\sup_{P\in\mathscr{P}}\sup_{\tau\in P}\#\{\sigma\in P:\sigma\sim\tau\} <\infty ,
\end{equation}
which in turn is equivalent to the {\em strong gradedness condition}:
\begin{equation}\label{e:graded}
\sup_{P\in\mathscr{P}}\big\{\frac{\mathrm{diam}\,\sigma}{\mathrm{diam}\,\tau}:\tau,\sigma\in P,\,\sigma\sim\tau\big\} <\infty .
\end{equation}
By defining the {\em support extension} $\hat\tau=P(\tau)$ of $\tau\in P$ as the interior of
\begin{equation}\label{e:supp-ext}
\bigcup \{\sigma\in P:\sigma\sim\tau\} ,
\end{equation}
we can also write the strong gradedness condition as
\begin{equation}
\sup_{P\in\mathscr{P}}\sup_{\tau\in P} \frac{\mathrm{diam}\,\hat\tau}{\mathrm{diam}\,\tau} < \infty .
\end{equation}
In what follows, the implicit constant in any of the aforementioned conditions will be referred to as an {\em admissibility constant}.
\begin{figure}[ht]
\centering
\begin{subfigure}{0.35\textwidth}
\includegraphics{nodes2}
\subcaption{Quadratic elements ($m=2$).}
\end{subfigure}
\qquad\qquad\qquad
\begin{subfigure}{0.35\textwidth}
\includegraphics{nodes3}
\subcaption{Cubic elements ($m=3$).}
\end{subfigure}
\caption{Examples of nodal sets.}
\label{f:Lagrange-nodes}
\end{figure}
Next, we introduce a basis dual to $\{\phi_z\}$.
For each $\tau\in P$, we let
\begin{equation}
N_{P,\tau}=\{z\in N_P:\tau\subset\mathrm{supp}\,\phi_z\} ,
\end{equation}
and define $\eta_{\tau,z}\in\mathbb{P}_m$, $z\in N_{P,\tau}$, by the condition
\begin{equation}
\int_\tau \eta_{\tau,z} \xi_{\tau,z'} = \delta_{z,z'},
\qquad z,z'\in N_{P,\tau} .
\end{equation}
Note that $\#N_{P,\tau}=\#N_\tau=\dim\mathbb{P}_m$, so that the set $\{\eta_{\tau,z}:z\in N_{P,\tau}\}$ is uniquely determined.
Then for $z\in N_P$, we let
\begin{equation}
\tilde\phi_z = \frac1{n_z} \sum_{\{\tau\in P:\,\tau\subset\mathrm{supp}\,\phi_z\}} \chi_\tau\eta_{\tau,z} ,
\end{equation}
where $n_z=\#\{\tau\in P:\tau\subset\mathrm{supp}\,\phi_z\}$, and $\chi_\tau$ is the characteristic function of $\tau$.
By construction, $\mathrm{supp}\,\tilde\phi_z=\mathrm{supp}\,\phi_z$ for $z\in N_P$, and we have the biorthogonality
\begin{equation}
\langle\tilde\phi_z,\phi_{z'}\rangle = \int_\Omega \tilde\phi_z \phi_{z'} = \delta_{z,z'},
\qquad z,z'\in N_P .
\end{equation}
Now we define the quasi-interpolation operator $Q_P:\Leb{1}(\Omega)\to S_P$ by
\begin{equation}\label{e:quasi-interpolator-std}
Q_Pu = Q_{P}^{(\Omega)}u = \sum_{z\in N_P} \langle u,\tilde\phi_z\rangle\phi_z .
\end{equation}
It is clear that $Q_P$ is linear and that $Q_Pv=v$ for $v\in S_P$.
\begin{lemma}\label{l:quasi-interp-std}
For $1\leq p\leq\infty$, we have
\begin{equation}\label{e:quasi-interp-global-best-std}
\|u - Q_{P} u\|_{\Leb{p}(\Omega)} \lesssim \inf_{v\in S_P} \|u-v\|_{\Leb{p}(\Omega)} ,
\qquad u\in \Leb{p}(\Omega) ,
\end{equation}
with the implicit constant depending only on the shape regularity and admissibility constants of $\mathscr{P}$.
Furthermore, for $0<p\leq\infty$ and $\tau\in P$, we have
\begin{equation}\label{e:quasi-interp-bdd-disc-std}
\|Q_{P} v\|_{\Leb{p}(\tau)} \lesssim \|v\|_{\Leb{p}(\hat\tau)} ,
\qquad v\in \bar S^m_P ,
\end{equation}
where
\begin{equation}\label{e:disc-pol-1}
\bar S^m_P = \{w\in \Leb{\infty}(\Omega):w|_\tau\in\mathbb{P}_m\,\forall\tau\in P\} ,
\end{equation}
and $\hat\tau=P(\tau)$ is the support extension of $\tau$ as defined in \eqref{e:supp-ext}.
\end{lemma}
\begin{proof}
For $1\leq p\leq\infty$ and $u\in \Leb{p}(\Omega)$, we have
\begin{equation}\label{e:quasi-interp-bdd-std-pf}
\|Q_Pu\|_{\Leb{p}(\tau)}
\leq \sum_{z\in N_{P,\tau}} |\langle u,\tilde\phi_z\rangle| \, \|\phi_z\|_{\Leb{p}}
\leq \|u\|_{\Leb{p}(\hat\tau)} \sum_{z\in N_{P,\tau}} \|\tilde\phi_z\|_{\Leb{q}} \|\phi_z\|_{\Leb{p}} ,
\end{equation}
where $\frac1p+\frac1q=1$.
It is clear that $\|\phi_z\|_{\Leb{p}}\leq|\mathrm{supp}\,\phi_z|^{1/p}$ and by a scaling argument one can deduce that
$\|\tilde\phi_z\|_{\Leb{q}}\lesssim|\mathrm{supp}\,\tilde\phi_z|^{1/q-1}$, with the implicit constant depending only on the shape regularity and admissibility constants of $\mathscr{P}$.
Consequently, for $1\leq p<\infty$, we infer
\begin{equation}
\|Q_Pu\|_{\Leb{p}(\Omega)}^p
= \sum_{\tau\in P} \|Q_Pu\|_{\Leb{p}(\tau)}^p
\lesssim \sum_{\tau\in P} \|u\|_{\Leb{p}(\hat\tau)}^p
\lesssim \|u\|_{\Leb{p}(\Omega)}^p ,
\end{equation}
by the strong local finiteness of the mesh.
The case $p=\infty$ can be handled similarly, and we have $\|Q_Pu\|_{\Leb{p}(\Omega)}\lesssim\|u\|_{\Leb{p}(\Omega)}$ for $1\leq p\leq\infty$.
Then a standard argument yields \eqref{e:quasi-interp-global-best-std}.
For $1\leq p\leq\infty$, \eqref{e:quasi-interp-bdd-std-pf} implies \eqref{e:quasi-interp-bdd-disc-std}.
The proof for $0<p<1$ follows exactly the same lines as those in the proof of \cite[Lemma 3.2]{GM13}.
\end{proof}
In the following, we fix $p_0>0$, and for $\tau\subset\mathbb{R}^n$ a domain,
let $\Pi_{p_0,\tau}:\Leb{p_0}(\tau)\to\mathbb{P}_m$ be the local polynomial approximation operator given in Definition 3.7 of \cite{GM13}.
We recall the following important properties of this operator, cf. \cite[Theorem 3.8]{GM13}.
\begin{enumerate}[(i)]
\item
There is a constant $C_{m,p_0}$ depending only on $m$ and $p_0$, such that
\begin{equation}\label{e:Pi-near-best}
\|u-\Pi_{p_0,\tau} u\|_{\Leb{p_0}(\tau)} \leq C_{m,p_0} \inf_{v\in\mathbb{P}_m} \|u-v\|_{\Leb{p_0}(\tau)} ,
\qquad u\in \Leb{p_0}(\tau) .
\end{equation}
In other words, $\Pi_{p_0,\tau} u$ is a near-best approximation of $u$ from $\mathbb{P}_m$ in $\Leb{p_0}(\tau)$.
\item
We have
\begin{equation}\label{e:Pi-bdd}
\|\Pi_{p_0,\tau} u\|_{\Leb{p_0}(\tau)} \lesssim \|u\|_{\Leb{p_0}(\tau)} ,
\qquad u\in \Leb{p_0}(\tau) ,
\end{equation}
i.e., the operator $\Pi_{p_0,\tau}:\Leb{p_0}(\tau)\to \Leb{p_0}(\tau)$ is bounded.
\item
For any $u\in \Leb{p_0}(\tau)$ and $v\in \mathbb{P}_m$, we have
\begin{equation}\label{e:Pi-linear}
\Pi_{p_0,\tau} (u+v) = \Pi_{p_0,\tau} u+v .
\end{equation}
In particular, $\Pi_{p_0,\tau}v=v$ for $v\in \mathbb{P}_m$.
\end{enumerate}
Finally, we let
\begin{equation}\label{e:quasi-interpolator-disc}
\Pi_Pu = \sum_{\tau\in P} \chi_\tau \Pi_{p_0,\tau} u ,
\end{equation}
and define the operator $\tilde Q_{P}:\Leb{p_0}(\Omega)\to S_P$ by
\begin{equation}\label{e:quasi-interpolator-Lagrange}
\tilde Q_Pu = Q_P\Pi_Pu = \sum_{z\in N_P} \langle \Pi_Pu,\tilde\phi_z\rangle\phi_z .
\end{equation}
It is easy to see that $\tilde Q_Pv=v$ for $v\in S_P$, and that $(\tilde Q_Pu)|_\tau$ depends only on $u|_{\hat\tau}$,
where $\hat\tau=P(\tau)$ is the support extension of $\tau$, as defined in \eqref{e:supp-ext}.
Furthermore, as a consequence of the linearity property \eqref{e:Pi-linear}, we have
\begin{equation}\label{e:quasi-interpolator-linearity}
( \tilde Q_P(u+v) ) |_{\tau} = ( \tilde Q_Pu ) |_{\tau} + v|_\tau ,
\qquad u,v\in \Leb{p_0}(\tau) , \quad v|_{\hat\tau} \in S_P ,
\end{equation}
for $\tau\in P$.
\begin{lemma}\label{l:quasi-interp-best-Lagrange}
Let $p_0\leq p\leq\infty$ and $P\in\mathscr{P}$.
Then for $\tau\in P$ we have
\begin{equation}\label{e:quasi-interp-stable}
\|\tilde Q_{P} u\|_{\Leb{p}(\tau)} \lesssim \|u\|_{\Leb{p}(\hat\tau)} ,
\qquad u\in \Leb{p}(\Omega) .
\end{equation}
As a consequence, we have
\begin{equation}\label{e:quasi-interp-local-best-Lagrange}
\|u - \tilde Q_{P} u\|_{\Leb{p}(\tau)} \lesssim \inf_{v\in S_P} \|u-v\|_{\Leb{p}(\hat\tau)} ,
\qquad u\in \Leb{p}(\Omega) ,
\end{equation}
and
\begin{equation}\label{e:quasi-interp-global-best-Lagrange}
\|u - \tilde Q_{P} u\|_{\Leb{p}(\Omega)} \lesssim \inf_{v\in S_P} \|u-v\|_{\Leb{p}(\Omega)} ,
\qquad u\in \Leb{p}(\Omega) .
\end{equation}
\end{lemma}
\begin{proof}
An application of \eqref{e:quasi-interp-bdd-disc-std} gives
\begin{equation}
\|\tilde Q_{P} u\|_{\Leb{p}(\tau)} = \|Q_{P} \Pi_P u\|_{\Leb{p}(\tau)} \lesssim \|\Pi_P u\|_{\Leb{p}(\hat\tau)} .
\end{equation}
On the other hand, for $\sigma\in P$, we have
\begin{equation}
\|\Pi_{p_0,\sigma} u\|_{\Leb{p}(\sigma)}
\lesssim |\sigma|^{\frac1p-\frac1{p_0}} \|\Pi_{p_0,\sigma} u\|_{\Leb{p_0}(\sigma)}
\lesssim |\sigma|^{\frac1p-\frac1{p_0}} \|u\|_{\Leb{p_0}(\sigma)}
\leq \|u\|_{\Leb{p}(\sigma)} ,
\end{equation}
where we have used scaling properties of polynomials in the first step,
the boundedness \eqref{e:Pi-bdd} of $\Pi_{p_0,\sigma}$ in the second step,
and the H\"older inequality in the final step.
Using this, with the usual modifications for $p=\infty$, we infer
\begin{equation}
\|\Pi_P u\|_{\Leb{p}(\hat\tau)}
= \left( \sum_{\{\sigma\in P:\sigma\subset\hat\tau\}}\|\Pi_{p_0,\sigma} u\|_{\Leb{p}(\sigma)}^p \right)^{\frac1p}
\lesssim \left( \sum_{\{\sigma\in P:\sigma\subset\hat\tau\}}\|u\|_{\Leb{p}(\sigma)}^p \right)^{\frac1p}
= \|u\|_{\Leb{p}(\hat\tau)} ,
\end{equation}
establishing \eqref{e:quasi-interp-stable}.
Then \eqref{e:quasi-interp-local-best-Lagrange} follows from the linearity property \eqref{e:quasi-interpolator-linearity}.
The estimate \eqref{e:quasi-interp-global-best-Lagrange} is proved by first deriving the stability
\begin{equation}
\|\tilde Q_{P} u\|_{\Leb{p}(\Omega)}
= \left( \sum_{\tau\in P} \|\tilde Q_{P} u\|_{\Leb{p}(\tau)}^p \right)^{\frac1p}
\lesssim \left( \sum_{\tau\in P} \|u\|_{\Leb{p}(\hat\tau)}^p \right)^{\frac1p}
\lesssim \|u\|_{\Leb{p}(\Omega)} ,
\end{equation}
and then invoking the linearity property \eqref{e:quasi-interpolator-linearity}.
\end{proof}
An important tool in approximation theory is the {\em Whitney estimate}
\begin{equation}\label{e:Whitney-est}
\inf_{v\in\mathbb{P}_m} \|u-v\|_{\Leb{p}(G)} \lesssim \omega_{m+1}(u,\mathrm{diam}\,G,G)_p,
\qquad u\in \Leb{p}(G),
\end{equation}
that holds for any convex domain $G\subset\mathbb{R}^n$,
with the implicit constant depending only on $n$, $m$, and $0<p\leq\infty$, see \cite{DL04a}.
The same estimate is also true when $G$ is the star around $\tau\in P$ for some partition $P\in\mathscr{P}$,
with the implicit constant additionally depending on the shape regularity constant of $\mathscr{P}$, see \cite{GM13}.
\begin{lemma}\label{l:quasi-interp-Lagrange}
Let $p_0\leq p\leq\infty$ and let $P\in\mathscr{P}$ be {\em conforming}.
Then we have
\begin{equation}\label{e:Jackson-Qj}
\|u - \tilde Q_{P} u\|_{\Leb{p}(\Omega)}
\lesssim \left( \frac{\max_{\tau\in P}\mathrm{diam}\,\tau}{\min_{\tau\in P}\mathrm{diam}\,\tau} \right)^{\frac{n}p} \omega_{m+1}(u,\max_{\tau\in P}\mathrm{diam}\,\tau,\Omega)_p ,
\qquad u\in \Leb{p}(\Omega) .
\end{equation}
\end{lemma}
\begin{proof}
We start with the special case $p=\infty$.
Note that since $P$ is conforming, the support extension $\hat\tau$ of $\tau$ coincides with the star around $\tau$.
It is immediate from \eqref{e:quasi-interp-local-best-Lagrange} and the Whitney estimate \eqref{e:Whitney-est} that
\begin{equation}
\begin{split}
\|u - \tilde Q_{P} u\|_{\Leb{\infty}(\Omega)}
&\lesssim \max_{\tau\in P} \|u - \tilde Q_{P} u\|_{\Leb{\infty}(\tau)}
\lesssim \max_{\tau\in P} \inf_{v\in S_P} \|u-v\|_{\Leb{\infty}(\hat\tau)} \\
&\lesssim \max_{\tau\in P} \omega_{m+1}(u,\mathrm{diam}\,\hat\tau,\hat\tau)_\infty
\lesssim \max_{\tau\in P} \omega_{m+1}(u,\mu^{-1}\mathrm{diam}\,\hat\tau,\hat\tau)_\infty ,
\end{split}
\end{equation}
with $\mu>0$ sufficiently large,
where in the last line we have used the property
\begin{equation}
\omega_r(u,\mu t,G)_p \leq (\mu+1)^r \omega_r(u,t,G)_p ,
\end{equation}
cf. \cite[\S2.7]{DL93}.
With $t=\displaystyle\mu^{-1}\max_{\tau\in P}\mathrm{diam}\,\hat\tau$, we proceed as
\begin{equation}
\begin{split}
\|u - \tilde Q_{P} u\|_{\Leb{\infty}(\Omega)}
&\lesssim \max_{\tau\in P} \omega_{m+1}(u,t,\hat\tau)_\infty
= \max_{\tau\in P} \sup_{|h|\leq t} \|\Delta^{m+1}_hu\|_{\Leb{\infty}(\hat\tau_{rh})} \\
&= \sup_{|h|\leq t} \max_{\tau\in P} \|\Delta^{m+1}_hu\|_{\Leb{\infty}(\hat\tau_{rh})}
\leq \sup_{|h|\leq t} \|\Delta^{m+1}_hu\|_{\Leb{\infty}(\Omega_{rh})} ,
\end{split}
\end{equation}
which establishes \eqref{e:Jackson-Qj} for $p=\infty$.
To handle the case $0<p<\infty$ we introduce the averaged $\Leb{p}$-modulus of smoothness
\begin{equation}
w_r(u,t,G)_p = \left( \frac1{t^n} \int_{[0,t]^n} \|\Delta_h^ru\|_{\Leb{p}(G_{rh})}^p \mathrm{d} h \right)^{1/p} ,
\end{equation}
for any domain $G\subset\mathbb{R}^n$.
When $G$ is Lipschitz, the averaged modulus is equivalent to the original one:
\begin{equation}\label{e:avg-modul-equiv}
w_r(u,t,G)_p \sim \omega_r(u,t,G)_p,
\qquad \textrm{for}\quad t\lesssim1 .
\end{equation}
This equivalence is also true when $G=\tau$ or $G=\hat\tau$ for $\tau\in P$ with $P\in\mathscr{P}$,
in the range $t\lesssim\mathrm{diam}\,G$, cf. Corollary 4.3 of \cite{GM13}.
In the latter case,
the implicit constants depend only on $p$, $r$, the shape regularity constant of $\mathscr{P}$, and the geometry of the underlying domain $\Omega$.
Let us get back to the proof of \eqref{e:Jackson-Qj} for $0<p<\infty$.
As in the case $p=\infty$, we have
\begin{equation}
\begin{split}
\|u - \tilde Q_{P} u\|_{\Leb{p}(\Omega)}^p
&\lesssim \sum_{\tau\in P} \|u - \tilde Q_{P} u\|_{\Leb{p}(\tau)}^p
\lesssim \sum_{\tau\in P} \inf_{v\in S_P} \|u-v\|_{\Leb{p}(\hat\tau)}^p \\
&\lesssim \sum_{\tau\in P} \omega_{m+1}(u,\mathrm{diam}\,\hat\tau,\hat\tau)_p^p
\lesssim \sum_{\tau\in P} \omega_{m+1}(u,\mu^{-1}\mathrm{diam}\,\hat\tau,\hat\tau)_p^p ,
\end{split}
\end{equation}
with $\mu>0$ sufficiently large.
Now we employ \eqref{e:avg-modul-equiv}, to get
\begin{equation}
\begin{split}
\|u - \tilde Q_{P} u\|_{\Leb{p}(\Omega)}^p
&\lesssim \sum_{\tau\in P} w_{m+1}(u,\mu^{-1}\mathrm{diam}\,\hat\tau,\hat\tau)_p^p \\
&= \sum_{\tau\in P} \frac1{t(\tau)^n} \int_{[0,t(\tau)]^n} \int_{\hat\tau_{rh}} |\Delta_h^ru(x)|^p \mathrm{d} x \, \mathrm{d} h ,
\end{split}
\end{equation}
where $t(\tau)=\mu^{-1}\mathrm{diam}\,\hat\tau$ and $r=m+1$.
With $t_0=\mu^{-1}\displaystyle\min_{\tau\in P}\mathrm{diam}\,\hat\tau$ and $t_1=\mu^{-1}\displaystyle\max_{\tau\in P}\mathrm{diam}\,\hat\tau$,
we can switch the sum with the outer integration as follows.
\begin{equation}
\begin{split}
\|u - \tilde Q_{P} u\|_{\Leb{p}(\Omega)}^p
&\lesssim \frac1{t_0^n} \int_{[0,t_1]^n} \sum_{\tau\in P} \int_{\hat\tau_{rh}} |\Delta_h^ru(x)|^p \mathrm{d} x \, \mathrm{d} h \\
&\lesssim \frac1{t_0^n} \int_{[0,t_1]^n} \int_{\Omega_{rh}} |\Delta_h^ru(x)|^p \mathrm{d} x \, \mathrm{d} h \\
&= \frac{t_1^n}{t_0^n} w_r(u,t_1,\Omega)_p^p .
\end{split}
\end{equation}
The proof is completed upon using the equivalence \eqref{e:avg-modul-equiv} for $G=\Omega$.
\end{proof}
\subsection{Multilevel approximation spaces}
\label{ss:multilevel}
In this subsection, we study approximation from uniformly refined Lagrange finite element spaces.
We keep the setting of the preceding subsection intact,
and define the partitions $P_j$ for $j=1,2,\ldots$ recursively as $P_{j+1}$ is the uniform refinement of $P_j$.
Let $G\subset\Omega$ be a domain consisting of elements from some $P_j$.
More precisely, let $G$ be the interior of $\bigcup_{\tau\in Q}\bar\tau$ for some $Q\subset P_j$ and $j$.
Then with $S_j=S_{P_j}$, and $0<p\leq\infty$, we let
\begin{equation}\label{e:Lp-best}
E(u,S_j)_{\Leb{p}(G)} = \inf_{v\in S_j} \|u-v\|_{\Leb{p}(G)} ,
\qquad u\in \Leb{p}(G) .
\end{equation}
Note that the infimum is achieved since $S_j$ is a finite dimensional space.
We define the {\em multilevel approximation spaces}
\begin{equation}\label{e:multilevel-approx}
A^\alpha_{p,q}(\{S_j\},G) = \left\{u\in \Leb{p}(G):|u|_{A^\alpha_{p,q}(G)}:=\left\| \left( \lambda^{j\alpha} E(u,S_j)_{\Leb{p}(G)} \right)_{j\geq0} \right\|_{\leb{q}}<\infty \right\} ,
\end{equation}
for $0<p,q\leq\infty$, and $\alpha>0$,
where $\lambda=2$ for red refinements and $\lambda=\sqrt[n]2$ for newest vertex bisections.
We will also use the shorthand notations
\begin{equation}
A^\alpha_{p,q}(G) = A^\alpha_{p,q;m}(G) = A^\alpha_{p,q}(\{S_j\},G) .
\end{equation}
These spaces are quasi-Banach spaces with the quasi-norms $\|\cdot\|_{\Leb{p}(G)}+|\cdot|_{A^\alpha_{p,q}(G)}$.
Since $\Omega$ is bounded, it is clear that $A^\alpha_{p,q}(\Omega)\hookrightarrow A^{\alpha}_{p',q}(\Omega)$
for any $\alpha\geq0$, $0<q\leq\infty$ and $\infty\geq p>p'>0$.
We also have the lexicographical ordering:
$A^\alpha_{p,q}(\Omega)\hookrightarrow A^{\alpha'}_{p,q'}(\Omega)$ for $\alpha>\alpha'$ with any $0<q,q'\leq\infty$,
and
$A^\alpha_{p,q}(\Omega)\hookrightarrow A^\alpha_{p,q'}(\Omega)$ for $0<q<q'\leq\infty$.
It is no coincidence that the aforementioned embedding relations are identical to those among Besov spaces.
When reading the following theorem, keep in mind that $B^\alpha_{p,q;m+1}(\Omega)$ is the classical Besov space $B^\alpha_{p,q}(\Omega)$ for $\alpha<m+\max\{1,\frac1p\}$.
\begin{theorem}\label{t:multilevel-Besov}
We have ${B^\alpha_{p,q;m+1}(\Omega)} \hookrightarrow {A^\alpha_{p,q;m}(\Omega)}$
for $0<p,q\leq\infty$, and $\alpha>0$.
In the other direction, we have ${A^\alpha_{p,q;m}(\Omega)} \hookrightarrow {B^\alpha_{p,q;m+1}(\Omega)}$
for $0<p,q\leq\infty$, and $0<\alpha<1+\frac1p$.
\end{theorem}
\begin{proof}
We follow the standard approach.
The inclusion $B^\alpha_{p,q;m+1}(\Omega)\hookrightarrow A^\alpha_{p,q;m}(\Omega)$ is a direct consequence of \eqref{e:Jackson-Qj} with $p_0\leq p$
and the norm equivalence \eqref{e:Besov-norm-disc}.
For the second part, we start with the estimate
\begin{equation}\label{e:phi-z-est}
\omega_{m+1}(\phi_z,t)_p \lesssim \lambda^{-jn/p} \min\{1,(\lambda^{j}t)^{1+1/p}\},
\end{equation}
which holds for all nodal basis functions $\phi_z$ of $S_{j}$ and for all $j\geq0$.
This is Proposition 4.7 in \cite{GM13}, which also holds for $p=\infty$.
Hence for $0<p<\infty$ and for all $u_j=\sum_zb_z\phi_z\in S_{P_j}$, we infer
\begin{equation}
\begin{split}
\omega_{m+1}(u_j,t)_p^p
&\lesssim \sum_{z} |b_z|^p \omega_{m+1}(\phi_z,t)_p^p
\lesssim \sum_{z} |b_z|^p \, \lambda^{-jn} \min\{1,(\lambda^{j}t)^{p+1}\} \\
&\lesssim \min\{1,(\lambda^{j}t)^{p+1}\} \|u_j\|_{\Leb{p}(\Omega)}^p,
\end{split}
\end{equation}
where we have used the finite overlap property of the nodal basis functions,
the $\Leb{p}$ stability of finite elements and the estimate $\|\phi_z\|_{\Leb{p}}\eqsim\lambda^{-jn/p}$.
The same ingredients are used to perform the corresponding computation for $p=\infty$, as
\begin{equation}
\begin{split}
\omega_{m+1}(u_j,t)_\infty
&\lesssim \max_{z} |b_z| \, \omega_{m+1}(\phi_z,t)_\infty
\lesssim \max_{z} |b_z| \min\{1,\lambda^{j}t\} \\
&\lesssim \min\{1,\lambda^{j}t\} \|u_j\|_{\Leb{\infty}(\Omega)} .
\end{split}
\end{equation}
Now we write $u=\sum_{j\geq0}(u_j-u_{j-1})$ with $u_j\in S_j$ a best approximation to $u$ from $S_j$ for $j\geq0$ and $u_{-1}=0$.
Note that the series converges in $\Leb{p}$ by \eqref{e:Jackson-Qj}.
With $p^*=\min\{1,p\}$, we have
\begin{equation}
\begin{split}
\omega_{m+1}(u,\lambda^{-k})_p^{p^*}
&\lesssim \sum_{j\geq0} \omega_{m+1}(u_j-u_{j-1},\lambda^{-k})_p^{p^*} \\
&\lesssim \sum_{j=0}^k \lambda^{(j-k)(1+1/p)p^*} \|u_j-u_{j-1}\|_{\Leb{p}(\Omega)}^{p^*} + \sum_{j=k+1}^\infty \|u_j-u_{j-1}\|_{\Leb{p}(\Omega)}^{p^*},
\end{split}
\end{equation}
and an application of the discrete Hardy inequality (Lemma \ref{e:Hardy-ineq-disc}) gives
\begin{equation}
|u|_{B^\alpha_{p,q;m+1}} \lesssim \left\| \left(\lambda^{j\alpha}\|u_j-u_{j-1}\|_{\Leb{p}(\Omega)} \right)_{j} \right\|_{\leb{q}},
\end{equation}
for $0<p,q\leq\infty$, and $0<\alpha<1+\frac1p$.
Finally, to go from $u_j-u_{j-1}$ to $u-u_j$ in the right hand side, we can apply the triangle inequality to $u_j-u_{j-1} = (u-u_{j-1}) - (u-u_j)$.
\end{proof}
\begin{figure}[ht]
\centering
\includegraphics{diag1}
\caption
The inverse embedding $A^\alpha_{p,q;m}\hookrightarrow B^\alpha_{p,q;m+1}$ holds below the dashed line.
The direct embedding $B^\alpha_{p,q;m+1}\hookrightarrow A^\alpha_{p,q;m}$ holds without restriction,
but the spaces $B^\alpha_{p,q;m+1}$ are nontrivial (and coincide with the classical Besov spaces $B^\alpha_{p,q}$) only below the solid line.}
\end{figure}
Notice the gap between the two inclusions: While $B^\alpha_{p,q;m+1}(\Omega)\hookrightarrow A^\alpha_{p,q;m}(\Omega)$ holds for all
$\alpha>0$, the reverse inclusion is proved only for $0<\alpha<1+\frac1p$.
In fact, if $\alpha\geq1+\frac1p$ and $p<\infty$, the forward inclusion is strict:
Any function from $S_{j}$ would be an element of all $A^\alpha_{p,q;m}(\Omega)$,
but there are functions in $S_{j}$ that are not in $B^\alpha_{p,q;m+1}(\Omega)$,
because the estimate \eqref{e:phi-z-est} is saturated for small $t$.
This leads to the expectation that for large $\alpha$, the difference $A^\alpha_{p,q;m}(\Omega)\setminus B^\alpha_{p,q;m+1}(\Omega)$
should be ``skewed'' considerably depending on the initial mesh $P_0$.
We will not pursue this issue here,
but we conjecture that the Besov space $B^\alpha_{p,q;m+1}(\Omega)$ coincides with the intersection of all $A^\alpha_{p,q;m}(\Omega)$
as one considers all possible initial triangulations $P_0$.
We quote the following standard result, in order to assure the reader of the fact that the multilevel approximation spaces $A^\alpha_{p,q}(\Omega)$
coincide with the spaces $\hat B^\alpha_{p,q}(\Omega)$ considered in \cite{GM13}, cf. Definition 7.1 and Corollary 4.14 therein.
\begin{theorem}\label{t:multiscale-equiv}
Let $p_0\leq p\leq\infty$, $0<q\leq\infty$ and $\alpha>0$.
Then we have
\begin{equation}
\begin{split}
|u|_{A^\alpha_{p,q}(\Omega)}
&\sim
\left\| \left( \lambda^{j\alpha} \|u-\tilde Q_ju\|_{\Leb{p}(\Omega)} \right)_{j\geq0} \right\|_{\leb{q}} \\
&\sim
\left\| \left( \lambda^{j\alpha} \|\tilde Q_{j+1}u-\tilde Q_ju\|_{\Leb{p}(\Omega)} \right)_{j\geq0} \right\|_{\leb{q}} ,
\end{split}
\end{equation}
for $u\in\Leb{p}(\Omega)$, where we have used the abbreviation $\tilde Q_j = \tilde Q_{P_j}$ for all $j$.
\end{theorem}
\begin{proof}
The first equivalence is immediate from \eqref{e:quasi-interp-global-best-Lagrange}.
The generalized triangle inequality
\begin{equation}
\|\tilde Q_{j+1}u-\tilde Q_ju\|_{\Leb{p}(\Omega)} \lesssim \|u-\tilde Q_ju\|_{\Leb{p}(\Omega)} + \|u-\tilde Q_{j+1}u\|_{\Leb{p}(\Omega)} ,
\end{equation}
implies one of the directions of the second equivalence,
while the other direction follows from applying the discrete Hardy inequality (Lemma \ref{e:Hardy-ineq-disc}) to
\begin{equation}
\|u-\tilde Q_ju\|_{\Leb{p}(\Omega)} \leq \left( \sum_{k=j}^\infty \|\tilde Q_{k+1}u-\tilde Q_ku\|_{\Leb{p}(\Omega)}^{p^*} \right)^{\frac1{p^*}} ,
\end{equation}
where $p^*=\min\{1,p\}$.
\end{proof}
The following technical result will be used later.
\begin{theorem}\label{t:interp-approx}
Let $0<\alpha_1<\alpha_2<\infty$ and $0<p,q,q_1,q_2\leq\infty$.
Then we have
\begin{equation}\label{e:interp-approx}
[A^{\alpha_1}_{p,q_1}(G),A^{\alpha_2}_{p,q_2}(G)]_{\theta,q}=A^\alpha_{p,q}(G),
\end{equation}
for $\alpha=(1-\theta)\alpha_1+\theta\alpha_2$ and $0<\theta<1$,
with the equivalence constants of quasi-norms depending only on the parameters $\alpha$, $\alpha_1$, $\alpha_2$, $p$, $q$, $q_1$ and $q_2$.
\end{theorem}
\begin{proof}
The equivalence \eqref{e:interp-approx} is standard,
but we want to keep track of the equivalence constants.
So we sketch a proof here.
First, for $v\in S_m$, we observe the inverse inequality
\begin{equation}\label{e:inverse-est-approx}
|v|_{A^{\alpha_2}_{p,q_2}(G)}^{q_2}
= \sum_{j=0}^{m-1} \lambda^{\alpha_2q_2j} E(v,S_j,G)_p^{q_2}
\leq \|v\|_{\Leb{p}(G)}^{q_2} \sum_{j=0}^{m-1} \lambda^{\alpha_2q_2j}
\leq \frac{\lambda^{\alpha_2q_2m}}{\lambda^{\alpha_2q_2}-1} \|v\|_{\Leb{p}(G)}^{q_2} .
\end{equation}
It is also true for $q_2=\infty$:
\begin{equation}
|v|_{A^{\alpha_2}_{p,\infty}(G)}
= \max_{0\leq j<m} \lambda^{\alpha_2j} E(v,S_j,G)_p
\leq \lambda^{\alpha_2m} \|v\|_{\Leb{p}(G)} .
\end{equation}
Another fact we will need is the following.
We have the generalized triangle inequality
\begin{equation}
|u+v|_{A^{\alpha_2}_{p,q_2}(G)} \leq c |u|_{A^{\alpha_2}_{p,q_2}(G)} + c|v|_{A^{\alpha_2}_{p,q_2}(G)} ,
\end{equation}
with $c\geq1$ depening only on $p$ and $q_2$.
Then the Aoki-Rolewicz theorem \cite[page 59]{BL76} implies that
\begin{equation}\label{e:-mu-triange-Apq2}
|v_1+\ldots+v_k|_{A^{\alpha_2}_{p,q_2}(G)}^\mu \leq 2 |v_1|_{A^{\alpha_2}_{p,q_2}(G)}^\mu +\ldots+ 2 |v_k|_{A^{\alpha_2}_{p,q_2}(G)}^\mu ,
\end{equation}
for any $v_1,\ldots,v_k\in A^{\alpha_2}_{p,q_2}(G)$, with $\mu$ given by $(2c)^\mu=2$.
With the abbreviation $K(u,t)=K(u,t;A^{\alpha_1}_{p,q_1}(G),A^{\alpha_2}_{p,q_2}(G))$, for $u\in A^{\alpha_1}_{p,q_1}(G)$, we have
\begin{equation}
K(u,\lambda^{-(\alpha_2-\alpha_1)m}) \leq |u-u_m|_{A^{\alpha_1}_{p,q_1}(G)} + \lambda^{-(\alpha_2-\alpha_1)m}|u_m|_{A^{\alpha_2}_{p,q_2}(G)} ,
\end{equation}
where $u_m\in S_m$ is an approximation satisfying $\|u-u_m\|_{\Leb{p}(G)}=E(u,S_m,G)_p$.
We estimate the first term in the right hand side as
\begin{equation}
\begin{split}
|u-u_m|_{A^{\alpha_1}_{p,q_1}(G)}^{q_1}
&= \sum_{j=0}^m \lambda^{\alpha_1q_1j}\|u-u_m\|_{\Leb{p}(G)}^{q_1} + \sum_{j=m+1}^\infty \lambda^{\alpha_1q_1j}\|u-u_j\|_{\Leb{p}(G)}^{q_1} \\
&\lesssim \sum_{j=m}^\infty \lambda^{\alpha_1q_1j}\|u-u_j\|_{\Leb{p}(G)}^{q_1} ,
\end{split}
\end{equation}
with the implicit constant depending only on $\lambda^{\alpha_1q_1}$, and the second term as
\begin{equation}
\begin{split}
|u_m|_{A^{\alpha_2}_{p,q_2}(G)}^\mu
&\leq 2\sum_{j=1}^m|u_j-u_{j-1}|_{A^{\alpha_2}_{p,q_2}(G)}^\mu
\lesssim \sum_{j=1}^m \lambda^{\alpha_2\mu j} \|u_j-u_{j-1}\|_{\Leb{p}(G)}^\mu \\
&\lesssim \sum_{j=0}^m \lambda^{\alpha_2\mu j} \|u-u_{j}\|_{\Leb{p}(G)}^\mu ,
\end{split}
\end{equation}
where we have used the $\mu$-triangle inequality \eqref{e:-mu-triange-Apq2} in the first step,
the inverse estimate \eqref{e:inverse-est-approx} in the second step,
and the (generalized) triangle inequality for the $\Leb{p}$-quasi-norm in the third step.
Note that the implicit constants depend only on $\lambda^{\alpha_2q_2}$, $\lambda^{\alpha_2\mu}$, and $p$.
Putting everything together, we have
\begin{equation}
\begin{split}
K(u,\lambda^{-(\alpha_2-\alpha_1)m})
&\lesssim
\left( \sum_{j=m}^\infty \lambda^{\alpha_1q_1j}\|u-u_j\|_{\Leb{p}(G)}^{q_1} \right)^{\frac1{q_1}} \\
&\quad + \lambda^{-(\alpha_2-\alpha_1)m}
\left( \sum_{j=0}^m \lambda^{\alpha_2\mu j} \|u-u_{j}\|_{\Leb{p}(G)}^\mu \right)^{\frac1\mu} ,
\end{split}
\end{equation}
and then the discrete Hardy inequalities (Lemma \ref{e:Hardy-ineq-disc}) give
\begin{equation}
\left\| [\lambda^{\gamma m} K(u,\lambda^{-(\alpha_2-\alpha_1)m})]_{m\geq0} \right\|_{\leb{q}}
\lesssim
\left\| [\lambda^{(\alpha_1+\gamma) m} \|u-u_{j}\|_{\Leb{p}(G)}]_{m\geq0} \right\|_{\leb{q}} ,
\end{equation}
for $0<\gamma<\alpha_2-\alpha_1$.
The left hand side of this inequality is the (quasi) norm for $[A^{\alpha_1}_{p,q_1}(G),A^{\alpha_2}_{p,q_2}(G)]_{\gamma/(\alpha_2-\alpha_1),q}$,
while the right hand side is the (quasi) norm for $A^{\alpha_1+\gamma}_{p,q}(G)$.
For the other direction, we start with
\begin{equation}
\|u-u_j\|_{\Leb{p}(G)} \leq \|u-w_j-v_j\|_{\Leb{p}(G)} \lesssim \|u-v-w_j\|_{\Leb{p}(G)} + \|v-v_j\|_{\Leb{p}(G)} ,
\end{equation}
where $u\in A^{\alpha_1}_{p,q_1}(G)$ and $u_j\in S_j$ are as before,
and $v\in A^{\alpha_2}_{p,q_2}(G)$, $v_j,w_j\in S_j$ are arbitrary.
Note that the implicit constant depends only on $p$.
Optimizing over $v_j$ and $w_j$ gives
\begin{equation}
\min_{w_j\in S_j} \|u-v-w_j\|_{\Leb{p}(G)} \leq \lambda^{-\alpha_1j}|u-v|_{A^{\alpha_1}_{p,q_1}(G)} ,
\end{equation}
and
\begin{equation}
\min_{v_j\in S_j} \|v-v_j\|_{\Leb{p}(G)} \leq \lambda^{-\alpha_2j}|v|_{A^{\alpha_2}_{p,q_2}(G)} ,
\end{equation}
and substituting these back, we get
\begin{equation}
\begin{split}
\|u-u_j\|_{\Leb{p}(G)}
&\lesssim \inf_{v\in A^{\alpha_2}_{p,q_2}(G)} \left( \lambda^{-\alpha_1j}|u-v|_{A^{\alpha_1}_{p,q_1}(G)} + \lambda^{-\alpha_2j}|v|_{A^{\alpha_2}_{p,q_2}(G)} \right) \\
&= \lambda^{-\alpha_1j} K(u,\lambda^{-(\alpha_2-\alpha_1)j}) .
\end{split}
\end{equation}
The proof is completed upon recalling the definition of $|\cdot|_{A^{\alpha}_{p,q}(G)}$.
\end{proof}
\subsection{Adaptive approximation}
In this subsection, we consider the approximation problem from adaptively generated Lagrange finite element spaces.
We study various approximation classes associated to the finite element spaces $S_P$, cf. \eqref{e:Lagrange-fem-space}.
In \cite{BDDP02,GM13}, among other things, it is proved that $B^{\alpha}_{q,q}(\Omega)\hookrightarrow\mathscr{A}^s_\infty(\Leb{p}(\Omega))$ with $s=\frac\alpha{n}$,
as long as $\frac\alpha{n}+\frac1p-\frac1q>0$ and $0<\alpha<m+ \max\{1,\frac1q\}$.
In the other direction,
the same references give $\mathscr{A}^s_q(\Leb{p}(\Omega))\hookrightarrow A^{\alpha}_{q,q}(\Omega)$ for $s=\frac\alpha{n}=\frac1q-\frac1p>0$ and $0<p,q<\infty$.
Below we complement these results by establishing direct embeddings of the form $A^{\alpha}_{q,q}(\Omega)\hookrightarrow\mathscr{A}^s_\infty(\Leb{p}(\Omega))$.
This is a genuine improvement, since $A^{\alpha}_{q,q}(\Omega)\supsetneq B^{\alpha}_{q,q}(\Omega)$ for $\alpha\geq1+\frac1q$.
Moreover, it seems natural to relate adaptive approximation to multilevel approximation first,
and then bring in the relationships between multilevel approximation and Besov spaces.
We also remark that while the existing results are only for the newest vertex bisection procedure and conforming triangulations,
we deal with possibly nonconforming triangulations, and therefore are able to handle the red refinement procedure,
as well as newest vertex bisections without the conformity requirement.
\begin{theorem}\label{t:direct-Lp}
Let $0<q\leq p\leq\infty$ and $\alpha>0$ satisfy $\frac\alpha{n}+\frac1p-\frac1q>0$ and $q<\infty$.
Then for any $0<p_0<q$ (Recall that $\tilde Q_P$ depends on $p_0$), we have
\begin{equation}\label{e:direct-Lagrange}
\|u-\tilde Q_Pu\|_{\Leb{p}(\Omega)} \lesssim \left( \sum_{\tau\in P} |\tau|^{p\delta} |u|_{A^\alpha_{q,q}(\hat\tau)}^p \right)^{\frac1p} ,
\qquad u\in A^\alpha_{q,q}(\Omega) , \quad P\in\mathscr{P} ,
\end{equation}
where $\delta=\frac\alpha{n}+\frac1p-\frac1q$.
In particular, we have $A^{\alpha}_{q,q}(\Omega)\hookrightarrow\mathscr{A}^s(\Leb{p}(\Omega))$ with $s=\frac\alpha{n}$.
\end{theorem}
\begin{proof}
We have the sub-additivity property
\begin{equation}
\sum_{k} \|u\|_{A^\alpha_{q,q}(P_k(\tau_k))}^q \lesssim \|u\|_{A^\alpha_{q,q}(\Omega)}^q,
\end{equation}
for $0<q<\infty$ and for any finite sequences $\{P_k\}\subset\mathscr{P}$ and $\{\tau_k\}$, with $\tau_k\in P_k$ and $\{\tau_k\}$ non-overlapping.
Recall that $\hat\tau=P(\tau)$ is the support extension of $\tau$, as defined in \eqref{e:supp-ext}.
Therefore the estimate \eqref{e:direct-Lagrange} would imply the second statement by Theorem \ref{t:direct-std}.
We shall prove \eqref{e:direct-Lagrange}.
Every element $\tau\in P$ of any partition $P\in\mathscr{P}$ is an element of a unique $P_j$,
with the number $j$ counting how many refinements one needs in order to arrive at $\tau$.
We call $j$ the {\em generation} or the {\em level} of $\tau$, and write $j=[\tau]$.
We will also need $j(\tau)=\min\{[\sigma]:\sigma\in P,\,\sigma\subset\hat\tau\}$.
Note that $|\tau|\sim\lambda^{-n[\tau]}\sim\lambda^{-nj(\tau)}$
and $S_{j(\tau)}|_{\hat\tau}\subset S_P|_{\hat\tau}$.
By invoking \eqref{e:quasi-interp-local-best-Lagrange}, we infer
\begin{equation}
\begin{split}
\|u - \tilde Q_{P} u\|_{\Leb{p}(\Omega)}^p
&= \sum_{\tau\in P} \|u - \tilde Q_{P} u\|_{\Leb{p}(\tau)}^p
\lesssim \sum_{\tau\in P} \inf_{v\in S_P} \|u-v\|_{\Leb{p}(\hat\tau)}^p \\
&\leq \sum_{\tau\in P} \|u-u_{j(\tau)}\|_{\Leb{p}(\hat\tau)}^p ,
\end{split}
\end{equation}
where $u_j\in S_j$ ($j\geq0$) is an approximation (that may depend on $\tau$) satisfying
\begin{equation}\label{e:def-uj-pf}
\|u-u_j\|_{\Leb{q}(\hat\tau)}\leq c E(u,S_j)_{\Leb{q}(\hat\tau)},
\end{equation}
with some constant $c\geq1$.
The same is true for $p=\infty$ with obvious modifications.
For an individual term in the right hand side, with $p^*=\min\{1,p\}$, we have
\begin{equation}
\begin{split}
\|u-u_{j(\tau)}\|_{\Leb{p}(\hat\tau)}^{p^*}
&\leq
\sum_{j=j(\tau)}^\infty \|u_{j+1}-u_j\|_{\Leb{p}(\hat\tau)}^{p^*}
\lesssim
\sum_{j=j(\tau)}^\infty \lambda^{(\frac1q-\frac1p)jnp^*}\|u_{j+1}-u_j\|_{\Leb{q}(\hat\tau)}^{p^*} \\
&\lesssim
\sum_{j=j(\tau)}^\infty \lambda^{(\frac1q-\frac1p)jnp^*}\|u-u_j\|_{\Leb{q}(\hat\tau)}^{p^*} ,
\end{split}
\end{equation}
where we have estimated $u-u_{j(\tau)}$ as a telescoping sum in the first step,
and used the estimate $\lambda^{jn/p}\|v\|_{\Leb{p}(\hat\tau)}\sim\lambda^{jn/q}\|v\|_{\Leb{q}(\hat\tau)}$ for $v\in S_{j+1}$ in the second step.
We continue by noting the relation $\frac1q-\frac1p=\frac\alpha{n}-\delta$, which yields
\begin{equation}\label{e:u-uj-pf}
\begin{split}
\|u-u_{j(\tau)}\|_{\Leb{p}(\hat\tau)}^{p^*}
&\lesssim
\sum_{j=j(\tau)}^\infty \lambda^{-j\delta np^*} \lambda^{j\alpha p^*} \|u-u_j\|_{\Leb{q}(\hat\tau)}^{p^*} \\
&\leq
\lambda^{-j(\tau)\delta np^*} \sum_{j=j(\tau)}^\infty \lambda^{j\alpha p^*} \|u-u_j\|_{\Leb{q}(\hat\tau)}^{p^*} \\
&\lesssim
|\tau|^{\delta p^*} |u|_{A^\alpha_{q,p^*}(\hat\tau)}^{p^*} ,
\end{split}
\end{equation}
by \eqref{e:def-uj-pf}.
This establishes the theorem for $q\leq1$, in which case we have $A^\alpha_{q,q}(\hat\tau)\hookrightarrow A^\alpha_{q,p^*}(\hat\tau)$.
If $q>1$, choose $0<\alpha_1<\alpha<\alpha_2$ satisfying $\alpha=\frac{\alpha_1+\alpha_2}{2}$ and $\delta_i=\frac{\alpha_i}{n}+\frac1p-\frac1q>0$ for $i=1,2$.
Moreover, we put $u_j=Q_{P_j}^{(\hat\tau)}u$, where $Q_{P_j}^{(\hat\tau)}:\Leb{1}(\hat\tau)\to S_j|_{\hat\tau}$ is the quasi-interpolation operator defined in \eqref{e:quasi-interpolator-std},
with $\hat\tau$ playing the role of $\Omega$.
Then Lemma \ref{l:quasi-interp-std} guarantees the property \eqref{e:def-uj-pf} with $c$ depending only on global geometric properties of $\mathscr{P}$.
In particular, $c$ is bounded independently of $\tau$.
Thus \eqref{e:u-uj-pf} gives
\begin{equation}
\|u-u_{j(\tau)}\|_{L^{p}(\hat\tau)} \lesssim |\tau|^{\delta_i} |u|_{A^{\alpha_i}_{q,1}(\hat\tau)} ,
\end{equation}
for $i=1,2$.
Since the operators $Q_{P_j}^{(\hat\tau)}$ are linear, so is the map $u\mapsto u-u_{j(\tau)}$,
and hence interpolation and Theorem \ref{t:interp-approx} yield
\begin{equation}
\|u-u_{j(\tau)}\|_{L^{p}(\hat\tau)}
\lesssim |\tau|^{(\delta_1+\delta_2)/2} |u|_{[A^{\alpha_1}_{q,1}(\hat\tau),A^{\alpha_2}_{q,1}(\hat\tau)]_{1/2,p}}
\lesssim |\tau|^{\delta} |u|_{A^{\alpha}_{q,p}(\hat\tau)}
\lesssim |\tau|^{\delta} |u|_{A^{\alpha}_{q,q}(\hat\tau)} ,
\end{equation}
with the implicit constants depending only on global geometric properties of $\mathscr{P}$ and on the indices of the spaces involved.
This completes the proof.
\end{proof}
\begin{figure}[ht]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.85\textwidth]{direct1}
\subcaption{If the space $A^\alpha_{q,q}$ is located above the solid line,
we have $A^\alpha_{q,q}\subset\mathscr{A}^s(\Leb{p})$ with $s=\frac\alpha{n}$.}
\end{subfigure}
\qquad
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.9\textwidth]{direct2}
\subcaption{If the space $A^\alpha_{q,q}$ is located above the solid line,
we have $A^\alpha_{q,q}\subset\mathscr{A}^s(A^\sigma_{p,p})$ with $s=\frac{\alpha-\sigma}{n}$.}
\end{subfigure}
\caption{Illustration of Theorem \ref{t:direct-Lp} and Theorem \ref{t:direct-App}.}
\label{f:direct-embeddings-std}
\end{figure}
Now we look at adaptive approximation in the space $A^\sigma_{p,p}(\Omega)$.
Recall from \cite{GM13} that $\mathscr{A}^s_q(A^\sigma_{p,p}(\Omega))\hookrightarrow A^{\alpha}_{q,q}(\Omega)$ for $s=\frac{\alpha-\sigma}{n}=\frac1q-\frac1p>0$ and $0<p,q<\infty$.
\begin{theorem}\label{t:direct-App}
Let $0<q\leq p\leq\infty$, and $\alpha,\sigma>0$ satisfy $\frac{\alpha-\sigma}{n}+\frac1p-\frac1q>0$ and $q<\infty$.
Then for any $0<p_0<q$, we have
\begin{equation}
\|u-\tilde Q_Pu\|_{A^\sigma_{p,p}(\Omega)} \lesssim \left( \sum_{\tau\in P} |\tau|^{p\delta} |u|_{A^\alpha_{q,q}(\hat\tau)}^p \right)^{\frac1p} ,
\qquad u\in A^\alpha_{q,q}(\Omega) , \quad P\in\mathscr{P} ,
\end{equation}
with $\delta=\frac{\alpha-\sigma}{n}+\frac1p-\frac1q$.
In particular, we have $A^{\alpha}_{q,q}(\Omega)\hookrightarrow\mathscr{A}^s(A^\sigma_{p,p}(\Omega))$ with $s=\frac{\alpha-\sigma}{n}$.
\end{theorem}
\begin{proof}
With $v=u-\tilde Q_Pu$ and $\tilde Q_j=\tilde Q_{P_j}$, we have
\begin{equation}\label{e:original-sum-pf}
\|v\|_{A^\sigma_{p,p}(\Omega)}
\leq \left( \sum_{j\geq0} \lambda^{j\sigma p} \|v-\tilde Q_jv\|_{\Leb{p}(\Omega)}^p \right)^{\frac1p}
= \left( \sum_{\tau\in P} \sum_{j\geq0} \lambda^{j\sigma p} \|v-\tilde Q_jv\|_{\Leb{p}(\tau)}^p \right)^{\frac1p} ,
\end{equation}
with the usual modification for $p=\infty$.
Let $j(\tau)=\max\{[\sigma]:\sigma\in P,\,\sigma\subset\tau\}$ for $\tau\in P$,
as in the preceding proof,
with $[\sigma]$ denoting the generation number (or the level) of $\sigma$.
Then for $\tau\in P$ and $j\geq j(\tau)$ we have $S_P|_{\hat\tau}\subset S_j|_{\hat\tau}$, and hence
\begin{equation}
\tilde Q_j(u-\tilde Q_Pu) = \tilde Q_ju-\tilde Q_Pu
\quad \textrm{on } \tau ,
\end{equation}
by the linearity property \eqref{e:quasi-interpolator-linearity}.
This implies that $v-\tilde Q_jv=u-\tilde Q_ju$ on $\tau$, for all $j\geq j(\tau)$.
Now,
proceeding exactly as in the preceding proof, with $p^*=\min\{1,p\}$, we infer
\begin{equation}
\begin{split}
\|u-\tilde Q_ju\|_{\Leb{p}(\tau)}^{p^*}
&\leq
\sum_{k=j}^\infty \|\tilde Q_{k+1}u-\tilde Q_ku\|_{\Leb{p}(\tau)}^{p^*}
\lesssim
\sum_{k=j}^\infty \lambda^{(\frac1q-\frac1p)knp^*} \|\tilde Q_{k+1}u-\tilde Q_ku\|_{\Leb{q}(\tau)}^{p^*} \\
&\lesssim
\sum_{k=j}^\infty \lambda^{(\frac1q-\frac1p)knp^*} \|u-\tilde Q_ku\|_{\Leb{q}(\tau)}^{p^*} .
\end{split}
\end{equation}
Then the discrete Hardy inequality yields
\begin{equation}
\begin{split}
\sum_{j\geq j(\tau)} \lambda^{j\sigma q}\|u-\tilde Q_ju\|_{\Leb{p}(\tau)}^{q}
&\lesssim
\sum_{k\geq j(\tau)} \lambda^{k\sigma q} \lambda^{(\frac1q-\frac1p)knq} \|u-\tilde Q_ku\|_{\Leb{q}(\tau)}^{q} \\
&\leq
\lambda^{-\delta n q j(\tau)} \sum_{k=j(\tau)}^\infty \lambda^{k\alpha q} \|u-\tilde Q_ku\|_{\Leb{q}(\tau)}^{q} \\
&\lesssim
|\tau|^{\delta q} |u|_{A^\alpha_{q,q}(\tau)}^q ,
\end{split}
\end{equation}
where we have taken into account the relation $\frac{\sigma}{n}+\frac1q-\frac1p=\frac{\alpha}{n}-\delta$.
Notice that the discrete Hardy inequality made the use of interpolation unnecessary, to compare the present arguments with the proof of the preceding theorem.
This takes care of one of the sums (or maximums) when we split the sum in the right hand side of \eqref{e:original-sum-pf} into two sums according to $j<j(\tau)$ or $j\geq j(\tau)$.
We rewrite the other sum (or maximum) as
\begin{equation}\label{e:arg-pf}
\begin{split}
\left( \sum_{\tau\in P} \sum_{\{j<j(\tau)\}} \lambda^{j\sigma p}\|v-\tilde Q_jv\|_{\Leb{p}(\tau)}^{p} \right)^{\frac1p}
&=
\left( \sum_{j\geq0} \sum_{\{\tau\in P:j(\tau)>j\}} \lambda^{j\sigma p}\|v-\tilde Q_jv\|_{\Leb{p}(\tau)}^{p} \right)^{\frac1p} \\
&=
\left( \sum_{j\geq0} \lambda^{j\sigma p} \|v-\tilde Q_jv\|_{\Leb{p}(\Omega_j)}^{p} \right)^{\frac1p} ,
\end{split}
\end{equation}
where $\Omega_j=\bigcup\{\tau\in P:j(\tau)>j\}$.
Note that $\Omega_j\supset\Omega_j^0$ with $\Omega_j^0=\bigcup\{\tau\in P:[\tau]>j\}$,
and that $\Omega_j^0$ consists of triangles from $P_j$, in the sense that there is $R_j^0\subset P_j$ such that $\Omega_j^0=\bigcup\{\tau\in R_j^0\}$ up to a zero measure set.
The triangles $\tau\in P$ with $\tau\not\subset\Omega_j^0$ are at the level $j$ or less, and
hence there is $R_j\subset P_j$ such that $\Omega_j=\bigcup\{\tau\in R_j\}$ up to a zero measure set.
Now, by the stability property \eqref{e:quasi-interp-stable}, we get
\begin{equation}
\|v-\tilde Q_jv\|_{\Leb{p}(\Omega_j)}
\lesssim
\|v\|_{\Leb{p}(\Omega_j)} + \|\tilde Q_jv\|_{\Leb{p}(\Omega_j)}
\lesssim
\|v\|_{\Leb{p}(\hat\Omega_j)} ,
\end{equation}
where $\hat\Omega_j = \bigcup\{\tau\in P_j:\bar\tau\cap\Omega_j\neq\varnothing\}$.
Obviously, $\hat\Omega_j$ is a subset of $\hat\Omega_j'=\bigcup\{\tau\in P:\bar\tau\cap\Omega_j\neq\varnothing\}$,
that can also be described as $\hat\Omega_j'=\bigcup\{\tau\in P:j^2(\tau)>j\}$, with
$j^2(\tau)=\max\{j(\sigma):\sigma\in P,\,\bar\sigma\cap\bar\tau\neq\varnothing\}$ for $\tau\in P$.
All this yields
\begin{equation}
\begin{split}
\left( \sum_{\tau\in P} \sum_{\{j<j(\tau)\}} \lambda^{j\sigma p}\|v-\tilde Q_jv\|_{\Leb{p}(\tau)}^{p} \right)^{\frac1p}
&\lesssim
\left( \sum_{\tau\in P} \sum_{\{j<j^2(\tau)\}} \lambda^{j\sigma p}\|u-\tilde Q_Pu\|_{\Leb{p}(\tau)}^{p} \right)^{\frac1p} \\
&\lesssim
\left( \sum_{\tau\in P} |\tau|^{\sigma p/n} \|u-\tilde Q_Pu\|_{\Leb{p}(\tau)}^{p} \right)^{\frac1p} ,
\end{split}
\end{equation}
where we have taken into account the geometric growth of $\lambda^{j\sigma p}$ in $j$,
and the fact that $\lambda^{j^2(\tau)}\sim|\tau|^{1/n}$.
Then once we recall from the proof of the preceding theorem that
\begin{equation}
\|u-\tilde Q_Pu\|_{\Leb{p}(\tau)}
\lesssim
|\tau|^{\delta'} |u|_{A^\alpha_{q,q}(\hat\tau)}
\end{equation}
with $\delta'=\frac\alpha{n}+\frac1p-\frac1q=\delta-\frac\sigma{n}$,
the proof is complete.
\end{proof}
\subsection{Discontinuous piecewise polynomials}
\label{ss:disc}
All that has been said on multilevel and adaptive approximation for continuous Lagrange finite elements have analogues in
the world of discontinuous polynomials subordinate to triangulations.
The theory is in fact much simpler due to the absence of the continuity requirement across elements.
Thus we will state here the relevant results and only sketch or omit the proofs.
The notations $\mathscr{P}$, $\{P_j\}$, etc., will mean the same things as before.
For $P\in\mathscr{P}$, let
\begin{equation}
\bar S_P = \bar S^d_P = \{v\in \Leb{\infty}(\Omega):v|_\tau\in\mathbb{P}_d\,\forall\tau\in P\} ,
\end{equation}
where $d$ is a nonnegative integer, and let $\bar S_j=\bar S_{P_j}$ for all $j$.
Then with $G\subset\Omega$ a domain consisting of elements from some $P_j$, we define the {multilevel approximation spaces}
$A^\alpha_{p,q}(\{\bar S_j\},G)$ by \eqref{e:multilevel-approx}, with the sequence $\{\bar S_j\}$ replacing $\{S_j\}$.
We will also use the shorthand notations
\begin{equation}
\bar A^\alpha_{p,q}(G) = \bar A^\alpha_{p,q;d}(G) = A^\alpha_{p,q}(\{\bar S_j\},G) .
\end{equation}
The analogue of Theorem \ref{t:multilevel-Besov} is the following.
\begin{theorem}\label{t:multilevel-Besov-disc}
We have ${B^\alpha_{p,q;d+1}(\Omega)} \hookrightarrow {\bar A^\alpha_{p,q;d}(\Omega)}$
for $0<p,q\leq\infty$, and $\alpha>0$.
In the other direction, we have ${\bar A^\alpha_{p,q;d}(\Omega)} \hookrightarrow {B^\alpha_{p,q;d+1}(\Omega)}$
for $0<p,q\leq\infty$, and $0<\alpha<\frac1p$.
\end{theorem}
Note that due to the lack of continuity the inverse inclusion holds in a very small range of indices.
We also have the analogue of Theorem \ref{t:interp-approx}.
\begin{theorem}
Let $0<\alpha_1<\alpha_2<\infty$ and $0<p,q,q_1,q_2\leq\infty$.
Then we have
\begin{equation}
[\bar A^{\alpha_1}_{p,q_1}(G),\bar A^{\alpha_2}_{p,q_2}(G)]_{\theta,q} = \bar A^\alpha_{p,q}(G),
\end{equation}
for $\alpha=(1-\theta)\alpha_1+\theta\alpha_2$ and $0<\theta<1$,
with the equivalence constants of quasi-norms depending only on the parameters $\alpha$, $\alpha_1$, $\alpha_2$, $p$, $q$, $q_1$ and $q_2$.
\end{theorem}
Finally, we want to record some results on adaptive approximation by discontinuous polynomials subordinate to the partitions in $\mathscr{P}$.
Given $0<p\leq\infty$, $\theta\in\mathbb{R}$, and $P\in\mathscr{P}$, we define the norm
\begin{equation}\label{e:Leb-disc}
\|u\|_{\Leb[\theta]{p}(\Omega)} = \left( \sum_{\tau\in P} |\tau|^{\frac{\theta p}n} \|u\|_{\Leb{p}(\tau)}^p \right)^{\frac1p} ,
\qquad \textrm{for} \quad u\in\Leb{p}(\Omega) ,
\end{equation}
with the obvious modification for $p=\infty$, and denote by $\Leb[\theta]{p}(\Omega)$ the space $\Leb{p}(\Omega)$ equipped with this norm.
Then we define the approximation class $\bar\mathscr{A}^s_{q;d}(\Leb[\theta]{p}(\Omega))$ exactly as $\mathscr{A}^s_q(\Leb{p}(\Omega))$,
by replacing $S^m_P$ with $\bar S^d_P$, and by using the distance function
\begin{equation}\label{e:rho-disc}
\rho(u,v,P) = \|u-v\|_{\Leb[\theta]{p}(\Omega)} .
\end{equation}
More precisely, recalling the definition \eqref{e:A-rho-def}, let
\begin{equation}\label{e:adapt-class-disc}
\bar\mathscr{A}^s_{q;d}(\Leb[\theta]{p}(\Omega)) = \mathscr{A}^s_q(\rho,\mathscr{P},\{\bar S^d_P\}) ,
\end{equation}
with $\rho$ given by \eqref{e:rho-disc}.
It is for later reference that we have introduced the mesh dependent weight in the distance function.
We write $\bar\mathscr{A}^s_{q;d}(\Leb{p}(\Omega))=\bar\mathscr{A}^s_{q;d}(\Leb[0]{p}(\Omega))$.
We have the following direct embedding result.
\begin{theorem}\label{t:direct-Lp-disc}
Let $0<q\leq p\leq\infty$, $\alpha>0$ and $\theta\geq0$ satisfy $\frac\alpha{n}+\frac1p-\frac1q>0$ and $q<\infty$, with $\frac\alpha{n}+\frac1p-\frac1q=0$ allowed if $\theta>0$.
Then we have $\bar A^{\alpha}_{q,q;d}(\Omega)\hookrightarrow\bar\mathscr{A}^s_{\infty;d}(\Leb[\theta]{p}(\Omega))$ with $s=\frac{\alpha+\theta}{n}$.
\end{theorem}
\begin{proof}
Let $u\in\Leb{p}(\Omega)$ and let $P\in\mathscr{P}$.
Then with $\Pi_P$ the projection operator defined in \eqref{e:quasi-interpolator-disc} with $m:=d$, we have
\begin{equation}
\|u - \Pi_P u\|_{\Leb[\theta]{p}(\Omega)}
= \left( \sum_{\tau\in P} |\tau|^{\frac{\theta p}n} \|u - \Pi_{P} u\|_{\Leb{p}(\tau)}^p \right)^{\frac1p}
\lesssim \left( \sum_{\tau\in P} |\tau|^{\frac{\theta p}n} \inf_{v\in \bar S_P} \|u-v\|_{\Leb{p}(\tau)}^p \right)^{\frac1p} .
\end{equation}
Now proceeding exactly as in the proof of Theorem \ref{t:direct-Lp}, we get
\begin{equation}
\|u - \Pi_P u\|_{\Leb[\theta]{p}(\Omega)}
\lesssim \left( \sum_{\tau\in P} |\tau|^{\frac{\theta p}n} |\tau|^{\delta p} |u|_{\bar A^{\alpha}_{q,q;d}(\tau)} \right)^{\frac1p} ,
\end{equation}
with $\delta=\frac\alpha{n}+\frac1p-\frac1q$.
Then an application of Theorem \ref{t:direct-std} finishes the proof.
\end{proof}
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{direct3}
\caption{Illustration of Theorem \ref{t:direct-Lp-disc} and Theorem \ref{t:inverse-Lp-disc}.
If the space $\bar A^\alpha_{q,q}$ is located above or on the solid line,
then $\bar A^\alpha_{q,q}\subset\bar\mathscr{A}^s(\Leb[\theta]{p})$ with $s=\frac{\alpha+\theta}n$.
It is as if the approximation is taking place in a space such as $B^{-\theta}_{p,p}$,
but instead of $\frac{\alpha+\theta}{n}>\frac1q-\frac1p$ (dashed line) we have the condition $\frac\alpha{n}\geq\frac1q-\frac1p$ (solid line).
On the other hand, the inverse embedding takes the form $\bar\mathscr{A}^s_{q}(\Leb[\theta]{p})\cap\Leb{p}\subset\bar A^{\alpha}_{q,q}$,
which holds on the part of the dashed line with $\alpha>0$.}
\label{f:direct-embeddings-disc}
\end{figure}
We close this section by stating an inverse embedding theorem (A proof can be found in arXiv version 1 of the current paper).
\begin{theorem}\label{t:inverse-Lp-disc}
Let $0<q\leq p<\infty$, $\alpha,\theta>0$, and let $s=\frac{\alpha+\theta}{n}=\frac1q-\frac1p$.
Then we have $\bar\mathscr{A}^s_{q;d}(\Leb[\theta]{p}(\Omega))\cap\Leb{p}(\Omega)\subset\bar A^{\alpha}_{q,q;d}(\Omega)$.
\end{theorem}
\section{Second order elliptic problems}
\label{s:2nd-order}
\subsection{Introduction}
In this section, we will apply the abstract theory of Section \ref{s:general} to second order elliptic boundary value problems.
As far as the domain $\Omega$ and the family of triangulations $\mathscr{P}$ are concerned,
we will keep the setting of the previous section intact.
In particular, we fix a refinement rule, which is either the newest vertex bisection or the red refinement,
and assume that the family $\mathscr{P}$ satisfies the admissibility criterion \eqref{e:finite-support}.
Let $\Gamma\subset\partial\Omega$ be an open piece (or the whole) of the boundary, consisting of faces of the initial triangulation $P_0$.
For $P\in\mathscr{P}$, the space $S_P$ will be the Lagrange finite element space of continuous piecewise polynomials of degree not exceeding $m$,
with the homogeneous Dirichlet condition on $\Gamma$.
We also define $H^1_\Gamma(\mathbb{R}^n)$ as the closure of $\mathscr{D}(\mathbb{R}^n\setminus\overline\Gamma)$ in $H^1(\mathbb{R}^n)$,
and $H^1_\Gamma=H^1_\Gamma(\Omega)$ as the restriction of functions from $H^1_\Gamma(\mathbb{R}^n)$ to $\Omega$.
Note that $S_P=H^1_\Gamma\cap S^m_P$.
The operator $T$ is given as
\begin{equation}
Tu = - a_{ij} \partial_i \partial_j u + b_{k} \partial_k u + cu,
\end{equation}
where the repeated indices are summed over.
The coefficients $a_{ij}$ are Lipschitz continuous, and $b_{k}, c\in \Leb{\infty}(\Omega)$.
The problem we consider is to find $u\in H^1_\Gamma$ satisfying
\begin{equation}\label{e:2nd-order-var-form}
\langle Tu, v\rangle = \langle f, v\rangle,
\qquad\textrm{for all}\quad
v\in H^1_\Gamma.
\end{equation}
Here $\langle\cdot,\cdot\rangle$ is the duality pairing between $(H^1_\Gamma)'$ and $H^1_\Gamma$,
and $f\in \Leb{2}(\Omega) \hookrightarrow (H^1_\Gamma)'$ is given.
This is of course the variational formulation of the mixed Dirichlet-Neumann problem with the homogenous Dirichlet data on $\Gamma$.
We will also denote by $T:H^1_\Gamma\to(H^1_\Gamma)'$ the operator defined in \eqref{e:2nd-order-var-form}.
Given an $n$-simplex $\tau$, let us denote by $E_\tau$ the union of the $(n-1)$-dimensional open faces of $\tau$.
In other words, $E_\tau$ is the boundary of $\tau$ with all but the $(n-1)$-dimensional faces removed.
Then for $P\in\mathscr{P}$, let
\begin{equation}
E_P=\{E_\tau\cap E_\sigma\cap(\bar\Omega\setminus\Gamma):\tau,\sigma\in P\} ,
\end{equation}
be the set of faces not intersecting the Dirichlet piece $\Gamma$.
Note that if $P$ is nonconforming, then only the faces of the ``smaller'' simplices go into $E_P$.
Given $P\in\mathscr{P}$, $u\in T^{-1}(\Leb{2}(\Omega))$, and $v\in S_P$, define the {\em element residual} $r_\tau=(Tu-Tv)|_\tau$ for $\tau\in P$,
and the {\em edge residual} $r_e\in \Leb{2}(e)$ for $e\in E_P$ as the jump of the normal component of the vector field $a_{ij}\partial_jv$ across the edge $e$.
Finally, we define the {\em a posteriori} error estimator
\begin{equation}\label{e:2nd-order-err-est}
(\eta(u,v,P))^2 = \sum_{\tau\in P} h_\tau^{2}\|r_\tau\|_{\Leb{2}(\tau)}^2 + \sum_{e\in E_P} h_e \|r_e\|_{\Leb{2}(e)}^2 .
\end{equation}
A typical adaptive finite element method that uses \eqref{e:2nd-order-err-est} as its error indicator converges optimally
with respect to the approximation classes $\mathscr{A}^s(\eta)$,
in the sense that if the solution $u$ of the problem \eqref{e:2nd-order-var-form} satisfies $u\in\mathscr{A}^s(\eta)$ for some $s>0$,
then the adaptive method reduces the quantity $\eta(u,u_P,P)$ with the rate $s$, where $u_P$ is the Galerkin approximation of $u$ from $S_P$,
cf. \cite*{FFP14}.
Moreover, it is well known that the estimator \eqref{e:2nd-order-err-est} is equivalent to the {\em total error}
\begin{equation}\label{e:2nd-order-tot-err}
(\rho_d(u,v,P))^2 = \|u-v\|_{H^{1}}^2 + (\mathrm{osc}_d(u,v,P))^2,
\end{equation}
when $v=u_P$ and for any fixed $d\geq m-2$,
where the {\em oscillation} is defined as
\begin{equation}
(\mathrm{osc}_d(u,v,P))^2 = \sum_{\tau\in P} h_\tau^{2}\|(1-\Pi_\tau)r_\tau\|_{\Leb{2}(\tau)}^2 + \sum_{e\in E_P} h_e \|(1-\Pi_e) r_e\|_{\Leb{2}(e)}^2 ,
\end{equation}
with $\Pi_\tau:\Leb{2}(\tau)\to\mathbb{P}_d$ and $\Pi_e:\Leb{2}(e)\to\mathbb{P}_{d+1}$ being $\Leb{2}$-orthogonal projections onto polynomial spaces,
see e.g., \cite*{NSV09}.
Optimality of adaptive finite element methods with respect to the approximation classes $\mathscr{A}^s(\rho_d)$ has also been proved, cf. \cite{CKNS08}.
Ideally, one would like to have optimality with respect to the classes $\mathscr{A}^s(H^1_\Gamma)$ that correspond to the energy error.
In particular, it is conceivable that for certain functions $u$, the energy error $\|u-u_P\|_{H^1}$ decays faster than the oscillation
$\mathrm{osc}_d(u,u_P,P)$, so that the class $\mathscr{A}^s(H^1_\Gamma)$ is strictly larger than both $\mathscr{A}^s(\rho_d)$ and $\mathscr{A}^s(\eta)$.
However, if the error estimator \eqref{e:2nd-order-err-est} is the only source of information used by the algorithm in its stopping criterion
(or in the marking of triangles for refinement),
then it is clear that one has to reduce the oscillation anyway.
It appears therefore that the approximation classes $\mathscr{A}^s(\rho_d)$ and $\mathscr{A}^s(\eta)$ are completely natural from the perspective of adaptive finite element methods.
\subsection{A characterization of adaptive approximation classes}
In this subsection, we give necessary and sufficient conditions for $u\in H^1_\Gamma$ to be in $\mathscr{A}^s(\rho_d)$.
These conditions will be in terms of memberships of $u$ and $Tu$ into suitable approximation classes,
which, in light of the preceding section, are related to Besov spaces.
The coefficients of $T$ are required to satisfy conditions of the form $g\in\bar\mathscr{A}^s_{\infty;d}(\Leb[\theta]{\infty}(\Omega))$,
where the latter space is defined in \eqref{e:adapt-class-disc},
and again these spaces can be cast in terms of Besov spaces with the help of Theorem \ref{t:direct-Lp-disc} and Theorem \ref{t:multilevel-Besov-disc}.
\begin{theorem}\label{t:elliptic-2nd}
Let $s>0$ and let $d\geq m-2$.
Assume that $a_{ij}\in\bar\mathscr{A}^s_{\infty;d+2-m}(\Leb{\infty}(\Omega))$, $b_{i}\in\bar\mathscr{A}^s_{\infty;d+1-m}(\Leb[1]{\infty}(\Omega))$, and $c\in\bar\mathscr{A}^s_{\infty;d-m}(\Leb[2]{\infty}(\Omega))$.
Then we have
\begin{equation}
\mathscr{A}^{s}(\rho_d) = \mathscr{A}^{s}(H^1_\Gamma)\cap T^{-1}(\bar\mathscr{A}^s_{\infty;d}(\Leb[1]{2}(\Omega))) .
\end{equation}
\end{theorem}
The proof of this theorem will be given below in Lemma \ref{l:elliptic-2nd-direct} and Lemma \ref{l:elliptic-2nd-inverse}.
Before proving those lemmata, let us make a few points on the conditions of the theorem.
First, recall from Theorem \ref{t:direct-Lp-disc} that $\bar A^{\sigma}_{q,q;d}(\Omega)\subset\bar\mathscr{A}^s_{\infty;d}(\Leb[1]{2}(\Omega))$
for $\sigma=sn-1$ and $0\leq\frac1q - \frac12\leq\frac\sigma{n}$,
and from Theorem \ref{t:multilevel-Besov-disc} that $B^{\sigma}_{q,q}(\Omega)\subset\bar A^{\sigma}_{q,q;d}(\Omega)$ for $\sigma<d+\max\{1,\frac1q\}$.
Second, while the approximation classes $\mathscr{A}^{s}(H^1_\Gamma)$ are associated to the finite element spaces $S_P=H^1_\Gamma\cap S^m_P$,
the approximation classes we considered in the preceding section are associated to the spaces $S^m_P$ with no boundary conditions.
In view of applying Theorem \ref{t:direct-App} and Theorem \ref{t:multilevel-Besov}, we need the latter type of approximation classes.
The following lemma provides a link between the two types.
\begin{lemma}
For $s>0$ we have
\begin{equation}\label{e:H1G}
\mathscr{A}^{s}(H^1_\Gamma)\equiv\mathscr{A}^{s}(H^1,\mathscr{P},\{H^1_\Gamma\cap S^m_P\})=H^1_\Gamma\cap\mathscr{A}^{s}(A^1_{2,2}(\Omega),\mathscr{P},\{S^m_P\}) .
\end{equation}
In particular, we have $H^1_\Gamma\cap A^\alpha_{p,p}(\Omega)\subset\mathscr{A}^{s}(H^1_\Gamma)$ for $\alpha=sn+1$ and $\frac1p<s+\frac12$.
\end{lemma}
\begin{proof}
Let $u\in H^1_\Gamma$, and let $u_P\in H^1_\Gamma\cap S^m_P$ be the Scott-Zhang interpolator of $u$ adapted to the Dirichlet boundary condition on $\Gamma$, cf. \cite{SZ90}.
We have
\begin{equation}
\inf_{v\in H^1_\Gamma\cap S^m_P} \|u-v\|_{H^1(\Omega)}
\leq \|u-u_P\|_{H^1(\Omega)}
\lesssim \inf_{v\in S^m_P} \|u-v\|_{H^1(\Omega)} ,
\end{equation}
by standard properties of the Scott-Zhang interpolator.
Since $H^1=B^1_{2,2}=A^1_{2,2}$ by Theorem \ref{t:multilevel-Besov},
this implies \eqref{e:H1G}.
Then the second assertion of the theorem follows from a direct application of Theorem \ref{t:direct-App}.
\end{proof}
The inclusion $\mathscr{A}^{s}(H^1_\Gamma)\cap T^{-1}(\bar\mathscr{A}^s_{\infty;d}(\Leb[1]{2}(\Omega)))\subset\mathscr{A}^{s}(\rho_d)$
of Theorem \ref{t:elliptic-2nd} is a consequence of the following lemma.
\begin{lemma}\label{l:elliptic-2nd-direct}
For $u\in H^1_\Gamma$ and $P\in\mathscr{P}$, there exists $v\in S_P$ such that
\begin{equation}\label{e:elliptic-2nd-direct}
\begin{split}
&\rho_d(u,v,P)^2
\lesssim E(u,S_P)_{H^1(\Omega)}^2
+ E(Tu,\bar S^d_P)_{\Leb[1]{2}(\Omega)}^2 \\
&\quad +
\left( E(a_{ij},\bar S^{d+2-m}_P)_{\Leb{\infty}(\Omega)}^2
+ E(b_{i},\bar S^{d+1-m}_P)_{\Leb[1]{\infty}(\Omega)}^2
+ E(c,\bar S^{d-m}_P)_{\Leb[2]{\infty}(\Omega)}^2 \right)
|u|_{H^1(\Omega)}^2 .
\end{split}
\end{equation}
\end{lemma}
\begin{proof
We take $v$ to be the Scott-Zhang interpolator of $u$ adapted to the Dirichlet boundary condition on $\Gamma$, cf. \cite{SZ90}.
We have
\begin{equation}
\|u-v\|_{H^1} \lesssim \inf_{w\in S_P} \|u-w\|_{H^1},
\end{equation}
for all $P\in\mathscr{P}$.
It remains to bound the oscillation term.
First, let us consider the special case where the coefficients of $T$ are piecewise polynomials subordinate to $P$.
More specifically, assume that $a_{ij}|_\tau\in\mathbb{P}_{d+2-m}$, $b_{i}|_\tau\in\mathbb{P}_{d+1-m}$, and $c|_\tau\in\mathbb{P}_{d-m}$ for each $\tau\in P$.
In this case, the oscillations associated to edges vanish, because the edge residuals $r_e$ are polynomials of degree not exceeding $d+1$.
For the element residuals, with the shorthand $f=Tu$, we have
\begin{equation}\label{e:element-residual-triangle}
\|(1-\Pi_\tau)(f-Tv)\|_{\Leb{2}(\tau)} \leq \|(1-\Pi_\tau)f\|_{\Leb{2}(\tau)} + \|(1-\Pi_\tau)Tv\|_{\Leb{2}(\tau)},
\end{equation}
and the last term is zero because $Tv\in \mathbb{P}_{d}$.
The remaining term gives rise to
\begin{equation}
\sum_{\tau\in P}h_\tau^2\|(1-\Pi_\tau)f\|_{\Leb{2}(\tau)}^2 = E(f,\bar S^d_P)_{\Leb[1]{2}(\Omega)}^2 ,
\end{equation}
which yields the desired result.
In the general case, the edge residuals and the terms $Tv|_\tau$ can be nonpolynomial.
Let us treat $Tv|_\tau=- a_{ij} \partial_i \partial_j v + b_{k} \partial_k v + c v$ term by term.
We have
\begin{equation}
(1-\Pi_\tau)a_{ij}\partial_i\partial_jv = (1-\Pi_\tau)(a_{ij} - \bar a_{ij})\partial_i\partial_jv,
\qquad \bar a_{ij} \in \mathbb{P}_{d+2-m} ,
\end{equation}
which implies
\begin{equation}
\|(1-\Pi_\tau)a_{ij}\partial_i\partial_jv \|_{\Leb{2}(\tau)}
= \|(a_{ij} - \bar a_{ij})\partial_i\partial_jv \|_{\Leb{2}(\tau)}
\leq \|a_{ij} - \bar a_{ij}\|_{\Leb{\infty}(\tau)} \|\partial_i\partial_jv \|_{\Leb{2}(\tau)} ,
\end{equation}
for any $\bar a_{ij} \in \mathbb{P}_{d+2-m}$.
Now we think of $\bar a_{ij}$ as a function in $\bar S^{d+2-m}_P$ that approximates $a_{ij}$ in each element $\tau\in P$ with the best $\Leb{\infty}(\tau)$-error.
As a result, we get
\begin{equation}\label{e:elliptic-2nd-direct-aij}
\begin{split}
\sum_{\tau\in P} h_\tau^2 \|(1-\Pi_\tau)a_{ij}\partial_i\partial_jv \|_{\Leb{2}(\tau)}^2
&\leq \sum_{\tau\in P} h_\tau^2 \|a_{ij} - \bar a_{ij}\|_{\Leb{\infty}(\tau)}^2 \|\partial_i\partial_jv \|_{\Leb{2}(\tau)}^2 \\
&\leq E(a_{ij},\bar S^{d+2-m}_P)_{\Leb{\infty}(\Omega)}^2 \sum_{\tau\in P} h_\tau^2 \|\partial_i\partial_jv \|_{\Leb{2}(\tau)}^2 \\
&\lesssim E(a_{ij},\bar S^{d+2-m}_P)_{\Leb{\infty}(\Omega)}^2 \|\nabla v \|_{\Leb{2}(\Omega)}^2 ,
\end{split}
\end{equation}
where we have used an inverse inequality in the last step.
In light of the $H^1$-stability of the Scott-Zhang projector,
this is one of the terms in the right hand side of \eqref{e:elliptic-2nd-direct}.
Similarly, let $\bar c$ be a function in $\bar S^{d-m}_P$ that approximates $c$ in each element $\tau\in P$ with the best $\Leb{\infty}(\tau)$-error.
Then we have
\begin{equation}\label{e:elliptic-2nd-direct-c}
(1-\Pi_\tau) c v = (1-\Pi_\tau) (c - \bar c) v = (1-\Pi_\tau) (c - \bar c) (v - \bar v),
\end{equation}
in each $\tau\in P$, where $\bar v$ is the average of $v$ over $\tau$.
This yields
\begin{equation}
\begin{split}
\sum_{\tau\in P} h_\tau^2 \|(1-\Pi_\tau)cv\|_{\Leb{2}(\tau)}^2
&\leq \sum_{\tau\in P} h_\tau^2 \|c - \bar c\|_{\Leb{\infty}(\tau)}^2 \|v - \bar v\|_{\Leb{2}(\tau)}^2 \\
&\lesssim \sum_{\tau\in P} h_\tau^4 \|c - \bar c\|_{\Leb{\infty}(\tau)}^2 \|\nabla v\|_{\Leb{2}(\tau)}^2 \\
&\leq E(c,\bar S^{d-m}_P)_{\Leb[2]{\infty}(\Omega)}^2 \|\nabla v\|_{\Leb{2}(\Omega)}^2 ,
\end{split}
\end{equation}
where we have used the Poincar\'e inequality in the second line.
Estimation of the term involving $b_i\partial_iv$ is more straightforward, which we omit.
As for the edge oscillations, let $\tau\in P$, and let $e$ be an edge of $\tau$.
Then we have
\begin{equation}\label{e:elliptic-2nd-direct-edge}
\begin{split}
\|(1-\Pi_e)a_{ij}\partial_jv\|_{\Leb{2}(e)}
&= \|(1-\Pi_e)(a_{ij}-\bar a_{ij})\partial_jv\|_{\Leb{2}(e)} \\
&\leq \|(a_{ij}-\bar a_{ij})\partial_jv\|_{\Leb{2}(e)} \\
&\leq \|a_{ij}-\bar a_{ij}\|_{\Leb{\infty}(e)} \|\partial_jv\|_{\Leb{2}(e)} \\
&\lesssim h_e^{-\frac12} \|a_{ij}-\bar a_{ij}\|_{\Leb{\infty}(\tau)} \|\nabla v\|_{\Leb{2}(\tau)} ,
\end{split}
\end{equation}
for any $\bar a_{ij}\in\mathbb{P}_{d+2-m}$,
which shows that
the contribution of the edge oscillations to the final estimate \eqref{e:elliptic-2nd-direct}
is identical to that of \eqref{e:elliptic-2nd-direct-aij}.
\end{proof}
\begin{remark}
By using the fact that the Scott-Zhang projector is bounded in $H^t(\Omega)$ for $t<\frac32$,
we could have introduced extra powers of $h_\tau$ or $h_e$ into the estimates \eqref{e:elliptic-2nd-direct-aij}, \eqref{e:elliptic-2nd-direct-c}, and \eqref{e:elliptic-2nd-direct-edge}.
This means that the regularity conditions on the coefficients $a_{ij}$, $b_k$, and $c$ in Theorem \ref{t:elliptic-2nd} can be relaxed slightly,
if the conclusion of the theorem is to be changed to $\mathscr{A}^{s}(H^1_\Gamma)\cap H^t(\Omega)\cap T^{-1}(\bar\mathscr{A}^s_{\infty;d}(\Leb[1]{2}(\Omega)))\subset\mathscr{A}^{s}(\rho_d)$ with $1<t<\frac32$.
\end{remark}
\begin{lemma}\label{l:elliptic-2nd-inverse}
For any $u\in H^1_\Gamma$, $P\in\mathscr{P}$ and $v\in S_P$, we have
\begin{equation}\label{e:elliptic-2nd-inverse}
\begin{split}
&E(Tu,\bar S^d_P)_{\Leb[1]{2}(\Omega)}^2 + \|u-v\|_{H^1(\Omega)}^2
\lesssim \rho_d(u,v,P)^2 \\
&\quad +
\left( E(a_{ij},\bar S^{d+2-m}_P)_{\Leb{\infty}(\Omega)}^2
+ E(b_{i},\bar S^{d+1-m}_P)_{\Leb[1]{\infty}(\Omega)}^2
+ E(c,\bar S^{d-m}_P)_{\Leb[2]{\infty}(\Omega)}^2 \right)
|v|_{H^1(\Omega)}^2 .
\end{split}
\end{equation}
In particular, under the hypotheses of Theorem \ref{t:elliptic-2nd}, we have the inclusion $\mathscr{A}^{s}(\rho_d)\subset\mathscr{A}^{s}(H^1_\Gamma)\cap T^{-1}(\bar\mathscr{A}^s_{\infty;d}(\Leb[1]{2}(\Omega)))$.
\end{lemma}
\begin{proof}
All the ingredients for establishing the estimate \eqref{e:elliptic-2nd-inverse} is already given in the proof of the preceding lemma.
Namely, we start with the bound
\begin{equation}
\begin{split}
(\mathrm{osc}_d(u,v,P))^2
&\lesssim \sum_{\tau\in P} h_\tau^{2} \|(1-\Pi_\tau)Tu\|_{\Leb{2}(\tau)}^2 + \sum_{\tau\in P} h_\tau^{2} \|(1-\Pi_\tau)Tv\|_{\Leb{2}(\tau)}^2 \\
&\quad + \sum_{e\in E_P} h_e \|(1-\Pi_e) r_e\|_{\Leb{2}(e)}^2 ,
\end{split}
\end{equation}
and use the estimates \eqref{e:elliptic-2nd-direct-aij}, \eqref{e:elliptic-2nd-direct-c}, and \eqref{e:elliptic-2nd-direct-edge}, etc.,
on the last two terms to get \eqref{e:elliptic-2nd-inverse}.
As for the second assertion, let $\{P_k\}\subset\mathscr{P}$ and $\{v_k\}$ be two sequences with $v_k\in S_{P_k}$
such that $\#P_k\lesssim2^k$ and $\rho_d(u,v_k,P_k)\lesssim2^{-ks}$.
Then since $\|u-v_k\|_{H^1}\leq\rho_d(u,v_k,P_k)$, we have $\|v_k\|_{H^1}\lesssim\|u\|_{H^1}$.
Hence, by employing overlay of partitions, without loss of generality, we can suppose that the right hand side of \eqref{e:elliptic-2nd-inverse} with $P=P_k$ and $v=v_k$ is bounded by
a constant multiple of $2^{-ks}$.
Looking at the left hand side then reveals that $Tu\in\mathscr{A}^s_{\infty;d}(\Leb[1]{2}(\Omega))$ and $u\in\mathscr{A}^s(H^1_\Gamma)$.
\end{proof}
\section*{Acknowledgements}
This work was supported by NSERC Discovery Grants Program and FQRNT New University Researchers Start Up Program.
\bibliographystyle{plainnat}
| {'timestamp': '2016-02-05T02:02:56', 'yymm': '1408', 'arxiv_id': '1408.3889', 'language': 'en', 'url': 'https://arxiv.org/abs/1408.3889'} |
\section{Introduction}
In the interstellar medium, dust grains and radiation fields affect one
another. Grains driven by radiation pressure can transfer momentum to nearby
gas, helping to regulate star formation and launch galactic outflows
\citep{Krumholz2009, Murray2010, Hopkins2012}. Dust scatters and absorbs
starlight and emits in the infrared \citep[IR;][]{Schlegel1998, Bernstein2002,
Schlafly2011}, influencing galactic spectral energy distributions
\citep{Silva1998, Dale2001}. The temperatures of dust grains are affected by
radiative heating, thermal emission, and collisional energy exchange with gas
\citep{Hollenbach1979, Dwek1986, Boulanger1988}, and small grains in particular
are susceptible to non-equilibrium temperature fluctuations \citep{Dwek1986,
Guhathakurta1989, Siebenmorgen1992}. Galaxy formation models must consider
these physical processes when accounting for the impact of dust and radiation.
Owing to the computational cost, many works do not directly treat dust or
radiation. Instead, they attempt to capture the effects of radiation pressure
on dust grains through physically-motivated subgrid methods \citep{Murray2005,
Hopkins2011, Agertz2013}. This typically involves injecting momentum or
thermal energy in gas surrounding sources of radiation like stars, avoiding the
need to evolve radiation fields. For example, to mimic the impact of radiation
multi-scattered by dust, momentum injection rates can increase as a function of
local IR optical depth. Such radiation pressure feedback models have been used
in cosmological settings and can reduce star formation \citep{Aumer2013, Hopkins2014, Roskar2014, Agertz2015}. To improve the accuracy of
these subgrid radiation pressure models, there have been renewed efforts to
develop numerical methods suited for simulations with limited resolution
\citep{Krumholz2018, Hopkins2019}.
In recent years, a variety of methods have been developed to more directly
model radiation and its effect on dust. For example, Monte Carlo methods track
the emission and absorption of individual photon packets but can be
computationally expensive to run \citep[e.g.][]{Bjorkman2001, Oxley2003,
Camps2015b, Tsang2015, Smith2019}. Other methods use long-characteristic ray
tracing to solve the exact radiative transfer equation \citep[e.g.][]{Abel1999,
Abel2002, Wise2012, Greif2014, Jaura2018}, although this can scale unfavourably
with the number of sources. Alternatively, some mesh-based radiation
hydrodynamics solvers combine moments of the radiative transfer equation with
the M1 closure relation \citep{Levermore1984, Dubroca1999}, in which the
Eddington tensor is calculated strictly from local quantities and is
independent of the number of sources. Various codes employ the M1 closure
\citep{Gonzalez2007, Rosdahl2013, Rosdahl2015b, Kannan2019a} and have been used
in radiation hydrodynamic simulations of isolated disc galaxies
\citep{Rosdahl2015, Kannan2019c}, quasar outflows via radiation pressure \citep{Bieri2017, Costa2018b, Barnes2018}, and reionisation calculations \citep[see for eg. ][]{sphinx, Wu2019a, Wu2019b}.
In this work, we evolve radiation using the radiation hydrodynamics solver
\textsc{arepo-rt} \citep{Kannan2019a} integrated into the unstructured
moving-mesh code \textsc{arepo} \citep{Springel2010}. \citet{Kannan2019a}
includes a simplified model for radiation pressure on, photon absorption by,
and IR emission from dust grains, treating dust as a passive scalar perfectly
coupled to hydrodynamic motion and in local thermodynamic equilibrium with gas.
In this work, we relax those assumptions and adopt a more general treatment of
dust in the context of radiation hydrodynamics. We do not treat dust as an
element of gas cells but instead using simulation particles
\citep{McKinnon2018}. This approach allows richer and more realistic dynamical
and thermal interactions among dust, gas, and radiation. We present our
methods and test problems in Section~\ref{SEC:methods} and conclude in
Section~\ref{SEC:conclusion}.
\section{Methods}\label{SEC:methods}
We start with a brief summary of the equation of motions for dust and gas components coupled through
aerodynamic drag and possibly subject to external accelerations. We refer the
reader to Section~2 in \citet{McKinnon2018} for background details. A dust
grain of mass $m_\text{d}$ feels an acceleration
\begin{equation}
\frac{\diff \vec{v}_\text{d}}{\diff t} = -\frac{K_\text{s} (\vec{v}_\text{d} - \vec{v}_\text{g})}{m_\text{d}} + \vec{a}_\text{d,ext},
\label{EQN:dvdt_dust}
\end{equation}
where $\vec{v}_\text{d}$ and $\vec{v}_\text{g}$ are the local dust and gas
velocities, respectively, $\vec{a}_\text{d,ext}$ accounts for additional
sources of dust acceleration (e.g.~gravity or radiation pressure), and $K_\text{s}$ is a drag coefficient. Similarly, the gas acceleration is given by
\begin{equation}
\frac{\diff \vec{v}_\text{g}}{\diff t} = -\frac{\nabla P}{\rho_\text{g}} + \frac{\rho_\text{d} K_\text{s} (\vec{v}_\text{d} - \vec{v}_\text{g})}{\rho_\text{g} m_\text{d}} + \vec{a}_\text{g,ext},
\label{EQN:dvdt_gas}
\end{equation}
where $P$ denotes gas pressure, $\rho_\text{d}$ and $\rho_\text{g}$ are the
dust and gas density, respectively, and $\vec{a}_\text{g,ext}$ is the external
gas acceleration. For use later, we define the dust-to-gas ratio $D \equiv
\rho_\text{d} / \rho_\text{g}$. Here, the drag backreaction force by dust on
gas is exactly opposite to the drag force by gas on dust. For drag in the
Epstein regime, the equations of motion can be rewritten in terms of
$t_\text{s} \equiv m_\text{d} / [K_\text{s} (1 + D)]$, the stopping time-scale
for drag, whose value is given by the approximation
\begin{equation}
t_\text{s} = \frac{\sqrt{\pi \gamma} a \rho_\text{gr}}{2 \sqrt{2} \rho c_\text{s}} \left( 1 + \frac{9 \pi}{128} \left|\frac{\vec{v}_\text{d} - \vec{v}_\text{g}}{c_\text{s}} \right|^2 \right)^{-1/2},
\label{EQN:t_s}
\end{equation}
for a given grain size $a$, internal solid density of dust grains
$\rho_\text{gr}$, total density $\rho \equiv \rho_\text{g} + \rho_\text{d}$,
and gas sound speed $c_\text{s}$ \citep{Paardekooper2006, Price2017}. The
dust-to-gas ratio $D$ affects the backreaction of dust dragging gas: when $D
\ll 1$, the gas drag acceleration is significantly smaller than the dust
acceleration. Since $D \approx 0.01$ for the Milky Way and other nearby
galaxies \citep[e.g.][]{Draine2007}, in galaxy simulations it is often
reasonable to neglect the backreaction of drag on gas.
However, dust dynamics are applicable in a wide variety of settings, and it is
worthwhile to relax the assumption of low dust-to-gas ratio. Below, we extend
the drag implementation of \citet{McKinnon2018}, which neglected drag
backreaction on gas, to handle drag coupling at arbitrary dust-to-gas ratio.
We adopt the second-order semi-implicit time integrator presented in
\citet{LorenAguilar2015}. Defining the functions
\begin{equation}
\xi(\Delta t) = \frac{1 - e^{- \Delta t / t_\text{s}}}{1 + D}
\end{equation}
and
\begin{equation}
\Lambda(\Delta t) = (\Delta t + t_\text{s}) \xi(\Delta t) - \frac{\Delta t}{1 + D},
\end{equation}
the velocity update of a dust particle from time $t$ to $t + \Delta t$ is given
by
\begin{equation}
\begin{split}
\vec{v}_\text{d}(t + \Delta t) &= \vectilde{v}_\text{d}(t + \Delta t) - \xi(\Delta t) \left[\vectilde{v}_\text{d}(t + \Delta t) - \vectilde{v}_\text{g}(t + \Delta t)\right] \\
&\quad + \Lambda(\Delta t) \left[\vec{a}_\text{d,ext}(t) - \vec{a}_\text{g,ext}(t) + \frac{\nabla P}{\rho_\text{g} }\right],
\end{split}
\label{EQN:dust_velocity_update}
\end{equation}
where $\vec{v}_\text{d}(t + \Delta t)$ is the dust particle velocity after the
drag update and $\vectilde{v}_\text{d}(t + \Delta t)$ and
$\vectilde{v}_\text{g}(t + \Delta t)$ are the dust and gas velocities at time
$t + \Delta t$ after accounting for non-drag accelerations. We apply
equation~(\ref{EQN:dust_velocity_update}) in the following SPH-like fashion to
conserve total momentum. After calculating the change in a dust particle's
momentum $\Delta \vec{p}_\text{d}$, we subtract momentum $w_i \Delta
\vec{p}_\text{d}$ from each of the $N_\text{ngb}$ closest gas cells, where
$w_i$ is a weight assigned to cell $i$ in this set. The amount of momentum
change in each neighboring gas cell is kernel-weighted so that closer gas cells
lose a greater fraction of $\Delta \vec{p}_\text{d}$. In our work, we adopt
the standard cubic spline kernel.
The dissipation of total dust and gas kinetic energy from drag leads to
frictional heating of the gas \citep[e.g.][]{Laibe2012}. When applying
equation~(\ref{EQN:dust_velocity_update}), we calculate the change in kinetic
energy of the dust particle $\Delta K_\text{d}$. Then, after updating the
momentum of local gas cell $i$, we determine its change in kinetic energy
$\Delta K_\text{g}$. The internal energy of the cell is then increased by
$-(w_i \Delta K_\text{d} + \Delta K_\text{g})$. Summing over all cells about a
dust particle, the gas internal energy increases by the amount total dust and
gas kinetic energy decreases.
\subsection{Interaction between dust grains and radiation}
Gravity and drag are not the only dynamical forces that act on interstellar
dust grains. Since dust provides a source of opacity to radiation, the
resulting radiation pressure force \citep[e.g.][]{Weingartner2001c} can also
affect dust and gas dynamics. This has potential implications for the
efficiency of stellar feedback and galactic outflows. For example, some models
predict that multi-scattering of IR radiation due to dust opacity can enhance
self-regulation of star formation \citep{Murray2010, Hopkins2011, Agertz2013}
and boost gas outflows in galaxies up to IR optical depths of $\tau_\text{IR}
\approx 10$ \citep{Thompson2015, Costa2018}. However, others suggest that dust
reprocessing of radiation does not strongly affect wind momentum flux
\citep{Krumholz2013, Reissl2018}. The interaction between dust and radiation can also
affect the CGM in addition to the ISM. Radiation pressure can efficiently
drive dust grains into galactic haloes, with grains of different chemical
composition feeling different strength forces \citep{Ferrara1991}. Given the
reliance of galaxy formation simulations on feedback physics, it is natural to
want to model the interaction between dust and radiation in a more direct
manner.
Recently, \citet{Kannan2019a} developed radiation hydrodynamics methods for the
moving-mesh code \textsc{arepo} \citep{Springel2010}, capable of tracking
multifrequency radiation transport and thermochemistry. Here, we briefly
summarise these methods and refer the reader to Sections~2 and~3 of
\citet{Kannan2019a} for a more complete description. In general, radiative transfer is
formulated in terms of specific intensity $I_\nu(\vec{x}, \vec{n}, t)$, which
at position $\vec{x}$ and time $t$ quantifies the rate of change of radiation
energy at frequency $\nu$ per unit time, per unit area in the direction
$\vec{n}$, per unit solid angle, and per unit frequency. This evolves
according to the radiative transfer equation,
\begin{equation}
\frac{1}{\tilde{c}} \frac{\partial I_\nu}{\partial t} + \vec{n} \cdot \nabla I_\nu = j_\nu - \kappa_\nu \rho I_\nu,
\end{equation}
where $\tilde{c}$ is the propagation speed of radiation, $j_\nu$ denotes the
emission coefficient, $\kappa_\nu$ is the gas opacity, and $\rho$ is the gas
density. Numerically solving the radiative transfer equation is challenging,
given the large number of variables. One approach is to take moments of the
radiative transfer equation, yielding
\begin{equation}
\frac{\partial E}{\partial t} + \nabla \cdot \vec{F} = S - \kappa_\text{E} \rho \tilde{c} E
\label{EQN:E_moment}
\end{equation}
and
\begin{equation}
\frac{\partial F}{\partial t} + \tilde{c}^2 \nabla \cdot \mathbb{P} = -\kappa_\text{F} \rho \tilde{c} \vec{F},
\label{EQN:F_moment}
\end{equation}
where $S$ denotes the emission coefficient integrated over frequency and solid
angle, $\kappa_\text{E}$ and $\kappa_\text{F}$ are the energy- and
flux-weighted mean opacities, and the radiation energy density $E$, flux $F$,
and pressure $\mathbb{P}$ are given by
\begin{equation}
\{ \tilde{c} E, F, \mathbb{P} \} = \int_{\nu_1}^{\nu_2} \int_{4\pi} \{ 1, \vec{n}, \vec{n} \otimes \vec{n} \} I_\nu \diff \Omega \diff \nu.
\end{equation}
All quantities are integrated over a frequency interval $[\nu_1, \nu_2]$. To
simplify this system, \textsc{arepo-rt} employs the M1 closure
\citep{Levermore1984, Dubroca1999} that calculates the pressure strictly in
terms of the local radiation energy density and flux. This approximation
yields a computational cost independent of the number of radiation sources.
Equations~(\ref{EQN:E_moment}) and~(\ref{EQN:F_moment}) are solved on an
unstructured moving mesh using an operator-split approach, breaking the moment
equations into transport and source and sink terms. To calculate intercell
fluxes, \textsc{arepo-rt} employs a Harten-Lax-van Leer or global
Lax-Friedrichs Riemann solver.
The propagation of radiation also affects hydrodynamics. The momentum
conservation equation takes the form
\begin{equation}
\frac{\partial (\rho \vec{v})}{\partial t} + \nabla \cdot (\rho \vec{v} \otimes \vec{v} + P \mathbb{I}) = \frac{\kappa_\text{F} \rho F}{c},
\end{equation}
where $\vec{v}$ is the gas velocity, $P$ is the thermal pressure, $\mathbb{I}$
is the identity tensor, and $c$ is the speed of light. In \citet{Kannan2019a},
the opacities entering the above equations are calculated for individual gas
cells using crude approximations or a simplified dust model treating dust as a
passive scalar perfectly coupled to hydrodynamic motion \citep{McKinnon2016}.
In this work, we instead model dust using live simulation particles
\citep{McKinnon2018} and not as a property of gas cells. We describe below how
the methods in \citet{Kannan2019a} can be extended to model radiation pressure
on and photon absorption by dust grains using this hybrid gas cell and dust
particle scheme. As a result, dust opacities are calculated self-consistently
using the local distribution of dust. We generalize these methods to
multifrequency radiation and distributions of grain sizes and discuss how dust
grains can exchange energy with IR radiation and gas.
\subsubsection{Calculating radiation pressure on populations of grains}\label{SEC:Q_pr_sizes}
The methods described in \citet{McKinnon2018} model dust using simulation
particles representing ensembles of grains of different sizes. Each particle
has a grain size distribution, discretised into $N_\text{GSD}$ grain size bins
with different typical sizes. Similarly, the radiative transfer scheme
introduced in \citet{Kannan2019a} solves for the propagation of radiation in
$N_\text{RT}$ different frequency bins (e.g.~to model different UV wavelength
ranges or simultaneous UV and IR radiation fields).
We calculate the radiation pressure force $\vec{f_\text{pr}}$ on a dust
particle with grains of different sizes in a kernel-smoothed manner,
interpolating over $N_\text{ngb}$ neighboring gas cells. In the general case
of multifrequency, anisotropic radiation \citep{Weingartner2001c}, this
radiation pressure force takes the form
\begin{equation}
\vec{f_\text{pr}} = \sum_{k = 0}^{N_\text{ngb} - 1} w_k \left[ \sum_{i = 0}^{N_\text{GSD} - 1} \sum_{j = 0}^{N_\text{RT} - 1} \left( \frac{\vec{F_{j,k}}}{c} \right) N_i \pi {a_i^\text{c}}^2 Q_\text{pr}(a_i^\text{c}, \lambda_j) \right],
\label{EQN:f_pr}
\end{equation}
where $\vec{F_{j,k}}$ and $\lambda_j$ are the energy flux and (mean) wavelength
in radiative transfer bin $j$ for neighboring gas cell $k$, and $N_i$ and
$a_i^\text{c}$ are the number of grains and midpoint of grain size bin $i$. We
use the same cubic spline kernel as in \citet{McKinnon2018} to calculate the
weight $w_k$ for gas cell $k$ as a function of its distance from the given dust
particle. These weights are normalised such that they sum to unity over all
neighboring gas cells. In equation~(\ref{EQN:f_pr}), the term in brackets
represents what the radiation pressure force would be if all grain cross
section were concentrated in gas cell $k$, summing over all grain sizes and
frequency bins.
This force depends on the radiation pressure efficiency $Q_\text{pr}(a,
\lambda)$, which is a dimensionless factor denoting the ratio of radiation
pressure cross section to geometric cross section for a dust grain of size $a$
and wavelength of incident radiation $\lambda$. This efficiency factor is
defined as $Q_\text{pr} \equiv Q_\text{abs} + (1 - \langle \cos \theta \rangle)
Q_\text{sca}$, where $Q_\text{abs}$ is the ratio of absorption cross section to
geometric cross section, $Q_\text{sca}$ is the ratio of scattering cross
section to geometric cross section, $\langle \cos \theta \rangle$ is the
average cosine of scattering angle $\theta$, and all three quantities are
functions of grain size and wavelength.
Following \citet{McKinnon2018}, we use tabulated values of $Q_\text{abs}$,
$Q_\text{sca}$, and $\langle \cos \theta \rangle$ for silicate and graphite
grains from \citet{Draine1984} and \citet{Laor1993}. These tabulations cover
the grain size range $0.001 \, \mu\text{m} < a < 10 \, \mu\text{m}$ and
wavelength range $0.001 \, \mu\text{m} < \lambda < 1000 \, \mu\text{m}$. We
perform two-dimensional logarithmic interpolation to calculate $Q_\text{pr}(a,
\lambda)$ values at arbitrary grain size and wavelength, averaging results for
silicate and graphite grains.
\subsubsection{Absorption of photons from dust grain opacity}
Dust grains absorb and reradiate in the IR a sizeable fraction of starlight,
often estimated around 30 per cent \citep{Soifer1991, Popescu2002}. We first
describe absorption in the single-scattering regime \citep[e.g.~for UV
frequency bins, as in Section~3.2.1 of][]{Kannan2019a}.
We account for photon absorption in a kernel-smoothed fashion, by spreading a
dust particle's absorption cross section among neighboring gas cells to update
energy densities and fluxes. This approach effectively replaces the dust
opacity terms in equations~47 and~48 in \citet{Kannan2019a} that are used in
simulations lacking live dust particles. Specifically, when processing an
active dust particle we update the energy density in radiation bin $j$ for
neighboring gas cell $k$ according to the rate
\begin{equation}
\frac{\partial E_{j,k}}{\partial t} = -\frac{w_k \tilde{c} E_{j,k}}{V_k} \sum_{i = 0}^{N_\text{GSD} - 1} N_i \pi {a_i^\text{c}}^2 Q_\text{abs}(a_i^\text{c}, \lambda_j),
\label{EQN:E_attenuation}
\end{equation}
where $w_k$ is the gas cell's kernel weight, $\tilde{c}$ is the reduced speed
of light, and $V_k$ is the gas cell volume. As in equation~(\ref{EQN:f_pr}),
$N_i$ is the number of grains in size bin $i$ for the dust particle,
$a_i^\text{c}$ is the grain size at the midpoint of size bin $i$, and
$\lambda_j$ is the (mean) wavelength in radiation bin $j$. The absorption
cross section efficiency $Q_\text{abs}$, described in
Section~\ref{SEC:Q_pr_sizes}, is a function of grain size and wavelength of
radiation. Likewise, the flux in frequency bin $j$ of gas cell $k$ is updated
using the rate
\begin{equation}
\frac{\partial \vec{F_{j,k}}}{\partial t} = -\frac{w_k \tilde{c} \vec{F_{j,k}}}{V_k} \sum_{i = 0}^{N_\text{GSD} - 1} N_i \pi {a_i^\text{c}}^2 Q_\text{abs}(a_i^\text{c}, \lambda_j).
\label{EQN:F_attenuation}
\end{equation}
\subsubsection{Multifrequency radiation and dust coupling}\label{SEC:multifrequency_radiation}
The radiation hydrodynamics methods in \citet{Kannan2019a} are capable of treating multifrequency radiation, divided
into several UV and optical bins (e.g.~corresponding to H\,\textsc{i},
He\,\textsc{i}, and He\,\textsc{ii} ionisation) and one IR bin. Photons in UV
and optical bins are ``single-scattered'': the energy absorbed by dust at these
wavelengths is reemitted as IR radiation. On the other hand, IR photons are
``multi-scattered'': dust grains can absorb IR photons but also thermally emit
in the IR.
In the notation below, we assume that the $N_\text{RT}$ radiation bins are
ordered by decreasing frequency, so that the IR bin is last. Then, the IR energy density in gas cell $k$, $E_{\text{IR},k}$, changes according to
\begin{equation}
\frac{\partial E_{\text{IR},k}}{\partial t} = \frac{\partial E_{\text{IR},k}^\text{reprocess}}{\partial t} + \frac{\partial E_{\text{IR},k}^\text{thermal}}{\partial t},
\end{equation}
the sum of reprocessing rates (i.e.~energy from UV and optical photons that is
reemitted in the IR) and thermal dust exchange rates (i.e.~dust emission and
absorption of IR photons), respectively. In this section, we focus on
reprocessing of UV and optical radiation and discuss thermal IR exchange next
in Section~\ref{SEC:thermal_coupling}. Then, the rate at which the IR energy
density increases is given by the total rate at which UV and optical energy
density decreases. That is,
\begin{equation}
\frac{\partial E_{\text{IR},k}^\text{reprocess}}{\partial t} = - \sum_{j = 0}^{N_\text{RT}-2} \frac{\partial E_{j,k}}{\partial t}
\end{equation}
where the sum is over all non-IR bins and the energy absorption rate in
radiation bin $j$, $\partial E_{j,k} / \partial t$, is computed via
equation~(\ref{EQN:E_attenuation}). Additionally, paralleling
equation~(\ref{EQN:F_attenuation}), the IR flux in gas cell $k$ evolves
according to
\begin{equation}
\frac{\partial \vec{F_{\text{IR},k}}}{\partial t} = -\frac{w_k \tilde{c} \vec{F_{\text{IR},k}}}{V_k} \sum_{i = 0}^{N_\text{GSD} - 1} N_i \pi {a_i^\text{c}}^2 Q_\text{abs}(a_i^\text{c}, \lambda_\text{IR}).
\end{equation}
\subsubsection{Thermal coupling for infrared radiation}\label{SEC:thermal_coupling}
Dust grains both emit and absorb IR photons \citep{Dwek1986, Krumholz2013} and
also collisionally exchange energy with gas \citep{Hollenbach1979, Omukai2000,
Goldsmith2001}. These competing processes affect the temperature of a
population of dust grains \citep{Goldsmith2001, Krumholz2014, Smith2017}.
Small grains in particular are susceptible to stochastic heating and
temperature fluctuations \citep{Guhathakurta1989, Siebenmorgen1992,
Draine2001}, and methods to calculate and evolve dust temperature probability
distributions are used in various applications
\citep[e.g.][]{Pavlyuchenkov2012, Camps2015}. However, given the limited
resolution our simulations, we instead follow the approach in
\textsc{despotic}~\citep{Krumholz2014} and \textsc{grackle}~\citep{Smith2017},
treating dust grains as being in thermal equilibrium and neglecting such
temperature fluctuations.
As with equation~(\ref{EQN:E_attenuation}), we perform the coupling between
dust, gas, and IR radiation in a kernel-smoothed manner, where dust particles
exchange energy with their surrounding neighbors. Since dust particles are
superpositions of grains of different sizes, we allow for the possibility that
the dust temperature varies from one grain size bin to another. When
considering the exchange between a dust particle and neighboring gas cell $k$,
the IR radiation energy density $E_{\text{IR},k}$, dust energy density
$u_\text{d}$, and gas energy density $u_{\text{g},k}$ evolve according to the
system
\begin{equation}
\begin{split}
&\frac{\partial E_{\text{IR},k}^\text{thermal}}{\partial t} = \Lambda_{\text{dr},k} \\
&\frac{\partial u_{\text{d}}}{\partial t} = -\Lambda_{\text{dr},k} - \Lambda_{\text{dg},k} \\
&\frac{\partial u_{\text{g},k}}{\partial t} = \Lambda_{\text{dg},k}.
\end{split}
\label{EQN:thermal_coupling_system}
\end{equation}
Here, the dust-radiation energy exchange rate per unit volume
\citep[e.g.][]{Krumholz2013} in gas cell $k$ is given by $\Lambda_{\text{dr},k}
= \sum_{i=0}^{N_\text{GSD}-1} \Lambda_{\text{dr},k,i}$, where the contribution
from grains in size bin $i$ is
\begin{equation}
\Lambda_{\text{dr},k,i} = \left[ \frac{w_k}{V_k} N_i \pi {a_i^\text{c}}^2 Q_\text{abs}(a_i^\text{c}, \lambda_\text{IR}) \right] (c a_\text{B} T_{\text{d},k,i}^4 - \tilde{c} E_{\text{IR},k}),
\label{EQN:Lambda_dr}
\end{equation}
where $w_k$ is the gas cell's kernel weight (see equation~\ref{EQN:f_pr}),
$V_k$ is the cell volume, $a_\text{B}$ is the radiation constant, and
$T_{\text{d},k,i}$ is the cell's dust temperature for grains in size bin $i$.
The prefactor in equation~(\ref{EQN:Lambda_dr}) is the cross section per unit
volume in gas cell $k$, obtained by assigning a weighted fraction of the dust
particle's grain population to the cell. Also, the dust-gas energy exchange
rate per unit volume \citep[e.g.][]{Burke1983, Hollenbach1989} is calculated
via $\Lambda_{\text{dg},k} = \sum_{i=0}^{N_\text{GSD}-1}
\Lambda_{\text{dg},k,i}$, using the exchange rate with grains in size bin $i$
of
\begin{equation}
\Lambda_{\text{dg},k,i} = \left[ \frac{w_k}{V_k} N_i \pi {a_i^\text{c}}^2 \right] n_{\text{H},k} v_{\text{th},k} \overline{\alpha}_T (2 k_\text{B}) (T_{\text{d},k,i} - T_{\text{g},k}).
\end{equation}
Here, $n_{\text{H},k}$ is the hydrogen number density in gas cell $k$,
$\overline{\alpha}_T \approx 0.5$ is an estimated ``accommodation coefficient''
\citep[see equation~9 in][]{Burke1983}, $k_\text{B}$ is the Boltzmann constant,
$T_{\text{g},k}$ is the gas temperature, and
\begin{equation}
v_{\text{th},k} \equiv \left( \frac{8 k_\text{B} T_{\text{g},k}}{\pi m_\text{p}} \right)^{1/2}
\end{equation}
is the gas thermal velocity.
\begin{figure*}
\centering
\includegraphics{figures/dustybox_backreaction_evolution.pdf}
\caption{Left panel: velocity profiles versus time for a dust and gas mixture
coupled via drag with an initial dust-to-gas ratio of $D = 0.5$. Coloured
points show simulation results as the dust and gas velocities approach the
barycentric velocity (dotted line), while the solid lines indicate the analytic
predictions. Time is given in units of the drag stopping time-scale
$t_\text{s}$. Right panel: change in different energy components versus time.
Dust kinetic energy (red) decreases, while gas kinetic energy (green)
increases. The net loss of kinetic energy from drag leads to an increase in
gas thermal energy (blue).}
\label{FIG:dustybox_backreaction_evolution}
\end{figure*}
Since dust grains reach thermal equilibrium on a rapid time-scale
\citep{Woitke2006, Krumholz2014, Smith2017}, we calculate the dust temperature
$T_{\text{d},k,i}$ that grains in size bin $i$ in gas cell $k$ have by solving
the instantaneous equilibrium condition
$\Lambda_{\text{dr},k,i}(T_{\text{d},k,i}) +
\Lambda_{\text{dg},k,i}(T_{\text{d},k,i}) = 0$ using Newton's method for
root-finding. Using this dust temperature, over the dust particle's time-step
$\Delta t$ we then add $\Lambda_{\text{dr},k,i} V_k \Delta t$ in IR radiation
energy and $\Lambda_{\text{dg},k,i} V_k \Delta t$ in thermal energy to gas cell
$k$. This process is repeated for all grain size bins and all gas cells within
a dust particle's smoothing length. An average dust temperature for size bin
$i$ in the dust particle can be calculated as $T_{\text{d},i} =
\sum_{k=0}^{N_\text{ngb}-1} w_k T_{\text{d},k,i}$.
\section{Test Problems}\label{SEC:result}
\subsection{Demonstration of drag and kinetic energy dissipation}
When the dust-to-gas ratio $D$ is of order unity, drag impacts both dust and
gas. To illustrate this in a simple test, we initialise a box of side length
$1 \, \text{kpc}$ with $32^3$ gas cells and $32^3$ dust particles arranged on
an equispaced lattice. Gas cell masses are set so that the box has a uniform
gas density of $n = 1 \, \text{cm}^{-3}$. We adopt a dust-to-gas ratio of $D =
0.5$, so that the mass of a dust particle is half the mass of a gas cell.
Initially, gas is at rest ($\vec{v}_\text{g}(t = 0) = \vec{0} \, \text{km} \,
\text{s}^{-1}$), and dust is given nonzero velocity along the $z$ axis
($\vec{v}_\text{d}(t = 0) \equiv \vec{v}_{0} = 1 \, \text{km} \, \text{s}^{-1}
\, \vechat{z}$).
Dust particles are assumed to consist of $0.1 \, \mu\text{m}$ grains, and the
internal energy per unit mass of the gas is initially $u = 1000 \,
\text{km}^{2} \, \text{s}^{-2}$. From the values above, one can calculate the
stopping time-scale $t_\text{s}$ over which the drag force acts. In this test,
the analytic prediction for gas velocity is
\begin{equation}
\vec{v}_\text{g}(t) = \left( \frac{D}{1 + D} \right) \left( 1 - e^{-t/t_\text{s}} \right) \vec{v}_{0},
\end{equation}
and the analytic prediction for dust velocity is
\begin{equation}
\vec{v}_\text{d}(t) = \left( \frac{1}{1 + D} \right) \left( D + e^{-t/t_\text{s}} \right) \vec{v}_{0}.
\end{equation}
As $t \to \infty$, both $\vec{v}_\text{g}$ and $\vec{v}_\text{d} \to D
\vec{v}_{0} / (1 + D) = \vec{v}_{0} / 3$, showing that gas and dust are
expected to move at the barycentric velocity of the mixture.
Figure~\ref{FIG:dustybox_backreaction_evolution} shows the time evolution of
dust and gas velocities and energies for this mixture coupled by drag. Since
the drag force acts in an equal and opposite manner on dust and gas components
and the dust-to-gas ratio is less than unity, dust velocities decrease more
quickly than gas velocities increase. The initial relative velocity between
dust and gas rapidly decays. After just two stopping time-scales, the dust and
gas velocities are already within thirty per cent of the barycentric velocity.
Over time, more dust kinetic energy is lost than gas kinetic energy is gained.
The net dissipation of kinetic energy leads to an increase in gas thermal
energy. In both panels of Figure~\ref{FIG:dustybox_backreaction_evolution},
simulation results for velocity and energy evolution are visually
indistinguishable from analytic predictions.
\subsection{Spherically symmetric radiation pressure test problem}
To demonstrate our radiation pressure scheme, we first consider a monochromatic
radiation source with constant luminosity $L$ and grains of a single fixed size
$a$. Analytically, a dust grain at distance $r$ from the radiation source
feels a radial force with magnitude
\begin{equation}
f_\text{pr} = \left( \frac{L}{4 \pi r^2 c} \right) \pi a^2 Q_\text{pr}.
\label{EQN:f_r}
\end{equation}
If the dust grain has internal density $\rho_\text{gr}$, we can also express
the radial force in terms of the grain mass $m$ and radial velocity $v_r$ as
\begin{equation}
f_\text{pr} = m \frac{\diff v_r}{\diff t} = \frac{4 \pi}{3} \rho_\text{gr} a^3 v_r \frac{\diff v_r}{\diff r},
\end{equation}
where in this problem we neglect the drag force and other external forces.
Equating these two expressions,
\begin{equation}
2 v_r \diff v_r = \left( \frac{3 L Q_\text{pr}}{8 \pi c \rho_\text{gr} a} \right) \left( \frac{\diff r}{r^2} \right).
\end{equation}
Integrating for a dust grain starting at rest at $r = r_0$, one obtains a
relation between dust grain radial velocity and radial distance,
\begin{equation}
v_r = \frac{\diff r}{\diff t} = \sqrt{\frac{3 L Q_\text{pr}}{8 \pi c \rho_\text{gr} a} \left( \frac{1}{r_0} - \frac{1}{r} \right)}.
\label{EQN:drdt_analytic}
\end{equation}
As $r \to \infty$, the dust grain approaches a constant terminal velocity.
Grains starting at smaller initial radii, closer to the radiation source, have
larger terminal velocities.
We place a monochromatic source of luminosity $L = 10^6 \, \text{L}_\odot$ at
the center of a three-dimensional box of side length $160 \, \text{pc}$. The
volume is tessellated by $256^3$ gas cells with uniform density $n = 1 \,
\text{cm}^{-3}$. The positions of mesh-generating points are initially
arranged on an equispaced Cartesian lattice. Then, points are uniformly
randomly displaced in each dimension by up to 20 per cent of the initial cell
size in order to produce an irregular mesh. This approximates the cell
geometries seen in typical simulations \citep{Vogelsberger2012, Kannan2019a}.
To model the luminosity source, during every time-step photons are injected
in equal amounts into gas cells within $1 \, \text{pc}$ of the box center. For
completeness, we assume photons have energy $13.6 \, \text{keV}$, though this
choice does not affect our results.
\begin{figure}
\centering
\includegraphics{figures/radiation_sphere_evolution.pdf}
\caption{Radial distance versus time for a sample of dust grains moving away
from a source of constant luminosity. Simulation results are shown in colour,
while analytic predictions are in black. Grains have fixed size but different
values of $r_0$, the initial radial distance from the radiation source. In
this test, we neglect drag between dust and gas and include only radiation
pressure on dust.}
\label{FIG:radiation_sphere_evolution}
\end{figure}
We place $256^3$ dust particles on an equispaced Cartesian lattice. Dust
particles are given equal mass to model an initially uniform dust distribution.
The dust particle mass is chosen so that the total dust-to-gas ratio is $0.5$.
However, to focus solely on the effect of radiation pressure, in this test we
neglect any drag coupling between dust and gas. As a result, the value of the
dust-to-gas ratio does not affect results. We also assume that only dust
particles provide opacity to radiation: the opacity of gas cells is set to
zero, and we turn off gas thermochemistry so that the gas begins and remains
purely neutral hydrogen. Thus, the gas is analytically expected to remain at
rest.
Dust particles are assumed to have grains of size $a = 0.01 \, \mu\text{m}$ and
internal density $\rho_\text{gr} = 2.4 \, \text{g} \, \text{cm}^{-3}$. For
simplicity, we assume that a grain's radiation pressure cross section equals
its geometric cross section and set $Q_\text{pr} = 1$. When calculating
radiation transport, we adopt the reduced speed of light approximation
\citep[in the notation of][we set $\tilde{c} / c = 1/25$]{Kannan2019a}. This
reduces the rate at which photons cross the simulation domain and in turn
lessens noise in photon fluxes near the box edges where photons meet.
\begin{figure}
\centering
\includegraphics{figures/radiation_layer_evolution.pdf}
\caption{Number of photons $N$ remaining as a function of time for a test in
which a burst of radiation is incident on a thin layer of dust. We normalize
by the initial number of photons $N_0$. Coloured lines show simulation results
at different resolutions, with the labels indicating number of gas cells. The
dotted line shows the expected analytic behaviour, with the photon fraction
dropping as it passes through the dust layer.}
\label{FIG:radiation_layer_evolution}
\end{figure}
Figure~\ref{FIG:radiation_sphere_evolution} shows the time evolution of a
sample of dust particles as they move away from the radiation source. We
select dust particles that begin at initial radial distances of $r_0 = 4, 6, \,
\text{and} \, 9 \, \text{pc}$. We compare the radial motion of these
simulation particles over a time period of $0.1 \, \text{Myr}$ with the
expected behaviour obtained by numerically integrating
equation~(\ref{EQN:drdt_analytic}). The initial radiation force is greater for
dust grains closer to the luminosity source. As a result, by $t = 0.1 \,
\text{Myr}$ the grain initially closest to the luminosity source ($r_0 = 4 \,
\text{pc}$) develops the largest velocity and is at the furthest radial
distance. In contrast, the grain starting at $r_0 = 9 \, \text{pc}$ lags
behind those two starting closer to the source. While radiation pressure does
drive the dust outwards, the resulting distribution does not preserve the
initial ordering of grains in terms of radial distance from the luminosity
source.
The simulation results largely agree with the expected behaviour. There are
some minor deviations above (e.g.~$r_0 = 4 \, \text{pc}$) and below (e.g.~$r_0
= 9 \, \text{pc}$) the analytic solutions, likely influenced by the fact that
we update dust particle positions using kernel-interpolated estimates of the
gas cell radiation flux. However, we have verified that these deviations
diminish as the resolution of the test problem increases.
Figure~\ref{FIG:radiation_sphere_evolution} demonstrates the suitability of our
hybrid gas cell and dust particle approach for modelling radiation pressure on
dust grains.
\subsection{Photon absorption by dust grains}
We demonstrate the ability of dust grains to absorb photons in our hybrid dust
particle and gas cell scheme using a simplified test problem, where radiation
is incident on a thin layer of dust.
We begin with a lattice of $N^3$ equispaced gas cells of uniform density $n = 1
\, \text{cm}^{-3}$ in a box of side length $L = 160 \, \text{pc}$. Cell
centers are uniformly randomly displaced by up to 20 per cent in each
coordinate, except for the two leftmost and two rightmost layers of cells along
the $x$ axis, where the mesh remains Cartesian. A layer of $N^2$ equispaced,
equal-mass dust particles is placed halfway through the box at $x = L/2$, with
the dust particle mass $m_\text{d}$ chosen so that the total dust-to-gas ratio
is $10^{-3}$. Both dust and gas components start at rest, and in this test of
radiation dynamics we switch off self-gravity and the drag force. Dust
particles are assumed to consist entirely of $a = 0.01 \, \mu\text{m}$ sized
grains of density $\rho_\text{gr} = 2.4 \, \text{g} \, \text{cm}^{-3}$, and we
take $Q_\text{abs} = 1$. This simplifies the expected analytic behaviour, but
in principle dust particles could be populated with a full distribution of
grain sizes.
For the initial conditions, we place radiation of uniform energy density in the
$N^2$ gas cells bordering the $x = 0$ edge of the box. The radiation flux
points in the positive $x$ direction and has its maximally allowed magnitude
\citep[following the notation of][$|\vec{F_r}| = \tilde{c} E_r$]{Kannan2019a}.
We use reduced speed of light $\tilde{c} = c/25$ for photon propagation. In
this test, we use one UV bin and switch off hydrogen and helium thermochemistry
so that dust grains provide the only source of opacity. Dust particles spread
their opacity over the $N_\text{ngb} = 64$ neighboring gas cells, though our
results are largely insensitive to this choice.
\begin{figure}
\centering
\includegraphics{figures/radiation_multifrequency_evolution.pdf}
\caption{Fraction of energy in UV (red) and IR (green) photons in a test where
UV radiation is incident on a thin layer of dust. Dust grains absorb UV
photons and reemit in the IR, and we turn off absorption of IR photons so that
the total energy in radiation is constant. Dotted lines show the expected
analytic result, given the optical depth $\tau \approx 1$.}
\label{FIG:radiation_multifrequency_evolution}
\end{figure}
Figure~\ref{FIG:radiation_layer_evolution} shows the number fraction of photons
remaining in the box as a function of time, for four resolution tests ($N =
32, 64, 128, \, \text{and} \, 256$). Analytically, the photons reach the layer of
dust at $x = L/2$ in time $t = L / (2 \tilde{c}) \approx 6.5 \, \text{kyr}$,
after which the number fraction should drop from $1$ to $\exp(-\tau)$, in terms
of the optical depth
\begin{equation}
\tau = \int_{0}^{L} n(x) \, \sigma \, \diff x
\end{equation}
where $n(x) = \delta(x - L/2) \times N^2 / L^2$ is the number density of dust
particles and $\sigma = (3 m_\text{d}) / (4 \rho_\text{gr} a)$ is the total
cross section of each dust particle. For our choice of parameters above, $\tau
\approx 0.26$. There is strong agreement between the analytic prediction and
simulation results, with the late-time photon number fraction dropping to
$\exp(-\tau)$ as expected. Increasing the resolution of the test produces a
sharper drop in photon number fraction, since the gas cells surrounding the
dust layer have smaller extent.
\subsection{Dust reprocessing}
To demonstrate the ability for dust grains to absorb UV photons and reemit in
the IR, we extend the test presented in
Figure~\ref{FIG:radiation_layer_evolution}, where UV photons incident upon a
thin dust layer were strictly absorbed and not reemitted. In this new test, we
adopt one UV and one IR radiation bin and convert absorbed UV energy to IR
energy. We neglect dust grain emission and absorption in the IR, so that the
total energy in radiation is constant and simply shifts between frequency bins.
The initial conditions are the same as in
Figure~\ref{FIG:radiation_layer_evolution}, except that we decrease the fixed
grain size to $a = 2.5 \, \times \, 10^{-3} \, \mu\text{m}$. Since the dust
particle mass is unchanged, the cross section per dust particle increases by a
factor of four, and the optical depth through the dust layer is $\tau \approx
1$. We pick a resolution of $N = 128$ for gas cells and dust particles.
\begin{figure}
\centering
\includegraphics{figures/thermal_coupling_evolution.pdf}
\caption{Temperature evolution for a test in which dust (red) is coupled to gas
(green) through collisional energy exchange and to IR radiation (blue) through
thermal IR emission and absorption. The test includes injecting IR photons
into every gas cell at a constant rate starting at $t = 1 \, \text{Myr}$ to
mimic a radiation source. Coloured points show simulation results, while black
lines indicate the expected solution by numerically integrating the analytic
energy exchange equations. Dust and radiation temperatures are highly coupled,
while the gas temperature is slower to change.}
\label{FIG:thermal_coupling_evolution}
\end{figure}
\begin{figure*}
\centering
\includegraphics{figures/radiation_momentum_evolution.pdf}
\caption{Left panel: mean velocity of gas (red) and dust (green) versus time
for a test in which dust grains feel radiation pressure but do not absorb
photons so that the radiation flux is constant. Analytic profiles are shown in
black. Radiation accelerates dust, which in turn drags gas. The rate of
momentum injection from radiation remains constant. Right panel: similar
results for a test with photon absorption included. As the photon flux drops,
dust and gas velocities equilibrate through drag. In both panels, simulation
output is subsampled to improve readability.}
\label{FIG:radiation_momentum_evolution}
\end{figure*}
Figure~\ref{FIG:radiation_multifrequency_evolution} shows the fraction of
radiation energy in UV and IR bins as a function of time, along with the
expected analytic behaviour. The optical depth $\tau \approx 1$ is chosen in
this test so that a majority of UV energy converts to IR as photons pass
through the dust layer. We note that, since IR photons are lower energy than
UV photons, this test implies an increase in the total number of photons. Our
results agree with the analytic prediction that the fraction of energy in UV
and IR bins reaches $e^{-\tau}$ and $1-e^{-\tau}$, respectively.
\subsection{Thermal coupling}
We next present a test in which dust, gas, and radiation are thermally coupled
through the system in equation~(\ref{EQN:thermal_coupling_system}). In a box
of side length $1 \, \text{kpc}$, we place an irregular mesh of $32^3$ gas
cells with uniform density $n = 1 \, \text{cm}^{-3}$ and an equispaced lattice
of $32^3$ dust particles. We set the dust-to-gas ratio $D = 1$, fixed grain
size $a = 0.005 \, \mu\text{m}$ (i.e.~here, we do not place grains in multiple
size bins), efficiencies $Q_\text{abs} = 1$, and internal grain density
$\rho_\text{gr} = 2.4 \, \text{g} \, \text{cm}^{-3}$. Dust and gas components
initially start at rest, although we do not focus on dynamics in this test.
The initial dust and gas temperatures are $T_\text{d} = 10 \, \text{K}$ and
$T_\text{g} = 100 \, \text{K}$, respectively, and we choose the initial IR
radiation density so that the radiation temperature $T_\text{r} \equiv
(E_\text{IR} / a_\text{B})^{1/4} = 10 \, \text{K}$ at the start. We do not use
a reduced speed of light, so that $\tilde{c} = c$. To mimic a radiation source
that emits in the IR, starting at $t = 1 \, \text{Myr}$ we inject IR photons
into gas cells at the constant rate $\dot{E_\text{IR}} = 1.2 \times 10^{57} \,
\text{erg} \, \text{Myr}^{-1} \, \text{kpc}^{-3}$. In this setup, we set
radiation in non-IR bins to zero and do not include other sources of gas
thermal change (e.g.~from photoheating) beyond dust-gas collisional energy
exchange.
Figure~\ref{FIG:thermal_coupling_evolution} shows the evolution of dust, gas,
and radiation temperatures, computed by averaging over simulation gas cells and
dust particles. Since the gas temperature initially is greater than the dust
temperature, the gas begins to collisionally cool. Because of the strong
dependence of dust-radiation energy exchange on dust temperature, the dust
temperature does not significantly change as the gas temperature drops. When
the IR radiation injection begins at $t = 1 \, \text{Myr}$, dust remains highly
coupled with radiation. In contrast, the gas temperature lags behind the
increase in dust and radiation temperatures. Our simulation results closely
follow the predictions obtained by integrating the analytic system in
equation~(\ref{EQN:thermal_coupling_system}), where we adopt the silicate grain
heat capacity from \citet{Dwek1986} to relate dust internal energy change to
dust temperature change. This test shows how dust, gas, and radiation
temperatures can evolve according to energy exchange processes and external
sources.
\subsection{Coevolution of radiation, dust, and gas through radiation pressure and drag}\label{SEC:drag_radiation}
Previous tests illustrated the effect of drag without radiation
(e.g.~Figure~\ref{FIG:dustybox_backreaction_evolution}) or radiation without
drag (e.g.~Figures~\ref{FIG:radiation_sphere_evolution},
\ref{FIG:radiation_layer_evolution},
\ref{FIG:radiation_multifrequency_evolution},
and~\ref{FIG:thermal_coupling_evolution}). We now demonstrate the ability of
our methods to simultaneously handle drag and radiation.
The initial conditions consist of uniform dust and gas distributions at rest
and are identical to those used in Section~\ref{SEC:thermal_coupling}, except
that we set the dust-to-gas ratio $D = 0.5$ and fixed grain size $a = 0.1 \,
\mu\text{m}$. We initialise a uniform monochromatic radiation field whose flux
has magnitude $F_0 = 10 \, \text{erg} \, \text{s}^{-1} \, \text{cm}^{-2}$
pointing in the $\vechat{x}$ direction. Dust particles contribute the only
opacity to radiation (i.e.~we neglect gas thermochemistry and opacity
contributed by gas cells), and photons propagate at the reduced speed of light
$\tilde{c} = c / 1000$. As a result, in this test dust grains accelerate from
photon radiation pressure and in turn accelerate the gas through drag. In
addition to our fiducial setup where dust grains both feel radiation pressure
and absorb photons, we also perform a test where photon absorption is turned
off so that analytically the radiation flux $\vec{F}(t) = F_0 \vechat{x}$ at
all times.
To ensure our simulation results can be compared with analytic predictions, we
require that the stopping time-scale $t_\text{s}$ is constant. For this test
only, we disable the injection of thermal energy into the gas from drag
dissipation of kinetic energy. Thus, the sound speed is expected to remain
constant. Additionally, in equation~(\ref{EQN:t_s}) we neglect the correction
term in parentheses. In Appendix~\ref{SEC:appendix_analytic}, we show how to
analytically integrate the dust and gas equations of motion.
The choice of parameters above yield a stopping time-scale $t_\text{s}$ and
photon decay time-scale $t_\text{d}$ both on the order of $40 \, \text{kyr}$.
If we had $t_\text{s} \ll t_\text{d}$, radiation pressure would essentially
move dust and gas together as one, while $t_\text{s} \gg t_\text{d}$ would see
photons rapidly deposit momentum into dust and decay away before drag
accelerated gas. In our setup, dust, gas, and radiation coevolve over similar
time-scales.
Figure~\ref{FIG:radiation_momentum_evolution} shows the mean velocity of gas
cells and dust particles as a function of time. In addition to the fiducial
setup with $t_\text{s} \approx t_\text{d}$, we also perform a test neglecting
photon absorption by dust grains. In this test, which is equivalent to taking
$t_\text{d} \to \infty$, radiation flux remains constant and continually
imparts momentum to the dust at a fixed rate. Runs with and without photon
absorption show different qualitative evolution, though in both cases
simulation results closely follow analytic results.
Without absorption, dust and gas velocities increase in time without bound.
Initially, dust accelerates more quickly than gas, since only dust feels
radiation pressure. At later times, dust and gas feel roughly the same
acceleration, with the velocity offset $v_\text{d} - v_\text{g} \approx 108 \,
\text{km} \, \text{s}^{-1}$ matching the analytic prediction of $F_0 \kappa
t_\text{s} / c$.
With absorption, dust still feels a larger initial acceleration. However, as
photons get absorbed, the rate of momentum injection decays to zero and dust
and gas velocities equilibrate via the drag force. Thus, there is no long-term
velocity offset between dust and gas components. The steady-state velocities
$v_\text{d} \approx v_\text{g} \approx 44 \, \text{km} \, \text{s}^{-1}$ agree
with the analytic expectation of $F_0 \kappa D t_\text{d} / [(1 + D) c]$.
\section{Conclusions}\label{SEC:conclusion}
We have developed a framework to couple dust physics and radiation
hydrodynamics in the moving-mesh code \textsc{arepo-rt}, where dust is modelled
using live simulation particles and radiation is handled on an unstructured
mesh. We first extend our implementation of the aerodynamic drag force that
couples dust and gas motion to properly capture dynamics at arbitrary
dust-to-gas ratios, even those well above Galactic values. We then describe
kernel-smoothed methods to model the interaction between dust and radiation,
where grain cross sections from simulation dust particles are interpolated onto
neighboring gas cells to affect radiation fields in a self-consistent manner.
We detail our approach for calculating radiation pressure forces on populations
of dust grains, including those covering a range of different grain sizes.
This method is tested using a constant luminosity source in a medium with an
initially uniform dust density, for which analytic behaviour is known. We
model absorption of photons by local dust grains and maintain the ability of
\textsc{arepo-rt} to treat multifrequency radiation, where UV and optical
photons are single-scattered and IR photons are multi-scattered. We illustrate
the ability for dust to absorb photons passing through a thin layer of dust of
known optical depth. Instead of assuming dust and gas are in local
thermodynamic equilibrium, we allow dust and gas to have separate temperatures
and model their energy exchange through collisional processes. Dust and
radiation also exchange energy through absorption and thermal dust emission at
IR wavelengths.
Finally, we demonstrate the evolution of a mixture of dust, gas, and radiation
using our hybrid dust particle and gas cell scheme. Dust and gas are coupled
through aerodynamic drag, and dust and radiation are coupled through radiation
pressure and photon absorption. In agreement with analytic predictions, we
verify that dust and gas velocities initially deviate as photons accelerate
dust but later equilibrate through drag as all photons get absorbed. This
behaviour more closely mirrors the expected physics compared to models that
assume a perfect coupling between dust and gas.
We emphasise the original nature of our methods in comparison to typical
radiation hydrodynamics schemes that model dust opacity using fixed values or
simplified functions of hydrodynamic quantities like temperature. By treating
dust using live simulation particles subject to dynamical forces, our approach
avoids the need to make assumptions about the local dust-to-gas ratio and dust
distribution and instead calculates dust opacity self-consistently based on the
local grain abundance. While there are codes to perform radiative transfer in
arbitrary dusty media, these usually run in post-processing and not
concurrently with hydrodynamic evolution. As radiation hydrodynamic
simulations of galaxy formation become more computationally tractable, our
model for the interaction of dust and radiation is an alternative to
simulations that neglect a direct treatment of dust.
\section*{Acknowledgements}
We thank Volker Springel for sharing access to \textsc{arepo}. MV acknowledges support through an MIT RSC award, a Kavli Research Investment Fund, NASA ATP grant NNX17AG29G, and NSF grants AST-1814053, AST-1814259 and AST-1909831. PT acknowledges support from NSF grant AST-1909933. The simulations
were performed on the joint MIT-Harvard computing cluster supported by MKI and
FAS. RM acknowledges support from the DOE CSGF under grant number
DE-FG02-97ER25308.
\bibliographystyle{mn2e}
| {'timestamp': '2019-12-09T02:00:20', 'yymm': '1912', 'arxiv_id': '1912.02825', 'language': 'en', 'url': 'https://arxiv.org/abs/1912.02825'} |
\section{Introduction}
Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$.
The Yamabe problem is concerned with finding metrics of constant scalar
curvature in the conformal class of $g$. This problem leads to a semi-linear elliptic PDE for the conformal factor. More precisely, a conformal metric of the form $u^{\frac{4}{n-2}} \, g$ has constant scalar curvature $c$ if and only if
\begin{equation}
\label{yamabe.pde}
\frac{4(n-1)}{n-2} \, \Delta_g u - R_g \, u + c \, u^{\frac{n+2}{n-2}} = 0,
\end{equation}
where $\Delta_g$ is the Laplace operator with respect to $g$ and $R_g$ denotes the scalar curvature of $g$. Every solution of (\ref{yamabe.pde}) is a critical point of the functional
\begin{equation}
\label{yamabe.functional}
E_g(u) = \frac{\int_M \big ( \frac{4(n-1)}{n-2} \, |du|_g^2 + R_g \, u^2
\big ) \, dvol_g}{\big ( \int_M
u^{\frac{2n}{n-2}} \, dvol_g \big )^{\frac{n-2}{n}}}.
\end{equation}
In this paper, we address the question whether the set of all solutions to the Yamabe PDE is compact in the $C^2$-topology. It has been conjectured that this should be true unless $(M,g)$ is conformally equivalent to the round sphere (see \cite{Schoen1},\cite{Schoen2},\cite{Schoen3}). The case of the round sphere $S^n$ is special in that (\ref{yamabe.pde}) is invariant under the action of the conformal group on $S^n$, which is non-compact. It follows from a theorem of Obata \cite{Obata} that every solution of the Yamabe PDE on $S^n$ is minimizing, and the space of all solutions to the Yamabe PDE on $S^n$
can be identified with the unit ball $B^{n+1}$. Note that the round sphere is the only compact manifold for which the set of minimizing solutions is non-compact.
The Compactness Conjecture has been verified in low dimensions and in the locally conformally flat case. If $(M,g)$ is locally conformally flat, compactness follows from work of R.~Schoen \cite{Schoen1},\cite{Schoen2}. Moreover, Schoen proposed a strategy, based on the Pohozaev identity, for
proving the conjecture in the non-locally conformally flat case. In \cite{Li-Zhu}, Y.Y.~Li and M.~Zhu followed this strategy to prove compactness in dimension
$3$. O.~Druet \cite{Druet} proved the conjecture in dimensions $4$ and $5$.
The case $n \geq 6$ is more subtle, and requires a careful analysis of
the local properties of the background metric $g$ near a blow-up point.
The Compactness Conjecture is closely related to the Weyl Vanishing
Conjecture, which asserts that the Weyl tensor should vanish to an order
greater than $\frac{n-6}{2}$ at a blow-up point (see \cite{Schoen3}). The
Weyl Vanishing Conjecture has been verified in dimensions $6$ and $7$ by
F.~Marques \cite{Marques} and, independently, by Y.Y.~Li and L.~Zhang
\cite{Li-Zhang1}. Using these results and the positive mass theorem, these
authors were able to prove compactness for $n \leq 7$. Moreover, Li and Zhang
showed that compactness holds in all dimensions provided that $|W_g(p)|
+ |\nabla W_g(p)| > 0$ for all $p \in M$. In dimensions $10$ and $11$, it
is sufficient to assume that $|W_g(p)| + |\nabla W_g(p)| + |\nabla^2 W_g(p)| > 0$ for all $p \in M$
(see \cite{Li-Zhang2}).
Very recently, M.~Khuri, F.~Marques and R.~Schoen \cite{Khuri-Marques-Schoen}
proved the Weyl Vanishing Conjecture up to dimension $24$. This result,
combined with the positive mass theorem, implies the Compactness Conjecture for
those dimensions. After proving sharp pointwise estimates, they reduce
these questions to showing a certain quadratic
form is positive definite. It turns out the quadratic form has negative
eigenvalues if $n \geq 25$.
In a recent paper \cite{Brendle1}, it was shown that the Compactness Conjecture
fails for $n \geq 52$. More precisely, given any integer $n \geq 52$, there
exists a smooth Riemannian metric $g$ on $S^n$ such that set of constant scalar
curvature metrics in the conformal class of $g$ is non-compact. Moreover, the
blowing-up sequences obtained in \cite{Brendle1} form exactly one bubble. The
construction relies on a gluing procedure based on some local model metric.
These local models are directions in which the quadratic form of
\cite{Khuri-Marques-Schoen} is negative definite. We refer to \cite{Brendle2} for a
survey of this and related results.
In the present paper, we extend these counterexamples to the dimensions
$25 \leq n \leq 51$. Our main theorem is:
\begin{theorem}
\label{main.theorem}
Assume that $25 \leq n \leq 51$. Then there exists a Riemannian metric
$g$ on $S^n$ (of class $C^\infty$) and a sequence of positive functions
$v_\nu \in C^\infty(S^n)$ ($\nu \in \mathbb{N}$) with the following
properties:
\begin{itemize}
\item[(i)] $g$ is not conformally flat
\item[(ii)] $v_\nu$ is a solution of the Yamabe PDE (\ref{yamabe.pde})
for all $\nu \in \mathbb{N}$
\item[(iii)] $E_g(v_\nu) < Y(S^n)$ for all $\nu \in \mathbb{N}$, and
$E_g(v_\nu) \to Y(S^n)$ as $\nu \to \infty$
\item[(iv)] $\sup_{S^n} v_\nu \to \infty$ as $\nu \to \infty$
\end{itemize}
(Here, $Y(S^n)$ denotes the Yamabe energy of the round metric on $S^n$.)
\end{theorem}
We note that O.~Druet and E.~Hebey \cite{Druet-Hebey1} have constructed
blow-up examples for perturbations of (\ref{yamabe.pde}) (see also \cite{Druet-Hebey2}).
In Section 2, we describe how the problem can be reduced to finding critical points of a certain function $\mathcal{F}_g(\xi,\varepsilon)$, where $\xi$ is a vector in $\mathbb{R}^n$ and $\varepsilon$ is a positive real number. This idea has been used by many authors (see, e.g., \cite{Ambrosetti}, \cite{Ambrosetti-Malchiodi}, \cite{Berti-Malchiodi}, \cite{Brendle1}). In Section 3, we show that the function
$\mathcal{F}_g(\xi,\varepsilon)$ can be approximated by an auxiliary
function $F(\xi,\varepsilon)$. In Section 4, we prove that the function
$F(\xi,\varepsilon)$ has a critical point, which is a strict local
minimum. Finally, in Section 5, we use a perturbation argument to
construct critical points of the function
$\mathcal{F}_g(\xi,\varepsilon)$. From this the non-compactness result
follows.
The authors would like to thank Professor Richard Schoen for constant support and encouragement. The first author was supported by the Alfred P. Sloan foundation and by the National Science Foundation under grant
DMS-0605223. The second author was supported by CNPq-Brazil, FAPERJ and the Stanford Mathematics Department. \\
\section{Lyapunov-Schmidt reduction}
In this section, we collect some basic results established in
\cite{Brendle1}. Let
\[\mathcal{E} = \bigg \{ w \in L^{\frac{2n}{n-2}}(\mathbb{R}^n) \cap
W_{loc}^{1,2}(\mathbb{R}^n): \int_{\mathbb{R}^n} |dw|^2 < \infty \bigg
\}.\] By Sobolev's inequality, there exists a constant $K$, depending
only on $n$, such that
\[\bigg ( \int_{\mathbb{R}^n} |w|^{\frac{2n}{n-2}} \bigg
)^{\frac{n-2}{n}} \leq K \, \int_{\mathbb{R}^n} |dw|^2\]
for all $w \in \mathcal{E}$. We define a norm on $\mathcal{E}$ by
$\|w\|_{\mathcal{E}}^2 = \int_{\mathbb{R}^n} |dw|^2$. It is easy to see
that $\mathcal{E}$, equipped with this norm, is complete. \\
Given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$,
we define a function $u_{(\xi,\varepsilon)}:
\mathbb{R}^n \to \mathbb{R}$ by
\[u_{(\xi,\varepsilon)}(x) = \Big ( \frac{\varepsilon}{\varepsilon^2 +
|x - \xi|^2} \Big )^{\frac{n-2}{2}}.\]
The function $u_{(\xi,\varepsilon)}$ satisfies the elliptic PDE
\[\Delta u_{(\xi,\varepsilon)} + n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} = 0.\]
It is well known that
\[\int_{\mathbb{R}^n} u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}} = \Big (
\frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}\]
for all $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$. We next
define
\[\varphi_{(\xi,\varepsilon,0)}(x) =
\Big ( \frac{\varepsilon}{\varepsilon^2 + |x - \xi|^2} \Big
)^{\frac{n+2}{2}} \, \frac{\varepsilon^2 - |x - \xi|^2}{\varepsilon^2 +
|x - \xi|^2}\]
and
\[\varphi_{(\xi,\varepsilon,k)}(x) =
\Big ( \frac{\varepsilon}{\varepsilon^2 + |x - \xi|^2} \Big
)^{\frac{n+2}{2}} \, \frac{2\varepsilon \, (x_k - \xi_k)}{\varepsilon^2
+ |x - \xi|^2}\]
for $k = 1,\hdots,n$. Finally, we define a closed subspace
$\mathcal{E}_{(\xi,\varepsilon)} \subset \mathcal{E}$ by
\[\mathcal{E}_{(\xi,\varepsilon)} = \bigg \{ w \in \mathcal{E}:
\int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \, w = 0 \quad
\text{for $k = 0,1,\hdots,n$} \bigg \}.\]
Clearly, $u_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$.
\begin{proposition}
\label{linearized.operator}
Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) =
\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on
$\mathbb{R}^n$ satisfying $h(x) = 0$ for $|x| \geq 1$. There exists a
positive constant $\alpha_0 \leq 1$, depending only on $n$, with the
following significance: if $|h(x)| + |\partial h(x)| + |\partial^2 h(x)|
\leq \alpha_0$ for all $x \in \mathbb{R}^n$, then, given any pair
$(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$ and any function
$f \in L^{\frac{2n}{n+2}}(\mathbb{R}^n)$, there exists a unique function
$w=G_{(\xi,\varepsilon)}(f) \in \mathcal{E}_{(\xi,\varepsilon)}$ such that
\[\int_{\mathbb{R}^n} \Big ( \langle dw,d\psi \rangle_g
+ \frac{n-2}{4(n-1)} \, R_g \, w \, \psi - n(n+2) \,
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w \, \psi \Big ) =
\int_{\mathbb{R}^n} f \, \psi\]
for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$.
Moreover, we have $\|w\|_{\mathcal{E}} \leq C \,
\|f\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}$, where $C$ is a constant that
depends only on $n$.
\end{proposition}
\begin{proposition}
\label{fixed.point.argument}
Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) =
\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on
$\mathbb{R}^n$ satisfying $h(x) = 0$ for $|x| \geq 1$. Moreover, let
$(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$. There exists a
positive constant $\alpha_1 \leq \alpha_0$, depending only on $n$, with
the following significance: if $|h(x)| + |\partial h(x)| + |\partial^2
h(x)| \leq \alpha_1$ for all $x \in \mathbb{R}^n$, then there exists a
function $v_{(\xi,\varepsilon)} \in \mathcal{E}$ such that
$v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} \in
\mathcal{E}_{(\xi,\varepsilon)}$
and
\[\int_{\mathbb{R}^n} \Big ( \langle dv_{(\xi,\varepsilon)},d\psi
\rangle_g + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \, \psi -
n(n-2) \, |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \,
v_{(\xi,\varepsilon)} \, \psi \Big ) = 0\] for all test functions $\psi
\in \mathcal{E}_{(\xi,\varepsilon)}$. Moreover, we have the estimate
\begin{align*}
&\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{\mathcal{E}} \\
&\leq C \, \Big \| \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)}
\, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)},
\end{align*}
where $C$ is a constant that depends only on $n$.
\end{proposition}
We next define a function $\mathcal{F}_g: \mathbb{R}^n \times (0,\infty)
\to \mathbb{R}$ by
\begin{align*}
\mathcal{F}_g(\xi,\varepsilon) &= \int_{\mathbb{R}^n}
\Big ( |dv_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \,
v_{(\xi,\varepsilon)}^2 - (n-2)^2 \,
|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} \Big ) \\
&- 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}.
\end{align*}
If we choose $\alpha_1$ small enough, then we obtain the following
result: \\
\begin{proposition}
\label{reduction.to.a.finite.dimensional.problem}
The function $\mathcal{F}_g$ is continuously differentiable. Moreover,
if $(\bar{\xi},\bar{\varepsilon})$ is a critical point of the function
$\mathcal{F}_g$, then
the function $v_{(\bar{\xi},\bar{\varepsilon})}$ is a non-negative weak
solution of the equation
\[\Delta_g v_{(\bar{\xi},\bar{\varepsilon})} - \frac{n-2}{4(n-1)} \, R_g
\, v_{(\bar{\xi},\bar{\varepsilon})} + n(n-2) \,
v_{(\bar{\xi},\bar{\varepsilon})}^{\frac{n+2}{n-2}} = 0.\]
\end{proposition}
\section{An estimate for the energy of a ``bubble"}
Throughout this paper, we fix a real number $\tau$ and a multi-linear
form $W: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \times
\mathbb{R}^n \to \mathbb{R}$. The number $\tau$ depends only on the
dimension $n$. The exact choice of $\tau$ will be postponed until
Section 4. We assume that $W_{ijkl}$ satisfies all the algebraic
properties of the Weyl tensor. Moreover, we assume that some components
of $W$ are non-zero, so that
\[\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 > 0.\]
For abbreviation, we put
\[H_{ik}(x) = \sum_{p,q=1}^n W_{ipkq} \, x_p \, x_q\]
and
\[\overline{H}_{ik}(x) = f(|x|^2) \, H_{ik}(x),\]
where $f(s) = \tau + 5s - s^2 + \frac{1}{20} \, s^3$. It is easy to see
that $H_{ik}(x)$ is trace-free, $\sum_{i=1}^n x_i \, H_{ik}(x) = 0$, and
$\sum_{i=1}^n \partial_i H_{ik}(x) = 0$ for all $x \in \mathbb{R}^n$. \\
We consider a Riemannian metric of the form $g(x) = \exp(h(x))$, where
$h(x)$ is a trace-free symmetric two-tensor on $\mathbb{R}^n$ satisfying
$h(x) = 0$ for $|x| \geq 1$,
\[|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha_1\]
for all $x \in \mathbb{R}^n$, and
\[h_{ik}(x) = \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x)\]
for $|x| \leq \rho$. We assume that the parameters $\lambda$, $\mu$, and
$\rho$ are chosen such that
$\mu \leq 1$ and $\lambda \leq \rho \leq 1$. Note that
$\sum_{i=1}^n x_i \, h_{ik}(x) = 0$ and $\sum_{i=1}^n \partial_i
h_{ik}(x) = 0$ for $|x| \leq \rho$.
Given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$,
there exists a unique function $v_{(\xi,\varepsilon)}$ such that $v_{(\xi,\varepsilon)}
- u_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$
and
\[\int_{\mathbb{R}^n} \Big ( \langle dv_{(\xi,\varepsilon)},d\psi
\rangle_g + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \, \psi -
n(n-2) \, |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \,
v_{(\xi,\varepsilon)} \, \psi \Big ) = 0\] for all test functions $\psi
\in \mathcal{E}_{(\xi,\varepsilon)}$ (see Proposition \ref{fixed.point.argument}).
For abbreviation, let
\[\Omega = \bigg \{ (\xi,\varepsilon) \in \mathbb{R}^n \times
\mathbb{R}: |\xi| < 1, \, \frac{1}{2} < \varepsilon < 2 \bigg \}.\]
The following result is proved in the Appendix A of \cite{Brendle1}. A similar formula is derived in \cite{Ambrosetti-Malchiodi}.
\begin{proposition}
\label{Taylor.expansion.of.scalar.curvature}
Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) =
\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on
$\mathbb{R}^n$ satisfying $|h(x)| \leq 1$ for all $x \in \mathbb{R}^n$.
Let $R_g$ be the scalar curvature of $g$. There exists a constant $C$,
depending only on $n$, such that
\begin{align*}
&\Big | R_g - \partial_i \partial_k h_{ik}
+ \partial_i(h_{il} \, \partial_k h_{kl}) - \frac{1}{2} \, \partial_i
h_{il} \, \partial_k h_{kl} + \frac{1}{4} \, \partial_l h_{ik} \,
\partial_l h_{ik} \Big | \\
&\leq C \, |h|^2 \, |\partial^2 h| + C \, |h| \, |\partial h|^2.
\end{align*}
\end{proposition}
\vspace{2mm}
\begin{proposition}
\label{estimate.for.error.term}
Assume that $(\xi,\varepsilon) \in \lambda \, \Omega$. Then we have
\begin{align*}
&\Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^8 \, \mu + C \, \Big ( \frac{\lambda}{\rho} \Big
)^{\frac{n-2}{2}}
\end{align*}
and
\begin{align*}
&\Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \\
&\hspace{10mm} + \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \,
|x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \Big (
\frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.
\end{align*}
\end{proposition}
\textbf{Proof.}
Note that $\sum_{i=1}^n \partial_i h_{ik}(x) = 0$ for $|x| \leq \rho$.
Hence, it follows from Proposition
\ref{Taylor.expansion.of.scalar.curvature} that
\[|R_g(x)| \leq C \, |h(x)|^2 \, |\partial^2 h(x)| + C \, |\partial
h(x)|^2 \leq C \, \mu^2 \, (\lambda + |x|)^{14}\]
for $|x| \leq \rho$. This implies
\begin{align*}
&\Big | \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \,
u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big | \\
&= \Big | \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik}) \,
\partial_k u_{(\xi,\varepsilon)} \big ]
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} \Big | \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda + |x|)^{8-n}
\end{align*}
and
\begin{align*}
&\Big | \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} + \sum_{i,k=1}^n
h_{ik} \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big | \\
&= \Big | \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik} +
h_{ik}) \, \partial_k u_{(\xi,\varepsilon)} \big ] - \frac{n-2}{4(n-1)}
\, R_g \, u_{(\xi,\varepsilon)} \Big | \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda + |x|)^{16-n} \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda +
|x|)^{\frac{8(n+2)}{n-2}-n}
\end{align*}
for $|x| \leq \rho$. From this the assertion follows. \\
\begin{corollary}
\label{estimate.for.v.1}
The function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}$ satisfies
the estimate
\[\|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \lambda^8 \, \mu + C \, \Big ( \frac{\lambda}{\rho} \Big
)^{\frac{n-2}{2}}\]
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{corollary}
\textbf{Proof.}
It follows from Proposition \ref{fixed.point.argument} that
\begin{align*}
&\|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\
&\leq C \, \Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)},
\end{align*}
where $C$ is a constant that depends only on $n$. Hence, the assertion
follows from Proposition \ref{estimate.for.error.term}. \\
We next establish a more precise estimate for the function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}$.
Applying Proposition \ref{linearized.operator} with $h = 0$, we conclude that there exists a unique function $w_{(\xi,\varepsilon)} \in
\mathcal{E}_{(\xi,\varepsilon)}$ such that
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle
- n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \,
w_{(\xi,\varepsilon)} \, \psi \Big ) \\
&= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, \psi
\end{align*}
for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$.
\begin{proposition}
\label{properties.of.w}
The function $w_{(\xi,\varepsilon)}$ is smooth. Moreover, if $(\xi,\varepsilon) \in \lambda \, \Omega$, then the function
$w_{(\xi,\varepsilon)}$ satisfies the estimates
\begin{align*}
&|w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \,
(\lambda + |x|)^{10-n} \\
&|\partial w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}}
\, \mu \, (\lambda + |x|)^{9-n} \\
&|\partial^2 w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}}
\, \mu \, (\lambda + |x|)^{8-n}
\end{align*}
for all $x \in \mathbb{R}^n$.
\end{proposition}
\textbf{Proof.}
There exist real numbers $b_k(\xi,\varepsilon)$ such that
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle
- n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \,
w_{(\xi,\varepsilon)} \, \psi \Big ) \\
&= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, \psi \\
&+ \sum_{k=0}^n b_k(\xi,\varepsilon) \, \int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \, \psi
\end{align*}
for all test functions $\psi \in \mathcal{E}$. Hence, standard elliptic regularity theory implies that $w_{(\xi,\varepsilon)}$ is smooth.
It remains to prove quantitative estimates for $w_{(\xi,\varepsilon)}$. To that end, we consider a pair $(\xi,\varepsilon) \in \lambda \, \Omega$. One readily verifies that
\[\Big \| \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \, \lambda^8 \, \mu.\] As a consequence, the function $w_{(\xi,\varepsilon)}$ satisfies $\|w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \lambda^8 \, \mu$. Moreover, we have $\sum_{k=0}^n |b_k(\xi,\varepsilon)| \leq C \, \lambda^8 \, \mu$. This implies
\begin{align*}
&\big | \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big | \\
&= \bigg | \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} - \sum_{k=0}^n b_k(\xi,\varepsilon) \, \int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \bigg | \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda+|x|)^{8-n}
\end{align*}
for all $x \in \mathbb{R}^n$. We claim that
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^8 \, \mu.\]
To show this, we fix a point $x_0 \in \mathbb{R}^n$. Let $r = \frac{1}{2} \, (\lambda + |x_0|)$. Then
\[u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \leq C \, r^{-2}\] and
\[\big | \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big | \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, r^{8-n}\]
for all $x \in B_r(x_0)$. Hence, it follows from standard interior estimates that
\begin{align*}
r^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x_0)| &\leq C \, \|w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(B_r(x_0))} \\
&+ C \, r^{\frac{n+2}{2}} \, \big \| \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big \|_{L^\infty(B_r(x_0))} \\
&\leq C \, \lambda^8 \, \mu + C \, \lambda^{\frac{n-2}{2}} \, \mu \, r^{-\frac{n-18}{2}} \\
&\leq C \, \lambda^8 \, \mu.
\end{align*}
Therefore, we have
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^8 \, \mu,\] as
claimed. Since $\sup_{x \in \mathbb{R}^n} |x|^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| < \infty$, we can express the function $w_{(\xi,\varepsilon)}$ in the form
\begin{equation}
\label{convolution.formula}
w_{(\xi,\varepsilon)}(x) = -\frac{1}{(n-2) \, |S^{n-1}|} \int_{\mathbb{R}^n} |x - y|^{2-n} \, \Delta w_{(\xi,\varepsilon)}(y) \, dy
\end{equation}
for all $x \in \mathbb{R}^n$.
We are now able to use a bootstrap argument to prove the desired estimate for $w_{(\xi,\varepsilon)}$. It follows from (\ref{convolution.formula}) that
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^\beta \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\beta+2} \, |\Delta w_{(\xi,\varepsilon)}(x)|\]
for all $0 < \beta < n-2$. Since
\begin{align*}
|\Delta w_{(\xi,\varepsilon)}(x)|
&\leq n(n+2) \, u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \, |w_{(\xi,\varepsilon)}(x)| \\
&+ C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda+|x|)^{8-n}
\end{align*}
for all $x \in \mathbb{R}^n$, we conclude that
\begin{align*}
\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^\beta \, |w_{(\xi,\varepsilon)}(x)| &\leq C \, \lambda^2 \, \sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\beta-2} \, |w_{(\xi,\varepsilon)}(x)| \\ &+ C \, \lambda^{\beta-\frac{n-18}{2}} \, \mu
\end{align*}
for all $0 < \beta \leq n-10$. Iterating this inequality, we obtain
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{n-10} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu.\]
The estimates for the first and second derivatives of $w_{(\xi,\varepsilon)}$ follow now from standard interior estimates. \\
\begin{corollary}
\label{estimate.for.v.2}
The function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}$ satisfies the estimate
\[\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}\]
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{corollary}
\textbf{Proof.}
Consider the functions
\[B_1 = \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik}) \,
\partial_k w_{(\xi,\varepsilon)} \big ] - \frac{n-2}{4(n-1)} \, R_g \,
w_{(\xi,\varepsilon)}\]
and
\[B_2 = \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \,
H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)}.\]
By definition of $w_{(\xi,\varepsilon)}$, we have
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle_g
+ \frac{n-2}{4(n-1)} \, R_g \, w_{(\xi,\varepsilon)} \, \psi - n(n+2) \,
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \, \psi
\Big ) \\
&= -\int_{\mathbb{R}^n} (B_1 + B_2) \, \psi
\end{align*}
for all functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Since
$w_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$, it follows
that
\[w_{(\xi,\varepsilon)} = -G_{(\xi,\varepsilon)}(B_1 + B_2).\]
Moreover, we have
\[v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} = G_{(\xi,\varepsilon)}
\big ( B_3 + n(n-2) \, B_4 \big ),\]
where
\[B_3 = \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \,
u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}}\]
and
\[B_4 = |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)}
- u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} - \frac{n+2}{n-2} \,
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, (v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}).\]
Thus, we conclude that
\[v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}
= G_{(\xi,\varepsilon)} \big ( B_1 + B_2 + B_3 + n(n-2) \, B_4 \big ),\]
where $G_{(\xi,\varepsilon)}$ denotes the solution operator constructed in Proposition \ref{linearized.operator}. As a consequence, we obtain
\[\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \,
\big \| B_1 + B_2 + B_3 + n(n-2) \, B_4 \big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}.\]
It follows from Proposition \ref{properties.of.w} that
\[|B_1(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda + |x|)^{16-n} \leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda+|x|)^{\frac{8(n+2)}{n-2}-n}\]
for $|x| \leq \rho$ and
\[|B_1(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, |x|^{8-n}\]
for $|x| \geq \rho$. This implies
\[\|B_1\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \,
\lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \rho^8 \, \mu \, \Big (
\frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.\]
Moreover, observe that
\[\|B_2 + B_3\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \,
\lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \Big ( \frac{\lambda}{\rho}
\Big )^{\frac{n-2}{2}}\]
by Proposition \ref{estimate.for.error.term}. Finally, Corollary \ref{estimate.for.v.1} implies that
\begin{align*}
\|B_4\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}
&\leq C \, \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}^{\frac{n+2}{n-2}}
\\
&\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n+2}{2}}.
\end{align*}
Putting these facts together, we obtain
\[\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.
\]
This completes the proof. \\
\begin{proposition}
\label{term.1}
We have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \Big ( |dv_{(\xi,\varepsilon)}|_g^2
- |du_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \,
(v_{(\xi,\varepsilon)}^2 - u_{(\xi,\varepsilon)}^2) \Big ) \\
&\hspace{10mm} + \int_{\mathbb{R}^n} n(n-2) \,
(|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} \\
&\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \,
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, w_{(\xi,\varepsilon)} \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \,
\lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} +
C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{proposition}
\textbf{Proof.}
By definition of $v_{(\xi,\varepsilon)}$, we have
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( |dv_{(\xi,\varepsilon)}|_g^2 - \langle
du_{(\xi,\varepsilon)},dv_{(\xi,\varepsilon)} \rangle_g
+ \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \,
(v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) \Big ) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \,
|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)} \,
(v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) = 0.
\end{align*}
Using Proposition \ref{estimate.for.error.term} and Corollary
\ref{estimate.for.v.1}, we obtain
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \Big ( \langle
du_{(\xi,\varepsilon)},dv_{(\xi,\varepsilon)} \rangle_g
- |du_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \,
u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)})
\Big ) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \, (v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)})
\bigg | \\
&\leq \Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \\
&\hspace{10mm} + \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \,
|x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\
&\hspace{5mm} \cdot \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \lambda^8 \, \mu \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} + C \, \Big (
\frac{\lambda}{\rho} \Big )^{n-2}.
\end{align*}
Moreover, we have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}
- w_{(\xi,\varepsilon)}) \bigg | \\
&\leq C \, \lambda^8 \, \mu \, \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \,
\lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}
\end{align*}
by Corollary \ref{estimate.for.v.2}. Putting these facts together, the
assertion follows. \\
\begin{proposition}
\label{term.2}
We have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} - \frac{2}{n} \int_{\mathbb{R}^n}
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \Big
( \frac{\lambda}{\rho} \Big )^n
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{proposition}
\textbf{Proof.}
We have the pointwise estimate
\begin{align*}
&\Big | (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} - \frac{2}{n} \,
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \Big | \\
&\leq C \, |v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}},
\end{align*}
where $C$ is a constant that depends only on $n$. This implies
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} - \frac{2}{n} \int_{\mathbb{R}^n}
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \bigg | \\
&\leq C \, \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}^{\frac{2n}{n-2}}
\\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \Big
( \frac{\lambda}{\rho} \Big )^n
\end{align*}
by Corollary \ref{estimate.for.v.1}. \\
\begin{proposition}
\label{term.3}
We have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \Big ( |du_{(\xi,\varepsilon)}|_g^2 +
\frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}^2 - n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}} \Big ) \\
&\hspace{10mm} - \int_{B_\rho(0)} \frac{1}{2} \, \sum_{i,k,l=1}^n
h_{il} \, h_{kl} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k
u_{(\xi,\varepsilon)} \\
&\hspace{10mm} + \int_{B_\rho(0)} \frac{n-2}{16(n-1)} \,
\sum_{i,k,l=1}^n (\partial_l h_{ik})^2 \,
u_{(\xi,\varepsilon)}^2 \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \Big (
\frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{proposition}
\textbf{Proof.}
Note that
\begin{align*}
&\Big | g^{ik}(x) - \delta_{ik} + h_{ik}(x) - \frac{1}{2} \,
\sum_{l=1}^n h_{il}(x) \, h_{kl}(x) \Big | \\
&\leq C \, |h(x)|^3 \leq C \, \mu^3 \, (\lambda + |x|)^{24} \leq C \,
\mu^3 \, (\lambda+|x|)^{\frac{16n}{n-2}}
\end{align*}
for $|x| \leq \rho$. This implies
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \big ( |du_{(\xi,\varepsilon)}|_g^2 -
|du_{(\xi,\varepsilon)}|^2 \big ) + \int_{\mathbb{R}^n} \sum_{i,k=1}^n
h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \\
&\hspace{10mm} - \int_{B_\rho(0)} \frac{1}{2} \sum_{i,k,l=1}^n h_{il}
\, h_{kl} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k
u_{(\xi,\varepsilon)} \bigg | \\
&\leq C \, \lambda^{n-2} \, \mu^3 \, \int_{B_\rho(0)} (\lambda+|x|)^{\frac{16n}{n-2}+2-2n}
+ C \, \lambda^{n-2} \, \int_{\mathbb{R}^n \setminus B_\rho(0)} (\lambda + |x|)^{2-2n} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.
\end{align*}
By Proposition \ref{Taylor.expansion.of.scalar.curvature}, the scalar
curvature of $g$ satisfies the estimate
\begin{align*}
&\Big | R_g(x) + \frac{1}{4} \sum_{i,k,l=1}^n (\partial_l h_{ik}(x))^2
\Big | \\
&\leq C \, |h(x)|^2 \, |\partial^2 h(x)| + C \, |h(x)| \, |\partial
h(x)|^2 \\
&\leq C \, \mu^3 \, (\lambda + |x|)^{22} \leq C \, \mu^3 \, (\lambda +
|x|)^{\frac{16n}{n-2}-2}
\end{align*}
for $|x| \leq \rho$. This implies
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} R_g \, u_{(\xi,\varepsilon)}^2 +
\int_{B_\rho(0)} \frac{1}{4} \sum_{i,k,l=1}^n (\partial_l h_{ik})^2
\, u_{(\xi,\varepsilon)}^2 \bigg | \\
&\leq C \, \lambda^{n-2} \, \mu^3 \, \int_{B_\rho(0)} (\lambda+|x|)^{\frac{16n}{n-2}+2-2n}
+ C \, \lambda^{n-2} \, \int_{\mathbb{R}^n \setminus B_\rho(0)} (\lambda + |x|)^{4-2n} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \rho^2 \, \Big (
\frac{\lambda}{\rho} \Big )^{n-2}.
\end{align*}
At this point, we use the formula
\begin{align*}
&\partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} -
\frac{n-2}{4(n-1)} \, \partial_i \partial_k (u_{(\xi,\varepsilon)}^2) \\
&= \frac{1}{n} \, \Big ( |du_{(\xi,\varepsilon)}|^2 - \frac{n-2}{4(n-1)}
\, \Delta (u_{(\xi,\varepsilon)}^2) \Big ) \, \delta_{ik}.
\end{align*}
Since $h_{ik}$ is trace-free, we obtain
\[\sum_{i,k=1}^n h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \,
\partial_k u_{(\xi,\varepsilon)} = \frac{n-2}{4(n-1)} \sum_{i,k=1}^n
h_{ik} \, \partial_i \partial_k (u_{(\xi,\varepsilon)}^2),\]
hence
\[\int_{\mathbb{R}^n} \sum_{i,k=1}^n h_{ik} \, \partial_i
u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} =
\int_{\mathbb{R}^n} \frac{n-2}{4(n-1)} \sum_{i,k=1}^n \partial_i
\partial_k h_{ik} \, u_{(\xi,\varepsilon)}^2.\]
Since $\sum_{i=1}^n \partial_i h_{ik}(x) = 0$ for $|x| \leq \rho$, it
follows that
\[\bigg | \int_{\mathbb{R}^n} \sum_{i,k=1}^n h_{ik} \, \partial_i
u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \bigg |
\leq C \int_{\mathbb{R}^n \setminus B_\rho(0)} u_{(\xi,\varepsilon)}^2 \leq C \, \rho^2 \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.\]
Putting these facts together, the assertion follows. \\
\begin{corollary}
\label{key.estimate}
The function $\mathcal{F}_g(\xi,\varepsilon)$ satisfies the estimate
\begin{align*}
&\bigg | \mathcal{F}_g(\xi,\varepsilon) - \int_{B_\rho(0)}
\frac{1}{2} \, \sum_{i,k,l=1}^n h_{il} \, h_{kl} \, \partial_i
u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \\
&\hspace{10mm} + \int_{B_\rho(0)} \frac{n-2}{16(n-1)} \,
\sum_{i,k,l=1}^n (\partial_l h_{ik})^2 \,
u_{(\xi,\varepsilon)}^2 \\
&\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i\partial_k
u_{(\xi,\varepsilon)} \, w_{(\xi,\varepsilon)} \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}}
+ C \, \lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big
)^{\frac{n-2}{2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{corollary}
\section{Finding a critical point of an auxiliary function}
We define a function $F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$
as follows: given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times
(0,\infty)$, we define
\begin{align*}
F(\xi,\varepsilon)
&= \int_{\mathbb{R}^n} \frac{1}{2} \sum_{i,k,l=1}^n \overline{H}_{il}(x)
\, \overline{H}_{kl}(x) \, \partial_i u_{(\xi,\varepsilon)}(x) \,
\partial_k u_{(\xi,\varepsilon)}(x) \\
&- \int_{\mathbb{R}^n} \frac{n-2}{16(n-1)} \, \sum_{i,k,l=1}^n
(\partial_l \overline{H}_{ik}(x))^2 \, u_{(\xi,\varepsilon)}(x)^2 \\
&+ \int_{\mathbb{R}^n} \sum_{i,k=1}^n \overline{H}_{ik}(x) \, \partial_i
\partial_k u_{(\xi,\varepsilon)}(x) \, z_{(\xi,\varepsilon)}(x),
\end{align*}
where $z_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$
satisfies the relation
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dz_{(\xi,\varepsilon)},d\psi \rangle
- n(n+2) \, u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \,
z_{(\xi,\varepsilon)} \, \psi \Big ) \\
&= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \overline{H}_{ik} \, \partial_i
\partial_k u_{(\xi,\varepsilon)} \, \psi
\end{align*}
for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Our
goal in this section is to show that the function $F(\xi,\varepsilon)$
has a critical point.
\begin{proposition}
\label{symmetry}
The function $F(\xi,\varepsilon)$ satisfies $F(\xi,\varepsilon) =
F(-\xi,\varepsilon)$ for all $(\xi,\varepsilon) \in \mathbb{R}^n \times
(0,\infty)$. Consequently, we have $\frac{\partial}{\partial \xi_p}
F(0,\varepsilon) = 0$ and $\frac{\partial^2}{\partial \varepsilon \,
\partial \xi_p} F(0,\varepsilon) = 0$ for all $\varepsilon > 0$ and $p =
1, \hdots, n$.
\end{proposition}
\textbf{Proof.}
This follows immediately from the relation $\overline{H}_{ik}(-x) =
\overline{H}_{ik}(x)$. \\
\begin{proposition}
\label{integral.identity.1}
We have
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \, x_p
\, x_q \\
&= \frac{2}{n(n+2)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+3} \\
&+ \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \, r^{n+3}
\end{align*}
and
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k=1}^n H_{ik}(x)^2 \, x_p \, x_q \\
&= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+5} \\
&+ \frac{1}{2n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \, r^{n+5}.
\end{align*}
\end{proposition}
\textbf{Proof.}
See \cite{Brendle1}, Proposition 16. \\
\begin{proposition}
\label{integral.identity.2}
We have
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \, x_p \, x_q \\
&= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{10mm} \cdot r^{n+3} \, \Big [ (n+4) \, f(r^2)^2 + 8r^2 \,
f(r^2) \, f'(r^2) + 4r^4 \, f'(r^2)^2 \Big ] \\
&+ \frac{1}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{10mm} \cdot r^{n+3} \, \Big [ (n+4) \, f(r^2)^2 + 4r^2 \,
f(r^2) \, f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ].
\end{align*}
\end{proposition}
\textbf{Proof.}
Using the identity
\[\partial_l \overline{H}_{ik}(x) = f(|x|^2) \, \partial_l H_{ik}(x) + 2
\, f'(|x|^2) \, H_{ik}(x) \, x_l\]
and Euler's theorem, we obtain
\begin{align*}
&\sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \\
&= f(|x|^2)^2 \, \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \\
&+ 4 \, f(|x|^2) \, f'(|x|^2) \, \sum_{i,k,l=1}^n H_{ik}(x) \, x_l \, \partial_l H_{ik}(x) \\
&+ 4 \, |x|^2 \, f'(|x|^2)^2 \, \sum_{i,k=1}^n H_{ik}(x)^2 \\
&= f(|x|^2)^2 \, \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \\
&+ \big [ 8 \, f(|x|^2) \, f'(|x|^2) + 4 \, |x|^2 \, f'(|x|^2)^2 \big ]
\, \sum_{i,k=1}^n H_{ik}(x)^2.
\end{align*}
Hence, the assertion follows from the previous proposition. \\
\begin{corollary}
\label{integral.identity.3}
We have
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \\ &= \frac{1}{n(n+2)} \, |S^{n-1}| \,
\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\
&\hspace{10mm} \cdot r^{n+1} \, \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \,
f(r^2) \, f'(r^2)
+ 2 \, r^4 \, f'(r^2)^2 \Big ].
\end{align*}
\end{corollary}
\vspace{2mm}
\begin{proposition}
\label{formula.for.F}
We have
\begin{align*}
F(0,\varepsilon) &= -\frac{n-2}{16n(n-1)(n+2)} \, |S^{n-1}| \,
\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{2-n} \, r^{n+1} \\
&\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2 \, r^4 \, f'(r^2)^2 \Big ] \, dr.
\end{align*}
\end{proposition}
\textbf{Proof.}
Note that $z_{(0,\varepsilon)}(x) = 0$ for all $x \in \mathbb{R}^n$.
This implies
\[F(0,\varepsilon) = -\int_{\mathbb{R}^n} \frac{n-2}{16(n-1)} \,
\varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{2-n} \, \sum_{i,k,l=1}^n
(\partial_l \overline{H}_{ik}(x))^2.\]
Using Corollary \ref{integral.identity.3}, we obtain
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{2-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \\
&= \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{2-n} \, r^{n+1} \\
&\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2 \, r^4 \, f'(r^2)^2 \Big ].
\end{align*}
This proves the assertion. \\
\begin{proposition}
\label{i}
The function $F(0,\varepsilon)$ can be written in the form
\begin{align*}
F(0,\varepsilon) &= -\frac{n-2}{16n(n-1)(n+2)} \, |S^{n-1}| \,
\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\
&\hspace{10mm} \cdot I(\varepsilon^2) \, \int_0^\infty (1 + r^2)^{2-n}
\, r^{n+7} \, dr,
\end{align*}
where
\begin{align*}
I(s) &= \frac{n-12}{n+6} \, \frac{n-10}{n+4} \, (n-8) \, \tau^2 \, s^2 +
10 \, \frac{n-12}{n+6} \, (n-10) \, \tau \, s^3 \\
&+ \Big ( 25 \, \frac{n-12}{n+6} \, (n+8) - 2(n-12) \, \tau \Big ) \,
s^4 + \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, s^5 \\
&+ \frac{n+8}{n-14} \, \frac{3n+52}{2} \, s^6 - \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+24}{10} \, s^7 \\
&+ \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+12}{n-18} \,
\frac{n+32}{400} \, s^8.
\end{align*}
\end{proposition}
\textbf{Proof.} It is straightforward to check that
\begin{align*}
&(n+2) \, f(s)^2 + 4s \, f(s) \, f'(s) + 2s^2 \, f'(s)^2 \\
&= (n+2)\tau^2 + 10(n+4)\tau \, s + \Big ( 25(n+8) - 2(n+6)\tau \Big )
\, s^2 \\
&+ \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, s^3 +
\frac{3n+52}{2} \, s^4 - \frac{n+24}{10} \, s^5
+ \frac{n+32}{400} \, s^6.
\end{align*}
This implies
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \,
r^{n+1} \\
&\hspace{10mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2r^4 \, f'(r^2)^2 \Big ] \, dr \\
&= (n+2)\tau^2 \, \varepsilon^4 \int_0^\infty (1+r^2)^{2-n} \, r^{n+1}
\, dr \\
&+ 10(n+4)\tau \, \varepsilon^6 \int_0^\infty (1+r^2)^{2-n} \, r^{n+3}
\, dr \\
&+ \Big ( 25(n+8) - 2(n+6)\tau \Big ) \, \varepsilon^8 \int_0^\infty
(1+r^2)^{2-n} \, r^{n+5} \, dr \\
&+ \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, \varepsilon^{10}
\int_0^\infty (1+r^2)^{2-n} \, r^{n+7} \, dr \\
&+ \frac{3n+52}{2} \, \varepsilon^{12} \int_0^\infty (1+r^2)^{2-n} \,
r^{n+9} \, dr \\
&- \frac{n+24}{10} \, \varepsilon^{14} \int_0^\infty (1+r^2)^{2-n} \,
r^{n+11} \, dr \\
&+ \frac{n+32}{400} \, \varepsilon^{16} \int_0^\infty (1+r^2)^{2-n} \,
r^{n+13} \, dr.
\end{align*}
Using the identity
\[\int_0^\infty (1+r^2)^{2-n} \, r^{\beta+2} \, dr
= \frac{\beta+1}{2n-\beta-7} \int_0^\infty (1+r^2)^{2-n}
\, r^\beta \, dr,\]
we obtain
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \,
r^{n+1} \\
&\hspace{10mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2r^4 \, f'(r^2)^2 \Big ] \, dr \\
&= I(\varepsilon^2) \, \int_0^\infty (1+r^2)^{2-n} \, r^{n+7} \, dr.
\end{align*}
This completes the proof. \\
In the next step, we compute the Hessian of $F$ at $(0,\varepsilon)$. \\
\begin{proposition}
\label{Hessian.of.F.1}
The second order partial derivatives of the function
$F(\xi,\varepsilon)$ are given by
\begin{align*}
\frac{\partial^2}{\partial \xi_p \, \partial \xi_q} F(0,\varepsilon)
&= \int_{\mathbb{R}^n} (n-2)^2 \, \varepsilon^{n-2} \, (\varepsilon^2 +
|x|^2)^{-n} \, \sum_{l=1}^n \overline{H}_{pl}(x) \, \overline{H}_{ql}(x) \\
&- \int_{\mathbb{R}^n} \frac{(n-2)^2}{4} \, \varepsilon^{n-2} \,
(\varepsilon^2 + |x|^2)^{-n} \, \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \, x_p \, x_q \\
&+ \int_{\mathbb{R}^n} \frac{(n-2)^2}{8(n-1)} \, \varepsilon^{n-2} \,
(\varepsilon^2 + |x|^2)^{1-n} \, \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \, \delta_{pq}.
\end{align*}
\end{proposition}
\textbf{Proof.}
See \cite{Brendle1}, Proposition 21. \\
\begin{proposition}
\label{Hessian.of.F.2}
The second order partial derivatives of the function
$F(\xi,\varepsilon)$ are given by
\begin{align*}
&\frac{\partial^2}{\partial \xi_p \, \partial \xi_q} F(0,\varepsilon) \\
&= -\frac{2(n-2)^2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n
(W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2
\Big ] \, dr \\
&- \frac{(n-2)^2}{2n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n
(W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+5} \, \Big [
2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\
&+ \frac{(n-2)^2}{4n(n-1)(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n
(W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{1-n} \, r^{n+5} \, f'(r^2)^2 \, dr.
\end{align*}
\end{proposition}
\textbf{Proof.}
Using the identity
\begin{align*}
&\int_{\partial B_r(0)} \sum_{l=1}^n \overline{H}_{pl}(x) \,
\overline{H}_{ql}(x) \\
&= \int_{\partial B_r(0)} \sum_{i,j,k,l,m=1}^n W_{ipkl} \, W_{jqml} \,
x_i \, x_j \, x_k \, x_m \, f(|x|^2)^2 \\
&= \frac{1}{n(n+2)} \, |S^{n-1}| \\
&\hspace{10mm} \cdot \sum_{i,j,k,l,m=1}^n W_{ipkl} \, W_{jqml} \,
(\delta_{ij} \, \delta_{km} + \delta_{ik} \, \delta_{jm} + \delta_{im}
\, \delta_{jk}) \, r^{n+3} \, f(r^2)^2 \\
&= \frac{1}{2n(n+2)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+3} \, f(r^2)^2,
\end{align*}
we obtain
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2}
\, (\varepsilon^2 + |x|^2)^{-n} \, \sum_{i,k,l=1}^n \overline{H}_{pl}(x)
\, \overline{H}_{ql}(x) \\
&= \frac{1}{2n(n+2)} \, |S^{n-1}| \,
\sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{5mm} \cdot \int_0^\infty
\varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3} \, f(r^2)^2
\, dr.
\end{align*}
Similarly, it follows from Proposition \ref{integral.identity.2} that
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, x_p \, x_q \\
&= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+3} \\
&\hspace{15mm} \cdot \Big [ (n+4) \, f(r^2)^2 + 8r^2 \, f(r^2) \,
f'(r^2) + 4r^4 \, f'(r^2)^2 \Big ] \, dr \\
&+ \frac{1}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+3} \\
&\hspace{15mm} \cdot \Big [ (n+4) \, f(r^2)^2 + 4r^2 \, f(r^2) \,
f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ] \, dr.
\end{align*}
Moreover, we have
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{1-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, \delta_{pq} \\
&= \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{1-n} \, r^{n+1} \\
&\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2)
\, f'(r^2) + 2 \, r^4 \, f'(r^2)^2 \Big ] \, dr
\end{align*}
by Corollary \ref{integral.identity.3}. A straightforward calculation yields
\begin{align*}
&(\varepsilon^2+r^2)^{1-n} \, r^{n+1} \, \big [ (n+2) \, f(r^2)^2 + 4r^2
\, f(r^2) \, f'(r^2) \big ] \\
&= 2(n-1) \, (\varepsilon^2+r^2)^{-n} \, r^{n+3} \, f(r^2)^2 +
\frac{d}{dr} \big [ (\varepsilon^2+r^2)^{1-n} \, r^{n+2} \, f(r^2)^2 \big ].
\end{align*}
This implies
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{1-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, \delta_{pq} \\
&= \frac{2(n-1)}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot
\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3}
\, f(r^2)^2 \, dr \\
&+ \frac{2}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{1-n} \, r^{n+5} \, f'(r^2)^2 \, dr.
\end{align*}
Putting these facts together, the assertion follows. \\
\begin{proposition}
\label{j}
We have
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \,
r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\
&= J(\varepsilon^2) \, \int_0^\infty (1+r^2)^{-n} \, r^{n+9} \, dr,
\end{align*}
where
\begin{align*}
J(s) &= 10 \, \frac{n-10}{n+8} \, \frac{n-8}{n+6} \, \tau \, s^2 +
\frac{n-10}{n+8} \, (75-4\tau) \, s^3 \\
&+ \Big ( \frac{3}{10} \, \tau - 50 \Big ) \, s^4 + \frac{23}{2} \,
\frac{n+10}{n-12} \, s^5 - \frac{11}{10} \, \frac{n+10}{n-12} \,
\frac{n+12}{n-14} \, s^6 \\
&+ \frac{3}{80} \, \frac{n+10}{n-12} \, \frac{n+12}{n-14} \,
\frac{n+14}{n-16} \, s^7.
\end{align*}
\end{proposition}
\textbf{Proof.}
Note that
\begin{align*}
&2 \, f(s) \, f'(s) + s \, f'(s)^2 \\
&= 10\tau + (75-4\tau) \, s + \Big ( \frac{3}{10} \, \tau - 50 \Big ) \,
s^2 + \frac{23}{2} \, s^3
- \frac{11}{10} \, s^4 + \frac{3}{80} \, s^5.
\end{align*}
This implies
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \,
r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2)
+ r^2 \, f'(r^2)^2 \Big ] \, dr \\
&= 10\tau \, \varepsilon^4 \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+5} \,
dr \\
&+ (75-4\tau) \, \varepsilon^6 \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+7} \, dr \\
&+ \Big ( \frac{3}{10} \, \tau - 50 \Big ) \, \varepsilon^8 \,
\int_0^\infty (1 + r^2)^{-n} \, r^{n+9} \, dr \\
&+ \frac{23}{2} \, \varepsilon^{10} \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+11} \, dr \\
&- \frac{11}{10} \, \varepsilon^{12} \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+13} \, dr \\
&+ \frac{3}{80} \, \varepsilon^{14} \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+15} \, dr.
\end{align*}
Hence, the assertion follows from the identity
\[\int_0^\infty (1+r^2)^{-n} \, r^{\beta+2} \, dr
= \frac{\beta+1}{2n-\beta-3} \int_0^\infty (1+r^2)^{-n}
\, r^\beta \, dr.\]
\begin{proposition}
Assume that $25 \leq n \leq 51$. Then we can choose $\tau \in
\mathbb{R}$ such that $I'(1) = 0$, $I''(1) < 0$, and $J(1) < 0$.
\end{proposition}
\textbf{Proof.}
The condition $I'(1) = 0$ is equivalent to
\[a_n \, \tau^2 + b_n \, \tau + c_n = 0,\]
where
\begin{align*}
a_n &= 2 \, \frac{n-12}{n+6} \, \frac{n-10}{n+4} \, (n-8) \\
b_n &= 30 \, \frac{n-12}{n+6} \, (n-10) - 8(n-12) + \frac{n+8}{2} \\
c_n &= 100 \, \frac{n-12}{n+6} \, (n+8) - 50(n+12) + 3 \,
\frac{n+8}{n-14} \, (3n+52) \\ &- 7 \, \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+24}{10} + \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+12}{n-18} \, \frac{n+32}{50}.
\end{align*}
By inspection, one verifies that $49 \, a_n - 7 \, b_n + c_n < 0$ for
$25 \leq n \leq 51$. Since $a_n$ is positive, there exists a unique real
number $\tau < -7$ such that $a_n \, \tau^2 + b_n \, \tau + c_n = 0$.
Moreover, we have
\[I''(1) - I'(1) = \alpha_n \, \tau + \beta_n\]
and
\[J(1) = \gamma_n \, \tau + \delta_n,\]
where
\begin{align*}
\alpha_n &= 30 \, \frac{n-12}{n+6} \, (n-10) - 16(n-12) +
\frac{3(n+8)}{2} \\
\beta_n &= 200 \, \frac{n-12}{n+6} \, (n+8) - 150(n+12) + 12 \,
\frac{n+8}{n-14} \, (3n+52) \\ &- 35 \, \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+24}{10} + 3 \, \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+12}{n-18} \, \frac{n+32}{25} \\[2mm]
\gamma_n &= 10 \, \frac{n-10}{n+8} \, \frac{n-8}{n+6} -
\frac{4(n-10)}{n+8} + \frac{3}{10} \\
\delta_n &= 75 \, \frac{n-10}{n+8} - 50 + \frac{23}{2} \,
\frac{n+10}{n-12} - \frac{11}{10} \, \frac{n+10}{n-12} \,
\frac{n+12}{n-14} \\ &+ \frac{3}{80} \, \frac{n+10}{n-12} \,
\frac{n+12}{n-14} \, \frac{n+14}{n-16}.
\end{align*}
By inspection, one verifies that $7\alpha_n > \beta_n > 0$ and
$7\gamma_n > \delta_n > 0$ for $25 \leq n \leq 51$. This implies
$I''(1) = \alpha_n \, \tau + \beta_n < -7\alpha_n + \beta_n < 0$ and
$J(1) = \gamma_n \, \tau + \delta_n < -7\gamma_n + \delta_n < 0$. This
completes the proof. \\
\begin{corollary}
\label{strict.local.minimum}
Assume that $\tau$ is chosen such that $I'(1) = 0$, $I''(1) < 0$, and
$J(1) < 0$. Then the function $F(\xi,\varepsilon)$ has a strict local
minimum at $(0,1)$.
\end{corollary}
\textbf{Proof.}
Since $I'(1) = 0$, we have $\frac{\partial}{\partial \varepsilon} F(0,1)
= 0$. Therefore, $(0,1)$ is a critical point of the function
$F(\xi,\varepsilon)$. Since $J(1) < 0$, we have
\[\int_0^\infty (1 + r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \,
f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr < 0\]
by Proposition \ref{j}. Hence, it follows from Proposition
\ref{Hessian.of.F.2} that the matrix $\frac{\partial^2}{\partial \xi_p
\, \partial \xi_q} F(0,1)$ is positive definite. Using
Proposition \ref{i} and the inequality $I''(0) < 0$, we obtain
$\frac{\partial^2}{\partial \varepsilon^2} F(0,1) > 0$. Consequently,
the function $F(\xi,\varepsilon)$ has a strict local minimum at $(0,1)$. \\
\section{Proof of the main theorem}
\begin{proposition}
\label{perturbation.argument}
Assume that $25 \leq n \leq 51$. Moreover, let $g$ be a smooth metric on
$\mathbb{R}^n$ of the form
$g(x) = \exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor
on $\mathbb{R}^n$ such
that $|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha \leq
\alpha_1$ for all $x \in \mathbb{R}^n$,
$h(x) = 0$ for $|x| \geq 1$, and
\[h_{ik}(x) = \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \,
H_{ik}(x)\] for $|x| \leq \rho$. If $\alpha$ and $\rho^{2-n} \, \mu^{-2}
\, \lambda^{n-18}$ are
sufficiently small, then there exists a positive function $v$ such that
\[\Delta_g v - \frac{n-2}{4(n-1)} \, R_g \, v + n(n-2) \,
v^{\frac{n+2}{n-2}} = 0,\]
\[\int_{\mathbb{R}^n} v^{\frac{2n}{n-2}} < \Big ( \frac{Y(S^n)}{4n(n-1)}
\Big )^{\frac{n}{2}},\] and
$\sup_{|x| \leq \lambda} v(x) \geq c \, \lambda^{\frac{2-n}{2}}$. Here,
$c$ is a positive constant that depends only on $n$.
\end{proposition}
\textbf{Proof.}
By Corollary \ref{strict.local.minimum}, the function
$F(\xi,\varepsilon)$ has a strict local minimum at $(0,1)$. It follows
from Proposition
\ref{formula.for.F} that $F(0,1) < 0$. Hence, we can find an open set
$\Omega' \subset \Omega$ such that $(0,1) \in \Omega'$ and
\[F(0,1) < \inf_{(\xi,\varepsilon) \in \partial \Omega'}
F(\xi,\varepsilon) < 0.\]
Using Corollary \ref{key.estimate}, we obtain
\begin{align*}
&|\mathcal{F}_g(\lambda\xi,\lambda\varepsilon) - \lambda^{16} \, \mu^2
\, F(\xi,\varepsilon)| \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \,
\lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} +
C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
for all $(\xi,\varepsilon) \in \Omega$. This implies
\begin{align*}
&|\lambda^{-16} \, \mu^{-2} \,
\mathcal{F}_g(\lambda\xi,\lambda\varepsilon) - F(\xi,\varepsilon)| \\
&\leq C \, \lambda^{\frac{32}{n-2}} \, \mu^{\frac{4}{n-2}} + C \,
\rho^{\frac{2-n}{2}} \, \mu^{-1} \, \lambda^{\frac{n-18}{2}} + C \,
\rho^{2-n} \, \mu^{-2} \, \lambda^{n-18}
\end{align*}
for all $(\xi,\varepsilon) \in \Omega$.
Hence, if $\rho^{2-n} \, \mu^{-2} \, \lambda^{n-18}$ is sufficiently
small, then we have
\[\mathcal{F}_g(0,\lambda) < \inf_{(\xi,\varepsilon) \in \partial
\Omega'} \mathcal{F}_g(\lambda\xi,\lambda\varepsilon) < 0.\]
Consequently, there exists a point $(\bar{\xi},\bar{\varepsilon}) \in
\Omega'$ such that
\[\mathcal{F}_g(\lambda\bar{\xi},\lambda\bar{\varepsilon}) =
\inf_{(\xi,\varepsilon) \in \Omega'}
\mathcal{F}_g(\lambda\xi,\lambda\varepsilon) < 0.\]
By Proposition \ref{reduction.to.a.finite.dimensional.problem}, the
function $v = v_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}$ is a
non-negative weak solution of the partial differential equation
\[\Delta_g v - \frac{n-2}{4(n-1)} \, R_g \, v + n(n-2) \,
v^{\frac{n+2}{n-2}} = 0.\]
Using a result of N.~Trudinger, we conclude that $v$ is smooth (see
\cite{Trudinger}, Theorem 3 on p. 271). Moreover, we have
\begin{align*}
2(n-2) \int_{\mathbb{R}^n} v^{\frac{2n}{n-2}}
&= 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}} +
\mathcal{F}_g(\lambda\bar{\xi},\lambda\bar{\varepsilon}) \\
&< 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}.
\end{align*}
Finally, it follows from Proposition \ref{fixed.point.argument} that
$\|v -
u_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \alpha$.
This implies
\[|B_\lambda(0)|^{\frac{n-2}{2n}} \, \sup_{|x| \leq \lambda} v(x) \geq
\|v\|_{L^{\frac{2n}{n-2}}(B_\lambda(0))} \geq
\|u_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}\|_{L^{\frac{2n}{n-2}}(B_\lambda(0))}
- C \, \alpha.\]
Hence, if $\alpha$ is sufficiently small, then we obtain
$\lambda^{\frac{n-2}{2}} \, \sup_{|x| \leq \lambda} v(x) \geq c$. \\
\begin{proposition}
Let $25 \leq n \leq 51$. Then there exists a smooth metric $g$ on
$\mathbb{R}^n$ with the following properties:
\begin{itemize}
\item[(i)] $g_{ik}(x) = \delta_{ik}$ for $|x| \geq \frac{1}{2}$
\item[(ii)] $g$ is not conformally flat
\item[(iii)] There exists a sequence of non-negative smooth functions
$v_\nu$ ($\nu \in \mathbb{N}$) such that
\[\Delta_g v_\nu - \frac{n-2}{4(n-1)} \, R_g \, v_\nu + n(n-2) \,
v_\nu^{\frac{n+2}{n-2}} = 0\]
for all $\nu \in \mathbb{N}$,
\[\int_{\mathbb{R}^n} v_\nu^{\frac{2n}{n-2}} < \Big (
\frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}\]
for all $\nu \in \mathbb{N}$, and $\sup_{|x| \leq 1} v_\nu(x) \to
\infty$ as $\nu \to \infty$.
\end{itemize}
\end{proposition}
\textbf{Proof.}
Choose a smooth cutoff function $\eta: \mathbb{R} \to \mathbb{R}$ such
that $\eta(t) = 1$ for $t \leq 1$ and $\eta(t) = 0$ for $t \geq 2$. We
define a trace-free symmetric two-tensor on $\mathbb{R}^n$ by
\[h_{ik}(x) = \sum_{N=N_0}^\infty \eta(4N^2 \, |x - y_N|) \, 2^{-4N} \,
f(2^{N} \, |x - y_N|^2) \, H_{ik}(x - y_N),\]
where $y_N = (\frac{1}{N},0,\hdots,0) \in \mathbb{R}^n$. It is
straightforward to verify that $h(x)$ is $C^\infty$ smooth.
Moreover, if $N_0$ is sufficiently large, then we have $h(x) = 0$ for
$|x| \geq \frac{1}{2}$ and $|h(x)| + |\partial h(x)| + |\partial^2 h(x)|
\leq \alpha$ for all $x \in \mathbb{R}^n$.
(Here, $\alpha$ is the constant that appears in Proposition
\ref{perturbation.argument}.) We now define a Riemannian metric $g$ by
$g(x) = \exp(h(x))$. The assertion is then a consequence of Proposition
\ref{perturbation.argument}. \\
| {'timestamp': '2009-05-23T20:39:20', 'yymm': '0905', 'arxiv_id': '0905.3841', 'language': 'en', 'url': 'https://arxiv.org/abs/0905.3841'} |
\section{Science objective of the Cherenkov telescope}
\label{sec:CTScience}
The CT is a novel instrument and no previous version was flown or even built before. Therefore, the main objective for the telescope is to raise the technical readiness level and to understand the occurring backgrounds in such an instrument before utilizing it in future space missions such as POEMMA. At any given time, the science goals of the CT depend on its pointing direction.
If the instrument is pointed below the limb, there are two objectives that EUSO-SPB2 aims to achieve: i) in principle, detect upwards going EAS sourced from neutrinos from transient astrophysical events (see \ref{subsec:ToO}) and ii) measure optical background signals for the observation of Earth-skimming neutrino events.\\
The background observations are particularly crucial for future missions as currently no measurements in this wavelength range or with this time resolution have been conducted from suborbital space. The night sky background has a significant impact on the detection threshold and the subsequent event reconstruction, which indicates that a detailed measurement of its variation over time and telescope pointing direction over the spectral response of the SiPMs is crucial for the event rate estimation of future experiments. In addition, EUSO-SPB2 will be able to identify known and unknown background sources that could influence the detection of the signal of upwards going EASs for example atmospheric events. Finally, the impact of direct hits on the focal surface cased by charged particles will be studied as they produce the same signature as Cherenkov light from upwards going EASs. EUSO-SPB2 will use the bi-focal technique to identify such events correctly.
\subsection{Target of Opportunity}
\label{subsec:ToO}
The sensitivity of EUSO-SPB2 to the diffuse cosmogenic neutrino flux was investigated but due to the limited field of view and relatively short mission time, EUSO-SPB2 is not sensitive to the diffuse neutrino flux even assuming the most optimistic UHECR source evolution (corresponding to the cosmological evolution of Active Galactic Nuclei), which is already excluded by the IceCube and Auger experiments. For more details how the sensitivity was estimated, see \cite{NeutrinoSensitivity}. One possibility for EUSO-SPB2 to still observe neutrinos is to point the instrument in the direction of a transient astrophysical event (e.g. binary neutron star mergers after receiving an alert). The EUSO-SPB2 sensitivity to such events, called "target of opportunity" (ToO) events, was investigated in detail in \cite{ToO_poemma}.\\
The ToO events can be divided in two classes based on their characteristic time duration. Short bursts are defined as events which last around 1000~s while long bursts can last from days to even weeks. For the latter, the neutrino flux is averaged over the duration of the event while for the former, it is assumed that the instrument is positioned at the ideal location for the source at the beginning of the event and the flux is integrated over the duration of the event.\\
The left panel of fig. \ref{fig:ToOSensitivity} shows EUSO-SPB2's sensitivity for Binary Neutron star mergers \cite{Fang_BNNMerger} scaled for a distance of 0.8~Mpc and 3~Mpc. Possible sources that may correspond to these distances could be located in M31, and NGC 253 and M82 (star burst) respectively. The red line shows a flight duration of 30 days while the black line is for 100 days. The right panel shows the sensitivity for emissions from short gamma-ray burst assuming a moderate emission as discussed in \cite{KMMK_sGRB}. The distances chosen are 3~Mpc and 40~Mpc which correspond to the distances of GW170817 and GRB170817A respectively.
\begin{figure}
\centering
\includegraphics[width=.9\textwidth]{fig/SPB2_ToO_Sen.pdf}
\caption{The EUSO-SPB2 all-flavor 90\% unified confidence level sensitivity to ToO neutrino flux per decade in energy. In blue the all-flavor fluency for different models, \cite{Fang_BNNMerger} for the long and \cite{KMMK_sGRB} for the short scenario. For the long duration events a 30 and 100 days flight is considered (red and black line respectively).}
\label{fig:ToOSensitivity}
\end{figure}
The sensitivities shown in fig. \ref{fig:ToOSensitivity} are not taking the balloon trajectory into account. For the long scenario the effect of the sun and moon is considered including a longer twilight period due to the balloon altitude. The short scenario does not contain these restrains. Further, although EUSO-SPB2 is sensitive to neutrinos produced in these events, the short mission duration of up to 100 days makes the probability of such an event occurring in close enough range during operations unlikely.
\subsection{Above-the-limb Extensive Air Showers}
\label{subsec:AboveLimb}
When the instrument is pointed above the limb, EUSO-SPB2 will be able to, for the first time, measure Cherenkov emission from EAS from suborbital space. The trajectory of the above-the-limb cosmic ray events is shown in panel A of fig. \ref{fig:AboveLimb}. The trajectories of these events provide enough atmosphere for a primary cosmic ray to produce an EAS which is detectable via its Cherenkov emission as long as the primary energy is larger than a few PeV. This is due to the highly forward beam nature of Cherenkov radiation. As the shower mostly develops in rarefied atmosphere, the atmospheric attenuation is minimized but the Cherenkov emission is also limited due to increased thresholds. Details of how to accurately simulate the Cherenkov signal of such showers is discussed in \cite{aboveLimb}.
\begin{figure}[ht]
\centering
\includegraphics[width=.9\textwidth]{fig/aboveLimb.png}
\caption{A) Schematic trajectory of upwards going EAS initiated by cosmic rays, observable at the detector via Cherenkov emission of the shower B) The cumulative, energy dependent, expected event rate of above-the-limb cosmic rays, assuming a detection threshold of 10~photons~m$^{-2}$ns$^{-1}$ (See \cite{aboveLimb} for details). The cosmic ray fluxes are taken from \cite{Ralf_book}.}
\label{fig:AboveLimb}
\end{figure}
The resulting event rate of the above-the-limb cosmic rays for EUSO-SPB2 is displayed in panel B of fig. \ref{fig:AboveLimb}. The cumulative event rate is more than 100 events per hour for primary energies above 1~PeV. Such a high rate makes it possible to collect a statistically relevant dataset in a relatively short observation time, thereby providing a guaranteed signal which has similar properties to those signals from air showers sourced from Earth-skimming neutrinos. Frequent observation of such events will help evaluate and refine the detection and reconstruction techniques of the experiment.
\section{Conclusion}
EUSO-SPB2 is the follow up project of EUSO-SPB1 and represents a pathfinder for the POEMMA mission and, as such, is equipped with two telescopes. One telescope is optimized for the fluorescence detection of UHECRs while the other is optimized to measure the Cherenkov emission from EAS sourced from either Earth skimming neutrinos or above-the-limb cosmic rays. EUSO-SPB2 is scheduled to fly as a NASA SPB payload from Wanaka, NZ in early 2023.
EUSO-SPB2 will measure, for the first time, EAS from above using the fluorescence technique. Extensive simulation studies have shown that the expected event rate is 0.12 UHECR events per hour with the FT and the possibility to search for ANITA event-like candidates are still under study.
For the first time, a Cherenkov Telescope based on SiPMs will be flown on a stratospheric balloon raising the technological readiness level. EUSO-SPB2 will study the background signals encountered while looking for signals from upward going EASs sourced from below the limb. Although studies have revealed that EUSO-SPB2 has very limited sensitivity towards the diffuse cosmogenic neutrino flux, being significantly less competitive than existing ground based experiments, the instrument still has the capability to detect neutrinos emitted by astrophysical events if they happen within our cosmic neighbourhood. Pointing the CT above the limb allows to record more than 100 events per hour of above-the-limb direct cosmic rays. As this signal is similar to that from potential neutrino induced showers, these events allow to evaluate the detection technique.
\footnotesize{\textit{Acknowledgment:}
This work was partially supported by Basic Science Interdisciplinary Research Projects of
RIKEN and JSPS KAKENHI Grant (22340063, 23340081, and 24244042), by
the Italian Ministry of Foreign Affairs and International Cooperation,
by the Italian Space Agency through the ASI INFN agreements n. 2017-8-H.0 and n. 2021-8-HH.0,
by NASA award 11-APRA-0058, 16-APROBES16-0023, 17-APRA17-0066, NNX17AJ82G, NNX13AH54G, 80NSSC18K0246, 80NSSC18K0473, 80NSSC19K0626, and 80NSSC18K0464 in the USA,
by the French space agency CNES,
by the Deutsches Zentrum f\"ur Luft- und Raumfahrt,
the Helmholtz Alliance for Astroparticle Physics funded by the Initiative and Networking Fund
of the Helmholtz Association (Germany),
by Slovak Academy of Sciences MVTS JEM-EUSO, by National Science Centre in Poland grants 2017/27/B/ST9/02162 and
2020/37/B/ST9/01821,
by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2094-390783311,
by Mexican funding agencies PAPIIT-UNAM, CONACyT and the Mexican Space Agency (AEM),
as well as VEGA grant agency project 2/0132/17, by the MEYS project CZ.02.1.01/0.0/0.0/17\_049/0008422, and by State Space Corporation ROSCOSMOS and the Interdisciplinary Scientific and Educational School of Moscow University "Fundamental and Applied Space Research".
}
\section{Science Objective of the Fluorescence Telescope}
\label{sec:FTscience}
The primary scientific goal of the FT is to achieve the first measurement of an EAS induced by a primary cosmic ray via the fluorescence technique from suborbital space as well as the accurate reconstruction of the properties of such an event (direction and energy). Such a measurement would prove the feasibility of measuring UHECR from space using fluorescence emission--a technique proposed also for the POEMMA mission. To evaluate the capability of the FT to obtain this goal, extensive simulation work was performed. A detailed discussion of the FT performance can be found in \cite{FT-performance}.
For these simulations, CONEX \cite{CONEX} is used to simulate the shower profiles using EPOS-LHC \cite{EPOSLHC} assuming proton primaries. A total of 1.6 million showers were isotropically simulated landing on a disk with \unit[100]{km} radius. The center of the disk is the detector location projected on ground. The 100~km disk is needed to ensure that all shower geometries that deposit light inside the field of view are taken into account. The shower azimuth angle is uniformly distributed between $0^{\circ}$ and $360^{\circ}$ while the zenith angle is distributed flat in $\mbox{sin}\theta\mbox{cos}\theta$ ($0^{\circ}$-80$^\circ$). The energy of the simulated showers is sampled between $10^{17.8}$~eV and $10^{19.7}$~eV in 20 bins evenly spaced in $\mathrm{log}_{10}(E/\mathrm{eV})$. The background used in the simulation is estimated using the measurements from previous experiments appropriately scaled to the new detector characteristics. The Pierre Auger Observatory \cite{Auger_energy} energy spectrum is used to convert a trigger rate to a hourly event rate as shown in fig. \ref{fig:FTEventRate}.
\begin{figure}[h]
\centering
\includegraphics[width=.6\textwidth]{fig/EventRate_jeser.pdf}
\caption{Expected event rate for the FT (See \cite{FT-performance} for more details).}
\label{fig:FTEventRate}
\end{figure}
Integrating over the entire energy span, we expect to record $0.12\pm0.01$ events per observation hour which equates to roughly 0.6 events per night. Machine learning techniques will be used on board to identify these events and download them with the highest priority. This is necessary as a recovery of the whole data set is not guaranteed for a stratospheric balloon flight. More information is provided in \cite{FT-performance}.
Another interesting science objective for the FT concerns the search for the candidate upwards going events as reported by the Antarctic Impulse Transient Antenna (ANITA) \cite{ANITA_upwards}. The feasibility of such a search is still under investigation but preliminary simulations look promising, mostly due to the apparent strong intensity of those events.
\section{The EUSO-SPB2 Instrument and Mission}
\label{sec:MissionInstrument}
To achieve the science goals outlined earlier, EUSO-SPB2 is equipped with two telescopes: one optimized for the fluorescence detection technique (nadir pointing), and a second one optimized for the Cherenkov detection technique (pointing near the Earth limb).
The \textit{Fluorescence Telescope (FT)} camera has a modular design following the layout of previous instruments, particularly EUSO-SPB1, which flew in 2017. While EUSO-SPB1 was equipped with only one Photo Detection Modules (PDM), each consisting of 2304 individual pixels capable of single photo electron counting with a double pulse resolution of 6~ns, EUSO-SPB2 will fly 3 PDMs. Also the integration time was shortened from \unit[2.5]{$\mu$s} to \unit[1]{$\mu$s} in EUSO-SPB2 to increase the signal to noise ratio. The optical system used in EUSO-SPB1 was a 2 Fresnel lens system. To increase the light collection onto the focal surface EUSO-SPB2 design was switched to a Schmidt system consisting of 6 mirror segments and a 1~m diameter Schmidt corrector plate at the aperture providing a field of view of roughly 12$^\circ$ by 36$^\circ$. A detailed description of the FT is given in \cite{SPB2-FT}.
The \textit{Cherenkov Telescope (CT)} is a brand new instrument. To accommodate the very fast signals of the Cherenkov emission, the camera is composed of 512 Silicon Photomultipliers (SiPMs) pixels with a 10ns integration time and a spectral range between 400~nm and 800~nm.
The optical system of the CT is a Schmidt system similar to the FT but with only 4 mirror segments. The field of view of the CT is 6.4$^\circ$ in zenith and 12.8 $^\circ$ in azimuth. The mirror segments are aligned in such a way that a parallel light pulse from outside the telescope produces two spots in the camera. This bi-focal alignment allows to distinguish triggers of interest from single spot events caused by low energy cosmic rays striking the SiPM camera directly. A detailed description of the CT is given in \cite{SPB2-CT}.
EUSO-SPB2 is designed as a payload of a NASA Super Pressure Balloon, allowing a mission duration of up to 100 days at a nominal altitude of 33~km. EUSO-SPB2's planned launch is in March/April of 2023 from Wanaka, New Zealand. This launch window and location is chosen to allow the balloon to travel easterly around the globe following a stratospheric air circulation that develops twice a year in these latitudes ($\sim 45^{\circ}$S). The azimuth rotator will allow to point the solar panels during the day towards the sun for maximum battery charging time for the night-time data taking. The average observation time per night is around 5 hours over the entire flight. While the FT points nadir during the entire flight for the observation of EAS via the fluorescence technique, the CT has a pointing mechanism allowing to aim the telescope both above and below the Earth limb. Above the limb, the CT will record direct Cherenkov light from PeV cosmic rays early in the flight to verify the functionality of the instrument. Below the limb, the CT will search for signals corresponding to the upward $\tau$-induced EAS. In case that an astrophysical event alert is issued during the mission, the zenith and azimuth pointing capability of the CT will allow for a follow up search (see \ref{subsec:ToO}).
\section{Introduction}
With a record energy of more than 100~EeV, Ultra High Energy Cosmic Rays (UHECRs) are the most energetic particles known to exist. Although ground-based observatories have observed these energetic particles for decades, their source and acceleration mechanisms remain largely unknown due to their extremely low flux at Earth's surface (see \cite{UHECRoverview} and references therein).
Very High Energy (VHE) neutrinos (E>10~PeV) can also help to explain the most energetic processes in the universe as well as the evolution of astrophysical sources. Few of these VHE neutrinos have been detected so far due to their minuscule interaction cross sections, requiring very large amounts of target material to allow for observation.
A space-based experiment such as the proposed Probe for Multi Messenger Astrophysics (POEMMA) \cite{POEMMA-JCAP} could overcome the shortcomings of ground-based observation (by observing the Earth and its atmosphere from above, allowing for gargantuan increase in acceptance at the highest energies), thereby representing the next frontier in UHECR and VHE neutrino physics. Before such a detector can be built and launched, it is highly advantageous to develop pathfinder missions to raise the technological readiness and to verify the targeted detection techniques. A stratospheric balloon allows for such an investigation in a near space environment without the risk and cost of a fully realized space mission.
The Extreme Universe Space Observatory on a Super Pressure Balloon 2 (EUSO-SPB2) is the third (and most advanced) balloon mission undertaken by the EUSO collaboration and will build on the experiences of previous missions \cite{EUSOBalloon, EUSOSPB1}. The timeline of the evolution of the EUSO ballooning missions is shown in Fig. \ref{fig:EUSOTimeline}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.75\textwidth]{fig/timeline.png}
\caption{Evolution of the EUSO ballooning program towards a space based mission, starting from the first proof of concept of the technology with EUSO-Balloon in 2014 and updated payload in EUSO-SPB1 in 2017. EUSO-SPB2 will launch in 2023 and is the next step towards a space based mission like K-EUSO or POEMMA.}
\label{fig:EUSOTimeline}
\end{figure}
EUSO-SPB2 has three main scientific objectives:
\begin{enumerate}
\item Observing the first extensive air showers via the fluorescence technique from suborbital space.
\item Observing Cherenkov light from upwards going extensive air showers initiated by cosmic rays.
\item Measuring the background conditions for the detection of neutrino induced upwards going air showers.
\item Searching for neutrinos from astrophysical transient events (e.g. binary neutron star mergers)
\end{enumerate}
The mission and the instrument will be detailed in section \ref{sec:MissionInstrument} before we discuss the different science objectives for the two telescopes in sections \ref{sec:FTscience} and \ref{sec:CTScience}, respectively.
\section*{Full Authors List: \Coll\ Collaboration}
\begin{sloppypar}
\scriptsize
\noindent
G.~Abdellaoui$^{ah}$,
S.~Abe$^{fq}$,
J.H.~Adams Jr.$^{pd}$,
D.~Allard$^{cb}$,
G.~Alonso$^{md}$,
L.~Anchordoqui$^{pe}$,
A.~Anzalone$^{eh,ed}$,
E.~Arnone$^{ek,el}$,
K.~Asano$^{fe}$,
R.~Attallah$^{ac}$,
H.~Attoui$^{aa}$,
M.~Ave~Pernas$^{mc}$,
M.~Bagheri$^{ph}$,
J.~Bal\'az{$^{la}$,
M.~Bakiri$^{aa}$,
D.~Barghini$^{el,ek}$,
S.~Bartocci$^{ei,ej}$,
M.~Battisti$^{ek,el}$,
J.~Bayer$^{dd}$,
B.~Beldjilali$^{ah}$,
T.~Belenguer$^{mb}$,
N.~Belkhalfa$^{aa}$,
R.~Bellotti$^{ea,eb}$,
A.A.~Belov$^{kb}$,
K.~Benmessai$^{aa}$,
M.~Bertaina$^{ek,el}$,
P.F.~Bertone$^{pf}$,
P.L.~Biermann$^{db}$,
F.~Bisconti$^{el,ek}$,
C.~Blaksley$^{ft}$,
N.~Blanc$^{oa}$,
S.~Blin-Bondil$^{ca,cb}$,
P.~Bobik$^{la}$,
M.~Bogomilov$^{ba}$,
E.~Bozzo$^{ob}$,
S.~Briz$^{pb}$,
A.~Bruno$^{eh,ed}$,
K.S.~Caballero$^{hd}$,
F.~Cafagna$^{ea}$,
G.~Cambi\'e$^{ei,ej}$,
D.~Campana$^{ef}$,
J-N.~Capdevielle$^{cb}$,
F.~Capel$^{de}$,
A.~Caramete$^{ja}$,
L.~Caramete$^{ja}$,
P.~Carlson$^{na}$,
R.~Caruso$^{ec,ed}$,
M.~Casolino$^{ft,ei}$,
C.~Cassardo$^{ek,el}$,
A.~Castellina$^{ek,em}$,
O.~Catalano$^{eh,ed}$,
A.~Cellino$^{ek,em}$,
K.~\v{C}ern\'{y}$^{bb}$,
M.~Chikawa$^{fc}$,
G.~Chiritoi$^{ja}$,
M.J.~Christl$^{pf}$,
R.~Colalillo$^{ef,eg}$,
L.~Conti$^{en,ei}$,
G.~Cotto$^{ek,el}$,
H.J.~Crawford$^{pa}$,
R.~Cremonini$^{el}$,
A.~Creusot$^{cb}$,
A.~de Castro G\'onzalez$^{pb}$,
C.~de la Taille$^{ca}$,
L.~del Peral$^{mc}$,
A.~Diaz Damian$^{cc}$,
R.~Diesing$^{pb}$,
P.~Dinaucourt$^{ca}$,
A.~Djakonow$^{ia}$,
T.~Djemil$^{ac}$,
A.~Ebersoldt$^{db}$,
T.~Ebisuzaki$^{ft}$,
L.~Eliasson$^{na}$,
J.~Eser$^{pb}$,
F.~Fenu$^{ek,el}$,
S.~Fern\'andez-Gonz\'alez$^{ma}$,
S.~Ferrarese$^{ek,el}$,
G.~Filippatos$^{pc}$,
W.I.~Finch$^{pc}$
C.~Fornaro$^{en,ei}$,
M.~Fouka$^{ab}$,
A.~Franceschi$^{ee}$,
S.~Franchini$^{md}$,
C.~Fuglesang$^{na}$,
T.~Fujii$^{fg}$,
M.~Fukushima$^{fe}$,
P.~Galeotti$^{ek,el}$,
E.~Garc\'ia-Ortega$^{ma}$,
D.~Gardiol$^{ek,em}$,
G.K.~Garipov$^{kb}$,
E.~Gasc\'on$^{ma}$,
E.~Gazda$^{ph}$,
J.~Genci$^{lb}$,
A.~Golzio$^{ek,el}$,
C.~Gonz\'alez~Alvarado$^{mb}$,
P.~Gorodetzky$^{ft}$,
A.~Green$^{pc}$,
F.~Guarino$^{ef,eg}$,
C.~Gu\'epin$^{pl}$,
A.~Guzm\'an$^{dd}$,
Y.~Hachisu$^{ft}$,
A.~Haungs$^{db}$,
J.~Hern\'andez Carretero$^{mc}$,
L.~Hulett$^{pc}$,
D.~Ikeda$^{fe}$,
N.~Inoue$^{fn}$,
S.~Inoue$^{ft}$,
F.~Isgr\`o$^{ef,eg}$,
Y.~Itow$^{fk}$,
T.~Jammer$^{dc}$,
S.~Jeong$^{gb}$,
E.~Joven$^{me}$,
E.G.~Judd$^{pa}$,
J.~Jochum$^{dc}$,
F.~Kajino$^{ff}$,
T.~Kajino$^{fi}$,
S.~Kalli$^{af}$,
I.~Kaneko$^{ft}$,
Y.~Karadzhov$^{ba}$,
M.~Kasztelan$^{ia}$,
K.~Katahira$^{ft}$,
K.~Kawai$^{ft}$,
Y.~Kawasaki$^{ft}$,
A.~Kedadra$^{aa}$,
H.~Khales$^{aa}$,
B.A.~Khrenov$^{kb}$,
Jeong-Sook~Kim$^{ga}$,
Soon-Wook~Kim$^{ga}$,
M.~Kleifges$^{db}$,
P.A.~Klimov$^{kb}$,
D.~Kolev$^{ba}$,
I.~Kreykenbohm$^{da}$,
J.F.~Krizmanic$^{pf,pk}$,
K.~Kr\'olik$^{ia}$,
V.~Kungel$^{pc}$,
Y.~Kurihara$^{fs}$,
A.~Kusenko$^{fr,pe}$,
E.~Kuznetsov$^{pd}$,
H.~Lahmar$^{aa}$,
F.~Lakhdari$^{ag}$,
J.~Licandro$^{me}$,
L.~L\'opez~Campano$^{ma}$,
F.~L\'opez~Mart\'inez$^{pb}$,
S.~Mackovjak$^{la}$,
M.~Mahdi$^{aa}$,
D.~Mand\'{a}t$^{bc}$,
M.~Manfrin$^{ek,el}$,
L.~Marcelli$^{ei}$,
J.L.~Marcos$^{ma}$,
W.~Marsza{\l}$^{ia}$,
Y.~Mart\'in$^{me}$,
O.~Martinez$^{hc}$,
K.~Mase$^{fa}$,
R.~Matev$^{ba}$,
J.N.~Matthews$^{pg}$,
N.~Mebarki$^{ad}$,
G.~Medina-Tanco$^{ha}$,
A.~Menshikov$^{db}$,
A.~Merino$^{ma}$,
M.~Mese$^{ef,eg}$,
J.~Meseguer$^{md}$,
S.S.~Meyer$^{pb}$,
J.~Mimouni$^{ad}$,
H.~Miyamoto$^{ek,el}$,
Y.~Mizumoto$^{fi}$,
A.~Monaco$^{ea,eb}$,
J.A.~Morales de los R\'ios$^{mc}$,
M.~Mastafa$^{pd}$,
S.~Nagataki$^{ft}$,
S.~Naitamor$^{ab}$,
T.~Napolitano$^{ee}$,
J.~M.~Nachtman$^{pi}$
A.~Neronov$^{ob,cb}$,
K.~Nomoto$^{fr}$,
T.~Nonaka$^{fe}$,
T.~Ogawa$^{ft}$,
S.~Ogio$^{fl}$,
H.~Ohmori$^{ft}$,
A.V.~Olinto$^{pb}$,
Y.~Onel$^{pi}$
G.~Osteria$^{ef}$,
A.N.~Otte$^{ph}$,
A.~Pagliaro$^{eh,ed}$,
W.~Painter$^{db}$,
M.I.~Panasyuk$^{kb}$,
B.~Panico$^{ef}$,
E.~Parizot$^{cb}$,
I.H.~Park$^{gb}$,
B.~Pastircak$^{la}$,
T.~Paul$^{pe}$,
M.~Pech$^{bb}$,
I.~P\'erez-Grande$^{md}$,
F.~Perfetto$^{ef}$,
T.~Peter$^{oc}$,
P.~Picozza$^{ei,ej,ft}$,
S.~Pindado$^{md}$,
L.W.~Piotrowski$^{ib}$,
S.~Piraino$^{dd}$,
Z.~Plebaniak$^{ek,el,ia}$,
A.~Pollini$^{oa}$,
E.M.~Popescu$^{ja}$,
R.~Prevete$^{ef,eg}$,
G.~Pr\'ev\^ot$^{cb}$,
H.~Prieto$^{mc}$,
M.~Przybylak$^{ia}$,
G.~Puehlhofer$^{dd}$,
M.~Putis$^{la}$,
P.~Reardon$^{pd}$,
M.H..~Reno$^{pi}$,
M.~Reyes$^{me}$,
M.~Ricci$^{ee}$,
M.D.~Rodr\'iguez~Fr\'ias$^{mc}$,
O.F.~Romero~Matamala$^{ph}$,
F.~Ronga$^{ee}$,
M.D.~Sabau$^{mb}$,
G.~Sacc\'a$^{ec,ed}$,
G.~S\'aez~Cano$^{mc}$,
H.~Sagawa$^{fe}$,
Z.~Sahnoune$^{ab}$,
A.~Saito$^{fg}$,
N.~Sakaki$^{ft}$,
H.~Salazar$^{hc}$,
J.C.~Sanchez~Balanzar$^{ha}$,
J.L.~S\'anchez$^{ma}$,
A.~Santangelo$^{dd}$,
A.~Sanz-Andr\'es$^{md}$,
M.~Sanz~Palomino$^{mb}$,
O.A.~Saprykin$^{kc}$,
F.~Sarazin$^{pc}$,
M.~Sato$^{fo}$,
A.~Scagliola$^{ea,eb}$,
T.~Schanz$^{dd}$,
H.~Schieler$^{db}$,
P.~Schov\'{a}nek$^{bc}$,
V.~Scotti$^{ef,eg}$,
M.~Serra$^{me}$,
S.A.~Sharakin$^{kb}$,
H.M.~Shimizu$^{fj}$,
K.~Shinozaki$^{ia}$,
T.~Shirahama$^{fn}$,
J.F.~Soriano$^{pe}$,
A.~Sotgiu$^{ei,ej}$,
I.~Stan$^{ja}$,
I.~Strharsk\'y$^{la}$,
N.~Sugiyama$^{fj}$,
D.~Supanitsky$^{ha}$,
M.~Suzuki$^{fm}$,
J.~Szabelski$^{ia}$,
N.~Tajima$^{ft}$,
T.~Tajima$^{ft}$,
Y.~Takahashi$^{fo}$,
M.~Takeda$^{fe}$,
Y.~Takizawa$^{ft}$,
M.C.~Talai$^{ac}$,
Y.~Tameda$^{fu}$,
C.~Tenzer$^{dd}$,
S.B.~Thomas$^{pg}$,
O.~Tibolla$^{he}$,
L.G.~Tkachev$^{ka}$,
T.~Tomida$^{fh}$,
N.~Tone$^{ft}$,
S.~Toscano$^{ob}$,
M.~Tra\"{i}che$^{aa}$,
Y.~Tsunesada$^{fl}$,
K.~Tsuno$^{ft}$,
S.~Turriziani$^{ft}$,
Y.~Uchihori$^{fb}$,
O.~Vaduvescu$^{me}$,
J.F.~Vald\'es-Galicia$^{ha}$,
P.~Vallania$^{ek,em}$,
L.~Valore$^{ef,eg}$,
G.~Vankova-Kirilova$^{ba}$,
T.~M.~Venters$^{pj}$,
C.~Vigorito$^{ek,el}$,
L.~Villase\~{n}or$^{hb}$,
B.~Vlcek$^{mc}$,
P.~von Ballmoos$^{cc}$,
M.~Vrabel$^{lb}$,
S.~Wada$^{ft}$,
J.~Watanabe$^{fi}$,
J.~Watts~Jr.$^{pd}$,
R.~Weigand Mu\~{n}oz$^{ma}$,
A.~Weindl$^{db}$,
L.~Wiencke$^{pc}$,
M.~Wille$^{da}$,
J.~Wilms$^{da}$,
D.~Winn$^{pm}$
T.~Yamamoto$^{ff}$,
J.~Yang$^{gb}$,
H.~Yano$^{fm}$,
I.V.~Yashin$^{kb}$,
D.~Yonetoku$^{fd}$,
S.~Yoshida$^{fa}$,
R.~Young$^{pf}$,
I.S~Zgura$^{ja}$,
M.Yu.~Zotov$^{kb}$,
A.~Zuccaro~Marchi$^{ft}$
}
\end{sloppypar}
{ \footnotesize
\noindent
$^{aa}$ Centre for Development of Advanced Technologies (CDTA), Algiers, Algeria \\
$^{ab}$ Dep. Astronomy, Centre Res. Astronomy, Astrophysics and Geophysics (CRAAG), Algiers, Algeria \\
$^{ac}$ LPR at Dept. of Physics, Faculty of Sciences, University Badji Mokhtar, Annaba, Algeria \\
$^{ad}$ Lab. of Math. and Sub-Atomic Phys. (LPMPS), Univ. Constantine I, Constantine, Algeria \\
$^{af}$ Department of Physics, Faculty of Sciences, University of M'sila, M'sila, Algeria \\
$^{ag}$ Research Unit on Optics and Photonics, UROP-CDTA, S\'etif, Algeria \\
$^{ah}$ Telecom Lab., Faculty of Technology, University Abou Bekr Belkaid, Tlemcen, Algeria \\
$^{ba}$ St. Kliment Ohridski University of Sofia, Bulgaria\\
$^{bb}$ Joint Laboratory of Optics, Faculty of Science, Palack\'{y} University, Olomouc, Czech Republic\\
$^{bc}$ Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic\\
$^{ca}$ Omega, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France\\
$^{cb}$ Universit de Paris, CNRS, AstroParticule et Cosmologie, F-75013 Paris, France\\
$^{cc}$ IRAP, Universit\'e de Toulouse, CNRS, Toulouse, France\\
$^{da}$ ECAP, University of Erlangen-Nuremberg, Germany\\
$^{db}$ Karlsruhe Institute of Technology (KIT), Germany\\
$^{dc}$ Experimental Physics Institute, Kepler Center, University of T\"ubingen, Germany\\
$^{dd}$ Institute for Astronomy and Astrophysics, Kepler Center, University of T\"ubingen, Germany\\
$^{de}$ Technical University of Munich, Munich, Germany\\
$^{ea}$ Istituto Nazionale di Fisica Nucleare - Sezione di Bari, Italy\\
$^{eb}$ Universita' degli Studi di Bari Aldo Moro and INFN - Sezione di Bari, Italy\\
$^{ec}$ Dipartimento di Fisica e Astronomia "Ettore Majorana", Universit di Catania, Italy\\
$^{ed}$ Istituto Nazionale di Fisica Nucleare - Sezione di Catania, Italy\\
$^{ee}$ Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Italy\\
$^{ef}$ Istituto Nazionale di Fisica Nucleare - Sezione di Napoli, Italy\\
$^{eg}$ UniversitaÕ di Napoli Federico II - Dipartimento di Fisica "Ettore Pancini", Italy\\
$^{eh}$ INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo, Italy\\
$^{ei}$ Istituto Nazionale di Fisica Nucleare - Sezione di Roma Tor Vergata, Italy\\
$^{ej}$ Universita' di Roma Tor Vergata - Dipartimento di Fisica, Roma, Italy\\
$^{ek}$ Istituto Nazionale di Fisica Nucleare - Sezione di Torino, Italy\\
$^{el}$ Dipartimento di Fisica, Universita' di Torino, Italy\\
$^{em}$ Osservatorio Astrofisico di Torino, Istituto Nazionale di Astrofisica, Italy\\
$^{en}$ Uninettuno University, Rome, Italy\\
$^{fa}$ Chiba University, Chiba, Japan\\
$^{fb}$ National Institutes for Quantum and Radiological Science and Technology (QST), Chiba, Japan\\
$^{fc}$ Kindai University, Higashi-Osaka, Japan\\
$^{fd}$ Kanazawa University, Kanazawa, Japan\\
$^{fe}$ Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Japan\\
$^{ff}$ Konan University, Kobe, Japan\\
$^{fg}$ Kyoto University, Kyoto, Japan\\
$^{fh}$ Shinshu University, Nagano, Japan \\
$^{fi}$ National Astronomical Observatory, Mitaka, Japan\\
$^{fj}$ Nagoya University, Nagoya, Japan\\
$^{fk}$ Institute for Space-Earth Environmental Research, Nagoya University, Nagoya, Japan\\
$^{fl}$ Graduate School of Science, Osaka City University, Japan\\
$^{fm}$ Institute of Space and Astronautical Science/JAXA, Sagamihara, Japan\\
$^{fn}$ Saitama University, Saitama, Japan\\
$^{fo}$ Hokkaido University, Sapporo, Japan \\
$^{fp}$ Osaka Electro-Communication University, Neyagawa, Japan\\
$^{fq}$ Nihon University Chiyoda, Tokyo, Japan\\
$^{fr}$ University of Tokyo, Tokyo, Japan\\
$^{fs}$ High Energy Accelerator Research Organization (KEK), Tsukuba, Japan\\
$^{ft}$ RIKEN, Wako, Japan\\
$^{ga}$ Korea Astronomy and Space Science Institute (KASI), Daejeon, Republic of Korea\\
$^{gb}$ Sungkyunkwan University, Seoul, Republic of Korea\\
$^{ha}$ Universidad Nacional Aut\'onoma de M\'exico (UNAM), Mexico\\
$^{hb}$ Universidad Michoacana de San Nicolas de Hidalgo (UMSNH), Morelia, Mexico\\
$^{hc}$ Benem\'{e}rita Universidad Aut\'{o}noma de Puebla (BUAP), Mexico\\
$^{hd}$ Universidad Aut\'{o}noma de Chiapas (UNACH), Chiapas, Mexico \\
$^{he}$ Centro Mesoamericano de F\'{i}sica Te\'{o}rica (MCTP), Mexico \\
$^{ia}$ National Centre for Nuclear Research, Lodz, Poland\\
$^{ib}$ Faculty of Physics, University of Warsaw, Poland\\
$^{ja}$ Institute of Space Science ISS, Magurele, Romania\\
$^{ka}$ Joint Institute for Nuclear Research, Dubna, Russia\\
$^{kb}$ Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Russia\\
$^{kc}$ Space Regatta Consortium, Korolev, Russia\\
$^{la}$ Institute of Experimental Physics, Kosice, Slovakia\\
$^{lb}$ Technical University Kosice (TUKE), Kosice, Slovakia\\
$^{ma}$ Universidad de Le\'on (ULE), Le\'on, Spain\\
$^{mb}$ Instituto Nacional de T\'ecnica Aeroespacial (INTA), Madrid, Spain\\
$^{mc}$ Universidad de Alcal\'a (UAH), Madrid, Spain\\
$^{md}$ Universidad Polit\'ecnia de madrid (UPM), Madrid, Spain\\
$^{me}$ Instituto de Astrof\'isica de Canarias (IAC), Tenerife, Spain\\
$^{na}$ KTH Royal Institute of Technology, Stockholm, Sweden\\
$^{oa}$ Swiss Center for Electronics and Microtechnology (CSEM), Neuch\^atel, Switzerland\\
$^{ob}$ ISDC Data Centre for Astrophysics, Versoix, Switzerland\\
$^{oc}$ Institute for Atmospheric and Climate Science, ETH Z\"urich, Switzerland\\
$^{pa}$ Space Science Laboratory, University of California, Berkeley, CA, USA\\
$^{pb}$ University of Chicago, IL, USA\\
$^{pc}$ Colorado School of Mines, Golden, CO, USA\\
$^{pd}$ University of Alabama in Huntsville, Huntsville, AL; USA\\
$^{pe}$ Lehman College, City University of New York (CUNY), NY, USA\\
$^{pf}$ NASA Marshall Space Flight Center, Huntsville, AL, USA\\
$^{pg}$ University of Utah, Salt Lake City, UT, USA\\
$^{ph}$ Georgia Institute of Technology, USA\\
$^{pi}$ University of Iowa, Iowa City, IA, USA\\
$^{pj}$ NASA Goddard Space Flight Center, Greenbelt, MD, USA\\
$^{pk}$ Center for Space Science \& Technology, University of Maryland, Baltimore County, Baltimore, MD, USA\\
$^{pl}$ Department of Astronomy, University of Maryland, College Park, MD, USA\\
$^{pm}$ Fairfield University, Fairfield, CT, USA
}
\end{document}
| {'timestamp': '2021-12-17T02:05:06', 'yymm': '2112', 'arxiv_id': '2112.08509', 'language': 'en', 'url': 'https://arxiv.org/abs/2112.08509'} |
\section{Introduction}
\label{sec:introduction}
The COVID-19 outbreak \cite{who_covid_definition}, which was declared as a pandemic by the World Health Organization (WHO) on 11 March 2020 \cite{who_covid_pandemic}, changed several aspects of our everyday life both in the online and offline sphere. For instance,
the news diet of users was remarkably modified in its structure by introducing a considerable amount of information referring to a new topic. This phenomenon was accelerated by social media platforms, which are known for shaping discussions on a wide range of issues, including politics, climate change, economics, migration, and health \cite{bessi2015trend,chou2018addressing,bovet2019influence,del2017news}.
The arising of the pandemic generated an overabundant flow of information and news, whose trustworthiness may not always be guaranteed, especially online. This phenomenon, referred as infodemic \cite{zarocostas2020fight,who_infodemic} reportedly affect people's behavior \cite{Sharot2020} in a harmful way. This aspect calls for urgent investigations of the turbulent dynamics of the online infosphere, complementary to the monitoring of the spreading of infections \cite{louis2019,Viboud2802,cinelli2020infodemic}. Indeed, the current infodemic may foster the tendency of users a) to acquire information adhering to their system of beliefs \cite{bessi2015science}, b) to ignore dissenting information \cite{zollo2017debunking}, c) to form polarized groups around a shared narrative \cite{del2016echo}. Two common factors to such behaviors carried on by users are opinion polarization \cite{vicario2019polarization}, one of the dominating traits of online social dynamics, and echo chambers \cite{cinelli2021echochambers}. Divided into echo chambers, users account for the coherence with their preferred narrative rather than the actual value of the information \cite{cinelli2019misinfo, delvicario2016, conti2017s}.
Such evidence for polarization and online echo chambers seems to be related to a feedback loop between individual choices and algorithm recommendations towards like-minded contents \cite{bakshy2015exposure, cinelli2020selective, cinelli2021echochambers}.
However, other presumably harmless factors like the enforcement of content regulation
may play a role in increasing online polarization.
Indeed, it was recently observed that moderation policies and removal actions/bans of users produce adverse effects in terms of online polarization \cite{berger2016occasional, hughes2015isis, siegel2019online}. Users who got banned often consider this action as a badge of honor, rejoining the same social media under new identities or migrating to more tolerant platforms. The result could be either a reinforcement of their (extreme) opinion or reduced exposure to opposing voices.
Therefore, raising awareness about the collateral costs of content policy and other interventions is crucial for making social media a less toxic environment.
\paragraph{Contribution}
This work provides a comparative analysis between two social media platforms that differ on how content moderation is applied. We select Twitter as a representative of content-regulated social media and Gab, a social network known for its willingness to ensure free speech by using little to no content moderation \cite{zannettou_2018}, like its counterpart. Despite their differences in how content policy is applied, both platforms are characterized by a similar platform design. Users are allowed to post and interact with content, together with their ability to create connections with other users.
We perform our analysis on a timespan between $1/1/2020$ and $30/09/2020$, covering the first global wave of COVID-19. The dataset includes about three million posts and comments related to the COVID-19 topic expressed from more than one million users. We investigate consumption patterns from a user and post perspective on the two social media, assessing differences in terms of engagement. We extend this analysis by taking into account the trustworthiness of the contents published, classifying news sources accordingly to a categorization based on Media Bias/Fact Check \cite{mbfc} and NewsGuard \cite{ng}. An akin type of classification was exploited in several papers \cite{bovet2019influence, cinelli2020infodemic, cinelli2021echochambers, zollo2017debunking} bringing essential insights on the circulation of misinformation online. Therefore, we employ this dichotomy by classifying posts as \textit{Questionable} or \textit{Reliable} depending on their credibility. The same labeling was used to model the persistence of users repeatedly commenting under a post of the same outlet category.
Finally, we investigate the presence of homophily, i.e., the tendency of users to aggregate around common interests, by measuring the relationship between users and their tendency to post questionable content. We find that the content moderation imposed by Twitter promotes the existence of two echo chambers of radically different sizes. In summary, the bulk of users on Twitter seems to share and interact with verified content.
Oppositely, users on Gab show a lack of a clear preference between the two types of outlets. Questionable posts are preferred in terms of commenting persistence. However, reliable posts are more likely to be commented on as time passes. Coherently, the existence of echo chambers on Gab is not as evident as observed in the case of Twitter due to the presence of users with a relatively heterogeneous leaning.
We conclude that a valid content regulation policy produces tangible results in contrasting misinformation spreading.
\paragraph{Organization}
This work is organized as follows. Section \ref{subsec:misinfo}
describes the recent developments in the study of misinformation dynamics and their relationship with the phenomenon of polarization. Section \ref{subsec:gab} introduces the structure of Gab, providing an overview of this particular social media. Section \ref{subsec:covid_research} describes the recent advances in the study of misinformation related to COVID-19 from a social dynamics and machine learning perspective. Section \ref{sec:preliminaries} introduces the methodology and the theoretical tools applied for this study. Section \ref{sec:results} describes the results obtained from the experiments. Section \ref{sec:conclusions} provides the final remarks of this work, summarizing the results obtained and the future developments. Finally, Section \ref{sec:supporting_information} provides additional information on the results presented in the paper.
\section{Related Works}
\label{sec:related_works}
\subsection{Misinformation and polarization}
\label{subsec:misinfo}
The study of misinformation and the spreading of fake news has received increasing interest in recent years, disentangling the role of news consumption on mainstream and niche social media \cite{delvicario2016,Vosoughi2017falsenews,lazer2018fakenews,bovet2019influence,cinelli2020infodemic}. The presence of psychological mechanisms that affect the way users choose which news to consume \cite{Sharot2020} has been attributed to the effect of online polarization \cite{delvicario2016, vicario2019polarization} and, consecutively, to the creation of the so-called echo chambers \cite{jamieson2008echo, garrett2009echo, garimella2018political, garimella2017effect, cota2019quantifying, cinelli2021echochambers}.
\subsection{The role of Gab}
\label{subsec:gab}
Gab \cite{gab} is an online social platform that aroused much controversy in recent years. It describes itself as ``A social network that champions free speech, individual liberty and the free flow of information online. All are welcome"~\cite{gab}.
Such a claim, together with the political leaning of its founders and developers, made Gab a safe place for the alt-right movement, playing a central role in the organizations of actions to harm the offline world \cite{pittsburgh_gab}. The lack of content regulation within Gab helped the proliferation of hate speech and fake news. The risks associated with this content policy led to a series of suspensions by its former service provider and the ban of its application from online stores~\cite{zannettou_2018}.
Gab attracted the interest of researchers due to its permissive content regulatory policy and the political leaning of its users. In \textit{Lima et al.} ~\cite{lima2018inside}, authors analyzed the content shared on Gab and the leaning of users, finding a homogeneous environment prone to share right biased content. \textit{Zannettou et al.} ~\cite{zannettou_2018} characterized Gab in terms of user leanings and their contents, suggesting that this platform better suits for a safe place for right-wing extremists rather than an environment where free speech is protected. Moreover, a topological analysis performed by \textit{Cinelli et al.} ~\cite{cinelli2021echochambers} reveals that Gab users form one relevant cluster biased to the right.
Overall, all these studies suggest that Gab can be considered as a homogeneous environment where biased content and misinformation may easily proliferate.
\subsection{Recent advances in COVID-19 misinformation studies}
\label{subsec:covid_research}
Research against misinformation during the COVID-19 outbreak produced a series of results to limit the spreading of harmful information.\\ \textit{Cinelli et al.}\cite{cinelli2020infodemic} analyzed posts obtained from 5 different social media platforms (Facebook, Twitter, Instagram, Gab, and Reddit), finding out that the spreading of information is mainly driven by the peculiar structure of the social media in exam that in turn shapes the interaction patterns between users.
In the field of machine learning, \textit{Elhadad et al.} \cite{Elhadad2020misinfocovid} introduced a framework that can identify, through a composed machine leaning approach, misleading health-related information based on a ground-truth dataset. Since their use case is related to COVID-19, the ground-truth dataset contained both epidemiological and textual data from organizations like WHO, UNICEF, UN and a range of fact-checking websites.\\
\textit{Sear et al.} \cite{sear2020opinionwar} provided a result that does not depend on a classification approach. Instead, they employed LDA-based algorithm to identify similar topics related to posts obtained from Facebook Pages belonging to pro-vax and anti-vax communities. Their findings describe how the anti-vax community develops a less focused debate on COVID-19 compared with the pro-vax counterpart. However, anti-vax seems to be more spread on the COVID-19 debate, with the result of being more positioned to attract new supporters than the pro-vax community.\\
In the context of Twitter, \textit{Jiang et al.} \cite{jiang2021social} proposed a machine-learning model based on BERT architecture which estimated user polarity within the U.S. debate by employing features related to language and network structures. They found that users belonging to the right-leaning are more active in the creation and spreading of news affiliated with their echo chamber if compared with their counterparts from the left-leaning.
\section{Preliminaries and Definitions}
\label{sec:preliminaries}
In this section, we present the methodology applied in this study. We start by introducing the data collection process of posts from Twitter and Gab together with its categorization. Then, we describe the theoretical tools behind the analysis of engagement patterns, homophily and survival lifetime.
\subsection{Data Collection}
\label{par:data_collection}
The collection of all posts concerning the COVID-19 was designed to capture its corresponding debate on social media, gathering posts and comments from both platforms. Therefore, as the first step of this process, we analyzed the most searched terms worldwide related to the aforementioned pandemic on Google Trends. The analysis period ranges from $1/1/2020$ to $30/09/2020$. We selected four terms based on their interest and significance over time, namely: \textit{coronavirus, corona, covid, covid19}. Those terms served as a proxy, in the form of hashtags, to retrieve posts on the two social media.
The collection of Twitter posts related to the COVID-19 pandemic relied on the existence of a public dataset \cite{chen2020tracking} covering this specific topic. It includes a collection of tweet IDs, starting from $28/01/2020$, posted by accounts with a recognized influence or including representative keywords.
To provide a categorization of the reliability of the tweets, we refined this dataset by retaining only those posts with a link, reducing the dimension to $1.1$M posts.
The same strategy was applied in the case of Gab. We queried their API to obtain posts that included at least one of the search hashtags. Due to some modifications made by the platform during the study, the API stopped providing results in chronological order since June 2020. Therefore, we started collecting all posts from the general stream until the end of the analysis period, filtering by hashtag as we previously described. This process produced a dataset of
$\sim130$K posts containing a link.
\subsection{Questionable and Reliable Sources}
\label{par:outlet_categorization}
To evaluate the reliability of information circulating on both social media, we employed a source-based approach. We built a dataset of news outlets' domains from our dataset where each domain is labeled either as \textit{Questionable} or \textit{Reliable}. The classification relied on two fact-checking organizations called MediaBias/FactCheck (MBFC, \url{https://mediabiasfactcheck.com}) and NewsGuard (NG, \url{https://www.newsguardtech.com/}).
On MBFC, each news outlet is associated with a label that refers to its political bias, namely: \textit{Right, Right-Center, Least-Biased, Left-Center, and Left}. Similarly, the website also provides a second label that expresses its reliability, categorizing outlets as \textit{Conspiracy-Pseudoscience, Pro-Science} or \textit{Questionable}. Noticeably, the \textit{Questionable} set includes a wide range of political biases, from \textit{Extreme Left }to \textit{Extreme Right}. For instance, the \textit{Right} label is associated with Fox News, the \textit{Questionable} label to Breitbart (a famous right extremist outlet), and the \textit{Pro-Science} label to \textit{Science}.
MBFC also provides a classification based on a \textit{ranking bias score} that depends on four categories: \textit{Biased Wording/Headlines, Factual/Sourcing, Story Choices,} and \textit{Political Affiliation}. Each category is rated on a $0-10$ scale, with $0$ indicating the absence of bias and $10$ indicating the presence of maximum bias. The \textit{bias outlet score} is computed as the average of the four score categories. Likewise, NG classifies news outlets into four categories based on nine journalistic criteria, each of them having a specific score whose sum ranges between $0$ and $100$. Outlets with a score of at least $60$ points are considered compliant with the basic standards of credibility and transparency. Otherwise, they are recognized as outlets that lack of credibility. A different characterization is provided for humor and platforms websites, not accounting for the categorization process.
Given the different ways of classifying information sources from the two organizations, the following heuristic was applied. On MBFC, all the outlets already classified as \textit{Questionable} or belonging to the category \textit{Conspiracy-Pseudoscience} were labeled as \textit{Questionable}. The remaining categories were labeled as \textit{Reliable}. Coherently, outlets on NG were classified based on their score, maintaining the dichotomy
provided by the website. We choose a score of 60 as threshold to consider an outlet as \textit{Reliable} (score $>60$), otherwise it is referred as \textit{Questionable} (score $\le60$).
Considering a total of $2738$ news outlets provided by the two organizations, $2701$ belonging to MBFC and $37$ to NG, we end up with $814$ outlets classified as Questionable and $1924$ outlets classified as Reliable.
\subsection{User Leaning}
\label{par:user_leaning}
To measure the extent to which a user is associated with the consumption of questionable or reliable contents, we introduce the \emph{user leaning} $q$. We define it in the range $q \in [0,1]$, where $0$ means that a user posts contents exclusively associated with reliable sources, and $1$ means that a user puts into circulation only questionable posts.
Formally, the user leaning can be defined as follows: let $\mathcal{P}$ be the set of all posts with a URL matching a domain in our dataset and $\mathcal{U}$ the set containing all the users with at least a categorized post. At each element $p_j \in \mathcal{P}$ is associated a binary value $l_j \in \{0,1\}$ based on the domain of the link contained: if the URL refers to a domain classified as questionable then $l_j=1$, otherwise $l_j=0$. Considering a user $u_i$ in a bipartite network between users and posts, then the user leaning $q_i$ of a user $u_i$ can be defined as:
\begin{equation}
q_i=\frac{1}{k_i}\sum_{j=1}^{k_i} l_j \enspace ,
\label{eq:user_leaning}
\end{equation}
where $l_j$ is the leaning score of the j-th neighbor of the user $u_i$, and $k_i$ is the number of categorized contents that the user posted.
\subsection{Comparison of power law distributions}
\label{par:statistical_tests}
Most quantities related to the activity of users on social media show a heavy tailed distribution of discrete variables. Given the discrete nature of such distributions, we could not rely on Kolmogorov-Smirnov \cite{clauset2009powerlaw} test to assess whether two distributions present significant differences between each other. Indeed, such a test assumes that distributions must be continuous, and the presence of a large number of ties in the long-tailed distributions that we want to compare may lead to the computation of biased p-values. To overcome this issue, we employed a methodology proposed in \textit{Zollo et al.}\cite{zollo2017debunking} which makes use of a Wald Test \cite{wald1943test} to assess significant differences between the scaling parameters of two long-tailed distributions.
\begin{figure*}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, scale = 0.2]{general_attention_patterns_paper_version_new.pdf}
\caption{Representation of the engagement collected on Gab (upper panel) and Twitter (bottom panel).
\textit{Left column}: frequency distribution of the interactions for posts, defined as \textit{Likes}, \textit{Reblogs} (or \textit{Retweets)} and \textit{Replies}. A like is generally considered positive feedback on a news item. A reblog indicates a desire to spread a news item to friends. A reply can have multiple features and meanings and can generate collective debate. Both social media shows a heavy-tailed distribution that allows room for large deviations, i.e., some posts go viral.
\textit{Middle column}: evolution of the cumulative number of interactions over time. The general trend shows a rapid increase during February 2020, in parallel with the spreading of the COVID-19 outbreak. The absence of replies on Twitter is due to the limitations provided by their API.
\textit{Right column:} frequency distribution of interactions received by users. Similarly to posts, the distribution is heavy-tailed, describing how users tend to collect similar values of different interactions as their number increases. }
\label{fig:overall_attention_pattern_general}
\end{figure*}
\subsection{Kaplan-Meier Estimator for lifetime analysis}
\label{par:km_definition}
Let $T \in [0, +\infty]$ be a random variable which represents the time an event takes place. The probability that a randomly selected subject lives up to time $t$ is called Survival Function $S(t) = P(T \le t)$. The estimation of this probability is provided by the Kaplan-Meier estimator \cite{kaplanmeier1958}, defined as
\begin{equation}
\hat{S}(t) = \prod_{t_i \le t}{\left(1 - \frac{ d_{i}}{n_{i}}\right)},
\label{eq:kaplan_meier_estimator}
\end{equation} where $d_i$ is the number of events that happened at time $t_i$ and $n_i$ are the numbers of subjects who survived up to time $t_i$.
In user lifetime analysis, we define as $d_i$ the number of users who have been commenting on a post up to $t_i$ days and $n_i$ the number of users who stopped commenting after $t_i$ days. Similarly, in post lifetime analysis, we define as $d_i$ the number of posts that have been receiving a comment up to $t_i$ days and $n_i$ the number of posts that stopped receiving comments at $t_i$ days. In both cases, our $N$ distinct events times $t_1, \small t_2,\small \dots,\small t_N \ge 0$ are independent, which is required by the assumptions of the Kaplan-Meier estimator.
\paragraph{Homophily Analysis in User Following Networks}
\label{par:homophily}
Results from the computation of the user leaning $q$ were generalized to measure user homophily based on his news consumption. This phenomenon was modeled from a network perspective based on the work originally proposed by \textit{Cota et al.} \cite{cota2019echo}. We considered a new network in which a user is represented as a node $i$ with a given leaning $q_i$. Each node can be connected to others through a \textit{following} relationship: if user $i$ follows user $j$ on the social media in exam, their corresponding representation in the adjacency matrix $A$ is $A_{ij} = 1$, meaning that there is a directed edge between $i$ and $j$. In case of no relationship between the two users, we have $A_{ij} = 0$. This representation allows the computation of a measure called \textit{average neighborhood leaning} that quantifies the mean leaning from all those users followed by a given one. It is defined as
\begin{equation}
q_i^N = \frac{1}{k_i^{\rightarrow}}\sum_jA_{ij}q_{j}\enspace,
\label{eq:average_neighborhood_leaning}
\end{equation}
where $k_i^{\rightarrow} = \sum_jA_{ij}$ is the out-degree of node $i$, i.e., the number of users followed by user $i$.
\begin{figure*}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, scale = 0.2]{categorized_post_consumption_pattern_paper_version_new.pdf}
\caption{
\textit{Column A-B}: categorized distribution of the number of posts
against the number of likes they received with its cumulative evolution. The distributions show some evidence about content preference on both platforms. Users on Gab show signs of better appreciation toward questionable posts, supported by the lack of clear content regulation. Twitter, oppositely, shows strong evidence towards the appreciation of reliable content, with a remarkable gap between the two categories. From a cumulative perspective, the regulation imposed by Twitter results in an increasing divergence between questionable and reliable posts, showing how the latter category is the most preferred one. The same does not apply to Gab, whose divergence seems not to increase during the analysis period.
\textit{Column C-D:} categorized distribution of the number of posts against the number of reblogs or retweets they received with its cumulative evolution. The previous considerations also apply to this kind of interaction, describing the willingness of users to inject the contents they support into the news feed of their followers.}
\label{fig:post_attention_pattern_categorized}
\end{figure*}
\section{Result and Discussion}
\label{sec:results}
This work aims at performing a comparative analysis of two social media, namely Twitter and Gab, in order to understand how news consumption and social dynamics change in presence of two radically different types of content regulation policies (more stringent in the case of Twitter, almost absent in the case of Gab). The following results will provide insights to explain this behavior from different perspectives. At first, we analyze the engagement of users with posts, which we consider as separated into two categories named questionable and reliable. Then, we quantify the commenting behavior of users and posts. Lastly, we provide a network analysis to measure the tendency of users to aggregate with like-minded peers, describing how the presence of content regulation may be correlated with the polarization towards specific narratives.
\subsection{Consumption Patterns}
\label{subsec:consumption_patterns}
We investigate how the engagement on the two social media differs in relationship with the COVID-19 topic. Fig.~\ref{fig:overall_attention_pattern_general} compares the engagement distribution for posts and users. Despite the differences in terms of scale that are attributable to the size of the platforms' user base, we observe that both frequency distributions are long-tailed. This feature provides a first evidence in the consumption of news, showing that interaction patterns are similar regardless of the content moderation imposed.
Next, we extend the analysis of consumption patterns by categorizing posts, based on their outlet leaning, into Questionable or Reliable. The resulting distributions from the application of this dichotomy are represented in Fig. \ref{fig:post_attention_pattern_categorized} and \ref{fig:user_attention_pattern_categorized}. Fig. \ref{fig:post_attention_pattern_categorized} displays the distribution of the number of likes and shares (reblogs or retweets) obtained by posts in our dataset, together with the corresponding cumulative. Similarly, Fig. \ref{fig:user_attention_pattern_categorized} describes the frequency distribution of the same kind of interactions from a user perspective. In general, we observe how Twitter users show a larger appreciation of reliable posts, establishing a clear gap from questionable ones that increases during the analysis period. This difference can be attributed to the commitment of Twitter to limit the spreading of unverified contents \cite{twitter_misinfo}. The opposite scenario happens on Gab, in which the consumption patterns do not show a clear sign of polarization towards a specific kind of narrative. This provides some evidence of how users belonging to segregated environments like Gab are not interested in the origin of the content itself. Instead, they tend to self-segregate within environments in which they can consume and spread questionable content. Therefore, the lack of regulation on this platform may allow them to perform information operations \cite{weedon2017information}, i.e., a category of actions
taken by organized actors (governments or non-state actors) to distort domestic or foreign
political sentiment, against other users who do not share the mainstream system of beliefs of the community.
In order to assess the similarity between the distributions deriving from the consumption patterns of questionable and reliable posts, we fit power-law distributions to such data and perform a statistical evaluation of their scaling parameters using the Wald test. For Gab, all the obtained p-values were significantly higher than $0.05$, describing how questionable and reliable distributions are comparable in terms of the engagement produced. The same behavior is found on Twitter, except for the likes distribution whose p-value is less than $0.001$, describing a significant difference in the way users
engage with questionable and reliable content.
We can conclude that the presence of content moderation is associated with a significant reduction of the flowing of misinformation. The avoidance of these countermeasures, as reported on Gab, seems to be associated with more heterogeneous news consumption in terms of outlet category. This particular shape of the news diet may be exploited to conduct information operations. Statistical tests indicate how questionable and reliable contents produce similar engagement behaviors within the same social media. In the end, such moderation helps the emergence of segregation, a condition in which users are affected by the restriction applied in terms of accessibility to the contents they are affiliated with but not in their engagement behaviors.
\begin{figure}
\centering
\includegraphics[trim = 65mm 0mm 0mm 0mm, scale = 0.18]{user_attention_pattern_categorized_new.pdf}
\caption{Distribution of likes (left column) and reblogs (right column) received by users posting Questionable or Reliable contents on Gab (upper panel) and Twitter (bottom panel). The figure shows how the presence of content regulations, performed by Twitter, results in a greater appreciation towards users who post reliable content. Gab, instead, shows a mixed endorsement pattern in which the appreciation towards users does not depend on the category of the content they post.}
\label{fig:user_attention_pattern_categorized}
\end{figure}
\subsection{Characterizing Commenting Behavior for Questionable and Reliable posts}
\label{subsec:commenting_lifetime}
\begin{figure}
\centering
\includegraphics[trim = 20mm 0mm 0mm 0mm, scale = 0.2]{plot_lifetime_compound_new.pdf}
\caption{Kaplan-Meier estimates for Gab (upper panel) and Twitter (lower panel), grouped by outlet category.\\
\textit{Left column}: estimates obtained through the computation of post lifetime, i.e., the period between the first and last comment a post received. \textit{Right column}: estimates obtained through the computation of post lifetime, i.e., the period between the user's first and last comment.\\
Gab shows how the lack of content regulation is associated with a commenting behavior that underlines a preference towards questionable content. This behavior is characterized by a discrepancy between the outlet category with the highest commenting persistence both on user and post lifetimes. By contrast, the introduction of content policies from Twitter makes reliable content those with the highest commenting persistence, which does not depend on the lifetime perspective.}
\label{fig:km_lifetime}
\end{figure}
To quantify the persistence of comments concerning users and posts, we employed Kaplan-Meier estimates of two survival functions. The first accounts for the period between the first and last comment received from posts. The second instead considers the period between the first and last comment made by a user. To characterize any significant difference in the two survival functions, we perform the Peto \& Peto \cite{peto1972test} test. The upper panel of Figure~\ref{fig:km_lifetime} shows the Kaplan-Estimates computed on Gab, grouped by outlet category. The test performed on its post and user lifetimes produces a p-value of $0.026$ and $0.001$, respectively. Therefore, we can conclude that the commenting persistence on Gab may be subjected to the outlet category of the post commented. Indeed, post lifetime on questionable posts reports a lower probability of being commented as time increases despite its longer persistence, reaching a maximum $340$ days. Results from user lifetime estimation, instead, describe how users are more likely to comment on questionable posts for the first $240$ days after post creation. After that time, the survival probability becomes higher on reliable posts.\\
In the end, we can conclude that the commenting behaviors on Gab reflect the general leaning of its community. Users are more likely to comment on questionable posts since their contents adhere to a common system of beliefs oriented to conspiracy theories. Coherently, the significant commenting persistence reported on reliable posts may describe the desire of users to express their dissent against the narratives introduced from such posts.
Next, we examine the commenting persistence on Twitter. Results from Peto \& Peto test on the post and user lifetimes report a p-value equal to $0.011$ and $0.0055$ respectively, stating how the survival functions on both lifetimes differentiate with respect to the outlet category of the posts commented. Indeed, such estimations on Twitter describe a uniformity in the commenting behavior for the reliable category. This fact also provides further evidence about how content moderation can discourage users from expressing their views under posts whose authority is not verified.
In summary, Gab demonstrates how the lack of content policy helps the emergence of the narratives that characterize this environment, resulting in a discrepancy between the outlet categories with the most commenting persistence on the two lifetimes. However, when the content policy is applied, like on Twitter, such discrepancy dissolves, resulting in a commenting behavior that favors reliable content.
\subsection{Quantifying Polarization}
\label{subsec:polarization}
The presence of content moderation may affect how users develop homophily, i.e., the tendency to surround themselves with other peers who share the same narratives or system of beliefs. To quantify this phenomenon, we build a network in which the nodes represent the users $i$ with their corresponding leaning $x_i$, while the edges represent the \textit{following} relationship with other users that occurs on the social media. This representation allows us to measure the neighborhood leaning $x_i^{N}$, i.e., a measure of the characteristic leaning of the network surrounding user $i$. Figure \ref{fig:echo_chambers} displays the joint distribution between the individual leaning of a user $x_i$ and its corresponding neighborhood leaning $x_i^N$, on Twitter and Gab. In addition to this, the marginal probability distributions $P(x)$ and $P^{N}(x)$, referring to the individual and average neighborhood leaning, are represented on their corresponding axis. Lastly, the density of users at point $(x, x^N)$ is represented as a contour map: the brighter the color in that point, the higher the user density. Results described in Figure \ref{fig:echo_chambers_twitter} show the presence of homophily on Twitter, characterized by a strong correlation of leanings in correspondence of low values. The existence of a second echo chamber of incomparable size made of users with high individual leaning, and therefore not represented in the main figure but only visible in the marginal distributions, signals strong segregation between two communities. This finding also indicates how content regulations actively affect the shape of the news diet of users in the context of the COVID-19 pandemic. Indeed, the concentration around small values for both leanings provides evidence about the effectiveness of the moderation imposed by the platform against posts and users that promote questionable content. On the other side, Gab shows a more heterogeneous behavior, as represented in Figure \ref{fig:echo_chambers_gab}. Indeed, the joint distribution spreads over different values of the individual leaning domain, with the highest mode represented in correspondence of the point $(0.6, 0.6)$. We observe that on average users, regardless of their leaning, are surrounded by a neighborhood skewed towards questionable contents. Only very few users have a reliable-based leaning, who are also likely to be those with a weaker activity since they could be on Gab just for curiosity or dissing. Furthermore, the outlet category of the news that users post is not relevant anymore since the user's peers share a leaning with a high value. Finally, these findings may suggest that questionable news is employed to support the narrative of the environment, whilst reliable ones are only used to perform information operations by changing the original meaning of the posts through a comment.
\begin{figure}%
\centering
\subfloat[\centering Twitter]{{\includegraphics[trim = 0mm 0mm 0mm 0mm, scale = 0.3]{twitter_echo_chambers.pdf} }
\label{fig:echo_chambers_twitter}
}%
\subfloat[\centering Gab]{{\includegraphics[trim = 10mm 0mm 0mm 0mm, scale = 0.3]{gab_echo_chamber.pdf} }
\label{fig:echo_chambers_gab}
}%
\caption{Joint distribution between individual and average neighborhood leaning of all users posting classifiable contents at least three times on Twitter (left) and Gab (right). The figure shows further evidence about the regulation imposed by Twitter which results in the creation of a unique echo chamber of users with strong posting habits towards reliable content. Oppositely, Gab shows the presence of an echo chamber in which both individual and neighborhood leanings are concentrated around high values of the intervals, with a greater dispersion due to the mixed posting habits of users.
}%
\label{fig:echo_chambers}%
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
In this work, we compared two social media, Twitter and Gab, to investigate the interplay between content regulation policies and news consumption. We provide quantitative measures of such differences by evaluating the engagement of users and posts. These measures are then extended by providing a categorization of news outlets. Next, we measure the commenting persistence of users and posts to describe their ability to express themselves under posts belonging to a specific outlet category. In the end, we characterize the presence of homophily, investigating how users with a specific leaning are more likely to surround themselves with users who share the same narratives.
Our results show how the application of content regulation, performed by Twitter, limits the diffusion of fake news and conspiracy theories, shaping the news consumption and the polarization of users towards reliable content. The avoidance of these countermeasures, carried on by Gab, provides results that underline the presence of patterns related to information operations. Indeed, users tend to engage with questionable and reliable content comparably. However, their commenting behavior and the assessment of the homophily in this environment describe a systematic affiliation towards questionable contents.
We conclude that content policies cover an important role against the circulation of harmful content, especially in the context of the COVID-19 pandemic. Our work provides meaningful evidence in this direction, indicating how a lack of content policy is associated with the emergence of harmful narratives that promote questionable content and mistrust everything that goes against them.
Future implementations of this work may focus on the dissing/endorsement behavior promoted by users in segregated environments like Gab, analyzing those mechanisms from a textual perspective. Furthermore, a topological analysis of users who perform information operations in such environments may be relevant to understand their inner dynamics and to promote specific countermeasures.
\appendices
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {'timestamp': '2021-06-09T02:02:46', 'yymm': '2106', 'arxiv_id': '2106.03924', 'language': 'en', 'url': 'https://arxiv.org/abs/2106.03924'} |
\section*{Introduction}\label{sec:intro}
In 1965 Richard Thompson introduced three groups that today are usually denoted $F$, $T$, and $V$. These have received a lot of recent attention for their interesting and often surprising properties. Most prominently, $T$ and $V$ are finitely presented, infinite, simple groups, and $F$ is torsion-free with infinite cohomological dimension and of type~$\text{F}_\infty$.
Numerous generalizations of Thompson's groups have been introduced in the literature; see for example \cite{higman74, stein92, guba97, roever99, brin04, hughes09, martinez-perez13, belk14}. Most of these constructions either generalize the way in which branching can occur, or mimic the self-similarity in some way. Here we describe a more algebraic construction of Thompson-like groups, which combines the usual branching of the group $F$ with a chosen family of groups. The construction is based on Brin's description on the braided Thompson group $\Vbr$ \cite{brin07}, which utilizes the family of braid groups. Another example is the pure braided Thompson group $\Fbr$ introduced by Brady, Burillo, Cleary and Stein in \cite{brady08}, using the pure braid groups. Classical examples include $F$, using the trivial group, and $V$, using the symmetric groups.
The input to our construction is a directed system of groups $(G_n)_{n \in \mathbb{N}}$ together with a \emph{cloning system}, which essentially determines how a group element is moved past a split. A cloning system consists of morphisms $G_n \to S_n$ (where $S_n$ is the symmetric group on $n$ symbols), and \emph{cloning maps} $\kappa^n_{k} \colon G_n \to G_{n+1}$, $1 \le k \le n$, subject to certain conditions (see Definition~\ref{def:filtered_cloning_system}). The output is a group $\Thomp{G_*}$:
\begin{main:thompson_construction}
Let $(G_n)_{n \in \mathbb{N}}$ be an injective directed system of groups equipped with a cloning system. Then there is a generalized Thompson group $\Thomp{G_*}$ that contains all of the $G_n$. There are homomorphisms $F \hookrightarrow \Thomp{G_*} \to V$ whose composition is the inclusion $F \hookrightarrow V$.
\end{main:thompson_construction}
The groups $F$, $V$, $\Fbr$ and $\Vbr$ are all examples of groups of the form $\Thomp{G_*}$.
One of our main motivations for constructing these new Thompson-like groups is the analysis of their finiteness properties. Recall that a group $G$ is of \emph{type~$\text{F}_n$} if there is a $K(G,1)$ with finite $n$-skeleton. For example, $\text{F}_1$ means finitely generated and $\text{F}_2$ means finitely presented. We are in particular interested in understanding how the finiteness properties of $\Thomp{G_*}$ depend on the finiteness properties of the groups $G_n$. Our main results are:
\begin{main:borel_thomp_full_fin_props}
Let $k$ be a global function field, let $S$ be a set of places of $k$, and let $\mathcal{O}_S$ be the ring of $S$-integers in $k$. Let $B_n$ denote the algebraic group of invertible upper triangular $n$-by-$n$ matrices. There is a generalized Thompson group $\Thomp{B_*(\mathcal{O}_S)}$ and it is of type~$\text{F}_{\abs{S}-1}$ but not of type~$\text{F}_{\abs{S}}$.
\end{main:borel_thomp_full_fin_props}
To put this into context it is important to know that the groups $B_n(\mathcal{O}_S)$ are themselves of type~$\text{F}_{\abs{S}-1}$ but not of type~$\text{F}_{\abs{S}}$ by \cite{bux04}. In particular, for every $n \in \mathbb{N}$, we get an example of a generalized Thompson group of type~$\text{F}_{n-1}$ but not of type~$\text{F}_n$.
\begin{main:thomp_abels_Finfty}
Let $\mathit{Ab}_n(\mathbb{Z}[1/p])$ be the $n$th Abels group (see Section~\ref{sec:matrix_groups}). There is a generalized Thompson group $\Thomp{\mathit{Ab}_*(\mathbb{Z}[1/p])}$ and it is of type~$\text{F}_\infty$.
\end{main:thomp_abels_Finfty}
The groups $\mathit{Ab}_n(\mathbb{Z}[1/p])$ are known to be of type~$\text{F}_{n-1}$ but not of type~$\text{F}_n$ by \cite{abels87,brown87}. To be of type~$\text{F}_\infty$ for a generalization of Thompson's groups is a relatively common phenomenon, but what is interesting about this example is that it organizes the groups $\mathit{Ab}_n(\mathbb{Z}[1/p])$, none of which is individually of type~$\text{F}_\infty$, into a group of type~$\text{F}_\infty$.
To formulate the above statements in a unified way, it is helpful to introduce the \emph{finiteness length} $\phi(G)$ of a group $G$, which is just the supremum over all $n$ for which $G$ is of type~$\text{F}_n$. Now Theorems~\ref{thrm:borel_thomp_full_fin_props} and \ref{thrm:thomp_abels_Finfty} can be formulated to say that
\begin{equation}\label{eq:limit}
\phi(\Thomp{G_*}) = \liminf_n \phi(G_n)
\end{equation}
for the respective groups.
This relation is not coincidental but is suggested by the structure of the groups. In fact, we give a general construction which reduces proving the inequality $\ge$ for~\eqref{eq:limit} to showing that certain complexes $\dlkmodel{G_*}{n}$ are asymptotically highly connected. This construction is an abstraction of the well developed methods from \cite{brown92,stein92,brown06,farley03,fluch13,bux14} (which were all used to prove that the respective groups are of type~$\text{F}_\infty$). For this reason, the proof of the inequality $\ge$ in Theorem~\ref{thrm:borel_thomp_full_fin_props} works without change for the groups $B_n(R)$ where $R$ is an arbitrary ring. This evidence leads us to ask:
\begin{main:finiteness_conjecture}
For which generalized Thompson groups $\Thomp{G_*}$ does~\eqref{eq:limit} hold?
\end{main:finiteness_conjecture}
The group $\Thomp{G_*}$ may be thought of as a limit of the groups $G_n$, for example since it contains all of them. From this point of view, it is rather remarkable that~\eqref{eq:limit} holds in such generality. For example compare this to an ascending direct limit of groups with good finiteness properties, which will not even be finitely generated.
Another reason why~\eqref{eq:limit} is interesting is that it describes how finiteness properties of groups change when they are subject to a certain operation (here Thompsonifying). A different such operation is braiding: when $V$ is ``braided,'' we get $\Vbr$, and similarly $F$ yields $\Fbr$. The question of the finiteness properties of $\Fbr$ and $\Vbr$ was answered in~\cite{bux14}; they are still of type~$\text{F}_\infty$, just like $F$ and $V$. When reinterpreting $F$, $V$, $\Fbr$ and $\Vbr$ as Thompsonifications (of the trivial group, the symmetric groups, the pure braid groups, and the braid groups, respectively), they provide more examples where~\eqref{eq:limit} holds: in all of these cases all the groups $G_n$ are of type~$\text{F}_\infty$ and so are the corresponding Thompson groups. This is in some cases related to a similar program carried out in \cite{bartholdi15} for wreath products, see Remark~\ref{rmk:wreath}.
\medskip
In addition to the groups discussed so far, we also construct generalized Thompson groups for more families of groups. All of them are relatives of the family of symmetric groups in some way and it is very natural to put them into a generalized Thompson group. The first is a family of mock reflection groups that were studied by Davis, Januszkiewicz and Scott \cite{davis03}. The groups naturally arise as blowups of symmetric groups and we call them mock symmetric groups. Constructing a generalized Thompson group for the mock symmetric groups was suggested to us by Januszkiewicz. The second family consists of loop braid groups, which are a melding of symmetric groups and braid groups.
\begin{main:existence_thompson_mock_loop}
There exist generalized Thompson groups $\Vmock$, $\Vloop$ and $\Floop$ built from (and thus containing) all mock symmetric groups, all loop braid groups, and all pure loop braid groups. The groups $\Vmock$ and $\Vloop$ surject onto $V$ and $\Floop$ surjects onto $F$.
\end{main:existence_thompson_mock_loop}
We expect that all of these groups belong to the list of groups that answer Question~\ref{ques:finiteness_conjecture} positively, and thus:
\begin{main:Vmock_Vloop_Floop_conj}
$\Vmock$, $\Vloop$ and $\Floop$ are of type~$\text{F}_\infty$.
\end{main:Vmock_Vloop_Floop_conj}
\medskip
To investigate the finiteness properties of a generalized Thompson group $\Thomp{G_*}$ we let it act on a contractible cube complex $\Stein{G_*}$ which we call the \emph{Stein--Farley complex}. This space exists for arbitrary cloning systems and in many cases has been used previously. When the cloning system is \emph{properly graded} (Definition~\ref{def:properly_graded}), the action has certain desirable properties: the cell stabilizers are subgroups of the groups $G_n$ and there is a natural cocompact filtration. To show that the generalized Thompson group is of type~$\text{F}_n$, assuming that all the $G_n$ are, (which gives one half of~\eqref{eq:limit}) thus amounts to showing that the descending links $\dlkmodel{G_*}{n}$ in this filtration are eventually $(n-1)$-connected. This is the only part of the proof that needs to be done for every properly graded cloning system individually and depends on the nature of the concrete example. This treats the positive case, which so far has been sufficient for most existing Thompson groups since they have been of type~$\text{F}_\infty$.
For the negative finiteness properties we have to develop new methods. For example we give a condition on a group homomorphism $G \to H$ that ensures that if the morphism factors through a group $K$ then $K$ cannot be of type~$\text{FP}_n$, see Theorem~\ref{thrm:relative_brown} (type~$\text{FP}_n$ is a homological, and slightly weaker, version of type~$\text{F}_n$). This is a similar idea to that of~\cite{krstic97} and may be of independent use. Unlike the proof that $\Thomp{B_*(\mathcal{O}_S)}$ is of type~$\text{F}_{\abs{S}-1}$, the proof that it is not of type~$\text{FP}_{\abs{S}}$ borrows large parts from the proof in~\cite{bux04} of the same fact for $B_n(\mathcal{O}_S)$. For example, the space for $\Thomp{B_*(\mathcal{O}_S)}$ is built out of the space for $B_2(\mathcal{O}_S)$ (which is a Bruhat--Tits tree).
\bigskip
The paper is organized as follows. In Section~\ref{sec:preliminaries} we recall some background on monoids and the Zappa--Sz\'ep product. In Section~\ref{sec:defining_data} we introduce cloning systems (Definition~\ref{def:filtered_cloning_system}) and explain how they give rise to generalized Thompson groups. Section~\ref{sec:basic_properties} collects some group theoretic consequences that follow directly from the construction. To study finiteness properties, the Stein--Farley complex is introduced in Section~\ref{sec:spaces}. The filtration and its descending links are described in Section~\ref{sec:finiteness_props}, and we discuss some background on Morse theory and other related techniques for proving high connectivity, including a new method in Section~\ref{sec:relative_brown}.
Up to this point everything is mostly generic. The following sections discuss examples. Section~\ref{sec:direct_prods} gives an elementary example where $G_n = H^n$ for some group $H$. Section~\ref{sec:matrix_groups} discusses cloning systems for groups of upper triangular matrices. In Section~\ref{sec:mtx_fin_props} we study their finiteness properties. The last two sections \ref{sec:mock} and \ref{sec:loop} introduce the groups $\Vmock$ and $\Vloop$ and $\Floop$.
\subsection*{Acknowledgments} We are grateful to Matt Brin and Kai-Uwe Bux for helpful discussions, to Tadeusz Januszkiewicz for proposing to us the group $\Vmock$, and to Werner Thumann and an anonymous referee for many helpful comments. Both authors were supported by the SFB~878 in M\"unster. The second author was also supported directly by the DFG through project WI~4079/2 and by the SFB~701 in Bielefeld. All of this support is gratefully acknowledged.
\numberwithin{equation}{section}
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\section{Motivation}\label{sec:motivation}
Starting with the first section we will spend some ten pages introducing notions and technical results from the theory of monoids. Before we dive into these preparations, we want to explain why they are precisely the ones needed to describe generalized Thompson groups. We illustrate this on the example of $\Vbr$.
\begin{figure}[t]
\centering
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(0,-2) -- (1,-1) -- (2,-2) (1,-2) -- (.5,-1.5);
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(1,-2) to [out=-90, in=90] (0,-4);
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\hspace{1cm}
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(1.25,0.25) to [out=-90, in=90, looseness=1] (0.75,-0.75);
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\end{tikzpicture}
\caption{On the left, an element of $\Vbr$ in its standard form consisting of splitting, braiding and merging. On the right, some relations: (a) splitting and then merging is trivial; (b) merging and then splitting is trivial; (c) splits and merges on different strands commute. The main relation, (d), which is encoded by the Zappa--Sz\'ep product, is how splits and group elements interact.}
\label{fig:vbr_element}
\end{figure}
We want to think of an element of a Thompson group as consisting of a tree of splittings, followed by a group element from a chosen group (a braid in the example), and finally an inverse tree of merges. An element of $\Vbr$ is illustrated in Figure~\ref{fig:vbr_element}. Two elements are multiplied by stacking them on top of each other and reducing, as in Figure~\ref{fig:reducing}. Among the relations available to reduce an element are the fact that splitting and then merging again is a trivial operation, as well as merging and then splitting (Figure~\ref{fig:vbr_element}(a),(b)). Another relation that is implicit in the pictures is that a group element followed by another group element is the same as the product. However, these relations are not typically sufficient to bring a diagram into the form that we want: splits, group element, merges. To move all the splits to the top (and all the merges to the bottom), we eventually will have to move a split $\lambda$ past a group element $g$. In Figure~\ref{fig:reducing} this point is reached in the third step. Expressed algebraically, we need to rewrite $g \lambda = \lambda' g'$ for some group element $g'$ and some split $\lambda'$ (Figure~\ref{fig:vbr_element}(d)). The algebraic operation that defines how a split is moved past a group element is the Zappa--Sz\'ep product.
The trees of splittings will be elements of the forest monoid $\mathscr{F}$. We will then form the Zappa-Sz\'ep product $\mathscr{F} \bowtie G$ with the chosen group $G$. To also obtain merges, we will pass to the group of fractions --- a merge is just the inverse of a split. For technical reasons, we will have started with infinitely many strands and in a final step have to reduce to elements that start and end with one strand. With this outline in mind, we hope the reader will find the following technical pages more illuminating.
\begin{figure}[htb]
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\end{scope}
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\node at (-.5,-1.5) {$5$};
\draw
(0,-2) -- (1,-1) -- (2,-2) (1,-2) -- (1.5,-1.5) (1.5,-2) -- (1.75,-1.75);
\draw
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\draw[white, line width=4pt]
(1.5,-2) to [out=-90, in=90] (2,-8);
\draw
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\draw[white, line width=4pt]
(1,-2) to [out=-90, in=90] (1.5,-8);
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(1,-2) to [out=-90, in=90] (1.5,-8);
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\caption{Computing the product of two elements of $\Vbr$. First, both elements are stacked onto each other. Second, pairs of merges and splits are resolved. Third, merges and splits are moved past each other. In the fourth and fifth step a merge and a split are moved past a group element (here a braid).}
\label{fig:reducing}
\end{figure}
\section{Preliminaries}\label{sec:preliminaries}
Much of the material in this section is taken from~\cite{brin07}.
\subsection{Monoids}\label{sec:monoids}
A \emph{monoid} is an associative binary structure with a two-sided identity. A monoid $M$ is called \emph{left cancellative} if for all $x,y,z\in M$, we have that $xy=xz$ implies $y=z$. Elements $x,y\in M$ have a \emph{common left multiple} $m$ if there exist $z,w\in M$ such that $zx=wy=m$. This is the \emph{least common left multiple} if for all $p,q\in M$ such that $px=qy$, we have that $px$ is a left multiple of $m$. There are the obvious definitions of \emph{right cancellative}, \emph{common right multiples} and \emph{least common right multiples}.
We say that $M$ \emph{has common right/left multiples} if any two elements have a common right/left multiple. It is said to \emph{have least common right/left multiples} if any two elements that have some common right/left multiple have a least common right/left multiple. Finally, we say $M$ is \emph{cancellative} if it is both left and right cancellative. The importance of these notions lies in the following classical theorem (see \cite[Theorems~1.23, 1.25]{clifford61}):
\begin{theorem}[Ore]\label{thm:ore}
A cancellative monoid with common right multiples has a unique group of right fractions.
\end{theorem}
Recall that for every monoid $M$ there exists a group $G_M$ and a monoid morphism $\omega \colon M \to G_M$ such that every monoid morphism from $M$ to a group factors through $\omega$ (namely the group generated by all the elements of $M$ subject to all the relations that hold in $M$). This is \emph{the group of fractions of $M$}. The morphism $\omega$ will be injective if and only if $M$ embeds into a group. A group $G$ is called \emph{a group of right fractions of $M$} if it contains $M$ and every element of $G$ can be written as $m \cdot n^{-1}$ with $m,n \in M$. A group of right fractions exists precisely in the situation of Ore's theorem and is unique up to isomorphism; see \cite[Section~1.10]{clifford61} for details. We call a monoid satisfying the hypotheses of Theorem~\ref{thm:ore} an \emph{Ore monoid}. The group of right fractions of an Ore monoid is its group of fractions (see for example \cite[Theorem~7.1.16]{kashiwara06}):
\begin{lemma}\label{lem:ore_fractions}
Let $M$ be an Ore monoid, let $G$ be its group of right fractions and let $H$ be any group. Let $\varphi \colon M \to H$ be a monoid morphism. Then the map $\tilde{\varphi} \colon G \to H$ defined by $\tilde{\varphi}(mn^{-1}) = \varphi(m) \cdot \varphi(n)^{-1}$ is a group homomorphism and $\varphi = \tilde{\varphi}|_M$.
\end{lemma}
\begin{proof}
That inverses map to inverses is clear. Let $m_1,m_2,n_1,n_2 \in M$ and let $n_1 \cdot x = m_2 \cdot y$ be a common right multiple so that $m_1 n_1^{-1} m_2 n_2^{-1} = m_1 x y^{-1}n_2^{-1}$. We have to check that
\begin{equation}
\label{eq:fraction_multiplication}
\varphi(m_1) \varphi(n_1)^{-1} \varphi(m_2) \varphi(n_2)^{-1} = \varphi(m_1 x) \varphi(n_2 y)^{-1}\text{.}
\end{equation}
The fact that $\varphi$ is a monoid morphism means that $\varphi(n_1)\varphi(x) = \varphi(m_2) \varphi(y)$ which entails $\varphi(n_1)^{-1}\varphi(m_2) = \varphi(x)\varphi(y)^{-1}$. Extending by $\varphi(m_1)$ from the left and by $\varphi(n_2)^{-1}$ from the right gives \eqref{eq:fraction_multiplication}.
\end{proof}
\subsection{Posets from monoids}\label{sec:posets_from_monoids}
Throughout this section let $M$ be an Ore monoid and let $G$ be its group of right fractions. The notions of left/right multiple/factor are uninteresting for $G$ as a monoid because it is a group. Instead we introduce these notions relative to the monoid $M$. Concretely, assume that elements $a,b,c \in G$ satisfy
\[
ab=c \text{.}
\]
If $a \in M$ then we call $b$ a \emph{right factor} of $c$ and $c$ a \emph{left multiple} of $b$. If $b \in M$ then we call $a$ a \emph{left factor} of $c$ and $c$ a \emph{right multiple} of $a$. If $g$ is a left factor (respectively right multiple) of both $h$ and $h'$ then we say that it is a \emph{common left factor} (respectively \emph{common right multiple}). If $g$ is a common left factor of $h$ and $h'$ and any other left factor of $h$ and $h'$ is also a left factor of $g$ then $g$ is called a \emph{greatest common left factor}. If $g$ is a common right multiple of $h$ and $h'$ and every other right multiple is also a right multiple of $g$ then $g$ is called a \emph{least common right multiple} of $h$ and $h'$. Thus we obtain notions of when $G$ has (least) common right/left multiples and (greatest) common right/left factors. We say that two elements have \emph{no common right factor} if they have greatest common right factor $1$.
Under a moderate additional assumption, having least common right multiples is inherited by $G$ from $M$:
\begin{lemma}
Let $M$ have least common right multiples. Let $n,n',m,m' \in M$ be such that $n$ and $m$ have no common right factor and neither do $n'$ and $m'$. Let $nv = n'u$ be a least common right multiple of $n$ and $n'$. Then $nv = n'u$ is a least common right multiple of $nm^{-1}$ and $n'{m'}^{-1}$.\qed
\end{lemma}
We call a monoid homomorphism $\operatorname{len} \colon M \to \mathbb{N}_0$ a \emph{length function} if every element of the kernel is a unit. It induces a length function $\operatorname{len} \colon G \to \mathbb{Z}$. Note that if $M$ admits a length function then every element of $G$ can be written as $mn^{-1}$ where $m$ and $n$ are elements of $M$ with no common right factor.
The following is an extension of \cite[Lemma~2.3]{brin07} to $G$.
\begin{lemma}
Assume that $M$ admits a length function. Then $G$ has least common right multiples if and only if it has greatest common left factors.\qed
\end{lemma}
One reason for our interest in least common right multiples and greatest common left factors is order theoretic. Define a relation on $G$ by declaring $g \le h$ if $g$ is a left factor of $h$. This relation is reflexive and transitive but fails to satisfy antisymmetry if $M$ has non-trivial units. We denote the relation induced on $G/M^\times$ also by $\le$. It is an order relation so $G/M^\times$ becomes a partially ordered set (poset). Spelled out, the relation is given by $gM^\times \le hM^\times$ if $g^{-1}h \in M$.
The algebraic properties discussed before immediately translate into order theoretic properties: recall that a poset $P$ is a \emph{join-semilattice} if any two elements of $P$ have a supremum (their join). We say that $P$ \emph{has conditional meets} if any two elements that have a lower bound have an infimum.
\begin{observation}
If $M$ has common right multiples, least common right multiples, and greatest common left factors then $M/M^\times$ is a join-semilattice with conditional meets. Similarly, if $G$ has common right multiples, least common right multiples and greatest common left factors then $G/M^\times$ is a join-semilattice with conditional meets.\qed
\end{observation}
Putting everything together, we find:
\begin{corollary}\label{cor:fraction_semilattice}
Let $M$ be a cancellative monoid with common right multiples, least common right multiples and length function. Let $G$ be its group of right fractions. Then $G/M^\times$ is a join-semilattice with conditional meets.\qed
\end{corollary}
\subsection{The monoid of forests}\label{sec:forest_monoid}
Since we are interested in Thompson's groups, an important monoid in all that follows will be the \emph{monoid of forests}, which we define in this section.
For us, a \emph{tree} is always a finite rooted full binary tree. In other words, every vertex has either no outgoing edges or a left and right outgoing edge, and every vertex other than the root has an incoming edge. The vertices without outgoing edges are called \emph{leaves}. The distinction between left and right induces a natural order on the leaves. If a tree has only one leaf, then the leaf is also its root and the tree is the \emph{trivial tree}.
By a \emph{forest} we mean a sequence of trees $E=(T_i)_{i\in\mathbb{N}}$ such that all but finitely many $T_i$ are trivial. The roots are numbered in the obvious way, i.e., the $i$th root of $E$ is the root of $T_i$. If all the $T_i$ are trivial we call $E$ \emph{trivial}. If the $T_i$ are trivial for $i > 1$ then the forest is called \emph{semisimple} (here we deviate from Brin's notation; what we call ``semisimple'' is called ``simple'' in \cite{brin07}, and what we will later call ``simple'', Brin calls ``simple and balanced''). The \emph{rank} of $E$ is the least index $i$ such that $T_j$ is trivial for $j > i$. So $E$ is semisimple if it has rank at most $1$. The leaves of all the $T_i$ are called the \emph{leaves} of $E$. The order on the leaves of the trees induces an order on the leaves of the forest by declaring that any leaf of $T_i$ comes before any leaf of $T_j$, whenever $i<j$. We may equivalently think of the leaves as numbered by natural numbers. The number of \emph{feet} of a semisimple forest $(T_i)_{i \in \mathbb{N}}$ is the number of leaves of $T_1$ .
Let $\mathscr{F}$ be the set of forests. Define a multiplication on $\mathscr{F}$ as follows. Let $E=(T_k)$ and $E'=(T'_k)$ be forests, and set $E E'$ to be the forest obtained by identifying the $i$th leaf of $E$ with the $i$th root of $E'$, for each $i$. This product is associative, and the trivial forest is a left and right identity, so $\mathscr{F}$ is a monoid. Some more details on $\mathscr{F}$ can be found in Section~3 of \cite{brin07}. Figure~\ref{fig:forest_mult} illustrates the multiplication of two elements.
\begin{figure}[t]
\centering
\begin{tikzpicture}[line width=0.8pt, scale=0.5]
\draw
(-3,-3) -- (0,0) -- (3,-3) (1,-3) -- (-1,-1) (-1,-3) -- (0,-2);
\filldraw
(-3,-3) circle (1.5pt) (0,0) circle (1.5pt) (3,-3) circle (1.5pt) (-1,-1) circle (1.5pt) (-1,-3) circle (1.5pt) (0,-2) circle (1.5pt) (1,-3) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt);
\node at (8,0) {$\dots$};
\begin{scope}[yshift=-5cm]
\draw
(-4,-1) -- (-3,0) -- (-2,-1) (1,-2) -- (3,0) -- (5,-2) (3,-2) -- (2,-1);
\filldraw
(-4,-1) circle (1.5pt) (-3,0) circle (1.5pt) (-2,-1) circle (1.5pt) (-1,0) circle (1.5pt) (1,0) circle (1.5pt) (1,-2) circle (1.5pt) (3,0) circle (1.5pt) (5,-2) circle (1.5pt) (3,-2) circle (1.5pt) (2,-1) circle (1.5pt) (5,0) circle (1.5pt);
\node at (7,0) {$\dots$}; \node at (10,2) {$=$};
\end{scope}
\begin{scope}[xshift=16cm]
\draw
(-3,-3) -- (0,0) -- (3,-3) (1,-3) -- (-1,-1) (-1,-3) -- (0,-2)
(-4,-4) -- (-3,-3) -- (-2,-4) (1,-5) -- (3,-3) -- (5,-5) (3,-5) -- (2,-4);
\filldraw
(-3,-3) circle (1.5pt) (0,0) circle (1.5pt) (3,-3) circle (1.5pt) (-1,-1) circle (1.5pt) (-1,-3) circle (1.5pt) (0,-2) circle (1.5pt) (1,-3) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt) (-4,-4) circle (1.5pt) (-2,-4) circle (1.5pt) (1,-5) circle (1.5pt) (5,-5) circle (1.5pt) (3,-5) circle (1.5pt);
\node at (8,0) {$\dots$};
\end{scope}
\end{tikzpicture}
\caption{Multiplication of forests.}
\label{fig:forest_mult}
\end{figure}
There is an obvious set of generators of $\mathscr{F}$, namely the set of single-caret forests. Such a forest can be characterized by the property that there exists $k\in\mathbb{N}$ such that for $i<k$, the $i$th root is also the $i$th leaf, and for $i>k$, the $i$th root is also the $(i+1)$st leaf. Denote this forest by $\lambda_k$. Every tree in $\lambda_k$ is trivial except for the $k$th tree, which is a single caret.
\begin{proposition}[Presentation of the forest monoid]\label{prop:monoid_pres}\cite[Proposition~3.3]{brin07}
$\mathscr{F}$ is generated by the $\lambda_k$, and defining relations are given by
\begin{equation}
\label{eq:forest_relation}
\lambda_j\lambda_i=\lambda_i\lambda_{j+1} \quad \text{for} \quad i<j \text{.}
\end{equation}
Every element of $\mathscr{F}$ can be uniquely expressed as a word of the form $\lambda_{k_1}\lambda_{k_2}\cdots\lambda_{k_r}$ for some $k_1 \le \cdots \le k_r$.
\end{proposition}
A consequence is that the number of carets is an invariant of a forest, and is exactly the length of the word in the $\lambda_k$ representing the forest. The following is part of \cite[Lemma~3.4]{brin07}.
\begin{lemma}\label{lem:fmonoid_properties}
The monoid $\mathscr{F}$ has the following properties.
\begin{enumerate}
\item It is cancellative.\label{eq:fmonoid_cancellative}
\item It has common right multiples.\label{eq:fmonoid_common_right_multiples}
\item It has no non-trivial units.\label{eq:fmonoid_no_units}
\item There is a monoid homomorphism $\operatorname{len} \colon \mathscr{F} \to \mathbb{N}_0$ sending each generator to $1$. \label{eq:fmonoid_length}
\item It has greatest common right factors and least common left multiples. \label{eq:fmonoid_greatest_common_right_factors}
\item It has greatest common left factors and least common right multiples.\label{eq:fmonoid_greatest_common_left_factors}
\end{enumerate}
\end{lemma}
In view of Theorem~\ref{thm:ore}, properties~\eqref{eq:fmonoid_cancellative} and~\eqref{eq:fmonoid_common_right_multiples} imply that $\mathscr{F}$ has a unique group of right fractions, which we denote $\Fhat$.
\subsection{Zappa--Sz\'ep products}\label{sec:zappa}
In this section we recall the background on Zappa--Sz\'ep products of monoids. Our main reference is \cite[Section~2.4]{brin07}, and also see \cite{brin05}. When the monoids are groups, Zappa--Sz\'ep products generalize semidirect products by dropping the assumption that one of the groups be normal.
The internal Zappa--Sz\'ep product is straightforward to define. Let $M$ be a monoid with submonoids $U$ and $A$ such that every $m\in M$ can be written in a unique way as $m=u\alpha$ for $u\in U$ and $\alpha\in A$. In particular, for $\alpha\in A$ and $u\in U$ there exist $u'\in U$ and $\alpha'\in A$ such that $\alpha u=u'\alpha'$, and the $u'$ and $\alpha'$ are uniquely determined by $\alpha$ and $u$, so we denote them $u'=\alpha\cdot u$ and $\alpha'=\alpha^u$, following \cite{brin07}. The maps $(\alpha,u)\mapsto \alpha\cdot u$ and $(\alpha,u)\mapsto \alpha^u$ should be thought of as mutual actions of $U$ and $A$ on each other. Then we can define a multiplication on $U\times A$ via
\begin{equation}\label{eq:zappa--szep_multiplication}
(u,\alpha)(v,\beta) \mathbin{\vcentcolon =} (u(\alpha\cdot v),\alpha^v \beta)\text{,}
\end{equation}
for $u,v\in U$ and $\alpha,\beta\in A$, and the map $(u,\alpha)\mapsto u\alpha$ is a monoid isomorphism from $U\times A$ (with this multiplication) to $M$; see \cite[Lemma~2.7]{brin07}. We say that $M$ is the (internal) Zappa--Sz\'ep product of $U$ and $A$, and write $M=U\bowtie A$.
\begin{example}[Semidirect product]\label{ex:semidirect}
Suppose $G$ is a group that is a semidirect product $G=U \ltimes A$ for $U,A\le G$. Then for $u\in U$ and $\alpha\in A$ we have $\alpha u=u(u^{-1} \alpha u)$, and $u^{-1} \alpha u\in A$, so the actions defined above are just $\alpha\cdot u = u$ and $\alpha^u=u^{-1} \alpha u$.
\end{example}
We actually need to use the \emph{external} Zappa--Sz\'ep product. This is discussed in detail in \cite[Section~2.4]{brin07} (and in even more detail in \cite{brin05}).
\begin{definition}[External Zappa--Sz\'ep product]\label{def:ext_ZS_prod_monoids}
Let $U$ and $A$ be monoids with maps $(\alpha,u)\mapsto \alpha\cdot u \in U$ and $(\alpha,u)\mapsto \alpha^u \in A$ satisfying the following eight properties for all $u,v\in U$ and $\alpha,\beta \in A$:
\smallskip
\begin{enumerate}[label={\arabic*)}]
\item $1_A \cdot u = u$ \hfill (Identity acting on $U$)
\item $(\alpha\beta)\cdot u = \alpha\cdot (\beta\cdot u)$ \hfill (Product acting on $U$)
\item $\alpha^{1_U} = \alpha$ \hfill (Identity acting on $A$)
\item $\alpha^{(uv)} = (\alpha^u)^v$ \hfill (Product acting on $A$)
\item $(1_A)^u = 1_A$ \hfill ($U$ acting on identity)
\item $(\alpha\beta)^u = \alpha^{(\beta\cdot u)}\beta^u$ \hfill ($U$ acting on product)
\item $\alpha \cdot 1_U = 1_U$ \hfill ($A$ acting on identity)
\item $\alpha \cdot (uv) = (\alpha\cdot u)(\alpha^u\cdot v)$. \hfill ($A$ acting on product)
\end{enumerate}
\smallskip
\noindent Then the maps are called a \emph{Zappa--Sz\'ep action}. The set $U \times A$ together with the multiplication defined by \eqref{eq:zappa--szep_multiplication} is called the \emph{(external) Zappa--Sz\'ep product} of $U$ and $A$, denoted $U \bowtie A$.
\end{definition}
It is shown in Lemma~2.9 in \cite{brin07} that the external Zappa--Sz\'ep product turns $U \bowtie A$ into a monoid and coincides with the internal Zappa--Sz\'ep product of $U$ and $A$ with respect to the embeddings $u\mapsto (u,1_A)$ and $\alpha\mapsto (1_U,\alpha)$.
Some pedantry about the use of the word ``action'' might now be advisable. The action of $U$ on $A$ is a right action described by a homomorphism of monoids $U \to \Symm(A)$, where $\Symm(A)$ is the symmetric group on $A$ (and is \emph{not} the group of monoid automorphisms). The action of $A$ on $U$ is a left action described by a homomorphism of monoids $A \to \Symm(U)$, again \emph{not} to $\Aut(U)$. In a phrase, both actions are actions \emph{of} monoids as monoids, but \emph{on} monoids as sets.
Brin \cite{brin07} regards the action $(\alpha,u) \mapsto \alpha^u$ of $U$ on $A$ as a family of maps from $A$ to itself parametrized by $U$ and defines properties of this family. For brevity we apply the same adjectives to the action itself but one should think of the family of maps. The action is called \emph{injective} if for all $u\in U$, $\alpha^u = \beta^u$ implies $\alpha = \beta$. It is \emph{surjective} if for every $\alpha \in A$ and $u \in U$ there exists a $\beta \in A$ with $\beta^u = \alpha$. The action is \emph{strongly confluent} if the following holds: if $u, v \in U$ have a least common left multiple $ru=sv$ and $\alpha = \beta^u = \gamma^v$ for some $\beta,\gamma \in A$ then there is a $\theta \in A$ such that $\theta^r = \beta$ and $\theta^s = \gamma$. Note that if the action is injective then for this to happen it is sufficient that $\theta^{ru} = \alpha$. The notions for the action of $A$ on $U$ are defined by analogy.
The following lemma can be found as Lemma~2.12 in \cite{brin07}, or as Lemma~3.15 in \cite{brin05}.
\begin{lemma}\label{lem:ZS_least_common_left_multiples}
Let $U$ be a cancellative monoid with least common left multiples and let $A$ be a group. Let $U$ and $A$ act on each other via Zappa--Sz\'ep actions. Assume that the action $(\alpha,u) \mapsto \alpha^u$ of $U$ on $A$ is strongly confluent. Then $M = U \bowtie A$ has least common left multiples.
A least common left multiple $(r,\alpha)(u,\theta) = (s,\beta)(v,\phi)$ of $(u,\theta)$ and $(v,\phi)$ in $M$ can be constructed so that $r(\alpha \cdot u) = s(\beta \cdot v)$ is the least common left multiple of $(\alpha \cdot u)$ and $(\beta \cdot v)$ in $U$. If $M$ is cancellative, every least common left multiple will have that property.
\end{lemma}
Being actions of monoids, Zappa--Sz\'ep actions are already determined by the actions of generating sets. It is not obvious, but also true, that they are often also determined by the actions of generating sets \emph{on} generating sets. This means that, in order to define the actions, we need only define $\alpha \cdot u$ and $u^\alpha$ where both $\alpha$ and $u$ come from generating sets. Brin \cite[pp.~768--769]{brin07} gives a sufficient condition for such partial actions to extend to well defined Zappa--Sz\'ep actions, which we restate here. Given sets $X$ and $Y$, let $X^*$ and $Y^*$ denote the free monoids generated respectively by them. Suppose maps $Y \times X \to Y^*, (\alpha,u) \mapsto \alpha^u$ and $Y \times X \to X, (\alpha,u) \mapsto \alpha \cdot u$ are given (so $\alpha\cdot u$ should be a single generator, but $\alpha^u$ may be a string of generators). Let $W$ be the set of relations $(\alpha u, (\alpha \cdot u)(\alpha^u))$ with $\alpha \in Y, u \in X$. Then
\[
\gen{X \cup Y \mid W}
\]
is a Zappa--Sz\'ep product of $X^*$ and $Y^*$. In particular, the above maps extend to Zappa--Sz\'ep actions $Y^* \times X^* \to Y^*$ and $Y^* \times X^* \to X^*$.
\begin{lemma}[{\cite[Lemma~2.14]{brin07}}]\label{lem:extend_zs-products}
Let $U = \gen{X \mid R}$ and $A = \gen{Y \mid T}$ be presentations of monoids (with $X \cap Y = \emptyset$). Assume that functions $Y \times X \to Y^*, (\alpha,u) \mapsto \alpha^u$ and $Y \times X \to X, (\alpha,u) \mapsto \alpha \cdot u$ are given. Let $\sim_R$ and $\sim_T$ denote the equivalence relations on $X^*$ and $Y^*$ imposed by the relation sets $R$ and $T$.
Extend the above maps to $Y^* \times X^*$ as above. Assume that the following are satisfied. If $(u,v) \in R$ then for every $\alpha \in Y$ we have $(\alpha \cdot u, \alpha \cdot v) \in R$ or $(\alpha \cdot v, \alpha \cdot u) \in R$, and also $\alpha^u \sim_T \alpha^v$. If $(\alpha,\beta) \in T$ then for all $u \in X$ we have $\alpha \cdot u = \beta \cdot u$ and $\alpha^u \sim_T \beta^u$.
Then the lifted maps induce well defined Zappa--Sz\'ep actions and the restriction of the map $A \times U \to U$ to $A \times X$ has its image in $X$. A presentation for $U \bowtie A$ is
\[
\gen{X \cup Y \mid R \cup T \cup W}
\]
where $W$ consists of all pairs $(\alpha u, (\alpha \cdot u)(\alpha^u))$ for $(\alpha,u) \in Y \times X$.
\end{lemma}
\section{Cloning systems and generalized Thompson groups}\label{sec:defining_data}
\subsection{Brin--Zappa--Sz\'ep products and cloning systems}\label{sec:BZS}
To construct Thompson-like groups we now consider Zappa--Sz\'ep products $\mathscr{F} \bowtie G$ of the forest monoid $\mathscr{F}$ with a group $G$.
\begin{definition}[BZS products]\label{def:BZS_product}
Suppose we have Zappa--Sz\'ep actions $(g,E)\mapsto g\cdot E$ and $(g,E)\mapsto g^E$ on $G \times \mathscr{F}$, for $G$ a group. For each standard generator $\lambda_k$ of $\mathscr{F}$ the map $\kappa_k=\kappa_{\lambda_k} \colon G\to G$ given by $g\mapsto g^{\lambda_k}$ is called the $k$th \emph{cloning map}. If every such cloning map is injective, we call the actions \emph{Brin--Zappa--Sz\'ep (BZS) actions} and call the monoid $\mathscr{F} \bowtie G$ the \emph{Brin--Zappa--Sz\'ep (BZS) product}.
\end{definition}
Since the action of $\mathscr{F}$ on $G$ is a right action we will also write the cloning maps $\kappa_k$ on the right.
The monoid $\mathscr{F}$ is cancellative and has common right multiples, and the same is true of $G$, being a group. Since $G$ is a group these properties are inherited by $\mathscr{F} \bowtie G$:
\begin{observation}\label{obs:BZS_cancellative_lcrms}
A BZS product $\mathscr{F} \bowtie G$ is cancellative and has (least) common right multiples. In particular it has a group of right fractions.
\end{observation}
\begin{proof}
This follows easily from the statements about $\mathscr{F}$ using the unique factorization in Zappa--Sz\'ep products and that $E$ is a right multiple and left factor of $(E,g)$.
\end{proof}
In Definition~\ref{def:BZS_product} we have already simplified the data needed to describe BZS products by using the fact that $\mathscr{F}$ is generated by the $\lambda_k$. In a similar fashion the following lemma reduces the data needed to describe the action of $G$ on $\mathscr{F}$. We denote by $S_\omega$ the group $\Symm(\mathbb{N})$ of permutations of $\mathbb{N}$ and by $S_\infty\le S_\omega$ the subgroup of permutations that fix almost all elements of $\mathbb{N}$.
\begin{lemma}[Carets to carets]\label{lem:shuffling_carets}
Let $\mathscr{F} \bowtie G$ be a BZS product. The action of $G$ on $\mathscr{F}$ preserves the set $\Lambda=\{\lambda_k\}_{k\in\mathbb{N}}$ and so induces a homomorphism $\rho \colon G\to S_\omega$. Conversely, the action of $G$ on $\mathscr{F}$ is completely determined by~$\rho$ and $(\kappa_k)_{k\in\mathbb{N}}$.
\end{lemma}
\begin{proof}
For $g\in G$ and $E,F \in \mathscr{F}$, we know that $g\cdot(EF)=(g\cdot E)(g^{E} \cdot F)$ by Definition~\ref{def:ext_ZS_prod_monoids}. We show that the action of $G$ preserves $\Lambda$. If $g\cdot \lambda_k=EF$ then $g^{-1} \cdot (EF)=\lambda_k$, so one of $g^{-1} \cdot E$ or $(g^{-1})^{E}\cdot F$ equals $1_\mathscr{F}$. Again by Definition~\ref{def:ext_ZS_prod_monoids}, we see that either $E=1_\mathscr{F}$ or $F=1_\mathscr{F}$. We conclude that $g\cdot \lambda_k$ equals $\lambda_\ell$ for some $\ell$ depending on $k$ and $g$. The map $\rho$ then is defined via $\rho(g)k=\ell$.
To see that the action of $G$ on $\mathscr{F}$ is determined by $\rho$ and $(\kappa_k)$, we use repeated applications of the equation $g\cdot (\lambda_k E)=\lambda_{\rho(g)k} ((g)\kappa_k \cdot E)$.
\end{proof}
As a consequence we see that the action of $G$ on $\mathscr{F}$ preserves the length of an element:
\begin{corollary}\label{cor:length}
There is a monoid homomorphism $\operatorname{len} \colon \mathscr{F} \bowtie G \to \mathbb{N}_0$ taking $(E,g)$ to the length of $E$ in the standard generators. The kernel of $\operatorname{len}$ is $G = (\mathscr{F} \bowtie G)^\times$. \qed
\end{corollary}
In particular, $\operatorname{len}$ is a length function in the sense of Section~\ref{sec:posets_from_monoids}. The induced morphism from the group of right fractions to $\mathbb{Z}$ (Lemma~\ref{lem:ore_fractions}) is also denoted $\operatorname{len}$.
The next result is a technical lemma that tells us that $\rho$ and the cloning maps always behave well together, in any BZS product.
\begin{lemma}[Compatibility]\label{lem:compatibility}
Let $\mathscr{F} \bowtie G$ be a BZS product. The homomorphism $\rho \colon G \to S_\omega$ and the maps $(\kappa_k)_{k \in \mathbb{N}}$ satisfy the following compatibility condition for $k < \ell$:
If $\rho(g)k < \rho(g)\ell$ then $\rho((g)\kappa_\ell)k = \rho(g)k$ and $\rho((g)\kappa_k)(\ell+1) = \rho(g)\ell + 1$.
If $\rho(g)k > \rho(g)\ell$ then $\rho((g)\kappa_\ell)k = \rho(g)k + 1$ and $\rho((g)\kappa_k)(\ell+1) = \rho(g)\ell$.
\end{lemma}
\begin{proof}
For $k < \ell$ we know that
\[
g \cdot (\lambda_\ell \lambda_k) = g \cdot(\lambda_k \lambda_{\ell+1}) \text{.}
\]
Writing this out using the axioms for Zappa--Sz\'ep products we obtain that
\[
(g \cdot \lambda_\ell) (g^{\lambda_\ell} \cdot \lambda_k) = (g \cdot \lambda_k)(g^{\lambda_k} \cdot \lambda_{\ell + 1})
\]
which can be rewritten using the action morphism $\rho$ as
\[
\lambda_{\rho(g)\ell} \lambda_{\rho(g^{\lambda_\ell})k} = \lambda_{\rho(g)k}\lambda_{\rho(g^{\lambda_k})(\ell + 1)}\text{.}
\]
Using the normal form for $\mathscr{F}$ (see Proposition~\ref{prop:monoid_pres}) we can distinguish cases for how this could occur. The first case is that both pairs of indices
\[
(\rho(g) \ell, \rho(g^{\lambda_\ell})k) \text{ and } (\rho(g)k,\rho(g^{\lambda_k})(\ell + 1))
\]
are ordered increasingly and coincide. But this is impossible because $\rho(g) \ell \ne \rho(g) k$. The second case is that both pairs are ordered strictly decreasingly and coincide, which is impossible for the same reason. The remaining two cases have that one pair is ordered increasingly and the other strictly decreasingly. In either case the monoid relation now yields a relationship among the indices, namely either
\[
\rho(g^{\lambda_k})(\ell+1)-1 = \rho(g)\ell > \rho(g^{\lambda_\ell})k = \rho(g)k
\]
or
\[
\rho(g)\ell = \rho(g^{\lambda_k})(\ell+1) < \rho(g)k = \rho(g^{\lambda_\ell})k - 1 \text{.}
\]
Finally, replacing the action of $\lambda_k$ by the map $\kappa_k$ yields the result.
\end{proof}
The compatibility condition can also be rewritten as
\begin{equation}
\label{eq:compatibility_cases}
\rho((g)\kappa_\ell)(k) =
\left\{
\begin{array}{ll}
\rho(g)(k) & k < \ell, \rho(g) k < \rho(g) \ell,\\
\rho(g)(k) + 1 & k < \ell, \rho(g) k > \rho(g) \ell,\\
\rho(g)(k-1) & k-1 > \ell, \rho(g)(k-1) < \rho(g) \ell,\\
\rho(g)(k-1)+1 & k-1 > \ell,\rho(g)(k-1) > \rho(g)\ell\text{.}
\end{array}
\right.
\end{equation}
\medskip
Lemma~\ref{lem:shuffling_carets} said that the action of $G$ on $\mathscr{F}$ is uniquely determined by $\rho$ and the cloning maps. The action of $\mathscr{F}$ on $G$ is also uniquely determined by the cloning maps, simply because $\mathscr{F}$ is generated by the $\lambda_k$. Our findings can be summarized as:
\begin{proposition}[Uniqueness]\label{prop:BZS_uniqueness}
A BZS product $\mathscr{F} \bowtie G$ induces a homomorphism $\rho \colon G \to S_\omega$ and injective maps $\kappa_k \colon G \to G, k \in \mathbb{N}$ satisfying the following conditions for $k, \ell \in \mathbb{N}$ with $k <\ell$ and $g,h \in G$:
\smallskip
\begin{enumerate}[label={(CS\arabic*)}, ref={CS\arabic*}, leftmargin=*]
\item $(gh) \kappa_k = (g)\kappa_{\rho(h)k}(h)\kappa_k$. \hfill (Cloning a product) \label{item:cs_cloning_a_product}
\item $\kappa_\ell \circ \kappa_k = \kappa_k \circ \kappa_{\ell+1} $. \hfill (Product of clonings) \label{item:cs_product_of_clonings}
\item If $\rho(g)k < \rho(g)\ell$ then $\rho((g)\kappa_\ell)k = \rho(g)k$ and \\
\indent $\rho((g)\kappa_k)(\ell+1) = \rho(g)\ell + 1$.\\
If $\rho(g)k > \rho(g)\ell$ then $\rho((g)\kappa_\ell)k = \rho(g)k + 1$ and \\
\indent $\rho((g)\kappa_k)(\ell+1) = \rho(g)\ell$. \hfill(Compatibility) \label{item:cs_compatibility}
\end{enumerate}
\smallskip
The BZS product is uniquely determined by these data. \qed
\end{proposition}
The converse is also true:
\begin{proposition}[Existence]\label{prop:BZS_existence}
Let $G$ be a group, $\rho \colon G \to S_\omega$ a homomorphism and $(\kappa_k)_{k \in \mathbb{N}}$ a family of injective maps from $G$ to itself. Assume that for $k < \ell$ and $g,h \in G$ the conditions \eqref{item:cs_cloning_a_product}, \eqref{item:cs_product_of_clonings} and \eqref{item:cs_compatibility} in Proposition~\ref{prop:BZS_uniqueness} are satisfied.
Then there is a well defined BZS product $\mathscr{F} \bowtie G$ corresponding to these data.
\end{proposition}
\begin{proof}
We will verify the assumptions of Lemma~\ref{lem:extend_zs-products}. This will produce a Zappa--Sz\'ep action, which will be a Brin--Zappa--Sz\'ep action by construction. We take $U$ to be $\mathscr{F}$ with the presentation
\[
\gen{\lambda_k \text{ for } k \in \mathbb{N} \mid (\lambda_\ell\lambda_k,\lambda_k\lambda_{\ell+1}) \text{ for } k < l}\text{.}
\]
Let $R$ denote the set of relations used here and let $R^{\text{sym}}$ be the symmetrization. We take $A$ to be $G$ with the trivial presentation
\[
\gen{g \text{ for } g \in G \mid (gh,g') \text{ for } gh=g'}\text{.}
\]
The maps on generators are defined as $g^{\lambda_k} \mathbin{\vcentcolon =} (g)\kappa_k$ and $g \cdot \lambda_k \mathbin{\vcentcolon =} \lambda_{\rho(g)k}$.
First, for $k < \ell$ and $g \in G$ we need to verify that
\[
(g \cdot (\lambda_\ell \lambda_k), g \cdot (\lambda_k \lambda_{\ell+1})) \in R^{\text{sym}} \quad \text{and} \quad g^{\lambda_\ell \lambda_k} = g^{\lambda_k \lambda_{\ell+1}} \text{.}
\]
The latter of these is just condition~\eqref{item:cs_product_of_clonings}. The former condition means that
\[
(\lambda_{\rho(g)\ell} \lambda_{\rho((g)\kappa_\ell)k}, \lambda_{\rho(g)k} \lambda_{\rho((g)\kappa_k)(\ell+1)})
\]
should lie in $R^{\text{sym}}$. If $\rho(g)k > \rho(g)\ell$ we can use condition~\eqref{item:cs_compatibility} to rewrite this as
\[
(\lambda_{\rho(g)\ell}\lambda_{\rho(g)k+1},\lambda_{\rho(g)k}\lambda_{\rho(g)\ell})
\]
which is in $R^{\text{sym}}$. If $\rho(g)k < \rho(g)\ell$ then the tuple is
\[
(\lambda_{\rho(g)\ell}\lambda_{\rho(g)k},\lambda_{\rho(g)k}\lambda_{\rho(g)\ell+1})
\]
which already lies in $R$.
Second, for every relation $(gh,g')$ of $G$ and every $k \in \mathbb{N}$ we have to verify that
\[
(gh) \cdot \lambda_k = g' \cdot \lambda_k \quad \text{and} \quad (gh)^{\lambda_k} = {(g')}^{\lambda_k}
\]
for $k \in \mathbb{N}$. The former is not really a condition because the partial action was already defined using $G$ (rather than the free monoid spanned by $G$). The latter means that we need
\[
{(g')}^{\lambda_k} = g^{\lambda_{\rho(h)k}}h^{\lambda_k}
\]
which is just condition~\eqref{item:cs_cloning_a_product}.
\end{proof}
\begin{definition}\label{def:cloning_system}
Let $G$ be a group, $\rho \colon G \to S_\omega$ a homomorphism and $(\kappa_k)_{k \in \mathbb{N}} \colon G \to G$ a family of maps, also denoted $\kappa_*$ for brevity. The triple $(G,\rho,\kappa_*)$ is called a \emph{cloning system} if the data satisfy conditions \eqref{item:cs_cloning_a_product}, \eqref{item:cs_product_of_clonings} and \eqref{item:cs_compatibility} above. We may also refer to $\rho$ and $\kappa_*$ as a forming a \emph{cloning system on} $G$.
\end{definition}
We now discuss an extended example, of the infinite symmetric group, and show that we have a cloning system. It is exactly the cloning system that gives rise to Thompson's group $V$.
\begin{example}[Symmetric groups]\label{ex:symm_gps}
Let $G=S_\infty$. Let $\rho \colon S_\infty \to S_\omega$ just be inclusion. The action of $G$ on $\mathscr{F}$ is thus given by $g \cdot \lambda_k = \lambda_{\rho(g)k} = \lambda_{gk}$.
Since we will use the specific cloning maps in this example even in the future general setting, we will give them their own name, $\varsigma_\ell$. They are defined by the formula
\begin{equation}
\label{eq:symmclone}
((g)\varsigma_k)(m) =
\left\{
\begin{array}{ll}
gm & m \le k, gm \le gk\text{,}\\
gm+1 & m < k, gm > gk\text{,}\\
g(m-1) & m > k, g(m-1) < gk\text{,}\\
g(m-1) + 1 & m > k, g(m-1) \ge gk\text{.}
\end{array}
\right.
\end{equation}
If we draw permutations as strands crossing each other, the word ``cloning'' becomes more or less literal: applying the $k$th cloning map creates a parallel copy of the $k$th strand, where we count the strands at the bottom. See Figure~\ref{fig:symm_clone} for an example.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[line width=0.8pt]
\draw (0,-2) -- (1,0); \draw (1,-2) -- (0,0); \draw (2,-2) -- (2,0);
\node at (3,-1) {$\stackrel{\varsigma_2}{\longrightarrow}$};
\begin{scope}[xshift=4cm]
\draw (0,-2) -- (2,0); \draw (1,-2) -- (0,0); \draw (2,-2) -- (1,0); \draw (3,-2) -- (3,0);
\end{scope}
\end{tikzpicture}
\caption{An example of cloning in symmetric groups. Here we see that $(1~2)\varsigma_2 = (1~3~2)$.}
\label{fig:symm_clone}
\end{figure}
We will prove that this defines a cloning system by verifying~\eqref{item:cs_cloning_a_product},~\eqref{item:cs_product_of_clonings} and~\eqref{item:cs_compatibility}. For this example we will just verify them directly, and not use any specific presentation for $S_\infty$. It is immediate from~\eqref{eq:symmclone} that the compatibility condition~\eqref{item:cs_compatibility} in the formulation~\eqref{eq:compatibility_cases} is satisfied.
To aid in checking condition~\eqref{item:cs_cloning_a_product}, we define two families of maps, $\pi_k \colon \mathbb{N} \to \mathbb{N}$ and $\tau_k \colon \mathbb{N} \to \mathbb{N}$, for $k \in \mathbb{N}$:
\begin{equation}
\label{eq:pi_tau}
\pi_k(m) = \left\{
\begin{array}{ll}
m & m \le k\text{,}\\
m-1 & m > k
\end{array}
\right.
\quad\text{and}\quad
\tau_k(m) = \left\{
\begin{array}{ll}
m & m \le k\text{,}\\
m + 1& m > k\text{.}
\end{array}
\right.
\end{equation}
Note that $\pi_k \circ \tau_k = \id$ and $\tau_k \circ \pi_k(m) = m$, unless $m = k+1$ in which case it equals $m-1$. In the $m=k+1$ case, we see that
\[
(gh)\varsigma_k(k+1) = gh(k) +1 = (g)\varsigma_{hk}(hk +1)= (g)\varsigma_{hk} (h)\varsigma_k(k+1) \text{,}
\]
by repeated use of the last case in the definition. It remains to check condition~\eqref{item:cs_cloning_a_product} in the $m \ne k+1$ case. According to the definitions, we have
\[
((g)\varsigma_k)(m) = \tau_{gk}(g\pi_k(m))
\]
whenever $m \ne k+1$. Using this we see that
\begin{align*}
((g)\varsigma_{hk}) \circ ((h)\varsigma_k)(m) &= \tau_{ghk} g \pi_{hk} \circ \tau_{hk} h \pi_k (m)\\
& = \tau_{ghk} gh \pi_k(m)\\
& = ((gh)\varsigma_k)(m)
\end{align*}
for $m \ne k+1$.
To check condition~\eqref{item:cs_product_of_clonings}, we consider $k<\ell$. We first verify, from the definition, the special cases
\begin{align*}
((g)\varsigma_\ell \circ \varsigma_k)(k+1) &= gk+1 = ((g)\varsigma_k \circ \varsigma_{\ell+1})(k+1)\quad \text{and}\\
((g)\varsigma_\ell \circ \varsigma_k) (\ell+2) &= g\ell+2 = ((g)\varsigma_k \circ \varsigma_{\ell+1})(\ell+2)\text{.}
\end{align*}
For the remaining case, when $m \ne k+1,\ell+2$, we have
\begin{align*}
((g)\varsigma_\ell \circ \varsigma_k) (m) &= \tau_k \tau_\ell g \pi_\ell \pi_k (m)\quad\text{and}\\
((g)\varsigma_k \circ \varsigma_{\ell+1}) (m) &= \tau_{\ell+1}\tau_k g \pi_k \pi_{\ell+1} (m)
\end{align*}
and it is straightforward to check that
\begin{equation}
\label{eq:pi_and_tau_relations}
\pi_\ell \pi_k = \pi_k \pi_{\ell+1} \text{ and }\tau_k \tau_\ell = \tau_{\ell+1}\tau_k\text{.}
\end{equation}
We conclude that $(S_\infty,\rho,(\varsigma_k)_k)$ is a cloning system.
\end{example}
\begin{remark}
Besides the example of symmetric groups there are two more examples of cloning systems previously existing in the literature (though of course not using this language): they are for the families of braid groups and pure braid groups and were used in \cite{brin07,brady08} to construct $\Vbr$ and $\Fbr$.
\end{remark}
\begin{observation}[Simplified compatibility]\label{obs:sync}
Condition~\eqref{item:cs_compatibility} in Proposition~\ref{prop:BZS_uniqueness} can equivalently be rewritten as
\[
\rho((g)\kappa_k)(i) = (\rho(g))\varsigma_k (i) \text{ for all } i\ne k,k+1 \text{.}
\]
\end{observation}
All the examples in the later sections satisfy the condition in Observation~\ref{obs:sync} even when $i=k,k+1$.
\begin{remark}\label{rmk:axioms_via_pres}
Proposition~\ref{prop:BZS_existence} is an application of Lemma~\ref{lem:extend_zs-products} to the trivial presentation. As this example demonstrates, it can be rather involved to verify the conditions for a cloning system. If the group in question comes equipped with a presentation involving only short relations, it may be easier to re-run the proof of Proposition~\ref{prop:BZS_existence} with that presentation by applying Lemma~\ref{lem:extend_zs-products}. In this case one has to check \eqref{item:cs_product_of_clonings} and \eqref{item:cs_compatibility} only on generators, but also has to check a variant of \eqref{item:cs_cloning_a_product} for every relation.
\end{remark}
We finish by discussing the case when we have least common left multiples. Let $\kappa_*$ be the cloning maps of a cloning system. For $E = \lambda_{k_1} \cdots \lambda_{k_r}$ define $\kappa_E \mathbin{\vcentcolon =} \kappa_{k_1} \circ \cdots \circ \kappa_{k_r}$. Note that this is well defined by condition~\eqref{item:cs_product_of_clonings} and is just the map $g \mapsto g^E$.
\begin{observation}\label{obs:BZS_lclm}
Let $G$ be a group and let $(\rho,\kappa_*)$ be a cloning system on $G$. The action of $\mathscr{F}$ on $G$ defines a strongly confluent family if and only if $\operatorname{im}(\kappa_{E_1}) \cap \operatorname{im}(\kappa_{E_2}) = \operatorname{im}(\kappa_{F})$ whenever $E_1$ and $E_2$ have least common left multiple $F$.
In particular the BZS product $\mathscr{F} \bowtie G$ has least common left multiples in that case.
\end{observation}
\begin{proof}
This is just unraveling the definition and using the remark before Lemma~\ref{lem:ZS_least_common_left_multiples}. Assume that the above condition holds. Write $F = F_1 E_1 = F_2 E_2$. Assume that $g = g_1^{E_1} = g_2^{E_2}$, that is, $g \in \operatorname{im}(\kappa_{E_1}) \cap \operatorname{im}(\kappa_{E_2})$. By assumption there is an $h \in G$ such that $g = (h)\kappa_{F}$. That is $g = h^{F} = h^{F_1 E_1} = g_1^{E_1}$. Injectivity of the action of $\mathscr{F}$ on $G$ now implies $h^{F_1} = g_1$. A similar argument shows $h^{F_2} = g_2$.
Conversely assume that the action of $\mathscr{F}$ on $G$ is strongly confluent and write $F$ as before. Let $g \in \operatorname{im}(\kappa_{E_1}) \cap \operatorname{im}(\kappa_{E_2})$. Write $g = (g_1)\kappa_{E_1}$ and $g = (g_2)\kappa_{E_2}$, that is $g = g_1^{E_1}$ and $g = g_2^{E_2}$. By strong confluence there is an $h \in G$ such that $h^{F_1} = g_1$ and $h^{F_2} = g_2$. Then $g = h^{F} = (h)\kappa_{F}$ as desired.
\end{proof}
To check this global confluence condition one either needs a good understanding of the action of $\mathscr{F}$ on $G$ (as was the case for $\Vbr$ \cite[Section~5.3]{brin07}) or one has to reduce it to local confluence statements.
\subsection{Interlude: hedges}\label{sec:hedges}
In the above example of the symmetric group, the action of $\mathscr{F}$ on $S_\infty$ factors through an action of a proper quotient. This amounts to a further relation being satisfied in addition to the product of clonings relation \eqref{item:cs_product_of_clonings}. The quotient turns out to be what Brin~\cite{brin07} called the monoid of \emph{hedges}. Without going into much detail we want to explain the action of the hedge monoid on $S_\infty$.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[line width=0.8pt, scale=0.5]
\draw
(-2,-2) -- (0,0) -- (1,-1) (-1,-1) -- (0,-2);
\filldraw
(-2,-2) circle (1.5pt) (-1,-1) circle (1.5pt) (0,-2) circle (1.5pt) (0,0) circle (1.5pt) (1,-1) circle (1.5pt);
\draw
(3,-1) -- (4,0) -- (5,-1);
\filldraw
(3,-1) circle (1.5pt) (4,0) circle (1.5pt) (5,-1) circle (1.5pt) (2,0) circle (1.5pt) (6,0) circle (1.5pt);
\node at (8,0) {$\dots$};
\begin{scope}[xshift=12cm]
\draw
(0,0) -- (0,-2) (0,0) -- (2,-2) (0,0) -- (4,-2)
(2,0) -- (6,-2)
(4,0) -- (8,-2) (4,0) -- (10,-2)
(6,0) -- (12,-2);
\filldraw
(0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt)
(0,-2) circle (1.5pt) (2,-2) circle (1.5pt) (4,-2) circle (1.5pt) (6,-2) circle (1.5pt) (8,-2) circle (1.5pt) (10,-2) circle (1.5pt) (12,-2) circle (1.5pt);
\node at (13,-.8) {$\dots$};
\end{scope}
\end{tikzpicture}
\caption{A forest and the corresponding hedge.}
\label{fig:forest_hedge}
\end{figure}
The \emph{hedge monoid} $\mathscr{H}$ is the monoid of monotone surjective maps $\mathbb{N} \to \mathbb{N}$. Multiplication is given by composition: $f \cdot h = f \circ h$. There is an action of $S_\infty$ on $\mathscr{H}$ given by the property that, for $g\inS_\infty$ and $f\in\mathscr{H}$, the cardinality of $(g \cdot f)^{-1}(i)$ is that of $f^{-1}(g^{-1}i)$. There is an obvious equivariant morphism $c \colon \mathscr{F} \to \mathscr{H}$ (see Figure~\ref{fig:forest_hedge}) given by $c(\lambda_k) = \eta_k$ where
\[
\eta_k(m) = \left\{
\begin{array}{ll}
m & m \le k\text{,}\\
m-1 & m > k\text{.}
\end{array}
\right.
\]
This morphism is surjective but not injective, in fact (see \cite[Proposition~4.4]{brin07}):
\begin{lemma}
The monoid $\mathscr{H}$ has the presentation
\[
\gen{\eta_k, k \in \mathbb{N}\mid \eta_\ell \eta_k = \eta_k \eta_{\ell+1}, \ell \ge k}\text{.}
\]
\end{lemma}
Observe that the only difference between this and the presentation of $\mathscr{F}$ is that the relation also holds for $\ell = k$, rather than only for $\ell > k$. It turns out that the action of $\mathscr{F}$ on $S_\infty$ defined in Example~\ref{ex:symm_gps} factors through $c$:
\begin{observation}
The maps $\varsigma_k$ defined in \eqref{eq:symmclone} satisfy $\varsigma_k \varsigma_k = \varsigma_k \varsigma_{k+1}$. Thus they define an action of $\mathscr{H}$ on $S_\infty$.
\end{observation}
\begin{proof}
The verification of \eqref{item:cs_product_of_clonings} above extends to the case $k = \ell$.
\end{proof}
\subsection{Filtered cloning systems}\label{sec:filtered_cloning_systems}
Typically one will want to think of Thompson's group $V$ not as built from $S_\infty$ but rather from the family $(S_n)_{n \in \mathbb{N}}$. We will now describe this approach. We regard $S_\infty$ as the direct limit $\varinjlim S_n$ where the maps $\iota_{m,n} \colon S_m \to S_n$ are induced by the inclusions $\{1,\ldots,m\} \hookrightarrow \{1,\ldots,n\}$.
Let $(G_n)_{n\in\mathbb{N}}$ be a family of groups with monomorphisms $\iota_{m,n} \colon G_m \to G_n$ for each $m\le n$. For convenience we will sometimes write $G_*$ for $(G_n)_{n \in \mathbb{N}}$; note that in this case the index set is always $\mathbb{N}$. The maps $\iota_{m,n}$ will be written on the right, e.g., $(g)\iota_{m,n}$ for $g\in G_m$. Suppose that $\iota_{m,m}=\id$ and $\iota_{m,n} \circ \iota_{n,\ell} = \iota_{m,\ell}$ for all $m\le n\le \ell$. Then $((G_n)_{n\in\mathbb{N}},(\iota_{m,n})_{m\le n})$ is a directed system of groups with a direct limit $G \mathbin{\vcentcolon =} \varinjlim G_n$. Since all the $\iota_{m,n}$ are injective, we may equivalently think of a group $G$ filtered by subgroups $G_n$.
Consider injective maps $\kappa^n_k \colon G_n \to G_{n+1}$ for $k,n \in \mathbb{N}, k \le n$. We call such maps a \emph{family of cloning maps} for the directed system $(G_n)_{n \in \mathbb{N}}$ if for $m,k \le n$ they satisfy
\begin{equation}
\label{eq:cloning_graded}
\iota_{m,n} \circ \kappa^n_k =
\left\{
\begin{array}{ll}
\kappa^m_k \circ \iota_{m+1,n+1} &\text{ if } k\le m \\
\iota_{m,n+1} &\text{ if } m<k\text{.}
\end{array}
\right.
\end{equation}
This amounts to setting $\kappa^n_k \mathbin{\vcentcolon =} \iota_{n,n+1}$ for $k > n$ and requiring that
\[
\iota_{m,n} \circ \kappa^n_k = \kappa^m_k \circ \iota_{m+1,n+1}\text{,}
\]
i.e., that the family $(\kappa_k^n)_{n \in \mathbb{N}}$ defines a morphism of directed systems of sets. From that it is clear that a family of cloning maps induces a family of injective maps $\kappa_k \colon G \to G$ by setting
\[
(g)\iota_{n} \circ \kappa_k = (g)\kappa_k^n \circ \iota_{n+1}
\]
for $g \in G_n$. Here $\iota_n \colon G_n \to G$ denotes the map given by the universal property of $G$.
\begin{definition}[Properly graded]\label{def:properly_graded}
We say that the cloning maps are \emph{properly graded} if the following strong confluence condition holds: if $g \in G_{n+1}$ can be written as $(h)\kappa_k^n = g = (\bar{g})\iota_{n,n+1}$ then there is an $\bar{h} \in G_{n-1}$ with $(\bar{h})\kappa_k^{n-1} = \bar{g}$ and $(\bar{h})\iota_{n-1,n} = h$.
\end{definition}
In view of the injectivity of all maps involved this is equivalent to saying that
\begin{equation}\label{eq:properly_graded}
\operatorname{im} \kappa_k^n \cap \operatorname{im} \iota_{n,n+1} \subseteq \operatorname{im} (\iota_{n-1,n} \circ \kappa_k^n)
\end{equation}
(where the converse inclusion is automatic) or to saying that the diagram
\begin{diagram}
G_{n-1} & \rTo^{\iota_{n-1,n}} & G_n\\
\dTo^{\kappa_k^{n-1}} && \dTo^{\kappa_k^n} \\
G_n & \rTo^{\iota_{n,n+1}} & G_{n+1}
\end{diagram}
is a pullback diagram of sets. A formulation in terms of the direct limit $G$ is that if $(h)\kappa_k \in G_n$ for $k\le n$ then $h \in G_{n-1}$. Note that a filtered cloning system satisfying the confluence condition of Observation~\ref{obs:BZS_lclm} is automatically properly graded.
\begin{example}
Take $G_n = S_n$ as in Example~\ref{ex:symm_gps}. A family of cloning maps $\varsigma^n_k$ is obtained by restriction of the maps from Example~\ref{ex:symm_gps}:
\begin{equation}
\label{eq:symmetric_cloning_maps}
\varsigma^n_k \mathbin{\vcentcolon =} \varsigma_k|_{S_n}^{S_{n+1}}\text{.}
\end{equation}
This family of cloning maps is properly graded: if $g \in \operatorname{im} \iota_{n,n+1}$ then $g$ fixes $n+1$; if moreover $g = (h)\varsigma_k$ then it follows from~\eqref{eq:symmclone} that $h$ fixes $n$ so $h \in \operatorname{im} \iota_{n-1,n}$.
\end{example}
Now suppose further that we have a family of homomorphisms $\rho_n \colon G_n \to S_n$ for each $n\in\mathbb{N}$ that are compatible with the directed systems, i.e., $\rho_n((g)\iota_{m,n})= (\rho_m(g))\iota_{m,n}$ for $m < n$ and $g \in G_m$. Let $\rho \colon G \to S_\infty$ be the induced homomorphism. We are of course interested in the case when $\rho$ and the family $(\kappa_k)_{k \in \mathbb{N}}$ define a cloning system on $G$. The corresponding defining formulas are obtained by adding decorations to the formulas from Section~\ref{sec:BZS}:
\begin{definition}[Cloning system]\label{def:filtered_cloning_system}
Let $((G_n)_{n \in \mathbb{N}}, (\iota_{m,n})_{m \le n})$ be an injective directed system of groups. Let $(\rho_n)_{n \in \mathbb{N}} \colon G_n \to S_n$ be a homomorphism of directed systems of groups and let $(\kappa^n_k)_{k \le n} \colon G_n \to G_{n+1}$ be a family of cloning maps. The quadruple
\[
((G_n)_{n \in \mathbb{N}},(\iota_{m,n})_{m \le n},(\rho_n)_{n \in \mathbb{N}},(\kappa^n_k)_{k \le n})
\]
is called a \emph{cloning system} if the following hold for all $k \le n$, $k<\ell$, and $g,h\in G_n$:
\begin{enumerate}[label={(FCS\arabic*)}, ref={FCS\arabic*}, leftmargin=*]
\item $(gh) \kappa_k^n = (g)\kappa_{\rho(h)k}^n(h)\kappa_k^n$. \label{item:fcs_cloning_a_product}\hfill (Cloning a product)
\item $\kappa_\ell^n \circ \kappa_k^{n+1} = \kappa_k^n \circ \kappa_{\ell+1}^{n+1}$.\label{item:fcs_product_of_clonings}\hfill (Product of clonings)
\item $\rho_{n+1}((g)\kappa^n_k)(i) = (\rho_n(g))\varsigma^n_k (i)$ for all $i\ne k,k+1$ \label{item:fcs_compatibility}\hfill(Compatibility)
\end{enumerate}
We may also refer to $\rho_*$ and $(\kappa_k^n)_{k \le n}$ as forming a \emph{cloning system on} the directed system $G_*$. The cloning system is \emph{properly graded} if the cloning maps are properly graded.
\end{definition}
Note that condition~\eqref{item:fcs_compatibility} is phrased more concisely than \eqref{item:cs_compatibility}, but this is just in light of Observation~\ref{obs:sync}. Again, condition~\eqref{item:fcs_compatibility} will in practice often be satisfied even when $i=k,k+1$.
\begin{observation}\label{obs:filtered_and_filtration_preserving_cloning_systems}
Let $(G_n)_{n \in \mathbb{N}}$ be an injective directed system of groups. A cloning system on $(G_n)_{n \in \mathbb{N}}$ gives rise to a cloning system on $G \mathbin{\vcentcolon =} \varinjlim G_n$. Conversely a cloning system on $G$ gives rise to a cloning system on $(G_n)_{n \in \mathbb{N}}$ provided $(G_n)\kappa^n_k \subseteq G_{n+1}$ and $\rho_n(G_n) \subseteq S_n$.
\end{observation}
We will usually not distinguish explicitly between a cloning system on $G_*$ and a cloning system on $\varinjlim G_*$ that preserves the filtration. In particular, given a cloning system on a directed system of groups we will implicitly define $\rho \mathbin{\vcentcolon =} \varinjlim \rho_n$ and $\kappa_k \mathbin{\vcentcolon =} \varinjlim \kappa_k^n$.
\subsection{Thompson groups from cloning systems}\label{sec:BZS_to_thomp}
Let $(G,\rho,(\kappa_k)_{k\in\mathbb{N}})$ be a cloning system and let $\mathscr{F} \bowtie G$ be the associated BZS product. We now define a group $\Thomphat{G}$ for the cloning system. This is a supergroup of the actual group $\Thomp{G_*}$ that we construct later in the case when $G$ arises as a limit of a family $(G_n)_n$ (Definition~\ref{def:thompson}).
\begin{definition}[Thompson group of a cloning system]\label{def:big_thompson}
The group of right fractions of $\mathscr{F} \bowtie G$ is denoted by $\Thomphat{G}$ and is called the \emph{large generalized Thompson group} of $G$. If more context is required we denote it $\Thomphat{G,\rho,(\kappa_k)_k}$ and call it the large generalized Thompson group of the cloning system $(G,\rho,(\kappa_k)_k)$.
\end{definition}
By Observation~\ref{obs:BZS_cancellative_lcrms} and Theorem~\ref{thm:ore} every element $t$ of of $\Thomphat{G}$ can be written as $t = (E_-,g)(E_+,h)^{-1}$ for some $E_-,E_+ \in \mathscr{F}$ and $g,h \in G$. If it can also be written $t = (E_-,g')(E_+,h')^{-1}$ then $gh^{-1} = g'{h'}^{-1}$. It therefore makes sense to represent it by just the triple $(E_-,gh^{-1},E_+)$. Of course, this representation is still not unique, for example $(E,1_G,E)$ represents the identity element for every $E \in \mathscr{F}$. We will denote the element represented by $(E_-,g,E_+)$ by $[E_-,g,E_+]$. Note that $[E_-,g,E_+]^{-1} = [E_+,g^{-1},E_-]$. We will call $(E_-(g\cdot F),g^F,E_+ F)$ an \emph{expansion} of $(E_-,g,E_+)$, and the latter a \emph{reduction} of the former, so any reduction or expansion of a triple $(E_-,g,E_+)$ represents the same element of $\Thomphat{G}$ as $(E_-,g,E_+)$.
\medskip
Now assume that $G = \varinjlim G_n$ is an injective direct limit of groups $(G_n)_{n \in \mathbb{N}}$ and that the cloning system is a cloning system on $(G_n)_{n \in \mathbb{N}}$. Recall from Section~\ref{sec:forest_monoid} that a forest $E$ is called semisimple if all but its first tree are trivial and in that case its number of feet is the number of leaves of the first tree.
We collect some facts about semisimple elements of $\mathscr{F}$.
\begin{observation}\label{obs:forest_semisimple_elements}
Let $E,E_1,E_2,F \in \mathscr{F}$.
\begin{enumerate}
\item The number of feet of a non-trivial semisimple element of $\mathscr{F}$ is its length plus one.
\item Any two semisimple elements of $\mathscr{F}$ have a semisimple common right multiple.\\
More generally, any two elements of rank at most $m$ have a common right multiple of rank at most $m$.\label{item:forest_semisimple_right_common_multiple}
\item If $E$ is semisimple with $n$ feet then $EF$ is semisimple if and only if $F$ has rank at most $n$.\\
More generally, if $E$ is non-trivial of rank $m$ and length $n-m$ then $EF$ has rank $m$ if and only if $F$ has rank at most $n$.\label{item:forest_right_multiple_rank}
\item If $E_1, E_2$ are semisimple with $n$ feet then $E_1E$ is semisimple if and only if $E_2E$ is.
\end{enumerate}
\end{observation}
Now we upgrade these facts to $\mathscr{F} \bowtie G$. We say that an element $(E,g) \in \mathscr{F} \bowtie G$ is \emph{semisimple} if $E$ is semisimple with $n$ feet (for some $n$) and $g\in G_n$. In this case we also say $(E,g)$ has \emph{$n$ feet}.
\begin{lemma}\label{lem:semisimple_elements}
Let $E,E_1,E_2,F \in \mathscr{F}$ and $g,h \in G$.
\begin{enumerate}
\item The number of feet of a semisimple element of $\mathscr{F} \bowtie G$ is its length plus one.\label{item:type_length}
\item Any two semisimple elements of $\mathscr{F} \bowtie G$ have a semisimple common right multiple.\label{item:semisimple_right_common_multiple}
\item If $(E,g)$ is semisimple then $(E,g) F = (E (g \cdot F),g^{F})$ is semisimple if and only if $E (g \cdot F)$ is semisimple.\label{item:monoid_semisimplicity_cancel_g}
\item If $(E,g)$ is semisimple with $n$ feet then $(E,g) F$ is semisimple if and only if $F$ has rank at most $n$.\label{item:monoid_semisimplicity_multiple}
\item If $(E_1,g)$ and $(E_2,h)$ are semisimple with same number of feet then $(E_1,g) E$ is semisimple if and only if $(E_2,g)E$ is semisimple.\label{item:monoid_semisimplicity_common_multiple}
\end{enumerate}
\end{lemma}
\begin{proof}
The first statement is clear by definition. The second statement can be reduced to the corresponding statement in $\mathscr{F}$ because $E$ is a right multiple of $(E,g)$.
In the third statement only the implication from right to left needs justification, namely that $g^F \in G_n$ where $n$ is the number of feet of $E (g \cdot F)$. This is because if $g \in G_m$ and $\operatorname{len} E = k$ then $g^E \in G_{m+k}$ as can be seen by induction on $\operatorname{len} E$ using $\kappa_k(G_n) \subseteq G_{n+1}$.
For \eqref{item:monoid_semisimplicity_multiple} note that $g \in G_n$. But $\rho(G_n) \subseteq S_n$ so having rank at most $n$ is preserved under the action of $G_n$, i.e., $\operatorname{rk} (g \cdot F) \le n \Leftrightarrow \operatorname{rk} F \le n$. Thus the statement follows from the one for $\mathscr{F}$. The last statement is immediate from \eqref{item:monoid_semisimplicity_multiple}.
\end{proof}
\begin{definition}[Simple]\label{def:simple}
A triple $(E_-,g,E_+)$ (and the element $[E_-,g,E_+]$ represented by it) is said to be \emph{simple} if $E_-$ and $E_+$ are semisimple, both of them with $n$ feet and $g \in G_n$. This is the case if it can be written as $(E_-,g) (E_+,h)^{-1}$ with both factors semisimple with the same number of feet.
\end{definition}
\begin{proposition}\label{prop:product_of_simple}
The set of simple elements in $\Thomphat{G}$ is a subgroup.
\end{proposition}
\begin{proof}
The proof closely follows \cite[Section~7]{brin07}.
Consider two simple elements $s = [E_-,g,E_+], t=[F_-,h,F_+]$. Let
\begin{equation}
\label{eq:common_simple_right_multiple}
E_+ E = F_- F
\end{equation}
be a semisimple common right multiple of $E_+$ and $F_-$ (Observation~\ref{obs:forest_semisimple_elements} \eqref{item:forest_semisimple_right_common_multiple}). Then
\begin{align}
st & = E_- g E F^{-1} h F_+^{-1}\nonumber\\
& = (E_-(g \cdot E),g^E)(F_+(h^{-1} \cdot F), (h^{-1})^F)^{-1}\label{eq:simple_representative}\\
& = [E_-(g \cdot E),g^E h^{h^{-1} \cdot F},F_+(h^{-1} \cdot F)] \text{.}\nonumber
\end{align}
In the last line we used that $(h^F)^{-1} = (h^{-1})^{h \cdot F}$ so that $((h^{-1})^{F})^{-1} = h^{h^{-1} \cdot F}$.
We claim that the last expression of~\eqref{eq:simple_representative} is simple. Indeed, $(E_-,g)$ and $E_+$ are semisimple with the same number of feet and $E_+ E$ is semisimple so $(E_-,g) E = (E_- (g \cdot E),g^E)$ is semisimple by Lemma~\ref{lem:semisimple_elements} \eqref{item:monoid_semisimplicity_common_multiple}. Similar reasoning applies to $(F_+(h^{-1} \cdot F), {h^{-1}}^F)$. Moreover, we can use Corollary~\ref{cor:length} to compute
\[
\operatorname{len} (E_-,g) + \operatorname{len} E \stackrel{s \text{ simple}}{=} \operatorname{len} E_+ + \operatorname{len} E \stackrel{\eqref{eq:common_simple_right_multiple}}{=} \operatorname{len} F_- + \operatorname{len} F \stackrel{t \text{ simple}}{=} \operatorname{len} (F_+,(h^{-1})^F) + \operatorname{len} F\text{.}
\]
By Lemma~\ref{lem:semisimple_elements}~\eqref{item:type_length} this shows that the last expression of~\eqref{eq:simple_representative} is simple.
\end{proof}
\begin{definition}[Thompson group of a filtered cloning system]\label{def:thompson}
The group of simple elements in $\Thomphat{G}$ is denoted $\Thomp{G_*}$ and called the \emph{generalized Thompson group} of $G_*$. If we need to be more precise, as with $\Thomphat{G}$, we can include other data from the cloning system in the notation as in $\Thomp{G_*,\rho_*,(\kappa^*_k)_k}$.
\end{definition}
Notationally, when we talk about a generalized Thompson group, the asterisk will always take the position of the index of the family. For instance, the generalized Thompson group for the family $(G^n)_{n \in \mathbb{N}}$ of direct powers in Section~\ref{sec:direct_prods} will be denoted $\Thomp{G^*}$; and the generalized Thompson group for the family of matrix groups $(B_n(R))_{n \in \mathbb{N}}$ in Section~\ref{sec:matrix_groups} will be denoted $\Thomp{B_*(R)}$.
Recall from the discussion after Corollary~\ref{cor:length} that there is a length morphism $\operatorname{len} \colon \Thomphat{G} \to \mathbb{Z}$ which takes an element $[E,g,F]$ to $\operatorname{len}(E) - \operatorname{len}(F)$. The group $\Thomp{G_*}$ lies in the kernel of that morphism, that is, simple elements have length $0$.
Given a simple element $[E,g,F]$ with $E=(T_i)_{i\in\mathbb{N}}$ and $F=(U_i)_{i\in\mathbb{N}}$, since all the $T_i$ and $U_i$ are trivial for $i>1$, we will often write our element as $[T_1,g,U_1]$ instead. In other words, we view an element of $\Thomp{G_*}$ as being a tree with $n$ leaves, followed by an element of $G_n$, followed by another tree with $n$ leaves.
\begin{remark}
Constructing $\Thomp{G_*}$ as the subgroup of simple elements of $\Thomphat{G}$ is somewhat artificial as can be seen in some of the proofs above. The more natural approach would be to have each element of $\mathscr{F}$ ``know'' on which level it can be applied. This amounts to considering the category of forests $\mathcal{P}$ that has objects the natural numbers and morphisms $\lambda_k^n \colon n \to n+1, 1 \le k \le n$ subject to the forest relations \eqref{eq:forest_relation}, cf.\ \cite[Section~7]{belk04}. Let $\mathcal{G}$ be another category that also has objects the natural numbers and morphisms from $n$ to $n$ that form a group $G_n$. So while $\mathcal{P}$ has only ``vertical'' arrows, $\mathcal{G}$ has only ``horizontal'' arrows. One would then want to form the Zappa--S\'zep product $\mathcal{P} \bowtie \mathcal{G}$ which would be specified by commutative squares of the form $\gamma \lambda_{k}^n = \lambda_{\rho(\gamma)k}^{n} \gamma^{\lambda_k}$ with $\gamma \in G_n$ and $\gamma^{\lambda_k} \in G_{n+1}$. Localizing everywhere one would obtain a groupoid of fractions $\mathcal{Q}$ and $\Thomp{G_*}$ should be just $\Hom_\mathcal{Q}(1,1)$.
The reason that we have not chosen that description is simply that Zappa--S\'zep products for categories are not well-developed to our knowledge, while for monoids all the needed statements were already available thanks to Brin's work \cite{brin05, brin07}.
Artifacts of this approach, which should be overcome by the general approach above, include the maps $\iota_{n,n+1}$ and the property of being properly graded. Not having to collect all the groups $G_n$ in a common group $G$ would also make it possible to construct, for example, the Thompson groups $T$ and $\Tbr$.
\end{remark}
\subsection{Morphisms}\label{sec:morphisms}
Let $(G,\rho^G,(\kappa^G_k)_{k \in \mathbb{N}})$ and $(H,\rho^H,(\kappa^H_k)_{k \in \mathbb{N}})$ be cloning systems. A homomorphism $\varphi \colon G \to H$ is a \emph{morphism of cloning systems} if
\begin{enumerate}
\item $(\varphi(g)) \kappa^H_k = \varphi((g) \kappa^G_k)$ for all $k\in\mathbb{N}$ and $g\in G$, and\label{item:morphism_group}
\item $\rho^H \circ \varphi = \rho^G$.\label{item:morphism_monoid}
\end{enumerate}
\begin{observation}\label{obs:thomphat_morphism}
Let $\varphi \colon G \to H$ be a morphism of cloning systems. There is an induced homomorphism $\Thomphat{\varphi} \colon \Thomphat{G} \to \Thomphat{H}$. If $\varphi$ is injective or surjective then so is $\Thomphat{\varphi}$. In particular, there is always a homomorphism $\Thomphat{G} \to \Thomphat{S_\omega}$.
\end{observation}
\begin{proof}
We show that a morphism of cloning systems induces a morphism $\mathscr{F} \bowtie G \to \mathscr{F} \bowtie H$. The statement then follows from Lemma~\ref{lem:ore_fractions}. Naturally, $\Thomphat{\varphi}$ is defined by $\Thomphat{\varphi}(E g) = E \varphi(g)$. Well definedness amounts to $\Thomphat{\varphi}((g \cdot E)g^E) = (\varphi(g) \cdot E)(\varphi(g)^E)$ which follows from \eqref{item:morphism_group} and \eqref{item:morphism_monoid} above by writing $E$ as a product of $\lambda_k$s and inducting on the length.
The injectivity and surjectivity statements are clear.
\end{proof}
Similarly let $(G_n)_{n \in \mathbb{N}}$ and $(H_n)_{n \in \mathbb{N}}$ be injective direct systems equipped with cloning systems. A morphism of directed systems of groups $\varphi_* \colon G_* \to H_*$ is a \emph{morphism of cloning systems} if
\begin{enumerate}
\item $(\varphi_n(g)) \kappa^{H,n}_k = \varphi_{n+1}((g) \kappa^{G,n}_k)$ for all $1\le k\le n$ and $g\in G_n$, and
\item $\rho^H_n \circ \varphi_n = \rho^G_n$ for all $n\in\mathbb{N}$.
\end{enumerate}
\begin{observation}\label{obs:thomp_morphism}
Let $\varphi_* \colon G_* \to H_*$ be a morphism of cloning systems. There is an induced homomorphism $\Thomp{\varphi} \colon \Thomp{G_*} \to \Thomp{H_*}$. If $\varphi$ is injective or surjective then so is $\Thomp{\varphi}$. In particular, there is always a homomorphism $\Thomp{G_*} \to \Thomp{S_*}$, the latter being Thompson's group $V$.
\end{observation}
\begin{proof}
We have to show that if $E g \in \mathscr{F} \bowtie G$ is semisimple with $n$ feet then so is $\Thomphat{\varphi}(E g) = E \varphi(g)$. But this follows since $E$ is semisimple with $n$ feet and $g \in G_n$, so $\varphi(g) \in H_n$.
\end{proof}
Functoriality is straightforward:
\begin{observation}\label{obs:functoriality}
If $\varphi \colon G_* \to H_*$ and $\psi \colon H_* \to K_*$ are morphisms of cloning systems then $\Thomphat{\psi \varphi} = \Thomphat{\psi} \Thomphat{\varphi} \colon \Thomphat{G} \to \Thomphat{K}$. If $\varphi$ and $\psi$ are morphisms of filtered cloning systems then $\Thomp{\psi \varphi} = \Thomp{\psi} \Thomp{\varphi} \colon \Thomp{G_*} \to \Thomp{K_*}$.\qed
\end{observation}
\section{Basic properties}\label{sec:basic_properties}
Throughout this section let $\Thomp{G_*}$ be the generalized Thompson group of a cloning system on an injective directed system of groups $(G_n)_{n \in \mathbb{N}}$ and let $G = \varinjlim G_n$. We collect some properties of $\Thomp{G_*}$ that follow directly from the construction.
\subsection{A short exact sequence}
\begin{observation}
Let $T \in \mathscr{F}$ be semisimple with $n$ feet. The map $g \mapsto [T,g,T]$ is an injective homomorphism $G_n \to \Thomp{G_*}$.
\end{observation}
\begin{proof}
The maps $G_n \to G \to \mathscr{F} \bowtie G \to \Thomphat{G}$ are all injective. The element $[T,g,T]$ is simple, so the image lies in $\Thomp{G_*}$. The map is visibly a homomorphism.
\end{proof}
In fact, this can be explained more globally. For a semisimple forest $T$ with $n$ feet let $G_T$ denote the subgroup (isomorphic to $G_n$) of $\Thomp{G_*}$ that consists of elements $[T,g,T]$. The cloning map $\kappa_k$ induces an embedding $G_T \hookrightarrow G_U$ where $U$ is obtained from $T$ by adding a split to the $k$th foot (so $U=T\lambda_k$). Finite binary trees form a directed set and the condition \eqref{item:fcs_product_of_clonings} (product of clonings) ensures that that the groups $(G_T)_T$ form a directed system of groups.
\begin{lemma}
Consider a cloning system satisfies condition~\eqref{item:fcs_compatibility} even for $i =k, k+1$ (this is the case in particular if $\rho = 0$). There is a directed subsystem $(K_T)_T$ of $(G_T)_T$ and a short exact sequence
\[
1 \to \varinjlim_T K_T \to \Thomp{G_*}\to W \to 1
\]
where the quotient morphism is the morphism $\Thomp{\rho_*}$ from Observation~\ref{obs:thomp_morphism} and $W$ is its image.
\end{lemma}
Note that $W$ contains Thompson's group $F$.
\begin{proof}
For each $T$, say with $n$ feet, let $K_T$ be the kernel of $\rho_n \colon G_T \to S_n$. The assumption on the cloning system implies that if $\rho(g) = 1$ then $\rho((g)\kappa_k) = 1$, showing that $(K_T)_T$ is indeed a subsystem of $(G_T)_T$. It remains to see that the direct limit is isomorphic to the kernel of $\Thomp{\rho_*}$. This is clear once one realizes that it consists of all elements that can be written in the form $[T,g,T]$, for some $T$ and $g\in K_T$.
\end{proof}
In what follows we will concentrate on the case where $\rho=0$ is the trivial morphism $\rho(g) = 1$, so $K_T=G_T$ for all $T$. Examples are $F$ and $\Fbr$ but not $V$ and $\Vbr$.
\begin{observation}\label{obs:thomp_kern}
Suppose $\rho = 0$. Then $\Thomp{G_*}=\Thkern{G_*}\rtimes F$.
\end{observation}
\begin{proof}
Since each $\rho_n=0$, we have $W=F$, which is $\Thomp{\{1\}}$. Then the splitting map $F\to \Thomp{G_*}$ is $\Thomp{\iota_*}$ where $\iota_* \colon \{1\} \to G_*$ is the trivial homomorphism.
\end{proof}
\begin{remark}
\label{rmk:wreath}
Bartholdi, Cornulier, and Kochloukova \cite{bartholdi15} studied finiteness properties of wreath products. Observation~\ref{obs:thomp_kern} shows how this relates to our groups. A wreath product is built by taking a direct product of copies of a group $H$, indexed by a set $X$, and combining this with another group $G$ acting on $X$. The generalized Thompson groups in Observation~\ref{obs:thomp_kern} can be viewed as the result of taking a direct limit (instead of product) of groups from a family $(G_T)_T$, indexed by a set of trees $T$ on which there is a \emph{partial} (instead of full) action of $F$, and combining these data into a group $\Thomp{G_*}$.
\end{remark}
The question of whether $F$ is amenable or not is probably the most famous question about Thompson's groups. The following observation does not purport to be deep, but it seems worth recording nonetheless.
\begin{observation}[Amenability]\label{obs:amenability}
Suppose $\rho = 0$. Then $\Thomp{G_*}$ is amenable if and only if $F$ and every $G_n$ is amenable.
\end{observation}
\begin{proof}
We have seen that $\Thkern{G_*}$ is a direct limit of copies of $G_n$. Since amenability is preserved under taking subgroups and direct limits, this tells us that $\Thkern{G_*}$ is amenable if and only if every $G_n$ is. Then since $\Thomp{G_*}=\Thkern{G_*}\rtimes F$, the conclusion follows since amenability is also closed under group extensions.
\end{proof}
\begin{observation}[Free group-free]\label{obs:free_gps}
Suppose $\rho = 0$. If none of the $G_n$ contains a non-abelian free group then neither does $\Thomp{G_*}$.
\end{observation}
\begin{proof}
Suppose $H\le \Thomp{G_*}$ is free. If $H\cap \Thkern{G_*}=\{1\}$ then $H$ embeds into $F$, and so $H$ must be cyclic, since $F$ does not contain a non-abelian free group. Now suppose there is some $1\neq x\in H\cap \Thkern{G_*}$. For any $y\in H$, the conjugate $x^y$ is in $H\cap \Thkern{G_*}$. Since $\Thkern{G_*}$ is a direct limit of copies of the $G_n$, it does not contain a non-abelian free group by assumption, and so $\langle x,x^y\rangle$ is abelian. But $y\in H$ was arbitrary, so $H$ must already be abelian.
\end{proof}
The next result does not require $\rho=0$. It fits into the context of this section but to prove it we need some of the tools of Section~\ref{sec:stein_space}.
\begin{lemma}[Torsion-free]\label{lem:torfree}
Assume that the cloning system is properly graded. If all the $G_n$ are torsion-free then so is $\Thomp{G_*}$.
\end{lemma}
\subsection{Truncation}\label{sec:truncation}
For $g \in G_n$ and $k \le n$ we have the equation $g \lambda_k = (g \cdot \lambda_k) g^{\lambda_k}$ in $\mathscr{F} \bowtie G$ where $g^{\lambda_k} \in G_{n+1}$. In $\Thomphat{G}$ this implies
\begin{equation}
\label{eq:lift_group_element}
g = (g\cdot \lambda_k) g^{\lambda_k} \lambda_k^{-1}\text{.}
\end{equation}
This elementary observation has an interesting consequence. Let $N \in \mathbb{N}$ be arbitrary and define a directed system of groups $(G'_n)_{n \in \mathbb{N}}$ by $G'_n \mathbin{\vcentcolon =} \{1\}$ for $n \le N$ and $G'_n \mathbin{\vcentcolon =} G_n$ for $n > N$. Define a cloning system on $G'_*$ by letting $(\kappa')^n_k \colon G'_n \to G'_{n+1}$ be the trivial homomorphism when $n\le N$, and $(\kappa')^n_k = \kappa^n_k$ and $\rho'_n = \rho_n$ when $n > N$. We call $G'_*$ the \emph{truncation} of $G_*$ at $N$ and $((\rho'_n)_n,((\kappa')^n_k)_{k \le n})$ the truncation of $((\rho_n)_n, ({\kappa}^n_k)_{k \le n})$ at $N$.
\begin{proposition}[Truncation isomorphism]\label{prop:truncation_isomorphism}
Let $G'_*$ be the truncation of $G_*$ at $N$. The morphism $\Thomp{G'_*} \to \Thomp{G_*}$ induced by the obvious homomorphism $G'_* \to G_*$ is an isomorphism.
\end{proposition}
\begin{proof}
The morphism $G'_* \to G_*$ is injective hence so is $\Thomp{G'_*} \to \Thomp{G_*}$. To show that it is surjective let $[T,g,U] \in \Thomp{G_*}$ be such that $T$ and $U$ have $n$ leaves. If $n > N$ there is nothing to show. Otherwise use \eqref{eq:lift_group_element} to write
\[
[T,g,U] = [T (g \cdot \lambda_k) , g^{\lambda_k}, U \lambda_k]
\]
for some $k \le n$. The trees in the right hand side expression have $n+1$ leaves. Proceeding inductively, we obtain an element whose trees have $N+1$ leaves and therefore the element is in $\Thomp{G'_*}$.
\end{proof}
This proposition is in line with treating $\Thomp{G_*}$ as a sort of limit of $G_*$ since it does not depend on an initial segment of data.
\section{Spaces for generalized Thompson groups}\label{sec:spaces}
The goal of this section is to produce for each generalized Thompson group $\Thomp{G_*}$ a space on which it acts. The space will be contractible and have stabilizers isomorphic to the groups $G_n$, assuming the cloning system on $G_*$ is properly graded. The ideas used in the construction were used before in \cite{stein92, brown92, farley03, brown06, fluch13, bux14}. Throughout let $G_*$ be an injective directed system of groups equipped with a cloning system and let $G = \varinjlim G_*$.
As a starting point we note that Corollary~\ref{cor:fraction_semilattice}, Observation~\ref{obs:BZS_cancellative_lcrms} and Corollary~\ref{cor:length} imply that $\Thomphat{G}/G$ is a join-semilattice with conditional meets, under the relation $xG \le yG$ if $x^{-1}y \in \mathscr{F} \bowtie G$. Later on it will be convenient to have a symbol for the quotient relation so we let $x \sim_G y$ if $x^{-1}y \in G$.
\subsection{Semisimple group elements}\label{sec:semisimple}
We generalize some of the notions that were introduced in Sections~\ref{sec:forest_monoid} and \ref{sec:BZS_to_thomp}. We say that an arbitrary (not necessarily semisimple) element $E$ of $\mathscr{F}$ has \emph{$n$ feet} if it has rank $m$ and length $n-m$. Visually this means that the last leaf that is not a root is numbered $n$. An element $(E,g)$ of $\mathscr{F} \bowtie G$ has \emph{$n$ feet} if $E$ has at most $n$ feet and $g \in G_n$. Finally, we call an element $[E,g,F]$ of $\Thomphat{G}$ \emph{semisimple} if $(E,g)$ is semisimple with $n$ feet and $F$ has at most $n$ feet (note $F$ need not be semisimple). This is consistent with the previous definition of ``semisimple'': If an element of the group $\Thomphat{G}$ is semisimple in this sense, and is an element of the monoid $\mathscr{F}\bowtie G$, then it must be semisimple in the monoid. We let $\widetilde{\mathcal{P}}_1$ denote the set of all semisimple elements of $\Thomphat{G}$.
\begin{lemma}\label{lem:simple_times_semisimple}
If $[E_1,g_1,F_1]$ is simple and $[E_2,g_2,F_2]$ is semisimple then $[E_1,g,F_1][E_2,g,F_2]$ is semisimple. As a consequence, $\Thomp{G_*}$ acts on $\widetilde{\mathcal{P}}_1$.
\end{lemma}
\begin{proof}
This is shown analogously to Proposition~\ref{prop:product_of_simple}.
\end{proof}
If $[E,g,F]$ is semisimple we say that it has $\operatorname{len}([E,g,F]) + 1 = \operatorname{len}(E) - \operatorname{len}(F) + 1$ \emph{feet}, which is well defined by Corollary~\ref{cor:length}. This can be visualized as the number of roots of $F$ that can be ``reached'' from the first root of $E$. We let $\widetilde{\mathcal{P}}_{1,n}$ denote the set of all semisimple elements with at most $n$ feet. We define $\mathcal{P}_{1,n}$ to be the quotient $\widetilde{\mathcal{P}}_{1,n}/{\sim}_G$ and call the passage from $\widetilde{\mathcal{P}}_{1,n}$ to $\mathcal{P}_{1,n}$ \emph{dangling}. Note that $\mathcal{P}_{1,n}$ is a subposet of $\Thomphat{G}/G$. We also denote $\widetilde{\mathcal{P}}_1/{\sim_G}$ by $\mathcal{P}_1$.
For context, the term ``dangling'' comes from the case when $G_*$ is the system of braid groups $B_*$, and the elements of $\mathcal{P}_{1,n}$ can be pictured as ``dangling braided strand diagrams'' \cite{bux14}, originating on one strand and ending on $n$ strands.
The next lemma is the reason for having introduced the notion of a cloning system being properly graded.
\begin{lemma}\label{lem:dangling}
Assume that the cloning system is properly graded. If $x,y \in \widetilde{\mathcal{P}}_{1,n}$ are semisimple then $x \sim_G y$ if and only if $x^{-1}y \in G_n$.
\end{lemma}
\begin{proof}
What needs to be shown is that if $x^{-1} y \in G$ then $x^{-1} y\in G_n$. Write $x = [E_1,g^{-1},F_1]$ and $y = [E_2,h^{-1},F_2]$. Let $E = E_1 E_1' = E_2 E_2'$ be a common right multiple so that $x^{-1}y = [F_1 (g \cdot E_1'),g^{E_1'} (h^{E_2'})^{-1},F_2 (h \cdot E_2')] \mathbin{=\vcentcolon} [A,b,C]$. For this to equal some $d \in G$ it is necessary that $Ab = dC$ in $\mathscr{F} \bowtie G$, that is, $A = d \cdot C$ and $b = d^C$.
Say that $E$ has length $m$. Then we compute that $\operatorname{len}(A) = \operatorname{len}(C) \ge m-n+1$. Since the cloning system is properly graded, the fact that $b=d^C$ implies that $d$ has to lie in $G_{m+1-\operatorname{len}(C)} \subseteq G_n$.
\end{proof}
\subsection{Poset structure}\label{sec:poset}
Consider the geometric realization $\realize{\mathcal{P}_1}$. This is the simplicial complex with a $k$-simplex for each chain $x_0\le\cdots\le x_k$ of elements of $\mathcal{P}_1$, and face relation given by subchains.
\begin{lemma}\label{lem:poset_semilattice}
The poset $\mathcal{P}_1$ is a join-semilattice with conditional meets, in particular $\realize{\mathcal{P}_1}$ is contractible.
\end{lemma}
\begin{proof}
We already know that $\Thomphat{G}/G$ is a join-semilattice with conditional meets so it suffices to show that $\mathcal{P}_1$ is closed under taking suprema and infima. In other words, it suffices to show that least common right multiples of semisimple elements are semisimple and that left factors of semisimple elements are semisimple. The first is similar to the proof of Proposition~\ref{prop:product_of_simple} and the second is easy.
\end{proof}
In $\realize{\mathcal{P}_1}$ every vertex is contained in a simplex of arbitrarily large dimension, which makes it too big for practical purposes. It has proven helpful to consider a subspace called the \emph{Stein--Farley complex}, which we introduce next.
\subsection{The Stein--Farley complex}\label{sec:stein_space}
The preorder on $\widetilde{\mathcal{P}}_1$ was defined by declaring that $x \le y$ if $y = x(E,g)$ for some $(E,g) \in \mathscr{F} \bowtie G$. The basic idea in constructing the Stein--Farley complex is to regard this relation as a transitive hull of a finer relation $\preceq$ and to use this finer relation in constructing the space. It is defined by declaring $x \preceq y$ if $y = x(E,g)$ for some $(E,g) \in \mathscr{F} \bowtie G$ with the additional assumption that $E$ is elementary. An \emph{elementary} forest is one in which every tree has at most two leaves. That is, a forest is elementary if it can be written as $\lambda_{k_1} \cdots \lambda_{k_r}$ with $k_{i+1} > k_i + 1$ for $i < r$. Note that if $x \in \widetilde{\mathcal{P}}_{1,n}$, in order for $x(E,g)$ to be in $\widetilde{\mathcal{P}}_1$ as well, it is necessary that $E$ has rank at most $n$ and that $g \in G_{n + \operatorname{len}(E)}$. Note also that if $E$ is elementary then so is $g\cdot E$ for any $g\in G$ because the action of $G$ (via $\rho \colon G \to S_\omega$) just permutes the trees of $E$.
As a consequence $\preceq$ is invariant under dangling and we also write $\preceq$ for the relation induced on $\mathcal{P}_1$. Note that $\preceq$ is not transitive, but it is true that if $x\preceq z$ and $x\le y\le z$ then $x\preceq y\preceq z$. Given a simplex $x_0\le\cdots\le x_k$ in $\realize{\mathcal{P}_1}$, call the simplex \emph{elementary} if $x_0\preceq x_k$. The property of being elementary is preserved under passing to subchains, so the elementary simplices form a subcomplex.
\begin{definition}[Stein--Farley complex]\label{def:stein_space}
The subcomplex of elementary simplices of $\realize{\mathcal{P}_1}$ is denoted by $\Stein{G_*}$ and called the \emph{Stein--Farley complex} of $\Thomp{G_*}$.
\end{definition}
The Stein--Farley complex has the structure of a cubical complex, which we now describe. The key point is:
\begin{observation}\label{obs:fmonoid_elementary_boolean}
If $E$ is elementary then the set of right factors of $E$ forms a boolean lattice under $\preceq$. \qed
\end{observation}
For $x \preceq y$ in $\mathcal{P}_1$ we consider the closed interval $[x,y]\mathbin{\vcentcolon =}\{z\in\mathcal{P}_1\mid x\le z\le y\}$ as well as the open and half open intervals $(x,y)$, $[x,y)$ and $(x,y]$ that are defined analogously. As a consequence of Observation~\ref{obs:fmonoid_elementary_boolean} we obtain that the interval $[x,y]\mathbin{\vcentcolon =}\{z\in\mathcal{P}_1\mid x\le z\le y\}$ is a boolean lattice and so $\realize{[x,y]}$ has the structure of a cube. The intersection of two such cubes $\realize{[x,y]}$ and $\realize{[z,w]}$ is empty if $y$ and $w$ do not have a common lower bound and is $\realize{[\sup(x,z),\inf(y,w)]}$ (which may be empty if the supremum is larger than the infimum) otherwise. In particular the intersection of cubes is either empty or is again a cube. Hence $\Stein{G_*}$ is a cubical complex in the sense of Definition~7.32 of~\cite{bridson99}.
\begin{observation}\label{obs:stein_space_locally_finite}
For any vertex $x$ in $\Stein{G_*}$, there are only finitely many vertices $y$ in $\Stein{G_*}$ with $x\preceq y$.
\end{observation}
\begin{proof}
If $\tilde{x} \in \widetilde{\mathcal{P}}_1$ is a vertex representative (modulo dangling) for $x$, it is clear using dangling that every vertex $y$ with $x \preceq y$ has a representative $\tilde{y}$ with $\tilde{y} = \tilde{x}(E,1)$ for some some elementary forest $E$. In order for $\tilde{y}$ to be semisimple, $E$ can have rank at most $\operatorname{len}(\tilde{x}) - 1$, and there are only finitely many elementary forests of a given rank, so the result follows.
\end{proof}
The next step is to show that $\Stein{G_*}$ is itself contractible. The argument is similar to that given in Section~4 of \cite{brown92}. We follow the exposition in \cite{bux14}.
\begin{lemma}\label{lem:cube_lemma}
For $x<y$ with $x\not\prec y$, $|(x,y)|$ is contractible.
\end{lemma}
\begin{proof}
For any $z\in(x,y]$ let $z_0$ be the unique largest element of $[x,z]$ such that $x\preceq z_0$. By hypothesis $z_0\in[x,y)$, and by the definition of $\preceq$ it is clear that $z_0\in(x,y]$, so in fact $z_0\in(x,y)$. Also, $z_0\le y_0$ for any $z\in(x,y)$. The inequalities $z\ge z_0\le y_0$ then imply that $|(x,y)|$ is contractible, by Section~1.5 of~\cite{quillen78}.
\end{proof}
\begin{proposition}\label{prop:stein_space_contractible}
$\Stein{G_*}$ is contractible.
\end{proposition}
\begin{proof}
We know that $|\mathcal{P}_1|$ is contractible by Lemma~\ref{lem:poset_semilattice}. We can build up from $\Stein{G_*}$ to $|\mathcal{P}_1|$ by attaching new subcomplexes, and we claim that this never changes the homotopy type, so $\Stein{G_*}$ is contractible. Given a closed interval $[x,y]$, define $r([x,y])\mathbin{\vcentcolon =} \operatorname{len}(y)-\operatorname{len}(x)$. As a remark, if $x\preceq y$ then $r([x,y])$ is the dimension of the cube given by $[x,y]$. We attach the contractible subcomplexes $|[x,y]|$ for $x\not\preceq y$ to $\Stein{G_*}$ in increasing order of $r$-value. When we attach $|[x,y]|$ then, we attach it along $|[x,y)|\cup|(x,y]|$. But this is the suspension of $|(x,y)|$, and so is contractible by the previous lemma. We conclude that attaching $|[x,y]|$ does not change the homotopy type, and since $|\mathcal{P}_1|$ is contractible, so is $\Stein{G_*}$.
\end{proof}
\begin{lemma}[Stabilizers]\label{lem:stabs}
Assume that the cloning system is properly graded. The stabilizer in $\Thomp{G_*}$ of a vertex in $\Stein{G_*}$ with $n$ feet is isomorphic to $G_n$. The stabilizer in $\Thomp{G_*}$ of an arbitrary cell is isomorphic to a finite index subgroup of some $G_n$.
\end{lemma}
\begin{proof}
First consider the stabilizer of a vertex $x$ with $n$ feet. We claim that $\Stab_{\Thomp{G_*}}(x)\cong G_n$. Choose $\tilde{x}\in\widetilde{\mathcal{P}}_1$ representing $x$ and let $g\in \Stab_{\Thomp{G_*}}(x)$. By the definition of dangling, and by Lemma~\ref{lem:dangling}, there is a (unique) $h \in G_n$ such that $g\tilde{x} = \tilde{x}h$. Then the map $g\mapsto h = \tilde{x}^{-1} g \tilde{x}$ is a group isomorphism.
Now let $\sigma = \realize{[x,y]}$, $x \preceq y$ be a an arbitrary cube. Since the action of $\Thomp{G_*}$ preserves the number of feet, the stabilizer $G_\sigma$ of $\sigma$ fixes $x$ and $y$. Hence $G_\sigma$ is contained in $G_x$ and contains the kernel of the map $G_x \to \Symm(\{w\mid x\preceq w \preceq y\})$, the image of which is finite by Observation~\ref{obs:stein_space_locally_finite}.
\end{proof}
We close this section by providing the proof of Lemma~\ref{lem:torfree}, left out in the last section, which says that $\Thomp{G_*}$ is torsion-free as soon as all the $G_n$ are.
\begin{proof}[Proof of Lemma~\ref{lem:torfree}]
The vertices in $\Stein{G_*}$ coincide with the vertices of $\realize{\mathcal{P}_1}$, and, as we just proved, any vertex has some $G_n$ as a stabilizer. Hence it suffices to prove that if $g\in\Thomp{G_*}$ has finite order then it fixes an element of the directed poset $\mathcal{P}_1$. By Lemma~\ref{lem:poset_semilattice}, $\mathcal{P}_1$ is a join-semilattice, so any finite collection of elements has a unique least upper bound. But then if $g$ has finite order, for any $x\in\mathcal{P}_1$ the unique least upper bound of the finite set $\gen{g}.x$ is necessarily fixed by $g$.
\end{proof}
\section{Finiteness properties}\label{sec:finiteness_props}
One of our main motivations for defining the functor $\Thomp{-}$ is to study how it behaves with respect to finiteness properties. Recall that a group $G$ if said to be \emph{of type~$\text{F}_n$} if there is a $K(G,1)$ whose $n$-skeleton is compact. Most of the known Thompson's groups are of type~$\text{F}_\infty$, that is, of type~$\text{F}_n$ for all $n$. To efficiently speak about groups that are not of type~$\text{F}_\infty$ recall that the \emph{finiteness length} of $G$, denoted $\phi(G)$, is the supremum over all $n \in \mathbb{N}$ such that $G$ is of type~$\text{F}_n$.
We will see below that proofs of the finiteness properties of $\Thomp{G_*}$ depend on the finiteness properties of the individual groups $G_n$ as well as on the asymptotic connectivity of certain descending links, which is infinite in many cases. Since finite initial intervals of $G_*$ can always be ignored by Proposition~\ref{prop:truncation_isomorphism} we ask:
\begin{question}\label{ques:finiteness_conjecture}
For which directed systems of groups $G_*$ equipped with properly graded cloning systems do we have
\[
\phi(\Thomp{G_*}) = \liminf \phi(G_*)\text{?}
\]
\end{question}
Note that for any directed system of groups $G_*$ one can take all $\rho_k$ to be trivial and all $\kappa_k^n$ to be $\iota_{n,n+1}$. In this case $\Thomp{G_*} = (\lim_n G_n) \times F$, which would seem to give a negative answer to Question~\ref{ques:finiteness_conjecture}. However, in order to be properly graded in this example we would need $\operatorname{im} \iota_{n,n+1} \subseteq \operatorname{im} \iota_{n-1,n+1}$, and this implies that the $\iota_{n,n+1}$ are all isomorphisms. Thus, in fact this does provide a positive answer to the question.
\subsection{Morse theory}\label{sec:morse}
One of the main tools to study connectivity properties of spaces, and thus to study finiteness properties of groups, is combinatorial Morse theory. We collect here the main ingredients that will be needed later on.
Let $X$ be a Euclidean cell complex. A map $h \colon X^{(0)} \to \mathbb{N}_0$ is called a \emph{Morse function} if the maximum of $h$ over the vertices of a cell of $X$ is attained in a unique vertex. We typically think of $h$ as assigning a \emph{height} to each vertex. If $h$ is a Morse function and $r \in \mathbb{R}$, the sublevel set $X_r = X^{\le r}$ consists of all cells of $X$ whose vertices have height at most $r$. For a vertex $x \in X^{(0)}$ of height $r$, the \emph{descending link} $\dlk(x)$ of $x$ is the subcomplex of $\lk (x)$ spanned by all vertices of strictly lower height. The main observation that makes Morse theory work is that keeping track of the connectivity of descending links allows one to deduce global (relative) connectivity statements:
\begin{lemma}[Morse Lemma]\label{lem:morse}
Let $X$ be a Euclidean cell complex and let $h \colon X^{(0)} \to \mathbb{N}_0$ be a Morse function on $X$. Let $s,t \in \mathbb{R} \cup \{\infty\}$ with $s < t$. If $\dlk(x)$ is $(k-1)$-connected for every vertex in $X_t \setminus X_s$ then the pair $(X_t,X_s)$ is $k$-connected.
\end{lemma}
The connection between connectivity of spaces and finiteness properties of groups is most directly made using Brown's criterion. A Morse function on $X$ gives rise to a filtration $(X_r)_{r \in \mathbb{N}_0}$ by subcomplexes. We say that the filtration is \emph{essentially $k$-connected} if for every $i \in \mathbb{N}_0$ there exists a $j \ge i$ such that $\pi_\ell(X_i \to X_j)$ is trivial for all $\ell \le k$.
Now assume that a group $G$ acts on $X$. If $h$ is $G$-invariant then so is the filtration $(X_r)_r$. We say that the filtration is \emph{cocompact} if the quotient $G \backslash X_r$ is compact for all $r$. This is the setup for Brown's criterion, see \cite[Theorems~2.2,~3.2]{brown87}.
\begin{theorem}[Brown's criterion]
Let $n \in \mathbb{N}$ and assume a group $G$ acts on an $(n-1)$-connected CW complex $X$. Assume that the stabilizer of every $p$-cell of $X$ is of type~$\text{F}_{n-p}$. Let $(X_r)_{r \in \mathbb{N}_0}$ be a $G$-cocompact filtration of $X$. Then $G$ is of type~$\text{F}_n$ if and only if $(X_r)_r$ is essentially $(n-1)$-connected.
\end{theorem}
Putting both statements together we obtain the version that we will mostly use.
\begin{corollary}\label{cor:brown_crit_use}
Let $G$ act on a contractible Euclidean cell complex $X$ and let $h \colon X^{(0)} \to \mathbb{N}_0$ be a $G$-invariant Morse function. Assume that the stabilizer of every $p$-cell of $X$ is of type~$\text{F}_{n-p}$ and that the sublevel sets $X_r$ are cocompact. If there is an $s \in \mathbb{R}$ such that $\dlk(x)$ is $(n-1)$-connected for all vertices $x \in X^{(0)} \setminus X_s$ then $G$ is of type~$\text{F}_n$.
\end{corollary}
If $G_*$ is a system of groups equipped with a properly graded cloning system then $\Thomp{G_*}$ acts on the Stein--Farley complex $\Stein{G_*}$, which is contractible (Proposition~\ref{prop:stein_space_contractible}) with stabilizers from $G_*$ (Lemma~\ref{lem:stabs}). Our next goal is to define an invariant, cocompact Morse function and to describe the descending links.
\subsection{The Morse function}\label{sec:morse_function}
Recall that the vertices of $\Stein{G_*}$ are classes $[E,g,F]$ of semisimple elements modulo dangling. The height function we will be using assigns to such a vertex its number of feet (see~Section~\ref{sec:semisimple}). That is, $\Stein{G_*}_n = \realize{\mathcal{P}_{1,n}}\cap \Stein{G_*}$. This height function is $\Thomp{G_*}$-invariant because it is induced by the morphism $\operatorname{len} \colon \Thomphat{G} \to \mathbb{Z}$ and every element of $\Thomp{G_*}$ has length $0$.
\begin{lemma}[Cocompactness]\label{lem:cocompact}
The action of $\Thomp{G_*}$ is transitive on vertices of $\Stein{G_*}$ with a fixed number of feet. Consequently the action of $\Thomp{G_*}$ on $\Stein{G_*}_n$ is cocompact for every $n$.
\end{lemma}
\begin{proof}
Let $\tilde{x} = [E_-,g,E_+]$ and $\tilde{y} = [F_-,h,F_+]$ be semisimple with $n$ feet. We know that $\tilde{x}\tilde{y}^{-1}$ takes $\tilde{y}$ to $\tilde{x}$, so it suffices to show that $\tilde{x}\tilde{y}^{-1}$ is simple. Note that $E_+$ and $F_+$ have rank at most $n$. By Observation~\ref{obs:forest_semisimple_elements}~\eqref{item:forest_semisimple_right_common_multiple} they admit a common right multiple $E_+E = F_+F$ of rank at most $n$. Let the length of this multiple be $m$, so it has at most $m+n$ feet. Then
\[
\tilde{x}\tilde{y}^{-1} = [E_-(g \cdot E),g^E(h^{F})^{-1},F_-(h \cdot F)]
\]
and both $E_-(g \cdot E)$ and $F_-(h \cdot F)$ are semisimple by Observation~\ref{obs:forest_semisimple_elements}~\eqref{item:forest_right_multiple_rank}. They have $m+n$ feet and both $g^E$ and $h^F$ lie in $G_{n+m}$. Thus $\tilde{x}\tilde{y}^{-1}$ is simple.
The second statement now follows from Observation~\ref{obs:stein_space_locally_finite}.
\end{proof}
\subsection{Descending links}\label{sec:dlks}
Let $x$ be a vertex in $\Stein{G_*}$, with $n$ feet. We want to describe the descending link of $x$. A vertex $y$ is in the link of $x$ if either $x \preceq y$ or $y \preceq x$. In the first case $y$ is ascending so the descending link is spanned by vertices $y$ with $y \preceq x$. These are by definition of the form $x(E,g)^{-1}$ for $E$ an elementary forest and $g \in G_n$. In particular, for a fixed $n$, the descending links of any vertices of height $n$ look the same, and are all isomorphic to the simplicial complex of products $g E^{-1}$ where $g \in G_n$ and $E$ is an elementary forest with at most $n$ feet, modulo the relation $\sim_G$.
It is helpful to describe this complex somewhat more explicitly. In doing so we slightly shift notation by making use of the fact that elementary forests can be parametrized by subgraphs of linear graphs.
Let $L_n$ be the graph with $n$ vertices, labeled $1$ through $n$, and a single edge connecting $i$ to $i+1$, for each $1\le i\le n-1$. This is the \emph{linear graph} with $n$ vertices. Denote the edge from $i$ to $i+1$ by $e_i$. We will exclusively consider \emph{spanning} subgraphs of $L_n$, that is, subgraphs whose vertex set is $\{1,\ldots,n\}$. We call the spanning subgraph without edges \emph{trivial}. A \emph{matching} on a graph is a spanning subgraph in which no two edges share a vertex. For an elementary forest $E$ with at most $n$ feet, define $\Gamma(E)$ to be the spanning subgraph of $L_n$ that has an edge from $i$ to $i+1$ if and only if the $i$th and $(i+1)$st leaves of $E$ are leaves of a common caret. Note that this is a matching. Conversely, given a matching $\Gamma$ of $L_n$, there is an elementary forest $E(\Gamma) = \lambda_{i_k} \cdots \lambda_{i_1}$ where $\Gamma$ has edges $e_{i_1}, \ldots,e_{i_k}$. Both operations are inverse to each other so we conclude:
\begin{observation}\label{obs:matchings_to_forests}
There is a one-to-one correspondence between matchings of $L_n$ and elementary forests with at most $n$ feet. \qed
\end{observation}
In particular, if $\Gamma$ is a matching with $m$ edges and $n$ vertices we obtain a cloning map $\kappa_\Gamma \colon G_{n-m} \to G_n$ which is just the cloning map of $E(\Gamma)$ as defined before Observation~\ref{obs:BZS_lclm}. We also get an action of $G_{n-m}$ on $\Gamma$ which is given by the action of $\rho(G_{n-m})$ on connected components. For future reference we also note:
\begin{observation}\label{obs:graphs_to_hedges}
There is a one-to-one correspondence between spanning subgraphs of $L_n$ and hedges with at most $n$ feet. \qed
\end{observation}
Now define a simplicial complex $\dlkmodel{G_*}{n}$ as follows. A simplex in $\dlkmodel{G_*}{n}$ is represented by a pair $(g,\Gamma)$, where $g\in G_n$ and $\Gamma$ is a non-trivial matching of $L_n$. Two such pairs $(g_1,\Gamma_1)$, $(g_2,\Gamma_2)$ are \emph{equivalent (under dangling)} if the following conditions hold:
\begin{enumerate}
\item $\Gamma_1$ and $\Gamma_2$ both have $m$ edges for some $1\le m\le n/2$,
\item $g_2^{-1} g_1$ lies in the image of $\kappa_{\Gamma_1}$, and
\item $\Gamma_2=(g_2^{-1} g_1)\kappa_{\Gamma_1}^{-1} \cdot \Gamma_1$.
\end{enumerate}
We make $\dlkmodel{G_*}{n}$ into a simplicial complex with face relation given by passing to subgraphs of the second term in the pair. Denote the equivalence class of $(g,\Gamma)$ under dangling by $[g,\Gamma]$. In summary,
\[
\dlkmodel{G_*}{n} \text{ has simplex set } \{[g,\Gamma]\mid \Gamma \text{ is a matching of } L_n \text{ and } g\in G_n\} \text{.}
\]
\begin{observation}\label{obs:descending_link_model}
If $x$ has $n$ feet, the correspondence $(g,\Gamma) \mapsto x g E(\Gamma)^{-1}$ induces an isomorphism $\dlkmodel{G_*}{n} \to \dlk (x)$. \qed
\end{observation}
In particular, the $\dlkmodel{G_*}{n}$ are indeed simplicial complexes as claimed, since $\Stein{G_*}$ is a cubical complex.
We now have all the pieces together to apply Brown's criterion to our setting.
\begin{proposition}\label{prop:generic_finiteness}
Let $G_*$ be equipped with a properly graded cloning system. If $G_k$ is eventually of type~$\text{F}_n$ and $\dlkmodel{G_*}{k}$ is eventually $(n-1)$-connected then $\Thomp{G_*}$ is of type~$\text{F}_n$.
\end{proposition}
\begin{proof}
Suppose first that all $G_k$ are of type~$\text{F}_n$. Let $X = \Stein{G_*}$, which is contractible by Proposition~\ref{prop:stein_space_contractible}. Our Morse function ``number of feet'' has cocompact sublevel sets by Lemma~\ref{lem:cocompact}. The stabilizer of any cell is a finite index subgroup of some $G_k$ by Lemma~\ref{lem:stabs}. Since finiteness properties are inherited by finite index subgroups, our assumption implies that all stabilizers are of type~$\text{F}_n$. By the second assumption there is an $s$ such that $\dlkmodel{G_*}{k}$ is $(n-1)$-connected for $k > s$, which by Observation~\ref{obs:descending_link_model} means that descending links are $(n-1)$-connected from $s$ on. Applying Corollary~\ref{cor:brown_crit_use} we conclude that $\Thomp{G_*}$ is of type~$\text{F}_n$.
If the $G_k$ are of type~$\text{F}_n$ only from $t$ on, we use Proposition~\ref{prop:truncation_isomorphism} to replace $\Thomp{G_*}$ by the isomorphic group $\Thomp{G_*'}$ where $G_k' = G_k$ for $k \ge t$ and $G_k = \{1\}$ for $k < t$. In particular, all of the $G_k'$ are of type~$\text{F}_n$.
Of course $\Stein{G_*'}$ is not isomorphic to $\Stein{G_*}$ and neither are the $\dlkmodel{G_*'}{m}$ isomorphic to the $\dlkmodel{G_*}{m}$. However, the $k$-skeleton of $\dlkmodel{G_*'}{m}$ is isomorphic to the $k$-skeleton of $\dlkmodel{G_*}{m}$ once $m > k + t$. Since $(n-1)$-connectivity only depends on the $n$-skeleton, if the $\dlkmodel{G_*}{m}$ are eventually $(n-1)$-connected then so are the $\dlkmodel{G_*'}{m}$.
\end{proof}
For a negative counterpart to this statement, this is, to show that $\Thomp{G_*}$ is not of type~$\text{F}_n$, we would need stabilizers with good finiteness properties and a filtration that is not essentially $(n-1)$-connected -- at least as long as we are trying to apply Brown's criterion. Hence if we have groups $G_n$ whose finiteness lengths do not have a limit inferior of $\infty$, we would need an action on a different space to show that $\Thomp{G_*}$ answers Question~\ref{ques:finiteness_conjecture} affirmatively.
Returning to the positive statement, we remark that inspecting the homotopy type of $\dlkmodel{G_*}{n}$ does not seem possible uniformly. Instead, in what follows we will focus on examples and in particular find some instances of $\dlkmodel{G_*}{n}$ being highly connected. In the case where the $G_n$ are braid groups, these complexes were modeled by arc complexes in \cite{bux14}. In Section~\ref{sec:matrix_groups} below, where the $G_n$ are matrix groups, we will directly work with the combinatorial description. General tools that have turned out to be helpful will be collected in Sections~\ref{sec:high_connectivity} and~\ref{sec:relative_brown}.
We can make one positive statement about finiteness properties without knowing much at all about $G_*$. Before stating this as a lemma, we need to define the \emph{matching complex} of $L_n$. This is a simplicial complex, denoted $\mathcal{M}(L_n)$, whose simplices are matchings on $L_n$ and with face relation given by passing to subgraphs. It is well-known and not hard to see that $\mathcal{M}(L_n)$ is $(\floor{\frac{n-2}{3}}-1)$-connected. A precise description of the homotopy type is given in \cite[Proposition~11.16]{kozlov08} where $\mathcal{M}(L_n)$ arises as the independence complex $\operatorname{Ind}(L_{n-1})$.
\begin{lemma}[Finite generation]\label{lem:fin_gen_case}
Let $G_*$ be a family of groups equipped with a properly graded cloning system, with cloning maps $\kappa_k^n$. Suppose that for $n$ sufficiently large, all $G_n$ are finitely generated and also are generated by the images of the cloning maps with codomain $G_n$. Then $\Thomp{G_*}$ is finitely generated.
\end{lemma}
\begin{proof}
By the above discussion, we need only show that the $\dlkmodel{G_*}{n}$ are connected, for large enough $n$. Suppose $n$ is large enough that: (a) $G_n$ is generated by images of cloning maps, and (b) $n\ge 5$ so $\mathcal{M}(L_n)$ is connected. We will show that every vertex can be connected by an edge path to the vertex $[1,J_1]$, where $J_i$ denotes the spanning graph whose only edge connects the $i$th vertex to the $(i+1)$st. So let $[g,\Gamma]$ be a vertex of $\dlkmodel{G_*}{n}$ and write $g=s_1\cdots s_r$, where the $s_i$ are generators coming from images of cloning maps $s_i\in \operatorname{im}(\kappa_{k_i})$ for some $k_i$. Since $\mathcal{M}(L_n)$ is connected, there is a path in $\dlkmodel{G_*}{n}$ from $[s_1\cdots s_r,\Gamma]$ to $[s_1\cdots s_r,J_{k_r}]=[s_1\cdots s_{r-1},((s_r)\kappa_{k_r}^{-1})\cdotJ_{k_r}]$. Repeating this $r$ times, we connect to $[1,J_k]$ for some $k$, and then to $[1,J_1]$.
\end{proof}
\subsection{Proving high connectivity}\label{sec:high_connectivity}
As we have seen, Morse theory is a tool that allows one to show that a pair $(X,X_0)$ is highly connected. We will eventually want to inductively apply this to the situation where $X = \dlkmodel{G_*}{n}$ and $X_0 = \dlkmodel{G_*}{n-k}$ for some $k \in \mathbb{N}$. This is insufficient to conclude that the connectivity tends to infinity though, because we would be trying to get $X$ to be more highly connected than $X_0$. The following lemma expresses the degree of insufficiency. The lemma is straightforward to prove but can be seen as a roadmap for the argument that follows.
\begin{lemma}\label{lem:relative_connectivity}
Let $(X,X_0)$ be a $k$-connected CW-pair. Assume that $X_0$ is $(k-1)$-connected. Then $X$ is $k$-connected if and only if $\pi_k(X_0 \to X)$ is trivial.
\end{lemma}
\begin{proof}
Consider the part of the homotopy long exact sequence associated to $(X,X_0)$:
\[
\pi_{j+1}(X,X_0) \to \pi_j(X_0) \stackrel{\iota_j}{\to} \pi_j(X) \to \pi_j(X,X_0) \text{.}
\]
For $j < k$ the map $\iota_j$ is an isomorphism and $\pi_j(X_0)$ trivial. For $j = k$ it is an epimorphism, so indeed $\pi_k(X)$ is trivial if and only if $\iota_k$ is.
\end{proof}
In our applications we will know $X_0$ to be $(k-1)$-connected by induction and $(X,X_0)$ will be seen to be $k$-connected using Morse theory. To show that $\pi_k(X_0 \to X)$ is trivial we will use a relative variant of the Hatcher flow for arc complexes that was shown to us by Andrew Putman (Proposition~\ref{prop:putman_flow} below). Before we can prove it we need some technical preliminaries.
A \emph{combinatorial~$k$-sphere (respectively~$k$-disk)} is a simplicial complex that can be subdivided to be isomorphic to a subdivision of the boundary of a $(k+1)$-simplex (respectively to a subdivision of a $k$-simplex). An $m$-dimensional \emph{combinatorial manifold} is an $m$-dimensional simplicial complex in which the link of every simplex~$\sigma$ of dimension $k$ is a combinatorial $(m-k-1)$-sphere. In an $m$-dimensional \emph{combinatorial manifold with boundary} the link of a $k$-simplex $\sigma$ is allowed to be homeomorphic to a combinatorial~$(m-k-1)$-disk; its \emph{boundary} consists of all the simplices whose link is indeed a disk.
A simplicial map is called \emph{simplexwise injective} if its restriction to any simplex is injective. The following is Lemma~3.8 of \cite{bux14}, cf.\ also the proof of Proposition~5.2 in \cite{putman12}.
\begin{lemma}\label{lem:injectifying}
Let $Y$ be a $k$-dimensional combinatorial manifold. Let $X$ be a simplicial complex and assume that the link of every $d$-simplex in $X$ is $(k-2d-2)$-connected for $d \ge 0$. Let $\psi \colon Y \to X$ be a simplicial map whose restriction to $\partial Y$ is simplexwise injective. Upon changing the simplicial structure of $Y$, $\psi$ is homotopic relative $\partial Y$ to a simplexwise injective map.
\end{lemma}
In practice $Y$ will be a sphere, so the lemma allows us to restrict attention to simplexwise injective combinatorial maps when collapsing spheres.
For the proposition, we need one more technical definition. Let $X$ be a simplicial complex and $w$ a vertex. We say that $X$ is \emph{conical at} $w$ if for any simplex $\sigma$, as soon as every vertex of $\sigma$ lies in the closed star $\overline{\operatorname{st}}(w)$ then so does $\sigma$ (that is, the star of $w$ is the cone over the link of $w$). In particular, if $X$ is a flag complex then it is conical at every vertex.
\begin{proposition}\label{prop:putman_flow}
Let $X_0 \subseteq X_1 \subseteq X$ be simplicial complexes. Assume that $(X,X_0)$ is $k$-connected, that $X_0$ is $(k-1)$-connected and that the link of every $d$-simplex is $(k-2d-2)$-connected for $d \ge 0$. Further assume the following ``exchange condition'':
\begin{enumerate}[label={(EXC)}, ref={EXC}, leftmargin=*]
\item There is a vertex $w \in X$ at which $X$ is conical, such that for every vertex $v \in X_0$ that is not in $\overline{\operatorname{st}} w$ there is a vertex $v' \in \overline{\operatorname{st}}_{X_1} w$ such that $\lk_{X_1} v \subseteq \lk_{X_1} v'$ and $\lk_{X_1} v$ is $(k-1)$-connected.\label{item:exchange1}
\end{enumerate}
Then $X$ is $k$-connected.
\end{proposition}
\begin{proof
Let $\iota \colon X_0 \to X$ denote the inclusion. In view of Lemma~\ref{lem:relative_connectivity}, all that needs to be shown is that if $\varphi \colon S^k \to X_0$ is a map from a $k$-sphere then $\bar{\varphi} \mathbin{\vcentcolon =} \iota \circ \varphi$ is homotopically trivial.
By simplicial approximation \cite[Theorem~3.4.8]{spanier66} we may assume $\varphi$ (and thus $\bar{\varphi}$) to be a simplicial map $Y \to X_0$ and by our assumptions and Lemma~\ref{lem:injectifying} we may assume it to be simplexwise injective. Our goal is to homotope $\bar{\varphi}$ to a map to $\overline{\operatorname{st}} w$. Once we have achieved that, we are done since $\overline{\operatorname{st}} w$ is contractible.
The simplicial sphere $Y$ contains finitely many vertices $x$ whose image $v = \bar{\varphi}(x)$ does not lie in $\overline{\operatorname{st}} w$. Pick one and define $\bar{\varphi}' \colon Y \to X$ to be the map that coincides with $\bar{\varphi}$ outside the open star of $x$ and takes $x$ to the vertex $v'$ from the statement. We claim that $\bar{\varphi}$ is homotopic to $\bar{\varphi}'$. Inductively replacing vertices then finishes the proof, since $X$ is conical at $w$.
It remains to show that $\bar{\varphi} |_{\overline{\operatorname{st}} x}$ and $\bar{\varphi}'|_{\overline{\operatorname{st}} x}$ are homotopic relative to $\lk x$. Note that $\bar{\varphi}(\lk x) \subseteq \lk v$ by simplexwise injectivity. Furthermore the complex spanned by $v$, $v'$ and $\lk v$ is the suspension $\Sigma(\lk v)$ of $\lk v$ (unless $v$ and $v'$ are adjacent in which case there is nothing to show). So both $\bar{\varphi}|_{\overline{\operatorname{st}} x}$ and $\bar{\varphi}'|_{\overline{\operatorname{st}} x}$ are maps $(D^k, S^{k-1}) \cong (\overline{\operatorname{st}} x, \lk x) \to (\Sigma(\lk v),\lk v)$. But $\lk v$ is $(k-1)$-connected by assumption so $(\Sigma(\lk v), \lk v)$ is $k$-connected and both maps are homotopic.
\end{proof}
\subsection{Proving negative finiteness properties}\label{sec:relative_brown}
We have already seen that if the $G_*$ are not eventually of type~$\text{F}_n$, then Brown's criterion applied to the Stein--Farley complex cannot be used to show that $\Thomp{G_*}$ is not of type~$\text{F}_n$. In Section~\ref{sec:mtx_neg_fin_props}, when the $G_n$ are matrix groups, we will instead use a different action, together with the following result. It is formulated in terms of the homological finiteness properties $\text{FP}_n$. The relationship is explained for example in \cite[Chapter~8]{geoghegan08}, but we mostly just need to know the fact that a group of type~$\text{F}_n$ is also of type~$\text{FP}_n$. Note that for $\Lambda = \Gamma$ the following theorem is essentially one half of Brown's criterion.
\begin{theorem}\label{thrm:relative_brown}
Let $\Lambda$ be a group and let $\Gamma$ be a subgroup. Let $Y$ be a CW complex on which $\Lambda$ acts. Assume that $Y$ is $(n-1)$-acyclic and that the stabilizer of every $p$-cell in $Y$ (in $\Lambda$ as well as in $\Gamma$) is of type~$\text{FP}_{n-p}$. Let $Z$ be a $\Gamma$-cocompact subspace of $Y$ . Let $(Y_\alpha)_{\alpha \in I}$ be a $\Lambda$-cocompact filtration of $Y$. Assume that there is no $\alpha$ with $Z\subseteq Y_\alpha$ such that the map $\tilde{H}_{n-1}(Z \hookrightarrow Y_\alpha)$ is trivial. Then no group $\Delta$ through which the inclusion $\Gamma \hookrightarrow \Lambda$ factors is of type~$\text{FP}_n$.
\end{theorem}
The application is similar in spirit to that of \cite{krstic97}, where a morphism $\Gamma \to \Lambda$ is constructed that cannot factor through a finitely presented group. The proof should be compared to \cite[Theorem~2.2]{brown87}.
\begin{proof}
For $n = 1$ suppose that $\Gamma$ is contained in a finitely generated subgroup $\gen{S}$ of $\Lambda$. Let $K$ be a compact subspace such that $\Gamma.K = Z$. Since $Y$ is connected, we can add finitely many edges to $K$ and take $Z$ to be its $\Gamma$-orbit, so without loss of generality $K$ is connected. For every $s \in S$ we may pick an edge path $p_s$ that connects $K$ to $s.K$. Let $P \mathbin{\vcentcolon =} \bigcup\{p_s \mid s \in S\}$. Now any two points in $Z$ can be connected in $\Gamma.(K \cup P)$. In other words, the map $\tilde{H}_0(Z \to \Gamma.(K \cup P))$ is trivial. But $K \cup P$ and thus $\Gamma.(K \cup P)$ is contained in some $Y_\alpha$, contradicting the assumption.
From now on we assume that $n > 1$. Our goal is to find an index set $J$ such that the map $H_{n-1}(\Gamma,\prod_J \mathbb{Z}\Gamma) \to H_{n-1}(\Lambda,\prod_J \mathbb{Z}\Lambda)$ is non-trivial. The result then follows from the Bieri--Eckmann criterion \cite[Proposition~1.2]{bieri74}, because if this map factors through $H_{n-1}(\Delta,\prod_J \mathbb{Z}\Delta)$ then the latter module cannot be zero. Note that $Z$ is contained in a subfiltration of $(Y_\alpha)_\alpha$ so we may assume without loss of generality that $Z$ is contained in all $Y_\alpha, \alpha \in I$.
Let $J$ be a cofinal set in $I$ (for instance all of $I$) and for $\alpha \in J$ let $c_\alpha \in H_{n-1}(Z)$ be such that the image in $H_{n-1}(Y_\alpha)$ is non-trivial. By the arguments in the proof of \cite[Theorem~2.2]{brown87} we have the isomorphisms in the diagram
\begin{diagram}
H_{n-1}(\Gamma,\prod_J \mathbb{Z}\Gamma) & \rTo & H_{n-1}(\Lambda,\prod_J \mathbb{Z}\Lambda)\\
\uTo^\cong && \uTo^\cong\\
H_{n-1}^\Gamma(Y, \prod_J \mathbb{Z}\Gamma) & \rTo & H_{n-1}^{\Lambda}(Y,\prod_J \mathbb{Z}\Lambda)\\
\uTo && \uTo^\cong\\
H_{n-1}^\Gamma(Z, \prod_J \mathbb{Z}\Gamma) & \rTo & \varinjlim H_{n-1}^{\Lambda}(Y_\alpha,\prod_J \mathbb{Z}\Lambda)\\
\dTo^\cong && \dTo^\cong\\
\prod_J H_{n-1}(Z) &\rTo & \varinjlim \prod_J H_{n-1}(Y_\alpha)\text{.}
\end{diagram}
(Essentially the two vertical arrows at the top are isomorphisms because $Y$ is $(n-1)$-acyclic and the two vertical arrows at the bottom are isomorphisms by cocompactness of the actions and the assumptions on the finiteness properties of the stabilizers.)
Assuming that the diagram commutes, the chain $(c_\alpha)_{\alpha \in J} \in \prod_J H_{n-1}(Z)$ has non-trivial image in $\varinjlim \prod_J H_{n-1}(Y_\alpha)$ and we are done.
The rest of the proof will be concerned with the commutativity of the diagram. The only square whose commutativity is not clear is the bottom one. In what follows, all products are taken over $J$ which we suppress from notation.
Let $C_*$, $C_*^\alpha$, and $D_*$ be the cellular chain complexes of $Y$, $Y_\alpha$, and $Z$ (respectively). Let $P_* \to \mathbb{Z}$ be a resolution by projective $\mathbb{Z}\Lambda$-modules (which are also projective $\mathbb{Z}\Gamma$-modules). The third horizontal map is induced by the maps $P_q \otimes_{\Gamma} (D_p \otimes \prod \mathbb{Z}\Gamma) \to P_q \otimes_{\Lambda} (C_p^\alpha \otimes \prod \mathbb{Z}\Lambda)$. (Or equivalently $(P_q \otimes D_p) \otimes_{\Gamma} \prod \mathbb{Z}\Gamma \to (P_q \otimes C_p^\alpha) \otimes_{\Lambda} \prod \mathbb{Z}\Lambda)$, which is the same since the tensor product is associative and, using the notation from \cite[p.~55]{brown82}, also commutative.) The bottom horizontal map is just induced by $D_* \to C_*^\alpha$. The lower vertical maps come from spectral sequences
\begin{align}
E^1_{pq} &= \Tor^\Gamma_q(D_p,\prod \mathbb{Z}\Gamma) \Rightarrow H_{p+q}^\Gamma(Z,\prod \mathbb{Z}\Gamma)\text{ and }\label{eq:spec_seq1}\\
E^1_{pq} &= \Tor^\Lambda_q(C_p^\alpha,\prod \mathbb{Z}\Lambda) \Rightarrow H_{p+q}^\Lambda(Y_\alpha,\prod \mathbb{Z}\Lambda)\text{.}\label{eq:spec_seq2}
\end{align}
The finiteness and cocompactness assumptions guarantee that $D_p$ is of type~$\text{FP}_{n-p}$ over $\mathbb{Z}\Gamma$ and $C_p^\alpha$ is of type~$\text{FP}_{n-p}$ over $\mathbb{Z}\Lambda$ so that the natural maps $\Tor^\Gamma_q(D_p,\prod \mathbb{Z}\Gamma) \to \prod \Tor^\Gamma_q(D_p,\mathbb{Z}\Gamma)$ and $\Tor^\Lambda_q(C_p^\alpha,\prod \mathbb{Z}\Lambda) \to \prod \Tor^\Lambda_q(C_p^\alpha,\mathbb{Z}\Lambda)$ are isomorphisms and the spectral sequences collapse on the second page. We have the commutative diagram of chain complexes
\begin{diagram}
&\Tor^\Gamma_0(D_*,\prod \mathbb{Z}\Gamma) & \rTo & \Tor^\Lambda_0(C_*^\alpha,\prod \mathbb{Z}\Lambda)&\\
&\dTo^\cong && \dTo^\cong&\\
\prod D_* = &\prod \Tor^\Gamma_0(D_*,\mathbb{Z}\Gamma) & \rTo & \prod\Tor^\Lambda_0(C_*^\alpha,\mathbb{Z}\Lambda)& = \prod C_*^\alpha
\end{diagram}
and taking homology in degree $n-1$ gives the commutative diagram
\begin{diagram}
H_{n-1}^\Gamma(Z,\prod \mathbb{Z}\Gamma) & \rTo & H_{n-1}^\Lambda(Y_\alpha,\prod \mathbb{Z}\Lambda)\\
\dTo^\cong && \dTo^\cong\\
\prod H_{n-1}(Z) & \rTo & \prod H_{n-1}(Y_\alpha)
\end{diagram}
that we were looking for.
\end{proof}
\section{A Thompson group for direct products of a group}\label{sec:direct_prods}
The examples in this section were constructed independently by Slobodan Tanusevski in his PhD thesis \cite{tanusevski14}, using entirely different techniques, and in discussions with him we have determined that his groups are identical to those discussed here.
Fix a group $G$. Let $G_n$ be the direct power $G^n$. We declare that $\rho_n$ is trivial for all $n$, and define cloning maps via $(g_1,\dots,g_k,\dots,g_n)\kappa^n_k \mathbin{\vcentcolon =} (g_1,\dots,g_k,g_k,\dots,g_n)$. This makes rather literal the word ``cloning.'' To verify that this defines a cloning system, observe that since the $\rho_n$ are trivial, we need only check that the cloning maps are homomorphisms (which they are) and that $\kappa_\ell^n \circ \kappa_k^{n+1} = \kappa_k^n \circ \kappa_{\ell+1}^{n+1}$ for $1\le k<\ell \le n$ (which is visibly true). These respectively handle conditions~\eqref{item:fcs_cloning_a_product} and~\eqref{item:fcs_product_of_clonings} of Definition~\ref{def:filtered_cloning_system}, and condition~\eqref{item:fcs_compatibility} is trivial. Lastly, the cloning system is visibly properly graded.
It turns out that this cloning system is an example answering Question~\ref{ques:finiteness_conjecture} positively, that is, the finiteness length of $\Thomp{G^*}$ is exactly that of $G$ (notationally, the asterisk is a superscript now because we are considering the family of direct powers $(G^n)_{n \in \mathbb{N}}$). The proof is due to Tanusevski and we sketch a version of it here, using our setup and language. For the positive finiteness properties, we just need that the complexes $\dlkmodel{G^*}{n}$ become increasingly highly connected. This follows by noting that every simplex fiber of the projection $\dlkmodel{G^*}{n} \to \mathcal{M}(L_n)$ is the join of its vertex fibers, and applying \cite[Theorem~9.1]{quillen78}. For the negative finiteness properties, we claim that there is a sequence of homomorphisms $G\to \Thomp{G^*}\to G$ that composes to the identity. This is sufficient by the Bieri--Eckmann criterion \cite[Proposition~1.2]{bieri74}; see \cite[Proposition~4.1]{bux04}. The first map in the claim is $g\mapsto [1,g,1]$, and the second is $[T_-,(g_1,\dots,g_n),T_+] \mapsto g_1$. One must check that this second map is well defined on equivalence classes under reduction and expansion, and is a homomorphism, but this is not hard to see.
A variation of these groups was recently studied using cloning systems, by Berns-Zieve, Fry, Gillings, and Mathews \cite{berns-zieve14}. With the above setup, they consider cloning maps of the form $(g_1,\dots,g_k,\dots,g_n)\kappa^n_k \mathbin{\vcentcolon =} (g_1,\dots,g_k,\phi(g_k),\dots,g_n)$ where $\phi\in\Aut(G)$. They prove that for $G$ finite, the resulting Thompson group is co$\mathcal{CF}$. If these groups turn out to not embed into $V$, which seems believable when $\phi\ne\id$, then they would be counterexamples to the conjecture that $V$ is universal co$\mathcal{CF}$.
\section{Thompson groups for matrix groups}\label{sec:matrix_groups}
Let $R$ be a unital ring and consider the algebra of $n$-by-$n$ matrices $M_n(R)$. We will define a family of injective functions $M_n(R) \to M_{n+1}(R)$, which will become cloning maps after we restrict to the subgroups of upper triangular matrices $B_n(R)$. Consider the map $\kappa_k$ defined by
\[
\left(\left(
\begin{array}{ccc}
A_{<,<} & A_{<,k} & A_{<,>}\\
A_{k,<} & A_{k,k} & A_{k,>}\\
A_{>,<} & A_{>,k} & A_{>,>}
\end{array}
\right)\right)\kappa_k
=
\left(
\begin{array}{cccc}
A_{<,<} & A_{<,k} & A_{<,k} & A_{<,>}\\
A_{k,<} & A_{k,k} & 0 & 0\\
0 & 0 & A_{k,k} & A_{k,>}\\
A_{>,<} & A_{>,k} & A_{>,k} & A_{>,>}
\end{array}
\right)
\]
where the matrix has a block structure under which the middle column and row are the $k$th column and row of the full matrix respectively. Given the block structure it is not hard to see that $\kappa_k$ is a morphism of monoids, but it generally fails to map invertible elements to invertible elements. We therefore restrict to the groups $B_n(R)$ of invertible upper triangular matrices. Let $B_\infty(R) = \varinjlim B_n(R)$.
\begin{lemma}\label{lem:up_tri_clone}
The trivial morphisms $\rho_n$ and the maps $\kappa_k^n$ defined above describe a properly graded cloning system on $B_*(R)$.
\end{lemma}
It may be noted that the action of $\mathscr{F}$ on $B_\infty(R)$ factors through $\mathscr{H}$, that is $\kappa_\ell \kappa_k = \kappa_k \kappa_{\ell+1}$ even for $\ell = k$.
\begin{proof}
Since $\rho_*$ is trivial, condition~\eqref{item:fcs_cloning_a_product} asks that the cloning maps be group homomorphisms. That $\kappa_k$ is multiplicative and takes $1$ to $1$ is straightforward to check. Also, $A$ is invertible if and only if all the $A_{i,i}$ are units, in which case $(A)\kappa_k$ is also invertible.
To check condition~\eqref{item:fcs_product_of_clonings} it is helpful to note that for any $A\in M_n(R)$, $((A)\kappa_k)_{i,j} = A_{\pi_k(i),\pi_k(j)}$ unless $i = k$ or $i>j$ (here $\pi_k$ is as in Example~\ref{ex:symm_gps}). One can now distinguish cases similar to Example~\ref{ex:symm_gps}. The compatibility condition \eqref{item:fcs_compatibility} is vacuous for trivial $\rho_*$.
To see that the cloning system is properly graded note that $g \in \operatorname{im} \iota_{n,n+1}$ if and only if the last column of $g$ is the vector $e_{n+1}$. If at the same time $g = (h)\kappa_k$ then by the definition of $\kappa_k$ the last column of $h$ has to be $e_n$. Hence $h \in \operatorname{im} \iota_{n-1,n}$.
\end{proof}
Having equipped $B_*(R)$ with a cloning system, we get a generalized Thompson group $\Thomp{B_*(R)}$. Elements are represented by triples $(T_-,A,T_+)$ for trees $T_\pm$ with $n$ leaves and matrices $A\in B_n(R)$, up to reduction and expansion. Figure~\ref{fig:borel_thomp_expansion} gives an example of an element of $\Thomp{B_*(R)}$, represented as a triple and an expansion of that triple.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[line width=0.8pt, scale=0.4]
\begin{scope}
\node at (-3,-1){$\Bigg[$};
\draw
(-2,-2) -- (0,0) -- (2,-2) (0,-2) -- (-1,-1);
\filldraw
(-2,-2) circle (1.5pt) (0,0) circle (1.5pt) (2,-2) circle (1.5pt) (-1,-1) circle (1.5pt) (0,-2) circle (1.5pt);
\node at (2.5,-2){,};
\node at (5,-1){$\begin{pmatrix}
1&2&3\\0&4&5\\0&0&6
\end{pmatrix}$};
\node at (7.5,-2){,};
\begin{scope}[xshift=4in]
\draw
(-2,-2) -- (0,0) -- (2,-2) (0,-2) -- (1,-1);
\filldraw
(-2,-2) circle (1.5pt) (0,0) circle (1.5pt) (2,-2) circle (1.5pt) (1,-1) circle (1.5pt) (0,-2) circle (1.5pt);
\node at (3,-1){$\Bigg]$};
\end{scope}
\end{scope}
\node at (14,-1){$=$};
\begin{scope}[xshift=18cm]
\node at (-3,-1){$\Bigg[$};
\draw
(-2,-2) -- (0,0) -- (2,-2) (0,-2) -- (-1,-1) (-1,-2) -- (-.5,-1.5);
\filldraw
(-2,-2) circle (1.5pt) (0,0) circle (1.5pt) (2,-2) circle (1.5pt) (-1,-1) circle (1.5pt) (0,-2) circle (1.5pt) (-1,-2) circle (1.5pt);
\node at (3,-2){,};
\node at (6.5,-1){$\begin{pmatrix}
1&2&2&3\\0&4&0&0\\0&0&4&5\\0&0&0&6
\end{pmatrix}$};
\node at (9.75,-2){,};
\begin{scope}[xshift=5in]
\draw
(-2,-2) -- (0,0) -- (2,-2) (0,-2) -- (1,-1) (.5,-1.5) -- (1,-2);
\filldraw
(-2,-2) circle (1.5pt) (0,0) circle (1.5pt) (2,-2) circle (1.5pt) (1,-1) circle (1.5pt) (0,-2) circle (1.5pt) (1,-2) circle (1.5pt);
\node at (3,-1){$\Bigg]$};
\end{scope}
\end{scope}
\end{tikzpicture}
\caption[Expansion for matrix groups]{An example of expansion in $\Thomp{B_*(\mathbb{Q})}$.}
\label{fig:borel_thomp_expansion}
\end{figure}
We are interested in finiteness properties of $\Thomp{B_*(R)}$ because of the following examples where the groups $B_*(R)$ themselves have interesting finiteness properties, see \cite[Theorem~A, Remarks~3.6,~3.7]{bux04}.
\begin{theorem}\label{thm:bux}
Let $k$ be a global function field, let $S$ be a finite nonempty set of places and $\mathcal{O}_S$ the ring of $S$-integers. Then $B_n(\mathcal{O}_S)$ is of type~$\text{F}_{\abs{S}-1}$ but not of type~$\text{F}_{\abs{S}}$ for any $n \ge 2$.
\end{theorem}
For instance, when $R=\mathbb{F}_p[t,t^{-1}]$ then $B_n(\mathbb{F}_p[t,t^{-1}])$ is finitely generated but not finitely presented, for $n\ge2$. What is particularly interesting about Theorem~\ref{thm:bux} is that the finiteness properties of $B_n(\mathcal{O}_S)$ depend on $\abs{S}$ but not on $n$.
A class of examples where the finiteness properties do depend on $n$ arises as subgroups of groups of the form $B_n(R)$. Let $\mathit{Ab}_n \le B_{n+1}$ be the group of invertible upper triangular $n+1$-by-$n+1$ matrices whose upper left and lower right entries are $1$. The groups $\mathit{Ab}_n(\mathbb{Z}[1/p])$ were studied by Abels and others and we call them the \emph{Abels groups}. Their finiteness length tends to infinity with $n$ \cite{abels87, brown87}:
\begin{theorem}\label{thm:abels}
For any prime $p$ the group $\mathit{Ab}_n(\mathbb{Z}[1/p])$ is of type~$\text{F}_{n-1}$ but not of type~$\text{F}_n$ for $n \ge 1$.
\end{theorem}
For any ring $R$, the cloning system described above for $B_n(R)$ preserves the Abels groups $\mathit{Ab}_{n-1}(R)$. By restriction we obtain a generalized Thompson group $\Thomp{\mathit{Ab}_{*-1}(R)}$ which we will just denote by $\Thomp{\mathit{Ab}_*(R)}$.
\section{Finiteness properties of Thompson groups for matrix groups}\label{sec:mtx_fin_props}
We will prove below that the finiteness length of $\Thomp{B_*(\mathcal{O}_S)}$ is the same as that of all the $B_n(\mathcal{O}_S)$. For consistency, we can state this as
\[
\phi(\Thomp{B_*(\mathcal{O}_S)}) = \liminf_n \phi(B_n(\mathcal{O}_S))\text{.}
\]
The inequality $\ge$, i.e., that $\Thomp{B_*(\mathcal{O}_S)}$ is of type~$\text{F}_{\abs{S}-1}$, is proved in Section~\ref{sec:mtx_pos_fin_props}, and follows the general strategy outlined in Sections~\ref{sec:spaces} and~\ref{sec:finiteness_props}. In fact, it applies to arbitrary rings. To show the inequality $\le$, i.e., that $\Thomp{B_*(\mathcal{O}_S)}$ is not of type~$\text{FP}_{\abs{S}}$, we develop some new tools in Section~\ref{sec:mtx_neg_fin_props}, and make use of the criterion established in Theorem~\ref{thrm:relative_brown}.
The proof showing the inequality $\ge$ above also applies to $\Thomp{\mathit{Ab}_*(R)}$. Since the right hand side is infinite this time, this directly gives the full equation
\[
\phi(\Thomp{\mathit{Ab}_*(\mathbb{Z}[1/p])}) = \liminf_n \phi(\mathit{Ab}_n(\mathbb{Z}[1/p]))\text{.}
\]
\subsection{Positive finiteness properties}\label{sec:mtx_pos_fin_props}
The first main result of this section is that the group $\Thomp{B_*(R)}$ has all the finiteness properties that the individual groups $B_*(R)$ eventually have:
\begin{theorem}\label{thrm:borel_thomp_pos_fin_props}
$\displaystyle\phi(\Thomp{B_*(R)}) \ge \liminf_n(\phi(B_n(R)))$.
\end{theorem}
In particular, together with Theorem~\ref{thm:bux} this implies:
\begin{corollary}
$\Thomp{B_*(\mathcal{O}_S)}$ is of type~$\text{F}_{\abs{S}-1}$.
\end{corollary}
In view of Proposition~\ref{prop:generic_finiteness}, to prove Theorem~\ref{thrm:borel_thomp_pos_fin_props} it suffices to show that the connectivity of $\dlkmodel{B_*(R)}{n}$ goes to infinity with $n$. In fact, we will induct, so we need to consider a slightly larger class of complexes.
For a spanning subgraph $\Delta$ of the linear graph $L_n$, define $\dlkmodel{B_*(R);\Delta}{n}$ to be the subcomplex of $\dlkmodel{B_*(R)}{n}$ whose elements only use graphs that are subgraphs of $\Delta$. Define $e(\Delta)$ to be the number of edges of $\Delta$. Define $\eta(m)\mathbin{\vcentcolon =} \lfloor\frac{m-1}{4}\rfloor$. Taking $\Delta = L_n$, Theorem~\ref{thrm:borel_thomp_pos_fin_props} will follow from:
\begin{proposition}\label{prop:borel_thomp_fin_props_strong}
$\dlkmodel{B_*(R);\Delta}{n}$ is $(\eta(e(\Delta))-1)$-connected.
\end{proposition}
The base case is that $\dlkmodel{B_*(R);\Delta}{n}$ is non-empty provided $e(\Delta) \ge 1$, which is clearly true.
We need to do a bit of preparation before we can prove the proposition. To work with simplices of $\dlkmodel{B_*(R)}{n}$ it will be helpful to have simple representatives for dangling classes. To define them we have to recall some of the origins of $\dlkmodel{B_*(R)}{n}$: by Observation~\ref{obs:matchings_to_forests} matchings $\Gamma$ of $L_n$ correspond to elementary forests. Using this correspondence, it makes sense to denote the corresponding cloning map by $\kappa_\Gamma$. In fact, since our cloning maps factor through the hedge monoid, we even get a cloning map $\kappa_\Gamma$ for any spanning subgraph $\Gamma$ of $L_n$ using Observation~\ref{obs:graphs_to_hedges}. For the sake of readability, we describe this map explicitly. Let $D_k(\lambda)$ be the $k$-by-$k$ matrix with all diagonal entries $\lambda$ and all other entries $0$. Let $F_{k,\ell}(\lambda)$ be the $k$-by-$\ell$ matrix whose bottom row has all entries $\lambda$ and all other entries are $0$ and let $C_{k,\ell}(\lambda)$ be defined analogously for the top row. Assume that $\Gamma$ has $m$ connected components which we think of as numbered from left to right. Then
\[
\kappa_\Gamma \colon M_m(R) \to M_n(R)
\]
can be described as follows. The image $\kappa_\Gamma(A)$ has a block structure where columns and rows are grouped together if their indices lie in a common component of $\Gamma$. More precisely, the $(i,j)$-block has $k$ rows and $\ell$ columns if the $i$th (respectively $j$th) component of $\Gamma$ has $k$ (respectively $\ell$) vertices. The block is $D_k(A_{i,i})$, $F_{k,\ell}(A_{i,j})$ or $C_{k,\ell}(A_{i,j})$ depending on whether $i = j$, $i<j$, or $i>j$ (see Figure~\ref{fig:cloning}).
\begin{figure}[th]
\begin{align*}
\left(
\begin{array}{ccc}
a_{1,1}&a_{1,2}&a_{1,3}\\
a_{2,1}&a_{2,2}&a_{2,3}\\
a_{3,1}&a_{3,2}&a_{3,3}
\end{array}
\right)
\stackrel{\kappa_\Gamma}{\mapsto}
&\left(
\begin{array}{ccc}
D_2(a_{1,1})&F_{2,4}(a_{1,2})&F_{2,3}(a_{1,3})\\
C_{4,2}(a_{2,1})&D_4(a_{2,2})&F_{4,3}(a_{2,3})\\
C_{3,2}(a_{3,1})&C_{3,4}(a_{3,2})&D_3(a_{3,3})
\end{array}
\right)\\[.5cm]
&
=\hspace{.5cm}
\left(
\begin{array}{c@{}cc|cccc|ccc}
& \tikzmark{t1} & \tikzmark{t2} & \tikzmark{t3} & \tikzmark{t4} & \tikzmark{t5} & \tikzmark{t6} &
\tikzmark{t7} & \tikzmark{t8}& \tikzmark{t9}\\[-.3cm]
\tikzmark{l1}&a_{1,1} & & &&&& &&\\
\tikzmark{l2}& &a_{1,1} & a_{1,2}&a_{1,2}&a_{1,2}&a_{1,2}& a_{1,3}&a_{1,3}&a_{1,3}\\
\hline
\tikzmark{l3}&a_{2,1} &a_{2,1} & a_{2,2}&&&& &&\\
\tikzmark{l4}&& & &a_{2,2}&&& &&\\
\tikzmark{l5}&& & &&a_{2,2}&& &&\\
\tikzmark{l6}&& & &&&a_{2,2}& a_{2,3}&a_{2,3}&a_{2,3}\\
\hline
\tikzmark{l7}&a_{3,1} &a_{3,1} & a_{3,2}&a_{3,2}&a_{3,2}&a_{3,2}& a_{3,3}&&\\
\tikzmark{l8}&& & &&&& &a_{3,3}&\\
\tikzmark{l9}&& & &&&& &&a_{3,3}
\end{array}
\right)
\end{align*}
\tikz[remember picture, overlay,yshift=.5cm]
\filldraw
(pic cs:t1) circle (1.5pt)
(pic cs:t2) circle (1.5pt)
(pic cs:t3) circle (1.5pt)
(pic cs:t4) circle (1.5pt)
(pic cs:t5) circle (1.5pt)
(pic cs:t6) circle (1.5pt)
(pic cs:t7) circle (1.5pt)
(pic cs:t8) circle (1.5pt)
(pic cs:t9) circle (1.5pt);
\tikz[remember picture, overlay,yshift=.5cm]
\draw
(pic cs:t1) -- (pic cs:t2) (pic cs:t3) -- (pic cs:t4) -- (pic cs:t5) -- (pic cs:t6) (pic cs:t7) -- (pic cs:t8) -- (pic cs:t9) ;
\tikz[remember picture, overlay,xshift=-.7cm]
\filldraw
(pic cs:l1) circle (1.5pt)
(pic cs:l2) circle (1.5pt)
(pic cs:l3) circle (1.5pt)
(pic cs:l4) circle (1.5pt)
(pic cs:l5) circle (1.5pt)
(pic cs:l6) circle (1.5pt)
(pic cs:l7) circle (1.5pt)
(pic cs:l8) circle (1.5pt)
(pic cs:l9) circle (1.5pt);
\tikz[remember picture, overlay,xshift=-.7cm]
\draw
(pic cs:l1) -- (pic cs:l2) (pic cs:l3) -- (pic cs:l4) -- (pic cs:l5) -- (pic cs:l6) (pic cs:l7) -- (pic cs:l8) -- (pic cs:l9) ;
\caption[Cloning map of a graph]{Visualization of the cloning map of a graph. The graph $\Gamma$ is drawn on top and to the left of the last matrix.}
\label{fig:cloning}
\end{figure}
Recall that we denote by $e_k$ the $k$th edge of $L_n$. We denote by $J_k$ the matching of $L_n$ whose only edge is $e_k$ (as we did in Lemma~\ref{lem:fin_gen_case}). For a spanning subgraph $\Gamma$ of $L_n$ we say that an index $i$ is \emph{fragile} if $e_i \in \Gamma$ and we say that $i$ is \emph{stable} otherwise. In other words, $i$ is stable if it is the rightmost vertex of its component in $\Gamma$. A matrix $A \in M_n(R)$ is said to be \emph{modeled on} $\Gamma$ if $A_{i,j} = 0$ whenever both $i$ and $j$ are stable in $\Gamma$ (see Figure~\ref{fig:modeled}).
\begin{figure}[th]
\[
\left(
\begin{array}{c@{}cc|cccc|ccc}
& \tikzmark{tt1} & \tikzmark{tt2} & \tikzmark{tt3} & \tikzmark{tt4} & \tikzmark{tt5} & \tikzmark{tt6} &
\tikzmark{tt7} & \tikzmark{tt8}& \tikzmark{tt9}\\[-.3cm]
\tikzmark{ll1}&* &* & * &*&*&*& * &*&*\\
\tikzmark{ll2}&* & 0& * &*&*&0& * &*&0\\
\hline
\tikzmark{ll3}&* &* & * &*&*&*& * &*&*\\
\tikzmark{ll4}&*& *& * &*&*&*& * &*&*\\
\tikzmark{ll5}&*& *& * &*&*&*& * &*&*\\
\tikzmark{ll6}&*&0 & * &*&*&0& * &*&0\\
\hline
\tikzmark{ll7}&* &* &* &*&*&*&* &*&*\\
\tikzmark{ll8}&*&* & * &*&*&*& * &*&*\\
\tikzmark{ll9}&*& 0& * &*&*&0& * &*& 0
\end{array}
\right)
\quad\quad\quad
\left(
\begin{array}{c@{}cc|cccc|ccc}
& \tikzmark{ttt1} & \tikzmark{ttt2} & \tikzmark{ttt3} & \tikzmark{ttt4} & \tikzmark{ttt5} & \tikzmark{ttt6} &
\tikzmark{ttt7} & \tikzmark{ttt8}& \tikzmark{ttt9}\\[-.3cm]
\tikzmark{lll1}&* &* & * &*&*&*& * &*&*\\
\tikzmark{lll2}& & 1& * &*&*&0& * &*&0\\
\hline
\tikzmark{lll3}& & & * &*&*&*& * &*&*\\
\tikzmark{lll4}&& & &*&*&*& * &*&*\\
\tikzmark{lll5}&& & &&*&*& * &*&*\\
\tikzmark{lll6}&& & &&&1& * &*&0\\
\hline
\tikzmark{lll7}& & & &&&&* &*&*\\
\tikzmark{lll8}&& & &&&& &*&*\\
\tikzmark{lll9}&& & &&&& && 1
\end{array}
\right)
\]
\tikz[remember picture, overlay,yshift=.5cm]
\filldraw
(pic cs:tt1) circle (1.5pt)
(pic cs:tt2) circle (1.5pt)
(pic cs:tt3) circle (1.5pt)
(pic cs:tt4) circle (1.5pt)
(pic cs:tt5) circle (1.5pt)
(pic cs:tt6) circle (1.5pt)
(pic cs:tt7) circle (1.5pt)
(pic cs:tt8) circle (1.5pt)
(pic cs:tt9) circle (1.5pt);
\tikz[remember picture, overlay,yshift=.5cm]
\draw
(pic cs:tt1) -- (pic cs:tt2) (pic cs:tt3) -- (pic cs:tt4) -- (pic cs:tt5) -- (pic cs:tt6) (pic cs:tt7) -- (pic cs:tt8) -- (pic cs:tt9) ;
\tikz[remember picture, overlay,xshift=-.7cm]
\filldraw
(pic cs:ll1) circle (1.5pt)
(pic cs:ll2) circle (1.5pt)
(pic cs:ll3) circle (1.5pt)
(pic cs:ll4) circle (1.5pt)
(pic cs:ll5) circle (1.5pt)
(pic cs:ll6) circle (1.5pt)
(pic cs:ll7) circle (1.5pt)
(pic cs:ll8) circle (1.5pt)
(pic cs:ll9) circle (1.5pt);
\tikz[remember picture, overlay,xshift=-.7cm]
\draw
(pic cs:ll1) -- (pic cs:ll2) (pic cs:ll3) -- (pic cs:ll4) -- (pic cs:ll5) -- (pic cs:ll6) (pic cs:ll7) -- (pic cs:ll8) -- (pic cs:ll9) ;
\tikz[remember picture, overlay,yshift=.5cm]
\filldraw
(pic cs:ttt1) circle (1.5pt)
(pic cs:ttt2) circle (1.5pt)
(pic cs:ttt3) circle (1.5pt)
(pic cs:ttt4) circle (1.5pt)
(pic cs:ttt5) circle (1.5pt)
(pic cs:ttt6) circle (1.5pt)
(pic cs:ttt7) circle (1.5pt)
(pic cs:ttt8) circle (1.5pt)
(pic cs:ttt9) circle (1.5pt);
\tikz[remember picture, overlay,yshift=.5cm]
\draw
(pic cs:ttt1) -- (pic cs:ttt2) (pic cs:ttt3) -- (pic cs:ttt4) -- (pic cs:ttt5) -- (pic cs:ttt6) (pic cs:ttt7) -- (pic cs:ttt8) -- (pic cs:ttt9) ;
\tikz[remember picture, overlay,xshift=-.7cm]
\filldraw
(pic cs:lll1) circle (1.5pt)
(pic cs:lll2) circle (1.5pt)
(pic cs:lll3) circle (1.5pt)
(pic cs:lll4) circle (1.5pt)
(pic cs:lll5) circle (1.5pt)
(pic cs:lll6) circle (1.5pt)
(pic cs:lll7) circle (1.5pt)
(pic cs:lll8) circle (1.5pt)
(pic cs:lll9) circle (1.5pt);
\tikz[remember picture, overlay,xshift=-.7cm]
\draw
(pic cs:lll1) -- (pic cs:lll2) (pic cs:lll3) -- (pic cs:lll4) -- (pic cs:lll5) -- (pic cs:lll6) (pic cs:lll7) -- (pic cs:lll8) -- (pic cs:lll9) ;
\caption[Matrix reduced relative to a graph]{A matrix that is modeled on a graph (left) and an upper triangular matrix that is reduced relative to a graph (right).}
\label{fig:modeled}
\end{figure}
\begin{lemma}\label{lem:borel_reduction}
Let $\Gamma$ be a spanning subgraph of $L_n$ with $m$ components and let $A \in B_n(R)$. There is a representative $B$ in the coset $A(B_m(R))\kappa_\Gamma$ such that $B-I_n$ is modeled on $\Gamma$. Moreover, rows of zeroes in $A$ (off the diagonal) can be preserved in $B$.
\end{lemma}
\begin{proof}
We inductively multiply $A$ on the right by matrices in $(B_m(R))\kappa_\Gamma$ to eventually obtain $B$. Let $E_{i,j}(\lambda)$ denote the matrix that coincides with the identity matrix in all entries but $(i,j)$ and is $\lambda$ there.
We begin by clearing the diagonal. Let $i$ be the (stable) rightmost vertex of the $k$th component of $\Gamma$ and let $\lambda = A_{i,i}^{-1}$. Then $A(E_{k,k}(\lambda))\kappa_\Gamma$ has $(i,i)$-entry one and no other diagonal entry with stable indices was affected.
Now we clear the region above the diagonal. We proceed inductively by rows and columns. Let $(i,j)$ be the (lexicographically) minimal pair of stable indices of $\Gamma$ such that $0 \ne A_{i,j} \mathbin{=\vcentcolon} - \lambda$. Let $i$ and $j$ lie in the $k$th respectively $\ell$th component of $\Gamma$. Then $A(E_{k,\ell}(\lambda))\kappa_{m}$ has $(i,j)$-entry zero and no other entry with stable indices was affected.
For the last statement assume that the $i$th row of $A$ was zero off the diagonal. Then none of the matrices by which we multiplied had a nonzero off-diagonal entry in the $i$th row. If $i$ is fragile no such matrix even lies in $(B_m(R))\kappa_\Gamma$. If $i$ is stable then the only matrices we might have used of this form were meant to clear the $i$th row, but since the entries there were zero, nothing happened in these steps.
\end{proof}
\begin{corollary}[Reduced form]\label{cor:borel_reduced_form}
Every simplex in $\dlkmodel{B_*(R)}{n}$ has a representative $(A,\Gamma)$ such that the matrix $A - I_n$ is modeled on $\Gamma$. \qed
\end{corollary}
We will refer to a matrix $A\in B_n(R)$ as being \emph{reduced relative $\Gamma$} if it satisfies the conclusion of Corollary~\ref{cor:borel_reduced_form}.
The next sequence of lemmas is a gradual checking of the hypotheses of Proposition~\ref{prop:putman_flow}, still in the context of an induction proof, ultimately leading to a proof of Proposition~\ref{prop:borel_thomp_fin_props_strong}.
\begin{lemma}[Flag complex]\label{lem:mtx_cpx_flag}
$\dlkmodel{B_*(R);\Delta}{n}$ is a flag complex.
\end{lemma}
\begin{proof}
We need to show that any collection of vertices $\{v_1,\dots,v_r\}$ that are pairwise connected by edges spans a simplex. We induct on $r$ (with the trivial base case of $r\le 2$). Each vertex $v_i$ in our collection is of the form $[A_i,J_{k_i}]$ for $J_{k_i}$ some single-edge subgraphs of $\Delta$. Assume without loss of generality that $k_1<k_i$ for all $1<i\le r$, so $v_1$ is the vertex whose lone merge occurs farthest to the left among all the $v_i$. By induction, $v_2,\dots,v_r$ span a simplex, $\sigma$. Thanks to the action of $B_n(R)$, without loss of generality $v_1$ is the vertex $[I_n,J_k]$, where we have set $k\mathbin{\vcentcolon =} k_1$ for brevity.
Represent $\sigma=[A,\Gamma]$ with $A$ reduced relative to $\Gamma$. Since $k$ is less than the index of any edge of $\Gamma$, we know that the $k$th column of $A-I_n$ is all zeros. Since $v_1$ shares an edge with every vertex of $\sigma$, we know that in fact $k$ is even less than the index of any edge of $\Gamma$, minus one. Hence the $(k+1)$st column of $A-I_n$ is similarly all zeros. Our goal is to show that $A\in \operatorname{im} \kappa_k$, since then $\sigma$ and $v_1$ will share a simplex. Thanks to the setup, it suffices to show that the $k$th row of $A-I_n$ is all zeros. Since $A$ is reduced relative $\Gamma$, non-zero entries of $A-I_n$ may only possibly occur in columns indexed by $k_2,\dots,k_r$.
For each vertex $[A,J_{k_i}]$, $2\le i\le r$, of $\sigma$, let $A_i$ be such that $[A_i,J_{k_i}]=[A,J_{k_i}]$ and $A_i$ is reduced relative $J_{k_i}$. Let $\ell\in \{k_2,\dots,k_r\}$. Observe that $A_\ell$ is obtained from $A$ by right multiplication by an element $D$ of $\operatorname{im}(\kappa_\ell)$. For $1\le i\le n$ denote by $M_{(i,*)}$ the $i$th row of an $n$-by-$n$ matrix $M$, and by $M_{(*,i)}$ the $i$th column. When we multiply by $D$ to get $AD=A_\ell$, the $(k,\ell)$-entry of $A_\ell$ is $A_{(k,*)}D_{(*,\ell)}$ and the $(k,\ell+1)$-entry is $A_{(k,*)}D_{(*,\ell+1)}$. Since $A_\ell$ is reduced relative $J_\ell$, we know that its $(k,\ell+1)$-entry must be $0$. Also since $D\in\operatorname{im}(\kappa_{J_\ell})$, we have $D_{(*,\ell)}=D_{(*,\ell+1)}+d(e_\ell-e_{\ell+1})$ for some $d\in R^\times$. Let $a$ denote the $(k,\ell)$-entry of $A$, and note that the $(k,\ell+1)$-entry of $A$ is $0$. We calculate that the $(k,\ell)$-entry of $A_\ell$ is
\begin{align*}
A_{(k,*)}D_{(*,\ell)} &= A_{(k,*)}(D_{(*,\ell+1)}+d(e_\ell-e_{\ell+1})) \\
&= A_{(k,*)}(d(e_\ell-e_{\ell+1})) \\
&= da \text{.}
\end{align*}
Since $d$ is a unit, this shows that the $(k,\ell)$-entry of $A_\ell$ is zero if and only if the $(k,\ell)$-entry of $A$ is zero. By the same argument just given, this statement remains true with $A_\ell$ replaced by $A_\ell D$ for any $D\in\operatorname{im}(\kappa_\ell)$. But by assumption $v_1$ shares an edge with $v_\ell$, and so some such $A_\ell D$ must have $(k,\ell)$-entry zero. We conclude that $A$ has $(k,\ell)$-entry zero. Since $\ell$ was arbitrary, the $k$th row of $A-I_n$ is all zeros and so $v_1$ and $\sigma$ share a simplex.
\end{proof}
Let $\Delta_0\mathbin{\vcentcolon =} \Delta\setminus\{e_1\cup e_2\}$, and consider $\dlkmodel{B_*(R);\Delta_0}{n}$ as a subcomplex of $\dlkmodel{B_*(R);\Delta}{n}$. For a vertex $[A,J_k] \in \dlkmodel{B_*(R);\Delta_0}{n}$ we write $\lk_0([A,J_k])$ for the link in $\dlkmodel{B_*(R);\Delta_0}{n}$, to differentiate from the link in $\dlkmodel{B_*(R);\Delta}{n}$ which is just denoted $\lk([A,J_k])$. To prove Proposition~\ref{prop:borel_thomp_fin_props_strong} we follow the strategy outlined by Proposition~\ref{prop:putman_flow}: we want to show that $\dlkmodel{B_*(R);\Delta_0}{n}$ is $(\eta(e(\Delta))-2)$-connected, that $(\dlkmodel{B_*(R);\Delta}{n},\dlkmodel{B_*(R);\Delta_0}{n})$ is $(\eta(e(\Delta))-1)$-connected and that there is a vertex satisfying condition \eqref{item:exchange1}. That vertex is $w \mathbin{\vcentcolon =} [I_n,J_1]$ in our case. The following statements (up to the proof of Proposition~\ref{prop:borel_thomp_fin_props_strong}) are part of an induction, so we assume that Proposition~\ref{prop:borel_thomp_fin_props_strong} has been proven for graphs $\Delta'$ with $e(\Delta') < e(\Delta)$ and intend to prove it for $\Delta$.
\begin{lemma}[Links are lower rank complexes]\label{lem:links_are_cpxes}
Let $\sigma$ be a simplex of dimension $d \ge 0$ in $\dlkmodel{B_*(R);\Delta}{n}$. Then $\lk(\sigma)$ is isomorphic to a complex of the form $\dlkmodel{B_{*}(R);\Delta'}{n-(d+1)}$ where $\Delta'$ is a spanning subgraph of $L_{n-(d+1)}$ with at least $e(\Delta)-3d-3$ edges. In particular, it is $(\eta(e(\Delta)-3d-3)-1)$-connected by induction.
\end{lemma}
\begin{proof}
The simplex $\sigma$ is of the form $[g,\Gamma]$ with $g \in B_n(R)$ and $\Gamma \subseteq \Delta$. If it has dimension $d$ then $\Gamma$ has $d+1$ edges, say $e_{i_1},\ldots,e_{i_{d+1}}$. Using the left action of $B_n(R)$ we may assume that $g = 1$. Then $\lk(\sigma)$ is $\dlkmodel{(B_{*}(R))\kappa_\Gamma;\Delta^\sharp}{n}$, where $\Delta^\sharp$ is $\Delta$ with the edges $e_{i_j-1}, e_{i_j}, e_{i_j+1}$ removed for each $1 \le j \le d+1$. In particular $\Delta^\sharp$ has at least $e(\Delta)-3d-3$ edges. Now consider the map $b_\Gamma \colon L_n\to L_{n-(d+1)}$ given by blowing down the edges of $\Gamma$. The image of $\Delta^\sharp$ under $b_\Gamma$ is what we will call $\Delta'$. Note that $\Delta'$ still has at least $e(\Delta)-3d-3$ edges. Since $\kappa_\Gamma$ is injective, we may now apply $\kappa_\Gamma^{-1}$ paired with $b_\Gamma$ to $\dlkmodel{(B_{*}(R))\kappa_\Gamma;\Delta^\sharp}{n}$ and get an isomorphism to $\dlkmodel{B_{*}(R);\Delta'}{n-(d+1)}$.
\end{proof}
\begin{lemma}\label{lem:borel_relative_connectivity}
The pair $(\dlkmodel{B_*(R);\Delta}{n},\dlkmodel{B_*(R);\Delta_0}{n})$ is $(\eta(e(\Delta))-1)$-connected.
\end{lemma}
\begin{proof}
Note that for any vertex of $\dlkmodel{B_*(R);\Delta}{n} \setminus \dlkmodel{B_*(R);\Delta_0}{n}$, the entire link of the vertex lies in $\dlkmodel{B_*(R);\Delta_0}{n}$. Hence the function sending vertices of the former to $1$ and vertices of the latter to $0$ yields a Morse function in the sense of Section~\ref{sec:finiteness_props}, and to prove the statement we need only show that links of vertices in $\dlkmodel{B_*(R);\Delta}{n} \setminus \dlkmodel{B_*(R);\Delta_0}{n}$ are $(\eta(e(\Delta))-2)$-connected. By Lemma~\ref{lem:links_are_cpxes}, each descending link is isomorphic to a complex of the form $\dlkmodel{B_{*}(R);\Delta'}{n-1}$ for $\Delta'$ a graph with at least $e(\Delta)-3$ edges. By induction, these are $(\eta(e(\Delta))-2)$-connected as desired.
\end{proof}
In addition to the subcomplex $\dlkmodel{B_*(R);\Delta_0}{n}$ we will now need to consider $\dlkmodel{B_*(R);\Delta_1}{n}$ where $\Delta_1\mathbin{\vcentcolon =} \Delta\setminus\{e_1\}$. We will write links in this complex using the symbol $\lk_1$.
\begin{lemma}[Shared links]\label{lem:mut_lks_are_lks}
Let $k>2$ and let $A$ be reduced relative $J_k$. Let $A'$ be obtained from $A$ by setting the $(1,k)$-entry to $0$. Then $\lk_1([A,J_k]) \subseteq \lk_1([A',J_k])$ and $[A',J_k] \in \lk w$.
\end{lemma}
\begin{proof}
As a first observation, note that since $A$ is reduced relative $J_k$ and $k>2$, the $(1,1)$-entry and $(2,2)$-entry of $A$ are both $1$, and the entries of the top row of $A$ past the first entry is all $0$'s except possibly in the $k$th column. Let $-\lambda$ be the $(1,k)$-entry of $A$, and note that $A' = A E_{1k}(\lambda)$. The first row of $A'$ is now $(1,0,\ldots,0)$ and the $(2,2)$-entry is $1$, which tells us that $A' \in (B_{n-1}(R))\kappa_1$. Hence $[A',J_k] \in \lk_0 w$.
To see that $\lk_1 ([A,J_k]) \subseteq \lk_1([A',J_k])$ we first multiply by $A^{-1}$ from the left and are reduced to showing that $\lk_1 ([I_n,J_k]) \subseteq \lk_1([E_{1k}(\lambda),J_k])$. An arbitrary simplex of $\lk_1([I_n,J_k])$ is of the form $[B,\Gamma]$, with $B\in\operatorname{im}(\kappa_k)$ and $\Gamma$ not containing any of $e_1$, $e_{k-1}$, $e_k$, or $e_{k+1}$. Note that the $k$th row of $B$ is zero off the diagonal. By Lemma~\ref{lem:borel_reduction} there is a $B' \in B \operatorname{im}(\kappa_\Gamma)$ that is reduced relative $\Gamma$ and has $k$th row zero off the diagonal. We have $[B',\Gamma]=[B,\Gamma]$. Since $e_1 \not\in \Gamma$ and $B'$ is reduced relative $\Gamma$, the first column of $B'$ is $e_1$.
We now claim that $B'$ commutes with $E_{1k}(\lambda)$. Indeed, left multiplication by $E_{1k}(\lambda)$ is the row operation $r_1\mapsto r_1+\lambda r_k$, and right multiplication by $E_{1k}(\lambda)$ is the column operation $c_k\mapsto c_k+\lambda c_1$. For our $B'$, both of these operations change the $(1,k)$-entry by adding $\lambda$ to it, and change no other entries. This proves the claim.
Now we have
\[
[B,\Gamma] =[B',\Gamma]\\
=[E_{1k}(\lambda)B'E_{1k}(-\lambda),\Gamma]\\
=[E_{1k}(\lambda)B',\Gamma]\\
=[E_{1k}(\lambda)B,\Gamma]\text{.}
\]
The second to last step works since $E_{1k}(-\lambda)\in\operatorname{im}(\kappa_\Gamma)$ by virtue of $e_{k-1},e_k \not \in \Gamma$. This shows that our arbitrary simplex of $\lk_1([I_n,J_k])$ is also in $\lk_1([E_{1k}(\lambda),J_k])$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:borel_thomp_fin_props_strong}]
We want to apply Proposition~\ref{prop:putman_flow}. The complexes are $X = \dlkmodel{B_*(R);\Delta}{n}$, $X_1 = \dlkmodel{B_*(R);\Delta_1}{n}$ and $X_0 = \dlkmodel{B_*(R);\Delta_0}{n}$, and $k = \eta(e(\Delta))-1$. We check the assumptions. The pair $(\dlkmodel{B_*(R);\Delta}{n},\dlkmodel{B_*(R);\Delta_0}{n})$ is $k$-connected by Lemma~\ref{lem:borel_relative_connectivity}. Since $X$ is a flag complex (Lemma~\ref{lem:mtx_cpx_flag}), it is conical at every vertex, in particular at our vertex $w=[I_n,J_1]$. The complex $\dlkmodel{B_*(R);\Delta_0}{n}$ is $(\eta(e(\Delta_0))-1)$-connected by induction. This is sufficient because $\eta(e(\Delta_0)) - 1 \ge \eta(e(\Delta) - 2) - 1 \ge \eta(e(\Delta)) - 2 = k -1$. The link of a $d$-simplex is $(\eta(e(\Delta)-3d-3)-1)$-connected by Lemma~\ref{lem:links_are_cpxes}. This is sufficient because $\eta(e(\Delta)-3d-3) - 1\ge \eta(e(\Delta)) - d - 2 = k - d - 1$. Finally condition \eqref{item:exchange1} is satisfied by Lemma~\ref{lem:mut_lks_are_lks} where $\lk_1([A,J_k])$ is at least $(\eta(e(\Delta)-4)-1)$-connected and $\eta(e(\Delta)-4) - 1 = \eta(e(\Delta)) - 2 = k - 1$ as desired.
\end{proof}
Shifting focus to the Abels groups, thanks to the flexibility of Lemma~\ref{lem:borel_reduction}, the above arguments also show high connectivity of $\dlkmodel{\mathit{Ab}_*(\mathbb{Z}[1/p])}{n}$, and using Proposition~\ref{prop:generic_finiteness} and Theorem~\ref{thm:abels} we conclude:
\begin{theorem}\label{thrm:thomp_abels_Finfty}
$\Thomp{\mathit{Ab}_*(\mathbb{Z}[1/p])}$ is of type~$\text{F}_\infty$.
\end{theorem}
This, despite none of the $\mathit{Ab}_n(\mathbb{Z}[1/p])$ individually being $\text{F}_\infty$.
\medskip
The remaining question is whether $\phi(\Thomp{B_*(R)})=\liminf_n(\phi(B_n(R)))$, that is whether negative finiteness properties of the $B_n(R)$ can impose negative finiteness properties on $\Thomp{B_*(R)}$. For $R$ the ring of $S$-integers of a global function field, we will answer this question affirmatively in the next section.
Before we do that, we need to treat one more relative of the family $B_n(R)$: Let $B_n^2$ be the normal subgroup of $B_n$ consisting of matrices that differ from the identity only from the second off-diagonal on (the second term of the lower central series), and let $\bar{B}_n \mathbin{\vcentcolon =} B_n/B_n^2$ be the quotient group. Set $\nu(n) = \lfloor \frac{n-2}{3} \rfloor$. One could check that the above proof for $B_*$ goes through for the family $\bar{B}_*$ as well, but instead we will prove directly:
\begin{proposition}
\label{prop:barb_thomp_pos_fin_props}
The descending link $\dlkmodel{\bar{B}_*(R)}{n}$ is $(\nu(n)-1)$-connected. Thus $\displaystyle \phi(\Thomp{\bar{B}_*(R)}) \ge \liminf_n \phi(\bar{B}_*(R))$.
\end{proposition}
\begin{proof}
Using reductions as in Lemma~\ref{lem:borel_reduction} one can see the following: every simplex in $\dlkmodel{\bar{B}_*(R)}{n}$ has a representative $[A,\Gamma]$ where the matrix $A$ has a diagonal block of the form
\[
\left(
\begin{array}{cc}
* & *\\
& 1
\end{array}
\right)
\]
above every edge of $\Gamma$ and otherwise equals the identity matrix (here the representative is modulo dangling as well as modulo $B_n^2(R)$). What makes this case particularly easy is that this representative is unique. That is, we may think of $\dlkmodel{\bar{B}_*(R)}{n}$ as consisting of pairs $(A,\Gamma)$ where $A$ is as above and the face relation is given by removing an edge of $\Gamma$ and turning the diagonal block above it into an identity block.
Let $sL_n$ denote the linear graph with vertices $\{1,\ldots,n\}$ and with every pair of adjacent vertices $i$ and $i+1$ connected by $s$ distinct edges. By what we just said, $\dlkmodel{\bar{B}_*(R)}{n}$ is isomorphic to the matching complex $\mathcal{M}(sL_n)$ where $s = \abs{R^* \times R}$. There is an obvious map $\mathcal{M}(sL_n) \to \mathcal{M}(L_n)$. The fiber of this map over a $k$-simplex is a $(k+1)$-fold join of $s$-element sets, thus $k$-spherical. Moreover $\mathcal{M}(L_n)$ is $(\nu(n)-1)$-connected by \cite[Proposition~11.16]{kozlov08} (and links in $\mathcal{M}(L_n)$ are highly connected as well, being joins of lower-rank copies of the complex). Thus we can apply \cite[Theorem~9.1]{quillen78} to conclude that $\mathcal{M}(sL_n)$ is $(\nu(n)-1)$-connected.
\end{proof}
As a remark, this simple approach for $\bar{B}_*(R)$ would not have worked for $B_*(R)$, since the analogous fibers are not joins of vertex fibers.
\subsection{Negative finiteness properties}\label{sec:mtx_neg_fin_props}
In the last section we saw that for any $R$, the generalized Thompson group $\Thomp{B_*(R)}$ is of type~$\text{F}_n$ if all but finitely many $B_k(R)$ are. In this section we prove the converse in the case we are most interested in (cf.~\cite{bux04}): Let $k$ be a global function field and let $S$ be a non-empty set of places. Denote by $\mathcal{O}_S$ the ring of $S$-integers in $k$.
\begin{theorem}\label{thrm:borel_thomp_neg_fin_props}
The group $\Thomp{B_*(\mathcal{O}_S)}$ not of type~$\text{FP}_{\abs{S}}$.
\end{theorem}
\begin{remark}
Unlike the positive statement from the previous section, for the proof of Theorem~\ref{thrm:borel_thomp_full_fin_props} we cannot just use the \emph{results} from \cite{bux04} but have to use parts of the \emph{proof}. By using the more substantial parts of the proof, it is quite possible that the setup of this section could be used to prove the positive finiteness properties as well, but we will not do so.
\end{remark}
We will actually prove first that $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ is not of type~$\text{FP}_{\abs{S}}$. We then use the result from Section~\ref{sec:relative_brown} to deduce Theorem~\ref{thrm:borel_thomp_full_fin_props}. Instead of the Stein--Farley complex on which $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ acts with stabilizers isomorphic to the $\bar{B}_*(\mathcal{O}_S)$ we will construct a new space $Y$ for which the stabilizers are themselves generalized Thompson groups of smaller cloning systems. In particular the stabilizers on $Y$ will have good finiteness properties and the negative finiteness properties of the $\bar{B}_*(\mathcal{O}_S)$ are reflected in bad connectivity properties.
\medskip
For any place $s \in S$ denote by $k_s$ the completion of $k$ at $s$, and by $\mathcal{O}_s$ the ring of integers of $k_s$. As before we let $B_n$ be the linear algebraic group of invertible upper triangular $n$-by-$n$ matrices, let $B_n^2$ be the normal subgroup of matrices that differ from the identity only from the second off-diagonal on, and let $\bar{B}_n \mathbin{\vcentcolon =} B_n/B_n^2$ be the quotient group. Let $Z_n \le B_n$ be the group of homotheties, i.e., scalar multiples of the identity matrix, and let $\mathit{PB}_2 = B_2/Z_2$.
All of this is relevant to us for the following reason: For any of the local fields $k_s$ the group $\PGL_2(k_s)$ admits a Bruhat--Tits tree $V_s$ on which it acts properly. Since $\mathcal{O}_S$ is discrete as a subset of $\prod_{s \in S} k_s$ when embedded diagonally, we get a properly discontinuous action of $\PGL_2(\mathcal{O}_S)$ on
$$V \mathbin{\vcentcolon =} \prod_{s \in S} V_s \text{.}$$
Our goal is to use this action to understand finiteness properties of $\Thomp{\bar{B}_*(\mathcal{O}_S)}$. Note that the group $\PGL_2(\mathcal{O}_s)$ is the stabilizer of a vertex in $V_s$, call it $z_s$. Define $z \mathbin{\vcentcolon =} (z_s)_{s \in S}$, so $z$ is a vertex in $V$.
Denote the quotient morphism from $B_n$ to $\bar{B}_n$ by $\bar{~} \colon B_n \to \bar{B}_n \text{, } g \mapsto \bar{g}$. For $1 \le i \le n-1$ let $\pi_i$ denote the homomorphism $\bar{B}_n \to \mathit{PB}_2, [A] \mapsto [A_i]$ where $A_i$ is the $i$th diagonal $2$-by-$2$ block of $A$. For brevity we denote the composition $\pi_i \circ \bar{~}$ by $\bar{\pi}_i$. Now for any $i$, $1 \le i \le n-1$ consider the composition
\[
\alpha_i \colon \bar{B}_n(\mathcal{O}_S) \to \prod_{s \in S} \bar{B}_n(k_s) \stackrel{\prod \pi_i(k_s)}{\to} \prod_{s \in S} \mathit{PB}_2(k_s)
\]
where the first morphism is induced by the diagonal inclusion $\mathcal{O}_S \to \prod_{s \in S} k_s$.
Define
$$K_n \mathbin{\vcentcolon =} \bigcap_{1 \le i \le n-1} \alpha_i^{-1}\left(\prod_{s \in S} \mathit{PB}_2(\mathcal{O}_s)\right)\text{.}$$
\medskip
\begin{lemma}\label{lem:good_stab}
The group $K_n$ is of type~$\text{F}_\infty$.
\end{lemma}
\begin{remark}
The importance of the Lemma lies in the fact that the groups $K_n$ will appear in stabilizers of an action of $\Thomp{\bar{B}_*(\mathcal{O}_S)}$. It is worth noting that the statement does not remain true if $\bar{B}_n(\mathcal{O}_S)$ is replaced by $B_n(\mathcal{O}_S)$ so that the strategy does not immediately carry over to $\Thomp{B_*(\mathcal{O}_S)}$. Instead we will have to apply Theorem~\ref{thrm:relative_brown} in the end to conclude that $\Thomp{B_*(\mathcal{O}_S)}$ is not of type~$\text{FP}_{\abs{S}}$.
\end{remark}
\begin{proof}[Proof of Lemma~\ref{lem:good_stab}]
We first study the map $\pi_i(k_s) \colon \bar{B}_n(k_s) \to \mathit{PB}_2(k_s)$. The kernel $N_i(k_s)$ is determined by the conditions that the $(i,i+1)$-entry of a matrix is $0$ and that the $(i,i)$ and the $(i+1,i+1)$-entry coincide. The inverse image of $\mathit{PB}_2(\mathcal{O}_s)$ under $\pi_i(k_s)$ is thus generated by $N_i(k_s)$ and a copy of $B_2(\mathcal{O}_s)$. Intersecting over all $i$, we find that $\bigcap_{i} \pi_i(k_s)^{-1}(\mathit{PB}_2(\mathcal{O}_s)) = Z_n(k_s) \bar{B}_n(\mathcal{O}_s)$.
The intersection of this group with $\bar{B}_n(\mathcal{O}_S)$ is $Z_n(\mathcal{O}_S)\bar{B}_n(\mathcal{O}_{S \setminus \{s\}})$. Intersecting over all $s\in S$ we find that $K_n = Z_n(\mathcal{O}_S)\bar{B}_n(\ell)$ where $\ell \mathbin{\vcentcolon =} \mathcal{O}_\emptyset$ is the coefficient field of $k$, which is finite. In particular $\bar{B}_n(\ell)$ is finite and of type~$\text{F}_\infty$. By the Dirichlet Unit Theorem, as extended to $S$-units by Hasse and Chevalley, $Z_n(\mathcal{O}_S)$ is finitely generated abelian and so of type~$\text{F}_\infty$. Since $K_n$ is a central product of these groups, this finishes the proof.
\end{proof}
Now consider the action of $\bar{B}_n(\mathcal{O}_S)$ on $V^{n-1}$ via the maps $\alpha_i$.
\begin{corollary}\label{cor:all_stabs_F_infty}
The stabilizers for the action of $\bar{B}_n(\mathcal{O}_S)$ on $V^{n-1}$ are all of type~$\text{F}_\infty$.
\end{corollary}
\begin{proof}
The group $K_n$ is precisely the stabilizer of $(z,\dots,z)\in V^{n-1}$. Since the product of trees $V^{n-1}$ is locally finite and the action is proper, every stabilizer is commensurable to $K_n$ and therefore of type~$\text{F}_\infty$ as well.
\end{proof}
We are about to define a space $Y$ for $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ to act on. The advantage over the Stein--Farley complex will be that the stabilizers have better finiteness properties. Let $D = \mathbb{Z}[1/2] \cap (0,1)$ be the set of dyadic points in $(0,1)$. Let $V^D$ be the set of all maps $D \to V$. We will usually regard these elements as tuples; that is, we write $x_q$ for the value of $x \in V^D$ at $q \in D$ and sometimes we write $x$ as $(x_q)_{q \in D}$. Let
\[
Y \mathbin{\vcentcolon =} V^{(D)}
\]
be the subset consisting of those maps that evaluate to $z$ at all but finitely many points. An alternative description is as a direct limit $\varinjlim_{I \subseteq D \text{ finite}} V^I$. Note that this set is naturally equipped with a (unique) topology: the topology induced from the product topology and the CW topology coincide.
Note that Thompson's group $F$ acts on $D$ from the right, via $q.f = f^{-1}(q)$ for $f \in F$ and $q \in D$. To describe this action in terms of paired tree diagrams, note that every point in $D$ corresponds to a caret in the leafless rooted binary tree. Thus every finite rooted binary tree $T$ determines a finite subset $D(T)$ of $D$, namely that consisting of points that correspond to its carets. An element $[T,U]$ of $F$ takes $D(T)$ to $D(U)$ (preserving the order) and is linear between these break points.
As a consequence, $F$ acts from the left on the set $V^D$ via $(f.x)_q = x_{q.f}$ where $x \in V^D$, $q \in D$ and $f \in F$. Clearly this induces an action of $F$ on $Y$. Explicitly, the action of $F$ on $Y$ satisfies
\[
([T,U].x)_{t_i} = x_{u_i}
\]
where $D(T) = \{t_1 < \ldots < t_{n-1}\}$ and $D(U) = \{u_1 < \ldots < u_{n-1}\}$. Away from the break points, the values are interpolated linearly: $[T,U].x_{s t_i + (1-s) t_{i+1}} = x_{s u_i + (1-s) u_{i+1}}$.
There is also an action of $\Thkern{\bar{B}_*(\mathcal{O}_S)}$ on $Y$ which is given as follows: if $T$ is a finite rooted binary tree and $D(T) = \{q_1 < \ldots < q_{n-1}\}$ then
\[
([T,g,T].x)_q = \left\{
\begin{array}{ll}
\alpha_i(g).x_{q_i} & \text{if }q = q_i\\
x_q & \text{else.}
\end{array}
\right.
\]
This is just the action obtained by taking the direct limit over the actions of $\bar{B}_T(\mathcal{O}_S)$ on $V^{D(T)}$.
These actions are compatible and so give an action of $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ on $Y$, which is given by
\[
([T,g,U].x)_{t_i} = \alpha_i(g).x_{u_i}
\]
and $([T,g,U].x)_t = ([T,U].x)_t$ for $t \not\in D(T)$; see Figure~\ref{fig:action_on_Y}.
\newsavebox{\barb}
\newsavebox{\barc}
\savebox{\barb}{$\bar{b} = \left(\begin{smallmatrix}1 & 2\\&4\end{smallmatrix}\right)b$}
\savebox{\barc}{$\bar{c} = \left(\begin{smallmatrix}4 & 5\\&6\end{smallmatrix}\right)c$}
\begin{figure}[htb]
\begin{tikzpicture}[scale=0.3,line width=0.8pt]
\begin{scope}
\draw
(-2,-2) -- (0,0) -- (2,-2) (0,-2) -- (-1,-1);
\filldraw
(-2,-2) circle (1.5pt) (2,-2) circle (1.5pt) (0,-2) circle (1.5pt);
\end{scope}
\begin{scope}[yshift=-3.5cm]
\node at (0,-1){$\begin{pmatrix}
1&2& \\ &4&5\\ & &6
\end{pmatrix}$};
\end{scope}
\begin{scope}[yshift=-9cm,yscale=-1]
\draw
(-2,-2) -- (0,0) -- (2,-2) (0,-2) -- (1,-1);
\filldraw
(-2,-2) circle (1.5pt) (2,-2) circle (1.5pt) (0,-2) circle (1.5pt);
\end{scope}
\begin{scope}[yshift=-9cm]
\draw
(-3,-3) -- (0,0) -- (3,-3) (1,-3) -- (2,-2) (-1,-3) -- (-2,-2);
\filldraw
(-3,-3) circle (1.5pt) (-1,-3) circle (1.5pt) (1,-3) circle (1.5pt) (3,-3) circle (1.5pt);
\end{scope}
\begin{scope}[yshift=-12.5cm]
\node at (0,-.5){$a \quad b \quad c$};
\end{scope}
\node at (4,-6.5) {$=$};
\begin{scope}[xshift=9cm,yshift=-3cm]
\draw
(-3.5,-3.5) -- (0,0) -- (3.5,-3.5) (-.5,-3.5) -- (-2,-2) (-2.5,-3.5) -- (-3,-3);
\node at (0,-4.5){$a~\, \bar{b}\quad\quad\bar{c}\quad\quad \phantom{ }$};
\end{scope}
\end{tikzpicture}
\hspace{2cm}
\begin{tikzpicture}[scale=2,line width=0.8pt]
\draw
(0,0) -- (1,0) -- (1,-1) -- (0,-1) -- cycle
(.5,0) -- (.75,-1)
(.25,0) -- (.5,-1);
\fill[color=white,xshift=.5cm,yshift=-.5cm] {(-.25,-.2) -- (.25,-.2) -- (.25,.2) -- (-.25,.2) -- cycle};
\node at (.5,-.5) {$\left(\begin{smallmatrix}1&2&\\&4&5\\&&6\end{smallmatrix}\right)$};
\draw[yshift=-1.1cm]
(0,0) -- (1,0) (0,.05) -- (0,-.05) (.25,.05) -- (.25,-.05) (.5,.05) -- (.5,-.05) (.75,.05) -- (.75,-.05) (1,.05) -- (1,-.05);
\node at (.5,-1.3) {$a \quad b \quad c$};
\node at (1.2,-.5) {$=$};
\draw[xshift=1.5cm, yshift= -.5cm]
(0,0) -- (1,0) (0,.05) -- (0,-.05) (.125,.05) -- (.125,-.05) (.25,.05) -- (.25,-.05) (.5,.05) -- (.5,-.05) (1,.05) -- (1,-.05);
\node at (2,-.7) {$a~\! \bar{b}\quad\bar{c}\quad\quad \phantom{ }$};
\end{tikzpicture}
\caption{Two points of view on the action of $\mathscr{T}(\bar{B}_*(\mathcal{O}_S))$ on $Y$. On the left the action is described in terms of tree diagrams, on the right in terms of piecewise linear homeomorphisms. In both pictures \usebox{\barb} and \usebox{\barc}. All unspecified values are $z$.}
\label{fig:action_on_Y}
\end{figure}
Next we want to understand stabilizers of this action. First observe that the action has a nontrivial kernel, namely the center of $\Thomp{\bar{B}_*(\mathcal{O}_S)}$, which is isomorphic to $\mathcal{O}_S^\times$.
\begin{observation}\label{obs:center}
Let $(G_*,(\kappa_k)_k)$ be a cloning system and let $H$ be a group. Define a new cloning system $(H \times G_*,(\hat{\kappa}_k)_k)$ by taking $\hat{\kappa}_k \mathbin{\vcentcolon =} \id \times \kappa_k$. Then $\Thomp{H \times G_*} = H \times \Thomp{G_*}$.
\end{observation}
\begin{proof}
The isomorphism is given by $(h,[T,g,U]) \mapsto [T,hg,U]$.
\end{proof}
We now turn to one particular stabilizer.
\begin{observation}\label{obs:stabilizer}
The cloning system on $\bar{B}_*(\mathcal{O}_S)$ induces a cloning system on $K_*$. The stabilizer in $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ of the point $(z)_q$ is $\Thomp{K_*}$.
\end{observation}
\begin{proof}
For the first statement it suffices to show that $(Z_n(\mathcal{O}_S))\kappa_k \subseteq Z_{n+1}(\mathcal{O}_S)$ and that $(\bar{B}_n(\ell))\kappa_k \subseteq \bar{B}_{n+1}(\ell)$ which is easy to see. The second statement is clear.
\end{proof}
\begin{corollary}\label{cor:Thomp(K_*)F_infty}
The group $\Thomp{K_*}$ is of type~$\text{F}_\infty$.
\end{corollary}
\begin{proof}
By Observation~\ref{obs:center} $\Thomp{K_*}$ is isomorphic to a central product $\mathcal{O}_S^\times\Thomp{\bar{B}_*(\ell)}$. The second factor is of type~$\text{F}_\infty$ by Proposition~\ref{prop:barb_thomp_pos_fin_props}.
\end{proof}
We now turn to general stabilizers. For a point $x \in V$ write $[x]$ for its $\mathit{PB}_2(\mathcal{O}_S)$-orbit. We call a point $(x_q)_q \in Y$ \emph{reduced} if $x_q = z$ whenever $[x_q] = [z]$.
\begin{lemma}\label{lem:reduced_points}
Every point of $Y$ has a reduced point in its $\Thomp{\bar{B}_*(\mathcal{O}_S)}$-orbit.
\end{lemma}
\begin{proof}
If $(x_q)_q \in Y$ is arbitrary, let $T$ be a tree such that $D(T)$ contains all of the finitely many indices $q \in D$ for which $x_q \ne z$. Write $D(T) = \{q_1,\ldots,q_{n-1}\}$ where the indices are in increasing order. For each $i$ pick $g_i \in \mathit{PB}_2(\mathcal{O}_S)$ such that $g_i.x_{q_i} = z$ whenever possible (i.e., when $[x_{q_i}] = [z]$) and arbitrarily otherwise. Take $g \in \bar{B}_n(\mathcal{O}_S)$ such that $\alpha_i(g) = g_i$ for all $i$. Then $[T,g,T].(x_q)_q$ is reduced.
\end{proof}
\begin{lemma}\label{lem:stabilizer_of_reduced_point}
The stabilizer in $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ of any reduced point is of type~$\text{F}_\infty$.
\end{lemma}
\begin{proof}
Let $(x_q)_{q \in D}$ be a reduced point and let $I = \{q \in D \mid x_q \ne z\} = \{q_1,\ldots,q_{n-1}\}$. Let $H$ be the stabilizer of $(x_q)_{q}$ in $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ and let $K$ be the kernel of the action of $H$. Since $[x_q] \ne [z]$ for $q \in I$, we see that the stabilizer has to fix $I$ (when acting on $D$ via the canonical homomorphism to $F$). Thus the action of the stabilizer $H$ on $Y=V^{(D)}$ decomposes into an action on $V^I$ and on $V^{(D \setminus I)}$.
Modulo $K$ we find that $H$ is a direct product of the pointwise stabilizer (in $H$) of $V^I$ and the pointwise stabilizer of $V^{(D \setminus I)}$. The action of the pointwise stabilizer of $V^{(D \setminus I)}$ in $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ is isomorphic to $\bar{B}_n(\mathcal{O}_S)$ acting on $V^I$. Thus its intersection with $H$ is isomorphic to a point stabilizer in $\bar{B}_n(\mathcal{O}_S)$, and hence is of type~$\text{F}_\infty$ by Corollary~\ref{cor:all_stabs_F_infty}.
The pointwise stabilizer of $V^I$ decomposes further. Let $D_1 \mathbin{\vcentcolon =} D \cap (0,q_1)$, $D_2 \mathbin{\vcentcolon =} D \cap (q_1,q_2)$, \ldots, $D_n \mathbin{\vcentcolon =} D \cap (q_{n-1},1)$. The pointwise stabilizer of $V^{(D \setminus D_j)}$ is itself isomorphic to a copy of $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ and therefore the stabilizer of $(z)_{q \in D_J}$ in this stabilizer is isomorphic to a copy of $\Thomp{K_*}$, which is of type~$\text{F}_\infty$ by Corollary~\ref{cor:Thomp(K_*)F_infty}.
Putting everything together we find that $H/K$ is a product of groups of type~$\text{F}_\infty$, and $K$ is of type~$\text{F}_\infty$ as well, so $H$ is of type~$\text{F}_\infty$.
\end{proof}
In summary we have:
\begin{proposition}\label{prop:Y_stabilizers}
The group $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ acts on $Y$ with stabilizers of type~$\text{F}_\infty$.
\end{proposition}
\begin{proof}
Every point is in the orbit of a reduced point by Lemma~\ref{lem:reduced_points} so every stabilizer is isomorphic to that of a reduced point. Those are of type~$\text{F}_\infty$ by Lemma~\ref{lem:stabilizer_of_reduced_point}.
\end{proof}
It remains to provide a cocompact filtration and determine its essential connectivity. For this purpose we will use the key result from \cite{bux04} used to show that $\mathit{PB}_2(\mathcal{O}_S)$ is not of type~$\text{FP}_{\abs{S}}$:
\begin{theorem}[\cite{bux04}]
There is a filtration $(V_r)_{r \in \mathbb{N}}$ of $V$ that is $\mathit{PB}_2(\mathcal{O}_S)$-invariant and -cocompact and is essentially $(\abs{S}-2)$-connected but not essentially $(\abs{S}-1)$-acyclic.
\end{theorem}
In fact, by Brown's criterion \emph{any} cocompact filtration of $V$ has that property just because $B_2(\mathcal{O}_S)$ is of type~$\text{F}_{\abs{S}-1}$ but not of type~$\text{FP}_{\abs{S}}$. We use this filtration to construct a cocompact filtration of $Y$ as follows. For $r \in \mathbb{N}$ let $Y^{(r)}$ be the set of all points $(x_q)_{q \in D}$ for which $\{q \in D \mid [x_q] \ne [z]\}$ has at most $r$ elements. Note that $Y^{(r)}$ is $\Thomp{\bar{B}_*(\mathcal{O}_S)}$-invariant. The filtration we want to consider is
\[
Y_r \mathbin{\vcentcolon =} Y^{(r)} \cap V_r^{(D)}\text{.}
\]
The last piece that is missing to conclude that $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ is not of type~$\text{FP}_{\abs{S}}$ is the following:
\begin{proposition}\label{prop:Y_essential_connectivity}
The filtration $(Y_r)_{r \in \mathbb{N}}$ is $\Thomp{\bar{B}_*(\mathcal{O}_S)}$-invariant and -cocompact. It is not essentially $(\abs{S}-1)$-acyclic.
\end{proposition}
Before we can prove the second part, we have to state a technical lemma which says that taking products does not help to kill cycles:
\begin{lemma}\label{lem:non-trivial_maps}
Let $(X_1,A_1)$ and $(X_2,A_2)$ be pairs of CW complexes and assume that the map $\tilde{H}_n(A_1 \to X_1)$ is non-trivial and that $A_2$ is non-empty. Then the map $\tilde{H}_n(A_1 \times A_2 \to X_1 \times X_2)$ is non-trivial as well.
\end{lemma}
\begin{proof}
The case $n = 0$ is clear so assume $n > 0$ from now on.
Let $c$ be an $n$-cycle in $A_1$ that is mapped non-trivially into $X_1$ and let $d$ be a non-trivial $0$-cycle in $A_2$. Consider the diagram
\begin{diagram}
H_n(A_1) \otimes H_0(A_2) & \rInto & H_n(A_1 \times A_2)\\
\dTo && \dTo\\
H_n(X_1) \otimes H_0(X_2) & \rInto & H_n(X_1 \times X_2) \text{.}
\end{diagram}
where the rows are parts of the K\"unneth formula (see \cite[Theorem~3B.6]{hatcher01}) and the columns are the maps induced from the inclusions. The diagram commutes by naturality of the K\"unneth formula. The cycle $c \otimes d$ in the upper left maps non-trivially into the lower left which injects into the lower right. Hence it has non-trivial image in the lower right. Since the diagram commutes, it follows that its image in the upper right also has non-trivial image in the lower right, which is what we want.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:Y_essential_connectivity}]
For cocompactness let $C_r \subseteq V$ be compact such that its $\mathit{PB}_2(\mathcal{O}_S)$-translates cover $V_r$. Let $\hat{C}_r \subseteq Y$ be the product of $r$ copies of $C_r$ (say at positions $q_1,\ldots,q_r$) and $\{z\}$ otherwise. We claim that the translates of $\hat{C}_r$ cover $Y_r$.
Indeed, let $(x_q)_q \in Y_r$ be arbitrary. Since it lies in $Y^{(r)}$ there are at most $r$ positions where $[x_q] \ne [z]$. Using the action of $F$ we can achieve that these positions are (some of) $q_1$ to $q_r$. Now, since each $x_{q_i}$ lies in $V_r$, we can move it into $C_r$, using an element of the form $[T,g,T]$, without moving any of the other $x_q$. At all other coordinates $q$, i.e., where $[x_q]=[z]$, we can move $x_q$ to $z$ using the same method. Since all but finitely many $x_q$ were $z$ to begin with, we have moved $(x_q)_q$ into $\hat{C}_r$ in finitely many steps.
For the second statement let $N = \abs{S}-1$, so we want to show that $(\tilde{H}_N(Y_r))_r$ is not essentially trivial. Let $k$ be such that the map $H_{\abs{S}-1}(V_k \to V_m)$ is non-trivial for every $m \ge k$. For arbitrary $m \ge k$ take $A_1 = V_k$, $A_2 = \prod_{\substack{q \in D\\q \ne 1/2}} \{z\}$, $X_1 = V_m$, and $X_2 = \prod_{\substack{q \in D\\q \ne 1/2}} V_m$. Then $A_1 \times A_2 \subseteq Y_k$ and $Y_m \subseteq X_1 \times X_2$ (on the infinite products we take the CW topology, not the product topology). By Lemma~\ref{lem:non-trivial_maps} the map $H_N(A_1 \times A_2 \to X_1 \times X_2)$ is non-trivial. But this factors through the map $H_N(Y_k \to Y_m)$ which is therefore non-trivial as well. This shows that $(H_N(Y_r))_r$ is not essentially trivial.
\end{proof}
\begin{theorem}
The group $\Thomp{\bar{B}_*(\mathcal{O}_S)}$ is not of type~$\text{FP}_{\abs{S}}$.
\end{theorem}
\begin{proof}
The group acts on $Y$, which is contractible, with stabilizers of type~$\text{F}_\infty$ (Proposition~\ref{prop:Y_stabilizers}). There is an invariant, cocompact filtration $(Y_r)_r$ which is not essentially $(\abs{S}-1)$-acyclic (Proposition~\ref{prop:Y_essential_connectivity}). We conclude using Brown's criterion.
\end{proof}
\begin{remark}
As far as we can tell, none of the established methods in the literature can be used now to show that $\Thomp{B_*(\mathcal{O}_S)}$ is not of type~$\text{FP}_{\abs{S}}$. The kernel of the morphism $\Thomp{B_*(\mathcal{O}_S)} \to \Thomp{\bar{B}_*(\mathcal{O}_S)}$ is very unlikely to be even finitely generated (or else one could apply \cite[Proposition~2.7]{bieri76} or the following exercise, see also \cite[Theorem~7.2.21]{geoghegan08}). Also the projection does not split (or else one could apply the retraction argument \cite[Proposition~4.1]{bux04}). For this reason we will now use the new methods established in Section~\ref{sec:relative_brown}, which can be regarded as a generalization of the retraction argument. We should mention that one \emph{can} deduce that $\Thomp{B_*(\mathcal{O}_S)}$ is not finitely generated if $\abs{S}=1$, without using this new machinery.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thrm:borel_thomp_neg_fin_props}]
We apply Theorem~\ref{thrm:relative_brown} to the inclusion homomorphism $B_2(\mathcal{O}_S) \hookrightarrow \Thomp{\bar{B}_*(\mathcal{O}_S)}$ that takes $g$ to $[\lambda_1,g,\lambda_1]$ where $\lambda_1$ is a single caret. We take $Z$ to be the subspace of $Y$ consisting of points $(x_q)_q$ with $x_q = z$ for $q \ne 1/2$, so $Z$ is $B_2(\mathcal{O}_S)$-cocompact. Our $\Thomp{\bar{B}_*(\mathcal{O}_S)}$-cocompact filtration of $Y$ is $(Y_r)_{r\in\mathbb{N}}$. Observe that $H_{\abs{S}-1}(Z \to Y_r)$ is not eventually trivial by the proof of Proposition~\ref{prop:Y_essential_connectivity}. Thus we can apply Theorem~\ref{thrm:relative_brown}.
Since the inclusion $B_2(\mathcal{O}_S) \hookrightarrow \Thomp{\bar{B}_*(\mathcal{O}_S)}$ clearly factors through $\Thomp{B_*(\mathcal{O}_S)}$ we conclude that this group is not of type~$\text{FP}_{\abs{S}-1}$.
\end{proof}
Combining Theorem~\ref{thrm:borel_thomp_pos_fin_props} and Theorem~\ref{thrm:borel_thomp_neg_fin_props}, we obtain:
\begin{theorem}\label{thrm:borel_thomp_full_fin_props}
The group $\Thomp{B_*(\mathcal{O}_S)}$ is of type~$\text{F}_{\abs{S}-1}$ but not of type~$\text{FP}_{\abs{S}}$. \qed
\end{theorem}
\section{Thompson groups for mock-symmetric groups}\label{sec:mock}
The groups discussed in this section are instances of what Davis, Januszkiewicz and Scott call ``mock reflection groups'' \cite{davis03}. These are groups generated by involutions, and act on associated cell complexes very much like Coxeter groups, with the only difference being that some of the generators may be ``mock reflections'' that do not fix their reflection mirror pointwise. Here we will only be concerned with one family of groups consisting of the minimal blow up of Coxeter groups of type $A_n$. These Coxeter groups are symmetric groups and so we call their blow ups \emph{mock symmetric groups}. For $n \in \mathbb{N}$ the mock symmetric group $\mockS_n$ is given by the presentation
\begin{align}
\mockS_n = \gen{s_{i,j}, 1 \le i < j \le n \mid{}& s_{i,j}^2 = 1 \text{ for all } i, j\nonumber\\
&s_{i,j}s_{k,\ell} = s_{k,\ell}s_{i,j} \text{ for } i < j < k < \ell\label{eq:mock_presentation}\\
&s_{k,\ell}s_{i,j} = s_{k+\ell-j,k+\ell-i}s_{k,\ell} \text{ for }k \le i < j \le \ell}\text{.}\nonumber
\end{align}
We also set $\mockS_\infty = \varinjlim \mockS_n$. See Figure~\ref{fig:mock_relation} for a visualization of elements of $\mockS_n$, and a visualization of the last relation.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[line width=0.8pt, scale=0.5]
\node at (10,-2.5) {$=$};
\draw
(4,0) -- (6,-2)
(6,0) -- (4,-2);
\filldraw[color=white]
(5,-1) circle (5pt);
\draw
(5,-1) circle (5pt);
\filldraw
(0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt) (8,0) circle (1.5pt)
(0,-2) circle (1.5pt) (2,-2) circle (1.5pt) (4,-2) circle (1.5pt) (6,-2) circle (1.5pt) (8,-2) circle (1.5pt);
\begin{scope}[yshift=-2cm]
\draw
(0,0) -- (8,-2)
(2,0) -- (6,-2)
(4,0) -- (4,-2)
(6,0) -- (2,-2)
(8,0) -- (0,-2);
\filldraw[color=white]
(4,-1) circle (5pt);
\draw
(4,-1) circle (5pt);
\filldraw
(0,-2) circle (1.5pt) (2,-2) circle (1.5pt) (4,-2) circle (1.5pt) (6,-2) circle (1.5pt) (8,-2) circle (1.5pt);
\end{scope}
\begin{scope}[xshift=12cm]
\draw
(0,0) -- (8,-2)
(2,0) -- (6,-2)
(4,0) -- (4,-2)
(6,0) -- (2,-2)
(8,0) -- (0,-2);
\filldraw[color=white]
(4,-1) circle (5pt);
\draw
(4,-1) circle (5pt);
\filldraw
(0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt) (8,0) circle (1.5pt)
(0,-2) circle (1.5pt) (2,-2) circle (1.5pt) (4,-2) circle (1.5pt) (6,-2) circle (1.5pt) (8,-2) circle (1.5pt);
\end{scope}
\begin{scope}[yshift=-2cm, xshift=12cm]
\draw
(2,0) -- (4,-2)
(4,0) -- (2,-2);
\filldraw[color=white]
(3,-1) circle (5pt);
\draw
(3,-1) circle (5pt);
\filldraw
(0,-2) circle (1.5pt) (2,-2) circle (1.5pt) (4,-2) circle (1.5pt) (6,-2) circle (1.5pt) (8,-2) circle (1.5pt);
\end{scope}
\end{tikzpicture}
\caption[Mock relation]{The relation $s_{i,j}s_{k,\ell} = s_{k,\ell} s_{k+\ell-j,k+\ell-i}$ of $\mockS_n$ in the case $i = 3$, $j=4$, $k=1$, $\ell=5$, $n = 5$.}
\label{fig:mock_relation}
\end{figure}
Let $\bar{s}_{i,j} \in S_n$ be the involution $(i\ j) ((i+1)\ (j-1)) \cdots (\floor{\frac{i+j}{2}}\ \ceil{\frac{i+j}{2}})$ (this is the longest element in the Coxeter group generated by $(i\ i+1), \ldots, (j-1\ j)$). Taking $s_{i,j}$ to $\bar{s}_{i,j}$ defines a surjective homomorphism $\rho_n \colon \mockS_n \to S_n$. We define cloning maps $\kappa^n_k \colon \mockS_n \to \mockS_{n+1}$ by first defining them on the generators:
\begin{equation}
\label{eq:cloning_mock}
(s_{i,j}) \kappa^n_k = \left\{
\begin{array}{ll}
s_{i,j} & \text{for }j < k\\
s_{i,j+1}s_{k,k+1} & \text{for }i \le k \le j\\
s_{i+1,j+1} & \text{for }k < i\text{.}
\end{array}
\right.
\end{equation}
Now we extend $\kappa^n_k$ to a map $\mockS_n \to \mockS_{n+1}$ as in the paragraph leading up to Lemma~\ref{lem:extend_zs-products}. See Figure~\ref{fig:mock_cloning} for an example of cloning.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[line width=0.8pt, scale=0.5,yscale=-1]
\node at (9,-2.5) {$=$};
\begin{scope}[yshift=-1cm]
\draw
(3,0) -- (4,-1)
(5,0) -- (4,-1);
\filldraw
(3,0) circle (1.5pt) (5,0) circle (1.5pt)
(0,-1) circle (1.5pt) (2,-1) circle (1.5pt) (4,-1) circle (1.5pt) (6,-1) circle (1.5pt);
(0,-1) circle (1.5pt) (2,-1) circle (1.5pt) (4,-1) circle (1.5pt) (6,-1) circle (1.5pt);
\end{scope}
\begin{scope}[yshift=-2cm]
\draw
(0,0) -- (6,-2)
(2,0) -- (4,-2)
(4,0) -- (2,-2)
(6,0) -- (0,-2);
\filldraw[color=white]
(3,-1) circle (5pt);
\draw
(3,-1) circle (5pt);
\filldraw
(0,-2) circle (1.5pt) (2,-2) circle (1.5pt) (4,-2) circle (1.5pt) (6,-2) circle (1.5pt);
\end{scope}
\begin{scope}[xshift=12cm,yshift=-2cm]
\draw
(0,0) -- (8,-2)
(2,0) -- (6,-2)
(4,0) -- (4,-2)
(6,0) -- (2,-2)
(8,0) -- (0,-2);
\filldraw[color=white]
(4,-1) circle (5pt);
\draw
(4,-1) circle (5pt);
\filldraw
(0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt) (8,0) circle (1.5pt)
(0,-2) circle (1.5pt) (2,-2) circle (1.5pt) (4,-2) circle (1.5pt) (6,-2) circle (1.5pt) (8,-2) circle (1.5pt);
\end{scope}
\begin{scope}[yshift=0cm, xshift=12cm]
\draw
(4,0) -- (6,-2)
(6,0) -- (4,-2);
\filldraw[color=white]
(5,-1) circle (5pt);
\draw
(5,-1) circle (5pt);
\filldraw
(4,0) circle (1.5pt) (6,0) circle (1.5pt);
\end{scope}
\begin{scope}[yshift=-4cm, xshift=12cm]
\draw
(2,0) -- (3,-1)
(4,0) -- (3,-1);
\filldraw
(2,0) circle (1.5pt) (4,0) circle (1.5pt)
(3,-1) circle (1.5pt);
\end{scope}
\end{tikzpicture}
\caption[Mock cloning]{The relation $s_{1,4}\lambda_3 = \lambda_2s_{1,5} s_{3,4}$ of $\mathscr{F} \bowtie \mockS_\infty$.}
\label{fig:mock_cloning}
\end{figure}
\begin{proposition}\label{prop:mock_clone}
The above data define a cloning system on $\mockS_*$.
\end{proposition}
\begin{proof}
Note first that \eqref{eq:mock_presentation} is a presentation for $\mockS_n$ as a monoid because all the generators are involutions by the first relation. Following the advice from Remark~\ref{rmk:axioms_via_pres}, we will apply Lemma~\ref{lem:extend_zs-products} with this presentation rather than the trivial presentation used in Proposition~\ref{prop:BZS_existence}.
We have to verify conditions coming from relations of $\mathscr{F}$ and conditions coming from relations of $\mockS_n$, after which the proof proceeds as that of Proposition~\ref{prop:BZS_existence}. For the relations of $\mathscr{F}$ we must verify the conditions~\eqref{item:fcs_product_of_clonings} (product of clonings) and~\eqref{item:fcs_compatibility} (compatibility)
\begin{align}
(s_{i,j})\kappa_\ell \kappa_k &= (s_{i,j})\kappa_k \kappa_{\ell + 1}&&\text{for }k < \ell\text{ and }i<j\label{eq:mock_product_of_clonings}\\
\rho((s_{i,j})\kappa_k) & = (\rho(s_{i,j}))\varsigma_k &&\text{for }i < j\text{.}\label{eq:mock_compatibility}
\end{align}
(Note that we verified \eqref{item:fcs_compatibility} for all $i$, which is not technically necessary; see the remark after Observation~\ref{obs:sync}).
For the relations of $\mockS_n$ we have to check that $\rho$ is a well defined homomorphism, and check that the following equations, standing in for~\eqref{item:cs_cloning_a_product} (cloning a product), are satisfied:
\begin{align}
(s_{i,j})\kappa_{\rho(s_{k,\ell})p}(s_{k,\ell})\kappa_p &= (s_{k,\ell})\kappa_{\rho(s_{i,j})p}(s_{i,j})\kappa_p &&\text{for }i < j < k < \ell\label{eq:mock_cloning_commutator}\\
(s_{k+\ell-j,k+\ell-i})\kappa_{\rho(s_{k,\ell})p}(s_{k,\ell})\kappa_p &= (s_{k,\ell})\kappa_{\rho(s_{i,j})p}(s_{i,j})\kappa_p&&\text{for } k \le i < j \le \ell\text{.}\label{eq:mock_cloning_mock_relation}
\end{align}
Note that the conditions coming from the relations $s_{i,j}^2 = 1$ are vacuous.
Condition \eqref{eq:mock_product_of_clonings} is easy to check if $k< i$ or $\ell > j$ so we consider the situation where $i \le k < \ell \le j$. In this case we have
\begin{multline*}
(s_{i,j})\kappa_\ell\kappa_k = (s_{i,j+1}s_{\ell,\ell+1})\kappa_k = (s_{i,j+1})\kappa_k(s_{\ell,\ell+1})\kappa_k = s_{i,j+2} s_{k,k+1} s_{\ell+1,\ell+2}\\
= s_{i,j+2} s_{\ell+1,\ell+2} s_{k,k+1} = (s_{i,j+1})\kappa_{\ell+1}(s_{k,k+1})\kappa_{\ell+1} = (s_{i,j+1}s_{k,k+1})\kappa_{\ell+1} = (s_{i,j})\kappa_k\kappa_{\ell+1}
\end{multline*}
since $\rho(s_{k,k+1})(\ell+1) = (\ell+1)$, $\rho(s_{\ell,\ell+1})k = k$ and $s_{k,k+1}$ and $s_{\ell+1,\ell+2}$ commute.
Condition \eqref{eq:mock_compatibility} amounts to showing that
\[
(\bar{s}_{i,j})\varsigma_k =
\left\{
\begin{array}{ll}
\bar{s}_{i+1,j+1}&k<i\\
\bar{s}_{i,j+1}\bar{s}_{k,k+1}&i \le k \le j\\
\bar{s}_{i,j}&k>j\text{.}
\end{array}
\right.
\]
The cases $k<i$ and $k > j$ are clear. For the remaining case we first note that
\[
\bar{s}_{i,j+1}\bar{s}_{k,k+1}(m) = \tau_{i+j-k} \bar{s}_{i,j} \pi_{k}(m) = ((\bar{s}_{i,j})\varsigma_k)(m)
\]
for $m \ne k,k+1$ (which is also the same as $\bar{s}_{i,j+1}(m)$). Here $\tau_k$ and $\pi_k$ are as in Example~\ref{ex:symm_gps}. Finally one checks that
\[
\bar{s}_{i,j+1}\bar{s}_{k,k+1}(k) = i+j-k = (\bar{s}_{i,j})\varsigma_k(k)
\]
and that
\[
\bar{s}_{i,j+1}\bar{s}_{k,k+1}(k+1) = i+j-k+1 = (\bar{s}_{i,j})\varsigma_k(k+1)\text{.}
\]
That $\rho$ is a well defined homomorphism amounts to saying that the defining relations of $\mockS_n$ hold in $S_n$ with $s_{i,j}$ replaced by $\bar{s}_{i,j}$, which they do.
Condition \eqref{eq:mock_cloning_commutator} is also easy to check unless $i \le p \le j$ or $k \le p \le \ell$. We treat the case $i \le p \le j$, the other remaining case being similar. We have
\[
(s_{i,j})\kappa_{\rho(s_{k,\ell})p}(s_{k,\ell})\kappa_p = s_{i,j+1}s_{p,p+1}s_{k+1,\ell+1}
= s_{k+1,\ell+1} s_{i,j+1} s_{p,p+1} = (s_{k,\ell})\kappa_{\rho(s_{i,j})p}(s_{i,j}) \kappa_p\text{.}
\]
Finally, the interesting case of condition \eqref{eq:mock_cloning_mock_relation} is when $i \le p \le j$. We have
\begin{multline*}
(s_{k,\ell})\kappa_{\rho(s_{i,j})p}(s_{i,j})\kappa_p = s_{k,\ell+1}s_{i+j-p,i+j-p+1}s_{i,j+1}s_{p,p+1} = s_{k,\ell+1}s_{i,j+1}\\
= s_{k+\ell-j,k+\ell-i+1} s_{k+\ell-p,k+\ell-p+1} s_{k,\ell+1}s_{p,p+1} = (s_{k+\ell-j,k+\ell-i})\kappa_{\rho(s_{k,\ell})p}(s_{k,\ell})\kappa_p
\end{multline*}
using the defining relations of $\mockS_n$ several times.
\end{proof}
As a consequence we get
\begin{theorem}\label{thm:existence_thompson_mock}
There is a generalized Thompson group $\Thomp{\mockS_*}$ which contains all the $\mockS_n$ and canonically surjects onto $V$. We denote it $\Vmock$.
\end{theorem}
\begin{conjecture}\label{conj:Vmock_conj}
$\Vmock$ is of type~$\text{F}_\infty$.
\end{conjecture}
Since each $\mockS_n$ is of type~$\text{F}_\infty$ \cite[Section~4.7, Corollary~3.5.4]{davis03}, to prove the conjecture it suffices to show that the the cloning system is properly graded and that the connectivity of the complexes $\dlkmodel{\mockS_*}{n}$ goes to infinity as $n$ goes to infinity.
\section{Thompson groups for loop braid groups}\label{sec:loop}
Our next example of a cloning system comes from the family of \emph{loop braid groups} $\mathit{LB}_n$, also known as groups $\SAut_n$ of \emph{symmetric automorphisms} of free groups, or as \emph{braid-permutation groups}. This will produce a generalized Thompson group $\Vloop$ that contains both $\Vbr$ and $V$ as subgroups. There is also a pure version of this cloning system, using the \emph{pure loop braid groups}, which we will discuss as well, yielding a group $\Floop$.
We first describe the family of groups in terms of free group automorphisms. Fix a set of generators $\{x_1,\dots,x_n\}$ for $F_n$, and call an automorphism $\phi\in \Aut(F_n)$ \emph{symmetric} if for every $1\le i\le n$ there exists $1\le j\le n$ such that $\phi(x_i)$ is conjugate to $x_j$. If every $\phi(x_i)$ is even conjugate to $x_i$, call $\phi$ \emph{pure symmetric}. The group of symmetric automorphisms of $F_n$ is denoted $\SAut_n$, and the group of pure symmetric automorphisms is denoted $\PSAut_n$. The latter is also denoted by $PLB_n$, for \emph{pure loop braid group}. The reader is cautioned that in the literature ``symmetric'' sometimes allows for generators to map to conjugates of \emph{inverses of} generators, but we do not allow this.
The $\mathit{LB}_n$ fit into a directed system. The map $\iota_{n,n+1} \colon \mathit{LB}_n \hookrightarrow \mathit{LB}_{n+1}$ is given by sending the automorphism $\phi$ of $F_n$ to the automorphism of $F_{n+1}$ that does nothing to the new generator and otherwise acts like $\phi$. This restricts to $\mathit{PLB}_n$ as well, and so we have directed systems $\mathit{LB}_*$ and $\mathit{PLB}_*$.
Our presentation for $\mathit{LB}_n=\SAut_n$ will be taken from~\cite{fenn97}. The generators are as follows, for $1\le i\le n$.
\begin{align*}
&\beta_i \colon \left\{\begin{array}{lll} x_i & \mapsto x_{i+1} \\ x_{i+1} & \mapsto x_{i+1}^{-1} x_i x_{i+1} \\ x_j & \mapsto x_j & (j\neq i,i+1) \end{array}\right. \\
&\sigma_i \colon \left\{\begin{array}{lll} x_i & \mapsto x_{i+1} \\ x_{i+1} & \mapsto x_i \\ x_j & \mapsto x_j & (j\neq i,i+1) \end{array}\right.
\end{align*}
\medskip
The $\beta_i$ together with the $\sigma_i$ generate $\SAut_n$. The $\beta_i$ by themselves generate a copy of $B_n$ in $\SAut_n$, and the $\sigma_i$ generate a copy of $S_n$. As seen in \cite{fenn97}, defining relations for $\SAut_n$ are as follows (with $1\le i\le n-1$):
\begin{align*}
\beta_i \beta_j &= \beta_j \beta_i \hfill (|i-j|>1)\\
\beta_i \beta_{i+1} \beta_i &= \beta_{i+1} \beta_i \beta_{i+1}\\
\sigma_i^2 &= 1\\
\sigma_i \sigma_j &= \sigma_j \sigma_i \hfill (|i-j|>1)\\
\sigma_i \sigma_{i+1} \sigma_i &= \sigma_{i+1} \sigma_i \sigma_{i+1}\\
\beta_i \sigma_j &= \sigma_j \beta_i \hfill (|i-j|>1)\\
\sigma_i \sigma_{i+1} \beta_i &= \beta_{i+1} \sigma_i \sigma_{i+1}\\
\beta_i \beta_{i+1} \sigma_i &= \sigma_{i+1} \beta_i \beta_{i+1} \text{.}
\end{align*}
\medskip
This is a group presentation, and it becomes a monoid presentation after adding generators $\beta_i^{-1}$ with relations $\beta_i \beta_i^{-1} = \beta_i^{-1} \beta_i = 1$.
Since we already have cloning systems on $S_*$ (from Example~\ref{ex:symm_gps}) as well as on $B_*$ (from \cite{brin07}), we already know how the cloning system on $\mathit{LB}_*=\SAut_*$ should be defined. The only thing to check is that it is actually well defined.
The homomorphism $\rho_n \colon \mathit{LB}_n \to S_n$ just takes $\beta_i$ as well as $\sigma_i$ to $\sigma_i \in S_n$. This is easily seen to be well defined.
The cloning maps are defined as they are defined for the symmetric groups and braid groups respectively: for $\varepsilon \in \{\pm 1\}$ this means that
\begin{align}
(\beta_i^\varepsilon)\kappa_k \mathbin{\vcentcolon =} \left\{\begin{array}{ll} \beta_{i+1}^\varepsilon &\text{ if } k<i \\
\beta_i^\varepsilon \beta_{i+1}^\varepsilon &\text{ if } k=i \\
\beta_{i+1}^\varepsilon \beta_i^\varepsilon &\text{ if } k=i+1 \\
\beta_i^\varepsilon &\text{ if } k>i+1
\end{array}
\right.\label{eq:loop_braid_braid_clone}\\
(\sigma_i)\kappa_k \mathbin{\vcentcolon =} \left\{\begin{array}{ll} \sigma_{i+1} &\text{ if } k<i \\
\sigma_i \sigma_{i+1} &\text{ if } k=i \\
\sigma_{i+1} \sigma_i &\text{ if } k=i+1 \\
\sigma_i &\text{ if } k>i+1
\end{array}
\right.\label{eq:loop_braid_transposition_clone}
\end{align}
\begin{lemma}\label{lem:loop_bd_cloning}
The above data $\rho_*$ and $\kappa^*_k$ define cloning systems on $\mathit{LB}_*$ and on $\mathit{PLB}_*$.
\end{lemma}
\begin{proof}
We already noted that $\rho$ is a well defined group homomorphism. We have to check~\eqref{item:cs_product_of_clonings} (product of clonings) and~\eqref{item:cs_compatibility} (compatibility) on generators of $\mathit{LB}_n$. But since every generator is a generator of either $S_n$ or of $B_n$, each verification needed has been performed in establishing the cloning systems on either $S_*$ or $B_*$.
It remains to check that cloning a relation is well defined, standing in for~\eqref{item:cs_cloning_a_product} (cloning a product). Again, the relations involving only elements of $S_n$ or $B_n$ are already verified. This leaves the last three kinds of relations.
For the first relation we have to check that
\[
(\beta_i)\kappa_{\rho(\sigma_j)k} (\sigma_j)\kappa_k = (\beta_i)\kappa_{\sigma_j k} (\sigma_j)\kappa_k = (\sigma_j)\kappa_{\sigma_i k} (\beta_i)\kappa_k = (\sigma_j)\kappa_{\rho(\beta_i)k} (\beta_i)\kappa_k
\]
which is easy to do case by case. For the other two relations we must show that
\begin{align*}
(\sigma_i)\kappa_{(i~i+2~i+1)k} (\sigma_{i+1})\kappa_{(i~i+1)k} (\beta_i)\kappa_k &= (\beta_{i+1})\kappa_{(i~i+1~i+2)k} (\sigma_i)\kappa_{(i+1~i+2)k} (\sigma_{i+1})\kappa_k\\
(\beta_i)\kappa_{(i~i+2~i+1)k} (\beta_{i+1})\kappa_{(i~i+1)k} (\sigma_i)\kappa_k &= (\sigma_{i+1})\kappa_{(i~i+1~i+2)k} (\beta_i)\kappa_{(i+1~i+2)k} (\beta_{i+1})\kappa_k
\end{align*}
which can be treated formally equivalently as long as we do not use either of the relations $\sigma_i^2 = 1$ or $\beta_i\beta_i^{-1} = \beta_i^{-1}\beta_i = 1$. The cases $k < i$ and $k > i+2$ are easy. For $k=i$ we apply only mixed relations to find
\begin{multline*}
(\sigma_i)\kappa_{i+2} (\sigma_{i+1})\kappa_{i+1} (\beta_i)\kappa_i = \sigma_i \sigma_{i+1} \sigma_{i+2} \beta_i \beta_{i+1}\\
= \beta_{i+1} \beta_{i+2} \sigma_i \sigma_{i+1} \sigma_{i+2} = (\beta_{i+1})\kappa_{i+1} (\sigma_i)\kappa_i (\sigma_{i+1})\kappa_i\text{.}
\end{multline*}
Similarly for $k = i+1$ we get
\begin{multline*}
(\sigma_i)\kappa_i (\sigma_{i+1})\kappa_i (\beta_i)\kappa_{i+1} = \sigma_i \sigma_{i+1} \sigma_{i+2} \beta_{i+1} \beta_i\\
= \beta_{i+2} \beta_{i+1} \sigma_i \sigma_{i+1} \sigma_{i+2} =(\beta_{i+1})\kappa_{i+2} (\sigma_i)\kappa_{i+2} (\sigma_{i+1})\kappa_{i+1} \text{.}
\end{multline*}
Lastly for $k = i+2$ we first apply a braid relation and then use the mixed relations to get
\begin{multline*}
(\sigma_i)\kappa_{i+1} (\sigma_{i+1})\kappa_{i+2} (\beta_i)\kappa_{i+2} = \sigma_{i+1} \sigma_i \sigma_{i+2} \sigma_{i+1} \beta_i = \sigma_{i+1} \sigma_{i+2} \sigma_{i} \sigma_{i+1} \beta_i\\
= \beta_{i+2} \sigma_{i+1} \sigma_i \sigma_{i+2} \sigma_{i+1} = (\beta_{i+1})\kappa_i (\sigma_i)\kappa_{i+1} (\sigma_{i+1})\kappa_{i+2} \text{.}
\end{multline*}
Finally, that the cloning system on $\mathit{LB}_*$ restricts to one on $\mathit{PLB}_*$ is straightforward.
\end{proof}
\begin{theorem}\label{thm:existence_thompson_loop}
There are generalized Thompson groups
\[
\Vloop\mathbin{\vcentcolon =}\Thomp{\mathit{LB}_*} \quad \text{and}\quad \Floop\mathbin{\vcentcolon =}\Thomp{\mathit{PLB}_*}
\]
containing the loop braid groups and the pure loop braid groups, respectively. The group $\Vloop$ canonically surjects onto $V$, and the group $\Floop$ canonically surjects onto $F$.
\end{theorem}
The group $\mathit{LB}_n$ is known to be of type~$\text{F}_\infty$, for instance it acts properly cocompactly on the contractible space of marked cactus graphs \cite{collins89}. For this reason understanding the finiteness properties of $\Vloop$ and $\Floop$ amounts to showing that the cloning systems are properly graded, and understanding the connectivity of $\dlkmodel{\mathit{LB}_*}{n}$ and $\dlkmodel{\mathit{PLB}_*}{n}$. We expect that these should be increasingly highly connected and thus:
\begin{conjecture}\label{conj:Vloop_Floop_conj}
$\Vloop$ and $\Floop$ are of type~$\text{F}_\infty$.
\end{conjecture}
We do not attempt to prove this conjecture here. However, we end by sketching a more geometric viewpoint of these cloning systems, which could be useful in the future. To do so, we will view $\mathit{LB}_n$ as a group of motions of loops (which is where the name comes from); see \cite{baez07}, \cite{brendle13} and \cite{wilson12}. Let $\mathbb{R}^3$ be Euclidean~$3$-space, and define a \emph{loop} $\gamma$ to be a smooth, unknotted, oriented embedded copy of the circle $S^1$ in $\mathbb{R}^3$. Now fix a set $L$ of $n$ pairwise disjoint, unlinked loops in $\mathbb{R}^3$, and let $C_n \mathbin{\vcentcolon =} \coprod\limits_{\gamma \in L}\gamma$. A \emph{motion} of $C_n$ is a path of diffeomorphisms $f_t\in\Diff(\mathbb{R}^3)$ for $t\in[0,1]$ such that $f_0$ is the identity and $f_1$ stabilizes $C_n$ set-wise, preserving orientations of the loops. Two motions $f_{t,0}$ and $f_{t,1}$ are considered equivalent if they are smoothly isotopic via an isotopy $f_{t,s}$ with $f_{0,s}$ and $f_{1,s}$ setwise stabilizing $C_n$. If $f_1$ also stabilizes each $\gamma \in L$ then the motion $f_t$ is a \emph{pure motion}. These constructions and the above ones yield isomorphic groups, that is to say $\mathit{LB}_n$ is the group of motions, and $\mathit{PLB}_n$ is the group of pure motions. This is explained, e.g., in \cite{goldsmith81} and \cite[Section~3]{wilson12}. One should picture $\sigma_i$ as the motion in which the $i$th and $(i+1)$st loops move around each other and take each other's old spots. Then $\beta_i$ is similar, except that during the motion the $(i+1)$st loop passes through the $i$th instead of around. See Figure~\ref{fig:loop_braid_gens} for an idea.
\begin{figure}[htb]\centering
\begin{tikzpicture}[line width=0.8pt]
\draw (0,0) circle (0.5cm) (3,0) circle (0.5cm);
\draw[->] (0.5,-0.5) to [out=-30, in=210] (3.5,-0.5);
\draw[->] (2.5,0.5) to [out=150, in=30] (-0.5,0.5);
\node at (1.5,-1.5) {$\sigma_i$}; \node at (-0.75,0) {$i$}; \node at (4,0) {$i+1$};
\begin{scope}[xshift=8cm]
\draw[->] (2.25,0) -- (0,0);
\draw[white,line width=4pt] (0,0) circle (0.5cm) (3,0) circle (0.5cm);
\draw (0,0) circle (0.5cm) (3,0) circle (0.5cm);
\draw[->] (0.5,-0.5) to [out=-30, in=210] (3.5,-0.5);
\draw[->] (0.5,0.5) to [out=30, in=-210] (3.5,0.5);
\node at (1.5,-1.5) {$\beta_i$}; \node at (-0.75,0) {$i$}; \node at (4,0) {$i+1$};
\end{scope}
\end{tikzpicture}
\caption{Generators of $\mathit{LB}_n$.}
\label{fig:loop_braid_gens}
\end{figure}
There is a bit of inconsistency in the literature: all that we have described here is as in, e.g., \cite{fenn97}, but in, e.g., \cite{brendle13}, instead of the generators $\beta_i$ their inverses are used (called $\rho_i$ there), and then the relevant relations look slightly different.
In \cite{baez07} there are some helpful diagrams, analogous to strand diagrams for braids, illustrating elements of $\mathit{LB}_n$. The pictures are four-dimensional, and show one loop passing through another in a sort of movie. Using a bit of artistic license, we can draw similar diagrams to demonstrate cloning; see Figure~\ref{fig:tube_cloning}.
\begin{figure}[htb]\centering
\begin{tikzpicture}[line width=0.8pt]
\coordinate (a) at (0,0); \coordinate (b) at (1,0); \coordinate (c) at (2,0); \coordinate (d) at (3,0); \coordinate (e) at (0,-3); \coordinate (f) at (1,-3); \coordinate (g) at (2,-3); \coordinate (h) at (3,-3); \coordinate (i) at (1.55,-0.725); \coordinate (j) at (2.13,-1.17); \coordinate (k) at (1.47,-0.85); \coordinate (l) at (0.9,-1.75); \coordinate (m) at (1.98,-1.33); \coordinate (n) at (1.33,-2.13); \coordinate (o) at (0.75,-1.5); \coordinate (p) at (1.5,-0.7); \coordinate (q) at (2.2,-1.3); \coordinate (r) at (2.17,-1.23); \coordinate (s) at (0.75,-1.88); \coordinate (t) at (1.19,-2.28); \coordinate (u) at (1.25,-4.5); \coordinate (v) at (2.25,-4.5); \coordinate (w) at (2.75,-4.5); \coordinate (x) at (3.75,-4.5); \coordinate (y) at (2.5,-4.1);
\draw (a) to[out=-90, in=-90] (b) to[out=90, in=90] (a); \draw (c) to[out=-90, in=-90] (d) to[out=90, in=90] (c); \draw (e) to[out=-90, in=-90] (f); \draw[gray,dotted] (f) to[out=90, in=90] (e); \draw (g) to[out=-90, in=-90] (h); \draw[gray,dotted] (h) to[out=90, in=90] (g);
\draw[lightgray] (i) to[out=-15, in=130] (j);
\draw[lightgray,dotted] (l) to[out=30, in=80, looseness=.4] (n); \draw[gray] (l) to[out=-150, in=-100, looseness=.4] (n);
\draw (c) to[out=-90, in=60] (k); \draw[gray] (k) to[out=-120, in=50] (l); \draw (s) to[out=-130, in=90] (e);
\draw (d) to[out=-90, in=50] (m); \draw[gray] (m) to[out=-130, in=50] (n); \draw (t) to[out=-130, in=90] (f);
\draw (a) to[out=-90, in=90] (o);
\draw (b) to[out=-90, in=140] (p); \draw (p) to[out=180, in=-90] (q) to[out=90, in=-50] (r); \draw (p) to[out=-15, in=160] (i); \draw (j) to[out=-50, in=130] (r);
\draw (o) to[out=-90, in=90] (g); \draw (q) to[out=-90, in=90] (h);
\draw (s) to[out=30, in=80, looseness=.4] (t); \draw (s) to[out=-150, in=-100, looseness=.4] (t);
\draw[gray,dotted] (u) to[out=90, in=90] (v); \draw[gray,dotted] (w) to[out=90, in=90] (x); \draw (g) to[out=-90, in=90] (u); \draw (h) to[out=-90, in=90] (x); \draw (u) to[out=-90, in=-90] (v); \draw (w) to[out=-90, in=-90] (x); \draw (v) to[out=90, in=180] (y) to[out=0, in=90] (w);
\node at (4,-2) {$=$};
\begin{scope}[xshift=5cm]
\coordinate (a) at (0,0); \coordinate (b) at (1,0); \coordinate (c) at (2,0); \coordinate (d) at (3,0); \coordinate (e) at (0,-4); \coordinate (f) at (1,-4); \coordinate (i) at (1.55,-0.725); \coordinate (j) at (2.13,-1.17); \coordinate (k) at (1.47,-0.85); \coordinate (l) at (1.32,-1.02); \coordinate (m) at (1.98,-1.33); \coordinate (n) at (1.79,-1.54); \coordinate (o) at (0.2,-0.9); \coordinate (p) at (1.5,-0.7); \coordinate (q) at (2.2,-1.3); \coordinate (r) at (2.17,-1.23); \coordinate (s) at (0.5,-2.3); \coordinate (t) at (1,-2.7); \coordinate (u) at (1.5,-4); \coordinate (v) at (2.5,-4); \coordinate (w) at (3,-4); \coordinate (x) at (4,-4); \coordinate (y) at (0.9,-0.9);
\coordinate (A) at (1,-1.6); \coordinate (B) at (1.4,-2.1); \coordinate (C) at (1.25,-1.25); \coordinate (D) at (1.65,-1.66); \coordinate (E) at (0.95,-1.7); \coordinate (F) at (1.35,-2.15); \coordinate (G) at (0.7,-2.1); \coordinate (H) at (1.05,-2.55);
\path[name path=AB] (A) to[out=160, in=-70, looseness=2] (B);
\path[name path=CG] (C) -- (G);
\path[name path=DH] (D) to[out=-110, in=58] (H);
\path[name intersections={of=AB and CG}];
\coordinate (E) at (intersection-1);
\path[name intersections={of=AB and DH}];
\coordinate (F) at (intersection-1);
\draw[lightgray,dotted] (G) to[out=30, in=80, looseness=0.4] (H);\draw (a) to[out=-90, in=-90] (b) to[out=90, in=90] (a); \draw (c) to[out=-90, in=-90] (d) to[out=90, in=90] (c); \draw (e) to[out=-90, in=-90] (f); \draw[gray,dotted] (f) to[out=90, in=90] (e);
\draw[lightgray] (i) to[out=-15, in=130] (j);
\draw[lightgray,dotted] (l) to[out=-20, in=120] (n); \draw[gray] (l) to[out=-70, in=160, looseness=0.7] (n);
\draw (c) to[out=-90, in=60] (k); \draw[gray] (k) to[out=-120, in=60] (l); \draw (s) to[out=-130, in=90] (e);
\draw (d) to[out=-90, in=50] (m); \draw[gray] (m) to[out=-130, in=50] (n); \draw (t) to[out=-130, in=90] (f);
\draw[gray] (E) to[out=-120, in=50] (G); \draw[gray] (F) to[out=-120, in=50] (H);
\draw (a) to[out=-90, in=90] (o);
\draw (b) to[out=-90, in=140] (p); \draw (p) to[out=180, in=-90] (q) to[out=90, in=-50] (r); \draw (p) to[out=-15, in=160] (i); \draw (j) to[out=-50, in=130] (r);
\draw (s) to[out=30, in=80, looseness=.4] (t); \draw (s) to[out=-150, in=-100, looseness=.4] (t);
\draw[gray,dotted] (u) to[out=90, in=90] (v); \draw[gray,dotted] (w) to[out=90, in=90] (x); \draw (o) to[out=-90, in=90] (u); \draw (q) to[out=-70, in=90] (x); \draw (u) to[out=-90, in=-90] (v); \draw (w) to[out=-90, in=-90] (x); \draw (v) to[out=90, in=-60] (B); \draw (A) to[out=120, in=180] (y); \draw (y) to[out=0, in=90,looseness=0.3] (w);
\draw[lightgray] (A) to[out=-50, in=120] (B);
\draw (A) to[out=160, in=-70, looseness=2] (B); \draw (C) to[out=180, in=-90] (D); \draw (C) to[out=-40, in=125] (D);
\draw (C) to[out=-155, in=60] (A) -- (E); \draw (D) to[out=-115, in=60] (B) -- (F);
\draw[gray] (G) to[out=-150, in=-100, looseness=0.4] (H);
\end{scope}
\end{tikzpicture}
\caption{An example of cloning, namely $(\beta_1)\kappa_2^2 = \beta_2 \beta_1$. The picture shows $\beta_1 \lambda_2 = \lambda_1 \beta_2 \beta_1$. The vertical direction is time, while the missing spatial direction is indicated by breaking the surfaces; see \cite[p.~717]{baez07} for a detailed explanation.}
\label{fig:tube_cloning}
\end{figure}
Alternatively we can draw cloning using the welded braid diagrams from \cite{fenn97}. See Figure~\ref{fig:weld_cloning}.
\begin{figure}[htb]\centering
\begin{tikzpicture}[line width=0.8pt]
\draw (2,0) to[out=-90, in=90] (1,-3);
\draw[white, line width=4pt] (0,0) to[out=-90, in=90] (2,-3);
\draw (0,0) to[out=-90, in=90] (2,-3) (1,0) to[out=-90, in=90] (0,-3);
\filldraw (0.65,-1.2) circle (3pt);
\draw (1.5,-3.5) -- (2,-3) -- (2.5,-3.5);
\node at (3,-1.5) {$=$};
\begin{scope}[xshift=4cm]
\draw (3,0) to[out=-90, in=90] (1,-3);
\draw[white, line width=4pt] (0,0) to[out=-90, in=90] (2,-3) (1,0) to[out=-90, in=90] (3,-3);
\draw (2,0) to[out=-90, in=90] (0,-3) (0,0) to[out=-90, in=90] (2,-3) (1,0) to[out=-90, in=90] (3,-3);
\filldraw (1,-1.5) circle (3pt) (1.5,-1.06) circle (3pt);
\draw (0,0) -- (0.5,0.5) -- (1,0);
\end{scope}
\end{tikzpicture}
\caption{Another example of cloning, now using welded braid diagrams. We see that $\sigma_1 \beta_2 \lambda_3 = \lambda_1 \sigma_2 \sigma_1 \beta_3 \beta_2$.}
\label{fig:weld_cloning}
\end{figure}
One might expect the descending links to be modeled on disjoint ``tubes'' in $3$-space with prescribed boundaries, or ``welded arcs'' of some sort. This is in analogy to the disjoint arcs in $2$-space with prescribed boundaries for descending links in the braid group case.
\newcommand{\etalchar}[1]{$^{#1}$}
| {'timestamp': '2016-03-31T02:10:09', 'yymm': '1405', 'arxiv_id': '1405.5491', 'language': 'en', 'url': 'https://arxiv.org/abs/1405.5491'} |
\section{Introduction}\label{sec:intro}
In the recent years, there has been a tremendous increase in the amount of digital multimedia data, especially the video content, both in the form of video archives and live streams. According to statistics~\cite{burgess2018youtube}, 300 hours of video is being uploaded every minute on the YouTube. A key factor responsible for this enormous increase is the availability of low-cost smart phones equipped with cameras. With such huge collections of data, there is a need to have efficient as well as effective retrieval techniques allowing users retrieve the desired content. Traditionally, videos are mostly stored with user assigned annotations or keywords which are called tags. When a content is to be searched, a keyword provided as query is matched with these tags to retrieve the relevant content. The assigned tags, naturally, cannot encompass the rich video content leading to a constrained retrieval. A better and more effective strategy is to search within the actual content rather than simply matching the tags i.e. Content based Image or Video Retrieval. CBVR systems have been researched and developed for long a time and allow a smarter way of retrieving the desired content. The term content may refer to the visual content (for example objects or persons in the video), audio content (the spoken keywords for instance) or the textual content (News tickers, anchor names, score cards etc.). Among these, the focus of our current study lies on textual content. More specifically, we target a smart retrieval system that exploits the textual content in videos as an index. \\
The textual content in video can be categorized into two broad classes, scene text and caption text. Scene text (Figure~\ref{fig:scenetext}) is captured through camera during the video recording process and may not always be correlated with the content. Examples of scene text include advertisement banners, sign boards, text on a T-shirt etc. Scene text is commonly employed for applications like robot navigation and assistance systems for the visually impaired. Artificial or caption text (Figure~\ref{fig:captext}) is superimposed on video and typical examples include News tickers, movie credits, score cards, names of anchors etc. Caption text is generally correlated with the video content and is mostly applied for semantic retrieval of videos. \\
\begin{figure}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{figure1.pdf}
\caption{Examples of Scene Text}
\label{fig:scenetext}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{figure2.pdf}
\caption{Examples of Caption Text}
\label{fig:captext}
\end{figure}
The key components of a textual content based indexing and retrieval system include detection of text regions~\cite{mirza2018urdu}, extraction of text (segmentation from background)~\cite{ali2019}, identification of script (for multi-script videos)~\cite{ali2016} and finally recognition of text (through a video OCR)~\cite{hayat2018ligature}. Among these, we focus on detection of text in the current study. Detection of text can be carried out using unsupervised~\cite{baran2018automated, dai2018scene, banerjee2013robust}, supervised~\cite{xu2018end, he2018single, liao2017textboxes} or hybrid~\cite{mirza2018urdu, shrivastava2017learning} approaches. Unsupervised text detection employs image analysis techniques to discriminate between text and non-text regions. Supervised methods, on the other hand, involve training a learning algorithm with examples of text and non-text regions to discriminate between the two. In some cases, a combination of the two techniques is employed where the candidate text regions identified by unsupervised methods are validated through a supervised approach. \\
This paper presents a comprehensive framework for video text detection in a multi-script environment. Though we primarily target cursive caption text, since video frames frequently contain text in more than one script, text in the Roman script is also detected by the proposed technique. The key highlights of this study are outlined in the following.
\begin{itemize}
\item Development of a comprehensive dataset of video images with ground truth information supporting evaluation of detection and recognition tasks.
\item Adaptation of various deep learning based object detectors for detection of textual content.
\item Combination of text detection and script identification in a single end-to-end system.
\item Validation of proposed technique through an extensive series of experiments and a comprehensive performance comparison of various detectors.
\end{itemize}
The paper is organized as follows. In the next section, we present an overview of the current state-of-the-art on detection of textual content in videos. In Section~\ref{sec:Dataset}, we introduce the dataset developed in our study along with the ground truth information. Section~\ref{sec:Methodology} presents the details of the proposed framework while Section~\ref{sec:Experiments} presents the experimental protocol, the realized results and the corresponding discussion. Finally, Section~\ref{sec:Conclusion} concludes the paper with a discussion on open challenges on this subject.
\section{Background}\label{sec:lit}
Detection of textual content in videos, images, documents and natural scenes has been investigated for more than four decades. The domain has matured progressively over the years starting with trivial image analysis based systems to complex end-to-end learning based systems. We discuss notable contributions to text detection in the following while detailed surveys on the problem (and related problems) can be found in~\cite{Ye2015,wang2016,yin2016text,zhang2013,sharma2012recent}. \\
Text detection refers to localization of textual content in images. Techniques proposed for detection of text are typically categorized into unsupervised and supervised approaches. While unsupervised approaches primarily rely on image analysis techniques to segment text from background, supervised methods involve training a learning algorithm to discriminate between text and non-text regions.\\
Unsupervised text detection techniques include edge-based methods~\cite{baran2018automated,dai2018scene,banerjee2013robust,huang2013scene,Jamil:2011} which (assume and) exploit the high contrast between text and its background; connected component based methods~\cite{Kiran:2012,koo2013scene,lee2010scene,pan2011hybrid} which mostly rely on the color/intensity of text pixels and texture-based methods~\cite{XiangBai2016,Huang:2011} which consider textual content in the image as a unique texture that distinguishes itself from the non-text regions. Texture based methods have remained a popular choice of researchers and features based on Gabor filters~\cite{gabor1946}, wavelets~\cite{ye2005fast}, curvelets~\cite{joutel2007curvelets}, local binary patterns (LBP)~\cite{anthimopoulos2010two}, discrete cosine transformation (DCT)~\cite{zhong2000automatic}, histograms of oriented gradients (HoG)~\cite{dalal2005histograms} and Fourier transformation~\cite{shivakumara2010new} have been investigated in the literature. Another common category of techniques includes color-based methods~\cite{yi2012localizing,Yi:2007,Shivakumara:2010} which are similar in many aspects to the component-based methods and employ color information of text pixels to distinguish it from non-text regions. \\
Supervised approaches for detection of textual content typically employ state-of-the-art learning algorithms which are trained on examples of text and non-text blocks either using pixel values or by first extracting relevant features. Classifiers like naive Bayes~\cite{shivakumara2012multioriented}, Support Vector Machine~\cite{Zhen:2009}, Artificial Neural Network~\cite{Yin:2013,mirza2018urdu} and Deep Neural Networks~\cite{lecun2015deep} have been investigated for this problem over the years. \\
In the recent years, deep learning based solutions have been widely applied to a variety of recognition problems and have outperformed the traditional techniques. Among deep learning based techniques adapted for text detection, Huang et al.~\cite{huang2014robust} employed sliding windows with CNNs to detect textual regions in low resolution scene images. Likewise, fully convolutional networks are explored for detection of textaul regions in~\cite{zhang2016multi} and the technique is evaluated on various ICDAR datasets. A similar work is presented by Gupta et al.~\cite{gupta2016synthetic} where CNNs are trained using synthetic data for detection of text at multiple scales from natural images. Another method called `SegLink', is proposed in~\cite{shi2017detecting} that relies on decomposing the text into segments (oriented boxes of words or lines) and links (connecting two adjacent segments). The segments and links are detected using fully convolutional networks at multiple scales and combined together to detect the complete text line. In~\cite{tian2016detecting} vertical anchor based method is reported that predicts text and non-text scores of fixed size regions and reports high detection performance on the ICDAR 2013 and ICDAR 2015 datasets. In another notable work, Wang et al.~\cite{wang2018crf} present a framework based on conditional random field (CRF) to detect text in scene images. Authors define a cost function by considering the color, stroke, shape and spatial features with CNN for effective detection of textual regions.\\
Among other end-to-end trainable deep neural networks based systems, Liao et al.~\cite{liao2017textboxes} present a system called `TextBoxes' which detects text in natural images in a single forward pass network. The technique was later extended to `TextBoxes++' and was evaluated on four public databases outperforming the state-of-the-art. He et al.~\cite{he2018single} improved the convolutional layer of CNNs to detect text with arbitrary orientation. EAST~\cite{zhou2017east}, is another well-known scene text detector that provides promising results in challenging scenarios. In another study~\cite{xu2018end}, an ensemble of Convolutional Neural Networks (CNNs) is trained on synthetic data to detect video text in East Asian languages.\\
The literature is relatively limited once it comes to detection of cursive caption text. Among one of the preliminary works, Jamil et al.~\cite{jamil2011edge} exploit edge based features with morphological processing to detect Urdu caption text from a small set of 150 video frames. The same study was extended by Raza et al.~\cite{siddiqi2012database} and evaluated on a larger set of 1000 video frames reporting a recall of 0.80. The dataset of images termed as IPC Artificial Text dataset~\cite{siddiqi2012database} was also made publicly available. In a later study~\cite{raza2013multilingual} by the same group, the authors proposed a cascaded framework of spatial transforms to detect caption text in five different scripts including Arabic and Urdu. In a relatively recent work on detection of Arabic caption text, Zayene et al.~\cite{Oussama2016} employ a combination of stroke width transform with a convolutional auto-encoder and evaluate the technique on a publicly available dataset AcTiV-DB~\cite{Oussama2015}. In one of our previous works~\cite{mirza2018urdu}, we investigated a combination of image analysis techniques with textural features to detect textual regions in video frames and realized an F-measure of 0.80 on 1000 images.\\
Summarizing, it can be concluded that the problem of text detection has been dominated by the application of different deep learning based techniques in the recent years. The availability of benchmark datasets has also contributed to the rapid developments in this area. While detection of text in languages based on the Latin alphabet has received significant research attention and is very much mature, detection of cursive text still remains a relatively less addressed and challenging issue. Development of a (generic) text detector that could work in multi-script environments also remains an open problem.\\
In the next section, we introduce the dataset that has been collected and labeled as a part of this study.
\section{Dataset}\label{sec:Dataset}
Availability of labeled datasets is of utmost importance for algorithmic development and evaluation of any computerized system. From the perspective of Urdu caption text detection, a dataset of 1000 labeled video frames has been made publicly available~\cite{siddiqi2012database}. A collection of 1000 frames, however, seems to be very small to generalize the findings for practical applications. We, therefore, collected and labeled a comprehensive dataset of video frames allowing evaluation of text detection and text recognition tasks. We collected a set of 46 videos from four different News channels in Pakistan. All videos are recorded at a resolution of $900 \times 600$ and a frame rate of 25 fps. Frames in these video contain textual content in two languages, (cursive) Urdu and English. The collected video frames are labeled from two perspectives, detection and recognition. For detection, the bounding rectangle of all text regions in a frame is labeled and stored. Similarly, for recognition, the transcription of each text line is stored as ground truth.
In the literature, several evaluation metrics have been proposed to evaluate the performance of text detection systems~\cite{jamil2011edge,lucas2005icdar,wolf2006object}. In our system, for evaluation of the text detection module, we employ the most commonly used area based precision and recall measures reported in~\cite{jamil2011edge} and defined as follows.\\
Let $A_E$ be the estimated text area given by the system and $A_T$ be the ground truth text area, then the precision $P$ and recall $R$ are defined as:
\begin{equation}
P~ =~\frac{~A_E~ \cap~ A_T}{A_E}
\end{equation}
\begin{equation}
R~ =~ \frac{~A_E~ \cap~ A_T}{A_T}
\end{equation}
The precision and recall measures can be combined in a single F-measure as follows.
\begin{equation}
F~=~ \frac{~2~ \times ~Precision~ \times~ Recall~ }{~Precision ~+~ Recall}
\end{equation}
The same idea can be extended to multiple images by simply summing up area of intersection and dividing by the total ground truth area (in $N$ images) for recall and the total detected area for precision. To compute these measures, for each frame, we need to store the actual location of the textual content. The text detected automatically by the system can then be compared with the ground truth text regions to compute precision, recall and F-measure. The idea is illustrated in Figure~\ref{fig:textreg}. Figure~\ref{fig:textreg}-a illustrates an example where the text regions detected by the system are shown while Figure~\ref{fig:textreg}-b illustrates the ground truth text locations for the given frame. The detected and ground truth text regions can be compared to compute the metrics defined earlier and quantify the detection performance.\\
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{figure3.pdf}
\caption{Text regions in an image and the corresponding ground truth image}
\label{fig:textreg}
\end{figure*}
To facilitate the labeling process and standardize the ground truth data, a comprehensive labeling tool has been developed that allows storing the location of each textual region in a frame along with its ground truth transcription. The location is stored in terms of the $x$ and $y$ coordinates of the top left of the bounding box along its $width$ and $height$. The ground truth information of each frame is stored as an XML file that comprises frame meta data and the information on textual regions. A screen shot of the labeling tool is presented in Figure~\ref{fig:GTTool} while the ground truth information of an example frame is illustrated in Figure~\ref{fig:XMLScreen}.\\
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{figure4.pdf}
\caption{Screen shot of ground truth labeling tool for text data}
\label{fig:GTTool}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{figure5.pdf}
\caption{An XML file containing ground truth information of a frame}
\label{fig:XMLScreen}
\end{figure*}
It is known that videos typically contain 25--30 frames per second; consequently, successive frames in a video contain redundant information (both visual and textual content). From the view point of automatic analysis systems, frames with unique content are of interest. Hence, each single video frame does not need to be labeled as major proportions of such frames will have exactly the same textual information. In our study, we have extracted more than 11,000 frames from videos with an attempt to have as much unique text as possible. The statistics of videos, frames and text lines of our dataset are presented in Table~\ref{tab:channelStats}.
Inspired from the Arabic caption text dataset AcTiV-DB~\cite{Oussama2015}, we have named the our dataset UTiV (Urdu Text in Video). The dataset along with its ground truth has also been made publicly available\footnote{http://cbvir.media-tics.net/} to support quantitative evaluation of text detection and recognition tasks.\\
\begin{table*}[!ht]
\centering
\caption{Statistics of labeled video frames}
\label{tab:channelStats}
\begin{tabular}{llcccc}
\hline
\textbf{S\#} & \textbf{Channel} & \textbf{Videos} & \textbf{Labeled Images} & \textbf{Urdu Lines} & \textbf{English Lines}\\
\hline
1 & Ary News & 7 & 3,206 & 10,250 & 3,605\\
2 & Samaa News & 13 & 2,503 & 10,961 & 4,411\\
3 & Dunya News & 16 & 3,059 & 10,723 & 8,861\\
4 & Express News & 10 & 2,424 & 8,536 & 6,755\\
& \textbf{Total} & \textbf{46}& \textbf{11,203} & \textbf{40,470} & \textbf{23,632}\\
\hline
\end{tabular}
\end{table*}
\section{Methods}\label{sec:Methodology}
This section presents the details of detecting textual content from video frames. Detection relies on adapting object detectors based on deep convolutional neural networks for text regions. Once the text is detected, script of the detected text is identified by employing the ConvNets in a classification framework. Subsequently, text detection and script identification are combined in a single end-to-end system that detects the textual content along with its script. Details are presented in the following sections.
\subsection{Deep Learning based Object Detectors}
Deep neural networks enjoy a renewed interest of the machine learning community thanks primarily to the availability of high performance computing hardware (GPUs) as well as large data sets to train these systems. A major development contributing to the current fame of deep learning was the application of ConvNets by Krizhevsky et al.~\cite{krizhevsky2012imagenet} on the ImageNet Large Scale Visual Recognition competition~\cite{russakovsky2015imagenet}, which greatly reduced the error rates. Since then, CNNs are considered to be state-of-the-art feature extractors and classifiers~\cite{simonyan2014very, szegedy2015going} and have been applied to a variety of recognition tasks~\cite{bouchain2006character, uijlings2013selective, farfade2015multi}. \\
While traditional CNNs are typically employed for object classification, Region-based Convolutional Networks (R-CNN)~\cite{girshick2014rich} and their further enhancements Fast R-CNN~\cite{girshick2015fast} and Faster R-CNN~\cite{ren2015faster} adapt CNNs for object detection. In addition to different variants of R-CNN, a number of new architectures have also been proposed in the recent years for real time object detection. The most notable of these include YOLO (You Only Look Once)~\cite{redmon2016you} and SSD (Single Shot Detector)~\cite{liu2016ssd}. Each of these object detectors can be trained to detect $C$ object classes (plus one for the background). The output of the detector is the location of the bounding box (four coordinates) containing one of the $C$ classes as well as the class confidence score.\\
In our study, for detection of textual content in a given frame, we investigated a number of CNN based object detectors. Although, many object detectors are trained with thousands of class examples and provide high accuracy in detection and recognition of different objects, these object detectors can not be directly applied to identify text regions in images. These models have to be tuned to the specific problem of discrimination of text from non-text regions. The convolutional base of these models can be trained from scratch or, known pre-trained models can be fine-tuned by training them on text and non-text regions. In our study, we investigated the following object detectors for localization of text regions.
\begin{itemize}
\item Faster R-CNN
\item Region-Based Fully Convolutional Networks (R-FCN)
\item Single Shot Detector (SSD)
\item You Only Look Once (YOLO)
\end{itemize}
For completeness, we provide a brief overview of these object detectors in the following sections.
\subsubsection{Faster R-CNN}
Faster R-CNN~\cite{ren2015faster} is an enhanced version of its predecessors R-CNN~\cite{girshick2014rich} and Fast R-CNN~\cite{girshick2015fast}. Each of these detectors exploits the powerful features of ConvNets for object localization as well as classification. R-CNN was one of the first attempts to apply ConvNets for object detection. An R-CNN scans the input image for potential objects using Selective Search~\cite{uijlings2013selective} that generates around 2,000 region proposals. Each of these region proposals is then fed to a CNN for feature extraction. The output of the CNN is finally employed by an SVM to classify the object and a linear regressor to tighten the bounding box. R-CNN was enhanced in terms of training efficiency by extending it to Fast R-CNN~\cite{girshick2015fast}. In Fast R-CNN, rather than separately feeding each region proposal to the ConvNet, convolution is performed only once on the complete image and the region proposals are projected on the feature maps. Furthermore, the SVM in R-CNN was replaced by a softmax layer extending the network to predict the class labels rather than using a separate model. While Fast R-CNN significantly reduced the time complexity of the basic R-CNN, a major bottleneck was the selective search algorithm to generate the region proposals. This was addressed through Region Proposal Network (RPN) in Faster R-CNN~\cite{ren2015faster} which shares convolutional features with the detection network. RPN predicts region proposals which are then fed to the detection network to identify the object class and refine the bounding boxes produced by the RPN. A summary of various R-CNN models in presented in Figure~\ref{fig:frcnn}.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{figure6.pdf}
\caption{Summary of R-CNN Family based Object Detectors}
\label{fig:frcnn}
\end{figure*}
\subsubsection{You Only Look Once (YOLO)}
YOLO~\cite{redmon2016you} takes a different approach to object detection primarily focusing on improving the detection speed (rather than accuracy). As the name suggests, YOLO employs a single pass of the convolutional network for localization and classification of objects from the the input images. The input image is divided into a grid and an object is expected to be detected by the grid which holds the center of the object. Each cell in the grid predicts up to two bounding boxes (and class probabilities). The network comprises 24 convolutional and fully connected layers. YOLO works in real time but in terms of accuracy, it is known to make significant localization errors in comparison to region based object detectors (Faster R-CNN for instance).
YOLO was later enhanced to YOLO9000~\cite{redmon2017yolo9000} by introducing batch normalization, increasing the resolution of the input image (by a factor of 2) and introducing the concept of anchor boxes. YOLO9000 employs Darknet 19 architecture with 19 convolutional layers, 5 max pooling layers and a softmax layer for classification objects. Incremental improvements in YOLO v2 resulted in YOLO v3~\cite{redmon2018yolov3} that uses logistic regerssion to predict the score of objectness for each bounding box. Furthermore, it employs class-wise logistic classifiers (rather than softmax) allowing multi-label classification.
\subsubsection{Single Shot Detector (SSD)}
Unlike the R-CNN series object detectors which require two shots to detect objects in an image, Single Shot Multi-box Detector~\cite{liu2016ssd}, as the name suggests, requires a single shot to detect objects (similar to YOLO). SSD relies on the idea of default boxes and multi-scale predictions and directly applies bounding box regression to the default boxes without generating the region proposals. Detection at multiple scales are handled by exploiting the feature maps of different convolutional layers corresponding to different receptive fields in the input image. The architecture has an input size of $300 \times 300 \times 3$ and primarily builds on the VGG-16 architecture discarding the fully connected layers. VGG-16 is used as base network mainly due to its robust performance of image classification tasks. The bounding box regression technique of SSD is inspired by~\cite{szegedy2015going} while the MultiBox relies on priors, the pre-computed fixed size bounding boxes. The priors are selected in such a way that their Intersection over Union ratio (with ground truth objects) is greater than 0.5. The MultiBox starts with the priors as predictions and attempt to regress closer to the ground truth bounding boxes. SSD works in real time but requires images of fixed square size and is known to miss small objects in the image.
\subsubsection{Region-Based Fully Convolutional Networks (R-FCN) }
R-FCN~\cite{dai2016r} builds on the idea of increasing the detection accuracy by maximizing the shared calculations. R-FCN generates position-sensitive score maps to represent different relative positions of an object. An object is represented by $k^2$ relative positions dividing it into a grid of size $k \times k$. A ConvNet (ResNet in the original R-FCN paper) sweeps the input image and an additional fully convoltional layer produces the position-sensitive scores in $k^2 \times (C+1)$ score maps where $C$ is the number of classes plus 1 class for the background. A fully convolutional proposal network generates regions of interest which are divided in $k^2$ bins and the corresponding class probabilities are obtained from the score maps. The scores are averaged to convert the $k^2 \times (C+1)$ values into a one dimensional $(C+1)$ sized vector which is finally fed to the softmax layer for classification. Localization is carried out using the bounding box regression similar to other object detectors. R-FCN speeds up the detection in comparison to Faster R-CNN but compared to other Single Shot methods, it requires more computational resources.
\subsection{Adapting Object Detectors for Text Detection}
In the context of object detection, the problem of text detection can be formulated as a two class problem. The text regions represent object of interest while the non-text regions need to be ignored. The object detectors discussed in the previous sections are adapted for text detection using two pre-trained models, ResNet 101~\cite{he2016deep} and Inception v2~\cite{ioffe2015batch}. These models are trained on the large scale Microsoft COCO (Common Objects in Context) database~\cite{lin2014microsoft}. The database contains images of 91 different object types with a total of 2.5 million labeled instances in 328K images. The pre-trained network serves as starting point rather than random weight initialization and the network is made to learn the specific class labels (text or non-text) by continuing back propagation. The ground truth localization information of the textual regions in the video frames is employed for training the models, the overall workflow being illustrated in Figure~\ref{fig:traintestmethod}. \\
A critical aspect in employing object detectors for text detection is the choice of anchor boxes. The anchor boxes in all the detectors have been designed to detect general object categories. Text appearing in videos has specific geometric properties in terms of size and aspect ratio hence the default anchor boxes of the detectors need to be adapted to detect textual regions. We carried out a comprehensive analysis of the textual regions in terms of width, height and aspect ratios of the bounding boxes. As a result of this analysis we have chosen a base anchor of size $256 \times 256$. To each anchor box we apply three scales ($1.0, 2.0, 5.0$) and five aspect ratios ($0.125, 0.1875, 0.25, 0.375, 0.50$) as illustrated in Figure~\ref{fig:Scales}. Models are fine-tuned using the proposed anchor boxes and the effectiveness of these anchor boxes is validated through experimental study as presented in Section~\ref{sec:Experiments} of the paper. \\
\begin{figure*}[!ht]
\centering
\includegraphics[width=1.0\textwidth]{figure7.pdf}
\caption{Overview of adapting object detectors for text detection}
\label{fig:traintestmethod}
\end{figure*}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.75\textwidth]{figure8.pdf}
\caption{Anchor boxes (base size $256 \times 256$) at three scales ($1.0, 2.0, 5.0$) and five aspect ratios ($0.125, 0.1875, 0.25, 0.375, 0.50$)}
\label{fig:Scales}
\end{figure}
\subsection{Script Identification}
As discussed earlier, we primarily target detection of cursive caption text. However, like many practical scenarios, video frames in our case contain bilingual textual content (Urdu \& English). Consequently, once the text is detected, we need to identify the script of each detected region (Figure~\ref{fig:scriptIden}) so that the subsequent processing of each type of script can be carried out by the respective recognition engine. For script identification, we employ CNNs in a classification framework (rather than detection). Urdu and English text lines are employed to fine-tune CNNs to discriminate between the two classes. Once trained, the model is able to separate text lines as a function of the script. Similar to detection, rather than training the networks from scratch, we fine-tune known pre-trained models (Inception and ResNet) to solve the two-class classification problem.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{figure9.pdf}
\caption{Script identification of detected text lines}
\label{fig:scriptIden}
\end{figure*}
\subsection{Hybrid Text Detector \& Script Identifier}
Detection of text and identification of script, as discussed previously, can be implemented in a cascaded framework where the output of text detector is fed to the script identifier. A deep learning framework can be tuned to discriminate between text and non-text regions and the extracted text regions can be fed to a separate script recognition model that identifies the script of the detected text. This, however, introduces a bottleneck of training two separate networks. Furthermore, the cascaded solution also implies that errors in detection are propagated to the next step as well. We, therefore, propose to combine the text detector and script identifier in a single hybrid model. Rather than treating detection as a two-class problem (text and non-text), we consider it as a three class problem, i.e. non-text regions, English text and Urdu text. This not only avoids training two separate models but also eliminates the accumulation of errors in a cascaded solution. The superiority of the combined text detector and script identifier is also supported through quantitative evaluations as discussed in the next section.\\
All detectors are trained in an end-to-end manner with a multi-task objective function that combines the classification and regression losses. The evolution of training loss for the investigated detectors (with Inception and ResNet) is illustrated in Figure~\ref{fig:alllosses} where it can be seen that the loss begins to stabilize from 30 epochs on wards.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.75\textwidth]{figure10.pdf}
\caption{Training loss of various detectors -- Hybrid text detector and script identifier}
\label{fig:alllosses}
\end{figure}
\section{Experiments and Results}\label{sec:Experiments}
The detection performance is evaluated through a series of experiments carried out on the collected set of video frames. We first present the experimental protocol followed by the detection results of various object detectors. We then present the script identification results and the performance of the combined text detector and script identifier. Furthermore, performance sensitivity of the system as well as a comparison with state-of-the-art is also presented.
\subsection{Experimental Settings}
As introduced in Section~\ref{sec:Dataset}, we collected a total of 11,203 video frames from four different News channel videos. The localization information of text regions in these frames is used to train and subsequently evaluate the text detection and script identification performance. The distribution of frames into training and test sets along with the number of text lines in each set is summarized in Table~\ref{tab:dataDist} while the details of detection performance are presented in the next section.
\begin{table}[!ht]
\centering
\caption{Data distribution for text detection experiments}
\label{tab:dataDist}
\begin{tabular}{lllll}
\hline
& \multicolumn{2}{c}{\textbf{Train}} & \multicolumn{2}{c}{\textbf{Test}} \\ \hlin
& \multicolumn{1}{c}{\textbf{Frames}} & \multicolumn{1}{c}{\textbf{Lines}} & \multicolumn{1}{c}{\textbf{Frames}} & \multicolumn{1}{c}{\textbf{Line}} \\ \hline
\multicolumn{1}{l}{\textbf{Urdu}} & \multirow{2}{*}{8,500} & 31,321 & \multirow{2}{*}{2,703} & 9,149 \\
\multicolumn{1}{l}{\textbf{English}} & & 16,207 & & 7,425 \\ \hline
\multicolumn{1}{l}{\textbf{Total}} & & 49,046 & & 11,056 \\ \hline
\end{tabular}
\end{table}
\subsection{Text Detection Results}
Object detectors including Faster R-CNN, YOLO, SSD and R-FCN are adapted to detect textual content by fine-tuning the Inception and ResNet pre-trained models and changing the anchor boxes as discussed previously. Performance of each of these detectors in terms of precision, recall and F-measure is summarized in Table~\ref{tab:detectResText}. It can be seen that in all cases, detectors pre-trained on Inception outperform those trained on ResNet. Among various detectors, Faster R-CNN reports the highest F-measure of 0.90. The lowest performance is reported by Yolo reading an F-measure of 0.66. A comprehensive study on the trade-off between speed and accuracy of various object detectors is presented in~\cite{huang2017speed} and our findings on detection of text are consistent with those of~\cite{huang2017speed}. It is also important to recap that precision and recall are computed using area based metrics. As a result, if the detected bounding box is larger (smaller) than the ground truth, it results in penalizing the precision (recall) of the detector as illustrated in Figure~\ref{fig:prerecall}. The output of the Faster R-CNN based text detector for few sample frames in our dataset is illustrated in Figure~\ref{fig:detectionResultsSamp}.
\\
\begin{table*}[!ht]
\centering
\caption{Text Detection Results}
\label{tab:detectResText}
\begin{tabular}{lcccccc}
\hlin
& \multicolumn{3}{c}{\textbf{RestNet}} & \multicolumn{3}{c}{\textbf{Inception}} \\ \hline
\multicolumn{1}{l}{\textbf{Model}} & \textbf{Precision} & \textbf{Recall} & \textbf{F-Measure} & \textbf{Precision} & \textbf{Recall} & \textbf{F-Measure} \\ \hline
\multicolumn{1}{l}{\textbf{SSD}} & 0.83 & 0.71 & 0.77 & 0.82 & 0.77 & 0.80 \\ \hline
\multicolumn{1}{l}{\textbf{R-FCN}} & 0.79 & 0.86 & 0.82 & 0.84 & 0.89 & 0.86 \\ \hline
\multicolumn{1}{l}{\textbf{Faster R-CNN}} & 0.82 & 0.90 & 0.85 & \textbf{0.86} & \textbf{0.95} & \textbf{0.90} \\ \hline
\multicolumn{1}{l}{\textbf{Yolo}} & - & - & - & 0.63 & 0.69 & 0.66 \\ \hline
\end{tabular}
\end{table*}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.75\textwidth]{figure11.pdf}
\caption{Computation of precision and recall (a):Ground Truth Bounding Box (b): Detected region is larger than ground truth (c):Detected region is smaller than ground truth (d):Detected region overlaps perfectly with the ground truth}
\label{fig:prerecall}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.85\textwidth]{figure12.pdf}
\caption{Text detection results on sample images (Faster R-CNN with Inception) }
\label{fig:detectionResultsSamp}
\end{figure}
In an attempt to provide an insight into the detection errors, few of the errors are illustrated in Figure~\ref{fig:mistakes1}. It can be seen that in most cases, the detector is able to detect the textual region but the localization is not perfect i.e. in some cases the bounding box is larger (shorter) than the actual content leading to a reduced precision (recall). \\
\begin{figure}[!ht]
\centering
\includegraphics[width=0.85\textwidth]{figure13.pdf}
\caption{Imperfect Localization of Text Regions}
\label{fig:mistakes1}
\end{figure}
\subsection{Script Identification Results}
For script identification, we employ the same distribution of frames into training and test sets as that of the detection protocol. Text lines from the video frames in the training set are employed to fine-tune the pre-trained ConvNets while the identification rates are computed on text lines from the frames in the test set. A total of $31,321$ Urdu and $16,207$ English text lines are used in the training set while the test set comprises $9,9149$ and $7,425$ text lines in Urdu and English respectively. The resulting confusion matrix is presented in Table~\ref{tab:siConMat} while the precision, recall and F-measure are summarized in Table~\ref{tab:sifmeasure}. It can be seen that the model was able to correctly identify the scripts with an accuracy of more than 94\%.\\
\begin{table}[!ht]
\centering
\caption{Script identification confusion matrix}
\label{tab:siConMat}
\begin{tabular}{lcc}
\hline
& \textbf{Urdu} & \textbf{English} \\
\hlin
\textbf{Urdu} & 8763 & 386 \\
\textbf{English} & 551 & 6874 \\
\hline
\end{tabular}
\end{table}
\begin{table}[!ht]
\centering
\caption{Performance of Script Identification }
\label{tab:sifmeasure}
\begin{tabular}{lccc}
\hline
& \textbf{Precision} & \textbf{Recall} & \textbf{F--Measure} \\ \hline
\textbf{Urdu} & 0.940 & 0.957 & 0.95\\
\textbf{English} & 0.946 & 0.925 & 0.94 \\
\hline
\end{tabular}
\end{table}
\subsection{Hybrid Text Detection \& Script Identification Results}
As discussed previously, text detection and script identification can be combined in a single model treating detection as a three (rather than two) class problem. The results of these experiments are summarized in Table~\ref{tab:detectRes} keeping the same distribution of training and test frames as in the previous experiments. Many interesting observations can be drawn from the results in Table~\ref{tab:detectRes}. Similar to the script independent detectors, models pre-trained on Inception outperform those trained on ResNet and the observation is consistent for all four detectors. Likewise, Faster R-CNN reports the highest F-measure both for detection of Urdu and English text reading 0.91 and 0.87 respectively. In all cases, the performances on detection of Urdu text are better than those on Engish text. This can be attributed to the fact that the data is collected primarily from Urdu News channels which have limited amount of English text. It is also interesting to note that by combining text detection and script identification in a single model, not only the cascaded solution is avoided, the detection F-measures have also improved (in most cases). Though the improvement is marginal, eliminating the separate processing of detected text regions to identify the script offers a much simplified (yet effective) solution. Detection outputs on sample frames for the four detectors are illustrated in Figure~\ref{fig:alldetectors}.
\begin{table*}[!ht]
\centering
\caption{Performance of hybrid text detector and script identifier}
\label{tab:detectRes}
\begin{tabular}{llcccccc}
\hlin
& & \multicolumn{3}{c}{\textbf{RestNet}} & \multicolumn{3}{c}{\textbf{Inception}} \\ \hline
\multicolumn{1}{l}{\textbf{Method}} & \textbf{Script} & \textbf{Precision} & \textbf{Recall} & \textbf{F-Measure} & \textbf{Precision} & \textbf{Recall} & \textbf{F-Measure} \\ \hline
\multicolumn{1}{l}{\multirow{3}{*}{\textbf{SSD}}} & \textbf{Urdu} & 0.83 & 0.72 & 0.77 & 0.82 & 0.78 & 0.80 \\
\multicolumn{1}{l}{} & \textbf{English} & 0.80 & 0.63 & 0.70 & 0.82 & 0.70 & 0.75 \\
\hline
\multicolumn{1}{l}{\multirow{3}{*}{\textbf{R-FCN}}} & \textbf{Urdu} & 0.80 & 0.87 & 0.83 & 0.85 & 0.90 & 0.87 \\
\multicolumn{1}{l}{} & \textbf{English} & 0.73 & 0.81 & 0.77 & 0.77 & 0.84 & 0.81 \\
\hline
\multicolumn{1}{l}{\multirow{3}{*}{\textbf{Faster R-CNN}}} & \textbf{Urdu} & 0.82 & 0.92 & 0.86 & \textbf{0.87} & \textbf{0.95} & \textbf{0.91} \\
\multicolumn{1}{l}{} & \textbf{English} & 0.80 & 0.81 & 0.80 & \textbf{0.81} & \textbf{0.94} & \textbf{0.87} \\
\hline
\multicolumn{1}{l}{\multirow{3}{*}{\textbf{Yolo}}} & \textbf{Urdu} & - & - & - & 0.64 & 0.70 & 0.67 \\
\multicolumn{1}{l}{} & \textbf{English} & - & - & - & 0.62 & 0.67 & 0.64 \\
\hline
\end{tabular}
\end{table*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.85\textwidth]{figure14.pdf}
\caption{Detection output of hybrid text detection and script identification for different detectors (a): SSD (b): R-FCN (c): Faster RCNN (d): Yolo }
\label{fig:alldetectors}
\end{figure*}
In an attempt to carry out an in-depth analysis of the detection performance and its evolution with respect to important system parameters, we carried out another series of experiments using Faster R-CNN (with Inception). In the first such experiment, we study the performance sensitivity to the amount of training data. We train the model by varying the number of text line images (from 10K to 49K) and compute the detector F-measure. Naturally, the detector performance enhances with the increase in the amount of training data (Figure~\ref{fig:impact}) and begins to stabilize from around 30K-35K training lines.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.75\textwidth]{figure15.pdf}
\caption{Impact of size of training data on text detection performance (Faster R-CNN with Inception)}
\label{fig:impact}
\end{figure}
Resolution of input video frames is an important parameter that might affect the detector performance. To study the detector sensitivity to image resolution, we varied the image resolution from $256 \times 144$ to $1920 \times 1080$. The resolution was varied only in the test set and all sets of images were evaluated on the detector trained on a single resolution ($900 \times 600$). The F-measures in Figure~\ref{fig:resolution} are more less consistent for varied image resolutions reflecting the robustness of the detector. The proposed anchor boxes adapted for textual content play a key role in achieving this scale invariance.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.75\textwidth]{figure16.pdf}
\caption{Impact of video resolution on text detection performance (Faster R-CNN with Inception)}
\label{fig:resolution}
\end{figure}
\subsection{Performance Comparison}
In an attempt to compare the performance of our detector with those reported in the literature, we present a comparative overview of various text detectors targeting cursive caption text in Table~\ref{tab:detectResCom}. It is important to note that since different studies are evaluated on different datasets, a direct comparison of these techniques is difficult. Most of the listed studies employ a small set of images ($\leq 1000$). Moradi et al.~\cite{moradi2013hybrid} and Zayene et al.~\cite{Oussama2016} report results on relatively larger datasets with F-measures of 0.89 and 0.84 respectively. In comparison to other studies, we employ a significantly larger set of images with an F-measure of 0.91. Furthermore, for a fair comparison, we also evaluated our system on the set 1000 images in the publicly available IPC dataset~\cite{siddiqi2012database}, the corresponding F-measure reads 0.92 validating the effectiveness of our detection technique,
\begin{table*}[!ht]
\centering
\caption{Performance comparison with other techniques}
\label{tab:detectResCom}
\footnotesize
\begin{tabular}{llllcccc}
\hline
\multicolumn{1}{l}{\textbf{Study}} &
\multicolumn{1}{l}{\textbf{Method}} &
\multicolumn{1}{l}{\textbf{Dataset}} &
\multicolumn{1}{l}{\textbf{Script}} & \multicolumn{1}{c}{\textbf{Video Frames}} & \multicolumn{1}{c}{\textbf{Precision}} & \multicolumn{1}{c}{\textbf{Recall}} & \multicolumn{1}{c}{\textbf{F-Measure}} \\ \hline
Jamil et al.(2011)~\cite{jamil2011edge} & Edge-based Features & IPC & Urdu & 150 & 0.77 & 0.81 & 0.79 \\
Siddiqi and Raza(2012)~\cite{siddiqi2012database} & Image Analysis & IPC & Urdu & 1,000 & 0.71 & 0.80 & 0.75 \\
Moradi et al.(2013)~\cite{moradi2013hybrid} & LBP with SVM & - & Farsi/Arabic & 4971 & 0.91 & 0.87 & 0.89 \\
Raza et al.(2013)~\cite{raza2013multilingual} & Cascade of Transforms & IPC & Urdu & 1,000 & 0.80 & 0.89 & 0.84 \\
Raza et al.(2013)~\cite{raza2013multilingual}& Cascade of Transforms & IPC & Arabic & 300 & 0.81 & 0.93 & 0.86 \\
Yousfi et al.(2014)~\cite{yousfi2014arabic} & ConvNet & - & Arabic & 201 & 0.75 & 0.80 & 0.77 \\
Zayene et al.(2015)~\cite{Oussama2015} & SWT & AcTiV & Arabic & 425 & 0.67 & 0.73 & 0.70 \\
Zayene et al.(2016)~\cite{Oussama2016} & SWT\&Conv Autoencoders & AcTiV & Arabic & 1843 & 0.83 & 0.85 & 0.84 \\
Shahzad et al.(2017)~\cite{shahzad2017oriental} & Image Analysis & - & Urdu/Arabic & 240 & 0.83 & 0.93 & 0.88 \\
Mirza et al.(2018)~\cite{mirza2018urdu} & Textural Features & UTiV & Urdu & 1,000 & 0.72 & 0.89 & 0.80 \\
Unar et al.(2018)~\cite{unar2018artificial} & Image Analysis+SVM & IPC & Urdu & 1,000 & 0.83 & 0.88 & 0.85 \\
\textbf{Proposed Method} & \textbf{Deep ConvNets} & UTiV & \textbf{Urdu}& \textbf{11,203} & \textbf{0.87} & \textbf{0.95} & \textbf{0.91}\\
& & IPC & \textbf{Urdu} & \textbf{1,000} & \textbf{0.91} & \textbf{0.93} & \textbf{0.92}\\
\hline
\end{tabular}
\end{table*}
\section{Conclusion}\label{sec:Conclusion}
This paper presented a system for detection of caption text appearing in video frames. The developed technique relies on exploiting deep learning based object detectors and adapting them for text detection. Since it is common in videos to have text in more than one script, we presented, as a case study, video frames with text in cursive (Urdu) and Roman (English) scripts. Since each script requires different processing, the detection is combined with script identification in an end-to-end fashion so that the system is able to not only localize the text but also identify its script. Among various investigated object detectors, Faster R-CNN with our proposed set of anchor boxes reported the highest detection rates.\\
The presented work is a part of a comprehensive video indexing and retrieval system and the current study focused on the detection of text. In our other~\cite{ali1,ali2} work, the detection module is integrated with the video OCR module so that detected text is recognized. Once recognized, videos are indexed based on keywords and retrieved on user provided queries. In addition to retrieval, automatic News summarization from ticker text and comparison of News across various News channels is also planned to be implemented. Furthermore, the textual content based retrieval will be combined with audio as well as visual objects appearing in videos.
| {'timestamp': '2023-01-10T02:16:39', 'yymm': '2301', 'arxiv_id': '2301.03164', 'language': 'en', 'url': 'https://arxiv.org/abs/2301.03164'} |
\section{Introduction}
Let $A\in B(\ell_2({\Bbb Z}))$ be a bounded operator on $\ell_2({\Bbb Z})$, we can write $A$ in its matrix presentation
$A=(a_{k,j})$ with $a_{k,j}=\langle Ae_k,e_j\rangle $.
Given a bounded function $m$ on ${\Bbb Z}\times {\Bbb Z}$,
we call the map
\begin{eqnarray}
S_m: (a_{k,j})\mapsto (m_{k,j}a_{k,j})\label{sm}
\end{eqnarray}
a Schur multiplier with symbol $m$.
A historic question is to find conditions on the symbol $m$ so that $S_m$ is bounded with respect to the Schatten $p$-norm of $A$. Here the Schatten-$p$ norm is defined as
\begin{eqnarray*}
\|A\|_p&=&(tr (A^*A)^{\frac p2})^{\frac1p}, \ \ p<\infty\\
\|A\|_\infty&=&\sup_ {\|x\|_{\ell_2}=1} \|Ax\|_{\ell_2}.
\end{eqnarray*}
Using the well-know connection between Fourier multiplier on operator-valued functions and Toeplitz type Schur multipliers, J. Bourgain (\cite{Bou86}) proved a Marcinkiewitz type theory for Toeplitz type Schur multpliers.
The main result of this article extends J. Bourgain's result to non-Toeplitz type Schur multipliers.
\begin{theorem}
\vskip.1cm
$S_m$ defined in (\ref{sm}) extends to a completely bounded map on $S_p$ for all $1<p<\infty$ with bounds $\lesssim (\frac{p^2}{p-1})^3$, if $m$ is bounded and there exists a constant $C$ such that
\begin{eqnarray}
\sum_{2^{N-1}\leq |k|< 2^{N}}|m(k+j+1,j)-m(k+j,j)| <C \label{Marc}\\
\sum_{2^{N-1}<|k|<2^{N}}|m(j,k+j+1)-m(j,k+j)| <C \label{Marr},
\end{eqnarray}
for all $N\in{\Bbb N}$.
\end{theorem}
Hyt\"onen and Weiss (\cite{HW08}) have showed that for operator valued multipliers on a Banach space $X$, the natural Marcinkiewicz Fourier multiplier theory holds if and only if the Banach space $X$ satisfies Pisier's property $\alpha$. While Schatten-$p$ class fails the property $\alpha$, this article shows that Marcinkiewicz type Schur multiplier theory still holds.
The writing of this article was motivated by the recent work by Conde etc. (\cite{CondeAlonso2022}), although the third author had known Theorem 1.1 for a while. Conde etc. proved in \cite{CondeAlonso2022} a remarkable H\"ormander-Mikhlin type criterion for Schur mulipliers on $S_p({\Bbb R}^n)$. The considering of multi-indexed $S_p$ is motivated by the recent work of Lafforgue/de la Salle (\cite{Lafforgue2011}) on lattices of higher rank Lie groups (e.g. $SL_3({\Bbb Z}$)), in which Schur multipliers on multi-indexed Schatten $p$-classes play important roles. Following this trend, we extend our result to higher dimension ${\Bbb Z}^n$ and the case of continuous indices ${\Bbb R}^n$ as well.
\section{Preliminaries}
Denote $\mathcal{B}(\ell^2(\mathbb{Z}^d))$ the set of bounded linear operators on $\ell^2(\mathbb{Z}^d)$. We present operator $A\in \mathcal{B}(\ell^2)$ as $A=(a_{i,j})_{(i,j)\in \mathbb{Z}^d\times \mathbb{Z}^d}$ with $a_{i,j}=\langle Ae_k,e_j\rangle $ for a fixed orthogonal basis $e_i$ of $\ell^2(\mathbb{Z}^d)$
The Schur multiplier $M=(m_{i,j})_{(i,j)\in \mathbb{Z}\times \mathbb{Z}}$ acting on the the $\mathcal{B}(\ell^2(\mathbb{Z}^d))$ is the weak* continuous map on $\mathcal{B}(\ell^2{(\mathbb{Z})}))$ extending the map
\[
S_M(A)=(m_{ij})\circ(a_{ij})=(m_{ij}a_{ij})
\]
for operators $A=(a_{i,j})$ on the Schatten class $S_2(\ell^2(\mathbb{Z}^d))$.
For $f\in L^2(\mathbb{T}^d)$ and $R$ a subset of ${\Bbb Z}^d$, denote $S_{R} f$ the partial Fourier sum
\[
S_{R} f(z)= \sum_{n\in R} \hat{f}(n) z^n.
\]
\begin{comment}
Suppose $\eta$ is a $C^{\infty}$ function of compact support on $\mathbb{R}$ with the properties that $\eta(x)=1$ for $|x|\leq 1$ and $\eta(x)=1$ for $|x|\geq 2$. Let $\delta(x)=\eta(x)-\eta(2x)$, then $\delta$ is also a $C^{\infty}$ function with compact support on $[-2,-\frac{1}{2}]\cup [\frac{1}{2}, 2]$. Then for any $x\in\mathbb{R}$, there is
\[
\sum_{k=-\infty}^{\infty} \delta(2^{-j}x)=1. \quad \text{all~}x\neq 0.
\]
\end{comment}
Choose $\delta_d\in C^{\infty}(\mathbb{R})$ such that $0\leq\delta_d\leq 1$, $supp(\delta)\subset [-2\sqrt d ,-\frac{1}{4}]\cup[\frac{1}{4},2\sqrt d]$ and
$\delta(x)=1$ when $\frac12\leq |x|\leq \sqrt d$. For $j\in\mathbb{Z}$, define $\delta_j(x):=\delta(2^{-j}x)$. For $f\in L^2(\mathbb{T}^d)$, we define
\[
S_{\delta_j}f(z)=\sum_{n\in\mathbb{Z}^d} \delta_j(|n|_{2}) \hat{f}(n) z^n.\]
Let $(E_j)_{j\geq 0}$ be the cubs with squared holes in $\mathbb{Z}^d$ given by
\begin{equation}\label{eq:dyadic}
E_j =
\begin{cases}
\{n\in\mathbb{Z}^d: 2^{j-1}< |n|_\infty\leq 2^j\} & j\geq 0,\\
\{0\} & j=0,
\end{cases}
\end{equation} Note our construction implies $ S_{E_j}S_{\delta_j}=S_{E_j}$ which we will use frequently.
We will need the following noncommutative Littlewood-Paley theorem on $\mathbb{Z}^d$.
\begin{lemma}\label{thm:Littwood-P}
Suppose $f(z)=\sum_{n\in \mathbb{Z}^d}\hat{f}(n)z^n\in L^{\infty}(\mathbb{T}^d)\otimes \mathcal{B}(\ell^2(\mathbb{Z}^d))$ where $\hat{f}(n)\in \mathcal{B}(\ell^2(\mathbb{Z}^d))$ and $p\in (1,\infty)$. Denote $\mathcal{M}= L^{\infty}(\mathbb{T}^d)\bar{\otimes} \mathcal{B}(\ell^2(\mathbb{Z}^d))$. There is
\begin{align}
\|f\|_{L^p(\mathbb{T}^d;S_p)} &\leq C \frac{p^2}{p-1} \left\| (S_{E_j} f )\right\|_{L^p(\mathcal{M},\ell^2_{cr})}, \\
\left\| (S_{\delta_j} f )\right\|_{L^p(\mathcal{M},\ell^2_{cr})} &\leq C \frac{p^2}{p-1}\|f\|_{L^p(\mathbb{T}^d;S_p)}.
\end{align}
Here $C$ is a constant independent of $p$ and $d$.
\end{lemma}
\begin{comment}
\begin{theorem}[Marcinkiewicz,\cite{Marcinkiewicz1939}]
Let $(\Delta_j)_{j\in\mathbb{Z}}$ be the dyadic decomposition of $\mathbb{Z}$. Suppose $M(n)$ is a function on $\mathbb{Z}$ such that
\[
\sup_{n\in\mathbb{Z}} |M(n)|\leq C,\quad \sup_{j\in\mathbb{Z}}\sum_{n\in\Delta_j}|\Delta M(n)|\leq C
\]
where $\Delta M(n)=M(n+1)-M(n)$. Then
\[
\|T_M f\|_p \leq C_p C \|f\|_p
\]
for every $p\in (1,\infty)$, where $C_p$ is a number depending only on $p$.
\end{theorem}
\end{comement}
\begin{comment}
{
\begin{lemma}\label{thm:Littwood-P}
Suppose $f(z)=\sum_{n\in \mathbb{Z}}\hat{f}(n)z^n\in L^{\infty}(\mathbb{T})\otimes \mathcal{B}(\ell^2(\mathbb{Z}))$ where $\hat{f}(n)\in \mathcal{B}(\ell^2(\mathbb{Z}))$ and $p\in (1,\infty)$, then
\[
\|f\|_{L^p(\mathbb{T};S_p)} \asymp
\begin{cases}
\inf\left\{ \left\|\left(\sum_{j\in\mathbb{Z}}g_j g_j^*\right)^{\frac12} \right\|_{L^p(\mathbb{T};S_p)}+\left\|\left(\sum_{j\in\mathbb{Z}}h_j^* h_j\right)^{\frac12} \right\|_{L^p(\mathbb{T};S_p)}:S_{j}f=g_j+h_j\right\} & p\in(1,2]\\
\left\| \left(\sum_{j\in\mathbb{Z}} (S_jf)^*(S_jf)\right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)}+ \left\| \left(\sum_{j\in\mathbb{Z}} (S_jf) (S_jf)^*\right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)} & p\in [2,\infty).
\end{cases}
\]
\end{lemma}
}
\end{comment}
\begin{lemma}\label{Hilbert}
Suppose $R_j$ is a family a rectangles with sides parallel to the axes in $\mathbb{R}^n$. Then there is a constant $C_d>0$ such that for all $1<p<\infty$, for all $\sigma$-finite measures $\mu$ on $\mathbb{R}^d$, and for all families of measurable functions $f_j$ on $\mathbb{R}^d$, we have
\begin{equation}
\left\| \sum_j | (\chi_{R_j} \widehat{f}_j)^{\vee}|^2 )^{\frac12}\right\|_{L^p(\mathbb{T};S_p)} \leq C_d \left(\frac{p^2}{p-1}\right)^d \left\| \left( \sum_j |f_j|^2\right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)}.
\end{equation}
\end{lemma}
Also we are going to use the following Cauchy-Schwarz type lemma.
\begin{lemma}\label{lem:Cauchy-S}
Suppose $a_n,c_n\in \mathcal{B}(\ell^2(\mathbb{Z}^d)), n\in\mathbb{Z}^d$. Then
\begin{equation}\label{eq:C-S}
\left|\sum_{n\in\mathbb{Z}^d} a_n^* c_n\right|^2 \leq \left\|\sum_{n\in\mathbb{Z}^d} |a_n|^2 \right\|_{\infty} \left( \sum_{n\in\mathbb{Z}^d} |c_n|^2\right)
\end{equation}
\end{lemma}
\begin{proof}
It is equivalent to prove that for any $v\in \ell^2(\mathbb{Z}^d)$, there is
\begin{equation}\label{eq:inner}
\langle |\sum_n a_n^* c_n|^2 v, v\rangle \leq \|\sum_n |a_n|^2\|_{\infty}\langle \sum_{n}|c_n|^2 v,v \rangle.
\end{equation}
Suppose $(e_{k,l})_{k,l\in\mathbb{Z}^d}$ is the canonic basis of $\mathcal{B}(\ell^2(\mathbb{Z}^d))$ and denote $A=\sum_{n} a_n^*\otimes e_{n,1}$, $C=\sum_n c_n v \otimes e_{1,n}$. Then
\[
\text{LHS of \eqref{eq:inner}}= \|\sum_{n} a_n^* c_n v \|^2= \|AB\|^2 \leq \|AA^*\| \|C^*C\|\leq \text{RHS of \eqref{eq:inner}}.
\]
Hence, the lemma is proved.
\end{proof}
}
\section{The Schur-Marcinkiewicz multiplier theorem on ${\mathbb{Z}}$}
\begin{theorem}\label{thm:main}
Given $M=(m_{ij})_{i,j\in \mathbb{Z}}\in \mathcal{B}(\ell^2(\mathbb{Z}))$.
Suppose $M$ satisfies that
\begin{enumerate}
\item[i)] $\sup_{i,j\in \mathbb{Z}}|m_{ij}|<C_1$ and
\item[ii)] For any $k\in\mathbb{N}, s\in\mathbb{Z}$, there are constants $C_1,C_2$ such that
\begin{equation}\label{eq:rowAndColumn}
\sum_{2^{k-1}\leq |j| <2^k}|m_{i+j+1,i}-m_{i+j,i}|<C_2, \quad \sum_{2^{k-1}\leq |j| <2^k} |m_{i,i+j+1}-m_{i,i+j}|<C_2,
\end{equation}
\end{enumerate}
Then $M$ is a (completely) bounded Schur multiplier on $S_p(\ell^2(\mathbb{Z}))$ for $p\in(1,\infty)$ with the bound $C_3 (\frac{p^2}{p-1})^3$, where $C_3$ is an absolute constant.
\end{theorem}
\begin{proof}Define the *-homomorphism $\pi: \mathcal{B}(\ell^2(\mathbb{Z}^d))\to L^{\infty}(\mathbb{T}^d)\otimes \mathcal{B}(\ell^2(\mathbb{Z}^d))$ by
\begin{equation}\label{eq:transference}
\pi: A=(a_{st})\longmapsto (a_{st} z^{s-t}).
\end{equation}
Given $M=(m_{ij})_{i,j\in \mathbb{Z}^d}, A=(a_{ij})_{i,j\in\mathbb{Z}}\in \mathcal{B}(\ell^2(\mathbb{Z}^d ))$, denote
\begin{align}\label{eq:muliplierNotation}
&M_{l}(n)=\sum_{s\in\mathbb{Z}^d}m_{s,s-n}\, e_{s,s},\quad M_r(n)=\sum_{s\in \mathbb{Z}^d } m_{s+n,s}\,e_{s,s},\\
&A_l(n)=\sum_{s\in\mathbb{Z}^d } a_{s,s-n} e_{s,s-n}, \quad A_r(n)=\sum_{s\in\mathbb{Z}^d } a_{s+n,s} e_{s+n,s}. \nonumber
\end{align}
By the notations in \eqref{eq:muliplierNotation}, the condition \eqref{eq:rowAndColumn} is equivalent to
\begin{equation}
\sup_{j\in \mathbb{Z}} \left\|\sum_{n\in E_j} |M_l(n+1)-M_l(n)| \right\|_{\infty}<C, \quad \sup_{j\in \mathbb{Z}} \left\|\sum_{n\in E_j} |M_r(n+1)-M_r(n)| \right\|_{\infty}<C,
\end{equation}
Let
\begin{equation}\label{f}
f(z)=\pi(A)
=\sum_{n\in\mathbb{Z}} A_l(n)z^n =\sum_{n\in\mathbb{Z}} A_r(n)z^n.
\end{equation}
Let
\begin{equation}\label{eq:transfer}
T_{\widetilde{M}} \, \pi (A)=\pi S_M (A) .
\end{equation}
It can be easily verified that
\begin{align}\label{eq:transf1}
T_{\widetilde{M}} \, \pi (A)&= \sum_{n\in\mathbb{Z}^d} M_l(n)A_l(n)z^n\\
&= \sum_{n\in\mathbb{Z}^d } A_r(n) M_r(n) z^n, \label{eq:transf2}
\end{align}
Since $\pi$ is a $*$-homomorphism then
\begin{equation}
\|A\|_{S_p}=\|f(z)\|_{L^p(\mathbb{T};S_p)}, \ \ \|S_MA\|_{S_p}=\|T_{\widetilde{M}}(f)\|_{L^p(\mathbb{T};S_p)}.
\end{equation}
In order to prove $M$ is a bounded Schur multiplier, we need to prove that $T_{\widetilde{M}}$ is bounded on the subspace of $L_p(\mathbb{T};S_p)$ consisting all $f$ in the form of (\ref{f}). By noncommutative Littlewood-Paley theorem \eqref{thm:Littwood-P} and the transference relation \eqref{eq:transf1},\eqref{eq:transf2}, it is sufficient to show that
\begin{equation}\label{eq:L-P1}
\left\| \left( \sum_{j\in\mathbb{Z}} |S_{E_j}(T_{\widetilde{M}}f)|^2 \right)^{\frac12} \right\|_{L_p(\mathbb{T};S_p)}\leq C \frac{p^2}{p-1} \left\| \left( \sum_{j\in\mathbb{Z}} |S_{\delta_j}f|^2 \right)^{\frac12} \right\|_{L_p(\mathbb{T};S_p)}
\end{equation}
and its adjoint form for $p\geq 2$, where $C$ is a positive constant.
Denote the frequency projection operator $S_{(a,b)}$ for the $g(z)=\sum_{n\in\mathbb{Z}} \hat{g}(n)z^n\in L^{\infty}(T)\otimes~\mathcal{B}(\ell^2(\mathbb{Z}))$ as follows.
\begin{equation}
S_{(a,b)}g(z)= \sum_{a<n<b} \hat{g}(n)z^n
\end{equation}
Write $E_j=E_{j,1}\cup E_{j_2}$ with $E_{j,1}=E_j\cap {\Bbb Z}_-, E_{j,2}=E_j\cap {\Bbb Z}_+$. Let $a_{j,1}=-2^{j},a_{j,2}=2^{j}$ and
\begin{eqnarray}
\Delta M(n)=sgn(n)(M(n+1)-M(n))
\end{eqnarray}
Then via summation by parts and notice that $S_{(2^{j-1}-1,2^{j}-1)}f=S_{E_j}f$ and $S_{(2^{j+1}-1,2^{j+1})}f=0$,
\begin{align}\label{eq:summationByPart1}
S_{E_j}(T_{\widetilde{M}}f) =\sum_{i=1,2}S_{E_{j,i}}(T_{\widetilde{M}}f) =\sum_{i=1,2} \left(M_l(a_{j,i}) (S_{E_{j,i}} f)+\sum_{n\in E_{j,i}} \Delta M_l(n) (S_{(n,a_{j,i})}f)\right)
\end{align}
We will use the presentation (\ref{eq:summationByPart1}) to prove (\ref{eq:L-P1}), and use (\ref{eq:summationByPart2})
\begin{align}
S_{E_j}(T_{\widetilde{M}}f) =\sum_{i=1,2}S_{E_{j,i}}(T_{\widetilde{M}}f) =\sum_{i=1,2} \left((S_{E_{j,i}} f)M_r(a_{j,i}) +\sum_{n\in E_{j,i}} (S_{(n,a_{j,i})}f)\Delta M_r(n)\right) \label{eq:summationByPart2}
\end{align}
to prove the adjoint version of (\ref{eq:L-P1}). This will help to reduce the usual difficulties caused by the noncommutativity.
Note $\Delta M_l(n)$ is a diagonal operator, we can write $\Delta M_l (n)=a^*_n b_n $ with $a_n,b_n$ diagonal operators and $|a_n|^2=|b_n|^2=|\Delta M_l(n)| $,
then by Lemma \ref{lem:Cauchy-S}, we have
\begin{align}
\left|\sum_{n\in E_j}\Delta M(n) S_{(n,a_{j,i})} f\right|^2 &\leq \left\| \sum_{n\in E_j} |\Delta M_l(n)| \right\|_{\infty} \left( \sum_{n\in E_j}|b_nS_{(n,a_{j,i})}f|^2 \right) \nonumber \\
&\leq C_2 \left( \sum_{n\in E_j}|S_{(n,a_{j,i})}(b_nS_{\delta_j}f)|^2 \right).\label{eq:sumP2}
\end{align}
\noindent Thus, by Lemma \ref{Hilbert}
\begin{align}
\left\| \left( \sum_{j\in\mathbb{Z}}\left|\sum_{n\in E_{j,i}}\Delta M(n) S_{(n,a_{j,i})} f\right|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T};S_p)}
&\leq C_2^{\frac{1}{2}} \left\| \left( \sum_{j\in\mathbb{Z}}\sum_{n\in E_{j,i}}\left| S_{(n,a_{j,i})} \left(b_n S_{\delta_j} f\right)\right|^2\right)^{\frac12} \right\|_{L^p(\mathbb{T};S_p)}
\nonumber \\
&\leq C_3\frac{p^2}{p-1} C_2^{\frac{1}{2}} \left\| \left( \sum_{j\in\mathbb{Z}}\sum_{n\in E_{j,i}}\left| \left(b_n S_{\delta_j} f\right)\right|^2\right)^{\frac12} \right\|_{L^p(\mathbb{T};S_p)}
\nonumber \\
&= C_3\frac{p^2}{p-1} C_2^{\frac{1}{2}} \left\| \left( \sum_{j\in\mathbb{Z}} (S_{\delta_j}f)^* \left(\sum_{n\in E_{j,i}} |b_n|^2\right) S_{\delta_j}f \right)^{\frac12} \right\|_{L^p(\mathbb{T};S_p)} \nonumber\\
&= C_3\frac{p^2}{p-1} C_2^{\frac{1}{2}} \left\| \left( \sum_{j\in\mathbb{Z}} (S_{\delta_j}f)^* \left(\sum_{n\in E_{j,i}} |\Delta M_l(n)|\right) S_{\delta_j}f \right)^{\frac12} \right\|_{L^p(\mathbb{T};S_p)} \nonumber\\
& \leq C_3\frac{p^2}{p-1}C_2 \left\|\left( \sum_{j\in\mathbb{Z}} | S_{\delta_j} f|^2\right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)} \label{eq:sumP3}
\end{align}
Hence, by \eqref{eq:summationByPart1}, \eqref{eq:sumP3}, and Lemma \ref{Hilbert}
\begin{align}\label{eq:normSum}
\left\|\left(\sum_{j\in\mathbb{Z}} |S_{E_{j,i}} (T_{\widetilde{M}} f)|^2 \right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)} &\leq
\left\| \left(\sum_{j\in\mathbb{Z}} \left| M_l(a_{j,i})(S_{E_{j,i}} f)\right|^2\right)^\frac12\right\|_{L^p(\mathbb{T};S_p)} +C_3\frac{p^2}{p-1}C_2 \left\|\left( \sum_{j\in\mathbb{Z}} | S_{\delta_j} f|^2\right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)} \nonumber\\
&\leq\left\|C_1 \left(\sum_{j\in\mathbb{Z}} \left| (S_{E_{j,i}} f)\right|^2\right)^\frac12\right\|_{L^p(\mathbb{T};S_p)} +C_3\frac{p^2}{p-1}C_2 \left\|\left( \sum_{j\in\mathbb{Z}} | S_{\delta_j} f|^2\right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)} \nonumber\\
&=\left\|C_1 \left(\sum_{j\in\mathbb{Z}} \left| (S_{E_{j,i}}S_{\delta_j} f)\right|^2\right)^\frac12\right\|_{L^p(\mathbb{T};S_p)} +C_3\frac{p^2}{p-1}C_2 \left\|\left( \sum_{j\in\mathbb{Z}} | S_{\delta_j} f|^2\right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)} \nonumber\\
&\leq C \frac{p^2}{p-1} \left\|\left(\sum_{j\in\mathbb{Z}} |S_{\delta_j} f)|^2 \right)^{\frac12}\right\|_{L^p(\mathbb{T};S_p)}
\end{align}
for $i=1,2$.
Therefore we finish the proof of (\ref{eq:L-P1}). Its adjoint form can be proved similarly. We then complete the proof Theorem \ref{thm:main}.
\end{proof}
\begin{comment}
by the approximation of de la Salle (need to cite!!!),ect, we have ...
\begin{theorem}
Continuous case for one dimension, to be added...
\end{theorem}
\begin{remark} to be added...
1. some examples (Bourgain's result)
2. connection with
\end{remark}
\end{comment}
\section{The Schur-Marcinkiewicz multiplier theorem on ${\mathbb{Z}}^d$}
In this part, we will generalize Theorem \ref{thm:main} to the $d$-dimensional case. Before we come to the main statement of the theorem, we need to borrow some notations from from calculus of finite differences.
\begin{definition} Let $\sigma:\mathbb{Z}^d\to \mathbb{C}$ and ${j}=(j_1,\cdots,j_n)\in\mathbb{Z}^d$. Let $\{e_j\}_{j=1}^d$ be standard basis of $\mathbb{Z}^d$, i.e., the $j$-th entry of $e_j$ is 1 and 0 for other entries, for $j=1,\cdots,d$. We define the forward and backward partial difference operators $\Delta_{\xi_j}$ and $\widebar{\Delta}_{\xi_j}$, respectively, by
\begin{equation}\label{eq:differenceNotion}i
\Delta_{\xi_j} \sigma (\xi) \coloneqq \sigma(\xi+ e_j)-\sigma(\xi), \quad
\widebar{\Delta}_{\xi_j} \sigma (\xi) \coloneqq \sigma (\xi)- \sigma(\xi-e_j).
\end{equation}
and for $\mathbf{\alpha}\in \mathbb{N}_0^n$, define
\begin{equation*}
\Delta_{\mathbf{\xi}}^{\mathbf{\alpha}} \coloneqq \Delta_{\xi_1}^{\alpha_1}\cdots \Delta_{\xi_n}^{\alpha_n},\quad
\widebar{\Delta}_{\mathbf{\xi}}^{\mathbf{\alpha}} \coloneqq \widebar{\Delta}_{\xi_1}^{\alpha_1}\cdots \widebar{\Delta}_{\xi_n}^{\alpha_n}.
\end{equation*}
\end{definition}
Readers can finde more information of calculus of finite differences in chapter 3 of \cite{Ruzhansky2010}. For convenience, we list the following properties of finite differences.
\begin{itemize}
\item $\Delta_{\xi}^{\alpha} \Delta_{\xi}^{\beta}= \Delta_{\xi}^{\beta}\Delta_{\xi}^{\alpha}=\Delta_{\xi}^{\alpha+\beta}$,
\item Let $\phi:\mathbb{Z}^D\to\mathbb{C}$, then
$
\Delta_{\xi}^{\alpha}(\xi)= \sum_{\beta\leq \alpha} (-1)^{|\alpha-\beta|} {\alpha \choose \beta} \phi(\xi+\beta)
$
\item (Discrete Leibniz formula) Let $\phi, \psi:\mathbb{Z}^d\to \mathbb{C} $. Then \[\Delta_{\xi}^{\alpha}(\phi\psi)(\xi)= \sum_{\beta\leq \alpha} {\alpha \choose \beta} (\Delta_{\xi}^{\beta}\phi(\xi)) \Delta_{\xi}^{\alpha-\beta} \psi(\xi+ \beta).\]
\end{itemize}
\subsection{two dimensional case}
Denote a partition $\mathscr{I}^2 \coloneqq \{ E_j: j\geq 0\} $ of $\mathbb{Z}^2$ as in the case of $\mathbb{Z}$, where $E_j$ is defined as follows
\begin{equation}\label{eq:partitionZ2}
E_j=\begin{cases}
\{(0,0) \} & j=0,\\
\{(n_1,n_2)\in\mathbb{Z}^2:2^{j-1}\leq |(n_1,n_2)|_{\infty}<2^j \} & j\geq 1.
\end{cases}
\end{equation}
\begin{comment}
Thus, we would have the following noncommutative Littlewood-Paley theorem on $\mathbb{Z}^2$.
\begin{lemma}\label{thm:Littwood-P2}
Suppose $f(z)=\sum_{n\in \mathbb{Z}^2}\hat{f}(n)z^n\in L^{\infty}(\mathbb{T}^2)\otimes \mathcal{B}(\ell^2(\mathbb{Z}))$ where $\hat{f}(n)\in \mathcal{B}(\ell^2(\mathbb{Z}^2))$ and $p\in (1,\infty)$. Denote $S_{E_j}$ the partial sum projection on $L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))$ given by
$
S_{E_j}f(z)=\sum_{n\in E_j} \widehat{f}(n) z^n
$, where $z\in \mathbb{T}^2$. Then
\[
\|f\|_{L^p(\mathbb{T}^2;S_p)} \asymp
\begin{cases}
\inf\left\{ \left\|\left(\sum_{j\in\mathbb{Z}^2}g_j g_j^*\right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p)}+\left\|\left(\sum_{j\in\mathbb{Z}^2}h_j^* h_j\right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p)}:S_{E_j}f=g_j+h_j\right\} & p\in(1,2]\\
\left\| \left(\sum_{j\in\mathbb{Z}^2} (S_{E_j} f)^*(S_{E_j} f)\right)^{\frac12}\right\|_{L^p(\mathbb{T}^2;S_p)}+ \left\| \left(\sum_{j\in\mathbb{Z}^2} (S_{E_j} ) (S_{E_j} f)^*\right)^{\frac12}\right\|_{L^p(\mathbb{T}^2;S_p)} & p\in [2,\infty).
\end{cases}
\]
\end{lemma}
\end{comment}
\begin{comment}
\begin{figure}[ht]
\centering
\def0.5\columnwidth{0.25\columnwidth}
\import{./figures/}{drawing.pdf_tex}
\caption{Drawing}
\label{fig:A drawing}
\end{figure}
\end{comment}
With the notations of difference and partition of $\mathbb{Z}^2$, we are ready to state the generalization of Theorem \ref{thm:main} in~$\mathbb{Z}^2$.
\begin{theorem} \label{thm:dimension2}
Given $M=(m_{s,t})_{s,t\in\mathbb{Z}^2}\in \mathcal{B}(\ell^2(\mathbb{Z}^2))$. Suppose $M$ satisfies that
\begin{itemize}
\item[i)] $\sup_{s,t\in\mathbb{Z}^2} |m_{s,t}|<C_1$
\item[ii)] For any $k\in\mathbb{N}, s\in\mathbb{Z}^2 $
there is
\begin{align}\label{eq:left}
\left(\sum_{\{t_1:t=(t_1,\pm 2^{k-1})\in E_k\}} |\Delta_{t_1} m_{s,s+t}| + \sum_{\{t_2:t=(\pm 2^{k-1}, t_2)\in E_k\}} |\Delta_{t_2} m_{s,s+t}| \right) &<C_2, \\
\sum_{\{t_1,t_2: t=(t_1,t_2)\in E_k\}} |\Delta_t m_{s,s+t}|&<C_3, \label{eq:leftDouble}
\end{align}
and
\begin{align}\label{eq:right}
\left(\sum_{\{t_1:t=(t_1,\pm 2^{k-1})\in E_k\}} |\Delta_{t_1} m_{s+t,s}| + \sum_{\{t_2:t=(\pm 2^{k-1}, t_2)\in E_k\}} |\Delta_{t_2} m_{s+t,s}| \right) &<C_2\\
\sum_{\{t_1,t_2: t=(t_1,t_2)\in E_k\}} |\Delta_t m_{s+t,s}|&<C_3, \label{eq:rightDouble}
\end{align}
\end{itemize}
Then $M$ is a (completely) bounded Schur multiplier on $S_p(\ell^2(\mathbb{Z}^2))$ for $p\in (1,\infty)$ with the bound $C(\frac{p^2}{p-1})^4$. Here $C_1, C_2$ and $C_3,C$ are some positive constants.
\end{theorem}
Now we come to the proof of Theorem \ref{thm:dimension2}. Denote $S_{E_j}$ the partial sum projection on $L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))$ given by
$
S_{E_j}f(z)=\sum_{n\in E_j} \widehat{f}(n) z^n
$, where $z\in \mathbb{T}^2$.
By the Lemma \ref{thm:Littwood-P} and the Schur and Fourier multiplier transference in $\mathbb{Z}^2$ which is defined similar to the one dimensional case, it is sufficient to prove
\begin{equation}\label{eq:SufficientCondition}
\left\| \left( \sum_{j = 0}^{\infty} \left| S_{E_{j }} T_{\widetilde{M}} f\right|^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \leq C \frac{p^2}{p-1} \left\| \left( \sum_{j = 0}^{\infty} \left| S_{\delta_{j }} f \right|^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}(\mathbb{T}^2;S_p(\mathbb{Z}^2))}.
\end{equation}
for $p\geq 2$ and $f(z)\in \pi(\mathcal{B}(\ell^2(\mathbb{Z}^2)))$.
\begin{comment}
To prove the ....
We need the following operator valued Littlewood-Paley theorem in $\mathbb{Z}^2$.
\begin{theorem}
Suppose $f(z)=$
\end{theorem}
\end{comment}
Notice that \eqref{eq:left} and \eqref{eq:leftDouble} are equivalent to
\begin{align} \label{eq:leftPartialMl}
\sup_{j\geq 0} \left( \left\| \sum_{\{n_1:n=(n_1,\pm 2^{j-1})\in E_j\}} |\Delta_{n_1} M_{l} (n)| \right\|_{\infty} +\left\| \sum_{\{n_2:n=(\pm 2^{j-1},n_2)\in E_j\}} |\Delta_{n_2} M_{l} (n )| \right\|_{\infty} \right) & < C_2, \\
\sup_{j\geq 0}\left\|\sum_{\{n_1,n_2: n=(n_1,n_2)\in E_j\}} \left| \Delta_n M_{l} (n) \right| \right\|_{\infty}& < C_3. \label{eq:leftDoublePartial}
\end{align}
To prove \eqref{eq:SufficientCondition}, we may cut $E_j$ into 4 rectangles $E_{j,k}, k=1,\cdots 4$ for $j\geq 1$. Let $I_{j}=[2^{j-1}, 2^j)\cap \mathbb{Z}, J_i=(-2^{j-1},2^j)\cap \mathbb{Z}$, denote
\begin{equation*}
E_{j,1}=J_j\times I_j,\qquad E_{j,2}=(-I_j)\times J_j, \qquad E_{j,3}= I_j\times (-J_{j}),\qquad E_{j,4}=(-J_j)\times (-I_{j}).
\end{equation*}
\begin{comment}
\begin{figure}[ht]
\centering
\def0.5\columnwidth{0.5\columnwidth}
\import{./figures/}{E2.pdf_tex}
\caption{Decomposition of $E_j$}
\label{fig:A drawing2}
\end{figure}
\end{comment}
Thus, we have
\begin{equation}\label{eq:combin}
S_{E_j}T_{\widetilde{M}} f= \sum_{k=1}^4 S_{E_{j,k}}T_{\widetilde{M}}f.
\end{equation}
Since $E_{j,k}$ can be obtained from $E_{j,1}$ by rotation for $k=2,3,4$. To prove \eqref{eq:SufficientCondition}, it is sufficient to prove
\begin{equation}\label{eq:sufficient2}
\left\| \left( \sum_{j = 0}^{\infty} \left| S_{E_{j,1 }} T_{\widetilde{M}} f\right|^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \leq C^{\prime} \left(\frac{p^2}{p-1}\right)^2 \left\| \left( \sum_{j = 0}^{\infty} \left| S_{\delta_j} f \right|^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}(\mathbb{T}^2;S_p(\mathbb{Z}^2))}
\end{equation}
for $p\geq 2$ and $f(z)\in \pi(\mathcal{B}(\ell^2(\mathbb{Z}^2)))$. By fundamental theorem of calculus,
\begin{align}\label{eq:abelianSummationDimension2}
S_{E_{j,1}} T_{\widetilde{M}} f
= &M_l(-2^{j-1}+1, 2^{j-1}) S_{E_{j,1}}f + \sum_{n_1\in J_j} \Delta_{n_1} M_l(n_1, 2^{j-1}) S_{(n_1,2^j)\times E_j} f \nonumber \\
&+\sum_{n_2\in E_j} \Delta_{n_2} M_l (-2^{j-1}+1,n_2) S_{J_j\times (n_2,2^j)} f + \sum_{n=(n_1,n_2)\in E_{j,1}} \Delta_n M_l (n_1,n_2) S_{(n_1,2^j)\times (n_2,2^j)}f \nonumber\\
&=: P^1_j + P^2_j + P^3_j + P^4_j
\end{align}
By the operator inequality $|\sum_{k=1}^n a_k|^2\leq n \sum_{k=1}^n |a_k|^2$, we have
\begin{equation}\label{eq:Dimension2Part2}
|S_{E_{j,1}} T_M f|^2 = |P^1_j + P^2_j + P^3_j + P^4_j|^2 \leq 4(|P^1_j|^2+ |P^2_j|^2+|P^3_j|^2+|P^4_j|^2).
\end{equation}
For part $P^1_j$, by assumption $i)$ of Theorem \ref{thm:dimension2}, there is
\begin{equation}\label{eq:partA}
|P^1_j|^2= | M_l(-2^{j-1+1}, 2^{j-1}) ~S_{E_{j,1}}f|^2 \leq C_1^2 |S_{E_{j,1}}f|^2 \leq C_1^2 |S_{\delta_j} f|^2 .
\end{equation}
For part $P^2_j$, following the similar arguments for \eqref{eq:sumP2} and \eqref{eq:sumP3} in the one dimension case and denote $R_{n_1,j}=(n_1, 2^j)\times E_j$. There is
\begin{align}\label{eq:partB1}
|P^2_j|^2 &= \left|\sum_{n_1\in J_j} \Delta_{n_1} M_l(n_1, 2^{j-1}) S_{(n_1,2^j)\times E_j} f \right|^2 \nonumber\\
& \leq \left\|\sum_{n_1\in J_j} |\Delta_{n_1}M_l(n_1,2^{j-1})| \right\|_{\infty} \left( \sum_{n_1\in J_j} \left|S_{R_{n_1,j}} (|\Delta_{n_1} M_l(n_1, 2^{j-1})|^{\frac12}S_{E_j}f) \right|^2 \right).
\end{align}
\begin{comment}
Before we proceed, we need to estimate $\left\|\sum_{n_1\in J_j} |\Delta_{n_1}M_l(n_1,2^{j-1})| \right\|_{\infty}$. By the assumption of \eqref{eq:left} and \eqref{eq:leftDouble},
\begin{align} \label{eq:partB2}
\left\|\sum_{n_1\in J_i} \left| \Delta_{n_1} M_l(n_1,2^{j-1}) \right| \right\|_{\infty} &= \left\| \sum_{n_1\in J_j} \left| \Delta_{n_1} M_{n_1}(n_1, 2^j-1)- \sum_{n_2\in E_j} \Delta_{n_2}\Delta_{n_1} M_l (n_1, n_2) \right| \right\|_{\infty} \nonumber\\
&\leq \left\| \sum_{n_1\in J_j} \left|\Delta_{n_1} M_l(n_1, 2^j-1) \right| \right\|_{\infty} + \left\| \sum_{n\in E_{j,1}} \left| \Delta_n M_l (n) \right| \right\|_{\infty} \nonumber\\
&\leq C_2+C_3=: C_4.
\end{align}
\end{comment}
Thus, by \eqref{eq:partB1}
\begin{align}\label{eq:partB3}
\left\| \left(\sum_{j\geq 0} |P^2_j|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2; S_p(\mathbb{Z}^2))} &\leq C_4^{\frac12} \left\| \left( \sum_{j\geq 0} \sum_{n_1\in J_j} \left| S_{R_{n_1,j}} S_{\delta_j} | \Delta_{n_1} M_l (n_1, 2^{j-1}) |^{\frac12} f \right|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \nonumber \\
\leq C_4^{\frac12} \left(\frac{p^2}{p-1}\right) & \left\| \left( \sum_{j\geq 0} \sum_{n_1\in J_j} \left| S_{\delta_j} | \Delta_{n_1} M_l (n_1, 2^{j-1}) |^{\frac12} f \right|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \nonumber \\
\leq C_4^{\frac12} \left(\frac{p^2}{p-1}\right) &\left\| \left( \sum_{j\geq 0} (S_{\delta_j})^* \left( \sum_{n_1\in J_j} \left| \Delta_{n_1} M_l(n_1, 2^{j-1}) \right| \right) S_{\delta_j}f \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \nonumber\\
\leq C_4 \left(\frac{p^2}{p-1}\right) &\left\| \left( \sum_{j\geq 0} |S_{\delta_j}f|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))}
\end{align}
For part $P^3_j$, similarly, we have
\begin{equation} \label{eq:partC}
\left\| \left(\sum_{j\geq 0} |C_j|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2; S_p(\mathbb{Z}^2))} \leq C_4 \left(\frac{p^2}{p-1}\right) \left\| \left( \sum_{j\geq 0} |S_{\delta_j}f|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))}.
\end{equation}
Now we come to the estimate of part $P^4_j$. Denote $R_{n,j}=(n_1,2^j)\times (n_2,2^j)$. Similarly,
\begin{align}\label{eq:partD}
\left\| \left( \sum_{j\geq 0} |P^4_j|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))} & \leq C_3^{\frac12} \left\| \left( \sum_{j\geq 0} \sum_{n=(n_1,n_2)\in E_{j,1}} \left| S_{R_{n,j}} S_{\delta_j} |\Delta_n M_l(n)|^{\frac12} f \right|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \nonumber\\
\leq C_3^{\frac12} \left(\frac{p^2}{p-1} \right)^2 & \left\| \left( \sum_{j\geq 0} \sum_{n=(n_1,n_2)\in E_{j,1}} \left| S_{\delta_j} |\Delta_n M_l(n)|^{\frac12} f \right|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \nonumber \\
\leq C_3 \left(\frac{p^2}{p-1} \right)^2 &\left\| \left(\sum_{j\geq 0} |S_{\delta_j}f|^2 \right)^{\frac12} \right\|_{L^p(\mathbb{T}^2;S_p(\mathbb{Z}^2))}.
\end{align}
Therefore, by \eqref{eq:combin}, \eqref{eq:abelianSummationDimension2} and \eqref{eq:partB1},\eqref{eq:partB3}, \eqref{eq:partC}, \eqref{eq:partD}, we have
\begin{align}
&\left\| \left( \sum_{j = 0}^{\infty} \left| S_{E_{j,1 }} T_{\widetilde{M}} f\right|^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \leq 2 \left\| \sum_{j = 0}^{\infty} ( |P^1_j|^2+ |P^2_j|^2 + |P^3_j|^2 + |P^4_j|^2 ) \right\|^{\frac12}_{L^{\frac{p}{2}}(\mathbb{T}^2;S_p(\mathbb{Z}^2))} \nonumber \\
& \leq 2 \left( \left\| \sum_{j\geq 0} |P^1_j|^2 \right\|_{L^{\frac{p}{2}} (\mathbb{T}^2;S_p(\mathbb{Z}^2))} + \left\| \sum_{j\geq 0} |P^2_j|^2 \right\|_{L^{\frac{p}{2}} (\mathbb{T}^2;S_p(\mathbb{Z}^2))} + \left\| \sum_{j\geq 0} |P^3_j|^2 \right\|_{L^{\frac{p}{2}} (\mathbb{T}^2;S_p(\mathbb{Z}^2))} + \left\| \sum_{j\geq 0} |P^4_j|^2 \right\|_{L^{\frac{p}{2}} (\mathbb{T}^2;S_p(\mathbb{Z}^2))} \right)^{\frac12} \nonumber \\
&\leq C^{\prime} \left(\frac{p^2}{p-1}\right)^2 \left\| \left( \sum_{j = 0}^{\infty} \left| S_{\delta_{j }} f\right|^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}(\mathbb{T}^2;S_p(\mathbb{Z}^2))}
\end{align}
Thus, \eqref{eq:sufficient2} is proved. Hence, we finish the proof of theorem \ref{thm:dimension2}.
\medskip
\noindent{ \bf Open Question.}
Assume $m$ is a bounded map on ${\Bbb Z}\times{\Bbb Z}$ such that
\begin{eqnarray*}
\sum_s |m(k,j_s)-m(k,j_{s+1})|^2&<&C\\
\sum_s |m(k_s,j )-m(k_{s+1},j)|^2&<&C
\end{eqnarray*}
for all possible increasing sequences $j_s,k_s\in {\Bbb Z}$. Does $M_m$ extend to a (completely) bounded map on $S_p$ for all $1<p<\infty$?
\begin{remark} \cite{MeXu07} claimed a positive answer to the question above. But the argument contains a serious mistake. The third author tried different methods without success but found Theorem 1.1 in the process.
\end{remark}
\subsection{Higher dimensional case}
We need some additional notations to deal with $d$-dimensional case. Borrowing the notations from \cite{Hytoenen2016}, we denote by
\[
\mathbb{Z}^{\alpha}:=\{(n_i)_{i:\alpha_i=1}:n_i\in \mathbb{Z} \}
\]
the space $\mathbb{Z}^{|\alpha|}$ for $\alpha\in \{0,1\}^d$ with an indexing of the components derived from $\alpha$. For any $n\in\mathbb{Z}^d$ and $E= I_1\times \cdots \times I_d\subseteq \mathbb{Z}^d$, let
\[
n_{\alpha}\coloneqq (n_i)_{i:\alpha_i=1}\in \mathbb{Z}^{\alpha}, \quad E_{\alpha}\coloneqq \prod_{i:\alpha_i=1} I_i\subseteq \mathbb{Z}^{\alpha}
\]
be their natural projections onto $\mathbb{Z}^{\alpha}$. In particular, we will use the splittings $n=(n_{\alpha},n_{1-\alpha})\in \mathbb{Z}^{\alpha}\times \mathbb{Z}^{\mathbf{1}-\alpha}$ and $E=E_{\alpha}\times E_{\mathbf{1}-\alpha}$ where $\mathbf{1}=(1,\dots,1)$. Suppose $s,t\in \mathbb{Z}^d$ and we abbreviate the interval notation $[s,t)\cap \mathbb{Z}^d$ as $[s,t)$.
Similarly, denoted a partition $\mathscr{I}^d \coloneqq \{E_j:j\geq 0\}$ of $\mathbb{Z}^d$, where
\begin{equation}\label{eq:partitionZd}
E_j=\begin{cases}
\{(0,\cdots,0) \} & j=0,\\
\{(n_1,\cdots,n_d)\in\mathbb{Z}^d:2^{j-1}\leq |(n_1,\cdots,n_d)|_{\infty}<2^j \} & j\geq 1.
\end{cases}
\end{equation}
Each $E_j$ can be further decomposed into $2^d(2^d-1)$ subsets and each of the subsets can be obtained by translation of the cube $F_j=[2^{j-1},2^j)\times \cdots \times[2^{j-1},2^j)$. Denote $s^{(j)}=(2^{j-1},\cdots, 2^{j-1}), j\geq 1, $ then $F_j$ can be denoted as $[{t}^{(j)},{t}^{(j+1)})$.
Following the similar procedures in the two dimensional case and using the discrete fundamental theorem formula which takes the following form
\begin{align}\label{eq:fundamentalCalculus}
\chi_{[s,t)} (n) M(n) &= \chi_{[s,t)} \sum_{\alpha\in \{0,1\}^d} \sum_{k_\alpha\in [s,n)_{\alpha}} \Delta^{\alpha} M(s_{\mathbf{1}-\alpha},k_{\alpha}) \nonumber \\
&= \sum_{\alpha\in \{0,1\}^d} \sum_{k_{\alpha}\in [s,t)_{\alpha}} \chi_{[k,t)_{\alpha}\times [s,t)_{\mathbf{1}-\alpha}}(n) \Delta^{\alpha}M(s_{\mathbf{1}-\alpha},k_{\alpha})
\end{align}
for the function $M:\mathbb{Z}^d\to \mathbb{C}$, $s,t\in\mathbb{Z}^d$, we obtain the following theorem. The details are left to the interested readers.
\begin{theorem}\label{higerd}
Given $M=(m_{s,t})_{s,t\in\mathbb{Z}^d}\in \mathcal{B}(\ell^2(\mathbb{Z}^d))$. Suppose $M$ satisfies that
\begin{itemize}
\item[i)] $\sup_{s,t\in\mathbb{Z}^d} |m_{s,t}|<C$,
\item[ii)] For any $k\in\mathbb{N}$, $s\in \mathbb{Z}^d, \alpha\in \{0,1\}^d$ and $\alpha\neq \mathbf{0}$, there is
\begin{equation}
\sum_{\{t_{\alpha}:t=(t_{\alpha}, t^{(k)}_{\mathbf{1}-\alpha})\in E_k \} } |\Delta^{\alpha} m_{s,s+t}|<C,\quad \sum_{\{t_{\alpha}:t=(t_{\alpha}, t^{(k)}_{\mathbf{1}-\alpha})\in E_k \} } |\Delta^{\alpha} m_{s+t,s}|<C ,
\end{equation}
\end{itemize}
where $C$ is some positive constant. Then $M$ is a (completely) bounded Schur multiplier on $S_p(\ell^2(\mathbb{Z}^d))$ for $p\in (1,\infty)$ with the bound $C_d (\frac{p^2}{p-1})^{d+2}$ where $C_d$ is some constant only dependent on the dimension $d$.
\end{theorem}
\section{Schur-Marcinkiewicz Multiplier theorems in continuous cases}
We would like to extend the Theorem \ref{thm:main} and Theorem \ref{higerd} to continuous case via the method of approximation in \cite{Lafforgue2011}.
Let $S_p(\mathbb{R}^n)$ denote the Schatten $p$-class acting on $L^2(\mathbb{R}^n)$.
\begin{theorem}
For $p\in (1,\infty)$, consider the Schur multiplier $S_M$ on $S_p(\mathbb{R})$ with symbol $M (\cdot,\cdot)\in L^{\infty} (\mathbb{R}^{2})$ whose partial derivatives are continuous in every dyadic interval set $(-2^{j+1},-2^j)\bigcup (2^j, 2^{j+1})$ for $j~\in~\mathbb{Z}$. If $M$ satisfies
\begin{equation} \label{eq: con 1}
\sup_{j \in \mathbb{Z}} \sup_{y \in \mathbb{R}} \left\{ \int_{- 2^{j + 1}}^{- 2^{j}} \left| \partial_{1} M (y + t, y) \right| \ d t + \int_{2^{j}}^{2^{j + 1}} \left| \partial_{1} M (y + t, y) \right| \ d t \right\} \leq A
\end{equation}
and
\begin{equation} \label{eq: con 2}
\sup_{j \in \mathbb{Z}} \sup_{x \in \mathbb{R}} \left\{ \int_{- 2^{j + 1}}^{- 2^{j}} \left| \partial_{2} M (x, x + t) \right| \ d t + \int_{2^{j}}^{2^{j + 1}} \left| \partial_{2} M (x, x + t) \right| \ d t \right\} \leq A.
\end{equation}
Then, the Schur multiplier $S_{M} : S_p(\mathbb{R})\to S_p(\mathbb{R})$ is bounded with $\| S_{M} \|\leq C \max \left\{ p^{3}, \frac{1}{(p - 1)^{3}} \right\}$.
\end{theorem}
\begin{proof}
Let $k \in \mathbb{N}$. Consider the measure space $\mathbb{R}^{2}$ with the Lebesgue measure. Let $\mathcal{B}_{k}$ be the $\sigma$-algebra generated by the half-open parallelogram $D_{k, a, b}$ with vertices $\left(\frac{a}{2^{k}}, \frac{b}{2^{k}} \right)$, $\left(\frac{a + 1}{2^{k}}, \frac{b + 1}{2^{k}} \right)$, $\left( \frac{a}{2^{k}}, \frac{b + 1}{2^{k}} \right)$, $\left( \frac{a + 1}{2^{k}}, \frac{b + 2}{2^{k}} \right)$, where $a$, $b \in \mathbb{Z}$.
\begin{center}
\begin{tikzpicture}
\node (n) [draw, minimum width=2cm, minimum height=2cm, xslant= - 1.1] {};
\draw (n.north west) node[above] {$\left( \frac{a}{2^{k}}, \frac{b}{2^{k}} \right)$}
(n.north east) node[above] {$\left( \frac{a}{2^{k}}, \frac{b + 1}{2^{k}} \right)$}
(n.south west) node[below] {$\left( \frac{a + 1}{2^{k}}, \frac{b + 1}{2^{k}} \right)$}
(n.south east) node[below] {$\left( \frac{a + 1}{2^{k}}, \frac{b + 2}{2^{k}} \right)$};
\end{tikzpicture}
\end{center}
let $\mathscr{D}_{k}$ be the collection of all half-open parallelogram of the above form. Note that $(\mathcal{B}_{k})_{k = 1}^{\infty}$ forms a filtration for $\mathbb{R}^{2}$ equipped with the Lebesgue measure. Given a Lebesgue measurable function $M$ on $\mathbb{R}^{2}$, the conditional expectation $M_k$ of $M$ with respect to the sub-$\sigma$-algebra $\mathcal{B}_{k}$ is given by
\[
M_{k} (x) = \sum_{Q \in \mathscr{D}_{k}} \text{Avg}_{Q} (M) \chi_{Q} (x).
\]
Note that $\left\| M_{k} \right\|_{M[S^{p} (L^{2} (\mathcal{B}_{k}))]} = \left\| \widetilde{M_{k}} \right\|_{M [S^{p} (\ell^{2} (\mathbb{Z}))]}$, where $\widetilde{M_{k}} = [ m_{k} (s, t)]_{s, t \in \mathbb{Z}}$ and
\[
m_{k} (s, t) = \text{Avg}_{ D_{k, s, t}} (M).
\]
First, we show that $\widetilde{M_{k}}$ satisfies the assumptions for the discrete case. For each $j \in \mathbb{N}$ and $s \in \mathbb{Z}$,
\begin{align} \label{eq:continue}
&\sum_{\ell = 0}^{2^{j} - 2} \left| m_{k} \left( s, s + 2^{j} + \ell + 1 \right) - m_{k} \left( s, s + 2^{j} + \ell \right) \right| \nonumber\\
&= 2^{2 k}\sum_{\ell = 0}^{2^{j} - 2} \left| \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{\frac{2^{j} + \ell + 1}{2^{k}}}^{{\frac{2^{j} + \ell + 2}{2^{k}}}} M \left( y, y + x \right) \ d x \ d y- \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{\frac{2^{j} + \ell}{2^{k}}}^{{\frac{2^{j} + \ell + 1}{2^{k}}}} M \left( y, y
+ x \right) \ d x \ d y \right| \nonumber\\
&= 2^{2 k}\sum_{\ell = 0}^{2^{j} - 2} \left| \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{\frac{2^{j} + \ell}{2^{k}}}^{{\frac{2^{j} + \ell + 1}{2^{k}}}} M ( y, y + x + \frac{1}{2^{k}} ) \ d x \ d y - \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{\frac{2^{j} + \ell}{2^{k}}}^{{\frac{2^{j} + \ell + 1}{2^{k}}}} M \left( y, y + x \right) \ d x \ d y \right| \nonumber\\
&= 2^{2 k}\sum_{\ell = 0}^{2^{j} - 2} \left| \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{\frac{2^{j} + \ell}{2^{k}}}^{{\frac{2^{j} + \ell + 1}{2^{k}}}} \int_{y + x}^{y + x + \frac{1}{2^{k}}} \partial_{2} M \left( y, t \right) \ d t \ d x \ d y \right| \nonumber\\
&\leq 2^{2 k}\sum_{\ell = 0}^{2^{j} - 2} \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{\frac{2^{j} + \ell}{2^{k}}}^{{\frac{2^{j} + \ell + 1}{2^{k}}}} \int_{y + x}^{y + x + \frac{1}{2^{k}}} \left| \partial_{2} M \left( y, t \right) \right| d t \ d x \ d y \nonumber\\
&= 2^{2 k} \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{\frac{2^{j}}{2^{k}}}^{{\frac{2^{j + 1} - 1}{2^{k}}}} \int_{y + x}^{y + x + \frac{1}{2^{k}}} \left| \partial_{2} M \left( y, t \right) \right| d t \ d x \ d y \nonumber\\
&= 2^{2 k} \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{y + \frac{2^{j}}{2^{k}}}^{{y + \frac{2^{j + 1}}{2^{k}}}} \int_{\frac{2^{j}}{2^{k}}}^{{\frac{2^{j + 1} - 1}{2^{k}}}} \chi_{\left( t - \frac{1}{2^{k}}, t \right)} (x) \ d x \left| \left[ \partial_{2} (M) \right] \left( y, t \right) \right| d t \ d y \nonumber\\
&\leq 2^{k} \int_{\frac{s}{2^{k}}}^{\frac{s + 1}{2^{k}}} \int_{y + \frac{2^{j}}{2^{k}}}^{{y + \frac{2^{j + 1}}{2^{k}}}} \left| \partial_{2} M \left( y, t \right) \right| d t \ d y
\leq \sup_{y \in \mathbb{R}} \int_{\frac{2^{j}}{2^{k}}}^{{\frac{2^{j + 1}}{2^{k}}}} \left| \left[ \partial_{2} (M) \right] \left( y, y + t \right) \right| d t \leq A.
\end{align}
The last inequality in \eqref{eq:continue} is from the assumption \eqref{eq: con 2}.Using the same method, we also obtain that for each $j \in \mathbb{N}$ and $s \in \mathbb{Z}$,
\[
\sum_{\ell = 0}^{2^{j} - 2} \left| m_{k} \left( s, s - 2^{j} - \ell - 1 \right) - m_{k} \left( s, s - 2^{j} - \ell \right) \right| \leq A.
\]
Similarly, by \eqref{eq: con 1}, we also have that for each $j \in \mathbb{N}$ and $s \in \mathbb{Z}$,
\[
\sum_{\ell = 0}^{2^{j} - 2} \left| m_{k} \left( s + 2^{j} + \ell + 1, s \right) - m_{k} \left( s + 2^{j} + \ell, s \right) \right| \leq A
\]
and
\[
\sum_{\ell = 0}^{2^{j} - 2} \left| m_{k} \left( s - 2^{j} - \ell - 1, s \right) - m_{k} \left( s - 2^{j} - \ell, s \right) \right| \leq A.
\]
By the Theorem \ref{thm:main}, $\left\| M_{k} \right\|_{M[S^{p} (L^{2} (\mathcal{B}_{k}))]} = \left\| \widetilde{M_{k}} \right\|_{M [S^{p} (\ell^{2} (\mathbb{Z}))]} \leq C \max \left\{ p^{3}, \frac{1}{(p - 1)^{3}} \right\}$
It then follows from Lemma 1.11 in \cite{Lafforgue2011} that
\[
\left\| M \right\|_{M[S^{p} (L^{2} (\mathbb{R}^{2}))]}= \lim_{k \to \infty} \left\| M_{k} \right\|_{M[S^{p} (L^{2} (\mathcal{B}_{k}))]} \leq C \max \left\{ p^{3}, \frac{1}{(p - 1)^{3}} \right\}.
\]\end{proof}
Similarly, we can extend Theorem \ref{higerd} to the continuous case using approximation property in \cite{Lafforgue2011} and obtain the following theorem.
\begin{theorem}
Denote $F_{j} := \{ t \in \mathbb{R}^{d} : 2^{j} < | t |_{\infty} \leq 2^{j + 1} \}$ for $j\in\mathbb{Z}$. For $p\in (1,\infty)$, consider the Schur multiplier $M$ on $S_p (\mathbb{R}^{d}) $ with symbol $M \in L^{\infty} (\mathbb{R}^{2d})$ whose partial derivatives are continuous up to the boundary of $F_j$ for all $j\in\mathbb{Z}$. Assume that $M$ satisfies
\begin{align} \label{eq:rightCondition}
&\sup_{j \in \mathbb{N}} \sup_{s \in \mathbb{R}^d} \int_{(t_{\alpha}, t_{1-\alpha}^{(k)}) \in F_{j}} \left| \partial_{\alpha} M ( s,s + t ) \right| dt_\alpha \leq A\\
&\sup_{j \in \mathbb{N}} \sup_{s \in \mathbb{R}} \int_{(t_{\alpha}, t_{1-\alpha}^{(k)}) \in F_{j}} \left| \partial_{\alpha} M ( s+t, s ) \right| dt_{\alpha}\leq A
\end{align}
Then, the Schur multiplier $S_{M} : S_p(\mathbb{R}^d)\to S_p(\mathbb{R}^d)$ is bounded with $\| S_{M} \|\leq C \max \left\{ p^{d+2}, \frac{1}{(p - 1)^{d+2}} \right\}$.
\end{theorem}
\begin{comment}
\begin{theorem}
continuous case dimension 2, to be added...
\end{theorem}
\begin{theorem}
higher dimension n, to be added...
\end{theorem}
\end{comment}
\bibliographystyle{abbrv}
| {'timestamp': '2022-09-28T02:08:31', 'yymm': '2209', 'arxiv_id': '2209.13108', 'language': 'en', 'url': 'https://arxiv.org/abs/2209.13108'} |
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{S}{egmentation} and/or classification of medical images plays a vital role in disease diagnosis, therapy planning and follow-up tracking of therapy efficiency. Therefore, segmentation/classification by using deep learning (DL) models, i.e. deep neural networks (NNs), is steadily in focus in biomedical image analysis \cite{shen2017deep,litjens2016deep,de2018clinically,mishra2022data,gu2021net,gunesli2020attentionboost,li2021pathal,yang2020guided,zhu2021hard,guo2021learn}, to cite a few. Thus, a constant need for performance improvement results in many specialized and extremely complex DL models, whereat some of them are cited above. Their decision-making process is often hard to explain and sometimes even harder to interpret in terms of features understandable to medical experts \cite{rudin2019stop,tjoa2020survey}. When high-stakes judgments must be made, it is critical to comprehend the reasons behind the decision-making process.
As a result, rather than explaining black box models, it is preferable to create interpretable models \cite{rudin2019stop}. Furthermore, there is currently a scarcity of expert-annotated datasets in many medical imaging modalities. For sophisticated models like convolutional NNs (CNNs), this causes a generalization challenge \cite{litjens2016deep,veta2014breast,qiu2018deep}. This issue is also present in histopathological image segmentation of intraoperatively collected frozen sections \cite{komura2018machine}. One reason is that sample collection during surgery is distracting. Also, another reason is highly time-consuming effort for pathologists to label cancerous pixels \cite{sitnik2021cocahis}.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{figures/LEFM-Net.png}
\caption{Proposed framework comprising the slightly modified existing deep networks such as UNet, UNet++, DeepLabv3+, and MA-net, embedded in a space induced by the learnable explicit feature map (LEFM) layer. Increased number of learnable weights in the LEFM layer is represented as $\mathbf{a}_{m}$. Additional deep network's weights are introduced through modification to accept expanded features. $\phi_{m}(\mathbf{X})$ is defined as Hadamard product ($\otimes$) of: $\psi_{m}(\mathbf{x_t})$ - the spatial location dependent map of monomials of the input data space features, and $\mathbf{a}_{m}$ - the spatial location invariant vector of coefficients. The first part contributes prior knowledge in terms of monomial-based predefined algebraic structure. The second part learns the importance of explicit features.}
\label{figure:adafel}
\end{figure*}
The issues outlined above motivated us to propose an inclusive framework for DL-based segmentation/classification of medical images in general, and color histopathological images of frozen sections in particular. Instead of designing specialized DL models, we aim to improve segmentation/classification performance of the existing NNs, such as UNet \cite{ronneberger2015u}, UNet++ \cite{zhou2018unet++,zhou2020unet++}, DeepLabv3+ \cite{chen2018encoder} and MA-net \cite{fan2020manet}. To achieve that goal, we embed the existing network in a space induced by the learnable explicit feature map (LEFM) layer (Fig \ref{figure:adafel}). The LEFM layer is an analog to learnable kernels \cite{wang2021bridging,pan2011learning,kulis2009low,li2015adaptive,bach2004multiple,rebai2016deep,wilson2016deep,bohn2019representer}, and its use is motivated by the need to create low-dimensional subspace in data-adaptable Hilbert space where existing network will yield improved performance. To reduce risk of overfitting, the aim was to only modestly increase the complexity relative to the original network. That is achieved by using the LEFM of low order, whereat the LEFM order is the only new hyperparameter introduced by the proposed framework. Its suggested value is either 2 or 3, which simplifies the cross-validation procedure.
The explanation why the LEFM layer improves performance of deep network is in representation of the LEFM. The LEFM is represented as Hadamard (entry-wise) product of the spatial location invariant vector of coefficients and spatial location dependent map composed of the monomials of features from the input data space. Thus, the predefined (known) algebraic structure of the EFM contains \textit{a priori} knowledge that contributes to improved performance of the proposed framework. For example, \textit{a priori} knowledge in a form of translation symmetry in images is incorporated in CNNs \cite{lecun1998gradient}, and enables them to achieve translation equivariance by re-using convolutional filters at all spatial locations. Such design highly reduces the number of learnable parameters in CNNs. Analogously, the learned vector of coefficients, in proposed LEFM, can be re-used at all spatial locations with location dependent map of monomials. Learnable EFM alleviates the analogous problem in kernel-based nonlinear methods: the choice of kernel. After learning, the coefficients emphasize the importance of the features in LEFM. Since all features are known polynomial functions of the original features from the input data space, that contributes to explainability and interpretability of the decision-making process of the proposed framework. The main contributions of this work are listed as follows:
\begin{itemize}
\item Framework for DL-based segmentation/classification of low-dimensional medical images comprising slightly modified existing deep network embedded into a subspace induced by learnable EFM layer.
\item Representation of learnable EFM in terms of spatial location dependent map of monomials of features from the input data space and spatial location invariant vector of coefficients. The first part brings prior knowledge in terms of predefined algebraic structure with monomial-based explicit features. The second part learns the importance of explicit features. That contributes to medically explainable and interpretabe decision-making. The overall representation modestly increases the number of learnable parameters compared to the original network.
\item Extensive validation of the proposed framework where we used existing deep architectures: DeepLabv3+, UNet, UNet++, and MA-net on two problems. The first one was image segmentation of adenocarcinoma of a colon in a liver from hematoxylin and eosin (H\&E) stained frozen sections. The second problem was the segmentation of nuclei from H\&E stained frozen section images of ten human organs. Proposed framework was compared to the original networks. In the first problem, statistically significant improvements in terms of micro balanced accuracy, micro $F_{1}$ score and micro precision are obtained. In the second, the proposed framework yielded only better results.
\end{itemize}
\newpage
\section{Related work}
\subsection{Brief overview of some specialized deep neural networks}
Herein, we briefly comment on recently proposed deep networks with task- or imaging modality specific architectures with an emphasis on histopathological image analysis. New deep CNN was proposed in \cite{mishra2022data} for medical image segmentation. It exploits specific attributes in the input datasets. That is achieved with auxiliary supervision implemented in terms of object perceptive field and layer-wise effective receptive fields. Comprehensive attention-based CNN (CA-Net) is proposed in \cite{gu2021net} for more accurate and more explainable medical image segmentation. That is achieved by using multiple attention modules in CNN architecture aware of the most important spatial positions, channels and scales\cite{gu2021net}. Although the explainability of decision is improved, the interpretability of the features is still problematic. AttentionBoost network is proposed in \cite{gunesli2020attentionboost} for gland instance segmentation in histopathological images. It is based on a multi-attention learning model that, based on adaptive boosting, adjusts loss of fully convolutional networks to adaptively learn what to attend at each stage. In \cite{li2015adaptive}, a new methodology, named PathAL, is presented aiming to improve classification performance in histopathology image analysis with a reduced number of expert annotations. Guided soft attention network for classification of histopathological breast cancer images is proposed in \cite{yang2020guided}. In particular, region-level supervision is used to guide attention of CNN. The attention mechanism activates neurons in diagnostically relevant regions and suppresses activation in diagnostically irrelevant regions. Hard sample aware noise robust learning deep network for classification of histopathology images is proposed in \cite{zhu2021hard}. The method is aimed at distinguishing informative hard samples from the harmful noisy ones. That lowers the noise rate and yields an almost clean dataset for training network in a self-supervised manner. Novel ThresholdNet \cite{guo2021learn} is designed for automatic segmentation of polyp in endoscopy images such that segmentation threshold is learnable. Proposed LEFM-Net framework is directly applicable to methods proposed in \cite{mishra2022data,gu2021net,gunesli2020attentionboost,li2021pathal,yang2020guided,zhou2020unet++,guo2021learn}. In this paper, however, we demonstrated performance improvement of the LEFM-Net framework using well know deep networks: DeepLabv3+, UNet, UNet++ and MA-net.
\subsection{Brief overview of kernel-based learning}
Let us assume there is a function $f:=\mathbb{R}^{d} \rightarrow \mathbb{R}$ that assigns the class label $y \in S_{C}$ to the sample $\mathbf{x} \in \mathbb{R}^{d}$. Here, $d$ stands for dimension of the ambient input space, $C$ stands for the number of classes and $S_{C}$ is a set containing all possible labels. As an example, for binary classification problem we have $C=2$ and $S_{2}=\{0,1\}$ or $S_{2}=\{-1,1\}$. The classifier learning problem is to estimate $f$ from the given set of training data $\left\{\left(\mathbf{x}_{t}, y_{t}\right)\right\}_{t=1}^{T}$. The representer theorem \cite{scholkopf2018learning,kimeldorf1971some}, states that each function $f$ in Hilbert space $\mathcal{H}$, a.k.a. feature space, that minimizes an arbitrary loss function, admits representation:
\begin{equation}
\label{eq:repre}
f(\mathbf{x})=\sum_{t=1}^{T} \alpha_{t} \kappa\left(\mathbf{x}_{t}, \mathbf{x}\right)
\end{equation}
\noindent where $\{ \alpha_t \}_{t=1}^{T}$, are estimated from the training data. That gave rise to the development of many machine learning algorithms, such as support vector machines and kernel principal component analysis, to name a few. These methods work well when the underlying problem fits $\mathcal{H}$ properly, i.e. when the given empirical values stem from $f \in \mathcal{H}$. In other words, the kernel function $\kappa(\circ, \circ)$ has to be chosen properly. If the appropriate $\mathcal{H}$ is not known, one can resort to data adaptive kernel methods \cite{pan2011learning,kulis2009low,li2015adaptive}, multiple kernel learning methods \cite{bach2004multiple}, multi-layer (deep) multiple kernel learning method \cite{rebai2016deep}, and deep kernel method \cite{wilson2016deep}. In particular, the last two approaches combine flexibility of deep networks. One where feature detection is done automatically with the approximation power of kernel methods. The second one is where the feature map is determined implicitly by the kernel function. Moreover, in \cite{bohn2019representer} the representer theorem is proven for a multilayer network of $L$ concatenated kernel functions. The power of kernel methods is their capability to implicitly learn potentially infinite dimensional nonlinear features in the reproducing kernel Hilbert space $\mathcal{H}$. The computation is made tractable by leveraging the kernel trick \cite{bohn2019representer}. However, implicitly learned features are hard to interpret.
\section{Learnable feature maps and deep networks}
As pointed out in \cite{wilson2016deep}, one of the main reasons to exploit deep kernel learning was growing frustration among researchers with too many design options associated with deep networks. Furthermore, the decision-making process of deep networks is also difficult to interpret. However, deep networks composed of multiple layers of nonlinear functions can approximate a rich set of naturally occurring input-output dependencies \cite{wang2021bridging}. Thus, they can achieve high classification performance. Therefore, we aim to propose a framework that further improves the performance of deep networks, contributes to the interpretability of the results and brings a modest increase in the number of learnable parameters compared to original networks. Towards that goal, we propose to embed existing deep networks in a subspace induced by the LEFM of small order (typically 2 or 3). Our approach relies on the well know kernel trick \cite{scholkopf2018learning}: for any positive definite kernel function $\kappa(\mathbf{x}, \mathbf{y})$ there exists a function $\phi(\mathbf{x})$ that maps data $\mathbf{x}$ to a Hilbert space $\mathcal{H}$, such that $\kappa(\mathbf{x}, \mathbf{y})=\langle\phi(\mathbf{x}), \phi(\mathbf{y})\rangle_{\mathcal{H}} \ \cdot \ \phi(\mathbf{x})$ is known as EFM. Thus, we can recast the representer theorem (\ref{eq:repre}) in terms of kernel associated EFM:
\begin{equation}
f(\mathbf{x})=\langle\mathbf{w}, \phi(\mathbf{x})\rangle_{\mathcal{H}}
\end{equation}
\noindent where $\mathbf{w}=\sum_{t=1}^{T} \alpha_t \phi\left(\mathbf{x}_{t}\right)$. Thus, the nonlinear approximation problem (\ref{eq:repre}) in the input data space becomes the linear one in the feature space $\mathcal{H}$. Hence, kernel-based nonlinear algorithms can be seen as linear algorithms operating in appropriate feature space \cite{vedaldi2012efficient}. That is formalized by the Cover’s theorem \cite{cover1965geometrical}. It states that the number of separating hyperplanes is proportional to the dimension of the feature space. Thus, it is important for the dimension of expanded feature space $D$ to satisfy $D>>d$. That is fulfilled easily since EFMs associated with many popular kernels are mostly high dimensional or even infinite dimensional (e.g. EFM associated with the Gaussian kernel). That is also a reason feature maps are not used explicitly in nonlinear algorithms \cite{vedaldi2012efficient}. Nevertheless, we can partially circumvent this obstacle by using the approximate EFM of order $m$, i.e. $\phi_{m}(\mathbf{x})\in\mathbb{R}^D$. In that regard, we emphasize that EFMs can be written in a common algebraic form:
\begin{equation}
\phi_{m}(\mathbf{x})=\left[\left\{a_{q_{1}, \ldots ,q_{d}} x_{1}^{q_{1}} \ldots x_{d}^{q_{d}}\right\}_{q_{1}, \ldots, q_{d}=0}^{m}\right]^{\mathrm{T}}
\end{equation}
\begin{equation*}
\text{such that} \sum_{i=1}^{d} q_{i} \leq m.
\end{equation*}
When $m \rightarrow \infty$ we have: $\phi_{m}(\mathbf{x}) \rightarrow \phi(\mathbf{x})$. Bearing in mind that the dimension of induced feature space $D$ depends on $d$ and $m$ through:
\begin{equation}
\label{eq:binom}
D = \binom{d+m}{m}
\end{equation}
\noindent it becomes clear that we need to keep $m$ low if we do not want to introduce too many learnable parameters in the proposed framework. Thus, in the experiments conducted in this paper, we have $m \in\{2,3\}$. Hence, the formulation (2) with approximate EFM $\phi_{m}(\mathbf{x})$ will only approximate the original nonlinear approximation problem (1). That is why we need to embed the deep network in a subspace $\mathcal{H}_{D} \subset \mathcal{H}$ induced by $\phi_{m}(\mathbf{x})$. By doing so, we expect to further boost segmentation/classification performance of the network. Nevertheless, the problem related to the proper fit of the underlying problem to $\mathcal{H}$ still exists. Thus, we propose the following computationally efficient representation of $\phi_{m}(\mathbf{x})$:
\begin{equation}
\label{eq:phim}
\phi_{m}(\mathbf{x}):= \psi_{m}(\mathbf{x}) \otimes \mathbf{a}_{m}
\end{equation}
\noindent where $\otimes$ denotes Hadamard (entry wise) product, and:
\begin{equation}
\label{eq:psim}
\psi_{m}(\mathbf{x}):=\left[\left\{x_{1}^{q_{1}} \ldots x_{d}^{q_{d}}\right\}_{q_{1}, \ldots, q_{d}=0}^{m}\right]^{\mathrm{T}} \text{s.t.} \sum_{i=1}^{d} q_{i} \leq m ,
\end{equation}
\begin{equation}
\label{eq:am}
\mathbf{a}_{m}:=\left[\left\{a_{q_{1}, \ldots , q_{d}}\right\}_{q_{1}, \ldots, q_{d}=0}^{m}\right]^{\mathrm{T}} \text {s.t.} \sum_{i=1}^{d} q_{i} \leq m .
\end{equation}
Hence, when existing deep network is trained with $\phi_{m}(\mathbf{x})$, the coefficients $\mathbf{a}_{m}$ are learned together with the rest of network parameters. Since $\mathbf{a}_{m}$ is spatial location invariant, learning is computationally efficient. As opposed to that, $\psi_{m}(\mathbf{x})$ is a spatial location dependent function of the input features and it is not learnable. Thus, when $d$ and $m$ are small, the number $D$ of learnable coefficients in $\mathbf{a}_{m}$ is modest when compared to the millions of parameters in existing deep networks. Number of trainable parameteres is additionally increased in amounts specific to the deep architecture and its backbone. Thus, all newly introduced parameters are directly related to the increase from $d$ to $D$ dimensional input. However, vast majority of the original network remains the same.
\noindent As an example, in the problem considered in the paper, we apply the proposed framework to segmentation of color (RGB) histopathological images of H\&E stained frozen sections. Thus, we have $d=3$. As we worked with approximate EFM of order $m \in\{2,3\}$, that respectively yields $D \in\{10,20\}$. The rationale why the deep network, embedded in a subspace $\mathcal{H}_{D} \subset \mathcal{H}$ induced by learnable EFM, outperforms the original network operating in the input data space is due to the explicit character of features up to the selected order $m$. Through learning, a deep network can determine their importance. Thus, in the proposed framework, the network will use more explicit features than it is available in the input data space. Since features in $\phi_{m}(\mathbf{x})$ are polynomials comprising the originals features from the input data space, it is possible to interpret them.
\section{Experiments and Results}
\label{sec:expRes}
\subsection{Data adaptive feature expansion layer}
\label{subsec:layer}
For implementing an LEFM layer, fundamental functionality is described in (\ref{eq:phim}) - (\ref{eq:am}). As mentioned, the coefficients are data adaptive (i.e. learnable) and, for LEFM layer construction, we are only interested in monomials. Thus, we need to find monomials (\ref{eq:psim}) using all combinations of $\boldsymbol{q}$ such that $\sum_{i=1}^d{q_i}\leq m,\ \forall q_i \geq 0$. Thus, the number of terms in this setting becomes (\ref{eq:binom}). Optionally, if the LEFM layer is followed by batch normalization layer \cite{ioffe2015batch}, two parameters per each term (mean and standard deviation) are introduced.
The core functionality of LEFM layer is developed using two matrices, term- and power mask. Both matrices are $D\times d$ dimensional. The term mask matrix $\mathbf{TM}$ is associated with selecting features $\{x_i \}^d_{i=1}$ and the power mask matrix $\mathbf{PM}$ with exponantiating selected features by $\{q_i \}^d_{i=1}$. A resulting monomial matrix $\mathbf{MM}$ is made by element-wise exponentiation $\mathbf{MM}=\mathbf{TM}^{\mathbf{PM}}$. Thus, when the resulting matrix is squeezed by elements-product across the $2^{nd}$ dimension, we get the monomials of the input features $\psi_m(\mathbf{x})$. Using the randomly initialized vector of learnable coefficients $\mathbf{a}_{m}$ we apply the Hadamard product as in (\ref{eq:phim}).
Thus, with $d$ features as an input, the LEFM layer gives $D$ output features with learnable coefficients $\{ a_i \}^D_{i=1}$. As illustrated in Fig. \ref{figure:adafel}, the expanded output is passed to the arbitrary deep NN to jointly train its weights and LEFM layer’s coefficients.
\begin{figure*}[ht!]
\centering
\includegraphics[width=\textwidth]{figures/fleiss_kappa_vis.png}
\caption{Fleiss kappa statistics for evaluating inter-annotators agreement. Left and right plots represent statistics per image in CryoNuSeg and CoCaHis datasets, respectively. The Fleiss kappa for the entire CryoNuSeg is 0.8012 and for CoCaHis 0.7405. Former indicates an almost perfect inter-annotator agreement, whereas the latter indicates a substantial inter-annotator agreement.}
\label{figure:fleisskappa}
\end{figure*}
\subsection{Deep neural networks}
\label{subsec:dnn}
Within the scope of this paper, the LEFM layer is combined with several types of slightly modified neural networks. In DeepLabv3+ \cite{chen2018encoder}, a deep network has the spatial pyramid pooling module combined with the encoder-decoder structure. Feature extraction at arbitrary resolution was enabled by atrous convolution layers contained in the encoder module \cite{chen2018encoder}. On the other hand, a simple decoder recovers detailed boundaries of an object. A popular UNet model \cite{ronneberger2015u} employs a symmetric U-shaped encoder-decoder for precise feature localization. Skip connections help in preserving context at the same level of the model’s depth \cite{ronneberger2015u}. Natural extension of the UNet is a nested model. Introduced in \cite{zhou2018unet++}, UNet++ model proposes additional nodes and skip connections to the original UNet. That facilitates predicting the output from different levels of depth. Training the entire network by combining all outputs, the UNet++ model outperforms UNet with a cost of a larger number of parameters. Finally, MA-net \cite{fan2020manet} incorporates a self-attention mechanism. Based on the attention mechanism, it collects rich contextual dependencies while adaptively integrating local features. Model implements Position-wise Attention Block and Multi-scale Fusion Attention Block. Prior is used to model the feature interdependencies spatially, and the former to capture channel dependencies using semantic feature fusion \cite{fan2020manet}.
All mentioned models can implement different backbones for feature extraction to their core. When comparing a model with and without the LEFM layer, it is important to keep the backbone fixed. MobileNetV2 \cite{sandler2018mobilenetv2} was implemented in DeepLabv3+ model, whereas other models have DenseNet201 \cite{huang2017densely} backbone. All backbone’s weights are pretrained on Imagenet classification problem \cite{deng2009imagenet} and fine-tuned with the rest of the model. Thus, the total number of trainable parameters in DeepLabv3+ model is \textasciitilde4.4M, in UNet \textasciitilde28.5M, in UNet++ \textasciitilde48.5M, and in MA-net \textasciitilde133.7M. Furthermore, LEFM-Nets, with mentioned architectures and $m=3$, contribute to an increase of the number of learnable parameters in the amount of: 0.11\%, 0.18\%, 0.11\%, and 0.04\% relative to the original networks. In the application considered herein, it is modest when compared to the reported total number of trainable parameters of original models.
\subsection{Datasets}
This paper is focused on segmentation of color images of H\&E stained frozen sections. Thereby, we used the only two publicly available datasets: CoCaHis \cite{sitnik2021cocahis,cocahis2021dataset} and CryoNuSeg \cite{mahbod2021cryonuseg}. Datasets are divided into the train and test sets. Furthermore, during the training process, 20\% of the train set is left for validation and early stopping purposes.
\subsubsection{CoCaHis}
Dataset comprises 82 images with dimensions $1037 \times 1388$. Training set has 58 images belonging to 13 patients. The rest of 24 images from 6 patients are contained in the test set. Sets are disjointed for both images and patients. Seven annotators generated the pixel-wise labels for each image. One and zero indicate cancerous and non-cancerous pixels, respectively. Resulting average Fleiss kappa statistic \cite{fleiss1971measuring}, presented in Section \ref{sec:results} and Fig. \ref{figure:fleisskappa}, supports usage of majority voting due to the substantial inter-annotator agreement \cite{landis1977measurement}.
\renewcommand{\thefootnote}{\arabic{footnote}}
\subsubsection{CryoNuSeg} Collected from The Cancer Genome Atlas\footnote[1]{https://www.cancer.gov/tcga}, the dataset contains 30 whole-slide frozen section images. Afterwards, they are cropped to the most representative window. Hence, the dataset comprises 30 images of the size $515\times 512$ pixels that correspond to 10 organs. The problem formulated by this dataset is segmentation of nuclei in different images. Therefore, the ground truth generation is similar to CoCaHis. The Fleiss kappa statistic, presented in Section \ref{sec:results} and Fig. \ref{figure:fleisskappa}, suggests an almost perfect agreement between three annotations, and that is why the majority vote is used. To ensure the representation of every organ in both train and test set, one image per each organ is left out for the test set (10 images), whereas the remaining images are included in the training set (20 images).
\subsection{Statistics and Metrics}
\subsubsection{Statistical significance analysis}
To determine if inclusion of the LEFM layer yields statistically better performance when compared to the original models, we used a one-way analysis of variance (ANOVA) test. All models were trained 10 times with different initial weights and different subsets for validation. Null hypothesis was that two groups of results, obtained by the LEFM-Net and the original deep network, have equal means. For all tests, a level of significance $\alpha=0.05$ was used.
\subsubsection{Evaluation metrics}
\label{sec:metrics}
All metrics used for evaluation were calculated using standard $TP$, $TN$, $FN$, and $FP$ notation. $TP$ stands for a number of correctly diagnosed cancerous pixels. $TN$ indicates a number of non-cancerous pixels correctly diagnosed as non-cancerous. $FN$ stands for a number of cancerous pixels incorrectly diagnosed as non-cancerous. Finally, $FP$ indicates the number of noncancerous pixels incorrectly diagnosed as cancerous. Thus, the following micro measures were used for models’ performance validation:
\begin{itemize}
\item a harmonic mean of precision and sensitivity (recall) presented as a $F_1$ score:
\begin{equation}
\label{eq:f1}
F_1 = \frac{2\cdot TP}{2\cdot TP+FP+FN},
\end{equation}
\item balanced accuracy (BACC) as an arithmetic mean of sensitivity (SE) and specificity (SP):
\begin{equation*}
SE = \frac{TP}{TP+FN},\ SP = \frac{TN}{{TN}+{FP}},
\end{equation*}
\begin{equation}
\label{eq:bacc}
BACC = \frac{SE+SP}{2},
\end{equation}
\item and Precision score, which measures the classifier's capacity not to categorize a sample as positive if it is negative:
\begin{equation}
\label{eq:prec}
PREC = \frac{TP}{TP+FP}.
\end{equation}
\end{itemize}
\noindent For all the metrics, 0 indicates the worst performance and 1 indicates the best performance.
\begin{figure}[H]
\centering
\includegraphics[width=0.35\textwidth]{figures/target2.png}
\caption{Target image for stain normalization chosen by pathologist. Selection is based on the quality-of-staining criterion.}
\label{figure:target_image}
\end{figure}
\subsection{Data preprocessing}
\label{sec:preprocessing}
As an important part of the computational pathology pipeline, stain normalization was performed to eliminate the variability of staining and experimental variations. The structure preserved color normalization method \cite{vahadane2016structure} was used on both CoCaHis and CryoNuSeg datasets. A target image, shown in Fig. \ref{figure:target_image}, was carefully chosen by an experienced pathologist, ensuring a well stained specimen for the mentioned algorithm. Fig. \ref{figure:panel} shows original and stain-normalized images from the mentioned datasets. Stain-transfer from the target image helps models during the training to cope with the color variation problem.
Prior to the training, every image was scaled to the range [0,1]. Additionally, random shifting, scaling, rotating and flipping (horizontal, vertical) were applied as part of the data augmentation procedure. Chosen parameters are presented in Section \ref{sec:implDetail}.
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/img_80_raw.png}
\caption{}
\label{figure:subplot1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/img_80_sn.png}
\caption{}
\label{figure:subplot2}
\end{subfigure}
\bigskip
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Human_LymphNodes_03.png}
\caption{}
\label{figure:subplot3}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/img_9_sn.png}
\caption{}
\label{figure:subplot4}
\end{subfigure}
\caption{Visualization of raw and stain normalized images. Subfigures (a) and (c) show original images from CoCaHis and CryoNuSeg, respectively. Their stain-normalized equivalents are shown in subfigures (b) and (d).}
\label{figure:panel}
\end{figure}
\begin{table*}[ht]
\caption{Performance scores of four different models on CryoNuSeg and CoCaHis datasets. Each model had 10 runs with different initial weights. Backbones were initialized on the pretrained ImageNet classification problem. Variable $m\in \{2,3\}$ indicates level of expansion. The best result for each metric and each dataset is in bold font.}
\label{tab:cryonusegPerf}
\resizebox{\textwidth}{!}{%
\begin{tabular}{c|l|cc|cc|cc|}
\cline{2-8}
\multicolumn{1}{l|}{{\color[HTML]{3D3D3D} \textbf{}}} &
\multicolumn{1}{c|}{{\color[HTML]{3D3D3D} }} &
\multicolumn{2}{c|}{{\color[HTML]{3D3D3D} \textbf{Original network}}} &
\multicolumn{2}{c|}{{\color[HTML]{3D3D3D} \textbf{LEFM-Net m=2}}} &
\multicolumn{2}{c|}{{\color[HTML]{3D3D3D} \textbf{LEFM-Net m=3}}} \\ \cline{3-8}
\textbf{} &
\multicolumn{1}{c|}{\multirow{-2}{*}{{\color[HTML]{3D3D3D} \textbf{Metric}}}} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{CryoNuSeg}} &
\cellcolor[HTML]{C0C0C0}\textbf{CoCaHis} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{CryoNuSeg}} &
\cellcolor[HTML]{C0C0C0}\textbf{CoCaHis} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{CryoNuSeg}} &
\cellcolor[HTML]{C0C0C0}\textbf{CoCaHis} \\ \hline
\multicolumn{1}{|c|}{{\color[HTML]{3D3D3D} }} &
{\color[HTML]{3D3D3D} BACC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 87.24 $\pm$ 0.54}} &
\cellcolor[HTML]{C0C0C0}86.06 $\pm$ 1.60 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 86.99 $\pm$ 0.64}} &
\cellcolor[HTML]{C0C0C0}87.24 $\pm$ 1.55 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 87.52 $\pm$ 0.74}} &
\cellcolor[HTML]{C0C0C0}88.19 $\pm$ 0.70 \\ \cline{2-8}
\multicolumn{1}{|c|}{{\color[HTML]{3D3D3D} }} &
{\color[HTML]{3D3D3D} F\textsubscript{1}} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 82.34 $\pm$ 0.60}} &
\cellcolor[HTML]{C0C0C0}80.75 $\pm$ 1.44 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 82.11 $\pm$ 0.67}} &
\cellcolor[HTML]{C0C0C0}81.89 $\pm$ 1.69 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 82.60 $\pm$ 0.69}} &
\cellcolor[HTML]{C0C0C0}82.85 $\pm$ 1.38 \\ \cline{2-8}
\multicolumn{1}{|c|}{\multirow{-3}{*}{{\color[HTML]{3D3D3D} \textbf{DeepLabv3+}}}} &
{\color[HTML]{3D3D3D} PREC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 85.49 $\pm$ 0.39}} &
\cellcolor[HTML]{C0C0C0}84.48 $\pm$ 4.70 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 85.74 $\pm$ 0.60}} &
\cellcolor[HTML]{C0C0C0}83.29 $\pm$ 2.96 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}{\color[HTML]{3D3D3D} 85.29 $\pm$ 0.79}} &
\cellcolor[HTML]{C0C0C0}82.76 $\pm$ 3.96 \\ \hline
\multicolumn{1}{|c|}{} &
{\color[HTML]{3D3D3D} BACC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.99 $\pm$ 0.87} &
\cellcolor[HTML]{C0C0C0}86.55 $\pm$ 1.46 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}89.05 $\pm$ 0.95} &
\cellcolor[HTML]{C0C0C0}85.58 $\pm$ 2.32 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.73 $\pm$ 0.95} &
\cellcolor[HTML]{C0C0C0}88.67 $\pm$ 1.64 \\ \cline{2-8}
\multicolumn{1}{|c|}{} &
{\color[HTML]{3D3D3D} F\textsubscript{1}} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}84.95 $\pm$ 0.82} &
\cellcolor[HTML]{C0C0C0}79.94 $\pm$ 1.70 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}85.02 $\pm$ 0.87} &
\cellcolor[HTML]{C0C0C0}78.53 $\pm$ 3.72 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}84.79 $\pm$ 0.95} &
\cellcolor[HTML]{C0C0C0}82.59 $\pm$ 1.67 \\ \cline{2-8}
\multicolumn{1}{|c|}{\multirow{-3}{*}{\textbf{UNet}}} &
{\color[HTML]{3D3D3D} PREC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.12 $\pm$ 0.74} &
\cellcolor[HTML]{C0C0C0}78.60 $\pm$ 5.71 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.17 $\pm$ 1.37} &
\cellcolor[HTML]{C0C0C0}77.35 $\pm$ 9.59 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{88.73 $\pm$ 1.28}} &
\cellcolor[HTML]{C0C0C0}80.08 $\pm$ 4.19 \\ \hline
\multicolumn{1}{|c|}{} &
{\color[HTML]{3D3D3D} BACC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.98 $\pm$ 0.57} &
\cellcolor[HTML]{C0C0C0}82.01 $\pm$ 6.57 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}89.29 $\pm$ 0.60} &
\cellcolor[HTML]{C0C0C0}86.99 $\pm$ 4.66 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.88 $\pm$ 0.56} &
\cellcolor[HTML]{C0C0C0}88.23 $\pm$ 2.13 \\ \cline{2-8}
\multicolumn{1}{|c|}{} &
{\color[HTML]{3D3D3D} F\textsubscript{1}} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}84.94 $\pm$ 0.52} &
\cellcolor[HTML]{C0C0C0}74.04 $\pm$ 9.71 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}85.27 $\pm$ 0.61} &
\cellcolor[HTML]{C0C0C0}79.94 $\pm$ 7.37 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}84.93 $\pm$ 0.61} &
\cellcolor[HTML]{C0C0C0}81.55 $\pm$ 4.38 \\ \cline{2-8}
\multicolumn{1}{|c|}{\multirow{-3}{*}{\textbf{UNet++}}} &
{\color[HTML]{3D3D3D} PREC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.09 $\pm$ 0.72} &
\cellcolor[HTML]{C0C0C0}75.64 $\pm$ 17.35 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.04 $\pm$ 0.57} &
\cellcolor[HTML]{C0C0C0}76.69 $\pm$ 11.94 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}88.48 $\pm$ 1.08} &
\cellcolor[HTML]{C0C0C0}78.23 $\pm$ 10.51 \\ \hline
\multicolumn{1}{|c|}{} &
{\color[HTML]{3D3D3D} BACC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}89.30 $\pm$ 0.44} &
\cellcolor[HTML]{C0C0C0}88.02 $\pm$ 1.22 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}89.31 $\pm$ 0.38} &
\cellcolor[HTML]{C0C0C0}88.69 $\pm$ 1.63 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{89.41 $\pm$ 0.29}} &
\cellcolor[HTML]{C0C0C0}\textbf{89.36 $\pm$ 1.27} \\ \cline{2-8}
\multicolumn{1}{|c|}{} &
{\color[HTML]{3D3D3D} F\textsubscript{1}} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}85.12 $\pm$ 0.51} &
\cellcolor[HTML]{C0C0C0}82.75 $\pm$ 1.10 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}85.27 $\pm$ 0.39} &
\cellcolor[HTML]{C0C0C0}84.33 $\pm$ 1.98 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{85.35 $\pm$ 0.25}} &
\cellcolor[HTML]{C0C0C0}\textbf{84.96 $\pm$ 1.14} \\ \cline{2-8}
\multicolumn{1}{|c|}{\multirow{-3}{*}{\textbf{MA-net}}} &
{\color[HTML]{3D3D3D} PREC} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}87.36 $\pm$ 0.58} &
\cellcolor[HTML]{C0C0C0}83.16 $\pm$ 2.44 &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}87.93 $\pm$ 0.46} &
\cellcolor[HTML]{C0C0C0}\textbf{86.52 $\pm$ 1.60} &
\multicolumn{1}{c|}{\cellcolor[HTML]{EFEFEF}87.76 $\pm$ 0.53} &
\cellcolor[HTML]{C0C0C0}86.15 $\pm$ 2.00 \\ \hline
\end{tabular}%
}
\end{table*}
\subsection{Implementation with training and test details}
\label{sec:implDetail}
LEFM layer is implemented in Python using the Pytorch framework \cite{paszke2019pytorch}. All existing deep networks are inherited and modified from \textit{segmentation-models-pytorch} library \cite{yakubovskiy2019segmentationmodels}. For faster convergence \cite{sitnik2021cocahis}, weights of backbones are initialized on ImageNet classification problem \cite{deng2009imagenet} and then fine-tuned with the rest of the LEFM-Net.
Before the training, images were preprocessed as described in Section \ref{sec:preprocessing}. For the evaluation of the statistical significance, all models were trained with 10 different random seeds. Thus, the initialization of newly added weights, coefficients, and choice of patches withheld for validation set were different in every run. In Table \ref{tab:cryonusegPerf}, we show the average performance of models on selected metrics (see Section \ref{sec:metrics}). For each dataset, the corresponding test sets are the same for every model.
We selected Adam \cite{kingma2014adam} optimizer for the training procedure. To equally punish the underperformance in terms of false positives and false negatives, we used Dice loss function \cite{seo2021closing,eelbode2020optimization}. The hyperparameters for every model were set as follows:
\begin{itemize}
\item maximum number of epochs equals to $3\cdot 10^4$
\item starting learning rate was set to $10^{-3}$
\item learning rate is reduced when the loss plateau is reached
\begin{itemize}
\item waiting patience for improvement is set to 20 epochs
\item reduction factor is set to 0.5
\item minimum learning rate equals $10^{-6}$
\end{itemize}
\item factor for $L_2$ weight decay is set to $10^{-4}$
\item Early stopping after 40 epochs when there is no improvement on the validation set
\item data augmentation
\begin{itemize}
\item shift limit = 0.2
\item scale limit = 0.2
\item rotation limit = 30$^{\circ}$
\item horizontal and vertical flip
\item probability = 0.5
\end{itemize}
\item LEFM order $m\in\{2,3\}$
\end{itemize}
\subsection{Results}
\label{sec:results}
Results of exhaustive training and testing of four different models, with and without the LEFM layer, on two datasets and 10 random seeds are shown in Table \ref{tab:cryonusegPerf}. Models are ordered by the number of parameters. As mentioned in Section \ref{subsec:dnn}, DeepLabv3+ has the least and MA-net the most trainable weights. When comparing these two models, a noticeable increase in scores can always be found in the model with larger capacity, i.e. MA-net.
It can be seen that four different versions of LEFM-Nets, compared to the original networks, yield better scores on two datasets. Higher order features are important for better end results. As it can be seen in Table \ref{tab:cryonusegPerf}, regardless of the level of expansion, LEFM-Net outperforms original deep models. Also, nearly half of the observed metrics for CryoNuSeg dataset and almost all of them for CoCaHis dataset are better when the level of expansion is $m=3$.
Tests performed on CryoNuSeg show that all LEFM-Nets are on average better than the original network. ANOVA statistical testing indicates that, with statistical significance $\alpha = 0.05$, all LEFM-Nets significantly outperform original models on CoCaHis dataset for $m=3$. Tests are performed on $F_1$ and BACC metrics. Hence, introduction of the LEFM layer seems to be justified for intraoperative pixel-based diagnostics with deep learning models.
The training time depends on the number of training parameters in the model, dataset, batch- and patch size, and on the degree of expansion $m$. With NVIDIA Tesla V100S GPU, training time for one run ranges from 2 hours, when training DeepLabv3+, to 20 hours when training MA-net.
\section{Discussion and Conclusion}
\label{sec:discAndCon}
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{figures/cryonuseg_d3.png}
\caption{Visualization of coefficients learned during the training on CryoNuSeg dataset. DeepLabv3+, UNet, UNet++, and MA-net were trained with expansion layer of degree $m=3$. During the training, $L_2$ norm was used for weight decay. Red ($R$), green ($G$) and blue ($B$) stand for $x_1$, $x_2$ and $x_3$ features, respectively.}
\label{figure:coefficientsD3}
\end{figure*}
The core idea behind the LEFM layer is generation of more explicit features in a data-driven manner, i.e. letting the network to decide on their importance for the image segmentation problem at hand. Towards that, we illustrate in Fig. \ref{figure:coefficientsD3} the relative importance of explicit features induced by the LEFM or order $m=3$ on the CryNuSeg dataset. For the sake of interpretability, we name the abstract features in the input space $\{x_1, x_2, x_3\}$ by the colors they represent \{red, green, blue\}. To interpret Fig. \ref{figure:coefficientsD3} it is important to notice that in images of H\&E stained specimens, nucleus, cytoplasm and glandular structures appear respectively blue-purple, pink and white. Intensely red structures may also represent red blood cells \cite{kothari2014removing}. Since red (R) and blue (B) colors are dominant in the purple color formation, they are dominantly expressed in Fig. \ref{figure:coefficientsD3}. However, justification for using the LEFM layer is because some explicit features, that do not exist in the input data space (monomials of the original features), are also important when the model is segmenting the image. As seen in Fig. \ref{figure:coefficientsD3}, the networks are using them up to certain extents. Hence, when the new features were added, i.e. number of channels is increased, LEFM-Net is expected to be guided towards better predictions while training. Thus, the level of expansion $m$ is a hyperparameter for cross-validation. However, there is a memory limitation to what degree can features be expanded (see Eq. (\ref{eq:binom})). Thus, we limited our experiments to the $m\in \{2,3\}$.
Interpretability of features in the domain of histopathological pixel-wise diagnostics is straightforward. Newly added features are nonlinear mixtures of the original features from the input space. Therefore, they can be interpreted as mixtures of colors needed for discriminating between cancer and non-cancer pixels. Using the previously explained coloring for nuclei, cytoplasm and glandular structures, it is clear that blue-purple, pink and white colors cannot be explained in terms of $R$, $G$ and $B$ features only. Thus, networks also used $RG$, $RB$ and $GB$ features, as seen in Fig. \ref{figure:coefficientsD3}. Since those features do not exist in the input data space, the LEFM layer helps the original network by providing the new medically meaningful features.
It is important to mention that feature expansion brings slightly increased computational complexity as there are more channels to the input of existing neural networks. Therefore, besides newly introduced coefficients in the LEFM layer, new neurons are added in every existing network to adapt for the expanded input. Thus, specific number of new learnable weights depends on the network's architecture. However, as discussed in Section IV.B the increase is modest.
To conclude, this work introduced a data-adaptive and learnable explicit feature map layer. Besides the features from the input data space, it generates higher order features and helps the existing network to improve segmentation/classification performance. New features are predefined by order of expansion and their importance is learned through the training process. Learned coefficients identify whether the feature is important for classification. Overall, if the original features are interpretable, then their monomials should be interpretable as well. The same applies to color mixtures of H\&E stains in the domain of histopathological image segmentation. By performing exhaustive validation on two different datasets and across multiple runs, introducing the LEFM layer to existing deep models seems to be justified. Presented statistical significance analysis further supports this statement.
\bibliographystyle{IEEEtran}
| {'timestamp': '2022-04-15T02:20:59', 'yymm': '2204', 'arxiv_id': '2204.06955', 'language': 'en', 'url': 'https://arxiv.org/abs/2204.06955'} |
\section{Introduction \label{sec:introduction}}
Recently, the use of a Wien filter operated at radio frequency (RF) has been proposed as a tool to search for electric dipole moments (EDMs) of protons and deuterons in storage rings~\cite{Rathmann:2013rqa,morse}. The occurrence of EDMs of elementary particles is intimately connected to the matter-antimatter asymmetry observed in the universe~\cite{doi:10.1146/annurev.nucl.49.1.35}, which the Standard Model of elementary particle physics fails to describe. A non-zero EDM measurement would point to new physics beyond the Standard Model~\cite{Dekens2014}.
A Wien filter provides orthogonal electric and magnetic fields, usually generated by a parallel plate capacitor and encircling coils, such that the Lorentz force for charged particles traveling with a specific velocity orthogonal to both fields vanishes~\cite{salomaa}. This principle has found widespread applications not only in mass spectro\-meters, but also in electron microscopes and ion optics~\cite{wienmicro}. A few reports have been dedicated to its use in accelerator facilities, but most of them describe static Wien filters~\cite{orloff2008handbook,steiner}. The electric and magnetic fields of an RF Wien filter can be used to manipulate the spins of particles in a storage ring, and, as shown in~\cite{Rathmann:2013rqa,morse}, open up the possibility for a measurement of the EDMs of protons and deuterons.
The objective of the present publication is to describe the design of a novel waveguide RF Wien filter for the search for the EDMs of protons and deuterons at COSY~\cite{Maier19971,PhysRevSTAB.18.020101}, pursued by the JEDI collaboration\footnote{JEDI collaboration \url{http://collaborations.fz-juelich.de/ikp/jedi}}. The fact that proton and deuteron EDMs are expected to be very small calls for precise design and manufacturing. We must admit that the spin-tracking tools required to provide specifications for the design of the RF Wien filter have not yet been fully developed. Recently, spin-tracking studies with an \textit{ideal} RF Wien filter were carried out~\cite{Rosenthal:2015jzr,doi:10.1142/S2010194516600995}, but these do not yet take into account a number of important systematic effects, such as fringe fields, unwanted field components, positioning errors, and non-vanishing Lorentz forces. Therefore, an approach was adopted here to try to realize the best possible device based on state-of-the-art technologies.
A prototype RF Wien filter, based on an already existing RF dipole with radial magnetic field ~\cite{PhysRevLett.93.224801,CERN:Courier}, was recently developed and used at COSY. (The use of RF dipoles and solenoids to manipulate stored polarized beams is discussed in~\cite{lee}.) The RF dipole was equipped with horizontal electric field plates in order to provide an RF Wien filter configuration with vertical electric and horizontal magnetic field~\cite{Mey:2015xbq,doi:10.1142/S2010194516600946}. The RF electromagnetic field was generated using two coupled resonators; one that generates the electric field and another one that generates the magnetic field. The approach of using separate systems to generate electric and magnetic fields, however, neglected the inherent coupling between the electric and magnetic fields. Therefore, the approach described here, is based on a novel waveguide system where by design the orthogonality between electric and magnetic fields is accomplished.
The paper is organized as follows:
\begin{itemize}
\item Section~\ref{sec:design} describes the mechanical design of the RF Wien filter. The parallel-plates wave\-guide and its mechanical structure are described in Section~\ref{sec:waveguide}, the driving circuit in Section~\ref{sec:driving_circuit}. The results of the electromagnetic field simulations, the Lorentz force compensation, and the optimization of the electric and magnetic field homogeneity by shaping the electrodes is discussed in Section~\ref{sec:field_homogenity}.
\item Section~\ref{sec:beam_dynamics_simulations} describes the beam dynamics simulations. A Monte Carlo simulation was carried out in order to quantify the unwanted field components of the RF Wien filter with a realistic phase-space distribution of the beam. A comparison of the prototype RF Wien filter~\cite{Mey:2015xbq,doi:10.1142/S2010194516600946} and the wave\-guide RF Wien filter is given in Section~\ref{sec:comparison}.
\item Section~\ref{sec:thermal} presents results of thermal simulations based on the power losses in the different materials of the device.
\end{itemize}
\section{Design \label{sec:design}}
\subsection{Waveguide design \label{sec:waveguide}}
The RF Wien filter shall be operated at frequencies of about 100\,kHz to 2\,MHz. The maximum acceptable length for the vacuum vessel, given by space restrictions at COSY, is 870\,mm, which corresponds to approximately $1/300$ of the length of the electromagnetic wave (at $\SI{1}{MHz}$). The transverse electromagnetic (TEM) mode of a parallel-plate wave\-guide fulfills the requirement of orthogonal electric and magnetic fields.
In order to maintain an overall vanishing Lorentz-force, and to provide minimal unwanted field components, it is a priori not clear whether a small or a large beam size is advantageous. Within the COSY ring, the PAX low-$\beta$ section~\cite{PhysRevSTAB.18.020101} offers the possibility to vary the beam size by about a factor of three, therefore the RF Wien filter will be installed at this location.
The design calculations for the waveguide RF Wien filter were carried out using a full-wave simulation with CST Microwave Studio\footnote{CST - Computer Simulation Technology AG, Darmstadt, Germany, \url{http://www.cst.com}}, and the electric and magnetic fields were modeled with an accuracy of $10^{-6}$. Because of the high expectations on field homogeneity, an approach was adopted that allows one to calculate the electric and magnetic fields without additional assumptions, such as quasi-static approximations or the like\footnote{Each simulation required up to 12 hours of computing time on a 4-Tesla C2075 GPU cluster\footnotemark[4], with 2 six-core Xeon E5 processors\footnotemark[5] and a RAM capacity of 94\,GB.}.
\footnotetext[4]{Nvidia Corporation, Santa Clara, California, USA, \url{http://www.nvidia.com/object/tesla-workstations.html}}\addtocounter{footnote}{+1}
\footnotetext[5]{Intel Corporation, Santa Clara, California, USA, \url{http://www.intel.com/content/www/us/en/processors/xeon/xeon-processor-e5-family.html}}\addtocounter{footnote}{+1}
The system comprises a power source, a parallel plate waveguide and a load. The plates are fed on their front and rear sides by an RF current, which distributes on the surfaces, thereby generating the required electromagnetic field in the enclosed space. Deviations from the ideal orthogonality and homogeneity of the fields are due to the finite size of the plates and their limited conductivities.
The intended working frequencies of the RF Wien filter are calculated according to
\begin{equation}
f_{\text{RF}}=f_{\text{rev}} \lvert {k+\gamma G} \rvert, k \in \mathbb{Z}\,
\end{equation}
where $k$ is the harmonic number, $G$ the gyromagnetic anomaly, $\gamma G$ the spin tune~\cite{PhysRevLett.115.094801}, and $f_{\text{rev}}$ the revolution frequency. With respect to the search for the EDMs of deute\-rons and protons, experiments at a number of harmonics shall be carried out for systematic reasons. Table~\ref{tab:harmonic} summarizes the resonance frequencies for deuterons at a momentum of $\SI{970}{MeV/c}$ and for protons at $\SI{520}{MeV/c}$, for which electron-cooled beams with compensated cooler solenoid are available at COSY. As indicated in Table~\ref{tab:harmonic} for deuterons (protons), five (four) harmonics are in the operating frequency range of $\SI{100}{kHz}$ to $\SI{2}{MHz}$. In the following, we restrict the discussion to deuterons at a momentum of $\SI{970}{MeV/c}$.
\begin{table*}[htb]
\renewcommand{\arraystretch}{1.2}
\centering
\caption{Operating frequencies $f_\text{RF}$ of the waveguide RF Wien filter for deuterons ($d$) at a momentum of 970\,MeV/c and for protons ($p$) at $\SI{520}{MeV/c}$ in COSY for the harmonic numbers $k$. The frequencies $f_\text{RF}$ shown in bold fit in the frequency range from $\SI{100}{kHz}$ to 2\,MHz. The revolution frequency $f_\text{rev}$, G factors, Lorentz $\beta$ and $\gamma$, and the spin tune $\gamma G$ are given as well.}
\label{tab:harmonic}
\begin{tabular}{cccccc|ccccccc}\hline
& & & & & & & \multicolumn{5}{c}{$f_\text{RF}$\,[kHz]}\\
& $f_\text{rev}$\,[kHz] & G & $\beta$ & $\gamma$ & $\gamma G$ & $\scriptstyle k=-4$ & $\scriptstyle k=-3$& $\scriptstyle k=-2$ & $\scriptstyle k=-1$ & $\scriptstyle k=0$ & $\scriptstyle k=+1$ & $\scriptstyle k=+2$ \\\hline
$d$ & $750.2$ & $-0.143$ & $0.459$ & $1.126$ & $-0.161$ & $3121.6$ & $2371.4$ & $\mathbf{1621.2}$ & $\mathbf{871.0}$ & $\mathbf{120.8}$ & $\mathbf{629.4}$ & $\mathbf{1379.6}$\\
$p$ & $791.6$ & $1.793$ & $0.485$ & $1.143$ & $2.050$ & $\mathbf{1543.9}$ & $\mathbf{752.2}$& $39.4$ & $\mathbf{831.0}$ & $\mathbf{1622.7}$ & $2414.3$ & $3206.0$ \\\hline
\end{tabular}
\end{table*}
The Lorentz force is given by
\begin{equation}
\vec F_\text{L} = q \left(\vec E + \vec v \times \vec B\right)\,,
\label{eq:lorentz}
\end{equation}
where $q$ is the charge of the particle, $\vec{v}=c(0,0,\beta)$ is the velocity vector, $\vec E=(E_x,E_y,E_z)$ and $\vec B=\mu_0(H_x,H_y,H_z)$ denote the components of the electric and magnetic fields, and $\mu_0$ the vacuum permeability. For a vanishing Lorentz force $\vec{F}_\text{L}=0$, the required field quotient $Z_q$ is determined, which yields
\begin{align}
E_x & = -c \cdot \beta \cdot \mu_0 \cdot H_y\,, \nonumber \\
Z_q=-\frac{E_x}{H_y} & = c \cdot \beta \cdot \mu_0 \approx 173 \, \, \Omega\,.
\label{eq:zq}
\end{align}
Figure~\ref{fig:structure} shows a cross section of the parallel-plates wave\-guide RF Wien filter. The axis of the waveguide points along the beam direction ($z$). The plates are separated by $\SI{100}{mm}$ along the $x$-direction. The width of the plates is $\SI{182}{mm}$. This setup ensures that during the EDM studies, the main component of the electric field ($E_x$) points radially inwards in $-x$-direction, and the main component of the magnetic field ($H_y$) upwards in $y$-direction with respect to the stored beam. The system is placed inside a cylindrical vacuum vessel. The electrodes are surrounded by ferrite blocks made of CMD5005\footnote{National Magnetics Group, Inc., Pennsylvania, USA, \url{http://www.magneticsgroup.com/m_ferr_nizn.htm}}. The copper electrodes are shaped in order to improve the homogeneity of the electric and magnetic fields and to minimize the Lorentz force, as explained in detail in Section~\ref{sec:EM-simulation}.
\begin{figure*}[!]
\centering
\includegraphics[width=11cm]{structure.eps}
\caption{\label{fig:structure} Design model of the RF Wien filter showing the parallel-plates waveguide and the support structure. 1: beam position monitor (BPM); 2: copper electrodes; 3: vacuum vessel; 4: clamps to hold the ferrite cage; 5: belt drive for $\ang{90}$ rotation, with a precision of $\ang{0.01}$ ($\SI{0.17}{mrad}$); 6: ferrite cage; 7: CF160 rotatable flange; 8: support structure of the electrodes; 9: inner support tube. The axis of the waveguide points along the $z$-direction, the plates are separated along $x$, and the plate width extends along $y$. During the EDM studies, the main field component $E_x$ points radially outwards and $H_y$ upwards with respect to the stored beam.}
\end{figure*}
A sophisticated support structure, shown in Fig.~\ref{fig:support}, was designed to ensure a high degree of rigidity, precision alignment and most importantly, low distortion of the generated electromagnetic field. The electrodes are made from double T-shaped copper plates for better stiffness and stability. They are mounted with 12 stainless-steel screws on the main support structure of the electrodes. An inner stainless-steel tube is used to hold, align and mount the structure with high precision ($\approx 10$\,$\mu$m).
The angular position of the RF Wien filter with respect to the beam axis can be chosen to allow for a rotation of the field direction by $\ang{90}$ with an angular precision of $\ang{0.01}$ ($\SI{0.175}{mrad}$) using a belt drive (see label 5 in Fig.~\ref{fig:structure}), without breaking the vacuum. This feature is foreseen in order to fine tune the orientation of the RF Wien filter~\cite{doi:10.1142/S2010194516600995}, and also to exchange the role of electric and magnetic fields during the EDM measurements for investigations of the systematics.
\begin{figure*}[!t]
\centering
\includegraphics[width=10cm]{support.eps}
\caption{\label{fig:support} Inner support structure of the RF Wien Filter. 1: copper electrodes with the trapezium shaping at the edges; 2: specially designed connector; 3: ceramic insulator between the electrodes and the support structure. A stainless-steel screw is located inside to connect the electrodes to the support structure; 4: support of the electrodes; 5: clamps to support the ferrite cage; 6: inner tube support structure.}
\end{figure*}
The parallel-plates waveguide constitutes a transmission line structure, which has been analyzed in detail in~\cite{slimpstp2015}. For the waveguide to be used as a Wien filter, wave mismatch must be introduced into the structure via a reflection, ensured by adding a resistor that controls the reflection coefficient. Thereby, the characteristic field quotient $Z_q$, given in Eq.~(\ref{eq:zq}), can be adjusted to the required value.
The multi-input feedthroughs ensure a homogeneous current distribution over the electrodes, the two parallel plates are connected to the amplifier and the load via four high-frequency, high-power CF 40 feedthroughs\footnote{VACOM Vakuum Komponenten $\&$ Messtechnik GmbH, Jena, Germany, \url{http://www.vacom.de}}. The electrodes are connected to the feedthroughs via specially designed connectors, and $3$\,dB power splitters are installed to feed each plate at the edges.
\begin{figure}[hbt]
\centering
\includegraphics[width=0.9\columnwidth]{circuit.eps}
\caption{\label{fig:circuit} Schematic of the driving circuit for the RF Wien filter. The condition for minimal Lorentz force is met using two adjustable devices, the resistor $R_\text{m}$ and the inductance $L_\text{p}$.}
\end{figure}
The RF Wien filter comprises a cage of ferrite blocks from CMD5005, which contains the electromagnetic fields within the Wien filter and enhances the field homogeneity. CMD5005 is a NiZn-type high-permeability ferrite material. Ref.~\cite{Hahn_rhicabort} provides data of a dispersive model of the complex permeability $\mu_r$ that have been used in the full-wave simulator at frequencies from $\SI{100}{kHz}$ to 2\,MHz. At a frequency of $\SI{871}{kHz}$, for instance, $\mu_r=1449 + i 467$. The relative permittivity $\epsilon_r$ is around 25~\cite{zhang}. CMD5005 elements come in blocks with a maximum length of 330\,mm. The ferrites are held together by clamps in a symmetric arrangement. To make a connection between the electrodes and the main support structure possible, 12 boreholes are drilled in the ferrites. The screws supporting the electrodes are insulated from the ferrites by ceramic cylindrical insulators.
On each side of the CF100 entry and exit ports, 4 CF16 feedthroughs are connected for the beam position monitors (BPMs)~\cite{hinder}, especially designed to ensure the alignment of the beam with respect to the axis of the RF Wien filter.
\subsection{Driving circuit \label{sec:driving_circuit}}
Ideally, the driving circuit connects a load resistor directly to the electrodes, as shown in~\cite{slimpstp2015}. The amplifier (with 50 $\Omega$ internal impedance) is connected to the Wien filter via a ($1:0.5$) Balun~\cite{hickman} which transforms the impedance and converts the unbalanced output of the amplifier into a balanced one. A schematic of the driving circuit is shown in Fig.~\ref{fig:circuit}. One of the main characteristics of Balun transformers is their broadband response. The return loss $L_\text{r}$ of the driving circuit (Fig.~\ref{fig:circuit}), calculated using CST Design Studio, is depicted in Fig.~\ref{fig:s_11}.
\begin{figure}[hbt]
\centering
\includegraphics[width=0.9\columnwidth]{s11.eps}
\caption{\label{fig:s_11} Return loss $L_\text{r}$ of the electromagnetic simulation of the circuit, shown Fig.~\ref{fig:circuit}. For the entire range of spin harmonics (from 0 to $\SI{2}{MHz}$), $L_\text{r}$ is below $\SI{-15}{dB}$. Wideband matching is achieved with a $1:0.5$ Balun.}
\end{figure}
Another ($1:1$) Balun is used to connect the Wien filter to the load resistor. Introducing the cables into the schematic induces a phase shift between the electric and magnetic field,
consequently, the field quotient $Z_q$, given in Eq.~(\ref{eq:zq}), becomes complex.
This effect can be compensated with an additional inductance $L_\text{p} = 8\,\mu\text{H}$ (as indicated in Fig.~\ref{fig:circuit}), and an almost purely real-valued field quotient is obtained, \textit{i.e.}, $Z_q = 173\,\angle 0.1 ^\circ \Omega$.
\subsection{Optimization of field homogeneity \label{sec:field_homogenity}}
\label{sec:EM-simulation}
For minimal Lorentz forces, the electric and magnetic forces must be matched at the center and the edges of the RF Wien filter. As mentioned before, the ratio between the electric and magnetic fields inside the RF Wien filter can be controlled via the load resistor $R_\text{m}$, and the inductance $L_\text{p}$ (see Fig.~\ref{fig:circuit}). A special solution, however, is required at the edges, where the slope of the curves of the electric and magnetic forces are not the same. According to our simulations, a simple parallel-plates waveguide deflects particles passing through the device in the same directions at the exit and entry points. Under these cir\-cum\-stances, the overall Lorentz force \textit{cannot} be zero when the Lorentz force in the center is zero.
Our solution aims at a decomposition of the kicks at each side of the RF Wien filter into two deflections of opposite sign in such a way that both deflections average out. Decoupling the fields at the edges, keeping the electric field unchanged while manipulating the magnetic field, is accomplished by retaining the plate separation (spacing between the plates), while altering the width of the plates~\cite{pozar}. In doing so, the slope of the curves of the electric field remains constant while the magnetic field crosses the electric field, as shown in Fig.~\ref{fig:lorentz}. This approach results in trapezoid-shaped edges of the parallel plates, where the field crossing can now be optimized via the geometrical properties of the trapezoid-shaped edges. After performing a series of simulations, an optimum trapezoid depth of $\SI{50}{mm}$, with a short base of $\SI{50.5}{mm}$ and a long base of $\SI{100}{mm}$ was found, as shown in Fig.~\ref{fig:support} (label 1).
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{lorentz.eps}
\caption{Electric force $F_\text{e}$, magnetic force $F_\text{m}$, and Lorentz force $F_\text{L}$ inside the RF Wien filter, for the geometry shown in Fig.~\ref{fig:structure}. The integral Lorentz force is of the order of $10^{-3}$\,eV/m. The trapezoid-shaped electrodes at the entrance and exit of the RF Wien filter determine the crossing of electric and magnetic forces.}
\label{fig:lorentz}
\end{figure}
Integrating and averaging the Lorentz force $\vec F_\text{L}$, given in Eq.~(\ref{eq:lorentz}), along the axis of the RF Wien filter for the geometry shown in Fig.~\ref{fig:structure} at an input power of $\SI{1}{kW}$, yields
\begin{equation}
\frac{q}{\ell}\int_{-\ell/2}^{\ell/2} \left( \begin{array}{ccc}
E_x - c \beta B_y\\
E_y + c \beta B_x\\
E_z
\end{array} \right) dz=
\left(\begin{array}{ccc}
5.97 \times 10^{-3}\\
7.97 \times 10^{-3}\\
1.27 \times 10^{-21}\\
\end{array}\right) \, \text{eV/m}\,,
\label{eq:lorentz-force-value}
\end{equation}
where $\ell=\SI{1550}{mm}$ denotes the active length of the RF Wien filter, defined as the region where the fields are non-zero. The momentum variation in the beam of about $\Delta p / p = 10^{-4}$ translates into a variation of the required field quotient $Z_q$ of the same order of magnitude [see Eq.~(\ref{eq:zq})], and therefore, the resulting values for the Lorentz forces, given in Eq.~(\ref{eq:lorentz-force-value}), are acceptable.
The waveguide RF Wien filter can be rated according to its ability to manipulate the spins of the stored particles, and as a figure of merit, the field integral of $\vec B$ along the beam axis is evaluated, yielding for an input power of $\SI{1}{\,kW}$,
\begin{equation}
\int_{-\ell/2}^{\ell/2} \vec B dz =
\left(\begin{array}{ccc}
2.73 \times 10^{-9}\\
2.72 \times 10^{-2}\\
6.96 \times 10^{-7}\\
\end{array}
\right)\,\text{T\,mm}\,.
\end{equation}
Under these conditions, the corresponding integrated electric field components are given by
\begin{equation}
\int_{-\ell/2}^{\ell/2} \vec E dz =
\left(\begin{array}{rrr}
3324.577 \\
0.018\\
0.006\\
\end{array}
\right)\,\text{V}\,.
\end{equation}
Figure~\ref{fig:main_fields} shows the main (wanted) components of the electric and magnetic fields, $E_x$ and $B_y$, in the $xz$ plane.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.45\textwidth]{e_field_em.eps}
\includegraphics[width=0.45\textwidth]{h_field_em.eps}
\caption{\label{fig:main_fields} Main components of the electric field $E_x$ (left panel) and magnetic field $H_y$ (right) of the waveguide RF Wien filter, for the geometry shown in Figs.~\ref{fig:structure} and \ref{fig:support}.}
\end{figure*}
In order to further increase the field homogeneity, the geometric parameters such as the width and the surface shape of the copper electrodes, and also the geometric parameters of the ferrite blocks and their distance to the metallic support structure, including the surrounding vacu\-um vessel were optimized. The simulations showed that parabolically-shaped electrode surfaces instead of flat ones substantially improve the local homogeneity of the electric field along the RF Wien filter. The parameters of the optimized parabolic electrodes are indicated in Fig.~\ref{fig:parabolic_design}, with a major radius of $\SI{91}{mm}$, and a minor radius of $\SI{6}{mm}$, where the sharp edges have been rounded using a $1$\,mm radius. In the case of flat electrodes, the electric field varies up to $\SI{8}{\text{V/m}}$, and by parabolically shaping the electrodes, as shown in Fig.~\ref{fig:parabolic_design}, the electric field variation does not exceed $\SI{0.1}{\text{V/m}}$.
\begin{figure}[hbt]
\centering
\includegraphics[width=1\columnwidth]{para.eps}
\caption{\label{fig:parabolic_design} Schematic view of the cross section of the bottom electrode. The double T-shaped support is used to fix the electrode to the support structure. The parabolic shape is constructed using an ellipse with a major radius of $a = \SI{91}{\,mm}$, and a minor radius of $b=\SI{6}{mm}$. Sharp edges are avoided using a rounding radius of $1\,$mm.}
\end{figure}
The advantage of using parabolic electrodes is illustrated in Fig.~\ref{fig:parabolic_vs_flat}, where the relative variation of $E_x$ and $H_y$, \textit{e.g.}, $\left| \frac{|E_x|-\langle E_x \rangle}{|E_x|}\right|$, is shown in the $xy$ plane in the range $x=\pm 5$\,mm and $y=\pm 5$\,mm in the center of the RF Wien filter. The results are summarized in Table~\ref{tab:c_variation}, where the relative standard deviation of panels a), b), d), and e) of Fig.~\ref{fig:parabolic_vs_flat} is listed. With respect to the Lorentz force $\vec F_\text{L}$, parabolically-shaped electrodes show a better homogeneity along the beam trajectory.
\begin{table}[b]
\renewcommand{\arraystretch}{1.1}
\centering
\caption{Calculated relative standard deviation (RSD) of the electric and magnetic fields in the cases of flat and parabolically-shaped electrodes from Fig.~\ref{fig:parabolic_vs_flat}, quantitatively indicating the achieved field homogeneity.}
\begin{tabular}{ccc}\hline
RSD & flat shape & parabolic shape\\ \hline
$|\frac{\sigma(E_x)}{\langle E_x\rangle}|$ & $4.74 \times 10^{-4}$ & $2.33 \times 10^{-5}$\\
$|\frac{\sigma(H_y)}{\langle H_y\rangle}|$ & $3.17 \times 10^{-5}$ & $ 3.53 \times 10^{-4}$\\\hline
\end{tabular}
\label{tab:c_variation}
\end{table}
The parabolically-shaped electrodes yield an improve\-ment of up to a factor of $20$ in terms of local electric field homogeneity while reducing the local homogeneity of the magnetic field by a factor of $11$. In total, the Lorentz forces are about a factor $5$ smaller for parabolically-shaped electrodes with respect to flat-shaped electrodes.
\begin{figure*}[htb]
\centering
\subfigure[Flat: $\left|\frac{|E_x|- \left\langle E_x \right\rangle}{|E_x|} \right|$]{\includegraphics[width=0.32\textwidth]{e_flat.eps}}
\subfigure[Flat: $\left|\frac{|H_y|- \left\langle H_y \right\rangle}{|H_y|} \right|$]{\includegraphics[width=0.32\textwidth]{h_flat.eps}}
\subfigure[Flat: Lorentz force in $\text{eV/m}$]
{\includegraphics[width=0.32\textwidth]{lorentz_flat.eps}}
\subfigure[Parabolic: $\left|\frac{|E_x|- \left\langle E_x \right\rangle}{|E_x|} \right|$]{\includegraphics[width=0.32\textwidth]{e_para.eps}}
\subfigure[Parabolic: $\left|\frac{|H_y|- \left\langle H_y \right\rangle}{|H_y|} \right|$]{\includegraphics[width=0.32\textwidth]{h_para.eps}}
\subfigure[Parabolic: Lorentz force in $\text{eV/m}$]
{\includegraphics[width=0.32\textwidth]{lorentz_para.eps}}
\caption{\label{fig:parabolic_vs_flat} Comparison of the field homogeneity of flat-shaped and parabolically-shaped electrodes across the beam extension in the range $\pm x=5$\,mm and $\pm y=5$\,mm around the beam axis in the center of the RF Wien filter ($z=0$). The top row shows the fields variation of $E_x$ (panel a) and $H_y$ (b), and the Lorentz force (c) for flat-shaped electrodes, and the bottom row the corresponding results for parabolically-shaped electrodes, $E_x$ (d), $H_y$ (e), and the Lorentz force (f). (Note the different in the graphs.)}
\end{figure*}
\section{Beam dynamics simulations \label{sec:beam_dynamics_simulations}}
\subsection{Definition of unwanted field components}
The finite beam size induces non-ideal field components of $\vec{E}$ and $\vec{H}$ in undesired directions. The system is designed such that \textit{unwanted} field components average out at the exit and entry of the RF Wien filter. The unwanted field components are given by
\begin{equation}
\vec{E}_\perp = \left(\begin{array}{ccc}
0\\
E_y \\
E_z\\
\end{array}\right) \,, \, \text{and}\,\,
\vec{H}_\perp = \left(\begin{array}{ccc}
H_x\\
0 \\
H_z\\
\end{array}\right) \label{eq:EH_perp} \,.
\end{equation}
The index $\perp$ is introduced to indicate that the unwanted electric and magnetic field components are perpendicular to the main field components $E_x$ and $H_y$.
The particle beam that enters the RF Wien filter has a defined phase-space distribution~\cite{PhysRevSTAB.18.020101}, therefore the particles do not travel along straight lines, parallel to the beam axis. In order to quantify the effect of unwanted field components, a Monte-Carlo simulation based on solving the relativistic equation of motion using the time- and space-dependent fields has been carried out, with subsequent integration of the field components along the trajectories. The electromagnetic simulation yields the complex fields $\tilde{\vec{E}}$ and $\tilde{\vec{H}}$. The real fields $\vec{E} \left(\vec{r},t\right)$ and $\vec{H} \left(\vec{r},t\right)$ are obtained from
\begin{eqnarray}
\vec{E} \left(\vec{r},t\right) & = & \Re\left( \tilde{\vec{E}}e^{i\omega t} \right) \,,\text{and} \nonumber\\
\vec{H} \left(\vec{r},t\right) & = & \Re\left( \tilde{\vec{H}}e^{i\omega t} \right)\,.
\end{eqnarray}
The relativistic equations of motion~\cite{humphries2013charged,wiedemann2015particle},
\begin{eqnarray}
\frac{d\vec{v}}{dt} &=&\frac{q}{m \gamma}\left[ \vec{E}(\vec r,t)+\vec{v}\times \vec{B}(\vec r,t)\right]\nonumber\\
&&-\frac{q}{m \gamma c^2} \vec{v} \left[\vec{v} \cdot \vec{E}(\vec r,t) \right]\,,\text{and} \nonumber\\
\frac{d\vec{r}}{dt} &=&\vec{v}\,,
\end{eqnarray}
were solved in Matlab$\footnote{Mathworks, Inc. Natick, Massachusetts, USA, \url{http://de.mathworks.com}}$ using the field maps imported from the full-wave simulations. The mesh accuracy in the ${xy}$ plane amounts to $\SI{0.1}{mm}$ and $\SI{11}{mm}$ in ${z}$-direction. The phase-space distribution of the beam at the entry of the RF Wien filter is known~\cite{PhysRevSTAB.18.020101}. The phase space is expressed in terms of positions $x$ and $y$, and transverse angles $x'= v_x/v_z$ and $y'= v_y/v_z$.
For each phase space, 5000 particles have been simulated with a $2\sigma$ beam emittance of $\varepsilon_{x,y}=1$\,$\mu$m. The initial ${(x,x')}$ phase-space distributions are shown in red in Fig.~\ref{fig:pse_beta}. The blue points represent the phase-space distribution of the particles after passing through the RF Wien filter. The $(y,y')$ distributions are not shown, but are very similar. With low-$\beta$ section ON and OFF, the areas of the initial and final ellipses are the same, and consistent with the behavior of a field free (drift) region. Thus the RF Wien filter does not appear to alter the phase-space distributions of the beam.
\begin{figure*}[htb]
\centering
\subfigure[Low-$\beta$ section ON: $\beta = \SI{0.4}{m}$]{\includegraphics[width=0.40\textwidth]{pse_x_bl.eps}}
\subfigure[Low-$\beta$ section OFF: $\beta = \SI{4}{m}$]{\includegraphics[width=0.40\textwidth]{pse_x_bh.eps}}
\caption{\label{fig:pse_beta} Phase-space distributions $(x,x')$ of a beam with an emittance of $\varepsilon_{x,y}=1$\,$\mu$m entering (red) and exiting (blue) the RF Wien filter with low-$\beta$ section ON: $\beta = \SI{0.4}{m}$ (panel a) and OFF: $\beta = \SI{4}{m}$ (panel b). The ellipses show the $2 \sigma$ results of a fit with a 2D Gaussian distribution, respectively.}
\end{figure*}
Based on the simulations, the effects of unwanted field components, given in Eq.~(\ref{eq:EH_perp}), are quantified by integration along the particle trajectories, and expressed as ratios with respect to the total field integrals, yielding
\begin{eqnarray}
f^\text{int}_{E_\perp} &=& \frac{\int |\vec E_\perp|ds}{\int |\vec E| ds} \,\,, \text{and} \label{eq:e_field_err} \\
f^\text{int}_{H_\perp} &=& \frac{\int |\vec H_\perp|ds}{\int |\vec H| ds} \label{eq:b_field_err}\,,
\end{eqnarray}
where $ds$ denotes a differential element of the particle's path, which does not necessarily point along the $z$ axis.
\subsection{Results of beam dynamics simulations \label{sec:results}}
The integrals in Eqs.~(\ref{eq:e_field_err}) and (\ref{eq:b_field_err}) are evaluated for each of the 5000 simulated particles and for the two cases with low-$\beta$ section ON ($\beta=\SI{0.4}{m}$) and OFF ($\beta=\SI{4}{m}$). The results are shown in the left panels of Fig.~\ref{fig:parasitic}, respectively, and summarized in Table~\ref{tab:RF-WF-comparison}. With low-$\beta$ section ON, the beam size is small, and the angular variation is large. As a consequence, the unwanted field components $E_\perp$ and $H_\perp$ do not completely cancel at the edges, leading to a wider distribution. When the low-$\beta$ section is OFF, the beam size is large, but with small angular spread, leading to a better cancellation of the unwanted field components at the edges.
\begin{figure*}[!]
\centering
\subfigure[Unwanted electric field components $f^\text{int}_{E_\perp}$ from Eq.~(\ref{eq:e_field_err}).]
{\includegraphics[width=0.48\textwidth]{e_err.eps}}
\hspace{0.5cm}
\subfigure[Unwanted magnetic field components $f^\text{int}_{H_\perp}$ from Eq.~(\ref{eq:b_field_err}).]
{\includegraphics[width=0.48\textwidth]{b_err.eps}}
\caption{\label{fig:parasitic} Comparison of the unwanted electric (a) and magnetic (b) field components with low-$\beta$ section ON (red) and low-$\beta$ section OFF (blue). The right panels display Gaussian fits to the projections of the simulated distributions, respectively, summarized in Table~\ref{tab:RF-WF-comparison}.}
\end{figure*}
The distributions in Fig.~\ref{fig:parasitic} exhibit a line structure that stems from the finite mesh ($\SI{0.1}{mm}$ in the $xy$ plane, and $\SI{11}{mm}$ along $z$).
Gaussian fits to the projections of the resulting distributions $f^\text{int}_{E_\perp}$ and $f^\text{int}_{H_\perp}$ are shown in the right panels of Fig.~\ref{fig:parasitic}, the fit results are summarized in Table~\ref{tab:RF-WF-comparison}. For the low-$\beta$ section either switched ON or OFF, the mean values of the integrated unwanted electric and magnetic field components are zero within the errors. The widths of the distributions, however, indicate that with low-$\beta$ section OFF, the unwanted electric and magnetic field components picked up by the beam are smaller by about a factor of two to three compared to when the low-$\beta$ section is ON. These findings should be further investigated using spin and particle tracking of a stored beam. In addition, once the device is installed, the predictions should be verified experimentally.
\subsection{Comparison of prototype and waveguide RF Wien filter \label{sec:comparison}}
The waveguide RF Wien filter is compared to the prototype RF Wien filter~\cite{Mey:2015xbq,doi:10.1142/S2010194516600946} (see Section~\ref{sec:introduction}) in terms of the unwanted field components, evaluated according to Eqs.~(\ref{eq:e_field_err}) and (\ref{eq:b_field_err}), listed in Table~\ref{tab:RF-WF-comparison}. With the novel waveguide RF Wien filter, unwanted electric and magnetic field components can be reduced by one to two orders of magnitude.
\begin{table*}[hbt]
\renewcommand{\arraystretch}{1.1}
\centering
\caption{Comparison of the prototype RF Wien filter~\cite{Mey:2015xbq,doi:10.1142/S2010194516600946} and the waveguide design in terms of unwanted field components $f^\text{int}_{E_\perp}$ and $f^\text{int}_{H_\perp}$ (see Eqs.~(\ref{eq:e_field_err}) and (\ref{eq:b_field_err})) using the simulated data, shown in Fig.~\ref{fig:parasitic}.
}
\begin{tabular}{lccccc}\hline
RF Wien filter & & \multicolumn{2}{c}{$f^\text{int}_{E_\perp}$} & \multicolumn{2}{c}{$f^\text{int}_{H_\perp}$}\\
& & mean $\mu$ & width $\sigma$ & mean $\mu$ & width $\sigma$ \\\hline
\multirow{ 2}{*}{Waveguide} & low-$\beta$ ON & $(-1.1 \pm 0.8) \times 10^{-5}$ & $5.7 \times 10^{-4}$ & $(1.4 \pm 0.8) \times 10^{-5}$ & $5.5 \times 10^{-4}$\\
& low-$\beta$ OFF & $(4.1 \pm 2.5) \times 10^{-6}$ & $1.8 \times 10^{-4}$ & $(0.8 \pm 2.6) \times 10^{-6}$ & $1.8 \times 10^{-4}$\\\hline
Prototype & & $(1.454 \pm 0.009) \times 10^{-2}$ & $8.9 \times 10^{-3}$ & $(2.927 \pm 0.003) \times 10^{-2}$ & $3.0\times 10^{-3}$\\
\hline
\end{tabular}
\label{tab:RF-WF-comparison}
\end{table*}
Spin tracking simulation are required to quantify the systematic errors induced by unwanted field components, and by other systematic effects, \textit{e.g.}, positioning errors, and non-vanishing Lorentz forces in the determination of proton and deuteron EDMs.
\section{Thermal response of the RF Wien filter during operation \label{sec:thermal}}
Because of the thermal insulation by the vacuum, it was investigated whether even small power losses could lead to a temperature rise during operation. Therefore, for the corresponding thermal simulations using the FEM software ANSYS\footnote{ANSYS, Inc. Canonsburg, USA \url{http://http://www.ansys.com}} only radiative heat exchange between the internal surfaces was considered. The entire power in the copper plates was assumed to be generated at the locations of the feedthroughs. The calculated power loss densities with an input power of $\SI{1}{kW}$ were integrated over the structural elements and are summarized in Table~\ref{tbl_2}. The temperature rise for the copper electrodes is $6$\,K, and $0.5$\,K for the ferrites, respectively. Based on the temperature distribution, the simulated thermal expansion of the copper plates amounts to $48$\,$\mu $m, and to $8.6$\,$\mu$m for the ferrite blocks. These values are small compared to the structure's dimensions, and we do not expect problems with respect to the electromagnetic performance and the suspension of the mechanical structure during operation of the RF Wien filter.
\begin{table}[htb]
\renewcommand{\arraystretch}{1.1}
\centering
\caption{\label{tbl_2} Losses per material computed using the electromagnetic solver of CST Microwave Studio at a frequency $f_\text{RF}=\SI{1}{MHz}$.}
\begin{tabular}{rl}
\hline
Material & Thermal losses [W] \\\hline
Copper Plates & 0.0957 \\
Ferrites & 0.0734 \\
Steel & 0.0255 \\
Ceramic & 0.0010 \\\hline
\end{tabular}
\end{table}
\section{Conclusion and Outlook \label{sec:conclusion}}
The paper presents the electromagnetic design calculations for a novel type of RF Wien filter that shall be used at the COSY storage ring to determine the EDMs of deuterons and protons. The main emphasis of the work was to design a device that exhibits a high level of homogeneity of the electric and magnetic fields. The optimization of the electromagnetic design was performed in close cooperation with the mechanical design, taking into account all the details of the mechanical construction.
A waveguide structure was selected because in this case the orthogonality of $E$ and $B$ fields, required for a Wien filter, can be ensured to a high level of precision. Minimizing the overall Lorentz force, while still providing sizable electric and magnetic field integrals, leads to parabolically-shaped electrodes, equipped with trapezoid-shaped entrance and exit partitions, surrounded by a closed box of ferrites. Using the above described electrodes, the overall Lorentz force is reduced by about a factor of five, compared to flat-shaped electrodes.
The RF Wien filter will be installed in a section at COSY, where it is possible to vary the beam size by adjusting the $\beta$ function between about $\beta=0.4$ and $\SI{4}{m}$. Single-pass tracking calculations were performed in order to quantify the effect of unwanted electric and magnetic field components picked up by the beam, realistically distributed in phase space. The calculations indicate that unwanted field components picked up by the beam can be reduced by about a factor of three when the RF Wien filter is operated at $\beta=\SI{4}{m}$.
Thermal simulations show that the heat load in the different materials of the device, when operated with an input power of $\SI{1}{kW}$ is tolerable. Thus during operation, the mechanical accuracy of the device does not appear to substantially deteriorate the performance.
Forthcoming work will address improvements of the driving circuit, which will allow us to reach larger field values with the same input power. In addition, a novel semi-analytical approach to assess quantitatively the effect of mechanical tolerances on the electromagnetic performance of the RF Wien filter is presently being developed. In addition, spin tracking studies are planned to quantify the impact of the waveguide RF Wien filter on the systematic error of the planned proton and deuteron EDM experiments.
\section*{Acknowledgement}
This work has been performed in the framework of the JEDI (J{\"u}lich Electric Dipole Moment Investigations) collaboration. The authors would like to thank Martin Gaisser, Volker Hejny, Kirill Grigoryev, and J{\"o}rg Pretz for valuable comments on the manuscript.
\section*{References}
| {'timestamp': '2016-03-07T02:16:09', 'yymm': '1603', 'arxiv_id': '1603.01567', 'language': 'en', 'url': 'https://arxiv.org/abs/1603.01567'} |
\section{Introduction}
We work over the field of complex numbers which shall be denoted by
${\mathbb C}$.
The motivation for the results of this article lie in the study of
certain conjectures of Griffiths and Harris on the structure of curves
in hypersurfaces in ${\mathbb P}^4$. These conjectures can be viewed as a
generalisation of the Noether-Lefschetz theorem which we recall now.
\begin{thm}[Noether-Lefschetz theorem]\label{nlt}
Let $X\subset{\mathbb P}^3$ be a smooth, very general hypersurface of degree
$d\geq 4$. Then any curve $C\subset X$ is a complete intersection,
i.e., $C=X\cap{S}$ where $S\subset {\mathbb P}^3$ is a hypersurface.
\end{thm}
Inspired by the above theorem, Griffiths and Harris (see \cite{GH})
made a series of conjectures in decreasing strength about the
structure of $1$-cycles in $X$, the strongest one of which is the
following.
\begin{conj}\label{ghconj}
Let $X\subset {\mathbb P}^4$ be a general, smooth hypersurface of degree
$d\geq 6$, and $C\subset X$ be any curve. Then $C=X\cap{S}$ where
$S\subset {\mathbb P}^4$ is a surface.
\end{conj}
This conjecture was proved to be false by Voisin (see \cite{V}). In
fact, she showed that pursuing a certain line of thought, which we
describe below, weaker versions of this conjecture are also false.
Notice that in the Noether-Lefschetz situation, for a smooth curve $C\subset
X$, the normal bundle sequence
\begin{eqnarray}\label{nbs}
0 \to N_{C/X} \to N_{C/{\mathbb P}^3} \to {\mathcal O}_C(d) \to 0
\end{eqnarray}
splits. Griffiths and Harris investigated the splitting of this normal bundle sequence and
proved (see \cite{GH1}) the following characterisation
\begin{thm}\label{nbs1}
Let $X \subset {\mathbb P}^3$ be a smooth hypersurface of degree $d$ and let
$C\subset X$ be a smooth curve. Then the normal bundle sequence \eqref{nbs}
splits if and only if $C\subset X$ is a complete intersection.
\end{thm}
Unfortunately, the situation is not as simple in higher
dimensions. Let $X\subset {\mathbb P}^4$ be a smooth hypersurface of degree
$d$ and $C\subset X$ be any smooth curve. It is not hard to see that
if $C=X\cap{S}$ where $S\subset{\mathbb P}^4$ is a surface, then the normal
bundle sequence for the inclusion $C\subset X \subset {\mathbb P}^{4}$ splits.
Furthermore if $X_2$ denotes the first order thickening of $X$ in
${\mathbb P}^4$, then the splitting of the above normal bundle sequence
implies the splitting of the sequence
$$0\to N_{C/X} \to N_{C/X_2} \to N_{X/X_2}|_C \to 0.$$
By Lemma 1 in \cite{MPRV}, the splitting of this normal bundle
sequence implies that there exists $D\subset X_2$, a one dimensional
subscheme ``extending'' $C$ i.e., $C=D\cap{X}$.
It is this weaker splitting that Voisin investigates. In \cite{V},
she proves the following
\begin{prop}
Let $X\subset {\mathbb P}^4$ be a smooth hypersurface of degree $d>1$. There
exist smooth curves $C\subset X$ such that the normal bundle sequence
for the inclusions $C\subset X \subset X_2$ (and hence for the
inclusions $C\subset X\subset {\mathbb P}^4$) does not split. In fact, $C$
does not extend to $X_2$. Consequently, it is not an intersection of
the form $C=X\cap{S}$ for any surface $S\subset {\mathbb P}^4$.
\end{prop}
At this point, what would seem to be missing in a more complete
understanding of the conjecture of Griffiths and Harris is firstly,
whether the existence of the ``special'' curves of Voisin which
disprove it are indeed as special as they seem. Secondly, one would
like to know that in spite of this conjecture being false, whether
there is a ``weaker'' {\it generalised Noether-Lefschetz theorem}.
As explained below, {\it Arithmetically Cohen-Macaulay} (ACM) vector
bundles and subvarieties on hypersurfaces provide answers to both
these questions. Let $(X,{\mathcal O}_X(1))$ be a smooth polarised variety and
$\mathcal{F}$ be any coherent sheaf. Let
$\HH^i_\ast(X,\mathcal{F}):=\oplus_{\nu\in{\mathbb Z}}\HH^i(X,\mathcal{F}(\nu))$.
Recall that a vector bundle $E$ on $X$ is said to be ACM if
$\HH^i_\ast(X,E)\comment{:=\oplus_{\nu\in{\mathbb Z}}\HH^i(X,E(\nu))}=0$ for
$0< i<\dim{X}$. A subvariety $Z\subset X$ is said to be ACM if
$\HH^i_\ast(X,I_{Z/X})\comment{:=\oplus_{\nu\in{\mathbb Z}}\HH^i(X,I_{Z/X}(\nu))}=0$
for $0<i\leq \dim{Z}$. Furthermore, a codimension two subvariety
$Z\subset X$ is said to be {\it arithmetically Gorenstein}, if it is
the zero locus of a section of a rank two ACM bundle $E$ on $X$.
Let $X\subset {\mathbb P}^n$ be a smooth hypersurface. Given a codimension
two ACM subvariety $Z\subset X$ with dualising sheaf $\omega_{Z}$, one
can associate an ACM vector bundle $E$ of rank $r+1$ where $r$ is the minimal number
of generators of the $\HH^{0}_{\ast}({\mathbb P}^{n}, {\mathcal O}_{{\mathbb P}^{n}})$-module
$\HH^{0}_{\ast}(Z, \omega_{Z})$. The isomorphism
$ \omega_{Z}\cong \mbox{${\sE}{xt}$}^{1}_{{\mathcal O}_{X}}(I_{W/X}, K_{X}),$
where $K_{X}$ is the canonical bundle of $X$, gives rise to an isomorphism
$$ \HH^{0}(X,\oplus_{i=1}^{r}\omega_{Z}(a_{i})) \cong
\HH^{0}(X, \mbox{${\sE}{xt}$}^{1}_{{\mathcal O}_{X}}(I_{W/X},\oplus_{i=1}^{r} K_{X}(a_{i})) )\cong
Ext^{1}(I_{W/X}, \oplus_{i=1}^{r}K_{X}(a_{i})).$$
This isomorphism takes a minimal set of generators to a rank $r+1$ ACM bundle (the fact that it
is a bundle follows from the Auslander-Buchsbaum formula since $Y$ is locally Cohen-Macaulay)
and hence in the case when $Z$ is arithmetically Gorenstein, this is just Serre's construction.
Conversely (see \cite{Kleiman}), given any ACM bundle $E$ of rank $r+1$ on $X$
and $r$ general sections in sufficiently high degree, one can obtain
an ACM subvariety $Z\subset X$ such that $E$ is the ACM bundle
associated to it. In \cite{MPRV}, using the above correspondence, the following was proved:
\begin{prop}
Let $X\subset {\mathbb P}^n$ be a smooth hypersurface of degree $d>1$. If a
codimension two ACM subvariety $Z\subset X$ extends to $X_2$, then
the associated ACM vector bundle $E$ splits into a sum of line
bundles.
\end{prop}
Examples of non-split ACM bundles on smooth hypersurfaces of
degree $d>1$ can be found in \cite{BGS} (see \cite{MPRV} for
another construction). The existence of such bundles, together with the above proposition,
immediately implies that there exist plenty of curves in $X$, which
disprove the conjecture of Griffiths and Harris. It can be easily
checked that Voisin's curves are in fact ACM, thus providing a
conceptual explanation why the conjecture is false.
\comment{
As regards a generalised Noether-Lefschetz theorem: our viewpoint here
is that we are interested in statements which can be viewed as a
generalisation of certain consequences of Theorem \ref{nlt}. We
mention a few of those here.
Using the correspondence between divisors and line bundles, the
Noether-Lefschetz theorem can be viewed as a statement about the
Picard group of $X$. Namely that $\Pic(X)\cong{\mathbb Z}[{\mathcal O}_X(1)]$ for
$X\subset{\mathbb P}^3$ a smooth, very general hypersurface of degree $d\geq
4$. In particular, any ACM line bundle on $X$ is the restriction of a
line bundle on the ambient projective space ${\mathbb P}^3$ (see \cite{R} for
more details). For curves in threefolds, though there is no such
correspondence with rank two bundles, one could restrict one's
attention to curves which by Serre's correspondence are associated to
rank two bundles and ask whether every ACM rank two vector bundle is a
restriction of a rank two bundle on the ambient projective
space. Since every such extension is also ACM, by Horrocks' theorem,
it would be a sum of line bundles and hence so would the original
bundle on X.
Theorem \ref{nlt} implies (see \cite{Beauville} for details) that a
general, degree $d$ homogeneous polynomial in four variables for
$d\geq 4$, cannot be expressed as the determinant of a minimal
$k\times k$, $2\leq k\leq d$, matrix whose entries are homogeneous
polynomials in those four variables. Here by minimal we mean that the
constant polynomial entries in the matrix are all zero. This is
equivalent to the fact that a general hypersurface $X\subset{\mathbb P}^3$ of
degree $d\geq 4$, does not support any non-trivial ACM line bundle. The
question regarding the triviality of rank two ACM bundles also admits
a similar reformulation in terms of matricial representations of the
defining polynomial of a general hypersurface $X\subset{\mathbb P}^4$ (for
details, see \cite{R} for instance).
Finally, a natural question which arises in the context of Theorem
\ref{nlt} being false in case of $X\subset{\mathbb P}^4$, is whether it is
possible to characterise complete intersection curves $C\subset X$.
}
The following theorem which can be viewed as a weak generalisation of
Theorem \ref{nlt} was proved in \cite{MPR2} and \cite{R}.
\begin{thm}\label{weaknlt}
Let $X\subset{\mathbb P}^4$ be a smooth, general hypersurface of degree
$d\geq 6$ with defining polynomial $f\in \HH^0({\mathbb P}^4,
{\mathcal O}_{{\mathbb P}^4}(d))$. The following equivalent statements hold true.
\begin{enumerate}
\item\label{weaknlt1} Any rank two ACM bundle on $X$ is a sum of line
bundles.
\item\label{weaknlt2} $f$ cannot be expressed as the {\it Pfaffian} of
a minimal skew-symmetric matrix of size $2k\times 2k$, $2\leq k \leq
d$, whose entries are homogeneous polynomials in five variables.
\item\label{weaknlt3} A curve $C\subset X$ is a complete intersection
if and only if $C$ is {\it arithmetically Gorenstein} i.e., $C$ is
the zero locus of a non-zero section of a rank two ACM bundle on
$X$.
\end{enumerate}
\end{thm}
ACM bundles on hypersurfaces have been studied earlier. To add some
history, Kleppe showed in \cite{Kleppe} that any rank two ACM bundle
$E$ on a smooth hypersurface $X\subset{\mathbb P}^n$, $n\geq 6$, splits as a
sum of line bundles. When $n=5$, $3\leq d \leq 6$ or $n=4$ and $d=6$,
and $X$ is a general smooth hypersurface, the above splitting result
was first obtained by Chiantini and Madonna (see \cite{CM1, CM2}).
\comment{while when $n=4$ and $d=5$, the infinitesimal rigidity was first
proved in \cite{CM3}.} The first general results on ACM bundles, which
subsumed these results \comment{of \cite{Kleppe} and \cite{CM1},} were
first proved in \cite{MPR1}. These in turn led to the proof of Theorem \ref{weaknlt},
as given in \cite{MPR2}.
\comment{
One of the main results of that paper is the
following which we record here for future reference.
\begin{thm}
Let $X\subset{\mathbb P}^{5}$ be a smooth, general hypersurface of degree
$d\geq 3$. Then any ACM vector bundle of rank two on $X$ splits as a
sum of line bundles.
\end{thm}
}
An important ingredient of the proof of Theorem \ref{weaknlt} used in
\cite{R}, is the following theorem of Green (see \cite{MG}) and Voisin
(unpublished).
\begin{thm}\label{aj}
Let $X\subset {\mathbb P}^4$ be a smooth, general hypersurface of degree
$d\geq 6$. Then the image of the Abel-Jacobi map
$$\CH^2(X)_{\mathbb Q} \to J^2(X)_{\mathbb Q}$$ from the (rational) Chow group of
codimension two cycles on $X$ to the intermediate Jacobian modulo
torsion, is zero.
\end{thm}
\comment{
As mentioned earlier, Conjecture \ref{ghconj} is one in a series of
conjectures made by Griffiths and Harris. The above theorem partially
answers another in this series.
}
Notice that Noether-Lefschetz type questions can be asked more
generally for complete intersection subvarieties in projective
space. Theorem \ref{nlt} for instance, is well understood in a more
general situation (see \cite{SGA}).
\begin{thm}
Let $Y$ be a smooth projective threefold and $L$ a sufficiently ample,
base point free line bundle on $Y$. Let $X\in\vert L\vert$ be a
smooth, very general member of the linear system $\vert L\vert$. Then
the restriction map $\iota^\ast:\Pic(Y)\to \Pic(X)$ is an isomorphism.
\end{thm}
In the above theorem, we need $L$ sufficiently ample to imply that the
map $\HH^2({\mathcal O}_Y) \to \HH^2({\mathcal O}_X)$ is not surjective. In case
$Y={\mathbb P}^3$, this translates to the condition $L={\mathcal O}_{{\mathbb P}^3}(d)$ with
$d\geq 4$. In particular, this gives an extension of the
Noether-Lefschetz theorem to complete intersection surfaces of
multi-degree $(d_1,\cdots,d_{n-2})$ in ${\mathbb P}^n$ (here the corresponding
condition on the multi-degree of $X$ is $\sum_{i=1}^{n-2}d_i\geq n+1$).
Using (the infinitesimal version of) this theorem and some explicit
analysis, \comment{(in the case where $X$ is the intersection of two quadrics in
${\mathbb P}^4$ which is the only case where $X$ is not a hypersurface and
the multi-degree does not satisfy the above inequality),} Harris and
Hulek (see \cite{HH}) extended Theorem \ref{nbs1} to smooth complete
intersection surfaces.
\comment{
\begin{thm}
Let $X\subset {\mathbb P}^n$ be a smooth complete intersection surface of
multidegree $(d_1,\cdots,d_{n-2})$ and $C\subset X$ be a smooth
curve. Then the normal bundle sequence
$$0 \to N_{C/X} \to N_{C/{\mathbb P}^n} \to \bigoplus_{i=1}^{n-2}{\mathcal O}_C(d_i)
\to 0$$ splits if and only if $C\subset X$ is a complete intersection.
\end{thm}
}
Finally, Green and M\"uller-Stach (see \cite{G-MS}) have proved a
generalisation of Theorem \ref{aj} to complete intersection
subvarieties of sufficiently high multidegree.
\begin{thm}\label{gms}
Let $X$ be a general complete intersection subvariety in ${\mathbb P}^n$ of
sufficiently high multi-degree and dimension at least three.\comment{Let
$\CH^2(X)_{\mathbb Q}$ be the (rational) Chow group of codimension $2$
cycles modulo rational equivalence.} The image of the cycle class map
$\CH^2(X)_{\mathbb Q} \to \HH^{4}_{\mathcal{D}}(X,{\mathbb Q})$ into Deligne
cohomology is just the image of the hyperplane class in ${\mathbb P}^n$. In
particular, the image of the Abel-Jacobi map of $X$ is contained in
the torsion points of $\JJ^2(X)$.
\end{thm}
In view of these above two theorems, it is natural to seek extensions
of Theorem \ref{weaknlt} to complete intersection subvarieties of
projective space.
\comment{Let $X\subset{\mathbb P}^n$ be a complete intersection with multi-degree
$(d_1,\cdots,d_k)$. If $d_i\gg 0$ $\forall$ $1\leq i \leq k$, then we
say that $X$ is a complete intersection of {\it sufficiently large
multi-degree}.
}
\section{Main results}
The main results of this note are the following.
\begin{thma}\label{hdims}
Let $X\subset {\mathbb P}^n$ be a general smooth complete intersection
subvariety of dimension at least four and sufficiently high
multidegree. Then any ACM vector bundle of rank two on $X$ is a direct
sum of line bundles.
\end{thma}
This can be viewed as a generalisation to complete intersections of
the main result of \cite{MPR1}.
Let $d_1\leq d_2\leq \cdots\leq d_{n-3}$ be a sequence of positive
integers, with $d_i\gg 0$ for $1\leq i \leq n-3$ and $d_{n-3}>
\mbox{max}\{d_i+d_j\}$ for $1\leq i,~j \leq n-4$. Recall that a bundle
$E$ is said to be {\it normalised} if $h^0(E(-1))=0$ but $h^0(E)\neq
0$. With this notation, we have
\begin{thmb}\label{3fold}
Let $X\subset {\mathbb P}^n$ be a general smooth complete intersection
threefold of sufficiently high multidegree $(d_1,\cdots,d_{n-3})$ as
above. Then any arithmetically Cohen-Macaulay, normalised rank two
vector bundle $E$ on $X$ is a direct sum of line bundles provided
$c_1(E)< d_{n-3}-1$.
\end{thmb}
Theorem A is obtained as a consequence of Theorem B. A word about the
inequality satisfied by the first Chern class and about the condition
$d_i\gg 0$ in the above theorems: since any bundle $E$ splits iff
$E(m):=E\otimes{\mathcal O}_X(m)$ splits for some $m$, we may assume that $E$
is normalised. Madonna showed in \cite{Madonna} that on any smooth
three dimensional complete intersection $X\subset {\mathbb P}^n$ a normalised
rank two ACM bundle splits as a direct sum of two line bundles unless
$-\sum_{i=1}^{n-3}d_i+n-1\leq c_1(E) \leq \sum_{i=1}^{n-3}d_i+3-n$. We
notice that when $n=4$ and $X$ is a general hypersurface of degree
$d\gg0$, combining our result and Madonna's bound, it follows that the
only possible first Chern class of a normalised and indecomposable
rank two ACM bundle on $X$ is $d-1$. This case is classical. Indeed,
the existence of an indecomposable ACM bundle of rank two on a general
smooth hypersurface in ${\mathbb P}^4$ is equivalent to the fact that a
general homogeneous polynomial in five variables can be obtained as
the Pfaffian of a (minimal) skew-symmetric matrix of linear forms. It
is well known (see \cite{Beauville}), by a simple dimension count,
that when $d\geq 6$, this is not possible. Thus Theorem B can be seen
as a generalisation of Theorem \ref{weaknlt} above. Finally, to prove
Theorem B, we argue as in \cite{R} for the cases of general three
dimensional hypersurfaces of degree at least six. Indeed we use here
the result of Green and M\"uller-Stach (Theorem \ref{gms}) which
generalised the corresponding result of Green and Voisin (Theorem
\ref{aj}). In doing so, we need to restrict to the cases of complete
intersections of sufficiently high multi-degree and dimension at least $3$.
We have then the following interesting:
\begin{corc}
Let $X\subset {\mathbb P}^n$ be a general smooth complete intersection
subvariety of sufficiently high multidegree.
\begin{enumerate}
\item Any arithmetically Gorenstein subvariety $T\subset X$ of
codimension two is a complete intersection in $X$ provided
$\dim{X}\geq 4$.
\item Suppose $X$ is a threefold, and $C\subset X$ is any
arithmetically Gorenstein curve. Then $C$ is the intersection of $X$
with a codimension two subscheme $S\subset {\mathbb P}^n$ if and only if
$C$ is a complete intersection in $X$. In addition, if the
rank two bundle $E$ associated to $C$ via Serre's correspondence is normalised,
and $c_1(E)<d_{n-3}-1$, then $C$ is a complete intersection in $X$.
\end{enumerate}
\end{corc}
Finally, we should mention that another motivation for the questions
on ACM vector bundles comes from the conjectures of
Buchweitz-Gruel-Schreyer (see \cite{BGS}, Conjecture B) on the
triviality of low rank ACM bundles on hypersurfaces (see \cite{R} for
more details).
\comment{
From the point of view of algebraic cycles, complete intersections (at
least those of sufficiently high multidegree) behave like
hypersurfaces; hence the result is not surprising. Indeed, we would
expect that the theorem above can be extended to other possible values
of $c_1$ when $X$ is a threefold.
A complete intersection curve $C$ is clearly arithmetically
Gorenstein. It seems likely in view of the result on hypersurfaces and
the above corollary that the property of arithmetic Gorenstein-ness
characterizes complete intersection curves when $X$ is has dimension
three for all possible values of $c_{1}$.
An outline of the proof of the Theorem B (from which Theorem A will
follow) is as follows. We shall suppose the contrary: that a
normalised, indecomposable ACM bundle $E$ of rank two exists on a
general $X$ as above. In such a situation (see section 3 of
\cite{MPR1} for details), there exists a rank two bundle $\mathcal{E}$
on the universal hypersurface $\mathcal{X} \subset Y \times S'$ where
$S'$ is a Zariski open subset of $S$, the moduli space of smooth,
degree $d$ hypersurfaces of $Y$, such that for a general point $s\in
S'$, $\mathcal{E}_{|X_s}$ is normalised, ACM and
non-split. Furthermore, from the construction of this family, one sees
that there exists a family of curves $\mathcal{C} \to S'$ such that
$\mathcal{C}_s$ is the zero locus of a section of
$\mathcal{E}_{|X_s}$. Let $\mathcal{Z}$ be a family of $1$-cycles with
fibre $\mathcal{Z}_s:=d\mathcal{C}_s-lD_s$ where, as before, $D_s$ is
plane section of $X_s$ and $l=l(s)$ is the degree of
$\mathcal{C}_s$. We shall show that under the hypotheses above, the
infinitesimal invariant of the normal function $\nu_{\mathcal{Z}}$,
associated to the cycle $\mathcal{Z}$, and denoted by
$\delta\nu_{\mathcal{Z}}$ (see section \ref{iinf} for definitions) is
not identically zero. This contradicts Theorem \ref{gms}; hence our
assumption that there exists a rank two vector bundle $\mathcal{E}$
satisfying the properties above does not hold. This proves Theorem B.
}
The present paper builds on results proved and techniques developed in
\cite{MPR1, MPR2, R, Wu1, Wu2}, some of which have been included here
for the sake of completeness. The non-degeneracy of the infinitesimal
invariant in particular, is shown by refining Xian Wu's proof in
\cite{Wu1}.
\section{Acknowledgements}
The authors would like to thank the referee for comments which helped make the paper more readable.
\section{The infinitesimal invariant associated to a normal
function}\label{iinf}
Let $Y$ be a smooth projective variety of dimension $2m$, $m\geq 1$.
Let $\mathcal{X} \to S$ be the universal family of smooth, degree $d$
hypersurfaces of in $Y$. Let $\mathcal{C}\subset\mathcal{X}$ be a
family of codimension $m$ subvarieties over $S$. If $l=l(s)$
is the degree of $C_s$ and $D_s$ is a codimension $m$ linear section
(i.e. $D_s$ is an intersection of $m$ hyperplanes in $X_s$), then the
family of cycles $\mathcal{Z}$ with fibre
$\mathcal{Z}_s:=d~\mathcal{C}_s-lD_s$ for $s\in S$ defines a
fibre-wise null-homologous cycle, i.e. an element in
$\CH^m(\mathcal{X}/S)_{hom}$. Let $\mathcal{J}:=\{J(X_s)\}_{s\in S}$
be the family of intermediate Jacobians. In such a situation,
Griffiths (see \cite{G}) has defined a holomorphic function
$\nu_{\mathcal{Z}}:S \to \mathcal{J}$, called the {\it normal
function}, which is given by $\nu_{\mathcal{Z}}(s)=\mu_s(Z_s)$ where
$\mu_s:\CH^m(X_s)_{\mbox{\small{hom}}} \to J(X_s)$ is the {\it
Abel-Jacobi} map from the group of null-homologous cycles to the
intermediate Jacobian.
This normal function satisfies a ``quasi-horizontal'' condition (see
\cite{Vbook}, Definition 7.4). Associated to the normal function
$\nu_{\mathcal{Z}}$ above, Griffiths (see \cite{G1} or \cite{Vbook}
Definition 7.8) has defined the infinitesimal invariant
$\delta\nu_{\mathcal{Z}}$. Later Green \cite{MG} generalised this
definition and showed that Griffiths' original infinitesimal invariant
is just one of the many infinitesimal invariants that one can
associate to a normal function. For a point $s_0\in S$, let
$X=X_{s_0}$, $C:=\mathcal{C}_{s_0} \subset X$ and $D:=D_{s_0}$. Green
showed that in particular $\delta\nu_{\mathcal{Z}}(s_0)$ is an element
of the dual of the middle cohomology of the following (Koszul) complex
\begin{equation}\label{koszul}
\ext{2}\HH^1(X,T_X)\otimes\HH^{m+1,m-2}(X)\to
\HH^1(X,T_X)\otimes\HH^{m,m-1}(X)\to
\HH^{m-1,m}(X).
\end{equation}
We now specialise to the case $m=2$ where $X\subset Y$ is a smooth
hypersurface of dimension $3$ and $C\subset X$ is a curve of degree
$l$. Then $Z:=dC-lD$ is a nullhomologous $1$-cycle with support
$W:=C\bigcup D$.
At a point $s\in S$, this infinitesimal invariant is therefore a functional
$$\delta\nu_{\mathcal{Z}}(s):
\ker\left(\HH^1(X,T_X)\otimes\HH^1(X,\Omega^2_X) \to
\HH^2(X,\Omega^1_X)\right)\to{\mathbb C}.$$
Consider the composite map
$$\gamma: \HH^1(X,T_X)\otimes\HH^1(X,\mathcal{I}_{W/X}\otimes\Omega^2_X) \to
\HH^1(X,T_X)\otimes\HH^1(X,\Omega^2_X) \to \HH^2(X,\Omega^1_X).$$
By abusing notation, we will let
$$\delta\nu_{\mathcal{Z}}(s):\ker\gamma\to {\mathbb C}$$
denote the composite map.
On the other hand, starting with the short exact sequence
$$ 0 \to I_{W/X}\to {\mathcal O}_{X} \to {\mathcal O}_{W} \to 0,$$
and tensoring with $\Omega^{1}_{X}$, yields a
long exact sequence of cohomology
$$ \cdots \to \HH^1(\Omega^1_X) \to \HH^1(\Omega^1_X\otimes{\mathcal O}_W)
\to\HH^2(I_{W/X}\otimes\Omega^1_X) \to
\HH^2(\Omega^1_X)
\to 0 .$$
Combining this sequence with the Koszul complex \eqref{koszul},
we get a commutative diagram:
\begin{equation}\label{formula}
\begin{diagram}{ccccccccc}
& & & & \HH^1( T_X)\otimes\HH^1(I_{W/X}\otimes\Omega_X^2) & &
& &
\\
& & & & \downarrow{\beta} & \searrow{\gamma}& & & \\
0 & \to & \frac{\HH^1(\Omega^1_X\otimes{\mathcal O}_W)}{\HH^1(\Omega^1_X)}
&
\by{\lambda} &\HH^2(I_{W/X}\otimes\Omega^1_X) & \to &
\HH^2(\Omega^1_X) &
\to & 0 \\
& & \downarrow{\chi} & & & & & & \\
& & {\mathbb C} & & & & & & \\
\end{diagram}
\end{equation}
where $\chi$ is given by integration over the cycle $Z$ and $\beta$ is
a Koszul map. As a result, one has an induced map
$$\ker\gamma \to \frac{\HH^1(\Omega^1_X\otimes{\mathcal O}_W)}{\HH^1(\Omega^1_X)} .$$
The following is the main result that we shall use in this paper.
\begin{thm}[Griffiths \cite{G1,G2}]
Let $\nu_{\mathcal{Z}}$ be the normal function as described
above. \comment{Consider the following diagram:
\begin{equation
\begin{diagram}{ccccccccc}
& & & & \HH^1( T_X)\otimes\HH^1(I_{W/X}\otimes\Omega_X^2) & &
& &
\\
& & & & \downarrow{\beta} & \searrow{\gamma}& & & \\
0 & \to & \frac{\HH^1(\Omega^1_X\otimes{\mathcal O}_W)}{\HH^1(\Omega^1_X)}
&
\by{\lambda} &\HH^2(I_{W/X}\otimes\Omega^1_X) & \to &
\HH^2(\Omega^1_X) &
\to & 0 \\
& & \downarrow{\chi} & & & & & & \\
& & {\mathbb C} & & & & & & \\
\end{diagram}
\end{equation}
where $\chi$ is given by integration over the cycle $Z$.} Then
$\delta\nu_{\mathcal{Z}}(s_0)$, the infinitesimal invariant evaluated at
the point $s_0\in S$, is the composite
$$ \ker{\gamma}\to \frac{\HH^1(\Omega^1_X\otimes{\mathcal O}_W)}
{\HH^1(\Omega^1_X)}\by{\chi} {\mathbb C}.$$
\end{thm}
\comment{We identify the tangent space $T_{s}=\HH^0(X,{\mathcal O}_X(d))$ and
$\HH^1(I_{W/X}\otimes T_X)$ with their respective images in $\HH^1(
T_X)$.}
\begin{remark}
The map $\chi$ can be understood as follows: Since $D$ is a
general plane section of $X$, by Bertini $C\cap{D}=\emptyset$. Thus
${\mathcal O}_W\cong{\mathcal O}_C\oplus{\mathcal O}_D$ and so
$$\HH^1(\Omega^1_X\otimes{\mathcal O}_W)\cong \HH^1(\Omega^1_X\otimes{\mathcal O}_C)
\oplus \HH^1(\Omega^1_X\otimes{\mathcal O}_D). $$ For any irreducible curve
$T\subset X$, let $r_T:\HH^1(\Omega^1_X\otimes{\mathcal O}_T) \to
\HH^1(\Omega^1_T)\cong {\mathbb C}$ be the natural restriction map.
For any element $(a,b)\in
\HH^1(\Omega^1_X\otimes{\mathcal O}_W)$, $\chi(a,b):=dr_C(a)-lr_D(b) \in
{\mathbb C}$. Clearly, this map factors via the quotient
$\HH^1(\Omega^1_X\otimes{\mathcal O}_W)/\HH^1(\Omega^1_X).$
\end{remark}
The main result in this situation is Theorem \ref{gms} which implies in
particular that the normal function is zero on an open subset of the
parameter space. By Theorem 1.1 in \cite{MG} (or Proposition 1.2.3 of \cite{Wu1}),
we then have the following.
\begin{thm}\label{gmseq}
Let $X$ be a general complete intersection subvariety in ${\mathbb P}^n$ of
sufficiently high multi-degree and dimension at least
three.
\comment{Let $\CH^2(X)\otimes{\mathbb Q}$ be the (rational) Chow group
of codimension $2$ cycles modulo rational equivalence. The image of
the cycle class map $\CH^2(X)\otimes{\mathbb Q} \to
\HH^{4}_{\mathcal{D}}(X,{\mathbb Q})$ into the (rational) Deligne cohomology
is just the image of the hyperplane class in ${\mathbb P}^n$. In
particular, the image of the Abel-Jacobi map of $X$ is contained in
the torsion points of $\JJ^2(X)$. Equivalently,} If $\mathcal{Z}\to
S$ is a family of codimension two, degree zero cycles contained in the
universal complete intersection $\mathcal{X}\subset {\mathbb P}^n\times S$,
then the infinitesimal invariant $\delta\nu_{\mathcal{Z}}$ associated
to $\mathcal{Z}$ vanishes at a general point $s\in S$.
\end{thm}
\comment{
\section{A criterion for the non-degeneracy of the infinitesimal invariant}
As mentioned in the introduction, we shall first assume that contrary
to the statement of Theorem B, a general, smooth complete intersection
threefold $X\subset {\mathbb P}^{n+3}$ supports an indecomposable ACM rank
two vector bundle with first Chern class $\alpha< d_{n-3}-1$. This
implies that there exists a rank two bundle $\mathcal{E}$ on the
universal hypersurface $\mathcal{X} \subset Y \times S'$ where $S'$ is
a Zariski open subset of $S$, the moduli space of smooth, degree $d$
hypersurfaces of $Y$, such that for a general point $s\in S'$,
$\mathcal{E}_{|X_s}$ is normalised, indecomposable, ACM and its first
Chern class $\alpha_s=\alpha$ satisfies the inequality
above. Furthermore, from the construction of this family, one sees
that there exists a family of curves $\mathcal{C} \to S'$ such that
$\mathcal{C}_s$ is the zero locus of a section of
$\mathcal{E}_{|X_s}$. Let $\mathcal{Z}$ be a family of $1$-cycles with
fibre $\mathcal{Z}_s:=d\mathcal{C}_s-lD_s$ where, as before, $D_s$ is
plane section of $X_s$ and $l=l(s)$ is the degree of
$\mathcal{C}_s$. We shall show that under the hypotheses above, the
infinitesimal invariant of the normal function $\nu_{\mathcal{Z}}$,
associated to the cycle $\mathcal{Z}$, and denoted by
$\delta\nu_{\mathcal{Z}}$ (see section \ref{iinf} for definitions) is
not identically zero.
We continue with notation in the above section where
\begin{prop}
Let $Y$ be a smooth projective variety with $\dim{Y}=4$, $X\in
\vert{\mathcal O}_Y(d)\vert$ a smooth general hypersurface of degree $d$. Let
$\mathcal{X} \subset Y\times S$ be a family of smooth hypersurfaces
and $\mathcal{C}\subset \mathcal{X}$ be a family of curves over
$S$. Let $\mathcal{Z}\to S$ denote the family of codimension two,
fibre-wise null-homologous cycles. For a point $s\in S$ parametrising
a smooth hypersurface $X\subset Y$, we shall denote by
$C:=\mathcal{C}_s$ and $W:=C\bigcup D$ where $D:=D_s\subset Y$ is a
codimension two linear section. Let }
\section{ACM bundles on a smooth subvariety $X\subset Y$}
Let $X=\bigcap_{i=1}^{n-3}Y_i$ be a general complete intersection of
smooth hypersurfaces $Y_i\subset{\mathbb P}^n$ of degree $d_i$. Let $E\to X$
be an indecomposable, normalised ACM bundle of rank two. In this
section, we shall establish several lemmas which will enable us to
prove the non-degeneracy of the infinitesimal invariant coming from a
family of arithmetically Gorenstein curves. The criterion is a
refinement of Wu's criterion (see \cite{Wu1}).
\begin{lemma}\label{c1inequality}
Let $E$ be as above with first Chern class $\alpha$. Then the zero
locus of every non-zero section of $E$ has codimension $2$ in $X$. If
$C\subset X$ is the zero locus of a section of $E$, then we have the
exact sequence
\begin{eqnarray}\label{serre}
0\to {\mathcal O}_X(-\alpha) \to E^\vee \to I_{C/X} \to 0.
\end{eqnarray}
Furthermore, $E$ is $((\sum_{i=1}^{n-3}d_i)-n+3-\alpha)$--regular,
$n-1-\sum_{i=1}^{n-3}d_i\leq \alpha\leq (\sum_{i=1}^{n-3}d_i)-n+3$, and
$K_C={\mathcal O}_C(\sum_{i=1}^{n-3}d_i-n-1+\alpha)$.
\end{lemma}
\begin{proof}
See \cite{R} for the proof of the first part of the Lemma. The
regularity $\rho$ of $E$ can be computed easily (see {\it
op.~cit.}). For the inequality satisfied by $\alpha$, see
\cite{Madonna}. \comment{ Thus the upper bound for $\rho$ is easily
seen. For the lower bound and the rest of the proof, see proof of
Lemma 3.3 in \cite{MPR1}.}
\end{proof}
\noindent{\bf Notation:} Let $d_1\leq d_2\leq \cdots\leq d_{n-3}$ be a
sequence of sufficiently large positive integers (i.e. $d_i\gg 0$), and
$d_{n-3}>\mbox{max}\{d_i+d_j\}$ for $1\leq i,~j\leq n-4$. For the rest of the
paper, we will denote by $$Y:=\bigcap_{i=1}^{n-4}Y_i, \hspace{2mm}
d=d_{n-3} \hspace{3mm} \mbox{and} \hspace{3mm} X\in|{\mathcal O}_Y(d)|$$ a
general member. Also, we will let $E$ denote a normalised, indecomposable ACM
bundle of rank two on $X$ and $C\subset X$ to denote the zero locus of a
non-zero section of $E$ (which is a curve by Lemma
\ref{c1inequality}).
\subsection{Towards the surjectivity of the map $\chi$}
The main result of this subsection is the surjectivity of the map $\chi$.
This is achieved by identifying a subspace of $\HH^{1}(W, \Omega^{1}_{X|W})$,
restricted to which $\chi$ is surjective (see Corollary \ref{chiisonto}). The proof
crucially depends on the following:
\begin{lemma}\label{firststep}
Let $C\subset X\subset Y\subset{\mathbb P}^{n}$ be as above. Then the natural map
$$\HH^{1}(X,\Omega^{2}_{{\mathbb P}^{n}}(d)_{|X}) \to\HH^{1}(C,\Omega^{2}_{{\mathbb P}^{n}}(d)_{|C}) $$
is zero.
\end{lemma}
Before we prove this lemma, we shall need several results which we shall prove now. Let
\begin{equation}\label{ses1}
0 \to G_Y \to F_{0,Y}\to E \to 0,
\end{equation}
be a minimal resolution of $E$ by vector bundles on $Y$. So
$F_{0,Y}:=\bigoplus{\mathcal O}_Y(-a_i)$, where $a_i\geq 0$ and the kernel
$G_Y$ is ACM. The fact that $G_Y$ is a bundle follows from the
Auslander-Buchsbaum formula (see \cite{Eisenbud}, Chapter
19).
Applying $\mbox{${\sH}{om}$}_{{\mathcal O}_Y}(\cdot\,,{\mathcal O}_Y)$ to sequence (\ref{ses1}),
we get
\begin{equation}\label{dualres}
0\to F_{0,Y}^\vee \to G_Y^\vee \to E^\vee(d) \to 0.
\end{equation}
(see \cite{MPR1} where it is proved when $X\subset{\mathbb P}^{n}$ is a hypersurface:
the same argument works on replacing ${\mathbb P}^{n}$ by $Y$)
\begin{lemma}[see also \cite{R}]\label{split}
There exists a commutative diagram
\begin{equation}
\begin{array}{ccccccccc} \label{c1}
0 & \to & F_{0,Y}^\vee & \to & G_Y^\vee & \to & E^\vee(d)
& \to & 0 \\
& & \downarrow{\phi} & & \downarrow & & \downarrow{s^\vee} & & \\
0 & \to & {\mathcal O}_{Y} & \to & {\mathcal O}_{Y}(d) & \to & {\mathcal O}_{X}(d) & \to & 0
\\
\end{array}
\end{equation}
where under the isomorphism $$\Hom(F_{0,Y}^\vee,
{\mathcal O}_Y)\cong\HH^0(F_{0,Y})\cong\HH^0(E)
\comment{\cong\Ext^1_{{\mathcal O}_Y}(E^\vee(d),{\mathcal O}_Y)}, \hspace{3mm} \phi
\mapsto s. $$ In addition, $F_{0,Y}^\vee\by{\phi}{\mathcal O}_Y$ is a split
surjection (i.e. $\phi$ is the projection onto one of the factors).
\end{lemma}
\begin{proof}
We consider the following push-out diagram:
\begin{equation}
\begin{array}{ccccccccc}
0 & \to & F_{0,Y}^\vee & \to & G_Y^\vee & \to & E^\vee(d)
& \to & 0 \\
& & \downarrow{\phi} & & \downarrow & & || & & \\
0 & \to & {\mathcal O}_{Y} & \to & \mathcal{K} & \to & E^\vee(d) & \to & 0
\\
\end{array}
\end{equation}
Since $F_{0,Y}^\vee=\oplus{\mathcal O}_Y(a_i)$, $a_i\geq 0$, any such diagram
corresponds to a section $\phi\in\HH^0(F_{0,Y})$.
Next consider the pull-back diagram:
\begin{equation}
\begin{array}{ccccccccc}
0 & \to & {\mathcal O}_{Y} & \to & \mathcal{K} & \to & E^\vee(d) & \to & 0\\
\comment{0 & \to & F_{0,Y}^\vee & \to & G_Y^\vee & \to & E^\vee(d)
& \to & 0 \\}
& & || & & \downarrow & & \downarrow{s^\vee} & & \\
0 & \to & {\mathcal O}_{Y} & \to & {\mathcal O}_{Y}(d) & \to & {\mathcal O}_{X}(d) & \to & 0
\\
\end{array}
\end{equation}
Any such diagram corresponds to a section $s\in\HH^0(E)$. Since
$\HH^0(F_{0,Y})\cong \HH^0(E)$, there is a bijective correspondence
between the diagrams above. Combining them, we get the desired
commutative diagram. The morphism $\phi$ is a split surjection since
$a_i\geq 0,$ $\forall~ i$.
\end{proof}
Tensoring the exact sequence (\ref{dualres}) with $\Omega^2_{{\mathbb P}^n}$
and taking cohomology, we get
\begin{equation}\label{cohdualres}
\to \HH^1(E^\vee(d)\otimes\Omega^2_{{\mathbb P}^n}) \to
\HH^2(F^\vee_{0,Y}\otimes\Omega^2_{{\mathbb P}^n}) \to
\HH^2(G^\vee_Y\otimes\Omega^2_{{\mathbb P}^n}) \to
\end{equation}
\begin{lemma}\label{mapiszero}
The map $\HH^2(F^\vee_{0,Y}\otimes\Omega^2_{{\mathbb P}^n}) \to
\HH^2(G^\vee_Y\otimes\Omega^2_{{\mathbb P}^n})$ in diagram (\ref{cohdualres}) is the
zero map.
\end{lemma}
\begin{proof}
Let $F_1\to G_Y^\vee$ be a surjection from a sum of line bundles on
${\mathbb P}^n$ to $G_Y^\vee$, induced by a minimal set of generators of
$G_Y^\vee$. Let $F_0$ be a sum of line bundles on ${\mathbb P}^n$ such that
$F_0\otimes{\mathcal O}_Y=F_{0,Y}$. \comment{Then
$$\HH^2(F^\vee_{0}\otimes\Omega^2_{{\mathbb P}^n}) \cong
\HH^2(F^\vee_{0,Y}\otimes\Omega^2_{{\mathbb P}^n}).$$} The map
$F_{0,Y}^\vee \to G_Y^\vee$ lifts to a map $\Phi:F_0^\vee \to F_1$,
since $G_Y^\vee$ is ACM. Hence we have a commuting square
\[
\begin{array}{ccc} \label{c3}
\HH^2(F^\vee_{0}\otimes\Omega^2_{{\mathbb P}^n}) & \to &
\HH^2(F_{1}\otimes\Omega^2_{{\mathbb P}^n}) \\
\downarrow{\cong} & & \downarrow \\
\HH^2(F^\vee_{0,Y}\otimes\Omega^2_{{\mathbb P}^n}) & \to &
\HH^2(G^\vee_Y\otimes\Omega^2_{{\mathbb P}^n}) \\
\end{array}
\]
To prove the lemma, it is enough to prove that the top horizontal map,
which is given by the matrix $\Phi$, is zero. In other words, we need
to show that $\Phi$ has no non-zero scalar entries.
Suppose there was such an entry, then we would have a diagram
\[
\begin{array}{ccccccccc} \label{conmin}
& & 0 & & 0 & &
& & \\
& & \downarrow & & \downarrow & & & & \\
& & {\mathcal O}_Y & = & {\mathcal O}_Y & &
& & \\
& & \downarrow & & \downarrow & & & & \\
0 & \to & F_{0,Y}^\vee & \to & G_Y^\vee & \to & E^\vee(d)
& \to & 0 \\
& & \uparrow\downarrow & & \downarrow & & || & & \\
0 & \to & \bar{F} & \to & \bar{G} & \to & E^\vee(d)
& \to & 0 \\
& & \downarrow & & \downarrow & & & & \\
& & 0 & & 0 & &
& & \\
\end{array}
\]
Here $\bar{F}$ (resp. $\bar{G}$) is defined as the cokernel of the
inclusion ${\mathcal O}_Y\hookrightarrow F_{0,Y}^\vee$ (resp. ${\mathcal O}_Y\hookrightarrow G_{Y}^\vee$).
Applying $\mbox{${\sH}{om}$}_{{\mathcal O}_Y}(.\, ,{\mathcal O}_Y)$ to the diagram above, we get
\[
\begin{array}{ccccccccc} \label{dualconmin}
& & 0 & & 0 & &
& & \\
& & \downarrow & & \downarrow & & & & \\
0 & \to & \bar{G}^\vee & \to & \bar{F}^\vee & \to &
Ext^1_{{\mathcal O}_Y}(E^\vee(d),{\mathcal O}_Y)=E
& \to & \\
& & \downarrow & & \downarrow & & || & & \\
0 & \to & G_Y & \to & F_{0,Y} & \to &
Ext^1_{{\mathcal O}_Y}(E^\vee(d),{\mathcal O}_Y)=E
& \to & 0 \\
& & \downarrow & & \uparrow\downarrow & & & & \\
& & {\mathcal O}_Y & = & {\mathcal O}_Y & &
& & \\
& & \downarrow & & \downarrow & & & & \\
& & & & 0 & &
& & \\
\end{array}
\]
Since ${\mathcal O}_Y$ is a summand of $F_{0,Y}$ which is in the image of the
map $G_Y \to F_{0,Y}$, the composite map ${\mathcal O}_Y \to F_{0,Y} \to E$ is
zero. In particular, this implies that $G_Y\to {\mathcal O}_Y$ is a surjection
and so
$$0 \to \bar{G}^\vee \to \bar{F}^\vee \to E \to 0 $$ is also a
resolution. This contradicts the minimality of sequence (\ref{ses1}).
\end{proof}
\begin{proof}[Proof of Lemma \ref{firststep}]
We have the following resolution for ${\mathcal O}_X$ on ${\mathbb P}^n$:
$$0 \to {\mathcal O}_{{\mathbb P}^n}(-\sum_{i=1}^{n-3}d_i) \to \cdots \to
\bigoplus_{i=1}^{n-3}{\mathcal O}_{{\mathbb P}^n}(-d_i) \to {\mathcal O}_{{\mathbb P}^n}\to {\mathcal O}_{X}
\to 0.$$ Using this resolution and the fact that
$d=d_{n-3}>\mbox{max}_{1\leq i,j\leq n-4}\{d_i+d_j\}$, one can show that
$$\HH^1({\mathcal O}_{X}(d)\otimes\Omega^2_{{\mathbb P}^n})\cong \bigoplus_{i=1}^{n-3}
\HH^2(\Omega^2_{{\mathbb P}^n}(d-d_i))\cong\HH^2(\Omega^2_{{\mathbb P}^n}).$$
Tensoring diagram (\ref{c1}) with $\Omega^2_{{\mathbb P}^n}$ yields a
commuting square
\begin{equation}
\begin{array}{ccccccccc} \label{c2}
& & \HH^1(E^\vee(d)\otimes\Omega^2_{{\mathbb P}^n}) & \twoheadrightarrow &
\HH^2(F^\vee_{0,Y}\otimes\Omega^2_{{\mathbb P}^n}) & & & & \\
& & \downarrow & & \downarrow & & & & \\
& &
\HH^1({\mathcal O}_{X}(d)\otimes\Omega^2_{{\mathbb P}^n}) & \cong &
\HH^2({\mathcal O}_{Y}\otimes\Omega^2_{{\mathbb P}^n}) & & & & \\
\end{array}
\end{equation}
Here the top horizontal map is a surjection by Lemma \ref{mapiszero},
and the right vertical arrow is a surjection by Lemma \ref{split}.
Hence the map
$\HH^1(E^\vee(d)\otimes\Omega^2_{{\mathbb P}^n})\to
\HH^1({\mathcal O}_{X}(d)\otimes\Omega^2_{{\mathbb P}^n})$ is a surjection. Since this
map factors via $\HH^1(I_{C/X}(d)\otimes\Omega^2_{{\mathbb P}^n})$, the
natural map $\HH^1(I_{C/X}(d)\otimes\Omega^2_{{\mathbb P}^n})\to
\HH^1({\mathcal O}_{X}(d)\otimes\Omega^2_{{\mathbb P}^n})$ is a surjection. Hence we
are done.
\end{proof}
Let $h_Y\in\HH^1(\Omega^1_Y)$ be the restriction of the generator
$h\in\HH^1(\Omega^1_{{\mathbb P}^n})$ and consider the class
$h_Y^2\in\HH^2(\Omega^2_Y)$. This is the image of $h_X$, the
hyperplane class in $X$, under the Gysin map ${\mathbb C}\cong
\HH^1(\Omega^1_X) \to\HH^2(\Omega^2_Y)$. Furthermore, since $d\gg0$,
by Serre vanishing, we have $\HH^i(\Omega^2_Y(d))=0$ for $i=1,2$. Hence the
coboundary map $\HH^1(\Omega^2_Y(d)_{|X})\to\HH^2(\Omega^2_Y)$ is an
isomorphism. By abuse of notation, we will denote the inverse image of $h^{2}_{Y}$
under this isomorphism by $h^{2}_{Y}$.
\begin{cor}
Under the natural map
$$\HH^1(\Omega^2_Y(d)_{|X}) \to \HH^1(\Omega^2_Y(d)_{|C}), \hspace{5mm}
h_Y^2\mapsto 0.$$
\end{cor}
\begin{proof}
One has a commutative square
\[
\begin{array}{ccccc}
\HH^1(I_{C/X}(d)\otimes\Omega^2_{{\mathbb P}^n}) & \twoheadrightarrow &
\HH^1({\mathcal O}_X(d)\otimes\Omega^2_{{\mathbb P}^n}) & \to &
\HH^1({\mathcal O}_C(d)\otimes\Omega^2_{{\mathbb P}^n}) \\
\downarrow & & \downarrow & & \downarrow \\
\HH^1(I_{C/X}(d)\otimes\Omega^2_{Y}) & \to &
\HH^1({\mathcal O}_X(d)\otimes\Omega^2_{Y}) & \to &
\HH^1({\mathcal O}_C(d)\otimes\Omega^2_{Y})\\
\end{array}
\]
The first horizontal arrow in the top row is surjection, and the middle
vertical map $\HH^1({\mathcal O}_X(d)\otimes\Omega^2_{{\mathbb P}^n})\to
\HH^1({\mathcal O}_X(d)\otimes\Omega^2_{Y})$ can be identified with the map
$\HH^2(\Omega^2_{{\mathbb P}^n}) \to \HH^2(\Omega^2_Y)$ which takes the
element $h^2\mapsto h_Y^2$. Hence $h_Y^2 \mapsto 0$ under the map
$\HH^2(\Omega^2_Y)\cong\HH^1(\Omega^2_Y(d)_{|X}) \to
\HH^1(\Omega^2_Y(d)_{|C})$.
\end{proof}
Now we are in a position to prove the first step i.e., the surjectivity of the map $\chi$.
Consider the exact sequence
$$0 \to {\mathcal O}_X(-d) \to \Omega^1_{Y{|X}} \to \Omega^1_X \to 0.$$
Taking second exteriors and tensoring the resulting sequence by
${\mathcal O}_X(d)$, we get a short exact sequence
\begin{equation}\label{2ndext}
0 \to \Omega^1_X \to \Omega^2_{Y}(d)_{|X}\to\Omega^2_X(d)\to 0.
\end{equation}
For the inclusion $C\subset X$, the natural map $\Omega^1_{X{|C}} \to
\Omega^1_C$ yields a push out diagram:
\[
\begin{diagram}{ccccccccc}
0 & \to & \Omega^1_{X{|C}} & \to & \Omega^2_{Y}(d)_{|C} &
\to &
\Omega^2_X(d)_{|C} & \to & 0 \\
& & \downarrow & & \downarrow & & || & & \\
0 & \to & \Omega^1_C & \to &
\mathcal{F} & \to & \Omega^2_X(d)_{|C} & \to & 0. \\
\end{diagram}
\]
where $\mathcal{F}$ is defined by the diagram.
\begin{lemma}
The map $\HH^1(C,\Omega^1_C) \to \HH^1(C,\mathcal{F})$ in the
associated cohomology sequence of the bottom row in the above diagram
is zero. Thus we have a surjection
$$\VV_C:=\ker[\HH^1(\Omega^1_{X|C}) \to
\HH^1(\Omega^2_{Y}(d)_{|C})] \twoheadrightarrow \HH^1(C,\Omega^1_C).$$
\end{lemma}
\begin{proof}
We have a commutative diagram
\[\begin{array}{ccc}
\HH^1(\Omega^1_X) & \to & \HH^1(\Omega^2_{Y}(d)_{|X}) \\
\downarrow & & \downarrow \\
\HH^1(\Omega^1_{X{|C}}) & \to &
\HH^1(\Omega^2_{Y}(d)_{|C}) \\
\downarrow & & \downarrow \\
\HH^1(\Omega^1_C) & \to & \HH^1(\mathcal{F}) \\
\end{array}\]
The composite of the vertical maps on the left is the map which takes the class
$h_{X} \mapsto h_{C}$. Since these are the respective generators of these cohomology groups both of which are one dimensional, this composite is an isomorphism. On the other hand, the composite
$$\HH^1(\Omega^1_X) \to \HH^1(\Omega^2_Y(d)_{|X}) \to
\HH^1(\Omega^2_Y(d)_{|C})$$ is zero: this is because the map
$\HH^1(\Omega^1_X) \to \HH^1(\Omega^2_Y(d)_{|X})$ can be
identified with the Gysin map $\HH^1(\Omega^1_X)\to\HH^2(\Omega^2_Y)$,
and so by the above Corollary, the generator $h_{X} \mapsto 0$ under the composite.
This implies that the map $\HH^1(C,\Omega^1_C) \to
\HH^1(C,\mathcal{F})$ is zero and so we have a surjection
$\VV_C \twoheadrightarrow \HH^1(C,\Omega^1_C)$.
\end{proof}
\begin{cor}[Surjectivity of $\chi$]\label{chiisonto}
The composite map
\[
\begin{array}{ccc}
\VV_C \hookrightarrow \ker[\HH^1(\Omega^1_{X|W}) \to
\HH^1(\Omega^2_{Y}(d)_{|W})] & \stackrel{\chi}\to & {\mathbb C} \\
\end{array}
\]
is a surjection. Hence $\chi$ is a surjection.
\end{cor}
\begin{proof}
This first inclusion follows from the fact that ${\mathcal O}_W \cong {\mathcal O}_C
\oplus {\mathcal O}_D$. The surjectivity of the composite follows from the
definition of $\chi$ and the above lemma.
\end{proof}
\subsection{Some vanishing lemmas} In this subsection, we shall prove vanishing of
certain cohomologies. The technical condition in Theorem B is required for these
vanishings to hold and that is the only reason for its appearance in in the statement
of the theorem. The main result here is Lemma \ref{tgtvanishing} and the reader
may skip the details which are pretty standard arguments if s/he so wishes.
\begin{lemma}\label{vanishings}
With notation as above and $\alpha<d-1$, we have
$$\HH^j(T_Y\otimes K_Y\otimes{I}_{C/X}(2d-j))=0, \hspace{3mm} j=1,~2.$$
\end{lemma}
\begin{proof}
From the exact sequence $$0 \to {\mathcal O}_X(-\alpha)\to E^\vee \to I_{C/X}
\to 0,$$ it is enough to prove the following
\begin{enumerate}
\item
$\HH^{j+1}(T_Y\otimes K_Y\otimes{\mathcal O}_X(2d-j-\alpha))=0 \hspace{3mm} j=1,~2.$
\item
$\HH^j(T_Y\otimes K_Y\otimes{E}(2d-j-\alpha))=0 \hspace{3mm} j=1,~2.$
\end{enumerate}
The vanishings in (1) above follow, on tensoring the exact sequence
$$0 \to T_Y\otimes{\mathcal O}_X \to T_{{\mathbb P}^n}\otimes{\mathcal O}_X\to
\oplus_{i=1}^{n-4}{\mathcal O}_X(d_i) \to 0,$$ with $K_Y\otimes{\mathcal O}_Y(2d-\alpha-j)$
and using the vanishing of the following terms
\begin{itemize}
\item[(A)] $\HH^j(K_Y(d_i+2d-\alpha-j)_{|X})$ for $j=1,~2$. Since
$K_Y\cong {\mathcal O}_Y(\sum_{i=1}^{n-4}d_i-n-1)$ and $X$ is a complete
intersection, this follows.
\item[(B)] $\HH^{j+1}(T_{{\mathbb P}^n}\otimes K_Y(2d-\alpha-j)_{|X})$ for $j=1,~2$.
\end{itemize}
Using the Euler sequence restricted to $X$:
$$0\to {\mathcal O}_{X} \to {\mathcal O}_{X}(1)^{\oplus{n+1}} \to T_{{\mathbb P}^n}\otimes{\mathcal O}_X
\to 0,$$ the vanishing in (B) follows from the vanishing of
\begin{itemize}
\item $\HH^{j+1}(K_Y(2d-\alpha-j+1)_{|X})$ for $j=1,~2$ and
\item $\HH^{j+2}(K_Y(2d-\alpha-j)_{|X})$ for $j=1,~2$.
\end{itemize}
The only non-trivial cases are when $j=2$ in the first case and $j=1$
in the second case. These vanishings hold provided $\alpha<d-1$.
For the vanishings in (2), we use the minimal resolution of $E$ on
$Y$:
$$0 \to G_Y \to F_{0,Y} \to E \to 0.$$
Then it suffices to show that
\begin{enumerate}
\item[(C)]$\HH^{j}(T_{Y}\otimes K_Y\otimes F_{0,Y}(2d-\alpha-j))=0$
for $j=1,~2$.
\item[(D)]$\HH^{j+1}(T_{Y}\otimes K_Y\otimes
G_Y(2d-\alpha-j))$ for $j=1,~2$.
\end{enumerate}
For (C): $F_{0,Y}=\bigoplus{\mathcal O}_Y(-a_i)$ where $-a_i+\reg{E}\geq 0$. So
the above term is $\bigoplus_i \HH^{j}(T_{Y}(b_i))$ where $b_i\geq
d-j-4$. Since $d>>0$, this is true by Serre vanishing.
For (D): From the tangent bundle sequence
$$0 \to T_Y \to T_{{\mathbb P}^n}\otimes{\mathcal O}_Y \to
\oplus_{i=1}^{n-4}{\mathcal O}_Y(d_i)\to 0,$$ the required vanishings follow
from
$\bullet$ $\HH^{j}({\mathcal O}_{Y}(d_i)\otimes K_Y\otimes G_Y(2d-\alpha-j))=0$ for
$j=1,~2$ since $G_Y$ is ACM, and
$\bullet$ $\HH^{j+1}(T_{{\mathbb P}^n}\otimes K_Y\otimes G_Y(2d-\alpha-j))=0$
for $j=1,~2$. For this use the Euler sequence to reduce this statement
to vanishing like the above and then use the fact that $G_Y$ is ACM.
\end{proof}
\comment{
\begin{remark}
The above vanishing is the only reason why we have the technical condition
$\alpha<d-1$ in the statement of Theorem B.
\end{remark}
}
Recall that $W=C\cup D$.
\begin{lemma}\label{tgtvanishing}
$\HH^1(T_Y\otimes K_Y\otimes\mathcal{I}_{W/Y}(2d))=0.$
\end{lemma}
\begin{proof}
From the exact sequence
$$0\to {\mathcal O}_Y(-d) \to \mathcal{I}_{W/Y}\to I_{W/X} \to 0$$ we have,
since $d>>0$, $\HH^1(T_Y\otimes
K_Y\otimes\mathcal{I}_{W/Y}(2d))\cong\HH^1(T_Y\otimes K_Y\otimes
I_{W/X}(2d)).$ We shall prove that the latter vanishes. For that we
use the exact sequence
$$0\to {\mathcal O}_{X}(-2)\to {\mathcal O}_X(-1)^{\oplus 2} \to I_{D/X} \to 0.$$
Tensoring this with $I_{C/X}(2d)$ yields
\comment{$T_Y\otimes K_Y\otimes I_{C/X}(2d)$ yields
$$0\to T_Y\otimes K_Y\otimes I_{C/X}(2d-2)\to T_Y\otimes K_Y\otimes
I_{C/X}(2d-1)^{\oplus 2} \to T_Y\otimes K_Y\otimes I_{W/X}(2d) \to 0.$$}
$$0\to I_{C/X}(2d-2)\to I_{C/X}(2d-1)^{\oplus 2} \to I_{W/X}(2d) \to
0.$$ Since $\HH^j(T_Y\otimes K_Y\otimes{I}_{C/X}(2d-j))=0$ for
$j=1,~2$ from Lemma \ref{vanishings} above, we are done.
\end{proof}
\subsection{An auxiliary vector space}
In this subsection, we shall construct an auxiliary vector space which surjects
onto the domain of the map $\chi$. This construction is a crucial refinement of condition (1) in \cite{Wu1}
which was first proved in \cite{R}.
\begin{lemma}\label{utov}
Let $U:=\ker[\HH^0( T_{Y}\otimes K_{Y}(2d)) \to
\HH^0(K_{Y}(3d)_{|W})]$ and $V:=\ker[\HH^1(\Omega^1_{X|W}) \to
\HH^1(\Omega^2_{Y}(d)_{|W})]$. Then the natural map $U \to V$ is
a surjection.
\end{lemma}
\begin{proof}
Tensoring the short exact sequence
$$ 0 \to T_X \to T_{{Y|X}} \to {\mathcal O}_X(d) \to 0$$ with
$K_{Y}(2d)_{|W}$ and taking cohomology, we get
$$0 \to \HH^0( T_{X}\otimes K_{Y}(2d)_{|W}) \to
\HH^0( T_{Y}\otimes K_{Y}(2d)_{|W}) \to
\HH^0(K_{Y}(3d)_{|W}) $$
Since $ T_X\otimes K_{Y}(d) \cong \Omega^2_X$, we have
the following commutative diagram:
\begin{equation}\label{unexplained}
\begin{array}{ccccccc}
0 & \to & \UU & \to & \HH^0( T_{Y}\otimes
K_{Y}(2d))
& \to & \HH^0(K_{Y}(3d)_{|W}) \\
& & \downarrow & & \downarrow & & || \\
0 & \to & \HH^0(\Omega^2_{X}(d)_{|W}) & \to &
\HH^0( T_{Y}\otimes K_{Y}(2d)_{|W}) & \to &
\HH^0(K_{Y}(3d)_{|W}).\\
\end{array}
\end{equation}
The middle vertical arrow can be seen to be a surjection by using the
fact that the cokernel of this map injects into $\HH^1( T_{Y}\otimes
K_Y\otimes\mathcal I_{W/Y}(2d))$ which vanishes by Lemma
\ref{tgtvanishing}. By the snake lemma, the first map is also a
surjection. Since
$$\im[\HH^0(\Omega^2_{X}(d)_{|W}) \to \HH^1(\Omega^1_{X|W})]
= \ker[\HH^1(\Omega^1_{X|W}) \to
\HH^1(\Omega^2_{Y}(d)_{|W})]=\VV,$$ we have a surjection
$ \UU \twoheadrightarrow \HH^0(\Omega^2_{X}(d)_{|W}) \twoheadrightarrow \VV.$
\end{proof}
\subsection{The final lifting}
All that remains to be done now is to lift the elements from the auxiliary
vector space $U$ constructed above to $\ker\gamma$ for which we need
the following
\begin{lemma}\label{multiplicationmap}
With notation as above, the multiplication map
$$\HH^0(\mathcal{I}_{W/Y}\otimes K_Y(2d))\otimes\HH^0({\mathcal O}_{Y}(d)) \to
\HH^0(\mathcal{I}_{W/Y}\otimes K_Y(3d)) $$ is surjective.
\end{lemma}
\begin{proof}
Tensoring the exact sequence
\begin{equation}\label{resforD}
0 \to {\mathcal O}_X(-2) \to {\mathcal O}_X(-1)^{\oplus 2} \to I_{D/X} \to 0 \, ,
\end{equation}
by $E$, we have
\begin{equation}\label{aa}
0 \to E(-2) \to E(-1)^{\oplus 2} \to I_{D/X}E \to 0 \, .
\end{equation}
Let $T_m:=\HH^0({\mathcal O}_X(m))$. The exact sequence above gives rise to a diagram
with exact rows where the vertical arrows are all multiplication maps:
\[
\begin{array}{ccccccccc}
0 & \to &\HH^0(E(k-2))\otimes T_m & \to &
\HH^0(E(k-1))^{\oplus{2}}\otimes T_m & \to &
\HH^0(I_{D/X}E(k))\otimes T_m & \to & 0 \\
& & \downarrow & & \downarrow & & \downarrow & & \\
0 & \to & \HH^0(E(m+k-2)) & \to &
\HH^0(E(m+k-1))^{\oplus{2}} & \to &
\HH^0(I_{D/X}E(m+k)) & \to & 0. \\
\end{array}
\]
Since $E$ is $(\sum_{i=1}^{n-3}d_i-\alpha -n+3)$-regular by Lemma
\ref{c1inequality}, the middle vertical arrow is a surjection for
$k\geq (\sum_{i=1}^{n-3}d_i-\alpha-n+4)$ and $m\geq 0$. It follows
that the multiplication map
$$\HH^0(I_{D/X}E(k))\otimes\HH^0({\mathcal O}_X(m))\to \HH^0(I_{D/X}E(m+k))$$
is surjective for $k\geq (\sum_{i=1}^{n-3}d_i-\alpha-n+4)$ and $m\geq
0$. Next consider the exact sequence $0 \to I_{D/X} \to I_{D/X}E \to
I_{W/X}(\alpha) \to 0$ obtained by tensoring sequence (\ref{serre}) by
$I_{D/X}(\alpha)$. Repeating the previous argument, it is easy to
check that the multiplication map
$$\HH^0(I_{W/X}(k))\otimes\HH^0({\mathcal O}_X(m))\to \HH^0(I_{W/X}(m+k))$$ is
surjective for $k\geq (\sum_{i=1}^{n-3}d_i-n+4)$ and $m\geq 0$. Now
using the exact sequence
$$ 0 \to {\mathcal O}_{Y}(-d) \to \mathcal{I}_{W/Y} \to I_{W/X} \to 0,$$ and
the fact that the regularity of ${\mathcal O}_Y$ is $\sum_{i=1}^{n-4}d_i-n-2$,
we can conclude by repeating the argument above, that the
multiplication map
$$\HH^0(\mathcal{I}_{W/Y}\otimes K_Y(2d))\otimes\HH^0({\mathcal O}_{Y}(d)) \to
\HH^0(\mathcal{I}_{W/Y}\otimes K_Y(3d))$$ is surjective.
\end{proof}
\section{Proofs of the main results}
Assume that a general, smooth complete intersection threefold $X\subset {\mathbb P}^{n}$
of sufficiently high multi-degree $(d_{1},\cdots, d_{n-3})$
supports an indecomposable ACM rank two vector
bundle $E$ with first Chern class $\alpha< d_{n-3}-1$. This implies
that there exists a rank two bundle $\mathcal{E}$ on the universal
hypersurface $\mathcal{X} \subset Y \times S'$ where $S'$ is a Zariski
open subset of $S$, the moduli space of smooth, degree $d$
hypersurfaces of $Y$, such that for a general point $s\in S'$,
$\mathcal{E}_{|X_s}$ is normalised, indecomposable, ACM and its first
Chern class $\alpha_s=\alpha$ satisfies the inequality
above. Furthermore, from the construction of this family, one sees
that there exists a family of curves $\mathcal{C} \to S'$ such that
$\mathcal{C}_s$ is the zero locus of a section of
$\mathcal{E}_{|X_s}$. Let $\mathcal{Z}$ be a family of $1$-cycles with
fibre $\mathcal{Z}_s:=d\mathcal{C}_s-lD_s$ where, as before, $D_s$ is
plane section of $X_s$ and $l=l(s)$ is the degree of
$\mathcal{C}_s$.
\begin{prop}\label{finale}
In the situation above, $\delta\nu_{\mathcal{Z}} \not\equiv 0$.
\end{prop}
\begin{proof}
We shall show that $\delta\nu_{\mathcal{Z}}(s)\neq 0$ at any point
$s\in S$ parametrising a smooth hypersurface $X\subset Y$. To do
this, we shall lift elements of $U$ to $\ker\gamma$ in diagram
\ref{formula}. Since we have surjections $U \twoheadrightarrow V \twoheadrightarrow {\mathbb C}$, we
will be done.
Let $\partial_f:\Omega^3_{Y}(2d)\to K_{Y}(3d)$ be the
derivative map where $f$ is the degree $d$ polynomial defining
$X$. Composing with the quotient $ K_{Y}(3d)/ K_{Y}(2d)$, we
get a map $\bar\partial_f:\Omega^3_{Y}(2d)\to
K_{Y}(3d)/K_{Y}(2d)$. Using the identification
$\Omega^3_{Y}\cong T_{Y}\otimes K_{Y}$, and taking
cohomology, we get
$$\HH^0( T_{Y}\otimes K_{Y}(2d))\by{\partial_f}
\HH^0(K_{Y}(3d)) \to
\frac{\HH^0(K_{Y}(3d))}{\HH^0(K_{Y}(2d))}.$$ The cokernel of
the composite map above can be identified with $\HH^2(\Omega^1_X)$
(see \cite{CGGH}, Page 174 or \cite{JL}, Chapter 9 for details).
The key ingredient in this lifting is the following commutative
diagram:
\begin{equation}
\begin{array}{ccc}
\HH^0({\mathcal O}_{Y}(d))\otimes\HH^0(\mathcal I_{W/Y}\otimes
K_{Y}(2d)) & \stackrel{\gamma'}\to &
\frac{\HH^0(K_{Y}(3d))}{\partial_f\HH^0( T_{Y}\otimes
K_{Y}(2d))}\\ \downarrow & & \downarrow \\
\HH^1( T_{X})\otimes
\HH^1(\mathcal{I}_{W/Y}\otimes\Omega^2_{X}) & \by{\gamma} &
\HH^2(\Omega^1_{X}).\\
\end{array}
\end{equation}
Here the right vertical map is the one explained above. The horizontal
maps $\gamma$ and $\gamma'$ are (essentially) cup product maps. The
vertical map on the left is a tensor product of two maps: The first
factor is the composite $\HH^0({\mathcal O}_{Y}(d))\to
\HH^0({\mathcal O}_X(d))\to\HH^1( T_X)$. The normal bundle of $X\subset
Y$ is ${\mathcal O}_X(d)$ and $\HH^0({\mathcal O}_X(d))\to \HH^1( T_X)$ is the
natural coboundary map in the cohomology sequence of the tangent
bundle sequence for this inclusion. The second factor $\HH^0(\mathcal
I_{W/Y}\otimes K_{Y}(2d))\to
\HH^1(\mathcal{I}_{W/Y}\otimes\Omega^2_{X})$ is also obtained
as above by observing that $T_X\otimes K_{Y}(d)\cong \Omega^2_X$.
This diagram yields a map $\ker\gamma' \to \ker\gamma$. To complete
the lifting, recall that by Lemma \ref{multiplicationmap}, the map
$\HH^0({\mathcal O}_{Y}(d))\otimes\HH^0(\mathcal I_{W/Y}\otimes K_{Y}(2d))\twoheadrightarrow
\HH^0(\mathcal{I}_{W/Y}\otimes{K_{Y}}(3d))$ is a surjection.
Restricting this map to $\ker{\gamma'}$, we get a surjection
$$\ker{\gamma'}\twoheadrightarrow \bar{U}:=\partial_f\HH^0( T_{Y}\otimes
K_{Y}(2d))\cap \HH^0(\mathcal{I}_{W/Y}\otimes
K_{Y}(3d)).$$
Let $\widetilde{U}$ be the kernel of the map
$\HH^0( T_{Y}\otimes K_{Y}(2d)) \to
\HH^0(K_{Y}(3d)_{|X})$. Looking at the diagram analogous to
(\ref{unexplained}) obtained by replacing $W$ by $X$, we see that
there is a map $\widetilde{U}\to \HH^0(\Omega^2_X(d))$. The boundary
map $\HH^0(\Omega^2_X(d))\to \HH^1(\Omega^1_X)$ in the cohomology
sequence associated to diagram (\ref{2ndext}) is the zero map (this is
because the composite map $\HH^1(\Omega^1_X) \to
\HH^1(\Omega^2_{Y}(d)_{|X})\cong \HH^2(\Omega^2_{Y})$ is the
Gysin inclusion). This implies that the surjection $U \twoheadrightarrow V$ of
Lemma \ref{utov} factors as $U \twoheadrightarrow \bar{U}\twoheadrightarrow U/\widetilde{U}\twoheadrightarrow V$
and thus we have surjections $\ker\gamma'\twoheadrightarrow \bar{U} \twoheadrightarrow V
\stackrel{\chi}\twoheadrightarrow {\mathbb C}$. By the compatibility of these maps with
the map $\ker\gamma'\to\ker\gamma$ and those in diagram
(\ref{formula}), we conclude (using Griffiths' formula) that
$\delta\nu_{\mathcal{Z}}(s)\neq 0$.
\end{proof}
\begin{proof}[Proof of Theorem B]
Assume that a general complete intersection threefold $X$ supports an
indecomposable normalised ACM bundle $E$, with $\alpha < d-1$. Let
$\mathcal{Z}$ be the family of degree zero $1$-cycles defined
earlier. By the refined Wu's criterion
$\delta\nu_{\mathcal{Z}}\not\equiv 0$: this contradicts the theorem of
Green and M\"uller-Stach. Thus we are done.
\end{proof}
\begin{proof}[Proof of Theorem A]
Let $X$ be a complete intersection subvariety of dimension four. Let
$E$ be an ACM bundle of rank two on $X$. As mentioned above, we may
assume $E$ to be normalised with first Chern class $\alpha$. Now
choose a general hypersurface $T\subset X$ of degree $d>>0$,
satisfying $d > \alpha +1$. Since $E\otimes{\mathcal O}_T$ is ACM, and $\alpha
< d-1$, it follows from Theorem B that $E\otimes{\mathcal O}_T$
splits. This implies by a standard argument that $E$ itself
splits. The case for dimension greater than four now follows in a
similar way.
\end{proof}
\begin{proof}[Proof of Corollary C]
The proof of the first part is trivial. The proof of the second part
is as follows: Suppose $C=X\cap{\tilde{S}}$ where
$\tilde{S}\subset{\mathbb P}^n$ is a codimension two subscheme. Then
$C=X\cap{S}$ where $S:=\tilde{S}\cap{Y}$ is a surface in $Y$. We have
a commutative diagram
\[
\begin{array}{ccccccccc}
0&\to&\Omega^2_Y & \to&\Omega^2_Y(d)&\to&\Omega^2_Y(d)_{|X}&\to & 0 \\
& & \downarrow & & \downarrow & & \downarrow & & \\
0&\to&\Omega^2_{Y}\, _{|S} & \to&\Omega^2_Y(d)_{|S}
&\to&\Omega^2_Y(d)_{{|C}}&\to & 0 \\
\end{array}
\]
Taking cohomology, we get a commutative diagram
\[
\begin{array}{ccc}
\HH^1(\Omega^2_Y(d)_{|X})& \cong & \HH^2(\Omega^2_Y)\\
\downarrow & & \downarrow \\
\HH^1(\Omega^2_Y(d)_{{|C}}) & \to & \HH^2(\Omega^2_{Y}\, _{|S}) \\
\end{array}
\]
The map $\HH^2(\Omega^2_Y) \to \HH^2(\Omega^2_Y\,_{|S})$ is non-zero
since the composite
$$\HH^2(\Omega^2_Y) \to \HH^2(\Omega^2_Y\,_{|S})\to
\HH^2(\Omega^2_S)$$ is a surjection which sends $h_Y^2 \mapsto h_S^2$
where $h_Y$ and $h_S$ are the classes of hyperplane sections in $Y$
and $S$ respectively. Thus the image of $h_X$ under the composite map
$$\HH^1(\Omega^1_X) \to \HH^1(\Omega^2_Y(d)_{|X}) \to
\HH^2(\Omega^2_{Y}\, _{|S})$$ is non-zero and hence its image under the
map
$$\HH^1(\Omega^1_X) \to \HH^1(\Omega^2_Y(d)_{|X}) \to
\HH^1(\Omega^2_Y(d)_{|C})$$ is also non-zero. By the proof of Lemma
\ref{chiisonto}, if $E$ were indecomposable, then the above map is
zero. This implies when $\alpha<d-1$, that the associated rank two bundle splits, hence
$C$ is a complete intersection.
\end{proof}
| {'timestamp': '2010-05-24T02:01:37', 'yymm': '1005', 'arxiv_id': '1005.3988', 'language': 'en', 'url': 'https://arxiv.org/abs/1005.3988'} |
\section*{Notation}
{\small
\begin{xtabular}{p{1cm}p{7.25cm}}
$j$ & The imaginary unit ($j^2 + 1 = 0$) \\
$a,A$ & (No boldface letter) scalar \\
$\bf a$ & (Boldface lowercase letter) column vector \\
$\bf A$ & (Boldface uppercase letter) matrix \\
$\mathcal{A}$ & (Calligraphic font uppercase letter) set \\
${\rm Re}(\,\cdot\,)$ & Element-wise real part operator \\
${\rm Im}(\,\cdot\,)$ & Element-wise imaginary part operator \\
$(\,\cdot\,)^*$ & Element-wise conjugate operator \\
$(\,\cdot\,)^T$ & Transpose operator \\
$(\,\cdot\,)^H$ & Conjugate transpose operator \\
${\bf 0}_{n \times m}$ & Zero matrix of size $n \times m$ \\
${\bf 0}$ & Zero matrix of appropriate size, determined from context \\
$\left\{{\bf a}\right\}_k$ & $k$-th element of vector $\bf a$ (scalar) \\
$\left\{{\bf A}\right\}_k$ & $k$-th row of matrix $\bf A$ (row vector) \\
$\left\{{\bf A}\right\}_{ik}$ & Element of matrix $\bf A$ in row $i$, column $k$ (scalar) \\
$\left|{a}\right|$ & Absolute value of scalar $a$ \\
$\left|{\mathcal{A}}\right|$ & Cardinality of set $\mathcal{A}$ \\
$\left\|{\bf a}\right\|_1$ & 1-norm of vector $\bf a$: $\left\|{\bf a}\right\|_1 = \sum\limits_k {\left| {\left\{ {\bf a} \right\}_k } \right|}$ \\
$\left\|{\bf a}\right\|$ & Euclidean norm of vector $\bf a$: $\left\|{\bf a}\right\| = (\sum\limits_k {{\left| {\left\{ {\bf a} \right\}_k } \right|}^2})^{1/2}$ \\
${\rm diag}\left({\bf a}\right)$ & Diagonal matrix such that $\left\{{{\rm diag}\left({\bf a}\right)}\right\}_{kk} = \left\{{\bf a}\right\}_k$. ${\rm diag}\left({\bf a}\right)$ has as rows and columns as the size of ${\bf a}$ \\
${\rm rank}\left({\bf A}\right)$ & Rank of matrix $\bf A$ (scalar) \\
${\rm Null}\left({\bf A}\right)$ & Null space (kernel) of matrix $\bf A$ \textcolor{black}{(The set of all vectors $\bf x$ such that ${\bf A}{\bf x} = {\bf 0}$. The null space is always a vector space.)} \\
${\rm dim}(\,\cdot\,)$ & Dimension of a vector space (scalar) \\
${\rm sym}\left({\bf B}\right)$ & Symmetric part of square matrix $\bf B$: ${\rm sym} \left({\bf B}\right) = \left({{\bf B} + {\bf B}^T}\right)/2$ \\
${\bf B} \succeq {\bf 0}$ & Square matrix $\bf B$ is positive-semidefinite (for all ${\bf x} \neq {\bf 0}$, ${\rm Re}\left( {{\bf x}^H {\bf Bx}} \right) \geq 0$), but not necessarily Hermitian \\
${\bf B} \succ {\bf 0}$ & Square matrix $\bf B$ is positive-definite (for all ${\bf x} \neq {\bf 0}$, ${\rm Re}\left( {{\bf x}^H {\bf Bx}} \right) > 0$), but not necessarily Hermitian
\end{xtabular}}
\section{Introduction}
\IEEEPARstart{T}{he} admittance matrix, which relates the current injections to the bus voltages, is one of the most fundamental concepts in power engineering. In the phasor domain, admittance matrices are complex-valued square matrices. These matrices are used in many applications, including system modeling, power flow, optimal power flow, state estimation, stability analyses, etc. \cite{kundur, crow}.
This paper thoroughly characterizes the invertibility of admittance matrices, which is a fundamental property for many power system applications.
Several applications directly rely on the invertibility of the admittance matrix. For instance, Kron reduction \cite{kron} is a popular technique for reducing the number of independent bus voltages modeled in a power system. The feasibility of Kron reduction is contingent on the invertibility of an appropriate sub-block of the admittance matrix. Many applications of Kron reduction assume that this procedure is feasible without performing further verification (e.g., \cite{efficient_thevenin, impedance_estimator, frequency_divider}). Additionally, various fault analysis techniques require the explicit computation of the inverse of the admittance matrix (the impedance matrix) \cite{anderson_faults}. The DC power flow \cite{stott2009} and its derivative applications \cite{ptdf, atc} also require the invertibility of admittance matrices for purely inductive systems. The invertibility of the admittance matrix is a requirement seen in both classical literature and recent research efforts (see, e.g., \cite{tenti2018, rahman2019}).
Checking invertibility of a matrix can be accomplished via rank-revealing factorizations \cite{rrlu_fact, rrqr_fact}. However, this approach is computationally costly for large matrices. Invertibility can also be checked approximately by computing the condition number via iterative algorithms that have lower complexity than matrix factorizations \cite{cond}. However, iterative estimation of the condition number can be inaccurate \cite{cond_reliable}. In some applications, such as transmission switching \cite{transmission_switching} and topology reconfiguration \cite{reconfiguration_1, reconfiguration_2}, the admittance matrix changes as part of the problem and checking invertibility for every case is intractable. Recent research has studied the theoretical characteristics of the admittance matrix in order to guarantee invertibility without the need for computationally expensive explicit checks \cite{rank_1phase, rank_3phase, gatsis, low_theorem}. \textcolor{black}{We note that these existing theoretical results have limited applicability to practical power system models, as we discuss in Section~\ref{sec:implementation}.}
One of the most important results regarding theoretical invertibility guarantees comes from~\cite{rank_1phase}. The authors of~\cite{rank_1phase} show that the admittance matrix is invertible for connected networks consisting of reciprocal branches without mutual coupling and at least one shunt element\textcolor{black}{\footnote{\textcolor{black}{A branch is said to be \emph{reciprocal} if its two-port admittance matrix is symmetrical. See~\cite{bakshi} for details.}}}. This result relies on additional modeling assumptions requiring that all admittances have positive conductances and prohibiting transformers with off-nominal tap ratios (including on-load tap changers which control the voltage magnitudes or phase shifters which control the voltage angles).
These requirements can be restrictive for practical power system models. While perfectly lossless branches do not exist in physical circuits, power system datasets often approximate certain branches as lossless.
For instance, out of the 41 systems with more than 1000 buses in the PGLib test case repository~\cite{pglib}, zero-conductance branches exist in 26 systems (63.4\%).
We further note that transformers with off-nominal tap ratios and non-zero phase shifts are also present in many practical datasets (e.g., 39 of the aforementioned 41 PGLib systems (95.1\%)).
In addition to these modeling restrictions, there is a technical issue with the proof presented in~\cite{rank_1phase}. This paper demonstrates that the result of \cite{rank_1phase} can still be achieved and generalized to a broader class of power system models. We first detail the technical issue in the proof in~\cite{rank_1phase}. We then prove invertibility of the admittance matrix under a condition that generalizes the requirements in~\cite{rank_1phase}. The condition holds for a broad class of realistic systems, including systems with lossless branches and transformers with off-nominal tap ratios. Next we show that the theorem condition holds for networks that can be decomposed into reactive components with simple structure. Finally, we present a proof-of-concept program that implements the theorem, and we show through numerical experiments that the theorem can be applied to a wide variety of realistic power systems.
The rest of the paper is organized as follows. Section~II describes the result of previous research and the technical issue in their proof. Section~III states the modifications and additional lemmas required to amend and generalize the previous result to systems with purely reactive elements and more general transformer models. Section~IV describes the implementation and numerical experiments. Section~V concludes the paper.
\section{Claims from Previous Literature \texorpdfstring{\\}{} and Limitations}
Borrowing the notation of \cite{rank_1phase}, the admittance matrix is (see \cite{stevenson}):
\begin{equation}
{\bf Y}_\mathcal{N} = {\bf A}_{\mathcal{L},\mathcal{N}}^T {\bf Y}_\mathcal{L} {\bf A}_{\mathcal{L},\mathcal{N}} + {\bf Y}_\mathcal{T},
\end{equation}
where \textcolor{black}{${\bf A}_{\mathcal{L},\mathcal{N}} \in \mathbb{R}^{|\mathcal{L}| \times |\mathcal{N}|}$} is the oriented incidence matrix of the network graph\textcolor{black}{\footnote{\textcolor{black}{The oriented incidence matrix relates the admittances of each branch with the nodes of that branch. The $ij$-th entry is $0$ if branch $i$ is not connected to node $j$, otherwise the entry is $\pm 1$, and the sign depends on the orientation of the branch. The orientation of the branches is arbitrary. See~\cite{stevenson} for details.}}} (excluding ground), \textcolor{black}{${\bf Y}_\mathcal{L} = {\rm diag \left( {{\bf y}_\mathcal{L}}\right)} \in \mathbb{C}^{|\mathcal{L}| \times |\mathcal{L}|}$} is the diagonal matrix with the series admittances of each branch, and \textcolor{black}{${\bf Y}_\mathcal{T} = {\rm diag \left( {{\bf y}_\mathcal{T}}\right)} \in \mathbb{C}^{|\mathcal{N}| \times |\mathcal{N}|}$} is the diagonal matrix with the total shunt admittances at each node. $\mathcal{N}$ is the set of nodes (excluding ground) and $\mathcal{L}$ is the set of branches. Reference~\cite{rank_1phase} states the following assumption and lemmas (presented here with some minor extensions as described below):
\textbf{Assumption 1.} \textit{The branches are not electromagnetically coupled and have nonzero admittance, hence ${\bf Y}_\mathcal{L}$ is full-rank.}
\textbf{Lemma 1.} \textit{The rank of the oriented incidence matrix of a connected graph with $\left| {\mathcal{N}} \right|$ nodes, ${\bf A}_{\mathcal{L},\mathcal{N}}$, is $\left| {\mathcal{N}} \right|-1$. The vector of ones ${\bf 1}$ forms a basis of the null space of ${\bf A}_{\mathcal{L},\mathcal{N}}$.}
While the second statement regarding the basis of the null space is not included in Lemma~1 as presented in~\cite{rank_1phase}, it is a well-known characteristic of oriented incidence matrices\footnote{The sum of the elements of each row of ${\bf A}_{\mathcal{N},\mathcal{L}}$ is always zero since every row has exactly one entry of 1 and one entry of -1 with the rest of the entries equal to zero; see~\cite{algebraic_graph}.} that we will use later in this paper.
\textbf{Lemma 2.} \textit{The sum of the columns of ${\bf Y}_\mathcal{N}$ equals the transpose of the sum of its rows, which also equals the vector of shunt elements ${\bf y}_\mathcal{T}$ (see \cite{arrillaga}).}
\textbf{Lemma 3.} \textit{For any matrix ${\bf M}$, ${\rm rank}\left({{\bf M}^T {\bf M}}\right)={\rm rank}\left({{\bf M}}\right)$.}
As we will discuss shortly, \emph{Lemma~3 as stated above is incorrect}. This is the technical issue in~\cite{rank_1phase} mentioned above.
\textbf{Lemma 4.} \textit{For square matrices ${\bf N}_L$ and ${\bf N}_R$ with full rank and matching size, ${\rm rank}\left({{\bf N}_L {\bf M}}\right)={\rm rank}\left({{\bf M}}\right)={\rm rank}\left({{\bf M}{\bf N}_R }\right)$. Furthermore, ${\rm Null}\left({{\bf N}_L {\bf M}}\right)={\rm Null}\left({{\bf M}}\right)$.}
While the second statement regarding the relationship between the null spaces is not included in Lemma~4 as presented in~\cite{rank_1phase}, it is a well-known result from matrix theory.\footnote{Since the only solution of ${\bf N}_L {\bf x} = {\bf 0}$ is ${\bf x} = {\bf 0}$, we make ${\bf x} = {\bf M} {\bf z}$ for some vector ${\bf z}$ and the result follows.}
One of the main results of~\cite{rank_1phase} is the following theorem:
\textbf{Theorem 1.} \textit{If the graph $\left( {\mathcal{N},\mathcal{L}} \right)$ defines a connected network and Assumption 1 holds, then:}
\begin{equation}
{\rm rank}\left( {{\bf Y}_\mathcal{N} } \right) = \left\{ {\begin{array}{*{20}l}
{\left| \mathcal{N} \right| - 1} & {\text{if} \; \; {\bf y}_\mathcal{T} = {\bf 0}}, \\
{\left| \mathcal{N} \right|} & \text{otherwise}. \\
\end{array}} \right.
\end{equation}
The authors of \cite{rank_1phase} prove Theorem 1 by cases. They first assume ${\bf y}_\mathcal{T} = {\bf 0}$ and use the fact that ${\bf Y}_\mathcal{L}$ is diagonal to write it as
\begin{equation}
{\bf Y}_\mathcal{L} = {\bf B}^T {\bf B},
\end{equation}
where ${\bf B} \in \mathbb{C}^{|\mathcal{N}| \times |\mathcal{N}|}$ is full-rank. Therefore:
\begin{subequations}
\begin{align}
{\bf Y}_\mathcal{N} &= {\bf A}_{\mathcal{L},\mathcal{N}}^T {\bf B}^T {\bf B} {\bf A}_{\mathcal{L},\mathcal{N}}, \\
{\bf Y}_\mathcal{N} &= \left({{\bf B} {\bf A}_{\mathcal{L},\mathcal{N}}}\right)^T {\bf B} {\bf A}_{\mathcal{L},\mathcal{N}}, \\
{\bf Y}_\mathcal{N} &= {\bf M}^T {\bf M},
\end{align}
\end{subequations}
where ${\bf M}={\bf B} {\bf A}_{\mathcal{L},\mathcal{N}}$. According to Lemma 1, ${\bf A}_{\mathcal{L},\mathcal{N}}$ has rank $\left| {\mathcal{N}} \right| - 1$. According to Lemma 4, ${\rm rank}\left({{\bf B} {\bf A}_{\mathcal{L},\mathcal{N}}}\right)={\rm rank}\left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right)$, so ${\rm rank}({\bf M})=\left| \mathcal{N} \right| - 1$. Finally, according to Lemma 3, ${\rm rank}\left({{\bf Y}_\mathcal{N}}\right)=\left| {\mathcal{N}} \right| - 1$.
There is a technical issue in the proof of Theorem~1 resulting from the fact that Lemma 3 only holds for real-valued matrices. A complex-valued counterexample is the following:
\begin{equation}
{\bf M} = \left[ {\begin{array}{*{20}c}
1 & 0 \\
j & 0 \\
\end{array}} \right], \qquad {\rm rank}\left( {\bf M} \right) = 1,
\end{equation}
\begin{equation}
{\bf M}^T {\bf M} = \left[ {\begin{array}{*{20}c}
0 & 0 \\
0 & 0 \\
\end{array}} \right], \qquad {\rm rank}\left( {{\bf M}^T {\bf M}} \right) = 0.
\end{equation}
\textcolor{black}{However, Lemma 3 holds if we use the \textit{conjugate transpose} operator $(\cdot)^H$ instead of using the transpose operator $(\cdot)^T$ (that is, we not only need to transpose the matrix, we also need to conjugate its entries as well). The corrected lemma is stated next.}
\textbf{Lemma 3 (Corrected).} \textit{For any matrix ${\bf M}$ with complex entries, ${\rm rank}\left({{\bf M}^H {\bf M}}\right)={\rm rank}\left({\bf M}\right)$. Furthermore, ${\rm Null}\left({{\bf M}^H {\bf M}}\right)={\rm Null}\left({\bf M}\right)$.}
\textbf{Proof.} Suppose a vector $\bf z$ is in the null space of $\bf M$, then:
\begin{equation}
{\bf 0} = {\bf M}{\bf z}, \quad \Longrightarrow \quad {\bf 0} = {\bf M}^H{\bf M}{\bf z},
\end{equation}
so $\mathbf{z}$ is also in the null space of ${\bf M}^H{\bf M}$. Moreover, suppose a vector $\bf z$ is in the null space of ${\bf M}^H{\bf M}$. Then, we have
\begin{subequations}
\begin{align}
{\bf 0} = {\bf M}^H{\bf M}{\bf z}, \quad &\Longrightarrow \quad 0 = {\bf z}^H{\bf M}^H{\bf M}{\bf z} = \left\|{{\bf M}{\bf z}}\right\|^2 \\
&\Longrightarrow \quad {\bf 0} = {\bf M}{\bf z},
\end{align}
\end{subequations}
so $\bf z$ is also in the null space of ${\bf M}$. In conclusion, $\bf z$ is in the null space of ${\bf M}$ if and only if it is in the null space ${\bf M}^H{\bf M}$; this means that ${\rm Null}\left({{\bf M}^H {\bf M}}\right)={\rm Null}\left({\bf M}\right)$. Now we apply the rank-nullity theorem (see \cite{linear_algebra}) to complete the proof. $\hfill\square$
With the corrected version of Lemma~3 and a modeling restriction to systems where all branches are strictly lossy (have positive conductances), we can fix the proof of Theorem~1 as stated above. More specifically, the assumptions of~\cite{rank_1phase} imply Theorem~2 stated in the next section.
We now turn our attention to the modeling restrictions of~\cite{rank_1phase}. Before generalizing Theorem~1, we need to understand why a system that violates the modeling restrictions may not satisfy the theorem. Consider the circuit modeling a transformer with an off-nominal tap ratio shown in Fig.~\ref{fig:transformer_circuit}. Let $y_t = 1 / z_t$. The transformer's turns ratio $a_t$ is an arbitrary complex number. The transformer's admittance matrix is:
\begin{align}
{\bf Y}_t = \left[ {\begin{array}{*{20}c}
{y_t} & {-a_t y_t} \\
{-a_t^* y_t} & {\left| {a_t} \right|^2 y_t} \\
\end{array}} \right] = \textcolor{black}{{\bf a}_t\, y_t\, {\bf a}_t^H}, \label{eq:single_trans}
\end{align}
where ${\bf a}_t^H = \left[{1, -a_t}\right]$. If $a_t$ is purely real, then $a_t^*=a_t$ and we can model the transformer with the $\pi$ circuit in Fig.~\ref{fig:transformer_circuit_pi}~\cite{kundur}.
The transformer's $\pi$ circuit is a two-port network with ${\rm rank}\left( {{\bf Y}_t} \right)=1$. This $\pi$ circuit violates the requirement of strictly lossy branches if $a_t \neq 1$, as then one of the shunts will always have non-positive conductance. Notice that the impedances around the loop in the $\pi$ circuit have the sum $\frac{-1}{a_t-1}z_t + \frac{1}{a_t} z_t + \frac{1}{a_t^2-a_t} z_t = 0$. With a zero-impedance loop (i.e., a closed path through the circuit where the sum of the impedances along the path equals zero), it is mathematically possible to have non-zero voltages even in the case of zero current injections. This means that the admittance matrix is singular. More generally, admittance matrix singularity can result from other power system models with zero-impedance loops besides those associated with transformers.
\begin{figure}[t]
\centering
\vspace{-1em}
\subfloat[Transformer circuit providing a counterexample to Theorem~1 in~\cite{rank_1phase}.]{
\scalebox{0.83}{
\begin{circuitikz}[american voltages,scale=0.6]
\draw
(-1.75,3) to [short] (-1,3)
(-1.75,3) to [short,o-,l^=$1$] (-1.75,3)
(-1,3) to [R,l=$z_t$] (1,3)
(1,3) to [short] (1.5,3)
to [L] (1.5,0)
node (N1) {}
(-1.75,0) to (-1.75,0) node[ground]{}
(1.5,0) to [short,-o] (-1.75,0)
(2.5,0) node (N2) {}
(2.5,0) to [L] (2.5,3)
(3.75,0) to (3.75,0) node[ground]{}
(2.5,0) to [short,-o] (3.75,0)
(2.5,3) to [short] (3,3)
(3.75,3) to [short] (3,3)
(3.75,3) to [short,o-,l_=$2$] (3.75,3)
;
\draw [fill=black] ($(N1) + (0.25,2.7)$) circle (0.6mm);
\draw [fill=black] ($(N1) + (0.75,2.7)$) circle (0.6mm);
\draw ($(N1) + (0.5,3.4)$) node (tap) {$a_t:1$};
\end{circuitikz}
}
\label{fig:transformer_circuit}
}\quad
\subfloat[$\pi$-equivalent circuit for the transformer in Fig.~\ref{fig:transformer_circuit} for a real-valued turns ratio $a_t$.]{
\scalebox{0.83}{
\begin{circuitikz}[american voltages,scale=0.6]
\draw
(2,3) to [short] (3.5,3)
(2,3) to [short,o-,l=$1$] (2,3)
(3.5,3) to [R,l_=$\frac{-1}{a_t-1}z_t$] (3.5,0)
(3.5,3) to [open,l=$\frac{1}{a_t} z_t$,yshift=0.1cm] (6.5,3)
(3.5,3) to [R] (6.5,3)
(6.5,3) to [R,l=$\frac{1}{a_t^2-a_t} z_t$] (6.5,0)
(8,3) to [short] (6.5,3)
(8,3) to [short,o-,l_=$2$] (8,3)
(2,0) to (2,0) node[ground]{}
(8,0) to (8,0) node[ground]{}
(2,0) to [short,o-o] (8,0)
;
\end{circuitikz}
}
\label{fig:transformer_circuit_pi}
}%
\caption{Transformer circuits.}
\vspace{-1em}
\end{figure}
The strict-lossiness restriction in~\cite{rank_1phase} requires all impedances in the power systems to have strictly positive real part. This means that the sum of the impedances over any possible loop will always have positive real part, thus being different from zero. Hence, the strict-lossiness restriction forbids the existence of zero-impedance loops. However, this also restricts the presence of transformers with off-nominal tap ratios and branches modeled as purely reactive elements, both of which appear in practical power system datasets as discussed in Section~I. To circumvent this issue, we will treat transformers as general series elements while modeling the shunt elements of the transformer $\pi$ circuit by employing an appropriate representation of the admittance matrix. \textcolor{black}{In this new representation, the branch admittances are related to the admittance matrix through a generalized version of the incidence matrix. Using this generalized incidence matrix, we can represent a transformer as a single series branch without shunts.} With this approach, the conditions we derive in this paper only forbid the existence of \textit{non-transformer} zero-impedance loops. Further, the new representation allows us to generalize Theorem~1 to systems with purely reactive elements and transformers with off-nominal tap ratios.
\section{Main Results}\label{sec:main_results}
This section describes our process for fixing and generalizing the main theorem. We first state and prove all necessary lemmas that will be used to prove the main result. We also declare an additional reasonable assumption that allow us to extend the result to systems with general transformer models. We then state the generalized version of the main theorem, which requires a relaxed condition in order to hold. We close this section by proving that the relaxed condition in the generalized Theorem~1 holds for power systems with reasonably common structures. \textcolor{black}{(These structures will be discussed in the conditions of Theorem 3; see Fig.~\ref{fig:alg_example} for an example system presenting these common structures).}
\subsection{Preliminaries}
We start by introducing the following assumption:
\textbf{Assumption 2.} \textit{For any series branch $l \in \mathcal{L}$ from node~$i$ to node~$k$, the admittance matrix associated with just this element can be written as ${\bf Y}_l = {\bf a}_l y_l^{\vphantom{H}} {\bf a}_l^H \in \mathbb{C}^{\left|{\mathcal{N}}\right| \times \left|{\mathcal{N}}\right|}$, where $\left\lbrace{{\bf a}_l}\right\rbrace_i = 1$, $\left\lbrace{{\bf a}_l}\right\rbrace_k = -a_l^*$ ($a_l$ is a non-zero complex number) and all other entries of ${\bf a}_l$ are zero.}
Transmission lines and transformers (including transformers with off-nominal tap ratios) satisfy Assumption 2. Transmission lines can be modeled as transformers with $a_l=1$ along with some shunt elements. This permits modeling, for instance, $\Pi$-circuit models of transmission lines. Using Assumption~2, the admittance matrix of the full system is:
\begin{equation}\label{eq:reformulated_admittance}
{\bf Y}_\mathcal{N} = \sum\limits_{l \in \mathcal{L}} {{\bf Y}_l } + {\bf Y}_\mathcal{T}.
\end{equation}
In~\eqref{eq:reformulated_admittance}, note that ${\bf Y}_\mathcal{T}$ does not include the shunt elements in the transformers' $\pi$ circuits as these elements are instead included in ${{\bf Y}_l }$. The sum of the matrices can be rewritten as:
\begin{equation}
{\bf Y}_\mathcal{N} = {\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf Y}_\mathcal{L} {\bf A}_{\mathcal{L},\mathcal{N}} + {\bf Y}_\mathcal{T}, \label{eq:YN_standard}
\end{equation}
where, in a slight abuse of notation relative to Section~II, \textcolor{black}{${\bf A}_{\mathcal{L},\mathcal{N}} \in \mathbb{C}^{|\mathcal{L}| \times |\mathcal{N}|}$} is the \textit{generalized incidence matrix}, whose $l$-th row is $\left\lbrace{{\bf A}_{\mathcal{L},\mathcal{N}}}\right\rbrace_l = {\bf a}_l^H$; \textcolor{black}{${\bf Y}_\mathcal{L} = {\rm diag \left( {{\bf y}_\mathcal{L}}\right)} \in \mathbb{C}^{|\mathcal{L}| \times |\mathcal{L}|}$} is the diagonal matrix containing the series admittances for each branch; and \textcolor{black}{${\bf Y}_\mathcal{T} = {\rm diag \left( {{\bf y}_\mathcal{T}}\right)} \in \mathbb{C}^{|\mathcal{N}| \times |\mathcal{N}|}$} is the diagonal matrix containing the total shunt admittances at each node. \textcolor{black}{In the case of a single transformer, notice that \eqref{eq:YN_standard} reduces to \eqref{eq:single_trans} with ${\bf A}_{\mathcal{L},\mathcal{N}} = {\bf a}_t^H$, ${\bf Y}_\mathcal{L} = y_t$ and ${\bf Y}_\mathcal{T} = {\bf 0}$. In this new representation, the effect of the off-nominal tap is not represented as a shunt in ${\bf Y}_\mathcal{T}$, but is instead contained within ${\bf A}_{\mathcal{L},\mathcal{N}}$. The representation stated in \eqref{eq:YN_standard} will be the default used in the rest of the paper.}
Parallel shunts or branches with the same tap ratio can be reduced to a single branch or shunt by adding the admittances, so we assume that this reduction is always performed:
\textbf{Remark 1.} \textit{There are no parallel shunts or parallel branches with the same tap ratio.}
The connectedness condition of the network is evaluated considering its representation with parallel branches reduced. Parallel transformers with different tap ratios cannot be represented as single branch in the form stated by Assumption~2, so they are not reduced (each parallel branch individually satisfies Assumption 2, so our results are also applicable to those cases). We next state the rank-nullity theorem as we will use it several times in the paper:
\textbf{Rank-nullity theorem ([Theorem~4.4.15] in\cite{linear_algebra}).} \textit{Let ${\bf M} \in \mathbb{C}^{m \times n}$ be an arbitrary matrix, then:}
\begin{equation}
{\rm rank}\left({\bf M}\right) + {\rm dim}\left({{\rm Null}\left({\bf M}\right)}\right) = n.
\end{equation}
\textcolor{black}{The main theoretical results of this paper are Theorems~1, 2, and 3. To prove these results, we need a series of lemmas that will be presented next. To clarify how the lemmas are related to the problem at hand, Fig.~\ref{fig:relationships} illustrates the multiple dependence relationships between the lemmas and theorems presented in this work. We start our endeavor by extending Lemma~1 to generalized incidence matrices:}
\begin{figure}[t]
\centering
\tikzstyle{process} = [rectangle, minimum width=1.0cm, minimum height=0.6cm, text centered, draw=black, text width=2.0cm]
\tikzstyle{arrow} = [thick,->,>=stealth]
\scalebox{0.9}{
\begin{tikzpicture}[color=black,x=1cm,y=1cm,node distance=1.2cm]
\node (le1) [process, text width=1.7cm] {Lemma 1 (extended)};
\node (le4) [process, text width=1.5cm, below of=le1, yshift=-1.0cm] {Lemma 4};
\node (le7) [process, text width=1.5cm, right of=le4, xshift=1.3cm, yshift=-0.9cm] {Lemma 7};
\node (le5) [process, text width=1.5cm, below of=le7, yshift=-1.0cm] {Lemma 5};
\node (le3) [process, text width=1.7cm, left of=le5, xshift=-0.9cm] {Lemma 3 (corrected)};
\node (le6) [process, text width=1.5cm, right of=le5, xshift=0.8cm] {Lemma 6};
\node (thm2) [process, text width=1.7cm, right of=le7, xshift=1.3cm] {Theorem 2};
\node (thm1) [process, above of=thm2, yshift=0.8cm] {Theorem 1 (generalized)};
\node (thm3) [process, text width=1.7cm, right of=le4, xshift=6.3cm] {Theorem 3};
\draw[black, thick, dashed] ($(le5.center)+(-3.3,-0.7)$) rectangle ($(le5.center)+(3.1,0.7)$) node[name=rec1_ne]{};
\draw [arrow,>=.] (le1) -- (le1 -| thm1) node(fork1){};
\draw [arrow] (fork1.center) -- (thm1);
\draw [arrow] (fork1.center) -| (thm3);
\draw [arrow,>=.] (le4) -- (le4 -| thm2) node(fork2){};
\draw [arrow] (fork2.center) -- (thm2);
\draw [arrow] (fork2.center) -- (thm3);
\draw [arrow] (le7) -- (thm2);
\draw [arrow] (le5.center |- rec1_ne) -- (le7);
\draw [arrow] ($0.5*(le5.center |- rec1_ne)+0.5*(le7.south)$) node(fork3){} -| (thm2);
\draw [arrow] (thm2) -| (thm3);
\fill[black]
(fork1) circle(2.0pt)
(fork2) circle(2.0pt)
(fork3) circle(2.0pt)
;
\draw[red, thick, dashed]
($(thm2.center)+(-1.4,-0.6)$) node[name=rec2_sw]{} rectangle ($(thm1.center -| thm3.center)+(1.2,0.8)$) node[name=rec2_ne]{}
($(rec2_sw -| rec2_ne)+(-1.0,0.0)$) node[anchor=north]{Main results}
;
\end{tikzpicture}
}
\caption{\textcolor{black}{Relationship diagram for the lemmas and theorems. An arrow going from X to Y indicates that X is used to prove Y. The three theorems are the main theoretical results of this paper.}}
\label{fig:relationships}
\vspace{-1em}
\end{figure}
\textbf{Lemma 1 (Extended).} \textit{The rank of the generalized incidence matrix of an arbitrary connected network with $\left| {\mathcal{N}} \right|$ nodes, ${\bf A}_{\mathcal{L},\mathcal{N}} \in \mathbb{C}^{\left|{\mathcal{L}}\right| \times \left|{\mathcal{N}}\right|}$, is at least $\left| {\mathcal{N}} \right|-1$. If ${\bf A}_{\mathcal{L},\mathcal{N}}$ is not full column rank, then none of the basis vectors of its null space have null entries.}
\textbf{Proof.} Let $\mathcal{S} \subseteq \mathcal{L}$ be a set of branches forming a spanning tree of the network graph\textcolor{black}{\footnote{\textcolor{black}{To make a power engineering analogy, a \emph{spanning tree} is a subsystem obtained by removing branches from the original system until the resulting network is radial and connected. Every connected network has a spanning tree (see \cite{graph_theory}).}}}. We can order the branches of $\mathcal{L}$ by numbering all the branches of $\mathcal{S}$ first. Thus we can write ${\bf A}_{\mathcal{L},\mathcal{N}}$ in blocks as follows:
\begin{equation}
{\bf A}_{\mathcal{L},\mathcal{N}} = \left[ {\begin{array}{*{20}c}
{\bf A}_{\mathcal{S},\mathcal{N}} \\
{\bf A}_{\mathcal{L} \setminus \mathcal{S},\mathcal{N}} \\
\end{array}} \right],
\end{equation}
where ${\bf A}_{\mathcal{S},\mathcal{N}}$ is the generalized incidence matrix of the branches in $\mathcal{S}$ and ${\bf A}_{\mathcal{L} \setminus \mathcal{S},\mathcal{N}}$ is the generalized incidence matrix of the remaining branches. For any vector \textcolor{black}{$\bf x \in \mathbb{C}^{|\mathcal{N}|}$} in the null space of ${\bf A}_{\mathcal{L},\mathcal{N}}$, $\bf x$ must be orthogonal to all rows of ${\bf A}_{\mathcal{L},\mathcal{N}}$:
\begin{equation}\label{eq:nullA}
{\bf a}_s^H {\bf x} = 0, \qquad \forall s \in \mathcal{S}.
\end{equation}
Take an arbitrary branch $s$ that goes from node $i$ to node $k$, then from \eqref{eq:nullA} we have:
\begin{equation}
\left\lbrace{\bf x}\right\rbrace_i - a_s \left\lbrace{\bf x}\right\rbrace_k = 0,
\end{equation}
where $a_s$ is the tap ratio of branch $s$. We can write:
\begin{subequations}
\begin{align}
\left\lbrace{\bf x}\right\rbrace_i &= a_s \left\lbrace{\bf x}\right\rbrace_k, \\
\left\lbrace{\bf x}\right\rbrace_k &= a_s^{-1} \left\lbrace{\bf x}\right\rbrace_i.
\end{align}
\end{subequations}
We generalize this result and say that if nodes $i$ and $k$ are connected through a branch $b \in \mathcal{S}$ we can write:
\begin{equation}\label{eq:xk_from_xi}
\left\lbrace{\bf x}\right\rbrace_k = a_b^{d\left( {b,i,k} \right)} \left\lbrace{\bf x}\right\rbrace_i,
\end{equation}
where $a_b$ is the tap ratio of branch $b$, $\mathcal{S}(i,k) \subseteq \mathcal{S}$ is the (unique) set of branches in $\mathcal{S}$ forming a path from node $i$ to node $k$ (in this case the only member of $\mathcal{S}(i,k)$ is $b$), and $d\left( {b,i,k} \right)$ is a function that returns either $1$ or $-1$ depending on the direction of branch $b$ relative to the path defined by $\mathcal{S}(i,k)$ (if branch $b$ goes from node $i$ to node $k$ then $d\left( {b,i,k} \right)=-1$, otherwise $d\left( {b,i,k} \right)=1$). As $\mathcal{S}$ is a spanning tree, there exists a unique path from node 1 to every other node $k \ne 1$. Define $p_{ik}(m)$ as a function returning the node in the $m$-th position along the path from node $i$ to node $k$ ($p_{ik}(1)=i$ and $p_{ik}(1+|\mathcal{S}(i,k)|)=k$), and let $b_{ik}(m) \in \mathcal{S}_{ik}$ be the branch connecting nodes $p_{ik}(m)$ and $p_{ik}(m+1)$. Let $D_k = \left| {\mathcal{S}(1,k)} \right|$. We write $\left\lbrace{\bf x}\right\rbrace_k$ in terms of $\left\lbrace{\bf x}\right\rbrace_1$ by chaining \eqref{eq:xk_from_xi} for each pair of consecutive nodes in the path between nodes $1$ and $k$:
\begin{equation*}
1 \xrightarrow{b_{1k}(1)} p_{1k}(2) \xrightarrow{b_{1k}(2)} \,\cdots\, \xrightarrow{b_{1k}(D_k-1)} p_{1k}(D_k) \xrightarrow{b_{1k}(D_k)} k.
\end{equation*}
We backtrack the chain of equations starting from node $k$ until we reach node 1:
\begin{subequations}
\begin{align}
\left\lbrace{\bf x}\right\rbrace_k &= a_{b_{1k}(D_k)}^{d\left( {b_{1k}(D_k), 1, k} \right)} \cdot \left\lbrace{\bf x}\right\rbrace_{p_{1k}(D_k)}, \\
\left\lbrace{\bf x}\right\rbrace_k &= a_{b_{1k}(D_k)}^{d\left( {b_{1k}(D_k), 1, k} \right)} \cdot a_{b_{1k}(D_k-1)}^{d\left( {b_{1k}(D_k-1), 1, k} \right)} \cdot \left\lbrace{\bf x}\right\rbrace_{p_{1k}(D_k-1)}, \\
&\; \; \vdots \nonumber \\
\left\lbrace{\bf x}\right\rbrace_k &= \left\lbrace{\bf x}\right\rbrace_1 \prod\limits_{m=1}^{D_k} {a_{b_{1k}(m)}^{d\left( {b_{1k}(m), 1, k} \right)}},
\end{align}
\end{subequations}
or written more succinctly (as the product is commutative):
\begin{equation}
\left\lbrace{\bf x}\right\rbrace_k = \left\lbrace{\bf x}\right\rbrace_1 \prod\limits_{s \in \mathcal{S}(1,k)} {a_s^{d\left( {s,1,k} \right)} }.
\end{equation}
Let $\left\lbrace{\bf x}\right\rbrace_1 = \alpha$, for an arbitrary \textcolor{black}{$\alpha \in \mathbb{C}$}. We can then write $\bf x$ as:
\begin{subequations}
\begin{align}
{\bf x} &= \alpha {\bf v}, \\
\left\lbrace{\bf v}\right\rbrace_1 &= 1, \\
\left\lbrace{\bf v}\right\rbrace_k &= \prod\limits_{s \in \mathcal{S}(1,k)} {a_s^{d\left( {s,1,k} \right)} }, \qquad k = 2,\ldots,|\mathcal{N}|.
\end{align}
\end{subequations}
Since $\bf x$ has only one free parameter ($\alpha$) and ${\bf v} \neq {\bf 0}$, the rank-nullity theorem implies that ${\rm rank} \left({{\bf A}_{\mathcal{S},\mathcal{N}}}\right) = \left| {\mathcal{N}} \right|-1$. Furthermore, as $a_l \ne 0$ for all $l \in \mathcal{L}$, then all entries of $\bf v$ are non-zero.
Since $\bf x$ must also be orthogonal to all rows of ${\bf A}_{\mathcal{L} \setminus \mathcal{S},\mathcal{N}}$, we have the following equation for each row of ${\bf A}_{\mathcal{L} \setminus \mathcal{S},\mathcal{N}}$:
\begin{equation}
\alpha \left({\prod\limits_{s \in \mathcal{S}(1,i)} {a_s^{d\left( {s,1,i} \right)} } - a_l \prod\limits_{s \in \mathcal{S}(1,k)} {a_s^{d\left( {s,1,k} \right)} }}\right) = 0,
\end{equation}
for any branch $l \in \mathcal{L} \setminus \mathcal{S}$ going from node $i$ to $k$. If the term inside the parentheses is null for all rows, then the (directed) product of tap ratios $a_l$ across branches in a cycle is $1$, for all cycles. In that case, $\alpha$ is a free parameter and ${\rm rank} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right) = \left| {\mathcal{N}} \right|-1$. Otherwise $\alpha = 0$, and so ${\rm rank} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right) = \left| {\mathcal{N}} \right|$ (i.e., ${\bf A}_{\mathcal{L},\mathcal{N}}$ is full column rank). $\hfill\square$
We also require some new lemmas. We start with Lemma~5, which is a simple extension of Lemma 1 from \cite{gatsis}:
\textbf{Lemma 5.} \textit{Consider a matrix ${\bf Y}={\bf G}+j{\bf B} \in \mathbb{C}^{n \times n}$ with ${\bf G},{\bf B} \in \mathbb{R}^{n \times n}$. Suppose ${\bf G} \succeq {\bf 0}$, then ${\rm Null}\left({\bf Y}\right) \subseteq {\rm Null}\left({{\rm sym}\left({\bf G}\right)}\right)$ and ${\rm rank} \left({{\rm sym}\left({\bf G}\right)}\right) \le {\rm rank} \left({\bf Y}\right)$.}
\textbf{Proof.} Consider a vector ${\bf x} \in \mathbb{C}^n$ in the null space of $\bf Y$. We can write $\bf x$ in rectangular form as ${\bf x}={\bf x}_R+j {\bf x}_I$ with ${\bf x}_R,{\bf x}_I \in \mathbb{R}^n$. Using the definition of the null space, we have:
\vspace*{-1em}
\begin{subequations}
\begin{align}
0 &= {\rm Re}\left( {{\bf x}^H {\bf Yx}} \right), \\
0 &= {\bf x}_R^T {\bf Gx}_R + {\bf x}_I^T {\bf Gx}_I + {\bf x}_I^T {\bf Bx}_R - {\bf x}_R^T {\bf Bx}_I.
\end{align}
\end{subequations}
The quadratic terms are real, so they only depend on the symmetric part of the matrices\footnote{For any real (possibly non-symmetric) matrix $\mathbf{A}$ and appropriately sized real vector ${\bf x}$, the following relationships hold: ${\bf x}^T\mathbf{A}{\bf x} = ({\bf x}^T\mathbf{A}{\bf x})^T = {\bf x}^T\mathbf{A}^T{\bf x} = {\bf x}^T(\mathbf{A}/2 + \mathbf{A}^T/2){\bf x} = {\bf x}^T {\rm sym}\left( {\bf A} \right) {\bf x}$. \textcolor{black}{See \cite{quadratic_forms} for more details about symmetric quadratic forms.}}:
\begin{subequations}
\begin{align}
0 &= {\bf x}_R^T {\rm sym}\left( {\bf G} \right){\bf x}_R + {\bf x}_I^T {\rm sym}\left( {\bf G} \right){\bf x}_I \nonumber \\
&\quad + {\bf x}_I^T {\rm sym}\left( {\bf B} \right){\bf x}_R - {\bf x}_R^T {\rm sym}\left( {\bf B} \right){\bf x}_I, \\
0 &= {\bf x}_R^T {\rm sym}\left( {\bf G} \right){\bf x}_R + {\bf x}_I^T {\rm sym}\left( {\bf G} \right){\bf x}_I \nonumber \\
&\quad + {\bf x}_I^T {\rm sym}\left( {\bf B} \right){\bf x}_R - {\bf x}_I^T {\rm sym}\left( {\bf B} \right){\bf x}_R, \\
0 &= {\bf x}_R^T {\rm sym}\left( {\bf G} \right){\bf x}_R + {\bf x}_I^T {\rm sym}\left( {\bf G} \right){\bf x}_I.
\end{align}
\end{subequations}
As ${\rm sym}\left( {\bf G} \right) \succeq {\bf 0}$, both terms must be non-negative. Equality only holds if both terms are zero, and hence both ${\bf x}_R$ and ${\bf x}_I$ belong to the null space of ${\rm sym}\left( {\bf G} \right)$. Therefore if ${\bf Y}{\bf x}={\bf 0}$ then ${\rm sym}\left( {\bf G} \right) {\bf x} = {\bf 0}$, so ${\rm Null}\left({\bf Y}\right) \subseteq {\rm Null}\left({{\rm sym}\left({\bf G}\right)}\right)$. We apply the rank-nullity theorem to conclude the proof. $\hfill\square$
\textbf{Lemma 6.} \textit{Let ${\bf A} \succeq {\bf 0}$ and ${\bf B} \succeq {\bf 0}$ be square matrices in $\mathbb{R}^{n \times n}$. Then the following equations hold:}
\begin{align}
&{\bf A} + {\bf B} \succeq {\bf 0}, \\
&{\rm Null}\left({{\rm sym} \left({{\bf A} + {\bf B}}\right)}\right) = \nonumber \\
&\qquad \qquad \qquad \quad \; {\rm Null}\left({{\rm sym} \left({\bf A}\right)}\right) \cap {\rm Null}\left({{\rm sym} \left({\bf B}\right)}\right), \\
&{\rm rank} \left({{\rm sym} \left({\bf A}\right)}\right), {\rm rank} \left({{\rm sym} \left({\bf B}\right)}\right) \le \nonumber \\
&\qquad \qquad \qquad \qquad \qquad \qquad \; \; \; {\rm rank} \left({{\rm sym} \left({{\bf A} + {\bf B}}\right)}\right).
\end{align}
\textbf{Proof.} Let us calculate the quadratic form of ${\bf A}+{\bf B}$:
\begin{equation}
{\bf x}^T \left({{\bf A}+{\bf B}}\right){\bf x}={\bf x}^T {\bf A}{\bf x}+{\bf x}^T {\bf B}{\bf x}.
\end{equation}
As both ${\bf A}$ and ${\bf B}$ are positive semi-definite, then ${\bf x}^T {\bf A}{\bf x}\ge 0$ and ${\bf x}^T {\bf B}{\bf x} \ge 0$, thus ${\bf x}^T \left({{\bf A}+{\bf B}}\right){\bf x} \ge 0$. Then by definition ${\bf A}+{\bf B} \succeq {\bf 0}$. Now let ${\bf z}$ be a vector in the null space of ${\rm sym} \left({{\bf A} + {\bf B}}\right)$. This implies that:
\begin{subequations}
\begin{align}
{\bf z}^T {\rm sym} \left({{\bf A} + {\bf B}}\right){\bf z}&=0, \\
{\bf z}^T {\rm sym} \left({\bf A}\right){\bf z}+{\bf z}^T {\rm sym} \left({\bf B}\right){\bf z}&=0. \label{eq:lemma6_proof}
\end{align}
\end{subequations}
Notice that ${\bf A} \succeq {\bf 0}$ implies that ${\rm sym} \left({\bf A}\right) \succeq {\bf 0}$, and similarly we have that ${\rm sym} \left({\bf B}\right) \succeq {\bf 0}$ as well. Thus both terms of \eqref{eq:lemma6_proof} are non-negative we get that
\begin{equation}
{\bf z}^T {\rm sym} \left({\bf A}\right){\bf z}=0, \qquad {\bf z}^T {\rm sym} \left({\bf B}\right){\bf z}=0,
\end{equation}
and hence ${\bf z}$ belongs to the null spaces of both ${\rm sym} \left({\bf A}\right)$ and ${\rm sym} \left({\bf B}\right)$. The converse can be proved trivially by reversing the steps, so ${\rm Null}\left({{\rm sym} \left({{\bf A} + {\bf B}}\right)}\right)={\rm Null}\left({{\rm sym} \left({\bf A}\right)}\right) \cap {\rm Null}\left({{\rm sym} \left({\bf B}\right)}\right)$. We then apply the rank-nullity theorem to conclude the proof of Lemma 6. $\hfill\square$
\textbf{Lemma 7.} \textit{Let ${\bf M} \succeq {\bf 0}$, ${\bf M} \in \mathbb{C}^{n \times n}$ be a Hermitian matrix, then ${\rm Re}\left({{\bf M}}\right) \succeq {\bf 0}$, ${\rm Null}\left({{\rm Re}\left({{\bf M}}\right)}\right) = {\rm Null}\left({\bf M}\right)$, and ${\rm rank}\left({{\rm Re}\left({{\bf M}}\right)}\right) = {\rm rank}\left({\bf M}\right)$.}
\textbf{Proof.} As $\bf M$ is Hermitian and positive-semidefinite, it can be factored as \textcolor{black}{${\bf M} = {\bf A}^H {\bf A}, {\bf A} \in \mathbb{C}^{n \times n}$}. Now we expand ${\rm Re}\left({\bf M}\right)$:
\begin{equation}
{\rm Re}\left({\bf M}\right) = {\rm Re}\left({{\bf A}}\right)^T {\rm Re}\left({{\bf A}}\right)+{\rm Im}\left({{\bf A}}\right)^T {\rm Im}\left({{\bf A}}\right).
\end{equation}
Note that each term in the right hand side is positive semidefinite, so from Lemma 6 we have that ${\rm Re}\left({\bf M}\right)$ is symmetric and positive semidefinite as well. Applying Lemmas 6 and 3:
\vspace{-1em}
\begin{subequations}
\begin{align}
{\rm Null}\left({{\rm Re}\left({\bf M}\right)}\right) &= {\rm Null}\left({{\rm Re}\left({{\bf A}}\right)}\right) \cap {\rm Null}\left({{\rm Im}\left({{\bf A}}\right)}\right), \\
{\rm Null}\left({{\rm Re}\left({\bf M}\right)}\right) &\subseteq {\rm Null}\left({{\bf A}}\right) = {\rm Null}\left({{\bf M}}\right).
\end{align}
\end{subequations}
Recall that ${\rm Re}\left({\bf M}\right)$ is symmetric and positive semidefinite, so applying Lemma~5 yields:
\begin{equation}
{\rm Null}\left({\bf M}\right) \subseteq {\rm Null}\left({{\rm Re}\left({\bf M}\right)}\right).
\end{equation}
Since ${\rm Null}\left({{\rm Re}\left({\bf M}\right)}\right) \subseteq {\rm Null}\left({{\bf M}}\right)$ and ${\rm Null}\left({\bf M}\right) \subseteq {\rm Null}\left({{\rm Re}\left({\bf M}\right)}\right)$, we have that ${\rm Null}\left({{\rm Re}\left({\bf M}\right)}\right) = {\rm Null}\left({\bf M}\right)$. The claim follows after applying the rank-nullity theorem. $\hfill\square$
\subsection{Admittance Matrix Invertibility Theorem}
We now have the tools to present the amended version of Theorem~1 and prove its validity under various conditions.
\textbf{Theorem 1 (Generalized).} \textit{Let the graph $\left( {\mathcal{N},\mathcal{L}} \right)$ define a connected network and let $\mathcal{T}$ define the shunts of the network. If Assumptions 1 and 2 hold and ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right)$, then:}
\begin{equation}
{\rm rank}\left( {{\bf Y}_\mathcal{N} } \right) = \left\{ {\begin{array}{*{20}l}
{{\rm rank}\left( {{\bf A}_{\mathcal{L},\mathcal{N}}} \right)} & {\text{if} \; \; \mathcal{T} = \emptyset}, \\
{\left| \mathcal{N} \right|} & \text{otherwise}. \\
\end{array}} \right. \label{eq:thm_rank}
\end{equation}
\textbf{Proof.}
First assume that $\mathcal{T} = \emptyset$, then
\begin{equation}
{\bf Y}_{\mathcal{N}} = {\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}}. \label{eq:YN_noshunt}
\end{equation}
Clearly, any vector $\bf w$ such that ${\bf A}_{\mathcal{L},\mathcal{N}}\, {\bf w} = {\bf 0}$ also satisfies ${\bf Y}_{\mathcal{N}}\, {\bf w} = {\bf 0}$. This means that
\begin{equation}
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \supseteq {\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right),
\end{equation}
so
\begin{equation}
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) = {\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right).
\end{equation}
Applying the rank-nullity theorem, we conclude that \eqref{eq:thm_rank} holds for this case. Next assume that $\mathcal{T} \neq \emptyset$, then
\begin{equation}
{\bf Y}_{\mathcal{N}} = {\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}} + {\bf Y}_{\mathcal{T}}.
\end{equation}
If ${\bf A}_{\mathcal{L},\mathcal{N}}$ is full rank, then the fact that ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right)$ and the rank-nullity theorem will imply that ${\bf Y}_{\mathcal{N}}$ is invertible, meaning that \eqref{eq:thm_rank} holds. If ${\bf A}_{\mathcal{L},\mathcal{N}}$ is not full rank, then we take an arbitrary vector ${\bf x} \in {\rm Null}\left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right)$. From Lemma~1 (extended) we have that ${\bf x} = \alpha {\bf u}$ where $\bf u$ is a vector with no null entries. We now calculate ${\bf Y}_{\mathcal{N}}\, {\bf x}$:
\begin{subequations}
\begin{align}
{\bf Y}_{\mathcal{N}}\, {\bf x}&= \alpha {\bf Y}_{\mathcal{N}}\, {\bf u}, \\
{\bf Y}_{\mathcal{N}}\, {\bf x}&= \alpha \left({{\bf A}_{\mathcal{L},\mathcal{N}}^T {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}} + {\bf Y}_{\mathcal{T}} }\right){\bf u}, \\
{\bf Y}_{\mathcal{N}}\, {\bf x}&= \alpha \left({{\bf A}_{\mathcal{L},\mathcal{N}}^T {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}} {\bf u}+{\bf Y}_{\mathcal{T}} {\bf u}}\right), \\
{\bf Y}_{\mathcal{N}}\, {\bf x}&= \alpha {\bf Y}_{\mathcal{T}} {\bf u}.
\end{align}
\end{subequations}
Since ${\bf Y}_{\mathcal{T}}={\rm diag}\left({{\bf y}_{\mathcal{T}}}\right)$, ${\bf y}_{\mathcal{T}} \ne {\bf 0}$, and $\bf u$ has no null entries, we observe that ${\bf Y}_{\mathcal{N}}\, {\bf u}$ cannot be $\bf 0$ unless $ \alpha = 0$. This means the only vector in the null space of ${\bf A}_{\mathcal{L},\mathcal{N}}$ that is also in the null space of ${\bf Y}_{\mathcal{N}}$ is $\bf 0$. This implies that ${\bf Y}_{\mathcal{N}}$ is full-rank, so \eqref{eq:thm_rank} holds. $\hfill \square$
We have recovered the results of \cite{rank_1phase}, at the cost of requiring that the condition ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right)$ holds. With the next theorem, we will show that the problem of verifying the condition for the whole network can be reduced to multiple smaller problems of the same nature.
\textbf{Theorem 2.} \textit{Let the graph $\left( {\mathcal{N},\mathcal{L}} \right)$ define a connected network and let $\mathcal{T}$ define the shunts of the network. Assumptions 1 and 2 hold, ${\rm Re}\left({y_l }\right) \ge 0$ for all $y_l$ of ${\bf y}_\mathcal{L}$ and ${\rm Re}\left({y_t}\right) \ge 0$ for all $y_t$ of ${\bf y}_\mathcal{T}$. Let $\mathcal{G}$ be the set containing the ground node, let $\mathcal{L}' \subseteq \mathcal{L}$ be the set of purely reactive branches, \textcolor{black}{let $\mathcal{N}' \subseteq \mathcal{N}$ be the set of non-isolated nodes\footnote{\textcolor{black}{An \emph{isolated node} of a graph is a node that does not have any graph branches connected to it.}} of the graph $\left({\mathcal{N}, \mathcal{L}'}\right)$}, and let $\mathcal{T}' \subseteq \mathcal{T}$ be the set of purely reactive shunts that are connected to some node in $\mathcal{N}'$. Let the reactive network $\left({\mathcal{N}', \mathcal{L}'}\right)$ have $K$ connected components ($K$ may be $0$), indexed as $\left({\mathcal{N}'(k) \cup \mathcal{G}, \mathcal{L}'(k) \cup \mathcal{T}'(k)}\right)$ for $k = 1, \ldots, K$, and let $\mathcal{T}'(k) \subseteq \mathcal{T}'$ be the set of shunts of component $k$. If the admittance matrices of all components, ${\bf Y}_{\mathcal{N}'(k)}$, satisfy that ${\rm Null}\left({{\bf Y}_{\mathcal{N}'(k)} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)}}\right)$ (for $K=0$ this is vacuously true), then:
\begin{equation*}
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right).
\end{equation*}
}
\textcolor{black}{For the sake of clarity, we split the proof of Theorem~2 into three steps. Informally, these steps are the following:
\begin{enumerate}[label=\arabic*.]
\item We prove that the effect of the purely reactive elements on the invertibility of the admittance matrix is independent of the elements with positive conductance. In particular, we may remove the positive conductance elements while retaining the relationship between the remaining elements and the original system.
\item We prove that each reactive component of the system affects the invertibility of the admittance matrix independently of other components.
\item We use steps 1 and 2 to prove the claim in Theorem~2.
\end{enumerate}
For convenience, we will also use the following conventions throughout the proof:
\begin{itemize}
\item The incidence matrix of an empty branch set is ${\bf A}_{\emptyset,\mathcal{N}} = {\bf 0} \in \mathbb{C}^{1 \times \left|{\mathcal{N}}\right|}$.
\item The series branch admittance matrix of an empty branch set is \textcolor{black}{${\bf Y}_{\emptyset} = 0 \in \mathbb{C}$}.
\item The shunt admittance matrix of an empty shunt set is ${\bf Y}_{\emptyset} = {\bf 0} \in \mathbb{C}^{\left|{\mathcal{N}}\right| \times \left|{\mathcal{N}}\right|}$.
\textcolor{black}{\item Any admittance matrix (branch, shunt, or node) has the rectangular form ${\bf Y}_{\mathcal{S}} = {\bf G}_{\mathcal{S}} + j {\bf B}_{\mathcal{S}}$. ${\bf G}_{\mathcal{S}}$ and ${\bf B}_{\mathcal{S}}$ are real matrices and $\mathcal{S}$ denotes a (branch, shunt, or node) set.}
\end{itemize}
We rely on context to discern between branch and shunt admittance matrices.
}
\textcolor{black}{\textbf{Proof, step 1.}} Assume that both $\mathcal{L}'$ and $\mathcal{L} \setminus \mathcal{L}'$ are non-empty. We can write ${\bf A}_{\mathcal{L},\mathcal{N}}$ in block form as follows:
\begin{equation}
{\bf A}_{\mathcal{L},\mathcal{N}} = \left[ {\begin{array}{*{20}c}
{\bf A}_{\mathcal{L}',\mathcal{N}} \\
{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}} \\
\end{array}} \right]. \label{eq:AVE_blocks}
\end{equation}
Writing ${\bf Y}_{\mathcal{N}}$ in terms of the block matrices yields:
\begin{subequations}
\begin{align}
{\bf Y}_{\mathcal{N}} &= {\bf A}_{\mathcal{L}',\mathcal{N}}^H {\bf Y}_{\mathcal{L}'} {\bf A}_{\mathcal{L}',\mathcal{N}} + {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf Y}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}} \nonumber \\
&\quad + {\bf Y}_{\mathcal{T}'} + {\bf Y}_{\mathcal{T} \setminus \mathcal{T}'}, \\
{\bf Y}_{\mathcal{N}} &= {\bf A}_{\mathcal{L}',\mathcal{N}}^H {\bf Y}_{\mathcal{L}'} {\bf A}_{\mathcal{L}',\mathcal{N}} + {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf Y}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}} \nonumber \\
&\quad + {\bf Y}_{\mathcal{T}'} + {\bf G}_{\mathcal{T} \setminus \mathcal{T}'} + j {\bf B}_{\mathcal{T} \setminus \mathcal{T}'}, \label{eq:YN_L_Lp}
\end{align}
\end{subequations}
Next we compute ${\rm sym}\left( {{\bf G}_{\mathcal{N}} } \right)$ as follows:
\begin{subequations}
\begin{align}
{\bf Y}_{\mathcal{N}} &= {\bf G}_{\mathcal{N}} + j{\bf B}_{\mathcal{N}}, \\
{\bf G}_{\mathcal{N}} &= {\rm Re} \left( {{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}} + {\bf Y}_{\mathcal{T}}} \right), \label{eq:GN_noshunt}
\end{align}
\end{subequations}
let ${\bf A}_R$ and ${\bf A}_I$ denote the real and imaginary parts of ${\bf A}_{\mathcal{L},\mathcal{N}}$, then
\begin{subequations}
\begin{align}
{\bf G}_{\mathcal{N}} &= {\bf A}_R^T {\bf G}_{\mathcal{L}} {\bf A}_R + {\bf A}_I^T {\bf G}_{\mathcal{L}} {\bf A}_I + {\bf A}_I^T {\bf B}_{\mathcal{L}} {\bf A}_R \nonumber \\
&\quad - {\bf A}_R^T {\bf B}_{\mathcal{L}} {\bf A}_I + {\bf G}_{\mathcal{T}}, \\
{\rm sym}\left( {{\bf G}_{\mathcal{N}} } \right) &= {\bf A}_R^T {\bf G}_{\mathcal{L}} {\bf A}_R + {\bf A}_I^T {\bf G}_{\mathcal{L}} {\bf A}_I + {\bf G}_{\mathcal{T}}. \label{eq:Sym_GN_semipos}
\end{align}
\end{subequations}
Notice that:
\begin{subequations}
\small
\begin{align}
{\rm Re} \left({{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf G}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}}}\right) &= {\rm Re} \left({\left({{\bf A}_R + j {\bf A}_I}\right)^H \cdot {\bf G}_{\mathcal{L}} \cdot \left({{\bf A}_R + j {\bf A}_I}\right) \phantom{{\bf A}_{\left( {{\mathcal{N}},{\mathcal{L}}} \right)}^H} \hspace*{-32pt}}\right), \\
{\rm Re} \left({{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf G}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}}}\right) &= {\rm Re} \left({\left({{\bf A}_R^T - j {\bf A}_I^T}\right) \cdot {\bf G}_{\mathcal{L}} \cdot \left({{\bf A}_R + j {\bf A}_I}\right) \phantom{\left({{\bf A}_R^T - j {\bf A}_I^T}\right)} \hspace*{-57pt}}\right), \\
{\rm Re} \left({{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf G}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}}}\right) &= {\bf A}_R^T {\bf G}_{\mathcal{L}} {\bf A}_R + {\bf A}_I^T {\bf G}_{\mathcal{L}} {\bf A}_I,
\end{align}
\end{subequations}
where ${\bf A}_R^T {\bf G}_{\mathcal{L}} {\bf A}_R \succeq {\bf 0}$ and \textcolor{black}{${\bf A}_I^T {\bf G}_{\mathcal{L}} {\bf A}_I \succeq {\bf 0}$} (because all conductances are non-negative), so from Lemma 6 we have that ${\rm Re} \left({{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf G}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}}}\right) \succeq {\bf 0}$. Replacing in \eqref{eq:Sym_GN_semipos}:
\begin{equation}\label{eq:Sym_GN_semipos_simplified}
{\rm sym}\left( {{\bf G}_{\mathcal{N}} } \right) = {\rm Re} \left({{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf G}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}}}\right) + {\bf G}_{\mathcal{T}}.
\end{equation}
From the definition of $\mathcal{L}'$, we have
{\setlength{\parskip}{-1em}
\begin{subequations}
\begin{align}
{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf G}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}} &= {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}, \\
{\rm Re} \left({{\bf A}_{\mathcal{L},\mathcal{N}}^H {\bf G}_{\mathcal{L}} {\bf A}_{\mathcal{L},\mathcal{N}}}\right) &= {\rm Re} \left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right), \\
{\bf G}_{\mathcal{T}} &= {\bf G}_{\mathcal{T} \setminus \mathcal{T}'}.
\end{align}
\end{subequations}
}%
Replacing in~\eqref{eq:Sym_GN_semipos_simplified} yields:
\begin{equation}
{\rm sym}\left( {{\bf G}_{\mathcal{N}} } \right) = {\rm Re} \left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right) + {\bf G}_{\mathcal{T} \setminus \mathcal{T}'}. \label{eq:Sym_GN}
\end{equation}
As all conductances are non-negative, we know that ${\bf G}_{\mathcal{T} \setminus \mathcal{T}'} \succeq {\bf 0}$. Applying Lemma 6, we conclude that ${\rm sym}\left( {{\bf G}_{\mathcal{N}} } \right) \succeq {\bf 0}$ and its null space is the intersection of the null spaces of ${\rm Re} \left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right)$ and ${\bf G}_{\mathcal{T} \setminus \mathcal{T}'}$. As ${\rm sym}\left( {{\bf G}_{\mathcal{N}} } \right) \succeq {\bf 0}$, then ${\bf G}_{\mathcal{N}} \succeq {\bf 0}$ as well, so we can apply Lemma~5:
\begin{subequations}
\label{eq:Sym_GN_null}
\begin{align}
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) &\subseteq {\rm Null}\left({{\rm sym}\left( {{\bf G}_{\mathcal{N}} } \right)}\right), \\
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) &\subseteq {\rm Null}\left({{\rm Re} \left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right)}\right) \nonumber \\
&\quad \cap {\rm Null}\left({{\bf G}_{\mathcal{T} \setminus \mathcal{T}'} }\right).
\end{align}
\end{subequations}
Applying Lemma 7 yields:
\begin{align}
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) &\subseteq {\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right) \nonumber \\
&\quad \cap {\rm Null}\left({{\bf G}_{\mathcal{T} \setminus \mathcal{T}'} }\right).
\end{align}
From the way $\mathcal{L}'$ is defined, we know that ${\bf G}_{\mathcal{L} \setminus \mathcal{L}'} \succ {\bf 0}$ so we can factor ${\bf G}_{\mathcal{L} \setminus \mathcal{L}'}$ as ${\bf G}_{\mathcal{L} \setminus \mathcal{L}'} = {\bf D}^H {\bf D}$, ${\bf D} \succ {\bf 0}$ (in particular, $\bf D$ is invertible). Next, we apply Lemma 3 and Lemma 4:
{\setlength{\parskip}{-1em}
\begin{subequations}
\small
\label{eq:Null_AGA}
\begin{align}
&{\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right) = \nonumber \\
&\hspace*{8.5em} {\rm Null}\left({\left({{\bf D} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right)^H \left({{\bf D} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right)}\right), \\
&{\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right) = {\rm Null}\left({{\bf D} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right), \\
&{\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf G}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right) = {\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right).
\end{align}
\end{subequations}
}%
Substituting into \eqref{eq:Sym_GN_null} yields:
\begin{equation}
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right) \cap {\rm Null}\left({{\bf G}_{\mathcal{T} \setminus \mathcal{T}'} }\right). \label{eq:Null_YN}
\end{equation}
With our established conventions, we note that \eqref{eq:Null_YN} holds even if $\mathcal{L} \setminus \mathcal{L}'$ is empty, so from now on we drop such assumption and only assume that $\mathcal{L}' \neq \emptyset$. Let \textcolor{black}{${\bf v} \in {\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq \mathbb{C}^{|\mathcal{N}|}$}, then ${\bf 0} = {\bf Y}_{\mathcal{N}} {\bf v}$. From \eqref{eq:YN_L_Lp} we have that
\begin{align}
{\bf 0} &= {\bf A}_{\mathcal{L}',\mathcal{N}}^H {\bf Y}_{\mathcal{L}'} {\bf A}_{\mathcal{L}',\mathcal{N}} {\bf v} + {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}^H {\bf Y}_{\mathcal{L} \setminus \mathcal{L}'} {\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}} {\bf v} + {\bf Y}_{\mathcal{T}'} {\bf v} \nonumber \\
&\quad + {\bf G}_{\mathcal{T} \setminus \mathcal{T}'} {\bf v} + j {\bf B}_{\mathcal{T} \setminus \mathcal{T}'} {\bf v}.
\end{align}
From \eqref{eq:Null_YN}, we conclude that ${\bf G}_{\mathcal{T} \setminus \mathcal{T}'} {\bf v} = {\bf 0}$. Both ${\bf G}_{\mathcal{T} \setminus \mathcal{T}'}$ and ${\bf B}_{\mathcal{T} \setminus \mathcal{T}'}$ are diagonal, and the position of the null columns of ${\bf G}_{\mathcal{T} \setminus \mathcal{T}'}$ also correspond to null columns of ${\bf B}_{\mathcal{T} \setminus \mathcal{T}'}$. We conclude that ${\rm Null}\left({{\bf G}_{\mathcal{T} \setminus \mathcal{T}'} }\right) \subseteq {\rm Null}\left({{\bf B}_{\mathcal{T} \setminus \mathcal{T}'} }\right)$ and so ${\bf B}_{\mathcal{T} \setminus \mathcal{T}'} {\bf v} = {\bf 0}$. We also have from \eqref{eq:Null_YN} that ${\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}} {\bf v} = {\bf 0}$. Removing these terms, the equation becomes:
\begin{equation}
{\bf 0} = {\bf A}_{\mathcal{L}',\mathcal{N}}^H {\bf Y}_{\mathcal{L}'} {\bf A}_{\mathcal{L}',\mathcal{N}} {\bf v} + {\bf Y}_{\mathcal{T}'} {\bf v}. \label{eq:YN_nullvec}
\end{equation}
\textcolor{black}{Notice that \eqref{eq:YN_nullvec} does not depend on the positive conductance elements of the system (these are the elements of $\mathcal{L} \setminus \mathcal{L}'$ and $\mathcal{T} \setminus \mathcal{T}'$). Thus, we have completed step 1.}
\textcolor{black}{\textbf{Proof, step 2.}} As we assumed that $\mathcal{L}'$ is non-empty then $K \geq 1$. We assume, without loss of generality, that the nodes of $\mathcal{N}$ and $\mathcal{N}'$ are sorted such that we can write:
\begin{subequations}
\label{eq:A_NLp}
\begin{align}
{\bf A}_{\mathcal{L}',\mathcal{N}} &= \left[ {\begin{array}{*{20}c}
{\bf A}_{\mathcal{L}'(1),\mathcal{N}} \\
{\vdots} \\
{\bf A}_{\mathcal{L}'(K),\mathcal{N}} \\
\end{array}} \right], \label{eq:A_NLp_rows} \\
{\bf A}_{\mathcal{L}'(k),\mathcal{N}} &= \left[ {{\bf 0}_{|\mathcal{L}'(k)| \times |\mathcal{N}'(1)|}, \ldots, {\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)}, \ldots,}\right. \nonumber \\
&\quad\quad \left.{ {\bf 0}_{|\mathcal{L}'(k)| \times |\mathcal{N}'(K)|}, {\bf 0}_{|\mathcal{L}'(k)| \times |\mathcal{N} \setminus \mathcal{N}'|}} \right], \label{eq:A_NLpk} \\
{\bf A}_{\mathcal{L}',\mathcal{N}} &= \left[ {\begin{array}{*{20}c}
{{\bf A}_{\mathcal{L}'(1),\mathcal{N}'(1)}} & {\cdots} & {\bf 0} & {\bf 0}\\
{\vdots} & {\ddots} & {\vdots} & {\vdots} \\
{\bf 0} & {\cdots} & {{\bf A}_{\mathcal{L}'(K),\mathcal{N}'(K)}} & {\bf 0} \\
\end{array}} \right].
\end{align}
\end{subequations}
Similarly:
\begin{subequations}
\label{eq:Yp_Tp_and_Y_Lp}
\begin{align}
{\bf Y}_{\mathcal{T}'} &= \left[ {\begin{array}{*{20}c}
{{\bf Y}_{\mathcal{T}'(1)}'} & {\cdots} & {\bf 0} & {\bf 0} \\
{\vdots} & {\ddots} & {\vdots} & {\vdots} \\
{\bf 0} & {\cdots} & {{\bf Y}_{\mathcal{T}'(K)}'} & {\bf 0} \\
{\bf 0} & {\cdots} & {\bf 0} & {\bf 0} \\
\end{array}} \right], \\
{\bf Y}_{\mathcal{L}'} &= \left[ {\begin{array}{*{20}c}
{{\bf Y}_{\mathcal{L}'(1)}} & {\cdots} & {\bf 0} \\
{\vdots} & {\ddots} & {\vdots} \\
{\bf 0} & {\cdots} & {{\bf Y}_{\mathcal{L}'(K)}} \\
\end{array}} \right],
\end{align}
\end{subequations}
where ${{\bf Y}_{\mathcal{L}'(k)}}$ has size $|\mathcal{L}'(k)| \times |\mathcal{L}'(k)|$ and ${\bf Y}_{\mathcal{T}'(k)}'$ has size $|\mathcal{N}'(k)| \times |\mathcal{N}'(k)|$. Notice that the admittance matrix of the network $\left({\mathcal{N}'(k) \cup \mathcal{G}, \mathcal{L}'(k) \cup \mathcal{T}'(k)}\right)$, using the node in $\mathcal{G}$ as ground, is
\begin{equation}
{\bf Y}_{\mathcal{N}'(k)} = {\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)}^H {\bf Y}_{\mathcal{L}'(k)} {\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)} + {\bf Y}_{\mathcal{T}'(k)}'. \label{eq:YNpk}
\end{equation}
Replacing \eqref{eq:A_NLp}, \eqref{eq:Yp_Tp_and_Y_Lp}, and \eqref{eq:YNpk} in \eqref{eq:YN_nullvec}, we have
\begin{equation}
{\bf 0} = \left[ {\begin{array}{*{20}c}
{{\bf Y}_{\mathcal{N}'(1)}} & {\cdots} & {\bf 0} & {\bf 0} \\
{\vdots} & {\ddots} & {\vdots} & {\vdots} \\
{\bf 0} & {\cdots} & {{\bf Y}_{\mathcal{N}'(K)}} & {\bf 0} \\
{\bf 0} & {\cdots} & {\bf 0} & {\bf 0} \\
\end{array}} \right] {\bf v}.
\end{equation}
Taking only the entries associated with nodes of $\mathcal{N}'(k)$ yields
\begin{equation}
{\bf 0} = {\bf R}_k {\bf v},
\end{equation}
where
\begin{align}
{\bf R}_k &= \left[ {{\bf 0}_{|\mathcal{N}'(k)| \times |\mathcal{N}'(1)|}, \ldots, {\bf Y}_{\mathcal{N}'(k)}, \ldots,}\right. \nonumber \\
&\quad\quad \left.{ {\bf 0}_{|\mathcal{N}'(k)| \times |\mathcal{N}'(K)|}, {\bf 0}_{|\mathcal{L}'(k)| \times |\mathcal{N} \setminus \mathcal{N}'|}} \right]. \label{eq:Rk}
\end{align}
Then, by definition:
\begin{subequations}
\begin{align}
{\bf v} &\in {\rm Null} ({\bf R}_k), \quad \forall k = 1, \ldots, K, \\
{\bf v} &\in \cap_{k=1}^{K} {\rm Null} ({\bf R}_k), \\
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) &\subseteq \cap_{k=1}^{K} {\rm Null} ({\bf R}_k).
\end{align}
\end{subequations}
\textcolor{black}{From \eqref{eq:Rk}, we conclude that each ${\bf R}_k$ is determined by each reactive component, independently of the others. The null space of ${\bf Y}_{\mathcal{N}}$ is contained in the null space of each ${\bf R}_k$, so we have completed step 2.}
\textcolor{black}{\textbf{Proof, step 3.}} The null space of ${\bf R}_k$ can be computed directly as the following Cartesian product:
\begin{align}
{\rm Null} ({\bf R}_k) &= \prod_{i=1}^{k-1}{\mathbb{R}^{|\mathcal{N}'(i)|}} \times {\rm Null} ({\bf Y}_{\mathcal{N}'(k)}) \times \prod_{i=k+1}^{K}{\mathbb{R}^{|\mathcal{N}'(i)|}} \nonumber \\
&\quad \times \mathbb{R}^{|\mathcal{N} \setminus \mathcal{N}'|} \label{eq:Null_Rk}
\end{align}
From the statement of Theorem 2, we have
\begin{equation}
{\rm Null} \left({{\bf Y}_{\mathcal{N}'(k)}}\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)}}\right). \label{eq:Null_YNpk}
\end{equation}
Replacing this in \eqref{eq:Null_Rk} yields
\begin{align}
{\rm Null} ({\bf R}_k) &\subseteq \hspace*{-0.2em} \prod_{i=1}^{k-1}{\mathbb{R}^{|\mathcal{N}'(i)|}} \times \hspace*{-0.2em} {\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)}}\right) \hspace*{-0.2em} \times \hspace*{-0.7em} \prod_{i=k+1}^{K}{\mathbb{R}^{|\mathcal{N}'(i)|}} \nonumber \\
&\quad \times \mathbb{R}^{|\mathcal{N} \setminus \mathcal{N}'|}. \label{eq:Null_Rk_bound}
\end{align}
From \eqref{eq:A_NLpk}, we have
\begin{subequations}
\begin{align}
{\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}}}\right) &= {\rm Null} \left({\left[ {{\bf 0}_{|\mathcal{L}'(k)| \times |\mathcal{N}'(1)|}, \ldots,}\right.}\right. \nonumber \\
&\quad\quad \left.{\left.{{\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)}, \ldots, {\bf 0}_{|\mathcal{L}'(k)| \times |\mathcal{N}'(K)|}, }\right.}\right. \nonumber \\
&\quad\quad \left.{\left.{{\bf 0}_{|\mathcal{L}'(k)| \times |\mathcal{N} \setminus \mathcal{N}'|}} \right]}\right) \\
{\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}}}\right) &= \prod_{i=1}^{k-1}{\mathbb{R}^{|\mathcal{N}'(i)|}} \times {\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}'(k)}}\right) \nonumber \\
&\quad \times \prod_{i=k+1}^{K}{\mathbb{R}^{|\mathcal{N}'(i)|}} \times \mathbb{R}^{|\mathcal{N} \setminus \mathcal{N}'|}.
\end{align}
\end{subequations}
Replacing this in \eqref{eq:Null_Rk_bound} yields
\begin{equation}
{\rm Null} \left({{\bf R}_k}\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}}}\right),
\end{equation}
therefore:
\begin{equation}
{\bf v} \in \cap_{k=1}^{K} {\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}}}\right).
\end{equation}
From \eqref{eq:A_NLp_rows}, we know that the matrices ${\bf A}_{\mathcal{L}'(k),\mathcal{N}}$ are the row blocks of ${\bf A}_{\mathcal{L}',\mathcal{N}}$, and hence
\begin{equation}
{\rm Null} \left({{\bf A}_{\mathcal{L}',\mathcal{N}}}\right) = \cap_{k=1}^{K} {\rm Null} \left({{\bf A}_{\mathcal{L}'(k),\mathcal{N}}}\right).
\end{equation}
Replacing:
\begin{subequations}
\begin{align}
{\bf v} &\in {\rm Null} \left({{\bf A}_{\mathcal{L}',\mathcal{N}}}\right), \\
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) &\subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}',\mathcal{N}}}\right). \label{eq:Null_YN_Lp}
\end{align}
\end{subequations}
Combining \eqref{eq:Null_YN} and \eqref{eq:Null_YN_Lp} yields
\begin{subequations}
\begin{align}
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) &\subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}',\mathcal{N}}}\right) \cap {\rm Null}\left({{\bf G}_{\mathcal{T} \setminus \mathcal{T}'} }\right) \cap {\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right), \\
{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) &\subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}',\mathcal{N}}}\right) \cap {\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right). \label{eq:Null_YN_ANLp_ANLmLp}
\end{align}
\end{subequations}
From \eqref{eq:AVE_blocks}, we know that ${\bf A}_{\mathcal{L}',\mathcal{N}}$ and ${\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}$ are the row blocks of ${\bf A}_{\mathcal{L},\mathcal{N}}$, and thus
\begin{equation}
{\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right) = {\rm Null} \left({{\bf A}_{\mathcal{L}',\mathcal{N}}}\right) \cap {\rm Null}\left({{\bf A}_{\mathcal{L} \setminus \mathcal{L}',\mathcal{N}}}\right). \label{eq:Null_ANL}
\end{equation}
We note that both \eqref{eq:Null_YN_ANLp_ANLmLp} and \eqref{eq:Null_ANL} hold even if $\mathcal{L}'$ is empty, so from now on we drop such assumption. Finally, from \eqref{eq:Null_YN_ANLp_ANLmLp} and \eqref{eq:Null_ANL} we conclude that in general:
\begin{align*}
&&\hspace*{5.4em} {\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right). &&\hspace*{5.4em} \square
\end{align*}
Qualitatively speaking, Theorem 2 is a recursive reduction: we can apply Theorem 1 to the network admittance matrix if we can also apply Theorem 1 to the reactive components of the network (defined by the subgraphs $\left({\mathcal{N}'(k), \mathcal{L}'(k)}\right)$). If there are no such reactive components, then we only require the standard condition of non-negative conductances in order to apply Theorem 1. We still need to prove that the conditions of Theorem 1 hold over the reactive components of the network. In the general case such proof may be too complex or even unattainable. However, we will prove that the conditions hold for common cases of reactive components with simple structures. Moreover, as we will see in the experiments section, the reactive components of practical power systems often have such structures, making the theory practically applicable. We next show the validity of Theorem 1 over several cases.
\textbf{Theorem 3.} \textit{Let the graph $\left( {\mathcal{N},\mathcal{L}} \right)$ define a connected network and let $\mathcal{T}$ define the shunts of the network. Moreover, Assumptions 1 and 2 hold. If the network satisfies at least one of the following conditions:
\begin{enumerate}[label=\arabic*)]
\item \textcolor{black}{$\left( {\mathcal{N},\mathcal{L}} \right)$ is a tree and there exists a root node $r \in \mathcal{N}$ such that equivalent admittance of any node to ground, under the condition that the parent node (if any) is grounded, is non-zero}\textcolor{black}{\footnote{\textcolor{black}{In power system terms, a \emph{tree} is a radial network, the \emph{root node} is the feeder node, the \emph{parent} of a node is the next node (the only one) when moving up towards the feeder, a \emph{child} of a node is one of the next nodes when going down from the feeder, and a \emph{leaf} is one of the end nodes when going down the feeder. For formal definitions of the terms, see \cite{graph_theory}.}}}.
\item $\left( {\mathcal{N},\mathcal{L}} \right)$ is a tree and $\mathcal{T} = \emptyset$.
\item There are only inductors or there are only capacitors.
\end{enumerate}
then ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right)$.
}
\textbf{Proof, Condition \textit{1)}.}
Let \textcolor{black}{${\bf v} \in {\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq \mathbb{C}^{|\mathcal{N}|}$} and define the vectors ${\bf i}_{\mathcal{L}} = {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L}, \mathcal{N}}\, {\bf v}$ and ${\bf i}_{\mathcal{T}} = {\bf Y}_{\mathcal{T}}\, {\bf v}$. We have that
\begin{subequations}
\begin{align}
{\bf 0} &= {\bf Y}_{\mathcal{N}}\, {\bf v} \\
{\bf 0} &= {\bf A}_{\mathcal{L}, \mathcal{N}}^H\, {\bf i}_{\mathcal{L}} + {\bf i}_{\mathcal{T}}. \label{eq:YN_loop}
\end{align}
\end{subequations}
We define $\mathcal{V}(0) \subseteq \mathcal{N}$ as the leaves of tree $\left( {\mathcal{N},\mathcal{L}} \right)$ (not including the root node $r$, see\cite{graph_theory}). For $l>0$, we define $\mathcal{V}(l) \subseteq \mathcal{N}$ as the set of nodes having all their children in $\cup_{k=0}^{l-1} \mathcal{V}(k)$, but do not belong to $\cup_{k=0}^{l-1} \mathcal{V}(k)$ themselves (i.e. $\mathcal{V}(l) \cap (\cup_{k=0}^{l-1} \mathcal{V}(k)) = \emptyset$). The \textit{height} of the tree is the unique integer $L$ such that $\{r\} = \mathcal{V}(L)$. The sets $\mathcal{V}(0), \ldots, \mathcal{V}(L)$ form a partition of $\mathcal{N}$. We also define
\begin{equation}
{\bf Y}^{i,k} = \left[ {\begin{array}{*{20}c}
{y_{11}^{i,k} } & {y_{12}^{i,k} } \\
{y_{21}^{i,k} } & {y_{22}^{i,k} } \\
\end{array}} \right],
\end{equation}
as the $2 \times 2$ admittance matrix formed by considering only nodes $i$ and $k$ (in that order), and all branches connecting them (shunts excluded). Lastly, we define $i^{i,k}$ as
\begin{equation}
i^{i,k} = y^{i,k}_{11} \left\{{\bf v}\right\}_i + y^{i,k}_{12} \left\{{\bf v}\right\}_k.
\end{equation}
Consider a node $n \in \mathcal{V}(l)$ for some $l < L$ (so $n \neq r$). Let $\mathcal{C}(n)$ be index set of all branches connecting $n$ to some child node, let $p$ be the parent node of $n$, and let $k$ be the index of the branch connecting $n$ and $p$. The scalar equation of \eqref{eq:YN_loop} associated with node $n$ is
\begin{equation}
0 = y^{p,n}_{21} \left\{{\bf v}\right\}_p + y^{p,n}_{22} \left\{{\bf v}\right\}_n + \left\{{{\bf i}_{\mathcal{T}}}\right\}_n + \sum\nolimits_{i \in \mathcal{C}(n)} i^{n,i}. \label{eq:iL_i}
\end{equation}
We assume for induction that for any $i^{n,i} \in \mathcal{C}(n)$ we can write
\begin{equation}
i^{n,i} = y^b_i \left\{{\bf v}\right\}_n, \qquad y^b_i \in \mathbb{C}, \label{eq:ybi}
\end{equation}
for some finite $y^b_i$. We recall that if $l = 0$ then $n$ is a leaf node, hence $\mathcal{C}(n) = \emptyset$ and the induction hypothesis holds vacuously. Let the shunt of node $n$ be $y^s_n = \left\{{{\bf Y}_{\mathcal{T}}}\right\}_{nn}$, then from \eqref{eq:ybi} and the definition of ${\bf i}_{\mathcal{T}}$ we get that
\begin{subequations}
\begin{align}
0 &= y^{p,n}_{21} \left\{{\bf v}\right\}_p + y^{p,n}_{22} \left\{{\bf v}\right\}_n + y^s_n \left\{{\bf v}\right\}_n + \sum\nolimits_{i \in \mathcal{C}(n)} y^b_i \left\{{\bf v}\right\}_n, \\
0 &= y^{p,n}_{21} \left\{{\bf v}\right\}_p + \left({y^{p,n}_{22} + y^{sb}_n}\right) \left\{{\bf v}\right\}_n, \label{eq:ysb_n_vn}
\end{align}
\end{subequations}
where
\begin{equation}
y^{sb}_n = y^s_n + \sum\nolimits_{i \in \mathcal{C}(n)} y^b_i. \label{eq:ysb_n}
\end{equation}
Multiplying by $y^{p,n}_{12}$ on both sides of \eqref{eq:ysb_n_vn} we get that
\begin{equation}
0 = y^{p,n}_{21} y^{p,n}_{12} \left\{{\bf v}\right\}_p + \left({y^{p,n}_{22} + y^{sb}_n}\right) y^{p,n}_{12} \left\{{\bf v}\right\}_n.
\end{equation}
Notice that:
\begin{subequations}
\begin{align}
i^{p,n} &= y^{p,n}_{11} \left\{{\bf v}\right\}_p + y^{p,n}_{12} \left\{{\bf v}\right\}_n, \\
y^{p,n}_{12} \left\{{\bf v}\right\}_n &= i^{p,n} - y^{p,n}_{11} \left\{{\bf v}\right\}_p,
\end{align}
\end{subequations}
hence
\begin{equation}
0 = y^{p,n}_{21} y^{p,n}_{12} \left\{{\bf v}\right\}_p + \left({y^{p,n}_{22} + y^{sb}_n}\right) \left({i^{p,n} - y^{p,n}_{11} \left\{{\bf v}\right\}_p}\right).
\end{equation}
The term $y^{p,n}_{22} + y^{sb}_n$ is the equivalent admittance between node $n$ and ground, under the condition of node $p$ being grounded. Hence $y^{p,n}_{22} + y^{sb}_n \neq 0$ according to Condition \textit{1)}, and
\begin{subequations}
\begin{align}
i^{p,n} &= \left({y^{p,n}_{11} - \frac{y^{p,n}_{12}y^{p,n}_{21}}{y^{p,n}_{22} + y^{sb}_n}}\right) \left\{{\bf v}\right\}_p, \\
i^{p,n} &= y^b_n \left\{{\bf v}\right\}_p,
\end{align}
\end{subequations}
where
\begin{equation}
y^b_n = y^{p,n}_{11} - \frac{y^{p,n}_{12}y^{p,n}_{21}}{y^{p,n}_{22} + y^{sb}_n}.
\end{equation}
We conclude that the induction hypothesis holds for any node $n \neq r$. We remark that $y^b_n$ is finite because $y^{p,n}_{22} + y^{sb}_n \neq 0$. Now we write the scalar equation of \eqref{eq:YN_loop} associated with the root node $r$:
\begin{subequations}
\begin{align}
0 &= \left\{{{\bf i}_{\mathcal{T}}}\right\}_{r} + \sum\nolimits_{i \in \mathcal{C}(r)} i^{r,i}, \\
0 &= y^s_r \left\{{\bf v}\right\}_r + \sum\nolimits_{i \in \mathcal{C}(r)} y^b_i \left\{{\bf v}\right\}_r, \\
0 &= y^{sb}_r \left\{{\bf v}\right\}_r,
\end{align}
\end{subequations}
\textcolor{black}{so $y^{sb}_r$ is the equivalent admittance of the root node $r$ to ground. As node $r$ has no parent, Condition \textit{1)} states that $y^{sb}_r$ is non-zero, so we conclude that $\left\{{\bf v}\right\}_r = 0$.} Now we propose a backward induction hypothesis: for every node $m \in \cup_{k=l}^{L} \mathcal{V}(k), l>0$ we have that $\left\{{\bf v}\right\}_m = 0$ (which trivially holds for $l=L$). We take any node $n \in \mathcal{V}(l-1)$, let $p$ be the parent node of $n$, then $p \in \cup_{k=l}^{L} \mathcal{V}(k)$ and so $\left\{{{\bf v}}\right\}_p = 0$. As $y^{p,n}_{21}$ is finite, we get from \eqref{eq:ysb_n_vn} that
\begin{equation}
0 = \left({y^{p,n}_{22} + y^{sb}_n}\right) \left\{{\bf v}\right\}_n,
\end{equation}
and as $y^{p,n}_{22} + y^{sb}_n \neq 0$ we conclude that $\left\{{\bf v}\right\}_n = 0$, proving the induction hypothesis. This means that ${\bf v} = {\bf 0}$, so
\begin{align*}
&&\hspace*{3.9em} {\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) = \{{\bf 0}\} \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right). &&\hspace*{3.9em} \square
\end{align*}
\textbf{Proof, Condition \textit{2)}.}
In this case we have that ${\bf Y}_{\mathcal{N}} = {\bf A}_{\mathcal{L}, \mathcal{N}}^H {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L}, \mathcal{N}}$ (see \eqref{eq:YN_noshunt}), and thus ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \supseteq {\rm Null} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right)$. We also know, since $\left( {\mathcal{N},\mathcal{L}} \right)$ is a tree, that the network has exactly $|\mathcal{N}|-1$ branches. This means that ${\bf A}_{\mathcal{L}, \mathcal{N}}$ has size $|\mathcal{N}|-1 \times |\mathcal{N}|$, so from Lemma~1 we have that ${\rm rank} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right) = |\mathcal{N}| - 1$. Applying the Frobenius inequality (see exercise 4.5.17 in \cite{linear_algebra}) to \eqref{eq:YN_noshunt}, we have
\begin{subequations}
\begin{align}
&{\rm rank} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}^H {\bf Y}_{\mathcal{L}}}\right) + {\rm rank} \left({{\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L}, \mathcal{N}}}\right) \leq \nonumber \\
&\hspace*{6.5em} {\rm rank} \left({{\bf Y}_{\mathcal{L}}}\right) + {\rm rank} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}^H {\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L}, \mathcal{N}}}\right), \\
&{\rm rank} \left({{\bf Y}_{\mathcal{N}}}\right) \geq {\rm rank} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}^H {\bf Y}_{\mathcal{L}}}\right) + {\rm rank} \left({{\bf Y}_{\mathcal{L}} {\bf A}_{\mathcal{L}, \mathcal{N}}}\right) \nonumber \\
&\hspace*{6.5em} - {\rm rank} \left({{\bf Y}_{\mathcal{L}}}\right).
\end{align}
\end{subequations}
Applying Lemma 4 and the fact that ${\bf Y}_{\mathcal{L}}$ is square and full rank, we get that
\begin{subequations}
\begin{align}
{\rm rank} \left({{\bf Y}_{\mathcal{N}}}\right) &\geq {\rm rank} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}^H}\right) + {\rm rank} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right) \nonumber \\
&\quad - {\rm rank} \left({{\bf Y}_{\mathcal{L}}}\right), \\
{\rm rank} \left({{\bf Y}_{\mathcal{N}}}\right) &\geq |\mathcal{N}| - 1.
\end{align}
\end{subequations}
Applying the rank-nullity theorem:
\begin{equation}
{\rm dim} \left({{\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right)}\right) \leq 1 = {\rm dim} \left({{\rm Null}\left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right)}\right),
\end{equation}
but ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \supseteq {\rm Null} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right)$, and as they have equal dimension then ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) = {\rm Null} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right)$. This trivially implies that ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right)$. $\hfill \square$
\textbf{Proof, Condition \textit{3)}.}
As the network is purely inductive (or purely capacitive) we can write
\begin{subequations}
\begin{align}
{\bf Y}_{\mathcal{N}} &= {\bf A}_{\mathcal{L}, \mathcal{N}}^H \left({jk {\bf B}_{\mathcal{L}}}\right) {\bf A}_{\mathcal{L}, \mathcal{N}} + \left({jk {\bf B}_{\mathcal{T}}}\right), \\
{\bf Y}_{\mathcal{N}} &= jk \left({{\bf A}_{\mathcal{L}, \mathcal{N}}^H {\bf B}_{\mathcal{L}} {\bf A}_{\mathcal{L}, \mathcal{N}} + {\bf B}_{\mathcal{T}}}\right),
\end{align}
\end{subequations}
where ${\bf B}_{\mathcal{L}}$ and ${\bf B}_{\mathcal{T}}$ are diagonal matrices with non-negative real entries, and $k=1$ if the network is purely capacitive or $k=-1$ if the network is purely inductive. Now we consider an alternative network with a set of nodes $\mathcal{N}'$ identical to $\mathcal{N}$, a set of branches $\mathcal{L}'$ such that ${\bf A}_{\mathcal{L}', \mathcal{N}'}={\bf A}_{\mathcal{L}, \mathcal{N}}$ and ${\bf Y}_{\mathcal{L}'}= {\bf B}_{\mathcal{L}}$, and a set of shunts $\mathcal{T}'$ such that ${\bf Y}_{\mathcal{T}'}= {\bf B}_{\mathcal{T}}$. The admittance matrix of the alternative network is:
\begin{equation}
{\bf Y}_{\mathcal{N}'} = {\bf A}_{\mathcal{L}, \mathcal{N}}^H {\bf B}_{\mathcal{L}} {\bf A}_{\mathcal{L}, \mathcal{N}} + {\bf B}_{\mathcal{T}},
\end{equation}
therefore
\begin{equation}
{\bf Y}_{\mathcal{N}} = jk {\bf Y}_{\mathcal{N}'}
\end{equation}
Notice that the alternative network satisfies Assumptions 1 and 2 and is purely resistive with no negative conductances. Hence ${\bf Y}_{\mathcal{N}'}$ satisfies Theorem 2. As $jk \neq 0$ we have that ${\rm Null} \left({{\bf Y}_{\mathcal{N}}}\right) = {\rm Null} \left({{\bf Y}_{\mathcal{N}'}}\right)$. Moreover, we know that ${\bf A}_{\mathcal{L}', \mathcal{N}'} = {\bf A}_{\mathcal{L}, \mathcal{N}}$, so ${\rm Null}\left({{\bf Y}_{\mathcal{N}} }\right) \subseteq {\rm Null} \left({{\bf A}_{\mathcal{L}, \mathcal{N}}}\right)$. $\hfill \square$
\begin{figure}[t]
\tikzstyle{startstop} = [rectangle, rounded corners, minimum width=3cm, minimum height=1cm,text centered, draw=black, text width=3.5cm, fill=blue!30]
\tikzstyle{io} = [trapezium, trapezium left angle=70, trapezium right angle=110, minimum width=0.5cm, minimum height=0.65cm, text centered, draw=black, fill=green!30]
\tikzstyle{process} = [rectangle, minimum height=0.65cm, text centered, draw=black, text width=3.5cm, fill=orange!15]
\tikzstyle{decision} = [diamond, minimum width=0.5cm, text centered, draw=black, aspect=1.5, text width=1.5cm, fill=cyan!20]
\tikzstyle{line} = [thick,-,>=stealth]
\tikzstyle{arrow} = [thick,->,>=stealth]
\scalebox{0.725}{
\begin{tikzpicture}[x=1cm,y=1cm,node distance=1.5cm]
\node (pro1) [process] {Sort lines and reduce parallels};
\node (in1) [io, right of=pro1, xshift=2.0cm, inner xsep=-0.1cm] {Get $(\mathcal{N,L,T})$};
\node (start) [startstop, right of=in1, xshift=2.0cm] {Start};
\node (dec1) [decision, below of=pro1, yshift=-0.5cm, minimum width=3.0cm, aspect=1.0] {} (dec1) node[text centered, text width=2.0cm] {Thm. 2 assumptions hold?};
\node (pro2) [process, below of=dec1, yshift=-0.5cm] {Compute $(\mathcal{N',L',T'})$};
\node (pro3) [process, below of=pro2] {Get all reac. comps. $(\mathcal{N}'(k),\mathcal{L}'(k),\mathcal{T}'(k))$ for $k=1,\cdots,K$};
\node (pro4) [process, below of=pro3] {Set $k \gets 1$};
\node (dec2) [decision, below of=pro4, yshift=0.0cm, minimum width=2.0cm, aspect=1.5] {} (dec2) node[text centered] {$k \leq K$?};
\node (pro5) [process, below of=dec2, text width=1.6cm] {$k \gets k+1$};
\node (dec3) [decision, below of=pro5, yshift=-0.75cm, minimum width=4.5cm, aspect=0.8] {} (dec3) node[text centered, text width=3.0cm, yshift=-0.1cm] {Thm. 3 holds for $(\mathcal{N}'(k),\mathcal{L}'(k),\mathcal{T}'(k))$ ?};
\node (dec4) [decision, right of=dec2, xshift=2.0cm, minimum width=2.0cm, aspect=1.5] {} (dec4) node[text centered] {$\mathcal{T} = \emptyset$?};
\node (dec5) [decision, above of=dec4, yshift=0.5cm, minimum width=2.5cm, aspect=1.25] {} (dec5) node[text centered, text width=2.0cm, yshift=-0.03cm] {Is $(\mathcal{N,L})$ a tree?};
\node (out1) [io, right of=dec5, xshift=2.0cm, inner xsep=-0.3cm] {``${\bf Y}_\mathcal{N}$ is invertible''};
\node (out2) [io, above of=dec5, yshift=0.5cm, text width=2.0cm, inner xsep=0.0cm] {``${\rm rank}\left({{\bf Y}_\mathcal{N}}\right) =$ $ {\rm rank}\left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right)$''};
\draw (out2 -| out1) node(end) [startstop] {End};
\draw (dec1 -| end) node(out3) [io, text width=2.0cm, inner xsep=-0.15cm] {``Theorems cannot be applied''};
\draw [arrow] (start) -- (in1);
\draw [arrow] (in1) -- (pro1);
\draw [arrow] (pro1) -- (dec1);
\draw [arrow] (pro2) -- (pro3);
\draw [arrow] (pro3) -- (pro4);
\draw [arrow] (pro4) -- (dec2);
\draw [arrow] (out1) -- (end);
\draw [arrow] (out2) -- (end);
\draw [arrow] (out3) -- (end);
\draw [arrow] (pro5) -- (dec2);
\draw [arrow] (dec1) -- node[anchor=east] {yes} (pro2);
\draw [arrow] (dec1) -- node[anchor=south] {no} (out3);
\draw [arrow] (dec2) -- node[anchor=north] {no} (dec4);
\draw [arrow] (dec3) -- node[anchor=west] {yes} (pro5);
\draw [arrow] (dec4) -- node[anchor=east] {yes} (dec5);
\draw [arrow] (dec5) -- node[anchor=south] {yes} (out1);
\draw [arrow] (dec5) -- node[anchor=east] {no} (out2);
\draw [arrow] (dec2) -- node[anchor=south] {yes} ++(-2.75,0.0) |- (dec3);
\draw [arrow] (dec4) -- node[anchor=north] {no} (dec4 -| out1) -- (out1);
\draw [arrow] (dec3) -- node[anchor=north] {no} ($(dec3 -| out3) + (2.25,0.0)$) |- (out3);
\end{tikzpicture}
}
\caption{\textcolor{black}{Flowchart describing an algorithm to certify the invertibility (or singularity) of an admittance matrix through the use of Theorems 1 to 3.}}
\label{fig:algorithm}
\vspace{-1em}
\end{figure}
\textcolor{black}{We now have enough tools to construct an algorithm to check the invertibility of the admittance matrix. First, we reduce any parallel lines in order to comply with Remark~1. Then we check if the network satisfies the assumptions of Theorem~2; if so, we compute the reactive subsystem $(\mathcal{N',L',T'})$ by removing all elements with positive resistance. Afterwards, we compute all the $K$ connected components of $(\mathcal{N',L',T'})$ (when $(\mathcal{N',L',T'})$ is empty, then $K=0$). These connected components are computed using the Breadth First Search (BFS) algorithm \cite{algorithms}, whose complexity is $\mathcal{O}\left({|\mathcal{N}| + |\mathcal{L}|}\right)$ (linear in the system size). For each connected component $(\mathcal{N}'(k),\mathcal{L}'(k),\mathcal{T}'(k)), k=1,\ldots,K$, we check if Theorem~3 can be applied to the component through any of its conditions. If Theorem~3 holds for all reactive components, then Theorem~2 holds for the networks and thus Theorem~1 holds as well. Finally, using Theorem~1, we can certify the invertibility of the admittance matrix if the network has shunts. Otherwise, ${\rm rank}\left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right)$ needs to be computed. In the special case that the network is radial, we have that ${\rm rank}\left({{\bf A}_{\mathcal{L},\mathcal{N}}}\right) = |\mathcal{N}| - 1$, and thus if there are no shunts we can certify that the admittance matrix is \textit{singular}. A flowchart of the algorithm is shown Fig.~\ref{fig:algorithm}. To illustrate the idea behind the algorithm, consider the example system of Fig.~\ref{fig:alg_example}. The one-line diagram of the system is shown in Fig.~\ref{fig:alg_example_circuit}. In Fig.~\ref{fig:alg_example_comps}, we have the circuit model of the system, where the loads are modelled as constant admittances and each transmission line is modeled using a $\pi$ circuit. The example system possesses two reactive components, outlined in the figure (shunt loads are not included in the components, as they have a resistive part). If the main condition of Theorem~2 can be proved for each component (by means of Theorem~3, for example), then Theorem~2 will hold for the system, and thus Theorem~1 will holds for the system as well. The branches of the first component, $(\mathcal{N}'(1),\mathcal{T}'(1))$, form a tree. Choosing node 6 as root, we obtain the node partition shown in Fig.~\ref{fig:alg_example_tree}. This partition can be used to check Condition \textit{1)} of Theorem~3.}
\begin{figure}[t]
\centering
\vspace{-1em}
\subfloat[One-line diagram of the example system.]{
\scalebox{1.0}{
\begin{circuitikz}[american voltages, scale=0.6]
\ctikzset{monopoles/vee/arrow={Triangle[width=0.4*\scaledwidth, length=0.8*\scaledwidth]}}
\draw
(0,0)
node[tlground]{}
to[sV, sources/scale=0.7, fill=white] ++(0,2) -- ++(0.5,0)
to[bus=0.6, l_=$1$] ++(1.5,0)
++(-0.5,0) to[oosourcetrans, fill=white] ++(2,0)
-- ++(0.3,0)
to[bus=0.6, l_=$2$, name=n2] ++(0.1,0)
++(0,-0.3)-- ++(3,0) ++(0,0.3)
to[bus=0.6, l_=$3$, name=n3] ++(0.1,0)
-- ++(0.3,0)
to[oosourcetrans, fill=white] ++(2,0)
++(-0.5,0) to[bus=0.6, l_=$4$] ++(1.5,0) -- ++(0.5,0)
++(0,-2) node[tlground]{}
to[sV, sources/scale=0.7, fill=white] ++(0,2)
;
\draw[color=white]
(n2.center) ++(0.5,0.3) to[ooosource, name=twt, color=black, fill=white] ++(0,3)
;
\draw
(n2.center) ++(0,0.3) -- ++(0.5,0) -- (twt.left)
(twt.sec1) -- ++(0,0.5) -- ++(-0.5,0) -- ++(0,1.1)
++(-0.3,0) to[bus=0.6, l=$5$] ++(0,0)
++(-0.3,0) -- ++(0,-0.5) node[vee]{}
(twt.tert1) -- ++(0,0.5) -- ++(0.5,0) -- ++(0,1.1)
++(0.3,0) to[bus=0.6, l_=$6$] ++(0,0)
++(0.3,0) -- ++(0,-0.5) -- ++(2,0)
++(0,-0.3) to[bus=0.6, l=$7$, name=n7] ++(0.1,0)
-- ++(0.5,0) -- ++(0,-0.5) node[vee]{}
(n7.center) ++(0,-0.3) -- ++(-1.5,0) |- ($(n3.center) + (0,0.3)$)
;
\end{circuitikz}
} \label{fig:alg_example_circuit}
} \\
\subfloat[Circuit model of the system with its reactive components outlined.]{
\scalebox{1.0}{
\begin{circuitikz}[american voltages, american resistors, scale=0.6]
\ctikzset{resistors/scale=0.6, inductors/scale=0.6, capacitors/scale=0.6}
\draw
(0,0) node[name=n1]{} node[anchor=east]{$1$}
to[L, *-*] ++(2,0) node[name=n2]{} node[anchor=south west]{$2$}
to[C] ++(0,-1.5) node[tlground]{}
(n2.center) -- ++(0,0.5) to[L, -*] ++(0,3.75) node[name=n8]{} node[anchor=south]{$8$}
to[L, -*, mirror] ++(-2,0) node[anchor=south]{$5$}
-- ++(-1.5,0) to[R, european resistors] ++(0,-2) node[tlground]{}
(n8.center) to[L, -*] ++(2,0) node[name=n6]{} node[anchor=south]{$6$}
to[C] ++(0,-1.5) node[tlground]{}
(n6.center) -- ++(1,0) to[R] ++(1.5,0)
to[L] ++(1.5,0) -- ++(0.5,0)
to[C] ++(0,-1.5) node[tlground]{} ++(0,1.5)
-- ++(1,0) node[circ, name=n7]{} node[anchor=south]{$7$}
-- ++(1.5,0) to[R, european resistors] ++(0,-2) node[tlground]{}
(n7.center) -- ++(0,-0.25) to[R] ++(0,-1.5)
to[L] ++(0,-1.5) -- ++(0,-1) node[circ, name=n3]{} node[anchor=south east]{$3$}
(n2.center) -- ++(3,0) to[R] ++(1.5,0)
to[L] ++(1.5,0) -- (n3.center)
to[C] ++(0,-1.5) node[tlground]{}
(n3.center) to[L] ++(2,0) node[circ]{} node[name=n4]{} node[anchor=west]{$4$}
;
\draw[red, thick, dashed]
($(n1.center)+(-0.75,-2)$) node[name=rec1_sw]{}
rectangle ($(n6.center)+(0.6,0.8)$) node[midway, name=rec1_center]{}
(rec1_center |- rec1_sw) node[anchor=north]{$\mathcal{N}'(1), \mathcal{L}'(1), \mathcal{T}'(1)$}
($(n3.center)+(-0.75,-2)$) node[name=rec2_sw]{}
rectangle ($(n4.center)+(0.6,0.8)$) node[midway, name=rec2_center]{}
(rec2_center |- rec2_sw) node[anchor=north]{$\mathcal{N}'(2), \mathcal{L}'(2), \mathcal{T}'(2)$}
;
\end{circuitikz}
} \label{fig:alg_example_comps}
} \\
\subfloat[Node partition for the tree $(\mathcal{N}'(1),\mathcal{L}'(1))$, using node 6 as root.]{
\scalebox{1.0}{
\begin{circuitikz}[american voltages, american resistors, scale=0.6]
\ctikzset{resistors/scale=0.6, inductors/scale=0.6, capacitors/scale=0.6}
\draw
(0,0) node[circ, name=n1]{} node[anchor=east]{$1$}
to[L] ++(3.5,0)
node[circ, name=n2]{} node[anchor=west]{$2$}
-- ++(0,-0.75) to[C] ++(0,-1.5) node[tlground]{}
(n2.center) -- ++(0,0.5) to[L, -*] ++(0,3.75) node[name=n8]{} node[anchor=south]{$8$}
to[L, mirror] ++(-3.5,0)
node[circ, name=n5]{} node[anchor=east]{$5$}
(n8.center) to[L, -*] ++(3.5,0) node[name=n6]{} node[anchor=west]{$6$}
-- ++(0,-0.75) to[C] ++(0,-1.5) node[tlground]{}
;
\draw[red, thick, dashed]
($(n1.center)+(-0.8,-0.8)$) node[name=rec0_sw]{}
rectangle ($(n5.center)+(0.8,0.8)$) node[midway, name=rec0_center]{}
(rec0_center |- rec0_sw) node[anchor=north]{$\mathcal{V}(0)$}
($(n2.center)+(-0.8,-0.8)$) rectangle ($(n2.center)+(0.8,0.8)$) node[midway, name=rec1_center]{} node[name=rec1_ne]{}
(rec1_center -| rec1_ne) node[anchor=west]{$\mathcal{V}(1)$}
($(n8.center)+(-0.8,-0.8)$) node[name=rec2_sw]{}
rectangle ($(n8.center)+(0.8,0.8)$) node[name=rec2_ne]{}
(rec2_sw -| rec2_ne) node[anchor=north west, shift=({-0.1,0.1})]{$\mathcal{V}(2)$}
($(n6.center)+(-0.8,-0.8)$) rectangle ($(n6.center)+(0.8,0.8)$) node[midway, name=rec3_center]{} node[name=rec3_ne]{}
(rec3_center -| rec3_ne) node[anchor=west]{$\mathcal{V}(3)$}
;
\end{circuitikz}
} \label{fig:alg_example_tree}
}%
\caption{Example system to illustrate how to apply the main theorems.}
\label{fig:alg_example}
\vspace{-1em}
\end{figure}
\section{Implementation and Test Cases}
\label{sec:implementation}
\begin{table}[t]
\centering
\caption{PGLib Test Cases Used for Checking the Theorems}
\label{tbl:test_cases}
\hspace*{-1em}
\textcolor{black}{
\scalebox{0.95}{
\begin{tabular}{|l|r|r|r|l|l|}
\hline
Test case & $|\mathcal{N}|$ & $|\mathcal{L}|$ & Reac. & Satisfy thm. & ${\bf Y}_{\mathcal{N}}$ non- \\
& & & line \% & conditions? & singular? \\
\hline
case3\_lmbd &3 &3 & 0.0\% &Yes &Yes \\
\hline
case5\_pjm &5 &6 & 0.0\% &Yes &Yes \\
\hline
case14\_ieee &14 &20 &25.0\% &No &- \\
\hline
case24\_ieee\_rts &24 &38 & 0.0\% &Yes &Yes \\
\hline
case30\_as &30 &41 &17.1\% &No &- \\
\hline
case30\_ieee &30 &41 &17.1\% &No &- \\
\hline
case39\_epri &39 &46 & 8.7\% &Yes &Yes \\
\hline
case57\_ieee &57 &80 &22.5\% &No &- \\
\hline
case60\_c &60 &88 &40.9\% &Yes &Yes \\
\hline
case73\_ieee\_rts &73 &120 & 0.8\% &Yes &Yes \\
\hline
case89\_pegase &89 &210 & 4.8\% &Yes &Yes \\
\hline
case118\_ieee &118 &186 & 4.8\% &Yes &Yes \\
\hline
case162\_ieee\_dtc &162 &284 &11.6\% &Yes &Yes \\
\hline
case179\_goc &179 &263 &27.4\% &Yes &Yes \\
\hline
case200\_activ &200 &245 & 0.0\% &Yes &Yes \\
\hline
case240\_pserc &240 &448 &20.8\% &Yes &Yes \\
\hline
case300\_ieee &300 &411 &15.6\% &No &- \\
\hline
case500\_goc &500 &733 & 0.0\% &Yes &Yes \\
\hline
case588\_sdet &588 &686 & 7.1\% &No &- \\
\hline
case793\_goc &793 &913 & 0.7\% &No &- \\
\hline
case1354\_pegase &1354 &1991 & 0.1\% &Yes &Yes \\
\hline
case1888\_rte &1888 &2531 & 9.8\% &Yes &Yes \\
\hline
case1951\_rte &1951 &2596 &13.1\% &Yes &Yes \\
\hline
case2000\_goc &2000 &3639 & 0.0\% &Yes &Yes \\
\hline
case2312\_goc &2312 &3013 & 0.0\% &No &- \\
\hline
case2383wp\_k &2383 &2896 & 6.7\% &Yes &Yes \\
\hline
case2736sp\_k &2736 &3504 & 1.1\% &Yes &Yes \\
\hline
case2737sop\_k &2737 &3506 & 1.1\% &No &- \\
\hline
case2742\_goc &2742 &4673 & 0.0\% &Yes &Yes \\
\hline
case2746wop\_k &2746 &3514 & 1.1\% &No &- \\
\hline
case2746wp\_k &2746 &3514 & 1.1\% &Yes &Yes \\
\hline
case2848\_rte &2848 &3776 & 5.5\% &Yes &Yes \\
\hline
case2853\_sdet &2853 &3921 & 9.7\% &No &- \\
\hline
case2868\_rte &2868 &3808 & 6.7\% &Yes &Yes \\
\hline
case2869\_pegase &2869 &4582 & 3.0\% &No &- \\
\hline
case3012wp\_k &3012 &3572 & 0.3\% &No &- \\
\hline
case3022\_goc &3022 &4135 & 5.2\% &No &- \\
\hline
case3120sp\_k &3120 &3693 & 0.3\% &No &- \\
\hline
case3375wp\_k &3374 &4161 & 0.6\% &No &- \\
\hline
case3970\_goc &3970 &6641 & 0.0\% &Yes &Yes \\
\hline
case4020\_goc &4020 &6988 & 0.0\% &Yes &Yes \\
\hline
case4601\_goc &4601 &7199 & 0.0\% &Yes &Yes \\
\hline
case4619\_goc &4619 &8150 & 0.0\% &Yes &Yes \\
\hline
case4661\_sdet &4661 &5997 & 1.6\% &No &- \\
\hline
case4837\_goc &4837 &7765 & 0.0\% &Yes &Yes \\
\hline
case4917\_goc &4917 &6726 & 2.6\% &No &- \\
\hline
case6468\_rte &6468 &9000 & 2.1\% &Yes &Yes \\
\hline
case6470\_rte &6470 &9005 & 2.4\% &Yes &Yes \\
\hline
case6495\_rte &6495 &9019 & 2.8\% &Yes &Yes \\
\hline
case6515\_rte &6515 &9037 & 2.9\% &Yes &Yes \\
\hline
case8387\_pegase &8387 &14561 & 4.3\% &No &- \\
\hline
case9241\_pegase &9241 &16049 & 5.3\% &No &- \\
\hline
case9591\_goc &9591 &15915 & 0.0\% &Yes &Yes \\
\hline
case10000\_goc &10000 &13193 & 0.0\% &Yes &Yes \\
\hline
case10480\_goc &10480 &18559 & 0.0\% &Yes &Yes \\
\hline
case13659\_pegase &13659 &20467 & 7.0\% &No &- \\
\hline
case19402\_goc &19402 &34704 & 0.0\% &Yes &Yes \\
\hline
case24464\_goc &24464 &37816 & 0.0\% &Yes &Yes \\
\hline
case30000\_goc &30000 &35393 & 0.0\% &Yes &Yes \\
\hline
\end{tabular}
}
}
\vspace*{-1em}
\end{table}
We developed MATLAB R2012b code that implements the algorithm described in Section~\ref{sec:main_results}. The code is publicly available at the following page:
\phantom{.}
\noindent
\href{https://github.com/djturizo/ybus-inv-check}{\textcolor{blue}{\texttt{https://github.com/djturizo/ybus-inv-check}}}
\phantom{.}
\noindent
This code is not optimized for performance, but rather serves as a proof-of-concept for the complexity of the algorithm. The interested reader can examine the code and its comments to see that the program has a time complexity of $\mathcal{O}\left({|\mathcal{N}| + |\mathcal{L}|}\right)$ (linear in the system size)\footnote{Our implementation relies on standard algorithms like counting sort, radix sort and BFS. These algorithms are known to have linear time complexity \cite{algorithms}, so the whole implementation has linear complexity as well.}. We remark that comparisons cannot be exact due to the finite-precision computations, so the program uses a user-defined tolerance for all comparisons.
\textcolor{black}{For the numerical experiments we ran the program using radial (distribution) and meshed (transmission) test cases. The radial test were taken from M{\sc atpower} \cite{matpower_manual}. The program successfully applied the theorems to certify the invertibility (or singularity) of the admittance matrix for all the readial test cases of M{\sc atpower}. The meshed test cases were selected from the Power Grid Library PGLib \cite{pglib} (from the OPF benchmarks, more specifically).} Some of the PGLib test cases have a small number of negative resistance elements, precluding the application of Theorem~1. This is the result of modeling choices associated with equivalenced networks~\cite{josz_pegase}. Since these non-passive branches are of an artificial (non-physical) nature, we focus on the other 44 PGLib test cases without negative resistance elements for our numerical experiments. With a tolerance of $10^{-12}$, we obtained the results shown in Table~\ref{tbl:test_cases}. \textcolor{black}{Note that the fourth column of this table refers to the percentage of branches in the system that are purely reactive.} Of the 44 test cases, we found that 6 of them did not satisfy the conditions of the theorems. Thus, the program could not certify the invertibility of the admittance matrix for those 6 cases. These cases are identified with a dash in the last column of Table~\ref{tbl:test_cases} to indicate that the theorems cannot certify whether or not the admittance matrix is invertible. The reason why the invertibility could not be certified for each of the 6 cases is because they have reactive components with topologies not covered by Theorem~3. Such components have inductors, capacitors and loops formed by branches. However, those complex topologies are uncommon, as for the other 38 cases (86\% of all cases) the conditions of the theorems hold, so the program can certify the whether the admittance matrix is invertible or not for each case. The admittance matrix is known to be invertible for the test cases, so we get positive results whenever the theorems were applicable. \textcolor{black}{In contrast, none of the 41 cases satisfy the conditions of the invertibility theorems developed in \cite{rank_1phase, rank_3phase, gatsis, low_theorem}. That is, while none of the existing theoretical results regarding the invertibility of the admittance matrix can be applied to any of the considered PGLib test cases, our results successfully certify the invertibility of the admittance matrix in 86\% of these test cases.} The results show that the theorems can be used to certify the invertibility of the admittance matrices for a wide range of practical and realistic power systems. Moreover, the cases where the theorems cannot be applied to a realistic power system are uncommon.
\section{Conclusions}
This paper studied the invertibility of the admittance matrix for balanced networks. First, we analyzed a theorem from the literature regarding conditions guaranteeing invertibility of the admittance matrix, and we found a technical issue in the proof of that theorem. Next, we developed a framework of lemmas and assumptions that allowed us to amend the proof of previous claims, developing relaxed conditions that guarantee the invertibility of the admittance matrix and generalizing the results to systems with branches modeled as purely reactive elements and transformers with off-nominal tap ratios. Finally, we implemented and publicly released a proof-of-concept program that uses the theorems to certify the invertibility of the admittance matrix. Numerical tests showed that the theorems are applicable in a large number of realistic power systems.
The theory developed in this paper has solely considered admittance matrices for balanced single-phase equivalent network representations. With rapidly increasing penetration of distributed energy resources in unbalanced distribution systems, extending the theory developed here to address the admittance matrices associated with polyphase networks is an important direction for future work. The authors of~\cite{rank_1phase} considered this topic in~\cite{rank_3phase}, where they generalize Theorem~1 to polyphase networks. However, the theory in~\cite{rank_3phase} also relies on the incorrectly stated Lemma~3 and hence may also benefit from amendments and extensions similar to those in this paper.
\section*{Acknowledgements}
The authors greatly appreciate technical discussions with Mario Paolone and Andreas Kettner.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {'timestamp': '2022-11-03T01:17:17', 'yymm': '2012', 'arxiv_id': '2012.04087', 'language': 'en', 'url': 'https://arxiv.org/abs/2012.04087'} |
\section{Introduction}
It is well known that the mapping class group $\M(S)$ of a closed orientable surface $S$ is generated by Dehn twists. This result was originally proved by Dehn \cite{Dehn} and rediscovered by Lickorish \cite{Lick0}. On a closed nonorientable surface $N$, however, there are elements of the mapping class group $\M(N)$ which cannot be expressed as a product of Dehn twists. The first example of such element is the crosscap slide, also called Y-homeomorphism, introduced by Lickorish \cite{Lick1}. He proved \cite{Lick1,Lick2} that the twist subgroup $\T(N)$, generated by Dehn twists about all two-sided simple closed curves, has index 2 in $\M(N)$, and that $\M(N)$ is generated by Dehn twists and one crosscap slide. Later, Chillingworth \cite{Chill} found finite generating sets for $\M(N)$ and $\T(N)$. Korkmaz \cite{KorkHom,Kork} and Stukow \cite{Stu2,Stu3} extended Chillingworth's results to surfaces with punctures and boundary and also computed the abelianizations of $\M(N)$ and $\T(N)$.
In this paper we consider the subgroup $\Y(N)$ of $\M(N)$ generated by all crosscap slides.
Our main result is Theorem \ref{main}, which asserts that for a closed nonorientable surface $N$ of genus $g\ge 2$, $\Y(N)$ is equal to the level 2 subgroup $\Gamma_2(N)$ of $\M(N)$ consisting of those mapping classes which act trivially on $H_1(N;\Z_2)$. In particular $\Y(N)$ is a subgroup of finite index. We also prove that $\Y(N)$ is generated by involutions (Theorem \ref{invol}).
It is an interesting problem to find a finite generating set for $\Y(N)$ and also compute its abelianization. For an orientable surface $S$, the abelianization of the level 2 subgroup of $\M(S)$ was computed by Sato \cite{Sato}.
The paper is organised as follows. After Preliminaries, we prove in Section \ref{s3} that certain elements of $\M(N)$ belong to $\Y(N)$ and that the last group is generated by involutions. In Section \ref{s4} we study the level $m$ mapping class groups $\Gamma_m(S)$ of an orientable surface $S$. We use known results about generators of Torelli groups and the Bass-Milnor-Serre's solution to the congruence subgroup problem for the symplectic group, to obtain convenient generating sets of $\Gamma_m(S)$. In Section \ref{s5} we use lemmas proved in previous sections, as well as some results of McCarthy-Pinkall \cite{McCP} and
Korkmaz \cite{KorkHom}, to prove our main result, which identifies the group $\Y(N)$ with the level 2 mapping class group $\Gamma_2(N)$.
\medskip
\noindent{\bf Acknowledgment.} I was informed by Professor Susumu Hirose that he found a shorter proof of my Theorem \ref{main} using Nowik's result \cite[Theorem 3.6]{Now}. I thank him for the discussion and for the reference \cite{Now}.
\section{Preliminaries}
Let $F$ be a connected compact surface, possibly with
boundary. Define $\Dif(F)$ to be the group of all, orientation
preserving if $F$ is orientable, homeomorphisms $h\colon F\to F$
equal to the identity on the boundary $\bdr{F}$. The {\it mapping class
group} $\M(F)$ is the group of isotopy classes in $\Dif(F)$. By
abuse of notation we will use the same symbol to denote a
homeomorphism and its isotopy class. If $g$ and $h$ are two
homeomorphisms, then the composition $gh$ means that $h$ is applied
first.
A surface of genus $g$ with $n$ boundary components will be denoted by $S_{g,n}$ if it is orientable, or by $N_{g,n}$ if it is nonorientable. The genus of a nonorientable surface is the number of projective planes in the connected sum decomposition.
\subsection{Simple closed curves and Dehn twists.}
By a {\it simple closed curve} in $F$ we mean an embedding
$\gamma\colon S^1\to F\backslash\bdr{F}$. Note that $\gamma$ has an orientation; the
curve with the opposite orientation but same image will be denoted by
$\gamma^{-1}$.
By abuse of notation, we will often identify a simple closed curve with its
oriented image and also with its isotopy class.
According to whether a regular neighborhood of $\gamma$ is an annulus or a M\"obius strip, we call $\gamma$ respectively {\it two-} or {\it one-sided}.
We say that $\gamma$ is {\it nonseparating} if $F\backslash\gamma$
is connected and {\it separating} otherwise. If $\gamma=\bdr{M}$ for some subsurface $M\subset F$ then we call $\gamma$ {\it bounding}. If $n\le 1$ then every separating curve is bounding.
Given a two-sided simple closed curve $\gamma$, $T_\gamma$ denotes a Dehn
twist about $\gamma$. On a nonorientable surface it is
impossible to distinguish between right and left twists, so the
direction of a twist $T_\gamma$ has to be specified for each curve
$\gamma$. Equivalently we may choose an orientation of a regular
neighborhood of $\gamma$. Then $T_\gamma$ denotes the right Dehn twist with
respect to the chosen orientation. Unless we specify which of the
two twists we mean, $T_\gamma$ denotes any of the
two possible twists. Recall that $T_\gamma$ does not depend on the orientation of $\gamma$.
{\it Bounding pair} is a pair $(\gamma,\gamma')$ of disjoint
| {'timestamp': '2011-08-22T02:01:13', 'yymm': '1006', 'arxiv_id': '1006.5410', 'language': 'en', 'url': 'https://arxiv.org/abs/1006.5410'} |
\section{Introduction}
The faint and metal-poor satellites of massive galaxies are among the oldest structures of the Universe, whether they are dwarf galaxies or globular clusters (\citealt{white_rees_78}, \citealt{beasley02}, \citealt{mo10}). Because the faintest, low-mass galaxies are, in a $\Lambda$CDM context, dominated by dark matter (DM), they can be the key to constraining the nature of the DM particle and more broadly the properties of our Universe (\citealt{klypin99}, \citealt{geringer-sameth15a}, \citealt{bullock17}, \citealt{nadler19}). The old age of the dwarf galaxies further implies that they have hosted Population III stars (\citealt{ishiyama16}), while their low mass and metallicity suggest that the pollution from successive supernovae is limited in these systems (\citealt{dekel86}, \citealt{frebel_norris15}). As a consequence, they are unique laboratories to study and quantify the different evolutionary pathways of stellar formation (\citealt{roederer16}, \citealt{webster16}, \citealt{ji19}).
On the other hand, globular clusters are usually not thought to be DM-dominated \citep{moore96} and the faintest of them are often modelled as simpler systems with a single stellar population (\citealt{gratton07}, \citealt{carretta09}, \citealt{cohen10}, \citealt{willman_strader12}). They are important for various reasons. They give us insight in the mode of stellar formation in the early Universe and its similarities and differences with star formation today (\citealt{gieles18},\citealt{gratton19}). The close link of their properties to those of their host galaxies has become clear over the years: for example, the number of clusters are related to the properties of their host galaxies (\citealt{brodie_huchra91}, \citealt{blakeslee97}). They are thereby direct probes of their galaxy' properties and therefore indirect probes for cosmology (\citealt{cote02_cluster}, \citealt{villaume19}, \citealt{riley20}). Old globular clusters are also witnesses of the build-up of massive galaxies halos, which are thought to have formed by disrupting and accreting faint stellar systems (\citealt{beasley18}, \citealt{pfeffer18}). They can also offer a unique insight into the observational validation of stellar population models \citep{chantereau16}.
The diversity of the properties of the MW faint satellites has been progressively unveiled over the past decades. For dwarf galaxies, it all started with Sculptor and Fornax, the very first dwarf spheroidal galaxies discovered (\citealt{shapley38a}, \citealt{shapley38b}) to the most recent discoveries (\citealt{willman05b}, \citealt{zucker06}, \citealt{belokurov06}, \citealt{laevens15}, \citealt{drlica-wagner15}, \citealt{kim15b}, \citealt{homma16}, \citealt{torrealba18}, \citealt{homma19}). Their observed diversity in size, luminosity and mass is not a surprise as it naturally arises in various simulations trying to reproduce our Local Universe (\citealt{springel08}, \citealt{vogelsberger14}, \citet{eagle15}, \citealt{wheeler19}). For the Milky Way's globular clusters, \citet{harris10} has compiled the properties of 157 systems and summarized years of efforts of the community to characterize and understand them. That list will only grow longer in the future considering the recent discoveries of new clusters (\citealt{martin16c}, \citealt{koposov17}, \citealt{mau19}), especially with the advent of the Legacy Survey of Space and Time (LSST) in the incoming years (\citealt{ivezic08}, \citealt{simon19}) \\
As increasingly faint MW satellites are discovered, the difficulties in classifying them have grown. Systems fainter than $ M_V\sim -6$ mag are often challenging to classify as globular cluster or galaxy. This issue was notably addressed by \citet{cote02} and \citet{gilmore07} and is commonly known as the ``valley of ambiguity". Distinguishing faint satellite galaxies from globular clusters relies on two main diagnostics: (i) evidence for the presence of substantial amounts of dark matter, and (ii) the presence of a substantial dispersion in metallicity, a feature usually associated with recurrent star formation. This requires combining deep photometry with extensive spectroscopic campaigns. Dynamical evidence for an excess of mass over that of the stellar component, would then be attributed, in the standard cosmological model, to the presence of a massive DM halo \citep{willman_strader12} and would favour a dwarf galaxy scenario The first measurement of the velocity dispersion of a non-classical MW dwarf galaxy, undertaken by \citet{kleyna05}, showed that Ursa Major~I is indeed DM-dominated. This effort was repeated for most systems known at the time, enriching our knowledge of the MW satellites' properties (\citealt{martin07}, \citealt{simon07}, \citealt{koposov11}, \citealt{simon11}, \citealt{martin16_tri}), an effort that would also rely on other tracers such as the metallicity and the metallicity dispersion of these faint satellites (\citealt{kirby08}, \citealt{koposov15}, \citealt{walker16}). However, in practice, the velocity and metallicity dispersions can be challenging to constrain due to the low number of member stars observed with spectroscopy for a significant fraction of the known faint MW satellites. This problem, that first arose with the study of the kinematics of Segue~2 \citep{kirby13a} still limits our understanding of some of them. The examples over the last few years are numerous, with Draco~II (\citealt{martin16_dra}, \citealt{longeard18}), Tucana~III \citep{simon17}, Colomba~I or Horologium~II \citep{fritz19} that are still not well understood, despite extensive studies with photometry and spectroscopy.
In that context, we present a new spectroscopic study of the faint MW satellite Sagittarius~II (Sgr~II) discovered by \citet{laevens15}, and studied in depth by \citet[L20]{longeard20}. L20 used deep photometry, multi-object spectroscopy and the metallicity-sensitive, narrow-band photometry provided by the Pristine survey \citep{starkenburg17} to refine the structural properties of Sgr~II and infer its dynamical and metallicity properties for the first time. Its velocity dispersion ($\sigma^\mathrm{L20}_\mathrm{vr} = 2.7^{+1.3}_{-1.0}$ km s$^{-1}$) favoured the existence of a low-mass DM halo and therefore a dwarf galaxy scenario. The metallicity of the satellite was found to be in good agreement with the metallicity-luminosity relation of dwarf galaxies of \citet{kirby13}, while its metallicity dispersion was deemed small ($\sigma^\mathrm{L20}_\mathrm{[Fe/H]} = 0.10^{+0.06}_{-0.04}$) but resolved. This last result, however, could be inflated by systematics, an underestimation of the individual uncertainties on the spectroscopic metallicities, or just by the small sample of six stars used to perform the metallicity analysis. If L20 marginally favour Sgr~II to be a very low-mass galaxy, an independent study of the satellite presented in an American Astronomical Society meeting \citep{fu_simon_19}, yet to be published, unequivocally found that Sgr~II is a globular cluster, based on an extremely low metallicity dispersion measurement from their own spectroscopic sample ($< 0.08$ at the 95\% confidence level). Furthermore, if the satellite was found to be physically compact by L20 (r$^\mathrm{L20}_h = 35.5 ^{+1.4}_{-1.2}$ pc), but still possibly within the realm of dwarf galaxies, an estimation of the heliocentric distance of the satellite based on the identification of five RR Lyrae stars potentially members of Sgr~II \citep[V20]{vivas20} would place the system closer than the estimate of L20 ($m-M = 18.97 \pm 0.20$ vs. $19.32^{+0.03}_{-0.02}$ mag, i.e. a difference of $\sim 10$ kpc). This would result in a slight overestimation of its physical size (r$^\mathrm{V20}_h = 30.7 ^{+2.7}_{-2.9}$ pc) and bring the system closer to the realm of old, metal-poor globular clusters. For the rest of this paper, the results of L20 will be used. In many ways, Sgr~II is shrouded in mystery, and the following work will attempt to close the case on the nature of this faint and elusive satellite.
\section{Data selection and acquisition}
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{spectrum.pdf}}
\caption{Non sky-substracted spectrum of a Sgr~II member star with a S/N of $\sim 65$ focusing on the Calcium II triplet lines. This star has a metallicity of [Fe/H] $= -2.32 \pm 0.18$ using the calibration of \citet{starkenburg10} as detailed in section 3.2.}
\label{spectrum}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\centerline{\includegraphics[width=\hsize]{CMDs_field.pdf}}
\caption{{\textit{Left panel: }} Spatial distribution of Sgr~II-like stars, i.e. stars with a CMD probability membership of 1 per cent or higher. The field is centered on ($\alpha_0 = 298.16628^\circ, \delta_0 = -22.89633^\circ$). The red contour defines two half-light radii ($r_h \sim 1.7'$) of the satellite, as inferred by L20. The locations of all stars in the spectroscopic dataset are indicated as large, coloured markers. Squares represent the DEIMOS observations from L20, while the circles stand for the new FLAMES data presented in this work. All stars observed spectroscopically are colour-coded according to their heliocentric velocities. Cyan stars have a radial velocity corresponding to Sgr~II at around $-177$ km s$^{-1}$. Filled squares and circles are the member stars of the satellite. {\textit{Right panel: }} CMD of Sgr~II within two half-light radii (grey) superimposed with the entire spectroscopic dataset. The best-fitting Darmouth isochrone from L20 ($12$ Gyr, [Fe/H] $= -2.35$, $[\alpha$/Fe] $= 0$, $m-M = 19.32$ mag) is shown as a black dashed line and is perfectly compatible with the identified members of Sgr~II.}
\label{cmd}
\end{center}
\end{figure*}
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{CaHK_FLAMES.pdf}}
\caption{Pristine colour-colour diagram of Sgr~II showing field stars (grey) and the spectroscopic dataset with large, coloured markers. Member stars of Sgr~II are shown with filled markers. While the x-axis is a temperature proxy, the y-axis contains the Pristine metallicity-sensitive, narrow-band photometry denoted $c_0$, and therefore the metallicity information. As a result, stars are distributed according to their metallicity: the ones around solar metallicity form the stellar locus where most of the grey field stars are located. As a star goes upwards in the diagram (i.e. lower y-axis value) for a fixed temperature, its metallicity decreases. This is represented more visually with the three iso-metallicity sequences in black dashed lines, showing where stars with a [Fe/H] of $-4.0$, $-3.0$ and $-2.0$ should be located. This plot shows the efficiency of Pristine in identifying Sgr~II members, as almost all DEIMOS stars dynamically compatible with the satellite come out as metal-poor.}
\label{cahk}
\end{center}
\end{figure}
The spectroscopy used in this work is a combination of two datasets. The first data were observed with the DEep Imaging Multi-Object Spectrograph (DEIMOS, \citealt{faber03}), already detailed and analysed in L20. The second dataset is composed of new observations performed with the Fibre Large Array Multi Element Spectrograph \citep[FLAMES]{pasquini02} mounted on the Very Large Telescope (VLT). The \mbox{GIRAFFE}/HR21 grating, with a resolution of R $\sim 18000$, a central wavelength of $8757$ $\; \buildrel \circ \over {\rm A}$ $\; $and a bandwidth between 8484 and 9001$\; \buildrel \circ \over {\rm A}$$\;$was used to resolve the Calcium II triplet infrared lines. These FLAMES observations were conducted during three different nights to attempt to detect binary stars: on the 22.06.16 ($1 \times 2610$s of integration), the 21.07.16 ($2 \times 2610$s) and the 30.07.16 ($2 \times 2610$s).
The data were reduced by the European Southern Observatory team using their pipeline specifically tailored to FLAMES data \citep{melo09}. The spectrum of a Sgr~II member star is displayed in Figure \ref{spectrum} to illustrate the quality of the data. In order to infer the radial velocities, the equivalent widths and their respective uncertainties, the method already used in L20 was used and consists of determining the combination of Gaussian profiles that best fits a continuum composed of the Calcium II triplet lines at rest as the unique spectral features. It also includes a correction to account for the non-gaussianity of the wings' lines. The typical uncertainty on the radial velocities for the FLAMES dataset is of the order of $1$ km s$^{-1}$, and all the stars with signal-to-noise (S/N) ratio below 3 per resolution element are discarded from the sample. In the same fashion as in \citet{simon_geha07} and L20, the systematic error on the velocities is determined from stars observed more than once in the FLAMES sample. The resulting systematics are a bias of $ b = 0.4 \pm 0.3$ km s$^{-1}$ and a standard deviation of $ \delta_\mathrm{thr} = 0.8 \pm 0.1$ km s$^{-1}$. \\
To build our list of FLAMES targets, we used the locations of stars in the colour-magnitude diagram (CMD) of Sgr II built from deep MegaCam photometry and shown in the right panel of Figure \ref{cmd}, along with their spatial distribution in the left panel. More importantly, this spectroscopic study benefits from the use of the narrow-band, metallicity-sensitive photometry provided by the Pristine survey \citep{starkenburg17}. All stars observed in Pristine have a photometric metallicity measurement, as illustrated in the colour-colour diagram shown in Figure \ref{cahk}. In this diagram, stars are distributed according to their metallicities, from the most metal-rich at the bottom of the plot, to the most metal-poor at the top, down to a metallicity of [Fe/H] $\sim -4.0$. Using the prior knowledge that Sgr~II is indeed a very metal-poor satellite ([Fe/H]$_\mathrm{SgrII} \sim -2.28$, L20), the target list was built so that the stars observed are confirmed to be very metal-poor according to the Pristine survey. In doing so, we increase the probability of finding new members (\citealt{youakim17}, \citealt{aguado19}). The DEIMOS spectroscopy supplementing this dataset consists only of the DEIMOS stars identified as non variables, non binaries and metal-poor according to Pristine ([Fe/H]$_\mathrm{CaHK} < -1.6$) by L20.\\
In order to clean the FLAMES data, we check for the existence of binaries in the sample. For a given star, each measurement from the $k$-th epoch is assumed to be reasonably well-described by a Gaussian distribution centered on the radial velocity measurement and a standard deviation corresponding to its uncertainty. Then, the probability that the measurement $k$ is discrepant from that of the same star at another epoch $j$ at the $2\sigma$ level is computed. This probability corresponds to the star being variable between epochs $k$ and $j$. The same procedure is used to compute the probability that each velocity measurement of a given epoch is compatible with all the others within $2\sigma$, i.e. the probability that it is not varying. If the product of the probabilities representing the hypothesis "variable" is greater than the hypothesis "non-variable", then the star is considered a binary star and discarded from the main sample so that their variability does not affect the dynamical analysis of Sgr~II. We find a total of 6 potential binaries in the sample, however, none have the right velocity to be a member of Sgr~II.
The final spectroscopic sample consists of 113 stars, with 47 new stars observed with FLAMES as detailed in Table \ref{table1} with four stars out of those 47 in common with L20's sample. As a consistency check, we compare the velocities obtained with FLAMES and DEIMOS for these four stars and find no statistical differences between the two samples.
\section{Results}
In this section, we constrain the systemic heliocentric velocity of Sgr~II and its associated velocity dispersion.
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{v_vs_r.pdf}}
\caption{Heliocentric velocities vs. radial distance to Sgr~II's centroid for FLAMES (blue circles) and DEIMOS (grey squares) datasets. The addition of the FLAMES data doubles the number of identified members for Sgr~II. All error bars not appearing in this plot are smaller than the size of the circles/squares.}
\label{histos_vel}
\end{center}
\end{figure}
\subsection{Dynamical properties}
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{pdfs_vel.pdf}}
\caption{1D marginalised PDFs of the systemic radial velocity (top left panel) and velocity dispersion (top right panel) of Sgr~II from this work as black solid lines, and L20 in grey dashed lines using only DEIMOS data. The expected velocity dispersion for a Sgr~II-like typical MW globular cluster ($1.1 \pm 0.1$ km s$^{-1}$) is indicated as the red shaded region. The bottom panel shows the observed velocity dispersions of most globular clusters (black) and confirmed dwarf galaxies (blue) of the MW as a function of their absolute magnitudes. The properties of the globular clusters were taken from the \citet{harris10} catalog and references therein. The dwarf galaxy measurements come from \citet{simon19} and references therein. In this plot, the location of Sgr~II is indicated by the red diamond.}
\label{pdfs_vel}
\end{center}
\end{figure}
All stars in our spectroscopic sample with a heliocentric velocity are shown in Figure \ref{histos_vel}, along with a more detailed view. As already detailed in section 2, the final sample consists of all non-binary, non-HB stars compatible with Sgr~II's sequence in the CMD and that appear as very metal-poor using the CaHK photometry. Stars with mediocre CaHK photometry from the DEIMOS catalog are also included as their photometric metallicities are not reliable. Thanks to the photometry provided by the Pristine survey, the FLAMES data are already quite clean with only a few clear contaminants discrepant from the Sgr~II population at around $-180$ km s$^{-1}$. When available, the proper motions provided by the Gaia Data Release 2 \citep{brown18} are also used to filter out contaminants. This procedure allows us to get rid of one star with CMD and metallicity properties strongly compatible with Sgr~II, but a proper motion unequivocally discrepant from that of the satellite (\citealt{massari18}, L20).
In order to derive the dynamical properties of Sgr~II, the velocity distribution of our dataset and shown in Figure \ref{histos_vel} is assumed to be the sum of two Gaussian distributions, one standing for Sgr~II's population and the other for the MW contamination. The favoured dynamical model is obtained through a Monte Carlo Markov Chain \citep[MCMC]{hastings70} algorithm. \\
The Probability Distribution Functions (PDFs) of the systemic heliocentric velocity and the velocity dispersion of Sgr~II are displayed in the top panels of Figure \ref{pdfs_vel}. The resulting systemic velocity $\langle \mathrm{v}_\mathrm{SgrII} \rangle$ is $-177.2_{-0.6}^{+0.5}$ km s$^{-1}$ and is perfectly compatible with the one of L20. The velocity dispersion found in this work is also compatible with L20 ($\sigma^\mathrm{SgrII}_\mathrm{v} = 2.7_{-1.0}^{+1.3}$ km s$^{-1}$), but with much tighter constraints, it is measured to be $\sigma^\mathrm{SgrII}_\mathrm{v} = 1.7 \pm 0.5$ km s$^{-1}$. Using the formalism of \citet{wolf10} which assumes dynamical equilibrium and a flat velocity dispersion profile, this results into a mass-to-light (M/L) ratio of $3.5^{+2.7}_{-1.6}$ M$_{\odot}$ L$_{\odot}^{-1}$. Sgr~II shows no sign of a velocity dispersion gradient as the determination of its dynamical properties by selecting stars inside and outside 1 arcminute yields no statistical difference. Furthermore, adding another contaminating population to the likelihood model does not change our results.
We classify as a member of Sgr~II any star with line-of-sight velocity and proper motion membership probabilities (when available) above 50\%. We identify $22$ member stars in the FLAMES data. 3 of them were already identified in the L20 dataset, leading to the identification of 19 new members. Combined with the spectroscopically confirmed members of L20, 43 stars are confirmed to belong to Sgr~II, including binaries and horizontal branch (HB) stars. This underlines the importance of the Pristine photometry. To estimate the success rate of the FLAMES data in identifying new members, we do not simply take the fraction of confirmed members over the overall number of stars observed since the sample of Sgr~II candidates identified before observation was not large enough to fill all the FLAMES fibers. Among the 47 new stars observed, only $32$ were considered as promising Sgr~II candidates. Therefore, it yields a success rate of $\sim 60 \%$ for the FLAMES data sample based on the Pristine selection. In comparison, the DEIMOS dataset of L20, solely based on a CMD-based selection, had a 20\% success rate. The contrast is even more striking when considering that the DEIMOS selection focuses on the central region of the satellite and is therefore more likely to find Sgr~II stars, while the FLAMES observations had a much wider field of view and aim to find stars located much further away than Sgr~II's centroid.
\subsection{Metallicity properties}
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{FeH_individual_metallicities.pdf}}
\caption{Individual metallicities of all bright stars in the spectroscopic sample used to derive the metallicity properties of Sgr~II using the calibration of \citet{starkenburg10}. Each measurement is modelled by a Gaussian with a mean corresponding to the favoured metallicity of the star, and a standard deviation equal to its uncertainty. Stars from the L20 sample are shown as blue dots, while the new FLAMES stars are represented in red.}
\label{pdfs_feh}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{pdfs_metal.pdf}}
\caption{One-dimensional PDFs of the systemic metallicity (left panel) and metallicity dispersion (right panel) of Sgr~II with spectroscopy. The 95\% C.L. on the metallicity dipersion is hown as a black dashed line. }
\label{metallicities}
\end{center}
\end{figure}
In the following section, the mean stellar metallicity of Sgr~II and its metallicity dispersion are derived using our new FLAMES data. To do so, we select only non-binary, non-HB stars with a spectral signal-to-noise ratio greater than 12 and with $g_0 < 20.5$ (i.e. $\sim$ 1 mag below the HB of Sgr~II). This yields a subsample of 17 stars to extract a measurement of the spectroscopic metallicity. This procedure is performed through the use of the empirical calibration of \citet{starkenburg10} that uses the equivalent widths (EWs) of the Calcium II triplet lines to deduce a measurement of the metallicity of a star. The uncertainties on the coefficients defining the polynomials to transform a set of EWs into a [Fe/H]$_\mathrm{spectro}$ are folded in with the uncertainties on the EWs through a Monte Carlo procedure to obtain the final uncertainties $\delta_\mathrm{[Fe/H]}$ on each individual spectroscopic metallicity measurement. We take the value of 8\% reported by \citet{starkenburg10} to be the uncertainty on each coefficients. We note that these uncertainties do not include several systematics (due to, for instance, the continuum placement, or the detailed chemical abundance pattern of the star when transforming a Ca II measurement into [Fe/H]). Figure \ref{pdfs_feh} shows the spectroscopic metallicities of the 17 stars and their uncertainties assuming each measurement can be modelled by a Gaussian centered on [Fe/H]$_\mathrm{spectro}$ and with a standard deviation corresponding to $\delta_\mathrm{[Fe/H]}$. For stars observed more than once, a metallicity measurement is derived for each epoch and we verify that each epoch yields a metallicity compatible with the others. This procedure highlights two stars with discrepant metallicity measurements at different epochs. We decide to discard those stars for the rest of the analysis, however, including them has a very limited impact on the results. None of the 4 stars in common between the FLAMES and DEIMOS samples have a spectroscopic metallicity measurement.
Since all 15 stars in the subsample are likely Sgr~II members, their metallicity distribution is assumed to be only reflective of Sgr~II population, which is therefore modelled with a Gaussian distribution weighted with the membership probability of each star according to L20. We find a spectroscopic systemic metallicity of $\langle$[Fe/H]$^\mathrm{SgrII}_\mathrm{spectro} \rangle = -2.23 \pm 0.06$, and an unresolved metallicity dispersion with $\sigma^\mathrm{SgrII}_\mathrm{spectro} < 0.20$ at the 95\% confidence limit. The PDFs corresponding to these results are shown in Figure \ref{metallicities}. The metallicity dispersion is unresolved, and is constrained to be below 0.20 dex at the 95\% confidence limit (C.L.).
\section{Discussion}
Throughout this paper, we analyse the dynamical and metallicity properties of the faint MW satellite Sgr~II. To this end, we combine the Keck II/DEIMOS spectroscopic dataset of our previous analysis of the satellite with new VLT/FLAMES data. These FLAMES observations are the first ones to be carried out by selecting \textit{a priori} the interesting candidates using the narrow-band, metallicity sensitive photometry of the Pristine survey for a faint satellite of the MW. The spectroscopic observations presented in this work are decisive as they double the overall sample of members stars known. Nine new stars are bright enough to estimate their spectroscopic metallicities with the empirical Calcium II triplet calibration of \citet{starkenburg10}, while the sample of L20 only consists of 6 stars. Therefore, this study considerably enlarges the statistics available for the satellite and allows us to put much tighter constraints on the dynamical and metallicity properties of SgrII.
The systemic radial velocity of Sgr~II is found to be $\langle \mathrm{v}_\mathrm{SgrII} \rangle = -177.2_{-0.6}^{+0.5}$ km s$^{-1}$. The velocity dispersion of $\sigma_\mathrm{v}^\mathrm{SgrII} = 1.7 \pm 0.5$ km s$^{-1}$ is well among the range observed for the MW globular clusters at the same luminosity, as shown in Figure \ref{pdfs_vel}. It translates into a M/L ratio of $3.5^{+2.7}_{-1.6}$ M$_{\odot}$ L$^{-1}_{\odot}$, thus suggesting that the dynamics of the satellite is not mainly driven by a DM halo. We identify 19 new members in the faint system, for a total of 43 confirmed spectroscopic Sgr~II members. With a fraction of member identified over the entirety of the FLAMES sample of $60$\%, the \textit{a priori} selection using Pristine performs three times better than the simple CMD-based selection of L20, despite a riskier observational strategy through the search for stars located beyond two half-light radii of the satellite. Four of these stars are identified, with the outermost one lying at $6.2$r$_{h}$. 15 out of the 43 members have good enough quality spectra to measure the EWs of their Calcium II triplet and measure their metallicity using the empirical calibration of \citet{starkenburg10}. We find a systemic metallicity of $-2.23 \pm 0.07$ and constrain the metallicity dispersion of the satellite to be less than $0.20$ at the $95$\% confidence limit.
The velocity dispersion as inferred by L20 is larger than the one found in the present analysis. Furthermore, L20's metallicity analysis indicates that there are multiple stellar populations in the system, as the metallicity dispersion is resolved both with CaHK photometry and spectroscopy. This is not the case anymore in this work as the spectroscopic metallicity distribution of our dataset is well-described with only one stellar population despite the addition of 9 new stars with spectroscopic metallicites. Moreover, Figure \ref{pdfs_feh} shows that no star with a metallicity below $-2.5$ is confidently detected, a limit commonly attributed to the lowest metallicity achievable by globular clusters (\citealt{harris10} and references therein, \citealt{beasley19}). Therefore, we are able to conclude that Sgr~II is a globular cluster.
Nonetheless, the satellite is still quite extended when compared to other MW globular clusters of the same luminosity. Within a range of one magnitude around Sgr~II's absolute magnitude, the largest MW cluster is Pal~5 (M$_V \sim -5.2 $), a fairly metal-poor system ([Fe/H] $\sim -1.4$) with a size of $\sim 19.2$ pc, while most of the others have a size below $10$ pc \citep{harris10}. Whether we consider the distance of L20 or \citet{vivas20}, Sgr~II is still at least 1.5 times more extended than Pal~5. Sgr~II is a case in point of the ambiguity surrounding the faint MW satellites discovered in recent years, with a somewhat larger half-light radius than expected from GCs at that luminosity, a metallicity perfectly compatible with the luminosity-metallicity relation of dwarf galaxies \citep{kirby13}, and velocity and metallicity dispersions challenging to resolve. Though we conclude that Sgr~II is an old and metal-poor globular cluster, the system remains interesting in many aspects, from its possible association to the Sgr stream to an even deeper understanding of the unusually large cluster, which could build a bridge to the still exclusive definitions of clusters and galaxies of the MW.
\section*{Acknowledgments}
We gratefully thank the CFHT staff for performing the observations in queue mode, for their reactivity in adapting the schedule, and for answering our questions during the data-reduction process.
NFM, RI, and NL gratefully acknowledge support from the French National Research Agency (ANR) funded project ``Pristine'' (ANR-18-CE31-0017) along with funding from CNRS/INSU through the Programme National Galaxies et Cosmologie and through the CNRS grant PICS07708. ES gratefully acknowledges funding by the Emmy Noether program from the Deutsche Forschungsgemeinschaft (DFG). This work has been published under the framework of the IdEx Unistra and benefits from a funding from the state managed by the French National Research Agency as part of the investments for the future program. The authors thank the International Space Science Institute, Berne, Switzerland for providing financial support and meeting facilities to the international team ``Pristine''.
Based on observations collected at the European Southern Observatory under ESO programme(s) 099.B-0690(A).
Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made pos- sible by the generous financial support of the W. M. Keck Foundation. Furthermore, the authors wish to recognize and acknowledge the very significant cultural role and rever- ence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.
This work has made use of data from the European Space Agency (ESA)
mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by
the {\it Gaia} Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding
for the DPAC has been provided by national institutions, in particular
the institutions participating in the {\it Gaia} Multilateral Agreement.
\clearpage
\newpage
\begin{table*}
\caption{Properties of the new FLAMES spectroscopic sample. The Pristine metallicity of a given star is indicated only if [Fe/H]$_\mathrm{CaHK} < -1.0$. The individual spectroscopic metallicity is reported for stars with S/N $>= 12$ and $g_0 <= 20.5$ mag. The systematic threshold $\delta_{\mathrm{thr}}$ is not included in the velocity uncertainties presented in this table.
\label{table1}}
\setlength{\tabcolsep}{2.5pt}
\renewcommand{\arraystretch}{0.3}
\begin{sideways}
\begin{tabular}{cccccccccccccc}
\hline
RA (deg) & DEC (deg) & $g_0$ & $i_0$ & $CaHK_0$ & $\mathrm{v}_{r} ({\rm\,km \; s^{-1}})$ & $\mu_{\alpha}^{*}$ (mas.yr$^{-1}$) & $\mu_{\delta}$ (mas.yr$^{-1}$) & S/N & [Fe/H]$_{\mathrm{CaHK}}$ & [Fe/H]$^\mathrm{S10}_\mathrm{spectro}$ & Member\\
\hline
298.37718 & $-$21.98293 & 21.05 $\pm$ 0.01 & 20.42 $\pm$ 0.01 & 21.74 $\pm$ 0.04 & -96.5 $\pm$ 1.4 & 5.0 $\pm$ 4.0 & -5.1 $\pm$ 2.1 & 3.5 & $-$2.50 $\pm$ 0.12 & --- & N \\ \\
298.28281 & $-$21.92996 & 19.57 $\pm$ 0.01 & 18.80 $\pm$ 0.01 & 20.45 $\pm$ 0.02 & -177.8 $\pm$ 0.6 & -0.2 $\pm$ 0.5 & -1.1 $\pm$ 0.3 & 24.4 & $-$2.58 $\pm$ 0.12 & -2.2 $\pm$ 0.11 & Y \\ \\
298.24280 & $-$21.97804 & 19.75 $\pm$ 0.01 & 19.05 $\pm$ 0.01 & 20.67 $\pm$ 0.02 & 34.1 $\pm$ 2.4 & -1.9 $\pm$ 2.7 & -1.9 $\pm$ 1.3 & 12.7 & $-$1.90 $\pm$ 0.12 & -1.5 $\pm$ 0.21 & N \\ \\
298.20794 & $-$21.86545 & 19.18 $\pm$ 0.01 & 18.21 $\pm$ 0.01 & 20.64 $\pm$ 0.02 & -58.3 $\pm$ 0.5 & -0.6 $\pm$ 0.2 & -1.0 $\pm$ 0.1 & 48.9 & $-$1.25 $\pm$ 0.12 & -1.26 $\pm$ 0.16 & N \\ \\
298.18438 & $-$21.93702 & 19.93 $\pm$ 0.01 & 19.29 $\pm$ 0.01 & 20.80 $\pm$ 0.02 & -195.5 $\pm$ 3.3 & --- $\pm$ --- & --- $\pm$ --- & 6.3 & $-$1.72 $\pm$ 0.12 & --- & N \\ \\
298.18958 & $-$21.84866 & 21.18 $\pm$ 0.01 & 20.66 $\pm$ 0.01 & 21.84 $\pm$ 0.04 & -268.8 $\pm$ 16.9 & 1.7 $\pm$ 0.2 & -2.0 $\pm$ 0.1 & 3.1 & $-$2.01 $\pm$ 0.12 & --- & N \\ \\
298.05410 & $-$21.87831 & 18.29 $\pm$ 0.01 & 17.48 $\pm$ 0.01 & 19.51 $\pm$ 0.01 & 91.0 $\pm$ 0.3 & --- $\pm$ --- & --- $\pm$ --- & 79.8 & $-$1.34 $\pm$ 0.12 & -1.43 $\pm$ 0.16 & N \\ \\
298.08415 & $-$21.96661 & 18.93 $\pm$ 0.01 & 18.12 $\pm$ 0.01 & 20.03 $\pm$ 0.01 & -181.7 $\pm$ 0.4 & -12.8 $\pm$ 0.7 & -18.6 $\pm$ 0.4 & 60.6 & $-$1.99 $\pm$ 0.12 & -1.8 $\pm$ 0.13 & N \\ \\
298.00202 & $-$21.92952 & 19.33 $\pm$ 0.01 & 18.50 $\pm$ 0.01 & 20.37 $\pm$ 0.02 & 139.8 $\pm$ 0.4 & --- $\pm$ --- & --- $\pm$ --- & 31.3 & $-$2.28 $\pm$ 0.12 & -2.12 $\pm$ 0.11 & N \\ \\
298.01422 & $-$21.89008 & 19.93 $\pm$ 0.01 & 19.30 $\pm$ 0.01 & 20.73 $\pm$ 0.02 & -208.9 $\pm$ 1.0 & -4.7 $\pm$ 0.2 & -6.0 $\pm$ 0.1 & 16.4 & $-$2.00 $\pm$ 0.12 & -2.24 $\pm$ 0.09 & N \\ \\
298.01803 & $-$21.92193 & 20.17 $\pm$ 0.01 & 19.47 $\pm$ 0.01 & 21.06 $\pm$ 0.03 & 145.7 $\pm$ 0.9 & 4.2 $\pm$ 1.2 & -6.5 $\pm$ 0.9 & 14.6 & $-$1.95 $\pm$ 0.12 & -1.13 $\pm$ 0.22 & N \\ \\
298.36550 & $-$22.00547 & 20.22 $\pm$ 0.01 & 19.60 $\pm$ 0.01 & 20.98 $\pm$ 0.02 & 112.9 $\pm$ 2.1 & -0.6 $\pm$ 0.6 & -1.4 $\pm$ 0.4 & 15.3 & $-$2.10 $\pm$ 0.12 & -2.36 $\pm$ 0.12 & N \\ \\
298.36878 & $-$22.01959 & 21.36 $\pm$ 0.01 & 20.76 $\pm$ 0.01 & 21.95 $\pm$ 0.04 & -43.6 $\pm$ 3.0 & 4.8 $\pm$ 3.1 & 2.9 $\pm$ 1.8 & 3.6 & $-$3.11 $\pm$ 0.12 & --- & N \\ \\
298.27562 & $-$22.01614 & 20.23 $\pm$ 0.01 & 19.42 $\pm$ 0.01 & 21.40 $\pm$ 0.03 & -82.3 $\pm$ 2.8 & -0.6 $\pm$ 1.3 & -11.3 $\pm$ 0.8 & 4.1 & $-$1.58 $\pm$ 0.12 & --- & N \\ \\
298.27457 & $-$22.00481 & 20.67 $\pm$ 0.01 & 20.10 $\pm$ 0.01 & 21.34 $\pm$ 0.03 & 54.2 $\pm$ 1.5 & --- $\pm$ --- & --- $\pm$ --- & 10.1 & $-$2.23 $\pm$ 0.12 & --- & N \\ \\
298.23845 & $-$22.11143 & 20.69 $\pm$ 0.01 & 20.06 $\pm$ 0.01 & 21.40 $\pm$ 0.03 & -175.5 $\pm$ 0.9 & -0.6 $\pm$ 1.3 & -0.0 $\pm$ 0.8 & 10.8 & $-$2.41 $\pm$ 0.12 & --- & Y \\ \\
298.25661 & $-$22.0404 & 20.73 $\pm$ 0.01 & 20.19 $\pm$ 0.01 & 21.42 $\pm$ 0.03 & -34.7 $\pm$ 1.4 & 0.4 $\pm$ 1.1 & 0.0 $\pm$ 0.7 & 6.1 & $-$2.03 $\pm$ 0.12 & --- & N \\ \\
298.23141 & $-$22.10773 & 20.75 $\pm$ 0.01 & 20.12 $\pm$ 0.01 & 21.53 $\pm$ 0.03 & -69.0 $\pm$ 1.7 & -1.1 $\pm$ 1.1 & -1.8 $\pm$ 0.6 & 4.4 & $-$2.13 $\pm$ 0.12 & --- & N \\ \\
298.19512 & $-$22.09244 & 18.18 $\pm$ 0.01 & 17.21 $\pm$ 0.01 & 19.37 $\pm$ 0.01 & -175.9 $\pm$ 0.3 & -13.5 $\pm$ 0.6 & -25.9 $\pm$ 0.4 & 95.7 & $-$2.31 $\pm$ 0.12 & -2.15 $\pm$ 0.12 & Y \\ \\
298.18145 & $-$22.07735 & 18.72 $\pm$ 0.01 & 17.92 $\pm$ 0.01 & 19.56 $\pm$ 0.01 & -177.7 $\pm$ 0.8 & -7.7 $\pm$ 1.3 & -9.1 $\pm$ 0.7 & 18.3 & $-$2.97 $\pm$ 0.12 & -2.18 $\pm$ 0.11 & N \\ \\
298.11270 & $-$22.04758 & 18.97 $\pm$ 0.01 & 18.23 $\pm$ 0.01 & 19.74 $\pm$ 0.01 & -178.0 $\pm$ 0.9 & -1.3 $\pm$ 1.1 & -8.7 $\pm$ 0.6 & 11.9 & $-$2.91 $\pm$ 0.12 & -2.13 $\pm$ 0.18 & N \\ \\
298.19043 & $-$22.03676 & 19.14 $\pm$ 0.01 & 18.32 $\pm$ 0.01 & 20.03 $\pm$ 0.01 & -180.2 $\pm$ 0.3 & -1.7 $\pm$ 1.5 & -1.9 $\pm$ 0.9 & 53.5 & $-$2.87 $\pm$ 0.12 & -2.23 $\pm$ 0.11 & Y \\ \\
298.15270 & $-$22.08463 & 19.25 $\pm$ 0.01 & 18.46 $\pm$ 0.01 & 20.11 $\pm$ 0.01 & -176.2 $\pm$ 0.5 & 12.3 $\pm$ 0.5 & -27.2 $\pm$ 0.3 & 27.6 & $-$2.83 $\pm$ 0.12 & -2.23 $\pm$ 0.11 & Y \\ \\
298.15185 & $-$22.09654 & 19.29 $\pm$ 0.01 & 18.52 $\pm$ 0.01 & 20.18 $\pm$ 0.01 & -177.6 $\pm$ 0.4 & 6.8 $\pm$ 3.3 & 0.1 $\pm$ 2.1 & 29.0 & $-$2.56 $\pm$ 0.12 & -2.25 $\pm$ 0.12 & Y \\ \\
298.18099 & $-$22.20728 & 19.58 $\pm$ 0.01 & 18.99 $\pm$ 0.01 & 20.55 $\pm$ 0.02 & 35.3 $\pm$ 0.8 & --- $\pm$ --- & --- $\pm$ --- & 11.7 & $-$1.14 $\pm$ 0.12 & --- & N \\ \\
298.21784 & $-$22.06409 & 19.66 $\pm$ 0.01 & 18.90 $\pm$ 0.01 & 20.47 $\pm$ 0.02 & -175.1 $\pm$ 0.4 & -2.4 $\pm$ 0.7 & -5.2 $\pm$ 0.4 & 34.3 & $-$99.0 $\pm$ 0.00 & -2.36 $\pm$ 0.1 & Y \\ \\
298.16019 & $-$22.09027 & 19.94 $\pm$ 0.01 & 19.34 $\pm$ 0.01 & 20.77 $\pm$ 0.02 & 93.9 $\pm$ 2.8 & -0.3 $\pm$ 0.8 & -1.3 $\pm$ 0.4 & 3.6 & $-$1.70 $\pm$ 0.12 & --- & N \\ \\
298.19280 & $-$22.05979 & 19.99 $\pm$ 0.01 & 19.28 $\pm$ 0.01 & 20.72 $\pm$ 0.02 & -174.3 $\pm$ 0.6 & -0.8 $\pm$ 0.3 & -1.0 $\pm$ 0.1 & 22.6 & $-$3.07 $\pm$ 0.12 & -2.3 $\pm$ 0.11 & Y \\ \\
298.19388 & $-$22.07918 & 20.01 $\pm$ 0.01 & 19.28 $\pm$ 0.01 & 20.79 $\pm$ 0.02 & -176.2 $\pm$ 0.6 & -7.1 $\pm$ 1.0 & -8.9 $\pm$ 0.6 & 17.7 & $-$2.67 $\pm$ 0.12 & -2.15 $\pm$ 0.13 & Y \\ \\
298.20960 & $-$22.07638 & 20.03 $\pm$ 0.01 & 19.33 $\pm$ 0.01 & 20.87 $\pm$ 0.02 & 121.4 $\pm$ 1.3 & 0.0 $\pm$ 0.5 & -0.4 $\pm$ 0.3 & 22.5 & $-$2.26 $\pm$ 0.12 & -1.78 $\pm$ 0.13 & N \\ \\
298.16641 & $-$22.06769 & 20.21 $\pm$ 0.01 & 19.51 $\pm$ 0.01 & 20.95 $\pm$ 0.02 & -172.2 $\pm$ 1.5 & -3.2 $\pm$ 0.6 & -4.5 $\pm$ 0.3 & 10.8 & $-$2.73 $\pm$ 0.12 & -2.72 $\pm$ 0.1 & Y \\ \\
298.13014 & $-$22.14273 & 20.24 $\pm$ 0.01 & 19.56 $\pm$ 0.01 & 21.01 $\pm$ 0.02 & -178.5 $\pm$ 1.0 & --- $\pm$ --- & --- $\pm$ --- & 14.5 & $-$2.40 $\pm$ 0.12 & -2.06 $\pm$ 0.14 & Y \\ \\
298.14121 & $-$22.06475 & 20.28 $\pm$ 0.01 & 19.62 $\pm$ 0.01 & 21.05 $\pm$ 0.02 & -175.2 $\pm$ 1.2 & -14.8 $\pm$ 1.5 & -20.4 $\pm$ 0.8 & 15.1 & $-$99.0 $\pm$ 0.00 & -1.89 $\pm$ 0.16 & Y \\ \\
298.19401 & $-$22.08638 & 20.38 $\pm$ 0.01 & 19.70 $\pm$ 0.01 & 21.09 $\pm$ 0.03 & -175.7 $\pm$ 1.2 & -1.2 $\pm$ 0.9 & -1.1 $\pm$ 0.5 & 7.7 & $-$2.78 $\pm$ 0.12 & -2.17 $\pm$ 0.11 & Y \\ \\
298.21019 & $-$22.05475 & 20.51 $\pm$ 0.01 & 19.82 $\pm$ 0.01 & 21.20 $\pm$ 0.03 & -188.7 $\pm$ 5.8 & -1.5 $\pm$ 0.5 & -0.8 $\pm$ 0.3 & 3.8 & $-$3.06 $\pm$ 0.12 & --- & Y \\ \\
298.14290 & $-$22.07784 & 20.53 $\pm$ 0.01 & 19.87 $\pm$ 0.01 & 21.23 $\pm$ 0.03 & -177.4 $\pm$ 1.3 & -1.4 $\pm$ 1.5 & -9.8 $\pm$ 0.7 & 14.3 & $-$2.68 $\pm$ 0.12 & --- & Y \\ \\
298.17751 & $-$22.16337 & 20.60 $\pm$ 0.01 & 19.93 $\pm$ 0.01 & 21.36 $\pm$ 0.03 & 89.3 $\pm$ 0.8 & -7.3 $\pm$ 0.4 & -10.1 $\pm$ 0.2 & 13.4 & $-$2.38 $\pm$ 0.12 & --- & N \\ \\
298.21550 & $-$22.08104 & 20.65 $\pm$ 0.01 & 20.00 $\pm$ 0.01 & 21.36 $\pm$ 0.03 & -177.4 $\pm$ 1.0 & -1.4 $\pm$ 1.0 & -1.2 $\pm$ 0.6 & 12.1 & $-$2.60 $\pm$ 0.12 & --- & Y \\ \\
298.14837 & $-$22.04551 & 20.64 $\pm$ 0.01 & 19.99 $\pm$ 0.01 & 21.26 $\pm$ 0.03 & -176.7 $\pm$ 1.3 & 1.0 $\pm$ 1.3 & 0.6 $\pm$ 0.7 & 12.6 & $-$99.0 $\pm$ 0.00 & --- & Y \\ \\
298.13415 & $-$22.04367 & 21.03 $\pm$ 0.01 & 20.46 $\pm$ 0.01 & 21.58 $\pm$ 0.04 & -267.1 $\pm$ 2.5 & -1.0 $\pm$ 1.5 & -2.2 $\pm$ 0.9 & 5.3 & $-$2.99 $\pm$ 0.12 & --- & N \\ \\
298.11950 & $-$22.0253 & 21.19 $\pm$ 0.01 & 20.56 $\pm$ 0.01 & 21.89 $\pm$ 0.05 & -279.0 $\pm$ 2.8 & -2.0 $\pm$ 2.4 & -7.7 $\pm$ 1.4 & 6.7 & $-$2.44 $\pm$ 0.12 & --- & N \\ \\
298.14955 & $-$22.10703 & 21.24 $\pm$ 0.01 & 20.62 $\pm$ 0.01 & 21.89 $\pm$ 0.05 & -174.5 $\pm$ 11.9 & 4.8 $\pm$ 3.1 & 2.9 $\pm$ 1.8 & 3.4 & $-$2.69 $\pm$ 0.12 & --- & Y \\ \\
298.03665 & $-$22.04132 & 18.13 $\pm$ 0.01 & 17.23 $\pm$ 0.01 & 19.54 $\pm$ 0.01 & 127.2 $\pm$ 0.5 & 0.2 $\pm$ 3.5 & -1.4 $\pm$ 1.8 & 105.2 & $-$1.23 $\pm$ 0.12 & -1.19 $\pm$ 0.17 & N \\ \\
298.09544 & $-$22.14968 & 19.71 $\pm$ 0.01 & 18.89 $\pm$ 0.01 & 20.98 $\pm$ 0.02 & -2.7 $\pm$ 0.9 & 0.1 $\pm$ 1.4 & -1.9 $\pm$ 0.7 & 31.0 & $-$1.25 $\pm$ 0.12 & -1.36 $\pm$ 0.15 & N \\ \\
298.06312 & $-$22.11093 & 20.13 $\pm$ 0.01 & 19.44 $\pm$ 0.01 & 21.05 $\pm$ 0.02 & -29.5 $\pm$ 1.1 & -0.3 $\pm$ 0.8 & -1.3 $\pm$ 0.4 & 12.8 & $-$99.0 $\pm$ 0.00 & -1.37 $\pm$ 0.17 & N \\ \\
298.05710 & $-$22.02942 & 20.59 $\pm$ 0.01 & 20.02 $\pm$ 0.01 & 21.35 $\pm$ 0.03 & -63.9 $\pm$ 0.8 & 2.7 $\pm$ 1.8 & 0.4 $\pm$ 1.0 & 10.9 & $-$1.89 $\pm$ 0.12 & --- & N \\ \\
297.96338 & $-$22.04654 & 20.35 $\pm$ 0.01 & 19.76 $\pm$ 0.01 & 21.11 $\pm$ 0.02 & -18.8 $\pm$ 1.5 & --- $\pm$ --- & --- $\pm$ --- & 12.1 & $-$1.99 $\pm$ 0.12 & -1.61 $\pm$ 0.16 & N \\ \\
298.15344 & $-$22.04959 & 17.76 $\pm$ 0.01 & 16.74 $\pm$ 0.01 & 18.98 $\pm$ 0.01 & -175.5 $\pm$ 0.3 & -0.9 $\pm$ 0.4 & -1.0 $\pm$ 0.2 & 125.2 & $-$2.31 $\pm$ 0.12 & -2.23 $\pm$ 0.12 & Y \\ \\
298.28018 & $-$22.12346 & 17.18 $\pm$ 0.01 & 16.40 $\pm$ 0.01 & 18.60 $\pm$ 0.01 & -14.2 $\pm$ 0.4 & -0.3 $\pm$ 1.3 & 0.8 $\pm$ 0.8 & 131.5 & --- & -1.19 $\pm$ 0.18 & N \\ \\
298.01474 & $-$21.91537 & 17.29 $\pm$ 0.01 & 16.29 $\pm$ 0.01 & 18.90 $\pm$ 0.01 & -48.0 $\pm$ 0.4 & -4.0 $\pm$ 1.7 & -8.8 $\pm$ 1.0 & 131.5 & --- & -1.23 $\pm$ 0.18 & N \\ \\
\end{tabular}
\end{sideways}
\end{table*}
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| {'timestamp': '2020-05-14T02:00:18', 'yymm': '2005', 'arxiv_id': '2005.05976', 'language': 'en', 'url': 'https://arxiv.org/abs/2005.05976'} |
\section{Introduction}
Self-supervised contrastive learning has been shown great potential in extracting general visual representations recently \cite{chen2020simple}. Models pre-trained in such a manner achieve better results in downstream tasks like object detection and segmentation, comparing to fully-supervised pre-trained models \cite{he2020momentum}. Besides, the gap of performance in the basic image classification is also narrowing. However, problems with self-supervised learning are discovered recently, including inductive bias and sensitivity to data augmentation. These are mainly caused by the weak-supervising signal and image pairs used in the contrastive mechanism. The goal of learning general representations has not been achieved yet.
We propose a novel framework FLAGS with a custom sampling strategy that incorporate features from different latent spaces. Our motivation is to combine the strengths of unsupervised contrastive learning and fully-supervised learning to generate richer representations. We argue that there are two levels of semantics in images, global semantics and local semantics. Global semantics refer to all entities in the image including the background and the foreground. Local semantics refer to the entity that is defined by the label of the image. Self-supervised contrastive learning methods perform instance discrimination which can be seen as learning global semantics. While the supervised learning methods on the other side, learn the local semantics. As shown in Table 1, We find a trend that tasks like object detection and segmentation benefit from global semantics, whereas image classification benefits more from local semantics. In details, for image classification, the supervised method which learns local semantics achieves much better accuracy than self-supervised methods which learn global semantics. On the other hand, for detection and segmentation which require more information besides the semantics of the entity of interest, self-supervised methods perform close to or even better than the supervised method. Thus intuitively a method learns both global and local semantics should be a better pre-training for larger variety of tasks, performing better than the methods learn either one. The authors in \cite{wei2020semantic} also argue that self-supervised methods and fully-supervised methods learn different kinds of representations. They try to alleviate the conflict between the fully-supervised and unsupervised learning objectives. Comparing to their method, our method aims to learn different level of semantics features thoroughly with equal attention on global and local semantics. Our contributions are summarized below:
\begin{enumerate}
\item We design a novel architecture which can effectively guide feature extractors to learn both global and local semantic features.
\item We avoid the conflict between the learning objectives, by using a sampling strategy and splitting the learning of two semantic features by projecting them into separate hyperspheres.
\item Our method of guiding contrastive networks opens a new direction of research for general representation learning.
\end{enumerate}
\begin{table*}
\begin{center}
\centering
\label{Table_1}
\caption{ Comparison of different pre-training methods on three downstream tasks. The standard backbone is ResNet-50. Linear evaluation protocol \cite{he2020momentum} is applied for image classification. For object detection and semantic segmentation, models are fine-tuned on the PASCAL VOC dataset \cite{pascal}.}
\begin{tabular}{c|c|c|c|c}
\hline
Type of semantics &pre-train & Classification (Accuracy\%) & Detection ($AP_{50}$) & Segmentation (mIOU) \\ \hline
\multirow{4}{*}{global} &Colorization \cite{zhang2016colorful} & 39.6 &- &- \\ \cline{2-5}
& Jigsaw \cite{noroozi2017unsupervised} & 44.6 &61.4 &- \\ \cline{2-5}
&SimCLR \cite{chen2020simple} & 69.3 & 79.4 \cite{wang2021dense} & 64.3 \cite{wang2021dense} \\ \cline{2-5}
&MoCo-v2 \cite{chen2020improved} & 71.1 & 82.5 & 67.5 \cite{wang2021dense} \\ \hline
local &supervised \cite{he2015deep} & 76.5 & 81.3 & 67.7 \cite{wang2021dense} \\ \hline
\end{tabular}
\end{center}
\end{table*}
\section{Related Work}
In this section, we summarize the current development in representation learning. The objective of representation learning is to abstract and disentangle underlying factors of variation existing in raw data \cite{bengio2014representation}. A robust representation contains unique properties that can be easily distinguished from others. To achieve this goal, different learning strategies are explored.
\subsection{Supervised Learning}
Fully-supervised methods heavily rely on labels as strong supervising signals when training. The objective is to learn a representation that maximizes the probability of finding the correct label of images. In terms of network architectures, different convolutional neural networks \cite{simonyan2015deep, he2015deep, tan2020efficientnet} are commonly used to encode input images and generate feature maps for classification and other tasks. Recently, transformer-based models have been shown to better encode visual information and achieved promising results in downstream tasks \cite{dosovitskiy2021image, carion2020endtoend, zheng2021rethinking}. Commonly, these methods train on large datasets such as ImageNet \cite{imagenet_cvpr09} and JFT-300M \cite{sun2017revisiting} which capture a great variation of objects. The extracted representations primarily encode specific features biased to the learning objective. Geirhos et al. \cite{geirhos2019imagenettrained} has shown through experiments that CNN models trained on ImageNet are biased to texture features that are beneficial to object recognition tasks. This indicates the learned representation only captures partial invariance of objects.
\subsection{Pretext Learning}
Learning to perform a predefined pretext task is a discriminative approach in self-supervised learning. Pretext tasks are hand-designed to learn desired features using self-generated pseudo labels. Specifically, image inpainting \cite{pathak2016context} and colorization \cite{zhang2016colorful} are generation-based tasks where the objective is to guide models to recover incomplete images. To achieve these tasks, both semantic and context information of scenes are necessary to be learned. Solving image jigsaw puzzle \cite{noroozi2017unsupervised, kim2018learning} and image rotation \cite{gidaris2018unsupervised} guide networks to encode positional and global context information of images. However, experiments from Chen et al. \cite{chen2020simple} have demonstrated the gap between the model performance on downstream tasks trained using pretext task methods and fully-supervised methods as shown in Table 1. The common problem is that a single pretext task cannot provide enough supervision signal when training and it limits networks to learn biased representations.
\subsection{Contrastive Learning}
Contrastive learning methods utilize instance discrimination as the pretext task and it shows the potential to better extract useful representation that transfers well in downstream tasks. In the beginning, image features are extracted by an encoder and they are projected into a latent space. Contrastive loss \cite{1640964} is then used to measure the similarity(distance) between those projected features. For supervised contrastive learning, SCL \cite{khosla2021supervised} utilizes images with the same label as positive samples to contrast which achieves competitive results as traditional supervised methods. In unsupervised contrastive learning, MoCo \cite{he2020momentum} introduces a dynamic dictionary with the momentum update mechanism which extracts representation by comparing with a diverse set of negative samples. SimCLR \cite{chen2020simple} proves the importance of data augmentation and large batch size in a simple contrastive setting. However, data augmentation introduces inductive bias that restricts the generalization of representations. Experiments show that both strength and selection of data augmentation affect the performance of pre-trained models. As shown by Purushwalkam and Gupta \cite{purushwalkam2020demystifying}, data augmentations like random cropping learn occlusion invariance. Besides, instead of contrasting with different images within the same class, such approaches do not use class labels and can only contrast with augmented images of themselves which fail to explore object invariance.
\begin{figure*}
\begin{center}
\includegraphics[width=0.64\linewidth]{positive_pair_example.jpg}
\end{center}
\caption{Illustration of the sorted images in the same class based on cosine similarity.
In this example, the top-2 images are selected as global positive pairs since they have similar background as the query image. While the two images in the middle of the list, do not have the similar background as the query image. So they are selected as local positive pairs.}
\label{fig:short}
\end{figure*}
\section{Method}
\subsection{Preliminary Knowledge}
\textbf{Supervision Signals} To learn a desired representation, specific supervision signals need to be supplied to guide networks during training. However, direct combination of supervision signals injects confusion into learning objectives. In self-supervised contrastive learning, methods are sensitive to data augmentation which serves as a weak supervision signal. To tackle this problem, LooC \cite{xiao2021contrastive} projects features into different latent subspaces for each augmentation so the invariance of raw data and the variance of augmentation are both preserved in the extracted representations. We argue that this concept of disentangling features into separate latent spaces can help to learn a more robust representation with stronger guidance.
\textbf{Adjustment In Positive Pair Selection} The selection of positive pairs plays an important role in contrastive learning. In self-supervised settings, augmented images of themselves are treated as positive sample whereas other images including those from the same classes are treated as negatives. In supervised settings, SCL \cite{khosla2021supervised} treats all samples from the same class as positives and other remaining samples in the batch as negative samples, which networks learn a richer representation within local regions. Recent development of SCAN \cite{wei2020semantic} first reveals the conflicting objectives in supervised and self-supervised learning and shows task-agnostic appearance information are learned in common self-supervised methods. To alleviate such conflict in contrastive learning, SCAN selects the top-k nearest neighbors of query image from the same class as positives in the feature space formulated by pre-trained MoCo-v2, which the pre-trained model has been proved to generate more robust representations.
\begin{figure}[t]
\label{figure_2}
\begin{center}
\includegraphics[width=0.97\linewidth]{model_structure_new.png}
\end{center}
\caption{\textbf{The main architecture of FLAGS} At each training step, a query image X is processed by the encoder. A local key pair \{$X_{l0}, X_{l1}$\}, a global key pair \{$X_{g0}, X_{g1}$\}, and an augmented version of query $X_{aug0}$ are fed into the momentum encoder. The output features \{$k_{l0}, k_{l1}, k_{g0}, k_{g1}, k_{aug0}$\} are then projected to corresponding local and global subspace via mlp\_l\_k and mlp\_g\_k. The encoded query q is projected into local and global subspace via mlp\_l\_q and mlp\_g\_q. Losses for each subspace are calculated separately and combined to form the total loss as shown in the loss function section. Two queues are maintained for local and global semantic pairs. }
\label{fig:long}
\label{fig:onecol}
\end{figure}
\subsection{Our Approach}
Our goal is to enable networks to learn rich representations that can smoothly adapt to various downstream tasks. We first define that self-supervised contrastive methods learn \textbf{global semantics} and fully-supervised methods learn \textbf{local semantics}. Specifically, global semantics encode rough context and semantic information of all objects throughout the region. Local semantics focus on extracting detailed representations of specific objects located at certain regions. Our intuitive is to guide models to encode both local and global semantics by fusing the learning objectives of fully-supervised learning and contrastive learning. Inspired by LooC \cite{xiao2021contrastive}, we propose a novel supervised contrastive framework FLAGS as shown in Figure 2. The core idea of FLAGS is to project pairs of features into either global or local subspace. Specifically, a query image with a local and a global positive pair (keys) are fed into the encoders, then the output features are projected into corresponding subspaces using two separate sets of MLPs. We maintain two queues to store local and global keys that are used to contrast during training. Losses are computed independently for each subspace and combined in the end.
\newpage
\begin{widetext}
\begin{equation}
\label{equation 1}
\mathcal{L}_{ALL} = \sum_{i\in I} \mathcal{L}_{global}(i) + \sum_{i\in I}\mathcal{L}_{local}(i)
\end{equation}
\begin{equation}
\label{equation 2}
\mathcal{L}_{global}(i) = \frac{-1}{|P_g(i)|}
\sum_{p\in P_g(i)}
\log \frac{\exp(z_g(i) \cdot z_g(p) / \tau )}
{\sum\limits_{a\in Q_g}\exp(z_g(i) \cdot z_g(a) / \tau) + {\exp(z_g(i) \cdot z_g(p) / \tau )}}
\end{equation}
\begin{equation}
\label{equation 3}
\mathcal{L}_{local}(i) = \frac{-1}{|P_l(i)|}
\sum_{p\in P_l(i)}
\log \,
\frac{\exp(z_l(i) \cdot z_l(p) / \tau )}
{\sum\limits_{a\in Q_l}\exp(z_l(i) \cdot z_l(a) / {\tau)+\exp(z_l(i) \cdot z_l(p) / \tau )}}
\end{equation}
\end{widetext}
\textbf{Image Pairs Generation} To form image pairs for each query image that will be fed into different subspaces, we adapt the image pair selection strategy in SCAN \cite{wei2020semantic} and modify it accordingly. In details, images are fed into the ResNet pre-trained in MoCo-v2 to generate features \(p\) with the size of [1, 1024]. The cosine similarity between each image within the same class is calculated. Then images are sorted using the similarity and stored into a list as shown in Figure 1. We select the top-2 similar images as a pair to project on global-semantic subspace. To alleviate the conflict between global and local learning objectives, two images at middle of lists are chosen to form a pair for local-semantic subspace. Since their extent of global semantics similarity with the query image is moderate.
\textbf{Loss Function}
Our loss function is based on the common contrastive loss \cite{1640964}. We call it combined contrastive loss \autoref{equation 1}. It is the summation of contrastive loss at the global branch and the contrastive loss at the local branch. \autoref{equation 2} shows the loss for one query image at the global branch. \autoref{equation 3} shows the loss for one query image at the local branch. The form of equation is inspired by the supervised contrastive loss introduced in SCL \cite{khosla2021supervised}. It allows the contrastive loss function to generalize to an arbitrary number of positives. In details, there are $N$ query images in one batch. Let $i$ be the index of images in the batch. For the $i$-th query image, it has global positive keys and its augmentation $P_g(i)$. It also has local positive keys $P_l(i)$. $P_g(i)$ has dimension $M+1$ where $M$ is the number of positive keys and one represents the augmentation. $P_l(i)$ has dimension $M$. $Q_g$ is the queue that contains negative images for the global semantic branch and $Q_l$ is the queue for the local semantic branch. $Q_g$ and $Q_l$ both has dimension of $K$. $Z_g(i)$ is the normalized projected features of image $i$ in the global subspace. Accordingly, $Z_l(i)$ is in the local subspace. $\tau \in \mathbb R$ is temperature parameter.
\section{Experiments}
\subsection{Pre-training}
\textbf{Dataset Preparation}
We use ImageNet-1M \cite{imagenet_cvpr09} for pre-training and validation. We use the proposed image pair generation strategy to prepare the global and local positive pairs for each image in the ImageNet-1M \cite{imagenet_cvpr09} train set. These positive pairs are keys for the corresponding query image during training.
\textbf{Training}
We perform three different pre-training methods. The first pre-training method is FLAGS with only the global branch. The second pre-training method is FLAGS with both the global and the local branch. The last is MoCo\_v2 \cite{chen2020improved} which using the same loss as FLAGS. The hyper parameters are basically the same as those used in MoCo \cite{he2020momentum}. We use SGD as the optimizer with a momentum of 0.9. The learning rate is 0.03. The batch size is 256. We use the loss function shown in Section 3.2. The checkpoint at the 200th epoch of each pre-training is used for the rest of experiments. Our train set contains about 1.23 million images rather than normal 1.28 million images. 50,000 images are evenly taken out of the train set from each class.
\subsection{Linear Evaluation}
We follow the common linear evaluation protocol where we freeze the weights of the pre-trained model except the fully-connected layers. Then, we train a supervised classifier using the ImageNet-1M \cite{imagenet_cvpr09}. The hyper parameters are the same as those used in MoCo \cite{he2020momentum}. The top-1 accuracy is recorded during training after each epoch and the top accuracy is shown in Table 2. As we can see, FLAGS models achieve much higher accuracy than MoCo \cite{he2020momentum}. This is expected because FLAGS models used more supervised signals during pre-training. At the same time, FLAGS with a local branch has a lower accuracy than FLAGS without a local branch. This might be due to the noise added when we select positive local pairs. The selection strategy for local positive pair should vary for different query images or different classes. Since currently, the local key might be too different from the global key which creates a conflicting signal that makes the model confused. In other words, we tell the model to encode two very different images in the same way. This may explains the accuracy difference between FLAGS with and without the local branch.
\begin{table}[t]
\begin{center}
\centering
\label{Table_2}
\caption{Linear evaluation accuracy of ResNet models trained using different methods}
\begin{tabular}{c|c}
\hline
$\ $ & Accuracy\%\\
\hline
MoCo & 67.16 \\ \hline
FLAGS aug+global & 78.45 \\ \hline
FLAGS aug+global+local & 77.75 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Object Detection and Segmentation}
We use object detection and segmentation as downstream tasks to evaluate how good the features are transferable. We perform the experiments with two datasets, PASCAL\_VOC\_2007 \cite{pascal} and COCO\_2017 \cite{coco}. In all experiments, we train detectors with the ResNet R50-C4 \cite{he2015deep} as the feature encoder, where the weights of encoder are initialized from the three pre-trained models, two of them using FLAGS and one using MoCo. We then fine-tune the whole detectors end-to-end. The same hyper parameters settings are used for all experiments.
\textbf{COCO Object Detection and Instance Segmentation}
We use Mask R-CNN \cite{mask} as the detector with the backbone of ResNet R50-C4 \cite{he2015deep}. The batch size is 8 and iteration is 180,000. These mean that the total epoch is about the same as 1*schedule defined in detectron2 \cite{detectron2}. The learning rate is 0.01. We fine-tune models using COCO train\_2017 and evaluate using COCO val\_2017. The results are shown in Table 3 and 4. As we can see, initialized with weights of FLAGS with global and local branch surpass with weights of a supervised pre-training in both detection and segmentation. This shows the effectiveness of our proposed method.
\begin{table}[t]
\begin{center}
\centering
\label{Table_3}
\caption{Object detection results of Mask R-CNN on COCO dataset with different pre-trained ResNet R50-C4 models.}
\begin{tabular}{c|c|c|c}
\hline
pre-train & AP & $AP_{50}$ & $AP_{75}$ \\ \hline
supervised 1*schedule & 38.200 & 58.200 & 41.200 \\ \hline
MoCo batch size 8 & 37.879 & 57.201 & 40.903 \\ \hline
FLAGS aug+global & 38.344 & 57.852 & 41.357 \\ \hline
FLAGS aug+global + local & 38.197 & 58.053 & 40.994 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[t]
\begin{center}
\centering
\label{Table_4}
\caption{Instance segmentation results of Mask R-CNN on COCO dataset with different pre-trained ResNet R50-C4 models.}
\begin{tabular}{c|c|c|c}
\hline
pre-train & AP & $AP_{50}$ & $AP_{75}$ \\ \hline
supervised 1*schedule & 33.300 & 54.700 & 35.200 \\ \hline
MoCo batch size 8 & 33.312 & 54.055 & 35.598 \\ \hline
FLAGS aug + global & 33.511 & 54.461 & 35.505 \\ \hline
FLAGS aug + global + local & 33.431 & 54.552 & 35.460 \\
\hline
\end{tabular}
\end{center}
\end{table}
\textbf{PASCAL VOC Object Detection}
The detector used is Faster R-CNN \cite{faster} with R50-C4 \cite{he2015deep} as backbone. The batch size is 4 and the number of iteration is 96,000. The learning rate is 0.005. The same feature normalization (sync norm) described in MoCo \cite{he2020momentum} is used. Other parameters are defaults defined in detectron2 \cite{detectron2}. We fine-tune models using VOC trainval\_2012 and VOC trainval\_2007. Evaluation is conducted on VOC test\_2007. The results are shown in Table 5. FLAGS with only global branch gets similar performance as MoCo \cite{he2020momentum}. When the local branch is added, the performance decreases. This is opposite to the results with COCO \cite{coco} where FLAGS with local branch achieves the best overall scores. We think this phenomenon is due to the difference between two datasets. As survey shows, in PASCAL\_VOC \cite{pascal} more than 40\% instances take above 50\% of whole image size. On the other side, in COCO \cite{coco} only 1\% instances take more than 50\% of whole image size. In addition, COCO \cite{coco} has 7.3 objects per image and PASCAL\_VOC \cite{pascal} has 2.3 objects per image. These differences mean that detection and segmentation requires more local semantics to get into details of part of images in COCO \cite{coco}. FLAGS’s local branch successfully captures local semantics, improving the detector’s performance on COCO \cite{coco}.
\begin{table}[t]
\begin{center}
\centering
\label{Table_5}
\caption{Object detection results of Mask R-CNN on PASCAL\_VOC dataset with different pre-trained ResNet R50-C4 models.}
\begin{tabular}{c|c|c|c}
\hline
pre-train & AP & $AP_{50}$ & $AP_{75}$ \\ \hline
MoCo & 50.571 & 78.879 & 55.193 \\ \hline
FLAGS aug+global & 50.701 & 79.328 & 54.310 \\ \hline
FLAGS aug+global+local & 48.141 & 78.348 & 51.034 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusion}
In this paper, we propose FLAGS that extracts rich and transferable representations for various downstream tasks. We suggest representations have two levels of semantics: global semantics and local semantics. By contrasting with local and global image pairs in different subspaces, models benefit from both learning objectives. Through a few experiments, we demonstrate the learned representations based on FLAGS improves the performance in image classification, object detection and segmentation. It has been shown that this direction is promising and we hope others can expand the research. The positive pair sampling strategy can be optimized to get more precise keys. In addition, more visualizations of learned representations are beneficial.
\newpage
\section*{Acknowledgement}
We would like to express our great appreciation to Dr. Hailin Hu for providing tremendous support and constructive suggestions. We also want to thank Xiangqian Wang and Lin Du for facilitating the publishing. Last but not least, thanks Ce Wang for the great effort in validating experiment results.
{\small
\bibliographystyle{ieee}
| {'timestamp': '2022-03-01T02:50:56', 'yymm': '2202', 'arxiv_id': '2202.13837', 'language': 'en', 'url': 'https://arxiv.org/abs/2202.13837'} |
\section{Introduction}
The matrix model of uncompactified M theory \cite{dhn}-\cite{bfss}
has been generalized to
arbitrary toroidal compactifications of type IIA and IIB string theory.
These models can be viewed as particular large $M$ limits of the
original matrix model, in the sense that
they may be viewed as the dynamics of a restricted class
of large $M$ matrices, with the original matrix model Lagrangian.
A separate line of reasoning has led to a description of the
Ho\v rava-Witten domain wall in terms of matrix quantum
mechanics \cite{orbi}. Here,
extra degrees of freedom have to be added to the original matrix model.
As we will review below, if these new variables, which tranform in the
vector representation of the gauge group, are not added, then the model
does not live in an eleven dimensional spacetime, but only on its
boundary.. Although it is, by
construction, a unitary quantum mechanics, it probably does not recover
ten dimensional Lorentz invariance in the large $M$ limit. Its nominal
massless particle content is the ten dimensional $N = 1$ supergravity
(SUGRA) multiplet, which is anomalous.
With the proper number of vector variables added, the theory does have
an eleven dimensional interpretation. It is possible to speak of states
far from the domain wall and to show that they behave exactly like the
model of \cite{bfss}. Our purpose in the present paper is to compactify
this model on spaces of the general form $S^1 / Z_2 \times T^d$. We
begin by reviewing the argument for the single domain wall quantum
mechanics, and generalize it to an $S^1 / Z_2$ compactification. The
infinite momentum frame Hamiltonian for this system is practically
identical to the static gauge $O(M)$ Super Yang Mills (SYM)
Hamiltonian for $M$ heterotic D strings
in Type I string theory. They differ only in the boundary conditions
imposed on the fermions which transform in the vector of $O(M)$. These
fermions are required for $O(M)$ anomaly cancellation in both models,
but the local anomaly does not fix their boundary conditions.
Along the moduli space of the $O(M)$ theory, the model exactly
reproduces the string field theory Fock space of
heterotic string theory. The inclusion of both Ramond and Neveu Schwarz
boundary conditions for the matter fermions, and the GSO projection, are
simple consequences of the $O(M)$ gauge invariance of the model.
Generalizing to higher dimensions,
we find that the heterotic matrix model on
$S^1 / Z_2 \times T^d$ is represented by a $U(M)$ gauge theory on
$S^1 \times T^d / Z_2$ . On the orbifold circles, the gauge
invariance reduces to $O(M)$. We are able to construct both the
heterotic and open string sectors of the model, which dominate in
different limits of the space of compactifications.
In the conclusions, we discuss the question of whether the heterotic
models which we have constructed are continuously connected to the
original uncompactified eleven dimensional matrix model. The answer to
this question leads to rather surprising conclusions, which inspire us
to propose a conjecture about the way in which the matrix model solves
the cosmological constant problem. It also suggests that string vacua
with different numbers of supersymmetries are really states of {\it
different} underlying theories. They can only be continuously connected
in limiting situations where the degrees of freedom which differentiate
them decouple.
\section{Heterotic Matrix Models in Ten and Eleven Dimensions}
In \cite{evashamit} an $O(M)$ gauged supersymmetric
matrix model for a single Ho\v rava-Witten domain
wall embedded in eleven dimensions was proposed. It was based on an
extrapolation of the quantum mechanics describing $D0$ branes near an
orientifold plane in Type IA string theory \cite{df}. The model was
presented as an orbifold of the original \cite{bfss} matrix model in
\cite{motl}.
In the Type IA context
it is natural to add degrees of freedom transforming in the vector of
$O(M)$ and corresponding to the existence
of $D8$ branes and the $08$ strings connecting them to the $D0$ branes.
Since $D8$ branes are movable in Type IA theory, there are consistent
theories both with and without these extra degrees of freedom. That is,
we can consistently give them masses, which represent the distances
between the $D8$ branes and the orientifold.
However,
as first pointed out by \cite{df}, unless the number of $D8$ branes sitting
near the orientifold is exactly $8$, the $D0$ branes feel a linear
potential which either attracts them to or repels them from the
orientifold. This is the expression in the quantum mechanical approximation,
of the
linearly varying dilaton first found by Polchinski and Witten \cite{jopo}.
This system was studied further in \cite{rk} and
\cite{bss}. In the latter work the supersymmetry and gauge structure of
model were clarified, and the linear potential was shown to correspond to
the fact that the ``supersymmetric ground state''
of the model along classical flat directions representing excursions
away from the orientifold was not gauge invariant.
{}From this discussion it is clear that the only way to obtain a model
with an eleven dimensional interpretation is to add sixteen massless
real fermions
transforming in the vector of $O(M)$, which is the model proposed in
\cite{evashamit}. In this case, $D0$ branes can move freely away from the
wall, and far away from it the theory reduces to the $U([{M\over 2}])$
model of \cite{bfss} \footnote{Actually there
is a highly nontrivial question
which must be answered in order to prove that the effects of the
wall are localized.
In \cite{bss} it was shown that supersymmetry allowed an arbitary
metric for the coordinate representing excursions away from the wall.
In finite orders of perturbation theory the metric falls off with
distance but, as in the discussion of the graviton scattering amplitude
in \cite{bfss}, one might worry that at large $M$ these could sum up to
something with different distance dependence. In \cite{bfss} a
nonrenormalization theorem was conjectured to protect the relevant term
in the effective action. This cannot be the case here.}.
Our task now is to construct a model representing two Ho\v rava-Witten
end of the world $9$-branes separated by an interval
of ten dimensional space.
As in \cite{motl} we can approach this task by attempting to mod out the $1+
1$ dimensional field theory \cite{bfss}, \cite{taylor}, \cite{motlb},
\cite{bs}, \cite{dvv} which describes M
theory compactified on a circle. Following the logic of \cite{motl}, this
leads to an $O(M)$ gauge theory. The $9$-branes are stretched around
the longitudinal direction of the infinite momentum frame (IMF)
and the $2-9$ hyperplane of the transverse
space. $X^1$ is the differential operator
$${R_1 \over i}{\partial \over \partial\sigma} - A_1$$
where $\sigma$ is in $[0,2\pi ]$, and $A_1$ is
an $O(M)$ vector potential. The other $X^i$ transform in the ${\bf
{M(M+1) \over 2}}$ of $O(M)$. There are two kinds of fermion multiplet.
$\theta$ is an ${\bf 8_c}$ of the spacetime $SO(8)$, a symmetric tensor
of $O(M)$ and is the superpartner of $X^i$ under the eight dynamical
and eight kinematical SUSYs which survive the projection. $\lambda$ is
in the adjoint of $O(M)$, the ${\bf 8_s}$ of $SO(8)$, and is the
superpartner of the gauge potential. We will call it the gaugino.
As pointed out in \cite{rk} and \cite{bss}, this model is anomalous.
One must add $32$ Majorana-Weyl fermions $\chi$ in the ${\bf M}$ of $O(M)$.
For sufficiently large $M$, this is the only fermion content which can
cancel the anomaly. The continuous $SO(M)$ anomaly does not fix the
boundary conditions of the $\chi$ fields. There are various consistency
conditions which help to fix them, but in part we must make a choice
which reflects the physics of the situation which we are trying to
model.
The first condition follows from the fact that our gauge group is $O(M)$
rather than $SO(M)$. That is, it should consist of the subgroup of
$U(M)$ which survives the orbifold projection. The additional $Z_2$
acts only on the $\chi$ fields, by reflection. As a consequence, the
general principles of gauge theory tell us that each $\chi$ field
might appear with either periodic or antiperiodic boundary conditions,
corresponding to a choice of $O(M)$ bundle. We must also make a
projection by the discrete transformation which reflects all the
$\chi$'s.
What is left undetermined by these
principles is choice of relative boundary
conditions among the $32$ $\chi$'s.
The Lagrangian for the $\chi$ fields is
\eqn{chilag}{\chi (\partial_t
+ 2\pi R_1\partial_{\sigma} - i A_0 - i A_1) \chi
.} In the large $R_1$ limit, the volume of the space on which the gauge
theory is compactified is small, and its coupling is weak, so we can
treat it by semiclassical methods. In particular, the Wilson lines
become classical variables. We will refer to classical values of the
Wilson lines as expectation values of the gauge potential $A_1$.
(We use the term expectation value loosely, for we are dealing with a
quantum system in finite volume. What we mean
is that these ``expectation values'' are the slow variables in
a system which is being treated by the Born-Oppenheimer approximation.)
An excitation of the system at some position in the direction tranverse
to the walls is represented by a wave function of
$n \times n$ block matrices in which $A_1$
has an expectation value breaking $O(n)$ to $U(1) \times U([n/2])$.
In the presence of a generic expectation value, in $A_0 = 0$
gauge, the $\chi$ fields will not have any zero frequency modes.
The exceptional positions where zero frequency modes exist are $A_1 = 0$
(for periodic fermions) and $A_1 = \pi R_1$ (for antiperiodic fermions).
These define the positions of the end of the world $9$-branes, which we
call the walls. When $R_1 \gg l_{11}$, all of the finite wavelength
modes of all of the fields have very high frequencies and can be
integrated out.
In this limit, an excitation
far removed from the walls has precisely the degrees of freedom of a
$U([{n\over2}])$ gauge quantum mechanics. The entire content of the
theory far from
the walls is $U([{M\over 2}])$ gauge quantum mechanics. It has no
excitations carrying the quantum numbers created by the $\chi$ fields,
and according to the conjecture of \cite{bfss} it reduces to eleven
dimensional M theory in the large $M$ limit. This reduction assumes
that there is no longe range interaction between the walls and the rest
of the system.
In order to fulfill this latter condition it must be true that at $A_1 =
0$, and in the large $R_1$ limit, the field theory reproduces
the $O(M)$ quantum mechanics described at the beginning of this section
(and a similar condition near the other boundary).
We should find $16$ $\chi$ zero modes near each wall. {\it Thus,
the theory must contain a sector in which
the $32$ $1+1$ dimensional $\chi$ fields are grouped
in groups of $16$ with opposite periodicity}. Half of the fields will
supply the required zero modes near each of the walls. Of course, the
question of which fields have periodic and which antiperiodic boundary
conditions is a choice of $O(M)$ gauge. However, in any gauge
only half of the $\chi$ fields will have zero modes located at any
given wall. We could of course consider sectors of the
fundamental $O(M)$ gauge theory in which there is a different
correlation between boundary conditions of the $\chi$ fields.
However, these would not have an eleven dimensional interpretation at
large $R_1$. The different sectors are not mixed by the Hamiltonian so
we may as well ignore them.
To summarize, we propose that M theory compactified on $S^1 / Z_2$ is
described by a
$1+1$ dimensional $O(M)$ gauge theory with $(0,8)$ SUSY. Apart from the
$(A_{\mu}, \lambda )$ gauge multiplet, it contains
a right moving $X^i, \theta$ supermultiplet in the symmetric tensor
of $O(M)$ and 32 left moving fermions, $\chi$, in the vector. The
allowed gauge bundles for $\chi$ (which transforms under the discrete
reflection which leaves all other multiplets invariant), are those in
which two groups of $16$ fields have opposite periodicities.
In the next section we will generalize this construction to
compactifications on general tori.
First let us see how heterotic strings emerge from this formalism in the
limit of small $R_1$. It is obvious that in this limit, the string
tension term in the SYM Lagrangian becomes very small. Let us rescale
our $X^i$ and time variables so that the quadratic part of the
Lagrangian is independent of $R_1$. Then, as in \cite{bs},
\cite{motlb}, \cite{dvv},
the commutator term involving the $X^i$ gets a coefficient
$R^{-3}$ so that we are forced onto the moduli space in that limit. In
this $O(M)$ system, this means that the $X^i$ matrices are diagonal,
and the gauge group is completely broken
to a semidirect product of $Z_2$ (or
$O(1)$) subgroups which reflect the individual components of the
vector representation, and an $S_M$ which permutes the eigenvalues of
the $X^i$. The moduli space of low
energy
fields\footnote{We use the term moduli space to refer to the space of
low energy fields whose effective theory describes the small $R_1$
limit (or to the target space of this effective theory).
These fields are in a Kosterlitz Thouless phase and do not have
expectation values, but the term moduli space is a convenient shorthand
for this subspace of the full space of degrees of freedom.}
consists of diagonal $X^i$ fields, their superpartners $\theta_a$ (also
diagonal matrices), and the $32$ massless left moving $\chi$ fields.
The gauge bosons and their superpartners $\lambda^{\dot{\alpha}}$
decouple in the small $R_1$ limit. All of the $\chi$ fields are light
in this limit.
\subsection{Screwing Heterotic Strings}
As first explained in \cite{motlb} and elaborated in
\cite{bs}, and \cite{dvv},
twisted sectors under $S_N$ lead to strings of arbitrary
length\footnote{These observations are mathematically identical to
considerations that arose in the counting of BPS states in black hole
physics \cite{BPS} . }.
The
strings of conventional string theory, carrying continuous values of the
longitudinal momentum, are obtained by taking $N$ to infinity
and concentrating on
cycles whose length is a finite fraction of $N$.
The new feature which arises in the heterotic string is that the
boundary conditions of the $\chi$ fields can be twisted by the discrete
group of reflections.
A string configuration of length $2\pi k$, $X_S^i (s)$, $0 \leq s \leq
2\pi k$, is represented by a diagonal matrix:
\eqn{screw}{
X^i(\sigma)=\tb{{cccc}
X_S^i(\sigma)&&&\\
&X_S^i(\sigma+2\pi)&&\\
&&\ddots&\\
&&&X_S^i(\sigma+2\pi(N-1))}.}
This satisfies the twisted boundary condition
$X^i(\sigma+2\pi)=E_O^{-1}X^i(\sigma) E_O$ with
\eqn{bcx}{
E_O=\tb{{ccccc}
&&&&\epsilon_k\\
\epsilon_1&&&&\\
&\epsilon_2&&&\\
&&\ddots&&\\
&&&\epsilon_{N-1}&},}
and $\epsilon_i = \pm 1$.
The latter represent the $O(1)^k$ transformations,
which of course do not effect $X^i$ at all.
To describe the possible twisted sectors of the matter fermions we
introduce the matrix $r^a_b = diag (1\ldots 1 ,-1 \ldots -1)$, which
acts on the $32$ valued index of the $\chi$ fields.
The sectors are then defined by
\eqn{chibc}{\chi^a (\sigma + 2\pi ) = r^a_b E_O^{-1} \chi^b
(\sigma )}
As usual, inequivalent sectors
correspond to conjugacy classes of the gauge group.
In this case, the classes can be described by a permutation
with a given set of cycle
lengths, corresponding to a collection of
strings with fixed longitudinal momentum fractions,
and the determinants of the $O(1)^k$ matrices inside each cycle.
In order to understand the various gauge bundles,
it is convenient to write the ``screwing
formulae'' which express the
components of the vectors $\chi^a$ in terms of string
fields $\chi_s^a$ defined on the interval $[0, 2\pi k]$.
The defining boundary conditions are
\eqn{bcchi}{\chi_i^a (\sigma + 2\pi ) =
\epsilon_i r^a_b \chi_{i + 1}^b (\sigma )}
where we choose the gauge in which $\epsilon_{i < k} =1$
and $\epsilon_k = \pm 1$ depending
on the sign of the determinant. The vector index $i$ is counted modulo $k$.
This condition is solved by
\eqn{soln}{\chi_i^a (\sigma ) = (r^{i - 1})^a_b \chi_S^b
(\sigma + 2\pi (i - 1))}
where $\chi_S$ satisfies
\eqn{hetbc}{\chi_S^a (\sigma + 2\pi k)
= (r^k)^a_b \epsilon_k \chi_S^b (\sigma )}
For $k$ even, this gives the PP and AA sectors
of the heterotic string, according to the
sign of the determinant.
Similarly, for $k$ odd, we obtain the AP and PA sectors.
As usual in a gauge theory,
we must project on gauge invariant states. It turns out that
there are only two independent kinds of
conditions which must be imposed. In a sector characterized by a
permutation $S$, one can be chosen to be
the overall multiplication of $\chi$ fields associated with a given
cycle of the permutation (a given string)
by $-1$.
This GSO operator anticommuting with all the 32 $\chi$ fields
is represented by the ${\bf -1}$ matrix from the gauge group $O(N)$.
The other is the projection associated with the cyclic
permutations themselves.
It is easy to verify that under the latter transformation
the $\chi_S$ fields transform as
\eqn{chitransf}{\chi_S^a (\sigma )
\rightarrow r^a_b \chi_S^b (\sigma + 2\pi)}
Here $\sigma \in [0,2\pi k]$ and we are
taking the limit $M\rightarrow \infty$,
$k/M$ fixed. In this limit the $2\pi$
shift in argument on the righthand side of
(\ref{chitransf}) is negligible,
and we obtain the second GSO projection of the heterotic string.
Thus, $1+1$ dimensional $O(M)$
SYM theory with $(0,8)$ SUSY, a left moving
supermultiplet in the symmetric
tensor representation and 32 right moving fermion
multiplets in the vector (half
with P and half with A boundary conditions)
reduces in the weak coupling, small (dual) circle limit to two copies of
the Ho\v rava-Witten domain wall
quantum mechanics, and in the strong coupling large (dual)
circle limit, to the string field theory of the $E_8 \times E_8$
heterotic string.
\section{Multidimensional Cylinders}
The new feature of heterotic compactification on $S^1 /Z_2 \times T^d$
is that the coordinates in the toroidal dimensions are represented by
covariant derivative operators with respect to new world volume
coordinates. We will reserve $\sigma$ for the periodic coordinate dual
to the interval $S^1 / Z_2$ and denote the other coordinates by
$\sigma^A$. Then,
\eqn{covder}{X^A = {2\pi R_A \over i}{\partial
\over \partial \sigma^A} - A_A
(\sigma ); \quad A = 2\ldots k+1.}
Derivative operators are antisymmetric, so in order to implement the
orbifold projection, we have to include the transformation $\sigma^A
\rightarrow - \sigma^A$, for $A = 2\ldots d + 1$,
in the definition of the orbifold symmetry.
Thus, the space on which SYM is compactified is $S^1 \times (T^d /
Z_2)$. There are $2^d$ {\it orbifold circles} in this space, which are
the fixed manifolds of the reflection.
Away from these singular loci, the gauge group
is $U(M)$ but it will be restricted to $O(M)$ at the singularities.
We will argue that there must be
a number of $1 + 1$ dimensional fermions living only on these circles.
When $d = 1$ these orbifold lines can be thought of as the boundaries of
a {\it dual cylinder}.
Note that if we take $d = 1$ and rescale the $\sigma^A$ coordinates
so that their lengths are $1/R_A$ then a long thin cylinder in spacetime
maps into a long thin cylinder on the world volume, and a short fat
cylinder maps into a short fat cylinder. As we will see, this
geometrical fact is responsible for the emergence of Type $IA$
and heterotic strings in the appropriate limits.
The boundary conditions on the world volume fields are
\eqn{bca}{X^i (\sigma ,\sigma^A ) = \bar X^i (\sigma , - \sigma^A ),\quad
A_a (\sigma ,\sigma^A ) = \bar A_a (\sigma , - \sigma^A ),}
\eqn{bcc}{A_1 (\sigma ,\sigma^A ) = -\bar A_1 (\sigma , - \sigma^A)}
\eqn{bcd}{\theta (\sigma ,\sigma^A ) = \bar\theta(\sigma ,-\sigma^A),\quad
\lambda (\sigma ,\sigma^A ) = -\bar\lambda (\sigma ,-\sigma^A )}
All matrices are hermitian,
so transposition is equivalent to complex conjugation.
The right hand side of the boundary condition \ref{bcc} can also be shifted
by $2\pi R_1$, reflecting the fact that $A_1$ is an angle variable.
Let us concentrate on the cylinder case, $d=1$.
In the limits $R_1 \ll l_{11} \ll R_2$ and
$R_2 \ll l_{11} \ll R_1$, we will find that
the low energy dynamics is completely
described in terms of the moduli space,
which consists of commuting $X^i$ fields.
In the first of these limits,
low energy fields have no $\sigma^2$ dependence, and
the boundary conditions restrict
the gauge group to be $O(M)$, and force
$X^i$ and $\theta$ to be real symmetric matrices.
Anomaly arguments
then inform us of the existence of $32$
fermions living on the boundary circles.
The model reduces to the $E_8 \times E_8$
heterotic matrix model described in the previous
section, which, in the indicated limit,
was shown to be the free string field theory of
heterotic strings.
\subsection{Type IA Strings}
The alternate limit produces something novel.
Now, low energy fields are restricted
to be functions only of $\sigma^2$.
Let us begin with a description of closed strings.
We will exhibit a solution of the boundary conditions for each
closed string field $X_S (\sigma )$ with periodicity $2\pi k$. Multiple
closed strings are constructed via the usual block diagonal procedure.
\eqn{uopen}{
X^i(\sigma^2)=U(\sigma^2)
D
U^{-1}(\sigma^2),}
\eqn{diagstring}{\begin{array}{c}D=\mbox{diag}(X_s^i(\sigma^2),\epsilon
X_s^i(2\pi-\sigma^2),X_s^i(2\pi+\sigma^2),
\epsilon X_s^i(4\pi-\sigma^2),\\
\dots,
X_s^i(2\pi(N-1)+\sigma^2),\epsilon X_s^i(2\pi N-\sigma^2)).
\end{array}}
where $\epsilon$ is $+1$ for $X^{2\dots 9}$ and $\theta$'s,
$-1$ for $A^1$ and $\lambda$'s.
{}From this form it is clear that the matrices will commute
with each other for any value of $\sigma^2$.
We must obey Neumann boundary conditions
for the real part of matrices and Dirichlet conditions
for the imaginary parts (or for $\epsilon=-1$ vice versa),
so we must use specific values of the unitary matrix
$U(\sigma^2)$ at the points $\sigma^2=0,\pi$.
Let us choose
\eq{U'(0+)=U'(\pi-)=0}
(for instance, put $U$ constant on a neighbourhood
of the points $\sigma^2=0,\pi$)
and for a closed string,
\eq{U(\pi)=\tb{{cccc}
m&&&\\
&m&&\\
&&\ddots &\\
&&&m},\qquad
U(0)=C\cdot
U(\pi)\cdot C^{-1},}
where $C$ is a cyclic permutation matrix
\eq{C=\tb{{ccccc}
&1&&&\\
&&1&&\\
&&&\ddots&\\
&&&&1\\
1&&&&}}
where $m$ are $2\times 2$ blocks (there are $N$ of them)
(while in the second matrix the $1$'s are $1\times 1$ matrices so that
we have a shift of the $U(\pi)$ along the diagonal
by half the size of the Pauli matrices.
The form of these blocks guarantees the conversion of $\tau_3$
to $\tau_2$:
\eq{m=\frac{\tau_2+\tau_3}{\sqrt 2}.}
This $2\times 2$ matrix causes two ends to be connected
on the boundary. It is easy to check that the right
boundary conditions will be obeyed.
To obtain open strings, we just change the
$U(0)$ and $U(\pi )$.
An open string of odd length is obtained by
throwing out the last element in (\ref{diagstring}) and taking
\eq{
U(0)=\tb{{ccccc}
1_{1\times 1}&&&&\\
&m&&&\\
&&m&&\\
&&&\dots &\\
&&&&m
},\quad\,
U(\pi)=\tb{{ccccc}
m&&&\\
&m&&\\
&&m&\\
&&&\dots&\\
&&&&1_{1\times 1}}}
Similarly, an open string of even length will have one of the matrices
$U(0),U(\pi)$ equal to what it was in the closed string case $m\otimes
1$ while
the other will be equal to
\eq{
U(0)={\small\tb{{cccccc}
1_{1\times 1}&&&&&\\
&m&&&&\\
&&m&&&\\
&&&\dots &&\\
&&&&m&\\
&&&&&1_{1\times 1}
}}}
Similar constructions for the fermionic coordinates are straightforward
to obtain. We also note that we have worked above with the original
boundary conditions and thus obtain only open strings whose ends are
attached to the wall at $R_1 = 0$. Shifting the boundary condition
\ref{bcc} by $2\pi R_1$ (either at $\sigma^2 =0$ or $\sigma^2 = \pi$ or
both) we obtain strings attached to the other wall, or with one end on
each wall. Finally, we note that we can perform the gauge
transformation $M \rightarrow \tau_3 M \tau_3$ on our construction.
This has the effect of reversing the orientation of the string fields,
$X_S (\sigma^2 ) \rightarrow X_S (- \sigma^2 )$. Thus we obtain
unoriented strings.
We will end this section with a brief comment about moving $D8$ branes
away from the orientifold wall. This is achieved by adding explicit
$SO(16) \times SO(16)$ Wilson lines to the Lagrangian of the $\chi^a$
fields. We are working in the regime $R_2 \ll l_{11} \ll R_1$, and we
take these to be constant gauge potentials of the form $\chi^a {\cal
A}_{ab} \chi^b$, with ${\cal A}$ of order $R_1$. In the presence of
such terms $\chi^a$ will not have any low frequency modes, unless we
also shift the $O(M)$ gauge potential $A_1$ to give a compensating shift
of the $\chi$ frequency. In this way we can construct open strings
whose ends lie on $D8$ branes which are not sitting on the orientifold.
In this construction, it is natural to imagine that $16$ of the $\chi$
fields live on each of the boundaries of the dual cylinder. Similarly,
for larger values of $d$ it is natural to put ${32 \over 2^d}$ fermions
on each orbifold circle, a prescription which clearly runs into problems
when $d > 4$. This is reminiscent of other orbifold constructions in M
theory in which the most symmetrical treatment of fixed points is not
possible (but here our orbifold is in the dual world volume). It is
clear that our understanding of the heterotic matrix model for general
$d$ is as yet quite incomplete. We hope to return to it in a future
paper.
\section{Conclusions}
We have described a class of matrix field theories which incorporate the
Fock spaces of the the $E_8 \times E_8$
heterotic/Type $IA$ string field theories
into a unified quantum theory. The underlying gauge dynamics provides a
prescription for string interactions. It is natural to ask what the
connection is between this nonperturbatively defined system and previous
descriptions of the nonperturbative dynamics of string theories with
twice as much supersymmetry. Can these be viewed as two classes of
vacua of a single theory? Can all of these be obtained as different
large $N$ limits of a quantum system with a finite number of degrees of
freedom?
The necessity of introducing the $\chi$ fields into our model suggests
that the original eleven dimensional system does not have all the
necessary ingredients to be the underlying theory. Yet we are used to
thinking of obtaining lower dimensional compactifications by restricting
the degrees of freedom of a higher dimensional theory in various ways.
Insight into this puzzle can be gained by considering the limit of
heterotic string theory which, according to string duality, is supposed
to reproduce M theory on $K3$. The latter theory surely reduces
to eleven dimensional M theory in a continuous manner as the radius of
$K3$ is taken to infinity.
Although we have not yet worked out the details of
heterotic matrix theory on higher dimensional tori, we think that it
is clear that the infinite $K3$
limit will be one in which the $\chi$ degrees
decouple from low energy dynamics.
The lesson we learn from this example is that {\it decompactification of
space time dimensions leads to a reduction in degrees of freedom}.
Indeed, this principle is clearly evident in the prescription for
compactification of M theory on tori in terms of SYM theory. The more
dimensions we compactify, the higher the dimension of the field theory we
need to describe the compactification. There has been some discussion
of whether this really corresponds to adding degrees of freedom since
the requisite fields arise as limits of finite matrices. However there
is a way of stating the principle which is independent of how one
chooses to view these constructions. Consider, for example, a graviton
state in M theory compactified on a circle. Choose a reference energy
$E$ and ask how the number of degrees of freedom with energy less than
$E$ which are necessary to describe this state, changes with the radius
of compactification. As the radius is increased, the radius of the dual
torus decreases. This decreases the number of states in the theory with
energy less than $E$, precisely the opposite of what occurs when we
increase the radius of compactification of a local field theory
\subsection{Cosmological Constant Problem}
It seems natural to speculate that this property, so counterintuitive
from the point of view of local field theory, has something to do with
the cosmological constant problem. In \cite{tbcosmo}
one of the authors
suggested that any theory which satisfied the 't Hooft-Susskind
holographic principle
would suffer a thinning out of degrees
of freedom as the universe expanded, and that this would lead to an
explanation of the cosmological constant problem. Although the
speculations there did not quite hit the mark, the present ideas suggest
a similar mechanism. Consider a hypothetical state of the matrix model
corresponding to a universe with some number of Planck size dimensions
and some other dimensions of a much larger size, $R$. Suppose also that
SUSY is broken at scale $B$, much less than
the (eleven dimensional) Planck scale.
The degrees of freedom
associated with the compactified dimensions all have energies
much higher than the
SUSY breaking scale. Their zero point fluctuations will lead
to a finite, small (relative to the Planck mass) $R$ independent,
contribution to the total vacuum
energy.
As $R$ increases, the number of degrees of freedom at scales less than
or equal to $B$ will decrease.
Thus, we expect a corresponding decrease in the
total vacuum energy.
The total vacuum energy in the large $R$ limit is thus bounded
by a constant, and is dominated by the
contribution of degrees of freedom associated with
the small, compactified dimensions.
Assuming only the minimal supersymmetric cancellation
in the computation of the vacuum energy,
we expect it to be of order $B^2 l_{11}$. This
implies a vacuum energy density of order
$B^2 l_{11} / R^3$, which is too small to be of
observational interest for any plausible values of the parameters.
If a calculation of this nature turns
out to be correct, it would constitute a prediction
that the cosmological constant is essentially zero in the matrix model.
It should not be necessary to emphasize how premature
it is to indulge in speculations
of this sort (but we couldn't resist the temptation).
We do not understand supersymmetry
breaking in the matrix model and we are even
further from understanding its cosmology.
Indeed, at the moment we do not even
have a matrix model derivation of the fact\footnote{Indeed
this ``fact'' is derived by rather indirect arguments in perturbative
string theory.} that
parameters like the radius of compactification are dynamical variables.
Perhaps the
most important lacuna in our understanding
is related to the nature of the large $N$
limit. We know that many states of the system
wander off to infinite energy as $N$
is increased.
Our discussion above was based on extrapolating results of the finite $N$
models, without carefully verifying that the
degrees of freedom involved survive
the limit. Another disturbing thing about our discussion is the absence
of a connection to Bekenstein's area law for the number of states. The
Bekenstein law
seems to be an integral part of the physical picture of the matrix model.
Despite these obvious problems, we feel
that it was worthwhile to present this preliminary discussion of
the cosmological constant problem
because it makes clear that the spacetime picture
which will eventually emerge from the matrix model is certain to be
very different from the one implicit in local field theory.
\section{Acknowledgements}
L.Motl is grateful to staff and students of Rutgers University
for their hospitality.
\newpag
| {'timestamp': '1997-03-31T11:52:45', 'yymm': '9703', 'arxiv_id': 'hep-th/9703218', 'language': 'en', 'url': 'https://arxiv.org/abs/hep-th/9703218'} |
\section{Introduction}
The concentration of both dark and baryonic matter in the cores of
clusters of galaxies has many profound implications for our
understanding of cluster growth and cosmology. Firstly, the structure
and evolution of the gravitational potential of a cluster of galaxies
depends on the nature of dark matter and thus allows direct comparison
with predictions from numerical simulations (e.g.\ Navarro, Frenk \&
White 1997). Secondly, the surface density of mass integrated through
the core of a cluster is often sufficiently high to strongly lens
background galaxies into gravitational arcs (Soucail et al.\ 1987;
Mellier et al.\ 1991; Kneib et al.\ 1996; Smith et
al.\ 2001). Detailed analysis of the location and shape of such arcs,
as well as of lens-generated multiple images, can be used to model the
projected mass in the cluster, and in-depth follow-up of the brightest
lensed features often yields valuable insights into the properties of
distant, faint galaxies (e.g.\ Kneib et al.\ 2004; Smail et
al.\ 2007). Thirdly, the high density and temperature of the gas in
the core of clusters leads to intense X-ray emission that current
instrumentation can detect out to redshifts well above unity. The
selection of massive clusters through X-ray emission has proved very
successful at providing cosmological constraints (Henry 2000; Borgani
et al.\ 2001; Allen et al.\ 2003; Pierpaoli et al.\ 2003; Allen et
al.\ 2008; Mantz et al.\ 2008) and follow-up observations of X-ray
luminous clusters have revealed many spectacular cases of
gravitational lensing (Gioia \& Luppino 1994; Smith et al.\ 2001;
Dahle et al.\ 2002; Covone et al.\ 2006).
In this paper we present a comprehensive multiwavelength study of a
complex gravitational arc and its host cluster MACS\,J1206.2$-$0847,
an X-ray selected system at intermediate redshift found by the Massive
Cluster Survey, MACS (Ebeling, Edge \& Henry 2001, 2007). We describe
our optical, NIR and X-ray observations in Sections 2 and 3,
investigate the properties of the giant arc, the cluster lens, and the
brightest cluster galaxy in Section 4 to 7, and derive mass estimates
for the cluster core and the entire system in Section 7. We present a
discussion of our results as well as conclusions in Section 8.
Throughout we use a $\Lambda$CDM cosmology ($\Omega_M=0.3$,
$\Omega_{\lambda}=0.7$) and adopt $H_0=$ 70 km s$^{-1}$ Mpc$^{-1}$.
\section{Observations}
The galaxy cluster MACS\,J1206.2$-$0847 was originally discovered in a
short two-minute R-band image taken on June 15, 1999 with the
University of Hawaii's 2.2m telescope (UH2.2m) on Mauna Kea. The
observation, performed as part of the MACS project, was triggered by
the presence of the X-ray source 1RXS\,J120613.0$-$084743 in the ROSAT
Bright Source Catalogue which had no obvious counterpart in the
standard astronomical databases and could also not trivially
be identified by inspection of the respective Digitized Sky Survey
image.
Following the initial, tentative identification of the RASS X-ray
source as a potentially massive galaxy cluster, we conducted a range
of follow-up observations to firmly establish the cluster nature of
this source and to characterize its physical properties.
\subsection{Optical}
\label{optspec}
Spectra of two galaxies in MACS\,J1206.2$-$0847, one of them the BCG,
were taken with the Wide-Field Prism Spectrograph on the UH2.2m on
July 4, 1999, using a 420 l/mm grism, a Tektronix 2048$^2$ CCD
yielding 0.355 arcsec/pixel, and a 1.6 arcsec slit. The two redshifts
were found to be concordant, establishing an approximate cluster
redshift of $z\approx 0.434$.
Moderately deep, multi-passband imaging observations ($3\times 240$s,
dithered by 10 arcsec, in each of the V, R, and I filters) of the
cluster were obtained with the UH2.2m on January 29, 2001, using again
the Tektronix 2048$^2$ CCD which provides a scale of 0.22 arcsec per
pixel and a $7.3\times7.3$ arcminute$^2$ field of view. The seeing was
variable throughout the night; we measure seeing values of 0.85, 1.05,
0.90 arcsec in the V, R, and I passbands, respectively, from the
final, co-added images.
Spectroscopic observations of presumed cluster galaxies as well as of
the giant arc in MACS\,J1206.2$-$0847 were performed with the FORS1
spectrograph in multi-object spectroscopy mode at the UT3 Melipal
telescope of the VLT on April 11, 2002. The G300V grism, an order
sorting filter (GG375), and a 1 arcsec slit were used, yielding a
wavelength coverage from $\sim$4000\AA\ to $\sim$8600\AA\ at a
resolution of $R=500$. The total exposure time was 38 minutes. A
single mask was designed, covering the $\sim$7' field of FORS1 with 19
slitlets of fixed length (22"). Credible redshifts could be measured
for 14 objects. During the exposure the seeing was 0.6 arcsec. Spectra
of the spectrophotometric standard star EG274 were obtained for
calibration.
Additional multi-object spectroscopy of colour-selected galaxies in
the field of MACS\,J1206.2$-$0847 was performed on May 8, 2003 using
the MOS spectrograph on the Canada-France-Hawaii Telescope (CFHT) on
Mauna Kea. We used the B300 grism and the EEV1 CCD, which provides a
resolution of 3.3\AA/pixel, and observed through a broadband filter
(\#4611, General Purpose set) to produce truncated spectra, covering
about 1800\AA\ centred on 6150\AA, such that spectra could be stacked
in three tiers along the dispersion direction. This choice of filter
and grism ensured that, for all cluster members, redshifts could be
obtained from the Ca H+K lines which fall at 5660\AA\ and 5710\AA\ at
the approximate cluster redshift of $z=0.44$. Weather conditions were
poor though, and only 48 of the 67 objects observed (total integration
time: one hour) yielded reliable redshifts.
Finally, MACS\,J1206.2$-$0847 was observed on December 6, 2005 with
the Advanced Camera for Surveys (ACS) aboard the Hubble Space
Telescope as part of program SNAP-10491 (PI Ebeling), for a total of
1200 seconds in the F606W filter, resulting in a high-resolution image
of the cluster core, including the giant arc.
\subsection{Near-infrared}
Near-infrared observations of the core of MACS\,J1206.2$-$0847 were
performed in the J and K bands using the United Kingdom Infra-Red
telescope (UKIRT) on April, 5 2001 using the UFTI imager during a
period of good seeing. The observations consisted of two iterations of
a nine-point dither pattern, each of 60 second exposures for a total
integration time of 1080 seconds. The seeing measured from these
observations was 0.47 and 0.59 arc-seconds in the J and K bands,
respectively.
\subsection{X-ray}
MACS\,J1206.2$-$0847 was observed on December 18, 2002 with the ACIS-I
detector aboard the Chandra X-ray Observatory for a nominal duration
of 23.5 ks, as part of a Chandra Large Programme awarded to the MACS
team. The target was placed about two arcmin off the standard aimpoint
to avoid flux being lost in the chip gaps of the ACIS-I detector, while
still maintaining good (sub-arcsec) angular resolution across the
cluster core. VFAINT mode was used in order to maximize the efficiency
of particle event rejection in the post-observation processing.
\section{Data reduction}
\subsection{Optical imaging}
Standard data-reduction techniques (bias-subtracting, flat-fielding,
image combination and registration) were applied to the V, R and I
band data using the relevant \textsc{iraf} packages. The data were
photometrically calibrated via the observation of Landolt
standard-star fields.
In order to measure aperture magnitudes, seeing-matched frames were
produced by applying a Gaussian smoothing to the V, I, J and K images
such that the seeing in these frames was degraded to match the 1.1
arc-seconds measured in the R band. Using SExtractor, the
seeing-matched (undegraded) images were then used to measure aperture
(total) magnitudes.
Photometry of the giant arc was performed by manually defining an
aperture mask fitted to the arc profile. The area defined by this
aperture was then masked out of the science frame and a background
image produced by median smoothing over the absent arc. This
background image was subtracted from the science frame and VRIJK
photometry obtained by applying the aperture to the resulting
sky-subtracted images.
\subsection{Optical spectroscopy}
For the reduction of our spectroscopic data we applied the same
standard techniques as for the imaging data, followed by straightening
of the skylines, extraction of spectra, and wavelength calibration
using the relevant \textsc{iraf} packages.
Preliminary redshifts were determined via visual inspection, typically
from the calcium H and K lines. Final refined redshifts were found
with a multi-template cross-correlation method using the \textsc{iraf}
task \textsc{fxcor}.
Redshifts with cross-correlation peak heights exceeding 0.70 are
typically accurate to 0.0002; for peak heights between 0.5 and 0.7 the
error is typically 0.0005. All redshifts were converted to the
heliocentric reference frame. Table~\ref{ztbl} lists the coordinates
of all galaxies observed, as well as the measured redshifts with their
uncertainties.
\subsection{Near-infrared imaging}
The J and K imaging data were reduced using the ORAC-DR data-reduction
pipeline and were calibrated through observation of observatory
standard stars.
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.vri.eps}
\caption{\label{imvri} Colour image (IVR mapped to RGB) of
MACSJ1206.2$-$0847 from $3\times 240$s observations (per filter)
obtained with the UH 2.2m telescope (see text for observational
details).}
\end{figure}
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{spec_gal_pos.eps}
\caption{\label{imgalpos}Locations of all galaxies for which
spectroscopic redshifts were measured with the VLT (circles) and
CFHT (diamonds), overlaid on the UH2.2m R-band image. Bold symbols
mark galaxies found to be cluster members. See Table~\ref{ztbl} for
coordinates and redshifts.}
\end{figure}
\subsection{X-ray}
We use {\sc Ciao} (version 3.3), the standard suite of software tools
developed for the analysis of Chandra data at the Chandra Science
Center, as well as the most recent calibration information, to
reprocess the raw ACIS-I data. Our inspection of the lightcurve of the
event count rate in the source-free regions of the ACIS-I detector
finds no significant flaring, leading to an effective (dead-time
corrected) total exposure time of 23.2 ks.
To investigate potential spatial variations in the cluster gas
temperature we define various source and background regions. For each
of these regions we generate auxiliary response files (ARF) and
response matrix files (RMF) which weigh the position-dependent
instrument characteristics by the observed count distribution in the
respective area. Following Markevitch \& Vikhlinin (2001), only the
0.5--2.0 keV band data are used to create these maps of spatial
weights, since the effects of vignetting are small in this energy
range. We also apply a correction of a factor of 0.93 to the effective
area distribution at energies below 2 keV in all ARFs as suggested by
a comparison of calibration results for the front- and
back-illuminated ACIS chips (Vikhlinin et al.\ 2002). Finally, we use
the {\sc acisabs} package to correct all ARFs for the effects of the
(time-dependent) buildup of a contaminating deposit on the optical
detector window which results in a reduction of the effective area at
low energies.
Background regions are defined by copying the respective source
regions to the same chip-y location on the other three ACIS-I CCDs.
This strategy minimizes the impact of any residual chip-y dependence
of the background on the data analysis, an effect that is unavoidable
if the background is selected as a source-free region on the same CCD
as the cluster.
\section{Cluster galaxy distribution}
\label{galprop}
Figure~\ref{imvri} shows a colour image of the cluster generated from
the V, R, I images obtained with the UH2.2m. MACS\,J1206.2$-$0847 is
found to be an optically very rich system with a single dominant
central galaxy and no obvious subclustering in the apparent, projected
cluster galaxy distribution. A giant arc is clearly visible about 20
arcsec to the West of the BCG. The astrometric solution used is based
on eight stars within the field of view of the V band image that have
accurate celestial coordinates in the Hubble Guide Star Catalogue
(GSC2).
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{z_hist.eps}
\caption{\label{zhist}Histogram of galaxy redshifts in the field of
MACS\,J1206.2$-$0847 as observed with multi-object spectrographs on
CFHT and the VLT (cf.\ Table~\ref{ztbl}). The overlaid Gaussian
curve is characterized by the best-fit values for the systemic
redshift and cluster velocity dispersion of $z=0.4385$ and
$\sigma=1575^{+191}_{-190}$ km/s, respectively. }
\end{figure}
Our spectroscopic observations of 85 galaxies in the field of
MACS\,J1206.2$-$0847 (Fig.~\ref{imgalpos}) yield redshifts as listed
in Table~\ref{ztbl}. Two galaxies were observed with both the VLT and
CFHT; their spectroscopic redshifts agree within the errors. Using
only the most accurate redshifts with correlation peak heights
exceeding 0.7, we apply iterative $3\sigma$ clipping to the redshift
histogram to obtain a systemic cluster redshift of $z=0.4384$ and a
very high velocity dispersion in the cluster rest frame of
$\sigma=1581$ km/s based on 38 redshifts. Entirely consistent values
of $z=0.4385$ and $\sigma=1575^{+191}_{-190}$ km/s are found using the
ROSTAT statistics package (Beers, Flynn \& Gebhardt 1990). The
resulting redshift histogram is shown in Fig.~\ref{zhist}. Despite the
extremely high velocity dispersion of the system we find no obvious
signs of substructure along the line of sight; a one-sided
Kolmogorov-Smirnov test finds the observed redshift distribution to be
only mildly inconsistent with a Gaussian ($2.05\sigma$ significance).
\section{Arc properties}
\begin{figure*}[h]
\vspace*{-0.1cm}
\parbox{0.79\textwidth}{
\epsfxsize=0.2\textwidth
\hspace*{9cm} \epsffile{cimage.hst.noastro.eps}\\
\epsfxsize=0.785\textwidth
\hspace*{5mm}\epsffile{MACSJ1206.2-0847.vik.eps}}
\parbox[c]{0.18\textwidth}{
\vspace*{1.8cm}
\epsfxsize=0.2\textwidth
\epsffile{arc.hst.noastro.eps}}\mbox{}\\*[-0.1cm]
\caption{\label{imvik}Colour image (KIV mapped to RGB) of MACSJ1206.2$-$0847
from $3\times 240$s observations (I,V) and $18\times 60$s observations (K)
obtained with the UH2.2m and UKIRT, respectively (see text for observational
details). An arrow points to the counter image of two of the four background
galaxies distorted by the cluster's gravitational field to create the
prominent giant arc 20 arcsec west of the BCG. Smaller panels on top and to
the right show enlarged high-resolution views of the arc and its counter image
as observed with HST/ACS in a 1200s snapshot with the F606W filter. }
\end{figure*}
The bright ($V=21.0$) giant arc is centered at $\alpha$ (J2000) =
12$^{\rm h}$ 06$^{\rm m}$ 10.75$^{\rm s}$, $\delta$ (J2000) =
$-$08$^{\circ}$ 48' 04.5'', about 20\arcsec west of the BCG. Its
unusually red colour is apparent in Fig.~\ref{imvik} which shows a
composite V, I, K image of the cluster core. Also prominent, and
marked by the arrow in Fig.~\ref{imvik}, is the counter image, clearly
identifiable by its colour, at $\alpha$ (J2000) = 12$^{\rm h}$
06$^{\rm m}$ 11.27$^{\rm s}$, $\delta$ (J2000) = $-$08$^{\circ}$ 47'
43.0''. Enlarged high-resolution views of arc and counter image
provided by HST are shown in the margins of Fig.~\ref{imvik}. The
photometric properties of arc and counter image are summarized in
Table~\ref{arctbl}; note however that at the resolution of our
ground-based images both the arc and its counter image is blend of
several objects. Figure~\ref{arcspec} shows the spectrum of the giant
arc in MACS\,J1206.2$-$0847 as observed with FORS1 on the VLT.
\begin{table*}
\centering
\caption{Extinction-corrected magnitudes and colours of the
giant arc in MACSJ1206.2$-$0847 and its counter image
(cf.\ Fig.~\ref{imvik}). \label{arctbl}}
\begin{tabular}{@{}lccccccccccc@{}}
\hline
object & I & R & V & J & K & R$-$I & V$-$I & V$-$R & R$-$J & R$-$K & J$-$K\\
\hline
giant arc & $19.27\pm 0.07$ & $20.27\pm 0.11$ & $20.99\pm 0.08$ & $17.82\pm 0.06$ & $15.95\pm 0.06$ & $1.00\pm 0.12$ & $1.72\pm 0.10$ & $0.72\pm 0.13$ & $2.45\pm 0.12$ & $4.32\pm 0.12$ & $1.87\pm 0.08$ \\
counter image & $20.41\pm 0.06$ & $21.34\pm 0.07$ & $21.94\pm 0.07$ & $18.88\pm 0.06$ & $17.14\pm 0.06$ & $0.93\pm 0.09$ & $1.52\pm 0.09$ & $0.59\pm 0.10$ & $2.46\pm 0.09$ & $4.20\pm 0.09$ & $1.74\pm 0.09$ \\
\hline
\end{tabular}
\end{table*}
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{arc_spectrum.eps}
\caption{\label{arcspec}Spectrum of the giant gravitational arc in
MACS\,J1206.2$-$0847 as observed with FORS1 on the VLT. We interpret
the single emission line observed at 7589\AA\ as being [O II], which
translates into a redshift of 1.036 for the lensed galaxy. Our
measurement is confirmed by independent observations conducted
almost simultaneously by Sand et al.\ (2003) using the ESI echelle
spectrograph on Keck-II which resolves the [O II] doublet thereby
making the identification unambiguous.}
\end{figure}
At $R-K=4.3$ the giant arc in MACS\,J1206.2$-$0847 is among the
reddest strongly lensed features currently known. Its colour is
comparable to that of the giant arc in Abell 370 ($R-K=4.1$,
Arag\'{o}n-Salamanca \& Ellis 1990) and only slightly bluer than the
red arc in Abell 2390 ($R-K=4.6$, Smail et al.\ 1993).
Our multi-band photometry allows the classification of the background
galaxy lensed into the giant arc. Standard template spectral energy
distributions (SEDs) of five galaxy types were redshifted to match
that of the giant arc. The filter response curves of each of the V, R,
I, J and K bands were then convolved with these SEDs and normalized to
the R band to produce predicted colours for each of the five galaxy
types, all relative to the R band. The resulting model
m$_{\lambda}$--m$_{R}$ colours, as well as the equivalent observed
colours of the arc, are shown in Fig.~\ref{arcsed}. We find the colour
distribution of the arc in MACS\,J1206.2$-$0847 to be consistent with
the background galaxy being a normal spiral of class Sbc.
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{arc_sed.eps}
\caption{\label{arcsed}Broad-band colours of the giant gravitational
arc in MACS\,J1206.2$-$0847, relative to the R band, compared to
predicted colours for standard galaxy types at the same
redshift. The best agreement is found for a spiral of type Sbc to
Scd.}
\end{figure}
\section{Intra-cluster gas properties}
\label{gas}
Figure~\ref{xcont} shows contours of the adaptively smoothed X-ray
emission from MACS\,J1206.2$-$0847, as observed with Chandra/ACIS-I in
the 0.5--7 keV band, overlaid on the UH2.2m R-band image\footnote{
For this overlay, as well as for any other comparisons of the spatial
appearance of the cluster in the X-ray and optical wavebands, we have
used three X-ray point sources with obvious optical counterparts to
slightly adjust the astrometric solution of the X-ray image, namely by
$-$0.03 seconds in right ascension and $-$0.4 arcseconds in
declination. We estimate the resulting, relative astrometry between
the optical and X-ray images to be accurate to better than 0.2
arcseconds.}.
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.r-x.eps}
\caption{\label{xcont}Iso-intensity contours of the adaptively
smoothed X-ray emission from MACS\,J1206.2$-$0847 in the 0.5--7 keV
range as observed with Chandra's ACIS-I detector, overlaid on the
UH2.2m R-band image. The algorithm used, {\textit Asmooth} (Ebeling,
White \& Rangarajan 2005), adjusts the size of the smoothing kernel
across the image such that the signal under the kernel has constant
significance (here 3$\sigma$) with respect to the local background.
The lowest contour is placed at twice the value of the X-ray
background in this observation; all contours are logarithmically
spaced by factors of 1.5.}
\end{figure}
At large distances from the center, the system's X-ray appearance is
close to spherical in projection. The central region at $r\la 250$
kpc, however, shows a pronounced ellipticity as well as non-concentric
X-ray flux contours in the cluster core. The observed elongation, as
well as the displacement of the innermost contours toward the very
compact X-ray core, are both in the direction of a group of galaxies
in the vicinity of the giant arc.
\subsection{Spatial analysis}
\label{xbeta}
Using {\sc Sherpa}, the fitting package provided with {\sc Ciao}, we
fit the observed X-ray surface brightness distribution within 2.5
arcmin of the cluster core with a two-dimensional spatial model,
consisting of an elliptical $\beta$ model
\[
S(r) = S_0 \left[ 1+\left(\frac{r}{r_0}\right)^2\right]^{-3\beta + \frac{1}{2}},
\]
(where $r=r(\phi)$ is the variable radius of an ellipse with
ellipticity $\epsilon$ and orientation angle $\Theta$), an additional
circular Gaussian component to account for the compact core, and a
constant background.
Point sources with detection significances exceeding 99\% (as measured with the
{\sc celldetect} algorithm) have been excised from the image, and a spectrally
weighted exposure map is used as a two-dimensional response function in the
fit. We use a composite exposure map to account for the differences between
photons of cosmic origin and high-energy particles, the latter being subject
neither to off-axis vignetting nor to variable detection rates due to spatial or
temporal variations in the CCD quantum efficiency (QE). At energies between 0.5
and 2 keV particles account for roughly 2/3 of the observed background; at
higher energies this fraction rises to over 90 per cent. Since, overall, more
than 80\% of the background events registered in the 0.5--2 keV band used here
are caused by particles we ignore the sky contribution altogether and compute a
background exposure map which incorporates the effects of bad pixels and
dithering but not those of a variable CCD QE and vignetting. A second exposure
map is computed using weights based on the spectrum of the target of our
observations. The spectral weights for this `cluster-weighted' exposure map are
created assuming a plasma model with $kT=12$ keV, a metal abundance of 0.3
solar, and the Galactic value of $4.23\times 10^{20}$ cm$^{-2}$ for the
equivalent Hydrogen column density value in the direction of the cluster,
consistent with the results of our spectral fits to the global cluster X-ray
spectrum (see below). The resulting exposure map shows significant vignetting of
more than 20\% across the ACIS-I field of view. The final exposure map used in
the following is then the weighted average of the background and cluster
exposure maps, with the weights being given by the fraction of counts from the
cluster and the background, and with the peak value set to that of the
cluster-weighted map.
Since the image used in the fit is relative coarsely binned ($2\times
2$ arcsec$^2$) and no small-scale spatial components are included in
the model, we do not convolve the model with the telescope
point-spread function (PSF). All parameters of the two-dimensional
spatial model are fit. Because of the low number of counts (zero or
one) in the large majority of image pixels, we use the C statistic
(Cash 1979) during the optimization process.
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.2dfit-im.eps}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.2dfit-res.eps}
\caption{\label{2dxfit}Iso-intensity contours (logarithmically spaced)
of the best-fitting analytic model (elliptical $\beta$ model plus
circular Gaussian component) of the cluster emission in the 0.5--7
keV range, overlaid on the observed image (top, logarithmic scaling)
and the residual image (bottom, linear scaling). The dashed ellipse
in the bottom panel highlights the excess emission in the direction
of the arc and marks the region excluded in the spatial fit of the
data to the two-dimensional surface-brightness model.}
\end{figure}
Our spatial fit yields best-fit values for the parameters of the $\beta$ model
of $r_0=(23.6\pm 0.8)$ arcsec (corresponding to ($134\pm 5)$ kpc) for the core
radius, $\beta=0.57\pm 0.01$ for the slope parameter, $\epsilon=0.17\pm 0.01$
for the ellipticity, and $\Theta=(56\pm 2)$ degrees (counted North through
West). Our two-dimensional spatial fit also finds coordinates of $\alpha$
(J2000) = 12$^{\rm h}$ 06$^{\rm m}$ 12.10$^{\rm s}$, $\delta$ (J2000) =
$-$08$^{\circ}$ 48' 01.7'' for the centroid of the compact cluster core, offset
by $(8.1\pm 1.7)$ arcsec from the position $\alpha$ (J2000) = 12$^{\rm h}$
06$^{\rm m}$ 12.50$^{\rm s}$, $\delta$ (J2000) = $-$08$^{\circ}$ 48' 07.1''
which marks the center of the elliptical component that describes the shape of
the X-ray emission on larger scales.
Figure~\ref{2dxfit} shows the contours of the best-fitting model
overlaid both on the observed exposure-corrected X-ray image and on
the residuals remaining when the model is subtracted from the
data. The residual image shows a clear excess of emission,
corresponding to about 300 photons, to the West of the cluster
core. Note that the best-fit model parameters quoted above were
obtained with the excess region excluded from the fit.
Since the two-dimensional fitting procedure allows no immediate
assessment of the goodness of fit other than via visual inspection of
the residuals (Fig.~\ref{2dxfit}), we also fit a one-dimensional,
spherical $\beta$ model to the radial X-ray surface brightness
profile. In this fit we adopt the center of the elliptical model
component as the overall center of the X-ray emission, and exclude a
60-degree-wide azimuthal section around the compact core and the
excess emission to the West-North-West. Again we account for
variations in the exposure time across the source and background
regions. Since all annuli contain at least 50 photons we are now
justified in using $\chi^2$ statistics in the fit.
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{radprof.eps}
\caption{\label{1dxfit}Radial X-ray surface brightness profile of
MACS\,J1206.2$-$0847 in the 0.5--7 keV band as observed with
Chandra/ACIS-I. The profile is centered on the peak of the
elliptical surface brightness component as determined in our
two-dimensional fit (see text for details). The solid line shows the
best-fitting $\beta$ model. The dashed horizontal line marks the
surface brightness corresponding to twice the background level; the
vertical dotted line shows the radial limit within which a $\beta$
model provides an acceptable description of the data.}
\end{figure}
The resulting one-dimensional radial profile as well as the
best-fitting $\beta$ model are shown in Figure~\ref{1dxfit}. When
fitting out to a radius of about 150 arcsec (850 kpc), where the
observed surface brightness profile begins to drop below twice the
background level, we find the $\beta$ model to provide an unacceptable
fit to the data at a reduced $\chi^2$ value of 1.9 (48 data points, 45
degrees of freedom (d.o.f.). However, essentially the same model fits
the data very well ($\chi^2$=1.1 for 19 d.o.f.) within a radius of 66
arcsec (375 kpc). At larger radii the observed slope varies such that
the best-fitting $\beta$ model first systematically underpredicts, then
systematically exceeds the observed values. The best-fit values
of $S_0=(3.23\pm 0.13)$ ct arcsec$^2$, $r_0=(20.1\pm 1.3)$ arcsec
(corresponding to ($114\pm 7$) kpc), and $\beta=0.57\pm 0.02$, all of
which are consistent with the results from our two-dimensional fit,
thus allow a credible parametrization of the observed emission out to
about 375 kpc, provided the compact core and the excess emission
region to the West are excluded. At larger radii, the X-ray morphology
of MACS\,J1206.2$-$0847 is again too complex to be described by a
simple $\beta$ model.
Although the one-dimensional model is too simplistic to adapt to the spatial
variations in the X-ray emission near the cluster core or at very large radii,
it provides an adequate global description of the cluster. Extrapolating the
model to $r_{\rm 200}$ (see Section~\ref{xmass}) yields values of $(4.4\pm 0.07)
\times 10^{-12}$ erg s$^{-1}$ cm$^{-2}$ and $(24.3\pm 0.5)\times 10^{44}$ erg
s$^{-1}$ for the total X-ray flux and X-ray luminosity of MACS\,J1206.2--0847,
respectively (0.1--2.4 keV). The quoted errors do, however, not account for any
systematic errors which are bound to be present in view of the fact that X-ray
emission is detected only out to about 1 Mpc from the cluster center and that,
at larger radii, the $\beta$ model tends to overpredict the observed X-ray
surface brightness (see Fig.~\ref{1dxfit}). A more robust measurement can be
obtained within $r_{\rm 1000}$ (1.04 Mpc) and yields lower limits to the total
X-ray flux and luminosity of $(3.8\pm 0.06) \times 10^{-12}$ erg s$^{-1}$
cm$^{-2}$ and $(21.0\pm 0.4)\times 10^{44}$ erg s$^{-1}$.
\subsection{Spectral analysis}
\label{xspec}
We measure a global temperature for the intra-cluster medium (ICM) in
MACS\,J1206.2$-$0847 by extracting the X-ray spectrum from $r=70$ kpc
to $r=1$ Mpc (r$_{1000}$) and using \textit{Sherpa} to fit a MEKAL
plasma model (Mewe et al.\ 1985) with the absorption term frozen at
the Galactic value. We find ${\rm k}T= (11.6\pm 0.66)$ keV, a high
value even for extremely X-ray luminous clusters (Chen et
al.\ 2007). Although the relatively high reduced-$\chi^2$ value of
this spectral fit of 1.4 is statistically acceptable, it could be
indicative of systematic effects such as spatial temperature
variations or the presence of multiphase gas. We find only mild
evidence of the former when fitting absorbed plasma models to the
X-ray spectra extracted from five concentric annuli. As shown in
Fig.~\ref{tprof}, the ICM temperature is consistently high ($\sim 12$
keV), with the exception of the core region where, at $r<130$ kpc, a
significant drop to about 9 keV (still a very high value) is
observed. The lower gas temperature measured around the cluster core
could be caused by the presence of a minor cool core, in agreement
with the results of our X-ray imaging analysis.
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{tprof.eps}
\caption{\label{tprof}Radial profile of the ICM temperature in
MACS\,J1206.2$-$0847. The profile is centered on the peak of the
elliptical surface brightness component as determined in our
two-dimensional fit (see text for details). Horizontal bars mark the
width of the respective annulus, defined such that each region
contains about 3,000 net photons.}
\end{figure}
\section{Properties of the central galaxy}
The spectrum of the central cluster galaxy taken at CFHT
(Fig.~\ref{cdspec}, see also Section~\ref{optspec}) covers only a
small wavelength range mostly redward of the 4000\AA\ break. While
the wavelength coverage is thus insufficient to check for the presence
of H$\beta$ and OIII in emission, we do detect faint OII emission
($\lambda_{\rm rest}=3727$ \AA), albeit at a much lower level than
typically observed in the central cluster galaxies of large cool-core
clusters (e.g., Allen et al.\ 1992, Crawford et al.\ 1995).
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{cd_spectrum.eps}
\caption{\label{cdspec}Low-resolution spectrum of the central cluster
galaxy in MACSJ1206.2$-$0847 as observed with CFHT (see text for
details). A tentative detection of OII in emission is highlighted, as
are a series of absorption features characteristic of the old stellar
population dominating cluster ellipticals.}
\end{figure}
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.r-x.core.hst.eps}
\caption{\label{corealign}As Figure~\ref{xcont} but using the HST
image and zoomed in to show only the cluster core.}
\end{figure}
Bright NVSS (162 mJy at 1.4~GHz) and Molonglo (900 mJy at 365~MHz)
radio sources are coincident with the central galaxy. The radio source
exhibits a relatively steep spectrum ($\alpha=-1.32\pm0.05$) and is
similar to the one found in MACSJ1621.3+3810 (Edge et al.\ 2003) but
an order of magnitude more powerful (6$\times 10^{26}$ W~Hz$^{-1}$ at
1.4~GHz). The radio source is also detected at 74 MHz in the VLA
Low-frequency Sky Survey (VLSS) with a measured flux of
$(6.67\pm0.7)$ Jy that is in excellent agreement with the power-law
prediction from the detections at 365 MHz and 1.4 GHz.
Although no clear signs of cavities are detected in the cluster core, we can,
from our Chandra data alone, not conclusively assess the degree of interaction
between the BCG and the surrounding intra-cluster medium. High-resolution radio
data would be required to map the radio morphology of the BCG. We do observe
though a slight, but significant displacement of $(1.7\pm 0.4)$ arcsec --
$(9.6\pm 2.3)$ kpc at the cluster redshift -- between the position of the
central cluster galaxy ($\alpha$ (J2000) = 12$^{\rm h}$ 06$^{\rm m}$ 12.14$^{\rm
s}$, $\delta$ (J2000) = $-$08$^{\circ}$ 48' 03.3'') and the X-ray centroid of
the compact cluster core (Fig.~\ref{corealign}). Having carefully aligned the
optical and X-ray images (see footnote in Section~\ref{gas}) we estimate that at
most 10\% of this offset can reasonably be attributed to residual astrometric
uncertainties; the majority of the misaligned is thus real. Similar offsets
($\sim$ 10 kpc) have been noted in previous Chandra studies of galaxy clusters
(Arabadjis, Bautz \& Garmire 2002) and may also have contributed to minor
optical/X-ray misalignments in the cores of cooling clusters observed with ROSAT
(Peres et al.\ 1998).
In addition, the velocity offset of 550 km s$^{-1}$ between the
central galaxy and the cluster mean velocity is comparable to the
largest peculiar velocities observed in local clusters (Zabludoff et
al.\ 1990; Hill \& Oegerle 1993; Oegerle \& Hill 2001), although we
note that the non-Gaussian velocity distribution in this cluster
(Fig.~\ref{zhist}) complicates the measurement of an accurate systemic
velocity.
Spatial and velocity offsets between central galaxy and X-ray peak are
not expected in the simplest cool-core scenario, although it is worth
noting that an offset of the same size as observed by us here (10 kpc)
has been seen in the best-studied cool-core cluster, Perseus
(B\"ohringer et al.\ 1993), and there has been a widespread
realization that the physics of cool cluster cores are more complex
and the role of AGN feedback more important than previously thought
(Peterson et al.\ 2001; Soker et al.\ 2002; Edge 2001; Mittal et
al.\ 2008). We conclude that MACSJ1206.2$-$0847 is likely to contain a
moderate cool core, as well as an extremely luminous radio galaxy (one
of the most powerful ones known in cluster cores at $z>0.4$), with the
observed disturbances being likely due to a recent or still ongoing
cluster merger.
\section{Mass measurements}
\subsection{Virial mass}
Using the measured redshifts of cluster members and their spatial distribution
as projected on the sky, we determined the virial cluster mass based on the
method of Limber \& Mathews (1960) in which the mass is calculated as
\begin{equation}
M_V = \frac{3\pi}{2} \frac{\sigma_{P}^{2}R_H}{G}.
\label{M_Veqn}
\end{equation}
Here $\sigma_P$ is the one-dimensional (radial) velocity dispersion and $R_H$ is
the projected mean harmonic point-wise separation (projected virial
radius). $R_H$ is defined by
\begin{equation}
R_H^{-1} = \frac{1}{N^2}\sum_{i<j} \frac{1}{|\mathbf{r}_i-\mathbf{r}_j|},
\label{R_Heqn}
\end{equation}
where N is the number of galaxies, $|\mathbf{r}_i-\mathbf{r}_j|$ is the
projected separation of galaxies $i$ and $j$, and the $ij$ sum is over all
pairs. Being a pairwise estimator this quantity is sensitive to close pairs and
quite noisy (Bahcall \& Tremaine 1981). It also systematically underestimates
the radius for a rectangular aperture typical of cluster redshift surveys
(Carlberg et al.\ 1996). Carlberg et al.\ therefore introduce a new radius
estimator, the ringwise projected harmonic mean radius $R_h$, given by
\begin{eqnarray}
R_h^{-1} &=& \frac{1}{N^2} \sum_{i<j}\frac{1}{2\pi} \int_{0}^{2\pi}\frac{d\theta}{\sqrt{r_{i}^{2}+r_{j}^{2}+2r_ir_j\cos\theta}} \nonumber \\
&=& \frac{1}{N^2} \sum_{i<j}\frac{2}{\pi (r_i+r_j) K(k_{ij})}.
\label{R_heqn}
\end{eqnarray}
Here $r_i$ and $r_j$ are the projected distances from the cluster center to
galaxies $i$ and $j$ respectively, $k_{ij}^{2} = 4r_ir_j/(r_i+r_j)^2$, and
$K(k)$ is the complete elliptic integral of the first kind in Legendre's
notation. This estimator requires an explicit choice of cluster center and
assumes the cluster is spherically symmetric with respect to this center. It
treats one of the particles in the pairwise potential $|r_i-r_j|^{-1}$ as having
its mass distributed in a ring around the cluster center. $R_h$ is less
sensitive to close pairs, less noisy, and tolerates non-circular apertures
better than $R_H$. If the cluster is significantly flattened or subclustered,
however, $R_h$ will systematically overestimate the true projected virial
radius.
We calculate both radius estimators in our mass determinations to investigate
the resulting systematic uncertainty in the virial mass. For our sample the
virial radius and mass derived using $R_h$ are 7\% larger than those based on
$R_H$. We choose to use $R_h$ as the more robust estimator in our analysis and
define the three-dimensional (deprojected) virial radius as
\begin{equation}
r_V = \frac{\pi}{2}R_h.
\label{r_Veqn}
\end{equation}
Our determination of the virial radius estimators $R_H$ and $R_h$ was made using
all 62 galaxies with redshifts within $3\sigma$ of the cluster mean (see
Section~\ref{galprop}). The resulting projected virial radius is $R_h = 1.176$
Mpc ($R_H = 1.096$ Mpc) with a virial mass of $3.861\times10^{15}$ M$_{\odot}$
and a three-dimensional virial radius of $r_V = 1.847$ Mpc.
\subsection{X-ray mass}
\label{xmass}
To estimate the total gravitational mass of the cluster from its X-ray emission,
we need to assume that the cluster is in hydrostatic equilibrium. In addition,
we need a description of the density as well as the temperature of the
intracluster gas, often assumed to be isothermal. For a cluster with a core
region as disturbed as the one of MACSJ1206.2$-$0847 such simplifying
assumptions are unlikely to be justified; however, an isothermal $\beta$ model
should provide an adequate description of the cluster outskirts and allow us to
obtain a crude mass estimate.
From the X-ray temperature (see Section~\ref{xspec}) we estimate the virial
radius $R_{200}$ using the formula of Arnaud et al.\ (2002),
\begin{eqnarray*}
R_{200} & = & 3.80~\beta_T^{1/2}\Delta_z^{-1/2}~(1+z)^{-3/2} \nonumber \\
& & \times(\frac{kT}{10{\rm keV}})^{1/2}h_{50}^{-1}~{\rm Mpc} \nonumber
\end{eqnarray*}
with
\begin{eqnarray*}
\Delta_z = (200\Omega_0)/(18\pi^2\Omega_z). \nonumber
\end{eqnarray*}
Here $\beta_T=1.05$ (Evrard et al.\ 1996) is the normalization of the virial relation, i.e.,
$GM_v/(2R_{200})= \beta_T~{\rm k}T$. Then, the total mass of the
cluster within a radius $r$ can be computed with the help of the $\beta$-model profile
discussed in Section~\ref{xbeta}:
\begin{eqnarray}
M(r) & = & 1.13\times10^{14}\beta\frac{T}{\rm{keV}}\frac{r}{\rm{Mpc}}\frac{(r/r_c)^2}{1+
(r/r_c)^2}M_{\odot}
\end{eqnarray}
(Evrard et al.\ 1996). Using the above equations we find $R_{\rm 200}=(2.3\pm
0.1)$ Mpc as an approximate value for the virial radius. The X-ray estimates for
the total mass within $R_{\rm 200}$ and the mass within the core region (defined
as the sphere interior to the giant arc, i.e.\ $r<119$ kpc) are then $(17.1\pm
1.2)\times 10^{14}$ M$_{\sun}$ and $(0.46\pm 0.05)\times 10^{14}$~M$_{\sun}$,
respectively.
\subsection{Lensing mass}
Using the high-resolution HST images, we have identified within the
giant arc two features A \& B that are replicated six times. The same
two features are also identified in the southern part of the counter
image as shown in Figure~\ref{fig:critics_closeup}. The northern part
of the counter image is most likely not multiply imaged. Altogether,
the giant arc and its counter image represent a seven-image multiple
system which we use to constrain a strong-lensing model of the cluster
mass distribution.
To model the cluster core we used \textsc{Lenstool}\footnote{publicly
available at http://www.oamp.fr/cosmology/lenstool} (Kneib et
al.\ 1996; Jullo et al.\ 2007) which now uses a Bayesian MCMC sampler
to optimize the cluster mass model and generate robust lens results.
We have followed the procedure of Limousin et al.\ (2007) to model the
mass distribution using one cluster-scale dark-matter halo described
by a truncated PIEMD (Pseudo Isothermal Elliptical Mass Distribution),
as well as an additional 84 truncated galaxy-scale PIEMD potentials to
describe the dark-matter halos associated with the brightest cluster
member galaxies selected from the cluster V--K red
sequence. Furthermore, to minimize the number of free parameters, we
assumed that the mass of galaxy-size halos scales with the K-band
luminosity of the associated galaxy (Natarajan \& Kneib 1997). The
obtained critical curves shown in figure~\ref{fig:critics_core}
display a winding shape in between the galaxies (see close-up in
Fig.~\ref{fig:critics_closeup}), which explains the extreme elongation
of the giant arc.
Using the most probably strong-lensing mass model, we estimate the mass
enclosed by the giant arc. We find $M(<21'') = (112.0 \pm 5) \times
10^{12}\ M_{\odot}$ and a mass-to-light ratio interior to the giant
arc of $M/L = 56\pm2.5$.
These numbers are very robust and depend little on the mass profile
assumed for the dark-matter distribution of the cluster. Caution is
advised though when extrapolating to larger radius, as the slope of
the cluster mass profile is not well constrained by only one
(multiple) arc. Deeper, high-resolution imaging (with, e.g., ACS or
WFC3), however, would likely detect a large number of multiple images
as was the case for Abell~1703 (Limousin et al.\ 2008), allowing us to
accurately measure the slope of the cluster dark-matter profile. Note
that the current best estimate of the Einstein radius at $z\sim 7$ is
nearly 45\arcsec, making this cluster a superb cosmological telescope
to probe the first galaxies in the Universe.
The total magnification of the system (both the giant arc and the part
of the counter image that is multiply imaged) is about 80$\pm$10, one
of the largest amplification factors known for a giant arc. A detailed
model of the arc surface brightness is beyond the scope of this study
but will be presented in a future paper (Clement et al., in
preparation).
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{newfinal.eps}
\caption{\label{fig:critics_core} HST/ACS image of the cluster core
observed through the F606 filter. The critical line computed at the
redshift of the giant arc ($z=1.036$) is shown in red. The yellow
curve shows the corresponding caustics in the source plane. }
\end{figure}
\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{poststamps.eps}
\caption{ \label{fig:critics_closeup} Close-up of the critical lines
at the positions of the giant arc and the counter image. The yellow
ellipses and identifiers mark components of the set of multiple
images used to constrain the lens model. }
\end{figure}
\section{Summary and Conclusions}
We present a comprehensive multi-wavelength analysis of the properties of the
massive galaxy cluster MACS\,J1206.2--0847. At a redshift of $z=0.4385$, the
system acts as a gravitational lens for a background galaxy at $z=1.04$,
resulting in spectacular gravitational arc of high surface brightness, 15\arcsec
in length, a total magnitude of V=21.0, and of unusual, very red colour of
$R-K=4.3$. Our X-ray analysis based on Chandra data yields global X-ray
properties ($L_{\rm X}=2.3\times 10^{45}$ erg s$^{-1}$, 0.1--2.4 keV, and ${\rm
k}T=11.6\pm 0.7$ keV) that make this cluster one of the most extreme systems
known at any redshift.
Belying its relaxed appearance at optical wavelengths, MACS\,J1206.2--0847
exhibits many signs of ongoing merger activity along the line of sight when
looked at more closely, including a disturbed X-ray morphology in the cluster
core, a small but significant offset of the peak of the X-ray emission from the
brightest cluster galaxy, and a very high velocity dispersion of 1580 km
s$^{-1}$. The strongest indication of recent or ongoing cluster growth, however,
is obtained from a comparison of X-ray, virial, and lensing mass estimates for
this system. A high-resolution image of the giant arc and its counter image
obtained with HST allows us to create a lens model that places tight constraints
on the mass distribution interior to the arc. The strong-lensing value of the
mass of the cluster core of $(11.2 \pm 0.5) \times 10^{13}\ M_{\odot}$ thus
obtained is higher by about a factor of two than the X-ray estimate of $(4\pm
0.4)\times10^{13}$ M$_{\odot}$. A similar discrepancy is found between the X-ray
estimate of the total mass within $r_{\rm 200}$ and the virial mass estimate
derived from radial-velocity measurements for 38 cluster galaxies.
Comparable discrepancies between X-ray and lensing mass estimates, in particular
for cluster cores, have been reported before for other systems, the perhaps most
famous example being A1689 (e.g., Miralda-Escud\'e \& Babul 1995; Xue \& Wu
2002; Limousin et al.\ 2007). In all cases, including the one presented here,
the mass derived using gravitational-lensing features is two to three times
higher than the one obtained by an X-ray analysis assuming hydrostatic
equilibrium. In agreement with simulations (Bartelmann \& Steinmetz 1996),
detailed observational studies of such discrepancies for individual clusters
find deviations from hydrostatic equilibrium and the presence of substructure
along the line of sight to be responsible, tell-tale signs being offsets between
the X-ray peak and the location of the BCG as well as extreme elongations and
structure in radial-velocity space (Allen 1998, Machacek et al.\ 2001). We
conclude that MACS\,J1206.2--0847 is a merging cluster with a merger axis that
is close to aligned with our line of sight. A modest cool core has either
survived the merger or is in the process of formation.
The discovery of a giant arc in this MACS cluster underlines yet again the
efficiency of X-ray luminous clusters as gravitational lenses. Much deeper,
high-resolution images of systems like MACS\,J1206.2--0847 will, owing to the
large number of multiple images detected, allow a detailed mapping of the mass
distribution in the cluster core, and, through the power of gravitational
magnification, provide an ultra-deep look at the very distant Universe.
\section*{Acknowledgments}
We thank Alexey Vikhlinin for very helpful advice and suggestions
concerning systematic effects in the Chandra data analysis. HE
gratefully acknowledges financial support from grants NAG 5-8253 and
GO2-3168X. ACE thanks the Royal Society for generous support during
the gestation phase of this paper. JPK acknowledges support from the
{\it Centre National de la Recherche Scientifique} (CNRS), and the ANR
grant 06-BLAN-0067. EJ acknowledges support from an ESO-studentship
and from CNRS.
| {'timestamp': '2009-01-15T00:45:31', 'yymm': '0901', 'arxiv_id': '0901.2144', 'language': 'en', 'url': 'https://arxiv.org/abs/0901.2144'} |
\section{Introduction}
\setcounter{equation}{0}
Let $d\mu$ be a measure on ${\mathbb R}^d$ with all finite moments and we assume that $d \mu$
is positive definite in the sense that $\int_{{\mathbb R}^d} p^2(x) d\mu >0$ for every $p \in \Pi^d$,
$p \ne 0$, where $\Pi^d$ denotes the space of real polynomials in $d$--variables. Let
$\langle \cdot,\cdot \rangle_\mu$ denote the inner product defined by
\begin{equation} \label{ipd1}
\langle p, q \rangle_\mu : = \int_{{\mathbb R}^d} p(x)q(x) d\mu(x), \qquad p, q \in \Pi^d.
\end{equation}
Then orthogonal polynomials of several variables with respect to $\langle p,q\rangle_\mu$
exist. Let $N \ge 1$ be a positive integer and let $\xi_1, \xi_2, \ldots, \xi_N$ be distinct
points in ${\mathbb R}^d$. Let $\Lambda$ be a positive definite matrix of size $N \times N$.
We define a new inner product $\langle \cdot,\cdot \rangle_\nu$ by
\begin{equation} \label{ipd2}
\langle p, q \rangle_\nu : = \langle p, q \rangle_\mu
+ (p(\xi_1), p(\xi_2), \ldots, p(\xi_N)) \Lambda (q(\xi_1), q(\xi_2), \ldots, q(\xi_N))^{\mathsf {tr}},
\end{equation}
where the superscript ${\mathsf {tr}}$ indicates the transpose, which can be defined via an integral
as in \eqref{ipd1} against a measure $d\nu$ that is obtained from adding $N$ mass points
to $d\mu$. A typical example is when $\Lambda$ is a diagonal matrix with positive entries.
The purpose of this paper is to study orthogonal polynomials with respect to the new
inner product $\langle \cdot,\cdot \rangle_\nu$.
In the case of one--variable, the first study of orthogonal polynomials for measures with
mass points was carried out, as far as we know, by Uvarov (\cite{U}), who gave a short
discussion on the case of adding a finite set of mass points to a measure and showed
how to express the orthogonal polynomials with respect to the new measure in terms
of those with respect to the old one. The problem was later revitalized by A. M. Krall
(\cite{K1}), who considered orthogonal polynomials for measures obtained by adding
mass points at the end of the interval on which a continuous measure lives. The case
of Jacobi measure with additional mass points at the end of $[-1,1]$ was studied in
\cite{Koor}, where explicit formulas of orthogonal polynomials were constructed. For
Jacobi weight with multiple mass points, it is possible to study asymptotic properties
of orthogonal polynomials \cite{GPRV}. In the case of several variables, however,
only the case of $N =1$ has been studied \cite{FPPX}.
Our main results contain explicit formulas that express orthogonal polynomials and
reproducing kernels with respect to ${\langle}\cdot, \cdot {\rangle}_\nu$ in terms of those with
respect to ${\langle}\cdot,\cdot{\rangle}_\mu$. These results are stated and proved in Section 2.
As an example, we consider the case of Jacobi weight function on the simplex in ${\mathbb R}^d$,
with mass points added at the vertices, for which our formulas can be further specified
and expressed in terms of the classical Jacobi polynomials. The result is then used to
study the asymptotic expansion of the Christoffel functions with respect to
${\langle}\cdot, \cdot{\rangle}_\nu$.
\section{Orthogonal polynomials for measures with mass points}
\setcounter{equation}{0}
We start with a short subsection on necessary definitions, for which we follow essentially
\cite{DX}, and prove our main results in the second subsection.
\subsection{Preliminary}
Through this paper, we will use the standard multi--index notation. Let $\mathbb{N}_0$
denote the set of nonnegative integers. For a multi--index $\alpha=(\alpha_1,\dots,\alpha_d)
\in\mathbb{N}_0^d$ and $x=(x_1,\dots,x_d) \in {\mathbb R}^d$, a monomial in $d$ variables is
defined as $x^{\alpha}=x_1^{\alpha_1}\cdots x_d^{\alpha_d}$. The integer
$|\alpha|=\alpha_1+ \dots+\alpha_d$ is called the \emph{total degree} of $x^{\alpha}$.
We denote by ${\mathcal P}_n^d$ the space of homogeneous polynomials of degree $n$ in
$d$--variables, ${\mathcal P}_n^d := \operatorname{span}\{x^{\alpha}: |{\alpha}| =n\}$, and denote by $\Pi_n^d$
the space of polynomials of total degree at most $n$. The collection of all polynomials
in $d$--variables is $\Pi^d$. It is well known that
$$
\dim \Pi_n^d = \binom{n+d}{n} \quad \hbox{\rm and}\quad \dim
\mathcal{P}_n^d = \binom{n+d-1}{n}:=r_n^d.
$$
Let $\langle \cdot,\cdot \rangle_\mu$ be the inner product defined in \eqref{ipd1}.
A polynomial $p \in \Pi_n^d$ is {\it orthogonal} with respect to \eqref{ipd1} if
$$
\langle p,\, q\rangle_\mu = 0, \qquad \forall q\in \Pi_{n-1}^d.
$$
Our assumption that $d\mu$ is positive definite implies that orthogonal polynomials
with respect to $\langle \cdot,\cdot \rangle_\mu$ exist. Let us denote by $\mathcal{V}_n^d$
the space of orthogonal polynomials of total degree $n$. It follows that
$\dim \mathcal{V}_n^d = r_n^d$. Let $\{P_\alpha^n\}_{|\alpha|=n}$ denote a basis of
$\mathcal{V}_n^d$. It is often convenient to use vector notations introduced in \cite{Ko1}
and \cite{X93}. Let $\{\alpha_1, {\alpha}_2,\ldots, {\alpha}_{r_n^d}\}$ be an enumeration of the set
$\{\alpha \in {\mathbb N}_0^d: |\alpha| =n\}$ according to a fixed monomial order, say the
lexicographical order or the reversed lexicographical order. Then the basis
$\{P_\alpha^n\}_{|\alpha|=n}$ can be written as
$$
\mathbb{P}_n= \left \{P^n_{\alpha_1}, P^n_{\alpha_2}, \ldots, P^n_{\alpha_{r_n^d}} \right \}.
$$
We will treat ${\mathbb P}_n$ both as a set of functions and as a {\it column} vector of functions.
As column vectors, the orthogonality of $\{P_{{\alpha}_j}^n\}$ can be expressed as
$$
{\langle} {\mathbb P}_n, {\mathbb P}_m^{\mathsf {tr}} {\rangle}_\mu = \int_{{\mathbb R}^d} {\mathbb P}_n(x) {\mathbb P}_m^{\mathsf {tr}}(x) d\mu = \begin{cases}
0, & \hbox{if $n \ne m$}, \\ H_n, & \hbox{if $n = m$}, \end{cases}
$$
where the superscript denotes the transpose (so that ${\mathbb P}^{\mathsf {tr}}$ is a row vector) and
$H_n$ is a matrix of size $r_n^d \times r_n^d$, necessarily symmetric, and in
fact a positive definite matrix by our assumption on $d\mu$. For convenience, we shall
call the system $\{\mathbb{P}_n\}_{n=0}^\infty = \{P_\alpha^n: |{\alpha}| =n, n=0, 1,\ldots\}$ an
{\it orthogonal polynomial system} (OPS). If $H_n$ is the identity matrix, then
$\{P_{\alpha}^n: |{\alpha}|=n\}$ is an orthonormal basis for ${\mathcal V}_n^d$ and the OPS is called an \emph{orthonormal} polynomial system.
Likewise, we can write ${\mathbf x}^n : = \{x^{\alpha}: |{\alpha}| =n\} = \{x^{{\alpha}_1}, x^{{\alpha}_2}, \ldots, x^{{\alpha}_{r_n^d}} \}$
and regard it as a {\it column} vector. Since each element in ${\mathbb P}_n$ is a polynomial of degree
$n$, it can be written as a sum of monomials, which, in vector notation, becomes
$$
\mathbb{P}_n = \sum_{j=0}^n G_{j,n}\, {\mathbf x}^j, \qquad \hbox{where} \quad G_{j,n} \in
\mathcal{M}_{r_n^d\times r_j^d},
$$
in which ${\mathcal M}_{p \times q}$ denotes the set of real matrices of size $p \times q$. In
particular, $G_{n,n}$ is a square matrix and it is necessarily invertible since ${\mathbb P}_n$
is a basis of ${\mathcal V}_n^d$. We call $G_{n,n}$ the leading coefficient of ${\mathbb P}_n$.
With respect to $d\mu$, the reproducing kernel of ${\mathcal V}_n^d$, denoted by $P_n(d\mu;x,y)$,
is defined by ${\langle} P_n(d\mu; x, \cdot), p{\rangle}_\mu = p(x)$, $p\in {\mathcal V}_n^d$. In terms of
a basis ${\mathbb P}_n$ of ${\mathcal V}_n^d$, it satisifes
$$
P_n(d\mu;x,y) = \mathbb{P}^{\mathsf {tr}}_n(x)\, H^{-1}_n\, \mathbb{P}_n(y) \quad\hbox{with}
\quad H_n = {\langle} {\mathbb P}_n, {\mathbb P}_n{\rangle}_\mu.
$$
Similarly, the reproducing kernel of $\Pi_n^d$, denoted by $K_n(d\mu;x,y)$, is defined by
${\langle} K_n(d\mu; x, \cdot), p{\rangle}_\mu = p(x)$, $p\in \Pi_n^d$, and satisfies
$$
K_n(d\mu;x,y) = \sum_{j=0}^n P_j(d\mu;x,y), \qquad n\ge 0.
$$
Since the definitions of $P_n(d\mu;x,y)$ and $K_n(d\mu;x,y)$ are independent of the
choice of a particular basis, (see \cite[Theorem 3.5.1]{DX}), it is often more
convenient to work with an orthonormal basis. The kernel $K_n(d\mu;x,y)$ plays an
important role in studying Fourier orthogonal expansions, as it is the kernel function of
the partial sum operator. The reciprocal of $K_n(d\mu;x,x)$ is called Christoffel function,
denoted by $\Lambda_n(x)$, and it satisfies
$$
\Lambda_n(x) : = \frac{1}{K_n(d\mu;x,x)} = \inf_{P(x) =1, P\in\Pi_n^d} \int_{{\mathbb R}^d} |P(y)|^2 d\mu(y).
$$
\subsection{Main results}
Our goal is to study orthogonal polynomials with respect to the inner product
${\langle} \cdot,\cdot {\rangle}_\nu$ defined in \eqref{ipd2}. Let us recall that $\Lambda$ is a given
positive definite matrix of order $N$ and $\{\xi_1, \xi_2, \ldots, \xi_N\}$ is a set of distinct
points in $\mathbb{R}^d$. Introducing the notation
$$
\mathbf{p}(\xi) = \left \{ p(\xi_1), p(\xi_2), \ldots, p(\xi_N) \right \},
$$
and regarding it also as a column vector, we can then rewrite the inner product
${\langle} \cdot,\cdot {\rangle}_\nu$ in \eqref{ipd2} as
\begin{equation*}
(1.2') \hspace{1.5in} \hfill \langle p, q \rangle_\nu = \langle p, q \rangle_\mu +\mathbf{p}(\xi)^{\mathsf {tr}} \, \Lambda \,\mathbf{q}(\xi), \hspace{1.5in} \hfill
\end{equation*}
where $\langle \cdot, \cdot \rangle_\mu$ denotes the inner product defined in \eqref{ipd1}.
In the case that $\Lambda$ is a diagonal matrix, $\Lambda = \mathrm{diag}
\{{\lambda}_1,\ldots,{\lambda}_N\}$, the inner product ${\langle} \cdot,\cdot {\rangle}_\nu$ takes the form
\begin{equation} \label{sum-mass}
\langle p, q \rangle_\nu = \langle p, q \rangle_\mu + \sum_{j=1}^N {\lambda}_j p(\xi_j) q(\xi_j).
\end{equation}
Our first result shows that orthogonal polynomials with respect to $\langle p, q \rangle_\nu$
can be derived in terms of those with respect to $\langle p, q \rangle_\mu$. The statement
and the proof of this result relies heavily on the vector--matrix notation. To facilitate the
study, we shall introduce several new notations.
Throughout this section, we shall fix ${\mathbb P}_n$ as an orthonormal basis for ${\mathcal V}_n^d$
associated with $d\mu$. We denote by $\mathsf{P}_n(\xi)$ the matrix that has ${\mathbb P}_n(\xi_i)$
as columns,
\begin{equation} \label{sP}
\mathsf{P}_n(\xi):= \left(\mathbb{P}_n(\xi_1) | \mathbb{P}_n(\xi_2) | \ldots
| \mathbb{P}_n(\xi_N) \right) \in \mathcal{M}_{r_n^d \times N},
\end{equation}
denote by ${\mathbf K}_{n-1}$ the matrix whose entries are $K_{n-1}(d\mu;\xi_i,\xi_j)$,
\begin{equation} \label{cK}
\mathbf{K}_{n-1} := \big(K_{n-1}(d\mu;\xi_i,\xi_j) \big)_{i,j=1}^N \in \mathcal{M}_{N\times N},
\end{equation}
and, finally, denote by $\mathbb{K}_{n-1}(\xi,x)$ the vector of functions
\begin{equation} \label{sK}
{\mathbb{K}_{n-1}(\xi,x)} = \left\{K _{n-1}(d\mu; \xi_1,x), K_{n-1}(d\mu;\xi_2,x),
\ldots, K_{n-1}(d\mu;\xi_N,x)\right \},
\end{equation}
which we again regard as a column vector.
From the fact that $K_n(d\mu; x,y) - K_{n-1}(d\mu;x,y) = P_n(d\mu;x,y)$, we have immediately
the following relations,
\begin{align}
\mathsf{P}_n^{\mathsf {tr}}(\xi) {\mathbb P}_n (x) & = {\mathbb K}_n(\xi,x) - {\mathbb K}_{n-1}(\xi,x), \label{P-K1}\\
\mathsf{P}_n^{\mathsf {tr}}(\xi) \mathsf{P}_n (\xi) & = {\mathbf K}_n - {\mathbf K}_{n-1}, \label{P-K2}
\end{align}
which will be used below. Let $I_N$ denote the identity matrix of order $N$.
\begin{lemma}
The matrix $I_N + \Lambda\,{\mathbf{K}_{n-1}}$ is invertible.
\end{lemma}
\begin{proof}
First we show that the matrix ${\mathbf K}_{n-1}$ is positive definite, By the definition of
$K_n(d\mu;\cdot,\cdot)$, for every ${\mathbf c} \in {\mathbb R}^N$, ${\mathbf c} \ne 0$, we have
$$
{\mathbf c}^{\mathsf {tr}} {\mathbf K}_{n-1} {\mathbf c} = \sum_{|{\alpha}| \le n-1} \sum_{i,j =1}^N c_i c_j P_{\alpha}(\xi_i) P_{\alpha}(\xi_j)
= \sum_{|{\alpha}| \le n-1} \Big| \sum_{j=0}^N c_j P_{\alpha}(\xi_j) \Big|^2 > 0,
$$
so that ${\mathbf K}_{n-1}$ is positive definite. The matrix $\Lambda$ is also positive definite,
by assumption, so that it is invertible. Since $\Lambda^{-1} (I_N + \Lambda {\mathbf K}_{n-1}) =
\Lambda^{-1} + {\mathbf K}_{n-1}$, we see that it is positive definite as well, hence invertible.
Consequently, $I_N + \Lambda {\mathbf K}_{n-1}$ is invertible.
\end{proof}
We are now ready to state and prove our first main result.
\begin{thm} \label{main-thm}
Define a polynomial system $\{\mathbb{Q}_n\}_{n\ge 0}$ by $\mathbb{Q}_0(x) := \mathbb{P}_0(x)$ and
\begin{equation} \label{ex-expl}
\mathbb{Q}_n(x) = \mathbb{P}_n(x) - {\mathsf{P}_n(\xi)}\,(I_N +
\Lambda\,{\mathbf{K}_{n-1}})^{-1}\,\Lambda \,
{\mathbb{K}_{n-1}(\xi,x)}, \qquad n\ge 1.
\end{equation}
Then $\{\mathbb{Q}_n\}_{n\ge 0}$ is a sequence of orthogonal polynomials
with respect to ${\langle} \cdot,\cdot{\rangle}_\nu$ defined in \eqref{ipd2}. Conversely, any
sequence of orthogonal polynomials with respect to \eqref{ipd2} can be expressed
as in \eqref{ex-expl}.
\end{thm}
\begin{proof}
Let us assume that $\{{\mathbb Q}_n\}_{n\ge 0}$ is an OPS with respect to ${\langle} \cdot, \cdot {\rangle}_\nu$
and ${\mathbb Q}_n$ has the same leading coefficient as ${\mathbb P}_n$, which implies, in particular, that
${\mathbb Q}_0$ is a constant and ${\mathbb Q}_0 = {\mathbb P}_0$. We show that ${\mathbb Q}_n$ satisfies \eqref{ex-expl}.
By the assumption, the components of $\mathbb{Q}_n-\mathbb{P}_n$ are elements in
$\Pi_{n-1}^d$ for $n\ge 1$. Since $\{\mathbb{P}_n\}_{n \ge 0}$ is a basis of $\Pi^d$, we
can express these components as linear combinations of orthogonal polynomials in
${\mathbb P}_0, {\mathbb P}_1, \ldots, {\mathbb P}_{n-1}$. In vector-matrix notation, this means that
$$
\mathbb{Q}_n(x) = \mathbb{P}_n(x) + \sum_{j=0}^{n-1} M_j^n\, \mathbb{P}_j(x),
$$
where $M_j^n$ are matrices of size $r_n^d\times r_j^d$. These coefficient matrices can
be determined from the orthonormality of $\mathbb{P}_n$ and ${\mathbb Q}_n$. Indeed, by the
orthogonality of ${\mathbb Q}_n$, ${\langle} {\mathbb Q}_n, {\mathbb P}_j {\rangle}_\nu =0$ for $0 \le j \le n-1$, which shows,
by the definition of ${\langle} \cdot,\cdot {\rangle}_\nu$ and the fact that ${\mathbb P}_j$ is orthonormal,
$$
M_j^n = \langle \mathbb{Q}_n, \mathbb{P}^{\mathsf {tr}}_j \rangle_\mu =
- \mathsf{Q}_n(\xi)^{\mathsf {tr}} \Lambda \mathsf{P}_n(\xi),
$$
where $\mathsf{P}_n(\xi)$ is defined as in \eqref{sP} and $\mathsf{Q}_n(\xi)=
\left\{\mathbb{Q}_n(\xi_1) | \mathbb{Q}_n(\xi_2) | \ldots | \mathbb{Q}_n(\xi_N)\right \}$ in
the analogous matrix with ${\mathbb Q}_n(\xi_i)$ as its column vectors. Consequently, we obtain
\begin{align} \label{QnPj}
\mathbb{Q}_n(x) & = \mathbb{P}_n(x) - \sum_{j=0}^{n-1} \mathsf{Q}_n(\xi)\,\Lambda
\mathsf{P}^{\mathsf {tr}}_j(\xi) \, \mathbb{P}_j(x) \\
& = \mathbb{P}_n(x) - \mathsf{Q}_n(\xi)\,\Lambda \,\mathbb{K}_{n-1}(\xi,x). \notag
\end{align}
where the second equation follows from the relation \eqref{P-K1}, which leads to a
telescoping sum that sums up to $\mathbb{K}_{n-1}(\xi,x)$. Setting $x=\xi_i$, we obtain
$$
\mathbb{Q}_n(\xi_i) = \mathbb{P}_n(\xi_i) - \mathsf{Q}_n(\xi)\,\Lambda \,\mathbb{K}_{n-1}(\xi,\xi_i), \quad 1 \le i \le N,
$$
which leads to, by the definition of ${\mathbf K}_{n-1}$ at (\ref{cK}), that
$$
\mathsf{Q}_n(\xi) = \mathsf{P}_n(\xi) - \mathsf{Q}_n(\xi)\,\Lambda \,\mathbf{K}_{n-1}.
$$
Solving for $\mathsf{Q}_n (\xi)$ from the above equation gives
\begin{equation}\label{qn(c)}
\mathsf{Q}_n(\xi) = \mathsf{P}_n(\xi) (I_N + \,\Lambda \,\mathbf{K}_{n-1})^{-1}.
\end{equation}
Substituting this expression into \eqref{QnPj} establishes (\ref{ex-expl}).
Conversely, if we define polynomials $\mathbb{Q}_n$ by \eqref{ex-expl}, the above proof
shows that $\mathbb{Q}_n$ is orthogonal with respect to $\langle \cdot, \cdot \rangle_\nu$.
Since $\mathbb{Q}_n$ and $\mathbb{P}_n$ have the same leading coefficient, it is evident
that $\{\mathbb{Q}_n \}_{n \ge 0}$ is an OPS in $\Pi^d$.
\end{proof}
Let $\{\mathbb{Q}_n\}_{n\ge0}$ be an OPS with respect to \eqref{ipd2} as in
Theorem \ref{main-thm}. In general, ${\mathbb Q}_n$ is not orthonormal. We denote, in the
rest of this section,
$$
H_n := \langle \mathbb{Q}_n, \mathbb{Q}^{\mathsf {tr}}_n \rangle_\nu.
$$
Then $H_n$ is a positive definite matrix. It turns out that both $H_n$ and $H_n^{-1}$ can
be expressed in terms of matrices that involve only $\{{\mathbb P}_j\}_{j \ge 0}$.
\begin{prop} \label{prop.3.2}
For $n \ge 0$,
\begin{align}
H_n & = I_{r_n^d} + \mathsf{P}_n(\xi) (I_N + \,\Lambda \,\mathbf{K}_{n-1})^{-1} \Lambda \mathsf{P}_n^{\mathsf {tr}}(\xi), \label{directa} \\
H^{-1}_n & = I_{r_n^d} - \mathsf{P}_n(\xi) (I_N + \,\Lambda \,\mathbf{K}_{n})^{-1} \Lambda \mathsf{P}_n^{\mathsf {tr}}(\xi). \label{inversa}
\end{align}
\end{prop}
\begin{proof}
Since ${\mathbb P}_n$ is orthonormal, $\langle \mathbb{P}_n, \mathbb{P}^{\mathsf {tr}}_n \rangle_\mu
= I_{r_n^d}$. From \eqref{ex-expl} and \eqref{qn(c)} we obtain
\begin{align*}
H_n =& \langle \mathbb{Q}_n , \mathbb{Q}_n^{\mathsf {tr}}\rangle_\nu =
\langle \mathbb{Q}_n , \mathbb{P}_n^{\mathsf {tr}}\rangle_\nu = \langle
\mathbb{Q}_n , \mathbb{P}_n^{\mathsf {tr}}\rangle_\mu + \mathsf{Q}_n(\xi) \Lambda \mathsf{P}_n^{\mathsf {tr}}(\xi) \\
=& I_{r_n^d} + \mathsf{P}_n(\xi) ( I_N + \,\Lambda \,\mathbf{K}_{n-1})^{-1}
\Lambda \mathsf{P}_n^{\mathsf {tr}}(\xi),
\end{align*}
which proves (\ref{directa}). In order to establish \eqref{inversa}, we need to verify that
$$
H_n ( I_{r_n^d} - \mathsf{P}_n(\xi) (I_N + \,\Lambda \,\mathbf{K}_{n})^{-1} \Lambda \mathsf{P}_n^{\mathsf {tr}}(\xi) ) = I_{r_n^d},
$$
which, by \eqref{directa} and after simplification, reduces to the following equation,
\begin{align}\label{identity}
(I_N + \Lambda\,\mathbf{K}_{n-1})^{-1}\, & \Lambda\, \mathsf{P}_n(\xi)^{\mathsf {tr}} \mathsf{P}_n(\xi)
(I_N + \Lambda\,\mathbf{K}_{n})^{-1} \\
& = (I_N + \Lambda\,\mathbf{K}_{n-1})^{-1} - (I_N +\Lambda\,\mathbf{K}_{n})^{-1}. \notag
\end{align}
Using \eqref{P-K2}, the above equation can be verified by a simple computation.
\end{proof}
Our next result gives explicit formulas for the reproducing kernels associated with
${\langle} \cdot,\cdot{\rangle}_\nu$, which we denote by
\begin{equation*}
P_j(d\nu; x,y):= \mathbb{Q}^{\mathsf {tr}}_j(x)\, H^{-1}_j\, \mathbb{Q}_j(y) \quad \hbox{and} \quad
K_n(d\nu; x,y):= \sum_{j=0}^n P_j(d\nu;x,y).
\end{equation*}
\begin{thm} \label{thm.3.3}
For $j\ge 0$,
\begin{align}
P_j(d\nu; x,y) = P_j(d\mu; x,y) & -
\mathbb{K}_j^{\mathsf {tr}}(\xi,x) \, (I_N + \,\Lambda \,\mathbf{K}_{j})^{-1} \Lambda \, \mathbb{K}_j(\xi,y) \label{Pk}\\
& + \mathbb{K}_{j-1}^{\mathsf {tr}}(\xi,x) \, (I_N + \,\Lambda \,\mathbf{K}_{j-1})^{-1} \Lambda \, \mathbb{K}_{j-1}(\xi,y). \nonumber
\end{align}
Furthermore, for $n \ge 0$,
\begin{equation}\label{kernel}
K_n(d\nu; x,y) = K_n(d\mu; x,y) -
\mathbb{K}_n^{\mathsf {tr}}(\xi,x) \, (I_N + \,\Lambda \,\mathbf{K}_{n})^{-1} \Lambda \, \mathbb{K}_n(\xi,y).
\end{equation}
\end{thm}
\begin{proof}
Since $\Lambda^{-1}\,(I_N + \Lambda\,\mathbf{K}_{j-1}) = \Lambda^{-1}\,+\,\mathbf{K}_{j-1}$
is a symmetric matrix, so is $(I_N + \Lambda\,\mathbf{K}_{j-1})^{-1}\,\Lambda$. Using this
fact, it follows from \eqref{ex-expl} and \eqref{inversa} that
\begin{align*}
\mathbb{Q}^{\mathsf {tr}}_j(x)\, H^{-1}_j = & \mathbb{P}^{\mathsf {tr}}_j(x) -
{\mathbb P}^{\mathsf {tr}}_j(x) \mathsf{P}_j(\xi)(I - \Lambda {\mathbf K}_j)^{-1} \Lambda \mathsf{P}^{\mathsf {tr}}_j(\xi) \\
& - \mathbb{K}_{j-1}^{\mathsf {tr}}(\xi,x) \,(I_N +\Lambda\,\mathbf{K}_{j-1})^{-1}\,\Lambda\,
\mathsf{P}_j^{\mathsf {tr}}(\xi) \\
& - \mathbb{K}_{j-1}^{\mathsf {tr}}(\xi,x) \,(I_N + \Lambda\,\mathbf{K}_{j-1})^{-1}\Lambda\,
\mathsf{P}^{\mathsf {tr}}_j(\xi) \mathsf{P}_j(\xi) (I+\Lambda {\mathbf K}_j)^{-1} \mathsf{P}_j^{\mathsf {tr}}(\xi),
\end{align*}
which simplifies to, upon using \eqref{identity} and \eqref{P-K1},
\begin{align*}
\mathbb{Q}^{\mathsf {tr}}_j(x)\, H^{-1}_j = & \mathbb{P}_j^{\mathsf {tr}}(x) -
{\mathbb P}_j^{\mathsf {tr}}(x) \mathsf{P}_j(\xi)(I - \Lambda {\mathbf K}_j)^{-1} \Lambda \mathsf{P}_j^{\mathsf {tr}}(\xi) \\
& - \mathbb{K}_{j-1}^{\mathsf {tr}}(\xi,x) \,(I_N +
\Lambda\,\mathbf{K}_{j-1})^{-1}\,\Lambda\, \mathsf{P}_j^{\mathsf {tr}}(\xi) \\
= & \mathbb{P}_j^{\mathsf {tr}}(x) - \mathbb{K}_{j}^{\mathsf {tr}}(\xi,x) \,(I_N +
\Lambda\,\mathbf{K}_{j})^{-1}\,\Lambda\, \mathsf{P}_j^{\mathsf {tr}}(\xi).
\end{align*}
Using again \eqref{ex-expl} and \eqref{P-K1}, we then obtain
\begin{align*}
\mathbb{Q}^{\mathsf {tr}}_j(x)\,& H^{-1}_j \mathbb{Q}_j(y) =
\mathbb{P}_j^{\mathsf {tr}}(x) \,\mathbb{P}_j(y) \\
&- [\mathbb{K}_{j}^{\mathsf {tr}}(\xi,x) - \mathbb{K}_{j-1}^{\mathsf {tr}}(\xi,x)] \,(I_N +
\Lambda\,\mathbf{K}_{j-1})^{-1}\,\Lambda\, \mathbb{K}_{j-1}(\xi,y) \\
&- \mathbb{K}_{j}^{\mathsf {tr}}(\xi,x) \,(I_N +
\Lambda\,\mathbf{K}_{j})^{-1}\,\Lambda\, [\mathbb{K}_{j}(\xi,y) - \mathbb{K}_{j-1}(\xi,y)] \\
&+ \mathbb{K}_{j}^{\mathsf {tr}}(\xi,x) \,(I_N +
\Lambda\,\mathbf{K}_{j})^{-1}\,\Lambda\,\mathsf{P}^{\mathsf {tr}}_j(\xi) \mathsf{P}_j(\xi) (I_N + \Lambda\,\mathbf{K}_{j-1})^{-1} \Lambda \mathbb{K}_{j-1}(\xi,y),
\end{align*}
which simplifies to \eqref{Pk} upon using the identity \eqref{identity}.
Finally, summing over \eqref{Pk} for $j = 0, 1, \ldots, n$, we obtain \eqref{kernel}.
\end{proof}
The result in this section can be extended without much difficulty to mass points with derivative
values. To be more precise, let $\partial^{\alpha} = \partial_1^{{\alpha}_1} \cdots \partial_d^{{\alpha}_d}$, where
$\partial_i = \frac{\partial}{\partial x_i}$, and for ${\alpha}_i \in {\mathbb N}_0^d$, $i = 1,2,\ldots, N$, define
\begin{equation} \label{lpd-Soblev}
D_{\alpha}\mathbf{p}(\xi) := \left \{ \partial^{{\alpha}_1} p(\xi_1), \partial^{{\alpha}_2}p(\xi_2), \ldots,
\partial^{{\alpha}_N}p(\xi_N) \right \},
\end{equation}
and regard it also as a column vector. Instead of requiring $\xi_i \ne \xi_j$, we only assume
that $\xi_i \ne \xi_j$ when ${\alpha}_i = {\alpha}_j$. In other word, $\xi_i$ and $\xi_j$ can be the same
as long as ${\alpha}_i \ne {\alpha}_j$. We then consider the inner product defined by
\begin{equation} \label{soblev}
\langle p, q \rangle_\nu = \langle p, q \rangle_\mu + D_{\alpha}\mathbf{p}(\xi)^{\mathsf {tr}} \, \Lambda \,
D_{\alpha} \mathbf{q}(\xi).
\end{equation}
When ${\alpha}_i =0$ for all $i$, this is the inner product in \eqref{ipd2}. Other interesting cases
include, for example,
$$
\langle p, q \rangle_\nu = \langle p, q \rangle_\mu +
\sum_{j =0}^N {\lambda}_j p(\xi_j) q(\xi_j) + \sum_{j=0}^N \lambda_j' \nabla p(\xi_j) \cdot \nabla q(\xi_j).
$$
Our results in Theorem~\ref{main-thm}, Proposition~\ref{prop.3.2} and Theorem~\ref{thm.3.3}
still hold in this setting, but we need to replace $\mathsf{P}_n$ in \eqref{sP}, ${\mathbf K}_{n-1}$ in \eqref{cK}, and ${\mathbb K}_{n-1}(\xi,x)$ in \eqref{sK} by
\begin{align*}
\mathsf{P}_n^*(\xi):= & \left(\partial^{{\alpha}_1}\mathbb{P}_n(\xi_1) | \partial^{{\alpha}_2}\mathbb{P}_n(\xi_2)
| \ldots | \partial^{{\alpha}_d}\mathbb{P}_n(\xi_N) \right) \in \mathcal{M}_{r_n^d \times N}, \\
\mathbf{K}_{n-1}^* := & \big(\partial^{{\alpha}_i}_{\{1\}} \partial^{{\alpha}_j}_{\{2\}}
K_{n-1}^*(d\mu;\xi_i,\xi_j) \big)_{i,j=1}^N \in \mathcal{M}_{N\times N}, \\
{\mathbb{K}_{n-1}^*(\xi,x)} = & \left\{\partial^{{\alpha}_1}_{\{1\}}K _{n-1}(d\mu; \xi_1,x),
\partial^{{\alpha}_2}_{\{1\}} K_{n-1}(d\mu;\xi_2,x),
\ldots, \partial^{{\alpha}_d}_{\{1\}} K_{n-1}(d\mu;\xi_N,x)\right \}
\end{align*}
respectively, where $\partial^{\alpha}_{\{1\}} K_n(u,v)$ means that the derivative is taken with respect
to $u$ variable.
\begin{thm}
The results in Theorem~\ref{main-thm} and Theorem~\ref{thm.3.3} hold for the inner
product defined in \eqref{lpd-Soblev} when $\mathsf{P_n}$, ${\mathbf K}_{n-1}$ and ${\mathbb K}_{n-1}(\xi,x)$
are replaced by $\mathsf{P}^*_n$, ${\mathbf K}_{n-1}^*$ and ${\mathbb K}_{n-1}^*(\xi,x)$, respectively.
\end{thm}
The proof follows as before almost verbatim with little additional difficulty.
\section{Orthogonal polynomials on the simplex}
\setcounter{equation}{0}
In this section we apply the general result in the previous section to orthogonal polynomials on
the simplex
$$
T^d : = \{x = ({x_1},\ldots,
{x_d})\in \mathbb{R}^d: {x_i} \ge 0, 1-|x|_1 \ge0 \}
$$
in $\mathbb{R}^d$, where $|x|_1 = x_1 + \ldots + x_d$.
\subsection{Jacobi polynomials on the simplex}
We consider the Jacobi weight function
$$
W_{\kappa}(x) = x_1^{\kappa_1 -1/2} \cdots x_d^{\kappa_d -1/2} (1 - |x|_1)^{\kappa_{d+1} -1/2}, \quad \kappa_i \ge 0,
$$
on the simplex, where
$$
w_{\kappa} = \frac{\Gamma(|\kappa|+ \frac{d+1}{2})}{\Gamma(\kappa_1+ \frac{1}{2}) \cdots \Gamma(\kappa_{d+1} + \frac{1}{2})}, \qquad
|\kappa| : = \kappa_1 + \kappa_2 + \cdots + \kappa_{d+1},
$$
is the normalization constant of $W_{\kappa}$ such that $w_{\kappa}\,\int_{T^d} W_{\kappa}(x)\,dx=1$. Associated with $W_{\kappa}$, we consider the inner product on the simplex
\begin{equation}\label{simplex-ipg}
\langle f, g \rangle = w_{\kappa} \, \int_{T^d} f(x) \,g(x) \, W_{\kappa}(x) \, dx,
\end{equation}
which plays the role of ${\langle} \cdot, \cdot {\rangle}_\mu$ when we deal with the settings of
the previous section. For $d=1$, $W_{\kappa}$ is the classical Jacobi weight function, which
has orthogonal polynomials $P_n^{({\kappa}_1,{\kappa}_2)}(2t -1)$, where $P_n^{(a,b)}$ is the
classical Jacobi polynomial of degree $n$ that is orthogonal with respect to
$(1-t)^a (1+t)^b$ on $[-1,1]$ and normalized by $P_n^{(a,b)}(1) = \binom{n+a}{n}$.
We shall also denote the orthonormal Jacobi polynomials by $p_n^{(a,b)}(t)$. Evidently,
$p_n^{(a,b)}(t) = c_n P_n^{(a,b)}(t)$, where the constant $c_n$ is given by \cite[(4.3.3)]{Sz}.
To state an orthonormal basis for ${\mathcal V}_n^d$ on the simplex, we follow \cite[p. 47]{DX}
and introduce the following notation. Associated with $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$,
we define by $\mathbf{x}_j$ the truncation of $x$, namely
$$
\mathbf{x}_0 = 0, \quad \mathbf{x}_j = (x_1, \ldots, x_j), \quad 1 \le j \le d,
$$
and associated with $\alpha = (\alpha_1, \ldots, \alpha_{d}) \in \mathbb{N}^d_0$ and
$\kappa = (\kappa_1, \ldots, \kappa_{d+1}) \in \mathbb{R}^{d+1}$, we introduce, respectively,
$$
\alpha^j := (\alpha_j, \ldots, \alpha_{d}), \quad 1 \le j \le d, \qquad
\kappa^j := (\kappa_j, \ldots, \kappa_{d+1}), \quad 1 \le j \le d+1.
$$
Then, an orthonormal basis associated with \eqref{simplex-ipg} is given explicitly by
\begin{equation} \label{simplex-base}
P_{\alpha}(W_\kappa; x) = h^{-1}_{\alpha}
\prod_{j=1}^d \left(\frac{1- |\mathbf{x}_{j}|_1}{1- |\mathbf{x}_{j-1}|_1} \right)^{ |\alpha^{j+1}|}
p_{\alpha_j}^{(a_j,b_j)}\left(\frac{2 x_j}{1- |\mathbf{x}_{j-1}|_1} - 1\right),
\end{equation}
where the parameters $a_j$ and $b_j$ are given by
$$
a_j = 2 |\alpha^{j+1}| + |\kappa^{j+1}| + \frac{d-j-1}{2}, \qquad b_j = \kappa_j - \frac{1}{2},$$
and $h_{\alpha}$ is the normalizing constant given by
$$
h^{2}_{\alpha} = \frac{(|\kappa|+ \frac{d+1}{2})_{2 |\alpha|}}
{\prod_{j=1}^d (2|\alpha^{j+1}|+|\kappa^j|+ \frac{d-j+2}{2})_{2\alpha_j}},$$
in which $(a)_k := a(a+1) \ldots (a+ k-1)$ denotes the shifted factorial.
In this case, we also have a compact formula for the reproducing kernels, given in
terms of the Gegenbauer polynomials $C^{\lambda}_{n}$, which are orthogonal with
respect to the weight function $(1-t^2)^{\lambda-1/2}$ and normalized by
$C_n^{\lambda}(1) = \binom{n+2{\lambda}-1}{n}$. The formula, first derived in \cite[Theorem 2.3]{X98},
is given by
\begin{align} \label{KnSimplex}
K_n(W_{\kappa}; x, y)= & \frac{1}{2^{d+1}} \, \int_{[-1,1]^{d+1}} C^{\lambda}_{2n}(\sqrt{x_1\,y_1}\,t_1 + \cdots + \sqrt{x_{d+1}\, y_{d+1}}\, t_{d+1}) \\
&\quad \times \prod_{j=1}^{d+1} c_{\kappa_j}\, (1-t_j^2)^{\kappa_j-1}\, dt, \notag
\end{align}
where $x_{d+1} = 1-|x|_1$, $y_{d+1} = 1-|y|_1$, $\lambda:= |\kappa|+\frac{d+1}{2}$, and $c_{\kappa_j} = \int_{-1}^1 (1-t_j^2)^{\kappa_j-1}\, dt_i$.
Let us denote the standard Euclidean basis of ${\mathbb R}^d$ by $\{e_1, \ldots, e_d\}$, where
$e_i = (0,\ldots,0,1,0\ldots, 0)$ with the single 1 in the $i$-th position. Furthermore, we
set $e_{d+1} = (0,\ldots, 0) \in {\mathbb R}^d$. Then $\{e_1,e_2,\ldots, e_{d+1}\}$ is the set of
vertices of $T^d$.
\begin{prop}
Let $\lambda= |\kappa|+\frac{d+1}{2}$. For $1\le i \le d+1$, we have
\begin{equation}\label{kernel-s}
K_n(W_{\kappa}; x, e_i) = \frac{1}{2^{d+1}} \, \frac{(\lambda)_n}{(\kappa_i+1/2)_n} \,
P^{(\lambda-\kappa_i-1/2,\kappa_i-1/2)}_{n}(2\, x_i-1).
\end{equation}
In particular, we have
\begin{align}
K_n(W_{\kappa}; e_i, e_i) &= \frac{1}{2^{d+1}} \frac{(\lambda)_n}{n!}
\frac{(\lambda-\kappa_i+1/2)_n}{(\kappa_i+1/2)_n}, \quad 1\le i \le d+1. \label{kernel-i-i} \\
K_n(W_{\kappa}; e_i, e_j) &= \frac{(-1)^n}{2^{d+1}} \frac{(\lambda)_n}{n!}, \quad 1\le i,j \le d+1, \label{kernel-k-i}
\end{align}
\end{prop}
\begin{proof}
Since $C^{\lambda}_{2n}$ is an even function, for $1\le i \le d +1 $ we deduce from
\eqref{KnSimplex} that
\begin{align*}
K_n(W_{\kappa}; x, e_i) =& \frac{c_{\kappa_i}}{2^{d+1}} \, \int_{-1}^1
C^{\lambda}_{2n}(\sqrt{x_i}\,t_i) \, (1-t_i^2)^{\kappa_i-1}\, dt_i\\
=& \frac{c_{\kappa_i}}{2^{d+1}} \, \int_{-1}^1 C^{\lambda}_{2n}(\sqrt{x_i}\,t_i) \, (1-t_i)^{\kappa_i-1}\, (1+t_i)^{\kappa_i}\,dt_i\\
=& \frac{1}{2^{d+1}} \, V^{(\kappa_i)} \, C^{\lambda}_{2n}(\sqrt{x_i}) \\
=& \frac{1}{2^{d+1}} \, \frac{(\lambda)_n}{(\kappa_i+1/2)_n} \, P^{(\lambda-\kappa_i-1/2,\kappa_i-1/2)}_{n}(2\, x_i-1),
\end{align*}
where $V^{(\kappa_i)}$ is the operator defined in \cite[Definition 1.5.1, p. 24]{DX} and the last equality comes from \cite[Proposition 1.5.6, p. 27]{DX}. In particular, setting $x = e_j$ in
\eqref{kernel-s} shows that
$$
K_n(W_{\kappa}; e_i, e_i) = \frac{1}{2^{d+1}} \, \frac{(\lambda)_n}{(\kappa_i+1/2)_n} \,
P^{(\lambda-\kappa_i-1/2,\kappa_i-1/2)}_{n}(1),
$$
and, for $i\neq j$,
$$
K_n(W_{\kappa}; e_j, e_i) = \frac{1}{2^{d+1}} \, \frac{(\lambda)_n}{(\kappa_i+1/2)_n} \,
P^{(\lambda-\kappa_i-1/2,\kappa_i-1/2)}_{n}(-1),
$$
from which \eqref{kernel-i-i} and \eqref{kernel-k-i} follow from \cite[(4.1.1) and (4.1.4)]{Sz}).
\end{proof}
\subsection{Orthogonal polynomials on the simplex with mass points}
We consider orthogonal polynomials on the simplex for the Jacobi measure with additional
mass at each of the vertices of the simplex. In order to preserve symmetry, we shall limit
ourself to the situation that every vertex has the same weight $M>0$. In other words, we
consider the inner product
\begin{equation} \label{simplex-ip}
\langle f, g \rangle_{\nu} = w_{\kappa} \, \int_{T^d} f(x) \, g(x) \, W_{\kappa}(x) \, dx +
M \, \sum_{i=1}^{d+1} f(e_i) \, g(e_i), \qquad M > 0.
\end{equation}
In the language of the inner product \eqref{ipd2}, we assume that $\Lambda$ is a diagonal
matrix $\Lambda = M\, I_{d+1}$ and the inner product take the form of \eqref{sum-mass}.
We will further limit ourself to the case that $\kappa_1 = \kappa_2 = \cdots =
\kappa_{d+1} = \varsigma \ge 0$. Under this assumption,
$$
\lambda = (d+1) (\varsigma + 1/2).
$$
We further denote
\begin{align*}
A_n &:= K_n(W_{\kappa};e_i,e_i) = \frac{1}{2^{d+1}} \frac{(\lambda)_n}{n!} \frac{(\lambda-\varsigma + 1/2)_n}{(\varsigma + 1/2)_n},\\
B_n &:= K_n(W_{\kappa};e_j,e_i) = \frac{(-1)^n}{2^{d+1}} \frac{(\lambda)_n}{n!}, \qquad j\neq i.
\end{align*}
As a result, we see that the matrix ${\mathbf K}_n$ defined in \eqref{cK} is given by
\begin{align*}
{\mathbf{K}_{n}} = \, & (K_{n}(W_{\kappa};e_i,e_j))_{i,j=1}^{d+1} = \begin{pmatrix}
{A_n} & {B_n} & \cdots & {B_n}\\
{B_n} & {A_n} & \cdots & {B_n}\\
\vdots & \vdots & \ddots & \vdots\\
{B_n} & {B_n} & \cdots & {A_n}
\end{pmatrix} \\
= \, & ({A_n}-{B_n})I_{d+1} +
{B_n} \begin{pmatrix} 1 & \cdots & 1\\\vdots & & \vdots\\
1 & \cdots & 1\end{pmatrix}\\
= \, & ({A_n}-{B_n})I_{d+1} +
{B_n} \begin{pmatrix} 1 \\ \vdots \\1\end{pmatrix} (1,\ldots,1).
\end{align*}
This shows that $\mathbf{K}_n$ is a rank one perturbation of the identity matrix and,
consequently, the inverse of the matrix $I_{d+1} + \Lambda \, {\mathbf K}_n$ can be easily
verified to be
\begin{align*}
(I_{d+1} + \Lambda \, \mathbf{K}_n)^{-1} \Lambda = & \,
\frac{M}{[1+M({A_n} - {B_n})][1+M\,{A_n} + d\,M\,{B_n}]} \\
& \times\left[(1+M\,{A_n} + d\,M\,{B_n})I_{d+1} - M\,{B_n}
\begin{pmatrix} 1 & \cdots & 1\\\vdots & & \vdots\\
1 & \cdots & 1\end{pmatrix}\right].
\end{align*}
As a result, we can now use Theorem \ref{main-thm} to derive an explicit orthogonal basis
for the inner product \eqref{simplex-ip}, which is given by
\begin{align*}
\mathbb{Q}_n(x) = \, & \mathbb{P}_n(x) +
\frac{M}{1+M(A_{n-1} - B_{n-1})} \sum_{i=1}^{d+1}
\mathbb{P}_n(e_i)\, K_{n-1}(W_{\kappa};x,e_i)\\
& - \frac{M^2 \,B_{n-1}}{[1+M(A_{n-1} - B_{n-1})][1 +M\,A_{n-1} + d\,M\,{B_{n-1}}]}\times\\
& \quad \times\sum_{i=1}^{d+1} \mathbb{P}_n(e_i) \,
\sum_{i=1}^{d+1} K_{n-1}(W_{\kappa};x,e_i),
\end{align*}
where $\{\mathbb{P}_n\}_{n\ge0}$ denotes the orthonormal polynomial system on
the simplex $T^d$ given by \eqref{simplex-base}. Furthermore, by Theorem \ref{thm.3.3},
the reproducing kernel $K_n(d\nu;x,y)$ for $\Pi_n^d$ under the inner product
\eqref{simplex-ip} is given by
\begin{align}
K_n(d\nu; x,y) = & K_n(W_{\kappa};x,y) +
\frac{M}{1+M({A_n} - {B_n})} \sum_{i=1}^{d+1}
K_n(W_{\kappa};x,e_i)\,K_n(W_{\kappa};y,e_i)\nonumber \label{kernel-simplex}\\
& - \frac{M^2 \,{B_n}}{[1+M({A_n} - {B_n})][1 +M\,{A_n} + d\,M\,{B_n}]}\times\\
& \quad \times\sum_{i=1}^{d+1} K_n(W_{\kappa};x,e_i)
\sum_{i=1}^{d+1} K_n(W_{\kappa};y,e_i). \nonumber
\end{align}
The explicit formula of the kernel allows us to derive a sharp estimate for the
kernel $K_n(d\nu;x,y)$ from those for $K_n(W_{\kappa};x,y)$ and for the Jacobi polynomials.
In the case of one variable ($d=1$), such an estimate has been carried out in \cite{GPRV}.
We shall give one result on the strong asymptotic of the Christoffel function with respect
to $d\nu$ on the simplex $T^d$. For this purpose, we will need the following estimate of
the Jacobi polynomials (\cite[(7.32.5) and (4.1.3)]{Sz}):
\begin{lemma} \label{lem:3.2}
For an arbitrary real number $\alpha$ and $t \in [0,1]$,
\begin{equation} \label{Est-Jacobi}
|P_n^{(\alpha,\beta)} (t)| \le c n^{-1/2} (1-t+n^{-2})^{-(\alpha+1/2)/2}.
\end{equation}
The estimate on $[-1,0]$ follows from the fact that $P_n^{(\alpha,\beta)}
(t) = (-1)^nP_n^{(\beta, \alpha)} (-t)$.
\end{lemma}
Note that \eqref{Est-Jacobi} shows that $|P_n^{({\alpha},{\beta})}| \le c n^{-1/2}$ uniformly inside
a compact subset of $(-1,1)$. We derive the asymptotic for the difference
$K_n(d\nu;x,x) - K_n(W_{\kappa};x,x)$.
\begin{thm} \label{thm:K-K}
For $x$ in $T^d$,
\begin{align} \label{Kn-Kn}
K_n(d\nu; x,x)- & {K}_n(W_{\kappa};x,x) =
\frac{1}{2^{d+1}} \frac{\Gamma(\lambda-\varsigma + 1/2)
\Gamma(\varsigma + 1/2)}{\Gamma(\lambda)} \\
& \times \sum_{i=1}^{d+1}
\left[P^{(\lambda-\varsigma-1/2,\varsigma-1/2)}_{n}(2\, x_i-1)\right]^2 \left (1+ {\mathcal O}(n^{-1})\right). \notag
\end{align}
In particular, for $x$ in the interior of $T^d$,
$$
\lim_{n\to \infty} \left[K_n(d\nu; x,x)- {K}_n(W_{\kappa};x,x) \right] = 0,
$$
and the convergence is uniform in any compact set in the interior of $T^d$.
\end{thm}
\begin{proof}
From \eqref{kernel-simplex} we deduce
\begin{align*}
K_n(d\nu; x,x)- & {K}_n(W_{\kappa};x,x) =
\frac{M}{1+M({A_n} - {B_n})} \sum_{i=1}^{d+1} {K}_n(W_{\kappa};x,e_i)^2 \\
&- \frac{M^2 \,{B_n}}{[1+M({A_n} - {B_n})][1 +M\,{A_n} +
d\,M\,{B_n}]}\times\left(\sum_{i=1}^{d+1}K_n(W_{\kappa};x,e_i)\right)^2\\
=\, & \frac{M C_n^2}{1+M({A_n} - {B_n})} \sum_{i=1}^{d+1}
\left[P^{(\lambda-\varsigma-1/2,\varsigma-1/2)}_{n}(2\, x_i-1)\right]^2 \\
& - \frac{M^2 C_n^2 B_n}{[1+M({A_n} - {B_n})][1 + M\,{A_n} + d\,M\,{B_n}]}\times\\
& \quad \times\left(\sum_{i=1}^{d+1}
P^{(\lambda-\varsigma-1/2,\varsigma-1/2)}_{n}(2\, x_i-1)\right)^2,
\end{align*}
where
$$
C_n = \frac{1}{2^{d+1}} \, \frac{(\lambda)_n}{(\varsigma+1/2)_n}.
$$
By the Stirling formula for the Gamma function (see \cite[(6.1.39), p. 257]{AS}), we have
$$
\frac{ \Gamma(n+a) }{\Gamma(n+1)} = n^{a-1} (1+ {\mathcal O}(n^{-1}))
$$
as $n\to\infty$. Consequently, it is easy to see that the following limit relations hold:
\begin{align*}
\frac{M C_n^2}{1+M({A_n} - {B_n})} = \, &
\frac{1}{2^{d+1}} \frac{\Gamma(\lambda-\varsigma + 1/2)
\Gamma(\varsigma + 1/2)}{\Gamma(\lambda)} \left (1+ {\mathcal O}(n^{-1})\right), \\
\frac{M B_n}{1+ M\,{A_n} + d\,M\,{B_n}} = \, & (-1)^n \frac{\Gamma(\lambda-\varsigma + 1/2)}
{\Gamma(\varsigma + 1/2)} n^{-({\lambda} - 2 \varsigma -1)} \left (1+ {\mathcal O}(n^{-1})\right).
\end{align*}
Since ${\lambda} - 2 \varsigma -1 = (d-1)(\varsigma +1/2) > 0$ for $d \ge 2$ and
$$
\left(\sum_{i=1}^{d+1}
P^{(\lambda-\varsigma-1/2,\varsigma-1/2)}_{n}(2\, x_i-1)\right)^2 \le (d+1)
\sum_{i=1}^{d+1} \left[P^{(\lambda-\varsigma-1/2,\varsigma-1/2)}_{n}(2\, x_i-1)\right]^2
$$
by the Cauchy--Schwarz inequality, it follows readily that
\begin{equation*}
K_n(d\nu; x,x) - K_n(W_{\kappa};x,x) = c_{{\lambda},\varsigma} \sum_{i=1}^{d+1}
\left[P^{(\lambda-\varsigma-1/2,\varsigma-1/2)}_{n}(2\, x_i-1)\right]^2 \left (1+ {\mathcal O}(n^{-1})\right),
\end{equation*}
where $c_{{\lambda},\varsigma}$ is the constant
$$
c_{{\lambda},\varsigma}= \frac{1}{2^{d+1}} \frac{\Gamma(\lambda-\varsigma + 1/2)
\Gamma(\varsigma + 1/2)}{\Gamma(\lambda)}.
$$
This is \eqref{Kn-Kn}. If $x$ is in the interior of $T^d$, then
$|P_n^{(\lambda-\varsigma-1/2,\varsigma-1/2)}(2\, x_i-1)| \le c n^{-1/2}$, so that
$K_n(d\nu; x,x) - K_n(W_{\kappa};x,x)$ goes to zero as $n \to \infty$.
\end{proof}
The asymptotic of the Christoffel function for $W_{\kappa}$ was studied in \cite{X99}, where
most of the results were for convergence in the interior of $T^d$. Such results carry over
to $K_n(d\nu; x,y)$ by Theorem \ref{thm:K-K}. In one particular case, $\kappa =0$, the
convergence holds for all $T^d$ as given in \cite[Theorem 2.3]{X99}:
\begin{equation}\label{limitK}
\lim_{n \to \infty} \frac{1}{\binom{n+d}{n}} K_n(W_0;x,x) = 2^{d- k}, \qquad x \in T_k^d, \quad
0 \le k \le d,
\end{equation}
where $T_k^d$ denotes the {\it $k$--dimensional face} of $T^d$, which contains elements
of $T^d$ for which exactly $d-k$ inequalities in $T^d=\{x: x_1\ge 0, \ldots, x_{d+1} \ge 0\}$
becomes equalities. In particular, $T_d^d$ (when none of the inequalities become equality)
is the interior of $T^d$, and $0$--dimensional face $T_0^d$ is the set of the vertices.
Setting $\kappa =0$, so that $\varsigma =0$ and ${\lambda} = (d+1)/2$, we see that
$$
P_n^{(\lambda-\varsigma-1/2,\varsigma-1/2)}(2\, x_i-1) = P_n^{(d/2, -1/2)}(2\, x_i-1)
= (-1)^n P_n^{(-1/2, d/2)}(1-2\, x_i),
$$
which is bounded by $c n^{-1/2}$ whenever $1- x_i \ge \varepsilon >0$. It follows then that
$$
\lim_{n\to \infty} \frac{1}{\binom{n+d}{n}}
\sum_{i=1}^{d+1} \left[P_n^{(d/2, -1/2)}(2\, x_i-1) \right]^2
= \begin{cases} 0 & x \in T_k^d, \,\, k >0 \\ 1 & x \in T_0^d, \end{cases}
$$
upon using the fact that $P_n^{(a,b)}(1) = \binom{n+a}{n}$.
By \eqref{Kn-Kn}, we then end up with the following corollary.
\begin{cor}
For ${\kappa} = 0$,
$$
\lim_{n \to \infty} \frac{1}{\binom{n+d}{n}} K_n(d\nu;x,x) = \begin{cases}
2^{d- k}, & x \in T_k^d, \,\, k > 0, \\
2^d + E_d, & x \in T_0^d, \end{cases}
$$
where $E_d = c_{(d+1)/2,0} = \Gamma(d/2+1) \sqrt{\pi} /(\Gamma(d+1/2) 2^{d+1})$.
\end{cor}
Comparing to \eqref{limitK}, the result shows the impact of the additional mass points at
the vertices. More generally, if $\kappa_i = \varsigma > 0$, then \eqref{Kn-Kn} shows that
\begin{align*}
K_n(d\nu;x,x) - K_n(W_{\kappa};x,x) = & c_{{\lambda},\varsigma} \left (1 + {\mathcal O}(n^{-1}) \right) \\
& \times \begin{cases} \binom{n + {\lambda} - \varsigma -1/2}{n}, & x \in T_0^d, \\
2^k \binom{n + \varsigma -1/2}{n}, & x \in T_k^d, \,\,1\le k \le d-1, \end{cases}
\end{align*}
since, for $x\in T^d$, $x_i=1$ only when $x = e_i$. In particular, we see that
$$
\lim_{n \to \infty} \frac{1}{\binom{n+d}{n}} \left[K_n(d\nu;x,x) - K_n(W_{\kappa};x,x) \right] = 0,
\quad x \in T_k^d, \quad 1 \le k \le d,
$$
if $d > 2 \varsigma -1$, whereas this limit is unbounded when $x \in T_0^d$.
| {'timestamp': '2009-11-15T01:03:30', 'yymm': '0911', 'arxiv_id': '0911.2818', 'language': 'en', 'url': 'https://arxiv.org/abs/0911.2818'} |
\section{Introduction}
\label{sec:Intro} During the last decade, Online
Social Networks (OSNs) have successfully attracted billions of
people who share a huge amount of personal information through the Internet,
such as their background, preferences and social connections. Owing to the increase of potential violations such as advertising
spam, online stalking and identity theft~\cite{Gross2005}, in recent
years, more and more users have concerns about their \textit{privacy}
in OSNs and become reluctant to publish all their personal
information~\cite{FacebookTrend_PV}. Consequently, users may not fill out their privacy-sensitive attributes (e.g., location, age, or phone number), or they hide this information from strangers and only allow their friends to view such information~\cite{Chen2013661}.
While hiding the privacy-sensitive attributes, users usually expose
some other information that appears to be less sensitive to them. It
has been reported that Facebook users publicly reveal four
attributes on average, and $63\%$ of them uncover their friends list
\cite{analysis_public_info}. Due to the correlations among various
attributes, some of the self-exposed information may indicate the
invisible privacy-sensitive attributes to some extent~\cite{CCP}\cite{ICDMW2012}. Hence, it is questionable whether the privacy-sensitive attributes that a user intends to hide are really hidden.
This work, using location information as a representative case, aims to assess what is the risk that a user's invisible information could be disclosed. There are several reasons that lead us to conduct this study based on location information. First, among various kinds of information, location is
usually one of the privacy-sensitive attributes for most users~\cite{chakraborty2013privacy}. In
real-life OSNs, we notice that users are quite careful to not reveal
their location information: $16\%$ of users in Twitter reveal home
city \cite{MLP} and $0.6\%$ of Facebook users publish home
address~\cite{find_me}. Second, location
information is a commercially valuable attribute which might even be misused by unscrupulous businesses to bombard a user with unsolicited marketing~\cite{duckham2006location}.
In addition, location information leakage may lead to a spectrum of intrusive inferences such as inferring a user's political view or personal preference~\cite{duckham2006location}\cite{han2015alike}.
Therefore, protecting the hidden location information for a user
becomes rather critical. In particular, as Facebook is the most
popular OSN~\cite{SM_report}, we concentrate on the attribute of
\textit{current city} in Facebook and investigate the following
issues:
\vspace{0.1cm} 1) \textit{Is the private current city that a user
expects to hide really hidden? In other words, if a user hides
his current city but exposes some other information, can we predict
a user's current city by using his self-exposed information?}
\vspace{0.15cm} 2) \textit{Can we help individual users
to understand the actual risk (probability) that their private current
city could be correctly predicted based on their self-exposed
information? Furthermore, can we provide some countermeasures to
increase the security of the hidden current city?} \vspace{0.1cm}
To address these issues, we first propose a current
city prediction approach to predict users' hidden current city.
Although many location prediction approaches have been developed for
Twitter~\cite{T_content_prediction}\cite{geo_twitter}\cite{insights}\cite{www2014}
and Foursquare~ \cite{ICDMW2012}\cite{foursquare_behavior}, they
cannot be appropriately implemented on Facebook because of the
different properties (e.g., obtainable information) in these OSNs.
For Facebook, Backstrom et al. predict users' locations based on
their friends' locations~\cite{find_me}. In addition to friends' locations,
users' profile attributes, such as hometown, school and workplace,
may also indicate their current city to some extent~\cite{CCP}. In
order to achieve high prediction accuracy in Facebook, we devise a
novel current city prediction approach by extracting location
indications from integrated self-exposed information including
profile attributes and friends list.
Second, based on the proposed prediction approach, we construct a
current city exposure estimator to estimate the exposure probability
that a user's invisible current city may be correctly inferred via
his self-exposed information. The exposure estimator can also
provide a user with some countermeasures to keep his hidden current
city hidden. To the best of our knowledge, this is the first
work that estimates the exposure probability of a user's invisible attribute by his self-exposed information.
It is a non-trivial task to construct either the current city
prediction approach or the exposure estimator. We encounter the
following challenges:
1) \textbf{\textit{How to extract and integrate different location
indications from a user's multiple self-exposed information?}} Since
the proposed prediction approach explores location
indications from both profile attributes and friends list, two
subproblems are considered. $(i)$~A user probably reveals multiple
attributes (e.g., hometown, workplace) which may indicate different
locations; besides, a certain attribute might indicate several
locations. For example, a user working in \textsc{Google} suggests
that the user could probably live in any city where \textsc{Google}
sets up an office, e.g., \textsc{California}, \textsc{Beijing} or
\textsc{Paris}. $(ii)$~The friends of a user, probably residing in
different cities, may be close to or far away from the user. These
strong or weak geographic relations may influence the significance
of the friends' location indications. Thus, it is challenging to
appropriately combine these various location indications into an
integrated model, so as to determine the probabilities of locations
where the user may live.\color{black}
2) \textbf{\textit{How to predict a user's current city when we
obtain the probabilities of the user being at various locations?}}
By overcoming \textit{challenge 1}, we can obtain a probability
vector which indicates the probabilities that a user resides at
certain locations. With this probability vector, a straight-forward prediction approach could select the location with the highest probability as the user's current city. However, this might not be the best option
when concerning the locations' geographic relations. Assume the
probability vector suggests that a user $u$ has $40\%$, $35\%$ and
$25\%$ probability of residing in \textsc{Beijing}, \textsc{Paris}
and \textsc{Evry} respectively. Then, $u$ is more likely to live in
the area around \textsc{Paris} and \textsc{Evry} than
\textsc{Beijing}, because \textsc{Paris} and \textsc{Evry} are only
$30km$ apart but they are thousands of kilometers away from
\textsc{Beijing}. Hence, a location selection method should be
carefully designed for a current city prediction approach.
3) \textbf{\textit{How to estimate the exposure risk of a user's
hidden current city?}} To help a user understand the exposure risk
of his hidden current city, a straight-forward method would be to provide a predicted location; thus the user can decide whether
his current city can be predicted correctly (risky) or incorrectly
(secure). However, this method may not meet users' expectations. A user, whose location is correctly predicted, may expect being able to know which of his self-exposed information primarily leads to the leakage of his private current city and how to increase its security. A user, whose hidden location is not predicted correctly, still needs to be aware of some leakage of location that may exist. For example, a
prediction approach may incorrectly infer a Parisian living in
\textsc{Lyon} according to probabilistic results: $55\%$ in
\textsc{Lyon} and $45\%$ in \textsc{Paris}; Even though the
prediction result is incorrect, the user still leaks some location
information. Therefore, how to estimate the current city exposure
risk and help a user achieve his desired privacy level is a challenging
objective.
This paper makes the following contributions:
1) \textbf{Profile and friend location indication model:} To
properly extract location indications from users' self-exposed
information, we construct an integrated probability model. We
capture location indications from two types of information:
\textit{location sensitive attributes} and \textit{friends list}.
Location sensitive attributes are the profile attributes that can
indicate one or multiple locations. In this paper, we use `Hometown'
and `Work and Education' as the location sensitive attributes. For
each location sensitive attribute, we set up a \textit{location
attribute indication matrix} from which we can index the locations
and the corresponding probabilities that a certain attribute value
indicates. Besides, considering a user and each of his friends who
publish current city, we estimate their location similarity
according to their attribute correlations, and assign a large weight
to a friend that has a high location similarity to the user. For
a friend who does not reveal current city, we predict the friend's current
city using his visible location sensitive attributes, and assign him a
very small weight. Finally, based on information from $371,913$ users collected from Facebook, we train an integrated model that can determine the probability for each potential city where a user may
reside.
2) \textbf{Current city prediction approach:} To address
\textit{Challenge 2}, we aggregate locations into clusters by
considering the locations' geographic relations. Then, based on the
proposed \textit{profile and friend location indication model}, we
predict a user's invisible current city in two steps: $(i)$
\textit{cluster-selection}: for each cluster, we sum up the probabilities of
locations inside the cluster; then we select the cluster with the
highest probability; $(ii)$ \textit{location-selection}: we determine a best
location within the selected cluster as the user's current city. The
evaluation results demonstrate that our proposed prediction approach
achieves lower error distance and higher accuracy than the
state-of-the-art approaches. Furthermore, for the users who reveal
their `Hometown' and `Work and Education', our proposed approach can
predict current city with an accuracy of $90\%$.
3) \textbf{Current city exposure estimator:} We define some
measurements to describe the characteristics of users' self-exposed
information.
Based on these measurements, we analyze how the users' self-exposed
information affects the probability that their current city may be
correctly inferred (i.e., current city exposure probability).
Furthermore, \textit{Random Decision Forest} method is employed to model the current city
exposure probability, and subsequently a current city exposure
estimator is constructed. Given a user's self-exposed information, the proposed
exposure estimator provides two estimators
--- \textit{Exposure Probability} and \textit{Risk Level} --- to
quantify the current city exposure risk. The exposure estimator can
also estimate the exposure risk assuming that the user hides some of
his self-exposed information. Consequently, the user can easily
decide which information he should hide to satisfy his privacy
intention.
The rest of this paper is organized as follows. We review the
literature in Sec.~\ref{sec:Literature}, formulate the current city
prediction problem in Sec.~\ref{sec:problem}, and overview our
solution to the prediction problem in Sec.~\ref{sec:overview}. Next, the profile and friend location indication model is devised in Sec.~\ref{sec:model}; the current city prediction approach is respectively presented and evaluated in Sec.~\ref{sec:approach} and Sec.~\ref{sec:Eva}. By inspecting the current city prediction results, Sec.~\ref{sec:exposure} proposes the exposure estimator. Finally, Sec.\ref{sec:Discussion} makes some discussions and points out future work. Sec.~\ref{sec:conclusion} concludes this work.
\section{Literature Review}
\label{sec:Literature} In this section, we briefly review the
related work from two perspectives: city-level
location prediction and privacy in OSNs.
\subsection{City-Level Location Prediction}
Existing city-level location prediction approaches can be classified into
four categories: \textit{relationship-based} prediction,
\textit{content-based} prediction, \textit{hybrid content-relationship} prediction and \textit{multi-indication} prediction.
\subsubsection{Relationship-based Prediction}
Based on the principle that the probability of being friends is
declining with geographic distance, this prediction category infers
a user's location according to the visible locations of his friends~\cite{find_me}. Researchers have studied the correlation between
geographic distance and social relationship on large-scale Facebook
users in United States. They reveal that the probability of being
friends falls down monotonically as the distance increases. Depending
on this observation, they build a maximum-likelihood location
prediction model and finally refine the prediction with an
iterative algorithm.
\subsubsection{Content-based Prediction}
The rise of Twitter has spawned a mass of tweets. As some tweets
contain location-specific data, this category of prediction
approaches~\cite{T_content_prediction}\cite{geo_twitter}\cite{insights}
infers a user's location relying on his location-related tweets. The
basic idea of these approaches is to detect the location-related
tweets and construct a probabilistic model to estimate the
distribution of location-related words used in tweets. In order to
raise the prediction accuracy, the basic idea is improved by various means, such as such as selecting the top K probable
cities~\cite{T_content_prediction}, identifying words with a strong
local geo-scope and refining the prediction with a neighborhood
smoothing model~\cite{geo_twitter}.
\subsubsection{Hybrid Content-Relationship Prediction}
Another compelling category combines the location indications from
relationships and tweet content. TweetHood identifies a user's
location by exploring both his tweets and his closest friends'
locations~\cite{TweetHood}. Tweecalization improves TweetHood by
employing a semi-supervised learning algorithm and introducing a new
measurement which combines trustworthiness and the number of common
friends to weight friends~\cite{Tweecalization}. Li et al. integrate
the location influences captured from both social network and
user-centric tweets into a unified discriminative probabilistic
model~\cite{UID}. By considering a user who may be related to multiple
locations, MLP model~\cite{MLP} proposes to set up a complete
`location profiles' prediction which infers not only a user's home
location but also his other related locations.
\subsubsection{Multi-Indication Prediction}
Besides users' relationships and content, multi-indication
prediction approaches explore multiple location indications from
other possible location resources to infer users' invisible
location. To resolve ambiguous toponymies in tweet content, besides
location indications extracted from tweets, existing work has
introduced location indications from websites' country code,
geocoded IP addresses, time zone and UTC24-offset
\cite{Multi_Indicator}. Such a multi-indication idea has also been
used to Foursquare, which specifically exploits mayorships, tips and
dones that users marked \cite{foursquare_behavior}. However, all
these multi-indication prediction approaches are proposed for either Twitter
or Foursquare, but not for Facebook. Our previous work reveals the
statistical analyzed correlation between users' current city and
other location sensitive attributes in Facebook~\cite{CCP}. It also
predicts a user's current city with city-level and country-level
results by using a neural network approach. However, this previous
work assumed that an attribute value could map to a specific
location, which is not true for many cases. Recall the example that
a user works in \textsc{Google} might work in \textsc{California},
\textsc{Paris} or \textsc{Beijing} (\textit{Challenge 1} of Sec.~\ref{sec:Intro}).
In this paper, we consider multiple location indications by
integrating relationship and profile attributes in Facebook.
Compared to our previous work where an attribute value only allows
to bind with one fixed location~\cite{CCP}, an attribute value can
be mapped to multiple locations with different probabilities in the
newly proposed model. In addition, we consider both the friends
whose current city is either visible or invisible; whereas the
existing work relies on the friends who reveal their
locations~\cite{find_me}\cite{UID}. Particularly, we propose a new
approach to bias the weights of friends whose current city is
visible.
\subsection{Privacy in OSNs}
In OSNs, users are more and more concerned with privacy of their
personal information~\cite{FacebookTrend_PV}. A majority of users
configure their privacy settings and hide some of their information
from strangers. Unfortunately, previous research has pointed out the
disparity between the expectation and the reality of users' privacy;
and it has also showed that much of users' private information is easily uncovered~\cite{fb_privacy}.
Much existing work ascribes the privacy leakage to the users themselves.
On one hand, users might incorrectly manage their privacy settings
due to the poor human-computer interaction or complex privacy
maintainability~\cite{fb_privacy}\cite{chakraborty2013privacy}. To address this issue, researchers have
designed a user-friendly interface for managing privacy settings
with an audience view~\cite{design_interface}.
On the other hand, users only hide some of the attributes that are
privacy-sensitive to them while make the others accessible to
public --- users on Facebook generally expose more than four attributes to
strangers and $63\%$ of users share their friend lists with the
public \cite{analysis_public_info}. As reported, such user
self-exposure behavior leaves a huge chance for inferring the
hidden attributes
\cite{age_estimate}\cite{infer_prof}\cite{Infer_PV}. Many tools have
been developed to infer users' invisible information by various
means such as inferring the private information through users' other
self-exposed information \cite{infer_prof}, their social connections
\cite{age_estimate}\cite{Infer_PV} and social
groups~\cite{Illusion_PV}\cite{User_Prof}.
Some papers claim that it is hard for a user to avoid privacy
leakages if he only hides the private attribute
~\cite{age_estimate}\cite{infer_prof}\cite{Strategies_struggles};
whereas many studies merely suggest users with a general idea of
hiding other attributes so as to become more secure (e.g., hide
relationships~\cite{protect_privacy}). Unlike the above
work, we provide an individual user with the exposure probability of
his private current city concerning his self-exposed information.
We also suggest some pointed rules for protecting users'
privacy on their current city.
\section{Formulation of Current City Prediction Problem}
\label{sec:problem} In this section, we formulate the current city
prediction problem. Facebook, as a social network containing
location information, can be viewed as an undirected graph
$\mathcal{G}=(\mathcal{U},\mathcal{E},\mathcal{L})$, where
$\mathcal{U}$ is a set of users; $\mathcal{E}$ is a set of edges
$e\langle u, v \rangle$ representing the friend relationship between
users $u$ and $v$, where $u$ and $v \in \mathcal{U}$; $\mathcal{L}$
is a candidate locations list composed of all the user-generated
locations.
Typically, a user $u$ in Facebook might contribute various items of information, e.g., basic profile information, friends, comments and photos. The core information of $u$ in this paper is the user's current
city, denoted as $l(u)$. The users are classified into two sets according to the accessibility of users'
current city: current city
available users (LA-users) and current city unavailable users
(LN-users). We, respectively, use $\mathcal{U}^{^{LA}}$ and
$\mathcal{U}^{^{LN}}$ to denote the sets of LA-users and LN-users,
where $\mathcal{U}=\mathcal{U}^{^{LA}} \cup \mathcal{U}^{^{LN}}$.
To predict users' current city, we exploit the users'
location sensitive attributes and friends list. Assume that there
exist $m$ types of location sensitive attributes, denoted as
$\mathcal{A} = \{a_1, a_2, \cdots, a_m\}$. Specifically, we denote a
user $u$'s location sensitive attributes as $\mathcal{A}(u) =
\{a_1(u), a_2(u), \cdots, a_m(u)\}$. The users may also have a
friends list, denoted as $\mathcal{F}(u)$, where $\mathcal{F}(u) =
\{f \in \mathcal{U}: e \langle u, f \rangle \in
\mathcal{E}\}$. Therefore, we use a tuple to represent a user as $u:
\langle l(u), \mathcal{A}(u), \mathcal{F}(u) \rangle$.
Additionally, each location is associated with a unified \textit{ID} ($l_{id}$). Then,
with this \textit{ID}, we can obtain each location's latitude and longitude
coordinate via Facebook Graph API Explorer. Therefore, a
location can also be written as a tuple: $l : \langle l_{id}, lat, lon
\rangle$ and the candidate locations list can be denoted as a set of
location tuples: $\mathcal{L} = \{l : \langle l_{id}, lat, lon
\rangle \}_N$, where $lat$ and $lon$ respectively stand for the
latitude and longitude of a location, and $N$ is the number of
candidate locations in the list.
Thus, the \textbf{\textit{current city prediction problem}} can be formally
stated as: \textit{Given, (i)~a graph $\mathcal{G}=(\mathcal{U}^{^{LA}}\cup \mathcal{U}^{^{LN}},\mathcal{E},\mathcal{L})$; (ii)~the public location $l(u)$ for LA-users $u \in U^{^{LA}}$; (iii)~the location sensitive attributes $\mathcal{A}(u)$ and the friends list $\mathcal{F}(u)$ for all the users $u \in (\mathcal{U}^{^{LA}} \cup \mathcal{U}^{^{LN}})$, we predict current city $\hat{l}(u)$ for each LN-user $u \in \mathcal{U}^{^{LN}}$, so as to make $\hat{l}(u)$ close to the user's real current city.}
Note that the current city of a user's friends can be either
available ($f \in \mathcal{U}^{^{LA}}$) or unavailable ($f \in \mathcal{U}^{^{LN}}$).
Thus, we introduce two notations to represent the two groups of
friends: current city available friends (LA-friends) and current
city unavailable friends (LN-friends). Let denote a user's
LA-friends as $\mathcal{F}^{^{LA}}(u)$ and LN-friends as
$\mathcal{F}^{^{LN}}(u)$, where $\mathcal{F}(u)
=\mathcal{F}^{^{LA}}(u) \cup \mathcal{F}^{^{LN}}(u)$.
\section{Overview of Current City Prediction}
\label{sec:overview} The goal of current city prediction is to correctly infer a coordinate point with latitude and longitude for a LN-user, given the candidate locations list $\mathcal{L}$ and the
user's self-exposed information including his location sensitive attributes and friends list.
Figure~\ref{fig:overview} illustrates the framework of the proposed current city prediction solution. To determine the current city of a LN-user, we first train an integrated profile and friend location indication (i.e., \textit{PFLI}) model to compute the probabilities of the candidate locations in which the LN-user may currently live. Next we take a two-step location selection strategy: cluster selection and location selection. Specifically, we aggregate the nearby locations into a location cluster and obtain a set of location clusters. We then calculate the probability of a user being in a cluster by summing up the probabilities of all the candidate locations belonging to this cluster; the cluster with the
highest probability is picked out as a candidate cluster. Finally, we try to select the `best' location from the candidate cluster as the predicted current city.
\begin{figure}[!htp]
\centering
\includegraphics[width=9cm, height=5.8cm]{./overview20150620.eps}
\caption{Framework of Current City Prediction.}
\label{fig:overview}
\vspace{-0.5cm}
\end{figure}
To train the integrated \textit{PFLI} model (see the right-hand part of Figure~\ref{fig:overview}), we separately consider the location indications from location sensitive attributes and friends, and consequently obtain two sub-models: profile location indication (\textit{PLI}) model and friend location indication (\textit{FLI}) model. Both \textit{PLI} model and \textit{FLI} model calculate a probability vector in which the element stands for the probability of a user being at a certain candidate location. Note that, \textit{FLI} model leverages the location indications from both LA-friends and LN-friends. By integrating the probability vectors that are generated by \textit{PLI} and \textit{FLI} models with appropriate parameters, a unified profile and friend location indication (\textit{PFLI}) model is derived.
Next, we will elaborate the \textit{PFLI} model and the current city prediction approach.
\section{Profile and Friend Location Indication Model}
\label{sec:model} In this section, we describe the design of the
probabilistic models that can suggest the probabilities of users
being at each of the candidate locations. We first introduce the
\textit{profile location indication} (\textit{PLI}) model; it
estimates the probability of each candidate location by merely relying on a user's location
sensitive attributes. Then, we describe the \textit{friend location
indication} (\textit{FLI}) model, which captures the location
indications from a user's friends. Finally, we integrate these
two models and obtain the integrated \textit{profile and
friend location indication} (\textit{PFLI}) model.
\subsection{Profile Location Indication Model}
\label{sec:PLI} According to \textit{Challenge 1} in Sec.
\ref{sec:Intro}, two problems should be considered in constructing
\textit{PLI} model. First, a certain value of a location sensitive
attribute may indicate several locations. For instance, \textsc{Google}, being a certain value of workplace, could indicate any city where \textsc{Google} sets up an office such as \textsc{California}, \textsc{Beijing} or \textsc{Paris}. Therefore, for each attribute value, we consider all possible location
indications with the corresponding probabilities. Second, a user may present multiple
location sensitive attributes (e.g., hometown, workplace, college). Thus we integrate
various location indications extracted from different location
sensitive attributes.
To capture the multiple possible location indications from
one attribute value, we define a \textit{location-attribute
indication matrix} for each ($k$-th) location sensitive attribute
$a_k \in \mathcal{A}$, denoted as $\mathcal{R}_{k}$. The rows of
this matrix represent the candidate locations ($l \in \mathcal{L}$),
while the columns stand for the possible values of $a_k$. We use
$l_i$ to represent the $i$-th candidate location and $a_{k_j}$ to
denote the $j$-th possible value of $a_k$. A cell
$\sigma_{k}^{ij}$ in the matrix calculates the \textit{indication
probability} of $a_{k_j}$ to $l_i$ --- the probability that a user,
whose $k$-th location sensitive attribute $a_k$ equals $a_{k_j}$,
currently lives in the city $l_i$. Specifically, the indication
probability equals the number of users who live in $l_i$ and have a
value of $a_{k_j}$ divided by the total number of users who have a
value of $a_{k_j}$. For instance, considering workplace, if $10$ out
of $100$ employees from \textsc{Telecom SudParis} in the whole data set state that they
live in \textsc{Evry}, then the indication
probability of \textsc{Telecom SudParis} to \textsc{Evry} is $0.1$.
Note that, the $j$-th column of $\mathcal{R}_{k}$ represents the
multiple location indications of $a_{k_j}$.
Assume that $a_k$ has $M$ possible values except \textit{null}; $N$
is the total number of the candidate locations. The $k$-th
location-attribute indication matrix can be written as:
\begin{align*}
\mathcal{R}_{k}=\{\sigma_{k}^{ij}\}_{N \times
M}= \{p(l(u)=l_i | a_k(u)=a_{k_j})\}_{N
\times M}=[R_{\cdot
k_1}, R_{\cdot k_2},...,R_{\cdot k_M}]
\end{align*}
where $R_{\cdot k_j}$ represents all the locations' probabilities for a user who presents $a_{k_j}$.
Based on the location-attribute indication matrix ($\mathcal{R}$),
we model the probability of a user's current city at $l_i$ by
combining all of a user's available location sensitive attributes in
his profile:
\begin{equation}
\begin{split}
\label{eq:PLM1}
p_{_{Prof}}(u,l_i) &= \sum_{a_k \in \mathcal{A}, a_{k}(u) \neq \textit{null}}{\alpha_k p(l(u)=l_i | a_{k}(u)=a_{k_j})} \\
&= \sum_{a_k \in \mathcal{A},a_{k}(u) \neq \textit{null}}{\alpha_k
\sigma_{k}(u,l_i)}
\end{split}
\end{equation}
where $\sigma_{k}(u,l_i)$ can be easily obtained by indexing the
corresponding location-attribute indication matrix
($\mathcal{R}_{k}$) according to $u$'s value of $a_k$
($a_k(u)=a_{k_j}$) and the given location ($l_i$), namely
$\sigma_{k}^{ij}$; $\alpha_k$ is a parameter to adjust the
significance of the different location sensitive attributes.
As we discussed in Sec. \ref{sec:problem}, a user may not reveal some attributes. Therefore, in Eq.
\ref{eq:PLM1}, the location indication from the attribute $a_k(u)$ at any location equals zero if the user's $a_k(u)$ is invisible. If all of a user's
location sensitive attributes are invisible, we rely on
his friends' information to infer his current city,
which we will discuss in the next section.
\subsection{Friend Location Indication Model}
In addition to a user's location sensitive attributes,
we explore location indications from users' friends to construct \textit{FLI} model. A user's friends can be either LA-friends (current city available) or LN-friends (current city unavailable). We build up \textit{FLI} model primarily depending on LA-friends' location indications and also considering LN-friends' location indications as a small regulator. Accordingly, \textit{FLI} model contains two components: LA-friends location indication (\textit{LA-FLI}) model and LN-friends location indication (\textit{LN-FLI}) model.
\subsubsection{LA-FLI Model} \textit{LA-FLI} model differentiates the weights of a user's LA-friends and estimates his probability of living in location $l_i$ by the weights of his friends who also live in $l_i$. \textit{LA-FLI} model expects to assign high weights to the LA-friends who live in the same city as the user does. However, since the user's city is unknown, whether or not a friend and the user live in the same city cannot be directly determined. Therefore, \textit{LA-FLI} model assesses the likelihood that two users live in a same city (i.e., \textit{location similarity}) based on the correlation between their location sensitive attributes.
\begin{figure}[!htp]
\centering
\includegraphics[width=6.5cm, height=3.5cm]{./example.eps}
\caption{An Example of Social Relations and Profile Information.}
\label{fig:example
\end{figure}
Figure \ref{fig:example} illustrates an example to show that
the location sensitive attributes can be used to distinguish the
weights among various LA-friends. Focusing on LN-user $u_2$ and his
LA-friends $u_3$, $u_4$ and $u_5$, we notice that $u_2$ and $u_3$,
$u_4$ work in the same institute, while $u_5$ works in another
company which is far away from $u_2$'s workplace. In this case, it
is natural to infer that $u_2$ is more likely to be living in the
same city with $u_3$ and $u_4$ than with $u_5$; then $u_3$ and $u_4$
should be assigned with higher weights than $u_5$ because of the
location similarity indicated by their workplace.
Inspired by the example, we construct an \textit{attribute-based
location similarity matrix} ($\mathcal{W}_{k}$) by each ($k$-th) location
sensitive attribute ($a_k \in \mathcal{A}$). In the matrix, a cell $w_{k}^{ij}$ calculates the probability that two users live in the same city (i.e., \textit{location similarity}) when they respectively have values of $a_{k_i}$ and $a_{k_j}$ regarding $a_k$.
Specifically, we compute the total number of friend pairs
where one user has a value of $a_{k_i}$ and the other has a value of
$a_{k_j}$, denoted as $|\{a_k(u)=a_{k_i} \wedge a_k(v)=a_{k_j}\}|$;
Among these friend pairs, we further count the pairs of friends who live in the same city, denoted as $|\{l(u)=l(v) \wedge a_k(u)=a_{k_i} \wedge a_k(v)=a_{k_j}\}|$. Then,
\begin{align*}
\mathcal{W}_{k} &= \{w_{k}^{ij}\}_{M \times M}\\
&= \{p(l(u)=l(v) | a_k(u)=a_{k_i} \wedge
a_k(v)=a_{k_j})\}_{M \times M}\\
&= \{\frac{|\{l(u)=l(v) \wedge a_k(u)=a_{k_i} \wedge a_k(v)=a_{k_j}\}|}{|\{a_k(u)=a_{k_i} \wedge a_k(v)=a_{k_j}\}|}\}_{M \times M}
\end{align*}
where $M$ is the number of possible values of attribute $a_k$ including $null$.
For a certain attribute $a_k$, assume that $u$ and his LA-friend $v$
have a value of $a_{k_i}$ and $a_{k_j}$ respectively. Then, the
$u$ and $v$'s location similarity on $a_k$ can be
easily obtained by indexing the $i$-th row and $j$-th column of
$\mathcal{W}_{k}$, denoted as $w_{k}(u,v) = w_{k}^{ij}, v \in
\mathcal{F}^{^{LA}}(u)$
We combine multiple location similarities on all the location sensitive
attributes (e.g., work, hometown) with a set of trained parameters ($\mathbf{\beta}$) to
measure $v$'s weight. This combined weight describes the probability that $u$ and $v$ live in the same city concerning all of their location sensitive
attributes.
Then, \textit{LA-FLI} model calculates the probability of $u$ living
in $l_i$ by integrating all the weights of $u$'s LA-friends who live
in $l_i$:
\begin{equation}
\label{eq:LA-FLI} p_{_{LA-F}}(u,l_i)=\sum_{v \in
\mathcal{F}^{^{LA}}(u)}\sum_{a_k \in
\mathcal{A}}{\beta_k{w_{k}(u,v)p_{_{LA-U}}(v,l_i)}}
\end{equation}
where $p_{_{LA-U}}(v,l_i)$ represents whether or not the LA-friend
$v$ living in $l_i$. It equals $1$ if $v$ states his current city is
$l_i$; otherwise, it is $0$:
$$
p_{_{LA-U}}(v,l_i) =
\begin{cases}
1 & \textit{if } l(v) = l_i \\
0 & \textit{otherwise}
\end{cases}
$$
\subsubsection{LN-FLI Model}
Before introducing \textit{LN-FLI} model, we inspect the potential benefit of
a user's LN-friends for his current city prediction with another
example shown in Figure~\ref{fig:example}. We observe that $u_2$,
being a LN-friend of $u_1$, does not expose his current city;
whereas, the workplace of $u_2$, \textsc{Telecom SudParis},
indicates two cities
--- \textsc{Paris} and \textsc{Evry}
--- according to the current cities of the users $u_3$ and $u_4$ who
are also the employees of \textsc{Telecom SudParis}. Thereby, a
user's LN-friends can also reveal some location indications in their
exposed attributes, which may help the prediction.
Therefore, for a LN-friend $v$, we first rely on his exposed
location sensitive attributes and use \textit{PLI} model (Sec.
\ref{sec:PLI}) to predict his current city, as:
$$
p_{_{Prof}}(v,l_i) = \sum_{a_k \in \mathcal{A},a_k(v) \neq
\textit{null}}{\alpha_k}p(l(v)=l_i | a_k(v)=a_{k_j})
$$
Treating all the LN-friends equally, \textit{LN-FLI} model integrates
LN-friends' location indications and computes the probability that
$u$ lives in $l_i \in \mathcal{L}$ as:
\begin{equation}
\label{eq:LN-FLI} p_{_{LN-F}}(u,l_i)=\sum_{v \in
F^{LN}(u)}p_{_{Prof}}(v,l_i)
\end{equation}
\subsubsection{FLI Model}
Finally, primarily relying on \textit{LA-FLI} model and being
adjusted by \textit{LN-FLI} model with a small regulator
parameter $\lambda$, \textit{FLI} model estimates the probability
that $u$ currently lives in $l_i$ as:
\begin{equation}
\label{eq:FLI} p_{_F}(u,l_i)=p_{_{LA-F}}(u,l_i)+\lambda
p_{_{LN-F}}(u,l_i)
\end{equation}
\subsection{Integrated Profile and Friend Location Indication Model}
Next, we discuss how to integrate \textit{PLI} model and \textit{FLI}
model into a unified probabilistic location indication model, so as to capture the complete
location indications. Specifically, \textit{PFLI}
model calculates the probability of $u$ living in $l_i \in
\mathcal{L}$ as:
\begin{equation}
\label{eq:PFLI} p(u,l_i)=\theta_{_{P}}p_{_{Prof}}(u,l_i)+\theta_{_F} p_{_F}(u,l_i)
\end{equation}
\textbf{Parameter Computation:} To obtain a set of good parameters
for the model, we first rewrite the model as:
\begin{equation}
\begin{split}
\label{eq:FLI_cmpt}
p(u,l_i)&= \sum_{a_k \in \mathcal{A}}{\theta_{_{P}} \alpha_k
\sigma_{k}(u,l_i)}\\
&+\sum_{a_k \in \mathcal{A}}{\theta_{_{F}} \beta_k \sum_{v \in \mathcal{F}^{^{LA}}(u)} {w_{k}(u,v) p_{_{LA-F}}(v,l_i)}}\\
&+ \sum_{a_k \in \mathcal{A}} \lambda \theta_{_{F}} \sum_{v \in \mathcal{F}^{^{LN}}(u)} \alpha_k \sigma_{k}(v,l_i)\\
&=\sum_{a_k \in \mathcal{A}} \{[\mu_{k} \sigma_{k}(u,l_i)+\nu_{k}
\delta_{k}(u,l_i)] + [\lambda_{\alpha} \alpha_k \eta_{k}(u,l_i)]\}
\end{split}
\end{equation}
where
\begin{itemize}
\item $\mu_{k}=\theta_{_P} \alpha_k$; $\nu_{k}=\theta_{_F} \beta_k$; $\lambda_{\alpha}=\lambda \theta_{_F}$
\vspace{+.1cm}
\item $\delta_{k}(u,l_i) = \sum_{v \in \mathcal{F}^{^{LA}}(u)} {w_{k}(u,v) p_{_{LA-F}}(v,l_i)}$
\vspace{+.1cm}
\item $\eta_{k}(u,l_i) = \sum_{v \in \mathcal{F}^{^{LN}}(u)} \sigma_{k}(v,l_i)$
\end{itemize}
The location indications extracted from a user's location sensitive
attributes and his LA-friends are considered as primary indications, while the location indication
captured from the LN-friends is only used to regulate the results. Therefore, we integrally train a good set of parameters $\mu_k$ and $\nu_{k}$; while we separately train $\alpha_{k}$.
To train the parameters $\mu_k$ and $\nu_{k}$, we generate a training data set with items $\langle \text{label}(l_i): \text{features}(u,l_i) \rangle$, if the probability that a LA-user $u$ lives in
$l_i$ is larger than zero, i.e., $\sum_{a_k \in \mathcal{A}}{[\sigma_k(u,l_i) + \delta_k(u,l_i)] > 0}$. In particular, $l_i$ is labeled as a \textsl{far} location ($\text{label}(l_i)=0$), if the distance between $l_i$ and $u$'s actual location is larger than a pre-defined threshold; otherwise, it is labeled as a \textsl{close} location ($\text{label}(l_i)=1$). Additionally, $\text{features}(u,l_i)$ is a vector consisting of $\sigma_{k}(u,l_i)$ and $\delta_{k}(u,l_i)$, where $k \in [1,m]$ represents the $k$-th location sensitive attribute. Based on the generated items, we use a logistic regression
method to train the model in the following format:
$$f(y|\mathbf{x};\sigma_1,\cdots,\sigma_m,\delta_1,\cdots,\delta_m)=h_{\mathbf{\sigma},\mathbf{\delta}}(\mathbf{x})^y(1-h_{\mathbf{\sigma},\mathbf{\delta}}(\mathbf{x}))^{1-y}$$
where $y$ is the $\text{label}(l_i)$, $\mathbf{x}$
stands for the $\text{features}(u,l_i)$ and
$h_{\mathbf{\sigma},\mathbf{\delta}}(\mathbf{x})$ is the hypothesis
function. Then we can apply the gradient descent method to maximize
$f(y|\mathbf{x};\mathbf{\sigma},\mathbf{\delta})$ and compute the
parameters. In the similar way, we can train a set of parameters $\alpha_{k}$.
\section{Current City Prediction Approach}
\label{sec:approach} To address \textit{Challenge 2} of Sec. \ref{sec:Intro}, we aggregate the close candidate locations into clusters and devise a two-step current city selection approach. In this section, referring to Figure \ref{fig:overview}, we elaborate the \textit{Candidate Locations Cluster}, \textit{Cluster Selector} and \textit{Location Selector} respectively. We summarize the prediction approach at the end of this section.
\subsection{Candidate Locations Cluster}
\label{sec:lcc}
We draw on the hierarchical clustering method, i.e., UPGMA (Unweighted Pair Group Method with Arithmetic Mean)~\cite{sokal1958statistical}\cite{hastie01statisticallearning}, to generate location clusters. This method arranges all the candidate locations in a hierarchy with a treelike structure based on the distance between two locations, and successively merges the closest locations into clusters. Algorithm \ref{alg:cluster} elaborates the clustering process.\color{black}
Figure \ref{fig:cluster} illustrates an example of the clustering results on $154$ candidate locations that are located in the area with latitude in $47^\circ N \sim 49^\circ N$ and longitude in $1^\circ W \sim 6^\circ E$.
By using the hierarchical clustering method, we divide these locations into $5$ clusters. We note several properties of our location clusters. First, instead of dividing areas with equal-sized grid cells~\cite{Grid_1}\cite{Grid_2}, the hierarchical clustering method only considers the user-generated locations while the areas that no user mentions are out of consideration. Second, the densities inside the clusters are different; however, the average distances between all the candidate locations in any two neighboring clusters are equal ($100km$ in Figure~\ref{fig:cluster}). Third, the complexity of the algorithm is $O(|\mathcal{L}|^3)$, where $|\mathcal{L}|$ is the total number of the candidate locations.
\begin{algorithm}[t]
\SetAlgoNoLine
\KwIn{All the candidate locations $l \in \mathcal{L}$\;}
\KwOut{Location clusters set $\mathcal{C} = \{c_1, c_2, \cdots, c_s\}$ ($s$ is the number of clusters)\;}
\textbf{\textit{Step 1}}: treat all $l \in \mathcal{L}$ as a cluster and calculate the distance between any two locations\;
\Repeat{all the candidate locations are organized into one cluster tree}{
\textbf{\textit{Step 2}}: find and merge the two closest location clusters into a new location cluster\;
\textbf{\textit{Step 3}}: compute the average distance between the new cluster and each of the old ones\;
}
\textbf{\textit{Step 4}}: cut the cluster tree into clusters with an ideal distance threshol
\caption{Clustering Locations}
\label{alg:cluster}
\end{algorithm}
\begin{figure}[!htp]
\centering
\includegraphics[width=8.5cm, height=5.5cm]{./cluster.eps}
\caption{Example of Candidate Locations Cluster.}
\label{fig:cluster}
\end{figure}
\subsection{Cluster Selector}
Given a location cluster and a LN-user's location probability vector obtained by \textit{PFLI} model, we sum up the user's probabilities of locations inside the cluster as the cluster probability. Cluster selector calculates the probabilities of all the clusters that the LN-user may reside in and then selects the cluster with the highest probability.
\subsection{Location Selector}
\label{sec:L_slt} Finally, we select a best point from
the selected cluster as the user's predicted location of the current city. Three alternatives are considered. First, we select the \textit{point of the highest probability} inside the selected cluster
as the best point. Second, we consider the \textit{geographic
centroid} of the selected cluster as the user's best point. The
geographic centroid is the average coordinate for all the points in
a cluster while the probability of each point is considered as its
weight. Third, we calculate the \textit{center of minimum distance}
which has the minimum overall distance from itself to all the rest of
locations in a cluster. We will further discuss and compare the three methods in Sec.~\ref{sec:Eva}.
\subsection{Implementation of Prediction Approach}
We summarize the current city prediction approach in Algorithm~\ref{alg:ccp}. In practice, to speed up the computation of location probability vector for a given LN-user $u$, we first compute location indications from $u$'s location sensitive attributes and LN-friends:
\begin{equation}
\begin{split}
\label{pro_init}
\mathbf{p}(u)= \sum_{a_k \in \mathcal{A}} (\mu_k R_{\cdot a_k(u)} \ + \lambda_{\alpha} \sum_{v \in \mathcal{F}^{^{LN}}} \alpha_k R_{\cdot a_k(v)})
\end{split}
\end{equation}
Assume $\mathcal{L}_{_{LA-F}}$ is the set of current cities of $u$'s LA-friends. We sum location indications from $u$'s LA-friends $p_{_{LA-F}}(u,l_i)$ (refer to Eq.\ref{eq:LA-FLI}) to $p(u,l_i)$, where $l_i \in \mathcal{L}_{_{LA-F}}$.
\begin{algorithm}[t]
\SetAlgoNoLine
\KwIn{A LN-user $u$'s location sensitive attributes\; $u$'s friends list and friends' location sensitive attributes\; Location clusters set $\mathcal{C} = \{c_1, c_2, \cdots, c_s\}$ ($s$ is the number of clusters)\;}
\KwOut{Predicted current city for $u$: $\langle \textit{lat},\textit{lon}\rangle$\;}
Compute location indications $\mathbf{p}(u)$ by $u$'s location sensitive attributes and LN-friends \textbf{\textit{(Eq. \ref{pro_init})}}\;
Obtain all of LA-friends' current city $\mathcal{L}_{_{LA-F}}$\;
\For{$l_i \in \mathcal{L}_{_{LA-F}}$}{
$p(u,l_i)\leftarrow p(u,l_i)+p_{_{LA-F}}(u,l_i)$\;
}
\For{$c_x \in \mathcal{C}$}{
$p(u)_{c_x}= \sum_{l\in c_x}p(u,l)$
}
Cluster selection: $c_h$ where $p(u)_{c_h} \geq p(u)_{c_x}, \forall c_x \in \mathcal{C}$\;
Location selection from $c_h$ \textbf{\textit{(Sec. \ref{sec:L_slt})}}\;
The predicted current city of $u$: $\langle \textit{lat},\textit{lon}\rangle $
\caption{Current City Prediction}
\label{alg:ccp}
\end{algorithm}
\section{Evaluation on Current City Prediction}
\label{sec:Eva} In this section, we first introduce the experiment setups including the used Facebook data
set, the compared approaches and the measurements. Then, we report
the experiment results.
\subsection{Experiment Setup}
\subsubsection{Data description}
We crawled Facebook by a Breadth First Search (BFS)~\cite{FB_UCIrvine} approach from March to June in $2012$ and collected $371,913$ users' information including profile (e.g., gender, current city, hometown) and friends. Among all these users, $153,909$ users publicly report their current city (LA-users) and $225,314$ users do not reveal their current city (LN-users). All these users generate $12,863$ different locations. For more details about this data set, please refer to our previous work~\cite{han2015alike}.
To evaluate the prediction approach,
a user's latest work or education experience is extracted as a location sensitive attribute, named `Work and Education'; we also exploit a user's `Hometown' as another location sensitive attribute. In our data set, $122,899$ LA-users show `Hometown', $54,097$ LA-users reveal `Work and Education' and $115,807$ LA-users publish their friend lists.
In addition to the exploited location sensitive information, some other information (e.g., a user's geo-tagged posts) in Facebook may also leak the location. Our prediction approach can be extended to consider other location sensitive information smoothly, which we will discuss more in Sec.~\ref{subsec:extensibility_of_prediction_approach}.
\subsubsection{Approaches}
We first compare the different location selection approaches introduced in Sec.~\ref{sec:L_slt} to finalize the prediction approach with a good location selector. We also evaluate the performance of non-cluster prediction approach to show the effectiveness of location cluster. Specifically, these approaches can be denoted as:
\begin{itemize}
\item $\textit{PFLI}_{prob}$ is a cluster based approach which selects the \textit{point of highest probability} from the selected cluster as the predicted location.
\item $\textit{PFLI}_{cent}$ is a cluster based approach which selects the \textit{geographic centroid}\footnote{Geographic centroid is the average coordinate for all the points in a cluster while the probability of each point is considered as its weight.} from the selected cluster as the predicted location.
\item $\textit{PFLI}_{dist}$ is a cluster based approach which selects the \textit{center of minimum distance} from the selected cluster as the predicted location.
\item $\textit{PFLI}_{noclst}$ is a non-cluster approach which selects the \textit{point of highest probability} from all candidate locations as the predicted location.
\end{itemize}
The proposed approaches are also compared to several state-of-the-art methods:
\begin{itemize}
\item $\textit{Base}_{dist}$ predicts a user's location based on the observation that the distance between two users decreases by the increase of their friendship~\cite{find_me}.
\item $\textit{Base}_{ann}$ maps any location sensitive attribute value to a certain location and applies artificial neural network to train a current city prediction model \cite{CCP}.
\item $\textit{Base}_{freq}$, borrowing the idea from the prior works based on the Twitter data set~\cite{T_content_prediction}\cite{geo_twitter}, counts the frequency of locations that emerge in a user's friends and predicts his current city by the most frequent location.
\item $\textit{Base}_{freq+}$ improves $\textit{Base}_{freq}$ by further using the neighborhood smoothing approach~\cite{geo_twitter}. Given a location $l$, the points that are less than $20km$ apart from $l$ are considered as $l$'s neighborhoods
\item $\textit{Base}_{knn}$ also relies on the frequency idea for Twitter; however, it merely counts on a user's $k$ closest friends who have the most common friends with him to compute the most frequent location \cite{TweetHood}\cite{Tweecalization}.
\end{itemize}
Among the above approaches, $\textit{Base}_{dist}$ and
$\textit{Base}_{ann}$ are originally devised for Facebook; while
$\textit{Base}_{freq}$, $\textit{Base}_{freq+}$ and $\textit{Base}_{knn}$ are on Twitter. We
utilize the main ideas from $\textit{Base}_{freq}$, $\textit{Base}_{freq+}$ and
$\textit{Base}_{knn}$, and adopt them to fit our data set. By
comparing our approach to $\textit{Base}_{dist}$,
$\textit{Base}_{freq}$, $\textit{Base}_{freq+}$ and $\textit{Base}_{knn}$ which mainly depend
on friendships, we test the effectiveness of integrating location sensitive attributes. By comparing to $\textit{Base}_{ann}$, we examine the newly introduced
one-attribute/multiple-locations mapping method.
\subsubsection{Measurement}
Two widely used measurements:
\textit{Average Error Distance} (\textit{AED}) and
\textit{Accuracy within K km} (\textit{ACC@K})~\cite{T_content_prediction}\cite{geo_twitter}\cite{UID} are exploited.
\textit{Error Distance} computes the distance in kilometers between a
user $u$'s real location and predicted location, i.e., $ErrDist(u)$. \textit{AED}
averages the \textit{Error Distances} of the overall evaluated
users, denoted as $\textit{AED} = \frac{\sum_{u \in
U}ErrDist(u)}{|U|}$. In addition, we rank the users by their \textit{Error Distance} in descending order and report
\textit{AED} of the top $60\%$, $80\%$ and $100\%$ of the evaluated
users in the ranked list, denoted as $\textit{AED@}60\%$, $\textit{AED@}80\%$ and
$\textit{AED@}100\%$ respectively \cite{UID}.
Given a predefined \textit{Error Distance $K$ km}, a prediction for a user is considered as a correct prediction, if the predicted \textit{Error Distance} is less than $K$ km; otherwise, the prediction is incorrect. Then, \textit{Accuracy within K km} is defined as the percentage of correct predictions (i.e., the percentage of users being predicted with an \textit{Error Distance} less than $K$ km), denoted as $\textit{ACC@K} = \frac{|\{u|u \in U
\wedge ErrDist(u)< K\}|}{|U|}$. \textit{ACC@K} shows the
prediction capability of an approach at a specific pre-established
\textit{Error Distance}.
\subsection{Experiment Results}
\begin{table}
\centering
\scriptsize
\begin{tabular}{c||c|c|c|c|c|c|c|c|c}
\hline
\bfseries & $\textit{Base}_{dist}$ & $\textit{Base}_{ann}$ & $\textit{Base}_{freq}$ & $\textit{Base}_{freq+} $ & $\textit{Base}_{knn}$ & $\textit{PFLI}_{noclst}$ & $\textit{PFLI}_{dist}$ & $\textit{PFLI}_{cent}$ & $\textit{PFLI}_{prob}$\\
\hline
$\textit{AED@}60\%$ & 8.6 & 5.7 & 5.9 & 4.9 & 10.8 & 2.5 & 49.5 & 5.6 & \bfseries 2.1\\
\hline
$\textit{AED@}80\%$ & 85.0 & 64.3 & 91.8 & 56.0 & 100.0 & 40.1 & 77.4 & 38.0 & \bfseries 36.9\\
\hline
$\textit{AED@}100\%$ & 1288.5 & 1129.0 & 1160.5 & 1123.7 & 1397.6 & 874.0 & 885.9 & 855.3 & \bfseries 854.4\\
\hline
\end{tabular}
\caption{Prediction Results (\textit{AED}) for Users with LA-Friends}
\label{table:AED_UF}
\end{table}
\begin{table}
\centering
\scriptsize
\begin{tabular}{c||c|c|c|c|c|c|c|c|c}
\hline
\bfseries & $\textit{Base}_{dist}$ & $\textit{Base}_{ann}$ & $\textit{Base}_{freq}$ & $\textit{Base}_{freq+} $ & $\textit{Base}_{knn}$ & $\textit{PFLI}_{noclst}$ & $\textit{PFLI}_{dist}$ & $\textit{PFLI}_{cent}$ & $\textit{PFLI}_{prob}$\\
\hline
$\textit{AED@}60\%$ & 102.8 & 6.7 & 73.9 & 66.6 & 119.5 & 3.5 & 50.6 & 6.3 & \bfseries 3.1\\
\hline
$\textit{AED@}80\%$ & 1368.8 & 74.7 & 1257.2 & 1243.1 & 1429.6 & 52.5 & 88.2 & 50.2 & \bfseries 49.1\\
\hline
$\textit{AED@}100\%$ & 2671 & 1204.0 & 2523.5 & 2498 & 2698.5 & 981.0 & 989.9 & 960.8& \bfseries 960.0\\
\hline
\end{tabular}
\caption{Prediction Results (\textit{AED}) for Overall Users}
\label{table:AED_Uall}
\end{table}
Many relationship-based methods (e.g., $\textit{Base}_{dist}$, $\textit{Base}_{freq}$, $\textit{Base}_{freq+}$ and $\textit{Base}_{knn}$) rely heavily on users' LA-friends whose locations are exposed. In general, such methods can work well for the users who have a certain number of LA-friends; but when they are applied to the overall users (who either have or do not have LA-friends), the performance notably decreases. We evaluate the prediction performance on two user sets: \textit{users with LA-friends} and \textit{overall users}, and report the evaluation results on \textit{AED} and \textit{ACC@K} subsequently.
\subsubsection{Evaluation on \textit{AED}}
Table \ref{table:AED_UF} and Table \ref{table:AED_Uall} show the \textit{AED}s of all the compared approaches for two user sets. The smallest \textit{AEDs}, which are generated by $\textit{PFLI}_{prob}$, have been highlighted in bold.
Let us first look at the \textit{PFLI} model based approaches (i.e., $\textit{PFLI}_{dist}$, $\textit{PFLI}_{cent}$, $\textit{PFLI}_{prob}$, and $\textit{PFLI}_{noclst}$). Among the first three cluster based approaches that are different at their location selectors, $\textit{PFLI}_{dist}$ generates the largest \textit{AEDs} while $\textit{PFLI}_{prob}$ achieves the smallest \textit{AEDs}. We also compare the non-cluster approach $\textit{PFLI}_{noclst}$ and the cluster approach $\textit{PFLI}_{prob}$, which both select location of the highest probability. We observe that $\textit{PFLI}_{prob}$ presents smaller \textit{AEDs} than $\textit{PFLI}_{noclst}$ and verify the effectiveness of the location cluster approach.
In addition, the results show that the \textit{PFLI} model based approaches present much smaller \textit{AEDs} than all the other baselines. In particular, the results demonstrate the \textit{PFLI} model based approaches mapping one-attribute to multiple locations reduce the \textit{AED} significantly compared to $\textit{Base}_{ann}$ which maps one-attribute to one-location.
By examining the results of $\textit{AED@60\%}$, $\textit{AED@80\%}$ and $\textit{AED@100\%}$, we observe that the \textit{PFLI} model based approaches can predict current city with relatively small $\textit{AED@60\%}$ and $\textit{AED@80\%}$; whereas, $\textit{AED@100\%}$ increases by $10$--$23$ times from $\textit{AED@80\%}$. This demonstrates the large \textit{Error Distance} only occurs at predictions for a small number of users.
Lastly, we compare the results in the two Tables and notice that the prior approaches ($\textit{Base}_{dist}$, $\textit{Base}_{freq}$, $\textit{Base}_{freq+}$ and $\textit{Base}_{knn}$) predict locations with much larger \textit{AEDs} for \textit{overall users} than for \textit{users with LA-friends}; however, for the \textit{PFLI} model based approaches, \textit{AEDs} differ slightly for two user sets. It demonstrates that a user's profile can significantly contribute to the location prediction when the user's friends' locations are unavailable.
\subsubsection{Evaluation on \textit{ACC@K}}
We study \textit{ACC@K} of the three proposed prediction approaches ($\textit{PFLI}_{prob}$, $\textit{PFLI}_{cent}$ and $\textit{PFLI}_{dist}$) for two user sets in Figure \ref{fig:Selt}. We observe that the accuracy of $\textit{PFLI}_{prob}$ goes up steadily with the increase of \textit{Error Distance}. $\textit{PFLI}_{cent}$ may lead to very low accuracy when the pre-established \textit{Error Distance} is quite small; but it can achieve higher accuracy than $\textit{PFLI}_{prob}$, when the pre-established \textit{Error Distance} is larger than $40$ km. This reveals the properties of these two prediction approaches: $\textit{PFLI}_{cent}$, which selects the geographic centroid of a cluster, generates a short average \textit{Error Distance} to all the locations in the cluster but fails to pick the user's exact coordinate once it is not the centroid; while $\textit{PFLI}_{prob}$ may produce a large \textit{Error Distance} if the location of the highest probability is not the user's real location. In addition, $\textit{PFLI}_{dist}$ is not competitive with the other two approaches.
\begin{figure}[!htbp]
\centering \subfigure[Users with LA-Friends]{
\includegraphics[width=5.5cm, height=5cm]{./label_AUC_slt_cmpl.eps}
\label{fig:lb_slt} } \subfigure[Overall Users]{
\includegraphics[width=5.5cm, height=5cm]{./AUC_slt_cmpl.eps}
\label{fig:slt} }\caption{\textit{ACC@K} of Different
Location Selectors.} \label{fig:Selt}
\end{figure}
Rather than solely using any one of the proposed approaches, we exploit a combined-approach strategy by flexibly selecting the best approach according to the pre-established \textit{Error Distance}. Specifically, this strategy uses $\textit{PFLI}_{prob}$ when the pre-established \textit{Error Distance} is smaller than $40$ km and otherwise applies $\textit{PFLI}_{cent}$. The combination is practical and can obtain a better performance than using any single approach. We plot the combination line in Figure \ref{fig:Selt} and call it $\textit{PFLI}_{cmb}$.
Figure \ref{fig:ACC} compares $\textit{PFLI}_{cmb}$ to various baseline methods in terms of \textit{ACC@K}. We observe that the proposed $\textit{PFLI}_{cmb}$ outperforms all the compared baselines for both user sets. Compared to $\textit{PFLI}_{noclst}$, $\textit{PFLI}_{cmb}$ increases around $1.5\%$ and $1.2\%$ of accuracy on average for \textit{users with LA-friends} and \textit{overall users}. This proves the effectiveness of the cluster strategy with successive cluster selection and location selection.
\begin{figure}[!htbp]
\centering \subfigure[Users with LA-Friends]{
\includegraphics[width=5.5cm, height=5cm]{./label_AUC_comp_cmpl_max.eps}
\label{fig:lb_acc} } \subfigure[Overall Users]{
\includegraphics[width=5.5cm, height=5cm]{./AUC_comp_cmpl_max.eps}
\label{fig:acc} } \caption{\textit{ACC@K} of the Proposed
Approach and Other Baselines.}
\label{fig:ACC}
\end{figure}
Comparing Figure~\ref{fig:lb_acc} and \ref{fig:acc}, we observe that the approaches $\textit{Base}_{freq}$, $\textit{Base}_{freq+}$, $\textit{Base}_{dist}$ and $\textit{Base}_{knn}$ perform much worse for \textit{overall users} than for \textit{users with LA-friends}. This observation again indicates that these approaches depend heavily on the friends' locations. However, in respect of the other approaches, which integrate location indications from both location sensitive attributes and friends (including our previous work $\textit{Base}_{ann}$~\cite{CCP}), the prediction performance for \textit{overall users} relatively approaches to the performance for \textit{users with LA-friends}.
\section{Current City Exposure Estimator}
\label{sec:exposure} In this section, we pay attention to estimating current city exposure probability for a user who hides his
current city. We formulate the \textbf\textit{{current city
exposure estimation}} problem as: \textit{Given, (i)~a graph $\mathcal{G}=(\mathcal{U}^{^{LA}}\cup \mathcal{U}^{^{LN}},\mathcal{E},\mathcal{L})$; (ii)~the public location $l(u)$ for LA-users $u \in \mathcal{U}^{^{LA}}$; (iii)~the location sensitive attributes $\mathcal{A}(u)$ and the friends list $\mathcal{F}(u)$ for all the users $u \in (\mathcal{U}^{^{LA}} \cup \mathcal{U}^{^{LN}})$; $(iv)$ a pre-established Error Distance $K$ km, we forecast the current city exposure probability within $K$
km and report the exposure risk level for each LN-user $u \in \mathcal{U}^{^{LN}}$}.
To solve this problem, we run the proposed prediction approach on an
aggregation of users and conduct analysis on the aggregated
prediction results. Furthermore, we apply a regression method to
construct the exposure model according to the analysis observations.
Relying on this model, we devise a current city exposure estimator
to inform users of their current city \textit{Exposure Probability within
$K$ km} and \textit{Exposure Risk Level}.
The \textit{Exposure Probability within $K$ km (EP@K)} represents
the probability that a user's current city could be inferred
correctly if the pre-established \textit{Error Distance} is $K$ km. As it
is conceptually similar to the metric \textit{ACC@K}, we compute
it by the same formula:
\begin{equation}
\label{eq:exp_pro}
EP@K = \frac{|\{u|u \in U \wedge ErrDist(u)<K\}|}{|U|}
\end{equation}
Additionally, we set up five \textit{Exposure Risk Levels} according
to the value of \textit{Exposure Probability}, shown in Table
\ref{table:Risk_Level}. \textit{Level 5} is defined as the most risky
level, which indicates an \textit{Exposure Probability} higher than
$0.9$, while \textit{Level 1} is the safest one, which represents a
small \textit{Exposure Probability} lower than $0.25$.
Next, we show some observations of inspections on the aggregated
prediction results. We then introduce the current
city exposure model and the model based estimator. Finally, we
illustrate some case studies to show the use of our proposed
exposure estimator. We also summarize some guidelines to reduce the
exposure risk.
\begin{table}
\centering
\scriptsize
\begin{tabular}{c||c|c|c|c|c}
\hline
\bfseries Exposure Probability & $[0.9,1]$ & $[0.75,0.9)$ & $[0.5,0.75)$ & $[0.5,0.25)$ & $ [0.25,0]$ \\
\hline
\bfseries Risk Level & Level 5 & Level 4 & Level 3 & Level 2 & Level 1\\
\hline
\end{tabular}
\caption{Risk Level vs. Exposure Probability}
\label{table:Risk_Level}
\end{table}
\subsection{Current City Exposure Inspection}
\label{sec:expo_inspection}
In this subsection, we extract several measurable characteristics from users' self-exposed information (e.g., User Category), and inspect the current city exposure probability by these characteristics.
First, we classify users into diverse categories with respect to the
combinations of visible/invisible properties of their location
sensitive attributes and friends list. Table \ref{table:category}
lists the obtained seven \textit{User Categories}. \textit{User
Category} measures the types and amount of users' self-exposed
information.
\begin{table}
\centering
\scriptsize
\begin{tabular}{c|c}
\hline
\bfseries User's Visible Attributes & \bfseries Abbreviation\\
\hline
`Hometown' & `HT'\\
`Work and Education' & `WE'\\
`Friends' & `F'\\
`Hometown' and `Work and Education' & `HT+WE'\\
`Hometown' and `Friends' & `HT+F'\\
`Work and Education' and `Friends' & `WE+F'\\
`Hometown', `Work and Education' and `Friends' & `HT+WE+F'\\
\hline
\end{tabular}
\caption{Users Categories by Visible Attributes Combination}
\label{table:category}
\end{table}
\begin{figure}[!htp] \centering
\includegraphics[width=6cm, height=5.5cm]{./AUC_reminder.eps}
\caption{Current City Exposure Probability by User Category.}
\label{fig:Exposure_category}
\end{figure}
Figure \ref{fig:Exposure_category} inspects the \textit{Exposure
Probabilities} for various \textit{User Categories}. From this
figure, we observe that different types of self-exposed information
may divulge users' current city to different extent. For instance,
users in `WE' category are normally more dangerous to disclose their
current city than users in `HT' or `F' categories. We also find that
the users who publish their `WE' (in `WE', `HT+WE', `WE+F'
or `HT+WE+F' categories) exhibit a high \textit{Exposure Probability}. This
means that `WE' is a very risky attribute to leak users' current
city. The results also reveal that `HT' is more sensitive to
disclose current city than `F', although `F' is generally regarded
as a significant location indication.
Figure \ref{fig:Exposure_category} also indicates that a user's current city generally could be predicted with a higher
probability if the user exposes more information. For example, users
who expose `HT+F' exhibit a higher exposure probability than users
only revealing either `HT' or `F'. Note that, for a user who exposes
`HT+WE', his current city exposure probability can be up to $90\%$,
which approaches to the exposure probability of users who expose
`HT+WE+F'. In other words, merely exposing `HT+WE' can
almost lead to the exposure of a user's current city. To conclude, \textit{User
Category}, which distinguishes users by the types and amount of their
self-exposed information, relates to \textit{Exposure Probability}.
In addition to \textit{User Category}, we study the influence of the percentage of friends with attributes (i.e., \textit{\%~Friends with Attributes}) on \textit{Exposure Probability}. \textit{\%~Friends with Attributes} is the ratio of a user's friends who present at least one attribute to his overall friends.
\begin{figure}[!htp] \centering
\includegraphics[width=6cm, height=5.5cm]{./EP_percentage_friends_of_attributes.eps}
\caption{Current City Exposure Probability by the Percentage of Friends with Attributes.}
\label{fig:Exposure_PF}
\end{figure}
Figure \ref{fig:Exposure_PF} displays the \textit{Exposure Probability} (i.e., EP, $Z$ axis) by \textit{\% Friends with Attributes} (i.e., FA, $X$ axis) at different \textit{Error Distances} (i.e., ED, $Y$ axis). As more than $95\%$ of the users have a \textit{\% Friends with Attributes} smaller than $45\%$, we only look at its value in a range of $0\%$ to $45\%$.
Generally speaking, \textit{Exposure Probability} grows by the increase of \textit{\%~Friends with Attributes}
\begin{figure*}[!htb]
\centering \subfigure[`HT']{
\includegraphics[width=3cm, height=3cm]{./AUC_reminder_Fw_ht.eps}
\label{fig:ht} } \subfigure[`WE']{
\includegraphics[width=3cm, height=3cm]{./AUC_reminder_Fw_em.eps}
\label{fig:em} } \subfigure[`F']{
\includegraphics[width=3cm, height=3cm]{./AUC_reminder_Fw_f.eps}
\label{fig:f} } \subfigure[`HT+WE']{
\includegraphics[width=3cm, height=3cm]{./AUC_reminder_Fw_htem.eps}
\label{fig:htem} } \subfigure[`HT+F']{
\includegraphics[width=3cm, height=3cm]{./AUC_reminder_Fw_htf.eps}
\label{fig:htf} } \subfigure[`WE+F']{
\includegraphics[width=3cm, height=3cm]{./AUC_reminder_Fw_emf.eps}
\label{fig:emf} } \subfigure[`HT+WE+F']{
\includegraphics[width=3.3cm, height=3cm]{./AUC_reminder_Fw_htemf.eps}
\label{fig:htemf} } \caption{Exposure Probability by Cluster Confidence in Different User Categories.}
\label{fig:Exposure_Coefficient}
\end{figure*}
In addition, we define a new metric named \textit{Cluster Confidence}. It estimates the ratio of the
probabilities of candidate locations in the selected cluster $c_h$
to the overall probabilities of all the candidate locations (equal
$1$), calculated as follows:
\begin{equation}
\label{eq:cluster_confidence}
CC(u) = \frac{\sum_{l\in c_h}p(u,l)}{\sum_{l\in \mathcal{L}}p(u,l) } = \sum_{l\in c_h}p(u,l)
\end{equation}
\textit{Cluster Confidence} represents the confidence of the users' location indications. For example, \textit{Cluster Confidence} with a value of $100\%$ means that all of a user's
location indications point to an exclusive location cluster. We
further look into the change of \textit{Exposure Probability} according to
\textit{Cluster Confidence} for each \textit{User Category}.
Figure \ref{fig:Exposure_Coefficient} reveals how \textit{Exposure
Probability} (i.e., EP, $Z$ axis) varies with diverse \textit{Cluster Confidence} (i.e., CC, $X$ axis) and
\textit{Error Distances} (i.e., ED, $Y$ axis) in different \textit{User Categories}. The results show that the \textit{Exposure Probability} normally grows up when the
\textit{Cluster Confidence} gets larger. When the \textit{Cluster Confidence} equals $100\%$, the \textit{Exposure Probability}
surpasses $90\%$ within a pre-established \textit{Error Distance} of $20$
km almost for all \textit{User Categories}. This observation
indicates that the current city is more dangerous to be predicted
when a user's location indications are more likely to point to one
city or to multiple cities that are in the same cluster. In other
words, a user's current city can be easily disclosed if the confidence
of the user's self-exposed information is high.
Note that, there exists an exception for the users only exposing
their `F': the decline of \textit{Exposure Probability} when the
\textit{Cluster Confidence} is larger than $0.9$. One reasonable
explanation is that only the users with an extremely small number of
friends (e.g., only one friend) can have the \textit{Cluster Confidence} higher than $0.9$, which might reduce the exposure risk of
current city due to the limited information.
\subsection{Estimating Current City Exposure Risk}
\label{expo_extimation}
\subsubsection{Current City Exposure Model}
In the previous section, we observe that a user's current city
\textit{Exposure Probability} is probably influenced by four
factors: \textit{Error Distance}, \textit{User Category}, \textit{\% Friends with Attributes} and
\textit{Cluster Confidence}. Taking these four factors as features, we respectively use Random Decision Forest and Linear Regression approaches to model \textit{Exposure Probability}. The performance of model is evaluated by two commonly used metrics, \textit{Mean Absolute Error} (MAE) and \textit{Root Mean Squared Error} (RMSE), with $10$-cross validation, shown in Table~\ref{table:EM_comp}. We observe that the Random Decision Forest based model outperforms the Linear Regression based model by presenting smaller MAE and RMSE. Therefore, we employ the Random Decision Forest based model to estimate current city exposure probability, denoted as \textit{RDF Exposure Model}.
\begin{table}
\centering
\scriptsize
\begin{tabular}{c||c|c}
\hline
& \bfseries Random Decision Forest & \bfseries Linear Regression \\
\hline
\bfseries MAE & $0.027$ & $0.061$ \\
\bfseries RMSE & $0.077$ & $0.146$ \\
\hline
\end{tabular}
\caption{Performance Comparison of Exposure Models}
\label{table:EM_comp}
\end{table}
\begin{table}
\centering
\scriptsize
\begin{tabular}{c||c|c|c|c|c}
\hline
& \bfseries RDF Exposure & \bfseries No Error & \bfseries No User & \bfseries No \% Friends & \bfseries No Cluster \\
& \bfseries Model & \bfseries Distance & \bfseries Category & \bfseries with Attributes & \bfseries Confidence\\
\hline
\bfseries MAE & $0.027$ & $0.052$ & $0.065$ & $0.045$ & $0.082$ \\
\bfseries RMSE & $0.077$ & $0.106$ & $0.131$ & $0.117$ & $0.166$ \\
\hline
\end{tabular}
\caption{Feature Verification of \textit{RDF Exposure Model}}
\label{table:EM_features}
\end{table}
Furthermore, `Leave-one-feature-out' approach is exploited to verify the effectiveness of the features. We use Random Decision Forest approach to train exposure models by taking out any one of the four features, namely \textit{No Error Distance}, \textit{No User Category}, \textit{No \% Friends with Attributes} and \textit{No Cluster Confidence}. Table~\ref{table:EM_features} compares these `Leave-one-feature-out' models to the \textit{RDF Exposure Model}. We observe that the \textit{RDF Exposure Model} presents the best performance with the smallest MAE and RMSE. The performance degradations when removing any one of the features just verify that all the four studied features contribute to the model. \textit{Cluster Confidence} is observed as the most sensitive feature for the model, because the performance of the \textit{RDF Exposure Model} drops most significantly when \textit{Cluster Confidence} is taken out.
\subsubsection{Current City Exposure Estimator}
By exploiting the proposed current city exposure model, we construct
an exposure estimator to forecast the exposure risk of a user's
private current city. Figure \ref{fig:Exposure_Reminder} illustrates
the framework of the current city exposure estimator. The exposure
estimator contains three main function modules: user information
handler, current city exposure model and exposure risk level
decision. The inputs of the exposure estimator include a user's
self-exposed information and a pre-established \textit{Error
Distance}. Given a user's self-exposure information, the user
information handler determines \textit{User Category}, and computes
\textit{Cluster Confidence} and \textit{\% Friends with Attributes}. Based on the pre-established
\textit{Error Distance}, the obtained \textit{User Category},
\textit{Cluster Confidence}, and \textit{\% Friends with Attributes}, the exposure model calculates the
current city exposure probability for the user. The exposure risk module
determines a risk level according to the exposure probability.
Finally, the exposure estimator outputs two risk measurements of
current city: \textit{Exposure Probability} and \textit{Risk Level}.
\begin{figure}[!htp] \centering
\includegraphics[width=8cm, height=3.8cm]{./exposure_reminder.eps}
\caption{Framework of Current City Exposure Estimator.}
\label{fig:Exposure_Reminder}
\end{figure}
\subsection{Case Studies: Exposure Estimator and Privacy Protection}
\begin{table*}[!htp]
\centering
\scriptsize
\begin{tabular}{c||c|c|c|c||c|c}
\hline
\multirow{2}*{\bfseries User} & \bfseries User & \bfseries Cluster & \bfseries Error & \bfseries \% Friends with & \bfseries Exposure &\bfseries Risk\\
& \bfseries Category & \bfseries Confidence& \bfseries Distance& \bfseries Attribute & \bfseries Probability & \bfseries Level\\
\hline
\bfseries $U1$ & `HT+WE+F' & $0.69$ &$100 km$ & $0.9\%$ & 0.967 & Level 5\\
\bfseries $U1$ & `HT+WE+F' & $0.69$ & $20 km$ & $0.9\%$ & 0.883 & Level 4\\
\bfseries $U2$ & `F' & $0.208$ & $100 km$ & $11.2\%$ & 0.564 & Level 3\\
\bfseries $U3$ & `F' & $0.208$ & $100 km$ & $0.2\%$ & 0.374 & Level 2\\
\bfseries $U4$ & `WE+F' & $0.281$ & $100 km$ & $2.1\%$ & 0.407 & Level 2\\
\bfseries $U5$ & `WE+F' & $0.57$ & $100 km$ & $2.1\%$ & 0.797 & Level 4\\
\bfseries $U6$ & `HT+F' & $0.332$ & $20 km$ & $20.1\%$ & 0.276 & Level 2\\
\bfseries $U7$ & `HT+WE' & $0.73$ & $100 km$ & $0\%$ & 0.903 & Level 5\\
\bfseries $U8$ & `HT' & $0.169$ & $20 km$ & $0\%$ & 0.059 & Level 1\\
\bfseries $U9$ & `WE' & $0.404$ & $20 km$ & $0\%$ & 0.834 & Level 4\\
\bfseries $U10$ & `F' & $0.891$ & $20 km$ & $17.2\%$ & 0.823 & Level 4\\
\hline
\end{tabular}
\caption{Exposure Estimator Cases Study}
\label{table:User_Case}
\end{table*}
Any LN-users who reveal their self-exposed information and pre-define an \textit{Error Distance} can use the proposed current city exposure estimator to assess their \textit{Exposure Probability} and \textit{Risk Level}. To better understand the use of exposure estimator, we illustrate several use cases in Table \ref{table:User_Case}. In this study, we observe that some of the
LN-users are not really safe to hide their current city if they
leave some other information visible. For instance, considering
$U9$, even though only `WE' is published, his current city is almost leaked
with an extremely high \textit{Exposure Probability} of $0.834$
within an \textit{Error Distance} of $20$ km. In addition, for users
in the same \textit{User Category}, the one with a higher
\textit{Cluster Confidence} is more likely to divulge his current
city. Looking at $U4$ and $U5$ who are both in `WE+F' category, the
current city of $U5$ who exhibits a higher \textit{Cluster Confidence} is more dangerous to be inferred, compared to $U4$'s current city.
In addition, the exposure estimator can offer some countermeasures on
privacy configuration against information leakage. Assume users
hide some part of their exposed information, the exposure estimator
estimates and reports the corresponding \textit{Exposure
Probability} and \textit{Exposure Risk Level}. Then users can decide on
a new privacy configuration accordingly. We take $U1$ as an example
and list some possible exposure risks assuming that he adjusts his
privacy configuration. The results shown in Table
\ref{table:Exa_Guide} reveal that the exposure risk could be significantly decreased
if $U1$ hides his `HT+WE', `WE+F' or `WE'. The results also point
out that merely hiding `F' or `HT' cannot protect $U1$'s current city
privacy.
\begin{table}
\centering
\scriptsize
\begin{tabular}{c||c|c|c|c|c|c}
\hline
\multicolumn{1}{c||}{\multirow{2}{*}{\bfseries $U1$}} & \multicolumn{1}{c|}{\bfseries Current status} & \multicolumn{5}{c}{\bfseries Hide} \\
\cline{3-7}
\multicolumn{1}{c||}{} & \multicolumn{1}{c|}{\bfseries `HT+WE+F'} & \multicolumn{1}{c|}{\bfseries `WE'} & \multicolumn{1}{c|}{\bfseries `F'} & \multicolumn{1}{c|}{\bfseries `HT'} & \multicolumn{1}{c|}{\bfseries `WE+F'} & \multicolumn{1}{c}{\bfseries `HT+WE'}\\ %
\hline
\bfseries Exposure & \multirow{2}*{$0.967$} & \multirow{2}*{$0.503$} & \multirow{2}*{$0.944$} & \multirow{2}*{$0.936$} & \multirow{2}*{$0.456$} & \multirow{2}*{$0.073$} \\
\bfseries Probability & & & & & & \\
\hline
\bfseries Risk Level & Level 5 & Level 3 & Level 5 & Level 5 & Level 2 & Level 1\\
\hline
\end{tabular}
\caption{Exposure Guidelines for $U1$: the exposure risks if he
adjusts some privacy configurations with an \textit{Error Distance}
of $100km$}
\label{table:Exa_Guide}
\end{table}
Finally, according to the studies on current city exposure
risk, we summarize the following general suggestions:
\begin{itemize}
\item As all the location indications may expose the hidden current city, close all of location sensitive information including `WE', `F' and `HT' so as to achieve a high current city security.
\item Hide the most sensitive exposed information (e.g., `WE') if users want to publicly share some personal information (e.g., `F'), since the most
sensitive information can independently lead to a quite high \textit{Exposure Probability}. For example, `WE' alone can lead to an \textit{Exposure Probability} higher than $80\%$.
\item According to the centrality principle which refers to the \textit{Cluster Confidence}, hide `F' if most friends indicate the same place where the user
lives. For instance, $U10$ in Table \ref{table:User_Case} is
necessarily advised to hide his `F'.
\end{itemize}
\section{Discussion and Future Work}
\label{sec:Discussion}
In this section, we discuss some issues which are not addressed in this work due to space limitations, and point out some future potential research directions.
\subsection{Extensibility of the Current City Prediction Approach}
\label{subsec:extensibility_of_prediction_approach}
Due to the data set limitation, we only use three features (i.e., `Hometown', `Work and Education' and `Friend') to evaluate our proposed current city prediction approach. However, our prediction approach can be extended to consider other location sensitive attributes. For instance, for the location sensitive pages that a user follows (e.g., the page of a favorite local restaurant) or the location sensitive posts that a user published (e.g., geo-tagged posts), we can regard one page or one post as a LA-Friend and refer to \textit{LA-FLI} model to explore the location indications.
\subsection{Adaptability of the Exposure Estimation Approach}
In addition, our exposure estimation approach can easily adapt to other current city prediction approaches by the two-step solution: (1) \textit{feature extraction} (Sec.~\ref{sec:expo_inspection}) and (2) \textit{exposure model training} (Sec.~\ref{expo_extimation}). In particular, we can first extract similar features for other city prediction approaches as the inspected features in Sec.~\ref{sec:expo_inspection}.
Take \textit{Cluster Confidence} as an example. For the cluster-based city prediction approaches like ours, \textit{Cluster Confidence} can be extracted in the same way, i.e., the largest \textit{cluster} prediction probability (Eq.\ref{eq:cluster_confidence}). For the other city prediction approaches without a clustering step~\cite{find_me}\cite{UID}, following the essence of \textit{Cluster Confidence}, a similar feature, \textit{Prediction Confidence}, can be computed as the largest \textit{city} prediction probability. Likewise, we can also obtain the other features presented in our exposure model for many other city prediction approaches, while we do not discuss them further for brevity. Once the features are derived, in the second step, we can directly apply the regression methods used in Sec.~\ref{expo_extimation} to train the exposure models for other prediction approaches.
\subsection{Generalizability of the Exposure Estimator}
Taking `current city' as a representative attribute to study the information exposure issue, this work gives further insights on how to assess the exposure risk of other privacy-sensitive attributes (e.g., age). Denoting the privacy-sensitive attribute as PSA, the process to assess its exposure risk can be generalized into three steps: $1)$ Explore PSA-sensitive attributes and construct a PSA prediction model; $2)$ Inspect the prediction results to extract features and train a PSA exposure model; $3)$ Based on the exposure model, implement an exposure estimator to notify users of the exposure risk and provide suggestions to lower the risk if necessary.
Moreover, our future work will consider integrating multiple exposure models into the exposure estimator, so as to construct an exposure estimation system that can provide reliable and multi-functional exposure risk estimations.
\section{Conclusion}
\label{sec:conclusion} This paper starts with two open questions
regarding the security of users' hidden privacy-sensitive
attributes. To answer these questions, we first propose a novel
current city prediction approach to infer users' current city by
leveraging users' self-exposed information including location
sensitive attributes and friends list. We validate the new
prediction approach on a Facebook data set containing $371,913$ users,
and the results reveal that the users' hidden current city may be
dangerous to be predicted. Then we apply the proposed prediction
approach to predict users' current city and model the exposure
probability by considering four measurable characteristics --- \textit{Cluster Confidence}, \textit{Error Distance}, \textit{User Category} and \textit{Percentage of Friends with Attributes}. Based on
the exposure model, we propose a current city exposure estimator to measure
the exposure probability and risk level of users' hidden current city
according to their self-exposed information. The exposure estimator
can also help users to adjust their privacy configuration to satisfy
their privacy requirements. While this work studies the
potential risk of users' privacy-sensitive attributes with a
representative attribute of current city in Facebook, the proposed
idea and approach could be extended to other attributes and
utilized by other OSNs.
\section*{Acknowledgment}
We would like to acknowledge Dr. Rebecca Copeland for the careful proof-reading. This work has been funded by the China Scholarship Council.
\nocite{*}
\bibliographystyle{elsarticle-harv}
| {'timestamp': '2015-08-05T02:10:26', 'yymm': '1508', 'arxiv_id': '1508.00784', 'language': 'en', 'url': 'https://arxiv.org/abs/1508.00784'} |
\section{Introduction}
Filippov introduced $n$-Lie algebras in 1985 \cite{filippov1985}. $n$-Lie algebras, in particular $3$-Lie algebras are important in mathematical physics. Lie superalgebras are the $\mathbb{Z}_2$-graded Lie algebras which was introduced by Kac \cite{kac1977}. These are too interesting from a purely mathematical point of view. The notion of 3-Lie superalgebras are generalization of 3-Lie algebras extending to a $\mathbb{Z}_2$-graded case. $n$-Lie superalgebras are more general structures including $n$-Lie algebras and Lie superalgebras whose definition was introduced by Cantarini $et ~al.$ \cite{cantarini2010}.
\par Derivation algebra is an important topic in Lie algebras which has widespread applications in physics and geometry. A superderivation of a Lie superalgebra is certain generalization of derivation of a Lie algebra. The structure of superderivation of Lie superalgebras was studied in \cite{nandi2021,wang2016}. Cohomology is an important tool in modern mathematics and theoretical physics, its range of applications contain algebra and topology as well as the theory of smooth manifolds or holomorphic functions. The cohomology of Lie algebras was defined by Chevalley $et~ al.$ \cite{chevalley1948}. Leites introduced cohomology of Lie superalgebras and extended some of the basic structures and results of classical theories to Lie superalgebras \cite{leites1975}. Further cohomology for $n$-Lie superalgebras was discussed in \cite{ma2014}.
\par Recently Tang $et ~al.$ studied a Lie algebra with a derivation from the cohomological point of view and construct a cohomology theory that controls, among other things, simultaneous deformations of a Lie algebra with a derivation \cite{tang2019}. These results have been extended to associative algebras \cite{das2020}, Leibniz algebras \cite{das2021}, $3$-Lie colour algebras \cite{zhang2014}, 3-Lie algebras \cite{xu2018}, Lie triple systems \cite{wu2022}, and $n$-Lie algebras \cite{sun2021}. Generalized representations of 3-Lie algebras and 3-Lie superalgebras was introduced in \cite{zhu2017,zhu2020}. Zhao $et~ al.$ studied a representation of a Lie superalgebra with a superderivation pair and its corresponding cohomologies \cite{zhao2021}.
\par The aim of this paper is to generalize the results of Xu \cite{xu2018} to $3$-Lie superalgebra case. First we take a representation $(\Phi,\mathcal{P})$ of a $3$-Lie superalgebra $\mathcal{Q}$ on $\mathcal{P}$ and construct $2$-cocycles by using superderivations of $\mathcal{P},\mathcal{Q}$ and hence first-order cohomologies. This construction develops a Lie superalgebra $\mathcal{T}_{\Phi}$ by the representation $\Phi$ and the space $\mathbb{H}^1(\mathcal{Q};\mathcal{P})$ of first-order cohomology class gives a representation $\Psi$ of the Lie superalgebra $\mathcal{T}_{\Phi}$. Further we consider representation of $3$-Lie superalgebras given by abelian extensions of $3$-Lie superalgebras of the form $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow \mathcal{Q}\rightarrow 0$ with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$ and construct an obstruction class to extensibility of a compatible pair of superderivations of $\mathcal{P},\mathcal{Q}$ to those of $\mathcal{L}$.
\section{Preliminaries}
In this section, we recall representations and cohomologies of $3$-Lie superalgebras and their relations to abelian extensions of $3$-Lie superalgebras of the form $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\xrightarrow{\pi} \mathcal{Q}\rightarrow 0$ with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$. We show that $\mathbb{H}^1(\mathcal{Q};\mathcal{P})=0$ for such extensions implies split property.
\par Let $\mathbb{Z}_2 =\{\overline{0},\overline{1}\}$ be the field of two elements. Throughout the paper, we denote $\mathbb{F}$ as a field of characteristic zero. A superspace is a $\mathbb{Z}_2$-graded vector space $\mathcal{V}=\mathcal{V}_{\overline{0}}\oplus \mathcal{V}_{\overline{1}}$. A $sub superspace$ is a $\mathbb{Z}_2$-graded vector space which is closed under bracket operation. Non-zero elements of $\mathcal{V}_{\overline{0}}\cup \mathcal{V}_{\overline{1}}$ are said to be $homogeneous$ and whenever the degree function occurs in a formula, the corresponding elements are supposed to be homogeneous.
A $superalgebra$ is a superspace $\mathcal{G}=\mathcal{G}_{\overline{0}}\oplus \mathcal{G}_{\overline{1}}$ endowed with an algebra structure such that $\mathcal{G}_\alpha \mathcal{G}_\beta \subseteq \mathcal{G}_{\alpha + \beta}$ for $\alpha , \beta \in \mathbb{Z}_2$.
\begin{definition}\label{d21}
A 3-$Lie ~superalgebra$ is a $\mathbb{Z}_2$-graded vector space $\mathcal{G}=\mathcal{G}_{\overline{0}}\oplus \mathcal{G}_{\overline{1}}$ equipped with a trilinear map $[.,.,.]:\wedge^3\mathcal{G}\rightarrow \mathcal{G}$ satisfying
\begin{enumerate}
\item $|[x_1,x_2,x_3]|=|x_1|+|x_2|+|x_3|$,
\item $[x_1,x_2,x_3]=-(-1)^{|x_1||x_2|}[x_2,x_1,x_3]=-(-1)^{|x_2||x_3|}[x_1,x_3,x_2]$,
\item $[x_1,x_2,[x_3,x_4,x_5]]=[[x_1,x_2,x_3],x_4,x_5]+(-1)^{|x_3|(x_1|+|x_2|)}[x_3,[x_1,x_2,x_4],x_5]\\
+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}[x_3,x_4,[x_1,x_2,x_5]]$,
\end{enumerate}
for $x_1,x_2,x_3,x_4,x_5\in \mathcal{G} $ and $|x_i|$ is the degree of homogeneous element $x_i$.
\end{definition}
\par A subsuperspace $\mathcal{N}$ of a $3$-Lie superalgebra $\mathcal{G}$ is said to be a $Lie~ subsuperalgebra$ if it is closed under the superbracket. If $\mathcal{G}$ and $\mathcal{M}$ are 3-Lie superalgebras then a $3$-$Lie~ superalgebra$ $homomorphism$ $\theta:\mathcal{G}\rightarrow \mathcal{M}$ is an even linear map satisfying $\theta([x,y,z])=[\theta(x),\theta(y),\theta(z)]$ for $x,y,z\in \mathcal{G}$.
\begin{definition}\label{d22}
A superderivation of a 3-$Lie ~superalgebra$ $\mathcal{G}$ is a linear map $\mathcal{D}:\mathcal{G}\rightarrow \mathcal{G}$ of degree $\beta$ satisfying:
$$\mathcal{D}([x,y,z])=[\mathcal{D}(x),y,z]+(-1)^{|\beta||x|}[x,\mathcal{D}(y),z]+(-1)^{|\beta|(|x|+|y|)}[x,y,\mathcal{D}(z)],$$ for $x,y,z\in \mathcal{G}.$
\end{definition}
\noindent We denote $Der(\mathcal{G})$ as the space of superderivations of $\mathcal{G}$. Define an even skew-supersymmetric bilinear map $ad:\wedge^2 \mathcal{G}\rightarrow gl(\mathcal{G})$ by $$ad(x_1,x_2)x_3=[x_1,x_2,x_3],$$ for $x_1,x_2,x_3\in \mathcal{G}$.
\begin{definition}\label{d23}
A representation of a $3$-Lie superalgebra $(\mathcal{G},[.,.,.])$ on a superspace $\mathcal{V}$ is a bilinear map $\Phi:\wedge ^2 \mathcal{G}\rightarrow gl(\mathcal{V})$ such that the following equalities hold:
\begin{enumerate}
\item $|\Phi(x_1,x_2)|=|x_1|+|x_2|$,
\item $\Phi(x_1,x_2)=-(-1)^{|x_1||x_2|}\Phi(x_2,x_1)$,
\item $\Phi(x_1,x_2) \Phi(x_3,x_4)=\Phi([x_1,x_2,x_3],x_4)+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(x_3,[x_1,x_2,x_4])\\
+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\Phi(x_3,x_4)\Phi(x_1,x_2)$,
\item $\Phi(x_1,[x_2,x_3,x_4])=(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\Phi(x_3,x_4)\Phi(x_1,x_2)\\-(-1)^{|x_1|(|x_2|+|x_4|)+|x_3||x_4|}\Phi(x_2,x_4)\Phi(x_1,x_3)+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(x_2,x_3)\Phi(x_1,x_4)$,
\end{enumerate} for $x_1,x_2,x_3,x_4\in \mathcal{G}.$
\end{definition}
\noindent We denote a representation of $\mathcal{G}$ on a superspace $\mathcal{V}$ by $(\Phi;\mathcal{V})$.
\par Now onwards we always assume that $(\mathcal{G},[.,.,.])$ is a $3$-Lie superalgebra and we shall write, for any $X=x_1\wedge x_2 \in \wedge ^2 \mathcal{G}$, $x_3\in \mathcal{G}$,
\begin{equation}\label{e21}
[X,x_3]:=[x_1,x_2,x_3]\in \mathcal{G}.
\end{equation}
We shall use the following bilinear operation $[.,.,.]_\mathbb{F}$ on $\wedge ^2 \mathcal{G}$ given by
\begin{equation}\label{e22}
[X,Y]_\mathbb{F}=[X,y_1]\wedge y_2 +(-1)^{|y_1||X|} y _1 \wedge[X,y_2] \in \mathcal{G},
\end{equation}
for $X=x_1\wedge x_2,~Y=y_1\wedge y_2$, and $|X|=|x_1|+|x_2|$. One can see that $\wedge ^2 \mathcal{G}$ is a Leibniz superalgebra with respect to $[.,.]_\mathbb{F}$.
\par Let $(\Phi;\mathcal{V})$ be a representation of $\mathcal{G}$. Cohomology groups of $\mathcal{G}$ with coefficients in $\mathcal{V}$ are defined as in \cite{ma2014}. At first, the space $C^{p-1}(\mathcal{G};\mathcal{V})$ of $p$-cochains is the set of multilinear maps of the form
\begin{equation}\label{e23}
f:\underbrace{\wedge ^2\mathcal{G} \otimes \wedge ^2\mathcal{G}\otimes \dots \otimes \wedge ^2 \mathcal{G}}_{p-1} \otimes \mathcal{G}\rightarrow \mathcal{V},
\end{equation}
while the coboundary operator $\delta_{\Phi}:C^{p-1}(\mathcal{G};\mathcal{V})\rightarrow C^{p}(\mathcal{G};\mathcal{V})$ is given by
\begin{equation}\label{e24}
\begin{split}
&(\delta_{\Phi}f)(X_1,X_2,\dots,X_p,z)\\
&=\sum_{1\leq j<k\leq p}(-1)^j (-1)^{|X_j|(|X_{j+1}|+\dots+|X_{k-1}|)}f(X_1,\dots,\hat{X_j},\dots,X_{k-1},[x_j^1,x_j^2,x_k^1]\wedge x_k^2,\\&~~~~~~X_{k+1},\dots,X_p,x)+\sum_{1\leq j<k\leq p}(-1)^j (-1)^{|X_j|(|X_{j+1}|+\dots+|X_{k-1}|)+|x_k^1||X_j|}f(X_1,\dots,\hat{X_j},\dots,X_{k-1},\\&~~~~~~x_k^1\wedge[x_j^1,x_j^2,x_k^2],X_{k+1},\dots,X_p,x)+\sum_{j=1}^{p}(-1)^j(-1)^{|X_j|(|X_{j+1}|+\dots+|X_p|)}f(X_1,\dots,\hat{X_j},\dots,\\& X_p, [X_j,x])+\sum_{j=1}^{p}(-1)^{j+1}(-1)^{|X_j|(|f|+|X_1|+\dots+|X_{j-1}|)}\Phi(X_j)f(X_1,\dots,\hat{X_j},\dots, X_p, x)\\
&+(-1)^{p+1}(-1)^{(|x_p^2|+|x|)(|f|+|X_1|+\dots+|X_{p-1}|+|x_p^1|)}\Phi(x_p^2,x)f(X_1,\dots, X_{p-1},x_p^1)\\
&+(-1)^{p+1}(-1)^{(|x_p^1|+|x|)(|f|+|X_1|+\dots+|X_{p-1}|)+|X_p||x|}\Phi(x,x_p^1)f(X_1,\dots, X_{p-1},x_p^2),
\end{split}
\end{equation}
for $X_i=x_i\wedge y_i\in \wedge ^2\mathcal{G}$ and $z\in \mathcal{G}$. The $p$$^{th}$ cohomology group is $\mathbb{H}^p(\mathcal{G};\mathcal{V})=\mathbb{Z}^p(\mathcal{G};\mathcal{V})/\mathbb{B}^p(\mathcal{G};\mathcal{V}),$ where $\mathbb{Z}^p(\mathcal{G};\mathcal{V})$ (respectively, $\mathbb{B}^p(\mathcal{G};\mathcal{V}))$ is the space of $(p+1)$-cocycles(respectively, $(p+1)$-coboundaries). We denote $(p+1)$-cocycles of even degree as $(\mathbb{Z}^p(\mathcal{G};\mathcal{V}))_{\overline{0}}$.\\
\noindent By using Eq \ref{e24}, for $f \in \mathbb{C}^0(\mathcal{G};\mathcal{V})$, $X_1=x_1\wedge x_2\in \wedge ^2 \mathcal{G}$, and $x_3\in \mathcal{G}$, we have
\begin{equation}\label{e25}
\begin{split}
&(\delta_{\Phi} f)(X_1,x_3)=-f([X_1,x_3])+(-1)^{|f|(|x_1|+|x_2|)}\Phi(X_1)f(x_3)\\
&+(-1)^{(|f|+|x_1|)(|x_2|+|x_3|)}\Phi(x_2,x_3)f(x_1)+(-1)^{|f|(|x_1|+|x_3|)+|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)f(x_2)\\
&=-f([x_1,x_2,x_3])+(-1)^{|f|(|x_1|+|x_2|)}\Phi(x_1,x_2)f(x_3)\\
&+(-1)^{(|f|+|x_1|)(|x_2|+|x_3|)}\Phi(x_2,x_3)f(x_1)+(-1)^{|f|(|x_1|+|x_3|)+|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)f(x_2),
\end{split}
\end{equation}
and for $f \in \mathbb{C}^1(\mathcal{G};\mathcal{V})$, $X_1=x_1\wedge x_2,~X_2=x_3\wedge x_4$, and $x_5\in \mathcal{G}$,
\begin{equation}\label{e26}
\begin{split}
&(\delta_{\Phi} f)(X_1,X_2,x_5)=-f([X_1,X_2]_\mathbb{F},x_5)-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}f(X_2,[X_1,x_5])+f(X_1,[X_2,x_5])\\
&+(-1)^{|f|(|x_1|+|x_2|)}\Phi(X_1)f(X_2,x_5)-(-1)^{(|f|+|x_1|+|x_2|)(|x_3|+|x_4|)}\Phi(X_2)f(X_1,x_5)\\
&-(-1)^{(|f|+|x_1|+|x_2|+|x_3|)(|x_4|+|x_5|)}\Phi(x_4,x_5)f(X_1,x_3)\\
&-(-1)^{(|f|+|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(|x_3|+|x_4|)}\Phi(x_5,x_3)f(X_1,x_4)\\
&=-f([x_1,x_2,x_3],x_4,x_5)-(-1)^{|x_3|(|x_1|+|x_2|)}f(x_3,[x_1,x_2,x_4],x_5) \\
&-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}f(x_3,x_4,[x_1,x_2,x_5])+f(x_1,x_2,[x_3,x_4,x_5])\\
&+(-1)^{|f|(|x_1|+|x_2|)}\Phi(x_1,x_2)f(x_3,x_4,x_5)-(-1)^{(|f|+|x_1|+|x_2|)(|x_3|+|x_4|)}\Phi(x_3,x_4)f(x_1,x_2,x_5)\\
&-(-1)^{(|f|+|x_1|+|x_2|+|x_3|)(|x_4|+|x_5|)}\Phi(x_4,x_5)f(x_1,x_2,x_3)\\
&-(-1)^{(|f|+|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(|x_3|+|x_4|)}\Phi(x_5,x_3)f(x_1,x_2,x_4),
\end{split}
\end{equation}
where $[.,.,.]_\mathbb{F}$ is given by Eq \ref{e22}.
\par Suppose that $\mathcal{L}$ and $\mathcal{P}$ are 3-Lie superalgebras. If $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\xrightarrow{\pi}\mathcal{Q}\rightarrow 0$ is an exact sequence of 3-Lie superalgebras and $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$ then we call $\mathcal{L}$ an abelian extension of $\mathcal{L}$ by $\mathcal{P}$. An even linear map $s:\mathcal{Q}\rightarrow \mathcal{L}$ is called a section if it satisfies $\pi s=1$. If there exists a section which is also a homomorphism between 3-Lie superalgebras, we say that the abelian extension splits. Now we construct a representation of $\mathcal{Q}$ on $\mathcal{P}$ and a cohomology class. Fix any section $s:\mathcal{Q}\rightarrow \mathcal{L}$ of $\pi$ and define $\Phi:\wedge^2 \mathcal{Q}\rightarrow End(\mathcal{P})$ by
\begin{equation}\label{e27}
\Phi(x,y)(v)=[s(x),s(y),v]_\mathcal{L},
\end{equation}
for $x,y \in \mathcal{Q}$ and $v\in \mathcal{P}$. It is easy to check that $\Phi$ is independent of the choice of $s$. Moreover, since
\begin{equation}\label{e28}
[s(x),s(y),s(z)]_\mathcal{L}-s([x,y,z]_\mathcal{Q})\in \mathcal{P},
\end{equation}
for $x,y,z\in \mathcal{Q}$,
we have a map $\Omega:\wedge^3 \mathcal{Q}\rightarrow End(\mathcal{P})$ given by
\begin{equation}\label{e29}
\Omega(x,y,z)=[s(x),s(y),s(z)]_\mathcal{L}-s([x,y,z]_\mathcal{Q})\in \mathcal{P},
\end{equation}
for $x,y,z\in \mathcal{Q}$.
\begin{lemma}\label{l1}
Let $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\xrightarrow{\pi} \mathcal{Q}\rightarrow 0$ be an extension of $3$-Lie superalgebras with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$. Then
\begin{enumerate}
\item $\Phi$ given by Eq \ref{e27} is a representation of $\mathcal{Q}$ on $\mathcal{P}$.
\item $\Omega$ given by Eq \ref{e29} is a 2-cocycle associated to $(\Phi;\mathcal{P})$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item By the equality
\begin{equation}
\begin{split}
&[s(x_1),u,[s(y_1),s(y_2),s(y_3)]_\mathcal{L}]_\mathcal{L}=[[s(x_1),u,s(y_1)]_\mathcal{L},s(y_2),s(y_3)]_\mathcal{L}\\
&~~~~~~+(-1)^{|y_1|(|x_1|+|u|)}[s(y_1),[s(x_1),u,s(y_2)]_\mathcal{L},s(y_3)]_\mathcal{L}\\&~~~~~~+(-1)^{(|x_1|+|u|)(|y_1|+|y_2|)}[s(y_1),s(y_2),[s(x_1),u,s(y_3)]_\mathcal{L}]_\mathcal{L},
\end{split}
\end{equation}
we have \begin{equation*}
\begin{split}
&[s(x_1),u,[s(y_1),s(y_2),s(y_3)]_\mathcal{L}]_\mathcal{L}=[[s(x_1),u,s(y_1)]_\mathcal{L},s(y_2),s(y_3)]_\mathcal{L}\\
&~~~~~~+(-1)^{|y_1|(|x_1|+|u|)}[s(y_1),[s(x_1),u,s(y_2)]_\mathcal{L},s(y_3)]_\mathcal{L}\\&~~~~~~+(-1)^{(|x_1|+|u|)(|y_1|+|y_2|)}[s(y_1),s(y_2),[s(x_1),u,s(y_3)]_\mathcal{L}]_\mathcal{L}\\
&\implies \Phi(x_1,[y_1,y_2,y_3]_\mathcal{Q})=[\Phi(x_1,y_1)(u),s(y_2),s(y_3)]_\mathcal{L}\\&~~~~~~+(-1)^{|y_1|(|x_1|+|u|)}[s(y_1),\Phi(x_1,y_2)(u),s(y_3)\\&~~~~~~+(-1)^{(|y_1|+|y_2|)(|x_1|+|u|)}[s(y_1),s(y_2),\Phi(x_1,y_3)(u)]_\mathcal{L}\\
&\implies \Phi(x_1,[y_1,y_2,y_3]_\mathcal{Q})=(-1)^{(|x_1|+|y_1|)(|y_2|+|y_3|)}\Phi(y_2,y_3)\Phi(x_1,y_1)\\&~~~~~-(-1)^{|x_1|(|y_1|+|y_3|)+|y_2| |y_3|}\Phi(y_1,y_3)\Phi(x_1,y_2) +(-1)^{|x_1|(|y_1|+|y_2|))}\Phi(y_1,y_2)\Phi(x_1,y_3).
\end{split}
\end{equation*}
Therefore $\Phi$ is a representation of $\mathcal{Q}$ on $\mathcal{P}$.
\item By the equality
\begin{equation}
\begin{split}
&[s(x_1),s(x_2),[s(y_1),s(y_2),s(y_3)]_\mathcal{L}]_\mathcal{L}=[[s(x_1),s(x_2),s(y_1)]_\mathcal{L},s(y_2),s(y_3)]_\mathcal{L}\\
&~~~~~~+(-1)^{|y_1|(|x_1|+|x_2|)}[s(y_1),[s(x_1),s(x_2),s(y_2)]_\mathcal{L},s(y_3)]_\mathcal{L}\\&~~~~~~+(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}[s(y_1),s(y_2),[s(x_1),s(x_2),s(y_3)]_\mathcal{L}]_\mathcal{L},
\end{split}
\end{equation}
\noindent we have
\begin{equation*}
\begin{split}
&[s(x_1),s(x_2),\Omega(y_1,y_2,y_3)]_\mathcal{L}+[s(x_1),s(x_2),s([y_1,y_2,y_3]_\mathcal{Q})]_\mathcal{L}\\
&~~~~~~=[\Omega (x_1,x_2,y_1),s(y_2),s(y_3)]_\mathcal{L}+[s([x_1,x_2,y_1]_\mathcal{Q}),s(y_2),s(y_3)]_\mathcal{L}\\
&~~~~~~+(-1)^{|y_1|(|x_1|+|x_2|)}[s(y_1),\Omega (x_1,x_2,y_2),s(y_3)]_\mathcal{L}\\&~~~~~~+(-1)^{|y_1|(|x_1|+|x_2|)}[s(y_1),s([x_1,x_2,y_2]_\mathcal{Q}),s(y_3)]\\
&~~~~~~+(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}[s(y_1),s(y_2),\Omega (x_1,x_2,y_3)]_\mathcal{L}\\&~~~~~~+(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}[s(y_1),s(y_2),s([x_1,x_2,y_3]_\mathcal{Q})]_\mathcal{L}\\
&\implies \Phi(x_1,x_2) (\Omega(y_1,y_2,y_3))+\Omega(x_1,x_2,[y_1,y_2,y_3]_\mathcal{Q})+s([x_1,x_2,[y_1,y_2,y_3]_\mathcal{Q}]_\mathcal{Q})\\
&~~~~~~=\Phi(y_2,y_3) (\Omega(x_1,x_2,y_1))+\Omega([x_1,x_2,y_1]_\mathcal{Q},y_2,y_3)+s([x_1,x_2,y_1]_\mathcal{Q},y_2,y_3)\\
&~~~~~~+(-1)^{|y_1|(|x_1|+|x_2|)}(\Phi(y_3,y_1) (\Omega(x_1,x_2,y_2))+(-1)^{|y_1|(|x_1|+|x_2|)}\Omega(y_1,[x_1,x_2,y_2]_\mathcal{Q},y_3)\\&~~~~~+(-1)^{ |y_1|(|x_1|+|x_2|)}s(y_1,[x_1,x_2,y_2]_\mathcal{Q},y_3))+(-1)^{ (|x_1|+|x_2|)(|y_1|+|y_2|)}(\Phi(y_1,y_2) (\Omega(x_1,x_2,y_3))\\&~~~~~~+(-1)^{ (|x_1|+|x_2|)(|y_1|+|y_2|)}\Omega(y_1,y_2,[x_1,x_2,y_3]_\mathcal{Q})+(-1)^{ (|x_1|+|x_2|)(|y_1|+|y_2|)}s[y_1,y_2,[x_1,x_2,y_3]_\mathcal{Q}]_\mathcal{Q})\\
&\implies \Phi(x_1,x_2) (\Omega(y_1,y_2,y_3))+\Omega(x_1,x_2,[y_1,y_2,y_3]_\mathcal{Q})=\Phi(y_2,y_3) (\Omega(x_1,x_2,y_1))\\&~~~~~~+\Omega([x_1,x_2,y_1]_\mathcal{Q},y_2,y_3)+(-1)^{|y_1|(|x_1|+|x_2|)}(\Phi(y_3,y_1) (\Omega(x_1,x_2,y_2))\\&~~~~~~~+(-1)^{ |y_1|(|x_1|+|x_2|)}\Omega(y_1,[x_1,x_2,y_2]_\mathcal{Q},y_3)+(-1)^{ (|x_1|+|x_2|)(|y_1|+|y_2|)}(\Phi(y_1,y_2) (\Omega(x_1,x_2,y_3))\\&~~~~~~~+(-1)^{ (|x_1|+|x_2|)(|y_1|+|y_2|)}\Omega(y_1,y_2,[x_1,x_2,y_3]_\mathcal{Q}).
\end{split}
\end{equation*}
\noindent Hence $\Omega$ is a 2-cocycle associated to $(\Phi;\mathcal{P})$.
\end{enumerate}
\end{proof}
\begin{corollary}\label{coro21}
Let $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\xrightarrow{\pi} \mathcal{Q}\rightarrow 0$ be an extension of $3$-Lie superalgebras with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$. Then the cohomology class $[\Omega]$ does not depend on the choice of $s$.
\end{corollary}
\begin{proof}
Let $s_1$ and $s_2$ be sections of $\pi$ and $\Omega_1,\Omega_2$ be defined by Eq \ref{e29} which are corresponding to $s_1,s_2$, respectively. For any $x\in \mathcal{Q}$, set $\lambda(x)=s_1(x)-s_2(x)$.\\
$(\pi \lambda)(x)=x-x=0$, $\lambda(x)\in \mathcal{P}$, and $\lambda\in (\mathbb{C}^0(\mathcal{G};\mathcal{V}))_{\overline{0}}$. Then
\begin{equation*}
\begin{split}
&\Omega_1(x,y,z)-\Omega_2(x,y,z)\\
&=[s_1(x),s_1(y),s_1(z)]_\mathcal{L}-s_1([x,y,z]_\mathcal{Q})-[s_2(x),s_2(y),s_2(z)]_\mathcal{L}+s_2([x,y,z]_\mathcal{Q})\\
&=[s_2(x)+\lambda(x),s_2(y)+\lambda(y),s_2(z)+\lambda(z)]_\mathcal{L}-s_2([x,y,z]_\mathcal{Q})+\lambda([x,y,z]_\mathcal{Q})\\
&~~~~~~-[s_2(x),s_2(y),s_2(z)]_\mathcal{L}+s_2([x,y,z]_\mathcal{Q})\\
&=[s_2(x),s_2(y),\lambda(z)]_\mathcal{L}+[\lambda(x),s_2(y),s_2(z)]_\mathcal{L}+[s_2(x),\lambda(y),s_2(z)]_\mathcal{L}-\lambda([x,y,z]_\mathcal{Q}) \\
&=-\lambda([x,y,z]_\mathcal{Q})+\Phi(x,y)(\lambda(z))+(-1)^{|x|(|y|+|z|)}\Phi(y,z)(\lambda(x))\\
&~~~~~~~+(-1)^{|z|(|x|+|y|)}\Phi(z,x)(\lambda(y))\\
&=(\delta_{\Phi}\lambda)(x,y,z),
\end{split}
\end{equation*}
which completes the proof.
\end{proof}
\begin{proposition}
If $(\Phi, \mathcal{P})$ is a representation of $\mathcal{Q}$ and $\Omega$ is a $2$-cocycle given by the representation $(\Phi, \mathcal{P})$, then $\mathcal{L}_{\Phi,\Omega}:=\mathcal{Q}\oplus \mathcal{P}$ is a 3-Lie superalgebra with the superbracket given by
\begin{equation}\label{e210}
\begin{split}
[x+u,y+v,z+w]_{\mathcal{L}_{\Phi,\Omega}}=&[x,y,z]_{\mathcal{Q}}+\Omega(x,y,z)+\Phi(x,y)(w)\\&~~~~~~+(-1)^{|y||z|}\Phi(z,x)(v)+(-1)^{|x|(|y|+|z|)}\Phi(y,z)(u),
\end{split}
\end{equation}
where $x,y,z\in \mathcal{Q}$ and $u,v,w\in \mathcal{P}$.
\end{proposition}
\begin{proof}
It is sufficient to verify the fundamental identity. So, we have
\begin{equation}\label{e211}
\begin{split}
&[x_1+u_1,x_2+u_2,[y_1+v_1,y_2+v_2,y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}]_{\mathcal{L}_{\Phi,\Omega}}\\
&=[x_1,x_2,[y_1,y_2,y_3]_\mathcal{Q}]_\mathcal{Q}+\Omega(x_1,x_2,[y_1,y_2,y_3]_\mathcal{Q})+\Phi(x_1,x_2)(\Omega(y_1,y_2,y_3)\\
&~~~~~~+\Phi(y_1,y_2)(v_3)+(-1)^{|y_2||y_3|}\Phi(y_3,y_1)(v_2)+(-1)^{|y_1|(|y_2|+|y_3|)}\Phi(y_2,y_3)(v_1))\\&~~~~~~~+(-1)^{|x_2|(|y_1|+|y_2|+|y_3|)}\Phi([y_1,y_2,y_3]_\mathcal{Q},x_1)(u_2)\\&~~~~~~+(-1)^{|x_1|(|x_2|+|y_1|+|y_2|+|y_3|)}\Phi(x_2,[y_1,y_2,y_3]_\mathcal{Q})(u_1),
\end{split}
\end{equation}
\begin{equation}\label{e212}
\begin{split}
&[[x_1+u_1,x_2+u_2,y_1+v_1]_{\mathcal{L}_{\Phi,\Omega}},y_2+v_2,y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}\\
&=[[x_1,x_2,y_1]_\mathcal{Q},y_2,y_3]_\mathcal{Q}+\Omega([x_1,x_2,y_1]_\mathcal{Q},y_2,y_3)\\&~~~~~~+(-1)^{(|x_1|+|x_2|+|y_1|)(|y_2|+|y_3|)}\Phi(y_2,y_3)(\Omega(x_1,x_2,y_1)\\
&~~~~~~+\Phi(x_1,x_2)(v_1)+(-1)^{|x_2||y_1|}\Phi(y_1,x_1)(u_2)+(-1)^{|x_1|(|x_2|+|y_1|)}\Phi(x_2,y_1)(u_1))\\&~~~~~~~+\Phi([x_1,x_2,y_1]_\mathcal{Q},y_2)(v_3)+(-1)^{|y_3|(|x_1|+|x_2|+|y_2|)}\Phi(y_3,[x_1,x_2,y_1]_\mathcal{Q})(v_2),
\end{split}
\end{equation}
\begin{equation}\label{e213}
\begin{split}
&(-1)^{|y_1|(|x_1|+|x_2|)}[y_1+v_1,[x_1+u_1,x_2+u_2,y_2+v_2]_{\mathcal{L}_{\Phi,\Omega}},y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}\\
&=(-1)^{|y_1|(|x_1|+|x_2|)}[y_1,[x_1,x_2,y_2]_\mathcal{Q},y_3]_\mathcal{Q}+(-1)^{|y_1|(|x_1|+|x_2|)}\Omega(y_1,[x_1,x_2,y_2]_\mathcal{Q},y_3) \\&~~~~~~+(-1)^{(|x_1|+|x_2|+|y_1|)(|y_2|+|y_3|)+|y_1|(|x_1|+|x_2|)}\Phi(y_3,y_1)(\Omega(x_1,x_2,y_2)\\
&~~~~~~+\Phi(x_1,x_2)(v_2)+(-1)^{|x_2||y_2|}\Phi(y_2,x_1)(u_2)+(-1)^{|x_1|(|x_2|+|y_2|)}\Phi(x_2,y_2)(u_1))\\&~~~~~~~+(-1)^{|y_1|(|x_1|+|x_2|)}\Phi(y_1,[x_1,x_2,y_2]_\mathcal{Q})(v_3)\\ &~~~~~~+(-1)^{|y_3|(|x_1|+|x_2|+|y_2|)+|y_1|(|x_1|+|x_2|)}\Phi([x_1,x_2,y_2]_\mathcal{Q},y_3)(v_1),
\end{split}\end{equation}
and
\begin{equation}\label{e214}
\begin{split}
&(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}[y_1+v_1,y_2+v_2,[x_1+u_1,x_2+u_2,y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}]_{\mathcal{L}_{\Phi,\Omega}}\\
&=(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}[y_1,y_2,[x_1,x_2,y_3]_\mathcal{Q}]_\mathcal{Q}\\&~~~~~~ +(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}\Omega(y_1,y_2,[x_1,x_2,y_3]_\mathcal{Q}) \\&~~~~~~+(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}\Phi(y_1,y_2)(\Omega(x_1,x_2,y_3)+\Phi(x_1,x_2)(v_3)\\
&~~~~~~+(-1)^{|x_2||y_3|}\Phi(y_3,x_1)(u_2)+(-1)^{|x_1|(|x_2|+|y_3|)}\Phi(x_2,y_3)(u_1))\\&~~~~~~~+(-1)^{|y_2|(|x_1|+|x_2|+|y_3|)+(|x_1|+|x_2|)(|y_1|+|y_2|)}\Phi([x_1,x_2,y_3]_\mathcal{Q},y_1)(v_2)\\ &~~~~~~+(-1)^{|y_1|(|x_1|+|x_2|+|y_2|+|y_3|)+(|x_1|+|x_2|)(|y_1|+|y_2|)}\Phi(y_2,[x_1,x_2,y_3]_\mathcal{Q})(v_1).
\end{split}\end{equation}
We will see that Eq $(\ref{e211})$=Eq $(\ref{e212})$+Eq $(\ref{e213})$+Eq $(\ref{e214})$, if $\Phi$ is a representation and $\Omega$ is a $2$-cocycle.
Hence, we have
\begin{equation}\label{e215}
\begin{split}
&[x_1+u_1,x_2+u_2,[y_1+v_1,y_2+v_2,y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}]_{\mathcal{L}_{\Phi,\Omega}}\\&=[[x_1+u_1,x_2+u_2,y_1+v_1]_{\mathcal{L}_{\Phi,\Omega}},y_2+v_2,y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}\\&~~~~~~+(-1)^{|y_1|(|x_1|+|x_2|)}[y_1+v_1,[x_1+u_1,x_2+u_2,y_2+v_2]_{\mathcal{L}_{\Phi,\Omega}},y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}\\&~~~~~~~+
(-1)^{(|x_1|+|x_2|)(|y_1|+|y_2|)}[y_1+v_1,y_2+v_2,[x_1+u_1,x_2+u_2,y_3+v_3]_{\mathcal{L}_{\Phi,\Omega}}]_{\mathcal{L}_{\Phi,\Omega}},
\end{split}
\end{equation}
which implies $\mathcal{L}_{\Phi,\Omega}$ is a 3-Lie superalgebra.
\end{proof}
\begin{proposition}\label{p2}
If $\mathbb{H}^1(\mathcal{Q};\mathcal{P})=0$ then the extension splits.
\end{proposition}
\begin{proof}
It is sufficient to show that there is a section of $\pi$ which is a 3-Lie superalgebra homomorphism. It is known that $(\Phi;\mathcal{P})$ given by Eq \ref{e27} is independent of the choice of the sections of $\pi$. Consider the 2-cocycle $\Omega$ given by Lemma \ref{l1}. Since $\mathbb{H}^1(\mathcal{Q};\mathcal{P})=0$, there exists an $\xi\in (\mathbb{C}^0(\mathcal{Q};\mathcal{P}))_{\overline{0}}$ such that $\Omega=\delta_{\Phi}\xi$. For any $x,y,z\in \mathcal{Q}$, it follows that
\begin{equation}
\begin{split}
\Omega(x,y,z)= &-\xi([x,y,z]_\mathcal{Q})+\Phi(x,y)(\xi(z))+(-1)^{|x|(|y|+|z|)}\Phi(y,z)(\xi(x))\\
&+(-1)^{|z|(|x|+|y|)}\Phi(z,x)(\xi(y)).
\end{split}
\end{equation}
Define an even linear map $s^{'}:\mathcal{Q}\rightarrow \mathcal{P}$ by
$$s^{'}(x)=s(x)-\xi(x).$$
Note that $s^{'}$ is also a section of $\pi$. Then for any $x,y,z\in \mathcal{Q}$, we have
\begin{equation*}
\begin{split}
&~~~~~~~~~[s^{'}(x),s^{'}(y),s^{'}(z)]_\mathcal{L}\\
&=[s(x)-\xi(x),s(y)-\xi(y),s(z)-\xi(z)]_\mathcal{L}\\
&=[s_1(x),s_1(y),s_1(z)]_\mathcal{L}-\Phi(x,y)(\xi(z))-(-1)^{|x|(|y|+|z|)}\Phi(y,z)(\xi(x))-(-1)^{|z|(|x|+|y|)}\Phi(z,x)(\xi(y))\\
&=s([x,y,z]_\mathcal{Q})+\Omega(x,y,z)-\Phi(x,y)(\xi(z))-(-1)^{|x|(|y|+|z|)}\Phi(y,z)(\xi(x))-(-1)^{|z|(|x|+|y|)}\Phi(z,x)(\xi(y))\\
&=s([x,y,z]_\mathcal{Q})-\xi([x,y,z]_\mathcal{Q})\\
&=s^{'}([x,y,z]_\mathcal{Q}).
\end{split}
\end{equation*}
Hence $s^{'}$ is a $3$-Lie superalgebra homomorphism.
\end{proof}
\section{Cohomology Classes and a Lie superalgebra}
Let $(\Phi;\mathcal{P})$ be a representation of a $3$-Lie superalgebra $\mathcal{Q}$ on $\mathcal{P}$. In this section, we use superderivations of $\mathcal{P}$, $\mathcal{Q}$ to construct first-order cohomology classes. By using this, we construct a Lie superalgebra and its representation on $\mathbb{H}^1(\mathcal{Q};\mathcal{P})$.
\par Let $\mathcal{P},\mathcal{Q}$ be $3$-Lie superalgebras. Given a representation $(\Phi;\mathcal{P})$ of $\mathcal{Q}$. Suppose $\Omega \in (\mathbb{C}^1(\mathcal{Q};\mathcal{P}))_{\overline{0}}$. For any pair $(\mathcal{D}_p,\mathcal{D}_q)\in Der(\mathcal{P})\times Der(\mathcal{Q})$, define a $2$-cochain $Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)} \in \mathbb{C}^1(\mathcal{Q};\mathcal{P})$ as
\begin{equation}\label{e31}
Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}=\mathcal{D}_p\Omega-\Omega(\mathcal{D}_q\otimes 1\otimes 1)-\Omega(1\otimes \mathcal{D}_q\otimes 1)-\Omega(1\otimes 1\otimes \mathcal{D}_q),
\end{equation}
where ``1'' denotes the identity map and the degree of identity map is always even. This is equivalent to
\begin{equation}\label{e32}
\begin{split}
Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x,y,z)=&\mathcal{D}_p(\Omega(x,y,z))-\Omega(\mathcal{D}_q(x),y,z)-(-1)^{|\alpha| |x|}\Omega(x,\mathcal{D}_q(y),z)\\&-(-1)^{|\alpha| (|x|+|y|)}\Omega(x,y,\mathcal{D}_q(z)),
\end{split}
\end{equation}
for $x,y,z\in \mathcal{Q}$ and $|Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}|=|\mathcal{D}_p|=|\mathcal{D}_q|=\alpha$ where $\alpha \in \mathbb{Z}_2$.
\par We begin with the following.
\begin{lemma}\label{l31}
Let $(\Phi;\mathcal{P})$ be a representation of $\mathcal{Q}$ and $\Omega \in (\mathbb{C}^1(\mathcal{G};\mathcal{V}))_{\overline{0}}$ associated to the representation $(\Phi;\mathcal{P})$. Assume that a pair $(\mathcal{D}_p,\mathcal{D}_q)\in Der(\mathcal{P})\times Der(\mathcal{Q})$ satisfies that
\begin{equation}\label{e33}
\mathcal{D}_p\Phi(x,y)-(-1)^{|\alpha|(|x|+|y|)}\Phi(x,y)\mathcal{D}_p=\Phi(\mathcal{D}_q(x),y)+(-1)^{|\alpha||x|}\Phi(x,\mathcal{D}_q(y)),
\end{equation}
for $x,y\in \mathcal{Q}$ and $|\mathcal{D}_p|=|\mathcal{D}_q|=\alpha$. If $\Omega$ is a $2$-cocycle then $Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}\in \mathbb{C}^1(\mathcal{G};\mathcal{V})$ given by Eq \ref{e32} is also a $2$-cocycle.
\end{lemma}
\begin{proof}
It is sufficent to show that $\delta_\Phi Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}=0$. Since $\Omega$ is a $2$-cocycle, so $\delta_\Phi \Omega=0$, by Eq \ref{e26} it follows that, for any $x_i\in \mathcal{Q}$
\begin{equation} \label{e34}
\begin{split}
0=&-\Omega ([x_1,x_2,x_3]_\mathcal{Q},x_4,x_5)-(-1)^{|x_3|(|x_1|+|x_2|)}\Omega(x_3,[x_1,x_2,x_4]_\mathcal{Q},x_5)\\&-(-1)^{(|x_3|+|x_4|)(|x_1|+|x_2|)}\Omega(x_3,x_4,[x_1,x_2,x_5]_\mathcal{Q})+\Omega(x_1,x_2,[x_3,x_4,x_5]_\mathcal{Q})\\
&+\Phi(x_1,x_2)\Omega(x_3,x_4,x_5)-(-1)^{(|x_3|+|x_4|)(|x_1|+|x_2|)}\Phi(x_3,x_4)\Omega(x_1,x_2,x_5)\\
&-(-1)^{(|x_4|+|x_5|)(|x_1|+|x_2|+|x_3|)}\Phi(x_4,x_5)\Omega(x_1,x_2,x_3)\\&-(-1)^{(|x_3|+|x_5|)(|x_1|+|x_2|)+(|x_3|+|x_4|)|x_5|}\Phi(x_5,x_3)\Omega(x_1,x_2,x_4).
\end{split}
\end{equation}
Now $|Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}|=|\mathcal{D}_p|=|\mathcal{D}_q|=\alpha$, we have
\begin{equation}\label{e35}
\begin{split}
\delta_\Phi &Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_1,x_2,x_3,x_4,x_5)\\&=-\underbrace{Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}([x_1,x_2,x_3]_\mathcal{Q},x_4,x_5)}_{(1)}-\underbrace{(-1)^{|x_3|(|x_1|+|x_2|)}Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_3,[x_1,x_2,x_4]_\mathcal{Q},x_5)}_{(2)}\\&
-\underbrace{(-1)^{(|x_3|+|x_4|)(|x_1|+|x_2|)}Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_3,x_4,[x_1,x_2,x_5]_\mathcal{Q})}_{(3)}
+\underbrace{Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_1,x_2,[x_3,x_4,x_5]_\mathcal{Q})}_{(4)}\\
&+\underbrace{(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_3,x_4,x_5)}_{(5)}\\&-\underbrace{(-1)^{(|x_3|+|x_4|)(\alpha+|x_1|+|x_2|)}\Phi(x_3,x_4)Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_1,x_2,x_5)}_{(6)}\\
&-\underbrace{(-1)^{(|x_4|+|x_5|)(\alpha+|x_1|+|x_2|+|x_3|)}\Phi(x_4,x_5)Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_1,x_2,x_3)}_{(7)}\\&-\underbrace{(-1)^{(|x_3|+|x_5|)(\alpha+|x_1|+|x_2|)+(|x_3|+|x_4|)|x_5|}\Phi(x_5,x_3)Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}(x_1,x_2,x_4)}_{(8)}.
\end{split}
\end{equation}
Applying the definition of $Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}$ as in Eq \ref{e32}, we get
\begin{equation}\label{e36}
\begin{split}
(1)&=-\mathcal{D}_p(\Omega([x_1,x_2,x_3]_\mathcal{Q},x_4,x_5))+\Omega([\mathcal{D}_q(x_1),x_2,x_3]_\mathcal{Q},x_4,x_5)\\
&~~~~~~+(-1)^{\alpha|x_1|}\Omega([x_1,\mathcal{D}_q(x_2),x_3]_\mathcal{Q},x_4,x_5)\\&~~~~~~+(-1)^{\alpha(|x_1|+|x_2|)}\Omega([x_1,x_2,\mathcal{D}_q(x_3)]_\mathcal{Q},x_4,x_5)\\
&~~~~~~+(-1)^{\alpha(|x_1|+|x_2|+|x_3|)}\Omega([x_1,x_2,x_3]_\mathcal{Q},\mathcal{D}_q(x_4),x_5)\\
&~~~~~~+(-1)^{\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega([x_1,x_2,x_3]_\mathcal{Q},x_4,\mathcal{D}_q(x_5)),
\end{split}
\end{equation}
\begin{equation}\label{e37}
\begin{split}
(2)
&=-(-1)^{|x_3|(|x_1|+|x_2|)}\mathcal{D}_p(\Omega(x_3,[x_1,x_2,x_4]_\mathcal{Q},x_5))\\&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)}\Omega(\mathcal{D}_q(x_3),[x_1,x_2,x_4]_\mathcal{Q},x_5)\\
&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)+\alpha|x_3|}\Omega(x_3,[\mathcal{D}_q(x_1),x_2,x_4]_\mathcal{Q},x_5)\\&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)+\alpha(|x_1|+|x_3|)}\Omega(x_3,[x_1,\mathcal{D}_q(x_2),x_4]_\mathcal{Q},x_5)\\&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)+\alpha(|x_1|+|x_2|+|x_3|)}\Omega(x_3,[x_1,x_2,\mathcal{D}_q(x_4)]_\mathcal{Q},x_5)\\
&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)+\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega(x_3,[x_1,x_2,x_4]_\mathcal{Q},\mathcal{D}_q(x_5)),
\end{split}
\end{equation}
\begin{equation}\label{e38}
\begin{split}
(3)
&=-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\mathcal{D}_p(\Omega (x_3,x_4,[x_1,x_2,x_5]_\mathcal{Q}))\\&~~~~~~+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\Omega (\mathcal{D}_q (x_3),x_4,[x_1,x_2,x_5]_\mathcal{Q})\\
&~~~~~~+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)+\alpha|x_3|}\Omega (x_3,\mathcal{D}_q (x_4),[x_1,x_2,x_5]_\mathcal{Q})\\
&~~~~~~+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)+\alpha(|x_3|+|x_4|)}\Omega (x_3,x_4,[\mathcal{D}_q (x_1),x_2,x_5]_\mathcal{Q})\\
&~~~~~~+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)+\alpha(|x_1|+|x_3|+|x_4|)}\Omega (x_3,x_4,[x_1,\mathcal{D}_q (x_2),x_5]_\mathcal{Q})\\
&~~~~~~+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)+\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega (x_3,x_4,[x_1,x_2,\mathcal{D}_q (x_5)]_\mathcal{Q}),
\end{split}
\end{equation}
\begin{equation}\label{e39}
\begin{split}
(4)&=\mathcal{D}_p(\Omega(x_1,x_2,[x_3,x_4,x_5]_\mathcal{Q}))-\Omega (\mathcal{D}_q(x_1),x_2,[x_3,x_4,x_5]_\mathcal{Q})\\
&~~~~~~-(-1)^{\alpha|x_1|}\Omega (x_1,\mathcal{D}_q(x_2),[x_3,x_4,x_5]_\mathcal{Q})\\&~~~~~~-(-1)^{\alpha(|x_1|+|x_2|)}\Omega (x_1,x_2,[\mathcal{D}_q(x_3),x_4,x_5]_\mathcal{Q})\\&~~~~~~-(-1)^{\alpha(|x_1|+|x_2|+|x_3|)}\Omega (x_1,x_2,[x_3,\mathcal{D}_q(x_4),x_5]_\mathcal{Q})\\&~~~~~~
-(-1)^{\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega (x_1,x_2,[x_3,x_4,\mathcal{D}_q(x_5)]_\mathcal{Q}),
\end{split}
\end{equation}
\begin{equation}\label{e310}
\begin{split}
(5)&=-(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)(\mathcal{D}_p(\Omega(x_3,x_4,x_5))-\Omega(\mathcal{D}_q(x_3),x_4,x_5)\\
&~~~~~~-(-1)^{\alpha|x_3|}\Omega(x_3,\mathcal{D}_q(x_4),x_5)-(-1)^{\alpha(|x_3|+|x_4|)}\Omega(x_3,x_4,\mathcal{D}_q(x_5))),
\end{split}
\end{equation}
\begin{equation}\label{e311}
\begin{split}
(6)&=-(-1)^{(|x_3|+|x_4|)(\alpha+|x_1|+|x_2|)}\Phi(x_3,x_4)(\mathcal{D}_p(\Omega(x_1,x_2,x_5))-\Omega(\mathcal{D}_q(x_1),x_2,x_5)\\
&~~~~~~-(-1)^{\alpha|x_1|}\Omega(x_1,\mathcal{D}_q(x_2),x_5)-(-1)^{\alpha(|x_1|+|x_2|)}\Omega(x_1,x_2,\mathcal{D}_q(x_5))),
\end{split}
\end{equation}
\begin{equation}\label{e312}
\begin{split}
(7)&=-(-1)^{(|x_4|+|x_5|)(\alpha+|x_1|+|x_2|+|x_3|)}\Phi(x_4,x_5)(\mathcal{D}_p(\Omega(x_1,x_2,x_3))-\Omega(\mathcal{D}_q(x_1),x_2,x_3)\\
&~~~~~~-(-1)^{\alpha|x_1|}\Omega(x_1,\mathcal{D}_q(x_2),x_3)-(-1)^{\alpha(|x_1|+|x_2|)}\Omega(x_1,x_2,\mathcal{D}_q(x_3))),
\end{split}
\end{equation}
\begin{equation}\label{e313}
\begin{split}
(8)&=-(-1)^{(|x_3|+|x_5|)(\alpha+|x_1|+|x_2|)+|x_5|(|x_3|+|x_4|)}\Phi(x_5,x_3)(\mathcal{D}_p(\Omega(x_1,x_2,x_4))-\Omega(\mathcal{D}_q(x_1),x_2,x_4)\\
&~~~~~~-(-1)^{\alpha|x_1|}\Omega(x_1,\mathcal{D}_q(x_2),x_4)-(-1)^{\alpha(|x_1|+|x_2|)}\Omega(x_1,x_2,\mathcal{D}_q(x_4))).
\end{split}
\end{equation}
\begin{equation}\label{e314}
\begin{split}
&-\mathcal{D}_p(\Omega([x_1,x_2,x_3]_\mathcal{Q},x_4,x_5))-(-1)^{|x_3|(|x_1|+|x_2|)}\mathcal{D}_p(\Omega(x_3,[x_1,x_2,x_4]_\mathcal{Q},x_5))\\&-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\mathcal{D}_p(\Omega(x_3,x_4,[x_1,x_2,x_5]_\mathcal{Q}))+\mathcal{D}_p(\Omega (x_1,x_2,[x_3,x_4,x_5]_\mathcal{Q}))\\
&=-\mathcal{D}_p(\Phi(x_1,x_2)(\Omega(x_3,x_4,x_5)))+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\mathcal{D}_p(\Phi(x_3,x_4)(\Omega(x_1,x_2,x_5)))\\&+(-1)^{(|x_1|+|x_2|+|x_3|)(|x_4|+|x_5|)}\mathcal{D}_p(\Phi(x_4,x_5)(\Omega(x_1,x_2,x_3)))\\& +(-1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(|x_3|+|x_4|)}\mathcal{D}_p(\Phi(x_5,x_3)(\Omega(x_1,x_2,x_4))).
\end{split}
\end{equation}
For Eq \ref{e36}-\ref{e314}, by suitable combinations and with the aid of Eq \ref{e34}, we get
\begin{equation}\label{e315}
\begin{split}
&\Omega([\mathcal{D}_q(x_1),x_2,x_3]_\mathcal{Q},x_4,x_5)+(-1)^{|x_3|(|x_1|+|x_2|)+\alpha|x_3|}\Omega(x_3,[\mathcal{D}_q(x_1),x_2,x_4]_\mathcal{Q},x_5)\\
&+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)+\alpha(|x_3|+|x_4|)}\Omega (x_3,x_4,[\mathcal{D}_q (x_1),x_2,x_5]_\mathcal{Q})\\
&-\Omega (\mathcal{D}_q(x_1),x_2,[x_3,x_4,x_5]_\mathcal{Q})+(-1)^{(|x_3|+|x_4|)(\alpha+|x_1|+|x_2|)}\Phi(x_3,x_4)\Omega(\mathcal{D}_q(x_1),x_2,x_5)\\
&+(-1)^{(|x_4|+|x_5|)(\alpha+|x_1|+|x_2|+|x_3|)}\Phi(x_4,x_5)\Omega(\mathcal{D}_q(x_1),x_2,x_3)\\&-(-1)^{(|x_3|+|x_5|)(\alpha+|x_1|+|x_2|)+|x_5|(|x_3|+|x_4|)}\Phi(x_5,x_3)\Omega(\mathcal{D}_q(x_1),x_2,x_4)\\
&=\Phi(\mathcal{D}_qx_1,x_2)\Omega(x_3,x_4,x_5),
\end{split}
\end{equation}
\begin{equation}\label{e316}
\begin{split}
&(-1)^{\alpha|x_1|}\Omega([x_1,\mathcal{D}_q(x_2),x_3]_\mathcal{Q},x_4,x_5)+(-1)^{(|x_3|+\alpha)(|x_1|+|x_2|)}\Omega(x_3,[x_1,\mathcal{D}_q(x_2),x_4]_\mathcal{Q},x_5)\\
&+(-1)^{(|x_1|+|x_2|+\alpha)(|x_3|+|x_4|)+\alpha|x_1|}\Omega (x_3,x_4,[x_1,\mathcal{D}_q (x_2),x_5]_\mathcal{Q})\\
&-(-1)^{\alpha|x_1|}\Omega(x_1,\mathcal{D}_q(x_2),[x_3,x_4,x_5]_\mathcal{Q})\\
&+(-1)^{(|x_3|+|x_4|)(\alpha+|x_1|+|x_2|)+\alpha|x_1|}\Phi(x_3,x_4)\Omega(x_1,\mathcal{D}_q(x_2),x_5)\\
&+(-1)^{(|x_4|+|x_5|)(\alpha+|x_1|+|x_2|+|x_3|)+\alpha|x_1|}\Phi(x_4,x_5)\Omega(x_1,\mathcal{D}_q(x_2),x_3)\\
&+(-1)^{(|x_3|+|x_5|)(\alpha+|x_1|+|x_2|)+|x_5|(|x_3|+|x_4|)+\alpha|x_1|}\Phi(x_5,x_3)\Omega(x_1,\mathcal{D}_q(x_2),x_4)\\
&=(-1)^{\alpha|x_1|}\Phi(x_1,\mathcal{D}_q(x_2))\Omega(x_3,x_4,x_5),
\end{split}
\end{equation}
\begin{equation}\label{e317}
\begin{split}
&(-1)^{\alpha(|x_1|+|x_2|)}\Omega([x_1,x_2,\mathcal{D}_q(x_3)]_\mathcal{Q},x_4,x_5)+(-1)^{|x_3|(|x_1|+|x_2|)}\Omega(\mathcal{D}_q(x_3),[x_1,x_2,x_4]_\mathcal{Q},x_5)\\
&+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\Omega (\mathcal{D}_q (x_3),x_4,[x_1,x_2,x_5]_\mathcal{Q})-(-1)^{\alpha(|x_1|+|x_2|)}\Omega (x_1,x_2,[\mathcal{D}_q(x_3),x_4,x_5]_\mathcal{Q})\\&+(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)\Omega(\mathcal{D}_q(x_3),x_4,x_5)\\&+(-1)^{(|x_4|+|x_5|)(\alpha+|x_1|+|x_2|+|x_3|)+\alpha(|x_1|+|x_2|)}\Phi(x_4,x_5)\Omega(x_1,x_2,\mathcal{D}_q(x_3))\\
&=-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\Phi(\mathcal{D}_q(x_3),x_4)\Omega(x_1,x_2,x_5)\\&-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(\alpha+|x_3|+|x_4|)}\Phi(x_5,\mathcal{D}_q(x_3))\Omega(x_1,x_2,x_4),
\end{split}
\end{equation}
\begin{equation}\label{e318}
\begin{split}
&(-1)^{\alpha(|x_1|+|x_2|+|x_3|)}\Omega([x_1,x_2,x_3]_\mathcal{Q},\mathcal{D}_q(x_4),x_5)\\
&+(-1)^{|x_3|(|x_1|+|x_2|)+\alpha(|x_1|+|x_2|+|x_3|)}\Omega(x_3,[x_1,x_2,\mathcal{D}_q(x_4)]_\mathcal{Q},x_5)\\
&+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)+\alpha|x_3|}\Omega (x_3,\mathcal{D}_q (x_4),[x_1,x_2,x_5]_\mathcal{Q})\\
&-(-1)^{\alpha(|x_1|+|x_2|+|x_3|)}\Omega (x_1,x_2,[x_3,\mathcal{D}_q(x_4),x_5]_\mathcal{Q})+(-1)^{\alpha(|x_1|+|x_2|+|x_3|)}\Phi(x_1,x_2)\Omega(x_3,\mathcal{D}_q(x_4),x_5)\\
&+(-1)^{(|x_3|+|x_5|)(\alpha+|x_1|+|x_2|)+|x_5|(|x_3|+|x_4|)+\alpha(|x_1|+|x_2|)}\Phi(x_5,x_3)\Omega(x_1,x_2,\mathcal{D}_q(x_4))\\
&=-(-1)^{(|x_3|+|x_4|)(|x_1|+|x_2|)+\alpha|x_3|}\Phi(x_3,\mathcal{D}_q(x_4))\Omega(x_1,x_2,x_5)\\&-(-1)^{(|x_4|+|x_5|)(|x_1|+|x_2|+|x_3|)}\Phi(\mathcal{D}_q(x_4),x_5)\Omega(x_1,x_2,x_3),
\end{split}
\end{equation}
and
\begin{equation}\label{e319}
\begin{split}
&(-1)^{\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega([x_1,x_2,x_3]_\mathcal{Q},x_4,\mathcal{D}_q(x_5))\\
&+(-1)^{|x_3|(|x_1|+|x_2|)+\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega(x_3,[x_1,x_2,x_4]_\mathcal{Q},\mathcal{D}_q(x_5))\\
&+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)+\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega (x_3,x_4,[x_1,x_2,\mathcal{D}_q (x_5)]_\mathcal{Q})\\
&-(-1)^{\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Omega(x_1,x_2,[x_3,x_4,\mathcal{D}_q(x_5)]_\mathcal{Q})\\&+(-1)^{\alpha(|x_1|+|x_2|+|x_3|+|x_4|)}\Phi(x_1,x_2)\Omega(x_3,x_4,\mathcal{D}_q(x_5))\\&+(-1)^{(|x_3|+|x_4|)(\alpha+|x_1|+|x_2|)+\alpha(|x_1|+|x_2|)}\Phi(x_3,x_4)\Omega(x_1,x_2,\mathcal{D}_q(x_5))\\
&=-(-1)^{(|x_4|+|x_5|)(|x_1|+|x_2|+|x_3|)}\Phi(x_4,\mathcal{D}_q(x_5))\Omega(x_1,x_2,x_3)\\&-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+(|x_3|+|x_4|)(|x_5|+\alpha)}\Phi(\mathcal{D}_q(x_5),x_3)\Omega(x_1,x_2,x_4).
\end{split}
\end{equation}
By inserting Eq \ref{e315}-\ref{e319} into Eq \ref{e34}, we get
\begin{equation*}\label{e320}
\begin{split}
&(\delta_\Phi Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)})(x_1,x_2,x_3,x_4,x_5)\\
&=-\mathcal{D}_p(\Phi(x_1,x_2)(\Omega(x_3,x_4,x_5)))+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\mathcal{D}_p(\Phi(x_3,x_4)(\Omega(x_1,x_2,x_5)))\\
&+(-1)^{(|x_1|+|x_2|+|x_3|)(|x_4|+|x_5|)}\mathcal{D}_p(\Phi(x_4,x_5)(\Omega(x_1,x_2,x_3)))\\
& +(-1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(|x_3|+|x_4|)}\mathcal{D}_p(\Phi(x_5,x_3)(\Omega(x_1,x_2,x_4)))\\
&+(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)(\mathcal{D}_p(\Omega(x_3,x_4,x_5)))\\
&-(-1)^{(|x_3|+|x_4|)(\alpha+|x_1|+|x_2|)}\Phi(x_3,x_4)(\mathcal{D}_p(\Omega(x_1,x_2,x_5))\\
&-(-1)^{(|x_4|+|x_5|)(\alpha+|x_1|+|x_2|+|x_3|)}\Phi(x_4,x_5)(\mathcal{D}_p(\Omega(x_1,x_2,x_3)))\\
&-(-1)^{(|x_3|+|x_5|)(\alpha+|x_1|+|x_2|)+|x_5|(|x_3|+|x_4|)}\Phi(x_5,x_3)(\mathcal{D}_p(\Omega(x_1,x_2,x_4)))\\
&+\Phi(\mathcal{D}_q(x_1),x_2)(\Omega(x_3,x_4,x_5))+ (-1)^{\alpha|x_1|}\Phi(x_1,\mathcal{D}_q(x_2))(\Omega(x_3,x_4,x_5))\\ &-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\Phi(\mathcal{D}_q(x_3),x_4)(\Omega(x_1,x_2,x_5))\\&-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(\alpha+|x_3|+|x_4|)}\Phi(x_5,\mathcal{D}_q(x_3))(\Omega(x_1,x_2,x_4))\\
&-(-1)^{(|x_3|+|x_4|)(|x_1|+|x_2|)+\alpha|x_3|}\Phi(x_3,\mathcal{D}_q(x_4))(\Omega(x_1,x_2,x_5))\\&-(-1)^{(|x_4|+|x_5|)(|x_1|+|x_2|+|x_3|)}\Phi(\mathcal{D}_q(x_4),x_5)(\Omega(x_1,x_2,x_3))\\&-(-1)^{(|x_4|+|x_5|)(|x_1|+|x_2|+|x_3|)+\alpha|x_4|}\Phi(x_4,\mathcal{D}_q(x_5))(\Omega(x_1,x_2,x_3))\\&-(-1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(|x_3|+|x_4|)}\Phi((\mathcal{D}_q(x_5),x_3)\Omega(x_1,x_2,x_4)
\\&=-(\mathcal{D}_p\Phi(x_1,x_2)-(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_1),x_2)\\&-(-1)^{\alpha|x_1|}\Phi(x_1,\mathcal{D}_q(x_2)))(\Omega(x_3,x_4,x_5))+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}(\mathcal{D}_p\Phi(x_3,x_4)\\&-(-1)^{\alpha(|x_3|+|x_4|)}\Phi(x_3,x_4)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_3),x_4)-(-1)^{\alpha|x_3|}\Phi(x_3,\mathcal{D}_q(x_4)))(\Omega(x_1,x_2,x_5))\\&
+(-1)^{(|x_1|+|x_2|+|x_3|)(|x_4|+|x_5|)}(\mathcal{D}_p\Phi(x_4,x_5)-(-1)^{\alpha(|x_4|+|x_5|)}\Phi(x_4,x_5)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_4),x_5)\\&-(-1)^{\alpha|x_4|}\Phi(x_4,\mathcal{D}_q(x_5)))(\Omega(x_1,x_2,x_3))
+(-1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+|x_5|(|x_3|+|x_4|)}(\mathcal{D}_p\Phi(x_5,x_3)\\&-(-1)^{\alpha(|x_3|+|x_5|)}\Phi(x_5,x_3)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_5),x_3)-(-1)^{\alpha|x_5|}\Phi(x_5,\mathcal{D}_q(x_3)))(\Omega(x_1,x_2,x_4))\\
&=0.
\end{split}
\end{equation*}
\end{proof}
\begin{definition}\label{d31}
Let $(\Phi; \mathcal{P})$ be a representation of $\mathcal{Q}$. A pair $(\mathcal{D}_p,\mathcal{D}_q)\in Der(\mathcal{P})\times Der(\mathcal{Q})$ is called compatible (with respect to $\Phi$) if Eq \ref{e33} holds.
\end{definition}
Now we are ready to construct a Lie superalgebra and its representation on the first cohomology group. Set
\begin{equation}\label{e321}
\mathcal{T}_\Phi=\{(\mathcal{D}_p,\mathcal{D}_q)\in Der(\mathcal{P})\times Der(\mathcal{Q})|(\mathcal{D}_p,\mathcal{D}_q)~ is~ compatible~ with ~respect~ to~ \Phi\}.
\end{equation}
We have the following.
\begin{lemma}
There is an even linear map $\Psi:\mathcal{T}_\Phi\rightarrow gl(\mathbb{H}^1(\mathcal{Q};\mathcal{P}))$ given by
\begin{equation}\label{e322}
\Psi(\mathcal{D}_p,\mathcal{D}_q)([\Phi])=[Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}]~~~~~for~\Omega\in (\mathbb{Z}^1(\mathcal{Q};\mathcal{P}))_{\overline{0}},
\end{equation}
where $[Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}]$ is given by Eq \ref{e32}.
\end{lemma}
\begin{proof}
Since $(\mathcal{D}_p,\mathcal{D}_q)$ is compatible with respect to $\Phi$, in Lemma \ref{l31} it is proved that $Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}$ is a 2-cocycle whenever $\Omega$ is a 2-cocycle. So it is sufficient to show that if $\delta_{\Phi} \lambda$ is a 2-coboundary then $\Phi(\mathcal{D}_p,\mathcal{D}_q)(\delta_{\Phi} \lambda)=0$ which implies that $\Psi$ is well-defined. Here $|\lambda|=0$ and $|\mathcal{D}_p|=|\mathcal{D}_q|=\alpha$.
\begin{equation*}
\begin{split}
&(\Psi(\mathcal{D}_p,\mathcal{D}_q)(\delta_{\Phi} \lambda))(x,y,z)\\
&=\mathcal{D}_p(\delta_{\Phi} \lambda)(x,y,z)-(\delta_{\Phi} \lambda)(\mathcal{D}_q(x),y,z)-(-1)^{\alpha|x|}(\delta_{\Phi} \lambda)(x,\mathcal{D}_q(y),z)\\
&~~~~~~-(-1)^{\alpha(|x|+|y|)}(\delta_{\Phi} \lambda)(x,y,\mathcal{D}_q(z))\\
&=\mathcal{D}_p(-\lambda([x,y,z]_\mathcal{Q})+\Phi(x,y)(\lambda(z))+(-1)^{|x|(|y|+|z|)}\Phi(y,z)(\lambda(x))\\
&~~~~~~+(-1)^{|z|(|x|+|y|)}\Phi(z,x)(\lambda(y)))-(-\lambda([\mathcal{D}_q(x),y,z]_\mathcal{Q})+\Phi(\mathcal{D}_q(x),y)(\lambda(z))\\
&~~~~~~+(-1)^{(\alpha+|x|)(|y|+|z|)}\Phi(y,z)(\lambda(\mathcal{D}_q(x)))+(-1)^{|z|(|x|+|y|+\alpha)}\Phi(z,\mathcal{D}_q(x))(\lambda(y)))\\
&~~~~~~+(-1)^{\alpha|x|}(-\lambda([x,\mathcal{D}_q(y),z]_\mathcal{Q})+\Phi(x,\mathcal{D}_q(y))(\lambda(z))+(-1)^{|x|(\alpha+|y|+|z|)}\Phi(\mathcal{D}_q(y),z)(\lambda(x))\\
&~~~~~~+(-1)^{|z|(|x|+|y|+\alpha)}\Phi(z,x)(\lambda(\mathcal{D}_q(y))))-(-1)^{\alpha(|x|+|y|)} (-\lambda([x,y,\mathcal{D}_q(z)]_\mathcal{Q})+\Phi(x,y)(\lambda(\mathcal{D}_q(z)))\\
&~~~~~~+(-1)^{|x|(\alpha+|y|+|z|)}\Phi(y,\mathcal{D}_q(z))(\lambda(x))+(-1)^{(|z|+\alpha)(|x|+|y|)}\Phi(\mathcal{D}_q(z),x)(\lambda(y))).
\end{split}
\end{equation*}
Since $\mathcal{D}_q$ is a superderivation,
\begin{equation*}
\begin{split}
&\lambda([\mathcal{D}_q(x),y,z]_{\mathcal{Q}})+(-1)^{\alpha|x|}\lambda([x,\mathcal{D}_q(y),z]_{\mathcal{Q}}))+(-1)^{\alpha(|x|+|y|)} \lambda([x,y,\mathcal{D}_q(z)]_{\mathcal{Q}})\\&=\lambda(\mathcal{D}_q([x,y,z]_{\mathcal{Q}})).
\end{split}
\end{equation*}
Then we have
\begin{equation*}
\begin{split}
&(\Psi(\mathcal{D}_p,\mathcal{D}_q)(\delta_{\Phi} \lambda))(x,y,z)\\
&=(\mathcal{D}_p\Phi(x,y)(\lambda(z))-\Phi(\mathcal{D}_q(x),y)(\lambda(z))-(-1)^{\alpha|x|}\Phi(x,\mathcal{D}_q(y))(\lambda(z)))\\
&+(-1)^{\alpha(|x|+|y|)} (\mathcal{D}_p\Phi(y,z)(\lambda(x))-\Phi(\mathcal{D}_q(y),z)(\lambda(x))-(-1)^{\alpha|y|}\Phi(y,\mathcal{D}_q(z))(\lambda(x)))\\
&+(-1)^{|z|(|x|+|y|)} (\mathcal{D}_p\Phi(z,x)(\lambda(y))-\Phi(\mathcal{D}_q(z),x)(\lambda(y))-(-1)^{\alpha|z|}\Phi(x,\mathcal{D}_q(x))(\lambda(y)))\\
&-(-1)^{(\alpha+|x|)(|y|+|z|)}\Phi(y,z)(\lambda(\mathcal{D}_q(x)))+(-1)^{\alpha|x|+|z|(|x|+|y|+\alpha)}\Phi(z,x)(\lambda(\mathcal{D}_q(y)))\\
&-(-1)^{\alpha(|x|+|y|)} \Phi(x,y)(\lambda(\mathcal{D}_q(z)))-\mathcal{D}_p(\lambda([x,y,z]_{\mathcal{Q}}))+\lambda(\mathcal{D}_q([x,y,z]_{\mathcal{Q}})).
\end{split}
\end{equation*}
By using Eq \ref{e33}, we have
\begin{equation*}
\begin{split}
&\mathcal{D}_p\Phi(x,y)(\lambda(z))-\Phi(\mathcal{D}_q(x),y)(\lambda(z))-(-1)^{\alpha|x|}\Phi(x,\mathcal{D}_q(y))(\lambda(z))\\&=(-1)^{\alpha(|x|+|y|)}\Phi(x,y)\mathcal{D}_p(\lambda(z)),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&(-1)^{|x|(|y|+|z|)}(\mathcal{D}_p\Phi(y,z)(\lambda(x))-\Phi(\mathcal{D}_q(y),z)(\lambda(x))-(-1)^{\alpha||y|}\Phi(y,\mathcal{D}_q(z))(\lambda(x)))\\&=(-1)^{|x|(|y|+|z|)+\alpha(|y|+|z|)}\Phi(y,z)\mathcal{D}_p(\lambda(x)),
\end{split}\end{equation*}
\begin{equation*}
\begin{split}
&(-1)^{|z|(|x|+|y|)}(\mathcal{D}_p\Phi(z,x)(\lambda(y))-\Phi(\mathcal{D}_q(z),x)(\lambda(y))-(-1)^{\alpha|z|}\Phi(z,\mathcal{D}_q(x))(\lambda(y)))\\&=(-1)^{|z|(|x|+|y|)+\alpha(|z|+|x|)}\Phi(z,x)\mathcal{D}_p(\lambda(y)).
\end{split}
\end{equation*}
\begin{equation}
\begin{split}
&(\Psi(\mathcal{D}_p,\mathcal{D}_q)(\delta_{\Phi} \lambda))(x,y,z)\\
&=(-1)^{\alpha(|x|+|y|)}\Phi(x,y)\mathcal{D}_p(\lambda(z))+(-1)^{(\alpha+|x|)(|y|+|z|)}\Phi(y,z)\mathcal{D}_p(\lambda(x))\\&+(-1)^{|z|(|x|+|y|)+\alpha(|z|+|x|)}\Phi(z,x)\mathcal{D}_p(\lambda(y))-(-1)^{(\alpha+|x|)(|y|+|z|)}\Phi(y,z)(\lambda(\mathcal{D}_q(x)))\\&-(-1)^{|z|(|x|+|y|)+\alpha(|z|+|x|)}\Phi(z,x)(\mathcal{D}_q(\lambda(y)))-(-1)^{\alpha(|x|+|y|)} \Phi(x,y)(\mathcal{D}_q(\lambda(z)))\\
& -\mathcal{D}_p(\lambda([x,y,z]_{\mathcal{Q}}))+\lambda(\mathcal{D}_q([x,y,z]_{\mathcal{Q}}))\\
&=\delta_{\Phi}(\mathcal{D}_p \lambda-\lambda \mathcal{D}_q)(x,y,z),
\end{split}
\end{equation}
which implies that $[Ob^{\Omega,1} _{(\mathcal{D}_p,\mathcal{D}_q)}]=[Ob^{\Omega,2} _{(\mathcal{D}_p,\mathcal{D}_q)}]\in \mathbb{H}^1(\mathcal{Q},\mathcal{P})$ as required.
\end{proof}
Below is the main result of this section.
\begin{theorem}
Keep notation as above. For any representation $(\Phi;\mathcal{P})$ of $\mathcal{Q}$, $\mathcal{T}_\Phi$ is a Lie subsuperalgebra of $Der(\mathcal{P})\times Der(\mathcal{Q})$ and the map $\Psi$ given by Eq \ref{e322} is a Lie superalgebra homomorphism.
\end{theorem}
\begin{proof}
Take $|\mathcal{D}_{p_1}|=\alpha_1$ and $|\mathcal{D}_{p_2}|=\alpha_2$.
\begin{equation}\label{e323}
\begin{split}
&(\mathcal{D}_{p_1}\mathcal{D}_{p_2}-(-1)^{\alpha_1\alpha_2}\mathcal{D}_{p_2}\mathcal{D}_{p_1})\Phi (x,y)- (-1)^{(\alpha_1+\alpha_2)(|x|+|y|)} \Phi (x,y)(\mathcal{D}_{p_1}\mathcal{D}_{p_2}-(-1)^{\alpha_1\alpha_2}\mathcal{D}_{p_2}\mathcal{D}_{p_1})\\
&=\underbrace{\mathcal{D}_{p_1}((-1)^{\alpha_2 (|x|+|y|)}\Phi(x,y)\mathcal{D}_{p_2}+\Phi (\mathcal{D}_{p_2}(x),y)+(-1)^{\alpha_2|x|}\Phi (x,\mathcal{D}_{p_2}(y)))}_{I_1}\\
&-\underbrace{(-1)^{\alpha_1\alpha_2}\mathcal{D}_{p_2}((-1)^{\alpha_1 (|x|+|y|)}\Phi(x,y)\mathcal{D}_{p_1}+\Phi (\mathcal{D}_{p_1}(x),y)+(-1)^{\alpha_1|x|}\Phi (x,\mathcal{D}_{p_1}(y)))}_{I_2}\\
&-(-1)^{(\alpha_1+\alpha_2) (|x|+|y|)}\Phi(x,y)\mathcal{D}_{p_1}\mathcal{D}_{p_2} (-1)^{(\alpha_1+\alpha_2) (x+y)+\alpha_1\alpha_2}\Phi(x,y)\mathcal{D}_{p_2}\mathcal{D}_{p_1},
\end{split}
\end{equation}
where
\begin{equation*}\label{e324}
I_1=\mathcal{D}_{p_1}((-1)^{\alpha_2 (|x|+|y|)}\Phi(x,y)\mathcal{D}_{p_2}+\Phi (\mathcal{D}_{p_2}(x),y)+(-1)^{\alpha_2|x|}\Phi (x,\mathcal{D}_{p_2}(y))),
\end{equation*}
\begin{equation*}\label{e325}
I_2=(-1)^{\alpha_1\alpha_2}\mathcal{D}_{p_2}((-1)^{\alpha_1 (|x|+|y|)}\Phi(x,y)\mathcal{D}_{p_1}+\Phi (\mathcal{D}_{p_1}(x),y)+(-1)^{\alpha_1 |x|}\Phi (x,\mathcal{D}_{p_1}(y))).
\end{equation*}
By Eq \ref{e33}, we get
\begin{equation}\label{e326}
\begin{split}
&I_1=(-1)^{\alpha_2(|x|+|y|)}((-1)^{\alpha_1(|x|+|y|)} \Phi(x,y)\mathcal{D}_{p_1}+\Phi (\mathcal{D}_{p_1}(x),y)\\
&+(-1)^{\alpha_1|x|}\Phi (x,\mathcal{D}_{p_1}(y)))\mathcal{D}_{p_2}+((-1)^{\alpha_1 (|x|+|y|+\alpha_2)}\Phi(\mathcal{D}_{p_2}(x),y)\mathcal{D}_{p_1}\\&+\Phi (\mathcal{D}_{p_1}\mathcal{D}_{p_2}(x),y)+(-1)^{\alpha_1(|\alpha_2|+|x|)}\Phi (\mathcal{D}_{p_2}(x),\mathcal{D}_{p_1}(y)))\\&+(-1)^{\alpha_2|x|}((-1)^{\alpha_1 (|x|+|y|+\alpha_2)}\Phi(x,\mathcal{D}_{p_2}(y))\mathcal{D}_{p_1}+\Phi (\mathcal{D}_{p_1}(x),\mathcal{D}_{p_2}(y)) \\&+(-1)^{\alpha_1|x|}\Phi (x,\mathcal{D}_{p_1}\mathcal{D}_{p_2}(y))),
\end{split}
\end{equation}
and
\begin{equation}\label{e327}
\begin{split}
&I_2=-(-1)^{\alpha_1(\alpha_2+|x|+|y|)}((-1)^{\alpha_2(|x|+|y|)} \Phi(x,y)\mathcal{D}_{p_2}+\Phi (\mathcal{D}_{p_2}(x),y)\\&+(-1)^{\alpha_2|x|}\Phi (x,\mathcal{D}_{p_2}(y)))\mathcal{D}_{p_1}-(-1)^{\alpha_1\alpha_2}((-1)^{\alpha_2 (|x|+|y|+\alpha_1)}\Phi(\mathcal{D}_{p_1}(x),y)\mathcal{D}_{p_2}\\&+\Phi (\mathcal{D}_{p_2}\mathcal{D}_{p_1}(x),y)+(-1)^{\alpha_2(\alpha_1+|x|)}\Phi (\mathcal{D}_{p_1}(x),\mathcal{D}_{p_2}(y)))\\&-(-1)^{\alpha_1(|x|+\alpha_2)}((-1)^{\alpha_2 (|x|+|y|+\alpha_1)}\Phi(x,\mathcal{D}_{p_1}(y))\mathcal{D}_{p_2}+\Phi (\mathcal{D}_{p_2}(x),\mathcal{D}_{p_1}(y)) \\&+(-1)^{\alpha_2|x|}\Phi (x,\mathcal{D}_{p_2}\mathcal{D}_{p_1}(y))).
\end{split}
\end{equation}
Then inserting Eqs \ref{e326} and \ref{e327} into Eq \ref{e323}, we get
\begin{equation}\label{e328}
\begin{split}
&(\mathcal{D}_{p_1}\mathcal{D}_{p_2}-(-1)^{\alpha_1\alpha_2}\mathcal{D}_{p_2}\mathcal{D}_{p_1})\Phi (x,y)- (-1)^{(\alpha_1+\alpha_2)(|x|+|y|)} \Phi (x,y)(\mathcal{D}_{p_1}\mathcal{D}_{p_2}\\
&-(-1)^{\alpha_1\alpha_2}\mathcal{D}_{p_2}\mathcal{D}_{p_1})=\Phi((\mathcal{D}_{q_1}\mathcal{D}_{q_2}-(-1)^{\alpha_1\alpha_2|}\mathcal{D}_{q_2}\mathcal{D}_{q_1})(x),y)\\
&+ (-1)^{|x|(\alpha_1+\alpha_2)}\Phi (x,(\mathcal{D}_{q_1}\mathcal{D}_{q_2}-(-1)^{\alpha_1\alpha_2}\mathcal{D}_{q_2}\mathcal{D}_{q_1})(y)),
\end{split}
\end{equation}
which implies that $[(\mathcal{D}_{p_1},\mathcal{D}_{q_1}),(\mathcal{D}_{p_2},\mathcal{D}_{q_2})]$ is compatible by Definition \ref{d31}.\\
\noindent To prove second part refer \cite{xu2018}.
\end{proof}
\section{Abelian Extensions and Extensibility of superderivations}
In this section, we construct obstruction classes for extensibility of superderivations by Lemma \ref{l31} and also give a representation of $\mathcal{T}_{\Phi}$ in terms of extensibility of superderivations.
\begin{lemma}\label{l41}
Keep notations as above. The cohomology class $[Ob^\Omega _{(\mathcal{D}_p,\mathcal{D}_q)}]\in \mathbb{H}^1(\mathcal{Q};\mathcal{P})$ does not depend on the choice of the section $\pi$.
\end{lemma}
\begin{proof}Let $s_1$ and $s_2$ be sections of $\pi$ and $\Omega_1,\Omega_2$ be defined by Eq \ref{e29} while $Ob^{\Omega,1} _{(\mathcal{D}_p,\mathcal{D}_q)},~Ob^{\Omega,2} _{(\mathcal{D}_p,\mathcal{D}_q)}$ are defined by Eq \ref{e32} with respect to $\Omega_1,\Omega_2$. Then
\begin{equation}\label{e41}
\begin{split}
&Ob^{\Omega,1} _{(\mathcal{D}_p,\mathcal{D}_q)}(x,y,z)- Ob^{\Omega,2} _{(\mathcal{D}_p,\mathcal{D}_q)}(x,y,z)\\
&=\mathcal{D}_p(\Omega_1(x,y,z))-\Omega_1(\mathcal{D}_q(x),y,z)-(-1)^{\alpha |x|}\Omega_1(x,\mathcal{D}_q(y),z)-(-1)^{\alpha (|x|+|y|)}\Omega_1(x,y,\mathcal{D}_q(z))\\
&~~~~~-\mathcal{D}_p(\Omega_2(x,y,z))+\Omega_2(\mathcal{D}_q(x),y,z)+(-1)^{\alpha |x|}\Omega_2(x,\mathcal{D}_q(y),z)+(-1)^{\alpha (|x|+|y|)}\Omega_2(x,y,\mathcal{D}_q(z))\\
&=\underbrace{\mathcal{D}_p(\Omega_1(x,y,z)-\Omega_2(x,y,z))}_{I_1}-\underbrace{(\Omega_1(\mathcal{D}_q(x),y,z)-\Omega_2(\mathcal{D}_q(x),y,z))}_{I_2}\\
&~~~~~-(-1)^{\alpha |x|}\underbrace{(\Omega_1(x,\mathcal{D}_q(y),z)-\Omega_2(x,\mathcal{D}_q(y),z))}_{I_3}\\
&~~~~~-(-1)^{\alpha (|x|+|y|)}\underbrace{(\Omega_1(x,y,\mathcal{D}_q(z))-\Omega_2(x,y,\mathcal{D}_q(z)))}_{I_4},
\end{split}
\end{equation}
for $x,y,z\in \mathcal{Q}$ and $|\mathcal{D}_p|=|\mathcal{D}_q|=\alpha$.
Define an even linear map $\lambda:\mathcal{Q}\rightarrow \mathcal{P}$ by $\lambda(x)=s_1(x)-s_2(x)$ where $x\in \mathcal{Q}$. From Corollary \ref{coro21}, we have
\begin{equation}\label{e42}
\begin{split}
\Omega_1(x,y,z)-\Omega_2(x,y,z)=&-\lambda([x,y,z]_\mathcal{Q})+\Phi(x,y)(\lambda(z))+(-1)^{|x|(|y|+|z|)}\Phi(y,z)(\lambda(x))\\
&~~~~~~~+(-1)^{|z|(|x|+|y|)}\Phi(z,x)(\lambda(y)),
\end{split}
\end{equation}
for any $x,y,z\in \mathcal{Q}$. Therefore, we get
\begin{equation*} \begin{split}
I_1&=\mathcal{D}_p(\Omega_1(x,y,z)-\Omega_2(x,y,z))\\
&=\mathcal{D}_p(-\lambda([x,y,z]_\mathcal{Q})+\Phi(x,y)(\lambda(z))+(-1)^{|x| (|y|+|z|)}\Phi(y,z)(\lambda(x))\\&+(-1)^{|z|(|x|+|y|)}\Phi(z,x)(\lambda(y))),
\end{split} \end{equation*}
\begin{equation*}
\begin{split}
I_2&=\Omega_1(\mathcal{D}_q(x),y,z)-\Omega_2(\mathcal{D}_q(x),y,z)\\
&=-\lambda([\mathcal{D}_q(x),y,z]_\mathcal{Q})+\Phi(\mathcal{D}_q(x),y)(\lambda(z))+(-1)^{(\alpha+|x|) (|y|+|z|)}\Phi(y,z)(\lambda(\mathcal{D}_q(x)))\\
&+(-1)^{|z| (\alpha+|x|+|y|)}\Phi(z,\mathcal{D}_q(x))(\lambda(y)),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
I_3&=\Omega_1(x,\mathcal{D}_q(y),z)-\Omega_2(x,\mathcal{D}_q(y),z)\\
&=-\lambda([x,\mathcal{D}_q(y),z]_\mathcal{Q})+\Phi(x,\mathcal{D}_q(y))(\lambda(z))+(-1)^{|x|(\alpha+|y|+|z|)}\Phi(\mathcal{D}_q(y),z)(\lambda(x))\\
&~~~~~+(-1)^{|z|(\alpha+|x|+|y|)}\Phi(z,x)(\lambda(\mathcal{D}_q(y))),
\end{split}
\end{equation*} and
\begin{equation*}
\begin{split}
I_4&=\Omega_1(x,y,\mathcal{D}_q(z))-\Omega_2(x,y,\mathcal{D}_q(z))\\
&=-\lambda([x,y,\mathcal{D}_q(z)]_\mathcal{Q})+\Phi(x,y)(\lambda(\mathcal{D}_q(z)))+(-1)^{|z| (\alpha+|x|+|y|)}\Phi(y,\mathcal{D}_q(z))(\lambda(x))\\
&+(-1)^{(\alpha+|z|) (|x|+|y|)}\Phi(\mathcal{D}_q(z),x)(\lambda(y)).
\end{split}
\end{equation*}
By $I_1,I_2,I_3,I_4$, and Eq \ref{e41}, we get
\begin{equation*}
\begin{split}
&Ob^{\Omega,1} _{(\mathcal{D}_p,\mathcal{D}_q)}(x,y,z)- Ob^{\Omega,2} _{(\mathcal{D}_p,\mathcal{D}_q)}(x,y,z)\\
&=(\mathcal{D}_p\Phi(x,y)(\lambda(z))-\Phi(\mathcal{D}_q(x),y)(\lambda(z))-(-1)^{\alpha |x|}\Phi(x,\mathcal{D}_q(y))(\lambda(z)))\\
&~~~~~~+(-1)^{|x|(|y|+|z|)}(\mathcal{D}_p\Phi(y,z)(\lambda(x))-\Phi(\mathcal{D}_q(y),z)(\lambda(x))-(-1)^{\alpha |y|}\Phi(y,\mathcal{D}_q(z))(\lambda(x)))
\\
&~~~~~~+(-1)^{|z|(|x|+|y|)}(\mathcal{D}_p\Phi(z,x)(\lambda(y))-\Phi(\mathcal{D}_q(z),x)(\lambda(y))-(-1)^{\alpha|z|}\Phi(z,\mathcal{D}_q(x))(\lambda(y)))\\
&~~~~~~-\mathcal{D}_p(\lambda([x,y,z]_\mathcal{Q}))-(-1)^{(\alpha+|x|)(|y|+|z|)}\Phi(y,z)\lambda(\mathcal{D}_q(x))\\
&~~~~~~-(-1)^{\alpha|x|+|z|(\alpha+|x|+|y|)}\Phi(z,x)\lambda(\mathcal{D}_q(y))-(-1)^{\alpha(|x|+|y|)}\Phi(x,y)\lambda(\mathcal{D}_q(z))+\lambda(\mathcal{D}_q([x,y,z]_\mathcal{Q})),
\end{split}
\end{equation*}
where $\mathcal{D}_q$ is a superderivation. Since $(\mathcal{D}_p,\mathcal{D}_q)$ is compatible, by Eq \ref{e33} it follows that
\begin{equation*}
\begin{split}
&(\mathcal{D}_p\Phi(x,y)(\lambda(z))-\Phi(\mathcal{D}_q(x),y)(\lambda(z))-(-1)^{\alpha |x|}\Phi(x,\mathcal{D}_q(y))(\lambda(z)))\\&=(-1)^{\alpha(|x|+|y|)}\Phi(x,y)(\mathcal{D}_p(\lambda(z))),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
(-1)^{|x|(|y|+|z|)}(\mathcal{D}_p\Phi(y,z)(\lambda(x))&-\Phi(\mathcal{D}_q(y),z)(\lambda(x))-(-1)^{\alpha |y|}\Phi(y,\mathcal{D}_q(z))(\lambda(x)))\\&=(-1)^{(|x|+\alpha)(|y|+|z|)}\Phi(y,z)(\mathcal{D}_p(\lambda(x))),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
(-1)^{|z|(|x|+|y|)}(\mathcal{D}_p\Phi(z,x)(\lambda(y))&-\Phi(\mathcal{D}_q(z),x)(\lambda(y))-(-1)^{\alpha |z|}\Phi(z,\mathcal{D}_q(x))(\lambda(y)))\\&=(-1)^{|z|(|x|+|y|+\alpha)+\alpha|x|}\Phi(z,x)\mathcal{D}_p(\lambda(y)).
\end{split}
\end{equation*}
Therefore we obtain
\begin{equation*}
\begin{split}
&Ob^{\Omega,1} _{(\mathcal{D}_p,\mathcal{D}_q)}(x,y,z)- Ob^{\Omega,2} _{(\mathcal{D}_p,\mathcal{D}_q)}(x,y,z)\\
&=(-1)^{\alpha(|x|+|y|)}\Phi(x,y)(\mathcal{D}_p(\lambda(z)))-\mathcal{D}_p(\lambda([x,y,z]_\mathcal{Q}))+(-1)^{(|x|+\alpha)(|y|+|z|)}\Phi(y,z)(\mathcal{D}_p(\lambda(x)))\\
&~~~~~~+(-1)^{|z|(|x|+|y|+\alpha)+\alpha|x|}\Phi(z,x)(\mathcal{D}_p(\lambda(y)))+(-1)^{(\alpha+|x|)(|y|+|z|)}\Phi(y,z)(\lambda(\mathcal{D}_q(x)))\\
&~~~~~~-(-1)^{|z|(\alpha+|x|+|y|)+\alpha|x|}\Phi(z,x)(\lambda(\mathcal{D}_q(y)))-(-1)^{\alpha(|x|+|y|)}\Phi(x,y)(\lambda(\mathcal{D}_q(z)))+\lambda(\mathcal{D}_q([x,y,z]_\mathcal{Q}))\\
&=\delta_{\Phi}(\mathcal{D}_p \lambda-\lambda \mathcal{D}_q)(x,y,z),
\end{split}
\end{equation*}
which implies that $[Ob^{\Omega,1} _{(\mathcal{D}_p,\mathcal{D}_q)}]=[Ob^{\Omega,2} _{(\mathcal{D}_p,\mathcal{D}_q)}]\in \mathbb{H}^1(\mathcal{Q};\mathcal{P})$ as required.
\end{proof}
\par Now we will define extensibility of superderivations.
\begin{definition}
Let $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow \mathcal{Q}\rightarrow 0$ be an extension of $3$-Lie superalgebras with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$. A pair $(\mathcal{D}_p,\mathcal{D}_q)\in Der(\mathcal{P})\times Der(\mathcal{Q})$ is called extensible if there is a superderivation $\mathcal{D}_l\in Der(\mathcal{L})$ such that the following diagram:
\begin{center}
\begin{tikzpicture}[>=latex]\label{diagram}
\node (A_{1}) at (0,0) {\(0\)};
\node (A_{2}) at (2,0) {\(\mathcal{P}\)};
\node (A_{3}) at (4,0) {\(\mathcal{L}\)};
\node (A_{4}) at (6,0) {\(\mathcal{Q}\)};
\node (A_{5}) at (8,0) {\(0\)};
\node (B_{1}) at (0,-2) {\(0\)};
\node (B_{2}) at (2,-2) {\(\mathcal{P}\)};
\node (B_{3}) at (4,-2) {\(\mathcal{L}\)};
\node (B_{4}) at (6,-2) {\(\mathcal{Q}\)};
\node (B_{5}) at (8,-2) {\(0\)};
\draw[->] (A_{1}) -- (A_{2});
\draw[->] (A_{2}) -- (A_{3}) node[midway,above] {$i$};
\draw[->] (A_{3}) -- (A_{4}) node[midway,above] {$\pi$};
\draw[->] (A_{4}) -- (A_{5});
\draw[->] (B_{1}) -- (B_{2});
\draw[->] (B_{2}) -- (B_{3}) node[midway,above] {$i$};
\draw[->] (B_{3}) -- (B_{4}) node[midway,above] {$\pi$};
\draw[->] (B_{4}) -- (B_{5});
\draw[->] (A_{2}) -- (B_{2}) node[midway,right] {$\mathcal{D}_p$};
\draw[->] (A_{3}) -- (B_{3}) node[midway,right] {$\mathcal{D}_l$};
\draw[->] (A_{4}) -- (B_{4}) node[midway,right] {$\mathcal{D}_q$};
\end{tikzpicture}\\
\end{center}
commutes, where $i:\mathcal{P}\rightarrow \mathcal{L}$ is the inclusion map.
\end{definition}
The following result means that extensibility implies compatibility.
\begin{proposition}\label{p42}
Let $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow \mathcal{Q}\rightarrow 0$ be an extension of $3$-Lie superalgebras with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$. If $(\mathcal{D}_p,\mathcal{D}_q)\in Der(\mathcal{P})\times Der(\mathcal{Q})$ is extensible, then $(\mathcal{D}_p,\mathcal{D}_q)$ is compatible with respect to $\Phi$ given by Eq \ref{e27}.
\end{proposition}
\begin{proof}
Since $(\mathcal{D}_p,\mathcal{D}_q)$ is extensible, there exists a superderivation $\mathcal{D}_l\in Der(\mathcal{L})$ such that $i\mathcal{D}_p=\mathcal{D}_li,~\pi \mathcal{D}_l=\mathcal{D}_q\pi$, and $|\mathcal{D}_p|=|\mathcal{D}_q|=|\mathcal{D}_l|=\alpha$. Then
$$\mathcal{D}_ls(x)-s(\mathcal{D}_q(x))\in \mathcal{P}~ {\rm{for}} ~x\in \mathcal{Q}.$$ So there is an even linear map $\mu:\mathcal{Q}\rightarrow \mathcal{P}$ given by
\begin{equation}\label{e43}
\mu(x)=\mathcal{D}_ls(x)-s(\mathcal{D}_q(x)).
\end{equation}
Since $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$, we have
\begin{equation*}
[\mu(x),s(y),v]_\mathcal{L}=[s(x),\mu(y),v]_\mathcal{L}=0,
\end{equation*}
for $x,y\in \mathcal{Q}$ and $v\in \mathcal{P}$. Since $i\mathcal{D}_p=\mathcal{D}_l i$ and $\mathcal{D}_l\in Der(\mathcal{L})$, we get
\begin{equation*}
\begin{split}
&\mathcal{D}_p(\Phi(x,y)(v))-(-1)^{\alpha(|x|+|y|)}\Phi(x,y)(\mathcal{D}_p(v))\\
&=\mathcal{D}_p([s(x),s(y),v]_\mathcal{L})-(-1)^{\alpha(|x|+|y|)}[s(x),s(y),\mathcal{D}_p(v)]_\mathcal{L}\\
&=\mathcal{D}_l([s(x),s(y),v]_\mathcal{L})-(-1)^{\alpha(|x|+|y|)}[s(x),s(y),\mathcal{D}_p(v)]_\mathcal{L}\\
&=[\mathcal{D}_l(s(x)),s(y),v]_\mathcal{L}+(-1)^{\alpha|x|}[s(x),\mathcal{D}_l(s(y)),v]_\mathcal{L}+(-1)^{\alpha(|x|+|y|)}[s(x),s(y),\mathcal{D}_l(v)]_\mathcal{L}\\
&~~~~~~-(-1)^{\alpha(|x|+|y|)}[s(x),s(y),\mathcal{D}_p(v)]_\mathcal{L}\\
&=[s(\mathcal{D}_q(x)),s(y),v]_\mathcal{L}+[\mu(x),s(y),v]_\mathcal{L}+(-1)^{\alpha|x|}([s(x),s(\mathcal{D}_q(y)),v]_\mathcal{L}\\
&~~~~~~+[s(x),\mu(y),v]_\mathcal{L})+(-1)^{\alpha(|x|+|y|)}[s(x),s(y),\mathcal{D}_l(v)]_\mathcal{L}-(-1)^{\alpha(|x|+|y|)}[s(x),s(y),\mathcal{D}_p(v)]_\mathcal{L}\\
&=[s(\mathcal{D}_q(x)),s(y),v]_\mathcal{L}+(-1)^{\alpha|x|}([s(x),s(\mathcal{D}_q(y)),v]_\mathcal{L}\\
&=\Phi(\mathcal{D}_q(x),y)(v)+(-1)^{\alpha|x|}\Phi(x,\mathcal{D}_q(y))(v).
\end{split}
\end{equation*}
\end{proof}
\begin{proposition}\label{p43}
Let $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow \mathcal{Q}\rightarrow 0$ be an extension of $3$-Lie superalgebras with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$. Assume that $(\mathcal{D}_p,\mathcal{D}_q)\in Der(\mathcal{P}_{\overline{0}})\times Der(\mathcal{Q}_{\overline{0}})$ is compatible with respect to $\Phi$ given by Eq \ref{e27}. Then $(\mathcal{D}_p,\mathcal{D}_q)_{\overline{0}}$ is extensible if and only if $[Ob^\mathcal{L}_{(\mathcal{D}_p,\mathcal{D}_q)}]\in \mathbb{H}^1(\mathcal{Q};\mathcal{P})$ is trivial.
\end{proposition}
\begin{proof}
Suppose that $(\mathcal{D}_p,\mathcal{D}_q)$ is extensible. Then there exists a superderivation $\mathcal{D}_l\in Der(\mathcal{L})$ such that the associative diagram \ref{diagram} is commutative. Since $\pi \mathcal{D}_l=\mathcal{D}_q \pi $ and $|\mathcal{D}_p|=|\mathcal{D}_q|=|\mathcal{D}_l|=\alpha$, we have $$\mathcal{D}_ls(x)-s(\mathcal{D}_q(x))\in \mathcal{P}~ {\rm{for}} ~x\in \mathcal{Q}.$$ So there is an even linear map $\mu:\mathcal{Q}\rightarrow \mathcal{P}$ given by Eq \ref{e43}. It is sufficient to show that
\begin{equation}\label{e44}
Ob^\mathcal{L}_{(\mathcal{D}_p,\mathcal{D}_q)}(x_1,x_2,x_3)=(\delta_{\Phi}(\mu))(x_1,x_2,x_3),~~~~x_i\in \mathcal{Q}.
\end{equation}
Now
\begin{equation}\label{e45}
\begin{split}
&\mathcal{D}_l([s(x_1)+v_1,s(x_2)+v_2,s(x_3)+v_3]_\mathcal{L})\\
&=[\mathcal{D}_l(s(x_1)+v_1)),s(x_2)+v_2,s(x_3)+v_3]_\mathcal{L}\\
&~~~~~~+(-1)^{\alpha|x_1|}[s(x_1)+v_1,\mathcal{D}_l(s(x_2)+v_2)),s(x_3)+v_3]_\mathcal{L}\\
&~~~~~~+(-1)^{\alpha(|x_1|+|x_2|)}[s(x_1)+v_1,s(x_2)+v_2,\mathcal{D}_l(s(x_3)+v_3)]_\mathcal{L},
\end{split}
\end{equation}
for $x_i\in \mathcal{Q}$ and $v_i\in \mathcal{P}$. Since $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$, we get
\begin{equation*}
\begin{split}
&[s(x_1)+v_1,s(x_2)+v_2,s(x_3)+v_3]_\mathcal{L}\\
&=[s(x_1),s(x_2),s(x_3)]_\mathcal{L}+[s(x_1),s(x_2),v_3]_\mathcal{L}+[v_1,s(x_2),s(x_3)]_\mathcal{L}+[s(x_1),v_2,s(x_3)]_\mathcal{L}\\
&=[s(x_1),s(x_2),s(x_3)]_\mathcal{L}+\Phi(x_1,x_2)(v_3)+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(x_2,x_3)(v_1)\\
&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)(v_2),
\end{split}
\end{equation*}
and hence the left-hand side of Eq \ref{e45} is
\begin{equation*}
\begin{split}
&\mathcal{D}_l([s(x_1),s(x_2),s(x_3)]_\mathcal{L}+\Phi(x_1,x_2)(v_3)+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(x_2,x_3)(v_1)\\
&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)(v_2))\\
&=\mathcal{D}_l(s([x_1,x_2,x_3]_\mathcal{Q})+\Omega(x_1,x_2,x_3)+\Phi(x_1,x_2)(v_3)+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(x_2,x_3)(v_1)\\
&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)(v_2)).
\end{split}
\end{equation*}
Since $\mathcal{D}_l i=i \mathcal{D}_p$ where $i$ is the inclusion map, and $\Omega(x_1,x_2,x_3)$, $\Phi(x_1,x_2)(v_3)$, $\Phi(x_2,x_3)(v_1)$, $\Phi(x_3,x_1)(v_2)\in \mathcal{P}$, by the definition of $\mu$ as in Eq \ref{e43}, it follows that
\begin{equation}\label{e46}
\begin{split}
&s(\mathcal{D}_q([x_1,x_2,x_3]_\mathcal{Q})+\mu([x_1,x_2,x_3]_\mathcal{Q})+\mathcal{D}_p(\Omega(x_1,x_2,x_3))+\mathcal{D}_p(\Phi(x_1,x_2)(v_3))\\
&~~~~~~+(-1)^{|x_1|(|x_2|+|x_3|)}\mathcal{D}_p(\Phi(x_2,x_3)(v_1))+(-1)^{|x_3|(|x_1|+|x_2|)}\mathcal{D}_p(\Phi(x_3,x_1)(v_2)))\\
&=s([\mathcal{D}_q(x_1),x_2,x_3]_\mathcal{Q})+(-1)^{\alpha|x_1|}s([x_1,\mathcal{D}_q(x_2),x_3]_\mathcal{Q})\\
&~~~~~~+(-1)^{\alpha(|x_2|+|x_3|)}s([x_1,x_2,\mathcal{D}_q(x_3)]_\mathcal{Q})+\mu([x_1,x_2,x_3]_\mathcal{Q})+\mathcal{D}_p(\Omega(x_1,x_2,x_3))\\
&~~~~~~+\mathcal{D}_p(\Phi(x_1,x_2)(v_3))+(-1)^{|x_1|(|x_2|+|x_3|)}\mathcal{D}_p(\Phi(x_2,x_3)(v_1))\\
&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)}\mathcal{D}_p(\Phi(x_3,x_1)(v_2)).
\end{split}
\end{equation}
Now we compute the right-hand side of Eq \ref{e45}. Since $\mathcal{D}
_l|_{\mathcal{P}}=\mathcal{D}_p$,
\begin{equation*}
\begin{split}
\mathcal{D}_l(s(x_i)+v_i)&=\mathcal{D}_l(s(x_i))+\mathcal{D}_p(v_i)\\
&=\mathcal{D}_l(s(x_i))-s(\mathcal{D}_q(x_i))+s(\mathcal{D}_q(x_i))+\mathcal{D}_p(v_i)\\
&=s(\mathcal{D}_q(x_i))+\mu(x_i)+\mathcal{D}_p(v_i)\in s(\mathcal{Q})\oplus \mathcal{P}.
\end{split}
\end{equation*}
By this and $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$, the right-hand side of Eq \ref{e45} is
\begin{equation}\label{e47}
\begin{split}
&[s(\mathcal{D}_q(x_1))+\mu(x_1)+\mathcal{D}_p(v_1),s(x_2)+v_2,s(x_3)+v_3]_\mathcal{L}\\
&~~~~~~+(-1)^{|\alpha||x_1|}[s(x_1)+v_1,s(\mathcal{D}_q(x_2))+\mu(x_2)+\mathcal{D}_p(v_2),s(x_3)+v_3]_\mathcal{L}\\
&~~~~~~+(-1)^{|\alpha|(|x_1|+|x_2|)}[s(x_1)+v_1,s(x_2)+v_2,s(\mathcal{D}_q(x_3))+\mu(x_3)+\mathcal{D}_p(v_3)]_\mathcal{L}\\
&=[s(\mathcal{D}_q(x_1)),s(x_2),s(x_3)]_\mathcal{L}+[s(\mathcal{D}_q(x_1)),s(x_2),v_3]_\mathcal{L}+[s(\mathcal{D}_q(x_1)),v_2,s(x_3)]_\mathcal{L}\\
&~~~~~~+[\mu(x_1),s(x_2),s(x_3)]_\mathcal{L}+[\mathcal{D}_p(v_1),s(x_2),s(x_3)]_\mathcal{L}+(-1)^{\alpha|x_1|}([s(x_1),s(\mathcal{D}_q(x_2)),s(x_3)]_\mathcal{L}\\
&~~~~~~+[s(x_1),s(\mathcal{D}_q(x_2)),v_3]_\mathcal{L}+[s(x_1),\mu(x_2),s(x_3)]_\mathcal{L}+[s(x_1),\mathcal{D}_p(v_2),s(x_3)]_\mathcal{L}\\
&~~~~~~+[v_1,s(\mathcal{D}_q(x_2)),s(x_3)]_\mathcal{L})+(-1)^{\alpha(|x_1|+|x_2|)}([s(x_1),s(x_2),s(\mathcal{D}_q(x_3))]_\mathcal{L}\\
&~~~~~~+[v_1,s(x_2),s(\mathcal{D}_q(x_3))]_\mathcal{L}+[s(x_1),s(x_2),\mu(x_3)]_\mathcal{L}+[s(x_1),s(x_2),\mathcal{D}_p(v_3)]_\mathcal{L}\\
&~~~~~~+[s(x_1),v_2,s(\mathcal{D}_q(x_3))]_\mathcal{L}).
\end{split}
\end{equation}
By Eqs \ref{e46} and \ref{e47} it follows that
\begin{equation*}
\begin{split}
&s([\mathcal{D}_q(x_1),x_2,x_3]_\mathcal{Q})+(-1)^{\alpha|x_1|}s([x_1,\mathcal{D}_q(x_2),x_3]_\mathcal{Q})+(-1)^{\alpha(|x_1|+|x_2|)}s([x_1,x_2,\mathcal{D}_q(x_3)]_\mathcal{Q})\\
&~~~~~~+\mu([x_1,x_2,x_3]_\mathcal{Q})+\mathcal{D}_p(\Omega(x_1,x_2,x_3))+\mathcal{D}_p(\Phi(x_1,x_2)(v_3))\\&~~~~~~+(-1)^{|x_1|(|x_2|+|x_3|)}\mathcal{D}_p(\Phi(x_2,x_3)(v_1))+(-1)^{|x_3|(|x_1|+|x_2|)}\mathcal{D}_p(\Phi(x_3,x_1)(v_2))\\
&=[s(\mathcal{D}_q(x_1)),s(x_2),s(x_3)]_\mathcal{L}+\Phi(\mathcal{D}_q(x_1),x_2)(v_3)+(-1)^{|x_3|(\alpha+|x_1|+|x_2|)}\Phi(x_3,\mathcal{D}_q(x_1))(v_2)\\
&~~~~~~~+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(x_2,x_3)(\mu(x_1))+(-1)^{(\alpha+|x_1|)(|x_2|+|x_3|)}\Phi(x_2,x_3)(\mathcal{D}_p(v_1))\\
&~~~~~~+(-1)^{\alpha|x_1|}([s(x_1),s(\mathcal{D}_q(x_2)),s(x_3)]_\mathcal{L}+(-1)^{\alpha|x_1|}\Phi(x_1,\mathcal{D}_q(x_2))(v_3)\\
&~~~~~~+(-1)^{\alpha(|x_1|+|x_3|)+|x_1|(|x_2|+|x_3|)}\Phi(x_3,x_1)(\mathcal{D}_p(v_2))+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)(\mu(x_2))\\
&~~~~~~+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(\mathcal{D}_q(x_2),x_3)(v_1))+(-1)^{\alpha(|x_1|+|x_2|)}([s(x_1),s(x_2),s(\mathcal{D}_q(x_3))]_\mathcal{L}\\
&~~~~~~+(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)(\mu(x_3))+(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)(\mathcal{D}_p(v_3))\\&~~~~~~+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(\mathcal{D}_q(x_3),x_1)(v_2)+(-1)^{\alpha(|x_1|+|x_2|)+|x_1|(\alpha+|x_2|+|x_3|)}\Phi(x_2,\mathcal{D}_q(x_3))(v_1)).
\end{split}
\end{equation*}
Then we get
\begin{equation*}
\begin{split}
0&=-\Omega(\mathcal{D}_q(x_1),x_2,x_3)-(-1)^{\alpha|x_1|}\Omega(x_1,\mathcal{D}_q(x_2),x_3)-(-1)^{\alpha(|x_1|+|x_2|)}\Omega(x_1,x_2,\mathcal{D}_q(x_3))\\
&~~~~~~+\mathcal{D}_p(\Omega(x_1,x_2,x_3))+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(x_2,x_3)(\mu(x_1))+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)(\mu(x_2))\\&~~~~~~+(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)(\mu(x_3))+\mu([x_1,x_2,x_3])+(\mathcal{D}_p\Phi(x_1,x_2)\\&~~~~~~-(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_1),x_2)-(-1)^{\alpha|x_1|}\Phi(x_1,\mathcal{D}_q(x_2)))(v_3)\\&~~~~~~+(-1)^{|x_1|(|x_2|+|x_3|)}(\mathcal{D}_p\Phi(x_2,x_3)-(-1)^{\alpha(|x_2|+|x_3|)}\Phi(x_2,x_3)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_2),x_3)\\&~~~~~~-(-1)^{\alpha|x_2|}\Phi(x_2,\mathcal{D}_q(x_3)))(v_1)+(-1)^{|x_3|(|x_1|+|x_2|)}(\mathcal{D}_p\Phi(x_3,x_1)\\&~~~~~~-(-1)^{\alpha(|x_3|+|x_1|)}\Phi(x_3,x_1)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_3),x_1)-(-1)^{\alpha|x_1|}\Phi(x_3,\mathcal{D}_q(x_1)))(v_2).
\end{split}
\end{equation*}
Since $(\mathcal{D}_p,\mathcal{D}_q)$ is compatible with respect to $\Phi$,
\begin{equation*}
\begin{split}
(\mathcal{D}_p\Phi(x_1,x_2)&-(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_1),x_2)\\&-(-1)^{\alpha|x_1|}\Phi(x_1,\mathcal{D}_q(x_2)))(v_3)=0,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
(\mathcal{D}_p\Phi(x_2,x_3)&-(-1)^{\alpha(|x_2|+|x_3|)}\Phi(x_2,x_3)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_2),x_3)\\&-(-1)^{\alpha|x_2|}\Phi(x_2,\mathcal{D}_q(x_3)))(v_1)=0,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
(\mathcal{D}_p\Phi(x_3,x_1)&-(-1)^{\alpha(|x_3|+|x_1|)}\Phi(x_3,x_1)\mathcal{D}_p-\Phi(\mathcal{D}_q(x_3),x_1)\\&-(-1)^{\alpha|x_1|}\Phi(x_3,\mathcal{D}_q(x_1)))(v_2)=0.
\end{split}
\end{equation*}
Thus we have
\begin{equation*}
\begin{split}
&-\Omega(\mathcal{D}_q(x_1),x_2,x_3)-(-1)^{\alpha|x_1|}\Omega(x_1,\mathcal{D}_q(x_2),x_3)-(-1)^{\alpha(|x_1|+|x_2|)}\Omega(x_1,x_2,\mathcal{D}_q(x_3))\\
&+\mathcal{D}_p(\Omega(x_1,x_2,x_3))+(-1)^{|x_1|(|x_2|+|x_3|)}\Phi(x_2,x_3)(\mu(x_1))+(-1)^{|x_3|(|x_1|+|x_2|)}\Phi(x_3,x_1)(\mu(x_2))\\&+(-1)^{\alpha(|x_1|+|x_2|)}\Phi(x_1,x_2)(\mu(x_3))+\mu([x_1,x_2,x_3])=0,
\end{split}
\end{equation*}
since $|\mathcal{D}_p|=|\mathcal{D}_q|=|\mathcal{D}_l|=|\alpha|=0$, hence we have $Ob^\mathcal{L}_{(\mathcal{D}_p,\mathcal{D}_q)}(x_1,x_2,x_3)=(\delta_{\Phi}(\mu))(x_1,x_2,x_3)$ due to Eqs \ref{e32} and \ref{e25}.\\
To prove the converse part refer \cite{xu2018}.
\end{proof}
| {'timestamp': '2022-07-26T02:05:43', 'yymm': '2207', 'arxiv_id': '2207.11443', 'language': 'en', 'url': 'https://arxiv.org/abs/2207.11443'} |
\section{Introduction}
In spite of being an attractive material with excellent electronic properties \cite{ahcn09}, practical applications of graphene as in conventional semiconductor devices are still questionable due to its gapless nature. In particular, the ON/OFF current ratio is low while the saturation of current is poor in pristine graphene transistors \cite{schw10}. Many efforts of bandgap engineering in graphene \cite{yhan07,khar11,lher13,jbai10,zhan09} have been made to solve these issues. The pioneer technique proposed \cite{yhan07} is to cut 2D graphene sheets into 1D narrow nanoribons. In 2D graphene sheets, some options as Bernal-stacking of graphene on hexagonal boron nitride substrate \cite{khar11}, nitrogen-doped graphene \cite{lher13}, graphene nanomesh lattice \cite{jbai10,berr13} and Bernal-stacking bilayer graphene \cite{zhan09} have been explored. However, the possibility to open a sizable bandgap in graphene as large as those of standard semiconductors is still very unlikely. In particular, it requires a very good control of lattice geometry and edge disorder in narrow graphene nanoribbons (GNRs) \cite{quer08} and in graphene nanomesh lattices \cite{hung13}, while the bandgap opening in bilayer graphene by a perpendicular electric field may not be large enough for realistic applications \cite{fior09}. Other methods should be further verified by experiments.
\begin{figure}[!t]
\centering
\includegraphics[width=2.8in]{Fig01.pdf}
\caption{Schematic of unstrained/strained graphene junctions investigated in this work.}
\label{fig_sim1}
\end{figure}
On the other hand, graphene was experimentally demonstrated to be able to sustain a much larger strain than conventional semiconductors, making it a promising candidate for flexible electronics (see in a recent review \cite{shar13}). Indeed, strain engineering has been suggested to be an alternative approach to modulating efficiently the electronic properties of graphene nanomaterials. In particular, the bandgap has periodic oscillations in the armchair GNRs \cite{ylu210} while the spin polarization at the ribbon edges (and also the bandgap) can be modulated by the strain in the zigzag cases. In 2D graphene sheets, a finite gap can open under large strains, otherwise, it may remain close to zero but the Dirac points are displaced \cite{cocc10,per209,pere09,huan10}. Many interesting electrical, optical, and magnetic properties induced by strain in graphene have been also explored, e.g. see in \cite{bunc07,pere09,kuma12,per010,pell10,guin10,tlow10,zhai11}.
Besides, local strain is a good option to improve the electrical performance of graphene devices \cite{pere09,ylu010,fuji10,juan11,baha13}. For instance, it has been shown to enhance the ON current in a GNR tunneling FET \cite{ylu010} and to fortify the transport gap in GNR strained junctions \cite{baha13}. In a recent work \cite{hung14}, we have investigated the effects of uniaxial strain on the transport in 2D unstrained/strained graphene junctions and found that due to the strain-induced shift of Dirac points, a significant conduction gap of a few hundreds meV can open with a small strain of a few percent. This type of strained junction was then demonstrated to be an excellent candidate to improve the electronic operation of graphene transistors. It hence motivates us to further investigate the properties of this conduction gap so as to optimize the performance of graphene devices. On the one hand, the effects of strain should be, in principle, dependent on its applied direction. On the other hand, because the appearance of conduction gap is a consequence of the shift of Dirac points along the $k_y$-axis, it is predicted that this gap should also depend on the transport direction. Note that here, Oy (Ox) - axis is assumed to be perpendicular (parallel) to the transport direction. The effects of both strain and transport directions will be clarified systematically in the current work.
\section{Model and calculations}
In this work, the $\pi$-orbital tight binding model constructed in \cite{per209} is used to investigate the electronic transport through the graphene strained junctions schematized in Fig. 1. The Hamiltonian is ${H_{tb}} = \sum\nolimits_{nm} {{t_{nm}}c_n^\dag {c_m}}$ where $t_{nm}$ is the hopping energy between nearest neighbor \emph{n}th and \emph{m}th atoms. The application of a uniaxial strain of angle $\theta$ causes the following changes in the $C-C$ bond vectors:
\begin{eqnarray}
{{\vec r}_{nm}}\left( \sigma \right) &=& \left\{ {1 + {M_s}\left( \sigma, \theta \right)} \right\}{{\vec r}_{nm}}\left( 0 \right) \\
{M_s}\left( \sigma, \theta \right) &=& \sigma \left[ {\begin{array}{*{20}{c}}
{{{\cos }^2}\theta - \gamma {{\sin }^2}\theta }&{\left( {1 + \gamma } \right)\sin \theta \cos \theta }\\
{\left( {1 + \gamma } \right)\sin \theta \cos \theta }&{{{\sin }^2}\theta - \gamma {{\cos }^2}\theta }
\end{array}} \right] \nonumber
\end{eqnarray}
where $\sigma$ represents the strain and $\gamma \simeq 0.165$ is the Poisson ratio \cite{blak70}. The hopping parameters are defined as $t_{nm} \left( \sigma \right) = t_0 \exp\left[-3.37\left(r_{nm} \left( \sigma \right) /r_0 - 1\right)\right]$, where the hopping energy $t_0 = -2.7$ $eV$ and the bond length $r_{nm} \left( 0 \right) \equiv r_0 = 0.142$ $nm$ in the unstrained case. Therefore, there are three different hoping parameters $t_{1,2,3}$ corresponding to three bond vectors ${\vec r}_{1,2,3}$, respectively, in the strained graphene part of the structure (see Fig. 1). Here, we assume a 1D profile of applied strain, i.e., the strain tensor is a function of position along the transport direction Ox while it is constant along the Oy-axis. The transport direction, $\phi$, and strain direction, $\theta$, are determined as schematized in Fig. 1. Based on this tight binding model, two methods described below can be used to investigate the conduction gap of the considered strained junctions.
\textbf{Green's function calculations.} First, we split the graphene sheet into the smallest possible unit cells periodically repeated along the Ox/Oy directions with the indices $p/q$, respectively (similarly, see the details in \cite{hung12}). The tight-binding Hamiltonian can therefore be expressed in the following form:
\begin{eqnarray}
{H_{tb}} = \sum\limits_{p,q} {\left( {{H_{p,q}} + \sum\limits_{{p_1},{q_1}} {{H_{p,q \to p_1,q_1}}} } \right)}
\end{eqnarray}
where $H_{p,q}$ is the Hamiltonian of cell $\{p,q\}$, and $H_{p,q \to p_1,q_1}$ denotes the coupling of cell $\{p,q\}$ to its nearest neighbor cell $\{p_1,q_1\}$. We then Fourier transform the operators in Eq. (2) as follows:
\begin{eqnarray}
{c_{p,q}} = \frac{1}{{\sqrt {{M_{cell}}} }}\sum\limits_{{\kappa_y}} {{e^{i{q\kappa_y}}}} {{\hat c}_{p,{\kappa_y}}},
\end{eqnarray}
where $M_{cell}$ is the number of unit cells and $\kappa_y \equiv k_y L_y$ with the size $L_y$ of unit cells along the Oy direction. The Hamiltonian (2) is finally rewritten as a sum of $\kappa_y$-dependent 1D-components:
\begin{eqnarray}
{H_{tb}} &=& \sum\limits_{{\kappa_y}} {\hat H\left( {{\kappa_y}} \right)} \\
\hat H\left( {{\kappa_y}} \right) &=& \sum\limits_p {{{\hat H}_{p \to p - 1}}\left( {{\kappa_y}} \right) + {{\hat H}_p}\left( {{\kappa_y}} \right) + {{\hat H}_{p \to p + 1}}}\left( {{\kappa_y}} \right) \nonumber
\end{eqnarray}
With this Hamiltonian form, the Green's function formalism can be easily applied to compute transport quantities in the graphene strained junction with different transport directions. In particular, the conductance at zero temperature is determined as:
\begin{eqnarray}
\mathcal{G} \left( \epsilon \right) = \frac{{e^2 W}}{{\pi h L_y}}\int\limits_{BZ} {d{\kappa_y} \mathcal{T}\left( {\epsilon, {\kappa_y}} \right)}
\end{eqnarray}
where $\mathcal{T}\left( {\epsilon,{\kappa_y}} \right)$ is the transmission probability computed from the Green's functions. The integration over $\kappa_y$ is performed in the whole first Brillouin zone. As in ref. \cite{hung13}, the gap of conductance (conduction gap) is then measured from the obtained data of conductance.
\textbf{Bandstructure analyses.} To determine the conduction gap of strained junctions, we find that another simple way based on the analysis of graphene bandstructures could be efficiently used. It is described as follows. Since the conductance is computed from Eq. (5), the appearance of conduction gap is essentially governed by the gaps of transmission probability, which is determined from the energy gaps in the unstrained and strained graphene sections. These energy gaps can be defined directly from the graphene bandstructures. Therefore, our calculation has two steps, similar to that in \cite{hung14}. From the graphene bandstructures obtained using the tight-binding Hamiltonian above, we first look for the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ for a given $\kappa_y$ of two graphene sections. The maximum of these energy gaps determines the gap $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ of transmission probability through the junction. Finally, the conduction gap $E_{cond.gap}$ is obtained by looking for the minimum value of $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ when varying $\kappa_y$ in the whole Brillouin zone.
In particular, the energy bands of strained graphene are given by
\begin{eqnarray}
E\left( {\vec k} \right) = \pm \left| {{t_1}{e^{i\vec k{{\vec a}_1}}} + {t_2}{e^{i\vec k{{\vec a}_2}}} + {t_3}} \right|
\end{eqnarray}
where the plus/minus sign corresponds to the conduction/valence bands, respectively. For a given direction $\phi$ of transport, in principle, the vectors $\vec L_{x,y}$ defining the sizes of unit cell along the Ox and Oy directions, respectively, can be always expressed as ${\vec L_x} = {n_1}{\vec a_1} + {n_2}{\vec a_2}$ and ${\vec L_y} = {m_1}{\vec a_1} + {m_2}{\vec a_2}$ with $\cos \phi = \frac{{{{\vec L}_x}\vec L_x^0}}{{{L_x}L_x^0}}$ and $\sin \phi = \frac{{{{\vec L}_x}\vec L_y^0}}{{{L_x}L_y^0}}$ while $\vec L_{x,y}^0 = {\vec a_1} \pm {\vec a_2}$. Note that $n_{1,2}$ and $m_{1,2}$ are integers while $\frac{{{m_1}}}{{{m_2}}} = - \frac{{{n_1} + 2{n_2}}}{{{n_2} + 2{n_1}}}$, i.e., ${\vec L_{x}} {\vec L_{y}} = 0$. In other words, we have the following expressions
\begin{eqnarray}
{{{\vec a}_1} = \frac{{ - {m_2}{{\vec L}_x} + {n_2}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}};\,\,\,{{\vec a}_2} = \frac{{{m_1}{{\vec L}_x} - {n_1}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}}}
\end{eqnarray}
On this basis, the energy bands can be rewritten in terms of $\kappa_{x, y} = \vec k \vec L_{x,y} \left( { \equiv {k_{x,y}}{L_{x,y}}} \right)$ by substituting Eqs. (7) into Eq. (6). This new form of energy bands is finally used to compute the conduction gap of strained junctions.
As a simple example, in the case of $\phi = 0$ (armchair direction), we calculate the conduction gap as follows. First, Eq. (6) is rewritten in the form
\begin{eqnarray}
E_{\phi = 0}\left( {\vec \kappa} \right) = \pm \left| {{t_1}{e^{i\kappa_y/2}} + {t_2}{e^{ - i\kappa_y/2}} + {t_3}{e^{ - i\kappa_x/2}}} \right|
\end{eqnarray}
with the vectors $\vec L_{x,y} \equiv \vec L_{x,y}^0$. Using this new form, the energy gap of strained graphene for a given $\kappa_y$ is determined as
\begin{equation}
{E_{strain}^{gap}}\left( {{\kappa_y}} \right) = 2 \left| {\sqrt {{{\left( {{t_1} - {t_2}} \right)}^2} + 4{t_1}{t_2}{{\cos }^2}\frac{{{\kappa_y}}}{2}} + {t_3}} \right|
\end{equation}
while ${E_{unstrain}^{gap}}\left( {{\kappa_y}} \right)$ is given by the same formula with $t_1$ = $t_2$ = $t_3$ $\equiv$ $t_0$. The gap of transmission probability through the junction is then determined as ${E_{junc}^{gap}}\left( {{\kappa_y}} \right) = \max \left[ {E_{unstrain}^{gap}\left( {{\kappa_y}} \right),E_{strain}^{gap}\left( {{\kappa_y}} \right)} \right]$ and, finally, the conduction gap is given by ${E_{cond.gap}} = \min \left[ {E_{junc}^{gap}\left( {{\kappa_y}} \right)} \right]$ for $\kappa_y$ in the whole Brillouin zone.
We would like to notice that the Green's function calculations and the banstructure analyses give the same results of conduction gap in the junctions where the transition region between unstrained and strained graphene sections is long enough, i.e., larger than about 5 to 6 nm. In the case of short length, as discussed in \cite{baha13,hung14}, this transition zone can have significant effects on the transmission between propagating states beyond the energy gaps and hence can slightly enlarge the gap of conductance, compared to the results obtained from the bandstructure calculations.
\section{Results and discussion}
\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{Fig02.pdf}
\caption{Dependence of graphene bandgap (in the unit of eV) on the applied strain and its direction: tensile (a) and compressive (b). The radius from the central point indicates the strain strength ranging from 0 (center) to 30 $\%$ (edge of maps) while the graphene lattice is superimposed to show visibly the strain direction. The orange circle corresponds to the strains of $\sigma = 23 \%$.}
\label{fig_sim2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{Fig03.pdf}
\caption{Conductance ($G_0 = e^2W/hL_y$) as a function of energy in graphene strained junctions for $\sigma = 4 \%$ with different strain directions. The transport along the armchair direction ($\phi = 0$) is considered. The data obtained in a uniformly strained graphene is displayed for the comparison.}
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.8in]{Fig04.pdf}
\caption{Local density of states (left panels) and corresponding transmission coefficient (right panels) for three different wave-vectors $k_y$ obtained in an unstrained/strained graphene junction of $\sigma = 4 \%$, and $\theta \equiv \phi = 0$. On the top is a schematic of graphene bandedges illustrating the strain-induced shift of Dirac points along the $k_y$-direction.}
\label{fig_sim4}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.6in]{Fig05.pdf}
\caption{Maps of conduction gap in unstrained/strained graphene junctions: tensile (a,c) and compressive cases (b,d). The transport is along the armchair $\phi = 0$ (a,b) and zigzag $\phi = 30^\circ$ directions (c,d). The strain strength ranges from 0 (center) to 6 $\%$ (edge of maps) in all cases.}
\label{fig_sim4}
\end{figure*}
First, we re-examine the formation of the bandgap of graphene under a uniaxial strain. From Eq. (9), it is shown that a strain-induced finite-bandgap appears only if ${E_{strain}^{gap}}\left( {{\kappa_y}} \right) > 0$ for all $k_y$ in the first Brillouin zone, i.e., ${k _y} \in \left[ { - \frac{\pi}{L_y}, \frac{\pi}{L_y}} \right]$, otherwise, the bandgap remains zero. Hence, the condition for the bandgap to be finite is either
\begin{equation*}
\left| {{t_1} - {t_2}} \right| > \left| {{t_3}} \right|\,\,\,\,\,{\rm{OR}}\,\,\,\,\,\left| {{t_3}} \right| > \left| {{t_1} + {t_2}} \right|
\end{equation*}
and the corresponding values of bandgap are
\begin{equation*}
{E_{gap}} = 2\left( {\left| {{t_1} - {t_2}} \right| - \left| {{t_3}} \right|} \right)\,\,\,\,\,{\rm{OR}}\,\,\,\,\,2\left( {\left| {{t_3}} \right| - \left| {{t_1} + {t_2}} \right|} \right)
\end{equation*}
This result was actually reported in \cite{per209,hase06}. We remind as displayed in Fig. 2(a) that a finite bandgap opens only for strain larger than $\sim 23 \%$ and the zigzag (not armchair) is the preferred direction for bandgap opening under a tensile strain \cite{per209}. We extend our investigation to the case of compressive strain and find (see in Fig. 2(b)) that (i) the same gap threshold of $\sigma \simeq 23 \%$ is observed but (ii) the preferred direction to open the gap under a compressive strain is the armchair, not the zigzag as the case of tensile strain. This implies that the properties of graphene bandstructure at low energy should be qualitatively the same when applying strains of $\left\{ {\sigma ,\theta } \right\}$ and of $\left\{ {-\sigma ,\theta + 90^\circ} \right\}$. This feature can be understood by considering, for example, strains of $\left\{ {\sigma , \theta = 0} \right\}$ and of $\left\{ {-\sigma , \theta = 90^\circ} \right\}$. Indeed, these strains result in the same qualitative changes on the bond-lengths, i.e., an increased bond-length $r_3$ and reduced bond-lengths $r_{1,2}$. However, for the same strain strength, because of the exponential dependence of hoping energies on the bond-lengths, the compressive strain generally induces a larger bandgap than the tensile one, as can be seen when comparing the data displayed in Figs. 2(a) and 2(b). To conclude, we would like to emphasize that a large strain is necessary to open a bandgap in graphene. This could be an issue for practical applications, compared to the use of graphene strained junctions explored in \cite{hung14}.
We now go to explore the properties of conduction gap in the graphene strained junctions. In Fig. 3, we display the conductance as a function of energy computed from Eq. (5) using the Green's function technique. As discussed above, a small strain of a few percent (e.g., 4 $\%$ here) can not change the gapless character of graphene, i.e., there is no gap of conductance in the case of uniformly strained graphene. However, similar to that reported in \cite{hung14}, a significant conduction-gap of a few hundreds meV can open in the unstrained/strained graphene junctions. The appearance of this conduction gap, as mentioned previously, is due to the strain-induced shift of Dirac points and is explained as follows. Actually, the strain causes the lattice deformation and can result in the deformation of graphene bandstructure. Therefore, the bandedges as a function of wave-vector $k_y$ in unstrained and strained graphene can be illustrated schematically as in the top panel of Fig. 4. As one can see, the shift of Dirac points leads to the situation where there is no value of $\kappa_y$, for which the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are simultaneously equal to zero. This means that the transmission probability always shows a finite gap for any $\kappa_y$. For instance, the energy gap is zero (or small) in the unstrained (resp. strained) graphene section but finite in the strained (resp. unstrained) one in the vicinity of Dirac point $k_y = K_{unstrain}$ (resp. $K_{strain}$). Accordingly, as illustrated in the pictures of LDOS in the left panels of Fig. 4 and confirmed in the corresponding transmissions in the right panels, clear gaps of transmission are still obtained. Far from these values of $k_y$, $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are both finite (e.g., see the LDOS plotted for $k_y = K_{gap}$) and hence a finite gap of transmission also occurs. On this basis, a finite gap of conductance is achieved. More important, Fig. 3 shows that besides the strength of strain, the strain effect is also strongly dependent on the applied direction. For instance, the conduction gap takes the values of $\sim$ 295, 172 and 323 meV for $\theta = 0$, $30^\circ$ and $90^\circ$, respectively.
Below, we will discuss the properties of the conduction gap with respect to the strain, its applied direction, and the direction of transport. Note that due to the lattice symmetry, the transport directions $\phi$ and $\phi + 60^\circ$ are equivalent while the applied strain of angle $\theta$ is identical to that of $\theta + 180^\circ$. Hence, the data obtained for $\phi$ ranging from $-30^\circ$ to $30^\circ$ and $\theta \in \left[ {0^\circ ,180^\circ } \right]$ covers the properties of conduction gap in all possible cases.
In Fig. 5, we present the maps of conduction gap with respect to the strain and its applied direction in two particular cases: the transport is either along the armchair ($\phi = 0$) or the zigzag ($\phi = 30^\circ$) directions. Both tensile and compressive strains are considered. Let us first discuss the results obtained in the armchair case. Figs. 5(a,b) show that (i) a large conduction gap up to about 500 meV can open with a strain of 6 $\%$ and (ii) again the conduction gap is strongly $\theta$-dependent, in particular, its peaks occur at $\theta = 0$ or $90^\circ$ while the gap is zero at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain. In principle, the conduction gap is larger if the shift of Dirac points in the $\kappa_y$-axis is larger, as discussed above about Figs. 3-4. We notice that the strain-induced shifts can be different for the six Dirac points of graphene \cite{kitt12} and the gap is zero when there is any Dirac point observed at the same $\kappa_y$ in the two graphene sections. From Eq. (9), we find that the Dirac points are determined by the following equations:
\begin{eqnarray*}
{\cos}\frac{\kappa_y}{2} &=& \pm \frac{1}{2}\sqrt{\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{{t_1}{t_2}}}}, \\
\cos \frac{{\kappa_x}}{2} &=& \frac{{{t_1} + {t_2}}}{{\left| {{t_3}} \right|}}\cos \frac{{\kappa_y}}{2},\,\,\,\sin \frac{{\kappa_x}}{2} = \frac{{{t_2} - {t_1}}}{{\left| {{t_3}} \right|}}\sin \frac{{\kappa_y}}{2},
\end{eqnarray*}
which simplify into ${\cos}\frac{\kappa_y}{2} = \pm \frac{1}{2}$ and, respectively, $\cos \left( {\frac{{{\kappa _x}}}{2}} \right) = \mp 1$ in the unstrained case. Hence, the zero conduction gap is obtained if
\begin{equation*}
\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{4{t_1}{t_2}}} = \frac{1}{4}
\end{equation*}
Additionally, it is observed that the effects of a strain $\{\sigma,\theta\}$ are qualitatively similar to those of a strain $\{-\sigma,\theta+90^\circ\}$, i.e., the peaks and zero values of conduction gap are obtained at the same $\theta$ in these two situations. To understand this, we analyze the strain matrix $M_s \left(\sigma,\theta\right)$ and find that in the case of small strains studied here, there is an approximate relationship between the bond lengths under these two strains, given by \[{r \left( \sigma, \theta \right)} - {r \left( -\sigma, \theta + 90^\circ\right)} \simeq \sigma \left( {1 - \gamma } \right) r_0,\] which is $\theta$-independent for all \emph{C-C} bond vectors. It implies that there is a fixed ratio between the hopping energies $t_i \left( \sigma, \theta \right)$ and $t_i \left( -\sigma, \theta + 90^\circ\right)$ and hence there is the similar shift of Dirac points in these two cases.
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{Fig06.pdf}
\caption{Map showing the dependence of conduction gap on the directions ($\theta,\phi$) for $\sigma = 4 \%$. The top is a diagram illustrating the rotation of Dirac points in the \emph{k}-space with the change in the transport direction $\phi$.}
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.5in]{Fig07.pdf}
\caption{Maps of conduction gap obtained in tensile/compressive strained junctions. The transport along the armchair/zigzag directions is considered in (a,b)/(c,d), respectively. The strains $\sigma_c = -2 \%$ and $\sigma_t = 2 \%$ are applied in (a,c) while $\sigma_c = -1 \%$ and $\sigma_t = 3 \%$ in (b,d).}
\label{fig_sim4}
\end{figure*}
We now go to analyze the properties of conduction gap shown in Figs. 5(c,d) where the transport is along the zigzag direction $\phi = 30^\circ$. In fact, the conduction gap in this case can reach a value as high as that of the case of $\phi = 0$ but has different $\theta$-dependence. In particular, the conduction gap has peaks at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain, where it is zero in the case of $\phi = 0$. It is also equal to zero at $\theta = 0$ and $\theta = 90^\circ$ where the peaks of conduction gap occur in the latter case of $\phi = 0$. The relationship between these two transport directions can be explained as follows. On the one hand, based on the analyses above for $\phi = 0$, we find that for a given strength of strain, a maximum shift of Dirac points along the $k_y$-axis corresponds to a minimum along the $k_x$-one and vice versa when varying the strain direction $\theta$. On the other hand, as schematized in the top of Fig. 6 below, the change in the transport direction results in the rotation of the first Brillouin zone, i.e., the $k_x$ (resp. $k_y$) axis in the case of $\phi = 30^\circ$ is identical to the $k_y$ (resp. $k_x$) axis in the case of $\phi = 0$. These two features explain essentially the opposite $\theta$-dependence of conduction gap for $\phi = 30^\circ$, compared to the case of $\phi = 0$ as mentioned. Again, we found the same qualitative behavior of conduction gap when applying the strains of $\{\sigma,\theta\}$ and $\{-\sigma,\theta+90^\circ\}$.
Next, we investigate the conduction gap with respect to different transport directions $\phi$. We display a ($\theta,\phi$)-map of conduction gap for $\sigma = 4 \%$ in Fig. 6 and, in the top, an additional diagram illustrating the rotation of Dirac points in the $k-$space with the change in the transport direction. It is clearly shown that (i) a similar scale of conduction gap is obtained for all different transport directions, (ii) there is a smooth and continuous shift of $E_{cond.gap}-\theta$ behavior when varying $\phi$, and (iii) the same behavior of $E_{cond.gap}$ is also observed when comparing the two transport directions of $\phi$ and $\phi+30^\circ$, similarly to the comparison above between $\phi = 0^\circ$ and $30^\circ$. The data plotted in Fig. 6 additionally shows that $E_{cond.gap}$ takes the same value in both cases of $\{\phi,\theta\}$ and $\{-\phi,-\theta\}$ with a remark that the strains of $-\theta$ and $180^\circ-\theta$ are identical. Moreover, the values of $\theta$ and $\phi$, for which the conduction gap has a peak or is equal to zero, have an almost linear relationship. In particular, the relationship for conduction gap peaks is approximately given by $\theta = \theta_A - \eta_s \phi$. For tensile strains, $\eta_s$ takes the values of $\sim 1.5667$ and $1.4333$ for $\theta_A = 0$ and $90^\circ$, respectively. On the opposite, it is about $1.4333$ and $1.5667$ for $\theta_A = 0$ and $90^\circ$, respectively, for compressive strain cases. All these features are consequences of the rotation of Dirac points in the $k$-space with respect to the transport direction $\phi$ as illustrated in the diagram on the top and the lattice symmetry of graphene.
Finally, we investigate other junctions based on compressive and tensile strained graphene sections. The idea is that in this type of strained junction, the shifts of Dirac points are different in two graphene sections of different strains, which offers the possibilities to use smaller strains to achieve a similar conduction gap, compared to the case of unstrained/strained junction. In Fig. 7, we display the maps of conduction gap with respect to the directions of compressive ($\theta_c$) and tensile ($\theta_t$) strains in two cases of transport direction $\phi = 0$ (armchair) and $30^\circ$ (zigzag) for given strain strengths. Indeed, as seen in Fig. 7(a,b), with smaller strains $\left\{ {{\sigma _c},{\sigma _t}} \right\} = \left\{ { - 2\% ,2\% } \right\}$ or $\left\{ { - 1\% ,3\% } \right\}$, similar conduction gap of about 310 meV can be achieved (see Figs. 7(a,b)) while it requires a strain of 4 $\%$ in the unstrained/strained junctions discussed above. However, since the shift of Dirac points is strongly dependent on the direction of applied strains and the transport direction, the properties of conduction gap are more complicated than in the latter case. In particular, our calculations show that the preferred transport directions to achieve a large conduction gap are close to the armchair one. Otherwise, the conduction gap is generally smaller, similarly to the data for $\phi = 30^\circ$ compared to $\phi = 0$, as shown in Fig. 7. Additionally, it is shown that the preferred directions of applied strains in the case of $\phi = 0$ are close to ${\theta _c} \equiv {\theta _t} = 0$ or $90^\circ$.
\section{Conclusion}
Based on the tight binding calculations, we have investigated the effects of uniaxial strain on the transport properties of graphene strained junctions and discuss systematically the possibilities of achieving a large conduction gap with respect to the strain, its applied direction and the transport direction. It has been shown that due to the strain-induced deformation of graphene lattice and hence of graphene bandstructure, a finite conduction gap higher than 500 meV can be achieved for a strain of only 6 $\%$. Moreover, as a consequence of the shift of Dirac points along the $k_y$-axis, the conduction gap is strongly dependent not only on the strain strength but also on the direction of applied strain and the transport direction. A full picture of these properties of conduction gap has been presented and explained. The study hence could be a good guide for the use of this type of unstrained/strained graphene junction in electronic applications.
\textbf{\textit{Acknowledgment.}} This research in Hanoi is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.02-1012.42. We also acknowledges the French ANR for financial support under the projects NANOSIM-GRAPHENE (Grant no. ANR-09-NANO-016) and MIGRAQUEL (Grant no. ANR-10-BLAN-0304).
| {'timestamp': '2014-04-09T02:01:59', 'yymm': '1403', 'arxiv_id': '1403.5310', 'language': 'en', 'url': 'https://arxiv.org/abs/1403.5310'} |
\section{Introduction}
Quantum Chromodynamics (QCD) suggests the possible
existence of hadrons with a substructure that is more complex
than the three quark baryons and the quark-antiquark mesons
of the Quark Parton Model (QPM). Possibilities for these
so-called {\it exotic} hadrons include
pentaquark baryons ($qqq\bar{q}q$), tetraquark mesons
($q\bar{q}q\bar{q}$) and quark-gluon hybrids ($q\bar{q}g$).
Although considerable theoretical and experimental effort
has gone into identifying exotic states, the situation
remains unclear. The interest in this subject is demonstrated
by the huge literature related to the purported observation
of the $\Theta(1535)$ strangeness=+1 pentaquark. According to
SPIRES, the experimental paper~\cite{nakano_theta}
that claimed first observation of the $\Theta(1530)$
has received over 785 citations.
There has been some recent progress in the identification
of what may be exotic mesons. The BaBar and Belle
$B$-factory experiments have, somewhat unexpectedly,
discovered a number of interesting charmonium-like meson states
that have defied assignment to any of the unfilled levels
of the $c\bar{c}$ spectrum and, thus, remain unclassified.
These have come to be known
collectively as the ``$XYZ$'' mesons, and
include: the experimentally well established
$X(3872)$~\cite{belle_x3872} and
$Y(4260)$~\cite{babar_y4260}, which decay to
$\pipiJ/\psi$; the $X(3940)$~\cite{belle_x3940}, seen
in $D^*\bar{D}$~\cite{charge_conjugate},
and the $X(4160)$~\cite{belle_x4160}
seen in $D^*\bar{D}^*$; the
$Y(3940)$~\cite{belle_y3940,babar_y3940}, seen in $\omegaJ/\psi$;
and the $Y(4350)$~\cite{babar_y4325}
and $Y(4660)$~\cite{belle_y4350} seen in
$\pi^+\pi^- \psi '$. In addition, Belle reported observations of similar
states but with non-zero electric charge: the
$Z(4430)$~\cite{belle_z4430}
seen in $\pi^+\psi '$ and the $Z_1(4040)$ \&
$Z_2(4240)$~\cite{belle_z4040} seen in $\pi^+\chi_{c1}$. These
$Z$ states have not yet been confirmed by other
experiments and remain somewhat controversial~\cite{babar_z4430}.
Table~{\ref{table1}} summarizes the
abovementioned $XYZ$ candidate states
as well as some other states discussed below.
In this report I will briefly review the reasons why these states
have eluded conventional $c\bar{c}$ assignments, discuss possible
alternative interpretations, and present some evidence for similar
states in the $s$- and $b$-quark sectors.
\begin{table}[htb!]
\begin{center}
\caption{\label{table1} Summary of the candidate $XYZ$ mesons discussed
in this talk.}
\footnotesize
\begin{tabular*}{180mm}{lcccccc}
\hline\hline
state & $M$~(MeV) &$\Gamma$~(MeV) & $J^{PC}$ & Decay Modes &
Production Modes & Observed by:\\\hline
$Y_s(2175)$& $2175\pm8$&$ 58\pm26 $& $1^{--}$ & $\phi f_0(980)$ &
$e^{+} e^{-}$~(ISR), $J/\psi\rightarrow\eta Y_s(2175)$ & BaBar, BESII, Belle\\
$X(3872)$& $3871.4\pm0.6$&$<2.3$& $1^{++}$ & $\pi^{+}\pi^{-}
J/\psi$,$\gamma J/\psi$,
$D\bar{D^*}$
& $B\rightarrow KX(3872)$, $p\bar{p}$ & Belle, CDF, D0, BaBar\\
$Z(3930)$& $3929\pm5$&$ 29\pm10 $& $2^{++}$ & $D\bar{D}$ &
$\gamma\gamma\rightarrow Z(3940)$ & Belle \\
$X(3940)$& $3942\pm9$&$ 37\pm17 $& $0^{?+}$ & $D\bar{D^*}$ (not
$D\bar{D}$ or $\omega J/\psi$) & $e^{+} e^{-}\rightarrow J/\psi X(3940)$ & Belle \\
$Y(3940)$& $3943\pm17$&$ 87\pm34 $&$?^{?+}$ & $\omega J/\psi$ (not
$D\bar{D^*}$) & $B\rightarrow K Y(3940)$ & Belle, BaBar \\
$Y(4008)$& $4008^{+82}_{-49}$&$ 226^{+97}_{-80}$ &$1^{--}$& $\pi^{+}\pi^{-} J/\psi$ &
$e^{+} e^{-}$(ISR) & Belle \\
$X(4160)$& $4156\pm29$&$ 139^{+113}_{-65}$ &$0^{?+}$& $D^*\bar{D^*}$
(not $D\bar{D}$) & $e^{+} e^{-} \rightarrow J/\psi X(4160)$ & Belle \\
$Y(4260)$& $4264\pm12$&$ 83\pm22$ &$1^{--}$& $\pi^{+}\pi^{-} J/\psi$ & $e^{+} e^{-}$(ISR)
&BaBar, CLEO, Belle \\
$Y(4350)$& $4361\pm13$&$ 74\pm18$ &$1^{--}$& $\pi^{+}\pi^{-} \psi '$ & $e^{+} e^{-}$(ISR)
& BaBar, Belle \\
$Y(4660)$& $4664\pm12$&$ 48\pm15 $ &$1^{--}$& $\pi^{+}\pi^{-} \psi '$ & $e^{+} e^{-} $(ISR)
& Belle \\
$Z_1(4050)$& $4051^{+24}_{-23}$&$ 82^{+51}_{-29}$ & ? &
$\pi^{\pm}\chi_{c1}$ & $B\rightarrow K Z_1^{\pm}(4050)$ & Belle \\
$Z_2(4250)$& $4248^{+185}_{-45}$&$ 177^{+320}_{-72}$ & ? &
$\pi^{\pm}\chi_{c1}$ & $B\rightarrow K
Z_2^{\pm}(4250)$ & Belle \\
$Z(4430)$& $4433\pm5$&$ 45^{+35}_{-18}$ & ? & $\pi^{\pm}\psi '$ & $B\rightarrow K
Z^{\pm}(4430)$ & Belle \\
$Y_b(10890)$ & $10,890\pm 3$ & $55\pm 9$ & $1^{--}$ &
$\pi^{+}\pi^{-}\Upsilon(1,2,3S)$
& $e^{+} e^{-}\rightarrow Y_b$ & Belle \\
\hline\hline
\end{tabular*}%
\end{center}
\end{table}
\section{Charmonium possibilities}
\label{charmonium}
The $c\bar{c}$ charmonium meson level diagram is shown in
Fig.~\ref{fig:charmonium_spectrum}~\cite{charmonium_spectrum}. Here the
states that have already been assigned are labeled by their commonly used
symbols and measured mass values. The solid lines indicate the
measured levels and the broken lines indicate masses derived from
QCD-motivated potential model calculations~\cite{BGS}. If any of the
$XYZ$ mesons are to be interpreted as simple quark-antiquark states,
they must be assigned to one of the figure's unlabeled levels.
All of the states with mass below the $M=2m_D=3.73$~GeV
``open-charm'' threshold (indicated by the horizontal line in
Fig.~\ref{fig:charmonium_spectrum}) have already been identified and
have properties that are in good agreement with potential model
expectations. In addition, all of the $1^{--}$ levels above the
open charm threshold have been assigned to peaks in
the total annihilation cross section for
$e^{+} e^{-}\rightarrow~hadrons$~\cite{bes_R-fit}.
Of the $XYZ$ states listed in Table~\ref{table1}, only the
$Z(3930)$~\cite{belle_z3930} has been convincingly assigned to a
charmonium
level;
there is general agreement that this is the ($2^3P_2$)
$\chi_{c2}^\prime$.
\begin{figure}[t]
\includegraphics[scale=0.43]{olsen_panic_fig1.eps}
\caption{The predicted and observed spectrum of $c\bar{c}$ charmonium
mesons. Already assigned states and their experimentally
measured masses are indicated by solid bars and their
commonly used names. The broken lines indicate
various theoretical predictions. The horizontal line
at 3.73~GeV indicates the mass threshold for decays
to $D\bar{D}$ ``open charm'' final states.}
\label{fig:charmonium_spectrum}
\end{figure}
\subsection{The $X(3872)$}
The experimentally preferred $J^{PC}$ value for the $X(3872)$
is $1^{++}$, although $2^{-+}$ has not been conclusively ruled
out~\cite{CDF_jpc}. The only unfilled $1^{++}$ level in
Fig.~\ref{fig:charmonium_spectrum} is the $\chi_{c1}^\prime$, the
$2^3P_1$ $c\bar{c}$ state. As mentioned above, the $J=2$
triplet partner state
for this level has been identified as the $Z(3930)$
with a mass of $3929\pm 5$~MeV. A
$\chi_{c1}^\prime$ assignment for the $X(3872)$ would imply
a $\chi_{c2}^{\prime}$-$\chi_{c1}^{\prime}$ mass splitting for
radial quantum number
$n=2$ ({\it i.e.} $\delta m \sim 57$~MeV) that is larger than that
for the $n=1$ splitting ($\delta m = 46$~MeV), contrary to
potential model expectations. A bigger difficulty with this
assignment is the fact that the
$X(3872)\rightarrow\rhoJ/\psi$ discovery mode would be an isospin violating
transition that should be strongly suppressed compared to
the $X(3872)\rightarrow \gammaJ/\psi$ mode; the latter is measured to
be much smaller than the former~\cite{x3872_2gammajpsi}.
Using the $2^{-+}$ assignment does not help; in this
case the $\pipiJ/\psi$ mode would also be isospin violating, and the
$\gammaJ/\psi$ transition, which would be a $\Delta L=2$ transition,
would be unmeasurably small, which it isn't.
\subsection{The $X(3940)$ and $X(4160)$}
The $X(3940)$ is seen in the $D\bar{D^*}$ system recoiling from
the $J/\psi$ in exclusive $ee\rtJ/\psi D\bar{D^*}$ annihilations;
the $X(4160)$ is seen in the $D^*\bar{D^*}$ system in
$J/\psi$ in $ee\rtJ/\psi D^*\bar{D^*}$. Neither are seen in the
experimentally more accessible $D\bar{D}$ channel. The
only known charmonium states that are seen recoiling from the $J/\psi$
in $e^{+} e^{-}\rtJ/\psi X$ processes have $J=0$. This, plus the
absence of the $D\bar{D}$ mode,
provides circumstantial evidence that favors $J^{PC}=0^{-+}$ assignments
for both states, which for charmonium would be the $\eta_c^{\prime\prime}$
and $\eta_c^{\prime\prime\prime}$. Such an assignment has difficulty
with the measured masses: the predicted $\eta_c^{\prime\prime}$ mass
is about 4050~MeV, over 100~MeV too high for the $X(3940)$; the
predicted $\eta_c^{\prime\prime\prime}$ is around 4400~MeV, more
than 200~MeV higher than the $X(4160)$,
\subsection{The $Y(3940)$}
The $Y(3940)$ was first seen by Belle
as a near-threshold peak in the $\omegaJ/\psi$
invariant mass spectrum in exclusive $B\rightarrow K\omegaJ/\psi$
decays~\cite{belle_y3940}. It was subsequently confirmed
by BaBar~\cite{babar_y3940}, although there remain some
($\sim 2\sigma$) discrepancies between the Belle \& BaBar
measurements of the mass and width.
It is unlikely
that the $Y(3940)$ (seen in $\omegaJ/\psi$) and the
$X(3940)$ (seen in $D\bar{D^*}$) are different
decay modes of the same state. Belle has searched for
$Y(3940)\rightarrow D\bar{D^*}$ in $B\rightarrow K D\bar{D^*}$
decays and finds a 90\%~CL lower limit of
$\mathcal{B}(Y\rightarrow\omegaJ/\psi)/
\mathcal{B}(Y\rightarrow D\bar{D^*})>0.71$~\cite{belle_y3940_ddstr}
that contradicts a 90\%~CL upper limit from a search for
$X(3940)\rightarrow\omegaJ/\psi$ in $e^{+} e^{-}\rightarrow J/\psi\jp\omega$ annihilations:
$\mathcal{B}(X\rightarrow\omegaJ/\psi)/
\mathcal{B}(X\rightarrow D\bar{D^*})<0.58$~\cite{belle_x3940}.
Possible charmonium assignments for the $Y(3940)$ are the
$\eta_c^{\prime\prime}$ ($0^{-+}$) --- although its mass
is a little low --- and the $\chi_{c0}^{\prime}$ ($0^{++}$)
for which its mass is too high. The primary
difficulty with a charmonium assignment for the $Y(3940)$ is
its large partial width to $\omegaJ/\psi$, which reasonable
estimates put above 1~MeV~\cite{godfrey_olsen}
and which may in fact be quite a bit higher. This is
well above the measured partial widths for any of the
observed hadronic transitions between charmonium states.
\subsection{The $J^{PC}=1^{--}$ $Y$ states}
The $Y(4260)$ was first seen by BaBar as a peak in the
$\pipiJ/\psi$ mass spectrum in the initial-state-radiation
(ISR) process $e^{+} e^{-}\rightarrow\gamma_{ISR}\pipiJ/\psi$.
The $Y(4350)$ and $Y(4660)$ are seen in the $\pi^{+}\pi^{-}\psi '$ mass
spectrum in the $ee\rightarrow\pi^{+}\pi^{-}\psi '$ ISR process. (Belle also sees
a broad $Y(4008)$ peak in $\pipiJ/\psi$~\cite{belle_y4008}, but this
has not been confirmed by BaBar~\cite{babar_y4008}.) Since these
states are produced via ISR, their $J^{PC}$ has to be $1^{--}$.
There are no unassigned $1^{--}$ slots for any of these states in
the $M<4.4$~GeV spectrum of Fig.~\ref{fig:charmonium_spectrum}.
Moreover, no hint of any of them in seen in any of the
$D^{(*)}\bar{D^{(*)}}$ channels~\cite{galina}. This implies
that their $\pipiJ/\psi(\psi ')$ decay widths must be quite large.
In the case of the $Y(4260)$, the $\pipiJ/\psi$ has been
established to be more that 1.6~MeV~\cite{mo_xiaohu}. This is much too
large for charmonium, where allowed $\pipiJ/\psi$ transitions have
measured partial widths of 100~keV or less.
\subsection{The charged $Z$ particles}
Belle reported a peak with a $\sim 6.5\sigma$ statistical significance
near 4430~MeV in the $\pi^{\pm}\psi '$ channel
in exclusive $B\rightarrow K\pi^{\pm}\psi '$ decays~\cite{belle_z4430}.
A peak at the observed $\pi\psi '$ invariant mass value
cannot be produced by reflections from the $K\pi$ system.
However, BaBar did not confirm this peak, finding at most a
signal of $\sim 1.7\sigma$ significance~\cite{babar_z4430}.
A subsequent full Dalitz-plot analysis
(see Fig.~\ref{fig:z_projections} left)
of the Belle $B\rightarrow
K\pi^{\pm}\psi '$ sample confirms their
original mass and significance determinations~\cite{chistov_qwg}.
Belle also reported two peaks with greater than $5\sigma$
statistical significance in the $\pi^{\pm}\chi_{c1}$ channel,
the $Z_1(4050)$ \& $Z_2(4250)$, in exclusive
$B\rightarrow K\pi^{\pm}\chi_{c1}$ decays, again from a Dalitz-plot
analysis (Fig.~\ref{fig:z_projections} right).
If these peaks are interpreted as meson states, they must have a
minimal tetraquark $c\bar{c} u\bar{d}$
substructure and there are no possible charmonium
or charmonium hybrid assignments.
\begin{figure}[h]
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.40]{olsen_panic_fig2a.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.55]{olsen_panic_fig2b.eps}
\end{minipage}
\caption{\label{fig:z_projections} {\bf Left:} ($B\rightarrow K\pi^{\pm}\psi '$)
The points with errors are data and the histograms show fits
to a Dalitz-plot projection that has the $K^*$ bands removed with \&
without the inclusion of the $Z(4430)$ resonance in the
$\pi^{\pm}\psi '$ channel.
{\bf Right:} ($B\rightarrow K\pi^{\pm}\chi_{c1}$)
A similar Dalitz-plot projection for data \& fits with \& without
the inclusion of two resonances in the $\pi^{\pm}\chi_{c1}$ channel.
}
\end{figure}
\section{Exotic possibilities}
\label{exotic}
In this section I discuss the possible interpretations of the
$XYZ$ peaks as $c\bar{c}q\bar{q}$ tetraquark states or $c\bar{c}$-gluon
hybrid states.
\subsection{Tetraquarks}
Two very distinct types of tetraquark mesons have been proposed:
{\it molecular states}, which are relatively
loosely bound structures comprised of deuteron-like
mesons-antimeson bound states~\cite{molecules},
and {\it diquark-diantiquark mesons} in which the two quarks form
an anticolor triplet state that binds tightly to a
color triplet that is formed from the two antiquarks~\cite{maiani}. These
two types of structures have very different phenomenologies.
\subsubsection{Molecules}
A molecular state is expected to have a mass that is
slightly below the sum of the masses of its
meson-antimeson constituents and exhibit large
isospin violations. The $X(3872)$, with a mass
that is within errors of the $m_D + m_{D^*}$ mass
threshold and has decay rates to $\pipiJ/\psi$ and
$\pi^{+}\pi^{-}\pi^0J/\psi$ that are nearly equal~\cite{belle_x23pijpsi}.
Thus, this is a nearly ideal candidate for a $D\bar{D^*}$ molecular
state, either real~\cite{tornqvist2004,swanson,voloshin,braaten}
or virtual~\cite{hanhart}. On the other hand, its proximity
to the $D\bar{D^*}$ threshold has also led to speculation
that it is some kind of a threshold effect~\cite{bugg,chao,zhang}.
In theses latter schemes, mixing with the $\chi_{c1}^{\prime}$
charmonium state can play an important role.
A new piece of experimental information, the significance of
which has yet to be commented on by any theorist, is a study
of $X(3872)$ production in exclusive $B\rightarrow K\pi X(3872)$
decays~\cite{belle_kpix3872}.
The left panel of Fig.~\ref{fig:mkpi} shows the $K\pi$ invariant
mass distribution, where it is evident that non-resonant
$K\pi$ production dominates, and the $K^*(890)$ contribution
is small and of marginal significance. This is in contrast to what is
seen in all other
$B\rightarrow K\pi$+charmonium decays in which the $K^*(890)$ contribution
dominates; for example,
the right panel of Fig.~\ref{fig:mkpi} shows the
$M(K\pi)$ distribution for $B\rightarrow K^-\pi^+\chi_{c1}$
decays~\cite{belle_kpichic1}, where a prominent $K^*(890)$ signal
is clearly evident.
Not all of the $XYZ$ states fit the molecule picture.
For example the $X(3940)$, $Y(3940)$ and $Y(4660)$ are not
near any $D^{(*)}\bar{D^{(*)}}$ mass threshold. (Note that
$\pi$-exchange, the dominant binding term in molecular models,
is absent in $D_s^{(*)}\bar{D_s^{(*)}}$ systems.)
\begin{figure}[h]
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.3]{olsen_panic_fig3a.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.3]{olsen_panic_fig3b.eps}
\end{minipage}
\caption{\label{fig:mkpi} {\bf Left:}
The $M(K\pi)$ distribution for $B\rightarrow K^-\pi^+ X(3872)$ decays.
The lower curves show backgrounds from
non-resonant and $K^*(890)$ $K\pi$ systems. {\bf Right:}
The $M(K\pi)$ distribution for $B\rightarrow K^-\pi^+ \chi_{c1}$ decays.
}
\end{figure}
\subsubsection{Diquark-diantiquarks}
Essentially all of the observed $XYZ$ states can be accommodated
by the diquark-diantiquark model. However, in this picture, each
of the assigned state is expected to have an associated flavor-$SU(3)$
multiplet of states. One prediction of this model is that there
should be two $X(3872)$ states --- $X_u = cu\bar{c}\bar{u}$ and
$X_d = cd\bar{c}\bar{d}$ --- with a mass difference of $8\pm 3$~MeV.
No evidence for such a pairing has been
found~\cite{belle_y3940_ddstr,belle_kpix3872}.
In addition, in this model one expects
a charged isospin partner of the $X(3872)$ to be produced
in $B$ decays. BaBar searched for such an
$X^+(3872)\rightarrow\rho^+J/\psi$ state in neutral $B$
meson decays and and set an upper limit that is well below
isospin-based expectations~\cite{babar_xplus3872}. No isospin
partners of any of the other $XYZ$ states have been reported.
\subsection{Hybrids}
The lattice QCD expectation for the mass of the lowest-lying
charmonium hybrid is around 4.3~GeV and the relevant open-charm
threshold is 4.29~GeV, the $D^{**}\bar{D}$ mass threshold, where
$D^{**}$ denotes the lowest mass $P$-wave charmed meson with mass
2.42~GeV. Since the $Y(4260)$ mass is near the LQCD value and
below the $D^{**}\bar{D}$ mass threshold --- which would explain its
relatively strong decay rate to $\pipiJ/\psi$ as opposed to open
charm states --- a charmonium hybrid interpretation is attractive.
However, the $Y(4260)$ is broad and $D^{**}\bar{D}$ decays are
accessible from its high mass side, but there is no sign of a lineshape
distortion that might be expected when a dominant new decay
channel opens up. Moreover, the $Y(4350)$ \& $Y(4660)$ are both
well above all $D^{**}\bar{D}$ thresholds and no open charm decays
have been seen. So, while the hybrid interpretation might work for
the $Y(4260)$, it does not seem to apply to the other $1^{--}$
ISR-produced states.
\section{Evidence for $XYZ$-like states in the $s$- and $b$-quark
sectors}
An obvious question is whether or not there are counterpart states
in the $s$- and $b$-quark sectors. Recent results
suggest that there are. In a BaBar study of the ISR process
$e^{+} e^{-}\rightarrow\gamma_{ISR}\pi^{+}\pi^{-}\phi$, where the $\pi^{+}\pi^{-}$ comes from
$f_0(980)\rightarrow\pi^{+}\pi^{-}$, a distinct $\pi^{+}\pi^{-}\phi$ mass peak is seen
at 2175~MeV~\cite{babar_y2175}. This peak was confirmed in an ISR
measurement by Belle~\cite{belle_y2175} (left panel of Fig.~\ref{fig:yb})
and seen in $J/\psi\rightarrow\eta f_0\phi$
decays by BES~\cite{bes_y2175}. Although a conventional $s\bar{s}$
assignment cannot be ruled out~\cite{mlyan_y2175}, this state has
properties similar to what one would expect for an $s$-quark sector
counterpart of the $Y(4260)$.
The Belle group recently reported measurements of the energy dependence
of the $e^{+} e^{-}\rightarrow\pi^{+}\pi^{-}\Upsilon(nS)~(n=1,~2~\&~3)$ cross section around
$E_{cm}\sim 10.9$~GeV and found peaks in all three channels at
10.899~GeV (right panel of Fig.~\ref{fig:yb})~\cite{belle_y10899}.
The peak mass
and width valuse are quite distinct from those of
the nearby $\Upsilon(5S)$ bottomonium ($b\bar{b})$
state, and the cross section values are more than two-orders-of-magnitude
above expectations for a conventional $b\bar{b}$ system. One
interpretation for this peak is that it is a $b$-quark sector
equivalent of the $1^{--}$ $Y$ states seen in the $c$-quark
sector~\cite{weishu}.
\begin{figure}[h]
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.35]{olsen_panic_fig4a.epsi}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.35]{olsen_panic_fig4b.eps}
\end{minipage}
\caption{\label{fig:yb} {\bf Left:}
The $M(f_0(980)\phi))$ distribution for $e^{+} e^{-}\rightarrow \gamma_{ISR} f_0\phi$.
{\bf Right:} the energy dependence of $e^{+} e^{-}\rightarrow \pi^{+}\pi^{-}
\Upsilon(nS)~(n=1,~2~\&~3)$
near $\sqrt{s}=10.9$~GeV. In both cases the data are from Belle.}
\end{figure}
\section{Summary}
There is a growing body of evidence for a new type of hadron spectroscopy
involving pairs of $c$-quarks that neither fits well to classic Quark
Parton Model expectations nor QCD-motivated extensions. A recurring
feature of these new state are large partial widths for decays to
charmonium plus light hadrons. There is some evidence for similar
structures in the $s$- and $b$-quark sectors.
\section{Acknowledgments}
I thank Professor Tserruya and the PANIC organizing committee for their
gracious hospitality. I also thank Avraham~Gal
and my colleagues Haibo~Li, Roman~Mizuk,
Chengping~Shen and Bruce~Yabsley
for their assistance in the preparation of this manuscript.
| {'timestamp': '2009-01-16T01:10:58', 'yymm': '0901', 'arxiv_id': '0901.2371', 'language': 'en', 'url': 'https://arxiv.org/abs/0901.2371'} |
\section*{Abstract}
\else
\begin{center}
{\bf Abstract\vspace{-.5em}\vspace{0pt}}
\end{center}
\quotation
\fi}
\renewcommand{\endabstract}{\if@twocolumn\else\endquotation\fi}
\newcommand{\preprintnumber}[1]
{\begin{flushright}
\begin{tabular}{l} #1 \end{tabular}
\end{flushright}}
\makeatother
\newcommand{\gsim}%
{\mbox{\raisebox{-1.0ex}
{$\ \stackrel{\textstyle >}{\textstyle \sim}\ $}}}
\newcommand{\lsim}%
{\mbox{\raisebox{-1.0ex}
{$\ \stackrel{\textstyle <}{\textstyle \sim}\ $}}}
\newcommand{{\it i.e.\ }}{{\it i.e.\ }}
\newcommand{{\it etc.\ }}{{\it etc.\ }}
\newcommand{{\it et al.\ }}{{\it et al.\ }}
\newcommand{\vev}[1]{\left\langle #1 \right\rangle}
\newcommand{{\rm e}}{{\rm e}}
\newcommand{\ {\rm tr}\ }{\ {\rm tr}\ }
\newcommand{\ {\rm Str}\ }{\ {\rm Str}\ }
\newcommand{\repr}[1]{{\boldmath$#1$}}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\ {\rm MeV}\ }{\ {\rm MeV}\ }
\newcommand{\ {\rm GeV}\ }{\ {\rm GeV}\ }
\newcommand{\ {\rm TeV}\ }{\ {\rm TeV}\ }
\newcommand{\Journal}[4]{{#1} {\bf #2}, {#4} {(#3)}}
\newcommand{\sl Phys.~Lett.}{\sl Phys.~Lett.}
\newcommand{\sl Phys.~Rep.}{\sl Phys.~Rep.}
\newcommand{\sl Phys.~Rev.}{\sl Phys.~Rev.}
\newcommand{\sl Phys.~Rev.~Lett.}{\sl Phys.~Rev.~Lett.}
\newcommand{\sl Nucl.~Phys.}{\sl Nucl.~Phys.}
\newcommand{\sl Prog.~Theor.~Phys.}{\sl Prog.~Theor.~Phys.}
\newcommand{\sl Sov.~J.~Nucl.~Phys.}{\sl Sov.~J.~Nucl.~Phys.}
\newcommand{\sl Z.~Phys.}{\sl Z.~Phys.}
\newcommand{\it ibid.}{\it ibid.}
\begin{document}
\baselineskip 18pt
\begin{titlepage}
\preprintnumber{ICRR-Report-317-94-12 \\ April 1994}
\vspace*{\titlesep}
\begin{center}
{\LARGE\bf Flavor mixing in the gluino coupling \\
and the nucleon decay}\\
\vspace*{\titlesep}
{\large Toru {\sc Goto}, Takeshi {\sc Nihei}} and
{\large Jiro {\sc Arafune}}\\
\vspace*{\titlesep}
{\it Institute for Cosmic Ray Research, University of Tokyo,\\
Midori-cho, Tanashi-shi, Tokyo 188 JAPAN}\\
\end{center}
\vspace*{\titlesep}
\begin{abstract}
Flavor mixing in the quark-squark-gluino coupling is studied for the
minimal SU(5) SUGRA-GUT model and applied to evaluation of the nucleon
lifetime.
All off-diagonal (generation mixing) elements of Yukawa coupling
matrices and of squark/slepton mass matrices are included in solving
numerically one-loop renormalization group equations for MSSM
parameters, and the parameter region consistent with the radiative
electroweak symmetry breaking condition is searched.
It is shown that the flavor mixing in the gluino coupling for a large
$\tan\beta$ is of the same order of magnitude as the corresponding
Kobayashi-Maskawa matrix element in both up-type and down-type sector.
There exist parameter regions where the nucleon decay amplitudes for
charged lepton modes are dominated by the gluino dressing process,
while for all the examined regions the neutrino mode amplitudes are
dominated by the wino dressing over the gluino dressing.
\end{abstract}
\end{titlepage}
\section{Introduction}
\label{sec:introduction}
SU(5) supersymmetric (SUSY) grand unified theory (GUT) is an
attractive candidate for the unified theory of strong and electroweak
interactions.
The analyses of the gauge coupling unification \cite{LL-ABF-M} suggest
the validity of the minimal supersymmetric standard model (MSSM)
just above the electroweak scale $\sim 100 \ {\rm GeV}\ $ and the unification
of SU(3)$\times$SU(2)$\times$U(1) gauge group into a simple SU(5) at
the GUT scale $M_X \sim 10^{16} \ {\rm GeV}\ $.
One of the features of MSSM/SUSY-GUT is the existence of soft SUSY
breaking.
It gives quarks (or leptons) and their superpartners different mass
matrices in the generation (flavor) space.
In results, due to the discrepancy between the mass diagonalizing
bases of quarks and those of squarks, a generation mixing occurs in
the quark-squark-gaugino coupling (gluino coupling, in particular),
which may give considerable contributions to the nucleon decay, flavor
changing neutral currents (FCNC) and other various phenomena in
SUSY-GUT.
The flavor mixing in the gaugino coupling plays an important role in
the nucleon decay, since its amplitude is dominated by the dimension
five interaction followed by the gaugino ``dressing'' process
\cite{SY-W} in the minimal SU(5) SUSY-GUT model.
However, only a simplified treatment is made in the previous analyses
of the nucleon decay \cite{HMY,MATS,NCA,ENR}, where the diagonal
(in the generation space) gluino coupling is assumed leading to the
negligible contribution from the gluino dressing process.
The purpose of the present paper is to study the flavor mixing in the
gaugino coupling extensively and evaluate the contribution of the
gluino dressing process to the nucleon decay amplitude.
Based on the minimal SU(5) supergravity (SUGRA-) GUT \cite{N-LN},
we assume that the soft SUSY breaking parameters are ``universal'' at
the GUT scale \cite{BFS-CAN-HLW}.
We include all off-diagonal elements of Yukawa coupling matrices and
of squark/slepton mass matrices in solving numerically one-loop
renormalization group equations (RGEs)%
\footnote{Two-loop RGEs for soft SUSY breaking parameters are obtained
recently \protect\cite{Y-MH}, which will be important for more
accurate analysis.}
for all MSSM parameters \cite{IKKT-APW,BKS} with the universal
boundary conditions.
We then evaluate the effective potential for the Higgs fields at the
electroweak scale to find a consistent SU(2)$\times$U(1) breaking
minimum in accordance with the radiative electroweak symmetry breaking
scenario \cite{IKKT-APW}.
The obtained mass matrices of all particles are diagonalized to
evaluate the flavor mixing in the gaugino couplings.
The mass spectrum and the mixing are then used to calculate the
nucleon decay amplitudes for various decay modes.
It is found that the flavor mixing in the gluino coupling depends on
$\tan\beta$ (ratio of vacuum expectation values of the Higgs doublets)
and is roughly of the same order of magnitude as the corresponding
Kobayashi-Maskawa matrix element.
As for the nucleon decay, we find the gluino dressing process
dominates the amplitude for the decay modes containing a charged
lepton (and a meson) if $\tan\beta$ is large and if the gluino mass is
much smaller than the squark masses.
Note that the flavor mixing in the gaugino couplings is studied
previously in a systematic analysis of FCNC \cite{BKS} with a
semi-analytic solution of the RGEs for small $\tan\beta$ (top Yukawa
coupling $\gg$ bottom Yukawa coupling).
On the contrary, our numerical method cover the whole range of
$\tan\beta$, since all Yukawa couplings are taken into account.
The remaining part of this paper is organized as follows.
After introducing in the next section the minimal SU(5) SUGRA-GUT
model which we consider, we formulate the flavor mixing in the gaugino
couplings in Sec.~\ref{sec:mixing}.
In Sec.~\ref{sec:decay} the nucleon decay amplitudes are obtained with
a careful treatment of the flavor mixing.
The outline of the numerical calculation and the results are presented
in Sec.~\ref{sec:calculations} and our conclusions are summarized in
Sec.~\ref{sec:conclusion}.
\section{Minimal SU(5) SUGRA-GUT model}
\label{sec:model}
Minimal SU(5) SUSY-GUT model contains three generations of matter
multiplets with \repr{10} and \repr{\ol{5}} representations,
$\Psi_i^{AB}$ and $\Phi_{jA}$ respectively, where suffices
$A,B=1,2,\cdots,5$ are SU(5) indices and $i,j=1,2,3$ are the
generation labels, and three kinds of Higgs multiplets with \repr{5},
\repr{\ol{5}} and \repr{24} representations, $H_5^A$, $H_{\ol{5}A}$
and $\Sigma^A_B$ respectively.
SU(5) and R-parity invariant superpotential $W_{\rm GUT}$ at the GUT
scale is written as
\begin{eqnarray}
W_{\rm GUT}(M_X) &=& f^{ij} \Psi_i^{AB} \Phi_{jA} H_{\ol{5}B}
+ \frac{1}{8} g^{ij} \epsilon_{ABCDE}
\Psi_i^{AB} \Psi_j^{CD} H_5^E
\nonumber\\
& & + \lambda H_{\ol{5}A}
\left( \Sigma^A_B + M \delta^A_B \right) H_5^B
+ W_\Sigma \left( \Sigma \right) ~.
\label{GUTsuperpotential}
\end{eqnarray}
Here, $\epsilon_{ABCDE}$ is the totally antisymmetric constant,
$f^{ij}$, $g^{ij}=g^{ji}$ and $\lambda$ are dimensionless couplings,
$M$ is a mass parameter and $W_\Sigma$ is a self-interaction
superpotential for $\Sigma^A_B$.
SU(5) symmetry is spontaneously broken down to SU(3) $\times$ SU(2)
$\times$ U(1) with the nonvanishing vacuum expectation value of the
adjoint Higgs $\Sigma^A_B$.
Below the GUT scale, the model is reduced to MSSM with effective
higher dimensional operators, which are obtained by integrating out
the superheavy particles in Eq.~(\ref{GUTsuperpotential}).
The effective superpotential is then written as
\begin{eqnarray}
W_{\rm eff} &=& W_{\rm MSSM} + W_5 + O(M_X^{-2}) ~,
\nonumber\\
W_{\rm MSSM} &=& f_D^{ij} Q_i^{a\alpha} D_{ja} H_{1\alpha}
+ f_L^{ij} \epsilon^{\alpha\beta} E_i L_{j\alpha} H_{1\beta}
+ g_U^{ij} \epsilon_{\alpha\beta} Q_i^{a\alpha} U_{ja} H_2^\beta
+ \mu H_{1\alpha} H_2^\alpha ~,
\nonumber\\
W_5 &=& \frac{1}{M_C} \left\{
\frac{1}{2} C_L^{ijkl} \epsilon_{abc} \epsilon_{\alpha\beta}
Q_k^{a\alpha} Q_l^{b\beta} Q_i^{c\gamma} L_{j\gamma}
+ C_R^{ijkl} \epsilon^{abc} U_{ia} D_{jc} E_k U_{lb} \right.
\nonumber\\
& & ~~~~~~~+ ( \mbox{baryon number/lepton number
conserving terms} ) \biggr\} ~.
\label{superpotential}
\end{eqnarray}
Here, $Q$, $U$ and $E$ are chiral superfields which contain
left-handed quark doublet, right-handed up-type quark and right-handed
charged lepton respectively, and are embedded in $\Psi$ (to be
specified in Eq.~(\ref{Embedding}));
$D$ and $L$, which are embedded in $\Phi$, contain right-handed
down-type quark and left-handed lepton doublet respectively;
$H_1$ and $H_2$ are Higgs doublets embedded in $H_{\ol{5}}$ and $H_5$,
respectively.
The suffices $a,b,c=1,2,3$ are SU(3) indices and $\alpha,\beta=1,2$
are SU(2) indices.
$M_C$ is the colored Higgs mass which is assumed to be $O(M_X)$, while
the supersymmetric mass of Higgs doublet $\mu$ is of the order of the
$Z$ boson mass $m_Z$.
This discrepancy is owing to a tree level fine-tuning in the GUT
superpotential.
At the GUT scale, $f_D$ and $f_L$ are unified and $C_L$ and $C_R$ are
written in terms of the Yukawa coupling constants (see
Sec.~\ref{sec:decay}).
Baryon number (and lepton number) violating terms in $W_5$ give
dominant contributions to the nucleon decay in this model.
In addition to the supersymmetric Lagrangian to be derived from
(\ref{superpotential}), the following soft SUSY breaking terms are
included:
\begin{eqnarray}
-{\cal L}_{\rm soft} &=&
\left(m_Q^2\right)_i^j \tilde{q}^{\dagger i} \tilde{q}_j
+ \left(m_D^2\right)_i^j \tilde{d}^{\dagger i} \tilde{d}_j
+ \left(m_U^2\right)_i^j \tilde{u}^{\dagger i} \tilde{u}_j
\nonumber\\
&& + \left(m_L^2\right)_i^j \tilde{l}^{\dagger i} \tilde{l}_j
+ \left(m_E^2\right)_i^j \tilde{e}^{\dagger i} \tilde{e}_j
+ \Delta_1^2 h_1^\dagger h_1 + \Delta_2^2 h_2^\dagger h_2
\nonumber\\
&& + \left\{ A_D^{ij} \tilde{q}_i \tilde{d}_j h_1
+ A_L^{ij} \tilde{e}_i \tilde{l}_j h_1
+ A_U^{ij} \tilde{q}_i \tilde{u}_j h_2
- B\mu h_1 h_2 + {\rm h.~c.} \right\}
\nonumber\\
&& + \frac{1}{2} \left\{ M_1 \tilde{B} \tilde{B}
+ M_2 \tilde{W} \tilde{W}
+ M_3 \tilde{G} \tilde{G}
+ {\rm h.~c.} \right\} ~,
\label{softbreaking}
\end{eqnarray}
where $\tilde{q}_i$, $\tilde{d}_i$, $\tilde{u}_i$, $\tilde{e}_i$,
$\tilde{l}_i$, $h_1$ and $h_2$ are scalar components of $Q_i$, $D_i$,
$U_i$, $E_i$, $L_i$, $H_1$ and $H_2$, respectively, and $\tilde{B}$,
$\tilde{W}$ and $\tilde{G}$ are U(1), SU(2) and SU(3) gauge fermion
fields (bino, wino and gluino), respectively.
SU(2) and SU(3) suffices are omitted in (\ref{softbreaking}) for
simplicity.
We assume that the soft SUSY breaking parameters satisfy simple
relations at the GUT scale:
\begin{eqnarray}
\left( m_Q^2 \right)_i^j &=& \left( m_D^2 \right)_i^j ~=~
\left( m_U^2 \right)_i^j ~=~ \left( m_L^2 \right)_i^j ~=~
\left( m_E^2 \right)_i^j ~\equiv~ m_0^2\ \delta_i^j ~,
\nonumber\\
\Delta_1^2 &=& \Delta_2^2 ~=~ m_0^2 ~,
\nonumber\\
A_D^{ij} &=& f_{DX}^{ij} A_X m_0 ~, ~~
A_L^{ij} ~=~ f_{LX}^{ij} A_X m_0 ~, ~~
A_U^{ij} ~=~ g_{UX}^{ij} A_X m_0 ~,
\nonumber\\
M_1 &=& M_2 ~=~ M_3 ~\equiv~ M_{gX} ~,
\label{boundaryconditions}
\end{eqnarray}
where the suffix ``$X$'' stands for the value at the GUT scale.
The boundary conditions (\ref{boundaryconditions}) are due to the
minimal SUGRA model, where local SUSY is spontaneously broken in the
hidden sector which couples to the observable sector (SUSY-GUT, in the
present case) only gravitationally, and hence universal soft SUSY
breaking terms are induced in the observable sector \cite{BFS-CAN-HLW}.
Below the GUT scale, radiative corrections modify all parameters in
the superpotential (\ref{superpotential}) and the soft SUSY breaking
terms (\ref{softbreaking}), as well as three gauge coupling constants
$g_1$, $g_2$ and $g_3$ for U(1), SU(2) and SU(3), respectively.
The evolution of the parameters are described by the RGEs \cite{BKS}.
According to the radiative SU(2) $\times$ U(1) breaking scenario
\cite{IKKT-APW}, we numerically solve the RGEs down to the electroweak scale
$m_Z$ and evaluate the effective potential for the neutral Higgs
fields:
\begin{eqnarray}
V({\rm Higgs}) &=& V_0 + V_1 ~,
\nonumber\\
V_0 &=& \left( \mu^2 + \Delta_1^2 \right) |h_1|^2
+ \left( \mu^2 + \Delta_2^2 \right) |h_2|^2
- \left( B\mu h_1 h_2 + {\rm h.~c.} \right)
\nonumber\\
& & + \frac{g_1^2 + g_2^2}{8} \left( |h_1|^2 - |h_2|^2 \right)^2 ~,
\nonumber\\
V_1 &=& \frac{1}{64\pi^2}
\ {\rm Str}\ {\cal M}^4 \left( \log \frac{{\cal M}^2}{m_Z^2} -
\frac{3}{2} \right) ~,
\label{Higgspotential}
\end{eqnarray}
where $\ {\rm Str}\ $ means the supertrace and ${\cal M}$ includes all
(s)quark and (s)lepton masses.
Then the electroweak symmetry breaking condition
\begin{eqnarray}
\vev{h_1} &=& v \cos \beta ~, ~~
\vev{h_2} ~=~ v \sin \beta ~,
\label{vev}
\\
m_Z^2 &=& \frac{g_2^2}{2\cos^2\theta_W} v^2 ~,
\nonumber
\end{eqnarray}
is imposed.
\section{Flavor mixing in the gluino coupling}
\label{sec:mixing}
In order to discuss the flavor mixing in the gluino coupling, we have
to diagonalize the mass matrices for quarks and squarks.
Throughout the calculation hereafter, we choose the basis in the
generation space for the superfields so that the Yukawa coupling
constants for up-type quarks and leptons should be diagonalized {\em
at the electroweak scale}.
The Yukawa terms in (\ref{superpotential}) are then written as
\begin{equation}
W_{\rm Yukawa}(m_Z) =
\hat{f}_D^{kj}
\left( V_{\rm KM}^\dagger \right)_k^{~i} Q_i D_j H_1
+ \hat{f}_L^{ij} E_i L_j H_1
+ \hat{g}_U^{ij} Q_i U_j H_2 ~,
\label{YukawaZ}
\end{equation}
where the notation ``$\hat{~~}$'' stands for a diagonal matrix and
$V_{\rm KM}$ is the Kobayashi-Maskawa matrix.
All eigenvalues of $\hat{f}_D$, $\hat{f}_L$ and $\hat{g}_U$ are taken
to be real positive.
Since this choice of the basis is different from that in the GUT
superpotential (\ref{GUTsuperpotential}), a re-diagonalization of the
Yukawa couplings at the GUT scale is needed in order to find the
unification condition of $f_D$ and $f_L$ and the relation between the
Yukawa coupling constants and the dimension-five coupling constants
$C_{L,R}$.
$W_{\rm Yukawa}$ at the GUT scale is diagonalized with appropriate
unitary matrices $U_Q^{(f)}$, $U_D$, $U_E$, $U_L$, $U_Q^{(g)}$ and
$U_U$:
\begin{subequations}
\begin{eqnarray}
W_{\rm Yukawa}(M_X) &=& f_{DX}^{ij} Q_i D_j H_1
+ f_{LX}^{ij} E_i L_j H_1
+ g_{UX}^{ij} Q_i U_j H_2
\label{YukawaX}
\\
&=& \hat{f}_{DX}^{kl} \left( U_Q^{(f)} \right)_k^{~i}
\left( U_D \right)_l^{~j} Q_i D_j H_1
+ \hat{f}_{LX}^{kl} \left( U_E \right)_k^{~i}
\left( U_L \right)_l^{~j} E_i L_j H_1
\nonumber\\
& & + \hat{g}_{UX}^{kl} \left( U_Q^{(g)} \right)_k^{~i}
\left( U_U \right)_l^{~j} Q_i U_j H_2 ~.
\end{eqnarray}
\end{subequations}
The unification condition of $f_D$ and $f_L$ is then written as%
\footnote{The ``unification'' of $\hat{f}_{DX}$ and $\hat{f}_{LX}$,
with quark masses given in Ref.~\protect\cite{GL}, however, is not
so satisfactory numerically in the first and the second generations
as the gauge coupling unification.
We ignore the difference in the present calculation, since we may
still have ambiguities of the renormalization effect in the very low
energy region.}
\begin{equation}
\hat{f}_{DX}^{ij} = \hat{f}_{LX}^{ij} ~.
\label{YukawaUnification}
\end{equation}
The matter multiplets are accommodated into the SU(5) multiptets as
\begin{eqnarray}
\Psi_i &\Leftarrow& \left\{ Q_i
,~ \left( U_Q^{(g)\dagger} P^\dagger U_U \right)_i^{~j} U_j
,~ \left( U_Q^{(f)\dagger} U_E \right)_i^{~j} E_j \right\} ~,
\nonumber\\
\Phi_i &\Leftarrow& \left\{ D_i
,~ \left( U_D^\dagger U_L \right)_i^{~j} L_j \right\} ~,
\label{Embedding}
\end{eqnarray}
and the GUT Yukawa coupling constants in Eq.~(\ref{GUTsuperpotential})
are expressed with those in Eq.~(\ref{YukawaX}) as
\begin{eqnarray}
f^{ij} &=& f_{DX}^{ij} ~,
\nonumber\\
g^{ij} &=& g_{UX}^{ik}
\left( U_U^\dagger P U_Q^{(g)} \right)_k^{~j} ~,
\label{GUTYukawa}
\end{eqnarray}
where $P$ is a diagonal phase matrix which cannot be absorbed by field
redefinitions in the colored Higgs coupling \cite{EGN}.
The origin of the flavor mixing in the gluino coupling lies in the
difference between the mass basis for quarks and that for squarks.
The mass matrix for up-type squarks is expressed as
\begin{eqnarray}
-{\cal L}(\mbox{s-up mass}) &=&
( \tilde{q}_{u} ,~ \tilde{u}^{\dagger} )
{\cal M}_{\tilde{u}}^2
\left( \begin{array}{c}
\tilde{q}_{u}^{\dagger} \\ \tilde{u}
\end{array}\right) ~,
\nonumber\\
&=& ( \tilde{q}_{ui} ,~ \tilde{u}^{\dagger i} )
\left( \begin{array}{cc}
\left( m_{LL}^2 \right)^i_j &
\left( m_{LR}^2 \right)^{ij} \\
\left( m_{RL}^2 \right)_{ij} &
\left( m_{RR}^2 \right)_i^j
\end{array}\right)
\left( \begin{array}{c}
\tilde{q}_{u}^{\dagger j} \\ \tilde{u}_j
\end{array}\right) ~,
\nonumber\\
\left( m_{LL}^2 \right)^i_j &=& \left( M_U M_U^\dagger \right)^i_j
+ \left( m_Q^2 \right)^i_j
+ m_W^2 \cos 2\beta \left( \frac{1}{2}
- \frac{1}{6}\tan^2 \theta_W \right)
\delta^i_j ~,
\nonumber\\
\left( m_{RR}^2 \right)_i^j &=& \left( M_U^\dagger M_U \right)_i^j
+ \left( m_U^2 \right)_i^j
+ m_W^2 \cos 2\beta \left( \frac{2}{3}\tan^2 \theta_W \right)
\delta_i^j ~,
\nonumber\\
\left( m_{LR}^2 \right)^{ij} &=& \mu M_U^{ij} \cot\beta
+ A_U^{ij} v \sin\beta ~,
\nonumber \\
m_{RL} &=& m_{LR}^{\dagger} ~,
\label{squarkmass}
\end{eqnarray}
where $M_U$ is the up-type quark mass matrix $M_U^{ij} = g_U^{ij} v
\sin\beta$ and $\tilde{q}_u$ is the up-type component of the SU(2)
doublet $\tilde{q}$.
The squark mass matrix ${\cal M}_{\tilde{u}}^2$ is not diagonalized
with the quark mass basis (\ref{YukawaZ}) since off-diagonal elements
are induced in the soft SUSY breaking parameter matrices due to the
renormalization effect.
Squark mass basis is obtained by diagonalizing (\ref{squarkmass}) with
a 6$\times$6 unitary matrix $\tilde{U}_U$:
\begin{eqnarray}
\tilde{u}'_I &=& \left( \tilde{U}_U \right)_I^J \tilde{u}_J ~,
~~ I ~=~ 1,\ 2,\ \cdots,\ 6 ~,
\nonumber\\
\tilde{u}_I &=&
\left\{ \begin{array}{lcl}
\tilde{q}_{uI} & \mbox{for} & I = 1,\ 2,\ 3 \\
\tilde{u}_{I-3} & \mbox{for} & I = 4,\ 5,\ 6
\end{array} \right. ~,
\nonumber\\
\tilde{U}_U^\dagger {\cal M}_{\tilde{u}}^2 \tilde{U}_U &=&
\mbox{diagonal} ~,
\label{DiagonalizeSquark}
\end{eqnarray}
where $\tilde{u}'_I$ is the mass eigenstate of up-type squark.
We define the numbering of $\tilde{u}'_I$ such that the mixing of
$\tilde{u}_I$ is the largest in $\tilde{u}'_I$.
Accordingly we call $\tilde{u}'_1$, $\tilde{u}'_2$, $\cdots$,
$\tilde{u}'_6$ as $\tilde{u}_L$, $\tilde{c}_L$, $\tilde{t}_L$,
$\tilde{u}_R$, $\tilde{c}_R$ and $\tilde{t}_R$ respectively in the
later discussions.
The mass bases of down-type squarks and charged sleptons are obtained
in the same way with 6$\times$6 unitary matrices $\tilde{U}_D$ and
$\tilde{U}_E$, respectively.
Notice that no generation mixing occurs in the lepton/slepton sector
since the right-handed (s)neutrino does not exist in the minimal
model; the nonvanishing off-diagonal elements of the slepton mass
matrix are left-right mixing components only.
Consequently, quark-squark-gluino coupling is written as
\begin{eqnarray}
{\cal L}_{\rm int}(\mbox{gluino}) &=&
-i \sqrt{2} g_3 \left\{ \tilde{d}'^{\dagger I}
\left(
\left( \tilde{U}_D \right)_I^k
\left( V_{\rm KM} \right)_k^j
\tilde{G} d_{Lj} +
\left( \tilde{U}_D \right)_I^{j+3}
\ol{\tilde{G}} \ol{d}_{Rj}
\right)
\right.
\nonumber\\
& & ~~~~~~~~~ + \left. \tilde{u}'^{\dagger I}
\left(
\left( \tilde{U}_U \right)_I^j
\tilde{G} u_{Lj} +
\left( \tilde{U}_U \right)_I^{j+3}
\ol{\tilde{G}} \ol{u}_{Rj}
\right)
\right\} + {\rm h.~c.}
\nonumber\\
&=&
-i \sqrt{2} g_3 \left\{ \tilde{d}'^{\dagger I}
\left(
\left( \tilde{U}'_D \right)_I^j
\tilde{G} d_{Lj} +
\left( \tilde{U}_D \right)_I^{j+3}
\ol{\tilde{G}} \ol{d}_{Rj}
\right)
\right.
\nonumber\\
& & ~~~~~~~~~ + \left. \tilde{u}'^{\dagger I}
\left(
\left( \tilde{U}_U \right)_I^j
\tilde{G} u_{Lj} +
\left( \tilde{U}_U \right)_I^{j+3}
\ol{\tilde{G}} \ol{u}_{Rj}
\right)
\right\} + {\rm h.~c.} ~,
\label{GluinoCoupling}
\end{eqnarray}
with the definition of $\tilde{U}'_D$ as
\begin{equation}
\left( \tilde{U}'_D \right)_I^j \equiv
\left( \tilde{U}_D \right)_I^k \left( V_{\rm KM} \right)_k^j ~.
\label{primedU}
\end{equation}
$u_{Li}$ and $d_{Li}$ in (\ref{GluinoCoupling}) are left-handed quarks
of mass eigenstates which compose the SU(2) doublet as
\begin{equation}
Q_i \ni \left( \begin{array}{c}
u_{Li} \\
\left( V_{\rm KM} \right)_i^j d_{Lj}
\end{array}
\right) ~,
\label{Doublet}
\end{equation}
and $u_{Ri}$ and $d_{Ri}$ are the fermion components of $U_i$ and
$D_i$, respectively.
Similar flavor mixing formulae are obtained for other gaugino (wino
and bino) coupling terms.
\section{Nucleon decay with dimension five operators}
\label{sec:decay}
As mentioned in Sec.~\ref{sec:model}, the nucleon decay amplitude in
the minimal SU(5) SUSY-GUT model is dominated by the dimension five
operators \cite{SY-W} induced by colored higgsino/Higgs exchanges.
Since the dimension five operators are made from two fermion
(quark/lepton) and two boson (squark/slepton) component fields,
effective baryon number violating four-fermion operators are generated
by one loop ``dressing'' diagrams which involve gauginos or higgsinos
(see Fig.~\ref{fig:diagrams}).
In the present calculation, only the gluino dressing and the charged
wino dressing diagrams are included;
contributions from higgsino dressing diagrams are negligibly small due
to the small Yukawa couplings of light quarks ($u,d,s$), compared to
the SU(2) gauge coupling $g_2$;
neutral wino and bino coupling have the same flavor mixing structure
as that in the gluino coupling, hence their contributions are smaller
than that from the gluino dressing.
The dimension five coupling constants $C_L$ and $C_R$ of
(\ref{superpotential}) at the GUT scale are written in terms of the
Yukawa coupling constants (see (\ref{Embedding}) and
(\ref{GUTYukawa})):
\begin{eqnarray}
C_{LX}^{ijkl} &=& f_{DX}^{im} \left( U_D^\dagger U_L \right)_m^j
g_{UX}^{kn} \left( U_U^\dagger P
U_Q^{(g)} \right)_n^l ~,
\nonumber\\
C_{RX}^{ijkl} &=& f_{DX}^{mj} \left( U_Q^{(g)\dagger} P^\dagger
U_U \right)_m^i
g_{UX}^{nl} \left( U_Q^{(f)\dagger}
U_E \right)_n^k ~.
\label{GUTDim5}
\end{eqnarray}
Note that this relation with the index ``$X$'' removed does not hold
true at the electroweak scale.
The effective baryon number violating four-fermion operators at the
electroweak scale are written as
\begin{eqnarray}
{\cal L}_{\rm eff} (\Delta B = \pm 1) &=&
\left( \tilde{C}_{\nu}^{ijkl}(\tilde{G})
+ \tilde{C}_{\nu}^{ijkl}(\tilde{W}) \right)
\epsilon_{abc} (u_{Lk}^a d_{Ll}^b) (d_{Li}^c \nu_{Lj})
\nonumber\\
& & + \left( \tilde{C}_{e}^{ijkl}(\tilde{G})
+ \tilde{C}_{e}^{ijkl}(\tilde{W}) \right)
\epsilon_{abc} (u_{Lk}^a d_{Ll}^b) (u_{Li}^c e_{Lj})
\nonumber\\
& & + (\mbox{right-handed quark/lepton}) + \mbox{h.~c.} ~,
\label{4Fermi}
\end{eqnarray}
where $\tilde{C}_{\nu,e}$ are calculated as follows with use of the
numerical values of $C_L$ and $C_R$ at the electroweak scale to be obtained
through their RGEs, squark/slepton mass eigenvalues and the mixing
matrices in the gaugino couplings%
\footnote{Contributions from $C_R$'s and higgsino/neutralino dressings
are estimated to be small and neglected in the present calculations.
}:
\begin{eqnarray}
\tilde{C}_{\nu}^{ijkl}(\tilde{G}) &=& -\frac{4 g_3^2}{3M_C}
\left\{ \left( C_L^{ijmn} - C_L^{njmi} \right)
\left( \tilde{U}_U^\dagger \right)_m^I
\left( \tilde{U}_D^\dagger \right)_n^J
F_{\tilde{G}}( \tilde{u}'_I, \tilde{d}'_J )
\left( \tilde{U}_U \right)_I^k
\left( \tilde{U}'_D \right)_J^l
\right.
\nonumber\\
& & ~~~~~~~~
\left. - \left( C_L^{mjkn} - C_L^{njkm} \right)
\left( \tilde{U}_D^\dagger \right)_m^I
\left( \tilde{U}_D^\dagger \right)_n^J
F_{\tilde{G}}( \tilde{d}'_I, \tilde{d}'_J )
\left( \tilde{U}'_D \right)_I^i
\left( \tilde{U}'_D \right)_J^l
\right\} ~,
\nonumber\\
\tilde{C}_{\nu}^{ijkl}(\tilde{W}) &=& -\frac{g_2^2}{M_C}
\left\{ \left( C_L^{ijmn} - C_L^{njmi} \right)
\left( \tilde{U}_U^\dagger \right)_m^I
\left( \tilde{U}_D^\dagger \right)_n^J
F_{\tilde{W}}( \tilde{u}'_I, \tilde{d}'_J )
\left( \tilde{U}'_U \right)_I^k
\left( \tilde{U}_D \right)_J^l
\right.
\nonumber\\
& & ~~~~~~~~
\left. + \left( C_L^{mnkl} - C_L^{knml} \right)
\left( \tilde{U}_U^\dagger \right)_m^I
\left( \tilde{U}_E^\dagger \right)_n^J
F_{\tilde{W}}( \tilde{u}'_I, \tilde{e}'_J )
\left( \tilde{U}'_U \right)_I^i
\left( \tilde{U}_E \right)_J^j
\right\} ~,
\nonumber\\
\tilde{C}_{e}^{ijkl}(\tilde{G}) &=& -\frac{4 g_3^2}{3M_C}
\left\{ \left( C_L^{ijmn} - C_L^{mjin} \right)
\left( \tilde{U}_U^\dagger \right)_m^I
\left( \tilde{U}_D^\dagger \right)_n^J
F_{\tilde{G}}( \tilde{u}'_I, \tilde{d}'_J )
\left( \tilde{U}_U \right)_I^k
\left( \tilde{U}'_D \right)_J^l
\right.
\nonumber\\
& & ~~~~~~~~
\left. - \left( C_L^{mjnl} - C_L^{njml} \right)
\left( \tilde{U}_U^\dagger \right)_m^I
\left( \tilde{U}_U^\dagger \right)_n^J
F_{\tilde{G}}( \tilde{u}'_I, \tilde{u}'_J )
\left( \tilde{U}_U \right)_I^i
\left( \tilde{U}_U \right)_J^k
\right\} ~,
\nonumber\\
\tilde{C}_{e}^{ijkl}(\tilde{W}) &=& -\frac{g_2^2}{M_C}
\left\{ \left( C_L^{ijmn} - C_L^{mjin} \right)
\left( \tilde{U}_U^\dagger \right)_m^I
\left( \tilde{U}_D^\dagger \right)_n^J
F_{\tilde{W}}( \tilde{u}'_I, \tilde{d}'_J )
\left( \tilde{U}'_U \right)_I^l
\left( \tilde{U}_D \right)_J^k
\right.
\nonumber\\
& & ~~~~~~~~
\left. + \left( C_L^{mjkl} - C_L^{ljkm} \right)
\left( \tilde{U}_D^\dagger \right)_m^I
F_{\tilde{W}}( \tilde{d}'_I, \tilde{\nu}'_j )
\left( \tilde{U}_D \right)_I^i
\right\} ~.
\label{4FermiCoupling}
\end{eqnarray}
Here, $( \tilde{U}'_D )_I^i$ is defined in (\ref{primedU}) and
$( \tilde{U}'_U )_I^i = ( \tilde{U}_U )_I^j (V_{\rm KM})_j^i$.
$F_{\tilde{G}}$ and $F_{\tilde{W}}$ are obtained by the loop integral
\cite{HMY,MATS,NCA}:
\begin{eqnarray}
F_{\tilde{G}}( \tilde{f}_1, \tilde{f}_2 ) &=&
\tilde{F}( m_{\tilde{f}_1}, m_{\tilde{f}_2}; M_3 ) ~,
\nonumber\\
F_{\tilde{W}}( \tilde{f}_1, \tilde{f}_2 ) &=&
\left( U_{-} \right)_1^\alpha
\tilde{F}( m_{\tilde{f}_1}, m_{\tilde{f}_2}; M_{\pm}^\alpha )
\left( U_{+}^{\dagger} \right)_\alpha^1 ~,
\label{Floop}
\\
\tilde{F}( m_1, m_2; M ) &=&
\frac{1}{16\pi^2} \frac{M}{m_1^2 - m_2^2}
\left( \frac{m_1^2}{m_1^2 - M^2}\log\frac{m_1^2}{M^2}
- \frac{m_2^2}{m_2^2 - M^2}\log\frac{m_2^2}{M^2}
\right) ~,
\nonumber
\end{eqnarray}
where $U_{-}$, $U_{+}$ are 2$\times$2 unitary matrices which diagonalize
the chargino mass matrix and $M_{\pm}^\alpha$ ($\alpha = 1, 2$) are its
eigenvalues:
\begin{eqnarray}
M(\mbox{chargino}) &=& \left( \begin{array}{cc}
M_2 & \sqrt{2} m_W \sin\beta \\
-\sqrt{2} m_W \cos\beta & -\mu
\end{array} \right)
\nonumber\\
&=& U_{-} \left( \begin{array}{cc}
M_{\pm}^1 & 0 \\
0 & M_{\pm}^2
\end{array} \right) U_{+}^\dagger ~.
\label{CharginoMass}
\end{eqnarray}
The low energy QCD correction between $m_Z$ and $1 \ {\rm GeV}\ $ is taken into
account in order to evaluate the four fermion operators in the next
section.
The quark Lagrangian at $\sim 1 \ {\rm GeV}\ $ is then converted to the hadron
chiral Lagrangian with $\Delta B = \pm 1$ terms \cite{HMY,CD-CWH} with
use of the matrix element
\begin{equation}
\vev{ 0 | \epsilon_{abc} (d_L^a u_L^b) u_L^c | p } = \beta_p N_L ~,
\label{Hadron}
\end{equation}
where $N_L$ is a left-handed proton wave function; it enables us to
evaluate the partial lifetimes of the nucleon decay.
\section{Numerical calculations}
\label{sec:calculations}
According to the framework described in the previous sections, we
calculate the flavor (and left-right) mixing in gaugino couplings and
the nucleon partial lifetimes in a five-dimensional parameter space
$\{m_{\rm top},~ \tan\beta,~ m_0,~ M_{gX},~ \tilde{A}_X\}$, where a
dimensionful $A$ parameter is defined as $\tilde{A}_X \equiv A_X m_0$.
Actual calculations are made in the following procedure.
At first, $m_{\rm top}$ and $\tan\beta$ (at the electroweak scale) are fixed.
Using the numerical values of light quark masses and the
Kobayashi-Maskawa mixing angles given in literatures \cite{GL,PDG}
with the above fixed $m_{\rm top}$ and $\tan\beta$, we evaluate the
Yukawa coupling constants at the electroweak scale (\ref{YukawaZ}).
QCD corrections below the electroweak scale for quark masses other than
$m_{\rm top}$ are included at the one-loop level.
Next, the RGEs for the dimensionless parameters {\it i.e.\ }, the gauge coupling
constants and the Yukawa coupling constants are solved upward to the
GUT scale with the boundary conditions at the electroweak scale.
At the GUT scale, the Yukawa coupling constants are re-diagonalized to
obtain the boundary conditions for the dimension-five coupling
constants (\ref{GUTDim5}).
Then the RGEs for the soft SUSY breaking parameters and dimension-five
coupling constants are solved downward with the boundary conditions
(\ref{boundaryconditions}) and (\ref{GUTDim5}).
Since the RGEs are linear for the dimensionful parameters, all soft
SUSY breaking parameters at the electroweak scale are written as linear
combinations of the initial parameters $(m_0,~ M_{gX},~ \tilde{A}_X)$
\cite{MATS}:
\begin{eqnarray}
\tilde{m}^2_I (m_Z) &=& c_{1I} m_0^2 + c_{2I} M_{gX}^2 +
c_{3I} \tilde{A}_X^2 + c_{4I} M_{gX} \tilde{A}_X ~,
\nonumber\\
\tilde{M}_J (m_Z) &=& d_{1J} M_{gX} + d_{2J} \tilde{A}_X ~,
\label{LinearCombinations}
\end{eqnarray}
where $\tilde{m}^2_I$ and $\tilde{M}_J$ are collective notations for
the soft SUSY breaking parameters of mass dimension two
($m^2_{Q,D,U,L,E}$, $\Delta^2_{1,2}$) and one ($M_{1,2,3}$,
$A_{D,U,L}$), respectively.
The coefficients $c$'s and $d$'s are implicit functions of the gauge
couplings and the Yukawa couplings and are determined numerically by
solving the RGEs with four cases of boundary conditions $(m_0,~
M_{gX},~ \tilde{A}_X) = (1,~0,~0)$, $(0,~1,~0)$, $(0,~0,~1)$ and
$(0,~1,~1)$.
Once the coefficients are obtained, the values of soft SUSY breaking
parameters at the electroweak scale for given $(m_0,~ M_{gX},~ \tilde{A}_X)$
are evaluated with the formulae (\ref{LinearCombinations}), and it is
easy to scan the three-dimensional parameter space $\{m_0,~ M_{gX},~
\tilde{A}_X\}$ for fixed $m_{\rm top}$ and $\tan\beta$ with this
method.
The next step is to evaluate the remaining two parameters $\mu$ and
$B$ with the electroelectroweak SU(2) $\times$ U(1) symmetry breaking
condition.
The requirement that the minimum of the Higgs potential
(\ref{Higgspotential}) gives the vacuum expectation values (\ref{vev})
leads to
\begin{eqnarray}
\mu^2 &=& \frac{\Delta_2^2 - \Delta_1^2}{2\cos 2\beta}
- \frac{\Delta_1^2 + \Delta_2^2}{2} - \frac{1}{2}m_Z^2
\nonumber\\
&& - \frac{1}{v\cos 2\beta} \left.
\left( \frac{\partial V_1}{\partial h_1} \cos\beta
- \frac{\partial V_1}{\partial h_2} \sin\beta
\right)
\right|_{\mathop{}^{\scriptstyle h_1 = v\cos\beta}
_{\scriptstyle h_2 = v\sin\beta} } ~,
\label{minimum1}
\\
B\mu &=& \left( \mu^2 + \frac{\Delta_1^2 + \Delta_2^2}{2}
\right) \sin 2\beta
\nonumber\\
&& + \frac{1}{v} \left.
\left( \frac{\partial V_1}{\partial h_1} \sin\beta
+ \frac{\partial V_1}{\partial h_2} \cos\beta
\right)
\right|_{\mathop{}^{\scriptstyle h_1 = v\cos\beta}
_{\scriptstyle h_2 = v\sin\beta} } ~.
\label{minimum2}
\end{eqnarray}
Notice that the solution of the equation (\ref{minimum1}) for
$\mu$ cannot be written in a simple formula since the one-loop part of
the Higgs potential $V_1$ depends on $\mu$.
We solve (\ref{minimum1}) numerically for both signs of $\mu$ and then
calculate $B$ with (\ref{minimum2}).
Since all the MSSM parameters and the dimension-five coupling
constants at the electroweak scale for a given parameter set $(m_{\rm top},~
\tan\beta,~ m_0,~ M_{gX},~ \tilde{A}_X)$ are thus determined, the mass
spectrum of all superparticles and the mixing matrices in the gaugino
couplings are obtained by diagonalizing their mass matrices.
We investigate the parameter space $\{m_0,~ M_{gX},~ A_X\}$ within the
range $10 \ {\rm GeV}\ \leq m_0 \leq 10 \ {\rm TeV}\ $, $10 \ {\rm GeV}\ \leq M_{gX} \leq 10
\ {\rm TeV}\ $ and $-5 \leq A_X \leq +5$ for each combination of $m_{\rm top}$
= 120, 150 or 180 GeV and $\tan\beta = 2$, 10, 30 or 50.
For $m_{\rm top}$ = 180 GeV and $\tan\beta = 2$, the top Yukawa
coupling diverges below the GUT scale when solving the RGEs.
For $\tan\beta = 50$, no consistent radiative breaking solution is
found for any $m_{\rm top}$.
Figs.~\ref{fig:mixingu2} -- \ref{fig:mixings3} are histograms for the
specified off-diagonal elements of the gluino coupling matrices
$\tilde{U}_U$ and $\tilde{U}'_D$ for $m_{\rm top}$ = 150 GeV.
The magnitudes of the generation mixing in the right-right and
left-right sectors are small compared to the corresponding left-left
elements.
For a small $\tan\beta$, the left-left elements of $\tilde{U}'_D$ are
approximately equal to the Kobayashi-Maskawa matrix elements:
$(\tilde{U}'_D)_2^1 \approx V_{cd}$, $(\tilde{U}'_D)_3^1 \approx
V_{td}$, {\it etc.\ } in most of the parameter space, while the corresponding
off-diagonal elements of $\tilde{U}_U$ are small: the mass matrices of
up-type quarks, up-type squarks and down-type squarks are diagonalized
in the same basis.
This agrees with the conclusion of Ref.~\cite{BKS} where those mixing
matrices are semi-analytically obtained with an assumption that the
bottom Yukawa coupling is much smaller than the top Yukawa coupling,
which is applicable for small $\tan\beta$.
On the other hand, for a large $\tan\beta$, we find that nonvanishing
off-diagonal elements in the left-left sector of $\tilde{U}_U$ arises
with the same order of magnitudes as $V_{\rm KM}$, which contributes
significantly to the charged lepton decay modes (see below).
Off-diagonal elements of $\tilde{U}'_D$ are smaller than those for
small $\tan\beta$.
The qualitative behavior of the mixing matrices is rather independent
of the different values of $m_{\rm top}$.
Using the obtained values of superpartner masses and mixing matrices,
we evaluate $\tilde{C}$'s in (\ref{4Fermi}) with the formula
(\ref{4FermiCoupling}).
We then take into account of the low energy QCD correction to
$\tilde{C}$'s at the one-loop level%
\footnote{This corresponds to taking $A_{\rm L} \approx 0.46$, where
$A_{\rm L}$ is the low energy QCD factor used in literatures
\cite{HMY,MATS,NCA,ENR,AN}. It is argued the two-loop analysis
gives $A_{\rm L}\approx 0.28$ in Ref.~\cite{AN} }.
We take the chiral Lagrangian factors given in Ref.~\cite{HMY} to
derive amplitudes of various decay modes from (\ref{4Fermi}).
A large uncertainty of the nucleon lifetime comes from the numerical
values of $\beta_p$ and $M_C$.
$\beta_p$ is calculated with various methods \cite{GKSMPT-BEHS}, which
give
\begin{displaymath}
0.003 \mbox{GeV}^3 \leq \beta_p \leq 0.03 \mbox{GeV}^3 ~,
\end{displaymath}
and Ref.~\cite{HMY,HY} shows
\begin{displaymath}
2.2 \times 10^{13} \mbox{GeV} \leq M_C \leq
2.3 \times 10^{17} \mbox{GeV} ~.
\end{displaymath}
Here, we take a small value of $\beta_p$ = 0.003 GeV$^3$ and a large
value of $M_C = 10^{17}$ GeV so that we have a longer nucleon lifetime
for the safety of the later arguments.
Fig.~\ref{fig:lifetime} shows the partial lifetime for each nucleon
decay mode with fixed $m_{\rm top}$ = 150 GeV and $\tan\beta = 2$.
The range of each lifetime comes mainly from the ranges of the soft
SUSY breaking parameters.
As can be seen in the figure, $K \ol{\nu}$ decay modes are dominant
and most severely constrained by the experiments \cite{Kamioka,IMB} as
\begin{displaymath}
\begin{array}{lcrr}
\tau( p \rightarrow K^+ \ol{\nu} ) & \geq & 1.0 \times 10^{32} ~
{\rm yrs} & ~~~~~ {\rm (KAMIOKANDE)} ~, \\
\tau( p \rightarrow K^+ \ol{\nu} ) & \geq & 6.2 \times 10^{31} ~
{\rm yrs} & ~~~~~ {\rm (IMB)} ~, \\
\tau( n \rightarrow K^0 \ol{\nu} ) & \geq & 8.6 \times 10^{31} ~
{\rm yrs} & ~~~~~ {\rm (KAMIOKANDE)} ~, \\
\tau( n \rightarrow K^0 \ol{\nu} ) & \geq & 1.5 \times 10^{31} ~
{\rm yrs} & ~~~~~ {\rm (IMB)} ~.
\end{array}
\end{displaymath}
Our main concern is to study the contribution of the gluino dressing
diagrams.
To do that, we compare $\tau$(wino) with $\tau$(total), where
$\tau$(wino) is a partial lifetime calculated
with only the wino dressing diagrams taken into account and
$\tau$(total) is that calculated with both wino and gluino dressing
diagrams combined.
The results for $m_{\rm top}$ = 150 GeV are presented in
Figs.~\ref{fig:ratio02} -- \ref{fig:ratio30}.
The nucleon decay amplitude is dominated by the wino dressing diagrams
if $\tau (\mbox{wino}) / \tau(\mbox{total}) \approx 1$, while it is
dominated by the gluino dressing diagrams if $\tau (\mbox{wino}) /
\tau(\mbox{total}) \gg 1$.
There occur cancellations between the wino dressing contributions and
gluino dressing contributions in $\tau (\mbox{wino}) /
\tau(\mbox{total}) \ll 1$ region.
The gluino contributions are small for any modes in the small
$\tan\beta$ case.
For large $\tan\beta$, however, gluino dominant region is realized in
the charged lepton modes.
This is brought about by the following two reasons:
(1) in the simplified analyses based on an assumption of the diagonal
gluino coupling, the gluino dressing diagrams contribute only to the
$K \ol{\nu}$ modes due to the color antisymmetry in the dimension-five
coupling \cite{NCA,ENR}.
In the present case, the gluino coupling is not diagonal any more and
hence the gluino dressing processes contribute to the decay modes
other than $K \ol{\nu}$ modes.
The nonvanishing off-diagonal gluino coupling in the {\em up sector\ }
for large $\tan\beta$ significantly contributes to the charged lepton
modes.
(2) furthermore, if the squark masses are much larger than the gaugino
masses, the amplitude from the gluino dressing diagrams is enhanced by
a factor of $(\alpha_3(m_Z)/\alpha_2(m_Z))^2$ compared to that from the
wino dressing diagrams, since $\tilde{F}$ in Eq.~(\ref{Floop}) is
asymptotically
\begin{equation}
\tilde{F}(m,m;M) \sim \frac{1}{16\pi^2} \frac{M}{m^2}
~~~~~ {\rm for} ~ m \gg M ~,
\label{asymptoticF}
\end{equation}
and $M_3(m_Z)/M_2(m_Z) = \alpha_3(m_Z)/\alpha_2(m_Z)$ because of the
GUT relation of gaugino masses.
Since (\ref{asymptoticF}) gives an overall suppression of the nucleon
decay amplitude for $m \gg M$ (see Fig.~\ref{fig:md1-M2}), the
dominant gluino contribution in Figs.~\ref{fig:ratio10} and
\ref{fig:ratio30} is realized in the long lifetime region (see
Fig.~\ref{fig:lifetimevsratio}).
Translating the scanned parameters $(m_0,~ M_{gX},~ A_X)$ into the
MSSM parameters $(m_{\tilde{d}_L},~ M_2,~ \mu)$ where we take
$m_{\tilde{d}_L}$, the down-type squark mass of the first generation
as the typical squark mass, we plot the calculated points in the MSSM
parameter space for $m_{\rm top}$ = 150 GeV and $\tan\beta = 2$ in
Figs.~\ref{fig:mu-M2} and \ref{fig:md1-M2}.
The region A in Fig.~\ref{fig:mu-M2} is excluded by LEP constraints on
charginos and neutralinos \cite{LEP}:
\begin{eqnarray}
m_{\chi^\pm} &>& 45 \ {\rm GeV}\ ~,
\nonumber\\
\Gamma ( Z \rightarrow \chi \chi ) &<& 22 \ {\rm MeV}\ ~,
\nonumber\\
B ( Z \rightarrow \chi \chi' ) &<& 5 \times 10^{-5} ~,
\nonumber\\
B ( Z \rightarrow \chi' \chi' ) &<& 5 \times 10^{-5} ~,
\nonumber
\label{lep}
\end{eqnarray}
where $\chi^\pm$ is a chargino, $\chi$ is the lightest neutralino and
$\chi'$ is a heavier neutralino.
No solution with radiative SU(2)$\times$U(1) breaking is found in
region B, which is a forbidden region.
Points plotted with small dots are excluded due to the present lower
bound for the proton lifetime $\tau( p \rightarrow K^+ \ol{\nu} ) >
10^{32}$ yrs, giving the constraint of $|\mu| \gsim$ 300 GeV.
This constraint for $\mu$ is roughly unchanged for different $m_{\rm
top}$ and/or $\tan\beta$.
Fig.~\ref{fig:md1-M2} shows the squark mass bound $m_{\tilde{d}_L}
\gsim 400 \ {\rm GeV}\ $.
If the lower bound for the proton lifetime is raised to $\tau( p
\rightarrow K^+ \ol{\nu} ) > 10^{33}$ yrs with the near future
experiment at Super-KAMIOKANDE, most of the parameter region with
$m_{\tilde{d}_L} \lsim$ 1 TeV will be excluded.
Lower bounds for other first and second generation squark masses are
found similar to that for $m_{\tilde{d}_L}$, while the bound for the
third generation squarks is lower in general due to the
renormalization effect and the left-right mixing.
Since the nucleon lifetime is approximately proportional to
$(\tan\beta)^{-2}$ \cite{HMY}, the lower bound for the squark mass is
raised for larger $\tan\beta$.
In fact, $\tau( p \rightarrow K^+ \ol{\nu} ) > 10^{32}$ yrs implies
$m_{\tilde{d}_L} \gsim 1 \ {\rm TeV}\ $ for $\tan\beta = 30$.
\section{Conclusion}
\label{sec:conclusion}
In this paper we have made a systematic analysis of the flavor mixing
in the gaugino couplings within the framework of the minimal SUGRA-GUT.
We have solved the one-loop RGEs for all MSSM parameters including
off-diagonal Higgs coupling matrices with five input parameters,
namely $(m_{\rm top},~ \tan\beta)$ at the electroweak scale $m_Z$ and
$(m_0,~ M_{gX},~ A_X)$ at the GUT scale $M_X$, and we have numerically
obtained full mass spectra and mixing matrices, which satisfy the
radiative electroweak symmetry breaking condition.
For a small $\tan\beta$ ($\tan\beta = 2$), we have obtained a result
consistent with the semi-analytic study \cite{BKS}, in which the top
Yukawa coupling is assumed to be much larger than other Yukawa
couplings: the left-left sector of the generation mixing matrix in the
down-type quark-squark-gluino coupling is approximately equal to the
Kobayashi-Maskawa matrix, while the off-diagonal mixing matrix
elements in the up-type gluino coupling are small.
On the other hand, for large $\tan\beta = 10$ and 30 where the bottom
(and tau, for extremely large $\tan\beta$) Yukawa coupling is not
negligibly small compared with the top Yukawa coupling, we have found
that nonvanishing generation mixing in the up-type gluino coupling
occurs with the magnitudes comparable to the corresponding
Kobayashi-Maskawa matrix elements.
The generation mixing in the down-type gluino coupling is also changed
considerably.
We have applied the generation mixing to the calculation of nucleon
decay widths to study the contributions from the gluino dressing
diagrams compared with the wino dressing diagrams.
In result, it is found that the gluino dressing diagrams give the
dominant contribution to the decay mode containing a charged lepton if
$\tan\beta \gg 1$ and $M_3 \ll m_{\tilde{q}}$ (typical squark mass).
For the charged lepton modes with small $\tan\beta$, or the (anti-)
neutrino emission modes with any $\tan\beta$, the gluino contributions
are relatively small.
In those cases, the contributions from the gluino dressing are at most
of the same order of magnitude as the wino dressing contributions.
We have scanned the MSSM parameter space to find allowed regions with
the present constraints given by the nucleon decay experiments and the
accelerator experiments%
\footnote{In addition, cosmological constraint will be given by the
analyses of the relic abundance of the lightest superprticle
\protect\cite{MY-DN-KM}.}.
The latter excludes the parameter region of small superpartner masses,
and the former gives a strict bound to the masses of first and second
generation squarks.
We argue that the whole parameter region with $m_{\tilde{d}_L} \lsim 1
\ {\rm TeV}\ $ in the minimal SU(5) SUGRA-GUT model can be tested by
Super-KAMIOKANDE.
Our method of calculations and the numerical result itself are
adaptable to the analyses of FCNC in the minimal SUGRA model, which
will be discussed elsewhere.
\subsection*{Acknowledgment}
One of the authors (T.~G.) would like to thank J.~Hisano for helpful
discussions.
\newpage
| {'timestamp': '1999-10-01T21:51:09', 'yymm': '9404', 'arxiv_id': 'hep-ph/9404349', 'language': 'en', 'url': 'https://arxiv.org/abs/hep-ph/9404349'} |
\section{Introduction}
In \cite{lodha2016nonamenable}, Lodha and Moore introduced the group $G_0$ consisting of piecewise projective homeomorphisms of the real projective line.
This group is a finitely presented torsion free counterexample to the von Neumann-Day problem \cite{Neumann1929, day1950means}, which asks whether every nonamenable group contains nonabelian free subgroups.
Although counterexamples of the von Neumann-Day problem are known \cite{MR586204, adian1979burnside, MR682486, ol2003non, ivanov2005embedding, monod2013groups}, it is still an open question whether the Thompson group $F$ can be a new counterexample.
The Lodha--Moore group $G_0$ has similar properties to the Thompson group $F$.
Indeed, the Lodha--Moore group can also be defined as a finitely generated group consisting of homeomorphisms of the space of infinite binary sequences, whose generating set is obtained by adding an element to the well-known finite generating set of $F$.
Both have (small) finite presentations \cite{brown1987finiteness, lodha2016nonamenable}, normal forms with infinite presentations \cite{brown1984infinite, lodha2020nonamenable}, simple commutator subgroups \cite{cannon1996introductory, burillo2018commutators}, trivial homotopy groups at infinity \cite{brown1984infinite, zaremsky2016hnn}, no nonabelian free subgroups \cite{brin1985groups, lodha2016nonamenable}, and are of type $F_\infty$ \cite{brown1984infinite, lodha2020nonamenable}.
On the other hand, there exist various generalizations of the Thompson group $F$.
One of the most natural ones is the $n$-adic Thompson group $F(n)$, which is obtained by replacing infinite binary sequences with infinite $n$-ary sequences.
Even now, it is still being actively studied whether what is true for the group $F$ is also true for the generalized group and what interesting properties can be obtained under the generalization.
In this paper, we generalize the Lodha--Moore group similarly and study its properties.
Namely, we define the $n$-adic group $G_0(n)$ of the Lodha--Moore group $G_0$ and show that several properties which hold for $G_0$ also hold for $G_0(n)$.
We remark that $G_0(2)$ is isomorphic to $G_0$.
Let $n, m \geq 2$.
We show the following:
\begin{theorem}
\begin{enumerate}[font=\normalfont]
\item The group $G_0(n)$ admits an infinite presentation with a normal form of elements.
\item The group $G_0(n)$ is finitely presented.
\item The group $G_0(n)$ is nonamenable.
\item The group $G_0(n)$ has no free subgroups.
\item The group $G_0(n)$ is torsion free.
\item The groups $G_0(n)$ and $G_0(m)$ are isomorphic if and only if $n=m$ holds.
\item The commutator subgroup of the group $G_0(n)$ is simple.
\item The center of the group $G_0(n)$ is trivial.
\item There does not exist any nontrivial direct product decomposition of the group $G_0(n)$.
\item There does not exist any nontrivial free product decomposition of the group $G_0(n)$.
\end{enumerate}
\end{theorem}
This paper is organized as follows.
In Section \ref{section_F(n)}, we generalize Dehornoy's infinite presentation of $F$ to $F(n)$, which will be used to construct that of $G_0(n)$.
To the best of the author's knowledge, this is a new presentation of $F(n)$.
In Section \ref{section_G0(n)}, we first recall the definition of the group $G_0$ and define the group $G_0(n)$.
Then by using the infinite presentation of $F(n)$ constructed in Section \ref{section_F(n)}, we generalize Lodha's method to obtain that of $G_0(n)$ and its normal form.
Finally, in Section \ref{section_G0(n)_properties}, we study several properties of $G_0(n)$.
Let us mention some open problems which are known to hold in the case of $G_0$.
First, it is an interesting question whether $G_0(n)$ can be realized as a subgroup of the group of piecewise projective homeomorphisms of the real projective line.
The second problem is whether this group is of type $F_{\infty}$, and all homotopy groups are trivial at infinity.
If it has these two properties, then ${G_0(n)}$ is an example of an (infinite) family of groups satisfying all Geoghegan's conjectures for the Thompson group $F$.
Furthermore, we can consider some groups related to $G_0(n)$.
The first one is constructed by using another definition of the map $y$ defined in Section \ref{subsection_def_G_0(n)}.
Although we define the map so that $G_0$ is naturally a subgroup of $G_0(n)$, we can consider several different generalizations.
We can also construct groups that contain $G_0(n)$.
In \cite{lodha2016nonamenable}, the group $G$ is defined, where $G$ contains $G_0$ as a subgroup.
For our group, by adding some of the generators $y_0$, $y_{(n-1)1}$, $\dots$, $y_{(n-1)(n-2)}$, $y_{(n-1)}$, we can define not only the group $G(n)$, which corresponds to $G$ but also the groups ``between'' $G_0(n)$ and $G(n)$.
\section{The generalized Thompson group $F(n)$}\label{section_F(n)}
\subsection{Definition} \label{subsection_F(n)_definition}
Let $n \geq 2$.
There exist several ways to define the generalized Thompson group $F(n)$.
In this paper, we define it as a group of homeomorphisms on the $n$-adic Cantor set.
We use tree diagrams to represent elements of the group visually.
We define $\bm{N}$ to be the set $\{0, 1, \dots, n-1 \}$.
We endow $\bm{N}$ with the discrete topology and endow $\N^\mathbb{N}$ with the product topology.
Note that $\N^\mathbb{N}$ and the Cantor set are homeomorphic.
We also consider the set of all finite sequences on $\bm{N}$ and write $\N^{<\mathbb{N}}$ for it.
For $s \in \N^{<\mathbb{N}}$ and $t \in \N^{<\mathbb{N}}$ (or $\N^\mathbb{N}$), the concatenation is denoted by $st$.
The group $F(n)$ is a finitely generated group that is generated by the following $n$ homeomorphisms on $\N^\mathbb{N}$:
\begin{align*}
x_0(\zeta)&=
\begin{cases}
0\eta & ( \zeta=00\eta )\\
1\eta & ( \zeta=01\eta ) \\
&\vdots \\
(n-2)\eta & (\zeta=0(n-2)\eta) \\
(n-1)0\eta & (\zeta=0(n-1)\eta) \\
(n-1)1\eta & (\zeta=1\eta) \\
&\vdots \\
(n-1)(n-1)\eta & (\zeta=(n-1)\eta),
\end{cases} \\
x_1(\zeta)&=
\begin{cases}
0\eta & (\zeta=0\eta) \\
1\eta & (\zeta=10\eta) \\
2\eta & (\zeta=11\eta) \\
&\vdots \\
(n-2)\eta & (\zeta=1(n-3)\eta) \\
(n-1)0\eta & (\zeta=1(n-2)\eta) \\
(n-1)1\eta & (\zeta=1(n-1)\eta) \\
(n-1)2\eta & (\zeta=2\eta) \\
&\vdots \\
(n-1)(n-1)\eta & (\zeta=(n-1)\eta),
\end{cases} \\
&\vdots \\
x_{n-2}(\zeta)&=
\begin{cases}
0\eta & (\zeta=0\eta) \\
&\vdots \\
(n-3)\eta & (\zeta=(n-3)\eta) \\
(n-2)\eta & (\zeta=(n-2)0\eta) \\
(n-1)0\eta & (\zeta=(n-2)1\eta) \\
&\vdots \\
(n-1)(n-2)\eta & (\zeta=(n-2)(n-1)\eta) \\
(n-1)(n-1)\eta & (\zeta=(n-1)\eta),
\end{cases}
\shortintertext{and}
\x{0}{(n-1)}(\zeta)&=
\begin{cases}
(n-1)x_0(\eta) & (\zeta=(n-1)\eta) \\
\zeta & (\zeta \neq (n-1)\eta).
\end{cases}
\end{align*}
These maps are represented by tree diagrams as in Figure \ref{generator_Fn}.
\begin{figure}[tbp]
\centering
\includegraphics[width=150mm]{generator_of_Fn.pdf}
\caption{Tree diagrams of the homeomorphisms. }
\label{generator_Fn}
\end{figure}
Here, we briefly review the definition of tree diagrams.
See \cite{burillo2001metrics} for details.
An \textit{$n$-ary tree} is a finite tree with a top vertex (\textit{root}) with $n$ edges, and all vertices except the root have degree only $1$ (\textit{leaves}) or $n+1$.
We define a \textit{caret} to be an $n$-ary tree with no vertices whose degree is $n+1$ (see Figure \ref{n-caret}).
\begin{figure}[tbp]
\centering
\includegraphics[width=30mm]{n-caret.pdf}
\caption{A caret. }
\label{n-caret}
\end{figure}
Then, each $n$-ary tree is obtained by attaching carets to a leaf of a caret.
We always assume that the root is the top and the others are descendants.
Each $n$-ary tree can be regarded as a finite subset of $\N^{<\mathbb{N}}$.
To do this, we label each edge of each caret by $0, 1, \dots, n-1$ from the left.
Since every leaf corresponds to a unique path from the root to the leaf, we can regard it as an element in $\N^{<\mathbb{N}}$.
Let $T_+$ and $T_-$ be $n$-ary trees with $m$ leaves.
Let $a_1, \dots, a_m$ be elements in $\N^{<\mathbb{N}}$ with lexicographic order corresponding to the leaves of $T_+$.
For $T_-$, define $b_1, \dots, b_m$ in the same way.
Then, for every $\zeta \in \N^\mathbb{N}$, there exists $i$ uniquely such that $\zeta=a_i\eta$ for some $\eta \in \N^\mathbb{N}$.
Thus we obtain a homeomorphism $a_i \eta \mapsto b_i \eta$.
It is known that every homeomorphism obtained from two $n$-ary trees with the same number of leaves in this way is generated by the composition of $x_0, x_1, \dots, x_{n-2}, \x{0}{(n-1)}$.
See \cite[Corollary 10.9]{meier2008groups} for the case $n=2$.
We define $\epsilon$ to be the empty word.
Let $i$ in $\{0, \dots, n-2 \}$ and $\alpha$ in $\N^{<\mathbb{N}} \cup \{\epsilon \}$.
We define the map $\x{i}{\alpha}: \N^\mathbb{N} \to \N^\mathbb{N}$ by
\begin{align*}
\x{i}{\alpha}(\zeta)&=
\begin{cases}
\alpha x_i(\eta) & (\zeta=\alpha \eta) \\
\zeta & (\zeta \neq \alpha \eta),
\end{cases}
\end{align*}
and we define
\begin{align*}
X(n)&:=
\left\{ \x{i}{\alpha} \mid i = 0, \dots, n-2, \alpha \in \N^{<\mathbb{N}} \cup \{ \epsilon \} \right\}.
\end{align*}
This set contains the well-known infinite generating set of $F(n)$, which we describe below, and is the key generating set in Section \ref{subsubsection_New_presentation_F(n)}.
Let $X^\prime(n):=\{ \x{0}{s}, \dots, \x{n-2}{s} \mid s=\epsilon, (n-1), (n-1)(n-1), \dots \}$.
We denote each element as $x_0=X_0, \dots, x_{n-2}=X_{n-2}, \x{0}{(n-1)}=X_{n-1}, \x{1}{(n-1)}=X_{n}, \dots$ only in this section for the sake of simplicity.
For $i<j$, we have $X_i^{-1}X_jX_i=X_{j+n-1}$.
This implies that every element in $F(n)$ has the following form:
\begin{align*}
\X{i_1}{r_1}\X{i_2}{r_2}\cdots \X{i_m}{r_m}\X{j_k}{-s_k}\cdots\X{j_2}{-s_2}\X{j_1}{-s_1}
\end{align*}
where $i_1<i_2<\cdots<i_m\neq j_k>\cdots>j_2>j_1$ and $r_1, \dots, r_m, s_1, \dots, s_k>0$.
We require that this form satisfies the following additional condition:
if there exist $X_i$ and $X_i^{-1}$, then there also exists one of
\begin{align*}
X_{i+1}, \X{i+1}{-1}, X_{i+2}, \X{i+2}{-1}, \dots, X_{i+n-1}, \X{i+n-1}{-1}.
\end{align*}
It is known that this form with the additional condition always uniquely exists.
Thus we call this form \textit{normal form} of elements of $F(n)$.
The proof for the case $n=2$ is in \cite[Section 1]{brown1984infinite}.
\subsection{A presentation of the generalized Thompson group}
While the group $F(n)$ has the well-known infinite and finite presentations,
we construct another presentation with respect to $X(n)$ for Section \ref{infinite_presentation}.
We list the relations of the elements in $F(n)$ as follows.
Here, $A_i$s are defined at the beginning of Section \ref{subsubsection_New_presentation_F(n)}, and the shift-map is defined in Definition \ref{Def_shiftmap}.
\begin{table}[H]
{\small
\begin{tabular}{ccccc}
$A_0A_1=\A{1}{(n-1)}A_0$, & $A_1A_2=\A{2}{(n-1)}A_1$, & $\dots$, & $A_{n-2}\A{0}{(n-1)}=\A{0}{(n-1)^2}A_{n-2}$, & $\dots$ \\
$A_0A_2=\A{2}{(n-1)}A_0$, & $A_1A_3=\A{3}{(n-1)}A_1$, & $\dots$, & $A_{n-3}\A{0}{(n-1)}=\A{0}{(n-1)^2}A_{n-3}$, & $\dots$ \\
$A_0A_3=\A{3}{(n-1)}A_0$, & $\dots$ \\
$\vdots$ & $\ddots$ \\
$A_0A_{n-2}=\A{n-2}{(n-1)}A_0$, \\
$A_0\A{0}{(n-1)}=\A{0}{(n-1)^2}A_0$, \\
$A_0\A{1}{(n-1)}=\A{1}{(n-1)^2}A_0$, \\
$\vdots$
\end{tabular}
}
\centering
\caption{Moving to a right column corresponds to the sift-map. }
\label{relations_1}
\end{table}
\begin{table}[H]
{\small
\begin{tabular}{cccc}
$A_0A_0=\A{0}{(n-1)}A_0\A{0}{0}$, & $A_1A_0=\A{0}{(n-1)}A_0\A{1}{0}$, & $\dots$, & $A_{n-2}A_0=\A{0}{(n-1)}A_0\A{n-2}{0}$, \\
$A_1A_1=\A{1}{(n-1)}A_1\A{0}{1}$, \\
$\vdots$ & $\ddots$ \\
\end{tabular}
}
\centering
\caption{Moving to a lower row corresponds to the sift-map. }
\label{relations_2}
\end{table}
\begin{table}[H]
{\small
\begin{tabular}{ccc}
$\A{k}{0\alpha}A_1=A_1\A{k}{0\alpha}$, & $\A{k}{1\alpha}A_2=A_2\A{k}{1\alpha}$, & $\dots$ \\
$\A{k}{0\alpha}A_2=A_2\A{k}{0\alpha}$, & $\A{k}{1\alpha}A_3=A_3\A{k}{1\alpha}$, & $\dots$ \\
$\A{k}{0\alpha}A_3=A_3\A{k}{0\alpha}$, \\
$\vdots$ & $\ddots$
\end{tabular}
}
\centering
\caption{For each $\alpha$ in $\N^{<\mathbb{N}} \cup \{\epsilon \}$ and $k$ in $\{0, \dots, n-2\}$, moving to a right column corresponds to the sift-map. }
\label{relations_3}
\end{table}
\begin{table}[H]
{\small
\begin{tabular}{ccc}
$\A{k}{0\alpha}A_0=A_0\A{k}{00\alpha}$, & $\A{k}{1\alpha}A_1=A_1\A{k}{10\alpha}$, & $\dots$ \\
$\A{k}{1\alpha}A_0=A_0\A{k}{01\alpha}$, & $\A{k}{2\alpha}A_1=A_1\A{k}{11\alpha}$, & $\dots$ \\
$\A{k}{2\alpha}A_0=A_0\A{k}{02\alpha}$, \\
$\vdots$ & $\ddots$ \\
$\A{k}{(n-2)\alpha}A_0=A_0\A{k}{0(n-2)\alpha}$, \\
$\A{k}{(n-1)0\alpha}A_0=A_0\A{k}{0(n-1)\alpha}$,
\end{tabular}
}
\centering
\caption{For each $\alpha$ in $\N^{<\mathbb{N}} \cup \{\epsilon \}$ and $k$ in $\{0, \dots, n-2\}$, moving to a right column corresponds to the sift-map. }
\label{relations_4}
\end{table}
\subsubsection{Dehornoy's results}
For the ``geometric'' presentation of $F(n)$, we generalize Dehornoy's method in \cite{dehornoy2005geometric}.
Section 1 of \cite{dehornoy2005geometric} gives the way to find a presentation of a group.
So we recall the general setting and the way.
\begin{definition}\cite[Section 1.1]{dehornoy2005geometric}
Let $G$ be a group (monoid) and $T$ be a set.
We define a \textit{partial} (\textit{right}) \textit{action} to be a map $\phi$ from $G$ to the set of injections $\{f: T^\prime \to T \mid T^\prime \subset T \}$ such that the following are satisfied (in the following, we write $t \cdot g$ for the image of $t$ under $\phi(g)$ if it is defined):
\begin{itemize}
\item[$({PA}_1)$] For every $t \in T$, $t \cdot e=t$ holds;
\item[$({PA}_2)$] For every $g, h \in G$, and $t \in T$, if $t \cdot g$ is defined, then $(t\cdot g)\cdot h$ is defined if and only if $t\cdot gh$ is defined, and if one of them is defined, we have $(t\cdot g)\cdot h=t \cdot gh$;
\item[$({PA}_3)$] For every finite family $g_1, \dots, g_m$, there exists $t \in T$ such that $t \cdot g_1, \dots, t \cdot g_m$ are defined.
\end{itemize}
\end{definition}
Dehornoy also introduced a stronger condition on partial actions.
\begin{definition}\cite[Section 1.3]{dehornoy2005geometric}
We assume that $G$ has a partial action on $T$.
Then we call a subset $S \subset T$ is \textit{discriminating} if:
\begin{enumerate}
\item In (${PA}_3$), we can take $t$ in the set $S \cdot G=\{s \cdot g \mid s \in S, \mbox{$g \in G$ such that $s \cdot g$ is defined} \}$;
\item Each $G$-orbit contains at most one element of $S$;
\item For every $s \in S$, its stabilizer is trivial.
\end{enumerate}
\end{definition}
\begin{remark}
In the above setting, for every $t \in S \cdot G$, there exist the unique $s \in S$ and $g \in G$ such that $t=s \cdot g$.
\end{remark}
When $R$ is a family of relations of a group, we write $w\equiv_R z$ if one can rewrite $w$ to $z$ by using elements in $R$.
The following theorem holds.
\begin{theorem}[{\cite[Proposition 1.4]{dehornoy2005geometric}}]
\label{dehornoy_presentation}
Let $G$ be a group with a partial action on a set $T$.
Let $X$ be a subset of $G$ and $R$ be a collection of relations satisfied in $G$ by the elements of $X$.
Assume that $S$ is a discriminating subset of $T$ and that, for each $s$ in $S$ and $t$ in the $G$-orbit of $s$, a word $w_t$ on $X$ is chosen so that $t=s \cdot w_t$ holds.
Then $\langle X \mid R \rangle$ is a presentation of $G$ if and only if for all $t, t^\prime$ in $S \cdot G$ and $x$ in $X$,
\begin{align} \label{condition}
\mbox{$t^\prime=t \cdot x$ implies $w_{t^\prime} \equiv_R w_t \times x$},
\end{align}
where $\times$ denotes the concatenation of words in $X$.
\end{theorem}
\subsubsection{A presentation of $F(n)$}\label{subsubsection_New_presentation_F(n)}
In this section, we construct a partial action of $F(n)$ and give a presentation by using Theorem \ref{dehornoy_presentation}.
Except for the construction of $w_t$, the discussions are almost the same as the case $n=2$.
Let $T(n)$ be a set consisting of $n$-ary trees and the root (the graph with a single vertex).
We first define partial actions of $n-1$ elements $A_0, \dots, A_{n-2}$ on $T(n)$ as illustrated in Figure \ref{partial_action}.
\begin{figure}[tbp]
\centering
\includegraphics[width=160mm]{partial_action.pdf}
\caption{Partial actions of $A_1, \dots, A_{n-2}$. }
\label{partial_action}
\end{figure}
We note that the actions of $A_i$ ($i=0, \dots, n-2$) on a single caret are not defined.
Let $\alpha$ in $\N^{<\mathbb{N}}$.
Then, for $n$-ary trees which contain $\alpha$ as a subpath, we define a partial action of $\A{i}{\alpha}$ to be the partial mapping obtained by applying $A_i$ to the element of $T(n)$ positioned just below $\alpha$ (if it is defined).
See Figure \ref{monoid_def} and observe that the bottom caret is the only one moved by the action.
\begin{figure}[tbp]
\centering
\includegraphics[width=60mm]{monoid_def_color.pdf}
\caption{Example of the action of $\A{1}{2}$ for $n=4$. }
\label{monoid_def}
\end{figure}
\begin{definition}
We define $\mathscr{G}_n(\mathscr{A})$ to be the monoid generated by $X(n):=\{\A{i}{\alpha} \mid i \in \{0, \dots, n-2\}, \alpha \in \N^{<\mathbb{N}} \cup \{ \epsilon \} \}$ and their inverse, where $\epsilon$ is the empty word and $\A{i}{\epsilon}:=A_i$.
\end{definition}
From the construction, the monoid $\mathscr{G}_n(\mathscr{A})$ naturally has a partial action on $T(n)$.
In order to make $\mathscr{G}_n(\mathscr{A})$ into a group, we introduce a congruence on $\mathscr{G}_n(\mathscr{A})$.
\begin{definition}
We assume that a group $G$ has a partial action on a set $T$.
If there exists $t \in T$ such that $t \cdot g$ and $t \cdot g^\prime$ are defined, and they coincide for all such $t$, then we define $g$ and $g^\prime$ to be \textit{near-equal} and write it as $g \approx g^\prime$.
\end{definition}
As in \cite[Corollary 2.4]{dehornoy2005geometric}, we can show that $\mathscr{G}_n(\mathscr{A})$ and the set of all-right trees $S(n)$ satisfy the assumptions in \cite[Lemma 2.2]{dehornoy2005geometric} (see Figure \ref{T_m} as an example of an all-right tree).
Thus, near-equality is a congruence on $\mathscr{G}_n(\mathscr{A})$, and the quotient monoid $\mathscr{G}_n(\mathscr{A}) / {\approx}$ is a group.
We write $G_n(\mathscr{A})$ for this group.
Moreover, $S(n)$ is discriminating for the induced partial action on $G_n(\mathscr{A})$.
We omit the proofs because they are the same for $n=2$, but we recall the induced action for the reader's convenience.
\begin{definition}
For $t, t^\prime$ in $T(n)$ and $x$ in $G_n(\mathscr{A})$, we define $t \cdot x=t^\prime$ if $t \cdot g =t^\prime$ holds for some $g$ in $\mathscr{G}_n(\mathscr{A})$ such that $g$ is a representative of $x$ (if there exists such $g$).
\end{definition}
By the definitions, it is easy to see that $G_n(\mathscr{A})$ and $F(n)$ are isomorphic.
We simply write $g$ for the class of $g$ in $G_n(\mathscr{A})$ for the sake of simplicity.
In order to give a presentation of $G_n(\mathscr{A})$, we first define the ``shift-map'' on $G_n(\mathscr{A})$.
\begin{definition}\label{Def_shiftmap}
Define the map $[1]$ by the following:
\begin{align*}
A_0 \overset{[1]}{\mapsto} &A_1 \overset{[1]}{\mapsto} \cdots \overset{[1]}{\mapsto} A_{n-2} \overset{[1]}{\mapsto} \A{0}{(n-1)} \overset{[1]}{\mapsto} \A{1}{(n-1)} \overset{[1]}{\mapsto} \cdots \\
&\overset{[1]}{\mapsto} \A{n-2}{(n-1)} \overset{[1]}{\mapsto} \A{0}{(n-1)(n-1)} \overset{[1]}{\mapsto} \cdots,
\end{align*}
and for each $k \in \{0, \dots, n-2\}$ and $\alpha \in \N^{<\mathbb{N}} \cup \{ \epsilon \}$,
\begin{align*}
\A{k}{0\alpha} \overset{[1]}{\mapsto} &\A{k}{1\alpha} \overset{[1]}{\mapsto} \cdots \overset{[1]}{\mapsto} \A{k}{(n-2)\alpha} \overset{[1]}{\mapsto} \A{k}{(n-1)0\alpha} \overset{[1]}{\mapsto} \A{k}{(n-1)1\alpha} \overset{[1]}{\mapsto} \cdots \\
&\overset{[1]}{\mapsto} \A{k}{(n-1)(n-2)\alpha} \overset{[1]}{\mapsto} \A{k}{(n-1)(n-1)0\alpha} \overset{[1]}{\mapsto} \cdots \overset{[1]}{\mapsto} \A{k}{(n-1)(n-1)1\alpha} \overset{[1]}{\mapsto} \cdots.
\end{align*}
For the empty word $\epsilon$, define $[1](\epsilon)=\epsilon$.
Furthermore, for an element $\A{k_1}{\alpha_1} \cdots \A{k_m}{\alpha_m}$ in $G_n(\mathscr{A})$, we define $[1](\A{k_1}{\alpha_1} \dots \A{k_m}{\alpha_m})=[1](\A{k_1}{\alpha_1}) \cdots [1](\A{k_m}{\alpha_m})$.
A map $[i]$ denotes the composition of $[1]$ $i$ times.
\end{definition}
\begin{remark}\label{shift_n-1}
By the definition, for each $\A{k}{\alpha}$ (even if $\alpha$ is empty), we have
\begin{align*}
[n-1](\A{k}{\alpha})=\A{k}{(n-1)\alpha}.
\end{align*}
This fact is useful to check the relations.
\end{remark}
Next, for an $n$-ary tree $t$, we define inductively two words $w_t$ and $w_t^\ast$ as follows:
\begin{definition}
\begin{enumerate}
\item If $t$ is a single vertex, we define
\begin{align*}
w_t=w_t^\ast=\epsilon.
\end{align*}
\item If $t$ is an $n$-ary tree, as in Figure \ref{inductive_tree}, we define
\begin{align*}
w_t&=w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times [2](w_{t_2}^\ast) \times \cdots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}), \\
w_t^\ast&=w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times [2](w_{t_2}^\ast) \times \cdots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}^\ast) \times A_0,
\end{align*}
where $\times$ denotes the concatenation.
\end{enumerate}
\end{definition}
\begin{figure}[tbp]
\centering
\includegraphics[width=30mm]{inductive_tree.pdf}
\caption{An $n$-ary tree $t$ with subtrees (or leaves) $t_0, \dots, t_{n-1}$}
\label{inductive_tree}
\end{figure}
Now, we show that $t=s \cdot w_t$ holds.
We note that $s$ is an $n$-ary tree with the same number of carets of $t$.
\begin{lemma} \label{lemma1_presentation}
Let $t$ in $T(n)$ with $m$ carets.
We define $T_m$ to be an all-right tree with $m$ carets if $m \geq 1$ and a single vertex if $m=0$.
Then, for every $t^\prime$ in $T(n)$, we have $T_m \cdot w_t=t$, and the equality in Figure $\ref{presentation_assumption}$ holds.
\begin{figure}[tbp]
\centering
\includegraphics[width=80mm]{presentation_assumption.pdf}
\caption{The second claim in Lemma \ref{lemma1_presentation}. }
\label{presentation_assumption}
\end{figure}
\begin{proof}
We show by induction on $m$.
For the base case, since $t, T_m$ are single vertices and, we have $w_t=w_t^\prime=\epsilon$, the equality is clear.
Let $t$ be in Figure \ref{inductive_tree} and $n_i$ be the number of carets of $t_i$ for $i=0, \dots, n-1$.
By the definitions,
\begin{align*}
w_t=w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times [2](w_{t_2}^\ast) \times \cdots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}})
\end{align*}
holds and $T_m$ is in Figure \ref{T_m}.
\begin{figure}[tbp]
\centering
\includegraphics[width=70mm]{n1nn-1carets.pdf}
\caption{The $n$-ary tree $T_m$. }
\label{T_m}
\end{figure}
By the inductive hypothesis for $w_{t_0}^\ast$, it acts on $T_m$ as in Figure \ref{wt0_action}.
\begin{figure}[tbp]
\centering
\includegraphics[width=125mm]{wt0_action.pdf}
\caption{The action of $w_{t_0}^\ast$. }
\label{wt0_action}
\end{figure}
Since $w_t, w_t^\ast$ are the words on $\{A_i, \A{i}{(n-1)}, \A{i}{(n-1)(n-1)}, \dots \mid i=0, \dots, n-2 \}$, the sift-map $[1]$ shits indices just one.
Thus, by applying the inductive hypothesis repeatedly, $w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times [2](w_{t_2}^\ast) \times \cdots \times [n-2](w_{t_{n-2}}^\ast)$ acts on $T_m$ as in Figure \ref{wto_wtn-2_action}, and we write $\tilde{t}$ for the resulting tree.
\begin{figure}[tbp]
\centering
\includegraphics[width=160mm]{wto_wtn-2_action.pdf}
\caption{The action of $w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times [2](w_{t_2}^\ast) \times \cdots \times [n-2](w_{t_{n-2}}^\ast)$. }
\label{wto_wtn-2_action}
\end{figure}
Recall Remark \ref{shift_n-1}, the action of $[n-1](w_{t_{n-1}})$ on $\tilde{t}$ is obtained from that of $w_{t_{n-1}}$ on the subtree $T_{n_{n-1}}$ in $\tilde{t}$.
By the inductive hypothesis for $w_{t_{n-1}}$, we have $T_{n_{n-1}} \cdot w_{t_{n-1}}=t_{n-1}$.
This completes the proof of the first claim that $w_t$ satisfies the equality.
By the similar argument as in the case of the action of $w_t$, that of $w_t^\ast$ is calculated as shown in Figure \ref{wtast}.
\begin{figure}[tbp]
\centering
\includegraphics[width=160mm]{wtast.pdf}
\caption{The action of $w_t^\ast$. }
\label{wtast}
\end{figure}
\end{proof}
\end{lemma}
It remains to prove that the condition \eqref{condition} in Theorem \ref{dehornoy_presentation} holds for a collection of relations.
Let $R(n)$ be the set of the elements in Tables \ref{relations_1}, \ref{relations_2}, \ref{relations_3}, and \ref{relations_4}.
It is easy to see that all elements in $R(n)$ are relations of $F(n)$.
\begin{lemma}
Let $t^\prime=t \cdot \A{k}{\alpha}$.
Then we have
\begin{align*}
w_{t^\prime} &\equiv_{R(n)} w_t \cdot \A{k}{\alpha}, \\
{w_{t^\prime}}^\ast &\equiv_{R(n)} {w_t}^\ast \cdot \A{k}{0\alpha}.
\end{align*}
\begin{proof}
When we rewrite a word $w$ to $z$ by applying a relation in Table $i$ ($1 \leq i \leq 4$), we denote by $w \equiv_i z$.
We show by induction on the length of $\alpha$ as an $n$-ary sequence.
For the base case, first, we assume $k=0$.
Then we can illustrate trees $t, t^\prime$ as in Figure \ref{t_tprime}.
\begin{figure}[tbp]
\centering
\includegraphics[width=100mm]{t_tprime.pdf}
\caption{The $n$-ary trees $t$ (left) and $t^\prime$ (right) for $\alpha=\epsilon$ and $k=0$. }
\label{t_tprime}
\end{figure}
Let $\tilde{t}$ be the subtree of $t^\prime$ consisting of $t_0, \dots, t_{n-1}$ (as in Figure \ref{inductive_tree}).
Then we have
\begin{align*}
w_{t^\prime}
&=w_{\tilde{t}}^\ast \times [1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}}) \\
&=w_{t_0}^\ast \times [1](w_{t_1}^\ast)\times \dots \times [n-1]({w_{t_{n-1}}}^\ast) \times A_0 \\
&\; \times [1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}}) \\
&\equiv_1 w_{t_0}^\ast \times [1](w_{t_1}^\ast)\times \dots \times [n-1]({w_{t_{n-1}}}^\ast) \\
&\; \times[n-1]\left([1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}})\right) \times A_0 \\
&= w_{t_0}^\ast \times [1](w_{t_1}^\ast)\times \cdots \\
&\; \times [n-1]\bigl(({w_{t_{n-1}}}^\ast) \times[1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}})\bigr) \times A_0 \\
&= w_t \cdot A_0,
\end{align*}
and
\begin{align*}
w_{t^\prime}^\ast
&=w_{t_0}^\ast \times [1](w_{t_1}^\ast)\times \dots \times [n-1]({w_{t_{n-1}}}^\ast) \times A_0 \\
&\; \times [1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}}^\ast) \times A_0 \\
&\equiv_1 w_{t_0}^\ast \times [1](w_{t_1}^\ast)\times \cdots \\
&\; \times [n-1]\bigl(({w_{t_{n-1}}}^\ast) \times[1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}}^\ast)\bigr) \times A_0A_0 \\
&\equiv_2 w_{t_0}^\ast \times [1](w_{t_1}^\ast)\times \cdots \\
&\; \times [n-1]\bigl(({w_{t_{n-1}}}^\ast) \times[1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}}^\ast)\bigr) \\
&\; \times \A{0}{(n-1)}A_0\A{0}{0} \\
&=w_{t_0}^\ast \times [1](w_{t_1}^\ast)\times \cdots \\
&\; \times [n-1]\bigl(({w_{t_{n-1}}}^\ast) \times[1](w_{t_n}^\ast) \times \dots \times [n-2](w_{t_{2n-3}}^\ast) \times [n-1](w_{t_{2n-2}}^\ast)\times A_0 \bigr) \\
&\; \times A_0\A{0}{0} \\
&=w_t^\ast \times \A{0}{0}.
\end{align*}
The case $k\geq1$ can be proved in the same way.
Let $\alpha=0\beta$.
That is, we consider the case when the condition $t_0^\prime=t_0 \cdot \A{k}{\beta}$ holds for $t$ and $t^\prime$, as shown in Figure \ref{inductive_t_tprime}.
\begin{figure}[tbp]
\centering
\includegraphics[width=80mm]{inductive_t_tprime.pdf}
\caption{The $n$-ary trees $t$ (left) and $t^\prime$ (right) for $\alpha=0\beta$. }
\label{inductive_t_tprime}
\end{figure}
When we rewrite a word by applying the inductive hypothesis, we use $\equiv_I$ to denote its equality.
We note that
\begin{align*}
[1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}})
\shortintertext{and}
[1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}^\ast)
\end{align*}
are words on the set
\begin{align*}
\{ A_1, \dots, A_{n-2}, \A{i}{\alpha} \mid i= 0, \dots, n-2, \alpha=(n-1), (n-1)^2, \dots \}.
\end{align*}
Then we have
\begin{align*}
w_{t^\prime}&=w_{t_0^\prime}^\ast \times [1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}) \\
&\equiv_I w_{t_0}^\ast \times \A{k}{0\beta} \times [1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}) \\
&\equiv_3 w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}) \times \A{k}{0\beta} \\
&=w_t \times \A{k}{\alpha},
\shortintertext{and}
w_{t^\prime}^\ast&=w_{t_0^\prime}^\ast \times [1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}^\ast) \times A_0 \\
&\equiv_I w_{t_0}^\ast \times \A{k}{0\beta} \times [1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}^\ast) \times A_0 \\
&\equiv_3 w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}^\ast) \times \A{k}{0\beta}A_0 \\
&\equiv_4 w_{t_0}^\ast \times [1](w_{t_1}^\ast) \times \dots \times [n-2](w_{t_{n-2}}^\ast) \times [n-1](w_{t_{n-1}}^\ast) \times A_0 \A{k}{00\beta} \\
&=w_{t}^\ast \times \A{k}{0\alpha}.
\end{align*}
Since the collection of relations is closed under the shift-map, we can apply the inductive hypothesis when considering any other $i \beta$ ($1 \leq i \leq n-1$).
Although the only case $w_t$ for $i=n-1$ is rewritten slightly differently (since $w_{t_{n-1}}$ appears in the word $w_t$ instead of $w_{t_{n-1}}^\ast$), all can be shown similarly.
This completes the proof.
\end{proof}
\end{lemma}
\section{The Lodha--Moore group and its generalization}\label{section_G0(n)}
\subsection{The Lodha--Moore group}
In this section, we briefly review the original Lodha--Moore group $G_0$.
In some papers, this group is denoted by $G$.
Let $x_0$, $\x{0}{1}$ be the maps defined in Section \ref{subsection_F(n)_definition} for $n=2$.
The group $G_0$ is generated by these two maps and one more generator called $y_{10}$.
To define this generator, we first define the homeomorphism called $y$.
\begin{definition}\label{definiton_y_2}
The map $y$ and its inverse $y^{-1}$ is defined recursively based on the following rule:
\begin{align*}
&y: 2^\mathbb{N} \to 2^\mathbb{N} & &y^{-1}: 2^\mathbb{N} \to 2^\mathbb{N} \\
&y(00\zeta)=0y(\zeta) & &y^{-1}(0\zeta)=00y^{-1}(\zeta) \\
&y(01\zeta)=10y^{-1}(\zeta) & &y^{-1}(10\zeta)=01y(\zeta) \\
&y(1\zeta)=11y(\zeta), & &y^{-1}(11\zeta)=1y^{-1}(\zeta).
\end{align*}
\end{definition}
For each $s$ in $2^{<\mathbb{N}}$, we also define the map $y_s$ by setting
\begin{align*}
y_s(\xi)&=
\left \{
\begin{array}{cc}
s y(\eta), & \xi=s\eta \\
\xi, & \mbox{otherwise}.
\end{array}
\right.
\end{align*}
We give an example of a calculation of the $y_{001}$ on $00101101 \cdots$:
\begin{align*}
y_{001}(00101101\cdots)&=001y(01101\cdots)\\
&=00110y^{-1}(101\cdots)\\
&=0011001y(1\cdots).
\end{align*}
\begin{definition}
The group $G_0$ is a group generated by $x_0$, $\x{0}{1}$, and $y_{10}$.
\end{definition}
The group $G_0$ is also realized as a group of piecewise projective homeomorphisms.
\begin{proposition}[{\cite[Proposition 3.1]{lodha2016nonamenable}}]\label{proposition_piecewiseprojective}
The group $G_0$ is isomorphic to the group generated by the following three maps of $\mathbb{R}$:
\begin{align*}
a(t)&=t+1,
&
b(t) &= \left \{
\begin{array}{cc}
t & \mbox{if $t \leq 0$} \\
\frac{t}{1-t} & \mbox{if $0 \leq t \leq \frac{1}{2}$} \\
3-\frac{1}{t} & \mbox{if $\frac{1}{2} \leq t \leq 1$} \\
t+1 & \mbox{if $1 \leq t$},
\end{array}
\right.
&
c(t) &= \left \{
\begin{array}{cc}
\frac{2t}{1+t} & \mbox{if $0 \leq t \leq 1$} \\
t & \mbox{otherwise}.
\end{array}
\right.
\end{align*}
\end{proposition}
This proposition is shown by identifying $2^\mathbb{N}$ with $\mathbb{R}$ by the following maps:
\begin{align*}
&\varphi: 2^\mathbb{N} \to [0, \infty] &
&\Phi: 2^\mathbb{N} \to \mathbb{R} \cup \{ \infty \} \\
&\varphi(0\xi)=\frac{1}{1+\frac{1}{\varphi(\xi)}} &
&\Phi(0\xi)=-\varphi(\tilde{\xi}) \\
&\varphi(1\xi)=1+\varphi(\xi) &
&\Phi(1\xi)=\varphi(\xi),
\end{align*}
where $\mathbb{R} \cup \{ \infty \}$ denotes the real projective line.
We note that for every $x \in \mathbb{R} \cup \{ \infty \}$, the inverse image $\Phi^{-1}(\{x\})$ is either a one-point set or a two-point set.
\subsection{$n$-adic Lodha--Moore group} \label{subsection_def_G_0(n)}
In order to define a generalization of the Lodha--Moore group, we first define a homeomorphism on $\N^\mathbb{N}={\{0, \dots, n-1 \}}^\mathbb{N}$ corresponding to $y$ for the case of $n=2$.
We fix $n \geq 2$ and also denote this map by $y$ as in the case of $n=2$.
\begin{definition}
The map $y$ and its inverse map $y^{-1}$ is defined recursively based on the following rule:
\begin{align*}
y: \N^\mathbb{N} &\to \N^\mathbb{N} & y^{-1}: \N^\mathbb{N} &\to \N^\mathbb{N} \\
y(00\zeta)&=0y(\zeta) & y^{-1}(0\zeta)&=00y^{-1}(\zeta) \\
y(01\zeta)&=1\zeta & y^{-1}(1\zeta)&=01\zeta \\
&\;\vdots & &\;\vdots \\
y(0(n-2)\zeta)&=(n-2)\zeta & y^{-1}((n-2)\zeta)&=0(n-2)\zeta \\
y(0(n-1)\zeta)&=(n-1)0y^{-1}(\zeta) & y^{-1}((n-1)0\zeta)&=0(n-1)y(\zeta) \\
y(1\zeta)&=(n-1)1\zeta & y^{-1}((n-1)1\zeta)&=1\zeta \\
&\;\vdots & &\;\vdots \\
y((n-2)\zeta)&=(n-1)(n-2)\zeta & y^{-1}((n-1)(n-2)\zeta)&=(n-2)\zeta \\
y((n-1)\zeta)&=(n-1)(n-1)y(\zeta) & y^{-1}((n-1)(n-1)\zeta)&=(n-1)y^{-1}(\zeta)
\end{align*}
\end{definition}
\begin{remark}
Note that if we restrict the domain to $\{0, n-1\}^\mathbb{N}$, $y$ is exactly the map defined in Definition \ref{definiton_y_2} under the identification of $n-1$ with $1$.
We will use this fact to reduce the discussion to the case of $n=2$.
\end{remark}
For each $s$ in $\N^\mathbb{N}$, define the map $y_s$ by setting
\begin{align} \label{n-adicy_definition}
y_s(\xi)&=
\left \{
\begin{array}{cc}
s y(\eta), & \xi=s\eta \\
\xi, & \mbox{otherwise}.
\end{array}
\right.
\end{align}
\begin{definition}
We define $G_0(n)$ to be the group generated by the $n+1$ elements $x_0, \dots, x_{n-2}, {x_0}_{[n-1]}$, and $y_{(n-1)0}$.
We call this group the \textit{$n$-adic Lodha--Moore group}.
\end{definition}
For the infinite presentation of $G_0(n)$ described in Section \ref{infinite_presentation}, we introduce an infinite generating set of this group.
Let $i$ in $\{0, \dots, n-2 \}$ and $\alpha$ in $\N^{<\mathbb{N}} \cup \{\epsilon \}$.
We recall that the map $\x{i}{\alpha}$ is defined as follows:
\begin{align*}
\x{i}{\alpha}(\zeta)&=
\begin{cases}
\alpha x_i(\eta) & (\zeta=\alpha \eta) \\
\zeta & (\zeta \neq \alpha \eta).
\end{cases}
\end{align*}
We also recall that the group $F(n)$ is generated by the following infinite set:
\begin{align*}
X(n)=
\left\{ \x{i}{\alpha} \mid i = 0, \dots, n-2, \alpha \in \N^{<\mathbb{N}} \cup \{ \epsilon \} \right\}.
\end{align*}
In addition, let
\begin{align*}
Y(n):=
\left\{ y_\alpha \;\middle|\;
\begin{array}{l} \alpha \in \N^{<\mathbb{N}}, \\
\alpha \neq 0\cdots0, (n-1)\cdots(n-1), \epsilon, \\
\mbox{the sum of each number in $\alpha$ is equal to $0 \bmod {n-1}$}
\end{array}
\right\}.
\end{align*}
We remark that for $\alpha=\alpha_1 \cdots \alpha_m \in \N^{<\mathbb{N}}$, the actions of $x_0, \dots, x_{n-2}, \x{0}{n-1}$, and $y$ preserve the value of $\alpha_1+\cdots+\alpha_m \bmod {n-1}$.
Then the set $Z(n):=X(n) \cup Y(n)$ also generates the group $G_0(n)$.
\subsection{Infinite presentation and normal form}\label{infinite_presentation}
In this section, we give the unique word with ``good properties'' for each element of $G_0(n)$ (Definition \ref{normal_form}).
In this process, we also give an infinite presentation of $G_0(n)$ (Corollary \ref{G_0(n)_presentation}).
Although almost all result in the rest of this section follows along the lines of the arguments in \cite{lodha2020nonamenable, lodha2016nonamenable}, we write them down for the convenience of the reader.
Let $s $ in $\N^{<\mathbb{N}}$ and $t$ in $\N^{<\mathbb{N}}$ or $\N^\mathbb{N}$.
We write $s \subset t$ if $s$ is a proper prefix of $t$ and write $s \subseteq t$ if $s \subset t$ or $s=t$.
For $s$, $t$, we say that $s$ and $t$ are independent if one of the following holds:
\begin{itemize}
\item $s, t \in \N^{<\mathbb{N}}$ and neither $s \subseteq t$ and $t \subseteq s$ holds.
\item $s \in \N^{<\mathbb{N}}$, $t \in \N^\mathbb{N}$ and $s$ is not any prefixes of $t$.
\item $s, t \in \N^\mathbb{N}$ and $s\neq t$.
\end{itemize}
In all cases, we write $s \perp t$.
Let $s(i), t(i)$ denote the $i$-th number of $s, t$, respectively.
Then we say $s<t$ if one of the following is true:
\begin{itemize}
\item[(a)] $t \subset s$;
\item[(b)] $s \perp t$ and $s(i)<t(i)$, where $i$ is the smallest integer such that $s(i) \neq t(i)$.
\end{itemize}
We note that this order is transitive.
For elements in $\N^\mathbb{N}$, we use the same symbol to denote the lexicographical order.
We claim that the following collection of relations gives a presentation of $G_0(n)$ (Corollary \ref{G_0(n)_presentation}):
\begin{enumerate}
\item the relations of $F(n)$ in Tables \ref{relations_1}, \ref{relations_2}, \ref{relations_3}, and \ref{relations_4};
\item $y_t\x{i}{s}=\x{i}{s}y_{\x{i}{s}(t)}$ for all $i$ and $s, t \in \N^{<\mathbb{N}}$ such that $y_t \in Y(n)$ and $\x{i}{s}(t)$ is defined;
\item $y_s y_t =y_t y_s$ for all $s, t \in \N^{<\mathbb{N}}$ such that $y_s, y_t \in Y(n)$ and $s\perp t$;
\item $y_s=\x{0}{s} y_{s0} y_{s(n-1)0}^{-1}y_{s(n-1)(n-1)}$ for all $s \in \N^{<\mathbb{N}}$ such that $y_s \in Y(n)$.
\end{enumerate}
We note that $\x{0}{n-1}((n-1)0)$ is not defined, for example.
All relations can be verified directly.
We write $R(n)$ for the collection of these relations.
We first define a form, which is easy to compute the composition of maps.
\begin{definition}
A word $\Omega$ on $Z(n)$ is in \textit{standard form} if $\Omega$ is a word such as $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ where $f$ is a word on $X(n)$ and $\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ is a word on $Y(n)$ with the condition that $s_i<s_j$ if $i<j$.
\end{definition}
In some cases, it is helpful to use the following weaker form.
\begin{definition}
A word $\Omega$ on $Z(n)$ is in \textit{weak standard form} if $\Omega$ is a word such as $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ where $f$ is a word on $X(n)$ and $\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ is a word on $Y(n)$ with the condition that if $s_j \subset s_i$, then $i<j$.
\end{definition}
We can always make a word in weak standard form into one in standard form.
\begin{lemma}[{\cite[Lemma 3.11]{lodha2020nonamenable}} for $n$=2]\label{weakstandard_standard}
We can rewrite a weak standard form into a standard form with the same length by just switching the letters $($i.e., relation $(3)$$)$ finitely many times.
\begin{proof}
Let $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ be a word in weak standard form.
We show this by induction on $m$.
It is obvious if $m=0, 1$.
For $m \geq 2$, by the induction hypothesis, we get a word $f\y{s_1^\prime}{t_1^\prime}\cdots\y{s_{m-1}^\prime}{t_{m-1}^\prime}\y{s_m}{t_m}$ where $f\y{s_1^\prime}{t_1^\prime}\cdots\y{s_{m-1}^\prime}{t_{m-1}^\prime}$ is in standard form.
Then one of the following holds:
\begin{enumerate}
\item $s_{m-1}^\prime \supset s_m$;
\item $s_{m-1}^\prime \perp s_m$ and $s_{m-1}^\prime(i)<s_m(i)$, where $i$ is the smallest number such that $s_{m-1}^\prime (i) \neq s_m(i)$;
\item $s_{m-1}^\prime \perp s_m$ and $s_{m-1}^\prime(i)>s_m(i)$, where $i$ is the smallest number such that $s_{m-1}^\prime(i)\neq s_m(i)$.
\end{enumerate}
If (1) or (2), $f\y{s_1^\prime}{t_1^\prime}\cdots\y{s_{m-1}^\prime}{t_{m-1}^\prime}\y{s_m}{t_m}$ is also in standard form.
If (3), by applying the relation (3) to $\y{s_{m-1}^\prime}{t_{m-1}^\prime}\y{s_m}{t_m}$ and using the induction hypothesis for the first $m-1$ characters again, we get a word in standard form.
\end{proof}
\end{lemma}
For an infinite $n$-ary word $w$ in $\N^\mathbb{N}$ and a word $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ in weak standard form, we define their \textit{calculation} as follows:
First, we apply $f$ to $w$. Then apply $\y{s_1}{t_1}, \dots, \y{s_m}{t_m}$ to $f(w)$ in this order, where the latter term ``apply'' means to rewrite each $y_{s_i}$ by using its definition in equation \eqref{n-adicy_definition}, and no rewriting can be done by the definition of the map $y$.
\begin{example}
Let $n=4$. For $w=3002\cdots$ and $y_{300}^{-1}y_{30}y_1$, we apply as follows:
\begin{align*}
3002\cdots \xrightarrow{y_{300}^{-1}}300y^{-1}(2\cdots) \xrightarrow{y_{30}} 30(y(0(y^{-1}(2\cdots)))) \xrightarrow{y_1} 30(y(0(y^{-1}(2\cdots)))).
\end{align*}
Therefore the calculation of $w=3002\cdots$ and $y_{300}^{-1}y_{30}y_1$ is $30(y(0(y^{-1}(2\cdots))))$.
\end{example}
We write such the element as $30y0y^{-1}2\cdots$ and sometimes regard it as a word on $\bm{N} \cup \{y, y^{-1}\}$.
We also define calculations for finite words in $\N^{<\mathbb{N}}$ and weak standard forms in the same way, although finite words are not in the domain of $y$.
We note that, unlike infinite words, not all calculations of finite words can be defined.
For example, in $n=4$, the calculation of $w=3002$ and $y_{300}^{-1}y_{30}y_1$ is $30y0y^{-1}2$.
However, that of $w=3$ and $y_{300}^{-1}y_{30}y_1$ is not defined.
We call the operation of rewriting a calculation by using the definition of $y$ once as \textit{substitution} and call the element in $\N^\mathbb{N}$ obtained by repeating the substitution \textit{output string}.
\begin{example}
For $w=0^4(n-1)0^4(n-1)\cdots$ and $y^2$, we have the following substitutions:
\begin{align*}
y^20^4(n-1)0^4(n-1)\cdots \to y 0 y 0^2(n-1)0^4(n-1)\cdots \to y0^2 y(n-1)0^4(n-1)\cdots.
\end{align*}
Their output string is $0(n-1)^40(n-1)^4 \cdots$.
\end{example}
As described in \cite[Section 3.6]{lodha2020nonamenable}, the map $y$ can be expressed as a finite state transducer.
For the sake of simplicity, assume $n\geq3$.
Then our transducer is illustrated in Figure \ref{transducer_y}.
\begin{figure}[tbp]
\centering
\includegraphics[width=120mm]{transducer_y.pdf}
\caption{The transducer of $y$, where $i=1, \dots, n-2$. }
\label{transducer_y}
\end{figure}
In this setting, each edge is labeled by a form $\sigma \mid \tau$, where $\sigma$ and $\tau$ are in $\N^{<\mathbb{N}}$.
The element $\sigma$ represents the input string, and $\tau$ represents the output string.
The difference from the case of $n=2$ is that there exists an accepting state (double circle mark).
This difference corresponds to the fact that $y$ vanishes in $y \sigma$ by substitutions if $\sigma$ is not in $\{0, n-1 \}^{\mathbb{N}}$ or $\{0, n-1 \}^{<\mathbb{N}}$.
\begin{definition}
A calculation contains a \textit{potential cancellation} if there exists a subword of the form
\begin{align*}
y^{t_1}\sigma y^{t_2}, \sigma \in \N^{<\mathbb{N}}, t_i \in \{1, -1 \}
\end{align*}
such that we get the word $\sigma^\prime y^{-t_2}$ by substituting $y^{t_1}\sigma$ finitely many times.
\end{definition}
\begin{example}
The subwords $y0(n-1)y$ and $y00y^{-1}$ are potential cancellations.
The subword $y0y$ is not a potential cancellation since we can not apply any substitutions.
The subword $y^{-1}(n-1)00(n-1)y$ is also not a potential cancellation since we have $y^{-1}(n-1)00(n-1)y \to 0(n-1)y0(n-1)y \to 0(n-1)(n-1)0y^{-1}y$.
\end{example}
\begin{remark} \label{remark_PCA}
If a calculation contains a potential cancellation, then $\sigma$ is in $\{0, n-1\}^{<\mathbb{N}}$.
Indeed, if not, $y$ vanishes.
\end{remark}
The following holds.
\begin{lemma}[{\cite[Lemma 5.9]{lodha2016nonamenable}} for $n=2$] \label{substitution_NPCA}
Let $\Lambda$ be a calculation that contains no potential cancellations.
Then the word $\Lambda^{\prime}$ obtained by substitution at any $y^{\pm1}$ again contains no potential cancellations.
\begin{proof}
By Remark \ref{remark_PCA}, it is sufficient to consider the case $\sigma \in \{0, n-1\}^{<\mathbb{N}}$.
Then we can identify $\{0, n-1 \}$ with $\{0, 1\}$.
By \cite[Lemma 5.9]{lodha2016nonamenable}, we have the desired result.
\end{proof}
\end{lemma}
We generalize the notion of potential cancellation to the case of weak standard forms.
\begin{definition}
A weak standard form $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ has a \textit{potential cancellation} if there exists $w$ in $\N^\mathbb{N}$ such that the calculation of $w$ by $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ contains a potential cancellation.
\end{definition}
To construct unique words, we define the moves obtained from the relations of $G_0(n)$.
\begin{definition} \label{definition_five_moves}
We assume that every map in the following is defined.
\begin{description}
\item[Rearranging move]
$\y{t}{i}\x{j}{s}^{\pm1} \to \x{j}{s}^{\pm1}\y{\x{j}{s}^{\pm1}(t)}{i}$;
\item[Expansion move]
$y_s \to \x{0}{s}y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}$ and $y_s^{-1} \to \x{0}{s}^{-1}\y{s00}{-1}y_{s0(n-1)}\y{s(n-1)}{-1}$;
\item[Commuting move] $y_uy_v\leftrightarrow y_v y_u$ (if $u \perp v$);
\item[Cancellation move] $\y{s}{\pm i}\y{s}{\mp i} \to \epsilon$;
\item[ER moves]
\begin{enumerate}
\item $f(\y{s_1}{t_1}\cdots\y{s_k}{t_k})y_u \to f\x{0}{u}(\y{\x{0}{u}(s_1)}{t_1}\cdots \y{\x{0}{u}(s_k)}{t_k})(y_{u0}\y{u(n-1)0}{-1}y_{u(n-1)(n-1)})$
\item $f(\y{s_1}{t_1}\cdots\y{s_k}{t_k})\y{u}{-1} \to f\x{0}{u}^{-1}(\y{\x{0}{u}^{-1}(s_1)}{t_1}\cdots \y{\x{0}{u}^{-1}(s_k)}{t_k})(\y{u00}{-1}y_{u0(n-1)}\y{u(n-1)}{-1})$.
\end{enumerate}
\end{description}
ER moves are a combination of an expansion move and rearranging moves.
\end{definition}
By the definition of potential cancellations, we note that if we apply ER move to a standard form that contains a potential cancellation, then either the resulting word also contains a potential cancellation, or we can apply a cancellation move to the resulting word.
Like the Thompson group $F$, each element of $G_0(n)$ can be represented by tree diagrams.
Then the expansion move for $y_s$ is the replacement from the left diagram to the right diagram in Figure \ref{image_expansion_move}.
\begin{figure}[tbp]
\centering
\includegraphics[width=110mm]{image_expansion_move.pdf}
\caption{The black and white circles represent that we apply $y$ and $y^{-1}$ to the words corresponding to the edges below them, respectively. }
\label{image_expansion_move}
\end{figure}
See \cite[Section 4]{lodha2016nonamenable} for details of the case $n=2$.
We introduce the following notion, which makes it easy to check whether the moves are defined.
\begin{definition}
For $s$ in $\N^{<\mathbb{N}}$, we write $\| s\|$ for the length of $s$.
We define the \textit{depth} of a weak standard form $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ to be the integer $\min_{1 \leq i \leq m} \|s_i \|$.
If a weak standard form is on $X(n)$, we define its depth to be $\infty$.
\end{definition}
We show that an arbitrary word on $Z(n)$ can be rewritten in standard form.
In order to do this, we prepare two lemmas.
\begin{lemma}[{\cite[Lemma 5.2]{lodha2016nonamenable}} for $n=2$] \label{standard_base_case}
Let $y_s$ in $Y(n)$ and $l$ in $\mathbb{N}\cup\{0\}$.
Then there exists a standard form $\Omega$ obtained by applying expansion move and rearranging move on $y_s^{\pm1}$ finitely many times that satisfies the following:
\begin{enumerate}[font=\normalfont]
\item If there exists $\x{i}{u}^t$ in $\Omega$, then $s \subset u$ holds.
\item If there exists $y_u$ or $\y{u}{-1}$, then it is the only one, and $s \subseteq u$ and $\|u \| \geq l$ hold.
\item If there exist $y_u, y_v$ with $u \neq v$, then $u \perp v$ holds.
\end{enumerate}
\begin{proof}
We show this by induction on $l-\|s\|$.
If $l-\|s\| \leq 0$, then $\Omega=y_s^{\pm1}$ satisfies all the conditions.
We assume that $l-\|s\|>0$ holds.
Because we can discuss $y_s^{-1}$ in almost the same way, we consider only $y_s$.
We apply expansion move on $y_s$ and get the word $\x{0}{s}y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}$.
Since we have
\begin{align*}
\max \{ l-\|s0\|,~ l-\|s(n-1)0\|,~ l-\|s(n-1)(n-1)\| \} \leq l-\|s\|,
\end{align*}
by the induction hypothesis, there exists three standard forms $\Omega_{s0}, \Omega_{s(n-1)0}, \Omega_{s(n-1)(n-1)}$ for $y_{s0}$, $y_{s(n-1)}^{-1}$, $y_{s(n-1)(n-1)}$, respectively, all of which satisfy the conditions.
Thus the word is rewritten from $\x{0}{s}y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}$ to $\x{0}{s}\Omega_{s0}\Omega_{s(n-1)0}\Omega_{s(n-1)(n-1)}$.
If there exists $\x{i}{u}^t$ in $\Omega_{s(n-1)0}$, then $u \supset s(n-1)0$ holds.
If there exists $\y{v}{\pm1}$ in $\Omega_{s0}$, then $v \supseteq s0$ holds.
Since $u \perp v$, $\x{i}{u}^t(v)$ is defined and equals to $v$.
Thus, we can apply rearranging moves on this $\x{i}{u}^t$.
A similar process can be done for the $\x{i}{u}^t$ in $\Omega_{s(n-1)(n-1)}$.
This completes the proof.
\end{proof}
\end{lemma}
\begin{lemma}[{\cite[Lemma 5.3]{lodha2016nonamenable}} for $n=2$] \label{standard_depth_lemma}
Let $\Lambda$ be a word on $X(n)$, and $k$ be the length of $\Lambda$.
Then there exists $l_0$ in $\mathbb{N}$ such that the following holds:
If $\Omega$ is a standard form with depth $l \geq l_0$, then we have a standard form $\Omega^\prime$ obtained from $\Omega \Lambda$ by rearranging moves, with the depth of $\Omega^\prime$ being at least $l-k$.
\begin{proof}
We show this by induction on the length of $\Lambda$.
For the base case, let $\Lambda=\x{i}{s}^{\pm1}$.
If $\x{i}{s}^{\pm1}(t)$ is not defined, then $t=si$ holds.
This means that if $t \in \N^{<\mathbb{N}}$ with $\|t \| \geq \|s\|+2$, then $\x{i}{s}(t)$ is defined.
Hence let $l_0=\|s\|+2$.
We note that
\begin{align*}
\|t\|-1 \leq \|\x{i}{s}^{\pm1}(t)\| \leq \| t\|+1
\end{align*}
holds.
Then we can apply rearranging moves on $\x{i}{s}^{\pm1}$ with every $y_t$ in $\Omega$.
The depth of the resulting standard form $\Omega^\prime$ is at least $l-1$.
We assume that the claim holds for $k-1$.
Let $\Lambda=\Lambda^\prime\x{i}{s}^{\pm1}$.
By the induction hypothesis, there exists $l_0^\prime$ for $\Lambda^\prime$ such that we can apply moves $\Omega\Lambda^\prime\x{i}{s}^{\pm1}\to\Omega^{\prime \prime}\x{i}{s}^{\pm1}$ where the depth of $\Omega$ is $l \geq l_0^\prime$ and that of $\Omega^{\prime \prime}$ is at least $l-(k-1)$.
Then, let $l_0=\max\{l_0^\prime, k+\|s\|+1\}$ and assume that the depth of $\Omega$ is $l \geq l_0$.
Since we have $l-(k-1) \geq \|s\|+2$, for the same reason as in the base case, we can apply rearranging moves and obtain a standard form $\Omega^\prime$ whose depth is at least $l-(k-1)-1=l-k$.
\end{proof}
\end{lemma}
Now we make arbitrary words into standard forms.
\begin{proposition}[{\cite[Lemma 5.4]{lodha2016nonamenable}} for $n=2$] \label{Prop_standard_forms_from_words}
Let $l$ be a natural number and $w$ be a word on $Z(n)$.
Then we can rewrite $w$ into a standard form whose depth is at least $l$ by moves.
\begin{proof}
If $w$ is in $X(n)$, there is nothing to do since the depth of $w$ is $\infty$.
Hence we assume that $w$ is not in $X(n)$.
We show the claim by induction on the length $m$ of $w$.
For the base case, since $w$ is not in $X(n)$, we have $w=\y{s}{\pm1}$.
By Lemma \ref{standard_base_case}, the claim holds.
Let $m>1$.
By dividing $a^{\pm(k+1)}$ in $w$ as $a^{\pm k}a^{\pm1}$ if necessary, we can decompose $w$ into two positive length words $\Omega_0\Omega_1$.
We assume that $\Omega_1$ is not on $X(n)$.
By the induction hypothesis, we obtain a standard form $\Lambda\Upsilon$ from $\Omega_1$, where $\Lambda$ is on $X(n)$, and $\Upsilon$ is on $Y(n)$, with the depth being at least $l$.
Let $k$ be the length of $\Lambda$ and $r=\max\{\|u\| \mid \mbox{$\y{u}{i}$ belongs to $\Upsilon$} \}+1$.
By the induction hypothesis, there exists a standard form $\Omega_0^\prime$ obtained from $\Omega_0$, with the depth being at least $\max\{l_0, r+k\}$, where $l_0$ is a natural number for $\Lambda$ in Lemma \ref{standard_depth_lemma}.
Then we rewrite $\Omega_0^\prime\Lambda \to \Omega_0^{\prime\prime}$ where $\Omega_0^{\prime\prime}$ is in standard form with the depth being at least $r$.
The form $\Omega_0^{\prime\prime}\Upsilon$ obtained through the previous transformations $w \to \Omega_0\Omega_1 \to \Omega_0^\prime\Lambda\Upsilon \to \Omega_0^{\prime\prime}\Upsilon$ is in weak standard form.
Indeed, since $\Omega_0^{\prime\prime}$ and $\Upsilon$ (on $Y(n)$) are in standard form, by the definition of $r$, the claim holds.
By Lemma \ref{weakstandard_standard}, we get the standard form without changing the depth.
If $\Omega_1$ is in $X(n)$, then we set $\Lambda=\Omega$, $\Upsilon=\epsilon$, and $r=l$.
Then we can apply the above argument.
\end{proof}
\end{proposition}
Next, we list three lemmas about ER moves.
These are useful for getting words in standard form without potential cancellations.
\begin{lemma}[{\cite[Lemma 3.21]{lodha2020nonamenable}} for $n$=2]\label{ER_move_preserve_wsf}
Let $f(\y{s_1}{t_1}\cdots\y{s_k}{t_k})y_u^{\pm1}(\y{p_1}{q_1}\cdots\y{p_m}{q_m})$ be in weak standard form.
Then the word obtained from $f(\y{s_1}{t_1}\cdots\y{s_k}{t_k})y_u^{\pm1}(\y{p_1}{q_1}\cdots\y{p_m}{q_m})$ by the ER move on $y_u^{\pm1}$ is also in weak standard form.
\begin{sproof}
We show that only the case $y_u$.
Let
\begin{align*}
f\x{0}{u}(\y{\x{0}{u}(s_1)}{t_1}\cdots \y{\x{0}{u}(s_k)}{t_k})(y_{u0}\y{u(n-1)0}{-1}y_{u(n-1)(n-1)})(\y{p_1}{q_1}\cdots\y{p_m}{q_m})
\end{align*}
be the word obtained from the given word by ER move.
We show this by considering whether each $s_i$ is independent with $u$ or not.
If $s_i \perp u$ holds, then $\x{0}{u}(s_i)=s_i$, and there is nothing to do.
If $s_i \supset u$ holds, since $\x{0}{u}(s_i)$ is defined, $\x{0}{u}(s_i) \supset u$ holds, and it can never be a proper prefix of $u0$, $u(n-1)0$, or $u(n-1)(n-1)$. Since $p_j$ satisfies either $p_j\perp u$ or $p_j \subseteq u$, the claim holds in both cases.
\end{sproof}
\end{lemma}
\begin{lemma}[{\cite[Lemma 3.22]{lodha2020nonamenable}} for $n$=2]\label{lem_ERmove_preserve_NPCA}
Let $f \lambda_1$ be in weak standard form with no potential cancellations and
$g\lambda_2$ be a word obtained from $f \lambda_1$ by an ER move.
Then $g\lambda_2$ is in weak standard form with no potential cancellations.
\begin{proof}
Let $\lambda_1=\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ with every $t_i$ in $\{\pm1\}$.
We apply ER move on $\y{u_i}{t_i}$.
Since $f$ does not affect the existence of potential cancellations, we assume that $f$ is the empty word.
We only consider the case $t_i=1$ since the cases $t_i=1$ and $t_i=-1$ are shown similarly.
Let $s_j^{\prime}:=\x{0}{s_i}(s_j)$ $(1 \leq j \leq i-1)$.
By the definition of ER move, we have
\begin{align*}
g\lambda_2=\x{0}{s_i}(\y{s_1^\prime}{t_1}\cdots\y{s_{i-1}^\prime}{t_{i-1}})(y_{s_i0}\y{s_i(n-1)0}{-1}y_{s_i(n-1)(n-1)})(\y{s_{i+1}}{t_{i+1}}\cdots\y{s_m}{t_m}).
\end{align*}
Since it is clear by Lemma \ref{ER_move_preserve_wsf} that $g\lambda_2$ is in weak standard form, suppose that $g\lambda_2$ has a potential cancellation.
Let $\tau \in \N^\mathbb{N}$ be an element such that the calculation $\Lambda$ of $\tau$ by $(\y{s_1^\prime}{t_1}\cdots\y{s_{i-1}^\prime}{t_{i-1}})(y_{s_i0}\y{s_i(n-1)0}{-1}y_{s_i(n-1)(n-1)})(\y{s_{i+1}}{t_{i+1}}\cdots\y{s_m}{t_m})$ contains a potential cancellation.
Then, $s_i$ is a prefix of $\tau$.
Indeed, we have the following:
\begin{enumerate}
\item if $s_j \perp s_i$, then $\y{s_j^\prime}{t_j}=\y{s_j}{t_j}$;
\item if $s_j \perp s_i$, then $\y{s_j}{t_j}(\tau^\prime)\perp s_i$ where $\tau^\prime$ is in $\N^\mathbb{N}$ with $\tau^\prime \perp s_i$;
\item if $s_j \supset s_i$, since $\x{0}{s_i}(s_j) \supset s_i$, we have $\y{s_j^\prime}{t_j}(\tau^\prime)=\y{\x{0}{s_i}(s_j)}{t_j}(\tau^\prime)=\tau^\prime$ and $y_{s_j}(\tau^\prime)=\tau^\prime$ where $\tau^\prime$ is in $\N^\mathbb{N}$ with $\tau^\prime \perp s_i$;
\item $y_{s_i}$, $y_{s_i0}$, $y_{s_i(n-1)0}^{-1}$, and $y_{s_i(n-1)(n-1)}$ all fix $\tau^\prime$ where $\tau^\prime$ is in $\N^\mathbb{N}$ with $\tau^\prime \perp s_i$.
\end{enumerate}
Since ER move is defined, each $y_j$ satisfies either $s_j\perp s_i$ or $s_j \supset s_i$.
If $s_i$ is not a prefix of $\tau$, $\tau \perp s_i$ holds.
Then the calculations of $\tau$ with $f\lambda_1$ and with $g\lambda_2$ are the same.
This contradicts the assumption that $f\lambda_1$ does not have a potential cancellation. Hence $s_i$ is a prefix of $\tau$.
Let $\Lambda^\prime$ be the calculation of $\x{0}{s_i}^{-1}(\tau)$ by $\y{s_1}{t_1}\cdots\y{s_m}{t_m}$.
By the assumption, this calculation does not contain a potential cancellation.
On the other hand, $\Lambda$ is obtained from $\Lambda^\prime$ by substituting once.
However, this also contradicts Lemma \ref{substitution_NPCA}.
\end{proof}
\end{lemma}
\begin{lemma} \label{lem_depth_ER_moves}
Let $l$ be an arbitrary natural number.
Then, by applying ER moves to the weak standard form $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ finitely many times, we obtain a weak standard form whose depth is at least $l$.
\begin{sproof}
If $\|s_i\|< l$ holds, we apply ER moves on $\y{s_i}{t_i}$ (if $t_i \neq \pm1$, we apply on the last one).
We note that the move may not be defined.
In that case, before we do that, we apply ER moves to $y_{s_j}$ $(j<i)$ which is the cause that the ER move on $y_{s_i}$ is not defined.
If it is also not defined, repeat the process.
Since we can always apply ER moves on $fy_{s_1}$, this process is finished.
We do this process repeatedly until the depth of the obtained word is at least $l$.
By Lemma \ref{ER_move_preserve_wsf}, it is also in weak standard form.
\end{sproof}
\end{lemma}
Since subwords play an essential role in the notion of potential cancellations, we introduce the following definitions for simplicity.
\begin{definition}
Let $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ be in weak standard form.
We say the pair $(\y{s_j}{t_j}, \y{s_i}{t_i})$ is \textit{adjacent} if the following two conditions hold:
\begin{enumerate}
\item $s_i \subset s_j$,
\item if $u$ in $\N^{<\mathbb{N}}$ satisfies $s_i \subset u \subset s_j$, then $u \notin \{ s_1, \dots, s_m\}$.
\end{enumerate}
\end{definition}
We define an adjacent pair $(\y{s_j}{t_j}, \y{s_i}{t_i})$ is a \textit{potential cancellation} if $y^{t_i}\sigma y^{t_j}$ is a potential cancellation, where $\sigma$ is the word satisfying $s_i\sigma=s_j$.
It is clear from the definition that a weak standard form contains a potential cancellation if and only if there exists an adjacent pair such that it is a potential cancellation.
By the definition, for example, if $(\y{j}{t}, y_s)$ is a potential cancellation, then either $\y{j}{t}=\y{s00}{-1}$, $y_{0(n-1)}$, or $\y{s(n-1)}{-1}$ holds, or $(\y{\x{0}{s}(j)}{t}, y_{s0})$, $(\y{\x{0}{s}(j)}{t}, \y{s(n-1)0}{-1})$, or $(\y{\x{0}{s}(j)}{t}, y_{s(n-1)(n-1)})$ is also a potential cancellation.
As mentioned in Remark \ref{remark_PCA}, we note that if an adjacent pair is a potential cancellation, then $\sigma$ is in $\{0, n-1\}^{<\mathbb{N}}$.
We introduce the moves which are the ``inverse'' of ER moves.
First, we define the conditions where the moves are defined.
\begin{definition}\label{def_potential_contraction}
We say that a weak standard form contains a \textit{potential contraction} if it satisfies either of the following:
\begin{enumerate}
\item there exists a subword $y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}$, but there does not exist $\y{s(n-1)}{\pm1}$;
\item there exists a subword $\y{s00}{-1}y_{s0(n-1)}\y{s(n-1)}{-1}$, but there does not exist $\y{s0}{\pm1}$.
\end{enumerate}
We also call the same when the condition above is satisfied by commuting moves.
\end{definition}
\begin{example} \label{example_potential_contraction}
Let $n=4$. A weak standard form $y_{300}y_{3030}^{-1}y_{3031}y_{3033}y_{30}$ contains a potential contraction.
\end{example}
We now define moves.
\begin{definition}\label{def_contraction_move}
Let $f\lambda$ be in standard form.
We assume that this form contains a potential contraction in the sense of (1).
We apply commuting move to all $y_u^{v}$ except the subword in the standard form satisfying $s0<u \leq s(n-1)(n-1)$ to the left of $y_{s0}$ while preserving the order, and we replace $y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}$ with $\x{0}{s}^{-1}y_s$.
Then, move $\x{0}{s}^{-1}$ to just next to $f$ by rearranging moves.
Finally, we apply cancellation moves or moves $x^ax^b \to x^{a+b}$ if necessary.
The rearranging moves are defined, and the word obtained by the above sequence of moves is in standard form.
We call this sequence of moves a \textit{contraction move}.
We also define the contraction move for (2) in the same way.
\end{definition}
The moves and the claim in this definition are justified by the following lemma.
We describe only case (1), but the similarity holds for case (2).
\begin{lemma}\label{lem_contraction_move}
In Definition $\ref{def_contraction_move}$, the rearranging moves are defined, and the resulting word is again in standard form.
\begin{proof}
By the definition of standard form, for $y_u^v$, we have either $s0\perp u$ or $s0 \subseteq u$.
In the former case, we have either $s \perp u$ or $s \subset u$.
In the latter case, we have $s \subset u$.
By the definition of potential contraction, there does not exist $\y{s(n-1)}{\pm1}$.
Hence $\x{0}{s}^{-1}(u)$ is defined, and the rearranging moves can be done.
Suppose that we have finished applying commuting move.
Let $s_j^\prime:=\x{0}{s}^{-1}(s_j)$, and $\x{0}{s}^{-1}(\y{s_1^\prime}{t_1}\cdots\y{s_k^\prime}{t_k})y_s(\y{p_1}{q_1}\cdots\y{p_m}{q_m})$ be the resulting word.
If $s_k^\prime=s_k$, then $s_k\perp s$ holds. Since the original word is in standard form, we have $s_k^\prime <s$.
If $s_k^\prime\neq s_k$, then $s_k^\prime \supset s$.
By the transitivity of this order, $\x{0}{s}^{-1}(\y{s_1^\prime}{t_1}\cdots\y{s_k^\prime}{t_k})y_s$ is in standard form.
We show that $y_s(\y{p_1}{q_1}\cdots\y{p_m}{q_m})$ is in standard form.
We note that $s(n-1)(n-1)<p_1$ holds.
If $s(n-1)(n-1) \supset p_1$, either $s=p_1$ or $s \supset p_1$ holds.
In the former case, by $y_s\y{p_1}{t_1}\to\y{s}{t_1+1}$ or $\to \epsilon$, we get the standard form.
In the latter case, we have $s<p_1$.
If $s(n-1)(n-1) \perp p_1$, since $(n-1)$ is the largest number, we have $s<p_1$.
By the transitivity, $\x{0}{s}^{-1}(\y{s_1^\prime}{t_1}\cdots\y{s_k^\prime}{t_k})y_s(\y{p_1}{q_1}\cdots\y{p_m}{q_m})$ is in standard form.
\end{proof}
\begin{example}
In example \ref{example_potential_contraction}, we obtain $\x{0}{30}^{-1}y_{301}\y{30}{2}$ by applying a contraction move.
\end{example}
\end{lemma}
Contraction moves have the following property.
\begin{lemma}\label{lem_contraction_move_preserve_NPCA}
Let $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ be in standard form which contains a potential contraction and no potential cancellations.
Then the word obtained by contraction move contains no potential cancellations.
\begin{proof}
We show only the case of potential contraction condition (1).
We assume that we have produced a potential cancellation after applying a contraction move.
If there exists such an adjacent pair, the pair must be $(\y{s}{k}, \y{s^\prime}{k^\prime})$, where $s^\prime$ satisfies $s^\prime \subset s$.
Indeed, if not, each adjacent pair is either $(\y{u_1}{j_1}, \y{u_2}{j_2})$ where $u_1, u_2 \neq s$, or $(y_{s^{\prime\prime}}^{k^{\prime\prime}}, y_s^k)$ where $s^{\prime\prime}$ satisfies $s^{\prime\prime} \supset s$.
In the former case, since either $\x{0}{s}(u_i)\neq u_i$ ($i=1, 2$) or $\x{0}{s}(u_i)= u_i$ ($i=1, 2$) holds, this contradicts the assumption that $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ contains no potential cancellations.
In the latter case, we apply ER move on $\y{s}{k}$.
Since the contraction move is the inverse of the ER move, if $y_{s^{\prime\prime}}^{k^{\prime\prime}}$ is $\y{s0}{-1}$, $y_{s(n-1)0}$, or $\y{s(n-1)(n-1)}{-1}$, then it contradicts the assumption that $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ is in standard form.
If $y_{s^{\prime\prime}}^{k^{\prime\prime}}$ is none of them, then it contradicts that $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ contains no potential cancellations.
There does not exist $\y{s}{k_1}$ in $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$.
Indeed, if there exists $y_s^{k_1}$ ($k_1<0$), since there exists $y_{s0}$ by the assumption of potential contraction, $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ contains a potential cancellation.
If there exists $\y{s}{k_1}$ ($k_1>0$), since $(\y{s}{k}, \y{s^\prime}{k^\prime})$ is a potential cancellation in the word after applying the contraction move, the adjacent pair $(y_s^{k_1}, \y{s^\prime}{k^\prime})$ is a potential cancellation in $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$.
This contradicts the assumption about no potential cancellations.
Now we consider the adjacent pair $(y_s, y_{s^\prime}^{k^\prime})$.
Let $\sigma$ be the finite word such that $s^\prime\sigma=s$ holds.
By the assumption, $y\sigma y$ or $y^{-1}\sigma y$ is potential cancellation.
In the former case, by substitutions, we have $y\sigma=\sigma^\prime y^{-1}$.
Then, we have $y\sigma 0y=\sigma^\prime y^{-1}0y=\sigma^\prime00y^{-1}y$.
This contradicts the assumption that the standard form contains no potential cancellations.
Similarly, in the latter case, we have $y^{-1}\sigma=\sigma^\prime y^{-1}$, which also contradicts the assumption.
\end{proof}
\end{lemma}
Our goal in the rest of this section is to give the unique word for each element in $G_0(n)$, which satisfies the following:
\begin{definition} \label{normal_form}
Let $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ be in standard form with no potential cancellations, no potential contractions, and $f$ is in the normal form in the sense of $F(n)$.
Then we say that $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ is in the \textit{normal form}.
\end{definition}
See Section \ref{subsection_F(n)_definition} for the definition of the normal form of elements in $F(n)$.
We obtain a normal form from an arbitrary word on $Z(n)$ through the following four steps:
\begin{description}
\item[Step 1] Convert an arbitrary word into a standard form (Proposition \ref{Prop_standard_forms_from_words});
\item[Step 2] Convert a standard form into a standard form that contains no potential cancellations;
\item[Step 3]Convert a standard form that contains no potential cancellations into a standard form $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ which contains no potential cancellations and no potential contractions;
\item[Step 4]Convert $f$ into $g$, where $g$ is the unique normal form in $F(n)$ (Section \ref{subsection_F(n)_definition}).
\end{description}
The remaining steps are only 2 and 3.
\begin{lemma}[{\cite[Lemma 4.5]{lodha2020nonamenable}} for $n$=2]\label{lem_for_step2}
Let $(\y{s_1}{t_1}\cdots\y{s_m}{t_m})\y{s}{t}$ be in standard form such that the following hold:
\begin{enumerate}[font=\normalfont]
\item $t_1, \dots, t_m, t \in \{1, -1\}$;
\item any two $s_i, s_j$ are independent;
\item for any $\y{s_i}{t_i}$, $\y{s_i}{t_i}$ and $\y{s}{t}$ is an adjacent pair with being potential cancellation.
\end{enumerate}
Then by applying ER moves and cancellation moves, we obtain a standard form $f\y{u_1}{v_1}\cdots\y{u_k}{v_k}$ such that any two $u_i, u_j$ are independent and $v_1, \dots, v_k \in \{1, -1\}$.
\begin{proof}
We consider only the case $t=1$.
We claim that $\x{0}{s}(s_i)$ is defined for every $i$.
Indeed, for $s_i$ such that $s \subset s_i$, the only case where $\x{0}{s}(s_i)$ is not defined is the case $s_i=s0$.
Then since the adjacent pair $(\y{s0}{t_i}, y_s)$ is not potential cancellation whether $t_i$ is $1$ or $-1$, this contradicts the assumption $(3)$.
Let $s_i^\prime=\x{0}{s}(s_i)$.
By applying ER move to $y_s$, we have
\begin{align*}
(\y{s_1}{t_1}\cdots\y{s_m}{t_m})y_s=\x{0}{s}(\y{s_1^\prime}{t_1}\cdots\y{s_m^\prime}{t_m})y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}.
\end{align*}
We note that $s0$, $s(n-1)0$, and $s(n-1)(n-1)$ are independent of each other.
For every $\y{s_i^\prime}{t_i}$, one of $y_{s0}$, $\y{s(n-1)0}{-1}$, or $y_{s(n-1)(n-1)}$ corresponds to it, either as the inverse or as an adjacent pair.
Let $\y{\sigma}{\tau}$ be the corresponding one.
If it is the inverse, we apply a cancellation move.
If it forms an adjacent pair, the distance in an $n$-ary tree between $s_i^\prime$ and $\sigma$ is smaller than the distance between $s_i$ and $s$.
Thus, by iterating this process, we obtain the desired result.
\end{proof}
\end{lemma}
The following lemma completes step 2.
\begin{lemma}[{\cite[Lemma 4.6]{lodha2020nonamenable}} for $n$=2]\label{lem_step2}
By applying moves, any weak standard form can be rewritten into a standard form that contains no potential cancellations.
\begin{proof}
Let $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ $(t_i \in \{-1, 1 \})$ be a weak standard form.
We show by induction on $m$.
If $m\leq1$, since there exists no adjacent pair, it is clear.
We assume that $m>1$ holds.
By applying the inductive hypothesis, Lemmas \ref{lem_ERmove_preserve_NPCA}, and \ref{lem_depth_ER_moves} to $f\y{s_1}{t_1}\cdots\y{s_{m-1}}{t_{m-1}}$, we obtain a weak standard form $g\y{p_1}{q_1}\cdots \y{p_k}{q_k}$ with at least depth $\|s_m\|+1$ and without potential cancellations.
Since $p_i \subset s_m$ does not hold by its depth, $g\y{p_1}{q_1}\cdots \y{p_k}{q_k}\y{s_m}{t_m}$ is in weak standard form.
If there exists an adjacent pair that is a potential cancellation, it is only of the form $(\y{p_i}{q_i}, \y{s_m}{t_m})$.
We record all such $\y{p_i}{q_i}$.
By applying commuting moves, we obtain a weak standard form $h(\y{v_1}{k_1}\cdots\y{v_l}{k_l})(\y{u_1}{v_1}\cdots\y{u_o}{v_o})\y{s_m}{t_m}$ which satisfies the following:
\begin{enumerate}
\item $v_1, \dots, v_o \in \{ -1, 1\}$;
\item For $j=1, \dots, o$, $\y{u_j}{v_j}$ and $\y{s_m}{t_m}$ is an adjacent pair which is a potential cancellation;
\item All other adjacent pairs are not potential cancellations.
\end{enumerate}
Indeed, since $s_m$ is the shortest word, each adjacent element is the ``second shortest.''
Hence we can apply commuting moves to get a word.
We again note that Lemmas \ref{lem_ERmove_preserve_NPCA} and \ref{lem_depth_ER_moves}, i.e., we can increase the depth by ER moves while preserving in weak standard form without potential cancellations.
We apply ER moves to $h(\y{v_1}{k_1}\cdots\y{v_l}{k_l})(\y{u_1}{v_1}\cdots\y{u_o}{v_o})\y{s_m}{t_m}$ in two phases.
First, we apply ER moves to the part of $h(\y{v_1}{k_1}\cdots\y{v_l}{k_l})$ so that we can apply ER moves to the word on $X(n)$ that appears when we apply Lemma \ref{lem_for_step2} to $(\y{u_1}{v_1}\cdots\y{u_o}{v_o})\y{s_m}{t_m}$.
Secondly, we apply Lemma \ref{lem_for_step2} to $(\y{u_1}{v_1}\cdots\y{u_o}{v_o})\y{s_m}{t_m}$.
By the same argument in Lemma \ref{lem_ERmove_preserve_NPCA}, no new potential cancellations are produced in this process.
Finally, by Lemma \ref{weakstandard_standard}, we have the desired result.
\end{proof}
\end{lemma}
The following lemma completes step 3.
\begin{lemma}[{\cite[Lemma 4.7]{lodha2020nonamenable}} for $n$=2]\label{lem_step3}
Any standard form which contains no potential cancellations can be rewritten into a standard form that contains no potential cancellations and no potential contractions.
\begin{proof}
We apply contraction moves repeatedly.
By Lemmas \ref{lem_contraction_move} and \ref{lem_contraction_move_preserve_NPCA}, the applied word is again in standard form and contains no potential cancellations.
Since this move makes the word on $Y(n)$ in the standard form strictly shorter, the process finishes.
\end{proof}
\end{lemma}
The only thing that remains to be proved is the uniqueness of the normal form.
We will show this by contradiction.
In order to do this, we define an ``invariant'' of the forms.
\begin{definition}
A calculation that contains no potential cancellations has \textit{exponent $m$} if $m$ is the number of $y^{\pm1}$ satisfying the condition that the number appearing after it is only $0$, $(n-1)$, or $y^{\pm1}$.
\end{definition}
\begin{example}
Let $n=4$.
The string $y0y02y^{-1}(n-1)y00\cdots$ has exponent $2$.
The string $y100\cdots$ has exponent $0$.
\end{example}
For the proof of uniqueness, we prepare two lemmas.
\begin{lemma}[{\cite[Lemma 5.10]{lodha2016nonamenable}} for $n=2$]\label{lem_exponent_1}
Let $\Lambda$ be a finite word on $\bm{N} \cup \{y, y^{-1}\}$ that contains no potential cancellations.
Let $m$ be the exponent of $\Lambda$.
Then there exists a finite word $u$ on $\{0, n-1\}$ and $v$ in $\N^{<\mathbb{N}}$ such that $\Lambda u$ can be rewritten into $vy^m$ by substitutions.
\begin{proof}
By substitutions, we rewrite $\Lambda$ into $v^\prime\Lambda^\prime$, where $v^\prime$ is in $\N^{<\mathbb{N}}$ and $\Lambda^\prime$ is in $\{0, (n-1), y, y^{-1}\}$.
By the definition of exponent, $\Lambda^\prime$ also has exponent $m$, and by Lemma \ref{substitution_NPCA}, this also contains no potential cancellations.
Hence, by identifying $\{0, n-1\}$ with $\{0, 1\}$, the claim comes down to the case $n=2$.
\end{proof}
\end{lemma}
For $u, s$ in $\N^{<\mathbb{N}}$, we say that \textit{$u$ dominates $s$} if the condition $u \perp s$ or $u \supset s$ is satisfied.
\begin{lemma}[{\cite[Lemma 4.8]{lodha2020nonamenable}} for $n=2$]\label{lem_exponent_2}
Let $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}$ and $g\y{p_1}{q_1}\cdots\y{p_m}{q_m}$ be in standard form which represent the same element in $G_0(n)$.
Let $u$ be in $\N^{<\mathbb{N}}$ such that the following hold:
\begin{enumerate}[font=\normalfont]
\item $f(u)=:u_1$ and $g(u)=:u_2$ are defined;
\item $u_1$ dominates $s_1, \dots, s_l$;
\item $u_2$ dominates $p_1, \dots, p_m$.
\end{enumerate}
Let $\Theta$ be the calculation of $u$ and $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}$, and let $\Lambda$ be the calculation of $u$ and $g\y{p_1}{q_1}\cdots\y{p_m}{q_m}$.
Assume that these calculations contain no potential cancellations.
Then two exponents of the calculations are the same.
\begin{proof}
Let $e(\Lambda)$ and $e(\Theta)$ be the two corresponding exponents.
We show this by contradiction.
We can assume that $e(\Lambda)>e(\Theta)$ without loss of generality.
Since exponents are non-negative integer, let $k:=e(\Lambda)>0$.
By Lemma \ref{lem_exponent_1}, we have $wy^k$ obtained from $\Lambda v$ by substitutions, where $v \in \N^{<\mathbb{N}}$.
Since $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}$ and $g\y{p_1}{q_1}\cdots\y{p_m}{q_m}$ are equal as elements of $G_0(n)$, the output strings of
\begin{align*}
\Theta v 0^{2^k}(n-1)0^{2^k}(n-1)\cdots
\shortintertext{and}
\Lambda v 0^{2^k}(n-1)0^{2^k}(n-1)\cdots
\end{align*}
are equal as elements of $\N^\mathbb{N}$.
Then the latter is
\begin{align*}
\Lambda v 0^{2^k}(n-1)0^{2^k}(n-1)\cdots = w y^k 0^{2^k}(n-1)0^{2^k}(n-1)\cdots=w0(n-1)^{2^k}0(n-1)^{2^k}\cdots.
\end{align*}
Since $e(\Lambda)>e(\Theta)$ holds, by the definition of $y$, this implies a contradiction.
\end{proof}
\end{lemma}
\begin{corollary} \label{G_0(n)_presentation}
The group obtained from the presentation $\langle Z(n) \mid R(n)\rangle$ is the $n$-adic Lodha--Moore group $G_0(n)$.
\begin{proof}
We show that any standard form which contains no potential cancellations and represents an element of $F(n)$ is always a word on $X(n)$.
Let $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}=g$ be standard forms, where $f, g \in F(n)$, and $m \geq 1$.
Assume that they contain no potential cancellations.
Let $u=s_100\cdots$ in $\N^\mathbb{N}$.
Then the exponent of the calculation of $f^{-1}(u)$ and $f\y{s_1}{t_1}\cdots\y{s_m}{t_m}$ is strictly greater than $0$, and that of the calculation of $f^{-1}(u)$ and $g$ is $0$.
By Lemma \ref{lem_exponent_2}, this is a contradiction.
In particular, for a word on $Z(n)$ representing the identity element of $G_0(n)$, we can rewrite it into a standard form that contains no potential cancellations by Proposition \ref{Prop_standard_forms_from_words} and Lemma \ref{lem_step2}.
Since this standard form is a word on $X(n)$, as shown above, we can reduce it to the empty word by using the relations of $F(n)$.
\end{proof}
\end{corollary}
\begin{remark}
From the above argument, the uniqueness of the normal form of the elements of $F(n)$ in $G_0(n)$ follows.
\end{remark}\label{remark_uniqueF(n)}
The following completes the proof of the uniqueness.
\begin{theorem}[{\cite[Theorem 4.4]{lodha2020nonamenable}} for $n=2$]\label{thm_normal_form_uniqueness}
For each element in $G_0(n)$, its normal form is unique.
\begin{proof}
By Remark \ref{remark_uniqueF(n)}, we can assume that every normal form in the following argument is not in $F(n)$.
We show this by contradiction.
Let $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}$ and $g\y{p_1}{q_1}\cdots\y{p_m}{q_m}$ be different normal forms representing the same element.
We can assume that $s_l \leq p_m$ without loss of generality.
One of the following three holds:
\begin{enumerate}
\item $s_l=p_m$ and $t_l=q_m$;
\item $s_l=p_m$ and $t_l \neq q_m$;
\item $s_l<p_m$.
\end{enumerate}
First, we show that it is sufficient to consider only case $(3)$.
In case (1), since $\y{s_l}{t_l}=\y{p_m}{q_m}$, we start from $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m-1}}{q_{m-1}}$.
They are two different standard forms representing the same element.
They contain no potential cancellations but possibly contain a potential contraction.
Since we only cancel $\y{s_l}{t_l}=\y{p_m}{q_m}$, the only case that contains a potential cancellation at this step is when $\y{s_l}{t_l}$ or $\y{p_m}{q_m}$ or both play the role of $\y{s(n-1)}{\pm1}$ for some $s$ (see Definition \ref{def_potential_contraction} (1)).
By the assumption of the standard form, the case with $\y{s0}{\pm 1}$ (Definition \ref{def_potential_contraction} (2)) does not occur.
We further divide case (1) into the following four subcases:
\begin{description}
\item[\rm (1-1)] Both $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m-1}}{q_{m-1}}$ contain a potential contraction, and $t_{l-1}=q_{m-1}$;
\item[\rm (1-2)] Both $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m-1}}{q_{m-1}}$ contain a potential contraction, and $t_{l-1}\neq q_{m-1}$;
\item[\rm (1-3)] $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ contains a potential contraction, and $g\y{p_1}{q_1}\cdots\y{p_{m-1}}{q_{m-1}}$ contains no potential contractions, or vice versa;
\item[\rm (1-4)] $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m-1}}{q_{m-1}}$ contain no potential contractions.
\end{description}
In case (1-1), we consider $f\y{s_1}{t_1}\cdots\y{s_{l-2}}{t_{l-2}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m-2}}{q_{m-2}}$ instead of the original words in order to eliminate the potential contraction part.
Since the assumption of containing a potential contraction and being in standard form, $s_l=p_m=s(n-1)$, $s_{l-1}=p_{m-1}=s(n-1)(n-1)$ and $t_{l-1}=q_{m-1}>0$ hold.
Then $f\y{s_1}{t_1}\cdots\y{s_{l-2}}{t_{l-2}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m-2}}{q_{m-2}}$ are different normal forms.
Indeed, the only case where a potential contraction is generated by canceling $\y{s_{l-1}}{t_{l-1}}=\y{p_{m-1}}{q_{m-1}}$ is when $\y{s_{l-1}}{t_{l-1}}=\y{p_{m-1}}{q_{m-1}}$ plays the role of $y_{s^\prime(n-1)}$ for $s^\prime=s(n-1)$, but this does not occur since $\y{s^\prime0}{k_1}=\y{s(n-1)0}{k_1}$ and $\y{s^\prime0}{k_2}=\y{s(n-1)0}{k_2}$ exist in two forms respectively and $k_1, k_2<0$ holds by the assumption of potential contractions.
Thus, we get ``shorter'' normal forms that satisfy all the assumptions.
In case (1-2), we can assume that $0<t_{l-1}<q_{m-1}$ without loss of generality.
We consider $f\y{s_1}{t_1}\cdots\y{s_{l-2}}{t_{l-2}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m-1}}{q_{m-1}-t_{l-1}}$ instead.
For the same reason as for case (1-1), the former is in normal form.
By the assumption of a potential contraction, $\y{s_{l-2}}{t_{l-2}}=\y{s(n-1)i\sigma}{t_{l-2}}$, where $i$ is in $\bm{N}$ and $\sigma$ is a word in $\N^{<\mathbb{N}} \cup \{\epsilon\}$.
Note that $s(n-1)0 \leq s(n-1)i \sigma$ holds.
For the latter, by performing contraction moves as in step 3, the last letter is $\y{u}{k}$ where $u \subseteq s$.
This is the situation in case (3).
In case (1-3), we only consider the former situation.
By the assumptions, $s_l=p_m=s(n-1)$ holds for some $s$.
By applying contraction moves to $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ as in step3, the last latter is $\y{u}{k}$ where $u \subseteq s \subset s_l=p_m$.
Since $g\y{p_1}{q_1}\cdots\y{p_{m-1}}{q_{m-1}}$ is in normal form, and in particular in standard form, we have $y_{p_{m-1}}\neq y_u$.
This is the situation in case (3).
In case (1-4), both are ``shorter'' normal forms that satisfy all the assumptions.
Therefore, any subcase of case (1) yields shorter words or in case (3).
In case (2), we can assume that neither $0<q_m<t_l$ nor $t_l<q_m<0$ without loss of generality.
Consider $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ and $g\y{p_1}{q_1}\cdots\y{p_{m}}{q_{m}-t_{l}}$ instead.
The latter is in normal form, but as in case (1), the former may contain a potential contraction.
Similarly, we divide case (2) into two subcases:
\begin{description}
\item[\rm (2-1)] $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ contains a potential contraction;
\item[\rm (2-2)] $f\y{s_1}{t_1}\cdots\y{s_{l-1}}{t_{l-1}}$ contains no potential contractions.
\end{description}
In case (2-1), the same argument as in (1-3) can be applied.
In case (2-2), since $s_{l-1}\neq p_m$, this is the situation in case (3).
We note that case (1-1) or (1-4) happens only finitely many times due to the uniqueness of the normal form of $F(n)$.
Thus, we consider case (3).
We only consider the case $q_m>0$.
We note that $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}\y{p_m}{-1}$ is in standard form since $s_l < p_m$.
One of the following holds:
\begin{enumerate}[label=(\roman*)]
\item there exists an infinite word $\sigma$ on $\{0, n-1\}$ such that for any finite word $\sigma_1 \subset \sigma$, $p_m\sigma_1$ is not in $\{s_1, \dots, s_l \}$;
\item there exists an adjacent pair of the form $p_m, s_i$ where $p_mu=s_i$ for $u$ on $\{0, n-1\}$ which is not a potential cancellation.
\end{enumerate}
Indeed, if both are false, it contradicts that $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}$ contains no potential contractions.
Then, in either case, there exists a finite word $w$ on $\{0, n-1\}$ such that the calculation $\Lambda$ of $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}\y{p_m}{-1}$ and $f^{-1}(p_mw)$ contains no potential cancellations.
By expanding $w$ if necessary, we can consider the following three calculations:
\begin{enumerate}[label=(\alph*)]
\item $\Theta$ is the calculation of $g\y{p_1}{q_1}\cdots\y{p_m}{q_m-1}$ and $f^{-1}(p_mw)$;
\item $\Lambda^\prime$ is the calculation of $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}$ and $f^{-1}(p_mw)$;
\item $\Theta^\prime$ is the calculation of $g\y{p_1}{q_1}\cdots\y{p_m}{q_m}$ and $f^{-1}(p_mw)$.
\end{enumerate}
All calculations contain no potential cancellations.
Indeed, for $\Lambda^\prime$ and $\Theta^\prime$, it is clear from the assumption of normal form.
For $\Theta$, either $q_m-1$ is zero or strictly greater than zero since $q_m>0$.
In both cases, it is clear from the definitions and the assumption that $g\y{p_1}{q_1}\cdots\y{p_m}{q_m}$ contains no potential cancellations.
Let $e(\Theta)$, $e(\Theta^\prime)$, $e(\Lambda)$, and $e(\Lambda^\prime)$ be the its exponent, respectively.
Since we have
\begin{align*}
f\y{s_1}{t_1}\cdots\y{s_l}{t_l}\y{p_m}{-1}=g\y{p_1}{q_1}\cdots\y{p_m}{q_m-1}
\end{align*}
as elements of $G_0(n)$, by Lemma \ref{lem_exponent_2}, $e(\Lambda)=e(\Theta)$ holds.
Similarly, we have $e(\Theta^\prime)=e(\Lambda^\prime)$.
By the construction of $\Lambda$ and $\Lambda^\prime$, we have $e(\Lambda)>e(\Lambda^\prime)$.
Similarly, we have $e(\Theta)\leq e(\Theta^\prime)$ since $q_m>0$.
Combining them, we obtain
\begin{align*}
e(\Theta)\leq e(\Theta^\prime)=e(\Lambda^\prime)<e(\Lambda),
\end{align*}
which is a contradiction.
If $q_m<0$, we can prove the same for $f\y{s_1}{t_1}\cdots\y{s_l}{t_l}y_{p_m}$ and $g\y{p_1}{q_1}\cdots\y{p_m}{q_m+1}$.
\end{proof}
\end{theorem}
\section{Several properties of $G_0(n)$}\label{section_G0(n)_properties}
We show that $G_0(n)$ has ``expected'' properties.
\subsection{The finite presentation of $G_0(n)$}
Using the infinite presentation given in Section \ref{infinite_presentation}, we construct a finite presentation of $G_0(n)$ with respect to the generating set $\{x_0, \dots, x_{n-2}, {x_0}_{[n-1]}, y_{(n-1)0}\}$.
As in \cite[Theorem 3.3]{lodha2016nonamenable}, we use two properties of $F(n)$.
Since $G_0=G_0(2)$ is already known to be finitely presented, we assume that $n>2$.
For $s$ in $\N^{<\mathbb{N}}$ that is neither $0\cdots0$ nor $(n-1)\cdots(n-1)$, we define a map $f_s$ in $F(n)$ as follows:
Let $a$ be the sum of each number in $s$, and let $d, k$ be the integers such that $a=(n-1)d+k$ holds, where $0\leq d$ and $0\leq k<n-1$.
Then $f_s$ is defined as in Figure \ref{definition_fs}.
\begin{figure}[tbp]
\centering
\includegraphics[width=100mm]{definition_fs.pdf}
\caption{The definition of $f_s$. The rightmost carets are for adjustment. The triangle labeled $s$ denotes the minimal $n$-ary tree containing $s$. }
\label{definition_fs}
\end{figure}
We fix a word on $\{x_0, \dots, x_{n-2}, \x{0}{(n-1)}\}$ representing $f_s$.
Since the leaf corresponding to $s$ in the minimal $n$-ary tree containing $s$ is the $a$-th leaf counting from the left (from $0$), we note that $f_s((n-1)k)=s$ holds.
\begin{lemma}\label{lem_finitely_presented1}
For the above $f_s$, we have $\x{i}{(n-1)k}f_s=f_s \x{i}{s}$ and $y_{(n-1)0}f_s=f_sy_s$.
\begin{sproof}
We note that we have $f_s^{-1}\x{i}{(n-1)k}f_s=\x{i}{f_s((n-1)k)}=\x{i}{s}$.
Indeed, if $w=sw^\prime \in \N^\mathbb{N}$, the left hand side sends $w$ to $sx_i(w^\prime)$ via $(n-1)kw^\prime$ and $(n-1)kx_i(w^\prime)$.
If $w \neq sw^\prime$, since $f_s^{-1}(w)$ does not contains $(n-1)k$ as a prefix, $\x{i}{(n-1)k}(f_s^{-1}(w))=f_s^{-1}(w)$.
Another statement can be shown in the same way.
\end{sproof}
\end{lemma}
\begin{lemma}\label{lem_finitely_presented2}
Let $u, v$ be in $\N^{<\mathbb{N}}$ such that $u<v$, $u \perp v$, and $y_u, y_v$ are in $Y(n)$.
Then there exists $g$ in $F(n)$ such that $g(0(n-1))=u$ and $g((n-1)0)=v$ hold.
\begin{sproof}
Like the construction of $f_s$, by adding some carets, we can obtain two trees (may no be tree diagram) such that $0(n-1) \mapsto u$.
Since $n>2$, we can add some carets between two leaves corresponding to $0(n-1)$ and $(n-1)0$ so that $(n-1)0 \mapsto v$.
Finally, we add some carets to the rightmost if necessary, to make it a tree diagram.
See Figure \ref{construction_g_uv} for the sketch of the construction of $g$.
\end{sproof}
\end{lemma}
\begin{figure}[tbp]
\centering
\includegraphics[width=80mm]{construction_g_uv.pdf}
\caption{The construction of $g$ in Lemma \ref{lem_finitely_presented2}. }
\label{construction_g_uv}
\end{figure}
\begin{theorem}[{\cite[Theorem 3.3]{lodha2016nonamenable}} for $n=2$]\label{Thm_finitely_presented_G0(n)}
$G_0(n)$ admits a finite presentation.
\begin{proof}
Since $F(n)$ is finitely presented \cite[Theorem 4.17]{brown1987finiteness},
the relation (1) in $R(n)$ is expressed by finite relations.
Thus we only consider the other three relations (2), (3), and (4).
For the relation $y_t\x{i}{s}=\x{i}{s}y_{\x{i}{s}(t)}$,
we can rewrite as follows:
\begin{align*}
y_t\x{i}{s}&=f_t^{-1}y_{(n-1)0}f_t \x{i}{s}, \\
\x{i}{s}y_{\x{i}{s}(t)}&=\x{i}{s}f^{-1}_{\x{i}{s}(t)}y_{(n-1)0}f_{\x{i}{s}(t)}.
\end{align*}
Thus, it is sufficient to show that
\begin{align}
y_{(n-1)0}f_t \x{i}{s}f^{-1}_{\x{i}{s}(t)}=f_t \x{i}{s}f^{-1}_{\x{i}{s}(t)}y_{(n-1)0}, \label{relation2_into_finite}
\end{align}
by using a finite number of relations.
We note that $f_t \x{i}{s}f^{-1}_{\x{i}{s}(t)}$ maps $(n-1)0$ to itself.
Therefore, $f_t \x{i}{s}f^{-1}_{\x{i}{s}(t)} \in F(n)$ is rewritten into a word on the finite set
\begin{align*}
X_{(n-1)0}(n):=
&\{x_0x_i^{-1}, \x{i}{(n-1)} \mid i=1, \dots, n-2 \} \\
&\cup \{\x{i}{0}, \x{i}{1}, \dots, \x{i}{n-2}, \x{i}{(n-1)1}, \dots, \x{i}{(n-1)(n-1)}\mid i=0, \dots, n-2 \} \\
&\cup \{ \x{0}{0(n-1)}, \dots, \x{0}{(n-2)(n-1)}, \x{0}{(n-1)1(n-1)}, \dots, \x{0}{(n-1)(n-1)(n-1)}\},
\end{align*}
by finite relations of $F(n)$.
Since each element is commute with $y_{(n-1)0}$, the collection of the finite relations $hy_{(n-1)0}=y_{(n-1)0}h$ where $h$ is in $X_{(n-1)0}(n)$ implies equation \eqref{relation2_into_finite}.
For the relation $y_sy_t=y_ty_s$ where $s \perp t$, we note that we have $y_{(n-1)0}y_{0(n-1)}=y_{0(n-1)}y_{(n-1)0}$.
By using this one and relation (2), we have
\begin{align*}
y_sy_t=f^{-1}_sy_{(n-1)0}f_sf^{-1}_ty_{(n-1)0}f_t=g^{-1}y_{0(n-1)}gg^{-1}y_{(n-1)0}g=g^{-1}y_{(n-1)0}gg^{-1}y_{0(n-1)}g=y_ty_s,
\end{align*}
where $g$ is the element such that $g(0(n-1))=s$ and $g((n-1)0)=t$.
Finally, for the relation $y_s=\x{0}{s}y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}$, we note that we have $y_{(n-1)0}=\x{0}{(n-1)0}y_{(n-1)00}\y{(n-1)0(n-1)0}{-1}y_{(n-1)0(n-1)(n-1)}$.
Since $f_s((n-1)0w)=sw$ for $w \in \N^{<\mathbb{N}}$, by using relation (2), we have
\begin{align*}
y_s&=f^{-1}_sy_{(n-1)0}f_s \\
&=f^{-1}_s\x{0}{(n-1)0}y_{(n-1)00}\y{(n-1)0(n-1)0}{-1}y_{(n-1)0(n-1)(n-1)}f_s \\
&=f^{-1}_s\x{0}{(n-1)0}f_sf^{-1}_sy_{(n-1)00}f_sf^{-1}_s\y{(n-1)0(n-1)0}{-1}f_sf^{-1}_sy_{(n-1)0(n-1)(n-1)}f_s \\
&=\x{0}{s}y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}.
\end{align*}
This completes the proof.
\end{proof}
\end{theorem}
\subsection{Nonamenability of $G_0(n)$}
In this section, we discuss embeddings of the groups $G_0(n)$.
For nonamenability, we only use the fact that a subgroup of an amenable group is also amenable \cite[Theorem 18.29 (1)]{dructu2018geometric}.
The idea for the following proposition comes from \cite{burillo2001metrics}.
\begin{theorem}\label{prop_embedding_G_0(p)G_0(q)}
Let $p, q \geq 2$ and assume that there exists $d$ in $\mathbb{N}$ such that $q-1=d(p-1)$ holds.
Then there exists an embedding $I_{p, q}: G_0(p) \to G_0(q)$.
Moreover, the equality $I_{p,r}=I_{p, q}I_{q, r}$ holds for the maps $I_{p, q}$, $I_{q, r}$ and $I_{p, r}$ defined for $r\geq q \geq p \geq2$ such that any two satisfy the condition.
\begin{proof}
For the sake of clarity, we label elements of $G_0(p)$ with ``tildes'' and elements of $G_0(q)$ with ``hats.''
We first recall the definition of the embedding $F(p) \to F(q)$ given in \cite[Section 3, example (3)]{burillo2001metrics}.
This homomorphism is defined by
\begin{align*}
&\tilde{x}_0\mapsto\hat{x}_0 & &\tilde{x}_1 \mapsto \hat{x}_d & &\cdots & &\tilde{x}_{p-2} \mapsto \hat{x}_{d(p-2)}& &\tx{0}{(p-1)} \mapsto \hx{0}{(q-1)},
\end{align*}
and extended to the (quasi-isometric) embedding \cite[Theorem 6]{burillo2001metrics}.
By considering $F(p)$ and $F(q)$ as pairs of $p$-ary trees and $q$-ary trees, this embedding is regarded as a ``caret replacement.''
Indeed, for every pair of $p$-ary trees, inserting $d-1$ edges between every pair of adjacent edges of each $p$-caret corresponds to the embedding.
See Figure \ref{embedding_F3F7}, for example.
\begin{figure}[tbp]
\centering
\includegraphics[width=120mm]{embedding_F3F7.pdf}
\caption{Example of embedding for $p=3$ and $q=7$. }
\label{embedding_F3F7}
\end{figure}
Define the map $i_{p, q}: \Pset^{<\mathbb{N}}=\{0, \dots, p-1\}^{<\mathbb{N}} \to \Qset^{<\mathbb{N}}=\{0, \dots, q-1\}^{\mathbb{N}}$ by setting
\begin{align*}
w_1 w_2\cdots w_k \mapsto (dw_1)(dw_2)\cdots (dw_k),
\end{align*}
namely, we multiply each number of a given word by $d$.
In addition, define the map $Y(p)\to Y(q)$ by setting $\tilde{y}_s \mapsto \hat{y}_{i_{p, q}(s)}$.
By the definition of $i_{p, q}$, we have that $\hat{y}_{i_{p, q}(s)}$ is in $Y(q)$.
By combining the two maps $F(p) \to F(q)$ and $Y(p) \to Y(q)$, we obtain the map $I_{p, q}: F(p)\cup Y(p) \to F(q)\cup Y(q)$.
This map can be extended to the homomorphism $I_{p, q}: G_0(p) \to G_0(q)$.
Indeed, since $i_{p, q}(\tx{i}{s}(t))=\hx{di}{i_{p, q}(s)}(i_{p, q}(t))$ and $I_{p, q}(\tilde{y}_s)=\hat{y}_{i_{p, q}(s)}$ hold, we can verify that the relations of the infinite presentation (Corollary \ref{G_0(n)_presentation}) are preserved under $I_{p, q}$ by directly calculations.
We claim that $I_{p, q}: G_0(p) \to G_0(q)$ is injective.
We show this by contradiction.
Assume that $\operatorname{Ker}(I_{p, q})$ is not trivial.
By the construction, the restriction $I_{p, q}\restr{F(p)}$ coincides with the embedding $F(p) \to F(q)$ mentioned above.
Hence if there exists $x$ in $\operatorname{Ker}(I_{p, q})$ that is not identity, then $x$ is not in $F(p)$.
We note that the map $I_{p, q}$ preserves being in normal form.
In particular, if $x$ is not in $F(p)$, then $I_{p, q}(x)$ is not in $F(q)$ by the uniqueness (Theorem \ref{thm_normal_form_uniqueness}).
This implies that $I_{p, q}(x)$ is not identity, as desired.
Let $d_1$, $d_2$ and $d_3$ be natural numbers such that $q-1=d_1(p-1)$, $r-1=d_2(q-1)$ and $r-1=d_3(p-1)$ holds respectively.
We note that $d_3=d_1d_2$ holds.
By considering the definitions of the map $i_{p, q}$ and the homomorphism from $F(p)$ to $F(q)$, the equality of $I_{p,r}$ and $I_{p, q}I_{q, r}$ follow immediately.
\end{proof}
\end{theorem}
\begin{corollary}
Let $n\geq2$. Then $G_0(n)$ is nonamenable.
\begin{proof}
Since we have $n-1=(n-1)(2-1)$ for every $n$, by Theorem \ref{prop_embedding_G_0(p)G_0(q)}, we have an embedding from $G_0(2)=G_0$ into $G_0(n)$.
This implies that $G_0(n)$ has a subgroup that is isomorphic to $G_0$.
Since $G_0$ is nonamenable \cite[Theorem 1.1]{lodha2016nonamenable}, $G_0(n)$ is also nonamenable.
\end{proof}
\end{corollary}
\begin{corollary}
Let $s_i:=2^{i-1}+1$.
Then the sequence $G(s_1), G(s_2), \dots$ forms an inductive system of groups.
\end{corollary}
\subsection{$G_0(n)$ has no free subgroups}
We assume $n \geq 3$.
In order to show that $G_0(n)$ does not contain the free groups, we will follow \cite[Section 3]{brin1985groups}.
The difference in this paper is that the domain and range set of the maps are $\N^\mathbb{N}$ instead of $\mathbb{R}$.
For an element $g$ in $G_0(n)$, we write the set-theoretic support $\supp{g}$ for the set $\{\xi \in \N^\mathbb{N} \mid g(\xi)\neq\xi\}$.
Although we equip $\N^\mathbb{N}$ with the topology that is homomorphic to Cantor set, which is totally disconnected, we can consider ``connected components'' of $\supp{g}$, using the total order of $\N^\mathbb{N}$.
For $a, b \in \N^\mathbb{N}$ with $a<b$, set $(a, b):=\{\xi \in \N^\mathbb{N} \mid a<\xi<b\}$.
\begin{lemma}\label{lem_supp_finite_component}
For $g \in G_0(n)$, there exists a sequence $a_1<a_2 \leq a_3<a_4\leq \cdots < a_{2m}$ in $\N^\mathbb{N}$ such that we have
\begin{align*}
\supp{g}=(a_1, a_2) \sqcup (a_3, a_4) \sqcup \cdots \sqcup (a_{2m-1}, a_{2m}).
\end{align*}
We call each $(a_{2i}, a_{2i+1})$ a \textit{connected component} of $\supp{g}$.
\begin{proof}
Let $f\y{s_1}{t_1}\cdots \y{s_m}{t_m}$ be in normal form of $g$.
We decompose $\N^\mathbb{N}$ into disjoint sets $\{n_1\eta \mid \eta \in \N^\mathbb{N} \}, \dots, \{n_p\eta \mid \eta \in \N^\mathbb{N}\}$ by using $n_1, \dots, n_p$ in $\N^{<\mathbb{N}}$ such that the following holds:
\begin{enumerate}
\item For each $n_l$, $f(n_l)$ is defined;
\item For each $s_l$, there exists $n_{l^\prime}$ such that $s_l \leq f(n_{l^\prime})$ holds.
\end{enumerate}
We show the claim in the lemma by contradiction.
Assume that there exist $a_1<b_1<\cdots$ such that each $a_i$ is a fixed point, and each $b_i$ is in $\supp{g}$.
Since $\N^\mathbb{N}=\{n_1\eta \mid \eta \in \N^\mathbb{N} \}\cup \dots \cup \{n_p\eta \mid \eta \in \N^\mathbb{N}\}$ holds, we can assume that each $a_i$ and $b_i$ is in some set $\{n_j\eta \mid \eta \in \N^\mathbb{N} \}$ without loss of generality.
Since $a_{i}<b_i<a_{i+1}$ holds, we can write each $a_i$ and $b_i$ as $n_ja_i^\prime$ and $n_jb_i^\prime$, respectively.
We claim that each $a_i^\prime$ and $b_i^\prime$ can be replaced by words in $\{0, n-1\}^{\mathbb{N}}$, respectively.
Indeed, if $a_i^\prime$ is not in $\{0, n-1\}^{\mathbb{N}}$, then there exist $\hat{a}_i \in \{0, n-1\}^{<\mathbb{N}}\cup \{\epsilon\}$, $k \in \{1, \dots, n-2\}$, and $a_i^{\prime \prime} \in \N^\mathbb{N}$ such that $a_i^\prime=\hat{a}_i k a_i^{\prime \prime}$ holds.
Then, by the definition of $y$, $g(n_j\hat{a}_i k)$ is defined and equals to $n_j\hat{a}_i k$ since $a_i=n_j \hat{a}_i k a_i^{\prime \prime}$ is a fixed point of $g$.
Moreover, for any other $k^\prime \in \{1, \dots, n-2\}$, we have $g(n_j\hat{a}_i k^\prime)=n_j\hat{a}_i k^\prime$.
This implies that we have $g(n_j\hat{a}_i 0 \overline{(n-1)})=n_j\hat{a}_i 0 \overline{(n-1)}$, where $\overline{(n-1)}$ denotes the element $(n-1)(n-1)(n-1)\cdots $ in $\N^\mathbb{N}$.
By a similar argument for $b_i^\prime$, we have $g(n_j\hat{b}_i k)=n_j\tilde{b_i}k$, where $\hat{b}_i \neq \tilde{b}_i$.
Since we have $g(n_j\hat{b}_i k^\prime)\neq n_j\hat{b}_i k^\prime$ for any other $k^\prime$, $n_j\hat{b}_i 0\overline{(n-1)}$ is in $\supp{g}$.
Even if each $a_i$ or $b_i$ is replaced by the above one, the order $a_1<b_1<\cdots$ is also preserved.
Let $w_1, \dots, w_t$ be in $\{0, \dots, n-1\}$ such that $f(n_j)=w_1 \cdots w_t$ holds.
For $n_j \zeta \in \N^\mathbb{N}$ and $f\y{s_1}{t_1}\cdots \y{s_m}{t_m}$, let $w_1y^{l_1}w_2 y^{l_2}\cdots w_t y^{l_t} \zeta$ be their calculation.
Then some $l_q$ may be zero.
Assume that $f(n_j)=w_1 \cdots w_t$ is not in $\{0, n-1\}^{<\mathbb{N}}$ and let $z$ be the maximal number such that $w_z$ is in $\{1, \dots, n-2\}$.
By applying finitely many substitutions to all $y^{l_{z^\prime}}$ ($z^\prime<z$), we obtain a word
$w y^{l_1^\prime} w_2^\prime y^{l_2^\prime} \cdots w_{t^\prime}^\prime y^{l_{t^\prime}^\prime}\zeta$,
where $w_j^\prime \in \{0, n-1\}$.
Since $n_j a_i$ is the output string of $w y^{l_1^\prime} w_2^\prime y^{l_2^\prime} \cdots w_{t^\prime}^\prime y^{l_{t^\prime}^\prime} a_i$, we have $w \leq n_j$.
Indeed, it is clear that $w \perp n_j$ does not hold, and if $w > n_j$ holds, then it contradicts the facts that $a_i$ is in $\{0, n-1\}^\mathbb{N}$, and the last number of $w$ is $w_j$.
Let $n_j=w n_j^\prime$.
Then we note that $n_j^\prime$ is in $\{0, n-1\}^{<\mathbb{N}} \cup \{\epsilon\}$ since $y^{l_1^\prime} w_2^\prime y^{l_2^\prime} \cdots w_{t^\prime}^\prime y^{l_{t^\prime}^\prime}a_i$ is in $\{0, n-1\}^\mathbb{N}$ and its output string equals to $n_j^\prime a_i$.
We recall that $G_0(n)$ has a subgroup that is isomorphic to $G_0$ (Theorem \ref{prop_embedding_G_0(p)G_0(q)}).
We construct an element $g^\prime$ which is in the subgroup and satisfies the assumption of $g$.
Let $g^\prime$ be the element represented by
\begin{align*}
f^\prime \y{(n-1)0w_2^\prime \cdots w_{t^\prime}^\prime}{l_{t^\prime}^\prime} \y{(n-1)0w_2^\prime \cdots w_{t^\prime-1}^\prime}{l_{t^\prime-1}^\prime} \cdots \y{(n-1)0w_2^\prime}{j_2^\prime} \y{(n-1)0}{j_1^\prime},
\end{align*}
where $f^\prime$ is a word on $\{x_0, \x{0}{(n-1)}\}$ satisfying $f^\prime((n-1)0n_j^\prime)=(n-1)0w_2^\prime \cdots w_{t^\prime}^\prime$.
Since $n_j^\prime$ and $w_2^\prime \cdots w_{t^\prime}^\prime$ are in $\{0, n-1\}^{<\mathbb{N}}$, there must exist such an element $f^\prime$.
By the construction, $g^\prime$ is in the subgroup that is isomorphic to $G_0$.
In addition, each $(n-1)0n_j^\prime a_i$ is a fixed point of $g^\prime$, and $(n-1)0n_j^\prime b_i$ is in $\supp{g^\prime}$.
By the construction of the map $I_{2, n}$ in Theorem \ref{prop_embedding_G_0(p)G_0(q)}, if we restrict the domain set of $g^\prime$ to $\{0, n-1\}^\mathbb{N}$ and identify $n-1$ with $1$, then we obtain the element of $G_0$ which satisfies all the assumptions about fixed points and elements of support.
However, since this does not happen by Proposition \ref{proposition_piecewiseprojective}, this is a contradiction.
Finally, assume that $f(n_j)=w_1 \cdots w_t$ is in $\{0, n-1\}^{<\mathbb{N}}$.
Since $w_1y^{l_1}w_2 y^{l_2}\cdots w_t y^{l_t} a_i$ is in $\{0, n-1\}^{\mathbb{N}}$, $n_j$ is also in $\{0, n-1\}^{<\mathbb{N}}$.
Therefore, $f$ is represented by a word on $\{x_0, \x{0}{(n-1)}\}$.
We again note that some $l_q$ may be zero.
Then $g^\prime=f\y{w_1\cdots w_t}{l_t} \cdots \y{w_1w_2}{l_2} \y{w_1}{l_1}$ is in $G_0(n)$, and an element of $G_0$ can be similarly constructed from $g^\prime$.
\end{proof}
\end{lemma}
\begin{remark}
By the construction, the sequence $a_1, \dots, a_{2m}$ in Lemma \ref{lem_supp_finite_component} is uniquely determined.
\end{remark}
For a group $G$, $[G, G]$ denotes its commutator subgroup.
For two elements of $x, y \in G$, $[x, y]$ denotes the commutator $xyx^{-1}y^{-1}$.
\begin{theorem}\label{Theorem_no_free_group}
The group $G_0(n)$ has no free subgroups.
\begin{proof}
Similarly to \cite{brin1985groups}, we will show that if $G_1$ is a subgroup of $[G_0(n), G_0(n)]$, then either $\mathbb{Z}^2$ is a copy of a subgroup of $G_1$ or $[G_1, G_1]$ is trivial.
Indeed, in that case, if $G$ is a subgroup of $G_0(n)$, then either $\mathbb{Z}^2$ is a copy of a subgroup of $G$ or $[G, G]$ is abelian, and the free group $F_2$ does not have such a property.
Let $G_1$ be a subgroup of $[G_0(n), G_0(n)]$, and assume that $[G_1, G_1]$ is not trivial.
Then there exist elements $f$ and $g$ in $G_1$ such that $z=fgf^{-1}g^{-1}\neq \mathrm{Id}$.
For $f$ and $g$, let $a_1, \dots, a_{2s}$ and $b_1, \dots, b_{2t}$ be sequences given in Lemma \ref{lem_supp_finite_component}, respectively.
By choosing elements appropriately from $a_1, \dots, a_{2s}, b_1, \dots, b_{2t}$, we obtain $c_1, \dots, c_{2m}$ such that
\begin{align*}
\supp{f} \cup \supp{g}=(c_1, c_2) \sqcup (c_3, c_4) \sqcup \cdots \sqcup (c_{2m-1}, c_{2m}).
\end{align*}
Let $d_1, \dots, d_{2l}$ be a sequence given by Lemma \ref{lem_supp_finite_component} for $[f, g]$.
Then we have that for each $x \in \supp{[f, g]}$, there exist $i$ and $i^\prime$ such that
\begin{align}
c_{2i^\prime-1}<d_{2i-1}<x<d_{2i}<c_{2i^\prime} \label{support_condition_commutator}
\end{align}
holds.
Indeed, it is sufficient to show that for each common fixed point $x$ of $f$ and $g$, there exists a prefix $u$ such that $[f, g]$ fixes all the element of $\{u\xi \mid \xi \in \N^\mathbb{N}\}$.
If $x$ is not in $\{0, n-1\}^{\mathbb{N}}$, then $y$ ``vanishes.''
Thus by regarding $f$ and $g$ as piecewise linear map of $F(n)$, it is clear.
If $x$ is in $\{0, n-1\}^{\mathbb{N}}$, by identifying $\{0, n-1\}$ with $\{0, 1\}$, it is also clear by Proposition \ref{proposition_piecewiseprojective}.
We write $\supp{[f, g]} \subsetneq \supp{f} \cup \supp{g}$ for the ``proper'' inclusion of each connected component of the support described above.
Let
\begin{align*}
W:=\{h \in \langle f, g \rangle \mid \supp{h} \subsetneq \supp{f} \cup \supp{g}, h \neq \mathrm{Id} \},
\end{align*}
where $\langle f, g \rangle$ is the subgroup generated by $f$ and $g$.
The set $W$ is not empty since $z$ is in $W$.
Define the function $\kappa: W \to \mathbb{Z}_{\geq 0}$ by setting
\begin{align*}
\kappa(h)= \# \left\{ i \in \{ 1, 3, \dots, 2m-1\} \mid \supp{h}\cap (c_i, c_{i+1}) \neq \emptyset \right\}.
\end{align*}
Here $\#$ denotes the cardinality of the set.
Let $z^\prime$ be a minimizer of $\kappa$ and let $e_1, \dots, e_{2p}$ denote a sequence given by Lemma \ref{lem_supp_finite_component} for $z^\prime$.
Since $z^\prime$ is not identity map, there exist $i$ and $i^\prime$ such that $c_{2i^\prime-1}<e_{2i-1}<e_{2i}<c_{2i^\prime}$ holds.
We note in the following that each element of $G_0(n)$ preserves the order of $\N^\mathbb{N}$.
Since there may exist more than one such $i$, we denote the smallest one by $i_-$ and the largest one by $i_+$.
Then there exists $w \in \langle f, g \rangle$ such that $w(e_{2{i_-} -1})>e_{2i_+}$ holds.
By construction of $w$, we have
\begin{align*}
\supp{z^\prime} \cap \supp{w^{-1}z^\prime w} \cap (c_{2i^\prime-1}, c_{2i^\prime})=\emptyset.
\end{align*}
This implies that we have
\begin{align*}
\supp{[z^\prime, w^{-1}z^\prime w]} \cap (c_{2i^\prime-1}, c_{2i^\prime})=\emptyset.
\end{align*}
Indeed, if $x$ is in $\supp{z^\prime}$, then $z^\prime(x)$ is in $\supp{z^\prime}$, thus $z^\prime(x)$ is not in $\supp{w^{-1}z^\prime w}$ and we have $[z^\prime, w^{-1}z^\prime w](x)=x$.
If $x$ is in $\supp{w^{-1}z^\prime w}$, then the image of $x$ by $w^{-1}z^\prime w$ is also in $\supp{w^{-1}z^\prime w}$. Since $z^\prime(x)=x$ holds, we have $[z^\prime, w^{-1}z^\prime w](x)=x$.
We note that
\begin{align*}
\supp{[z^\prime, w^{-1}z^\prime w]} \subsetneq \supp{z^\prime} \cup \supp{w^{-1}z^\prime w} \subsetneq \supp{f} \cup \supp{g}.
\end{align*}
However, $[z^\prime, w^{-1}z^\prime w]$ is not in $W$.
Indeed, if $\supp{z^\prime} \cap (c_{2j-1}, c_{2j}) =\emptyset$ for any other $j$, then we have
\begin{align*}
\supp{[z^\prime, w^{-1}z^\prime w]}\cap (c_{2j-1}, c_{2j})=\emptyset,
\end{align*}
since if $x$ is in $(c_{2j-1}, c_{2j})$ then $w(x)$ is also in $(c_{2j-1}, c_{2j})$.
Thus if $[z^\prime, w^{-1}z^\prime w]$ is in $W$, it contradicts the minimality of $\kappa(z^\prime)$.
This implies that we have
\begin{align*}
z^\prime(w^{-1}z^\prime w)=(w^{-1}z^\prime w)z^\prime.
\end{align*}
Therefore, these two maps $z^\prime$ and $(w^{-1}z^\prime w)$ generate the group which is isomorphic to $\mathbb{Z}^2$.
We have the desired result.
\end{proof}
\end{theorem}
\begin{remark} \label{remark_torsion_free}
From this proof, we can see that $G_0(n)$ is torsion free.
Indeed, let $g$ $(g \neq \textrm{Id})$ be in $G_0(n)$ and assume that $g^m= \textrm{Id}$ holds.
Since $g$ preserves the order of $\N^\mathbb{N}$, for $x$ in $\supp{g}$, we have either
\begin{align*}
x < g(x)<g^2(x)< \cdots < g^m(x)=x
\shortintertext{or}
x>g(x)>g^2(x)<\cdots <g^m(x)=x.
\end{align*}
This is a contradiction.
\end{remark}
\begin{remark}
If there exists an embedding from $G_0(n)$ into either $G_0$ or the Monod group $H$ \cite{monod2013groups}, the theorem follows immediately because $G_0$ and $H$ contain no free subgroups.
\end{remark}
\subsection{The abelianization of $G_0(n)$ and simplicity of the commutator subgroup}
The idea for the theorems in this section comes from \cite{brown1987finiteness} and \cite{burillo2018commutators}.
We note that the group $F(n)$ is called $F_{n, 1}$ in \cite[section 4D]{brown1987finiteness}.
Thus there exists a surjective homomorphism $\phi: F(n) \to \mathbb{Z}^n$.
We briefly recall its definition to compute the abelianization of $G_0(n)$.
Let $A_{n+1}$ be the free abelian group generated by $e_-$, $e_+$, and $e_i$ ($i \in \mathbb{Z}$) and satisfying the relations $e_i =e_j$ if $i \equiv j \pmod {(n-1)}$.
Then the rank of $A_{n+1}$ is $n+1$.
From an $n$-ary tree $Y$ and an $n$-ary tree Z which contains $Y$ as a rooted subtree, we define the element $\delta(Z, Y)$ in $A_{n+1}$ as follows:
\begin{enumerate}
\item Label the leftmost leaf of Y as $-$, the rightmost as $+$, and the other leaves as
$1, 2, \dots $ from left to right.
\item To construct $Z$ from $Y$, add a caret to some leaf of $Y$.
Then record the label of the leaf.
\item Regard the obtained tree as $Y$ again.
\item Repeat the process (1) to (3) until the tree $Z$ is obtained.
\item Add up all the $e_i$ that have the recorded labels as indices.
\end{enumerate}
Since $e_i=e_j$ if $i \equiv j$ mod $(n-1)$, we can add carets in any order in process (2).
For example, for the trees $Y$ and $Z$ in Figure \ref{definition_Y_and_Z}, by adding carets from left to right, $\delta(Y, Z)=e_-+e_7+e_{10}+e_+=e_-+2e_1+e_+$.
\begin{figure}[tbp]
\centering
\includegraphics[width=80mm]{definition_Y_and_Z.pdf}
\caption{The trees $Y$ and $Z$. }
\label{definition_Y_and_Z}
\end{figure}
The process of this example is in Figure \ref{calculation_delta_Y_Z}.
\begin{figure}[tbp]
\centering
\includegraphics[width=150mm]{calculation_delta_Y_Z.pdf}
\caption{A process of the calculation of $\delta(Y, Z)$. }
\label{calculation_delta_Y_Z}
\end{figure}
For the group $F(n)$, we define the homomorphism $\phi: F(n) \to A_{n+1}$ as follows:
Let $x$ be in $F(n)$ and $(T_+, T_-)$ be a tree diagram that represents $x$.
Then we set
\begin{align*}
\phi(x):=\delta(W, T_-)-\delta(W, T_+),
\end{align*}
where $W$ is an $n$-ary tree that contains both $T_+$ and $T_-$.
From the construction, this map is independent of the choice of $W$ and $(T_+, T_-)$, and this is a homomorphism.
By calculating $\phi(x_0), \dots, \phi(x_{n-2})$, and $\phi(\x{o}{(n-1)})$, we have that $\operatorname{Im} \phi =\{\Sigma\lambda_ie_i \mid \Sigma \lambda_i=0 \} \cong \mathbb{Z}^n$.
Thus we obtain a surjective homomorphism $F(n) \to \mathbb{Z}^n$.
\begin{theorem}[{\cite[Lemma 2.1]{burillo2018commutators}} for $n=2$] \label{Th_abelianization_G0(n)}
The abelianization of $G_0(n)$ is isomorphic to $\mathbb{Z}^{n+1}$.
\begin{proof}
We define the map $\pi: Z(n) \to \mathbb{Z}^n \bigoplus \mathbb{Z}$ by
\begin{align*}
&\x{i}{s} \mapsto (\phi(\x{i}{s}), 0), & &y_s \mapsto (\bm{0}, 1).
\end{align*}
Since $G_0(n)$ is $(n+1)$-generated group, its abelianization is a quotient of $\mathbb{Z}^{n+1}$.
So if we obtain a surjective homomorphism $G_0(n) \to \mathbb{Z}^{n+1}$, then this map must be its abelianization map.
Thus it is sufficient to show that $\pi$ extends to a homomorphism $G_0(n) \to \mathbb{Z}^{n+1}$.
In order to do this, we only need to check that the relations in $R(n)$ are satisfied, which is clear except for (4).
By the definition of $\phi$, we have $\phi(\x{0}{s})=\bm{0}$ for each $\x{0}{s}$, where $s$ is in $\N^{<\mathbb{N}}$ such that $y_s$ is in $Y(n)$.
Indeed, since $s$ is neither $0\cdots0$ nor $(n-1)\cdots(n-1)$, $\phi(\x{0}{s})$ is calculated as in Figure \ref{phi_x0s=0}.
\begin{figure}[tbp]
\centering
\includegraphics[width=140mm]{phi_x0s=0.pdf}
\caption{Note that $e_{s}=e_{s+(n-1)}$. }
\label{phi_x0s=0}
\end{figure}
Thus the relation (4) is also satisfied.
\end{proof}
\end{theorem}
\begin{corollary}
Let $n, m \geq 2$.
Then the groups $G_0(n)$ and $G_0(m)$ are isomorphic if and only if $n=m$ holds.
\end{corollary}
In the following, we show that the commutator subgroup $G_0(n)^\prime=[G_0(n), G_0(n)]$ is simple.
As in the case of $n=2$, first we show that the second derived subgroup $G_0(n)^{\prime \prime}=[G_0(n)^\prime, G_0(n)^\prime]$ is simple by using Higman's Theorem, and
then we show that $G_0(n)^{\prime \prime}=G_0(n)^{\prime}$.
Let $\Gamma$ be a group of bijections of a set $E$.
For $\alpha \in \Gamma$, we write the set-theoretic support $\supp{\alpha}$ for the set $\{x \in E \mid \alpha(x) \neq x\}$.
\begin{theorem}[{\cite[Theorem 1]{MR72136}}] \label{Theorem_Higman_simple}
Suppose that for every $\alpha, \beta, \gamma \in \Gamma \setminus \{ 1_\Gamma \}$, there exists $\rho$ such that
\begin{align*}
\gamma \Bigl( \rho \bigl(\supp{\alpha}\cup \supp{\beta}) \Bigr) \cap \rho \bigl(\supp{\alpha}\cup \supp{\beta}\bigr) = \emptyset
\end{align*}
holds.
Then the commutator subgroup $\Gamma^\prime$ is simple.
\end{theorem}
\begin{theorem}[{\cite[Theorem 2]{burillo2018commutators}} for $n=2$] \label{theorem_commutator_simple}
The commutator subgroup of the group $G_0(n)$ is simple.
\end{theorem}
The following two lemmas complete the proof.
\begin{lemma}
The second derived subgroup $G_0(n)^{\prime \prime}$ is simple.
\begin{proof}
Let $\alpha, \beta, \gamma \in G_0(n)^\prime$.
Choose $x \in \supp{\gamma}$.
If $\gamma(x)>x$, then let $I$ be the set $\{x^\prime \in \N^\mathbb{N} \mid x<x^\prime<\gamma(x)\}$.
If $\gamma(x)<x$, then let $I$ be the set $\{x^\prime \in \N^\mathbb{N} \mid \gamma(x)<x^\prime<x\}$.
Then $\gamma(I) \cap I = \emptyset$ holds.
We note that $\supp{\alpha}$ and $\supp{\beta}$ are not $\N^\mathbb{N} \setminus \{00\cdots, (n-1)(n-1) \cdots\}$, by the argument in Theorem \ref{Theorem_no_free_group} (in particular, see the order \eqref{support_condition_commutator}).
Therefore there exists $\rho \in F(n)$ such that
\begin{align*}
\rho \bigl(\supp{\alpha} \cup \supp{\beta}\bigr) \subset I
\end{align*}
holds.
Then we have
\begin{align*}
\rho \bigl(\supp{\alpha} \cup \supp{\beta} \bigr) \cap \gamma \Bigl(\rho \bigl(\supp{\alpha} \cup \supp{\beta} \bigr) \Bigr) \subset I \cap \gamma(I)=\emptyset.
\end{align*}
By Theorem \ref{Theorem_Higman_simple}, $G_0(n)^{\prime \prime}$ is simple.
\end{proof}
\end{lemma}
\begin{lemma}[{\cite[Proposition 2.5]{burillo2018commutators}} for $n=2$]
$G_0(n)^\prime=G_0(n)^{\prime \prime}$.
\begin{proof}
We assume that $n \geq 3$.
Since $G_0(n)^{\prime \prime} \subset G_0(n)^\prime$ holds, we show $G_0(n)^\prime \subset G_0(n)^{\prime \prime}$.
Let $g \in G_0(n)^\prime$ and $f \y{s_1}{t_1}\cdots \y{s_m}{t_m}$ be its normal form.
By the definition of the map $\pi$ in Theorem \ref{Th_abelianization_G0(n)}, $t_1+\cdots +t_m=0$ holds and $f$ is in $F(n)^\prime$.
By \cite[Theorem 4.13]{brown1987finiteness}, we have $F(n)^\prime = F(n)^{\prime \prime} \subset G_0(n)^{\prime \prime}$.
Therefore, it is sufficient to show that $\y{s_1}{t_1}\cdots \y{s_m}{t_m}$ is in $G_0(n)^{\prime \prime}$.
We show this by induction on $k=|t_1|+\cdots + |t_m|$ for words $\y{s_1}{t_1}\cdots \y{s_m}{t_m}$ that satisfy $t_1+\cdots t_m=0$ (which may not be in normal).
We note that $k$ is an even number.
Since the base case $k=0$ is clear, we assume $k>0$.
Then, since $G_0(n)^{\prime \prime}$ is a normal subgroup of $G_0$, by cyclically conjugating, we can assume that the word starts with a subword of the form $y_sy_t^{-1}$.
For example, we can do the following:
\begin{align*}
y_{s_1}y_{s_2}\y{s_3}{-1}\y{s_4}{-1} \mapsto \y{s_1}{-1}(y_{s_1}y_{s_2}\y{s_3}{-1}\y{s_4}{-1})y_{s_1}=y_{s_2}\y{s_3}{-1}\y{s_4}{-1}y_{s_1}.
\end{align*}
By the inductive hypothesis, it is sufficient to show that $y_s \y{t}{-1}$ is in $G_0(n)^{\prime \prime}$.
We divide the proof into two cases:
Case (1): $s \perp t$.
Since $(y_s y_t^{-1})^{-1}=y_t y_s^{-1}$ holds, we can assume that $s<t$ without loss of generality.
By the definition of $Y(n)$, $s$ and $t$ are neither $000\cdots$ nor $(n-1)(n-1)(n-1)\cdots$.
Then since $n \geq 3$, there exists an element $h^\prime$ in $F(n)$ such that
\begin{align*}
h^\prime((n-1)000)=s
\shortintertext{and}
h^\prime((n-1)00(n-1)0)=t
\end{align*}
hold.
Indeed, we can construct it similarly as in Figure \ref{construction_g_uv}.
Then we have
\begin{align*}
y_s y_t^{-1}=({h^\prime}^{-1}y_{(n-1)000} h^\prime)({h^\prime}^{-1}\y{(n-1)00(n-1)0}{-1} h^\prime)={h^\prime}^{-1}y_{(n-1)000} \y{(n-1)00(n-1)0}{-1} h^\prime.
\end{align*}
Thus it is sufficient to show that $y_{(n-1)000}\y{(n-1)00(n-1)0}{-1}$ is in $G_0(n)^{\prime \prime}$.
Let $w=y_{(n-1)00}\y{(n-1)0(n-1)}{-1} \in G_0(n)^\prime$.
We note that there exists an element $h$ in $F(n)^\prime$ such that
\begin{align*}
h\bigl((n-1)00(n-1)(n-1)\bigr)=(n-1)00
\shortintertext{and}
h\bigl((n-1)0(n-1)\bigr)=(n-1)0(n-1)
\end{align*}
hold.
Indeed, let $h$ be as in Figure \ref{definition_h_in_simple}.
Then since $\phi(h)=\bm{0}$ holds where $\phi$ is the abelianization map of $F(n)$, the element $h$ is in $F(n)^\prime$.
\begin{figure}[tbp]
\centering
\includegraphics[width=80mm]{definition_h_in_simple.pdf}
\caption{The element $h$. }
\label{definition_h_in_simple}
\end{figure}
Since $w$ is in $G_0(n)^\prime$ and $h$ is in $F(n)^\prime \subset G_0(n)^\prime$, the commutator $[w, h]=whw^{-1} h^{-1}$ is in $G_0(n)^{\prime \prime}$.
By the construction of $h$, we have $h w h^{-1}=y_{(n-1)00(n-1)(n-1)}\y{(n-1)0(n-1)}{-1}$.
Thus we have
\begin{align*}
[w, h]&=w (hwh^{-1})^{-1} \\
&=y_{(n-1)00}\y{(n-1)0(n-1)}{-1} (y_{(n-1)00(n-1)(n-1)}\y{(n-1)0(n-1)}{-1})^{-1}\\
&=y_{(n-1)00} \y{(n-1)00(n-1)(n-1)}{-1}.
\end{align*}
By applying expansion move (in Definition \ref{definition_five_moves}) to $y_{(n-1)00}$, we have
\begin{align*}
y_{(n-1)00} \y{(n-1)00(n-1)(n-1)}{-1}
&=\x{0}{(n-1)00}y_{(n-1)000}\y{(n-1)00(n-1)0}{-1}y_{(n-1)00(n-1)(n-1)}\y{(n-1)00(n-1)(n-1)}{-1} \\
&=\x{0}{(n-1)00}y_{(n-1)000}\y{(n-1)00(n-1)0}{-1}.
\end{align*}
Since $\x{0}{(n-1)00}$ is in $F(n)^\prime$ (see Figure \ref{phi_x0s=0}), we note that this element is in $F(n)^{\prime \prime} \subset G_0(n)^{\prime \prime}$.
Therefore we have
\begin{align*}
y_{(n-1)000}\y{(n-1)00(n-1)0}{-1}=\x{0}{(n-1)00}^{-1} [w, h] \in G_0(n)^{\prime \prime},
\end{align*}
as required.
Case (2): $s$ is a prefix of $t$ or vice versa.
Since $(y_s y_t^{-1})^{-1}=y_t y_s^{-1}$ holds, we can assume that $s$ is a prefix of $t$ without loss of generality.
Let $t=su$.
By expansion move, we have $y_s \y{su}{-1}=\x{0}{s}y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}\y{su}{-1}$.
Then since $\x{0}{s}$ is in $F(n)^\prime=F(n)^{\prime \prime} \subset G_0(n)^{\prime \prime}$ (see Figure \ref{phi_x0s=0}), it is enough to show that $y_{s0}\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}\y{su}{-1}$ is in $G_0(n)^{\prime \prime}$.
We further divide the proof of case (2) into two subcases.
Case (2-1): $0$ is a prefix of $u$.
We note that $s0 \perp s(n-1)0$ and $s(n-1)(n-1)\perp su$ hold.
Thus $y_{s0}\y{s(n-1)0}{-1}$ and $y_{s(n-1)(n-1)}\y{su}{-1}$ are in $G_0(n)^{\prime \prime}$, by case (1).
Case (2-2): $0$ is not a prefix of $u$.
By cyclically conjugating, it is sufficient to show that $\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}\y{su}{-1}y_{s0}$ is in $G_0(n)^{\prime \prime}$.
Since $su \perp s0$ holds, $\y{s(n-1)0}{-1}y_{s(n-1)(n-1)}$ and $\y{su}{-1}y_{s0}$ are in $G_0(n)^{\prime \prime}$, by case (1).
\end{proof}
\end{lemma}
\subsection{The center of $G_0(n)$}\label{subsec_center_G0(n)}
In this section, we show that the center of the group $G_0(n)$ is trivial.
The idea for the theorem comes from \cite[Section 4]{cannon1996introductory}.
Let $D(n):=\{s \overline{0} \mid s \in \N^{<\mathbb{N}}\}$, where $\overline{0}$ denotes the element $000\cdots $ in $\N^\mathbb{N}$.
Then the following holds.
\begin{lemma} \label{lemma_rich_Thompson_F(n)}
For any $s\overline{0}$ in $D(n)$, there exists $x$ in $F(n)$ such that
\begin{align*}
\supp{x}=(s\overline{0}, \overline{(n-1)})=\{\xi \in \N^\mathbb{N} \mid s\overline{0}<\xi< \overline{(n-1)}\}
\end{align*}
holds.
\begin{proof}
If $s=0, \dots, (n-2)$, then $x_0, \dots, x_{n-2}$ satisfy the claim, respectively.
By regarding $(n-1)\overline{0}$ as $(n-1)0\overline{0}$, we can assume that the length of $s$ is greater than or equal to $2$.
Let $s=s^\prime i \overline{0}$ ($i \in \{0, \dots, n-2 \}$).
If $s^\prime=(n-1)\cdots(n-1)$, then $\x{i}{s^\prime}$ satisfies the claim.
If $s^\prime \neq (n-1)\cdots(n-1)$, by using $\x{i}{s^\prime}$, we can define an element in $F(n)$ as in Figure \ref{rich_elements_Fn}, which satisfies the claim
\begin{figure}[tbp]
\centering
\includegraphics[width=90mm]{rich_elements_Fn.pdf}
\caption{The construction of an element of $F(n)$ from $\x{i}{s^\prime}$. We add a caret to each of the leaf $s(n-1)$ of the domain tree and the rightmost leaf of the range tree. }
\label{rich_elements_Fn}
\end{figure}
\end{proof}
\end{lemma}
We note that $D(n)$ is a dense subset of $\N^\mathbb{N}$.
\begin{theorem}[{\cite[Proposition 2.7]{burillo2018commutators}} for $n=2$] \label{theorem_center_trivial}
The center of $G_0(n)$ is trivial.
\begin{proof}
Let $f$ be an element of the center of $G_0(n)$.
For $g \in G_0(n)$, assume that $supp(g)=(b_1, \overline{(n-1)})$ holds.
Then we have $f(b_1)=b_1$.
Indeed, if not, either $f(b_1)>b_1$ or $f^{-1}(b_1)>b_1$ holds.
Since $g(b_1)=b_1$, in both cases, this contradicts that $fg=gf$ holds.
By Lemma \ref{lemma_rich_Thompson_F(n)}, for every $s\overline{0} \in D(n)$, there exists $g \in F(n) \subset G_0(n)$ such that $b_1=s \overline{0}$ holds.
Thus we have $f(s\overline{0})=s\overline{0}$ for every $s\overline{0} \in D(n)$.
Since $D(n)$ is a dense subset of $\N^\mathbb{N}$, we conclude that $f$ is the identity map.
\end{proof}
\end{theorem}
\subsection{Indecomposability with respect to direct products and free products}
In this section, we show that there exist no nontrivial ``decompositions'' using theorems of $G_0(n)$.
\begin{theorem}
There exists neither nontrivial direct product decompositions nor nontrivial free product decompositions.
\begin{proof}
Suppose that $G_0(n)$ is isomorphic to $K \times H$ for some groups $K$ and $H$.
We first assume that $H$ (or $K$) is abelian.
Then the center of $G_0(n)$ contains $\{1\} \times H$.
Since the center of $G_0(n)$ is trivial (Theorem \ref{theorem_center_trivial}), $H$ must be the trivial group.
We assume that $K$ and $H$ are not abelian.
We note that the commutator subgroup of $G_0(n)=K \times H$ is $[K, K] \times [H, H]$.
Since $[K, K]$ and $[H, H]$ are not trivial, the group $\{1\} \times [H, H]$ is a nontrivial normal subgroup of $[K, K] \times [H, H]$.
However, this contradicts that $[G_0(n), G_0(n)]$ is simple (Theorem \ref{theorem_commutator_simple}).
Finally, we assume that $G_0(n)=K \star H$ for nontrivial groups $K$ and $H$.
Let $k \in K \setminus \{1\}$ and $h \in H \setminus \{1\}$.
Since $G_0(n)$ is torsion free (Remark \ref{remark_torsion_free}), both $h$ and $k$ generate infinite cyclic groups $\langle k \rangle$ and $\langle h \rangle$, respectively.
This implies that $G_0(n)$ has a subgroup
\begin{align*}
\langle k \rangle \star \langle h \rangle \cong \mathbb{Z} \star \mathbb{Z}=F_2.
\end{align*}
By Theorem \ref{Theorem_no_free_group}, this is a contradiction.
\end{proof}
\end{theorem}
\subsection*{Acknowledgments}
I would like to appreciate Professor Motoko Kato and Professor Shin-ichi Oguni for several comments and suggestions.
I would also like to thank my supervisor, Professor Tomohiro Fukaya, for his comments and careful reading of the paper.
\bibliographystyle{plain}
| {'timestamp': '2022-04-19T02:40:14', 'yymm': '2204', 'arxiv_id': '2204.08230', 'language': 'en', 'url': 'https://arxiv.org/abs/2204.08230'} |
\section{Introduction}
\section{Thermodynamic formalism, introductory notions}\label{Introduction}
Among founders of this theory are
\cite{Sinai}, \cite{Bowen1975} and David Ruelle, who wrote in \cite{Ruelle}: ``thermodynamic formalism has been developed since G. W. Gibbs to describe [...] physical systems consisting of a large number of subunits''. In particular one considers a {\it configuration space} $\Om$ of functions $\Z^n\to \mbA$ on the lattice $\Z^n$ with interacting values in $\mbA$ over its sites, e.g.~``spin'' values in the Ising model of ferromagnetism. One considers probability distributions on $\Om$, invariant under translation, called {\it equilibrium states} for potential functions on $\Om$.
Given a mapping $f:X\to X$ one considers as a configuration space the set of trajectories $n\mapsto (f^n(x))_{n\in \Z_+}$ or $n\mapsto \Phi(f^n(x))_{n\in \Z_+}$ for a test function $\Phi:X\to Y$.
\
The following simple fact \cite[Lemma 1.1]{Bowen1975} and \cite[Introduction]{Ruelle}, \cite[Introduction]{PUbook}, resulting from Jensen's inequality applied to the function logarithm, stands at the
heart of thermodynamic formalism.
\begin{lemma}[Finite Variational Principle]\label{finite}
For given real
numbers $\phi_1,\dots,\phi_d$, the function
$F(p_1,\dots p_d):=\sum_{i=1}^n -p_i\log p_i+
\sum_{i=1}^d p_i \phi_i$ defined
on the simplex $\{(p_1,\dots ,p_d):p_i\ge 0, \sum_{i=1}^d p_i=1\}$
attains its maximum value
\noindent $P(\phi_1,\dots, \phi_d)=\log\sum_{i=1}^de^{\phi_i}$ at
and only at
$\hat p_j=e^{\phi_j}\bigl(\sum_{i=1}^de^{\phi_i}\bigr)^{-1}.$
\end{lemma}
We can read $i\mapsto \phi_i, i=1,\dots ,d$ as a {\it potential} function
and $\hat p_i$
as the equilibrium probability
distribution on the finite space $\{1,\dots ,d\}$.
$P(\phi_1,\dots, \phi_d)$ is called the {\it pressure} or
{\it free energy}, see \cite{Ruelle}.
\
Let $f:X\to X$ be a continuous mapping of a compact metric space $X$ and $\phi:X\to \mathbb{R}$ be a continuous function (the potential). We define the {\it topological pressure} or free energy by
\begin{definition}\label{top_pres}
\begin{equation}\label{var_pres}
P_{\rm var}(f,\phi)=
\sup_{\mu\in{\cM}(f)}\left( h_\mu(f)+\int_X \phi \,d\mu\right),
\end{equation}
where ${\cM}(f)$ is the set of all $f$-invariant Borel probability measures on $X$ and $h_\mu(f)$ is measure theoretical entropy. Sometimes we write $\cM(f,X)$.
\end{definition}
Recall that $h_\mu(f)=\sup_{\mbA} \lim_{n\to\infty} \frac1{n+1} \sum_{A\in {\mbA}^n} -\mu(A)\log\mu(A)$, where the supremum is taken over
finite partitions ${\mbA}$
of $X$, where ${\mbA}^n:=\bigvee_{j=0,\dots, n} f^{-j}{\mbA}$.
Notice that this resembles the sum $\sum_{i=1}^n -p_i\log p_i$ in Lemma \ref{finite}.
\smallskip
Topological pressure can also be defined in other ways, e.g.~by \eqref{Psep}, and then its equality to the one given by \eqref{var_pres} is called the variational principle. This explains the notation
$P_{\rm var}$.
Any $\mu\in{\cM}(f)$ for which the supremum in \eqref{var_pres} is attained is called {\it equilibrium}, {\it equilibrium measure} or {\it equilibrium state}.
\smallskip
A model case is any map $f:U\to \mathbb{R}^n$ of class $C^1$, defined on a neighbourhood $U$ of a compact set $X\subset \mathbb{R}^n$, {\it expanding} (another name: {\it uniformly expanding} or {\it hyperbolic} in dimension 1)
that is there exist $C>0, \lambda>1$
such that for all positive integers $n$ all $x\in X$ and all $v$ tangent to $\mathbb{R}^n$ at $x$,
\begin{equation}\label{expanding}
||Df^n(v)||\ge C\lambda^n ||v||,
\end{equation}
and {\it repelling} that is every forward trajectory sufficiently close to $X$ must be entirely in $X$. Not assuming the differentiability of $f$ one uses the notion of {\it distance expanding} meaning the increase of distances
under the action of $f$ by a factor at least $\lambda>1$ for pairs of distinct points sufficiently close to each other. Repelling happens to be equivalent to the internal condition: $f|_X$ being an open map,
provided $f$ is open on a neighbourhood of $X$, see \cite[Lemma 6.1.2]{PUbook}. Then the classical theorem holds, here in the version from \cite[Section 5.1]{PUbook}:
\begin{theorem} \label{Gibbs}
Let $f:X\to X$ be a distance expanding, topologically transitive continuous open map of a compact metric space $X$ and $\phi:X\to\mathbb{R}$ be a H\"older continuous potential. Then, there exists
exactly one measure $\mu_\phi\in\cM(f,X)$, called the {\it Gibbs} measure, satisfying
\begin{equation}\label{Gibbs-eq}
C<\frac{\mu_\phi(f_x^{-n}(B(f^n(x),r_0))}{\exp (S_n\phi(x)-nP(\phi))}<C^{-1}
\end{equation}
where $f_x^{-n}$ is the branch of $f^{-n}$ mapping $f^n(x)$ to $x$ (locally making sense, since $f$ is a local homeomorphism) and $S_n\phi(x):=\sum_{j=0}^{n-1}\phi (f^j(x))$.
The measure $\mu_\phi$ is the only equilibrium state for $\phi$.
It is equivalent to the unique $\phi$-conformal measure $m_\phi$, that is a forward quasi-invariant Borel probability measure $m_\phi$ with Jacobian
$\exp -(\phi-P(\phi))$. Moreover, the limit
$P(\phi)=P(f,\phi):=$
\noindent $\lim_{n\to\infty}\frac1n\log \sum_{x\in f^{-n}(x_0)}\exp S_n\phi(x)$ exists
and is equal to $P_{\var}(f,\phi)$ for every $x\in X$.
\end{theorem}
This $P(\phi)$ is a normalizing quantity corresponding to $P(\phi_1,\dots, \phi_d)$ in Lemma \ref{finite}
and the sum in the definition of $P(\phi)$
corresponds to the so called {\it statistical sum} over the space $\Om_n$ of all admissible configurations over $\{0,1,\dots,n-1\}$, as in the Ising model. Compare to the {\it tree pressure} defined in Definition \ref{treep}.
So $\varsigma:\Sigma^d\to\Sigma^d$, the shift to the left on the space
$\Sigma^d=\{(\a_0,\a_1,\dots ): \a_j\in\{1,\dots,d\}\}$, defined by $\varsigma((\a_n))=(\a_{n+1})$, is an example where Theorem \ref{Gibbs} holds. The sets $f_x^{-n}(B(f^n(x),r_0)$ correspond to {\it cylinders} of fixed
$\{\a_j\in\{1,\dots,d\}, j=0,\dots,n-1\}$. One can impose an admissibility condition:
$\a_i\a_{i+1}$ admissible if the pair has the digit 1 attributed in a defining 0,1 $d\times d$ matrix. Then one calls the system a {\it one-sided topological Markov chain}.
The condition of openness of $f$ can be replaced by a weaker one: the existence of a finite Markov partition, see \cite{PUbook}.
\smallskip
The existence of a conformal measure follows from the existence of a fixed point in the
convex weakly*-compact set of probability measures for the dual operator to the transfer (Perron-Frobenius-Ruelle) operator $\cL$ divided by the norm, where for $u:X\to\mathbb{R}$ continuous one defines
\begin{equation}\label{transfer}
\cL(u)(x):= \sum_{y\in f^{-1}(x)} u(y)\exp\phi(y).
\end{equation}
Indeed,
for every Borel set $Y\subset X$ on which $f$ is injective, denoting by $I_Y$ indicator function: 1 on $Y$, 0 outside $Y$, due to an approximation by continuous functions, one has for every finite Borel measure $\nu$ on $X$
\begin{equation}
(\cL^*(\nu))(Y))=\int_X \cL(I_Y) \,d\nu =\int_{f(Y)}\exp\phi\circ f|_Y^{-1}\,d\nu.
\end{equation}
Hence the (positive) eigen-measure $m_\phi$ has Jacobian for $(f|_Y)^{-1}$ equal to
$\exp(\phi\circ f|_Y^{-1})/\lambda$, hence $f$ has Jacobian
$\exp(-\phi)$ multiplied by an eigenvalue $\lambda:=\exp P(\phi)$.
The proof of the existence of an invariant Gibbs measure equivalent to $m_\phi$ is harder.
One first proves the existence of a positive eigenfunction $u_\phi$ for $\cL$ and then
defines $\mu_\phi=u_\phi m_\phi$.
For a more complete introduction to this theory, see e.g.~\cite{PUbook}.
\section{Introduction to dimension 1}\label{dim1}
Thermodynamic formalism is useful for studying properties of the underlying space $X$.
In dimension 1, for $f$ real of class $C^{1+\e}$ or $f$ holomorphic, for an expanding repeller $X$, considering $\phi=\phi_t:=-t\log |f'|$ for $t\in\mathbb{R}$, \eqref{Gibbs-eq} gives
\begin{align
\mu_{\phi_t}(f_x^{-n}(B(f^n(x),r_0)))\approx \exp (S_n\phi(x)-nP(\phi_t))\approx \label{diam}
\\
\diam f_x^{-n}(B(f^n(x),r_0))^{t}\exp -nP(\phi_t). \nonumber
\end{align}
The latter follows from a comparison of the diameter with the inverse of the absolute value of the derivative of $f^n$ at $x$, due to {\it bounded distortion}. Here, the symbol ``$\approx$'' denotes that the mutual ratios are bounded by a constant.
\smallskip
When $t=t_0$ is a zero of the function $t\mapsto P(\phi_t)$, this gives
\begin{equation}\label{Bowen}
\mu_{\phi_{t_0}} (B)\approx (\diam B)^{t_0}
\end{equation}
for all small balls $B$ (the $t_0$-Ahlfors measure property).
We obtain the so-called Bowen's formula for Hausdorff dimension:
\begin{equation}\label{HDformula}
\HD(X)=t_0.
\end{equation}
Moreover, the Hausdorff measure of $X$ in this dimension is finite and nonzero.
\smallskip
A model example of application is the proof of
\begin{theorem}\label{z2+c}
For $f_c(z):= z^2+c$ for an arbitrary complex number $c\not=0$ sufficiently close to 0,
the invariant Jordan curve $J$ (Julia set for $f_c$) is a fractal, i.e.~has Hausdorff dimension bigger than 1.
\end{theorem}
\begin{proof}[Sketch of Proof] $t_0>1$ yields $\HD(J)=t_0>1$ by \eqref{Bowen} (one does not need to use the invariance of $\mu_{\phi_{t_0}}$).
The case $t_0=1$ yields by \eqref{Bowen} finite Hausdorff measure in dimension 1, i.e.~the rectifiability of $J$.
To conclude that $J$ is a circle and $c=0$, one can use ergodic invariant measures in the classes of harmonic ones on $J$ from inside and outside. They must coincide.
This relies on Birkhoff's Ergodic Theorem, the heart of ergodic theory. This is an ``echo'' of the celebrated Mostov Rigidity Theorem. See \cite{Sullivan:82} and \cite[Theorem 9.5.5]{PUbook}.
\end{proof}
\bigskip
In dimension 1 (real or complex), we call $c$ a critical point if the derivative $f'(c)=0$. The set of critical points will be denoted by $\Crit(f)$.
\smallskip
In this survey, we allow for the presence of critical points and concentrate mainly on two cases:
\smallskip
1. (Complex case) $f$ is a rational mapping of degree at least 2 of the Riemann sphere $\ov{\C}$. We consider $f$ acting on its Julia set $K=J(f)$.
\smallskip
For entire or meromorphic maps see e.g.~\cite{BKZ1, BKZ2}, compare Definition \ref{hypdim}.
\smallskip
2. (Real case) $f$ is a {\it generalized multimodal map} defined on a neighbourhood $U_K\subset\bR$ of its compact invariant subset $K$. We assume that $f\in C^2$,
is non-flat at all of its turning and inflection critical points, satisfies the {\it bounded distortion} property for iterates, abbr. BD, see \cite{PrzRiv:14}, is topologically transitive and has positive
topological entropy on $K$.
We assume that $K$ is a maximal invariant subset of a finite union of pairwise disjoint closed intervals $\hat I=I^1\cup \dots \cup I^k \subset U_K$ whose endpoints are in $K$. (This maximality corresponds to the Darboux property, compare \cite[Appendix A]{PrzRiv:14} and \cite[page 49]{MiSzlenk}.)
We write $(f,K)\in {\cA}^{\BD}_+$, with the subscript + to mark positive entropy. In place of BD one can assume $C^3$ (and write $(f,K)\in {\cA}^3_+$), and assume that all periodic orbits in $K$ are hyperbolic repelling. Indeed, changing $f$ outside $K$ if necessary, one can get the corrected $(f,K)$ in ${\cA}^{\BD}_+$
\smallskip
Recall the notions concerning periodic orbits: {\it Parabolic} means $f^n(p)=p$ with $(f^n)'(p)$ being a root of unity. For $|(f^n)'(p)|=1$ the term {\it indifferent periodic} is used and for $|(f^n)'(p)|>1$ the term {\it hyperbolic repelling}. If $|(f^n)'(p)|<1$ the orbit is called {\it hyperbolic attracting}.
\smallskip
For the real setting, see \cite{PrzRiv:14}, \cite{GPR2} and \cite{Prz:16}. Examples are provided by basic sets in the spectral decomposition \cite{dMvS}.
\smallskip
{\bf Question.} Are there any other examples?
\smallskip
{\bf Problem}. Generalize the real case theory, see further sections, to the piecewise continuous maps, that is allow the intervals $I^j$ to have common ends (see \cite{HU} for some results in this direction).
\smallskip
In this survey, we compare equilibrium states to (refined) Hausdorff measures in the complex case. For the real case, we refer the reader to \cite{HKe} and the references therein.
\section{Hyperbolic potentials}\label{Hyperbolic potential}
For general $f:X\to X$ and $\phi:X\to \mathbb{R}$ as in Definition \ref{top_pres} the following conditions
are of special interest \cite{InoRiv:12},
1) $P(f,\phi)>\sup\phi$,
2) $P(f^n,S_n\phi)>\sup_{X} S_n\phi$
for an integer $n$,
3) $P(f,\phi)>\sup_{\nu\in\cM(f)}\int\phi\, d\nu$,
4) For each equilibrium state $\mu$ for the potential $\phi$, the entropy $h_\mu(f)$ is positive.
\smallskip
The conditions 2) -- 4) are equivalent, see \cite[Proposition 3.1]{InoRiv:12}.
Potentials $\phi$ satisfying them have been called in \cite{InoRiv:12} {\it hyperbolic}.
The condition 1) has longer traditions, see \cite{DenUrb:91}.
The intuitive meaning is that no minority of trajectories carries the full pressure.
For every $f:\ov\C\to\ov\C$ rational of degree at least 2 and $\phi:J(f)\to\mathbb{R}$ H\"older continuous,
the
following condition is also equivalent to 2) -- 4), see \cite{InoRiv:12}:
\smallskip
5) For each ergodic equilibrium state $\mu$ for $\phi$, the Lyapunov exponent $\chi(\mu):=\int \log |f'|\,d\mu$ is positive, that is for $\mu$-a.e. $x$, \
$\chi(\mu)=\chi(x):=\lim_{n\to\infty}\frac1n\log |(f^n)'(x)|>0.$
\smallskip
The conditions 2)-5) are also equivalent in the real case for $(f,K)\in\sA^{\BD}_+$ or $(f,K)\in\sA^3_+$ and all periodic orbits hyperbolic repelling. The arguments in \cite{InoRiv:12} work. See also \cite{RivLi2}.
\smallskip
\begin{theorem}\label{equi-hyp}
Let $f:\ov\C\to\ov\C$ be a rational mapping as above. If $\phi$ is a H\"older continuous hyperbolic potential on $J(f)$, then there exists a unique equilibrium state $\mu_\phi$. For every H\"older $u:J(f)\to\mathbb{R}$, the Central Limit Theorem (abbr. CLT) for the sequence of random variables $u\circ f^n$ and $\mu_\phi$ holds.
\end{theorem}
For a proof, see \cite{Prz:90} and preceding \cite{DenUrb:91}.
To find this equilibrium one can iterate the transfer operator proving
$\cL^n(1\!\!1)/\exp nP(f,\phi) \to u_\phi$. The convergence is uniformly $\exp-\sqrt{n}$ fast and the limit is H\"older continuous, \cite{DPU}. Finally, define $\mu_\phi:=u_\phi\cdot m_\phi$, as at the end of
Section \ref{Introduction}.
\begin{remark}\label{inducing}
Given $\mu_\phi$ a priori, an efficient way
to study it is an inducing method, see \cite{SzoUrbZdu:15}, i.e.~the use of a return map $A\ni x\mapsto f^{n(x)}(x)\in A$ for $A$ and $n(x)$ adequate to $\mu_\phi$. Then one proves even an exponential convergence
(with any $u$ H\"older in place of $1\!\!1$),
which yields exponential mixing, hence stochastic laws for $u\circ f^n$ for H\"older $u$,
e.g.~CLT, LIL, compare Sections \ref{LIL} and \ref{sec:acc}. See also Remark \ref{ind2}. The key feature is the exponential decay of $\mu_\phi(A_n)$, where
$A_n:=\{x\in A: n(x)\ge n\}$.
\smallskip
See also \cite{BT2} for the real case, and stronger \cite{RivLi1} and \cite{RivLi2} including also the complex case proving the exponential convergence to $u_\phi$, hence CLT and LIL.
See also \cite{SzoUrbZdu:14} for endomorphisms $f$ of higher dimensional complex projective space, where 1) is replaced by a stronger ``gap'' assumption.
\end{remark}
\
\section{Non-uniform hyperbolicity in real and complex dimension 1}
Here we discuss a set of conditions, valid in both the real and complex situations. Below we concentrate on the case of complex rational maps with $K=J(f)$, only remarking differences in the real case.
\smallskip
\noindent
(a) \ CE. \ {\it Collet-Eckmann condition.}\
There exist $\la_{CE} > 1$ and $C>0$ such that for every critical point $c$ in $J(f)$, whose forward orbit does not meet other critical points, for every $n \ge 0$ we have
$$
|(f^n)'(f(c))|\ge C \la_{CE}^n.
$$
Moreover, there are no parabolic (indifferent) periodic orbits.
\smallskip
\noindent
(b) \ CE2$(z_0)$. \ {\it Backward} or {\it second Collet-Eckmann condition at $z_0 \in J(f)$.}
There exist $\la_{CE2}>1$ and $C>0$ such that for every $n \ge 1$ and
every $w \in f^{-n}(z_0)$ (in a neighbourhood of $K$ in the real case)
$$
|(f^n)'(w)|\ge C \la_{CE2}^n.
$$
\smallskip
\noindent
(b') \ CE2. \ {\it The second Collet-Eckmann condition.} $\CE2(c)$ holds for all critical points $c$ not in the forward orbit of any other critical point.
\smallskip
\noindent
(c) \ TCE. \ {\it Topological Collet-Eckmann condition.}
There exist $M \ge 0, P \ge 1$, $r>0$ such that for every
$x\in K$ there exists a strictly increasing sequence of positive integers
$n_j$, $j=1,2,\dots$, such that $n_j \le P \cdot j$ and for each $j$ (and discs $B(\cdot)$ below understood in $\ov\C$ or $\mathbb{R}$)
\begin{equation}\label{TCE}
\#\{0 \le i < n_j: \Comp_{f^i(x)}f^{-(n_j-i)}B(f^{n_j}(x),r))
\cap\Crit(f) \not= \emptyset \} \le M,
\end{equation}
where in general $\Comp_z V$ means for $z\in V$ the component of $V$ containing $z$.
\smallskip
In the real case, one adds the condition that there are no parabolic periodic orbits, which is automatically true in the case of complex rational maps.
\smallskip
\noindent
(d) \ ExpShrink. \ {\it Exponential shrinking of components.}
There exist $\lambda_{\Exp}>1$ and $r>0$ such that for every $x \in K$,
every $n > 0$ and every connected component $W_n$ of $f^{-n}(B(x, r))$ for the disc (interval) $B(x,r)$ in $\ov\C$ (or $\mathbb{R}$), intersecting $K$
\begin{equation}\label{ExpShrink}
\diam (W_n) \le \lambda_{\Exp}^{-n}.
\end{equation}
\noindent
(e) LyapHyp ({\it Lyapunov hyperbolicity}).
There is a constant $\lambda_{\Lyap} > 1$ such that the Lyapunov exponent $\chi(\mu)$
of any ergodic measure $\mu\in\cM(f,K)$
satisfies $\chi(\mu)\ge \log \la_{\Lyap}$.
\smallskip
\noindent
(f) \ UHP. \ {\it Uniform Hyperbolicity on periodic orbits.}
There exists $\lambda_{\Per} > 1$ such that every periodic point $p
\in K$ of period $k \ge 1$ satisfies
$$
|(f^k)'(p)|\ge \lambda_{\Per}^k.
$$
\
We distinguish LyapHyp as the most adequate among these conditions to carry the name (strong) non-uniform hyperbolicity.\footnote{Then all H\"older continuous potentials are hyperbolic, see Condition 5) in Section \ref{Hyperbolic potential} and \cite{InoRiv:12}.}
\begin{theorem}\label{Nonunifhyp}
1. The conditions (c)--(f) and else (b) for some $z_0$
are equivalent in the complex case. In the real case, the equivalence also holds under the assumption of weak isolation (see the definition below).
2. In the complex case, the suprema over all possible constants $\lambda_{\Exp}$,
$\lambda_{CE2}$ (supremum over all $z_0$),
$\lambda_{\Per}$ and $\lambda_{\Lyap}$ coincide.
3. Both CE and CE2 imply (c)--(f).
4. If there is only one critical point in the Julia set in the complex case or if $f$ is $S$-unimodal on $K=I$ in the real case, i.e.~has just one turning critical point $c$ and negative Schwarzian derivative on $I\setminus \{c\}$,
then all
conditions above are equivalent to each other.
\end{theorem}
For more details, see \cite{PrzRivSmi:03}, \cite{Riv:12} and \cite{PrzRiv:14}.
\begin{definition}\label{weak-isolation}
$(f,K)\in\cA$ is said to be {\it weakly isolated}
if there exists
an open neighbourhood $U$ of $K$ in the domain of $f$
such that for every $f$-periodic orbit $O(p)\subset U$ is contained in $K$.
\end{definition}
In the complex case, we can replace \eqref{TCE} by
$$
\deg \Bigl (f^{n_j} \bigl |_{\Comp_x f^{-n_j}(B(f^{n_j}(x),r)) } \Bigr )
\le M'
$$
for a constant $M'$. In the real case, this condition is weaker than \eqref{TCE} since $f$ mapping $W_{n+1}$ into $W_n$ may happen not surjective. It can have folds, thus
truncating backward trajectories of critical points acquired before when pulling back.
\smallskip
In the real case, the proof of CE$\Rightarrow$TCE can be found in \cite{NP}. For the complex case, we refer the reader to \cite{PrzRoh1}.
\smallskip
The implication TCE$\Rightarrow$CE was proved in the complex case in \cite[Theorem 4.1]{P-Holder}. The proof used the idea of the ``reversed telescope'' by \cite{GraSmi1}.
In the real case, this implication was proved for $S$-unimodal maps in \cite{NowSands}.
In presence of more than one critical point this implication may be false, see \cite[Appendix C]{PrzRivSmi:03}.
\smallskip
{\bf Question.} Is this implication true for every $(f,K)\in\cA_+^{\BD}$ with one critical point, provided it is weakly isolated? See Definition \ref{weak-isolation}. It seems that the answer is yes.
\smallskip
Since the condition TCE is stated in purely topological terms (in the class of maps without indifferent periodic orbits), it is invariant under topological conjugacy.
So we obtain the following immediate corollary.
\begin{corollary}
All equivalent conditions listed above are invariant under topological conjugacies between $(f,K)$'s).
\end{corollary}
Another proof of the topological invariance of CE in the complex case was provided in \cite{PrzRoh2} with the use of Heinonen and Koskela criterion for quasi-conformality, \cite{HeiKos:95}.
Note that this topological invariance is surprising, as all the conditions except TCE are expressed in geometric-differential terms. I do not know how to express CE for unimodal maps of interval in the (topological-combinatorial)
kneading sequence terms.
\smallskip
An important lemma used here has been an estimate of an average distance in the logarithmic scale of every orbit from $\Crit(f)$, see \cite{DPU}. Namely
\begin{lemma}\label{av_dist}
\begin{equation}\label{average distance}
{\sum_{j=0}^n}{}' -\log |f^j(x)-c| \le Qn
\end{equation}
for a constant $Q>0$ every $c\in\Crit(f)$, every $x\in K$ and every integer
$n>0$. $\Sigma'$ means that we omit in the sum an index $j$ of smallest distance $|f^j(x)-c|$.
\end{lemma}
An order of proving the equivalences in Theorem \ref{Nonunifhyp} is
\noindent CE2$(z_0)\Rightarrow$ExpShrink$\Rightarrow$LyapHyp$\Rightarrow$UHP$\Rightarrow$CE2$(z_0)$ and separately
\noindent ExpShrink$\Leftrightarrow$TCE.
E.g. assumed UHP one proves CE2$(z_0)$ by ``shadowing'', compare the beginning
of Section \ref{sec:other}.
\section{Geometric pressure and equilibrium states}
We go back to topological pressure, Definition \ref{top_pres}, but for $\phi=-t\log |f'|$, $t\in \mathbb{R}$ in the complex $K=J(f)$ or real cases, where $\phi$ can attain the values $\pm\infty$ at the critical points of $f$. See the beginning of Section \ref{dim1}. We call it the geometric pressure, because it is useful in studying of geometry of the underlying space, e.g.~as in \eqref{HDformula} via equilibrium states for all $t$.
The definition of $P_{\var}(f, -t\log|f'|)$ in Definition \ref{top_pres} makes sense due to $\chi(\mu)\ge 0$ for all $\mu\in\cM(f)$, in particular due to the integrability of $\log |f'|$, see
\cite{Prz:93} and \cite[Appendix A]{Riv:12} for a simpler proof. We conclude that it is convex and monotone decreasing. We start by defining a quantity occurring equal to $P(t)=P_{\var}(t):=P_{\var}(f,-t\log|f'|)$, to explain its geometric meaning, compare with Section \ref{dim1}.
\begin{definition}[Hyperbolic pressure]\label{hyperbolic pressure}
$$
P_{\hyp}(t):= \sup_{X\in\cH(f,K)} P(f|_X,-t\log|f'|) ,
$$
where $\cH(f,K)$ is defined as the space of all compact forward $f$-invariant (that is $f(X)\subset X$)
hyperbolic subsets of $K$, repellers in $\mathbb{R}$.
\end{definition}
From this definition, it immediately follows that:
\begin{proposition}\label{hypdim}{\rm (Generalized Bowen's formula, compare \eqref{HDformula})}
The first zero $t_0$ of $t\mapsto P_{\hyp}(K,t)$ is equal
to the hyperbolic dimension $\HD_{\hyp} (K)$ of $K$, defined by
$
\HD_{\hyp} (K):=\sup_{X\in\cH(f,K)} \HD(X).
$
\end{proposition}
For the discussion $\HD_{\hyp}(J(f))$ vs $\HD(J(f))$, see \cite[Section 2.13]{Lyubich_ICM14}.
\medskip
Below we state Theorem \ref{equi2} proved in \cite{PrzRiv:11} in the complex setting and in
\cite{PrzRiv:14} in the real setting.
It extends \cite{BT1, PS} and \cite{IT}. See also impressive \cite{DT}.
\begin{figure}
\begin{minipage}[c]{\linewidth}
\centering
\begin{overpic}[scale=.35]{P_TCE1.pdf}
\put(102,23){\small$t$}
\put(52,32){\small$t_0$}
\put(30,93){\small$P(t)$}
\put(-15,63){\tiny$-\chi_{\rm sup}$}
\put(50,4){\tiny$-\chi_{\rm inf}$}
\end{overpic}
\hspace{0.5cm}
\begin{overpic}[scale=.35]{P_TCE2.pdf}
\put(102,23){\small$t$}
\put(44,32){\small$t_0$}
\put(60,32){\small$t_+$}
\put(30,93){\small$P(t)$}
\put(-15,63){\tiny$-\chi_{\rm sup}$}
\put(60,4){\tiny$-\chi_{\rm inf}$}
\end{overpic}
\hspace{0.5cm}
\begin{overpic}[scale=.35]{P_TCE3.pdf}
\put(102,23){\small$t$}
\put(44,32){\small$t_0=t_+$}
\put(30,93){\small$P(t)$}
\put(-15,63){\tiny$-\chi_{\rm sup}$}
\put(50,15){\tiny$-\chi_{\rm inf}$}
\end{overpic}
\caption{The geometric pressure: LyapHyp with $t_+=\infty$, \newline LyapHyp with $t_+<\infty$, and non-LyapHyp. This Figure is taken from \cite{GPR2}, see notation in Remarks below.}
\label{geometric-pressure}
\end{minipage}
\end{figure}
\begin{theorem}\label{equi2}
1. Real case, \cite{PrzRiv:14}. Let
$(f,K)\in {\sA}_+^3$
and let all $f$-periodic orbits in $K$ be hyperbolic repelling.
Then $P(t)$ is real analytic on the open interval bounded by the ``phase transition parameters'' $t_-$ and $t_+$. For every $t\in (t_-,t_+)$,
the domain where
\begin{equation}\label{hyplog}
P(t)>\sup_{\nu\in\cM(f)} -t\int\log|f'|\, d\nu,
\end{equation}
there is a unique invariant equilibrium state. It is ergodic and absolutely continuous with respect to an adequate conformal measure $m_{\phi_t}$ with the density bounded from below by a positive constant almost everywhere.
If furthermore $f$ is topologically exact on $K$ (that is for every $V$ an open subset of $K$ there exists $n\ge 0$ such that $f^n(V)=K$), then this measure is mixing, has exponential decay of correlations and it satisfies the Central Limit Theorem for Lipschitz gauge functions.
\smallskip
2. Complex case, \cite{PrzRiv:11}. The assertion is the same. One assumes a very weak expansion: the existence of arbitrarily small nice, or pleasant, couples and hyperbolicity away from critical points.
\end{theorem}
{\bf Remarks.}
1) $t_-$ and $t_+$ are called the phase transition parameters. Since $P(0)=h_{\rm top}(f)>0$, $t_-<0<t_+$, they need not exist; we say then they are equal to $-\infty$ and/or $+\infty$ respectively. $P(t)$ is linear to the left of $t_-$ and to the right of $t_+$, equal to $t\mapsto -t\chi_{\sup}$ where $\chi_{\sup}:=\sup_\nu \chi(\nu)$ and
$t\mapsto -t\chi_{\inf}$, where $\chi_{\inf}:=\inf_\nu\chi(\nu)$, respectively. Of course, $P(t)$ is not real-analytic at finite $t_-$ and $t_+$.
2) For $f(z)=z^2-2$, $f:[-2,2]\to[-2,2]$ (the Tchebyshev polynomial), we have $f(2)=2, f'(2)=4,
\chi(l)=\log 2$, where $l$ is the normalized length measure. We have $P(t)=\log 2-t\log 2$ for $t\ge -1$ and $P(t)=-t\log 4$ for $t\le -1$, so $t_-=-1$, $P(t)$ is non-differentiable at $t_-$ and for $t=-1$ there are two ergodic equilibrium states: Dirac at $z=2$ and $l$.
3) For any $f$ non-LyapHyp, $t_+=t_0<\infty$. However $t_+<\infty$ can happen even for $f$ LyapHyp, see \cite{MS2} and \cite{CR1, CR2}.
4) Notice that the condition \eqref{hyplog} is similar to the condition 3) from Section \ref{Hyperbolic potential}. For $f$ LyapHyp and $t>t_+$, no equilibrium state can exist, see \cite{InoRiv:12}.
5) For real $f$ as in Theorem \ref{equi2} satisfying LyapHyp and $K=\hat I$, we have $t_0=1$ and for $-\log|f'|$ we conclude that
a unique equilibrium state exists which is a.c.i.m.( that is: invariant absolutely continuous with respect to Lebesgue measure). In fact this assertions hold even for $t=t_0=t_+=1$ with very weak hyperbolicity properties e.g.~$|(f^n)'(f(c))|\to\infty$ for all $c\in\Crit(f)$, see
\cite{BRSvS} and \cite{ShenStrien}. For the complex case, see \cite{GraSmi:09} and stronger \cite{RivShen:14}.
\smallskip
\begin{remark}\label{ind2}
In the proof of Theorem \ref{equi2},
we use (compare with the Remark \ref{inducing}) a return map $F(x)=f^{n(x)}$ to a ``nice'' (Markov) domain. However unlike in \cite{SzoUrbZdu:15}, we do not use in the construction of this set the equilibrium measure $\mu_\phi$ because we do not know a priori that it exists. The construction is geometric. $F$ is an infinite Iterated Function System, more precisely the family of all branches of $F^{-1}$ is,
see \cite{MU} and \cite{Pesin} and references therein,
expanding due to the ``acceleration'' from $f$ to $F$. Then we consider an equilibrium state ${\tt P}$ for $(F,\Phi)$ where $\Phi(x):=\sum_{j=0}^{n(x)-1}\phi_t(f^j(x))$, and consider an equivalent conformal measure.
We propagate these measures to the Lai-Sang Young tower $\{(x,j): 0\le j<n(x)\}$ and
project by $(x,j)\mapsto f^j(x)$ to $K$.\footnote{For applications to decide the existence or nonexistence of a finite a.c.i.m. for maps of interval with flat critical points or for entire or meromorphic maps depending on the {\tt P}-integrability of the first return time, see papers by N.~Dobbs, B.~Skorulski, J.~Kotus, G.~\'Swi\c atek.}
Stochastic properties of ${\tt P}$ stay preserved along the construction to $\mu_\phi$. The analyticity of $P(t)$ follows from expressing $P(t)$ as
zero of a pressure for $F$ with potential depending on two parameters and Implicit Function Theorem. The latter idea came from \cite{StratUrb}.
\end{remark}
\begin{remark}
For probability measures $\mu_n$ weakly* convergent to some $\hat\mu$, in presence of critical points
$\int\log |f'|\,d\mu_n$ need not converge to $\int\log |f'|\,d\hat\mu$. Only upper semicontinuity holds. Therefore, for $t>0$, the equilibrium states for $t_n\to t$ need not converge to an equilibrium state for $t$. A priori, the free energy in the Definition \ref{top_pres} can jump down. However, a modification of this method to prove existence of equilibria works, see \cite{DT}.
Notice also that passing to a weak*-limit with averages of Dirac measures on $\{x, \dots, f^n(x)\}$ proves
$\limsup_{n\to\infty}\sup_{x\in K}\frac1nS_n(\log|f'|)(x) \le \chi_{\max}$. However an analogous inequality $\liminf \dots \ge\chi_{\inf}$ is obviously false. These observations contribute to the understanding of Lyapunov spectrum.
\end{remark}
\smallskip
{\bf Remarks on the Lyapunov spectrum.} Theorem \ref{equi2} allows us to express the so-called dimension spectrum for Lyapunov exponents
with the use of Legendre transform, that is for all $\a>0$ and $\cL(\a):=\{x\in K: \chi(x)=\a\}$
\begin{equation}\label{Lyap_spec}
\HD(\cL(\a))=
\frac{1}{|\alpha|} \inf_{t\in\bR}
\left(P(t)+\alpha t \right).
\end{equation}
An ingredient is {\bf Ma\~n\'e's equality}
\begin{equation}\label{mane}
\HD(\mu)=h_\mu(f)/\chi(\mu)
\end{equation}
provided $\chi(\mu)>0$, \cite{PUbook}, where $\HD(\mu):=\sup\{\HD(X): \mu(X)=1\}$, applied to $\mu_{\phi_t}$.
The equality
\eqref{Lyap_spec} concerns regular $x$'s, where $\chi(x)=\lim_{n\to\infty} \frac1n\log |(f^n)'(x)|$
exists. It is also possible to provide formulas or at least estimates for Hausdorff dimension of
the sets of irregular points $\cL(\a,\b):=\{x\in K: \underline{\chi}(x)=\a, \overline{\chi}(x)=\b\}$
for lower and upper Lyapunov exponents where we replace lim by $\liminf$ and $\limsup$ respectively.
See \cite{GPR1} and \cite{GPR2} for this theory in complex and real settings.
However, these papers give no information about the size of sets with zero (upper) Lyapunov exponent.
Note at least that if $J(f)\not=\ov\C$ then ${\rm Leb}_2\{x\in J(f):\ov\chi(x)>0\}=0$. This is so because $\ov\chi(x)>0$ implies there exists $\sN\subset \Z_+$ of positive upper density, such that for $n\in\sN$, \eqref{ExpShrink} and \eqref{TCE} hold, see \cite[Section 3]{LPS}.
We do not know whether $\chi(x)=-\infty$ can happen for $x$ not pre-critical, except there is only one critical point in $K$, where $\chi(x)>-\infty$ follows from \eqref{average distance}, see \cite[Lemma 6]{GPR1}.
For $x$ being a critical value we can prove (in analogy to $\chi(\mu)\ge 0$):
\begin{theorem}[\cite{LPS}]\label{LPShen}
If for a rational function $f:\ov\C\to\ov\C$ there is only one critical point $c$ in $J(f)$ and no parabolic periodic orbits, then $\underline\chi(f(c))\ge 0$.
\end{theorem}
For $S$-unimodal maps of interval this was proved
by \cite{NowSands}.
\section{Other definitions of geometric pressure}\label{sec:other}
\begin{definition}[safe]\label{safe} See \cite[Definition 12.5.7]{PUbook}. We call $z\in K$
\textit{safe} if $z\notin \bigcup_{j=1}^\infty(f^j(\Crit(f)))$ and for every $\delta>0$ and all $n$ large enough
$B(z, \exp (-\delta n))\cap \bigcup_{j=1}^n(f^j(\Crit(f)))=\emptyset$.
\end{definition}
Notice that this definition implies that all points except at most a set of Hausdorff dimension 0, are safe.
\begin{definition}[Tree pressure]\label{treep}
For every $z\in K$ and $t\in \mathbb{R}$ define
\begin{equation}\label{treep-formula}
P_{\tree}(z,t)=\limsup_{n\to\infty}\frac1n\log\sum_{f^n(x)=z,\, x\in K} |(f^n)'(x)|^{-t}.
\end{equation}
\end{definition}
Compare with $P(f,\phi)$ from Theorem \ref{Gibbs}. Under suitable conditions, e.g.~for $z$ ``safe''
the limit exists, it is independent of $z$ and equal to $P(t)$. See \cite{Prz:99}, \cite{PrzRivSmi:03} and \cite{PUbook} for the complex case and \cite{PrzRiv:14} and \cite{Prz:16} for the real case.
A key is to extend all trajectories $T_n(x)=\{x,\dots,z\}$ backward and forward by time $m\ll n$ to get an Iterated Function System for $f^{n+m}$ and to consider its limit set. Its trajectories for time $n$ ``shadow'' $T_n(x)$. This proves $P_{\tree}(z,t)\le P_{\hyp}(t)$. The opposite inequality is immediate.
(Similarly one proves $P_{\var}(t)\le P_{\hyp}(t)$. Given $\mu$ with $\chi(\mu)>0$ one captures a hyperbolic $X$ by Pesin-Katok method.)
\medskip
For a continuous potential $\phi:X\to\mathbb{R}$, consider
\begin{equation}\label{Psep}
P_{\rm {sep}}(f,\phi):=\lim_{\e\to 0}\limsup_{n\to\infty}\frac1n \log \big(\sup_Y \sum_{y\in Y}\exp S_n\phi(y)\big),
\end{equation}
where the supremum is taken over all $(n,\e)$-separated sets $Y\subset X$, that is such $Y$ that for every distinct $y_1,y_2\in Y$, $\rho_n(y_1,y_2)\ge \e$, where $\rho_n$ is the metric defined by $\rho_n(x,y)=\max\{\rho(f^j(x),f^j(y)): j=0,\dots,n\}$.
For $\phi=-t\log|f'|$ for positive $t$, in presence of critical points for $f$, $P_{\rm {sep}}$ is always equal to $\infty$ by putting a point of a separated set at a critical point.
So we replace it by the tree pressure. One can however use infimum over $(n,\e)$-spanning sets, thus defining \underline{$P_{\rm {spanning}}(f,\phi)$}. This is a valuable notion, often coinciding with other pressures. See \cite{Prz:16} for an
outline of a respective theory. Let me mention only that this is equal to $P(f,-t\log|f'|)$ for $t>0$ in the complex case if
\begin{definition}\label{wbls}
$f$ is {\it weakly backward Lyapunov stable} which means that for every $\delta>0$ and $\e>0$ for all $n$ large enough and every disc $B=B(x,\exp -\delta n)$ centered at $x\in K$, for every $0\le j \le n$ and every component $V$ of $f^{-j}(B)$ intersecting $K$, it holds that $\diam V\le\e$.
\end{definition}
This holds for all rational maps with at most one critical point whose forward trajectory is in $J(f)$ or is attracted to $J(f)$, due to Theorem \ref{LPShen}.
\smallskip
{\bf Question.} Does backward weak Lyapunov stability hold for all rational maps?
\smallskip
Finally, {\it periodic pressure} $P_{\Per}$ is defined as $P_{\tree}$ with $x\in \Per_n$ (periodic of period $n$) rather than $f^n$-preimages of $z$. In \cite{PrzRivSmi:04}, this was proved for rational $f$ (see also \cite{BMS} for a class of polynomials) on $K=J(f)$ that $P_{\Per}(t)=P(t)$ provided
\smallskip
{\bf Hypothesis H}.\ For every $\d>0$ and all $n$ large enough, if for a set
$\cP\subset\Per_n$ for all
$p,q\in P$ and all $i:0\le i<n$ \
$\dist (f^i(p),f^i(q))< \exp -\d n$, then
$\# \cP\le \exp\d n$.
\smallskip
{\bf Question.}
Does this condition always hold? In particular, can large bunches of periodic orbits exist with orbits exponentially close to a Cremer fixed point?
\section{Geometric coding trees, limit sets, Gibbs meets Hausdorff}\label{sec:gct}
The notion of geometric coding tree, g.c.t., already appeared in the work \cite{Ja}, where in the expanding case the finite-to-one property of the resulting coding was proved. It was used later
in \cite{Prz:85, Prz:86} and in a full strength in \cite{PUZ:89, PUZ:91} and papers following them. Similar graphs have since been constructed to analyse the topological aspects of
non-invertible dynamics, see for instance \cite{Nek, HaPi}.
\begin{definition}\label{gct}
Let $U$ be an open connected subset of the Riemann sphere $\ov\C$.
Consider a holomorphic mapping $f:U\to \ov\C$ such that $f(U)\supset U$ and
$f:U\to f(U)$ is a proper map.
Suppose that $\Crit(f)$ is finite.
Consider an arbitrary $z\in f(U)$. Let $z^1,z^2,\dots,z^d$ be some
of the $f$-preimages of $z$ in $U$ with $d\ge 2$. Consider smooth
curves $\g^j:[0,1]\to f(U)$, \ $j=1,\dots,d$, joining $z$ to $z^j$ respectively
(i.e.~$\g^j(0)=z, \g^j(1)=z^j$), intersections allowed, such that $\g^j\cap f^n(\Crit(f))=\emptyset$ for every $j$ and $n>0$.
For every sequence
$\a=(\a_n)_{n=0}^\infty \in \S^d$ (shift space with left shift map $\varsigma$ defined
in Section \ref{Introduction}) define $\g_0(\a):=\g^{\a_0}$. Suppose that for some $n\ge
0$, for every $ 0\le m\le n$, and all $\a\in\S^d$, curves $\g_m(\a): [0.1] \to U$ are already defined.
Suppose that for $1\le m\le n$ we have $f\circ \g_m(\a)=\g_{m-1}(\varsigma(\a))$,
and $\g_m(\a)(0)=\g_{m-1}(\a)(1)$.
Define the curves $\g_{n+1}(\a) $ so that the previous
equalities hold by taking respective $f$-preimages of curves $\g_n$.
For every $\a\in\S^d$ and $n\ge 0$ denote $z_n(\a):=\g_n(\a)(1)$.
The graph $\sT={\sT}(z,\g^1,\dots,\g^d)$ with the vertices $z$ and $z_n(\a)$ and edges $\g_n(\a)$ is
called a {\it geometric coding tree} with the root at $z$. For every $\a\in\S^d$ the subgraph
composed of $z,z_n(\a)$ and $\g_n(\a)$ for all $n\ge 0$ is called an {\it infinite geometric branch}
and denoted by $b(\a)$.
It is called {\it convergent} if the sequence $\g_n(\a)$
is convergent to a point in $\cl U$. We define the {\it coding map}
$z_\infty :{\sD}(z_\infty)\to \cl U$ by $z_\infty(\a):=\lim_{n\to\infty}z_n(\a)$ on the
domain ${\sD}={\sD}(z_\infty)$ of all such $\a$'s for which $b(\a)$ is convergent.
Denote $\La:=z_\infty({\sD}(z_\infty))$. If the map $f$ extends
holomorphically to a neighbourhood of its closure $\cl\Lambda$ in $\ov\C$, then $\La$ is called a {\it quasi-repeller},
see \cite{PUZ:89}.
A set formally larger than $\cl \Lambda$ is of interest, namely $\widehat\Lambda$ being the set of all accumulation points of $\{z_n(\alpha): \alpha\in\Sigma^d\}$ as $n\to\infty$. If our g.c.t. is in $\Omega$ being an RB-domain, see Section \ref{sec:bound}, or $f$ is just $R\circ g \circ R^{-1}$ defined only on $\Omega$, see Remarks below, then it is easy to see that $\cl \Lambda=\widehat\Lambda$. I do not know how general this equality is.
\end{definition}
{\bf Remarks.}
Given a Riemann map $R:\D\to \Om$ to a connected simply connected domain $\Om\subset\C$, (i.e. holomorphic bijection)
we can consider a branched covering map, say $g(z)=z^d$ on $\D$,
and $f=R\circ g\circ R^{-1}$. Then, chosen $z\in \Om$ and $\g^j$ joining it with its preimages in $\Om$ (close to $\Fr\Om$) we can consider the respective tree $\sT$.
Then instead of considering $R$ and its radial limit $\ov R$, we can consider
the limit (along branches) $z_\infty: \S^d\to\Fr\Om$.
This provides a structure of symbolic dynamics useful to verify stochastic laws.
This is especially useful if considered measures come from $\partial \D$ via $\ov R$, rather than being some equilibrium states for potentials living directly on $\Fr \Om$.
This is the case of harmonic measure $\omega$ which is the $\ov R_*$-image of a length measure $l$.
We can consider the lift of $l$ to $\Sigma^d$ via coding by the tree $\sT'=R^{-1}(\sT)$ and next its projection by $(z_\infty)_*$ to $\Fr\Om$.
Our g.c.t.'s are always available in presence of adequate holomorphic $f$, even in the absence of
$\Om$,
i.e.~in the absence of a Riemann map. The tree with the coding it induces yields a discrete generalization/replacement of a Riemann map.
It was proved in \cite{PSkrzy} that $\sD$ is the whole $\Sigma^d$ except a ``thin'' set.
In particular, for a Gibbs measure $\nu$ for a H\"older potential, $z_\infty(\a)$ exists for $\nu$-a.e.~$\a$, hence the push forward measure $(z_\infty)_*(\nu)$ makes sense. Moreover, our codings $\z_\infty$ are always ``thin''-to-one. This is a discrete generalization of Beurling's Theorem concerning the boundary behaviour of Riemann maps. ``Thin'' means of zero logarithmic capacity type, depending on the properties of the tree
(the speed of the accumulation of $\gamma^j$ by critical trajectories; the speed does not matter if we replace ``thin'' by zero Hausdorff dimension).
In particular this coding preserves the entropies.
\smallskip
For appropriate $\nu\in \cM(\varsigma,\Sigma^d)$ and $\psi:\Sigma^d\to\mathbb{R}$ with $\int\psi\,d\nu=0$,
consider the {\it asymptotic variance} (of course one can consider spaces more general than $\Sigma^d$)
\begin{equation}\label{s2}
\s^2=\sigma^2_\nu(\psi):=\lim_{n\to\infty}\frac1n\int (S_n\psi)^2\, d\nu.
\end{equation}
\begin{theorem}\label{tree-HD}
Let $\La$ be a quasi-repeller for a geometric coding tree
for a
holomorphic map $f:U\to\ov\C$. Let $\nu$ be a $\varsigma$-invariant Gibbs
measure on $\S^d$
for a H\"older continuous real-valued function $\phi$ on $\S^d$. Assume $P(\varsigma,\phi)=0$.
Consider $\mu:= (z_\infty)_*(\nu)$.
Then, for $\psi:= -\HD(\mu)(\log|f'|\circ z_\infty))-\phi$, we have $\int\psi\,d\nu=0$.
If the asymptotic variance $\s^2=\s^2_\nu(\psi)$ is positive,
then there exists a compact $f$-invariant hyperbolic repeller $X$ being a subset of $\La$ such that
$\HD (X)>\HD(\mu)$. In consequence $\HD_{\hyp}(\La)>\HD(\mu)$ (defined after \eqref{Lyap_spec}).
If $\sigma^2=0$ then $\psi$ is cohomologous to 0. Then for each $x,y\in \cl\La$ not postcritical, if
$z=f^n(x)=f^m(y)$ for some positive integers $n,m$, the orders of criticality of $f^n$ at $x$ and $f^m$ at $y$ coincide. In particular all critical points in $\cl\Lambda$ are pre-periodic.
\end{theorem}
The latter condition
happens only in special situations, see e.g. Theorem \ref{Z1-maximal} below.
See \cite{SzoUrbZdu:15}
for more details; $\phi$ lives there directly on $J(f)$, but it does not make substantial difference. See also Section \ref{sec:acc}.
\smallskip
Given a mapping $f:X\to X$, given two functions $u,v:X\to\mathbb{R}$ we call $u$ {\it cohomologous} to $v$ in class $\cC$ if there exists $h:X\to \mathbb{R}$ belonging to $\cC$ such that $u-v=h\circ f-h$.
An important \cite[Lemma 1]{PUZ:89} says that $\sigma^2=0$ above implies $\psi$ cohomologous to 0 in $L^2(\mu)$ and often in a smaller class depending on $\psi$ (Liv\v{s}ic type rigidity).
\smallskip
Notice that $\int\psi\,d\nu=-\HD(\mu)\chi(\mu)-\int\phi\,d\nu=-h_\mu(f)-\int\phi\,d\nu=
-h_\nu(\varsigma)-\int\phi\,d\nu=P(\varsigma,\phi)=0$.
Now, to prove Theorem \ref{tree-HD} note
$2\chi(\mu)\ge h_{\mu}(f)=h_\nu(\varsigma)>0$, see \cite[Ruelle's inequality]{PUbook} (used also to 3)$\Rightarrow$5) in Section \ref{Hyperbolic potential}) and \cite{Prz:85}.
So considering the natural extension of $(\Sigma^d,\nu,\varsigma)$ (here two-sided shift space) and Katok-Pesin theory, we find hyperbolic $X$ with
$\HD(X)\ge \HD(\mu)-\e$ for an arbitrary $\e>0$. Compare comments on shadowing in Section \ref{sec:other}.
\smallskip
$\bullet$ \ The positive $\s^2$ yields by Central Limit Theorem large fluctuations of the sums $\sum_{j=0}^{n-1}\psi\circ\varsigma^j$ from $n\int\psi\,d\nu$ (here 0), allowing to find $X$ with $\HD(X)>\HD(\mu)$.
A special care is needed to get $X\subset \La$,
see \cite{Prz:05} (originated in \cite{PZ:94}).
\smallskip
The above fluctuations were used by A. Zdunik to prove for constant $\phi$
\begin{theorem}[\cite{Z1}]\label{Z1-maximal} Let $f:\ov\C\to\ov\C$ be a rational mapping of degree
$d\ge 2$. If $\sigma^2>0$,
then for $\mu_{\max}(f)$ the measure of maximal entropy (equal $\log d$), $\HD(J(f))>\HD(\mu_{\max}(f))$. Otherwise, $f$ is postcritically finite with a parabolic orbifold, \cite{Milnor}.
\end{theorem}
She proved in fact the existence of a hyperbolic $X\subset J(f)$ satisfying
$\HD (X)>\HD(\mu_{\max}(f))$, hence $\HD_{\hyp}(J(f))>\HD(\mu_{\max}(f))$.
\smallskip
$\bullet$ \ In the $\sigma^2=0$ case,
$v:J(f)\to \mathbb{R}$ satisfying the \underline{cohomology equation} $\log|f'|=v\circ f -v +\Const$ on $J(f)$ extends to a harmonic function beyond $J(f)$ (Liv\v{s}ic rigidity) giving this equality on the union of real analytic curves containing $J(f)$ (called {\it real case}) or to $\ov\C$. In Theorem \ref{tree-HD} on $\Lambda$ and for the extension beyond, in Theorem \ref{Z1-maximal},
the ``orders'' of growth of $-\log |(f^n)'|$ at $x$ and of $-\log |(f^m)'|$ at $y$ must by cohomology equation be equal to the ``order'' of growth of $v$ at $z$, so they
must coincide (a phenomenon ``conjugated'' to the presence of an invariant line field).
This implies parabolic orbifold for Theorem \ref{Z1-maximal}.
\smallskip
Theorem \ref{Z1-maximal} applied to a polynomial $f$ with connected Julia set, by
\noindent $\HD(\mu_{\max}(f))=1$ \cite{Manning},
implies the following celebrated result:
\begin{theorem}[ A. Zdunik \cite{Z1}]\label{Z1-polynomial}
For every polynomial $f$ of degree at least 2, with connected Julia set,
either $J(f)$ is a circle or an interval or else
it is fractal, namely $\HD(J(f))>1$.
\end{theorem}
\section{Boundaries, radial growth, harmonic vs Hausdorff}\label{sec:bound}
For polynomials with connected Julia set the measure $\mu_{\max}(f)$ coincides with harmonic measure $\omega$ (viewed from $\infty$).
This led to another proof of Theorem \ref{Z1-polynomial}, especially the $\sigma^2=0$ part, see \cite{Z2}, in the language of boundary behaviour of Riemann map and harmonic measure (compare also model Theorem \ref{z2+c} ).
Theorem \ref{Z1-polynomial} has been strengthened from this point of view in
\cite{Prz:06}, preceded by \cite{PZ:94}, as follows.
\begin{theorem}\label{HD>1}
Let $f:\ov\C\to\ov\C$ be a rational map of degree at least 2 and $\Om$ be
a simply connected immediate basin of attraction to an attracting periodic orbit (that is a connected component of the set attracted to the orbit, intersecting it).
Then, provided $f$ is not a finite Blaschke product in some holomorphic coordinates, or
a two-to-one holomorphic factor of a Blaschke product,
$\HD_{\hyp}(\Fr\Om)>1$.
\end{theorem}
The novelty was to show how to ``capture'' a large hyperbolic $X$ in $\Fr\Om$ in the case it was not the whole $J(f)$.
In fact the following ``local'' version of this theorem was proved in \cite{Prz:06}
\begin{theorem}\label{local-HD>1}
Assume that $f$ is defined and holomorphic on
a neighbourhood $W$ of $\Fr\Om$, where $\Omega$ is a connected, simply connected domain in $\ov\C$ whose
boundary has at least 2 points. We assume that $f(W\cap \Om)\subset \Om$,\
$f(\Fr\Om)\subset\Fr\Om$ and $\Fr\Om$ repels
to the side of $\Om$, that is $\bigcap_{n=0}^\infty
f^{-n}(W\cap\cl\Om)=\Fr\Om$. Then either $\HD_{\hyp}(\Fr(\Om))>1$ or
$\Fr\Om$ is a real-analytic Jordan curve or arc.
\end{theorem}
$\Om$ with $f$ as above has been called an {\it RB-domain} (repelling boundary), introduced in \cite{Prz:86, PUZ:89}. Theorem \ref{local-HD>1} (at least the $\sigma^2>0$ part) follows directly from Theorem \ref{tree-HD}. Let $R:\D\to\Omega$ be a Riemann map and $g: W'\to\D$ be defined by $g:=R^{-1}\circ f \circ R$ on $W'=R^{-1}(W\cap\Om)$.
We consider a g.c.t. ${\sT}=\sT(z,\g^1,\dots,\g^d)$ with $z$ and $\g^j$ in $W\cap\Om$, sufficiently close to $\Fr\Omega$ that the definition makes sense, and with $d=\deg f|_{W\cap\Om}$,
(the situation is the same as in Remarks in Section \ref{sec:gct} above, but the order of defining $f$ and $g$ is different). Consider the g.c.t. ${\sT}'=R^{-1}(\sT)$. The function $g$ extends holomorphically beyond the circle $\partial \D$ and it is expanding.
Hence
$\phi:\Sigma^d\to \mathbb{R}$ defined by $\phi(\alpha)=-\log |g'|\circ (R^{-1}(z))_\infty(\a)$ for the tree ${\sT}'$ is H\"older continuous. Let $\nu=\nu_\phi$.
Note that here $P(\phi)=0$, e.g. since by expanding property of $g$ on $\partial\D$ there exists
$\hat l\in\cM(g)$, equivalent to length measure $l$ (a.c.i.m.).
Then $\nu$ is the lift of $\hat l$ to $\Sigma^d$ with the use of $\sT'$.
Note that our $\mu=z_\infty(\nu)$ is equal to $\hat\omega=\ov{R}_*(\hat l)$ which is $f$-invariant, equivalent to harmonic measures $\omega$ on $\Fr \Om$ viewed from $\Om$.
Note that $\HD(\hat\omega)=1$ due to Ma\~n\'e's equality, \eqref{mane}, $h_{\hat\omega}(f)=h_{\hat l}(g)$, see \cite{Prz:85, Prz:86}, and the equality of Lyapunov exponents $\int\log|f'|\,d\hat\omega = \int \log |g'|\,d{\hat l} > 0$. The latter equality holds due to the equality for almost every $\zeta\in \partial\D$:
\begin{equation}\label{radial1}
\lim_{r\to 1}\frac {\log |(f^n)'(R(r\z))|-\log |(g^n)'(r\z))}{\log (1-r)} =\lim_{r\to 1}
\frac{-\log |R'(r\z)|}
{\log (1-r)}=0.
\end{equation}
The first equality is proved using $f\circ R=R\circ g$ in $\D$, first applying $R$ close to $\partial \D$, next by iterating $f$ applying $R^{-1}$ well inside $\Om$, finally iterating $g$ back. The latter equality relies on the harmonicity of $\log|R'|$ allowing to replace its integral on circles by its value at the origin. For details see \cite{Prz:86}. Remind however that in fact $\HD(\omega)=1$ holds in general, see \cite{Makarov}.
The sketch of Proof of Theorem \ref{local-HD>1} for $\sigma^2>0$ is over. That $\sigma^2=0$ implies the analyticity of $\Fr\Om$ was already commented at the beginning of this Section.
\section{Law of Iterated Logarithm refined versions}\label{LIL}
Applying Law of Iterated Logarithm (abbr. LIL) to $\psi: \Sigma^d\to \mathbb{R}$ the fluctuations of $S_n\psi$ from 0 which follow lead to, see \cite{PUZ:89} and \cite{PUbook},
\begin{theorem}\label{LIL-refined-HD}
In the setting of Theorem \ref{tree-HD}
if $\s^2=\s^2_\nu(\psi)>0$, for $c(\mu):=\sqrt{2\s^2/\chi(\mu)}$, $\kappa:=\HD(\mu)$ and
$\a_c(r):=r^\kappa\exp(c\sqrt{\log 1/r\log\log\log 1/r})$
1) $\mu \bot H_{\a_c}$, that is singular with respect to the refined Hausdorff measure, \cite[Section 8.2]{PUbook} for the gauge function $\a_c$), for all $0<c<c(\mu)$;
2) $\mu \ll H_{\a_c}$, that is absolutely continuous, for all $c>c(\mu)$.
\end{theorem}
Indeed, substituting in LIL $n\sim (\log 1/r_n)/\chi(\mu)$
for $r_n = |(f^n)'(z)|^{-n}$, we get for $\mu$-a.e. $z$
\begin{equation}\label{sing-vs-cont}
\limsup_{n\to\infty}\frac{\mu(B(z,r_n)}{\a_c(r_n)}=\inft
\ {\rm for} \ 0<c<c(\mu) \
\ \ {\rm and}\ \ \dots =0 \ {\rm for} \ c>c(\mu).
\end{equation}
This is called the Refined Volume Lemma, \cite[Section 4]{PUZ:89} and, the harder case: $c>c(\mu)$, \cite[Section 5]{PUZ:91}.
\medskip
We can apply the assertion of Theorem \ref{LIL-refined-HD} for $\mu=\hat\omega\in\cM(f,\Fr\Om)$ equivalent to a harmonic measure $\omega$ as in Section \ref{sec:bound}.
\smallskip
This yields refined information about the radial growth of the derivative of Riemann maps, following the proof of \eqref{radial1}:
\begin{theorem}\label{radial growth}
Let $\Om$ be a simply connected RB-domain in $\ov\C$ with non-analytic boundary and $R:\D\to\Om$ be a Riemann map. Then there exists $c(\Om)>0$ such that for Lebesgue a.e. $\z\in\partial \D$
\begin{equation}\label{radial}
G^+(\z):=\limsup_{r\to 1}\frac{\log |R'(r\z)|}{\sqrt{\log(1/1-r)\log\log\log(1/1-r)}}=c(\Omega).
\end{equation}
Similarly $G^-(\z):=\liminf \dots= -c(\Omega)$. Finally $c(\Omega)=c(\hat\omega)$ in
Theorem \ref{LIL-refined-HD}.
\end{theorem}
In fact Theorem \ref{LIL-refined-HD} for $\mu=\hat\omega$ and Theorem \ref{radial growth} hold for every connected, simply connected open $\Om\subset\C$, together with $c(\Omega)=c(\hat\omega)$. No dynamics is needed. Of course one should add to both definitions
${\rm ess}\sup$ over $\z\in\partial\D$ and over $z\in \Fr\Om$ (for $c(z)=c(\omega)$ calculated from \eqref{sing-vs-cont}, see \cite[Th. 8.6.1]{PUbook} ) respectively, since in the absence of ergodicity these functions need not be constant. See \cite[Th. VIII.2.1 (a)]{GaMa} and references to Makarov's breakthrough papers therein, in particular \cite{Makarov}.
There is a universal Makarov's upper bound $C_{\rm M}<\infty$ for all $c(\Omega), c(\hat\omega)$'s in \eqref{radial}.
The best upper estimate I found in literature is $C_{\rm M}\le 1.2326$, \cite{HK}. I proved in \cite{Prz:89} a much weaker estimate, using a natural method of representing $\log |R'|$ by a series of weakly dependent random variables leading to a martingale on $\partial\D$, thus satisfying LIL. Unfortunately consecutive approximations resulted with looses in the final estimate.
For a holomorphic expanding repeller $f:X\to X$ and a H\"older continuous potential $\phi:X\to X$, the asymptotic variance for the equilibrium state $\mu=\mu_{t_0\phi}$ for every $t_0\in \mathbb{R}$ satisfies Ruelle's formula (see \cite{PUbook}):
\begin{equation}\label{second_derivative}
\sigma^2_\mu(\phi-\int\phi\,d\mu)=\frac{d^2P(t\phi)}{dt^2} \biggl |_{t=t_0}.
\end{equation}
{\bf Question.} Does \eqref{second_derivative} hold for all rational maps and hyperbolic potentials on Julia sets? For all simply connected RB-domains, $f:\Fr\Om\to\Fr\Om$ and $\mu=\hat\omega$?
\smallskip
For a simply connected RB-domain $\Om$ for $f$ and for $\phi=-\log|f'|$, if $g(z)=z^d$ (e.g.~$\Om$ being the basin of $\infty$ for a polynomial $f$), one considers the {\it integral means spectrum} depending only on $\Om$,
\begin{equation}
\beta_{\Om}(t):= \limsup_{r\to 1}\frac1{|\log (1-r)|}\log\int_{\z\in\partial\D}|R'(r\z)|^t\, |d\z|
\end{equation}
which happens to satisfy $\beta_{\Om}(t)=t-1+\frac{P(t\phi)}{\log d}$, see e.g.~\cite[Eq. (9.6.2.)]{PUbook}.
For $t_0=0$ we have $\mu=\hat\omega$ and the left hand side of \eqref{second_derivative} can be written as $(\frac12 \log d)\sigma^2(\log R')$,
see \eqref{s2} and \eqref{radial1}, where
$$\sigma^2(\log R'):=
\limsup_{r\to 1} \frac{\int_{\partial\D}|\log R'(t\z)|^2\, |d\z|}{-2\pi \log(1-r)|}.
$$
So \eqref{second_derivative} changes to $\sigma^2(\log R')=2\frac{d^2\beta_{\Om}(t))}{dt^2}|_{t=0}$,
compare \cite{Ivrii}.
It has an analytic, non-dynamical, meaning. It is also
related to the Weil-Petersson metric, see \cite{McM}.
\section{Accessibility}\label{sec:acc}
Let us recall the following theorem from \cite{Prz:94}.
\begin{theorem}\label{access}
Let
$\La$ be a quasi-repeller for a geometric coding tree
for a
holomorphic map $f:U\to\ov\C$. Suppose that
\begin{equation}\label{shrink}
\diam (\g_n(\a)) \to 0 , \ \hbox{as} \ n\to
\infty
\end{equation}
uniformly with respect to $\a\in\S^d$.
Then every {\it good} $q\in\widehat\La$ (defined in Section \ref{sec:gct}) is a limit of a convergent
branch $b(\a)$. So $q\in\Lambda$. In particular, this holds for every $q$ with $\underline{\chi}(q)>0$ and the local backward inviariance (explained below).
\end{theorem}
For the definition of ``good'', see \cite[Definition 2.5]{Prz:94}. It roughly says that there are many integers $n$ (positive lower density) for which $f^n$ properly map small domains $D_{n,0}$ in $U$ close to $q$ onto large $D_n\subset U$, giving ``telescopes'' ${\rm Tel}_k$ with ``traces'' $D_{n_k,0}\subset D_{n_{k-1},0}\subset\dots\subset D_{n_1,0}\subset D_0$; for each $k$ the choices may be different.
A part of this condition that $D_{n,0}\subset U$ can be called a ``local backward invariance'' of $U$ along the forward trajectory of $q$.
When $U$ is an immediate basin of attraction of an attracting fixed point for a rational map $f$ or just an RB-domain then this theorem asserts that $q$ is an endpoint of a continuous curve in $U$. This is a generalization of the Douady-Eremenko-Levin-Petersen theorem where $q$ is a repelling periodic point and the domain is completely invariant, e.g. basin of attraction to $\infty$ for $f$ a polynomial.
Due to this theorem we can prove that invariant measures of positive Lyapunov exponents lift to $\Sigma^d$.
More precisely, the following holds:
\begin{corollary} Every non-atomic hyperbolic probability measure $\mu$ \ (i.e. $\chi(\mu)>0$), on $\widehat\La$, is the $(z_\infty)_*$ image of a probability $\varsigma$-invariant measure $\nu$ on $\Sigma^d$, assumed \eqref{shrink}, $\sT$ has no self-intersections and else $\mu$-a.e. local backward invariance of $U$,.
In particular, $\nu$ exists for every RB-domain which is completely (i.e. backward) invariant.
\end{corollary}
\begin{proof} (the lifting part missing in \cite{Prz:94} and \cite{Prz:06}). By Theorem \ref{access} $\mu$ is supported on $\Lambda$ i.e. on $z_\infty(\sD(z_\infty))$. The lift of $\mu$ to $\mu'$ on the pre-image $\cB'$ under $z_\infty$ of the Borel $\sigma$-algebra of subsets of $\La$ can be extended to a $\varsigma$-invariant $\nu$ on $\cB$ the Borel $\sigma$-algebra of the subsets of $\Sigma^d$ by using the fact
that the set of at least triple points (limit points of at least three infinite branches of $\sT$) is countable,
hence $z_\infty^{-1}(x)$ of $\mu$-a.e $x$ contains at most 2 points. More precisely, let $A_1$ be the set
of points having one $z_\infty$-preimage, $A_2$ two preimages.
They are both $f$-invariant (except measure 0), so are their $z_\infty$-preimages $A'_1$ and $A'_2$ under $\varsigma$.
We extend $\mu'$ by distributing conditional measures on the two points preimages of points in $A_2$ half-half and Dirac on one point preimages.
\end{proof}
This allows to conclude Theorem \ref{LIL-refined-HD} (a part relying on CLT) and Theorem \ref{tree-HD}
for equilibrium states for rational maps and H\"older
potentials on $J(f)$ by lifting $\mu_\phi$ to $\Sigma^d$ as in \cite{Prz:06}.
However, this
seems useless since the proof of CLT in \cite{Prz:06} is done directly on $J(f)$ (seemingly also for LIL, for which one should however refer to the proofs in \cite{PUZ:89}) and there are direct proofs of LIL in \cite{RivLi2} and \cite{SzoUrbZdu:15}.
\subsection*{Acknowledgments}
Thanks to H. Hedenmalm, O. Ivrii, J. Rivera-Letelier, M. Sabok, M. Urba\'nski, and A. Zdunik for comments and corrections.
\bibliographystyle{amsplain}
| {'timestamp': '2018-06-19T02:05:06', 'yymm': '1806', 'arxiv_id': '1806.06186', 'language': 'en', 'url': 'https://arxiv.org/abs/1806.06186'} |
\section{Introduction}\label{sec: intro}
Let us consider the following continuous multidimensional white noise model:
\begin{equation}\label{eq:Mdl}
{\small Y(t_1,\ldots,t_d)=\sqrt{n}\int_{0}^{t_1}\ldots\int_{0}^{t_d} f(s_1,\ldots,s_d)\; ds_d \ldots ds_1 + W(t_1,\ldots,t_d),}
\end{equation}
for $(t_1,\ldots,t_d) \in [0,1]^d$ ($d \ge 1$), where $\{Y(t_1,\ldots,t_d): (t_1,\ldots,t_d)\in[0,1]^d \}$ is the observed data, $f \in L_1([0,1]^d)$ is the unknown (regression) function of interest and $W(\cdot)$ is the unobserved $d$-dimensional Brownian sheet (see Definition~\ref{sec:BS}), and $n$ is a known scale parameter. Estimation and inference in this model is closely related to that of nonparametric regression based on sample size $n$. We work with this white noise model as this formulation is more amiable to rescaling arguments; see e.g.,~\citet{Donoho1992},~\citet{DC01},~\citet{Carter2006}.
In this paper we develop {\it optimal} tests (in an asymptotic minimax sense) based on a newly proposed {\it multidimensional multiscale statistic} (i.e., $d \ge 1$) for testing:
\begin{itemize}
\item[(i)] $f=0$ versus a H\"{o}lder class of functions with unknown degree of smoothness;
\item[(ii)] $f=0$ against alternatives of the form $f=\mu_n \mathbb{I}_{B_n}$, where $B_n$ is an unknown hyperrectangle in $[0,1]^d$ with sides parallel to the coordinate axes (i.e., axis-aligned) and $\mu_n \in \R$ is unknown (for different regimes of $\mu_n$ and $B_n$).
\end{itemize}
Scenario (i) arises quite often in nonparametric regression where the goal is to test whether the underlying $f$ is 0 versus $f \ne 0$ with unknown smoothness; see e.g.,~\citet{Lepski2000},~\citet{Horowitz2001},~\citet{Ingster2009multi} and the references therein. Our proposed multiscale statistic, which extends the work of~\citet{DC01} that considered the analogous statistic for $d=1$, leads to rate optimal detection in this problem. Moreover, with the knowledge of the smoothness of the underlying $f$, we construct a {\it asymptotically minimax test} which even attains the exact separation constant (see Section~\ref{Sec 1.2} for formal definitions and related concepts).
Setting (ii) is a prototypical problem in signal detection --- an unknown (constant) signal spread over an unknown hyperrectangular region --- and the goal is to detect the presence of such a signal; see e.g., \citet{Castro2005},~\citet{chan2009},~\citet{Walther10},~\citet{munkchangepoint},~\citet{Butucea2013},~\citet{scanalr},~\citet{Glaz2004},~\citet{Munk2018} for a plethora of examples and applications.
Although several minimax rate optimal tests have been proposed in the literature for this problem (see e.g.,~\citet{Castro2005}, \citet{chan2009}, \citet{Butucea2013} and \citet{Munk2018}), as far as we are aware, our proposed multiscale test is the only test that attains the exact separation constant --- this leads to simultaneous optimal detection of signals both at small and large scales.
We first motivate and introduce our multiscale statistic below (Section~\ref{secmultiscale}) and briefly describe the asymptotic minimax testing framework and our main optimality results in Section~\ref{Sec 1.2}.
\subsection{Multiscale statistic when $d \ge1$}\label{secmultiscale}
To motivate our multiscale statistic let us first look at the following testing problem:
\begin{equation} \label{test}
H_0: f=0 \quad \mbox{ versus } \quad H_1: f \neq 0 \in \mathbb{H}_{\beta,L},
\end{equation}
where $\mathbb{H}_{\beta,L}$ is the H\"{o}lder class of function with parameters $\beta >0$ and $L>0$. For $\beta \in (0,1]$ and $L>0$ the H\"{o}lder class $\mathbb{H}_{\beta,L}$ is defined as
{\small \begin{equation}\label{eq:H_b_L}
\mathbb{H}_{\beta,L}:= \Big \{f\in L_1([0,1]^d): |f(x) - f(y)| \leq L \norm{x-y}^\beta \mbox{ for all } x,y \in [0,1]^d \Big\}.
\end{equation} }
For $\beta > 1$ the H\"{o}lder class $\mathbb{H}_{\beta,L}$ is defined similarly; see Definition~\ref{def:holder}.
Our multiscale statistic is based on the idea of {\it kernel averaging}. Suppose that $\psi:\mathbb{R}^d \to \R$ is a measurable function such that (i) $\psi$ is $0$ outside $[-1,1]^d$; (ii) $\psi \in L_2(\R^d)$, i.e., $\int_{\R^d} \psi^2(x) dx <\infty$; (iii) $\psi$ is of bounded (HK)-variation (see Definition~\ref{totalvar}); and (iv) $\int_{\R^d} \psi(x) dx > 0$. We call such a function a {\it kernel}. For any $h:=(h_1,\ldots,h_d) \in (0,1/2]^d$ we define \begin{equation}\label{ah}
A_h:=\{t\in\mathbb{R}^d : h_i \leq t_i \leq 1-h_i \;\;\mbox{ for } i =1,\ldots,d\}.
\end{equation}
For any $t \in A_h$ we define the centered (at $x$) and scaled kernel function $\psi_{t,h}:[0,1]^d \to \R$ as
{\small \begin{equation}\label{kerest}
\psi_{t,h}(x) :=\psi \left(\frac{x_1-t_1}{h_1},\ldots,\frac{x_d-t_d}{h_d}\right), \quad \mbox{for }\; x=(x_1,\ldots, x_d) \in [0,1]^d.
\end{equation} }
For a fixed $t \in A_h$ we can construct a kernel estimator $\hat{{f_h}}(t)$ of $f(t)$ based on the data process $Y(\cdot)$ as
\begin{equation*}\label{kerequ-0}
\hat{{f_h}}(t):= \frac{1}{n^{1/2} (\Pi_{i=1}^dh_i) \langle 1,\psi\rangle} \int_{[0,1]^d} \psi_{t,h}(x)dY(x),
\end{equation*}
where for any two functions $g_1,g_2 \in L_2(\R^d)$, we define $\langle g_1,g_2 \rangle:=\int_{\R^d} g_1(x)g_2(x) dx.$
We consider the {\it normalized} version of the above kernel estimator $\hat{{f_h}}(t)$:
\begin{equation}\label{kerest2}
\hat{\Psi}(t,h):=\frac{1}{(\Pi_{i=1}^dh_i)^{1/2}\norm{\psi}}\int_{[0,1]^d} \psi_{t,h}(x) dY(x),
\end{equation}
where $\norm{\psi}^2 := \int_{\R^d} \psi^2(x) dx <\infty$. We can use $\hat{\Psi}(t,h)$ to test $$H_0: f(t)=0 \quad \mbox{ versus } \quad H_1: f(t) \neq 0 $$ where we would reject the null hypothesis for extreme values of $\hat{\Psi}(t,h)$. So, a naive approach to testing \eqref{test} could be to consider $\sup_{ t\in A_h} |\hat{\Psi}(t,h)|$. As this test statistic crucially depends on the choice of the smoothing bandwidth vector $h$, an approach that bypasses the choice of the tuning parameter $h$ and also combines information at various bandwidths would be to consider the test statistic
\begin{equation}\label{eq:Scan}
\sup_{h>0} \sup_{ t\in A_h} |\hat{\Psi}(t,h)|,
\end{equation}
where $h>0$ is a short-hand for $h \in (0,\infty)^d$. However, under the null hypothesis $$\sup_{h>0} \sup_{ t\in A_h} |\hat{\Psi}(t,h)|=\infty \qquad \mbox{almost surely (a.s.).}$$ This is because, for a fixed scale $h$, $\sup_{t \in A_h}|\hat{\Psi}(t,h)| = $ $O_p(\sqrt{2\log(1/(2^d h_1 \cdots h_d))})$; see~\citet{gine02}. Thus, to use the above approach to construct a valid test for~\eqref{test} we need to put the test statistics $\sup_{ t\in A_h} |\hat{\Psi}(t,h)|$ at different scales (i.e., $h$) in the same footing --- this leads to the following definition of the {\it multiscale statistic} in $d$-dimensions:
\begin{equation}\label{eq:TZ}
T(Y,\psi):=\sup_{h\in(0,1/2]^d} \sup_{t \in A_{h}} \frac{\lvert \hat{\Psi}(t,h)\rvert - \Gamma(2^dh_1 \ldots h_d)}{D(2^dh_1 \ldots h_d)}
\end{equation}
where $\Gamma,D:(0,1] \to (0,\infty)$ are two functions defined as
\begin{equation*}
\Gamma(r):=(2 \log(1/r))^{1/2}
\end{equation*}
and
\begin{equation*}
D(r):= (\log(e/r))^{-1/2} \log \log (e^e/r).
\end{equation*}
In Theorem \ref{Thm 1}, a main result in this paper, we show that the above multivariate multiscale statistic $T(Y,\psi)$ is well-defined and finite a.s.~for any kernel function $\psi$, when $f\equiv0$. This result immediately extends the main result of~\citet[Theorem 2.1]{DC01} beyond $d=1$. Although there has been several proposals that extend the definition and the optimality properties of the multiscale statistic of~\citet{DC01} beyond $d=1$ (see e.g.,~\citet{Munk2018},~\citet{Walther10},~\citet{scanalr}), we believe that our proposed multiscale statistic is the right generalization. Further, the exact form of $T(Y,\psi)$ leads to optimal tests for~\eqref{test} and other alternatives (which the other competing procedures do not necessarily yield; see Remarks~\ref{rem:v} and~\ref{v3} for more details).
To show the finiteness of the proposed multiscale statistic we prove a general result about a stochastic process with sub-Gaussian increments (Theorem~\ref{Thm0}) on a pseudometric space which may be of independent interest. This result has the same conclusion as that of~\citet[Theorem 6.1]{DC01} but assumes a weaker condition on the packing numbers of the pseudometric space on which the stochastic process is defined. This weaker condition on the packing numbers is crucial to the proof of Theorem~\ref{Thm 1}; see Remark~\ref{connection to dumbgen} where we compare our result with the existing result of~\citet[Theorem 6.1]{DC01}. Moreover, Lemma~\ref{lem1} gives a tighter bound on the packing numbers of the pertinent (to our application) pseudometric space, which we believe is also new; see Remarks~\ref{rem:pack} and \ref{rem:v} where we compare our result with some relevant recent work.
\subsection{Optimality of the multiscale statistic}\label{Sec 1.2}
Before we describe our main results let us first introduce the asymptotic minimax hypothesis testing framework. There is an extensive literature on nonparametric testing of the simple hypothesis $\{0\}$. As a staring point we refer the readers to \citet{Ingsterbook}. In the nonparametric setting it is usually assumed that $f$ belongs to a certain class of functions $\mathbb{F}$ and its distance from the null function $f = 0$ is defined by a seminorm $ \norm{\cdot}$. In this setting, given $\alpha \in (0,1)$, the goal is to find a level $\alpha$ test $\phi_n$ (i.e., $\E_0 [\phi_n(Y)] \le \alpha$) such that
\begin{equation}\label{testinginf}
\inf_{g \in \mathbb{F} : \norm{g} \geq \delta \rho_n} \E_{g} [\phi_n(Y)]
\end{equation}
is as large as possible for some $\delta >0$ and $\rho_n>0$ where $\rho_n \to 0$ as $n \to \infty$ ($\rho_n$ is a function of the sample size $n$); in the above notation $\E_g$ denotes expectation under the alternative function $g$. However, it can be shown that given $\mathbb{F}$ and $\norm{\cdot}$, the constants $\delta$ and $\rho_n$ cannot be chosen arbitrarily if one wants to have a statistically meaningful framework (see the survey papers \citet{Ingster19931}, \citet{Ingster19932}, \citet{Ingster19933} for $d=1$ and \citet{Ingster2009multi} for $d>1$). It turns out that if $\delta \rho_n$ is too small then it is not possible to test the null hypothesis with nontrivial asymptotic power ({i.e., the infimum in~\eqref{testinginf} cannot be strictly larger than $\alpha + o(1)$}). On the other hand if $\delta \rho_n$ is very large many procedures can test $f\equiv 0$ with significant power (i.e., the infimum in~\eqref{testinginf} goes to $1$ as $n \to \infty$).
The hypothesis testing problem then reduces to: (a) finding the largest possible $\delta \rho_n$ such that no test can have {nontrivial asymptotic power} (i.e., under the alternative $f$ such that $\norm{f}\leq \delta \rho_n $, the asymptotic power is less than or equal to the level $\alpha$), and (b) trying to construct test procedures that can detect signals $f$, with $\norm{f} \geq \delta \rho_n$, with considerable power (power going to $1$ as $n \to \infty$). More specifically, $\delta$ and $\rho_n$ are defined such that $\delta \rho_n$ is the largest for which, for all $\epsilon>0$, we have \begin{equation*}
\sup_{\phi_n} \limsup_{n \to \infty} \inf_{g \in \mathbb{F} : \norm{g} \geq (1-\epsilon)\delta \rho_n} \E_{g} [\phi_n(Y)] \leq \alpha,
\end{equation*}
where the supremum is taken over all sequence of level $\alpha$ tests $\phi_n$. In this case $\rho_n$ is called the {\it minimax rate of testing} and {$\delta$} is called the {\it exact separation constant} (see \citet{Lepski2000}, \citet{Ingster2011sobolev} for more details about minimax testing). On the other hand, we want to find a test $\tilde{\phi}_n$ such that
\begin{equation*}
\lim_{n \to \infty}\inf_{g \in \mathbb{F} : \norm{g} \geq (1+\epsilon)\delta \rho_n} \E_{g} [\tilde\phi_n (Y)]=1.
\end{equation*}
In such a scenario, $\tilde{\phi}_n$ is called an {\it asymptotically minimax test}. Here we would also like to point out that if there exists a test $\hat{\phi}_n$ and a constant $\hat{\delta} > \delta$ such that $$\lim_{n \to \infty}\inf_{g \in \mathbb{F} : \norm{g} \geq \hat\delta \rho_n} \E_{g} [\hat\phi_n(Y)]=1$$ then the test $\hat\phi_n$ is called a {\it rate optimal test}.
In Section~\ref{sec 2.1} we show that our proposed multiscale statistic yields an asymptotically minimax test for the following scenarios:
\begin{enumerate}
\item (Optimality for H\"{o}lderian alternatives). Consider testing hypothesis~\eqref{test}. If $$\norm{f}_\infty \geq c_*(1+\epsilon_n)(\log (en)/n)^{\frac{\beta}{2\beta+d}},$$ where $f$ belongs to the H\"{o}lder class $\mathbb{H}_{\beta,L}$ with $ \beta>0$ and $L >0$, $\norm{f}_\infty := \sup_{x \in[0,1]^d} |f(x)|$ denotes the sup-norm of $f$, and $c_*$ is a constant (defined explicitly in Theorem~\ref{multopt}), we show that we can construct a level $\alpha$ test based on the multiscale statistic~\eqref{eq:TZ} that has power converging to 1, as $n \to \infty$, provided $\epsilon_n$ does not go to $0$ too fast (see Theorem~\ref{multopt} for the exact order of $\epsilon_n$). We note that this multiscale statistic would require the knowledge of $\beta$ but not of $L$. $\vspace{0.05in}$
Moreover, we show that if $\norm{f}_\infty \leq c_*(1-\epsilon_n)(\log (en)/n)^{{\beta}/{2\beta+d}}$ no test of level $\alpha \in (0,1)$ can have nontrivial asymptotic power; see Theorem~\ref{multopt} for the details. {This shows that our proposed multiscale test is asymptotically minimax with rate of testing $\rho_n = (\log(en)/n)^{\beta/(2\beta+d)}$ and exact separation constant $\delta = c_*$. As far as we are aware this is the first instance of an asymptotically minimax test for the H\"older class $\mathbb{H}_{\beta,L}$ when $d>1$ (under the supremum norm). Moreover, if the smoothness $\beta$ of the H\"older class $\mathbb{H}_{\beta,L}$ is unknown (but $\beta \le 1$) then we can still construct a rate optimal test for this problem; see Proposition~\ref{prop:triangle} for the details.} \vspace{0.07in}
\item (Optimality for detecting signals at {large/small scales}). Consider testing the hypothesis
\begin{equation}\label{boxtest}
H_0: f=0 \quad \mbox{ versus } \quad H_1: f=\mu_n \mathbb{I}_{B_n} ,
\end{equation}
where $\mu_n\neq 0 \in \R$ and $$B_n \equiv B_{\infty}(t^{(n)},h^{(n)}):=\{x \in [0,1]^d : |x_i - t_i^{(n)}| < h_i^{(n)} \mbox{ for all } i=1,\ldots, d\}$$ are unknown, for some $h^{(n)} \in (0,1/2]^d$ and $t^{(n)} \in A_{h^{(n)}}$, and $\mathbb{I}_{B_n}$ denotes the indicator of the hyperrectangle ${B_n}$. First, consider the scenario $\liminf_{n\to \infty} |B_n| > 0 $ where $|B_n|$ denote the Lebesgue measure of $B_n$. Then, if $\lim_{n\to \infty} \sqrt{n} |\mu_n| \to +\infty$, we can construct a level $\alpha$ test based on the multiscale statistic~\eqref{eq:TZ} that has power converging to 1 as $n \to \infty$; see Theorem \ref{boxthm}. Further, we show that, if $\limsup_{n \to \infty} \sqrt{n} |\mu_n| < \infty $, no test of level $\alpha$ can detect the alternative with power going to 1. Thus, the multiscale test is optimal for detecting signals on large scales. \vspace{0.07in}
On the other hand, let us now consider the case $\lim_{n \to \infty} |B_n|=0$. If $$|\mu_n|\sqrt{n|B_n|} \geq (1+\epsilon_n) \sqrt{2\log(1/|B_n|)},\quad \mbox{ for all } n,$$ we can construct a test of level $\alpha$, based on the proposed multiscale statistic, that has power converging to 1 as $n \to \infty$, provided $\epsilon_n$ does not go to $0$ too fast (see Theorem \ref{boxthm}). Furthermore, we can show that if $$|\mu_n|\sqrt{n|B_n|} = (1-\epsilon_n) \sqrt{2\log(1/|B_n|)},\quad \mbox{ for all } n,$$ no test can detect the signal reliably with nontrivial power (i.e., for any level $\alpha$ test $\phi_n$ there exists a signal $f_n$ of the above described strength such that $\phi_n$ will fail to detect $f_n$ with asymptotic probability at least $1-\alpha$); see Theorem \ref{boxthm} for the details. This shows that our multiscale test is asymptotically minimax for signals at small scales.
\end{enumerate}
\subsection{Literature review and connection to existing works}
Our multiscale statistic~\eqref{eq:TZ} can be thought of as a penalized scan statistic, as it is based on the maximum of an ensemble of local test statistics $|\hat{\Psi}(t,h)|$, penalized and properly scaled. Scan-type procedures have received much attention in the literature over the past few decades. Examples of such procedures can be found in~\citet{Seigmund1995}, \citet{Seigmund2000}, \citet{Naus}, \citet{Kulldorffscan1997}, \citet{haiman2006}, \citet{Jiang2002}, etc. All the above mentioned papers consider $d=1$ and no penalization term (like $\Gamma(\cdot)$ in our case) was used. Asymptotic properties of the scan statistic have been studied expensively. In~\citet{Naus} and~\citet{Pozdnyakov} the authors give asymptotic approximations of the distribution of the scan statistic when $d=1$. For $d=2$, similar results can be found in~\citet{Glaz2004}, \citet{haiman2006},~\citet{WangGlaz2014}, among others. Recently in~\citet{Sharpnack} the authors give exact asymptotics for the scan statistic for any dimension $d$.
In all of the above papers it is noted that the scan statistic is dominated by small scales; this creates a problem for detecting large scale signals. One common proposal to fix this problem is to modify the scan statistic so that instead of the maximum over all scales we look at the maximum over scales that are in an appropriate interval containing the true scale of the signal; see e.g.,~\citet{Sharpnack},~\citet{Naus}. In particular, the last two papers show that if the extent of the signal is of a certain order ($\log n$) then this approach leads to power comparable to an oracle. An obvious drawback with the above approach is that we need to have some prior knowledge on which scales the signal(s) may be present. In contrast, our multiscale method does not require any such knowledge.
Another approach that has been proposed to optimally detect signals on both large and small scales is to use different critical values (of the scan statistic) to test for signals at different scales separately (see e.g.,~\citet{scanalr},~\citet{Walther10}) and use multiple testing procedures (see~\citet{Hall08} and the references within) to calibrate the method. However, note that a vast majority of the multiple testing literature either assume that the test statistics are independent (which is not the case here) or are too generic and generally quite conservative.
Conceptually, our work is most related to that of~\citet{DC01}, where the authors proposed our multiscale statistic for $d=1$. Thus, our work can be thought of as a generalization of~\citet{DC01} to multidimension ($d>1$).
\subsection{Organization of the paper}
The proposed multiscale statistic is studied in Section~\ref{sec:2}. In Section~\ref{sec 2.1} we construct {\it optimal} tests for: (i) $f=0$ versus H\"{o}lderian alternatives; (ii) $f=0$ versus alternatives of the form $f=\mu_n \mathbb{I}_{B_n}$, where $B_n$ is an axis-aligned hyperrectangle in $[0,1]^d$ and $\mu_n \in \R$ (for different regimes of $\mu_n$ and $B_n$, both unknown). We compare the performance of our multiscale based test with other competing methods in Section~\ref{simulation studies}. In Section~\ref{future} we discuss some open problems and possible applications/extensions of our work. Section~\ref{proofs of result} gives the proof of Theorem~\ref{Thm 1}. The proofs of the other results are relegated to Appendix~\ref{proofs of result2}.
\section{Multidimensional multiscale statistic}\label{sec:2}
Let us first recall the definition of the multivariate multiscale statistic $T(Y,\psi)$ given in~\eqref{eq:TZ}. The following theorem, our main result in this section, shows that the multiscale statistic $T(Y,\psi)$ is well-defined and finite a.s.~for any (reasonable) kernel function $\psi$; see Section~\ref{sec:Thm 1} for a proof.
\begin{theorem}\label{Thm 1}
Let $\psi$ be a kernel function. For a positive vector $h:=(h_1,\ldots,h_d)>0$ let $A_{h}$ be as defined in~\eqref{ah}. For $t \in A_h$, let $ \psi_{t,h}(\cdot)$ and $\hat{\Psi}(t,h)$ be as defined in~\eqref{kerest} and~\eqref{kerest2}, respectively.
Consider the statistic $T(W,\psi)$ as defined in~\eqref{eq:TZ}, where $W(\cdot)$ is the Brownian sheet on $[0,1]^d$. Then, almost surely, $T(W,\psi)<\infty$, i.e., $T(W,\psi)$ is a tight random variable.
\end{theorem}
Theorem~\ref{Thm 1} immediately extends the main result of~\citet[Theorem 2.1]{DC01} beyond $d=1$. The proof of the above theorem crucially relies on the following two results. We first introduce some notation.\begin{Def}[Packing number] \label{packing}
For any pseudometric space $(\mathscr{F},\rho)$ and $\epsilon>0$, the packing number $N(\epsilon,\mathscr{F})$ is defined as the supremum of the number of elements in $\mathscr{F}^\prime$ where $\mathscr{F}^\prime \subseteq \mathscr{F}$ and for all $a \ne b \in \mathscr{F}^{\prime}$ we have $\rho(a,b)> \epsilon.$
\end{Def}
We will prove Theorem~\ref{Thm 1} as a consequence of the following more general result about stochastic processes with sub-Gaussian increments on some pseudometric space (see Section~\ref{pf:Thm0} for its proof).
\begin{theorem}\label{Thm0}
Let $X$ be a stochastic process on a pseudometric space $(\mathscr{F},\rho)$ with continuous sample paths. Suppose that the following three conditions hold:
\begin{itemize}
\item[(a)] There is a function $\sigma : \mathscr{F} \to (0,1]$ and a constant $K \geq 1$ such that $$\ensuremath{{\mathbb P}} \big(X(a) > \sigma (a) \eta \big) \leq K \exp(-\eta^2/2) \qquad \forall \, \eta > 0,\; \forall \, a\in \mathscr{F}.$$
Moreover, $\sigma^2(b) \leq \sigma^2(a) + \rho^2(a,b),\;\; \forall \;a,b \in \mathscr{F}$.
\item[(b)] For some constants $L,M \geq 1$, $$\ensuremath{{\mathbb P}}\big(|X(a)-X(b)| > \rho(a,b)\eta\big) \leq L \exp(-\eta^2/M)\quad \forall \, \eta > 0, \; \forall\, a,b\in \mathscr{F} .$$
\item[(c)] For some constants $A,B,V, p > 0$,
$${\small N((\delta u)^{1/2}, \{a \in \mathscr{F}: \sigma^2(a) \leq \delta\})\leq A u^{-B} \delta^{-V} (\log(e/\delta))^{p} \;\; \forall \, u, \delta \in (0,1].}$$
Then the random variable
\begin{equation}\label{eq:S(X)}
S(X):= \sup_{a \in \mathscr{F}} \frac{X^2(a)/\sigma^2(a) - 2V\log(1/\sigma^2(a))}{\log \log (e^e/\sigma^2(a))}
\end{equation} is finite almost surely. More precisely, $\ensuremath{{\mathbb P}}(S(X) > r) \le \xi(r)$ for some function $\xi:\R_+ \to \R$ depending only on the constants $K,L,M,A,B,p,V$ such that $\lim_{r \to \infty} \xi(r) = 0$.
\end{itemize}
\end{theorem}
\begin{rem}[Connection to~\citet{DC01}]\label{connection to dumbgen}
A similar result to Theorem~\ref{Thm0} above appears in~\citet[Theorem 6.1]{DC01}. However note that there is a subtle and important difference: The bound on the packing number in (c) of Theorem~\ref{Thm0} involves the additional logarithmic factor $(\log(e/\delta))^{p}$ which is not present in~\citet[Theorem 6.1]{DC01}. In fact, we show that even with this additional logarithmic factor, the random variable $S(X)$, defined in~\eqref{eq:S(X)}, involves the same penalization term $2V\log(1/\sigma^2(a))$ as in~\citet[Theorem 6.1]{DC01}. Hence, we can think of Theorem~\ref{Thm0} as an generalization of~\citet[Theorem 6.1]{DC01}.
\end{rem}
To apply Theorem~\ref{Thm0} to prove Theorem~\ref{Thm 1} we need to define a suitable pseudometric space $(\mathscr{F},\rho)$ and a stochastic process, and verify that conditions (a)-(c) in Theorem~\ref{Thm0} hold. In that vein, let us define the following set
$$\mathscr{F} := \left\{(t,h) \in \mathbb{R}^d \times (0,1/2]^d: h_i\leq t_i\leq 1-h_i, \mbox{ for all } i=1,2,\ldots,d \right\}$$
with the following pseudometric $$\rho^2((t,h),(t^\prime,h^\prime)):=|B_\infty(t,h) \bigtriangleup B_\infty(t^\prime,h^\prime)|, \quad \quad \mbox{for } (t,h), (t^\prime,h^\prime) \in \mathscr{F},$$
where $B_\infty(t,h) :=\Pi_{i=1}^d(t_i-h_i,t_i+h_i)$, $A \bigtriangleup B := (A \cap B^c)\cup (A^c \cap B)$ denotes the symmetric difference of the sets $A$ and $B$, and $|A|$ denotes the Lebesgue measure of the set $A$. Also, define $$\sigma^2(t,h):= |B_\infty(t,h)|=2^d \Pi_{i=1}^d h_i, \qquad \mbox{for } (t,h) \in \mathscr{F}.$$
The following important result shows that indeed for the above defined pseudometric space $(\mathscr{F},\rho)$ condition (c) of Theorem~\ref{Thm 1} holds.
\begin{lem}\label{lem1}
Let $\mathscr{F},\rho(\cdot,\cdot)$ and $\sigma(\cdot)$ be as described above. Then
\begin{equation*}
N\left((u\delta)^{1/2},\{(t,h) \in \mathscr{F}: \sigma^2(t,h)\leq \delta\}\right) \leq K u^{-2d}\delta^{-1} (\log(e/\delta))^{d-1} \;\; \forall \, u,\delta \in(0,1],
\end{equation*}
for some constant $K$ depending only on $d$.
\end{lem}
\begin{rem}\label{rem:pack}
Here we would like to point out that Lemma~\ref{lem1} shows that condition (c) of Theorem~\ref{Thm0} holds with $B=2d$, $p=d-1$ and most importantly for $V=1$, which was also the case when $d=1$ (as shown in~\citet{DC01}).
\end{rem}
\begin{rem}\label{rem:v}
Compare the numerator of our multiscale statistic~\eqref{eq:TZ} with the multiscale statistic proposed in~\citet[Equation (6)]{Munk2018}. In \citet{Munk2018} the authors propose a penalization term $\Gamma_V(2^dh_1\ldots h_d)$ where $\Gamma_V:(0,1]\to (0,\infty)$ is defined as $$\Gamma_V(r):=(2V \log(1/r))^{1/2}$$
instead of the penalization $\Gamma(2^dh_1\ldots h_d)$ as in~\eqref{eq:TZ}. Further, in that paper the authors recommend the choice of $V=(2d-1+\epsilon)$ for any $\epsilon>0$; see~\citet[Lemma 5.1]{Munk2018}. Thus, Theorem~\ref{Thm 1} and Lemma~\ref{lem1}, improve on the existing results in the literature. Our penalization term $\Gamma(\cdot)$ results in optimal detection properties for testing \eqref{test} and \eqref{boxtest} which cannot be achieved if the penalization term $\Gamma_V(\cdot)$, for $V >1$, is used.
\end{rem}
It is well-known that we should choose the constant $V$ in the penalization term $\Gamma_V $ as small as possible (see e.g.,~\citet[Section 1.1]{Munk2018}) for optimal testing. In our proposed multiscale statistic we take $V=1$. The following proposition shows that indeed $V=1$ is the smallest possible permissible value; see Section~\ref{sec:Prop_V} for a proof.
\begin{prop}\label{v<1}
Suppose $V<1$. Let $\Gamma_V$ and $\mathscr{F}$ be as defined above. Then we have $$\sup_{(t,h)\in \mathscr{F}} |\hat{\Psi}(t,h)|-\Gamma_V(2^dh_1\ldots h_d)=\infty\quad \mbox{a.s.}$$
Thus, $\sup_{(t,h)\in \mathscr{F}} \frac{|\hat{\Psi}(t,h)|-\Gamma_V(2^dh_1\ldots h_d)}{D(2^dh_1\ldots h_d)}=\infty\quad \mbox{a.s.}$
\end{prop}
\section{Optimality of the multiscale statistic in testing problems}\label{sec 2.1}
In this section we prove that we can construct tests based on the multiscale statistic that are optimal for testing~\eqref{test} and~\eqref{boxtest}. For both the testing problems we can define a multiscale test based on kernel $\psi $ as follows:
Let $$\kappa_{\alpha,\psi}=\inf\{c \in \R: \ensuremath{{\mathbb P}}(T(W,\psi)> c) \leq \alpha\},$$
where $W$ is the standard Brownian sheet on $[0,1]^d$. For notational simplicity we would denote $\kappa_{\alpha,\psi}$ by $\kappa_\alpha$ from now on.
For testing~\eqref{test} and~\eqref{boxtest} a test of level $\alpha$ can be defined as follows: $$\mbox{Reject $H_0\quad\quad$ if and only if $\qquad T(Y,\psi)>\kappa_{\alpha}$}.$$ Let us call this testing procedure the multiscale test. Although any kernel $\psi$ can be used to construct the above test, in Sections~\ref{sec 3.1} and~\ref{sec 3.2} we show that specific choices of the kernel function $\psi$ leads to asymptotically minimax tests.
\subsection{Optimality against H\"older classes of functions}\label{sec 3.1}
Let us recall the definition of the H\"{o}lder class of functions $\mathbb{H}_{\beta,L}$, for $\beta \in (0,1]$ and $L>0$, as in~\eqref{eq:H_b_L}; see Definition~\ref{def:holder} for the formal definition of $\mathbb{H}_{\beta,L}$ for any $\beta >0$. Let $\psi_\beta:\R^d \to \R$, for $0<\beta< \infty$, be the unique solution of the following optimization problem:
\begin{equation}\label{minimize} \mbox{Minimize} \norm{\psi} \mbox{ over all } \psi \in \mathbb{H}_{\beta,1} \mbox{ with } \psi(0) \geq 1.\end{equation} Elementary calculations show that for $0< \beta \leq 1$, we have $$\psi_\beta(x)=(1-\norm{x}^\beta) \mathbb{I}(\norm{x} \leq 1);$$ see Section~\ref{lem:Ele} for a proof. For $ \beta>1$, $\psi_\beta$ can be calculated numerically. We consider the kernel $\psi_\beta$, for $\beta>0$, described above and state our first optimality result for testing \eqref{test}; see Section~\ref{multiopt} for a proof.
\begin{theorem}\label{multopt}
Let $T_\beta \equiv T(Y,\psi_\beta)$ be the multiscale statistic defined in~\eqref{eq:TZ} with kernel $\psi_{\beta}$, for $0<\beta<\infty$. Define $$\rho_n :=\left(\frac{\log n}{n}\right)^{\frac{\beta}{2\beta+d}}$$ and $$c_* \equiv c_*(\beta,L) :=\left(\frac{2dL^{d/\beta}}{(2\beta+d)\norm{\psi_\beta}^2}\right)^{\frac{\beta}{2\beta+d}}.$$
Then, for arbitrary $\epsilon_n > 0 $ with $\epsilon_n \to 0$ and $\epsilon_n \sqrt{\log n }\to \infty$ as $n \to \infty$, the following hold:
\begin{itemize}
\item[(a)] For any arbitrary sequence of tests $\phi_n$ with level $\alpha$ for testing~\eqref{test}, we have
$$\limsup_{n \to \infty} \inf_{g\in \mathbb{H}_{\beta,L}: \norm{g}_{\infty}= (1-\epsilon_n)c_*\rho_n} \E_g [\phi_n(Y)] \leq \alpha;$$
\item[(b)] for $J_n :=[(c_*\rho_n/L)^{1/\beta},1-(c_*\rho_n/L)^{1/\beta}]^d$, we have
$$\lim_{n\to \infty} \inf_{g\in \mathbb{H}_{\beta,L}: \norm{g}_{J_n,\infty}\geq (1+\epsilon_n)c_*\rho_n} \ensuremath{{\mathbb P}}_g(T_\beta > \kappa_\alpha)=1$$
where $\norm{g}_{J_n,\infty}:= \sup_{t \in J_n}|g(t)|$.
\end{itemize}
\end{theorem}
The above result generalizes~\citet[Theorem 2.2]{DC01} beyond $d=1$. Theorem~\ref{multopt} can be interpreted as follows: (a) for every test $\phi_n$ there exists a function with supremum norm $(1-\epsilon_n)c_*\rho_n$ which cannot be detected with nontrivial asymptotic power; whereas (b) when we restrict to functions with signal strengths (i.e., supremum norm in the interior of $[0,1]^d$) just a bit larger than the above threshold, our proposed multiscale test is able to detect every such function with asymptotic power 1. In this sense our proposed test is optimal in detecting departures from the zero function for H\"{o}lder classes $\mathbb{H}_{\beta,L}$. We note here that to calculate $T_\beta$ we need the knowledge of $\beta$ but we do not need to know $L$.
If $\beta$ is unknown, but is less than or equal to 1, we can use $T_1$ as a test statistic for testing~\eqref{test}. Although the resulting test is not asymptotically minimax, the test is still rate optimal. The following result formalizes this; see Section~\ref{proof:proptriangle} for its proof.
\begin{prop}\label{prop:triangle}
Consider testing~\eqref{test} where $\beta\leq 1$ is unknown. Let us recall the definition of $\psi_1$ in~\eqref{minimize}. Let $T_1 \equiv T(Y,\psi_1)$ be the multiscale statistic defined in~\eqref{eq:TZ} with kernel $\psi_{1}$. Define $$\rho_n :=\left(\frac{\log n}{n}\right)^{\frac{\beta}{2\beta+d}}$$ and let $M$ be any constant such that $M >\left(\frac{2dL^{d/\beta}\norm{\psi_1}^2}{(2\beta+d)\langle \psi_1,\psi_\beta \rangle^2}\right)^{\frac{\beta}{2\beta+d}}.$
Let $J_n :=[(M\rho_n/L)^{1/\beta},1-(M\rho_n/L)^{1/\beta}]^d$. Then we have
$$\lim_{n\to \infty} \inf_{g\in \mathbb{H}_{\beta,L}: \norm{g}_{J_n,\infty}\geq M\rho_n} \ensuremath{{\mathbb P}}_g(T> \kappa_\alpha)=1$$
where $\kappa_\alpha$ is the $(1-\alpha)$ quantile of the multiscale statistic $T(Y,\psi_1)$ under the null hypothesis.
\end{prop}
\begin{rem}
Instead of using the test statistic $T_\beta$ if we use the test statistic
\begin{equation}\label{eq:T^*}T_\beta^\star:=\sup_{h\in(0,1/2]^d} \sup_{t \in A_h} \big[\lvert \hat{\Psi}(t,h)\rvert - \Gamma(2^dh_1 \ldots h_d)\big]
\end{equation} with the kernel $\psi_\beta$, then the same conclusions as that of Theorem~\ref{multopt} and Proposition~\ref{prop:triangle} would hold. Thus the multiscale statistic $T_\beta^\star$ is also optimal against H\"{o}lderian alternatives.
\end{rem}
\begin{rem}\label{rem:Gamma_V}
Note that in~\citet{Munk2018} the authors propose a multiscale statistic like $T_\beta^\star$, with a slightly different penalization term
\begin{equation}\label{eq:Gamma_V}
\Gamma_V: r \mapsto (2V \log(1/r))^{1/2}
\end{equation}
instead of $\Gamma(\cdot)$. A close inspection of our proof of Theorem~\ref{multopt} reveals that for such a statistic, only signals with $\norm{g}_{J_n,\infty}\geq \sqrt{V}(1+\epsilon_n)c_*\rho_n$ will be detected with power converging to 1. This shows how a proper penalization (as in our multiscale statistic) can lead to the testing procedure attaining the exact separation constant for testing~\eqref{test}.
\end{rem}
\subsection{Optimality against axis-aligned hyperrectangular signals}\label{sec 3.2}
In Theorem~\ref{multopt} we proved the optimality of the multiscale test when the supremum norm of the signal is large. A natural question that arises next is: ``What if the signal is not peaked but distributed evenly on some subset of $[0,1]^d$?". To answer this question we look at the testing problem~\eqref{boxtest}, and establish below the optimality of our multiscale test in this setting (see Section~\ref{multiopt} for a proof of Theorem~\ref{boxthm}). Note that when $d=1$ similar optimality results are known for the multiscale statistic; see~\citet[Theorem 2.6]{munkchangepoint} and~\citet{scanalr}. For $h=(h_1,\ldots,h_d) \in (0,1/2]^d$, let us first define $$\mathscr{B}_{h}:=\{B \subseteq [0,1]^d: B=\Pi_{i=1}^d[t_i - h_i,t_i+h_i] \mbox{ for some } t=(t_1,\ldots,t_d) \in A_h \}.$$
\begin{theorem}\label{boxthm}
Let $T \equiv T(Y,\psi_0)$ where $\psi_0= \mathbb{I}_{[-1,1]^d}$. Let $f_n=\mu_n\mathbb{I}_{B_n}$ where $B_n$ is an axis-aligned hyperrectangle and let $|B_n|$ denote the Lebesgue measure of the set $B_n$. Then we have the following results:
\begin{itemize}
\item[(a)] Suppose that $\liminf_{n \to \infty }|B_n| > 0$. Let $\phi_n$ be any test of level $\alpha\in(0,1)$ for \eqref{boxtest}. Then, for any $f_n = \mu_n\mathbb{I}_{B_n}$ such that $\limsup_n |\mu_n|\sqrt{n |B_n|}< \infty$, we have $$\limsup_{n \to \infty} \E_{f_n} [\phi_n(Y)] <1.$$ Moreover, for the proposed multiscale test based on $T$, we have $$\lim_{n \to \infty} \inf_{f_n:\lim |\mu_n|\sqrt{n |B_n|}=\infty} \ensuremath{{\mathbb P}}_{f_n}(T>\kappa_\alpha)=1.$$
\item[(b)]Now let us look at the case $\lim_{n\to \infty} |B_n|=0$. Let $h_n=(h_{1,n},\ldots,h_{d,n}) \in (0,1/2]^d$ be any sequence of points such that $\lim_{n \to \infty} \Pi_{i=1}^d h_{i,n} \to 0$. Let $$\mathcal G_n^{-}:=\{f_n=\mu_n\mathbb{I}_{B_n}:|\mu_n|\sqrt{n|B_n|} = (1-\epsilon_n)\sqrt{2\log(1/|B_n|)} , B_n \in \mathscr{B}_{h_n}\}$$ with $\epsilon_n \to 0$ and $\epsilon_n \sqrt{2\log(1/|B_n|)} \to \infty$. (Here we have omitted the dependence of $h_n$ in the notation $\mathcal G_n^{-}$). If $\phi_n$ be any test of level $\alpha\in(0,1)$ for \eqref{boxtest} then we have $$ \limsup_{n \to \infty} \inf_{f_n \in \mathcal G_n^{-}} \E_{f_n} [\phi_n(Y)] \leq \alpha.$$
Moreover, let $$\mathcal G_n^{+}:=\{f_n=\mu_n\mathbb{I}_{B_n}:|\mu_n|\sqrt{n|B_n|} \geq (1+\epsilon_n)\sqrt{2\log(1/|B_n|)},B_n \in \mathscr{B}_{h_n}\}.$$ Then for our multiscale test we have $$\lim_{n \to \infty} \inf_{f_n \in \mathcal G_n^{+}} \ensuremath{{\mathbb P}}_{f_n}(T>\kappa_\alpha)=1.$$
\end{itemize}
\end{theorem}
\begin{rem}
If we use the test statistic $T^\star$, as defined in~\eqref{eq:T^*} (with the kernel $\psi_0$), instead of $T$ in Theorem~\ref{boxthm}, the optimality results described in the theorem still hold.
\end{rem}
Our first result in Theorem~\ref{boxthm} shows that as long as $\liminf_{n \to \infty }|B_n| > 0$, for any test to have power converging to $1$ we need to have $\lim |\mu_n|\sqrt{n |B_n|}=\infty$, in which case our multiscale test achieves asymptotic power 1. Thus our multiscale test is optimal for detecting large scale signals. The next result can be interpreted as follows: (i) For signals with small spatial extent (i.e., $\lim_{n \to \infty} |B_n|=0$) if the signal strength is too small ($|\mu_n|\sqrt{n|B_n|} \le (1-\epsilon_n)\sqrt{2\log(1/|B_n|)}) $ no test can detect the signal reliably with nontrivial probability (i.e., for every test $\phi_n$ there exist a signal such that $\phi_n$ will fail to detect it with probability $1-\alpha+o(1)$); (ii) on the other hand, if the signal strength is a bit larger than the threshold (i.e., the exact separation constant) described above our multiscale test will detect the signal with asymptotic power 1. This shows that our multiscale test achieves optimal detection for signals with small spatial footprint. We would like to emphasize here that by using the same exact test (using the same kernel $\psi_0$) we are able to optimally detect both large and small scale signals.
\begin{rem}\label{v3}
As we mentioned in Remark~\ref{rem:Gamma_V} if we used $\Gamma_V(\cdot)$ (see~\eqref{eq:Gamma_V}), for $V>1$, instead of $\Gamma(\cdot)$, in defining the multiscale statistic then we would only be able to detect signals (when $|B_n| \to 0$) if $|\mu_n|\sqrt{n|B_n|} \geq \sqrt{V}(1+\epsilon_n) \sqrt{2\log(1/|B_n|)}$ which is not the exact separation constant as mentioned in Theorem~\ref{boxthm}. This agains illustrates the importance of choosing the right penalization term $\Gamma(2^dh_1\ldots h_d)$ in defining the multiscale statistic.
\end{rem}
\begin{rem}
Here we would like to point out that proofs for the minimax lower bound that have been derived for the two scenarios in Theorems ~\ref{multopt} and~\ref{boxthm} follows the standard techniques that have been used in \citet{Ingster19931}, \citet{Ingster19932}, \citet{Ingster19933},~\citet{Lepski2000}, \citet{DC01}, \citet{Ingster2009multi} etc.
\end{rem}
\subsubsection{Comparison with the scan and average likelihood ratio statistics when $d=1$}
When $d=1$ there exists an extensive literature on the optimal detection threshold for signals of the form $f_n=\mu_n\mathbb{I}_{B_n}$, where now $B_n \subseteq [0,1]$ is an interval. In~\citet{scanalr} the authors compare the performance of the scan statistic (i.e., the statistic~\eqref{eq:Scan} in the discrete setup with $\psi = \mathbb{I}_{[-1,1]}$) and the average likelihood ratio (ALR) statistic (which is the discrete analogue of $\int_0^{1/2} \int_{h}^{1-h} \exp[ |\hat{\Psi}(t,h)|^2/2] dt \, dh$); see Section~\ref{simulation studies} for a description and comparison of the two competing methods with our multiscale test when $d=2$.
When $\liminf_{n \to \infty} |B_n|>0$ the scan statistic can only detect the signal, with asymptotic power 1, when $|\mu_n|\sqrt{n} \geq (1+\epsilon_n)\sqrt{2\log n}$, whereas the ALR statistic (and the proposed multiscale statistic) can detect the signal whenever we have $|\mu_n|\sqrt{n} \to \infty$ (which is a less stringent condition). Note that $|\mu_n|\sqrt{n} \to \infty$ is also required for any test to detect the signal with asymptotic power $1$. This shows that the scan statistic is not optimal for detecting large scale signals.
On the other hand if $\lim_{n \to \infty} |B_n|=0$, the scan statistic can detect the signal if $|\mu_n|\sqrt{n|B_n|} \geq (1+\epsilon_n) \sqrt{2\log n}$ whereas the ALR statistic can detect the signal when $|\mu_n|\sqrt{n|B_n|} \geq \sqrt{2}(1+\epsilon_n) \sqrt{2\log(1/|B_n|)}$. The optimal detection threshold in this scenario is $ |\mu_n|\sqrt{n|B_n|} \geq (1+\epsilon_n) \sqrt{2\log(1/|B_n|)}$, which is attained by the multiscale statistic. Thus that scan statistic is optimal in detecting signals only when $|B_n|=O(1/n)$. The ALR statistic requires the signal to be at least $\sqrt{2}$ times the (detectable) threshold. This shows that neither the standard scan or the ALR is able to achieve the optimal threshold for detecting small scale signals.
\citet[Theorem 2.6]{munkchangepoint} shows the optimality of the multiscale statistic (which is a modification of the scan statistic) in detecting signals in both cases when $d=1$. In \citet{Rivera13} and \citet{scanalr} the authors propose a condensed ALR statistic which, much like the multiscale statistic, is able to attain the optimal threshold for detection in both regimes of $B_n$. As far as we are aware the condensed ALR statistic has not been extended beyond $d=1$ and therefore whether it achieves the optimal threshold for $d>1$ is not known.
In summary, Theorem~\ref{boxthm} shows that our multidimension multiscale test is asymptotically minimax even when $d>1$.
\section{Simulation studies}\label{simulation studies}
\begin{table}
\begin{tabular}{| c | c || c| c|}
\hline
\multicolumn{4}{|c|}{Critical values}\\
\hline
$m$ & 95\% quantile & $m$ & 95\% quantile \\
\hline
25 & 3.02 & 75 & 3.27 \\
40 & 3.12 & 100 & 3.31 \\
50 & 3.18 & 125 & 3.32 \\
60 & 3.22 & 150 & $~3.30^\star$ \\
\hline
\end{tabular}
\caption{Critical values $\kappa_{0.05}$ for different $n = m^2$.}\label{tab:Crit}
{\it $^\star$Note that 0.95 quantiles necessarily increase as $n$ increases. But in our simulations the 0.95 quantile for $n=150^2$ turned out to be slightly less than that of $n=125^2$ due to sampling variability.}
\end{table}
In this section we demonstrate the performance of the multiscale testing procedure described in Section~\ref{sec 2.1} and compare it with other competing methods through simulation studies. For computational tractability, we replace the continuous white noise model~\eqref{eq:Mdl} with a discrete one and consider the case $d=2$. More specifically, we consider data on the $m\times m$ grid $S_n = \{(i/m,j/m): 1\leq i,j\leq m\}$ (here $n =m^2$), where the model is
\begin{equation*}
Y\left(\frac{i}{m},\frac{j}{m}\right)=f\left(\frac{i}{m},\frac{j}{m}\right) + \epsilon\left(\frac{i}{m},\frac{j}{m}\right), \qquad \mbox{for }\; i, j = 1,\ldots, m,
\end{equation*}
with $\epsilon(i/m,j/m)$'s being i.i.d.~standard normal random variables. In our simulation experiments we vary our bandwidth parameter $h=(h_1,h_2)$ in the $m \times m$ grid $S_n$. For the simulations we have used the kernel function $\psi =\mathbb{I}_{[-1,1]^d}$. In Table~\ref{tab:Crit} we give the empirical 0.95-quantile of the multiscale statistic $T(W,\psi)$ (see~\eqref{eq:TZ}) for different values of $n$; the computation of the empirical quantiles were based on 3000 replications. Observe that the empirical quantiles seem to stabilize as $m$ increases beyond 100. Figure~\ref{multi dist} shows the empirical distribution function estimates, based on 3000 replications, of the multiscale statistic for different values of $n$.
\begin{figure}
\includegraphics[scale=.50]{distributionfunc.png}\caption{The empirical distribution functions of the multiscale statistic for different values of $n$.}\label{multi dist}
\end{figure}
In Tables~\ref{Tab: com1} and~\ref{Tab: com2} we compare the powers of the multiscale test, a test based on a scan-statistic, and the ALR test (see \citet{scanalr} for the details). Formally, we consider testing~\eqref{boxtest} against alternatives of the form $H_1: f=\mu_n \mathbb{I}_{B_n}$, for both small and large scale signals ($B_n$). We briefly describe the above two competing procedures. Let $\mathscr{B}$ be the set of all axis-aligned rectangles on $[0,1]^2$ with corner points of the form $({i}/{m},{j}/{m})$, for $i,j \in \{1,\ldots, m\}$. For every $B \in \mathscr{B}$ define $$\hat{\Psi}(B) := \frac{1}{\sqrt{|B|}} \sum_{(i/m,j/m)\in B} Y\left(\frac{i}{m},\frac{j}{m}\right).$$ Note that $\hat{\Psi}(\cdot)$ is the discrete analogue of the normalized kernel estimator as defined in~\eqref{kerest2}. The scan test statistic (see \citet[Chapter 5]{Scanbook}) for this problem is defined as $$ M_n:= \max_{B \in \mathscr{B}} |\hat{\Psi}(B)|. $$ The ALR test statistic (see \citet{chan2009}) is defined as $$A_n:= \frac{1}{{m \choose 2}^2} \sum_{B \in \mathscr{B}} \exp(\hat{\Psi}(B)^2/2).$$ The scan test (ALR test) rejects the null hypothesis if the observed $M_n$ ($A_n$) exceeds the 0.95-quantile for $M_n$ ($A_n$) under the null hypothesis. In Tables~\ref{Tab: com1} and \ref{Tab: com2} we compare the performance of the three procedures. Here $\mu$ denotes the signal strength, and $k/m$ denotes the length of each side of the square signal $B_n$ (here $m=40$ and $100$ for the two cases). The power of the tests were calculated using 1000 replications.
\begin{table}
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=1$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
3.5 & 0.23 & 0.08 & 0.07 \\
4.0 & 0.34 & 0.13 & 0.08 \\
4.5 & 0.50 & 0.18 & 0.08 \\
5.0& 0.71 & 0.30 & 0.08 \\
5.5 & 0.86 & 0.53 & 0.09 \\
\hline
\end{tabular} \quad
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=4$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
1.00 & 0.22 & 0.14 & 0.11 \\
1.20 & 0.43 & 0.31 & 0.30 \\
1.35 & 0.60 & 0.48 & 0.44 \\
1.50 & 0.74 & 0.55 & 0.52 \\
1.65 & 0.86 & 0.72 & 0.61 \\
\hline
\end{tabular}
\vspace{.3cm}
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=18$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
0.20 & 0.15 & 0.21 & 0.19 \\
0.30 & 0.49 & 0.68 & 0.67 \\
0.35 & 0.65 & 0.80 & 0.82 \\
0.40 & 0.80 & 0.90 & 0.89 \\
\hline
\end{tabular} \quad
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=40$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
0.040 & 0.15 & 0.32 & 0.31 \\
0.043 & 0.30 & 0.56 & 0.54 \\
0.047 & 0.45 & 0.78 & 0.78 \\
0.050 & 0.68 & 0.94 & 0.95 \\
\hline
\end{tabular}
\caption{Power of the scan, the multiscale and the ALR tests for $m=40$ (i.e., $n=40^2$) as $\mu$ changes.}
\label{Tab: com1}
\end{table}
We make the following observations. For both the cases ($m=40$ and $100$) when the signal is at the smallest scale, e.g., $k=1$, the scan statistic outperforms everything else. However, when $m=100$, even in relatively small scales, e.g., $k=8$ (i.e., about $0.6\%$ of the observations contain the signal) our multiscale test starts to outperform the scan test. Note that in this setting (small scales) the ALR performs the worst. As the spatial extent of the signal increases, our multiscale procedure and the ALR procedure starts performing favorably whereas the performance of the scan statistics deteriorates. Thus, the simulation experiments corroborates our theoretical findings.
\begin{table}
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=1$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
4.5 & 0.34 & 0.11 & 0.06 \\
5.0 & 0.52 & 0.28 & 0.06 \\
5.5 & 0.75 & 0.43 & 0.09 \\
6.0 & 0.95 & 0.61 & 0.13 \\
\hline
\end{tabular} \quad
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=8$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
0.25 & 0.08 & 0.17 & 0.07 \\
0.30 & 0.35 & 0.46 & 0.13 \\
0.35 & 0.60 & 0.72 & 0.22 \\
0.40 & 0.82 & 0.96 & 0.50 \\
\hline
\end{tabular}
\vspace{.3cm}
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=30$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
0.040 & 0.07 & 0.22 & 0.22 \\
0.050 & 0.17 & 0.42 & 0.45\\
0.055 & 0.42 & 0.74 & 0.75 \\
0.060 & 0.58 & 0.93 & 0.96 \\
\hline
\end{tabular} \quad
\begin{tabular}{| c | c | c| c| c|}
\hline
\multicolumn{4}{|c|}{$k=100$}\\
\hline
$\mu$ & Scan & Multiscale & ALR\\
\hline
0.014 & 0.08 & 0.42 & 0.42 \\
0.018 & 0.17 & 0.62 & 0.63 \\
0.020 & 0.22 & 0.84 & 0.86 \\
0.025 & 0.45 & 0.96 & 0.95 \\
\hline
\end{tabular}
\caption{Power of the scan, the multiscale and the ALR tests for $m=100$ (i.e., $n=100^2$) as $\mu$ changes.}
\label{Tab: com2}
\end{table}
\section{Discussion}\label{future}
In this paper we have proposed a multidimensional multiscale statistic in the continuous white noise model and used this statistic to construct asymptotically minimax tests for testing $f=0$ against (i) H\"{o}lder classes of functions; and (ii) alternatives of the form $f=\mu_n \mathbb{I}_{B_n}$, where $B_n$ is an unknown axis-aligned hyperrectangle in $[0,1]^d$ and $\mu_n \in \R$ is unknown. However, there are many open questions in this area. We briefly delineate a few of them below and in the process describe some important papers in related areas of research.
We have shown that for the H\"older class $\mathbb{H}_{\beta,L}$, if the smoothness parameter $\beta$ is known, we can construct an asymptotically minimax test. However, if $\beta$ is unknown (and $\beta \le 1$) we can only construct a rate optimal test. A natural question that arises is whether a test can be constructed that is asymptotically minimax (for the H\"{o}lder class of functions with the supremum norm) without the knowledge of the smoothness parameter $\beta$ (and $L>0$); see~\citet[Section 1.3]{Ji2017}. Another interesting question would be to try to extend our results to other smoothness classes like Sobolev/Besov classes; in \citet{Ingster2011sobolev} the authors gave the minimax rate of testing for Sobolov class, but no test was proposed that achieves the exact separation constant.
Note that we have shown that our multiscale test is asymptotically minimax for detecting the presence of a signal on an axis-aligned hyperrectangle in $[0,1]^d$. One obvious extension of our work would be to correctly identify the hyperrectangle on which the signal is present. Further, we could go beyond hyperrectangles and try to identify signals that are present on some other geometric structures $A \subset [0,1]^d$ (i.e., $f=\mu \mathbb{I}_A$ where $A$ is not necessarily an axis-aligned hyperrectangle). Examples of such geometric structures could be: $(i)$ $A$ is an hyperrectangle which is not necessarily axis-aligned, $(ii)$ $A$ is a $d$-dimensional ellipsoid, $(iii)$ $A=\bigcup_{i=1}^k A_i$ where each $A_i \subseteq [0,1]^d$ is an (axis-aligned) hyperrectangle, etc.~\citet{munkchangepoint} and the references therein investigated the problem of finding change points in $d=1$ which can be thought of as detection of multiple intervals. In~\citet{Castro2005} the authors use the scan statistic to detect regions in $\mathbb{R}^d$ where the underlying function is non-zero.~\citet{Arias2010} considers the problem of finding a cluster of signals (not necessarily rectangular) in a network using the scan statistic. Although the method they propose achieves the optimal boundary for detection, it requires the knowledge of whether the signal shape is ``thick" or ``thin". For hyperrectangles this refers to whether or not the minimum side length is of order $\log n /n$ or not. We believe that the multiscale statistic, with proper modifications, can be used to find asymptotically minimax/rate optimal tests in such problems.
In our white noise model~\eqref{eq:Mdl} we assume that the distribution of the response variables is (homogeneous and independent) Gaussian. Similar questions about signal detection can be asked when the response is non-Gaussian; see e.g.,~\citet{Munk2018},~\citet{ChanWalther2015}, \citet{Rivera13}, \citet{Walther10} etc. In~\citet{Munk17heterochange} the authors looked at the problem of detecting change points under heterogeneous variance of the response variable (when $d=1$).~\citet{Rohde2008} looked at this problem where the error distribution is known to be symmetric (when $d=1$). \citet{Walther10} studied a similar problem where the response variable is binary. A multiscale approach could be used to tackle such problems as well.
Several interesting applications of the multiscale approach exist when $d=1$ (following the seminal paper of~\citet{DC01}): In~\citet{Dumbgen2008} the authors propose a multiscale test statistic to make inference about a probability density on the real line given i.i.d.~observations;~\citet{Hieber2013} use multiscale methods to make inference in a deconvolution problem;~\citet{Rivera13} use multiscale methods to detect a jump in the intensity of a Poisson process, etc. We believe that our extension beyond $d=1$ will also lead to several interesting multidimensional applications.
\section*{Acknowledgements}
The authors would like to thank Lutz D\"{u}mbgen and Sumit Mukherjee for several helpful discussions.
\section{Proofs of our main results}\label{proofs of result}
\subsection{Some useful concepts}\label{useful concepts}
In this subsection we formally define some technical concepts that we use in this paper.
\begin{Def}[Brownian sheet]\label{sec:BS}
By a $d$-dimensional Brownian sheet we mean a mean-zero Gaussian process $\{W(t): t \in [0,1]^d\}$ with covariance
\begin{equation*}
\Cov(W(t_1,\ldots,t_d),W(s_1,\ldots,s_d))=\Pi_{i=1}^d \min({t_i,s_i}),
\end{equation*}
for $(t_1,\ldots,t_d), (s_1,\ldots,s_d)\in [0,1]^d$. The Brownian sheet is the $d$-dimensional counterpart of the standard Brownian motion; see e.g.,~\citet{WZ01},~\citet[Chapter 5]{Bsheetbook} for detailed properties of the Brownian sheet.
\end{Def}
In the following we give some useful properties of a Brownian sheet $W(\cdot)$.
\begin{itemize}
\item If $g \in L_2([0,1]^d) $ then $\int g dW:=\int_{[0,1]^d} g(t) dW(t) \sim N(0,\norm{g}^2).$
\item If $g_1,g_2 \in L_2([0,1]^d) $ then $\Cov \left(\int g_1 dW,\int g_2 dW\right)= \int_{[0,1]^d} g_1(t)g_2(t) dt. $
\item { {\it Cameron-Martin-Girsanov Theorem for Brownian sheet:} Let us state the simplest version of the Cameron-Martin-Girsanov Theorem that we will use in this paper (see~\citet[Chapter 3]{Protterbook} for detailed discussion about change of measure and the result). $\vspace{0.05in}$
Assume $f \in L_1([0,1]^d)$ and let $\{{W(t):t\in[0,1]^d}\}$ be a standard Brownian sheet. Let $\Omega$ be the set of all real-valued continuous functions defined on $[0,1]^d$. Let $P$ denote the measure on $\Omega$ induced by the Brownian sheet $\{{W(t):t\in[0,1]^d}\}$ and let $Q$ denote the measure induced by $\{Y(t): t \in [0,1]^d\} $ where $Y(t)$ is defined as in \eqref{eq:Mdl}.
Then $Q$ is absolutely continuous with respect to $P$ and the Radon-Nikodym derivative is given by $$\frac{dQ}{dP}(Y)=\exp\left(\sqrt{n}\int fdW - \frac{n}{2}\norm{f}^2 \right).$$ This, in turn, implies that for any measurable function $\phi$ we have $$ \E_Q\left(\phi(Y)\right) = \E_P \left(\phi(Y)\frac{dQ}{dP}(Y) \right).$$}
\end{itemize}
Let us now define the H\"{o}lder class of functions $\mathbb{H}_{\beta,L}$, for $\beta>0$ and $L>0$.
\begin{Def}\label{def:holder}
Fix $\beta>0$ and $L>0$. Let $\lfloor \beta \rfloor$ be the largest integer which is strictly less than $\beta$ and for $k=(k_1,k_2,\ldots ,k_d)\in \mathbb{N}^d$ set $\norm{k}_1 := \sum_{i=1}^d k_i$. The H\"{o}lder class $\mathbb{H}_{\beta,L}$ on $[0,1]^d$ is the set of all functions $f:[0,1]^d \to \R$ having all partial derivatives of order $\lfloor \beta \rfloor$ on $[0,1]^d$ such that $$\sum_{0\leq \norm{k}_1 \leq \lfloor \beta \rfloor} \sup_{x \in [0,1]^d} \left| \frac{\partial^{\norm{k}_1} f(x) }{\partial x_1^{k_1} \ldots \partial x_d^{k_d}} \right| \leq L$$ and $$\sum_{\norm{k}_1 = \lfloor \beta \rfloor} \left| \frac{\partial^{\norm{k}_1} f(y) }{\partial x_1^{k_1} \ldots \partial x_d^{k_d}} - \frac{\partial^{\norm{k}_1} f(z) }{\partial x_1^{k_1} \ldots \partial x_d^{k_d}} \right| \leq L \norm{y-z}^{\beta -\lfloor \beta \rfloor } \quad \forall \, y,z \in [0,1]^d. $$
\end{Def}
\begin{rem} One of the most important properties of $\mathbb{H}_{\beta,L}$ that we will use is the following: If $f \in \mathbb{H}_{\beta,1}$ then, for any $h=(h_1,\ldots,h_d)>0$ and $t\in A_h$, $$g(x_1,\ldots, x_d) := L \min(h)^\beta f\left(\frac{x_1-t_1}{h_1}, \ldots ,\frac{x_d-t_d}{h_d} \right) \in \mathbb{H}_{\beta,L}$$
where $\min(h):= \min_{i=1,\ldots,d} h_i$.
\end{rem}
\begin{Def}[Hardy-Krause variation]\label{totalvar}
The notion of bounded variation for a function $f: \R^d \to \R$, where $d \ge 2$, is more involved than when $d=1$. In fact there is no unique notion of bounded variation for a function when $d \ge 2$. Below we describe the notion of Hardy and Krause variation as given in~\citet{Variationpaper}, which suffices for our purpose.
Let $f: [-1,1]^d \to \R$ be a measurable function. Let $a=(a_1,\ldots,a_d)$ and $b=(b_1,\ldots,b_d)$ be elements of $[-1,1]^d$ such that $a<b$ (coordinate-wise). We introduce the d-dimensional difference operator $\Delta^{(d)}$ which assigns to the axis-aligned box $A:=[a,b]$ a d-dimensional quasi-volume
$$\Delta^{(d)}(f;A)=\sum_{j_1=0}^1\cdots \sum_{j_d=0}^1 (-1)^{j_1+\cdots+j_d}f(b_1+j_1(a_1-b_1),\ldots,b_d+j_d(a_d-b_d)).$$
Let $m_1, \ldots, m_d \in \N$. For $s=1,\ldots,d$, let $-1=:x_0^{(s)}<x_1^{(s)}<\cdots < x_{m_s}^{(s)}:=1$ be a partition of $[-1,1]$ and let $\mathsf{P}$ be a partition of $[-1,1]^d$ which is given by
$$\mathsf{P}:=\left\{[x_{l_1}^{(1)},x_{l_1+1}^{(1)}] \times \cdots \times [x_{l_d}^{(d)},x_{l_d+1}^{(d)}]: \; l_s=0,1,\ldots,m_s-1, \;\mbox{for } s=1,\ldots,d \right\}.$$
Then the variation of $f$ on $[-1,1]^d$ in the sense of {\it Vitali} is given by
$$V^{(d)}(f;[-1,1]^d):=\sup_{\mathsf{P}} \sum_{A \in \mathsf{P}}|\Delta^{(d)}(f;A)|$$
where the supremum is extended over all partitions of $[-1,1]^d$ into axis-parallel boxes generated by $d$ one-dimensional partitions of $[-1,1]$. For $1 \leq s \leq d$ and $1\leq i_1 < \ldots < i_s \leq d$, let $V^{(s)}(f;i_1,\ldots,i_s;[-1,1]^d)$ denote the $s$-dimensional variation
in the sense of Vitali of the restriction of $f$ to the face $$U_d^{(i_1,\ldots,i_s)}=\left\{ (x_1,\ldots,x_d) \in [-1,1]^d : x_j = 1 \mbox{ for all } j \neq i_1,\ldots, i_s \right\}$$ of $[-1,1]^d$. Then the variation of $f$ on $[-1,1]^d$ in the sense of Hardy and Krause anchored at 1, abbreviated by HK-variation, is given by $$TV(f):= \sum_{i=1}^d \sum_{1 \leq s \leq d} V^{(s)}(f;i_1,\ldots,i_s;[-1,1]^d).$$
We say a function $f$ has bounded HK-variation if $TV(f) < \infty.$
\end{Def}
The main property of a bounded HK-variation function that we will need in this paper is stated below.
\begin{rem}
If $f$ is a right continuous function on $[-1,1]^d$ which has bounded HK-variation then there exists a unique signed Borel measure $\nu$ on $[-1,1]^d$ for which $$f(x)=\nu([-1,x]), \quad x \in [-1,1]^d;$$see \citet{Variationpaper} for details.
\end{rem}
\subsection{Proof of Theorem~\ref{Thm0}}\label{pf:Thm0}
In the following proofs $K$ would be used to denote a generic constant whose value would change from line to line.
For every $v > 0$, we define $$\Gamma(X,v):=\sup_{a,b \in \mathscr{F},\rho(a,b)\leq v} |X(a)-X(b)|.$$ For simplicity we divide the proof in three steps.\newline
\noindent \textbf{Step 1:} In this step we will prove that \begin{equation}\label{tailsup}
\ensuremath{{\mathbb P}}\big(\Gamma(X,v) > \eta \big) \leq K \exp \left( -\frac{\eta^2}{Kv^2\log(e/v)} \right) \quad \forall \, \eta >0 \mbox{ and } v \in (0,1],
\end{equation} where $K>0$ is a positive constant not depending on $v$. We will prove the above result by introducing the notion of Orlicz norm. Let $\lambda:\mathbb{R}_+ \to \mathbb{R}$ be a nondecreasing convex function with $\lambda(0)=0$. For any random variable $X $ the Orlicz norm $\norm{X}_{\lambda}$ is defined as $$\norm{X}_{\lambda}= \inf\bigg\{C>0: \E\lambda \left(\frac{|X|}{C}\right)\leq 1\bigg\}.$$ The Orlicz norm is of interest to us as any Orlicz norm easily yields a bound on the tail probability of a random variable i.e., $\ensuremath{{\mathbb P}}(|X|>x) \leq [\lambda(x/\norm{X}_{\lambda})]^{-1}, \mbox{ for all } x \in \R.$
Let us define $\lambda(x) :=\exp(x^2)-1$, $x >0$.
Hence,
\begin{equation}\label{eq:Orlicz}
\ensuremath{{\mathbb P}} \big(|X|>x\big) \leq \min \bigg\{1,\frac{1}{\exp(x^2/\norm{X}^2_{\lambda})-1}\bigg\} \leq 2\times \exp(-x^2/\norm{X}^2_{\lambda}).
\end{equation} Hence, it is enough to bound the Orlicz norm of $\Gamma(X,v)$. A bound on the Orlicz norm of $\Gamma(X,v)$ can be shown by appealing to \citet[Theorem 2.2.4]{VW01} which we state below.
\begin{lem}\label{lem:Cov_no}
Let $\lambda:\R_+\to \R$ be a convex, nondecreasing, non-zero function with $\lambda(0)=0$ and for some constant $c>0$, $\limsup_{x,y \to \infty} \frac{\lambda(x)\lambda(y)}{\lambda(cxy)} < \infty $. Let $\{X_a, a \in \mathscr{F}\}$ be a separable stochastic process with $$\norm{X_a - X_b}_{\lambda} \leq C \rho(a,b) \mbox{ for all } a,b \in \mathscr{F} $$ for some pseudometric $\rho$ on $ \mathscr{F}$ and constant C. Then for any $\zeta , v > 0$,
$$ \norm{\Gamma(X,v)}_{\lambda} \leq K \left[\int_{0}^{\zeta} \lambda^{-1}(N(\epsilon,\mathscr{F})) d\epsilon + v \lambda^{-1}(N^2(\zeta,\mathscr{F}))\right] $$ for some constant $K$ depending only on $\lambda$ and $C$.
\end{lem}
We apply the above lemma with $\lambda(x) :=\exp(x^2)-1$ (i.e., $\lambda^{-1}(y)=\sqrt{\log(1+y)}$). Note that condition (b) of Theorem~\ref{Thm0} directly implies that $\norm{X_a - X_b}_{\lambda} \leq C \rho(a,b)$ by an application of~\citet[Lemma 2.2.1]{VW01}.
By taking $\delta=1,\epsilon=u^{1/2}$, condition (c) of Theorem~\ref{Thm0} yields $N(\epsilon,\mathscr{F}) \leq A\epsilon^{-2B}$. Thus, Lemma~\ref{lem:Cov_no} gives (with $\zeta = v$)
$$ \norm{\Gamma(X,v)}_{\lambda} \leq K \left[ \int_{0}^{v} \sqrt{\log(1+A\epsilon^{-2B})} d\epsilon + v \sqrt{\log(1+A^2v^{-4B})} \right].$$
The expression on the right side of the above display can be easily shown to be less than or equal to $ K v \sqrt{\log(e/v)}$ for some constant $K$. This result along with an application of~\eqref{eq:Orlicz} with $\Gamma(X,v)$ instead of $X$ imply
$$\ensuremath{{\mathbb P}}\big(\Gamma(X,v) > \eta \big) \leq K \exp\left(-\frac{\eta^2}{Kv^2\log(e/v)}\right) \qquad \mbox{for all } \eta >0, \; 0<v\leq1,$$ for some constant $K$.\newline
\noindent \textbf{Step 2:} Let us define $\mathscr{F(\delta)}:= \{a \in \mathscr{F}: \delta/2 < \sigma^2(a) \leq \delta\}$, for $\delta \in (0,1]$, and
\begin{equation}\label{eq:Pi-delta}\Pi(\delta):=\ensuremath{{\mathbb P}}\left(\frac{X^2(a)}{\sigma^2(a)} > 2V\log(\frac{1}{\delta})+S\log\log(\frac{e^e}{\delta}) \mbox{ for some } a\in \mathscr{F}(\delta)\right)
\end{equation}
for $S \geq 4p+1$. In this step we will prove that $$\Pi(\delta) \leq K \exp((K-S/K)\log\log(e^e/\delta))$$ for some constant $K$.
Fix $u < 1/2$. Let $\mathscr{F}(\delta,u)$ be a $\sqrt{u \delta}$-packing set of $\mathscr{F(\delta})$. By our assumption the cardinality of $\mathscr{F}(\delta,u)$ is less than or equal to $Au^{-B}\delta^{-V} (\log({e/\delta}))^p$. Fix $a \in \mathscr{F}(\delta)$. From the definition of $\mathscr{F}(\delta,u)$ we can associate $\hat{a} \in \mathscr{F}(\delta,u)$ (corresponding to $a \in \mathscr{F}(\delta)$) such that $ \rho^2(a,\hat{a}) \leq u\delta$. Using assumption (a) of Theorem~\ref{Thm0} we have \begin{equation}\label{equ:ineq}
\sigma^2(a) \geq \sigma^2(\hat{a}) - u\delta \geq \sigma^2(\hat{a})(1-2u)\end{equation} where the last inequality follows from the fact that $\hat{a} \in \mathscr{F(\delta})$ (thus $\sigma^2(\hat{a}) > \delta/2$).
We want to study the event
\begin{equation}\label{event}
\frac{X^2(a)}{\sigma^2(a)} > r
\end{equation} for some $r>0$. Obviously,
for any $\lambda \in (0,1)$, either (i) $|X(a) - X(\hat{a})|^2 > \lambda^2 X^2(a)$ or (ii) $|X(a) - X(\hat{a})|^2 \le \lambda^2 X^2(a)$ (which, in particular implies $|X(\hat{a})| \ge (1 - \lambda) |X(a)|$). The above two cases reduce to:
\begin{equation}\label{eventcase1}
\Gamma(X,(u\delta)^{1/2})^2 \geq |X(a) - X(\hat{a})|^2 > \lambda^2 X^2(a) \geq \lambda^2 r \sigma^2(a) \geq\lambda^2 r \frac{\delta}{2}
\end{equation}
(here the first inequality follows from the definition of $\Gamma(X,(u\delta)^{1/2})$ and the third inequality follows from condition \eqref{event}), and
\begin{equation}\label{eventcase2}
X^2(\hat{a}) \geq (1-\lambda)^2 X^2(a) \geq (1-\lambda)^2 r \sigma^2(a) \geq (1-\lambda)^2 r (1-2u) \sigma^2(\hat{a})
\end{equation} (here the second inequality follows from \eqref{event} and last inequality follows from \eqref{equ:ineq}). Therefore, for any $r >0$,
\begin{eqnarray*}
\Pi_r(\delta) &:= &\ensuremath{{\mathbb P}}\left(\frac{X^2(a)}{\sigma^2(a)} > r \mbox{ for some } a \in \mathscr{F(\delta)}\right)\\
& \leq & \ensuremath{{\mathbb P}}\left(\Gamma(X,(u\delta)^{1/2})^2 > \lambda^2 \delta r/2\right) \\
& & \qquad \qquad \qquad + \sum_{\hat{a} \in \mathscr{F}(\delta,u)} \ensuremath{{\mathbb P}}\left(X^2(\hat{a})/\sigma^2(\hat{a}) > (1-\lambda)^2 r (1-2u)\right)
\end{eqnarray*}
where we have used the fact that if $X^2(a)/\sigma^2(a) > r$ for some $a \in \mathscr{F}$, then either~\eqref{eventcase1} holds or~\eqref{eventcase2} is satisfied for some $\hat{a} \in \mathscr{F}(\delta,u)$. The first term on the right side of the above display can be bounded by appealing to~\eqref{tailsup} with $\eta=\sqrt{\lambda^2\delta r /2}$ and $v=\sqrt{u\delta}$ and the second term can be bounded by using conditions (a) and (c) of Theorem~\ref{Thm0}. Hence we get
\begin{align}
\Pi_r(\delta) & \leq K \exp\left(- \frac{\lambda^2 \delta r/2}{Ku\delta \log(e/\sqrt{u\delta})} \right) \nonumber \\
& \qquad \qquad \qquad+ A u^{-B}\delta^{-V} \big(\log(\frac{e}{\delta})\big)^p \exp\left(-\frac{(1-\lambda)^2 r (1-2u)}{2}\right) \nonumber \\
& \leq K \Big[\exp\left(-\frac{\lambda^2 r}{K u \log(e/(u\delta))}\right) \nonumber \\
& \qquad + \exp\big(B \log(1/u)+V\log(1/\delta)+p\log\log(e/\delta)+ur- (1/2-\lambda)r \big)\Big]. \label{eq1}
\end{align}
Fix $S \geq 8p+1$ and set $$r:= 2V \log(1/\delta) + S \log\log\big(\frac{e^e}{\delta}\big)$$ and $$\lambda:=\frac{1}{r} \Big((S/4)\log\log(e^e/\delta) - p\log \log (e/\delta)\Big).$$ Observe that $r >1$ and $0<\lambda< 1/4$. Moreover, we have $$(1/2-\lambda)r= V\log(1/\delta)+p\log\log(e/\delta)+ (S/4)\log\log(e^e/\delta).$$
Putting these values in~\eqref{eq1} gives us
\begin{eqnarray}
\Pi(\delta) \equiv \Pi_r(\delta)
& \leq & K \Bigg[\exp\left(-\frac{(S-4p)^2 (\log\log(e^e/\delta))^2}{Kur\log(e/(u\delta))}\right) \nonumber \\
& &\quad \quad \quad+ \exp\Big(B\log(1/u)+ ur- (S/4)\log\log(e^e/\delta)\Big) \Bigg] \qquad \label{eq2}
\end{eqnarray}
where we have used the fact that $\lambda^2 r^2 = ((S/4)\log\log(e^e/\delta) - p\log \log (e/\delta))^2 \ge (S-4p)^2 (\log\log(e^e/\delta))^2/16$.
Now, let us pick $$u:= \frac{S}{8r \log(e/\delta)} < \frac{1}{2}.$$
Then we have $\frac{1}{u} \leq K \log^2(e/\delta)$ for some constant $K$. Let us consider the two terms on the right side of~\eqref{eq2} separately. For the first term, using $ur = S[ \log(e/\delta)]^{-1}/8$, and that $\frac{1}{u} \leq K \log^2(e/\delta)$, we have
{\small \begin{eqnarray*}
\frac{(S-4p)^2 (\log\log(e^e/\delta))^2}{Kur\log(e/(u\delta))} & = &\frac{8(S-8p + 16p^2/S) (\log\log(e^e/\delta))^2\log(e/\delta)}{K\big(\log(e/\delta) + \log (u^{-1})\big)}\\
& \ge &(S-8p)(\log\log(e^e/\delta))\Big(\frac{(\log\log(e^e/\delta))\log(e/\delta)}{K\big(\log(e/\delta) + \log K + 2\log \log (e/\delta)\big)}\Big)\\
& \geq & (1/K')(S-8p)(\log\log(e^e/\delta)).
\end{eqnarray*}}
Here the last inequality follows from the following fact: As $$\tau(\delta) := \frac{(\log\log(e^e/\delta))\log(e/\delta)}{K\big(\log(e/\delta) + \log K + 2\log \log (e/\delta)\big)} \to \infty, \qquad \mbox{ as }\delta \to 0,$$ we can find a lower bound $K'>0$ such that $\tau(\delta) \ge 1/K'$ for all $\delta \in (0,1]$.
For the second term on the right side of~\eqref{eq2} we have
\begin{eqnarray*}
&& B\log(1/u)+ur -(S/4)\log\log(e^e/\delta) \\
& \leq & B\log K+2B\log\log(e/\delta)+S/8-(S/4)\log\log(e^e/\delta)\\
& \leq & B\log K+2B\log\log(e/\delta) - (S/8)\log\log(e^e/\delta)\\
& \leq & B\log K+(2B-S/8)\log\log(e^e/\delta).\end{eqnarray*}
Thus, both the terms on the right side of~\eqref{eq2} have the form $K \exp[(C - S/K') \log\log(e^e/\delta)]$ for some constants $K,C,K' >0$. Putting these values in~\eqref{eq2} gives us, for suitable constant $K >0$, we get $$\Pi(\delta)\leq K \exp\left((K-S/K)\log\log(e^e/\delta)\right).$$
\noindent \textbf{Step 3:} In this step we will prove that as $S \to \infty$ $$\ensuremath{{\mathbb P}}\left(X^2(a)/\sigma^2(a) > 2V\log(1/\sigma^2(a)) + S \log\log\big(\frac{e^e}{\sigma^2(a)}\big)\mbox{ for some } a \in \mathscr{F} \right) \to 0.$$
First let us define $$\tilde{\Pi}(\delta) :=\ensuremath{{\mathbb P}}\left(X^2(a)/\sigma^2(a) > 2V\log(1/\sigma^2(a)) + S \log\log\big(\frac{e^e}{\sigma^2(a)}\big)\mbox{ for some } a \in \mathscr{F}(\delta) \right).$$
Comparing with~\eqref{eq:Pi-delta} we can see that for any $\delta \in (0,1]$, $$\tilde{\Pi}(\delta) \leq \Pi(\delta) $$ as: If $a \in \mathscr{F}(\delta)$ then $\sigma^2(a) \leq \delta$ and $x \longmapsto 2V\log(1/x)+ S \log \log (e^e/x)$ is a decreasing function of $x$. Hence, we have
$$\tilde{\Pi}(\delta)\leq K \exp\left((K-S/K)\log\log(e^e/\delta)\right).$$
Therefor, as $\mathscr{F}=\bigcup_{l\geq 0} \mathscr{F}(2^{-l})$,
\begin{align*}
\ensuremath{{\mathbb P}} \Big(X^2(a)/\sigma^2(a) & > 2V \log(1/\sigma^2(a)) + S \log\log(\frac{e^e}{\sigma^2(a)})\mbox{ for some } a \in \mathscr{F} \Big)\\
& \leq \sum_{l=0}^\infty \tilde{\Pi}(2^{-l})\\
& \leq K \sum_{l=0}^\infty \exp((K-S/K)\log\log(e^e2^l))\\
&= K \sum_{l=0}^\infty (e+l\log 2)^{-(S/K-K)} \quad
\to 0 \quad \mbox{ as } S\to \infty.
\end{align*}
This proves that $S(X) :=\sup_{a \in \mathscr{F}} \frac{X^2(a)/\sigma^2(a) - 2V\log(1/\sigma^2(a))}{\log \log (e^e/\sigma^2(a))} < \infty$ a.s.\qed
\subsection{Proof of Lemma~\ref{lem1}}\label{pf:lem1}
First let us define the following sets:
\begin{eqnarray*}
\mathscr{F}_{\delta,(l_1,\ldots ,l_d)} & := &\big\{(t,h) \in \mathscr{F}: \delta/2 < \sigma^2(t,h)\leq \delta, \; 2^{l_i-1} < \frac{h_i}{\delta^{1/d}} \leq 2^{l_i}, \;\forall \; i=1,\ldots ,d \big\} \\
& & \hspace{2.5in} \mbox{ for some }(l_1,\ldots,l_d) \in \mathbb{Z}^d, \\
\mathscr{F}(\delta) & := & \big\{(t,h) \in \mathscr{F}: \delta/2 < \sigma^2(t,h)\leq \delta \big\}.
\end{eqnarray*}
We note that $\mathscr{F}_{\delta,(l_1,\ldots ,l_d)}$ is empty unless we have
$$ \mathrm{(i)} \quad l_i \leq (1/d) \log_2(1/\delta) \qquad \mbox{for all } i=1,\ldots ,d;$$ (this restriction is a consequence of the fact that $h_i \leq 1/2$) and $$ \mathrm{(ii)} \quad -(d+1) < \sum_{i=1}^d l_i \leq 0$$ (this restriction is a consequence of the fact that $\delta/2 < \sigma^2(t,h) \leq \delta$).\newline
\noindent{\bf Step 1:}
First, we will show that for any $(l_1,\ldots , l_d) \in \mathbb{Z}^d$, and $\delta,u \in (0,1]$,
\begin{equation}\label{eq3}
N\left((u\delta)^{1/2},\mathscr{F}_{\delta,(l_1,\ldots,l_d)}\right) \leq K u^{-2d}\delta^{-1}.
\end{equation}
Let $\mathscr{F}^{\prime}$ be a subset of $\mathscr{F}_{\delta,(l_1,\ldots,l_d)}$ such that for any two elements $(t,h),(t^\prime,h^\prime) \in \mathscr{F}^\prime$ we have \begin{equation}\label{packingcond} \rho^2((t,h),(t^\prime,h^\prime)) > u\delta.\end{equation} Our aim is to show that $$|\mathscr{F}^\prime| \leq Ku^{-2d}\delta^{-1} ,$$ for some constant $K$ independent of $(l_1,\ldots,l_d)$, $u$ and $\delta$. If $\mathscr{F}_{\delta,(l_1,\cdots,l_d)} $ is empty then the assertion is trivial. So assume that $\mathscr{F}_{\delta,(l_1,\cdots,l_d)} $ is non-empty which imposes bounds on the $l_i$'s as shown above.
Let us define the following partition of $[0,1]^d$ into disjoint hyperrectangles:
\begin{eqnarray*}
R:=\Big\{M_{(i_1,\ldots, i_d)}\cap [0,1]^d: M_{(i_1,\ldots, i_d)} :=\Pi_{k=1}^d \Big((i_k-1)\frac{u\delta^{\frac{1}{d}}2^{l_k}}{c}, i_k\frac{u\delta^{\frac{1}{d}}2^{l_k}}{c} \Big], \\
\qquad \qquad \: 1 \le i_k \le \lceil cu^{-1}\delta^{-\frac{1}{d}}2^{-l_k} \rceil \Big\}
\end{eqnarray*}
where we take $c :=d4^d$.
We would like to point out that in the above definition when $i_k=1$, for any $k=1,\ldots,d$, by $\big((i_k-1)c^{-1}u\delta^{1/d}2^{l_k},$ $ i_kc^{-1}u\delta^{1/d}2^{l_k}\big]$ we mean the closed interval $\big[0, c^{-1}u\delta^{1/d}2^{l_k}\big]$. Observe that all the sets in $R$ are disjoint and moreover
\begin{equation*}
\bigcup_{M\in R} M =[0,1]^d.
\end{equation*}
Observe that \begin{eqnarray*}
2^{l_i-1}\delta^{1/d} < h_i \leq 1/2 \;\; \Rightarrow \;\;2^{l_i} \delta^{1/d} <1 &\Rightarrow & cu^{-1}\delta^{-1/d}2^{-l_i} > 1 \\ & \Rightarrow & \lceil cu^{-1}\delta^{-1/d}2^{-l_i} \rceil \leq 2cu^{-1}\delta^{-1/d}2^{-l_i}.
\end{eqnarray*}
Hence we can easily see that $$|R|=\Pi_{i=1}^d \lceil cu^{-1}\delta^{-1/d}2^{-l_i} \rceil \leq 2^d c^d u^{-d}\delta^{-1} 2^{-\sum_{i=1}^d l_i}\leq 2^{2d+1} c^d u^{-d}\delta^{-1}. $$ Here the last inequality follows from the fact that $\sum_{i=1}^d l_i \geq -(d+1)$.
Let us define the following set:
\begin{eqnarray*}
R_2:=\Big\{(M_{\underset{\sim}{i}},M_{\underset{\sim}{i}^\prime})\in R \times R: \exists \; (t,h)\in \mathscr{F}^{\prime} \mbox{ such that } t-h \in M_{\underset{\sim}{i}} \mbox{ and } t+h \in M_{\underset{\sim}{i}^\prime} \Big\}.
\end{eqnarray*}
Note that if $(t,h)\in \mathscr{F}^{\prime}$ then $h_k \leq 2^{l_k}\delta^{1/d}$ for all $k=1,\ldots,d$. This implies that if $(M_{\underset{\sim}{i}},M_{\underset{\sim}{i}^\prime}) \in R_2$, where $\underset{\sim}{i} = (i_1,\ldots, i_d)$ and $\underset{\sim}{i}^\prime = (i^\prime_1, \ldots, i^\prime_d)$, then
\begin{equation}\label{eq:i'_k-i_k}
(i^\prime_k- i_k) \leq (1+ 2c u^{-1} ), \qquad \mbox{ for all } \;k=1,\ldots,d,
\end{equation}
as (i) $(i^\prime_k - 1) u\delta^{{1}/{d}}2^{l_k} c^{-1} \le t_k + h_k$, and (ii) $i_k u\delta^{\frac{1}{d}}2^{l_k} c^{-1} \ge t_k - h_k$. Thus for each hyperrectangle $M_{\underset{\sim}{i}} \in R$ the number of hyperrectangles $M_{\underset{\sim}{i}^\prime} \in R$ such that $(M_{\underset{\sim}{i}},M_{\underset{\sim}{i}^\prime}) \in R_2$ is less than or equal to $(1+2c u^{-1})^d \leq 4^d c^d u^{-d}$. Hence we have $$|R_2|\leq |R| \times 4^d c^d u^{-d} \leq 2^{4d+1} c^{2d} u^{-2d}\delta^{-1} \leq d^{2d} 2^{4d^2+4d+1} u^{-2d}\delta^{-1}.$$
Thus, our proof will be complete if we can show that $|R_2|=|\mathscr{F}^\prime|$. From the definition of $R_2$ and the fact that elements in $R$ are disjoint it is easy to observe that $|R_2|\leq |\mathscr{F}^\prime|$. \\
Therefore, the only thing left to show is that $|\mathscr{F}^\prime|\leq |R_2|$. Let us assume the contrary, i.e., $|R_2| < |\mathscr{F}^\prime|$. This implies that there exist two elements $(t,h) $ and $(t^\prime,h^\prime) \in \mathscr{F}^{\prime}$ and $(M_{\underset{\sim}{i}},M_{\underset{\sim}{i}^\prime}) \in R_2 $ such that both $t-h$ and $t^\prime-h^\prime$ belong to $M_{\underset{\sim}{i}}$ and, also, $t+h$ and $t^\prime+h^\prime$ belong to $M_{\underset{\sim}{i}^\prime}$. Let us first define the following two hyperrectangles:
$$B_1:=\Pi_{k=1}^d (i_k-1,i^\prime_k] \times c^{-1}u\delta^{1/d}2^{l_k} \qquad \mbox{ and } \qquad B_2:=\Pi_{k=1}^d (i_k,i^\prime_k-1] \times c^{-1}u\delta^{1/d}2^{l_k}.$$ Our goal is to show that
\begin{equation}\label{eq:B_infty}
B_\infty(t,h) \bigtriangleup B_\infty(t^\prime,h^\prime)\subseteq B_1 \setminus B_2
\end{equation}
which is implied by the following two assertions:
\begin{enumerate}
\item[(1)] $B_\infty(t,h) \cup B_\infty(t^\prime,h^\prime) \subseteq B_1$ and
\item [(2)] $B_2 \subseteq B_\infty(t,h)\cap B_\infty(t^\prime,h^\prime)$.
\end{enumerate}
See the figure below for a visual illustration of~\eqref{eq:B_infty} when $d=2$.
\begin{figure}
\includegraphics[scale=.48]{grid.png}\caption{The figure shows how the symmetric difference of the hyperrectangles $B_\infty(t,h)$ (denoted by the green border) and $B_\infty(t^\prime,h^\prime)$ (denoted by the blue border) is contained in the set $B_1 \setminus B_2$ (denoted by the shaded region). }
\end{figure}
Now, as $t-h\in M_{\underset{\sim}{i}}$, this implies $t_k-h_k \ge (i_k-1)c^{-1}u\delta^{1/d}2^{l_k}$, for all $k=1,\ldots,d$. Also $t+h \in M_{\underset{\sim}{i}^\prime}$ implies that $t_k+h_k \leq i^\prime_{k}c^{-1}u\delta^{1/d}2^{l_k} $, for all $k=1,\ldots,d.$ Therefore, $B_\infty(t,h) = \Pi_{i=1}^d(t_i-h_i,t_i+h_i) \subseteq B_1 $. A similar argument shows that $B_\infty(t^\prime,h^\prime) \subseteq B_1$. Hence assertion $(1)$ above holds. \newline
Now as $t-h\in M_{\underset{\sim}{i}}$, we have $t_k-h_k \leq i_kc^{-1}u\delta^{1/d}2^{l_k} $, for all $k=1,\ldots,d.$ Also $t+h \in M_{\underset{\sim}{i}^\prime}$ implies that $t_k+h_k \ge (i^\prime_{k}-1)c^{-1}u\delta^{1/d}2^{l_k} $, for all $k=1,\ldots,d$. Hence we have $B_2 \subseteq B_{\infty}(t,h)$. A similar argument shows that $B_2 \subseteq B_\infty(t^\prime,h^\prime)$. Therefore, assertion $(2)$ is also satisfied. Now let us define the following set \begin{eqnarray*}I:=\big\{{\underset{\sim}{j}}=(j_1,\ldots,j_d) \in \mathbb{N}^d & : & j_k \in (i_k-1,i^\prime_k], \mbox{ for all } k=1,\ldots,d, \\
& & \exists \: l\in\{1,\ldots,d\} \mbox{ such that } j_l= i_l \mbox{ or } i_l^\prime \big\}.\end{eqnarray*}
Clearly, using~\eqref{eq:i'_k-i_k}, $$ |I| \leq 2d(2+2cu^{-1})^{d-1}.$$
Also see that $w = (w_1,\ldots, w_d) \in B_1\setminus B_2$ if and only if
\begin{enumerate}
\item[(1)] for every $k=1,\ldots,d$, we have $w_k \in \big(i_k-1, i_k^{\prime}\big] \times c^{-1}u\delta^{1/d}2^{l_k}$ (this is true as $w\in B_1$),
\item[(2)] there exists $l \in \{1,2,\ldots,d\}$ such that either $w_l \in \big(i_l-1, i_l\big] \times c^{-1}u\delta^{1/d}2^{l_l}$ or $w_l \in \big(i_l^\prime -1, i_l^\prime \big] \times c^{-1}u\delta^{1/d}2^{l_l}$ (this is true as $w\not \in B_2$ implies that there exist $l$ such that $w_l \not\in (i_l,i_l^\prime-1]\times c^{-1}u\delta^{1/d}2^{l_l}$ and $w \in B_1$ implies that $w_l \in (i_l-1,i_l^\prime] \times c^{-1}u\delta^{1/d}2^{l_l} $).
\end{enumerate}
Therefore, we see that $$B_1\setminus B_2= \bigcup_{{\underset{\sim}{j}} \in I} M_{{\underset{\sim}{j}}}.$$ Also, note that, $|M_{\underset{\sim}{j}}| \le u^d\delta c^{-d}2^{\sum_{i=1}^d l_i} \leq u^d\delta c^{-d}$ for all ${\underset{\sim}{j}}$.
Therefore, using~\eqref{eq:B_infty} and the fact that $c=d4^d$, we easily see that $$\rho^2((t,h),(t^\prime,h^\prime)) \leq |B_1\setminus B_2| \leq 2d (2+2cu^{-1})^{d-1} \times \frac{u^d\delta}{c^d} \le 2^d d (1 + c^{-1})^{d-1} u \delta c^{-1} < u\delta $$
which contradicts \eqref{packingcond}. This proves that two elements of $\mathscr{F}^{\prime}$ cannot correspond to the same pair of hyperrectangles $(M_{\underset{\sim}{i}},M_{\underset{\sim}{i}^\prime}) \in R_2$. Hence we have proved \eqref{eq3}.\newline
\noindent{\bf Step 2: }
In this part of the proof we show that
\begin{equation}\label{eq4}
N\left((u\delta)^{1/2},\mathscr{F}(\delta)\right) \leq K u^{-2d}\delta^{-1} (\log(e/\delta))^{d-1}.
\end{equation}
Let us define the set {\small $$S:=\Big\{(l_1,\ldots,\l_d) \in\mathbb{Z}^d:-(d+1) < \sum_{k=1}^d l_k \leq 0 \mbox{ and } l_k \leq \frac{1}{d} \log_2(1/\delta) \mbox{ for all } k=1,\ldots ,d \Big\}.$$}
Now it can be easily seen that $l:=(l_1,\cdots,\l_d) \in S$ implies $l_k \geq -(d+1) - (d-1)(1/d)\log_2(1/\delta)$, for all $k=1,\ldots,d$. This shows that each $l_k$ can only take at most $(d+2)+ \log_2(1/\delta) \leq (d+2)+\log(1/\delta) \log_2(e) \leq d+2(\log(e/\delta))$ many values. This shows that $$|S| \leq (d+1) (d+2\log(e/\delta))^{d-1} \leq (d+2)^d(\log(e/\delta))^{d-1}.$$
Note that the power of $(d+2\log(e/\delta))$ in the above display is $d-1$ because if we fix the values of $l_1,l_2,\ldots, l_{d-1}$ then $l_d$ can only take at most $(d+1)$ values such that $(l_1,l_2,\ldots l_{d}) \in S$ (as $\sum_{k=1}^d l_k$ can take at most $d+1$ distinct values). Also note that
\begin{equation*}
\mathscr{F}(\delta)\subseteq \bigcup_{l \in S} \mathscr{F}_{\delta,l}.
\end{equation*}
The above representation of $\mathscr{F}(\delta)$ along with the trivial fact that $N(\epsilon,\bigcup_{i=1}^n A_i)\leq \sum_{i=1}^n N(\epsilon,A_i)$ gives us \eqref{eq4}.\\[.4cm]
{\bf Step 3: }
In this step we will complete the proof of Lemma~\ref{lem1}. We want control the $\sqrt{u\delta}$-packing number of the set $\{(t,h)\in \mathscr{F}: \sigma^2(t,h) \leq \delta\} $ which can be decomposed in the following way: for $u \in (0,1]$,
$$\{(t,h)\in \mathscr{F}: \sigma^2(t,h) \leq \delta\} = \left(\bigcup_{l=0}^{\lfloor 1+\log_2(1/u)\rfloor} \mathscr{F}(\delta2^{-l})\right) \cup \{a \in \mathscr{F}: \sigma^2(a) \leq u\delta/2\}.$$
Now we can control the $\sqrt{u\delta}$-packing number of each of the above sets. First observe that
$$N((u\delta)^{1/2},\{(t,h) \in \mathscr{F}: \sigma^2(t,h) \leq u\delta/2\})=1. $$
Also, for any $u\in(0,2)$ and $\delta\in(0,1]$ we have
\begin{equation}\label{eq4.2}
N((u\delta)^{1/2},\mathscr{F}(\delta)) \leq N((u\delta/2)^{1/2},\mathscr{F}(\delta))\leq K u^{-2d}\delta^{-1} (\log(e/\delta))^{d-1}
\end{equation}
for some constant $K$. Putting $\delta\leftarrow\delta/2^l$ and $u\leftarrow2^lu$ for $0\le l\leq \lfloor 1+\log_2(1/u)\rfloor$ in \eqref{eq4.2} we get $$N((u\delta)^{1/2},\mathscr{F}(\delta2^{-l}) ) \leq K 2^{-(2d-1)l} u^{-2d} \delta^{-1}(\log(e/\delta))^{d-1}.$$ Now from the trivial fact that $N(\epsilon,\bigcup_{i=1}^m A_i)\leq \sum_{i=1}^m N(\epsilon,A_i)$ we get
\begin{eqnarray*}
&& N\left(\sqrt{u\delta},\{(t,h)\in \mathscr{F}: \sigma^2(t,h) \leq \delta\}\right) \\
& \leq & \sum_{l=0}^{\lfloor 1+\log_2(1/u)\rfloor}N\left(\sqrt{u\delta},\mathscr{F}(\delta2^{-l})\right) + N\left(\sqrt{u\delta},\{(t,h) \in \mathscr{F}: \sigma^2(t,h) \leq u\delta/2\}\right)\\
& \leq & 1+ Ku^{-2d}\delta^{-1} (\log(e/\delta))^{d-1} \sum_{l=0}^\infty 2^{-(2d-1)l}\\
& \leq & 1+ 2Ku^{-2d}\delta^{-1} (\log(e/\delta))^{d-1}\\
& \leq &(2K+1)u^{-2d}\delta^{-1} (\log(e/\delta))^{d-1},
\end{eqnarray*}
which proves Lemma~\ref{lem1}. \qed \newline
\subsection{Proof of Theorem~\ref{Thm 1}}\label{sec:Thm 1}
We use Theorem~\ref{Thm0} to prove Theorem~\ref{Thm 1}.
Let us recall the definitions of $\mathscr{F}, \sigma$ and $\rho$ as introduced just before Lemma~\ref{lem1}. Without loss of generality we assume that $\norm{\psi}=1$. For $h \in (0,1/2]^d$, let us define the stochastic process
$$X(t,h) :=2^{d/2}(h_1h_2 \ldots h_d)^{1/2} \hat{\Psi}(t,h)=2^{d/2} \int \psi_{t,h}(x) dW(x), \qquad t \in A_h,$$ where $W(\cdot)$ is the standard Brownian sheet on $[0,1]^d$. This defines a centered Gaussian process with $\mathrm{Var}\big(X(t,h)\big)=\sigma^2(t,h)$. Also by a standard calculation on the variance we have $\mathrm{Var}\big(X(t,h)-X(t^\prime,h^\prime)\big) \leq 2^d TV^2(\psi)\rho^2((t,h),(t^\prime,h^\prime))$.
As $X(t,h)$ and $X(t,h)-X(t^\prime,h^\prime)$ have normal distributions this shows that conditions (a) and (b) of Theorem \ref{Thm0} are satisfied. Condition (c) is also satisfied because of Lemma \ref{lem1}. Thus, by an application of Theorem \ref{Thm0} we have
$$\sup_{0<h\leq1/2}\sup_{t\in A_{h}} \frac{\hat{\Psi}^2(t,h)- 2\log(1/2^dh_1h_2...h_d)}{\log\log(e^e/2^dh_1h_2...h_d)} < \infty.$$
For notational simplicity, let us define $\kappa_1 :=2\log(1/\sigma^2(t,h))$~and~$\kappa_2 :=2\sqrt{2} S\log\log({e^e/\sigma^2(t,h)})$. Therefore,
\begin{eqnarray*}
& & {\small \ensuremath{{\mathbb P}} \left(|\hat{\Psi}(t,h)| \leq \sqrt{2\log\left(\frac{1}{\sigma^2(t,h)}\right)}+ S \Bigg(\frac{ \log\log(e^e/\sigma^2(t,h))}{\log^{\frac{1}{2}}(1/\sigma^2(t,h)) } \Bigg) \;\; \forall \; (t,h) \in \mathscr{F} \right)}\\
&=&\ensuremath{{\mathbb P}}\left(|\hat{\Psi}(t,h)| \leq \kappa_1^{1/2}+ \kappa_1^{-1/2} \kappa_2/2 \quad \forall \; (t,h) \in \mathscr{F} \right)\\
&=& \ensuremath{{\mathbb P}} \left(\hat{\Psi}(t,h)^2 \leq \Big(\kappa_1^{1/2}+ \kappa_1^{-1/2} \kappa_2/2\Big)^2 \: \forall (t,h) \in \mathscr{F} \right)\\
&\geq & \ensuremath{{\mathbb P}} \left(\hat{\Psi}(t,h)^2 \leq \kappa_1+\kappa_2 \quad \forall (t,h) \in \mathscr{F} \right)\\
&=& \ensuremath{{\mathbb P}} \left(\sup_{t,h\in \mathscr{F}} \frac{\hat{\Psi}^2(t,h)- 2\log(1/2^dh_1h_2...h_d)}{\log\log(e^e/2^dh_1h_2...h_d)} < 2\sqrt{2}S \right) \; \to \; 1 \quad \mbox{ as }S\to \infty. \qed
\end{eqnarray*}
| {'timestamp': '2018-06-07T02:14:19', 'yymm': '1806', 'arxiv_id': '1806.02194', 'language': 'en', 'url': 'https://arxiv.org/abs/1806.02194'} |
\section{Introduction}
In this essay we introduce fractal transformations. The main examples are
fascinating mappings between diverse subsets of $\mathbb{R}^{2}$; they can be
readily illustrated by using the chaos game. Fractal transformations can be
quicky grasped because they rely on basic notions in topology, probability,
dynamical systems, and geometry. They may be applied to computer graphics to
produce digital content with new look-and-feel \cite{barnhutch}; they may also
be relevant to image compression and biological modelling.
\section{\label{hypersec}Hyperbolic IFS}
\begin{definition}
Let $\mathbb{(X},d_{\mathbb{X}})$ be a complete metric space. Let
$\{f_{1},f_{2},...,f_{N}\}$ be a finite sequence of strictly contractive
transformations, $f_{n}:\mathbb{X\rightarrow X}$, for $n=1,2,...,N$. Then
\[
\mathcal{F}:=\{\mathbb{X};f_{1},f_{2},...,f_{N}\}
\]
is called a hyperbolic iterated function system or hyperbolic IFS.
\end{definition}
A transformation $f_{n}:\mathbb{X\rightarrow X}$ is strictly contractive iff
there exists a number $l_{n}\in\lbrack0,1)$ such that $d(f_{n}(x),f_{n}%
(y))\leq l_{n}d(x,y)$ for all $x,y\in\mathbb{X}$. The number $l_{n}$ is called
a contractivity factor for $f_{n}$ and the number%
\[
l=\max\{l_{1},l_{2},...,l_{N}\}
\]
is called a contractivity factor for $\mathcal{F}$.
Let $\Omega$ denote the set of all infinite sequences of symbols $\{\sigma
_{k}\}_{k=1}^{\infty}$ belonging to the alphabet $\{1,...,N\}$. We write
$\sigma=\sigma_{1}\sigma_{2}\sigma_{3}...\in\Omega$ to denote a typical
element of $\Omega$, and we write $\omega_{k}$ to denote the $k^{th}$ element
of $\omega\in\Omega$. Then $(\Omega,d_{\Omega})$ is a compact metric space,
where the metric $d_{\Omega}$ is defined by $d_{\Omega}(\sigma,\omega)=0$ when
$\sigma=\omega$ and $d_{\Omega}(\sigma,\omega)=2^{-k}$ when $k$ is the least
index for which $\sigma_{k}\neq\omega_{k}$. We call $\Omega$ the code space
associated with the IFS\ $\mathcal{F}$.
Let $\sigma\in\Omega$ and $x\in\mathbb{X}$. Then, using the contractivity of
$\mathcal{F}$, it is straightfoward to prove that
\[
\phi_{\mathcal{F}}(\sigma):=\lim_{k\rightarrow\infty}f_{\sigma_{1}}\circ
f_{\sigma_{2}}\circ...f_{\sigma_{k}}(x)
\]
exists, uniformly for $x$ in any fixed compact subset of $\mathbb{X}$, and
depends continuously on $\sigma$. See for example \cite{BaDe}, Theorem 3. Let
\[
A_{\mathcal{F}}=\{\phi_{\mathcal{F}}(\sigma):\sigma\in\Omega\}\text{.}%
\]
Then $A_{\mathcal{F}}\subset\mathbb{X}$ is called the attractor of
$\mathcal{F}$. The continuous function
\[
\phi_{\mathcal{F}}:\Omega\rightarrow A_{\mathcal{F}}%
\]
is called the address function of $\mathcal{F}$. We call $\phi_{\mathcal{F}%
}^{-1}(\{x\}):=\{\sigma\in\Omega:\phi_{\mathcal{F}}(\sigma)=x\}$ the set of
addresses of the point $x\in A_{\mathcal{F}}$.
Clearly $A_{\mathcal{F}}$ is compact, nonempty, and has the property
\[
A_{\mathcal{F}}=f_{1}(A_{\mathcal{F}})\cup f_{2}(A_{\mathcal{F}})\cup...\cup
f_{N}(A_{\mathcal{F}})\text{.}%
\]
Indeed, if we define $\mathbb{H(X)}$ to be the set of nonempty compact subsets
of $\mathbb{X}$, and we define $\mathcal{F}:\mathbb{H(X)\rightarrow H(X)}$ by%
\begin{equation}
\mathcal{F}(S)=f_{1}(S)\cup f_{2}(S)\cup...\cup f_{N}(S)\text{,}
\label{fseteq}%
\end{equation}
for all $S\in\mathbb{H(X)}$, then $A_{\mathcal{F}}$ can be characterized as
the unique fixed point of $\mathcal{F}$, see \cite{hutchinson}, section 3.2,
and \cite{williams}.
IFSs may be used to represent diverse subsets of $\mathbb{R}^{2}$. For
example,\ let $A$, $B$, and $C$, denote three noncollinear points in
$\mathbb{R}^{2}$. Let $a$ denote a point on the line segment $AB$, let $b$
denote a point on the line segment $BC$ and let $c$ denote a point on the line
segment $CA$, such that $\{a,b,c\}\cap\{A,B,C\}=\varnothing$, see panel (i) of
Figure \ref{homtrisel}.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.0237in,
width=4.8845in
]%
{HomTriSel.eps}%
\caption{(i) The points used to define the affine transformations
$h_{n}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ for $n=1,2,3,4$; (ii) sketch
of the attractor of the IFS $\{\mathbb{R}^{2};h_{1},h_{2},h_{3},h_{4}\}$;
(iii) sketch of the attractor of the IFS $\{\mathbb{R}^{2};h_{1},h_{2}%
,h_{3}\}$. Here $\alpha=0.65$, $\beta=0.3$, and $\gamma=0.4$.}%
\label{homtrisel}%
\end{center}
\end{figure}
Let $h_{1}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ denote the unique affine
transformation such that
\[
h_{1}(ABC)=aBc\text{,}%
\]
by which we mean that $h_{1}$ maps $A$ to $a$, $B$ to $B$, and $C$ to $c$.
Using the same notation, let affine transformations $h_{2}$, $h_{3}$, and
$h_{4}$ be uniquely defined by%
\[
h_{2}(ABC)=abC\text{, }h_{3}(ABC)=Abc\text{, and }h_{4}(ABC)=abc\text{.}%
\]
Let $\mathcal{F}_{\alpha,\beta,\gamma}=\{\mathbb{R}^{2};h_{1},h_{2}%
,h_{3},h_{4}\}$ where $\alpha=|Bc|/|AB|,\beta=|Ca|/|BC|$ and $\gamma
=|Ab|/|CA|$. The attractor of $\mathcal{F}_{\alpha,\beta,\gamma}$ is the
filled triangle with vertices at $A$, $B$, and $C$. The attractor of the IFS
$\{\mathbb{R}^{2};h_{1},h_{2},h_{3}\}$ is an affine Sierpinski triangle, as
illustrated in (iii) in Figure \ref{homtrisel}.
For reference we note that when $A=(0,0)$, $B=(0,1),$ and $C=(0.5,1),$ the
transformations of the IFS $\mathcal{F}_{\alpha,\beta,\gamma}$ are given by
\[
h_{n}(x,y)=(a_{n}x+b_{n}y+c_{n},d_{n}x+e_{n}y+l_{n})
\]
with the parameters specified in Table \ref{ProjTable}. We will write
$\blacktriangle$ to denote the filled triangle $ABC$.%
\begin{table}[tbp] \centering
\begin{tabular}
[c]{|c|c|c|c|c|c|c|}\hline
$n$ & $a_{n}$ & $b_{n}$ & $c_{n}$ & $d_{n}$ & $e_{n}$ & $l_{n}$\\\hline
$1$ & $-1+\beta$ & $-\frac{1}{2}+\frac{1}{2}\beta+\frac{1}{2}\alpha$ &
$1-\beta$ & $0$ & $\alpha$ & $0$\\
$2$ & $\beta+\frac{1}{2}\gamma-\frac{1}{2}$ & $\frac{1}{2}\beta-\frac{1}%
{4}\gamma+\frac{1}{4}$ & $1-\beta$ & $1-\gamma$ & $\frac{1}{2}\gamma-\frac
{1}{2}$ & $0$\\
$3$ & $\frac{1}{2}\gamma$ & $-\frac{1}{2}+\frac{1}{2}\alpha-\frac{1}{4}\gamma$
& $\frac{1}{2}$ & $-\gamma$ & $-1+\alpha+\frac{1}{2}\gamma$ & $1$\\
$4$ & $\beta+\frac{1}{2}\gamma-\frac{1}{2}$ & $-\frac{3}{4}+\frac{1}{2}%
\beta+\frac{1}{2}\alpha-\frac{1}{4}\gamma$ & $1-\beta$ & $1-\gamma$ &
$\alpha-\frac{1}{2}+\frac{1}{2}\gamma$ & $0$\\\hline
\end{tabular}
\caption{ \label{ProjTable}}%
\end{table}%
\section{Chaos game}
When the underlying space is the euclidean plane, one way to sketch the
attactor $A_{\mathcal{F}}$ of an IFS $\mathcal{F}$ is to plot the set of
points%
\[
\widetilde{A}_{\mathcal{F}}=\{f_{\sigma_{1}}\circ f_{\sigma_{2}}%
\circ...f_{\sigma_{K}}(x):\sigma_{k}\in\{1,2,...,N\},k=1,2,...K\},
\]
for some $x\in\mathbb{X}$ and some integer $K$. The Hausdorff distance between
$A_{\mathcal{F}}$ and $\widetilde{A}_{\mathcal{F}}$ is bounded above by
$C\cdot l^{K}$ where the constant $C$ depends only on $\mathcal{F}$ and $x$.
A more efficient method is by means of a type of Markov Chain Monte Carlo
algorithm which we refer to as the chaos game. Starting from any point
$(x_{0},y_{0})\in\mathbb{R}^{2}$, a sequence of a million or more points
$\{(x_{k},y_{k})\}_{k=0}^{K}$ is computed recursively; at the $k^{th}$
iteration one of the functions of $\mathcal{F}$ is chosen at random,
independently of all other choices, and applied to $(x_{k-1},y_{k-1})$ to
produce $(x_{k},y_{k})$ which is plotted when $k\geq100$. The result will be
usually a sketch of the attractor of the IFS, accurate to within viewing resolution.
The reason that the chaos game yields, almost always, a "picture" of the
attractor of an IFS depends on Birkhoff's ergodic theorem, see for example
\cite{vrscay1}. The scholarly history of the chaos game is discussed in
\cite{kaijser} and \cite{stenflo3}, and appears to begin in 1935 with the work
of Onicescu and Mihok, \cite{onicescu}. Mandelbrot used a version of it to
help compute pictures of certain Julia sets, \cite{mandelbrot} pp.196-199; it
was introduced to IFS theory and developed by the author and coworkers, see
for example \cite{BaDe},\ \cite{barnsley}, \cite{marcaBerg}, and
\cite{JEergodic}, where the relevant theorems and much discussion can be
found. Its applications to fractal geometry were popularized initially by the
author and others, see for example \cite{BaDe}, \cite{devaney}, \cite{peak},
and \cite{peitgen}.
The sketches in panels (ii) and (iii) of Figure \ref{homtrisel} were computed
using the chaos game. At each iteration the function $h_{n}$ was selected with
probability proportional to the area of the triangle $h_{n}(ABC)$, for
$n=1,2,3,4$.
In section \ref{topsec} we show how the chaos game may be modified to
calculate examples of the fractal transformations that are the subject of this
article. Hopefully you will be inspired to try this new application of the
chaos game.
\section{\label{topsfnsec}The tops function}
We order the elements of $\Omega$ according to
\[
\sigma<\omega\text{ iff }\sigma_{k}>\omega_{k}%
\]
where $k$ is the least index for which $\sigma_{k}\neq\omega_{k}$. This is a
linear ordering, sometimes called the lexicographic ordering.
Notice that all elements of $\Omega$ are less than or equal to $\overline
{1}=11111...$ and greater than or equal to $\overline{N}=NNNNN....$. Also, any
pair of distinct elements of $\Omega$ is such that one member of the pair is
strictly greater than the other. In particular, the set of addresses of a
point $x\in A_{\mathcal{F}}$ is both closed and bounded above by $\overline
{1}$. It follows that $\phi_{\mathcal{F}}^{-1}(\{x\})$ possesses a unique
largest element. We denote this element by $\tau_{\mathcal{F}}(x)$.
\begin{definition}
Let $\mathcal{F}$ be a hyperbolic IFS with attractor $A_{\mathcal{F}}$ and
address function $\phi_{\mathcal{F}}:\Omega\rightarrow A_{\mathcal{F}}$. Let
\[
\tau_{\mathcal{F}}(x)=\max\{\sigma\in\Omega:\phi_{\mathcal{F}}(\sigma
)=x\}\text{ for all }x\in A_{\mathcal{F}}\text{.}%
\]
Then
\[
\Omega_{\mathcal{F}}:=\{\tau_{\mathcal{F}}(x):x\in A_{\mathcal{F}}\}
\]
is called the tops code space and
\[
\tau_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow\Omega_{\mathcal{F}}%
\]
is called the tops function, for the IFS $\mathcal{F}$.
\end{definition}
Notice that the tops function $\tau_{\mathcal{F}}$ is one-to-one. It provides
a right-hand inverse to the address function, according to%
\[
\phi_{\mathcal{F}}\circ\tau_{\mathcal{F}}=i_{A_{\mathcal{F}}}%
\]
where $i_{A_{\mathcal{F}}}$ denotes the identity function on $A_{\mathcal{F}}$
and $\circ$ denotes composition of functions. Let
\[
\Phi_{\mathcal{F}}:\Omega_{\mathcal{F}}\rightarrow A_{\mathcal{F}}%
\]
denote the restriction of $\phi_{\mathcal{F}}$ to $\Omega_{\mathcal{F}}$,
defined by $\Phi_{\mathcal{F}}(\sigma)=\phi_{\mathcal{F}}(\sigma)$ for all
$\sigma\in\Omega_{\mathcal{F}}$. Then $\Phi_{\mathcal{F}}$ is the inverse of
$\tau_{\mathcal{F}}$, namely%
\[
\Phi_{\mathcal{F}}=\tau_{\mathcal{F}}^{-1}\text{.}%
\]
We note that although $\Phi_{\mathcal{F}}$ is one-to-one, onto, and
continuous, $\tau_{\mathcal{F}}$ may not be continuous. Let $\overline{\Omega
}_{\mathcal{F}}$ denote the closure of $\Omega_{\mathcal{F}}$, treated as a
subset of the metric space $(\Omega,d_{\Omega})$. Let
\[
\overline{\Phi}_{\mathcal{F}}:\overline{\Omega}_{\mathcal{F}}\rightarrow
A_{\mathcal{F}}%
\]
denote the restriction of $\phi_{\mathcal{F}}$ to $\overline{\Omega
}_{\mathcal{F}}$. Then $\overline{\Phi}_{\mathcal{F}}$ is continuous and onto.
Notice that the ranges of $\overline{\Phi}_{\mathcal{F}}$ and $\Phi
_{\mathcal{F}}$ are both equal to $A_{\mathcal{F}}$ because $A_{\mathcal{F}}$
is closed.
\section{Fractal transformations}
Let $\mathcal{G}$ denote a hyperbolic IFS that consists of $N$ functions. Then
$\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow
A_{\mathcal{G}}$ is a mapping from the attractor of $\mathcal{F}$ into the
attractor of $\mathcal{G}$. We refer to $\phi_{\mathcal{G}}\circ
\tau_{\mathcal{F}}$ as a fractal transformation.
In order to illustrate transformations between subsets of $\mathbb{R}^{2}$ we
use pictures. We define a picture function to be a function of the form
\[
\mathfrak{P}:D_{\mathfrak{P}}\subset\mathbb{R}^{2}\rightarrow\mathfrak{C}%
\]
where $\mathfrak{C}$ is a color space. A picture function $\mathfrak{P}$
assigns a unique color to each point in its domain $D_{\mathfrak{P}}$. For
example we may have $\mathfrak{C=}\{0,1,...255\}^{3}$ and each point of
$\mathfrak{C}$ may specify the red, green, and blue components of a color. A
picture in the non-mathematical sense may be thought of as a physical
representation of the graph of a picture function.
If $T:D_{\mathfrak{Q}}\subset\mathbb{R}^{2}\rightarrow D_{\mathfrak{P}}$ then
$\mathfrak{Q}=\mathfrak{P}\circ T$ denotes a picture whose domain is
$D_{\mathfrak{Q}}$. We can obtain insights into the nature of $T$ by comparing
the picture functions $\mathfrak{P}$ and $\mathfrak{P}\circ T$, where
$\mathfrak{P}$ represents a given picture which may be varied. We will use
this method to illustrate fractal transformations.
Let
\[
\mathcal{F}=\{\mathbb{C};f_{1}(z)=sz-1,f_{2}(z)=sz+1,\text{ for all }%
z\in\mathbb{C}\},
\]%
\[
\mathcal{G}=\{\mathbb{C};g(z)=s_{1}z-1,g(z)=s_{1}z+1,\text{ for all }%
z\in\mathbb{C}\},
\]%
\[
\mathcal{H}=\{\mathbb{C};h(z)=s_{2}z-1,h(z)=s_{2}z+1,\text{ for all }%
z\in\mathbb{C}\},
\]
where $\mathbb{C}$, denotes the complex plane, $s=0.5(1+i)$, $s_{1}%
=0.44(1+i)$, and $s_{2}=0.535(1+i)$. We denote the attractors of these IFSs by
$A_{\mathcal{F}}$, $A_{\mathcal{G}}$, and $A_{\mathcal{H}}$. In the top row of
Figure \ref{boatdragonsgray} we illustrate, from left to right, the three
picture functions, $\mathfrak{P}_{\mathcal{G}}:A_{\mathcal{G}}\rightarrow
\mathfrak{C}$, $\mathfrak{P}_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow
\mathfrak{C}$, and $\mathfrak{P}_{\mathcal{H}}:A_{\mathcal{H}}\rightarrow
\mathfrak{C}$. These pictures were obtained by masking a single original
digital picture, whose domain we took to be $\{z=x+iy\in\mathbb{C}:$ $-3.5\leq
x\leq3.5,$ $-3.5\leq y\leq3.5\}$, by the complement of each of the sets
$A_{\mathcal{G}}$, $A_{\mathcal{F}}$, and $A_{\mathcal{H}}$.
The attractor $A_{\mathcal{F}}$ is a so-called twin-dragon fractal. It is an
example of a just-touching attractor: that is, $f_{1}(A_{\mathcal{F}})\cap
f_{2}(A_{\mathcal{F}})$ is non-empty and equals $f_{1}(\partial A_{\mathcal{F}%
})\cap f_{2}(\partial A_{\mathcal{F}})$ where $\partial A_{\mathcal{F}}$
denotes the boundary of $A_{\mathcal{F}}$. This contrasts with $A_{\mathcal{G}%
}$ which is totally disconnected, perfect, and in fact homeomorphic to the
classical cantor set. This also contrasts with $A_{\mathcal{H}}$ which is such
that there exists a disk in $\mathbb{R}^{2}$, of non-zero radius, which is
contained in $h_{1}(A_{\mathcal{H}})\cap h_{2}(A_{\mathcal{H}})$.
The bottom row of Figure \ref{boatdragonsgray} illustrates the pictures, from
left to right, $\mathfrak{P}_{\mathcal{G}}\mathfrak{\circ}\phi_{\mathcal{G}%
}\circ\tau_{\mathcal{F}},$ $\mathfrak{P}_{\mathcal{F}}\mathfrak{\circ}%
\phi_{\mathcal{F}}\circ\tau_{\mathcal{F}},$ and $\mathfrak{P}_{\mathcal{H}%
}\mathfrak{\circ}\phi_{\mathcal{H}}\circ\tau_{\mathcal{F}}$. They were
computed by a variant of the chaos game as explained in section \ref{topsec}.
The domain of each of these pictures is $A_{\mathcal{F}}$. We notice that
$\mathfrak{P}_{\mathcal{F}}\mathfrak{\circ}\phi_{\mathcal{F}}\circ
\tau_{\mathcal{F}}=\mathfrak{P}_{\mathcal{F}}$, which is true regardless of
the choice of IFS $\mathcal{F}$ since $\phi_{\mathcal{F}}\circ\tau
_{\mathcal{F}}$ is the identity on $A_{\mathcal{F}}$. We notice that both
$\mathfrak{P}_{\mathcal{G}}\mathfrak{\circ}\phi_{\mathcal{G}}\circ
\tau_{\mathcal{F}}$ and $\mathfrak{P}_{\mathcal{H}}\mathfrak{\circ}%
\phi_{\mathcal{H}}\circ\tau_{\mathcal{F}}$ have features in common with the
underlying digital picture; for example, $\mathfrak{P}_{\mathcal{G}%
}\mathfrak{\circ}\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ displays something
like the texture of the hat, near the middle of the bottom-left image. The
bottom right image shows parts of the hat, repeated several times, and some
clearly delineated small twin-dragon tiles.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.2085in,
width=4.8075in
]%
{BoatDragonsGray.eps}%
\caption{Examples of fractal transformations are illustrated using the three
picture functions $\mathfrak{P}_{\mathcal{G}}$, $\mathfrak{P}_{\mathcal{F}}$,
and $\mathfrak{P}_{\mathcal{H}}$ shown in the top row, from left to right. The
domains of these functions are the attractors $A_{\mathcal{G}}$,
$A_{\mathcal{F}}$, $A_{\mathcal{H}}$ of three IFSs $\mathcal{F}$,
$\mathcal{G}$, and $\mathcal{H}$, defined in the text. The bottom row
illustrates the pictures $\mathfrak{P}_{\mathcal{G}}\mathfrak{\circ}%
\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}},$ $\mathfrak{P}_{\mathcal{F}%
}\mathfrak{\circ}\phi_{\mathcal{F}}\circ\tau_{\mathcal{F}},$ and
$\mathfrak{P}_{\mathcal{H}}\mathfrak{\circ}\phi_{\mathcal{H}}\circ
\tau_{\mathcal{F}}$.}%
\label{boatdragonsgray}%
\end{center}
\end{figure}
We are led to consider the following questions. Under what conditions on
general IFSs $\mathcal{F}$ and $\mathcal{G}$ is the fractal transformation
$\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ continuous? When does it provide a
homeomorphism between $A_{\mathcal{F}}$ and $A_{\mathcal{G}}$?
\begin{definition}
The address structure of\textbf{ }$\mathcal{F}$ is defined to be the set of
sets
\[
\mathcal{C}_{\mathcal{F}}=\{\phi_{\mathcal{F}}^{-1}(\{x\})\cap\overline
{\Omega}_{\mathcal{F}}:x\in A_{\mathcal{F}}\}\text{.}%
\]
\end{definition}
The address structure of an IFS is a certain partition of $\overline{\Omega
}_{\mathcal{F}}$. Let $\mathcal{C}_{\mathcal{G}}$ denote the address structure
of $\mathcal{G}$. Let us write $\mathcal{C}_{\mathcal{F}}$ $\prec
\mathcal{C}_{\mathcal{G}}$ to mean that for each\ $S\in\mathcal{C}%
_{\mathcal{F}}$ there is $T\in\mathcal{C}_{\mathcal{G}}$ such that $S\subset
T$. Notice that if $\mathcal{C}_{\mathcal{F}}=\mathcal{C}_{\mathcal{G}}$ then
$\Omega_{\mathcal{F}}=$ $\Omega_{\mathcal{G}}$. Some examples of address
structures are given in section \ref{AddressSec}.
\begin{theorem}
Let $\mathcal{F}$ and $\mathcal{G}$ be two hyperbolic IFSs such that
$\mathcal{C}_{\mathcal{F}}\prec\mathcal{C}_{\mathcal{G}}$. Then the fractal
transformation $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}:A_{\mathcal{F}%
}\rightarrow A_{\mathcal{G}}$ is continuous. If $\mathcal{C}_{\mathcal{F}%
}=\mathcal{C}_{\mathcal{G}}$ then $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$
is a homeomorphism.
\end{theorem}
The proof relies on a standard result in topology, Lemma 2 below, which we
present in the context of metric spaces.
\begin{lemma}
(cf. \cite{Mendelson}, bottom of p.194.) Let $F:X\rightarrow Y$ be a
continuous mapping from a compact metric space $X$ onto a metric space $Y$.
Then $S\subset Y$ is open if and only if $F^{-1}(S)\subset X$ is open.
\end{lemma}
\begin{proof}
If $S\subset Y$ is open then $F^{-1}(S)\subset X$ is open because
$F:X\rightarrow Y$ is continuous. Suppose that $F^{-1}(S)$ is open. Then
$X\backslash F^{-1}(S)$ is closed. But a closed subset of a compact metric
space is compact. The continuity of $F$ now implies that $F(X\backslash
F^{-1}(S))$ is compact and hence closed. But $F(X\backslash F^{-1}%
(S))=Y\backslash S$. Hence $S$ is open.
\end{proof}
\begin{lemma}
(cf. \cite{Mendelson}, Proposition 7.4 on p.195.) Let $F:X\rightarrow Y$ be a
continuous mapping from a compact metric space $X$ onto a metric space $Y$.
Let $H:Y\rightarrow Z$ where $Z$ is a metric space.\ Let $H\circ
F:X\rightarrow Z$ be continuous. Then $H:Y\rightarrow Z$ is continuous.
\end{lemma}
\begin{proof}
Let $O\subset Z$ be open. Then $(H\circ F)^{-1}(O)=F^{-1}(H^{-1}(O))$ is open.
But then by Lemma 1 $H^{-1}(O)$ is open. Hence $H:Y\rightarrow Z$ is continuous.
\end{proof}
\begin{proof}
[Proof of Theorem 2]In Lemma 2 we set $X=\overline{\Omega}_{\mathcal{F}}$,
$Y=A_{\mathcal{F}}$, and $Z=A_{\mathcal{G}}$. We choose $F:X\rightarrow Y$ to
be $\overline{\Phi}_{\mathcal{F}}:\overline{\Omega}_{\mathcal{F}}\rightarrow
A_{\mathcal{F}}$. Then $F:X\rightarrow Y$ is a continuous mapping from a
compact metric space $X$ onto a metric space $Y$. We also choose
$H:Y\rightarrow Z$ to be
\[
H=\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow
A_{\mathcal{G}}\text{.}%
\]
Now look at the function
\[
G:=H\circ F=\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}\circ\overline{\Phi
}_{\mathcal{F}}:\overline{\Omega}_{\mathcal{F}}\rightarrow A_{\mathcal{G}%
}\text{.}%
\]
If $\sigma\in\overline{\Omega}_{\mathcal{F}}$, then both $\sigma$ and
$(\tau_{\mathcal{F}}\circ\overline{\Phi}_{\mathcal{F}})(\sigma)$ belong to the
same set in the $\mathcal{C}_{\mathcal{F}}$. Since $\mathcal{C}_{\mathcal{F}%
}\prec\mathcal{C}_{\mathcal{G}}$ it follows that both $\sigma$ and
$(\tau_{\mathcal{F}}\circ\overline{\Phi}_{\mathcal{F}})(\sigma)$ belong to the
same set in $\mathcal{C}_{\mathcal{G}}$. It follows that%
\[
(\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}\circ\overline{\Phi}_{\mathcal{F}%
})(\sigma)=\phi_{\mathcal{G}}(\sigma)\text{ for all }\sigma\in\overline
{\Omega}_{\mathcal{F}}\text{.}%
\]
But $\phi_{\mathcal{G}}:\Omega\rightarrow A_{\mathcal{G}}$ is continuous.
Hence $G$ is continuous.
We have shown that the conditions in Lemma 2 hold. It follows that
$H=\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ is continuous.
When $\mathcal{C}_{\mathcal{F}}=\mathcal{C}_{\mathcal{G}}$ it is readily
verified that $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}:A_{\mathcal{F}%
}\rightarrow A_{\mathcal{G}}$ is one-to-one and onto and that its inverse is
$\phi_{\mathcal{F}}\circ\tau_{\mathcal{G}}$. Also $\mathcal{C}_{\mathcal{F}%
}=\mathcal{C}_{\mathcal{G}}$ implies $\mathcal{C}_{\mathcal{G}}\prec
\mathcal{C}_{\mathcal{F}}$ and so, by the first part of the theorem,
$\phi_{\mathcal{F}}\circ\tau_{\mathcal{G}}$ is continuous. Hence
$\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow
A_{\mathcal{G}}$ is a homeomorphism.
\end{proof}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.1465in,
width=4.8075in
]%
{TriHomBefAndAft2.eps}%
\caption{Before, on the left, and after, on the right, a fractal
homeomorphism. See text.}%
\label{trihombefoandaft2}%
\end{center}
\end{figure}
\section{\label{AddressSec}Examples of address structures}
\subsection{Backwards orbits}
Let $x\in\mathcal{A}_{\mathcal{F}}$. Let $\sigma\in\Omega$ be such that
$\phi_{\mathcal{F}}(\sigma)=x$. Assume that the $f_{n}$'s are one-to-one. Then
define $x_{0}=x$ and $x_{k}=f_{\sigma_{k}}^{-1}(x_{k-1})$ for $k=1,2,3,...$.
Notice that $x_{k}=\phi_{\mathcal{F}}(\sigma_{k}\sigma_{k+1}\sigma_{k+2}...)$.
We call $\{x_{k}\}_{k=0}^{\infty}$ a backwards orbit of $x$ (under the IFS
$\mathcal{F)}$.
The set of all addresses of $x$ can be calculated by following all possible
backwards orbits of $x$. Define a sequence of points $\{\widetilde{x}%
_{k}\}_{k=0}^{\infty}$ in $A_{\mathcal{F}}$ and an address $\widetilde{\sigma
}=\widetilde{\sigma}_{1}\widetilde{\sigma}_{2}\widetilde{\sigma}_{3}%
...\in\Omega$, as follows. Let $\widetilde{x}_{0}=x$. For each $k=1,2,3,...$
first choose%
\[
\widetilde{\sigma}_{k}\in\{n\in\{1,2,...,N\}:\widetilde{x}_{k-1}\in
f_{n}(A_{\mathcal{F}})\}
\]
and then define%
\[
\widetilde{x}_{k}=f_{\widetilde{\sigma}_{k}}^{-1}(\widetilde{x}_{k-1})\text{.}%
\]
Then $\widetilde{\sigma}\in\phi_{\mathcal{F}}^{-1}(\{x\})$ and all
$\widetilde{\sigma}\in\phi_{\mathcal{F}}^{-1}(\{x\})$ can be obtained in this manner.
\subsection{Some notation}
We use the notation $(PQ)=PQ\backslash\{P,Q\}$ to denote the straight line
segment which connects the two points $P$ and $Q$ in $\mathbb{R}^{2},$ without
its endpoints. We write $\Omega^{\prime}$ to denote the set of all finite
length strings of symbols from the alphabet $\{1,2,...,N\}$, including the
empty string "$\varnothing$" . We write $\left\vert \sigma\right\vert $ to
denote the length of $\sigma\in\Omega^{\prime}$. We define $\omega
\sigma=\omega_{1}\omega_{2}...\omega_{\left\vert \omega\right\vert }\sigma
_{1}\sigma_{2}...\sigma_{\left\vert \sigma\right\vert }$ for all
$\omega,\sigma\in\Omega^{\prime}$. Similarly we define $\omega\sigma
=\omega_{1}\omega_{2}...\omega_{\left\vert \omega\right\vert }\sigma_{1}%
\sigma_{2}...$ for all $\omega\in\Omega^{\prime},\sigma\in\Omega$. We write
$S:\Omega^{\prime}\rightarrow\Omega^{\prime}$ to denote the shift operator
defined by $S(\sigma)=\sigma_{2}\sigma_{3}...\sigma_{\left\vert \sigma
\right\vert }$ when $\left\vert \sigma\right\vert \geq1$ and $S($%
"$\varnothing$"$)=$"$\varnothing$". We write $f_{\sigma}=f_{\sigma_{1}}\circ
f_{\sigma_{2}}\circ...\circ f_{\sigma_{\left\vert \sigma\right\vert }}$ for
all $\sigma\in\Omega^{\prime}$ with $\left\vert \sigma\right\vert \geq1$ and
$f_{"\varnothing"}$ denotes the identity function.
\subsection{\label{example1}Example 1}
An interesting example of address structures is provided by the IFS
$\mathcal{F}=\mathcal{F}_{\alpha,\beta,\gamma}=\{\mathbb{R}^{2};h_{1}%
,h_{2},h_{3},h_{4}\}$, introduced at the end of section \ref{hypersec}. Here
we prove that
\begin{equation}
\mathcal{C}_{\mathcal{F}_{\alpha,\beta,\gamma}}=\mathcal{C}_{\mathcal{F}%
_{\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}}}
\label{equalstructs}%
\end{equation}
for all $\alpha,\beta,\gamma,\widetilde{\alpha},\widetilde{\beta}%
,\widetilde{\gamma}\in(0,1)$ by calculating the address structure
$\mathcal{C}_{\mathcal{F}_{\alpha,\beta,\gamma}}$.
In this case there is only one backwards orbit $\{x_{k}\}_{k=0}^{\infty}$ of
each $x\in\mathcal{A}_{\mathcal{F}}$. This is because of the form of
$\mathcal{F}$, and because the $h_{n}$'s are affine and so preserve ratios of
distances between points which lie on any given straight line: for example if
$x\in ab=h_{2}(\blacktriangle)\cap h_{4}(\blacktriangle)$ then $h_{2}%
^{-1}(x)=h_{4}^{-1}(x)$. Indeed, the mapping $T:\blacktriangle\rightarrow
\blacktriangle$ defined as in equations \ref{partition} is continuous and the
backwards orbit of $x$ is the same as the orbit of $x$ under $T$ treated as a
dynamical system.
Any point $x_{K}\in\{x_{k}\}_{k=0}^{\infty}$ on the backwards orbit of $x,$
such that more than one map $h_{n}^{-1}$ may be applied, is such that $x_{K}$
belongs to the set
\[%
{\textstyle\bigcup\limits_{i\neq j}}
h_{i}(\blacktriangle)\cap h_{j}(\blacktriangle)=(ab)\cup(bc)\cup
(ca)\cup\{a,b,c\}\text{.}%
\]
If $x_{K}\in(ab)$ then $\sigma_{K}\in\{2,4\}$, if $x_{K}=a$ then $\sigma
_{K}\in\{1,2,4\}$, and so on. For example, if $x=\phi_{\mathcal{F}}(\omega
_{1}\omega_{2}\omega_{3}...)$ and the only point on the backwards orbit of $x$
which lies in $(ab)\cup(bc)\cup(ca)\cup\{a,b,c\}$ is $x_{K}\in(ab)$ then
$\phi_{\mathcal{F}}^{-1}(\{x\})=\{\sigma\in\Omega:$ $\sigma_{k}=\omega_{k}$
for all $k\neq K$, and $\sigma_{K}\in\{2,4\}\}$.
Let $\triangle$ denote the boundary of $\blacktriangle$ as a subset of
$\mathbb{R}^{2}$, $\triangledown=(ab)\cup(bc)\cup(ca),$ and
\[
\Xi=\triangle\cup\triangledown\cup(%
{\textstyle\bigcup\limits_{\{\sigma\in\Omega^{\prime}:\left\vert
\sigma\right\vert \geq1\}}}
h_{\sigma}(\triangledown))\text{.}%
\]
Then
(i) each of the sets $\triangle,\triangledown,$ $h_{1}(\triangledown),$
$h_{2}(\triangledown),$ $h_{3}(\triangledown),$ $h_{4}(\triangledown),$
$h_{11}(\triangledown),$ $h_{12}(\triangledown),...$ is disjoint;
(ii) $T(\Xi)=\mathcal{F}(\Xi)=\Xi$;
(iii) $T$ is one-to-one on $h_{\sigma}(\triangledown)$ and $T(h_{\sigma
}(\triangledown))=h_{S(\sigma)}(\triangledown)$ for all $\sigma\in
\Omega^{\prime}$ with $\left\vert \sigma\right\vert \geq1$;
(iv) $T$ is one-to-one on $\triangledown$ and $T(\triangledown)=\triangle
\backslash\{A,B,C\}$;
(v) $T$ is two-to-one on $\triangle$ and $T(\triangle\backslash
\{A,B,C\})=T(\triangle)=\triangle$.
The transformation $T$ maps $\triangle$ continuously onto itself. If $x$ goes
around $\triangle$ clockwise once, then $T(x)$ goes around $\triangle$
anticlockwise twice. It does so in such a way that $T(\{A\})=\{A\},$
$T((Ac))=(Ab)\cup\{b\}\cup(bC),$ $T(\{c\})=\{C\}$ and so on. This information
provides us with the directed graph, with labelled edges, shown in Figure
\ref{markov}. We denote this graph by $G$. It is such that that there is a
bijective correspondence between the points of $\triangle$ and the set of all
paths in $G$. A path in $G$ is obtained by starting at any node and
successively following edges in the directions of the arrows, yielding an
infinite sequences of edges. The set of addresses of the point represented by
a path in $G$ consists of all sequences of the numbers $\{1,2,3,4\}$ which can
be read off successivey from the path, with one symbol from each edge. For
example, the only possible address for $\{A\}$ is $\overline{3}=3333..$, and
the set of addresses of a point in $(Ab)$ may be $3\{1,2,$ or $3\}222...,$ or
$33\{2,3,$ or $4\}111..,.$ or one or more which begin $332222$.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=4.6726in,
width=4.8845in
]%
{markov.eps}%
\caption{This graph can be used to compute the IFS addresses of all points
which lie on the boundary of the triangle $\blacktriangle$.}%
\label{markov}%
\end{center}
\end{figure}
Now suppose $x\in h_{\sigma}(\triangledown)$ for some $\sigma\in\Omega
^{\prime}$ with $\left\vert \sigma\right\vert \geq1.$ Then by repeated
application of (iii) above we find that the first $\left\vert \sigma
\right\vert $ terms in any address of $x$ are precisely $\sigma$, and
$T^{\circ\left\vert \sigma\right\vert }(x)\in\triangledown$. Since
$T^{\circ\left\vert \sigma\right\vert }$ maps $h_{\sigma}(\triangledown)$
one-to-one onto $\triangledown$, it follows that the set of all sets of
addresses of all points in $h_{\sigma}(\triangledown)$ is the same as the set
of all sets of addresses of all points in $\triangledown$ after $\sigma$ has
been appended to the front of each of the latter addresses. So what is the set
of all sets of addresses of all points in $\triangledown$?
We have $\triangledown=(ab)\cup(bc)\cup(ca)$. Let us deal with $(ab)$. The
transformation $T$ maps $(ab)$ one-to-one onto $(AB)=(Ac)\cup\{c\}\cup(cB)$.
It follows that the set of all sets of addresses of all points in $(ab)$,
which we denote by $C(ab),$ is determined by the set of sets of addresses of
all points in $(Ac)\cup\{c\}\cup(cB)$, which we denote by $C((Ac)\cup
\{c\}\cup(cB))$. Specifically,
\[
C(ab)=\{\{\eta\sigma:\eta\in\{2,4\},\sigma\in\pi\}:\pi\in C((Ac)\cup
\{c\}\cup(cB))\}
\]
The set of sets of addresses $C((Ac)\cup\{c\}\cup(cB))$ corresponds to the set
of paths in $G$ which start at a node labelled $(Ac)$, $\{c\}$, or $(cB)$. We
can similarly describe the address structures of $(bc)$, and $(ca)$. We are
thus able, in principle, to write down the set of addresses of each $x\in$
$\Xi$; in particular, the set of all sets thus obtained does not depend on
$\alpha,\beta,$ or $\gamma$.
Next we deal with $\Xi^{C}=\blacktriangle\backslash\Xi$. Since all points with
multiple addresses lie in $\Xi$ and the backwards orbit of each point in
$\Xi^{C}$ lies in $\Xi^{C}$ it follows that each point in $\Xi^{C}$ has a
unique address, and in particular $\phi_{\mathcal{F}}^{-1}(\Xi^{C}%
)=\Omega\backslash\phi_{\mathcal{F}}^{-1}(\Xi)$ does not depend on
$\alpha,\beta,$ or $\gamma$.
Finally we note that $\overline{\phi_{\mathcal{F}}^{-1}(\Xi)}=\Omega$. Hence
$\Omega_{\mathcal{F}}=\Omega$. Hence $\Omega_{\mathcal{F}}=\phi_{\mathcal{F}%
}^{-1}(\Xi)\cup\phi_{\mathcal{F}}^{-1}(\Xi^{C}).$ Since the equivence class
structures of both $\phi_{\mathcal{F}}^{-1}(\Xi)$ and $\phi_{\mathcal{F}}%
^{-1}(\Xi^{C})$ do not depend on $\alpha,\beta,$ or $\gamma,$ it follows that
equation \ref{equalstructs} is true.
So, for example, let $\mathcal{F}=\mathcal{F}_{0.5,0.5,0.5}=\{\blacktriangle
;h_{1},h_{2},h_{3},h_{4}\}$ and $\mathcal{G=F}_{0.65,0.3,0.4}=\{\blacktriangle
;g_{1},g_{2},g_{3},g_{4}\}$. Then $\mathcal{C}_{\mathcal{F}}=\mathcal{C}%
_{\mathcal{G}}$ and, by Theorem 1, the fractal transformation $\phi
_{\mathcal{G}}\circ\tau_{\mathcal{F}}:\blacktriangle\rightarrow\blacktriangle$
is a homeomorphism. Figure \ref{trihombefoandaft2} illustrates the action of
this homeomorphism. The figure on the left shows the set $S$, defined to be
the union of the attractors of the two IFSs $\{\blacktriangle;h_{1}%
,h_{3},h_{4}\}$ and $\{\blacktriangle;h_{2},h_{3},h_{4}\}$. The image on the
right shows the set $\widetilde{S}$, defined to be the union of the attractors
of the IFSs $\{\blacktriangle;g_{1},g_{3},g_{4}\}$ and $\{\blacktriangle
;g_{2},g_{3},g_{4}\}$. The two sets are related by $\phi_{\mathcal{G}}%
\circ\tau_{\mathcal{F}}(S)=\widetilde{S}$. Figure \ref{threebirds} illustrates
two other homeomorphisms associated with the family $\mathcal{F}_{\alpha
,\beta,\gamma}$. These examples were computed as described in section
\ref{topsec}.
\subsection{\label{example2}Example 2}
An example of address structures $\mathcal{C}_{\mathcal{F}}$ and
$\mathcal{C}_{\mathcal{G}}$ such that $\mathcal{C}_{\mathcal{F}}%
\prec\mathcal{C}_{\mathcal{G}}$ and $\mathcal{C}_{\mathcal{F}}\neq
\mathcal{C}_{\mathcal{G}}$ is provided by taking $\mathcal{F}=\{\square
;f_{1},f_{2},f_{3},f_{4}\}$ and $\mathcal{G}=\{\square;g_{1},g_{2},g_{3}%
,g_{4}\}$ to be the IFSs of affine maps specified in Tables \ref{FernTable}
and \ref{CtsTable} respectively. Here $\square\subset\mathbb{R}^{2}$ denotes
the filled square with vertices at $I=(1,1)$, $J=(1,0)$, $K=(0,0)$, $J=(0,1)$.
The attractor $A_{\mathcal{F}}$ of $\mathcal{F}$ is represented by the fern
image in Figure \ref{fernmaps}. The attractor $A_{\mathcal{G}}$ of
$\mathcal{G}$ is $\square$.
The transformations of $\mathcal{F}$ are such\ that
\begin{align}
f_{1}(i) & =m,f_{1}(k)=k,f_{2}(i)=i,f_{2}(k)=m,\label{Faddress}\\
f_{3}(i) & =m,f_{3}(k)=l,f_{4}(i)=m,f_{4}(k)=j,\nonumber
\end{align}
where the points $i,j,k,l,m\in A_{\mathcal{F}}$ are approximately as labelled
in Figure \ref{fernmaps}. Furthermore $f_{p}(A_{\mathcal{F}})\cap
f_{q}(A_{\mathcal{F}})=m$ whenever $p,q\in\{1,2,3,4\}$ with $p\neq q$. It is
readily deduced that $k=\phi_{\mathcal{F}}(\overline{1}),i=\phi_{\mathcal{F}%
}(\overline{2}),m=\phi_{\mathcal{F}}(1\overline{2})=\phi_{\mathcal{F}%
}(2\overline{1})=\phi_{\mathcal{F}}(3\overline{2})=\phi_{\mathcal{F}%
}(4\overline{2})$, that
\[
\Omega_{\mathcal{F}}=\{\sigma\in\Omega:S^{\circ n}(\sigma)\notin
\{2\overline{1},3\overline{2},4\overline{2}\}\text{ for all }n\in
\{0,1,2,...\}\},
\]
that $\overline{\Omega}_{\mathcal{F}}=\Omega$ and that the address structure
of $\mathcal{F}$ is
\[
\mathcal{C}_{\mathcal{F}}=\mathcal{C}_{\mathcal{F}}^{(1)}\cup\mathcal{C}%
_{\mathcal{F}}^{(2)}%
\]
where
\begin{align*}
\mathcal{C}_{\mathcal{F}}^{(1)} & =\{\{\sigma\}:\sigma\in\Omega,S^{\circ
n}(\sigma)\notin\{1\overline{2},2\overline{1},3\overline{2},4\overline
{2}\}\text{ for all }n\in\{0,1,2,...\}\},\\
\mathcal{C}_{\mathcal{F}}^{(2)} & =\{\{\sigma^{\prime}1\overline{2}%
,\sigma^{\prime}2\overline{1},\sigma^{\prime}3\overline{2},\sigma^{\prime
}4\overline{2}\}:\sigma^{\prime}\in\Omega^{\prime}\}.
\end{align*}
To determine the address structure of $\mathcal{G}$, we note that $\square$ is
the union of four rectangular tiles $g_{n}(\square)$ which share portions of
their boundaries. The transformations of $\mathcal{G}$ are such that
\begin{align}
g_{1}(I) & =M,g_{1}(K)=K,g_{2}(I)=I,g_{2}(K)=M,\label{Gaddress}\\
g_{3}(I) & =M,g_{3}(K)=L,g_{4}(I)=M,g_{4}(K)=J,\nonumber
\end{align}
where the points $I,J,K,L,M\in A_{\mathcal{G}}$ are approximately as labelled
in Figure \ref{fernmaps}.
Note that equations \ref{Gaddress} are the same as equations \ref{Faddress}
upon substitution of $f_{1},f_{2},f_{3},f_{4},i,j,k,l,$ and $m$, by
$g_{1},g_{2},g_{3},g_{4},I,J,K,L,$ and $M$ respectively. It is readily deduced
that $K=\phi_{\mathcal{G}}(\overline{1}),I=\phi_{\mathcal{G}}(\overline
{2}),M=\phi_{\mathcal{G}}(1\overline{2})=\phi_{\mathcal{G}}(2\overline
{1})=\phi_{\mathcal{G}}(3\overline{2})=\phi_{\mathcal{G}}(4\overline{2})$, and
that $\overline{\Omega}_{\mathcal{G}}=\Omega$. As a consequence $\mathcal{C}%
_{\mathcal{F}}\prec\mathcal{C}_{\mathcal{G}}$: if $s\in\mathcal{C}%
_{\mathcal{F}}$ then either $s\in\mathcal{C}_{\mathcal{F}}^{(1)}$ or
$s\in\mathcal{C}_{\mathcal{F}}^{(2)}$; if $s\in\mathcal{C}_{\mathcal{F}}%
^{(1)}$ then $s$ is a singleton and, since $\mathcal{C}_{\mathcal{G}}$ is a
partition of $\Omega$, there must be $t\in\mathcal{C}_{\mathcal{G}}$ such that
$s\subset t$; if $s\in\mathcal{C}_{\mathcal{F}}^{(2)}$ then $s=\{\sigma
^{\prime}1\overline{2},\sigma^{\prime}2\overline{1},\sigma^{\prime}%
3\overline{2},\sigma^{\prime}4\overline{2}\}$ for some $\sigma^{\prime}%
\in\Omega^{\prime}$, and since $M=\phi_{\mathcal{G}}(1\overline{2}%
)=\phi_{\mathcal{G}}(2\overline{1})=\phi_{\mathcal{G}}(3\overline{2}%
)=\phi_{\mathcal{G}}(4\overline{2}),$ it follows that $\mathcal{C}%
_{\mathcal{G}}$ contains a set that contains $s$. Hence $\mathcal{C}%
_{\mathcal{F}}\prec\mathcal{C}_{\mathcal{G}}$ and, by Theorem 1, the fractal
transformation $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ from the
fern-shaped set onto $\square$ is continuous. This transformation is
illustrated in Figure \ref{ferns}, as described at the start of section
\ref{topsec}. Note however that in this case $\mathcal{C}_{\mathcal{F}}%
\neq\mathcal{C}_{\mathcal{G}}$ because there is a set in $\mathcal{C}%
_{\mathcal{G}}$ which consist of a pair of distinct addresses, whereas all
sets in $\mathcal{C}_{\mathcal{F}}$ contain either one or four distinct addresses.
If, in this example, we change $\mathcal{G}$ to $\widetilde{\mathcal{G}}$
specified in Table \ref{DiscTable} then the attractor is still the filled
square, that is $A_{\widetilde{\mathcal{G}}}=\square$, but Equation
\ref{Gaddress} no longer holds and we can show that the fractal transformation
$\phi_{\widetilde{\mathcal{G}}}\circ\tau_{\mathcal{F}}$ from the fern-shaped
set onto $\square$ is not continuous. This lack of continuity is illustrated
in Figure \ref{ferns}, as described in section \ref{topsec}.
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5356in,
width=4.8075in
]%
{fernmaps.eps}%
\caption{(i) Shows the points $i,j,k,l,m$ and (ii) shows the points
$I,J,K,L,M$. }%
\label{fernmaps}%
\end{center}
\end{figure}
\begin{table}[tbp] \centering
\begin{tabular}
[c]{|c|c|c|c|c|c|c|}\hline
$n$ & $a_{n}$ & $b_{n}$ & $c_{n}$ & $d_{n}$ & $e_{n}$ & $l_{n}$\\\hline
$1$ & $0.85$ & $-0.05$ & $0.125$ & $0.05$ & $0.85$ & $-0.039$\\
$2$ & $0.06$ & $0.02$ & $0.45$ & $0.0$ & $0.165$ & $0.835$\\
$3$ & $0.17$ & $0.22$ & $0.195$ & $-0.22$ & $0.17$ & $0.776$\\
$4$ & $-0.17$ & $-0.22$ & $0.805$ & $-0.22$ & $0.17$ & $0.776$\\\hline
\end{tabular}
\caption{ \label{FernTable}}%
\end{table}%
\begin{table}[tbp] \centering
\begin{tabular}
[c]{|c|c|c|c|c|c|c|}\hline
$n$ & $a_{n}$ & $b_{n}$ & $c_{n}$ & $d_{n}$ & $e_{n}$ & $l_{n}$\\\hline
$1$ & $0.8$ & $0.0$ & $0.0$ & $0.0$ & $0.8$ & $0.0$\\
$2$ & $0.2$ & $0.0$ & $0.8$ & $0.0$ & $0.8$ & $0.2$\\
$3$ & $-0.2$ & $0.0$ & $1.0$ & $0.0$ & $0.8$ & $0.0$\\
$4$ & $0.8$ & $0.0$ & $0.0$ & $0.0$ & $-0.2$ & $1.0$\\\hline
\end{tabular}
\caption{ \label{CtsTable}}%
\end{table}%
\begin{table}[tbp] \centering
\begin{tabular}
[c]{|c|c|c|c|c|c|c|}\hline
$n$ & $a_{n}$ & $b_{n}$ & $c_{n}$ & $d_{n}$ & $e_{n}$ & $l_{n}$\\\hline
$1$ & $-0.8$ & $0.0$ & $0.8$ & $0.0$ & $-0.8$ & $0.8$\\
$2$ & $-0.2$ & $0.0$ & $1.0$ & $0.0$ & $-0.2$ & $1.0$\\
$3$ & $0.8$ & $0.0$ & $0.0$ & $0.0$ & $0.2$ & $0.8$\\
$4$ & $0.2$ & $0.0$ & $0.8$ & $0.0$ & $0.8$ & $0.0$\\\hline
\end{tabular}
\caption{ \label{DiscTable}}%
\end{table}%
\section{\label{topsec}Pictures of tops functions}
When the underlying space is $\mathbb{R}^{2}$ we can use the chaos game to
compute illustrations of fractal transformations.
Let two\ hyperbolic IFSs
\[
\mathcal{F}:=\{\square;f_{1},...,f_{N}\}\text{ and }\mathcal{G}:=\{\square
;g_{1},...,g_{N}\}
\]
and a picture function%
\[
\mathfrak{P}:\square\rightarrow\mathfrak{C}%
\]
be given, where%
\[
\square:=\{(x,y)\in\mathbb{R}^{2}:0\leq x,y\leq1\}\text{.}%
\]
Let $\mathfrak{P}_{\mathcal{G}}$ denote $\mathfrak{P}$ restricted to
$A_{\mathcal{G}}$, that is $\mathfrak{P}_{\mathcal{G}}=\mathfrak{P}|_{A_{G}}$.
Then we define a new picture%
\[
\mathfrak{P}_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow\mathfrak{C}%
\]
by
\[
\mathfrak{P}_{\mathcal{F}}=\mathfrak{P}_{\mathcal{G}}\circ\phi_{\mathcal{G}%
}\circ\tau_{\mathcal{F}}\text{.}%
\]
We say that $\mathfrak{P}_{\mathcal{F}}$ is defined by\textbf{\ }tops plus color-stealing.
In order to make a physical picture of $\mathfrak{P}_{\mathcal{F}}$ and thus
illustrate the tops function $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ we
use a variant of the chaos game. To work at finite precision\ we partition the
set $\square\subset\mathbb{R}^{2}$ into a finite set of small rectangles, say
ten thousand of them, which we refer to as pixels. Each point $(x,y)\in
\square$ belongs to exactly one pixel, which we denote by $p((x,y))$.
Start from an arbitrary pair of points $(x_{0}^{\mathcal{F}},y_{0}%
^{\mathcal{F}})\in\square$ and $(x_{0}^{\mathcal{G}},y_{0}^{\mathcal{G}}%
)\in\square$. Let $K$ be a large number such as ten million. For $k=1,2,...K$
let $\sigma_{k}$ denote an element of $\{1,2,...,N\}$ chosen at random,
independently of all other choices. Let
\[
(x_{k}^{\mathcal{F}},y_{k}^{\mathcal{F}})=f_{\sigma_{k}}(x_{k-1}^{\mathcal{F}%
},y_{k-1}^{\mathcal{F}})\text{ and }(x_{k}^{\mathcal{G}},y_{k}^{\mathcal{G}%
})=g_{\sigma_{k}}(x_{k-1}^{\mathcal{G}},y_{k-1}^{\mathcal{G}})\text{.}%
\]
For each iterative step $k>100$, if the color of the pixel $p((x_{k}%
^{\mathcal{F}},y_{k}^{\mathcal{F}}))$ was not assigned at an earlier step
$l<k$ such that $\sigma_{l}\sigma_{l-1}\sigma_{l-2}...\sigma_{1}\overline
{1}>\sigma_{k}\sigma_{k-1}\sigma_{k-2}...\sigma_{1}\overline{1}$, then plot
the pixel $p((x_{k}^{\mathcal{F}},y_{k}^{\mathcal{F}}))$ in the color
$\mathfrak{P}_{\mathcal{G}}(x_{k}^{\mathcal{G}},y_{k}^{\mathcal{G}})$.
The reason this algorithm converges in practice to produce a stable physical
picture that approximates $\mathfrak{P}_{\mathcal{G}}\circ\phi_{\mathcal{G}%
}\circ\tau_{\mathcal{F}}$ is described in Chapter 4 of \cite{Bsuperfractals}.
Again, it depends on Birkhoff's ergodic theorem. Intuitively, ergodicity of
the shift transformation ensures that, almost always, the sequences
$\{(x_{k}^{\mathcal{F}},y_{k}^{\mathcal{F}})\}$ and $\{(x_{k}^{\mathcal{G}%
},y_{k}^{\mathcal{G}})\}$ repeatedly visit all of the pixels that represent
the points of $A_{\mathcal{F}}$ and $A_{\mathcal{G}}$ respectively. Let
$\sigma^{(k)}=\sigma_{k}\sigma_{k-1}\sigma_{k-2}...\sigma_{1}\overline{1}$.
Then the point $(x_{k}^{\mathcal{F}},y_{k}^{\mathcal{F}})$ is very close to
$\phi_{\mathcal{F}}(\sigma^{(k)})$ when $k$ is sufficiently large; indeed%
\[
d_{\mathbb{R}^{2}}(\phi_{\mathcal{F}}(\sigma^{(k)}),(x_{k}^{\mathcal{F}}%
,y_{k}^{\mathcal{F}}))\leq l^{k}d_{\mathbb{R}^{2}}(\phi_{\mathcal{F}%
}(\overline{1}),(x_{0}^{\mathcal{F}},y_{0}^{\mathcal{F}}))\text{.}%
\]
Similarly $(x_{k}^{\mathcal{G}},y_{k}^{\mathcal{G}})$ is very close to
$\phi_{\mathcal{G}}(\sigma^{(k)})$ when $k$ is sufficiently large. Hence, to a
good approximation, the color of the pixel $p(\phi_{\mathcal{F}}(\sigma
^{(k)}))$ is updated to become the color of the pixel $p(\phi_{\mathcal{G}%
}(\sigma^{(k)}))$ except when $\sigma^{(l)}>\sigma^{(k)}$ for some $l<k$ for
which $p(\phi_{\mathcal{F}}(\sigma^{(l)}))=p(\phi_{\mathcal{F}}(\sigma
^{(k)}))$. Let
\[
100<k_{1}<k_{2}<k_{3}<...<k_{M}\leq K
\]
denote the sequence of successive values of $k$ at which such updates occur.
Then $\{\sigma^{(k_{l})}\}_{l=1}^{M}$ is an increasing sequence of addresses,
each associated with a point in the pixel $p(\phi_{\mathcal{F}}(\sigma
^{(k_{1})}))$. Hence, again invoking ergodicity, $\{\sigma^{(k_{l})}%
\}_{l=1}^{M}$approaches the highest address of all points in the pixel
$p(\phi_{\mathcal{F}}(\sigma^{(k_{1})}))$. The address $\sigma^{(k_{M})}$ is
our approximation to $sup\{\tau_{\mathcal{F}}(\sigma):\sigma\in\tau
_{\mathcal{F}}(\phi_{\mathcal{F}}(\sigma^{(k_{1})}))\}$. In general we expect
it to become increasingly accurate with increasing $K$. According to this
approximation, the pixel $p(\phi_{\mathcal{F}}(\sigma^{(k_{1})}))$ is assigned
the colour of the pixel $p(\phi_{\mathcal{G}}(\tau_{\mathcal{F}}%
(\sigma^{(k_{M})})))$. Thus we obtain a sensible pixel-based approximation to
$\mathfrak{P}_{\mathcal{G}}\circ\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$.
In Figure \ref{ferns} we illustrate two different fractal transformations from
a fern-like fractal to a filled square, computed using this algorithm. For the
picture on the left $\mathcal{F}$ and $\mathcal{G}$ are as discussed in
section \ref{example1}, with $\mathcal{C}_{\mathcal{F}}\prec\mathcal{C}%
_{\mathcal{G}}$, $\mathcal{C}_{\mathcal{F}}\neq\mathcal{C}_{\mathcal{G}}$, so
that $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ is continuous. The picture
$\mathfrak{P}_{\mathcal{G}}$ is represented in the center of Figure
\ref{ferns}. It has been chosen to have apparently continuously varying
intensity so that the continuity of $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}%
}$ is illustrated by the smooth variation of intensity in the left-hand fern
image, which represents a close-up on $\mathfrak{P}_{\mathcal{F}}=$
$\mathfrak{P}_{\mathcal{G}}(\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})$. To
produce the picture on the right the IFS $\mathcal{G}$ has been switched, from
the one in Table \ref{CtsTable} to the one in Table \ref{DiscTable}, so that
$\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ is not continuous and
$\mathfrak{P}_{\mathcal{F}}(\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})$ is no
longer smoothly varying.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.9in,
width=4.8845in
]%
{ferns.eps}%
\caption{The ferns on the left and right are both obtained by fractal
transformations. The one on the left is continuous image of the central
image.}%
\label{ferns}%
\end{center}
\end{figure}
In Figure \ref{threebirds} we illustrate two examples, computed using the
modified chaos game described here, in each of which the fractal
transformation $\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}:A_{\mathcal{F}%
}\rightarrow A_{\mathcal{G}}$ is a homeomorphism. The homeomorphisms are
constructed using IFSs of the form $\mathcal{F}_{\alpha,\beta,\gamma}$
discussed in sections \ref{hypersec} and \ref{example1}. In both examples
$A_{\mathcal{F}}=A_{\mathcal{G}}=\blacktriangle$, the filled triangle with
vertices at $A=(0,0),$ $B=(1,0)$, and $C=(0.5,1)$. Also in both cases,
$\mathfrak{P}_{\mathcal{G}}:\blacktriangle\rightarrow\mathfrak{C}$ corresponds
to the grayscale picture of a caged bird in the top triangle in Figure
\ref{threebirds}. The image at bottom left shows $\mathfrak{P}_{\mathcal{G}%
}\circ\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ when $\mathcal{F}%
=\mathcal{F}_{0.525,0.525,0.525}$ and $\mathcal{G}=\mathcal{F}%
_{0.475,0.475,0.475}$. In this case the corresponding subtriangles have the
same areas at all levels with the consequence that the fractal transformation
$\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}$ is area-preserving. To produce the
image at the bottom right we used $\mathcal{F}=\mathcal{F}_{0.4,0.6,0.475}$
and $\mathcal{G}=\mathcal{F}_{0.5,0.5,0.5}$.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.2987in,
width=3.0606in
]%
{ThreeBirds.eps}%
\caption{Two examples of fractal homeomorphisms applied to the picture at the
top. The tranformations from the top image to the one at bottom left is
area-preserving. }%
\label{threebirds}%
\end{center}
\end{figure}
\section{\label{tdssec}The tops dynamical system}
In general, to determine the nature of the fractal transformation
$\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow
A_{\mathcal{G}}$ we need to know the tops code space $\Omega_{\mathcal{F}}$.
Here we prove that $\Omega_{\mathcal{F}}$ is shift invariant. Consequently it
may be described in terms of the orbits of an associated dynamical system
$T_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow A_{\mathcal{F}}$.
Throughout this section we assume that the transformations of the IFS
$\mathcal{F}$ are one-to-one. Let $\mathcal{S}_{\mathcal{F}}:\Omega
_{\mathcal{F}}\rightarrow\Omega$ denote the shift transformation, defined by
\[
\mathcal{S}_{\mathcal{F}}(\sigma_{1}\sigma_{2}\sigma_{3}...)=\sigma_{2}%
\sigma_{3}\sigma_{4}...
\]
for all $\sigma_{1}\sigma_{2}\sigma_{3}...\in\Omega_{\mathcal{F}}$. Let
\[
G_{\mathcal{F}}:=\{(x,\tau_{\mathcal{F}}(x)):x\in A_{\mathcal{F}}\}
\]
denote the graph of the the tops function $\tau_{\mathcal{F}}$.
\begin{lemma}
\label{lemma1}Let $(x,\sigma)\in G_{\mathcal{F}}$. Then $(f_{\sigma_{1}}%
^{-1}(x),\mathcal{S}_{\mathcal{F}}(\sigma))\in G_{\mathcal{F}}$.
\end{lemma}
\begin{proof}
$(x,\sigma)\in G_{\mathcal{F}}$ implies $x\in A_{\mathcal{F}}$, $\sigma
\in\Omega_{\mathcal{F}}$ and $\tau_{\mathcal{F}}(x)=\sigma$. In particular,
$\phi_{\mathcal{F}}(\sigma)=x$ for any $z\in\mathbb{X}$,
\[
\lim_{k\rightarrow\infty}f_{\sigma_{1}}\circ f_{\sigma_{2}}\circ
...f_{\sigma_{k}}(z)=x\text{.}%
\]
Using the continuity and invertibility of $f_{\sigma_{1}}$ it follows that
\[
\lim_{k\rightarrow\infty}f_{\sigma_{2}}\circ f_{\sigma_{3}}\circ
...f_{\sigma_{k}}(z)=f_{\sigma_{1}}^{-1}(x)\text{.}%
\]
This says that $\phi_{\mathcal{F}}(\mathcal{S}_{\mathcal{F}}(\sigma
))=f_{\sigma_{1}}^{-1}(x)$ which tells us that $\mathcal{S}_{\mathcal{F}%
}(\sigma)\in\phi_{\mathcal{F}}^{-1}(\{f_{\sigma_{1}}^{-1}(x)\})$.
Now suppose that there is $\omega\in\phi_{\mathcal{F}}^{-1}(\{f_{\sigma_{1}%
}^{-1}(x)\})$ with $\omega>\mathcal{S}_{\mathcal{F}}(\sigma)$. Then
$\phi_{\mathcal{F}}(\omega)=f_{\sigma_{1}}^{-1}(x)$ which implies
$f_{\sigma_{1}}(\phi_{\mathcal{F}}(\omega))=\phi_{\mathcal{F}}(\sigma
_{1}\omega)=x$. Let $\widetilde{\sigma}=\sigma_{1}\omega$. Then $\widetilde
{\sigma}>\sigma$ and $\phi_{\mathcal{F}}(\widetilde{\sigma})=x$ which
contradicts the assertion that $\sigma$ is the largest element of $\Omega$
such that $\phi_{\mathcal{F}}(\sigma)=x$. Hence $\mathcal{S}_{\mathcal{F}%
}(\sigma)\in\Omega_{\mathcal{F}}$ and $\tau_{\mathcal{F}}(f_{\sigma_{1}}%
^{-1}(x))=\mathcal{S}_{\mathcal{F}}(\sigma)$.
\end{proof}
\begin{lemma}
\label{lemma2}Let $(x,\sigma)\in G_{\mathcal{F}}$. Then $(f_{1}(x),1\sigma)\in
G_{\mathcal{F}}$.
\end{lemma}
\begin{proof}
$(x,\sigma)\in G_{\mathcal{F}}$ implies $\tau_{\mathcal{F}}(x)=\sigma$. Hence
$x=\phi_{\mathcal{F}}(\sigma)$ and so $f_{1}(x)=\phi_{\mathcal{F}}(1\sigma)$.
Now suppose that $(f_{1}(x),1\sigma)\notin G_{\mathcal{F}}$. Then there is
$\omega>1\sigma$ such that $\phi_{\mathcal{F}}(\omega)=f_{1}(x)$. But then
$\omega=1\widetilde{\sigma}$ where $\widetilde{\sigma}>\sigma$ and
$\phi_{\mathcal{F}}(1\widetilde{\sigma})=f_{1}(x)$. This implies
$\phi_{\mathcal{F}}(\widetilde{\sigma})=x$ with $\widetilde{\sigma}>\sigma$
which implies $\tau_{\mathcal{F}}(x)>\sigma$ which is a contradiction. Hence
$(f_{1}(x),1\sigma)\in G_{\mathcal{F}}$.
\end{proof}
It follows from Lemmas \ref{lemma1} and \ref{lemma2} that the mapping
$\widehat{T}_{\mathcal{F}}:G_{\mathcal{F}}\rightarrow G_{\mathcal{F}}$
specified by
\[
\widehat{T}_{\mathcal{F}}(x,\sigma)=(f_{\sigma_{1}}^{-1}(x),\mathcal{S}%
_{\mathcal{F}}(\sigma))\text{ for all }(x,\sigma)\in G_{\mathcal{F}}%
\]
is well-defined and onto.
In particular, the projection of $\widehat{T}_{\mathcal{F}}$ on $\Omega
_{\mathcal{F}}$ yields the symbolic dynamical system $\mathcal{S}%
_{\mathcal{F}}:\Omega_{\mathcal{F}}\rightarrow\Omega_{\mathcal{F}}$, because
from Lemma \ref{lemma2} we have
\[
\mathcal{S}_{\mathcal{F}}(\Omega_{\mathcal{F}})=\Omega_{\mathcal{F}}\text{.}%
\]
The projection of $\widehat{T}_{\mathcal{F}}:G_{\mathcal{F}}\rightarrow
G_{\mathcal{F}}$ onto $A_{\mathcal{F}}$ yields what we call the tops dynamical
system
\[
T_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow A_{\mathcal{F}}%
\]
where
\begin{equation}
T_{\mathcal{F}}(x)=\left\{
\begin{array}
[c]{ccc}%
f_{1}^{-1}(x) & if & x\in D_{1}:=f_{1}(A_{\mathcal{F}}),\\
f_{2}^{-1}(x) & if & x\in D_{2}:=f_{2}(A_{\mathcal{F}})\backslash
f_{1}(A_{\mathcal{F}}),\\
. & . & .\\
f_{N}^{-1}(x) & if & x\in D_{N}:=f_{N}(A_{\mathcal{F}})\backslash%
{\textstyle\bigcup\limits_{n=1}^{N-1}}
f_{n}(A_{\mathcal{F}}),
\end{array}
\right. \label{partition}%
\end{equation}
for all $x\in A_{\mathcal{F}}$. Lemma \ref{lemma2} implies
\[
T_{\mathcal{F}}(A_{\mathcal{F}})=A_{\mathcal{F}}\text{.}%
\]
\begin{theorem}
\label{TDSthm}The tops dynamical systems $T_{\mathcal{F}}:A_{\mathcal{F}%
}\rightarrow A_{\mathcal{F}}$ is related to the symbolic dynamical system
$\mathcal{S}_{\mathcal{F}}:\Omega_{\mathcal{F}}\rightarrow\Omega_{\mathcal{F}%
}$ by the tops function $\tau_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow
\Omega_{\mathcal{F}}$, according to
\[
\mathcal{S}_{\mathcal{F}}=\tau_{\mathcal{F}}\circ T_{\mathcal{F}}\circ
\tau_{\mathcal{F}}^{-1}\text{.}%
\]
If $\Omega_{\mathcal{F}}\subset\Omega_{\mathcal{G}}$ then%
\[
(\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}\circ T_{\mathcal{F}}%
)(x)=(T_{\mathcal{G}}\circ\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})(x)\text{
for all }x\in A_{\mathcal{F}}\text{.}%
\]
If $\mathcal{C}_{\mathcal{F}}=\mathcal{C}_{\mathcal{G}}$ then the tops
dynamical systems $T_{\mathcal{F}}:A_{\mathcal{F}}\rightarrow A_{\mathcal{F}}$
and $T_{\mathcal{G}}:A_{\mathcal{G}}\rightarrow A_{\mathcal{G}}$ are
topologically conjugate.
\end{theorem}
\begin{proof}
Let $\Phi_{\mathcal{F}}=\tau_{\mathcal{F}}^{-1}$ be as discussed at the end of
section \ref{topsfnsec}. Then we claim that%
\[
T_{\mathcal{F}}\circ\Phi_{\mathcal{F}}=\Phi_{\mathcal{F}}\circ\mathcal{S}%
_{\mathcal{F}}\text{.}%
\]
Since $\mathcal{S}_{\mathcal{F}}$ maps $\Omega_{\mathcal{F}}$ onto itself and
$\Phi_{\mathcal{F}}$ maps $\Omega_{\mathcal{F}}$ onto $A_{\mathcal{F}}$ it
follows that the mapping $\Phi_{\mathcal{F}}\circ\mathcal{S}_{\mathcal{F}}$
takes $\Omega_{\mathcal{F}}$ onto $A_{\mathcal{F}}$. (Similarly,
$T_{\mathcal{F}}\circ\Phi_{\mathcal{F}}$ maps $\Omega_{\mathcal{F}}$ onto
$A_{\mathcal{F}}$.)
Let $\sigma=\sigma_{1}\sigma_{2}\sigma_{3}...\in\Omega_{\mathcal{F}}$. Then
$\mathcal{S}_{\mathcal{F}}(\sigma)=\sigma_{2}\sigma_{3}...\in\Omega
_{\mathcal{F}}$ and
\[
(\Phi_{\mathcal{F}}\circ\mathcal{S}_{\mathcal{F}})(\sigma)=\lim_{k\rightarrow
\infty}(f_{\sigma_{2}}\circ f_{\sigma_{3}}\circ...\circ f_{\sigma_{k}%
})(z)\text{.}%
\]
On the other hand
\begin{align*}
\Phi_{\mathcal{F}}(\sigma) & =\lim_{k\rightarrow\infty}(f_{\sigma_{1}}\circ
f_{\sigma_{2}}\circ...\circ f_{\sigma_{k}})(z)\\
& =f_{\sigma_{1}}(\lim_{k\rightarrow\infty}(f_{\sigma_{2}}\circ f_{\sigma
_{3}}\circ...\circ f_{\sigma_{k}})(z))=f_{\sigma_{1}}(\Phi_{\mathcal{F}%
}(\sigma_{2}\sigma_{3}...))\text{,}%
\end{align*}
belongs to $A_{\mathcal{F}}$ and lies in the range of $f_{\sigma_{1}}$ and so
must belong to $D_{\sigma_{1}}$ as defined in Equation \ref{partition}. Hence
\[
(T_{\mathcal{F}}\circ\Phi_{\mathcal{F}})(\sigma)=\Phi_{\mathcal{F}}(\sigma
_{2}\sigma_{3}...)=(\Phi_{\mathcal{F}}\circ\mathcal{S}_{\mathcal{F}})(\sigma)
\]
for all $\sigma\in\Omega_{\mathcal{F}}$.
We now apply $\tau_{\mathcal{F}}$ to both sides of this last equation to
complete the proof of the first assertion in the theorem.
Now assume that $\Omega_{\mathcal{F}}\subset\Omega_{\mathcal{G}}$. Then,
since
\[
\mathcal{S}_{\mathcal{F}}(\sigma)=\mathcal{S}_{\mathcal{G}}(\sigma)
\]
for all $\sigma\in\Omega_{\mathcal{F}}$, it follows from the first part of the
theorem that%
\[
(\tau_{\mathcal{F}}\circ T_{\mathcal{F}}\circ\Phi_{\mathcal{F}})(\sigma
)=(\tau_{\mathcal{G}}\circ T_{\mathcal{G}}\circ\Phi_{\mathcal{G}})(\sigma)
\]
for all $\sigma\in\Omega_{\mathcal{F}}$. It follows that
\[
(\tau_{\mathcal{F}}\circ T_{\mathcal{F}}\circ\Phi_{\mathcal{F}}\circ
\tau_{\mathcal{F}})(x)=(\tau_{\mathcal{G}}\circ T_{\mathcal{G}}\circ
\Phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})(x)
\]
for all $x\in A_{\mathcal{F}}$. But $\Phi_{\mathcal{F}}\circ\tau_{\mathcal{F}%
}=i_{A_{\mathcal{F}}}$ and
\[
(\Phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})(x)=(\phi_{\mathcal{G}}\circ
\tau_{\mathcal{F}})(x)
\]
for all $x\in A_{\mathcal{F}}$. Hence%
\[
(\tau_{\mathcal{F}}\circ T_{\mathcal{F}})(x)=(\tau_{\mathcal{G}}\circ
T_{\mathcal{G}}\circ\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})(x)
\]
for all $x\in A_{\mathcal{F}}$. Applying $\phi_{\mathcal{G}}$ to both sides we
obtain%
\[
(\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}}\circ T_{\mathcal{F}})(x)=(\phi
_{\mathcal{G}}\circ\tau_{\mathcal{G}}\circ T_{\mathcal{G}}\circ\phi
_{\mathcal{G}}\circ\tau_{\mathcal{F}})(x)\text{ }%
\]
for all $x\in A_{\mathcal{F}}$. But $\phi_{\mathcal{G}}\circ\tau_{\mathcal{G}%
}=i_{A_{\mathcal{G}}}$. This completes the proof of the second assertion in
the theorem.
Finally, let us suppose that $\mathcal{C}_{\mathcal{F}}=\mathcal{C}%
_{\mathcal{G}}$. Then Theorem 1 implies that $\phi_{\mathcal{G}}\circ
\tau_{\mathcal{F}}$ is a homeomorphism from $A_{\mathcal{F}}$ onto
$A_{\mathcal{G}}$. Also $\mathcal{C}_{\mathcal{F}}=\mathcal{C}_{\mathcal{G}}$
implies $\Omega_{\mathcal{F}}=\Omega_{\mathcal{G}}$ which implies, via the
previously proven part of this theorem,
\[
T_{\mathcal{F}}(x)=(\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})^{-1}\circ
T_{\mathcal{G}}\circ\phi_{\mathcal{G}}\circ\tau_{\mathcal{F}})(x)
\]
for all $x\in A_{\mathcal{F}}$.
\end{proof}
If the domains $\{D_{n}:n=1,2,..,N\}$ are known then it is easy to compute the
tops function. Just follow the orbit of $x$ under the tops dynamical system
and keep track of the sequence of indices $\sigma_{1}\sigma_{2}\sigma
_{3}\sigma_{4}...$ \ visited by the orbit.
In the special case where the IFS is totally disconnected and the $f_{n}$s are
one-to-one then $T:A\rightarrow A$ is defined by $T(x)=f_{n}^{-1}(x)$ where
$n$ is the unique index such that $x\in f_{n}(A)$. This dynamical system has
been considered elsewhere, for example in \cite{BaDe} and \cite{kieninger}. In
this case $\phi_{\mathcal{F}}:\Omega\rightarrow A_{\mathcal{F}}$ is a
homeomorphism, $\tau_{\mathcal{F}}=\phi_{\mathcal{F}}^{-1}$, and
$T_{\mathcal{F}}$ it is conjugate to the shift transformation according
to$T_{\mathcal{F}}=\phi_{\mathcal{F}}\circ\mathcal{S}_{\mathcal{F}}\circ
\phi_{\mathcal{F}}^{-1}$.
Theorem \ref{TDSthm} says\ in particular that $T_{\mathcal{F}}:A_{\mathcal{F}%
}\rightarrow A_{\mathcal{F}}$ is a factor of $\mathcal{S}_{\mathcal{F}}%
:\Omega_{\mathcal{F}}\rightarrow\Omega_{\mathcal{F}}$, and as defined for
example in \cite{katok} p.68, because $\Phi_{\mathcal{F}}\circ\mathcal{S}%
_{\mathcal{F}}=T_{\mathcal{F}}\circ\Phi_{\mathcal{F}}$ where $\Phi
_{\mathcal{F}}=\tau_{\mathcal{F}}^{-1}$ is continuous; this tells us that the
topological entropy of $T_{\mathcal{F}}$ is less than or equal to the
topological entropy of $\mathcal{S}_{\mathcal{F}}$, \cite{katok} Proposition
3.1.6, p.111. If $\tau_{\mathcal{F}}$ is continuous then Theorem \ref{TDSthm}
says that the two dynamical systems $T_{\mathcal{F}}:A_{\mathcal{F}%
}\rightarrow A_{\mathcal{F}}$ and $\mathcal{S}_{\mathcal{F}}:\Omega
_{\mathcal{F}}\rightarrow\Omega_{\mathcal{F}}$ are topologically conjugate,
see \cite{katok} p. 60, and it follows that the two systems must have the same
topological entropy.
This suggests that we may compare the complexity of some subsets of
$\mathbb{R}^{2}$ by assigning to them the topological entropy of a
corresponding shift dynamical system. Let $\mathcal{M}$ denote the set of all
attractors of hyperbolic IFSs in $\mathbb{R}^{2}$, whose transformations are
all affine and invertible, such that the associated tops function is
continuous. Then we can define the topological entropy of each $A_{\mathcal{F}%
}$ to be the infimum of the entropies of the set of corresponding shift
dynamical systems. In this way we arrive at a geometry-based definition of the
topological entropy of some subsets of $\mathbb{R}^{2}$. Is it useful?
| {'timestamp': '2007-03-14T13:44:54', 'yymm': '0703', 'arxiv_id': 'math/0703398', 'language': 'en', 'url': 'https://arxiv.org/abs/math/0703398'} |
\section{Introduction}
\IEEEPARstart{T}{erahertz} (THz) and millimeter-wave (mmWave) are the key technologies for beyond 5G and next generation (6G) wireless communications, which can enable ultra-high data-rate communications and improve date security \cite{mmwaveSurvey,NatureShuping}.
However, the severe propagation loss and blockage-prone nature in such high-frequency bands also pose greater challenges to information security, such as short secure propagation distance and unreliable secure communications \cite{Ma2018Nature,Ela2020Open}.
These phenomena are more serious for THz bands due to higher frequency. Thus a new approach for information secure transmission in high frequency bands is urgently needed.
Recently, the intelligent reflecting surface (IRS) \cite{Renzo2019J,Nad2020open,Nad2020TWC}
has emerged as an invaluable way for widening signal coverage and overcome high pathloss of mmWave and THz systems \cite{Wang201908arXiv,Chen2019ICCC,Pan2020TWC} and has drawn increasing attention in secure communications.
In IRS-assisted secure systems, the IRS intelligently adjusts its phase shifts to steer the signal power to desired user, and reduce information leakage \cite{RuiZhang2019IEEEWCL}.
To maximize the secrecy rate, the active transmit beamforming and passive reflecting beamforming were jointly designed in \cite{Shen2019CL,Dong2020WCL,Robert2019GLOBECOM}.
However, the above works mainly focus on microwave systems, while IRS-assisted secure mmWave/THz systems still remain unexplored.
Moreover, extremely narrow beams of mmWave/THz waves can cause serious information leakage of beam misalignment and the costly implementation of continuous phase control.
Besides, the blockage-prone nature of high frequency bands may lead to a serious secrecy performance loss, since eavesdroppers can not only intercept but also block legal communications.
Motivated by the aforementioned problems, this letter investigates the IRS-assisted secure transmission in mmWave/THz bands.
Under the discrete phase-shift assumption, a joint optimization problem of transmit beamforming and reflecting matrix is formulated to maximize the secrecy rate. It is proved that under the rank-one channel model, the transmit beamforming design is independent of reflecting matrix design.
Thus the formulated non-convex problem is solved by converting it into two subproblems.
The closed-form transmit beamforming is derived and the the hybrid beamforming structure is designed adopting orthogonal matching pursuit (OMP) method.
Meanwhile, the semidefinite programming (SDP)-based method and the element-wise block coordinate descent (BCD) method are proposed to obtain the optimal discrete phase shifts.
Simulation results demonstrate that the proposed methods can achieve near-optimal secrecy rate performance
with discrete phase shifts.
\section{System Model and Problem Formulation}
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth
{figure/THz_IRS_systemmodel_vNoDT3
\caption{System model for IRS-assisted secure mmWave/THz system, where BS communicates with desired user Bob, in the presence of an eavesdropper.
\label{fig_1}
\end{figure}
The IRS-assisted secure mmWave/THz system is considered in this letter. As shown in \figref{fig_1}, one base station (BS) with $M$ antennas communicates with a single-antenna user Bob in the presence of a single-antenna eavesdropper Eve, which is an active user located near Bob.
To protect confidential signals from eavesdropping, an IRS with a smart controller is adapted to help secure transmission.
Note that due to deep path loss or obstacle blockage, there is no direct link between source and destination or eavesdropper.
\subsection{Channel Model}
It is assumed that the IRS with $N$ reflecting elements is installed on some high-rise buildings around the desired receiver Bob.
Thus, the LoS path is dominant for the BS-IRS channel and the rank-one channel model is adopted
\begin{equation}
\mathbf{H}_{BI}^{H}=\sqrt{MN}\alpha_B G_{r}G_{t}\mathbf{a}\mathbf{b}^{H},
\end{equation}
where $\alpha_B$ is the complex channel gain \cite{Bar2017TVT}, and $G_{r}$ and $G_{t}$ are receive and transmit antenna gain.
$\mathbf{a}\in \mathbb{C}^{N\times 1}$ and $\mathbf{b}\in \mathbb{C}^{M\times 1}$ denote the array steering vector at IRS and BS, respectively.
The IRS-Bob/Eve channel is assumed as\footnote{All channels are assumed to be perfectly known at BS, and the results derived can be considered as the performance upper bound. }
\begin{equation}
\mathbf{g}_k=\sqrt{\frac{N}{L}}\sum^L_{i=1}\alpha_{k,i}G_{r}^kG_{I}^{k}\mathbf{a}_{k,I},
\end{equation}
where $k=\{D,E\}$, $L$ is the number of paths from IRS to $k$ node, $\mathbf{a}_{k,I}$ denotes the transmit array steering vector at IRS.
\subsection{Signal Model}
In the IRS-assisted secure mmWave/THz system, the BS transmits signal $s$ with power $P_s$ to an IRS, and the IRS adjusts phase shifts of each reflecting element to help reflect signal to Bob.
We assume $\mathbf{\Theta}\!=\!\text{diag}\{ \!e^{j\theta_{1}}\!, e^{j\theta_{2}}\!,\dots,e^{j\theta_{N}}\!\}$ as the reflecting matrix, where $\theta_{i}$ is the phase shift of each element.
Different from traditional IRS-assisted strategies with continuous phase shifts, the discrete phase shifts are considered in terms of IRS hardware implementation.
Specifically, $\theta_{i}$ can only be chosen from a finite set of discrete values $\mathcal{F}\!=\!\{0,\Delta\theta,..., (L_P\!-\!1)\Delta\theta\}$, $L_P$ is the number of discrete values, and $\Delta\theta=2\pi/L_P$.
The received signal at user Bob can be expressed as
\begin{equation}\label{eq_3}
y_D=\mathbf{g}_{D}^{H}\mathbf{\Theta}\mathbf{H}_{BI}^{H}\mathbf{w}s+n_D,
\end{equation}
where $\mathbb{E}\{|s|^2\}\!\!=\!\!1$,
$n_D\!\sim\!\mathcal{CN}(0, \sigma_D^2)$ is the noise at destination.
$\mathbf{w}=\mathbf{F}_{RF}\mathbf{f}_{BB}$ is the transmit beamforming at the BS, where $\mathbf{F}_{RF}\!\in\!\mathbb{C}^{M\times R}$ is analog beamformer and $\mathbf{f}_{BB}\!\in\! \mathbb{C}^{R\times 1}$ is digital beamformer, $R$ is the number of radio frequency (RF) chains.
Similarly, the received signal at eavesdropper is written as
\begin{equation}\label{eq_4}
y_E=\mathbf{g}_E^{H}\mathbf{\Theta}\mathbf{H}_{BI}^{H}\mathbf{w}s+n_E,
\end{equation}
where $n_E \sim~\mathcal{CN}(0, \sigma_E^2)$ denotes the noise at eavesdropper.
Then the system secrecy rate can be written as
\begin{equation}\label{eq_Rs}
R_s=\left[\log_2\left(\frac{1+\frac{1}{\sigma_D^2}|\mathbf{g}_{D}^{H}\mathbf{\Theta}\mathbf{H}_{BI}^{H}\mathbf{w}|^2}
{1+\frac{1}{\sigma_E^2}|\mathbf{g}_{E}^{H}\mathbf{\Theta}\mathbf{H}_{BI}\mathbf{w}|^2}\right)\right]^+,
\end{equation}
where $[x]^+=\max\{0,x\}$.
\vspace{-0.2cm}
\subsection{Problem Formulation}
To maximize the system secrecy rate, the joint optimization problem of transmit beamforming and reflecting matrix is formulated as
\begin{equation*}
\setlength{\abovedisplayskip}{2pt}
\setlength{\belowdisplayskip}{3pt}
(\mathrm{P1}):~
\max_{\mathbf{w},\mathbf{\Theta}}
\frac{\sigma_D^2+|\mathbf{g}_{D}^{H}\mathbf{\Theta}\mathbf{H}_{BI}^{H}\mathbf{w}|^2}
{\sigma_E^{2}+|\mathbf{g}_{E}^{H}\mathbf{\Theta}\mathbf{H}_{BI}^{H}\mathbf{w}|^2}
\end{equation*}
\begin{equation}
\begin{array}{ll}
\text{s.t.} &\|\mathbf{w}\|^2\leq P_s,\\
& \theta_i\in \mathcal {F}, \forall i.\\
\end{array
\end{equation}
The formulated problem is non-convex and is quite challenging to be solved directly, because of coupled variables $(\mathbf{w},\mathbf{\Theta})$ and the non-convex constraint of $\theta_i$.
To cope with this difficulty, the original problem $\mathrm{P1}$ is converted into two subproblems, which can be solved alternatively.
\section{Secrecy Rate Maximization}
To find out solutions, we propose to convert problem $\mathrm{P1}$ into two subproblems. This idea is based on the fact that two subproblems are independent of each other.
Specifically, the closed-form solution of $\mathbf{w}$ will be first derived. Then the SDP-based method and element-wise BCD method will be proposed to obtain the solution of reflecting matrix.
\vspace{-0.2cm}
\subsection{Transmit Beamforming Design}
Since the rank-one channel is assumed from BS to IRS,
the subproblem with respect to beamformer $\mathbf{w}$ is expressed as
\begin{align}
(\mathrm{P2.1}):~&
\max_{\mathbf{w}}
\frac{\sigma_D^2+|\alpha_{B}G_{r}G_{t}\mathbf{g}_{D}^{H}\mathbf{\Theta}\mathbf{a}|^2|\mathbf{b}^{H}\mathbf{w}|^2}
{\sigma_E^{2}+|\alpha_{B}G_{r}G_{t}\mathbf{g}_{E}^{H}\mathbf{\Theta}\mathbf{a}|^2|\mathbf{b}^{H}\mathbf{w}|^2}\nonumber\\
&~\text{s.t.} ~~\|\mathbf{w}\|^2\leq P_s.
\end{align}
\begin{prop}\label{prop_1}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
Under the positive secrecy rate constraint, the suboptimal problem of $\mathbf{w}$ is equivalent to
\begin{align}
(\mathrm{P2.1}^{'}):
\max_{\mathbf{w}}
|\mathbf{b}^{H}\mathbf{w}|^2,\nonumbe
~~~\text{s.t.} ~\|\mathbf{w}\|^2\leq P_s.
\end{align}
It is independent of reflecting matrix design.
For any value of $\mathbf{\Theta}$, the transmit beamformer solution is $\mathbf{w}^{opt}=\sqrt{P_s}\frac{\mathbf{b}}{\|\mathbf{b}\|}$.
\end{prop}
\begin{proof}
See Appendix A.
\end{proof}
After obtaining $\mathbf{w}^{opt}$ from Proposition 1, $(\mathbf{F}_{RF}^{opt}, \mathbf{f}_{BB}^{opt})$ with a full-connected architecture can be easily derived by typical hybrid precoding methods, such as OMP algorithm \cite{Tropp2007TIT}.
\subsection{Reflecting Matrix Design}
To facilitate the following mathematical operations, we first define $\hat{\pmb{\theta}}=[e^{j\theta_1},e^{j\theta_2},...,
e^{j\theta_N}]^{T}$, then we have $\mathbf{\Theta}=\text{diag}\{\hat{\pmb{\theta}}\}$.
The subproblem of reflecting matrix can be rewritten as
\vspace{-0.2cm}
\begin{align}
(\mathrm{P2.2}):~&
\max_{\hat{\pmb{\theta}}}
\frac{1+\frac{1}{\sigma_D^2}|\hat{\pmb{\theta}}^{T}\text{diag}\{\mathbf{g}_{D}^{H}\}\mathbf{H}_{BI}^{H}\mathbf{w}|^2}
{1+\frac{1}{\sigma_E^2}|\hat{\pmb{\theta}}^{T}\text{diag}\{\mathbf{g}_{E}^{H}\}\mathbf{H}_{BI}^{H}\mathbf{w}|^2},\nonumber\\
&\text{s.t.} ~ \hat{\pmb{\theta}}=[e^{j\theta_1},e^{j\theta_2},...,e^{j\theta_N}]^{T},
\theta_i\in \mathcal {F}, \forall i.\nonumber
\end{align}\label{P_2.2}
It is obvious that the variable $\theta_i$ only takes a finite number of values from $\mathcal{F}$.
Thus, problem $\mathrm{P2.2}$ is feasible to be solved with an exhaustive search method.
However, due to a large feasible set of $\hat{\pmb{\theta}}$ ($N^{L_P}$ possibilities), the complexity of such method is considerably high.
To cope with this, the SDP-based algorithm and element-wise BCD algorithm are proposed.
\subsubsection{SDP-based Algorithm}
Define $\mathbf{\Phi}=\hat{\pmb{\theta}}^{*}(\hat{\pmb{\theta}}^{*})^{H}$ and relax discrete variables $\theta_i$ into continuous $\theta_i\in[0,2\pi)$, i.e., $|e^{j\theta_i}|=1, \forall i$, then we can rewrite problem $\mathrm{P2.2}$ as $\max_{\mathbf{\Phi}\succeq 0} \frac{\text{tr}(\hat{\mathbf{R}}_{RD}\mathbf{\Phi})}{\text{tr}(\hat{\mathbf{R}}_{RE}\mathbf{\Phi})}$ with rank-one constraint, $\text{rank}(\mathbf{\Phi})=1$.
Since $\text{rank}(\mathbf{\Phi})=1$ is a non-convex constraint, the semidefinite relaxation (SDR) is adopted to relax this constraint.
Using Charnes-Cooper transformation approach, we define $\mathbf{X}=\mu\mathbf{\Phi}$ and $\mu=1/\text{tr}(\hat{\mathbf{R}}_{RE}\mathbf{\Phi})$. Then problem $\mathrm{P2.2}$ is rewritten as
\begin{align}\label{eq_11}
(\mathrm{P2.2^{'}}):~& \max_{\mu\geq 0,\mathbf{X}\succeq 0}
\text{tr}(\hat{\mathbf{R}}_{RD}\mathbf{X}) \nonumber\\
&\begin{array}{ll}
\text{s.t.}&
\text{tr}(\hat{\mathbf{R}}_{RE}\mathbf{X})=1,\\
&\text{tr}(\mathbf{E}_{n}\mathbf{X})=\mu, \forall n.\\
\end{array
\end{align}
where $\hat{\mathbf{R}}_{RD}=\frac{1}{N}\mathbf{I}_{N}+\frac{1}{\sigma_{D}^2}\text{diag}\{\mathbf{g}_{D}^{*}\}
\mathbf{H}_{BI}^{H}\mathbf{w}\mathbf{w}^{H}\mathbf{H}_{BI}
\text{diag}\{\mathbf{g}_{D}\}$,
$\hat{\mathbf{R}}_{RE}\!=\!\!\frac{1}{N}\mathbf{I}_{N}\!+\!\frac{1}{\sigma_{E}^2}\text{diag}\{\mathbf{g}_{E}^{*}\}
\mathbf{H}_{BI}^{H}\mathbf{w}\mathbf{w}^{H}\mathbf{H}_{BI}
\text{diag}\{\mathbf{g}_{E}\}$, and $\mathbf{E}_n$ means the element on position $(n,n)$ is $1$ and 0 otherwise.
Problem $\mathrm{P2.2^{'}}$ is a standard SDP problem, and can be solved by adopting Interior-point method or CVX tools. \cite{QingWu2019TCOM}
Then the rank-one solution $\hat{\pmb{\theta}}^{opt}\!\!=\![
e^{j\tilde{\theta}_1^{opt}},e^{j\tilde{\theta}_2^{opt}},
...,e^{j\tilde{\theta}_N^{opt}}]^{T}$ can be achieved by the Gaussian randomization method.
To obtain the discrete solution of problem $\mathrm{P2.2}$, we quantify the continuous solution $\hat{\pmb{\theta}}^{opt}$ as the nearest discrete value in set $\mathcal{F}$, and the following principle is adopted
\begin{equation}\label{eq_12}
\theta_{i}^{opt}=\arg\min_{\theta_i\in \mathcal{F}} |e^{j\tilde{\theta}_i^{opt}}-e^{j\theta_i}|, ~~\forall i.
\end{equation}
Note that since the closed-form solution of $\theta_i$ is not obtained, the entire process of SDP-based method needs to be done for each transmission block, which leads to high complexity.
\subsubsection{Element-Wise BCD Algorithm}
To obtain the closed-form solution of reflecting matrix, the element-wise BCD method \cite{Robert2019GLOBECOM} is employed in this section.
Taking each $\theta_i$ as one block in the BCD, we can iteratively derive the continuous solutions of phase shifts using Proposition \ref{prop_2}.
\begin{prop}\label{prop_2}
There exists one and only one optimal $\theta_i^{opt}$ to maximize the secrecy rate, and
\begin{equation}\label{eq_18}
\small
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
\theta_i^{opt}\!\!=\!\!
\left\{\begin{array}{ll}
\!\!\tilde{\theta}_i^{opt},\!\!&\!\!\! c_{D,i}d_{E,i}\cos(p_{E,i})\!<\!c_{E,i}d_{D,i}\!\cos(p_{D,i})\\
\!\!\tilde{\theta}_i^{opt}\!+\!\pi,\!\!&\!\!\! \text{otherwise}
\!\!\end{array}\right.
\end{equation}
where $\tilde{\theta}_i^{opt}$ is shown in (\ref{eq_13})
on the top of next page, and
\newcounter{mytempeqncnt}
\begin{figure*}[!t]
\setcounter{mytempeqncnt}{\value{equation}}
\setcounter{equation}{12}
\begin{equation}\label{eq_13}
\tilde{\theta}_i^{opt}=-\arctan\!\left(\frac{c_{D,i}d_{E,i}\sin(p_{E,i})\!-\!c_{E,i}d_{D,i}\sin(p_{D,i})}{c_{D,i}d_{E,i}\cos(p_{E,i})\!-\!c_{E,i}d_{D,i}\cos(p_{D,i})}\right)
-\arcsin\left(\!\frac{-d_{D,i}d_{E,i}\sin(p_{E,i}-p_{D,i})}{\sqrt{c_{D,i}^{2}d_{E,i}^{2}+\!c_{E,i}^{2}d_{D,i}^{2}
\!-\!2c_{D,i}c_{E,i}d_{D,i}d_{E,i}\cos(p_{E,i}\!-\!p_{D,i})}}\!\right)
\end{equation}
\setcounter{equation}{\value{mytempeqncnt}}
\hrulefill
\vspace*{-15pt}
\end{figure*}
\setcounter{equation}{13}
\begin{align
&c_{k,i}=1+\frac{1}{\sigma_k^2}\left|g_{k,i}^{*}\mathbf{h}_{BI,i}^{H}\mathbf{w}\right|^2+\frac{1}{\sigma_k^2}\left|\sum_{m\neq i}e^{j\theta_{m}}g_{k,m}^{*}\mathbf{h}_{BI,m}^{H}\mathbf{w}\right|^2,\nonumber\\
&d_{k,i}=\frac{2}{\sigma_k^2}\left|\left(g_{k,i}^{*}\mathbf{h}_{BI,i}^{H}\mathbf{w}\right)
\cdot
\left(\sum_{m\neq i}e^{-j\theta_{m}}g_{k,m}\mathbf{w}^{H}\mathbf{h}_{BI,m}\right)\right|,\nonumber\\
&p_{k,i}=\angle\left(g_{k,i}^{*}\mathbf{h}_{BI,i}^{H}\mathbf{w}\sum_{m\neq i}
e^{-j\theta_{m}}g_{k,m}\mathbf{w}^{H}\mathbf{h}_{BI,m}\right).\nonumber
\end{align
in which $k\in\{D,E\}$.
\end{prop}
\begin{proof}
See Appendix B.
\end{proof}
With the above proposition, the optimal discrete solution of $\theta_i$ can be chosen using the same quantization principle in (\ref{eq_12}), and the entire element-wise BCD-based secrecy rate maximization algorithm can be summarized as Algorithm \ref{alg_1}.
Since the objective function in $\mathrm{P2.2}$ is non-decreasing after each iteration and upper-bounded by a finite value of generalized eigenvalue problem \cite{Qiao2018TVT}, the convergence is guaranteed.
\begin{algorithm}
\caption{E-BCD based Secrecy Rate Maximization
}
\label{alg_1}
\begin{algorithmic}[1]
\STATE
Initialize
$\mathbf{\Theta}^{0}=\text{diag}\{e^{j\theta_{1}^{0}},e^{j\theta_{2}^{0}},...,e^{j\theta_{N}^{0}}\}$, $\varepsilon$, and set $n=0$.
\STATE Find $\mathbf{w}^{opt}$ using Proposition \ref{prop_1}, then calculate optimal $\mathbf{F}_{BF}^{opt}$ and $\mathbf{f}_{BB}^{opt}$ using OMP method.
\REPEAT
\STATE $n=n+1$
\STATE
\textbf{for} $i=1,2,...,N$
\textbf{do}
\STATE calculate $\theta_{i}^{n}$ using Proposition 2, and quantify it as discrete solution.
\STATE
\textbf{end for}
\UNTIL
{$\|\mathbf{\Theta}^{n}-\mathbf{\Theta}^{n-1}\|\leq \varepsilon$}.
\RETURN $(\mathbf{w}^{opt})^{'}=\mathbf{F}_{BF}^{opt}\mathbf{f}_{BB}^{opt}$ and $\mathbf{\Theta}^{opt}=\mathbf{\Theta}^{n}$
\end{algorithmic}
\end{algorithm}
\subsection{Complexity Analysis}
The total complexity of the SDP-based method is about $\mathcal{O}(N_{gaus}N^8)$,
which is mainly determined by the complexity of solving SDP problem and the number of rank-one solutions to construct the feasible set of Gaussian randomization method `$N_{gaus}$'.
While the complexity of the proposed element-wise BCD method is about $\mathcal{O}(N(NM\!+\!L_P)N_{iter})$, which relies on the complexity of $\theta_i$ calculation and the iteration number $N_{iter}$ for $\mathbf{\Theta}^{opt}$.
Intuitively, these methods have lower complexity than exhaustive research method $\mathcal{O}(N^{L_P+1}(N^2\!+\!NM))$, and the complexity of the element-wise BCD method is the lowest.
\section{Simulation Results and Analysis}
In this section, simulation results are presented to validate the secrecy rate performance in mmWave and THz bands.
All results are obtained by averaging over $1000$ independent trials.
We consider a scenario where the BS employs a uniform linear array (ULA) and the IRS is a uniform rectangular array (URA).
Unless otherwise specified, the transmitter frequency is $f=0.3$~THz, and the transmit power is $P_s=25$~dBm, $M=16$. Due to a full-connected hybrid beamforming structure at BS, the RF chain number is less than $M$, $M_{RF}=10$.
The number of discrete values in $\mathcal{F}$ is $L_P=2^3$, and $\Delta \theta=\frac{\pi}{4}$.
Additionally, the antenna gain is set to $12$~dBi.
The complex channel gain $\alpha_B$ and $\alpha_{k,i}$ can be obtained on the basis of \cite{Bar2017TVT}.
\begin{figure}[!t]
\centering
\subfigure[$R_s$ versus $L_P$.]{
\begin{minipage}{0.2\textwidth
\centering
\includegraphics[width=1\textwidth]{figure/fig_1_1_THz_Lp}\\
\label{fig_2}
\end{minipage}}
\subfigure[$R_s$ versus $P_s$.]{
\begin{minipage}{0.2\textwidth}
\centering
\includegraphics[width=1\textwidth]{figure/fig_1_2_THz_Ps}\\
\label{fig_3}
\end{minipage}}
\caption{Secrecy rate performance for IRS-assisted mmWave and THz system, in which $N=4$, the BS-to-IRS and IRS-to-Bob distances are $d_{sr}=d_{rd}=5$~m, and Eve is located near Bob with IRS-Eve distance $d_{re}=5$~m.}
\end{figure}
The secrecy rate with different number of discrete values of set $\mathcal{F}$ is shown in \figref{fig_2}. With the increase of $L_P$, the secrecy rate of proposed methods approaches that of exhaustive search method (`Exh')\footnote{The solution of exhaustive search method is the optimal solution of discrete case, since all feasible solutions of phase shifts are searched.}.
This is rather intuitive, since an increased number of discrete values leads to a reduction in quantization error in (\ref{eq_12}), thereby enhancing secrecy rate.
Besides, it is obvious that discrete schemes are upper-bounded by the continuous solution (`Cont').
The secrecy rate versus transmit power $P_s$ is shown in \figref{fig_3}.
Based on the optimal beamformer obtained from Proposition \ref{prop_1}, it is easily proved that the secrecy rate is an increasing function of $P_s$.
Thus as $P_s$ increase,
the secrecy rate increases monotonously. Similarly as \figref{fig_2}, \figref{fig_3} reveals that our proposed methods can achieve near optimal performance as exhaustive search method.
Besides, results also demonstrate that the more antennas are equipped at BS, the higher secrecy rate is achieved.
The effect of the number of reflecting elements on the secrecy rate of IRS-based and BS-based interception is also investigated.\footnote{Here, optimal $(\mathbf{w}, \mathbf{\Theta})$ for BS-based interception can be derived iteratively.}.
As shown in \figref{fig_4}, as N increases from $10$ to $100$, the secrecy rate increases monotonously. This is because the more reflecting elements result in sharper reflecting beams, thereby enhancing information security.
In particular, when Eve is located within the reflecting/transmit beam of IRS/BS,
we assume that it can intercept and block $\rho$ portion of confidential signals.
Intuitively, the more information is blocked, the worse secrecy rate can be achieved.
Thus, compared with interception without blocking, this case is more serious for mmWave and THz communications.
However, since the IRS-assisted secure transmission scheme is designed, the secrecy rate is significantly improved compared with secure oblivious approach (only maximizing information rate at the legal user).
\begin{figure}[!t]
\centering
\subfigure[Eve intercepts IRS.]
\begin{minipage}{0.2\textwidth
\centering
\includegraphics[width=1\textwidth]{figure/fig_resp_4_2IRS}\\%fig_4_1THz_N_nearr
\label{fig_4a}
\end{minipage}}
\subfigure[Eve intercepts BS.]
\begin{minipage}{0.2\textwidth}
\centering
\includegraphics[width=1\textwidth]{figure/fig_resp_4_1BS}\\%fig_4_2THz_N_blockrr2
\label{fig_4b}
\end{minipage}}
\caption{Secrecy rate versus the number of reflecting elements from $10$ to $100$.
(a) Eve intercepts IRS, $d_{re}\!=\!5$~m for non-blocking, $d_{re}\!=\!2$~m for blocking, (b) Eve intercepts BS, $d_{se}\!=\!5$~m for non-blocking, $d_{se}\!=\!2$~m for blocking.
}
\label{fig_4}
\end{figure}
\section{Conclusion}
This letter investigated the secrecy performance of IRS-assisted mmWave/THz systems.
Considering the hardware limitation at the IRS, the transmit beamforming at the BS and discrete phase shifts at the IRS have been jointly designed to maximize the system secrecy rate.
To deal with the formulated non-convex problem, the original problem was divided into two subproblems under the rank-one channel assumption.
Then the closed-form beamforming solution was derived,
and the reflecting matrix was obtained by the proposed SDP-based method and element-wise BCD method.
Simulations demonstrated that our proposed methods can achieve the near optimal secrecy performance, and can combat eavesdropping occurring at the BS and the IRS.
\appendices
\section{Proof of Proposition 1}
In $\mathrm{P2.1}$, the beamforming vector $\mathbf{w}$ is always coupled with $\mathbf{b}^{H}$, thus the secrecy rate can be seen as a function of $|\mathbf{b}^{H}\!\mathbf{w}|^2$.
Under the positive secrecy rate constraint, i.e., $\scriptsize{|\mathbf{g}_{D}^{H}\!\mathbf{\Theta}\mathbf{a}|^2\!/\sigma_D^2\!\!>\!\!|\mathbf{g}_{E}^{H}\!\mathbf{\Theta}\mathbf{a}|^2\!/\sigma_E^2}$,
it is easily proved that $R_s$ is an increasing function of $|\mathbf{b}^{H}\!\mathbf{w}|^2$.
That is, $\mathrm{P2.1}$ is equivalent to $\max|\mathbf{b}^{H}\mathbf{w}|^2$ and is intuitively independent of $\mathbf{\Theta}$.
\section{Proof of Proposition 2}
Choosing each element at the IRS $\theta_i$ as one block of BCD, the objective function of $\mathrm{P2.2}$ can be reformulated as
\begin{equation}
f(\theta_i)=\frac{c_{D,i}+d_{D,i}\cos(\theta_i+p_{D,i})}{c_{E,i}+d_{E,i}\cos(\theta_i+p_{E,i})},
\end{equation}
in which all parameters can be found in section III-B-2, and $c_{D,i},c_{E,i}\geq 1$, $c_{D,i}>d_{D,i}>0$, $c_{E,i}>d_{E,i}>0$.
The sign of the derivative of $f(\theta_i)$ is determined by\footnote{To reduce complexity, $\sin(x)$-based expression is used instead of $\cos(x)$.}
\begin{equation}
h(x)=\sqrt{A_i^2+B_i^2}\sin(x)+C_i,
\end{equation}
where $x\!=\!\theta_i\!+\!\phi$,
$A_{i}\!=\!c_{D,i}d_{E,i}\cos(p_{E,i})\!-\!c_{E,i}d_{D,i}\cos(p_{D,i})$, $B_{i}\!= \!c_{D,i}d_{E,i}\sin(p_{E,i})\!-c_{E,i}d_{D,i}\sin(p_{D,i})$,
and $ C_i=d_{D,i}d_{E,i}\sin(p_{E,i}-p_{D,i})$, $\sin(\phi)=B_i\!/\!\sqrt{A_i^2\!+\!B_i^2}$, $\cos(\phi)=A_i/\sqrt{A_i^2+B_i^2}$.
Then the main problem in deriving the unique optimal solution is to determine the value of $\phi$ or $\phi^{'}$.
\begin{itemize}
\item For $A_i>0$, we have $\cos(\phi)>0$ and $\phi=\arctan(\frac{B_i}{A_i})$. Thus, $\theta_i^{opt}=\pi-\arctan(\frac{B_i}{A_i})-\arcsin(\frac{-C_i}{\sqrt{A_i^2+B_i^2}})$.
\item For $A_i\!<\!0$, we have $\cos(\phi)\!<\!0$ and $\phi=\pi+\arctan(\frac{B_i}{A_i})$. Thus $\theta_i^{opt}=-\arctan(\frac{B_i}{A_i})-\arcsin(\frac{-C_i}{\sqrt{A_i^2+B_i^2}})$.
\end{itemize}
\vspace{-0.2cm}
\scriptsize
\bibliographystyle{IEEEtran}
| {'timestamp': '2020-05-28T02:20:30', 'yymm': '2005', 'arxiv_id': '2005.13451', 'language': 'en', 'url': 'https://arxiv.org/abs/2005.13451'} |
\subsubsection*{Organisation of the paper}
The paper is organised as follows. In Section~\ref{sec-def}, we briefly recall the basic properties of hyperbolic groups which we will need, and we introduce Markov compacta. Section~\ref{sec-typy} contains auxiliary facts regarding mainly the \textit{conical} and \textit{ball types} in hyperbolic groups (defined in~\cite{CDP}), which will be the key tool in the proof of Theorem~\ref{tw-kompakt}.
The main claim of Theorem~\ref{tw-kompakt} is obtained by constructing an appropriate family of covers of $\partial G$ in Section~\ref{sec-konstr}, considering the corresponding inverse sequence of nerves in Section~\ref{sec-engelking-top} and finally verifying the Markov property in Section~\ref{sec-markow}. An outline of this reasoning is given in the introduction to Section~\ref{sec-konstr}, and its summary appears in Section~\ref{sec-markow-podsum}. Meanwhile, we give the proof of Theorem~\ref{tw-bi-lip-0} (as a~corollary of Theorem~\ref{tw-bi-lip}) in Section~\ref{sec-bi-lip}, mostly by referring to the content of Section~\ref{sec-engelking-top}.
While the Markov system obtained at the end of Section~\ref{sec-markow} will be already barycentric and have mesh property, in Sections~\ref{sec-abc} and \ref{sec-wymd} we focus respectively on ensuring distinct types property and bounding the dimensions of involved complexes. This will lead to a complete proof of Theorem~\ref{tw-kompakt}, summarised in Section~\ref{sec-wymd-podsum}.
Finally, Section~\ref{sec-sm} contains the proof of Theorem~\ref{tw-semi-markow-0}; its content is basically unrelated to Sections~\ref{sec-konstr}--\ref{sec-wymd}, except for that we re-use the construction of \textit{$B$-type} from Section~\ref{sec-sm-abc-b}.
\subsubsection*{Acknowledgements}
I would like to thank my supervisors Jacek Świątkowski and Damian Osajda for recommending this topic, for helpful advice and inspiring conversations, and Aleksander Zabłocki for all help and careful correcting.
\section{Introduction}
\label{sec-def}
\subsection{Hyperbolic groups and their boundaries}
\label{sec-def-hip}
Throughout the whole paper, we assume that $G$ is a~hyperbolic group in the sense of~Gromov~\cite{G}. We implicitly assume that $G$ is equipped with a~fixed, finite generating set~$S$, and we identify~$G$ with its Cayley graph~$\Gamma(G, S)$. As a~result, we will often speak about ``distance in~$G$'' or ``geodesics in~$G$'', referring in fact to the Cayley graph. Similarly, the term ``dependent only on~$G$'' shall be understood so that dependence on~$S$ is also allowed. By~$\delta$ we denote some fixed constant such that $\Gamma(G, S)$ is a $\delta$-hyperbolic metric space; we assume w.l.o.g. that $\delta \geq 1$.
We denote by~$e$ the identity element of~$G$, and by~$d(x, y)$ the distance of elements $x, y \in G$. The distance $d(x, e)$ will be called the \textit{length} of~$x$ and denoted by~$|x|$.
We use a notational convention that $[x, y]$ denotes a geodesic segment between the points $x, y \in G$, that is, an isometric embedding $\alpha : [0, n] \cap \mathbb{Z} \rightarrow G$ such that $\alpha(0) = x$ and $\alpha(n) = y$, where $n$ denotes $d(x, y)$. In the sequel, geodesic segments as well as geodesic rays and bi-infinite geodesic paths (i.e. isometric embeddings resp. of~$\mathbb{N}$ and $\mathbb{Z}$) will be all refered to as ``geodesics in $G$''; to specify which kind of geodesic is meant (when unclear from context), we will use adjectives \textit{finite}, \textit{infinite} and \textit{bi-infinite}.
We denote by~$\partial G$ the Gromov boundary of~$G$, defined as in~\cite{Kap}. We recall after \cite[Chapter 1.3]{zolta} that, as a~set, it is the~quotient of the set of all infinite geodesic rays in~$G$ by the relation of being close:
\[ (x_n) \sim (y_n) \qquad \Leftrightarrow \qquad \exists_{C > 0} \ \forall_{n \geq 0} \ d(x_n, y_n) < C; \]
moreover, in the above definition one can equivalently assume that $C = 4\delta$. It is also known that the topology defined on $\partial G$ is compact, preserved by the natural action of~$G$, and compatible with a~family of \textit{visual metrics}, defined depending on a~parameter~$a > 1$ with values sufficiently close to~$1$. Although we will not refer directly to the definition and properties of these metrics, we will use an estimate stated as (P2) in~\cite[Chapter~1.4]{zolta} which guarantees that, for every sufficiently small~$a > 1$, the visual metric with parameter~$a$ (which we occasionally denote by $d_v^{(a)}$) is bi-Lipschitz equivalent to the following \textit{distance function}:
\[ d_a \big( p, q \big) = a^{-l} \qquad \textrm{ for } p, q \in \partial G, \]
where $l$ is the largest possible distance between $e$ and any bi-infinite geodesic in~$G$ joining $p$ with $q$. As we will usually work with a~fixed value of~$a$, we will drop it in the notation.
For $x, y \in G \cup \partial G$, the symbol $[x, y]$ will denote \textit{any} geodesic in~$G$ joining~$x$ with~$y$. We will use the following fact from~\cite[Chapter~1.3]{zolta}:
\begin{fakt}
\label{fakt-waskie-trojkaty}
Let $\alpha, \beta, \gamma$ be the sides of a~geodesic triangle in~$G$ with vertices in~$G \cup \partial G$. Then, $\alpha$ is contained in the $4(p+1)\delta$-neighbourhood of $\beta \cup \gamma$, where $p$ is the number of vertices of the triangle which lie in~$\partial G$.
\end{fakt}
\subsection{Markov compacta}
\label{sec-def-markow}
\begin{df}[{\cite[Definition~1.1]{Dra}}]
\label{def-kompakt-markowa}
Let $(K_i, f_i)_{i \geq 0}$ be an inverse system consisting of the spaces $K_i$ and maps $f_i: K_{i + 1} \rightarrow K_i$ for $i \geq 0$.
Such system will be called \textit{Markov} (or said to satisfy \textit{Markov property}) if the following conditions hold:
\begin{itemize}
\item[(i)] $K_i$ are finite simplicial complexes which satisfy the inequality $\sup \dim K_i < \infty$;
\item[(ii)] for every simplex $\sigma$, in $K_{i+1}$ its image $f_i(\sigma)$ is contained in some simplex belonging to $K_i$ and the restriction $f_i|_\sigma$ is an affine map;
\item[(iii)] simplexes in $\amalg_i K_i$ can be assigned finitely many \textit{types} so that for any simplexes $s \in K_i$ and~$s' \in K_j$ of the same type there exist isomorphisms of subcomplexes $i_k : (f^{i+k}_i)^{-1}(s) \rightarrow (f^{j+k}_j)^{-1}(s')$ for $k \geq 0$ such that the following diagram commutes:
\begin{align}
\label{eq-markow-drabinka}
\xymatrix@+3ex{
s \ar[d]_{i_0} & \ar[l]_{f_i} f_i^{-1}(s) \ar[d]_{i_1} & \ar[l] \ldots & \ar[l] (f^{i+k}_i)^{-1}(s) \ar[d]_{i_k} & \ar[l]_{f_{i+k}} (f^{i+k+1}_i)^{-1}(s) \ar[d]_{i_{k+1}} & \ar[l] \ldots \\
s' & \ar[l]_{f_j} f_j^{-1}(s') & \ar[l] \ldots & \ar[l] (f^{j+k}_j)^{-1}(s') & \ar[l]_{f_{j+k}} (f^{j+k+1}_j)^{-1}(s') & \ar[l] \ldots
}
\end{align}
where $f^a_b$ (for $a \geq b$) means the composition $f_b \circ f_{b+1} \circ \ldots \circ f_{a-1} : K_a \rightarrow K_b$.
\end{itemize}
\end{df}
\begin{df}[{\cite[Definition~1.1]{Dra}}]
\label{def-kompakt-markowa2}
A topological space $X$ is a \textit{Markov compactum} if it is the inverse limit of a Markov system.
\end{df}
\begin{df}[{cf.~\cite[Lemma~2.3]{Dra}}]
\label{def-mesh}
\
\begin{itemize}
\item[\textbf{(a)}] A sequence $(\mathcal{A}_n)_{n \geq 0}$ of families of subsets in a compact metric space has \textit{mesh property} if \[ \lim_{n \rightarrow \infty} \, \max_{A \in \mathcal{A}_n} \diam A = 0. \]
\item[\textbf{(b)}] An inverse system of polyhedra $(K_n, f_n)$ has \textit{mesh property} if, for any $i \geq 0$, the sequence $(\mathcal{F}_n)_{n \geq i}$ of families of subsets in $K_i$ has mesh property, where
\[ \mathcal{F}_n = \big\{ f^n_i(\sigma) \ \big|\ \sigma \textrm{ is a simplex in }K_n \big\}. \]
\end{itemize}
\end{df}
\begin{uwaga}
\label{uwaga-mesh-bez-metryki}
We can formulate Definition \ref{def-mesh}a in an equivalent way (regarding only the topology): for any open cover $\mathcal{U}$ of $X$ there exists $n \geq 0$ such that, for every $m \geq n$, every set $A \in \mathcal{A}_m$ is contained in some $U \in \mathcal{U}$.
In particular, this means that the sense of Definition \ref{def-mesh}b does not depend on the choice of a metric (compatible with the topology) in $K_i$.
\end{uwaga}
\begin{df}
\label{def-kompakt-barycentryczny}
A Markov system $(K_i, f_i)$ is called \textit{barycentric} if, for any $i \geq 0$, the vertices of $K_{i+1}$ are mapped by $f_i$ to the vertices of the first barycentric subdivision of $K_i$.
\end{df}
\begin{df}
\label{def-kompakt-wlasciwy}
A Markov system $(K_i, f_i)$ has \textit{distinct types property} if for any $i \geq 0$ and any simplex $s \in K_i$ all simplexes in the pre-image $f_i^{-1}(s)$ have pairwise distinct types.
\end{df}
\begin{uwaga}
\label{uwaga-sk-opis}
A motivation for the above two definitions is the observation that barycentric Markov systems with distinct types property are \textit{finitely describable}. In more detail, if the system $(K_i, f_i)_{i \geq 0}$ satisfies the conditions from Definitions ~\ref{def-kompakt-markowa}, \ref{def-kompakt-barycentryczny} and~\ref{def-kompakt-wlasciwy}, and if $N$ is so large that complexes $K_0, \ldots, K_N$ contain simplexes of all possible types, then the full system $(K_i, f_i)_{i = 0}^\infty$ can by rebuilt on the base of the initial part of the system (which is finitely describable because of being barycentric).
\[
\xymatrix@C+3ex{
K_0 & \ar[l]_{f_0} K_1 & \ar[l]_{f_1} \ldots & \ar[l]_{f_{N-1}} K_N & \ar[l]_{f_N} K_{N+1}.
}
\]
The proof is inductive: for any $n \geq N+1$ the complex $K_{n+1}$ with the map $f_n : K_{n+1} \rightarrow K_n$ is given uniquely by the subsystem $K_0 \longleftarrow \ldots \longleftarrow K_n$. This results from the following:
\begin{itemize}
\item for any simplex $s \in K_n$ there exists a model simplex $\sigma \in K_m$ of the same type, where $m < n$, and then the pre-image $f_n^{-1}(s)$ together with the types of its simplexes and the restriction $f_n \big|_{f_n^{-1}(s)}$ is determined by the pre-image $f_{m}^{-1}(\sigma)$ and the restriction $f_m \big|_{f_m^{-1}(\sigma)}$ (which follows from Definition \ref{def-kompakt-markowa});
\item for any pair of simplexes $s' \subseteq s \in K_n$ the choice of a type preserving injection $f_n^{-1}(s') \rightarrow f_n^{-1}(s)$ is uniquely determined by the fact that vertices in $f_n^{-1}(s)$ have pairwise distinct types (by Definition \ref{def-kompakt-wlasciwy});
\item since $K_{n+1}$ is the union of the family of pre-images of the form $f_n^{-1}(s)$ for $s \in K_n$, which is closed with respect to intersecting, the knowledge of these pre-images and the type preserving injections between them is sufficient to recover $K_{n+1}$; obviously we can reconstruct~$f_n$ too, by taking the union of the maps $f_n^{-1}(s) \rightarrow s$ determined so far.
\end{itemize}
\end{uwaga}
\section{Types of elements of~$G$}
\label{sec-typy}
The goal of this section is to introduce the main properties of the \textit{cone types} (Definition \ref{def-typ-stozkowy}) and \textit{ball types} (Definition \ref{def-typ-kulowy}) for elements of a hyperbolic group $G$. These classical results will be used in the whole paper.
The connection between cone types (which describe the natural structure of the group and its boundary) and ball types (which are obviously only finite in number) in the group $G$ was described for the first time by Cannon in \cite{Cannon} and used to prove properties of the growth function for the group. This result turns out to be an important tool in obtaining various finite presentations of Gromov boundary: it is used in \cite{CDP} to build an automatic structure on $\partial G$ and in \cite{zolta} to present $\partial G$ as a semi-Markovian space in torsion-free case (the goal of Section \ref{sec-sm} is to generalise this result to all groups). Therefore it is not surprising that we will use this method to build the structure of Markov compactum for the space $\partial G$.
\subsection{Properties of geodesics in~$G$}
\begin{fakt}
\label{fakt-geodezyjne-pozostaja-bliskie}
Let $\alpha = [e, x]$ and $\beta = [e, y]$, where $|x| = |y| = n$ and $d(x, y) = k$. Then, for $0 \leq m \leq n$, the following inequality holds:
\[ d \big( \alpha(m), \beta(m) \big) \ \leq \ 8\delta + \max \big( k + 8\delta - 2(n-m), \, 0 \big). \]
In particular, for $0 \leq m \leq n - \tfrac{k}{2} - 4\delta$, we have $d(\alpha(m), \beta(m)) \leq 8\delta$.
\end{fakt}
\begin{proof}
Let us consider the points $\alpha(m), \beta(m)$ lying on the sides of a $4\delta$-narrow geodesic triangle $[e, x, y]$. We will consider three cases.
If $\alpha(m)$ lies at distance at most $4\delta$ from $\beta$, then we have $d(\alpha(m), \beta(m')) \leq 4\delta$ for some $m'$, so from the triangle inequality in the triangle $[e, \alpha(m), \beta(m')]$ we obtain $|m' - m| \leq 4\delta$, so
\[ d(\alpha(m), \beta(m)) \leq d(\alpha(m), \beta(m')) + |m' - m| \leq 8\delta, \]
which gives the claim.
If $\beta(m)$ lies at distance at most $4\delta$ from $\alpha$, the reasoning is analogous.
It remains to consider the case when $\alpha(m)$, $\beta(m)$ are at distance at most $4\delta$ respectively from $a, b \in [x, y]$. Then, $|a|, |b| \leq m + 4\delta$, so $a$, $b$ are at distance at least $D = n - m - 4\delta$ from the both endpoints of $[x, y]$. Therefore, $d(a, b) \leq k - 2D$, and so
\[ d \big( \alpha(m), \beta(m) \big) \leq d \big( \alpha(m), a \big) + d(a, b) + d \big( b, \beta(m) \big) \leq 8\delta + k - 2D = 16\delta + k - 2(n - m). \qedhere \]
\end{proof}
\begin{wn}
\label{wn-krzywe-geodezyjne-pozostaja-bliskie}
Let $\alpha = [e, x]$ and $\beta = [e, y]$, with $|x| = n$ and $d(x, y) = k$. Then, for $0 \leq m \leq \min(n, |y|)$, we have:
\[ d \big( \alpha(m), \beta(m) \big) \ \leq \ 8\delta + \max \big( 2k + 8\delta - 2(n-m), \, 0 \big). \]
\end{wn}
\begin{proof}
From the triangle inequality we have $\big| n - |y| \big| = \big| |x| - |y| \big| \leq k$. Let $n' = \min(n, |y|)$; we claim that $d(\alpha(n'), \beta(n')) \leq 2k$. Indeed: if $n' = n$, we have
\[ d(\alpha(n'), \beta(n')) \leq d(x, y) + d(y, \beta(n)) \leq k + \big| |y| - n \big| \leq 2k; \]
otherwise $n' = |y|$ and so
\[ d(\alpha(n'), \beta(n')) \leq d(\alpha(|y|), x) + d(x, y) \leq \big| |y| - n \big| + k \leq 2k. \]
It remains to use Lemma~\ref{fakt-geodezyjne-pozostaja-bliskie} for geodesics $\alpha, \beta$ restricted to the interval $[0, n']$ and the doubled value of~$k$.
\end{proof}
\begin{fakt}
\label{fakt-geodezyjne-przekatniowo}
Let $(\alpha_k)_{k \geq 0}$ be a sequence of geodesic rays in~$G$ which start at~$e$. Denote $x_k = \lim_{n \rightarrow \infty} \alpha_k(n)$. Then, there exists a subsequence $(\alpha_{k_i})_{i \geq 0}$ and a geodesic $\alpha_{\infty}$ such that $\alpha_{k_i}$ coincides with~$\alpha_{\infty}$ on the segment~$[0, i]$. Moreover, the point $x_{\infty} = \lim_{n \rightarrow \infty} \alpha_{\infty}(n)$ is the limit of $(x_{k_i})$.
\end{fakt}
\begin{proof}
The first part of the claim is obtained from an easy diagonal argument: since, for every $n \geq 0$, the set $\{ x \in G \,|\, |x| \leq n \}$ is finite, the set of possible restrictions $\{ \alpha_k \big|_{[0, n]} \,|\, k \geq 0 \}$ must be finite too. This allows to define inductively $\alpha_{\infty}$: we take $\alpha_\infty(0) = e$, and for the consecutive $n > 0$ we choose $\alpha_\infty(n)$ so that $\alpha_\infty$ coincides on $[0, n]$ with infinitely many among the~$\alpha_k$'s. Such choice is always possible and guarantees the existence of a subsequence $(\alpha_{k_i})$.
The obtained sequence $\alpha_\infty$ is a geodesic because every its initial segment $\alpha_\infty \big|_{[0, i]}$ coincides with an initial segment $\alpha_{k_i} \big|_{[0, i]}$ of a geodesic. (We note that we can obtain an increasing sequence $(k_i)$). In this situation, from Lemma~5.2.1 in~\cite{zolta} and the definition of the topology in $G \cup \partial G$ it follows that $x_{\infty} = \lim_{i \rightarrow \infty} x_{k_i}$ holds in~$\partial G$.
On the other hand, we have $\gamma_\infty(k) = g$, and so $x = [\gamma_\infty] \in \sppan(g)$.
\end{proof}
\subsection{Cone types and their analogues in~$\partial G$}
\begin{df}[cf.~\cite{CDP}]
\label{def-typ-stozkowy}
We define the \textit{cone type} $T^c(x)$ of $x \in G$ as the set of all $y \in G$ such that there exists a geodesic connecting $e$ to $xy$ and passing through $x$.
\end{df}
Elements of the set $xT^c(x)$ will be called \textit{descendants} of~$x$.
\begin{fakt}[{\cite[Chapter~12.3]{CDP}}]
\label{fakt-przechodniosc-potomkow}
The relation of being a descendant is transitive: if $y \in T^c(x)$ and $w \in T^c(xy)$, then $yw \in T^c(x)$.
\end{fakt}
\begin{fakt}
\label{fakt-synowie-typy-stozkowe}
If $y \in T^c(x)$, then the cone type $T^c(xy)$ is determined by $T^c(x)$ and $y$.
\end{fakt}
\begin{proof}
This results from multiple application of Lemma 12.4.3 in~\cite{CDP}.
\end{proof}
\begin{df}
The \textit{span} of an element $g \in G$ (denoted $\sppan(g)$) is the set of all $x \in \partial G$ such that there exists a geodesic from $e$ to $x$ passing through $g$.
\end{df}
\begin{fakt}
The set $\sppan(g)$ is closed for every $g \in G$.
\end{fakt}
\begin{proof}
Denote $|g| = k$ and let $x_i$ be a sequence in $\sppan(g)$ converging to $x \in \partial G$. We will show that $x$ also belongs to $\sppan(g)$. Let $\gamma_i$ be a geodesic in~$G$ starting in~$e$, converging to~$x_i$ and such that $\gamma_i(k) = g$. By Lemma~\ref{fakt-geodezyjne-przekatniowo}, there is a subsequence $(\gamma_{i_j})$ which is increasingly coincident with some geodesic~$\gamma_\infty$; in particular, we have $\gamma_\infty(0) = e$ and $\gamma_\infty(k) = g$. Moreover, Lemma~\ref{fakt-geodezyjne-przekatniowo} ensures that~$[\gamma_\infty] \in \partial G$ is the limit of~$x_{i_j}$, so it is equal to~$x$. This means that $x \in \sppan(g)$.
\end{proof}
\begin{fakt}
\label{fakt-stozek-a-span}
For any $g \in G$, $\sppan(g)$ is the set of limits in $\partial G$ of all geodesic rays in $G$ starting at $g$ and contained in $gT^c(g)$.
\end{fakt}
\begin{proof}
Denote $|g| = k$. Let $\alpha$ be a geodesic starting at $g$ and contained in $gT^c(g)$. From the definition of the set $T^c(g)$ it follows that for any $n > 0$ we have $|\alpha(n)| = n + k$. This shows that for any geodesic $\beta$ connecting $e$ with $g$ the curve $\beta \cup \alpha$ is geodesic, because for any $m > k$ its restriction to $[0, m]$ connects the points $e$ and $\alpha(m - k)$ which have distance exactly $m$ from each other. Therefore $\lim_{n \rightarrow \infty} \alpha(n) = \lim_{m \rightarrow \infty} \beta(m)$ belongs to $\sppan(g)$.
The opposite inclusion is obvious.
\end{proof}
Let us fix a constant $a > 1$ (depending on the group $G$) used in the definition of the visual metric on $\partial G$.
\begin{fakt}
\label{fakt-spany-male}
Let $g \in G$. If $|g| = n$, then $\diam \sppan(g) \leq C \cdot a^{-n}$, where $C$ is a constant depending only on $G$.
\end{fakt}
\begin{proof}
Let $x, y \in \sppan(g)$ and $\alpha, \beta$ be geodesics following from $e$ through $g$ correspondingly to $x$ and $y$. By Lemma 12.3.1 in~\cite{CDP}, the path $\overline{\beta}$ built by joining the restrictions $\alpha \big|_{[0, n]}$ and $\beta \big|_{[n, \infty)}$ is a geodesic converging to~$y$. On the other hand, $\overline{\beta}$ coincides with~$\alpha$ on the interval~$[0, n]$. Then, if $\gamma$ is a bi-infinite geodesic connecting~$x$ with~$y$, from Lemma~5.2.1 in~\cite{zolta} we obtain $d(e, \gamma) \geq n - 12\delta$, which finishes the proof.
\end{proof}
\subsection{Ball $N$-types}
\label{sec-typy-kulowe}
\begin{ozn}
For any $x \in G$ and $r > 0$, we denote by $B_r(x)$ the set $\{ y \in G \,|\, d(x, y) \leq r \}$.
\end{ozn}
\begin{df}[{\cite[Chapter~12]{CDP}}]
\label{def-typ-kulowy}
Let $x \in G$ and $N > 0$. We define the \textit{ball $N$-type} of an element $x$ (denoted $T^b_N(x)$) as the function $f^b_{x,\,N} : B_N(e) \rightarrow \mathbb{Z}$, given by formula
\begin{align}
\label{eq-def-n-typu}
f^b_{x,\,N}(y) = |xy| - |x|.
\end{align}
\end{df}
\begin{lem}[{\cite[Lemma~12.3.3]{CDP}}]
\label{lem-kulowy-wyznacza-stozkowy}
There exists a constant $N_0$, depending only on $G$, such that for any $N \geq N_0$ and $x, y \in G$, the equality $T^b_N(x) = T^b_N(y)$ implies that $T^c(x) = T^c(y)$.
\end{lem}
\begin{fakt}
\label{fakt-kulowy-duzy-wyznacza-maly}
Let $x, y \in G$, $N, k > 0$ and $|y| \leq k$. Then, $T^b_N(xy)$ depends only on $T^b_{N + k}(x)$, $y$ and~$N$. \qed
\end{fakt}
\begin{proof}
Let $f$, $f'$ denote the functions of $(N+k)$-type for $x$ and $N$-type for $xy$, respectively. Let $z \in B_N(e)$.
Then, $yz$ and $y$ both belong to $B_{N + k}(e)$, which is the domain of~$f$, and moreover
\[ f'(z) = |xyz| - |xy| = |xyz| - |x| - (|xy| - |x|) = f(yz) - f(y). \qedhere \]
\end{proof}
\begin{lem}
\label{lem-potomkowie-dla-kulowych}
Let $N_0$ be the constant from Proposition~\ref{lem-kulowy-wyznacza-stozkowy}. Let $N > N_0 + 8\delta$, $M \geq 0$, $x \in G$ and $y \in T^c(x)$, where $|y| \geq M + 4\delta$. Then, $T^b_M(xy)$ depends only on $T^b_N(x)$, $y$ and $N$, $M$.
\end{lem}
Note that the value $M \geq 0$ in this proposition can be chosen arbitrarily.
\begin{proof}
Let $x, x' \in G$ be such that $T^b_N(x) = T^b_N(x')$. Denote $n = |x|$.
Let $z \in B_M(e)$. We need to prove that
\begin{align}
\label{eq-potomkowie-do-spr}
|xyz| - |xy| = |x'yz| - |x'y|.
\end{align}
Let $\alpha, \beta$ be geodesics connecting $e$ respectively with $xy$ and $xyz$; we can assume that $\alpha$ passes through $x$. Denote $w = x^{-1}\beta(n)$. Since $n \leq |xy| - \tfrac{2M}{2} - 4\delta$, by applying Corollary \ref{wn-krzywe-geodezyjne-pozostaja-bliskie} for geodesics $\alpha$, $\beta$, we obtain
\[ |w| = d(x, xw) = d(\alpha(n), \beta(n)) \leq 8\delta. \]
Then, by the equality $T^b_N(x) = T^b_N(x')$, we deduce from Lemma~\ref{fakt-kulowy-duzy-wyznacza-maly} that $T^b_{N - 8\delta}(xw) = T^b_{N - 8\delta}(x'w)$.
By Proposition \ref{lem-kulowy-wyznacza-stozkowy}, we obtain
\[ T^c(x) = T^c(x'), \qquad T^c(xw) = T^c(x'w), \]
where $y$ belongs to the first set and $w^{-1}yz$ to the second one. This gives \eqref{eq-potomkowie-do-spr} because
\[ |xyz| - |xy| = |xw| + |w^{-1}yz| - (|x| + |y|) = |w^{-1}yz| - |y| = |x'w| + |w^{-1}yz| - (|x'| + |y|) = |x'yz| - |x'y|. \qedhere \]
\end{proof}
\begin{fakt}
\label{fakt-kuzyni-lub-torsje}
For any $r > 0$ there is $N_r > 0$ such that for any $N \geq N_r$ and $g, h \in G$, the conditions
\[ |h| \leq r, \qquad |gh| = |g|, \qquad T^b_N(gh) = T^b_N(g) \]
imply that $h$ is a torsion element.
\end{fakt}
\begin{proof}
If $h$ is not a torsion element, then the remark following Proposition~1.7.3 in~\cite{zolta} states that it must be of hyperbolic type (which means that the sequence $(h^n)_{n \in \mathbb{Z}}$ is a bi-infinite quasi-geodesic in $G$). In this situation, a contradiction follows from the proof of Proposition~7.3.1 in~\cite{zolta}, provided that we replace in this proof the constant $4\delta$ by $r$. (This change may increase the value of~$N_r$ obtained from the proof but the argument does not require any other modification).
\end{proof}
\section{Quasi-invariant systems}
\label{sec-konstr}
The presentation of $\partial G$ as a Markov compactum will be obtained in the following steps:
\begin{itemize}[nolistsep]
\item[(i)] choose a suitable system $\mathcal{U}$ of open covers of~$\partial G$;
\item[(ii)] build an inverse system of nerves of these covers (and appropriate maps between them);
\item[(iii)] prove that $\partial G$ is the inverse limit of this system;
\item[(iv)] verify the Markov property (see~Definition~\ref{def-kompakt-markowa}).
\end{itemize}
The steps (ii-iii) and (iv) will be discussed in Sections~\ref{sec-engelking} and \ref{sec-markow} respectively. In this section, we focus on step~(i). We begin with introducing in Section~\ref{sec-konstr-quasi-niezm} the notion of a \textit{quasi-$G$-invariant system of covers} in $\partial G$ (or, more generally, in a compact metric $G$-space), which summarises the conditions under which we will be able to execute steps (ii-iv). Section~\ref{sec-konstr-gwiazda} contains proof an additional \textit{star property} for such systems; we will need it in Section~\ref{sec-engelking}. Finally, in Section~\ref{sec-konstr-pokrycia} we construct an example quasi-$G$-invariant system in~$\partial G$, which will serve as the basis for the construction of the Markov system representing $\partial G$.
\subsection{Definitions}
\label{sec-konstr-quasi-niezm}
Let $(X, d)$ be a metric space equipped with a homeomorphic action of a hyperbolic group $G$ (recall that we assume that $G$ is equipped with a fixed set of generators).
Definitions \ref{def-quasi-niezm} and~\ref{def-quasi-niezm-pokrycia} summarise conditions which --- as we will prove in Sections \ref{sec-konstr-gwiazda} and~\ref{sec-markow} --- are sufficient to make the construction of Section \ref{sec-engelking-top} (and in particular Theorem~\ref{tw-konstr}) applicable to the sequence $(\mathcal{U}_n)_{n \geq 0}$, and to guarantee that the constructed inverse system has Markov property (in the sense of Definition~\ref{def-kompakt-markowa}). In the next subsection, we will construct, for a given hyperbolic group $G$, a~sequence of covers of $\partial G$ with all properties introduced in this subsection.
\begin{ozn}
\label{ozn-suma-rodz}
For any family $\mathcal{C} = \{ C_x \}_{x \in G}$ of subsets of a~space $X$, we denote:
\[ \mathcal{C}_n = \big\{ C_x \ \big|\ x \in G, \ |x| = n \big\}, \qquad |\mathcal{C}|_n = \bigcup_{C \in \mathcal{C}_n} C. \]
We will usually identify the family $\mathcal{C}$ with the sequence of subfamilies $(\mathcal{C}_n)_{n \geq 0}$.
\end{ozn}
\dzm{
\begin{df}
\label{def-funkcja-typu}
By a \textit{type function} on~$G$
we will mean any function~$T$ on~$G$
with values in a~finite set. For $x \in G$, the value $T(x)$ will be called the \textit{($T$-)type} of~$x$.
Analogously, by a \textit{type function} on a system $(K_n)_{n \geq 0}$ of simplicial complexes we will mean any function $T$ mapping simplexes of all $K_n$ to a finite set; the value $T(\sigma)$ will be called the \textit{($T$-)type} of $\sigma$.
For two type functions $T_1, T_2$ on $G$ (resp. on a system $(K_n)_{n \geq 0}$), we will call $T_1$ \textit{stronger} than $T_2$ if the $T_2$-type of any element (resp. simplex)
can be determined out of its $T_1$-type.
\end{df}
}
\begin{df}
\label{def-quasi-niezm}
A family $\mathcal{C} = \{ C_x \}_{x \in G}$ of subsets of a $G$-space $X$ is a~\textit{quasi-$G$-invariant system} \dzm{(with respect to a type function $T : G \rightarrow \mathcal{T}$)} if there exists a \textit{neighbourhood constant} $D > 0$ and a~\textit{jump constant} $J > 0$ such that:
\begin{itemize}[leftmargin=1.5cm]
\qhitem{c}{QI1}{QI1} the sequence of subfamilies $(\mathcal{C}_n)_{n \geq 0}$, where $\mathcal{C}_n = \big\{ C_x \ \big|\ x \in G, \ |x| = n \big\}$, has mesh property (in the sense of Definition \ref{def-mesh}a);
\qhitem{d}{QI2}{QI2} for every $n$ and $x, y \in G$, the following implication holds:
\[ |x| = |y| = n, \quad C_x \cap C_y \neq \emptyset \qquad \Rightarrow \qquad d(x, y) \leq D; \]
\qhitem{e}{QI3}{QI3} for every $x \in G$ and $0 < k \leq \tfrac{|x|}{J}$, there exists $y \in G$ such that $|y| = |x| - kJ$ and $C_y \supseteq C_x$;
\qhitem{f}{QI4}{QI4} whenever $T(x) = T(gx)$ for $g, x \in G$, we have:
\begin{itemize}
\qhitem{f1}{a}{QI4a} $C_{gx} = g \cdot C_x$;
\qhitem{f2}{b}{QI4b} for every $y \in G$ such that $|y| = |x|$ and $C_x \cap C_y \neq \emptyset$, we have
\[ C_{gy} = g \cdot C_y, \qquad |gy| = |gx|; \]
\qhitem{f3}{c}{QI4c}
for every $y \in G$ such that $|y| = |x| + kJ$ for some $k > 0$ and $\emptyset \neq C_y \subseteq C_x$, we have
\[ |gy| = |gx| + kJ, \qquad T(gy) = T(y), \qquad \textrm{and \ so} \qquad C_{gy} = g \cdot C_y. \]
\end{itemize}
\end{itemize}
\end{df}
\begin{uwaga}
\label{uwaga-quasi-niezm-jeden-skok}
Let us note that if \qhlink{e} is satisfied for $k = 1$, then by induction it must hold for all $k > 0$, and that the same applies to \qhlink{f3}.
\end{uwaga}
\begin{uwaga}
\label{uwaga-quasi-niezm-rozne-poziomy}
From now on, we adopt the convention that the sets belonging to $\mathcal{C}_n$ are implicitly equipped with the value of~$n$; this would matter only if some subsets $C_1 \in \mathcal{C}_{n_1}$, $C_2 \in \mathcal{C}_{n_2}$ with $n_1 \neq n_2$ happen to consist of the same elements. In this case, we will treat $C_1$, $C_2$ as \textit{not} equal; in particular, any condition of the form $C_1 = C_g$ will implicitly imply $|g| = n_1$. This should not lead to confusion since, although we will often consider an inclusion between an element of $\mathcal{C}_{n_1}$ and an element of $\mathcal{C}_{n_2}$ with $n_1 \neq n_2$, we will be never interested whether set-equality holds between these objects.
\end{uwaga}
\begin{df}
\label{def-quasi-niezm-pokrycia}
A system $\mathcal{C} = \{ C_x \}_{x \in G}$ of subsets of $X$ will be called \textit{a system of covers} if $\mathcal{C}_n$ is an open cover of $X$ for every $n \geq 0$.
\end{df}
\dzm{
\begin{df}
\label{def-quasi-niezm-system-wpisany}
Let $\mathcal{C} = \{ C_x \}_{x \in G}$, $\mathcal{D} = \{ D_x \}_{x \in G}$ be two quasi-$G$-invariant systems of subsets of~$X$. We will say that $\mathcal{C}$ is \textit{inscribed} in~$\mathcal{D}$ if $C_x \subseteq D_x$ for every $x \in G$, and if the type function associated to~$\mathcal{C}$ is stronger than the one associated to~$\mathcal{D}$.
\end{df}
}
\subsection{The star property}
\label{sec-konstr-gwiazda}
\begin{df}
Let $\mathcal{U}$ be an open cover of~$X$, and $U \in \mathcal{U}$. Then, the \textit{star} of $U$ in~$\mathcal{U}$ is the union $\bigcup\{U_i \, | \, U_i \in \mathcal{U}, \ U_i \cap U \neq \emptyset\}$.
\end{df}
\begin{df}
\label{def-wl-gwiazdy}
Let $(\mathcal{U}_n)$ be a family of open covers of~$X$. We say that $(\mathcal{U}_n)$ has \textit{star property} if, for every $n > 0$, every star in the cover $\mathcal{U}_n$ is contained in some element of the cover $\mathcal{U}_{n-1}$; more formally:
\[ \forall_{n > 0} \ \forall_{U \in \mathcal{U}_n} \ \exists_{V \in \mathcal{U}_{n-1}} \ \bigcup_{U' \in \mathcal{U}_n; \, U \cap U' \neq \emptyset} U' \subseteq V. \]
\end{df}
\begin{lem}
\label{lem-gwiazda}
Let $(\mathcal{U}_n)$ be a quasi-$G$-invariant system of covers of a compact metric $G$-space $X$ and let $J$ denote its jump constant. Then, there exists a constant $L_0$ such that, for any $L \geq L_0$ divisible by~$J$, the sequence of covers $(\mathcal{U}_{Ln})_{n \in \mathbb{N}}$ has star property.
\end{lem}
\begin{proof}
Let $L_{(i)}$ be constant such that, for every $j \geq i + L_{(i)}$, every element of~$\mathcal{U}_j$ together with its star is contained in some set from $\mathcal{U}_i$. Its existence is an immediate result of the existence of a Lebesgue number for $\mathcal{U}_i$, and from the mesh condition for the system~$(\mathcal{U}_n)$.
Since there exist only finite many $N$-types in~$G$, there exists $S > 0$ such that for any $g \in G$ there is $g' \in G$ such that $|g'| < S$ and $T(g) = T(g')$.
We will show the claim of the proposition is satisfied by
\[ L_0 = 1 + \max \{ L_{(i)} \,|\, i < S \}. \]
Let $|g| = L(k + 1)$ and $L \geq L_0$ be divisible by $J$; we want to prove that there exists $\tilde{f} \in G$ of length $Lk$ such that $U_{\tilde{f}}$ contains $U_g$ together with all its neighbours in $\mathcal{U}_{L(k+1)}$. If $Lk < S$, this holds by the inequality $L \geq L_0 \geq L_{(Lk)}$ and the definition of the constant $L_{(Lk)}$.
Otherwise, by the property \qhlink{e} there exists $f$ of length $Lk$ such that $U_g \subseteq U_{f}$. Let $f' \in G$ of length $j < S$ satisfy $T(f') = T(f)$.
Denote $h = f' f^{-1}$. Then, since $J \mathrel{|} L$, by~\qhlink{f3} we have
\begin{align}
\label{eq-gwiazda-sukces-na-malej}
U_{hg} = h \cdot U_g \subseteq h \cdot U_{f} = U_{f'}, \qquad T(hg) = T(g), \qquad |hg| = j + L.
\end{align}
Therefore, since $j < S$, there exists some $\tilde{f}'$ of length $j$ such that $U_{\tilde{f}'}$ contains $U_{hg}$ together with its whole star. Then, by \eqref{eq-gwiazda-sukces-na-malej}, we have $U_{\tilde{f}'} \cap U_{f'} \neq \emptyset$, and so from \qhlink{f2} we obtain
\[ \qquad U_{h^{-1}\tilde{f}'} = h^{-1} \cdot U_{\tilde{f}'}, \qquad |h^{-1} \widetilde{f}'| = |h^{-1} f'| = Lk. \]
Now, let $|x| = |g|$ and $U_x \cap U_g \neq \emptyset$. Then, from \eqref{eq-gwiazda-sukces-na-malej} and~\qhlink{f2} we have $U_{hx} = h \cdot U_x$; in particular, $U_{hx}$ is contained in the star of the set $U_{hg} = h \cdot U_g$, and so it is contained in $U_{\tilde{f}'}$. Then, by \qhlink{f3}:
\[ U_x = h^{-1} \cdot U_{hx} \subseteq h^{-1} \cdot U_{\tilde{f}'} = U_{h^{-1} \tilde{f}'}. \]
This means that the element $\tilde{f} := h^{-1}\tilde{f}'$ has the desired property.
\end{proof}
\subsection{The system of span-star interiors}
\label{sec-konstr-pokrycia}
\begin{df}
\label{def-konstr-towarzysze}
For every element $g \in G$ and $r > 0$, we denote
\[ P(x) = \big\{ y \in G \,\big|\, |xy| = |x| \big\}, \qquad P_r(x) = P(x) \cap B_r(e). \]
If $y \in P(x)$ (resp. $P_r(x)$), we call $xy$ a \textit{fellow} (resp. \textit{$r$-fellow}) of $x$.
\end{df}
From the definition of the ball type, we obtain the following property.
\begin{fakt}
\label{fakt-kulowy-wyznacza-towarzyszy}
If $N \geq r > 0$, then the set $P_r(x)$ depends only on $T^b_N(x)$ and $r$, $N$. \qed
\end{fakt}
\begin{df}
\label{def-span-star}
We define the set $S_g$ as the interior of span-star in $\partial G$ around $\sppan(g)$:
\[ S_g = \innt \Big( \bigcup_{h \in I(g)} \sppan(gh) \Big), \qquad \textrm{ gdzie } \quad I(g) = \big\{ h \in P(g) \,\big|\, \sppan(gh) \cap \sppan(g) \neq \emptyset \big\}. \]
For any $k > 0$, we define the family
\[ \mathcal{S}_k = \{ S_g \ |\ g \in G, \, |g| = k, \, S_g \neq \emptyset \}. \]
\end{df}
\begin{fakt}
\label{fakt-span-w-pokryciu}
For every $g \in G$, we have $\sppan(g) \subseteq S_g$.
\end{fakt}
\begin{proof}
Let us consider the equality
\[ \partial G = \bigcup_{h \in P(g)} \sppan(gh) = \Big( \bigcup_{h \in I(g)} \sppan(gh) \Big) \cup \Big( \bigcup_{h \in P(g) \setminus I(g)} \sppan(gh) \Big). \]
The second summand is disjoint with $\sppan(g)$, and moreover closed (as a finite union of closed sets), which means that $\sppan(g)$ must be contained in the interior of the first summand, which is exactly $S_g$.
\end{proof}
\begin{wn}
\label{wn-konstr-pokrycie}
For every $k > 0$, the family $\mathcal{S}_k$ is a cover of $\partial G$.
\end{wn}
\begin{proof}
This is an easy application of the above lemma and of the equality $\partial G = \bigcup_{g \in G \,:\, |g| = k} \sppan(g)$.
\end{proof}
\begin{fakt}
\label{fakt-pokrycie-male}
Under the notation of Lemma \ref{fakt-spany-male}, for every $k > 0$ and $U \in \mathcal{S}_k$, we have $\diam U \leq 3 C \cdot a^{-k}$.
\end{fakt}
\begin{proof}
Let $U = S_g$ for some $g \in G$, where $|g|=k$, and let $x, y \in S_g$. Then, $x \in \sppan(gh_1)$ and $y \in \sppan(gh_2)$ for some $h_1, h_2 \in I(g)$. By Lemma \ref{fakt-spany-male}, we obtain
\[ d(x, y) \leq \diam \sppan(gh_1) + \diam \sppan(g) + \diam \sppan(gh_2) \leq 3 C \cdot a^{-k}. \qedhere \]
\end{proof}
\begin{fakt}
\label{fakt-sasiedzi-blisko}
Let $h \in P(g)$. Then:
\begin{itemize}
\item[\textbf{(a)}] If $\sppan(g) \cap \sppan(gh) \neq \emptyset$, then $|h| \leq 4\delta$ (so: $I(g) \subseteq P_{4\delta}(g)$);
\item[\textbf{(b)}] If $S_g \cap S_{gh} \neq \emptyset$, then $|h| \leq 12\delta$.
\end{itemize}
\end{fakt}
\begin{proof}
\textbf{(a)} Let $|g| = |gh| = k$ and $x \in \sppan(g) \cap \sppan(gh)$. Then, there exist geodesics $\alpha, \beta$ stating at $e$ and converging to $x$ such that $\alpha(k) = g$, $\beta(k) = gh$.
By inequality (1.3.4.1) in~\cite{zolta}, this implies that $d(g, gh) \leq 4\delta$.
\textbf{(b)} Let $x \in S_g \cap S_{gh}$. Then, by definition, we have $x \in \sppan(gu) \cap \sppan(ghv)$ for some $u \in I(g)$, $v \in I(gh)$. Using part~\textbf{a)}, we obtain
\[ |h| \leq |u| + |u^{-1}hv| + |v^{-1}| \leq 4\delta + 4\delta + 4\delta = 12\delta. \qedhere \]
\end{proof}
\begin{fakt}
\label{fakt-wlasnosc-gwiazdy-bez-gwiazdy}
Let $g \in G$ and $k < |g|$. Then:
\begin{itemize}
\item[\textbf{(a)}] there exists $f \in G$ of length $k$ such that $g \in fT^c(f)$;
\item[\textbf{(b)}] for any $f \in G$ with the properties from part~\textbf{(a)}, we have $\sppan(g) \subseteq \sppan(f)$;
\item[\textbf{(c)}] for any $f \in G$ with the properties from part~\textbf{(a)}, we have $S_g \subseteq S_f$.
\end{itemize}
\end{fakt}
\begin{proof}
\textbf{(a)} Let $\alpha$ be a geodesic from $e$ to $g$. Then, $f = \alpha(k)$ has the desired properties.
\textbf{(b)} If $f$ has the properties from part~\textbf{(a)}, then, by Lemma \ref{fakt-przechodniosc-potomkow}, we have $gT^c(g) \subseteq fT^c(f)$, so it remains to apply Lemma \ref{fakt-stozek-a-span}.
\textbf{(c)} By the parts~\textbf{(a)} and~\textbf{(b)}, for any $h \in I(g)$ there exists some element $f_h$ of length $k$ such that $\sppan(gh) \subseteq \sppan(f_h)$; here $f_e$ can be chosen to be $f$. In particular, we have:
\[ \emptyset \neq \sppan(g) \cap \sppan(gh) \subseteq \sppan(f) \cap \sppan(f_h), \]
so $f^{-1} f_h \in I(f)$. Since $h \in I(g)$ is arbitrary, we obtain
\[ \bigcup_{h \in I(g)} \sppan(gh) \subseteq \bigcup_{h \in I(g)} \sppan(f_h) \subseteq \bigcup_{x \in I(f)} \sppan(f x). \]
By taking the interiors of both sides of this containment, we get the claim.
\end{proof}
\begin{lem}
\label{lem-typy-pokrycia-niezmiennicze}
Let $N_0$ denote the constant from Proposition~\ref{lem-kulowy-wyznacza-stozkowy}. Assume that $N, r \geq 0$ and $g, x \in G$ satisfy $T^b_N(gx) = T^b_N(x)$. Then:
\begin{itemize}
\item[\textbf{(a)}] if $N \geq N_0$, then $\sppan(gx) = g \cdot \sppan(x)$;
\item[\textbf{(b)}] if $N \geq N_0 + r$, then $\sppan(gxy) = g \cdot \sppan(xy)$ for $y \in P_r(x)$;
\item[\textbf{(c)}] if $N \geq N_1 := N_0 + 4\delta$, then $S_{gx} = g \cdot S_x$;
\item[\textbf{(d)}] if $N \geq N_1 + r$, then $S_{gxy} = g \cdot S_{xy}$ for $y \in P_r(x)$;
\item[\textbf{(e)}] if $N \geq N_2 := N_0 + 16\delta$ and $y \in G$ satisfy $|y| = |x|$ and $S_x \cap S_y \neq \emptyset$, then
\[ S_{gy} = g \cdot S_y, \qquad \textrm{ and moreover } \quad |gy| = |gx|; \]
\item[\textbf{(f)}] if $N \geq N_3 := N_0 + 21\delta$, $k \geq 0$, $L > N + k + 4\delta$ and $y \in G$ satisfy $|y| = |x| + L$ and $\emptyset \neq S_y \subseteq S_x$, then:
\[ S_{gy} = g \cdot S_y, \qquad \textrm{ and moreover } \quad |gy| = |gx| + L \quad \textrm{and} \quad T^b_{N + k}(gy) = T^b_{N + k}(y). \]
\end{itemize}
\end{lem}
\begin{proof}
\textbf{(a)} If $N \geq N_0$, by Proposition \ref{lem-kulowy-wyznacza-stozkowy} we have $T^c(gx) = T^c(x)$ and so $gxT^c(gx) = g \cdot xT^c(x)$. In particular, the left action by $g$, which is an isometry, gives a unique correspondence between geodesics in $G$ starting at $x$ and contained in $xT^c(x)$ and geodesics in $G$ starting at $gx$ and contained in~$gxT^c(gx)$. Then, the claim holds by Lemma~\ref{fakt-stozek-a-span} and by continuity of the action of $g$ on~$G \cup \partial G$.
\textbf{(b)} If $N \geq N_0 + r$, then, by Lemma~\ref{fakt-kulowy-duzy-wyznacza-maly}, we have $T^b_{N_0}(gxy) = T^b_{N_0}(xy)$ for every $y \in P_r(x)$; it remains to apply~\textbf{(a)}.
\textbf{(c)} Let $y \in I(x)$. By Lemma~\ref{fakt-sasiedzi-blisko}a, we have $y \in P_{4\delta}(x)$. Since $N \geq N_0 + 4\delta$, from~\textbf{(b)} and Lemma~\ref{fakt-kulowy-wyznacza-towarzyszy} we obtain that
\[ \sppan(gx) = g \cdot \sppan(x), \qquad \sppan(gxy) = g \cdot \sppan(xy), \qquad y \in P_{4\delta}(gx). \]
Since $\sppan(x) \cap \sppan(xy) \neq \emptyset$, by acting with~$g$ we obtain $\sppan(gx) \cap \sppan(gxy) \neq \emptyset$, so $y \in I(gx)$. Then, we have
\[ g \cdot \bigcup_{y \in I(x)} \sppan(xy) = \bigcup_{y \in I(x)} \sppan(gxy) \subseteq \bigcup_{y \in I(gx)} \sppan(gxy). \]
By an analogous reasoning for the inverse element $g^{-1}$, we prove that the above containment is in fact an equality. Moreover, since the left action of~$g$ is a homeomorphism, it must map the interior of the left-hand side sum (which is~$S_x$) exactly onto the interior of the right-hand side sum (resp.~$S_{gx}$).
\textbf{(d)} This follows from~\textbf{(c)} in the same way as \textbf{(b)} was obtained from~\textbf{(a)}.
\textbf{(e)} By Lemma~\ref{fakt-sasiedzi-blisko}b, we have $x^{-1} y \in P_{12\delta}(x)$. Then, the first part of the claim follows from~\textbf{(d)}. For the second part, note that from $N \geq 16\delta$ we obtain that $x^{-1} y \in P_N(x)$ which is contained in the domain of $T^b_N(x)$ (as a~function); hence, the assumption that $T^b_N(x) = T^b_N(gx)$ implies that $|gxy| - |gx| = |xy| - |x| = 0$, as desired.
\textbf{(f)} Let $|y| \geq |x|$ and $S_y \subseteq S_x$. By Lemma~\ref{fakt-wlasnosc-gwiazdy-bez-gwiazdy}, there exists $z \in G$ such that
\[ |xz| = |x|, \qquad y \in xzT^c(xz), \qquad S_y \subseteq S_{xz}. \]
In particular, $S_{xz} \cap S_x \neq \emptyset$, and so by Lemma~\ref{fakt-sasiedzi-blisko} we have $z \in P_{12\delta}(x)$. By Lemmas~\ref{fakt-kulowy-duzy-wyznacza-maly} and~\ref{fakt-kulowy-wyznacza-towarzyszy}, we obtain
\begin{align}
\label{eq-niezm-z-tow-gx}
T^b_{N - 12\delta}(gxz) = T^b_{N - 12\delta}(xz), \qquad z \in P_{12\delta}(gx).
\end{align}
From the first of these properties and from Proposition \ref{lem-kulowy-wyznacza-stozkowy}, we have $T^c(gxz) = T^c(xz)$. Since $(xz)^{-1}y$ belongs to $T^c(xz)$ and is of length
\begin{align}
\label{eq-niezm-dlugosci}
|(xz)^{-1}y| = |y| - |xz| = |y| - |x| = L > N + k + 4\delta,
\end{align}
by applying Proposition~\ref{lem-potomkowie-dla-kulowych} for \eqref{eq-niezm-z-tow-gx} and the action of $(xz)^{-1}y$ (with parameters $N - 12\delta > N_0 + 8\delta$, $N + k$) we obtain
\[ T^b_{N + k}(gy) = T^b_{N + k}(y), \]
and then from \textbf{(c)}
\[ S_{gy} = g \cdot S_y. \]
Moreover, the conditions $(xz)^{-1}y \in T^c(gxz)$, \eqref{eq-niezm-dlugosci} and \eqref{eq-niezm-z-tow-gx} imply that
\[ |gy| = |gxz| + |(xz)^{-1}y| = |gxz| + L = |gx| + L, \]
which finishes the proof.
\end{proof}
\begin{wn}
\label{wn-spanstary-quasi-niezm}
For $N \geq N_3$, the sequence of covers $(\mathcal{S}_n)$ together with the type function $T = T^b_N$ is a quasi-$G$-invariant system of covers.
\end{wn}
\begin{proof}
We have checked in Corollary \ref{wn-konstr-pokrycie} that every $\mathcal{S}_n$ is a~cover of $\partial G$; obviously it is open. The subsequent conditions from Definition \ref{def-quasi-niezm} hold correspondingly by \ref{fakt-pokrycie-male}, \ref{fakt-sasiedzi-blisko}b and \ref{fakt-wlasnosc-gwiazdy-bez-gwiazdy} and Proposition~\ref{lem-typy-pokrycia-niezmiennicze}c,e,f (for $k = 0$). Here, we take the following constants:
\[ D = 12\delta, \qquad J_0 = 0, \qquad J = N_3 + 4\delta. \qedhere \]
\end{proof}
\section{Inverse limit construction}
\label{sec-engelking}
\dzm{
In this section, we present a classical construction (see Theorem~\ref{tw-konstr} below) which presents --- up to a~homeomorphism --- every compact metric space~$X$ as the inverse limit of the sequence of nerves of an appropriate system of covers of~$X$ (which we will call \textit{admissible}; see Definition~\ref{def-konstr-admissible}). We will also show (in Lemma~\ref{fakt-konstr-sp-zal}) that admissible systems can be easily obtained from any quasi-$G$-invariant systems of covers~$\mathcal{U}$.
In Section~\ref{sec-bi-lip}, we investigate this construction in the particular case when~$X = \partial G$ and $\mathcal{U}$ is inscribed in the system~$\mathcal{S}$ from Section~\ref{sec-konstr-pokrycia}. As we will show in Theorem~\ref{tw-bi-lip}, in such case the construction allows as well to describe certain metric properties of $\partial G$. (See the introduction to Section~\ref{sec-bi-lip} for more details).
}
\dzm{
\subsection{A topological description by limit of nerves}
\label{sec-engelking-top}
}
Let $X$ be a~compact metric space.
\begin{df}
Recall that the \textit{rank} of a family $\mathcal{U}$ of subspaces of a space $X$ is the maximal number of elements of $\mathcal{U}$ which have non-empty intersection.
\end{df}
\begin{df}
\label{def-konstr-admissible}
A sequence $(\mathcal{U}_i)_{i \geq 0}$ of open covers $X$ will be called an \textit{admissible system} if the following holds:
\begin{itemize}
\item[(i)] for every $i \geq 0$, the cover $\mathcal{U}_i$ is finite and does not contain empty sets;
\item[(ii)] there exists $n \geq 0$ such that $\rank \mathcal{U}_i \leq n$ for every $i \geq 0$;
\item[(iii)] the sequence $(\mathcal{U}_i)_{i \geq 0}$ has mesh property (in the sense of Definition \ref{def-mesh}a);
\item[(iv)] the sequence $(\mathcal{U}_i)_{i \geq 0}$ has star property (see Definition \ref{def-wl-gwiazdy}).
\end{itemize}
\end{df}
There is an easy connection between this notion and the contents of the previous section:
\begin{fakt}
\label{fakt-konstr-sp-zal}
Let $(\mathcal{U}_n)$ be a quasi-$G$-invariant system of covers of a $G$-space $X$. Define
\[ \widetilde{\mathcal{U}}_n = \{ U \in \mathcal{U}_n \,|\, U \neq \emptyset \}. \]
Let $L_0$ denote the constant obtained for the system $(\mathcal{U}_n)$ from Proposition~\ref{lem-gwiazda}. Then, for any $L \geq L_0$, the sequence of the covers $(\widetilde{\mathcal{U}}_{nL})_{n \geq 0}$ is admissible.
\end{fakt}
\begin{proof}
Clearly, for every $n \geq 0$ the family $\widetilde{\mathcal{U}}_n$ is an open cover of $X$. The condition (i) follows from the definition of~$\widetilde{\mathcal{U}}_n$. The mesh and star properties result correspondingly from the property \qhlink{c} and Proposition~\ref{lem-gwiazda}.
Finally, the condition (ii) follows from \qhlink{d}: whenever $U_x \cap U_y \neq \emptyset$, we have $d(x, y) \leq D$, so $x^{-1}y$ belongs to the ball in $G$ centred in $e$ of radius $D$. This means that the rank of the cover $\mathcal{U}_n$ (and thus also of $\widetilde{\mathcal{U}}_n$) does not exceed the number of elements in this ball, which is finite and independent from $n$.
\end{proof}
\begin{ozn}
Let $\mathcal{U}$ be an open cover of~$X$. For $U \in \mathcal{U}$, we denote by~$v_U$ the vertex in the nerve of~$\mathcal{U}$ corresponding to~$U$. We also denote by~$[v_1, \ldots, v_n]$ the simplex in~this nerve spanned by vertices $v_1, \ldots, v_n$.
\end{ozn}
\begin{df}
\label{def-konstr-nerwy}
For an admissible system $(\mathcal{U}_i)_{i \geq 0}$ in~$X$, we define the \textit{associated system of nerves} $(K_i, f_i)_{i \geq 0}$, where $f_i : K_{i+1} \rightarrow K_i$ for $i \geq 0$, as follows:
\begin{itemize}
\item[(i)] for $i \geq 0$, $K_i$ is the nerve of the cover~$\mathcal{U}_i$;
\item[(ii)] for $U \in K_{i+1}$, $f_i(v_U)$ is the barycentre of the simplex spanned by $\{ v_V \,|\, V \in K_i, \, V \supseteq U \}$;
\item[(iii)] for other elements of $K_{i+1}$, we extend $f_i$ so that it is affine on every simplex.
\end{itemize}
For any $j \geq 0$, we denote by $\pi_j$ the natural projection from the inverse limit $\mathop{\lim}\limits_{\longleftarrow} K_i$ to $K_j$.
\end{df}
\begin{uwaga}
If $v_{U_1}, \ldots, v_{U_n}$ span a~simplex in $K_{i+1}$, then $U_1 \cap \ldots \cap U_n \neq \emptyset$; this implies that the family $\mathcal{A} = \{ V \in \mathcal{U}_i \,|\, V \supseteq U_1 \cap \ldots \cap U_n \}$ has a non-empty intersection and therefore the vertices $\{ v_V \,|\, V \in \mathcal{A} \}$ span a simplex in~$K_i$ which contains all the images $f_i(v_{U_j})$ for $1 \leq j \leq n$. This ensures that the affine extension described in condition (iii) of Definition~\ref{def-konstr-nerwy} is indeed possible.
\end{uwaga}
The following theorem is essentially an adjustment of Theorem~1.13.2 in~\cite{E} to our needs (see the discussion below).
\begin{tw}
\label{tw-konstr}
Let $(\mathcal{U}_i)_{i \geq 0}$ be an admissible system in~$X$, and $(K_i, f_i)$ be its associated nerve system. For any $x \in X$ and $i \geq 0$, denote by $K_i(x)$ the simplex in $K_i$ spanned by the set $\{ v_U \,|\, U \in \mathcal{U}_i \, x \in U \}$. Then:
\begin{itemize}
\item[\textbf{(a)}] The system $(K_i, f_i)$ has mesh property;
\item[\textbf{(b)}] For every $x \in X$, the space $\mathop{\lim}\limits_{\longleftarrow} K_i(x) \subseteq \mathop{\lim}\limits_{\longleftarrow} K_i$ has a~unique element, which we will denote by~$\varphi(x)$;
\item[\textbf{(c)}] The map $\varphi : X \rightarrow \mathop{\lim}\limits_{\longleftarrow} K_i$ defined above is a~homeomorphism.
\end{itemize}
\end{tw}
A proof of Theorem~\ref{tw-konstr} can be obtained from the proof of Theorem~1.13.2 given in~\cite{E} as follows:
\begin{itemize}
\item Although our assumptions are different than those in~\cite{E}, they still imply all statements in the proof given there, except for the condition labelled as~(2). However, this condition is used there only to ensure the mesh and star properties of~$(\mathcal{U}_i)$ which we have assumed anyway.
\item The theorem from~\cite{E} does not state the mesh property for the nerve system.
However, an inductive application of the inequality labelled as~(6) in its proof gives (in our notation) that:
\begin{align}
\label{eq-konstr-szacowanie-obrazow}
\diam f^j_i(\sigma) \leq \left( \tfrac{n}{n + 1} \right)^{j - i} \qquad \textrm{ for every simplex $\sigma$ in~$K_j$},
\end{align}
where $n$ denotes the upper bound for the rank of covers required by Definition~\ref{def-konstr-admissible}. The right-hand side of~\eqref{eq-konstr-szacowanie-obrazow} does not depend on~$\sigma$, but only on~$i$, and tends to zero as $i \rightarrow \infty$, which proves mesh property for the nerve system. (Although the above estimate holds only for the particular metric on~$K_i$ used in~\cite{E}, this suffices to deduce the mesh property in view of Remark~\ref{uwaga-mesh-bez-metryki}).
\end{itemize}
\subsection{A metric description for systems inscribed in~$\mathcal{S}$}
\label{sec-bi-lip}
\dzm{
Let~$G$ by a hyperbolic group, and let~$\mathcal{U}$ be a quasi-$G$-invariant system of covers of~$\partial G$, inscribed in the system~$\mathcal{S}$ defined in Section~\ref{sec-konstr-pokrycia}. We will now prove that, under such assumptions, the homeomorphism $\varphi : \partial G \rightarrow \mathop{\lim}\limits_{\longleftarrow} K_i$ obtained from Theorem~\ref{tw-konstr} on the basis of~$\mathcal{U}$ (through Lemma~\ref{fakt-konstr-sp-zal}) is a bi-Lipschitz equivalence --- when $\partial G$ is considered with the visual metric $d_v^{(a)}$ for sufficiently small value of~$a$, and $\mathop{\lim}\limits_{\longleftarrow} K_i$ with the natural \textit{simplicial metric} (see Definition~\ref{def-metryka-komp} below) for the same value of~$a$.
}
To put this in a context, let us recall the known properties of visual metrics on~$\partial G$. The definition of the visual metric given in~\cite{zolta} depends not only on the choice of~$a$, but also on the choice of a basepoint in the group (in this paper, we always set it to be~$e$) and a set of its generators. It is known that the visual metrics obtained for different choices of these parameters do not have to be bi-Lipschitz equivalent, however, they all determine the same quasi-conformal structure (\cite[Theorems~2.18 and~3.2]{Kap}). In this situation, Theorem~\ref{tw-bi-lip} shows that this natural quasi-conformal structure on~$\partial G$ can be as well described by means of \dzm{the inverse limit of polyhedra which we have built so far. This will enable us, in view of Theorems~\ref{tw-kompakt-ogolnie} and~\ref{tw-sk-opis} (to be shown in the next sections), to} give (indirectly) a description of quasi-conformal structures on the boundaries of hyperbolic groups in terms of appropriate Markov systems.
\subsubsection{The simplicial metric}
Let us recall the definition of the metric on simplicial complexes used in the proof of Theorem 1.13.2 in~\cite{E} (which serves as the base for Theorem \ref{tw-konstr}). For any $n \geq 0$, we denote
\[ e_i = (\underbrace{0, \ldots, 0}_{i-1}, 1, 0, \ldots, 0) \in \mathbb{R}^n. \]
\begin{df}
\label{def-metryka-l1}
Let $K$ be a simplicial complex with $n$ vertices. Let $m \geq n$ and $f : K \rightarrow \mathbb{R}^m$ be an injective affine map sending vertices of $k$ to points of the form $e_i$ (for $1 \leq i \leq m$). We define the \textit{$l^1$ metric} on $K$ by the formula:
\[ d_K(x, y) = \| f(x) - f(y) \|_1 \qquad \textrm{ for } x, y \in K. \]
\end{df}
\begin{uwaga}
\label{uwaga-metryka-l1-sens}
The metric given by Definition \ref{def-metryka-l1} does not depend on the choice of $m$ and~$f$ because any other affine inclusion $f' : K \rightarrow \mathbb{R}^{m'}$ must be (after restriction to $K$) a composition of $f$ with a linear coordinate change which is an isometry with respect to the norm $\| \cdot \|_1$.
\end{uwaga}
\begin{uwaga}
\label{uwaga-metryka-l1-ogr}
Since in Definition~\ref{def-metryka-l1} we have $f(K) \subseteq \{ (x_i) \,|\, x_i \geq 0 \textrm{ for } 1 \leq i \leq m, \ \sum_{i=1}^m x_i = 1 \}$, it can be easily deduced that any complex $K$ has diameter at most $2$ in the $l^1$ metric.
\end{uwaga}
\begin{df}
\label{def-metryka-komp}
Let $(K_i, f_i)_{i \geq 0}$ be an inverse system of simplicial complexes. For any real $a > 1$, we define \dzm{the \textit{simplicial metric} (with parameter~$a$)} $d^M_a$ on~$\mathop{\lim}\limits_{\longleftarrow} K_i$ by the formula
\[ d^M_a \big( (x_i)_{i \geq 0}, (y_i)_{i \geq 0} \big) = \sum_{i = 0}^\infty a^{-i} \cdot d_{K_i}(x_i, y_i). \]
\end{df}
\begin{uwaga}
In the case when $a = 2$, Definition \ref{def-metryka-komp} gives the classical metric used in countable products of metric spaces (and hence also in the limits of inverse systems); in particular, it is known that the metric~$d^M_2$ is compatible with the natural topology on the inverse limit (i.e. the restricted Tichonov's product topology). However, this fact holds, with an analogous proof, for any other value of $a>1$ (see~\cite[the remark following Theorem~4.2.2]{ET})
\end{uwaga}
\subsubsection{Bi-Lipschitz equivalence of both metrics}
In the following theorem, we use the notions \textit{quasi-$G$-invariant}, \textit{system of covers}, \textit{inscribed} defined respectively in Definitions~\ref{def-quasi-niezm}, \ref{def-quasi-niezm-pokrycia} and \ref{def-quasi-niezm-system-wpisany}, as well as the system $\mathcal{S}$ defined in Definition~\ref{def-span-star}.
\begin{tw}
\label{tw-bi-lip}
Let $G$ be a hyperbolic group. \dzm{Let~$\mathcal{U}$ be a quasi-$G$-invariant system of covers of~$\partial G$, inscribed in the system~$\mathcal{S}$ (see Section~\ref{sec-konstr-pokrycia}), and let $\varphi : \partial G \rightarrow \mathop{\lim}\limits_{\longleftarrow} K_i$ be the homeomorphism obtained for~$\mathcal{U}$ from Theorem~\ref{tw-konstr}.}
Then, there exists a constant $a_1 > 1$ (depending only on~$G$) such that, for any $a \in (1, a_1)$,
$\varphi$ is a bi-Lipschitz equivalence between the visual metric on~$G$ with parameter~$a$ (see Section~\ref{sec-def-hip}) and the simplicial metric $d^M_a$ on~$\mathop{\lim}\limits_{\longleftarrow} K_i$.
\end{tw}
\begin{uwaga}
Theorem~\ref{tw-bi-lip} re-states the second claim of Theorem~\ref{tw-bi-lip-0}, which is sufficient to deduce the first claim in view of the introduction to Section~\ref{sec-bi-lip}.
\end{uwaga}
\begin{uwaga}
\label{uwaga-bi-lip-wystarczy-d}
To prove the above theorem, it is clearly sufficient to check a bi-Lipschitz equivalence between the simplicial metric $d_a^M$ and the \textit{distance function}~$d_a$ which has been introduced in Section~\ref{sec-def-hip} as a bi-Lipschitz approximation of the visual metric.
\end{uwaga}
\begin{fakt}
\label{fakt-rozlaczne-symp-daleko}
If $s_1,s_2$ are two disjoint simplexes in a complex $K$, then for any $z_1 \in s_1, z_2 \in s_2$, we have $d_K(z_1, z_2) = 2$.
\end{fakt}
\begin{proof}
Let $f : K \rightarrow \mathbb{R}^m$ satisfy the conditions from Definition \ref{def-metryka-l1}. For $j = 1, 2$, let~$A_j$ denote the set of indexes $1 \leq i \leq m$ for which $e_i = f(v)$ for some vertex~$v \in s_j$. Then we have
\[ f(s_j) = \Big\{ (x_i) \in \mathbb{R}^m \ \Big|\ \ x_i \geq 0 \textrm{ for } 1 \leq i \leq m, \ \ x_i = 0 \textrm{ for } i \in A_j, \ \ \sum_{i \in A_j} x_i = 1 \Big\}. \]
However, since $f$ is an inclusion, the sets $A_1$, $A_2$ are disjoint, from which it results that, for any $p_j \in f(s_j)$ (for $j = 1, 2$), we have $\| p_1 - p_2 \|_1 = 2$.
\end{proof}
\begin{lem}
\label{lem-bi-lip-geodezyjne}
There exist constants $E_1, N_4$ (depending only on~$G$) such that if $k, l \geq 0$, $N > N_4$, $g, x \in G$ and $p, q \in \partial G$ satisfy the conditions:
\[ |x| = k, \qquad |gx| = l, \qquad T^b_N(x) = T^b_N(gx), \qquad p \in \sppan(x), \qquad d(p, q) \leq a^{-(k+E_1)}, \]
then in $\partial G$ we have
\begin{align}
\label{eq-bi-lip-geodezyjne-teza}
d(g \cdot p, \, g \cdot q) \leq a^{-(l-k)} \cdot d(p, q).
\end{align}
\end{lem}
\begin{uwaga}
As soon as we prove the inequality \eqref{eq-bi-lip-geodezyjne-teza} in general, it will follow that it can be strengthened to an equality. This is because if the elements $g, x, p, q$ satisfy the assumptions of the proposition, then its claim implies that the elements~$g^{-1}, gx, g \cdot p, g \cdot q$ also satisfy these assumptions. By using the proposition to these elements, we will then obtain that $d(p, q) \leq a^{-(k-l)} \cdot d(g \cdot p, g \cdot q)$, ensuring that an equality in \eqref{eq-bi-lip-geodezyjne-teza} holds. We do not include this result it in the claim of the proposition because it is not used in this article.
\end{uwaga}
\begin{proof}[Proof of Proposition~\ref{lem-bi-lip-geodezyjne}]
We set
\[ E_1 = 13\delta, \qquad N_4 = N_0 + 64\delta, \]
where $N_0$ denotes the constant from Proposition \ref{lem-kulowy-wyznacza-stozkowy}.
\textbf{1. }Let $\gamma$ be some geodesic connecting $p$ with~$q$ for which the distance $d(e, \gamma)$ is maximal. Note that then
\[ d(e, \gamma) = - \log_a d(p, q) \geq k + 13\delta. \]
Since the left shift by $g$ is an isometry in $G$, the sequence $g \cdot \gamma$ determines a bi-infinite geodesic which, by definition, connects the points $\gamma \cdot p, \gamma \cdot q$ in~$\partial G$. Then, to finish the proof it suffices to estimate from below the distance $d(e, g \cdot \gamma)$.
\textbf{2. }Let $\alpha, \beta$ be some geodesics connecting $e$ correspondingly with~$p$ and~$q$; we can require in addition that $\alpha(k) = x$. Denote $y = \beta(k)$.
Since $\alpha, \beta, \gamma$ form a geodesic triangle (with two vertices in infinity), by Lemma~\ref{fakt-waskie-trojkaty}, there exists an element $s \in \beta \cup \gamma$ in a distance $\leq 12\delta$ from~$x$.
Then, $\big| |s| - k \big| \leq 12\delta$, so in particular $s \notin \gamma$, and so $s \in \beta$. In this situation, we have
\[ d(x, y) \leq d(x, s) + d(s, y) \leq 12\delta + \big| |s| - k \big| \leq 24\delta. \]
\textbf{3. }For any $i \in \mathbb{Z}$, we choose a geodesic $\eta_i$ connecting $e$ with~$\gamma(i)$. Since $\gamma(i)$ must lie in a distance $\leq 12\delta$ from some element of~$\alpha$ or~$\beta$, by using Corollary \ref{wn-krzywe-geodezyjne-pozostaja-bliskie} for the geodesic $\eta_i$ and correspondingly $\alpha$ or~$\beta$, we obtain that the point $z_i = \eta(k)$ lies in a distance $\leq 40\delta$ correspondingly from $x$ or~$y$. Therefore, in any case we have
\[ d(x, z_i) \leq 64 \delta. \]
\textbf{4. }We still consider any value of $i \in \mathbb{Z}$. Since $N > N_4$ and $T^b_N(x) = T^b_N(gx)$, as well as $|x| = |z_i| = k$, from Lemmas \ref{fakt-kulowy-duzy-wyznacza-maly} and~\ref{fakt-kulowy-wyznacza-towarzyszy} we obtain that
\[ T^b_{N_0}(z_i) = T^b_{N_0}(gz_i), \qquad |gx| = |gz_i| = l. \]
Then, $z_i$ and~$gz_i$ have the same cone types by Proposition \ref{lem-kulowy-wyznacza-stozkowy}, so from $\gamma(i) \in z_iT^c(z_i)$ we deduce that $g\gamma(i) \in gz_iT^c(gz_i)$, and then
\[ |g\gamma(i)| = |gz_i| + |z_i^{-1}\gamma(i)| = |gz_i| + |\gamma(i)| - |z_i| = |\gamma(i)| + (l - k). \]
By taking the minimum over all $i \in \mathbb{Z}$, we obtain that
\[ d(e, g \cdot \gamma) = d(e, \gamma) + (l - k), \]
and then
\[ d(g \cdot p, g \cdot q) \leq d(p, q) \cdot a^{-(l-k)}. \qedhere \]
\end{proof}
\begin{fakt}
\label{fakt-duze-gwiazdy}
\dzm{Under the assumptions of Theorem~\ref{tw-bi-lip}, }there exists a constant $E$ (depending only on $G$ \dzm{and~$\mathcal{U}$}) such that, for any $k \geq 0$, the Lebesgue number of the cover $\mathcal{U}_k$ is at least $E \cdot a^{-k}$.
\end{fakt}
\begin{proof}
Let $N > N_4 \dzm{+ D}$, where $N_4$ is the constant from Proposition~\ref{lem-bi-lip-geodezyjne} \dzm{and $D$ is the neighbourhood constant of the system~$\mathcal{S}$. Denote by~$T$ the type function associated with~$\mathcal{U}$.}
Let $M>0$ be chosen so that for any $g \in G$ there exists $h \in G$ such that \dzm{$T(g) = T(h)$} and $|h| < M$. Let $L_j$ denote the Lebesgue constant for the cover $\mathcal{U}_j$ for $j < M$. We will prove that the claim of the lemma is satisfied by the number
\[ E = a^{-E_1} \cdot \min_{j<M} \, (a^j L_j), \]
where $E_1$ is the constant from Proposition~\ref{lem-bi-lip-geodezyjne}.
Let $k \geq 0$ and~$B \subset \partial G$ be a non-empty subset with diameter at most $E \cdot a^{-k}$. Let $x$ be any element of $B$, then
\dzm{
there exist elements $g, \widetilde{g} \in G$ of length~$k$ such that
\[ x \in \sppan(g), \qquad x \in U_{\widetilde{g}}. \]
Then, we have $x \in \sppan(g) \cap U_{\widetilde{g}} \subseteq S_g \cap S_{\widetilde{g}}$, so by~\qhlink{d} it follows that $d(g, \widetilde{g}) \leq D$.
}
By the definition of $M$, there exists $h \in G$ such that
\[ |h| < M, \qquad \dzm{T(g) = T(h)}. \]
Denote $\gamma = hg^{-1}$
\dzm{
and $\widetilde{h} = \gamma \widetilde{g}$.
By Definition~\ref{def-quasi-niezm-system-wpisany}, the type function~$T$ is stronger than the ball type~$T^b_N$ (in the sense of Definition~\ref{def-funkcja-typu}). Therefore, $T^b_N(g) = T^b_N(h)$, which together with Lemma~\ref{fakt-kulowy-duzy-wyznacza-maly} and $d(g, \widetilde{g}) \leq D$ implies that $T^b_{N-D}(\widetilde{g}) = T^b_{N-D}(\widetilde{h})$.
Since $N - D > N_4$, we obtain
}
from Proposition \ref{lem-bi-lip-geodezyjne} that
\[ \diam (\gamma \cdot B) \leq E \cdot a^{-k} \cdot a^{-(|h| - k)} \leq a^{-|h|} \cdot \min_{j < M} (a^j L_j) \leq L_{|h|}, \]
which means that there exists $h' \in G$ such that $|h'| = |h|$ and
$\gamma \cdot B \subseteq U_{h'}$.
Let us note that it follows from \qhlink{f1} that
\[ \gamma \cdot x \in \dzm{\gamma \cdot U_{\widetilde{g}} = U_{\widetilde{h}}}, \]
so $\gamma \cdot x$ is a common element of \dzm{$U_{\widetilde{h}}$ and~$U_{h'}$}. Then, from \qhlink{f2} we obtain that
\[ U_{\gamma^{-1} h'} = \gamma^{-1} \cdot U_{h'} \supseteq B. \qedhere \]
\end{proof}
\begin{proof}[{\normalfont \textbf{Proof of Theorem~\ref{tw-bi-lip}}}]
Denote by $n$ the \dzm{maximal rank of all the covers~$\mathcal{S}_t$, for $t \geq 0$. Then, for every $t \geq 0$ we have $\rank \, \mathcal{U}_t \leq n$ and hence $\dim K_t \leq n$.} Denote also by $a_0$ a (constant) number such that the visual metric, considered for values $1 < a < a_0$, has all the properties described in Section~\ref{sec-def}. We define
\[ a_1 = \min \big( a_0, \tfrac{n+1}{n} \big). \]
Let $1 < a < a_1$. Denote by $M$ the diameter of $\partial G$ with respect to the visual metric (which is finite due to compactness of $\partial G$), and by $C_1$ --- the multiplier of bi-Lipschitz equivalence between the distance function $d$ and the visual metric.
Let $p, q$ be two distinct elements~of $\partial G$ and let $k \geq 0$ be the minimal natural number such that $d(p, q) > a^{-k}$. Observe that $d(p, q) \leq a^{-(k-1)}$ if $k > 0$, while $d(p, q) \leq MC_1 \leq MC_1 \cdot a^{-(k-1)}$ in the other case, so in general we have:
\begin{align}
\label{eq-bi-lip-lapanie-ujemnych}
d(p, q) \leq M' \cdot a^{-(k-1)}, \qquad \textrm{ where } \qquad M' = \max(MC_1, 1).
\end{align}
\dzm{As in Definition~\ref{def-konstr-nerwy}, we let~$\pi_n$ denote the projection from~$\mathop{\lim}\limits_{\longleftarrow} K_i$ to $K_n$. Our goal is to estimate $d^M_a(\overline{p}, \overline{q})$, where $\overline{p}$, $\overline{q}$ denote correspondingly the images of $p, q$ under $\varphi$.}
First, we will estimate $d^M_a(\overline{p}, \overline{q})$ from above. Let $l$ be the maximal number not exceeding $k - \log_a E$. We consider two cases:
\begin{itemize}
\item If $l < 0$, then $k < \log_a E$, and then, by Remark~\ref{uwaga-metryka-l1-ogr},
\[ d^M_a(\overline{p}, \overline{q}) = \sum_{t = 0}^\infty a^{-t} \cdot d_{K_t} \big( \pi_t(\overline{p}), \pi_t(\overline{q}) \big) \leq \sum_{t = 0}^\infty a^{-t} \cdot 2 \leq \frac{2a}{a-1} \leq \frac{2a}{(a-1)MC_1} \cdot d(p, q). \]
\item If $l \geq 0$, then by Lemma~\ref{fakt-duze-gwiazdy} there exists $U \in \mathcal{U}_l$ containing both $p$ and $q$.
Then, in the complex $K_l$, the points $\pi_l(\overline{p})$ and~$\pi_l(\overline{q})$ must lie in some (possibly different) simplexes containing the vertex $v_U$. Then, we have
\[ d_{K_l} \big( \pi_l(\overline{p}),v_U \big) \leq 2, \qquad d_{K_l} \big( \pi_l(\overline{q}), v_U \big) \leq 2 \]
by Remark \ref{uwaga-metryka-l1-ogr}, and moreover
\[ d_{K_t} \big( \pi_t(\overline{p}), \pi_t(\overline{q}) \big) \leq 2 \cdot 2 \cdot \big( \tfrac{n}{n+1} \big)^{l-t} \qquad \textrm{ for } \quad 0 \leq t \leq l. \]
by the condition \eqref{eq-konstr-szacowanie-obrazow} from the proof of theorem~\ref{tw-konstr} (which we may use here because we are now working with the same metric in $K_i$ which was used in~\cite{E}). Then, since $\diam K_t \leq 2$ for $t \geq 0$ (by Remark \ref{uwaga-metryka-l1-ogr}) and $\tfrac{an}{n+1} < 1$, we have
\begin{align*}
d^M_a(\overline{p}, \overline{q}) & = \sum_{t = 0}^\infty a^{-t} \cdot d_{K_t} \big( \pi_t(\overline{p}), \pi_t(\overline{q}) \big) \leq \sum_{t = 0}^l a^{-t} \cdot 4 \cdot \big( \tfrac{n}{n+1} \big)^{l-t} + \sum_{t = l+1}^\infty a^{-t} \cdot 2 \leq \\
& \leq 4 a^{-l} \cdot \sum_{t = 0}^l \big( \tfrac{an}{n+1} \big)^{l-t} + 2 \cdot \sum_{t = l+1}^\infty a^{-t} \leq C_2 \cdot a^{-l} \leq (C_2Ea) \cdot a^{-k} \leq (C_2Ea) \cdot d(p, q),
\end{align*}
where $C_2$ is some constant depending only on $a$ and~$n$ (and so independent of~$p, q$).
\end{itemize}
The opposite bound will be obtained by Lemma \ref{fakt-pokrycie-male}. Let $C$ denote the constant from that lemma and let $l'$ be the smallest integer greater than $k + \log_a(3C)$. Then, Lemma \ref{fakt-pokrycie-male} ensures that, for any $t \geq l'$ and \dzm{$x \in G$ of length~$t$}, we have
\[ \dzm{ \diam_d U_x \leq \diam_d S_x } \leq 3C \cdot a^{-t} \leq a^{-k} < d(p, q), \]
so the points $p, q$ cannot belong simultaneously to any element of the cover $\mathcal{U}_t$. Then, \dzm{by the definition of $\varphi: \partial G \simeq \mathop{\lim}\limits_{\longleftarrow} K_i$,} the points $\pi_t(\overline{p})$, $\pi_t(\overline{q})$ lie in some two disjoint simplexes in $K_t$, and so, by Lemma \ref{fakt-rozlaczne-symp-daleko}, their distance is equal to $2$. Then, by \eqref{eq-bi-lip-lapanie-ujemnych}, we have:
\[ d^M_a(\overline{p}, \overline{q}) = \sum_{t = 0}^\infty a^{-t} \cdot d_{K_t} \big( \pi_t(\overline{p}), \pi_t(\overline{q}) \big) \geq \sum_{t = l'}^\infty a^{-t} \cdot 2 \geq \frac{2a^{-l'} \cdot a}{a-1} \geq \tfrac{2}{3C(a-1)} \cdot a^{-k} \geq \tfrac{2}{3CM'a(a-1)} \cdot d(p, q). \]
In view of Remark \ref{uwaga-bi-lip-wystarczy-d}, this finishes the proof.
\end{proof}
\section{Markov property}
\label{sec-markow}
The main goal of this section is to prove the following theorem:
\begin{tw}
\label{tw-kompakt-ogolnie}
Let $(\mathcal{U}_n)_{n \geq 0}$ be a quasi-$G$-invariant system of covers of a compact, metric $G$-space-$X$. Let $L_0$ denote the constant given by Proposition \ref{lem-gwiazda} for this system, $L \geq L_0$ and let $(K_n, f_n)$ be the associated inverse system of nerves obtained for the sequence of the covers $(\widetilde{\mathcal{U}}_{nL})_{n \geq 0}$ (see Definition~\ref{def-konstr-nerwy}).
Then, the system $(K_n, f_n)$ is barycentric, Markov and has the mesh property.
\end{tw}
The proof of this theorem appears --- after a number of auxiliary definitions and facts --- in Section~\ref{sec-markow-podsum}.
\subsection{Simplex types and translations}
\label{sec-markow-typy}
Below (in Definition \ref{def-typ-sympleksu}) we define simplex \textit{types} which we will use to prove the Markov property of the system $(K_n, f_n)$. Intuitively, we would like the type of a simplex $s = [v_{U_{g_1}}, \ldots, v_{U_{g_k}}]$ to contain the information about types of elements $g_i$ (which seems to be natural), but also about their relative position in $G$ (which, as we will see in Section \ref{sec-markow-synowie}, will significantly help us in controlling the pre-images of the maps $f_n$).
However, this general picture becomes more complicated because we are not guaranteed a unique choice of an element $g$ corresponding to a given set $U_g \in \widetilde{\mathcal{U}}_n$.
Therefore, in the type of a simplex, we will store information about relative positions of \textit{all} elements of~$G$ representing its vertices.
As an effect of the above considerations, we will obtain a quite complicated definition of type (which will be only rarely directly referred to). An equality of such types for given two simplexes can be conveniently described by existence of a \textit{shift} between them, preserving the simplex structure described above (see Definition~\ref{def-przesuniecie-sympleksu}). This property will be used in a number of proofs in the following sections.
We denote by $Q_n$ the nerve of the cover $\widetilde{\mathcal{U}}_n)$. (Then, $K_n = Q_{nL}$).
\begin{df}
\label{def-graf-typu-sympleksu}
For a simplex $s$ in $Q_n$, we define a directed graph $G_s = (V_s, E_s)$ in the following way:
\begin{itemize}
\item the vertices in $G_s$ are \textit{all} the elements $g \in G$ for which $v_{U_g}$ is a vertex in~$s$ (and so $|g| = n$ by Remark~\ref{uwaga-quasi-niezm-rozne-poziomy}); thus, $G_s$ may possibly have more vertices than $s$ does;
\item every vertex $g \in V_s$ is labelled with its type $T(g)$;
\item the edges in~$G_s$ are all pairs $(g, g')$ for $g, g' \in V_s$, $g \neq g'$;
\item every edge $(g, g')$ is labelled with the element $g^{-1} g' \in G$.
\end{itemize}
\end{df}
\begin{df}
\label{def-typ-sympleksu}
We call two simplexes $s \in Q_n$ and $s' \in Q_{n'}$ \textit{similar} if there exists an isomorphism of graphs $\varphi: G_s \rightarrow G_{s'}$ preserving all labels of vertices and edges.
The \textit{type} of a simplex $s \in Q_n$ (denoted by $T^\Delta(s)$) is its similarity class. \\
(Hence: two simplexes are similar if and only if they have the same type).
\end{df}
\begin{df}
\label{def-przesuniecie-sympleksu}
A simplex $s' \in Q_{n'}$ will be called the \textit{shift} of a simplex $s \in Q_n$ by an element~$\gamma$ (notation: $s' = \gamma \cdot s$) if the formula $\varphi(g) = \gamma \cdot g$ defines an isomorphism $\varphi$ which satisfies the conditions of Definition \ref{def-typ-sympleksu}.
\end{df}
\begin{fakt}
\label{fakt-przesuniecie-skladane}
Shifting simplexes satisfies the natural properties of a (partial) action of $G$ on a set:
\[ \textrm{ if } \qquad s' = \gamma \cdot s \quad \textrm{ and } \quad s'' = \gamma' \cdot s', \qquad \textrm{ then } \qquad s = \gamma^{-1} \cdot s' \quad \textrm{ and } \quad s'' = (\gamma' \, \gamma) \cdot s. \pushQED{\qed}\qedhere\popQED \]
\end{fakt}
\begin{fakt}
\label{fakt-przesuniecie-istnieje}
Two simplexes $s \in Q_n$, $s' \in Q_{n'}$ have equal types $\ \Longleftrightarrow\ $ $s' = \gamma \cdot s$ for some $\gamma \in G$.
\end{fakt}
\begin{proof}
The implication $(\Leftarrow)$ is obvious. On the other hand, let $\varphi : G_s \rightarrow G_{s'}$ be an isomorphism satisfying the conditions from Definition \ref{def-typ-sympleksu}. We choose arbitrary $g_0 \in V_s$ and define $\gamma = \varphi(g_0) \, g_0^{-1}$. Since $\varphi$ preserves the labels of edges, for any $g \in V_s \setminus \{ g_0 \}$ we have
\[ \varphi(g_0)^{-1} \, \varphi(g) = g_0^{-1} \, g \quad \Rightarrow \quad \varphi(g) \, g^{-1} = \varphi(g_0) \, g_0^{-1} = \gamma \quad \Rightarrow \quad \varphi(g) = \gamma \cdot g. \qedhere \]
\end{proof}
\begin{fakt}
\label{fakt-przesuwanie-symp-zb}
If $s' = \gamma \cdot s$ and $v_{U_x}$ is a vertex in $s$, then $v_{U_{\gamma x}}$ is a vertex in~$s'$ and moreover
\[ U_{\gamma x} = \gamma \cdot U_x, \qquad T(\gamma x) = T(x). \]
In particular, shifting the sets from $\widetilde{\mathcal{U}}$ by~$\gamma$ gives a bijection between the vertices of $s$ and~$s'$.
\end{fakt}
\begin{proof}
This follows from Definitions \ref{def-graf-typu-sympleksu} and~\ref{def-przesuniecie-sympleksu}, and from property~\qhlink{f1}.
\end{proof}
\begin{lem}
The total number of simplex types in all of the complexes $Q_n$ is finite.
\end{lem}
\begin{proof}
Let us consider a simplex $s \in K_n$. If $g, g' \in V_s$, then the vertices $v_{U_g}, v_{U_{g'}}$ belong to $s$, which means by definition that $U_g \cap U_{g'} \neq \emptyset$, and then, by \qhlink{d} and the definition of $V_s$, we have $|g^{-1} g'| \leq D$.
Then, the numbers of vertices in the graphs $G_s$, as well as the number of possible edge labels appearing in all such graphs, are not greater than the cardinality of the ball $B(e, D)$ in the group~$G$. This finishes the proof because the labels of vertices are taken by definition from the finite set of types of elements in~$G$.
\end{proof}
\subsection{The main proposition}
\label{sec-markow-synowie}
\begin{lem}
\label{lem-przesuwanie-dzieci-sympleksow}
Let $s \in K_n$, $s' \in K_{n'}$ be simplexes of the same type and $s' = \gamma \cdot s$ for some $\gamma \in G$.
Then, the maps $I : s \rightarrow s'$ and $J : f_n^{-1}(s) \rightarrow f_n^{-1}(s')$, defined on the vertices of the corresponding subcomplexes by the formulas
\[ I(v_U) = v_{\gamma \cdot U} \quad \textrm{ for } v_U \in s, \qquad J(v_U) = v_{\gamma \cdot U} \quad \textrm{ for } v_U \in f_n^{-1}(s), \]
and extended affinely to the simplexes in these subcomplexes, have the following properties:
\begin{itemize}
\item they are well defined (in particular, $\gamma \cdot U$ is an element of the appropriate cover);
\item they are isomorphisms of subcomplexes;
\item they map simplexes to their shifts by $\gamma$ (in particular, they preserve simplex types).
\end{itemize}
Moreover, the following diagram commutes:
\begin{align}
\label{eq-diagram-do-spr}
\xymatrix@+3ex{
s \ar[d]_{I} & \ar[l]_{f_n} f_n^{-1}(s) \ar[d]_{J} \\
s' & \ar[l]_{f_{n'}} f_{n'}^{-1}(s').
}
\end{align}
\end{lem}
\begin{proof}
Let
\begin{align}
\label{eq-markow-wstep-1}
s = [v_{U_1}, \ldots, v_{U_k}], \qquad U_i = U_{g_i}, \qquad g_i' = \gamma \, g_i, \qquad U_i' = U_{g_i'}.
\end{align}
Then, from the assumptions (using the definitions and Lemma \ref{fakt-przesuwanie-symp-zb}) we obtain that
\[ s' = [v_{U'_1}, \ldots, v_{U'_k}], \qquad U_i' = \gamma \cdot U_i, \qquad T(g_i) = T(g_i'), \qquad |g_i| = nL, \qquad |g'_i| = n'L. \]
In particular, for every $v_U \in s$ the value $I(v_U)$ is correctly defined and belongs to $s'$; also, $I$ gives a bijection between the vertices of $s$ and $s'$, so it is an isomorphism. Moreover, for any subsimplex $\sigma = [v_{U_{i_1}}, \ldots, v_{U_{i_l}}] \subseteq s$ and $g \in G_s$, by Lemma \ref{fakt-przesuwanie-symp-zb} we have an equivalence
\[ v_{U_g} \in \sigma \quad \Leftrightarrow \quad U_g \in \{ U_{i_j} \,|\, 1 \leq j \leq l \} \quad \Leftrightarrow \quad U_{\gamma g} \in \{ \gamma \cdot U_{i_j} \,|\, 1 \leq j \leq l \} \quad \Leftrightarrow \quad v_{U_{\gamma g}} \in I(\sigma), \]
so the isomorphism $G_s \simeq G_{s'}$ given by $\gamma$ restricts to an isomorphism $G_\sigma \simeq G_{I(\sigma)}$, so $I(\sigma) = \gamma \cdot \sigma$.
It remains to check the desired properties of the map $J$, and commutativity of the diagram \eqref{eq-diagram-do-spr}.
First, we will check that $J$ is correctly defined. Let $v_U$ be a vertex in ~$f_n^{-1}(s)$ and let $U = U_h$ for some $h \in G$ of length $(n + 1)L$. From the definition of $f_n$ we obtain that $U_h \subseteq U_{g_i}$ for some $1 \leq i \leq k$. Then, denoting $h' = \gamma h$ and using~\qhlink{f3}, we have
\begin{align}
\label{eq-markov-wlasnosci-h'}
U_{h'} = \gamma \cdot U_h \subseteq \gamma \cdot U_{g_i} = U_{g'_i}, \qquad T(h') = T(h), \qquad |h'| = (n' + 1)L,
\end{align}
so in particular $\gamma \cdot U_h \in \widetilde{\mathcal{U}}_{(n'+1)L}$, and then $J(v_U) = v_{\gamma \cdot U_h}$ is a vertex in~$K_{n' + 1}$.
Now, we will prove that the vertex $J(v_U)$ belongs to $f_{n'}^{-1}(s')$ and that the diagram~\eqref{eq-diagram-do-spr} commutes.
From the definition of maps $f_n$, $f_{n'}$ it follows that, for both these purposes, it is sufficient to prove that
\begin{align}
\label{eq-markov-zgodnosc-rodzicow-ogolnie}
\big\{ U' \,\big|\, U' \in \mathcal{U}_{n'L}, \, U' \supseteq U_{h'} \big\} = \big\{ \gamma \cdot U \,\big|\, U \in \mathcal{U}_{nL}, \, U \supseteq U_h \big\}.
\end{align}
Let us check the inclusion $(\supseteq)$. Let $U_g = U \supseteq U_h$ for some $g \in G$ of length~$nL$. Then in particular $U_g \cap U_{g_i} \supseteq U_h \neq \emptyset$, so from property \qhlink{f2} we have
\[ U_{\gamma g} = \gamma \cdot U_g \supseteq \gamma \cdot U_h = U_{h'}, \qquad |\gamma g| = n' L. \]
This proves the inclusion $(\supseteq)$ in~\eqref{eq-markov-zgodnosc-rodzicow-ogolnie}. Since $U_{g_i'} = \gamma \cdot U_{g_i} \supseteq \gamma \cdot U_h = U_{h'}$, the opposite inclusion can be proved in an analogous way, by considering the shift by $\gamma^{-1}$. Hence, we have verified \eqref{eq-markov-zgodnosc-rodzicow-ogolnie}. In particular, we obtain that $J(v_U) \in f_{n'}^{-1}(s')$ for every $v_U \in f_n^{-1}(s)$, and so $J$ is correctly defined on the vertices of the complex $f_n^{-1}(s)$. From~\eqref{eq-markov-zgodnosc-rodzicow-ogolnie} we also deduce the commutativity of the diagram \eqref{eq-diagram-do-spr} when restricted to the vertices of the complexes considered.
Now, let $\sigma = [v_{U_{h_1}}, \ldots, v_{U_{h_l}}]$ be a simplex in~$f_n^{-1}(s)$. Then, we have $\bigcap_{i = 1}^l U_{h_i} \neq \emptyset$, so also $\bigcap_{i=1}^l U_{\gamma h_i} = \gamma \cdot \bigcap_{i=1}^l U_{h_i} \neq \emptyset$. This implies that in $f_{n'}^{-1}(s')$ there is a simplex $[J(v_{U_{h_1}}), \ldots, J(v_{U_{h_l}})]$. This means that $J$ can be affinely extended from vertices to simplexes, leading to a correctly defined map of complexes. The commutativity of the diagram \eqref{eq-diagram-do-spr} is then a result from the (already checked) commutativity for vertices.
By exchanging the roles of $s$ and $s'$, and correspondingly of $g_i$ and $g_i'$, we obtain an exchange of roles between $\gamma$ and $\gamma^{-1}$. Therefore, the map $\widetilde{J} : f_{n'}^{-1}(s') \rightarrow f_n^{-1}(s)$, obtained analogously for such situation, must be inverse to $J$. Hence, $J$ is an isomorphism.
It remains to check the equality $J(\sigma) = \gamma \cdot \sigma$ for any simplex $\sigma$ in~$f_n^{-1}(s)$. For this, we choose any element $h \in V_\sigma$ (i.e. a vertex from the graph $G_\sigma$ from Definition~\ref{def-graf-typu-sympleksu}) and take $\varphi(h) = \gamma h$; this element was previously denoted by $h'$. Then, from \eqref{eq-markov-wlasnosci-h'} and the already checked properties of $J$, we deduce that
\[ v_{U_{\gamma h}} = v_{\gamma \cdot U_h} = J(v_{U_h}) \, \textrm{ is a vertex in } J(\sigma), \qquad T(\gamma h) = T(h). \]
The first of these facts means that $\varphi(h)$ indeed belongs to $V_{J(\sigma)}$; the second one ensures that $\varphi$ preserves the labels of vertices in the graphs.
Preserving the labels of edges follows easily from the definition of $\varphi$. Hence, it remains only to check that $\varphi$ gives a bijection between $V_\sigma$ and ~$V_{J(\sigma)}$, which we obtain by repeating the above reasoning for the inverse map $J^{-1}$ (with $s'$, $J(\sigma)$ playing now the roles of $s$, $\sigma$).
\end{proof}
\subsection{Conclusion: $\partial G$ is a Markov compactum}
\label{sec-markow-podsum}
Although we will be able to prove Theorem~\ref{tw-kompakt} in its full strength only at the end of Section~\ref{sec-wymd}, we note that the results already obtained imply the main claim of this theorem, namely that Gromov boundaries of hyperbolic groups are always Markov compacta (up to homeomorphism). Before arguing for that, we will finish the proof of Theorem~\ref{tw-kompakt-ogolnie}.
\begin{proof}[{\normalfont \textbf{Proof of Theorem~\ref{tw-kompakt-ogolnie}}}]
Barycentricity and the mesh property for the system $(K_n, f_n)$ follow from Theorem \ref{tw-konstr}; it remains to check the Markov property for this system. The condition (ii) from Definition \ref{def-kompakt-markowa} follows from the way in which the maps $f_n$ are defined in the claim of Theorem~\ref{tw-konstr}, while (i) is a result of the assumption (ii) in Definition \ref{def-konstr-admissible} (admissibility of $\mathcal{U}$). It remains to check (iii).
The type which we assign to simplexes is the $T^\Delta$-type from Definition \ref{def-typ-sympleksu}. If two simplexes $s \in K_i$, $s' \in K_j$ have the same type, then by Lemma \ref{fakt-przesuniecie-istnieje} we know that $s' = \gamma \cdot s$ for some $\gamma \in G$. Then, by an inductive application of Proposition \ref{lem-przesuwanie-dzieci-sympleksow}, we deduce that for every $k \geq 0$ the simplexes in the pre-image $(f^{j+k}_j)^{-1}(s')$ coincide with the shifts (by $\gamma$) of simplexes in the pre-image $(f^{i+k}_i)^{-1}(s)$, and moreover, by letting
\[ i_k (v_U) = v_{\gamma \cdot U} \qquad \textrm{ for } \quad k \geq 0, \ v_U \in (f^{i+k}_i)^{-1}(s), \]
we obtain correctly defined isomorphisms of subcomplexes which preserve simplex types and make the diagram from Definition \ref{def-kompakt-markowa} commute. This finishes the proof.
\end{proof}
\begin{proof}[{\normalfont \textbf{Proof of the main claim of Theorem~\ref{tw-kompakt}}}]
Let $G$ by a hyperbolic group and let~$\mathcal{S}$ be the system of covers of~$\partial G$ defined in Section~\ref{sec-konstr-pokrycia}. By~Corollary~\ref{wn-spanstary-quasi-niezm}, $\mathcal{S}$ is quasi-$G$-invariant; hence, by Lemma~\ref{fakt-konstr-sp-zal}, there is $L \geq 0$ such that the system $(\widetilde{S}_{nL})_{n \geq 0}$ is admissible. By applying Theorem~\ref{tw-konstr}, we obtain an inverse system $(K_n, f_n)$ whose inverse limit is homeomorphic to~$\partial G$; on the other hand, Theorem~\ref{tw-kompakt-ogolnie} ensures that this system is Markov. This finishes the proof.
\end{proof}
\begin{uwaga}
Theorem~\ref{tw-kompakt-ogolnie} ensures also barycentricity and mesh property for the obtained Markov system.
\end{uwaga}
\section{Strengthenings of type}
\label{sec-abc}
In this section, we will construct new \textit{type functions} in the group $G$ (in the sense of Definition~\ref{def-funkcja-typu}), or in the inverse system $(K_n)$ constructed in Section~\ref{sec-engelking} (in an analogous sense, i.e. we assign to every simplex its \textit{type} taken from a finite set), with the aim of ensuring
certain regularity conditions of these functions which will be needed in the next sections.
The basic condition of our interest, which will be considered in several flavours, is ``children determinism'': the type of an element (resp. a simplex) should determine the type of its ``children'' (in an appropriate sense), analogously to the properties of the ball type $T^b_N$ described in Proposition \ref{lem-potomkowie-dla-kulowych}. The most important result of this section is the construction of a new type (which we call \textit{$B$-type} and denote by $T^B$) which, apart from being children-deterministic in such sense, returns different values for any pair of \textit{$r$-fellows} in $G$ (see Definition~\ref{def-konstr-towarzysze}) for some fixed value of $r$. (To achieve the goals of this article, we take $r = 16\delta$). This property will be crucial in three places of the remaining part of the paper:
\begin{itemize}
\item In Section \ref{sec-sk-opis}, we will show that, by including the $B$-type in the input data for Theorem \ref{tw-kompakt-ogolnie}, we can ensure the resulting Markov system has distinct types property (see Definition~\ref{def-kompakt-wlasciwy}), which will in turn guarantee its finite describability (see Remark \ref{uwaga-sk-opis}).
\item In Section \ref{sec-wymd}, this property will allow us a~kind of ``quasi-$G$-invariant control'' of simplex dimensions in the system $(K_n)$.
\item In Section \ref{sec-sm}, the $B$-type's property of distinguishing fellows will be used to present the boundary $\partial G$ as a semi-Markovian space (see Definition~\ref{def-sm-ps}).
\end{itemize}
Let us note that this property of $B$-type is significantly easier to be achieved in the case of torsion-free groups (see the introduction to Section \ref{sec-sm-abc-b}).
A natural continuation of the topic of this section will also appear in Section \ref{sec-sm-abc-c}, in which we will enrich the $B$-type to obtain a new \textit{$C$-type}, which will serve directly as a basis for the presentation of $\partial G$ as a semi-Markovian space. (We postpone discussing the $C$-type to Section \ref{sec-sm} because it is needed only there, and also because we will be able to list its desired properties only as late as in Section~\ref{sec-sm-zyczenia}).
\subsubsection*{Technical assumptions}
In Sections \ref{sec-abc}--\ref{sec-wymd}, we assume that $N$ and~$L$ are fixed and sufficiently large constants; explicit bounds from below will be chosen while proving consecutive facts. (More precisely, we assume that $N$ satisfies the assumptions of Corollary \ref{wn-sm-kulowy-wyznacza-potomkow} and Proposition \ref{lem-sm-kuzyni}, and that $L \geq \max(N + 4\delta, 14\delta)$ satisfies the assumptions of Lemma \ref{fakt-sm-rodzic-kuzyna}; some of these bounds will be important only in Section \ref{sec-wymd}). Let us note that, under such assumptions, Proposition \ref{lem-potomkowie-dla-kulowych} ensures that the ball type $T^b_N(x)$ determines the values of $T^b_N$ for all descendants of $x$ of length $\geq |x| + L$.
The types which we will construct --- similarly as the ball type $T^b$ --- will depend on the value of a parameter $N$ (discussed in the above paragraph) which, for simplicity, will be omitted in the notation.
Also, we assume that the fixed generating set~$S$ of the group~$G$ (which we are implicitly working with throughout this paper) is closed under taking inverse, and we fix some enumeration $s_1, \ldots, s_Q$ of all its elements (This will be used in Section \ref{sec-abc-pp}).
\subsection{Prioritised ancestors}
\label{sec-abc-pp}
\begin{df}
Let $x, y \in G$. We call $y$ a \textit{descendant} of $x$ if $|y| = |x| + d(x, y)$. (Equivalently: if $y \in xT^c(x)$). In such situation, we say that $x$ is an \textit{ancestor} of~$y$.
If in addition $d(x, y) = 1$, we say that $y$ is a \textit{child} of~$x$ and $x$ is a \textit{parent} of~$y$.
\end{df}
\begin{df}
\label{def-sm-nic-sympleksow}
The \textit{prioritised parent} (or \textit{p-parent}) of an element $y \in G \setminus \{ e \}$ is the element $x \in G$ such that $x$ is a parent of~$y$ and $x = y s_i$ with $i$ least possible. The p-parent of $y$ will be denoted by $y^\uparrow$.
An element $g' \in G$ is a \textit{priority child} (or \textit{p-child}) of $g$ if $g$ is its p-parent.
As already suggested, a given element of $G \setminus \{ e \}$ must have exactly one p-parent but may have many p-children.
The relation of \textit{p-ancestry} (resp. \textit{p-descendance}) is defined as the reflexive-transitive closure of p-parenthood (resp. p-childhood); in particular, for any $g \in G$ and $k \leq |g|$, $g$ has exactly one p-ancestor $g'$ such that~$|g'| = |g| - k$, which we denote by $g^{\uparrow k}$.
\end{df}
\begin{df}
\label{def-sm-p-wnuk}
Let $x, y \in G$. We call $x$ a \textit{p-grandchild} (resp. \textit{p-grandparent}) of $y$ if it is a p-descendant (resp. p-ancestor) of $y$ and $\big| |x| - |y| \big| = L$. For simplicity of notation, we denote $x^\Uparrow = x^{\uparrow L}$ (for $|x| \geq L$) and analogously $x^{\Uparrow k} = x^{\uparrow Lk}$ (for $|x| \geq Lk$).
\end{df}
Let $N_0$ denote the constant coming from Proposition \ref{lem-kulowy-wyznacza-stozkowy}.
\begin{fakt}
\label{fakt-sm-kulowy-wyznacza-dzieci}
Let $x, y, s \in G$ satisfy $T^b_{N_0+2}(x) = T^b_{N_0+2}(y)$ and $|s| = 1$. Then, $(xs)^\uparrow = x$ if and only if $(ys)^\uparrow = y$.
\end{fakt}
\begin{proof}
Since the set $S$ is closed under taking inverse, we have $s = s_i^{-1}$ for some~$i$. Assume that $(xs_i^{-1})^\uparrow = x$ but $(ys_i^{-1})^\uparrow = z \neq y$. Then, $z = ys_i^{-1}s_j$ for some $j < i$. Note that $d(y, z) \leq 2$. Define $z' = xs_i^{-1}s_j$; then by Lemma \ref{fakt-kulowy-duzy-wyznacza-maly} and Proposition \ref{lem-kulowy-wyznacza-stozkowy} we have
\[ T^b_{N_0}(z') = T^b_{N_0}(z), \qquad s_j^{-1} \in T^c(z) = T^c(z'). \]
This means that the element $xs_i^{-1} = z's_j^{-1}$ is a child of $z'$; then, since $j < i$, it cannot be a p-child of $x$.
\end{proof}
\begin{wn}
\label{wn-sm-kulowy-wyznacza-potomkow}
Let $N \geq N_5 := 2N_0 + 8\delta + 2$. Then, if $x, y \in G$ satisfy $T^b_N(x) = T^b_N(y)$, the left translation by $\gamma = yx^{-1}$ gives a bijection between the p-descendants of $x$ and the p-descendants of $y$.
\end{wn}
\begin{proof}
Let $z$ be a p-descendant of $x$; we want to prove that $\gamma z$ is a p-descendant of~$y$. We will do this by induction on the difference $|z| - |x|$.
If $|z| = |x|$, the claim is obvious. Assume that $|z| > |x|$ and denote $w = z^\uparrow$; then $w$ is a p-descendant of $x$ for which we may apply the induction assumption, obtaining that $\gamma w$ is a p-descendant of $y$. To finish the proof, it suffices to check that
\begin{align}
\label{eq-sm-typy-potomkow}
T^b_{N_0 + 2}(w) = T^b_{N_0 + 2}(\gamma w),
\end{align}
since then from Lemma \ref{fakt-sm-kulowy-wyznacza-dzieci} it will follow that $\gamma z$ is a p-child of $\gamma w$, and then also a p-descendant of $y$. We consider two cases:
\begin{itemize}
\item If $d(z, x) \leq N - (N_0 + 2)$, then \eqref{eq-sm-typy-potomkow} holds by equality $T^b_N(x) = T^b_N(y)$ and Lemma \ref{fakt-kulowy-duzy-wyznacza-maly}.
\item If $|z| - |x| \geq N_0 + 4\delta + 2$, then \eqref{eq-sm-typy-potomkow} follows (in view of the equality $T^b_N(x) = T^b_N(y)$ and the inequality $N \geq N_0 + 8\delta$) from Proposition \ref{lem-potomkowie-dla-kulowych}.
\end{itemize}
Since $d(z, x) = |z| - |x|$ (because $z$ is a descendant of $x$) and $N \geq 2N_0 + 4\delta + 2$, at least one of the two cases must hold. This finishes the proof.
\end{proof}
\subsection{The $A$-type}
\label{sec-sm-abc-a}
Let $\mathcal{T}^b_N$ be the set of values of the type $T^b_N$. As an introductory step, we define the \textit{$Z$-type} $T^Z$ in $G$ so that:
\begin{itemize}
\item $T^Z$ is compatible with $T^b_N$ for all elements $g \in G$ with length $\geq L$;
\item $T^Z$ assigns pairwise distinct values, not belonging to $\mathcal{T}^b_N$, to all elements of length $< L$.
\end{itemize}
Let us note that such a strengthening preserves most of the properties of $T^b_N$, in particular those described in Proposition \ref{lem-potomkowie-dla-kulowych} and Corollary \ref{wn-sm-kulowy-wyznacza-potomkow}.
\begin{uwaga}
\label{uwaga-pomijanie-N}
Although the values of $T^Z$ depend on the value of $N$, for simplicity we omit this fact in notation, assuming $N$ to be fixed. (We are yet not ready to state our assumptions on $N$; this will be done below in Proposition \ref{lem-sm-kuzyni}).
\end{uwaga}
Let $\mathcal{T}^Z$ be the set of all possible values of $T^Z$. For every $\tau \in \mathcal{T}^Z$, choose some \textit{representative} $g_\tau \in G$ of this type. For convenience, we denote by $\gamma_g$ the element of~$G$ which (left-) translates $g$ to the representative (chosen above) of its $Z$-type:
\[ \gamma_g = g_{T^Z(g)} \, g^{-1} \qquad \textrm{ for } \quad g \in G. \]
Let us recall that, by Corollary \ref{wn-sm-kulowy-wyznacza-potomkow}, the left translation by $\gamma_g$ gives a bijection between p-descendants of $g$ and p-descendants of $g_{T^Z(g)}$, and thus also a bijection between the p-grandchildren of these elements.
For every $\tau \in \mathcal{T}$, let us fix an (arbitrary) enumeration of all p-grandchildren of $g_\tau$.
\begin{df}
\label{def-sm-numer-dzieciecy}
The \textit{descendant number} of an element $g \in G$ with $|g| \geq L$ (denoted by~$n_g$) is the number given (in the above enumeration) to the element $\gamma_{g^\Uparrow} \cdot g$ as a p-grandchild of $g_{T^Z(g^\Uparrow)}$. If $|g| < L$, we set $n_g = 0$.
\end{df}
\begin{df}
\label{def-sm-typ-A}
We define the \textit{$A$-type} of an element $g \in G$ as the pair $T^A(g) = (T^Z(g), n_g)$.
\end{df}
We note that the set $\mathcal{T}^A$ of all possible $A$-types is finite because the number of p-grandchildren of $g$ depends only on $T^Z(g)$.
\begin{fakt}
\label{fakt-sm-istnieje-nss}
For any two distinct elements $g, g' \in G$ of equal length, there is $k \geq 0$ such that $g^{\Uparrow k}$ and $g'^{\Uparrow k}$ exist and have different $A$-types.
\end{fakt}
\begin{proof}
Let $n = |g| = |g'|$ and let $k \geq 0$ be the least number for which $g^{\Uparrow k} \neq g'^{\Uparrow k}$. If $n - kL < L$, then by definition $g^{\Uparrow k}$ and $g'^{\Uparrow k}$ have different $Z$-types. Otherwise, we have $g^{\Uparrow k+1} = g'^{\Uparrow k+1}$, which means that $g^{\Uparrow k}$ and~$g'^{\Uparrow k}$ have different descendant numbers.
\end{proof}
\begin{fakt}
\label{fakt-sm-przen-a}
If $g, h \in G$ have the same $Z$-type, then the left translation by $\gamma = hg^{-1}$ maps p-grandchildren of~$g$ to p-grandchildren of~$h$ and preserves their $A$-types.
\end{fakt}
\begin{proof}
Denote $\tau = T^Z(g) = T^Z(h)$. Let $g'$ be a p-grandchild of $g$; then $\gamma g'$ is a p-grandchild of $h$ by Corollary \ref{wn-sm-kulowy-wyznacza-potomkow}. Moreover, from Proposition \ref{lem-potomkowie-dla-kulowych} we know that $g'$ and $\gamma g'$ have equal ball types $T^b_N$, and since they both have the same length $\geq L$, it follows that they have equal $Z$-types. They also have equal descendant numbers because
\[ \gamma_{(\gamma g')^\Uparrow} \, \gamma g' = g_\tau \, h^{-1} \, \gamma \, g' = g_\tau \, g^{-1} \, g' = \gamma_{g'^\Uparrow} \, g'. \qedhere \]
\end{proof}
\subsection{Cousins and the $B$-type}
\label{sec-sm-abc-b}
The aim of this subsection is to strengthen the type function to distinguish any pair of neighbouring elements in $G$ of the same length. In the case of a torsion-free group, there is nothing new to achieve as the desired property is already satisfied by the ball type $T^b_N$ (by Lemma \ref{fakt-kuzyni-lub-torsje}). In general, the main idea is to remember within the type of $g$ the ``crucial genealogical difference'' between $g$ and every it \textit{cousin} (i.e. a neighbouring element of the same length --- see Definition \ref{def-sm-kuzyni}) --- where, to be more precise, the ``genealogical difference'' between $g$ and $g'$ consists of the $A$-types of their p-ancestors in the oldest generation in which these p-ancestors are still distinct. However, it turns out that preserving all the desired regularity properties (in particular, children determinism) requires remembering not only the genealogical differences between $g$ and its cousins, but also similar differences between any pair of its cousins.
\begin{df}
\label{def-sm-sasiedzi}
Two elements $x, y \in G$ will be called \textit{neighbours} (denotation: $x \leftrightarrow y$) if $|x| = |y|$ and $d(x, y) \leq 8\delta$.
\end{df}
Denote
\begin{align}
\label{eq-sm-zbiory-torsji}
\begin{split}
Tor & = \big\{ g \in G \,\big|\, |g| \leq 16\delta, \, g \textrm{ is a torsion element} \big\}, \\
R & = \max \Big( \big\{ |g^n| \,\big|\, g \in Tor, \, n \in \mathbb{Z} \big\} \cup \{ 16 \delta \} \Big).
\end{split}
\end{align}
Since~$|Tor| < \infty$, it follows that $R < \infty$.
\begin{df}
\label{def-sm-kuzyni}
Two elements $g, g' \in G$ will be called \textit{cousins} if $|g| = |g'|$ and $d(g, g') \leq R$. The set of cousins of $g$ will be denoted by $C_g$. (It is exactly the set $g P_R(g)$ using the notation of Section \ref{sec-konstr-pokrycia}).
\end{df}
Let us note that if $g, h \in G$ are neighbours, then they are cousins too.
\begin{fakt}
\label{fakt-sm-rodzic-kuzyna}
If $L \geq \tfrac{R}{2} + 4\delta$, then for any cousins $g, g' \in G$ their p-grandparents are neighbours (and hence also cousins).
\end{fakt}
\begin{proof}
This is clear by Lemma \ref{fakt-geodezyjne-pozostaja-bliskie}.
\end{proof}
For $g \in G$ of length~$n$ and any two distinct $g', g'' \in C_g$, let $k_{g', g''}$ be the least value $k \geq 0$ for which $g'^{\Uparrow k}$ and $g''^{\Uparrow k}$ have different $A$-types (which is a correct definition by Lemma \ref{fakt-sm-istnieje-nss}). Let the sequence $k^{(g)} = (k^{(g)}_i)_{i = 1}^{M_g}$ come from arranging the elements of the set $\{ k_{g', g''} \,|\, g', g'' \in C_g \} \cup \{ 0 \}$ in a decreasing order.
\begin{df}
\label{def-sm-typ-B}
The \textit{$B$-type} of an element $g$ is the set
\[ T^B(g) = \Big\{ \big( g^{-1} g', \, W_{g, \, g'} \big) \ \Big|\ g' \in C_g \Big\}, \qquad \textrm{ where } \qquad W_{g, \, g'} = \Big( T^A \big( {g'}^{\Uparrow k^{(g)}_i} \big) \Big)_{i=1}^{M_g}. \]
(We recall that the notation hides the dependence on a fixed parameter $N$, whose value will be chosen in Proposition \ref{lem-sm-kuzyni}).
\end{df}
\begin{uwaga}
\label{uwaga-sm-B-wyznacza-A}
Since $g$ is a cousin of itself and $0$ is the last value in the sequence $(k^{(g)}_i)$, the set $T^B(g)$ contains in particular a pair of the form $\big( e, \, (\ldots, T^A(g)) \big)$, which means that the $B$-type determines the $A$-type (of the same element).
\end{uwaga}
\begin{uwaga}
\label{uwaga-sm-indeksy-z-tabelki}
For any two distinct $g', g'' \in C_g$, let $i^{(g)}_{g', g''}$ be the position on which the value $k_{g', g''}$ appears in sequence~$k^{(g)}$. Then from definition it follows that this index depends only on the sequences $W_{g, g'}$ and~$W_{g, g''}$, and more precisely it is equal to the greatest $i$ for which the $i$-th coordinates of these sequences differ. (At the same time, we observe that the whole sequences $W_{g, g'}$, $W_{g, g''}$ must be different). This fact will be used in the proofs of Proposition \ref{lem-sm-B-dzieci} and~\ref{lem-sm-kuzyni}.
\end{uwaga}
\begin{fakt}
\label{fakt-sm-B-skonczony}
There exist only finitely many possible $B$-types in~$G$.
\end{fakt}
\begin{proof}
The finiteness of $A$-type results from its definition. If $g' \in C_g$, then the element $g^{-1} g'$ belongs to the ball $B(e, R)$ in~$G$, whose size is finite and independent of $g$. Since the number of cousins of $g$ is also globally limited, we obtain a limit on the length of the sequence $(k^{(g)}_i)$, which ensures a finite number of possible sequences $W_{g, g'}$.
\end{proof}
\begin{lem}
\label{lem-sm-B-dzieci}
Let $g_1, h_1 \in G$ have the same $B$-types and $\gamma = h_1g_1^{-1}$. Then, the left translation by $\gamma$ maps p-grandchildren of $g_1$ to p-grandchildren of $h_1$ and preserves their $B$-type.
\end{lem}
\begin{proof}
\textbf{1. }Let $g_2$ be a p-grandchild of $g_1$ and $h_2 = \gamma g_2$. Then, $h_1 = h_2^\Uparrow$ by Remark~\ref{uwaga-sm-B-wyznacza-A} and Lemma~\ref{fakt-sm-przen-a}; it remains to check that $T^B(g_2) = T^B(h_2)$.
Since the left multiplication by $\gamma$ clearly gives a bijection between cousins of $g_2$ and cousins of~$h_2$, it suffices to show that for any $g_2' \in C_{g_2}$ we have
\begin{align}
\label{eq-sm-B-dzieci-cel}
W_{g_2, \, g_2'} = W_{h_2, \, h_2'}, \qquad \textrm{ where } \quad h_2' = \gamma g_2'.
\end{align}
\textbf{2. }Let $g_2'$, $h_2'$ be as above. Denote $g_1' = {g_2'}^\Uparrow$; by Lemma~\ref{fakt-sm-rodzic-kuzyna}, $g_1'$ is a cousin of $g_1$. Then, the element $h_1' = \gamma g_1'$ is a cousin of $h_1$ and from the equalities $T^B(g_1) = T^B(h_1)$ and $g_1^{-1} g_1' = h_1^{-1} h_1'$ it follows that
\begin{align}
\label{eq-sm-B-dzieci-znane-war-t1}
W_{g_1, \, g_1'} = W_{h_1, \, h_1'}.
\end{align}
Since $0$ is the last element in the sequence $(k^{(g_1)}_i)$, as well as in $(k^{(h_1)}_i)$, we obtain in particular that $T^A(g_1') = T^A(h_1')$. By Lemma \ref{fakt-sm-przen-a}, we have:
\begin{align}
\label{eq-sm-B-dzieci-zachowane-A-t2}
h_1' = h_2'^\Uparrow, \qquad T^A(g_2') = T^A(h_2').
\end{align}
\textbf{3. }For arbitrarily chosen $g_2', g_2'' \in C_{g_2}$, we denote:
\[ h_2' = \gamma \, g_2', \qquad h_2'' = \gamma \, g_2'', \qquad \tau' = T^A(g_2') = T^A(h_2'), \qquad \tau'' = T^A(g_2'') = T^A(h_2''). \]
Moreover, we denote by $g_1', g_1'', h_1', h_1''$ the p-grandparents correspondingly of $g_2', g_2'', h_2', h_2''$.
Then, by definition:
\begin{align}
\label{eq-sm-B-dzieci-nss}
k_{g_2', g_2''} = \begin{cases}
k_{g_1', g_1''} + 1, & \textrm{ when } \tau' = \tau'', \\
0, & \textrm{ when } \tau' \neq \tau''
\end{cases}
\qquad \textrm{ and } \qquad
k_{h_2', h_2''} = \begin{cases}
k_{h_1', h_1''} + 1, & \textrm{ when } \tau' = \tau'', \\
0, & \textrm{ when } \tau' \neq \tau''.
\end{cases}
\end{align}
This implies that the sequence $(k^{(g_2)}_j)$ (resp. $(k^{(h_2)}_j)$) is obtained from $(k^{(g_1)}_i)$ (resp. $(k^{(h_1)}_i)$) by removing some elements, increasing the remaining elements by $1$, and appending the value $0$ to its end. Then, for any $g_2' \in C_{g_2}$, the sequences $W_{g_2, g_2'}$, $W_{h_2, h_2'}$ are obtained respectively from $W_{g_1, g_1'}$, $W_{h_1, h_1'}$ by removing the corresponding elements and appending respectively the values $T^A(g_2')$, $T^A(h_2')$; these two appended values must be the same by \eqref{eq-sm-B-dzieci-zachowane-A-t2}. (Note that increasing the values in the sequence $(k^{(g_2)}_j)$ by~$1$ translates to making no change in the sequence $W_{g_2, g_2'}$ due to the equality $g_2'^{\Uparrow k+1} = g_1'^{\Uparrow k}$)
Therefore, to finish the proof (i.e. to show \eqref{eq-sm-B-dzieci-cel}) it is sufficient, by \eqref{eq-sm-B-dzieci-znane-war-t1}, to check that both sequences $(k^{(g_1)}_i)$ and $(k^{(h_1)}_i)$ are subject to removing elements exactly at the same positions.
\textbf{4. }From \eqref{eq-sm-B-dzieci-nss} we know that the value $k^{(g_1)}_i + 1$ appears in the sequence $(k^{(g_2)}_j)$ if and only if there exist $g_2', g_2'' \in C_{g_2}$ such that (with the above notations ):
\begin{align}
\label{eq-sm-B-dzieci-ii}
T^A(g_2') = T^A(g_2''), \qquad i = i^{(g_1)}_{g_1', g_1''}.
\end{align}
Then, from Remark \ref{uwaga-sm-indeksy-z-tabelki} and \eqref{eq-sm-B-dzieci-znane-war-t1} used for the pairs $(g_1', h_1')$ and~$(g_1'', h_1'')$, we obtain that $i^{(h_1)}_{h_1', h_1''} = i^{(g_1)}_{g_1', g_1''} = i$, while from \eqref{eq-sm-B-dzieci-ii} and~\eqref{eq-sm-B-dzieci-zachowane-A-t2} we have $T^A(h_2') = T^A(h_2'')$, so analogously we prove that the value $k^{(h_1)}_i + 1$ appears in the sequence $(k^{(h_2)}_j)$. The proof of the opposite implication --- if $k^{(h_1)}_i + 1$ appears in~$(k^{(h_2)}_j)$, then $k^{(g_1)}_i + 1$ must appear in~$(k^{(g_2)}_j)$ --- is analogous.
The obtained equivalence finishes the proof.
\end{proof}
\begin{lem}
\label{lem-sm-kuzyni}
If $N$ is sufficiently large, then, for any $g, h \in G$ satisfying $|g| = |h|$ and $d(g, h) \leq 16\delta$, the condition $T^B(g) = T^B(h)$ holds only if $g=h$.
\end{lem}
\begin{proof}
Suppose that $g \neq h$ while $T^B(g) = T^B(h)$. Then, by Remark \ref{uwaga-sm-B-wyznacza-A}, $T^A(g) = T^A(h)$, and so $T^b_N(g) = T^b_N(h)$. Denote $\gamma = g^{-1}h$ and
assume that $N$ is greater than the constant $N_r$ given by Lemma \ref{fakt-kuzyni-lub-torsje} for $r = 16\delta$. Then, the lemma implies that $\gamma$ is a torsion element, i.e. (using the notation of \eqref{eq-sm-zbiory-torsji}) $\gamma \in Tor$, and so $|\gamma^i| \leq R$ for all $i \in \mathbb{Z}$. Then, the set $A = \{ g \gamma^i \,|\, i \in \mathbb{Z} \}$ has diameter not greater then $R$, so any two of its elements are cousins. Moreover, $A$ contains $g$ and $h$.
We obtain that all elements of the set $K = \{ k_{g', g''} \,|\, g', g'' \in A, \, g' \neq g'' \}$ appear in the sequence $k^{(g)}$ as well as in $k^{(h)}$; moreover, from Remark \ref{uwaga-sm-indeksy-z-tabelki} we obtain that they appear in $k^{(g)}$ exactly at the following set of positions:
\[ I_1 = \Big\{ \max \big\{ j \,\big|\, (W)_j \neq (W')_j \big\} \ \Big| \ (\gamma^i, W), \, (\gamma^{i'}, W') \in T^B(g), \ i, i' \in \mathbb{Z}, \, \gamma^i \neq \gamma^{i'} \Big\}, \]
where by $(W)_j$ we denoted the $j$-th element of the sequence $W$. Similarly, the elements of $K$ appear in $k^{(h)}$ exactly on the positions from the set $I_2$ defined analogously (with $g$ replaced by $h$). However, by assumption we have $T^B(g) = T^B(h)$, and so $I_1 = I_2$.
Now, consider the pair $(\gamma, W_{g, h})$ which appears in $T^B(g)$; from the equality $T^B(g) = T^B(h)$ we obtain that
\[ (\gamma, W_{g, h}) = (h^{-1} h', W_{h, h'}) \qquad \textrm{ for some } h' \in C_h. \]
Since the sets of indexes $I_1, I_2$ are equal, from $W_{g, h} = W_{h, h'}$ we obtain in particular that
\[ T^A(h^{\Uparrow k}) = T^A({h'}^{\Uparrow k}) \qquad \textrm{ for every } k \in K. \]
However, from $h^{-1} h' = \gamma$ it follows that $h' \in A \setminus \{ h \}$ and then $k_{h, h'} \in K$, which contradicts the above equality.
\end{proof}
\subsection{Stronger simplex types}
\label{sec-sk-opis}
\begin{tw}
\label{tw-sk-opis}
Let $(\mathcal{U}_n)$ be a quasi-$G$-invariant system of covers of a~space $X$, equipped with a type function stronger than $T^B$ and a neighbourhood constant $D$ not greater than $16\delta$. Let $(K_n, f_n)$ be the inverse system obtained from applying Theorem \ref{tw-kompakt-ogolnie} for $(\mathcal{U}_n)$. Then, the simplex types in the system $(K_n, f_n)$ can be strengthened to ensure the distinct types property for this system, without losing Markov property.
\end{tw}
\begin{proof}[Organisation of the proof]
We will consider the inverse system with a new simplex type $T^{\Delta + A}$, defined in Definition \ref{def-sk-opis-typ-delta-a}. Then, Lemmas \ref{fakt-sk-opis-markow} and~\ref{fakt-sk-opis-bracia} will ensure that this system satisfies the conditions from Definitions \ref{def-kompakt-markowa} and~\ref{def-kompakt-wlasciwy}, respectively.
\end{proof}
We now start the proof.
Let $T$ denote the type function associated with the system $(\mathcal{U}_n)$, and $T^\Delta$ --- the corresponding simplex type function in the system $(K_n)$, as defined in Section \ref{sec-markow-typy}.
\begin{fakt}
\label{fakt-sk-opis-podsympleksy-rozne-typy}
For any simplex $s \in K_n$, all subsimplexes $s' \subseteq s$ have pairwise distinct $T^\Delta$-types.
\end{fakt}
\begin{proof}
Let $v = v_{U_x}$, $v' = v_{U_{x'}}$ be two distinct vertices joined by an edge in $K_n$. Then, by using the definition of the complex $K_n$, then the property \qhlink{d} and Proposition \ref{lem-sm-kuzyni}, we have:
\begin{align}
\label{eq-sm-sk-opis-rozne-typy-wierzch}
U_x \cap U_{x'} \neq \emptyset \quad \Rightarrow \quad d(x, x') \leq D \leq 16\delta \quad \Rightarrow \quad T^B(x) \neq T^B(x') \quad \Rightarrow \quad T(x) \neq T(x').
\end{align}
Recall that, by Definition \ref{def-typ-sympleksu}, for any $s = [v_1, \ldots, v_k] \in K_n$ the value $T^\Delta(s)$ determines in particular the set of labels in the graph $G_s$ (defined in Definition \ref{def-graf-typu-sympleksu}). This set can be described by the formula:
\[ A_s = \big\{ T(x) \ \big|\ x \in G, \, |x| = n, \, v_{U_x} = v_i \textrm{ for some } 1 \leq i \leq k \big\} \]
However, it follows from \eqref{eq-sm-sk-opis-rozne-typy-wierzch} that if two simplexes ~$s', s'' \subseteq s$ differ in that some vertex~$v_{U_x}$ belongs only to~$s'$, then the value $T(x)$ belongs to $A_{s'} \setminus A_{s''}$, which means that $T^\Delta(s') \neq T^\Delta(s'')$.
\end{proof}
\begin{fakt}
\label{fakt-sk-opis-przesuniecie-jedyne}
Let $s \in K_n$, $s' \in K_{n'}$ have the same types. Then, there exists a unique $\gamma \in G$ such that $s' = \gamma \cdot s$.
\end{fakt}
\begin{uwaga}
The claim of Lemma~\ref{fakt-sk-opis-przesuniecie-jedyne} is stronger than what was stated in Section \ref{sec-markow-typy} in that $\gamma$ should be unique. The stronger claim follows, as we will show below, from the additional assumption that the type function~$T$ is stronger than~$T^B$.
\end{uwaga}
\begin{proof}[Proof of Lemma~\ref{fakt-sk-opis-przesuniecie-jedyne}]
By Lemma \ref{fakt-przesuniecie-istnieje}, a desired element $\gamma$ exists, it remains to check its uniqueness. Let $s' = \gamma \cdot s = \gamma' \cdot s$; then, by Lemma \ref{fakt-przesuniecie-skladane}, we have~$s = (\gamma^{-1} \gamma') \cdot s$. By Definition \ref{def-przesuniecie-sympleksu}, this means that if $v_{U_x}$ is a vertex in $s$, then setting $x' = \gamma^{-1} \gamma' x$ we have $T(x') = T(x)$ and moreover $v_{U_{x'}}$ is also a vertex in $s$. By reusing the argument from \eqref{eq-sm-sk-opis-rozne-typy-wierzch}, we conclude that $x = x'$ holds, and so $\gamma = \gamma'$.
\end{proof}
We will now define a strengthening of the $T^\Delta$-type, in a way quite analogous to the definition of the $T^A$-type for simplexes. (Some differences will occur in the proofs, and also in Definition ~\ref{def-sk-opis-typ-delta-a}).
\begin{df}[cf.~Definition~\ref{def-sm-nic-sympleksow}]
For $n > 0$, we call the \textit{prioritised parent} of a simplex $s \in K_n$ the minimal simplex in $K_{n-1}$ containing~$f_n(s)$ (which will be denoted by $s^\uparrow$). A Simplex $s$ is a \textit{prioritised child} of a simplex $s'$ if $s' = s^\uparrow$.
\end{df}
\begin{fakt}[cf.~Lemma~\ref{fakt-sm-kulowy-wyznacza-dzieci}, and also Lemma~\ref{fakt-przesuniecie-istnieje}]
\label{fakt-sk-opis-przesuwanie-dzieci}
Let $s \in K_n$, $s' \in K_{n'}$ and $\gamma \in G$ be such that $s' = \gamma \cdot s$ (in the sense of Definition \ref{def-przesuniecie-sympleksu}). Then, the translation by $\gamma$ gives a bijection between prioritised children of $s$ and prioritised children of $s'$.
\end{fakt}
\begin{proof}
This is an easy corollary of Proposition \ref{lem-przesuwanie-dzieci-sympleksow} (and Lemma \ref{fakt-przesuniecie-skladane}). The proposition ensures that the translation by $\gamma$ maps simplexes contained in $f_n^{-1}(s)$ to simplexes contained in $f_{n'}^{-1}(s')$. Moreover, if $\sigma \subseteq s$ and $\gamma \cdot \sigma$ is not a prioritised child of $s'$, then we have $\gamma \cdot \sigma \subseteq f_{n'}^{-1}(s'')$ for some $s'' \subsetneq s'$ and then $\sigma \subseteq f_n^{-1}(\gamma^{-1} \cdot s'')$, which means that $\sigma$ is not a prioritised child of $s$. The reasoning in the opposite direction is analogous because $s = \gamma^{-1} \cdot s'$.
\end{proof}
Let $\mathcal{T}$ be the set of values of $T^\Delta$. For every $\tau \in \mathcal{T}$, choose some \textit{representative} of this type $s_\tau \in K_{n_\tau}$. For any simplex $s \in K_n$ of type~$\tau$, let $\gamma_\tau \in G$ be the unique element such that $\gamma_s \cdot s = s_\tau$ (the uniqueness follows from Lemma \ref{fakt-sk-opis-przesuniecie-jedyne}).
\begin{df}[cf. Definitions~\ref{def-sm-numer-dzieciecy} and~\ref{def-sm-typ-A}]
\label{def-sk-opis-typ-delta-a}
The $T^{\Delta + A}$-type of a simplex $s \in K_n$ is defined by the formula
\begin{align}
\label{eq-sk-opis-typ-delta-a}
T^{\Delta + A}(s) = \big( T^\Delta(s), \ T^\Delta(s^\uparrow), \ \gamma_{s^\uparrow} \cdot s \big).
\end{align}
\end{df}
Note that, by Proposition \ref{lem-przesuwanie-dzieci-sympleksow}, the translation $\gamma_{s^\uparrow} \cdot s$ (which plays an analogous role to the descendant number) exists and it is one of the simplexes in the pre-image of $f_{n_\tau}^{-1}(s_\tau)$, where $\tau = T^\Delta(s^\uparrow)$. This ensures the correctness of the above definition, as well as finiteness of the resulting type $T^{\Delta + A}$.
Also, note that the component $T^\Delta(s^\uparrow)$ in the formula~\eqref{eq-sk-opis-typ-delta-a} has no equivalent in Definition \ref{def-sm-typ-A}. It will be used in the proof of Lemma \ref{fakt-sk-opis-bracia}.
\begin{fakt}
\label{fakt-sk-opis-markow}
The inverse system $(K_n, f_n)$, equipped with the simplex type function $T^{\Delta + A}$, satisfies the conditions from Definition \ref{def-kompakt-markowa}.
\end{fakt}
\begin{proof}
Since the system $(K_n)$ has the Markov property when equipped a type function $T^\Delta$, weaker than $T^{\Delta + A}$, it suffices to check the condition (iii). To achieve this --- by Proposition \ref{lem-przesuwanie-dzieci-sympleksow} --- we need only to check that, if for some $s \in K_n$, $s' \in K_{n'}$, $\gamma \in G$ the equality $s' = \gamma \cdot s$ holds and $s$, $s'$ have the same $T^{\Delta + A}$-type, then the translation by $\gamma$ preserves the values of $T^{\Delta + A}$ for all simplexes contained in $f_n^{-1}(s)$.
Let then $\sigma$ be any such simplex. Then, $\sigma^\uparrow \subseteq s$, so from the claim of Proposition \ref{lem-przesuwanie-dzieci-sympleksow} (more precisely: from the fact that the translation by $\gamma$ preserves $T^\Delta$ and that it commutes with $f_n$ and~$f_{n'}$) we deduce that:
\begin{align}
\label{eq-sk-opis-delta-typy-zgodne}
T^\Delta \big( (\gamma \cdot \sigma)^\uparrow \big) = T^\Delta \big( \gamma \cdot (\sigma^\uparrow) \big) = T^\Delta( \sigma^\uparrow ).
\end{align}
Moreover, it is clear that $T^\Delta(\sigma) = T^\Delta(\gamma \cdot \sigma)$, so it only remains to verify the equality of the last coordinates in the types $T^{\Delta + A}(\sigma)$, $T^{\Delta + A}(\gamma \cdot \sigma)$.
Denote by $\tau$ the formula~\eqref{eq-sk-opis-delta-typy-zgodne}. Then, using Lemma \ref{fakt-przesuniecie-skladane}, we have
\[ \gamma_{\sigma^\uparrow} \cdot \sigma^\uparrow = s_\tau = \gamma_{(\gamma \cdot \sigma)^\uparrow} \cdot (\gamma \cdot \sigma)^\uparrow = \gamma_{(\gamma \cdot \sigma)^\uparrow} \cdot \big( \gamma \cdot (\sigma)^\uparrow \big) = ( \gamma_{(\gamma \cdot \sigma)^\uparrow} \, \gamma ) \cdot \sigma^\uparrow, \]
so, by Lemma \ref{fakt-sk-opis-przesuniecie-jedyne}, we obtain $\gamma_{\sigma^\uparrow} = \gamma_{(\gamma \cdot \sigma)^\uparrow} \, \gamma$. This in turn implies that:
\[ \gamma_{\sigma^\uparrow} \cdot \sigma = ( \gamma_{(\gamma \cdot \sigma)^\uparrow} \, \gamma ) \cdot \sigma = \gamma_{(\gamma \cdot \sigma)^\uparrow} \cdot (\gamma \cdot \sigma), \]
which finishes the proof.
\end{proof}
\begin{fakt}
\label{fakt-sk-opis-bracia}
For any simplex $s \in K_n$, all simplexes in the pre-image $f_n^{-1}(s)$ have pairwise distinct $T^{\Delta + A}$-types.
\end{fakt}
\begin{proof}
Let $\sigma, \sigma' \in f_n^{-1}(s)$ satisfy $T^{\Delta + A}$. Then in particular $T^\Delta(\sigma^\uparrow) = T^\Delta({\sigma'}^\uparrow)$, and since $\sigma^\uparrow$, ${\sigma'}^\uparrow$ are subsimplexes of $s$, from Lemma \ref{fakt-sk-opis-podsympleksy-rozne-typy} we obtain that they are equal. Then, the equality of the third coordinates in types $T^{\Delta + A}(\sigma)$, $T^{\Delta + A}(\sigma')$ implies that $\sigma = \sigma'$.
\end{proof}
\section{Markov systems with limited dimension}
\label{sec-wymd}
In this section, we assume that $\dim \partial G \leq k < \infty$, and we discuss how to adjust the construction of a Markov system to ensure that all the complexes in the inverse system also have dimension $\leq k$. Since $\partial G$ is a compact metric space, its dimension can be understood as the covering dimension, or equivalently as the small inductive dimension (cf.~\cite[Theorem~1.7.7]{E}). In the sequel, we denote the space $\partial G$ by~$X$, and the symbol $\partial$ will always mean the topological frontier taken in~$X$ or in some its subset.
The main result of this section is given below.
\begin{lem}
\label{lem-wym}
Let $k \geq 0$ and let $G$ by a hyperbolic group such that $\dim \partial G \leq k$. Then, there exists a quasi-$G$-invariant system of covers of~$\partial G$ of rank~$\leq k + 1$.
\end{lem}
Since the rank of a cover determines the dimension of its nerve, this result will indeed allow to limit the dimension of the complexes in the Markov system for~$\partial G$ (see Section~\ref{sec-wymd-podsum}).
\begin{uwaga}
Although the proof of Proposition~\ref{lem-wym} given below will involve many technical details, let us underline that --- in its basic sketch --- it resembles an elementary result from dimension theory stating that every open cover $\mathcal{U} = \{ U_i \}_{i = 1}^n$ of a compact metric space~$X$ of dimension~$k$ contains an open subcover of rank~$\leq k + 1$. Below, we present the main steps of a proof of this fact, and pointing out the analogies between these steps and the contents of the rest of this section.
\begin{itemize}
\item[(i)] We proceed by induction on $k$. For convenience, we work with a slightly stronger inductive claim: the cover~$\{ U_i \}$ contains an open subcover~$\{ V_j \}$ such that the closures $\overline{V_j}$ form a family of rank~$\leq k + 1$.
(For the proof of Proposition~\ref{lem-wym}, the inductive reasoning is sketched in more detail in Proposition~\ref{lem-wym-cala-historia}).
\item[(ii)] Using the auxiliary Theorem~\ref{tw-wym-przedzialek} stated below, we choose in each $U_i \in \mathcal{U}$ an open subset $U_i'$ with frontier of dimension $\leq k-1$ so that the sets~$U_i'$ still form a cover of~$X$.
(Similarly we will define the sets~$D_x$ in the proof of Proposition~\ref{lem-wym-bzdziagwy}).
\item[(iii)] We define the sets $U_i''$ by the condition:
\[ x \in U_i'' \qquad \qquad \Longleftrightarrow \qquad \qquad x \in U_i' \quad \textrm{ and } \quad x \notin U'_j \quad \textrm{ for } j < i. \]
(Analogously we will define the sets~$E_x$ in the proof of Proposition~\ref{lem-wym-bzdziagwy})
\item[(iv)] The space~$X$ is now covered by the interiors of the sets $U_i''$ (which are pairwise disjoint) together with the set~$\widetilde{X} = \bigcup_i \partial U_i''$, which is a closed subset of~$X$ of dimension~$\leq k - 1$. In~$\widetilde{X}$, we consider an open cover formed by the sets $\widetilde{U}_i = U_i \cap \widetilde{X}$. By the inductive hypothesis, this cover must contain an open subcover formed by some sets~$\widetilde{V}_j$ ($1 \leq j \leq m$) whose closures form a family of rank~$\leq k$. Also, we may require that $\widetilde{V}_j$ is open in~$\widetilde{X}$ --- but not necessarily in~$X$.
(The sets $\partial U_i''$, $\innt U_i''$ correspond to the sets $F_x$, $G_x$ appearing in the formulation of Proposition~\ref{lem-wym-bzdziagwy}).
\item[(v)] Let $\varepsilon > 0$ be the least distance between any \textit{disjoint pair} of closures $\overline{\widetilde{V}_{j_1}}$, $\overline{\widetilde{V}_{j_2}}$. For $1 \leq j \leq m$, we define $V_j$ as the $\tfrac{\varepsilon}{4}$-neighbourhood (in~$X$) of~$\widetilde{V}_j$. Then, it is easy to verify that the sets $V_j$ are open in~$X$ and cover~$\widetilde{X}$, and moreover the rank of the family $\{\overline{ V_j }\}$ does not exceed the rank of $\{\overline{ \widetilde{V}_j }\}$ which is $\leq k$.
Let $U_i'''$ denote $\innt U_i''$ minus the closed $\tfrac{\varepsilon}{8}$-neighbourhood of $\widetilde{X}$.
Then, the family
\[ \mathcal{V} = \{ V_j \}_{j = 1}^m \cup \{ U_i''' \}_{i = 1}^n \]
is an open cover of~$X$. Moreover, the rank of the family of closures of all elements of $\mathcal{V}$ is at most the sum of ranks of the families $\{ \overline{ V_j } \}$ and $\{ \overline{ U_i''' } \}$, which are respectively $k$ and $1$ (the latter because for every $i \neq i'$ we have $\overline{U_i'''} \cap \overline{U_{i'}'''} \subseteq \overline{U_i''} \cap \overline{U_{i'}''} \subseteq \widetilde{X}$ which is disjoint from both $\overline{U_i'''}$ and~$\overline{U_{i'}'''}$). Hence, $\mathcal{V}$ satisfies all the desired conditions.
(In our proof of Proposition~\ref{lem-wym}, the construction of appropriate neighbourhoods takes place in Proposition~\ref{lem-wym-kolnierzyki}, and the other of the above steps have their counterparts in the proof of Proposition~\ref{lem-wym-cala-historia}).
\end{itemize}
In comparison to the above reasoning, the main difficulty in proving Proposition~\ref{lem-wym} lies in ensuring quasi-$G$-invariance of the adjusted covers, which we need for preserving the Markov property for the system of their nerves (using Theorem~\ref{tw-kompakt-ogolnie}). For this, instead of defining each of the sets $U_i'$ independently, we will first choose a finite number of \textit{model sets}, one for each possible value of type in~$G$, and translate these model sets using Proposition~\ref{lem-potomkowie-dla-kulowych}. The inductive argument will now require special care for preserving quasi-$G$-invariance; nevertheless; the main idea remains unchanged.
\end{uwaga}
We will use the following auxiliary result from dimension theory:
\begin{tw}[{\cite[Theorem~1.5.12]{E}}]
\label{tw-wym-przedzialek}
Let~$Y$ be a separable metric space of dimension~$k$ and $A, B$ be disjoint closed subsets of~$Y$. Then, there exist open subsets $\widetilde{A}, \widetilde{B} \subseteq Y$ such that
\[ A \subseteq \widetilde{A}, \qquad B \subseteq \widetilde{B}, \qquad \widetilde{A} \cap \widetilde{B} = \emptyset \qquad \textrm{ and } \qquad \dim \big(Y \setminus (\widetilde{A} \cup \widetilde{B})\big) \leq k - 1. \]
\end{tw}
\subsection{$\theta$-weakly invariant subsystems}
\begin{ozn}
For any two families $\mathcal{C} = \{ C_x \}_{x \in G}$, $\mathcal{D} = \{ D_x \}_{x \in G}$, we denote:
\[ \mathcal{C} \sqcup \mathcal{D} = \{ C_x \cup D_x \}_{x \in G}. \]
\end{ozn}
\begin{df}
A system $\mathcal{C} = \{ C_x \}_{x \in G}$ of subsets of~$X$ will be called:
\begin{itemize}
\item \textit{of dimension $\leq k$} if $|\mathcal{C}|_n$ is of dimension $\leq k$ for all $n \geq 0$;
\item \textit{of rank $\leq k$} if, for every $n \geq 0$, the family $\mathcal{C}_n$ is of rank $\leq k$ (i.e. if the intersection of any $k+1$ pairwise distinct members of $\mathcal{C}_n$ must be empty); in particular, \textit{disjoint} if it is of rank~$\leq 1$.
\end{itemize}
\end{df}
\begin{fakt}
\label{fakt-wym-suma-pokryc}
If two systems of subsets $\mathcal{C}$, $\mathcal{D}$ are correspondingly of rank~$\leq a$ and~$\leq b$, then $\mathcal{C} \sqcup \mathcal{D}$ is of rank $\leq a+b$. \qed
\end{fakt}
\begin{df}
Let $X$ be a topological space and $A, B \subseteq X$. We will say that~$A$ is a \textit{separated} subset of~$B$ (denotation: $A \Subset B$) if $\overline{A} \subseteq \innt B$.
\end{df}
\begin{df}
Let $\mathcal{C} = \{ C_x \}$, $\mathcal{D} = \{ D_x \}$ be two quasi-$G$-invariant systems of subsets of~$X$. We will say that:
\begin{itemize}
\item $\mathcal{C}$ is an \textit{(open, closed) subsystem} in~$\mathcal{D}$ if, for every $n \geq 0$ and $C_x \in \mathcal{C}_n$, $C_x$ is an (open, closed) subset in~$|\mathcal{D}|_n$ \\ (recall from \ref{ozn-suma-rodz} that $|\mathcal{D}|_n$ denotes $\bigcup_{C \in \{C_x \,|\, x \in G, \, |x|=n\}} C$ )
\item $\mathcal{C}$ is a \textit{semi-closed subsystem} in~$\mathcal{D}$ if $|\mathcal{C}|_n$ is a closed subset in~$|\mathcal{D}|_n$ for $n \geq 0$;
\item $\mathcal{C}$ \textit{covers} $\mathcal{D}$ if $|\mathcal{C}|_n \supseteq |\mathcal{D}|_n$ for $n \geq 0$.
\end{itemize}
A system $\mathcal{C}$ will be called \textit{semi-closed} if $|\mathcal{C}|_n$ is a closed subset in~$X$ for $n \geq 0$.
\end{df}
\begin{df}
For any integer $\theta \geq 0$, we define the type function $T^B_\theta$ in~$G$ as the extension (in the sense of Definition~\ref{def-sm-typ-plus}) of the type function $T^B$ (defined in Definition~\ref{def-sm-typ-B}) by~$r = \theta \cdot 12\delta$:
\[ T^B_\theta = (T^B)^{+ \theta \cdot 12\delta} \]
\end{df}
\begin{fakt}
\label{fakt-wym-poglebianie-B+}
Let $g, x, y \in G$ and~$\theta \geq 0$, $k > 0$ satisfy
\[ y \in xT^c(x), \qquad T^B_\theta(x) = T^B_\theta(gx), \qquad |y| = |x| + kL, \]
where $L$ denotes the constant from Section~\ref{sec-sm} (defined in Section~\ref{sec-sm-nici}).
Then:
\[ T^B_{\theta + 1}(y) = T^B_{\theta + 1}(gy), \qquad |gy| = |gx| + kL. \]
\end{fakt}
\begin{proof}
It suffices to prove the claim for $k = 1$; for greater values of~$k$, it will then easily follow by induction.
Let $z \in yP_{(\theta + 1) \cdot 12\delta}(y)$. Denote $w = z^\Uparrow$ (see Definition~\ref{def-sm-p-wnuk}). Since $L \geq 14\delta$, Lemma~\ref{fakt-geodezyjne-pozostaja-bliskie} implies that
\[ d(x, w) \leq \max \big( (\theta+1) \cdot 12\delta + 16\delta - 2L, \ 8\delta \big) \leq \theta \cdot 12\delta, \]
so $w \in xP_{12\delta}(x)$, and then $T^B(w) = T^B(gw)$ and $|gx| = |gw|$. Therefore, by Proposition~\ref{lem-sm-B-dzieci} we know that $gw = (gz)^\Uparrow$ and $T^B(z) = T^B(gz)$. The first of these equalities implies in particular that $gz$ is a descendant of $gw$, that is,
\[ |gz| = |gw| + d(gz, gw) = |gx| + d(z, w) = |gx| + L. \]
in particular, setting $z = y$ we obtain that $|gy| = |y| + L$. Considering again an arbitrary~$z$, we deduce that $|gz| = |gy|$, so $P_{(\theta+1) \cdot 12\delta}(y) \subseteq P_{(\theta+1) \cdot 12\delta}(gy)$; the opposite inclusion can be proved analogously (by exchanging the roles between $g$ and $g^{-1}$). In this situation, the equality $T^B(z) = T^B(gz)$ for an arbitrary~$z$ implies that $T^B_{\theta+1}(y) = T^B_{\theta+1}(gy)$.
\end{proof}
\begin{df}
A family of subsets $\mathcal{C} = \{ C_x \}$ will be called a \textit{$\theta$-weakly invariant system} if:
\begin{itemize}
\item for every $x \in G$, $C_x$ is a subset of the set $S_x$ defined in Section~\ref{sec-konstr-pokrycia};
\item the family $\mathcal{C}$, together with the type function $T^B_\theta$, satisfies the condition~\qhlink{f1} of Definition~\ref{def-quasi-niezm}.
\end{itemize}
\end{df}
Although we only require the condition~\qhlink{f1} to hold, we will show in Proposition~\ref{lem-wym-slabe-sa-qi} that this suffices to force the other conditions of Definition~\ref{def-quasi-niezm} to hold under some appropriate assumptions.
Let us observe that the system $\mathcal{S}$ described in Section~\ref{sec-konstr-pokrycia} is $0$-weakly (and then also $\theta$-weakly for $\theta \geq 0$) invariant. This is because the values of $T^B_\theta$ determine uniquely the values of $T^b_N$ (by Remark~\ref{uwaga-sm-B-wyznacza-A} and Definition~\ref{def-sm-typ-A}), while the system $\mathcal{S}$ equipped with the latter type function is quasi-$G$-invariant by Corollary~\ref{wn-spanstary-quasi-niezm}.
\begin{fakt}
\label{fakt-wym-typ-wyznacza-sasiadow}
Let $\mathcal{C} = \{ C_x \}_{x \in G}$ be a $\Theta$-weakly invariant system for some $\Theta \geq 0$. Let $n \geq 0$, $\theta \geq 1$ and $x, y \in G$ be of length~$n$, and suppose that $C_x \cap C_y \neq \emptyset$. Then:
\begin{itemize}
\item[\textbf{(a)}] $T^B_\theta(x) \neq T^B_\theta(y)$;
\item[\textbf{(b)}] For every $g \in G$, the equality $T^B_\theta(x) = T^B_\theta(gx)$ implies that $T^B_{\theta-1}(y) = T^B_{\theta-1}(gy)$ and $|gx| = |gy|$.
\end{itemize}
\end{fakt}
\begin{proof}
Since $\emptyset \neq C_x \cap C_y \subseteq S_x \cap S_y$, it follows from Lemma~\ref{fakt-sasiedzi-blisko} that $d(x, y) \leq 12\delta$. Then, Proposition~\ref{lem-sm-kuzyni} implies that $T^B(x) \neq T^B(y)$, so $T^B_\theta(x) \neq T^B_\theta(y)$. Moreover, since $|x| = |y|$, we have $x^{-1} y \in P_{\theta \cdot 12\delta}(x) = P_{\theta \cdot 12\delta}(gx)$, and so $|gx| = |gy|$.
Note that the triangle inequality gives:
\[ P_{(\theta-1) \cdot 12\delta}(y) = \big( y^{-1} x \, P_{\theta \cdot 12\delta}(x) \big) \cap B \big( e, (\theta - 1) \cdot 12\delta \big) \]
and analogously for~$gx, gy$, which shows that $P_{(\theta-1) \cdot 12\delta}(y) = P_{(\theta-1)\cdot 12\delta}(gy)$. Moreover, for any $z \in yP_{(\theta-1) \cdot 12\delta}(y)$ the above equality implies that $x^{-1}z \in P_{\theta \cdot 12\delta}(x) = P_{\theta \cdot 12\delta}(gx)$, and so $T^B(z) = T^B(gz)$. Since~$z$ was arbitrary, we deduce that $T^B_{\theta-1}(y) = T^B_{\theta-1}(gy)$.
\end{proof}
\begin{lem}
\label{lem-wym-slabe-sa-qi}
Let $\theta \geq 0$ and $\mathcal{C}$ be a $\theta$-weakly invariant system of open covers of~$X$. Then, $\mathcal{C}$ is quasi-$G$-invariant when equipped with the type function~$T^B_{\theta+1}$.
\end{lem}
\begin{proof}
Let $\mathcal{C} = \{ C_x \}_{x \in G}$. The conditions~\qhlink{c}, \qhlink{d} for $\mathcal{C}$ follow directly from the same conditions for~$\mathcal{S}$. Also, \qhlink{f2} can be easily translated: whenever we have
\[ T^B_{\theta+1}(x) = T^B_{\theta+1}(gx), \qquad |x| = |y|, \qquad C_x \cap C_y \neq \emptyset, \]
then by Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow}b it follows that $|gx| = |gy|$ and $T^B_{\theta}(y) = T^B_{\theta}(gy)$, so --- by the property~\qhlink{f1} for~$\mathcal{C}$ --- we have $C_{gy} = g \cdot C_y$.
We will now verify the property \qhlink{f3}.
Let $L$ denote the constant coming from Lemma~\ref{fakt-wym-poglebianie-B+}.
Suppose that
\[ T^B_{\theta+1}(x) = T^B_{\theta+1}(gx), \qquad |y| = |x| + L, \qquad \emptyset \neq C_y \subseteq C_x. \]
Let $\alpha$ be a geodesic joining~$e$ with~$y$ and let $z = \alpha(|x|)$. Then, by Lemma~\ref{fakt-wlasnosc-gwiazdy-bez-gwiazdy}c we have $C_y \subseteq S_y \subseteq S_z$; on the other hand, $C_y \subseteq S_x$, so $S_x \cap S_z \neq \emptyset$, and then by Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow} we have
\[ T^B_\theta(z) = T^B_\theta(gz), \qquad |gx| = |gz|. \]
Since $y \in zT^c(z)$ and $|y| = |x| + L = |z| + L$, using Lemma~\ref{fakt-wym-poglebianie-B+} and then the property~\qhlink{f1} for~$\mathcal{C}$ we obtain that
\[ T^B_{\theta+1}(y) = T^B_{\theta+1}(gy), \qquad |gy| = |gz| + L = |gx| + L, \qquad C_{gy} = g \cdot C_y. \]
By Remark~\ref{uwaga-quasi-niezm-jeden-skok}, this means that $\mathcal{C}$ satisfies~\qhlink{f3} for the jump constant~$L$.
It remains to verify the property \qhlink{e}. Let $\mathcal{T}$ be the set of all possible values of~$T^B_{\theta + 1}$. Choose $N > 0$ such that, for every $\tau \in \mathcal{T}$, there is $x \in G$ of length less than~$N$ such that $T^B_{\theta + 1}(x) = \tau$. Let $\varepsilon > 0$ be the minimum of all Lebesgue numbers for the covers $\mathcal{C}_1, \ldots, \mathcal{C}_N$, and choose $J' > N$ so that $\max_{S \in \mathcal{S}_n} \diam S_n < \varepsilon$ for every $n \geq J'$. We will show that the system~$\mathcal{C}$ together with $J'$ satisfies~\qhlink{e}. By Remark~\ref{uwaga-quasi-niezm-jeden-skok}, it suffices to verify this in the case when $k = 1$.
Let $x \in G$ satisfy $|x| \geq J'$. By Lemma~\ref{fakt-wlasnosc-gwiazdy-bez-gwiazdy}, we have $S_x \subseteq S_y$ for some $y \in G$ of length~$|x| - J'$. Let $y' \in G$ be an element such that $T^B_{\theta+1}(y') = T^B_{\theta+1}(y)$ and $|y'| < N$. Denote $\gamma = y'y^{-1}$ and $x' = \gamma x$; then, by~\qhlink{f3}, we have
\[ S_{x'} = \gamma \cdot S_x, \qquad T^B_{\theta+1}(x') = T^B_{\theta+1}(x), \qquad |x'| \, = \, |y'| + |x| - |y| \, = \, |y'| + J'. \]
The last equality implies that $\diam S_{x'} < \varepsilon$, which means (since $|y'| < N$) that $S_{x'}$ must be contained in~$C_{z'}$ for some $z' \in G$ such that $|z'| = |y'|$. Denote $z = \gamma^{-1} z'$. Then, using the property~\qhlink{f2} and then Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow}b, we obtain:
\[ S_z = \gamma^{-1} \cdot S_{z'}, \qquad T^B_\theta(z) = T^B_\theta(z'), \qquad |z| = |y| = |x| - J', \]
and so, since $\mathcal{C}$ is $\theta$-weakly invariant, it follows that:
\[ C_x \subseteq S_x = \gamma^{-1} \cdot S_{x'} \subseteq \gamma^{-1} \cdot C_{z'} = C_z. \]
Altogether, we obtain that $\mathcal{C}$ is quasi-$G$-invariant with $\gcd(L, J')$ as the jump constant.
\end{proof}
\begin{fakt}
\label{fakt-wym-homeo-na-calej-kuli}
Let $\theta \geq 0$, $\mathcal{C}$ be a $\theta$-weakly invariant system, and $x, y \in G$ with $T^B_{\theta + 1}(x) = T^B_{\theta + 1}(y)$. Then, we have
\[ |\mathcal{C}|_{|y|} \cap S_y = yx^{-1} \cdot \big( |\mathcal{C}|_{|x|} \cap S_x \big). \]
In particular, the left translation by $yx^{-1}$, when applied to separated subsets of~$S_x$, preserves the interiors and closures taken in $|\mathcal{C}|_{|x|}$ and respectively $|\mathcal{C}|_{|y|}$.
\end{fakt}
\begin{proof}
Denote $n = |x|$, $m = |y|$ and $\gamma = yx^{-1}$.
By symmetry, it suffices to show one inclusion. Let~$p \in |\mathcal{C}|_n \cap S_x$; then~$p$ belongs to some $C_{x'} \in \mathcal{C}_n$. In particular, we have:
\[ p \ \in \ S_x \cap C_{x'} \ \subseteq \ S_x \cap S_{x'}. \]
Denote $y' = \gamma x'$.
Then, we have $T^B_\theta(y') = T^B_\theta(x')$ by Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow}b, so~\qhlink{f1} implies that $C_{y'} = \gamma \cdot C_{x'}$, as well as $S_y = \gamma \cdot S_x$. Therefore,
\[ \gamma \cdot p \ \in \ \gamma \cdot (C_{x'} \cap S_x) \ = \ C_{y'} \cap S_y \ \subseteq \ |\mathcal{C}|_m \cap S_y. \qedhere \]
\end{proof}
\subsection{Disjoint, weakly invariant nearly-covers}
\label{sec-wym-wyscig}
Before proceeding with the construction, we will introduce notations and conventions used below.
In the proofs of Propositions~\ref{lem-wym-bzdziagwy} and \ref{lem-wym-kolnierzyki}, we will use the following notations, dependent on the value of a~parameter~$\theta$ (which is a~part of the input data in both propositions). Let $\tau_1, \ldots, \tau_K$ be an enumeration of all possible $T^B_{\theta+1}$-types. For simplicity, we identify the value $\tau_i$ with the natural number~$i$. For every~$1 \leq i \leq K$, we fix an arbitrary $x_i \in G$ such that $T^B_{\theta+1}(x_i) = i$, and we set $S_i = _S{x_i}$, $n_i = |x_i|$.
Similarly, let $1, \ldots, \widetilde{K}$ be an enumeration of all possible~$T^B_{\theta+2}$-types, and for every $1 \leq \widetilde{\imath} \leq \widetilde{K}$ let $\widetilde{x}_{\widetilde{\imath}} \in G$ be a~fixed element such that $T^B_{\theta+2}(\widetilde{x}_{\widetilde{\imath}}) = \widetilde{\imath}$. We denote also $M = \max_{{\widetilde{\imath}}=1}^{\widetilde{K}} |\widetilde{x}_{\widetilde{\imath}}|$.
In the remaining part of Section~\ref{sec-wymd}, we will usually consider sub-systems of a given semi-closed system in~$X$ (which is given the name~$\mathcal{C}$ in Propositions~\ref{lem-wym-bzdziagwy} and~\ref{lem-wym-kolnierzyki}), and more generally --- subsets of~$X$ known to be contained in~$|\mathcal{C}|_n$ for some~$n$ (known from the context). Unless explicitly stated otherwise, the basic topological operators for such sets (closure, interior, frontier) will be performed within the space~$|\mathcal{C}|_n$ (for the appropriate value of~$n$). This will not influence closures, since~$|\mathcal{C}|_n$ is a closed subset of~$X$, but will matter for interiors and frontiers.
This subsection contains the proof of the following result.
\begin{lem}
\label{lem-wym-bzdziagwy}
Let $k \geq 0$, $\theta \geq 1$ and $\mathcal{C} = \{ C_x \}$ be a semi-closed, $\theta$-weakly invariant system of dimension $\leq k$ in~$X$. Then, there exist:
\begin{itemize}
\item a disjoint, open, $(\theta + 2)$-weakly invariant subsystem $\mathcal{G} = \{ G_x \}$ in $\mathcal{C}$;
\item a closed, $(\theta + 2)$-weakly invariant subsystem $\mathcal{F} = \{ F_x \}$ in $\mathcal{C}$ of dimension $\leq k - 1$
\end{itemize}
such that $\mathcal{F} \sqcup \mathcal{G}$ covers $\mathcal{C}$ and $\partial G_x \subseteq |\mathcal{F}|_n$ for every $n \geq 0$ and~$G_x \in \mathcal{G}_n$.
\end{lem}
\begin{proof}
Let $\varepsilon > 0$ be the minimum of all Lebesgue numbers for the covers $\mathcal{S}_1, \ldots, \mathcal{S}_M$. Define:
\[ I_x = \{ p \in S_x \,|\, B(p, \varepsilon) \subseteq S_x \} \qquad \qquad \textrm{ for } x \in G, \, |x| \leq M \]
and
\begin{align*}
G_i = \bigcup_{|x| \leq M, \, T^B_{\theta+1}(x) = i} x_i x^{-1} \cdot I_x, \qquad H_i = X \setminus S_i \qquad \qquad \textrm{ for } 1 \leq i \leq K.
\end{align*}
Fix some $1 \leq i \leq K$. We observe that, for every~$x \in G$, we have $d(I_x, X \setminus S_x) \geq \varepsilon$, which implies that~$G_i$ is a finite union of separated subsets of~$S_i$; then,
$\overline{G_i} \cap \overline{H_i} = \emptyset$ (in~$X$). Then, the intersections $\overline{G_i} \cap |\mathcal{C}|_{n_i}$, $\overline{H_i} \cap |\mathcal{C}|_{n_i}$ are disjoint closed subsets of~$|\mathcal{C}|_{n_i}$, so by Theorem~\ref{tw-wym-przedzialek} there are open subsets of~$|\mathcal{C}|_{n_i}$:
\[ \widetilde{G}_i \supseteq \overline{G_i} \cap |\mathcal{C}|_{n_i}, \qquad \widetilde{H}_i \supseteq \overline{H_i} \cap |\mathcal{C}|_{n_i}, \qquad \qquad (\textrm{ closures of } G_i, H_i \textrm{ taken in } X) \]
which cover $|\mathcal{C}|_{n_i}$ except for some subset of dimension~$\leq k - 1$ (which must contain $\partial \widetilde{G}_i$).
For any $x \in G$, we denote $|x| = n$ and $T^B_{\theta+1}(x) = i$, and then we define:
\begin{align*}
D_x & = x x_i^{-1} \cdot \widetilde{G}_i, \\
E_x & = D_x \setminus \, \bigcup_{y \in G, \, |y| = n, \, T^B_{\theta+1}(y) < T^B_{\theta+1}(x)} D_y, \\
F_x & = \partial E_x, \\
G_x & = \intr E_x.
\end{align*}
Note that $\partial G_x = \overline{G_x} \setminus G_x \subseteq \overline{E_x} \setminus \intr E_x = \partial E_x = F_x$, as desired in the claim.
The remaining part of the proof consists of verifying the following claims (of which \textbf{(a)} and \textbf{(b)} are auxiliary):
\begin{itemize}[nolistsep]
\item[\textbf{(a)}] $\mathcal{D} = \{ D_x \}_{x \in G}$ covers~$\mathcal{C}$;
\item[\textbf{(b)}] $\mathcal{E} = \{ E_x \}_{x \in G}$ is disjoint and covers~$\mathcal{C}$;
\item[\textbf{(c)}] $\mathcal{F} = \{ F_x \}_{x \in G}$ is of dimension~$\leq k - 1$;
\item[\textbf{(d)}] $\mathcal{G} = \{ G_x \}_{x \in G}$ is disjoint;
\item[\textbf{(e)}] $F_x \subseteq S_x$ for every~$x \in G$;
\item[\textbf{(f)}] $\mathcal{F} \sqcup \mathcal{G}$ covers~$\mathcal{C}$;
\item[\textbf{(g)}] $\mathcal{F}$ and~$\mathcal{G}$ are $(\theta+2)$-weakly invariant.
\end{itemize}
\textbf{(a)}
To verify that $\mathcal{D}$ covers $\mathcal{C}$, choose an arbitrary $p \in |\mathcal{C}|_n$; then $p$ lies in some $S_x \in \mathcal{S}_n$. Denote
\[ \widetilde{\imath} = T^B_{\theta + 2}(x), \qquad \gamma = x\widetilde{x}_{\widetilde{\imath}}^{-1}, \qquad p' = \gamma^{-1} \cdot p. \]
Then, $p' \in S_{\widetilde{x}_{\widetilde{\imath}}}$. By the definition of $\varepsilon$, we have $B(p', \varepsilon) \subseteq S_y$ for some $S_y \in \mathcal{S}_{|\widetilde{x}_{\widetilde{\imath}}|}$; then $p'$ lies in~$I_y$. Now, let
\[
j = T^B_{\theta+1}(y), \qquad \beta = yx_j^{-1}, \qquad p'' = \beta^{-1} \cdot p'. \]
By definition, $p''$ lies in $G_j$; moreover, by applying Lemma~\ref{fakt-wym-homeo-na-calej-kuli} twice we obtain that $p'' \in |\mathcal{C}|_{|x_j|}$. On the other hand, since $S_y$ intersects non-trivially with $S_{x_i}$ and $T^B_{\theta+2}(\gamma x_i) = T^B_{\theta+2}(x_i)$, by~\qhlink{f2} and Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow}b we have:
\[ S_{\gamma \beta x_j} \ = \ S_{\gamma y} \ = \ \gamma \cdot S_y \ \ni \ \gamma \cdot p' \ = \ p, \qquad T^B_{\theta+1}(\gamma \beta x_j) = T^B_{\theta+1}(\gamma y) = T^B_{\theta+1}(y) = j, \]
which implies that
\[ p \ = \ \gamma \beta \cdot p'' \ \in \ \gamma \beta \cdot (G_j \cap |\mathcal{C}|_{|x_j|}) \ \subseteq \ \gamma \beta \cdot \widetilde{G}_j \ \subseteq \ D_{\gamma \beta x_j}. \]
\textbf{(b)}
Suppose that $p \in E_x \cap E_y$ for some $x \neq y$. Then, $S_x \cap S_y \neq \emptyset$, so by Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow}a we have $T^B_{\theta+1}(x) \neq T^B_{\theta+1}(y)$; assume w.l.o.g. that $T^B_{\theta+1}(x)$ is the smaller one. Then, by definition, $p \in E_x \subseteq D_x$ cannot belong to $E_y$. This proves that $\mathcal{E}$ is disjoint.
Now, let~$p \in |\mathcal{C}|_n$ and let $x \in G$ of length~$n$ be chosen so that $p \in D_x$ and $T^B_{\theta+1}(x)$ is the lowest possible. Then, by definition, $p \in E_x$. This means that $\mathcal{E}$ covers $\mathcal{C}$.
\textbf{(c)}
First, note that, if $x \in G$ and $T^B_{\theta+1}(x) = i$, then, by Lemma~\ref{fakt-wym-homeo-na-calej-kuli} and the definitions of~$D_x$ and~$\widetilde{G}_i$, we have:
\[ \dim \partial D_x = \dim \partial \big( xx_i^{-1} \cdot \widetilde{G}_i \big) = \dim \big( xx_i^{-1} \cdot \partial \widetilde{G}_i \big) = \dim \partial \widetilde{G}_i \leq k - 1. \]
Fix some $n \geq 0$. Recall that, for any subsets $Y, Z$ in any topological space, we have $\partial(Y \setminus Z) \subseteq \partial Y \cup \partial Z$. By applying this fact finitely many times in the definition of every $E_x$ with $|x| = n$, we obtain that $|\mathcal{F}|_n = \bigcup_{|x| = n} \partial E_x$ is contained in $\bigcup_{|x| = n} \partial D_x$, i.e. in a~finite union of closed subsets of~$X$ of dimension~$\leq k - 1$. By Theorem 1.5.3 in~\cite{E}, such union must have dimension $\leq k-1$, which proves that $\mathcal{F}$ is of dimension~$\leq k - 1$.
\textbf{(d)} follows immediately from~\textbf{(b)}.
\textbf{(e)} Note first that, for every $1 \leq i \leq K$, we have
\[ \overline{D_{x_i}} = \overline{\widetilde{G}_i} \subseteq |\mathcal{C}|_{n_i} \setminus \widetilde{H}_i \subseteq X \setminus \overline{H_i} = \intr S_{x_i}. \]
Now, let $x \in G$ and denote $T^B_{\theta+1}(x) = i$ and $\gamma = xx_i^{-1}$. The left translation by~$\gamma$ is clearly a~homeomorphism mapping $S_{x_i}$ to~$S_x$ and $D_{x_i}$ to $D_x$; moreover, by Lemma~\ref{fakt-wym-homeo-na-calej-kuli}, it preserves interiors and closures computed within the appropriate spaces $|\mathcal{C}|_n$. Hence, we have:
\[ F_x \subseteq \overline{E_x} \subseteq \overline{D_x} = \gamma \cdot \overline{D_{x_i}} \subseteq \gamma \cdot \intr S_{x_i} = \intr S_x. \]
\textbf{(f)}
follows easily from \textbf{(b)}:
\[ |\mathcal{C}|_n \setminus |\mathcal{G}|_n = |\mathcal{C}|_n \setminus \bigcup_{|x| = n} G_x \subseteq \bigcup_{|x| = n} \big( E_x \setminus G_x \big) \subseteq \bigcup_{|x| = n} \partial E_x = |\mathcal{F}|_n. \]
\textbf{(g)}
Let $T^B_{\theta+2}(x) = T^B_{\theta+2}(y)$ for some $x, y \in G$ and denote $\gamma = yx^{-1}$. By Lemma~\ref{fakt-wym-homeo-na-calej-kuli}, it is sufficient to check that $E_y = \gamma \cdot E_x$. We will show that $\gamma \cdot E_x \subseteq E_y$; the other inclusion is analogous.
Let $i = T^B_{\theta+1}(x) = T^B_{\theta+1}(y)$ and let $p \in E_x$. Then, in particular, $p \in D_x$, so
\[ \gamma \cdot p = yx_i^{-1} \cdot ( x_ix^{-1} \cdot p ) \in D_y. \]
Suppose that $\gamma \cdot p \notin E_y$; then, there must be some $y' \in G$ such that
\[ |y'| = |y|, \qquad T^B_{\theta+1}(y') < i, \qquad \gamma \cdot p \in D_{y'}. \]
In particular, we have $\emptyset \neq D_y \cap D_{y'} \subseteq S_y \cap S_{y'}$. By~Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow}b, setting $x' = \gamma^{-1} y'$ we obtain that
\[ |x'| = |x|, \qquad T^B_{\theta+1}(x') = T^B_{\theta+1}(y'). \]
In particular, since $T^B_{\theta+1}(x') = T^B_{\theta+1}(y')$ and $x' = \gamma^{-1} y'$, we must have $D_{x'} = \gamma^{-1} \cdot D_{y'} \ni p$. This contradicts the assumption that $p \in E_x$ because $T^B_{\theta+1}(x') < T^B_{\theta+1}(x)$.
\end{proof}
\subsection{Weakly invariant neighbourhoods}
\label{sec-wym-kolnierzyki}
\begin{lem}
\label{lem-wym-kolnierzyki}
Let $k \geq 0$, $\theta \geq 0$ and suppose that:
\begin{itemize}
\item $\mathcal{C} = \{ C_x \}_{x \in G}$ is a semi-closed, $\theta$-weakly invariant system;
\item $\mathcal{D} = \{ D_x \}_{x \in G}$ is a closed, $(\theta+1)$-weakly invariant subsystem in~$\mathcal{C}$ of rank~$\leq k$.
\end{itemize}
Then, there exists an open, $(\theta+1)$-weakly invariant subsystem $\mathcal{G} = \{ G_x \}_{x \in G}$ in~$\mathcal{C}$ such that $\mathcal{G}$ covers $\mathcal{D}$ and moreover the system of closures $\overline{\mathcal{G}} = \{ \overline{G_x} \}_{x \in G}$ is $(\theta+1)$-weakly invariant and of rank~$\leq k$.
\end{lem}
\begin{uwaga}
Since $\mathcal{G}$ itself is claimed to be $(\theta+1)$-weakly invariant, the condition that the system of closures $\overline{\mathcal{G}}$ is $(\theta+1)$-weakly invariant reduces to the condition that $\overline{G_x} \subseteq S_x$ for every $x \in G$.
\end{uwaga}
\begin{proof}[Proof of Proposition~\ref{lem-wym-kolnierzyki}]
In the proof, we use the notations and conventions introduced in the beginning of~Section~\ref{sec-wym-wyscig}.
Also, we will frequently (and implicitly) use Lemma~\ref{fakt-wym-homeo-na-calej-kuli} to control the images of interiors/closures (taken ``in~$\mathcal{C}$'') under translations by elements of~$G$.
\textbf{1. }We choose by induction, for $1 \leq i \leq K$, an open subset $G_i \Subset S_{x_i}$ containing $D_{x_i}$ such that:
\begin{gather}
\begin{split}
\label{eq-wym-war-laty}
\textrm{for every } x, y \in G \textrm{ with } |x| = |y| \leq M, \, T^B_{\theta+1}(x) = i, \, T^B_{\theta+1}(y) = j \textrm{ and } D_x \cap D_y = \emptyset, \textrm{ we have:} \\
(xx_i^{-1} \cdot \overline{G_i}) \cap D_y = \emptyset, \textrm{ and moreover } (xx_i^{-1} \cdot \overline{G_i}) \cap (yx_j^{-1} \cdot \overline{G_j}) = \emptyset \textrm{ if } j < i. \qquad \
\end{split}
\end{gather}
Such choice is possible because we only require $G_i$ to be an open neighbourhood of~$D_{x_i}$ such that the closure $\overline{G_i}$ is disjoint with the union of sets of the following form:
\[ |\mathcal{C}|_{n_i} \setminus S_{x_i}, \qquad x_i x^{-1} \cdot D_y, \qquad x_i x^{-1} y x_j^{-1} \cdot \overline{G_j}. \]
Since this is a~finite union of closed sets (as we assume $|x|, |y| \leq M$), and we work in a~metric space, it suffices to check that each of these sets is disjoint from~$D_{x_i}$.
In the case of~$|\mathcal{C}_{n_i}| \setminus S_{x_i}$, this is clear.
If we had $D_{x_i} \cap (x_i x^{-1} \cdot D_y) \neq \emptyset$ for some $x, y$ as specified above, it would follow that
\[ D_x \cap D_y = (x x_i^{-1} \cdot D_{x_i}) \cap D_y \neq \emptyset, \]
contradicting one of the assumptions in~\eqref{eq-wym-war-laty}. Similarly, if we had $D_{x_i} \cap (x_i x^{-1} y x_j^{-1} \cdot \overline{G_j}) \neq \emptyset$ for some $j < i$, then it would follow that $D_x \cap (y x_j^{-1} \cdot \overline{G_j}) \neq \emptyset$, which contradicts the assumption that we have (earlier) chosen $G_j$ to satisfy~\eqref{eq-wym-war-laty}.
\textbf{2. }Now, let
\[ G_x = x x_i^{-1} \cdot G_i \qquad \textrm{ for } \quad x \in G, \, T^B_{\theta+1}(x) = i. \]
Then, the system $\mathcal{G} = \{ G_x \}_{x \in G}$ is obviously open, $(\theta+1)$-weakly invariant and covers~$\mathcal{D}$. The fact that $\overline{\mathcal{G}}$ is also $(\theta+1)$-weakly invariant follows then from Lemma~\ref{fakt-wym-homeo-na-calej-kuli} (because $G_i \Subset S_{x_i}$, and hence $G_x \Subset S_x$ for $x \in G$). It remains to check that $\overline{\mathcal{G}}$ is of rank~$\leq k$.
\textbf{3. }Let $x, y \in G$ be such that $|x| = |y|$ and $\overline{G_x} \cap \overline{G_y} \neq \emptyset$. Denote $T^B_{\theta+1}(x) = i$, $T^B_{\theta+1}(y) = j$. Let $x'$ be such that $|x'| \leq M$ and $T^B_{\theta+2}(x') = T^B_{\theta+2}(x)$. Since $\overline{G_x} \cap \overline{G_y} \neq \emptyset$ implies $S_x \cap S_y \neq \emptyset$, by Lemma~\ref{fakt-wym-typ-wyznacza-sasiadow}b (combined with~\qhlink{f1} for $\mathcal{G}$) we obtain that
\[ T^B_{\theta+1}(y') = j, \qquad |y'| = |x'| \leq M, \qquad \overline{G_{x'}} \cap \overline{G_{y'}} = x'x^{-1} \cdot (\overline{G_x} \cap \overline{G_y}) \neq \emptyset, \qquad \quad \textrm{ where } \quad y' = x'x^{-1}y. \]
Then it follows that
\[ \emptyset \neq \overline{G_{x'}} \cap \overline{G_{y'}} = (x'x_i^{-1} \cdot \overline{G_i}) \cap (y'x_j^{-1} \cdot \overline{G_j}), \]
so the condition~\eqref{eq-wym-war-laty} implies that $D_{x'} \cap D_{y'} \neq \emptyset$. Then, since $\mathcal{D}$ is $(\theta+1)$-weakly invariant, we have
\[ D_x \cap D_y = xx'^{-1} \cdot (D_{x'} \cap D_{y'}) \neq \emptyset. \]
This means that, for $x, y \in G$ of equal length, $\overline{G_x}$ and $\overline{G_y}$ can intersect non-trivially only if $D_x$ and $D_y$ do so, whence it follows that the rank of~$\overline{\mathcal{G}}$ is not greater than that of~$\mathcal{D}$. This finishes the proof.
\end{proof}
\subsection{The overall construction}
The following proposition describes the whole inductive construction --- analogous to the one presented in the introduction to this section --- of a cover satisfying the conditions from Proposition~\ref{lem-wym}.
\begin{lem}
\label{lem-wym-cala-historia}
Let $k \geq -1$, $\theta \geq 1$ and let $\mathcal{C}$ be a semi-closed, $\theta$-weakly invariant system of dimension $\leq k$.
Then, there exist $(\theta + 3(k+1))$-weakly invariant subsystems $\mathcal{D}$, $\mathcal{E}$ in~$\mathcal{C}$ of rank $\leq k+1$ which both cover~$\mathcal{C}$. Moreover, $\mathcal{D}$ is closed and $\mathcal{E}$ is open.
\end{lem}
\begin{proof}
We proceed by induction on~$k$. If $k = -1$, the system $\mathcal{C}$ must consist of empty sets, so we can set~$\mathcal{D} = \mathcal{E} = \mathcal{C}$.
Now, let $k > -1$. Denote $\Theta = \theta + 3(k+1)$. We perform the following steps:
\textbf{1. }By applying Proposition~\ref{lem-wym-bzdziagwy} to the system $\mathcal{C}$, we obtain some $(\theta + 2)$-weakly invariant systems $\mathcal{G} = \{ G_x \}_{x \in G}$ and $\mathcal{F}$ with additional properties described in the claim of the proposition.
\textbf{2. }Since $\mathcal{F}$ is closed (and then also semi-closed), $(\theta+2)$-weakly invariant and has dimension~$\leq k - 1$, it satisfies the assumptions of the current proposition (with parameters $k-1$ and $\theta+2$). Therefore, by the inductive hypothesis, there exists a $(\Theta-1)$-weakly invariant closed system $\mathcal{D}'$ of rank $\leq k$ which covers $\mathcal{F}$.
\textbf{3. }Since $\mathcal{C}$ is $\theta$-weakly invariant, it is also $(\Theta-2)$-weakly invariant, which means that the systems $\mathcal{C}$, $\mathcal{D}'$ satisfy the assumptions of Proposition~\ref{lem-wym-kolnierzyki} (with parameters $k$ and $\Theta-2$). Then, there exists an open, $(\Theta-1)$-weakly invariant subsystem $\mathcal{G}' = \{ G'_x \}_{x \in G}$ in~$\mathcal{C}$ which covers~$\mathcal{D}'$ and such that the system of closures $\mathcal{F}' = \{ \overline{G'_x} \}_{x \in G}$ is $(\Theta-1)$-weakly invariant and of rank~$\leq k$.
\textbf{4. }Now, we define two subsystems $\mathcal{D} = \{ D_x \}_{x \in G}$ and $\mathcal{E} = \{ E_x \}_{x \in G}$ as follows:
\begin{align}
\label{eq-wym-konstr-wycinanka}
D_x = (G_x \setminus |\mathcal{G}'|_n) \cup F'_x, \qquad E_x = G_x \cup G'_x \qquad \textrm{ for } \quad x \in G, \, |x| = n.
\end{align}
Observe that $G_x \setminus |\mathcal{G}'|_n$ is closed because the claim of Proposition~\ref{lem-wym-bzdziagwy} implies that $\partial G_x \subseteq |\mathcal{F}|_n \subseteq |\mathcal{D}'|_n \subseteq |\mathcal{G}'|_n$, so $G_x \setminus |\mathcal{G}'|_n = \overline{G_x} \setminus |\mathcal{G}'|_n$ is a difference of a closed and an open subset (in~$|\mathcal{C}|_n$). Hence, $D_x$ is closed. On the other hand, $E_x$ is clearly open in~$|\mathcal{C}|_n$.
Since $\mathcal{G}$, $\mathcal{F}'$ and~$\mathcal{G}'$ are all $(\Theta-1)$-weakly invariant, $\mathcal{D}$ and~$\mathcal{E}$ must both be $\Theta$-weakly invariant (more precisely: $\mathcal{E}$ is obviously $(\Theta-1)$-weakly and then also $\Theta$-weakly invariant, while for~$\mathcal{D}$ we apply Lemma~\ref{fakt-wym-homeo-na-calej-kuli}). It is also easy to see that
\[ |\mathcal{E}_n| \ = \ |\mathcal{G}|_n \cup |\mathcal{G}'|_n \ \supseteq \ |\mathcal{G}|_n \cup |\mathcal{F}|_n \ = \ |\mathcal{C}|_n,
\qquad
|\mathcal{D}|_n \ = \ \big( |\mathcal{G}|_n \setminus |\mathcal{G}'|_n \big) \cup |\mathcal{F}'|_n \ = \ |\mathcal{G}|_n \cup |\mathcal{F}'|_n \ \supseteq \ |\mathcal{E}|_n, \]
so $\mathcal{D}$ and~$\mathcal{E}$ both cover~$\mathcal{C}$. Finally, since $\mathcal{G}$ is disjoint and $\mathcal{F}'$ (and so also $\mathcal{G}'$) is of rank $\leq k$, it follows that $\mathcal{D}$ and~$\mathcal{E}$ must be of rank~$\leq k +1$ by Lemma~\ref{fakt-wym-suma-pokryc}.
\end{proof}
\subsection{Conclusion: The complete proof of Theorem~\ref{tw-kompakt}}
\label{sec-wymd-podsum}
\begin{proof}[{\normalfont \textbf{Proof of Proposition~\ref{lem-wym}}}]
The claim follows from applying Proposition~\ref{lem-wym-cala-historia} to the system~$\mathcal{S}$ (defined in Section~\ref{sec-konstr-pokrycia}). This is a semi-closed and~$0$-weakly (and hence also $1$-weakly) invariant system, so the proposition ensures that there exists an open $(3k+4)$-weakly invariant subsystem $\mathcal{E}$ of rank~$\leq k + 1$ which covers~$\mathcal{S}$.
Since $\mathcal{S}$ is a system of covers, while~$\mathcal{E}$ is open and covers~$\mathcal{S}$, it follows that $\mathcal{E}$ is also a system of covers. Then, it follows from Proposition~\ref{lem-wym-slabe-sa-qi} that $\mathcal{E}$ is quasi-$G$-invariant (with the type function~$T^B_{3k+5}$). This means that $\mathcal{E}$ has all the desired properties.
\end{proof}
\begin{proof}[{\normalfont \textbf{Proof of Theorem~\ref{tw-kompakt}}}]
We use the quasi-$G$-invariant system of covers~$\mathcal{E}$ obtained in the proof of Proposition~\ref{lem-wym}. By Lemma~\ref{fakt-konstr-sp-zal}, there is $L \geq 0$ such that the system $(\widetilde{\mathcal{E}}_{Ln})_{n \geq 0}$ (where $\widetilde{\mathcal{E}}_n$ denotes $\mathcal{E}_n$ with empty members removed) is admissible. Then, by Theorems~\ref{tw-konstr} and~\ref{tw-kompakt-ogolnie}, the corresponding system of nerves $(K_n, f_n)$ is Markov, barycentric and has mesh property.
Since the type function $T^B_{3k+5}$ associated with this system is stronger than $T^B$, Theorem~\ref{tw-sk-opis} ensures that the simplex types used in the system $(K_n, f_n)$ can be strengthened to make this system simultaneously Markov and has the distinct types property. (Barycentricity and mesh property are clearly preserved as the system itself does not change). Moreover, for every $n \geq 0$ we have
\[ \dim K_n = \rank \widetilde{\mathcal{E}}_{Ln} - 1 = \rank \mathcal{E}_{Ln} - 1 \leq \dim \partial G, \]
where the last inequality follows from the property of~$\mathcal{E}$ claimed by Proposition~\ref{lem-wym}. Finally, since $\mathcal{E}$ is $(3k+4)$-weakly invariant, it is in particular inscribed into~$\mathcal{S}$, which means in view of Theorem~\ref{tw-bi-lip} that the homeomorphism $\varphi : \partial G \simeq \mathop{\lim}\limits_{\longleftarrow} K_n$ obtained from Theorem~\ref{tw-konstr} is in fact a bi-Lipschitz equivalence (in the sense specified by Theorem~\ref{tw-bi-lip}).
This shows that the system~$(K_n, f_n)_{n \geq 0}$ has all the properties listed in Theorem~\ref{tw-kompakt}, which finishes the proof.
\end{proof}
\section{$\partial G$ as a semi-Markovian space}
\label{sec-sm}
The aim of this section is to show that the boundary $\partial G$ of a hyperbolic group~$G$ is a \textit{semi-Markovian space} (see Definition \ref{def-sm-ps}). In Section \ref{sec-sm-def}, we introduce notions needed to formulate the main result, which appears at its end as Theorem \ref{tw-semi-markow-0}. The remaining part of the section contains the proof of this theorem.
\subsection{Semi-Markovian sets and spaces}
\label{sec-sm-def}
Let $\Sigma$ be a finite alphabet and $\Sigma^\mathbb{N}$ denote the set of infinite words over $\Sigma$.
In the set $\Sigma$ we define the operations of \textit{shift} $S : \Sigma^\mathbb{N} \rightarrow \Sigma^\mathbb{N}$ and \textit{projection} $\pi_F : \Sigma^\mathbb{N} \rightarrow \Sigma^F$ (where $F \subseteq \mathbb{N}$) by the formulas:
\[ S \big( (a_0, a_1, \ldots) \big) = (a_1, a_2, \ldots), \qquad \pi_F \big( (a_0, a_1, \ldots) \big) = (a_n)_{n \in F}. \]
\begin{df}[{\cite[Chapter~2.3]{zolta}}]
\label{def-sm-cylinder}
A subset $C \subseteq \Sigma^\mathbb{N}$ is called a \textit{cylinder} if $C = \pi_F^{-1}(A)$ for some finite $F \subseteq \mathbb{N}$ and for some $A \subseteq \Sigma^F$.
(Intuitively: the set $C$ can be described by conditions involving only a finite, fixed set of positions in the sequence $(a_n)_{n \geq 0} \in \Sigma^\mathbb{N}$).
\end{df}
\begin{df}[{\cite[Definition~6.1.1]{zolta}}]
\label{def-sm-zb}
A subset $M \subseteq \Sigma^\mathbb{N}$ is called a \textit{semi-Markovian set} if there exist cylinders $C_1$, $C_2$ in~$\Sigma^\mathbb{N}$ such that $M = C_1 \cap \bigcap_{n \geq 0}^\infty S^{-n}(C_2)$.
\end{df}
\begin{uwaga}
\label{uwaga-sm-proste-zbiory}
In particular, for any subset $\Sigma_0 \subseteq \Sigma$ and binary relation $\rightarrow$ in $\Sigma$, the following set is semi-Markovian:
\[ M(\Sigma_0, \, \rightarrow) = \big\{ (a_n)_{n \geq 0} \,\big|\, a_0 \in \Sigma_0, \, a_n \rightarrow a_{n+1} \textrm{ for } n \geq 0 \big\}. \]
\end{uwaga}
We consider the space of words $\Sigma^\mathbb{N}$ with the natural Cantor product topology (generated by the base of cylinders). In this topology, all semi-Markovian sets are closed subsets of $\Sigma^\mathbb{N}$.
Before formulating the next definition, we introduce a natural identification of pairs of words and words of pairs of symbols:
\[ J : \quad \Sigma^\mathbb{N} \times \Sigma^\mathbb{N} \quad \ni \quad \Big( \big( (a_n)_{n \geq 0}, \, (b_n)_{n \geq 0} \big) \Big) \qquad \mapsto \qquad \big( (a_n, \, b_n) \big)_{n \geq 0} \quad \in \quad (\Sigma \times \Sigma)^\mathbb{N}. \]
\begin{df}
\label{def-sm-rel}
A binary relation $R \subseteq \Sigma^\mathbb{N} \times \Sigma^\mathbb{N}$ will be called a \textit{semi-Markovian relation} if its image under the above identification $J(R) \subseteq (\Sigma \times \Sigma)^\mathbb{N}$ is a semi-Markovian set (over the product alphabet $\Sigma \times \Sigma$).
\end{df}
\begin{df}[{\cite[Definition~6.1.5]{zolta}}]
\label{def-sm-ps}
A topological Hausdorff space $\Omega$ is called a \textit{semi-Markovian space} if it is the topological quotient of a semi-Markovian space (with the Cantor product topology) by a semi-Markovian equivalence relation.
\end{df}
We can now re-state the main result of this section:
\begin{twsm}
The boundary of any hyperbolic group $G$ is a semi-Markovian space.
\end{twsm}
The proof of Theorem \ref{tw-semi-markow-0} --- preceded by a number of auxiliary facts --- is given at the end of this section. Roughly, it will be obtained by applying Corollary~\ref{wn-sm-kryt} to the \textit{$C$-type} function which will be defined in Section~\ref{sec-sm-abc-c}.
\begin{uwaga}
Theorem~\ref{tw-semi-markow-0} has been proved (in~\cite{zolta}) under an additional assumption that $G$ is torsion-free.
We present a proof which does not require this assumption; the price for it is that our reasoning (including the results from Section~\ref{sec-abc}, which will play an important role here) is altogether significantly more complicated.
However, in the case of torsion-free groups, these complications mostly trivialise (particularly, so does the construction of $B$-type) --- and the remaining basic structure of the reasoning (summarised in Lemma \ref{fakt-sm-kryt-zbior}) is analogous to that in the proof from~\cite{zolta}. Within this analogy, a key role in our proof is played by Proposition~\ref{lem-sm-kuzyni}, corresponding to Lemma 7.3.1 in~\cite{zolta} which particularly requires $G$ to be torsion-free. More concrete remarks about certain problems related with the proof from \cite{zolta} will be stated later in Remark~\ref{uwaga-zolta}.
\end{uwaga}
\begin{uwaga}
Theorem \ref{tw-semi-markow-0} can be perceived as somewhat analogous to a known result stating automaticity of hyperbolic groups (described for example in \cite[Theorem~12.7.1]{CDP}).
The relation between those theorems seems even closer if we notice that --- although the classical automaticity theorem involves the Cayley graph of a group --- it can be easily translated to an analogous description of the Gromov boundary. Namely, the boundary is the quotient of some ``regular'' set of infinite words by some ``regular'' equivalence relation (in a sense analogous to Definition \ref{def-sm-rel}) where ``regularity'' of a set $\Phi \subseteq \Sigma^\mathbb{N}$ means that there is a finite automaton $A$ such that any infinite word $(a_n)_{n \geq 0}$ belongs to $\Phi$ if and only if $A$ accepts all its finite prefixes.
However, such regularity condition is weaker than the condition from Definition \ref{def-sm-zb}, as the following example shows:
\[ \Phi = \big\{ (x_n)_{n \geq 0} \in \{ a, b, c \}^\mathbb{N} \ \big|\ \forall_n \ (x_n = b \ \Rightarrow \ \exists_{i < n} \ x_i = a) \big\}, \qquad \quad A: \ \
\raisebox{-3.5ex}{
\begin{tikzpicture}[scale=0.15]
\node[draw, circle, double] (s0) at (0, 0) {};
\node[draw, circle, double] (s1) at (10, 0) {};
\node[draw, circle] (s2) at (20, 0) {};
\draw[->] (-3,0) -- (s0);
\draw[->] (s0) -- (s1) node [midway, above, draw=none, inner sep=2] {\footnotesize $a$};
\draw (s0) edge [in=110,out=70,loop] node [midway, above, inner sep=2] {\footnotesize $c$} ();
\draw (s1) edge [in=110,out=70,loop] node [midway, above, inner sep=2] {\footnotesize $a,b,c$} ();
\draw (s2) edge [in=110,out=70,loop] node [midway, above, inner sep=2] {\footnotesize $a,b,c$} ();
\draw[->] (s0) edge[bend right=20] node [midway, sloped, below, inner sep=2] {\footnotesize $b$} (s2);
\end{tikzpicture}
}
\]
It is easy to check that the set $\Phi$ corresponds to the automaton $A$ in the sense described above, while it is not semi-Markovian.
The latter claim can be shown as follows. Assume that there exists a presentation $\Phi = C_1 \cap \bigcap_{n \geq 0} S^{-n} (C_2)$ as required by Definition \ref{def-sm-zb}, and let the cylinder $C_1$ have form $\pi_F^{-1}(A)$, according to Definition \ref{def-sm-cylinder}. Denote by $N$ the maximal element of $F$. Then, the word $\alpha_1 = \underbrace{cc\ldots{}cc}_{N+1}\underbrace{aa\ldots{}aa}_{N+1}\underbrace{cc\ldots{}cc}_{N+1}bb\ldots$ belongs to $\Phi$, while $\alpha_2 = \underbrace{cc\ldots{}cc}_{N+1}bb\ldots$ does not belong to $\Phi$. However, these words have a common prefix of length $N+1$ (so $\alpha_2$ cannot be rejected by $C_1$) and moreover $\alpha_2$ is a suffix of $\alpha_1$ (so $\alpha_2$ cannot be rejected by $C_2$).
\end{uwaga}
\subsection{Compatible sequences}
\label{sec-sm-nici}
In the remainder of Section \ref{sec-sm}, we work under the assumptions formulated in the introduction to Section \ref{sec-abc}.
\begin{ozn}
For any $n \geq 0$, we denote
\[ G_n = \big\{ x \in G \,\big|\, |x| = n \big\}. \]
\end{ozn}
\begin{df}
An infinite sequence $(g_n)_{n \geq 0}$ in~$G$ will be called \textit{compatible} if, for all $n \geq 0$, we have $g_n \in G_{Ln}$ and $g_n = g_{n+1}^\Uparrow$. We denote the set of all such sequences by $\mathcal{N}$.
\end{df}
Note that to any compatible sequence $(g_n)_{n \geq 0}$ we can naturally assign a geodesic $\alpha$ in~$G$, defined by the formula
\[ \alpha(m) = g_n^{\uparrow Ln-m} \qquad \textrm{ for } \quad m, n \geq 0, \ Ln \geq m. \]
(It is easy to check that the value $g_n^{\uparrow Ln-m}$ does not depend on the choice of $n$). This means that every compatible sequence $(g_n)$ has a \textit{limit}: it must converge in~$G \cup \partial G$ to the element $\lim_{m \rightarrow \infty} \alpha(m) = [\alpha] \in \partial G$.
\begin{fakt}
\label{fakt-sm-nici-a-punkty}
Let the map $I : \mathcal{N} \rightarrow \partial G$ assign to a compatible sequence $(g_n)_{n \geq 0}$ its limit in $\partial G$. Then:
\begin{itemize}
\item[\textbf{(a)}] $I$ is surjective;
\item[\textbf{(b)}] for every $(g_n), (h_n) \in \mathcal{N}$, we have
\[ I \big( (g_n) \big) = I \big( (h_n) \big) \qquad \Longleftrightarrow \qquad g_n \leftrightarrow h_n \quad \textrm{ for every } n \geq 0. \]
\end{itemize}
\end{fakt}
\begin{proof}
\textbf{(a)} Let $x \in \partial G$ and let $\alpha$ be any infinite geodesic going from $e$ towards $x$. For $k \geq 0$, we define a geodesic $\alpha_k$ in~$G$ by the formula
\[ \alpha_k(n) = \begin{cases}
\alpha(k)^{\uparrow k-n} & \textrm{ for } n \leq k, \\
\alpha(n) & \textrm{ for } n \geq k.
\end{cases}
\]
(For $n = k$, both branches give the same result). Then, for any $n \geq 0$, we have $|\alpha_k(n)| = n$ and moreover $d \big( \alpha_k(n), \alpha_k(n+1) \big) = 1$, which proves that $\alpha_k$ is a geodesic. Moreover, we have $\alpha_k(0) = e$ and $\lim_{n \rightarrow \infty} \alpha_k(n) = x$ because $\alpha_k$ ultimately coincides with $\alpha$.
Using Lemma~\ref{fakt-geodezyjne-przekatniowo} to the sequence $(\alpha_k)$, we obtain some subsequence $(\alpha_{k_i})$ and a geodesic $\alpha_\infty$ such that $\alpha_\infty$ coincides with~$\alpha_{k_i}$ on the segment~$[0, i]$. By choosing a subsequence if necessary, we can assume that $k_i \geq i$; then we have $\alpha_\infty(i-1) = \alpha_\infty(i)^\uparrow$ for every $i \geq 1$, so the sequence $\big( \alpha_\infty(Ln) \big)_{n \geq 0}$ is compatible. On the other hand, Lemma \ref{fakt-geodezyjne-przekatniowo} also ensures that $I \big( (g_n) \big) = [\alpha_\infty] = \lim_{i \rightarrow \infty} [\alpha_{k_i}] = x$, which proves the claim.
\textbf{(b)} The implication $(\Rightarrow)$ follows directly from the inequality (1.3.4.1) in~\cite{zolta}. On the other hand, if $g_n \leftrightarrow h_n$ for every $n \geq 0$, and if $\alpha, \beta$ are the geodesics corresponding to the compatible sequences $(g_n)$ and~$(h_n)$, then we have
\[ d \big( \alpha(Ln), \beta(Ln) \big) = d(g_n, h_n) \leq 8\delta \qquad \textrm{ for } \quad n \geq 0, \]
so from the triangle inequality we deduce that $d \big( \alpha(m), \beta(m) \big) \leq 2L + 8\delta$ for all $m \geq 0$, and so in $\partial G$ we have $[\alpha] = [\beta]$.
\end{proof}
\subsection{Desired properties of the type function}
\label{sec-sm-zyczenia}
The presentation of $\partial G$ as a semi-Markovian space will be based on an appropriate type function (see the introduction to Section \ref{sec-abc}). Since the ball type $T^b_N$ used in the previous sections has too weak properties for our needs, we will use some its strengthening. In this section, we state (in Lemma \ref{fakt-sm-kryt-zbior}) a list of properties of a type function which are sufficient (as we will prove in Corollary \ref{wn-sm-kryt}) to give a semi-Markovian structure on $\partial G$. The construction of a particular function $T^C$ satisfying these conditions will be given in Section \ref{sec-sm-abc-c}.
\begin{df}
Let $T$ be any type function in $G$ with values in a~finite set $\mathcal{T}$. For a compatible sequence $\nu = (g_n)_{n \geq 0} \in \mathcal{N}$, we call its \textit{type} $T^*(\nu)$ the sequence $\big( T(g_n) \big)_{n \geq 0} $.
\end{df}
Then, using the definition of a semi-Markovian space, it is easy to show the following lemma.
\begin{fakt}
\label{fakt-sm-kryt-zbior}
Let $T$ be a type function in $G$ with values in $\mathcal{T}$. Then:
\begin{itemize}
\item[\textbf{(a)}] If, for every element of~$G$, all its p-grandchildren have pairwise distinct types, then the function $T^* : \mathcal{N} \rightarrow \mathcal{T}^{\mathbb{N}}$ is injective;
\item[\textbf{(b)}] If the set of p-grandchildren of $g \in G$ depends only on the type of $g$, then the image of $T^*$ is a semi-Markovian set over $\mathcal{T}$;
\item[\textbf{(c)}] Under the assumptions of parts \textbf{(a)} and \textbf{(b)}, if for any $g, g' \in G_{L(n+1)}$, $h, h' \in G_{L(m+1)}$ the conditions
\begin{gather*}
T(g) = T(h), \qquad T(g') = T(h'), \qquad T(g^\Uparrow) = T(h^\Uparrow), \qquad T({g'}^\Uparrow) = T({h'}^\Uparrow), \\
g^\Uparrow \leftrightarrow g'^\Uparrow, \qquad g \leftrightarrow g', \qquad h^\Uparrow \leftrightarrow {h'}^\Uparrow,
\end{gather*}
imply that $h \leftrightarrow h'$, then the equivalence relation $\sim$ in the set $T^*(\mathcal{N})$, given by the formula \linebreak \mbox{$T^*(\nu) \sim T^*(\nu') \ \Leftrightarrow \ I(\nu) = I(\nu')$}, is a semi-Markovian relation.
\end{itemize}
\end{fakt}
\begin{proof}
Part \textbf{(a)} is clear. If the assumption of part~\textbf{(b)} holds, it is easy to check that
$T^*(\mathcal{N}) = M(\Sigma_0, \rightarrow)$, where $\Sigma_0 = \{ T(e) \}$ and $\tau \rightarrow \tau'$ if and only if $\tau = T(g^\Uparrow)$ and $\tau' = T(g)$ for some $n \geq 0$ and $g \in G_{L(n+1)}$.
Analogously, it is easy to check that, under the assumptions of part~\textbf{(c)}, the relation $\sim$ has the form $M(A_0, \leadsto)$, where
\begin{gather*}
A_0 = \big\{ \big( T(e), T(e) \big) \}, \\
\big( T(g^\Uparrow), T({g'}^\Uparrow) \big) \ \leadsto \ \big( T(g), T(g') \big) \qquad \textrm{ for } \quad g, g' \in G_{L(n+1)}, \ g^\Uparrow \leftrightarrow g'^\Uparrow, \, g \leftrightarrow g', \ n \geq 0.
\end{gather*}
Indeed: the containment $\sim \subseteq M(A_0, \leadsto)$ results from Lemma \ref{fakt-sm-nici-a-punkty}b. On the other hand, if a sequence $\big( (\tau_n, \tau_n') \big)_{n \geq 0}$ belongs to $M(A_0, \leadsto)$, then the sequences $(\tau_n)_{n \geq 0}$ and $(\tau'_n)_{n \geq 0}$ belong to the set $M(\Sigma_0, \rightarrow)$
defined in the previous paragraph, so they are types of some compatible sequences $(h_n)_{n \geq 0}$ and correspondingly $(h'_n)_{n \geq 0}$. Moreover, it is easy to check by induction that $h_n \leftrightarrow h_n'$ for every $n \geq 0$: for $n = 0$ this holds since $h_0 = h'_0 = e$, and for $n > 0$ one can use the relation $h_{n-1} \leftrightarrow h'_{n-1}$, the condition $(\tau_{n-1}, \tau'_{n-1}) \leadsto (\tau_n, \tau'_n)$ and the assumptions of part~\textbf{(c)}. Therefore, we obtain that $h_n \leftrightarrow h_n'$ for $n \geq 0$, and so by Lemma~\ref{fakt-sm-nici-a-punkty}b we deduce that $(\tau_n) \sim (\tau_n')$.
\end{proof}
\begin{wn}
\label{wn-sm-kryt}
Under the assumptions of parts~\textbf{(a-c)} in Lemma~\ref{fakt-sm-kryt-zbior}, $\partial G$ is a semi-Markovian space.
\end{wn}
\begin{proof}
Since the map $I \circ (T^*)^{-1} : T^*(\mathcal{N}) \rightarrow \partial G$ is surjective by Lemma \ref{fakt-sm-nici-a-punkty}a, to verify that it is a homeomorphism we need only to check its continuity. Let $(\tau_n^{(i)}) \mathop{\longrightarrow}\limits_{i \rightarrow \infty} (\tau_n)$ in the space $T^*(\mathcal{N})$; this means that there exists a sequence $n_i \rightarrow \infty$ such that for every $i \geq 0$ the sequences $(\tau_n^{(i)})$ and $(\tau_n)$ coincide on the first $n_i$ positions. Then, the assumptions of part~\textbf{(a)} imply that the corresponding compatible sequences $(g_n^{(i)})$, $(g_n)$ also coincide on the first $n_i$ positions; in particular, $g_{n_i}^{(i)} = g_{n_i}$. Then, also the geodesics $\alpha^{(i)}$ corresponding to the sequences $(g_n^{(i)})$ are increasingly coincident with the geodesic $\alpha$ corresponding to the sequence $(g_n)$, which means by the definition of $\partial G$ that $I \big( (g_n^{(i)}) \big) = [\alpha^{(i)}] \rightarrow [\alpha] = I \big( (g_n) \big)$ in $\partial G$.
\end{proof}
\begin{uwaga}
\label{uwaga-zolta}
The main ``skeleton'' of the proof of Theorem \ref{tw-semi-markow-0} presented in Lemma \ref{fakt-sm-kryt-zbior} is taken from \cite{zolta}. The proof given there uses the ball type $T^b_N$ (defined in Section~\ref{sec-typy-kulowe}) as the type function, and $1$ as the value of~$L$. However, in the case of a torsion group, this type function does not have to satisfy the assumptions of part~\textbf{(a)} in Lemma \ref{fakt-sm-kryt-zbior}. Moreover, even in the torsion-free case, the verification of the assumptions of part~\textbf{(b)} --- given in \cite{zolta} on page 125 (Chapter~7, proof of Proposition~2.4) --- contains a~defect in line 13. More precisely, it is claimed there that if one takes $N$ sufficiently large, $L = 1$ and $x', y' \in G$ such that $y'$ ``follows'' (in our terms: is a~child of) $x'$, then every element of the set $B_N(y') \setminus B_N(x')$ ``can be considered as belonging to the tree $T_{geo,x'}$'' (in our terms: to $x' T^c(x')$).
Our approach avoids this problem basically by choosing $L$ and $N$ so large that the analogous claim must indeed hold, as one could deduce e.g. from Lemma~\ref{fakt-kulowy-duzy-wyznacza-maly} combined with the proof of Proposition~\ref{lem-potomkowie-dla-kulowych}.
\end{uwaga}
\subsection{Extended types and the $C$-type}
\label{sec-sm-abc-c}
\begin{df}
\label{def-sm-typ-plus}
Let $T$ be an arbitrary type function in $G$ with values in a finite set $\mathcal{T}$ and let $r \geq 0$. Let $g \in G$, and let $P_r(g)$ denote the set of $r$-fellows of $g$ (see Definition \ref{def-konstr-towarzysze}). We define the \textit{extended type} of element $g$ as the function $T^{+r}(g) : P_r(g) \rightarrow \mathcal{T}$, defined by the formula:
\[ \big( T^{+r}(g) \big)(h) = T(gh) \qquad \textrm{ for } \quad h \in P_r(g). \]
\end{df}
Since $P_r(g)$ is contained in a bounded ball $B(e, r)$, the extended type function $T^{+r}$ has, in an obvious way, finitely many possible values.
\begin{df}
\label{def-sm-typ-C}
We define the \textit{$C$-type} of an element $g \in G$ as its $B$-type extended by $8\delta$:
\[ T^C(g) = (T^B)^{+8\delta}(g) \qquad \textrm{ for } g \in G. \]
\end{df}
Note that, by comparing Definitions~\ref{def-konstr-towarzysze} and~\ref{def-sm-sasiedzi}, we obtain that the set $P_{8\delta}(g)$ contains exactly these $h \in G$ for which $g \leftrightarrow gh$. This means that the $C$-type of $g$ consists of the $B$-type of~$g$ and of $B$-types of its neighbours (together with the knowledge about their relative location).
\begin{fakt}
\label{fakt-sm-C-bracia}
For every $g \in G$, all p-grandchildren of $g$ have pairwise distinct $C$-types.
\end{fakt}
\begin{proof}
By definition, the $C$-type of an element $h \in G$ contains its $B$-type, which in turn contains its $A$-type and finally its descendant number $n_h$, which by definition distinguishes all the p-grandchildren of a fixed element $g \in G$.
\end{proof}
\begin{lem}
\label{lem-sm-C-dzieci}
The set of $C$-types of all p-grandchildren of a given element $g \in G$ depends only on $T^C(g)$.
\end{lem}
\begin{proof}
Let $g_1, h_1 \in G$ satisfy $T^C(g_1) = T^C(h_1)$; denote $\gamma = h_1g_1^{-1}$. By Lemma \ref{fakt-sm-przen-a} we know that the left translation by $\gamma$
gives a bijection between p-grandchildren of $g_1$ and p-grandchildren of $h_1$. Let $g_2$ be a p-grandchild of $g_1$ and $h_2 = \gamma g_2$; our goal is to prove that $T^C(g_2) = T^C(h_2)$. For this, choose any $g_2' \in G$ such that $g_2 \leftrightarrow g_2'$; we need to prove that $T^B(g_2') = T^B(h_2')$, where $h_2' = \gamma g_2'$.
Denote $g_1' = g_2'^\Uparrow$ and $h_1' = \gamma g_1'$. Since $g_2 \leftrightarrow g_2'$, by Lemma~\ref{fakt-sm-rodzic-kuzyna} we have $g_1 \leftrightarrow g_1'$; then from the equality $T^C(g_1) = T^C(h_1)$ we obtain that $T^B(g_1') = T^B(h_1')$. In this situation, Proposition \ref{lem-sm-B-dzieci} ensures that $T^B(g_2') = T^B(h_2')$, q.e.d.
\end{proof}
\begin{lem}
\label{lem-sm-C-klejenie}
The type function $T^C$ satisfies the condition stated in part~\textbf{(c)} of Lemma \ref{fakt-sm-kryt-zbior}.
\end{lem}
\begin{proof}
Let $g, g', h, h'$ be as in part~\textbf{(c)} of Lemma \ref{fakt-sm-kryt-zbior}. In particular, we assume that $T^C(g^\Uparrow) = T^C(h^\Uparrow)$. By the definition of $C$-type, this means that the left translation by $\gamma = h^{-1}g$ neighbours of $g$ to neighbours of $h$, preserving their $B$-type, which in turn implies by Proposition \ref{lem-sm-B-dzieci} that this shift preserves the children of these neighbours, together with their $B$-types. In particular:
\begin{itemize}
\item the element ${g'}^\Uparrow$ must be mapped to ${h'}^\Uparrow$ since by assumption we have $T^B({g'}^\Uparrow) = T^B({h'}^\Uparrow)$, and moreover ${h'}^\Uparrow$ is the only neighbour of $h^\Uparrow$ with the appropriate $B$-type (by Proposition~\ref{lem-sm-kuzyni});
\item the elements $g$, $g'$ must be mapped correspondingly to $h$, $h'$ since by assumption the corresponding $B$-types coincide, and moreover $h$, $h'$ are the only p-grandchildren of $h^\Uparrow$, ${h'}^\Uparrow$ with the appropriate $B$-types (because by Remark~\ref{uwaga-sm-B-wyznacza-A} the $B$-type determines the $A$-type, which in turn distinguishes all the p-grandchildren of a given element).
\end{itemize}
Therefore, we have $d(h, h') = d(\gamma g, \gamma g') = d(g, g') \leq 8\delta$, q.e.d.
\end{proof}
\begin{proof}[{\normalfont \textbf{Proof of Theorem~\ref{tw-semi-markow-0}}}]
By Corollary~\ref{wn-sm-kryt} it suffices to ensure that the type~$T^C$ (which is finitely valued by Lemma~\ref{fakt-sm-B-skonczony} and Definition~\ref{def-sm-typ-C}) satisfies the conditions \textbf{(a-c)} from Lemma~\ref{fakt-sm-kryt-zbior}. The conditions of parts~\textbf{(b)} and~\textbf{(c)} follow correspondingly from Propositions~\ref{lem-sm-C-dzieci} and~\ref{lem-sm-C-klejenie}, while the condition of part~\textbf{(a)} follows from the fact that, by definition, the value $T^C(x)$ for a given $x \in G$ determines $T^B(x)$ and further $T^A(x)$, while the $A$-types of all p-grandchildren of a given element are pairwise distinct by definition. This finishes the proof.
\end{proof}
\bibliographystyle{plain}
| {'timestamp': '2015-03-17T01:14:04', 'yymm': '1503', 'arxiv_id': '1503.04577', 'language': 'en', 'url': 'https://arxiv.org/abs/1503.04577'} |
\chapter{Assembly, Installation and Engineering Issues}
\label{ch:AssemblyAndInstallation}
\input{AssemblyAndInstallation/chapter12.1.tex}
\input{AssemblyAndInstallation/chapter12.2.tex}
\input{AssemblyAndInstallation/chapter12.3.tex}
\input{AssemblyAndInstallation/chapter12.4.tex}
\input{AssemblyAndInstallation/chapter12.5.tex}
\section{Introduction}
After the completion of civil construction, through systems integration, assembly, installation and commissioning, all subsystems will be formed as a whole JUNO detector.
In this process, we need to establish various engineering
regulations and standards, and to coordinate subsystems' assembly, installation, testing and commissioning, especially their onsite work.
\subsection{Main Tasks}
Main tasks of the integration work include:
\begin{itemize}
\item To prepare plans and progress reports of each phase;
\item To establish a project technology review system;
\item To standardize the executive technology management system;
\item To have strictly executive on-site work management system;
\item To develop and specify security management system on-site;
\item To prepare common tools and equipment for each system, and to guarantees project progress;
\item To coordinate the installation progress of each system according to the on-site situation.
\end{itemize}
\subsection{Contents}
The work contents mainly include:
\begin{itemize}
\item To summarize the design, and review progress of each subsystem;
\item To organize preparation work for installation in the experiment region;
\item To inspect and certify Surface Buildings, underground Tunnels, and Experiment Hall with relevant utilities;
\item To coordinate technology interfaces between key parts;
\item To coordinate the procedure of assembly and installation both on surface and underground;
\end{itemize}
\section{ Design Standards, Guidelines and Reviews}
\subsection{Introduction}
There will be a document to outline the mechanical design standards or guidelines that will be applied to the design work. It also describes the review process of engineering design that will be implemented to ensure that experimental equipment meets all requirements for performance and safety. The following is a brief summary of the guidance that all mechanical engineers/designers should follow in the process. For reasons ranging from local safety requirements to common sense practices, the following information should be understood and implemented in the design process.
\subsection{Institutional Standards}
When specific institutional design standards or guidelines exist, they should be followed. The guidelines outlined are not meant to replace but instead to supplement institutional guidelines. The majority of equipment and components built for the JUNO Experiment will be engineered and designed at home institutions, procured or fabricated at commercial vendors, then eventually delivered, assembled, installed, tested and operated at the JUNO Experimental facilities, Jiangmen, China. The funding agencies in other countries as well as Jiangmen, China will have some guidelines in this area, as would your home institutions. Where more than one set of guidelines exist, use whichever are the more stringent or conservative approaches.
\subsection{Design Loads}
The scope of engineering analysis should take handling, transportation, assembly and operational loads into account as well as thermal expansion considerations caused by potential temperature fluctuations.
For basic stability and a sensible practice in the design of experimental components an appropriate amount of horizontal load (0.10~g) should be applied.
In addition, seismic requirements for experimental equipment is based on a National Standard of People's Republic of China Code of Seismic Design of Buildings GB 50011-2010. At the location of the JUNO Experiment the seismic fortification intensity is grade 7 with a basic seismic acceleration of 0.10~g horizontal applied in evaluating structural design loads. A seismic hazard analysis should be performed and documented based on this local code. The minimum total design lateral seismic base shear force should be determined and applied to the component. The direction of application of seismic forces would be applied at the CG of the component that will produce the most critical load effect, or separately and independently in each of the two orthogonal directions. A qualitative seismic performance goal defines component functionality as: the component will remain anchored, but no assurance it will remain functional or easily repairable. Therefore, a seismic design factor of safety F.S. > 1 based on the Ultimate Strength of component materials would satisfy this goal.
Where an anticipated load path as designed above, allows the material to be subjected to stresses beyond the yield point of the material, redundancy in the support mechanism must be addressed in order to prevent a collapse mechanism of the structure from being formed.
The potential for buckling should also be evaluated. It should be
noted that a rigorous analytical seismic analysis may be performed
in lieu of the empirical design criteria. This work should be
properly documented for review by the Chief Engineer and appropriate
Safety Committee personnel.
\subsection{Materials Data}
All materials selected for component design must have their engineering data sources referenced along with those material properties used in any structural or engineering analysis; this includes fastener certification. There are many sources of data on materials and their properties that aid in the selection of an appropriate component material. There are many national and international societies and associations that compile and publish materials test data and standards. Problems frequently encountered in the purchasing, researching and selection of materials are the cross-referencing of designations and specification and matching equivalent materials from differing countries.
It is recommended that the American association or society standards be used in the
materials selection and specification process, or equivalency to these
standards must be referenced. Excellent engineering materials data have been
provided by the American Society of Metals or Machine Design Handbook Vol. 1-5 and
Mechanical Industry Publishing Co. in PRC, Vol 1-6, 2002, which are worth investigating.
\subsection{Analysis Methods}
The applicable factors of safety depend on the type of load(s) and how well they can be estimated, how boundary conditions have been approximated, as well as how accurate your method of analysis allows you to be.
\paragraph{Bounding Analyses:}
Bounding analysis or rough scoping analyses have proven to be
valuable tools. Even when computer modeling is in your plans, a
bounding analysis is a nice check to avoid gross mistakes. Sometimes
bounding analyses are sufficient. An example of this would be for
the case of an assembly fixture where stiffness is the critical
requirement. In this case where deflection is the over-riding
concern and the component is over-designed in terms of stress by a
factor of 10 or more, then a crude estimation of stress will
suffice.
\paragraph{Closed-Form Analytical Solutions:}
Many times when boundary conditions and applied loads are simple to
approximate, a closed-form or handbook solution can be found or developed. For
the majority of tooling and fixture and some non-critical experimental
components, these types of analyses are sufficient. Often, one of these
formulas can be used to give you a conservative solution very quickly, or a
pair of formulas can be found which represent upper and lower bounds of the
true deflections and stresses. Formulas for Stress and Strain by Roark and
Young is a good reference handbook for these solutions.
\paragraph{Finite Element Analysis:}
When the boundary conditions and loads get complex, or the correctness of the solution is critical, computer modeling is often required. If this is the case, there are several rules to follow, especially if you are not intimately familiar with the particular code or application.
\begin{enumerate}
\item Always bound the problem with an analytical solution or some other approximate means.
\item If the component is critical, check the accuracy of the code and application by modeling a similar problem for which you have an analytical or handbook solution.
\item Find a qualified person to review your results.
\item Document your assumptions and results.
\end{enumerate}
\subsection{Failure Criteria}
The failure criterion depends upon the application. Many factors such as the rate or frequency of load application, the material toughness (degree of ductility), the human or monetary risk of component failure as well as many other complications must be considered.
Brittle materials (under static loads, less than 5\% yield prior to failure), includes ceramics, glass, some plastics and composites at room temperature, some cast metals, and many materials at cryogenic temperatures. The failure criterion chosen depends on many factors so use your engineering judgment. In general, the Coulomb-Mohr or Modified Mohr Theory should be employed.
Ductile materials (under static loads, greater than 5\% yield prior to failure), includes most metals and plastics, especially at or above room temperature. The failure criterion chosen again ultimately rests with the cognizant engineer because of all the adverse factors that may be present. In general, the Distortion- Energy Theory, or von Mises-Hencky Theory (von Mises stresses), is most effective in predicting the onset of yield in materials. Slightly easier to use and a more conservative approach is the Maximum-Shear-Stress Theory.
\subsection{Factor of Safety}
Some institutions may have published guidelines which specifically discuss factors of safety for various applications. For the case where specific guidelines do not exist, the following may be used.
Simplistically, if F is the applied load (or S the applied stress),
and $F_f$ is the load at which failure occurs (or $S_s$ the stress
at which failure occurs), we can then define the factor of safety
(F.S.) as:
\begin{displaymath}
F.S. = F_{f} / F \quad\mathrm{or}\quad S_{s} / S
\end{displaymath}
The word failure, as it applies to engineering elements or systems, can be defined in a number of ways and depends on many factors. Discussion of failure criteria is presented in the previous section, but for the most common cases it will be the load at which yielding begins.
\subsection{Specific Safety Guidelines for JUNO}
Lifting and handling fixtures, shipping equipment, test stands, and fabrication
tooling where weight, size and material thickness do not affect the physical
capabilities of the detector, the appropriate F.S. should be at least 3. When
life safety is a potential concern, then a F.S. of 5 may be more appropriate.
Note that since the vast majority of this type of equipment is designed using
ductile materials, these F.S.'s apply to the material yield point. Experimental hardware that does not present a life safety or significant cost/schedule risk if failure occurs, especially where there is the potential for an increase in physics capabilities, the F.S. may be as low as 1.5. Many factors must be taken into account if a safety factor in this low level is to be employed: a complete analysis of worst case loads must be performed; highly realistic or else conservative boundary conditions must be applied; the method of analysis must yield accurate results; reliable materials data must be used or representative samples must be tested. If F.S.'s this low are utilized, the analysis and assumptions must be highly scrutinized. Guidelines for F.S. for various types of equipment are:
\begin{center}
\begin{tabular}{|p{3.5cm}|c|p{3.5cm}|}
\hline
Type of Equipment & Minimum F.S. & Notes \\
\hline
Lifting and handling & 3 - 5 & Where there is a risk to life
safety or to costly hardware,
choose F.S closer to 5. \\
\hline
Test stands, shipping and assembly fixtures. & 3 & \\
\hline
Experimental hardware & 1.5 - 3 & 1.5 is allowable for physics
capability and analysis where
method is highly refined \\
\hline
\end{tabular}
\end{center}
\subsection{Documentation}
It is not only good engineering practice to document the analysis, but it is an
ESH\&Q requirement for experimental projects. For this reason all major
components of JUNO Experiment will have their engineering analyses documented as Project Controlled Documents. Utilize your institutional documentation formats or use the following guidelines.
Calculations and analyses must
\begin{itemize}
\item Be hard copy documented.
\item Follow an easily understood logic and methodology.
\item Be legible and reproducible by photocopy methods.
\item Contain the following labeling elements.
\begin{itemize}
\item Title or subject
\item Originators signature and date
\item Reviewers signature and date
\item Subsystem WBS number.
\item Introduction, background and purpose.
\item Applicable requirements, standards and guidelines.
\item Assumptions (boundary conditions, loads, materials properties, etc.).
\item Analysis method (bounding, closed form or FEA).
\item Results (including factors of safety, load path and location of critical areas).
\item Conclusions (level of conservatism, limitations, cautions or concerns).
\item References (tech notes, textbooks, handbooks, software code, etc.).
\item Computer program and version (for computer calculations)
\item Filename (for computer calculations).
\end{itemize}
\end{itemize}
\subsection{Design Reviews}
All experimental components and equipment whether engineered or
procured as turn-key systems will require an engineering design
review before procurement, fabrication, assembly or installation can
proceed. The L2 subsystem manager will request the design review
from the project management office, which will appoint a chair for
the review and develop a committee charge. The selected chair should
be knowledgeable in the engineering and technology, and where
practical, should not be directly involved in the engineering design
effort. With advice from the L2, the technical board and project
office chair appoints members to the review committee that have
experience and knowledge in the engineering and technology of the
design requiring review. At least one reviewer must represent ESH\&Q
concerns. The committee should represent such disciplines as:
\begin{itemize}
\item Science requirements
\item Design
\item Manufacturing
\item Purchasing
\item Operational user
\item Maintenance
\item Stress analysis
\item Assembly/installation/test
\item Electrical safety
\item ESH\&Q
\end{itemize}
For the JUNO project there will be a minimum of two engineering design reviews: Preliminary and Final. The preliminary usually takes place towards the end of the conceptual design phase when the subsystem has exhausted alternative designs and has made a selection based on value engineering. The L2 along with the chair ensures that the preliminary design review package contains sufficient information for the review along with:
\begin{itemize}
\item Agenda
\item Requirements documents
\item Review committee members and charge
\item Conceptual layouts
\item Science performance expectations
\item Design specifications
\item Supportive R\&D or test results
\item Summaries of calculations
\item Handouts of slides
\end{itemize}
A final design review will take place before engineering drawings or specifications are released for procurement or fabrication. The L2 along with the chair ensures that the final design review package contains sufficient information for the final review along with:
\begin{itemize}
\item Changes or revisions to preliminary design
\item Completed action items
\item Final assembly and detailed drawings
\item Final design specifications
\item Design calculations and analyses
\end{itemize}
The committee chair will ensure that meeting results are taken and recorded along with further action items. The committee and chair shall prepare a design review report for submittal to the project office in a timely manner which should include:
\begin{itemize}
\item Title of the review
\item Description of system or equipment
\item Copy of agenda
\item Committee members and charge
\item Presentation materials
\item Major comments or concerns of the reviewers
\item Action items
\item Recommendations
\end{itemize}
The project office will file the design review report and distribute copies to all reviewers and affected groups.
\section{On-site management}
According to the experience from the Daya Bay Experiment, an effective technology review system has been well practiced. In JUNO, we will take it as a good reference and carry out for standardized review from the start of all system design, scheme
argument, technology checks, we should also establish management system, to cover engineering drawing control, engineering change control procedure, and mechanical design standards, guidelines and reviews, etc.
To control on-site work process, proper process should be introduced as well as a management structure. Safety training and safety management structure should be also introduced.
\section{Equipment and Tools Preparation on Site}
All of key equipment, devices and tools should be in place and get acceptance
for installation on site, including
\begin{itemize}
\item Cranes:
\begin{itemize}
\item 2 sets of bridge crane with 12T/5T load capacity, and lifting range should
cover any point for components delivery and installation in EH;
\item Several lifting equipment are needed in Surface Assembly areas, Assembly
Chamber underground, Storage area for Liquid Scintillation, and other Chambers
accordingly;
\end{itemize}
\item Removable lifting equipment: Forklifts, manual hydraulic carriages, carts,
etc.
\item Mobile vehicle for smaller items: Pickup van, battery car, and so on;
\item Several work aloft platform: Scissor lifts, Boom lifts, Scaffoldings and
Floating platforms, etc.
\item Common Tooling: complete kits of tool, fixtures, jigs, models, work-bench,
cabinets, and tables, etc.
\item Machine shop, Power equipment rooms, Control rooms are ready to be put into
use.
\item Safety measures and relevant equipment in place.
\end{itemize}
\section{Main Process of Installation}
Since there still are several optional design versions which could
be decided later, and different design could have different
requirements for installation, therefore, the specific installation
procedures will be established and developed later, here a very
rough installation procedure is given.
\begin{itemize}
\item Mounting, debugging, testing of basic utilities, such as, cranes, vertical
elevator, scaffoldings, test devices, etc.
\item Survey, alignment, and adjustment for tolerances of form and position, include,
embedded inserts, anchor plates, positioning holes, etc.
\item Mounting Bottom Structure of Veto Cherenkov detector on the floor of Water Pool.
\item Mounting anchor-plate for support poles of CD on the floor of Water Pool
\item Pre-mounting of Top Structure of Veto system along wall of Water Pool
\item Mount Substrate and Rails for Bridge over the water Pool edges
\item Mount Tyvek with support structure for Water Cherenkov Detector, and with PMT
positioning in the Water Pool
\item Installation for CD with its PMT system and cabling in Water Pool details will be established until design final selection.
\item Mount Calibration System
\item Mount Geomagnetic Shielding system around Central Detector in the Pool
\item Final installation for Tyvek and PMT of Veto system in the Pool
\item Mount a temporary protective cover on top of Pool, to prevent from anything
falling into Pool
\item Mount Bridge on the rails, and drive it closed
\item Mount Top Track Detector on Bridge and establish Electronic Room there
\item Have Cabling completion, and test to get normal signals
\item Check Grounding Electrode
\item Dry Run
\item Final Cleaning for Water Pool, then, drain out
\item Filling Water, LS, LAB (or Oil) evenly into Water Pool and CD
\item Mount Final Pool Cover, closed, and get fixation
\end{itemize}
\chapter{AssemblyAndInstallation}
\label{ch:AssemblyAndInstallation}
\input{AssemblyAndInstallation/chapter12.1.tex}
\input{AssemblyAndInstallation/chapter12.2.tex}
\input{AssemblyAndInstallation/chapter12.3.tex}
\input{AssemblyAndInstallation/chapter12.4.tex}
\input{AssemblyAndInstallation/chapter12.5.tex}
\chapter{The Central Detector Calibration System}
\label{ch:Calibration}
\section{Requirements on the Calibration System}
\label{sec:cal_req}
The neutrino oscillation signal in the JUNO experiment will be
quantified by the measurement of the positron energy in the inverse
beta decay (IBD) interaction.
The oscillation patterns are driven by the atmospheric mass-squared
splitting $\Delta m^2_{32}$ and the third neutrino mixing angle $\theta_{13}$,
and the solar mass-squared splitting $\Delta m^2_{21}$ and the solar mixing
angle $\theta_{12}$. The interplay of the two
contains the neutrino mass hierarchy (MH) information
through the energy dependence of the
effective atmospheric mass-squared splitting.
Therefore, one needs to determine the positron energy spectrum with high
precision in order to determine the neutrino MH and
to carry out the precision measurements of the oscillation parameters.
As stressed in Chapter~\ref{chap:intro} as well as in
independent studies~\cite{Qian:2013}, the JUNO central detector energy
resolution needs to be better than 3\% at 1~MeV (1.9\% at 2.5 MeV where
the MH signal lies), and the absolute energy
scale uncertainty to be much better than 1\% for the MH determination.
On the other hand, the requirement on the
energy resolution and the energy scale uncertainty is more relaxed for
the sub-percent precision measurements of the oscillation parameters.
There are two main contributors for the energy resolution of the liquid
scintillator (LS) detectors. The first is the number of collected scintillation photons,
which is influenced by the intrinsic light yield and attenuation length
of the LS, the photocathode coverage of the detector,
and the quantum and collection efficiencies of the photon detectors.
The second is the non-uniformity of the detector response, governed by
the detector geometry as well as the optical properties
of the relevant detector components (e.g., the attenuation of the liquid
scintillator and acrylic vessel). Nevertheless, the non-uniformity can in principle
be reliably evaluated by deploying radioactive sources with known
energies inside the detector during the calibration data taking.
Based on experiences from the Daya Bay experiment,
liquid scintillator detector has two major sources of energy non-linearity.
The first one is the intrinsic energy non-linearity of the liquid scintillator,
which includes the quenching of the scintillation light and the amount
of the Cerenkov light within the energy ranges of interest.
The second one is the energy
non-linearity introduced by the electronics. A precise determination of both
effects to sub-percent level is the primary challenge
that the JUNO calibration system has to confront.
The most reliable handle on the non-linearity effects
is through the \emph{in-situ} calibrations, since the
dependence on particle types as well as the Cerenkov radiation is difficult
to evaluate in a bench setup elsewhere.
To accurately address both the the non-uniformity
and non-linearity in the detector energy response, the calibration system
is required to deploy multiple sources (light or radioactive)
to a wide range of positions inside the detector.
Specifically, a summary of different types of detector response calibration
envisioned
is shown in Table~\ref{table:calib}. In addition to actively
deploying sources, the reconstructed background events can also provide
critical inputs to the detector response.
\begin{table}
\caption{\label{table:calib}Summary of the different types of calibration goals, methods and requirements}
\begin{tabular}{|p{1.25in}|p{2.1in}|p{2.1in}|}
\hline
Calibration Goal & Calibration Method & Requirements\tabularnewline
\hline
\hline
PMT gain & Low intensity light sources at the center & Periodical, low intensity, automatic\tabularnewline
\hline
Light yield (energy scale) & Radioactive sources at the center & Periodical and automatic\tabularnewline
\hline
PMT timing, time walk & Light sources at the center & Periodical and variation in the light intensity\tabularnewline
\hline
Optical property of scintillator & Radioactive sources at various positions & Periodical with optimized positions\tabularnewline
\hline
Boundary effect of energy response & Pre-installed guide tube system with radioactive sources & Scan the
detector boundary and 1-2 times during whole period compare source and cosmogenic data\tabularnewline
\hline
Detector response non-uniformity & Radioactive sources at various positions and full volume cosmogenic data
& Need based calibration frequency, targeted positions, compare source
and cosmogenic data\tabularnewline
\hline
Capture time non-uniformity & Neutron source at various positions & Need based calibration frequency, targeted positions, compare source and spallation neutrons \tabularnewline
\hline
Energy non-linearity & Various radioactive sources & Full calibration with various sources 1-2 times during the entire period, compare source and cosmogenic
data, periodical check for 2-3 types of sources at targeted positions, \tabularnewline
\hline
Energy non-linearity of positrons & Mono-energetic positrons to the center and various position along
the central axis & Need based calibration frequency \tabularnewline
\hline
Position-dependent energy non-linearity & Various radioactive sources at fine position coverage & 1-2 times during the whole period, compare source and cosmogenic data, periodic check at targeted positions with 1-2 sources\tabularnewline
\hline
\end{tabular}
\end{table}
Based on the considerations above, we set the following
requirements to the calibration system,
\begin{enumerate}
\item In order to carry out periodical calibrations, the most frequently used
calibration system should be fully automated (similar to the ACU system in
the Daya Bay experiment). It should be simple and extremely reliable.
Such a subsystem should be able to deploy multiple
radioactive and light sources into the detector, likely along the central
axes;
\item The calibration system is required to access the off-center positions in
order to control the position non-uniformity. The source deployment to such
locations is likely to occur, on the other hand, less frequently compared to
that of the central axis;
\item Gamma source energies should cover a significant part of the IBD energy
spectrum (1-7~MeV). Positron and
neutron sources are also needed to calibrate the detection efficiencies;
\item A dedicated deployment system that has nearly $\sim$ 4$\pi$ coverage
to the central detector should be considered;
\item The absolute source positions should be controlled to better than
5~cm (preliminary Monte Carlo simulation has shown that the event
vertex reconstruction accuracy can be $\sim$ 10~cm);
\item An accelerator that can supply mono-energetic $e^{+}$ and $e^-$ beams
is important to measure the positron non-linearity directly;
\item Multiple types of particles such as $\alpha$, $e^-$, $\gamma$,
and $e^{+}$ should be deployed to understand the LS quenching and
Cerenkov light contribution to the energy non-linearity;
\item With the photo-luminescence effect in liquid scintillator,
a UV laser and optical fiber system has superior properties than those of an LED.
Such more stable and versatile laser systems should be considered as
light sources to calibrate the nonlinearity of the photo-sensors and the
electronics.
\end{enumerate}
Considering all these requirements and the mechanical constraints, we
selected the following mutually complementary calibration options
for further investigations.
\begin{enumerate}
\item A central automated calibration unit (ACU) for vertical source deployment;
\item A rope loop system for off-center source deployment;
\item A pelletron system which can provide mono-energetic positron beams;
\item Remotely operated under-liquid-scintillator vehicles (ROV) for
``4$\pi$'' coverage;
\item Pre-installed guide tubes; and
\item A diffuse system that can introduce short-lived radioactive isotopes
into the central detector.
\end{enumerate}
The ACU, the rope loop system and the ROV can all scan the interior of the
central detector but with different ranges and frequencies - we will describe
them together. The pre-installed guide tube system is uniquely
designed to understand the
boundary effects of the central detector. The pelletron calibration system
provides an excellent bench mark of the energy non-linearity for the IBD.
A diffuse source system can validate the energy non-linearity and its
position dependence. We will describe these three systems separately.
\section{Conceptual Designs}
\subsection{Housing and Interfaces}
Calibration sources are deployed into the detector through the top
chimney. To avoid Rn contamination, the entire calibration system should be
enclosed in a clean and Rn free volume (``calibration house''),
as shown schematically in Fig.~\ref{fig:calib_house}.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.7\textwidth]{Calibration/figs/calib_house.png}
\par\end{centering}
\caption{\label{fig:calib_house}Illustration of a seal calibration house
on top of the detector.}
\end{figure}
A concept of the calibration house is shown in Fig.~\ref{fig:cal_overview}. The ACU
system is located at the top. Two rope loop systems are placed
inside the house. The source changing area can be accessed via two glove
boxes to allow manual operation. The ROV system is also kept inside the
house and moved along the rails attached to the roof.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.7\textwidth]{Calibration/figs/overview.jpg}
\par\end{centering}
\caption{\label{fig:cal_overview}Concept of a calibration house.}
\end{figure}
\subsection{Sources}
\subsubsection{Routine Sources}
To set the energy scales and to calibrate the positron and neutron detection
efficiencies, radioactive sources for JUNO include
routine gamma and positron sources together with neutron
sources with correlated high energy gamma ray emissions, shown in
Table~\ref{tab:sources}.
\begin{table}
\caption{\label{tab:sources}Radioactive sources under consideration in JUNO.}
\begin{tabular}{ccc}
\hline
Source & Type & Radiation \\\hline
$^{40}$K & $\gamma$ & 1.461 MeV\\
$^{54}$Mn & $\gamma$ & 0.835 MeV \\
$^{60}$Co & $\gamma$ & 1.173 + 1.333 MeV \\
$^{137}$Cs & $\gamma$ & 0.662 MeV\\
$^{22}$Na & e$^{+}$ & annil + 1.275 MeV\\
$^{68}$Ge & e$^{+}$ & annil 0.511 + 0.511 MeV\\
$^{241}$Am-Be & n, $\gamma$ & neutron + 4.43 MeV \\
$^{241}$Am-$^{13}$C or $^{241}$Pu-$^{13}$C & n, $\gamma$ & neutron + 6.13 MeV \\
$^{252}$Cf & multiple n, multiple $\gamma$ & prompt $\gamma$'s, delayed n's\\
\hline
\end{tabular}
\end{table}
To minimize the risk due to contamination of the radioactivities,
the enclosure of these sources should be thin-walled ($\sim$ mm)
stainless steel (SS) capsule, enclosed by a round-headed acrylic
shell to ensure chemical compatibility with the liquid scintillator.
The deployment rope, made with SS, shall be robustly
attached to the SS source enclosures. Two steel weights are attached
above and below the source to maintain a minimum tension in the rope. A
typical source/weight assembly used in Daya Bay is shown in Fig.~\ref{fig:source}.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.5\textwidth]{Calibration/figs/source_weight.png}
\par\end{centering}
\caption{\label{fig:source}Illustration of a typical source/weight assembly
used in Daya Bay, envisioned for JUNO as well.}
\end{figure}
Simulations are underway to optimize the geometry in order to
minimize impacts of the source assembly to the energy scale. In
Fig.~\ref{fig:Co60} a typical $^{60}$Co spectrum is shown. The ``dead volume''
of the source assembly affects the low energy shoulder, but has a much
smaller effect on the full absorption peak. The optical
shadowing of the source assembly, on the other hand, will bias the location
of the full absorption peak.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.5\textwidth]{Calibration/figs/Co60.png}
\par\end{centering}
\caption{\label{fig:Co60}an example JUNO Monte Carlo
spectrum in photoelectrons for the $^{60}$Co source with source geometry.}
\end{figure}
\subsubsection{Mini-balloon}
In order to study the liquid scintillator response to charged particles
(electrons, positrons, $\alpha$'s) thereby to control the quenching effects,
we are pursuing a ``mini-balloon'' concept (Fig.~\ref{fig:balloon}).
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.5\textwidth]{Calibration/figs/balloon.png}
\par\end{centering}
\caption{\label{fig:balloon}The mini-balloon source concept.}
\end{figure}
Radioactive isotope is loaded
into the liquid scintillator inside a small, thin-walled,
and transparent balloon ($\sim$ 10 cm OD). The balloon is further
enclosed by an acrylic cylinder ($\sim$ 20 cm OD) filled with undoped
liquid scintillator.
Such a design will minimize the energy loss of the charged particles
across the balloon wall to simulate real events, while maintaining
a double encapsulation of the radioactive isotopes. A comparison of the
$^{40}$K-loaded 10~$\mu$m thick balloon and that without the balloon from
the Monte Carlo is shown
in Fig.~\ref{fig:K40}, in which the bias in beta energy is estimated to be less than
0.3\%.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.5\textwidth]{Calibration/figs/K40.png}
\par\end{centering}
\caption{\label{fig:K40}Comparison of $^{40}$K spectra with a 10 $\mu$m balloon
and without.}
\end{figure}
One important
advantage is that the choice of the dopant in the balloon is very versatile,
for example $^{222}$Rn ($\alpha$'s), $^{40}$K ($\gamma$'s and $\beta$'s), $^{137}$Cs
($\gamma$'s, $\beta$'s and conversion electrons), $^{68}$Ge (positrons), and
even short-lived isotopes. The source deployment system can then take
the source into given locations inside the detector for calibration.
\subsubsection{Pulsed Light Source}
Unlike the radioactive sources, the pulsed light source can generate
photons at a given time with tunable intensity. These features are important
to calibrate the gains of the photomultipliers as well as
the response of the electronics system. Based on the experience in the Daya Bay
calibration system, the stability of the pulses from light emitting diodes or
LEDs ($\sim$5 ns timing) is not as good as pulsed laser
system (which could achieve a ns level timing precision). Furthermore,
a UV laser of $\sim$260~nm
wavelength can excite the LAB molecules thereby producing photons with
similar timing characters as those generated by real particle interactions.
In addition, commercial light sensors, e.g., Si diode can monitor
the intensity of the laser pulses to very high precision.
Based on these considerations, we are developing
a UV laser system coupled to a flexible optical fiber with
a diffuser sphere at the end to achieve a stable, uniform, tunable,
and deployable light source for JUNO.
\subsection{Source Deployment Systems for the Detector Interior}
As mentioned in Sec.~\ref{sec:cal_req}, three complementary subsystems are
under consideration, the ACU system, the rope loop system, as well as
the ROV system. They cover different ranges inside the central detector,
and are envisioned to be used at different frequencies. In addition to
the mechanical means to position a source, independent source locating
systems will be critical to ensure the required high position accuracy.
\subsubsection{Source Locating Systems}
Although the rope loop can position itself via the lengths of different
rope sections, mechanical uncertainty such as deformation of the ropes
would introduce uncertainty to the absolute source position. Similar
uncertainty arises in the ROV system as well.
A dedicated source locating system is required to
determine the source position to better than 5~cm.
Two options are
being considered. The first option is an ultrasonic system
with an array of ultrasonic receivers and an ultrasonic emitter attached to the
source. The emitter generates pulsed sonar waves and the receivers
reconstruct the origin via the timing or phase difference. The wavelength of
150~kHz sonar wave is $\sim$ 1~cm, introducing negligible uncertainty.
The dominating systematics is the positioning accuracy of the receivers
as well as the speed of sound in LAB under different temperature/pressure
conditions. Reflections at the interface are also systematics under study.
To allow package of the sound waves, the receivers have to be installed in the
inner wall of the vessel if the central detector housing is acrylic,
and not so critical if the balloon option is adopted.
In principle, the larger the receiver array, the higher the positioning
accuracy. Studies are undergoing to optimize the array. A typical
arrangement is shown in Fig.~\ref{fig:ultra}, with 9 redundant
receivers mounted on the wall.
\begin{figure}
\begin{centering}
\includegraphics[height=2.5in]{Calibration/figs/ultra.png}
\par\end{centering}
\begin{centering}
\caption{\label{fig:ultra}A typical placement of the ultrasonic receivers.}
\par\end{centering}
\end{figure}
An alternative technology is to locate the source via optical cameras (CCD)
and one reconstruct source position via image analysis.
Such a system is used by the Borexino experiment. The infrared
lighting of the CCDs poses minimal risk to the PMTs. Alternatively,
a battery driven LED can be attached to the source assembly to ease the
analysis.
\subsubsection{ACU}
The Automated Calibration Unit (ACU) system is a unit very similar to
that used in Daya Bay, capable of deploying a few different
sources along the central axis of the detector. As illustrated in
Fig.~\ref{fig:ACU}, four source deployment
units will be arranged on a turntable, which can select the source
to be deployed. A hole on the bottom plate of the ACU will be aligned to
the center of the chimney to the central detector.
Three of the sources (one light source
and two radioactive sources) will be attached permanently to three deployment
units. The fourth unit will have a changeable source fixture, providing
flexibility for the attachment of special sources.
The operation of the system will be fully automated to ensure routine (weekly)
deployment of calibration sources with high reliability. Based on the
experience from the Daya Bay experiment, a weekly deployment of the light
source allows PMT gain calibration to subpercent precision. A weekly
deployment of a gamma source, e.g., $^{60}$Co, is sufficient to track the
slow drift ($\sim$1\% per year) of the overall energy scale due to changes
in detector properties. Lastly, a routine deployment of the neutron source
can ensure the stability of the detection efficiency.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.6\textwidth]{Calibration/figs/ACU.png}
\par\end{centering}
\caption{\label{fig:ACU}A sketch of the ACU system.}
\end{figure}
\subsubsection{Rope Loop System}
The concept of a rope loop system is similar to that used in the SNO
experiment, illustrated in Fig.~\ref{fig:Rope-loop-solution}.
The source deployment position can be controlled
by adjusting the lengths of sections A and B of the rope.
In this design, one of the hanging points is through the central
chimney, and the other anchor is located at around 30$\circ$
latitude, allowing a theoretically 90\% 2-dimensional coverage
of a half vertical plane.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[height=2in]{Calibration/figs/rope-SphereInside.png}
\includegraphics[height=2in]{Calibration/figs/rope-SphereSide.png}
\par\end{centering}
\caption{\label{fig:Rope-loop-solution}Rope loop solution and its 2-dimensional coverage}
\end{figure}
A sketch to illustrate a deployment sequence is shown in Fig.~\ref{fig:dep_seq}.
With section A going through the central chimney
sources can be taken out of the detector and get changed. We are developing
a scheme which allows both an automated source change while maintaining
the option of manually changing the source via glove box.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[height=2in]{Calibration/figs/dep_seq.png}
\par\end{centering}
\caption{\label{fig:dep_seq}3-point hang design and volume coverage}
\end{figure}
The overall layout of the rope loop system on top of the central detector
is shown in Fig.~\ref{fig:rope_layout}. Two independent rope loop systems
are being considered, each covering a half vertical plane to allow
some control of the azimuthal symmetry of the detector response. Each system
has two spool drives A and B to adjust the lengths of the A/B rope in a
synchronous fashion. Rope A goes through a pulley, which is
attached at an end of an extendable lever arm. During the deployment,
the level arm is extended to move the pulley towards the center of the
chimney to lower the source into the detector. Once a deployment is completed,
the source is retrieved by extending B rope and shortening A rope. Once the
source is out, the pulley arm gets retracted to the side so that the source
change operation will be performed away from the chimney.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.8\textwidth]{Calibration/figs/rope_layout.png}
\par\end{centering}
\caption{\label{fig:rope_layout}An overview sketch of the rope loop
system.}
\end{figure}
Alternative designs allowing three-dimensional coverage are also under
considerations.
For example, if the anchor point is allowed to move on the latitude
circle, a 3-dimensional deployment can be made possible. However, with a
large diameter circle attached to the inner
surface of the detector, maintaining a smooth rail and reliable drive
of the anchor point is a non-trivial engineering challenge. Another
alternative 4-point rope design is shown in
Fig.~\ref{fig:alter_rope_system}. Three hang points, separated by 120$\circ$,
are placed
around the equator of the central detector. Three ropes, each going
through one hanging point and
the central calibration port, will connect to the source. By adjusting the
length of these three ropes,
one can achieve a volume calibration in the bottom half of the sphere.
With a fourth rope
connecting the source directly from the central calibration port,
the volume calibration on the upper sphere can be performed. In general,
three dimensional rope system would introduce significantly
more mechanical complexity. In such a case, a ROV system described below would become
a viable alternative.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[height=3in]{Calibration/figs/alternative_rope.png}
\par\end{centering}
\caption{\label{fig:alter_rope_system}Illustration of a 4-rope design.}
\end{figure}
Details of a single spool drive in the rope loop system
are shown in Fig.~\ref{fig:spool}. The
spool in the ACU will be under an identical design. The spool will be
made either with acrylic or PTFE. Helical grooves with $\sim$mm
separation will be machined on the spool,
and the deployment cable of less than 1.5~mm diameter will be wound into
the grooves without overlap. The rope capacity of each spool is set to be
50~m. Several additional measures will be implemented to avoid rope slipping
out of the groove, including a spring loaded Teflon press to constrain
the rope, a load cell to constantly monitor the tension in the cable, as
well as a co-moving spooling tracker so that the rope always
gets unwound perpendicular to the spool.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.6\textwidth]{Calibration/figs/spool.png}
\par\end{centering}
\caption{\label{fig:spool}An illustration of a spool drive.}
\end{figure}
To achieve automated source changing, a key component in the rope
system is a ``quick connection'' connector, which allows an automated
system to attach and detach a source to the rope. One of the options
under considerations is illustrated in Fig.~\ref{fig:quick1}.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.6\textwidth]{Calibration/figs/quick1.png}
\par\end{centering}
\caption{\label{fig:quick1}One of the quick connection option.}
\end{figure}
The upper piece is attached to the rope loop permanently.
We envision that an ultrasonic emitter can be attached to it for accurate
positioning with the SS rope being the core of a coaxial cable
carrying the electrical signals. The middle piece is the female connector,
which is connected permanently to the upper piece with a rotary joint.
Gravity will keep the middle piece horizontal to ease automated grabbing.
The lower piece is the male connector, permanently attached to the source
and the lower weight.
The lock between the male and female is achieved with a spring-loaded key.
During the attachment (detachment), both the male and female pieces have
to be held by mechanical hands and pushed (pulled) relatively.
One additional degree of freedom is needed to
actuate the key in order to lock or unlock the connector. A manual
key is located on the female connector to easy the manual operation through
a glove box.
An alternative option of the connection is illustrated in Fig.~\ref{fig:quick2},
where the upper piece is the same and omitted in the drawing. The connection
is made doubly safe with two spring loaded locks. The head of the male key
has a dumb bell shape, which can be pushed in and pulled out of the upper
lock with force. The lower lock requires a hand to press the side buttons
in order to get unlocked.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.4\textwidth]{Calibration/figs/quick2.png}
\par\end{centering}
\caption{\label{fig:quick2}An alternative quick connection option.}
\end{figure}
All sources will be placed in a circular storage ring surrounding the
chimney (Fig.~\ref{fig:source_ring}), which can be driven automatically
in order to select a source to get attached/detached to the rope loop.
Such a design allows the two rope loops to share the same source storage.
For illustration, a
source assembly placed in the storage is also shown in the figure. For
each rope loop, all sources share the same mechanical hand system to perform
the source changes.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[width=0.7\textwidth]{Calibration/figs/source_ring.png}
\par\end{centering}
\caption{\label{fig:source_ring}The source storage ring.}
\end{figure}
\subsubsection{Remotely Operated Under-liquid-scintillator Vehicles}
A ROV system can deploy the source to nearly everywhere inside the central
detector. Such a system is used in the SNO experiment to deploy the
$^3$He neutron tubes. A conceptual design of the ROV is shown in
Fig.~\ref{fig:ROV}. It can be made into an egg-shaped capsule with
a diameter less than 300~mm and a height less than 500~mm.
The ROV is driven by pump jet propulsion without external propeller
in order to maintain a simple external geometry and a compact size.
The ROV motion speed is set to be about 1 m per min, and the actual
position is feedback via an ultrasonic emitter. The power as well as
signals are transmitted to the ROV via a umbilical cord adjusted to
nearly zero buoyancy. The ROV can also
carry lighting, CCD camera and a magnet to allow emergency rescue or
monitoring the interior of the detector.
Radioactive source assembly is attached below the ROV, with a quick
connection which would allow automated source change.
For material compatibility with LAB, the enclosure of the ROV will be made
with Teflon, which is also highly reflective to
avoid photon losses. Monte Carlo studies are underway to optimize
the ROV geometry in order to minimize impact due to dead materials or
optical shadowing.
\begin{figure}
\begin{centering}
\includegraphics[height=3in]{Calibration/figs/ROV.png}
\par\end{centering}
\caption{\label{fig:ROV}ROV, its umbilical cable and the loaded radioactive
source}
\end{figure}
Based on its current location and destination location, the ROV software
will automatically select the optimal trajectory. The pump jet propulsion
engine controls the speed and direction of the device. The ROV should
approach the designated locations with a speed less than 1~mm/s or less,
then shuts off the engine. A piston driven buoyancy adjustment
mechanism is used as the depth control. In case of system failure such as
power failure, the control will shut down the pump propeller and pull the
piston of the depth control via a spring to increase the buoyant force to
ensure the ROV float to the surface.
\subsection{Conceptual Design of the Guide Tube System}
Based on positive experiences of the Double Chooz experiment,
a guide tube system
is important to understand detector response from anti-neutrino
interactions occurring at the
boundaries of the central detector volume. Guide tubes will be
positioned along the surface of
the central detector vessel. The tubes could be installed at the
inner and/or outer surfaces of the vessel wall.
Figure~\ref{fig:guide_tube} shows the conceptual design of the guide tube system
with the tubes at the outer surface
(left), and with tubes at the inner surface (right) of the
central detector vessel. The number
of tubes shown is for illustrative purposes only, and the final
number and distribution
of the tubes will be based on detailed simulations of the guide
tube system and studies of detector energy response, currently
under development.
The tube can be made with stainless steel to provide a good
mechanical strength, or it could
be manufactured from an acrylic tube to be compatible with the
central detector vessel materials.
Radioactive sources as small as a few mm in diameter and $\sim$2~cm
in length could be positioned within the tube with use of a stainless
steel wire driven by a stepper motor.
Figure~\ref{fig:guide_tube1} shows one considered design of source capsule inside the
stales steel guide tube, with the source connected on both sides.
Similar designs are considered with the acrylic guide tube solution.
Source positioning accuracy of $\sim$1~cm is anticipated.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[height=3.0in]{Calibration/figs/juno-outer-GT.png}\includegraphics[height=3.0in]{Calibration/figs/juno_inner_GT-1.pdf}
\par\end{centering}
\centering{}\caption{\label{fig:guide_tube}The guide tube system. }
\end{figure}
\begin{figure}[~!htbp]
\begin{centering}
\includegraphics[height=3.0in]{Calibration/figs/guide_tube1.png}
\par\end{centering}
\centering{}\caption{\label{fig:guide_tube1} The conceptual design
of the source capsule inside the guide tube.}
\end{figure}
\subsection{A Pelletron-based Beam Calibration System}
A pelletron is one type of electrostatic accelerator. The electric charge
is transported mechanically to its high voltage terminals by a chain
of pellets, which are connected with insulating materials (such as nylon).
Figure~\ref{fig:Diagram-of-Pelletron} shows a diagram of a pelletron,
taken from the website of National Electrostatics Corporation
(NEC)~\cite{NEC:2014}. Compared with a LINAC, the electrostatic
accelerator has the advantage of being more stable. This is crucial
for the intended application in JUNO, as the calibration of the Pelletron
energy and the JUNO detector response calibration cannot be
performed at the same time. Furthermore, below
\textasciitilde{}5 MeV in kinetic energy~\cite{Hinterberger:1997ur},
the pelletron solution is
compact and economical enough for an underground installation.
\begin{figure}
\begin{centering}
\includegraphics[width=0.8\textwidth]{Calibration/figs/chargingsys}
\par\end{centering}
\caption{\label{fig:Diagram-of-Pelletron}Diagram illustrating the
principle of a pelletron. As the drive pulley and the terminal pulley
rotate clockwise, the chain transports the positive charge to the terminal
shell and builds up a high voltage that could be higher than 25 MV.
By reversing the polarities of the charging voltages, the pelletron can
easily switch to accelerate electrons and negative ions.}
\end{figure}
Expected positron spectra is shown in Fig.~\ref{fig:Espec}.
We see that most events are between 1 and 6~MeV
prompt energy. Due to finite energy resolution, the mass hierarchy signal
only shows up above 2~MeV prompt energy. Therefore, a pelletron system,
which can provide 1-5~MeV kinetic energy corresponding to 2-6~MeV
prompt energy, can satisfy the requirement of precision energy calibration
for mass hierarchy determination. Figure~\ref{fig:sys-diagram} shows
the conceptual design of the pelletron system, following the ideas
from Refs.~\cite{Bauer:1990zz,Huomo:1988jw}. The high purity Ge (HPGe)
detector, which is calibrated by mono-energetic gamma sources,
will be used to control the beam energy precisely. A precision
of $10^{-4}$, which is well below the 0.1\% goal of the JUNO experiment,
has been achieved with existing facilities~\cite{Bauer:1990zz,Berg:1992jr,Huomo:1988jw}.
\begin{figure}
\begin{centering}
\includegraphics[width=0.85\textwidth]{Calibration/figs/calibsys-diagram}
\par\end{centering}
\caption{\label{fig:sys-diagram} The conceptual design of the pelletron
system}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[height=4.0in]{Calibration/figs/beammonitors-2}
\par\end{centering}
\caption{\label{fig:endcap-concept}A conceptual design of the beam pipe endcap
inside the LS detector. The transparent acrylic pipe and the Mylar
window make sure the scintillation light can largely pass through.
Two beam monitors near the top and bottom of the beam will allow diagnostics of
beam position and area when the beam first enters the detector vertically downward
and near the exit point of the beam out of the beam.}
\end{figure}
The calibration of the JUNO detector with a pelletron requires the delivery of the
positron beam into the detector center through an evacuated beam pipe. The positron
beam will exit the beam pipe through a window designed to minimize energy losses in
the beam while maintaining the vacuum of the beam pipe at the depth of the detector.
The design and instrumentation of the pelletron beam pipe inside the detector is an
engineering and R\&D challenge.
\underline{Detector beam pipe requirements:} The technical requirements for the
positron beam pipe inside the detector can be summarized as follows:
\begin{itemize}
\item Retractable, telescoping beam pipe that can be deployed into the detector during
the time of calibration and retracted during normal data taking to avoid shadowing
effects in the detector. During normal data taking the beam pipe is stored completely
out of the detector volume.
\item Beam pipe is compatible with the scintillator. To first order, the exposure of
the scintillator to the beam pipe materials is limited to the duration and time
period of the calibration. However, one has to account for possible long-term
interaction of the scintillator with the beam pipe material after the beam pipe
has been retracted from the detector. Excess scintillator may drip back into the
detector or lead to unwanted contamination that may be introduced during the time
of the next calibration. Compatibility issues include possible interaction with the
materials of the beam pipe, damage of the deployment mechanism, leaching out of beam
pipe materials into the scintillator, and degradation of the scintillator itself. The
amount of beam line material to be deployed into the detector poses a challenge for
the purity of the scintillator and to JUNO physics goals that require ultra low backgrounds.
\item Beam pipe and its deployment mechanism have to be leak tight against the
liquid scintillator to maintain the vacuum inside the beam pipe.
\item Beam pipe deployment mechanism is failsafe and can always be retracted out of
the detector. Since the beam pipe does interfere with the normal data taking the
deployment mechanism must allow the retraction of the beam pipe even under unusual
circumstances such as power failure etc. The system has to be tested to be mechanically
reliable for the 10+ year lifetime of the experiment.
\item It is desirable to be able to deploy the beam pipe to different depths. This would
allow calibration of the detector with positrons at different points along the z-axis.
The interaction region of the beam with the scintillator detector is defined by the exit
point of the beam out of the mylar window. A continuous deployment along the z-axis may
not be necessary, discrete steps in z are sufficient.
\item Deployment of the beam pipe is automated as much as possible to allow for
regular calibration and minimize the demands on the on-site personnel.
\end{itemize}
Inside the beam pipe instrumentation to monitor the beam position and profile are
required to help determine the exit point of the beam and the interaction region of
the calibration beam inside the detector. This monitoring instrumentation will also
aid in diagnostics of the beam during tuning and setup. One of the challenges of the
proposed calibration scheme is that a positron beam has to be delivered some 17~m into
the detector center without active steering. We expect that from the entrance point of
the beam into the detector region (at the top of the detector) to the point where it
exits the beam pipe through the mylar window no active steering components can be used.
As a result the beam will spread and its final position at the end of the beam pipe will
depend on its direction when it enters the detector and possible divergence. As a result,
we obtain the requirements for the monitoring instrumentation inside the
detector beam pipe (see Fig.~\ref{fig:endcap-concept}):
\begin{itemize}
\item \underline{Top beam monitor:} Very precise beam position monitor at the top of the
detector where the beam enters the detector to determine its direction and profile into the
detector right after the last active magnetic steering and focusing. This beam position monitor
can be fixed along the z-axis. It can be flipped or moved into the beam during tuning and
setup and removed from the beam path during the beam delivery and calibration. The position
resolution requirements for this beam monitor are set by the distance of the beam delivery
into the detector and the width of the beam pipe. Depending on the position of the 90-degree
bending magnet of the beam into the detector this monitoring setup can still be outside
the detector but already in the vertical region of the beam pipe.
\item \underline {Exit beam monitor:} A second beam position monitor near the end of the
beam pipe that determines the position and area of the beam spot before it exits through
the mylar window. Again, this beam monitor may be designed to be movable so that it can
be placed in the beam during monitoring and setup and removed (rotated or flipped) during
calibration. It may be desirable to make this beam monitor movable along the z-axis to
allow for a vertical scan of the beam position. Since this beam monitor will be located
towards the end of the telescoping beam pipe all cabling and supplies have to be routed
through the retractable beam pipe structure. This be may not required for the beam monitor
at the top of the detector. With a beam position monitor near the top and the bottom of
the beam pipe it will be possible to diagnose potential issues in the delivery of the
position beam. More than two position monitors will likely lead to unnecessary and
increased complexity.
\end{itemize}
In the following, we summarize various factors that can induce a bias in the beam energy
calibration. The first one is the energy loss in beam window. At the center
of the detector, the pressure is about 3 atm. A 76~$\mu$m Mylar window is
more than enough to handle this pressure. The resulting energy loss is
about 12~keV and can be calibrated with the HPGe detector.
The variation of the energy loss due to curved window
is well below the 0.1\% of the prompt energy. The second factor
is the shadowing effect of the calibration pipe and endcap.
In order to minimize the impact of these two factors, a transparent
endcap design using acrylic pipe (Fig.~\ref{fig:endcap-concept})
is proposed. The residual shadowing effect is about 1-2\%. The range of
the shadowing effect is due to the kinetic energy range of positrons. A high
energy positron would in general travel longer inside the liquid scintillator,
thus has less shadowing effect due to smaller solid angle. This shadowing
effect can be calibrated by injecting both electron and positron beams into
the central detector and a bench setup. The principle is illustrated in the following:
\begin{itemize}
\item The energy responses of electron and positron are strongly correlated.
The positron energy response is essentially the sum of the
electron energy response and positron annihilation energy. The latter contains two
0.511~MeV gammas and can be calibrated using dedicated $^{68}$Ge radioactive source.
\item The shadowing effect for the positron annihilation energy can be calibrated
with the dedicated $^{68}$Ge source deployed together with the calibration tube.
\item The shadowing effect can be calibrated by comparing a
dedicated bench measurement with the results from the central detector calibration.
\item The light pattern observed by JUNO detector provides additional handle
for the shadowing effect.
\item The obtained shadowing correction for positron ionization
can then be applied to obtain the positron energy response together with
the shadowing of the annihilation gammas.
\item The resulting positron energy non-linearity model can then compared with that
of the electron to further validate the energy model.
\end{itemize}
The following formula summarizes the correction strategy.
\begin{equation}
E_{e+}^{corr} = (E_{e+}^{rec} - E_{1.022}\cdot S_{gamma}^{shadowing})\cdot S_{ionization}^{shadowing} + E_{1.022} + C_{window}^{eloss}.
\end{equation}
Here, $E_{e+}^{rec}$ and $E_{e+}^{corr}$ are the energies before and after
the corrections, respectively. The $E_{1.022}$ is the annihilation
gamma energy for positron at rest, which can be calibrated directly with
dedicated positron source. The $S_{gamma}^{shadowing}$ is the shadowing
correction for annihilation gamma, which can be calibrated by combining
dedicated source with the calibration tube. The $C_{window}^{eloss}$
represents the energy loss correction in the window, it can be calibrated
directly with the HPGe detector. The $S^{shadowing}_{ionization}$ represents
the shadowing effect for the scintillation light for positron ionization energy. This piece
will be calibrated by i) comparing the bench data vs. calibration data vs. MC,
and ii) comparing the observed light pattern vs. MC. The initial simulation
shows that the residual uncertainty can be controlled to about
0.1\% with this strategy.
Simulation of Pelletron calibration has been carried out in the general JUNO
simulation framework to demonstrate the feasibility of reaching 0.1\% targeted
energy scale uncertainty. First, for a 20 m long calibration tube inside the liquid
scintillator, the buoyancy force needs to be taken into account. The left panel
of Fig.~\ref{fig:ct_force} shows the required wall thickness of the stainless steel
calibration tube to balance the buoyancy force. The right panel of
Fig.~\ref{fig:ct_force} shows the weight of the 20~m calibration tube.
Figure~\ref{fig:tube} shows the implemented geometry of calibration tube including
i) $\sim$17.3~m long stainless steel calibration tube, ii) 30~cm long
acrylic endcap, and iii) 7.6 $\mu$m thick curved Mylar window.
The energy loss of the electrons and positrons inside the Mylar window was
simulated and shown in Fig.~\ref{fig:ct_eloss}. For positrons at off-center
locations, the energy loss is slightly higher than those at the center of
calibration tube. This is consistent with the fact that the vertical thickness
of the window at off-center location is bigger than that in the center.
The energy loss of electron is slightly bigger than that of positron
due to additional annihilation process that positrons can go through.
The difference is much smaller than the targeted 0.1\% energy scale
uncertainty band.
A beam of positrons are shooting through the calibration tube to study the
energy response. The positron beam cross section is assumed to be a circle
with radius of 1~cm. The spread in momentum is assumed to be 0.1\%.
The results are compared with those of injecting positrons directly inside
the detector center without any calibration tube (Fig.~\ref{fig:ct_result1}).
The increase of the difference between two cases with respect of the true
prompt energy reflects the shadowing effect of the calibration tube.
The bias without any correction (top panel of Fig.~\ref{fig:ct_result2})
indicate the magnitude of bias is about 1-2\%. The bias becomes smaller at
high energy, because high-energy positrons can penetrate into liquid scintillator
further so that the shadowing effect is smaller. The bottom panel of
Fig.~\ref{fig:ct_result2} shows the residual bias after the energy loss
and shadowing correction. The residual bias can be controlled to below 0.1\%.
\begin{figure}
\begin{centering}
\includegraphics[height=2.5in]{Calibration/figs/calibration_tube_force}
\par\end{centering}
\caption{\label{fig:ct_force} (Left panel) Calculated required wall thickness
of the stainless steel calibration tube is plotted as a function of the radius
of the calibration tube to balance the buoyancy force. (Right panel) The corresponding
weight of the 20 m long calibration tube is plotted.}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[height=1.5in]{Calibration/figs/tube1}
\includegraphics[height=1.5in]{Calibration/figs/tube2}
\includegraphics[height=1.5in]{Calibration/figs/tube3}
\par\end{centering}
\caption{\label{fig:tube} (Left) Illustration of calibration tube of Pelletron
inside the JUNO central detector.
(Middle) Curved Mylar window for the calibration tube.
(Right) Endcap of the calibration tube with the curved Mylar window. }
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[height=3.5in]{Calibration/figs/eloss}
\par\end{centering}
\caption{\label{fig:ct_eloss} Energy loss of the electrons and positrons
inside the curved Mylar window. For electron, the prompt energy was shifted
to match that of positron. The traipzoid represents 0.1\% band of the energy scale.}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[height=1.6in]{Calibration/figs/ct_result1}
\par\end{centering}
\caption{\label{fig:ct_result1} (Left) The energy response of positron
injecting through the calibration tube (red) is compared to that
of the positron in the detector without the calibration tube(black).
(Right) The difference in energy is shown with respect of the true
positron prompt energy. The increase of difference reflects the shadowing
effect of the calibration tube}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[height=2.5in]{Calibration/figs/ct_result2}
\includegraphics[height=2.5in]{Calibration/figs/ct_result3}
\par\end{centering}
\caption{\label{fig:ct_result2}(Top) The bias of energy response (1-2\%)
due to the shadowing effect of the calibration tube is shown with respect of
the true prompt energy.
(Bottom) The residual bias of energy response ($<$ 0.1\%) after the
correction of energy loss and the shadowing effect is shown with respect of
the true prompt energy.
}
\end{figure}
\subsection{Diffused Short-lived Isotope Calibration}
Homogeneous energy calibration of a large liquid scintillator detector throughout
the entire volume can only be achieved with distributed sources in
the scintillator. The use of uniformly distributed sources was successfully
demonstrated by the Sudbury Neutrino Observatory using $^{16}$N,
$^{8}$Li, as well as \emph{in-situ} spikes from $^{24}$Na and $^{222}$Rn.
Injection and deployment of such sources in a 20 kt-detector pose challenges
including the production and injection of isotopes over long distances
and their distribution throughout the detector volume. We will survey
suitable short-lived isotopes, their production method and develop
concepts for the injection with a gas or scintillator stream into
the detector. We will pay particular attention to the possible use
of positron sources. We will identify possible isotopes and develop
a technical concept for the injection and distribution of sources.
The principles of the isotope injection will be studied with the test
chamber developed for various R\&D purposes or a re-purposed Daya Bay
detector.
\section{R\&D Status and Plans}
\subsection{Sources}
Source selection and conceptual design of the geometry
need to be determined via Monte Carlo. Mini-balloon source requires
prototype both mechanically as well as test loading of radioactive
isotopes. A robust and simple UV laser system needs to be developed
allowing $\sim$ns level timing and sub-percent intensity control.
\subsection{Rope Loop System}
We have learned much from the Daya Bay calibration experiences. The
challenges of the rope loop approach lie in the automatic source swapping
devices, interfacing with the central detector and the installation
procedure. A 1:1 prototype for the mechanical system is required before
the final design. We shall also find high bay area to performance deployment
tests similar to that in the real detector. Additional key considerations
are:
\begin{itemize}
\item {The rope loop subsystem will be installed after the cleaning of the
central detector, or even after LS filling, thus installation
procedures should be simple and they must be carried out quickly to avoid
contamination;}
\item {LS detector with such a large size is unprecedented. Calibration
system should be designed so it can accommodate potential structural
distortions; and}
\item {The reliability of the source changing devices and life span should be
understood well with destructive tests.}
\end{itemize}
\subsection{ROV}
To achieve good physics performance, we need
to optimize the shape and the dimension of the ROV via extensive
simulations.
The ROV shall be designed with high cleanliness and be fully compatible with
liquid scintillator. Prototype studies are needed in a realistic deployment
condition.
\subsection{Source Positioning System}
An ultrasonic emitter's diameter is roughly 30~mm and its working frequency
150~kHz with a pulse width of $\sim$1 ms.
The emitter can be installed either on the source or on
the ROV.
The receiver array is mounted on the detector sphere and at least
three receivers can
realize 3D positioning. The actual arrangement of the array need to be
optimized. To increase the reliability, we plan to have $\sim$10
receivers so when some malfunction or break down, the system can still
function. Longer array baselines means better positioning
precision, which implies receivers are better placed near the equator from
the perspective of positioning. However, this also means the cables serving
these receivers are longer and it might cause more potential issues. The
material compatibility between cables and LS needs to be checked and
addressed. Compared with the oceanic environment, JUNO central detector
is an isolated environment. Impact to positioning uncertainty
due to reflections and refraction shall be well studied and understood.
\subsection{Pre-installed Guide Tubes}
How to mount the guide tubes next to the acrylic panels is quite
challenging due to the distortion after the detector is filled. The maximal
distortion of the acrylic panel is expected to be $\sim$5~cm thus if the guide
tubes are fully attached to acrylic panels, this level of distortion is
highly harmful for the smoothness of the tube even the joint does not get
destroyed. We need to allow certain level of distortion thus probably the
two should not be fully coupled. Instead we should allow the relative
movement to some level while still keep the positioning accurate
enough.
In addition to the mechanical concerns, the impact of these additional,
tube materials to the detector performance needs to be
understood better. It should not make much difference they are deployed either inside or
outside of the acrylic sphere if refraction indices match well with LS
and/or mineral oil. Acrylic, stainless steel, and Teflon tubes tubes are considered.
Right now, the system realization and impact to the detector
performance are undergoing discussion. Argonne National Lab has a
high bay area, suitable for performing deployment
tests of the rope loop system, and tests of the guide tube systems.
\subsection{Pelletron System}
The need of space and transportation underground have been checked and no
obvious obstacles were observed. Currently, we need to detail out the beam
line design and the interfacing with the central detector.
\subsection{Diffused Short-lived Isotope Calibration}
Various sources are being studied. The concept of source injection is being
investigated.
\section{System Reliability and Safety Concerns}
It would be difficult to make large scale repair to the
calibration system during the life span of the
experiment. This places very stringent challenges on the reliability and
longevity of the calibration system, especially that there should absolutely
be no source dropping.
For the automatic source swapping component of the rope loop system, we have
considered the following multi-layer safety protection mechanism,
\begin{enumerate}
\item {The position and the clamping force of the robotic hands are
constantly monitored after each motion;}
\item {Robotic hands and other clamping devices are designed to be
normally-close to make sure that sources are still secured during power or
control signal loss;}
\item {To make sure of the reliability of the quick-connects, they are
pull-tested each time they engage;}
\item {A fall-prevention shutter is installed beneath the source-swapping
device. During source swapping, the shutter is closed; and}
\item {A piece of annealed iron will be installed inside the source. In case
of dropping, a ROV with a strong magnet can be deployed to recover it.}
\end{enumerate}
The ROV umbilical cable can take certain load besides functioning as a power
and control signal cable. Thus in case of malfunction, it can be pulled out
of the central detector. In addition, the design of the ROV also has some
built-in safety considerations and the ability of self test.
The effective density of the ROV should be roughly 95\% of LAB. During power
loss or other situations of losing motion, the ROV would float
automatically. The liquid dragging force shall be chosen so that the rising
speed is not too high, avoiding damaging the vessel. The internal pressure
of the ROV should be either positive or negative and should be constantly
monitored. Larger-than-normal fluctuations would indicate a leak might have
developed and an alarm signal shall be sent out.
\section{Schedule}
A major contribution to the calibration system and additional contributions in support
of central detector, liquid scintillator and PMT QA are being investigated for US JUNO
deliverables.
\clearpage
\begin{longtable}{|p{1.5cm}|p{4cm}|p{3cm}|p{5cm}|}
\caption{Yearly targets} \\
\hline
Year & Goals & Deadlines & Milestones \tabularnewline
\endfirsthead
\multicolumn{4}{r}{(See Next Pages.)}
\endfoot
\bottomrule
\endlastfoot
\hline
2013 &Form a working group, start conceptual designs and
simulation &Form a few preliminary mechanical designs &Form 3 alternative
conceptual mechanical designs before November\tabularnewline
\hline
2014 &Continue design and gradually converge; Technique R\&D; Design light
sources and build mock-ups &Finish the simulation of source selection;
finish the design of a stable light source &Finish the stable light source
design before June; Finish the simulation and analyses of the 3 alternative
designs; Select 2 out 3 in September\tabularnewline
\hline
2015 &Build and test a prototype; finalize both the mechanical and the
electronic designs &Finish the R\&D of selecting radioactive sources;
finalize the choice of radioactive sources & Finalize the mechanical design
before June; finalize the construction and testing of the prototype, and the
radioactive source designs before December\tabularnewline
\hline
2016 &Bid, order and start the construction of calibration systems & finish
one complete calibration system and testing & finish bidding before June and
complete one calibration system and finish its testing before
December\tabularnewline
\hline
2017 &Continue the construction of the calibration system & Complete the
remaining half of the project &Complete the remaining half of the project
\tabularnewline
\hline
2018 &Complete the construction of the calibration system; order all
radioactive sources; complete DAQ and the interface with DCS &Compelete the
last half of the project & Complete all remaining constructions, DAQ and the
interface between DAQ and DCS before December\tabularnewline
\hline
2019-2020 &Installation and commissioning &Complete the installation and
commissioning of the system & complete the calibration system before the
completion of the central detector installation\tabularnewline
\hline
\end{longtable}
\chapter{Central Detector}
\label{ch:Central Detector}
\section{Introduction and requirement of the central detector}
The central detector of JUNO aims to measure the neutrino energy spectrum using
$\sim$20~kt liquid scintillator (LS) and $\sim$17,000 PMTs. An inner sphere with a diameter
of around $\sim$35.4~m is designed to contain the huge volume of LS, and an outer
structure with a diameter of around $\sim$40~m is needed to support the inner sphere
as well as PMTs. In order to get the $3\%/\sqrt{E}$ energy resolution, the
central detector is required to maximize the collection of optical signals from
LS meanwhile minimize the background from a variety of radioactive sources.
Since it is not possible to repair during the operation, this detector
must have a long life time and a high reliability.
Since the LS and PMTs are separately documented in Chapter 4 and Chapter 6, this chapter is mainly focused on the challenges:
1) Construction of the large structure of the sphere and its support;
2) Test and installation of the $\sim$17,000~PMTs;
3) Filling of LS and long term operation of the detector.
Throughout the design effort, the strength and stability of the
structure is the drive. In addition, the working conditions in the
underground hall, safety issues during the construction, cost, and the overall
time schedule should be taken into account. The ideal design of the JUNO
central detector is a stainless-steel tank plus an acrylic sphere, where the
stainless-steel tank is used to separate the shielding liquid (mineral oil or
LAB, to be decided) from the pure water in the water pool, and the acrylic
sphere is to hold the $\sim$20~kt LS, as indicated in Fig.~\ref{fig:intro:det}. Due to the limitation
of space, it is very difficult to simultaneously construct the two spheres and
there is also a high risk and significant schedule delay if the two spheres were
constructed in series. As a result, alternative options could be:
1) the stainless-steel tank is replaced by an open space truss or other steel supporting structures;
2) the acrylic sphere is replaced by an off-site fabricated balloon.
Currently, the baseline is the first one, i.e. the so called acrylic sphere
plus stainless-steel support option, which will be detailed in this document.
The second option of the balloon plus stainless-steel sphere is a backup, and
will be described also in this chapter.
The main requirements for the JUNO central detector are the following:
1) The detector should meet the physics requirements of the JUNO experiment,
with $\sim$20~kt high purity LS and $\sim$17000 high Q.E. PMTs to reach an energy
resolution of $3\%/\sqrt{E}$;
2) The detector should minimize the radioactive background from different
sources, including environment, structure, materials and the pollution from
the construction process;
3) The detector structure should be reliable and its design should meet the
standards in fields of large vessel, civil architecture and engineering. Leakage is not allowed between different spheres. The structure should be safe up to a seismic intensity of Richter scale 5.5, and should not be
sensitive to temperature variation. The structure must have no single point of failure;
4) Materials used for the detector should have a long-term compatibility (about $\sim$30~years) with the liquid scintillator and pure water;
5) The central detector should provide a proper interface to other systems, i.e. VETO, calibration, electronics and so on;
6) The total construction duration should be reasonably short and not longer than 18~months;
7) The lifetime of the detector should be longer than 20~years, and during this period, no significant repair is needed;
8) The cost of the central detector should also be at a reasonable level.
\section{Baseline option: Acrylic Sphere with Stainless-Steel Support}
\subsection{Introduction and the design requirements}
In the baseline design the inner sphere is made of acrylic with a thickness of
$\sim$12~cm and an inner diameter of 35.4~m. Surrounding the acrylic sphere there is a
stainless-steel structure which supports the acrylic sphere and also the PMTs.
Currently the stainless steel support structure has three options: the double
layer truss, the single layer truss and the equator supporting method. For the
first two options, the truss can also be used to fix the PMTs, but for the
equator supporting method, an extra frame is needed for PMT installation. The
inner diameter of the stainless-steel truss is ~40~m in the present design,
supported by a number of columns which are built on the base of the water pool.
For the acrylic sphere, a chimney with an inner diameter of $\sim$1~m will extend up
from the top of the sphere, serving as the interface for the calibration system.
The relative pressure between the LS and the water has a great effect on the sphere. The
chimney will be a few meters higher than the water level in order to provide
flexibility in setting the LS relative height and the resulting stress in the
sphere. Due to the height of the chimney above the sphere, optical isolation is
needed to block the background induced light from being detected by the
PMTs. Between the acrylic sphere and the truss, there are about $\sim$17000~
inward-facing PMTs to collect optical signals produced by the LS. There is
an opaque layer behind the PMTs to separate the central detector from outside
veto detector. As an example, Fig.~\ref{fig:cd3-2} shows the acrylic sphere and the double
layer stainless-steel truss.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=15cm]{CentralDetector/figures/CD3-2.png}
\caption[Logo in CentralDetector]{Schematic view of acrylic sphere plus stainless-steel double layer truss}
\label{fig:cd3-2}
\end{center}
\end{figure}
Many factors need to be considered for the baseline design, such as:
$\bullet$ Stability and load status of the structure at different stages, such as during the construction, after the construction and
installation of PMTs, during LS and water filling, and long-term operation of the
detector.
$\bullet$ The acrylic sphere under different conditions such as earthquake, temperature variation and so on.
$\bullet$ The maximum stress on the sphere for long term operation < 5~MPa and for shorter durations < 10~MPa.
$\bullet$ Structure of the stainless-steel support should be reliable, and should meet the specifications of related fields.
$\bullet$ The joint of the acrylic and stainless steel should be reliable, and
the impact induced by local failures should be minimal.
\subsection{Stainless-steel Supporting Structure}
The acrylic sphere is the most critical part of the whole central detector and
is supported by the stainless-steel structure on its outer sphere. The design
of the joint of the acrylic and stainless steel structure is critical and the
ball head connection for flexibility and a rubber layer for buffering are
adopted in the design. Currently, there are three design options under
consideration for the stainless steel structure: double layer truss, single
layer truss and equator support. The stainless steel is chosen to be the 316
type and the main parameters are the following: density of 8.0~$g/cm^2$, elastic modulus of
200~GPa, poisson ratio of 0.33 and the yield stress of 240~MPa.
\subsubsection{Double Layer Stainless-Steel Truss}
The space truss has been a popular choice in the civil construction domain given its
properties of light-weight, small-size truss member, ease of handling and
transportation, high rigidity, short construction time, low cost and good
seismic resistance,etc. For the central detector, the truss is selected to be
the outer structure which will not only support the acrylic sphere but also the
PMTs. For the design of the truss, load-carrying capacity and stability were
considered under different conditions. The truss will be the square pyramid
space grid which is a multiple statically indeterminate structure. This type of
truss can withstand the load coming from different directions, and has better
load-carrying capability than the plane truss. The truss members have good
regularity and high stiffness, and they are connected to each other by
bolt-sphere joints which transmit only axial compression or tension, hence
there are no moments or torsional resistance. Since the truss can be built up
from simple, prefabricated units of standard size and shape which will be
mass-produced industrially, the units can be assembled on-site easily and
rapidly which greatly reduces the time for construction. With the present
design, the diameter of the inner layer truss is $\sim$38.5~m, while that of the
outer layer is $\sim$42.5~m at the two poles and $\sim$40.5~m at the equator. This
shrinkage from poles to equator can significantly reduce the size of
experimental hall hence the cost and time for civil construction. There are
hundreds of supporting rods between the acrylic sphere and the inner layer of
the truss. These rods will be connected to the truss at one end and inserted
into the acrylic sphere at the other end. The acrylic sphere is supported by
those rods directly. To reduce the axial load on each rod and hence the stress
in acrylic, some other rods are added between the sphere and the outer layer of
the truss. The truss itself will installed on the bottom of the water pool by
supporting columns at the lower hemisphere. Fig.~\ref{fig:cd3-3} shows the truss structure
and some of its details.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=12cm]{CentralDetector/figures/CD3-3.png}
\caption[Logo in CentralDetector]{Double layer stainless-steel truss}
\label{fig:cd3-3}
\end{center}
\end{figure}
According to JGJ7-2010, which is the technical specification for space frame
structures, the truss members will be designed as the compression members and
the slenderness ratio should be less than 150. Referring to the preliminary
finite element analysis, the size of the chord members is selected to be
$\phi$273~mm $\times$ 8~mm, and the ventral members are $\phi$219~mm $\times$ 8~mm. The
supporting rods (brace members) between acrylic and truss are $\phi$102~mm $\times$
12~mm, and the columns for supporting truss are designed to be $\phi$400~mm $\times$
20~mm. The final size will be determined in engineering design after further stress analysis.
\subsubsection{Single Layer Stainless-Steel Truss}
As shown in Fig.~\ref{fig:subfig:siglelayer} (a), this single layer truss is made of I-shaped unistrut
in both longitudinal and latitudinal directions. Similar to the double layer
truss, the supporting rods are also used to connect the sphere to the truss.
The truss itself is supported on the base of the water pool by a number of
columns. To improve the stability of the single layer truss and to avoid the
possible torsion, a ring of spiral bracings are added in the truss grids to
prevent any occurrence of torsional vibration shape. In addition, due to space
limitation in the pole region of the truss, the square shaped structure is
replaced by a triangle shaped one, so the number of truss members is reduced,
as sketched in Fig.~\ref{fig:subfig:siglelayer}(b). This optimization gives more space for PMT
installation and keeps the grid size of the truss in a reasonable range.
\begin{figure}[!htbp]
\centering
\subfigure[Single layer stainless-steel truss with spiral support]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-4a.png}}
\subfigure[Reduction of the number of truss members in pole region]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-4b.png}}
\caption{Single layer steel truss}
\label{fig:subfig:siglelayer}
\end{figure}
Compared to the double layer truss, the single layer truss can save a
significant amount of space and hence the civil construction cost. Another
advantage of this option is that PMT installation is much easier since there is
no interference caused by the truss members.
\subsubsection{Equator Supporting Method}
This supporting structure is indicated in Fig.~\ref{fig:subfig:equator}. The acrylic sphere is
supported at the equator area with stainless-steel rings which connect to the
wall or bottom of the water pool by rods. A radial extension growing from the
sphere at the equator functions as supporting points where the stainless-steel
rings are clamped to it. A rubber layer is used as buffer between the acrylic
and steel ring to avoid any stress concentration, and this buffer layer is also
very useful to improve the stability of the whole structure, especially under
seismic conditions. The rings are separated into several parts along the
circumference, each part anchored to the water pool by an H-shaped unistrut and a tilted rod.
\begin{figure}[!htbp]
\centering
\subfigure[Overall view of acrylic sphere supported at equator]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm,height=6.5cm]{CentralDetector/figures/CD3-5a.png}}
\subfigure[Detailed view of the support at equator by steel ring]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm,height=7.5cm]{CentralDetector/figures/CD3-5b.png}}
\caption{Acrylic sphere supported at equator}
\label{fig:subfig:equator}
\end{figure}
To minimize the light blocking by the steel ring, another method is using a
number of separated supporting points which are distributed at the equator. In
this design there are 32~blocks cast into the sheets that are assembled at the
equator. Into each of these blocks will be cast a steel plate. The sphere is
supported at each of these steel plates so there is no direct bearing on the
acrylic. Fig.~\ref{fig:subfig:equator-design} shows details of this concept. The 32 support points on the
sphere will be supported by an external steel structure that is made of columns
braced against the concrete wall. A beam will project forward from the columns
and through the layer of PMTs which will surround the sphere. At the end of this
beam a plate is welded to the profile. The height and
depth of the "C" will be much larger than the plate that is cast into the
acrylic sphere. The resulting gap between the plate and the "C" accommodates
the assembly tolerances. The sphere will be built up from the bottom using
temporary supports until the equator ring with the 32 support points is
completed. The external steel structure will be in place and the "C" plate
will be surrounding the plate cast into the sphere. At this point shims will be
placed around the "C" which will lock the cast plate into place. Once shims
have been installed on all 32 support points the bottom temporary supports can
be removed and the entire load of the sphere will be transferred to the "C"
plates and external steel structure. The top half of the sphere can then be
assembled.
\begin{figure}[!htbp]
\centering
\subfigure[]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm,height=7cm]{CentralDetector/figures/CD3-6a.png}}
\subfigure[]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm,height=7cm]{CentralDetector/figures/CD3-6b.png}}
\caption{Detailed view of the support at equator by separated points}
\label{fig:subfig:equator-design}
\end{figure}
\subsection{The Acrylic Sphere}
The inner structure of the central detector is a transparent acrylic sphere
with $\sim$35.4~m inner diameter and 120~mm in shell thickness. This acrylic sphere
will be assembled and bonded with bulk polymerization by a number of acrylic
sheets, as shown in Fig.~\ref{fig:cd3-7}. Considering the production capacity and
transportation limit, the acrylic shell will be divided into more than 170
sheets, each is about 3~m~$\times$~8~m in dimension. The final number and
size of the sheet may be modified after further consideration and discussion
with manufacturers. To reduce the stress on the sphere at the supporting
points, some appended acrylic pieces will be bonded on top of the
sheets. The supporting structure of the sphere will be connected to these
appended acrylics. A chimney of about $\sim$1~m diameter is designed on top of the
acrylic sphere, which will be used as the filling port and interface to the
calibration system. An outlet may be designed on the bottom of the sphere for
cleaning of the sphere and LS recycling during detector running.
\begin{figure}[htp]
\begin{center}
\includegraphics[width=10cm,height=11cm]{CentralDetector/figures/CD3-7.png}
\caption{Schematic view of the acrylic sphere}
\label{fig:cd3-7}
\end{center}
\end{figure}
\subsection{Joint of the Acrylic and Truss}
In the baseline design for the truss the concept of the joint between the acrylic
sphere and the steel truss comes from the idea of the connecting structure used
in the glass curtain wall. For each acrylic sheet, there will be one or two
stainless steel disks which are embedded in it as a connecting structure, as
shown in Fig.~\ref{fig:cd3-8}. At the location of each steel disk, there is an appended
acrylic piece with $\sim$100~mm thickness on top of it and bonded to the sphere, and
a rubber layer is placed between the acrylic and steel for buffering. The steel
disk is designed to have a ball-type head, and a supporting rod will connect it
to the truss. The load on the acrylic sphere will be transferred to the truss
through these rods. The dimensions of the appended acrylic and the steel disk
may be changed according to a further analysis of the stress.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=14cm]{CentralDetector/figures/CD3-8.png}
\caption[Logo in CentralDetector]{Supporting joint of the acrylic sphere and steel truss}
\label{fig:cd3-8}
\end{center}
\end{figure}
In addition, there is another type of joint structure under design, as shown in
Fig.~\ref{fig:cd3-9}. Instead of using the steel disk which is heavy and of large size, a
steel ring is embedded into the acrylic, and connected to a steel plate on top
of the appended acrylic by bolts. Rubber is placed in between the different
components, and the bolt is surrounded by plastic bushing to avoid direct contact with the acrylic.
This new structure is easier for construction and also can reduce the amount of stainless steel and hence the
radioactive background.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=14cm]{CentralDetector/figures/CD3-9.png}
\caption[Logo in CentralDetector]{Structure of the joint with bolt}
\label{fig:cd3-9}
\end{center}
\end{figure}
Currently the two design options are both under optimization, with a main goal of
reducing the stress level to less than 5~MPa. Several prototype tests have been
finished for the first option, and test for the second option will be done
in the near future.
\subsection{Finite Element Analysis and Testing}
\subsubsection{Global FEA}
We have finished a global FEA for the central detector under the following conditions.
1) sphere is empty after the completion of acrylic sphere and steel truss: the load is just the self-weight of the structure.
2) All PMTs are installed on the truss: the weight of total $\sim$17000 PMTs is about $\sim$1700 KN, which will load on the truss.
3) Sphere is filled with liquid and running for long term: the linear liquid
pressure will be loaded on the sphere, and the total buoyance of the sphere is
about $\sim$3100~t; the PMT's buoyancy is about $\sim$1000~t, which will be distributed on the joint nodes of the truss.
During the analysis a load factor of 1.35 was used for the dead load. The FEA
results include stress, deflection and stability for the three loading
conditions. The effect caused by seismic load, temperature or relative liquid
level difference has also been analyzed. Following are the main conclusions:
(1) Stress and deflection
From Table~\ref{table1}, Fig.~\ref{fig:cd3-10} to Fig.~\ref{fig:cd3-12}, we can see that the maximum
stress on the acrylic sphere is $\sim$8.5~MPa in condition (3). If the load factor
of 1.35 is removed, the stress is $\sim$6.3~MPa. The maximum displacement is $\sim$35~mm,
occurring at the bottom of the structure, which is only $\sim$1/1000 of the sphere
span and meets the technical specification for space frame structures
(JGJ7-2010).
\begin{table}[!htbp]
\centering
\caption{Maximum stress and deformation for each loading condition\label{table1}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{3cm}<{\centering}|p{3cm}<{\centering}|p{3cm}<{\centering}|p{3cm}<{\centering}}
\hline
Loading condition & Max. stress on sphere(MPa) & Max. stress on truss(MPa) & Max. general deflection (mm) \\
\hline
\tabincell{c}{Condition 1} & 1.1 & 23.7 & 8.2 \\
\hline
\tabincell{c}{Condition 2} & 1.3 & 16.3 & 6.3 \\
\hline
\tabincell{c}{Condition 3} & 8.5 & 83.4 & 35.1 \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\subfigure[condition 1]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-10a.png}}
\subfigure[condition 2]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-10b.png}}
\subfigure[condition 3]{
\label{fig:subfig:c}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-10c.png}}
\caption{Stress distribution on the acrylic sphere}
\label{fig:cd3-10}
\end{figure}
\begin{figure}
\centering
\subfigure[condition 1]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-11a.png}}
\subfigure[condition 2]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-11b.png}}
\subfigure[condition 3]{
\label{fig:subfig:c}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-11c.png}}
\caption{Stress distribution of the steel truss}
\label{fig:cd3-11}
\end{figure}
\begin{figure}
\centering
\subfigure[condition 1]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-12a.png}}
\subfigure[condtion 2]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-12b.png}}
\subfigure[condition 3]{
\label{fig:subfig:c}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-12c.png}}
\caption{Deflection of the central detector}
\label{fig:cd3-12}
\end{figure}
(2) Analysis of the seismic load
Based on the local geological condition, a seismic fortification
intensity at Richter scale 5.5 was considered, and a seismic load of 0.1~g was taken in our
analysis. Adding 0.1~g seismic load into the finite element model, the detector
structure was analyzed for condition 3 and the result is shown in Fig.~\ref{fig:cd3-13}.
We can see that the stress on the acrylic sphere is increased by only $\sim$1.6\%
and on the steel truss by only $\sim$4.4\%. The small change shows that the structure
of the detector is safe under the seismic load.
\begin{figure}
\centering
\subfigure[Stress on the acrylic sphere under seismic condition]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-13a.png}}
\subfigure[Stress on the steel truss under seismic condition]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-13b.png}}
\caption{Stress on the acrylic sphere and steel truss under seismic condition}
\label{fig:cd3-13}
\end{figure}
(3)Stability analysis
The safety factor for the buckling of the acrylic sphere is increased by the
support from the steel truss. The double non-linearity of the material and the
geometry were taken into account for the global stability analysis, and the
first elastic buckling mode of the structure was taken as the initial
imperfection in calculation. Fig.~\ref{fig:cd3-14}(a) shows the results of the instability
mode. Fig.~\ref{fig:cd3-14}(b) shows the deflection-load coefficient curve and we
can see from it the stability factor is 2.61, which meets the specification of JGJ7-2010.
\begin{figure}
\centering
\subfigure[Instability mode]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-14a.png}}
\subfigure[Deflection-load coefficient curve]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-14b.png}}
\caption{Stability analysis of the central detector}
\label{fig:cd3-14}
\end{figure}
(4) Effect of liquid level difference
Liquid level of the LS in the acrylic sphere will affect the stress status of
the sphere. We did the analysis for several cases of the relative liquid level.
A difference of -2~m, -1~m, 0~m, 1~m, 2~m and 3~m between the LS and water( LS
level - water level) were taken into account. The results are shown in Table
\ref{table2}. We can see that the variation of the relative liquid level impacts stress
significantly. If the LS level in the sphere is higher than the level of water
outside, the stress will be reduced significantly. Raising the LS level may be
considered in our final design.
\begin{table}[!htbp]
\centering
\caption{Impact analysis of liquid level\label{table2}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{4cm}<{\centering}|p{4cm}<{\centering}|p{4cm}<{\centering}}
\hline
Liquid level difference H(m)& Maximum stress on sphere (MPa) & Stress on supporting rod (MPa) \\
\hline
\tabincell{c} {-2} & 10.2 & 67.7 \\
\hline
\tabincell{c} {-1} & 9.4 & 61.6 \\
\hline
\tabincell{c} { 0} & 8.4 & 57.6 \\
\hline
\tabincell{c} { 1} & 7.4 & 53.7 \\
\hline
\tabincell{c} { 2} & 6.4 & 49.7 \\
\hline
\tabincell{c} { 3} & 5.6 & 46.5 \\
\hline
\end{tabular}
\end{table}
(5) Analysis of temperature impact
The detector will be constructed in a hall $\sim$728~m below the ground and the
yearly temperature variation is rather small. Since the thermal expansion
coefficients of the stainless steel and acrylic is different, the impact of
temperature on the stress still needs to be considered. The impact of
increasing or decreasing temperature by 10$^{\circ}$C has been analyzed. Table
\ref{table3} shows the results. For the stainless steel truss, the stress is always less
than 100~MPa and the safety level will not be affected by temperature change.
For the acrylic sphere, temperature elevation is helpful while decreasing of
temperature will lead to a larger stress.So avoiding temperature decrease is
necessary during detector running period.
\begin{table}[!htbp]
\centering
\caption{Temperature impact on stress\label{table3}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{5cm}<{\centering}|p{3.5cm}<{\centering}|p{3.5cm}<{\centering}}
\hline
Condition& Maximum stress on sphere (MPa) & Maximum stress on truss (MPa) \\
\hline
\tabincell{c} {Room temperature} & 8.5 & 83.4 \\
\hline
\tabincell{c} {Room temperature +10$^{\circ}$C} & 8.0 & 77.5 \\
\hline
\tabincell{c} {Room temperature -10$^{\circ}$C} & 10.1 & 95.0 \\
\hline
\end{tabular}
\end{table}
(6) Failure analysis
The central detector is required to have a life-time of at least 20 years, and
any crucial failure is not allowed during this period. For the acrylic sphere
and steel truss, the key component is the support points on the sphere. The
effect of one or more failure (for example, if a rod is broken) was
evaluated. Four kinds of possible failures were analyzed, which are:
$\bullet$ condition 1: failure appeared at the maximum stress point;
$\bullet$ condition 2: failure appeared at all points on the same latitudinal layer where the maximum stress is located;
$\bullet$ condition 3: failure appeared at the points located in one
longitudinal band in the lower hemisphere;
$\bullet$ condition 4. failure appeared at several points randomly distributed.
The analysis results of those cases are shown in Fig.~\ref{fig:cd3-15}.
\begin{figure}[!htbp]
\centering
\includegraphics[width=14cm]{CentralDetector/figures/CD3-15.png}
\caption{Failure analysis of the central detector for (a) condition 1,
(b)condition 2, (c)condition 3 and (d)condition 4.}
\label{fig:cd3-15}
\end{figure}
In the four failure cases, the maximum stress on the acrylic sphere is 10.8~Mpa
(8~Mpa in dead load), an increase of 27.3\% after failure. In fact,
the possibility is very small for failures appearing at the same latitudinal or
longitudinal band. One point failure is the most probable case and the stress
variation is 7\% in this case. The above analysis shows that the risk to the
central detector is controllable.
\subsubsection{Local FEA of the Supporting Point of the Acrylic Sphere}
The grid size of the FEA model affects the accuracy of the calculation,
especially for a singular point and the area around it. The above analysis
shows that the maximum stress of the acrylic sphere appears at the joint of
the stainless steel supporting rod and the acrylic sphere.
Since this node is an intersection of the beam element and the shell element,
it's a typical singular point. For FEA, a local fine model around the singular
point is often employed in order to improve the accuracy. A validation test is
necessary to check the analysis.
Fig.~\ref{fig:cd3-16} shows the structure of the connecting point between the acrylic
sphere and stainless steel truss. The structure at this joint involves the base
acrylic shell, the appended acrylic piece and the stainless steel part.
\begin{figure}[!htbp]
\centering
\includegraphics{CentralDetector/figures/CD3-16.png}
\caption{the structure of the connecting joint}
\label{fig:cd3-16}
\end{figure}
For the local finite model with 3D solid elements, the base acrylic shell is
2.6~m $\times$ 2.6~m with a thickness of 120~mm, the appended acrylic is 100~mm in
thickness and 900~mm in outer diameter, and the diameter of the stainless-steel
disk is about 400~mm. The axial load of 14~t in the stainless-steel rod was
taken into account for local FEA. Two different boundary conditions around the
base acrylic were applied for analysis respectively, one being a fixed
constraint and the other a simple constraint. Fig.~\ref{fig:cd3-17} shows the analysis
results of the fixed constraint and Fig.~\ref{fig:cd3-18} the results of the simple
constraint.
\begin{figure}
\centering
\subfigure[stress distribution on the base acrylic. (Max: 4.2~MPa)]{
\label{fig:subfig:a}
\includegraphics[width=7cm,height=6cm]{CentralDetector/figures/CD3-17a.png}}
\subfigure[stress distribution on the appended acrylic. (Max: 7.2~MPa)]{
\label{fig:subfig:b}
\includegraphics[width=7cm,height=6cm]{CentralDetector/figures/CD3-17b.png}}
\subfigure[stress distribution on the steel disk. (Max: 68.4~MPa)]{
\label{fig:subfig:c}
\includegraphics[width=7cm,height=6cm]{CentralDetector/figures/CD3-17c.png}}
\subfigure[deformation distribution on the whole prototype. (Max: 2.6~mm)]{
\label{fig:subfig:c}
\includegraphics[width=7cm,height=6cm]{CentralDetector/figures/CD3-17d.png}}
\caption{Analysis results for fixed constraint}
\label{fig:cd3-17}
\end{figure}
\begin{figure}
\centering
\subfigure[stress distribution on the base acrylic. (Max: 11.6~MPa)]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-18a.png}}
\subfigure[stress distribution on the appended acrylic. (Max: 7.9~MPa)]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-18b.png}}
\subfigure[stress distribution on the steel disk. (Max: 69.8~MPa)]{
\label{fig:subfig:c}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-18c.png}}
\subfigure[deformation distribution on the whole prototype. (Max: 4.1~mm)]{
\label{fig:subfig:c}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-18d.png}}
\caption{Analysis results for simple constraint}
\label{fig:cd3-18}
\end{figure}
The results show that there is no large stress difference for the appended
acrylic and stainless steel disk under the two boundary conditions for the base
acrylic. The maximum stress in the simply supported model is 63.6\% higher than
that for the fixed constraint. The main reason for this is the stress concentration
at boundary line due to the applied constraint. Removing these singular points,
the stress difference for the two conditions is going to be small. For the
central detector, the real loading status is between the fixed and simple
constraint, so the maximum stress on the acrylic should be less than 10~Mpa.
\subsubsection{Prototype and Testing of the Supporting Structure of the Acrylic Sphere}
A prototype of the supporting structure was constructed to study the technique
of fabricating the acrylic with steel built-in. Tests of the prototype were
performed to understand the load-carrying capacity of the structure and to
measure the stress and deformation to check FEA results. The prototype and the
test can provide a reference and useful parameters for further design.
Fig.~\ref{fig:cd3-19} shows the prototype and some pictures taken during the test.
\begin{figure}
\centering
\subfigure[the joint prototype]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-19a.png}}
\subfigure[Installation of the prototype]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-19b.png}}
\subfigure[Preparation of the prototype test]{
\label{fig:subfig:c}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-19c.png}}
\subfigure[load-carrying test of the prototype]{
\label{fig:subfig:d}
\includegraphics[width=7.5cm]{CentralDetector/figures/CD3-19d.png}}
\caption{the prototype and test}
\label{fig:cd3-19}
\end{figure}
The stress and deformation on the measured points in this test are shown in
Fig.~\ref{fig:cd3-20}. The measured points are distributed in the edge of the base
acrylic, the border between the base acrylic and the appended acrylic, and the
opening area of the appended acrylic. Fig.~\ref{fig:cd3-21} is the load-strain curve of
each measured point, which shows that all points have good elastic
characteristics. All the measured strain data were analyzed to get the stress
status of the prototype.
\begin{figure}
\begin{minipage}[t]{0.5\textwidth}
\centering
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-20.png}
\caption{layout of the measuring points}
\label{fig:cd3-20}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}\centering\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-21.png}
\caption{load-strain curve of the measurement points}
\label{fig:cd3-21}
\end{minipage}
\end{figure}
Under the axial load of 14~t, the maximum stress of the measured points is
8.6~Mpa which is located on the top of the appended acrylic. The measured
stress was compared with the result from FEA as shown in Fig.~\ref{fig:cd3-22}. The
deviation is less than 8\%, which demonstrates that the FEA is reasonable and
can simulate the real stress on the joint by using the local refinement method.
\begin{figure}[!htbp]
\centering
\includegraphics{CentralDetector/figures/CD3-22.png}
\caption{Comparison of stress between measurement and FEA}
\label{fig:cd3-22}
\end{figure}
The above prototypes and tests have verified that the supporting structure of
the sphere is safe under 14~t load as well as the reliability of FEA. Since
defects existed in fabrication of those prototypes(such as polymerization not
being completely finished due to the too low working temperature, the ball
joint not working as expected due to being accidentally glued to the appended
acrylic), a third prototype has been designed and tested with the known
problems fixed. In addition, to test the maximum load-carrying capacity of the
joint structure, an orthogonal load was applied to the prototype. The new
prototype and test are shown in Fig.~\ref{fig:cd3-23}.
\begin{figure}
\centering
\subfigure[the third prototype of joint]{
\label{fig:subfig:a}
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-23a.png}}
\subfigure[Test of the third prototype]{
\label{fig:subfig:b}
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-23b.png}}
\subfigure[The crack pattern of the third prototype]{
\label{fig:subfig:c}
\includegraphics[width=7cm,height=6cm]{CentralDetector/figures/CD3-23c.png}}
\caption{Third prototype test}
\label{fig:cd3-23}
\end{figure}
Given the improvements in the new prototype, the maximum load that the
prototype can bear is up to 51 tons, which is significantly larger than the 28
tons obtained from the previous prototypes. Since the maximum design load is 14
tons, a 3.6 safety factor is reached by the third prototype. Additionally, the
measurements confirm the FEA results, and the maximum stress is about 8.4~MPa,
as shown in Table~\ref{table4}:
\begin{table}[!htbp]
\centering
\caption{Comparison of FEA and measurement for the third prototype\label{table4}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{2.5cm}<{\centering}|p{3.5cm}<{\centering}|p{2cm}<{\centering}|p{2cm}<{\centering}|p{2cm}<{\centering}}
\hline
Load & Properties for comparison & Measured point & data & FEA\\
\hline
\tabincell{c}140KN & Stress/MPa & A7-3 & 8.467 &8.392 \\
\hline
\tabincell{c}{ } &{ } & B7-3 & 3.479 & 2.963 \\
\hline
\tabincell{c}{ } &{ } & C1-3 & 3.262 & 3.093 \\
\hline
\tabincell{c}{ }& Displacement /mm& W4-3 & 0.512 & 0.630 \\
\hline
\end{tabular}
\end{table}
Since we require that the maximum stress on the acrylic should not be larger
than 5~MPa, optimization of the supporting structure is still needed to further
reduce the stress and get a larger safety factor.
\subsubsection{Optimization of the Detector Structure}
From the global and local FEA of the detector structure as well as the
prototype test, we can see that the maximum stress in the acrylic shell (not
including the appended acrylic) is less than 5~MPa and meet our requirement,
but stress in the joints (on the appended acrylic) is still larger than 8~MPa
and further optimization is needed. The optimization and modification can be
done by:
$\bullet$ reducing the axial load on the connecting rod;
$\bullet$ raising the LS liquid level in the sphere;
$\bullet$ improving the uniformity of the whole structure to get a better loading status;
$\bullet$ reducing the number of connecting rods in the top and bottom area of the sphere to simplify the construction (see Fig.~\ref{fig:cd3-24}).
The results after those optimizations are shown in Table~\ref{table5}.
\begin{figure}
\centering
\subfigure[Inner layer truss before sparsifying the truss members]{
\label{fig:subfig:a}
\includegraphics[width=6cm,height=6cm]{CentralDetector/figures/CD3-24a.png}}
\subfigure[Inner layer truss after sparsifying]{
\label{fig:subfig:b}
\includegraphics[width=7.4cm,height=6cm]{CentralDetector/figures/CD3-24b.png}}
\caption{Inner layer truss after sparsifying the truss members}
\label{fig:cd3-24}
\end{figure}
\begin{table}[!htbp]
\centering
\caption{Results from FEA after optimization\label{table5}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{3cm}<{\centering}|p{2.7cm}<{\centering}|p{2.7cm}<{\centering}|p{1.3cm}<{\centering}|p{1.3cm}<{\centering}|p{2.5cm}<{\centering}}
\hline
Conditions & Max-stress on sphere(Singular point excluded) /MPa& Max-stress on sphere(Singular point included) /MPa & Max-stress on truss /MPa & Max-axial load /KN & Max-displacement /mm\\
\hline
\tabincell{c}{Optimization 1 }& 9.6 &6.4 & 74.7 & 164.7 & 47.4 \\
\hline
\tabincell{c} {Optimization 2 }& 6.9 &5.2& 115.1 & 135.9 & 27.3\\
\hline
\tabincell{c} {Optimization 3 }& 6.5 &4.9& 91.3 & 126.1 & 28.4\\
\hline
\tabincell{c} { No 1.35 factor}& 4.8& 3.6 & 67.6 & 93.4 & 21.0 \\
\hline
\end{tabular}
\end{table}
Note for Table~\ref{table5}:
$\bullet$ In all conditions, the liquid level difference is $\sim$3~m;
$\bullet$ Before optimization: the thickness of the upper acrylic hemisphere is
8~cm and the lower hemisphere is 12~cm. The number of supporting rods is 503
and no support comes from the outer layer of the truss to the upper hemisphere.
$\bullet$ Optimization step 1: the thickness of the sphere is changed to 10~cm.
The number of the supporting rods is increased to 646 and symmetrically placed
on the sphere;
$\bullet$ Optimization step 2: the supporting rods at both poles are sparsified, and the total number is 610.
From Table~\ref{table5}, we can see the stress in the acrylic is 3.6~MPa if the support
point is excluded, and the maximum axial load in the supporting rods is only
93.4~KN. Since the stress is proportional to the load, it means that the stress
should be also reduced a lot, and the FEA modeling to confirm this is currently
underway.
\subsection{Study of the Acrylic Properties}
Acrylic is the main material used for the central detector construction. Since
the detector will involve LS, LAB and high-purity water during long-time operation,
the acrylic materials in the liquids should be studied. At present, a number of tests have been finished or are in progress.
(1) Normal mechanical properties test
Referring to the standards of ASTM-D638, ASTM-D695 and ASTM-D790, the tensile
strength, compressive strength and flexural strength of the acrylic had been
tested and the results are 68.2~MPa, 108.3~MPa and 88.0~MPa, respectively.
Since the thickness of the acrylic sphere is 120~mm, it needs 2 or 3 layers of
thinner sheet to be bonded together. In order to know the strength after
bonding, we also did the test for the double-layer material. The test result
is 66.8~MPa for the tensile strength and 101.7~MPa for the flexural strength.
(2) Aging test in LS
The acrylic samples were submerged into LS at temperatures of 50$^{\circ}$C,
60$^{\circ}$C, and 70$^{\circ}$C. The testing time lasted 30, 60, 120, and 180
days for each temperature point. After that the mechanical properties of the
aged acrylic was measured to compare with the fresh one. Results are shown
in Fig.~\ref{fig:cd3-25}. Testing shows that the material has different aging behavior for
different mechanical properties. Here, we mainly focus on the tensile strength
since it is the weakest strength compared to others. It is known that 60
days at 70 degrees corresponds to 16 years at 20 degrees, so we can predict
that the tensile strength will drop by 18\% after 16 years of running. To
validate this conclusion, further tests are in progress.
\begin{figure}
\centering
\subfigure[Tensile strength]{
\label{fig:subfig:a}
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-25a.png}}
\subfigure[Compressive strength]{
\label{fig:subfig:b}
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-25b.png}}
\subfigure[Flextual strength]{
\label{fig:subfig:c}
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-25c.png}}
\caption{Performance of the acrylic after aging}
\label{fig:cd3-25}
\end{figure}
(3) Test of the old acrylic pieces
To study the influence of the surrounding environment such as ultraviolet
radiation, sunlight, temperature and humidity, measurements have been made for
old acrylic pieces after 14 and 17 years in outdoor. Results are shown in Table~\ref{table6}.
\begin{table}[!htbp]
\centering
\caption{Strength of the acrylic after outdoor exposure\label{table6}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{4cm}<{\centering}|p{4cm}<{\centering}|p{4cm}<{\centering}}
\hline
Explosure time& 14 years & 17 years \\
\hline
\tabincell{c} {Elastic modulus /MPa} & 803 & 942 \\
\hline
\tabincell{c} {Poisson ration} & 0.405& 0.405 \\
\hline
\tabincell{c} { Tensile strength} & 56.6 & 53.57 \\
\hline
\end{tabular}
\end{table}
The tensile strength of the normal acrylics is about 68~MPa. From the table,
we can see that it drops by 17\% and 21\% respectively, for 14 years and 17 years in outer door.
(4) Test for the new bonding techniques
The construction of the acrylic sphere will require a large amount of bonding.
Since the normal bonding technique needs a long time for curing, and therefore a
very long construction time, the manufacturers are studying two new bonding
techniques(ultraviolet irradiation and fast bonding) to reduce the curing time.
The tensile strength with those new techniques has been tested and compared
with the normal ones. The strength for ultraviolet irradiation is 46.5~MPa and
for fast bonding is 55.3~MPa. The latter corresponds to 80\% of the full
strength of the acrylic. Details are shown in Table~\ref{table7}. Fig.~\ref{fig:cd3-26} shows the
cross-section where the break happens; it can be seen that the cross-section of normal bonding
(A) and fast bonding (B) are similar, while for the other two
(C and D) it has a very flat surface, indicating that the bonding syrup is not
completely adhering with the original acrylic, hence giving a lower strength.
\begin{table}[!htbp]
\centering
\caption{Comparison of the different bonding techniques\label{table7}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{4cm}<{\centering}|p{2.5cm}<{\centering}|p{3cm}<{\centering}|p{2.5cm}<{\centering}}
\hline
Bonding techniques& time /hours & Tensile strength /MPa & Ratio \\
\hline
\tabincell{c} {Without bonding} &{ }& 68.2 & 100\% \\
\hline
\tabincell{c} {Normal bonding} & 12& 56.4& 82.7\% \\
\hline
\tabincell{c} {Ultraviolet irradiation \\ without annealing} &3 & 37.2 & 54.5\% \\
\hline
\tabincell{c} {Ultraviolet irradiation \\ with annealing} &7 & 46.5 & 68.2\% \\
\hline
\tabincell{c} {Fast bonding} &4 & 55.3 & 81.1\% \\
\hline
\end{tabular}
\end{table}
\begin{figure}[!htbp]
\centering
\includegraphics[width=12cm]{CentralDetector/figures/CD3-26.png}
\caption{cross section of the broken acrylic}
\label{fig:cd3-26}
\end{figure}
(5) Crazing test
Users and manufacturers of plastics have been aware of a phenomenon that glassy
plastics will develop micro-cracks, which start at the surface of the material
and grow perpendicularly to the direction of stress. Studies have shown that
there are at least five factors which may cause glassy plastics to craze, as listed in the following:
$\bullet$ Applied or residual tensile stress;
$\bullet$ Stress generated on the surface of the acrylic due to differential temperature;
$\bullet$ Stress generated on the surface of the acrylic due to absorption and desorption of moisture;
$\bullet$ Weathering;
$\bullet$ Contact with organic solvents.
Crazes are initiated and propagated only when the tensile stress on the
material surface exceeds some critical value.
For the central detector of JUNO, the acrylic sphere will be immersed in LS and
water under stress loading. The problems of crazing become important concerning
the reliability and longevity of the structural parts manufactured from
acrylic.
We did the following tests on acrylic samples in laboratory. Samples with 4.8~MPa
stress applied were immersed in the LS and the liquid mixture of 80\% LS+20\% C$_{9}$H$_{12}$, at the room temperature
and 60$^{\circ}$C. The results are summarized as the following:
1) No crazing appeared by submerging in LS for 90 days at the room temperature;
2) No crazing appeared by submerging in the mixture of 80\% LS+20\% C$_{9}$H$_{12}$ for 90
days at the room temperature;
3) By submerging samples in LS at 60$^{\circ}$C, two of the
samples showed initial crazing after 30 days, and the other two had no crazing.
After 90 days, the initial crazing had grown along the direction of thickness. More tests and studies on crazing will be made in future.
(6) Test of light transmittance
The test results indicate that the light transmittance of the acrylic with
4~cm-8~cm thickness is over 96\%, and the thickness has no significant effect
on it. The measured attenuation length in air is $\sim$110~cm at 410~nm and $\sim$278~cm
at 450~nm.
(7) Creep test
The creep test has been partially finished up to now for the acrylic submerged in LS
at 20 degrees under a stress of 30~MPa, 25~MPa, 20~MPa, and 15~MPa. The
results are shown in Fig.~\ref{fig:cd3-27}. This measured curve is consistent with the
theoretical prediction, and the time before breaking increases dramatically at lower
stress level. To understand the allowable maximum stress, more data are needed.
Currently the creep tests are ongoing, including the test in pure water.
\begin{figure}[!htbp]
\centering
\includegraphics{CentralDetector/figures/CD3-27.png}
\caption{Creep curve of the acrylic under different stress }
\label{fig:cd3-27}
\end{figure}
\section{Backup option: Balloon in a steel tank}
\subsection{Introduction}
The backup option of the central detector is a balloon with a stainless-steel tank.
The tank, located in the center of the experimental hall, is about $\sim$40~m in diameter.
A balloon of $\sim$35.4~m in diameter is installed in the steel tank.
This balloon, welded from transparent films, is filled with $\sim$20~kt of LS as a
target. The buffer liquid between the tank and balloon is LAB or MO. The
density of LS in the balloon is 0.3-0.5\% larger than the buffer liquid. The
soft balloon needs a reliable supporting structure to keep it spherical and
reduce the stress, so a thin acrylic layer surrounding the balloon is designed for
this purpose. Both the balloon and the acrylic structure have the same
diameter. The inner wall of the steel tank is covered by more than $\sim$17000
20~inch PMTs. Fig.~\ref{fig:cd3-28} shows the structural sketch of this option.
\begin{figure}[[!htbp]]
\centering
\includegraphics[width=14cm]{CentralDetector/figures/CD3-47.png}
\caption{Schematic view of stainless-steel tank and the balloon}
\label{fig:cd3-28}
\end{figure}
\subsection{Balloon}
The balloon is the most important structure in this option. The candidate
materials for the balloon include PA, ETFE, PET, PE/PA and FEP. There are two
flanges with a pipe on the top and bottom of the balloon, which will provide
the interface for filling and recycling. As the balloon is supported by the
acrylic layer, the stress of the balloon is very small, and we do not have to worry
about the balloon's shape. Therefore, the main study on the balloon is as
follows:
$\bullet$ the compatibility of the balloon material in the liquid;
$\bullet$ the radioactivity of the material;
$\bullet$ the transparency and aging properties of the material;
$\bullet$ the cleanliness of the balloon during its production and installation;
$\bullet$ sealing of the balloon.
The primary task is to understand the material properties. Present studies show that the yield strength of ETFE film is about 18~MPa. This film has
a good compatibility with all the liquids, good aging resistance, strength and impact
resistance. It is self-cleaning, but its light transmittance is poor.
For example, the transmittance of a 50 micron ETFE film is only 93\%. The test
result shows that PE/PA film is compatible with MO and the compatibility test with
liquid scintillator is ongoing. PET film has good compatibility and light
transmittance, but its flexibility is slightly worse. FEP film has good light
transmittance, but there is no wide application for this film so far.
In order to keep the cleanness and minimize the dust, the balloon must be made and
installed in a clean room. The cleanliness of the workshop in the factory
should hopefully be at the 1000 level. Constant temperature and humidity
are also needed during the production in the workshop. Providing clean gas to
remove radon is necessary during the balloon production. The balloon should be
carefully packaged, not only for safe transportation and ease of opening in the experimental hall, but also for avoiding contamination and damage.
Leak tests of the balloon should be done in the workshop and at the
experimental site. Assuming that the concentration of PPO penetrating into the
buffer liquid is less than 10~ppm in 10~years, the leakage of the balloon
should be about 5~$\times$~$10^{-2}$~cc/s at 3~mbar hydrostatic pressure. In the workshop, the
method of leak test can refer to that of the Borexino experiment based on SF6 gas
\cite{Borexino}.Sealed into another balloon, pumping the SF6 gas into
the inner balloon and getting it into every corner, we can measure and obtain the curve of concentration-time of
SF6 in the outer balloon and calculate the leakage. A vacuum leak test is also very useful. After the transportation to the underground hall,
the balloon will be installed into the stainless-steel tank and the final leak test will be done onsite. The balloon
can be filled with purified gas mixed with a certain concentration of SF6. The balloon's
leakage is determined by testing the concentration of SF6 in the volume between tank and balloon. Further studies and tests should be done
to get a detailed solution for the leak test.
\subsection{Acrylic Structure for the Balloon Support}
Acrylic here is designed as the supporting structure for the
balloon. It is a sphere with $\sim$35.4~m diameter, fixed on the steel tank by supporting legs, as shown in Fig.~\ref{fig:cd3-29}. The acrylic layer is made
up of many acrylic sheets and it is not necessary to seal it. In other words, it
will reduce the difficulty and working time for onsite installation.
The thickness of acrylic sheets in this option is 30~mm, which allows us to have a wide choice from commercial products.
\begin{figure}[[!htbp]]
\centering
\includegraphics[width=8cm]{CentralDetector/figures/CD3-48.png}
\caption{The acrylic supporting structure for the balloon}
\label{fig:cd3-29}
\end{figure}
The density of liquid inside and outside the balloon has a
small difference, expected to be 0.3-0.5\%. In stress analysis, we set a
0.5\% density difference as a starting condition, and the others conditions
being hydrostatic pressure towards the internal surface of the balloon and
external surface of the supporting structure. In this preliminary analysis, we used
tetrahedral finite elements to analyze a 9$^{\circ}$ partial spherical layer
with 25~mm thickness, and acrylic supporting legs were set as fixed ends. The
results showed that the stress of the main part is below 2~MPa and the total
displacement is about 9~mm, as shown in Fig.~\ref{fig:cd3-30}. Since the element density
affects the local stress on the supporting points, further analysis with more
detailed supporting structure will be done.
\begin{figure}
\centering
\subfigure[the stress distribution]{
\label{fig:subfig:a}
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-49a.png}}
\subfigure[the deflection distribution]{
\label{fig:subfig:b}
\includegraphics[width=7cm,height=7cm]{CentralDetector/figures/CD3-49b.png}}
\caption{the stress and deformation of the acrylic support}
\label{fig:cd3-30}
\end{figure}
\subsection{Stainless-steel Spherical Tank}
In the backup option, the outer structure is a spherical stainless-steel tank.
It will not only bear the weight of the liquid inside and support the
balloon and the acrylic structure, but also provide an installation
structure for the PMTs. The tank is not the same as a conventional pressure
tank; its design should consider the installation process and on-site
conditions as well as issues like cost and reliability. Based on the requirements
above, the preliminary structure of the tank is designed as shown in Fig.~\ref{fig:cd3-31}.
\begin{figure}[[!htbp]]
\centering
\includegraphics[width=8cm]{CentralDetector/figures/CD3-50.png}
\caption{A preliminary structure of the stainless-steel tank}
\label{fig:cd3-31}
\end{figure}
The tank is supported by landing legs, which are welded to the tank at its
equatorial belt. The tank has a chimney on the top, and its size should satisfy requirements such as the installation of a calibration
facility, the entrance for lifting the PMTs, ventilation and lighting of the
tank, arrangement of the cables, etc. The chimney reaches the liquid level, so it is
easy to seal. At the bottom of the tank, there is a hole to be used to
dispose the waste liquid from cleaning and as a manhole during the installation stage.
The tank is made of $\sim$170 welded steel plates, which are located in 9 belts of
the spherical tank. These plates can be produced in the factory and then be
assembled and welded after being transported to the site. Based on our
investigation, there are many professional companies who have experience in making large
spherical tanks. In addition, the largest spherical tank in China is up to 50~m
in diameter, while the largest stainless-steel spherical tank is only 17
meters in diameter. Although there is a certain degree of difficulty and
challenge for the $\phi$40~m stainless-steel tank in this option, there are many examples to follow.
\subsection{Installation Issues}
After the construction of the stainless-steel tank in the experimental hall, the PMTs and acrylic support sheets will be installed.
To reduce the time for installation, several PMTs will be pre-assembled as modules. The bottom of the PMT module is made of stainless steel plate and the PMTs can be mounted
on it. The ribs on the bottom plate are designed to increase its rigidity, as
each PMT suffers about $\sim$70~kg buoyancy. There has to be about 10~cm clearance
between the bottom plate of the PMT module and the inner wall of the
stainless-steel tank, which will provide space for cable arrangement.
During the installation, scaffolding will be installed in the stainless-steel
tank. After the tank is completed, the scaffolding can be used to wash the tank
from the top to the bottom. If the cable and PMT modules can be installed
separately, the pre-layout of the cables can be finished in the inner wall of
the tank before the PMT installation is started. A sling can also be hung along
the inner wall of the tank for lifting the PMT module. After the tank cleaning
and cable arrangement, the scaffolding can be removed. The pipe for filling the
liquid scintillator and LAB will be installed outside the tank, and
the pipe outlet, which is also the entrance into the steel tank, is at the
bottom of the tank. The PMT modules will be installed from bottom to top one by
one. The PMT modules can be moved by a manual chain which is hung by the sling.
The cable can be connected to each PMT after it is installed. The PMTs in each
module will be tested after the installation of each layer.
After the PMT modules are installed, acrylic rods will be installed at the back
plate of each module, and the acrylic sheet will be mounted on the rod to support
the balloon. Once PMTs are tested, water can be filled into the tank. When
water reaches the top of that layer of PMTs, installation of the next layer
of PMTs can be started. After installing all PMTs and acrylic sheets, the water can be drained from the bottom of the tank. During this process, it
is necessary to guarantee that the air coming into the tank is clean. After all
water is removed, the balloon will be lifted into the tank from the top, and the
filling pipe can be connected to the bottom flange of the balloon through the
manhole. The top hole and bottom manhole will then be sealed to prevent
dust getting into the tank. The flange on the top of the balloon will be
sealed, and the gas for the leak test will go through the pipeline for filling
liquid scintillator into the balloon to replace the air in it. Once the balloon leak test completed,
the manhole at the bottom will be sealed by the flange.
After making sure that every part works well, liquid
scintillator will be filled into the balloon. During the process, the filling speed of LS, LAB and water outside the tank should be
controlled strictly to keep the level of different liquids the same.
\subsection{Conclusions}
Since the balloon can be made in the factory and then delivered to underground lab for
the final installation and leak check, the time needed onsite is relatively
short. The cost of fabricating a balloon is much lower compared to the acrylic
sphere so more than one balloon can be made. If one balloon has
trouble in the leak checking or any other damage, it can be replaced in a short
time. On the other hand, the balloon is a soft structure and not easily
damaged under a shock load (such as seismic load). The use of the acrylic layer
as support for the balloon reduces the stress and greatly improves the safety.
Since the acrylic sheets in this option do not need to be sealed, the
difficulty and time needed for on-site construction is reduced. Because the
detector construction in this option involves the tank welding onsite,
installation of the acrylic sheets and the balloon in the tank, cleaning and
leak test of the balloon, the procedure will be more complex. Therefore this option is
considered as a backup.
\section{PMT assembly}
\subsection{PMT testing and validation}
The JUNO central detector will use 20~inch PMTs with highest possible quantum efficiency and large photo-cathode.
The main specifications of the PMT include:
$\bullet$ Response to single photoelectrons
$\bullet$ Quantum efficiency of photocathode
$\bullet$ Gain-HV function and gain nonlinearity
$\bullet$ Transit time spread, ratio of after pulse and time structure
$\bullet$ Dark noise rate
$\bullet$ Uniformity of detection efficiency
To precisely measure those parameters, a test bench integrated together with
software and hardware needs to be setup. Based on the requirements, a dark room
with shielding to earth's magnetic field is necessary. Efficient testing can be
implemented by using a light source with multiple optical fiber coupling.
Testing of PMTs consists of several steps: single sample test, test of a group of samples and batch test.
Details will be finalized later.
\subsection{High Voltage System of PMTs}
A high voltage system is needed to provide stable power to the PMTs. This
includes high voltage power supply, cable and water-proof structure. The three
options below are under consideration:
1) One to one option: one HV channel supplies one PMT, and a divider is needed
to produce different voltages for each electrode. This option can have a control
to each PMT HV independently but a huge number of cables are needed.
2) Multi-core cable option: a group of HV channels are carried by a single
cable with multiple cores, via an exchange box close to the PMTs, the HV supplies a bunch of PMTs. In this option: the number of
cables are decreased, while each PMT is still controlled independently. However, a dry box is needed for HV exchange and distribution, and a
connector with water-proof design is needed.
3) Front-end HV module option: for this option, the low voltage is firstly
delivered to the front end near the detector, then the low voltage is
transformed to high voltage which is supplied to the PMTs. The advantage of this
option is to save cost and significantly reduce the number of cables.
The voltage of the PMTs is provided by a Cockroft Walton capacitor diode, which
converts the low voltage carried by a flat cable to high voltage and finally
supplies the whole power grid. Compared to the traditional HV power based on
dividers, this method of voltage multiplier has some clear advantages:
1) the expensive high voltage cable and connector is replaced by a cheap flex cable. This is an efficient way to save cost, reduce the number of cable and improve the overall quality.
2) the high efficiency of the voltage multiplier can reduce heat power
consumption significantly. In general this kind of device will consume power
less than 50~mW without light load, while under maximum light load, the power
consumption is less than 200~mW. As a comparison, the option using dividers
will consume 3.6~W in the form of heat loss.
3) the high voltage is very stable due to a deep negative feedback designed in the module, and complete load characters are provided.
4) the cost of the high voltage power supply with the multiplier itself is low,
about 3-4 times lower than the other options.
The main parameters of the JUNO 20'' PMT high voltage power supply are shown in Table~ref{table8}, Table~ref{table9} and Table~ref{table10} below:
\begin{table}[!htbp]
\centering
\caption{USB bus adapter\label{table8}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{7cm}<{\centering}|p{6cm}<{\centering}}
\hline
Maximum number of channel& 127 \\
\hline
\tabincell{c} {Power supply} & computer USB port \\
\hline
\tabincell{c} {BUS voltage/ V} & LV - 5V,BV - 24 \\
\hline
\tabincell{c} {Interface} & RS-485 \\
\hline
\tabincell{c} {Communication line to computer} & USB-2.0 \\
\hline
\tabincell{c} {Working temperature/degree} & (- 10) - (+40) \\
\hline
\tabincell{c} {Working humidity/ \%} & 0 - 80 \\
\hline
\tabincell{c} {size/ mm x mm x mm} & 70x50x22 \\
\hline
\tabincell{c} {Weight/ kg} & 0.15 \\
\hline
\end{tabular}
\end{table}
\begin{table}[!htbp]
\centering
\caption{20'' PMT HV module\label{table9}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{7cm}<{\centering}|p{6cm}<{\centering}}
\hline
PMT connection, grounding& cathode \\
\hline
\tabincell{c} {Range of HV for anode / V} & 1500-2500* \\
\hline
\tabincell{c} {Operating voltage/ V} & +2300* \\
\hline
\tabincell{c} {Precision of anode HV/ V} & ~ 0.25 \\
\hline
\tabincell{c} {Error of HV output} & 3\% \\
\hline
\tabincell{c} {Stability of HV / \%} & 0.05\% \\
\hline
\tabincell{c} {Temperature coefficient/ ppm/degree} & 100 \\
\hline
\end{tabular}
\end{table}
\begin{table}[!htbp]
\centering
\caption{HV value of PMT dynode\label{table10}}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{8cm}<{\centering}|p{5cm}<{\centering}}
\hline
focus / V& +300 \\
\hline
\tabincell{c} {D1/ V} & +400 \\
\hline
\tabincell{c} {D2/ V} & +1200 \\
\hline
\tabincell{c} {D3/ V} & +1300 \\
\hline
\tabincell{c} {D4/ V} & +2100\\
\hline
\tabincell{c} {Anode/ V} & +2300 \\
\hline
\tabincell{c} {Up limit of anode current/ mA} & 100 \\
\hline
\tabincell{c} {Cross talk/ mV} & 20 \\
\hline
\tabincell{c} {Nominal voltage/ V} & +5,+24 \\
\hline
\tabincell{c} {Power consumption of single module /W} & 0.1 \\
\hline
\tabincell{c} {Communication protocol } & RS-485 \\
\hline
\end{tabular}
\end{table}
\subsection{PMT related Structure Design}
PMT related structures include PMT voltage divider, implosion protection
structure, potting structure and mechanical supporting structure. JUNO will use
20'' PMTs, each one weighting about 10kg and the buoyancy in water of 70~kg.
The PMTs are mounted on the stainless-steel truss.
A preliminary design and test of the MCP-PMT divider has been carried out. The
method of positive HV powering has been adopted, to reduce the number of cables
and connectors, and the signal is readout by a splitter.
A potting structure is needed since the PMTs are working in water, and a
conceptual view is shown as Fig.~\ref{fig:cd3-32}. The potting will use a plastic cover and
the water-proof adhesive to seal the PMT base, electronics and cable from
contact with water. The waterproof design needs to consider long-term
compatibility with high purity water and 4~atm hydrostatic pressure. After
finishing the design and related tests, we will set up a batch test system to
test each of the PMTs before installation.
\begin{figure}[!htbp]
\centering
\includegraphics{CentralDetector/figures/CD3-28.png}
\caption{conceptual view of PMT potting structure}
\label{fig:cd3-32}
\end{figure}
The following sections will discuss the implosion protection structure of the
PMTs. Currently there are three ideas under consideration to protect the PMTs
from implosion in 40~m depth of water:
A. the PMT is covered by an acrylic shell for the upper half sphere and
the lower part by another shell made from FRP, PE or ABS.
A few small holes are opened on the shell to allow air flow and liquid
circulation. When the PMT collapses, the amount of ingoing water is minimized
by those holes, hence the strength of the induced shock wave is suppressed;
B. an air chamber is placed around the lower half of the PMT. When
collapse happens, the air chamber will quickly expand due to a sudden drop of
the local water pressure, hence the amount of ingoing water is also reduced and the shock wave is weakened;
C. a thin layer of film is attached to the PMT surface, this film will prolong the PMT collapse time, and hence the intensity of inrushing water.
The film is transparent so no light will be blocked.
(1) Study of PMT implosion mechanism
The shell of the PMT is made of fragile glass, so an implosion could happen
when damaged glass shell is immersed in water. As was experienced in Super-K, implosion of one PMT could lead to a
cascaded implosion and eventually all the PMTs would be destroyed. A simulation
of the PMT implosion is shown in Fig.~\ref{fig:cd3-33}. The PMT glass shell crashes in 5~ms
due to an outer pressure of 5 atm, the water then rushes inward suddenly. After
about 40~ms, due to the collision of inrushing water, the implosion happens,
and a shock wave is produced which propagates outwards.
\begin{figure}[!htbp]
\centering
\subfigure[]{
\label{fig:subfig:a}
\includegraphics[width=6.3cm,height=7cm]{CentralDetector/figures/CD3-29a.png}}
\subfigure[]{
\label{fig:subfig:b}
\includegraphics[width=8.4cm,height=7cm]{CentralDetector/figures/CD3-29b.png}}
\caption{Simulation of PMT implosion}
\label{fig:cd3-33}
\end{figure}
A preliminary study of the implosion behaviors such as shock wave strength as a
function of time, distance, PMT volume and water pressure has been
performed. As shown in Fig.~\ref{fig:cd3-34}, the strength from simulation is consistent
with the real test. The strength follows a 1/r relation to distance as seen from
Fig.~\ref{fig:cd3-35}, and is proportional to PMT volume and water pressure as shown in
Fig.~\ref{fig:cd3-36} and Fig.~\ref{fig:cd3-37}, respectively.
\begin{figure}[!htbp]
\begin{minipage}[t]{0.5\textwidth}
\centering
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-30.png}
\caption{Peak pressure of the shock wave}
\label{fig:cd3-34}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}\centering\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-31.png}
\caption{strength of the shock wave as a function of distance}
\label{fig:cd3-35}
\end{minipage}
\end{figure}
\begin{figure}[!htbp]
\begin{minipage}[t]{0.5\textwidth}
\centering
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-32.png}
\caption{strength of the shock wave as a function of PMT volume}
\label{fig:cd3-36}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}\centering\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-33.png}
\caption{strength of the shock wave as a function of pressure}
\label{fig:cd3-37}
\end{minipage}
\end{figure}
(2) PMT implosion test and simulation
The purpose of the PMT implosion test is to measure the shock wave parameters,
including production, propagation and relation to space and time. In addition,
the test can verify the function of the acrylic cover. Fig.~\ref{fig:cd3-38} shows the
schematic view of the implosion test and the pressurized tank.
The tests are currently underway.
\begin{figure}[!htbp]
\centering
\subfigure[Schematic view of the implosion test]{
\label{fig:subfig:a}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-35a.png}}
\subfigure[the pressure tank for implosion test]{
\label{fig:subfig:b}
\includegraphics[width=7.5cm,height=6cm]{CentralDetector/figures/CD3-35b.png}}
\caption{the setup for implosion test}
\label{fig:cd3-38}
\end{figure}
\begin{figure}[!htbp]
\centering
\subfigure[the 2D model for simulation]{
\label{fig:subfig:a}
\includegraphics[width=7cm,height=4cm]{CentralDetector/figures/CD3-36a.png}}
\subfigure[the pressure-time curve for point 1]{
\label{fig:subfig:b}
\includegraphics[width=7cm,height=4cm]{CentralDetector/figures/CD3-36b.png}}
\subfigure[Pressure-time curve of different points]{
\label{fig:subfig:c}
\includegraphics[width=7cm,height=4cm]{CentralDetector/figures/CD3-36c.png}}
\subfigure[Peak pressure as a function of distance]{
\label{fig:subfig:d}
\includegraphics[width=7cm,height=4cm]{CentralDetector/figures/CD3-36d.png}}
\caption{Simulation of the PMT implosion}
\label{fig:cd3-39}
\end{figure}
To get the correct implosion model, simulation have been done following the
real boundary conditions in the test. Fig.~\ref{fig:cd3-39}(a) shows a 2D simulation of single PMT without protection enclosure, in which the diameter of the water
tank is 1.5~m, the diameter of the air bubble (PMT volume) is 0.5~m, the
hydrostatic pressure is 4 atms, and the gas pressure in PMT is about $10^{-5}$ Pa. The
boundary is set as a rigid border. There are 6 points in simulation which are
uniformly distributed from the center to the border, in which point 2 is
floating and movable to measure the size variation of the air bubble, and the
other 5 points are fixed to measure the dynamic pressure induced by the shock
wave. Fig.~\ref{fig:cd3-39}(b) shows that the peak pressure appears at t=35.2~ms. From
Fig.~\ref{fig:cd3-39}(c), multiple shock waves are emitted during the process of
compression and rebound of the air bubble, and the waves frequently happen at
beginning or the end of the compression. The air bubble is finally compressed
to a minimum size with the pressure reaching a maximum value of 20~MPa
inside of the bubble, then the strongest shock wave is initiated. Fig.~\ref{fig:cd3-39}(d)
shows the pressure for different points, where the pressure drops
along with the distance. At the border of the PMT (d=25cm), the strength of the
shock wave is about 11~MPa.
The data from the real test will validate the simulation described above. Tests of multiple PMT implosion, and simulation of the effect of the
acrylic cover are also in progress.
\section{Prototype}
\subsection{Motivations of the Prototype Detector}
Following the progress and schedule of the JUNO experiment, considering the requirements of each sub-system, a prototype detector was proposed to test key technical issues:
1) Test and study of the PMT candidates:
The JUNO experiment is designed to use large area, high quantum efficiency PMTs to realize the highest photo-cathode coverage. Candidate PMTs include Hamamatsu, Hainan Zhan Chuang (HZC), and a newly developed MCP-PMT. It is necessary to compare their performance in a real scintillator detector. From the prototype, we expect to extract the effective PMT parameters and compare them.
2) Test and study of the liquid scintillator:
Liquid scintillator is another key component of the JUNO detector, a lot of experiences show that the final LS parameters in a detector, such as light yield, attenuation length, stability (or aging) and background level, are different from what were measured in the lab. With the prototype, many of those parameters can be extracted and optimized.
3) Test and study of electronics:
The planned PMT waveform readout electronics will be very flexible for different physics requirements. The electronics system should be tested in a real LS detector.
4) Waveform data analysis algorithms and detector performance study:
With the prototype, algorithms of PMT waveform readout scheme will be developed and detector performances will be understood.
\subsection{Prototype Design}
The prototype detector is shown in Fig.~\ref{fig:cd3-40}, which re-uses the Daya Bay prototype as the main container. An acrylic sphere locates at the center of the stainless-steel tank (SST) as a LS vessel, and is viewed by 51 PMTs dipped in the pure water. The expected photo-cathode coverage is ~55\%. The water tankand the PP/Lead layer is designed to have 1 m.w.e. shielding, aiming to reduce the radioactivity coming from the outside of SST.
The expected trigger rate is less than [email protected], and the energy resolution is ~4\%@1MeV.
\begin{figure}[!htbp]
\centering
\subfigure[]{
\label{fig:subfig:a}
\includegraphics[width=6cm,height=5cm]{CentralDetector/figures/CD3-37a.png}}
\subfigure[]{
\label{fig:subfig:b}
\includegraphics[width=5cm,height=5cm]{CentralDetector/figures/CD3-37b.png}}
\caption{the prototype design}
\label{fig:cd3-40}
\end{figure}
As shown in Fig.~\ref{fig:cd3-41}, the diameter of the acrylic sphere is 50 cm with a thickness of ~1 cm, and a tube with a diameter of 5 cm and a length of 70 cm is located at the top of the acrylic sphere for filling and calibration. The 51 high quantum efficiency PMTs, in the diameter of 8'', 9'' and 20'', are uniformly arrayed in 4 layers facing to the center of the acrylic sphere. As follows is the detailed arrangement of the PMTs:
$\bullet$ Top layer: which includes 4 MCP-PMTs of 20'' and 2 Hamamatsu PMTs of 20'';
$\bullet$ Middle layers: which are divided into two layers and each layer including 8 MCP-PMTs of 8'', 4 HZC-PMTs of 9'' and 4 Hamamatsu PMTs of 8'';
$\bullet$ Bottom layers: which are divided into two rings, the inner ring includes 3 MCP-PMTs of 8'', 2 HZC-PMTs of 9''and 2 Hamamatsu PMTs of 8'', and the outer ring has the same arrangement as the top layer.
\begin{figure}[[!htbp]]
\centering
\includegraphics[width=6cm,height=6cm]{CentralDetector/figures/CD3-38.png}
\caption{The side view of the prototype detector}
\label{fig:cd3-41}
\end{figure}
Fig.~\ref{fig:cd3-42} is the shielding system. The bottom and top of this system are covered by 10 cm thick lead plus 10 cm thick pp plate, while each of the other 4 sides is shielded by a customized water tank in 1m x 5m x 3m dimension constructed by standard stainless-steel elements which are widely used by water and conditioning system. One of the 4 sides is movable to allow the internal detector installation.
\begin{figure}[!htbp]
\centering
\subfigure[schematics of the shielding structure]{
\label{fig:subfig:a}
\includegraphics[width=5.5cm,height=4cm]{CentralDetector/figures/CD3-39a.png}}
\subfigure[one side of the water tank opened]{
\label{fig:subfig:b}
\includegraphics[width=5.5cm,height=4cm]{CentralDetector/figures/CD3-39b.png}}
\subfigure[bottom and top shielding design with lead and pp plate]{
\label{fig:subfig:c}
\includegraphics[width=5.5cm,height=4cm]{CentralDetector/figures/CD3-39c.png}}
\caption{the shielding system}
\label{fig:cd3-42}
\end{figure}
\subsection{Expected Performance}
With preliminary Geant4 simulation (simulation geometry is shown in Fig.~\ref{fig:cd3-43}), we confirm that the detector will have a good measurement of electron and alpha for LS study. Gammas have a large energy leakage due to the limited detector dimension, we will need a spectral fitting to extract more parameters. Measurements of Muon and calibration will provide more opportunities for detailed study.
According to the prototype design, we expect to measure PMTs' parameters, liquid scintillator, electronics and background, to achieve the goals.
\begin{figure}[[!htbp]]
\centering
\includegraphics[width=8cm]{CentralDetector/figures/CD3-40.png}
\caption{Geant4 simulation geometry}
\label{fig:cd3-43}
\end{figure}
The prototype is now under preparation and will be operational soon.
\section{Construction of the central detector}
\subsection{Assembly of the Stainless-steel Truss}
The stainless-steel truss used in the central detector has a bolted connection,
which makes the on-site assembly easy and safe without any welding. The design
and the construction of the truss should follow the technical standards, design
specifications and procedures. In China, the space truss technique is very popular for building construction.
Many companies have capability and rich experience in design and production of steel trusses.
There are two main stages for the construction of the stainless-steel truss.
The first stage is the off-site preparation in the factory, and the second stage is assembly
onsite. All the truss members, bolts and joints will be finished for the
machining and production in factory before transportation, and the machining
precision and quality should be ensured at this stage. When all parts are
transported to the experimental hall, some important tests need to be done
onsite to make sure that all conditions meet the requirements for construction
before truss assembly. Theses tests include: checking the foundation of the
truss area; double checking the parts of the truss; checking the height mark,
axial and gradient of the embedded part, and so on. Since the complexity of
truss assembly depends on the accuracy and roundness of the first three
circles, the beginning preparation of axis positioning and height level is more
important for the whole assembly of the truss. To control and check the
coordinate position of each truss joint, some reference and checking points
should be installed at different areas of the hall. When all conditions for the construction
meets the requirements of design and standards, the truss assembly can be
started following the detailed construction procedure. Theodolites or other
survey instruments will be used during the truss assembly to control the
error.
For the central detector, the stainless-steel truss and acrylic sphere are not
independent and need overall consideration of the construction sequence and
procedure. It should be better to make the truss and acrylic sphere layer by
layer alternately. The detailed procedure needs further discussion with the
factory. In general, the whole procedure needs to ensure the safety and
reliability of the sphere and truss.
\subsection{Construction of the Acrylic Sphere}
As Fig.~\ref{fig:cd3-7} shows, the acrylic sphere is made up of many acrylic sheets which
are bonded together to form the sphere. Satisfying the production capacity and
transportation limit, the size of each acrylic sheet should be as big as
possible to reduce the total bonding length onsite. For the preliminary
consideration and design of the sphere, there are a total of $\sim$170 acrylic sheets
and divided into 17 layers. Each sheet is less than 3~m $\times$ 8~m in size. The
total bonding length of the sphere is about 1.8~km.
Production and machining of the acrylic sheets will be finished in the factory
before construction onsite. In the factory, the first step is to produce a flat
sheet of acrylic with 120~mm thickness. The next step is to thermoform the flat
acrylic into a spherical sheet with 35.4~m inner diameter in a concave
spherical mold, and these thermo-formed sheets should be stored carefully to
prevent damage to the acrylic surface. After that are the steps of machining,
milling, polishing and annealing. Some measurements should be done for each
step to control the quality in the factory. One such step is the thickness
measurement of each flat sheet used as the thermoforming blank. The second
measurement is for the thickness of the sheet after thermoforming. In general,
the acrylic blank increases in thickness around its border and decreases in
thickness at the center after the thermoforming operation in the female mold.
The third measurement is the acrylic thickness after annealing. The curvature
of the acrylic blank should also be measured to ensure the assembly requirement
\cite{Handbook}. In the factory, the fixtures for positioning and assembly
should be machined.
After all the part production and fixture preparation is finished by the
manufacturer, they will be delivered to the underground to do the construction
of the acrylic sphere onsite. The acrylic sphere can be constructed with the
help of the truss. The acrylic sectors can be supported by the truss joints
during assembling, positioning and bonding layer by layer. The truss can also
be used as the scaffolding or platform for sphere construction. The acrylic
sphere will be built from the bottom to the top, layer by layer. The quality
control and check of dimension, positioning, roundness, bonding status and
annealing need to be ensured for each layer throughout the period of
construction. Inspections of the finished spherical sphere consist of careful
visual checks, dimensional measurements, and stress checking.
For the construction of the detector, two cranes will be needed in the hall.
One is to carry and lift the components of the detector, and the other is a
lift to carry people to access the working area in the pool. The working
platforms should be designed and installed inside and outside the acrylic
sphere for the processes of positioning, assembly, sanding and polishing.
Safety nets should be installed surrounding the sphere during construction to
prevent any dropping of objects which will damage the detector or people.
Special requirements may be needed for the ventilation system of the assembly
cavern. There are special rules and regulations governing working underground
to which the project must conform.
\subsection{PMT Installation}
When the acrylic sphere and stainless-steel truss are finished, about 16000
PMTs need to be installed on the truss facing inwards to receive signal from
the LS. These PMTs should be arranged as close to each other as possible to
provide more than 75\% optical coverage. The PMT support structure must provide
a stable and accurate platform for the mounting and positioning of the PMT
assemblies to view the active detector regions. The PMT support structure
should also provide a mechanism to protect adjacent PMTs from the risks of
cascaded implosion and permit the replacement of the PMT assemblies. To reduce
the installation time, PMTs will be installed by PMT module. Several PMTs are
assembled to be one module in the surface assembly building. The PMT modules
will then be delivered underground for on-site installation. At present, two
options for PMT installation are considered.
One option is to use stainless steel as the support panel of the PMT module,
and each PMT with its protecting cover will be mounted using a clamp on the
panel as Fig.~\ref{fig:cd3-44} shows. Some slots on the truss should be left for the
lifting and installation of the PMT module. The circular rails along the
latitude will be installed on the joint of the inner layer of the truss. There
are pulleys at the back of the stainless-steel panel to allow the PMT element
to slide to the expected location and finish the mounting. Fig.~\ref{fig:cd3-45} shows the
sketch of this option. The design of the module size and shape should consider
the location and number of the joints and supporting rods between acrylic
sphere and truss to get the maximum optical coverage for the central detector.
\begin{figure}[!htbp]
\begin{minipage}[t]{0.5\textwidth}
\centering
\includegraphics[width=8cm]{CentralDetector/figures/CD3-43.png}
\caption{PMT module with the stainless-steel backplate}
\label{fig:cd3-44}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}\centering\includegraphics[width=8cm]{CentralDetector/figures/CD3-44.png}
\caption{Installation of the PMT module}
\label{fig:cd3-45}
\end{minipage}
\end{figure}
The other option is to use profile stainless steel as the supporting structure.
PMTs will be mounted on the profile steel one by one. This profile steel will
be pre-installed on the inner layer of the truss. Fig.~\ref{fig:cd3-46} shows the sketch of
this option. This option allows many work groups to do the installation at
different working areas simultaneously.
\begin{figure}[[!htbp]]
\centering
\includegraphics[width=10cm]{CentralDetector/figures/CD3-45.png}
\caption{Using profile steel for PMT installation}
\label{fig:cd3-46}
\end{figure}
\subsection{Filling system for the Liquid Scintillator}
Filling will start when the construction of central detector is finished. The
filling liquid involves LS and water for the main option and LAB needs to be
considered for the backup option. The veto system will be responsible for water
filling and the central detector system is responsible for LS and LAB filling.
The two systems should coordinate their filling speed to balance the pressure.
During filling, cleanness, background, temperature, liquid storage tank, pipe
design and process monitoring should be considered and controlled.
1) Controlling the Cleanness and low background
During filling, the detector and LS need to be kept clean, especially to
prevent the radioactive background and radon gas from pollution. The central
detector will be cleaned using purified water before filling. For the acrylic
sphere plus stainless-steel option, a special cleaning robot will be designed
and used for cleaning the acrylic sphere and the other parts such as the steel
truss. The PMTs will be washed first. The acrylic sphere can be washed with LS
at the end. For the backup option, the inner wall of the stainless-steel tank
and the outer surface of balloon will be water flushed and the inner surface
of the balloon can be cleaned by filling and pumping water repeatedly. The
filling pipes and pump, which will come into contact with LS, need special care
and cleaning, and they should be compatible with LS. Before filling, the whole
filling system will be cleaned with LS several times and the used LS will be
disposed of. Spaces where the LS is exposed to air, such as inside the acrylic
or balloon and the LS storage tank, will be filled with nitrogen to prevent
radon pollution and contact with oxygen.
2) Pure water exchange or nitrogen exchange for filling
Pure water exchange means we need to firstly fill the acrylic sphere or balloon
with water and then fill LS into the vessel while draining the water out.
Because of its lower density, the LS will be float at the top of the water. This
process filters the LS and is helpful to remove the background. The other
method is exchanging with pure nitrogen, which means filling the acrylic sphere
or balloon with nitrogen first and then putting pure LS into the vessel to
replace the nitrogen.
3) Controlling the liquid temperature
The LS temperature will be high if it is produced and stored on the ground.
Since we hope the liquid for filling can be kept at 20$^{\circ}$C, the liquid
needs to be cooled before filling the detector. The place for cooling should be
selected on ground or underground according to the general deployment of the
civil construction, and the cooling method can be water cooling or
air-conditioning.
4) The preliminary plan for the LS pipe, storage tank, overflow tank and LS cycling system.
As shown in Fig.~\ref{fig:cd3-47}, the filling port of the central detector is designed at
the bottom of the sphere. The filling pipe comes from the LS storage tank which
sits in the storage room, and goes down along the water pool wall to connect to
the filling port. When the LS filling is finished, the pipes between the
storage tank and central detector make them communicating vessels and then the
storage tank can be used as the overflow tank. If the LS needs recycling during
running, we will remove the LS from top of the detector and transport it to the
LS recycling equipment, and then pump it back into LS storage tank. Finally the
LS will be filled back into the central detector through filling pipes from the storage tank.
5) Filling process control
Filling process control involves the speed control, monitoring of liquid
levels, monitoring of stress and deformation of the detector, monitoring of
stress, temperature, and so on. The design of filling speed and capacity of the
filling system should consider the requirements of total filling time, the
capacity of LS storage and production, and cost. The design of the relative
liquid level difference should consider the load stress on the detector.
\begin{figure}[!htb]
\centering
\includegraphics[width=14cm]{CentralDetector/figures/CD3-46.png}
\caption{Sketch of the LS filling system}
\label{fig:cd3-47}
\end{figure}
\section{Reliability and Risk analysis}
$\bullet$ Structural reliability analysis
The baseline option takes large glass buildings and techniques used for glass
curtain walls as an important reference, given that those buildings and
techniques have been matured for long time in the field of architecture. In our
case, the material of glass is replaced by acrylic, and the crucial parts
include bulk-polymerization of the acrylic sheets, different liquid located
inside and outside of the acrylic and long term water pressure and buoyancy.
Reliability of the stainless-steel structure is less important than that of the
acrylic sphere in this option. We should consider:
1) The aging properties of acrylic under stress and submersion in organic or
inorganic liquid needs further tests, to make sure it will be stable during the
20-year running.
2) Acrylic is a brittle material, if there is a crack it will grow under
stress. Cracks can be repaired during the construction but will be fatal after
the liquid filling is finished.
3) The stress is generally larger in the joints of the acrylic and truss than
other areas, so needs to be controlled to keep it low enough.
4) As for the requirements of the SNO and Daya Bay, the stress on the acrylic
should be less than 5~MPa, and more work is needed to reach this goal.
$\bullet$ Construction reliability
Fabrication of a thick acrylic sheet in a spherical shape is not a mature technique, and on-site construction of the large acrylic sphere also needs more R$\&$D effort.
$\bullet$ The reliability of FEA
The analysis needs to be compared with test results, and the accuracy of
analysis should be verified by joint tests and small prototype tests. Maximum errors should be taken as the input parameters for the analysis.
$\bullet$ Construction risk
Construction of the truss should be under control since there is a mature specification. For construction of the acrylic sphere, since there will be a large mount of work done onsite, a standard procedure should be followed and strict quality control should be applied especially on the key parameters of the structure such as stress, deformation and stability.
$\bullet$ Local failure
Local failure means a failure of the joint between the acrylic sphere and
truss, so the stress of the whole structure would need to be re-analysed.
According to the current analysis, if the failure happens on the truss, it will
not be too serious. If it happens on the acrylic, however it will result in
severe damage, especially when the acrylic sphere has been filled with liquid
scintillator. This kind of failure should be avoided at all costs.
$\bullet$ Long term compatibility and leakage problem
It has been proved that acrylic is compatible with liquid scintillator, and
bulk-polymerization will not lead to leakage problems.
$\bullet$ Long term performance
The lifetime of acrylic is up to 30 years and the acrylic sphere for the SNO
\cite{SNO} experiment has already been running for nearly 20 years. In our
case, however the aging effect of the acrylic still needs more study.
\section{Schedule}
The schedules of the central detector are the following:
$\bullet$ 2013
1) Conceptual design of the detector, including FEA and prototype test
2) Consideration of the central detector schedule, budget, manpower related issues.
$\bullet$ 2014
1) Review existing options, select one main option and one candidate option. Perform extensive study on the two options, solve the key design parameters by analysis and prototype.
2) Preliminary design and test of implosion-proof structure of PMT.
3) Start to study PMT potting and LS filling.
4) Start to study batch test for PMT.
$\bullet$ 2015
1) Optimize the detector design, conduct prototype design, fabrication and test. Start to consider on-site construction of the detector.
2) Further study the water-proof of PMT
3) Finish a preliminary design of the implosion-proof structure of PMT
4) Study the batch test system of PMT, including electronics performance test, water-proof and pressure test.
5) Determine the final central detector structure.
$\bullet$ 2016
1) Continue detector structure design and on-site construction study
2) Determination of the batch test design for PMT
3) Start to design the liquid filling system and monitoring system.
4) Determination of the high voltage option for PMT
5) Start to manufacture the implosion-proof structure for PMT
6) Finish the engineering design of central detector.
$\bullet$ 2017
1) Finish bidding for the central detector structure, determine the
construction company and start to produce the components.
2) Determine the high voltage option and water-proof design of PMT.
3) Start PMT batch test.
4) Determine liquid filling option, bidding for the system.
5) Design of nitrogen protection system for liquid scintillator.
$\bullet$ 2018
1) Start to construct the central detector.
2) Continue PMT batch test.
3) Start to install detector monitor and other necessary devices.
$\bullet$ 2019
1) Finish central detector structure construction.
2) Finish PMT installation.
3) Finish detector monitor installation.
4) Finish installation of liquid filling system, start to fill the liquid.
$\bullet$ 2020
1) Finish liquid filling.
2) Finish detector commissioning and start data taking.
\vbox{}
The total time for the central detector construction is estimated about 18
months, as follows:
$\bullet$ Construction of the stainless-steel truss: 3 months (January 2018 -
April 2018). This includes interface(column supports on the base of the water
pool), fixtures, hoisting, measurement, fixing, surveys etc.
$\bullet$ Construction of the acrylic sphere: 8 months (May 2018 - November 2018). This includes acrylic sheet hoisting, positioning, fixture installation, joint connection and adjustment, bulk-polymerization, sanding, stress monitoring, installation of the LS filling device and the interface of calibration.
$\bullet$ Survey of the acrylic sphere and leak test: about 1 month (December 2018)
$\bullet$ Cleaning of the acrylic sphere. The total area of the inner and outer
surface of the sphere is about 8000 $m^{2}$. A high cleanliness level is needed
for the inner surface and it is hard for humans to clean, so a robot cleaner is
under consideration. An outlet need to be reserved in the bottom of the sphere. It will be closed after cleaning.
$\bullet$ Installation of PMTs and monitors: 5.5 months (February 2019 - June
2019). A number of work groups are considered to work in parallel at a rate of
2 PMTs/hour and 8 working hours/day, so 144 PMTs can be installed in one day.
In total this requires about 120 working days, meaning 5.5 months, for the $\sim$17000 PMTs.
$\bullet$ Cleaning of the whole detector after installation. This is not
included in the timetable of the central detector since it can be done together with VETO.
\chapter{Civil Design and Facility}
\label{ch:CivilConstruction}
\section{ Experimental Site Location and Layout}
The location of the JUNO experiment and its supporting facilities should meet the following criteria:
\begin{enumerate}
\item The site should have an equal distance from both the Yangjiang and the Taishan Nuclear Power Station with a largest possible overburden;
\item The site plan and support facilities should be comprehensive for safety, access to the site, and ease in managing the logistics of equipment and personnel above and below the ground for construction and operational considerations of the experiment;
\item The site engineering design and construction plan should be comprehensive in its consideration for energy conservation and emissions reduction, along with minimizing the impact on the local ecological environment;
\item The site should have readily available access to adequate utility water and electric power;
\item The surrounding transportation infrastructure of road should be reasonable for transportation of experimental equipment to the site. The road on site should be comprehensive in its consideration for easy equipment transportation;
\item The site plan for the above ground campus should be comprehensive in its design consideration for seamless and harmonious integration with the local topography, landforms and foliage for civil construction, experimental installation and onsite workforce;
\item All the design should meet applicable national building and construction codes and specifications for civil construction should be followed.
\end{enumerate}
\subsection{ Experiment Site Location}
The site location is driven by the physics requirements to be
optimally equidistant from both nuclear power complexes, along a
central axis determined by the latitudes and longitudes of the (6)
reactors at Yanjiang Station and (4) reactors in Taishan Station.
The site, with surface and underground facilities, and experimental
detector should be located in an area approximate 2,000~m in length
and 200~m in width along this central axis at a distance approximate
53~km from the Yangjiang and Taishan Nuclear Power stations for the
baseline physics requirements. In addition, the experimental hall
should be at a depth greater than 700~m. The lithology and rock
formation should be carefully considered for optimum stability,
along with mountain topography to maximize rock overburden.
Prospective site locations along the central axis are shown in
Fig.~\ref{Fig10-1} and Fig.~\ref{Fig10-2}, at points (1-3) with
elevations of 230~m, 210~m and 205~m respectively. In view of the
changes of boundary condition of the Mesozoic granite stocks
intruding into the Paleozoic rock mass, point (4) at 268.8 m
elevation away from this boundary of granite was selected. This
preliminary location for the experimental hall is far enough away
from the granite boundary to offer adequate safety margin.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{CivilConstruction/figures/Fig10-1.jpg}
\caption{ The site region (red dash line) required by Physics,
orange shadow is granite area } \label{Fig10-1}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{CivilConstruction/figures/Fig10-2.jpg}
\caption{ Experimental Hall Location (position 4) in the granite
area with green dash line} \label{Fig10-2}
\end{center}
\end{figure}
\subsection{ Experiment Hall Layout}
With site location determined, access options to the underground
experimental hall must be considered. Life safety and emergency
egress require that at least two independent access paths from the
underground experimental hall to the surface should be available.
With this requirement in mind, the site plan layout for underground
access is either: "vertical shaft and inclined shaft" or "vertical
shaft and normal tunnel". Both options would have the same vertical
shaft with cage-lift, but the normal tunnel with an 8\% grade slope
would be 6~km in length as compared to an inclined shaft at a grade
slope of 42.6\% and 1.34~km in length. The normal tunnel
construction cycle is too long and costly with transportation safety
issues using motorized vehicles for access. The inclined shaft
option was selected for its short construction cycle, lower cost,
and the use of rail cars for equipment transport underground
provides a safer and controlled environment.
Fig.~\ref{Fig10-3} is a general layout of the site and facilities
with the "vertical shaft + inclined shaft" scheme. Here the
vertical shaft is 581~m in depth; the inclined shaft is 1,340~m in
length with a slope of 42.6\%; and the overburden at the
experimental hall is 728.8~m. The access entry to the inclined shaft
will be close to the experimental hall, located in the granite
mountain area. All of these factors are favorable for civil
construction and stability of the rock. A site survey was conducted
to determine the optimal location of both access shafts. The
vertical shaft is located at the abandoned quarry. Impact of this
site during the construction and the operation to the surrounding
area, traffic conditions and local residents, have to be studied
well, as well as the construction cycle. To facilitate the
construction and reduce the impact to the local residents, the
construction yard and rock disposal area should be arranged near the
entry of shaft construction site. With full consideration of these
site conditions and other factors, such as difficulties in land
acquisition, etc., the entrance portal of the inclined shaft was
determined to be at Shenghe Village, Jinji Town, Kaiping City.
\begin{figure}[Fig10-3]
\begin{center}
\includegraphics[width=0.8\textwidth]{CivilConstruction/figures/Fig10-3.jpg}
\caption{ Layout of the Site }
\label{Fig10-3}
\end{center}
\end{figure}
The entrance portal of the inclined shaft (Fig.~\ref{Fig10-3})
includes permanent buildings, a construction yard area, and a waste
disposal area. All stone debris will be disposed in the waste
disposal area, along with that from the vertical shaft. The waste
transportation route to the waste disposal area does not pass
through neighboring villages or poultry farms, to limit the impact
to the local life and economy.
\subsubsection{ Underground Cavities }
The layout of the underground facilities (Fig.\ref{Fig10-4}) includes a cavity for the main experimental hall, ancillary rooms with connecting tunnels.
\begin{figure}[Fig10-4]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-4.jpg}
\caption{ Layout of the Underground Facility }
\label{Fig10-4}
\end{center}
\end{figure}
The experimental hall contains a spherical detector tank 38.5~m in diameter located in a cylindrical water pool with inner diameter of 42.5m, as shown in Fig\ref{Fig10-5}. Shape of the experimental hall ceiling, either a dome or a vault scheme, was studied. Criteria such as workflow, construction cost and schedule, rock stability, experimental equipment installation and operation were considered. Vault scheme was selected with two crane bridges on a common rail to span installation hall to water pool. A 2.75~m wide walkway on both side of pool is chosen, and the span of upper vault structure becomes 48~m. A 10~m wide area at the entry of experiment hall will be used for equipment un-loading, resulting in a hall 55.25~m long. The total height of the hall from vault apex to the bottom of the water pool will be 69.5m.
The underground assembly hall, pump room, and electrical equipment room are arranged at the same elevation as the inclined shaft entry into the experimental hall. The assembly hall is right next to the experimental hall. This hall is 40~m in length, 12~m span, and 12.5~m height and connected with the inclined shaft through a tunnel with cross-section of 5.7~m in width and 6.5~m in height. The rail car can transport the equipment directly to the experimental hall or to the door of assembly hall. Two bridge cranes on the same rail, each equipped with two 12.5~ton hooks, are arranged in the experimental hall, for handling of experimental equipment. An additional 10~ton crane is located in the assembly hall, for handling of the cargo and equipment from the inclined shaft rail car.
Liquid scintillator storage room, processing room, and filling room are located at the same elevation and opposite to the assembly hall towards the vertical shaft at a distance far enough away from the experimental hall for fire safety consideration. Electronics room, water purification room, gas room, and refuge room are also arranged around the experimental hall. A single access/drainage tunnel with 4.5~m in width that encompasses the experimental hall interconnects these ancillary rooms.
The inclined shaft being 1,340~m in length, 5.7~m in width, and 5.6~m height, bottoms out at elevation -460~m. From there it proceeds at the same level to the experimental hall by way of an 85~m long interconnecting tunnel with a turning radius of 35~m. The slope of the inclined shaft at 42.6\% ($23.08^{\circ}$) is consistent with requirements for debris transportation during the excavation and rail car vehicle specification, while minimizing the length of the inclined shaft. The transport equipment in the inclined shaft is a single-drum cable winch, which operates at a speed of 4~m/s. A walkway is located on one side of the inclined shaft for the maintenance personnel to carry out patrol check and for underground evacuation in case of an emergency. Utilities such as cable bridges, fire water pipeline, air conditioning water pipeline, and liquid scintillator pipeline, etc. are secured on the side walls of the inclined shaft, while air conditioning conduit and air supply conduit, etc., are located under the upper crown of the shaft.
The access entry to the vertical shaft is at 130~m elevation. The
shaft is 581~m in depth, with the bottom access entry connected to
the experimental hall through an interconnection tunnel of 300~m in
length at 3\% slope. The cross-section of this tunnel is 4.5~m
$\times$ 5.2~m. A lift-cage elevator is used in the vertical shaft
with a maximum lifting speed of 6~m/s, and a load capacity of 8-10
people. In addition, a vertical ladder stairway is installed for
personnel evacuation in case of emergencies and for equipment
maintenance access. Vent duct and cable utilities also pass through
this vertical shaft.
\begin{figure}[Fig10-5]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-5.jpg}
\caption{ Sideview of the Experimental Hall }
\label{Fig10-5}
\end{center}
\end{figure}
\subsubsection{ Layout of Surface Constructions }
The ground campus comprises inclined shaft entry area and vertical
shaft entry area. The two areas are connected by a 2.5 km road.
Exhaust room, electrical power room, and an access room are arranged near the vertical shaft entry.
The inclined shaft entry area is a comprehensive area for assembly,
office, dorms, and other service. A rock disposal area is also
arranged in this area. The landscape at the rock disposal area will
be recovered after the construction is completed. The living area is
near the entrance of the campus, and is provided with two houses and
one dorm building with 54 rooms. There is a canteen on site to
provide food service. The experimental office building consists of a
control room, a computer room, meeting rooms, and offices. The
assembly area is arranged at the inclined shaft entry, and comprises
one assembly building (3000 $m^{2}$ ) and temporary storage
buildings. The assembly building is air-conditioned and provided
with a crane to facilitate equipment transportation. In addition,
power station, utility buildings for pure water, LN2, et al are all
in the area. The liquid scintillation area is arranged away from the
living area in the valley area at the access entry. It includes
liquid scintillation storage tanks, purification and nitrogen
facilities.
\input{CivilConstruction/section2}
\section{Utility Systems}
\subsection{\bf Power distribution and grounding system}
To meet the requirements of having redundant, reliable electrical
power to the site, two 10~kV commercial power transmission lines
will be introduced to the power distribution centers at the entrance
to the inclined shaft and the vertical shaft from 110~kV transformer
substations in Jinji Town and in Chishui Town, respectively. They
belong to different public grids. An underground 10~kV power
equipment room is provided to distribute power through the
experimental hall and ancillary rooms. In addition to these
commercial power sources, the surface power distribution center will
be equipped with a self-starting diesel generator, which will supply
power for emergency lighting, ventilation or waste water sump-pumps
during disruptions of the main power supply.
The total expected demand for power distribution is approx. 5,000~kVA from the combined power supply sources, each of which can provide 4,000~kVA of power. Steady-state power loads include ventilation, air-conditioning, and lighting systems along with operational experimental equipment. However, peak power capability is required during the LS distillation process and its transfer to the detector. An isolated transformer will be installed in the underground power equipment room to distribute clean electrical power on a separate grid to experimental equipment to minimize system noise.
The electrical grounding systems include both a clean ground grid for experimental equipment and a safety ground grid for utility systems. The clean ground grid for experimental equipment on clean electrical power will be connected to the secondary winding of the source isolation transformer. The two grounding grids will have no electrical connection between them. The surface safety ground grid is distributed and transferred to underground with power transmission cables, and is connected to the underground safety ground grid. The clean ground for experimental equipment is distributed locally in the underground experimental hall. It is understood that the underground geological formation mainly consists of granite, having a high electrical resistivity, not conductive for electrical grounding. Based on our experience, the electrical grounding systems should include appropriate measures, such as high-efficiency electrolytic ion grounding electrodes to mitigate electrical safety concerns. This grid should also connect to the water pool steel-bar structure, to improve the connectivity with the ground and to increase the overall capacity, hence further stabilize the ground.
\subsection{\bf Ventilation and air-conditioning system} The temperature in the underground experimental hall shall be maintained at a nominal 22$^{\circ}C$ and the relative humidity shall be lower than 70\%, with adequate ventilation to sustain at least 50 people working 24/7 underground. Air quality in the experimental hall can be normal room air that meets cleanliness requirements for personnel safety. The cleanliness in ancillary halls related to
liquid scintillator storage or processing should meet class-100,000-level which is comparable to the requirements for the Experimental Hall-5 at Daya Bay. Special measures (e.g., high efficiency filters) can be taken during liquid scintillator operations to improve the cleanliness to class-10,000-level.
The air exchange rate in the experimental hall is designed for 6~volumes/day, to reduce radon in the air and maintain a comfortable environment for underground workers.
Water chillier units and fresh air handling systems for supplying dehumidified cool air underground are located at the entrances to both the inclined and vertical shafts. This conditioned air supply flows through insulated ducts in each shaft for dispersion in the experimental hall. Air is then drawn from the experimental hall via exhaust ducts to a main duct in the inclined shaft, and then exhausted to the atmosphere using exhaust fans.
\subsection{\bf Water supply and drainage system} The underground utility water supply comes from the fire-fighting water system. Normal water usage is obtained from the fire-fighting water system main pipeline, so both water systems are from a common source.
Ultra-pure water for experimental use either in the detector water pool or in scintillator processing will have its own supply by stainless steel pipes from the surface to underground facilities.
The underground wastewater comes from natural rock seepage, utility
water or fire-fighting water. It is collected in a series of
grate-covered tunnel floor trenches and diverted to a number of
sump-pits for pump-out to the surface. Since most of wastewater
during normal operations comes from rock seepage, five sump-pit
drainage systems have been strategically located through the
underground facilities, based on the distribution of water seepage.
Sump-pit-1, the main sump-pit, is located at the bottom of the
inclined shaft, and is distributed through the curved
interconnecting tunnel to the experimental areas. Sump-pit-2 is
located at elevation of -200~m in a side portal area of the inclined
shaft. Sump-pit-3 is located at the bottom area of the water pool,
and is used during pool cleaning. Sump-pit-4 is located at bottom of
vertical shaft. Sump-pit-5 is located at an elevation of -100~m in a
side portal area of the inclined shaft where the rock formation is
poor and water seepage is severe.
The water from whole underground facilities is pumped up to surface via sump-pit-1 and then sump-pit-2. This design may choose less power pump for sump-pit-1, which have less impact on the experiment during its operation.
Water pool draining will install additional dedicated pump system to sump-pit-1 temporarily.
\subsection{\bf Fire control system} The fire control system consists of automatic fire alarms and interlocks, fire-fighting water distribution to hydrant-hose units, numerous portable fire extinguishers, emergency smoke exhaust ventilation, emergency lighting, personnel evacuation planning, and refuge-rooms. The automatic fire alarm system covers smoke detection and alarming for all underground cavities, tunnels and shafts, along with surface facilities. All alarm signals will be sent to the surface fire control center to be monitored by dedicated fire-watch personnel around the clock. Relevant zones are interlocked to respond to alarm signals by taking the following actions: cut off main power supply sources and start up emergency power generator, close fire partition doors and relevant air exhaust valves to isolate the ignition source, and start emergency exhaust fans. Fire hydrant-hose units and portable fire extinguishers are the preferred method for fighting fires in the underground complex over that of sprinklers systems. The emergency smoke exhaust ventilation systems can be energized based on the location of the incident and the personnel evacuation planned route. For example, smoke can be exhausted through the vertical shaft while personnel escape via the inclined shaft; alternatively, smoke can be exhausted through the inclined shaft while personnel escape via the vertical shaft. In the case of a fire/smoke emergency the main power feeds will be shut off automatically by interlocks, and the emergency power generator will start, to supply power for smoke exhaust ventilation and lighting to ensure safe personnel evacuation.
\subsection{\bf Communication system} A variety of personnel communication systems will be employed throughout the surface and underground facilities of the experimental complex. All fire emergency communication systems are directly connected to the surface fire control center monitored by dedicated fire-watch personnel around the clock. An intercom communication system that has broadcast, point-to-point calling, and central command functionality will be used for working communications throughout the experimental complex. Local telephone systems are furnished at several key locations at the surface construction site for off-site communication.
High-bandwidth and private optical fiber channels will be acquired from local network providers for the experimental complex, to meet the demand for transmission of experimental data and general office communication.
\section{Risk Analysis and Measures}
The risk assessment for the project depends on the results of the detailed geological survey (section 10.2). Special risk assessment of the civil construction project include: risk assessment of geological hazards, seismic studies, water and soil conservation, and ecological impact on the surrounding environment. Geological survey and recommendations will be fully complied. Only major risks during the construction and operation phase, especially unpredictable risks and mitigation measures, are discussed here.
The underground cavities in this project are deep with large open
spans, and the inclined shaft passes through complex geological
area. Therefore, unpredictable collapse of the cavities, shafts or
tunnels, along with water seepage are the major risks in the
construction phase, and they will have direct impact on the project
schedule and construction cost. The geological survey also indicates
that faults may exist in the interconnecting tunnels at the
experimental hall elevation from the inclined shaft or the vertical
shaft, to the experimental hall. This represents the greatest
uncertainty to the construction and worker safety. Possible
mitigation measures include: perform geological forecast in advance
to make appropriate engineering plans, and prepare structural design
solutions based on geological conditions as the project is
implemented. Another major risk in the construction phase is rock
burst. Although the rock burst intensity in the experimental hall
predicted by the geological survey is not high, plans should be
there to prepare for the worst.
During the operational phase of the project, underground collapse and flooding are unpredictable risks that must be considered. It is recommended that a real-time rock monitoring system, such as a 3-D visual early warning system, should be established. This would monitor the stress and deformation of the cavity enclosing rock in real-time, to provide automatic early warning so appropriate and timely measures can be taken. Additionally, rock-safety consultants can be employed to analyze the data taken from monitoring system to provide hazard analysis and timely safety warnings. Dual-egress design also provides safety assurance for underground evacuation of personnel. Underground water conditions will be monitored and alarmed for accumulation and daily discharge to the surface. In conjunction with patrol checks of underground cavities and tunnels, any water abnormalities can be acted upon.
\section{Schedule}
The total civil construction period for the entire site complex is
36 months. This work has started on January 2015, and the civil
construction phase of the project should be finished and handed over
to the experimental users by December 2017.
\section{Geological Survey}
\subsection{Geographical Conditions of the Region}
The site is 5~km away to Jinji Town, which is 40 km away to Kaiping
City (Fig. (\ref{Fig10-6})). There are provincial road S367, S275
and village roads connecting the towns and villages. A new road will
be built to connect the site to the outside.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-6.jpg}
\caption{ Map of the Project Region }
\label{Fig10-6}
\end{center}
\end{figure}
The underground facility is underneath the Dashi Mountain area in Jinji Town, which has an elevation of 326 meters. The main mountain ridge extending approximately in south-north direction. The elevation of the hill area is about 30m${\sim}$300m while the topographic slope is usually 10$^{\circ}$ to 20$^{\circ}$. The main gulch exists and extends approximately in east-west direction. The depth of gulch cuts is 50 meters to 100 meters, and there is perennial water in the gulch. The largest river in the area, Dongkeng River, originates from Dongkeng Forest Farm and flows across the project area and Shenghe Village. Rivers in the region usually flow approximately in south-north direction. All of rivers are branches of Chishui River. Niuao Water Reservoir (a small reservoir) and several ponds are distributed at the southeast side of the mountain.
The project area is of Southeast Asian tropical maritime monsoon
climate, with frequently strong typhoons in the transition period
from summer to autumn, which bring abundant rainfall. Kaiping City
is surrounded by the plenty of rivers with wide water area. It is
not severely cold in winter and very hot in summer. The climate is
mild and abundant in rain. The annual mean air temperature is
21.7$^{\circ}$C while the annual mean humidity is as high as 82\%.
Annual rainfall varies from 1,700mm to 2,400mm. The rainy season
mainly concentrates from April to September while the dry climate
distributes from October to February. The perennial dominant wind
direction is east. Affected by the subtropical monsoon, the period
from June to October in every year is a strong wind season with a
force of 6-9 from east.
The main active fracture zone in the region is the Enping-Xinfeng
zone, which is at approx. 60 km distance to the project area. A
magnitude-6 earthquake once happened near Mingcheng long fracture
zone. Magnitude-4 or lower earthquakes happened near Kaiping, and a
magnitude-4 earthquake happened in Hecheng area. The rmoluminescence
survey value of the fracture near Gaoming is 245,200 years, and the
the rmoluminescence survey value of the Hecheng-Jinji fracture is
153,300 to 350,000 years. The fractures belong to Mid-Pleistocene
active faults.
According to the earthquake catalogue of China Seismic Network
(CSN), although there were more than 64 earthquakes happened within
150 km range of project area with magnitude-3.0 or higher since
1970, wherein, only 3 moderate earthquakes higher than magnitude
4.75 happened. No earthquake higher than 4.0 happened near the
project area (within 25 km range). The earthquake distribution map
in the project region shows that the earthquakes are mainly
distributed in Yangjiang region in southwest. The highest earthquake
is a magnitude-4.9 in Nanhai region on Mar. 26, 1995, which is 100
km away from the project area. Therefore, the impact of earthquakes
on the project area is low.
According to the "Seismic ground motion parameter zonation map of China", the seismic ground peak acceleration is 0.05g in 50 years. The characteristic time of the seismicresponse spectrum is 0.35 s, and the corresponding basic seismic intensity is degree VI.
\subsection{Geological Survey}
A detailed geological survey for the inclined shaft, vertical shaft, and experimental hall areas of the experimental station was carried out in 2013 to ascertain the geological conditions of the main constructions in the project region and provide a geological basis for engineering design and construction and raise proposals for handling main geological problems.
The survey result is summarized in Fig. {\ref{Fig10-9}}. There are
Yanshanian granite stocks at the location of the experimental hall.
The granite mass intruded into the Palaeozoic sand rocks. The
contact zone between them is a hornfelsic zone. Details are
described in the following.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-9.png}
\caption{ Engineering Geological survey in the Experiment Region }
\label{Fig10-9}
\end{center}
\end{figure}
\subsubsection{Engineering Geological Conditions of the Inclined Shaft}
The access entry of the inclined shaft is in a mild gulch with width of 400~m in northeast to Shenghei Village. Vegetation near the access entry is dense. The gulch in front of the access entry is open and favorable for organization of construction and arrangement of surface buildings (see Fig. \ref{Fig10-10}). The formation penetrated by the inclined shaft mainly consists of Ordovician Xinchang ($O_{1x}$), Cambrian Bacun (${\in}bc^c$), hornfels and granitic intrusions.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-10.png}
\caption{ Topography near the Access Entry of Inclined Shaft. The underground facility is at left. The entry of inclined shaft is at right. }
\label{Fig10-10}
\end{center}
\end{figure}
There is a water reservoir (Shantan Water Reservoir) at the upstream
of the main gulch at the access entry of the inclined shaft (see
Fig. \ref{Fig10-10}). The water reservoir has only $5\times10^4 m^3$
storage capacity, which has no feeding river at its upstream. The
water is mainly supplied by rainfall and ground water discharge. The
catchment area of the water reservoir is approximately $0.3 km^2$.
Therefore, the water inflow into the water reservoir is not
dangerous even in the rainy season, and will not cause severe flood
discharge. There are two floodways behind the dam of the water
reservoir. The minimum distance from the floodways to the access
entry of the inclined shaft is approx. 130 meter. The elevation of
the floodways is lower than that of the access entry by approx. 5
meter. The area at the downstream of the flood discharge area is a
lower and wider plain terrain. In summary, the flood discharge from
Shantang Water Reservoir in the rainy season will have no impact on
the access entry of the inclined shaft.
From the geological survey, the covering layer at the access entry is silty clay gravel soil in $1m\sim3m$ thickness. The bedrocks are mainly Xinchang Fm($O_{1x}$) gray and dark gray siltstone mixed with feldspar-quartz sandstones and clay shale stones in $N6^{\circ}E$, $SE{\angle}58^{\circ}$ occurrence. The bedrocks are completely weathered, locally intensely weathered, and fractured in fractured-loose structure. The condition is not perfect for shaft construction. Slope protection and water drainage should be considered.
The classification of the tunnel enclosing rocks is predicted and evaluated as follows (see Fig. \ref{Fig10-11}):
\begin{enumerate}
\item 1413m${\sim}$1350m section: The enclosing rocks near the tunnel entrance are mainly silt rocks with locally distributed quartz sandstones and clay shale stones. They are fully or intensely weathered while some are moderately weathered. Joints and fractures are developed and filled with silt. The rocks are mainly in silt filled cataclastic structure while some are in mosaic cataclastic structure. The crown and side walls have water seepage or water dripping.
The quality rating of the enclosing rocks is grade V. The cavity stability is very poor without self-stabilization time or with very short self-stabilization time.
\item 1350m${\sim}$1283m section: the tunnel enclosing rocks in the region are mainly silt rocks and quartz sandstones with moderately or slightly weathered Joints and fractures. The rocks are mainly in mosaic cataclastic structure while some are in layer structure. The crown and side walls have water seepage or water dripping.
The quality rating of the enclosing rocks is grade IV. The crown enclosing rocks are unstable, and the self-stabilization time is short. Various large-size deformations and damages may occur.
\item 1283m${\sim}$808m section: The tunnel enclosing rocks are mainly quartz sandstones, with silt rocks distributed locally, and are slightly weathered. Joints and fractures are developed. The rocks are mainly in mosaic cataclastic structure while some are in layer structure. The crown and side walls have water seepage or water dripping.
The quality rating of the enclosing rocks is grade III. The crown enclosing rocks have poor local stability.
\item 808m${\sim}$418m section: The rocks are hornstones and slightly weathered. Joints are developed or undeveloped. The rocks are mainly in layer structure. The crown and side walls have water seepage or water dripping.
the quality rating of the enclosing rocks is mainly grade III, and grade II locally. The cavity enclosing rocks have high overall stability.
\item 418m${\sim}$0m section: the rocks are granite rocks and slightly weathered. The joints are undeveloped. The rocks are mainly in block structure. The crown and side walls have water seepage or water dripping.
The quality rating of the enclosing rocks is grade II. The cavity enclosing rocks have high stability and will not have plastic deformation.
\end{enumerate}
Attention should be paid to water abundance in the sandstones as
indicated on the cross section of geophysical prospecting along the
route (A) near the ditch of Dongkeng River in the original inclined
shaft route (see Fig. \ref{Fig10-11}).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-11.jpg}
\caption{ Geophysical Investigation of Inclined Shaft }
\label{Fig10-11}
\end{center}
\end{figure}
\subsubsection{ Engineering Geological Conditions of the Vertical Shaft}
The vertical shaft is located in the Dongkeng Quarry. It is 589 m in depth and 6 meter in diameter.
\begin{figure}[Fig10-12]
\begin{center}
\includegraphics[width=0.8\textwidth]{CivilConstruction/figures/Fig10-12.jpg}
\caption{ Topography at the Entry of the Vertical Shaft }
\label{Fig10-12}
\end{center}
\end{figure}
The vertical shaft is located in a quarry and at the end of the
gulch (see Fig. \ref{Fig10-12}). The upstream catchment area of the
gulch is approx. $0.1 km^2$. In the rainy season, severe torrential
flood will be produced in the gulch. Water drainage and special care
must be considered during construction and operation.
The vertical shaft is in granite mountain. An exploring borehole ZK1
was drilled at the location of vertical shaft for the geological
survey. The borehole was deeper than bottom of water pool in the
experimental hall. Various borehole tests were carried out,
including terrestrial stress test, high pressure water test,
specific resistance test, natural gamma test, borehole televiewer
test, acoustic wave test, and ground temperature test, etc.
The vertical shaft is near the granite intrusion boundary. In the area of vertical shaft, the joints, fractures, and faults are developed. 5 exposed faults in different sizes are distributed nearby. Eleven or more faults or compresso-crushed zones in different sizes were unveiled in the borehole.
The classification of the vertical shaft enclosing rocks (Fig. \ref{Fig10-13}) is predicted and evaluated as follows:
The rock quality of vertical shaft entry section with depth of 10
meters is grade IV. The granite rocks are moderately weathered.
Joints and fractures are well developed. The section is in mosaic
cataclastic structure or block structure.
The shaft depth to the -118 meter, the rocks are mainly slightly weathered or moderately weathered rocks in some regions. As unveiled in the borehole surveying work, there are several small developed faults and compressive structural planes (grade IV structural planes). The rocks are mainly in sub-block structure or in block structure locally at some places. The quality rating of the enclosing rocks is grade III.
The rocks of shaft bottom section are slightly weathered or fresh. The complete section is in perfect block structure. The quality rating of the enclosing rocks is grade II. The shaft wall enclosing rocks have high overall stability and will not have plastic deformation.
The water inflow in the vertical shaft is predicted with Dupuit formula. According to the water pressure test result, the permeability factor of the rock mass of the vertical shaft is approximately 0.009 m/d. The shaft is approx. 589 meters in depth and 3 meters in radius. The calculated water inflow in the vertical shaft is estimated $1473m^3/d$.
There is only one concern that a fault of F2 may cross the horizontal tunnel between bottom of vertical shaft and experimental hall. The rock in the F2 region is grade IV. According to the survey, the F2 fault is a compressional-shear shifted reversed fault, in $N20^{\circ}{\sim}55^{\circ}E$ and $SE{\angle}59^{\circ}{\sim}90^{\circ}$ occurrence. The occurrence variation is severe. The fault is mainly developed in granite mass, in 0.5 meters${\sim}$3.0 meters width, and extending approximately 400 meters. On the ground surface, the fault zone mainly consists of flake rocks, mylonite, cataclastic rocks, and lens with inclined scratches. The cementation is poor to moderate. As unveiled in the ZK1 borehole, the deep fault zone is mainly filled with quartz. The cementation is good to very good and is usually in healed form. The buried depth of the F2 fault is approx. 660 meters. It is speculated that the cementation in the deep part of the fault is good to very good. The F2 fault area has no surface water and is close to the dividing ridge in which the rock mass is thin and has weak water permeability. In summary, it is speculated that the F2 fault zone has weak water permeability while the water inflow is low. The fault will not have severe water inflow at the level the tunnel is excavated to the F2 fault.
\begin{figure}[Fig10-13]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-13.png}
\caption{ Some Core Samples from the exploring drill well in the
Vertical Shaft } \label{Fig10-13}
\end{center}
\end{figure}
\subsubsection{ Engineering Geological Conditions of the Experimental Hall }
The experimental hall is located under a small hill at the southeast side of the vertical shaft. The surface elevation is 268 m, and the buried depth is approx. 763 m. The profile of hill is a mountain ridge. The experimental hall is in the granite mass, which consists of gray medium-fine grained adamellite. In the plan view, the experimental hall is relatively close to the central zone of the intruding small stocks.
A geophysical prospecting diagram of the experimental hall and the vertical shaft area obtained with high-frequency magnetotelluric method is shown in Fig.~\ref{Fig10-14}. Both the vertical shaft area and the experimental hall area consist granite mass. It is speculated that there are faults in the horizonal connecting tunnel between the vertical shaft and the experimental hall.
\begin{figure}[Fig10-14]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-14.jpg}
\caption{ Geologic Diagram Obtained with the High-Frequency
Magnetotelluric Method near the Experimental Hall } \label{Fig10-14}
\end{center}
\end{figure}
The major engineering geological issues include:
{\bf Terrestrial Stress and Rock Burst Analysis}
According to the laboratory test, the volumetric weight of slightly weathered (or fresh) granite is approximately $26.6{\times}10^3kN/m^3$, the saturated uniaxial compressive strength is 80 MPa${\sim}$120 MPa as usual. Since the maximum buried depth of the experimental hall is 763 meter, according to the terrestrial stress test, it is estimated that the terrestrial stress at the experimental hall is not higher than 20MPa and the rock strength/stress ratio is 4.85. The deep core obtained in the borehole ZK1 has high integrality in whole. The core near the borehole bottom (651m buried depth) is still in long column shape. Since the experimental hall is relatively close to the center of the intruding rock stocks, it is speculated that the rock mass there has high integrality. Through comprehensive analysis, it is speculated that the stress in the experimental hall is moderated, and will not result in severe rock burst.
{\bf External Water Pressure Analysis}
According to the water pressure test and the high-pressure water test, the permeable rate of the granite rock is usually lower than 1 Lu and considered as weak permeability. The difference between the ground water level and the bottom of experimental hall is 577.79 meter. According to the "Code for Engineering Geological Investigation of Water Resources and Hydropower", the external water pressure reduction factor of weak permeable rocks is 0.1${\sim}$0.2. Thus, the effective external water pressure of the experimental hall is only 58 meters${\sim}$116 meters, corresponding pressure is 0.58MPa${\sim}$1.16MPa. The high-pressure water test shows that most of the joints and fracture structure of the granite did not change under the water pressure and are mainly of turbulence type while some of them are of filling type. Dilatation or erosion phenomenon is not found. In summary, the external water pressure at the experimental hall is relatively low, and its impact on the cavity stability of the experimental hall is trivial.
{\bf Cavity Water Inflow Analysis} The water inflow in the
experimental hall is predicted with underground water dynamics
methods. The maximum water inflow in the early stage and the
long-term stable water inflow are predicted with several methods.
According to the water pressure test, the deep granite is slightly
permeable. The maximum water inflow in early stage calculated with
HirishiOshima empirical equation is approximately $395 m^3/d$. It is
$527 m^3/d$ with the empirical formula method. The long-term stable
water inflow was estimated to be $244 m^3/d$ and $120 m^3/d$
respectively with Ochiai Toshiro formula and empirical formula
methods.
In summary, the maximum water inflow in early stage and the long-term water inflow in the experimental hall are low.
{\bf Underground Temperature} According to the underground
temperature measurement in the borehole, ZK1, the underground
temperature at 450 meter depth (-314 meter elevation) or deeper is
$31{\sim}32 ^{\circ}C$ (see Fig. \ref{Fig10-15}). Therefore,
temperature is not a big issue for the experiment.
\begin{figure}[Fig10-15]
\begin{center}
\includegraphics[width=\textwidth]{CivilConstruction/figures/Fig10-15.png}
\caption{ Measured Underground Temperature}
\label{Fig10-15}
\end{center}
\end{figure}
{\bf pH value}
The pH value of underground water in borehole ZK1 at the location of the vertical shaft is analyzed to be 7.25${\sim}$7.75 while the pH value of ground water is near 7. It indicates that the deep ground water is medium corrosive.
{\bf Rock Radioactivity}
Table~\ref{tab:rockrad} shows the result of rock radioactivity measurement at different depths in borehole ZK1. All sample results meet the requirements for main building materials (as per "Code for Radionuclides Limits in Building Materials" (GB6566-2010)) except for the external exposure index Ir of three samples, which are slightly greater than 1.0. The rock material meets the criteria for Class B products according to the "Code for Radiation Protection Classification and Control of Natural Stone Material Products".
\begin{table}[H]
\caption{Result of Rock radioactivity Test in Borehole ZK1}
\label{tab:rockrad}
\centering
\scalebox {0.8} {
\begin{tabular}{|c|c|p{1.8cm}|p{1.8cm}|p{1.8cm}|p{1.8cm}|p{1.8cm}|p{1.8cm}|}
\hline
Site No. & Sampling & \multicolumn{6}{|c|}{Radioactivity} \\
\cline{3-8}
& Depth &
Specific Activity CRa of $^{226}Ra$ (Bq/kg) &
Specific Activity CTh of $^{232}Th$ (Bq/kg) &
Specific Activity CK of $^{40}K$ (Bq/kg) &
Radium Equivalent Concentration C[e]Ra &
Internal Irradiation Index IRa &
External Irration Index Ir \\
\hline
FSYY-1 & 299.18$\sim$300.28 & 166.6 & 122.1 & 1066.4 & 425.2782 & 0.8 & 1.2 \\ \hline
FSYY-2 & 403.11$\sim$404.11 & 155.9 & 110.0 & 1106.3 & 401.7544 & 0.8 & 1.1 \\ \hline
FSYY-3 & 557.28$\sim$558.38 & 88.6 & 83.6 & 1239.7 & 310.5536 & 0.4 & 0.9 \\ \hline
FSYY-4 & 616.47$\sim$617.12 & 173.1 & 129.5 & 536.6 & 395.1458 & 0.9 & 1.1 \\ \hline
FSYY-5 & 633.02$\sim$633.77 & 116.4 & 111.2 & 342.2 & 296.6336 & 0.6 & 0.8 \\ \hline
FSYY-6 & 650.23$\sim$650.93 & 130.1 & 113.0 & 1062.4 & 376.1412 & 0.7 & 1.0 \\ \hline
\hline
\end{tabular}
}
\end{table}
\subsubsection{ Stability Assessment of the Enclosing Rock }
The experimental hall is 48 meters wide, and can be divided into two parts: the upper cavity is enclosed by the crown and high side walls (vertical section) that connect the crown part, and is mainly used to accommodate lifting appliances and other facilities (e.g., cable channel, etc.), and, the lower cavity is a cylindrical pool, 42.5 meters in diameter and 42.5 meter in depth. Utility halls are arranged around the experimental hall. The experimental hall is large and complex in structure. The rock types and long term stabilibty of underground facility should be assessed thouroughly.
The length of the experimental hall is 55.25 meters. The axial is directed in $N37^{\circ}W$. The cavity enclosing rocks are granite, slightly weathered, in block structure. The developed joints are mainly steeply inclined while locally developed joints are slowly inclined and moderately inclined. The joint planes are straight, coarse, and are usually enclosed without filled material. Some joints are slightly open or fully open and filled with rock debris or kaolin film. There exists local water seepage or water dripping at some places.
The quality rating of the enclosing rock is considered as grade II.
Overall, the cavity enclosing rocks are stable. However, local unstable blocks may be produced under the inter-cutting action of the three groups of steeply inclined joints because of the large span of the experimental hall.
According to the terrestrial stress test, the direction of maximum horizontal stress is $N10^{\circ}W{\sim}N54^{\circ}W$. Since the upper cavity is approximately in a square shape, the terrestrial stress has little influence on the selection of the axial direction. The terrestrial stress difference among different directions are small and has little impact on the stability of the side walls. Since the cavity span is large, the terrestrial stress is in static state while the redistributed stress of the cavities is complex. Stability verification must be carried out and appropriate engineering measures must be taken.
The geological conditions (e.g., lithology of enclosing rocks, joint development, and underground water, et al) in the pool are the same as those of the upper cavity. However, because the main joints are steeply inclined, water pool may form an unfavorable combination with the side walls. However, the pool enclosing rocks are overall stable.
The geological conditions of the ancillary cavities are the same as
those of the experimental hall. The cavity enclosing rocks have high
overall stability. Although the utility cavities are small, at
intersections among the experimental hall and ancillary cavities,
care must be taken for the blasting control in order to keep the
integrality and stability of structure.
\chapter{DAQ and DCS}
\label{ch:DAQAndDCS}
The data acquisition (DAQ) system and detector control system (DCS) are introduced separately as followed.
\section{Data Acquisition System}
The main task of the data acquisition (DAQ) system is to record antineutrino candidate events observed in the antineutrino detectors. In order to understand the backgrounds, other types of events are also recorded, such as cosmic muon events, low energy radioactive backgrounds etc. Therefore, the DAQ must record data from the electronics of antineutrino and muon detectors with precise timing and charge information. DAQ should build event with separated data fragment from all electronics devices. Then DAQ needs to analyze and process data to compress data and monitor data quality. At the last step, DAQ should save most relevant data to disk.
\subsection{System Requirements}
This section presents the DAQ main design requirements.
\subsubsection{Event Rate}
The central detector (CD) of JUNO is composed of about $\sim$17,000~PMTs. According to MC simulation, order of tens antineutrino candidates per day can be acquired by the CD. Most events of the CD come from radioactive backgrounds of PMT glasses, steel and liquid scintillator material. The trigger system could reduce PMT dark noise and radioactive backgrounds event rate to order of 1~kHz. DAQ system will be designed to handle a maximum trigger rate of 1~kHz.
The water shielding detector is composed of about 1500~PMTs and adopt the same electronics design. But the trigger rate could be less than 100~Hz with radioactive backgrounds and muon events. The top muon detector is under design and the bandwidth requirements should be small and ignored.
The calibration system could bring several hundreds to 1~kHz additional event rate. But the trigger rate could be set to a reasonable level. A supernova explosion will generate many neutrino events over a short time scale. For example, about $\sim$10k events will be generated in $\sim$10~seconds from a 10~kpc distance supernova explosion.
\subsubsection{Readout Data Rate Estimation}
The electronics system plans to use 12 bits FADC at 1 GHz sample rate with 2 ranges to acquire signal waveform in maximum 1~$\mu$s time window. In case of 16 bits occupation of one sample, the data size of single PMT signal could be estimated about 2k bytes including bytes of time stamp and header. Table \ref{DAQRate} shows DAQ readout data estimation, and the total data rate of CD is about $\sim$2.4~GBytes/s which means data rate per PMT is $\sim$140~kBytes/s in Physics mode. The high energy cosmic muon events of CD are estimated to be about 3 Hz and will fire all $\sim$17,000~PMTs of the CD. Neutrino and background low energy events are estimated to be less than 1 kHz with less than 3,000 and 1,000 fired PMTs in average. Water shielding detector is estimated to be with $\sim$100~Hz event rate and $\sim$10\% channels fired per event.
We can set up both high and low energy calibration with similar data rate. The data rate of calibration run modes is 4.6 GB/s and means 273 KByte/s per PMT, which is about 2 times of the data rate of normal physics run mode. According to the table, 3,000~PMTs are assumed to be fired by one supernova event. The additional supernova event data rate is about 3 times larger than that of normal physics running modes. So DAQ was required 11.6 GB/s of total readout bandwidth at least and means 675~KB/s per PMT as 5 times physics mode when supernova happens with calibration together. But it will make remarkable increase for closed supernova explosion.
Supernova events will only continue about 10~seconds, LED based calibration run can be paused to improve the performance. So DAQ only needs to keep about 4.6 GB/s data processing performance for calibration run mode and buffer additional 60 GB supernova data in memory to process latter.
Due to the signal width is very small than 1~$\mu$s data window, we can assume to compress waveform to about 50~ns effective width with about 100~Bytes per signal. As estimated in the table about 95\% low energy event data will be removed. Then only about 200~MB/s data in Physics run mode and 300~MB/s in calibration run mode need to be built to full events and save to disk. The maximum data storage requirements is 600`MB/s when supernova happens during calibration run.
\begin{table}[!hbp]
\caption{JUNO DAQ Data Rate Estimation}
\label{DAQRate}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{p{3.5cm}|p{1.5cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}}
\hline
& Channels & Data size per channel(Bytes) & Fired nPMT & Trigger rate(Hz) & Data Rate (MBytes/s) \\
\hline
\tabincell{c}{CD\\(Low energy events)} & 17,000 & 100(50ns)/ 2,000(1us) & 1,000 & 1k & 100/2,300 \\
\hline
\tabincell{c}{CD\\(High energy events)} & 17,000 & 2,000(1us) & 17,000 & 3 & 96 \\
\hline
Water Shelf Detector & 1,500 & 100(50ns)/ 2,000(1us) & 150 & 100 & 1.5/30 \\
\hline
\tabincell{c}{Sum\\(Physics mode)} & & & & & 197.5/2426 \\
\hline
\hline
\tabincell{c}{CD\\(Calibration events)} & 17,000 & 100(50ns)/ 2,000(1us) & 1,000-17,000 & 1k &100/2,300\\
\hline
\tabincell{c}{CD\\(Supernova events)} & 17,000 &100(50ns)/ 2,000(1us) & 3000 & 1k & 300/6,900\\
\hline
\tabincell{c}{Sum\\(Maximum)} & & & & & 597.5/11,626\\
\hline
\end{tabular}
\end{table}
\subsubsection{Data Process Requirements}
There are several types of read out data links between DAQ and electronic. Ethernet with TCP/IP protocol could be a simple option. DAQ could organize readout branch with groups of PMTs. To deal with tens GB/s readout bandwidth, DAQ could use multiple 10 Gbps link to readout. Then DAQ should assemble completed events with data fragment from different links by trigger number or time stamp.
DAQ needs to finish waveform compression if electronics cannot do that. If we want to reduce data rate further, we need software trigger to remove more background events.
DAQ also needs to merge different detectors' events together and sorts events by time stamp.
\subsubsection{Other Functional Requirements}
DAQ also needs provide common functions like run control, run monitoring, information sharing, distributed process manager, software configure, bookkeeping, Elog, data quality monitoring, remote monitoring and so on.
\subsection{Conceptual Design Schema}
The JUNO experiment has many similar cases with BESIII and the Daya Bay experiments in DAQ part. We can design and develop JUNO DAQ based on BESIII, Daya Bay and ATLAS DAQ\cite{Bes3TDR,DybTDR,DybDaqTDR}. Figure \ref{daqArc} is the conceptual design schema of JUNO DAQ, DAQ readout electronics data through network. Except that network switches are placed in underground experiment hall, all other DAQ computers are deployed at ground computer room. DAQ constructs a blade servers' farm to process data, and the farm is connected with underground switches through multiple 10 gigabit fibers.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{DAQAndDCS/figures/daqArc.jpg}
\caption{DAQ Conceptual Architecture Design Diagram}
\label{daqArc}
\end{center}
\end{figure}
Due to electronics systems are not easy to dynamically distribute different DAQ hosts, it is better to design a two level event building at DAQ. The first level (ROS, read out system) reads out front-end electronics (FEE) data and builds event fragments and then the second level finishes full event building. It is better that DAQ compresses waveform data at the first level to reduce unuseful data transferring. The simplest waveform compression algorithm is zero compression with a configurable threshold. Only sample data exceeding the threshold can be reserved.
JUNO DAQ can use the same two level event building data flow scheme refer to BESIII event building as shown in Fig. \ref{daqEB} \cite{LifeiThesis}:
\begin{enumerate}
\item front-end electronics (FEE) send data to ROS(read out system) through network.
\item ROS receive all data slice of one event and send event id to TSM(trigger synchronizing manager).
\item TSM send event id to DFM(data flow manager) when TSM get all same event id from all ROSs.
\item DFM assign event id to a free EBN(event building node).
\item EBN send data request to each ROSs.
\item ROSs send requested data to EBN.
\item EBN receive all ROSs data fragments of one event and finish full event building, then send event id back to DFM.
\item DFM send event id to ROSs to clear data buffer. \ldots
\end{enumerate}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{DAQAndDCS/figures/daqEB.pdf}
\caption{DAQ Event Building Collaboration Diagram}
\label{daqEB}
\end{center}
\end{figure}
Software trigger can be performed as BESIII event filter\cite{LiuyjThesis}. The figure \ref{daqEF} shows event filter collaboration diagram, each EF (event filter) node requests events from EB node, then sends them to PT (process task) to analyze data for software trigger and data quality monitoring, at the end sends triggered event to DS (data storage) node for storage. Another option is performing software trigger before full event building as Atlas Level 2 trigger. It is better if DAQ needs compress waveform.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{DAQAndDCS/figures/daqEF.pdf}
\caption{DAQ Event Filter Collaboration Diagram}
\label{daqEF}
\end{center}
\end{figure}
All three detectors can share one DS node to save data to disks. Then event merging and sorting can be performed at DS node.
\subsection{R$\&$D Plan}
JUNO DAQ needs the following technical research for detail design:
\begin{enumerate}
\item Waveform compression algorithms and read out host performance.
\item Software trigger algorithms.
\item Event building network and performance.
\item Event sorting and storage.
\item Big scale computer farm computing and software developing.
\item Integrating and test with dedicated electronics and detector system.
\end{enumerate}
\subsection{Manufacture, Developing and Installation}
DAQ will not rely on custom made DAQ hardware, so there are no manufacture issues. But we need to investigate and survey different brands and types to integrate design schema. Hardware installation will be outsourced to vendors or system integration company. DAQ software will be designed and developed based on BESIII, Daya Bay and ATLAS DAQ software.
\subsection{Risk Analysis and Safety Evaluation}
The waveform data compression ratio could not reach expectation of detectors and electronics real performance.
There are another two challenges for DAQ up to now. One is that electronics might adopt none hardware trigger schema, the other is that supernova explosion could happen in a closer location.
\subsection{Schedule}
Major technical issues research according to electronics design. Finish DAQ technical design in 2013-2015.
Software developing, test and debug with electronics in 2016-2017.
Installation, deployment and integration in 2018-2020.
\section{Detector Control System}
\subsection{Requirements}
The main task of the Detector Control System is to establish long-term monitoring of the parameters affecting the performance of the experimental equipment. The parameters include pressure, temperature, humidity, liquid level of the scintillation, electronics, gas pressure and the pressure in the lobby and the entire electrical and mechanical environment working status of the devices. Some subsystems need to provide device control such as calibration system, gas system, water cycle system and power system. The real time operation states of the devices will be monitored and recorded into database. When an exception occurs the system can issue timely warnings at the same time through a secure interlock. Meanwhile, the devices can be protected by the safety interlock automatically to prevent equipment damage and personal protection.
\subsubsection{System Requirement}
The system will meet the requirements of an effective mass of 20,000 tons of central detector, which contains about 17,000 photomultiplier tubes. Due to the large scale of the detector there are about $\sim$1000 temperature and humidity monitoring points, about $\sim$20,000 channels of high voltages, and thousands of power supplies etc. There is also pure water system, gas system as well. The collection of the system requirements is based on the design of the detector, electronics and trigger system. The design requirements include the hardware and software design, the test bed of the subsystem design and the system integration.
\subsubsection{Function Requirement}
According to the experimental requirements the general software framework will be designed. The high voltage, the temperature monitoring module, the hardware interface, and the hardware and the software protocols will be defined. Drivers and data acquisition framework of the exchange process will be developed. The function modules of the DCS system are shown below.
\begin{figure}[htb]
\label{dcsFW}
\begin{center}
\includegraphics{DAQAndDCS/figures/dcsFW.JPG}
\caption{Framework Design of Detector Control System}
\end{center}
\end{figure}
\subsubsection{System Schema Design}
\begin{itemize}
\item Content
According to the actual hardware and software requirements of the experiment the system uses a hierarchical design framework. The system will be divided into the global control layer, the control layer and the data acquisition layer. The global control layer will realize the overall experimental equipment information collection, safety interlock, remote control, data storage, network information release and permissions management. Local control layer will realize the local control of the local equipment in the experiment hall. And support the local device monitoring, data recording, data upload and alarm. The data acquisition layer is response for the various hardware interfaces. It can support for embedded systems such as ARM, FPGA and standard industry interface hardware such as PLC, USB and serial RS232 interface and acquisition, which based on the network devices such as the TCP/IP data acquisition interface.
The system will be built on a distributed system development method. According to the experimental equipment distribution characteristics the distributed data exchange platform will be used for the development. Global control systems share the data and the interactively control commands by information sharing pool. Configuration files will use the text format specification which can realize the remote configuration, distributed sharing and management.
The system uses a module based approach of development. This method can achieve rapid integration of complex systems. From a functional view the system will be divided into data acquisition, control module, alarm module, memory module, data sharing and processing, system configuration, privilege management and user interface. Each subsystem can choose module assembly interface based on the actual system requirements. The modules can be divided by subsystem as following:
\begin{enumerate}
\item High voltage monitoring system, including the central detector PMT high voltage system, RPC high voltage etc.
\item Detector electronics chassis monitoring system, including the central detector electronics, RPC detector electronics, both inside and outside the pool detector electronics, monitoring content includes electronic temperature, power supply, fan, over current protection, low-voltage power supply.
\item Temperature and humidity monitoring system, including liquid temperature detector, temperature monitoring room.
\item Gas monitoring system, including the gas support system and center detector cover gas system
\item Center detector overflow tank monitoring, oil monitoring, liquid level clarity, camera monitoring and the calibration system of monitoring center.
\item Experimental hall of environmental temperature, humidity, pressure monitoring system, the radon monitoring system, video monitoring.
\item Control room monitoring system, database system and Webpage remote monitoring system.
\item Water system
\end{enumerate}
\item Key Technology
The key technology of detector control system is the integration framework development of the software. The system will develop a set of integrated management module according to the detector hardware requirements. The system functions will be realized by the conceptual design and the detailed modular design. For the hardware system using commercial framework or special design system the data acquisition module will be designed for the integration. The commercial software will interface with the DCS framework by the integration. The software design is completed after repeated assembly test and performance test before put into use. The user feedback will be fully considered during the software design and the commissioning period. The software will enter the formal running when the stage of training finished.
The traditional design of the JUNO central detector is a stainless-steel tank plus an acrylic sphere, where stainless-steel tank is used to separate the shielding liquid (mineral oil or LAB, to be decided) from the pure water in the water pool, and the acrylic sphere is to hold the 20kt LS. The design principle and key technology of the Daya Bay neutrino experiment could be a reference. The LS separated by an organic glass container is 35.4~meters in diameter. For the quality of the target substance which is one of the most important factors affecting the precision of experiment the quality change of the target material this requires for long-term supervisory. At the same time, the temperature inside the detector needs to be monitored. Environmental temperature variation not only impact on detector performance, the detector energy scale but also on photomultiplier tube noise, amplification and electronics precision. This requires many point of monitoring from the experimental hall. Therefore, the design of reliable, compatible of monitoring system is needed. Various experimental parameters such as temperature, humidity, voltage, electronics currents, radon density accurately level need to be achieved in the long-term and stable running of the experiment.
Software tools and applications which provide a software infrastructure for use in building distributed control systems will be used to operate devices. Such distributed control systems typically comprise tens or even hundreds of computers, networked together to allow communication between them and to provide control and feedback of the various parts of the device from a central control room, or even remotely over the internet. System will use Client/Server and Publish/Subscribe techniques to communicate between the various computers. Most servers (called Input/Output Controllers or IOCs) perform real-world I/O and local control tasks, and publish this information to clients using the Channel Access (CA) network protocol.
\item Development Process
Top development platform can be used to configure software such as LabVIEW or open source platforms such as EPICS (a widely used software platform for the large-scale control system physical device). The device driver can be developed by hardware I/O controller (IOC) server. Subsystems in the pre-design stage will build a testbed for the development of hardware drivers to test the hardware and software functions. Each detector subsystem model will be tested in testbed platform, as well as data acquisition interface. For the hardware which has no testbed device simulation models should be made with the software development team and a common definition of the interface specification, including data format, the transmission frequency, control flow, the interface distribution, database table and so on.
\end{itemize}
\chapter{ExecutiveSummary}
\label{ch:ExecutiveSummary}
\chapter{Introduction}
\label{chap:intro}
\section{Neutrino Physics: Status and Prospect}
\subsection{Fundamentals of Massive Neutrinos}
\label{subsec:mixing}
In the Standard Model \cite{Weinberg} of particle physics, the weak
charged-current interactions of leptons and quarks can be written as
\begin{eqnarray}
{\cal L}^{}_{\rm cc} = &-&\frac{g}{\sqrt{2}} \left[ \overline{\left(e
\hspace{0.3cm} \mu \hspace{0.3cm} \tau \right)^{}_{\rm L}} \
\gamma^\mu \ U \left(\begin{matrix} \nu^{}_1 \cr \nu^{}_2 \cr
\nu^{}_3 \cr \end{matrix}\right)^{}_{\rm L} W^-_\mu \right. \nonumber \\ &+&
\left.\overline{\left(u \hspace{0.3cm} c \hspace{0.3cm} t \right)^{}_{\rm
L}} \ \gamma^\mu \ V \left(\begin{matrix} d \cr s \cr b \cr
\end{matrix}\right)^{}_{\rm L} W^+_\mu \right] + {\rm h.c.} \; ,
\end{eqnarray}
where all the fermion fields are the mass eigenstates, $U$ is the
$3\times 3$ Maki-Nakagawa-Sakata-Pontecorvo (MNSP) matrix
\cite{MNS}, and $V$ denotes the $3\times 3$
Cabibbo-Kobayashi-Maskawa (CKM) matrix \cite{CKM}.
Given the basis in which the flavor eigenstates of the three charged
leptons are identified with their mass eigenstates, the flavor
eigenstates of the three active neutrinos and $n$ sterile neutrinos read
as
\begin{eqnarray}
\left(\begin{matrix} \nu^{}_e \cr \nu^{}_\mu \cr \nu^{}_\tau \cr
\vdots \cr
\end{matrix}
\right) = \left(\begin{matrix} U^{}_{e1} & U^{}_{e2} & U^{}_{e3} &
\cdots \cr U^{}_{\mu 1} & U^{}_{\mu 2} & U^{}_{\mu 3} & \cdots \cr
U^{}_{\tau 1} & U^{}_{\tau 2} & U^{}_{\tau 3} & \cdots \cr \vdots &
\vdots & \vdots & \ddots \cr \end{matrix} \right)
\left(\begin{matrix} \nu^{}_1 \cr \nu^{}_2 \cr \nu^{}_3 \cr \vdots
\cr
\end{matrix}
\right) \; .
\end{eqnarray}
where $\nu^{}_i$ is a neutrino mass eigenstate with the physical
mass $m^{}_i$ (for $i=1,2, \cdots, 3+n$). Equation (1.1) tells us that a
$\nu^{}_\alpha$ neutrino can be produced from the $W^+ +
\ell^-_\alpha \to \nu^{}_\alpha$ interaction, and a $\nu^{}_\beta$
neutrino can be detected through the $\nu^{}_\beta \to W^+ +
\ell^-_\beta$ interaction (for $\alpha, \beta = e, \mu, \tau$). So
oscillation may happen if the $\nu^{}_i$ beam with energy $E \gg
m^{}_i$ travels a proper distance $L$. In vacuum, the oscillation
probability of the $\bar\nu^{}_e \to \bar\nu^{}_e$ transition turns
out to be
\begin{eqnarray}
P(\overline{\nu}^{}_e \to \overline{\nu}^{}_e) = 1 -
\frac{4}{\displaystyle \left(\sum_i |U^{}_{e i}|^2 \right)^2}
\sum^{}_{i<j} \left(|U^{}_{e i}|^2 |U^{}_{e j}|^2 \sin^2
\frac{\Delta m^{2}_{ij} L}{4 E} \right) \; ,
\end{eqnarray}
with $\Delta m^2_{ij} \equiv m^2_i - m^2_j$ being the mass-squared
difference. Note that the denominator on the right-hand side of Equation
(1.3) is not equal to one if there are heavy sterile antineutrinos
which mix with the active antineutrinos but do not take part in the
flavor oscillations. Note also that the terrestrial matter effects
on $P(\overline{\nu}^{}_e \to \overline{\nu}^{}_e)$ are negligibly
small, because the typical value of $E$ is only a few MeV and that
of $L$ is usually less than several hundred km for a realistic reactor-based
$\overline{\nu}^{}_e \to \overline{\nu}^{}_e$ oscillation
experiment.
If the $3\times 3$ MNSP matrix $U$ is exactly unitary, it can be
parameterized in terms of three flavor mixing angles and three
CP-violating phases in the following standard way \cite{PDG}:
\begin{eqnarray}
U & \hspace{-0.2cm} = \hspace{-0.2cm} & \left( \begin{matrix} 1 & 0
& 0 \cr 0 & c^{}_{23} & s^{}_{23} \cr 0 & -s^{}_{23} & c^{}_{23} \cr
\end{matrix} \right) \left( \begin{matrix} c^{}_{13} & 0 & s^{}_{13}
e^{-{\rm i}\delta} \cr 0 & 1 & 0 \cr -s^{}_{13} e^{{\rm i}\delta} &
0 & c^{}_{13} \cr
\end{matrix} \right)
\left( \begin{matrix} c^{}_{12} & s^{}_{12} & 0 \cr -s^{}_{12} &
c^{}_{12} & 0 \cr 0 & 0 & 1 \cr \end{matrix} \right) P^{}_\nu
\nonumber \\
& \hspace{-0.2cm} = \hspace{-0.2cm} & \left( \begin{matrix}
c^{}_{12} c^{}_{13} & s^{}_{12} c^{}_{13} & s^{}_{13} e^{-{\rm i}
\delta} \cr -s^{}_{12} c^{}_{23} - c^{}_{12} s^{}_{13} s^{}_{23}
e^{{\rm i} \delta} & c^{}_{12} c^{}_{23} - s^{}_{12} s^{}_{13}
s^{}_{23} e^{{\rm i} \delta} & c^{}_{13} s^{}_{23} \cr s^{}_{12}
s^{}_{23} - c^{}_{12} s^{}_{13} c^{}_{23} e^{{\rm i} \delta} &
-c^{}_{12} s^{}_{23} - s^{}_{12} s^{}_{13} c^{}_{23} e^{{\rm i}
\delta} & c^{}_{13} c^{}_{23} \cr
\end{matrix} \right) P^{}_\nu \; ,
\end{eqnarray}
where $c^{}_{ij} \equiv \cos\theta^{}_{ij}$ and $s^{}_{ij} \equiv
\sin\theta^{}_{ij}$ (for $ij = 12, 13, 23$) are defined, and
$P^{}_\nu = {\rm Diag}\{e^{{\rm i}\rho}, e^{{\rm i}\sigma}, 1\}$
denotes the diagonal Majorana phase matrix which has nothing to do
with neutrino oscillations. In this case,
\begin{eqnarray}
P(\overline{\nu}^{}_e \to \overline{\nu}^{}_e) &=& 1 - \sin^2
2\theta^{}_{12} c^4_{13} \sin^2 \frac{\Delta m^{2}_{21} L}{4 E}\nonumber \\ &-&
\sin^2 2\theta^{}_{13} \left[ c^2_{12} \sin^2 \frac{\Delta
m^{2}_{31} L}{4 E} + s^2_{12} \sin^2 \frac{\Delta m^{2}_{32} L}{4 E}
\right] \; ,
\end{eqnarray}
in which $\Delta m^2_{32} = \Delta m^2_{31} - \Delta m^2_{21}$. The
oscillation terms driven by $\Delta m^2_{21}$ and $\Delta m^2_{31}
\simeq \Delta m^2_{32}$ can therefore be used to determine
$\theta^{}_{12}$ and $\theta^{}_{13}$, respectively.
\subsection{Open Issues of Massive Neutrinos}
\label{subsec:openissue}
\begin{table}[t]
\vspace{-0.25cm}
\begin{center}
\caption{The best-fit values, together with the 1$\sigma$
and 3$\sigma$ intervals, for the six three-flavor neutrino
oscillation parameters from a global analysis of current
experimental data \cite{GF1}.} \vspace{0.5cm}
\begin{tabular}{c|c|c|c}
\hline \hline
Parameter & Best fit & 1$\sigma$ range & 3$\sigma$ range \\
\hline \multicolumn{4}{c}{Normal neutrino mass hierarchy $(m^{}_1 <
m^{}_2 < m^{}_3$)} \\ \hline
$\Delta m^2_{21}/10^{-5} ~{\rm eV}^2$
& $7.54$ & 7.32 --- 7.80 & 6.99 --- 8.18 \\
$\Delta m^2_{31}/10^{-3} ~ {\rm eV}^2$~ & $2.47$
& 2.41 --- 2.53 & 2.27 --- 2.65 \\
$\sin^2\theta_{12}/10^{-1}$
& $3.08$ & 2.91 --- 3.25 & 2.59 --- 3.59 \\
$\sin^2\theta_{13}/10^{-2}$
& $2.34$ & 2.15 --- 2.54 & 1.76 --- 2.95 \\
$\sin^2\theta_{23}/10^{-1}$
& $4.37$ & 4.14 --- 4.70 & 3.74 --- 6.26 \\
$\delta/180^\circ$ & $1.39$ & 1.12 --- 1.77 & 0.00 --- 2.00 \\ \hline
\multicolumn{4}{c}{Inverted neutrino mass hierarchy $(m^{}_3 < m^{}_1
< m^{}_2$)} \\ \hline
$\Delta m^2_{21}/10^{-5} ~{\rm eV}^2$
& $7.54$ & 7.32 --- 7.80 & 6.99 --- 8.18 \\
$\Delta m^2_{31}/10^{-3} ~ {\rm eV}^2$~ & $2.34$
& 2.28 --- 2.40 & 2.15 --- 2.52 \\
$\sin^2\theta_{12}/10^{-1}$
& $3.08$ & 2.91 --- 3.25 & 2.59 --- 3.59 \\
$\sin^2\theta_{13}/10^{-2}$
& $2.40$ & 2.18 --- 2.59 & 1.78 --- 2.98 \\
$\sin^2\theta_{23}/10^{-1}$ & $4.55$ & 4.24 --- 5.94
& 3.80 --- 6.41 \\
$\delta/180^\circ$ & $1.31$ & 0.98 --- 1.60 & 0.00 --- 2.00 \\ \hline\hline
\end{tabular}
\end{center}
\end{table}
Although we have known quite a lot about massive neutrinos,
where the current status of neutrino oscillation measurements can be
summarized in Table~2.1, we have many open questions about their
fundamental
properties and their unique roles in the Universe
\cite{XZ}. In the following we concentrate on some intrinsic flavor issues
of massive neutrinos which may related to future neutrino
experiments:
(a) The nature of neutrinos and their mass spectrum;
{\it Question (1): Dirac or Majorana nature?}
{\it Question (2): Normal or inverted mass hierarchy?}
{\it Question (3): The absolute mass scale?}
(b) Lepton flavor mixing pattern and CP violation;
{\it Question (4): The octant of $\theta^{}_{23}$?}
{\it Question (5): The Dirac CP-violating phase $\delta$?}
{\it Question (6): The Majorana CP-violating phases $\rho$ and
$\sigma$?}
(c) Extra neutrino species and unitarity tests;
{\it Question (7): Extra light or heavy sterile neutrinos?}
{\it Question (8): Direct and indirect non-unitary effects?}
\section{JUNO Experiment}
\label{sec:juno}
The Jiangmen Underground Neutrino Observatory (JUNO) is a multi-purpose neutrino experiment.
It was proposed in 2008 for neutrino mass hierarchy (MH) determination by detecting reactor
antineutrinos from nuclear power plants (NPPs)~\cite{zhanl2008,
yfwang2008,caoj2009}. The site location is optimized to have the
best sensitivity for mass hierarchy determination, which is at 53~km
from both the Yangjiang and Taishan NPPs. The neutrino detector is a
liquid scintillator (LS) detector with a 20~kton fiducial mass,
deployed in a laboratory 700 meters underground. The JUNO experiment
is located in Jinji town, Kaiping city, Jiangmen city, Guangdong
province which is shown in Fig.~\ref{fig:intro:location}. The
thermal power and baselines are listed in Table~\ref{tab:intro:NPP}.
\begin{figure}[htb!]
\centering
\includegraphics[width=0.7\textwidth]{Introduction/Figs/JUNO_location.pdf}
\caption{Location of the JUNO site. The distances to the nearby
Yangjiang NPP and Taishan NPP are both 53 km. Daya Bay NPP is 215 km
away. Huizhou and Lufeng NPPs have not been approved yet. Three
metropolises, Hong Kong, Shenzhen, and Guangzhou, are also shown.
\label{fig:intro:location} }
\end{figure}
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}\hline\hline
Cores & YJ-C1 & YJ-C2 & YJ-C3 & YJ-C4 & YJ-C5 & YJ-C6 \\
\hline Power (GW) & 2.9 & 2.9 & 2.9 & 2.9 & 2.9 & 2.9 \\ \hline
Baseline(km) & 52.75 & 52.84 & 52.42 & 52.51 & 52.12 & 52.21 \\
\hline\hline
Cores & TS-C1 & TS-C2 & TS-C3 & TS-C4 & DYB & HZ \\
\hline Power (GW) & 4.6 & 4.6 & 4.6 & 4.6 & 17.4 & 17.4 \\ \hline
Baseline(km) & 52.76 & 52.63 & 52.32 & 52.20 & 215 & 265 \\
\hline
\end{tabular}
\caption{Summary of the thermal power and baseline to the JUNO
detector for the Yangjiang (YJ) and Taishan (TS) reactor cores, as
well as the remote reactors of Daya Bay (DYB) and Huizhou
(HZ).\label{tab:intro:NPP}}
\end{table}
JUNO consists of a central detector, a water Cherenkov
detector and a muon tracker (shown in Fig.~\ref{fig:intro:det}). The
central detector is a LS detector of 20~kton target mass and
$3\%/\sqrt{E{\rm (MeV)}}$ energy resolution. The central detector is
submerged in a water pool to be shielded from natural
radioactivities from the surrounding rock and air. The water pool is
equipped with PMTs to detect the Cherenkov light from muons. On top
of the water pool, there is another muon detector to accurately
measure the muon track.
\begin{figure}[htb!]
\centering
\includegraphics[width=0.7\textwidth]{Introduction/Figs/JUNO_generic_detector.png}
\caption{A schematic view of the JUNO detector.
\label{fig:intro:det} }
\end{figure}
It is crucial to achieve a $3\%/\sqrt{E{\rm (MeV)}}$ energy resolution for the determination of the MH.
A Monte Carlo simulation has been developed based on
the Daya Bay Monte Carlo. The photoelectron yield has been tuned
according to the real data of Daya Bay. The required energy resolution can
be reached with the following improvements from Daya Bay \cite{DYB}:
\begin{itemize}
\item The PMT photocathode covergage $\geq 75$\%.
\item The PMT photocathode quantum efficiency $\geq 35$\%.
\item The attenuation length of the liquid scintillator $\geq 20$~m at 430~nm,
which corresponds to an absorption length of 60~m with a Rayleigh
scattering length of 30~m.
\end{itemize}
For the real experimental environments, there are many other factors beyond the photoelectron
statistics
that can alter the energy resolution, including the dark noise from the PMTs and electronics,
the detector non-uniformity and vertex resolution, and the PMT charge resolution. A generic parametrization
for the detector energy resolution is defined as
\begin{equation}
\frac{\sigma_{E}}{E} = \sqrt{\left(\frac{a}{\sqrt{E}}\right)^2+b^2+\left(\frac{c}{E}\right)^2}\,\;,\label{eq:mh:abcterms}
\end{equation}
where the visible energy $E$ is in MeV.
Based on the numerical simulation as shown in the next section, a requirement for the resolution of
${a}/{\sqrt{E}}$ better than $3\%$ is approximately equivalent to the following requirement,
\begin{equation}
\sqrt{\left({a}\right)^2+\left({1.6\times b}\right)^2+\left(\frac{c}{1.6}\right)^2}\leq 3\%\;.\label{eq:mh:abc}
\end{equation}
\section{Physics Potentials}
\label{sec:potentials}
\subsection{Mass Hierarchy}
\label{subsubsec:MH}
\begin{figure
\begin{center}
\begin{tabular}{c}
\includegraphics*[width=0.6\textwidth]{Introduction/Figs/Espec.pdf}
\end{tabular}
\end{center}
\caption{Neutrino energy spectra (upper panel) and their ratio (lower panel) with the true (NH) and fit (IH) MHs.}
\label{fig:Espec}
\end{figure}
The neutrino mass hierarchy (MH) answers the question whether the
third generation ($\nu_3$ mass eigenstate) is heavier or lighter
than the first two generations ($\nu_1$ and $\nu_2$). The normal
mass hierarchy (NH) refers to $m_3
> m_1$ and the inverted mass hierarchy (IH) refers to $m_3 < m_1$.
JUNO is designed to resolve the neutrino MH using precision spectral
measurements of reactor antineutrino oscillations, where the general
principle is shown in Fig.~\ref{fig:Espec}.
In the JUNO simulation, we assume a 20~kton LS detector and
a total thermal power of two reactor complexes of 36~GW. We assume
nominal running time of six years, 300 effective days per year,
$80\%$ detection efficiency and a detector energy resolution
$3\%/\sqrt{E{\rm (MeV)}}$ as a benchmark. The simulation details can
be found in Ref.~\cite{JUNO}. In Fig.~\ref{fig:Espec},
the expected neutrino energy spectrum of the true normal MH, and best-fit
neutrino spectrum of the wrong inverted MH are shown in the upper panel, and
the ratio of two spectra is illustrated in the lower panel. One can observe that
the distinct feature of two MHs lies in the fine structures of the neutrino spectrum.
To quantify the sensitivity of MH determination, we define the following quantity as the MH discriminator,
\begin{equation}
\Delta \chi^2_{\text{MH}}=|\chi^2_{\rm min}(\rm NH)-\chi^2_{\rm
min}(\rm IH)|, \label{eq:mh:chisquare}
\end{equation}
where the minimization process is implemented for the oscillation parameters and systematics.
The discriminator defined in Eq.~(\ref{eq:mh:chisquare}) can be used
to obtain the optimal baseline, which is shown in the left panel of
Fig.~\ref{fig:mh:baseline}. An optimal sensitivity of $\Delta
\chi^2_{\text{MH}}\simeq16$ can be obtained for the ideal case with
identical baseline at around 50~km, where the oscillation effect of $\delta m^2_{21}$ is
maximal.
\begin{figure
\begin{center}
\begin{tabular}{cc}
\includegraphics*[bb=20 20 290 232, width=0.43\textwidth]{Introduction/Figs/MH_Baseline.pdf}
&
\includegraphics*[bb=20 16 284 214, width=0.42\textwidth]{Introduction/Figs/MH_disL.pdf}
\end{tabular}
\end{center}
\caption{The MH discrimination ability as the function of the
baseline (left panel) and as the function of the baseline difference of two
reactors (right panel).} \label{fig:mh:baseline}
\end{figure}
The impact of the baseline
difference due to multiple reactor cores is shown in the right panel
of Fig.~\ref{fig:mh:baseline}, by keeping the baseline of one
reactor unchanged and varying that of the other. A rapid oscillatory
behavior is observed, which demonstrates the importance of baseline
differences for the reactor cores. The worst case is at $\Delta L
\sim 1.7$~km, where the $|\Delta m^2_{ee}|$ related oscillation is
canceled between the two reactors. Taking into account the reactor power
and baseline distribution of the real experimental site of JUNO, we
show the reduction of the MH sensitivity in Fig.~\ref{fig:mh:ideal},
which gives a degradation of $\Delta \chi^2_{\text{MH}}\simeq5$.
There are also reactor and detector related uncertainties that
affect the MH sensitivity. Rate uncertainties are
negligible, because most of the MH sensitivity is derived from the spectral
information. On the other hand, the energy-related uncertainties are
important, including the reactor spectrum error, the detector
bin-to-bin error, and the energy non-linearity error. By considering
realistic spectral uncertainties and taking into account the
self-calibration of oscillation patterns of reactor antineutrino
oscillations, we obtain the nominal MH sensitivity of JUNO as shown
with the dashed lines in Fig.~\ref{fig:mh:chi2eemumu}. In addition,
due to the difference between $|\Delta m^2_{ee}|$ and $|\Delta
m^2_{\mu\mu}|$, precise measurements of the two mass-squared difference
can provide additional sensitivity to MH, besides the sensitivity
from the interference effects. In Fig.~\ref{fig:mh:chi2eemumu}, we
show with the solid lines the improvement by adding a $|\Delta
m^2_{\mu\mu}|$ measurement of $1\%$ precision, where an increase of
$\Delta \chi^2_{\text{MH}}\simeq8$ is achieved for the MH
sensitivity.
\begin{figure
\begin{center}
\begin{tabular}{c}
\includegraphics*[bb=26 22 292 222, width=0.5\textwidth]{Introduction/Figs/MH_Real_Ideal.pdf}
\end{tabular}
\end{center}
\caption{The comparison of the MH sensitivity for the ideal and
actual distributions of the reactor cores and baselines. The real
distribution gives a degradation of $\Delta
\chi^2_{\text{MH}}\simeq5$.} \label{fig:mh:ideal}
\end{figure}
\begin{figure
\begin{center}
\begin{tabular}{c}
\includegraphics*[bb=25 20 295 228, width=0.5\textwidth]{Introduction/Figs/MH_Prior_NMH_10.pdf}
\end{tabular}
\end{center}
\caption{the reactor-only (dashed) and combined (solid)
distributions of the $\Delta\chi^2$ function, where a $1\%$ relative
error of $\Delta m^2_{\mu\mu}$ is assumed and the CP-violating phase
($\delta$) is assigned to be $90^\circ/270^\circ$ ($\cos\delta=0$)
for illustration. The black and red lines are for the true (normal)
and false (inverted) neutrino MH, respectively.
\label{fig:mh:chi2eemumu}}
\end{figure}
\subsection{Precision Measurement of Mixing Parameters}
\label{subsec:prec}
JUNO is a precision experiment in terms of huge statistics (${\cal
O}(100\rm k)$ inverse beta decay (IBD) events), optimal baseline, unprecedented energy
resolution ($3\%/\sqrt{E}$) and accurate energy response (better
than $1\%$). Therefore, besides the determination of the neutrino
mass hierarchy (MH) \cite{JUNO}, the ${\cal O}(100\rm k)$ IBD events
allow JUNO to probe the fundamental properties of neutrino
oscillations and access four additional parameters $\theta_{12}$,
$\theta_{13}$, $\Delta m^2_{21}$, and $|\Delta m^2_{ee}|$.
\begin{table}[!htb]\footnotesize
\begin{center}
\begin{tabular}[c]{l|l|l|l|l|l} \hline\hline
& $\Delta m^2_{21}$ & $|\Delta m^2_{31}|$ & $\sin^2\theta_{12}$ &
$\sin^2\theta_{13}$ & $\sin^2\theta_{23}$ \\ \hline
Dominant Exps. & KamLAND & MINOS & SNO & Daya Bay & SK/T2K \\ \hline
Individual 1$\sigma$ & 2.7\% \cite{KLloe} & 4.1\% \cite{MINOS} & 6.7\% \cite{SNO}
& 6\% \cite{DBloe} & 14\% \cite{SKth23,T2Kth23} \\ \hline
Global 1$\sigma$ & 2.6\% & 2.7\% & 4.1\% & 5\% & 11\% \\
\hline\hline
\end{tabular}
\caption{\label{tab:prec:current} Current precision for the five
known oscillation parameters. The dominant experiments and their
corresponding 1$\sigma$ accuracy and global precision from global
fitting groups \cite{GF1} are shown in the first, second and third
row, respectively.}
\end{center}
\end{table}
\begin{table}[!htb]
\begin{center}
\begin{tabular}[c]{l|l|l|l|l|l} \hline\hline
& Nominal & + B2B (1\%) & + BG &
+ EL (1\%) & + NL (1\%) \\ \hline
$\sin^2\theta_{12}$ & 0.54\% & 0.60\% & 0.62\% & 0.64\% & 0.67\% \\ \hline
$\Delta m^2_{21}$ & 0.24\% & 0.27\% & 0.29\% & 0.44\% & 0.59\% \\ \hline
$|\Delta m^2_{ee}|$ & 0.27\% & 0.31\% & 0.31\% & 0.35\% & 0.44\% \\
\hline\hline
\end{tabular}
\caption{\label{tab:prec:syst} Precision of $\sin^2\theta_{12}$,
$\Delta m^2_{21}$ and $|\Delta m^2_{ee}|$ from the nominal setup to
those with more systematic uncertainties. The systematics are added
one by one from the left cell to right cell.}
\end{center}
\end{table}
Current precision for the five known oscillation parameters is
summarized in Table~\ref{tab:prec:current}, where both the results
from individual experiments and from global analyses \cite{GF1} are
presented. We notice that most of the oscillation parameters have
been measured to better than $10\%$.
Among all the four parameters that are accessible in JUNO, the $\theta_{13}$
measurement from JUNO is less accurate than that of Daya Bay because the
designed baseline is much larger than the optimized one ($\sim2\,\rm
km$) and only one single detector is considered in the design
concept. The ultimate $\sin^22\theta_{13}$ sensitivity of Daya Bay will be about $3\%$, and would lead this precision level in
the foreseeable future. Therefore, we shall consider the precision
measurements of $\theta_{12}$, $\Delta m^2_{21}$ and $|\Delta
m^2_{ee}|$\footnote{There will be two degenerate solutions for
$|\Delta m^2_{ee}|$ in case of undetermined MH. We consider
the correct MH in the following studies.}.
With the nominal setup as in the study of the MH
measurement \cite{JUNO}, we estimate the precision of the three relevant
parameters, $\sin^2\theta_{12}$, $\Delta
m^2_{21}$ and $\Delta m^2_{ee}$, which can achieve the level of $0.54\%$, $0.24\%$
and $0.27\%$, respectively.
Moreover, the effects of important systematic errors, such as
the bin-to-bin (B2B) energy uncorrelated uncertainty,
the energy linear scale (EL) uncertainty and the energy non-linear
(NL) uncertainty, and the influence of background
(BG) are presented. As a benchmark, a $1\%$ precision for all the
systematic errors considered is assumed. The background level and
uncertainties are the same as in the previous chapter for the MH
determination. In Table~\ref{tab:prec:syst}, we show the precision
of $\sin^2\theta_{12}$, $\Delta m^2_{21}$ and $|\Delta m^2_{ee}|$
from the nominal setup to those with more systematic uncertainties.
We can see that the energy-related uncertainties are more
important because most of the sensitivity is derived from the spectrum
distortion due to neutrino oscillations.
In summary, we can achieve the precision of $0.5\% - 0.7\%$ for
the three oscillation parameters $\sin^2\theta_{12}$, $\Delta m^2_{21}$
and $|\Delta m^2_{ee}|$. Precision tests of the unitarity
of the lepton mixing matrix and mass sum rule are possible with
the unprecedented precision of these measurements.
\subsection{Supernova Neutrinos}
\label{subsec:sn}
Measuring the neutrino burst from the next nearby supernova (SN) is
a premier goal of low-energy neutrino physics and astrophysics.
According to current understanding and numerical simulation, one
expects a neutrino signal with three characteristic phases as shown
in Fig.~\ref{fig:sn:SNburst}. In a high-statistics observation one
should consider these essentially as three different experiments,
each holding different and characteristic lessons for particle and
astrophysics.
\begin{itemize}
\item[\em 1.]{\em Infall, Bounce and Shock Propagation}.---Few tens of ms
after bounce. Prompt $\nu_e$ burst, emission of $\bar\nu_e$ at first
suppressed and emission of other flavors begins.
\item[\em 2.]{\em Accretion Phase (Shock Stagnation)}.---Few tens to few
hundreds of ms, depending on progenitor properties and other
parameters. Neutrino emission is powered by accretion flow.
Luminosity in $\nu_e$ and $\bar\nu_e$ perhaps as much as a factor of
two larger than each of the $\nu_x$ fluxes
\item[\em 3.]{\em Neutron-star cooling}.---Lasts until 10--20~s, powered by
cooling and deleptonization of the inner core on a diffusion time scale.
No strong asymmetries. Details depend on final neutron-star mass and
nuclear equation of state
\end{itemize}
\begin{figure
\centering
\includegraphics[width=1\textwidth]{Introduction/Figs/SNburst.pdf}
\caption{Three phases of neutrino emission from a core-collapse SN,
from left to right: (1)~Infall, bounce and initial shock-wave
propagation, including prompt $\nu_e$ burst. (2)~Accretion phase
with significant flavor differences of fluxes and spectra and time
variations of the signal. (3)~Cooling of the newly formed neutron
star, only small flavor differences between fluxes and spectra.
\label{fig:sn:SNburst}}
\end{figure}
In order to estimate the expected neutrino signals at JUNO, we
assume a fiducial mass of 20~kton LS. For a typical galactic SN at 10~kpc,
we take the time-integrated neutrino
spectra as $f_\nu(E_\nu) \propto E^\alpha_\nu
\exp[-(1+\alpha)E_\nu/\langle E_\nu \rangle]$ with a nominal index
$\alpha = 3$ and $\langle E_\nu \rangle$ being the average neutrino
energy. Furthermore, a total neutrino energy of $E_{\rm tot} = 3 \times
10^{53}~{\rm erg}$ is assumed to be equally distributed among neutrinos
and antineutrinos of three flavors. As average neutrino energies are
both flavor- and time-dependent, we calculate the event rates for
three representative values $\langle E_\nu \rangle = 12~{\rm MeV}$,
$14~{\rm MeV}$ and $16~{\rm MeV}$, and in each case the average
energy is assumed to be equal for all flavors. The total numbers of
neutrino events in JUNO are summarized in Table~\ref{table:events}.
Some comments on the detection channels are presented as follows.
(1) The inverse beta decay (IBD) is the dominant channel for
supernova neutrino detection at both scintillator and
water-Cherenkov detectors. In the IBD reaction
\begin{equation}
\overline{\nu}_e + p \to e^+ + n \; , \label{eq: IBD}
\end{equation}
the neutrino energy threshold is $E^{\rm th}_\nu \approx 1.8~{\rm MeV}$.
The deposition of
positron kinetic energy and the annihilation of the positron
give rise to a prompt signal. In addition, the
neutron is captured on free protons with a lifetime of about $200~{\rm \mu s}$,
producing a $2.2~{\rm MeV}$ $\gamma$. Hence the coincidence of
prompt and delayed signals increases greatly the power of background rejection.
\begin{table}[!t]
\centering
\begin{tabular}{ccccccccc}
\hline
\multicolumn{1}{c}{\multirow {2}{*}{Channel}} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{\multirow {2}{*}{Type}} & \multicolumn{1}{c}{} & \multicolumn{5}{c}{Events for different $\langle E_\nu \rangle$ values} \\
\cline{5-9} \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$12~{\rm MeV}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$14~{\rm MeV}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$16~{\rm MeV}$} \\
\hline
\multicolumn{1}{l}{$\overline{\nu}_e + p \to e^+ + n$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{CC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$4.3\times 10^3$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$5.0\times 10^3$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$5.7\times 10^3$} \\
\multicolumn{1}{l}{$\nu + p \to \nu + p$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{NC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$6.0\times 10^2$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$1.2\times 10^3$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$2.0\times 10^3$} \\
\multicolumn{1}{l}{$\nu + e \to \nu + e$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{NC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$3.6\times 10^2$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$3.6\times 10^2$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$3.6\times 10^2$} \\
\multicolumn{1}{l}{$\nu +~^{12}{\rm C} \to \nu +~^{12}{\rm C}^*$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{NC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$1.7\times 10^2$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$3.2\times 10^2$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$5.2\times 10^2$} \\
\multicolumn{1}{l}{$\nu_e +~^{12}{\rm C} \to e^- +~^{12}{\rm N}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{CC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$4.7\times 10^1$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$9.4\times 10^1$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$1.6\times 10^2$} \\
\multicolumn{1}{l}{$\overline{\nu}_e +~^{12}{\rm C} \to e^+ +~^{12}{\rm B}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{CC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$6.0\times 10^1$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$1.1\times 10^2$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$1.6\times 10^2$} \\
\hline
\end{tabular}
\caption{Numbers of neutrino events in JUNO for a SN at a galactic distance
of 10 kpc.}
\label{table:events}
\end{table}
(2) As an advantage of the LS detector, the
charged-current interaction on $^{12}{\rm C}$ takes place for both
$\nu_e$ and $\overline{\nu}_e$ via
\begin{eqnarray}
&& \nu_e +~^{12}{\rm C} \to e^- +~^{12}{\rm B} \; , \label{eq: CCnue}\\
&& \overline{\nu}_e +~^{12}{\rm C} \to e^+ +~^{12}{\rm N} \; .
\label{eq: CCnueb}
\end{eqnarray}
The energy threshold for $\nu_e$ is $17.34~{\rm MeV}$, while that
for $\overline{\nu}_e$ is $14.39~{\rm MeV}$. The subsequent beta
decays of $^{12}{\rm B}$ and $^{12}{\rm N}$ with a $20.2~{\rm ms}$
and $11~{\rm ms}$ half-life, respectively, lead to a prompt-delayed
coincident signal. Hence the charged-current reactions in
Eqs.~(\ref{eq: CCnue}) and (\ref{eq: CCnueb}) provide a possibility
to detect separately $\nu_e$ and $\overline{\nu}_e$.
(3) Neutral-current interaction on $^{12}{\rm C}$ is of crucial
importance to probe neutrinos of non-electron flavors, i.e.,
\begin{equation}
\nu +~^{12}{\rm C} \to \nu +~^{12}{\rm C}^* \; , \label{eq: NCnux}
\end{equation}
where $\nu$ stands for neutrinos or antineutrinos of all three
flavors. A $15.11~{\rm MeV}$ $\gamma$ from the de-excitation of
$^{12}{\rm C}^*$ to its ground state is a clear signal of SN
neutrinos. Since neutrinos of non-electron flavors $\nu_x$ have
higher average energies, the neutral-current interaction is more
sensitive to $\nu_x$, providing a possibility to pin down the flavor
content of supernova neutrinos. However,
it is impossible to reconstruct the neutrino energy in this channel.
(4) Elastic scattering of neutrinos on electrons will carry the
directional information of incident neutrinos, and thus can be used
to locate the SN. This is extremely important if a SN is hidden by
other stars and the optical signal is obscured by galactic dust. The
elastic scattering
\begin{equation}
\nu + e^- \to \nu + e^- \label{eq:ESe}
\end{equation}
is most sensitive to $\nu_e$ because of the larger cross section.
However, it is difficult to determine the direction of the scattered
electron in the scintillator detector. At this point,
large water-Cherenkov detectors, such as Super-Kamiokande,
are complementary to scintillator detectors.
(5) Elastic scattering of neutrinos on protons has been proposed as
a promising channel to measure supernova neutrinos of non-electron
flavors~\cite{Beacom:2002hs,Dasgupta:2011wg}:
\begin{equation}
\nu + p \to \nu + p \; . \label{eq: ESp}
\end{equation}
Although the total cross section is about four times smaller than
that of IBD reaction, the contributions from all the neutrinos and
antineutrinos of three flavors will compensate for the reduction of
cross section. As shown in
Refs.~\cite{Beacom:2002hs,Dasgupta:2011wg}, it is possible to
reconstruct the energy spectrum of $\nu_x$ at a large scintillator
detector, which is very important to establish flavor conversions or
spectral splits of SN neutrinos.
For a realistic measurement of $\nu_x$ spectrum, a low-energy
threshold and a satisfactory reconstruction of proton recoil energy
are required. Taking into account the quenching effect, the total number of events is about
$2240$ above a threshold $0.2~{\rm MeV}$.
In summary, we show the time-integrated neutrino event spectra of SN neutrinos
with respect to the visible energy $E^{}_{\rm d}$ in the JUNO detector for a SN at 10 kpc,
where no neutrino flavor conversions are assumed for illustration
and the average neutrino energies are $\langle E^{}_{\nu_e}\rangle = 12~{\rm MeV}$,
$\langle E^{}_{\overline{\nu}_e}\rangle = 14~{\rm MeV}$ and $\langle E^{}_{\nu_x}\rangle = 16~{\rm MeV}$.
The main reaction channels are shown together with the threshold of neutrino energies:
(1) IBD (black and solid curve), $E_{\rm d} = E^{}_\nu - 0.8~{\rm MeV}$;
(2) Elastic $\nu$-$p$ scattering (red and dashed curve), $E_{\rm d}$ stands for the recoil energy of proton;
(3) Elastic $\nu$-$e$ scattering (blue and double-dotted-dashed curve), $E_{\rm d}$ denotes the recoil energy of electron;
(4) Neutral-current reaction ${^{12}{\rm C}}(\nu, \nu^\prime){^{12}{\rm C}^*}$ (orange and dotted curve), $E_{\rm d} \approx 15.1~{\rm MeV}$;
(5) Charged-current reaction ${^{12}{\rm C}}(\nu_e, e^-){^{12}{\rm N}}$ (green and dotted-dashed curve), $E_{\rm d} = E_\nu - 17.3~{\rm MeV}$;
(6) Charged-current reaction ${^{12}{\rm C}}(\overline{\nu}_e, e^+){^{12}{\rm B}}$ (magenta and double-dotted curve), $E_{\rm d} = E_\nu - 13.9~{\rm MeV}$.
\begin{figure}[!t]
\centering
\includegraphics[width=0.7\textwidth]{Introduction/Figs/spectra.pdf}
\vspace{-0.4cm}
\caption{The neutrino event spectra with respect to the visible energy $E^{}_{\rm d}$ in the JUNO detector for a SN at 10 kpc, where no neutrino flavor conversions are assumed for illustration and the average neutrino energies are $\langle E^{}_{\nu_e}\rangle = 12~{\rm MeV}$, $\langle E^{}_{\overline{\nu}_e}\rangle = 14~{\rm MeV}$ and $\langle E^{}_{\nu_x}\rangle = 16~{\rm MeV}$. (See the text for details)}
\label{fig:spectra}
\end{figure}
\subsection{Diffuse Supernova Neutrino Background}
\label{subsec:DSNB}
The integrated neutrino flux from all past core-collapse events in
the visible universe forms the diffuse supernova neutrino background
(DSNB), holding information on the cosmic star-formation rate, the
average core-collapse neutrino spectrum, and the rate of failed
supernovae. The Super-Kamiokande water Cherenkov detector has
provided the first limits \cite{Malek:2002ns,Bays:2011si} and
eventually may achieve a measurement at the rate of a few events per
year, depending on the implementation of its gadolinium upgrade.
The JUNO detector has the potential to achieve a measurement comparable to Super-Kamiokande,
benefiting from the excellent intrinsic capabilities of
liquid scintillator detectors for antineutrino tagging and
background rejection.
Sources of correlated background events must be taken into account
when defining the energy window, fiducial volume and pulse shape selection
criteria for the DSNB detection.
Reactor and atmospheric neutrino IBD signals define an observational window reaching from 11 to $\sim$30\,MeV.
Taking into account the JUNO detector simulation of the expected signal and background rates,
a $3\sigma$ signal is conceivable after 10~years of running for typical assumptions of DSNB parameters.
A non-detection would strongly improve current limits and exclude a significant
range of the DSNB parameter space.
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|cc|cc|}
\hline
Syst. uncertainty BG & \multicolumn{2}{c|}{5\,\%}& \multicolumn{2}{c|}{20\,\%}\\
\hline
$\mathrm{\langle E_{\bar\nu_e}\rangle}$ & rate only & spectral fit & rate only & spectral fit \\
\hline
12\,MeV & $2.3\,\sigma$ & $2.5\,\sigma$ & $2.0\,\sigma$ & $2.3\,\sigma$\\
15\,MeV & $3.5\,\sigma$ & $3.7\,\sigma$ & $3.2\,\sigma$ & $3.3\,\sigma$\\
18\,MeV & $4.6\,\sigma$ & $4.8\,\sigma$ & $4.1\,\sigma$ & $4.3\,\sigma$\\
21\,MeV & $5.5\,\sigma$ & $5.8\,\sigma$ & $4.9\,\sigma$ & $5.1\,\sigma$\\
\hline
\end{tabular}
\caption{The expected detection significance after 10 years of data taking for different DSNB models
with $\langle E_{\bar\nu_e} \rangle$ ranging from 12\,MeV to 21\,MeV ($\Phi=\Phi_0$). Results are given based on either a rate-only or spectral fit analysis and assuming 5\% or 20\% for background uncertainty.}
\label{tab:snd:det_sig}
\end{center}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.66\textwidth]{Introduction/Figs/exclusion_plot_10y.pdf}
\caption{Exclusion contour at 90\%~C.L.\ as a function of the mean energy of the SN spectrum $\langle E_{\bar\nu_e}\rangle$ and
the flux amplitude (SN rate times total emitted energy per SN).
We assume 5\% background uncertainty and no DSNB signal detection ($N_ {\rm det}=\langle N_{\rm bg} \rangle$) after~10\,yrs.}
\label{fig:snd:exclusion}
\end{figure}
Following the method described in Ref.~\cite{Rolke:2004mj}, we show in
Tab.~\ref{tab:snd:det_sig} the expected detection significance after
10 years of data taking for different DSNB models
with $\langle E_{\bar\nu_e} \rangle$ ranging from 12\,MeV to 21\,MeV.
Results are given based on either a rate-only or spectral fit
analysis and assuming 5\% or 20\% for the background uncertainty.
For each DSNB model it was assumed that the number of
detected events equals the sum of the expected signal and background
events. After 10\,years the DSNB can be detected with $>3\,\sigma$
significance if $\mathrm{\langle E_{\bar\nu_e} \rangle}\ge 15\,$MeV.
If there is no positive detection of the DSNB, the current limit can be significantly improved.
Assuming that the detected event spectrum equals the background expectation in the
overall normalization and shape, the upper limit on the DSNB flux above 17.3\,MeV would be
$\sim 0.2\,{\rm cm}^{-2}{\rm s}^{-1}$ (90\%~C.L.) after 10 years for $\langle E_{\bar\nu_e}\rangle=18$\,MeV.
This limit is almost an order of magnitude better than the current value from Super-Kamiokande~\cite{Bays:2011si}.
In Fig.~\ref{fig:snd:exclusion} we show the corresponding exclusion contour at 90\%~C.L. as a function of the mean energy
with $\langle E_{\bar\nu_e} \rangle$ ranging from 12\,MeV to 21\,MeV.
\subsection{Solar Neutrinos}
\label{subsec:solar}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\centerline{\includegraphics[width=0.6\linewidth]{Introduction/Figs/solar-fig-singles_clean2.pdf}}
\\
\centerline{\includegraphics[width=0.6\linewidth]{Introduction/Figs/solar-fig-singles_clean1.pdf}}
\end{tabular}
\end{center}
\caption{(a)On the top, the expected singles spectra at JUNO with the radio purity assumption
$10^{-16} {\rm g/g}$ for $^{238}{\rm U}$ and $^{232}{\rm Th}$, $10^{-17} {\rm g/g}$ for $^{40}{\rm K}$
and $^{14}{\rm C}$. (b) On the bottom, assuming a background level one order of magnitude better for
$^{238}{\rm U}$, $^{232}{\rm Th}$ and for $^{40}{\rm K}$, $^{14}{\rm C}$.}
\label{fig:solar:simul2}
\end{figure}
More than forty years of experiments and phenomenological analyses brought to a crystal
clear solution of the Solar neutrino problem in terms of oscillating massive neutrinos
interacting with matter, with only the LMA region surviving in the
mixing parameter space.
The rather coherent picture emerging from all of these solar neutrino experiments is in
general agreement with the values of the mixing parameters extracted from
KamLAND data~\cite{KL-recent}.
A global three flavor analysis, including all the solar neutrino experiments and KamLAND data and
assuming CPT invariance, gives the following values for the mixing angles and the differences
of the squared mass eigenvalues~\cite{Borexino-fase-I}:
$
{\rm tan}^2 \, \theta_{12} = 0.457^{+0.038}_{-0.025} \, ;
{\rm sin}^2 \, \theta_{13} = 0.023^{+0.014}_{-0.018} \, ;
\Delta m_{21}^2 = 7.50^{+0.18}_{-0.21} \, \times \, 10^{-5} \, {\rm eV}^2\, .
$
The measured values of the neutrino fluxes produced in different
phases of the $pp$ chain and of the CNO cycle are consistent with the Standard Solar Model (SSM).
But they are not accurate enough to solve the metallicity
problem~\cite{Villante-Serenelli-2014}, discriminating between the
different (high Z versus low Z) versions of the SSM.
Because most of the experimental results fall somehow in the middle between the high and
the low Z predictions and the uncertainties are still too high.
In order to operate this discrimination, a future experimental challenge would be an even more
accurate determination of the $^8{\rm B}$ and $^7{\rm Be}$ fluxes,
combined with the measurement of CNO neutrinos.
We present in Fig.~\ref{fig:solar:simul2}(a) the expected singles spectra at JUNO with the
radio purity assumption of $10^{-16} {\rm g/g}$ for $^{238}{\rm U}$ and $^{232}{\rm Th}$, $10^{-17} {\rm g/g}$ for $^{40}{\rm K}$ and $^{14}{\rm C}$, and Fig.~\ref{fig:solar:simul2}(b) the expected singles spectra at JUNO with an ideal radio purity assumption of $10^{-17} {\rm g/g}$ for $^{238}{\rm U}$ and $^{232}{\rm Th}$, $10^{-18} {\rm g/g}$ for $^{40}{\rm K}$ and $^{14}{\rm C}$. The $^7{\rm Be}$ spectrum clearly stands out all backgrounds. More studies on the capability sensitivity deserve further simulations.
\subsection{Atmospheric Neutrinos}
\label{subsubsec:atm}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\centerline{\includegraphics[width=0.6\linewidth]{Introduction/Figs/Atm_NH.pdf}}
\\
\centerline{\includegraphics[width=0.6\linewidth]{Introduction/Figs/Atm_IH.pdf}}
\end{tabular}
\end{center}
\caption{The MH sensitivities of atmospheric neutrinos as a function of livetime for the true NH (upper panel) and IH (lower panel) cases.}
\label{fig:atm:MHyears}
\end{figure}
Atmospheric neutrinos are one of the most important neutrino sources for
neutrino oscillation studies. Atmospheric neutrinos have broad
ranges in the baseline length (15~km $\sim$ 13000~km) and energy (0.1
~GeV $\sim$ 10~TeV). When atmospheric neutrinos pass through the
Earth, the matter effect will affect the
oscillations of the neutrinos. The
Mikheyev-Smirnov-Wolfenstein resonance enhancement may occur in the
normal mass hierarchy (NH) for neutrinos and inverted mass hierarchy
(IH) for antineutrinos. Therefore, JUNO has the capability to measure
the neutrino MH through detecting the upward atmospheric neutrinos.
This is complementary to the JUNO reactor antineutrino results.
JUNO's ability in MH determination is at the 1$\sigma$-2$\sigma$ level
for 10 years of data taking. An optimistic estimation of the MH sensitivity is obtained based
on several reasonable assumptions:
\begin{itemize}
\item[$\bullet$] An angular resolution of $10^\circ$ for the neutrino direction and
$\sigma_{E_{vis}} = 0.01 \sqrt{E_{\rm vis}/{\rm GeV}}$ for the visible energy of neutrino events are assumed.
\item[$\bullet$] The $\nu_e/\bar{\nu}_e$ CC events are identified and
reconstructed in the $e^\pm$ visible energy range with $E^e_{\rm vis} >1$ GeV
and $Y_{\rm vis} < 0.5$, where $Y_{\rm vis}$ is defined as the ratio of the hadron visible energy
in the visible energy of an atmospheric neutrino event.
\item[$\bullet$] The selected $\nu_e/\bar{\nu}_e$ events are divided into two samples in terms of the numbers
of Michael electrons with $N_e = 0$ ($\bar\nu_e$-like events) or $N_e \geq 1$ ($\nu_e$-like events).
\item[$\bullet$] The $\nu_\mu/\bar{\nu}_\mu$ CC events are selected with $L_{\mu} > 3 $ m,
where $L_{\mu}$ is the track length inside the LS region for the charge lepton $\mu^\pm$.
\item[$\bullet$] The selected $\nu_\mu/\bar{\nu}_\mu$ events are divided into four samples. First, these events are
grouped as partially contained (PC) and fully contained (FC) events depending on whether $\mu^\pm$ can escape from the LS region or not.
Second, FC events with $N_e \geq 2$ or $\mu^{-}$ capture on $^{12}$C or $Y_{\rm vis}>0.5$ are
defined as FC $\nu_\mu$-like events, and the residual FC events are defined as FC $\bar\nu_\mu$-like events.
Finally, PC events with $N_e \geq 1$ or $Y_{\rm vis}>0.5$ are classified as PC $\nu_\mu$-like events, and all the other PC events are PC $\bar\nu_\mu$-like events.
\end{itemize}
The MH sensitivity of the separate contribution of muon and electron neutrino events and their combinations
in the cases of both normal and inverted MHs are presented in Fig.~\ref{fig:atm:MHyears}.
We can observe from the figure that
the $\nu_e/\bar{\nu}_e$ events have better sensitivity than $\nu_\mu/\bar{\nu}_\mu$ events because
more $\nu_e/\bar{\nu}_e$ events with higher energies can deposit their whole visible energy in
the LS region. The combined sensitivity can reach 1.8$\sigma$ (2.6$\sigma$) after 10 (20) years of running.
Finally, we want to mention that
atmospheric neutrinos can also be used to search for CP violation effects and
precisely measure the atmospheric mixing angle $\theta_{23}$.
\subsection{Geo-Neutrinos}
\label{subsubsec:geo}
For half a century we have established with considerable precision the Earth's surface
heat flow $46 \pm 3 $~TW ($10^{12}$~watts), however we are vigorously debating what fraction of
this power comes from primordial versus radioactive sources. This debate touches on the composition
of the Earth, the question of chemical layering in the mantle, the nature of mantle convection,
the energy needed to drive Plate Tectonics, and the power source of the geodynamo, the magnetosphere
that shields the Earth for the harmful cosmic ray flux.
Over the last decade particle physicists have detected the Earth's geoneutrino flux, the
planetary emission of electron anti-neutrinos that are derived from naturally occurring,
radioactive beta-decay events inside the Earth \cite{bib3,bib4}. Matter, including the Earth,
is mostly transparent to these elusive messengers that reveal the sources of heat inside the
Earth. By detecting a few events per years we are now measuring the geoneutrino flux from Thorium
and Uranium inside the planet, which in turn allows us to determine the amount of radiogenic
power driving the Earth's engine.
Predicting the geoneutrino signal at JUNO demands that we accumulate the
basic geological, geochemical and geophysical data for the regional area
surrounding the detector. Experience tells us that in the continents the closest 500~km to the
detector contributes half of the signal and it is this region that needs to be critically
evaluated \cite{bib5}. This goal demands that the physical (density and structure) and
chemical (abundance and distribution of Th and U) nature of the continent must be specified
for the region. The main tasks include surveys and descriptions of the geology, seismology,
heat flow, and geochemistry of the regional lithosphere.
The dominate background for the geo-neutrino observation is the reactor antineutrinos.
In addition, other backgrounds include the cosmic-muons spallation
products ($^{9}$Li-$^{8}$He isotopes, fast neutrons), accidental coincidences of non-correlated
events, and backgrounds induced by radioactive contaminants of scintillator and
detector materials (i.e., $^{13}$C ($\alpha$, n)$^{16}$O).
In Tab.~\ref{tab:geo:Nev}, we summarize the predicted geo-neutrino signal
and backgrounds considered in the sensitivity study.
\begin{table
\label{tab:exp}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{ll}
\hline \hline
Source &
Events/year
\\ \hline
Geoneutrinos & $408\pm 60$ \\
U chain & $311\pm 55$ \\
Th chain & $92\pm 37$ \\
Reactors & $16100\pm 900$ \\
Fast neutrons & $3.65\pm 3.65$ \\
$^{9}$Li - $^{8}$He & $657\pm 130$ \\
$^{13}$C$(\alpha,n)^{16}$O & $18.2\pm 9.1$ \\
Accidental coincidences & $401\pm 4$ \\
\hline \hline
\end{tabular}
\end{center}
\caption{Signal and backgrounds considered in the geoneutrino sensitivity study:
the number of expected events for all components contributing to the IBD spectrum in the 0.7 - 12 MeV energy region
of the prompt signal. We have assumed 80\% antineutrino detection efficiency and 17.2\,m radial cut (18.35\,kton of liquid scintillator).}
\label{tab:geo:Nev}
\end{table}
Precision of the reconstruction of geoneutrino signal is shown in
Tab.~\ref{tab:geo:Fit}, for the running time in 1, 3, 5, and 10 years.
Different columns refer to the measurement of geo-neutrino signal with fixed Th/U ratio, and
U and Th signals fit as free and independent components.
The given numbers are the position and root mean square (RMS) of the Gaussian fit to the distribution of the
ratios between the number of reconstructed and generated events.
It can be seen that while RMS is decreasing with longer data acquisition time, there are
some systematic effects which do not depend on the acquired statistics.
With 1, 3, 5, and 10 years of data, the statistical error amounts to 17\%, 10\%, 8\%, and 6\% respectively with
the fixed chondritic Th/U ratio.
\begin{table
\footnotesize
\label{tab:exp}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{cccc}
\hline \hline
Number of years & U + Th (fixed chondritic Th/U ratio) & U (free) & Th (free) \\
\hline
1 & $0.96\pm 0.17$ & $1.01\pm 0.32$ & $0.79\pm 0.66$ \\
3 & $0.96\pm 0.10$ & $1.03\pm 0.19$ & $0.80\pm 0.37$ \\
5 & $0.96\pm 0.08$ & $1.03\pm 0.15$ & $0.80\pm 0.30$ \\
10 & $0.96\pm 0.06$ & $1.03\pm 0.11$ & $0.80\pm 0.21$ \\
\hline \hline
\end{tabular}
\end{center}
\caption{Precision of the reconstruction of geoneutrino signal. See the text for details.}
\label{tab:geo:Fit}
\end{table}
\subsection{Nucleon Decay}
\label{subsubsec:nucleon}
Being a large underground LS detector, JUNO is
in an excellent position in search for nucleon decays. In
particular, in the SUSY favored decay channel $p \to K^+ + \bar\nu$,
JUNO will be competitive and complementary to other experiments
using water Cherenkov and liquid argon detectors ~\cite{kearns:isoup}.
The protons in the JUNO detector are provided by both the hydrogen
nuclei and the carbon nuclei. Using the Daya Bay liquid scintillator
as a reference, the H to C molar ratio is 1.639. For a 20~kt
fiducial volume detector, the number of protons from hydrogen (free
protons) is $1.45\times10^{33}$ and the number of protons from
carbon (bound protons) is $5.30\times10^{33}$.
If the proton is from hydrogen, it decays at rest. The kinetic energy of the $K^+$ is fixed by kinematics to
be 105~MeV, which gives a prompt signal in the liquid scintillator.
The $K^{+}$ has a lifetime of 12.4 nanoseconds and would quickly decay
into five different channels,
$K^{+} \to \mu^+ \nu_{\mu}$ and $K^{+} \to \pi^+ \pi^0$ are two most probably decay modes.
In either case, there is a short-delayed ($\sim$12~ns) signal from the $K^{+}$ daughters.
If the $K^+$ decays into $\mu^+ \nu_{\mu}$, the delayed signal comes
from the $\mu^+$, which has a fixed kinetic energy of 152~MeV from
the kinematics. The $\mu^+$ itself decays 2.2~$\mu s$ later into
$e^+ \nu_e \bar\nu_{\mu}$, which gives a third long-delayed signal
with well known (Michel electron) energy spectrum. If the $K^+$
decays into $\pi^+ \pi^0$, the $\pi^+$ deposits its kinetic energy
(108~MeV) and the $\pi^0$ instantaneously decays ($\tau =
8.4\times10^{-17} s$) into primarily two gamma rays (98.80\%) with
the sum of deposited energy equal to the total energy of $\pi^0$
(246~MeV). The delayed signal includes all of the aforementioned
deposited energy. Then, the $\pi^+$ decays ($\tau=26$ ns) primarily
into $\mu^+ \nu_{\mu}$ (99.99\%). The $\mu^+$ itself has very low
kinetic energy (4.1 MeV), but it decays 2.2~$\mu s$ later into $e^+
\nu_e \bar\nu_{\mu}$, which gives the third long-delayed decay
positron signal.
The simulated hit time distribution of a $K^{+} \to \mu^+
\nu_{\mu}$ event is shown in Fig.~\ref{fig:nd:pdk_event}, which
displays a clear three-fold coincidence:
\begin{itemize}
\item A prompt signal from $K^+$ and a delayed signal from its decay daughters with a time coincidence of 12 ns.
\item Both the prompt and delayed signals have well-defined energy.
\item One and only one decay positron with a time coincidence of 2.2 $\mu$s from the prompt signals.
\end{itemize}
The time coincidence and the well-defined energy provide a powerful
tool to reject background, which is crucial in the proton decay
search.
\begin{figure
\centering
\includegraphics[width=0.7\textwidth]{Introduction/Figs/pdk_event.pdf}
\caption{The simulated hit time distribution of photoelectrons (PEs)
from a $K^{+} \to \mu^+ \nu_{\mu}$ event at JUNO.}
\label{fig:nd:pdk_event}
\end{figure}
The sensitivity to proton lifetime, can be calculated as $\tau (p
\to K^+ + \bar\nu) = N_p T R \epsilon / S$, where $N_p$
($6.75\times10^{33}$) is the total number of protons, $T$ is the
measuring time where we assumed 10 years, $R$ (84.5\%) is the $K^+$
decay branching ratio included in the analysis, and $\epsilon$
(65\%) is the total signal efficiency. $S$ is the upper limit of
number of signal events at certain confidence interval, which
depends on the number of observed events as well as the expected
number of background events. The expected background is 0.5 events
in 10 years. If no event is observed, the 90\% C.L upper limit is $S
= 1.94$. The corresponding sensitivity to proton lifetime is $\tau >
1.9\times10^{34}$ yr. This represents a factor of three improvement
over the current best limit from Super-Kamiokande, and starts to
approach the region of interest predicted by various GUTs models. In
a real experiment, the sensitivity may decrease if background
fluctuates high. In the case that one event is observed (30\%
probability), the 90\% C.L upper limit is $S = 3.86$. The
corresponding sensitivity to proton lifetime is $\tau >
9.6\times10^{33}$ yrs. If two events are observed (7.6\%
probability), the sensitivity is further reduced to $\tau >
6.8\times10^{33}$ yrs.
\begin{figure
\centering
\includegraphics[width=0.7\textwidth]{Introduction/Figs/pdk_sens.pdf}
\caption{The 90\% C.L. sensitivity to the proton lifetime in the decay
mode $p \to K^+ + \overline\nu$ at JUNO as a function of time. In
comparison, Super-Kamiokande's sensitivity is also projected \cite{kearns:isoup}.} \label{fig:nd:pdk_sens}
\end{figure}
In Fig.~\ref{fig:nd:pdk_sens} we plot the 90\% C.L. sensitivity
to the proton lifetime in the decay mode $p \to K^+ + \overline\nu$ at JUNO
as a function of the running time. Due to the high efficiency in measuring this
mode, JUNO's sensitivity will surpass Super-Kamiokande's after
3 yrs of data taking.
\subsection{Light Sterile Neutrinos}
\label{subsubsec:sterile}
Motivated from the anomalies of LSND \cite{Aguilar:2001ty}, MiniBooNE \cite{Aguilar-Arevalo:2013pmq},
the reactor antineutrino anomaly \cite{Mention:2011rk}, and Gallium anomaly \cite{Giunti:2010zu},
the light sterile neutrino \cite{Giunti:2013aea,Kopp:2013vaa} is regarded as one of the most promising possibilities for new physics
beyond the three neutrino oscillation paradigm. Therefore, future experimental oscillation searches at short baselines are required
to test the light sterile neutrino hypothesis \cite{Lasserre:2014ita}.
Several possible methods of sterile neutrino studies are considered at JUNO. The first one is the use of existing reactor antineutrinos,
which can test the active-sterile mixing with the mass-squared difference ranging from $10^{-5}$ to $10^{-2}$ eV$^{2}$. This parameter space
is irrelevant to the short baseline oscillation, but could be tested as the sub-leading effect of solar neutrino oscillations.
The direct test of short baseline oscillations at JUNO requires additional neutrino sources placed near or inside the detector.
Using 50 kCi $^{144}$Ce-$^{144}$Pr as the antineutrino source at the detector center, JUNO can reach the $10^{-2}$ level of active-sterile mixing
at 95\% C.L. after 1.5 yrs of data taking. On the other hand, a cyclotron-driven $^8$Li source can be employed as the decay-at-rest (DAR) neutrino source near
the detector. The high DAR flux coupled with the large size of the JUNO detector
allows us to make an extremely sensitive search for antineutrino disappearance in the region of short baseline oscillation anomalies.
With a 60 MeV/amu cyclotron accelerator, JUNO can reach the $10^{-3}$ level of active-sterile mixing at 5$\sigma$ C.L.
after 5 yrs of data taking. IsoDAR$@$JUNO will be able to map out definitively the oscillation wave.
There is no other planned experiment that matches this sensitivity.
\subsection{Indirect Dark Matter Search}
\label{subsubsec:dark}
The existence of non-baryonic dark matter (DM) in the Universe has been
well established by astronomical observations. DM can also be detected indirectly by looking for the neutrino
signature from DM annihilation or decays in the Galactic halo, the
Sun or the Earth. In particular, the search for the DM-induced
neutrino signature from the Sun has given quite tight constraints on
the spin-dependent (SD) DM-proton scattering cross section $\sigma^{\rm
SD}_{\chi p}$~\cite{Aartsen:2012kia}.
\begin{figure
\centering
\includegraphics[width=0.7\textwidth]{Introduction/Figs/LS-SD.pdf}\\
\caption{ The JUNO $2\sigma$ sensitivity to the spin-dependent
cross section $\sigma^{\rm SD}_{\chi p}$ after 5 yrs of data taking. The
constraints from the direct detection experiments are also
shown for comparison. } \label{fig:SD}
\end{figure}
In general, DM inside the Sun can annihilate into leptons, quarks
and gauge bosons. The neutrino flux results from the decays of such
final-state particles.
Here we consider two annihilation modes $\chi\chi\to \tau^+\tau^-$ and $\chi\chi\to \nu\bar{\nu}$ as a
benchmark. The sensitivity calculations for $\sigma^{\rm SD}_{\chi p}$ are given in Fig.~\ref{fig:SD},
where one can see that the JUNO sensitivity is much better than current direct
detection constraints set by COUPP~\cite{Behnke:2012ys},
SIMPLE~\cite{Felizardo:2011uw} and PICASSO~\cite{Archambault:2012pm}
experiments. Notice that the sensitivity for $m_{\chi}< 3$ GeV
becomes poor due to the DM evaporation from the Sun.
\subsection{Other Exotic Searches}
\label{subsubsec:exotic}
The standard three-neutrino mixing paradigm can describe most of the phenomena in
solar, atmospheric, reactor, and accelerator neutrino oscillations \cite{PDG}.
However, other new physics mechanisms can operate at a sub-leading level,
and may appear at the stage of precision measurements of neutrino oscillations.
Therefore, other exotic searches can be performed at JUNO using reactor neutrinos,
solar neutrinos, atmospheric neutrinos and supernova neutrinos.
With the reactor neutrino oscillation, JUNO can:
\begin{itemize}
\item test the nonstandard neutrino interactions at the source and detector \cite{ohlsson:14a,Li:2014mlo};
\item test the Lorentz and CPT invariance violation \cite{Li:2014rya};
\item test the mass-varying properties of neutrino propagation \cite{Schwetz:2005fy}.
\end{itemize}
Meanwhile, using solar and atmospheric neutrino oscillations, JUNO can
\begin{itemize}
\item test the nonstandard neutrino interactions during the propagation \cite{Ohlsson:2012kf,Bolanos:2008km};
\item test the long range forces \cite{Joshipura:2003jh};
\item test the super light sterile neutrinos \cite{deHolanda:2010am};
\item test the anomalous magnetic moment \cite{Arpesella:2008mt,Giunti:2014ixa}.
\end{itemize}
Therefore, with precision measurements JUNO will be a powerful playground to test different exotic phenomena
of new physics beyond the Standard Model.
\chapter{Liquid Scintillator}
\label{ch:LiquidScintillator}
\section{Introduction}
The innermost part of the JUNO detector is formed by 20,000 tons of liquid scintillator (LS) contained inside an acrylic sphere of 35\,m diameter. The LS serves as target medium for the detection of neutrinos and antineutrinos. The primary reactions for reactor electron antineutrinos ($\bar\nu_e$) is the inverse beta decay on free protons, $\bar\nu_e+p\to n+e^+$, resulting in a prompt positron and a delayed signal from the neutron capture on hydrogen ($\tau\sim200\,\mu$s).
The scintillator itself is a specific organic material containing molecules featuring benzene rings that can be excited by ionizing particles. Compared to other materials, this process is very efficient: About $10^4$ photons in the near UV and blue are emitted per MeV of deposited energy. The time profile of the light emission is dictated by the decay constants of the excited states of the organic molecules but also depends on the type and energy of the incident particle.
The LS is composed of several materials: The solvent liquid, linear alkyl benzene (LAB), forms the bulk of the target material and is excited by the ionizing particles. This passes on the excitation to a two-component system of a fluor (PPO) and a wavelength-shifter (Bis-MSB), that are added at low concentration (a few g/l resp.~mg/l) and by subsequent Stokes' shifts increase the wavelength of the emitted photons to $\sim 430$\,nm. This shift is crucial as the long wavelength avoids spectral self-absorption by the solvent and allows the photons generated by a reaction at the very center of the target volume to reach the photosensors (PMTs) mounted on a scaffolding outside the target volume. At 430\,nm, the transparency of the compound liquid will be largely governed by the Rayleigh scattering of photons on the solvent molecules. It can be reduced by the presence of unwanted organic molecules featuring absorption bands in the wavelength region of interest.
The energy resolution of JUNO is specified with 3\,\% at 1\,MeV, corresponding to at least 1,100 photoelectrons (pe) per MeV of deposited energy. Compared to Borexino, this corresponds to more than twice the pe yield. To meet this demanding requirements, both the initial light yield and the transparency of the liquid have to be optimized simultaneously.
In spite of the large number of neutrinos crossing the detector and the huge target mass, the minuscule cross-section of weak interaction allows the detection of only a few tens of neutrino events per day. Low-background conditions are therefore absolutely crucial. From the point of view of the LS, this means that the concentration of radioactive impurities inside the liquid should result in an activity on the same level or below the rate of neutrino events. This is less critical in case of the inverse beta decay channel for $\bar\nu_e$ detection. The reaction provides a fast coincidence signature that can be used to suppress single-event background.
As a consequence, light yield, fluorescence time profile, transparency and radiopurity are the key features of the LS. These quantities will be mainly defined by the original materials, but as well by the presence of organic and radioactive impurities, temperature and aging effects. For the next years, we foresee a broad spectrum of laboratory measurements to characterize different brands of LAB and wavelength shifters. Multiple experimental methods will be used, like GC-MS, LS-MS, UV-VIS, and ICP-MS, amongst others. A close contact with the producing companies has been established which will allow the optimization of the production quality within certain limits. Moreover, we will study a variety of purification techniques to optimize both optical performance and radiopurity of the LS. Finally, a large-scale test will be conducted in one of the Antineutrino Detectors (ADs) of the Daya-Bay experiment, which will allow a to study the effect of purification on a sample of about 20 tons of LS.
The final aim is to set up a scheme for the mass production of 20,000 tons of LS. This study will comprise all steps from the initial production, the transport on site, the local facilities for liquid-handling, plants for purification and instrumentation for quality control as well as safety and cleaning methods~\cite{An:2012eh}.
\section{Specifications}
The basic specifications and requirements for the liquid scintillator are:
\begin{itemize}
\item {\bf Total mass:} 20,000 tons.
\item {\bf Composition:}\\
Scintillator solvent: Linear alkyl benzene (LAB)\\
Scintillating fluor: PPO (2,5- diphenyloxazole) at 3 g/L\\
Wavelength shifter: Bis-MSB at 15~mg/L
\item {\bf Light yield:}\\
Minimum of 1,100 photoelectrons (pe) per MeV
\item {\bf Transparency:}\\
Attenuation length @ 430\,nm: > 22~m\\
Absorption coefficient of less than $3\times10^{-3}$ in spectral range 430$-$600\,nm
\item {\bf Radiopurity:}\\
$\bar\nu_e$ detection: Concentrations of $^{238}$U, $^{232}$Th $\leq10^{-15}$\,g/g, $^{40}$K $\leq10^{-16}$\,g/g. \\*
$\nu_e$ detection: Concentrations of $^{238}$U, $^{232}$Th $\leq10^{-17}$\,g/g, $^{40}$K $\leq10^{-18}$\,g/g.
\end{itemize}
\section{Study of Optical Properties}
\subsection{Light Yield}
The light yield and emission time profile of the LS are key aspects of JUNO. For a given solvent (LAB), they depend mainly on the the concentrations of the fluor PPO and the wavelength-shifter Bis-MSB. The resulting light yield and fluorescence times of the composite LS will be measured as a function of these concentrations. While for both quantities higher concentrations will in general increase the performance, the optimum concentration of PPO and Bis-MSB will be determined taking into regard the detrimental effects of self-absorption on the attenuation length and the radioimpurities introduced into the scintillator.
The studies presented in the following section mainly concentrate on the characterization of the light yield as a function of fluor concentration and temperature. Moreover, the non-linearity of the light yield as a function of the energy of incident electrons will be studied as it has implications on the linearity in the energy response of positrons. Beyond the setups described here, first studies of the fluorescence properties in laboratory experiments have already been performed at the INFN institutes in Milano and Perugia and will be intensified in the future.
\subsubsection{Effect of Fluor Concentrations}
\label{sec:ls:ly-fluor}
The light yield (LY) of an LS sample depends greatly on the concentration of fluor(s) added to the solvent. At low concentrations, it will increase almost linearly with the fluor concentration. However, the increase will become less steep once a concentration of 1$-$2\,g/L is reached. To optimize the amount of collected light with regard to self-absorption effects which are expected to play a role for concentrations of several g/L and more, the LY of a variety of LAB samples featuring varying concentrations of PPO and Bis-MSB will be studied. This will be done by a setup based on a Compton coincidence detector, an established technique for a comparative measurement of the LY of LS samples.
Figure~\ref{fig:ls:ly} shows the basic principle of the experimental setup: A PMT is attached to a vessel containing the LS sample that is excited by gamma-rays from a nearby source. Without a coincidence detector, the pulse height spectrum observed by the PMT will correspond to a Compton shoulder, which will require a relatively complicated fit to evaluate the light out. Adding a coincidence detector to record the scattered gamma-rays allows to define a fixed energy deposition in the LS sample by fixing the scattering geometry. In these circumstances, the pulse height spectrum can be fitted by a simple Gaussian, providing a more accurate result.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=5cm]{LiquidScintillator/figures/LS4-1.png}
\caption[Schematic of the setup for light-out measurements]{Schematic setup for the relative determination of the light out}
\label{fig:ls:ly}
\end{center}
\end{figure}
The experimental setup shown in Fig.~\ref{fig:ls:ly2} has been designed to determine the LY with less than 2\,\% uncertainty. The panels on the right display the pulse-height spectra obtained in a sample measurement and demonstrate the impact of the coincidence measurement. Beyond fluor concentrations, this setup will be used in the long term to test the effects of purification techniques, LS temperature and aging.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-2.png}
\caption[Foto of the setup for LY measurements]{Setup for measuring the LS light out and preliminary result}
\label{fig:ls:ly2}
\end{center}
\end{figure}
\subsubsection{Temperature Dependence}
An increase in scintillation LY has been observed when lowering the temperature of the LS. This will be studied in a somewhat modified setup shown in Fig.~\ref{fig:ls:ly3} that has been adjusted for the application of different temperature levels to the LS sample. The $^{137}$Cs $\gamma$-source, the LS vessel, the PMT and the coincidence detector are put into an enclosed thermostat, allowing to vary the temperature from -40~$^{\circ}$C to 30~$^{\circ}$C. The signals from the PMT and the coincidence detector are recorded by a CAEN DT5751 FADC unit. A temperature-resistant optical fiber is used to monitor the temperature response of the PMT.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{LiquidScintillator/figures/LS4-3.png}
\caption[Setup for measuring temperature-dependence of LY]{Experimental setup to determine the temperature dependence of the LY}
\label{fig:ls:ly3}
\end{center}
\end{figure}
\subsubsection{Energy Non-linearity of Electron Signals}
While the light output of LS is an almost linear function of the deposited energy, additional small non-linear terms will introduce an additional systematic uncertainty for the energy reconstruction in JUNO. Of special relevance is the non-linearity in case of positron signals that might affect the sensitivity of the mass hierarchy measurement. To take these effects correctly into account, it is important to study the energy response in small-scale laboratory setups. For practical reasons, these experiments will rely on electrons instead of positrons.
A corresponding experimental setup has been realized at IHEP, designed to measure the energy non-linearity of electron signals below 1\,MeV at a precision of better than 1\,\%. The measurement is performed by inducing low-energy recoil electrons by Compton scattering of $\gamma$-quanta in a small LS sample. As the incident energy $E_\gamma$ is known, a specific electron energy $E_e$ can be selected by fixing the scattering angle $\theta$ (see above). It holds
\begin{equation}
E_e=E_\gamma\bigg[1-\frac{1}{1+\alpha(1-\cos\theta)}\bigg],
\end{equation}
where $\alpha=\frac{E\gamma}{m_e}$, with $m_e$ the electron mass. The non-linearity of the energy response is obtained by determining the ratio of observed light output and computed electron energy as a function of $\theta$.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=14cm]{LiquidScintillator/figures/LS4-4.png}
\caption[Conceptual drawing of the electron non-linearity setup]{Conceptual drawing of the electron energy non-linearity setup}
\label{fig:ls:enl}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-5.png}
\caption[Photograph of the electron non-linearity setup]{Photograph of the electron energy non-linearity setup}
\label{fig:ls:enl2}
\end{center}
\end{figure}
Figure~\ref{fig:ls:enl} shows a conceptual drawing of the experiment, while Fig.~\ref{fig:ls:enl2} displays the actual laboratory setup.The gamma source used in the setup is $^{22}$Na with an activity of 0.3\,mCi. From $\beta^+$ decay to an excited state, it provides two $\gamma$-rays of 0.511\, MeV and one at 1.275 MeV. After passing through a lead collimator of 9\,mm diameter, the $\gamma$-rays scatter in the LS sample, which is held in a cylindrical silica cup of 5\,cm diameter and height. The light generated by the recoil electron is collected by a PMT (XP2020) attached to the LS cup. Seven coincidence detectors are placed at various angles ($20^{\circ}$, $30^{\circ}$, $50^{\circ}$, $60^{\circ}$, $80^{\circ}$, $100^{\circ}$, $110^{\circ}$) at 60\,cm distance from the LS cup.These coincidence detectors consist of an inorganic crystal scintillator (LaBr) and a PMT (XP2020). The signals acquired simultaneously by all PMTs pass a Fan-In-Fan-Out unit (CAEN N625) before being sent to a trigger board (CAENN405) and a FADC (CAEN N6742). Using an array of PMTs rather than a single PMT mounted to a rotatable arm is reducing the required acquisition time and avoids variations of the PMT response induced by the rotation of the dynode chain in magnetic stray fields.
Preliminary results are expected for late 2015. At the present state, the uncertainty on electron energy non-linearity is estimated to be smaller than 2\,\%. The results will be used as well in the data analysis of the Daya Bay experiment.
\subsection{Optical Transparency}
Compared with other neutrino experiments, the optical transparency of the LS in JUNO is absolutely crucial. By scaling up the target volume from several hundreds to 20,000 tons, the diameter of the target volume in JUNO will be 35.4~m. If light transmission in the liquid scintillator is too low, scintillation photons from a neutrino interaction in the center of the detector will be absorbed by the liquid itself before reaching the photomultipliers positioned at the verge of the volume.
The light transmission is described by the attenuation length which quantifies the distance over which the original light intensity is reduced to a fraction of $1/e$. This attenuation is due to two kind of processes: Light absorption on organic impurities, in which case the photon can either be fully absorbed or re-emitted, and light scattering off the solvent molecules. In the latter case, light is not lost but merely re-directed and may still be detected.
Several laboratory scale experiments will be conducted to characterize LS samples regarding their transparency: Measurements include the wavelength-dependent attenuation spectrum of LS in relatively short cells (10\,cm), precise measurements of the LS attenuation length over large samples (several meters) and independent measurements of the scattering length of LS. Moreover, the fraction of organic impurities of different samples will be determined and the long-term stability of LS samples tested.
\subsubsection{Characterization of Attenuation Spectrum}
The attenuation spectrum of a material corresponds to the fraction of incident radiation absorbed by the sample over a certain thickness as a function of the wavelength of the incident light. The measurement is performed by commercial UV-Vis spectrometers that usually are sensitive to a wavelength range from 190 to 900\,nm. These measurements require only a low amount of time and a relatively small amount of LS as typical sample cells contain only 30\,ml or less. Therefore, this method is very suitable for the real-time monitoring of solvent or LS samples. Periodic measurements can be used to study the long-term stability of LS samples.
A first characterization of samples can be done based only on the solvent LAB (without adding the fluors). It is found that the resulting UV-Vis spectra vary for LAB samples from different providers. Fig.~\ref{fig:ls:specsolvent} shows the attenuation curves of LAB samples produced by several companies that apply different manufacturing technologies. Substantial divergences are present in the wavelength region below 500\,nm which is most important for the propagation of scintillation light.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-6.png}
\caption[UV-Vis attenuation spectra of LAB samples]{The UV-Vis attenuation spectra of LAB from different manufacturers}
\label{fig:ls:specsolvent}
\end{center}
\end{figure}
When loading the solvent with the fluors necessary to form the liquid scintillator, the attenuation spectrum changes. The shift of the absorption band to longer wavelengths is illustrated in Fig.~\ref{fig:ls:specfluors}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-7.png}
\caption[UV-Vis attenuation spectrum: Impact of fluors]{The UV-Vis spectra of LAB and LAB-based liquid scintillators}
\label{fig:ls:specfluors}
\end{center}
\end{figure}
\subsubsection{Long-range Measurement of Attenuation Length}
As stated above, the light intensity generated by particle interactions in the bulk of the LS volume will be attenuated by scattering and absorption before reaching the PMTs. The relation between the intensity $I(d)$ remaining after propagation by a distance $d$ in the LS and the initial light intensity $I_0$ can be expressed by
\begin{equation}
\label{eq:ls:att}
I(d)=I_0 \exp(-d/\ell_{\rm att}).
\end{equation}
The length scale $\ell_{\rm att}$ quantifying the distance after which $I(d)$ is reduced to $1/e$ of the initial value $I_0$ is the attenuation length.
The accuracy that can be reached by UV-vis spectrometers on the attenuation length is limited if $\ell_{\rm att}\geq10$\,m. This is due to the very small decrease of the light intensity when passing through a sample cell of less than 10\,cm in length and thus considerably shorter than the expected attenuation length of the LS ($\ell_{\rm att}>20$\,m). Considering the importance of exact knowledge of $\ell_{\rm att}$ for light propagation in the large LS volume, dedicated measurements are performed in the wavelength range around 430\,nm which corresponds to the emission band of Bis-MSB and at the same time represents the optimum intersection of scintillator transparency and PMT photosensitivity. Figure~\ref{fig:ls:att} shows the corresponding laboratory device at IHEP and a schematic sketch of the setup. A similar device with an even longer sample cell is currently developed at TU Munich (Germany). Measurements of this kind will be especially valuable to assess the effect of purification techniques on the optical transparency of the LS.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-8.png}
\caption[Setup for attenuation length measurement]{Setup for a long-range measurement of the attenuation length}
\label{fig:ls:att}
\end{center}
\end{figure}
\subsubsection{Determination of Scattering Lengths}
The attenuation of the initial scintillation light can be caused by a variety of processes: Absorption by organic impurities, self-absorption by the fluors (primarily Bis-MSB), Mie scattering off dust or particulates in suspension in the LS, or Rayleigh scattering of the solvent molecules. In general, the relation between the corresponding optical length scales can be given as:
\begin{equation}
\frac{1}{\ell_{\rm att}}=\frac{1}{\ell_{\rm abs}}+\frac{1}{\ell_{\rm are}}+\frac{1}{\ell_{\rm scat}},
\end{equation}
where $\ell_{\rm abs}$ describes the absorption length without re-emission, $\ell_{\rm are}$ means absorption followed by re-emission, and $\ell_{\rm scat}$ is the Rayleigh scattering length. Only the light absorbed without re-emission will be fully lost for event reconstruction. All other processes merely divert a photon from its original path so that it can still be detected by a PMT.
For an ideal scintillator without impurities, $\ell_{\rm att}\approx \ell_{\rm Ray}$ as Rayleigh scattering is the only irreducible process. In comparison, a conceivable contribution from Mie scattering on dust particles is expected to be negligible because the LS will be virtually dust-free after purification. Absorption-reemission might play a role as the long-wavelength tail of the Bis-MSB absorption band reaches into its emission spectrum centered around 430\,nm and might thus be relevant for light propagation.
The physics requirements of JUNO necessitate to come at least close to this ideal state in order to maximize the light output of the target LS. On the other hand, light emitted in a neutrino interaction and scattered while traveling to the PMTs will be deflected from a straight line of sight, influencing both spatial reconstruction and pulse-shaping capabilities of the experiment.
Therefore, a precise measurement of the Rayleigh scattering length will allow to determine the ultimate transparency limit for an LS based on the given solvent (LAB in case of JUNO), and at the same time provide valuable input for MC simulations that study the expected detector performance. Moreover, the impact of different purification techniques on the composition of the scintillator can be monitored by studying the combined results of attenuation and scattering lengths measurements (see below).
The Rayleigh scattering length expected for a given material can be calculated based on the Einstein-Smoluchowski-Cabannes (ESC) formula,
\begin{equation}
\frac{1}{l_{ray}}=\frac{8\pi^3}{3\lambda^4}kT\bigg[\rho\bigg(\frac{\partial\epsilon}{\partial\rho}\bigg)_T\bigg]\beta_T\bigg(\frac{6+3\delta}{6-7\delta}\bigg)
\end{equation}
where $\lambda$ is the wavelength of scattered light, $\rho$ is the density and $\epsilon$ the average dielectric constant of the liquid, $k$ is the Boltzmann constant, $T$ is the temperature, $\beta_T$ is the isothermal compressibility and $\delta$ is the depolarization ratio. Based on the ESC formula, the Rayleigh scattering of LAB can be obtained by measuring the reflectivity, isothermal compressibility, and the depolarization. Comparing to direct measurements of the scattered intensity (see below), this relative measurement can lower the systematic error on the depolarization ratio and directly derive the scattering length as a function of the incident wavelengths $\lambda$ of the scattered lights. Figure~\ref{fig:ls:scat1} shows a schematic drawing of the setup located at Wuhan University.
Complementarily to the described measurements, a direct measurement of the amount of scattered light will be performed at the University of Mainz (Germany). By investigating the dependence of the scattered intensity on the incident wavelength, polarization and scattering angle, contributions from different scattering processes as well as absorption/re-emission can be resolved. On the one hand, this will provide an important cross-check of the Rayleigh scattering length measurement. On the other hand, it will allow to scrutinize purified samples for residual dust particles and organic impurities and, more importantly, to assess the impact of Bis-MSB self-absorption on light propagation. Measurements of this kind have been already performed in Ref.~\cite{Wurm:2010ad}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-10.png}
\caption[Schematic setup for scattering length measurements]{Schematic diagram of Rayleigh scattering measuremens}
\label{fig:ls:scat1}
\end{center}
\end{figure}
\subsubsection{Organic Impurities from GC-MS Measurements}
\label{sec:ls:gcms}
Gas chromatography combined with mass spectrometry (GC-MS) is a commonly used technique to analyze the composition of complex organic mixtures. GC-MS analysis combines the high resolution of gas chromatography with the high sensitivity of mass spectrometry. Type and concentration of impurities in LAB as well as the primary LAB components can be resolved and identified by the GC-MS method. The results will provide useful information for improving the purification scheme.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=14cm]{LiquidScintillator/figures/LS4-9.png}
\caption[GC-MS Chromatogram of LAB]{Total ion chromatogram of LAB and impurities in LAB from GC-MS measurements}
\label{fig:ls:gcms}
\end{center}
\end{figure}
We have analyzed a sample of commercially available LAB produced by the Nanjing LAB factory. In Fig.~\ref{fig:ls:gcms}, the black curve indicates the components of LAB. The remaining curves stem from the impurities. The result suggests that
\begin{enumerate}
\item There are many impurities in LAB, which can be divided into several categories: fluorene, naphathelene derivatives, biphenyl derivatives, diphenyl alkane, and small amounts of alcohol, ketone, and ester.
\item The retention times of impurities coincide with LAB components. The impurities in LAB must be separated and enriched before GC-MS analysis.
\item The resolution of the chromatogram is not satisfying. The analytical conditions of GC-MS must be optimized in order to perform precise qualitative and quantitative analyses.
\end{enumerate}
Therefore, both the LAB pre-treatment methods and instrumental conditions have to be optimized in order to establish the GC-MS method for studying type and level of impurities in LAB samples.
GC-MS is an important addition to the characterization of the LS as it provides data on the chemical composition complementary to the optical measurements. We expect that following the adaptation of the method we will be able to clearly identify the main organic impurities in different brands of LAB. This will allow us to cooperate with the LAB manufacturers in order to achieve an optimization of the chemical purity of the LS raw materials.
\subsection{Long-term Stability}
JUNO is supposed to be in operation for at least 10 years. This poses the question of the long-term stability of the composite liquid and especially of its scintillation and optical properties. A common method of investigation often applied in chemistry are aging tests relying on heating the samples. According to the Van't Hoff equation, the rate of possible deteriorating processes will increase by a factor 2 to 4 for every 10~C$^{\circ}$ over room temperature (25~$^{\circ}$C). Therefore, a 6-months aging test performed at 40~$^{\circ}$C is equivalent to 1.5 - 4 years at 25~$^{\circ}$C. The temperature-treated samples will be characterized by their light out, attenuation length and absorption spectrum.
During the tests, the LS samples will be filled into containers of stainless steel. This will allow the performance of a material compatibility test with the LS at the same time. Different types of stainless steel (SS316 or SS304) will be investigated. Some ageing experiments are being conducted on LS. Figure~\ref{fig:ls:aging} shows two 15-liter stainless steel vessels.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-11.png}
\caption[Setup for LS aging tests]{Laboratory setup for LS aging tests}
\label{fig:ls:aging}
\end{center}
\end{figure}
\subsection{Selection of Raw Materials}
\subsubsection{Solvent}
LAB is a family of organic compounds with the formula C$_6$H$_5$C$_n$H$_{2n+1}$. When used as a detergent, $n$ lies typically between 10 and 16. For JUNO, a specific product selected for $n=10-13$ will be chosen as it shows better optical properties.
In China, the list of LAB manufacturers includes the Nanjing LAB factories of Jinling Petrochemical Company, the Fushun wash \& chemical factory of the Fushun Petrochemical Company, and the Jintung Petrochemical Corporation Ltd. The annual output of each factory is very large. Only several months will be needed to produce 20 ktons of LAB for JUNO.
In general, LAB production in China is based on a HF catalyst (HF-acid method). Outside China, an alternative technology based on a non-corrosive solid acid catalyst (Solid-acid method) is more common. The advantages of the HF acid method are the high conversion efficiency and the long-term stability in operation. The disadvantage is that it generates more by-products because the syntheisizing reaction is more violent than in case of the solid acid method. Therefore, the general expectation is that production by the solid-acid method should result in better optical properties of LAB.
Instead, our investigations based on the attenuation spectra of LAB samples suggest that the method of synthesis is not the critical factor for the resulting optical properties. Figure~\ref{fig:ls:optpur1} compiles the attenuation spectra of LAB samples produced both by the HF-acid method in the Nanjing LAB factory (specially ordered by optimizing production flow, not commercially available) and by the solid-acid method in an Egyptian plant (Helm AG). The sample from Nanjing shows much lower absorption for the spectral range below 410\,nm. As the spectral measurement is not sufficiently sensitive at the most interesting wavelength of 430\,nm, the sample was also tested in the long cell at IHEP. Again, the Nanjing sample showed a much longer attenuation length $\ell_{\rm att}\approx 20$\,m, while the LAB by Helm featured only $\ell_{\rm att}\approx12$\,m.
Based on these results, it seems that Chinese manufacturers using the HF-acid method should be able to provide LAB of sufficient quality for JUNO as long as the quality of the raw materials is assured and the production flow is optimized. For this, we foresee a cooperation with domestic LAB factories. In addition, we will further investigate samples of LAB from foreign companies.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-12.png}
\caption[UV-Vis attenuation spectra of LAB from different providers]{UV-Vis attenuation spectra of LAB produced by HF acid method and solid acid method}
\label{fig:ls:optpur1}
\end{center}
\end{figure}
\subsubsection{Fluor (PPO) and Wavelength Shifter (bis-MSB)}
As described above, a system of two fluors will be used in JUNO: LAB will non-radiatively pass on the excitations to the primary fluor PPO, which will in turn perform a mainly non-radiative transfer to the secondary wavelength shifter Bis-MSB. As a consequence, the effective light emission (including optical propagation effects) will be shifted to 430\,nm.
Due to the self-absorption of the fluors, the absorption spectra of the complete LS will differ significantly from pure LAB. This might be further enhanced by optical impurities introduced along with the fluors. Figure~\ref{fig:ls:fluor} illustrates the influence of PPO on the absorption spectrum of LS. If PPO concentration is increased from the baseline value of 3\,g/L to 10\,g/L, an effect induced by the tails of the absorption band of PPO becomes visible that extends up to 460\,nm. This clearly demonstrates that the optimum PPO concentration must be balanced between light yield and optical transparency in the 430\,nm range, taking re-emission processes into account.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-16.png}
\caption[UV-vis attenuation spectra of LAB doped with PPO]{UV-Vis attenuation spectra of LAB doped with PPO}
\label{fig:ls:fluor}
\end{center}
\end{figure}
Organic impurities introduced with PPO will as well have an impact on attenuation. Figure~\ref{fig:ls:fluor2} shows the attenuation spectra of LAB samples containing 10\,g/L PPO. The PPO was obtained from two different sources: one from rpi (Research Products International Corp), one from the Ukraine. For the latter, spectra for both the original and purified PPO (by Haiso pharmaceutical chemical Co., Ltd., Hubei, China) are shown separately. The positive impact of purification on transparency is clearly visible. The original quality of PPO from rpi seems to be comparable to purified PPO from Ukraine.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-17.png}
\caption[UV-vis attenuation spectra with PPO from selected providers]{UV-vis attenuation spectra of LAB with PPO from selected providers}
\label{fig:ls:fluor2}
\end{center}
\end{figure}
A similar or greater impact on the attenuation spectra is expected from the addition of the secondary fluor Bis-MSB. Like in the case of PPO, the chosen concentration has to be optimized and the impact of impurities studied. Two measured attenuation spectra of Bis-MSB in LAB are given in Fig.~\ref{fig:ls:fluor3}. While the same concentration of Bis-MSB was realized and the effective light yield of both solutions is identical, the absorption spectra differ.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-18.png}
\caption[UV-vis attenuation spectra of LAB with Bis-MSB]{UV-vis attenuation spectra of Bis-MSB in LAB solutions}
\label{fig:ls:fluor3}
\end{center}
\end{figure}
\subsection{Purification Techniques}
\label{sec:ls:opt_purification}
Beyond the choice of suitable raw materials, optical properties can be further improved by dedicated purification methods. Several methods of purification have been investigated at IHEP, including distillation, column purification, water extraction and nitrogen stripping. Figure~\ref{fig:ls:optpur2} displays a compilation of attenuation spectra of LAB samples by different manufacturers. For comparison, the spectrum of Nanjing LAB after purification in an aluminum column is shown. It clearly shows the best performance in the wavelength region of interest. Similar studies are currently performed at INFN and German institutes~\cite{Ford:2011zza}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-13.png}
\caption[UV-vis attenuation spectra before and after purification]{UV-vis attenuation spectra of LAB samples from different providers. In case of Nanjing LAB spectra taken before and after purification in an aluminum column are shown.}
\label{fig:ls:optpur2}
\end{center}
\end{figure}
\subsubsection{Fractional Distillation of LAB}
Distillation relies on the (partial) evaporation of the initial liquid to separate its constituents during vapor condensation. However, simple one-stage distillation usually does not meet the requirements for separating complex mixtures. This is usually achieved by multi-stage or fractional distillation: The technique relies on a heat exchange between upward-streaming vapor and downward-streaming condensate inside the distillation column. By this re-heating, the concentration of volatile components in the upper vapor increases, while less volatile components are enriched in the lower condensate. As the vapor keeps rising and the condensate keeps falling, several equilibrium states between gas and liquid are established along the column. Therefore, distillation occurs in multiple stages. If the fractional column is sufficiently long, an efficient separation of volatile from solid components is achieved.
In a first step, a setup including a single-stage distillation column was built up at IHEP. However, the resulting purification efficiency shown in Fig.~\ref{fig:ls:dist} proved insufficient. Therefore, the setup was upgraded to fractional distillation by the addition of a Vigreux column. This resulted in a substantial improvement due to the increased number of theoretical plates.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10.3cm]{LiquidScintillator/figures/LS4-14.png}
\caption[UV-Vis spectra of LAB before and after distillation]{UV-Vis spectra of LAB before and after distillation}
\label{fig:ls:dist}
\end{center}
\end{figure}
Compared to single-stage distillation, fractional distillation is much more efficient in the removal of the impurities causing absorption lines in the region from 330 to 410\,nm. However, in case of the IHEP setup, neither method provides a visible improvement in region around 430\,nm. Possible explanation comprise:
\begin{itemize}
\item The boiling points of the organic impurities influencing absorption at 430\,nm are too close to LAB to be removed by distillation.
\item New organic impurities are generated by distillation, e.g.~by the break-up of LAB molecules.
\item UV-vis spectroscopy is not sufficiently accurate to resolve an improvement.
\end{itemize}
While the unexpected absence of a beneficial affect of distillation in the 430-nm range requires further exploration, this method clearly improves transparency in the wavelength region below 410~nm. However, First results from a similar test setup at INFN Perugia in fact show an increase of the attenuation length in the region of interest.
The Vigreux column proved to be simple and easy to operate for laboratory-scale operations. However, a system based on fractional columns will be necessary for the distillation of large amounts of LAB for JUNO.
\subsubsection{Purification of LAB by Aluminum Columns}
The impact of purification in an aluminum (Al$_2$O$_3$) column on the optical properties of LAB has been studied in detail at IHEP. The results show that column chromatography is very effective in removing optical impurities.
The attenuation spectra of Fig.~\ref{fig:ls:column} clearly demonstrate that column purification does not only remove the impurities featuring absorption in the region from 330 to 410\,nm, but also improves transparency in the critical range around 430~nm. These findings have been confirmed by long-range attenuation length measurements. Column purification increased the attenuation length of an LAB sample originally featuring $\ell_{\rm att}\approx 9$\,m to 20\,m.
Moreover, the most transparent raw product, Nanjing special LAB, featured an attenuation length of 25\,m after purification. Provided the manufacturer can produce this high-quality LAB in large quantities, aluminum column purification will be sufficient to obtain a LS meeting the experimental specifications.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10.1cm]{LiquidScintillator/figures/LS4-15.png}
\caption[UV-Vis spectra of LAB before and after column purification]{UV-Vis spectra of LAB before and after purification in an aluminum column}
\label{fig:ls:column}
\end{center}
\end{figure}
Further laboratory studies are foreseen for the future:
\begin{enumerate}
\item Optimization of the Al$_2$O$_3$ column purification
\item Study of alternative column packings, such as a molecular sieve
\item Regeneration of the used Al$_2$O$_3$
\item Types and concentration of impurities in LAB which are removed by the Al$_2$O$_3$ column.
\end{enumerate}
\subsubsection{Further Purification Techniques Applied to LS}
Distillation cannot completely remove the radioactive contaminants dissolved in LAB. Thus, water extraction and nitrogen stripping are applied after mixing of distilled LAB, PPO and Bis-MSB.
\medskip\\
{\bf Water extraction} utilizes the polarity of water molecules to extract polarized impurities and radioactive free-state metal ions from LAB. The method efficiently removes metallic radionuclides such as $^{238}$Th, $^{232}$U, $^{210}$Bi and $^{40}$K.
\medskip\\
{\bf Gas stripping} is used for purging the LS from radioactive gases, mainly $^{85}$Kr and $^{39}$Ar, oxygen (which will decrease the light yield due to photon quenching) and water (introduced by water extraction).
\subsubsection{Purification of Fluors}
The purification of the fluors has to be carefully considered for the JUNO experiment. In the Daya Bay experiment, purification of PPO was performed in cooperation with a domestic chemical company. A domestic producer has also been identified for Bis-MSB: The provided sample meets the requirements concerning light yield, while it will have to be purified to obtain the necessary optical transparency. Moreover, experience from Borexino suggests that the radiopurity of the fluors is an important issue that has to be studied by laboratory experiments.
\section{Radiopurity Studies}
An important prerequisite for the detection of low-energy neutrinos is the radiopurity of the detector and especially the target material, i.e.~the LS. The residual contamination aimed for is on the level of the neutrino event rate, so in the order of hundreds of counts per day within the target volume. In case of antineutrino detection, the inverse beta decay coincidence signal somewhat relaxes the requirement on radiopurity as the fast double signature allows to discriminate most of the single event backgrounds.
\begin{table}
\begin{center}
\begin{tabular}{ccc}
\hline
Isotope & Antineutrinos & Solar neutrinos\\
\hline
$^{232}$Th & $\leq 10^{-15}$\,g/g & $\leq 10^{-17}$\,g/g \\
$^{238}$U & $\leq 10^{-15}$\,g/g & $\leq 10^{-17}$\,g/g \\
$^{40}$K & $\leq 10^{-16}$\,g/g & $\leq 10^{-18}$\,g/g \\
\hline
\end{tabular}
\caption[Requirements for LS radiopurity]{Requirements for LS radiopurity concerning uranium, thorium and potassium, listed both for antineutrino and for solar neutrino detection.}
\label{tab:ls:radiopurity}
\end{center}
\end{table}
Radiopurity levels are usually specified by the concentration of $^{232}$Th, $^{238}$U and $^{40}$K in the LS. The basic requirements for the JUNO LS are listed in Tab.~\ref{tab:ls:radiopurity}: The baseline scenario assumes a contamination on the level of $10^{-15}$ gram of U/Th and $10^{-16}$ gram of $^{40}$K per gram of LS, which will be sufficient for the detection of reactor antineutrinos. More stringent limits have to be met for the detection of solar neutrinos by elastic neutrino-electron scattering. Here, $10^{-17}$\,g/g resp.~$10^{-18}$\,g/g should be reached. These requirements are more than a factor 1000 resp.~10 above the current world-record held by the LS of the Borexino experiment. However, due to the scale of the project, achieving these radiopurity levels is a demanding task and will have to be studied and planned in considerate detail.
\medskip\\
While members of the natural $^{232}$Th and $^{238}$U decay chains are the most common contaminants, also other sources of radioactive impurities for the LS have to be taken into account. Moreover, the contamination may arise from different sources that have to be avoided or at least controlled:
\begin{itemize}
\item {\bf Dust particles} containing elements of the natural U/Th decay chains as well as radioactive potassium $^{40}$K.
\item {\bf Radon emanation}, especially the more long-lived $^{222}$Rn that is released from building materials (such as granite, brick sand, cement, gypsum, etc.) but also plastics, cables etc.
\item The {\bf radioactive noble gases $^{85}$Kr and $^{39}$Ar} can be introduced as residual contamination of the nitrogen used to create an inert atmosphere in the liquid handling system or by exposure to the ambient air
\item {\bf Surface depositions of $^{210}$Pb:} $^{222}$Rn will settle on surfaces exposed to air and decay to the long-lived isotope $^{210}$Pb ($\tau \sim 30$\,yrs), which in turn decays into $^{210}$Po and $^{210}$Bi.
\item {\bf $^{210}$Po} from surface contaminations, distilled water or other unknown sources.
\item {\bf Radioimpurities of the fluors}, mostly $^{40}$K.
\item {\bf Radioactive carbon $^{14}$C } intrinsic to the hydrocarbons of the LS.
\end{itemize}
While the majority of these studies is currently carried out at Chinese institutes, the European collaborators and especially the fraction that is involved in the Borexino experiment has a considerable experience in radio-purification. Several lab-scale studies are foreseen for the immediate future.
\subsection{Experimental Setups}
The aimed-for radiopurity levels of the JUNO specifications are too low to be measured by standard laboratory-scale experiments. Gamma spectrometers can only establish upper limits on the level of contamination. However, the effectiveness of purification techniques can be tested by artificially loading LS samples with greater amounts of radioactive elements and measuring the activity before and after applying a purification step. Moreover, the close relations with the Daya Bay experiment will allow a test of LS radiopurity by filling a large sample of LS ($\sim$20 tons) into one of the subvolumes of a Daya Bay Antineutrino Detector (AD). The AD will both offer the necessary capacity to hold a significant quantity of LS and a suitable low-background environment to test radiopurity to the level of the specificationsf~\cite{Ford:2011zza,Mark:2008gc,Mike:2008gc}.
\subsubsection{Low-background Gamma Spectrometer}
A standard laboratory technique to measure level and type of radioactive impurities inside a LS sample are gamma spectrometers. The device currently in use has been employed to measure LS samples before purification. Corresponding to the minimum sensitivity of the setup, only an upper limit can be set on the sample activity, which is smaller than 0.1\,Bq/kg. The corresponding concentration of $^{238}$U should be smaller than $8.1\cdot10^{-9}$\,g/g, so 6$-$7 orders of magnitude above the JUNO specification levels. The activity corresponding to the specified level of $10^{-15}$\,g/g is on the order of $10^{-8}$\,Bq/kg and thus clearly beyond the sensitivity of this and comparable setups.
\subsubsection{Assays with Loaded Scintillator Samples}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-19.png}
\caption[Laboratory setup for radon-loading of LS samples]{Laboratory setup for radon-loading of LS samples}
\label{fig:ls:rnload}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=6cm]{LiquidScintillator/figures/LS4-20.png}
\caption[Natural thorium decay chain]{The natural decay chain of $^{232}$Th}
\label{fig:ls:rnchain}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{LiquidScintillator/figures/LS4-21.png}
\caption[Setup for measuring radio-purification efficiency]{Laboratory setup for measuring the efficiency of radio-purification}
\label{fig:ls:counter}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=14cm]{LiquidScintillator/figures/LS4-22.png}
\caption[Sample pulse of a fast $^{212}$Bi-$^{212}$Po coincidence]{Sampel pulse of a fast $^{212}$Bi-$^{212}$Po coincidence.}
\label{fig:ls:bipo}
\end{center}
\end{figure}
In order to investigate the effect of purification schemes on LS in laboratory experiments, the only solution is an artificial pollution of the samples with radioactivity, bringing the activity to a level accessible by small-scale counting experiments. This is achieved by loading the LS with suitable radioisotopes.
The experience gained in the purification campaigns carried out in Borexino suggests that the radioisotope most difficult to remove from the LS is the long-lived $^{210}$Pb. Therefore, it is crucial that purification tests will be carried out using$^{212}$Pb as a tracer. While easier to tag in lab experiments, its chemical properties correspond to $^{210}$Pb. However, direct deposition of lead in LS is difficult as metallic powders will not dissolve in the organic liquid. Instead, the method commonly chosen for loading is the exposition of an LS sample to radon which will in turn decay into lead.
The corresponding setup inside a glove box is shown in Fig.~\ref{fig:ls:rnload}: A solid $^{232}$Th source contained in a small box is used to produce the noble gas $^{220}$Rn ($T_{1/2}=56\,$sec) by emanation (cf.~Fig.~\ref{fig:ls:rnchain}). This box is constantly ventilated by an air pump, creating a stream of radon-enriched air that is lead by a hose to a further container holding the LS sample. As a noble gas, radon is easily dissolved into the LS by purging the sample. It rapidly decays to $^{216}$Po and then $^{212}$Pb, which is $\beta$-unstable with a half-life of $T_{1/2}=10.6\,$h and is thus well-adjusted for performing purification tests. Its daughters $^{212}$Bi and $^{212}$Po decay in a fast coincidence that provides a further experimental tag.
The activity is measured by the experimental setup depicted in Fig.~\ref{fig:ls:counter}. In a light-tight box, a pair of 2'' PMTs is placed on both sides of an LS sample cell to detect the light produced by the $^{212}$Pb decays. Gamma rays from ambient radioactivity are attenuated by a shielding of low-activity lead bricks. Thus, the background rate for the $^{212}$Pb measurement is 0.5\,s$^{-1}$. If the fast coincidence of $^{212}$Bi and $^{212}$Po is used, the background rate is further reduced to only 2\,d$^{-1}$. A sample waveform of a Bi-Po event acquired in this setup is shown in Fig.~\ref{fig:ls:bipo}.
\subsubsection{Purification Test with a Daya Bay AD}
Once different purification schemes have been tested on the laboratory scale, the test in a medium-sized setup containing several tons of LS will be an important next step to assure that purification techniques are efficient down to the extremely low radiopurity levels of the JUNO specifications. The Antineutrino Detectors (ADs) of the Daya Bay Experiment are ideally suited to do such a study. The low-background conditions provided by the underground labs (the Daya Bay Near Experimental Halls are more than 100 meters underground) as well as the low-background material used in constructing the ADs themselves provide favorable conditions to measure the residual radioactivity of up to 20 tons of LS filled into one of the acrylic volumes of the AD in question.
Moreover, an AD test setup allows the practicability of the purification scheme on the scale of a system able to handle $\sim$100 liters of LS per hour. For this purpose, a circular system will be set up. After initial filling with an LAB-based LS replacing the original LS in the AD, purification techniques will be probed in loop mode by monitoring the change in radiopurity levels. This setup will provide important information for the design of a system able to handle the 20,000 tons of LS for the filling of the JUNO detector.
Beyond radiopurity, the filled AD will also provide information on the scintillation and optical properties of the LS. The large-scale setup allows to study the effect of purification on both the light yield and the light attenuation in the scintillator under realistic conditions.
\subsection{Purification Methods}
\label{sec:ls:radiopurity}
For the moment, the methods investigated for their purification efficiency concerning radioactive contaminants comprise aluminum columns, distillation, water extraction, and nitrogen stripping (cf.~Sec.~\ref{sec:ls:opt_purification}).
\subsubsection{Aluminum Columns}
Purification in aluminum Al$_2$O$_3$ columns proves to be very efficient in removing organic impurities from LAB, resulting in increased transparency. Beyond this, laboratory studies also show a decrease in radioactive impurities by adsorption on Al$_2$O$_3$.
The Al$_2$O$_3$ best suited for radio-purification was selected by gamma spectroscopy. Results of a comparative study are shown in Table~\ref{tab:ls:al2o3}. The Al$_2$O$_3$ featuring the lowest activity levels in $^{238}$U, $^{232}$Th, and $^{40}$K is produced by the Zibo Juteng chemical company ($\gamma$-type).
\begin{table}[!htb]
\begin{center}
\begin{tabular}{cccccc}
\hline
Nuclide & energy & XP type & $\gamma$ type & (GY)Al$_2$O$_3$ & hamamasu low \\
& & Al$_2$O$_3$ & Al$_2$O$_3$ & specific activity & background glass \\
& keV & Bq/kg & Bq/kg & Bq/kg & Bq/kg \\
\hline
$^{232}$Th & 583.1 & 3.94 & 0.07 & 0.52 & 2.02 \\
$^{238}$U & 609.4 &3.93 & 0.16 & 0.60 & 5.46 \\
$^{40}$K & 1460.8 & 5.49 & 0.54 & 4.30 & 24.14 \\
\hline
\end{tabular}
\caption[Comparison of Al$_2$O$_3$ radioactivity levels]{Comparison of Al$_2$O$_3$ radioactivity levels.}
\label{tab:ls:al2o3}
\end{center}
\end{table}
The radiopurity assay was based on a $^{220}$Rn-loaded LAB sample of 211\,ml volume. Measurements based on counting of $^{212}$Bi/Po coincidences were performed before and after column purification. A spin-type column was used, with a fill height of 20\,cm of Al$_2$O$_3$. Based on the $^{212}$Bi/Po count rate, a radio-purification efficiency of 99.4\,\% could be determined.
For the future, an extension of these studies is foreseen at European institutes, especially INFN and JINR Dubna, which will also comprise adsorbants other than aluminum oxide.
\subsubsection{Distillation of LAB}
Distillation relies on the difference in boiling points and volatility of the LS components and possible impurities (see above). In particular, excellent efficiency is expected for the removal of radioactive metal ions because of the vast difference in boiling points compared to LS components. However, there are also large differences between the boiling points of the solvent LAB and the solutes PPO and Bis-MSB, which make the application of this technique very demanding once the three components are mixed for forming the LS. Thus, distillation is most suited for a purifications stage before mixing.
The effectiveness of distillation regarding radio-purification has been tested in the laboratory setup depicted in Fig.~\ref{fig:ls:dist2}. Radioactivity levels of a $^{220}$Rn-loaded LAB sample have been determined before and after distillation. Based on $^{212}$Bi/Po coincidence counting the efficiency has been determined to 98.4\,\%. Further tests will be performed after adding a reflux unit that should further improve the purification efficiency.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=14cm]{LiquidScintillator/figures/LS4-25.png}
\caption[Setup for distillation tests]{Laboratory setup for distillation tests}
\label{fig:ls:dist2}
\end{center}
\end{figure}
In the current setup, we find that after distillation with high temperatures or long heating times the LS samples feature a reduced attenuation length, change color and release a strong smell. A possible reason is the chemical break-up of organic impurities by the heating process. Possible solutions are an optimization of the distillation process or a reduction of heat-sensitive impurities in the LAB raw materials. It should be noted that similar laboratory-scale distillation tests at INFN Perugia did not show a degradation of the LS.
In the final large-scale setup for JUNO, vacuum distillation will be used for a reduction of the necessary heating temperature and the related energy consumption. To reduce the energy consumption on the experimental site, the distillation plant could be realized directly at the LAB factory. The preliminary design of the installation comprises multistage plates and a vacuum
distillation tower for LAB with a processing capacity of up to 12,000 liters per hour, operated at a temperature of $220\,^{\circ}$C. These preliminary working parameters will be refined by further studies carried out at INFN institutes.
\subsubsection{Water Extraction}
Purification by water extraction relies on the polarity of water molecules to separate polarized impurities, e.g. free-state ions of radioactive metals, from the non-polar LAB and fluor molecules. Mixing of water and LS is performed in a purification tower that contains a counter-current water extraction column. Pure water is filled at the top while LS enters at the bottom of extraction column. The mixing of the two liquids is supported by a porous filling. As soon as a state of full mixing has been reached, the LS leaves at the top of the column while the waste water is collected by the drainage system at the bottom.
The water extraction method is highly efficient for removing metal ions: Radium is removed at the level of 96.5\%. For lead and polonium, the purification efficiency of 82$-$87\,\% is somewhat lower as both can form chemical bonds with the organic molecules of the solvent that are fairly stable at room temperature.
\subsubsection{Nitrogen Stripping}
Purging the LS by nitrogen aims both at the removal of dissolved radioactive noble gases (argon, krypton, xenon, and radon) in the liquid and the removal of oxygen, which acts as a quencher lowering the light yield by oxidation of the fluor molecules. Beyond active purification by purging, ultrapure nitrogen will be used for establishing a nitrogen atmosphere or blanket inside the liquid handling system and in the final detector to avoid gas contamination of the LS.
For laboratory purposes, industrial-grade high-purity nitrogen is mostly sufficient. However, the high solubility of noble gases in the LS makes it necessary to add a further purification step for the nitrogen before bringing it in contact with the liquid. For this, high-purity nitrogen of grade 6.0 (99.9999\,\% pure nitrogen) is passing through a cold trap at liquid argon temperatures. The resulting ultrapure nitrogen can reach radioactive background levels of 10$^{-17}$\,g/g, corresponding to 0.36\,ppm of $^{39}$Ar and 0.16\,ppt of $^{85}$Kr. Figure~\ref{fig:ls:coldtrap} shows a schematic drawing of a laboratory-scale setup for the production of ultrapure nitrogen based on an active carbon absorber submerged in a bath of liquid argon. The resulting radiopurity will be tested by GC-MS measurements (Agilent 7890A/5975, cf.~Sec.~\ref{sec:ls:gcms}).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=11cm]{LiquidScintillator/figures/LS4-26.png}
\caption[Schematic drawing of ultrapure N2 production unit]{Schematic setup of the production unit for ultrapure nitrogen.}
\label{fig:ls:coldtrap}
\end{center}
\end{figure}
\section{Mass Production of 20 kton LS}
{\bf Production. } The production of 20 kilotons of LS will be achieved in three or four cycles over a period of 1$-$2 years. Based on the current status of laboratory studies (see above), the solvent LAB will most likely be provided by the company from Nanjing. Criteria for quality control will be developed jointly by IHEP and the company. Similar arrangements will be made for the solutes PPO and Bis-MSB.
\medskip\\
{\bf Transport.} A professional logistics company will take care of the transport of the LAB from the production line at Nanjing to the JUNO experimental site. Suitable transport containers will be manufactured by a company specifically for this purpose, meeting specifications set by IHEP concerning radiopurity, air tightness and material compatibility with LAB.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=15cm]{LiquidScintillator/figures/LS4-27.png}
\caption[Schematic drawing of LS purification plants]{Schematic drawing of the system of LS purification plants}
\label{fig:ls:filling}
\end{center}
\end{figure}
\bigskip
\noindent The on-site LS handling system is depicted in Fig.~\ref{fig:ls:filling}. The facilities include:
\medskip\\
{\bf Storage tanks.} Since the LAB will arrive in 3 to 4 batches of 5\,kt each, the foreseen storage area comprises about 5,000 square meters, featuring several tanks with a volume of 3 ktons and another set of smaller tanks on the scale of several hundred tons. This includes not only tanks for LAB, but also for white oil and temporal containers needed during unloading. In addition, an area of 30\,m$^2$ is to be reserved for storage of the fluors.
\medskip\\
{\bf Surface purification plants.} Fig.~\ref{fig:ls:filling} depicts the basic scheme for the on-site system of purification plants: The current design mainly relies on an initial purification of the individual LS components (solvents and solutes) before mixing them to obtain the final LS. The major task of purifying LAB is to be performed on surface. The foreseen purification steps include distillation, adsorption in an aluminum oxide column, water extraction and nitrogen purging (cf.~sec.~\ref{sec:ls:radiopurity}).
Assuming a total mass of 20\,kt of LS and a period of 200 days reserved for continuous purification and detector filling, the required processing capacity of the purification plants is 100 tons per day. Based on the power consumption of similar (but smaller-scale) purification plants realized for the Borexino, KamLAND and SNO+ experiments, the corresponding power consumption is about 2\,MW.
In addition, a supply of ultrapure nitrogen on a rate of 200\,m$^3$/h as well as ultrapure water of 6 tons per hour has to be assured. The system will require a surface clean room of 200\,m$^2$ area and 10\,m height for its installation. In addition, rooms for the production plants of ultrapure water and nitrogen are needed (30\,m$^2$ each).
\medskip\\
{\bf LS mixing plants.} After purification of the individual components, LS mixing will be performed. A complication arises from the low solubility of Bis-MSB on LAB. To reduce the processing time and assure a constant amount of the fluor in the LS, batches of highly concentrated LAB-Bis-MSB solution (150\,mg/l) will be prepared beforehand and then watered down with pure LAB during mixing.
The mixing tanks will be made from organic glass to ensure both the material compatibility of the LS and to allow for a visual inspection of the mixing product. The mixing system will be housed in a room of 300\,m$^2$.
\medskip\\
{\bf Quality assurance and control.} Constant control of the LS leaving the purification and mixing plants is of uttermost importance for the success of JUNO. As the LS is produced in cycles, even a single low-quality batch passing the controls unnoticed and filled into the detector could potentially spoil the high-quality of all LS produced up to this point. Therefore, a careful program of QA/QC measures before and after purification and mixing has to be developed. Separate teams of operators and inspectors should be formed and be supervised by an on-site manager, following a strict protocol.
Two inspection rooms of a total area of 60\,m$^2$ will house systems for QC, including GC-MS for detection of organic impurities and small-scale experiments to monitor attenuation length and spectrum as well as light out of the LS.
\medskip\\
{\bf Underground installations.} Approved batches of LS will be sent by a pipeline to an underground storage area where it remains until detector filling. The pipe system should allow the flow of LS in both directions and feature high-pressure resistance to allow for a safe transport of the LS. Two storage tanks of 200 tons for LS and white oil are foreseen close to the detector cavern.
Beyond the surface plants for the initial purification of the LS, a secondary system is foreseen underground to perform loop-mode purification of the LS while the experiment is running. This online purification system will rely on water extraction and nitrogen stripping. The system will require two underground halls of 500\,m$^2$ area and 10\,m in height. The necessary supply of water, electricity and gas will be roughly equivalent to the requirements of the surface system.
\medskip\\
{\bf Cleanliness conditions.} Ensuring the cleanliness and purity of all materials in contact with the LS and its raw materials will be decisive for meeting the radiopurity requirements of JUNO. All pipes, pumps storage and processing vessels through which the LS passes from arrival on-site until filling of the detector will have to meet the requirements for chemical compatibility, cleanliness of all surfaces and air-tightness.
All surfaces have to be thoroughly cleaned and probably rinsed with LAB before being brought into contact with the LS. Removal of dust on surfaces is mandatory: At an average activity of 200 Bq/kg in U and Th, not more than 10\,g of dust are allowed to be dissolved in the total mass of 20\,kt of LS. What is more, piping and tank materials will be containing U and Th themselves and thus will potentially emanate radon that will diffuse into the liquid. This process can be mitigated by applying suitable surface liners stopping the diffusion.
The expected radon level in the underground air is about 100\,Bq/m$^3$. Contact of LS and process water to air has to be strictly avoided as the solubility of radon in LS is far greater than in water. To reach this goal, all the liquid handling system must be absolutely air-tight, and an atmosphere of ultrapure nitrogen will be established inside the system.
\medskip\\
All surface and underground facilities will be equipped with cameras, fire protection, monitors for oxygen and radon as well as alarm systems for poisonous gases.
\section{Risk Assessment}
\begin{table}
\begin{center}
\begin{tabular}{lcc}
\hline
Properties & LAB & Mineral Oil\\
\hline
Quantity & 20\,kt & 7\,kt \\
\hline
Appearance & odorless, & odorless,\\
& colorless, & colorless,\\
& transparent & transparent \\
Flammable & yes & yes \\
Toxic & no & no \\
Corrosive & no & no \\
\hline
Chemical formula & C$_6$H$_5$C$_n$H$_{2n+1}$ & C$_n$H$_{2n+2}$ \\
& ($n=10-13$) & \\
Density [g/cm$^3$] & 0.855$-$0.870 & 0.815$-$0.840\\
Freezing point & -50\,$^{\circ}$C & \\
Flash point & 135\,$^{\circ}$C & 145\,$^{\circ}$C\\
Vapor pressure (20$^\circ$C) & 0.1\,mmHg & 0.1\,mmHg \\
\hline
\end{tabular}
\caption[Table of material properties]{Table of material properties}
\label{tab:ls:matprop}
\end{center}
\end{table}
The JUNO detector will contain both LS and and mineral oil. Material properties are listed in Table~\ref{tab:ls:matprop}.
\medskip\\
{\bf LAB. } The main component of LS is linear alkyl benzene (LAB), of which 20 ktons are required. In industry, it is mainly used in the production of detergents. For this, LAB is sulfonated to form linear alkylbenzene sulfonate (LAS), which in turn serves as raw material for washing powders, liquid detergents, pesticide emulsifiers, lubricants and dispersants. It is non-poisonous and odorless. For use in JUNO, the only safety concern is its flammability.
\medskip\\
{\bf Mineral oil.} The buffer volume surrounding the acrylic sphere might be filled with 7,000 tons of mineral oil. The foreseen type is food grade mineral oil 10\#. It is a liquid by-product of the distillation of crude oil which is performed during the production of gasoline or other petroleum-based products. Again, the only safety concern is flammability. Mineral oil is widely used in biomedicine, cell cultures, veterinary uses, cosmetics, mechanical, electrical and industrial engineering, food preparation etc.
\medskip\\
Due to the flammability of both liquids, the main risk to be taken into account is fire safety. The necessary precautions have to be taken into account during transport, on-site storage, purification and mixing of the LS components as well as during detector operation.
From an experimental point of view, the intactness of the LS poses additional constraints on all operations. Cleanliness and tightness of the liquid handling system have to be considered carefully.
Further risks will be induced by the installation of the LS purification plants and storage tanks in an underground cavern at a depth of 700\,m. This will result in complicated hoisting equipment, high-altitude operations, and a need for accurate positioning. Purification underground will pose additional risks because a possible spillage of nitrogen are more serious under these confined conditions. All operations of this kind will have to be carefully planned beforehand.
\section{Schedule}
\begin{itemize}
\item[2013] Realization of laboratory setups for LS properties.
First explorative measurements of LS properties.
Measurement of the depolarization ratio to determine the Rayleigh scattering length.
\item[2014] Realization of laboratory-scale setups for testing different purification method:\\
Low-pressure distillation, aluminum-oxide column, nitrogen stripping and water extraction.
First steps towards the optimization of the purification scheme:
Tests of optical purification: Measurements of light out, attenuation length and spectrum, and water content of the purified LAB.
Tests of radio-purification: Establish the technology for injecting $^{212}$Pb and $^{224}$Ra in LS samples and setup of the corresponding measurement device to study the effect of purification on radioactive contamination levels.
Start of joint research programs with companies providing LAB.
\item[2015] Design and realization of a purification system.
Final determination of purification parameters.
Start of measurements for electron energy non-linearity.
Design and setup of the quality control devices.
Start of joint research activities with providers of PPO and Bis-MSB.\\
Conception and setup of devices for quality testing.
\item[2016] Jan$-$Jun: Setup and testing of purification plants at Daya Bay.
Jul$-$Dec: Installation and debugging the purification system.
\item[2017] Determination of the final recipe of LS production based on the experimental data and the purification methods.
Final measurement of light out and attenuation length of the LS.
Signing of contracts with the companies providing the purification plants and start of the design.
Signing of contracts with the providers for PPO and Bis-MSB.
Start of the batch production at the end of the year.
Finalization of the design of storage tanks for LAB. Design and build the LS purificaiton hall on the ground.
\item[2018] Signing of all contracts for LAB storage area.
Setup of the large storage tanks before the middle of the year:\\
3$-$5 of 3000$-$1000\,m$^3$ tanks for LAB, LS and white oil
Installation of the purification system until end of the year.
Setup of two 200-ton underground buffer tank.
Setup and testing of the LS QA/QC systems.
\item[2019] Testing of all purification plants.
Setup and testing of ultrapure water and nitrogen plants.
Design and build the LS mixture system.
Production of 8 ktons of LS of sufficient quality for the central detector.
\end{itemize}
\newpage
\chapter{Offline Software And Computing}
\label{ch:OfflineSoftwareAndComputing}
\section{Design Goals and Requirements}
\input{OfflineSoftwareAndComputing/requirements.tex}
\input{OfflineSoftwareAndComputing/system_introduction.tex}
\section{Offline Software Design}
\input{OfflineSoftwareAndComputing/platform.tex}
\input{OfflineSoftwareAndComputing/event_model.tex}
\input{OfflineSoftwareAndComputing/geometry.tex}
\input{OfflineSoftwareAndComputing/generator.tex}
\input{OfflineSoftwareAndComputing/detector_sim.tex}
\input{OfflineSoftwareAndComputing/digitization_evtMixing.tex}
\input{OfflineSoftwareAndComputing/detector_rec.tex}
\input{OfflineSoftwareAndComputing/event_display.tex}
\input{OfflineSoftwareAndComputing/database_service.tex}
\section{Computing System Design}
\input{OfflineSoftwareAndComputing/computing_network.tex}
\input{OfflineSoftwareAndComputing/data_trans.tex}
\input{OfflineSoftwareAndComputing/com_storage.tex}
\input{OfflineSoftwareAndComputing/management_and_service.tex}
\input{OfflineSoftwareAndComputing/dev_flow_quality_control.tex}
\subsection{Computing Network and Public Network Environment}
A 1 Gbps dedicated link, named IHEP-JUNO-LINK, is planned between the JUNO experiment site and IHEP. It is proposed that this link be used for data transfer between the experiment site and the IHEP data center as well as for connecting the onsite office network and the external network. The 1 Gbps bandwidth is sufficient to fully meet the experiment's requirements for stable data transfer, based on an estimated data volume of 2 PB per year, which should take an estimated average bandwidth of 544 Mbps. The remaining bandwidth can be used for network crash recovery, experiment remote control and office network connectivity for JUNO onsite researchers.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.8\textwidth]{OfflineSoftwareAndComputing/figures/network.png}
\caption[WAN topology]{\label{fig:storage}WAN Topology }
\end{center}
\end{figure}
The JUNO network link is planned to be provided by the Chinese Science and Technology Network (CSTNet) \cite{CSTNet}. The data will first be transferred from the experiment site to IHEP through IHEP-JUNO-LINK, then relayed to collaborating sites through CSTNet. At present, IHEP is connected to the CSTNet core network through two 10 Gbps links, one of which supports IPv4 and the other IPv6. The bandwidth from IHEP to the USA is 10 Gbps and from IHEP to Europe is 5 Gbps, both of which are through CSTNet and have good network performance.
\subsubsection{Network Information System Security}
Network security is becoming more and more important in all kinds of information systems. The JUNO application system and even the computing environment also face security risks, so we are planning to apply risk control and safety protection in the following aspects: physical layer, network layer, system layer, transport layer, virus threats layer and management layer.
The safety of the physical layer is the foundation of the whole information system security. In general, the security risks to the physical layer mainly include: system boot failure and database information loss due to power failure, whole systematic destruction caused by natural disasters, probable information loss caused by electromagnetic radiation and so on. In order to ensure the robustness of the physical layer, a powerful UPS should be deployed for the IT system and devices; proper access control policies should also be implemented for some special areas.
The risks to the network layer consist of risks to the network boundaries and risks to safety risk of the data transfer. A powerful network firewall system will be deployed in the JUNO network to avoid risks to the network boundaries. To ensure the safety of data transfers, there are two things we must consider: data integrity and confidentiality. To ensure data integrity, we can add checksums and provide multi-layer data buffering during transfers.To ensure data confidentiality, the information system should use encrypted transmission.
The system layer risk usually refers to the safety of operation systems and applications. In JUNO, a rigorous hierarchical control policy and auditing operation logs will be used to reduce and prevent security threats to the system layer.
For virus threats, while traditional viruses mainly spread through a variety of storage media, modern viruses mainly spread through the network. When a computer in the LAN is infected, the virus will quickly spread through the network to hundreds or thousands of machines in the same network. In JUNO, rigorous access control policy and real-time detection for the network traffic will be implemented for all the network nodes and IT systems to provide a safe network environment.
The risk to the management layer is usually caused by unclear responsibilities and illegal operation of the IT system, so the most feasible solution is to combine management policies and technical solutions together to achieve the integration of technology and policy.
\subsection{Computation and Storage}
\subsubsection{Local Cluster}
About 10,000 CPU cores should be used for raw data simulation, reconstruction and data analysis to meet JUNO offline computing requirement. All the resources will be managed by the batch system of the IHEP Computing Center. Several job queues with different priorities will be established to share the resources in the most efficient way.
An optimized scheduling algorithm will be developed based on the features of the the JUNO computing environment and hardware performance. The scheduler should dispatch jobs to the most suitable computing resource so that high overall computing efficiency is obtained. Job types can be defined based on the computing model, characteristics and I/O bandwidth consumption for different jobs. For this, it is necessary to study and test different types of jobs running on different machines with different CPUs. It is also necessary to analyze the effect of the distributed storage system on job running time. New scheduling optimization algorithms based on the results of this study should be implemented as a plug-in which can be integrated with the batch job system. This will be used for resource management and job scheduling to improve the efficiency of single jobs and the overall resource usage.
\subsubsection{Computing Virtualization}
Local clusters are managed by job management systems, with Torque \cite{Torque}, Condor \cite{HTCondor} and LSF \cite{LSF} being popular choices in HEP systems. These are responsible for scheduling jobs to a physical machine. With ever more powerful CPUs available, virtual machine clusters - cloud computing - can be used to relieve the pressure on computing resources at peak times. Cloud computing also reduces the burden of system management in applying upgrades to operating system and offline software, as well as satisfying requests for different software versions. Running jobs on virtual machines (VMs) can also avoid the trouble caused by hardware heterogeneity and give more flexibility in software configuration, without requiring changes in the job itself. Virtual clusters can therefore improve the availability and reliability of the computing resources as well as providing elastic resource expansion and flexible resource allocation. Rather than allocating a job to a node, as for physical cluster resource allocation, jobs are allocated to a CPU core. One CPU core supports one VM running a JUNO job. The lifetime of the VM depends on the job running time. This makes it possible to pre-empt jobs based on priority, and to migrate jobs. The VM can be migrated online and the job running on the VM can be hung up before the migration and released after the migration. These features of VMs will make the computing platform more robust.
\subsubsection{Distributed Storage Management and Data Sharing}
JUNO data processing is a typical massive data computation task. Distributed storage management is necessary for its large scale of data. The following figure shows the architecture of the storage.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.8\textwidth]{OfflineSoftwareAndComputing/figures/com_storate_f1.png}
\caption[Storage architecture]{\label{fig:storage}The storage architecture }
\end{center}
\end{figure}
The logical device layer, which is over the physical layer, provides stable and reliable storage for the upper layers. The logical storage devices are independent of the physical devices and isolated from them. They can be elastically extended or narrowed, copied and be migrated among physical devices. The features of logical devices, including high availability, high performance and high security, make it easy to satisfy the storage demands of JUNO. The logical device layer gives the storage system strong scalability with PB level capacity.
The storage management layer provides the functions of resource allocation, scheduling and load balancing. It provides a global unified name space, called the virtual storage layer. This can shield the heterogeneity of physical devices and integrate a unified virtual storage resource for users.
The distribution process layer is over the storage management, providing fault tolerance, consistency, distributed task scheduling support (such as map-reduce \cite{mapreduce}) and metadata management.
The application interface layer provides rich interfaces compatible with Posix semantics, and is transparent to users.
This virtual distributed storage system should fully satisfy JUNO's requirement of 10 PB storage, providing a high-reliability, high-performance storage service which can be extended elastically as needed.
.
\subsubsection{Distributed computation}
Since JUNO has a large computation scale, it needs cooperative work among the
group members. Distributed computation integrates heterogeneous resources from
different sites, which could be shared by all the sites located in different
region. A unified user interface would be provided to receive user jobs and job
is dispatched to the suitable sites by the system. Job ``pushing'' and
``pulling'' are the two main scheduling methods.
\subsection{Database Service}
Databases play more and more important roles in offline data processing. They are used to store many kinds of information, such as, sizes and positions of complicated detector geometry, optics information, running parameters, calibration constants, schema evolution, bookkeeping and so on. At the same time databases provide access to all kinds of information through their management services, such as creation, query, modification and deletion, which will be frequently used during offline data processing.
Many types of database have been used in particle physics experiments, of which the most popular are Oracle \cite{Oracle}, Mysql \cite{MySQL}, PostgreSQL \cite{postgreSQL} and SQLite \cite{SQLite}. Many of them provide user-friendly APIs interfaces for different programming languages. Recently, NoSQL databases have become more and more popular and powerful in some other fields, so we are also investigating the possibility of using some NoSQL database.
Regardless of which kind of underlying database will be chosen,the general schema of database services will be implemented in three levels. The lowest level is the APIs of the underlying database; the middle level is the new extension of the APIs according to the requirements of the JUNO experiment; the upper level is to provide user-friendly services for different applications such as detector simulation, event reconstruction and physics analysis.
Since Daya Bay already has one good database interface, we will implement it, optimizing and extending the current interface to provide more flexible and more powerful access to the different types of information stored by the databases. We will also set up two types of database servers, master servers,and the slave servers. The former are used to store and manage all information and can only be accessed by the slave servers, while the latter are responsible for retrieving information from the master servers and providing access for all applications.
\subsection{Data Process Requirement}
Data taking period of JUNO will start from 2019 and raw data it generates each year would be about 2PB. Raw data would be transferred to computing center of IHEP via dedicate network connection. A large scale offline computing platform is necessary for the computation of simulation and reconstruction. Besides, it will support analysis computation. The platform will also provide enough storage space for the raw data, Monte Carlo data and data archive. With the rough estimation, the scale of the platform would be 10,000 cpus/cores, 10PB disk storage and 30PB archive. 40Gbps bone switching network is the connection among computing nodes and storage servers. To promote the data process speed, the platform would integrate computing resources of collaboration group members via distributed computation. Data would be shared and computation tasks would be dispatched to the sites included in the platform cooperatively. So JUNO puts forward high demands to data analysis process, data storage, data transfer and data share as the following four aspects.
\subsection{Data transferring requirement}
Data transfer and exchange system with features of high performance, high security and high availability is necessary. To guarantee the experiment data transferring from experiment site to IHEP data center in time, the system should monitor and trace the online status of data transferring and exchange and it should provide data transfer visualization interface. This could also guarantee the data dispatch and share in the collaborative group member sites smoothly.
\subsection{Computing Environment}
To guarantee JUNO data process smoothly, a 10,000 cpus/cores local cluster would be established. The collaboration group member would also provide computing resources according to the different cases. The platform includes not only local cluster, but remote resources from collaboration group member, who is responsible for the JUNO computation task either.
\subsection{Data Storage}
JUNO will generate huge experiment data. Data from experiment site should be transferred and stored at computing center of IHEP. The capacity of storage in IHEP would be 10 PB. It could support 10,000 cpus/cores concurrence access. Data migration between tape library and disk is transparent to users.
\subsection{Maintenance Monitoring and Service Support}
To guarantee the platform stable, an intelligent monitoring system would be developed and an information system aimed at infrastructure management should be established. The status of both platform and network connection should be monitored in a very fine grain control. The monitor tool could give the status assessment of the whole platform and give a tendency prediction and pre-warning for the relative services of the platform. A set of information management tools should be developed including conference support, document management and online-shift management etc.
\subsection{Data transfer and sharing}
Data transfer and sharing is the foundation of physical analysis in JUNO. In steady operations mode, raw data will be transferred over the network from the experiment site to the Computing Center for long term data storage, then distributed to the collaboration group members according to their requirements. The raw data from the JUNO experiment is open to all collaboration group members for local or remote processing and analysis.
The raw data from JUNO will be acquired by the online data acquirement system (DAQ) which will be deployed onsite, and then stored in a local disk cache with sufficient storage to keep one month's raw data. After that, the raw data will be transferred to the IHEP Computing Center in Beijing. During the transfers, the checksum of the data will also be transferred to make sure of the data integrity. If the data integrity check is failed, the raw data should be retransferred. After the data transfer is done and the data integrity check is ok, the status of the data in the DAQ local disk cache will be marked as TRANSFERRED. Moreover, a high/low water line deletion algorithm will be used to clear the outdated data in DAQ local disk cache.
Most of the data will be transferred automatically by the data transfer system, so some possible human errors can be reduced. To ensure the stability and robustness of the data transferring system, a monitoring system will be developed and deployed which will provide the data transfer and sharing status in real-time and track the efficiency of the data transfer system. Combined with the status of the IT infrastructure (including network bandwidth), the data transfer system will optimize the transfer path and recover transfer failures automatically to improve the performance and stability of the system.
\subsection{Event Reconstruction}
\subsubsection{PMT waveform reconstruction}
The total number of photoelectrons and the first hit time of each PMT can be reconstructed based on the waveform sampling from the readout electronics with two algorithms, charge integration or waveform fitting. A fast reconstruction algorithm has been developed using the charge integration with appropriate baseline subtraction. The first hit time is determined as the time above threshold with time walk correction. A waveform fitting algorithm based on a PMT response model is under development, which is expected to give better timing and can reduce the distortion of the charge measurement caused by the PMT and electronics effects such as after-pulse and overshoot. The fast reconstruction is designed for event selection, while the waveform fitting is expected to be used for small subset samples such as IBD candidates or calibration data.
\subsubsection{Vertex reconstruction in the Central Detector}
Since the energy of a reactor antineutrino is usually below 10 MeV, it can be approximated as a point source in the JUNO detector. Then the event vertex can be reconstructed based on the relative arrival time from the point source to different PMTs, by minimizing a likelihood function:
\begin{equation}
F(x_0,y_0,z_0,T_0)=-\sum_i \log(f_t(t_i-T_0-\frac{L(x_0,y_0,z_0,x_i,y_i,z_i}{c_{eff}})),
\end{equation}
where $x_0,y_0,z_0,T_0$ are the event vertex position components and start time to be fitted, $x_i,y_i,z_i,t_i$ are the position components and hit time of the $i_{th}$ PMT, $c_{eff}$ is the speed of light in the detector, and $f_t$ is the probability density function of the PMT hit time with the time-of-flight being subtracted, which contains the decay time of scintillation light and the time resolution of the PMT. If the multiple photoelectrons in the same PMT can be clearly separated, $f_t$ can be expressed analytically, otherwise it relies on MC simulation. A charge center method is used to calculate the event vertex to the first order, which can be used as the initial value of the likelihood fitting.
\subsubsection{Visible energy reconstruction}
The visible energy in the central detector of a point source is reconstructed using the maximum likelihood fitting:
\begin{equation}
\mathcal{L}(E_{vis})=\prod_{no-hit}{P_{no-hit}(E_{vis},i)}\times\prod_{hit}{P_{hit}(E_{vis},i,q_i)},
\end{equation}
where $E_{vis}$ is the visible energy to be fitted, $P_{no-hit}(E_{vis},i)$ and $P_{hit}(E_{vis},i,q_i)$ are the probability that the $i_{th}$ PMT has no hit and has the total charge of $q_i$, respectively. This probability is calculated based on the knowledge of the detector response model, including the light yield of the liquid scintillator, the attenuation length of the liquid scintillator and the buffer, the angular response function of the PMT, and the PMT charge resolution, as well as the reconstructed vertex as input. Another fast reconstruction algorithm has also been developed, which is based on the total number of photoelectrons collected from all PMTs with corrections using the calibration sources.
\subsubsection{Muon tracking}
The muon tracking in the central detector is based on the PMT hit time. Since the energy deposit from cosmic muons is at the level of GeV, the average number of photoelectrons at each PMT will be greater than 100, therefore the first hit time of the PMTs is dominated by the fast component of the scintillation light, which approximately follows a Gaussian distribution. A $\chi^2$ function is defined:
\begin{equation}
\chi^2=\sum_i(\frac{t_i^{exp}-t_i^{mea}}{\sigma})^2,
\end{equation}
where $t_i^{exp}$ and $t_i^{mea}$ are the expected and measured first hit time of the $i_{th}$ PMT, respectively, $\sigma$ is the time resolution of the PMT, and $t_i^{exp}$ is calculated based on the tracking parameters and the optical model in the detector. As shown in Fig.~\ref{fig:rec:muon_tracking}, since the scintillation light is isotropic, the intersection angle $\theta$ can be calculated analytically. For muons passing through the detector, the PMTs around the injection point and the outgoing point see more light and form two clusters, which can be used to estimate the initial tracking parameters. A muon tracking algorithm combined with the veto detectors is still under development.
\begin{figure}[htb!]
\centering
\includegraphics[width=.5\textwidth]{OfflineSoftwareAndComputing/figures/muon_tracking.pdf}
\vspace{-0.15cm}
\caption{An illustration of the muon tracking. The red line is the muon track to be fitted, $R_c$ is the light source which provides the first hit to the PMT at $R_i$, determined by the intersection angle $\theta$. If $R_c$ is calculated to be outside the detector, it will be set as the injection point $R_0$.}
\label{fig:rec:muon_tracking}
\end{figure}
\subsection{Detector Simulation}
The detector simulation package for JUNO is based on Geant4. Detector simulation includes detector geometry description, physics processes, hit recording, and user interfaces.
\subsubsection{Description of detector geometry}
An accurate description of detector geometry is the foundation of detector simulation. In JUNO simulation, the geometry of the liquid scintillator, acrylic/nylon ball, PMTs, stainless steel tank or trusses need to be described in simulation software. About 17,000 PMTs are placed on a spherical surface. Geant4 is very sensitive to the geometry problem of volume overlaps or extrusions, which in general reduce the simulation speed or waste time in infinite loops. Some built-in or external tools can help to find such problems. At the stage of detector design, the detector geometry keeps changing. It is preferable that the geometry and event data can be exported in the same data file after simulation, so that the reconstruction job can read both information at the same time to ensure geometry consistency between simulation and reconstruction. In addition, the detector description method should allow easy modifications to the geometry with changes made to the detector design.
In simulation of the liquid scintillator detector, the simulation of optical photon processes is a key issue. The optical parameters for materials such as liquid scintillator, acrylic, mineral oil and PMT should be carefully defined. The optical parameters include refractive index, absorption length, light yield factor, quantum efficiency and so on. Each optical parameter varies with the wave-length of optical photons and may be time-dependent in the future. Due to the production technique, the parameters of each individual PMT may be different. How to effectively manage these optical parameters is an important issue in the detector data description. A possible method is to combine Geant4 with GDML/XML and database technology to manage the detector geometry and optical parameters.
In addition, simulation of the optical processes requires the definition of an optical surface, on which optical photons can have different behavior like refraction, reflection and absorption.
\subsubsection{Physics processes}
The physics processes in the JUNO detector simulation include standard electromagnetic, ion, hadronic and optical photon processes and optical model for the PMTs. Optical photon processes include scintillation, Cerenkov effect, absorption and Rayleigh scattering.
When a charged particle passes through liquid scintillator, ionization energy loss is converted to scintillation light. The non-linearity between scintillation light and the ionization energy loss is called the quenching effect. During the propagation of light in the liquid scintillator, the re-emission effect needs to be considered, which is not included in standard Geant4 processes and needs to be implemented in our experiment.
For such a large liquid scintillator detector as JUNO, the simulation of high energy muons in liquid scintillator can be too CPU-time consuming to be affordable. Reducing light yield or using a parameterized model for fast simulation are possible solutions to speed up the muon simulation.
\subsubsection{Hit recording}
The hit information, including the number of hits in each PMT and the hit time, is the output at the stage of Geant4 tracking. This information will be used as input to electronics simulation in the next step. In addition, particle history information should be saved, including the information of the initial primary particles, neutron capture time and position, energy deposition and position, direction and position of the hit on the PMT, and the relationship between track and hits.
\subsubsection{User interface}
A friendly user interface is necessary to use the simulation package, which should provide easy configuration of simulation parameters such as radius of LS volume, buffer thickness, vertex position, particle type and flags to switch on/off a specific physics process.
\section{Software Development and Quality Control}
The JUNO offline software system is composed of several sub-systems such as the framework, event generator, detector simulation, calibration, reconstruction, physics analysis and so on.
Each sub-system contains one or more packages,which are divided according to their functionalities. In principle, they are relatively independent in order to allow easy dynamic loading at the run time. The relationship between the necessary dependencies are implemented using the Configuration Management Tool (CMT) which is also used to compile each package.
Developers from different institutes and universities are collaborating together to write code in parallel. Subversion (SVN) \cite{svn} is deployed as the code repository and version control system to track the development of the offline software and tag it for releases. One central SVN server has been set up. All users and developers can easily check out the required release version and commit modifications according to their privileges.
Trac \cite{Trac} is an enhanced wiki and issue tracking system for software development projects. Trac and SVN can be used successfully working together to provide cross searching and references.
Some specific tools, testing algorithms/packages will be created with the development of the offline software in order to test and monitor its functionality and make sure that the whole software is going in the correct direction. Every time a new version is released, a complete test will be performed and a report given showing the results of some characteristic quantities, including CPU consumption and memory usage.
Several kinds of documentation, such as Wiki \cite{Wiki} pages and DocDB \cite{DocDB} will be provided.The Wiki pages will generally record the objectives and progress of each subsystem, while the DocDB will store technical documentation and status reports.
\subsection{Digitizer Simulation}
The digitizer simulation consists of electronics simulation, trigger simulation and readout simulation. The main goal is to simulate the real response of the three systems and apply their effects to physics results. The readout simulation transforms Monte Carlo data into the same format as experimental data and simplifies the analysis procedure. The digitizer simulation will be implemented in the JUNO offline framework, SNiPER.
Electronics simulation plays an important role in the digitizer simulation. The waveform from each PMT will be recorded using FADC and analyzed offline to get the final amplitude and timing information for each PMT. In electronics simulation, to transform the input hit information (given by Geant4) into electronic signals which are as real as possible, we need to build a model to exactly describe the single photo-electron response, which includes the effects of signal amplitude, rising time, pre-pulse, after-pulse, dark rate, etc. Electronics simulation should also have the ability to handle background noise, overlapping waveforms from multiple hits on the same PMT, and the effects from waveform distortion and non-linear effects. For very large signals like cosmic muons, however, it is difficult to save all FADC information so only integrated charge will be read out. At the current stage, since the detailed JUNO electronics design has not been finally determined, the electronics simulation imported from the Daya Bay experiment is implemented as a temporary substitute for the JUNO electronics simulation. It will be updated to the real case when the JUNO electronics design is finalized.
Trigger simulation takes the output from the electronics simulation as its input to simulate the real trigger logic and clock system, and decides whether or not to send a trigger signal if the current event passes a trigger. Currently, we have implemented a simple nHit trigger simulation.
Based on the trigger signals provided by the trigger simulation, FADC information in the readout window will be saved for each PMT and form an event. At the same time, the time stamp will be tagged. The final data format will be the same as for real data.
\subsection{Background mixing}
In real data, most events come from background, like natural radioactive events and cosmic muon induced events. In order to simulate the real situation, a background mixing algorithm needs to be developed to mix signal events with background events. It is an essential module if we want to make Monte Carlo data match well with real data. There are two options for background mixing: hit level mixing and readout level mixing. For hit level mixing, the hits from both signal and background are sorted by time first, and then handled by the electronics simulation. Hit level mixing is closer to the real case, but requires a lot of computing resources. Readout level mixing is much easier to implement and requires less computing resources, but it can not accurately model the overlapping between multiple hits. Since both options have advantages and disadvantages, more studies are necessary before an option is finally selected as the official JUNO background mixing algorithm.
\subsection{Event Display}
The event display is an indispensable tool in high energy physics experiments. It helps to develop simulation and reconstruction packages in the offline software, to analyze the physics in a recorded interaction and to illustrate the whole situation to general audiences in an imaginable way.
\subsubsection{Detector description and display}
The display of the detector is provided by the geometry service in the offline software. The visual attributes of every detector unit are associated and controlled by identifiers. The event display software will support two display modes -- 3D display and 2D projection display. The 2D projection will be realized by the 2D histograms that are provided by ROOT. For 3D display, several popular 3D graphic engines, like OpenGL \cite{OpenGL}, OpenInventor \cite{OpenInventor} and X3D \cite{X3D}, have been proven as practical choices for event display software for some other HEP experiments. More studies are necessary to determine which infrastructure 3D graphic engine will work best for JUNO and its software environment. Currently a solution based on OpenGL is under study to realize the 3D display mode.
\subsubsection{Event data and display}
The event data in the offline system has various formats, including raw data, simulation data, calibrated data, reconstruction data and data analysis results. Their data formats are uniformly defined by the JUNO offline event model. In processing the event data for display, we need to study how to display the particle trajectories, their interactions with materials and hit response in simulation data, the reconstructed event vertices, energy and display attributes, as well as the relationships between these.
\subsubsection{Graphical User Interface}
A Graphical User Interface (GUI) provides a convenient interface for users to execute complicated commands with simple actions such as pushing buttons, selections, sliding a ruler or pushing hot keys. Frequently used functions, like event control, 3D view rotation and translation, zooming and time controls can be realized through widgets. A ROOT-based GUI framework has been implemented to realize these functions, giving the advantages of seamless binding with the ROOT analysis platform, ROOT-based geometry service and event model.
\subsubsection{Graphic-based analysis and tuning}
Graphic-based analysis will further provide some useful functions such as reconstruction algorithm tuning, event trigger and event type discrimination, which can help users to quickly filter, analyze and understand the events in which they are interested.
\subsection{Event Model}
The event model defines the key data units to be processed in the offline data processing. It not only defines the information included in one event at different data processing steps such as event generation, detector simulation, calibration, reconstruction and physics analysis, but also provides the ways for events to be correlated between different processes.
Since ROOT \cite{ROOT} has been extensively used as the data storage format in most particle physics experiments, it is decided that the JUNO event model is based on the ROOT TObject. So all the event classes are inherited from TObject. In this way, we can directly take advantage of the powerful functionalities provided by ROOT, such as schema evolution, input/ouput streamer, runtime type information (RTTI), inspection and so on.
In order to quickly view event information and make the decision whether or not a given event is selected, the event information is divided into two levels: Event Header and Event. The Event Header defines the sequential information (such as RunId, EventId and Timestamp etc.) and characteristic information (such as energy of events), while Event defines more detailed information about the event. In this way, the event header can be read in from data files or other storage and its characteristic information can be used for fast event selection without taking extra CPU time or memory required to read in the contents of a full event.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.8\textwidth]{OfflineSoftwareAndComputing/figures/DataModel_04_02.png}
\caption[Event Model]{\label{fig:EventModel} Design Schema of JUNO Event Model}
\end{center}
\end{figure}
Event correlations are very important for neutrino physics analysis since inverse beta decay (IBD) events will be regarded as two separate event signals: prompt signal and delayed signal. Both of them actually refer to the same physics event. In the JUNO experiment, we design SmartRef to fully meet this requirement and to avoid the duplication in writing these information into ROOT files. SmartRef is based on the TRef class in ROOT but has more powerful functions. It not only provides correlations between events but also provides the information for lazy loading of the data.
For the JUNO event model, the class EvtNavigator is designed to organize all the event information at different processing steps and acts as an index when processing events. In Figure \ref{fig:EventModel}, we can see that EvtNavigator has several SmartRefs which refer to the headers of this event at different processes. Event model classes are usually simple. Only some quality variables and functions need to be defined. These functions can be divided into three types: Set-functions to set new values, Get-functions to return the variable's value, and streamer functions for reading/writing data from/to files. Most of the code is similar event by event, so an XOD (Xml Object Description) tool is developed and used to describe the event information with a more readable XML file. XOD can automatically produce the event header file, the event source file and the dictionary file needed by ROOT.
\subsection{Generator}
Generators produce simulated particles with desired distribution of momentum, direction, time, position and particle ID.
The generators for the JUNO experiment include inverse beta decay (IBD) generator, radioactivity generator, muon generator, neutron generator, and calibration source generator. Some of the generators can be imported from the existing Daya Bay software, while the others need to be developed for JUNO specifically. The generators are grouped into three kinds according to their development platform. One is based on Geant4 ParticleGun for shooting particles with specified particle ID, momentum and position distribution. The second kind, such as ${}^{238}U$ and ${}^{232}Th$ radioactivity generators, is developed with existing Fortran libraries. The third kind of generators, such as IBD and GenDecay, will be developed in C++ language,. The output data of generators should be in the format of HepEvt or HepMC \cite{HepMC} for better communication with other applications. The design of particle vertex position generation need to be flexible and allow the distribution to be geometry related, for example, a random distribution on the surface of a PMT, in liquid scintillator or in acrylic.
The muon generator particularly depends on the muon flux, energy spectrum and angular distribution at the JUNO site. A digitized description of the actual landform at the JUNO site need to be acquired first, and then used as input to MUSIC \cite{MUSIC} to simulate muons passing through the rock, getting the flux, energy spectrum and angular distribution of the muons finally reaching the JUNO detector. The effect of muon bundles also needs to be considered for such a large detector as JUNO.
\subsection{Geometry}
The geometry system describes and manages the detector data and structure in software, providing a consistent detector description for applications like simulation, reconstruction, calibration, alignment, event display and analysis. All subsystems, including the Central Detector, Top Veto Tracker, Water Cherenkov Pool and the Calibration system, will be described in the geometry system. The design of the JUNO detector also requires the geometry system to be flexible and able to handle time-dependent geometry and the co-existence of different designs.
Several detector description tools have been developed and implemented in high energy physics experiments, with some of the most popular languages being XML (eXtensible Markup Language) \cite{XML}, HepRep \cite{HepRep}, VRML(Virtual Reality Modeling Language, updated to X3D) \cite{VRML} \cite{X3D} and GDML (Geometry Description Markup Language) \cite{GDML}. At the stage of conceptual design for the JUNO detector, the detector description geometry is based on GDML, including geometry data, detector structure, materials and optical parameters. In consideration of the co-existence of multiple major designs and frequent minor modifications to the designs, we adopt GDML as the main tool to transfer detector description data between different applications, because GDML requires less human work and has the advantage of automatic translation of detector data between Geant4 \cite{Geant4} and ROOT, the two most popular HEP programs, which are widely used in simulation, reconstruction and analysis. However, it is noteworthy that GDML also has some limitations. For example, its Geant4 and ROOT interfaces are not completely consistent, so some information like optical surface description and matrix elements may be lost in the detector data translation. The GDML interfaces do not support complicated (user defined) shapes and require more work to implement the extensible functions. These small problems need to be solved in future geometry software development.
At the current stage, the detector geometry is described in each individual Geant4 simulation code, which also serves as the unique source of detector description. When a simulation job is running, it constructs the detector structure in Geant4 and exports it in GDML format; at the end of the simulation job, the GDML detector data is read back and converted into ROOT geometry object format, which is bundled with the output event data ROOT file while being written out. When another application such as reconstruction or event display wants to read in the geometry information, it searches the bundled ROOT file for the geometry object to initialize the geometry offline, and uses the geometry service package to retrieve the detector unit information it requires. This mechanism gives every detector designer the maximum flexibility to modify his specific design, uses a single module to handle multiple detector designs at the same time and guarantees the consistency of detector description between different applications.
In the future, when the detector design is fixed, it is preferred to generate the detector description within the geometry system and provide it to all applications through interfaces, rather than the current solution of generating geometry data in Geant4.
\subsection{Maintenance and service support}
\subsubsection{Network support platform}
Network support platform provides professional and comprehensive solutions for JUNO network and public services. On the one hand, it integrates related resources on the network side, and then transfers them to the expert of the user service, which can realize quick response to network complaints, and effectively raise user service front-end settlement ratio, reduce complaint processing links, reduce complaint processing duration, and enhance user satisfaction.On the other hand,as for the information related to network complaint received from user sides, the network side will obtain it automatically and pay active attention to it, and realize the comprehensive correlation between user complaint and network execution situation, and the quality degradation warning of terminal-to-terminal network, so as to guide network optimization, enhance network quality, and further realize active care for users and enhance service quality.
\subsubsection{Fine-grained monitoring and maintenance}
Since JUNO data process is a large scale task including many devices, a fine-grained monitoring system is necessary to guarantee smooth running of the platform. It is required that the monitoring system be real-time, easy to use and can recover from some unexpected errors itself. For those errors which can not be recovered, a warning should be sent in time.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.8\textwidth]{OfflineSoftwareAndComputing/figures/monitor_arc.png}
\caption[Monitoring tool architecture]{\label{fig:monitor}Fine-grained monitoring architecture }
\end{center}
\end{figure}
\subsubsection{ Information Service Platform}
The information Service Platform is an information system which provides services for the research and management activities of the JUNO experiment. The Information Service Platform will provide system services for many aspects of the experimental research and management, such as the JUNO Experiment website, wiki, conference management, document management, collaboration management, video conference, workshop live broadcasting/recording, remote shifts, personal data sharing, science advocacy and so on.
\subsection{Software Platform}
The offline software system is designed to meet the various requirements of data processing. As JUNO software developers are dispersed all over the world, a unified software platform is required, which provides a formal working environment. This has advantages for resource optimization and manpower integration, which can improve the software development, usage and maintenance.
Based on our experience, Linux OS and GNU compiler are the first choice for the JUNO software platform. However other popular OSes and development environments should be considered for the sake of compatibility. The CMT \cite{CMT} tool, which can calculate package dependencies and generate Makefiles, is used for the software configuration and management. An automated installation tool can ease the deployment of the JUNO software, and is also helpful for daily compilation and testing. Users are able to concentrate on the implementation of software functionality, without suffering from different development environments.
Nowadays, mixed programming with multiple languages is practical, so we can choose different technologies for different parts of our software. The main features of the application are implemented via C++ to guarantee efficiency. User interfaces are provided in Python for additional flexibility. Boost.Python, as a widely used library, is a good choice to integrate C++ and Python. If it is included properly at the beginning of the system design, most users will be able to enjoy the benefits of mixed programming without knowing the details of Boost.Python.
We will implement a software framework, SNiPER, as the core software of the JUNO platform. SNiPER stands for "Software for Non-collider Physics ExpeRiments". As shown in Fig.\ref{fig:OfflinePlatform}, components of the JUNO software, such as simulation and reconstruction, are executed as plug-ins based on SNiPER. The principle of this new framework is simplicity and high efficiency. It depends on a minimal set of external libraries, fully meeting our requirements without over-engineering or loss of efficiency.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.8\textwidth]{OfflineSoftwareAndComputing/figures/Offline_Software_Platform.png}
\caption[Offline Software Platform]{\label{fig:OfflinePlatform}The structure of the offline software system}
\end{center}
\end{figure}
The SNiPER kernel is the foundation of the JUNO software system, and determines the vitality of the whole platform. We design and manage SNiPER modularly. Every functional element is implemented as a module, and can be dynamically loaded and configured. Modules are designed as high cohesion units with low couplings between each other. They communicate only through interfaces. We can replace or modify one module without affecting any of the others. Since it's flexible and expandable, users can contribute to it independently.
The model is inspired by other pioneering software frameworks such as Gaudi \cite{Gaudi}. Expandable modules are strategically distinguished as algorithms and services. An algorithm provides a specific procedure for data processing. A sequence of algorithms composes the whole data processing chain. A service provides useful features that can be called by users when necessary. Algorithms and services are plugged and executed dynamically. They can be selected and combined flexibly for different requirements.
Data to be processed may come from different places and be in different types and formats. The results may also be stored in different ways. In the framework we reserve interfaces for multi-stream I/O support, so that the I/O service can communicate conveniently with other modules via the interface. At present, the disk access could be a serious bottleneck in massive data processing. This should be considered in I/O service developing. Many techniques, such as lazy loading, are adopted to optimize the I/O efficiency.
The framework also involves many frequently used functions, such as the logging mechanism, particle property lookup, system resource loading, database access and histogram booking, etc. These contents are wrapped in services in SNiPER, which will ease the development procedure. Users can configure a job interactively with the Python command line or in batch mode with script files. The flexibility of Python will ease the job execution procedure.
The JUNO experiment has a long lifetime and software technology is still evolving rapidly, so the framework has to be very expandable. It should be able to integrate with new tools and libraries in future. This will save the manpower needed for developing, and improve the system robustness.
It is expected that JUNO will have a huge amount of data. In order to make use of all possible computing resources, it is necessary to consider parallel computing techniques. Modern CPUs are generally multi-core. Multi-threaded programming extracts the capacity of multi-core CPUs, and speeds up a single job significantly. In PC clusters, a server/client distributed system can be implemented and deployed, too. The latest computing technology and HEP computing techniques, including GPUs, Grid and Cloud, should be explored and used as appropriate.
\subsection{Requirement Analysis}
The offline software plays an important role in improving physics analysis quality and efficiency. From the point of view of physics analysis, the following requirements need to be applied to the offline software system.
\begin{itemize}
\item Compared to collider experiments, neutrino experiments have two specific characteristics: 1). There is time correlation between events. 2).There is only a very small fraction of signal events among a large number of backgrounds. The software framework should therefore provide a mechanism of flexible data I/O and event buffering to enable high efficiency data access and storage, as well as the capability to retrieve the events within a user-defined time window.
\item An interface should be provided for different algorithms to interchange event data. There should be a unified geometry management. Software and physics analysis parameters should be managed by a conditional database. An interface to retrieve correct parameters according to software versions should also be provided.
\item There are many kinds of events in the JUNO experiment. Apart from reactor neutrino events, they are supernova neutrinos, geo-neutrinos, solar neutrinos, atmospheric neutrinos, radioactivity etc. The offline software should provide accurate simulations to describe interactions of neutrinos and background events with the detector.
\item The detector simulation software should be able to simulate detector performance and guide detector design. In addition, the calculation of detector efficiencies and systematic errors, which are needed by physics analysis, also relies on detector simulation.
\item The detector performance depends on event reconstruction. In order to get better energy resolution, optical model in liquid scintillator detector should be constructed, using PMT charge and time information to get event vertex and energy. Event vertex can also be used to veto the natural radioactivity background events from outside the liquid scintillator. To reduce the isotope background from muons, accurate muon track reconstruction is needed. For atmospheric neutrinos, the reconstruction of short tracks inside the liquid scintillator and identification of different charged particles are needed.
\item Event display software is needed in order to show the detector structure, physics processes in the detector, and reconstruction performance.
\end{itemize}
The JUNO detector will produce about 2 PB raw data every year, which will be transferred back to the Computing Center at the Institute of High Energy Physics (IHEP) in Beijing through a dedicated network connection. Due to the large data volume, a big scale of offline computing platform is required by Monte Carlo production, raw data processing and physics analysis, as well as data storage and archiving. A rough estimation suggests a level of 10,000 CPU cores, 10 PB disk storage and 30 PB archive in the future. The computing nodes and storage servers will be connected to each other by a 40-Gbps backbone high-speed switching network. In order to improve data processing capacity, the platform will integrate the computing resources contributed by outside members via a distributed computing environment. So the experiment data can be shared among collaboration members and computation tasks can be dispatched to all of the computing sites managed by the platform. Considering the above needs, requirements on data processing, storage, data transfer and sharing, can be summarized as follows:
\paragraph{Data Transfer}
A data transfer and exchange system with high performance, security and reliability is required. To guarantee the timely transfer of raw data from the experiment site to the IHEP data center, the system should monitor and trace the online status of data transfers and it should provide a data transfer visualization interface. This should also guarantee data is dispatched and shared among different data sites within the collaboration smoothly.
\paragraph{Computing Environment}
A computing farm with 10,000 CPU cores will be established at the IHEP Computing Center to facilitate successful processing of the JUNO data. The collaboration members will provide additional computing resources. The computing environment includes not only the local computing farm, but also those remote resources contributed by outside collaboration members.
\paragraph{Data Storage}
JUNO will generate a huge amount of experiment data. Data from the experiment site will be transferred and stored at the IHEP Computing Center. The capacity of storage at IHEP will be 10 PB for storing both real and simulated data. It should support concurrent access by applications running on 10,000 CPU cores. Data migration between tape repository and disk pools should be transparent to users.
\paragraph{Maintenance, Monitoring and Service Support}
To improve the stability of the JUNO computing resources, an intelligent monitoring system will be developed. With this system, the status of both platform and network connection can be monitored through a fine-grained control. The monitor system can not only give the status assessment of the whole platform but also give predictions and prior warnings for various platform services. A set of information management tools should be developed including conference support, document management and online-shift management etc.
\subsection{Introduction}
The offline software and computing system consists of two separate parts: offline software and computing. It not only closely connects data processing and physics analysis, but also it can be regarded as a bridge between detector operations and physics analysis. The primary tasks of this system are to process raw data collected by the detector and produce reconstructed data, to produce Monte Carlo data, to provide software tools for physics analysis, and to provide the networking and computing environment needed by data processing and analysis.
The offline software includes software framework, event generator, detector simulation, event reconstruction and event display etc. The framework is the underlying software supporting the whole system. Based on this framework, experiment related software will be developed such as software for event data model, data I/O, event generation, detector simulation, event reconstruction, physics analysis, geometry service, event display, and database service etc. To facilitate the development, the infrastructure of offline software provides various useful tools such as tools for compiling, debugging, deployment and installation etc.
The offline computing is designed to build a high efficiency and reliable computing environment for the JUNO experiment. It has the following major parts: a data transfer platform to transfer raw data from the experiment site to the Computing Center at IHEP, a large scale of computing farm for data processing and data analysis, a massive data storage subsystem and a monitoring and management platform providing fine-grained monitoring and management of the computing resources.
\chapter{Assembly, Installation and Engineering Issues}
\label{ch:AssemblyAndInstallation}
\input{AssemblyAndInstallation/chapter12.1.tex}
\input{AssemblyAndInstallation/chapter12.2.tex}
\input{AssemblyAndInstallation/chapter12.3.tex}
\input{AssemblyAndInstallation/chapter12.4.tex}
\input{AssemblyAndInstallation/chapter12.5.tex}
\section{Introduction}
After the completion of civil construction, through systems integration, assembly, installation and commissioning, all subsystems will be formed as a whole JUNO detector.
In this process, we need to establish various engineering
regulations and standards, and to coordinate subsystems' assembly, installation, testing and commissioning, especially their onsite work.
\subsection{Main Tasks}
Main tasks of the integration work include:
\begin{itemize}
\item To prepare plans and progress reports of each phase;
\item To establish a project technology review system;
\item To standardize the executive technology management system;
\item To have strictly executive on-site work management system;
\item To develop and specify security management system on-site;
\item To prepare common tools and equipment for each system, and to guarantees project progress;
\item To coordinate the installation progress of each system according to the on-site situation.
\end{itemize}
\subsection{Contents}
The work contents mainly include:
\begin{itemize}
\item To summarize the design, and review progress of each subsystem;
\item To organize preparation work for installation in the experiment region;
\item To inspect and certify Surface Buildings, underground Tunnels, and Experiment Hall with relevant utilities;
\item To coordinate technology interfaces between key parts;
\item To coordinate the procedure of assembly and installation both on surface and underground;
\end{itemize}
\section{ Design Standards, Guidelines and Reviews}
\subsection{Introduction}
There will be a document to outline the mechanical design standards or guidelines that will be applied to the design work. It also describes the review process of engineering design that will be implemented to ensure that experimental equipment meets all requirements for performance and safety. The following is a brief summary of the guidance that all mechanical engineers/designers should follow in the process. For reasons ranging from local safety requirements to common sense practices, the following information should be understood and implemented in the design process.
\subsection{Institutional Standards}
When specific institutional design standards or guidelines exist, they should be followed. The guidelines outlined are not meant to replace but instead to supplement institutional guidelines. The majority of equipment and components built for the JUNO Experiment will be engineered and designed at home institutions, procured or fabricated at commercial vendors, then eventually delivered, assembled, installed, tested and operated at the JUNO Experimental facilities, Jiangmen, China. The funding agencies in other countries as well as Jiangmen, China will have some guidelines in this area, as would your home institutions. Where more than one set of guidelines exist, use whichever are the more stringent or conservative approaches.
\subsection{Design Loads}
The scope of engineering analysis should take handling, transportation, assembly and operational loads into account as well as thermal expansion considerations caused by potential temperature fluctuations.
For basic stability and a sensible practice in the design of experimental components an appropriate amount of horizontal load (0.10~g) should be applied.
In addition, seismic requirements for experimental equipment is based on a National Standard of People's Republic of China Code of Seismic Design of Buildings GB 50011-2010. At the location of the JUNO Experiment the seismic fortification intensity is grade 7 with a basic seismic acceleration of 0.10~g horizontal applied in evaluating structural design loads. A seismic hazard analysis should be performed and documented based on this local code. The minimum total design lateral seismic base shear force should be determined and applied to the component. The direction of application of seismic forces would be applied at the CG of the component that will produce the most critical load effect, or separately and independently in each of the two orthogonal directions. A qualitative seismic performance goal defines component functionality as: the component will remain anchored, but no assurance it will remain functional or easily repairable. Therefore, a seismic design factor of safety F.S. > 1 based on the Ultimate Strength of component materials would satisfy this goal.
Where an anticipated load path as designed above, allows the material to be subjected to stresses beyond the yield point of the material, redundancy in the support mechanism must be addressed in order to prevent a collapse mechanism of the structure from being formed.
The potential for buckling should also be evaluated. It should be
noted that a rigorous analytical seismic analysis may be performed
in lieu of the empirical design criteria. This work should be
properly documented for review by the Chief Engineer and appropriate
Safety Committee personnel.
\subsection{Materials Data}
All materials selected for component design must have their engineering data sources referenced along with those material properties used in any structural or engineering analysis; this includes fastener certification. There are many sources of data on materials and their properties that aid in the selection of an appropriate component material. There are many national and international societies and associations that compile and publish materials test data and standards. Problems frequently encountered in the purchasing, researching and selection of materials are the cross-referencing of designations and specification and matching equivalent materials from differing countries.
It is recommended that the American association or society standards be used in the
materials selection and specification process, or equivalency to these
standards must be referenced. Excellent engineering materials data have been
provided by the American Society of Metals or Machine Design Handbook Vol. 1-5 and
Mechanical Industry Publishing Co. in PRC, Vol 1-6, 2002, which are worth investigating.
\subsection{Analysis Methods}
The applicable factors of safety depend on the type of load(s) and how well they can be estimated, how boundary conditions have been approximated, as well as how accurate your method of analysis allows you to be.
\paragraph{Bounding Analyses:}
Bounding analysis or rough scoping analyses have proven to be
valuable tools. Even when computer modeling is in your plans, a
bounding analysis is a nice check to avoid gross mistakes. Sometimes
bounding analyses are sufficient. An example of this would be for
the case of an assembly fixture where stiffness is the critical
requirement. In this case where deflection is the over-riding
concern and the component is over-designed in terms of stress by a
factor of 10 or more, then a crude estimation of stress will
suffice.
\paragraph{Closed-Form Analytical Solutions:}
Many times when boundary conditions and applied loads are simple to
approximate, a closed-form or handbook solution can be found or developed. For
the majority of tooling and fixture and some non-critical experimental
components, these types of analyses are sufficient. Often, one of these
formulas can be used to give you a conservative solution very quickly, or a
pair of formulas can be found which represent upper and lower bounds of the
true deflections and stresses. Formulas for Stress and Strain by Roark and
Young is a good reference handbook for these solutions.
\paragraph{Finite Element Analysis:}
When the boundary conditions and loads get complex, or the correctness of the solution is critical, computer modeling is often required. If this is the case, there are several rules to follow, especially if you are not intimately familiar with the particular code or application.
\begin{enumerate}
\item Always bound the problem with an analytical solution or some other approximate means.
\item If the component is critical, check the accuracy of the code and application by modeling a similar problem for which you have an analytical or handbook solution.
\item Find a qualified person to review your results.
\item Document your assumptions and results.
\end{enumerate}
\subsection{Failure Criteria}
The failure criterion depends upon the application. Many factors such as the rate or frequency of load application, the material toughness (degree of ductility), the human or monetary risk of component failure as well as many other complications must be considered.
Brittle materials (under static loads, less than 5\% yield prior to failure), includes ceramics, glass, some plastics and composites at room temperature, some cast metals, and many materials at cryogenic temperatures. The failure criterion chosen depends on many factors so use your engineering judgment. In general, the Coulomb-Mohr or Modified Mohr Theory should be employed.
Ductile materials (under static loads, greater than 5\% yield prior to failure), includes most metals and plastics, especially at or above room temperature. The failure criterion chosen again ultimately rests with the cognizant engineer because of all the adverse factors that may be present. In general, the Distortion- Energy Theory, or von Mises-Hencky Theory (von Mises stresses), is most effective in predicting the onset of yield in materials. Slightly easier to use and a more conservative approach is the Maximum-Shear-Stress Theory.
\subsection{Factor of Safety}
Some institutions may have published guidelines which specifically discuss factors of safety for various applications. For the case where specific guidelines do not exist, the following may be used.
Simplistically, if F is the applied load (or S the applied stress),
and $F_f$ is the load at which failure occurs (or $S_s$ the stress
at which failure occurs), we can then define the factor of safety
(F.S.) as:
\begin{displaymath}
F.S. = F_{f} / F \quad\mathrm{or}\quad S_{s} / S
\end{displaymath}
The word failure, as it applies to engineering elements or systems, can be defined in a number of ways and depends on many factors. Discussion of failure criteria is presented in the previous section, but for the most common cases it will be the load at which yielding begins.
\subsection{Specific Safety Guidelines for JUNO}
Lifting and handling fixtures, shipping equipment, test stands, and fabrication
tooling where weight, size and material thickness do not affect the physical
capabilities of the detector, the appropriate F.S. should be at least 3. When
life safety is a potential concern, then a F.S. of 5 may be more appropriate.
Note that since the vast majority of this type of equipment is designed using
ductile materials, these F.S.'s apply to the material yield point. Experimental hardware that does not present a life safety or significant cost/schedule risk if failure occurs, especially where there is the potential for an increase in physics capabilities, the F.S. may be as low as 1.5. Many factors must be taken into account if a safety factor in this low level is to be employed: a complete analysis of worst case loads must be performed; highly realistic or else conservative boundary conditions must be applied; the method of analysis must yield accurate results; reliable materials data must be used or representative samples must be tested. If F.S.'s this low are utilized, the analysis and assumptions must be highly scrutinized. Guidelines for F.S. for various types of equipment are:
\begin{center}
\begin{tabular}{|p{3.5cm}|c|p{3.5cm}|}
\hline
Type of Equipment & Minimum F.S. & Notes \\
\hline
Lifting and handling & 3 - 5 & Where there is a risk to life
safety or to costly hardware,
choose F.S closer to 5. \\
\hline
Test stands, shipping and assembly fixtures. & 3 & \\
\hline
Experimental hardware & 1.5 - 3 & 1.5 is allowable for physics
capability and analysis where
method is highly refined \\
\hline
\end{tabular}
\end{center}
\subsection{Documentation}
It is not only good engineering practice to document the analysis, but it is an
ESH\&Q requirement for experimental projects. For this reason all major
components of JUNO Experiment will have their engineering analyses documented as Project Controlled Documents. Utilize your institutional documentation formats or use the following guidelines.
Calculations and analyses must
\begin{itemize}
\item Be hard copy documented.
\item Follow an easily understood logic and methodology.
\item Be legible and reproducible by photocopy methods.
\item Contain the following labeling elements.
\begin{itemize}
\item Title or subject
\item Originators signature and date
\item Reviewers signature and date
\item Subsystem WBS number.
\item Introduction, background and purpose.
\item Applicable requirements, standards and guidelines.
\item Assumptions (boundary conditions, loads, materials properties, etc.).
\item Analysis method (bounding, closed form or FEA).
\item Results (including factors of safety, load path and location of critical areas).
\item Conclusions (level of conservatism, limitations, cautions or concerns).
\item References (tech notes, textbooks, handbooks, software code, etc.).
\item Computer program and version (for computer calculations)
\item Filename (for computer calculations).
\end{itemize}
\end{itemize}
\subsection{Design Reviews}
All experimental components and equipment whether engineered or
procured as turn-key systems will require an engineering design
review before procurement, fabrication, assembly or installation can
proceed. The L2 subsystem manager will request the design review
from the project management office, which will appoint a chair for
the review and develop a committee charge. The selected chair should
be knowledgeable in the engineering and technology, and where
practical, should not be directly involved in the engineering design
effort. With advice from the L2, the technical board and project
office chair appoints members to the review committee that have
experience and knowledge in the engineering and technology of the
design requiring review. At least one reviewer must represent ESH\&Q
concerns. The committee should represent such disciplines as:
\begin{itemize}
\item Science requirements
\item Design
\item Manufacturing
\item Purchasing
\item Operational user
\item Maintenance
\item Stress analysis
\item Assembly/installation/test
\item Electrical safety
\item ESH\&Q
\end{itemize}
For the JUNO project there will be a minimum of two engineering design reviews: Preliminary and Final. The preliminary usually takes place towards the end of the conceptual design phase when the subsystem has exhausted alternative designs and has made a selection based on value engineering. The L2 along with the chair ensures that the preliminary design review package contains sufficient information for the review along with:
\begin{itemize}
\item Agenda
\item Requirements documents
\item Review committee members and charge
\item Conceptual layouts
\item Science performance expectations
\item Design specifications
\item Supportive R\&D or test results
\item Summaries of calculations
\item Handouts of slides
\end{itemize}
A final design review will take place before engineering drawings or specifications are released for procurement or fabrication. The L2 along with the chair ensures that the final design review package contains sufficient information for the final review along with:
\begin{itemize}
\item Changes or revisions to preliminary design
\item Completed action items
\item Final assembly and detailed drawings
\item Final design specifications
\item Design calculations and analyses
\end{itemize}
The committee chair will ensure that meeting results are taken and recorded along with further action items. The committee and chair shall prepare a design review report for submittal to the project office in a timely manner which should include:
\begin{itemize}
\item Title of the review
\item Description of system or equipment
\item Copy of agenda
\item Committee members and charge
\item Presentation materials
\item Major comments or concerns of the reviewers
\item Action items
\item Recommendations
\end{itemize}
The project office will file the design review report and distribute copies to all reviewers and affected groups.
\section{On-site management}
According to the experience from the Daya Bay Experiment, an effective technology review system has been well practiced. In JUNO, we will take it as a good reference and carry out for standardized review from the start of all system design, scheme
argument, technology checks, we should also establish management system, to cover engineering drawing control, engineering change control procedure, and mechanical design standards, guidelines and reviews, etc.
To control on-site work process, proper process should be introduced as well as a management structure. Safety training and safety management structure should be also introduced.
\section{Equipment and Tools Preparation on Site}
All of key equipment, devices and tools should be in place and get acceptance
for installation on site, including
\begin{itemize}
\item Cranes:
\begin{itemize}
\item 2 sets of bridge crane with 12T/5T load capacity, and lifting range should
cover any point for components delivery and installation in EH;
\item Several lifting equipment are needed in Surface Assembly areas, Assembly
Chamber underground, Storage area for Liquid Scintillation, and other Chambers
accordingly;
\end{itemize}
\item Removable lifting equipment: Forklifts, manual hydraulic carriages, carts,
etc.
\item Mobile vehicle for smaller items: Pickup van, battery car, and so on;
\item Several work aloft platform: Scissor lifts, Boom lifts, Scaffoldings and
Floating platforms, etc.
\item Common Tooling: complete kits of tool, fixtures, jigs, models, work-bench,
cabinets, and tables, etc.
\item Machine shop, Power equipment rooms, Control rooms are ready to be put into
use.
\item Safety measures and relevant equipment in place.
\end{itemize}
\section{Main Process of Installation}
Since there still are several optional design versions which could
be decided later, and different design could have different
requirements for installation, therefore, the specific installation
procedures will be established and developed later, here a very
rough installation procedure is given.
\begin{itemize}
\item Mounting, debugging, testing of basic utilities, such as, cranes, vertical
elevator, scaffoldings, test devices, etc.
\item Survey, alignment, and adjustment for tolerances of form and position, include,
embedded inserts, anchor plates, positioning holes, etc.
\item Mounting Bottom Structure of Veto Cherenkov detector on the floor of Water Pool.
\item Mounting anchor-plate for support poles of CD on the floor of Water Pool
\item Pre-mounting of Top Structure of Veto system along wall of Water Pool
\item Mount Substrate and Rails for Bridge over the water Pool edges
\item Mount Tyvek with support structure for Water Cherenkov Detector, and with PMT
positioning in the Water Pool
\item Installation for CD with its PMT system and cabling in Water Pool details will be established until design final selection.
\item Mount Calibration System
\item Mount Geomagnetic Shielding system around Central Detector in the Pool
\item Final installation for Tyvek and PMT of Veto system in the Pool
\item Mount a temporary protective cover on top of Pool, to prevent from anything
falling into Pool
\item Mount Bridge on the rails, and drive it closed
\item Mount Top Track Detector on Bridge and establish Electronic Room there
\item Have Cabling completion, and test to get normal signals
\item Check Grounding Electrode
\item Dry Run
\item Final Cleaning for Water Pool, then, drain out
\item Filling Water, LS, LAB (or Oil) evenly into Water Pool and CD
\item Mount Final Pool Cover, closed, and get fixation
\end{itemize}
\chapter{AssemblyAndInstallation}
\label{ch:AssemblyAndInstallation}
\input{AssemblyAndInstallation/chapter12.1.tex}
\input{AssemblyAndInstallation/chapter12.2.tex}
\input{AssemblyAndInstallation/chapter12.3.tex}
\input{AssemblyAndInstallation/chapter12.4.tex}
\input{AssemblyAndInstallation/chapter12.5.tex}
\section{Design Goals and Specifications}
\subsection{Introduction}
Photon detectors measure the scintillation light created by interactions of the
neutrinos with liquid scintillator and are key components for accomplishing the
physics goals of JUNO. Important requirements for photon detectors used in JUNO
include high detection efficiency for scintillation photons, large area, low
cost, low noise, high gain, stable and reliable, long lifetime. JUNO plans to
use approximately 20,000 tons or 23175~m$^{3}$ liquid scintillator hosted in a
spherical container of diameter 35.4~m. Photon detectors will be mounted on a
sphere 40~m in diameter covering at least 75\% of the 4654~m$^{2}$ surface area
of the sphere according to the reference design of the central detector. Large
area vacuum photomultipliers are the only viable choice for the JUNO photon detectors.
Table~\ref{tab:PMTsrequiredbyJUNO} shows the numbers and sizes of PMTs required for JUNO in two scenarios with photocathode coverage ratio of 75\% and 70\%. Assuming 508~mm (20~inch) diameter PMTs with a photocathode diameter 500~mm are used, $\sim$17,000 such PMTs are required by the central detector of JUNO to cover 75\% of the detector surface.
\begin{table}[!hbp]
\centering
\caption{PMTs required by JUNO\label{tab:PMTsrequiredbyJUNO}}
\begin{tabular}{|c|c|c|}
\hline
\hline
PMT mounting diameter & photocathode coverage & 508 mm \\
\hline
40 m & 70\% & 16,000 \\
\hline
40 m & 75\% & 17,000 \\
\hline
\end{tabular}
\end{table}
\subsection{PMT Spectications}
The main specifications for the 508~mm PMTs in JUNO are given in
Fig.~\ref{fig:specificationOfPMTs}. Bialkali (BA) photocathode will be used. The
spectral response of the BA photocathode matches the light emission spectrum of
the liquid scintillator well. One of the most critical requirements for JUNO
PMTs is high photon detection efficiency (PDE), which is equal to the photocathode
quantum efficiency (QE) multiplied by the efficiency of photoelectron
collection efficiency (PCE) in the PMT vacuum. In order to achieve the
required peak PDE of 35\%, the peak photocathode QE is set to 38\% by assuming
the PCE to be 93\%. We hope the average PDE for scintillation photons,
which have a rather broad spectrum, can be > 30\%.
Note that PMTs are measured by manufacturers in air and with incident light
beams perpendicular to the surface of the glass entrance while PMTs are
submerged either in water or in oil when used in JUNO, and the scintillation
light can penetrate the curved glass window of a PMT at any angle. Small
discrepancies regarding the photocathode QE between the manufacturers,
measurements and those experimentally observed can occur as a result of
differences in light reflections at the medium/glass interface.
JUNO PMTs will work in photon counting mode and the typical gain required is
$10^{7}$. The PMT noise pulse level must be low so that the single
photoelectron peak can be clearly separated from the PMT noise. In JUNO cosmic
events the minimum momentum of cosmic ray muons is 218~GeV and the penetrating
muons' rate is 3.8~Hz. Energy deposition by muons that do not generate showers
would be as high as 7~GeV and a total of $7\times10^{7}$ scintillation photons
would be produced. The required dynamic range of JUNO PMTs is given as 0.1-4000~pe's.
The photocathode noise rate, rate of prepulse and after-pulses are given in
Fig.~\ref{fig:specificationOfPMTs}. The radioactivity levels of the PMTs,
which mostly comes from the glass, are also given.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.1.png}
\caption[ Specifications of PMTs for JUNO]{\label{fig:specificationOfPMTs} Specifications of PMTs for JUNO }
\end{center}
\end{figure}
In addition to the 508~mm diameter PMTs for measuring scintillation light
generated by neutrino reactions in the detector, approximately 2000 of
203~mm (8~inch) diameter PMTs are needed for cosmic ray muon veto by detecting
Cherenkov light generated by cosmic muons. These PMTs will be mounted on the
outside surface of the stainless steel structure that supports the liquid
scintillator vessel. The total area to be covered is approximately
5000~m$^{2}$. The requirements for these PMTs are similar to those used in the
Daya Bay neutrino detector.
\section{MCP-PMT R\&D }
In order to ensure optimum performance of the detector, the PDE of PMTs in the
detector is preferred to be $\ge$~35\%, averaging over the entire photocathode
and scintillation spectrum. No commercial large PMTs can achieve this. A R\&D
effort intended to accomplish such an ambitious goal was started in early 2009.
This effort was led by IHEP (Beijing) and joined by research institutes,
university groups and related companies in China, and a formal R\&D group was
established in early 2011. Significant progress in all aspects of PMT R\&D has
been made. The goal of the design and manufacturing 508~mm PMTs that meet the
requirements listed in Fig.~\ref{fig:specificationOfPMTs} and in particular to achieve the 35\% PDE is
not easy and cannot be taken for granted. If we are unable to accomplish our
stated goal in time, we will make corresponding adjustments to the design of
the experiment. Depending on the status of our R\&D effort by the time of the
decision on the PMT choice, commercially available PMTs that may or may not
have the desired photon detection efficiency can be considered.
It is worth noting that in early 2009, a team from IHEP lead by Yifang Wang,
then the deputy director of IHEP visited a former PMT manufacturer, the No.
741 factory of the Chinese Electronics Administration in Nanjing, China. The
manager of that factory became interested after hearing of our effort to
develop large PMTs for JUNO (then Daya bay II). He made an effort to line up
financial support from two entrepreneurs in Beijing and successfully negotiated
a deal with PHOTONIS France S.A.S. and bought their stalled PMT production line
and all related technologies. A Chinese PMT company, SCZ PHOTONIS, was
established in 2011 and have now started to make 12 inch PMTs, with a plan in
place to make 20 inch PMT samples soon.
Meanwhile, IHEP representatives have been in contact with Hamamatsu, Japan,
which have been developing high efficiency 20 inch PMTs for the future Hyper
Kamiokande experiment in Japan and made good progress in the last few years.
Hamamatsu has agreed to deliver 20 inch PMT samples to IHEP for testing.
We will discuss the R\&D effort for JUNO PMTs by the collaboration led by IHEP (Beijing) in this section. Options provided by the two commercial companies will be discussed in section 6.4.2.
\subsection{MCP-PMT R\&D at IHEP}
The R\&D effort to develop PMTs for JUNO started in early 2009 when the Daya
bay II experiment was in the very early conceptual development stage. The main
obstacle for achieving the physics goals of the experiment was the lack of
commercially available PMTs that can reach the photon detection efficiency
(PDE) required. Of the two major PMT manufacturers in China, the aforementioned
No. 741 Factory in Nanjing had stopped PMT production for a number of years and
another PMT manufacturer, the No. 261 Factory in Beijing was bought by
Hamamatsu and the resulted Hamamatsu Photonics (Beijing) produced only low-end
small PMTs. The R\&D work on PMT related technologies such as Bialkali
photocathode had ceased for more than a decade in China.
The quantum efficiency (QE) of a vacuum PMT is conventionally defined as the
number of photoelectrons emitted from the photocathode layer into vacuum by the
number of incident photons. When the QE of a PMT is measured, the device is
placed in air and the incident light beam is perpendicular to its glass window
action. Typically 4\% of photons are reflected by the glass surface and more
light is reflected at the interface of the glass substrate and the photocathode
layer, which has a high index of refraction. The cathode sensitivity and QE as
functions of wavelength of the incident photons are plotted in Fig.~\ref{fig:Fig.5.2} (taken
from the PMT manual published by Philips Photonics). The blue curve is QE of
conventional bialkali photocathode (SbK2Cs) with a peak value of about 30\%.
The spectral response of the conventional BA photocathode matches the spectrum
of scintillation light from the liquid scintillators well and the estimated
average QE for the entire scintillation light emission spectrum is about 25\%
based on this plot. However for large PMTs, the QE is normally lower than the
values shown in Fig.~\ref{fig:Fig.5.2}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.2.png}
\caption[ The photocathode sensitivity and quantum efficiencies of various photocathodes]{\label{fig:Fig.5.2} The photocathode sensitivity and quantum efficiencies of various photocathodes}
\end{center}
\end{figure}
For the 20 inch Hamamatsu R3600-02 used in the KamLand experiment, the peak QE
is about 20\% and the peak photon detection efficiency (PDE) is only about 16\%
since the photoelectron collection efficiency is 80\% as stated by Hamamatsu
for the R3600-02. The 16\% peak PDE is far lower than the desired 35\% PDE for
JUNO. Another factor that also needs to be considered is the effective
photocathode area. The R3600-02 PMT has an outside diameter of 508~mm and its
effective photocathode diameter is only 460~mm. This factor puts a limit on the
capability for JUNO to maximize its photocathode coverage. Given these factors,
it became apparent that we need to start our own large PMT R\&D program mainly
to improve the photon detection efficiency. Note that new 20 inch PMT (R12860
Under development) samples with super bialkali photocathode from Hamamatsu have
shown great improvement for the PDE.
Our PMT R\&D was based on a concept proposed by IHEP (Beijing) in early 2009.
The conceptual schematic is shown in Fig.~\ref{fig:Fig.5.3}. The concept calls
for a spherical PMT with top hemisphere, as shown in the figure, used as
transmission photocathode and its bottom hemisphere used as reflective
photocathode which converts the photons that passed through the transmission
photocathode. It is hoped that by optimizing the thickness of the photocathode
in the top and bottom halves, the QE of such a PMT can be significantly
improved compared to the conventional design that has only the top transmission
photocathode. This new design requires two back-to-back compact electron
multipliers, since conventional dynode chains would be too bulky. The proposed
electron multipliers are two back-to-back pairs of microchannel plates (MCPs),
so that photoelectrons from the top and bottom photocathodes are focused on the
two sets of MCPs facing left and right, as shown in Fig.~\ref{fig:Fig.5.3}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.3.png}
\caption[The conceptual drawing of the high efficiency spherical PMT with two sets of MCP electron multipliers]{\label{fig:Fig.5.3} The conceptual drawing of the high efficient spherical PMT with two sets of MCP electron multipliers }
\end{center}
\end{figure}
As a starting point, a contract for a 5 inch MCP-PMT prototype was signed with
the Nanjing Electronic Devices Institute, which was the only place in China
that had made MCP PMTs. The progress was slow at the beginning and after about
two years, working prototypes were successfully made. While working with the
Nanjing Electronic Devices Institute, it became apparent that we needed to
expand our efforts in order to broaden our technology bases to solve many
complex and difficult technical issues. The Large Area MCP-PMT Development
Collaboration was formally established in 2011. Coordinated by IHEP, the
collaboration consists of the CNNC Beijing Nuclear Instrument Factory, the
Xi'An Institute of Optics and Precision Mechanics (XIOPM) of the CAS, Shanxi
Baoguang Vacuum Electric Device Co., Ltd., China North Industries Group
Corporation (NORINCO GROUP), Nanjing University, and the PMT Division of the
CNNC Beijing Nuclear Instrument Institute (formally No. 261 Factory). Great
progress has been made since the start of the collaboration almost three years
ago.
\subsection{Large Area High Efficiency MCP-PMT Project}
Many critical components and technologies need to be developed for the success
of the Large Area High Efficiency MCP-PMT Project. First, the raw materials and
methods to produce large glass bulbs with high strength, high optical quality
and extremely low radioactivity must be identified. Among many requirements,
the glass bulbs must be able to work in pure water under high pressure. The
technology for vacuum sealing between the glass and the Kovar metal leads must
be developed. Technologies required for making large area high QE photocathodes
are also needed. The alkali metals and Antimony sources and thin film
deposition technologies necessary for photocathode fabrication also require
development work. Fabricating MCP pairs as electron multipliers with low noise
and gain exceeding 10$^{7}$ is very complex and requires a lot of development
work. Other key technologies include electro-optic, mechanic and electric
designs for MCP-PMTs and the testing of such PMTs. At the time when the
collaboration was started, most of the required technologies did not exist in
China. After three years of collective work, 8 inch and 20 inch MCP-PMT
prototypes have been successfully made and evaluated.
\subsubsection{Basic Concept of Large Area MCP-PMT}
1) Effect of combining the transmission and reflection photocathodes
Conventional wisdom says that the QE of a reflective photocathode, that can be
made thicker, has much higher QE than that of a transmission photocathode.
Coating the lower half hemisphere of a spherical PMT bulb to convert the light
that has passed through the transmission photocathode on the top hemisphere
into photoelectrons can in principle improve the QE of the device. Scientists
at XIOPM have done theoretical analysis and computer simulation, showing that
by optimizing the thicknesses of transmission and reflection photocathodes, the
QE of the spherical PMT can indeed be improved. Such improvement has also been
seen in prototypes of MCP-PMTs.
2) Electro-optical Simulation
The shapes of photocathodes of the focusing electrodes required for effective
photo-electron collections, and of the other structures inside the PMT bulb,
require careful electro-optical calculations. Commercial computer software
specially written for this purpose are extensively used by XIOPM and NORINCO to
optimize the MCP-PMT design both for 8 inch and 20 inch MCP-PMTs. An example of
the computer simulation results is given in Fig.~\ref{fig:Fig.5.4}. The
simulation includes the photocathode, MCP assembly, its support structure and
metal leads, focusing electrodes and the Earth's magnetic field. Electric
potentials for various electrodes were optimized.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.4.png}
\caption[MCP-PMT electro-optic simulation with Earth magnetic field]{\label{fig:Fig.5.4} MCP-PMT electro-optic simulation with Earth magnetic field }
\end{center}
\end{figure}
3) MCP computer simulation
XIOPM has performed computer simulations for photo-electron collection and
amplification in the microchannels of MCPs. Electrode arrangement, gap size
between two MCP plates, size of microchannels and inclination angles were
studied. A diagram of such simulation is given in Fig.~\ref{fig:Fig.5.5}. The
MCP assembly has several advantages such as compactness insensitivity of the
electron amplification to the Earth's magnetic field, high dynamic range,
etc. Additional concerns, including the photoelectron collection efficiency and
lifetime, etc. still need to be addressed.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.5.png}
\caption[Simulation for MCP functionality]{\label{fig:Fig.5.5} Simulation for MCP functionality }
\end{center}
\end{figure}
\subsubsection{MCP and MCP Assembly}
NORINCO has decades of extensive experience in MCP production and development.
The company has organized a special group to develop the MCP and MCP assembly
required for the MCP-PMT project. Large capital and manpower investments have
been made. MCP parameters and processing procedures have been adjusted. High
gain MCP assemblies with gain exceeding 10$^{7}$ for a pair of MCPs have been
developed after the design of the MCP assembly was optimized. This level of
gain is critical for single photoelectron detection.
\subsubsection{Glass Tube and Raw Materials}
Our 8 inch MCP-PMTs use the traditional electro-vacuum glass, whose coefficient
of thermal expansion (CTE) matches the CTE of Kovar, which is used for vacuum
feedthrough pins. The vacuum seal is easy to make. However, the electro-vacuum
glass is difficult to blow into 20 inch glass tubes due to its poor mechanical
properties above the softening point. Also there are other issues that make the
electro-vacuum glass unsuitable for 20 inch PMT tubes even if it can be used to
make 20 inch glass tubes. Its mechanical strength is not good enough to
withstanding the high hydraulic pressure, while oxides of alkaline metal in the
glass can slow leach out and affect the optical transparency of the glass.
Therefore, the Pyrex type of borosilicate glass that has the required
mechanical strength and chemical stability is preferred for making 20 inch
glass tubes. Unfortunately, the CTE of Pyrex glass does not match the CTE of
Kovar so a section of glass tube that joins the Perex to electro-vacuum glass
needs to be made by using glasses with different CTEs. Techniques and glass
materials must be developed for making this multiple transition glass tube
section. The aforementioned technical difficulties have been solved by the
collaborative effort of our collaboration and industrial partners. The
resulting 20 inch glass bulb has a good appearance and excellent quality. Its
mechanical accuracy and thickness uniformity are also excellent. Some completed
glass bulbs, including the Pyrex sphere and neck, multiple transient section
and Kovar seal, as shown in Fig.~\ref{fig:Fig.5.6}, have been tested in a pressure test
set-up with pressure up to 10 bars.
Since the neutrino event rate in JUNO is extremely low, radioactivity of the
PMTs for JUNO must be tightly controlled. The radiative decay of radioactive
materials in PMT can potentially be a major source of background for a low
energy neutrino experiment. The radioactive elements $^{238}$U, $^{232}$TH, and
$^{40}$K in the glass are the major contributors of PMT radiation backgrounds.
The glass of the Hamamatsu 8 inch R5912 used in Daya bay detector has rather
high radioactivity level, which is acceptable for a relatively high rate
experiment like Daya bay, but far exceeds the level that can be tolerated by
JUNO. To control the radioactive level in the PMT glass tubes, care must be
taken to select the raw materials and also to control the glass melting process
preventing the radioactive materials contained in the crucibles to be dissolved into the molten glass.
The main component of Pyrex glass is SiO2. We have identified sources of high
purity quartz sand in China and indeed the measured $^{238}$U, $^{232}$TH, and
$^{40}$K levels are satisfactory for JUNO. Low radioactivity raw materials for
other ingredients of the Pyrex glass have also been identified. Costs of high
purity raw materials are significantly higher than those of materials commonly
used to make Pyrex glassware. However, the cost of raw materials for glass,
even the high grade materials used, are still an insignificant part of the total PMT production cost.
Although we believe we know how to do it, so far low radioactivity glass tubes
have not yet been manufactured. The 20 inch glass tubes for our prototype PMTs
have been made with regular materials and using regular procedures in the
factory. This is because even just to make a test run for low radioactivity
glass requires the glass factory to stop their regular production for an
extended period to purge their ovens, and the test would cost a significant
amount of money. We will do a low radioactive test production when we are ready
to make a small batch of 20 inch PMTs.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.6.png}
\caption[Vacuum sealed 20inch glass bulb on a CMM for mechanical accuracy measurements]{\label{fig:Fig.5.6} Vacuum sealed 20inch glass bulb on a CMM for mechanical accuracy measurements }
\end{center}
\end{figure}
\subsubsection{Photocathode}
The Photocathode is a critical component of our PMT R\&D project. Typically to
make bialkali photocathode, a very thin Antimony layer is first deposited on
the inner surface of an evacuated glass tube, then layers of the alkali metals
Potassium and Cesium are deposited on top. Under high temperature a
semiconductor layer of K$_{2}$CsSb photocathode is formed. Both XIOPM and
NORINCO, on the basis of their previous experiences, have tried to make
bialkali photocathode for 8 inch MCP-PMTs successfully. Recently NORINCO has
also made a 20 inch photocathode with good results as shown in
Fig.~\ref{fig:sec2QE}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/QE.jpg}
\caption[QE]{\label{fig:sec2QE} QE }
\end{center}
\end{figure}
\section{Organization of the MCP-PMT R\&D Project}
\subsection{Introduction}
The required effort to develop a brand new product such as 20 inch MCP-PMT is
very technically demanding, and the R\&D process is complex and time consuming
both for skilled technicians and scientists. The cost of the R\&D, including
the basic equipment and the investment for developing the various technologies
involved, is quite high. IHEP (Beijing) would be unable to take on this task by
itself. The approach we have taken is to organize a concerted effort by
factories that have experience of producing PMTs and MCPs, as well as by
research institutions and university groups that have strong theoretical and
practical background in opto-electronic devices. This approach led to creating
a collaboration able to exploit effectively the expertise of each individual
member institution toward the final goal of designing and fabricating 20 inch
PMTs that meet the requirements for JUNO.
Unlike other electronics devices, PMTs often cannot be tested at every
production stage; instead meaningful tests can only be made after all the
production steps are completed and the tube is evacuated to high vacuum level.
We must do as much as possible of the testing and Q\&A work before the mass PMT
fabrication is started. Therefore, we divide our R\&D project into the
following categories, as shown in Fig.~\ref{fig:Fig.5.7}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.7.png}
\caption[Flow chart of the large area MCP-PMT R\&D process]{\label{fig:Fig.5.7} Flow chats of the large area MCP-PMT R\&D process}
\end{center}
\end{figure}
The development process includes many different aspects that must proceed in parallel as listed below.
1. Alkali sources: Develop and produce standardized alkali sources that ensure high QE and stable photocathode.
2. Glass: Develop 20 inch high quality and low radioactivity glass bulbs.
3. MCP: Develop and produce low noise, high gain MCPs and optimize the performance of MCP assemblies
4. Electronics: Optimize the signal propagation from the anode to readout electronics
5. R\&D to fabricate large high vacuum equipment for high throughput PMT production and testing .
6. Vacuum seal: Techniques for hot Indium seal and glass seal
7. Electro-optic: Design and performance simulation
8. Performance tests: Platform for 20 inch PMT testing that can be used in the PMT factory and other places.
\subsection{Capability of the Collaboration}
\subsubsection{IHEP (Beijing)}
The Institute of High Energy Physics, Chinese Academy of Sciences, is the
initiator and coordinator of the Large Area MCP-PMT Project. IHEP has extensive
experience in nuclear and particle instrumentation, vacuum technology, PMT
testing, and PMT applications in large particle physics experiments. The Large
Area MCP-PMT Project is organized within the Time-of-Flight detector group,
which is a subgroup of the State Key Laboratory of Particle Detection and
Electronics at IHEP. Funded partly by the Major Research Instrumentation
Initiatives of CAS, the IHEP group has built a PMT Performance Testing Lab.
Many PMT parameters such as the single photoelectron (SPE) spectrum, absolute
QE and photocathode sensitivity, dark current and dark pulse rate, linearity
and gain curves, etc. can be measured quickly and accurately. Figure~\ref{fig:Fig.5.8} shows graphically the functionality of the PMT testing lab.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.8.png}
\caption[ Organization of the existing PMT test lab at IHEP]{\label{fig:Fig.5.8} Organization of the existing PMT test lab at IHEP}
\end{center}
\end{figure}
The PMT Performance Lab is equipped with a 10000 class clean room, a dark room
with electromagnetic shielding, an optic testing system based on LEDs, a
variable frequency laser, light flux calibration systems and associated optic
components. Advance testing instrumentation, including pA meters and digital
oscilloscopes has been installed. With a VME based automated PMT input surface
scanning system and testing system for photo gain curves, the absolute
photocathode sensitivity (accurate to 1~$\mu$A/lm), QE (relative uncertainty <
2\%) and single photon transit time spread TTS can be measured in the frequency
range 230~nm - 1000~nm (accurate to 50~ps). The dark current can be measured to
pA level in the range of 1~pA - 10~nA. This test system is completely
automated and the HV, laser wavelength, data recording and data analysis can be
controlled by computers.
Moreover, a VME based automated single photoelectron (SPE) spectrum recording
system has been built. The SPE testing usually requires the PMT gain to be set
to gain > 10$^{7}$, but with a preamplifier the SPE spectrum can be acquired
also at a reduced gain > 10$^5$. Fig.~\ref{fig:Fig.5.9} shows measurement results by such a system.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.9.png}
\caption[PMT measurement results at IHEP]{\label{fig:Fig.5.9}PMT measurement results at IHEP}
\end{center}
\end{figure}
In addition, IHEP has recently assembled a small group to perform fundamental
studies on MCP optimization to prepare for the MCP-PMT production and to study
optic thin films. An ultra low radioactivity analysis system based on a
Canberra High-purity Germanium (HPGe) Detector with low background shielding
and BGO veto counter has been built at IHEP. This system is used to study the
radioactivity of PMT glass and raw material samples.
\subsubsection{XIOPM}
The XiAn Institute of Optics and Precision Mechanics (XIOPM) of CAS has
extensive experience in research and fabrication of various opto-electronic
devices and related technologies. With significant government funding, the
State Key Laboratory of Transient Optics, Photonics and Optoelectronics
Laboratory and the Photoelectric Measurement and Control Technology Research
Department at XIOPM have recently upgraded laboratories with new clean rooms
and new equipment, including three research platforms.
1. Platform for opto-electronic devices: Design software for opto-electronic devices, high precision assembly table, plasma surface cleaning, vacuum seal and welding machines, vacuum leak detection systems, photocathode activation station, photocathode testing station, and others. This platform provides basic tools for high sensitivity, high quality opto-electronic devices R\&D and fabrication.
2. Platform for high speed electronic fabrication and testing: Software for designing radiofrequency circuits, circuit fabrication facility.
3. Platform for opto-electronics testing: static and dynamic opto-electronics testing system, image intensify and fast DAQ system.
The 8 inch MCP-PMT design and prototype fabrication were started at XIOPM in
2011, followed by the design and fabrication of the equipment needed for the 20
inch MCP-PMTs after some success in their 8 inch MCP-PMT program. In this
process, they have performed extensive opto-electronics simulation, theoretical
analysis for high QE photocathode, photocathode fabrication and measurements,
MCP functionality studies and optimization. Part of their work is shown in
Fig.~\ref{fig:Fig.5.10}. The 20 inch MCP-PMT fabrication equipment,
both the non-photocathode transfer and photocathode transfer types, is now almost
ready at XIOPM and they are essentially ready to produce some 20 inch MCP-PMT
prototypes.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.10.png}
\caption[MCP-PMT fabrication process at XIOPM]{\label{fig:Fig.5.10}MCP-PMT fabrication process at XIOPM}
\end{center}
\end{figure}
\subsubsection{NORINCO Group}
China North Industries Group Corporation (NORINCO GROUP) is one of five
producers of image intensifier tubes based on MCPs. NORINCO has teams of
engineers to design and fabricate special equipment for fabricating and testing
vacuum opto-electronic devices. In particular it has more than 30 years
experiences in R\&D, design, fabrication and testing for various photocathodes,
MCP and MCP assemblies. Their original technology came from PHOTONIS in the
Netherlands. After many years refinement, they are now able to routinely
realize multialkali photocathodes in small image tubes with sensitivity
exceeding 900~$\mu$A/lm, which is higher than that reached by any other
manufacturer in the world. For the Large Area MCP-PMT R\&D Project, their
expertise in bialkali photocathode is assisted by the expertise of No. 741 PMT
factory.
For the R\&D of Large Area MCP-PMT project, NORINCO GROUP has organized a team of experienced engineers, made available workshops, lab spaces and equipment, and invested a large amount of capital. Three of their platforms, some new and some existing with additional equipment, can be used for various MCP-PMT R\&D tasks.
1. MCP Research Platform: Glass ovens, glass fiber pulling machines, MCP
slicing machines, automated etching machines, electrode deposition stations,
MCP testing station.
2. Platform for low level opto-electronic devices: Software for opto-electronics design and simulation, instrument for high sensitivity photocurrent measurements, photocathode production station, including plasma cleaning, monitoring during photocathode fabrication, thin film deposition stations for high QE photocathodes, photcathode sensitivity testing station, light transmission and reflection testing station, HV power supply design and fabrication and testing, etc.
3. Platform for Large Area PMT R\&D and prototyping: Glass lathe, large
annealing oven, RF welding machines, large ultrasound cleaning stations, clean
baking oven, point welding machines, thin film deposition stations, 20 inch PMT
fabrication station using non-vacuum transfer technology, 20 inch PMT
fabrication station using vacuum transfer technology (shown in
Fig.~\ref{fig:Fig.5.11}), photocathode activation monitoring station, PMT
dynamic testing system, PMT static testing system, opto-electronics design
software, etc.
These platforms provide a solid foundation for large area MCP-PMT design, R\&D, prototype testing and industrialization.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.11.png}
\caption[ Equipment for fabricating 20 inch MCP-PMTs developed by NORINCO using photocathode transfer technology]{\label{fig:Fig.5.11} Equipment for fabricating 20 inch MCP-PMTs developed by NORINCO using the photocathode transfer technology}
\end{center}
\end{figure}
\section{R\&D Results and Plan}
\subsection{Status of MCP-PMT R\&D}
The main achievements up to June of 2014 made by the Large Area MCP-PMT
Collaboration are summarized in Table~\ref{Achievementsandstatus}.
\begin{table}[!hbp]
\centering
\caption{Achievements and status\label{Achievementsandstatus}}
\begin{tabular}{|c|c|}
\hline
\hline
Items & Status \\
\hline
8 inch MCP-PMT prototypes & Completed \\
\hline
PMT testing system & 8 inch system completed at IHEP and XIOPM \\
& 20 inch system near completion at NORINCO \\
\hline
20 inch fabrication equipment & Near completion at XIOPM; Completion at NORINCO \\
\hline
20 inch glass bulb & Completed and met expectations \\
\hline
20 inch Prototypes & Three produced at NORINCO \\
\hline
\end{tabular}
\end{table}
\subsubsection{8 inch MCP-PMT Prototype R\&D}
The Large Area MCP-PMT collaboration had its effort focused on the 8 inch
MCP-PMT prototypes in the first two years in order to develop the necessary
technologies for the 20 inch MPC-PMT. The opto-electronic design and
simulation, photocathode fabrication techniques, design of the MCP assembly,
MCP scrubbing, electrode design, etc. have gradually become mature and many
difficult problems have been solved. The highest QE achieved in 8 inch MCP-PMT
slightly exceeds 30\%. The required gain 10$^{7}$ has been achieved and the
dark current can be as low as 10~nA. The Peak/Valley (P/V) ratios of SPE spectra for
prototype tubes have been in the range of 1.6 to 2.5, with one prototype tube
reaching 3.8. QE and SPE spectra of the 8 inch MCP-PMT made at NORINCO and
tested at IHEP are given in Fig.~\ref{fig:Fig.5.12}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.12.png}
\caption[QE values and SPE spectra of the 8 inch MCP-PMT]{\label{fig:Fig.5.12}QE values and SPE spectra of the 8 inch MCP-PMT}
\end{center}
\end{figure}
\subsubsection{20 inch Glass Bulb and Transition Section}
As discussed previously, we use the so-called Beijing hard glass (Chinese
standard CG-17), that is equivalent to the Pyrex borosilicate glass, to make
the 20 inch glass bulbs. The CTE of the CG-17 glass is very low at
0.0000033/\SI{}{\degreeCelsius}. It has very high mechanical strength and can
withstand various strong acids, alkalis and pure water. Glass bulbs as large as
508 mm are hand-blown by skilled glass technicians. Very few people have such
skills and experience in the country. The collaboration was able to identify a
glass factory that was very cooperative, which successfully made more than 50
sample bulbs with very good quality. In particular, the achieved mechanical
tolerance (< 0.6 mm) is outstanding.
We have tried three methods to make the glass transition section match the low
CTE of the Pyrex glass with the slightly higher CTE of the glass used to make
the vacuum seal with the Kovar pins and the end flanges. After being sealed and
annealed, the evacuated glass bulb was placed in a test container filled with
water at 1~MPa pressure (equivalent to 100 meters of water) and observed for 24
hours without any problem. Photos of a 20 inch glass bulb, its visual defect
inspection and transition sections are shown in Fig.~\ref{fig:Fig.5.13}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.13.png}
\caption[20 inch glass bulb, its defect inspection and transition sections]{\label{fig:Fig.5.13}20 inch glass bulb, its defect inspection and transition sections}
\end{center}
\end{figure}
20 inch MCP-PMT prototypes were successfully made at the Nanjing factory of the
NORINCO in June, 2014 by using the photocathode vacuum transfer equipment.
After increasing the pumping capacity of the fabrication station, a QE level of
26\% has been reached. More detailed testing is in progress and we expect
there will be further improvements when more prototypes will be made. In
addition, prototypes of 20 inch MCP-PMT will be fabricated by using the
non-vacuum transfer station and comparison will be made of the 20 inch
prototypes made by both technologies.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.14.png}
\caption[ 20 inch MCP-PMT Prototype]{\label{fig:Fig.5.14} 20 inch MCP-PMT Prototype}
\end{center}
\end{figure}
\subsubsection{Improving the Testing System}
The testing system for large MCP-PMTs with spherical photocathodes are
different in many aspects from the system used for conventional PMTs. IHEP,
XIOPM and NORINCO are all in the process to improve their current testing
system. The existing testing system at IHEP is shown in
Fig.~\ref{fig:Fig.5.15}. Photocathode sensitivity, QE, dark current, SPE
spectra, gain, etc. can be measured. The automated scanning station can
accommodate 8 inch PMTs. Test results using this system agree well with
parameters provided by Hamamatsu and Photonis for their standard PMTs.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{PMT/figures/Fig.5.15.png}
\caption[ PMT test system at IHEP]{\label{fig:Fig.5.15} PMT test system at IHEP}
\end{center}
\end{figure}
\subsection{Status of Commercial PMTs}
\subsubsection{Hamamatsu}
In recent years, some PMT manufacturers have been developing so called SBA
photocathode and try to use such photocathode in large size PMTs.
Hamamatsu has made a new SBA 20 inch R3600-02 with Venetian Blind type dynode.
Five such tubes have been recently tested. \cite{Nishimura2014Jan}. Hamamatsu is also developing
a 20 inch PMT (R12860) with box and line type dynodes, expecting 93\% photoelectron
collection efficiency compared to about 80\% for the R3600-02. The TTS will
also be improved with the box and line type dynodes. The QE of the Hamamatsu
20 inch SBA is expected to be about 30\% (Private communication with Hamamatsu)
at the sensitivity peak. The sensitivity peak of the current SBA photocathode
is located at 390~nm, which matches the requirement of the HyperK water Cherenkov
detector, but slight mismatchs the scintillation light, which is mainly emitted
at 390~nm to 450~nm. Based on what we know now, we can expect that the PDE of
Hamamatsu R12860 may reach 25\% for scintillation photons. Another problem of
using theR12860 for JUNO is that the size of its effective photocathode is
somewhat smaller than the 508 mm outside diameter , which would affect the
global useful coverage. Hamamatsu has promised to deliver samples of the R12860
to IHEP in June 2014.
\subsubsection{HZC Photonics}
HZC Photonics in Hainan Province, China acquired PMT production line and
related technologies, including the SBA technology, from PHOTONIS, France in
2011. HZC Photonics started their PMT production in late 2013 and now have a
12 inch PMT XP1807 available. Its stated photocathode sensitivity 60~\SI{}{\uA} /lm is
rather low. Their preliminary product, the XP53B20 3 inch circular tube, has a
very high peak photocathode sensitivity, listed as 160~\SI{}{\uA} /lm, but with rather
low gain of $ 6.25 \times10^{5}$ . HZC Photonics have promised IHEP to produce
samples of 20 inch SBA PMT for JUNO soon.
\section{Risk Analysis}
\subsection{Risk due to Low Photon Detection Efficiency}
PMTs are critical components for JUNO. The current MCP-PMT R\&D plan and backup
options carry certain risks that need to be considered. In order to achieve the
required energy resolution of less than 3\% at 1 MeV, which is expected to be
dominated by photon statistics, high efficiency liquid scintillator with a long
attenuation length must be used to detect enough photons when neutrinos
interact in the liquid scintillator. In addition, the fraction of the detector
surface that is covered by PMT photocathodes must be as large as possible and
the PDE of PMTs must be as high as possible. For a 508 mm diameter spherical
PMT, assuming glass wall thickness of 4~mm, the maximum diameter of the
photocathode would be 500~mm, or 97\% of the area of the PMT is actually
covered by photocathode. Space needed for anti-implosion covers and gaps among
PMTs futher reduce the maximum possible photocathode coverage.
We also must make the QE of PMTs as large as possible. As stated earlier, the
PDE of PMTs equals the QE of the photocathode multiplied by the photoelectron
collection efficiency (PCE). The goal of the peak photocathode QE for the JUNO
PMTs is 38\%, which is a very demanding goal. In Fig.~\ref{fig:Fig.5.16} (taken
from Hamamatsu PMT handbook), the solid black curve is the typical cathode
radiance sensitivity in unit of mA/W. The typical QE curve can be calculated
from it. The computed QE as a function of wavelength is given as the dashed
curve in Fig.~\ref{fig:Fig.5.16} and the scintillation light spectrum is plotted as the red
solid curve (linear scale) with double peaks at 400~nm and 425~nm.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.16.png}
\caption[Spectral response of common bialkali photocathode and the emission spectrum of liquid scintillator (red curve) ]{\label{fig:Fig.5.16}Spectral response of common bialkali photocathode and the emission spectrum of liquid scintillator (red curve) }
\end{center}
\end{figure}
It can be seen from Fig.~\ref{fig:Fig.5.17} that the typical QE of regular
bialkali photocathode is 20 - 25\% within the wavelength range 350~nm to
450~nm. The value of the peak QE typically is no more than 30\% in commercial
PMTs. In recent years, Hamamatsu has developed SBA and UBA photocathodes with
higher QE. The QEs as functions of wavelengths of these new photocathodes are
given in Fig.~\ref{fig:Fig.5.17}. We will not discuss the UBA photocathode since the UBA is
only available in small flat faced PMTs made by Hamamatsu. The peak QE of the
SBA photocathode shown in Fig.~\ref{fig:Fig.5.16} is about 35\% and the peak
sensitivity occurs at ~390~nm. There are two important issues to be considered.
It can be seen that the spectral response curve of the Hamamatsu SBA is rather
narrow and there is a slight mismatch between the SBA spectrum response peak at
390 nm and the scintillation light emission peak at ~ 425~nm. In Fig.~\ref{fig:Fig.5.17}, the
black curve that is shifted slightly toward longer wavelengths indicates the
desired spectrum response, which matches the spectrum of emitted scintillation
light better. The second issue is that according to our conversation with
Hamamatsu, the typical peak sensitivity they expect for the future 20 inch SBA
PMT is 30\%, not the 35\% given in their published documents. The average QE
for scintillation light is expected to be less than 30\%. With the stated 93\%
photoelectron collection effiiency of the new SBA R12860, we speculate that its
average PDE may be less than 25\% due to the spectral response mismatch
discussed above.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.17.png}
\caption[QEs vs. wavelengths for conventional bialkali, SBA and UBA photocathodes]{\label{fig:Fig.5.17}QEs vs. wavelengths for conventional bialkali, SBA and UBA photocathodes}
\end{center}
\end{figure}
The specified peak QE for JUNO PMTs as listed in
Table~\ref{tab:PMTsrequiredbyJUNO} is 38\% and we hope the average PCE of JUNO
PMTs can be higher than 30\% in the wavelength range 400~nm to 450~nm.
Therefore one of the critical risks is whether or not we can achieve this. Our
approaches to limit this risk are the following.
1. Transmission + Reflection Photocathode
Use the spherical photocathode, combining the power of transmission and
reflection photocathodes. We hope the targeted peak QE 38\% and average QE >
30\% within 400 - 450~nm can be achieved by optimizing the photocathode design
and fabrication process.
2. Transmission photocathode + reflection film
Coat a layer of thin film with high reflectivity in the bottom half of the sphere to reflect photons that have penetrated the transmission photocathode back and convert these photons into photoelectrons.
\subsection{Risk Related to MCP Aging}
The operational lifetime of JUNO is 20 years and the working lifetime of the
PMTs must also be greater than 20 years. It is expected that as long as the
vacuum in the MCP-PMTs can be maintained, their lifetime will mainly be
determined by the gain reduction due to MCP aging. It is expected that in the
completely dark environment of JUNO, accumulated charges will mainly be caused
by the photocathode thermal noise and the dark current of MCPs. At gain 10$^{7}$
and dark counting rate 10~kHz (equivalent to dark current 16~nA), the
accumulated charge due to photocathode thermal emissions will be ~10~C for 20
years operation. The dark counting rate of a PMT is typically measured by
counting dark pulses with a threshold of 0.25~pe. The true dark current,
including the MCP dark current and dark current from other sources, can be much
higher than the 16~nA used in the above estimations. Some of this
additional dark current should be considered when the MCP aging effect is
evaluated.
At this level of charge accumulation, MCP aging can become a problem for
MCP-PMT operation. The charge accumulation level of the MCPs developed by
NORINCO can reach 30~C. In addition, PMT anode assembly typically has a gain of
3 - 5 and the charge accumulation is reduced by the same factor if such an
anode design is used. This implies that the accumulated charge that the MCP-PMT
can be operated with can reach the level of 100~C.
In order to control the dark pulse rate and dark current due to MCPs, we are
testing two different methods to fabricate the MCP-PMTs. The photocathode
vacuum transfer method has separate vacuum chambers for photocathode
fabrication and MCP processing and can in principle lower the risk of MCP
contamination by alkali materials, thus achieving lower dark current.
In the past few years, MCP manufacturers and research labs have developed a new
type of MCP that use the ALD technique to deposit a thin secondary electron
emission layer on top of a resistive layer on the wall of the microchannels of
the MCPs. It has been shown that the lifetime of ALD coated MCPs can be an
order of magnitude higher than the conventional MCPs. IHEP has started a R\&D
program to develop MCPs using ALD technology.
\subsection{Risk due to PMT Radiation Background}
\subsubsection{Radiation Background Requirements}
The PMT radiation backgrounds are dominated by the radioactive elements, most
importantly $^{238}$U, $^{232}$Th and $^{40}$K in the glass bulb. In the current 20 inch spherical
MCP-PMT design, the weight of the glass bulb is ~9~kg. According to the current
detector physics simulation, the upper limits for $^{238}$U, $^{232}$Th and $^{40}$K contents
in the PMT glass are 12.1 ppb, 26.1 ppb and 1.5 ppb, respectively. For
comparison, the contents of these elements in Hamamatsu R5912 8 inch PMT glass
are 153 ppb, 335 ppb and 18.5 ppb, while they are 540.3 ppb, 568.3 ppb and 75.2 ppb in the
20 inch R3600-02 PMTs. The radioactivity levels of Hamamatsu PMTs are many times
higher than that allowed by JUNO. In a sample of regular CG-17 borosilicate
glass, the type of glass to be used to fabricate the 20inch bulbs for MCP-PMTs,
$^{238}$U, $^{232}$Th and $^{40}$K contents are measured to be $349.1 \pm 18.7$ ppb,
$851.2 \pm 72.8$ ppb and $26\pm3.$1 ppb, which are 29, 32 and 17.5 times higher
than the limits for JUNO PMTs. Clearly new glass materials with lower
radioactive contents must be developed.
In order to meet the required extra low radiation background requirements, we
must use ultra-pure raw materials for making the glass and also control the
possible contamination occurring in the material handling and glass melting
process. Chemical composition of the GG-17 glass and of raw materials are
listed in Table~\ref{tab:PMTglass}.
\begin{table}[!hbp]
\centering
\caption{Chemical composition of the GG-17 glass and raw materials\label{tab:PMTglass}}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\hline
Items & Status \\
\hline
Compostion & SiO2 & B2O3 & Na2O & Al2O3 & NaCl \\
\hline
Content (\%) & 80.0 & 13.0 & 4.0 & 2.5 & 0.5 \\
\hline
Raw materials & Quartz sand & Boric acid & Na2CO3 & AlOH & Salt \\
\hline
\end{tabular}
\end{table}
We have acquired a large number of raw material samples from different
manufacturers in different regions of China and measured contents of the three
radioactive elements in these samples carefully. We have identified suppliers
who can provide the necessary raw materials with $^{238}$U, $^{232}$Th and
$^{40}$K contents that are low enough to meet JUNO PMT requirements. Obviously
these high purity materials must be handled properly to prevent any
contamination. We have also investigated the possibilities for contamination
during the glass melting process. The most worrisome concern is that at high
temperature radioactive elements can be introduced from crucibles and other
debris into the molten glass in the oven. A Platinum crucible can mostly
eliminate this problem, but it would be a very expensive solution. We believe
corundum crucibles commonly used in Pyrex glass factories should be good
enough based on the radiation background test for AlOH. So far, for the 50 or
so 20 inch glass bulbs that have been made, regular raw materials were used
and we know their radiation levels are very high. We have not attempted to
make low radiation background PMT bulbs yet.
\subsection{Risk due to Nonuniformity of MCP PCE}
The photon detection efficiency (PDE) of PMTs can be expressed as $PDE=QE*PCE$,
where QE is the photocathode quantum efficiency and PCE is the photoelectron
collection efficiency of the electron multiplier. In our case the electron
multiplier is the MCP assembly. With a spherical glass bulb and the
photocathode vacuum transfer approach, we believe it is possible to fabricated
completely uniform photocathode since we can place the evaporation alkali
sources at the center of the sphere. To make the photoelectron collection
completely uniform in our current MCP-PMT design is more difficult, since
photoelectrons generated by a spherical photocathode arrive at the flat MCP
input surface from different angles and they may have different efficiencies of
collection and amplification. We are following several possible approaches to
find remedies for this potential problem.
1. Optimize the focusing design to limit the incident angular spread of photoelectrons.
2. Optimize the MCP assembly design to add a metal mesh in front of MCP and
coating the MCP with MgO or Al2O3 thin film, which has a high secondary
electron emission coefficient, or use other schemes to improve the MCP
photoelectron collection. Fig.~\ref{fig:Fig.5.18} shows the metal mesh and MCP
coating schemes.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{PMT/figures/Fig.5.18.png}
\caption[The MCP coating scheme]{\label{fig:Fig.5.18} The MCP coating scheme}
\end{center}
\end{figure}
3. Further increase the gain saturation level of the second MCP such that the
output pulse heights are less dependent on the numbers of secondary electrons
produced by the photoelectron striking the first MCP.
\section{Schedule}
The global schedule from prototyping to mass production is shown in
Table~\ref{the schedule of the mass production}.
\begin{table}[!hbp]
\centering
\caption{the schedule of the mass production\label{the schedule of the mass production}}
\begin{tabular}{|c|c|c|}
\hline
\hline
Year & Tasks & Parameters \\
\hline
2013 & 8inch prototype & QE $\ge$ 25\%, P/V $\ge$ 1.5 \\
\hline
2014 & 20 inch prototype & QE $\ge$ 30\%, P/V $\ge$ 2.0,\\
& & other parameters meet specs \\
\hline
2015 & Engineering design & QE $\ge$ 30\%, P/V $\ge$ 2.0, \\
& & other parameters meet specs and Yield $\ge$ 80\% \\
\hline
2016 & Preproduction & 1,000 20inch PMTs \\
\hline
2017 & Mass production & 5,000 20inch PMTs \\
\hline
2018 & Mass production & 5,000 20inch PMTs \\
\hline
2019 & Mass production & 5,000 20inch PMTs \\
\hline
\end{tabular}
\end{table}
\chapter{PMT}
\label{ch:PMT}
\input{PMT/CH5.1.PMTDesigngoalsandspecifications.tex}
\input{PMT/CH5.2.MCP-PMTRandDforJUNO.tex}
\input{PMT/CH5.3.OrganizationofMCP-PMTRandDproject.tex}
\input{PMT/CH5.4.RandDResultsandPlans.tex}
\input{PMT/CH5.5.Riskanalysis.tex}
\input{PMT/CH5.6.ScheduleandCost.tex}
\chapter{Readout electronics and trigger}
\label{ch:ReadoutElectronicsAndTrigger}
The JUNO readout electronics system will have to cope with the signals of 17,000 PMTs of the central detector as well as 1,500 PMTs installed in the surrounding water pool. The average number of photoelectrons being registered by an individual PMT will reach from below one for low-energy events up to several thousands in case of showering muons or muon bundles. In both extreme cases, the number of photoelectrons has to be counted and their arrival time profiles have to be determined. A dedicated trigger system will be necessary to perform a pre-selection of correlated PMT hits caused by physical events from a sea of random dark noise hits. Based on the selected arrival time and pe pattern, software algorithms will reconstruct energy, type and position of the event vertex. Thus, the main task of the readout electronics is to receive and digitize the analog signals from the PMTs and transmit all the relevant information to storage without a significant loss of data quality. The basic concepts considered in the design of the system as well as a list of specifications are given in Sec.~\ref{sec:roe:specs}.
The PMTs will be submerged in the water pool surrounding the acrylic sphere that contains the LS. Based on this layout, three different approaches for the realization of the read-out electronics are discussed: A mostly "dry scheme" for which individual cables carry the analog signals out of the water pool to an external electronics hut containing front-ends and digitization units. A "wet scheme" for which digitization happens at the PMT bases and the digital signals are bundled in submerged central underwater units before being passed on to an external online computer-farm. And finally, an intermediate readout scheme where a considerable part of the electronics is submerged inside the water pool but still accessible for replacements. The three schemes are presented in Secs.~\ref{sec:roe:dry}$-$\ref{sec:roe:wet}.
Sec.~\ref{sec:roe:dev} will provide an overview of further R\&D efforts to be undertaken. A first time line for the realization of the system and the most important aspects to be dealt with is laid down in Sec.~\ref{sec:roe:rea}. Possible risks are summarized in Sec.~\ref{sec:roe:risk}.
\section{Design considerations and specifications}
\label{sec:roe:specs}
The concept for the design of the electronics is driven by the following goals, which are listed here in order of their priority:
\begin{enumerate}
\item {\bf Energy reconstruction:} Optimization of the energy measurement of $\bar\nu_e$, especially at the lowest energies. A basic limitation on the energy resolution arises from the statistics of detected photoelectrons (pe). This limit must not be worsened significantly by the effect of electronics. At a photoelectron yield of 1100 pe per MeV, the achievable relative resolution is $\sim$3\,\% at 1\,MeV.
\item {\bf Reconstruction of the photon arrival time pattern:} A precise measurement of the arrival times of the photons is required to reconstruct the position of the events. This is particularly important for the IBD events induced by $\bar\nu_e$'s because (a) the energy response of the detector will be position-dependent and (b) the formation of spatial coincidences between prompt positron and delayed neutron events will support background suppression. Moreover, track reconstruction for cosmic muons and especially for muon bundles is very sensitive to timing and will be an important tool for the veto of cosmogenic backgrounds.
\item {\bf Low dead time and dynamic acquisition rate:} During usual data taking conditions, all neutrino events from reactor, solar or geological origin should be acquired without in-efficiency, i.e. with a minimum or zero dead time. In addition, readout electronics have to be able to cope with short episodes of extremely high trigger rate that might be caused by the neutrino burst from a nearby galactic Supernova. Dependent on the distance, thousands of neutrino events are expected within a period of 10 seconds. Therefore, the system has to be able to bear a dramatical increase in its acquisition rate for short periods without significant data loss or dead time.
\end{enumerate}
\subsection{Energy reconstruction}
For the low-energy events, the deposited energy can be almost directly inferred from the overall amount of photons or pe detected by the PMTs. However, there are several ways of performing this integral measurement of collected pe (see below). Additional complications arise for high energy events, where the accuracy of determining the number of pe detected for a given PMT will depend on the dynamic range of the readout electronics.
\subsubsection{Photoelectron counting}
There are two basic methods to determine the visible energy created by an interaction in the detector. One can either integrate the charge created by the PMTs or one can try to identify individual photons and count them. The reconstruction is complicated by the spread in the arrival time of the photons, which is caused both by the fluorescence time of the LS, photon time-of-flight and photon scattering which all we contribute to delays in the order of several hundred nanoseconds\footnote{Note that once the vertex position is known, the delay from the time-of-flight can be approximately corrected offline. Based on this, most of the light will be concentrated towards the start of the event.}. An additional distortion of the time pattern will be introduced in case of cosmic muons where the light is generated along the extended particle track, causing a delay by the time-of-flight of the muon ($\leq$120\,ns). The trigger system will have to take into account this spread in arrival times when defining a threshold condition.
\medskip\\
\noindent{\bf Charge Integration} is technically the simpler solution. During a certain time window the signal of the PMTs is integrated and the integrals are summed up into a global charge. This global charge is proportional to the visible energy of the event. For an online measurement the time window has to be in the order of 300\,ns from the arrival of the first photon at a given PMT. If time-of-flight effects are taken into account, e.g. for an off-center event close to the surface of the acrylic sphere or a cosmic muon, the acquisition gate has to be extended to 500\,ns. The length of the time window must be optimized between two aspects. With longer time windows more and more of the scintillation light is included, but the contribution from the electronics noise increases. Due to the electronics noise the potential of the baseline will vary around zero. Integrating such a baseline even without a signal will not give exactly zero. The contribution from the baseline noise will scatter around zero with a width of the distribution that increases with the square root of the integration time.
There are two subversions of the charge integration. One can either sum all the PMTs or just those PMTs above a certain threshold (e.g.~0.25\,pe). Due to fluctuations in the amplification process of the PMTs (in the dynode structures of a conventional PMT or in the MCP) the charge produced by a single photoelectron varies. The variation is usually classified by the so-called peak-to-valley ratio of the PMT. Due to these fluctuations it can happen that the signal of a single pe is not visible above the baseline noise level. But the information is not completely lost. In the absence of electronics noise the sum of all PMTs will give a better estimate of the visible energy than just the sum of those PMTs above a certain threshold. For a real system one has to optimize the threshold. It largely depends on the noise of the system and on the average number of pe per PMT for the relevant events . For JUNO it is not yet clear what the best value is. For Daya Bay it turned out to be 0.25\,pe. Note that this charge integration threshold is not necessarily the same as the trigger threshold applied for self-triggering of a channel.
\medskip\\
{\bf Photon Counting} is usually a self-triggered method. One analyzes the wave-form of each PMT individually in an attempt to identify the arrival of individual photons. Then the number of these photons is counted over at time window comparable to the above. The total number is proportional to the visible energy. The advantage of this method is that the number of photons is more directly related to the energy than the total charge. Some photoelectrons produce higher charges and some lower. This introduces an additional fluctuation into the charge measurement which is unrelated to the visible energy. With photon counting this additional fluctuation is eliminated.
With the method of photon counting a threshold for each PMT is unavoidable. Photo electrons with very low charge due to fluctuations in the amplification process will be lost. This degrades the resolution by an amount that strongly depends on the peak-to-valley ratio of the PMTs. Whether photon counting is superior or inferior to charge integration depends on the details of the detector and the electronics. For JUNO the answer is not yet known. A detailed simulation is necessary to produce it. For Borexino photon counting turned out to be the preferred method at low energies and also DoubleChooz is using a method of approximate photon counting. For Daya Bay the electronics does not allow to do photon counting.
To extract the number of photons from the waveform of a PMT, algorithms of different levels of sophistication are possible. In case of a software trigger, a very simple algorithm that counts the number of transitions of the photon pulse(s) across a threshold is probably sufficient. A more sophisticated algorithm will identify the charge of a pulse and its shape. Effectively at very low occupancies any pulse consistent with the charge of a single pe will be counted as one photon. At very high occupancies, the shape of the pulse will not contain any significant information. Therefore the number of photons will be derived from the integrated charge. The two basic methods become identical at very high occupancies. At intermediate occupancies, when the pulses of a few photons partially overlap in time, shape information like satellite peaks in the wave-form combined with the integrated charge will give the best answer
\subsubsection{Dynamic range}
The electronic system must be able to cope with a large range of signal heights (and also durations). For low-energy neutrino events, the average occupancy is below 1\,pe per PMT. Therefore the electronics must have good resolution for single pe's. Cosmic muon events are forming the high-energy end of the the dynamic range. These events can reach hit occupancies of 4000 pe for individual PMTs, and this has been defined as the maximum occupancy the electronics will have to cover. However, even larger signals are expected in case the muons arrive in bundles or create hadronic showers inside the target volume.
\subsection{Photon timing}
\label{sec:roe:timing}
For all event types, the arrival time of the first photons at the individual PMTs will be the primary information available for the reconstruction of the vertex position (in case of neutrino events) or the particle track (in case of muons).
This will be of particular importance for the energy reconstruction of antineutrino events because the effective pe yield will be position-dependent. A precise reconstruction of the vertex position will allow for low systematic uncertainties when applying correction terms.
\subsubsection{Reconstruction of point-like events}
Reconstruction algorithms will start from the hypothesis of a simultaneous emission of all photons from a single event vertex inside the detector. Then, the assumed vertex position will be adjusted to fit the observed photon arrival time pattern when considering the time-of-flight of the photons. The assumption of simultaneous emission is usually well fulfilled for the first photons detected for an event. More sophisticated algorithms will take into account the time delays introduced by the finite fluorescence times, photon scattering in the LS and the transit time spread of the PMTs by using appropriate probability density functions. Moreover, also the density of photon hits distributed over the surface of the detector sphere bears information on the vertex position.
What is more, the precision of the photon timing will be decisive for the practicability of pulse shape discrimination. Heavy particles like $\alpha$'s or neutrons (proton recoils) create more light than electrons in the slow components of the fluorescence profile. Based on the information of the vertex position, individual hit times can be corrected for the time-of-flight of the photons. The resulting sum pulse of all detected pe corresponds to the original scintillation profile of the event (plus impact of light scattering). Profiles can be characterized for their long-lived decay component, allowing to quite clearly distinguish electrons from $\alpha$-particles and proton recoils and with lower efficiency even positrons from electrons.
The quality of the separation will largely depend on the gate length over which the photoelectrons are acquired. The time window should cover a significant portion of the slow fluorescence component which is of the order of several 100\,ns. In this respect, a decentralized trigger schemes featuring a variable gate start time and gate width for each channel seems preferable. In this case, acquisition will stop only when the acquired pulse becomes indistinguishable from dark noise (see below).
\subsubsection{Reconstruction of cosmic muons}
Reconstruction of cosmic muon tracks is more complex. For single, minimum-ionizing muon tracks, the emitted light front will closely resemble a Cherenkov cone in the forward-running direction\footnote{This shape is due to the relative speeds of muon (speed of light in vacuum) and photons (speed of light in the medium) which results in the same geometrical condition as in case of the Cherenkov effect.}, plus a superposition of spherical back\-ward-running light fronts. Such events are reconstructed in a similar fashion as point-like events, relying on a fit of the muon time-of-flight plus the photon time-of-flight for a given track hypothesis to the arrival time pattern of first photons observed at the surface of PMT photocathodes. Also charge information from the many hundred photons per channel may be included as the collected number of photoelectron at a given PMT will depend on the distance from the muon track.
However, showering muons as well as muon bundles create much more complex event topologies that lead in effect to a superposition of several light fronts arriving at the photocathodes with short time-delays. First attempts at the reconstruction of such events suggest that the full time profile of photon arrival times have to be recorded for each individual PMT in order to be able to discern secondary maxima or at least substructures that are created by the delayed arrival of subsequent light fronts. While a variety of algorithms are currently investigated, none has reached a state from which hard requirements on the necessary photon timing information can be formulated.
Note that especially for these events, the ability for a separate detection of the true Cherenkov cone might provide valuable information for disentangling the contributions of individual particles. However, it seems unlikely that conventional or MCP-PMTs will reach the necessary time resolution.
\subsubsection{Limitations in timing}
While accurate timing information is for sure of great value, there are intrinsic limitations to the accuracy that can be reached in JUNO that are inherent to the LS detection technique. The first one arises from the finite time width of photon emission from the scintillator that is described by the fluorescence time profile. The fast component that is most relevant for timing questions features a decay time in the order of a few ns. This leads to a smearing of the timing information especially in the regime of low-photon statistics, i.e. at low energies. The second and more severe constraint arises from the transient time spread (tts) of the PMTs. The tts characterizes the variation of the time between the arrival of the photon and the creation of of the electrical pulse. As the final PMT is not yet available, the precise tts profile is not known. But the ($1\sigma$) uncertainty introduced is expected to be in the range of 3$-$4\,ns.
Taking both inaccuracies into account, MC simulations of scintillation signals have been performed to obtain realistic PMT signals that were then "digitized" with different sampling rates. These studies indicate that a time resolution of 1\,ns resp.~a sampling rate of 1\,GS/s will be sufficient to record the signals without loss of information.
\subsection{Triggering schemes}
A further important consideration is the triggering scheme: It may either be global, i.e. based on the information of all channels, or de-centralized, i.e. based on self-triggering channels that send all their digitized signals (including all dark noise photoelectron) to a computing farm. In the latter case, a (online) software trigger is used to identify physical events. Both options are discussed more closely in the following.
\medskip\\
In case of a {\bf Global Trigger}, the information of all PMTs is combined in order to form a global triggering decision that starts the readout of all PMT signals. The information from a limited local group of PMTs will not be sufficient to form the trigger decision. Information from all of the PMTs is required. Monte Carlo studies have shown that a simple trigger logic based on the total number of active PMTs (PMTs with at least one hit) within a 300\,ns window is sufficient to suppress the background from random coincidences of dark noise and to guarantee 100\,\% efficiency for the detection of $\bar\nu_e$ events. Threshold would be set to 500 hits ($\sim0.5\,$MeV) which is well above the average number of 225 dark noise hits within the same time span\footnote{This number is based on a trigger coincidence window of 300\,ns and 17,000 PMTs featuring a dark noise rate of 50\,kHz each.}.
Such a trigger can easily be realized in the more conventional scheme of the external readout electronics (sec.~\ref{sec:roe:dry}) that relies on the collection of the analog signals in a physically confined space. In this case, the delay introduced by the decision for triggering reaching the digitization units is fairly small. However, a global trigger is much harder to apply for a de-centralized digitization as it is propagated by the underwater electronics scheme (sec.~\ref{sec:roe:wet}). Hit information from the spread-out PMTs would have to be transmitted either by individual cables/fibers or by combining the information from a group of PMTs underwater into one cable/fiber per group. The trigger decision would then be sent back along the same network to the PMTs to start the readout. During the latency of the trigger (several ms), the wave forms of the PMTs would have to be buffered either on the PMTs or in the underwater units.
An important aspect regarding energy reconstruction that potentially favors a global trigger is the fact that this is the only configuration that will allow the recording of waveforms for channels in which the PMT signals are below an individual triggering threshold, i.e. the detection of low-charge photoelectron. On the other hand, the existence of a global trigger threshold at $\sim$0.5\,MeV poses a special challenge for data acquisition during a Supernova. Besides a large number of $\bar\nu_e$ events, a substantial amount of the resulting neutrino interactions in JUNO will be from elastic scattering of protons. Due to reaction kinematics and quenching, the visible light will be fairly low and necessitate a temporal lowering of the threshold below 500\,pe. While this does not pose a problem from point of view of radioactive background (there might be an interference with dark noise, though), the only information available to make the system aware of the on-going neutrino burst is the sudden increase in event rate. It will take a certain number of triggers to realize that neutrinos from a nearby supernova are arriving. Then the threshold will have to be lowered in retrospective. This is not impossible with an appropriate amount of buffers, but it does impose a significant complication of the trigger scheme.
\medskip\\
\noindent{\bf Decentralized triggering.} In principle this "untriggered" scheme is much simpler. The signal from each PMT is continuously monitored. If it exceeds the acquisition threshold for a single photoelectron (SPE), i.e. probably around 0.25 of the single photoelectron charge, the signal is digitized and transmitted to the online farm. On the online farm the signals from the individual PMTs are combined into events and a decision is formed whether to store the event or to drop it. It is mainly a question of data reduction and storage capacity to decide which fraction of events or even individual PMT waveforms to store.
The advantage is that much more information is available for the decision on storage of the event. Time-of-flight information can be taken into account to associate true photon hits inside a 300\,ns window to the emission by a single vertex. This potent mean for the suppression of background from random dark noise coincidences will allow to lower the threshold below 0.5\,MeV during normal data taking.
For the underwater readout electronics, this scheme bears the considerable advantage of reducing the required cabling. There is no need for a network to send trigger information to the central trigger and to return the trigger decision. The disadvantage is a higher load on the data links to the farm and on the required computing power. In this scheme not only the waveforms of signal but also background events have to be transferred to the farm in order to form a trigger decision. The difference in the data transmission rates between the two trigger schemes is somewhat reduced by the length of the time windows required for the readout. For the global trigger scheme it must cover the full events ($\geq300$\,ns) for all PMTs while in the untriggered scheme the gate width can be adjusted to the number of photoelectrons detected, e.g. 32\,ns per single photoelectron pulse. If for a given event multiple photoelectrons are detected by a single PMT, several internal triggers will occur and several such windows will be transferred.
Also the timing between the PMTs is simpler in the untriggered scheme. There is no need to synchronize the clocks between the PMTs. $t_0$-corrections can be applied in the farm. A White Rabbit system as proposed for the global triggering scheme is not necessary. Only the synchronization of the FADC frequencies has to be ensured, which will be achieved by locking the internal oscillators by a centrally distributed clock.
Last, but not least one should mention that a software algorithm for data reduction on a farm is much more flexible to react to unforeseen complications or opportunities than a hardware-based trigger.
\subsection{List of specifications}
Based on the above discussion, the following list of specifications has been compiled:
\begin{itemize}
\item {\bf Waveform sampling} should be available over the whole energy range with a sampling rate of 1\,GS/s.
\item {\bf Photoelectron resolution:}
In the signal range from 1$-$100\,pe, the charge resolution should increase linearly from 0.1$-$1\,pe.\\
In the background range of 100$-$4000\,pe, the charge resolution should be 1\,\%.\\
Signal range and background range should overlap.
\item {\bf Dynamic range} should reach from 1 to 4000\,pe per channel.
\item {\bf Arrival time resolution}, e.g. by fitting of the signal leading edge, should be $\sigma_t\approx100ps$.
\item {\bf Noise level} should be below 0.1\,pe for single photoelectron detection.
\item {\bf Maximum acquisition rate} should be on the order of $10^3-10^4$ triggers per second.
\item Acquisition should be low in or without any dead time.
\end{itemize}
\noindent Beyond these performance requirement, it has to be ensured that the system can be reliably and continuously operated for at least 10 years with no or minimal access to the parts of the readout system submerged inside the water pool. It is also important to facilitate the installation by minimize the cabling between the PMT matrix and the outside electronics.
\section{External readout scheme}
\label{sec:roe:dry}
The philosophy of the external read-out scheme foresees to place as large a part of the electronics as possible in a dedicated electronics area or hut outside the detector. In particular, the analog-to-digital converters are to be located outside the water pool.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{ReadoutElectronicsAndTrigger/figures/OutwaterScheme.jpg}
\caption[Scheme for external readout electronics]{\label{fig:out water scheme}Schematic drawing of the external readout scheme.}
\end{center}
\end{figure}
\subsection{System layout}
The general layout is shown in Fig.~\ref{fig:out water scheme}: The electronics attached directly to the PMTs is kept to a minimum. At the base of the PMT, the signal is decoupled from high voltage and increased by a pre-amplifier. From there, the analog signal travels along a 100\,m long cable to the input at the external front-end electronics (FEE). These will be arrayed in crates of the ATCA chassis specification. Each crate will hold ten FEE modules and two trigger modules. Each FEE module will provide for 32 channels backend amplification, signal distribution and processing, analog-to-digital conversion and data processing, as well as signal processing for the trigger system.
The FEE modules will send the digitized data to the trigger modules via the backplane of the ATCA crate. These will create the trigger information that will be sent to the central trigger crate by fiber links. In case of a positive trigger decision, the trigger signal will be sent back to each signal processing crate and then fanned out to each FEE module. Each FEE module buffers the data and sends the necessary data asynchronously to the DAQ. Trigger and slow control system will follow the White Rabbit protocol to transfer the clock and other service signals to each crate. The clock signal will be distributed together with the DAQ and slow control through an optical fiber network.
This has several advantages: The simple structure of the external scheme allows for an easy implementation. A global trigger can be easily formed. Maybe most importantly, broken components of the FEE and trigger logic are easy to access for repairs or replacement, and an upgrade of the system is easy to realize during the 10 years or more of detector operation time. The main disadvantage is the transmission of the analog signals from each individual PMT over the 100\,m long cables. While signal attenuation can be compensated by placing the preamplifier at the PMT, some shaping of the leading edge of the analog pulses will be unavoidable. In addition, the sheer amount of underwater cables complicates the installation and is a severe cost driver.
\subsection{Signal amplification and digitization}
In the external electronics scheme, pre-amplification will be performed directly at the PMTs and a second amplification stage is located directly after the input of the FEE modules. The bandwidth of the amplification circuit has to be chosen appropriately to minimize information loss by shaping. From there, as displayed in Fig.~\ref{fig:Amplification}, the signal will be divided into two paths: One is fed to a discriminator in order to provide a trigger flag, the other is sent to an FADC for charge measurement. Due to the large dynamic range of the PMT signals (1\,pe to $\geq$4,000\,pe), the use of a single FADC for the digitization seems hard to achieve while meeting the resolution requirements for low pulses. Therefore, the amplification will be divided into two or three paths with different amplification gains \cite{Qiuju}. In each path, signal amplitude will be adjusted to match the input range of the FADCs. For most of the fast, high-resolution FADCs \cite{AD9234} the maximum range of the input voltage is only 0.5$-$1\,V. Furthermore, minimization of the input electronics noise is quite important to keep up the specified voltage resolution. This can be achieved both by design and by filtering.
The FADCs will be operated in differential mode to increase the performance. The foreseen sampling rate is 1\,GS/s, the required voltage resolution 14 bit. There are two possibilities: Either to purchase commercial chips or to design a customized ASIC. In the first case, the ADS5409 from TI and AD9680 from Analog Device would fulfill the requirements. If relying on a self-designed ASIC, the most practicable approach is a multi-channel time-interleaved architecture which will feature a low power consumption compared to a realization based on a single channel. For each channel, a 12\,bit, 500\,MS/s pipeline quantizer (or a 12\,bit, 250\,MS/s SAR quantizer) will be used for meeting a good balance between power, speed and accuracy. Offsets and gain mismatches between channels can be calibrated offline. A differential clock input will be used to control all internal conversion cycles.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/amplification.png}
\caption[Scheme for signal amplification and digitization]{\label{fig:Amplification}Scheme for signal amplification and digitization.}
\end{center}
\end{figure}
\subsection{Data processing and transmission}
In a first step the FEE system will process the data and provide some simple information to the trigger system. Only when a trigger decision is formed and a corresponding signal is sent back to the crates, the full data (including waveforms) of all channels will be read out, bundled to a combined output in units of modules and crates and then transferred to the DAQ.
There are 10 FEE modules plus 2 trigger modules in each crate. Data is transmitted between the modules via the backplane BUS system of the crates. Communication is done by a dual-star structure. Each FEE module can receive the trigger signal from either of the two trigger modules via a pair of high-speed serial connections. The digitized data is transferred via ethernet connection to a switchboard and from there to the DAQ. A block diagram of the system is shown in Fig.~\ref{fig:DataProcessingAbove}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{ReadoutElectronicsAndTrigger/figures/DataProcessingAbove.jpg}
\caption[Data processing and transmission in the external scheme]{\label{fig:DataProcessingAbove}Block Diagram of the data processing and transmission of the external electronics scheme}
\end{center}
\end{figure}
\subsection{Trigger system}
The design goals for the trigger system are the following:
\begin{itemize}
\item The trigger efficiency for IBD events induced by reactor $\bar\nu_e$ should be high, i.e.~reach 99\,\% detection efficiency for an energy deposit of $\sim$0.5\,MeV, so well below the minimum energy deposit of a prompt positron.
\item The system should be able to suppress detector-related backgrounds as PMT dark noise coincidences and surface radioactivity to reduce the overall event rate.
\item The latency introduced for the processing of 17,000 PMTs should not exceed a few $\mu$s.
\end{itemize}
\subsubsection{Trigger rates and threshold}
The expected trigger rates generated by neutrino signals and backgrounds were studied by simulations. Signal sources considered were IBD signals, residual radioactive background events in the LS, cosmic-ray muons, and the dark noise of PMTs. We assumed that the buffer was thick enough to shield the gamma-ray background from the PMTs and support structures.
The details on signal and background assumed are summarized in Tab.~\ref{tab:trigger}: Radioactive background levels in the LS were estimated conservatively based on the conditions in the Daya Bay detectors \cite{Qiuju}. However, radiopurity requirements for JUNO are much stricter. If i.e.~levels from KamLAND \cite{K.Eguchi} are used, the rates will be lower by several orders of magnitude. Muon background was estimated with 1,500\,mwe of rock overburden, taking into account a coarse image of the three-dimensional surface topology. Muon-induced spallation isotopes (i.e.~$^9$Li etc.) were not considered.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{llc}
\hline
Signal or background & Calculation standard & Event rate before trigger \\
\hline
IBD & 80\,/day & 80\,/day \\
Background in LS & ${}^{238}$U: 2.0$\cdot$10$^{-5}$\,ppb & 69\,Hz \\
& ${}^{232}$Th: 4.0$\cdot$10$^{-5}$\,ppb & 32\,Hz \\
Muons & 1,500\,mwe & 3\,Hz \\
PMT dark noise & 50\,kHz/PMT & 3.3\,MHz \\
\hline
\end{tabular}
\caption[Input for trigger rate MC]{\label{tab:trigger}Input for trigger rate simulations}
\end{center}
\end{table}
For the parameters characterizing the detector, we assumed an LS light yield of 11,000 photons per MeV, an attenuation length of 20\,m, a PMT coverage of 78\,\%, and a quantum efficiency of 35\,\% \cite{Dayabay}.
In the simplest approach, the trigger decision will be based on the hit multiplicity inside a coincidence window of 300\,ns (see above),~i.e. on the number of simultaneously occurring PMT hits. Fig.~\ref{fig:trigger coincidence} shows the event rate as a function of this number of coinciding PMT hits. Based on these spectra, it seems that a simple hit multiplicity trigger will return both sufficiently low trigger rates and high trigger efficiency for IBD events if the trigger threshold is placed at $\sim$500 coincident hits. This coincidence trigger scheme is technically easy to implement and will create only low latencies.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/TriggerCoincidence.jpg}
\caption[Event spectrum for coinciding PMT hits]{\label{fig:trigger coincidence} Event rates as a function of the number of PMT hits coinciding in a 300\,ns trigger window.}
\end{center}
\end{figure}
\subsubsection{Trigger setup}
Each FEE will handle 32 PMT signals. The hit flags of ten FEEs inside a crate will be bundled by one Level 1 Trigger Collection Board (L1TCB). In turn the information of ten L1TCB will be fanned to one level-2 board (L2TCB). The Trigger Board (TB) itself will receive data from six L2TCBs. This will be sufficient to read out the trigger flags of 17,000 channels. The L1TCBs and L2TCBs will also work as a fan-out for re-transmitting the trigger decision from the TB to individual FEEs.
\subsection{Clock system}
The clock system for JUNO has two major functions. It will provide:
\begin{itemize}
\item a standard reference frequency to all FEE modules to support the waveform sampling, TDC measurement and process logic;
\item an absolute timestamp for each event to correlate the events of the central and veto detectors and for comparison with other experiments.
\end{itemize}
Based on the experience gained in the Daya Bay experiment and following the latest developments, a distributed clock network based on the "White Rabbit" standard will be applied for JUNO.
The "White Rabbit" (WR) system is a technology originally developed by CERN and GSI for the use at accelerators. It is based on IEEE1588 (PTP) including two further enhancements: The precise knowledge of the link delay and a possibility of clock synchronization over the physical layer with Synchronous Ethernet (SyncE). Applying WR in JUNO will offer the following advantages:
\begin{itemize}
\item The low-jitter recovery clock from SyncE provides a reference frequency for the electronics.
\item Sub-ns clock phase alignment can be achieved among all nodes by phase measurement and phase compensation.
\item The precise time protocol (PTP) synchronizes the timestamp among all nodes.
\item An external Rubidium oscillator and GPS can be used for frequency and UTC reference.
\end{itemize}
The system will synchronize all electronics crates. Each crate will contain a custom-designed clock module to recover the reference frequency and the absolute timestamp. This will then be broadcasted to all FEE and trigger modules in the crate via the dedicated high-speed clock traces on the back-plane for ATCA crates and via front-panel fan-out cables for non-ATCA crates.
\subsection{Power supply and electronics crates}
Conventionally, high voltage for the PMTs would be generated by a dedicated HV supply outside the water pool. However, the preferred solution is the generation of the HV by a Cockroft-Walton type power supply either attached to the PMT base or in underwater units. In this case, these generators have to be fed with low or medium voltage cables from outside the tank. Solutions with voltages of 5\,V or 24\,V are conceivable, the former requiring a larger cable diameter to transfer the larger current without loss. In order to avoid the power loss by using a linear power supply, a DC/DC module should be used for high efficiency. In order to reduce the ripple, the power supply should be independent.
The FEE and trigger modules will be designed according to the specifications of the ATCA standard. Fig.~\ref{fig:ATCA Crate} shows the ATCA crate that will house the modules. It provides a large bandwidth for digital transmission which allows to pass trigger information between FEE and trigger modules via the bus system of the crate.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/ATCACrate0.JPG}
\caption[ATCA crate]{\label{fig:ATCA Crate}The ATCA crate}
\end{center}
\end{figure}
\section{Intermediate readout scheme}
\label{sec:roe:int}
The primary disadvantage of the external electronics scheme (Sec.~\ref{sec:roe:dry}) is the length of the coaxial cables needed for the transmission of the analog PMT signals. This will lead to a shaping of the leading edge of the signals, deteriorating the time resolution. To avoid this problem, the intermediate scheme shown in Fig.~\ref{fig:under water scheme} places not only the pre-amplifier but most of the FE electronics including digitization in a water-tight box inside the tank, considerably shortening the signal transmission length for the analog signals. Each of these 500 submerged processing units will supply an array of 32 PMTs. From the box, only few cables carrying the digitized signals as well as trigger and clock information will run to the external electronics.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{ReadoutElectronicsAndTrigger/figures/UnderwaterScheme.jpg}
\caption[Intermediate scheme for readout electronics]{\label{fig:under water scheme}Schematic drawing of the intermediate, partially submerged readout scheme.}
\end{center}
\end{figure}
\subsection{Signal processing}
In this partially submerged configuration, the analog PMT signal will be transmitted only over a short cable to the amplification stage included in the underwater processing units (cf.~figure\ref{fig:Amplification}). Therefore, the signal shape will not change significantly. Otherwise, the design of the FEE boards inside the processing units will be mostly analogous to the external scheme. The digitized signals will be bundled for all channels connected to one box and passed by an optical fibers to external modules collecting the acquired data. From there, data will be passed on via ethernet. The block diagram is shown in figure\ref{fig:DataProcessingUnder}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/DataProcessingUnder.jpg}
\caption[Readout Electronics]{\label{fig:DataProcessingUnder} Block Diagram of the data processing and transmission of the under-water scheme}
\end{center}
\end{figure}
\subsection{Trigger and clock systems}
To simplify the design and maintenance of the trigger system, the trigger modules will be located with the other external electronics. The trigger flags from the FEE discriminators as well as the trigger signals can in principle be transmitted on the same optical fibers as the data. It is possible to transmit the same optical cable the data. However, separate cables might be used for the trigger system in order to increase the flexibility.
As for the external scheme, a White Rabbit system can be applied for synchronization of the FEEs in the watertight boxes, providing both a reference frequency for FADCs and an absolute timestamp. In case the optical fibers required by White Rabbit are not compatible to oil, a dedicated clock and data recovery (CDR) link could be designed to provide the reference frequency. The physical calibration system could be exploited to determine the absolute timestamps of events relative to calibration pulses fed in from outside for which GPS time would be known.
\subsection{Submerged processing units}
An important aspect for the long-term stability of the system is the possibility to replace the part of the electronics submerged in the water, i.e.~the processing units. The structure of a readout strand including one processing unit is shown in Figure\ref{fig:underwater changeable system}. The enclosing box will be 30\,cm high, 40\,cm wide and 60\,cm long. There will be mounting holes and the possibility to remove the top lid. Fig.~\ref{fig:watertight case appearance} shows a three-dimensional representation, Fig.~\ref{fig:watertight case} displays the side panels: The left panel of each watertight case is connected to coaxial-cables by 32 connectors, while the right panel is connected to a composite cable including an optical fibre for signal transmission and a power cable for HV generation inside the underwater box.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{ReadoutElectronicsAndTrigger/figures/UnderwaterChangeableSystem.jpg}
\caption[Representation of the intermediate scheme]{\label{fig:underwater changeable system}Structural representation of the readout chain in the intermediate scheme.}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/WatertightCaseAppearance.jpg}
\caption[Readout Electronics]{\label{fig:watertight case appearance}The diagram of the watertight case appearance}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/WatertightCase.jpg}
\caption[Side panels of submerged processing units]{\label{fig:watertight case}Diagrams of the side panels of the submerged processing units}
\end{center}
\end{figure}
The further components shown in Fig.~\ref{fig:underwater changeable system} have to fulfill a number of requirements:
\begin{itemize}
\item Cable $Aa$ should be coaxial, 10$-$20 meters long, and with a characteristic resistance of 50\,$\Omega$.
\item Connector $A$ is watertight.
\item Connector $a$ is watertight and has to be pluggable under water.
\item Cable $Bb$ is a composite cable of 100\,m length containing at least 4 optical fibres and 4 power wires (24-48V, 10-20A) usable for underwater environments.
\item Connector $B$ is watertight and does not have to be pluggable under water.
\item No special requirements are set for connector $b$.
\item Parts $B$, $Bb$ and $b$ can be replaced by non-watertight cables covered by a corrugated pipe.
\end{itemize}
\section{Underwater readout scheme}
\label{sec:roe:wet}
The underwater concept places almost the entire readout electronics inside the water pool, as close as possible to the PMTs themselves. Front-end electronics (FEE) and HV generation are done at the PMT base, all data transfer is done based on digitized data and bundled in central underwater units (CUU). Thus, the underwater scheme minimizes the deteriorating effects of analog signal transmission over long cables and the overall amount of cables needed. On the other hand, it seems unlikely that broken FEE channels can be replaced during operation, while CUUs probably can.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/Scheme2.png}
\caption[Readout Electronics]{\label{fig:Scheme 2}Schematic drawing of the underwater readout scheme.}
\end{center}
\end{figure}
\subsection{Overview}
The layout of the fully-submerged readout scheme is sketched in Fig.~\ref{fig:Scheme 2}. At the end of each PMT there is a small water-tight housing that contains the front-end electronics (FEE) with all the essential electronics of the system. It contains the base of the PMT and a module that generates the HV from a low-voltage AC input. And it contains a PC-board with a system on a chip (SoC) that includes the digitization of the signal, the discriminator to detect photoelectrons, and other functionalities.
The FEE is connected to the online computer farm through data links. It transmits the data to the farm and receives commands from it. It also receives a clock signal from a central clock unit and the power for the PCB and the HV from corresponding power supplies.
In order to reduce the cabling the PMTs are grouped together under water. We estimate that up to 128 PMTs might be grouped together. In this case the full detector would consist of 128 groups. The grouping of the PMTs takes place in central underwater units (CUU) mounted between the backend of the PMTs. To reduce the length of the cables from the PMTs to the CUUs the groups will be formed from nearby PMTs. The cables from the PMTs to the CUUs will have a unique length somewhere between 5 and 10 meters. They contain a by-directional data link, the clock signal, the low voltage power for the PCB (probably 3V), a 24 V AC power for the generation of the HV and a ground potential. On the PMT end the cables will be glued into the housing of the FEE. On the end of the CUU there will be a water-tight connector to allow for the mounting. From the CUU there will be a by-directional data link to the farm (probably two for redundancy) and power cables and the connection to the clock on top of the detector.
\subsubsection{The `intelligent' PMT}
With the FEE the PMT is turned into an `intelligent' PMT that can be connected directly to a computer. No further electronics is needed. Besides the module that generates the HV and contains also the voltage divider to derive all potentials there is a PCB in the FEE with an ASIC. It contains the SoC with the following functionalities:
\begin{itemize}
\item Amplification of the signal.
\item Shaping of the signal, should it be necessary.
\item Regulation of the baseline to zero.
\item Detection of single photon signals through a leading-edge discriminator.
\item Generation of a time-stamp for each readout window.
\item Digitization of the input signal with an FADC.
\item If necessary, a TDC to measure the precise start of each pulse (relative to the time-stamp of the readout window.
\item Feature extraction, i.e. determination of the number of photons contributing to a pulse and their arrival-time.
\item Self-calibration, i.e.~adjustment of the charge integral of single photons to a predefined value through a regulation of the HV and the amplification of the input stage.
\item Creation of test pulses and injection into the input of the chip and into the PMT.
\end{itemize}
These functionalities are provided for each of the two MCPs of one PMT individually (2 channels). The following additional functionalities are common to both channels:
\begin{itemize}
\item Creation of an optical test pulse through an LED on the PCB that illuminates the photo cathode.
\item Creation of an internal clock signal with the appropriate frequency locked to the input of the external clock.
\item Internal functionality tests.
\item Data exchange with the CUU.
\item Limited data buffer.
\end{itemize}
A sketch of the SoC components is displayed in Fig.~\ref{fig:SoC}. Its functionalities are described in more detail in Sec.~\ref{sec:roe:ipmt}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{ReadoutElectronicsAndTrigger/figures/SoC.png}
\caption[Readout Electronics]{\label{fig:SoC}The SoC scheme}
\end{center}
\end{figure}
\subsubsection{Central Underwater Units}
The CUU functions as the connection between the PMTs of the assorted array to the infrastructure on top of the detector. It receives the data from the PMTs. It combines the data and sends it by a single optical link to the computer farm. It must have a sufficient data buffer to compensate fluctuations in the data rate. At the moment it is envisioned to use a commercial PC-board for the CUU. For the data-links to the PMTs the USB-2 standard might be appropriate. Commercial USB-routers could combine the data from the PMTs into a single data-stream which is then sent to the farm through an optical 10\,GBit/s link. If this works out, no development is necessary for the hardware of the CUU. The CPU power of the CUU could be used for monitoring, pre-sorting of the data, data reduction, or other purposes.
The most critical parts of the CUU are the many connections attached to the CUU in an under-water environment (at least one connector per PMT plus the up-links). The distribution of the power and the clock to the PMTs will also be done inside the housing of the CUU, so that only one combined cable with connector is necessary for each PMT. For the uplink this is less critical. One could use a power cable separate from the fiber(s).
A substantial amount of monitoring will be necessary to control the detector. The CUUs are also the proper place where the data can be injected into the data stream. Sensors for temperatures, voltages, scintillator flows, can be mounted in or near the CUUs and connected to dedicated inputs of the PC. Monitoring data on the PMTs will be collected by the SoC in the FEE and transmitted through the data link to the CUU.
An interesting alternative to the routing of the data as described above is a passive routing of many data streams through one fiber. Such systems transmit each data stream with light of a different color. They are commercially available. They are potentially more reliable than a CUU based on a PC in a water-tight housing. In this case we would need a connector on the PMT to connect its fiber and a separate power cable to connect to a power distribution box under water.
\subsubsection{The Online Computer-Farm}
The online computer farm will receive the data through 128 optical links with 10\,GBit/s capacity. The data will arrive asynchronous. First is has to be sorted into proper time order. Then events are created as time-slices with a minimum activity of the PMTs. These proto-events are subjected to a reconstruction algorithm and from there the decision is derived whether to keep the event or to drop it (or to keep a reduced amount of data). After another sorting in time the accepted events are permanently stored. More details need to be worked out by the DAQ group.
\subsubsection{Expected data rates and buffer sizes}
As PMT pulses are converted early into digitized data, the amount of data to be transferred via fibers and data links as well as local buffers for temporary storage are important characteristics in this scheme. An estimate of the data volume to be processed starts with the amount of digital data caused by the detection of a single photoelectron (SPE). We assume a sampling rate of 500 MHz with one out of three 7-bit FADCs. With a typical rise time of 5\,ns and a fall time not much longer a time window of 32\,ns per spe is quite conservative. With the MCP-PMTs we will always digitize both channels at the same time. The data volume per spe is summarized in the table\ref{tab:Data Volume}: A single waveform is recorded with 144\,bits. For two channels and with the auxiliary information this adds up to 360 bits per photoelectron. For the potential TDC we have assumed 8\,bits with 125\,ps per unit to cover the full length of the readout window.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{lcl}
\hline
{\bf Data volume of a single waveform}\\
sampling frequency & 500\,MHz \\
readout window & 32\,ns \\
number of time-slices per readout & 16 \\
number of bits & 7 \\
range bits & 2 \\
number of channels & 2 \\
\hline
Intermediate sum & 288\,bit \\
\hline
{\bf Additional information} \\
address (PMT-ID within group, channel) & 8 \\
time stamp & 40 \\
control flags & 8 \\
TDC (2 x 8 bit) & 16 \\
\hline
{\bf Total sum} & 360 bit \\
\hline
\end{tabular}
\caption{\label{tab:Data Volume}Data volume needed per single photoelectron.
\label{Data volume per single photoelectron.}}
\end{center}
\end{table}
Based on the spe volume, the data rate to be transmitted between a PMT and the CUU can be derived. It is completely dominated by the dark rate of the PMT consisting of single photons. We assume a conservative 50\,kHz for the dark rate, which corresponds to 18.2\,Mbit/s. Any data volume of physics events is negligible compared to the dark rate, even if the physics events contain much more than single photons. For comparison the maximum data rate of a USB-2 link is 280\,Mbit/s.
In the event of a supernova the data rate will probably exceed the capacity of the link to the CUU. Therefore a buffer is needed that allows a temporary storage of the data on the FEE itself. A nearby ($\sim$3\,kpc) supernova will create 60,000 neutrino events in the detector within $\sim$10 seconds. The event energy ranges up to 50~100\,MeV and will create relatively large pulses in almost all of the PMTs. Therefore, we assume that the average sampling period per neutrino event corresponds to five 32\,ns readout-windows for each PMT. This produces a data volume of $60,000\times5\times360\,{\rm bit} = 13.7$\,MByte for each FEE channel. On top of this there is the data volume from the dark noise which adds up to 22.8\,MByte in 10 seconds. The total sum is 36.4\,MByte. This is the minimum size of the buffer on each FEE.
Further along the readout chain, the uplink from the CUU to the computer farm has to be considered. A maximum of 128 PMTs will be combined into one group read out by a single CUU. For smaller numbers the requirements are less severe. To obtain rates and data volumes, the requirements for of a single PMT have to be multiplied by 128, corresponding to a data rate of 2.3\,Gbit/s for the link. A link capacity of 10\,Gbit/s which is available for standard PCs today will be sufficient. The overall buffer storage needed for a supernova event amounts to 4.7\,GByte. The data can potentially be stored in the RAM of the CUU.
\subsection{The `intelligent' PMT}
\label{sec:roe:ipmt}
This section describes the functionalities of the single elements contributing to the underwater readout scheme.
\subsubsection{High Voltage}
The high voltage (HV) will be generated directly on the PMT from a low-voltage input. A single HV potential will be generated for each PMT from which all voltages required for the photo cathode, the field shaping electrodes, and the two MCPs are derived through a voltage divider. The final PMTs will have their cathodes on ground potential. Consequently the output of the signal will be on positive HV. Fig.~\ref{fig:MCP base} shows the corresponding circuit proposed by the Dubna group. The HV will be derived with a Cockcroft-Walton multiplier, also known as a Greinacher cascade, displayed in Fig.~\ref{fig:Greinacher}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{ReadoutElectronicsAndTrigger/figures/MCPbase.jpg}
\caption[Voltage divider for the MCP-PMT base]{\label{fig:MCP base}Circuit for the voltage divider at the MCP-PMT base}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/Greinacher.png}
\caption[Cockroft-Walton multiplier]{\label{fig:Greinacher}The Cockroft-Walton multiplier for HV generation.}
\end{center}
\end{figure}
The Dubna group has already developed a working module for the 8'' MCP-PMTs, which is displayed in Fig.~\ref{fig:MCP-PMT prototype}. The input is an AC voltage of 24V. For the final module the input voltage might be higher (up to 200V). The value still needs to be decided. The cascade is embedded in a HV-module that allows regulation of the output voltage. The module already includes the voltage divider and the capacitor necessary to decouple the signal from the HV. It also has the ability to monitor and limit some of the parameters, i.e.~the output current. The current prototype is controlled through a USB interface. The final version will be controlled by the SoC on the PMT.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/MCPprototype.jpg}
\caption[Prototype for HV module]{\label{fig:MCP-PMT prototype}Prototype of the HV module for the 8'' MCP-PMT prototype}
\end{center}
\end{figure}
\subsubsection{Clock System}
It is essential that all PMTs are running with the same frequency. This frequency will be provided by a central clock in the cavern above the detector. The clock signal is distributed to all PMTs. For most physics topics knowledge of the absolute time is not required, with the exception of the signals from a nearby supernova. A GPS-system that allows absolute timing should be considered for this case. The GPS receiver must be mounted on the surface, probably in one of the entrance areas.
The cables or fibers that connect a PMT to the central clock generate a time delay between the clock in the PMT and the central clock. These are called time-offsets $t_0$. For the reconstruction of events in the detector these offsets must be calibrated on the level of 100\,ps. If the system runs with a global trigger, a rough calibration is necessary already during running (maybe on the level of 1\,ns). For the untriggered system online-calibration is not required. The values of the $t_0$'s can be measured from calibration runs with a light-pulser (LED) in the center of the detector and from the data itself. The electronics must ensure stable $t_0$'s over longer running periods to reduce the calibration runs to a reasonable frequency.
A White Rabbit system has been proposed to calibrate the offsets between the PMTs and the central clock . Such a system seems well suited for a triggered system. For the untriggered system such an effort is not necessary. Offsets ($t_0$'s) from the previous calibration run will be more than sufficient. In the event of a nearby supernova all offsets (including the offset between the GPS receiver and the central clock) must be calibrated in absolute terms (maybe on the level of 1\,ns). The calibration can be done before or after the supernova event. A calibration after the event would save the effort, if no supernova appears, but a plan must be available in any case.
The simplest method to distribute the clock to the PMTs is through individual cables. Distribution of the clock signal could use the same infrastructure as the CUUs, i.e.~a single cable transmitting the signal from the surface to the CUU that than spreads it out to the whole group of PMTs. Alternatively one could add the clock signal to the power cables or distribute it through the data links. In principle a wireless distribution is also possible. The central clock will produce a reference frequency from which the PMTs derive their frequency through a phase lock loop (PLL). Most likely the reference frequency will be lower than the PMTs frequency. To give an example: For a 500 MHz digitization on the PMTs one could distribute a 62.5 MHz reference frequency and divide the clock cycles by a factor 8 on the PMTs.
\subsubsection{Baseline Adjustment}
The output of the PMTs will be on the potential of a positive high voltage. A capacitor will be used to decouple the signal from the HV. This capacitor will be integrated into the HV module. The potential behind the decoupling capacitor will be referenced to ground. For the determination of the charge integral of signal pulses a precise knowledge of this potential (i.e.~the signal baseline) is essential. One possible approach uses the FADC to measure the baseline directly prior to the pulse. The digitization of the input is started already before the arrival of the pulse. It is important not to under-sample the baseline. It must be determined with the same precision as the pulse itself. The baseline must be sampled for the as long as the pulse itself, doubling the data output.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/BaselineAdjustment.png}
\caption[Circuit for baseline adjustment]{\label{fig:Baseline adjustment}Circuit for the baseline adjustment}
\end{center}
\end{figure}
Here we propose to use a different approach. We want to measure the baseline continuously and adjust it to ground potential through a feedback loop (Fig.~\ref{fig:Baseline adjustment}). The concept is very simple. A discriminator compares the input against ground potential with the frequency of the clock. Due to noise on the input the input potential will fluctuate between positive and negative potentials. On average, time slices with positive ($+$'s) and negative ($-$'s) voltage bits will be balanced. The discriminator will produce as many $+$'s as $-$'s. If the input potential changes, the balance between $+$'s and $-$'s will be distorted. For example an excess of $+$'s will indicate a deviation of the input potential to positive values. A simple logic will determine the difference of the number of time slices with $+$'s and $-$'s. The result will be used to adjust the potential in one of the input amplifiers until the balance is restored.
Two special situations must be considered to avoid a bias in the baseline:
\begin{itemize}
\item If a true pulse arrives the feedback loop will detect it as a positive deviation of the baseline and counter-correct the deviation. To avoid this bias the feedback loop has to be stopped if an input pulse is detected.
\item The return of the signal to the baseline after a pulse will take some time. If a second pulse arrives before the signal has returned to zero, its charge integral will be measured with a too large value. This effect can be corrected offline. All photon pulses will be read out. Time and charge of the previous pulse can be determined independently. An algorithm on the online-farm can either correct for the previous pulse, or store the previous pulse for later correction, or do both.
\end{itemize}
There is, however, one special case, where this will not work. If the pulseheight of the previous pulse is below detection threshold, the correction cannot be applied. But one should keep in mind, that close pre-pulses will be rare, the undetected fraction of pre-pulses even rarer, and the bias will be very small, especially for pulses below threshold.
A number of parameters of the baseline adjustment need to be optimized. The difference of $+$'s and $-$'s outputs of the discriminator needs to be averaged over a certain amount of time slices. The larger the number, the more precise the adjustment becomes, but it also gets slower. This parameter should be kept flexible to adapt to the situation in the real experiment. Another important parameter is the step-size with which the baseline is adjusted.
The overall concept of the intelligent PMT will include a random trigger. That is, at adjustable time intervals the electronics will digitize a readout window and store the output. These readout slices should be empty and can be used to check the performance of the baseline adjustment.
\subsubsection{Internal Trigger}
A simple level discriminator will be used to trigger the readout of a single PMT. It will continuously monitor the output of the PMT. The threshold must be programmable. It will be set directly above the noise, as low as acceptable in terms of rate of noise triggers. The threshold can be adjusted individually for each PMT. We envisage a threshold at a level of 0.3 photoelectrons. In the case of the MCP-PMTs there will be separate triggers for each MCP. Most likely the electronics will be configured in such a way that a signal above threshold from one of the MCPs will trigger the readout of both channels.
The readout window will be adjusted to the length of a pulse from a single photoelectron, probably around 32\,ns. In the case of high energy events such as cosmic muons or atmospheric neutrinos several or even many photons will be detected by a PMT. The output signal will extend over more than 32\,ns. If the photons are separated in time, they will be read out as individual triggers. If not, the full signal will be covered by reading several consecutive 32\,ns windows. The trigger logic monitors the threshold discriminators. If at a certain time during the readout (i.e. in the middle of the 32\,ns window) the signal has not returned to below threshold, the logic will continue the readout for another 32\,ns. The readout will be sustained until the signal finally falls below threshold. No information is lost.
\subsubsection{Flash-ADC}
A FADC will digitize the input signal. For the MCP-PMTs two separate FADCs will be needed for the two MCPs. The required dynamic range of amplitude has been estimated to 1000\,pe for a whole 20'' MCP-PMT, or 500\,pe per MCP. The prototypes of the MCP-PMTs show an electronic noise level which is in the order of 10\,\% of the amplitude of a single photo-electron. Therefore a 10\,\% resolution on the amplitude of a single pe will be appropriate. The corresponding resolution on the spe charge integral will be better, on the level of a few percent. If the amplitude of a single pe corresponds to 10 ADC units, a dynamic range of 5,000 to 10,000 ADC units will be needed, corresponding to 13 bits. It is technically not possible to derive 13 bits from a single ADC. Multiple ADCs with staggered ranges are required. We propose to use 3 ADCs with 7 bits each, featuring the following scheme:
\begin{center}
\begin{tabular}{ccc}
\hline
amplitude & range & resolution \\
\hline
$\times1$ & 0$-$13\,pe & 0.1\,pe \\
$\times1/8$ & 0$-$100\,pe & 0.8\,pe \\
$\times1/64$ & 0$-$800\,pe & 6.4\,pe \\
\hline
\end{tabular}
\end{center}
\subsubsection{Time Measurements}
For the reconstruction of vertices or tracks it is necessary to measure the time of arrival of the photons on the PMTs. A resolution of 1\,ns is required. The arrival time can be extracted from the waveform recorded by the FADC. With the proposed 500-MHz FADC (2\,ns time slices) it is possible to extract the starting time with 1\,ns resolution.
More precise timing information is encoded in the digital pulse of the level discriminator of the internal trigger. The discriminator triggers the first photon of the pulse. This is the most valuable information for the reconstruction. A TDC could be included to measure the time of the trigger relative to the preceding clock pulse. For example to get a 500\,ps resolution a 2-bit TDC would be sufficient.
\subsubsection{Self-Calibration}
The ASIC will contain a mechanism to calibrate its PMT to a predefined charge integral of a single photoelectron (SPE). The basic scheme is shown in Fig.~\ref{fig:Self-calibration}. Whenever the input signal crosses the trigger threshold for a single photoelectron the flash-ADC will digitize the input and write the data into an internal memory. The logic in the digital part of the chip will analyze the pulses within a single photoelectron window and determine the average charge integral, $\langle q_{\rm pe}\rangle$. The $\langle q_{\rm pe}\rangle$ of each channel is compared to a default charge value, and the HV applied to the PMT adjusted until the predefined value is reached.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/Selfcalibration.png}
\caption[Self-calibration scheme]{\label{fig:Self-calibration}Schematic drawing of the self-calibration feedback loop}
\end{center}
\end{figure}
The presence of two MCPs with different gains inside one PMT adds a complication because both MCPs are powered from a common high voltage. The algorithm on the chip will compare the $\langle q_{\rm pe}\rangle$ of both MCPs to the predefined value. It will first adjust the gain of the input amplifiers until the $\langle q_{\rm pe}\rangle$ of the two MCPs agree with each other. Then it will adjust the high voltage to match the $\langle q_{\rm pe}\rangle$ to the predefined value.
\subsubsection{Internal test devices}
The circuitry will incorporate extensive possibilities for testing. This is important to guarantee the functionality of the system and to understand how to react in case of problems. The details still need to be worked out. Two test installations that will be integrated for sure are:
\begin{itemize}
\item A test pulser within the SoC: The pulser can be used to inject charge into the input of the chip. It could either be an analogue circuit which produces pulses similar to the real pe's or a digital pulse generator.
\item A light pulser: A LED will be mounted in the electronics in a position where the light illuminates the transmission cathode from the backside (through the glass neck of the PMT).
\end{itemize}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/PMTMechanics.png}
\caption[Mechanical realization of the intelligent PMT ]{\label{fig:PMT Mechanics}Mechanical realization of the intelligent PMT design.}
\end{center}
\end{figure}
\subsubsection{Mechanics}
The scheme for mounting the FEE directly to the base of the PMT is shown in Fig.~\ref{fig:PMT Mechanics}: The HV module plugs directly onto the pins that connect to the electrodes in the PMT. It includes the wire(s) that carries the output signal. A capacitor is needed to decouple the signal from the high voltage. This capacitor will be included in the high voltage module. On top of the high voltage module there will be a PCB that carries the ASIC with the SoC and some external components (voltage regulator for the low voltage, etc.). The cable to the central underwater units is connected to this PCB. The whole electronics is inside a watertight metal cap that connects to the mechanical fixture of the PMT. The details on how to seal the housing still need to be worked out.
For the default scenario, where data is routed through the central underwater units (CUU) there is a single cable from each PMT to the CUU. It contains power, clock and the data line. It is firmly attached to the electronics housing on the PMT without a connector. The cable length will be between 5 and 10 meters. It has a watertight connector that connects to the CUU. For redundancy there might be two such cables per PMT connecting to two different CUUs.
In case passive data links will be used, an optical fiber will be connected to each PMT. The driver circuit and the laser will be included on the PCB. There must be a watertight optical connector on the fiber where it connects to the PMT. For each PMT there will be a separate cable for the power that connects the PMT to the power distribution box.
\subsubsection{Power}
Each PMT needs a low voltage DC-power for the SoC and a low voltage AC-power for the high voltage module. The exact values have not yet been defined. For the time being we assume a +3V DC-voltage for the SoC and a +24V AC-voltage for the high voltage module. The voltages are generated in power supplies in the cavern on top of the detector. They are accessible and can be replaced in case of failures. From the power supplies, cables will transmit the power to distribution boxes inside the water pool. Each PMT is connected to one of the boxes with a cable and a water tight connector on the power distribution box. The cable contains the DC and the AC voltage, the ground potential and maybe the clock. In case the scheme with active data links is adopted, the power distribution will be integrated into the CUUs. In the case of passive data links there will be distribution boxes just for the power.
One power distribution box or CUU will supply power to a group of 128 PMTs. This system must be protected against failure of the whole group. If a cable to one of the PMTs breaks, a single PMT will be lost which is not catastrophic. However, a short in one PMT could take out the whole group. Therefore a serial resistor RP has to be introduced in the PMT contacting scheme shown in Fig.~\ref{fig:MCP base}. These resistors must be large enough to limit the current through a PMT with a short, at the same time avoiding a too large power consumption in the resistor if the voltage on the PMT changes. For example, if 3\,V is required for the SoC, a 6\,V power supply could be used. We choose a value for RP that creates a 1\,V voltage drop. A voltage regulator located on the PCB at the PMT base will reduce the remaining 5\,V to 3\,V. The protection resistor increases the power consumption by 20\,\%. which will still be acceptable. In case of a short the power of this channel goes up by a factor of 5. This is still negligible compared to the power of the whole group.
Another critical item is the cable from the power supply to the distribution box. It might break or shorten out. We propose to have two identical cables, but only one is connected to the power supply. Inside the distribution box there are switches on the power. Without power the switches are closed. Let's assume cable A is connected to the power supply. If we turn on the power, it will open the switch on cable B and disconnect it completely from the system. If cable A fails, it will be disconnected from the power supply. This closes the switch to cable B. Cable B can now be connected. It will open the switch on cable A and disconnect it from the system, bringing the system is back into operation.
\section{Development plan}
\label{sec:roe:dev}
\subsection{Characterization of PMT response}
To get a better understanding of the dynamic range and voltage span required, the PMT response and in particular the pulse height and linearity will be studied as a function of the incident number of photo-electrons. For this, a laboratory test stand is currently realized. As the MCP-PMTs are not yet available, characteristics will be first determined for the 20'' PMT produced by Hamamatsu. For covering the whole range specified, waveforms containing 1$-$4000 pe have to be recorded.
To match this requirement, the tests will be performed following this procedure:
\begin{enumerate}
\item Record {\bf single photoelectron pulses} at a gain level of $~$10$^7$. The right gain is assured by integrating the single photoelectron waveform (cf.~Fig.~\ref{fig:waveform for SPE}), subtracting the baseline pedestal and dividing by the elementary charge. An additional factor of 2 has to be taken into account as a double-end termination is used in the PMT base.
\item {\bf Ramp up of LED intensity} until 4000\,pe are reached, while continuously recording the waveforms for different LED output and thus multiple photoelectron levels. As before, the equivalent charge will be calculated based on waveform integration.
\end{enumerate}
At low hit multiplicities, a discret cascade of broadening peaks will be observed that corresponds to defined numbers of photoelectron. At higher LED intensities, the charge spectrum will become a Gaussian distribution. The number of incident photoelectrons can be calculated based on the gain.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/SPEWaveform.jpg}
\caption[Readout Electronics]{\label{fig:waveform for SPE}The waveform for SPE}
\end{center}
\end{figure}
\subsection{Signal amplification}
A careful study of the amplification requirements will be performed for the three readout schemes, including pre-amplification, gain division, and ASIC design. The relevant design specifications are the dynamic range of 1$-$4000\,pe and an equivalent input noise RMS of less than 0.1\,pe. The following aspects should be investigated:
\begin{itemize}
\item {\bf Impedance matching for long cable.} Reduce the reflection between transmission lines and other devices, considering the effect of parasitics and couplings of cables and transmission lines.
\item {\bf Gain division.} The dynamic range is divided into three overlapping FADC acquisition ranges, 1$-$67\,pe, 1$-$529\, and 1$-$4000\,pe. As the original amplification by the PMT dynode chain or MCP will depend on the PMT chosen in the final design, so will the amplification stages.
\item {\bf Amplifier selection.} Commercial amplifiers have to be selected for large dynamic range and low noise. Signal clipping and overload protection will be studied carefully for MCP signal.
\item {\bf Amplifier for FADC driver.} Since the PMT signal amplitude is large, the amplification factor will be small (less than 2), which makes the noise of ADC driver important to the resolution of the total system. Low noise of this amplifier is thus mandatory. Other performance parameters such as settling time have to be studied.
\item {\bf Discriminator selection}. A discrimination level of less than 1/4 pe is required.
\item {\bf ASIC design.} Amplifiers and discrimators have to be integrated in an ASIC holding 16 or 32 channels.
\end{itemize}
\subsection{Cables and connectors}
\subsubsection{External readout scheme}
The external scheme will employ 20,000 coaxial cables of 80$-$100\, in length and compatible with the underwater environment. Due to the great length, special attention should be given to low signal delay and attenuation to ensure a high quality for the transmission of the analog signals.
A selection of coaxial cables from the have been surveyed Tianjin 609 Cable Co. Ltd.. Samples of 100\,m length have been tested for attenuation. Results are listed in Table\ref{tab:Coaxial Cable}. While the Type III cable clearly shows the best performance in terms of signal dampening, it is also the most expensive.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{lccc}
\hline
Parameter & Type I & Type II & Type III \\
\hline
Product Model & SYVF -50-3-1 & 609C5021A & 609C5019A \\
Outer Diameter & 10.5\,mm & 3.5\,mm & 4.5\,mm \\
outer sleeve & ordinary sleeve & FEP & FEP \\
Attenuation @ 100\,MHz & 17.7\,dB & 15.9\,dB & 12.3\,dB \\
Attenuation @ 200\,MHz & 24.8\,dB & 23.4\,dB & 17.7\,dB \\
Price (RMB/m) & 10 & 14 & 18 \\
\hline
\end{tabular}
\caption{\label{tab:Coaxial Cable} Survey and attenuation results for three samples of 100\,m long coaxial cables from Tianjin 609 Cable co.}
\end{center}
\end{table}
The sleeves have also been tested for compatibility with the LS. The effects on the attenuation spectra of LS samples that have been left for some time in contact with the cables are shown in Fig.~\ref{fig:compatibility of the sleeves and LS}. The resulting changes for cables of Type II and III are less severe than for Type I.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{ReadoutElectronicsAndTrigger/figures/CompatibilityOfSleevesAndLS.jpg}
\caption[Compatibility tests of cables and LS]{\label{fig:compatibility of the sleeves and LS}Effect of the exposure of LS to the cable sleeves on the LS attenuation spectrum.}
\end{center}
\end{figure}
\subsubsection{Intermediate readout scheme}
The requirements for the cabling of the intermediate readout scheme are somewhat different. As the FEE are moved into the submerged processing units, watertight boxes as well as underwater connectors are needed. In detail, this scheme needs:
\begin{itemize}
\item About 20,000 coaxial cables of 10$-$20\,m length for the signal transmission from PMT to the processing units. Cables should be covered by a sleeve from fluoroplastics.
\item About 20,000 connectors that can be plugged under water. Fig.~\ref{fig:pluggable connector} displays suitable products by the Shanghai Rock-firm Co.~Ltd.
\item About 500 watertight cases for the central processing units
\item About 500 composite cables for data transmission via optical fibers and low-voltage power supply
\end{itemize}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/PluggableConnector.jpg}
\caption[Underwater connectors]{\label{fig:pluggable connector}Under-water connectors from Shanghai Rock co.}
\end{center}
\end{figure}
The feasibility of this scheme is currently evaluated at IHEP. Contacts have been established to a variety of providers for watertight connectors and cases, including Beijing the Great Wall Electronic Equipment Co., Ltd. China aviation optical-electrical technology CO., Ltd. Shanghai Rock-firm CO., Ltd. No.23 Research Institute of China Electronics Technology Group Corporation and others. While watertight cables can in any case be provided, the sleeve material and the fluorination treatment of the rubbers are still under investigation. This is an important aspect as fluorination will protect the rubber against the corrosive strength of pure water, allowing for a life time of 10$-$20 years. Further important questions are the impermeability of the watertight cases, the characteristics of the signal transmission and last but not least the additional cost introduced by this scheme.
\subsection{ADC selection and circuit design}
A further important step is the selection of the FADC chip. Several FADC chips have been included in a preliminary survey. Tab.~\ref{tab:ADC Select} summarizes the main characteristics for a variety of commercial and self-developed chips. Circuit designs are currently studied to evaluate the FADC performance and its compatibility.
\begin{table}[hbtp]
\begin{tabular}{l|lcccc}
\hline
& & Sampling & No.~of & Resolution & \\
Type & Model & rate [GS/s] & channels & bits & ENOB \\
\hline
FADC & ADC10D100D & 2/1 & 2 & 10 & 9.1 \\
& ADC12D1000 & 2/1 & 2 & 12 & 9.6 \\
& EV10AQ190A & 1.25 & 4 & 10 & 8.6 \\
& ADS5409 & 0.9 & 2 & 12 & 9.8 \\
& ADS5407 & 0.5 & 2 & 12 & 10.3 \\
& Stefan's (ASIC) & 1 & 1 & N/A & N/A\\
& Fule's (ASIC) & 1 & 1 & 12 & >10\\
\hline
Sampling & DRS4 + AD9252 & 1-5 & 8 & 11 & 8(?) \\
+ ADC & Weiw's (ASIC) & 1 & 1 & N/A & N/A\\
\hline
\end{tabular}
\caption[Overview of FADC chips]{\label{tab:ADC Select}Overview of selected (Flash-)ADC chips}
\end{table}
\subsection{Design of waveform digitization ASIC }
To fulfill the requirements of JUNO concerning timing and pulse height resolution, the acquisition and digitization of PMT waveforms at a sampling rate of 1\,GS/s and an acquisition range of 12\,bit is mandatory. However, commercial high-speed FADCs meeting these specifications are currently banned from import to mainland China. While possibilities exist to obtain suitable chips by transfer via a Sino-American collaboration, the resulting price will be undoubtedly high. In order to reduce the cost per electronics channel, design efforts for implementing a new form of ADC on an ASIC or for a self-made high-speed FADC have been started.
\subsubsection{Design of an ASIC for high-speed waveform sampling}
The bottleneck of the design of the high speed-waveform sampler is the huge data throughput. It is very difficult to store a huge data volume within a very short time as it would be encountered during a supernova. However, as long as the event rate is limited, continuous storage will not be necessary as most of the acquired date is either baseline or background signals.
The waveform-sampling chip will rely on the online selection and caching of the analog data. The chip will first sample the waveform at high speed, and save the resulting waveform information in an analog memory. When there are no physics triggers, the sampling process will keep running, thus refreshing the cached analog data and overwriting the old one; when a trigger is issued, the waveform stored in the analog cache will be digitized immediately. The post-processing logic therefore only has to deal with waveforms of the length of the analog buffer. This method allows for both high-speed sampling and caching. Moreover, the conventional approach of simultaneous analog-to-digital conversion and readout are thus separated. The analog memory first acquires the pulse, then the ADC is converting the stored waveform and sends only a short piece of waveform to the backend system for storage. Therefore, the data processing load for the backend systems is greatly reduced.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/waveformASIC.png}
\caption[Readout Electronics]{\label{fig:Waveform ASIC}The architecture of one channel of the waveform sampling ASIC}
\end{center}
\end{figure}
The architecture of the waveform sampling chip is given in Fig.~\ref{fig:Waveform ASIC}. The high speed sampling and analog memory is realized by a large switched-capacitor array (SCA). When no trigger comes, the SCA will keep sampling in a cycling style, and the voltage values held on the capacitors are consequently refreshed; when a trigger is issued, the held voltage will be compared with the common ramp signal of the chip by an exclusive comparator for each capacitor. Meanwhile, a counter will monitor the time duration from the start of the common ramp signal until stop signals are issued by the comparators. The stored analog information is then converted to digital output in a single step. The individual start-to-stop time counts will be stored in the corresponding latches. When the the ramp signal ends, the latched data will be transmitted by the serializer.
In order to be able to cope with the expected dark rate per channel of 50\,kHZ, each ASIC channel will include 256 storage cells.
\subsubsection{Design of a high-speed FADC}
A further solution is a self-designed high-speed FADC. The idea is based on a more conventional architecture, for which a high-speed ADC will take charge of sampling the waveform and digitization in a single step. All the converted digital data will then be transmitted to the backend system for data buffering, trigger selection processes and so on. Due to the import prohibitions, a self-designed FADC put into mass production will greatly decrease the cost of the system.
The ADC will be based on a hybrid architecture combining a pipeline A/D convertor and flash type. The signal will be sampled through an input buffer and a sample-and-hold circuit, and then be pre-converted by a four-stage pipeline conversion. A flash conversion stage will finally digitize the signal. In this way, the stress of high sampling speed and high voltage resolution are distributed to multiple stages. Also power dissipation per area will be reduced. The chip is expected to achieve asampling rate of 1\,GS/s and a resolution of 12\,bit, with a guaranteed effective number of 10\,bits.
\subsection{Data processing and transmission}
\subsubsection{External readout scheme: ATCA modules}
The FEE modules used in this electronics scheme will be based on the ATCA standard. The structure block diagram is shown in figure\ref{fig:ATCA Structure}: The modules will be based on a daughter-mother board structure. The function of the daughter board is to receive the PMT signals and to perform the digitization. The mother board is a standard ATCA module. As the main unit of data transmission and processing, it will aggregate the data from all daughter boards, transfer the clock and trigger signals via the high-speed backplane bus, transfer the final data to the DAQ via a Gigabit ethernet connection and take care of system control and configuration. The FPGA of the module will perform the digital signal processing and information extraction.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{ReadoutElectronicsAndTrigger/figures/ATCAStructure.jpg}
\caption[FEE block diagram for ATCA modules]{\label{fig:ATCA Structure}Standard ATCA module structure block diagram}
\end{center}
\end{figure}
\subsubsection{Intermediate readout scheme}
{\bf Submerged processing unit.} In the partially submerged scheme, the signal processing unit will still be based on a daughter-mother board structure. The FEE daughter board will be mostly identical to the external case. The mother board will package the data and transfer it to the external electronics via an optical fiber. For reliable data transmission, some redudancy will be introduced for the mother board:two FPGAs will serve as backup for each other. When one of the FPGAs is not working properly, the system will switch to the other one. A corresponding diagram is shown in Fig.~\ref{fig:UnderWaterSignalProcess}.
\medskip\\
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{ReadoutElectronicsAndTrigger/figures/UnderWaterSignalProcessing.jpg}
\caption[Block diagram of processing unit]{\label{fig:UnderWaterSignalProcess} Diagram of the submerged processing unit}
\end{center}
\end{figure}
\noindent {\bf Signal collection unit.} The data sent by the underwater processing units via optical fibers will be received by a collecting ACTA module based on a daughter-mother board structure. The motherboard is an ATCA standard module, while the daughter board completes some simple function like the collection of the optical data and the transmission to the mother board.
\subsection{Design of trigger and clock}
The White Rabbit technology has been used extensively in the LHAASO experiments. A compact FMC-form White Rabbit node (cute-wr) has been developed which can be easily integrated as a network mezzanine card on other electronics system. It will provide a uniform 125\, MHz clock signal, a Pulse-Per-Second signal and an encoded UTC timestamp signal. First tests with a prototype system show a 100\,ps accuracy and 21\,ps precision among several cute-wr cards.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{ReadoutElectronicsAndTrigger/figures/WhiteRabbit.png}
\caption[White Rabbit system]{\label{fig:WhiteRabbit}Compact FMC-form White Rabbit nodes and the measurement of PPS synchronization}
\end{center}
\end{figure}
\subsection{Design for system reliability}
\subsubsection {Specifications for reliability}
The reliability requirement for the electronic system is a failure rate of less than 0.1\,\% per year of operational time for each of the 16,500 detector channels. The design life time of the system is 20 years. At the desired failure rate, it should be able to run for at least 10 years without need of maintenance action.
In the intermediate and underwater schemes, most of the electronics will be put into sealed boxes at a maximum water depth of 40\,m. The temperature of the water will be in the range of +10 to +20 Centigrade. The ambient radiation levels will be normal.
\subsubsection {Reliability engineering}
As an important aspect of product assurance management, reliability engineering will cover the entire electronics design and development for JUNO. This will include:
\begin{itemize}
\item {\bf Management of components.} The used parts will extremely affect the quality and reliability of the final product. There will be a management plan for the components, and a list of preferred components to provide guidance in their selection and quality control.
\item {\bf Reliability design,} including reliability definition, system reliability analysis, reliability allocation, reliability design and prediction
\item {\bf Analysis of failure modes and effects}, including a fault tree analysis.
\item Analysis of {\bf sneaks}
\item Analysis of {\bf ASIC reliability}
\item Assurance of {\bf firmware reliability}.
\end{itemize}
For achievement of the high quality and reliability requirements for the electronics, all steps of production and installation of the JUNO electronics (except the reliability design in R\&D procedure) should be controlled by the overall Product Assurance (PA) management plan. This will include
\begin{itemize}
\item Risk management
\item Safety management
\item Control of critical items
\item Procurement, verification, screening, handling and storage of components
\item Control of techniques and production procedure
\item Verification and Environmental tests.
\item Software assurance, etc.
\end{itemize}
There will be a PA organization supervised by PA manager to establish and implement the management plan.
\section{Manufacture and assembly}
\label{sec:roe:rea}
\subsection{Production}
Production of $\sim$20,000 channels will be a demanding task that must be subject to a strict quality management and continuous supervision. The final design will be chosen after a phase of elaboration and prototyping of the three design concepts that have been introduced above and a careful evaluation of their respective merits and disadvantages. Once the design is fixed, production will start in several steps:
\begin{itemize}
\item {\bf Ordering of electronic components}, including cables and crates. Each supplier and batch will be strictly controlled to ensure the reliability.
\item {\bf PCB manufactory and assembly} will be carried out by professional manufacturers. A close personal contact with the companies will be established to allow for quality supervision and sampling at the production site.
\item {\bf Aging tests} with the components that should be performed well in advance of the start of JUNO data taking. Components failing the tests can be repaired and re-introduced after the performance has been tested.
\end{itemize}
\subsection{Aging and long-term tests}
According to typical reliability curves for electronic devices, the failure rate will be relatively high in the beginning while forming a plateau of constant low failure rate after. Therefore, all critical components should be installed and running for some time before the experiment starts, allowing to pick out and repair malfunctioning components.
Aging tests will be performed by exposing electronic parts for a period of the order of one week to an environment of increased temperature. If 1000 channels can be tested simultaneously, the whole process will require about 20 weeks. In order to optimize the conditions of the aging tests, pre-studies risking the destruction of some of the components should be conducted.
Before installation, the system needs to undergo a long-term test run at a dedicated test stand at a laboratory that allows for debugging, identification of failing components and further aging control. This thorough test of at least part of the system should last at least for 3$-$4 months, and should be performed with real PMTs connected.
\subsection{Installation}
The readout system will be installed along with the central detector and PMTs. The foreseen time span for installation is 6$-8$ months. It will start from the bottom and end at the top cap. The external testing stand should be completed before installation to allow to perform the tests during the installation phase. After completing the installation, a debugging phase to test the stability of the whole system is foreseen to ensure optimal working conditions at the start of the experiment.
\subsection{Reliability}
All steps require rigorous quality and reliability management and control, from ordering the components to the finalization of the installation. This will ensure the system stability, reduce losses in the production process, and minimize the consumption.
\section{Risk analysis}
\label{sec:roe:risk}
\subsection{Three read-out schemes}
Some risk obviously exists for such a 20,000-channel high-speed sampling system, in which all of the PMTs are located inside the water pool and therefore not immediately accessible to repairs. The "external readout scheme" (Sec.~\ref{sec:roe:dry}) will place most of the electronics outside the water pool. This will greatly simplify the replacement of broken modules but potentially reduces the measurement quality and increases the costs. In the "underwater readout scheme" (Sec.~\ref{sec:roe:wet}), almost all the critical components of the electronics will be placed underwater and are possible without reach for replacement (especially the FEE). This assures optimum measurement performance, but will increase the risk of failure unless special attention is given to an increased redundancy of vital system components. In case of the "intermediate readout scheme" (Sec.~\ref{sec:roe:int}), a large part fraction of the electronics is located inside the water. All vital electronics are concentrated inside the central processing units that are designed to be replaceable. Compared to the other schemes, both performance and repairability are optimized but costs and design difficulty are increased. The final design will be chosen after all three designs have been properly elaborated and prototypes have been tested.
\subsection{Design risks}
Design risks comprise mainly four aspects:
\begin{itemize}
\item Quality of signal measurement
\item Reliability
\item Cost control
\item Design integrity
\end{itemize}
The biggest risk is the signal measurements, which followed by the risk of reliability and design integrity. The reliability is associated with cost control. Risks in design integrity will be mostly caused by two aspects: Insufficient communication of the groups designing different parts of the system and a lack of understanding of the system itself. These risks can be mitigated by a good collaboration in between the participating groups.
\subsection{Production risks}
Production risks are mainly arising in the procurement of electronic components, PCB manufactory and assembly, aging tests and cost control. A very strict quality control system will be established.
\subsection {Installation risks}
Risk in installation will also be minimized by a strict management and quality control, including three aspects: installation quality, time and safety.
\section{Schedule}
\begin{itemize}
\item[2013] Start of design of the readout schemes.
\item[2014] R\&D for key components.
\item[2015] Test of a prototype system of 200 channels for the "external readout scheme" at Daya Bay.
Finalization of a 32-channel demonstrator for the "intermediate readout scheme".
Review of all three schemes and choice of the final scheme.
\item[2016] Prototype system for the final scheme, about 200 channels
\item[2017] Mass production
\item[2018] Mass production, testing ans start of installation
\item[2019] Installation and testing
\end{itemize}
\chapter{ScheduleAndBudget}
\label{ch:ScheduleAndBudget}
\chapter{Veto Detector}
\label{ch:VetoDetector}
\section{Experiment Requirements}
The main goal of JUNO is to determine the neutrino mass hierarchy, which is one of the most important unsolved problems related to neutrinos.
The neutrinos are detected via the inverse beta decay by measuring the correlated positron and neutron signals.
With a careful design of the detector, the neutrino spectrum can be measured precisely, and we are looking for the distortion of the spectrum for the sign of neutrino mass ordering. Compared to the small number of signal events (60/day), the number of background events is still very high due to the large volume of detector (20~kton). The cosmic ray muon induced backgrounds are the main backgrounds
and they are hard to remove. The cosmic ray generated backgrounds are:
1. $^9$Li/$^8$He background from muon spallation and muon shower particles.
2. Fast neutron background in the detector from muon induced high energy neutrons.
The cosmic ray induced backgrounds also effect the study of the diffuse supernova neutrino flux.
In order to reduce the experimental backgrounds, the neutrino detector must be placed in deep
underground and a veto system is used to tag muons. The muons should be detected with
high efficiency for the purpose of background reduction. This chapter is
mainly about research and design of the veto system.
Due to the strict requirements on background suppression, larger overburden of rocks on top of the detector is needed to reduce the cosmic ray muon flux.
The experiment is located at a site of about 53~km equal distance to the Yangjiang and Taishan nuclear power plants.
The height of the mountains is 270 meters and the experimental hall is located underground at the depth of 460 meters.
Therefore, there is about 700-meter rock on top of the experimental hall. Muon rate is estimated at about 0.003~Hz/$m^{2}$ and the average muon energy at about 214~GeV from simulation.
The cosmic ray muon flux is reduced by $\sim$60,000 times compared to that at the ground surface. The remaining energetic cosmic ray muons can still produce a large number of neutrons in
the rocks and other detector materials surrounding the central detector. These neutrons can produce fast neutron background in the central detector which mimics the inverse beta decay signal. This kind of background
cannot be ignored. In order to shield the neutrons and the natural radioactivities from the surrounding rocks, at least 2 meters of water surrounding the central detector is needed. The water
is effective for shielding against neutrons and gamma (Fig.~\ref{gamma_neutron_shield}). And when being instrumented with photomultiplier tubes (PMTs), the water pool can serve as a
water Cherenkov detector to tag muons. The Daya Bay experiment shows that the veto system utilizing water Cherenkov detector can be very successful \cite{DYB:muon2014}.
From simulation, muons with relatively long track in the detector can be detected with very high efficiency, and the undetected muons are mainly of short track lengths, which would induce
less background because they are relatively away from central detector. Based on the Daya Bay experimental results (fast neutron background about $\sim$0.2\%), if the water shield thickness is at least 2.5 meters in JUNO, fast neutron background to
signal ratio is $\sim$0.3\%, after taking into consideration the large detector volume and geometry effects.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/Gamma_neutron_attenu_water.jpg}
\caption[Thickness of water vs gamma and neutron attenuation]{Left: Gamma attenuation vs thickness of water shield; Right: Neutron attenuation vs thickness of water shield.}
\label{gamma_neutron_shield}
\end{center}
\end{figure}
Another background is produced from $^9$Li/$^8$He resulting from muon spallation in the
scintillator and muon shower particles. The beta-n decay of $^9$Li/$^8$He would mimic inverse beta events. This kind of background can not be reduced from increasing
the thickness of the water shield. The number of $^9$Li/$^8$He background events is estimated to be $\sim$80/day. Their existence would greatly reduce JUNO's capability
to determine the neutrino mass hierarchy.
To reduced the $^9$Li/$^8$He background, we need precise muon track
information. The $^9$Li/$^8$He background is reduced by excluding a
certain cylindrical region along the muon track within a certain period of
time after the muon had passed through the detector. Therefore, the $^9$Li/$^8$He background reduction depends on precise muon track
reconstruction. We will build a tracking detector on top of the water
Cherenkov detector to help muon tagging as well as track reconstruction. The top tracker can tag muons and reconstruct muon tracks, independent of the
central detector and water Cherenkov detector. The tracking results from the top tracker can be extrapolated
to the central detector. Therefore, it can provide an independent measurement of the
$^9$Li/$^8$He background. The muons which pass through the water pool but not the central detector can also
produce high energy particles (pions, gamma, neutrons, etc.), which could migrate into
the central detector and produce $^9$Li/$^8$He there. A simple
simulation shows that this kind of background which cannot be detected by the
central detector is $\sim$1.2/day. A top tracker covering a large area can help directly measure such background.
Through optimization of the top detector area and its layout, the detector placed at the top with a long strip
structure provides better results. This configuration can identify up to 1/4 of this kind of
background, which would be helpful for the study of such background.
It can also be used to measure the rock neutron related background.
Also the top tracker and water Cherenkov detector can help identify multi-muon events for the central detector to reduce the $^9$Li/$^8$He background.
\section{Detector Design}
The JUNO veto system is shown in Fig.~\ref{vetodet}. The system consists of the following components:
1. Water Cherenkov detector system, a pool filled with purified water and instrumented with PMTs. When energetic muons pass through the water, they can produce Cherenkov light. The Cherenkov
photons can be detected by PMTs. The arrangements of the PMTs as well as the number of PMTs needed in the water pool is currently under study to facilitate the high efficiency detection of muons as
well as good muon track reconstruction. One of the options is to have the water pool surface and central detector outer surface covered with reflective Tyvek to increase the light detection by PMTs without using a large number of PMTs.
2. Water circulation system. There will be 20,000-30,000 tons of water in the pool depending on the different central detector designs. This system will include a water production system on the ground and
a purification/circulation system underground in the experimental hall.
3. Top tracker system.
It can provide independent muon information to help muon tagging and track reconstruction. The OPERA top tracker are going to be transported and installed in JUNO.
4. Geomagnetic field shielding system.
Though small, the Earth's magnetic field can affect the performance of PMTs. Either compensation coils or magnetic shields will be used to reduce the effect on PMTs.
5. Mechanical system.
The system includes top tracker support structure, water pool PMT support structure as well as a light and air tight cover for the water pool.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{VetoDetector/Figures/VetoDetector.png}
\caption[Veto systems of JUNO]{Veto systems of JUNO.}
\label{vetodet}
\end{center}
\end{figure}
\subsection{The Central Detector as a Muon Tracker}
The large liquid scintillator central detector itself is one of the most
important muon trackers in the veto system. The muon rate for
the central detector is about 3~Hz due to 700-meter rock overburden above.
A minimum-ionizing muon deposits about 2~MeV cm$^{2}$/g.
On average, a muon will pass through 23~m of LAB based liquid scintillator
(density of about 0.85~g/cm$^{3}$) inside the central detector, resulting in
about 3.9~GeV energy deposition.
A muon generates Cherenkov photons and scintillation photons along its
track in the detector. The Cherenkov photons are emitted in a cone with an opening angle $\theta = \cos^{-1}(1/n)$ relative to the muon track trajectory,
where $n$ is the index of refraction for liquid scintillator $\approx 1.45$.
Although the scintillation photons are emitted isotropically,
the path for the fastest photon arriving at any PMT
follows the same angle as that of the Chrenkov light angle $\theta$ when
the velocity of the muon is nearly the speed of light.
Based on the earliest hit time from each PMT, it is straightforward
to reconstruct the muon track. Such a method, called the ``Fastest-Light''
muon reconstruction, has been proved to be very successful in
reconstructing more than 99\% of the non-showering muons in the KamLAND
experiment.
Based a preliminary muon tracking study on the simulated JUNO muon tracks,
the bias of this method on the zenith angle is less than 5$^{\circ}$ and
the distance to the detector center bias is less than 20~cm.
Although the current study has not considered many real detector effects,
such as the resolution of PMT transition time, given the much better energy resolution and comparable or better PMT transition time resolution
for JUNO detector compared with KamLAND,
the fitting quality is expected to be further improved.
However, the fitting quality of this algorithm becomes worse
for the corner clipping muons, which are close to the edge of the detector.
And more importantly, this reconstruciton method is also not appropriate for
multiple muons or showering muons inside the detector, which is estimated to be about 10-20\% of all muons. Since the muon rate for the JUNO central detector is about 10 times higher
than that of the KamLAND detector, it is not acceptable to veto the whole detector for a few milliseconds, which will significant reduce the detector live time.
In order to reduce the $^{9}$Li/$^{8}$He background introduced by those muons, we have to rely on the other veto system, such as the water Cherekov detector and the top tracker to track those muons.
\subsection{Water Cherenkov Detector}
PMTs are placed in the water pool to tag cosmic ray muons by detecting water Cherenkov light that is produced. The number of photons produced is proportional to the muon track length in the water.
Due to the limited thickness of water as well as the presence of the central detector, the majority of the muon tracks going through the water pool are not very long. Therefore, a large photo-coverage
in the water pool is desired to detect muons with high efficiency. For the purpose of tagging muon, normally the pool surface would be covered with high reflectivity Tyvek film to help collect the photons
without using a large number of PMTs.
Based on the experience of the Daya Bay experiment and with Geant4 for simulation, muon detection efficiency is about 98\% in (Fig.~\ref{junoEffi}) by using 1,600 8-inch PMTs with at least 2.5 meter
thickness of water in the pool, if we can keep the noise level as low as that of the Daya Bay experiment. The long track muons can be detected with extremely high efficiency, as shown in
Fig.~\ref{iwsEffi}. Since the central detector size is larger than that of the Daya Bay detector, the large surface area of stainless steel tank might have a big effect on light transmission. We carried out a
simulation and found that there is no major impact on detector performance if the surface of the steel tank is covered with reflective Tyvek.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=6cm]{VetoDetector/Figures/JUNO_vetoEffi.png}~
\caption[Muon detection efficiency vs Number of PMT threshold]{Muon detection efficiency vs threshold in number of PMT (1,600 PMTs, with a number of PMTs threshold at 20, the efficiency is larger than 97\%).}
\label{junoEffi}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{VetoDetector/Figures/IWS-effi.png}
\caption[Muon efficiency vs time]{Detection efficiency (vs time) for long track muons going through antineutrino detector of Daya Bay. }
\label{iwsEffi}
\end{center}
\end{figure}
The challenge for the JUNO water Cherenkov detector is that we not only need to tag the muon with high efficiency but also want to reconstruct the tracks of those muons.
Ideally full photo-coverage on the water pool surface would be the best choice. Yet the budget constraint limit the number of PMTs available for the water pool.
Considering the directional nature of the Cherenkov light and that all muons are coming from outside, it would be natural to have PMTs installed on the central detector.
Only few clipping muons at certain particular incident angles would not have direct photons projected on the central detector if not utilizing reflective water pool surface.
Some of these clipping muons would be detected by the top tracker. Most of these clipping muons are further away from central detector ($>$ 2 m) which would less likely to
produce cosmogenic backgrounds in the central detector.
The possibilities of different configurations of the central detector as well as the uncertainty to utilize the outer surface of the central detector, require careful study on PMT arrangements,
placement as well as the number of PMTs in the water pool. Large photo coverage enhances the muon detection but also drives the cost high. Therefore, it needs study of
utilizing larger size PMTs or wavelength shifting plates around the PMTs as done in the Super-Kamiokande experiment \cite{SK:detector2003} and find the optimal solution.
Since the muon track reconstruction is important for $^9$Li/$^8$He background reduction, we would consider to increase the water pool muon track reconstruction capability for the
purpose of background reduction. For JUNO, the shower muons will induce non-negligible dead time if the central detector can't reconstruct the shower muon tracks due to detector
saturation. If the water Cherenkov detector can help reconstruct those muon tracks, it would reduce the central detector dead time.
Reflective surfaces can enhance the Cherenkov photon detection, and therefore help tag muons. But the reflection would smear out the position information. We can use more PMTs to build
optically segmented water Cherenkov detectors to improve muon reconstruction resolution. But these segmented detectors could normally provide only one point along the muon track, therefore
it might need to have multiple layers of these kind of detectors installed to reconstruct the muon tracks, which would pose engineering as well as installation challenges.
In the mean time, developing better algorithms for track reconstruction utilizing both the PMT charge and timing information would be useful. For this purpose, a better water pool PMT calibration
system is essential to not only calibrate the PMT gains but also the PMT timing. Now, all these options are under study.
The simplest option would be to have a single water Cherenkov detector by
installing PMTs on the surface of the water pool. There are other options, such as having multiple layers of PMTs or modular
PMT boxes installed. Since there is not much space in the water pool and there
will be structures for the central detector in the pool, therefore from the
engineering point of view, a simple water Cherenkov
detector is desired. A simulation was performed to study the muon
reconstruction capability of the simple water Cherenkov detector. PMTs are
uniformly distributed on the surface of the water pool
as well as the outer surface of the central detector
(Figure~\ref{fig:watertank_config1}). Of the 2,000 PMTs, 200 PMTs are on the top
facing toward the outside of the pool, 200 PMTs are at the bottom and
800 PMTs are on the barrel all facing inward. Another 800 PMTs are on the outer
surface of the central detector. This gives roughly 2.5~m distance
between PMTs. All the surfaces are covered by Tyvek with reflectivity of 95\%.
Figure~\ref{fig:watertank_config2} shows various muon tracks with points on
different kinds of surface where the PMTs are installed
\begin{figure}[htpb]
\centering
\includegraphics[width=0.36\textwidth]{VetoDetector/Figures/watertank_config1.png}
\caption{Simple water Cherenkov detector option. Uniformed distribution of PMTs.
PMTs at the top of water pool are facing outward. PMTs at the bottom and on the
barrel are facing inward.}
\label{fig:watertank_config1}
\end{figure}
\begin{figure}[htpb]
\centering
\includegraphics[width=0.42\textwidth]{VetoDetector/Figures/watertank_config2.png}
\caption{Various muon tracks with reconstructed points at pool surface or
central detector. }
\label{fig:watertank_config2}
\end{figure}
\begin{figure}[htpb]
\centering
\includegraphics[height=2.in,
width=0.25\textwidth]{VetoDetector/Figures/watertank_CDresolX.png}
\includegraphics[height=2.in,
width=0.25\textwidth]{VetoDetector/Figures/watertank_CDresolY.png}
\includegraphics[height=2.in,
width=0.25\textwidth]{VetoDetector/Figures/watertank_CDresolZ.png}
\caption{Resolution of the reconstructed points on the central detector}
\label{fig:watertank_CDresol}
\end{figure}
\begin{table}[htdp]
\caption{Resolution of reconstructed points on the muon tracks.}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& X & Y & Z \\
\hline
\multicolumn{4}{|c|}{Points on the top of the pool}\\
\hline
Resolution & $\sim$1.5 m & $\sim$1.5 m & \\
\hline
\hline
\multicolumn{4}{|c|}{Points on the bottom of the pool}\\
\hline
Resolution & $\sim$0.8 m & $\sim$0.75 m & \\
\hline
\hline
\multicolumn{4}{|c|}{Points on the barrel of the pool}\\
\hline
Resolution & $\sim$0.3 m & $\sim$0.4 m & $\sim$0.8 m \\
\hline
\hline
\multicolumn{4}{|c|}{Points on the surface of CD}\\
\hline
Resolution & $\sim$0.4 m & $\sim$0.4 m & $\sim$0.4 m \\
\hline
\end{tabular}
\end{center}
\label{tab:watertank_muonpoints}
\end{table}
The muon tracks are represented by the interceptive points on either the top,
barrel, bottom of the pool or on the central detector. The points are obtained
using charge center method.
The reconstructed points on the barrel and bottom are the exit points of the
muon track. And the reconstructed points on the top and central detector are the
entry points.
Table \ref{tab:watertank_muonpoints} shows the resolution of the reconstructed
points at various location. The reconstructed points on the top are poor due to
thin water the muons going
through and thus having fewer photons for PMTs. The reconstructed points on the
barrel are good for X and Y and average with a bias of $\sim$0.5~m for Z, this
could be the results of all the tracks
are going downward. The reconstructed points are very good on the central detector
for all coordinates. It might be possible to use the reconstructed points from
top tracker for muons passing through the top of the pool.
It is essential to have good reconstructed muon track in central detector. Though the central detector can be excellent muon tracker when the muon path is close
to the center of the central detector, the muon
track reconstruction gets worse when the muon is close to edge of the central
detector. Therefore, the presence of good reconstructed points from the water
Cherenkov detector would definitely help and cross
check with the central detector muon track reconstruction. Optimizing the number
of PMTs, the PMT arrangement as well as taking into account the PMT hit timing
information in reconstruction definitely warrant
further study.
Of all the muons, about 10\% are multiple muons. For double muons, it was
estimated that the central detector could distinguish the muons with distance
greater than 4~m. Water Cherenkov detector with good resolution
of muon entry points on the central detector has the potential to be able to
separate multiple muon tracks by looking for clusters of hit PMTs. This would be
helpful to reduce the dead time caused by multiple
muons.
\subsubsection{Tyvek Reflector Film}
As proved in the Daya Bay experiment, reflective surface is effective in tagging muons without larger number of PMTs.
Tyvek film as a reflective material has the merits of high reflectivity and stable performance, therefore it is widely used in water Cherenkov detectors. The Daya Bay experiment uses Tyvek film as to
make reflective surface in the water Cherenkov detector. In JUNO, the Tyvek reflective film could be used to cover the inner wall of the pool and the outer surface of the central detector.
The Cherenkov photons produced in the pool can be detected by the PMTs after multiple reflections to improve the detection efficiency. Figure~\ref{MakeTyvek} shows the process of making
reflective Tyvek. Tyvek pieces with a width of 1 meter and short lengths can be welded together to form a larger piece of reflector for large detector installation.
The performance of various Tyvek reflective films is shown in Fig.~\ref{TvyekReflect}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{VetoDetector/Figures/Tyvek_made.png}
\caption[Production of the Tyvek reflector]{The process of making the tyvek reflector.}
\label{MakeTyvek}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{VetoDetector/Figures/Tyvek_reflectivity.png}
\caption[Reflectivity of different types of Tyvek]{Reflectivity of different type Tyvek films .}
\label{TvyekReflect}
\end{center}
\end{figure}
The reflectivity of the Tyvek depends on the thickness of Tyvek. The Daya Bay experiment uses two layers of thick Tyvek (1082 D) with PE between them to form a multi-layer Tyvek. The reflectivity
of the multilayer Tyvek is shown as the red points in Fig.~\ref{TvyekReflect}. The reflectivity is >95\% when the wavelength > 300~nm.
\subsubsection{Calibration system}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{VetoDetector/Figures/LED_diffuse_ball.png}
\caption[LED]{LED and diffuse ball for PMT calibration.}
\label{LED}
\end{center}
\end{figure}
The gain and the timing calibration of the PMTs will be monitored by a LED system. No radioactive sources will be required.
As Fig.~\ref{LED} shows, we will use LED or diffuse ball by LED for the PMT calibration. LED flashing is triggered by pluse generator. The diffuse balls could be put at differnt positions of water pool. So every PMT can receive the photons from diffuse ball. PMTs gain and timing calibration could be done once a week.
\subsection{Top Tracker}
The JUNO cosmic muon tracker will help enormously to evaluate the contamination of the cosmogenic background to the signal.
The OPERA detector has to be dismounted soon and the OPERA TT ~\cite{Adam:2007ex}($\sim4.5$M EUR) will become available by mid--2016.
It will be placed on top of the JUNO water Cherenkov detector to be used as a cosmic muon tracker.
\subsubsection{OPERA Target Tracker}
The TT is a plastic scintillating detector which had several critical roles in OPERA: it was used to trigger the neutrino events, to identify the brick in which the neutrino interaction took place and to reconstruct muons therefore reducing the Charm background.
Its performances well met the expectations: only limited aging was observed over the 2007 - 2012 data taking period, and the detector understanding was well demonstrated by showing very good data/MC agreement in particular on the muon identification and energy reconstruction~\cite{Agafonova:2011zz}.
The TT is composed of 62 walls each with a sensitive area of 6.7$\times$6.7~m$^2$.
Each wall is formed by four vertical ($x$) and four horizontal ($y$) modules (Fig.~\ref{wall_schematic}).
The TT module is composed of 64 scintillating strips, 6.7~m length and 26.4~mm wide.
Each strip is read on both sides by a Hamamatsu 64-channel multi-anode PMT.
The total surface which could be covered by the 62 x-y walls is 2783~m$^2$.
All TT walls in OPERA are hanged by the top part of the detector and are thus in vertical position (Fig.~\ref{hanging}).
In the case of JUNO, all TT modules have to be placed in horizontal position, in which case more supportive mechanical structure is needed.
\begin{figure}[hbt]
\begin{minipage}[b]{.45\linewidth}
\centering
\includegraphics[width=7cm]{VetoDetector/Figures/wall_schematic.pdf}
\caption{\small Schematic view of a plastic scintillator strip wall.} \label{wall_schematic}
\end{minipage} \hspace{1.cm}
\begin{minipage}[b]{.45\linewidth}
\centering
\includegraphics[width=7cm]{VetoDetector/Figures/hanging.pdf}
\caption{\small Target Tracker walls hanging in between two brick walls inside the OPERA detector.} \label{hanging}
\end{minipage}
\end{figure}
The particle detection principle used by the TT is depicted by Fig.~\ref{principle}.
The scintillator strips have been produced by extrusion, with a $TiO_2$ co-extruded reflective and diffusing coating for better light collection.
A long groove running on the whole length and at the center of the scintillating strips, houses the wavelength shifting (WLS) fiber which is glued inside the groove using a high transparency glue.
This technology is very reliable due to the robustness of its components.
Delicate elements, like electronics and PMTs are located outside the sensitive area where they are accessible (Fig.~\ref{endcap_schematic}).
\begin{figure}[hbt]
\begin{minipage}{.45\linewidth}
\centering
\includegraphics[width=7cm]{VetoDetector/Figures/wls_en.pdf}
\caption{\small Particle detection principle in a scintillating strip.}\label{principle}
\end{minipage} \hspace{1.cm}
\begin{minipage}{.45\linewidth}
\centering
\includegraphics[width=7cm]{VetoDetector/Figures/endcap_schematic.pdf}
\caption{\small Schematic view of an end--cap of a scintillator
strip module.}\label{endcap_schematic}
\end{minipage}
\end{figure}
Figure~\ref{ttelectronics} presents details about the end-caps of the TT modules hosting all electronics, front end and acquisition.
These electronics are composed of:
\begin{itemize}
\item a front end card located just behind the multi-anode Hamamatsu PMT, hosting the two OPERA ROC chips (32 channels each, BiCMOS 0.8 microns)~\cite{Lucotte:2004mi},
\item an acquisition card (DAQ)~\cite{Marteau:2009ct}, also hosting an ADC for charge digitization,
\item a light injection card located on the DAQ card,
\item two LEDs able to inject light at the level of the WLS fibers near the PMT and driven by the light injection card, this system is used to regularly calibrate the detector (PMT gain, stability etc.),
\item an ISEG High Voltage module located on the DAQ card.
\end{itemize}
\begin{figure}[htb]
\centering
\includegraphics[width=12cm]{VetoDetector/Figures/electronics.pdf}
\caption{TT electronics including DAQ card. All elements in red will be replaced.}
\label{ttelectronics}
\end{figure}
The TT electronics record the triggered channels and their charge, thanks to the OPERA-ROC chip.
A schematic view of this chip is given by Fig.~\ref{operaroc}.
Each channel has a low noise variable gain preamplifier that feeds both a trigger and a charge measurement arms.
The adjustable gain allows an equalization of all PMT channel gains which can vary from channel to channel by a factor 3.
The auto-trigger (lower part) includes a fast shaper followed by a comparator.
The trigger decision is provided by the logical ``OR" of all 32 comparator outputs, with a threshold set externally.
A mask register allows disabling externally any malfunctioning channel.
The charge measurement arm (upper part) consists of a slow shaper followed by a Track \& Hold buffer.
Upon a trigger decision, charges are stored in 2~pF capacitors and the 32 channels outputs are readout sequentially at a 5~MHz frequency, in a period of 6.4~$\mu$s.
All charges are digitized by an external ADC (12--bit AD9220) placed on the DAQ cards.
\begin{figure}[htb]
\centering
\includegraphics[width=12cm]{VetoDetector/Figures/opera_roc.pdf}
\caption{Architecture of a single channel of the OPERA--ROC chip.}
\label{operaroc}
\end{figure}
{\bf TT dismounting in Gran Sasso}\
The TT dismounting in Gran Sasso underground laboratory will start in summer 2015 (first OPERA Super Module) and will end in Spring 2016 (second OPERA Super Module).
The OPERA detector dismounting and cost sharing among the funding agencies are defined in a special MoU.
The cost of the TT dismounting up to its storage area is part of this MoU and thus it is not considered in the IN2P3 JUNO requests.
All TT modules will be stored in Gran Sasso in 10 containers before sending them to China.
The shipping of the containers will be done when storage halls are ready near the JUNO underground laboratory.
This is expected to take place in 2016.
In all the cases the TT will not be mounted on top of JUNO detector before 2019.
This implies that the TT containers will be stored somewhere for about three years.
The best place, in order to avoid big temperature variations and scintillator aging, is the Gran Sasso underground laboratory.
Negotiations are engaged with LNGS on this possibility.
If this is not possible, the TT will be temporarily stored in a hall in the surface LNGS laboratory waiting to be shipped to China.
{\bf TT in JUNO}\
This muon tracker, called now Top Tracker (again TT), will be needed in JUNO in order to well study the cosmogenic background production.
The most dangerous background is induced by cosmic muons generating $^9$Li and $^8$He unstable elements, and fast neutrons, which could fake an IBD interaction inside the central detector.
Figure~\ref{ttnoise} presents schematically the most important noise configurations.
The first two, (a) and (b), mainly concern $^9$Li and $^8$He production directly in the central detector (a) and in the veto water pool (b) while the last one (c) concerns the neutron production in the surrounding rock.
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{VetoDetector/Figures/Cosmogenic_background.png}
\caption{Configurations considered for the induced background.}
\label{ttnoise}
\end{figure}
The surface on the top of the JUNO detector is of the order of $40\times 40$~m$^2$.
The total surface that the TT could cover depends on the number of superimposed $x-y$ layers (composed by consecutive TT walls).
In any case it will never be able to cover the entire surface, half of cosmic muons crossing the JUNO detector can pass by the sides.
The number of layers will depend on several parameters:
\begin{itemize}
\item the minimum statistics needed to well measure the cosmogenic background ($^9$Li and $^8$He production),
\item the muon tracking accuracy needed up to the bottom part of the central detector (of the order of the one induced by the multiple scattering),
\item the noise rate reduction using coincidences (affordable by the acquisition system),
\item the rate reduction of fake tracks.
\end{itemize}
First studies show that 3 to 4 layers will be needed that means that the total surface on top of JUNO liquid scintillator detector will be 690~m$^2$ to 920~m$^2$, respectively.
The noise rate in JUNO underground laboratory is expected to be significantly higher than that in Gran Sasso.
During OPERA operation the noise rate for 1/3 p.e. threshold was of the order of 10~Hz.
This low rate was mainly due to the low PMT dark current (2.5~Hz) and the fact that the TT walls were shielded by the OPERA lead/emulsion bricks.
Before the insertion of the bricks the noise rate was of the order of 25~Hz/channel.
In JUNO this rate is expected to be significantly higher for several reasons:
\begin{itemize}
\item The JUNO overburden is lower (2,000~m.w.e.) than for Gran Sasso (4,200~m.w.e.) implying more cosmic muons crossing the detector.
\item There will be no shielding between the TT walls.
\item The radioactivity of the rock, measured by the Daya Bay experiment 200~km away, is expected to be significantly higher than the one observed in the Gran Sasso underground laboratory.
\item There will be no concrete on the walls surrounding the JUNO underground laboratory (contrary to the Gran Sasso underground laboratory), concrete that could absorb part of the $\gamma$'s emitted by the rock.
\end{itemize}
In the case of 4-layers cover, the configurations of Fig.~\ref{coverage} have been considered.
The first configuration has the advantage compared to the others of covering the maximum surface of the central detector region.
The second one is more representative of all detector configurations and noise measurements could be extrapolated to the total detector top surface.
The third configuration is best for rock muon production estimation but is not very representative of the rest of the detector.
Finally, as baseline it has been defined the second configuration.
In all the cases, the TT has to well cover the chimney region (central region not covered by the veto water Cherenkov) from where radioactive sources will be introduced to calibrate the detector.
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{VetoDetector/Figures/TT_coverage.pdf}
\caption{Configurations considered for the Top Tracker coverage.}
\label{coverage}
\end{figure}
{\bf Noise rate estimations}\
Table~\ref{noise} presents the rock noise rate for one $x$ layer on the top, for one $y$ layer just below and for the case of one $x-y$ coincidence of these two layers ($x-y$ correlated coincidence).
For the $x-y$ correlated coincidences the $x$ and $y$ hits must come from the same radioactivity event while for the $x-y$ accidental coincidences come from different events occurring during 200~ns time--window.
The slight noise reduction from the $x$ layer to the $y$ one is due to the fact that the top layer shields a bit the layer above.
A strong noise rate reduction is observed when an $x-y$ coincidence is required.
For all the cases a threshold of one and two photoelectrons is considered.
The results are shown considering no concrete on the JUNO cavern walls and with 10~cm concrete.
The presence of concrete would reduce the radioactivity noise by about a factor of 4.
\begin{table}[htdp]
\caption{Noise rate for several TT conditions. (a) is under 1 p.e. threshold and (b) is under 2 p.e. threshold.
(1)no concrete; (2)10~cm concrete.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
Config. & \multicolumn{2}{c|}{Rate (Hz/m$^2$),(a)} & \multicolumn{2}{c|}{Rate (Hz/m$^2$),(b)} \\
\cline{2-5} & (1) & (2) & (1) & (2) \\
\hline $1-x$ layer (top) & 6800 & 1780 & 2700 & 710 \\
\hline $1-y$ layer (bottom) & 6500 & 1780 & 1200 & 310 \\
\hline $x-y$ layer (correlated) & 540 & 150 & 29 & 7 \\
\hline $x-y$ layer (accidentals) & 400 & 29 & 29 & 2 \\
\hline
\end{tabular}
\end{center}
\label{noise}
\end{table}
The noise rate per channel is not the only parameter to take care during the design of the new TT configuration.
The number of $x-y$ layers will mainly be determined by the number of ``fake'' muons (virtual tracks due to many hits in a time--window of 200~ns) produced by the environmental radioactivity.
Table~\ref{fake} presents the ``fake'' muons' rate estimation considering two and three TT $x-y$ layers.
For the case of the two layers, the distance between is considered to be 2~m, while for the case of three layers the third layer is inserted at the middle between the two considered layers.
It can be observed that the rate is considerably reduced for the case of three layers compared to the one of only two layers.
From this table the case of two layers can be excluded.
The rate being of the order of few Hz and less for the case of three layers, the case of four layers initially considered probably will not be necessary.
A fourth layer would provide some redundancy in case of inefficiencies due to scintillator aging, dead time or dead zones.
It has also to be noted that each ``fake'' muon will bias the cosmogenic $^9$Li and $^8$He production and thus this rate has to be very low.
\begin{table}[htdp]
\caption{Rate of ``fake'' tracks for several TT conditions.(a) is under 1 p.e. threshold and (b) is under 2 p.e. threshold.
(1)no concrete; (2)10~cm concrete.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
Config. & \multicolumn{2}{c|}{Fake muon rate (Hz), (a)} & \multicolumn{2}{c|}{Fake muon rate (Hz), (b)} \\
\cline{2-5} & (1) & (2) & (1) & (2) \\
\hline 2 TT $x-y$ layers & 340,000 & 12,500 & 1300 & 31 \\
\hline 3 TT $x-y$ layers & 2.6 & 0.02 & $6\times 10^{-4}$ & $2\times 10^{-6}$ \\
\hline
\end{tabular}
\end{center}
\label{fake}
\end{table}
In order to study the background induced by $^9$Li, $^8$He and fast neutrons, enough statistics has to be collected using the TT.
To estimate the background rate induced by these three source for the three configurations of Fig.~\ref{ttnoise}, the following parameters have been considered:
\begin{itemize}
\item $E_\mu$: the muon energy,
\item $L_\mu$: the average track length of muons in the liquid scintillator (case a),
\item $R_\mu$: the muon rate in the liquid scintillator (case a),
\item $F_{nCap}$: the neutron capture ratio (1 for JUNO, case a),
\item $N$: the muon number (case b),
\item $L_{att}$: the attenuation length of $^9$Li/$^8$He in water pool (0.5~m, case b),
\item $S_{wp}$: surface of water pool (case c).
\end{itemize}
The following formulas give the background rate of the three considered configurations:
\begin{itemize}
\item ${R_{Li,He}} \propto E_\mu ^{0.74} \cdot {L_\mu } \cdot {R_\mu } \cdot {f_{nCap}}$ for case (a),
\item ${R_{Li,He}} \propto \frac{{\sum\limits_{i = 1}^N {E_{{\mu ^i}}^{0.74}(\sum\limits_{j = 1}^M {L_{{\mu ^i}}^j{e^{ - \frac{{{d_j}}}{{{L_{att}}}}}}} )} {f_{nCap}}}}{N} \cdot {R_{wp}}$ for case (b),
\item $R_n^{rock} \propto E_{{\mu ^i}}^{0.74} \cdot {R_\mu } \cdot {S_{wp}} \cdot {f_{nCap}}$ for case (c)
\end{itemize}
Table~\ref{casea} presents the expected background rate from $^9$Li and $^8$He for all three TT coverage configurations of Fig.~\ref{coverage}.
From these results, the TT configuration (4XY, ``O'') can be excluded because of the poor rate observed of the $^9$Li and $^8$He induced background. It has to be noted that in absence of the TT the $^9$Li and $^8$He rate per day is of the order of 90 to be compared to $\sim$50~IBD interactions induced by nuclear reactors.
Obviously, this background has to be reduced or at least its rate must be well known.
\begin{table}[htdp]
\caption{Noise rate induced by $^9$Li and $^8$He for case (a) of Fig.~\ref{ttnoise}.}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
& $L_\mu$ (m) & $R_\mu$ (Hz) & $^9$Li/$^8$He rate/day \\
\hline
all muons & 22.5 & 3.5 & 90 \\
TT (4XY, Mid) & 23.5 & 0.94 & 27 \\
TT (4XY, Rtg) & 23.4 & 0.80 & 23 \\
TT (4XY, ``O") & 21.8 & 0.30 & 9 \\
\hline
\end{tabular}
\end{center}
\label{casea}
\end{table}
Table~\ref{caseb} presents the expected background rate from $^9$Li and $^8$He for all three TT coverage configurations of Fig.~\ref{coverage} where the muons only cross the water pool without passing through the central detector.
As already said, the TT configuration (4XY, Rtg) is the most representative of all parts of the detector and, not seen a significant difference between this configuration and the configuration (4XY, Mid), the configuration (4XY, Rtg) has been considered as baseline.
\begin{table}[htdp]
\caption{Noise rate induced by $^9$Li and $^8$He for case (b) of Fig.~\ref{ttnoise}.}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
& $^9$Li/$^8$He rate/day & Back./signal (\%) \\
\hline
all muons & 1.2 & 3.0 \\
TT (4XY, Mid) & 0.35 & 0.9 \\
TT (4XY, Rtg) & 0.3 & 0.7 \\
TT (4XY, ``O") & 0.16 & 0.4 \\
\hline
\end{tabular}
\end{center}
\label{caseb}
\end{table}
Table~\ref{casec} presents the expected background rate from fast neutrons obtained by extrapolations for Daya Bay observations for case (c) of Fig.~\ref{ttnoise} and for the baseline configuration of TT (4XY, Rtg) of Fig.~\ref{coverage}.
It is expected to have a fast neutron induced background rate of the order of 0.13 per day which represents 0.3\% of the expected signal.
\begin{table}[htdp]
\caption{Expected noise rate from fast neutrons for case (c) of Fig.~\ref{ttnoise} and for the configuration of TT (4XY, Rtg) of Fig.~\ref{coverage}.}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
& Daya Bay & Far detector & JUNO \\
\hline
$E_\mu$ (GeV) & 57 & 137 & 215 \\
$R_\mu$ (Hz) & 1.2 & 0.055 & 0.003 \\
$S_{wp}$ (m$^2$) & 724 & 1032 & 3740 \\
$f_{nCap}$ (\%) & 45 & 45 & 100 \\
Rock neutron bkg. rate/day & 1.7 & 0.22 & 0.13 (B/S$\sim$0.3\%)\\
\hline
\end{tabular}
\end{center}
\label{casec}
\end{table}
The noise measured with the help of the TT will be extrapolated to the whole detector and will be introduced in the simulations in order to well estimated the related systematic errors.
{\bf Modifications to the Target Tracker}\
The present OPERA TT acquisition system can afford up to 20--25~Hz trigger rate per channel.
For higher rates (as those expected in JUNO) this system would have severe limitations inducing huge dead time and thus significantly reducing the TT efficiency.
Even if the $x-y$ coincidences already reduce the noise rate (see above), the acquisition system specifically developed for OPERA needs will not be able to be used for the TT installed in JUNO cavern.
The high noise rate expected in JUNO
obliges the replacement of the acquisition system (DAQ) by a new one.
By the same way, the front end electronics based on OPERA-ROC~\cite{Lucotte:2004mi} chip will be replaced.
Indeed, this chip is now obsolete and not enough spares are available.
The new generation chip MAROC3~\cite{Blin:2010tsa} (AMS SiGe 0.35~$\mu$m technology, developed by IN2P3 Omega laboratory) disposing of more functionalities can replace the OPERA-ROC chip.
On OPERA-ROC the word registering the triggered channels was not functioning properly while on MAROC3 this has been corrected.
This gives the possibility to only read the triggered channels and not all of them as it was done in OPERA TT, thereby gaining in readout speed.
The MAROC3 chip has a similar architecture (Fig.~\ref{maroc}) than the OPERA--ROC (fast part for triggering and slow one for charge measurement).
The difference is that in MAROC3 the ADC is already integrated in the chip while for OPERA--ROC the ADC was externally implemented on the DAQ cards.
The MAROC3 chip has two ``OR''s, one could be used for the timestamp while the second one could be used for fast coincidences with other sensors.
This would allow to do fast coincidences with at least $x$ and $y$ strips to reduce the DAQ trigger rate.
For all coincidences among TT sensors a dedicate electronics card has to be developed receiving the ``OR'' of all sensors and making coincidences.
The timestamp can be provided by a special clock card to be developed.
OPERA institutes participating now in JUNO have expressed a strong interest to develop the DAQ cards which will be located just behind the multi--anode PMTs, as done in OPERA.
The front end card where the MAROC3 chip will be located could be produced by the same institutes, which would also have the responsibility to check and characterize all MAROC3 chips produced for this application (992 plus spares).
These institutes could also develop the clock card and the coincidence card.
The drawings for the installation of the TT in JUNO will be produced by IN2P3 and IHEP--Beijing.
It is already agreed with IHEP that the TT supporting structure will be financed and produced by Chinese institutes.
This structure, very different from the one used in OPERA due to the fact that in JUNO all modules will be in horizontal position instead of vertical position in OPERA, has to be drawn with care and in a close collaboration between IN2P3 and IHEP.
A special care has also to be taken in order to leave accessibility to all electronics of all TT modules.
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{VetoDetector/Figures/MAROC_architecture.pdf}
\caption{Architecture of the MAROC3 chip.}
\label{maroc}
\end{figure}
\section{Water System}
For the JUNO water Cherenkov detector, it requires ultrapure water
for high muon detection efficiency. There will be different kinds of
materials submerged in the water, including stainless steel, Tyvek,
PMT glass, cables, etc. The complexity of the underground
environment makes it difficult to seal the pool completely and
almost impossible to passively keep the water quality good for a
long period of time.
Therefore, it is necessary to build a reliable
ultrapure water production, purification and circulation system.
Another important function of the water system is to keep the
overall detector temperature stable. The stability of the
temperature of the central detector is critical for the entire
experiment.
JUNO is located in southern China, and the temperature
of the surrounding rocks can reach 32 degrees Celsius. We intend to
maintain the temperature of the central detector at 20 degrees
Celsius.
Water is a good conductor of heat. Lowering the temperature
of the water flowing into the pool is a natural choice to offset the
heat from the surrounding rocks.
The Daya Bay experiment has 1,300 tons of water in the near
experimental hall water pool and 2,000 tons of water in the far
experimental hall water pool. The water production rate is
8~tons/hour and
the purification rate is 5$\sim$8 tons/hour. It can
circulate one volume of water in two weeks. The water Cherenkov
detector's muon detection efficiency is high to 99.9\% and its
long-term performance is stable.
The designed water purification
system satisfies the experiment requirements~\cite{Wilhelmi:2014rwa}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{VetoDetector/Figures/WaterSystemFlowRate.png}
\caption[Flow rate vs one cycle time]{Flow rate vs one cycle time.}
\label{flowRate}
\end{center}
\end{figure}
The JUNO water Cherenkov detector will have 20,000 - 30,000 tons of
ultrapure water. Based on the experience of Daya Bay experiment,
circulating one volume of water in two weeks requires a
flow rate of
about 80~tons/hour at JUNO. With this flow rate, we can keep good
water quality as well as maintain pool water at stable temperature.
Fig.~\ref{flowRate} shows the number of days to
circulate one volume
of water as function of the flow rates of the proposed water system
for the two candidate designs of the central detector. The red line
indicates the flow rate at 80~tons/hour.
It would take about 16 to
20 days to circulate one volume of water and this would satisfy the
experiment requirements.
In the proposed design, we divide the water system into two parts.
One part is for the production of ultrapure water and the other part
is for the circulation/purification of water. For the water
production
system, one part of the system would be on the surface
ground and the other part of the system would be underground. The
reason for this design is that there is a strict requirement of
waste water rate
to be less than 5~tons/hour at 500 meters
underground. Therefore, we have the part of the water production
system which generates the most waste water on the surface ground.
\subsection{Water Production on Surface}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/WaterProductionGround.png}
\caption[General view of water production system on ground]{General view of water production system on surface ground.}
\label{WaterProdGround}
\end{center}
\end{figure}
As Fig.~\ref{WaterProdGround} shows, water flows through the disk
filter first, which can block solid sediments and other large
particles from flowing into the raw tank before goes through the
ultrafiltration hosts. The ultrafiltration hosts can filter out
small particles of insoluble material in the water before it goes
into ultrafiltration tank. The water then passes through a reverse
osmosis
(RO) system, which can remove most of the ions in the water.
Finally the water flows into the primary pure water tank. After the
RO process, a third of the original water is produced as
waste
water with high ion concentration at a rate of 15~tons/hour. The
waste water is of no harm to the environment and can be directly
drained.
\subsection{Water Production Underground}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/WaterProductionUnderGround.png}
\caption[General view of water production system underground]{General view of water production system underground.}
\label{WaterProdUnderGround}
\end{center}
\end{figure}
From primary pure water tank on the surface ground, the water is
pumped through pipes at about 85~tons/hour going down a height of
500 meters into the underground raw water tank.
From there the
water will go through another RO system as well as other water
treatment systems, like reduction of Total Organic Carbons (TOC) and
Electrodeionization (EDI) systems.
The resulted ultrapure water with
resistance greater than 16~MOhm flows into experiment hall
at a rate
of 80~tons/hour.
\subsection{Water Circulation/Purification Underground}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/CirculationUnderground.png}
\caption[General view of water circulation/purification system on ground]{General view of water circulation/purification system on ground.}
\label{WaterCircuUnderGround}
\end{center}
\end{figure}
We would start the circulation process after the pool is fully
filled with water. The water will be drawn drained from the pool at
80~tons/hour. The water will first go through the heat/cooling
equipment . Then the
water will flow through the TOC system (to
remove organic carbon), the polishing mixed bed (to remove ions in
the water to improve water quality), UV sterilization system,
membrane filters,
the degassing apparatus (to remove oxygen and
other gases in water), and finally a cooling device (to control the
water temperature). In the end, the purified water will return to
the pool.
For the whole water production, purification and circulation system,
we will use a water control system to automate the running of the
system. The people on duty will only need to replace filters and
other supplies on a regular basis.
\section{Geomagnetic Field Shielding System}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/jiangmen_EarthMagnet.png}
\caption[Earth magnetic in Jiangmen]{Earth magentic in Jiangmen.}
\label{JUNO_filed}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{VetoDetector/Figures/PMT_CEvsMagnet.png}
\caption[PMT collection efficiency vs the external magnetic field]{PMT collection efficiency vs the external magnetic field.}
\label{CEvsField}
\end{center}
\end{figure}
At the experimental site, the horizontal component of the geomagnetic field is about 40~$\mu$T and vertical component is about 24~$\mu$T. Since JUNO will use 20 inch PMTs as the main device
for central detector, the geomagnetic field will have a big effect on large size PMT. It will deteriorate the whole detector performance. Figure~\ref{CEvsField} shows the relationship between the PMT collection efficiency and the external magnetic field intensity. We need to establish an effective magnetic shielding system to ensure good performance of central detector.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{VetoDetector/Figures/OneGroupCoils.png}
\caption[One group earth magnetic shield system]{One group earth magnetic shield system.}
\label{OneGroupCoil}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/MagDis_onegroup.PNG}
\caption[Magnetic field distribution with one group system]{Magnetic field distribution with one group system.}
\label{MagDis}
\end{center}
\end{figure}
We intend to ensure that a spherical region with a radius of 37.5~m would be well shielded from geomagnetic field.
We plan to use coils with current flowing through them to compensate the geomagnetic field. The residual field intensity could be reduced to be below 10~$\mu$T.
The baseline design is of one coil system. We can perform accurate measurement of the geomagnetic field to determine its main direction (mapping the geomagnetic field in the pool)
and then do compensation along its main direction. As Fig.~\ref{OneGroupCoil} shows, this system is easy to control.
Figure~\ref{MagDis} is the magnetic field distribution in the sphere from one coil system. It can have a good shielding. The uniformity within the 37.5~m volume is 1.8\%.
The PMTs in central detector will not be effected by the Earth's magnetic field after compensation.
Careful selection of non-magnetized materials used in the detector for parts such as the supporting structure for PMTs as well as stainless steel for the detector, etc. is also important.
They could cause an additional local magnetic field in additional to the Earth's field, which could affect the performance of the PMTs. The compensation coils would be installed on the central detector which means the PMTs of the water Cherenkov detector would
still be subject to the Earth's magnetic field plus some additional field generated from the coils. Therefore, it needs to have the pool PMTs shielded against magnetic field with passive magnetic shield like the one used in the Daya Bay experiment \cite{DeVore:2014nim}.
\section{Mechanical Structures and Installation}
\subsection{Support Structure over the Pool}
The tracking detectors over the pool are divided into two groups, each
consisting of several layers of detectors stacked together. One group
is placed on the steel bridge, while the other is put three meters above
the first group. Hence a two-story bridge is needed, as shown in
Fig.~\ref{SupportStr}.
The parameters of the two-story bridge are as follows: bridge width
is between 17~m and 20~m and the distance between two layers of the
bridge is 3 meters. The support structure has three functions:
(1) supporting the tracking detectors;
(2) supporting the calibration systems of the central detector;
(3) possibly supporting the electronics rooms.
The diameter of pool is 42 meters and the supports of bridge are located
at both ends of the bridge. Considering the height limit of the underground
space and the structure stiffness, a box structure is adopted for the bridge.
Two columns of tracking detectors, whose width is about 15~m, are placed
on the bridge. Considering the passageway and the width of the electronics
room (about 3 meters), the bridge width is between 17~m and 20~m. The load
of the bridge includes the weight of the central detector cable and
electronics rooms (about 100 tons), weight of the tracking detectors
(about 70 tons), weight of the electronics room (20 tons) and weight
of the supports of the tracking detectors.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/SupportStructure.png}
\caption[The top support structure]{The top support structure.}
\label{SupportStr}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{VetoDetector/Figures/PMT.png}
\caption[PMT support structure of DayaBay]{PMT support structure of DayaBay.}
\label{PMTstr}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/cover.png}
\caption[The cover design of Daya Bay]{The cover design of Daya Bay. a) gastight zipper b) Th connection between pool cover and pool dege}
\label{DYBcover}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{VetoDetector/Figures/wire.png}
\caption[The wire design option for cover installation of Daya Bay experiment]{The wire design option for cover installation of Daya Bay experiment.}
\label{cover1}
\end{center}
\end{figure}
\subsection{Support Structure for PMTs}
The support structure of PMT adopts the design for the Daya Bay veto PMTs,
as shown in Fig.~\ref{PMTstr}. The PMT support structure is then attached
to a steel-frame fixed on the waterpool wall. The design of the steel-frame structure will be similar to that for the central detector. The tyvek
film will also be fixed to the steel frame.
\subsection{Sealing of the Pool}
A black rubber cloth served as the pool cover in the Daya Bay experiment.
The pool cover is connected to the pool edge through an air-tight zipper
which shields the pool from the external air and light, as shown in
Fig.~\ref{DYBcover}. With the seal zipper, the pool is easily accessible
for routine maintenance, such as replacing the liquidometer. We will
adopt the same technique for JUNO's pool seal.
As the JUNO pool is very large with a 40-meter span, it is difficult
to unfold the black rubber cloth. Therefore, we propose two solutions:
the first is the tight-wire scheme used in the Daya Bay experiment;
the second is a slide-guide scheme. The two schemes are similiar.
For the slide-guide scheme shown in Fig.~\ref{DYBcover}, the cloth is
unfolded along the guide, instead of the tight-wire in the tight-wire
scheme shown in Fig.\ref{cover1}. For comparison, the tight-wire scheme
is more difficult to operate. In order to prevent the cloth from touching
the ground, we will adjust the height of the wire rope pillar.
The adjustment process is shown in Fig.\ref{cover1}. Firstly, wire is
put on the higher rope pillar and pulled tight; secondly, unfold the
black rubber cloth and fix the edge of the cloth; finally, put the
tight-wire on the lower pillar slowly.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{VetoDetector/Figures/DetView.png}
\caption[Lead rail for rubberized fabric cover option installation]{Lead rail for rubberized fabric cover option installation.}
\label{Detview}
\end{center}
\end{figure}
The slide-guide scheme costs more, but is more easily implemented.
The main structure contains the main beams, columns, slide guides
and pulleys. A couple of slide guides is fixed on the main beams
under bridge. One end of the main beam is supported on the bridge,
and the other end is connected to the pillar fixed on the ground.
Each of the two schemes have advantages and disadvantages, and the
final design choice is still under discussion.
\subsection{Lining and Thermal Insulation Layer of the Pool}
The cylindrical pool with 42-meter diameter and height will be filled
with 20,000-30,000 tons of ultra-pure water. Since the laboratory is located 400 meters
underground, the pressure of underground water is high. A powerful lining
is needed to prevent water from the surrounding rock seeping into the pool.
We propose to adopt high-density polyethylene (HDPE) material as the
waterproof lining between rock and pool. HDPE has good corrosion resistance,
high strength, good ductility, and does not pollute ultrapure water.
The design can be similar to that for landfill, but the lining for the JUNO
water pool contains many through holes and further studies are needed.
The operating temperature of the JUNO detector system is 20 degrees,
while the temperature near the laboratory rock is around 32 degrees,
significantly higher than the operating temperature. The granites
surrounding the pool have poor permeability, which has certain heat
preservation effect. The concrete wall of the pool is a poor heat conductor
with a heat transfer coefficient of only 1.63 W/(mK).
According to an independent heat transfer calculation and the example of calculation for
the Daya Bay experiment, the pool thermal insulation may not require
additional insulating layer. However, to reduce operating costs,
it is proposed to add an insulating layer.
\subsection{Installation}
The detector installation includes the assembling of PMT and bracket, and
the fixation of PMT bracket. The assembling of PMT and bracket is shown
in Fig.~\ref{PMTstr}. To make full use of the embedded parts in concrete
pool wall, the PMT bracket is considered to be fixed on the embedded parts.
In addition, special tools and operation management are needed to ensure
safe and smooth detector installation.
\section{Prototype}
\subsection{ Geomagnetic Shielding Prototype }
Through numerical calculation, we provide an option for the geomagnetic
shielding system for JUNO. A prototype of coil system is required for
validation and material property study. After the prototype study, we
will have a more accurate estimate for the shielding effect and
can then optimize the design of the geomagnetic shielding system.
\section{Detector Construction and Transportation}
The veto system mainly consists of the water Cherenkov detector and
the tracking detector. For the water Cherenkov detector, the most important
tasks are PMT test and transportation, water purification system construction
and transportation, and support structure manufacturing and transportation.
The veto PMT test and transportation will be done together with the
central detector PMTs. Water system manufacturing and transportation
and support structure construction await the final design decisions before
the bidding and other processes can be started. The option for the
top trackers also needs to be finalized before the detailed plans for the
constructions and transporting can be made.
\section{Risk and Safety Issues}
The large size of the JUNO detector limits the available space
in the experimental hall. For the veto system, it is important to
carefully coordinate the installations of various detectors and
hardwares. If the assembly coordination is not optimal, it could
influence the whole veto systems installation. To avoid this risk,
a careful coordination and strict schedule should be performed
for various assembly tasks. Any potential conflicts in the
installation schedule should be discussed and resolved
among the detector coordinators to ensure a smooth installation process.
Most of the safty issues occur during the detector installation stage.
The safety procedure must be strictly followed when the large JUNO
detector is installed.
\section{Schedule}
The plan for the period 2014-2020 is as follows:
\begin{itemize}
\item
2014:
Setup the following two kinds of veto detector prototypes and
take data with cosmic muons:
1) liquid scintillator + fiber prototype;
2) a multilayer resistive plate chamber (RPC).
\item
2015: Finalize the top tracker option.
\item
2016: Determine the pool liner design and begin to prepare lining
installation;
Complete the design of water pool insulating layer design and start
installation preparation;
Determine geomagnetic shielding design and start the installation preparation.
\item
2017: Start to install water pool insulating layer;
Start to install water pool lining;
Start to install the geomagnetic shielding system.
\item
2018: Complete the production of water Cherenkov detector PMT
support structure and top track detector supporting structure;
Start the installation of the top
track detector and water Cherenkov
detector PMT installation.
\item
2019: Complete water Cherenkov detector installation and top track
detector installation;
Start pump ultra pure water into water pool.
\item
2020: Complete pump
process and start veto system commissioning.
\end{itemize}
| {'timestamp': '2015-09-29T02:17:39', 'yymm': '1508', 'arxiv_id': '1508.07166', 'language': 'en', 'url': 'https://arxiv.org/abs/1508.07166'} |
\section{Introduction}
``\emph{Sometimes a picture is worth the proverbial thousand words; sometimes a few well-chosen words are far more effective than a picture}'' -- \cite{feiner1991automating}.
Modeling how visual and linguistic information can jointly contribute to coherent and effective communication is a longstanding open problem with implications across cognitive science. As \cite{feiner1991automating} already observe, it is particularly important for automating the understanding and generation of text--image presentations.\\
Theoretical models have suggested that images and text fit together into integrated presentations via \emph{coherence relations} that are analogous to those that connect text spans in discourse; see \cite{8397019} and Section~\ref{sec:related}.
This paper follows up this theoretical perspective through systematic corpus investigation.
We are inspired by research on text discourse, which has led to large-scale corpora with information about discourse structure and discourse semantics. The Penn Discourse Treebank (PDTB) is one of the most well-known examples \cite{MiltsakakiPJW04,DBLP:conf/lrec/PrasadDLMRJW08}. However, although multimodal corpora increasingly include discourse relations between linguistic and non-linguistic contributions, particularly for utterances and other events in dialogue \cite{cuayahuitl2015strategic,hunter:etal:2015},
to date there has existed no dataset describing the coherence of text--image presentations. In this paper, we describe the construction of an annotated corpus that fills this gap, and report initial analyses of the communicative inferences that connect text and accompanying images in this corpus.
As we describe in Section \ref{img--txt}, our approach asks annotators to identify the presence of specific inferences linking text and images, rather than to use a taxonomy of coherence relations. This enables us to deal with the distinctive discourse contributions of photographic imagery. We describe our data collection process in Section~\ref{data}, showing that our annotation scheme allows us to get reliable labels by crowdsourcing. We present analyses in Section~\ref{analysis} that show that our annotation highlights a range of cases where text and images work together in distinctive and theoretically challenging ways, and discuss the implications of our work for the understanding and generation of multimodal documents. We conclude in Section \ref{conclusion} with a number of problems for future research.
\section{Introduction}
``\emph{Sometimes a picture is worth the proverbial thousand words; sometimes a few well-chosen words are far more effective than a picture}'' --\cite{feiner1991automating}.
Modeling how visual and linguistic information can jointly contribute to coherent and effective communication is a longstanding open problem with implications across cognitive science. As \cite{feiner1991automating} already observe, it is particularly important for automating the understanding and generation of text--image presentations.\\
Theoretical models have suggested that images and text fit together into integrated presentations via \emph{coherence relations} that are analogous to those that connect text spans in discourse; see \cite{8397019} and Section~\ref{sec:related}. \\
This paper follows up this theoretical perspective through systematic corpus investigation.
We are inspired by research on text discourse, which has led to large-scale corpora with information about discourse structure and discourse semantics. The Penn Discourse Treebank (PDTB) is one of the most well-known examples \cite{MiltsakakiPJW04,DBLP:conf/lrec/PrasadDLMRJW08}. However, although multimodal corpora increasingly include discourse relations between linguistic and non-linguistic contributions, particularly for utterances and other events in dialogue \cite{cuayahuitl2015strategic,hunter:etal:2015}, to date there has existed no dataset describing the coherence of text--image presentations. In this paper, we describe the construction of an annotated corpus that fills this gap, and report initial analyses of the communicative inferences that connect text and accompanying images in this corpus.
As we describe in Section \ref{img--txt}, our approach asks annotators to identify the presence of specific inferences linking text and images, rather than to use a taxonomy of coherence relations. This enables us to deal with the distinctive discourse contributions of photographic imagery. We describe our data collection process in Section~\ref{data}, showing that our annotation scheme allows us to get reliable labels by crowdsourcing. We present analyses in Section~\ref{analysis} that show that our annotation highlights a range of cases where text and images work together in distinctive and theoretically challenging ways, and discuss the implications of our work for the understanding and generation of multimodal documents. We conclude in Section \ref{conclusion} with a number of problems for future research.
\vspace{3mm}
\section{Discourse Coherence and Text--Image Presentations}
\label{img--txt}
\label{sec:related}
We begin with an example to motivate our approach and clarify its relationship to previous work.
Figure~\ref{fig:example} shows two steps in an online recipe for a ravioli casserole from the RecipeQA data set \cite{DBLP:conf/emnlp/YagciogluEEI18}.
The image of Figure~\ref{fig:example}a shows a moment towards the end of carrying out the ``covering'' action of the accompanying text; that of Figure~\ref{fig:example}b shows one instance of the result of the ``spooning'' actions of the text.
\begin{figure*}[h!]
\centering
\begin{subfigure}[t]{0.5\columnwidth}
\centering
\includegraphics[width=4.7cm]{step2.png}
\caption{\textsc{Text:} Cover with a single layer of ravioli.}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\columnwidth}
\centering
\includegraphics[width=4.7cm]{step7.png}
\caption{\textsc{Text:} Let cool 5 minutes before spooning onto individual plates.}
\end{subfigure}
\caption{Two steps in a recipe from \protect{\cite{DBLP:conf/emnlp/YagciogluEEI18}} illustrating diverse inferential relationships between text and accompanying imagery in instructions. The recipe is from Autodesk Inc. www.instructables.com and is contributed by www.RealSimple.com.}
\label{fig:example}
\end{figure*}
Cognitive scientists have argued that such images are much like text contributions in the way their interpretation connects to the broader discourse.
In particular, inferences analogous to those used to interpret text seem to be necessary with such images to recognize their spatio-temporal perspective \cite{cumming2017conventions}, the objects they depict \cite{abusch2012applying}, and their place in the arc of narrative progression \cite{mccloud1993understanding,cohn2013visual}.
In fact, such inferences seem to be a general feature of multimodal communication, applying also in the coherent relationships of utterance to co-speech gesture \cite{lascarides2009discourse} or the coherent relationships of elements in diagrams \cite{alikhani2018arrows,inproceedings}.
In empirical analyses of text corpora, researchers in projects such as the Penn Discourse Treebank \cite{MiltsakakiPJW04,DBLP:conf/lrec/PrasadDLMRJW08} have been successful at documenting such effects by annotating discourse structure and discourse semantics via coherence relations.
We would like to apply a similar strategy to text--image documents like that shown in Figure~\ref{fig:example}.
However, existing discourse annotation guidelines depend on the distinctive ways that coherence is signaled in text.
In text, we find syntactic devices such as structural parallelism, semantic devices such as negation, and pragmatic elements such as discourse connectives, all of which can help annotators to recognize coherence relations in text.
Images lack such features.
At the same time, characterizing the communicative role of imagery, particularly photographic imagery, involves a special problem: distinguishing the content that the author specifically aimed to depict from merely incidental details that happen to appear in the scene \cite{stone2015meaning}.
Thus, rather than start from a taxonomy of discourse relations like that used in PDTB, we characterize the different kinds of inferential relationships involved in interpreting imagery separately.
\begin{itemize}
\itemsep0em
\parsep0em
\parskip0em
\item To characterize temporal relationships between imagery and text, we ask if the image gives information about the preparation, execution or results of the accompanying step.
\item To characterize the logical relationship of imagery to text, we ask if the image shows one of several actions described in the text, and if it depicts an action that needs to be repeated.
\item To characterize the significance of incidental detail, we ask a range of further questions (some relevant specifically to our domain of instructions), asking about what the image depicts from the text, what it leaves out from the text, and what it adds to the text.
\end{itemize}
Our approach is designed to elicit judgments that crowd workers can provide quickly and reliably.
This approach allows us to highlight a number of common patterns that we can think of as prototypical coherence relations between images and text.
Figure \ref{fig:example}a, for example, instantiates a natural \textbf{Depiction} relation: the image shows the action described in the text in progress; the mechanics of the action are fully visible in the image, but the significant details in the imagery are all reported in the text as well.
Our approach also lets us recognize more sophisticated inferential relationships, like the fact that Figure~\ref{fig:example}b shows an \textbf{Example:Result} of the accompanying instruction.
Many of the relationships that emerge from our annotation effort involve newly-identified features of text--image presentations that deserve further investigation: particularly, the use of loosely-related imagery to provide background and motivation for a multimodal presentation as a whole, and depictions of action that seem simultaneously to give key information about the context, manner and result of an action.
\vspace{5mm}
\section{Annotation Effort}
\label{data}
Work on text has found that text genre heavily influences both the kinds of discourse relations one finds in a corpus and the way those relations are signalled \cite{webber2009genre}.
Since our focus is on developing methodology for consistent annotation, we therefore choose to work within a single genre. We selected instructional text because of its concrete, practical subject matter and because of its step-by-step organization, which makes it possible to automatically group together short segments of related text and imagery.
\paragraph{Text--Image Pairs.} We base our data collection on an existing instructional dataset, RecipeQA~\cite{DBLP:conf/emnlp/YagciogluEEI18}. This is the only publicly available large-scale dataset of multimodal instructions. It consists of multimodal recipes---textual instructions accompanied by one or more images.
We excluded documents that either have multiple steps without images or that have multiple images per set. This was so that we could more easily study the direct relationship between an image and the associated text.
There are 1,690 documents with this characteristic in the RecipeQA train set. To avoid overwhelming crowd workers, we further filtered those to retain only recipes with 70 or fewer words per step, for a final count of 516 documents (2,047 image--text pairs).
\paragraph{Protocol.} We recruit participants through Amazon Mechanical Turk. All subjects were US citizens, agreed to a consent form approved by Rutgers's institutional review board, and were compensated at an estimated rate of USD 15 an hour.
\paragraph{Experiment Interface.} Given an image and the corresponding textual instruction from the dataset, participants were requested to answer the following 10 questions.
For Question 1, participants were asked to highlight the relevant part of the text. For the others, we solicited True/False responses.
\begin{enumerate}
\itemsep0em
\parsep0em
\parskip0em
\item Highlight the part of the text that is most related to the image.
\item The image gives visual information about the step described in the text.
\item You need to see the image in order to be able to carry out the step properly.
\item The text provides specific quantities (amounts, measurements, etc.) that you would not know just by looking at the picture.
\item The image shows a tool used in the step but \emph{not} mentioned in the text.
\item The image shows how to prepare before carrying out the step.
\item The image shows the results of the action that is described in the text.
\item The image depicts an action in progress that is described in the text.
\item The text describes several different actions but the image only depicts one.
\item One would have to repeat the action shown in the image many times in order to complete this step.
\end{enumerate}
The interface is designed such that if the answer to Question 8 is \tag{True}, the subject will be prompted with Question 9 and 10. Otherwise, Question 8 is the last question in the list.\footnote{The dataset and the code for the machine learning experiments are available at https://github.com/malihealikhani/CITE}
\paragraph{Agreement.} To assess the inter-rater agreement, we determine Cohen's $\kappa$ and Fleiss's $\kappa$ values. For Cohen's $\kappa$, we randomly selected 150 image--text pairs and assigned each to two participants, obtaining a Cohen's $\kappa$ of 0.844, which indicates almost perfect agreement. For Fleiss's $\kappa$ \cite{fleiss1973equivalence,cocos2015effectively,banerjee1999beyond}, we randomly selected 50 text--image pairs, assigned them to five subjects, and computed the average $\kappa$. We obtain a score of 0.736, which indicates substantial agreement \cite{viera2005understanding}.
\section{Analysis}
\label{analysis}
\paragraph{Overall Statistics.} Table~\ref{raw} shows the rates of true answers for questions Q2--Q10.
Subjects reported that in 17\% of cases the images did not give any information about the step described in the accompanying text. Such images deserve further investigation to characterize their interpretive relationship to the document as a whole. Our anecdotal experience is that such images sometimes provide context for the recipe, which may suggest that imagery, like real-world events \cite{hunter:etal:2015}, creates more flexible discourse structures than linguistic segments on their own.
\begin{table}[h!]
\centering
\begin{tabularx}{0.81\linewidth}{p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|}
\cline{2-10}
& Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Q8 & Q9 & Q10 \\ \hline
\multicolumn{1}{|l|}{True} & 0.829 & 0.058 & 0.211 & 0.131 & 0.056 & 0.491 & 0.209 & 0.289 & 0.133 \\ \hline
\end{tabularx}
\vspace{3mm}
\caption{Rate of true answers for annotation questions Q2--Q10 across the corpus.}
\label{raw}
\end{table}
\begin{table}[h]
\centering
\begin{tabularx}{0.89\linewidth}{p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|}
\cline{2-11}
& Q1 & Q2** & Q3** & Q4** & Q5 & Q6** & Q7** & Q8** & Q9* & Q10** \\ \hline
\multicolumn{1}{|l|}{F1} & 0.74 & 0.86 & 0.76 & 0.85 & 0.88 & 0.92 & 0.64 & 0.83 & 0.77 & 0.92 \\ \hline
\end{tabularx}
\vspace{4mm}
\caption{SVM classification accuracy: bag-of-words features; 80-20 train-test split; 5-fold cross validation. For the first question, this distinguishes highlighted text vs.\ its complement (excluded vs.\ included). For the rest of the questions, this distinguishes text of true instances from text of false instances, and is different from majority class baseline $^*$ at $p < 0.04$, $t= -3.5$ and $^{**}$ at $p < 0.01$, $t>|2.49|$.}
\label{svm}
\end{table}
Subjects reported that the image was required in order to carry out the instruction only for 6\% of cases.
This suggests that subjects construe imagery as backgrounded or peripheral to the document, much as speakers regard co-speech iconic gesture as peripheral to speech \cite{schlenker2017gestural}. Note, by contrast, that subjects characterized 12.7\% of images as introducing a new tool: this includes many cases where the same subjects say the image is not required. In other words, subjects' intuitions suggest that coherent imagery typically does not contribute instruction content, but rather serves as a visual signal that facilitates inferences that have to be made to carry out the instruction regardless. Our annotated examples, where imagery is linked to specific kinds of inferences, provide materials to test this idea.
\begin{figure}[h!]
\captionsetup[subfigure]{labelformat=empty}
\centering
\begin{subfigure}[t]{0.99\columnwidth}
\centering
\includegraphics[width=4.7cm]{step5-temporal.png}
\caption{\textsc{Text:} Top with another layer of ravioli and the remaining sauce not all the ravioli may be needed. Sprinkle with the Parmesan.}
\vspace{-2mm}
\end{subfigure}%
\caption{The image depicts both the action and the result of the action. The recipe is from Autodesk Inc.\ www.instructables.com and was contributed by www.RealSimple.com.}
\label{fig:complex}
\vspace{-4mm}
\end{figure}
\paragraph{The Complex Coherence of Imagery.} Our annotation reveals cases where a single image does include more information than could be packaged into a single textual discourse unit (the proverbial thousand words). In particular, such imagery participates in more complex coherence relationships than we find between text segments. Multiple temporal relationships show this most clearly: 12\% of images that have any temporal relation have more than one. For example, many images depict the action that is described in the text, while also showing preparations that have already been made by displaying the scene in which the action is performed. Figure \ref{fig:complex} depicts the action and the result of the action. It also shows how to prepare before carrying out the action. Other images show an action in progress but nearing completion and thereby depict the result. For instance, the image that accompanies ``mix well until blended'' can show both late-stage mixing and the blended result. Looking at a few such cases closely, the circumstances and composition of the photos seem staged to invite such overlapping inferences.
Such cases testify to the richness of multimodal discourse, and help to justify our research methodology. The True/False questions characterize the relevant features of interpretation without necessarily mapping to single discourse relations. For instance, Q4 and Q5 indicate inferences in line with an \textbf{Elaboration} relation; Q9 and Q10 indicate inferences in line with an \textbf{Exemplification} relation, as information presented in images show just one case of a generalization presented in accompanying text. However, our data shows that these inferences can be combined in productive ways, in keeping with the potentially complex relevant content of images.
\begin{table*}[h]
\centering
\begin{tabularx}{0.5\linewidth}{|X|X|X|X|}
\hline
\multicolumn{4}{|>{\centering\arraybackslash}m{60mm}|}{Q4. Text has quantities not in image} \\ \hline
\multicolumn{2}{|c|}{True} & \multicolumn{2}{c|}{False} \\ \hline
1 & -4.1 & add & -4.5 \\ \hline
cup & -4.4 & place & -4.9 \\ \hline
minutes & -4.7 & put & -5.0 \\ \hline
2 & -4.7 & make & -5.1 \\ \hline
1/2 & -4.9 & mix & -5.1 \\\hline
\multicolumn{4}{c}{} \\ \hline
\multicolumn{4}{|>{\centering\arraybackslash}m{60mm}|}{Q8. Image depicts action in progress} \\ \hline
\multicolumn{2}{|c|}{True} & \multicolumn{2}{c|}{False} \\ \hline
add & -5.0 & 1 & -4.6 \\ \hline
mix & -5.2 & cup & -4.7 \\ \hline
place & -5.3 & minutes & -4.9 \\ \hline
bread & -5.5 & 160 & -5.1 \\ \hline
make & -5.6 & put & -5.2 \\ \hline
\end{tabularx}
\caption{Top five features of Multimodal Naive Bayes classifier for two classification problems and their corresponding log--probability estimates.}
\label{features}
\end{table*}
\paragraph{Information across modalities.} We carried out machine learning experiments to assess what information images provide and what textual cues can guide image interpretation. We use SVM classifiers for performance, and Multinomial Naive Bayes classifiers to explain classifier decision making, both with bag-of-words features.
\begin{table}[]
\centering
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|>{\centering\arraybackslash}m{80mm}|}{Q1. Information in text} \\ \hline
1 & do it clearly & on which \\ \hline
2 & let cool for & favorite toppings \\ \hline
3 & recipe with directions & after an \\ \hline
4 & how slowly the & lightly season \\ \hline
5 & 7 minutes on & the 2 \\ \hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|>{\centering\arraybackslash}m{80mm}|}{Q1. Information in images } \\ \hline
1 & added a beautiful & cover with \\ \hline
2 & put as much & scrapping the \\ \hline
3 & skin off of & finally fold \\ \hline
4 & cut side toward & after an \\ \hline
5 & blend and blend & add a \\ \hline
\end{tabular}
\vspace{5pt}
\caption{Top five bigram and trigram features of NBSVM for the first question. The highlighted text that is most relevant to the image describes depicted actions, while the complement descriptions describe quantities or modifications of the actions that are described in the highlighted segments.}
\label{table:bigram-trigram}
\end{table}
Table~\ref{svm} reports the F1 measure for instance classification with SVMs (with 5-fold cross validation). In many cases, machine learning is able to find cues that reliably help guess inferential patterns.
Table~\ref{features} looks at two effective Naive Bayes classifiers, for Q4 (text has quantities) and Q8 (image depicts action in progress). It shows the features most correlated with the classification decision and their log probability estimates. For Q4, not surprisingly, numbers and units are positive instances.
More interestingly, verbs of movement and combination are negative instances, perhaps because such steps normally involve material that has already been measured. For Q8, a range of physical action verbs are associated with actions in progress; negative features correlate with steps involved in actions that don't require ongoing attention (e.g., baking).
Table~\ref{table:bigram-trigram} reports top SVM with NB (NBSVM) \cite{wang2012baselines} features for Q1 that asks subjects to highlight the part of the text that is most related to the image. Action verbs are part of highlighted text, whereas adverbs and quantitative information that cannot be easily depicted in images are part of the remaining segments of the text.
Such correlations set a direction for designing or learning strategies to select when to include imagery.
\section{Conclusions}
\label{conclusion}
In this paper, we have presented the first dataset describing discourse relations across text and imagery. This data affords theoretical insights into the connection between images and instructional text, and can be used to train classifiers to support automated discourse analysis. Another important contribution of this study is that it presents a discourse annotation scheme for cross-modal data, and establishes that annotations for this scheme can be procured from non-expert contributors via crowd-sourcing.
Our paper sets the agenda for a range of future research. One obvious example is to extend the approach to other genres of communication with other coherence relations, such as the distinctive coherence of images and caption text \cite{alikhani:sivl19}. Another is to link coherence relations to the structure of multimodal discourse. For example, our methods have not yet addressed whether image--text relations have the same kinds of subordinating or coordinating roles that comparable relations have in structuring text discourse \cite{asher2003logics}.
Ultimately, of course, we hope to leverage such corpora to build and apply better models of multimodal communication.
\section{Acknowledgement}
The research presented here is supported by NSF Award IIS-1526723 and through a fellowship from the Rutgers Discovery Informatics Institute. Thanks to Gabriel Greenberg, Hristiyan Kourtev and the anonymous reviewers for helpful comments. We would also like to thank the Mechanical Turk annotators for their contributions.
\bibliographystyle{unsrt}
| {'timestamp': '2019-04-17T02:17:52', 'yymm': '1904', 'arxiv_id': '1904.06286', 'language': 'en', 'url': 'https://arxiv.org/abs/1904.06286'} |
\section{Introduction}
\label{sect:introduction}
Due to the application in error-correction in randomized network coding \cite{KK08}, $q$-analogs of combinatorial designs have gained a lot of interest lately.
Arguably the most important open problem in this field is the question of the existence of a $2$-$(7,3,1)_q$ design, as it has the smallest admissible parameter set of a non-trivial $q$-Steiner system with $t \geq 2$.
It is known as a $q$-analog of the Fano plane and has been tackled in many articles~\cite{EV11,HS11,KP14,Met99,MMY95,Tho87,Tho96}.
In this paper we investigate the automorphism group of a putative binary $q$-analog of the Fano plane.
The following result will be proven in Section~\ref{sect:autfano}.
\begin{theorem}\label{tm:main}
A binary $q$-analog of the Fano plane is either rigid or its automorphism group is cyclic of order $2$, $3$ or $4$.
Representing the automorphism group as a subgroup of $\GL(7,2)$, up to conjugacy it is contained in the following list:
\begin{enumerate}[(a)]
\item The trivial group\text{.}
\item The group of order $2$
\[
\left\langle\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0 \\
1& 0& 0& 0& 0& 0& 0 \\
0& 0& 0& 1& 0& 0& 0 \\
0& 0& 1& 0& 0& 0& 0 \\
0& 0& 0& 0& 0& 1& 0 \\
0& 0& 0& 0& 1& 0& 0 \\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\right\rangle\text{.}
\]
\item One of the following two groups of order $3$:
\[
\left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 1& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)
\right\rangle
\quad
\text{and}
\quad
\left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
\item The cyclic group of order $4$
\[
\left\langle
\left(
\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)
\right\rangle\text{.}
\]
\end{enumerate}
\end{theorem}
The idea of eliminating automorphism groups has been
used in the existence problems for other notorious combinatorial objects. Probably the most prominent example is the projective plane of order $10$. In \cite{W79a,W79b} it was shown that the order of an automorphism is either $3$ or $5$. Those two orders had been eliminated later in \cite{AHT80,JT81}, implying that the automorphism group of a projective plane of order $10$ must be trivial.
Finally, the non-existence has been shown in \cite{LTS89} in an extensive computer search.
There are other examples of existence problems where the same idea has been used, that remain still open.
In coding theory, the question of the existence of a binary doubly-even self-dual code with the parameters $[72,36,16]$ was raised more than 40 years ago in \cite{S73}.
Its automorphisms have been heavily investigated in a series of articles.
After the latest result \cite{YY14}, it is known that its automorphism group is either trivial, cyclic of order $2$, $3$ or $5$, or a Klein four group.
In classical design theory, the smallest $v$ for which the existence of a $3$-design of order $v$ is undecided is $16$; indeed a $3$-$(16, 7, 5)$ design is still unknown.
Recent results show that if such design exists, then its automorphism group is either trivial or a $2$-group \cite{E10, N15}.
The proof of Theorem~\ref{tm:main} is partially based on the following more general result on automorphisms of order $2$ of binary $q$-Steiner triple systems, which will be shown in Section~\ref{sect:triple}.
\begin{theorem}\label{thm:aut2}
Let ${\mathcal D}$ be a binary $S_2[2,3,v]$ $q$-Steiner triple system and $A\in\GL(v,2)$ the matrix representation of an automorphism of ${\mathcal D}$ of order $2$.
For $s\in\{1,\ldots,\lfloor v/2\rfloor\}$ let $A_{v,s}$ denote the $v\times v$ block diagonal matrix built from $s$ blocks $\left(\begin{smallmatrix}0 & 1\\1 & 0\end{smallmatrix}\right)$ followed by a $(v-2s)\times (v-2s)$ unit matrix.
\begin{enumerate}[(a)]
\item In the case $v\equiv 1\bmod {6}$, $A$ is conjugate to a matrix $A_{v,s}$ with $3\mid s$.
\item In the case $v\equiv 3\bmod {6}$, $A$ is conjugate to a matrix $A_{v,s}$ with $s\not\equiv 2\bmod {3}$.
\end{enumerate}
\end{theorem}
\section{Preliminaries}
\subsection{Group actions}
Suppose that a finite group $G$ is acting on a finite set $X$.
For $x \in X$ and $\alpha \in G$, the image of $x$ under $\alpha$ is denoted by $x^{\alpha}$.
The set $x^G = \{x^{\alpha} \mathrel{:} \alpha \in G\}$ is called the \emph{orbit of $x$ under $G$} or the \emph{$G$-orbit of $x$}, in short.
The set of all orbits is a partition of $X$.
We say that $x \in X$ is \emph{fixed} by $G$ if $x^{G} = \{x\}$.
So the fixed elements correspond to the orbits of length $1$.
The set $G_x = \{\alpha \in G \,:\, x^{\alpha} = x \}$ is a subgroup of $G$, called the \emph{stabilizer} of $x$ in $G$.
For all $g\in G$ we have
\begin{equation}
\label{eq:stab_conj}
G_{x^{g}} = g^{-1} G_x g\text{.}
\end{equation}
In particular the stabilizers of elements in the same orbit are conjugate, and any conjugate subgroup $H$ of $G_x$ is the stabilizer of some element in the orbit of $x$.
The Orbit-Stabilizer Theorem states that $\#x^G = [G : G_x]$,%
\footnote{The symbol $\#$ denotes the cardinality of a set.}
implying that $\#x^G$ divides $\#G$.
The action of $G$ on $X$ extends in the obvious way to an action on the set of subsets of $X$:
For $S\subseteq X$, we set $S^\alpha = \{x^{\alpha} \,:\, x \in S\}$.
For further reading on the theory of finite group actions, see \cite{Ker99}.
\subsection{Linear and semilinear maps}
Let $V,W$ be vector spaces over a field $F$.
A map $f : V \to W$ is called \emph{semilinear} if it is additive and if there is a field automorphism $\sigma\in\Aut(F)$ such that $f(\lambda\mathbf{v}) = \sigma(\lambda) f(\mathbf{v})$ for all $\lambda\in F^\times$ and all $\mathbf{v}\in V$.
The map $f$ is \emph{linear} if and only if $\sigma = \id_F$.
In fact, the \emph{general linear group} $\GL(V)$ of all linear bijections of $V$ is a normal subgroup of the \emph{general semilinear group} $\GammaL(V)$ of all semilinear bijections of $V$.
Moreover, $\GammaL(V)$ decomposes as a semidirect product $\GL(V) \rtimes \Aut(F)$.
In this representation, the natural action on $V$ takes the form $(\phi,\sigma)(\mathbf{v}) = \phi(\sigma(\mathbf{v}))$, where $\sigma(\mathbf{v})$ is the component-wise application of $\sigma$ to the coordinate vector of $\mathbf{v}$ with respect to some fixed basis.
The center $Z(\GL(V))$ of $\GL(V)$ consists of all diagonal maps $\mathbf{v} \mapsto \lambda\mathbf{v}$ with $\lambda\in F^\times$.
It is a normal subgroup of $\GL(V)$ and of $\GammaL(V)$.
The quotient group $\GL(V)/Z(\GL(V))$ is known as the \emph{projective linear group} $\PGL(V)$, and the quotient group $\GammaL(V)/Z(\GL(V))$ is known as the \emph{projective semilinear group} $\PGammaL(V)$.
\subsection{The subspace lattice}
Let $V$ denote a $v$-dimensional vector space over the finite field ${\mathbb F}_q$ with $q$ elements.
For simplicity, a subspace of $V$ of dimension $r$ will be called an \emph{$r$-subspace}.
The set of all $r$-subspaces of $V$ is called the \emph{Gra{\ss}mannian} and is denoted by $\gauss{V}{r}{q}$.
As in projective geometry, the $1$-subspaces of $V$ are called \emph{points}, the $2$-subspaces \emph{lines} and the $3$-subspaces \emph{planes}.
Our focus lies on the case $q = 2$, where the $1$-subspaces $\langle\mathbf{x}\rangle_{{\mathbb F}_2}\in \gauss{V}{1}{2}$ are in one-to-one correspondence with the nonzero vectors $\mathbf{x} \in V\setminus\{\mathbf{0}\}$.
The number of all $r$-subspaces of $V$ is given by
$$
\#\gauss{V}{r}{q} = \gauss{v}{r}{q} = \frac{(q^v-1)\cdots(q^{v-r+1}-1)}{(q^r-1)\cdots(q-1)}.
$$
The set $\mathcal{L}(V)$ of all subspaces of $V$ forms the subspace lattice of $V$.
By the fundamental theorem of projective geometry, for $v\neq 2$ the automorphism group of $\mathcal{L}(V)$ is given by the natural action of $\PGammaL(V)$ on $\mathcal{L}(V)$.
In the case that $q$ is prime, the group $\PGammaL(V)$ reduces to $\PGL(V)$, and for the case of our interest $q = 2$, it reduces further to $\GL(V)$.
After a choice of a basis of $V$, its elements are represented by the invertible $v\times v$ matrices $A$, and the action on $\mathcal{L}(V)$ is given by the vector-matrix-multiplication $\mathbf{v} \mapsto \mathbf{v} A$.
\subsection{Designs}
\begin{definition}
Let $t,v,k$ be integers with $0 \leq t \leq k\leq v$, $\lambda$ another positive integer and $\mathcal{P}$ a set of size $v$.
A $t$-$(v,k,\lambda)$ \emph{(combinatorial) design} ${\mathcal D}$ is a set of $k$-subsets of $\mathcal{P}$ (called \emph{blocks}) such that every $t$-subset of $\mathcal{P}$ is contained in exactly $\lambda$ blocks of ${\mathcal D}$.
When $\lambda = 1$, ${\mathcal D}$ is called a \emph{Steiner system} and denoted by $S(t,k,v)$.
If additionally $t=2$ and $k=3$, ${\mathcal D}$ is called a \emph{Steiner triple system}.
\end{definition}
For the definition of a $q$-analog of a combinatorial design, the subset lattice on $\mathcal{P}$ is replaced by the subspace lattice of $V$.
In particular, subsets are replaced by subspaces and cardinality is replaced by dimension.
So we get \cite{Cam74,Cam74:a}:
\begin{definition}
Let $t,v,k$ be integers with $0 \leq t \leq k\leq v$, $\lambda$ another positive integer and $V$ an ${\mathbb F}_q$-vector space of dimension $v$.
A $t$-$(v,k,\lambda)_q$ \emph{design} ${\mathcal D}$ over the field ${\mathbb F}_q$ is a set of $k$-subspaces of $V$ (called \emph{blocks}) such that every $t$-subspace of $V$ is contained in exactly $\lambda$ blocks of ${\mathcal D}$.
When $\lambda = 1$, ${\mathcal D}$ is called a \emph{$q$-Steiner system} and denoted by $S_q[t,k,v]$.
If additionally $t=2$ and $k=3$, ${\mathcal D}$ is called a \emph{$q$-Steiner triple system}.
\end{definition}
There are good reasons to consider the subset lattice as a subspace lattice over the unary \enquote{field} ${\mathbb F}_1$ \cite{Cohn-2004}.
Thus, classical combinatorial designs can be seen as the limit case $q=1$ of a design over a finite field.
Indeed, quite a few statements about combinatorial designs have a generalization to designs over finite fields, such that the case $q = 1$ reproduces the original statement \cite{BKKL,KL,KP14,NP14}.
One example of such a statement is the following \cite[Lemma~4.1(1)]{Suzuki-1990}:
If ${\mathcal D}$ is a $t$-$(v, k, \lambda_t)_q$ design, then ${\mathcal D}$ is also an $s$-$(v,k,\lambda_s)_q$ for all $s\in\{0,\ldots,t\}$, where
\[
\lambda_s := \lambda_t \frac{\gauss{v-s}{t-s}{q}}{\gauss{k-s}{t-s}{q}}.
\]
In particular, the number of blocks in ${\mathcal D}$ equals
\[
\#{\mathcal D} = \lambda_0 = \lambda_t \frac{\gauss{v}{t}{q}}{\gauss{k}{t}{q}}.
\]
So a necessary condition on the existence of a design with parameters $t$-$(v, k, \lambda)_q$ is that for all $s\in\{0,\ldots,t\}$ the numbers $\lambda \gauss{v-s}{t-s}{q}/\gauss{k-s}{t-s}{q}$ are integers (\emph{integrality conditions}).
In this case, the parameter set $t$-$(v,k,\lambda)_q$ is called \emph{admissible}.
It is further called \emph{realizable} if a $t$-$(v,k,\lambda)_q$ design actually exists.
For designs over finite fields, the action of $\Aut(\mathcal{L}(V)) \cong \PGammaL(V)$ on $\mathcal{L}(V)$ provides a notion of isomorphism.
Two designs in the same ambient space $V$ are called \emph{isomorphic} if they are contained in the same orbit of this action (extended to the power set of $\mathcal{L}(V)$).
The \emph{automorphism group} $\Aut({\mathcal D})$ of a design ${\mathcal D}$ is its stabilizer with respect to this group action.
If $\Aut({\mathcal D})$ is trivial, we will call ${\mathcal D}$ \emph{rigid}.
Furthermore, for $G \leq \PGammaL(V)$, ${\mathcal D}$ will be called $G$-invariant if it is fixed by all $g\in G$ or equivalently, if $G\leq \Aut({\mathcal D})$.
Note that if ${\mathcal D}$ is $G$-invariant, then ${\mathcal D}$ is also $H$-invariant for all subgroups $H \leq G$.
\subsection{Admissibility and realizability}
The question of the realizability of an admissible parameter set is very hard to answer in general.
In the special case $t = 1$, $q$-Steiner systems are called \emph{spreads}.
It is known that the spread parameters $S_q[1,k,v]$ are realizable if and only if they are admissible if and only if $k$ divides $v$.
However for $t\geq 2$, the problem tantalized many researchers \cite{Beu78, BEO+, EV11,Met99,Tho87,Tho96}.
Only recently, the first example of such a $q$-Steiner system was constructed \cite{BEO+}.
It is a $q$-Steiner triple system with the parameters $S_2[2,3,13]$.
As a direct consequence of the integrality conditions, the parameter set of a Steiner triple system $S(2,3,v)$ or a $q$-Steiner triple system $S_q[2,3,v]$ is admissible if and only if $v\equiv 1,3\bmod 6$ and $v\geq 7$.
In the ordinary case $q=1$, the existence question is completely answered by the result that a Steiner triple system is realizable if and only if it is admissible \cite{Kirkman}.
However in the $q$-analog case, our current knowledge is quite sparse.
The only decided case is given by the above mentioned existence of an $S_2[2,3,13]$.
The smallest admissible case of a $q$-Steiner triple system is $S_q[2,3,7]$, whose existence is open for any prime power value of $q$.
It is known as a \emph{$q$-analog of the Fano plane}, since the unique Steiner triple system $S(2,3,7)$ is the Fano plane.
It is worth noting that there are cases of Steiner systems without a $q$-analog, as the famous large Witt design with parameters $S(5,8,24)$ does not have a $q$-analog for any prime power $q$ \cite{KL}.
\subsection{The method of Kramer and Mesner}
The method of Kramer and Mesner \cite{KM76} is a powerful tool for the computational construction of combinatorial designs.
It has been successfully adopted and used for the construction of designs over a finite field \cite{MMY95,BKL05}.
For example, the hitherto only known $q$-analog of a Steiner triple system in \cite{BEO+} has been constructed by this method.
Here we give a short outline, for more details we refer the reader to \cite{BKL05}.
As another computational construction method we mention the use of tactical decompositions~\cite{JT85}, which has been adopted to the $q$-analog case in~\cite{NP14}.
The \emph{Kramer-Mesner matrix} $M_{t,k}^G$ is defined to be the matrix whose rows and columns are indexed by the $G$-orbits on the set $\gauss{V}{t}{q}$ of $t$-subspaces and on the set $\gauss{V}{k}{q}$ of $k$-subspaces of $V$, respectively.
The entry of $M_{t,k}^G$ with row index $T^G$ and column index $K^G$ is defined as $\#\{K'\in K^G \mid T\subseteq K'\}$.
Now there exists a $G$-invariant $t$-$(v,k,\lambda)_q$ design if and only if there is a zero-one solution vector $\mathbf{x}$ of the linear equation system
\begin{equation}\label{eq:km}
M_{t,k}^{G}\mathbf{x}=\lambda \mathbf{1},
\end{equation}
where $\textbf{1}$ denotes the all-one column vector.
More precisely, if $\mathbf{x}$ is a zero-one solution vector of the system~\eqref{eq:km}, a $t$-$(v,k,\lambda)_q$ design is given by the union of all orbits $K^G$ where the corresponding entry in $\mathbf{x}$ equals one.
If $\mathbf{x}$ runs over all zero-one solutions, we get all $G$-invariant $t$-$(v,k,\lambda)_q$ designs in this way.
\subsection{Normal forms for square matrices}
Let $F$ be a field and $V$ a vector space over $F$ of finite dimension $v$.
Two elements of $\GL(V)$ are conjugate if and only if their transformation matrices are conjugate in the matrix group $\GL(v,F)$ if and only if the transformation matrices are similar in the sense of linear algebra.
If $F$ is algebraically closed, representatives of the matrices in $F^{v\times v}$ up to similarity are given by the Jordan normal forms.
In the following, we discuss two common normal forms for the more general case that $F$ is not algebraically closed, the Frobenius normal form and the generalized Jordan normal form.
For a monic polynomial
\[
f = a_0 + a_1 X + \ldots + a_{n-1} X^{n-1} + X^n\in F[X]\text{,}
\]
the \emph{companion matrix} $A_f$ of $f$ is the $n\times n$ matrix over $F$ defined as
\[
A_f
= \begin{pmatrix}
0 & 1 & 0 & \cdots & 0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots & \ddots & 0 \\
0 & \cdots & \cdots & 0 & 1 \\
-a_0 & -a_1 & -a_2 & \cdots & -a_{n-1}
\end{pmatrix}\text{.}
\]
It is known that for any square matrix $A$ over $F$ there are unique monic \emph{invariant factors} $f_1 \mid \ldots \mid f_s \in F[X]$ such that $A$ is conjugate to a block diagonal matrix consisting of the blocks $A_{f_1},\ldots,A_{f_s}$.
This matrix is called the \emph{Frobenius normal form} or the \emph{rational normal form} of $A$.
The last invariant factor $f_s$ equals the minimal polynomial of $A$ and the product of all the invariant factors equals the characteristic polynomial of $A$.
The Frobenius normal form corresponds to a decomposition of $V$ into a minimum number of $A$-cyclic subspaces. (A subspace $U$ is called \emph{$A$-cyclic} if there exists a vector $\mathbf{v}\in V$ such that $U = \langle\mathbf{v}, A\mathbf{v}, A^2\mathbf{v},\ldots\rangle_F$.)
By further decomposing the invariant factors into powers of irreducible polynomials, one arrives at an alternative matrix normal form:
For a positive integer $m$ and a monic irreducible polynomial $f$, we define the \emph{Jordan block}
\[
J_{f,m}
= \begin{pmatrix}
A_f & U & \mathbf{0} & \cdots & \mathbf{0} \\
\mathbf{0} & A_f & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & \mathbf{0} \\
\vdots & & \ddots & A_f & U \\
\mathbf{0} & \cdots & \cdots & \mathbf{0} & A_f
\end{pmatrix}
\]
with $m$ consecutive blocks $A_f$ on the diagonal.
Here, $\mathbf{0}$ denotes the $n\times n$ zero matrix and $U$ denotes the $n\times n$ matrix whose only nonzero entry is an entry $1$ in the lower left corner.
Each square matrix over $F$ is conjugate to a \emph{(generalized) Jordan normal form}, which is a block diagonal matrix consisting of Jordan blocks $J_{g_1,m_1},\ldots,J_{g_s,m_s}$ with monic irreducible polynomials $g_i$.
Furthermore, the Jordan normal form is unique up to permuting the Jordan blocks.
The Jordan normal form corresponds to a decomposition of $V$ into a maximal number of $A$-invariant subspaces.
For that reason, the matrix blocks of the Jordan normal form are typically smaller than those in the Frobenius normal form.
Depending on the application, this might be an advantage.
\section{Automorphisms of order $2$ of binary $q$-Steiner triple systems}
\label{sect:triple}
In this section we prove Theorem~\ref{thm:aut2}, which is a result on the automorphisms of order $2$ of a binary $q$-Steiner triple system.
\begin{lemma}
\label{lma:order2}
In $\GL(v,2)$ there are exactly $\lfloor v/2\rfloor$ conjugacy classes of elements of order $2$.
Representatives are given by the block-diagonal matrices $A_{v,s}$ with $s\in\{1,\ldots,\lfloor v/2\rfloor\}$, consisting of $s$ consecutive $2\times 2$ blocks $\left(\begin{smallmatrix}0 & 1\\1 & 0\end{smallmatrix}\right)$, followed by a $(v-2s)\times (v-2s)$ unit matrix.
\end{lemma}
\begin{proof}
Let $A\in\GL(v,2)$.
The matrix $A$ is of order $2$ if and only if its minimal polynomial is $X^2 - 1 = (X+1)^2$.
Equivalently, all the invariant factors of $A$ are $X+1$ or $(X+1)^2$, and the invariant factor $(X+1)^2$ must appear at least once.
So for the number $s$ of the invariant factor $(X+1)^2$, there are the possibilities $s\in\{1,\ldots,\lfloor v/2\rfloor\}$.
Representatives of the elements of order $2$ are now given by the associated Frobenius normal forms.
The invariant factor $X+1$ gives $1\times 1$ blocks $(1)$, and the invariant factor $(X+1)^2 = X^2 + 1$ gives $2\times 2$ blocks $\left(\begin{smallmatrix}0 & 1\\1 & 0\end{smallmatrix}\right)$.
By putting the $1\times 1$ blocks at the end rather than at the beginning, we attain the matrices $A_{v,s}$.
\end{proof}
For a matrix $A$ of order $2$, the unique conjugate $A_{v,s}$ given by Lemma~\ref{lma:order2} will be called the \emph{type} of $A$.
If $A$ is an automorphism of a design $\mathcal{D}$, by equation~\eqref{eq:stab_conj} there is an isomorphic design having the automorphism $A_{v,s}$.
Therefore, for the investigation of the action of $\langle A\rangle$ on $\mathcal{D}$, we may assume $A = A_{v,s}$ without loss of generality.
\begin{lemma}
\label{lem:fixp}
Let $A$ be a matrix of order $2$ and type $A_{v,s}$.
The action of $\langle A\rangle$ partitions the point set $\gauss{{\mathbb F}_2^v}{1}{q}$ into $2^{v-s} - 1$ fixed points and $2^{v-s-1} (2^s - 1)$ orbits of length $2$.
\end{lemma}
\begin{proof}
Me may assume $A = A_{v,s}$.
By the Orbit-Stabilizer Theorem and $\#\langle A_{v,s}\rangle = 2$, all orbits are of length $1$ or $2$.
From the explicit form of the matrices $A_{v,s}$, it is clear that a vector $\mathbf{x} = (x_1,x_2,\ldots,x_v)\in{\mathbb F}_2^v$ is fixed by $A_{v,s}$ if and only if
\[
x_1 = x_2\text{,}\quad x_3 = x_4\text{,}\ldots\text{,}\quad x_{2s - 1} = x_{2s}\text{.}
\]
Thus, each of these $2^{v-s} - 1$ vectors forms an orbit of length $1$, and the remaining $(2^v - 1) - (2^{v-s} - 1) = 2\times 2^{v-s-1} (2^s - 1)$ vectors are paired into $2^{v-s-1}(2^s - 1)$ orbits of length $2$.
\end{proof}
\begin{example}
\label{ex:con2:v3}
A model of the classical Fano plane is given by the points and the planes in $\PG(2,2) = \PG({\mathbb F}_2^3)$.
Its automorphism group is $\GL(3,2)$.
By Lemma~\ref{lma:order2}, there is a single conjugacy class of automorphism types of order $2$, represented by the matrix
\[
A_{3,1} = \begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}\text{.}
\]
The action of $\langle A_{3,1}\rangle$ partitions the point set $\gauss{{\mathbb F}_2^3}{1}{2}$ into the $3$ fixed points
\[
\langle(0, 0, 1)\rangle_{{\mathbb F}_2}\text{,}\quad
\langle(1, 1, 0)\rangle_{{\mathbb F}_2}\text{,}\quad
\langle(1, 1, 1)\rangle_{{\mathbb F}_2}\text{,}
\]
and the two orbits of length $2$
\[
\{
\langle(1, 0, 0)\rangle_{{\mathbb F}_2},\;
\langle(0, 1, 0)\rangle_{{\mathbb F}_2}
\}
\quad\text{and}\quad
\{
\langle (1, 0, 1)\rangle_{{\mathbb F}_2},\;
\langle (0, 1, 1)\rangle_{{\mathbb F}_2}
\}
\text{.}
\]
\end{example}
\begin{example}
\label{ex:con2:v7}
By Lemma~\ref{lma:order2}, the conjugacy classes of elements of order $2$ in $\GL(7,2)$ are represented by
\begin{align*}
A_{7,1}
& = \left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0 \\
1& 0& 0& 0& 0& 0& 0 \\
0& 0& 1& 0& 0& 0& 0 \\
0& 0& 0& 1& 0& 0& 0 \\
0& 0& 0& 0& 1& 0& 0 \\
0& 0& 0& 0& 0& 1& 0 \\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\text{,}
& A_{7,2}
& = \left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0 \\
1& 0& 0& 0& 0& 0& 0 \\
0& 0& 0& 1& 0& 0& 0 \\
0& 0& 1& 0& 0& 0& 0 \\
0& 0& 0& 0& 1& 0& 0 \\
0& 0& 0& 0& 0& 1& 0 \\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\text{,}
& A_{7,3}
& = \left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0 \\
1& 0& 0& 0& 0& 0& 0 \\
0& 0& 0& 1& 0& 0& 0 \\
0& 0& 1& 0& 0& 0& 0 \\
0& 0& 0& 0& 0& 1& 0 \\
0& 0& 0& 0& 1& 0& 0 \\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\text{.}
\end{align*}
The number of fixed points is $63$, $31$ and $15$, and the number of orbits of length~$2$ is $32$, $48$ and $56$, respectively.
\end{example}
Now we investigate the automorphisms of order $2$ of an $S_2[2,3,v]$ $q$-Steiner triple system ${\mathcal D}$.
For the remainder of the article, we will assume that $V = {\mathbb F}_2^v$.
This allows us to identify $\GL(V)$ with the matrix group $\GL(v,2)$.
\begin{lemma}\label{lem:fix_2}
Let ${\mathcal D}$ be an $S_2[2,3,v]$ $q$-Steiner system with an automorphism $A$ of order $2$ and type $A_{v,s}$.
Then each block of ${\mathcal D}$ fixed under the action of $\langle A\rangle$ contains either $3$ or $7$ fixed points.
The number of fixed blocks of the first type is
\begin{equation}\label{eq:f3}
F_3 = 2^{v-s-2}(2^s - 1)
\end{equation}
and the number of fixed blocks of the second type is
\begin{equation}\label{eq:f7}
F_7 = \frac{2^{2v-2s-1}+1-3\cdot 2^{v-s-2}(2^s+1)}{21}\text{.}
\end{equation}
\end{lemma}
\begin{proof}
Without restriction $A = A_{v,s}$.
Let ${\mathcal F}$ denote the set of blocks of ${\mathcal D}$ fixed by $G = \langle A_{v,s}\rangle$.
The restriction of $A_{v,s}$ to any fixed block $K\in{\mathcal F}$ is an automorphism of $\mathcal{L}(K)\cong \mathcal{L}({\mathbb F}_2^3)$ whose order divides $2$.
By Example~\ref{ex:con2:v3}, the restriction is either of the unique conjugacy type of order $2$ or the identity.
So the number of fixed points in $K$ is either $3$ or $7$.
Let ${\mathcal F}_3 \subseteq {\mathcal F}$ denote the set of all fixed blocks with $3$ fixed points and ${\mathcal F}_7$ those having $7$ fixed points.
We double count the set $X$ of all pairs $(\{P_1,P_2\},B)$ where $\{P_1,P_2\}$ is a point-orbit of length $2$ and $B$ is a fixed block of ${\mathcal D}$ passing through $P_1$ and $P_2$.
By Lemma~\ref{lem:fixp}, the number of choices for $\{P_1,P_2\}$ is $2^{v-s-1}(2^s - 1)$.
By the design property, there is exactly $\lambda = 1$ block $B$ of ${\mathcal D}$ passing through the line $L = \langle P_1,P_2\rangle_{{\mathbb F}_2}$.
Since $\{P_1,P_2\}$ is fixed under the action of $A_{v,s}$, so is $L$.
This implies that every block $B'$ in the orbit of $B$ passes through $L$, too.
By $\lambda = 1$, we get $B' = B$, so $B$ is a fixed block.
Thus $\#X = 2^{v-s-1}(2^s - 1)$.
Now we first count the possibilities for the fixed block $B$.
Since $B$ contains a point-orbit of length $2$, necessarily $B\in{\mathcal F}_3$.
By Example~\ref{ex:con2:v3}, each such $B$ contains exactly two point-orbits of length $2$.
So $\#X = 2F_3$, which verifies equation~\eqref{eq:f3}.
Now we double count the set $Y$ of all pairs $(\{P_1,P_2\},B)$ where $P_1,P_2$ are two distinct fixed points and $B$ is a fixed block of ${\mathcal D}$ passing through $P_1$ and $P_2$.
By Lemma~\ref{lem:fixp}, the number of choices for $\{P_1,P_2\}$ is $\binom{2^{v-s}-1}{2} = (2^{v-s} - 1)(2^{v-s-1} - 1)$.
As above, there is a single fixed block of ${\mathcal D}$ passing through $P_1$ and $P_2$, which yields $\#Y = (2^{v-s} - 1)(2^{v-s-1} - 1)$.
Vice versa, for $B\in\mathcal{F}_3$ there are $\binom{3}{2} = 3$ choices for $\{P_1,P_2\}$ and for $B\in\mathcal{F}_7$ the number of choices is $\binom{7}{2} = 21$.
So $\#Y = 3F_3 + 21 F_7$.
This shows that $(2^{v-s} - 1)(2^{v-s-1} - 1) = 3F_3 + 21 F_7$.
Replacing $F_3$ by formula~\eqref{eq:f3}, we obtain formula~\eqref{eq:f7}.
\end{proof}
\begin{corollary}
\label{cor:ord2}
Let ${\mathcal D}$ be an $S_2[2,3,v]$ $q$-Steiner system.
\begin{enumerate}[(a)]
\item In the case $v \equiv 1\bmod 6$, ${\mathcal D}$ does not have an automorphism of type $A_{v,s}$ with $3\nmid s$.
\item In the case $v \equiv 3\bmod 6$, ${\mathcal D}$ does not have an automorphism of type $A_{v,s}$ with $s\equiv 2\bmod 3$.
\end{enumerate}
\end{corollary}
\begin{proof}
By definition, the number $F_7$ of Lemma~\ref{lem:fix_2} is an integer.
So the numerator of the right hand side of \eqref{eq:f7} must be a multiple of $7$.
We compute its remainder modulo $7$.
The multiplicative order of $2$ modulo $7$ equals $3$, so the remainder only depends on the values of $v$ and $s$ modulo $3$.
We get:
\[
\begin{array}{c|ccc}
& s\equiv 0\bmod 3 & s\equiv 1\bmod 3 & s\equiv 2\bmod 3\\
\hline
v\equiv 1\bmod 6 & 0 & 1 & 1 \\
v\equiv 3\bmod 6 & 0 & 0 & -1
\end{array}
\]
This concludes the proof.
\end{proof}
Now Theorem~\ref{thm:aut2}, which was stated in Section~\ref{sect:introduction}, follows as a combination of Corollary~\ref{cor:ord2} and Lemma~\ref{lma:order2}.
\section{Automorphisms of a binary $q$-analog of the Fano plane}
\label{sect:autfano}
This section is dedicated to the proof of Theorem~\ref{tm:main}.
We prove this assertion by eliminating all other subgroups of $\GL(7,2)$.
This elimination is obtained by a combination of the theoretical results of Section~\ref{sect:triple} and a computer aided search based on the Kramer-Mesner method.
In the following, we shall denote by ${\mathcal D}$ a putative $S_2[2,3,7]$ $q$-Steiner system.
Its automorphism group $\Aut({\mathcal D})$ is a subgroup of the group $\GL(7,2) \cong \PGammaL({\mathbb F}_2^7)$ which has the order
\begin{equation}
\label{eq:ordgl27}
\#\GL(7,2) = 2^{21}\cdot 3^4\cdot 5^1\cdot 7^2\cdot 31^1\cdot 127^1\text{.}
\end{equation}
According to the Sylow theorems, for each prime power $p^r$ with $p^r \mid \#\Aut({\mathcal D})$ there exists a subgroup $G \le \GL(7,2)$ such that ${\mathcal D}$ is $G$-invariant.
Hence for each prime factor $p \in \{2,3,5,7,31,127\}$, we strive to find a (preferably small) exponent $r$ such that there is no subgroup $G \le \GL(7,2)$ of order $p^r$ admitting a $G$-invariant $S_2[2,3,7]$ Steiner system.
Then, we can conclude that $\Aut({\mathcal D})$ is not divisible by $p^r$.
Since the automorphism groups of isomorphic $q$-Steiner systems are conjugate, it is sufficient to restrict the search to representatives of the subgroups of $\GL(7,2)$ up to conjugacy.
Representatives of the conjugacy classes of the elements are given by the Jordan normal forms (with a fixed order of the Jordan blocks).
This provides an efficient way to create the cyclic subgroups up to conjugacy.%
\footnote{We remark that two non-conjugate elements may generate conjugate subgroups.}
For general subgroup generation, the software package Magma is used \cite{BCP97}.
For all the cases where subgroups were excluded computationally using the Kramer-Mesner method, details are given in Table~\ref{tbl:km}.
The column \enquote{group} contains the group in question.
Its isomorphism type as an abstract group is described in column \enquote{type}.
The columns \enquote{$T$-orb} and \enquote{$K$-orb} contain information on the induced partition of the $2$-subsets and $3$-subsets, respectively.
For example, the entry $4^{644} 2^{42} 1^{7}$ for the group $G_{4,1}$ means that $\gauss{{\mathbb F}_2^7}{2}{2}$ is partitioned into $644$ orbits of length $4$, $42$ orbits of length $2$, and $7$ orbits of length $1$.
If a column of the Kramer-Mesner matrix $M_{t,k}^G$ contains an entry $> \lambda = 1$, then the corresponding orbit $K^G$ cannot be contained in ${\mathcal D}$.
So we can remove all those columns and the respective variables from the equation system.
The orbit lengths and the size of the reduced system are given in the table columns \enquote{red. $K$-orb} and \enquote{size}.
The reduced system is fed into a solver based on the dancing links algorithm \cite{Knu00}, which is executed on a single core of an Intel Xeon E3-1275 V2 CPU.
The resulting computation time to show the insolvability of the system is given in the last table column \enquote{runtime}.
An entry \enquote{\emph{open}} means that the solver did not terminate within the time limit of a few days.
There are two possibilities where the insolvability can be seen immediately without the need to run the solver:
If the reduced system remains with a zero row (indicated by \enquote{\emph{zero row!}}) or if it is impossible to get the size $\#{\mathcal D} = 381$ as a sum of the lengths of the $K$-orbits (indicated by \enquote{\emph{orbits!}}).
\subsection{Groups of order $2^r$}
From Theorem~\ref{thm:aut2}, we get:
\begin{lemma}\label{lem:2}
If ${\mathcal D}$ is invariant under an automorphism group $G$ of order $2$, then $G$ is conjugate to the cyclic group
\[
G_2
= \langle A_{7,3}\rangle
= \left\langle\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0 \\
1& 0& 0& 0& 0& 0& 0 \\
0& 0& 0& 1& 0& 0& 0 \\
0& 0& 1& 0& 0& 0& 0 \\
0& 0& 0& 0& 0& 1& 0 \\
0& 0& 0& 0& 1& 0& 0 \\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\right\rangle\text{.}
\]
\end{lemma}
\begin{remark}
We attempted to construct a $G_2$-invariant $q$-Steiner triple system $S_2[2,3,7]$ by the Kramer-Mesner method.
The resulting reduced equation system matrix has size $1379 \times 4947$.
Furthermore, from Lemma~\ref{lem:fix_2} we know that ${\mathcal D}$ has exactly $29$ blocks fixed by $G_2$.
However, even with these constraints, the Kramer-Mesner system turned out to be too large to be solved in a reasonable amount of time.
\end{remark}
\begin{lemma}\label{lem:4}
If ${\mathcal D}$ is invariant under a group $G$ of order $4$, then $G$ is a conjugate to the cyclic group
\[
G_{4,1} =
\left\langle
\left(
\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)
\right\rangle
\text{.}
\]
\end{lemma}
\begin{proof}
Using Magma, we got that there are $42$ subgroup classes of $\GL(7,2)$ of order $4$, falling into $7$ cyclic and $35$ non-cyclic ones.
After removing all groups containing a subgroup of order $2$ of a conjugacy type which is excluded by Lemma~\ref{lem:2}, there remains a single cyclic group $G_{4,1}$ and $7$ non-cyclic groups
\begin{align*}
G_{4,2} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} &
G_{4,3} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 1 \\
1 & 1 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} \\
G_{4,4} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} &
G_{4,5} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} \\
G_{4,6} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 & 1 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 1 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} &
G_{4,7} & =
\left\langle
A_{7,3},
\left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{ and} \\
G_{4,8} & =
\left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 1 & 1 & 1 \\
0 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{.}
\end{align*}
The latter $7$ groups could be excluded computationally.
\end{proof}
\begin{lemma}\label{lem:8}
The design ${\mathcal D}$ is not invariant under a group of order $8$.
\end{lemma}
\begin{proof}
There are $867$ subgroup classes of $\GL(7,2)$ of order $8$.
After removing all groups containing a subgroup of order $2$ or $4$ of a conjugacy type which is excluded by Lemma~\ref{lem:2} or Lemma~\ref{lem:4}, there remains the single cyclic group
\[
G_{8,1} = \left\langle \left(\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&1&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)\right\rangle
\]
and the two quaternion groups
\begin{align*}
G_{8,2} & = \left\langle
\left(\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1&0&1&0&1&1&0 \\
0&1&1&0&0&1&0 \\
0&0&1&0&0&0&1 \\
0&1&0&1&1&0&0 \\
0&0&1&0&1&0&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1 \\
\end{smallmatrix}\right)
\right\rangle\text{ and} \\
G_{8,3} & = \left\langle
\left(\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1 \\
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1&1&0&0&0&1&0 \\
0&1&0&0&0&0&1 \\
0&0&1&0&0&0&0 \\
1&0&1&1&1&1&0 \\
0&1&1&0&1&0&1 \\
0&0&1&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)
\right\rangle\text{.}
\end{align*}
These $3$ groups are excluded computationally.
\end{proof}
\subsection{Groups of order $3^r$}
\begin{lemma}\label{lem:3}
If ${\mathcal D}$ is invariant under a group $G$ of order $3$, then $G$ is conjugate to one of the cyclic groups
\[
G_{3,1} = \left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 1& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)
\right\rangle
\quad
\text{and}
\quad
G_{3,2} = \left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
\end{lemma}
\begin{proof}
There are $3$ conjugacy classes of subgroups of $\GL(7,2)$ of order $3$.
They are represented by $G_{3,1}$, $G_{3,2}$ and
\[
G_{3,3} = \left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
The group $G_{3,3}$ could be excluded computationally.
\end{proof}
\begin{lemma}\label{lem:9}
The design ${\mathcal D}$ is not invariant under a group of order $9$.
\end{lemma}
\begin{proof}
There are four conjugacy classes of subgroups of $\GL(7,2)$ of order $9$, a single cyclic and three non-cyclic ones.
After removing all groups containing a subgroup of order $3$ of a conjugacy type which is excluded by Lemma~\ref{lem:3}, there remains the cyclic group
\[
G_{9,1}
= \left\langle
\left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 1 & 0 & 0
\end{smallmatrix}\right)
\right\rangle
\]
and the non-cyclic group
\[
G_{9,2}
= \left\langle\left(
\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1 & 0 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
These two groups could be excluded computationally.
\end{proof}
\subsection{Groups of other prime power orders}
\begin{lemma}\label{lem:5}
The design ${\mathcal D}$ is not invariant under a group of order $5$.
\end{lemma}
\begin{proof}
As the Sylow $5$-group of $\GL(7,2)$, there is a unique conjugacy class of subgroups of order $5$.
A representative is the cyclic group
\[
G_5 = \left\langle
\left(
\begin{smallmatrix}
0& 0& 0& 1& 0& 0& 0\\
1& 0& 0& 1& 0& 0& 0\\
0& 1& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
The action of this group partitions the set of $3$-subspaces into a single fixed point and $2362$ orbits of length $5$.
Because of $\#{\mathcal D} = 381 = 76\cdot 5 + 1$, the fixed $3$-subspace must be contained in ${\mathcal D}$.
After fixing the corresponding variable to the value $1$, the solver returned the insolvability of the system.
\end{proof}
\pagebreak
\begin{lemma}\label{lem:7}
The design ${\mathcal D}$ is not invariant under a group of order $7$.
\end{lemma}
\begin{proof}
The general linear group $\GL(7,2)$ has $3$ conjugacy classes of subgroups of order $7$, represented by
\begin{multline*}
G_{7,1} =
\left\langle\left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
1& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1\\
\end{smallmatrix}\right)\right\rangle
\text{,}\quad
G_{7,2} =
\left\langle\left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
1& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 1& 1& 0& 0\\
0& 0& 0& 0& 0& 0& 1\\
\end{smallmatrix}\right)\right\rangle \\
\text{and}\quad
G_{7,3} =
\left\langle\left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
1& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 1& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1\\
\end{smallmatrix}\right)\right\rangle
\text{.}
\end{multline*}
For the first two groups, the Kramer-Mesner system is immediately seen to be contradictory.
The third system was excluded computationally.
\end{proof}
\begin{lemma}\label{lem:31}
The design ${\mathcal D}$ is not invariant under a group of order $31$.
\end{lemma}
\begin{proof}
As the Sylow $31$-subgroup, $\GL(7,2)$ has a unique conjugacy class of subgroups of order $31$.
A representative is given by the group
\[
G_{31} = \left\langle\left(
\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 & 0
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
Its action on the set of $3$-subspaces is semiregular, which means that all the orbits are of full length $31$.
Because of $31 \nmid 381 = \#{\mathcal D}$, the design ${\mathcal D}$ cannot arise as a union of a selection of these orbits.
\end{proof}
\begin{lemma}\label{lem:127}
The design ${\mathcal D}$ is not invariant under a group of order $127$.
\end{lemma}
\begin{proof}
As the Sylow $127$-subgroup, $\GL(7,2)$ has a unique conjugacy class of subgroups of order $127$.
Because of $127 = \gauss{7}{1}{2}$, it is the Singer subgroup of $\GL(7,2)$, which has already been excluded in \cite[p.~242]{Tho87}.
\end{proof}
\subsection{Groups of non-prime power order}
The above results on prime-power orders can be combined into the following restriction on the order of $\Aut({\mathcal D})$:
\begin{lemma}
\label{lem:1234612}
\[
\#\Aut({\mathcal D})\in\{1,2,3,4,6,12\}
\]
\end{lemma}
\begin{proof}
Assume $\#\Aut({\mathcal D}) \neq 1$ and let $q$ be a prime power dividing $\#\Aut({\mathcal D})$.
By the Sylow theorems, $\Aut({\mathcal D}) \leq \GL(7,2)$ contains a subgroup $G$ of order $q$ and ${\mathcal D}$ is $G$-invariant.
By Equation~\eqref{eq:ordgl27} and Lemma~\ref{lem:8}, \ref{lem:9}, \ref{lem:5}, \ref{lem:7}, \ref{lem:31}, $\ref{lem:127}$ we get $q\in\{2,3,4\}$.
\end{proof}
\pagebreak
\begin{lemma}\label{lem:6}
The design ${\mathcal D}$ is not invariant under a group of order $6$.
\end{lemma}
\begin{proof}
There are $12$ subgroup classes of $\GL(7,2)$ of order $6$, falling into $6$ cyclic ones and $6$ of isomorphism type $S_3$.
After removing all groups containing a subgroup of order $2$ or $3$ of a conjugacy type which is excluded by Lemma~\ref{lem:2} or Lemma~\ref{lem:3}, there remains the single cyclic group
\[
G_{6,1} = \left\langle \left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1
\end{smallmatrix}\right)\right\rangle
\]
and the two groups of isomorphism type $S_3$
\begin{align*}
G_{6,2} & = \left\langle
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 0
\end{smallmatrix}\right)
\right\rangle\text{ and} \\
G_{6,3} & = \left\langle
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{.}
\end{align*}
These $3$ groups are excluded computationally.
\end{proof}
\begin{lemma}\label{lem:12}
The design ${\mathcal D}$ is not invariant under a group of order $12$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:6}, we only need to consider subgroups of $\GL(7,2)$ of order $12$ which do not have a subgroup of order $6$.
The only isomorphism type of a group of order $12$ with this property is the alternating group $A_4$.
However, the Sylow $2$-subgroup of $A_4$ is a Klein four group, which is not possible by Lemma~\ref{lem:4}.
\end{proof}
Now Theorem~\ref{tm:main} follows as a combination of Lemmas~\ref{lem:2}, \ref{lem:4}, \ref{lem:1234612}, \ref{lem:6} and \ref{lem:12}.
\begin{table}[t]
\caption{Kramer-Mesner equation systems}
\label{tbl:km}
\noindent
\noindent\resizebox{\linewidth}{!}{
$\begin{array}{llllllll}
G & \text{type} & T\text{-orb} & K\text{-orb} & \text{red. }K\text{-orb} & \text{size} & \text{runtime} \\
\hline
G_{2} & {\mathbb Z}/2{\mathbb Z} & 2^{1288} 1^{91} & 2^{5800} 1^{211} & 2^{4736} 1^{211} & 1379 \times 4947 & \text{\emph{open}} \\
G_{3,1} & {\mathbb Z}/3{\mathbb Z} & 3^{882} 1^{21} & 3^{3930} 1^{21} & 3^{3720} 1^{21} & 903 \times 3741 & \text{\emph{open}} \\
G_{3,2} & {\mathbb Z}/3{\mathbb Z} & 3^{885} 1^{12} & 3^{3925} 1^{36} & 3^{3710} 1^{36} & 897 \times 3746 & \text{\emph{open}} \\
G_{3,3} & {\mathbb Z}/3{\mathbb Z} & 3^{837} 1^{156} & 3^{3875} 1^{186} & 3^{2170} 1^{186} & 993 \times 2356 & <\text{ 1s} \\
G_{4,1} & {\mathbb Z}/4{\mathbb Z} & 4^{644} 2^{42} 1^{7} & 4^{2900} 2^{98} 1^{15} & 4^{2352} 2^{72} 1^{15} & 693 \times 2439 & \text{\emph{open}} \\
G_{4,2} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{616} 2^{84} 1^{35} & 4^{2816} 2^{252} 1^{43} & 4^{2032} 1^{43} & 735 \times 2075 & \text{\emph{zero row!}} \\
G_{4,3} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{616} 2^{84} 1^{35} & 4^{2816} 2^{252} 1^{43} & 4^{1792} 1^{43} & 735 \times 1835 & \text{\emph{zero row!}} \\
G_{4,4} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2824} 2^{228} 1^{59} & 4^{2032} 2^{144} 1^{59} & 735 \times 2235 & <\text{ 1s} \\
G_{4,5} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{616} 2^{84} 1^{35} & 4^{2816} 2^{252} 1^{43} & 4^{1840} 1^{43} & 735 \times 1883 & \text{\emph{zero row!}} \\
G_{4,6} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2816} 2^{252} 1^{43} & 4^{2000} 2^{96} 1^{43} & 735 \times 2139 & \text{\emph{zero row!}} \\
G_{4,7} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2808} 2^{276} 1^{27} & 4^{1600} 2^{144} 1^{27} & 735 \times 1771 & \text{\emph{zero row!}} \\
G_{4,8} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2816} 2^{252} 1^{43} & 4^{1632} 2^{192} 1^{43} & 735 \times 1867 & \text{1398m 57s}\\
G_5 & {\mathbb Z}/5{\mathbb Z} & 5^{532} 1^{7} & 5^{2362} 1^{1} & 5^{2107} 1^{1} & 539 \times 2108 & \text{1977m 20s} \\
G_{6,1} & {\mathbb Z}/6{\mathbb Z} & 6^{428} 3^{29} 2^{4} 1^{4} & 6^{1928} 3^{69} 2^{16} 1^{4} & 6^{1464} 3^{54} 2^{14} 1^{4} & 465 \times 1536 & \text{21m 39s} \\
G_{6,2} & S_3 & 6^{400} 3^{85} 2^{3} 1^{6} & 6^{1862} 3^{201} 2^{13} 1^{10} & 6^{998} 3^{156} 2^{7} 1^{10} & 494 \times 1171 & <\text{ 1s} \\
G_{6,3} & S_3 & 6^{399} 3^{84} 2^{7} 1^{7} & 6^{1863} 3^{204} 2^{7} 1^{7} & 6^{904} 3^{162} 2^{7} 1^{7} & 497 \times 1080 & <\text{ 1s} \\
G_{7,1} & {\mathbb Z}/7{\mathbb Z} & 7^{376} 1^{35} & 7^{1685} 1^{16} & 7^{1200} 1^{16} & 411 \times 1216 & \text{\emph{zero row!}}\\
G_{7,2} & {\mathbb Z}/7{\mathbb Z} & 7^{381} & 7^{1687} 1^{2} & 7^{1620} 1^{2} & 381 \times 1622 & \text{\emph{orbits!}} \\
G_{7,3} & {\mathbb Z}/7{\mathbb Z} & 7^{381} & 7^{1686} 1^{9} & 7^{1668} 1^{9} & 381 \times 1677 & \text{2443m 17s} \\
G_{8,1} & {\mathbb Z}/8{\mathbb Z} & 8^{322} 4^{21} 2^{3} 1^{1} & 8^{1450} 4^{49} 2^{7} 1^{1} & 8^{1144} 4^{34} 2^{6} 1^{1} & 347 \times 1185 & \text{29s} \\
G_{8,2} & Q & 8^{322} 4^{19} 2^{6} 1^{3} & 8^{1450} 4^{42} 2^{21} 1^{1} & 8^{1160} 4^{28} 2^{12} 1^{1} & 350 \times 1201 & \text{8m 14s} \\
G_{8,3} & Q & 8^{322} 4^{18} 2^{9} 1^{1} & 8^{1450} 4^{43} 2^{18} 1^{3} & 8^{1160} 4^{16} 2^{18} 1^{3} & 350 \times 1197 & \text{1m 50s} \\
G_{9,1} & {\mathbb Z}/9{\mathbb Z} & 9^{294} 3^{7} & 9^{1310} 3^{7} & 9^{1177} 3^{7} & 301 \times 1184 & \text{57s} \\
G_{9,2} & ({\mathbb Z}/3{\mathbb Z})^2 & 9^{291} 3^{15} 1^{3} & 9^{1299} 3^{39} 1^{3} & 9^{1077} 3^{27} 1^{3} & 309 \times 1107 & \text{11m 50s} \\
G_{31} & {\mathbb Z}/31{\mathbb Z} & 31^{86} 1^{1} & 31^{381} & 31^{270} & 87 \times 270 & \text{\emph{orbits!}} \\
\end{array}$}
\end{table}
\section{Conclusion}
We have shown that a binary $q$-analog of the Fano plane can only have very few automorphisms.
This result immediately nullifies many natural approaches for the construction, which would inherently imply too much symmetry.
From the point of view of computational complexity though, the vast part of the search space is still untouched, as it consists of the structures without any symmetry.
We believe that further theoretical insight is needed to reduce the complexity to a computationally feasible level.
After all, the question for the existence of a binary $q$-analog of the Fano plane is still wide open.
\section*{Acknowledgements}
The authors are grateful to the organizers of the conference \enquote{Conference on Random Network Codes and Designs over GF$(q)$} held at the Department of Mathematics of Ghent University, Belgium, from September 18--20, 2013, where the first steps towards the present article were initiated.
This conference was organized in the framework of the COST action IC1104, titled \enquote{Random Network Coding and Designs over GF$(q)$}.
Furthermore, we would like to thank the anonymous referees for valuable suggestions improving the readability of the paper.
| {'timestamp': '2015-07-17T02:08:21', 'yymm': '1501', 'arxiv_id': '1501.07790', 'language': 'en', 'url': 'https://arxiv.org/abs/1501.07790'} |
\section{Introduction}
Spin Glasses (SGs) are models whose mean field (MF) version \cite{SK} undergoes a phase transition, crossing
a critical line in the temperature-field $(T-h)$ plane.
The solution of the MF problem sees the introduction of replicas of the original system as a mathematical trick to perform computations. The resulting Hamiltonian is symmetric under
replica exchanges. However, quite surprising, one finds that in the low-temperature spin-glass phase the replica symmetry is broken.
While the MF behavior of the model is completely under control \cite{FRSB}, also from a rigorous viewpoint \cite{PAN}, we still do not have a confirmed theory for the finite dimensional version.
In particular, there is not agreement both on the upper and lower critical dimension, looking at theoretical, numerical and experimental data
\cite{Moore2011,ReplyMoore2011,SG_MK,Yaida,LR1,LR2,LR3,Janus,experimental}.
The project to perform a renormalization group (RG) analysis is an old one.
The spin glass transition in zero-field was already studied within the RG by Harris et al. \cite{Harris}, and in field by Bray and Roberts \cite{BR}, limiting at the
sector associated with the critical eigenvalue, the so called \textit{replicon}.
Their one-loop analysis was then repeated adding the other sectors, longitudinal and anomalous ones, in refs. \cite{Temesvari1,Temesvari2,Temesvari3,Temesvari4}.
In a recent work also the two loop computation in a field has been performed \cite{Yaida}, suggesting the possibility of a non-perturbative fixed point.
These works are expansions around the Fully Connected (FC) mean field model.
They start studying the symmetric phase, approaching the transition from the high-temperature side, where replica symmetry holds.
Thus the replica symmetric Lagrangian is written, that in its most complete version has three bare masses and eight cubic couplings involving the
replica fields, which correspond to all the possible invariants under the replica symmetry.
At this point one can perform a renormalization \'a la Wilson, integrating the degrees of freedom over an infinitesimal momentum shell, extracting the leading, one-loop, order approximation
in $\epsilon=6-d$.
Although the scheme is clear, the computation is highly technical also for the algebraic viewpoint.
Recently a new loop expansion around the mean field Bethe solution was proposed in ref. \cite{Mlayer}. The new method can be applied to each model that is well defined on a Bethe lattice.
In this paper, we apply for the first time this new expansion to the SG in a field.
We restrict ourself to the limit of high connectivity $z\rightarrow\infty$ to perform computations analytically.
We compute the 1th order correction, and we show that in the $T>0$ region this new expansion is completely equivalent to the field theoretical one,
recovering the results of Bray and Roberts \cite{BR}.
However it has the advantage that the starting point is the original Hamiltonian of the model, with no need to define an associated field theory, nor to
know the initial values of the couplings, and the computations have a clear and simple
physical meaning: while in standard field theory Feynmann diagrams have no special meaning, here the important diagrams have a geometrical interpretation.
Moreover, the expansion is around Bethe lattice that has finite connectivity, an important characteristic shared with finite dimensional systems.
Even if in this work we obtain the same results as standard RG around the fully-connected model,
we will discuss the differences that could arise in the two methods in particular situations.
This work is first of all a verification of the correctness of the method proposed in ref. \cite{Mlayer} and it is a first step towards the generalization
of this new expansion to more complicated cases: finite small connectivity and zero temperature.
\section{Expanding around the Bethe lattice solution}
In this section we just recap the results of ref. \cite{Mlayer}, while in following sections we will apply for the first time these results to the SG model in a field.
Starting from a $D$ dimensional system, the $M$-layer construction of ref. \cite{Mlayer} consists of taking $M$ copies of the original model and rewire them.
In the $M\rightarrow\infty$ limit, the rewiring procedure leads to the Bethe lattice. One can then expand the observables in powers of $\frac{1}{M}$.
In particular, we will focus our attention on the correlation functions, connected over the disorder, let's name them $G$.
The $\frac{1}{M}$ expansion results in a topological expansion in the number of loops.
At order $\frac{1}{M}$, the correlation function between the origin and a point $d$ in the $D$ dimensional model results to be:
$G(d)=\sum_{L=1}^{\infty}{\cal B}(d,L) g^B(L)$, where ${\cal B}(d,L)$ is the number of non-backtracking walks that go from the origin to the point $d$ in the original $D$ dimensional model
and $g^B(L)$ is the correlation between points at distance $L$ on a Bethe lattice.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{DiagramBethe}
\caption{\label{Fig:loop} Spatial loop $\mathcal{L}$ that gives the first correction to the bare correlation functions in the expansion around the Bethe solution.}
\end{figure}
At order $\frac{1}{M^2}$, the correlation function on the original system receive a leading contribution that is
the product of the so-called \textit{line-connected} observable $g^{lc}(\cal L)$ computed on a Bethe lattice,
in which it has been manually injected a loop ${\cal L}$ of the type in Fig. \ref{Fig:loop},
multiplied by the number of such a structure ${\cal L}$ present on the original model.
The line-connected observable is just the observable computed on a Bethe lattice with the loop minus the observable computed on the two paths $L_0,L_1+1,L_3$ and $L_0,L_2+1,L_3$
considered as independent.
The quantities $g^B(L)$ and $g^{lc}(\cal L)$ are model dependent.
In the following we will compute them for the Spin Glass in a field.
To make the computation analytically feasible, we will compute things on a Bethe lattice in the high connectivity limit, at temperature $T>0$.
However things can be computed in finite connectivity and even at $T=0$ using numerically the Belief Propagation equations. This will be the subject of a subsequent paper.
We just want to recall that one could perform the same $M$-layer construction around the fully connected model instead of the Bethe lattice. In the former approach, the leading
divergences at each order are exactly given by the corresponding terms in the loop expansion of the continuum field theory.
In ref. \cite{Mlayer} it is claimed that, if the critical behaviour of the Fully connected model and of the Bethe lattice model is the same, than also the two expansions will lead to the
same results. We will see that this is exactly the case for the SG with field at $T>0$.
\section{Model and definitions}
To be concrete, we are interested in the SG model in field, that has the following Hamiltonian:
$$
H(\{{\sigma\}})=-\sum_{i=1,N} \sigma_i h_i^R -\sum_{ij\in E} \sigma_i \sigma_j J_{ij}\,,
$$
where $E$ is the set of the edges of the lattice. On this model we should compute the quantities $g^B(L)$ and $g^{lc}(\cal L)$ introduced in the previous section.
We will consider the model on a Bethe lattice (for definiteness on a random $z$-regular lattice) in the large connectivity limit, i.e.\ $z$ large:
we will keep the leading terms and we will neglect the $1/z$ corrections. Only at the end we will perform the limit $z\to\infty$.
In this limit, computations are easier and the model has the same properties of the Sherrington Kirkpatrick one \cite{SK}.
The procedure of first computing the results for finite $z$ in the thermodynamic limit ($N\to\infty$) and later send $z$ to infinity makes the physical approach much clearer.
The couplings are i.i.d.\ random variables extracted from a distribution with the following properties: $\overline{J_{ij}}=0$, $\overline{J_{ij}^2}=1/z$.
Higher order moments are irrelevant in the $z\to\infty$ limit. We have indicated by $h_i^R$ the field on the site $i$.
It can be either a local random field extracted from a given distribution or a spatially uniform field. The physics is
equivalent in the two cases.
For simplicity here we consider the case where the fields $h_i^R$ are Gaussian variables with zero average and finite variance $v_h$
\footnote{The attentive reader could notice that the variance $v_h$ never appears in the following. This is not because things do not depend on $v_h$, but because the dependence is
hidden in the definitions of the magnetizations: $m^2\equiv \overline{\langle \sigma\rangle^2}$ and higher moments will implicitly depend on $v_h$.}.
With standard notation, we indicate with $\langle\cdot\rangle$ the thermal average and with $\overline{\:\cdot\:}$ the average over the disorder (random couplings and fields).
In the thermodynamic limit we will compute different kinds of correlation functions first between point at distance $L$ on a standard Bethe lattice;
this will lead to the \textit{bare} propagator
and we will compute its exact expression with two different methods in the high-temperature region: the replica method and the cavity method.
Then we will compute the first correction to this result due to the presence of one spatial loop.
The limit $z\rightarrow\infty$ allows us to compute all the quantities analytically. For finite connectivity, one could
compute everything numerically using Belief-Propagation as usually done on Bethe lattices.
\section{The replica computation of line-correlations}\label{Sec:line_rep}
In the high connectivity limit (that corresponds to small couplings limit) we can expand the replicated partition function as:
\begin{align}
\nonumber
\overline{Z^n}=&\overline{\sum_{\{{\sigma\}}}e^{\beta\sum_a\sum_i\sigma_i^ah_i^R}e^{\beta\sum_a\sum_{ij}\sigma_i^a\sigma_j^aJ_{ij}}}=\\
\simeq&\overline{\sum_{\{{\sigma\}}}e^{\beta\sum_a\sum_i\sigma_i^ah_i^R}\prod_{ij}\((1+\beta\sum_a\sigma_i^a\sigma_j^aJ_{ij}+\frac{\beta^2}{2}\sum_{a,b}\sigma_i^a\sigma_i^b\sigma_j^a\sigma_j^bJ^2_{ij}\))}
\label{eq:Zn_largez}
\end{align}
The neglected terms give sub-leading contributions for large $z$.
As usual in SG computations, $a,b,c,\ldots\in[1,n]$ indicate the replica index, where the replicas are independent copies of the system with the same disorder realization.
A Bethe random regular graphs in the $N\to\infty$ limit becomes locally loop-less. The distance on the graph between two generic points (i.e. the length of the shortest path between them)
is of order $\log(N)$. We are interested in the computation of the correlation functions of spins that are on points at a distance $L$ between them,
in the limit where $N$ goes to infinity at fixed $L$.
In this limit with probability one there is an unique path (of finite length) connecting them so the computation can be done on a single line.
In general we will be interested in correlation functions that are connected
with respect to the disorder. At this end it is convenient to compute
\begin{equation}
G_{a,b;c,d}(L)\equiv\overline{\langle\sigma_0^a\sigma_0^b\sigma_L^c\sigma_L^d\rangle}-\overline{\langle\sigma_0\rangle^2}\cdot \overline{\langle\sigma_L\rangle^2}\,,
\end{equation}
from which we can extract connected and disconnected (with respect to thermal average) correlation functions.
In the following, for simplicity of notation, we will indicate with $\overline{ (\cdot) }^c$ the correlations connected with respect to the disorder:
$\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c\equiv
\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}-\overline{\langle\sigma_0\rangle^2}\cdot\overline{\langle\sigma_d\rangle^2}$.
We define the matrix $T\in \mathbb{R}^{n(n-1)\times n(n-1)}$ such as $T_{ab,cd}(i)\equiv\frac{\beta^2}{2}\overline{\langle\sigma_i^a\sigma_i^b\sigma_i^c\sigma_i^d\rangle}$.
We define $T$ only for $a\neq b$, $c\neq d$, following what is usually done for the matrix $Q_{ab}(i)=\langle\sigma_i^a\sigma_i^b\rangle$ in replica calculations
\footnote{Please be careful to not confuse $T$ with a tensor. We could use two superindices $i,j\in[1,n(n-1)]$ instead of the couples $ab$, $cd$. However we choose this notation
because it will be useful to define the different types of correlation functions in the following.}.
Remembering that $\overline{J_{ij}}=0$, $\overline{J_{ij}^2}=\frac{1}{z}$,
from eq. (\ref{eq:Zn_largez}) we find that:
\begin{equation}
G_{a,b;c,d}(L)=\frac{2}{\beta^2z^L}\left[\prod_{i=0}^LT(i)\right]_{ab,cd}.
\label{eq:corr_d}
\end{equation}
Please notice that all the $i\in[0,L]$ are present in eq. (\ref{eq:corr_d}). In fact on a Bethe lattice, two spins are linked just by a path. If the link between two spins is cut,
the spins become disconnected. This means that if a coupling is 0, all the correlation functions between the two spins linked by that coupling are zero. This automatically implies that
a correlation function should be proportional to the product of all the couplings on the path between the two spins.
From eq. (\ref{eq:corr_d}), we need to compute powers of the matrix $T$.
It is easy to show that:
\small
\begin{align}
\label{eq:T1}
T_{ab,cd}(i)=&\frac{\beta^2}{2}\overline{\langle\sigma_i^a\sigma_i^b\sigma_i^c\sigma_i^d\rangle}=\\
\nonumber
=&\frac{\beta^2}{2}\cdot
\begin{cases}
1 &\text{ if }a=c\text{ , }b=d\text{ or if }a=d\text{ , }b=c \\
m_2 &\text{ if }a=c\text{ or }b=d\text{ or }a=d\text{ or } b=c \\
m_4 &\text{ if } a\neq b\neq c\neq d
\end{cases}
\end{align}
\normalsize
with $m_2=\overline{\langle\sigma_i\rangle^2}$ and $m_4=\overline{\langle\sigma_i\rangle^4}$.
Let us just mention that for eq. (\ref{eq:corr_d}) to hold, $T$ should be defined as the so-called ``cavity'' average:
the average over the rest of the system with the exception of the neighbouring spins on the considered line. However, in the large $z$ limit, cavity averages are equal to standard averages (see
also the Supplementary Material).
Eq. (\ref{eq:T1}) can be written in the form:
\begin{align}
\nonumber
T_{ab,cd}=&\frac{\beta^2}{2}\cdot\left[m_4+(m_2-m_4)(\delta_{ad}+\delta_{bc}+\delta_{bd}+\delta_{ac})+\right.\\
&\left.+(1-2m_2+m_4)(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{cb})\right].
\label{Eq:T}
\end{align}
In order to compute correlation functions, we need to compute powers of $T$; For this reason, we proceed to the diagonalization of $T$.
The whole calculation is explained in the Supplementary Material. Here we just sketch the main steps and state the final result.
First of all we look for eigenvalues and eigenvectors of $T_{ab,cd}$, of the form:
$\sum_{cd}T_{ab,cd}\psi_{cd}=\lambda\psi_{ab}$.
Because of the symmetry of the matrix $T$ under permutations of the replica indices, we know that there are three symmetry classes of eigenvectors
(in an analogous way to what one does when looking to the stability of the Sherrington Kirkpatrick solution for the FC model \cite{dAT}) and three associated eigenvalues:
In the limit $n\rightarrow 0$ the first two eigenvalues (longitudinal and anomalous) are $\lambda_{L/A}=\beta^2(3m_4-4m_2+1),$
while the third one (replicon) is $\lambda_R=\beta^2(1-2m_2+m_4).$
We just want to point out that the eigenvalues are not the same ones as the usual ``replicon, anomalous, longitudinal'' eigenvalues
that comes out from the diagonalization of the Hessian in ref. \cite{dAT}, but we called them in the same way because they identify the same sub-spaces with the same replica symmetries.
In particular at the spin-glass transition the usual replicon goes to zero, while $\lambda_R$ as defined in this paper goes to $\lambda_R=1$ leading to the
divergence of the spin-glass susceptibility (see the following Section).
At this point, we construct the projectors on the sub-spaces of the eigenvectors and write $T$ as a combination of the projectors. In this representation, it is
easy to compute powers of $T$. A special care should be taken in performing the limit $n\rightarrow0$, because of the degeneration of $\lambda_L$ and $\lambda_A$.
The final result is:
\small
\begin{align*}
T^L(n=0)=&\frac{\beta^2}{2}L\lambda_{L/A}^{L-1}\((3m_4-2m_2\))R+\\
&+\lambda_{L/A}^L\((-\frac{R}{2}-Q\))+\lambda_R^L\((\frac{R}{2}+Q+P\)).
\end{align*}
\normalsize
where we defined the matrices $R_{ab,cd}=1$, $Q_{ab,cd}=\frac{1}{4}\left[\delta_{ac}+\delta_{ad}+\delta_{bc}+\delta_{bd}\right]$,
$P_{ab,cd}=\frac{1}{2}\left[\delta_{ac}\cdot\delta_{bd}+\delta_{bc}\cdot\delta_{ad}\right]$.
For a spin-glass model, for each realization of the the system different correlation functions can be defined.
Because of the symmetry of the coupling distribution, the average over the realizations of all the ``linear'' correlations will be zero, and the relevant ones
will be the squared correlations. We will define three main correlations averaged over the thermal noise:
\begin{itemize}
\item The total correlation: $\langle\sigma_0 \sigma_L\rangle$
\item The disconnected correlation: $\langle\sigma_0\rangle\langle\sigma_L\rangle$
\item The connected correlation: $\langle\sigma_0\sigma_L\rangle_c=\langle\sigma_0\sigma_L\rangle-\langle\sigma_0\rangle\langle\sigma_L\rangle$
\end{itemize}
Obviously only two of these correlations are linearly independent.
With these two-spins correlations, we can build different squared correlations:
\begin{itemize}
\item The total-total correlation at distance $L$:
\begin{align}
\nonumber
\overline{\langle\sigma_0\sigma_{L}\rangle^2}^c=&\frac{2}{\beta^2z^{L}}\lim_{n\rightarrow 0}\left[\frac{1}{n(n-1)}\sum_{a\neq b}\((T^{L+1}\))_{ab,ab}\right]=\\
=&\frac{2}{\beta^2z^{L}}\((\frac{3}{2}\lambda_R^{L+1}-\lambda_{L/A}^{L+1}+(L+1)\lambda_{L/A}^{L}\frac{\beta^2}{2}(3m_4-2m_2)\)).
\label{Eq:corrTT_rep}
\end{align}
\item The disconnected-disconnected correlation at distance $L$:
\begin{align}
\nonumber
\overline{\langle\sigma_0\rangle^2\langle\sigma_{L}\rangle^2}^c=&\frac{2}{\beta^2z^{L}}\lim_{n\rightarrow 0}\left[\frac{1}{n(n-1)(n-2)(n-3)}\sum_{a\neq b \neq c\neq d}\((T^{L+1}\))_{ab,cd}\right]=\\
=&\frac{2}{\beta^2z^{L}}\((\frac{\lambda_R^{L+1}}{2}-\frac{\lambda_{L/A}^{L+1}}{2}+(L+1)\lambda_{L/A}^{L}\frac{\beta^2}{2}(3m_4-2m_2)\)).
\label{Eq:corrDD_rep}
\end{align}
\item The total-disconnected correlation at distance $L$:
\begin{align}
\nonumber
\overline{\langle\sigma_0\sigma_{L}\rangle\langle\sigma_0\rangle\langle\sigma_{L}\rangle}^c=&\frac{2}{\beta^2z^{L}}\lim_{n\rightarrow 0}\left[\frac{1}{n(n-1)(n-2)}\sum_{a\neq b\neq c}\((T^{L+1}\))_{ab,ac}\right]=\\
=&\frac{2}{\beta^2z^{L}}\((\frac{3 \lambda_R^{L+1}}{4}-\frac{3\lambda_{L/A}^{L+1}}{4}+(L+1)\lambda_{L/A}^{L}\frac{\beta^2}{2}(3m_4-2m_2)\)).
\label{Eq:corrTD_rep}
\end{align}
\item The connected-connected correlation at distance $L$, whose expression can be obtained as a combination of the previous ones:
\begin{align}
\nonumber
\overline{\langle\sigma_0\sigma_{L}\rangle_c^2}=&\overline{\((\langle\sigma_0\sigma_{L}\rangle-\langle\sigma_0\rangle\langle\sigma_{L}\rangle\))^2}=\overline{\langle\sigma_0\sigma_{L}\rangle^2}+\\
&\nonumber -2\overline{\langle\sigma_0\sigma_{L}\rangle\langle\sigma_0\rangle\langle\sigma_{L}\rangle}+\overline{\langle\sigma_0\rangle^2\langle\sigma_{L}\rangle^2}=\\
=&\frac{1}{\beta^2z^{L}}\lambda_R^{L+1}.
\label{Eq:corrCC_rep}
\end{align}
\end{itemize}
Also in this case only three of these correlations are linearly independent.
\footnote{The expression for the connected correlation in the Bethe lattice is obtained in this paper from a large $z$ expansion.
However we numerically checked that also for small $z$ eq. (\ref{Eq:corrCC_rep}) is valid substituting $z$ with $z-1$ (that is equivalent in the large $z$ limit). }
Others correlations can be readily obtained from the previous one by integration by part.
In the Supplementary Material, we show how to obtain the connected and disconnected bare correlation functions in a cavity approach leading to the same results.
\section{The dominant contribution in the correlation functions} \label{Sec:susc}
To build the susceptibility associated with a given correlation function $C(0,L)$ in a Bethe lattice (where $C$ can be one among the correlation functions considered in the previous section),
one should sum over all the spins that are at distance $L$ from the spin 0, and then over all the distances $L$:
\begin{equation}
\chi_C^B\propto\sum_{L=1}^{\infty}\mathcal{N}_L C(0,L),
\label{eq:chi}
\end{equation}
where $\mathcal{N}_L=z(z-1)^{L-1}$ is the number of spins at distance $L$ from a given spin in a Bethe lattice with connectivity $z$.
The SG transition line is commonly associated to the divergence of the SG susceptibility, that is the susceptibility associated to the connected correlation function.
Substituting eq. (\ref{Eq:corrCC_rep}) and the expression for ${\cal N}_L$ in eq. (\ref{eq:chi}), we discover that the critical line is identified by $\lambda_R=1$
\footnote{Indeed one can check that in the limit of $z\rightarrow\infty$ $\lambda_R$, as defined in this paper, is deeply related to the replicon eigenvalue $\lambda$
of eq. (15) from ref. \cite{dAT}. Using eq. (11) of ref. \cite{dAT}, it can be demonstrated that the relation $\big(\frac{KT}{J}\big)^2 \lambda=-\lambda_R+1$ holds.
The usual SG line associated with $\lambda=0$ translates in $\lambda_R=1$}.
Looking at the eigenvalues, we can numerically check that $\lambda_{L/A}<\lambda_R$.
All the bare correlation functions, as shown in the two precedent sections, have a term proportional to $\lambda_R^L$, that is thus the dominant one.
Recovering the result of standard theory, the critical behavior of all the correlation functions is the same because all depend on the only critical eigenvalue \cite{DedominicisGiardina}.
The susceptibility associated to the different correlations, computed at the critical point, is divergent at the critical line.
On the Bethe lattice, this divergence is a consequence of the exponential decay of the correlations multiplied by the exponential numbers of neighbours at a given distance.
Until now we have computed $\chi_C^B$ on the Bethe lattice.
If now we want to use the Bethe approximation to compute the susceptibility in a $D$ dimensional lattice,
following ref. \cite{Mlayer} (recall Sec. ``Expanding around the Bethe lattice solution''), we should replace $\mathcal{N}_L$ in eq. (\ref{eq:chi}) with the total
number ${\cal B}(d,L)$ of non-backtracking paths of length $L$ between two points at distance $d$ on the original lattice, and add a sum over $d$.
For large $d$ and $L$
\begin{equation}
{\cal B}(d,L) \propto (2D-1)^L |L|^{-D/2}\exp(-|d|^2/(2L))
\end{equation}
implying that at this order the divergence of the susceptibility in a finite dimensional lattice
takes place in correspondence with the divergence in a Bethe lattice with connectivity $z=2D$.
\section{One spatial loop in the RS phase}
In a Bethe lattice in the thermodynamic limit, the density of loops of finite length vanishes, while spatial loops of finite length are common in finite dimensional lattices. In ref.~\cite{Mlayer} a new expansion around the Bethe lattice is performed.
As a result, the first correction to the bare propagator computed in the precedent sections comes from one spatial loop.
In this section, we will compute this contribution confirming that it is totally equivalent to the first correction computed in the usual field theoretical loop expansion, as stated in ref. \cite{Mlayer}.
Following the prescriptions of ref. \cite{Mlayer}, we construct a spatial loop structure $\mathcal{L}$, shown in Fig. \ref{Fig:loop},
formed by two paths of length $L_1+1$ and $L_2+1$ between the points $x$ and $y$ (the length of the internal paths are defined of length $L_1+1$ and $L_2+1$
because in this way results are more compact), plus two external legs of length $L_0$ and $L_3$
to the external spins $\sigma_0$ and $\sigma_d$. The rest of the lattice is a Bethe lattice without loops in the thermodynamic limit.
We will compute the correction to the ``bare'' correlation functions computed in the previous section, that are those connected over the disorder.
Analogously to the definition of $T$, we define the vertex
\small
\begin{align*}
V_{ab,cd,ef}(x)=&\overline{\langle\sigma_x^a\sigma_x^b\sigma_x^c\sigma_x^d\sigma_x^e\sigma_x^f\rangle}=\\
=&\begin{cases}
1 &\mbox{if three pairs of indices are equal}\\
m_2 &\mbox{if two pairs of indices are equal}\\
m_4 &\mbox{if one pair of indices is equal}\\
m_6 &\mbox{if the indices are all different}\\
\end{cases}
\end{align*}
\normalsize
As in the case of $T$, $a\neq b$, $c\neq d$, $e\neq f$.
We write the partition function in the presence of one loop $\mathcal{L}$ and we expand it for small $J$s analogously to eq. (\ref{eq:Zn_largez}).
From it, following the same reasoning of Sec. "The replica computation of line-correlations", we obtain the form of the correlation function when a structure $\mathcal{L}$ is present:
\begin{align}
\nonumber
\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{\mathcal{L}}=&\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{L_0,L_1+1,L_3}+\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{L_0,L_2+1,L_3}+\\
&+\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}
\label{eq:oneloopcorr}
\end{align}
where the first and second terms turn out to be exactly the ``bare'' correlations computed as the two paths $L_0+(L_1+1)+L_3$ and $L_0+(L_2+1)+L_3$ were independent.
They have respectively $L_0+(L_1+1)+L_3$ and $L_0+(L_2+1)+L_3$ couplings, that are the minimal number of couplings to have a non zero correlation.
The last term has $L_0+(L_1+1)+(L_2+1)+L_3$ couplings and turns out to be
\begin{align}
\nonumber
\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}=&\((\frac{\beta^2}{2}\))^2\frac{1}{z^{L_0+(L_1+1)+(L_2+1)+L_3}}\times\\
&\times\sum_{q,r,s,t}\((T^{L_0}\))_{ab,qr}\left[\sum_{e,f,g,h,l,m,o,p}
V_{qr,ef,gh}\((T^{L_1}\))_{ef,lm}\((T^{L_2}\))_{gh,op}V_{st,lm,op}\right]\((T^{L_3}\))_{st,cd}.
\label{eq:loop_corr}
\end{align}
Thus in our case, from eq. (\ref{eq:oneloopcorr}) we see that $\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}$ is exactly the line-connected correlation function,
that gives the one-loop correction in the $\frac{1}{M}$ expansion around the Bethe solution.
The one loop contribution takes a very intuitive form.
To compute the explicit form for $\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}$,
we performed sums and products in eq. (\ref{eq:loop_corr}) using Mathematica
\footnote{The same results can be concluded from Eq.(62)
of Ref. \cite{Temesvari1} after a suitable correspondence between quantities like the
propagators and 3-point vertex.}.
At this point we can compute the one-loop contribution to the different correlation functions
as in Sec. ``The replica computation of line-correlations``. The whole expressions are reported in the Supplementary Material.
As for the bare term, the dominant terms are those with the highest power of $\lambda_R$, that are:
\begin{align}
\overline{\((\langle\sigma_0\sigma_d\rangle_c\))^2}_{lc}\simeq(\overline{\langle\sigma_0\rangle^2\langle\sigma_d\rangle^2}^c)_{lc}\simeq32 \lambda_R^{L_0+L_1+L_2+L_3} [&1+44 m_2^2+101 m_4^2+m_4 (22-90 m_6)+\\
&-2 m_2 (7+67 m_4-30 m_6)-10 m_6+20 m_6^2]
\label{eq:DominantTermLoop}
\end{align}
As explained in Sec. ``The dominant contribution in the correlation functions'', following ref. \cite{Mlayer},
we can now compute the correction to the susceptibility summing over all the length $L_0, L_1, L_2, L_3$, once we have multiplied
by the number of non-backtracking walks.
\section{Relation with previous RG studies}
In ref.~\cite{BR}, the authors perform standard RG calculations in $6-\epsilon$ dimensions for SG with a field.
They write the field-theoretic Hamiltonian in the vicinity of the critical line projected on the replicon eigenspace as
a function of the order parameter $q_{\alpha\beta}$ as:
\begin{align*}
H=&\frac{1}{4}r\sum q_{\alpha\beta}^2+\frac{1}{4}\sum (\nabla q_{\alpha\beta})^2+\\
&-\frac{1}{6} w_1\sum q_{\alpha\beta}q_{\beta\gamma}q_{\gamma\alpha}-\frac{1}{6} w_2\sum q^3_{\alpha\beta}
\end{align*}
with $r$ the reduced temperature, and $w_1$, $w_2$ being coupling constants.
The correlation functions in the momentum space $k$ (projected on the replicon eigenspace) are proportional to $(r+k^2)^{-1}$.
Integrating over an infinitesimal shell $e^{-dl}<k<1$ in the momentum space, they obtain the recursion relation for $r$:
\begin{equation}
\frac{\text{d}r}{\text{d}l}=(2-\eta)r-\frac{K_d}{(1+r)^2}(4 w_1^2-16 w_1w_2+11w_2^2),
\label{Eq:BR}
\end{equation}
with $\eta=\frac{1}{3}K_d(4 w_1^2-16 w_1w_2+11w_2^2)$ and $K_d$ the usual geometrical factor, together with analogous recursion relations for $w_1$ and $w_2$.
The first term in Eq. (\ref{Eq:BR}) is the contribution connected to a renormalization of the critical temperature,
and that we can compute in our approach computing the correction given by a spatial tadpole structure.
The second term is the one coming from the non-trivial loop.
Following ref. \cite{w1w2Parisi},\cite{w1w2Sompolinsky}, in a Bethe lattice in finite and infinite connectivity, as well as in the fully-connected model,
$w_1\propto1-3m_2+3m_4-m_6$ and $w_2\propto2m_2-4m_4+2m_6$.
Inserting these expressions in the loop term of eq. (\ref{Eq:BR}), we obtain:
\small
\begin{align*}
4 w_1-16 w_1w_2+11w_2^2\propto &4 - 56 m_2 + 176 m_2^2 + 88 m_4 +\\
&- 536 m_2 m_4 + 404 m_4^2 - 40 m_6 +\\
&+240 m_2 m_6 - 360 m_4 m_6 + 80 m_6^2
\end{align*}
that is exactly proportional to the coefficient of the dominant term
for the spatial-loop correction to the connected correlation function eq.~(\ref{eq:DominantTermLoop})\footnote{Please note that the replicon contribution, that is the one multiplied by the coefficient $b_4$, that corresponds to the
one found by Bray and Roberts, is the dominant one for the correlation functions. In our
approach, we obtain also the sub-dominant corrections coming from the other sectors, as in ref. \cite{Temesvari1,Temesvari2}.}.
Our approach permits to find the same results in a clearer, simpler and more physically intuitive way.
\section{Conclusions and perspectives}
In ref.~\cite{Mlayer} a new expansion is introduced around the Bethe lattice. This expansion can be applied to all the models that can be defined on a Bethe lattice.
It is supposed to give the same results as standard perturbation theory for fully connected models when the physics of fully connected and finite connectivity MF models is the same.
We test the new expansion of ref. \cite{Mlayer} for the first time in the case of the spin glass model at finite temperature and with an external field in the limit of high connectivity.
We analytically find the same results of standard RG \cite{BR}, confirming the validity of the new $M$-layer topological expansion.
The new method has, however, the advantage that the equations are physically very intuitive:
the bare correlations are just the correlation on a Bethe lattice (i.e.\ on a line), and
the first-order correction to the correlation function is just
the value of the correlation function computed on a spatial loop (finite loops are absent in the Bethe lattice and present in finite dimensional systems) once
the contributions of the two lines forming the loop, considered as independent graphs, are subtracted.
Even if for the case of the SG in a field in the limit $z\rightarrow \infty$ and finite temperature
the results of the new expansion add nothing to what already known about the SGs in finite dimensions, there are cases in which things should be different.
It was already underlined how the Bethe lattice is more similar to finite dimensional systems \cite{CammarotaBiroliTarziaTarjus} with respect to the FC version
for different disordered spin models. In the particular case of SG in a field, the critical line on the FC model tends to infinite field when the temperature goes to zero,
while in the Bethe lattice it ends at a finite field $h_c$ at $T=0$ \cite{w1w2Parisi}.
If the critical point for SGs in field in finite dimension is a zero temperature one, as supposed by some authors \cite{ParisiTemesvariT0dFinite}\cite{SG_MK}, it is crucial
to perform an expansion around the Bethe solution, instead of around FC model, since the latter model is always in the SG phase at $T=0$.
This paper is a first step in this direction. The application of the topological expansion to the SG model in a field at $T=0$ and finite connectivity is at the moment under study.
It could be done according to lines similar to what done in the case of the random field Ising model \cite{RFIM_Mlayer}.
\vspace{.5cm}
We thank Tommaso Rizzo for very useful discussions. We acknowledge funding from the European Research Council (ERC) under
the European Unions Horizon 2020 research and innovation programme
(grant agreement No [694925]).
| {'timestamp': '2018-02-01T02:07:14', 'yymm': '1708', 'arxiv_id': '1708.09581', 'language': 'en', 'url': 'https://arxiv.org/abs/1708.09581'} |
\section{Introduction: a brief overview on cold molecules}
\label{sec:introduction_a_brief_overview_on_cold_molecules}
The research field on translationally cold ($\approx 1$K and lower) and ultracold ($\approx 1$mK and lower) molecules is continuously expanding in many directions, involving an increasing number of groups throughout the world. The availability of gaseous samples of cold neutral molecules opens entirely new avenues for fascinating researches, offering the possibility to control all degrees of freedom of a quantum system. Molecular ions can also be trapped and cooled down inside atomic ion traps behaving like ionic crystals, where each ion is kept at a specific position with a residual motion equivalent to a temperature smaller than 1~K \cite{gerlich2008a}. Cold neutral molecules brought new perspectives in high-resolution molecular spectroscopy \cite{stwalley1999,jones2006,meerakker2005,gilijamse2007}. The expected accuracy of the envisioned measurements with ultracold molecules makes them appear as a promising class of quantum systems for precision measurements related to fundamental issues: the existence of the permanent electric dipole moment of the electron \cite{demille2000,hudson2002,kozlov2002,kawall2004} related to CP-parity violation \cite{hinds1980,cho1991,ravaine2005}, and the time-independence of the electron-to-nuclear and nuclear-to-nuclear mass ratios \cite{veldhoven2004,karr2005,schiller2005,chin2006,demille2008,zelevinsky2008}, or of the fine-structure constant \cite{hudson2006a}. Various proposals have been suggested for achieving quantum information devices \cite{demille2002,rabl2006,kotochigova2006,rabl2007} based on cold polar molecules (i.e. molecules exhibiting a permanent electric dipole moment). Elementary chemical reactions at very low temperatures could be manipulated by external electric or magnetic fields, therefore offering an extra flexibility for their control \cite{krems2005}. The large anisotropic interaction between cold polar molecules is also expected to give rise to quantum magnetism \cite{barnett2006}, and to novel quantum phases \cite{baranov2002,goral2002}. The achievement of quantum degeneracy with cold molecular gases \cite{jaksch2002,donley2002,herbig2003,jochim2003,greiner2003,zwierlein2003,regal2003,bourdel2004} together with the mastering of optical lattices \cite{bloch2005} built a fantastic bridge between condensed matter and dilute matter physics. Indeed, the smooth crossover between the Bose-Einstein condensation (BEC) of fermionic atomic pairs and the Bardeen-Cooper-Schrieffer (BCS) delocalized pairing of fermions related to superconductivity and superfluidity, has been observed experimentally \cite{bourdel2004,chin2004,bartenstein2004,zwierlein2004}.
Cold and ultracold neutral molecules can be formed along two paths:
\begin{itemize}
\item Existing molecules in their vibrational ground state can be slowed down and cooled by various means, such as interactions with external electric and magnetic fields, or by collision with surrounding particles (see for instance the review articles in refs.\cite{bethlem2003,krems2005,hutson2006}. This approach concerns a broad variety of small molecules (OH, NH, NO, SO$_2$, ND$_3$, CO, YbF, C$_7$H$_5$N, H$_2$CO, LiH, CaH,...), but is currently limited to the production of molecules with a translational temperature around 10~mK.
\item Pairs of trapped ultracold atoms can be associated using laser fields (photoassociation, or PA) \cite{jones2006}, or time-varying magnetic fields (magneto- or Feshbach- association) \cite{kohler2006}. Translational temperatures as low as a few tens of microKelvins can be reached, and even lower when quantum degeneracy is achieved. In contrast to the previous case, these results are up to now limited to alkali diatomic molecules which are most often formed in highly-excited vibrational levels, i.e. with high internal energy.
\end{itemize}
In the latter case, a breakthrough occurred in 2008, when the possibility to create ultracold bialkali molecules in the lowest vibrational level of their ground state or their lowest metastable triplet state, has been demonstrated. Caesium dimers have been created in their ground state $v=0$ level after a PA step followed by spontaneous emission and further vibrational pumping, using a sequence of shaped laser pulses \cite{viteau2008}. Dipolar molecules, namely LiCs, have been observed in their absolute ground state rovibrational level $v=0, J=0$ after PA and spontaneous emission \cite{deiglmayr2008a}. Samples close to the degeneracy regime of ultracold molecules in the $v=0$ level of their ground state have been observed for Cs$_2$ \cite{danzl2008,mark2009}, and KRb \cite{ni2008}, and in the $v=0$ level of their lowest triplet state for Rb$_2$ \cite{lang2008}, using the STIRAP technique (Stimulated Rapid Adiabatic Passage) to transfer the population from initial high-lying vibrational levels.
\section{Motivation of the present work}
\label{sec:motivation_of_the_present_work}
The growing availability of ultracold samples of alkali diatomics brings the possibility to observe and study their interactions with neighboring ultracold atoms and molecules. In two independent - and almost simultaneous- experiments, inelastic rate constants for atom-molecule and molecule-molecule collisions have been extracted, using optically trapped Cs$_2$ molecules created by PA and spontaneous emission \cite{staanum2006,zahzam2006}. The rate constants have been found independent from the initially populated rovibrational level. A resonant feature observed in the loss rate after collisions between trapped ultracold Cs$_2$ molecules created by Feshbach association has been interpreted as Cs$_{2}$-Cs$_{2}$ bound states \cite{chin2005}. Inelastic atom-molecule collisions in a trapped sample of RbCs molecules have been studied for both RbCs-Cs and RbCs-Rb cases~\cite{hudson2008}. No systematic dependence on the internal state of the molecule has been probed within the experimental precision.
Theoretical knowledge of the electronic structure of alkali triatomic systems is strongly needed for theoretical dynamical studies, which would support these experiments. There has been a significant research activity concerning the theoretical study of the structure and dynamical properties of homonuclear alkali-metal trimers. Non-additive effects in spin-polarized alkali-metal trimers were studied by Sold\'{a}n \textit{et.al.} \cite{soldan2003} and the three-dimensional potential energy surfaces for the lowest quartet states were constructed for Li$_{3}$~\cite{colavecchia2003a,cvitas2007}, Na$_{3}$~\cite{higgins2000,simoni2009}, K$_{3}$ \cite{quemener2005} and Rb$_{3}$ \cite{soldan2008u}. These were then used to study the corresponding spin-polarized reactive atom-dimer collisions at very low temperatures \cite{cvitas2007,quemener2005,soldan2002,quemener2004,cvitas2005a,cvitas2005,quemener2007,li2008a,li2008b,li2008c}. General trends for the lowest quartet state of homonuclear alkali trimers - correlated to three spin-polarized alkali atoms - have been described by Hutson and Sold\'an \cite{hutson2007}. One of the main characteristic of these systems is the importance of non-additive three-body forces, which are predicted to be large especially close to the equilibrium geometries \cite{soldan2003}. Much less has been done in the case of heteronuclear alkali-metal trimers. One of us has actually started with a systematic study the lowest quartet states of Li$_{2}$A mixed systems (with A=Na, K, Rb, Cs) \cite{soldan2008}. He concluded that the single-reference coupled-cluster approach, which has been successfully used for homonuclear alkali-metal trimers \cite{quemener2005,soldan2008}, would be for various reasons very difficult to employ for calculations of the potential energy surfaces of heteronuclear alkali-metal trimers, and that an alternative approach to the problem should be sought.
In the present paper, we present such an alternative approach. We extend our previous works on effective two-electron diatomic molecules like alkali dimers \cite{aymar2005,aymar2006,deiglmayr2008} and alkali hydrides \cite{aymar2009} to alkali trimers. We propose a preliminary study of the potential energy surfaces of the heaviest alkali trimer, Cs$_3$, modeled as an effective three-electron molecule. Our approach is based on large-core effective core potentials (ECP) with core polarization potentials (CPP). We focused our work on two aspects: (i) the estimation of repulsion effects between the three Cs$^+$ ionic cores at short distances within the present ECP+CPP approach; (ii) the comparison of the results with the results obtained by the MOLPRO package for {\it ab-initio} calculations.
\section{Method of calculation}
\label{sec:method_of_calculation}
\subsection{\emph{Ab initio} methods and basis sets}
\label{sub:ab_initio_methods_and_basis_sets}
The potential energies were calculated making use of the CIPSI package (Configuration Interaction by Perturbation of a multiconfigurational wave function Selected Iteratively)~\cite{huron1973}. As in our previous studies on alkali dimers, the alkali atom A is described by an $\ell$-dependent Effective Core Potential (ECP) for the ionic core A$^+$ \cite{durand1974,durand1975} including effective core polarization potential (CPP) terms \cite{muller1984,foucrault1992}, and by a large set of uncontracted Gaussian functions for the valence electron. The atom is modeled as an effective one-electron system, permitting us to perform Full Configuration Interaction (FCI) calculations for any alkali dimer or trimer, considered as an effective two-electron or three-electron system, respectively.
As the distance between different cores becomes smaller than the equilibrium distance of the system, it is well known that their electrostatic repulsion is not accounted for by this effective large-core potential. At the same time, the addition of the CPP term leads to an non-physical attractive behavior at short distances. For diatomic molecules, an empirical repulsion term is usually added to the calculation (see e.g. Refs.~\cite{spiegelmann1989,pavolini1989,magnier1993,magnier1996}). If this short-range repulsive interaction in the triatomic molecule can be treated as the sum of pairwise interaction terms, computation of the potential surfaces would be greatly simplified. In order to check if such an approximation could be justified, we calculated the core-core repulsion at the Hartree-Fock level making use of the MOLPRO 2006.1 Quantum Chemistry Package~\cite{MOLPRO_brief}. For lithium and sodium, the all-electron correlation-consistent polarized core-valence quadruple-$\zeta$ cc-pCVQZ basis sets~\cite{iron2003} were used. For potassium, rubidium, and caesium, the small-core scalar relativistic effective core potentials ECP10MDF, ECP28MDF, and ECP46MDF, respectively, together with the corresponding uncontracted valence basis sets~\cite{lim2005a} were employed.
For the caesium dimer and trimer, the counterpoise-corrected dimer interaction energies were optimized using an algorithm implemented in MOLPRO. In this algorithm the interaction energies were calculated at the coupled-clusters level making use of a single-reference restricted open-shell variant \cite{knowles1993} of the coupled cluster method \cite{cizek1966} with single, double and non-iterative triple excitations [RCCSD(T)]. The small-core scalar relativistic effective core potentials ECP46MDF was employed together with its uncontracted valence basis set, which was augmented by one set of diffuse function in an even-tempered manner (aug-ECP46MDF). All electrons from the ``outer-core'' $5s5p$ orbitals were included in the RCSSD(T) calculations. The full counterpoise correction of Boys and Bernardi \cite{boys1970} was applied to all the interaction energies in order to compensate for the basis set superposition errors.
\subsection{Core Polarization Potential for a triatomic molecule}
\label{sub:core_polarization_potential}
The core polarization potential (CPP) is given by~\cite{muller1984}:
\begin{equation}
\label{eq:VCPP}
V_{CPP}=-\frac{1}{2} \sum_{c} \alpha_{c}\,\mathbf{f}_{c}\cdot\mathbf{f}_{c}
\end{equation}
The electric field $\mathbf{f}_{c}$ is the one produced at point $\mathbf{r}_{c}$ by all the other electrons and the ionic cores with static dipole polarizability $\alpha_{c}$:
\begin{equation}
\label{eq:fc}
\mathbf{f}_{c}=\sum_{i}\frac{\mathbf{r}_{ci}}{r_{ci}^{3}}h(r_{ci},\rho_{c})\
-\sum_{c^{\prime}\neq c}\frac{\mathbf{R}_{cc^{\prime}}}{R_{cc^{\prime}}^{3}}Z_{c^{\prime}}
\end{equation}
where $\mathbf{r}_{ci}$ is the position vector from core $c$ to electron $i$, $\mathbf{R}_{cc^{\prime}}$ the one between cores $c$ and $c^{\prime}$. The cutoff function $h(r,\rho)$ ensures the convergence of the integral over electronic coordinates. The $V_{CPP}$ term contains a purely geometrical part $v_{nn}$ depending only on the relative positions of the nuclei:
\begin{equation}
\label{eq:vnn}
v_{nn}=-\frac{1}{2}\sum_{c}\alpha_{c}\sum_{c^{\prime},c^{\prime\prime}\neq c}\
\frac{\mathbf{R}_{cc^{\prime}}\cdot \mathbf{R}_{cc^{\prime\prime}}}{R_{cc^{\prime}}^{3}\
R_{cc^{\prime\prime}}^{3}}Z_{c^{\prime}}Z_{c^{\prime\prime}}
\end{equation}
For a diatomic molecule, $v_{nn}$ is given by the familiar formula: $v_{nn}=-\frac{Z_{2}^{2}\alpha_{1}+Z_{1}^{2}\alpha_{2}}{2 R^{4}}$. For a triatomic system, we have:
\begin{equation}
\label{eq:CPP_tri}
\begin{split}
&v_{nn}=-\frac{Z_{2}^{2}\alpha_{1}+Z_{1}^{2}\alpha_{2}}{2 R_{12}^{4}}-\
\frac{Z_{3}^{2}\alpha_{1}+Z_{1}^{2}\alpha_{3}}{2 R_{13}^{4}}\
-\frac{Z_{3}^{2}\alpha_{2}+Z_{2}^{2}\alpha_{3}}{2 R_{23}^{4}}\\
&-\frac{Z_{2}Z_{3}\alpha_{1}\,\mathbf{R}_{12}\cdot\mathbf{R}_{13}}{R_{12}^{3}R_{13}^{3}}\
-\frac{Z_{1}Z_{3}\alpha_{2}\,\mathbf{R}_{12}\cdot\mathbf{R}_{23}}{R_{12}^{3}R_{23}^{3}}\
-\frac{Z_{1}Z_{2}\alpha_{3}\,\mathbf{R}_{13}\cdot\mathbf{R}_{23}}{R_{13}^{3}R_{23}^{3}}
\end{split}
\end{equation}
In those formulas, $Z_{i}$ and $\alpha_{i}$ are, respectively, the net charge and static polarizability of ion core $i$. This term is independent of the \emph{ab initio} calculation itself and is added \emph{a posteriori}.
\section{Volume effect for triatomic systems}
\label{sec:volume_effect_for_triatomic_systems}
The part of the interaction between two nuclei other than the point charge repulsion has been dubbed “volume effect” by Jeung in his 1997 paper on alkali diatomics~\cite{jeung1997}. Here, we are interested in this volume effect for alkali triatomic systems. More precisely, we investigate whether it can be reasonably approximated as a sum of two-body terms.
The two-body repulsion energies $V_{cc}$ are calculated for all the alkali homonuclear diatomic molecules. The three-body repulsion energies $V_{ccc}$ are calculated for particular geometries, namely for D$_{\infty h}$ and D$_{3h}$ symmetries, to provide a representative investigation. Calculations ar done at the Hartree-Fock level of theory starting with the orbitals of the free atoms. Those orbitals are kept frozen as the internuclear distance is shortened. In this respect, our calculations for the two-body term are similar to those of the “A” column from the tables of ref.~\cite{jeung1997}. This comparison is shown in Figure~\ref{fig:compJeung} in logarithmic scale. Apart from a slight discrepancy in the case of Na$_{2}$, there is a good agreement between both calculations, and it is seen that the two-body core repulsion term could be well fitted by an exponential form, as already stated in ref.\cite{pavolini1989}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=15cm]{CompJeung.eps}
\end{center}
\caption{Two-body core repulsion energies for alkali dimers. Blue : our work. Red dashed : Ref.~\cite{jeung1997}. Note the logarithmic scale used for the potential energy axis.}
\label{fig:compJeung}
\end{figure}
Three-body core repulsion energies have been calculated for all homonuclear alkali trimers at linear symmetric and equilateral geometries according to the schemes of Figure~\ref{fig:geom}. In Figure~\ref{fig:addLin}, we compare the three-body core repulsion term for linear geometries $V_{ccc}^{lin}(R)$ (the internuclear distance $R$ being defined in Figure~\ref{fig:geom}), with the sum of two-body term $2 V_{cc}(R)+V_{cc}(2R)$. In the same manner, we compare in Figure~\ref{fig:addEqui} the three-body repulsion term for equilateral geometries $V_{ccc}^{eq}(R)$ to $3 V^{(2)}(R)$. In both cases, it is seen that the three-body core repulsion term is well described by a pairwise additive lemma, up to the region where it becomes negligible. Therefore, just like for alkali dimers, the $V_{ccc}$ term can be added {\it a posteriori} to the potential surface calculations, whatever the chosen grid is for them.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10cm]{geometries.eps}
\end{center}
\caption{Linear (D$_{\infty h}$) and equilateral (D$_{3h}$) geometries at which the three-body core repulsion term have been calculated. The Jacobi coordinates $\{r',R',\theta\}$ used in the calculation of the potential energy surfaces of Cs$_{3}$ are also presented.}
\label{fig:geom}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=15cm]{CoreCoreCore_linear.eps}
\end{center}
\caption{Additivity of the three-body core repulsion term for linear geometries.
Blue : Three-body term $V_{ccc}^{lin}(R)$. Red dashed : sum of the relevant two-body terms $2 V_{cc}(R)+V_{cc}(2R)$ (see text).}
\label{fig:addLin}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=15cm]{CoreCoreCore_equilateral.eps}
\end{center}
\caption{Additivity of the three-body core repulsion term for equilateral geometries.
Blue : Three-body term $V_{ccc}^{eq}(R)$. Red dashed : sum of the relevant two-body terms $3 V^{(2)}(R)$ (see text).}
\label{fig:addEqui}
\end{figure}
\newpage
\section{Preliminary application to caesium trimer}
\label{sec:preliminary_application_to_caesium_trimer}
The inclusion of the three-body core repulsion term is considered in preliminary calculations concerning the $^{4}A_{2}^{\prime}$ state of Cs$_{3}$ which correlates to three spin-polarized caesium atoms. The used basis set is kept relatively small to maintain reasonable size for the FCI, reaching $152\,532$ Slater determinants in C$_{s}$ symmetry in which the calculation is performed. The basis set $(5s3p4d/4s3p3d)$ is detailed in Table~\ref{tab:basisSet}. We use $\ell$-dependent CPP's~\cite{foucrault1992} which allow us to adjust the calculated atomic energies to the experimental ones for the $6s\,^{2}S$, $6p\,^{2}P$ and $5d\,\,^{2}D$ atomic states of caesium. This is achieved by tuning the cutoff radii $\rho_{\ell}$ of Eq.~\ref{eq:fc} to the values displayed in Table~\ref{tab:cutoffRadii}.
\begin{table}
\begin{center}
\begin{tabular}{ccc}
\hline
Angular momentum & Exponent & Contraction coefficients\\
\hline
s & 0.347926 & 0.411589\\
s & 0.239900 & -0.682422\\
s & 0.050502 & 1.\\
s & 0.036900 & 1.\\
s & 0.00515 & 1.\\
p & 0.1837 & 1.\\
p & 0.0655 & 1.\\
p & 0.0162 & 1.\\
d & 0.2106894 & 0.18965\\
d & 0.065471 & 0.22724\\
d & 0.021948 & 1.\\
d & 0.011200 & 1.\\
\hline
\end{tabular}
\end{center}
\caption{Gaussian basis set used on each caesium atom in the ECP+CPP-FCI approach.}
\label{tab:basisSet}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{cc}
\hline
$\ell$ & $\rho_{\ell}$\\
\hline
s & 2.6248\\
p & 1.87\\
d & 2.8111\\
\hline
\end{tabular}
\end{center}
\caption{$\ell$-dependent cutoff radii $\rho_{\ell}$ (in units of $a_{0}$) used in the core polarization potential. The polarization for Cs$^{+}$ ionic core of 16.33 $a_{0}^{3}$ is taken from Ref~\cite{coker1976}.}
\label{tab:cutoffRadii}
\end{table}
\subsection{The triplet state of Cs$_{2}$}
\label{sub:the_triplet_state_of_cs2}
With these basis set and cut-off radii, we first calculate the lowest $^{3}\Sigma_{u}^{+}$ potential energy curve for Cs$_{2}$ dissociating into Cs($6s$)+Cs($6s$) in the framework of our ECP+CC-FCI method. We found an equilibrium internuclear distance R$_{e}=11.82\,a_{0}$ and a well depth of D$_{e}=380.6\,\text{cm}^{-1}$. These numbers can be compared to our optimized results obtained at the RCCSD(T)/aug-ECP46MDF level R$_{e}=12.19\,a_{0}$ and D$_{e}=256.4\,\text{cm}^{-1}$ and the results from Ref.~\cite{soldan2003,hutson2007}, who obtained R$_{e}=12.44\,a_{0}$ and D$_{e}=246.8\,\text{cm}^{-1}$ at the RCCSD(T)/ECP46MWB level of theory. The MOLPRO calculation yields a potential well which is deeper by is a more than $140\,\text{cm}^{-1}$ than the one of the ECP+CC-FCI calculation. Apparently, the small-core ECP in combination with a rich valence basis set in the RCCSD(T) calculations provides more realistic description of the interaction energy than the large-core ECP in combination with a smaller valence basis set used in the FCI calculations (see also the discussion in Section \ref{sec:conclusion}). The potential energy well of the triplet state results from the competition between the exchange energy at short distances and the attractive dispersion forces at long distances and is very sensitive to the quality of the basis. For this $^{3}\Sigma_{u}^{+}$ curve, the inclusion of the two-body core repulsion term does not change the characteristics of the well but ensures a correct exponential behavior at short internuclear distances (Figure~\ref{fig:cs2Triplet}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=13cm]{Different_Quartet_Curves.eps}
\end{center}
\caption{Calculated potential energy curves for the lowest $^{3}\Sigma_{u}^{+}$ state of Cs$_{2}$. Black plain curve : ECP+CPP-FCI calculation with the basis detailed in Table~\ref{tab:basisSet}. Red dashed curve : obtained at the RCCSD(T)/aug-ECP46MDF level. Green chain dotted curve : ECP+CPP full CI result obtained with basis set “A” from Ref.~\cite{aymar2005} (see the discussion in Section~\ref{sec:conclusion}). Blue dotted curve : Multiparameter Morse Long Range fitting to spectroscopic data measured by Xie \emph{et al}~\cite{xie2009}.}
\label{fig:cs2Triplet}
\end{figure}
\subsection{Quartet state of Cs$_{3}$}
\label{sub:quartet_state}
We then calculate the potential energy surface of the lowest quartet state of Cs$_{3}$. Calculations are carried out in Jacobi coordinates $\{r',R',\theta\}$ (depicted in Figure~\ref{fig:geom}) with the angle $\theta$ fixed at $\pi/2$ in order to explore C$_{2v}$ nuclear geometries, as a representative geometry. We have computed 984 points on a grid where $4.4\,a_{0}<r'<40\,a_{0}$ and $4\,a_{0}<R'<40\,a_{0}$. The minimum of the surface is found at a D$_{3h}$ geometry with a bond length of $R'_{e}=10.72\,a_{0}$ and a well depth of $D_{e}=1518\,\text{cm}^{-1}$ with the inclusion of the pairwise three-body core repulsion term, which ensures a realistic short-range repulsive wall. These numbers can be compared to our optimization results obtained at the RCCSD(T)/aug-ECP46MDF level $R'_{e}=11.18\,a_{0}$ and D$_{e}=1208\,\text{cm}^{-1}$ and to the results from Ref.~\cite{soldan2003,hutson2007}, who obtained $R'_{e}=11.33\,a_{0}$ and D$_{e}=1139\,\text{cm}^{-1}$ at the RCCSD(T)/ECP46MWB level of theory. It is worth noting that as the minimum of the quartet potential well is located at shorter distances than the minimum of the Cs$_2$ triplet state, the $V_{ccc}$ term indeed contributes to the depth of the well, and not only in the region of the repulsive wall. However, the magnitude of $V_{ccc}$ is still small, as the equilibrium internuclear distance is $R'_{e}=10.7\,a_{0}$ and the well depth is $D_{e}=1522\,\text{cm}^{-1}$ if we neglect it. As expected from the diatomic calculations above, we obtain a difference of about $300\,\text{cm}^{-1}$ on the well depth of the quartet state of Cs$_{3}$ compared to the MOLPRO calculations. This confirms the limited quality of our basis set at the current level of our computations.
\subsection{Doublet states of Cs$_{3}$}
\label{sub:doublet_states}
The three lowest potential surfaces for Cs$_{3}$ are presented in Figure~\ref{fig:cs3Surfaces}. The $^{2}B_{2}$ and $^{2}A_{1}$ states are the two components of a $^{2}E^{\prime}$ state at D$_{3h}$ symmetry subjected to Jahn-Teller effect. The left (resp. right) column shows the surfaces without (resp. with) the addition of the core repulsion term. Note the strongly unphysical attractive behavior in the region $r'<5\,a_{0}$, which is removed when $V_{ccc}$ is added. The global ground state $^{2}B_{2}$ is characterized by a well depth of $5437.1\,\text{cm}^{-1}$ at the geometry $\{r'=10.58\, a_{0},R'=7.36\, a_{0}\}$ with the inclusion of the core repulsion term. This changes to a well depth of $5489\,\text{cm}^{-1}$ at $\{r'=10.57\, a_{0},R'=7.3\, a_{0}\}$ without the core repulsion term. Once again, the effect of $V_{ccc}$ is larger than in the dimer, as the equlibrium distance is shorter. The other component $^{2}A_{1}$ is characterized by a well depth of $5258.2\,\text{cm}^{-1}$ at $\{r'=8.75\, a_{0},R'=9.28\, a_{0}\}$ with the core repulsion term ($5312\,\text{cm}^{-1}$ at $\{r'=8.66\, a_{0},R'=9.26\, a_{0}\}$ without it).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=13cm]{Cs3_Jacobi_surfaces.eps}
\end{center}
\caption{Lowest three calculated surfaces for Cs$_{3}$ in Jacobi coordinates with fixed $\theta=\pi/2$. Left (resp. right) column : without (resp. with) the inclusion of the cores repulsion term $V_{ccc}$. The red dashed curve locates the equilateral D$_{3h}$ geometry $R'=\frac{\sqrt{3}}{2}r'$. Contours are distant of $500\,\text{cm}^{-1}$ and the dashed contour is the energy of the dissociation $\text{Cs}(^{2}S)+\text{Cs}(^{2}S)+\text{Cs}(^{2}S)$.}
\label{fig:cs3Surfaces}
\end{figure}
We present in Figures~~\ref{fig:cs3CurvesQuartet} and \ref{fig:cs3CurvesDoublet} the cut through the D$_{3h}$ geometry of the previous surfaces. We recall that in the D$_{3h}$ symmetry point group, the two doublet states $^{2}B_{2}$ and $^{2}A_{1}$ discussed before correlates to the two components of a doubly degenerate $^{2}E'$ state. We clearly see in these figures the abrupt attractive behavior which occurs at short distances and that the addition of the core repulsion term gives us back a realistic repulsive wall of potential.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=13cm]{Cs3_Jacobi_curves_quartet.eps}
\end{center}
\caption{Cut through the D$_{3h}$ geometry for the lowest quartet states of Cs$_{3}$. Plain black (resp. red dashed) curve : with (resp. without) the core repulsion term $V_{ccc}^{eq}(R)$. Green chain dotted curve : quartet state on the RCCSD(T)/aug-ECP46MDF level of theory (see the discussion in section \ref{sec:conclusion}.}
\label{fig:cs3CurvesQuartet}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=13cm]{Cs3_Jacobi_curves_doublet.eps}
\end{center}
\caption{Cut through the D$_{3h}$ geometry for the lowest doublet states of Cs$_{3}$. Plain black (resp. red dashed) curve : with (resp. without) the core repulsion term $V_{ccc}^{eq}(R)$. Both components of the $^{2}E'$ state are represented.}
\label{fig:cs3CurvesDoublet}
\end{figure}
\section{Future prospects}
\label{sec:conclusion}
Preliminary potential energy surfaces for doublet and quartet states of the Cs$_{3}$ molecule have been computed in the framework of a ECP+CPP-FCI quantum chemistry approach. The core repulsion interaction has been taken into account as an additional empirical correction as it is routinely done for diatomic molecules calculations. We showed that this term is well described by a sum of two-body terms. As mentioned before in ref.~\cite{pavolini1989}, it can be fitted by an exponential formula. Therefore, this term needs not to be calculated at each point of the three dimensional grid spanned by the degrees of freedom of a triatomic molecule on which the \emph{ab initio} calculations are performed. It can easily be added \emph{a posteriori} to cancel any non-physical behavior which might occur at short internuclear distances.
To estimate the quality of this basis set, we have compared our calculations to high level ones which uses a different method than ours, with the MOLPRO package. Such a comparison between different methods is crucial to assess their accuracy and their consistency, as no spectroscopic data are available for the Cs$_3$ system, as well as for most of the triatomic alkali systems. The calculations presented here are exploratory in nature: the small basis set used (Table~\ref{tab:basisSet}) makes the calculation time short enough to generate over a thousand \emph{ab initio} points. Figure~\ref{fig:cs2Triplet} shows the limitation of this basis set where the well depth of the triplet state of Cs$_{2}$ is overestimated by more than 100~cm$^{-1}$. For the quartet state of Cs$_{3}$, our calculated well depth is consequently 300~cm$^{-1}$ lower than the RCCSD(T) calculations. If we employ the basis set previously used for the caesium dimer~\cite{aymar2005} in the ECP+CPP-FCI method, this yields a very good agreement with the RCCSD(T) calculations, as the difference is now found at about $20\,\text{cm}^{-1}$ between them (Figure~\ref{fig:cs2Triplet}). The agreement is even more spectacular with the potential curve extracted from the spectroscopy of the lowest triplet state of Cs$_{2}$ \cite{xie2009}.
We are currently calculating a comprehensive set of \emph{ab initio} points with the ECP+CPP-FCI with this extended basis for the Cs$_{3}$ molecule which will be the subject of a future paper. Such FCI calculations now involve more than 500,000 Slater determinants. In Figure~\ref{fig:cs3Dinfh}, we display the first calculated points for the D$_{\infty h}$ symmetry of the lowest Cs$_{3}$ quartet state, compared to the MOLPRO calculation as described in Section~\ref{sec:preliminary_application_to_caesium_trimer}. The figure suggests that the results are in good agreement with each other as the position of the minimum looks similar. The point calculated around $12.3\,a_{0}$ with the ECP+CCP full CI method is deeper than the minimum of the MOLPRO curve by about $40\,\text{cm}^{-1}$, while the difference between corresponding triplet potential curves in Figure~\ref{fig:cs2Triplet} is seen to be about $20\,\text{cm}^{-1}$ This suggests that pairwise additivity of the forces is a reasonably good approximation in this case.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=15cm]{Cs3_Quartet_Dinfh.eps}
\end{center}
\caption{Quartet $^{4}\Sigma_{u}^{+}$ state of Cs$_{3}$ in D$_{\infty h}$ symmetry. Green chain dotted curve : calculation at the RCCSD(T)/aug-ECP46MDF level of theory. Plus signs : preliminary calculations with the ECP+CPP-FCI approach using basis set “A” from Ref.~\cite{aymar2005}.}
\label{fig:cs3Dinfh}
\end{figure}
\section*{Acknowledgments}
This work is performed in the framework of the network "Quantum Dipolar Molecular Gases" (QuDipMol) of the EUROCORES/EUROQUAM program of the European Science Foundation. J.D. acknowledges partial support of the French-German University (http://www.dfh-ufa.org), and R.G. support from {\it Institut Francilien de Recherches sur les atomes froids - IFRAF} (http://www.ifraf.org).
\newpage
\section*{References}
\label{sec:references}
| {'timestamp': '2009-02-17T08:25:47', 'yymm': '0902', 'arxiv_id': '0902.2845', 'language': 'en', 'url': 'https://arxiv.org/abs/0902.2845'} |
\section{Introduction}
In the evolution of a star, when the radius of a star becomes less than a
certain limit, called the Schwarzschild radius $R_{Sch}=2GM/c^2$,
where M is the mass of the object; $G$, Newton's universal
gravitational constant, and $c$, velocity of light; neither
light nor material particles can escape from the star. Thus, one finds
from above, the Schwarzschild radius corresponds to the situation when the
escape velocity is equal to the velocity of light, a particle (or a
photon) coming from a large distance when passes near by the black
hole\cite{raine,wheel},
it is not only attracted towards it but also its orbit diverges from a
straight line.It may so happen that if the particle goes too close to a
black hole it is likely to be trapped and hence, it cannot escape to
infinity. As a special case, if the particle travels straight to the
center of the black hole, it falls inside it and is lost forever.
An interesting
problem that is associated with the formation of a black hole is the final
collapse of a massive star. This happens when the nuclear fuel inside the
central core of the star gets exhausted. At this stage, it is the dominance
of the gravitational attraction among the particles within a star over the
internal outward pressure which makes the star to collapse. A black hole,
though may be formed from baryons, leptons etc; the exterior observer
cannot
have access to the details of the inside of the black hole. The observer
probes the black hole mass M, electromagnetic charge Q and angular
momentum J. This is refered to as baldness of the black hole or as Wheeler
describes, a black hole has no hair\cite{wheel}. But actually it has only
three hairs,
M,\ Q,\ J. We in this paper derive the Schwarzschild radius of a system of
gravitating particles quantum mechanically and show that when quantum
exchange correction is taken into account, there is a thin
correction to this radius, which we designate as the skin of the black
hole.
It has been shown previously\cite{duff} that using the quantum field
theoretic method, by
including a single-closed-loop in the self energy, a quantum
correction to
the classical Schwarzschild solution of the order of $\sim G^2$ can be
found. This
comes from the gravity sector. The correction that we find, comes from the
exchange part of the matter-energy sector of the black hole. Our correction is
also of the
order of $\sim G^2$.
We here consider a black hole with mass only, a Schwarzchild\cite{raine}
type black
hole. As
described above, since the black hole forms out of particles due to
immense gravitational attraction, we consider the system of particles as a
many particle system that ultimately forms a black hole.
We have succeeded in developing a quantum mechanical
approach\cite{sm1,sm2} for a system of self-gravitating particles using
Newtonian gravity. We calculate the
energy content of the system including exchange interaction. In order to
calculate the total energy of a star, we choose
a trial single-particle density to account for the distribution of
particles within star. The form of our single-particle density is such
that it has a singularity at the origin. Applying it to the case of a
neutron star, we not only arrive at a compact expression for the radius of
the neutron star, but also obtain an expression for the binding energy of
the star which varies with the particle number\cite{levy} as $N^{7/3}$,
where N is
the particle number. Such a dependence with $N$ is in agreement with those
of the earlier workers.\cite{levy} The aim of the present work is to give
a
derivation of the socalled Schwarzschild radius even without using GTR and
relativistic quantum mechanics. By accounting for the exchange effects due
to the interparticle correlations to the total ground state energy of the
system, we find a quantum correction to the Schwarzschild radius.
\section{Mathematical formulation}
In order to describe a system of N self-gravitating particles in absence
of any source for radiation, we use a Hamiltonian of the form:
\begin{equation}
H=\sum_{i=1}^N\frac{-\hbar^2\nabla_i^2}{2m}+\frac{1}{2}\sum_{i=1}^N
\sum_{j=1,i\ne j}^Nv(|\vec X_i-\vec X_j|),
\end{equation}
where $v(|\vec X_i-\vec X_j|)=-g^2/|\vec X_i-\vec X_j|$, is the
interparticle interaction between pair of gravitating particles
and $g^2=Gm^2$,
$m$ being the mass of particle and G being the universal gravitational
constant. In the present case we confine ourselves to the system of
neutrons only. Since the wavefunction of a neutron star is not known, we
proceed\cite{sm1,sm2} to evaluate the total kinetic energy of the system
using a model
density distribution function for particles (neutrons) within the neutron
star. The particles being fermions, we use the Thomas-Fermi formula for
calculating the total kinetic energy of the system. For an infinite
many-fermion system, the average particle density and the fermi momentum
of
a particle are related to each other as:
\begin{equation}
n=\frac{k_F^3}{3\pi^2}.
\end{equation}
For a finite system like the star, since the density distribution is a
function of radius vector $\vec r$, the fermi energy of a particle is
supposed to be dependent on $r$. For an infinite many-fermion system, the
total kinetic energy of the system\cite{fetter} is given as
\begin{eqnarray}
<KE>_{inf}=\frac{3}{5}n\epsilon_F=\frac{3}{5}n(3\pi^2n)^{2/3}\frac{\hbar^2}{2m}
\nonumber\\
=\frac{3\hbar^2}{10m}(3\pi^2)^{2/3}n^{5/3}\label{ke},
\end{eqnarray}
In analogy with
Eq.(\ref{ke}),
for a finite system\cite{land}, we write,
\begin{equation}
<KE>=\frac{3\hbar^2}{10m}(3\pi^2)^{2/3}\int d\vec r [\rho(\vec
r)]^{5/3}\label{kee},
\end{equation}
The total energy E of the system is given as
\begin{equation}
E=<H>=<KE>+<PE>,
\end{equation}
where
\begin{equation}
<PE>=-\frac{g^2}{2}\int d\vec r d\vec r'\frac{\rho(\vec r)\rho(\vec r')}
{|\vec r-\vec r'|}\label{pe}
\end{equation}
The expression given in Eq.(\ref{pe}) is written in the Hartree
approximation. In order to find E, we choose a trial single-particle
density of the form
\begin{equation}
\rho(\vec r)=A\frac{exp[-(\frac{r}{\lambda})^{1/2}]}
{(\frac{r}{\lambda})^{3/2}}\label{den}
\end{equation}
where A is the normalization constant, which is determined using the
relation
\begin{equation}
\int\rho(\vec r)d\vec r=N .
\end{equation}
As one can see from Eq.(\ref{den}), $\rho(\vec r)$ is singular at $r=0$.
In general one could choose a single particle density of the form
\begin{equation}
\rho(\vec r)=A\frac{exp[-(\frac{r}{\lambda})^{\nu}]}
{(\frac{r}{\lambda})^{3\nu}},\label{deng}
\end{equation}
where $\nu=1,2,3,4...or\ \frac{1}{2},\frac{1}{3},\frac{1}{4},...$. Integer
values of $\nu$ are not permissible because they make the normalization
constant infinite. Out of the fractional values, $\nu=\frac{1}{2}$ is
found to be most appropriate, because, as we shall see later, it gives
the expected upper limit for the critical mass of a neutron
star\cite{sm1}, beyond which black hole formation takes place. Any other
value of $\nu$ would give rise to a different value for the critical mass.
Also because if $\nu$ goes to zero (like $1/n$, $n\rightarrow\infty$),
$\rho(r)$ would tend to the case of a constant density as found in an
infinite many-fermion system. Having accepted the value $\nu=\frac{1}{2}$,
the parameter $\lambda$ associated with $\rho(r)$ is determined after
minimizing $E(\lambda)=<H>$ with respect to $\lambda$. This is how, we are
able to find the total energy of the system corresponding to its lowest
energy state.
After evaluating the integral shown in Eq.(\ref{kee}) and Eq.(\ref{pe}),
we
obtain
\begin{equation}
E(\lambda)=\frac{\hbar^2}{m}\frac{12}{25\pi}(\frac{3\pi
N}{16})^{5/3}\frac{1}{\lambda^2}-\frac{g^2N^2}{16}\frac{1}{\lambda}\label{bige}
\end{equation}
Differentiating this with respect to $\lambda$ and then equating it with
zero, we obtain the value of $\lambda$ at which the minimum occurs. This
is found as:
\begin{equation}
\lambda_0=\frac{72}{25}\frac{\hbar^2}{mg^2}(\frac{3\pi}{16})^{2/3}
\frac{1}{N^{1/3}}\label{lam}
\end{equation}
Here we are only concerned with the total kinetic energy of the system. At
$\lambda=\lambda_0$, we have,
\begin{equation}
<KE>=0.015441\frac{mg^4}{\hbar^2}N^{7/3}\label{kes}
\end{equation}
The total energy E of the system is found to be just negative of this.
\section{Derivation of Schwarzschild Radius}
We
now try to calculate the average velocity of a particle within the neutron
star using Eq.(\ref{kes}). Let us denote it by $<\vec v^2>$. If M denotes
the total mass of the neutron star, one writes
\begin{equation}
<KE>=\frac{1}{2}M<\vec v^2>\label{ket},
\end{equation}
where $M=Nm$. By comparing Eq.(\ref{ket}) with Eq.(\ref{kes}), one
obtains,
\begin{equation}
<\vec v^2>=0.030882\frac{g^4}{\hbar^2}N^{4/3}.
\end{equation}
From the expression for the total kinetic energy of an infinite
many-fermion system\cite{fetter}, one finds that the average velocity of a
particle within the system is $\sim 0.77v_f$, $v_f$ being the fermi
velocity\cite{fetter} of the particle within the system, which is maximum
velocity of
that particle. From this, one clearly sees that the maximum velocity of a
particle belonging to an infinite many-fermion system is greater than the
average particle velocity. In view of this fact, we could write the
maximum velocity of a particle within a neutron star as
\begin{equation}
<\vec v^2>_{max}=\alpha <v^2>=0.030882\alpha\frac{g^4}{\hbar^2}N^{4/3}
\end{equation}
where $\alpha$ is a constant whose value is to be greater than unity and
it is to be calculated later. $v_{max}$ can be identified as the escape
velocity of a particle within a neutron star.
According to special theory of relativity, $\vec v^2_{max}$ is to be less
than $c^2$, c being the velocity of light. That is,
\begin{equation}
0.030882\alpha\frac{g^4}{\hbar^2}N^{4/3}\le c^2,
\end{equation}
From this it follows that
\begin{equation}
N\le\frac{13.574409}{\alpha^{3/4}}(\frac{\hbar
c}{g^2})^{3/2}=N_c\ (say),\label{nu}
\end{equation}
having $g^2=Gm^2$. Substituting Eq.(\ref{nu}) in Eq.(\ref{lam}), one
finds
that,
\begin{equation}
\lambda_0\ge\lambda_c=\frac{Gm}{c^2}\alpha^{1/4}[0.8483718(\frac{\hbar
c}{g^2})^{3/2}].
\end{equation}
If we define the radius of a neutron star as $R_0=2\lambda_0$, we have
the expression for the critical radius as,
\begin{equation}
R_c=2\lambda_c=2\frac{Gm}{c^2}\alpha^{1/4}[0.8483718(\frac{\hbar
c}{g^2})^{3/2}]=\frac{2GM}{c^2}\label{rad}
\end{equation}
Our identification about the radius $R$ of the star with $2\lambda_0$ is
based on the use of socalled quantum mechanical tunneling\cite{karp}
effect. Classically, it is well known that a particle has a turning
point
where the potential energy becomes equal to the total energy. Since the
kinetic energy and therefore the velocity are equal to zero at such a
point, the classical particle is expected to be turned around or reflected
by the potential barrier. From the present theory it is seen that the
turning point occurs at a distance $R=2\lambda_0$. This is the reason why
we identify $2\lambda_0$ with the radius of a star. For $R>2\lambda_0$,
a particle, belonging to the system, may have an access to the region beyond
$R>2\lambda_0$, because of quantum mechanical tunneling, but is
forbidden by classical theory.
$R_c$ as given in Eq.(\ref{rad}) is being identified as
the so called Schwarzschild radius which we have derived here by treating
the system as a quantum many-body system. When $R_0\le R_c$, the
corresponding
neutron star becomes a black hole. From Eq.(\ref{nu}), we therefore find
that the lowest mass of the neutron star beyond which black hole formation
takes place is given as
\begin{equation}
M_c=mN_c=\frac{13.574409}{\alpha^{3/4}}m(\frac{\hbar c}{g^2})^{3/2}
\end{equation}
In order to determine $\alpha$, we now try to evaluate the limiting mass
of a neutron star following the general expression for the radius of a
star. Beyond this mass, the black hole formation is likely to take place.
For that, we consider the situation when
\begin{equation}
(R_0=2\lambda_0)= (R_{sch}=\frac{2GM}{c^2}),
\end{equation}
where $M=Nm$. From this, we arrive at
\begin{equation}
N\ge(1.696758)(\frac{\hbar c}{g^2})^{3/2}=N_c.\label{nc}
\end{equation}
Since the expression in the right hand side of Eq.(\ref{nc}) should be
equal to the one given in right hand side of Eq.(\ref{nu}), we must have
$\alpha=16$. Under this situation, we have
\begin{equation}
v^2_{max}=0.494112\frac{g^4}{\hbar^2}N_c^{4/3},
\end{equation}
where $N_c=1.696758(\frac{\hbar c}{g^2})^{3/2}$, which, when evaluated,
becomes $3.7390777\times 10^{57}$. For a neutron star in which the number
of neutrons exceeds $N_c$, it has the tendency of forming a black hole. In
that case, its mass must exceed $M=M_c=mN_c=3.12213\ M_{\odot}$,
$M_{\odot}$
being the solar mass.
\section{Quantum Correction}
So far we have been discussing about the quantum mechanical derivation of
the
Schwarzschild radius $R_{Sch}$. The very form of $R_{Sch}$ shows that it
is a classical result, leaving aside the fact the number of particles N
within a neutron star\cite{wes} is to be less than $N_c$ where
$N_c=1.70(\frac{\hbar
c}{Gm_n^2})^{3/2}$, which involves the Planck's constant $\hbar$.
Now, inorder to account for the
quantum corrections to $R_{Sch}$, we go beyond the Hartree
contribution to the total energy of the system. That is the exchange
correction or Hartree-Fock(HF) term\cite{beth} over the Hartree result
(direct contribution). Since
the
HF-correction term is non-local we make use of the local density
approximation\cite{beth} to write it as,
\begin{equation}
<PE>_{ex}=\frac{3}{2\pi}(3\pi^2)^{1/3}g^2\int d\vec r[\rho(\vec r)]^{4/3}.
\end{equation}
This when evaluated gives
\begin{equation}
<PE>_{ex}=\frac{27}{4}(\frac{1}{16\pi})^{4/3}(3\pi^2)^{1/3}g^2
\frac{N^{4/3}}{\lambda}.
\end{equation}
With the inclusion of this extra term, the expression for $E(\lambda)$,
Eq.(\ref{bige}) is minimized with respect to $\lambda$ and we arrive at
\begin{equation}
\lambda_0'=\frac{72}{25}(\frac{3\pi}{16})^{2/3}\frac{\hbar^2}{mg^2}
\frac{1}{N^{1/3}}[1+\frac{1.8010}{N^{2/3}}]\label{lamor}
\end{equation}
Following the argument discussed earlier, we identify the radius of the
neutron star by $R'_0=2\lambda'_0$. As before writing
${v'}^2_{max}=16<{v'}^2>$
and using the condition that ${v'}^2_{max}\le c^2$, we obtain
\begin{equation}
N\le N'_c=1.696758(\frac{\hbar c}{g^2})^{3/2}[1+0.4747761(\frac{g^2}{\hbar
c})]
\end{equation}
Corresponding to $N'_c$ , the new expression for the critical radius
$R'_c$ becomes
\begin{equation}
R'_c=2\lambda'_c=2\frac{GM_c}{c^2}[1+0.7912723(\frac{g^2}{\hbar
c})]\label{rp}
\end{equation}
where $M_c=mN_c=1.696758(\frac{\hbar c}{g^2})^{3/2}$. The above
expression, Eq. (\ref{rp}) is obtained by keeping terms upto order
$(\frac{g^2}{\hbar c})$ only in Eq. (\ref{lamor}).
For $N>N'_c$, the neutron star is likely to go over the black hole stage.
From Eq.(\ref{rp}), we find that the second term within the square
bracket, forms
the quantum correction to the Schwarzschild radius. As expected, it
involves the gravitational fine structure constant $(\frac{g^2}{\hbar
c})$. Since it is of the order $10^{-39}$, obviously it makes an
extremely small correction to $R_{Sch}$.
It has been shown earlier\cite{duff} that using the quantum field
theoretic method and by
including a single-closed-loop in the self energy, a quantum correction to
the classical Schwarzschild solution of the order of $\sim G^2$ can be
found. This
comes from the gravity sector. The correction that we get is also of the
order $\sim G^2$ but it comes from the
exchange part of the matter-energy sector of the black hole.
\section{Conclusion}
We in this paper derive the Schwarzschild radius of a black hole from a
condensed matter point of view by using a single particle density
distribution for the many-body self-gravitating system which ultimately
forms a black hole. By incorporating the quantum mechanical exchange
interaction, we also find a thin correction to the Schwarzschild radius
which we designate as the skin of the black hole.
We thank F. Hehl for critically reading the manuscript and bringing to our
notice the Ref.3.
| {'timestamp': '2007-03-18T15:31:53', 'yymm': '0703', 'arxiv_id': 'physics/0703173', 'language': 'en', 'url': 'https://arxiv.org/abs/physics/0703173'} |
\section{Introduction}
\label{sec:intro}
Let $G$ be a simply-connected, simple compact Lie group.
Principal $G$-bundles over $S^{4}$ are classified by the value
of the second Chern class, which can take any integer value. Let
\(\namedright{P_{k}}{}{S^{4}}\)
represent the equivalence class of principal $G$-bundle whose
second Chern class is $k$. Let \ensuremath{\mathcal{G}_{k}}\ be the \emph{gauge group}
of this principal $G$-bundle, which is the group of $G$-equivariant
automorphisms of $P_{k}$ which fix $S^{4}$.
Crabb and Sutherland~\cite{CS} showed that, while there are countably
many inequivalent principal $G$-bundles, the gauge groups
$\{\ensuremath{\mathcal{G}_{k}}\}_{k\in\mathbb{Z}}$ have only finitely many distinct homotopy
types. There has been a great deal of interest recently in determining
the precise number of possible homotopy types. The following
classifications are known. For two integers $a,b$, let $(a,b)$ be
their greatest common divisor. If $G=SU(2)$ then
$\ensuremath{\mathcal{G}_{k}}\simeq\ensuremath{\mathcal{G}_{k^{\prime}}}$ if and only if $(12,k)=(12,k^{\prime})$~\cite{K};
if $G=SU(3)$ then $\ensuremath{\mathcal{G}_{k}}\simeq\ensuremath{\mathcal{G}_{k^{\prime}}}$ if and only if
$(24,k)=(24,k^{\prime})$~\cite{HK}; if $G=SU(5)$ then
$\ensuremath{\mathcal{G}_{k}}\simeq\ensuremath{\mathcal{G}_{k^{\prime}}}$ when localized at any prime $p$ or
rationally if and only if $(120,k)=(120,k^{\prime})$~\cite{Th2}; and if
$G=Sp(2)$ then $\ensuremath{\mathcal{G}_{k}}\simeq\ensuremath{\mathcal{G}_{k^{\prime}}}$ when localized at any prime $p$
or rationally if and only if $(40,k)=(40,k^{\prime})$~\cite{Th1}. Partial
classifications that are potentially off by a factor of $2$ have been
worked out for $G_{2}$~\cite{KTT} and $Sp(3)$~\cite{Cu}.
The $SU(4)$ case is noticeably absent. The $SU(5)$ case was easier
since elementary bounds on the number of homotopy types matched at
the prime $2$ but did not at the prime $3$, and it is typically easier to
work out $3$-primary problems in low dimension than $2$-primary
problems. In the $SU(4)$ case the elementary bounds do not match at $2$, and
the purpose of this paper is to resolve the difference, at least after looping.
\begin{theorem}
\label{types}
For $G=SU(4)$, there is a homotopy equivalence
$\Omega\ensuremath{\mathcal{G}_{k}}\simeq\Omega\ensuremath{\mathcal{G}_{k^{\prime}}}$ when localized at any prime $p$ or
rationally if and only if $(60,k)=(60,k^{\prime})$.
\end{theorem}
Two novel features arise in the methods used, as compared to the other
known classifications. One is the use of Miller's stable splittings of Stiefel
manifolds in order to gain some control over unstable splittings, and the
other is showing that a certain ambiguity which prevents a clear
classification statement for $\ensuremath{\mathcal{G}_{k}}$ vanishes after looping. It would be
interesting to know if these ideas give access to classifications for
$SU(n)$-gauge groups for $n\geq 6$.
One motivation for studying $SU(4)$-gauge groups is their connection
to physics, in particular, to $SU(n)$-extensions of the standard model.
For instance, the group $SU(4)$ is gauged in the Pati-Salam model~\cite{PS}
and the flavour symmetry it represents there plays a role in several other
grand unified theories~\cite{BH}. The progression of results from $SU(2)$
to $SU(5)$ and possibly beyond would be of interest to physicists studying
the $SU(n)$-gauge groups in t'Hooft's large $n$ expansion~\cite{tH}.
\section{Determining homotopy types of gauge groups}
\label{sec:prelim}
We begin by describing a context in which homotopy theory can be applied
to study gauge groups. This works for any simply-connected, simple
compact Lie group $G$ and so is stated that way. Let $BG$ and $B\ensuremath{\mathcal{G}_{k}}$
be the classifying spaces of $G$ and $\ensuremath{\mathcal{G}_{k}}$ respectively.
Let $\ensuremath{\mbox{Map}}(S^{4},BG)$ and $\mapstar(S^{4},BG)$ respectively
be the spaces of freely continuous and pointed continuous maps
between~$S^{4}$ and $BG$. The components of each space are
in one-to-one correspondence with the integers, where the
integer is determined by the degree of a map
\(\namedright{S^{4}}{}{BG}\).
By~\cite{AB,G}, there is a homotopy equivalence
$B\ensuremath{\mathcal{G}_{k}}\simeq\ensuremath{\mbox{Map}}_{k}(S^{4},BG)$ between $B\ensuremath{\mathcal{G}_{k}}$ and the component
of $\mbox{Map}(S^{4},BG)$ consisting of maps of degree~$k$.
Evaluating a map at the basepoint of $S^{4}$, we obtain a map
\(ev\colon\namedright{B\ensuremath{\mathcal{G}_{k}}}{}{BG}\)
whose fibre is homotopy equivalent to $\mapstar_{k}(S^{4},BG)$.
It is well known that each component of $\mapstar(S^{4},BG)$ is
homotopy equivalent to $\Omega^{3}_{0} G$, the component of
$\Omega^{3} G$ containing the basepoint. Putting all this together,
for each $k\in\mathbb{Z}$, there is a homotopy fibration sequence
\begin{equation}
\label{evfib}
\namedddright{G}{\partial_{k}}{\Omega^{3}_{0} G}{}{B\ensuremath{\mathcal{G}_{k}}}{ev}{BG}
\end{equation}
where $\partial_{k}$ is the fibration connecting map.
The order of $\partial_{k}$ plays a crucial role. By~\cite{L}, the triple adjoint
\(\namedright{S^{3}\wedge G}{}{G}\)
of $\partial_{k}$ is homotopic to the Samelson product
$\langle k\cdot i,1\rangle$, where $i$ is the inclusion of
$S^{3}$ into $G$ and $1$ is the identity map on~$G$. This implies
two things. First, the order of $\partial_{k}$ is finite. For, rationally,
$G$ is homotopy equivalent to a product of Eilenberg-MacLane spaces
and the homotopy equivalence can be chosen to be one of $H$-maps.
Since Eilenberg-MacLane spaces are homotopy commutative, any Samelson
product into such a space is null homotopic. Thus, rationally, the
adjoint of $\partial_{k}$ is null homotopic, implying that
the same is true for $\partial_{k}$ and therefore the order of
$\partial_{k}$ is finite. Second, the linearity of the Samelson product
implies that $\langle k\cdot i,1\rangle\simeq k\circ\langle i,1\rangle$,
so taking adjoints we obtain $\partial_{k}\simeq k\circ\partial_{1}$.
Thus the order of~$\partial_{k}$ is determined by the order of $\partial_{1}$.
When $G=SU(n)$, Hamanaka and Kono~\cite{HK} gave the following lower bound
on the order of $\partial_{1}$ and the number of homotopy types of \ensuremath{\mathcal{G}_{k}}.
\begin{lemma}
\label{HKlemma}
Let $G=SU(n)$. Then the following hold:
\begin{letterlist}
\item the order of $\partial_{1}$ is divisible by $n(n^{2}-1)$;
\item if $\ensuremath{\mathcal{G}_{k}}\simeq\ensuremath{\mathcal{G}_{k^{\prime}}}$ then
$(n(n^{2}-1),k)=(n(n^{2}-1),k^{\prime})$.
\end{letterlist}
\end{lemma}
\vspace{-0.8cm}~$\hfill\Box$\medskip
The calculation in~\cite{HK} that established
Lemma~\ref{HKlemma} was that a composite
\(\nameddright{\Sigma^{2n-5}\mathbb{C}P^{2}}{}{SU(n)}{\partial_{1}}{\Omega_{0}^{3} SU(n)}\)
has order~$n(n^{2}-1)$. The adjoint of this map has the same order, so
we obtain the following alternative formulation.
\begin{lemma}
\label{loopHKlemma}
Let $G=SU(n)$. Then the following hold:
\begin{letterlist}
\item the order of $\Omega\partial_{1}$ is divisible by $n(n^{2}-1)$;
\item if $\Omega\ensuremath{\mathcal{G}_{k}}\simeq\Omega\ensuremath{\mathcal{G}_{k^{\prime}}}$ then
$(n(n^{2}-1),k)=(n(n^{2}-1),k^{\prime})$.
\end{letterlist}
\end{lemma}
\vspace{-0.8cm}~$\hfill\Box$\medskip
In particular, if $G=SU(4)$ then $60$ divides the order of $\Omega\partial_{1}$
and a homotopy equivalence $\Omega\ensuremath{\mathcal{G}_{k}}\simeq\Omega\ensuremath{\mathcal{G}_{k^{\prime}}}$ implies that
$(60,k)=(60,k^{\prime})$. In Section~\ref{sec:proof} we will find an upper bound
on the order of $\Omega\partial_{1}$ that matches the lower bound.
\begin{theorem}
\label{looppartialorder}
The map
\(\namedright{\Omega SU(4)}{\Omega\partial_{1}}{\Omega^{4}_{0} SU(4)}\)
has order~$60$.
\end{theorem}
Granting Theorem~\ref{looppartialorder} for now, we can prove
Theorem~\ref{types} by using the following general result from~\cite{Th1}.
If $Y$ is an $H$-space, let
\(k\colon\namedright{Y}{}{Y}\)
be the $k^{th}$-power map.
\begin{lemma}
\label{ptypecount}
Let $X$ be a space and $Y$ be an $H$-space with a homotopy inverse.
Suppose there is a map
\(\namedright{X}{f}{Y}\)
of order $m$, where $m$ is finite. Let $F_{k}$ be the homotopy fibre
of $k\circ f$. If $(m,k)=(m,k^{\prime})$ then $F_{k}$ and $F_{k^{\prime}}$
are homotopy equivalent when localized rationally or at any prime.~$\hfill\Box$
\end{lemma}
\noindent
\begin{proof}[Proof of Theorem~\ref{types}]
By Theorem~\ref{looppartialorder}, the map
\(\namedright{\Omega SU(4)}{\Omega\partial_{1}}{\Omega^{4}_{0} SU(4)}\)
has order~$60$. So Lemma~\ref{ptypecount} implies that if
$(60,k)=(60,k^{\prime})$, then $\Omega\ensuremath{\mathcal{G}_{k}}\simeq\Omega\ensuremath{\mathcal{G}_{k^{\prime}}}$ when
localized at any prime $p$ or rationally. On the other hand,
by Lemma~\ref{loopHKlemma}, if $\Omega\ensuremath{\mathcal{G}_{k}}\simeq\Omega\ensuremath{\mathcal{G}_{k^{\prime}}}$ then
$(60,k)=(60,k^{\prime})$. Thus there is a homotopy equivalence
$\Omega\ensuremath{\mathcal{G}_{k}}\simeq\Omega\ensuremath{\mathcal{G}_{k^{\prime}}}$ at each prime $p$ and rationally if and
only if $(60,k)=(60,k^{\prime})$.
\end{proof}
It remains to prove Theorem~\ref{looppartialorder}. In fact, the odd primary
components of the order of $\partial_{1}$ (and hence $\Omega\partial_{1}$
by Lemma~\ref{loopHKlemma}) are obtained as special cases of a more general
result in~\cite{Th3}.
\begin{lemma}
\label{oddbounds}
Localized at $p=3$, $\partial_{1}$ has order $3$; localized at $p=5$,
$\partial_{1}$ has order $5$; and localized at~$p>5$, $\partial_{1}$ has
order~$1$.~$\hfill\Box$
\end{lemma}
Thus to prove Theorem~\ref{looppartialorder} we are reduced to proving
the following.
\begin{theorem}
\label{looppartialorder2}
Localized at $2$ the map
\(\namedright{\Omega SU(4)}{\Omega\partial_{1}}{\Omega^{4}_{0} SU(4)}\)
has order~$4$.
\end{theorem}
\section{An initial upper bound on the $2$-primary order of $\partial_{1}$}
\label{sec:initialbound}
For the remainder of the paper all spaces and maps will be localized at $2$
and homology will be taken with mod-$2$ coefficients. Note that some statements
that follow are valid wilthout localizing, such as Theorems~\ref{Miller}
and~\ref{Millernat}, but rather than dancing back and forth between local
and non-local statements we simply localize at $2$ throughout.
In~\cite{B}, or by different means in~\cite{KKT},
it was shown that there is a homotopy commutative square
\[\diagram
SU(n)\rto^-{\partial_{1}}\dto^{\pi} & \Omega^{3}_{0} SU(n)\ddouble \\
SU(n)/SU(n-2)\rto^-{f} & \Omega^{3}_{0} SU(n)
\enddiagram\]
for some map $f$, where $\pi$ is the standard quotient map.
In our case it is well known that there is a homotopy equivalence
$SU(4)/SU(2)\simeq S^{5}\times S^{7}$. Thus there is a homotopy
commutative square
\begin{equation}
\label{su4su2}
\diagram
SU(4)\rto^-{\partial_{1}}\dto^{\pi} & \Omega^{3}_{0} SU(4)\ddouble \\
S^{5}\times S^{7}\rto^-{f} & \Omega^{3}_{0} SU(4).
\enddiagram
\end{equation}
Taking the triple adjoint of $f$, we obtain a map
\[f'\colon\nameddright{S^{8}\vee S^{10}\vee S^{15}}{\simeq}
{\Sigma^{3}(S^{5}\times S^{7})}{}{SU(4)}.\]
Mimura and Toda~\cite{MT} calculated the homotopy groups of $SU(4)$
through a range. The $2$-primary components of $\pi_{8}(SU(4))$,
$\pi_{10}(SU(4))$ and $\pi_{15}(SU(4))$ are
$\mathbb{Z}/8\mathbb{Z}$, $\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$
and $\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, respectively.
Consequently, the order of $f'$ is bounded above by $8$. The order of~$f$
is therefore also bounded above by $8$. The homotopy commutativity
of~(\ref{su4su2}) then implies the following.
\begin{lemma}
\label{initialbound}
The order the map
\(\namedright{SU(4)}{\partial_{1}}{\Omega^{3}_{0} SU(4)}\)
is bounded above by $8$.~$\hfill\Box$
\end{lemma}
Ideally it should be possible to reduce this upper bound by a factor of two.
The remainder of the paper aims to show that this can be done after looping.
\section{A cofibration}
\label{sec:Cprops}
The homotopy groups of spheres will play an important role. In all
cases except one we follow Toda's notation~\cite{To}. Specifically,
(i) for $n\geq 3$, $\eta_{n}=\Sigma^{n-3}\eta_{3}$ represents the
generator of $\pi_{n+1}(S^{n})\cong\mathbb{Z}/2\mathbb{Z}$;
(ii) for $n\geq 5$, $\nu_{n}=\Sigma^{n-3}\nu_{5}$ represents the
generator of $\pi_{n+3}(S^{n})\cong\mathbb{Z}/8\mathbb{Z}$;
and (iii) differing from Toda's notation, for $n\geq 3$, $\nu'_{n}=\Sigma^{n-3}\nu'_{3}$
represents the $n-3$ fold suspension of the generator~$\nu'_{3}$ of
$\pi_{6}(S^{3})\cong\mathbb{Z}/4\mathbb{Z}$. Note that for $n\geq 5$,
$\nu'_{n}=2\nu_{n}$.
\medskip
By~(\ref{su4su2}), the map
\(\namedright{SU(4)}{\partial_{1}}{\Omega^{3}_{0} SU(4)}\)
factors as a composite
\(\nameddright{SU(4)}{\pi}{S^{5}\times S^{7}}{f}{\Omega^{3}_{0} SU(4)}\).
In this section we determine properties of the map $\pi$ and its homotopy cofibre.
To prepare, first recall some properties of $SU(4)$. There is an
algebra isomorphism $\cohlgy{SU(4)}\cong\Lambda(x,y,z)$,
where the degrees of $x,y,z$ are $3,5,7$ respectively. A $\mathbb{Z}/2\mathbb{Z}$-module
basis for $\rcohlgy{SU(4)}$ is therefore $\{x,y,z,xy,xz,yz,xyz\}$
in degrees $\{3,5,7,8,10,12,15\}$ respectively, so $SU(4)$ may be given a
$CW$-structure with one cell in each of those dimensions. There is a canonical map
\(\namedright{\Sigma\ensuremath{\mathbb{C}P^{3}}}{}{SU(4)}\)
which induces a projection onto the generating set in cohomology. Notice
that $\Sigma\ensuremath{\mathbb{C}P^{3}}$ is homotopy equivalent to the $7$-skeleton of $SU(4)$,
and there is a homotopy cofibration
\begin{equation}
\label{CP3cofib}
\llnameddright{S^{4}\vee S^{6}}{\eta_{3}\vee\nu'_{3}}{S^{3}}{}{\Sigma\ensuremath{\mathbb{C}P^{3}}}.
\end{equation}
Miller~\cite{M} gave a stable decomposition of Stiefel manifolds which
includes the following as a special case.
\begin{theorem}
\label{Miller}
There is a stable homotopy equivalence
\[SU(4)\simeq_{S}\Sigma\ensuremath{\mathbb{C}P^{3}}\vee M\vee S^{15}\]
where $M$ is given by the homotopy cofibration
\[\llnameddright{S^{11}}{\nu'_{8}+\eta_{10}}{S^{8}\vee S^{10}}{}{M}.\]
\end{theorem}
\vspace{-1cm}~$\hfill\Box$\medskip
Crabb~\cite{C} and Kitchloo~\cite{Kitch} refined the stable decomposition
of Stiefel manifolds and proved that it was natural for maps
\(\namedright{SU(n)}{}{SU(n)/SU(m)}\).
In our case, this gives the following.
\begin{theorem}
\label{Millernat}
Stably, there is a homotopy commutative diagram
\[\diagram
SU(4)\rto^-{\simeq_{S}}\dto^{\pi}
& \Sigma\ensuremath{\mathbb{C}P^{3}}\vee M\vee S^{15}\dto^{\overline{\pi}} \\
S^{5}\times S^{7}\rto^-{\simeq_{S}} & S^{5}\vee S^{7}\vee S^{12}.
\enddiagram\]
where $\overline{\pi}$ is the wedge sum of: (i) the map
\(\namedright{\Sigma\ensuremath{\mathbb{C}P^{3}}}{}{S^{5}\vee S^{7}}\)
that collapses the bottom cell, (ii) the pinch map
\(\lnamedright{M}{}{S^{12}}\)
to the top cell, and (iii) the trivial map
\(\namedright{S^{15}}{}{\ast}\).~$\hfill\Box$
\end{theorem}
Now define the space $C$ and maps $j$ and $\delta$ by the homotopy cofibration
\begin{equation}
\label{Ccofib}
\namedddright{SU(4)}{\pi}{S^{5}\times S^{7}}{j}{C}{\delta}{\Sigma SU(4)}.
\end{equation}
Since $\pi^{\ast}$ is an inclusion onto the subalgebra $\Lambda(y,z)$
of $\Lambda(x,y,z)\cong\cohlgy{SU(4)}$, the long exact sequence
in cohomology induced by the cofibration sequence~(\ref{Ccofib}) implies that
a $\mathbb{Z}/2\mathbb{Z}$-module basis for $\rcohlgy{C}$
is given by $\{\sigma x,\sigma xy, \sigma xz, \sigma xyz\}$
in degrees $\{4,9,11,16\}$ respectively, where the elements of
$\rcohlgy{C}$ have been identified with the image of $\delta^{\ast}$.
So as a $CW$-complex, $C$ has one cell in each of the dimensions
$\{4,9,11,16\}$. The stable homotopy type of $C$ and the stable class
of the map $j$ follow immediately from Theorem~\ref{Millernat}.
\begin{proposition}
\label{Cstable}
Stably, there is a homotopy commutative diagram
\[\diagram
S^{5}\times S^{7}\rto^-{\simeq_{S}}\dto^{j}
& S^{5}\vee S^{7}\vee S^{12}\dto^{\overline{j}} \\
C\rto^-(0.6){\simeq_{S}} & S^{4}\vee S^{9}\vee S^{11}\vee S^{16}
\enddiagram\]
where $\overline{j}$ is the wedge sum of: (i)
\(\llnamedright{S^{5}\vee S^{7}}{\eta_{4}+\nu'_{4}}{S^{4}}\),
and (ii) \(\llnamedright{S^{12}}{\nu'_{9}+\eta_{11}}{S^{9}\vee S^{11}}\).
\end{proposition}
\vspace{-1cm}~$\hfill\Box$\medskip
The stable decomposition of $C$ will be useful but we will ultimately need
to work with unstable information in the form of the homotopy type
of $\Sigma^{3} C$ and the homotopy class of $\Sigma^{3} j$. We start
with the homotopy type of $\Sigma^{3} C$. In general, for a $CW$-complex $X$
and positive integer~$m$, let~$X_{m}$ be the $m$-skeleton of $X$. In our case,
the $CW$-structure for $C$ implies that there are homotopy cofibrations
\begin{align}
\label{Ccofib1}
& \nameddright{S^{8}}{g_{1}}{S^{4}}{}{C_{9}} \\
\label{Ccofib2}
& \nameddright{S^{10}}{g_{2}}{C_{9}}{}{C_{11}} \\
\label{Ccofib3}
& \nameddright{S^{15}}{g_{3}}{C_{11}}{}{C}
\end{align}
\begin{lemma}
\label{C9decomp}
There is a homotopy equivalence $\Sigma^{2} (C_{9})\simeq S^{6}\vee S^{11}$.
\end{lemma}
\begin{proof}
By~\cite{To}, $\pi_{10}(S^{6})=0$, so the map $\Sigma^{2} g_{1}$
in~(\ref{Ccofib1}) is null homotopic. The asserted homotopy equivalence
for $\Sigma^{2} (C_{9})$ follows immediately.
\end{proof}
\begin{lemma}
\label{C11decomp}
There is a homotopy equivalence
$\Sigma^{2} (C_{11})\simeq S^{6}\vee S^{11}\vee S^{13}$.
\end{lemma}
\begin{proof}
Substituting the homotopy equivalence in Lemma~\ref{C9decomp} into the
double suspension of~(\ref{Ccofib2}) gives a homotopy cofibration
\(\nameddright{S^{12}}{\Sigma^{2} g_{2}}{S^{6}\vee S^{11}}{}{\Sigma^{2} (C_{11})}\).
By the Hilton-Milnor Theorem, $\Sigma^{2} g_{2}\simeq a+b$ where
$a$ and $b$ are obtained by composing $\Sigma^{2} g_{2}$ with the
pinch maps to $S^{6}$ and $S^{11}$ respectively. We claim that each of $a$
and $b$ is null homotopic, implying that $\Sigma^{2} g_{2}$ is null
homotopic, from which the asserted homotopy equivalence for $\Sigma^{2} (C_{11})$
follows immediately.
By Proposition~\ref{Cstable}, $C$ is stably homotopy equivalent to a
wedge of spheres, and therefore $C_{11}$ is too. Thus $g_{2}$ is stably
trivial, implying for dimensional reasons that $a$ and $b$ are as well. On
the other hand, $a$ and $b$ are represented by classes in
$\pi_{12}(S^{6})\cong\mathbb{Z}/2\mathbb{Z}$
and $\pi_{12}(S^{11})\cong\mathbb{Z}/2\mathbb{Z}$ respectively. By~\cite{To},
these groups are generated by $\nu_{6}^{2}$ and $\eta_{11}$, both of which
are stable. Thus the only way that~$a$ and $b$ can be stably trivial is if
both are already trivial. Hence $\Sigma^{2} g_{2}$ is null homotopic.
\end{proof}
\begin{lemma}
\label{Cdecomp}
There is a homotopy equivalence
$\Sigma^{3} C\simeq E\vee S^{12}\vee S^{14}$
where $E$ is given by a homotopy cofibration
\(\llnameddright{S^{18}}{s\cdot\bar{\nu}_{7}\nu_{15}}{S^{7}}{}{E}\)
for some $s\in\mathbb{Z}/2\mathbb{Z}$.
\end{lemma}
\begin{proof}
Substituting the homotopy equivalence in Lemma~\ref{C11decomp} into the
double suspension of~(\ref{Ccofib3}) gives a homotopy cofibration
\(\nameddright{S^{17}}{\Sigma^{2} g_{3}}{S^{6}\vee S^{11}\vee S^{13}}{}{\Sigma^{2} C}\).
By the Hilton-Milnor Theorem, $\Sigma^{2} g_{3}\simeq a+b+c+d$ where
$a$, $b$ and $c$ are obtained by composing $\Sigma^{2} g_{3}$ with the
pinch maps to $S^{6}$, $S^{11}$ and~$S^{13}$ respectively, and $d$ is a composite
\(\nameddright{S^{17}}{}{S^{16}}{w}{S^{6}\vee S^{11}\vee S^{13}}\).
Here, $w$ is the Whitehead product of the identity maps on $S^{6}$ and $S^{11}$.
As $\Sigma w$ is null homotopic, we instead consider
\[\nameddright{S^{18}}{\Sigma^{3} g_{3}}{S^{7}\vee S^{12}\vee S^{14}}
{}{\Sigma^{3} C}\]
where $\Sigma^{3} g_{3}\simeq\Sigma a+\Sigma b+\Sigma c$.
By Proposition~\ref{Cstable}, $C$ is stably homotopy equivalent to a wedge
of spheres, so $\Sigma^{3} g_{3}\simeq\Sigma a+\Sigma b+\Sigma c$ is
stably trivial. Thus, for dimensional reasons, each of $\Sigma a$, $\Sigma b$
and $\Sigma c$ is stably trivial. Observe that both $\Sigma b$ and $\Sigma c$
are in the stable range, impling that they are null homotopic. On the other hand,
$\Sigma a$ represents a class in $\pi_{18}(S^{7})$. By~\cite{To},
$\pi_{18}(S^{7})\cong\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$
where the order $8$ generator is the stable class $\zeta_{7}$ and the
order~$2$ generator is the unstable class $\bar{\nu}_{7}\nu_{15}$. Note
too that the stable order of $\zeta_{7}$ is $8$, so the only nontrivial unstable
class in $\pi_{18}(S^{7})$ is $\bar{\nu}_{7}\nu_{15}$. As $\Sigma a$ is stably
trivial, we obtain $\Sigma a=s\cdot\bar{\nu}_{7}\nu_{15}$ for some
$s\in\mathbb{Z}/2\mathbb{Z}$. Hence $\Sigma^{2} g_{3}$ factors as the composite
\(\llnamedright{S^{18}}{s\cdot\bar{\nu}_{7}\nu_{15}}{S^{7}}\hookrightarrow
S^{7}\vee S^{12}\vee S^{14}\),
from which the asserted homotopy decomposition of $\Sigma^{3} C$ follows.
\end{proof}
Next, we identify $\Sigma^{3} j$. Let
\[\iota\colon\namedright{S^{7}}{}{E}\]
be the inclusion of the bottom cell.
\begin{lemma}
\label{jclass}
There is a homotopy commutative diagram
\[\diagram
S^{8}\vee S^{10}\vee S^{15}\rto^-{\simeq}\dto^{a+b+c}
& \Sigma^{3}(S^{5}\times S^{7})\dto^{\Sigma^{3} j} \\
E\vee S^{12}\vee S^{14}\rto^-{\simeq} & \Sigma^{3} C
\enddiagram\]
where $a$, $b$ and $c$ respectively are the composites
\begin{align*}
& a\colon\nameddright{S^{8}}{\eta_{7}}{S^{7}}{\iota}{E}\hookrightarrow
E\vee S^{12}\vee S^{14} \\
& b\colon\nameddright{S^{10}}{\nu'_{7}}{S^{7}}{\iota}{E}\hookrightarrow
E\vee S^{12}\vee S^{14} \\
& c\colon\lllnameddright{S^{15}}{\psi+\nu'_{12}+\eta_{14}}{S^{7}\vee S^{12}\vee S^{14}}
{\iota\vee 1\vee 1}{E\vee S^{12}\vee S^{14}}
\end{align*}
and $\psi=t\cdot\sigma'\eta_{14}$ for some $t\in\mathbb{Z}/2\mathbb{Z}$.
\end{lemma}
\begin{proof}
By Proposition~\ref{Cstable}, the diagram in the statement of the lemma
stably homotopy commutes if~$c$ is replaced by the composite
\(c'\colon\lllnameddright{S^{15}}{\ast+\nu'_{12}+\eta_{14}}{S^{7}\vee S^{12}\vee S^{14}}
{\iota\vee 1\vee 1}{E\vee S^{12}\vee S^{14}}\).
Since $a$ and $b$ are in the stable range, the diagram in the statement of the
lemma therefore does homotopy commute when restricted to $S^{8}\vee S^{10}$.
However, $c'$ is not in the stable range. It fails to be so only by a map
\(\psi''\colon\namedright{S^{15}}{}{S^{7}}\).
Thus if $c''$ is the composite
\(c''\colon\lllnameddright{S^{15}}{\psi''+\nu'_{12}+\eta_{14}}{S^{7}\vee S^{12}\vee S^{14}}
{\iota\vee 1\vee 1}{E\vee S^{12}\vee S^{14}}\)
then the diagram in the statement of the lemma homotopy commutes
with $c$ replaced by $c''$.
More can be said. By~\cite{To} (stated later also in~(\ref{SU4groups})),
$\pi_{15}(S^{7})\cong\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\oplus
\mathbb{Z}/2\mathbb{Z}$
with generators $\sigma'\nu_{14}$, $\bar{\nu}_{7}$ and $\epsilon_{7}$.
Thus $\psi''=t\cdot\sigma'\nu_{14}+u\cdot\bar{\nu}_{7} + v\cdot\epsilon_{7}$
for some $t,u,v\in\mathbb{Z}/2\mathbb{Z}$. The generators $\bar{\nu}_{7}$
and $\epsilon_{7}$ are stable while $\sigma'\nu_{14}$ is unstable. So as
$c''$ stabilizes to $c$, we must have $\psi''$ stabilizing to the trivial map.
Thus $u$ and $v$ must be zero. Hence $\psi''=t\cdot\sigma'\nu_{14}$.
Now $c''$ is exactly the map $c$ described in the statement of the lemma.
\end{proof}
\section{Preliminary information on the homotopy groups of $SU(4)$}
\label{sec:htpygroups}
This section records some information on the homotopy groups
of $SU(4)$ which will be needed subsequently. There is a homotopy fibration
\[\nameddright{S^{3}}{i}{SU(4)}{q}{S^{5}\times S^{7}}.\]
This induces a long exact sequence of homotopy groups
\[\cdots\longrightarrow\namedddright{\pi_{n+1}(S^{3}\times S^{5})}{}
{\pi_{n}(S^{3})}{i_{\ast}}{\pi_{n}(SU(4))}{q_{\ast}}{\pi_{n}(S^{5}\times S^{7})}
\longrightarrow\cdots\]
Following~\cite{MT}, the notation $[\alpha\oplus\beta]\in\pi_{n}(SU(4))$ means that
$[\alpha\oplus\beta]$ is a generator of $\pi_{n}(SU(4))$ with the property
that $q_{\ast}([\alpha\oplus\beta])=\alpha\oplus\beta$ for
$\alpha\in\pi_{n}(S^{5})$ and $\beta\in\pi_{n}(S^{7})$. The homotopy groups
of $SU(4)$ in low dimensions were determined by Mimura and Toda~\cite{MT}.
The information presented will be split into two parts, the first corresponding
to subsequent calculations involving $\pi_{8}(SU(4))$ and $\pi_{10}(SU(4))$,
and the second corresponding to calculations involving $\pi_{15}(SU(4))$.
First, for $r\geq 1$, let
\(\underline{2}^{r}\colon\namedright{S^{7}}{}{S^{7}}\)
be the map of degree $2^{r}$. In general, the degree two map on $S^{2n+1}$
need not induce multiplication by $2$ in homotopy groups. However,
as $S^{7}$ is an $H$-space, the degree $2$ map on $S^{7}$ is homotopic
to the $2^{nd}$-power map, implying that it does in fact induce multiplication
by $2$ in homotopy groups. We record this for later use.
\begin{lemma}
\label{S72}
The map
\(\namedright{S^{7}}{\underline{2}}{S^{7}}\)
induces multiplication by $2$ in homotopy groups.~$\hfill\Box$
\end{lemma}
\subsection{Dimensions $8$ and $10$}
The relevant table of homotopy groups from~\cite{MT} is:
\begin{equation}
\label{SU4groups1}
\begin{tabular}{|c|c|c|c|}\hline
& $\pi_{7}(SU(4))$ & $\pi_{8}(SU(4))$ & $\pi_{10}(SU(4))$ \\ \hline
$2$-component & $\mathbb{Z}$ & $\mathbb{Z}/8\mathbb{Z}$
& $\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ \\ \hline
generators & $[\eta_{5}^{2}\oplus\underline{2}]$
& $[\nu_{5}\oplus\eta_{7}]$
& $[\nu_{7}]$, $[\nu_{5}\eta_{8}^{2}]$ \\ \hline
\end{tabular}
\end{equation}
In addition, Mimura and Toda~\cite[Lemma 6.2(i)]{MT} proved that
\(\namedright{\pi_{n+1}(S^{5}\times S^{7})}{}{\pi_{n}(S^{3})}\)
is an epimorphism for $n\in\{8,10\}$, implying the following.
\begin{lemma}
\label{pi810inj}
The map
\(\namedright{\pi_{n}(SU(4))}{q_{\ast}}{\pi_{n}(S^{5}\times S^{7})}\)
is an injection for $n\in\{8,10\}$.~$\hfill\Box$
\end{lemma}
Toda~\cite{To} proved the following relations in the homotopy groups of spheres.
\begin{lemma}
\label{Todarelns1}
The following hold:
\begin{letterlist}
\item $2\nu'_{3}\simeq\eta_{3}^{3}$;
\item $4\nu_{5}\simeq\eta_{5}^{3}$;
\item $\eta_{3}\nu'_{3}\simeq\ast$.
\end{letterlist}
\end{lemma}
\vspace{-1cm}~$\hfill\Box$\medskip
For convenience, let
\[d\colon\namedright{S^{7}}{}{SU(4)}\]
represent the generator $[\eta_{5}^{2}\oplus\underline{2}]$ of $\pi_{7}(SU(4))$.
\begin{lemma}
\label{S810dgrms}
There are homotopy commutative diagrams
\[\diagram
S^{8}\rto^-{[\nu_{5}\oplus\eta_{7}]}\dto^{\eta_{7}} & SU(4)\dto^{4}
& & S^{10}\rto^-{[\nu_{7}]}\dto^{\nu'_{7}} & SU^{4}\dto^{4} \\
S^{7}\rto^-{d} & SU(4) & & S^{7}\rto^-{d} & SU(4).
\enddiagram\]
\end{lemma}
\begin{proof}
By Lemma~\ref{pi810inj},
\(\namedright{\pi_{n}(SU(4))}{q_{\ast}}{\pi_{n}(S^{5}\times S^{7})}\)
is an injection for $n\in\{8,10\}$. So in both cases it suffices to show that the
asserted homotopies hold after composition with
\(\namedright{SU(4)}{q}{S^{5}\times S^{7}}\).
Since the composite
\(\nameddright{S^{7}}{d}{SU(4)}{q}{S^{5}\times S^{7}}\)
is $\eta_{5}^{2}\times\underline{2}$, the two assertions will follow if we prove:
(i) $(\eta_{5}^{2}\times\underline{2})\circ\eta_{7}\simeq q\circ 4\circ[\nu_{5}\oplus\eta_{7}]$;
(ii) $(\eta_{5}^{2}\times\underline{2})\circ\nu'_{7}\simeq q\circ 4\circ[\nu_{7}]$.
By Lemma~\ref{S72}, $\underline{2}\circ\eta_{7}\simeq 2\eta_{7}$ and
$\underline{2}\circ\nu'_{7}\simeq 2\nu'_{7}$. Since $\eta_{7}$ has order~$2$
we obtain $\underline{2}\circ\eta_{7}\simeq\ast$. By Lemma~\ref{Todarelns1}~(a),
$2\nu'_{7}\simeq\eta_{7}^{3}$. Thus (i) and (ii) reduce to proving:
(i$^{\prime}$) $\eta_{5}^{3}\simeq q\circ 4\circ[\nu_{5}\oplus\eta_{7}]$;
(ii$^{\prime}$) $\eta_{7}^{3}\simeq q\circ 4\circ[\nu_{7}]$.
Consider the diagram
\[\diagram
S^{8}\rto^-{[\nu_{5}\oplus\eta_{7}]}\dto^{\underline{4}} & SU(4)\dto^{4} \\
S^{8}\rto^-{[\nu_{5}\oplus\eta_{7}]}\drto_{\nu_{5}\times\eta_{7}}
& SU(4)\dto^{q} \\
& S^{5}\times S^{7}.
\enddiagram\]
The top square homotopy commutes since the multiplications in
$[S^{8},SU(4)]$ induced by the $H$-structure on $SU(4)$ and the
co-$H$-structure on $S^{8}$ coincide. The bottom square homotopy
commutes by definition of $[\nu_{5}\circ\eta_{7}]$. Since $\eta_{7}$
has order $2$ and, by Lemma~\ref{Todarelns1}, $4\nu_{5}\simeq\eta_{5}^{3}$,
we obtain $(\nu_{5}\times\eta_{7})\circ\underline{4}\simeq\eta_{5}^{3}$.
Therefore $q\circ 4\circ [\nu_{5}\circ\eta_{7}]\simeq\eta_{5}^{3}$, and
so (i$^{\prime}$) holds.
Next, consider the diagram
\[\diagram
S^{10}\rto^-{[\nu_{7}]}\dto^{\underline{4}} & SU(4)\dto^{4} \\
S^{10}\rto^-{[\nu_{7}]}\drto_{\ast\times\nu_{7}}
& SU(4)\dto^{q} \\
& S^{5}\times S^{7}.
\enddiagram\]
The two squares homotopy commute as in the previous case.
By Lemma~\ref{Todarelns}, $4\nu_{7}\simeq\eta_{7}^{3}$. Therefore
$q\circ 4\circ\beta\simeq\eta_{7}^{3}$, and so (ii$^{\prime}$) holds.
\end{proof}
\subsection{Dimension $15$}
The relevant table of homotopy groups from~\cite{MT} is:
\begin{equation}
\label{SU4groups}
\begin{tabular}{|c|c|c|c|}\hline
& $\pi_{12}(SU(4))$ & $\pi_{14}(SU(4))$ & $\pi_{15}(SU(4))$ \\ \hline
$2$-component & $\mathbb{Z}/4\mathbb{Z}$
& $\mathbb{Z}/16\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$
& $\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ \\ \hline
generators & $[\sigma^{'''}]$
& $[\eta_{5}\epsilon_{6}\oplus\sigma']$, $[\nu_{5}^{2}]\circ\nu_{11}$
& $[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$, $[\sigma'\eta_{14}]$ \\ \hline
\end{tabular}
\end{equation}
In addition, Mimura and Toda~\cite[Lemma 6.2(i)]{MT} proved that
\(\namedright{\pi_{16}(S^{5}\times S^{7})}{}{\pi_{15}(S^{3})}\)
is an epimorphism, implying the following.
\begin{lemma}
\label{pi15inj}
The map
\(\namedright{\pi_{15}(SU(4))}{q_{\ast}}{\pi_{15}(S^{5}\times S^{7})}\)
is an injection.~$\hfill\Box$
\end{lemma}
The next table gives some information on the $2$-primary components of
selected homotopy groups of spheres that were determined by Toda~\cite{To}:
\begin{equation}
\label{S15groups}
\begin{tabular}{|c|c|c|c|}\hline
& $\pi_{15}(S^{7})$ & $\pi_{15}(S^{12})$ & $\pi_{15}(S^{14})$ \\ \hline
$2$-component & $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}
\oplus\mathbb{Z}/2\mathbb{Z}$
& $\mathbb{Z}/8\mathbb{Z}$ & $\mathbb{Z}/2\mathbb{Z}$ \\ \hline
generators & $\sigma'\eta_{14}$, $\bar{\nu}_{7}$, $\epsilon_{7}$
& $\nu_{12}$ & $\eta_{14}$ \\ \hline
\end{tabular}
\end{equation}
In addition, Toda~\cite{To} proved the following relations (the proofs are scattered
through Toda's book but a summary list can be found in~\cite[Equations 1.1 and 2.1]{O}).
\begin{lemma}
\label{Todarelns}
The following hold:
\begin{letterlist}
\item $\eta_{5}\bar{\nu}_{6}=\nu_{5}^{3}$;
\item $\eta_{3}\nu_{4}=\nu'_{3}\eta_{6}$;
\item $\eta_{6}\sigma'=4\bar{\nu}_{6}$;
\item $\eta_{6}\nu_{7}=\nu_{6}\eta_{9}=0$;
\item $\sigma^{'''}\nu_{12}=4(\nu_{5}\sigma_{8})$.
\end{letterlist}
\end{lemma}
\vspace{-1cm}~$\hfill\Box$\medskip
Lemma~\ref{Todarelns} is used to obtain two more relations.
\begin{lemma}
\label{sphererelns}
The following hold:
\begin{letterlist}
\item $\eta_{5}^{2}\bar{\nu}_{7}=0$;
\item $\eta_{5}^{2}\sigma'=0$.
\end{letterlist}
\end{lemma}
\begin{proof}
In what follows, we freely use the fact that the relations in Lemma~\ref{Todarelns}
imply analogous relations for their suspensions; for example,
$\eta_{5}\bar{\nu}_{6}=\nu_{5}^{3}$ implies that $\eta_{6}\bar{\nu}_{7}=\nu_{6}^{3}$.
For part~(a), the relations in Lemma~\ref{Todarelns}~(a), (b) and (d) respectively imply
the following string of equalities:
$\eta_{5}^{2}\bar{\nu}_{7}=\eta_{5}\nu_{6}^{3}=\nu'_{5}\eta_{8}\nu_{9}^{2}=0$.
For part~(b), Lemma~\ref{Todarelns}~(c) and the fact that $\eta_{5}$ has order~$2$
imply that there are equalities $\eta_{5}^{2}\sigma'=\eta_{5}(4\bar{\nu}_{6})=0$.
\end{proof}
We now determine the homotopy classes of two maps into $SU(4)$.
\begin{lemma}
\label{SU4relns}
The following hold:
\begin{letterlist}
\item the composite
\(\nameddright{S^{15}}{\bar{\nu}_{7}}{S^{7}}{d}{SU(4)}\)
is null homotopic;
\item the composite
\(\lnameddright{S^{15}}{\sigma'\eta_{14}}{S^{7}}{d}{SU(4)}\)
is null homotopic.
\end{letterlist}
\end{lemma}
\begin{proof}
By Lemma~\ref{pi15inj},
\(\namedright{\pi_{15}(SU(4))}{q_{\ast}}{\pi_{15}(S^{5}\times S^{7})}\)
is an injection. So in both cases it suffices to show that the assertions hold
after composition with
\(\namedright{SU(4)}{q}{S^{5}\times S^{7}}\).
Since the composite
\(\nameddright{S^{7}}{d}{SU(4)}{q}{S^{5}\times S^{7}}\)
is $\eta_{5}^{2}\times\underline{2}$, the two assertions will follow if we prove:
(a$^{\prime}$) $(\eta_{5}^{2}\times\underline{2})\circ\bar{\nu}_{7}\simeq\ast$;
(b$^{\prime}$) $(\eta_{5}^{2}\times\underline{2})\circ\sigma'\eta_{14}\simeq\ast$.
\noindent
By Lemma~\ref{S72}, the degree two map on $S^{7}$ induces multiplication
by $2$ on homotopy groups, so as both $\bar{\nu}_{7}$
and $\sigma'\eta_{14}$ have order~$2$, it suffices to prove:
(a$^{\prime\prime}$) $\eta_{5}^{2}\bar{\nu}_{7}\simeq\ast$;
(b$^{\prime\prime}$) $\eta_{5}^{2}\sigma'\eta_{14}\simeq\ast$.
\noindent
Part~(a$^{\prime\prime}$) is the statement of Lemma~\ref{sphererelns}~(a) and
part~(b$^{\prime\prime}$) is immediate from Lemma~\ref{sphererelns}~(b).
\end{proof}
One consequence of Lemma~\ref{SU4relns} is the existence of an extension
involving the space $E$ appearing in the homotopy decomposition of $\Sigma^{3} C$
in Lemma~\ref{Cdecomp}.
\begin{lemma}
\label{Eext}
There is an extension
\[\diagram
S^{7}\rto^-{d}\dto^{\iota} & SU(4) \\
E\urto_-{e} &
\enddiagram\]
for some map $e$.
\end{lemma}
\begin{proof}
By Lemma~\ref{Cdecomp}, there is a homotopy cofibration
\(\llnameddright{S^{18}}{t\cdot\bar{\nu}_{7}\nu_{15}}{S^{7}}{}{E}\)
for some $t\in\mathbb{Z}/2\mathbb{Z}$. By Lemma~\ref{SU4relns}~(a),
$d\circ\bar{\nu}_{7}$ is null homotopic. Therefore $d\circ (t\cdot\bar{\nu}_{7}\nu_{15})$
is null homotopic, implying that the asserted extension exists.
\end{proof}
\section{The proof of Theorem~\ref{looppartialorder2}}
\label{sec:proof}
Recall from~(\ref{su4su2}) that
\(\namedright{SU(4)}{\partial_{1}}{\Omega^{3}_{0} SU(4)}\)
factors as the composite
\(\nameddright{SU(4)}{\pi}{S^{5}\times S^{7}}{f}{\Omega^{3}_{0} SU(4)}\).
Let
\[f'\colon\namedright{\Sigma^{3}(S^{5}\times S^{7})}{}{SU(4)}\]
be the triple adjoint of $f$. Let $f'_{1}$, $f'_{2}$ and $f'_{3}$ be the
restrictions of the composite
\[\nameddright{S^{8}\vee S^{10}\vee S^{15}}{\simeq}{\Sigma^{3}(S^{5}\vee S^{7})}
{f'}{SU(4)}\]
to $S^{8}$, $S^{10}$ and $S^{15}$ respectively. We wish to identify
$f'_{1}$, $f'_{2}$ and $f'_{3}$ more explicitly. Let
\(t_{1}\colon\namedright{S^{5}}{}{SU(4)}\)
and
\(t_{2}\colon\namedright{S^{7}}{}{SU(4)}\)
represent generators of $\pi_{5}(SU(4))\cong\mathbb{Z}$ and
$\pi_{7}(SU(4))\cong\mathbb{Z}$ respectively. By~\cite{MT} these generators
can be chosen so that $\pi\circ t_{1}$ is homotopic to $\underline{2}\oplus\ast$
and $\pi\circ t_{2}$ is homotopic to $\eta^{2}_{5}\oplus\underline{2}$.
So there are homotopy commutative diagrams
\begin{equation}
\label{2Bottdgrms}
\diagram
S^{5}\rto^-{t_{1}}\drto_{\underline{2}\oplus\ast}
& SU(4)\rto^-{\partial_{1}}\dto^{\pi} & \Omega^{3}_{0} SU(4)\ddouble
& & S^{7}\rto^-{t_{2}}\drto_{\eta^{2}_{5}\oplus\underline{2}}
& SU(4)\rto^-{\partial_{1}}\dto^{\pi} & \Omega^{3}_{0} SU(4)\ddouble \\
& S^{5}\times S^{7}\rto^-{f} & \Omega^{3}_{0} SU(4)
& & & S^{5}\times S^{7}\rto^-{f} & \Omega^{3}_{0} SU(4).
\enddiagram
\end{equation}
On the other hand, since the triple adjoint of $\partial_{1}$ is the Samelson
product $\langle i,1\rangle$, the triple adjoint of $\partial_{1}\circ t_{j}$
is $\langle t_{j},1\rangle$ for $j=1,2$. Bott~\cite{B} calculated that both of
these maps have order $4$. Thus the left diagram in~(\ref{2Bottdgrms})
implies that the restriction of $f$ to $S^{5}$ has order~$8$, and the right
diagram in~(\ref{2Bottdgrms}) implies that the restriction of $f$ to $S^{7}$
has order~$8$. Thus, taking triple adjoints, $f'_{1}$ and $f'_{2}$ both
have order~$8$.
The order of $f'_{3}$ is not as clear. By~(\ref{SU4groups}),
$\pi_{15}(SU(4))\cong\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$,
so $f'_{3}$ may have order~$8$. This ambiguity will be reflected in the alternative
possibilities worked out below.
Recall from Lemma~\ref{jclass} that there is a homotopy commutative diagram
\[\diagram
S^{8}\vee S^{10}\vee S^{15}\rto^-{\simeq}\dto^{a+b+c}
& \Sigma^{3}(S^{5}\times S^{7})\dto^{\Sigma^{3} j} \\
E\vee S^{12}\vee S^{14}\rto^-{\simeq} & \Sigma^{3} C
\enddiagram\]
where $a$, $b$ and $c$ respectively are the composites
\begin{align*}
& a\colon\nameddright{S^{8}}{\eta_{7}}{S^{7}}{\iota}{E}\hookrightarrow
E\vee S^{12}\vee S^{14} \\
& b\colon\nameddright{S^{10}}{\nu'_{7}}{S^{7}}{\iota}{E}\hookrightarrow
E\vee S^{12}\vee S^{14} \\
& c\colon\lllnameddright{S^{15}}{\psi+\nu'_{12}+\eta_{14}}{S^{7}\vee S^{12}\vee S^{14}}
{\iota\vee 1\vee 1}{E\vee S^{12}\vee S^{14}}
\end{align*}
and $\psi=t\cdot\sigma'\eta_{14}$ for some $t\in\mathbb{Z}/2\mathbb{Z}$.
Let $c'$ be the composite
\[c'\colon\lllnameddright{S^{15}}{\psi'+\nu'_{12}+\eta_{14}}{S^{7}\vee S^{12}\vee S^{14}}
{\iota\vee 1\vee 1}{E\vee S^{12}\vee S^{14}}\]
where $\psi'=t\cdot\sigma'\eta_{14}+\eta_{7}\sigma_{8}$. Let $\xi$
be the composite
\[\xi\colon\nameddright{E\vee S^{12}\vee S^{14}}{}{E}{e}{SU(4)}\]
where the left map is the pinch onto the first wedge summand and $e$ is
the map from Lemma~\ref{Eext}.
\begin{lemma}
\label{alternative}
There is a homotopy commutative diagram
\[\diagram
S^{8}\vee S^{10}\vee S^{15}\rrto^-{f'_{1}+f'_{2}+f'_{3}}\dto^{a+b+\gamma}
& & SU(4)\dto^{4} \\
E\vee S^{12}\vee S^{14}\rrto^-{\xi} & & SU(4)
\enddiagram\]
where $\gamma$ may be chosen to be $c$ if the order of $f'_{3}$ is at most $4$
and $\gamma$ may be chosen to be $c'$ if the order of $f'_{3}$ is $8$. Further,
in the latter case, the composite
\(\namedddright{S^{15}}{\eta_{7}\sigma_{8}}{S^{7}}{\iota}{E}{e}{SU(4)}\)
represents $4[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$.
\end{lemma}
\begin{proof}
First, consider the diagram
\begin{equation}
\label{dgrm1}
\diagram
S^{8}\vee S^{10}\rto^-{f'_{1}+f'_{2}}\dto^{\eta_{7}+\nu'_{7}}
& SU(4)\dto^{4} \\
S^{7}\rto^-{d}\dto^{\iota} & SU(4)\ddouble \\
E\rto^-{e} & SU(4).
\enddiagram
\end{equation}
Since $\pi_{8}(SU(4))\cong\mathbb{Z}/8\mathbb{Z}$ is generated by
$[\nu_{5}\oplus\eta_{7}]$ and $f'_{1}$ has order~$8$, we must have
$f'_{1}=u\cdot[\nu_{5}\oplus\eta_{7}]$ for some unit $u\in\mathbb{Z}/8\mathbb{Z}$.
Thus $4f'_{1}\simeq 4[\nu_{5}\oplus\eta_{7}]$, so the restriction of the upper square
in~(\ref{dgrm1}) to~$S^{8}$ homotopy commutes by Lemma~\ref{S810dgrms}. Similarly,
since $\pi_{10}(SU(4))\cong\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$
with $[\nu_{7}]$ being the generator of order~$8$, and $f'_{2}$ has order $8$, we
must have $4f'_{2}\simeq 4[\nu_{5}]$, so the restriction of the upper square
in~(\ref{dgrm1}) to $S^{10}$ homotopy commutes by Lemma~\ref{S810dgrms}.
The lower square in~(\ref{dgrm1}) homotopy commutes by Lemma~\ref{Eext}.
Now observe that the lower direction around~(\ref{dgrm1}) is the definition of $a+b$.
Thus~(\ref{dgrm1}) implies that the diagram in the statement of the lemma
homotopy commutes when restricted to $S^{8}\vee S^{10}$.
Second, consider the diagram
\begin{equation}
\label{dgrm2}
\diagram
S^{15}\dto^{(t\cdot\sigma'\eta_{14}+\theta)+\nu'_{12}+\eta_{14}}\rrto^-{f'_{3}}
& & SU(4)\ddto^{4} \\
S^{7}\vee S^{12}\vee S^{15}\dto^{\iota\vee 1\vee 1} & & \\
E\vee S^{12}\vee S^{14}\rrto^-{\xi} & & SU(4)
\enddiagram
\end{equation}
where two possibilities for $\theta$ will be considered. In the lower direction
around the diagram, by definition, $\xi$ is the composite
\(\nameddright{E\vee S^{12}\vee S^{14}}{}{E}{e}{SU(4)}\)
where the left map is the pinch onto the first wedge summand. By
Lemma~\ref{Eext}, $\iota\circ e=d$. Thus the lower direction around the
diagram is homotopic to the composite
\(\lllnameddright{S^{15}}{t\cdot\sigma'\eta_{14}+\theta}{S^{7}}{d}{SU(4)}\).
By Lemma~\ref{SU4relns}~(b), $d\circ t\cdot\sigma'\eta_{14}$ is null homotopic.
Thus the lower direction around the diagram is in fact homotopic to the composite
\(\nameddright{S^{15}}{\theta}{S^{7}}{d}{SU(4)}\).
If $f'_{3}$ has order at most $4$ then $4f'_{3}$ is null homotopic. Taking
$\theta$ to be the constant map shows that~(\ref{dgrm2}) homotopy
commutes. Observe also that with this choice of $\theta$ the left column
in~(\ref{dgrm2}) is the definition of $c$, so we obtain the diagram in the
statement of the lemma when restricted to $S^{15}$. Now combining~(\ref{dgrm1})
and~(\ref{dgrm2}) we obtain the diagram asserted by the lemma.
Suppose that $f'_{3}$ has order~$8$. Since
$\pi_{15}(SU(4))\cong\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$
with the order~$8$ generator being $[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$,
we obtain $4f'_{3}\simeq 4[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$. Take
$\theta=\eta_{7}\sigma_{8}$. We claim that
$d\circ\theta\simeq 4[\nu_{5}\circ\eta_{7}]\circ\sigma_{8}$. If so then~(\ref{dgrm2})
homotopy commutes with this choice of $\theta$ and, as the left column of~(\ref{dgrm2})
is the definition of $c'$, we obtain the diagram in the statement of the lemma
when restricted to $S^{15}$. Therefore combining~(\ref{dgrm1}) and~(\ref{dgrm2})
we obtain the diagram asserted by the lemma.
It remains to show that
$d\circ\eta_{7}\sigma_{8}\simeq 4[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$.
By Lemma~\ref{pi15inj} it suffices to compose with
\(\namedright{SU(4)}{q}{S^{5}\times S^{7}}\)
and check there. On the one hand,
$q\circ d\circ\eta_{7}\sigma_{8}\simeq
(\eta_{5}^{2}\times\underline{2})\circ\eta_{7}\sigma_{8}\simeq
\eta_{5}^{3}\sigma_{8}$,
where the left homotopy holds by definition of $d$ and the right homotopy is
due to the fact that $\eta_{7}$ has order~$2$ and, by Lemma~\ref{S72},
$\underline{2}$ induces multiplication by $2$ on homotopy groups. On
the other hand,
$q\circ 4[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}\simeq
4(\nu_{5}\times\eta_{7})\circ\sigma_{8}\simeq 4\nu_{5}\sigma_{8}\simeq
\eta_{5}^{3}\sigma_{8}$.
Here, from left to right, the first homotopy holds by definition of $[\nu_{5}\oplus\eta_{7}]$,
the second holds since $\eta_{7}$ has order~$2$, and the third holds by~\cite{To}.
Thus $d\circ\eta_{7}\simeq_{8}\simeq 4[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$,
as claimed.
\end{proof}
Now return to the map
\(\namedright{SU(4)}{\partial_{1}}{\Omega_{0}^{3} SU(4)}\).
\begin{proposition}
\label{order4options}
The following hold:
\begin{letterlist}
\item if $f'_{3}$ has order at most $4$ then $4\circ\partial_{1}$ is null homotopic;
\item if $f'_{3}$ has order $8$ then $4\circ\partial_{1}$ is homotopic to the composite
\(\namedddright{SU(4)}{\pi}{S^{5}\times S^{7}}{}{S^{12}}{4\chi}
{\Omega^{3}_{0} SU(4)}\),
where the middle map is the pinch map to the top cell and $\chi$
is the triple adjoint of the order~$8$ generator
$[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$ in $\pi_{15}(SU(4))$.
\end{letterlist}
\end{proposition}
\begin{proof}
If the order of $f'_{3}$ is at most~$4$, then in Lemma~\ref{alternative} we
may take $\gamma=c$. Doing so, observe that by using the inverse
equivalences in Lemma~\ref{jclass} we obtain a homotopy commutative diagram
\begin{equation}
\label{Cjdgrm}
\diagram
\Sigma^{3}(S^{5}\times S^{7})\rto^-{f'}\dto^{\Sigma^{3} j} & SU(4)\dto^{4} \\
\Sigma^{3} C\rto^-{\xi'} & SU(4)
\enddiagram
\end{equation}
where $\xi'$ is the composite
\(\nameddright{\Sigma^{3} C}{\simeq}{E\vee S^{12}\vee S^{14}}{\xi}{SU(4)}\).
Now consider the diagram
\[\diagram
SU(4)\rto^-{\partial_{1}}\dto^{\pi} & \Omega^{3}_{0} SU(4)\ddouble \\
S^{5}\times S^{7}\rto^-{f}\dto^{j} & \Omega^{3}_{0} SU(4)\dto^{4} \\
C\rto & \Omega^{3}_{0} SU(4)
\enddiagram\]
The top square homotopy commutes by~(\ref{su4su2}) while the bottom
square is the triple adjoint of~(\ref{Cjdgrm}). Since the left
column consists of two consecutive maps in a homotopy cofibration sequence
it is null homotopic. The homotopy commutativity of the diagram therefore
implies that~$4\circ\partial_{1}$ is null homotopic.
If the order of $f'_{3}$ is $8$, then in Lemma~\ref{alternative} we may
take $\gamma=c'$. Doing so, since $c'=c+\eta_{7}\sigma_{8}$, instead
of~(\ref{Cjdgrm}) we obtain a homotopy commutative diagram
\begin{equation}
\label{Cj2dgrm}
\diagram
\Sigma^{3}(S^{5}\times S^{7})\rto^-{f'}\dto^{\Sigma^{3} j+\ell} & SU(4)\dto^{4} \\
\Sigma^{3} C\rto^-{\xi'} & SU(4)
\enddiagram
\end{equation}
where $\ell$ is the composite
\(\nameddright{\Sigma^{3}(S^{5}\times S^{7})}{}{S^{15}}{\eta_{7}\sigma_{8}}
{S^{7}}\hookrightarrow\Sigma^{3} C\).
Now consider the diagram
\[\diagram
\Sigma^{3} SU(4)\rto^-{\partial'_{1}}\dto^{\Sigma^{3}\pi} & SU(4)\ddouble \\
\Sigma^{3}(S^{5}\times S^{7})\rto^-{f'}\dto^{\Sigma^{3} j+\ell} & SU(4)\dto^{4} \\
\Sigma^{3} C\rto^-{\xi'} & SU(4)
\enddiagram\]
where $\partial'_{1}$ is the triple adjoint of $\partial$. The top square homotopy
commutes by~(\ref{su4su2}) while the bottom square homotopy commutes
by~(\ref{Cj2dgrm}). Since $\Sigma^{3} j\circ\Sigma^{3}\pi$ are consecutive
maps in a homotopy cofibration, their composite is null homotopic. Thus
this diagram implies that $4\circ\partial'_{1}$ is homotopic to the composite
\(\namedddright{\Sigma^{3} SU(4)}{\Sigma^{3}\pi}{\Sigma^{3}(S^{5}\times S^{7})}
{}{S^{15}}{\eta_{7}\sigma_{8}}{S^{7}}\hookrightarrow\namedright{\Sigma^{3} C}
{\xi'}{SU(4)}\).
Notice that the pinch map to the top cell
\(\namedright{\Sigma^{3}(S^{5}\times S^{7})}{}{S^{15}}\)
is a triple suspension, while by Lemma~\ref{alternative} the composite
\(\namedright{S^{15}}{\eta_{7}\sigma_{8}}{S^{7}}\hookrightarrow\namedright{\Sigma^{3} C}
{\xi'}{SU(4)}\)
represents $4[\nu_{5}\oplus\eta_{7}]\circ\sigma_{8}$. Thus, taking triple adjoints,
$4\circ\partial_{1}$ is homotopic to the composite
\(\namedddright{SU(4)}{\pi}{S^{5}\times S^{7}}{}{S^{12}}{4\chi}{SU(4)}\),
as asserted.
\end{proof}
\begin{remark}
It can be checked that if $f'_{3}$ has order~$8$ then there is no different
choice of the map~$\xi$ which makes $\xi\circ(a+b+c)\simeq 4f'$ in
Lemma~\ref{alternative}. The argument is to check all possible cases; it is
not included as it is not needed. However, it does imply that
$4\circ\partial_{1}$ is nontrivial; for if it were trivial then
$4\circ\partial_{1}\simeq 4\circ f\circ\pi_{1}$
would have to factor through the cofibre $C$ of $\pi$, implying that there
has to be a choice of $\xi$ such that $\xi\circ(a+b+c)\simeq 4f'$.
\end{remark}
\begin{theorem}
\label{partialorder4}
The following hold:
\begin{letterlist}
\item if $f'_{3}$ has order $4$ then $\partial_{1}$ has order $4$;
\item if $f'_{3}$ has order $8$ then $\Omega\partial_{1}$ has order $4$.
\end{letterlist}
\end{theorem}
\begin{proof}
By Proposition~\ref{order4options}, if $f'_{3}$ has order $4$ then $4\circ\partial_{1}$
is null homotopic, implying that $\partial_{1}$ has order at most $4$. On the
other hand, by Lemma~\ref{HKlemma}, the order of $\partial_{1}$ is divisible by $4$.
Thus $\partial_{1}$ has order $4$.
Next, in general, the quotient map
\(\namedright{X\times Y}{Q}{X\wedge Y}\)
is null homotopic after looping. For if
\(i\colon\namedright{X\vee Y}{}{X\times Y}\)
is the inclusion of the wedge into the product, then $Q\circ i$ is null
homotopic, but by the Hilton-Milnor Theorem $\Omega i$ has a right
homotopy inverse. In our case, if $f'_{3}$ has order $8$ then
Proposition~\ref{order4options} states that $4\circ\partial_{1}$
factors through the quotient map
\(\namedright{S^{5}\times S^{7}}{Q}{S^{5}\wedge S^{7}\simeq S^{12}}\).
Thus $4\Omega\partial_{1}$ is null homotopic. Consequently, $\Omega\partial_{1}$
has order at most $4$. By Lemma~\ref{loopHKlemma}, the order
of~$\Omega\partial_{1}$ is divisible by $4$. Thus $\Omega\partial_{1}$ has order $4$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{looppartialorder2}]
Proposition~\ref{partialorder4} implies that in any case the
$2$-primary component of the order of $\Omega\partial_{1}$ is $4$.
\end{proof}
\bibliographystyle{amsplain}
| {'timestamp': '2019-09-12T02:11:50', 'yymm': '1909', 'arxiv_id': '1909.04643', 'language': 'en', 'url': 'https://arxiv.org/abs/1909.04643'} |
\section{Introduction}\label{intro}
Geometric Langlands duality as originally formulated by Beilinson and Drinfeld \cite{BD} is a relationship
between categories associated to moduli spaces of fields on a Riemann surface $C$.
Many ingredients that enter in formulating and analyzing this
duality are familiar in quantum field theory. In particular, two-dimensional conformal field theory plays a prominent role, as reviewed in \cite{F}.
The relation of geometric Langlands duality to quantum field theory can be understood more fully by formulating the subject
in terms of a twisted version of ${\mathcal N}=4$ super Yang-Mills theory \cite{KW}. From that point of view, geometric Langlands duality is
deduced from electric-magnetic duality between the ${\mathcal N}=4$ theory for a compact gauge group $G$
and the same theory based on the Langlands or GNO dual group $G^\vee$ (the complexifications of these groups will be called $G_{\Bbb C}$ and $G^\vee_{\Bbb C}$).
Twisting of ${\mathcal N}=4$ super Yang-Mills
theory produces a four-dimensional topological field theory, which naturally \cite{BaDo,L1,Fr} assigns a number (the partition function) to a four-manifold,
a vector space (the space of physical states) to a three-manifold, and a category of branes or boundary conditions to a two-manifold.\footnote{\label{partial}
This idealized description
does not take into account the fact that the complex Lie groups $G_{\Bbb C}$ and $G^\vee_{\Bbb C}$ and the associated moduli spaces are not compact.
Because of this noncompactness, one likely gets only a partial topological field theory.
Branes and spaces of physical states can be defined, but
it is not clear that the integrals that formally would define partition functions associated
to four-manifolds can really be defined satisfactorily. (A somewhat similar situation arises in Donaldson theory of four-manifolds, though the
details are quite different: one does not get a complete topological field theory, as the partition function cannot be suitably defined for all four-manifolds.)
The noncompactness also means that to define branes and spaces of physical states,
one has to specify the allowed asymptotic behavior of a brane or a wavefunction. Each (reasonable) choice leads to a different version of the duality.
Possibilities include the Betti and de Rham versions \cite{BN}.}
The usual geometric Langlands duality is a duality between the categories associated to two-manifolds.
Recently an analytic version of geometric Langlands duality has been discovered by Etingof, Frenkel, and Kazhdan \cite{EFK,EFK2,EFK3}, stimulated in part by
a number of mathematical \cite{L} and physical \cite{T} developments. Rather than categories and functors acting on categories, one considers
a Hilbert space of quantum states and self-adjoint operators such as quantum Hitchin Hamiltonians acting on this Hilbert space. Very roughly, the usual formulation of geometric Langlands duality involves deformation quantization of the algebra of holomorphic functions on
the moduli space ${\mathcal M}_H(G,C)$ of $G$ Higgs bundles on a Riemann surface $C$, while the analytic version
of the theory involves ordinary quantization of the same moduli space, viewed now as a real symplectic manifold. As usual, quantization means that
a suitable class of smooth functions -- in general neither holomorphic
nor antiholomorphic -- become operators on a Hilbert space. See also \cite{Teschner:2010je,Balasubramanian:2017gxc,Nekrasov:2010ka,Nekrasov:2009rc,Nekrasov:2011bc,Nekrasov:2014yra,Jeong:2018qpc,Bonelli:2011na} for prior work on the gauge theory interpretation of the spectrum of quantum Hitchin Hamiltonians.
The goal of the present article is to place the analytic version of geometric Langlands duality in the gauge theory framework. The first step is simply to
understand what one should do in that framework to study the quantization of ${\mathcal M}_H$ viewed as a real symplectic manifold, as opposed
to its deformation quantization when viewed as a complex symplectic manifold. The basic idea here is that the problem of quantization of a real symplectic manifold $M$
is part of the $A$-model of a suitable complexification $Y$ of $M$ (if such a $Y$ exists) \cite{GuW}. We have explored this construction in more detail elsewhere
as background to the present article \cite{GW}. For the application to geometric Langlands,
we want to quantize the Higgs bundle moduli space ${\mathcal M}_H$ viewed as a real symplectic
manifold with one of its real symplectic structures. A suitable complexification of ${\mathcal M}_H$ is simply the product of two copies of ${\mathcal M}_H$ with opposite complex
structures. With this as the starting point for understanding the quantization of ${\mathcal M}_H$, we will show that the analytic version of geometric Langlands
can be understood by assembling in a novel fashion the same gauge theory
ingredients that have been used previously for understanding the more traditional version of geometric Langlands.
The organization of this article is as follows. In Section \ref{basic}, we explain the basic setup for formulating the analytic version of geometric Langlands
duality in terms of ${\mathcal N}=4$ super Yang-Mills theory. In Section \ref{hitchval}, we explain the predictions of electric-magnetic duality for the joint eigenvalues of the
Hitchin Hamiltonians, and in Section \ref{wtw}, we analyze Hecke or 't Hooft operators and the dual Wilson operators. In Section \ref{wkb}, we show that the
joint eigenfunctions of the Hitchin Hamiltonians satisfy a quantum-deformed WKB condition. In Section \ref{realb}, we discuss the quantization of real forms
of the Higgs bundle moduli space. Starting in Section \ref{avatar}, we use chiral algebras that arise at junctions between supersymmetric boundary conditions
to explore the analytic version of geometric Langlands duality. In Section \ref{sec:chiral}, we study Hecke operators from the point of view of these chiral algebras.
In Section \ref{sec:symp}, we explore in detail an example with remarkable properties.
Some further issues are treated in appendices. In Appendix \ref{bmodel}, we explain that electric-magnetic duality together with positivity of the Hilbert space
inner product in quantization of ${\mathcal M}_H$ imply that the intersections of the varieties $L_{\mathrm{op}}$ and $L_{\overline{\mathrm{op}}}$ that parametrize holomorphic and antiholomorphic
opers must be isolated and transverse (as conjectured in \cite{EFK} and proved in some cases). We further point out that electric-magnetic duality implies
a natural normalization for the joint eigenfunctions of the Hitchin Hamiltonians and argue that the Hilbert space norm of a normalized wavefunction is given
by, roughly speaking, the torsion of the associated oper bundle. In Appendix \ref{somex}, we explore some examples of differential equations satisfied by line
operators. In Appendix \ref{localmodel}, we construct a local model to analyze the singular behavior of the eigenfunctions of the Hitchin Hamiltonians
along the divisor of not very stable bundles.
Throughout this article, $C$ is a Riemann surface of genus $g>1$. All considerations can be naturally extended to the case of bundles with parabolic
structure, but for simplicity we will omit this generalization (which has been developed in \cite{EFK,EFK2,EFK3}). With sufficiently many parabolic
points, there is a quite similar theory for genus $g=0,1$. The cases $g=0,1$ without parabolic structure (or with too few parabolic points for $g=0$)
require a different treatment because any low energy description requires gauge fields, not just $\sigma$-model fields.
\section{Basic Setup}\label{basic}
This section is devoted to an explanation of the basic framework in which we will study the analytic version of geometric Langlands,
and a review of part of the background.
\subsection{Quantization Via Branes}\label{background}
Here we briefly summarize some things that have been described more fully elsewhere \cite{GuW,GW}.
Let $Y$ be a complex symplectic manifold, with complex structure $I$ and
holomorphic symplectic form $\Omega$. We view $Y$ as a real symplectic manifold with the real symplectic
form $\omega_Y={\mathrm{Im}}\,\Omega$. We assume that $Y$ is such that a quantum $\sigma$-model with target $Y$ exists (as an ultraviolet-complete
quantum field theory), and we consider the $A$-model obtained
by twisting this $\sigma$-model in a standard way. It is not known in general for what class of $Y$'s the $\sigma$-model does exist, but a sufficient condition is believed
to be that the complex symplectic structure of $Y$ can be extended to a complete hyper-Kahler structure. The example important for the present article is
the case that $Y$ is the Higgs bundle moduli space ${\mathcal M}_H$, which indeed admits a complete hyper-Kahler metric \cite{H}.
In general, the $A$-model of a symplectic manifold $Y$, in addition to the usual Lagrangian branes whose support is middle-dimensional in $Y$,
can have coisotropic branes, supported on a coisotropic submanifold of $Y$ that is above the middle dimension \cite{KO}. The simplest case and the only
case that we will need in the present article is a rank 1 coisotropic $A$-brane whose support is all of $Y$. Let ${\sf B}$ be the $B$-field of the $\sigma$-model,
and consider a brane with support $Y$ whose Chan-Paton or ${\mathrm{CP}}$ bundle is a line bundle ${\mathcal L}\to Y$, with a unitary connection ${\sf A}$ of curvature ${\sf F}=\mathrm d{\sf A}$.
The Kapustin-Orlov condition
for this data to define an $A$-brane is that ${\mathcal I}=\omega^{-1}({\sf F}+{\sf B})$ should be an integrable complex structure on $Y$. Two solutions
of this condition immediately present themselves. One choice is ${\sf F}+{\sf B}={\mathrm{Re}}\,\Omega$, ${\mathcal I}=I$. The brane constructed this way is what we will call
the canonical coisotropic $A$-brane, ${\mathscr B}_{\mathrm{cc}}$. A second choice is ${\sf F}+{\sf B}=-{\mathrm{Re}}\,\Omega$, ${\mathcal I}=-I$. This leads to what we will call the conjugate
canonical coisotropic $A$-brane, ${\bar {\mathscr B}}_{\mathrm{cc}}$. To treat the two cases symmetrically, in the present article it is convenient to take\footnote{This choice was made
in \cite{GuW}. However, when only one of ${\mathscr B}_{\mathrm{cc}}$, ${\bar {\mathscr B}}_{\mathrm{cc}}$ is relevant, it can be simpler to take ${\sf F}=0$, ${\sf B}={\mathrm{Re}}\,\Omega$.}
${\sf B}=0$, ${\sf F}=\pm {\mathrm{Re}}\,\Omega$. This choice is only possible if there exists a complex line bundle ${\mathcal L}\to Y$ with curvature ${\mathrm{Re}}\,\Omega$. In our
application to the Higgs bundle moduli space, ${\mathrm{Re}}\,\Omega$ is cohomologically trivial, so ${\mathcal L}$ exists and can be assumed to be topologically trivial.
Now consider the algebra ${\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$ (which in physical terms is the space of $({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$ open strings, with an associative
multiplication that comes by joining of open strings). Suppose that $\Omega=\Omega_0/\hbar$, where we keep $\Omega_0$ fixed and vary $\hbar$.
For $\hbar\to 0$, ${\mathscr A}$ reduces to the commutative algebra ${\mathscr A}_0$ of holomorphic functions on $Y$ in complex structure $I$. In order $\hbar$,
the multiplication law in ${\mathscr A}$ differs from the commutative multiplication law in ${\mathscr A}_0$ by the Poisson bracket $\{f,g\}=(\Omega^{-1})^{ij}\partial_i f\partial_j g$.
So ${\mathscr A}$ can be viewed as a deformation quantization of ${\mathscr A}_0$. $\bar{\mathscr A}={\mathrm{Hom}}({\bar {\mathscr B}}_{\mathrm{cc}},{\bar {\mathscr B}}_{\mathrm{cc}})$ is related in the same way to the commutative algebra $\bar{\mathscr A}_0$
of holomorphic functions on $Y$ in complex structure $-I$, or equivalently antiholomorphic functions in complex structure $I$, and can be viewed as a deformation
quantization of $\bar{\mathscr A}_0$.
Generically, deformation quantization is a formal procedure that has to be defined over a ring of formal power series in $\hbar$. When a quantum $\sigma$-model
of $Y$ exists, it is expected that $\hbar$ can be set to a complex value (such as 1), rather than being treated as a formal power series variable.
In the present example, one can give a more direct explanation of this. The Higgs bundle moduli space ${\mathcal M}_H(G,C)$ has a ${\Bbb C}^*$ symmetry, rescaling
the Higgs field by $\varphi\to \lambda\varphi$, $\lambda\in {\Bbb C}^*$. This operation rescales $\Omega$ in the same way. Of course, this is possible only
because $\Omega$ is cohomologically trivial. The ring ${\mathscr A}_0$ is generated by functions that scale with a definite degree
(namely the Hitchin Hamiltonians) and the scaling symmetry implies
that all the relations in the deformed ring ${\mathscr A}$ are polynomials in the deformation parameter $\hbar$. Hence it makes sense to set $\hbar=1$.
Similar remarks will apply when we consider quantization rather than deformation quantization, given that the important branes considered are ${\Bbb C}^*$-invariant:
the scaling symmetry will imply that certain semiclassical formulas are actually exact.
Deformation quantization is particularly interesting if $Y$ is an affine variety, with lots of holomorphic functions. The example of $Y={\mathcal M}_H$ studied in the present
article is far from being an affine variety.
The ring ${\mathscr A}_0$ in this case is simply the ring of functions on the base of the Hitchin fibration; in other words, the global holomorphic functions
on ${\mathcal M}_H$ are simply the functions of Hitchin's Poisson-commuting Hamiltonians. Hitchin's Poisson-commuting Hamiltonians can be quantized
to commuting differential operators, acting on sections of the line bundle $K^{1/2}$, where $K$ is the canonical bundle of ${\mathcal M}_H$.
This was shown by Hitchin \cite{H2}
for $G={\mathrm{SU}}(2)$ and by Beilinson and Drinfeld \cite{BD} in general. In particular, the ring ${\mathscr A}$ is commutative.
The fact that the Poisson-commuting classical Hamiltonians can be quantized to
commuting differential operators can also be seen in the gauge theory language, as we will discuss in Section \ref{hh}.
In the present article, we are really interested in quantization rather than deformation quantization. How to modify the story just described
to encompass quantization has been explained in \cite{GuW,GW}. Suppose that $M$ is a real symplectic manifold, with symplectic form $\omega_M$,
that we wish to quantize. If $M$ has a complexification $Y$ that obeys certain conditions, then quantization of $M$ is part of the $A$-model of $Y$.
$Y$ should be a complex symplectic manifold with a holomorphic symplectic form $\Omega$ whose restriction to $M$ is $\omega_M$. Moreover,
$Y$ should have an antiholomorphic involution\footnote{An involution is an automorphism whose square is 1.} $\tau$ with $M$ as a component of its fixed point set. These conditions imply that $\tau^*\Omega=\overline\Omega$.
Finally, the quantum $\sigma$-model of $Y$ must exist. Under these conditions, a choice of a prequantum line bundle ${{{\mathfrak L}}}\to M$, in the
sense of geometric quantization,\footnote{A prequantum line bundle over a symplectic manifold $M$ with symplectic form $\omega$ is a complex line
bundle ${{\mathfrak L}}\to M$ with a unitary connection of curvature $\omega$.}
determines an $A$-brane ${\mathscr B}$ with support $M$. (The details of this depend on the choice that was made in satisfying
the condition ${\sf F}+{\sf B}={\mathrm{Re}}\,\Omega$ to
define the brane ${\mathscr B}_{\mathrm{cc}}$. We will be specific in Section \ref{cm}.)
The $A$-model answer for the Hilbert space obtained by
quantizing $M$ with symplectic structure $\omega_M$ and prequantum line bundle ${{\mathfrak L}}$ is ${\mathcal H}={\mathrm{Hom}}({\mathscr B},{\mathscr B}_{\mathrm{cc}})$.
The definition of the hermitian inner product on ${\mathcal H}$ is described in Section \ref{cm}.
${\mathcal H}$ is always a module for ${\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$.
This can be described by saying that those functions on $M$ that can be analytically continued to holomorphic functions on $Y$ are quantized
to give operators on ${\mathcal H}$. In addition, under mild conditions, a correspondence\footnote{A correspondence between $M$ and itself is simply
a Lagrangian submanifold of $M_1\times M_2$, where $M_1$ and $M_2$ are two copies of $M$, with respective symplectic structures $\omega_M$ and
$-\omega_M$. A holomorphic correspondence between $Y$ and itself is defined similarly.}
between $M$ and itself that can be analytically continued to a holomorphic correspondence between
$Y$ and itself can be quantized in a natural way to give an operator on ${\mathcal H}$.
In the problem that we will be studying with $M={\mathcal M}_H$, the ring ${\mathscr A}$
will be relatively small, consisting of polynomials in the holomorphic and antiholomorphic Hitchin Hamiltonians. By way of compensation, there is an ample
supply of correspondences -- the Hecke correspondences -- that will provide additional operators on ${\mathcal H}$. Quantizing these correspondences is simple
because of the ${\Bbb C}^*$ scaling symmetry that was invoked in the discussion of deformation quantization.
We have described this in the context of two-dimensional $\sigma$-models, as is appropriate for quantization of a fairly general real symplectic manifold $M$.
However, in the particular case that $M$ is the Higgs bundle moduli space ${\mathcal M}_H(G,C)$ for gauge group $G$ on a Riemann surface $C$, a four-dimensional
picture is available and gives much more complete understanding. For this, the starting point is ${\mathcal N}=4$ super Yang-Mills theory, in four dimensions,
with gauge group $G$. One restricts to four-manifolds of the form $\Sigma\times C$, where $\Sigma$ is an arbitrary two-manifold
but $C$ is kept fixed. At low energies, the four-dimensional gauge theory reduces for many purposes, assuming that $G$ has trivial center,
to a supersymmetric $\sigma$-model on $\Sigma$ with target ${\mathcal M}_H(G,C)$
\cite{BJSV,JWS}. A certain twisting of the ${\mathcal N}=4$ theory in four dimensions produces a topological field
theory\footnote{This is really a partial topological field theory, as remarked in footnote \ref{partial}.} that, when we specialize
to four-manifolds of the form $\Sigma\times C$, reduces to an $A$-model on $\Sigma$ with target ${\mathcal M}_H(G,C)$. The brane ${\mathscr B}_{\mathrm{cc}}$ of the $\sigma$-model
originates in the four-dimensional gauge theory as a boundary condition that is a simple deformation of Neumann boundary
conditions for the gauge field, extended to the whole supermultiplet in a half-BPS fashion; see Section 12.4 of \cite{KW}.
The four-dimensional lift of the ${\mathscr B}_{\mathrm{cc}}$ boundary condition has an interesting and important feature. Though it is a brane in an $A$-model that
has full four-dimensional topological symmetry, the definition of the brane ${\mathscr B}_{\mathrm{cc}}$ depends on a choice of complex structure of $C$.
The deformed Neumann boundary condition is a {\it holomorphic-topological} local boundary condition for the four-dimensional
topological field theory. \footnote{Concretely, the boundary condition breaks some of the bulk supersymmetries
of the physical theory. As a consequence, some translation generators which are $Q$-exact in the bulk cease to be $Q$-exact in
the presence of the boundary and the boundary condition is not topological in the four-dimensional sense. Another explanation of the dependence of ${\mathscr B}_{\mathrm{cc}}$
on a choice of complex structure is simply that ${\mathscr B}_{\mathrm{cc}}$
is defined with ${\sf B}=\omega_J$, whose definition depends on the complex structure of $C$. See the beginning of Section \ref{cm} for a statement of which
structures on ${\mathcal M}_H$ do or do not depend on a choice of complex structure. }
Boundary local operators supported
at a point $p\in C$ can depend holomorphically on $p$, even though they depend topologically on the one remaining boundary
direction. This will be important whenever we
discuss the four-dimensional lift of our constructions. Although the $A$-model has full four-dimensional topological invariance, in the presence of the brane
${\mathscr B}_{\mathrm{cc}}$, only two-dimensional topological invariance is available. For example, in proving the commutativity of the Hitchin Hamiltonians and the Hecke
operators, we will use only two-dimensional topological invariance.
If $G$ has a nontrivial center ${\mathcal Z}(G)$, then the assertion about a reduction to a $\sigma$-model must be slightly modified. A more precise statement is that upon
compactification on $C$, the
four-dimensional gauge theory reduces, for many purposes, to the product of a $\sigma$-model with target ${\mathcal M}_H(G,C)$ and a gauge theory with the finite
gauge group ${\mathcal Z}(G)$ (acting trivially on ${\mathcal M}_H(G,C)$). This finite gauge group will play no role until we want to compute the eigenvalues of Hitchin Hamiltonians
and Hecke operators, so in much of the following it will not be mentioned.
A purely two-dimensional description via a $\sigma$-model with target ${\mathcal M}_H(G,C)$ (possibly extended by the finite gauge group) is useful for many purposes. But information is lost in the reduction to
two dimensions, and the four-dimensional picture is needed for a complete account of the duality.\footnote{To be more precise, a complete formulation of the duality
is possible in four dimensions. A fuller explanation of the duality comes from a certain starting point in six
dimensions \cite{WittenLecture}.}
In standard mathematical treatments, an analogous statement is that geometric Langlands duality must be formulated
in terms of the ``stack'' of $G_{\Bbb C}$-bundles or of $G_{\Bbb C}^\vee$ local systems, rather than in terms of a finite-dimensional moduli space.
There is a simple relation between the two statements.
It was shown by Atiyah and Bott \cite{AB} that the space of all $G$-valued connections on a smooth $G$-bundle $E\to C$, with the action
of the group of complexified gauge transformations, provides a model of the stack of holomorphic $G_{\Bbb C}$-bundles.
That is because the $(0,1)$ part of any connection gives $E$ (or more precisely its complexification) a complex structure, making it a holomorphic
$G_{\Bbb C}$ bundle over $E$.
A gauge field on a
$G$-bundle over $ \Sigma\times C$
determines, in particular, a family, parametrized by $\Sigma$, of gauge fields on $C$. So any such connection determines a map from $\Sigma$ to the stack of
$G_{\Bbb C}$ bundles over $C$. Thus ${\mathcal N}=4$ super Yang-Mills theory on $\Sigma\times C$ can be understood as a supersymmetric $\sigma$-model on $\Sigma$
with the target being the stack of $G_{\Bbb C}$ bundles over $C$. (In this formulation, the theory is a two-dimensional supersymmetric
gauge theory on $\Sigma$ coupled to matter fields.
The matter
fields are gauge fields on $C$ and their superpartners, and the gauge group is the group $\widehat G$ of maps of $C$ to the finite-dimensional group $G$. A gauge transformation in the two-dimensional theory is a map from $\Sigma$ to $\widehat G$; in the four-dimensional description, this is interpreted
as a map of the four-manifold $\Sigma\times C$ to $G$.
To keep these statements simple, we have assumed that all bundles are trivialized.)
The mathematical statement that the correct formulation involves stacks corresponds to the
quantum field theory statement that the correct formulation is in four dimensions.
\subsection{Quantizing A Complex Manifold}\label{cm}
We view ${\mathcal M}_H(G,C)$ as a complex manifold in the complex structure, called $I$ by Hitchin \cite{H}, in which it parametrizes Higgs bundles over $C$.
$I$ is one of a triple of complex structures $I,J,K$ that, along with the corresponding Kahler forms $\omega_I,\omega_J,\omega_K$,
make a hyper-Kahler structure on $C$. In complex structure $J$, ${\mathcal M}_H(G,C)$ parametrizes flat bundles over $C$ with structure group $G_{\Bbb C}$.
Complex structure $J$ and the corresponding holomorphic symplectic form $\Omega_J=\omega_K+{\mathrm i}\omega_I$ are topological in the sense that they
depend on $C$ only as an oriented two-manifold, while the other structures $I$, $K$, and $\omega_J$ depend on a choice of complex structure on $C$.
The complex symplectic form of ${\mathcal M}_H(G,C)$, in complex structure $I$, is $\Omega_I=\omega_J+{\mathrm i}\omega_K$. We want to view ${\mathcal M}_H$ as a real symplectic
manifold with the real symplectic structure $\omega=\omega_J={\mathrm{Re}}\,\Omega_I$, and quantize it.
As we have explained in Section \ref{background}, the first step in quantizing any real symplectic manifold $M$ via branes is to pick a suitable complexification
of it. Here we are in a special situation: $M$ is actually a complex symplectic manifold $Y$, with complex structure $I$ and complex symplectic form $\Omega$,
and we want to quantize $Y$ with the real symplectic structure $\omega_Y={\mathrm{Re}}\,\Omega$. In such a case, there
is a standard way to proceed, described in Section 5 of \cite{GW}.
We set $\widehat Y=Y_1\times Y_2$, where $Y_1$ and $Y_2$ are two copies of $Y$, with opposite complex structures $I$ and $-I$. The complex
structure of $\widehat Y$ is thus a direct sum ${\mathcal I}=I\oplus (-I)$. The involution $\tau:\widehat Y\to\widehat Y$
that exchanges the two factors is antiholomorphic, and its fixed point set is a copy of $Y$, embedded as the diagonal in $\widehat Y=Y_1\times Y_2$.
We endow $\widehat Y$ with the complex symplectic form $\widehat\Omega=\frac{1}{2}\Omega\boxplus \frac{1}{2}\overline\Omega$; in other words, the symplectic
form of $\widehat Y=Y_1\times Y_2$ is $\frac{1}{2}\Omega$ on the first factor and $\frac{1}{2}\overline\Omega$ on the second factor. This definition ensures
that the restriction of $\widehat\Omega$ to the diagonal is ${\mathrm{Re}}\,\Omega$. Suppose that the complex symplectic structure of $Y$ can be extended to a complete
hyper-Kahler metric. Then the complex symplectic structure of $\widehat Y$ can likewise be extended to a complete hyper-Kahler metric,
namely a product hyper-Kahler metric on $\widehat Y=Y_1\times Y_2$.
So all conditions are satisfied, and the $A$-model of $\widehat Y$ with real symplectic form
${\mathrm{Im}}\,\widehat\Omega=\frac{1}{2}{\mathrm{Im}}\,\Omega\boxplus (-\frac{1}{2}{\mathrm{Im}}\,\Omega)$
is suitable for quantizing $Y$ with the symplectic structure $\omega_Y={\mathrm{Re}}\,\Omega$.
To define coisotropic branes, we have to satisfy
the Kapustin-Orlov condition that $\omega^{-1} ({\sf F}+{\sf B})$ should be an integrable complex structure. In doing so, we will take ${\sf B}=0$, as this will make
it possible to treat the two factors of $\widehat Y$ symmetrically.
Since the complex structures
of $Y_1$ and $Y_2$ are respectively $I$ and $-I$ and the $A$-model symplectic structures are respectively $\frac{1}{2}{\mathrm{Im}}\,\Omega$ and $-\frac{1}{2}{\mathrm{Im}}\,\Omega$,
we can define canonical coisotropic branes ${\mathscr B}_{{\mathrm{cc}},1}$ and ${\mathscr B}_{{\mathrm{cc}},2}$ on $Y_1$ and $Y_2$ by taking in each case a ${\mathrm{CP}}$ bundle ${\mathcal L}$ with curvature
${\sf F}=\frac{1}{2}\omega_J$. (In our application to the Higgs bundle moduli space, ${\mathrm{Re}}\,\Omega$ is exact and $b_1(Y)=0$, so a topologically trivial line bundle over $Y$
with curvature $\frac{1}{2}\omega_J$ exists and is unique up to isomorphism.)
One then defines on $\widehat Y$ the product brane $\widehat{\mathscr B}_{\mathrm{cc}}=\widehat{\mathscr B}_{{\mathrm{cc}},1}\times
\widehat{\mathscr B}_{{\mathrm{cc}},2}$, with ${\mathrm{CP}}$ bundle $\widehat{\mathcal L}={\mathcal L}\boxtimes{\mathcal L}$.
We also define a Lagrangian brane ${\mathscr B}$ supported on $Y$ with trivial ${\mathrm{CP}}$ bundle. Following the general logic, the Hilbert space for quantization of $Y$ with the real symplectic structure ${\mathrm{Re}}\,\Omega$ is ${\mathcal H}={\mathrm{Hom}}({\mathscr B},\widehat{\mathscr B}_{\mathrm{cc}})$. The prequantum line bundle in this situation is ${{\mathfrak L}}=\widehat{\mathcal L}|_Y\cong {\mathcal L}^2$. Since ${\mathcal L}$ has curvature
$\frac{1}{2}{\mathrm{Re}}\,\Omega$, ${\mathcal L}^2$ has curvature ${\mathrm{Re}}\,\Omega$, so it is an appropriate prequantum line bundle for quantization of $Y$ with symplectic
structure ${\mathrm{Re}}\,\Omega$.
\begin{figure}
\begin{center}
\includegraphics[width=4in]{Folding.pdf}
\end{center}
\caption{\small (a) In the folded construction, we have two copies of a $\sigma$-model on a strip. The two copies are decoupled except on the
right boundary, where they are glued together by a brane ${\mathscr B}$ that is supported on the diagonal in $\widehat Y=Y_1\times Y_2$. ${\mathscr B}$ has trivial ${\mathrm{CP}}$
bundle so this gluing of the two copies is its only effect. (b) After unfolding,
we have a single copy of the $\sigma$-model on a strip of twice the width. No trace of ${\mathscr B}$ remains. \label{folding}}
\end{figure}
However,
this definition has a useful variant. To compute ${\mathrm{Hom}}({\mathscr B},\widehat{\mathscr B}_{\mathrm{cc}})$, we study the $\sigma$ model with target $\widehat Y$ on
a strip $\Sigma$, with boundary conditions set by $\widehat{\mathscr B}_{\mathrm{cc}}$ of the left boundary of the strip and by ${\mathscr B}$ on the right boundary, as in fig. \ref{folding}(a).
The two factors of $\widehat Y=Y_1\times Y_2$ are decoupled in the bulk of the $\sigma$-model, since the metric on $\widehat Y$ is a product; they are also
decoupled on the left boundary, since the brane $\widehat{\mathscr B}_{\mathrm{cc}}={\mathscr B}_{{\mathrm{cc}},1}\times {\mathscr B}_{{\mathrm{cc}},2}$ is likewise a product. So away from the right boundary of the strip, we can
think of $\Sigma$ as having two sheets, one of which is mapped to $Y_1$ and one to $Y_2$, as in the figure. The two sheets are coupled only
on the right boundary, where, as ${\mathscr B}$ is supported on the diagonal in $Y_1\times Y_2$, the two sheets are ``glued together'' and map to the same point in $Y$.
This suggests that we should ``unfold'' the picture (fig. \ref{folding}(b)). After this unfolding, we simply have a single sheet of twice the width that is
mapped to a single copy of $Y$. Unfolding reverses the orientation of one of the two sheets of the folded picture, and this orientation reversal
changes the sign of the $A$-model
symplectic form. In the folded picture, the $A$-model symplectic form was $\frac{1}{2}\omega_K$ on $Y_1$ and $-\frac{1}{2}\omega_K$ on $Y_2$;
hence after unfolding, the $A$-model symplectic form is $\frac{1}{2}\omega_K$ everywhere. In other words, the unfolded picture involves the ordinary
$A$-model of a single copy of $Y$ with symplectic form $\frac{1}{2}\omega_K$. Before unfolding, the branes ${\mathscr B}_{{\mathrm{cc}},1}$ and ${\mathscr B}_{{\mathrm{cc}},2}$ both
have ${\mathrm{CP}}$ bundles with curvature $\frac{1}{2} \omega_J$. Reversing the orientation of $\Sigma$ replaces the ${\mathrm{CP}}$ bundle of a brane with its dual
(or its inverse, in the rank 1 case), and so reverses the sign of the ${\mathrm{CP}}$ curvature. Hence in the unfolded picture, the boundaries are labeled
by branes ${\mathscr B}_{\mathrm{cc}}$ and ${\bar {\mathscr B}}_{\mathrm{cc}}$ whose respective ${\mathrm{CP}}$ bundles are lines bundles ${\mathcal L}$ and ${\mathcal L}^{-1}$ with curvatures
$\frac{1}{2}\omega_J$ and $-\frac{1}{2}\omega_J$. These are
the conjugate canonical coisotropic branes that were introduced in Section \ref{background}. The prequantum line bundle
is still ${{\mathfrak L}}={\mathcal L}^2$.
${\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$ is a deformation
quantization of the commutative algebra ${\mathscr A}_0$ of holomorphic functions on $Y$, and $\overline{\mathscr A}={\mathrm{Hom}}({\bar {\mathscr B}}_{\mathrm{cc}},{\bar {\mathscr B}}_{\mathrm{cc}})$ is similarly a deformation
quantization of the algebra $\bar{\mathscr A}_0$ of antiholomorphic functions on $Y$. What in the folded picture was ${\mathcal H}={\mathrm{Hom}}({\mathscr B},\widehat{\mathscr B}_{\mathrm{cc}})$ becomes in the
unfolded picture ${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$.
With either description of ${\mathcal H}$, we need to define a hermitian product on ${\mathcal H}$. For definiteness
we use the folded language.\footnote{For more detail on the following, see Section 2.7 of \cite{GW}.} Topological
field theory would give us in general a nondegenerate bilinear (not hermitian) pairing $(~,~)$ between ${\mathcal H}={\mathrm{Hom}}({\mathscr B},\widehat{\mathscr B}_{\mathrm{cc}})$ and its dual space
${\mathcal H}'={\mathrm{Hom}}(\widehat{\mathscr B}_{\mathrm{cc}},{\mathscr B})$. To get a hermitian pairing on ${\mathcal H}$, we need to compose this bilinear pairing with an antilinear map from ${\mathcal H}$ to ${\mathcal H}'$.
Such a map in the underlying physical theory is provided by the $\sf{CPT}$ symmetry $\Theta$. But $\Theta$ is not an $A$-model symmetry;
it maps the $A$-model to a conjugate $A$-model with the opposite sign of the symplectic form. The involution $\tau$ that exchanges the two factors
of $\widehat Y$
also exchanges the $A$-model with its conjugate, since it is antisymplectic
(it reverses the sign of the $A$-model symplectic form), so $\Theta_\tau=\Theta\tau$ is an antilinear symmetry
of the $A$-model. Finally, because the branes ${\mathscr B}$ and ${\mathscr B}_{\mathrm{cc}}$ are $\Theta_\tau$-invariant, $\Theta_\tau$ maps ${\mathcal H}$ to ${\mathcal H}'$ and
we can define a nondegenerate hermitian pairing on ${\mathcal H}$ by
\begin{equation}\label{hermdef}\langle \psi,\psi'\rangle= (\Theta_\tau\psi,\psi').\end{equation}
For general $\Theta_\tau$-invariant branes,
such a pairing is not positive-definite. For the specific case of quantizing a cotangent bundle, which is our main example on the $A$-model
side, one expects positivity. The $B$-model analog of this construction uses an antiholomorphic (not antisymplectic) involution $\tau$.
Positivity of the pairing in this case is subtle and is discussed in Appendix \ref{bmodel}.
In the folded picture, $\tau$ and therefore also
$\Theta_\tau$ exchanges the two factors of $\widehat Y=Y_1\times Y_2$; in the unfolded picture, they exchange the two ends of the strip. Exchanging the
two ends of the strip reverses the orientation of the strip and therefore would change the sign of ${\sf B}$. Hence in a description with ${\sf B}\not=0$,
the definition of the inner product is less natural (one would need to accompany $\Theta_\tau$ with a $B$-field gauge transformation). That is why we took
${\sf B}=0$ in solving the Kapustin-Orlov conditions for rank 1 coisotropic branes.
\subsection{Quantizing The Higgs Bundle Moduli Space}\label{aphiggs}
For our application to the case that $Y$ is the Higgs bundle moduli space ${\mathcal M}_H(G,C)$ for some gauge group $G$ and
Riemann surface $C$, we really want to study the four-dimensional
version of this construction. This means that we study the ${\mathcal N}=4$ super Yang-Mills theory, with gauge group $G$, on $\Sigma\times C$, where
$\Sigma$ is the strip of fig \ref{folding}(b). The boundary conditions on the left and right of the strip are set by the gauge theory versions of ${\mathscr B}_{\mathrm{cc}}$ and
${\bar {\mathscr B}}_{\mathrm{cc}}$. A detailed description of ${\mathscr B}_{\mathrm{cc}}$ in four-dimensional gauge theory language was given in Section 12.4 of \cite{KW}. We can describe
a Higgs bundle on $C$ by a pair $(A,\phi)$, where $A$ is gauge field, that is, a connection on a $G$-bundle $E\to G$,
and $\phi$ is a 1-form valued in the adjoint bundle ${\mathrm{ad}}(E)$. In this description, ${\bar {\mathscr B}}_{\mathrm{cc}}$ is obtained from ${\mathscr B}_{\mathrm{cc}}$ by $(A,\phi)\to (A,-\phi)$ (suitably extended
to the rest of the four-dimensional supermultiplet). This is a familiar involution of the Higgs bundle moduli space that acts holomorphically in complex
structure $I$ and antiholomorphically in complex structures $J$ and $K$.
Although we have used machinery of gauge theory and branes to construct a Hilbert space ${\mathcal H}$ associated to quantization of ${\mathcal M}_H(G,C)$, the actual
output of this construction is completely unsurprising. A dense open set in ${\mathcal M}_H(G,C)$ is a cotangent bundle $T^*{\mathcal M}(G,C)$, where ${\mathcal M}$ is the moduli
space of semistable holomorphic $G$-bundles over $C$. Geometric quantization -- or simply elementary quantum mechanics -- suggests that the
Hilbert space that we should associate to quantization of $T^*{\mathcal M}(G,C)$ should be the space of ${\mathrm L}^2$ half-densities on ${\mathcal M}(G,C)$.
The reason to speak of ${\mathrm L}^2$ half-densities rather than ${\mathrm L}^2$ functions is that on a general smooth manifold $N$ without some choice of
a measure,\footnote{The space ${\mathcal M}(G,C)$ actually does have a natural measure, namely the one associated to its real symplectic structure when viewed
as a moduli space of flat bundles over $C$ with compact structure group $G$. This is also the measure induced by its embedding in the hyper-Kahler
manifold ${\mathcal M}_H(G,C)$. However, this measure is not part of the $A$-model and does not naturally appear
in $A$-model constructions such as the definition of the Hilbert space ${\mathcal H}$.} there is no way to integrate a function
so there is no natural Hilbert space of ${\mathrm L}^2$ functions. A density on a manifold $N$ is a section of a trivial real line bundle ${\sf K}$
and can be written locally in any
coordinate system as $|\mathrm d x^1\mathrm d x^2\cdots \mathrm d x^w| \,f(x^1,x^2,\cdots, x^w)$, where $f(x^1,\cdots, x^w)$ is a function and
$|\mathrm d x^1\mathrm d x^2\cdots \mathrm d x^w|$ is a measure, not a differential form. ${\sf K}$ has a square root ${\sf K}^{1/2}$, also trivial, whose sections are locally
described in a given coordinate system by functions $g(x^1,x^2,\cdots, x^w) $ that transform under a change of coordinates in such a way
that $|\mathrm d x^1\mathrm d x^2\cdots \mathrm d x^w| \,g(x^1,x^2,\cdots, x^w)^2$ is invariant. It is convenient to formally write
\begin{equation}\label{formw} h=|\mathrm d x^1\mathrm d x^2\cdots \mathrm d x^w|^{1/2} g(x^1,x^2,\cdots, x^w)\end{equation}
for a section of ${\sf K}^{1/2}$. Complex-valued half-densities, which are described by the same formula where locally $g$ is a complex-valued function,
form a Hilbert space in an obvious way: \begin{equation}\label{ob} ||h||^2=\int |\mathrm d x^1\mathrm d x^2\cdots\mathrm d x^w|\, |g(x^1,x^2\cdots x^w)|^2.\end{equation}
Now, motivated by the application to the
complex manifold ${\mathcal M}(G,C)$, let us describe the bundle of densities or half-densities on a complex manifold $N$. If $N$ has complex
dimension $n=w/2$ and local holomorphic coordinates $z^1,z^2,\cdots, z^n$, then $|\mathrm d z^1 \mathrm d z^2 \cdots \mathrm d z^n \mathrm d \overline z^1\mathrm d\overline z^2\cdots \mathrm d\overline z^n|$
is a density on $N$, in other words a section of ${\sf K}\to N$. On the other hand, $\mathrm d z^1 \mathrm d z^2 \cdots \mathrm d z^n $ is a section of the holomorphic canonical bundle $K\to N$,
and $ \mathrm d \overline z^1\mathrm d\overline z^2\cdots \mathrm d\overline z^n$ is a section of the complex conjugate line bundle $\overline K\to N$ (which can also be viewed as the canonical
line bundle of $N$ if $N$ is endowed with the opposite complex structure). So ${\sf K}$ can be identified with $K\otimes \overline K$; more precisely
$K\otimes \overline K\cong {\sf K}\otimes _{\mathbb{R}}{\Bbb C}$, that is, $K\otimes \overline K$ is the complexification of ${\sf K}$, the bundle of complex-valued densities. Similarly $K$ always has a square root at least locally, and for any choice of local square
root of $K$, we have ${\sf K}^{1/2}\cong K^{1/2}\otimes \overline K{}^{1/2}$; more precisely $K^{1/2}\otimes \overline K^{1/2}\cong {\sf K}\otimes_{\mathbb{R}} {\Bbb C}$, that is,
$K^{1/2}\otimes \overline K^{1/2}$ is the bundle of complex-valued half-densities. As long as $\overline K{}^{1/2}$ is the complex conjugate of $K^{1/2}$, this relation holds
for any choice of $K^{1/2}$.
In Section 3 of \cite{GW}, criteria were described under which brane quantization of $M=T^*N$, with its standard symplectic structure as a cotangent
bundle, leads to a Hilbert space of ${\mathrm L}^2$ half-densities on $N$.
Beyond requiring that $M=T^*N$ has a complexification $Y$ that is suitable for brane quantization, the necessary condition
is that $Y$ should be the cotangent bundle of a complexification $W$ of $N$ (and $Y$ should have the natural complex symplectic structure
of a cotangent bundle). This condition is automatically satisfied if $M$ and $N$ are already complex manifolds and $Y$ is defined
as the product of two copies of $M$ with opposite complex structures.
The Higgs bundle moduli space ${\mathcal M}_H(G,C)$ contains $T^*{\mathcal M}(G,C)$ as a dense open set, but is not actually isomorphic to $T^*{\mathcal M}(G,C)$. One would not
expect a measure zero discrepancy to be important in the definition of a Hilbert space of ${\mathrm L}^2$ wavefunctions. The construction in \cite{GW} maps
the Hilbert space ${\mathcal H}$ obtained in quantizing ${\mathcal M}_H(G,C)$ to a space of half-densities on ${\mathcal M}(G,C)$ without requiring that $T^*{\mathcal M}(G,C)$ is literally all of ${\mathcal M}_H(G,C)$.
The Hilbert space ${\mathcal H}$ of half-densities on ${\mathcal M}(G,C)$
was already introduced in \cite{EFK} without any reference to branes and $\sigma$-models or gauge theories. The interpretation via branes
makes it possible to apply electric-magnetic duality and other methods of gauge theory. We will see an example next in discussing the Hitchin
Hamiltonians.
\subsection{Hitchin Hamiltonians}\label{hh}
As we have seen, in brane quantization of a complex manifold such as $Y={\mathcal M}_H(G,C)$, the Hilbert space has an unfolded description as
${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$. ${\mathcal H}$ admits a left action of\footnote{In the folded picture, we have instead a left action of both algebras
${\mathrm{Hom}}({\mathscr B}_{{\mathrm{cc}},1},{\mathscr B}_{{\mathrm{cc}},1})$ and ${\mathrm{Hom}}({\mathscr B}_{{\mathrm{cc}},2},{\mathscr B}_{{\mathrm{cc}},2})$. A left action of an algebra is the same as a right action of the opposite algebra. (The notion
of the opposite algebra
is explained in Appendix C of \cite{GW}.) Unfolding reverses
the orientation of one sheet in fig. \ref{folding} and hence replaces one of the two algebras with its opposite. Of course, which algebra
acts on the left and which on the right depends on some choices. None of this will be important in the present article as ${\mathscr A}$ and $\overline{\mathscr A}$ will be commutative and
hence isomorphic to their opposites.}
${\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$ and a right action of $\overline{\mathscr A}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},\overline{\mathscr B}_{\mathrm{cc}})$.
${\mathscr A}$ and $\overline{\mathscr A}$ are quantum deformations of the commutative rings ${\mathscr A}_0$ and $\overline{\mathscr A}_0$ of holomorphic functions on $Y$. For a general $Y$, these
deformations can be noncommutative, but
in the particular case of ${\mathcal M}_H(G,C)$, it turns out that ${\mathscr A}$ and $\overline{\mathscr A}$ are commutative and hence there is no distinction between a left action and a right action.
Concretely, the ring ${\mathscr A}_0$ of holomorphic functions on ${\mathcal M}_H(G,C)$ in complex structure $I$
is simply the ring of functions on the base of the Hitchin fibration \cite{H,NH}.
Consider a solution
$(A,\phi)$ of Hitchin's equations, where $A$ is a connection on a $G$-bundle $E\to C$ and $\phi$ is a $1$-form valued in ${\mathrm{ad}}(E)$. Let $\varphi$ be the holomorphic
Higgs field, that is, the $(1,0)$ part of $\phi$. Hitchin's equations give $\overline\partial_A\varphi=0$.
So if ${\mathcal P}$ be an invariant polynomial on the Lie algebra ${{\mathfrak g}}_{\Bbb C}$ of
$G_{\Bbb C}$, homogeneous of some degree $s$, then ${\mathcal P}(\varphi)$ is a holomorphic section of $K_C^s$ (with $K_C$ the canonical bundle of $C$; we will
also set $T_C=K_C^{-1}$). Given
any $(0,1)$-form $\alpha$ on $C$ with values in $T_C^{s-1}$, we can define
\begin{equation}\label{juno} H_{{\mathcal P},\alpha}=\int_C \alpha \,{\mathcal P}(\varphi). \end{equation}
This is a holomorphic function on ${\mathcal M}_H$ and depends only on the cohomology class of $\alpha$ in $H^1(C,T_C^{s-1})$.
The dimension of $H^1(C,T_C^{s-1})$ is $(s+1)(g-1)$, and this is the number of linearly independent functions
$H_{{\mathcal P},\alpha}$ for a given ${\mathcal P}$. For $G={\mathrm{SU}}(2)$, the ring ${\mathscr A}_0$ of holomorphic functions on ${\mathcal M}_H(G,C)$ is generated by the $H_{{\mathcal P},\alpha}$
where ${\mathcal P}(\varphi)={\rm Tr}\,\varphi^2$. More generally, a simple Lie group $G$ of rank $r$ has $r$ independent Casimir operators, corresponding to $r$
homogeneous polynomials ${\mathcal P}_j$, $j=1,\cdots,r$ of various degrees, and ${\mathscr A}_0$ is generated by the $H_{{\mathcal P}_j,\alpha_j}$. For example, if $G={\mathrm{SU}}(N)$, we can
take the generating polynomials to be ${\rm Tr}\,\varphi^{k}$, $k=2,3,\cdots,N$.
The functions $H_{{\mathcal P}_j,\alpha_j}$ are Poisson-commuting, since the holomorphic symplectic structure of ${\mathcal M}_H(G,C)$ in complex
structure $I$ is such that $\varphi_z$ and $A_{\overline z}$ are conjugate variables, and in particular
any functions constructed from $\varphi$ only (and not $A$) are Poisson-commuting. These Poisson-commuting functions
are the Hamiltonians of Hitchin's classical integrable system.
The quantum deformation from ${\mathscr A}_0$ to ${\mathscr A}$ is unobstructed in the sense that every element of ${\mathscr A}_0$ can be quantum-deformed to an element of ${\mathscr A}$.
This statement means, concretely, that if ${\mathcal P}$ is an invariant polynomial on ${\mathfrak g}$ homogeneous
of some
degree $s$, and $H_{{\mathcal P},\alpha}$ is a corresponding Hitchin Hamiltonian, then there is a differential operator $D_{{\mathcal P},\alpha}$, acting on sections of $K^{1/2}\to {\mathcal M}(G,C)$,
whose leading symbol is equal to $H_{{\mathcal P},\alpha}$. The passage from $H_{{\mathcal P},\alpha}$ to $D_{{\mathcal P},\alpha}$ is not entirely canonical, since specifying the desired leading
symbol of $D_{{\mathcal P},\alpha}$ leaves one free to add to $D_{{\mathcal P},\alpha}$ a globally-defined holomorphic differential operator of degree less than $s$.
For $G_{\Bbb C}={\mathrm{SL}}(2,{\Bbb C})$, one
can take ${\mathcal P}$ to be of degree 2, and then the only globally-defined holomorphic differential operators of lower degree are the operators of degree 0 -- the complex
constants. For groups of higher rank, in general there are more possibilities.
Mathematically, the fact that the deformation is unobstructed
follows from the fact that, for any simple $G$, $H^1({\mathcal M}_H,{\mathcal O})=0$, by virtue of which there is no potential obstruction in the deformation.
A gauge theory version of this argument was given in Section 12.4 of \cite{KW}.
From the point of view of the $\sigma$-model, or the underlying gauge theory, the deformation from $H_{{\mathcal P},\alpha}$ to $D_{{\mathcal P},\alpha}$ arises from an
expansion in powers of $\hbar$. This expansion terminates after finitely many steps, since we define ${\mathscr A}_0$ to consist of functions whose restriction to
a fiber of the cotangent bundle is a polynomial. A specific definition of the $\sigma$-model or the gauge theory gives a specific recipe for passing from $H_{{\mathcal P},\alpha}$
to $D_{{\mathcal P},\alpha}$, but this depends on data (such as a Riemannian metric on $C$, not just a complex structure) that is not part of the $A$-model.
\begin{figure}
\begin{center}
\includegraphics[width=4in]{Timing.pdf}
\end{center}
\caption{\small (a) As a general statement in two-dimensional topological field theory, ${\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$ commutes with $\overline{\mathscr A}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},\overline{\mathscr B}_{\mathrm{cc}})$
in acting on ${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$, because elements $a\in{\mathscr A}$ and $\overline a\in \overline{\mathscr A}$ are inserted on opposite boundaries. Diffeomorphism
inariance does not allow any natural notion of which is inserted ``first.'' (b) In general, in two-dimensional topological field theory, ${\mathscr A}$ (and similarly
$\overline{\mathscr A}$) can be noncommutative, because elements $a,a'\in{\mathscr A}$ are inserted on the same boundary with a well-defined order, relative
to the boundary orientation. However, in the present context there are two additional dimensions, comprising the Riemann surface $C$, not drawn
in the two-dimensional picture. One can assume that $a$ and $a'$ have disjoint support in $C$. Hence they can be moved up and down past
each other without singularity and must commute. \label{sliding}}
\end{figure}
Next we would like to explain why ${\mathscr A}$ is commutative, like ${\mathscr A}_0$. This was originally proved for ${\mathrm{SU}}(2)$ by Hitchin \cite{H2} and for general simple $G$ by
Beilinson and Drinfeld \cite{BD}. We will give a four-dimensional explanation similar to that in \cite{KW}. Of course the same considerations apply to $\overline{\mathscr A}$.
As a warmup, we first explain why ${\mathscr A}$ commutes with $\overline{\mathscr A}$. This is clear from the fact (fig. \ref{sliding}(a)) that an element
$a\in {\mathscr A}$ is inserted on the left boundary of the strip, while an element $\overline a\in \overline{\mathscr A}$ is inserted on the right boundary. In two-dimensional
topological field theory, we are free to slide these insertions up and down along the boundary independently. There is no meaningful relative time-ordering
between the two boundary insertions and they must commute. By contrast, consider the insertion of a pair of elements $a,a'\in{\mathscr A}$ (fig \ref{sliding}(b)). Here, as
a general statement in two-dimensional topological field theory, there is a meaningful time-ordering between $a$ and $a'$.
If we try to slide one of them past the other in time, there may be a discontinuity when they cross, and therefore in general we may have $aa'\not=a'a$.
However, in the present problem, we are really not in two dimensions but in four dimensions; there are two
extra dimensions, making up the Riemann surface $C$, that are not shown in the figure. The definition of $H_{{\mathcal P},\alpha}$ in eqn. (\ref{juno}) depended only
on the cohomology class of $\alpha$ in $H^1(C,T_C^{s-1})$. We can choose a representative with support in an arbitrarily selected small open ball in $C$.
Therefore, when we consider a pair of elements $a,a'\in {\mathscr A}$, we can assume that they are represented by operators that
have disjoint support in $C$. Hence we can slide the two operators
up and down past each other in the two-dimensional picture of fig. \ref{sliding}(b) without any singularity. Accordingly, they commute.
We can elaborate slightly on the four-dimensional origin of the quantum Hitchin hamiltonians. The integrands ${\mathcal P}(\varphi)$ in the classical Hitchin Hamiltonians depend holomorphically on $C$. As we review in Section \ref{sec:chiral}, the quantum Hitchin Hamiltonians $D_{{\mathcal P},\alpha}$ can also be written as \begin{equation}\label{qjuno} D_{{\mathcal P},\alpha}=\int_C \alpha \,{\cal D}_{{\mathcal P}} \end{equation}
in terms of certain differential operators ${\cal D}_{{\mathcal P}}$ which act on the bundle locally at a point $p\in C$ and depend holomorphically on $p$.
We identify ${\cal D}_{{\mathcal P}}(p)$ as the action of a four-dimensional boundary local operator $O_{{\mathcal P}}(p)$. The holomorphic-topological nature of the boundary condition insures that such boundary local operators commute in the topological direction and have non-singular operator product expansion (OPE) with each other.
As was explained in Section \ref{aphiggs}, the quantum Hilbert space ${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$ is the space of
${\mathrm L}^2$ half-densities on ${\mathcal M}(G,C)$, or equivalently the space of ${\mathrm L}^2$ sections of $K^{1/2}\otimes \overline K{}^{1/2}$.
We recall that this happens because brane quantization of ${\mathcal M}_H(G,C)$ is equivalent to quantizing it as a cotangent bundle $T^*{\mathcal M}(G,C)$. In quantizing a cotangent bundle $T^*W$, a function whose restriction to a fiber of the cotangent bundle is a polynomial of degree $s$
becomes a differential operator of degree $s$ acting on half-densities on $W$. If, as in the case of interest here, $W$ is a complex manifold, then more
specifically holomorphic functions on $T^*W$ become holomorphic differential operators on $W$. From a holomorphic point of view, one usually says
that holomorphic functions on $T^*W$ (with polynomial dependence on the fibers) become holomorphic differential operators acting on sections of $K^{1/2}$.
(The role of $K^{1/2}$ is explained from a $\sigma$-model point of view in \cite{GW}, Section 3.2 and Appendix C.)
In the case of the Hitchin Hamiltonians, the fact that they can be quantum deformed to differential operators acting on sections of $K^{1/2}$, and not on
sections of any other holomorphic line bundle, is part of the standard story \cite{H2,BD}.
However, the antiholomorphic line bundle $\overline K{}^{1/2}$ is invisible
to a holomorphic differential operator, since its transition functions are antiholomorphic and commute with holomorphic differential operators. So the holomorphic
differential operators that are obtained by deformation quantization of holomorphic functions on $T^*W$ can naturally act on $K^{1/2}\otimes \overline K{}^{1/2}$,
or equivalently on the bundle ${\sf K}^{1/2}$ of half-densities. Similarly, under deformation quantization, antiholomorphic functions on $T^*W$ become
antiholomorphic differential operators that can act on ${\sf K}^{1/2}$.
So ${\mathscr A}$ and $\bar{\mathscr A}$ become, respectively, algebras of holomorphic and antiholomorphic differential operators acting on half-densities on ${\mathcal M}(G,C)$.
From this point of view, the statement that ${\mathscr A}$ and $\bar{\mathscr A}$ commute just reflects the fact that holomorphic differential operators commute with antiholomorphic ones.
\subsection{The Duals Of The Coisotropic Branes}\label{dualco}
In order to be able to apply duality to this problem, we need one more ingredient. We need
to understand the duals of the $A$-branes ${\mathscr B}_{\mathrm{cc}}$ and ${\bar {\mathscr B}}_{\mathrm{cc}}$ in the $B$-model of ${\mathcal M}_H(G^\vee,C)$. To be specific, here we mean
the $B$-model in the
complex structure that is called $J$ in \cite{H}, in which ${\mathcal M}_H(G^\vee,C)$ parametrizes flat bundles over $C$ with structure group $G^\vee_{\Bbb C}$. We denote
the connection on the flat bundle as ${\mathcal A}=A+{\mathrm i}\phi$, where $(A,\phi)$ are the unitary connection and Higgs field that appear in Hitchin's equations.
A general $B$-brane is a coherent
sheaf, or a complex of coherent sheaves, on ${\mathcal M}_H(G^\vee,C)$.
However, the $A$-branes ${\mathscr B}_{\mathrm{cc}}$ and ${\bar {\mathscr B}}_{\mathrm{cc}}$ have additional properties that imply that the dual $B$-branes must be rather special.
To explain this, we recall that the Higgs bundle moduli spaces are hyper-Kahler manifolds,
with complex structures $I,J,K$ that obey the usual quaternion relations, and a triple of corresponding Kahler forms $\omega_I,\omega_J,\omega_K$ and
complex symplectic forms $\Omega_I=\omega_J+{\mathrm i} \omega_K$, etc. Geometric Langlands duality in general maps the $A$-model of ${\mathcal M}_H(G,C)$
with symplectic structure $\omega_K$ to the $B$-model of ${\mathcal M}_H(G^\vee,C)$ in complex structure $J$. When we speak of the $A$-model or the $B$-model
without further detail, these are the models we mean.
A generic brane in either of these models is merely an $A$-brane or $B$-brane
of the appropriate type. However, many of the branes that are most important in geometric Langlands have additional properties. For example, a brane
supported on a point in ${\mathcal M}_H(G^\vee,C)$ is a brane of type $(B,B,B)$, that is, it is a $B$-brane in each of complex structures $I,J$, and $K$ (or any linear
combination). The dual of a brane of type $(B,B,B)$ is a brane of type $(B,A,A)$; in the case of the Higgs bundle moduli space,
the dual of a rank 1 brane supported at a point is a brane
supported on a fiber of the Hitchin fibration, with a rank 1 flat ${\mathrm{CP}}$ bundle. These branes are the Hecke eigensheaves which are central objects of
study in the geometric Langlands program; they
will be discussed in Section \ref{wkb}. In the case at hand,
${\mathscr B}_{\mathrm{cc}}$ and ${\bar {\mathscr B}}_{\mathrm{cc}}$ are branes of type $(A,B,A)$; they are $A$-branes of types $I$ and $K$, by virtue of the Kapustin-Orlov conditions for
coisotropic branes, and they are $B$-branes of type $J$, because the curvature $\pm\frac{1}{2}\omega_J$ of their ${\mathrm{CP}}$ bundles is of type $(1,1)$
in complex structure $J$, so that those bundles are holomorphic in complex structure $J$. In general, the dual of a brane of type
$(A,B,A)$ is a brane of type $(A,B,A)$, so the duals of ${\mathscr B}_{\mathrm{cc}}$ and ${\bar {\mathscr B}}_{\mathrm{cc}}$ will be branes of that type. The simplest kind of brane of type $(A,B,A)$
is given by the structure sheaf of a complex Lagrangian submanifold in complex structure $J$. In more physical language, these are branes supported
on a complex Lagrangian submanifold with a trivial ${\mathrm{CP}}$ bundle. And indeed, the duals of ${\mathscr B}_{\mathrm{cc}}$ and ${\bar {\mathscr B}}_{\mathrm{cc}}$ are of this type. These duals
were first identified (in a different formulation) by Beilinson and Drinfeld \cite{BD}, with the help of conformal field theory at critical level $k=-h^\vee$.
A gauge-theory explanation involves duality between the D3-NS5 and D3-D5 systems of string theory \cite{GWknots}.
The complex Lagrangian submanifold supporting the dual of ${\mathscr B}_{\mathrm{cc}}$ parametrizes flat $G^\vee_C$ bundles which satisfy a ``holomorphic oper''
condition. We will denote it as $L_{\mathrm{op}}$. Similarly, the dual of ${\bar {\mathscr B}}_{\mathrm{cc}}$ is supported on a complex Lagrangian submanifold $L_{\overline{\mathrm{op}}}$ that parametrizes
flat $G^\vee_C$ bundles which satisfy an ``antiholomorphic oper'' condition.
The holomorphic oper condition can be stated rather economically as a global constraint on the holomorphic type of the bundle, i.e. on the $(0,1)$ part of ${\mathcal A}$,
as we will do here, or in a more local way, as we will do in Section \ref{localoper}. Both formulations are standard mathematically.
In a four-dimensional gauge theory, $S$-duality maps a deformed Neumann boundary condition to a deformed ``Nahm pole'' boundary condition,
which imposes directly the local constraints \cite{GWknots}.
One general way to define a complex Lagrangian submanifold of ${\mathcal M}_H(G^\vee,C)$ is to consider all flat
$G^\vee_C$ bundles $E^\vee\to C$ with some fixed holomorphic type. Specifying the holomorphic type of a bundle is
equivalent to specifying the $(0,1)$ part of ${\mathcal A}$. Here and in several
analogous cases considered momentarily, we place no constraint
on the $(1,0)$ part of the connection except that the full connection should be flat. While preserving the flatness of the connection on $E^\vee$,
we are free to add to the $(1,0)$ part of the connection an arbitrary $\overline\partial_{\mathcal A}$-closed form representing an
element of $H^0(C,K_C\otimes {\mathrm{ad}}(E^\vee))$. The dimension of this space is half the dimension of ${\mathcal M}_H(G^\vee,C)$, so flat $G^\vee_{\Bbb C}$
bundles
of a specified holomorphic type are a middle-dimensional submanifold $L$ of ${\mathcal M}_H(G^\vee,C)$. $L$ is a complex Lagrangian submanifold, because
the holomorphic symplectic structure $\Omega_J$ of ${\mathcal M}_H(G^\vee,C)$ in complex structure $J$ is a pairing between the $(1,0)$ and $(0,1)$ parts of ${\mathcal A}$
and vanishes if the $(0,1)$ part is specified. Once one picks a base point in $L$, $L$ is isomorphic to the vector space $H^0(C,K_C\otimes {\mathrm{ad}}(E^\vee))$.
We can define a second family of complex Lagrangian submanifolds, in the same complex structure on ${\mathcal M}_H(G^\vee,C)$, by specifying the antiholomorphic
structure of a flat $G^\vee_{\Bbb C}$ bundle. This amounts to specifying the $(1,0)$ part of ${\mathcal A}$, and letting the $(0,1)$ part vary. It leads to a complex
Lagrangian submanifold for the same reasons as before. It may come as a slight surprise that fixing either the $(1,0)$ or the $(0,1)$ part of ${\mathcal A}$ is
a holomorphic condition in complex structure $J$. Indeed, complex structure $J$ on the Higgs bundle moduli space is not sensitive to the complex
structure of the two-manifold $C$, and treats the $(1,0)$ and $(0,1)$ parts of ${\mathcal A}$ in a completely symmetric way.
The submanifolds $L_{\mathrm{op}}$ and $L_{\overline{\mathrm{op}}}$ can be defined by specifying a particular choice of the holomorphic or antiholomorphic structure of a flat $E^\vee$ bundle.
First we explain the definition for the case $G_{\Bbb C}={\mathrm{SL}}(2,{\Bbb C})$. For this group, an oper is a flat bundle $E^\vee\to C$ that, as a holomorphic bundle,
is a nontrivial extension of $K_C^{-1/2}$ by $K_C^{1/2}$:
\begin{equation}\label{nonex}0\to K_C^{1/2}\to E^\vee\to K_C^{-1/2}\to 0. \end{equation}
There is a unique such nontrivial extension, up to isomorphism. The family of flat bundles of this holomorphic type
is therefore a complex Lagrangian submanifold
that we will call $L_{\mathrm{op}}$. In making this definition, we have made a choice of $K_C^{1/2}$, or equivalently a choice of spin structure on $C$.
Indeed, for ${\mathrm{SL}}(2,{\Bbb C})$, the definition of an oper depends on such a choice of spin structure (though we do not indicate this in the notation for $L_{\mathrm{op}}$).
We return to this point in Section \ref{toposubt}.
It is possible to give a simple description of $L_{\mathrm{op}}$ once one picks a base point, that is, a particular ${\mathrm{SL}}(2,{\Bbb C})$ bundle $E^\vee$
of oper type with flat connection ${\mathcal A}_0$. In deforming $E^\vee$ as an oper, we may as well keep the $(0,1)$ part of ${\mathcal A}_0$ fixed, since we are required to
keep it fixed up to a complex gauge transformation. But we can modify the $(1,0)$ part of ${\mathcal A}_0$. To do this while preserving the flatness of ${\mathcal A}_0$,
we add to ${\mathcal A}_0$ a $\overline\partial_{{\mathcal A}_0}$-closed $(1,0)$-form, that is, an element of $H^0(C,K_C\otimes {\mathrm{ad}}(E^\vee))$. Using the exact sequence
(\ref{nonex}), one can show that $H^0(C,K_C\otimes {\mathrm{ad}}(E^\vee))$ is isomorphic to the space of quadratic differentials on $C$. This is the base of the Hitchin
fibration for ${\mathrm{SL}}(2,{\Bbb C})$, so $L_{\mathrm{op}}$ is isomorphic
to the base of the Hitchin fibration. This isomorphism is not entirely canonical as it depends on the choice of a base point in $L_{\mathrm{op}}$. A similar
reasoning applies for other groups.
Similarly, an antiholomorphic oper for $G_{\Bbb C}={\mathrm{SL}}(2,{\Bbb C})$, or anti-oper for short, is a flat bundle $E^\vee$ that, antiholomorphically, is a nontrivial extension of
$\overline K_C^{-1/2}$ by $\overline K_C^{1/2}$:
\begin{equation}\label{onex} 0\to\overline K_C^{1/2}\to E^\vee \to \overline K_C^{-1/2}\to 0. \end{equation}
The family of such flat bundles is another complex Lagrangian submanifold, which we will call $L_{\overline{\mathrm{op}}}$. It is noncanonically isomorphic to the base
of the Hitchin fibration, with the opposite complex structure.
In general, if $G_{\Bbb C}^\vee$ is a simple complex Lie group, there is a notion of a ``principal embedding'' of Lie algebras ${{\mathfrak {su}}}(2)\to {{\mathfrak g}^\vee}$. For example, if $G^\vee={\mathrm{SL}}(n,{\Bbb C})$,
the principal embedding is such that the $n$-dimensional representation of ${\mathfrak g}^\vee$ transforms irreducibly under ${\mathfrak {su}}(2)$; the corresponding principal
subgroup is a copy of ${\mathrm{SL}}(2,{\Bbb C})$ or ${\mathrm{SO}}(3,{\Bbb C})$ depending on whether $n$ is even or odd. For brevity we will sometimes ignore this subtlety and
refer simply to a principal
${\mathrm{SL}}(2,{\Bbb C})$ subgroup, though the global form of the group is sometimes ${\mathrm{SO}}(3,{\Bbb C})$.
A $G_{\Bbb C}^\vee$
oper is a flat $G_{\Bbb C}^\vee$ bundle $E_{\Bbb C}^\vee$ that, as a holomorphic bundle, is equivalent to a principal embedding of an ${\mathrm{SL}}(2,{\Bbb C})$ oper bundle,
that is, a principal embedding of a rank two bundle that is a nontrivial extension
of the form in eqn. (\ref{nonex}). For $G^\vee_{\Bbb C}={\mathrm{SL}}(n,{\Bbb C})$, this means that an oper bundle is, holomorphically,
the $(n-1)^{th}$ symmetric tensor power of such a nontrivial extension, and therefore
has a subbundle isomorphic to $K_C^{(n-1)/2}$:
\begin{equation}\label{nex}0\to K_C^{(n-1)/2}\to E^\vee\to \cdots \end{equation}
(and a filtration by powers of $K_C$).
Again,
the $(1,0)$ part of the connection on $E_{\Bbb C}^\vee$ is not restricted except by requiring the full connection to be flat. Similarly an antiholomorphic $G_{\Bbb C}^\vee$ oper
is a flat $G_{\Bbb C}^\vee$ bundle that, as an antiholomorphic bundle, is equivalent to a principal embedding of an antiholomorphic ${\mathrm{SL}}(2,{\Bbb C})$ oper bundle.
At the $\sigma$-model level, the duals of ${\mathscr B}_{\mathrm{cc}}$ and ${\bar {\mathscr B}}_{\mathrm{cc}}$ are the structure sheaves of $L_{\mathrm{op}}$ and $L_{\overline{\mathrm{op}}}$; that is, they are
rank 1 branes ${\mathscr B}_{\mathrm{op}}$ and ${\mathscr B}_{\overline{\mathrm{op}}}$ supported on $L_{\mathrm{op}}$ and $L_{\overline{\mathrm{op}}}$ with trivial ${\mathrm{CP}}$ bundles.
\subsection{The Local Constraints}\label{localoper}
To describe a more local characterization of an oper, we consider first the case $G^\vee_{\Bbb C}={\mathrm{SL}}(2,{\Bbb C})$. The extension structure of $E^\vee$ implies the existence of a
global holomorphic section $s$ of $E^\vee \otimes K_C^{-1/2}$. Denote as $D$ the $(1,0)$ part of the connection.
The ${\mathrm{SL}}(2,{\Bbb C})$-invariant combination $s \wedge D s$ is a global holomorphic function on $C$. This function must be non-zero:
if it vanished, we could write $D s= a s$ and $a$ would define a holomorphic flat connection on $K_C^{-1/2}$, which does not exist (for $C$ of genus greater than 1 or
in lower genus in the presence of parabolic structure).
Without loss of generality, we can normalize $s$ so that $s \wedge D s=1$. This fixes $s$ up to sign.
The local version of the holomorphic oper condition for $G^\vee_{\Bbb C}={\mathrm{SL}}(2,{\Bbb C})$ is precisely the condition that $E^\vee \otimes K_C^{-1/2}$ admits a
globally defined holomorphic section such that $s \wedge D s=1$.
Taking a derivative, we have $s \wedge D^2 s =0$ and thus $s$ satisfies a second order differential equation
\begin{equation}\label{stresst}
D^2 s + t s=0
\end{equation}
for some ``classical stress tensor'' $t$ on $C$.
Under a change of local coordinate on $C$, $t$ transforms as a stress tensor, not as a quadratic differential. Not coincidentally, eqn. (\ref{stresst})
can be viewed as a classical limit of the
Belavin-Polyakov-Zamolodchikov (BPZ) differential equations for the correlator of a degenerate field in two-dimensional
conformal field theory. The classical stress tensor can be used to define a set of generators of the algebra of holomorphic
functions on the oper manifold, consisting of functions of the form
\begin{equation}\label{opjuno2} f_{t,\alpha}=\int_C \alpha \,t ,\end{equation}
with $\alpha$ being a $(0,1)$-form with values in $T_C$.
The case of $G^\vee_{\Bbb C}={\mathrm{SL}}(n,{\Bbb C})$ can be analyzed similarly. In this case, the oper structure of $E^\vee$ (eqn. (\ref{nex})) implies the existence of a
global holomorphic section $s$ of $E^\vee \otimes K_C^{(1-n)/2}$. Then $s \wedge D s \wedge \cdots D^{n-1} s$ is a global holomorphic function on $C$ which cannot vanish, for a similar reason to what we explained for $n=2$. We can normalize $s$ so that $s \wedge D s \wedge \cdots D^{n-1} s=1$; this uniquely fixes $s$,
up to the possibility of multiplying by an $n^{th}$ root of 1. Since $0=D(s \wedge D s \wedge \cdots D^{n-1} s)=s\wedge Ds \wedge \cdots \wedge D^{n-2}s\wedge D^ns$,
we learn that $s$ satisfies a degree $n$ differential equation
\begin{equation}
D^n s + t_2 D^{n-2} s+ \cdots + t_n s =0.
\end{equation}
We can define a set of generators of the algebra of holomorphic functions on the oper manifold, consisting of functions of the form
\begin{equation}\label{opjunon} f_{t_k,\alpha}=\int_C \alpha \,t_k \end{equation}
with $\alpha$ being a $(0,1)$-form with values in $T_C^{k-1}$, $k=2,\cdots,n$.
For general $G^\vee_{\Bbb C}$, there is no distinguished representation as convenient as the $n$-dimensional representation of ${\mathrm{SL}}(n,{\Bbb C})$. However,
given an oper bundle $E^\vee$,
we can consider associated bundles $E^\vee_R$ in any irreducible representation $R$ of $G^\vee_{\Bbb C}$.
By the definition of an oper, the structure group of $E^\vee_R$ as a holomorphic bundle reduces to a rank 1 subgroup $H_{\Bbb C}\subset G^\vee_{\Bbb C}$;
this subgroup is a copy of either ${\mathrm{SL}}(2,{\Bbb C})$ or ${\mathrm{SO}}(3,{\Bbb C})$, depending on $G^\vee_{\Bbb C}$ and $R$. Let $R_n$ be the $n$-dimensional irreducible representation of $H_{\Bbb C}$ ($n$ is any positive integer
or any odd positive integer for ${\mathrm{SL}}(2,{\Bbb C})$ or ${\mathrm{SO}}(3,{\Bbb C})$, respectively). As a representation of $H_{\Bbb C}$, we have $R\cong \oplus_{n=0}^\infty Q_n\otimes R_n$, where $Q_n$ are some vector
spaces, almost all of which vanish. Actually, if $N$ is the largest integer for which $Q_n$ is nonzero, then $Q_N$ is 1-dimensional and we can replace $Q_N\otimes
R_N$ with $R_N$. So $R\cong R_N\oplus_{n=0}^{N-1}Q_n\otimes R_n$. In this decomposition, a highest weight vector of $R_N$ with respect to a Borel
subgroup $B_H$ of $H$ is a highest weight vector of $G^\vee$ with respect to the Borel subgroup $B_{G^\vee}$ of $G^\vee$ that contains $B_H$.
The associated bundle $E^\vee_R$
has a similar decomposition as holomorphic bundle
\begin{equation}\label{regasso} E^\vee_{R}= E^\vee_{R,N}\oplus \left(\oplus_{n=1}^{N-1} Q_n\otimes E^\vee_{R_n}\right), \end{equation}
where $E^\vee_{R_n}$ is the holomorphic bundle associated to an $H_{\Bbb C}$ oper in the $n$-dimensional representation.
For each $n$ we get from eqn. (\ref{nex}) a canonical image $s_{R,n}$ of the vector space $Q_n$ into the space of global holomorphic sections
of $E^\vee_{R} \otimes K_C^{(1-n)/2}$, or equivalently a holomorphic map
\begin{equation}\label{holmap} s_{R,n}: Q_n \otimes K_C^{(n-1)/2} \to E^\vee_{R}. \end{equation}
Of particular importance here is the ``highest weight'' object $s_{R,N}$, which we will just denote as $s_R$:
\begin{equation}\label{linmap}s_R:K_C^{(N-1)/2}\to E^\vee_R. \end{equation}
Here $N$ is defined by saying that a highest weight vector of the representation $R$ (for some Borel subgroup $B$) is in an $N$-dimensional
representation of a principal ${\mathrm{SL}}(2,{\Bbb C})$ subgroup (which has a Borel subgroup contained in $B$). For $G^\vee={\mathrm{SL}}(n,{\Bbb C})$ and $R$ the $n$-dimensional
representation or its dual, $N=n$.
A partial analogue to the $s \wedge D s=1$ condition is the condition that the collection $D^m s_{R,n}$ for $m< n\leq N$ should
span $E^\vee_{R}$ at each point of $C$. The derivatives $D^n s_{R,n}$ can then be expanded out in terms of the $D^m s_{R,n}$ with $m<n$, giving rise to an intricate collection of differential equations. (See Appendix \ref{somex} for some examples.) The coefficients of the differential equations
are holomorphic functions on the oper manifold and can be expressed as polynomials in derivatives of a collection of observables $T_{\cal P}$
which have the same labels as the integrands for the quantum Hitchin Hamiltonians for $G^\vee_{\Bbb C}$. Tensor products of the form $D^m s_{R,n} \otimes D^{m'} s_{R',n'}$
can also be expanded in the basis of $D^{m''} s_{R'',n''}$ for all $R''$ that appear in the decomposition of the tensor product $R \otimes R'$.
Coefficients in this expansion
are also holomorphic functions on the oper manifold and can be expressed as polynomials in derivatives of a collection of observables $T_{\cal P}$.
The collection of observables $T_{\cal P}$ is sometimes called the classical $W$-algebra for $G^\vee_{\Bbb C}$. In that language, the differential equations satisfied by the
$s_{R,n}$ are a classical analogue of the BPZ equations,
and the tensor product expansion is analogous to the operator product expansion (OPE) of degenerate fields.
In four-dimensional gauge theory, the deformed Neumann boundary condition is $S$-dual to the deformed Nahm pole boundary condition,
which is also holomorphic-topological. This boundary condition involves a certain prescribed singularity for the gauge theory fields at the boundary.
Effectively, the singular boundary conditions of the physical theory impose an oper boundary condition in the topologically twisted theory.
The gauge-invariant information contained in the subleading behaviour of the fields is captured by boundary local operators which match the $T_{\cal P}$
observables and are $S$-dual to the corresponding local operators at the deformed Neumann boundary condition which are employed in
the definition of the quantum Hitchin Hamiltonians. The $s_{R,n}$ also appear naturally in gauge theory, as we will illustrate in Section \ref{wilop}.
Finally, in order to gain further intuition on the various relations satisfied by the $s_{R,n}$, we
note that the oper manifold has a simpler ``classical'' cousin given by Hitchin's section of the Hitchin fibration.
For ${\mathrm{SL}}(2,{\Bbb C})$, the Hitchin section parametrizes Higgs bundles $(E,\varphi)$ such that $E$ is a direct sum $K_C^{1/2} \oplus K_C^{-1/2}$.
In other words, the Hitchin section is what we get if we work in complex structure $I$ (rather than $J$) and ask for the extension in eqn. (\ref{nonex}) to be split
(as opposed to a non-split extension, leading to an oper bundle). For any $G^\vee_{\Bbb C}$, the Hitchin section parametrizes pairs $(E,\varphi)$ such that $E$
is induced by the principal embedding of $K_C^{1/2} \oplus K_C^{-1/2}$.
For the Hitchin section, one can deduce local conditions analogous to what we have explained
for opers, but using the
Higgs field $\varphi$ instead of the holomorphic derivative $D$.
For example, the ${\mathrm{SL}}(2,{\Bbb C})$ Hitchin section of Higgs bundle moduli space is characterized locally by the existence of a holomorphic section $s$ of $E$
which satisfies $s \wedge \varphi s=1$ along with $\varphi^2 s = \frac12 {\rm Tr}\, \varphi^2 s$ (the latter equation holds simply because $\varphi^2=\frac12 {\rm Tr}\, \varphi^2$
for ${\mathrm{SL}}(2,{\Bbb C})$).
Comparing to the oper case, $D$ is replaced by $\varphi$ and the stress tensor $t$ is replaced by the quadratic differential
$\frac12{\rm Tr}\,\varphi^2$.
In general, the associated bundle $E^\vee_R$
in a representation $R$ of $G^\vee$ will have sections $s_{R,n}$, $n\leq N$, such that $\varphi^m s_{R,n}$ for $m<n$
span $E^\vee_{R}$. Hence $\varphi^n s_{R,n}$ can be expanded in terms of $\varphi^m s_{R,n}$ for $m<n$; likewise
for two representations $R$, $R'$, $\varphi^m s_{R,n} \otimes \varphi^{m'} s_{R',n'}$
can be expanded out in terms of $\varphi^{m''} s_{R'',n''}$ for all $R''$ in $R \otimes R'$, with coefficients built from the
gauge-invariant polynomials ${\mathcal P}(\varphi)$. The oper relations are a deformation of these. With an extension of this analysis, one can recover the assertion of section
\ref{dualco} that $L_{\mathrm{op}}$ is noncanonically isomorphic to the base of the Hitchin fibration.
\subsection{Some Topological Subtleties}\label{toposubt}
The definition of a $G_{\Bbb C}$ oper depends on the choice of a spin structure on $C$ if the image of the principal embedding is an ${\mathrm{SL}}(2,{\Bbb C})$ subgroup of $G_{\Bbb C}^\vee$,
but not if it is an ${\mathrm{SO}}(3,{\Bbb C})$ subgroup. (For example, for $G_{\Bbb C}^\vee={\mathrm{SL}}(n,{\Bbb C})$, the notion of an oper depends on a choice of spin structure precisely if
$n$ is even.) This dependence on spin structure for some groups is in tension with the claim that ${\mathscr B}_{\mathrm{op}}$ is the dual of ${\mathscr B}_{\mathrm{cc}}$, since the definition of ${\mathscr B}_{\mathrm{cc}}$
did not seem to depend on
a choice of spin structure. The resolution of this point was essentially described in Section 8 of \cite{FG}. We will explain the details for groups of
rank 1. In the context of the twisted version of ${\mathcal N}=4$ super
Yang-Mills theory that is relevant to geometric Langlands, the electric-magnetic dual of ${\mathrm{SO}}(3)$ gauge theory is not standard ${\mathrm{SU}}(2)$ gauge theory,
but what is sometimes called ${\mathrm{Spin}}\cdot{\mathrm{SU}}(2)$ gauge theory. For any $d>0$, the group ${\mathrm{Spin}}(d)\cdot{\mathrm{SU}}(2)$ is a double cover of ${\mathrm{SO}}(d)\times {\mathrm{SO}}(3)$
that restricts to a nontrivial double cover of either factor. In particular, ${\mathrm{Spin}}(d)\cdot{\mathrm{SU}}(2)$ has projections to ${\mathrm{SO}}(d)$ and to ${\mathrm{SO}}(3)$:
\begin{equation}\label{dobl} \begin{matrix}& & {\mathrm{Spin}}(d)\cdot{\mathrm{SU}}(2) && \cr
&\swarrow &&\searrow &\cr
{\mathrm{SO}}(d) &&&& {\mathrm{SO}}(3). \end{matrix}\end{equation}
By a ${\mathrm{Spin}}(d)\cdot {\mathrm{SU}}(2)$ structure on a $d$-manifold $M$, we mean a principal bundle over $ M$ with that structure group such that the projection to the first
factor gives the frame bundle of $M$ (the principal bundle associated to the tangent bundle $TM$ of $M$). Likewise a ${\mathrm{Spin}}(d)\cdot{\mathrm{SU}}(2)$ connection
on a Riemannian manifold $M$ is a connection with that structure group that when restricted to the first factor is the Levi-Civita connection of the tangent bundle of $M$.
Assuming that $M$ is spin, a down-to-earth
description of a ${\mathrm{Spin}}(d)\cdot{\mathrm{SU}}(2)$ structure on $M$ is as follows: once a spin structure is picked on $M$, a ${\mathrm{Spin}}(d)\cdot {\mathrm{SU}}(2)$ bundle is equivalent
to an ${\mathrm{SU}}(2)$ bundle $E^\vee\to M$; if the spin structure of $M$ is twisted by a line bundle $\ell$ such that $\ell^2$ is trivial, then $E^\vee$ is replaced by $E^\vee\otimes \ell$. With this
characterization, it is evident that although the variety $L_{\mathrm{op}}$ of opers is not canonically defined in ${\mathrm{SU}}(2)$ gauge theory, it is canonically defined in
${\mathrm{Spin}}(4)\cdot {\mathrm{SU}}(2)$ gauge theory. Indeed, bearing in mind that $\ell\cong\ell^{-1}$, if $E^\vee$ appears in the exact sequence defining an oper
with some choice of $K^{1/2}$, then $E^\vee\otimes \ell$ appears in a similar exact sequence with $K^{1/2}$ replaced by $K^{1/2}\otimes \ell$.
What is the dual of ${\mathrm{SU}}(2)$ gauge theory, as opposed to ${\mathrm{Spin}}(4)\cdot {\mathrm{SU}}(2)$ gauge theory?
The answer \cite{FG} is that the dual is ${\mathrm{SO}}(3)$ gauge theory, but with an extra factor $\Delta=(-1)^{\int_M w_2(M) w_2(E)}$ included in the definition of the path
integral. Here $w_2(M)$ and $w_2(E)$ are respectively the second Stieffel-Whitney classes of $TM$ and of an ${\mathrm{SO}}(3)$ bundle $E\to M$.
If the $B$-model description is by ${\mathrm{SU}}(2)$ gauge theory, and therefore the definition of $L_{\mathrm{op}}$ requires a spin structure on $C$, then the $A$-model
description is by ${\mathrm{SO}}(3)$ gauge theory with the additional factor $\Delta$, and this must ensure that the definition of ${\mathscr B}_{\mathrm{cc}}$ similarly requires a choice of
spin structure on $C$. That happens as follows.
If $M=\Sigma\times C$ where $\Sigma$ and $C$ are oriented two-manifolds without boundary, then $w_2(M)=0$ and $\Delta=1$, so the factor of $\Delta$ in
the path integral
has no consequence. But suppose $M$ has a boundary $\partial M$ with ${\mathscr B}_{\mathrm{cc}}$ boundary conditions. To define the topological invariant $\int_M w_2(M) w_2(E)$,
one needs a trivialization of the class $w_2(M) w_2(E)$ along $\partial M$. ${\mathscr B}_{\mathrm{cc}}$ boundary conditions are a version of free boundary conditions
for the gauge field, so with ${\mathscr B}_{\mathrm{cc}}$ boundary conditions, there is no restriction on $w_2(E)$ along $\partial M$. But we can trivialize $w_2(M)w_2(E)$
along $\partial M$ by trivializing $w_2(M)$, that is, by picking a spin structure along $\partial M$. In our application, $\partial M$ is the product of a Riemann
surface $C$ with a contractible one-manifold (the boundary of the strip) and what is needed is a spin structure on $C$. Thus the $A$-model
dual of ${\mathrm{SU}}(2)$ gauge theory is an ${\mathrm{SO}}(3)$ gauge theory in which, despite appearances, the definition of ${\mathscr B}_{\mathrm{cc}}$ (or similarly $\overline{\mathscr B}_{\mathrm{cc}}$) requires a choice of spin structure on $C$.
These remarks
have analogs for all groups such that the definition of an oper requires a choice of spin structure.
They do not have analogs in the usual physics of ${\mathcal N}=4$ super Yang-Mills theory because they only come into play after topological twisting. Untwisted
${\mathcal N}=4$ super Yang-Mills theory has fermion fields whose definition requires a spin structure on $M$. When a spin structure is present, the difference between
${\mathrm{Spin}}(4)\cdot {\mathrm{SU}}(2)$ gauge theory and ${\mathrm{SU}}(2)$ gauge theory disappears. Likewise, the choice of a spin structure trivializes $\Delta$.
\subsection{Topological Aspects of the Oper Boundary Condition In Gauge Theory}\label{opergauge}
When the center ${\mathcal Z}(G^\vee)$ of $G^\vee$ is nontrivial, the description of the dual of ${\mathscr B}_{\mathrm{cc}}$ as the brane ${\mathscr B}_{\mathrm{op}}$
(and the analogous
statement for $\overline{\mathscr B}_{\mathrm{cc}}$) needs a slight refinement. ${\mathscr B}_{\mathrm{op}}$ is the dual of ${\mathscr B}_{\mathrm{cc}}$ in the $\sigma$-model of ${\mathcal M}_H(G^\vee,C)$, but we should recall
that the low energy description also contains a ${\mathcal Z}(G^\vee)$ gauge field. Along a boundary labeled by ${\mathscr B}_{\mathrm{op}}$, the ${\mathcal Z}(G^\vee)$ gauge field is trivialized.
As explained momentarily, this
condition ensures that when quantized on $I\times C$, where $I$ is an interval with ${\mathscr B}_{\mathrm{op}}$ and ${\mathscr B}_{\overline{\mathrm{op}}}$ boundary conditions, the theory supports
a discrete electric charge along $I$.
Following the logic of Section 7 of \cite{KW}, this condition is dual to the fact that on the $A$-model side, a $G$-bundle over $C$ has a characteristic
class $\zeta \in H^2(C,\pi_1(G))$ and is classified by $\int_C\zeta$ (for $G={\mathrm{SO}}(3)$, $\zeta$ is the second Stieffel-Whitney class $w_2(E)$).
The Nahm pole boundary condition, which is a gauge theory version of ${\mathscr B}_{\mathrm{op}}$ \cite{GWknots}, reduces the gauge group along the boundary
from $G^\vee$ to its center
${\mathcal Z}(G^\vee)$. The trivialization of the center along the boundary is an additional condition.
For $G^\vee={\mathrm{SL}}(n,{\Bbb C})$, the oper condition, or the Nahm pole boundary condition,
ensures the existence of the object $s$ that was introduced in Section \ref{localoper}, and
was normalized there, up to an $n^{th}$ root of 1, by the condition condition $s\wedge Ds \wedge \cdots \wedge D^{n-1}s=1$.
An $n^{th}$ root of 1 is an element of ${\mathcal Z}(G^\vee)$, so a convenient way to express the fact that the ${\mathcal Z}(G^\vee)$ gauge invariance
is trivialized along the boundary is to say that the boundary is equipped with a particular choice of normalized section $s$.
When we quantize the theory on a strip ${\mathbb{R}}\times I$ (times the Riemann surface $C$), with oper and anti-oper boundary conditions, we have
such trivializations $s_\ell$ and $s_r$ at the left and right boundaries of the strip. We are free to make a global gauge transformation by an element $b$
of ${\mathcal Z}(G^\vee)$. This acts on the pair of trivializations by $(s_\ell,s_r)\to (b s_\ell, b s_r)$, so pairs differing in that way should be considered equivalent.
However, ${\mathcal Z}(G^\vee)$ acts on the
equivalence classes of pairs $(s_\ell,s_r)$ by
$(s_\ell,s_r)\to (b s_\ell,s_r)$, $b\in {\mathcal Z}(G^\vee)$, and this leads to an action of ${\mathcal Z}(G^\vee)$ on the physical Hilbert space. This action of ${\mathcal Z}(G^\vee)$ on ${\mathcal H}$
in the $B$-model description is dual to the fact that, on the $A$-model side, ${\mathcal H}$ is graded by $\int_C \zeta$.
For any $G^\vee$, the trivialization of the ${\mathcal Z}(G^\vee)$ gauge field on the boundary can be expressed in terms of the objects $s_{R,n}$ that were introduced
in Section \ref{localoper}, but this is less simple than for ${\mathrm{SL}}(n,{\Bbb C})$.
\section{The Eigenvalues Of The Hitchin Hamiltonians}\label{hitchval}
\subsection{The Case That The Center Is Trivial}\label{centertriv}
Now we can start to deduce interesting consequences of electric-magnetic duality.
Once one identifies the $B$-model dual of ${\mathscr B}_{\mathrm{cc}}$ as ${\mathscr B}_{\mathrm{op}}$,
as we have done in Section \ref{dualco}, one immediately has a dual description of ${\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$:
it is ${\mathrm{Hom}}({\mathscr B}_{\mathrm{op}},{\mathscr B}_{\mathrm{op}})$. Since ${\mathscr B}_{\mathrm{op}}$ is a rank 1 Lagrangian brane supported on $L_{\mathrm{op}}$, ${\mathrm{Hom}}({\mathscr B}_{\mathrm{op}},{\mathscr B}_{\mathrm{op}})$ is just the sum of the $\overline\partial_{\mathcal A}$ cohomology groups $H^i(L_{\mathrm{op}}, {\mathcal O})$.
In Section \ref{dualco}, we learned that $L_{\mathrm{op}}$ is noncanonically
isomorphic to a vector space. Hence the cohomology $H^i(L_{\mathrm{op}},{\mathcal O})$ vanishes for $i>0$, and ${\mathrm{Hom}}(L_{\mathrm{op}},L_{\mathrm{op}})$
is simply the (undeformed!) commutative algebra of holomorphic functions on $L_{\mathrm{op}}$. Thus duality with the $B$-model
gives another explanation that ${\mathscr A}$ must be commutative.
Moreover it shows that the ``spectrum'' of the algebra ${\mathscr A}$, in the abstract sense of the space of its 1-dimensional complex representations, is the
``variety''\footnote{For our purposes, ``variety'' is just a synonym for ``complex manifold.''} $L_{\mathrm{op}}$ that parametrizes opers, as originally shown in \cite{BD}. In the language of previous sections, this is a canonical identification between differential operators $D_{\mathcal P}$ and functions $T_{\mathcal P}$.
Precisely the same argument shows that $\overline{\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{op}},{\mathscr B}_{\mathrm{op}})$ is the algebra of holomorphic functions on the variety $L_{\overline{\mathrm{op}}}$
of antiholomorphic opers.
The variety of opers is noncanonically isomorphic to the base of the Hitchin fibration, as explained in Section \ref{dualco}.
So the fact that ${\mathscr A}$ is the algebra of holomorphic functions on $L_{\mathrm{op}}$ is a sort of quantum deformation of the
fact that ${\mathscr A}_0$ is the algebra of holomorphic functions on the base of the Hitchin fibration. A similar statement holds for $\overline{\mathscr A}$, of course.
However, we want to understand the spectrum of ${\mathscr A}\times \bar{\mathscr A}$ not in the abstract sense already indicated but as concrete operators on
${\mathcal H}={\mathrm{Hom}}({\bar {\mathscr B}}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$. The dual theory gives a dual description by ${\mathcal H}={\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_{\mathrm{op}})$. If the center of $G^\vee$ is trivial,
this can be analyzed just in a $\sigma$-model (rather than a $\sigma$-model with ${\mathcal Z}(G^\vee)$ gauge fields). Let us consider this case first.
Matters are simple because
the branes involved are rank 1 Lagrangian branes, supported on the complex Lagrangian manifolds $L_{\mathrm{op}}$ and $L_{\overline{\mathrm{op}}}$. In analyzing the problem,
we will assume that $L_{\mathrm{op}}$ and $L_{\overline{\mathrm{op}}}$ have only transverse intersections at isolated points. This is known to
be true for ${\mathrm{SL}}(2,{\Bbb C})$ and in general is one of the conjectures of Etingof, Frenkel, and Kazhdan \cite{EFK,EFK2}. For the intersections to be isolated
and transverse is actually a prediction of the duality; it is needed in order for the hermitian form on ${\mathcal H}$ to be positive-definite, as expected from the $A$-model
construction in which ${\mathcal H}$ is a Hilbert space of ${\mathrm L}^2$ half-densities. Unfortunately, to explain this requires a fairly detailed discussion of $B$-model quantum
mechanics, which has been relegated to Appendix \ref{bmodel}. (In this appendix, we learn that there is actually a further, unproved necessary condition for positivity.)
Let $\Upsilon=L_{\mathrm{op}}\cap L_{\overline{\mathrm{op}}}$. Assuming that the intersection points are isolated and transverse,
${\mathcal H}={\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_{\mathrm{op}})$ simply has a basis with one
basis vector $\psi_u$ for every $u\in \Upsilon$. That is a general statement about intersections of Lagrangian branes in the $B$-model. Concretely,
since $L_{\mathrm{op}}$ is the subvariety of ${\mathcal M}_H(G,C)$ that parametrizes flat bundles that
are holomorphic opers, and $L_{\overline{\mathrm{op}}}$ is the subvariety that parametrizes flat bundles that are antiholomorphic opers, it follows that
an intersection point represents a flat $G_{\Bbb C}^\vee$ bundle that is an oper both holomorphically and antiholomorphically.
We recall that the definition of a hermitian form on ${\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_{\mathrm{op}})$ makes use of an antiholomorphic involution $\tau$ that acts by $(A,\phi)\to
(A,-\phi)$. Hence $\tau$ transforms a complex flat connection ${\mathcal A}=A+{\mathrm i}\phi$ to the complex conjugate flat connection $\overline{\mathcal A}=A-{\mathrm i}\phi$. Recall that $A$ is
a gauge field in a theory in which the gauge group is the compact form $G^\vee$. Mathematically, the involution of $G^\vee_{\Bbb C}$ that leaves fixed $G^\vee$ is called the
Chevalley involution, so $\tau$ acts on ${\mathcal A}$ via the Chevalley involution (up to an inner automorphism, which here means a $G^\vee$-valued gauge transformation).
In the folded construction of the state space ${\mathcal H}$, $\tau$ acts antiholomorphically on $\widehat Y=Y_1\times Y_2$ by exchanging the two factors. That means
that in the unfolded construction, $\tau$ exchanges the two ends of the strip of fig. \ref{folding}(b). It is not difficult to see explicitly why this happens.
If ${\mathcal A}$ is a complex flat connection that is a holomorphic oper, then $\overline{\mathcal A}$ is a complex flat connection
that is an antiholomorphic oper, and similarly, if ${\mathcal A}$ is an antiholomorphic oper, then $\overline{\mathcal A}$ is a holomorphic one. Thus
$\tau$ exchanges $L_{\mathrm{op}}$ with $L_{\overline{\mathrm{op}}}$ and likewise exchanges\footnote{The statement that $\tau$ exchanges ${\mathscr B}_{\mathrm{op}}$ with ${\mathscr B}_{\overline{\mathrm{op}}}$ holds in the underlying physical
$\sigma$-model. Since $\tau$ acts antiholomorphically on ${\mathcal M}_H(G^\vee,C)$ in the relevant complex structure, it exchanges the $B$-model with a conjugate
$B$-model and is not a $B$-model symmetry (${\mathscr B}_{\mathrm{op}}$ and ${\mathscr B}_{\overline{\mathrm{op}}}$ are valid branes in both the $B$-model and its conjugate).
The $B$-model symmetry that exchanges
${\mathscr B}_{\mathrm{op}}$ with ${\mathscr B}_{\overline{\mathrm{op}}}$ and is used in defining the hermitian structure (eqn. (\ref{hermdef})) is $\Theta_\tau=\Theta\tau$, where $\Theta=\sf{CPT}$.}
${\mathscr B}_{\mathrm{op}}$ with ${\mathscr B}_{\overline{\mathrm{op}}}$.
Suppose that a point $u\in L_{\mathrm{op}}\cap L_{\overline{\mathrm{op}}}$ corresponds to a complex flat bundle $E^\vee_{\Bbb C}$ that is an oper both holomorphically and antiholomorphically. Then
its complex conjugate $\overline E^\vee_{\Bbb C}$ is also an oper both holomorphically and antiholomorphically. If $\overline E^\vee_{\Bbb C}$ is not gauge-equivalent to $E^\vee_{\Bbb C}$
as a flat bundle, then $\overline E^\vee_{\Bbb C}$ corresponds to a point $\overline u\in L_{\mathrm{op}}\cap L_{\overline{\mathrm{op}}}$ that is distinct from $u$. If so, $u$ and $\overline u$ will correspond to distinct
basis vectors $\psi_u$ and $\psi_{\overline u}$ of ${\mathcal H}$, and moreover these will be exchanged by $\Theta_\tau$. The natural $B$-model pairing is diagonal
in the basis of intersection points: the basis vectors can be normalized so that for $u,u'\in \Upsilon$, $(\psi_u,\psi_{u'})=\delta_{uu'}$. Therefore, if $\Theta_\tau$
exchanges two distinct basis vectors $\psi_u$ and $\psi_{\overline u}$, then $\psi_u$ and $\psi_{\overline u}$ are both null vectors for the hermitian inner product that was defined
in eqn. (\ref{hermdef}). The duality predicts that this hermitian inner product should be positive-definite, since on the $A$-model side, ${\mathcal H}$ is obtained by quantizing
a cotangent bundle and is a Hilbert space of half-densities. So we expect that a flat bundle that is an oper both holomorphically and antiholomorphically is
actually real. This was conjectured in \cite{EFK,EFK2} and was proved by an explicit (but surprisingly non-trivial) computation for $G^\vee={\mathrm U}(1)$; the result
is also known for $G^\vee={\mathrm{SU}}(2)$ \cite{Fal,Go}.
Finally, we can use the duality to predict the spectrum of the holomorphic and antiholomorphic Hitchin Hamiltonians as operators on ${\mathcal H}$. Let $H_{{\mathcal P},\alpha}$
be a quantized Hitchin Hamiltonian, that is, an element of ${\mathscr A}={\mathrm{Hom}}({\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$. The duality identifies ${\mathscr A}$ with ${\mathrm{Hom}}({\mathscr B}_{\mathrm{op}},{\mathscr B}_{\mathrm{op}})$ and therefore
identifies $H_{{\mathcal P},\alpha}$ with a holomorphic function $f_{{\mathcal P},\alpha}$ on $L_{\mathrm{op}}$. Acting on a basis vector $\psi_u$ that corresponds to a point $u\in L_{\mathrm{op}}\cap
L_{\overline{\mathrm{op}}}$, $H_{{\mathcal P},\alpha}$ simply acts by multiplication by the corresponding value $f_{{\mathcal P},\alpha}(u)$. Similarly, if $H_{\overline{\mathcal P},\overline\alpha}\in \overline{\mathscr A}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},\overline{\mathscr B}_{\mathrm{cc}})$
is an antiholomorphic quantized Hitchin Hamiltonian, then it corresponds under the duality to a holomorphic function $f_{\overline{\mathcal P},\overline\alpha}$
on $L_{\overline{\mathrm{op}}}$, and it acts on $\psi_u$ as multiplication by $f_{\overline{\mathcal P},\overline\alpha}(u)$. This completes the description of the eigenvalues of the quantized
Hitchin Hamiltonians.
\subsection{Including the Center}\label{inccenter}
It is not difficult to modify this description to take into account the center of $G_{\Bbb C}^\vee$.
Consider as usual the $B$-model on $M=\Sigma\times C$. It localizes on flat bundles over $\Sigma\times C$.
In our application, $\Sigma={\mathbb{R}}\times I$ is contractible, so a flat bundle on $M$ is the pullback of a flat bundle on $C$.
In the case of oper and anti-oper boundary conditions at the two ends of the strip, the flat bundle on $C$ is an oper both
holomorphically and antiholomorphically; thus it is a real oper.
An oper bundle, real or not, is irreducible and its automorphism group consists only of the center ${\mathcal Z}(G^\vee)$ of the gauge group.
However, as explained in Section \ref{opergauge}, the boundary conditions also give trivializations $s_\ell$ and $s_r$ of the ${\mathcal Z}(G^\vee)$
gauge symmetry on the two boundaries, modulo gauge transformations that act by $(s_\ell, s_r)\to (b s_\ell,b s_r)$, $b\in {\mathcal Z}(G^\vee)$.
For a given real oper corresponding to a point $u\in \Upsilon$, let ${\mathcal T}_u$ be the set of pairs $s_\ell,s_r$ modulo the
action of ${\mathcal Z}(G^\vee)$.
The $B$-model localizes on the isolated set of points $u,\varepsilon$ with $u\in \Upsilon$, $\varepsilon\in {\mathcal T}_u$.
So the Hilbert space ${\mathcal H}$ in the general case with a nontrivial center has a basis $\psi_{u,\varepsilon}$ for such $u,\varepsilon$.
One can think of $\varepsilon\in{\mathcal T}_u$ as a sort of global holonomy between the left and right boundaries of the strip.
This refinement involving the torsor ${\mathcal T}_u$ is not important in the dual description of
the algebras ${\mathscr A}$ and $\bar{\mathscr A}$ via holomorphic functions on $L_{\mathrm{op}}$ or $L_{\overline{\mathrm{op}}}$, since each algebra acts on only one side of the strip.
It is similarly not important in the determination of the eigenvalues of the Hitchin
Hamiltonians, which only depends on the interpretation of ${\mathscr A}$ and $\bar{\mathscr A}$ in terms of functions on $L_{\mathrm{op}}$ and $L_{\overline{\mathrm{op}}}$, and is not sensitive
to global holonomy across the strip. It does affect the multiplicity of the eigenvalues,
since eigenvectors $\psi_{u,\varepsilon}$ with the same $u$
and different $\varepsilon$ have the same eigenvalues of the Hitchin Hamiltonians. And it will be relevant
in describing the eigenvalues of the 't Hooft or Hecke operators, to which we turn next.
\section{Hecke, 't Hooft, and Wilson Operators}\label{wtw}
\subsection{Line Operators}
In the usual formulation of geometric Langlands \cite{BD}, the main objects of study include the Hecke functors acting on the category of $A$-branes and
the ``eigenbranes'' of these Hecke functors.
In the gauge theory picture \cite{KW}, the Hecke functors are interpreted in terms of 't Hooft line operators. Electric-magnetic duality maps 't Hooft
line operators to Wilson line operators, leading to some of the usual statements about geometric Langlands duality.
In general, in two-dimensional topological
field theory, line operators give functors acting on the category of boundary conditions because a line operator $T$ that runs parallel to a boundary
labeled by a brane ${\mathscr B}$ can be moved to the boundary, making a composite boundary condition $T{\mathscr B}$ (fig. \ref{example1}(a)).
Here we assume that the two-manifold and the line operator (or more precisely the one-manifold on which it is supported)
are oriented and that the orientation of the line operator agrees with the orientation of the boundary on which it acts.
The same
figure also makes clear the notion of the adjoint of a line operator. The adjoint $T'$ of a line operator is the same line operator with opposite orientation.
In fig. \ref{example1}(a), we could move the line operator $T$ to the right of the figure. As its orientation is opposite to the orientation of the right
boundary, this gives an action of the dual line operator $T'$ on the brane ${\mathscr B}'$ that defines the boundary condition on the right boundary.
So we get ${\mathrm{Hom}}({\mathscr B}',T{\mathscr B})={\mathrm{Hom}}(T'{\mathscr B}',{\mathscr B})$ for any ${\mathscr B},{\mathscr B}'$. (Some line operators have the property that $T$ is isomorphic to $T'$; their support
can be an unoriented 1-manifold.)
These statements hold in any two-dimensional topological field theory. Our actual application involves a four-dimensional theory with two additional dimensions
that comprise a Riemann surface $C$. Although it is possible to consider an 't Hooft operator (or a dual Wilson operator)
whose support is an arbitrary curve $\gamma$ in the four-manifold
$\Sigma\times C$, we will only consider the special case that $\gamma=\ell\times p$, where $p$ is a point in $C$ and $\ell$ is a curve in $\Sigma$. So
our line operators will be defined in part by the choice of $p$.
In addition, in the application to geometric Langlands, an 't Hooft operator is labeled by a finite-dimensional irreducible
representation $R$ of $G^\vee$ (or equivalently of $G^\vee_{\Bbb C}$). When we want to indicate
this data, we denote the 't Hooft operator as $T_{R,p}$.
Similarly the dual Wilson operator depending on the representation $R$ and the point $p$ will be denoted as $W_{R,p}$.
\begin{figure}
\begin{center}
\includegraphics[width=6.2in]{Example1.pdf}
\end{center}
\caption{\small (a) A line operator $T$ parallel to the left boundary of the strip, and oriented compatibly. Moving $T$ to the left, it maps
the boundary condition labeled ${\mathscr B}$ to a composite boundary condition $T{\mathscr B}$. This is a line operator viewed as a functor on the category
of branes or boundary conditions, as in the usual formulation of geometric Langlands. (b) In the analytic approach to geometric
Langlands, the same line operator $T$, running horizontally across the strip, and with some additional data at the endpoints, becomes
an operator acting on physical states. (c) and (d) The purpose of these drawings is to elucidate the additional data
that is needed at the left and right endpoints in (b). At the left endpoint, we have an element $\alpha\in {\mathrm{Hom}}({\mathscr B},T{\mathscr B} )$, and at the right endpoint,
an element $\beta\in{\mathrm{Hom}}(T{\mathscr B}',{\mathscr B}')$. \label{example1}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{Example2.pdf}
\end{center}
\caption{\small This picture illustrates an algebraic manipulation described in the text. Reading the drawing on the right from bottom to top,
one first encounters the operations sketched in figs. \ref{example1}(c,d), followed by the fusion of the product $TT'$ of line operators to the ``identity,'' that is,
to a trivial line operator. \label{example2}}
\end{figure}
In the analytic approach to geometric Langlands \cite{EFK,EFK2}, Hecke operators becomes ordinary operators acting on a Hilbert space of quantum states,
rather than more abstract functors acting on a category. Not surprisingly, the gauge theory interpretation of Hecke operators in this sense is based on the
same 't Hooft line operators as before, used somewhat differently. In fig. \ref{example1}(b), we consider the same line operator as before, but now
running from left to right of the figure. Some additional data must be supplied at the left and right endpoints where the line operator terminates on a
boundary of the strip. Let us assume for the moment that this has been done. Then the line operator becomes an ordinary operator acting on quantum states.
Reading the figure from bottom to top, an element of ${\mathrm{Hom}}({\mathscr B}',{\mathscr B})$ enters at the bottom and after the action of the line operator, a possibly different
element of ${\mathrm{Hom}}({\mathscr B}',{\mathscr B})$ emerges at the top. (If we read the figure from top to bottom, we see the transpose operator acting on the dual
vector space ${\mathrm{Hom}}({\mathscr B},{\mathscr B}')$.) It is because line operators that are supported on a one-manifold in space at a fixed time
can act in this way as ordinary quantum operators that they are traditionally\footnote{In traditional applications in particle physics, there
are no boundaries and the support of the line operator is taken to be a closed loop. The operator is then often called a loop operator.} called
line ``operators.''
The purpose of figs. \ref{example1}(c,d) is to explain what is happening where the line operator of fig. \ref{example1}(b) ends on the left or right boundary.
In fig. \ref{example1}(c), we see that the left endpoint of the line operator corresponds to an element $\alpha\in{\mathrm{Hom}}({\mathscr B},T{\mathscr B})$, and in fig.
\ref{example1}(d), we see that the right endpoint corresponds to an element $\beta\in {\mathrm{Hom}}(T{\mathscr B}',{\mathscr B}')$. Algebraically, the operator
$\widehat T:{\mathrm{Hom}}({\mathscr B}',{\mathscr B})\to {\mathrm{Hom}}({\mathscr B}',{\mathscr B})$ associated to a line operator $T$ with the additional data $\alpha,\beta$ can be described as follows. For $\psi\in {\mathrm{Hom}}({\mathscr B}',{\mathscr B})$,
we have $\alpha\circ\psi\circ\beta\in {\mathrm{Hom}}(T{\mathscr B}',T{\mathscr B})={\mathrm{Hom}}({\mathscr B}',T'T{\mathscr B})$. Then using the fact that line operators
form an algebra and that the trivial line operator appears in the product $T'T$, we get a map $w:{\mathrm{Hom}}({\mathscr B}',T'T{\mathscr B})\to {\mathrm{Hom}}({\mathscr B}' ,{\mathscr B})$. Finally
$\widehat T(\psi)=w\circ\alpha\circ\psi\circ\beta$. This sequence of algebraic manipulations corresponds to the picture of fig. \ref{example2}.
We will sometimes write $\widehat T_{\alpha,\beta}$ or $\widehat T_{R,p,\alpha,\beta}$ for the operator on ${\mathcal H}$ that is constructed from a line operator $T$ or $T_{R,p}$ with endpoint
data $\alpha,\beta$.
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{Example3.pdf}
\end{center}
\caption{\small The argument showing that quantized Hitchin Hamiltonians commute can be adapted to show that line operators (viewed as actual operators
on a space of quantum states) commute with the quantized Hitchin Hamiltonians and with each other. In each case, as sketched in (a) and (b) respectively,
the key point is that because of the existence of additional dimensions, the two operators can slide up and down past each other without singularity. \label{example3}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=2.1in]{Example4b.pdf}
\end{center}
\caption{\small To compute the eigenvalues of the 't Hooft/Hecke operators, one considers dual Wilson operators that describe parallel transport from
$a\times p$ to $b\times p$, where $p$ is a point in $C$ and $a,b$ are points on the left and right boundaries of $\Sigma$. \label{example4}}
\end{figure}
We will be particularly interested in elements $(\alpha,\beta)$ which originate from local endpoints of a four-dimensional line defect onto four-dimensional boundary conditions which lift ${\mathscr B}$ and ${\mathscr B}'$. As remarked in Section \ref{background}, the four-dimensional lifts of the boundary conditions we are considering are not topological. Instead, they are respectively holomorphic-topological and antiholomorphic-topological. As a consequence,
the local endpoint lifting $\alpha$ will depend holomorphically on $p$ while the local endpoint lifting $\beta$ will depend antiholomorphically on $p$.
Notice that the actual path of the line defect in four-dimensions is immaterial, as long as it is topologically equivalent to a straight path.
Only the positions of the endpoints in $C$ matters.
One of the main properties of 't Hooft or Hecke operators, when regarded as in \cite{EFK,EFK2} as operators on quantum states, is that they commute with each
other and with the quantized Hitchin Hamiltonians. This follows from the same reasoning that we used to show that the Hitchin Hamiltonians commute
with each other. An 't Hooft operator $T_{R,p}$ commutes with a Hitchin Hamiltonian $H_{{\mathcal P},\alpha}$ because one can assume that the support
of $\alpha$ is disjoint from the point $p$, so that one can slide $T_{R,p}$ and $H_{{\mathcal P},\alpha}$ up and down past each other (fig. \ref{example3}(a)) without
singularity.
Likewise, for distinct points $p,p'\in C$, 't Hooft operators $T_{R,p}$ and $T_{R',p'}$ commute (fig. \ref{example3}(b)). Taking the limit $p'\to p$, it follows that
$T_{R,p}$ and $T_{R',p}$ commute as well, even for $R\not= R'$.
\subsection{Wilson Operators And Their Eigenvalues}\label{wilop}
Since the 't Hooft operators commute with the Hitchin Hamiltonians, they can be
diagonalized in the same basis, namely the basis of states $\psi_{u,\varepsilon},$ $u\in \Upsilon,$
$\varepsilon\in {\mathcal T}_u$,
where $\Upsilon=L_{\mathrm{op}}\cap L_{\overline{\mathrm{op}}}$ (Section \ref{inccenter}). In fact, we can use electric-magnetic duality to determine the eigenvalues of the 't Hooft operators. An 't Hooft operator
$T_{R,p}$ is dual to a Wilson operator $W_{R,p}$, labeled by the same representation $R$ of $G^\vee_{\Bbb C}$
and supported at the same point $p\in C$.
While the 't Hooft operator is a ``disorder'' operator, whose microscopic definition involves a certain sort of singularity, Wilson operators are defined classically
in terms of holonomy, as follows. In the relevant gauge theory on a four-manifold $M$ (for our purposes, $M=\Sigma\times C$), one has a
$G^\vee_{\Bbb C}$ bundle $E^\vee_{\Bbb C}\to M$,
with connection ${\mathcal A}=A+{\mathrm i} \phi$, to which we can associate a vector bundle $E^\vee_R=E^\vee_{\Bbb C}\times_{G^\vee_{\Bbb C}} R$. We denote the induced connection
on this bundle simply as ${\mathcal A}$. The Wilson operator is constructed from the holonomy of the connection ${\mathcal A}$ on $E^\vee_R$, integrated in general
along some oriented path $\gamma\subset M$. If $\gamma$ is a closed loop, we take the trace of the holonomy around $\gamma$, and this gives
a version of the Wilson operator that is important in many physical applications.
However, to compute the eigenvalues of an operator defined by an 't Hooft line operator that stretches across the strip
(fig. \ref{example1}(b)), we need to consider a dual Wilson operator that similarly stretches across the strip (fig. \ref{example4}).
This is a Wilson operator supported, not on a closed loop, but on a path
$\gamma\subset \Sigma\times p$ from $a\times p$ on the left boundary of the strip to $b\times p$ on the right boundary.
In this case, the holonomy
is best understood as a linear transformation from the fiber $E^\vee_R$ at $a\times p$ to the fiber of this bundle at $b\times b$. Thus with an obvious
notation for these fibers, $W_{R,p}$ is a linear transformation
\begin{equation}\label{lintr} W_{R,p}:E^\vee_{R,a\times p}\to E^\vee_{R, b\times p}.\end{equation}
In order to treat the left and right edges of the strip more symmetrically, it is convenient to introduce the representation $R'$ dual to $R$ and view
$W_{R,p}$ as a linear function on a representation. Then we have
\begin{equation}\label{bintr} W_{R,p}\in {\mathrm{Hom}}(E^\vee_{R,a\times p}\otimes E^\vee_{R',b\times p},{\Bbb C}). \end{equation}
So far, we have a linear function on a vector space, rather than a complex-valued function of connections, which could be quantized to get a quantum
operator. To get a complex-valued function of connections, we need to supply vectors $v\in E^\vee_{R,a\times p}$, $w\in E^\vee_{R',b\times p}$.
Then $W_{R,p}(v\otimes w)$ is a complex valued function that can be quantized to get an operator.
A natural construction of suitable vectors was described in Section \ref{localoper}. In eqn. (\ref{linmap}), we described, for a holomorphic
oper with associated bundle $E^\vee_R$, a ``highest weight section'' $s_R:K_C^{(N-1)/2}\to E^\vee_R$. Thus, if we are given a vector
$v\in K_{C,p}^{(N-1)/2}$, then we can define $s_R(v)\in E^\vee_{R,p}$. Similarly, if $E^\vee_{R'}$ is an antiholomorphic oper, we have
$\overline s_{R'}:\overline K_C^{(N-1)/2}\to E^\vee_{R'}$, and hence, for $w\in K_{C,p}^{(N-1)/2}$, we have $\overline s_{R'}(w)\in E^\vee_{R',p}$. Note
that a dual pair of representations $R,R'$ have the same value of $N$.
In the case of a bundle $E^\vee_R\to \Sigma\times C$ that is a holomorphic oper on the left boundary and an antiholomorphic oper on the right
boundary, we can apply the holomorphic version of this construction on the left boundary and the antiholomorphic version on the right boundary, to get
\begin{equation}\label{wintr}W_{R,p,v\otimes w}=W_{R,p}(s_{R}(v)\otimes \overline s_{R'}(w)). \end{equation}
This finally is a complex-valued function of connections that can be quantized to get a Wilson operator on physical states. We will call this
operator $W_{R,p,v\otimes w}$. In the notation, we make use of the fact that the right hand side of eqn. (\ref{wintr}) depends on $v$ and $w$ only in the
combination $v\otimes w$. Hopefully it will cause no serious confusion that we use the same notation for a classical holonomy (or its matrix element)
and the corresponding quantum operator.
Because of the way the $B$-model localizes on flat connections, it is trivial to diagonalize this operator. The flat connections that satisfy the boundary
conditions, with trivializations of the center on the boundary and modulo gauge transformations, are in one-to-one correspondence with
the usual basis of states $\psi_{u,\varepsilon}$, $u\in\Upsilon,$ $\varepsilon\in {\mathcal T}_u$ that diagonalize the Hitchin Hamiltonians.
The Wilson operators are diagonal in this basis.
The eigenvalue of the quantum operator $ W_{R,p,v\otimes w}$ on a given
basis vector $\psi_{u,\varepsilon}$ is just the value of the corresponding classical function on the classical solution corresponding to $\psi_{u,\varepsilon}$. That value
is simply the natural dual pairing $(s_R(v), \overline s_{R'}(w))$ of the vectors $s_{R}(v)$ and $\overline s_{R'}(w)$ in the dual vector spaces $E^\vee_{R,p}$ and
$E^\vee_{R',p}$. This is true because a flat connection on $\Sigma\times C$ is actually a pullback from $C$.
The center ${\mathcal Z}(G^\vee)$ acts on the physical Hilbert space ${\mathcal H}$ by $(s_\ell, s_r)\to (b s_\ell, s_r)$ (Section \ref{opergauge}). This transforms
the eigenvalue of $\widehat W_{R,p}$ by a root of unity, which is simply the value of the central element $b$ in the representation $R$. For example,
if $G^\vee={\mathrm{SL}}(n,{\Bbb C})$ and $R$ is the $n$-dimensional representation, eigenvectors of $\widehat W_{R,p}$ come in $n$-plets of the form
$\lambda\exp(2\pi {\mathrm i} k/n)$, $\lambda\in{\Bbb C}$, $k=0,1,\cdots, n-1$. This is dual to the fact that on the $A$-model side, the Higgs bundle moduli space
has components labeled by a characteristic class $\zeta\in H^2(C,\pi_1(G))$. For $G^\vee={\mathrm{SL}}(n,{\Bbb C})$, there are $n$ components,
which are cyclically permuted by the 't Hooft operator $\widehat T_{R,p}$ dual to $\widehat W_{R,p}$, leading to the same structure of the spectrum.
In general, the action of ${\mathcal Z}(G^\vee)$ on $R$ mirrors the way $\widehat T_{R,p}$ permutes the components of the moduli space.
The holomorphic and antiholomorphic
corner data needed to define the operators $W_{R,p}$ has consisted precisely of the vectors $v\in K_C^{(N-1)/2}$, $w\in \overline K_C^{(N-1)/2}$. For the $n$-dimensional
representation of ${\mathrm{SL}}(n,{\Bbb C})$, we have $N=n$.
We expect the same data to be needed to define holomorphic and antiholomorphic corner data for the dual 't Hooft operators.
These are arguably the simplest Wilson operators and we will call them principal Wilson
operators. However, if the representation $R$ is reducible when restricted to a principal ${\mathfrak {su}}(2)$ subalgebra of ${\mathfrak g}^\vee$, then it is possible
to use the more general objects $s_{R,n}:Q_n\otimes K_C^{(n-1)/2}\to E^\vee_R$ (eqn. (\ref{holmap})). So picking $v\in Q_n \otimes K_C^{(n-1)/2}$,
$w\in \overline Q_{n'}\otimes K_C^{(n'-1)/2}$ (where $n$ and $n'$ can be chosen independently), we can define $W_{R,p,n,n',v\otimes w}=W_{R,p}(s_{R,n}(v)\otimes
\overline s_{R',n'}(w))$, which can again be interpreted as a quantum mechanical operator. Even more generally, we can consider holomorphic and antiholomorphic derivatives with respect to $p$ of $W_{R,p,n,n',v\otimes w} $. It is enough to consider $n-1$ holomorphic derivatives and $n'-1$ antiholomorphic
ones; this suffices to define a complete set of Wilson operators,
since it amounts to applying the linear form $W_{R,p}:E^\vee_{R,a\times p}\otimes E^\vee_{R',b\times p}\to {\Bbb C} $
(eqn. (\ref{bintr})) to a set of vectors that according to the analysis
in Section \ref{localoper}
form a basis of the finite-dimensional vector space on which $W_{R,p}$ is acting.
What happens if we continue to differentiate?
With $n$ holomorphic derivatives or $n'$ antiholomorphic ones, we will run into differential equations satisfied by the Wilson operators $W_{R,p,n,n',v\otimes w} $.
The holomorphic and antiholomorphic sections $ s_{R,r}$ and $\overline s_{R',n'}$ that were used to define these Wilson operators
obey certain holomorphic and antiholomorphic
differential equations (Section \ref{localoper} and Appendix \ref{somex}) as a function of $p$, and the corresponding Wilson operators obey the same equations.
Having defined operators that act on the physical Hilbert space ${\mathcal H}$, it is natural to ask what algebra they obey. In bulk, the product of line operators
mimics the tensor product of representations of $G^\vee$. Thus, if $R_i\otimes R_j\cong \oplus_k N^k_{ij}R_k$, with vector spaces $N^k_{ij}$, then
the corresponding decomposition of parallel Wilson operators is $W_{R_i} W_{R_j}=\oplus_k N^k_{ij} W_k$. This is the appropriate relation for Wilson
operators understood as functors acting on boundary conditions, as illustrated in fig. \ref{example1}(a). From this algebra, the structure of the nonabelian
group $G^\vee $ can be reconstructed, in principle.
However, for Wilson operators as operators on quantum states, as we are discussing here, the picture is different.
Matters are simplest if we multiply two principal Wilson operators, associated to highest weight vectors in the corresponding representations.
Since the tensor product of highest
weight vectors in two representations $R_1$ and $R_2$ is a highest weight vector in the tensor product $R_1\otimes R_2$,
the product of two principal Wilson line operators is another principal Wilson line operator, for a representation $R_3$ whose highest weight is the
sum of the highest weights of $R_1$ and $R_2$. We will see the structure of the nonabelian group if we multiply the more general Wilson
operators $W_{R,p,n,n',v\otimes w}$ and their derivatives.
A dual 't Hooft operator defined using the $S$-dual data will, of course, have the same eigenvalues as the Wilson
operators. The main challenge is to identify precisely the appropriate $A$-model endpoints. We turn to that problem in Section \ref{dualhooft}.
We conclude this discussion of Wilson operators with the following remark.
The most illuminating realization of the oper boundary condition in four-dimensional gauge theory involves the Nahm pole \cite{GWknots}.
Compared to a more direct approach that was assumed earlier \cite{KW} (in which the boundary condition is defined by just specifying the $(0,1)$ or
$(1,0)$ part of ${\mathcal A}$ along the boundary), the Nahm pole description of the oper boundary condition has two
advantages: it leads directly to the local constraints discussed in Section \ref{localoper}; and it also leads to a simple explanation of the duality
with the $A$-model description via ${\mathscr B}_{\mathrm{cc}}$. If one uses the Nahm pole description of the oper boundary condition, then the complex connection ${\mathcal A}$
is singular along the boundary, and some renormalization is involved in defining the classical holonomy $W_R$ across the strip and the corresponding
quantum operator. The renormalization amounts to a complex gauge transformation that removes the Nahm pole singularity.
\subsection{The Dual 't Hooft Operators}\label{dualhooft}
\begin{figure}
\begin{center}
\includegraphics[width=3.6in]{Example7.pdf}
\end{center}
\caption{\small (a) An 't Hooft operator viewed as an interface from the $A$-model of ${\mathcal M}_H(G,C)$ to itself. (b) After ``folding,'' the same 't Hooft
operator is interpreted as a brane defining a boundary condition in the product of two copies of ${\mathcal M}_H(G,C)$, with equal and opposite symplectic
form. \label{example7}}
\end{figure}
\subsubsection{'t Hooft Operators and Hecke Modifications}\label{THO}
An 't Hooft operator $T_{R,p}$ (fig. \ref{example7}(a))
will produce a jump in the fields $(A,\phi)$ and in the associated Higgs bundle $(E,\varphi)$, in the sense that the Higgs bundle $(E,\varphi)$ just
below the 't Hooft operator is generically not isomorphic to the Higgs bundle $(E',\varphi')$ just above it. They differ by what is known as
a Hecke modification. The type of Hecke modification is
determined by the magnetic singularity of the 't Hooft operator, which is classified by a choice of
irreducible representation $R$ of the dual group $G^\vee$.
Hecke modifications of bundles and Higgs bundles were described for physicists in \cite {KW} and in more detail in Section 4 of
\cite{More}. Here we will give a very brief synopsis.
A Hecke modification of a holomorphic $G_{\Bbb C}$
bundle $E$ at a point $p$ is a new bundle $E'$ that is presented with an isomorphism to $E$ away from
$p$, but such that this isomorphism does not extend over $p$. A section $w$ of $E'$ is a section of $E$ that is allowed to have poles of a specified
type at $p$, or that is constrained so that some components have zeroes of a specified type, or both.
For example, if $E$ is a rank 2 holomorphic
vector bundle, trivialized near $p$ so that a section of $E$ is a pair of holomorphic functions $\begin{pmatrix}f\cr g\end{pmatrix}$, then an example of a Hecke modification of $E$ at $p$
is a new bundle $E'$ whose sections are a pair $\begin{pmatrix}f\cr g\end{pmatrix}$,
where $f$ is holomorphic at $p$ but $g$ is allowed to have a simple pole at $p$. This example
can be slightly generalized to give a family of Hecke modifications of $E$ at $p$ that are parametrized by ${\Bbb{CP}}^1$. We simply pick a pair of complex numbers
$u,v$, not both zero, representing a point in ${\Bbb{CP}}^1$, and allow a section of $E'$ to have a polar part proportional to this pair:
\begin{equation}\label{thep} w= \frac{1}{z}\begin{pmatrix}u\cr v\end{pmatrix}+{\mathrm{regular}}, \end{equation}
where $z$ is a local parameter at $p$.
The relation between $E$ and $E'$ is reciprocal: instead of saying that we obtain $E'$ from $E$ by allowing a pole of a certain type, we could say
that we obtain $E$ from $E'$ by requiring a certain type of zero. If an 't Hooft operator $T_{R,p}$ can map $E$ to $E'$, then the dual 't Hooft operator
(which is associated to the dual representation) can map $E'$ to $E$.
A Hecke modification of a Higgs bundle $(E,\varphi)$ actually does ``nothing'' to $\varphi$. This means the following. A Hecke modification of $(E,\varphi)$
is just a Hecke modification $E'$ of $E$ such that $\varphi:E\to E\otimes K_C$ is holomorphic
as a map $E'\to E'\otimes K_C$. In the example of the rank 2 bundle, the Higgs field is locally $\varphi=\varphi_z\,\mathrm d z$, where $\varphi_z$ is a $2\times 2$ matrix of
holomorphic functions.
For $\varphi$ to be holomorphic as a map $E'\to E'\otimes K_C$, the necessary and sufficient condition is that, if $w$ is a section of $E'$
as characterized in eqn. (\ref{thep}), then the polar part of $\varphi w$ should be a multiple of $\begin{pmatrix}u\cr v\end{pmatrix}$; in other
words, $\begin{pmatrix}u\cr v\end{pmatrix}$ must be an eigenvector of the matrix $\varphi_z(p)$.
Generically, this condition is not satisfied and hence most Hecke modifications of $E$ are not valid as Hecke modifications of $(E,\varphi)$.
If $\varphi_z(p)$ is not nilpotent, it has two distinct eigenvectors and
the Higgs bundle $(E,\varphi)$ has two
possible Hecke modifications of this type at the point $p$;
if $\varphi_z(p)$ is nilpotent but not zero, it has only one eigenvector and there is just one possible Hecke modification of this type; only if $\varphi_z(p)=0$
does $(E,\varphi)$ has the same ${\Bbb{CP}}^1$ family of possible Hecke modifications of this type at $p$ that $E$ would have by itself.
Hecke modifications of the type just described can be viewed in two different ways.
They are dual to the two-dimensional representation of $G^\vee={\mathrm U}(2)$. This group is self-dual, so the discussion
is applicable in gauge theory of $G={\mathrm U}(2)$. Alternatively, the same Hecke modification is dual to the two-dimensional representation of $G^\vee={\mathrm{SU}}(2)$.
In this case, the dual group is $G={\mathrm{SO}}(3)$. In this application, since the rank two bundles $E$ and $E'$ do not have ${\mathrm{SO}}(3)$ structure group, the
preceding discussion should be restated in terms of the corresponding adjoint bundles ${\mathrm{ad}}(E)$ and ${\mathrm{ad}}(E')$
In this example, let $\delta$ be the eigenvalue of $\varphi(p)$ acting on $\begin{pmatrix}u\cr v\end{pmatrix}$:
\begin{equation}\label{defdelta} \varphi(p) \begin{pmatrix}u\cr v\end{pmatrix}=\delta \begin{pmatrix}u\cr v\end{pmatrix}.\end{equation}
As we vary the choice of $E$ and $E'$, $\delta$ varies holomorphically. It defines a holomorphic function on the space of Hecke modifications of this type. Since
the eigenvalues of $\varphi(p)$ are $\pm \delta$, we have
\begin{equation}\label{delchar} \delta^2=-\det \varphi(p),\end{equation}
where $\det\varphi(p)$ is a linear combination of the Hitchin Hamiltonians.
The sign of $\delta$ distinguishes the two choices of
Hecke modifications compatible with a given Higgs field.
\subsubsection{Hecke Correspondences}\label{HC}
The 't Hooft operator $T_{R,p}$ can be viewed as an interface between the $A$-model of ${\mathcal M}_H(G,C)$
and itself. This is a tautology; in fig. \ref{example7}(a), we see the $A$-model of ${\mathcal M}_H(G,C)$ above and below $T_{R,p}$, so we can view
$T_{R,p}$ as an interface between two copies of this $A$-model. In fact, this interface is of type $(B,A,A)$, because of the supersymmetric properties of $T_{R,p}$.
It is convenient to use a folding trick similar to the one of fig. \ref{folding}. Instead of associating to $T_{R,p}$
an interface in the $A$-model of ${\mathcal M}_H(G,C)$, we can associate to it a brane or boundary condition
${\mathscr B}_{R,p}$ of type $(B,A,A)$
in the $A$-model of a product ${\mathcal M}_H(G,C)\times {\mathcal M}_H(G,C)$
(fig. \ref{example7}(b)). Here the symplectic structure of ${\mathcal M}_H(G,C)\times {\mathcal M}_H(G,C)$ is $\omega_K\boxplus (-\omega_K)$, with a minus
sign in one factor because folding reverses the sign of the symplectic structure.
Let us consider explicitly what ${\mathscr B}_{R,p}$ will look like for the basic example, described in Section \ref{THO}, that $R$ is the two-dimensional representation of $G^\vee={\mathrm{SU}}(2)$.
What is its complex dimension? If $C$ has genus $g$, then the choice of a $G_{\Bbb C}$ bundle $E$ depends on $3g-3$ complex parameters. Choosing
$E'$'s that can be made from $E$ by action of $T_{R,p}$ adds one more complex parameter. But we have to constrain the Higgs field $\varphi$
so that $\begin{pmatrix}u\cr v\end{pmatrix}$ is an eigenvector of $\varphi(p)$. So the choice of $\varphi$ involves $3g-4$ parameters,
not $3g-3$. The upshot is that the support $Z_{R,p}$ of ${\mathscr B}_{R,p}$ has dimension $6g-6$, and thus $Z_{R,p}$ is middle-dimensional in ${\mathcal M}_H(G,C)\times {\mathcal M}_H(G,C)$.
Because $Z_{R,p}$ is middle-dimensional and is the support of a brane of type $(B,A,A)$, it must be a complex Lagrangian submanifold.
The brane ${\mathscr B}_{R,p}$ has rank 1 because the Hecke transformation by which $T_{R,p}$ produces $E'$ from $E$ is
generically unique, if it exists. In short, ${\mathscr B}_{R,p}$ is a rank 1 Lagrangian brane of type $(B,A,A)$.
We can be more specific, because ${\mathscr B}_{R,p}$ is manifestly invariant under scaling of $\varphi$, which corresponds to
the ${\Bbb C}^*$ symmetry of ${\mathcal M}_H(G,C)\times {\mathcal M}_H(G,C)$. We can give a simple
description in the same sense in which ${\mathcal M}_H(G,C)$ can be approximated by its dense open set
$T^*{\mathcal M}(G,C)$. (As in the general discussion of quantization, we expect that this approximation is sufficient in an ${\mathrm L}^2$ theory.)
The intersection of $Z_{R,p}$ with ${\mathcal M}(G,C)\times {\mathcal M}(G,C)\subset T^*{\mathcal M}(G,C)\times T^*{\mathcal M}(G,C)$ is the variety $X_{R,p}$
that parametrizes pairs $E,E'$ such that $E'$ can be reached from $E$ by a Hecke transformation of type $R$ at the point $p$. ${\Bbb C}^*$ invariance
of $Z_{R,p} $ means that it can identified as the conormal bundle\footnote{If a submanifold $U\subset M$ is defined locally by vanishing of
some coordinates $q_1,\cdots, q_r$, then its conormal bundle in $T^*M$ is defined by setting to zero those coordinates and the momenta that
Poisson-commute with them.} of $X_{R,p}$. $X_{R,p}$ is called the Hecke correspondence in this situation, and $Z_{R,p}$ is the Hecke correspondence
for Higgs bundles.
The particular example of an 't Hooft operator dual to the two-dimensional representation of $G^\vee={\mathrm{SU}}(2)$ is relatively simple because the space
of possible Hecke modifications of a given bundle $E$ at a given point $p$ is 1-dimensional. In general, the space of Hecke modifications of $E$ that
can be made at $p$ by $T_{R,p}$ has a dimension that depends on $R$ and becomes arbitrarily large if $R$ is a representation of $G^\vee$ of large highest weight.
When the dimension is sufficiently large, every $E'\in {\mathcal M}(G,C)$ can be made from $E$ by $T_{R,p}$ and the ways to do so form a complex manifold $\Phi_{E',E,R,p}$
of positive dimension.
In such a situation, $Z_{R,p}$ is rather complicated. It has a component on which the Higgs field vanishes (if $E'$ is produced from $E$ by a generic
Hecke modification of very high weight, then no nonzero $\varphi:E\to E\otimes K$ is holomorphic as a map $E'\to E'\otimes K$), and other
components with nonzero Higgs field (it is possible to pick a Hecke modification of $E$ of arbitrarily high weight
such that $\varphi:E'\to E'\otimes K$ is holomorphic).
The ${\mathrm{CP}}$ bundle of ${\mathscr B}_{R,p}$ restricted to the various components is not just of rank 1. For example, restricted to the component on which the
Higgs field vanishes, this ${\mathrm{CP}}$ bundle is formally (that is, modulo a proper treatment of singularities), the cohomology of $\Phi_{E',E,R,p}$.
\subsubsection{'t Hooft Line Operators as Operators on Quantum States}\label{lineq}
Under fairly general conditions, if a symplectic manifold $M$ can be quantized by branes to get a Hilbert space ${\mathcal H}$, an $A$-brane ${\mathscr B}$ in $M\times M$ (with
additional data at the ``corners,'' as discussed presently) can be interpreted
as a quantum operator on ${\mathcal H}$. This is discussed in general in Section 4 of \cite{GW}. In general it is difficult to get an explicit description of such
an operator. But here we are in a special situation with a simple answer. That is because $M={\mathcal M}_H(G,C)$ is effectively a cotangent bundle $T^*{\mathcal M}(G,C)$,
and the brane of interest is supported on the conormal bundle of a subvariety $X_{R,p}\subset {\mathcal M}(G,C) \times {\mathcal M}(G,C)$.
A fairly general operator ${\mathcal O}$ acting on the quantization of $T^*{\mathcal M}(G,C)$ can be represented by an integral kernel $F(x,y)$ which is a half-density
on ${\mathcal M}(G,C)\times {\mathcal M}(G,C)$, The action of ${\mathcal O}$ on a state $\Psi$ is
\begin{equation}\label{actst} {\mathcal O}\Psi(x)=\int_{{\mathcal M}(G,C)}\mathrm d y \, F(x,y) \Psi(y). \end{equation}
What sort of integral kernel should we expect for the quantum operator $\widehat T_{R,p}$ associated to the
't Hooft operator $T_{R,p}$? The points $y$ and $x$ in eqn. (\ref{actst}) correspond in fig. \ref{example7}(a) to the fields $(A,\phi)$ just below
and just above the line operator $T_{R,p}$. So they correspond to bundles $E,$ $E'$, such that $E'$ can be reached from $E$ by a Hecke modification of type $R$
at $p$. Hence in a classical limit, $F(x,y)$ is a distribution supported on $X_{R,p}$, which parametrizes such pairs $(E,E')$ . The simplest case, which
we will consider first,
is that $F(x,y)$ is a delta function in the directions normal to $X_{r,p}$. More generally, in its dependence on the normal directions, $F(x,y)$ can be proportional
to arbitrary derivatives of a delta function in the normal variables.
In general, one would expect
quantum corrections to the claim that $F(x,y)$ is supported on the underlying classical correspondence $X_{R,p}$. However, in the present situation, there are no corrections, because the symplectic manifold
that is being quantized is a cotangent bundle $T^*{\mathcal M}(G,C)$, and the correspondence $Z_{R,p}$ is a conormal bundle in $T^*{\mathcal M}(G,C)\times T^*{\mathcal M}(G,C)$.
The scale invariance of the cotangent bundle and the conormal bundle imply that the kernel $F(x,y)$ cannot depend on $\hbar$ and can be evaluated
in a semiclassical limit.
We should add a note on why it is valid here to argue based on scaling symmetry.
Brane quantization is based, as always, on studying $T^*{\mathcal M}(G,C)$ in the context of a suitable complexification. Similarly to study $Z_{R,p}$ in brane
quantization involves
complexifying it in a complexification of $T^*{\mathcal M}(G,C)\times T^*{\mathcal M}(G,C)$.
A scaling argument in brane quantization really involves scaling of the complexifications. Such an argument is valid in the present setting because
the structure of $T^*{\mathcal M}(G,C)$ as a cotangent bundle and of $Z_{R,p}$ as a conormal bundle do extend to holomorphic stuctures of the same
type for their complexifications.
We should also clarify what we mean by ``semiclassical limit.'' A one-loop correction is built into the assertion that a wavefunction is a half-density rather than a function and that $F(x,y)$ is correspondingly a half-density on ${\mathcal M}(G,C) \times {\mathcal M}(G,C)$. The assertion that $F(x,y)$ can be computed semiclassically
means that there is no quantum correction beyond this fact.
According to Etingof, Frenkel, and Kazhdan \cite{EFK2}, the Hecke operator dual to the two-dimensional representation of ${\mathrm{SU}}(2)$ is defined by an integral
kernel that can be factored as the product of holomorphic and antiholomorphic factors. Such a holomorphic factorization
is expected in the $A$-model. A holomorphic factor $f$ will come from
the left endpoint of $T_{R,p}$ in fig. \ref{example7}(a), or equivalently the lower left corner in fig.
\ref{example7}(b), and an antiholomorphic factor $\widetilde f$ will come from the right endpoint or the lower right corner.
We view the 't Hooft operator as a rank 1 brane of type $(B,A,A)$ in ${\mathcal M}_H(G,C)\times {\mathcal M}_H(G,C)$, with trivial ${\mathrm{CP}}$ bundle. In general, the space of corners between a brane of this type, supported on a Lagrangian submanifold $L$, and the canonical coisotropic $A$-brane is $H^0(L,K_L^{1/2})$. The 't Hooft operator corresponds to a brane whose support
is the Hecke correspondence $Z_{R,p}$. So in this case, a holomorphic corner is a holomorphic section $f\in H^0(Z_{R,p},K^{1/2}_{Z_{R,p}})$.
Similarly, an antiholomorphic corner is an element $\widetilde f\in H^0(\overline Z_{R,p},\overline K^{1/2}_{\overline Z_{R,p}})$, where
$\overline Z_{R,p}$ is $Z_{R,p}$ with opposite complex structure.
The product $\mu = f \widetilde f$ of $f$ and $\widetilde f$ will be a half-density on $Z_{R,p}$. We will show that this data is precisely what is needed to define a
distributional kernel $F(x,y)$. If $f$ and $\widetilde f$ are pull-backs from $X_{R,p}$, we will get a delta function kernel. If they have a polynomial dependence on the fiber of $Z_{R,p} \to X_{R,p}$, we will get a linear combination of derivatives of a delta function.
In order to describe a delta function distribution, we pick some local coordinates. We parametrize the input to the Hecke transformation
by coordinates $\vec x=x_1,\cdots, x_{3g-3}$ on ${\mathcal M}(G,C)$. We will write $|\mathrm d \vec x|$ for the half-density $(\mathrm d x_1 \cdots \mathrm d x_{3g-3} \mathrm d \overline x_1
\cdots \mathrm d \overline x_{3g-3})^{1/2}$, and similarly for other variables introduced momentarily.
For a given $\vec x$, the output of the Hecke transformation ranges over a copy of ${\Bbb{CP}}^1$ that we will call ${\Bbb{CP}}^1_x$; we parametrize it
by a complex variable $z$. ${\Bbb{CP}}^1_x $ is of complex codimension $3g-4$ in ${\mathcal M}(G,C)$. ${\Bbb{CP}}^1_x$ can be defined locally by a condition $\vec n=0$, where $\vec n=(n_1,\cdots, n_{3g-4})$ are local holomorphic coordinates on the normal bundle $N$ to ${\Bbb{CP}}^1_x$ in ${\mathcal M}(G,C)$. We can write the kernel as
\begin{equation}\label{nuff}
F(x,y)= b(z,\overline z,x, \overline x)|\mathrm d\vec x ||\mathrm d z \mathrm d \vec n |\delta(\vec n,\vec{\overline n})
\end{equation}
where $\mathrm d \vec n=\mathrm d n_1 \mathrm d n_2\cdots \mathrm d n_{g-4}$ and the delta function is defined by $\int \mathrm d \vec n \mathrm d \vec{ \overline n}\delta(\vec n,\vec{\overline n})=1$.
In view of that last relation, the delta function transforms under a change of coordinates on the normal bundle as $(\mathrm d \vec n \mathrm d \vec{\overline n})^{-1}$, which
means that the possible kernels are in one-to-one correspondence with objects
\begin{equation}\label{luff} \mu = b(z,\overline z,x, \overline x)|\mathrm d\vec x| |\mathrm d z(\mathrm d \vec n)^{-1}|. \end{equation}
The Hecke correspondence $Z_{R,p}$ for Higgs bundles is parametrized locally by the coordinates $\vec x$ and $z$, introduced above, which parametrize the Hecke correspondence $X_{R,p}$ for bundles, and additional variables $\vec m$ that parametrize the choice of Higgs field. Since $Z_{R,p}$ is the conormal bundle of $X_{R,p}$, the variables $\vec m$ are dual to the normal bundle coordinates $\vec n$ that appear in eqn. (\ref{nuff}).
Therefore, we can replace $(\mathrm d\vec n)^{-1}$ with $\mathrm d \vec m$, and the possible delta function kernels are in one-to-one correspondence with half-densities
\begin{equation}\label{luff1} \mu = b(z,\overline z,x, \overline x)|\mathrm d\vec x \mathrm d z\mathrm d \vec m|^2. \end{equation}
on $Z_{R,p}$ such that $b$ is independent of $\vec m$.
This discussion is immediately generalized to linear combinations of normal derivatives of a delta function. A kernel that involves normal derivatives of a delta
function
\begin{equation}\label{nuff2}
F(x,y)= b(x, \overline x,z,\overline z, \partial_{\vec n}, \partial_{\vec {\overline n}}) |\mathrm d\vec x | |\mathrm d z \mathrm d \vec n |\delta(\vec n,\vec{\overline n})
\end{equation}
corresponds to a half-density
\begin{equation}\label{luff2} \mu = b(x, \overline x, z,\overline z,\vec m, \vec {\overline m})|\mathrm d\vec x \mathrm d z\mathrm d \vec m|. \end{equation}
on $Z_{R,p}$ such that $b$ depends polynomially on $\vec m$.
A holomorphically factorized kernel will take the form $\mu = f \widetilde f$ where $f$ and $\widetilde f$ are respectively holomorphic and antiholomorphic.
We can now see what kind of holomorphic object $f$ must be. $f$ must be a half-density on the Hecke correspondence $Z_{R,p}$
in the holomorphic sense: $f=v(\vec x, z,\vec m) (\mathrm d \vec x\mathrm d z\mathrm d\vec m)^{1/2}$. In more standard language, $f$ must be
an element of $H^0(Z_{R,p}, K^{1/2}_{Z_{R,p}})$. As explained earlier, this is the expected form of the answer in the $A$-model for a left endpoint of $T_{R,p}$.
Similarly, $\widetilde f$ is an antiholomorphic section of the anticanonical bundle of $Z_{R,p}$, again in accord with the $A$-model expectation.
If and only if $Z_{R,p}$ is a Calabi-Yau manifold, there is a particular holomorphic section $\lambda_0$ of $K_{Z_{R,p}}^{1/2}$
that is everywhere nonzero. If such a $\lambda_0$
exists, then the data that defines any other holomorphic corner is $f= g \lambda_0$, where $g$ is a holomorphic function on $Z_{R,p}$. In particular, such
a function $g$ (with only polynomial growth)
is a polynomial in the Hitchin Hamiltonians $H_{{\mathcal P},\alpha}$ and the holomorphic function $\delta$ that was defined in eqn. (\ref{defdelta}), up to the relation
$\delta^2 = -\det \varphi(p)$, which expresses $\delta^2$ as a linear combination of the $H_{{\mathcal P},\alpha}$.
A nonconstant polynomial $g(H_{{\mathcal P},\alpha},\delta)$ has nontrivial zeroes, so $g\lambda_0$
is everywhere nonzero only if $g$ is a constant, showing that $\lambda_0$ is unique, up to a constant multiple, if it exists.\footnote{More generally, if $Z$ is a complex manifold with $b_1(Z)=0$, then an everywhere nonzero
holomorphic function $g$ on $Z$, with no exponential growth, is constant. Indeed, since $b_1(Z)=0$, the closed 1-form $\mathrm d g/g$ is exact, $\mathrm d g/g=\mathrm d w$
for some $w$, so $g=C e^w$ (with a nonzero constant $C$) and $g$ has exponential growth unless it is constant.
The Hecke correspondence for $G_{\Bbb C}$-bundles satisfies
$b_1(X_{R,p})=0$. The Hecke correspondence for $G_{\Bbb C}$ Higgs bundles is the conormal bundle of $X_{R,p}$ and hence $b_1(Z_{R,p})=0$.}
Corners of the form $g \lambda_0$, where $g$ is a polynomial in the Hitchin Hamiltonians and $\delta$, precisely match
the corners for Wilson lines, built from $s$, $Ds$ and
polynomials in the observables dual to the Hitchin Hamiltonians.
Such a $\lambda_0$ does indeed exist, by virtue of a result of Beilinson and Drinfeld \cite{BD} that was important
in the work of Etingof, Frenkel, and Kazhdan \cite{EFK2}.
The properties of $\lambda_0$ mirror those of the simplest Wilson line corner $s$ of Section \ref{wilop}. In particular, according to the result of Beilinson and Drinfeld, $\lambda_0$, like $s$, varies with $p$ as a section of $K_p^{-1/2}$. Of course, the existence of such a mirror of $s$ is expected from electric-magnetic duality.
A slightly different formulation was useful in \cite{EFK2}. To explain this, let us go back to
eqn. (\ref{luff}), from which we see that if $\mu$ can be holomorphically factorized, then the holomorphic
factor is a holomorphic form $k = w(\vec x,z) (\mathrm d \vec x \mathrm d z)^{1/2} (\mathrm d \vec n)^{-1/2}$. We can replace $\mathrm d z^{1/2} (\mathrm d\vec n)^{-1/2}$ with
$\mathrm d z (\mathrm d \vec y)^{-1/2}$ where $\vec y=(z,\vec n)$ parametrizes the output of the Hecke transformation. So in other words a holomorphically factorized kernel
will come from a holomorphic object\footnote{Since $\vec y$ is determined by $\vec x$ and $z$, and reciprocally $\vec x$ is determined by $\vec y$ and $z$,
we could equally well write $w(\vec y,z)$ instead of $w(\vec x,z)$.}
\begin{equation}\label{conf} k=w(\vec x,z) (\mathrm d \vec x)^{1/2} (\mathrm d\vec y)^{-1/2}\mathrm d z. \end{equation}
If we multiply $k$ by its complex conjugate, we get
\begin{equation}\label{onf}|k|^2= |w(\vec x,z)|^2 |\mathrm d\vec x| |\mathrm d z\mathrm d\overline z| |\mathrm d\vec y|^{-1}.\end{equation}
This leads directly to the definition used in \cite{EFK2}. The quantity $|k|^2$ can be regarded as a map from half-densities in $\vec y$ to half-densities in $\vec x$,
valued in differential forms $|\mathrm d z \mathrm d\overline z|$ that can be integrated over ${\Bbb{CP}}^1_x$. That integral gives the Hecke operator at the point $p\in C$:
\begin{equation}\label{tonf} H_p=\int_{{\Bbb{CP}}^1_x} |k|^2. \end{equation}
In Section \ref{sec:chiral}, we will interpret some of these statements via two-dimensional chiral algebras.
In particular, the considerations about adjoint-valued chiral fermions in Section \ref{fft} are a physicist's interpretation of the original analysis of Beilinson
and Drinfeld.
The chiral algebra approach is local on $C$ and and can potentially be extended to
situations where the space of Hecke modifications relating two given bundles has positive dimension.
\subsubsection{The Affine Grassmannian}\label{affgr}
So far, we have considered the simplest examples of Hecke modifications. But to develop the theory
further, one wants a more systematic approach.
As motivation, we consider first the case of a holomorphic vector bundle of rank $n$.
Let $z$ be a local holomorphic parameter that vanishes at a point $p\in C$. Let $U$ be a small neighborhood of $p$. $C$ has an open cover
with two open sets, namely $U$ and $C'=C\backslash p$ ($C$ with $p$ removed).
Pick a trivialization of $E$ in a small neighborhood $U$ of the point $p$ and restrict $E$ to $C'$. Let $E_0\to U$ be
a trivial rank $n$ vector bundle. We have an open cover of $C$ by open sets $C', U$ with vector bundles $E\to C'$ and $E_0\to U$.
So we can define a new vector bundle $E'\to C$ by gluing together $E$ and $E_0$ over $U'=C'\cap U$
via a gauge transformation. For example, we can use the diagonal gauge transformation from $E_0$ to $E$
\begin{equation}\label{gtr} g(z) =\begin{pmatrix} z^{d_1} &&&\cr
& z^{d_2} &&\cr
&&\ddots & \cr
&&& z^{d_n}\end{pmatrix},\end{equation} with integers $d_1,\cdots, d_n$.
For $G={\mathrm U}(n)$, with suitable choices of the $d_i$, this gives an example of a Hecke modification of a $G_{\Bbb C}$-bundle dual to an arbitrary irreducible finite-dimensional
representation of $G={\mathrm U}(n)$.
For $G={\mathrm{SU}}(n)$, one modifies this by requiring $\sum_i d_i=0$ so that $g(z)$ is valued in ${\mathrm{SL}}(n,{\Bbb C})$, and for $G={\mathrm{PSU}}(n)$, one considers the
$d_i$ to be valued in $\frac{1}{n}{\mathbb Z}$, with $d_i-d_j\in {\mathbb Z}$ (this is equivalent to saying that if $g(z)$ is written in a representation of ${\mathrm{PSU}}(n)$, then only
integer powers of $z$ appear). A constant shift of all $d_i$ by $d_i\to d_i+c$, where $c\in {\mathbb Z}$ (or $c\in \frac{1}{n}{\mathbb Z}$ for $G={\mathrm{PSU}}(n)$) does not affect the space
of Hecke modifications, since one can compensate for it by $E'\to E'\otimes {\mathcal O}(p)^c$.
For any $G$, one can make a similar construction replacing diagonal matrices whose entries are powers of $z$ with a homomorphism $g(z)= z^{\sf m} :{\Bbb C}^*\to T_{\Bbb C}$
where ${\Bbb C}^*$ is the punctured $z$-plane, $T_{\Bbb C}$ is a complex maximal torus of $G_{\Bbb C}$, and ${\sf m}$, which generalizes the $d$-plet of integers $(d_1,\cdots, d_n)$
in the previous paragraph, is an integral weight of the dual group $G$, corresponding physically to the magnetic charge of an 't Hooft operator.
What we have described so far is a standard example of a Hecke transformation dual to an arbitrary finite-dimensional representation $R$ of $G$; $R$ is encoded
in the integers $d_i$ or the choice of ${\sf m}$.
This construction depended on the initial choice of a trivialization of $E$ over $U$. By varying the choice of trivialization, one can
obtain a whole space of Hecke modifications of the same type. Once we pick a reference trivialization, any other trivialization of $E$
over the set $U$ would be obtained from the reference
one by applying
some gauge transformation $g'(z)$ that is holomorphic in $U$. The standard Hecke modification associated to this alternative trivialization of $E$ is described in the reference trivialization by the modified singular gauge transformation $g'(z) z^{\sf m}$.
Two singular gauge transformations lead to the same $E'$ if they can be related by composition from the right with a gauge transformation $g''(z)$ defined on $U$. Two trivializations of $E$ will thus give the
same standard Hecke modification if the corresponding gauge transformations $g_1'(z)$ and $g_2'(z)$ satisfy
\begin{equation}
g_1'(z)z^{\sf m} = g_2'(z)z^{\sf m} g''(z)
\end{equation}
for some $g''(z)$.
These relations can be formalized with the help of the {\it affine Grassmannian} ${\mathrm{Gr}}_{G_{\Bbb C}}$. It is customary to denote as $G_{\Bbb C}[{\cal K}]$ the (infinite-dimensional) space of $G_{\Bbb C}$-valued gauge transformations defined on $U'$ and as $G_{\Bbb C}[{\cal O}]\subset G_{\Bbb C}[{\cal K}]$ the subspace of such gauge transformations which extend holomorphically to $U$. The affine Grassmannian
\begin{equation}
{\mathrm{Gr}}_{G_{\Bbb C}} \equiv G_{\Bbb C}[{\cal K}]/G_{\Bbb C}[{\cal O}]
\end{equation}
parameterizes equivalent singular gauge transformations. We define ${\mathrm{Gr}}^{\sf m}_{G_{\Bbb C}}$ to be the orbit in ${\mathrm{Gr}}_{G_{\Bbb C}}$ of the standard
Hecke modification by the gluing function $z^{\sf m}$:
\begin{equation}
{\mathrm{Gr}}^{\sf m}_{G_{\Bbb C}} \equiv \left[G_{\Bbb C}[{\cal O}] z^{\sf m} \right].
\end{equation}
Every point in ${\mathrm{Gr}}^{\sf m}_{G_{\Bbb C}}$ is on such an orbit, for some ${\sf m}$.
For generic choices of $G$ and ${\sf m}$, one runs into a phenomenon of ``monopole
bubbling'' in which a downward jump can occur in the magnetic charge of an 't Hooft or Hecke operator
(this was introduced in \cite{Kron,Pauly}; see Section 10.2 of \cite{KW} for a short introduction).
Essentially, the orbit ${\mathrm{Gr}}^{\sf m}_{G_{\Bbb C}}$ is not closed in the affine Grassmannian, and the (possibly singular) closure of the orbit
includes other orbits ${\mathrm{Gr}}^{{\sf m}'}_{G_{\Bbb C}}$ with smaller charge. This can considerably complicate the analysis.
For $G={\mathrm U}(n)$ or a related group ${\mathrm{SU}}(n)$ or $\mathrm{PSU}(n)$, the condition to avoid monopole bubbling is that $|d_i-d_j|\leq 1$ for all $i,j$.
Up to a constant shift of all the $d_i$ (which does not affect the space of Hecke modifications, as explained in the discussion of eqn. (\ref{gtr})),
to avoid monopole bubbling we can assume that $k$ of the $d_i$ are $-1$ and the others 0.
Explicitly, this corresponds to a Hecke modification of the following sort.
For $E\to C$ a rank $n$ holomorphic vector bundle and $p\in C$, one chooses a $k$-dimensional subspace $V\subset E_p$ and defines a new vector
bundle $E'\to C$ whose sections are sections of $E$ that are allowed to have a simple pole at $p$ with residue in $V$. The space of Hecke
modifications of this type is parametrized by the choice of $V$, that is, by the Grassmannian of $k$-dimensional subspaces of $E_p$.
Such Hecke modifications are dual to the $k^{th}$ antisymmetric power of the fundamental representation of $G={\mathrm U}(n)$ or ${\mathrm{SU}}(n)$.
In this situation, $\varphi$ will act as a $k \times k$ matrix on the polar part of $w$. The characteristic polynomial of the $k \times k$ restriction of $\varphi$
takes the general form $u(x) = x^k + \delta_1 x^{k-1} + \cdots \delta_k$, for some holomorphic functions $\delta_1,\cdots,\delta_k$. These functions
satisfy polynomial relationships with the Hitchin Hamiltonians ${\mathcal H}_{{\mathcal P},\alpha}(\varphi)$ which encode the
constraint that $u(x)$ divides the characteristic polynomial of $\varphi$.
\section{Quantum-Deformed WKB Condition}\label{wkb}
\subsection{Quantum States And The Hitchin Fibration}\label{hitchinwkb}
\begin{figure}
\begin{center}
\includegraphics[width=4.7in]{Example6.pdf}
\end{center}
\caption{\small (a) A $B$-brane ${\mathscr B}_x$ supported at a point $x$, defining (with the help of
some data at the corners) a state
in ${\mathcal H}={\mathrm{Hom}}({\mathscr B}_{{\overline{\mathrm{op}}}},{\mathscr B}_{{\mathrm{op}}})$. (b) A dual $A$-brane ${\mathscr B}_F$ supported on a fiber $F$ of the
Hitchin fibration, defining a state in ${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$. (c) A pairing between states associated
to $A$-branes ${\mathscr B}$ and ${\mathscr B}'$ (together with data at corners). If at least one of ${\mathscr B}$ and ${\mathscr B}'$ has compact support,
this pairing is well-defined; otherwise, it may not be. \label{example6}}
\end{figure}
A slightly different way to think about an eigenfunction of the Wilson operators and the Hitchin Hamiltonians is as follows. Let $x$ be a point in
${\mathcal M}_H(G^\vee,C)$ corresponding to a flat $G^\vee_{\Bbb C}$ bundle $E^\vee_x\to C$,
and let ${\mathscr B}_x$ be a $B$-brane supported at $x$, with a rank 1 (and inevitably trivial) ${\mathrm{CP}}$ bundle. Then the part of ${\mathrm{Hom}}({\mathscr B}_x,{\mathscr B}_{\mathrm{op}})$ of degree zero\footnote{The $B$-model has a conserved fermion number symmetry, with the differential $Q$ having fermion number or degree 1. When $x\in L_{\mathrm{op}}$,
${\mathrm{Hom}}({\mathscr B}_x,{\mathscr B}_{\mathrm{op}})$ is also nonzero in positive degrees, but the positive degree states do not contribute in this discussion because ${\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_{\mathrm{op}})$ is
entirely in degree 0. A similar remark applies later when we discuss the $A$-model.}
is a copy of ${\Bbb C}$ if $x\in L_{\mathrm{op}}$, that is if $E^\vee_x$ is a holomorphic oper. Otherwise ${\mathrm{Hom}}({\mathscr B}_x,{\mathscr B}_{\mathrm{op}})=0$.
Similarly, the degree 0 part of ${\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_x)$ is ${\Bbb C}$ if $x\in L_{\overline{\mathrm{op}}}$, that is if $E^\vee_x$ is an antiholomorphic oper, and otherwise zero. If and only
if $E^\vee$ is both an oper and an anti-oper, we can pick nonzero elements $\alpha\in {\mathrm{Hom}}({\mathscr B}_x,{\mathscr B}_{\mathrm{op}})$, $\beta\in {\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_x)$,
and then define the element $\alpha\circ\beta \in {\mathcal H}={\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_{\mathrm{op}})$. A picture representing this situation is fig. \ref{example6}(a).
The brane ${\mathscr B}_x$ is used to provide a boundary condition at the bottom of the strip; $\alpha$ and $\beta$ provide the ``corner data'' needed to
define boundary conditions at the corners of the picture. The path integral with the ``initial conditions'' set by ${\mathscr B}_x,\alpha,\beta$ defines a physical state of the system.
This state is the desired eigenstate of the Hitchin Hamiltonians and the Wilson operators.
Via electric-magnetic duality, we can get a dual picture. After compactification on an oriented two-manifold $C$, electric-magnetic duality of
${\mathcal N}=4$ super Yang-Mills theory reduces at low energies to a mirror symmetry of the Higgs bundle moduli space \cite{BJSV,JWS}. As explained in
\cite{HT}, this is a rare instance in which the SYZ interpretation \cite{SYZ} of mirror symmetry as $T$-duality on the fibers of a family of Lagrangian tori can
be made very explicit. The Hitchin fibration is the map that takes a Higgs bundle $(E,\varphi)$ to the characteristic poiynomial of $\varphi$.
The fibers of the map are abelian varieties that are
complex Lagrangian submanifolds in complex structure $I$. This in particular means that they are Lagrangian submanifolds
from the point of view of the real symplectic structure $\omega_K={\mathrm{Im}}\,\Omega_I$ of the Higgs bundle moduli space. Hence, in the $A$-model
with symplectic structure $\omega_K$, the Hitchin fibration can be viewed as an SYZ fibration by Lagrangian submanifolds that generically are tori.
$T$-duality on the fibers of this fibration maps the $A$-model of symplectic structure $\omega_K$ to the $B$-model of complex structure $J$.
The relation of electric-magnetic duality to this instance of mirror symmetry was an important input in \cite{KW}.
In particular, mirror symmetry in this situation maps a rank 1 brane ${\mathscr B}_x$ supported at a point $x$ to a brane ${\mathscr B}_F$ supported on a fiber $F$ of the
Hitchin fibration, with a ${\mathrm{CP}}$ bundle that is a flat line bundle\footnote{\label{canon} In general, the ${\mathrm{CP}}$ bundle of a rank 1 brane is more canonically a ${\mathrm{Spin}}_c$
structure rather than a line bundle. In the present context, as $F$ is an abelian variety and so has a canonical spin structure, the distinction is not important.}
${\mathcal S}\to F$. The duals of $\alpha\in {\mathrm{Hom}}({\mathscr B}_x,{\mathscr B}_{\mathrm{op}})$ and $\beta\in{\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_x)$ are elements $\alpha'\in {\mathrm{Hom}}({\mathscr B}_F,{\mathscr B}_{\mathrm{cc}})$ and $\beta'\in {\mathrm{Hom}}(
\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_F)$. The element $\alpha'\circ\beta'\in {\mathcal H}$, which corresponds to the picture of fig. \ref{example6}(b), represents an element of ${\mathcal H}$ in the magnetic
description, in which the Hitchin Hamiltonians are differential operators acting on half-densities on ${\mathcal M}(G,C)$ and Wilson operators are replaced by
't Hooft operators.
The branes ${\mathscr B}_x$ and ${\mathscr B}_F$ both have compact support, consisting of either a point $x$ or a fiber $F$ of the Hitchin fibration. Compact support
makes it manifest that the states created by these branes (plus corner data) are normalizable. More than that, compact support
means that these states have well-defined pairings with states that are constructed similarly using an arbitrary brane ${\mathscr B}'$, possibly
with noncompact support, again with suitable corner data. In other words, the pairing constructed from the rectangle of fig. \ref{example6}(c) is always
well-defined, regardless of the brane at the top of the rectangle, as long as the brane at the bottom of the rectangle has compact support. This means roughly that an arbitrary brane, with a choice of corner data, defines a distributional state, not
necessarily a normalizable vector in the Hilbert space ${\mathcal H}$,
while a brane of compact support defines a vector that can be paired with any distribution.
The Hilbert space is contained in the space of distributional states, and contains a subspace, roughly analogous to a Schwartz space, spanned by states
associated to branes of compact support.
The eigenfunctions of the Hitchin Hamiltonians lie in this subspace.
The $B$-model gives a clear answer to the question of which points $x\in {\mathcal M}_H(G^\vee,C)$ are associated to physical states: these are points in
$\Upsilon=L_{\mathrm{op}}\cap L_{\overline{\mathrm{op}}}$.
It is more difficult to extract directly from the $A$-model a prediction
for which pairs $F,{\mathcal S}$ are similarly associated to physical states. We will only be able to get a sort of semiclassical answer, which we expect to be valid
asymptotically, in a sense that will be explained.
We recall that the ${\mathrm{CP}}$ bundles of the branes ${\mathscr B}_{\mathrm{cc}}$ and $\overline{\mathscr B}_{\mathrm{cc}}$ are dual line bundles ${\mathcal L}$ and ${\mathcal L}^{-1}$, and that the prequantum line bundle
of ${\mathcal M}_H(G,C)$, in the sense of geometric quantization, is ${{\mathfrak L}}={\mathcal L}^2$. We want the condition on $F$ and ${\mathcal S}\to F$ such that corners
$\alpha'\in {\mathrm{Hom}}({\mathscr B}_F,{\mathscr B}_{\mathrm{cc}})$ and $\beta'\in {\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_F)$ exist.
We will describe the condition for $\alpha'$ and $\beta'$ to exist to lowest order in $\sigma$-model perturbation theory, and we will also explain to what
extent higher order corrections can or cannot change the picture, in the regime where they are small. So let us first explain the regime in which $\sigma$-model
perturbation theory is valid. This is the case that the Higgs field $\varphi$ is parametrically large and far away from the discriminant locus (on which
the spectral curve becomes singular). Concretely if $(E,\varphi)$ is any Higgs bundle with a smooth spectral curve, and we rescale $\varphi$ by
a large factor
\begin{equation}\label{kcf} \varphi\to t\varphi,~~|t|\gg 1,\end{equation}
with any fixed value of $\mathrm{Arg}\,t$, then $\sigma$-model perturbation theory becomes valid. For $t\to\infty$, the Higgs bundle
moduli space has a concrete ``semi-flat'' description \cite{GMN}, leading to semiclassical results and asymptotic expansions as $t\to\infty$ for a variety of questions, including what we will consider here. We will call the region $t\to\infty$ the WKB limit, for reasons that will emerge.
The brane ${\mathscr B}_F$ is of type $(B,A,A)$. For such a brane, with support $F$ and ${\mathrm{CP}}$ bundle ${\mathcal S}$, the leading $\sigma$-model approximation to
${\mathrm{Hom}}({\mathscr B}_F,{\mathscr B}_{\mathrm{cc}})$, where ${\mathscr B}_{\mathrm{cc}}$ has ${\mathrm{CP}}$ bundle ${\mathcal L}$, is the $\overline\partial$ cohomology of $F$ with values in\footnote{As remarked in footnote \ref{canon}, ${\mathcal S}$ and likewise
${\mathcal S}^{-1}$ is canonically a ${\mathrm{Spin}}_c$ structure on $F$, not a line bundle. Likewise, $K_F^{1/2}$ is not canonically defined as a line bundle (and for general may not
exist at all as a line bundle) but on any complex manifold, $K_F^{1/2}$ is canonically defined as a ${\mathrm{Spin}}_c$ structure. Hence the product ${\mathcal S}^{-1}\otimes
K_F^{1/2}$ exists canonically as an ordinary line bundle and the answer stated in the text makes sense. In our application,
$K_F$ is trivial, and we can likewise take $K_F^{1/2}$ to be trivial and define ${\mathcal S}$ as a line bundle.} ${\mathcal L}|_F\otimes {\mathcal S}^{-1}\otimes K_F^{1/2}$,
where $K_F^{1/2}$ is a square root of the canonical bundle of $F$ (see Appendix B of \cite{GW}).
This comes about because the leading $\sigma$-model approximation to the $A$-model
differential $Q$ is
\begin{equation}\label{jno}Q\sim \overline\partial+a , \end{equation}
where $a$ is a $(0,1)$-form on $F$ that defines the complex structure of the line bundle ${\mathcal L}\otimes {\mathcal S}^{-1}\otimes K_F^{1/2}$.
The line bundle ${\mathcal L}$ is flat when restricted to $F$, because $F$ is a complex Lagrangian submanifold. The line bundle ${\mathcal S}$ is flat, because it is the
${\mathrm{CP}}$ bundle of a Lagrangian brane. And as $F$ is a complex torus, we can take $K_F^{1/2}$ to be trivial and omit this factor. The $\overline\partial$ cohomology
of a complex torus with values in a flat line bundle ${\mathcal L}|_F\otimes {\mathcal S}^{-1}$ vanishes unless this line bundle is trivial. So in the leading $\sigma$-model approximation,
${\mathrm{Hom}}({\mathscr B}_F,{\mathscr B}_{\mathrm{cc}})$ vanishes unless we pick ${\mathcal S}\cong {\mathcal L}|_F$. For ${\mathcal S}={\mathcal L}|_F$,we can pick a nonzero $\alpha' \in H^0(F,{\mathcal L}|_F\otimes {\mathcal S}^{-1})\cong {\Bbb C}$.
Similarly, as the ${\mathrm{CP}}$ bundle of ${\mathscr B}_{\overline{\mathrm{op}}}$ is ${\mathcal L}^{-1}$,
the leading $\sigma$-model approximation to ${\mathrm{Hom}}({\mathscr B}_{\overline{\mathrm{op}}},{\mathscr B}_F)$ is the $\overline\partial$ cohomology of $F$ with values in ${\mathcal S}\otimes {\mathcal L}|_F\otimes K_F^{1/2}$, or,
taking $K_F^{1/2}$ to be trivial, just ${\mathcal S}\otimes {\mathcal L}|_F$.
This cohomology vanishes if ${\mathcal S}\otimes {\mathcal L}|_F$ is nontrivial; if it is trivial, which is so precisely if ${\mathcal S}={\mathcal L}^{-1}|_F$, we can choose a nonzero $\beta'\in H^0(F, {\mathcal S}\otimes{\mathcal L}|_F)\cong {\Bbb C}$.
In short, we can use the brane ${\mathscr B}_F$ with suitable corner data to define a state in ${\mathcal H}$ if and only if we can choose ${\mathcal S}$ to be isomorphic
to both ${\mathcal L}|_F$ and ${\mathcal L}^{-1}|_F$. In other words, the condition is that ${\mathcal L}^2|_F$ must be trivial. As the prequantum line bundle over ${\mathcal M}_H(G,C)$
is ${{\mathfrak L}}={\mathcal L}^2$, the condition is that ${{\mathfrak L}}$ must be trivial when restricted to $F$.
This is actually the WKB condition of elementary quantum mechanics, which also is part of the theory of geometric quantization. To put the condition
in a more familiar form, recall that the symplectic form $\omega_K$ of ${\mathcal M}_H(G,C)$ is cohomologically trivial, so it can be written as $\omega_K=\mathrm d\lambda_K$
for a 1-form $\lambda_K$. The prequantum line bundle ${{\mathfrak L}}$ is supposed to be a unitary line bundle with a connection of curvature $\omega_K$.
We can take ${{\mathfrak L}}$ to be a trivial line bundle with the connection $D=\mathrm d+{\mathrm i}\lambda_K$. ${{\mathfrak L}}$ is flat when restricted to $F$ because $F$ is a Lagrangian
submanifold. The condition that ${{\mathfrak L}}$ is trivial is that its global holonomies vanish. In other words, the condition is that if $\gamma\subset F$ is a 1-cycle,
then the holonomy of ${{\mathfrak L}}$ around $\gamma$ must vanish: $\exp({\mathrm i} \oint_\gamma\lambda_K)=1$ or in other words $\oint_\gamma \lambda_K\in 2\pi{\mathbb Z}$.
To put this condition in a perhaps more familiar form, we can approximate ${\mathcal M}_H(G,C)$ as a cotangent bundle $T^*{\mathcal M}(G,C)$
and choose $\lambda_K=\sum_i p_i\mathrm d q^i$, where $q^i$ are coordinates on the base of the cotangent bundle and $p_i$ are fiber coordinates. Then
the condition is that
\begin{equation}\label{weffo}\sum_i\oint_\gamma p_i \mathrm d q^i\in 2\pi {\mathbb Z},\end{equation}
which may be recognizable as the WKB condition for associating a quantum state to the Lagrangian submanifold $F$.
We have reached this conclusion to lowest order in $\sigma$-model (or gauge theory) perturbation theory, and we do not claim that the result is exact.
However, it is possible to argue that, at least sufficiently near the WKB limit, there is a quantum-corrected WKB condition that leads to qualitatively
similar results. Consider correcting the
computation of ${\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_F)$ or ${\mathrm{Hom}}({\mathscr B}_F,{\mathscr B}_{\mathrm{cc}})$ in perturbation theory in inverse powers of the parameter $t$ that was introduced in
eqn. (\ref{kcf}). This has the effect of shifting the $(0,1)$-form $a$ by a $(0,1)$-form $c_1/t+c_2/t^2+\cdots$, leading to
\begin{equation}\label{zeflo}Q=\overline\partial+a +\frac{c_1}{t}+\frac{c_2}{t^2}+\cdots .\end{equation}
Whatever the $c_k$ are, we can compensate for them by shifting $a$. An arbitrary shift in $a$ can be interpreted as the sum of a $\overline\partial$-exact
term, which does not affect the cohomology of $Q$, plus a term that can be interpreted as resulting from a shift in the line bundle ${\mathcal S}$. Thus, instead of needing
${\mathcal L}|_F\otimes {\mathcal S}^{-1}$ and ${\mathcal L}|_F\otimes {\mathcal S}$ to be trivial in order to associate a quantum state with $F$, we need ${\mathcal S}$ to satisfy conditions
that are asymptotically close to these. Correspondingly, the classical WKB condition for ${{\mathfrak L}}|_F$ to be trivial is modified, at least for
sufficiently large $t$, to a quantum WKB condition that determines which fibers of the Hitchin fibration are associated to quantum states.
\subsection{WKB Condition and Special Geometry}
In order to better understand the quantization condition, it is useful to recall some facts about the special geometry which
governs the structure of ${\mathcal M}_H(G,C)$, as well as complex integrable systems that arise in Seiberg-Witten theory \cite{Seiberg:1994rs,GKMMM,DoW,Freed:1997dp, Gaiotto:2008cd}.
The basic ingredients of the geometry are
\begin{itemize}
\item The base ${{\mathcal B}}$ of the Hitchin fibration, of complex dimension $r$. We denote a point in the base as $u$ and the discriminant locus as ${\cal D}$.
\item A local system $\Gamma$ of lattices of rank $2r$ defined over ${\cal B}\backslash {\cal D}$ ($\cal B$ with $\cal D$ removed), equipped with a symplectic form $\langle\cdot,\cdot \rangle$.\footnote{In a true Seiberg-Witten geometry the symplectic form is integer-valued and $\Gamma$ is self-dual. The Seiberg-Witten geometry is self-mirror. Hitchin systems are related to Seiberg-Witten geometries by discrete orbifold operations which relax these conditions.} We will denote a charge (an
element of $\Gamma$) as $\gamma$.
\item A collection of central charges $Z: \Gamma \to {\Bbb C}$ which vary holomorphically on ${\cal B}\backslash{\cal D}$. We will denote the central charge evaluated on a charge $\gamma$ as $Z_\gamma$. The $Z_\gamma$ are also identified with periods of the canonical 1-form on the spectral curve of the Hitchin system.
\item Real angular coordinates $\theta: \Gamma \to S^1$ on the fibers of the complex integrable system. We will denote the coordinates evaluated on a charge $\gamma$ as $\theta_\gamma$. They are dual to the $Z_\gamma$ under the complex Poisson bracket
\begin{equation}
\{Z_\gamma, \theta_{\gamma'} \} = \langle \gamma, \gamma' \rangle
\end{equation}
\end{itemize}
The complex symplectic form in complex structure $I$ is defined with the help of the inverse pairing:
\begin{equation}
\Omega = \langle \mathrm d Z, \mathrm d\theta \rangle
\end{equation}
Correspondingly, we have a 1-form
\begin{equation}
\lambda = \langle Z, \mathrm d\theta \rangle\equiv \lambda_J + {\mathrm i} \lambda_K
\end{equation}
satisfying $\mathrm d\lambda=\Omega$, with periods $2 \pi Z_\gamma$.
We will use this special coordinate system for ${\mathcal M}_H(G^\vee,C)$. The parameters $u$ specify a fiber $F$ of the Hitchin fibration, which is an abelian
variety, and the angles $\theta_\gamma$ parameterize
the choice of a flat line bundle ${\mathcal S}\to F$. The WKB conditions for the existence of corners are thus that $\theta_\gamma = \pm \mathrm{Im}\,Z_\gamma$
and the quantization condition becomes $\mathrm{Im}\, Z_\gamma \in \pi \mathbb{Z}$.
The ${\mathscr B}_F$ branes as $A$-branes are supposed to depend on the data $(u, \theta)$ holomorphically in complex structure $J$,
as the $A_K$ twist on ${\mathcal M}_H(G,C)$ is mirror to the $B_J$ twist on ${\mathcal M}_H(G^\vee,C)$. Writing functions of $(u, \theta)$ which are holomorphic in complex structure $J$ is essentially as challenging as computing the hyper-K\"ahler metric on the moduli space. In the WKB region, though,
the functions
\begin{equation} \label{eq:semiflat}
X_\gamma = \exp \left(\mathrm{Re}\,Z_\gamma + i \theta_\gamma \right)
\end{equation}
are an excellent ``semiflat'' approximation to $J$-holomorphic functions. The Cauchy-Riemann equations fail by corrections suppressed exponentially in the WKB region \cite{Gaiotto:2008cd}.\footnote{One can define locally some corrected $X_\gamma$ which are truly $J$-holomorphic, but non-trivial ``wall crossing'' coordinate transformations are required in different patches
\cite{GMN}. This subtlety will not be important here. These statements have a transparent $A$-model interpretation. The semiclassical $A$-brane moduli combine the ${\mathrm{CP}}$ data with the deformation data associated to the same 1-forms on the brane support to give the $({\Bbb C}^*)^{2r}$ coordinates $X_\gamma$. The only corrections are non-perturbative and due to disk instantons ending on a cycle $\gamma$. These only exist at codimension one loci where $Z_\gamma$ is real
and lead to the wall-crossing transformations. } The complex symplectic form in complex structure $J$ is approximately
\begin{equation}
\Omega_J = \langle \mathrm d \log X, \mathrm d \log X \rangle
\end{equation}
Stated in this language, the WKB condition for the existence of a corner becomes
\begin{equation} \label{eq:wkb}
X_\gamma = e^{Z_\gamma(u)}
\end{equation}
This can be interpreted as the parametric definition of an $r$-dimensional complex Lagrangian submanifold, as expected.
Surprisingly, the parameters coincide with the coordinates $u$ on ${\cal B}\backslash{\cal D}$ and thus we get an approximate holomorphic identification
between ${\cal B}\backslash{\cal D}$ and the space of branes equipped with a corner.
This identification is only valid in a semiclassical approximation. To make such an expansion, as in Section \ref{hitchinwkb}, we replace $u$ with $tu$, and
take $t$ to be large. The corner condition receives perturbative corrections, which
will take a systematic form
\begin{equation}
X_\gamma = e^{t Z_\gamma(\widetilde u) + t^{-1} c_{1,\gamma}(\widetilde u) + \cdots}
\end{equation}
We introduced parameters $\widetilde u$ that cannot be taken to coincide with
the coordinates $u$ that are holomorphic in complex structure $I$. The relation between the two follows from the comparison of (\ref{eq:wkb}) and (\ref{eq:semiflat}):
\begin{equation}
\mathrm{Re}\,Z_\gamma(u) = \mathrm{Re}\,Z_\gamma(\widetilde u) + t^{-2} \mathrm{Re}\,c_{1,\gamma}(\widetilde u) + \cdots
\end{equation}
The perturbatively-corrected quantization condition to have both types of corners will become
\begin{equation}
\mathrm{Im}\,Z_\gamma(\widetilde u) + t^{-2} \mathrm{Im}\,c_{1,\gamma}(\widetilde u) + \cdots \in \pi \mathbb{Z}.
\end{equation}
\subsection{Match with real WKB opers}
There is another natural occurrence of the periods $Z_\gamma$ of the canonical 1-form on the spectral curve. If we attempt a WKB analysis on $C$ of the oper differential equation, the monodromy data of the oper flat connection will be computed at the leading order in terms of the exponentiated periods $e^{Z_\gamma(\widetilde u) }$ \cite{Voros1,Voros2,Voros3,Voros4,Gaiotto:2014bza}. Here we employed a non-canonical identification between $L_{{\mathrm{op}}}$ and ${{\mathscr B}}$. For example, for $G^\vee = {\mathrm{SU}}(2)$ we would write the oper differential operator as a reference operator deformed by a large quadratic differential $U(z;\widetilde u)$
\begin{equation}
\partial^2_z - t_2(z) \equiv \partial^2 - t^2 U(z;\widetilde u)- t_2^{0}(z)
\end{equation}
and the leading WKB approximation would involve the periods $Z_\gamma(\widetilde u)$ of the WKB 1-form $\sqrt{U(z;\widetilde u)} \mathrm d z$. Here $t_2(z)$ is the classical stress tensor, $t_2^0(z)$ a reference choice of classical stress tensor and $t$ is the scaling parameter.
The WKB calculation is really a combination of a topological and an analytic problem. The topological problem involves a careful Stokes analysis of the asymptotic behaviour of the flat sections of the connection. The analytic problem involves the computation of the Voros symbols, which are periods of the all-orders WKB 1-form. Remarkably, the Voros symbols precisely compute the corrected $X_\gamma$ coordinates of the oper in the space of flat $G^\vee$ connections \cite{Gaiotto:2014bza}:
\begin{equation}
X_\gamma = e^{t Z_\gamma(\widetilde u) + t^{-1} c_{1,\gamma}(\widetilde u) + \cdots}
\end{equation}
The WKB analysis of the oper differential equation thus gives directly the $B$-model analogue of the WKB corner condition in the mirror $A$ model.
\section{Real Bundles}\label{realb}
\subsection{The Setup}\label{setup}
So far we have studied the quantization of ${\mathcal M}_H(G,C)$ as a real symplectic manifold.
An alternative is to view ${\mathcal M}_H(G,C) $ as a complex symplectic manifold with complex structure $I$ and holomorphic symplectic structure
$\Omega_I=\omega_J+{\mathrm i} \omega_K$, and quantize a real symplectic submanifold ${\mathcal M}_H^{\mathbb{R}} \subset {\mathcal M}_H$. For this, as explained in Section \ref{background}, we look for an antiholomorphic involution $\tau$ of ${\mathcal M}_H(G,C)$
that satisfies $\tau^*\Omega_I=\overline\Omega_I$. In our application, ${\mathcal M}_H^{\mathbb{R}}$ will be a cotangent bundle, leading to a simple description of the Hilbert space
and of the action of the Hecke operators, and a real integrable system, equipped with a real version of the Hitchin fibration.
The classification of the possible antiholomorphic involutions is the same whether one considers holomorphic $G_{\Bbb C}$ bundles as in \cite{Hurtu,Hurtu2}
or Higgs bundles, as in
\cite{BaSc,BaSc2,BG}. Those references provide much more detail than we will explain here. We also note that
the three-manifold $U_\tau=(\Sigma\times \widehat I)/\{1,\tau\}$ that we will use in studying the duality was introduced in Section 11 of \cite{BaSc}. That paper also contains a duality proposal based on the structure of ${\mathcal M}_H^{\mathbb{R}}(G,C)$ as a real integrable system.
A suitable involution of ${\mathcal M}_H$ can be constructed starting with an antiholomorphic involution $\tau$ of the Riemann surface $C$, which exists
for suitable choices of $C$. An antiholomorphic map reverses
the orientation of $C$; conversely, if $\tau$ is an orientation-reversing smooth involution of an oriented two-manifold $C$, then one can pick a complex
structure on $C$ such that $\tau$ acts antiholomorphically.\footnote{Pick any Riemannian metric $\mathrm g$ on $C$. Then $\mathrm g'
=({\mathrm g}+\tau^*({\mathrm g}))/2$ is a $\tau$-invariant metric, and $\tau$ acts antiholomorphically in the complex structure determined by $\mathrm g'$.}
Topologically, the possible choices of $\tau$ can be classified as follows. If $C$ has genus $g$, then its Euler characteristic
is $2-2g$ and the quotient $C'=C/\{1,\tau\}$ will have Euler characteristic $1-g$. $C'$ can be any possibly unorientable two-manifold, possibly with boundary,
of Euler characteristic $1-g$. The boundary of $C'$ comes from the fixed points of $\tau:C\to C$. These fixed points make up a certain number of circles;
any integer number of circles from $0$ to $g+1$ is possible.
For example, if $g=0$, $C'$ can be a disc or $\mathbb{RP}^2$; if $g=1$, $C'$ can be a cylinder, a Mobius strip, or a Klein bottle.
Having fixed $\tau$, we then choose the topological type of a lift of $\tau$ to act on a smooth $G$-bundle $E\to C$. We choose the lift
to preserve $\tau^2=1$. If $s$ is a fixed point of $\tau:C\to C$, then $\tau$ acts on the fiber $E_s$ of $E$ over $s$
by an automorphism $x_s$ of $G$. Here $x_s$ can be an inner automorphism, that is, conjugation by an element $g_s$ of $G$,
but more generally, if $G$ has non-trivial outer automorphisms, $x_s$ may be an outer automorphism.
The conjugacy class of $x_s$ is constant on each fixed circle $S$, and we denote it as $x_S$.
The condition $\tau^2=1$ places a condition on $x_S^2$.
Since a Higgs bundle is defined by fields $(A,\phi)$ that are adjoint-valued, purely to define an involution of the Higgs bundle moduli space
${\mathcal M}_H(G,C)$, we require only that $x_S^2=1$ in the adjoint form of $G$; as an element of $G$, $x_S^2$ might be a central element not equal to 1.
The precise condition that should be imposed on $x_S^2$ depends on topological subtleties that were reviewed in Section \ref{toposubt}; we will return to this point in Section \ref{nonfree}.
If $\tau$ acts freely and $G$ is not simply-connected, other issues
come into play in lifting $\tau$ to act on $E$. For example, for $G={\mathrm{SO}}(3)$, a lift of $\tau$ to act on $E$ only exists if $\int_C w_2(E)=0$.
In that case, there are two topologically
inequivalent lifts, parametrized by $\int_{C'}w_2(E')$, where $E'\to C'$ is the quotient of $E\to C$ by $\{1,\tau\}$.
A Riemann surface $C$ endowed with
antiholomorphic involution $\tau$ can be viewed as an algebraic curve defined over ${\mathbb{R}}$. The boundary points of $C'$, if any,
correspond to the real points of $C$ over ${\mathbb{R}}$. Once it has been lifted to act on the smooth bundle $E$,
$\tau$ also acts on ${\mathcal M}(G,C)$, the moduli space of flat connections on $E$, and on the
corresponding Higgs bundle moduli space ${\mathcal M}_H(G,C)$. Whether or not there are fixed points in the action of $\tau$ on $C$, there always
are fixed points in the action of $\tau$ on\footnote{If $A$ is any connection on $E$, then $\frac{1}{2}(A+\tau^*(A))$ is a $\tau$-invariant connection.
Generically, the $(0,1)$ part of this connection defines a stable $G_{\Bbb C}$-bundle, which corresponds to a $\tau$-invariant point in ${\mathcal M}(G,C)$. The
same connection with zero Higgs field defines a $\tau$-invariant point in ${\mathcal M}_H(G,C)$.}
${\mathcal M}(G,C)$ and on ${\mathcal M}_H(G,C)$. Each component of the fixed point set of $\tau$ on ${\mathcal M}(G,C)$ or ${\mathcal M}_H(G,C)$ is middle-dimensional. This
is a general property of antiholomorphic involutions of complex manifolds.
We will write ${\mathcal M}^{\mathbb{R}}(G,C)$ or ${\mathcal M}_H^{\mathbb{R}}(G,C)$ for a component of the fixed point set of $\tau$ acting on ${\mathcal M}(G,C)$ or ${\mathcal M}_H(G,C)$, respectively.
The components in general are classified by additional data that we have not introduced so far. In particular, viewing ${\mathcal M}(G,C)$ or ${\mathcal M}_H(G,C)$
as moduli spaces of flat $G$-valued or $G_{\Bbb C}$-valued connections,
the monodromy $h_S$ of a flat bundle around $S$ will be invariant under $x_S$,
so it will lie in the $x_S$-invariant subgroup $H_S$ or $H_{S,{\Bbb C}}$ of $G$ or $G_{\Bbb C}$. $H_S$ and $H_{S,{\Bbb C}}$ are
connected if $G$ is simply-connected, but in general not otherwise
(for example, if $G={\mathrm{SO}}(3)$ and $g_S=\mathrm{diag}(1,-1,-1)$, then $H_S$ contains the component of the identity and another component
that contains the element $\mathrm{diag}(-1,-1,1)$). So in general, to specify a component of the fixed point sets requires an additional choice for each $S$.
Associated to each fixed circle $S$ is a real form $G_{{\mathbb{R}},S}$ of the complex Lie group $G_{\Bbb C}$. Writing $g\to \overline g$ for the antiholomorphic
involution of $G_{\Bbb C}$ that leaves fixed the compact form $G$, $G_{{\mathbb{R}},S}$ is defined by the condition
\begin{equation}\label{ralform} g=x_S(\overline g). \end{equation}
In case $x_S$ is conjugation by $g_S\in G$, the condition becomes
\begin{equation}\label{realform} g=g_S \overline g g_S^{-1}.\end{equation}
${\mathcal M}_H^{\mathbb{R}}(G,C)$, with symplectic structure $\omega_J$, is the real symplectic manifold that we want to quantize. The definition of ${\mathcal M}_H^{\mathbb{R}}(G,C)$ has
depended on various choices which we are not indicating in the notation, but regardless of those choices, ${\mathcal M}_H^{\mathbb{R}}(G,C)$ has ${\mathcal M}_H(G,C)$ as a complexification
and this complexification has the appropriate properties for brane quantization of ${\mathcal M}_H^{\mathbb{R}}(G,C)$.
${\mathcal M}_H^{\mathbb{R}}(G,C)$ has a real polarization that is defined
as follows: a leaf of the polarization consists of $\tau$-invariant Higgs pairs $(A,\phi)$ with fixed $A$, but varying $\phi$. The same condition, without
the requirement of $\tau$-invariance, defines a holomorphic polarization of the complexification ${\mathcal M}_H(G,C)$ of ${\mathcal M}^{\mathbb{R}}_H(G,C)$. Hence brane quantization
of ${\mathcal M}^{\mathbb{R}}(G,C)$ is equivalent to its quantization using this real polarization.
Concretely, in this real polarization, ${\mathcal M}_H^{\mathbb{R}}(G,C)$ can be approximated by the cotangent bundle $T^*{\mathcal M}^{\mathbb{R}}(G,C)$
for the same reason that such a statement holds for the full Higgs bundle moduli space: if $(A,\phi)$ is a $\tau$-invariant
Higgs bundle representing a point in ${\mathcal M}_H^{\mathbb{R}}(G,C)$, then generically $\phi$ represents a cotangent vector to ${\mathcal M}^{\mathbb{R}}(G,C)$ at the point in ${\mathcal M}^{\mathbb{R}}(G,C)$
corresponding to $A$.
Therefore, the Hilbert space ${\mathcal H}_\tau$ that arises in quantization of ${\mathcal M}^{\mathbb{R}}_H(G,C)$ is the space of ${\mathrm L}^2$ half-densities on ${\mathcal M}^{\mathbb{R}}(G,C)$,
the usual answer for geometric quantization of a cotangent bundle.
We can also restrict the Hitchin fibration to the $\tau$-invariant locus. $\tau$ acts on the base ${{\mathcal B}}$ of the Hitchin fibration with a fixed point
set ${{\mathcal B}}^{\mathbb{R}}$. Functions on ${{\mathcal B}}^{\mathbb{R}}$ are Poisson-commuting (with respect to the real symplectic structure $\omega_J$ of ${\mathcal M}_H^{\mathbb{R}}(G,C)$).
So ${\mathcal M}_H^{\mathbb{R}}(G,C)$ is a real integrable system.
The generic fiber of the map ${\mathcal M}_H^{\mathbb{R}}(G,C)\to {{\mathcal B}}^{\mathbb{R}}$ is a real torus.
Baraglia and Schaposnik \cite{BaSc} proposed to define a mirror symmetry between ${\mathcal M}_H^{\mathbb{R}}(G,C)$ and a similar moduli space with $G$ replaced by
$G^\vee$ by $T$-duality on the fibers of the
real integrable system. In general, one would expect such a definition to give reliable results at least asymptotically, far from the discriminant locus of the
real integrable system. At least when $\tau$ acts freely, one can be more precise, as we discuss next.
\subsection{Four-Dimensional Picture And Duality}\label{fourdpic}
To learn something interesting about this construction, one wants to apply duality, and for this, as usual, a four-dimensional picture is helpful.
For a four-dimensional picture, we start with the usual four-manifold $M=\Sigma\times C$, where $\Sigma={\mathbb{R}}\times I$ is a strip. Here ${\mathbb{R}}$ labels the ``time''
and $I$ is an interval, say the interval $0\leq w\leq 1$. We then modify $M$ by imposing an equivalence relation on the right boundary at $w=1$:
we declare that, for $p\in C$, points $(t,1,p)$ and $(t,1,\tau(p))$ are equivalent. (No equivalence is imposed except at $w=1$.)
Imposing this equivalence relation amounts to requiring that the data at $w=1$ should be $\tau$-invariant (up to a gauge transformation). So this
is a way to implement in four dimensions what in the two-dimensional picture of Section \ref{setup} was the brane ${\mathscr B}_{\mathbb{R}}$ at the right boundary.
We write $M_\tau$ for the quotient of $M$ by this equivalence relation. It is convenient to factor out the time and define $M_\tau={\mathbb{R}}\times U_\tau$.
There is an alternative construction of $U_\tau$ as a quotient by the ${\mathbb Z}_2$ group generated by $\tau$. For this, we start with a doubled interval
$\widehat I: 0\leq w\leq 2$. Then we divide $\widehat U= \widehat I\times C$ by the symmetry that acts by $(w,p)\to (2-w,\tau(p))$. A fundamental domain is the region
$0\leq w\leq 1$, so the quotient is the same as before.
Simplest is the case that $\tau$ acts freely on $C$. $U_\tau$ is then actually an orientable manifold,
whose boundary is a single copy of $C$, at $w=0$. $U_\tau$ is the total space of a real line bundle over $C'=C/\{1,\tau\}$.
$C'$ itself and the real line bundle are both unorientable, but the total space $U_\tau$ is orientable.
The four-dimensional picture associated to quantization of ${\mathcal M}_H^{\mathbb{R}}(G,C)$ is just the $A$-model on $M_\tau={\mathbb{R}}\times U_\tau$, with ${\mathscr B}_{\mathrm{cc}}$ boundary
conditions at $w=0$. There is no need for an explicit boundary condition at $w=1$, as there is no boundary there.
What in the two-dimensional description was a Lagrangian brane ${\mathscr B}_{\mathbb{R}}$ supported on ${\mathcal M}_H^{\mathbb{R}}(G,C)$ has been absorbed into the geometry of $M_\tau$.
Therefore, without further ado we can describe a dual description. The dual is just the $B$-model with gauge group $G^\vee$
on the same four-manifold $M_\tau$, but now with
${\mathscr B}_{\mathrm{op}}$ boundary conditions at $w=0$. In a two-dimensional language, the brane ${\mathscr B}^\vee_{\mathbb{R}}$ dual to ${\mathscr B}_{\mathbb{R}}$ is supported on ${\mathcal M}_H^{\mathbb{R}}(G^\vee,C)$.
In particular, it is supported on the same locus ${{\mathcal B}}_{\mathbb{R}}$ on the base of the Hitchin fibration. This is in accord with the duality proposal of
Baraglia and Schaposnik \cite{BaSc}, which appears to be valid at least when $\tau$ acts freely on $C$.
As usual, it is relatively easy to describe the physical states of the $B$-model in quantization on $U_\tau$ and the eigenvalues of the Hitchin Hamiltonians.
The first step is localization of the $B$-model on complex-valued flat connections, which here means $G^\vee_{\Bbb C}$-valued flat connections on $U_\tau$
that satisfy the oper boundary condition.
To put it differently, the localization is on flat $G_{\Bbb C}$ bundles $E^\vee\to U_\tau$ whose restriction to $C=\partial U_\tau$ is
a holomorphic oper. Let $\Upsilon_\tau$ be the set of isomorphism classes of such bundles.
For positivity of the Hilbert space inner product, we expect $\Upsilon_\tau$ to be a discrete set of nondegenerate points (see
Appendix \ref{bmodel}),
similarly to the analogous set $\Upsilon=L_{\mathrm{op}}\cap L_{\overline{\mathrm{op}}}$ encountered in the quantization of the full Higgs bundle moduli space.
Assuming this, the Hilbert space ${\mathcal H}_\tau$ has a basis $\psi_u$ labeled by $u\in \Upsilon_\tau$.
The $\psi_u$ are eigenfunctions of the Hitchin Hamiltonians. The eigenvalue of a Hitchin Hamiltonian $H_{{\mathcal P},\alpha}$ on $\psi_u$ is the value
of the corresponding function $f_{{\mathcal P},\alpha}:L_{\mathrm{op}}\to {\Bbb C}$ at the point in $L_{\mathrm{op}}$ that corresponds to the boundary values of the flat bundle
associated to $u$.
If an oper bundle over $C=\partial U_\tau$ extends over $U_\tau$ as a flat bundle, then in particular this
implies that the antihomomorphic involution $\tau:C\to C$ lifts to an action on $E^\vee$
and therefore that $E^\vee$ is an antiholomorphic oper, as well as a holomorphic one. So the set $\Upsilon_\tau$
has a natural map to $\Upsilon=L_{\mathrm{op}}\cap L_{\overline{\mathrm{op}}}$. However, in general the map from $\Upsilon_\tau$
to $\Upsilon$ is not an embedding, since a $\tau$-invariant flat bundle on the boundary of $U_\tau$ that is a holomorphic oper may have more
than one extension as a flat bundle over the interior of $U_\tau$. To analyze this situation, note that flat $G^\vee_{\Bbb C}$ bundles over $U_\tau$ correspond
to homomorphisms $\varrho:\pi_1(U_\tau)\to G^\vee_{\Bbb C}$, up to conjugation. As $U_\tau$ is contractible to $C'=C/\{1,\tau\}$, we can equally well
consider $\varrho:\pi_1(C')\to G^\vee_{\Bbb C}$. The group $\pi_1(C')$ has an index 2 subgroup $\pi_{1,+}(C')$ consisting of orientation-preserving loops in $C'$.
Such loops can be deformed to the boundary of $U_\tau$, so once a flat $G^\vee_{\Bbb C}$ bundle is given on the boundary of $U_\tau$, the restriction of the corresponding
homomorphism $\varrho$ to $\pi_{1,+}(C')$ is uniquely determined. However, there is some freedom in the extension of $\varrho$ to the rest of $\pi_1(C')$.
$\pi_1(C')$ is generated by the index 2 subgroup $\pi_{1,+}(C')$ together with any orientation-reversing element $\sigma\in \pi_1(C')$. To complete the description of
$\varrho$, we need to specify $\varrho(\sigma)$. $\varrho$ is supposed to be a homomorphism, so we require $\varrho(\sigma)^2=\varrho(\sigma^2)$,
where, since $\sigma^2\in \pi_{1,+}(C')$, $\varrho(\sigma^2)$ is determined by the boundary data. If $\varrho(\sigma^2)$ is a regular element of $G^\vee_{\Bbb C}$,
then there are only finitely many choices for $\varrho(\sigma)$, but $\varrho(\sigma)$ is not uniquely determined. In particular, we are always free
to transform $\varrho(\sigma)\to \varepsilon \varrho(\sigma)$, where $\varepsilon$ is an element of order 2 of the center of $G^\vee$. Thus the subgroup
${\mathcal Z}_2(G^\vee)$ of the center of $G^\vee$ consisting of elements of order 2 acts freely on $\Upsilon_\tau$. (For simple $G^\vee$, ${\mathcal Z}_2(G^\vee)$ is
1, ${\mathbb Z}_2$, or ${\mathbb Z}_2\times {\mathbb Z}_2$, depending on $G^\vee$.)
The $A$-model dual of the action of ${\mathcal Z}_2(G^\vee)$ is the following. Topologically, for simple $G$, a $G$-bundle on the three-manifold
$U_\tau$ is classified by $H^2(U_\tau,\pi_1(G))$. In the present instance, $U_\tau$ is contractible to $C'=C/\tau$ so $H^2(U_\tau,\pi_1(G))=H^2(C',\pi_1(G))$.
As $C'$ is unorientable, one has $H^2(C',\pi_1(G))=\pi_{1;2}(G)$, where $\pi_{1;2}(G)$ is the subgroup of $\pi_1(G)$ consisting of elements of
order 2. Thus on the $A$-model side, there is a grading of the Hilbert space by $\pi_{1;2}(G)$. As $\pi_{1;2}(G)={\mathcal Z}_2(G^\vee)$, this
matches the ${\mathcal Z}_2(G^\vee)$ action in the $B$-model.
Finally let us discuss the natural line operators in this problem.
As in Section \ref{wtw}, we could in principle consider Wilson and 't Hooft line operators supported on an arbitrary 1-manifold $\gamma\subset M_\tau$.
But a natural special case is the following. Pick a point $p\in C$. In $\widehat I\times C $, there is a natural path $\widehat\gamma_p$ between the points $0\times p$ and $2\times p$
on the left and right boundaries: we simply set $\widehat\gamma_p = \widehat I\times p$. Upon dividing by the group generated by $\tau$,
$\widehat\gamma_p$ descends to a path $\gamma_p\subset
U_\tau$, between the boundary points $0\times p$ and $0\times \tau(p)$. In a purely two-dimensional description, one would have needed endpoint or
corner data associated to the real brane ${{\mathscr B}}_{\mathbb{R}}$, but in the four-dimensional description, this is not needed as $\gamma$ has no endpoint at $w=1$.
As in Section \ref{wilop}, we can choose an arbitrary representation $R$ of $G^\vee$
and consider the holonomy $W_{R,\gamma}$
of the bundle $E^\vee_R=E^\vee\times_{G^\vee_{\Bbb C}}R$ on the curve $\gamma_p$. However, to turn this holonomy into a quantum operator $ W_{R,p}$,
we need, as before, to supply endpoint data.
For example, if we choose canonical endpoints $s_{R,n}$ at $0\times p$ and
$s_{\overline R, m}$ at $0\times \tau(p)$, the Wilson operator will take the form of the inner product between $s_{R,n}$ transported to $\tau(p)$ and $s_{\overline R, m}$. As
in Section \ref{wilop}, the Wilson operators $W_{R,p}$ constructed this way are diagonal
on the basis of states $\psi_u$, with eigenvalues given by evaluating the inner product on the concrete canonical sections in the flat bundle corresponding to $u$.
The duality predicts that a dual 't Hooft operator associated to the same representation $R$, supported on the same curve
$\gamma_p$, and with $S$-dual endpoints, has the same eigenvalues. The dual 't Hooft operator is associated to a real Hecke correspondence $Z^{\mathbb{R}}_{R,p}$ of ${\mathcal M}_H^{\mathbb{R}}(G,C)$ with itself;
$Z^{\mathbb{R}}_{R,p}$ is just the $\tau$-invariant locus of the ordinary Hecke correspondence $Z_{R,p;\overline R,\tau(p)}$ of ${\mathcal M}_H(G,C)$ with itself,
for a $\tau$-conjugate pair of points labeled by the conjugate (or dual)
pair of representations $R,\overline R$.
As in Section \ref{lineq}, because of the scaling symmetry of the cotangent bundle, the quantum operator associated to the
Hecke correspondence can be defined by a semiclassical formula. Beyond the ingredients that were used in Section \ref{lineq}, one needs one further
fact. Suppose that $X$ is a complex manifold with canonical bundle $K_X$; let $\tau:X\to X$ be an antiholomorphic involution with fixed point set $X^{\mathbb{R}}$,
and assume thast $X^{\mathbb{R}}$ is orientable.
Then a holomorphic section of $K_X^{1/2}\to X$ restricts on $X^{\mathbb{R}}$ to a complex-valued half-density. Hence a holomorphic endpoint or corner that one
would use (together with an antiholomorphic one)
in defining an 't Hooft operator in the quantization of ${\mathcal M}_H$ restricts on the real locus to a half-density that defines an 't Hooft
operator in the quantization of ${\mathcal M}_H^{\mathbb{R}}$. In our application, $Z^{\mathbb{R}}_{R,p}$ is orientable because $Z_{R,p;\overline R,\tau(p)}$ is a Calabi-Yau manifold,
as discussed in Section \ref{lineq}; the Calabi-Yau form of a Calabi-Yau manifold $X$ that has a real structure can be chosen to be real and restricts
on the real locus to a top-degree differential form that defines an orientation of $X^{\mathbb{R}}$.
\subsection{The Case That $\tau$ Does Not Act Freely}\label{nonfree}
Now we consider the case that $\tau$ does not act freely on $C$. Suppose that the action of $\tau$ leaves fixed a circle $S\subset C$.
Then $U_\tau=\widehat U/{\mathbb Z}_2$ contains $S$ as a locus of ${\mathbb Z}_2$ orbifold fixed points. The local behavior near $S$ looks like
\begin{equation}\label{wofo} S\times {\mathbb{R}}^2/\{1,\tau\}, \end{equation}
where $\tau$ acts on ${\mathbb{R}}^2$ as a $\pi$ rotation.
In the four-manifold $M_\tau={\mathbb{R}}\times U_\tau$,
the fixed point set is ${\mathbb{R}}\times S$.
We recall that in general the $\tau$ action on $S$ is accompanied by an action of an automorphism $x_S$ that satisfies $x_S^2=1$ at least
in the adjoint form of $G$. In $G$ gauge theory, one might expect to require $x_S^2=1$ in $G$, but one has to take into account
the topological subtleties that were reviewed in Section \ref{toposubt}.
To illustrate the issues, we consider the case that $G$ has rank 1 and thus is ${\mathrm{SU}}(2)$ or ${\mathrm{SO}}(3)$.
In this case, $G$ has no outer automorphisms and $x_S$ is conjugation by an element
$g_S$ of $G$.
As discussed in Section \ref{toposubt}, there
are two versions of ${\mathrm{SU}}(2)$ gauge theory. In ordinary ${\mathrm{SU}}(2)$
gauge theory, the most obvious condition is to require $g_S^2=1$ acting on an ${\mathrm{SU}}(2)$ bundle $E\to M$. Then the only options are $g_S=1$ and $g_S=-1$.
On the other hand, in
${\mathrm{Spin}}\cdot {\mathrm{SU}}(2)$ gauge theory, the most obvious condition is to ask for
$g_S^2=1$ acting on ${\mathcal S}\otimes E$, where ${\mathcal S}$ is a $\tau$-invariant spin bundle on $M$ (defined
at least locally near the fixed point set) and $E$ is an ${\mathrm{SU}}(2)$ bundle (defined wherever ${\mathcal S}$ is). Since $\tau$ acts as a $\pi$ rotation of the normal plane at the fixed
point set, $\tau^2$ is a $2\pi$ rotation and acts as $-1$ on ${\mathcal S}$. This would suggest that we require
$\tau^2=-1$ on $E$, leading to $g_S=\mathrm{diag}({\mathrm i},-{\mathrm i})$,
up to conjugation.\footnote{A role for this conjugacy class was suggested by D. Baraglia.}
However, we believe that it may also be possible to reverse these choices. For example, in ${\mathrm{SU}}(2)$, before trying to divide by $\tau$, we could assume
that there is a monodromy defect with monodromy $-1$ along the fixed point locus of $\tau$. Then in taking the quotient we would want $g_S^2=-1$.
Similarly, including such a monodromy defect before taking the quotient would motivate $g_S^2=+1$ for ${\mathrm{Spin}}\cdot {\mathrm{SU}}(2)$.
Similarly, there are two versions of ${\mathrm{SO}}(3)$ gauge theory, with or without a factor
$\Delta=(-1)^{\int_M w_2(M) w_2(E)}$ in the integrand of the path integral. With or without this factor, we want $g_S^2=1$, which gives two
possibilities, namely $g_S=1$ and $g_S=\mathrm{diag}(-1,-1,1)$. In the presence of a codimension 2 singularity with $g_S=\mathrm{diag}(-1,-1,1)$,
$\Delta$ is not well-defined topologically, so it appears that in this case we want $g_S=1$. Without the factor $\Delta$, both possibilities for $g_S$ are viable.
For ${\mathrm{SO}}(3)$, as the center is trivial, we do not have the option of including a defect with central monodromy, but we can include the dual of this, which is
a defect that senses the topology of the gauge bundle $E$ restricted to the fixed point set. For ${\mathrm{SO}}(3)$, this defect is a factor in the path integral of
$(-1)^{\int_W w_2(E)}$, where $W$ is the fixed point set.
In general, it is a subtle question to find the dual of a singularity of this nature. Part of the reason for the subtlety is that one {\it cannot} assume
that the dual of an ${\mathbb{R}}^2/{\mathbb Z}_2$ orbifold singularity, defined by a condition of $\tau$-invariance, is another ${\mathbb{R}}^2/{\mathbb Z}_2$
orbifold singularity, defined by a dual condition of $\tau$-invariance. In general, one only knows that the
dual of an ${\mathbb{R}}^2/{\mathbb Z}_2$ orbifold singularity is a codimension 2 defect that preserves the same supersymmetry as the ${\mathbb{R}}^2/{\mathbb Z}_2$ orbifold singularity.
In a somewhat similar problem of rigid surface defects, it proved difficult to get a general understanding of the action
of duality \cite{GWr}.
It is tempting to claim a simple answer if $g_S$ is central for all $S$, on the following grounds.
Suppose that $g_S=1$. Then the singularity is only in the geometry, not the gauge field.
The ${\mathbb{R}}^2/{\mathbb Z}_2$ orbifold singularity is not a singularity at all topologically, as ${\mathbb{R}}^2/{\mathbb Z}_2$ is equivalent topologically to ${\mathbb{R}}^2$. So if $g_S=1$ for all
fixed circles, $U_\tau$ is actually a manifold topologically, and one can ``round off'' the orbifold singularities to give it a smooth geometry. Let us call the rounded
version $\widetilde U_\tau$. If it is correct in the $A$-model to replace the orbifold with the smooth manifold $\widetilde U_\tau$, then the dual is the $B$-model on the same
manifold. This reasoning has a potential analog for the more general case that $g_S$ is central but not equal to the identity.
We can still round off the defect to get the smooth
manifold $\widetilde U_\tau$, but now $\widetilde U_\tau$ contains a defect with central monodromy, supported on the orbifold locus $W$.
As remarked earlier, the dual of this defect is a defect that senses
the topology of the bundle, for example a defect defined by a factor $(-1)^{\int_W
w_2(E)}$ in the case of gauge group ${\mathrm{SO}}(3)$.
The following is a strategy, in principle, to analyze the general case. Surround the fixed point locus by a two-torus. Then the orbifold defect defines
a boundary condition for the 2d theory which arises from $T^2$ compactification of four-dimensional super Yang-Mills theory.
This is not quite a $\sigma$-model, because of the
large unbroken gauge symmetry. If we can identify the mirror of the boundary condition, it will provide boundary conditions for the
$G^\vee_{\Bbb C}$ flat connection restricted to the two-torus on the dual side. This would be sufficient to characterize how the oper flat connection
at the boundary of $U_\tau$ can extend to the interior and thus determine the spectrum on the $B$-model side of the duality.
\subsection{Real Hecke Operators}\label{rho}
The most important novelty of the case that $\tau$ does not act freely on $C$ may be the existence of line operators supported on a real point in $C$,
as opposed to the line operators considered in Section \ref{fourdpic} that are supported on a $\tau$-conjugate pair of points.
First we describe a gauge theory picture for an 't Hooft operator supported at a real point $p\in C$. We work on the four-manifold $M={\mathbb{R}}\times \widehat I\times C$,
with ${\mathbb{R}}$ parametrized by $t$, and $\widehat I$ by $w$. A local picture suffices, so we take $C$ to be simply the complex $z$-plane ${\Bbb C}$, and we consider the involution $\tau$
that acts by $(t,w,z)\to (-t,w,\overline z)$. Acting just on ${\Bbb C}$, $\tau$ has the fixed line $S$ defined by ${\mathrm{Im}\, z}=0$; we acccompany $\tau$ with
an automorphism $x_S$ satisfying $x_S^2=1$ (or a slightly more general condition discussed in Section \ref{nonfree}). An 't Hooft operator supported at $z=t=0$
is described by a Dirac monopole solution of the $G$ gauge theory. The solution has a structure group that reduces to a maximal torus $T\subset G$
and can be characterized by its curvature:
\begin{equation}\label{curv} F=\frac{{\sf m}}{2}\star_3 \mathrm d \frac{1}{(t^2+|z|^2)^{1/2}}, \end{equation}
where $\star_3 $ is the Hodge star for the metric $\mathrm d t^2+|\mathrm d z|^2$ on ${\mathbb{R}}\times{\Bbb C}$, and ${\sf m}$ is a constant element of the Lie algebra $\mathfrak t$ of $T$.
For a connection with this curvature to exist, ${\sf m}$ must be an integral coweight, dual to a representation $R$ of $G^\vee$ (it coincides with the
object that was called ${\sf m}$ in Section \ref{affgr}). Now we ask whether
this solution is invariant under $\tau$, accompanied by the automorphism $x_S$. Since $\star_3$ is odd under $\tau$, a necessary
and sufficient condition is that ${\sf m}$ should be odd under $x_S$:
\begin{equation}\label{oddurv} x_S( {\sf m} )=-{\sf m}. \end{equation}
When and only when it is possible to choose ${\sf m}$ in its conjugacy class so that it is odd under $x_S$, the solution constructed this way is $\tau$-invariant
and descends to a solution on $M_\tau=M/\{1,\tau\}$. It describes a real 't Hooft operator, supported on a real point in $C$ and associated to the representation
$R$ of $G^\vee$.
Using this model solution, we can define a space of ``real'' Hecke modifications which can be implemented by a real 't Hooft operator.
There are real versions $G_{{\mathbb{R}},S}({\cal K})$ and $G_{{\mathbb{R}},S}({\cal O})$ of $G_{{\Bbb C}}({\cal K})$ and $G_{{\Bbb C}}({\cal O})$ which consist of
gauge transformations which lie in $G_{{\mathbb{R}},S}$ along $S$. We can thus define a real version
\begin{equation}
{\mathrm{Gr}}_{G_{{\mathbb{R}},S}} = G_{{\mathbb{R}},S}({\cal K})/G_{{\mathbb{R}},S}({\cal O})
\end{equation}
of the affine Grassmannian and the orbits $\left[G_{{\mathbb{R}},S}({\cal O}) z^{\sf m}\right]$ of real Hecke modifications of type ${\sf m}$.
A knowledge of which real 't Hooft operators are possible for a given $x_S$ puts a very strong constraint on the dual of a ${\mathbb Z}_2$ orbifold singularity with a given $x_S$.
Geometrically, a real 't Hooft operator stretches from the boundary of $U_\tau$ to a fixed point $p$ in the interior.
A full analysis of real 't Hooft operators, which we will not attempt here, would include a discussion of the possible endpoints of the 't Hooft operator
on a fixed point and a derivation of the corresponding integral operators.
On the $S$-dual side, we will have a Wilson operator stretched from the boundary to $p$. A specific endpoint of the Wilson line will give a vector $v_{\overline R,m}$ in the space of flat sections of the gauge bundle in a neighborhood of $p$. We cannot characterize this vector more precisely without knowing the $S$-dual of the orbifold
singularity; in particular, the flat bundle in the $B$-model may not extend over the fixed point set.
The Wilson operator expectation value will take the form of a pairing $(v_{\overline R,m}, s_{R,n})$,
selecting a specific solution of the oper differential equations.
More generally, one can consider an arbitrary oriented three-manifold $U$ with boundary
$C$, and study the dual $A$- and $B$-models on ${\mathbb{R}}\times U$. Modulo technical difficulties (the moduli space of flat bundles on $U$ may be very
singular), one can hope to define a space ${\mathcal H}_U$ of physical states, with an action of the Hitchin Hamiltonians on the $A$-model side and a prediction
for their eigenvalues in terms of classical data on the $B$-model side. One can also study line operators, though in general there will be no close
analogs of the ones that we have considered in this article. The state space ${\mathcal H}_U$ will have a hermitian inner product, but it is not
clear that this inner product will be positive-definite in general, since it is no longer obtained by quantizing a cotangent bundle.
\section{Four-Dimensional Avatars of BAA Boundary Conditions and Corners}\label{avatar}
Four-dimensional super Yang-Mills theory admits many half-BPS boundary conditions \cite{Gaiotto:2008sa}
which are topological in the four-dimensional $A$-twist and descend to BAA boundary conditions upon twisted compactification
on $C$ \cite{KW,Gaiotto:2016hvd,Gaiotto:2016wcv}.
Such a boundary condition, along with its ``corners'' with ${\mathscr B}_{\mathrm{cc}}$ and $\overline{\mathscr B}_{\mathrm{cc}}$, can be used to define a quantum state in
${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$. We have already made use of this construction; see fig. \ref{example6} of Section \ref{wkb}.
Apart from studying additional examples, what we will add in the present section is the use of two-dimensional chiral algebras to study the
corners and the associated quantum states.
In four dimensions, a junction or corner between two boundary conditions occurs on a two-manifold, which for our purposes is a copy of
the Riemann surface $C$. In the case of a brane of type BAA that can be engineered in four-dimensional gauge theory, its corners
with ${\mathscr B}_{\mathrm{cc}}$, if they can likewise be engineered in four-dimensional gauge theory, are frequently holomorphic-topological and support
holomorphic chiral algebras. Chiral algebras that arise this way were studied in \cite{Gaiotto:2016hvd,FG}.
Corners with $\overline{\mathscr B}_{\mathrm{cc}}$, if they can be engineered in four dimensions, likewise typically support antiholomorphic
chiral algebras. This is the situation that we will study in the present section.
\subsection{The Analytic Continuation Perspective}\label{ACP}
We begin by recalling a construction that simplifies the analysis of the relevant junctions. The brane ${\mathscr B}_{\mathrm{cc}}$ can be derived
from a deformed Neumann boundary condition in four dimensions.\footnote{\label{deformed} Ordinary Neumann boundary conditions for a gauge field assert that
$n^i F_{ij}=0$, where $F$ is the Yang-Mills curvature and $n$ is the normal vector to the boundary. Deformed Neumann boundary conditions
express $n^i F_{ij}$ in terms of the boundary values of some other fields. Note that a different, undeformed, Neumann boundary condition,
with a different extension to the rest of the supermultiplet, will enter the story in Section \ref{Enriched}.} We will call this the deformed Neumann or ${\mathscr B}_{\mathrm{cc}}$ boundary condition.
The path integral of the $A$-twisted 4d gauge theory in the presence of such a deformed Neumann boundary can be interpreted as
a slightly exotic path integral for a three-dimensional theory defined on the boundary \cite{Witten:2010zr,Witten:2011zz}.
The action of this three-dimensional theory is a holomorphic function of complex
variables, and the path integral is taken on a middle-dimensional integration cycle in the space of fields. The integration cycle is defined
by $A$-model localization in four dimensions, but the details of this are not important for our purposes. We will only discuss properties that
do not depend on the choice of integration cycle. We will call this type of path integral loosely a contour path integral.
The relevant three-dimensional auxiliary theory is most familiar not in the case of conventional geometric
Langlands but for what is known mathematically as ``quantum'' geometric Langlands. This corresponds in gauge theory to working at a generic
value of the canonical parameter $\Psi$ that was introduced in \cite{KW}.
At generic $\Psi$, the boundary theory associated to suitably deformed Neumann boundary conditions is a Chern-Simons theory with a complex
connection ${\mathcal A}$, with curvature ${\mathcal F}=\mathrm d {\mathcal A}+{\mathcal A}\wedge {\mathcal A}$, and action
\begin{equation}\label{csact} I_{\mathrm{CS}} =\frac{\Psi}{4\pi}\int_N \left( {\rm Tr}\,{\mathcal A}\wedge \mathrm d {\mathcal A} +\frac{2}{3}{\mathcal A}^3\right). \end{equation}
Here $N$ is the boundary or a portion of the boundary of a four-manifold $M$. The boundary condition that leads to the theory $I_{\mathrm{CS}}$ on $N$ is
``topological'' in the sense that the only structure of $N$ that is required to define it is an orientation. If $N=\partial M$, then the $A$-model path integral
on $M$ with boundary condition that leads to the boundary coupling $I_{\mathrm{CS}}$ is a Chern-Simons path integral on $N$ with a non-standard integration cycle (which depends on $M$).
More generally, $N$ itself may have a boundary; along $\partial N$, we consider a junction or corner between the deformed Neumann boundary condition that leads
to $I_{\mathrm{CS}}$ and some other boundary condition. In this case, as in conventional Chern-Simons theory on a three-manifold with boundary, for suitable
choices of the second boundary condition, a current algebra or Kac-Moody symmetry will appear along $\partial N$. The level of these currents is $\Psi-h$, where $h$ is the dual Coxeter number of $G$.
The contribution $\Psi$ to the level can be computed classically from the failure of $I_{\mathrm{CS}}$ to be gauge-invariant on a manifold with boundary, and the $-h$ is a 1-loop correction, which will be described in Section \ref{Dbc}.
For the present article, we are interested in ``ordinary'' geometric Langlands at $\Psi=0$. We will not get anything sensible if we simply set $\Psi=0$
in the action (\ref{csact}), since a contour path integral with zero action will not make sense,
and in fact there is no way to take the limit $\Psi\to 0$ while preserving topological invariance along $N$. That is one way
to understand the fact that the ${\mathscr B}_{\mathrm{cc}}$ boundary condition that has been important in the present article is holomorphic-topological rather than
topological. To take the limit $\Psi\to 0$ to get a holomorphic-topological boundary condition, we can do the following. Let $C$ be a complex Riemann
surface with local complex coordinate $z$, and assume that $N=S\times C$, where $S$ is a 1-manifold parametrized by $t$.
Then take the limit $\Psi\to 0$ keeping fixed $\varphi=\frac{\Psi}{4\pi} {\mathcal A}_z\mathrm d z$.
The Chern-Simons action goes over to
\begin{equation}\label{degac}
I_{\varphi{\mathcal F}}=
\int _N{\rm Tr}\, \varphi_z {\mathcal F}_{t \overline z} \mathrm d t \mathrm d^2z.
\end{equation}
The degeneration of the Chern-Simons action (\ref{csact}) to the action of eqn. (\ref{degac}) is somewhat analogous to the degeneration from a complex
flat connection to a Higgs bundle as the complex structure of the Higgs bundle moduli space ${\mathcal M}_H(G,C)$ is varied.
The action $I_{\varphi{\mathcal F}} $ describes what can be interpreted as a topological gauged quantum mechanics on the cotangent bundle to the space of $(0,1)$ connections on $C$.
The Higgs field $\varphi$ is the momentum conjugate to ${\mathcal A}_{\overline z}$. The analogous statement in two-dimensional terms
is that the $A$-model of ${\mathcal M}_H(G,C)$, with a ${\mathscr B}_{\mathrm{cc}}$ boundary, is related to
an analytically continued quantum mechanics on
${\mathcal M}_H(G,C)$ \cite{Witten:2010zr}.
This formulation makes it obvious that the deformed Neumann boundary condition is not topological in the $A$-twist. Instead,
it depends on a choice of complex structure on $C$ and it admits local operators which vary holomorphically along $C$ and topologically along $S$: it is a holomorphic-topological boundary condition. The equations of motion derived
from $I_{\varphi{\mathcal F}}$ imply that $\varphi$ is holomorphic in $z$ and independent of $t$.
Away from the boundary of $N$, the gauge-invariant local operators on $N$ are the
gauge-invariant polynomials ${\cal P}[\varphi](z)$ of $\varphi$, which descend to the Hitchin Hamiltonians in the 2d $A$-model. (At generic $\Psi$, there are no
gauge-invariant local operators on a boundary characterized by the Chern-Simons action $I_{\mathrm{CS}}$.)
What happens if $N$ has a boundary?
A junction between the deformed Neumann boundary condition that leads to the $\varphi{\mathcal F}$ theory and a topological 4d boundary condition can in many cases be described by a boundary condition in the $\varphi{\mathcal F}$ theory, encoding both the topological boundary condition and the choice of junction.
In many important examples, the topological boundary condition is a half-BPS boundary condition of type BAA.
With suitable choices, as we discuss further in Section \ref{Dbc}, $\varphi$ can behave along $\partial N$ as a holomorphic current generating a Kac-Moody symmetry.
But now the level of the Kac-Moody symmetry comes entirely from the 1-loop
correction and is $-h$.
In general, the boundary conditions which appear in the $\varphi{\mathcal F}$
auxiliary gauge theory are holomorphic as well and may support holomorphic local operators. This reflects the same property of the corresponding junctions between the deformed Neuman boundary and the half-BPS boundary: they are holomorphic in the $A$-twisted theory. The appearance of holomorphic junctions in the GL-twisted theory at general $\Psi$ and the relation to the Chern-Simons level were analyzed in \cite{Gaiotto:2017euk}.
\subsection{The Role of Chiral Algebras} \label{RoleChiral}
A holomorphic junction between a holomorphic-topological boundary condition and a topological one
may support local operators which depend holomorphically on their position on $C$. Essentially by definition, these operators define a chiral algebra. Although the chiral algebra depends on the choice both of the topological boundary condition and of the junction, we will suppress that dependence for notational convenience and denote the
chiral algebra simply as ${\cal V}$. The chiral algebra is akin to the chiral algebras of holomorphic local operators which can be
found in a 2d CFT, or at the boundary of a 3d topological field theory (TFT)
such as Chern-Simons theory. There are some differences \cite{Costello:2020ndc} due to the fact that the $\varphi {\mathcal F}$ theory is holomorphic-topological.
A single junction in the $A$-twisted theory can be used to build a variety of different corners in the 2d $A$-model, depending on the choice of
local operators $O_i(z_i)$ placed at points $z_i$ in $C$. These corners are not all independent: they depend on the $z_i$ holomorphically, with singularities as $z_i \to z_j$ controlled by the OPE of the chiral algebra. The OPE or the associated Ward identities imply recursion relations between different corners.\footnote{A solution of the Ward identities for a chiral algebra is usually called a {\it conformal block}. In a physical 2d CFT, conformal blocks can usually be obtained by some sewing procedure on the Riemann surface. This may not be possible for a general chiral algebra such as ${\cal V}$, but the notion of conformal blocks is still available and can be used to characterize the space of $A$-model corners which can be produced from a given junction.}
Recall the gauge-invariant local operators ${\cal P}[\varphi](z)$ on the deformed Neumann boundary condition which
give rise to the Hitchin Hamiltonians. These operators can be brought to the junction along the topological direction $t$ along the boundary. The resulting boundary-to-junction OPE is not singular and produces a collection of operators
$S_{\cal P}(z)$ in ${\cal V}$. These operators are {\it central}, i.e. they have non-singular OPE with the other operators in ${\cal V}$,
for the same reason that the ${\cal P}[\varphi](z)$ have non-singular OPE with each other: they can be freely displaced along the $t$ direction.
That property has an important corollary: $A$-model corners labelled by a collection of operators $S_{\cal P}(z) O_1(z_1) \cdots O_n(z_n)$
satisfy the same Ward identities as a function of the $z_i$ as corners labelled only by $O_1(z_1) \cdots O_n(z_n)$. This insures that the
the action of ${\cal P}[\varphi](z)$ on this space of $A$-model corners is well-defined.
\begin{figure}
\begin{center}
\includegraphics[width=1.7in]{Example8.pdf}
\end{center}
\caption{\small An 't Hooft operator $T$, of type BAA, acting on the state created by a brane ${\mathscr B}$ of type BAA, together with suitable chiral corners.
We can fuse $T$ with ${\mathscr B}$ to
make a new brane $T{\mathscr B}$, again of type BAA. This involves the same action of line operators on branes that we started with in fig. \ref{example1}(a). In addition
we have to consider the composition of the chiral corners at the left and right endpoints or boundaries of $T$ and ${\mathscr B}$. \label{example8}}
\end{figure}
We can also discuss the chiral algebra ingredients which occur in a four-dimensional description of the action of 't Hooft operators on the states created by branes
with chiral algebra corners. In fig. \ref{example8}, we sketch an 't Hooft operator $T$, of type BAA, acting on the state created by a brane ${\mathscr B}$ of the same type, with
suitable chiral and antichiral corners.
The action of $T$ on the state can be described as the fusion of the interface represented by $T$ with the boundary ${\mathscr B}$, accompanied by a composition of the corresponding corners both with ${\mathscr B}_{\mathrm{cc}}$ and with $\overline {\mathscr B}_{\mathrm{cc}}$.
In four dimensions, in contrast to the two-dimensional picture of fig. \ref{example8}, there is an obvious difference between the 't Hooft line defect and the half-BPS boundary: the former is supported at a point $p\in C$ while the latter wraps the whole $C$.
So the composition $T{\mathscr B}$ coincides with ${\mathscr B}$ away from $p$; it can be described as the brane ${\mathscr B}$ enriched with a line defect.
In order to describe the action of 't Hooft operators we should thus first generalize the discussion in this section to allow for
boundary line defects ending on the junction at some point $p\in C$. As usual, this setup will depend holomorphically on $p$. It depends only on $p$ because in the A-twisted theory, the boundary line defect itself is the image of a topological 't Hooft line defect and is thus also topological.
The presence of a boundary line defect $\ell$ and its endpoint does not affect the choice of chiral algebra operators available elsewhere on $C$, nor the Ward identities they satisfy away from $p$. It affects, though, their behavior near $p$. For each $\ell$, there is a vector space of possible endpoints $O_{\ell,i}(p)$ and they form a module ${\cal V}_\ell$ for ${\cal V}$: the module structure encodes the OPE between chiral algebra
operators and endpoints and controls the Ward identities satisfied at $p$ by the ${\cal V}_\ell$ insertions.
As we bring an 't Hooft operator with endpoint $\alpha_{R,n}(p)$ to the half-BPS boundary, we will produce a boundary defect $\ell$ as well as
a specific endpoint $S_{R,n}(p) \in {\cal V}_\ell$ at the junction. We can predict two general properties of such endpoints: they will have non-singular OPE with the chiral algebra ${\cal V}$, just like the $S_{\cal P}(z)$ associated to Hitchin Hamiltonians, and they will satisfy the same differential equations in $p$ as $\alpha_{R,n}(p)$ do, with coefficients controlled by the $S_{\cal P}(z)$. These properties can be derived immediately by separating the 't Hooft line from the boundary along the topological direction and transforming $S_{R,n}(p)$ back to $\alpha_{R,n}(p)$.
Again, the centrality property guarantees that $A$-model corners labelled by a collection of operators $S_{R,n}(p) O_1(z_1) \cdots O_n(z_n)$
satisfy the same Ward identities as a function of the $z_i$ as corners labelled only by $O_1(z_1) \cdots O_n(z_n)$. This insures that the
the action of the 't Hooft operator on the state created by the brane is well-defined.
We conclude this general discussion with some comments on the description of the 't Hooft lines in the auxiliary $\varphi F$ theory.
The 't Hooft lines perpendicular to $N$ can be interpreted as monopole local operators in the $\varphi F$ theory. Operators such as ${\cal P}[\varphi](z)$
appear as local operators both in the $\varphi F$ theory and in the four-dimensional theory because they are polynomial of the elementary fields and
their definition does not depend on a choice of contour for the path integral. The definition of disorder operators such as monopole operators, instead,
requires one to modify the space of field configurations allowed in the path integral and thus affects the possible choices of integration contours. This modification
is encoded in the presence of an actual line defect in the four-dimensional theory, ending on $p$ in $N$. These considerations also apply to boundary disorder operators
at $\partial N$, which will appear as $O_{\ell,i}(p)$ endpoints of a boundary line defect in the four-dimensional theory.
It would be interesting to analyze directly the space of monopole operators available in the $\varphi F$ theory, as well as their images at the boundary.
Some of the tools were developed in \cite{Gaiotto:2016wcv} and applied to Chern-Simons theory there. They involve cohomology calculations on the
affine Grassmanian which are likely to give a local version of the analysis in Section \ref{lineq}. We leave this exercise to an enthusiastic reader.
\subsection{From Corners to States} \label{fcs}
Now consider a strip with ${\mathscr B}_{{\mathrm{cc}}}$ and $\overline {\mathscr B}_{{\mathrm{cc}}}$ at the two ends, and with boundary conditions set at the bottom of the strip
by some other brane ${\mathscr B}$ (fig. \ref{example6}). At the ``corners,'' the construction that we have just described produces chiral algebras ${\mathcal V}$ and $\overline {\mathcal V}$.
With suitable operator insertions at the corners, the path integral on the strip produces a (possibly distributional) state
\begin{equation}\label{chidef}
\chi = \left|O_1(z_1) \cdots O_n(z_n) \overline O_{n+1}(\overline z_{n+1}) \cdots \overline O_{n+\overline n}(\overline z_{n+ \overline n})\right\rangle
\end{equation}
in the usual Hilbert space ${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$.
An important special case is that the chiral algebras ${\mathcal V}$ and $\overline{\mathcal V}$ at the two corners are complex conjugate,
though we are not restricted to this case. Operators sitting on different corners cannot have OPE singularities, so the $\overline O_j(\overline z_j)$
operators at the $\overline {\mathscr B}_{{\mathrm{cc}}}$ corner do not affect the Ward identities for the $O_i(z_i)$ and vice versa.
We can also consider the pairing of $\chi$ with a test state $\Psi \in {\cal H}$.
The resulting inner product may in general be ill-defined, but it is well-defined if $\Psi$ is sufficiently nice, for example
if $\Psi$ is created in a similar way using a brane of compact support at the top of the strip (fig. \ref{example6}(c)).
If the inner product is well-defined for all choices of the operator insertions, it gives a collection of correlation functions
\begin{equation}
\langle \Psi| O_1(z_1) \cdots O_n(z_n) \overline O_{n+1}(\overline z_{n+1}) \cdots \overline O_{n+\overline n}(\overline z_{n+ \overline n}) \rangle
\end{equation}
on $C$ which satisfy the Ward identities for ${\cal V}$ and $\overline {\cal V}$.
Half-BPS boundary conditions decorated by boundary line defects will produce a larger collection of states
\begin{equation}
\left|\prod_i O_i(z_i) \prod_j \overline O_{j}(\overline z_{j}) \prod_k O_{\ell_k,k}(p_k) \overline O_{\ell_k,k}(\overline p_k) \right\rangle,
\end{equation}
with corner data including endpoints for the boundary line defects.
In the remainder of this section we will make these structures explicit for some basic examples of half-BPS boundary conditions.
\subsection{Dirichlet Boundary Conditions}\label{Dbc}
Four-dimensional half-BPS Dirichlet boundary conditions fix the gauge connection $A$ to vanish at the boundary, or more generally to take some specified
value at the boundary,
extended to other fields in a supersymmetric fashion. In particular, three of the six scalar fields satisfy Dirichlet boundary conditions,
and other three, which include the Higgs field $\phi$
of the 2d $\sigma$-model, satisfy Neumann boundary conditions.
From the perspective of the auxiliary $\varphi {\mathcal F}$ theory on a portion $N$ of $\partial M$, a junction between the half-BPS Dirichlet boundary condition
and the deformed Neumann boundary is represented by a Dirichlet boundary condition on the field ${\mathcal A}_{\overline z}$ along $\partial N$.
Gauge transformations are restricted to be trivial at the boundary; otherwise it would not be possible to specify the value of ${\mathcal A}_{\overline z}$ along
the boundary. We do not impose any boundary condition on $\varphi=\varphi_z\mathrm d z$, the conjugate of ${\mathcal A}_{\overline z}$.
Since gauge transformations are constrained to be trivial along $\partial N$, local operators on $\partial N$, that is, on the junction,
are not required to be gauge-invariant. Instead, there is a $G$ global symmetry\footnote{\label{defglobal} A global symmetry transformation by an element $g\in G$
is a gauge transformation $g(x):N\to G$ whose restriction to $\partial N$ is constant, $g(x)|_{\partial N}=g$. This preserves the condition ${\mathcal A}_{\overline z}|_{\partial N}=0$,
so it is a symmetry of the theory defined with that boundary condition. How the constant $g$ is extended over the interior of $N$ as a gauge transformation
does not matter. Any two choices differ by
a gauge transformation that is trivial on the boundary and acts trivially on physical observables.}
acting on operators on $\partial N$; the boundary value $a$ of ${\mathcal A}$ can be interpreted as a background connection for that global $G$ symmetry. In the 4d setup, the $G$ symmetry acts on the whole half-BPS Dirichlet boundary condition, but in the $A$-twist there are no local operators it can act on at interior points of $N$. The global
symmetry acts on junction operators only.
The scalar field $\varphi$ is a valid local operator at the junction, identified with the same operator in the auxiliary $\varphi {\mathcal F}$ theory.
In the $A$-twisted theory, the operator $\varphi_z$ is actually the conserved current associated to the boundary $G$ symmetry.
One way to derive this result is as follows. If we vary the action $I_{\varphi{\mathcal F}}=\int_N {\rm Tr}\,\varphi_z F_{t\overline z}\mathrm d t \mathrm d^2z$ with ``free''
boundary conditions, we find that the variation of $I_{\varphi {\mathcal F}}$ contains a boundary term $\int_{\partial N} {\rm Tr}\,\varphi_z\delta A_{\overline z}$,
and therefore the Euler-Lagrange equations include a boundary condition $\varphi_z|_{\partial N}=0$, with no restriction on $A_{\overline z}$. If instead
we want the boundary condition to be $A_{\overline z}=a_{\overline z}$ (where $a$ is some specified connection on $C$), we can add a boundary
term to the action, so that the full action becomes
\begin{equation}\label{fullact} I'_{\varphi{\mathcal F}}=I_{\varphi{\mathcal F}}-\int_{\partial N}{\rm Tr}\,\varphi_z(A_{\overline z}-a_{\overline z}).\end{equation}
Then the Euler-Lagrange equations give a boundary condition $(A_{\overline z}-a_{\overline z})|_{\partial N}=0$, with no constraint on $\varphi$.
But now the current is $J_z=\partial I'_{\varphi{\mathcal F}}/\partial a_{\overline z}=\varphi_z$, as claimed.
As explained in Section \ref{ACP} and in \cite{Gaiotto:2017euk}, in a generalization of this problem with generic $\Psi$, the fact that $J_z$ is a Kac-Moody
current can be seen classically, though a 1-loop computation similar to what we are about to describe is needed to show that the level is $\Psi-h$ rather than
$\Psi$. At $\Psi=0$, the Kac-Moody level comes entirely from a 1-loop calculation. The necessary computation can be done very easily with the help
of
a simple shortcut, which has an analogue in gauge theory in any dimension.
We place the $\varphi{\mathcal F}$ theory on a slab $I\times C$, where $I$ is an interval $[0,L]$ with the same Dirichlet boundary conditions at each end.
Anomalies can always be computed from the low energy limit of a theory, so in this case we can drop the modes that are nonconstant along $I$
and reduce to a purely two-dimensional theory. This is equivalent to taking a naive $L\to 0$ limit.
The resulting 2d theory has action
\begin{equation}\label{belaction}
\int {\rm Tr} \, \varphi_z D_{\overline z} {\mathcal A}_t\end{equation}
and is a bosonic $\beta \gamma$ system with fields $\varphi_z$ and ${\mathcal A}_t$ (which now depend only on $z$ and $\overline z$, not on $t$) of spins 1 and 0, valued in the adjoint representation.
A similar fermionic chiral $bc$ system with adjoint-valued fields has an anomaly coefficient $2h$, so the bosonic $\beta\gamma$ system has anomaly $-2h$.
In gauge theory on the slab $I\times C$, the anomaly is localized on the boundaries of the slab since the bulk theory is not anomalous.
By symmetry, half the anomaly comes from one end of the slab and half from the other end, so the anomaly coefficient at either end is $-h$. This value of the level
has very special properties, as we will review momentarily. The value $-h$ of the Kac-Moody level is often called the critical level and denoted as $\kappa_c$.
We will now determine the image in ${\cal V}$ of the bulk local operators
${\cal P}[\varphi](z)$
of the $\varphi{\mathcal F}$ gauge theory.
Naively, the image would just be
${\cal P}[J](z)$. This expression, though, is ill-defined because of the OPE singularity of $J_z$ with itself. We can regularize this expression by point splitting, carefully subtracting singular terms. This is actually a familiar exercise in 2d CFT.
The simplest example is the image of the quadratic Hamiltonian ${\rm Tr}\, \varphi_z^2$. The regularized version of the operator
is the Sugawara operator $S_2(z)$. Recall that for general level $\kappa$, the Sugawara operator is proportional to the stress tensor: $T(z) = \frac{1}{\kappa + h} S_2(z)$. In particular, the singular part of the OPE of $S_2(z)$
\begin{equation}
S_2(z) J(w) \sim (\kappa + h)\frac{J(w)}{(z-w)^2} + (\kappa + h) \frac{D_w J(w)}{z-w} + \cdots
\end{equation}
is proportional to the critically-shifted level $\kappa+h$. When $\kappa=\kappa_c=-h$, $S_2(z)$ has non-singular OPE with the Kac-Moody currents, i.e. it is central. This is precisely the property we expect for the image of ${\rm Tr} \varphi_z^2$ in ${\cal V}$ and fully characterizes it
among operators with the same scaling dimension. So $S_2(z)$ corresponds to the quadratic Hitchin Hamiltonians.
The regularization of higher Hamiltonians takes more work, but we can invoke a general theorem \cite{BD}: the center of the critical Kac-Moody algebra for $G$ is generated by a collection of central elements $S_{{\mathcal P}}(z)$ which regularize ${\cal P}[J](z)$. As an algebra, the center of the critical Kac-Moody algebra is isomorphic to the space of holomorphic functions on the oper manifold $L_{\mathrm{op}}$ for the Langlands dual group $G^\vee$. We conclude that $S_{{\mathcal P}}(z)$ is the image at the junction of the 3d operators ${\cal P}[\varphi](z)$, which are also identified via $S$-duality with the generators of the algebra of holomorphic functions on $L_{\mathrm{op}}$.
The presence of a global $G$ symmetry at a Dirichlet boundary adds an extra ingredient to the construction of holomorphic-topological BAA branes. As already
remarked, we can generalize
Dirichlet boundary conditions by setting the boundary value of the connection to any fixed connection $a$ along $C$, rather than setting it to zero.
We thus produce a whole family of
BAA branes $\mathrm{Dir}(a)$ parameterized by the choice of background $G$ connection $a$. In the $A$-twisted theory, the
system only depends on $a_{\overline z}$ at the ${\mathscr B}_{\mathrm{cc}}$ corner and on $a_z$ at the $\overline {\mathscr B}_{\mathrm{cc}}$ corner. The dependence is encoded respectively in the holomorphic currents $J_z$ at the ${\mathscr B}_{\mathrm{cc}}$ corner and anti-holomorphic currents $\overline J_{\overline z}$ at the $\overline {\mathscr B}_{\mathrm{cc}}$ corner.
In the absence of local operator insertions at the junctions, Dirichlet boundary conditions thus define a family of (distributional) states
$\Delta(a) \in {\mathcal H}$. The insertion of Kac-Moody currents $J_z(z_i)$ or $\overline J_{\overline z}(\overline z_j)$ at the two corners gives functional derivatives
\begin{equation}
\prod_i \frac{\delta}{\delta a_{\overline z}}(z_i) \prod_j \frac{\delta}{\delta a_{ z}}(\overline z_j) \Delta(a)
\end{equation}
of the state with respect to the background connection.
The Kac-Moody nature of the holomorphic and anti-holomorphic connections has an important consequence:
the two currents are separately conserved. Recall the anomalous conservation laws:
\begin{align}
D_{\overline z} J_z &= - \frac{\kappa_c}{2 \pi} f_{z \overline z} \cr
D_z J_{\overline z} &= \frac{\kappa_c}{2 \pi} f_{z \overline z}
\end{align}
where $f$ is the curvature of $a$. These imply that the state $\Delta(a)$ transforms covariantly but anomalously under infinitesimal
complexified gauge transformations of $a$. To first order in $\lambda$,
\begin{equation}
\Delta(a_z + D_z \overline \lambda, a_{\overline z} + D_{\overline z} \lambda )=\left[1+\frac{\kappa_c}{2 \pi} \int_C {\rm Tr}\left(\overline \lambda - \lambda \right)f_{z \overline z} \right] \Delta(a_z,a_{\overline z})
\end{equation}
A nice enough abstract state $\Psi \in {\mathcal H}$ paired with Dirichlet boundary conditions at $t=0$ gives a functional
$\Psi(a)$ that transforms similarly under infinitesimal complexified gauge transformations of $a$:
\begin{equation} \label{eq:cogauge}
\Psi(a_z + D_z \overline \lambda, a_{\overline z} + D_{\overline z} \lambda )=\left[1+\frac{\kappa_c}{2 \pi} \int_C {\rm Tr}\left(\overline \lambda - \lambda \right)f_{z \overline z} \right] \Psi(a)
\end{equation} Note that $\Psi$ is invariant under real gauge transformations, which correspond to the special case $\lambda=\overline\lambda$.
Functional derivatives of $\Psi(a)$ with respect to $a$ give correlation functions of critical Kac-Moody currents coupled to $a$.
\subsection{Connections vs. Bundles}\label{CB}
Let $a$ be a connection on a Riemann surface $C$ with structure group the compact gauge group $G$. Consider a function $\Psi(a)$ which
is invariant not just under $G$-valued gauge transformations, but under $G_{\Bbb C}$-valued gauge transformations, acting at the infinitesimal level by
\begin{equation}\label{actfor}\delta a_{\overline z}=-D_{\overline z}\lambda, ~~\delta a_z=-D_z\overline\lambda. \end{equation}
Such a function determines a function on ${\mathcal M}(G,C)$, because ${\mathcal M}(G,C)$ can be viewed as the quotient of the space of all $G$-valued connections by
the group of complex gauge transformations.\footnote{This description is slightly imprecise as one needs to take account of considerations of stability
to realize ${\mathcal M}(G,C)$ as a quotient. But actually, there is no difficulty: as long as the function $\Psi(A)$ is continuous as well as invariant under $G_{\Bbb C}$-valued
gauge transformations, it does descend to a function on ${\mathcal M}(G,C)$. To get from the space of all connections to ${\mathcal M}(G,C)$, one throws away connections
that define unstable holomorphic bundles, imposes an equivalence relation on the semistable ones, and then takes the quotient. A function $\Psi(a)$
that is continuous and $G_{\Bbb C}$-invariant is always invariant under the equivalence relation. In any event, these considerations are unimportant for an ${\mathrm L}^2$
theory. Somewhat similar remarks apply in the next paragraph.}
Conversely, given a function $f$ on ${\mathcal M}(G,C)$, to define a function $\Psi(a)$ on the space of connections, we simply declare $\Psi(a)$, for a given $a$,
to equal $f$ at the point in ${\mathcal M}(G,C)$ that is associated to the holomorphic bundle $E\to C$ that is determined by the $(0,1)$ part of $a$. The function $\Psi(a)$
defined this way is automatically invariant under $G_{\Bbb C}$-valued gauge transformations.
This correspondence between functions on ${\mathcal M}(G,C)$ and $G_{\Bbb C}$-invariant functions of connections can be extended
to functions $\Psi(a)$ that are not $G_{\Bbb C}$-invariant, but rather transform covariantly under $G_{\Bbb C}$-valued gauge transformations, with an anomaly.
These functionals can be identified with sections of some line bundle over ${\mathcal M}(G,C)$.
For us, the most important case is a function $\Psi(a)$ that transforms with holomorphic and antiholomorphic anomaly coefficients $\kappa_c=-h$,
as in
eqn. (\ref{eq:cogauge}).
In this case, the line bundle is actually the bundle of half-densities on ${\mathcal M}(G,C)$, as we will show momentarily.
More generally, if $\kappa_c$ is replaced by some other level $\kappa$ (the same both holomorphically and antiholomorphically),
$\Psi(a)$ would represent a section of $|K_{{\mathcal M}(G,C)}|^{\frac{\kappa}{\kappa_c}}$.
There are many ways to demonstrate that a function that transforms with holomorphic and antiholomorphic
anomaly coefficient $\kappa_c$ correponds to a half-density on ${\mathcal M}(G,C)$ (as shown originally by Beilinson and Drinfeld \cite{BD}).
We will proceed by showing that a function that transforms with the anomaly coefficient $2\kappa_c$ is a density on ${\mathcal M}(G,C)$. A density
on ${\mathcal M}(G,C)$ is something that can be integrated over ${\mathcal M}(G,C)$ in a natural way, without using any structure of ${\mathcal M}(G,C)$ beyond the fact
that it is the quotient of the space of connections by the group of $G_{\Bbb C}$-valued gauge transformations.
To decide what kind of object $\Psi(a)$ can be integrated over ${\mathcal M}(G,C)$, we will use
a construction which is somewhat analogous to the definition of the bosonic string path integral.
Consider a two-dimensional $G$ gauge theory with connection $a$, coupled to some matter system with holomorphic and anti-holomorphic Kac-Moody symmetry at levels $(\kappa, \kappa)$. As the path integral of this theory is formally invariant under
complexified gauge transformations, we may hope to gauge-fix the path integral to an integral over ${\mathcal M}(G,C)$. In order to do so, we need a family of gauge-fixing conditions. We simply pick a representative 2d connection $a[m, \overline m] = (a_z[\overline m], a_{\overline z}[m])$ for every point $m$ in ${\mathcal M}(G,C)$ and gauge-fix $a = a[m, \overline m]$.
The aim is to reduce the integral over $a$ to an integral over $m,\overline m$.
We introduce Faddeev-Popov ghosts for this gauge-fixing in the customary manner. For this, we introduce adjoint-valued ghosts $c$, $\overline c$ associated to complexified gauge parameters $\lambda$ and $\overline \lambda$, and an adjoint-valued 1-form $(b_z, \overline b_{\overline z})$
associated to the gauge-fixing condition. The ghost action is
\begin{equation}
\int\left( {\rm Tr} \,b_z D_{\overline z} c + {\rm Tr} \,b_{\overline z} D_z c\right)\mathrm d^2z.
\end{equation}
This ghost system has holomorphic and anti-holomorphic Kac-Moody symmetries, with levels $(2h,2h)=(-2\kappa_c, -2\kappa_c)$.
The BRST current is known to be nilpotent if and only if the total anomaly of matter plus ghosts vanishes, that is, if and only if $\kappa - 2 \kappa_c=0$.
The integrand over ${\mathcal M}(G,C)$ is prepared with the help of the $b$ zero modes. If we denote the matter partition function as $\Psi(a)$, the gauge-fixed path integral becomes
\begin{equation}
\int_{{\mathcal M}(G,C)} \Psi\left(a[m,\overline m]\right) \left\langle \prod_i \left[\int_C b_z \frac{\partial a_{\overline z}[m]}{\partial m_i} \mathrm d m_i \right] \left[\int_C b_{\overline z} \frac{\partial a_{z}[\overline m]}{\partial \overline m_i} \mathrm d \overline m_i \right] \right\rangle
\end{equation} Here we have simply imitated the usual definition of the path integral of the bosonic string coupled to a conformal field theory of
holomorphic and antiholomorphic central charge $c=26$.
We thus learn that gauge-covariant functionals with level $2 \kappa_c$ correspond to densities on ${\mathcal M}(G,C)$.
Gauge-covariant functionals with level $\kappa_c$, as in (\ref{eq:cogauge}), thus correspond to half-densities on ${\mathcal M}(G,C)$. We can write down explicitly the Hilbert space inner product in this presentation:
\begin{equation}\label{eq:innergauge}
\left( \Psi', \Psi \right) \equiv \int_{{\mathcal M}(G,C)} \overline{\Psi}'\left(a[m,\overline m]\right) \Psi\left(a[m,\overline m]\right) \left\langle \prod_i \left[\int_C b_z \frac{\partial a_{\overline z}[m]}{\partial m_i} \mathrm d m_i \right] \left[\int_C b_{\overline z} \frac{\partial a_{z}[\overline m]}{\partial \overline m_i} \mathrm d \overline m_i \right] \right\rangle
\end{equation}
We now have two ways to associate a functional $\Psi(a)$ to a (nice enough) state $\Psi \in {\mathcal H}$: we identify an abstract state with a half-density on ${\mathcal M}(G,C)$ and promote
it to a functional of connections, or we contract $\Psi$ with the states $\Delta(a)$ produced by the shifted Dirichlet boundary condition $\mathrm{Dir}(a)$. To show
that these two procedures are equivalent, we can reason as follows.
Classically, the BAA brane $\mathrm{Dir}(a)$ is supported on the fiber of $T^*{\mathcal M}(G,C)$ at the bundle $E$ defined by $a_{\overline z}$.
This is the simplest type of conormal Lagrangian submanifold and the corresponding state $\Delta(a)$ has delta function support
at $E$. So the pairing $ (\Delta(a), \Psi)$
just evaluates the functional corresponding to $\Psi$ at the connection $a$.
As a final exercise, we can return to the definition of the quantum Hitchin Hamiltonians. We would
like to derive a formula expressing the functional $[D_{\mathcal P}(z) \circ \Psi](a)$ resulting from the action of a
Hamiltonian on some $\Psi$ in terms of the functional $\Psi(a)$ associated to $\Psi$. We consider a strip with ${\mathscr B}_{\mathrm{cc}}$ boundary conditions on the left boundary and some initial condition at the
bottom of strip that, together with data at the corners, defines the state $\Psi$. (This setup was sketched in fig. \ref{example6}(b), where the brane at the bottom
of the strip is called ${\mathscr B}_F$.)
By definition, the functional
$[D_{\mathcal P}(z) \circ \Psi](a)$ is computed by inserting ${\mathcal P}[\varphi_z]$ along the left boundary and moving it to the lower left corner of
the strip. When we do this, ${\mathcal P}[\varphi_z]$ is converted to
the central element $S_{{\mathcal P}}(z)$ of the chiral algebra, which is a regularized polynomial in $J_z$ and its derivatives. In turn,
the $J_z$ insertions can be traded for functional derivatives with respect to $a_{\overline z}$.
As a result, $[D_{\mathcal P}(z) \circ \Psi](a)$ is expressed as a certain differential operator $D^{(a)}_{\mathcal P}(z)$ acting on $\Psi(a)$. The operator
$D^{(a)}_{\mathcal P}(z)$ is a regularization of ${\mathcal P}\left[\frac{\delta}{\delta a_{\overline z}} \right]$. It maps gauge-covariant functionals to
gauge-covariant functionals precisely because $S_{{\mathcal P}}(z)$ is central: the $S_{{\mathcal P}}(z)$ insertion does not modify the
Ward identities of the currents and thus the differential operator ${\cal D}^{(a)}_{{\mathcal P}}(z)$ commutes with the
gauge-covariance constraints (\ref{eq:cogauge}). This is actually how the quantum Hitchin Hamiltonians
are defined mathematically \cite{BD}: they encode the effect of an $S_{{\mathcal P}}(z)$ insertion in a conformal block
for the critical Kac-Moody algebra.
A similar presentation of Hecke operators requires a discussion of boundary 't Hooft operators
at Dirichlet boundary conditions and their endpoints at the junction. In the auxiliary 3d perspective,
the boundary 't Hooft operators map to boundary monopole operators. The classical moduli space of such disorder operators
was discussed in a similar setting in \cite{Costello:2020ndc}: it coincides with the affine Grassmannian ${\mathrm{Gr}}_{G_{\Bbb C}}$.
In the presence of the disorder operator, the gauge bundle at some small distance from the boundary is a specific Hecke modification
of whatever fixed bundle is determined by the boundary value of the connection. Correspondingly, in order for $\varphi_z$
to be non-singular at some distance from the boundary it must have some prescribed poles and zeroes at the boundary.
This bare boundary monopole configuration can be dressed by local functionals of $\varphi_z$.
In Section \ref{sec:chiral} we will discuss the ``spectral flow operators'' $\Sigma_g$ in the chiral algebra, which are labelled by a point in
${\mathrm{Gr}}_{G_{\Bbb C}}$ and enforce an appropriate version of the constraint on $J_z$. The spectral flow operators and their Kac-Moody descendants
can play the role of endpoints of boundary 't Hooft operators. We will observe the existence of certain (continuous) linear combinations of
spectral flow operators which are central and can thus play the role of the images $S_{R,n}(z)$ of endpoints of bulk 't Hooft operators.
This will allow us to formulate Hecke operators in a 2d chiral algebra language.
\subsection{Nahm Pole Boundary Conditions}\label{nbc}
Half-BPS Dirichlet boundary conditions can be generalized to a larger collection of Nahm pole boundary conditions labelled by an embedding
$\rho:\mathfrak{su}(2)\to{\mathfrak g}$. These boundary conditions allow for a choice of background connection $a_\rho$ whose structure group commutes with $\rho$.
Following the analogy with the $\Psi \neq 0$ results in \cite{Gaiotto:2017euk}, we will tentatively identify the chiral algebra
associated to these boundary conditions with the critical level limit of the ${\cal W}^G_{\rho;\kappa}$ chiral algebras,
which are in turn defined as the Drinfeld-Sokolov reduction associated to $\rho$ of a $G$ Kac-Moody algebra at level $\kappa$.
As a basic check of this proposal, we observe that ${\cal W}_{\rho;\kappa_c}$ has the same large center as critical Kac-Moody,
generated by appropriate $S_{\cal P}(z)$.
A particularly interesting case is the Nahm pole associated to a regular embedding $\rho$. The corresponding chiral algebra is
the classical limit of a $W$-algebra and is completely central. It is generated by the $S_{\cal P}(z)$: all local operators on the junction
are specializations of local operators on the deformed Neumann boundary.
The regular Nahm pole is associated to a BAA brane ${\mathscr B}_N$ supported on the Hitchin section of the Hitchin fibration. This section is a complex Lagrangian
submanifold of ${\mathcal M}_H$, of type BAA, but it lies completely outside $T^*{\mathcal M}(G,C)$.
If an eigenstate $\Psi$ of the Hitchin Hamiltonians is viewed purely as a square-integrable half-density on ${\mathcal M}(G,C)$, then it would appear not to make any sense to compute
the inner product of $\Psi$ with a state created by ${\mathscr B}_N$ (with appropriate corners), as the space of square-integrable half-densities comes by quantization of
$T^*{\mathcal M}(G,C)$, which is completely disjoint from the support of ${\mathscr B}_N$. However, as discussed in Section \ref{wkb}, $\Psi$ is actually associated to a brane in ${\mathcal M}_H$
of compact support, and therefore should have a well-defined pairing with the state created by any brane. In fact, in the dual $B$-model, the computation is
straightforward. We return to this point at the end of Appendix \ref{bmodel}.
The regular Nahm pole boundary supports boundary 't Hooft lines which were studied in \cite{Witten:2011zz}. They are in natural correspondence
with bulk 't Hooft lines, and they are indeed the image of bulk 't Hooft lines brought to the boundary. Nahm pole boundary conditions decorated by boundary
't Hooft operators thus give rise to BAA branes supported on the Hecke modification of the Hitchin section
at a collection of points. If the number of points is large enough, these BAA branes are nice submanifolds in $T^*{\mathcal M}(G,C)$. The associated
states should play a role in the separation of variables analysis of \cite{T}.
\subsection{Enriched Neumann Boundary Conditions}\label{Enriched}
A basic BAA boundary condition in the 2d $\sigma$-model of ${\mathcal M}_H(G,C)$ is the Lagrangian boundary condition associated
to the Lagrangian submanifold ${\mathcal M}(G,C)\subset {\mathcal M}_H(G,C)$. In 4d terms, this comes
from a half-BPS boundary condition of type BAA in in which the gauge field $A$ satisfies Neumann boundary conditions and the Higgs field $\phi$
satisfies Dirichlet boundary conditions.\footnote{The brane ${\mathscr B}_{\mathrm{cc}}$ comes instead from a deformation of Neumann boundary conditions for $A$, in a
sense described in footnote \ref{deformed}, extended to the rest
of the supermultiplet in a different fashion and preserving a different symmetry (ABA rather than BAA).} We will refer to this boundary condition as BAA Neumann.
In terms of the $\varphi{\mathcal F}$ theory, this is simply the boundary condition defined by $\varphi|_{\partial N}=0$, with no constraint on $A_{\overline z}|_{\partial N}$.
We showed in Section \ref{Dbc} that this is the boundary condition one gets from the Euler-Lagrange equations of the action $I_{\varphi{\mathcal F}}$, with ``free''
variations of all fields.
With this boundary condition, no restriction is placed on a gauge transformation on the boundary. That is consistent, because the boundary
condition $\varphi|_{\partial N}=0$ is gauge-invariant. However, there is a gauge anomaly on a boundary of $N$ that has this boundary condition.
The anomaly coefficient is $+h$.
An easy way to see this is to consider the $\varphi{\mathcal F}$ theory on a slab $N=I\times C$, with the ${\mathcal A}_{\overline z}=0$ boundary condition at the left end
of the slab and the $\varphi_z=0$ boundary condition at the right end. These boundary conditions are invariant under constant gauge transformations
by an element $g\in G$. At the left end of the slab, a constant gauge transformation is interpreted as a global symmetry.
This gives an action of $G$ as a group of global symmetries of the theory on the slab.
The theory with $\varphi_z=0$
at one end of the slab and ${\mathcal A}_{\overline z}=0$ at the other end is completely trivial: up to a gauge transformation, the only classical solution
is $\varphi_z={\mathcal A}_{\overline z}=0$ everywhere, and there are no low energy excitations. So the $G$ action is anomaly free. As it acts by a constant gauge
transformation, its anomaly coefficient is the sum of the anomaly coefficient of the global symmetry at the left end of the slab and of the gauge symmetry at the right end.
We learned in Section \ref{Dbc} that the global symmetry has an anomaly coefficient $-h$ at the left end of the slab. So
the gauge symmetry must have an anomaly coefficient $+h$ at the right end.
So in short, the $\varphi_z=0$ boundary condition in the $\varphi{\mathcal F}$ theory has anomaly $+h$.
A possible cure for the anomaly is to add extra degrees of freedom at the junction with anomaly $-h$, the critical level. Unitary degrees of freedom at the junction
will not help, as they have a positive anomaly coefficient.
Instead, we can do the following.
BAA Neumann boundary conditions can be enriched, preserving the supersymmetry of type BAA, by adding to the boundary
3d matter degrees of freedom that make a 3d superconformal quantum
field theory (SQFT) with ${\mathcal N}=4$ supersymmetry.
We will call the resulting boundary condition
an enriched Neumann boundary condition (of type BAA, if it is necessary to specify this). The interesting case is that the SQFT has
$G$ symmetry and is coupled to the gauge field $A$ of the bulk ${\cal N}=4$ theory; this is possible, because with Neumann boundary conditions, $A$
is unconstrained on the boundary.
The $A$-twist of the bulk 4d theory induces an $A$-twist of the boundary
3d SQFT. The twisted boundary theory can contribute a negative amount to the anomaly at the junction. For our application, we want holomorphic boundary conditions
for the 3d SQFT that support a $G$ Kac-Moody algebra at critical level $\kappa_c=-h$.
For this purpose, we can employ one of the holomorphic boundary conditions defined in \cite{Costello:2018fnz}. The effect of ``enrichment'' is
that the boundary condition for the $\varphi{\mathcal F}$ theory ending on an enriched Neumann boundary is no longer $\varphi_z|_{\partial N}=0$. Rather, $\varphi_z|_{\partial N}$
equals the critical Kac-Moody currents of the boundary chiral algebra of the SQFT. In particular, the images $S_{{\mathcal P}}(z)$ of ${\mathcal P}[\varphi_z]$ are identified with the central elements built from the critical Kac-Moody currents for the matter.
The simplest example is the case that the SQFT is a theory of free 3d hypermultiplets transforming in a symplectic representation $R$ of $G$. Let $Z$ be
the bosonic field in the hypermultiplets. Twisting turns the components of
$Z$ into spinors, still valued in the representation $R$.
As analyzed in \cite{Gaiotto:2016hvd}, with the appropriate sort of boundary condition, the twisted hypermultiplet path integral on a three-manifold
with boundary is a 2d contour path integral, with a holomorphic action, of the general sort described in Section \ref{ACP}. In this case, the holomorphic action is
\begin{equation}
S[Z,{\mathcal A}] = \int_C \langle Z, \overline \partial_{\mathcal A} Z \rangle .
\end{equation}
where $\langle \cdot,\cdot \rangle$ denotes the symplectic pairing on the representation $R$, and we have included a coupling to the complex gauge
field ${\mathcal A}$. Fields $Z$ with such an action are sometimes called symplectic bosons and do have a negative Kac-Moody level; see Section \ref{sb} for more about them. The simplest possibility is to select an $R$ for which the level is precisely$-h$. It is also possible to select an $R$
for which the level is more negative and make up the difference with some extra 2d chiral fermions in a real representation $R_f$ of $G$ placed at the junction. We will discuss the simplest possibility here and briefly comment on the general case at the end.
Now consider a junction between the deformed Neumann boundary condition that supports the $\varphi{\mathcal F} $ theory and the
BAA Neumann boundary condition enriched by hypermultiplets. The appropriate holomorphic action is the sum of $I_{\varphi{\mathcal F}}$ and $S[Z,{\mathcal A}]$:
\begin{equation}\label{welli}\widehat I=\int_N {\rm Tr}\,\varphi {\mathcal F} +\int_{C=\partial N} \langle Z, \overline \partial_{\mathcal A} Z \rangle .\end{equation}
The Euler-Lagrange equation for ${\mathcal A}_{\overline z}$ gives a boundary condition
\begin{equation}\label{nell}\varphi_z|_{\partial N}=\mu(Z), \end{equation}
where $\mu(Z)$ is the holomorphic moment map for the action of $G_{\Bbb C}$ on the representation $R$. Here the components of $\mu(Z)$ become
(after quantization) the Kac-Moody
currents of the matter system, so this formula illustrates the statement that after enrichment, the appropriate boundary condition sets $\varphi$ equal to the Kac-Moody currents.
We also have the classical equations of motion
\begin{align}
0=\overline\partial_{\mathcal A}\varphi =\overline\partial_{\mathcal A} Z.
\end{align}
Triples $({\mathcal A},\varphi,Z)$ satisfying these conditions along with eqn. (\ref{nell}) describe a brane over ${\mathcal M}_H(G,C)$ of type BAA. The simplest case is that for given ${\mathcal A},\varphi$, there is at most one $Z$ satisfying the conditions. If so, the pairs $({\mathcal A},\varphi)$ for which such a $Z$ does exist furnish a complex Lagrangian submanifold of ${\mathcal M}_H(G,C)$, in complex structure $I$, corresponding to a brane of type BAA.
These Lagrangian submanifolds are of conormal type, since if a suitable $Z$ exists for one Higgs pair $({\mathcal A},\varphi)$, then a suitable $Z$ likewise exists after any rescaling of $\varphi$.
The natural quantization of these BAA branes is a path integral over $Z$ \cite{Gaiotto:2016wcv}:
\begin{equation}\label{eq:ZPsi}
\Psi(a) = \int DZ D\overline Z \,e^{\int_C \left[\langle Z, \overline \partial_a Z \rangle - \langle
\overline Z, \partial_a \overline Z \rangle\right]}
\end{equation}
possibly modified by the insertion of a non-trivial corner in the form of a collection of $Z$ and $\overline Z$ insertions in the path integral \cite{Costello:2018fnz}.
From the point of view of the present paper, the meaning of this formula is as follows. We place the enriched Neumann brane at the bottom of a strip,
playing the role of the brane denoted as ${\mathscr B}_x$ in fig. \ref{example6}(b). Assuming no operator insertions are made at the bottom corners of the strip,
the state in ${\mathcal H}={\mathrm{Hom}}(\overline{\mathscr B}_{\mathrm{cc}},{\mathscr B}_{\mathrm{cc}})$ defined by this picture is $\Psi(a)$. The statement makes sense, because the $Z$ and $\overline Z$ fields support
current algebras at critical level $\kappa_c=-h$, so that the path integral of these fields does indeed define a half-density on ${\mathcal M}(G,C)$.
The chiral algebra at the junction in this construction consists of the
subalgebra of gauge-invariant operators within the boundary chiral algebra of the 3d matter theory, i.e. it consists of operators built from the $Z$'s and their derivatives which have trivial OPE with the critical Kac-Moody currents. One can modify the construction just described by including chiral and antichiral operators at the
bottom corners of the strip; to describe the resulting state, one just includes the corresponding factors in eqn. (\ref{eq:ZPsi}).
We will discuss this construction further in Section \ref{sec:symp}.
\begin{figure}
\begin{center}
\includegraphics[width=3.2in]{Example9.pdf}
\end{center}
\caption{\small (a) A rectangle with an enriched Neumann brane ${\mathscr B}$ at the bottom and a generalized Dirichlet brane ${\mathscr B}'$ at the top. (b) In topological field
theory, the ``height'' and ``width'' of the rectangle are arbitrary. In a limit in which the height is small, we reduce to a purely three-dimensional computation on
a product $I\times C$. As always, $C$ is not drawn. \label{example9}}
\end{figure}
There is an alternative way to understand (\ref{eq:ZPsi}) directly in 4d. The alternative perspective can be applied as well to a more general
situation where the corresponding BAA brane has a non-trivial ${\mathrm{CP}}$ bundle or where extra chiral fermions are added at the junctions. In order to read off $\Psi(a)$, we can contract the state $\Psi$ created by the enriched Neumann boundary with the state $\Delta(a)$ created by a Dirichlet boundary condition $\mathrm{Dir}(a)$. The inner product between these two states is represented by
the path integral on the rectangle of fig. \ref{example9}(a) with $\mathrm{Dir}(a)$ boundary conditions at the top and enriched Neumann at the bottom. In two-dimensional topological field theory, the ``height'' and ``width'' of the rectangle are arbitrary.
Take the limit that the height is much less than the width (fig. \ref{example9}(b)). In this limit, the path integral reduces to a path integral in a 3d theory on $I\times C$.
The 3d theory is produced by compactification from four to three dimensions on an interval with Dirichlet boundary conditions
at one end and enriched Neumann boundary conditions at the other end.
This compactification gives a simple answer, because the 4d fields are all frozen at one boundary or the other: one just gets back the
same 3d theory which was employed to construct the enriched Neumann boundary conditions. In our example, this is the theory of the same free hypermultiplets
that we started with, with the global $G$ symmetry now identified
with the $G$ symmetry that acts at the Dirichlet boundary. The answer of (\ref{eq:ZPsi}) is just the partition function of the 3d theory on $I\times C $, with the boundary conditions which give rise to the symplectic bosons or their complex conjugates. We can thus apply (\ref{eq:ZPsi}) to a situation where the corresponding BAA brane is complicated, bypassing the 2d derivation. Any (anti)chiral fermions added at the junctions would just contribute their partition function, i.e.
\begin{equation}\label{eq:ZPsifer}
\Psi(a) = \int DZ D\overline Z D\psi D\overline \psi \,e^{\int_C \left[\langle Z, \overline \partial_a Z \rangle - \langle
\overline Z, \partial_a \overline Z \rangle\right]+\left[( \psi, \overline \partial_a \psi ) - (
\overline \psi, \partial_a \overline \psi)\right]}
\end{equation}
As long as the combined level of the symplectic bosons and fermions is $-h$, this represents a half-density on ${\mathcal M}(G,C)$.
In this section we described states associated to elementary boundary conditions. The construction can be easily generalized to describe operators associated to analogous elementary interfaces. The composition of elementary interfaces can produce a vast collection of BAA boundary conditions and interfaces, which are associated to the composition of the corresponding operators. This would allow, among other things, the calculation of $C \times [0,1]$ partition functions for A-twisted 3d ${\cal N}=4$ gauge theories with chiral and antichiral boundary conditions at the two ends of the segment. We leave a detailed analysis of this problem, as well as the B-model analogue,
to future work.
\section{Hecke Operators and Spectral Flow Modules}\label{sec:chiral}
\subsection{Preliminaries}
The quantization of BAA branes associated to enriched Neumann boundary conditions has given us examples (\ref{eq:ZPsi}) of wavefunctions which are defined as partition functions of 2d CFTs with chiral and antichiral critical Kac-Moody symmetry. In this section we describe how to compute the action of quantum Hitchin Hamiltonians and Hecke operators on such partition functions, directly in a 2d CFT language. At the same time, we will gain a better appreciation of the mathematical results we invoked in Section \ref{wtw} to define the Hecke operators.
We have already discussed briefly the 2d CFT interpretation of the quantum Hitchin Hamiltonians. The critical Kac-Moody chiral algebra
has a large center, generated by certain local operators $S_{\cal P}(z)$ which have non-singular OPE with the currents.
The transformation of a correlation function under complexified gauge transformations is described
by the Ward identities for the currents.
The statement that $S_{\cal P}(z)$ has non-singular OPE with the currents means that a correlation function with insertions of such operators only
\begin{equation}
\langle S_{{\cal P}_1}(z_1) \cdots S_{{\cal P}_n}(z_n) \rangle_a
\end{equation}
satisfies the same transformation properties (\ref{eq:cogauge}) as a partition function. It thus also defines a half-density on ${\mathcal M}(C,G)$.
Furthermore, the $S_{\cal P}(z)$ are assembled from Kac-Moody currents, which can be traded for functional derivatives with respect to
the connection. We can thus expand recursively
\begin{equation}
\langle S_{{\cal P}_1}(z_1) \cdots S_{{\cal P}_n}(z_n) \rangle_a = {\cal D}^{(a)}_{{\mathcal P}_1}(z_1 )\langle S_{{\cal P}_2}(z_2) \cdots S_{{\cal P}_n}(z_n) \rangle_a
\end{equation}
and the final answer will be independent of the order of the operators to which we apply the recursion. The differential operators ${\cal D}^{(a)}_{{\mathcal P}_1}(z_1 )$ thus commute.
Although here we referred to correlation functions of some 2d CFT, this is unnecessary: given a half-density on ${\mathcal M}(C,G)$ represented by
a gauge-covariant functional $\Psi(a)$ on the space of connections, the functional derivatives with respect to $a$ behave just as Kac-Moody currents.
The differential operators ${\cal D}^{(a)}_{{\mathcal P}}(z)$ represent in a gauge-covariant manner the action of the quantum Hitchin Hamiltonians
${\cal D}_{{\mathcal P}}(z)$ on the half-density $\Psi$.
When doing calculations in a neighborhood $U$ of a point $p$ in $C$, it is usually helpful to choose a representative connection for the bundle
which vanishes on $U$. This is always possible because ${\mathcal A}_{\overline z}$ can be set to zero locally by a complex-valued
gauge transformation. That amounts to trivializing the bundle over $U$, as we did in discussing general Hecke transformations in Section
\ref{affgr}. Then the Kac-Moody currents are meromorphic on $U$ and satisfy the Kac-Moody OPE in a standard form
\begin{equation}
J^a(z) J^b(w) \sim \frac{\kappa_c \delta^{ab}}{(z-w)^2} + \frac{f^{ab}_d J^d(w)}{z-w}.
\end{equation}
Recall the definition of the Fourier modes of the Kac-Moody algebra
\begin{equation}
J^a_n \equiv \oint_{|z|=\epsilon} \frac{\mathrm d z}{2 \pi {\mathrm i}} z^n J^a(z)
\end{equation}
The insertion of such a Fourier mode represents an infinitesimal deformation of $a_{\overline z}$ supported on the loop $|z|=\epsilon$,
or a deformation of the bundle which modifies the gluing of a bundle over $U$ to a bundle over the rest of the surface by an infinitesimal gauge transformation in $U'=U\backslash p$.
In the absence of other operator insertions in the disk $|z|<\epsilon$ (or in the presence of central operator insertions) correlation functions with insertions of
the non-negative Fourier modes vanish. The corresponding infinitesimal gauge transformations can be extended to $U$ and do not change the bundle.
They represent changes in the original trivialization over $U$. In the presence of a generic operator insertion in the disk,
the non-negative modes act non-trivially: the insertion of a general local operator requires some choice of trivialization of the bundle and
the result depends on the choice.
The negative Fourier modes can act non-trivially even in the absence of other operator insertions
and represent infinitesimal gauge transformations which can change the bundle.
Repeated action of the negative modes builds the image at $z=0$ of the vacuum module for the Kac-Moody algebra.
The operator
$S_{\cal P}(0)$ and other central elements in the chiral algebra correspond by the operator-state correspondence to the vectors in the
vacuum module that are annihilated by all the non-negative Fourier modes of the currents. For example, the Sugawara vector is
\begin{equation}
|S_2\rangle \equiv {\rm Tr} \,J_{-1} J_{-1} |0 \rangle,
\end{equation} with similar formulas for other central elements.
\subsection{Hecke Operators as Central Vertex Operators}\label{cvo}
The Hecke integral operators can also be analyzed with 2d chiral algebra technology. We would like to lift the Hecke operators to operators acting on gauge-covariant functionals and give them a 2d chiral algebra interpretation in terms of the insertion of local operators which have trivial OPE with the Kac-Moody currents. Such a formulation immediately guarantees that the Hecke operators commute with the quantum Hitchin Hamiltonians
and with other Hecke operators.
We can follow verbatim the definition of Hecke modifications from Section \ref{affgr}. First, we trivialize the bundle $E$ on a small neighborhood $U$ of a point $p$. We can then think of $E\to C$
as built by gluing a trivial bundle over $U$ to the bundle $E$ over $C\backslash p$ with a trivial gluing map. Then we produce a new bundle $E'$ by modifying the gluing map to $z^{\sf m}$ (where ${\sf m}$ is an integral weight of the dual group and $z$ is a local parameter at $p$). The bundles $E$ and $E'$ can be described by the same connection away from $U$. The connection $a$ which describes $E$ vanishes on $U$, while the connection $a'$ which describes $E'$ coincides with $a$ outside of $U$
and can be taken in $U$ to be some specific reference connection supported on an annulus in $U'$, and proportional to ${\sf m}$.
Take the functional $\Psi$ which represents the input wavefunction, and evaluate it on $a'$. This gives a new functional $\Psi_{{\sf m}}(a)$.
Crucially, $\Psi_{\sf m}(a)$ is not covariant under complexified gauge transformations: the new bundle $E'$ depends on the original choice
of trivialization of $E$. Formally, $\Psi_{\sf m}(a)$ and its functional derivatives can be interpreted as correlation functions of Kac-Moody currents in the presence of a ``spectral flow operator'' $\Sigma_{{\sf m}}(0)$.
The term ``spectral flow'' refers to a certain automorphism of the Kac-Moody algebra:
\begin{align}
J^\alpha_n &\to J^{\alpha}_{n+({\sf m}, \alpha)} \cr
J^h_n &\to J^h_n - {\sf m} \kappa \delta_{n,0}
\end{align}
where $J^\alpha$ is the current associated to a root $\alpha$ and $J^h$ are the Cartan currents.
This is precisely the effect of a $z^{\sf m}$ gauge transformation on the Fourier modes of the currents.
By definition, a spectral flow module is the image of the vacuum module under the spectral flow. In particular, the image of the vacuum vector under spectral flow is annihilated by $J^{\alpha}_{n+({\sf m}, \alpha)}$ with non-negative $n$ and is an eigenvector of $J^h_0$ with a nontrivial eigenvalue.
Correspondingly, a spectral flow operator $\Sigma_{{\sf m}}(0)$ is a local operator such that the OPE with the Kac-Moody currents become non-singular after a $z^{\sf m}$ gauge transformation. The $J^{\alpha}(z)$ will have a pole/zero of order $({\sf m}, \alpha)$ at $z=0$ and $J^h(z)$
will have a simple pole of residue\footnote{Bosonization offers a convenient way to describe $\Sigma_{{\sf m}}$.
Schematically, if the Cartan currents are bosonized as $J^h = \partial \phi^h$ and the remaining currents as vertex operators $J^\alpha = e^{\frac{\alpha}{\kappa} \cdot \varphi}$, then the spectral flow operator can be represented by a vertex operator $\Sigma_{{\sf m}} = e^{{\sf m} \cdot \varphi}$ as well. This representation can be useful for some calculations, but behaves poorly under general $G_{\Bbb C}$ gauge transformations.} ${\sf m}\kappa \Sigma_{{\sf m}}(0)$.
We stress again that the functional $\Psi_{{\sf m}}(a)$ does not represent a half-density on ${\mathcal M}(G,C)$, as it depends on the
choice of trivialization of $E$. The properties of the spectral flow operator $\Sigma_{{\sf m}}(0)$ characterize the precise failure of the
gauge-covariance constraints (\ref{eq:cogauge}). Our objective is to build from $\Sigma_{{\sf m}}(0)$ some local operator insertion which
is central and can thus represent the action of a Hecke operator on $\Psi(a)$.
Before continuing with the general discussion, we present the reference example of $G={\mathrm{SO}}(3)$ and minimal charge. The basic spectral flow automorphism is
\begin{equation}
J^\pm_n \to J^{\pm}_{n \pm 1} \qquad \qquad J^0_n \to J^0_n - \delta_{n,0}
\end{equation}
The spectral flow module is built from a vector $|1\rangle$ which satisfies
\begin{align}\label{laterref}
J^{\pm}_{n \pm 1} |1\rangle &=0 \qquad \qquad n\geq 0 \cr
J^0_n |1\rangle &=0 \qquad \qquad n> 0 \cr
J^0_0|1\rangle &= |1\rangle.
\end{align}
For this case of the basic ``charge 1'' spectral flow operator, we will write $\Sigma_1(0)$ for $\Sigma_{\sf m}(0)$.
Following our discussion of the affine Grassmannian in Section \ref{affgr}, we can
replace the gluing map $z^{\sf m}$ by another gluing map $g$ in the same orbit ${\mathrm{Gr}}^{\sf m}$.
The same construction with $z^{\sf m}$ replaced by $g$ produces a functional $\Psi_{g}(a)$. The functional derivatives of $\Psi_{g}(a)$ can be interpreted as correlation functions in the presence of a modified spectral flow operator $\Sigma_{g}(0)$. As a change of trivialization is implemented by the non-negative modes of the currents, we can express the action of these modes on $\Sigma_{g}(0)$ as certain differential operators along ${\mathrm{Gr}}^{\sf m}$.
We should stress that the definition of $\Psi_{g}(a)$ really requires a choice of reference connection supported within $U'=U\backslash p$ which realizes the gluing map $g$. Different connections describing the same $g$ are related by complex gauge transformations and
thus may lead to a different normalization for $\Psi_{g}(a)$ and $\Sigma_{g}(0)$. As a result, $\Psi_{g}(a)$ and $\Sigma_{g}(0)$ are
actually sections of a certain line bundle on ${\mathrm{Gr}}^{\sf m}$. We will indentify this line bundle in Section \ref{fft}.
The non-negative modes of the currents will act as vector fields on sections of
this line bundle. This is the chiral algebra manifestation of the mismatch between the bundles of half-densities before and after the Hecke modification.
The line bundle on ${\mathrm{Gr}}^{\sf m}$ is controlled by the level of the Kac-Moody algebra. In the next section, we will show that
at critical level, this line bundle coincides with the bundle of densities on ${\mathrm{Gr}}^{\sf m}$. This means that in a theory that has holomorphic and antiholomorphic
Kac-Moody levels that are both critical, the spectral flow operator is a density on ${\mathrm{Gr}}^{\sf m}$ and can be naturally integrated:
\begin{equation}\label{tendef}
\widehat\Sigma_{\sf m}(0) \equiv \int_{{\mathrm{Gr}}^{\sf m}}\Sigma_{g}(0) |\mathrm d g|^2.
\end{equation}
$\widehat\Sigma_{\sf m}(0)$ has the appropriate properties for the Hecke operator of charge ${\sf m}$ dual to a Wilson operator
with minimal corners $s_{R}$ in the language of Section \ref{wilop} (that is, a Wilson
operator defined using wavefunctions built from highest weight vectors). The integral over ${\mathrm{Gr}}^{\sf m}$ generalizes the integral over ${\Bbb{CP}}^1_x$ in eqn. (\ref{tonf}).
The action of a non-negative mode of the currents on $\widehat\Sigma_{\sf m}(0)$ can be traded for a
Lie derivative of $\Sigma_{g}(0)$ along the corresponding vector field on ${\mathrm{Gr}}^{\sf m}$. As long as no boundary terms appear upon integration by parts (this may
require a technical analysis when monopole bubbling is possible), $\widehat\Sigma_{\sf m}(0)$ will be annihilated by the non-negative modes of the Kac-Moody algebra and is thus central. Correspondingly, the averaged functional
\begin{equation} \label{eq:chirhecke}
\int_{{\mathrm{Gr}}^{\sf m}}\Psi_{g}(a)|\mathrm d g|^2
\end{equation}
obtained by acting with $\widehat\Sigma_{\sf m}(0)$ on $\Psi(a)$
is gauge-covariant and can represent the action of the principal Hecke operator of charge ${\sf m}$.
We can readily apply this construction to our illustrative example of $G={\mathrm{SO}}(3)$ and minimal charge.
We can define a ${\Bbb{CP}}^1$ family of spectral flow operators $\Sigma_{1;\mu}(0)$ as a global ${\mathrm{SO}}(3,{\Bbb C})$ rotation of $\Sigma_1(0)$.
Formally, we can write the corresponding states as
\begin{equation}
|1;\mu\rangle = e^{\mu J^+_0 }|1\rangle
\end{equation}
It is straightforward to express the action of the non-negative Fourier modes on $|1;\mu\rangle$ as differential operators in $\mu$
and verify that they are total derivatives. It is clear that
\begin{equation}
J^+_0 |1;\mu\rangle = \partial_\mu e^{\mu J^+_0 }|1\rangle= \partial_\mu |1;\mu\rangle
\end{equation}
The action of $J^0_0$ is also straightforward
\begin{equation}
J^0_0 |1;\mu\rangle = e^{\mu J^+_0 }(J^0_0 + \mu J^+_0)|1\rangle = \partial_\mu \left(\mu |1;\mu\rangle\right)
\end{equation}
Computing the action of $J^-_0$ requires only a bit more work:
\begin{equation}
J^-_0 |1;\mu\rangle = e^{\mu J^+_0 }(J^-_0 + 2 \mu J^0_0 + \mu^2 J^+_0)|1\rangle = \partial_\mu \left(\mu^2 |1;\mu\rangle\right)
\end{equation}
This makes the insertion of
\begin{equation} \label{eq:averso}
\int \Sigma_{1;\mu, \overline \mu}(0)|\mathrm d \mu|^2
\end{equation}
central, as long as boundary terms for the integration by parts vanish.
The natural way to show that boundary terms vanish is to show that (\ref{eq:averso}) is really the integral of a density on
${\Bbb{CP}}^1$. In order to do so, we need to cover ${\Bbb{CP}}^1$ with a second patch, starting from the opposite spectral flow operator
$\Sigma_{-1}(0)$ and deforming it to $\Sigma_{-1;\mu}(0)$ as
\begin{equation}
|-1;\mu\rangle = e^{-\mu^{-1} J^-_0 }|-1\rangle
\end{equation}
The action of the non-negative Fourier modes on this family involves the same differential operators in $\mu$ as for $\mu^2 |1;\mu\rangle$.
Including the antichiral modes we find that we can consistently identify
\begin{equation}
\Sigma_{-1;\mu}(0) = |\mu|^4 \Sigma_{1;\mu}(0)
\end{equation}
and combine them into a density $\Sigma_{1;\mu}(0)$ defined on the whole ${\Bbb{CP}}^1$.
We explain a different and more general approach to this result in Section \ref{fft}.
The $\Sigma_{1;\mu}(0)$ insertion, by construction,
corresponds to a very specific modification $a \to a'[a;\mu]$ of the background connection.
Recall that we work in a gauge where $a$ vanishes inside the open patch $U$ and
$a'[a;0]$ differs from $a$ by some reference connection supported on an annular region in $U'$. The insertion of the exponentiated Fourier mode $e^{\mu J^+_0}$ adds a further specific modification to the connection on a wider annular region, producing $a'[a;\mu]$.
The integral operator (\ref{eq:chirhecke}) corresponding to (\ref{eq:averso}) is thus
\begin{equation}\label{eq:chirhesutwo}
\int_{{\Bbb{CP}}^1}\Psi(a'[a;\mu])|\mathrm d \mu|^2
\end{equation}
Compare this with (\ref{actst}). We should write $\Psi(y)$ there as $\Psi(a(y))$ here, with $a(x)$ denoting our gauge-fixing
choice of a representative connection for every bundle $x$. There is no reason for $a'[a(x);\mu]$
to be already in a gauge-fixed form. A complexified gauge transformation will be needed to bring it to the gauge-fixed form $a(x_\mu)$ for the modified bundle $x_\mu$. The anomaly will give some rescaling factor which we can write as the absolute value of an holomorphic quantity $\omega$:
\begin{equation} \label{eq:omega}
\Psi(a'[a(x);\mu]) = |\omega(x;\mu)|^2 \Psi(a(x_\mu))
\end{equation}
The integral operator becomes
\begin{equation}
\int_{{\Bbb{CP}}^1}|\omega(x;\mu)|^2 \Psi(a(x_\mu))|\mathrm d \mu|^2
\end{equation}
We obtain:
\begin{equation}
F(x,y) = \int_{{\Bbb{CP}}^1} |\omega(x;\mu)|\delta(y;x_\mu)|\mathrm d \mu|^2
\end{equation}
where $\delta(y;x)$ is a delta function supported on the diagonal in ${\mathcal M} \times {\mathcal M}$.
The left hand side of (\ref{eq:omega}) is a half-density
in $x$ and a density in $\mu$. The right hand side involves a half-density in $y=x_\mu$.
We can thus identify $\omega(x;\mu)$ with $w(\vec x;\mu)$ in $(\ref{conf})$.
The factor $\omega(x;\mu)$ encodes the anomalous rescaling of $\Psi$ under a complexified gauge transformation
and is thus non-vanishing. This allows us to identify it with the holomorphic factor $k$ introduced by \cite{BD}.\footnote{We will see momentarily that $\omega(x;\mu)$ could be computed in the theory of adjoint free fermions.} We identify (\ref{eq:chirhesutwo}) with the Hecke operator $H_{p=0}$ associated to the two-dimensional representation of $G^\vee$ with a minimal choice of corners corresponding to $|k|^2$, as in eqn. (\ref{tonf}).
\subsection{Free Fermion Trick}\label{fft}
In the last section, we observed that in a CFT with Kac-Moody symmetry, the operator $\Sigma_g(0)$, where $g$ is a gauge transformation associated to a Hecke transformation of weight
${\sf m}$, is a section of a line bundle over ${\mathrm{Gr}}^{\sf m}$. This line bundle, since it is determined by the anomaly, depends only on the central charge $\kappa$ of the CFT.
We would like to compute this line bundle for a CFT of critical level $\kappa_c=-h$, but it turns out that it is particularly simple to compute it for a CFT
whose level is $-\kappa_c=+h$. This will give us the inverse of the line bundle over ${\mathrm{Gr}}^{\sf m}$ that we actually want.
After picking a spin structure on $C$, or equivalently a choice of $K_C^{1/2}$, we consider a system of chiral (Majorana-Weyl) fermions $\psi^a$ of spin 1/2
valued in the adjoint representation of the gauge group. The Kac-Moody currents are
constructed as normal ordered fermion bilinears and have anomalous gauge transformation due to
the normal ordering. The central charge is exactly $h=-\kappa_c$.
We claim that the spectral flow operators in the theory of adjoint free fermions are sections of $K^{-1}_{{\mathrm{Gr}}^{\sf m}}$. This means that the spectral flow
operators in a CFT at the critical level $\kappa_c$ are sections of the inverse of this or $K_{{\mathrm{Gr}}^{\sf m}}$.
We will illustrate the case of $G={\mathrm{SO}}(3)$ and minimal ${\sf m}$, and briefly indicate the generalization to other $G$ and ${\sf m}$.
After picking a Cartan subalgebra of ${\mathrm{SO}}(3)$, we have
chiral fermions $\psi^\pm$ and $\psi^0$. The basic Hecke modification at $z=0$ results in $\psi^+$ having a pole at $z=0$ and $\psi^-$ having a zero. This is implemented simply by a $\psi^-$ insertion at $z=0$. A Hecke modification associated to a point
$(u,v)\in{\Bbb{CP}}^1$ is implemented by an ${\mathrm{SO}}(3,{\Bbb C})$ rotation of $\psi^-$, i.e. by $\psi^-_{(u,v)} \equiv u^2 \psi^- + 2 u v \psi^0 + v^2 \psi^+$. An
insertion of $\psi^-_{(y,v)}(0)$ imposes the vanishing of $\psi^-_{(u,v)}$ at $z=0$, while giving a pole to other linear combinations of the components of $\psi$.
In the theory of adjoint fermions, we thus have $\Sigma_{1;(u,v)}(0) = \psi^-_{(u,v)}$.
This is quadratic in homogeneous coordinates of ${\Bbb{CP}}^1={\mathrm{Gr}}^{\sf m}$,
so it is a global section of ${\mathcal O}(2) = K_{{\mathrm{Gr}}^{\sf m}}^{-1}$, as claimed.
Hence at critical level, the spectral flow operator in this example is a section of $K_{{\mathrm{Gr}}^{\sf m}}$, a fact that was exploited in Section \ref{cvo}.
For a general gauge group and charge, the reference Hecke modification results in the fermions labelled by a root $\alpha$ having extra poles or zeroes of order $(\lambda, \alpha)$ at $z=0$. This is implemented by a very simple vertex operator:
\begin{equation}\label{opspin}
\prod_{\alpha | (\lambda, \alpha)<0} \prod_{n_\alpha =0}^{-(\lambda, \alpha)-1}\partial_z^{n_{\alpha}}\psi^{\alpha}
\end{equation}
It is straightforward to see that this product transforms as a section of $K_{{\mathrm{Gr}}^{\sf m}}^{-1}$: each fermion derivative in the product matches one of the
non-negative Fourier modes $J_{n_\alpha}^\alpha$ which act non-trivially on $\Sigma_{\sf m}$; these modes provide a basis of the tangent bundle to ${\mathrm{Gr}}^{\sf m}$.
This computation could be expressed as a comparison of the Pfaffian of the Dirac operator acting on $\psi$, before and after
the Hecke modification. This Pfaffian is analyzed in detail in \cite{BD}.
The operator $\Sigma_{g}(z,\overline z)$ fails to be a true function of $z$ because of the gauge anomaly. Indeed, even a rescaling of the local coordinate $z \to \lambda z$ changes the singular gauge transformation from $z^{\sf m}$ to $(\lambda z)^{\sf m}$ and thus results in the action of the Cartan zero modes ${\sf m} \cdot J_0$ on the spectral flow operator, resulting in a non-trivial scaling dimension proportional to $({\sf m},{\sf m})$.
We can study this anomalous dependence on $z$ with the help of the free fermion trick.
For example, for ${\mathrm{SO}}(3)$ and minimal ${\sf m}$ we have an insertion $\psi^-$ which behaves as a section of $K_C^{1/2}$.
Correspondingly, for critical level the spectral flow operator is a section of $K_C^{-1/2}\otimes \overline K_C^{-1/2}=|K_C|^{-1}$.
This remains true for the averaged Hecke operator $\widehat\Sigma_{\sf m}(0)$ because in this example, the coordinates on ${\mathrm{Gr}}^{\sf m}={\Bbb{CP}}^1$ have scaling dimension $0$ and thus the measure $\mathrm d \mu$ does not contribute to the scaling dimension. The fact that $\widehat\Sigma_{\sf m}(0)$ is a section of $|K_C|^{-1}$ is expected from $S$-duality. It matches
the fact that the holomorphic and antiholomorphic sections $s$ and $\overline s$ used to define the dual Wilson operator (Section \ref{wilop}) are sections of $K_C^{-1/2}$
and $\overline K_C^{-1/2}$, respectively.
For general groups and representations, matching the scaling dimension of $\widehat\Sigma_{\sf m}(0)$ with the behavior of the corresponding Wilson
operator is more subtle. The scaling dimension of the fermionic insertion grows quadratically in the charge, but so does the
negative scaling dimension of the measure $\mathrm d\mu$ on ${\mathrm{Gr}}^{\sf m}$. There is a nice cancellation between the derivatives on the fermions and the scaling
dimension of the measure, so that the scaling dimension of $\widehat\Sigma_{\sf m}$ is linear in ${\sf m}$. The scaling dimensions of dual Wilson operators were
described in Section \ref{wilop}.
\subsection{Integral-differential Hecke Operators and the Oper Differential Equation}\label{opde}
In Section \ref{lineq}, as well as integral Hecke operators, whose kernel has delta function support on the Hecke correspondence,
we considered integral-differential Hecke operators,
whose kernel is a derivative of a delta function.
The natural way to build such more general Hecke operators is to consider the insertion of Kac-Moody descendants of
$\Sigma_{g}$, which represent functional derivatives $\frac{\delta}{\delta a}$ taken in a neighbourhood of the
location of the Hecke modification. We can restrict ourselves to descendants by the negative modes of the Kac-Moody currents,
as the non-negative modes can be traded for $g$ derivatives which would be integrated by parts.
The action of non-negative modes on a descendant of $\Sigma_{g}$ will produce some linear combination of $g$ derivatives of other descendants. We need some $g$-dependent combination of descendants which transform as a density on $\mathrm{Gr}^{\sf m}$ and such that the action of non-negative Kac-Moody modes will produce total $g$ derivatives. We can produce a simple example of that: $\partial_z \widehat\Sigma_{\sf m}(z, \overline z)$. Indeed, the $z$ derivative of a basic spectral flow operator $\Sigma_{\sf m}(z)$ coincides with the Cartan Kac-Moody descendant :
\begin{equation}\label{firstone} \partial_z \Sigma_{\sf m}(z)= \frac{1}{2 \pi {\mathrm i}} \oint \frac{\mathrm d w}{w-z} {\sf m} \cdot J (w) \Sigma_{\sf m}(z, \overline z)\equiv {\sf m} \cdot J_{-1} \circ \Sigma_{\sf m}(z, \overline z). \end{equation}
To demonstrate this relation, recall that the insertion of $\Sigma_{\sf m}(z)$ in a correlation function represents a specific modification of the background connection in a neighbourhood $U$ of $z$. As we vary $z$, the modified connection changes. The change is supported in the annular region $U'$ and can be described by the insertion of a current integrated against the variation of the background connection.
The entire comparison occurs within the ${\mathrm U}(1)$ subgroup of the gauge group determined by ${\sf m}$.
As a small shortcut, we can compare the effect of the $z$ derivative and of the integrated current insertion at the level of the bundle modifications they implement. If $\Sigma_{\sf m}(z)$ implements the gauge transformation $g(w) = (w-z)^{\sf m}$ on $U'$, the $z$ derivative $\partial_z \Sigma_{\sf m}(z)$ implements $\partial_z (w-z)^{\sf m} = \frac{{\sf m}}{z-w} (w-z)^{\sf m}$. The $\frac{{\sf m}}{z-w}$ part is identified with the gauge transformation produced by the
${\sf m} \cdot J_{-1}$ Fourier mode and $(w-z)^{\sf m}$ represents $\Sigma_{\sf m}(z)$ again. As we are working at the level of the bundle modification instead of the connection, we could be missing effects due to the anomaly. A simple check in the free fermion theory can exclude that.\footnote{The bosonized description of $\Sigma_{\sf m}(z)$ is also an effective way to verify the computation.}
Inserting this relation into the definition of $\widehat\Sigma_{\sf m}(z, \overline z)$, we find that $\partial_z \widehat\Sigma_{\sf m}(z, \overline z)$ can be written as an integral over ${\mathrm{Gr}}^{\sf m}$ of a specific Kac-Moody descendant of $\Sigma_{g}(z, \overline z)$.
Another natural way to produce well-defined integral-differential operators of this type is to consider descendants of
$\widehat\Sigma_{\sf m}$ by modes of the Sugawara vector or other central elements. The resulting local operators are clearly gauge-invariant.
We expect that the classification of $g$-dependent combinations of Kac-Moody descendants of $\Sigma_{g}(z, \overline z)$ which are a total derivative on $\mathrm{Gr}^{\sf m}$ will match the corresponding classification of 't Hooft line endpoints $\alpha_{R_{\sf m},n}$.
\subsection{Wakimoto Realization}
We will give here an alternative derivation of the properties of $\Sigma_1$ with the help of the Wakimoto construction at critical level. As a bonus, we will recover in a different way the oper differential equation.
The critical-level Wakimoto construction presents the Kac-Moody currents as the symmetry currents for a twisted $\beta\gamma$ system for ${\Bbb{CP}}^1$:
\begin{align}\label{waki}
J^+ &= \beta \cr
J^0 &= - \beta \gamma + \partial \alpha \cr
J^- &= -\beta \gamma^2 - 2 \partial \gamma + \partial \alpha \gamma \cr
\end{align}
where $\alpha$ is a locally-defined holomorphic function. The Sugawara vector simplifies to a Miura form $S_2 = (\partial\alpha)^2 + \partial^2 \alpha$ and is thus manifestly a multiple of the identity operator, with trivial OPE with the currents.
The spectral flow automorphism extends naturally to the $\beta\gamma$ system, so that the spectral flow operator gives a zero to $\gamma$ and a pole to $\beta$. An
operator that does this is usually indicated as $\delta(\gamma)$. Comparison with the expected form of $\partial \Sigma_1$ from eqn. (\ref{firstone}) gives
\begin{equation}
\Sigma_1 = e^{\alpha+ \overline \alpha} \delta(\gamma)\delta(\overline \gamma).
\end{equation}
The exponential prefactor provides the $\partial \alpha$ part of $J^0_{-1} \circ \Sigma_1$.
An ${\mathrm{SO}}(3,{\Bbb C})$ rotation of this expression gives
\begin{equation}
\Sigma_1(z;\mu) = e^{\alpha+ \overline \alpha} \delta(\gamma -\mu)\delta(\overline \gamma- \overline \mu).
\end{equation}
where $\mu$ is an inhomogeneous coordinate on ${\Bbb{CP}}^1$.
The integral over $\mu$ is easily done, resulting in
\begin{equation}
\widehat\Sigma_1(z) = e^{\alpha+ \overline \alpha} .
\end{equation}
This is a multiple of the identity and thus annihilated by all non-negative modes of the currents. Furthermore, the oper differential equation manifestly holds: $\partial_z^2 \widehat\Sigma_1(z) = S_2(z)\widehat\Sigma_1(z)$.
We expect this pattern to persist for all $G$ and ${\sf m}$. The critical Wakimoto realization gives central
elements which take the form of a Miura oper built from the Cartan-valued $\alpha$. The spectral flow operators will take the form of spectral flow operators for the $\beta \gamma$ system combined with some function of $\alpha$. The averaged spectral flow operators will give multiples of the identity for the $\beta \gamma$ system, multiplied by certain functions of $\alpha$
which give the Miura expression for solutions of the oper differential equation.
\section{Wavefunctions from Symplectic Bosons} \label{sec:symp}
\subsection{Basics of Symplectic Bosons}\label{sb}
As we discussed in the Section \ref{Enriched}, Neumann boundary conditions enriched by 3d hypermultiplets create states described by a path integral
\begin{equation}
\Psi[a] = \int DZ D\overline Z\, e^{\int_C \left[\langle Z, \overline \partial_a Z \rangle - \langle
\overline Z, \partial_a \overline Z \rangle\right]}
\end{equation}
This is a non-chiral version of the path integral for {\it symplectic bosons}. Chiral symplectic bosons are the Grassmann-even analogue of chiral fermions. They are a special case of $\beta \gamma$ systems where the conformal dimension of both $\beta$ and $\gamma$ is set to $1/2$.
Concretely, chiral symplectic bosons are a collection of $2 n$ two-dimensional spin $1/2$ chiral bosonic fields $Z^a$ with action
\begin{equation}
\int_{C}\left( \omega_{ab} Z^a \overline\partial Z^b + {\cal A}_{ab} Z^a Z^b\right)
\end{equation}
where $\omega_{ab}$ is a constant symplectic form and we included a coupling to a background
connection ${\cal A}_{ab}$ of type $(0,1)$ defining an ${\mathrm{Sp}}(2n)$ bundle on the Riemann surface $C$.\footnote{As the symplectic bosons are spinors, we do not strictly need to separately define a spin structure and an ${\mathrm{Sp}}(2n)$ bundle. Instead, we can specify a ${\mathrm{Spin}}\cdot {\mathrm{Sp}}(2n)$ bundle, a notion that is precisely analogous
to the ${\mathrm{Spin}}\cdot {\mathrm{SU}}(2)$ bundles of Section \ref{toposubt}.} We will employ Einstein summation convention in this section unless otherwise noted.
The analogy to chiral fermions is somewhat imperfect. Chiral fermions are a well-defined two-dimensional (spin)CFT.
Chiral symplectic bosons are mildly anomalous. The anomaly manifests itself as a sign ambiguity of the chiral partition function
\begin{equation}
\int DZ \,e^{\int_{C}\left( \omega_{ab} Z^a \overline\partial Z^b + {\cal A}_{ab} Z^a Z^b\right)} = \frac{1}{\sqrt{\det \overline \partial_{\cal A} }}
\end{equation}
where we denote as $\overline \partial_{\cal A}$ the $\overline \partial$ operator acting on sections of $K_C^{1/2} \otimes E$ and $E$ is the rank $2n$ bundle associated to the ${\mathrm{Sp}}(2n)$ bundle.\footnote{Notice that generically $\overline \partial_{\cal A}$ has no zero modes and the functional determinant is well-defined. The partition function diverges for special choices of ${\mathrm{Sp}}(2n)$ bundle where zero modes appear. }
The non-chiral partition function, though,
\begin{equation}
\int DZ D\overline Z\, e^{\int_C \left[\langle Z, \overline \partial_A Z \rangle - \langle
\overline Z, \partial_A \overline Z \rangle\right]} = \frac{1}{|\det \overline \partial_{\cal A} |}
\end{equation}
is unambiguous: it can be defined by an actual the path integral along the cycle $\overline Z{}^a = (Z^a)^*$ (that is, the integration cycle is defined by saying $\overline Z{}^a$ is the complex conjugate of $Z^a$).
We record here the OPE
\begin{equation}
Z^a(z) Z^b(w) \sim \frac{\omega^{ab}}{z-w},
\end{equation}
where $\omega^{ab}$ is the inverse symplectic form. The corresponding algebra of modes is
\begin{equation}
[Z^a_m, Z^b_n] = \omega^{ab} \delta_{n+m,0}.
\end{equation}
Because of the half-integral spin, the mode indices $n$, $m$ are half-integral in the Neveu-Schwarz sector of the chiral algebra and in particular in the vacuum module. The vacuum satisfies
\begin{equation}
Z^a_n |0\rangle =0 \qquad \qquad n>0.
\end{equation}
The mode indices are integral in Ramond sector modules, which we will discuss in Section \ref{sec:Ramond}.
The variation of the action with respect to the ${\mathrm{Sp}}(2n)$ connection ${\cal A}$ gives Kac-Moody currents
\begin{equation}
J^{ab} = \frac12 :Z^a Z^b:
\end{equation}
of level $- \frac12$. The fractional level is another manifestation of the global anomaly of the chiral theory.\footnote{The $J^{ab}$ currents actually generate a quotient of the $\widehat{\mathfrak{sp}}(2n)_{-\frac12}$ chiral algebra: some linear combinations of current bilinears and derivatives of the currents vanish. The number of level $2$ descendants in the symplectic boson vacuum module is smaller than the number of level $2$ descendants in the Kac-Moody vacuum module. } The partition function of the non-chiral theory defines a well-defined section of a bundle $|{\cal L}|$ on the space of ${\mathrm{Sp}}(2n,{\Bbb C})$ bundles, where ${\cal L}$ is the line bundle corresponding to Kac-Moody level $-1$.
Once we specialize to the gauge group $G \subset {\mathrm{Sp}}(2n)$ of the 4d theory,
we will obtain $G$ currents of a level which may not be critical. This signals a gauge anomaly obstructing the
existence of a 2d junction between the deformed Neumann boundary condition and the enriched Neumann boundary condition. If the level is more negative than the critical level, we may attempt to cancel the anomaly by
some auxiliary 2d system, such as a collection of free fermions. In the example we discuss momentarily, the anomaly will be absent from the outset.
\subsection{The Trifundamental Example}\label{tri}
We now specialize to $n=4$ and focus on a $G\equiv {\mathrm{SL}}(2) \times {\mathrm{SL}}(2)\times {\mathrm{SL}}(2)$ subgroup of ${\mathrm{Sp}}(8)$.
In other words, we identify $\mathbb{C}^8$ with $\mathbb{C}^2 \otimes\mathbb{C}^2 \otimes\mathbb{C}^2$
and we couple the theory to a connection $a$ for ${\mathrm{SL}}(2) \times {\mathrm{SL}}(2)\times {\mathrm{SL}}(2)$.
This gives enriched Neumann boundary conditions which are conjecturally $S$-dual to a tri-diagonal interface, a BBB brane supported on the
diagonal of ${\mathcal M}_H \times {\mathcal M}_H \times {\mathcal M}_H$ with trivial ${\mathrm{CP}}$ bundle \cite{Gaiotto:2016hvd,Benini:2010uu}.\footnote{The $S$-duality statement can be generalized to other $G$ or diagonal interfaces between more than three copies. The corresponding enriched Neumann boundary conditions employ the theories defined in \cite{Benini:2010uu}. The boundary chiral algebras for these theories are known from work by Arakawa \cite{Arakawa:2018egx} and have $G$ currents of critical level. The corresponding states would be the partition function of a non-chiral 2d CFT built from Arakawa's chiral algebras, which is currently unknown.} The basic consequence of this $S$-duality identification is that the state $\Psi[a]$ produced by the partition function should intertwine the action of the three copies of ${\mathrm{SL}}(2)$ Hitchin Hamiltonians and quantum Hecke operators. Our goal in the rest of this section is to confirm this.
We denote the symplectic boson fields as $Z^{\alpha \beta \gamma}(z)$ and the symplectic form as
$\epsilon_{\alpha \alpha'}\epsilon_{\beta \beta'}\epsilon_{\gamma \gamma'}$. We get three copies of
$\widehat{\mathfrak{sl}}(2)_{-2}$ Kac-Moody currents such as
\begin{equation}
J^{\alpha \alpha'} =\frac12 \epsilon_{\beta \beta'}\epsilon_{\gamma \gamma'} :Z^{\alpha \beta \gamma}Z^{\alpha' \beta' \gamma'}:
\end{equation}
We will denote the three sets of currents simply as $J$, $J'$, $J''$, avoiding indices when possible.
Crucially, these currents have critical level. Accordingly, the Sugawara vectors are central. A remarkable observation is that the three Sugawara vectors actually coincide here:
\begin{equation} \label{eq:sugathree}
:JJ: = :J'J': = :J'' J'':
\end{equation}
As the three Sugawara vectors coincide, the correlation functions of Sugawara vectors also coincide. These are obtained from the action of the
$\mathfrak{sl}(2)$ quantum Hitchin Hamiltonians on the partition function $\Psi(a)$, seen as a half-density on ${\mathcal M}\times {\mathcal M} \times {\mathcal M}$.
Concretely, the the kinetic operator of the symplectic bosons $Z$ is the $\overline \partial_{a}$ operator acting on the bundle $E \otimes E' \otimes E'' \otimes K_C^{\frac12}$.
The partition function is:
\begin{equation}\label{sqr}
\Psi(a) = \frac{1}{|\det \overline \partial_a|}.
\end{equation}
We have thus given a chiral algebra derivation of an intertwining property
\begin{equation}
\boxed{H_i \Psi = H'_i \Psi = H''_i \Psi}
\end{equation}
where $H_i$ run over the $\mathfrak{sl}(2)$ quantum Hitchin Hamiltonians acting on the three spaces of ${\mathrm{SL}}(2)$ bundles.
Similarly
\begin{equation}
\boxed{\overline H_i \Psi = \overline H'_i \Psi = \overline H''_i \Psi}
\end{equation}
for the conjugate quantum Hitchin Hamiltonians, acting as antiholomorphic differential operators on the space of ${\mathrm{SL}}(2)$ bundles.
These relations match the relations expected on the B-model side for the tri-diagonal interface.
Our next objective is to demonstrate the analogous intertwining relations for Hecke operators.
For simplicity, we will work with Hecke operators of minimal charge for ${\mathrm{SO}}(3)$, even though the symplectic bosons are coupled to ${\mathrm{SL}}(2)$ bundles rather than ${\mathrm{PSL}}(2)$. Minimal Hecke modifications map ${\mathrm{SL}}(2)$ bundles to ${\mathrm{SL}}(2)$ bundles twisted by a gerbe
and viceversa, so we can consistently describe the action of pairs of Hecke operators on half-densities on ${\mathcal M}(C,{\mathrm{SU}}(2))$.
A minimal Hecke operator will create the endpoint of a $Z \to -Z$ cut for the symplectic bosons. This leads us to consider Ramond vertex operators.
\subsection{The Ramond Sector} \label{sec:Ramond}
The symplectic boson chiral algebra admits Ramond modules which are associated to a circle with
non-bounding spin structure. In such a module, the mode expansion of the $Z^a$ fields involves modes $Z^a_n$ with integral $n$.
The corresponding vertex operators introduce a cut across which the $Z^a$ flip sign.
The zero modes $Z^a_0$ form a Weyl algebra. There is a rich collection of {\it highest weight} Ramond modules for the chiral symplectic bosons which is induced from a module for the zero mode Weyl algebra. Every element of the Weyl module is promoted to a highest weight vector/vertex operator which is annihilated by the positive Fourier modes $Z^a_n$. The negative Fourier modes act freely and the zero modes act as in the Weyl module.
The Kac-Moody current zero modes $J_0^{ab}$ act on a highest weight vector as ${\mathrm{Sp}}(2n)$ generators $Z^{(a}_0 Z^{b)}_0$.
All Weyl modules break to some degree the ${\mathrm{Sp}}(2n)$ symmetry of the VOA. In other words, there is no Weyl module
equipped with an ${\mathrm{Sp}}(2n)$-invariant vector. Thus the insertion of any such vertex operator into a correlation function will always reduce ${\mathrm{Sp}}(2n)$ gauge invariance at that point.
The simplest way to produce a Ramond vertex operator is to consider a spectral flow operator of minimal charge in ${\mathrm{PSp}}(2n)={\mathrm{Sp}}(2n)/{\mathbb Z}_2$. Select a Lagrangian splitting $\mathbb{C}^{2n} = V \oplus V^\vee$. Pick a singular gauge transformation which acts as $z^{\frac12}$ on $V$ and $z^{-\frac12}$ on $V^\vee$. The resulting spectral flow operator $S_V(0)$ is a Ramond module. Linear combinations of $Z^a$ in $V$ vanish as $z^{\frac12}$ as they approach $S_V(0)$, while linear combinations in $V^\vee$ diverge as $z^{-\frac12}$. This means that $S_V(0)$ is a highest-weight Ramond module annihilated by linear combinations of $Z_0^a$ in $V$. In particular, it only depends on $V$.
$S_V(0)$, for any $V$, can be obtained by
an ${\mathrm{Sp}}(2n)$ rotation from some particular $S_{V_0}(0)$, which we choose as a reference. Without loss of generality, pick a basis where
\begin{equation}
\omega^{ab} = \delta^{a-b-n}- \delta^{b-a-n},~~a,b=1,\cdots, 2n,\end{equation}
and choose the reference vertex operator $S_{V_0}(z)$ to be annihilated by $Z^{n+1}_0, \cdots, Z^{2n}_0$.
Denote the remaining zero modes, which act as creation operators, as $u^a=Z_0^a$, $a\leq n$ and denote the corresponding descendants of $S_{V_0}(z)$ as $S_{V_0}[u^a](z)$, $S_{V_0}[u^a u^b](z)$, etcetera. Annihilation zero modes act as $Z_0^{a+n}=\partial_{u^a}$.
Consider a coherent state in the Weyl module:
\begin{equation}
S_{V_0}[e^{\frac12 B_{ab} u^a u^b}](z)
\end{equation}
This is annihilated by linear combinations $Z^{a+n}_0 - B_{ac} Z^c_0$. That condition defines a rotated
Lagrangian subspace $V = B \circ V_0$. We thus identify
\begin{equation}
S_{B \circ V_0}(z) = S_{V_0}[e^{\frac12 B_{ab} u^a u^b}](z)
\end{equation}
Indeed, we have $\partial_{B_{ab}}S_{B \circ V_0}(z) = J_0^{ab}S_{B \circ V_0}(z)$.
\subsection{Non-chiral Ramond Modules}
Next, we can consider the combined theory of chiral and antichiral symplectic bosons. In the Ramond sector, we now have
an action of the chiral zero modes $Z_0^a$ and the antichiral zero modes $\overline Z_0^a$. As the path integration contour
relates $\overline Z$ to the conjugate of $Z$, it is natural to consider a space of Ramond states such that the
zero modes are adjoint to each other.
Furthermore, the 2d theory in the zero momentum sector is essentially a quantum mechanics with target $\mathbb{C}^{2n}$.
It is thus natural to pick a polarization and set the zero mode Hilbert space to be ${\mathrm L}^2(\mathbb{C}^{n})$. This answer is actually independent of the choice of polarization, as we can use generalized Fourier transform operations to relate different polarizations.\footnote{The metaplectic anomaly cancels out because we act simultaneously on holomorphic and anti-holomorphic variables.}
The full Ramond Hilbert space ${\mathcal R}$ is induced from ${\mathrm L}^2(\mathbb{C}^{n})$ in the usual way, by having $Z^a_k$ and $\overline Z^a_k$ annihilate vectors in ${\mathrm L}^2(\mathbb{C}^{n})$ when $k>0$ and act freely with $k<0$.
A variety of different highest weight Ramond vertex operators are realized as distributions on $\mathbb{C}^{n}$.
Pick the same polarization as in the definition of $S_{V_0}(0)$. Then the reference spectral flow operator $S_{V_0}(0)$ is represented by the distribution ``$1$''. The rotated $S_{B \circ V_0}(z)$ is represented by the Gaussian
\begin{equation}
e^{\frac12 B_{ab} u^a u^b-\frac12 \overline B_{ab} \overline u^a \overline u^b }
\end{equation}
The distribution $\delta^{(2n)}(u)$, instead, represents a Ramond vertex operator annihilated by $Z^{1}_0, \cdots Z^{n}_0$,
which is a spectral flow operator with charge opposite to $S_{V_0}(0)$.
We can now specialize to $n=4$ and to the spectral flow operators associated to the ${\mathrm{SL}}(2) \times {\mathrm{SL}}(2) \times {\mathrm{SL}}(2)$ subgroup of ${\mathrm{Sp}}(8)$.
We pick our polarization of $\mathbb{C}^{8}$ to be invariant under the second and third ${\mathrm{SL}}(2)$ groups in the product. Concretely, we identify
$Z_0^{2 \beta \gamma} = u^{\beta \gamma}$ and $Z_0^{1 \beta \gamma}=\epsilon^{\beta\beta'}\epsilon^{\gamma\gamma'}\frac{\partial}{\partial u^{\beta'\gamma'}} .$
In this framework, the spectral flow operators $\Sigma_{1;\mu, \overline \mu}$, $\Sigma'_{1;\mu', \overline \mu}$ and $\Sigma''_{1;\mu'', \overline \mu}$ for the three ${\mathrm{SL}}(2)$'s are all special cases of ${\mathrm{Sp}}(8)$ spectral flow operators of minimal charge, i.e. of highest weight Ramond vertex operators. They are represented by distributions, which we can average over $\mu$ to obtain representations of $\widehat \Sigma_{1}$, $\widehat \Sigma'_{1}$, $\widehat \Sigma''_{1}$. The calculations are straightforward:
\begin{itemize}
\item $\Sigma_1$ is represented by the distribution ``1''. $\Sigma_{1;\mu, \overline \mu}$ is represented by the Gaussian
\begin{equation}
e^{\mu \epsilon_{\beta \beta'}\epsilon_{\gamma \gamma'} u^{2\beta \gamma} u^{2 \beta' \gamma'}- \mathrm{c.c.}}.
\end{equation} Averaging over $\mu$, $\widehat \Sigma_{1}$ is represented by the distribution
\begin{equation}
\delta^{(2)}(\epsilon_{\beta \beta'}\epsilon_{\gamma \gamma'} u^{2\beta \gamma} u^{2 \beta' \gamma'}).
\end{equation}
\item $\Sigma'_1$ is represented by the distribution
\begin{equation}
\delta^{(2)}(u^{211})\delta^{(2)}(u^{212}).
\end{equation}
$\Sigma'_{1;\mu', \overline \mu'}$ is represented by the distribution
\begin{equation}
\delta^{(2)}(u^{211}- \mu' u^{221})\delta^{(2)}(u^{212}- \mu' u^{222}).
\end{equation}
Averaging over $\mu'$, we find that $\widehat \Sigma'_{1}$ is represented by the same distribution as $\widehat \Sigma_{1}$.
\item $\Sigma''_1$ is represented by the distribution
\begin{equation}
\delta^{(2)}(u^{211})\delta^{(2)}(u^{221}).
\end{equation}
Similarly, $\Sigma''_{1;\mu'', \overline \mu''}$ is represented by the distribution
\begin{equation}
\delta^{(2)}(u^{211}- \mu'' u^{212})\delta^{(2)}(u^{221}- \mu'' u^{222}).
\end{equation}
Averaging over $\mu''$, we find that $\widehat \Sigma''_{1}$ is represented by the same distribution as $\widehat \Sigma_{1}$
\end{itemize}
We conclude that
\begin{equation}
\widehat \Sigma_{1} = \widehat \Sigma'_{1} =\widehat \Sigma''_{1}
\end{equation}
in the theory of trifundamental symplectic bosons. These relations play an analogous role to (\ref{eq:sugathree}): inserted in correlation functions they prove that the partition function $\Psi(a)$ intertwines the action of minimal Hecke operators for the three ${\mathrm{SL}}(2)$ groups:
\begin{equation}
\boxed{H_z \Psi = H'_z \Psi = H''_z \Psi}
\end{equation}
\subsection{A Marvelous Module}\label{marvel}
It is worth elaborating on the properties of the Weyl module generated by the distribution
\begin{equation}
M=\delta^{(2)}(\epsilon_{\beta \beta'}\epsilon_{\gamma \gamma'} u^{2\beta \gamma} u^{2 \beta' \gamma'}).
\end{equation}
We learned some surprising properties which follow from $M$ representing averaged spectral flow operators:
$M$ is invariant under ${\mathrm{SL}}(2) \times {\mathrm{SL}}(2) \times {\mathrm{SL}}(2)$ and the module treats the three ${\mathrm{SL}}(2)$ groups in a completely symmetric manner. The latter property is somewhat hidden in the analysis, so we can spell it out in detail here: we can change polarization
by a Fourier transform and go to representations of the Weyl module in terms of functions of $u^{\alpha 2 \gamma}$ or $u^{\alpha \beta 2}$. The Fourier transform of $M$ produces distributions $\delta^{(2)}(\epsilon_{\beta \beta'}\epsilon_{\gamma \gamma'} u^{\beta 2 \gamma} u^{\beta' 2\gamma'})$ and $\delta^{(2)}(\epsilon_{\beta \beta'}\epsilon_{\gamma \gamma'} u^{\beta \gamma 2} u^{\beta' \gamma' 2})$ respectively.
Acting with the Weyl algebra on $M$ we generate a remarkable module with an explicit action of ${\mathrm{SL}}(2) \times {\mathrm{SL}}(2) \times {\mathrm{SL}}(2)$.
As a vector space, the module decomposes as
\begin{equation}\label{marmod}
\oplus_{d=1}^\infty V_d\otimes V_d \otimes V_d
\end{equation}
where $V_d$ is the $d$-dimensional irreducible representation of ${\mathrm{SL}}(2)$. The Weyl generators act as a sum of two terms: one term raises $d$ by 1 and the other lowers it by $1$.
\subsection{Some Generalizations}
The ${\mathrm{SU}}(2) \times {\mathrm{SU}}(2) \times {\mathrm{SU}}(2)$ gauge group can be seen as a special case of ${\mathrm{Sp}}(2n) \times \mathrm{Spin}(2n+2)$.
Trifundamental hypermultiplets are a special case of bifundamental hypermultiplets for ${\mathrm{Sp}}(2n) \times \mathrm{Spin}(2n+2)$.
Bifundamental hypermultiplets engineer an ``NS5'' interface between ${\mathrm{Sp}}(2n)$ and $\mathrm{Spin}(2n+2)$ 4d gauge theories.
The levels of the corresponding Kac-Moody currents in the theory of bifundamental symplectic bosons are critical, so the NS5 interface
has non-anomalous corners with deformed Neumann boundaries and our construction applies.
The NS5 interface is $S$-dual to a ``D5'' interface between $\mathrm{{\mathrm{Spin}}(2n+1)}$ and $\mathrm{Spin}(2n+2)$ 4d gauge theories,
at which the $\mathrm{Spin}(2n+2)$ gauge group is reduced to $\mathrm{{\mathrm{Spin}}(2n+1)}$. The D5 interface will descend in the B-model to a BBB interface supported on the space of $\mathrm{Spin}(2n+1)$ flat connections embedded in the space of $\mathrm{Spin}(2n+2)$ flat connections. This gives simple predictions for the action of quantum Hitchin Hamiltonians and 't Hooft operators on the NS5 interface.
Our calculations in this section should be generalized to verify these predictions.
Bifundamental hypermultiplets can also be used to engineer an ``NS5'' interface between ${\mathrm U}(n)$ 4d gauge theories which has a relatively simple dual and should be amenable to a 2d chiral algebra analysis. The Kac-Moody currents for the ${\mathrm{SU}}(n)$ subgroups are critical, and the ${\mathrm U}(1)$ anomalies can be cancelled by a single complex chiral fermion of charge $(1,-1)$ under the diagonal ${\mathrm U}(1)$ gauge symmetries.
Again, our calculations in this section should be generalized to verify these predictions. For example, the minimal spectral flow operators
are represented by the distribution $\delta^{(2)}(\det u)$.
\noindent{\it Acknowledgment}
Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Research $\&$ Innovation.
Research of EW supported in part by NSF Grant PHY-1911298. We thank D. Baraglia, P. Etingof, E. Frenkel, D. Kazhdan, and L. Schaposnik for discussions.
We also thank P. Etingof for a careful reading of the manuscript and suggesting a number of clarifications.
| {'timestamp': '2021-08-10T02:35:07', 'yymm': '2107', 'arxiv_id': '2107.01732', 'language': 'en', 'url': 'https://arxiv.org/abs/2107.01732'} |
\section{\label{sec:intro}Introduction}
Dispersive shock waves~(DSWs) are widespread nonlinear phenomena in physics, from hydrodynamics~\cite{1966Peregrine,1988Smyth,2016Maiden} to acoustics~\cite{1970Taylor}, from Bose-Einstein condensates~\cite{2004Damski,2004Kamchatnov,2004Perez,2005Simula,2006Hoefer,2008Chang} to plasma physics~\cite{2008Romagnani}, and from quantum liquids~\cite{2006Bettelheim} to optics~\cite{1967Akhmanov,1989Rothenberg,2007El,2007HoeferPhysD,2007Ghofraniha,2007Wan,2008Conti,2012Ghofraniha,2013Garnier,2013Gentilini,2014Gentilini,2014Smith1,2015GentiliniPRA,2015GentiliniSciRep,2015Xu,2016Braidotti,2016Wetzel,2016XuMussot,2016XuGarnier,2017Zannotti,2019Gautam,2019Marcucci,2019MarcucciReview}.
In optics and photonics, when light propagates in a medium whose refractive index depending on the beam intensity, (e.g., light experiences the Kerr effect~\cite{2008Boyd}), the interplay between diffraction (or dispersion) and nonlinearity can lead to steep gradients in the phase profile, and, in some cases, to a wave breaking~\cite{2019MarcucciReview}.
Such discontinuity is regularized by very rapid oscillations, both in phase chirp and in intensity outlines, called undular bores~\cite{2007Wan}.
If we slightly mutate this third-order nonlinearity, making it nonlocal (or noninstantaneous), then the phenomenology changes very little. When the phase chirp reaches the discontinuity and starts to oscillate, the intensity does not develop undular bores, but rather the annular collapse singularity~(ACS) (or M-shaped singularity if the beam is $1+1$-dimensional)~\cite{2015Xu,2016Braidotti,2018Xu,2019MarcucciReview}. Spatial collapse-DSWs occur in thermal media, where the radiation-matter interaction is led by a thermo-optic effect, and the refractive index perturbation depends on the whole intensity profile~\cite{2008Boyd,2019MarcucciReview}.
Theoretically, such modification to the Kerr nonlinearity has significant consequences. Laser beam propagation in a standard Kerr medium is ruled by the nonlinear Schr\"odinger equation~(NLSE), exactly solvable by the inverse scattering transform method~\cite{1967Gardner,1972Zacharov,2015Fibich}. However, the NLSE with a nonlocal potential cannot be solved by inverse scattering transform (despite some recent progress in two-dimensional~(2D) media~\cite{2017Hokiris,2019Hokiris}), but only through the Whitham modulation and the hydrodynamic approximation~\cite{1999Whitham}. Moreover, the dynamics beyond the shock appears to be intrinsically irreversible, as recently shown by applying the time asymmetric quantum mechanics~\cite{1981Bohm,1989Bohm,1998Bohm,1999Bohm,2002Chruscinski,2002Delamadrid,2003Gadella,2003Chruscinski,2004Chruscinski,2004Civitarese,2016Celeghini,2016Marcucci} to the description of DSWs in highly nonlocal approximation~\cite{2015GentiliniPRA,2015GentiliniSciRep,2016Braidotti,2017Marcucci,2019MarcucciReview}.
In highly nonlocal Kerr media, the ACS is modeled by the simplest Hamiltonian of time asymmetric quantum mechanics: the reversed harmonic oscillator~(RHO). This Hamiltonian has a real complete continuous spectrum, which corresponds to a basis of eigenfunctions that have simple poles in their analytical prolongations to the complex plane. Since the RHO is a harmonic oscillator~(HO) with a pure imaginary frequency, starting from the HO complete point spectrum, it turns out that the RHO point spectrum is the set of the mentioned simple poles, and the related eigenfunctions are non-normalizable eigenvectors belonging to a rigged Hilbert space, called Gamow vectors~(GVs)~\cite{1928GamowDE,1928GamowENG}.
Light propagation beyond the collapse is then expressed as a superposition of GVs, which exponentially decay with quantized decay rates. Such laser beam evolution is the outcome of a phenomenon, the shock, that is intrinsically irreversible: in the absence of absorption and interaction with an external thermal bath, the dynamics cannot be inverted, i.e., it is time asymmetric.
Can this theoretical model be used to describe much more complex scenarios? To answer this question, we need to access regimes with much stronger nonlinearity
In recent experiments, it is showed that M-Cresol/Nylon solutions exhibit an isotropic giant self-defocusing nonlinearity, tunable by varying the nylon concentration~\cite{2014Smith}.
M-Cresol/Nylon is a thermal chemical mixture, consisting of an organic solvent (m-cresol) and a synthetic polymeric solute (nylon).
The nonlinear Kerr coefficient $n_2$ of pure m-cresol is $-9\times10^{-8} cm^2/W$, but it was found that, in such mixtures, $n_2=-1.6\times10^{-5} cm^2/W$ for a nylon mass concentration of $3.5\%$, higher than other thermal nonlinear materials in which ACSs have been observed~\cite{2014Smith1,2019MarcucciReview}.
In this Letter, we report on our theoretical discovery and experimental evidence of optical DSWs with an \textit{anisotropic zero-singularity}~(ZS) (i.e., a gap in the intensity profile along only one direction) in M-Cresol/Nylon solutions. Fixing $z$ as the longitudinal and $x,y$ as the transverse directions, we consider an initial beam which is even in the $y$ direction, and odd along the $x$ direction. This initial condition causes a new phenomenon: the shock does develop an annular collapse, but around the ZS it presents an abrupt intensity discontinuity.
We theoretically analyze this anisotropic wave breaking. We model the beam propagation beyond the shock point by time asymmetric quantum mechanics and uncover the mechanism that determines how such an abrupt intensity discontinuity is generated. We numerically simulate these results and find remarkable agreement between experimental outcomes and theoretical predictions.
For a laser beam propagating in a thermal medium with refractive index $n=n_0+\Delta n[|A|^2](\mathbf{R})$, where $\mathbf{R}=(\mathbf{R_{\perp}},Z)=(X,Y,Z)$, the NLSE describes the evolution of the envelope $A(\mathbf{R})$ of the monochromatic field $\mathbf{E}(\mathbf{R})=\hat{\mathbf{E}}_0 A(\mathbf{R})e^{\imath k Z}$, and it reads as follows:
\begin{equation}
2\imath k \partial_Z A+\nabla_{\mathbf{R_{\perp}}}^2 A+2k^2 \frac{\Delta n[|A|^2]}{n_0} A=-\imath\frac{k}{ L_{loss}}A,
\label{eq:NLSE}
\end{equation}
where $\nabla_{\mathbf{R_{\perp}}}^2=\partial_X^2+\partial_Y^2$, $k=\frac{2\pi n_0}{\lambda}$ is the wavenumber, $\lambda$ is the wavelength, and $L_{loss}$ is the linear loss length. By defining $I=|A|^2$ the intensity, $\bar{P}(Z)=\int\int\mathrm{d}\mathbf{R_{\perp}}I(\mathbf{R})$ the power, $L_d=k W_0^2$ the diffraction length, with $W_0$ the initial beam waist, and $\alpha=\frac{L_d}{L_{loss}}$, it turns out that $\bar{P}$ is not conserved only if $\alpha\neq0$. Indeed, if $\alpha\sim0$, then $\partial_Z \bar{P}\sim0$~\cite{1991Lisak}.
In low absorption regime, the refractive index perturbation in Eq.~(\ref{eq:NLSE}) is~\cite{2019MarcucciReview}
\begin{equation}
\Delta n[|A|^2](\mathbf{R_{\perp}})=n_2 \int\int\mathrm{d}\mathbf{R_{\perp}'}K(\mathbf{R_{\perp}}-\mathbf{R_{\perp}'})I(\mathbf{R_{\perp}'}),
\label{eq:n1}
\end{equation}
with $n_2$ the Kerr coefficient and $K(\mathbf{R_{\perp}})$ is the kernel function describing the nonlocal nonlinearity, normalized such that $\int\int\mathrm{d}\mathbf{R_{\perp}}K(\mathbf{R_{\perp}})=1$.
For $K(\mathbf{R_{\perp}})=\delta(\mathbf{R_{\perp}})$ we attain the well-known local Kerr effect, i.e., $n=n_0+n_2I$~\cite{2008Boyd}. In our nonlocal case, where the laser beam produces a thermo-optic effect that generates an isotropic variation of the medium density distribution, the response function is
\begin{equation}
K(X,Y)=\tilde{K}(X)\tilde{K}(Y),\;\; \tilde{K}(X)=\frac{e^{-\frac{|X|}{L_{nloc}}}}{2L_{nloc}}.
\label{eq:kernel}
\end{equation}
Here $L_{nloc}$ is the nonlocality length~\cite{1997Snyder,2003Conti,2007Minovich,2007Ghofraniha,2019Marcucci}.
We rescale Eq.~(\ref{eq:NLSE}), with $\alpha\sim0$, by defining the dimensionless variables $x=X/W_0$, $y=Y/W_0$ and $z=Z/L_d$, and obtain
\begin{equation}
\imath \partial_z \psi+\frac{1}{2}\nabla_{\mathbf{r_{\perp}}}^2\psi+\chi P K_0*|\psi|^2 \psi=0,
\label{eq:NLSEnorm}
\end{equation}
where $\mathbf{r}=(\mathbf{r_{\perp}},z)=(x,y,z)$, $\nabla_{\mathbf{r_{\perp}}}^2=\partial_x^2+\partial_y^2$, $\psi(\mathbf{r})=\frac{W_0}{\sqrt{\bar{P}}}A(\mathbf{R})$, $\chi=\frac{n_2}{|n_2|}$ and $P=\frac{\bar{P}}{P_{REF}}$ with $P_{REF}=\frac{\lambda^2}{4\pi^2 n_0 |n_2|}$.
The asterisk in Eq. (\ref{eq:NLSEnorm}) stands for the convolution product, while $K_0(x,y)=\tilde{K_0}(x)\tilde{K_0}(y)$ with $\tilde{K_0}(x)=W_0 \tilde{K}(X)=\frac{e^{-\frac{|x|}{\sigma}}}{2\sigma}$, $\tilde{K_0}(y)$ of the same form, and $\sigma=\frac{L_{nloc}}{W_0}$ the nonlocality degree.
In highly nonlocal approximation ($\sigma>>1$), once the initial conditions are fixed, $|\psi|^2$ mimics a delta function (or a narrow superposition of delta functions), and the nonlocal potential looses its $I$-dependence, becoming a simple function of the transverse coordinates~\cite{1997Snyder,2012Folli}:
\begin{equation}
\begin{array}{l}
K_0*|\psi|^2\simeq \kappa(\mathbf{r_{\perp}})\simeq\kappa(\mathbf{0})+\left(\partial_x\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)x+\left(\partial_y\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)y+\\ \\
+\frac{1}{2}\left(\partial_x^2\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)x^2+\left(\partial_x\partial_y\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)xy+\frac{1}{2}\left(\partial_y^2\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)y^2,
\end{array}
\label{eq:nonlocpot}
\end{equation}
after a Taylor second-order expansion.
This approximation maps the NLSE in Eq.~(\ref{eq:NLSEnorm}) into a linear Schr\"odinger equation $\imath \partial_z\psi(\mathbf{r})=\hat{H}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})\psi(\mathbf{r})$, with $\hat{H}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\frac{1}{2}\hat{\mathbf{p_{\perp}}}^2+\hat V(\mathbf{r_{\perp}})$ the Hamiltonian, $\mathbf{\hat p_{\perp}}=(\hat p_x, \hat p_y)=(-\imath\partial_x, -\imath\partial_y)$ the transverse momentum and $\hat V(\mathbf{r_{\perp}})=-\chi P\kappa(\mathbf{r_{\perp}})\mathbb{1}$ the multiplicative potential ($\mathbb{1}$ is the identity operator). Let us consider the initial condition
\begin{equation}
\psi_{\mathrm{ISO}}(\mathbf{r_{\perp}})=\psi_{even}(x)\psi_{even}(y),\;\;\psi_{even}(x)=\frac 1{\sqrt[4]{\pi}}e^{-\frac{x^2}2},
\label{eq:isoini}
\end{equation}
and $\psi_{even}(y)$ of the same form.
The shape of $\kappa(\mathbf{r_{\perp}})$ depends on $\psi_{\mathrm{ISO}}(\mathbf{r_{\perp}})$. Indeed, since $\psi_{\mathrm{ISO}}$ is an even, separable function, all the first derivatives in Eq.~(\ref{eq:isoini}) vanish, hence $\kappa(\mathbf{r_{\perp}})=\kappa_0^2-\frac{1}{2}\kappa_2^2\left|\mathbf{r_{\perp}}\right|^2$, where $\kappa_0^2=\frac1{4\sigma^2}$ and $\kappa_2^2=\frac{1}{2\sqrt{\pi}\sigma^3}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{figISO.eps}
\end{center}
\caption{Solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:isoini}), for $P=4\times10^6$ and $\sigma=120$, in arbitrary units: (a) shows the intensity transverse profile at $z=0$, (b) exhibits the symmetric response function derived from Eq.~(\ref{eq:isoini}), and (c) reports the intensity longitudinal outline, here on the plane $(x,z)$, equal to one on the plane $(y,z)$.
\label{fig:iso}}
\end{figure}
In the defocusing case ($n_2<0$), the transverse profile of the solution of Eq.~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:isoini}) is shown in Fig.~\ref{fig:iso}(a). Figure~\ref{fig:iso}(b) exhibits the central part of the symmetric response function $K_0(x,y)$, while the longitudinal profile on $x,z$ (same of $y,z$) is reported in Fig.~\ref{fig:iso}(c). The corresponding Hamiltonian reads
$\hat{H}=P\kappa_0^2 +\hat{H}_{\mathrm{RHO}}(p_x,x)+\hat{H}_{\mathrm{RHO}}(p_y,y)$,
where
\begin{equation}
\hat{H}_{\mathrm{RHO}}(p_x,x)=\frac 12\hat{p_x}^2-\frac{\gamma^2}{2}\hat{x}^2
\label{eq:RHO}
\end{equation}
is the 1D-RHO Hamiltonian of frequency $\gamma=\sqrt{P}\kappa_2$.
Once moved to $\phi(\mathbf{r})=e^{\imath P\kappa_0^2}\psi(\mathbf{r})$, the Schr\"odinger equation becomes
$\imath \partial_z\phi(\mathbf{r})=\left[\hat{H}_{\mathrm{RHO}}(p_x,x)+\hat{H}_{\mathrm{RHO}}(p_y,y)\right]\phi(\mathbf{r})$, which is completely separable.
In bra-ket notation
\begin{equation}
\begin{array}{l}
\imath \frac{\mathrm{d}}{\mathrm{d}z}|\phi(z)\rangle=\hat{H}_{\mathrm{ISO}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})|\phi(z)\rangle,\\ \\
\hat{H}_{\mathrm{ISO}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\hat{H}_{\mathrm{RHO}}(p_x,x)\otimes\mathbb{1}_y+\mathbb{1}_x\otimes\hat{H}_{\mathrm{RHO}}(p_y,y),\\ \\
|\phi(z)\rangle=|\phi_{even}(z)\rangle_x\otimes|\phi_{even}(z)\rangle_y,
\end{array}
\label{eq:iso}
\end{equation}
with $\otimes$ the tensorial product, no more explicitly written hereafter. The solution of Eq.~(\ref{eq:iso}) lives in a tensorial product between two 1D rigged Hilbert spaces. Indeed, if we consider the evolution operator $\hat{U}(z)=e^{-\imath\hat{H}z}$ such that $|\phi(z)\rangle=\hat{U}(z)|\phi(0)\rangle$, for Eq.~(\ref{eq:iso}) $|\phi(z)\rangle=e^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_xe^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_y$. The representation of $|\phi_{even}(z)\rangle_{x,y}=e^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_{x,y}$ in terms of GVs was already studied and was also already demonstrated to describe 1D DSWs in thermal media~\cite{2015GentiliniPRA,2015GentiliniSciRep,2016Braidotti,2017Marcucci,2019MarcucciReview}. It is $|\phi_{even}(z)\rangle_{x,y}=|\phi_N^G(z)\rangle+|\phi_N^{BG}(z)\rangle$, with
\begin{equation}
|\phi_N^G(z)\rangle=\sum_{n=0}^N e^{-\frac{\gamma}2(2n+1)z}|\mathfrak{f}_n^-\rangle\langle \mathfrak{f}_n^+|\psi_{even}\rangle
\label{eq:GV}
\end{equation}
the decaying superposition of Gamow states $|\mathfrak{f}_n^-\rangle$, corresponding to the energy levels $E_n^{RHO}=\imath\frac{\gamma}2(2n+1)$, and $|\phi_N^{BG}(z)\rangle$ the background function, both belonging to the same 1D rigged Hilbert space.
Our experiments are performed in an isotropic medium with asymmetric initial conditions for the beam, designed through a phase mask, to attain an anisotropic light propagation. The setup is illustrated in Fig.~\ref{fig:exp}(a). A laser beam with wavelength $\lambda=532$nm passes through two collimating lenses (L1 and L2), and then through a beam splitter (BS1), which divides the beam into two arms, one used for the nonlinear experiment, and the other for getting a reference beam for interference measurements. The beam outcoming from the first arm is transformed into an asymmetric input by a phase mask, and then is focused (via L3) onto the facet of a $2$mm-long cuvette, which contains a M-Cresol/Nylon solution (with $3.5\%$ Nylon). The output is imaged (via L4 and BS2) onto a CCD-camera.
Figure~\ref{fig:exp}(b) reports the input beam (zoom-in intensity and phase patterns at initial power $\bar{P}=2$mW and waist $W_0=15.8\mu$m) and its outputs at different initial powers. The input beam presents a phase discontinuity of $\pi$ along $x=0$.
The output beam exhibits diffraction at a low power, but evolves into an overall shock pattern with two parts at high powers. It's important to note that, while the whole pattern expands, the gap between two parts remains constant. This represents the first realization of what we define as \textit{anisotropic DSWs}: ACSs with an initial ZS, which generates two barriers of light intensity around a constant gap in the middle of the beam. Despite the medium isotropy, the oddity of the initial condition generates an anisotropic final transverse profile.
\begin{figure}[h!]
\begin{center}
\textbf{(a)
\raisebox{-1\height}{\includegraphics[width=0.95\linewidth]{Setup.eps}}
\\\vspace{3mm}
\textbf{(b)}\raisebox{-0.85\height}{\includegraphics[width=\linewidth]{anisotropic.png}}
\end{center}
\caption{
(a) Experimental setup. A $\lambda=532$nm CW-laser beam is collimated through two lenses (L1 and L2). A beam splitter (BS1) divides the beam in two arms. the first is made asymmetric by a phase mask and propagates in a $2$mm-long cuvette filled with M-Cresol/Nylon $3.5\%$-solution. The second is a reference beam for interference measurements. The output is imaged (via L4 and BS2) onto a CCD-camera.
(b) Input and outputs observed. The phase mask in (a) generates a $\pi$ discontinuity in the input phase, here reported with initial power $\bar{P}=2$mW and waist $W_0=15.8\mu$m, together with the intensity profile. Several output at different initial powers are shown. The low power cannot distinguish nonlinear effects from diffraction, but the higher the power is, the stronger the nonlinear effects are, leading to the formation of anisotropic DSWs.
\label{fig:exp}}
\end{figure}
We model the initial asymmetric beam-shape as follows:
\begin{equation}
\psi_{\mathrm{ANI}}(\mathbf{r_{\perp}})=\psi_{odd}(x)\psi_{even}(y),\;\;\psi_{odd}(x)=-\frac{\sqrt{2}}{\sqrt[4]{\pi}}xe^{-\frac{x^2}2},
\label{eq:aniini}
\end{equation}
$\psi_{even}(y)=\frac 1{\sqrt[4]{\pi}}e^{-\frac{y^2}2}$ as in Eq.~(\ref{eq:isoini}).
In this case, Eq.~(\ref{eq:nonlocpot}) is reduced to $\kappa(\mathbf{r_{\perp}})=\kappa_0^2+\frac{1}{2}\kappa_1^2 x^2-\frac{1}{2}\kappa_2^2 y^2$, with $\kappa_0^2=\frac1{4\sigma^2}$, $\kappa_1^2=\frac{1}{4\sigma^4}$ and $\kappa_2^2=\frac{1}{2\sqrt{\pi}\sigma^3}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{figANI.eps}
\end{center}
\caption{
Solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:aniini}), for $P=4\times10^6$ and $\sigma=120$, in arbitrary units: (a) shows the intensity transverse profile at $z=1$, (b) exhibits the asymmetric response function derived from Eq.~(\ref{eq:isoini}), and (c) reports the intensity longitudinal outline on the plane $(x,z)$, with the zero-singularity.
\label{fig:ani}}
\end{figure}
The anisotropy appears evident: not only the initial condition presents a ZS, but also the response function has two different behaviors along $x,y$ directions. Numerical simulations are illustrated in Fig.~\ref{fig:ani}. Figure~\ref{fig:ani}(a) shows the anisotropic DSWs, solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:aniini}). Figure~\ref{fig:ani}(b) gives numerical proof of the response function anisotropy: the $(x,y)$-plane origin corresponds to a saddle point, with a locally increasing profile along $x>0$, $y<0$ and a locally decreasing outline along $x<0$, $y>0$. Figure~\ref{fig:ani}(c) reports the intensity-ZS in a neighborhood of $x=0$ during propagation.
The presence of the saddle point in the response function has direct consequences through highly nonlocal approximation in mapping the NLSE in the quantum-like linear Schr\"odinger equation. From the expression of $\kappa(\mathbf{r_{\perp}})$ above, for $\phi(\mathbf{r})=e^{\imath P\kappa_0^2}\psi(\mathbf{r})$ we obtain
\begin{equation}
\begin{array}{l}
\imath \frac{\mathrm{d}}{\mathrm{d}z}|\phi(z)\rangle=\hat{H}_{\mathrm{ANI}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})|\phi(z)\rangle,\\ \\
\hat{H}_{\mathrm{ANI}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\hat{H}_{\mathrm{HO}}(p_x,x)\mathbb{1}_y+\mathbb{1}_x\hat{H}_{\mathrm{RHO}}(p_y,y),\\ \\
|\phi(z)\rangle=|\phi_{odd}(z)\rangle_x|\phi_{even}(z)\rangle_y,
\end{array}
\label{eq:ani}
\end{equation}
where $\hat{H}_{\mathrm{HO}}(p_x,x)=\frac 12\hat{p_x}^2+\frac{\omega^2}{2}\hat{x}^2$ is the one-dimensional harmonic oscillator Hamiltonian with $\omega=\sqrt{P}\kappa_{odd}$, and $\hat{H}_{\mathrm{RHO}}$ is the one-dimensional RHO Hamiltonian in Eq.~(\ref{eq:RHO}).
The solution of Eq.~(\ref{eq:ani}) is the tensorial product of $|\phi_{odd}(z)\rangle_{x}=\sum_{n=0}^{+\infty} e^{\imath\frac{\omega}2(2n+1)z}|\Psi_n^{HO}\rangle\langle\Psi_n^{HO}|\psi_{odd}\rangle$, where $|\Psi_n^{HO}\rangle$ are $\hat{H}_{\mathrm{HO}}$-eigenstates corresponding to the energy levels $E_n^{HO}=\frac{\omega}2(2n+1)$~\cite{2016Marcucci}, and $|\phi_{even}(z)\rangle_{y}=|\phi_N^G(z)\rangle+|\phi_N^{BG}(z)\rangle$, explicitly written in Eq.~(\ref{eq:GV}).
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{GVSign.eps}
\end{center}
\caption{
GVs signature. From Fig.~\ref{fig:ani}, in the same conditions, (a) is the $y,z$ profile at fixed $x=2.29$. Intensity along the pink line, i.e., $|\phi(x=2.29,y=2.29,z)|^2=\langle\phi_{even}(z)|\phi_{even}(z)\rangle_{y}=\sum_{n=0}^Ne^{-\Gamma_n z}\left|\langle\mathfrak{f}_n^+|\phi_{even}\rangle\right|^2$, is in (b) [$\Gamma_n=\gamma(2n+1)$ quantized decay rates]. The continuous lines represent the two first exponential functions of the summation above that fit the decaying part: the red line is the fundamental GV, with decay rate $\Gamma_0$, and the blue line is the first excited GV, with decay rate $\Gamma_2$.
\label{fig:GV}}
\end{figure}
Evidence of the presence of GVs is given in Fig.~\ref{fig:GV}. By defining $\Gamma_n=\gamma(2n+1)$, we look for the first two quantized decay rates $\Gamma_{0,2}$ (the even Gaussian initial function only leads to even energy levels) in the longitudinal propagation in $y$-direction. Indeed, if one computes the intensity of the $y$-part, one finds $\langle\phi_{even}(z)|\phi_{even}(z)\rangle_{y}\stackrel{N>>0}{\simeq}\langle\phi_N^G(z)|\phi_N^G(z)\rangle=\sum_{n=0}^Ne^{-\Gamma_n z}\left|\langle\mathfrak{f}_n^+|\phi_{even}\rangle\right|^2$. Figure~\ref{fig:GV}(a) shows the theoretical section of the nonlinear sample where we seek decaying states. We fix $x=2.29$, a little distant from the shock-gap, and report the corresponding intensity in $y,z$ plane. The pink line is equivalent to $x=y=2.29$. Figure~\ref{fig:GV}(b) exhibits $|\phi(x=2.29,y=2.29,z)|^2$, exponentially decaying. Two exponential fits demonstrate the GV occurrence: the fundamental Gamow state represents the plateau with decay rate $\Gamma_0=1.51$, whereas the first excited one interpolates the peak, with decay rate $\Gamma_2=7.51$.
We stress that the rule $\frac{\Gamma_2}{\Gamma_0}=5$ is respected.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{GapExp.eps}
\end{center}
\caption{
Observation of the shock-gap versus the initial power $\bar{P}$. The red line shows the theoretical fit with a functions $\propto\frac{1}{\sqrt{\bar{P}}}$.
\label{fig:gap}}
\end{figure}
The analysis of the barriers, due to the HO component, and the corresponding shock-gap is also examinated. Since $\hat{H}_{\mathrm{HO}}$ has potential $\hat{V}_{\mathrm{HO}}(x)=\frac{\omega^2}2\hat{x}^2$, we expect a shock-gap with the same behavior of the potential width $\Delta x\propto\frac1{\sqrt{\omega}}=\frac{2\sigma^2}{\sqrt{P}}$, that is, a shock-gap that varies among different experiments by changing the initial power but, in a single observation with a given $P$, does not change by varying $z$ [as shown in Fig.~\ref{fig:ani}(c)].
This is experimentally proven and reported in Fig.~\ref{fig:gap}, which shows measurements of the shock-gap at variance of initial power $\bar{P}$. A theoretical fit with a function $\propto\frac{1}{\sqrt{\bar{P}}}$ is drawn in the red line. The agreement between observations and numerical simulations confirms the theoretical statement.
We have proved that the interplay of a trapping (harmonic oscillator) and an antitrapping (reversed harmonic oscillator) potential generates a novel kind of dispersive shock waves, with the simultaneous presence of annular collapse singularities and a shock-gap enclosed by very intense light barriers.
The use of a thermal medium with a giant Kerr coefficient, the M-Cresol/Nylon solution with $3.5\%$ of nylon concentration, let us access an extremely-nonlinear highly-nonlocal regime and perform accurate experiments with negligible loss.
We modeled the outcoming dynamics through an advanced theoretical description in rigged Hilbert spaces, by means of time asymmetric quantum mechanics, proving its intrinsic irreversibility.
Our results not only confirm previous studies on the giant nonlinear response of M-Cresol/Nylon, but also disclose fundamental insights on propagation of dispersive shock waves with a singular initial intensity profile.
We believe that this work can be a further step towards a complete description of optical nonlinear phenomena, where inverse scattering transform, Whitham modulation, hydrodynamic approximation and time asymmetric quantum mechanics cooperate in establishing one uniform theory of dispersive shock waves.
We are pleased to acknowledge support from the QuantERA ERA-NET Co-fund 731473 (Project QUOMPLEX), H2020 project grant number 820392, Sapienza Ateneo, PRIN 2015 NEMO, PRIN 2017 PELM, Joint Bilateral Scientic Cooperation CNR-Italy/RFBR-Russia 2018-2020, the NSF Award DMR-1308084, and the National Key R\&D Program of China (2017YFA0303800).
| {'timestamp': '2019-09-11T02:17:54', 'yymm': '1909', 'arxiv_id': '1909.04506', 'language': 'en', 'url': 'https://arxiv.org/abs/1909.04506'} |
\section{Introduction}
\noindent
Let $(M,g)$ be a Lorentzian spacetime and $(N,h)$ a Riemannian manifold. In this paper, we study wave maps $u:(M,g) \longrightarrow (N,h)$, that is, critical points of the geometric action functional
\begin{align*}
S_{g}[u]:=\frac{1}{2} \int _{M} |d_{g} u|^{2}~d \mu_{g}.
\end{align*}
Here,
\begin{align*}
|d_{g}u(x)|^{2} \equiv |d_{g}u (x)|_{ T^{\star}_{x}M \otimes T_{u(x)} N }^{2}:= \text{tr}_{g} \left ( u^{\star} \left( h \right) \right)
\end{align*}
is the trace (with respect to $g$) of the pullback metric on $(M,g)$ via the map $u$. The integral is understood with respect to the standard measure $d\mu _{g}$ on the domain manifold. In local coordinates $(x_{\mu})$ on $(M,g)$, this expression reads
\begin{align*}
S_{g}[u]= \int_{M} g ^{\mu \nu} (\partial _{\mu} u^{a}) (\partial _{\nu} u^{b}) h_{a b} \circ u~ d \mu_{g}
\end{align*}
where the Einstein summation convention is used. The Euler-Lagrange equations associated to this functional are
\begin{align} \label{WM}
\Box _{g} u^{a} + g ^{\mu \nu} (\Gamma ^{a}_{b c} \circ u) (\partial _{\mu} u^{b} ) (\partial _{\nu} u^{c}) =0
\end{align}
and they constitute a system of semi-linear wave equations. Here, $\Box _{g}$ is the Laplace-Beltrami operator on $(M,g)$
\begin{align*}
\Box_{g} := \frac{1}{|g|} \partial _{\mu} (g^{\mu \nu} |g| \partial _{\nu}),\quad |g|:=\sqrt{ \left| \text{det}(g_{\mu \nu}) \right| }
\end{align*}
and $\Gamma ^{a}_{b c}$ are the Christoffel symbols associated to the metric $h$ on the target manifold. Eq.~\eqref{WM} is called the wave maps equation (known in the physics literature as non-linear $\sigma$ model) and is the analog of harmonic maps between Riemannian manifolds in the case where the domain is a Lorentzian manifold instead. For more details, we refer the reader to \cite{Ren08} and \cite{Str97}.
\subsection{Intuition} Recently, the wave maps equation has attracted a lot of interest. On the one hand, the wave maps equation is a rich source for understanding nonlinear geometric equations since it is a nonlinear generalization of the standard wave equation on Minkowski space. In addition, the wave maps equation has a pure geometric interpretation: it generalizes the notion of geodesic curves. Notice that, if $M = (\alpha, \beta)$ is an open interval and $(N,h)$ any curved Riemannian manifold, the wave maps equation is the geodesic equation
\begin{align*}
\frac{d^2 u^{a}}{dt ^2} (t) + (\Gamma ^{a}_{b c} \circ u(t)) \frac {d u^{b} }{dt} (t) \frac{d u^{c} }{dt} (t)=0.
\end{align*}
On the other hand, the Cauchy problem for the wave maps system provides an attractive toy-model for more complicated relativistic field equations. Specifically, wave maps contain many features of the more complex Einstein equations but are simple enough to be accessible for rigorous mathematical analysis.
Further details on the correlation between the wave maps system and the Einstein equations can be found in \cite{Mis78, Mon89, Wei90, Kla97}.
Being a time evolution equation, the fundamental problem is the Cauchy problem: given specified smooth initial data, does there exist a unique smooth solution to the wave maps equation with this initial data? Furthermore, does the solution exist for all times? On the other hand, if the solution only exists up to some finite time $T$, how does the solution blow up as $t$ approaches $T$?
The investigation of questions of global existence and formation of singularities for the wave maps equation can give insight into the analogous, but much more difficult, problems in general relativity.
\subsection{Equivariant wave maps} Now, we turn our attention to the Cauchy problem in the case where the domain is the Minkowski spacetime $(\mathbb{R}^{1+d},g)$ and the target manifold is the sphere $(\mathbb{S}^{d},h)$ for $d \geq 3$. Hence, we pick $g =$diag$(-1,1,\dots,1)$ and $h$ to be the standard metric on the sphere. Furthermore, we choose standard spherical coordinates on Minkowski space and hyper-spherical coordinates on the sphere. The respective metrics are given by
\begin{align*}
g = - dt^2 + dr^2 + r^2 d \omega ^2, \quad h= d \Psi ^2 + \sin^2(\Psi) d \Omega ^2,
\end{align*}
where $d\omega ^2$ and $d \Omega ^2$ are the standard metrics on $\mathbb{S}^{d-1}$. Moreover, a map $u:(\mathbb{R}^{1+d},g) \longrightarrow (\mathbb{S}^{d},h)$ can be written as
\begin{align*}
u (t,r,\omega) = \big( \Psi (t,r,\omega ), \Omega (t,r,\omega) \big).
\end{align*}
We restrict our attention to the special subclass known as 1-equivariant or co-rotational, that is
\begin{align*}
\Psi (t,r,\omega ) \equiv \psi (t,r),~~~ \Omega (t,r, \omega ) = \omega.
\end{align*}
Under this ansatz, the wave maps system for functions $u:(\mathbb{R}^{1+d},g) \longrightarrow (\mathbb{S}^{d},h)$ reduces to the single semi-linear wave equation
\begin{equation}
\label{eq:main} \psi_{tt}-\psi_{rr}-\frac{d-1}{r}\psi_r+\frac{d-1}{2}\frac{\sin(2\psi)}{r^2}=0.
\end{equation}
By finite speed of propagation and radial symmetry it is natural to
study this equation in backward light-cones with vertex $(T,0)$, that is
\begin{align*}
C_{T} :=\left \{ (t,r) : 0<t<T,~0 \leq r \leq T-t \right \}
\end{align*}
where $T>0$.
Consequently, we consider the Cauchy problem \\
\begin{align} \label{cauchy}
\begin{cases}
\psi _{tt} (t,r) - \Delta ^{ \text{rad} }_{r,d} \psi (t,r) = - \frac{d-1}{2} \frac{ \sin ( 2 \psi (t,r) ) }{r^2}, &\quad \text{in } C_{T} \\
\psi (0,r)= f(r),~~~ \psi _{t} (0,r)= g(r), &\quad \text{on } \{ t=0 \} \times [0,+\infty) \
\end{cases}
\end{align}
where $\Delta ^{ \text{rad} }_{r,d} $ stands for the radial Laplacian
\begin{align*}
\Delta ^{ \text{rad} }_{r,d} \psi (t,r) := \psi _{rr} (t,r) + \frac{d-1}{r} \psi _{r} (t,r).
\end{align*}
To ensure regularity of solutions, equations $\eqref{cauchy}$ must be supplemented by the boundary condition
\begin{align} \label{reg}
\psi (t,0)=0,\quad \text{~for all~} t \in (0,T).
\end{align}
\subsection{Self-similar solutions}
A basic question for the Cauchy problem $\eqref{cauchy}$ is whether solutions starting from smooth initial data
\begin{align*}
(f,g)=\left( \psi (0, \cdot), \partial _{t} \psi (0, \cdot) \right)
\end{align*}
can become singular in the future. Note that Eq.~\eqref{eq:main} has the conserved energy
\begin{align*}
E[\psi]:=\int_{0}^{\infty} \left( \psi _{t}^2 + \psi _{r}^2 + (d-1)\frac{\sin ^2(\psi) }{r^2} \right) r^2 dr .
\end{align*}
However, the energy cannot be used to control the evolution since Eq.~\eqref{cauchy} is not well-posed at energy regularity, cf.~\cite{ShaTah94}.
Indeed, Eq.~\eqref{eq:main} is invariant under dilations
\begin{align} \label{dilation}
\psi _{\lambda} (t,r):=\psi \left( \frac{t}{\lambda},\frac{r}{\lambda} \right),~ \lambda >0
\end{align}
and the critical Sobolev space for the pair $(\psi(t,\cdot),\partial_t \psi(t,\cdot))$ is $\dot H^{\frac{d}{2}}\times \dot H^{\frac{d}{2}-1}$.
Consequently, Eq.~\eqref{eq:main} is energy-supercritical for $d\geq 3$.
In fact, due to the scaling $\eqref{dilation}$ and the supercritical character it is natural to expect self-similar solutions and indeed, it is well known that there exist smooth initial data which lead to solutions that blowup in finite time in a self-similar fashion. Specifically, Eq.~\eqref{eq:main} admits the self-similar solution
\begin{align*}
\psi ^{T} (t,r) := f_{0} \Big ( \frac{r}{T-t } \Big) = 2 \arctan \Bigg( \frac{r}{\sqrt{d-2}(T-t) } \Bigg),\qquad T>0.
\end{align*}
This example is due to Shatah \cite{Sha88}, Turok-Spergel \cite{TurSpe90} for $d=3$, and Bizo\'n-Biernat \cite{BizBie15} for $d \geq 4$ and provides an explicit example for singularity formation from smooth initial data. Indeed, the self-similar solution $\psi^T$ is perfectly smooth for all $0<t < T$ but breaks down at $t = T$ in the sense that
\begin{align*}
\partial_r\psi ^{T} (t,r)|_{r=0}
\simeq \frac{1}{T-t} \longrightarrow + \infty,~~~\text{as}~ t \longrightarrow T^{-}.
\end{align*}
We note in passing that for $d\in \{3,4,5,6\}$, $\psi^T$ is just one member of a countable family of self-similar solutions, see \cite{Biz00, BieBizMal16}.
\subsection{The main result}
By finite speed of propagation one can use $\psi ^{T}$ to construct smooth, compactly supported initial data which lead to a solution that blows up as $t \longrightarrow T$. Our main theorem is concerned with the asymptotic nonlinear stability of $\psi^T$. In other words, we prove the existence of an open set of radial data which lead to blowup via $\psi ^{T}$. In this sense, the blowup described by $\psi ^{T}$ is stable. To state our main result, we will need the notion of the blowup time at the origin.
From now on we use the abbreviation $\psi[t]=(\psi(t,\cdot),\partial_t \psi(t,\cdot))$.
\begin{definition}
Given initial data $(\psi_{0},\psi_{1})$, we define
\begin{align*}
T_{(\psi_{0},\psi_{1})} := \sup \left\{
T >0 \middle|
\begin{subarray}{c}
\exists \text{~solution~} \psi : C_{T} \longrightarrow \mathbb{R} \text{~to~} \eqref{cauchy} \text{~in the sense of} \\
\text{~Definition~} \ref{def} \text{~with initial data~} \psi[0]=(\psi_{0},\psi_{1}) |_{\mathbb{B}_{T}^{d}}
\end{subarray} \right\} \cup \{0\}.
\end{align*}
In the case where $T_{(\psi_{0},\psi_{1})} < \infty$, we call $T \equiv T_{(\psi_{0},\psi_{1})}$ the blowup time at the origin.
\end{definition}
We remark that the effective spatial dimension for the problem \eqref{cauchy} is $d+2$.
To see this, recall that, by regularity, we get the boundary condition $\eqref{reg}$. Therefore, it is natural to switch to the variable $\widehat \psi(t,r):=r^{-1}\psi (t,r)$. Then \eqref{cauchy} transforms into
\begin{align*}
\begin{cases}
\widehat{\psi} _{tt} (t,r) - \Delta ^{ \text{rad} }_{r,d+2} \widehat{\psi} (t,r) = - \frac{d-1}{2} \frac{ \sin ( 2 r \widehat{\psi} (t,r) ) -2r \widehat{\psi} (t,r)}{r^3}, &\quad \text{in } C_{T} \\
\widehat{\psi} (0,r)= \frac{f(r)}{r},~~~ \widehat{\psi} _{t} (0,r)= \frac{g(r)}{r}, &\quad \text{on } \{ t=0 \} \times [0,+\infty) \
\end{cases}
\end{align*}
Note that the nonlinearity is now generated by a smooth function and the radial Laplacian is in $d+2$ dimensions.
\begin{theorem}
Fix $T_{0}>0$ and $d\geq 3$ odd. Then there exist constants $M,\delta,\epsilon >0$ such that for any radial initial data
$\psi [0]$
satisfying
\begin{align*}
\Big \| |\cdot|^{-1} \Big( \psi[0] -\psi^{T_{0}}[0] \Big) \Big \|_{ H^{\frac{d+3}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2}) \times H^{\frac{d+1}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2} )} \leq \frac{\delta}{M}
\end{align*}
the following statements hold:
\begin{enumerate}
\item $T \equiv T_{\psi[0]} \in [T_{0}-\delta,T_{0}+\delta]$,
\item the solution $\psi :C_{T} \longrightarrow \mathbb{R}$ satisfies
\begin{align*}
(T-t)^{k-\frac{d}{2}} \Big \| |\cdot|^{-1} \Big( \psi (t,\cdot) - \psi ^{T} (t, \cdot) \Big) \Big \|_{ \dot{H}^{k}(\mathbb{B}^{d+2}_{T-t} ) } &\leq\delta (T-t)^{\epsilon} \\
(T-t)^{\ell+1-\frac{d}{2}} \Big \| |\cdot|^{-1} \Big( \partial_t \psi (t,\cdot) - \partial_t \psi ^{T} (t, \cdot) \Big) \Big \|_{ \dot{H}^{\ell}(\mathbb{B}^{d+2}_{T-t} ) } &\leq \delta (T-t)^{\epsilon}
\end{align*}
for all $k=0,1,2, \dots, \frac{d+3}{2}$ and $\ell=0,1,2\dots,\frac{d+1}{2}$.
\end{enumerate}
\end{theorem}
\begin{remark}
Note that the normalizing factors on the left-hand sides appear naturally and reflect the behavior of the self-similar solution $\psi^T$ in the respective homogeneous Sobolev norms,
i.e.,
\begin{align*}
\||\cdot|^{-1}\psi^T(t,\cdot)\|_{\dot H^k(\mathbb B^{d+2}_{T-t})}&=\left \||\cdot|^{-1} f_0 \left(\frac{|\cdot|}{T-t} \right) \right \|_{ \dot{H}^{k}( \mathbb{B}^{d+2}_{T-t} ) } =(T-t)^{\frac{d}{2}-k} \||\cdot|^{-1} f_0 \left(|\cdot| \right) \|_{ \dot{H}^{k}( \mathbb{B}^{d+2}_{1} ) }
\end{align*}
and
\begin{align*}
\||\cdot|^{-1}\partial_t \psi^T(t,\cdot)\|_{\dot H^\ell(\mathbb B^{d+2}_{T-t})}&=(T-t)^{-2}\left \|f_0' \left(\frac{|\cdot|}{T-t} \right) \right \|_{ \dot{H}^{\ell}( \mathbb{B}^{d+2}_{T-t} ) } \\
& =(T-t)^{\frac{d}{2}-\ell-1} \| f_0' \left(|\cdot| \right) \|_{ \dot{H}^{\ell}( \mathbb{B}^{d+2}_{1} ) } .
\end{align*}
\end{remark}
\subsection{Related results}
The question of singularity formation for the wave maps equation attracted a lot of interest in the recent past, in particular in the energy-critical case $d=2$.
Bizo\'n-Chmaj-Tabor \cite{BizChmTab01} were the first to provide numerical evidence for the existence of blowup for critical wave maps with $\mathbb S^2$ target.
Rigorous constructions of blowup solutions for this model are due to Krieger-Schlag-Tataru \cite{KriSchTat08}, Rodnianski-Sterbenz \cite{RodSte10}, and Rapha\"el-Rodnianski \cite{RapRod12}. Struwe \cite{Str03} showed that blowup for equivariant critical wave maps takes place via shrinking of a harmonic map.
This result was considerably generalized to the nonequivariant setting by Sterbenz-Tataru \cite{SteTat10a, SteTat10b}, see also Krieger-Schlag \cite{KriSch12} for a different approach to the large-data problem and e.g.~\cite{CotKenLawSch15a, CotKenLawSch15b, Cot15,
CanKri15, Sha16, LawOh16} for more recent results on blowup and large-data global existence.
The energy-supercritical regime $d\geq 3$ is less understood. The small-data theory at minimal regularity is due to Shatah-Tahvildar-Zadeh \cite{ShaTah94} in the equivariant setting whereas Tataru \cite{Tat98, Tat01} and Tao \cite{Tao01a, Tao01b} treat the general case, see also \cite{KlaRod01, ShaStr02, NahSteUhl03, Kri03, Tat05}.
Self-similar blowup solutions were found by Shatah \cite{Sha88}, Turok-Spergel \cite{TurSpe90}, Cazenave-Shatah-Tahvildar-Zadeh \cite{CazShaTah98}, and Bizo\'n-Biernat \cite{BizBie15}.
The stability of self-similar blowup was investigated numerically in \cite{BizChmTab00, BizBie15, BieBizMal16} and proved rigorously in \cite{Don11, DonSchAic12, CosDonXia16, CosDonGlo16} in the case $d=3$. Furthermore, Dodson-Lawrie \cite{DodLaw15} proved that solutions with bounded critical norm scatter.
Finally, concerning the method, we remark that our proof relies on the techniques developed in the series of papers \cite{Don11, DonSchAic12, DonSch12, DonSch14, Don14, DonSch16, DonSch16a}.
However, we would like to emphasize that the present paper is not just a straightforward continuation of these works. In fact, new interesting issues arise, e.g.~in the spectral theory part, see Proposition \ref{projection} below.
\section{Radial wave equation in similarity coordinates}
\label{sec:sim}
\noindent
To start our analysis, we rewrite the initial value problem $\eqref{cauchy}$ as an abstract Cauchy problem in a Hilbert space. First, we rescale the variable $\psi \equiv \psi (t,r)$ and switch to similarity coordinates. Then, we linearize around the rescaled blowup solution and derive the evolution problem satisfied by the perturbation.
\subsection{Rescaled variables}
We define
\begin{align*}
\chi _{1} (t,r) := \frac{T-t}{r} \psi (t,r),\qquad \chi _{2} (t,r) := \frac{(T-t)^2}{r} \psi _{t} (t,r).
\end{align*}
Using the fact that $\psi$ is a solution to \eqref{cauchy}, we get
\begin{align*}
\partial _{t} \chi _{1} (t,r) = & - \frac{1}{T-t} \chi _{1} (t,r) + \frac{1}{T-t} \chi _{2} (t,r), \\
\partial _{t} \chi _{2} (t,r) = & - \frac{2}{T-t} \chi _{2} (t,r) + (T-t) \Delta ^{ \text{rad} }_{r,d} \chi _{1} (t,r) + \frac{2(T-t)}{r} \partial _{r} \chi _{1} (t,r) \\
& + (d-1) \frac{T-t}{r^2} \chi _{1} (t,r) - \frac{d-1}{2} (T-t)^2 \frac{ \sin \big( \frac{2r}{T-t} \chi _{1} (t,r) \big) }{r^3}.
\end{align*}
We introduce similarity coordinates
\begin{align*}
\mu: C_{T} \longrightarrow \mathcal{C} ,~~ (t,r) \longmapsto \mu (t, r) =(\tau,\rho) := \Big( \log \Big ( \frac{T}{T-t} \Big ), \frac{r}{T-t} \Big ),
\end{align*}
which map the backward light-cone $C_T$ to the cylinder $\mathcal{C}:=(0,+ \infty) \times [0,1]$.
By the chain rule, the derivatives transform according to
\begin{align*}
\partial _{t} = \frac{e^{\tau}}{T} (\partial _{\tau} + \rho \partial _{\rho}),~ \partial _{r} = \frac{e^{\tau}}{T} \partial _{\rho},~ \partial ^{2}_{r} = \frac{e^{2 \tau}}{T^{2}} \partial ^{2}_{\rho},~ \Delta ^{ \text{rad} }_{r,d} = \frac{e^{2 \tau}}{T^{2}} \Delta ^{ \text{rad} }_{\rho,d}.
\end{align*}
Finally, setting
\begin{align*}
\psi _{j} (\tau, \rho) := \chi _{j} (t(\tau,\rho),r(\tau,\rho) ) = \chi _{j} (T(1-e^{-\tau}), T \rho e^{-\tau}),
\end{align*}
for $j=1,2$, we obtain the system
\begin{align} \label{free}
& \begin{pmatrix}
\partial _{\tau} \psi _{1} (\tau,\rho) \\
\partial _{\tau} \psi _{2} (\tau,\rho)
\end{pmatrix}
=
\begin{pmatrix}
- \psi _{1} (\tau,\rho) + \psi _{2} (\tau,\rho) - \rho \partial _{\rho} \psi _{1} (\tau,\rho) \\
\Delta ^{ \text{rad} }_{\rho,d+2} \psi _{1} (\tau,\rho) - \rho \partial _{\rho} \psi _{2} (\tau,\rho) - 2 \psi _{2} (\tau,\rho)
\end{pmatrix} \\ \nonumber
& \quad \quad \quad \quad \quad \quad - \frac{d-1}{ 2 \rho ^3}
\begin{pmatrix}
0 \\
\sin ( 2 \rho \psi _{1} (\tau,\rho) ) -2 \rho \psi _{1} (\tau,\rho)
\end{pmatrix},
\end{align}
for $(\tau,\rho) \in \mathcal{C}$.
Note that the linear part is the free operator of the $(d+2)-$dimensional wave equation in similarity coordinates and the nonlinearity is perfectly smooth. Furthermore, the initial data transform according to
\begin{align*}
\begin{pmatrix}
\psi _{1} (0,\rho) \\
\psi _{2} (0,\rho)
\end{pmatrix}
= \frac{1}{\rho}
\begin{pmatrix}
f(T\rho) \\
T g (T\rho)
\end{pmatrix}
= \frac{1}{\rho}
\begin{pmatrix}
\psi^{T_{0}}(0,T\rho) \\
T\partial_0 \psi ^{T_{0}} (0,T\rho)
\end{pmatrix}+
\frac{1}{\rho}
\begin{pmatrix}
F(T\rho) \\
T G(T\rho)
\end{pmatrix},
\end{align*}
for all $\rho \in [0,1]$. Here, $T_{0}>0$ is a fixed parameter and
\begin{align*}
& \psi ^{T_{0}} (0,T\rho) = 2 \arctan \left( \frac{T }{T_{0}} \frac{\rho}{ \sqrt{d-2} } \right),\quad \rho \equiv \rho (t,r):=\frac{r}{T-t}, \\
& F:=f-\psi ^{T_{0}}(0,\cdot), \quad G:=g-\partial_0 \psi^{T_{0}}(0,\cdot).
\end{align*}
We emphasize that the only trace of the parameter $T$ is in the initial data.
\subsection{Perturbations of the rescaled blowup solution} We linearize around the rescaled blowup solution and use the initial value problem for $(\psi_{1},\psi_{2})^{T}$ to obtain an initial value problem for the perturbation as an abstract Cauchy problem in a Hilbert space. For notational convenience we set
\begin{align*}
\mathbf{ \Psi } (\tau) (\rho) :=
\begin{pmatrix}
\psi _{1} (\tau, \rho) \\
\psi _{2} (\tau, \rho)
\end{pmatrix} .
\end{align*}
The blowup solution is given by
\begin{align*}
\mathbf{ \Psi } ^{ \text{res} } (\tau) (\rho)
=
\begin{pmatrix}
\frac{T-t}{r} \psi ^{T} (t,r) \\
\frac{(T-t)^2}{r} \psi _{t}^{T} (t,r)
\end{pmatrix} \Bigg | _{
(t,r)=\mu ^{-1} (\tau,\rho)
} =
\begin{pmatrix}
\frac{1}{\rho} f_0 (\rho) \\
f_0'(\rho)
\end{pmatrix},
\end{align*}
i.e., it is static.
We linearize around $\mathbf{\Psi}^{\text{res}}$ by inserting the ansatz $\mathbf{ \Psi }= \mathbf{ \Psi } ^{ \text{res} } + \mathbf{ \Phi } $ into \eqref{free}.
For brevity we write
\begin{align*}
\eta (x):= \sin(2x) - 2x,\quad x \in \mathbb{R}
\end{align*}
and use Taylor's theorem to expand the nonlinearity around
$\frac{1}{\rho}f_0(\rho)$. We get
\begin{align*}
\sin \left( 2 \rho \psi _{1} \right) - 2 \rho \psi _{1} & = \eta \left( \rho \psi _{1} \right) = \eta \left( f_0 + \rho \phi _{1} \right) = \eta \left( f_0 \right) + \eta ^{\prime} \left( f_0 \right) \rho \phi _{1} + N(\rho \phi _{1}),
\end{align*}
where, by definition,
\begin{align*}
N (\rho \phi _{1} ) := \eta( f_0 + \rho \phi _{1}) - \eta ( f_0 ) - \eta ^{\prime} ( f_0 ) \rho \phi _{1}.
\end{align*}
We plug the ansatz and the Taylor expansion into Eq.~\eqref{free} which yields the abstract evolution equation
\begin{align} \label{Evolution}
\left\{
\begin{array}{ll}
\partial _{\tau} \mathbf{ \Phi } (\tau) = \widetilde{\mathbf{ L } } \big( \mathbf{ \Phi } (\tau) \big) + \mathbf{ N } \big( \mathbf{ \Phi }( \tau ) \big ), & \mbox{for } \tau \in (0,+\infty) \\
\mathbf{ \Phi } (0)= \mathbf{U}(\mathbf{v},T),
\end{array}
\right.
\end{align}
for the perturbation
\begin{align*}
\mathbf{ \Phi } (\tau) (\rho)=
\begin{pmatrix}
\phi _{1} (\tau, \rho) \\
\phi _{2} (\tau, \rho)
\end{pmatrix}
=
\begin{pmatrix}
\psi _{1} (\tau, \rho) -\frac{1}{\rho} f_0 (\rho) \\
\psi _{2} (\tau, \rho) - f_0' (\rho)
\end{pmatrix}
\end{align*}
where
\begin{align}
&\widetilde{ \mathbf{ L } } := \widetilde{ \mathbf{ L } }_{0} + \mathbf{ L ^{\prime} }, \label{1} \\
& \widetilde{ \mathbf{ L } }_{0} \mathbf u (\rho):=
\begin{pmatrix}
- \rho u_1'(\rho)-u_1(\rho)+ u_2(\rho) \\
\Delta _{\rho,d+2}^{ \text{rad} } u_1(\rho) - \rho u_2'(\rho) - 2 u_{2}(\rho)
\end{pmatrix}, \label{2} \\
& \mathbf{ L ^{\prime} } \mathbf u(\rho):=
\begin{pmatrix}
0 \\
- \frac{d-1}{2} \frac{\eta' ( f_0(\rho) ) }{\rho ^2} u_{1}(\rho),
\end{pmatrix}, \label{3} \\
& \mathbf{ N }(\mathbf u) (\rho):=
\begin{pmatrix}
0 \\
- \frac{d-1}{2} \frac{N( \rho u_{1}(\rho) )}{\rho ^3}
\end{pmatrix}, \label{4}
\end{align}
for $\mathbf u=(u_1,u_2)$ and
\begin{align*}
\eta'(f_0(\rho))=2 \cos (2f_0(\rho)) -2 = -16(d-2) \frac{\rho ^2}{ \left( \rho^2 +d-2 \right)^2 }.
\end{align*}
Furthermore, the initial data are given by
\begin{align}
\label{5}
\mathbf \Phi(0)(\rho)=\mathbf U(\mathbf v,T)(\rho)=
\left (\begin{array}{c}
\frac{1}{\rho}f_0(\frac{T}{T_0}\rho) \\
\frac{T^2}{T_0^2}f_0'(\frac{T}{T_0}\rho) \end{array} \right )
-\left ( \begin{array}{c}
\frac{1}{\rho}f_0(\rho) \\ f_0'(\rho) \end{array} \right )
+\mathbf V(\mathbf v,T)(\rho)
\end{align}
where
\[ \mathbf{ V } (\mathbf{v},T) (\rho):=
\begin{pmatrix}
\frac{1}{\rho} F (T \rho) \\
\frac{T}{\rho} G (T \rho)
\end{pmatrix},~ \mathbf{v} :=
\begin{pmatrix}
F \\
G
\end{pmatrix} .
\]
\subsection{Strong light-cone solutions}
To proceed, we need to define what it means to be a solution to the evolution problem $\eqref{Evolution}$. We introduce the Hilbert space
\begin{align*}
\mathcal{H} := H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} ) \times H_{\text{rad}}^{\frac{d+1}{2}} (\mathbb{B}^{d+2} ).
\end{align*}
Below we prove that the closure of the operator $\widetilde{\mathbf L}$, augmented with a suitable domain, generates a semigroup $\mathbf S(\tau)$ on $\mathcal H$.
This allows us to formulate $\eqref{Evolution}$ as an abstract integral equation via Duhamel's formula,
\begin{align} \label{Duhamel}
\Phi(\tau)= \mathbf{S}(\tau) \mathbf{U}(\mathbf{v},T)+ \int _{0}^{\tau} \mathbf{S}(\tau - s) \mathbf{N} \big( \Phi (s) \big) ds.
\end{align}
Eq.~\eqref{Duhamel} yields a natural notion of strong solutions in light-cones.
\begin{definition} \label{def}
We say that $\psi:C_{T} \longrightarrow \mathbb{R}$ is a solution to $\eqref{cauchy}$ if the corresponding $\Phi:[0,\infty) \longrightarrow \mathcal{H}$ belongs to $C \big( [0,\infty);\mathcal{H} \big)$ and satisfies $\eqref{Duhamel}$ for all $\tau \ge 0$.
\end{definition}
\section{Proof of the theorem}
\subsection{Notation} Throughout we denote by $\sigma(\mathbf{L}),~\sigma_{p}(\mathbf{L})$ and $\sigma_{e}(\mathbf{L})$ the spectrum, point spectrum, and essential spectrum, respectively, of a linear operator $\mathbf{L}$. Furthermore, we write $\mathbf{R}_{\mathbf{L}}(\lambda):=\left(\lambda - \mathbf{L} \right)^{-1}$, $\lambda \in \rho(\mathbf{L})$, for the resolvent operator where $\rho (\mathbf{L}):=\mathbb{C} \setminus \sigma(\mathbf{L})$ stands for the resolvent set. As usual, $a \lesssim b$ means $a \leq cb$ for an absolute, strictly positive constant $c$ which may change from line to line. Similarly, we write $a \simeq b$ if $a \lesssim b$ and $b \lesssim a$.
\subsection{Functional setting}
In the following we consider radial Sobolev functions $\hat{u} : \mathbb{B}_{R}^{d+2} \rightarrow \mathbb{C}$, that is, $\hat{u} (\xi)=u(|\xi|)$ for all $\xi \in \mathbb{B}_{R}^{d+2}$ where
$u:(0,R) \rightarrow \mathbb{C}$. In particular, we define
\begin{align*}
u \in
H_{\text{rad}}^{m} ( \mathbb{B}_{R}^{d+2} ) \iff \hat{u} \in H^{m} (\mathbb{B}_{R}^{d+2} ) := W^{m,2} (\mathbb{B}_{R}^{d+2} ).
\end{align*}
The function space $H_{\text{rad}}^{m} (\mathbb{B}_R^{d+2} )$ becomes a Banach space endowed with the norm
\begin{align*}
\| u \|_{{H^{m}_{\text{rad}}} ( \mathbb{B}_{R}^{d+2} )} = \| \hat{u} \|_{ {H}^{m} (\mathbb{B}_{R}^{d+2} )}.
\end{align*}
From now, we shall not distinguish between $u(|\cdot|)$ and $\hat{u}$. In addition, we introduce the Hilbert space
\begin{align} \label{H}
\mathcal{H} := H_{\text{rad}}^{m} (\mathbb{B}^{d+2} ) \times H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ),\quad m \equiv m_{d}:=\frac{d+3}{2}
\end{align}
associated with the induced norm
\begin{align*}
\| \mathbf{u} \|^{2} = \left \| (u_{1},u_{2}) \right \|^{2} := \| u_{1} \|_{H_{\text{rad}}^{m} (\mathbb{B}^{d+2} )}^{2} + \| u_{2} \|_{H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) }^{2}.
\end{align*}
\subsection{Well-posedness of the linearized problem}
We start with the study of the linearized problem and we convince ourselves that it is well-posed. Recall that the linear operator is given by $\eqref{1}$. To proceed, we follow \cite{DonSch16} and define the domain of the free part by
\begin{align*}
\mathcal{D} (\widetilde{ \mathbf{L} }_{0}) := \Big \{ \mathbf{u} \in C^{\infty}(0,1)^2 \cap \mathcal H: w_{2} \in C^{2}\left([0,1]\right),~w_{1} \in C^{3} \left([0,1] \right),~ w_{1}^{\prime \prime} (0)=0 \Big \},
\end{align*}
where, for all $\rho \in [0,1]$ and $j=1,2$,
\begin{align*}
w_{j} (\rho) := D_{d+2} u_{j} (\rho) := \Big ( \frac{1}{\rho} \frac{d}{d\rho} \Big )^{ \frac{d-1}{2} } \big( \rho ^{d} u_{j}(\rho) \big) = \sum _{n=0}^{ \frac{d-1}{2} } c _{n} \rho^{n+1} u_{j}^{(n)} (\rho),
\end{align*}
for some strictly positive constants $c_{n}~ (n=0,1,\dots,\frac{d-1}{2})$. Note that the density of $C^{\infty} ( \overline{ \mathbb{B}^{d+2} } )$ in $H^{m}(\mathbb{B}^{d+2})$ implies the density of
\begin{align*}
\big( C_{\text{even}}^{\infty} [0,1] \big)^2 := \Big \{ \mathbf{u} \in \big( C^{\infty} [0,1] \big)^2:~~\mathbf{u}^{(2k+1)}(0)=0,~~k=0,1,2,\dots \Big \} \subset \mathcal{D} ( \widetilde{ \mathbf{L} }_{0})
\end{align*}
in $\mathcal{H}$ which in turn proves the density of $\mathcal{D} ( \widetilde{ \mathbf{L} }_{0})$ in $\mathcal{H}$. In other words, $\overline{ \mathcal{D} ( \widetilde{ \mathbf{L} }_{0}) } = \mathcal{H}$ and $\widetilde{ \mathbf{L} }_{0}$ is densely defined.
\begin{prop} \label{growthestimatelinear}
The operator $\widetilde{ \mathbf{L} }_{0}: \mathcal{D}(\widetilde{ \mathbf{L} }_{0}) \subset \mathcal{H} \longrightarrow \mathcal{H} $ is closable and its closure $ \mathbf{L} _{0} : \mathcal{D}(\mathbf{L} _{0}) \subset \mathcal{H} \longrightarrow \mathcal{H} $ generates a strongly continuous one-parameter semigroup $( \mathbf{S}_{0}(\tau) )_{\tau \ge 0}$ of bounded operators on $\mathcal{H}$ satisfying the growth estimate
\begin{align} \label{growth}
\| \mathbf{S} _{0} (\tau) \| \leq M e^{-\tau}
\end{align}
for all $\tau\geq 0$ and some constant $M\geq 1$.
In addition, the operator $\mathbf{L}:= \mathbf{L}_{0} + \mathbf{L}^{\prime}: \mathcal{D}(\mathbf{L}) \subset \mathcal{H} \longrightarrow \mathcal{H},~\mathcal{D}(\mathbf{L}) =\mathcal{D}(\mathbf{L}_{0})$, is the generator of a strongly continuous semigroup $( \mathbf{S}(\tau) )_{\tau \ge 0}$ on $\mathcal H$
and $\mathbf L': \mathcal H\to \mathcal H$ is compact.
\end{prop}
\begin{proof}
The fact that $\widetilde{ \mathbf{L} }_{0}$ is closable and its closure generates a semigroup satisfying the growth estimate $\eqref{growth}$ follows from Proposition 4.9 in \cite{DonSch16} by replacing $d$ in \cite{DonSch16} with $d+2$ and setting $p=3$. It remains to apply the Bounded Perturbation Theorem to show that $\mathbf{L}:=\mathbf{L}_{0} + \mathbf{L}^{\prime}$ is the generator of a strongly continuous semigroup $( \mathbf{S}(\tau) )_{\tau \ge 0}$. In fact, we prove that $\mathbf{L}^{\prime}: \mathcal{H} \longrightarrow \mathcal{H}$, defined in $\eqref{3}$, is compact. We pick an arbitrary sequence $(\mathbf{u}_{n})_{n \in \mathbb{N}} \subseteq \mathcal{H}$ that is uniformly bounded. By Lemma 4.2 in \cite{DonSch16}, $(D_{d+2} u_{1,n} )_{n \in \mathbb{N}}$ is uniformly bounded in $H^{2} (0,1)$ and the compactness of the Sobolev embedding $H^{2}(0,1) \xhookrightarrow{} H^{1}(0,1)$ implies the existence of a subsequence, again denoted by $(D_{d+2} u_{1,n} )_{n \in \mathbb{N}}$, which is Cauchy in $H^{1}(0,1)$. Hence, for any $n,m \in \mathbb{N}$ sufficiently large, we get
\begin{align*}
\| \mathbf{L}^{\prime} \mathbf{u}_{n} - \mathbf{L}^{\prime}
\mathbf{u}_{m} \| & \lesssim \left \| \frac{ \eta'
\circ f_0 }{|\cdot|^2} \right \|_{W^{1,\infty}(0,1)} \| D_{d+2} u_{1,n} - D_{d+2} u_{1,m} \|_{H^{1}(0,1)} \\
& \simeq \left \| \frac{ 1}{ (|\cdot|^{2} + d-2)^2 }\right \|_{W^{1,\infty}(0,1)} \|D_{d+2} u_{1,n} - D_{d+2} u_{1,m} \|_{H^{1}(0,1)} \\
& \simeq \| D_{d+2} u_{1,n} - D_{d+2} u_{1,m} \|_{H^{1}(0,1)},
\end{align*}
which shows that $(\mathbf{L}^{\prime} \mathbf{u}_{n})_{n \in \mathbb{N}}$ is Cauchy in $\mathcal{H}$. This proves that $\mathbf{L}^{\prime}$ is compact.
\end{proof}
\subsection{The spectrum of the free operator} \label{O1}
We can use the previous decay estimate for the semigroup $( \mathbf{S}_{0}(\tau) )_{\tau \ge 0}$ to locate the spectrum of the free operator $\mathbf L_0$. Indeed, by \cite{EngNag00}, p.~55, Theorem 1.10, we immediately infer
\begin{equation}
\label{spectrumLo}
\sigma(\mathbf L_0)\subseteq \{\lambda\in \mathbb C: \text{Re}\lambda\leq -1\}.
\end{equation}
\subsection{The spectrum of the full linear operator} Next, we need to derive a suitable growth estimate for the semigroup $\mathbf{S}(\tau)$ and therefore turn our attention to the spectrum of the operator $\mathbf{L}$. To begin with, we consider the point spectrum.
\begin{prop} \label{OhaioProp}
We have
\begin{align*}
\sigma_p (\mathbf{L}) \subseteq \{ \lambda \in \mathbb{C}:~~\mathrm{Re} \lambda <0 \} \cup \{1\}.
\end{align*}
\end{prop}
\begin{proof}
We argue by contradiction and assume there exists a $\lambda \in \sigma_p(\mathbf{L}) \setminus \{ 1\}$ with $\mathrm{Re}\lambda \geq 0$. The latter means that there exists an element $\mathbf{u}=(u_{1},u_{2}) \in \mathcal{D} (\mathbf{L}) \setminus \{ 0 \}$ such that $\mathbf{u} \in$ $\ker(\lambda-\mathbf{L})$. A straightforward calculation shows that the spectral equation $(\lambda-\mathbf{L}) \mathbf{u} =0$ implies
\begin{small}
\begin{align*}
\big (1-\rho ^2 \big) u_{1}^{\prime \prime} (\rho) + \Bigg( \frac{d+1}{\rho} - 2(\lambda +2) \rho \Bigg)u_{1}^{\prime} (\rho)- \Bigg( (\lambda + 1) (\lambda +2)+ \frac{d-1}{2} V(\rho) \Bigg ) u_{1} (\rho) =0,
\end{align*}
\end{small}
for $\rho \in (0,1)$, where
\begin{align*}
V(\rho):=\frac{\eta'(f_0(\rho))}{\rho ^2} = \frac{-16(d-2)}{(\rho^2+d-2 )^2}.
\end{align*}
Since $\mathbf u\in \mathcal H$, we see that $u_{1}$ must lie in $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})$. To proceed, we set $v_{1}(\rho):=\rho u_{1}(\rho)$. A straightforward computation implies that $v_{1}$ solves the second order ordinary differential equation
\begin{align}
\label{eq:specode}
\big (1-\rho ^2 \big) v_{1}^{\prime \prime} (\rho) + \Bigg( \frac{d-1}{\rho} - 2(\lambda +1) \rho \Bigg)v_{1}^{\prime} (\rho)- \Bigg( \lambda (\lambda +1)+ \frac{d-1}{2}\hat{V}(\rho) \Bigg ) v_{1} (\rho) =0,
\end{align}
for $\rho \in (0,1)$, where
\begin{align*}
\hat{V}(\rho):= 2\frac{ \rho ^4 -6(d-2)\rho ^2 +(d-2)^2 }{\rho ^2 ( \rho^2 +d-2 )^2 }.
\end{align*}
We remark that this is the spectral equation studied in \cite{CosDonXia16, CosDonGlo16}.
Since all coefficients in \eqref{eq:specode} are smooth functions in $(0,1)$, we immediately get the a priori regularity $v_{1} \in C^{\infty}(0,1)$.
We claim that $v_1\in C^\infty[0,1]$. To prove this, we employ Frobenius' method. The point $\rho =0$ is a regular singularity with Frobenius indices $s_{1}=1$ and $s_{2}=-(d-1)$. Therefore, by Frobenius theory, there exists a solution of the form
\begin{align*}
v^{1}_{1} (\rho)=\rho \sum _{i=0}^{\infty} x_{i} \rho ^{i} = \sum _{i=0}^{\infty} x_{i} \rho ^{i+1},
\end{align*}
which is analytic locally around $\rho=0$. Moreover, since $s_{1}-s_{2}=d \in \mathbb{N}_{\text{odd}}$, there exists a second linearly independent solution of the form
\begin{align*}
v^{2}_{1} (\rho) &=C \log(\rho) v^{1}_{1} (\rho) + \rho ^{-(d-1)} \sum _{i=0}^{\infty} y_{i} \rho ^{i}
\end{align*}
for some constant $C\in \mathbb C$ and $y_0=1$.
However, $v^{2}_{1}(\rho)/\rho$ does not lie in the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ due to the strong singularity in the second term, no matter the value of the constant $C$. Consequently, $v_1$ must be a multiple of $v_1^1$ and we infer $v_1\in C^\infty[0,1)$. Similarly, the point $\rho =1$ is a regular singularity with Frobenius indices
$s_{1}=0$ and $s_{2}=\frac{d-1}{2}-\lambda$.
Now we need to distinguish different cases. If $\frac{d-1}{2} -\lambda \notin \mathbb{Z}$, we have two linearly independent solutions of the form
\begin{align*}
& v_{1} ^{1}(\rho) =\sum_{i=0}^\infty x_i (1-\rho)^i, \\
& v_{1} ^{2}(\rho)=(1-\rho)^{\frac{d-1}{2}-\lambda}\sum_{i=0}^\infty y_i (1-\rho)^i
\end{align*}
with $x_0=y_0=1$. The solution $v_{1} ^{2}(\rho)/\rho$ does not belong to the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ and thus, $v_1\in C^\infty[0,1]$. In the case
$\frac{d-1}{2} - \lambda:=k \in \mathbb{N}_{0}$, we have two fundamental solutions of the form
\begin{align*}
v_{1} ^{1}(\rho)&=(1-\rho)^{k}\sum_{i=0}^\infty x_i(1-\rho)^i,\qquad x_0=1 \\
v_{1} ^{2}(\rho)&= \sum_{i=0}^\infty y_i (1-\rho)^i+C\log (1-\rho)v_1^1(\rho),\qquad y_0=1
\end{align*}
near $\rho= 1$. By assumption, $\mathrm{Re}\lambda\geq 0$ and thus, $k \leq \frac{d-1}{2}$. Hence, $v_1^2(\rho)/\rho$ does not lie in the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ unless $C=0$ and we conclude $v_1\in C^\infty[0,1]$.
Finally, if $\frac{d-1}{2}-\lambda=:-k$ is a negative integer, the fundamental system around $\rho=1$ has the form
\begin{align*}
v_1^1(\rho)&=\sum_{i=0}^\infty x_i(1-\rho)^i \\
v_1^2(\rho)&=C\log(1-\rho)v_1^1(\rho)+(1-\rho)^{-k}\sum_{i=0}^\infty y_i (1-\rho)^i
\end{align*}
with $x_0=y_0=1$.
Again, $v_1^2(\rho)/\rho$ does not belong to $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ and we infer $v_1\in C^\infty[0,1]$ also in this case.
In summary, we have found a nontrivial solution $v_1\in C^\infty[0,1]$ to Eq.~\eqref{eq:specode} with $\mathrm{Re}\lambda \geq 0$, $\lambda\not=1$, but this contradicts \cite{CosDonXia16, CosDonGlo16}.
\end{proof}
The fact that $\mathbf L'$ is compact implies that the result on the point spectrum from Proposition \ref{OhaioProp} is already sufficient to obtain the same information on the full spectrum.
\begin{cor}
\label{cor:spec}
We have
\begin{align*}
\sigma (\mathbf{L}) \subseteq \{ \lambda \in \mathbb{C}:~~\mathrm{Re} \lambda <0 \} \cup \{1\}.
\end{align*}
\end{cor}
\begin{proof}
Suppose there exists a $\lambda\in \sigma(\mathbf L)\setminus \{1\}$ with $\mathrm{Re}\lambda\geq 0$.
Then $\lambda\notin\sigma(\mathbf L_0)$ and thus, $\mathbf R_{\mathbf L_0}(\lambda)$ exists.
From the identity $\lambda-\mathbf L=[1-\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)](\lambda-\mathbf L_0)$ we see that $1\in \sigma(\mathbf L'\mathbf R_{\mathbf L_0}(\lambda))$. Since $\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)$ is compact, it follows that $1\in \sigma_p(\mathbf L'\mathbf R_{\mathbf L_0}(\lambda))$ and thus, there exists a nontrivial $\mathbf f \in \mathcal H$ such that $[1-\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)]\mathbf f=0$.
Consequently, $\mathbf u:=\mathbf R_{\mathbf L_0}(\lambda)\mathbf f\not= 0$
satisfies $(\lambda-\mathbf L)\mathbf u=0$ and thus, $\lambda\in \sigma_p(\mathbf L)$.
This contradicts Proposition \ref{OhaioProp}.
\end{proof}
Next, we provide a uniform bound on the resolvent.
To this end, we define
\begin{align*}
\Omega _{\epsilon,R} := \{ \lambda \in \mathbb{C}:~~~\text{Re}\lambda \geq -1+\epsilon, |\lambda|\geq R \}
\end{align*}
for $\epsilon, R >0$.
\begin{prop} \label{O2}
Let $\epsilon >0$. Then there exist constants $R_{\epsilon}, C_{\epsilon}>0$ such that the resolvent $\mathbf{R}_{\mathbf{L}}$ exists on $\Omega _{\epsilon,R_\epsilon}$ and satisfies
\begin{align*}
\| \mathbf{R} _{\mathbf{L} } (\lambda)\| \leq C_{\epsilon}
\end{align*}
for all $\lambda\in \Omega_{\epsilon,R_\epsilon}$.
\end{prop}
\begin{proof}
Fix $\epsilon>0$ and take $\lambda\in \Omega_{\epsilon,R}$ for an arbitrary $R>0$.
Then $\lambda\in \rho(\mathbf L_0)$ and the identity $(\lambda-\mathbf L)=[1-\mathbf L' \mathbf R_{\mathbf L_0}(\lambda)](\lambda-\mathbf L_0)$ shows that
$\mathbf R_{\mathbf L}(\lambda)$ exists if and only if
$1-\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)$ is invertible.
By a Neumann series argument this is the case if $\|\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)\|<1$.
To prove smallness of $\mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L}_{0}} (\lambda)$, we recall the definition of $\mathbf L'$, Eq.~\eqref{3},
\begin{align*}
\mathbf{ L ^{\prime} } \mathbf{u} (\rho)=
\begin{pmatrix}
0 \\
- \frac{d-1}{2} V(\rho )u _{1} (\rho)
\end{pmatrix},
\quad V(\rho)=\frac{\eta'(f_0(\rho))}{\rho ^2} = \frac{-16(d-2)}{(\rho^2+d-2 )^2}.
\end{align*}
Let $\mathbf u=\mathbf R_{\mathbf L_0}(\lambda)\mathbf f$ or, equivalently,
$(\lambda - \mathbf{L}_{0}) \mathbf{u} =\mathbf{f}$. The latter equation implies
\begin{align*}
(\lambda +1) u_{1} (\rho ) =u_{2} (\rho)- \rho u_{1}^{\prime} (\rho) + f_{1} (\rho).
\end{align*}
Now we use Lemma 4.1 from \cite{DonSch16} and $\|V^{(k)}\|_{L^\infty(0,1)}\lesssim 1$ for all $k\in \{0,1,\dots,m-1\}$ to obtain
\begin{align*}
|\lambda +1| \| \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) \mathbf{f} \| & = |\lambda +1| \| \mathbf{L}^{\prime} \mathbf{u} \| \simeq \big \| V \big( u_{2} - (\cdot) u_{1}^{\prime} +f_{1} \big) \big \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } \\
& \lesssim \| u_{2} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } + \| (\cdot) u_{1}^{\prime} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } + \| f_{1} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } \\
& \lesssim \| u_{2} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } + \| u_{1} \|_{ H_{\text{rad}}^{m} (\mathbb{B}^{d+2} ) } + \| f_{1} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } \\
& \simeq \| \mathbf{u} \| + \| \mathbf{f} \| \lesssim \Big( \frac{1}{\text{Re}\lambda +1} + 1 \Big) \| \mathbf{f} \| \\
&\lesssim \|\mathbf f\|,
\end{align*}
where we have used the bound
\[ \|\mathbf u\|=\|\mathbf R_{\mathbf L_0}(\lambda)\mathbf f\|\leq \frac{1}{\mathrm {Re}\lambda+1}\|\mathbf f\| \]
which follows from semigroup theory, see \cite{EngNag00}, p.~55, Theorem 1.10.
In other words,
\begin{align*}
\| \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) \| \lesssim \frac{1}{ |\lambda +1| } \leq \frac{1}{|\lambda| -1} \leq \frac{1}{R-1}
\end{align*}
and by choosing $R$ sufficiently large, we can achieve the desired $\|\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)\|<1$.
As a consequence, $[1- \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) ]^{-1}$
exists and we obtain the bound
\begin{align*}
\| \mathbf{R}_{\mathbf{L}} (\lambda) \| & = \| \mathbf{R}_{\mathbf{L}_{0}} (\lambda) [1- \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) ]^{-1} \| \\
& \leq \| \mathbf{R}_{\mathbf{L}_{0}} (\lambda) \| \| [1- \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) ]^{-1} \| \\
& \leq \| \mathbf{R}_{\mathbf{L}_{0}} (\lambda) \| \sum _{i=0}^{\infty} \| \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) \|^{i} \\
& \leq C_\epsilon.
\end{align*}
\end{proof}
\subsection{The eigenspace of the isolated eigenvalue} In this section, we convince ourselves that the eigenspace of the isolated eigenvalue $\lambda=1$ for the full linear operator $\mathbf{L}$ is spanned by
\begin{align} \label{g}
\mathbf{g} (\rho):=
\begin{pmatrix}
g_{1} (\rho) \\
g_{2} (\rho)
\end{pmatrix}
=
\begin{pmatrix}
\frac{1}{\rho ^2 +d-2} \\
\frac{2(d-2)}{(\rho ^2 + d-2)^2}
\end{pmatrix},
~~\rho \in [0,1].
\end{align}
Consequently, we are looking for all $\mathbf{u}=(u_{1},u_{2}) \in \mathcal{D} (\mathbf{L}) \setminus \{ 0 \}$ such that $\mathbf{u} \in \ker(1-\mathbf{L})$. A straightforward calculation shows that the spectral equation $(1-\mathbf{L}) \mathbf{u} =0$ is equivalent to the following system of ordinary differential equations,
\begin{align*}
\begin{cases}
u_{2} (\rho)= \rho u_{1}^{\prime} (\rho)+ 2 u_{1} (\rho), & \\
\big (1-\rho ^2 \big ) u_{1}^{\prime \prime} (\rho) + \Big( \frac{d+1}{\rho} - 6\rho \Big)u_{1}^{\prime} (\rho)- \Big( 6+ \frac{d-1}{2} \frac{\eta'(f_0(\rho))}{\rho ^2} \Big ) u_{1} (\rho) =0, \
\end{cases}
\end{align*}
for $\rho \in (0,1)$. One can verify that a fundamental system of the second equation is given by
\begin{align*}
\Big \{ \frac{1}{\rho ^2 + d-2},~~\frac{Q_{d-1}(\rho)}{\rho ^d (\rho ^2 + d-2)} \Big \}
\end{align*}
where $Q_{d-1}$ is a polynomial of degree $d-1$ with non-vanishing constant term. We can write the general solution for the second equation as
\begin{align*}
u_{1} (\rho) = C_{1} \frac{1}{\rho ^2 + d-2} + C_{2} \frac{Q_{d-1}(\rho)}{\rho ^d (\rho ^2 + d-2)}.
\end{align*}
We must ensure that $\mathbf{u} \in \mathcal{D}( \mathbf{L})$ which in particular implies that $u_{1}$ must lie in the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$. This requirement yields $C_{2}=0$ which in turn gives $\mathbf u \in \langle \mathbf{g} \rangle$. In conclusion,
\begin{align} \label{ker}
\text{ker}(1-\mathbf{L}) = \langle \mathbf{g} \rangle,
\end{align}
as initially claimed.
\subsection{Time evolution for the linearized problem}
We now focus on the time evolution for the linearized problem \eqref{Evolution}. Due to the presence of the eigenvalue $\lambda =1$, there exists a one dimensional subspace $\langle \mathbf{g} \rangle$ of initial data for which the solution grows exponentially in time. We call this subspace the unstable space. On the other hand, initial data from the stable subspace lead to solutions that decay exponentially in time. As we will show now, this time evolution estimates can be established using semigroup theory together with the previous results on the spectrum of the linear operators $\mathbf{L}_{0}$ and $\mathbf{L}$. To make this rigorous, we follow \cite{DonSch16} and use the fact that the unstable eigenvalue $\lambda=1$ is isolated to introduce a (non-orthogonal) projection $\mathbf{P}$. This projection decomposes the Hilbert space of initial data $\mathcal{H}$ into the stable and the unstable space.
Most importantly, we must ensure that $\langle \mathbf{g}\rangle$ is the only unstable direction in $\mathcal{H}$. This is the key statement of the following proposition and it is equivalent to the fact that
the algebraic multiplicity of the isolated eigenvalue $\lambda =1$,
\begin{align*}
m_{a} (\lambda =1):=\mathrm{rank}\, \mathbf{P} =\dim\mathrm{rg}\, \mathbf{P},
\end{align*}
is equal to one. We denote by $\mathcal{B}(\mathcal{H})$ the set of bounded operators from $\mathcal{H}$ to itself and prove the following result.
\begin{prop} \label{projection}
There exists a projection
\begin{align*}
\mathbf{P} \in \mathcal{B} (\mathcal{H}),\quad \mathbf{P}: \mathcal{H} \longrightarrow \langle \mathbf{g} \rangle,
\end{align*}
which commutes with the semigroup $\big( \mathbf{S}(\tau) \big) _{\tau \ge 0}$. In addition, we have
\begin{equation}
\mathbf{S}(\tau) \mathbf{P} \mathbf{f} = e ^{\tau} \mathbf{P} \mathbf {f}, \label{semigroup1}
\end{equation}
and there exist constants $C, \epsilon>0$ such that
\begin{equation}
\| (1-\mathbf{P}) \mathbf{S}(\tau) \mathbf{f} \| \leq C e^{-\epsilon \tau } \| (1- \mathbf{P}) \mathbf{f} \|,\label{semigroup2}
\end{equation}
for all $\mathbf {f} \in \mathcal{H}$ and $\tau \geq 0$.
\end{prop}
\begin{proof}
We argue along the lines of \cite{DonSch16}. Since the eigenvalue $\lambda =1$ is isolated, we can define the spectral projection
\begin{align*}
\mathbf{P}: \mathcal{H} \longrightarrow \mathcal{H}, \quad \mathbf{P} := \frac{1}{2 \pi i} \int _{\gamma} \mathbf{R}_{\mathbf{L}} (\mu) d \mu,
\end{align*}
where $\gamma : [0,2\pi] \longrightarrow \mathbb{C}$ is a positively orientated circle around $\lambda =1$ with radius so small that $\gamma \big( [0,2 \pi] \big) \subseteq \rho (\mathbf{L})$, see e.g.~\cite{Kat95}. The projection $\mathbf P$ commutes with the operator $\mathbf{L}$ and thus with the semigroup $\mathbf S(\tau)$. Moreover, $\mathbf{P}$ decomposes the Hilbert space as $\mathcal{H} =\mathcal M \oplus \mathcal N$, where $\mathcal M:=\rg \mathbf P$ and $\mathcal N:=\rg(1-\mathbf P)=\ker \mathbf{P}$. Most importantly, the operator $\mathbf{L}$ is decomposed accordingly into the parts $\mathbf{L}_{\mathcal M}$ and $\mathbf{L}_{\mathcal N}$ on $\mathcal M$ and $\mathcal N$, respectively. The spectra of these operators are given by
\begin{align} \label{spectrum}
\sigma \left( \mathbf L_{\mathcal N} \right ) = \sigma (\mathbf{L}) \setminus \{1\},\qquad \sigma \left( \mathbf{L}_{\mathcal M} \right ) = \{1\}.
\end{align}
We refer the reader to \cite{Kat95} for these standard results.
To proceed, we break down the proof into the following steps: \\ \\
Step 1: We prove that $\rank\mathbf{P}:=\dim\rg\mathbf{P}<+\infty$. We argue by contradiction and assume that $\rank\mathbf{P}=+\infty$. Using \cite{Kat95}, p.~239, Theorem 5.28, the fact that $\mathbf{L}^{\prime}$ is compact (see Proposition $\ref{growthestimatelinear}$), and the fact that the essential spectrum is stable under compact perturbations (\cite{Kat95}, p.~244, Theorem 5.35), we obtain
\begin{align*}
\mathrm{rank}\,\mathbf{P} = +\infty \Longrightarrow 1 \in \sigma _{e} (\mathbf{L}) = \sigma _{e} (\mathbf{L} -\mathbf{L}^{\prime})=\sigma _{e}(\mathbf{L}_{0}) \subseteq \sigma (\mathbf{L}_{0}).
\end{align*}
This contradicts \eqref{spectrumLo}.
\\ \\
Step 2: We prove that $\langle \mathbf{g} \rangle=\mathrm{rg}\,\mathbf{P}$. It suffices to show $\mathrm{rg}\,\mathbf{P} \subseteq \langle \mathbf{g} \rangle$ since the reverse inclusion follows from the abstract theory. From Step 1, the operator $1-\mathbf{L}_{\mathcal M}$ acts on the finite-dimensional Hilbert space $\mathcal M=\rg \mathbf P$ and, from $\eqref{spectrum}$, $\lambda =0$ is its only spectral point. Hence, $1-\mathbf{L}_{\mathcal M}$ is nilpotent, i.e., there exists a $k\in \mathbb N$ such that
\begin{align*}
\big( 1-\mathbf{L}_{\mathcal M} \big)^{k} \mathbf{u}= 0
\end{align*}
for all $\mathbf{u} \in \mathrm{rg}\, \mathbf{P}$
and we assume $k$ to be minimal.
Recall $\eqref{ker}$ to see that the claim follows immediately if $k=1$. We proceed by contradiction and assume that $k\geq 2$. Then, there exists a nontrivial function $\mathbf{u} \in \rg \mathbf{P} \subseteq \mathcal{D}( \mathbf{L})$ such that
$(1-\mathbf L_{\mathcal M})\mathbf u$ is nonzero and belongs to $\ker(1-\mathbf L_{\mathcal M})\subseteq \ker(1-\mathbf L)=\langle\mathbf g\rangle$.
This means that $\mathbf{ u } \in \rg\mathbf{P} \subseteq \mathcal{D} (\mathbf{L})$ satisfies $(1- \mathbf{ L }) \mathbf{ u } = \alpha \mathbf{ g }$, for some $\alpha \in \mathbb{ C }\setminus \{0\}$. Without loss of generality we set $\alpha=-1$ and a straightforward computation shows that the first component of $\mathbf u$ solves the second order differential equation
\begin{align*}
\left(1-\rho ^2\right) u_{1}^{\prime \prime} (\rho) + \left( \frac{d+1}{\rho} -6 \rho \right ) u_{1} ^{\prime} (\rho) - \left ( 6 + \frac{d-1}{2} \frac{\eta'(f_0(\rho) ) }{\rho ^2} \right ) u_{1} (\rho) = G(\rho),
\end{align*}
for $\rho \in (0,1)$, where
\begin{align*}
G(\rho):= \frac{\rho ^2 +5(d-2)}{ (\rho ^2 + d-2)^2},~~\rho \in [0,1].
\end{align*}
In order to find the general solution to this equation, recall $\eqref{g}$ to see that
\begin{align*}
\hat{u}_{1} (\rho):= g_{1} (\rho) = \frac{1}{\rho ^2 +d-2},~~~\rho \in (0,1)
\end{align*}
is a particular solution to the homogeneous equation
\begin{align*}
\left(1-\rho ^2 \right) u_{1}^{\prime \prime} (\rho) + \left( \frac{d+1}{\rho} -6 \rho \right ) u_{1} ^{\prime} (\rho) - \left ( 6 + \frac{d-1}{2} \frac{\eta'(f_0(\rho) ) }{\rho ^2} \right ) u_{1} (\rho) = 0.
\end{align*}
To find another linearly independent solution, we use the Wronskian
\begin{align*}
\mathcal{W} (\rho) := (1-\rho ^2)^{\frac{d-5}{2} } \rho ^{-d-1}
\end{align*}
to obtain
\begin{align*}
\hat{u}_{2} (\rho) := \hat{u}_{1} (\rho) \int _{\rho _1}^{\rho} (1- x ^2)^{\frac{d-5}{2}} x^{-d-1} (x^2 + d-2)^2 dx,
\end{align*}
for some constant $\rho_{1} \in (0,1)$ and for all $\rho \in (0,1)$.
Note that we have the expansion
\[ \hat u_2(\rho)=\rho^{-d}\sum_{j=0}^\infty a_j\rho^j,\quad a_0\not= 0 \]
near $\rho=0$. Furthermore, if $d\geq 5$, $\hat u_2\in C^\infty(0,1]$ and we choose $\rho_1=1$ which yields the expansion
\[ \hat u_2(\rho)=(1-\rho)^{\frac{d-3}{2}}\sum_{j=0}^\infty b_j (1-\rho)^j,\qquad b_0\not= 0 \]
near $\rho=1$.
For $d=3$, we set $\rho_1=\frac12$ and the expansion of $\hat u_2$ near $\rho=1$ contains a term $\log(1-\rho)$.
We invoke the variation of constants formula to see that $u_1$ can be expressed as
\begin{align*}
u_{1} (\rho) & = c_{1} \hat{u}_{1} (\rho) + c_{2} \hat{u}_{2} (\rho) \\
& + \hat{u}_{2} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2} } } dy - \hat{u}_{1} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{2}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2} } } dy,
\end{align*}
for some constants $c_{1}, c_{2} \in \mathbb{C}$ and for all $\rho \in (0,1)$.
The fact that $u_1\in H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})$ implies
$c_2=0$ and we are left with
\begin{equation}
\label{eq:expru1}
u_{1} (\rho) = c_{1} \hat{u}_{1} (\rho)
+ \hat{u}_{2} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2} } } dy - \hat{u}_{1} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{2}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2} } } dy.
\end{equation}
If $d=3$, $\hat u_2(\rho)\simeq \log(1-\rho)$ near $\rho=1$ and thus, the last term in Eq.~\eqref{eq:expru1} stays bounded as $\rho\to 1-$ whereas the second term diverges unless
\[ \int_0^1 \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2} } } dy=0, \]
which, however, is impossible since the integrand is strictly positive on $(0,1)$.
This contradicts $u_1\in H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})$ and we arrive at the desired $k=1$.
Next, we focus on $d\geq 5$, where the last term in Eq.~\eqref{eq:expru1} is smooth on $[0,1]$. To analyze the second term, we set
\begin{align}
\label{def:Id}
\mathcal{I} _{d}(\rho) := \hat{u}_{2} (\rho) \int _{0}^{\rho} \frac{ F_{d}(y) }{(1-y)^{ \frac{d-3}{2} } } dy, \quad \text{~~~~~} ~ F_{d}(y):= \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1+y)^{\frac{d-3}{2}}}=\frac{y^{d+1}(y^2+5(d-2))}{(1+y)^{\frac{d-3}{2}}(y^2+d-2)^3}.
\end{align}
Note that $F_{5}(1)\not= 0$ and thus, the expansion of $\mathcal I_5(\rho)$ near $\rho=1$ contains a term of the form $(1-\rho)\log(1-\rho)$.
Consequently, $\mathcal I_5''\notin L^2(\frac12,1)$ and this is a contradiction to $u_1\in H^4_{\mathrm{rad}}(\mathbb B^7)$.
The general case is postponed to the appendix (Proposition \ref{prop:Id}) where it is shown that the function $\mathcal I_d$ is not analytic at $\rho=1$. This implies that the expansion of $\mathcal{I}_d(\rho)$ near $\rho =1$ contains a term $(1-\rho)^{\frac{d-3}{2}}\log(1-\rho)$ which again contradicts $u_1\in H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})$.\\ \\
Step 3: Finally, we prove the estimates $\eqref{semigroup1}$ and $\eqref{semigroup2}$ for the semigroup. First, note that $\eqref{semigroup1}$ follows immediately from the facts that $\lambda =1$ is an eigenvalue of $\mathbf{L}$ with eigenfunction $\mathbf{g}$ and $\rg\mathbf{P}=\langle \mathbf{g} \rangle$.
Furthermore, from Corollary \ref{cor:spec} and Proposition \ref{O2} we infer the existence of $C,\epsilon>0$ such that
\[ \|\mathbf R_{\mathbf L}(\lambda)(1-\mathbf P)\|\leq C \]
for all $\lambda \in \mathbb C$ with $\mathrm{Re}\lambda\geq -2\epsilon$.
Consequently, the Gearhart-Pr\"uss Theorem, see \cite{EngNag00}, p.~302, Theorem 1.11, yields the bound \eqref{semigroup2}.
\end{proof}
\subsection{Estimates for the nonlinearity}
The aim of this section is to establish a Lipschitz-type estimate for the nonlinearity. Recall that the nonlinear term in $\eqref{Evolution}$ is given by
\begin{align*}
\mathbf{ N } ( \mathbf{u} ) (\rho) =
\begin{pmatrix}
0 \\
\hat N (\rho, u_{1} (\rho))
\end{pmatrix}
:=
\begin{pmatrix}
0 \\
- \frac{d-1}{2} \frac{N( \rho u _{1} (\rho) )}{\rho ^3}
\end{pmatrix}.
\end{align*}
To begin with, we claim that
\begin{align*}
&\hat N(\rho,u_1(\rho)) \\
&=4 (d-1) u_{1}^{2} (\rho) \int _{0}^{1} \int _{0}^{1}\int _{0}^{1} \cos \left( 2z \left( f_0(\rho) +xy \rho u_{1} (\rho) \right) \right) \left( \frac{ f_0 (\rho)}{\rho} + xy u_{1} (\rho)\right) x dz dy dx. \\
\end{align*}
To see this, we use the fundamental theorem of calculus and the fact that $\eta ^{\prime \prime} (0)=0$ to write
\begin{align*}
N( \rho u _{1} (\rho) ) & = \eta( f_0 (\rho) + \rho u _{1} (\rho) ) - \eta ( f_0 (\rho)) - \eta ^{\prime} ( f_0 (\rho)) \rho u _{1} (\rho) \\
&= \int _{f_0 (\rho)}^{f_0(\rho) + \rho u_{1} (\rho)} \eta ^{\prime} (s) ds - \eta ^{\prime} ( f_0 (\rho)) \rho u _{1} (\rho) \\
&=\rho u_{1} (\rho) \int _{0}^{1} \eta ^{\prime} (f_0 (\rho) + x \rho u_{1} (\rho)) dx - \eta ^{\prime} ( f_0 (\rho)) \rho u _{1} (\rho) \\
&=\rho u_{1} (\rho) \int _{0}^{1}\left( \eta ^{\prime} (f_0 (\rho) + x \rho u_{1} (\rho)) - \eta ^{\prime} ( f_0(\rho)) \right) dx \\
&=\rho u_{1} (\rho) \int _{0}^{1}\left( \int _{f_0(\rho)}^{f_0 (\rho) + x\rho u_{1} (\rho)} \eta ^{\prime \prime} (s) ds \right) dx \\
&=\rho^{2} u_{1}^{2} (\rho) \int _{0}^{1}x \int _{0}^{1} \eta ^{\prime \prime} (f_0 (\rho) + xy \rho u_{1} (\rho)) dy dx \\
&=\rho^{2} u_{1}^{2} (\rho) \int _{0}^{1}x \int _{0}^{1} \int _{0}^{f_0(\rho) + xy \rho u_{1}(\rho)} \eta ^{\prime \prime \prime } (s) ds dy dx \\
&=\rho^{2} u_{1}^{2} (\rho) \int _{0}^{1}x \int _{0}^{1} \int _{0}^{1} \eta ^{\prime \prime \prime } \left( (f_0(\rho) +xy \rho u_{1} (\rho))z \right) \left( f_0 (\rho) + xy \rho u_{1} (\rho)\right) dz dy dx \\
&=\rho^{3} u_{1}^{2} (\rho) \int _{0}^{1} x \int _{0}^{1} \int _{0}^{1} \eta ^{\prime \prime \prime } \left( (f_0(\rho) +xy \rho u_{1} (\rho))z \right) \left(\frac{ f_0 (\rho)}{\rho} + xy u_{1} (\rho)\right) dz dy dx. \\
\end{align*}
For later purposes, we note that the function
\begin{align*}
\hat N (\rho, \zeta)=4 (d-1) \zeta ^{2} \int _{0}^{1} \int _{0}^{1} \int _{0}^{1} \cos \left( 2z \left( f_0(\rho) +xy \rho \zeta \right) \right) \left( \frac{ f_0 (\rho)}{\rho} + xy \zeta \right) x dz dy dx,
\end{align*}
defined for all $(\rho, \zeta) \in [0,1] \times \mathbb{R},$ is perfectly smooth in both variables since
\begin{align*}
\frac{f_0 (\rho)}{\rho} = \frac{2}{\rho} \arctan \left ( \frac{\rho}{\sqrt{d-2}} \right)
\end{align*}
is smooth at $\rho=0$.
Moreover, we define
\begin{align} \label{DefM}
M (\rho, \zeta) := \partial _{\zeta} \hat N (\rho, \zeta) = 4 (d-1) \left ( A(\rho, \zeta) + B(\rho, \zeta) +C(\rho, \zeta) +D(\rho, \zeta) \right),
\end{align}
where
\begin{align*}
& A(\rho,\zeta):= 2 \frac{f_0(\rho)}{\rho} \zeta \int _{0}^{1} \int_{0}^{1} \int _{0}^{1} \cos \left( 2z \left( f_0 (\rho) +xy \rho \zeta \right) \right) x dz dy dx, \\
& B(\rho, \zeta):= -2 f_0(\rho) \zeta ^{2} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \sin \left( 2z \left(f_0 (\rho) + xy \rho \zeta \right) \right) x^{2} y z dz dy dx, \\
& C(\rho, \zeta):= 3 \zeta ^{2} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \cos \left( 2z \left( f_0(\rho) + xy \rho \zeta \right) \right) x^{2} y dz dy dx, \\
& D(\rho, \zeta):= -2 \rho \zeta ^{3} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \sin \left( 2z \left( f_0 (\rho) + xy \rho \zeta \right) \right) x^{3} y^{2} zdz dy dx.
\end{align*}
We denote by $\mathcal{B}_{\delta} \subseteq \mathcal{H}$ the ball of radius $\delta$ in $\mathcal{H}$ centered at zero, i.e.,
\begin{align*}
\mathcal{B}_{\delta}:= \left \{\mathbf{u} \in \mathcal{H}:~\left \| \mathbf{u} \right \|= \left \| (u_{1},u_{2}) \right \|_{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) \times H_{\text{rad}}^{\frac{d+1}{2}} (\mathbb{B}^{d+2}) }
\leq \delta \right \}.
\end{align*}
The main result of this section is the following Lipschitz-type estimate.
\begin{lemma}
Let $\delta>0$.
Then we have
\begin{align} \label{Lipschitz}
\big \| \mathbf{N (u)} - \mathbf{N(v)}\big \| \lesssim (\| \mathbf{u} \| +\| \mathbf{v} \| ) \| \mathbf{u}-\mathbf{v} \|
\end{align}
for all $\mathbf{u}, \mathbf{v} \in \mathcal{B}_{\delta}$.
\end{lemma}
\begin{proof}
We start by fixing a $\delta>0$, we pick two elements $\mathbf{u}, \mathbf{v} \in \mathcal{B}_{\delta}$ and define the auxiliary function
\begin{align*}
\zeta (\sigma)(\rho)= \sigma u_{1}(\rho) + (1-\sigma) v_{1} (\rho),
\end{align*}
for $\rho \in (0,1)$ and $\sigma \in [0,1]$. The triangle inequality implies
\begin{align*}
\mathbf{u},\mathbf{v} \in \mathcal{B}_{\delta} \Longrightarrow \left \| u_{1} \right \|_{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})} \leq \delta ,~\left \| v_{1} \right \|_{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})} \leq \delta
\Longrightarrow \left \| \zeta (\sigma) \right \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \leq \delta,
\end{align*}
for all $\sigma \in [0,1]$. In other words,
\begin{align*}
\zeta (\sigma) \in \mathscr{B}_{\delta}:= \left \{f \in H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) :~\left \| f \right \|_{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \leq \delta \right \},
\end{align*}
for all $\sigma \in [0,1]$. Now, we claim that to show $\eqref{Lipschitz}$, it suffices to establish the estimate
\begin{align} \label{M}
\left \| M(\cdot, f(\cdot) ) \right \|_{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \lesssim \left \| f \right \|_{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }
\end{align}
for all $f \in \mathscr{B}_{\delta}$,
where $M$ is given by $\eqref{DefM}$. To see this, we use the algebra property
\begin{align*}
\| fg \|_{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \lesssim \| f \|_{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \| g \|_{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } ,
\end{align*}
which holds since $\frac{d+3}{2}>\frac{d+2}{2}$,
to estimate
\begin{align*}
\big \| \mathbf{N(u)} -\mathbf{N(v)} \big\| &= \big \| \hat N (\cdot, u_{1}(\cdot) ) - \hat N (\cdot , v_{1}(\cdot)) \big \| _{H_{\text{rad}}^{\frac{d+1}{2}} (\mathbb{B}^{d+2}) } \\
&\leq
\big \| \hat N (\cdot, u_{1}(\cdot) ) - \hat N (\cdot , v_{1}(\cdot)) \big \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \\
&=\left \| \int _{v_{1}(\cdot)}^{u_{1}(\cdot)} \partial _{2} \hat N (\cdot, \zeta ) d \zeta \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})} \\
&=\left \| \left( u_{1}(\cdot) - v_{1}(\cdot) \right) \int _{0}^{1} \partial _{2} \hat N (\cdot , \underbrace{ \sigma u_{1}(\cdot)+(1- \sigma) v_{1}(\cdot)}_{\zeta(\sigma) }) d \sigma \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \\
\nonumber
&\lesssim \left \| u_{1} - v_{1} \right \| _{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \left \| \int _{0}^{1} \partial _{2} \hat N (\cdot , \zeta (\sigma) ) d \sigma \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \\
& \lesssim \left \| u_{1} - v_{1} \right \| _{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \int _{0}^{1} \left \| M (\cdot , \zeta (\sigma)(\cdot) ) \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } d \sigma \\
& \lesssim \left \| u_{1} - v_{1} \right \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \int _{0}^{1} \left \| \zeta (\sigma) \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } d \sigma \\
& \lesssim \left \| u_{1} - v_{1} \right \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \int _{0}^{1} \left( \sigma \left \| u_{1} \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } +(1-\sigma) \left \| v_{1} \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \right) d \sigma \\
& \lesssim \left \| u_{1} - v_{1} \right \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \left( \left \| u_{1} \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } + \left \| v_{1} \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \right) \\
&\lesssim \left \| \mathbf{u} - \mathbf{v} \right \| \left( \left \| \mathbf{u} \right \| + \left \| \mathbf{v} \right \| \right).
\end{align*}
It remains to prove $\eqref{M}$. To this end we use a simple extension argument (see e.g.~Lemmas B.1 and B.2 in \cite{DonSch16}) and Moser's inequality (\cite{Rau12}, p.~224, Theorem 6.4.1) to infer the existence of a smooth function $h: [0,\infty)\to [0,\infty)$ such that
\[ \|M(\cdot,f(\cdot))\|_{H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})}\leq h\left (\|f\|_{L^\infty(\mathbb B^{d+2})}\right )\|f\|_{H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})} \]
for all $f\in \mathscr B_\delta$.
By Sobolev embedding we have $\|f\|_{L^\infty(\mathbb B^{d+2})}\lesssim \|f\|_{H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})}\leq \delta$
for all $f\in \mathscr B_\delta$ and \eqref{M} follows.
This concludes the proof.
\end{proof}
\subsection{The abstract nonlinear Cauchy problem} In this section, we focus on the existence and uniqueness of solutions to the Cauchy problem $\eqref{Evolution}$. In fact, by appealing to Definition $\ref{def}$, we consider the integral equation
\begin{align} \label{integralequation}
\Phi(\tau)= \mathbf{S}(\tau) \mathbf{u} + \int _{0}^{\tau} \mathbf{S}(\tau - s) \mathbf{N} \big( \Phi (s) \big) ds,
\end{align}
for all $\tau \ge 0$ and $\mathbf u\in \mathcal H$. We introduce the Banach space
\begin{align*}
\mathcal{X}:= \{ \Phi \in C( [0,\infty);\mathcal{H}) : ~~\| \Phi \|_{\mathcal{X}} := \sup _{\tau >0} e^{\epsilon \tau} \| \Phi(\tau) \| < + \infty \}
\end{align*}
with $\epsilon>0$ from Proposition \ref{projection}. Moreover, we denote by $\mathcal{X}_{\delta}$ the closed ball
\begin{align*}
\mathcal{X}_{\delta} :=\left \{ \Phi \in \mathcal{X}: \| \Phi \|_{\mathcal{X}} \leq \delta \right \} = \left \{ \Phi \in C( [0,\infty);\mathcal{H}): \| \Phi \| \leq \delta e^{-\epsilon \tau},~~\forall \tau >0 \right \}.
\end{align*}
In the following, we will only sketch the rest of the proof and discuss the main arguments since they are analogous to \cite{Don11, DonSch12, DonSch14, Don14, DonSch16}. To prove the main theorem, we would like to apply a fixed point argument to the integral equation $\eqref{integralequation}$. However, the exponential growth of the solution operator on the unstable subspace prevents from doing this directly. We overcome this obstruction by subtracting the correction term\footnote{All integrals here exist as Riemann integrals over continuous functions.}
\begin{align} \label{Correction}
\mathbf{C}(\Phi,\mathbf{u}) := \mathbf{P} \left( \mathbf{u} + \int_{0}^{\infty} e^{-s} \mathbf{N} \big( \Phi (s) \big) ds \right)
\end{align}
from the initial data. Consequently, we consider the fixed point problem
\begin{align} \label{modified}
\Phi (\tau)= \mathbf{K} ( \Phi, \mathbf{u})(\tau)
\end{align}
where
\begin{align} \label{K}
\mathbf{K} (\Phi, \mathbf{u}) (\tau):=\mathbf{S} (\tau) [\mathbf{u} - \mathbf{C}(\Phi,\mathbf{u})] + \int_{0}^{\tau} \mathbf{S} (\tau - s) \mathbf{N} \big( \Phi(s) \big) ds.
\end{align}
This modification stabilizes the evolution as the following result shows.
\begin{theorem} \label{th2}
There exist constants $\delta,C>0$ such that for every $\mathbf{u} \in \mathcal{H}$ with $\| \mathbf{u} \| \leq \frac{\delta}{C}$, there exists a unique $\mathbf{\Phi} (\mathbf{u}) \in \mathcal{X}_{\delta}$ that satisfies
\begin{align*}
\mathbf{\Phi} (\mathbf{u}) = \mathbf{K} (\mathbf{\Phi} (\mathbf{u}),\mathbf{u}).
\end{align*}
In addition, $\mathbf{\Phi}(\mathbf u)$ is unique in the whole space $\mathcal X$ and the solution map $\mathbf{u} \mapsto \mathbf{\Phi }(\mathbf{u})$ is Lipschitz continuous.
\end{theorem}
\begin{proof}
The proof is based on a fixed point argument and the essential ingredient is the Lipschitz estimate \eqref{Lipschitz} for the nonlinearity. Although the proof coincides with the one of Theorem 4.13 in \cite{DonSch16}, we sketch the main points for the sake of completeness. We pick $\delta>0$ sufficiently small and fix $\mathbf{u} \in \mathcal{H}$ with $\| \mathbf{u} \| \leq \frac{\delta}{C}$, where $C>0$ is sufficiently large. First, note that the continuity of the map
\begin{align*}
\mathbf{K} (\Phi, \mathbf{u}): [0,\infty) \longrightarrow \mathcal{H},\quad \tau \longmapsto \mathbf{K} (\Phi, \mathbf{u})(\tau)
\end{align*}
follows immediately from the strong continuity of the semigroup $\left( \mathbf{S}(\tau) \right)_{\tau >0}$. Next, to show that $\mathbf K(\cdot,\mathbf u)$ maps $\mathcal X_\delta$ to itself,
we pick an arbitrary $\Phi \in \mathcal{X}_{\delta} $ and decompose the operator according to
\begin{align*}
\mathbf{K} ( \Phi, \mathbf{u})(\tau) = \mathbf{P} \mathbf{K} ( \Phi, \mathbf{u})(\tau) + (1-\mathbf{P} ) \mathbf{K} ( \Phi, \mathbf{u})(\tau).
\end{align*}
The Lipschitz bound \eqref{Lipschitz} implies
\begin{align*}
\left \| \mathbf{N} \left( \Phi (\tau) \right) \right \| \lesssim \delta^2 e^{-2 \epsilon \tau}
\end{align*}
and together with the time evolution estimates for the semigroup on the unstable and stable subspaces (see Proposition \ref{projection}), we get
\begin{align*}
\left \| \mathbf{P} \mathbf{K} \left( \Phi, \mathbf{u} \right) (\tau) \right \| \lesssim \delta ^{2} e^{-2 \epsilon \tau}, \quad \left \| \left(1- \mathbf{P} \right) \mathbf{K} \left( \Phi, \mathbf{u} \right) (\tau) \right \| \lesssim (\tfrac{\delta}{C}+\delta^2) e^{-\epsilon \tau} .
\end{align*}
Clearly, these estimates imply that $ \mathbf{K} ( \Phi, \mathbf{u}) \in \mathcal{X}_{\delta}$ for sufficiently small $\delta$ and sufficiently large $C>0$. Finally, we need to show the contraction property. To this end, we pick two elements $\Phi,\widetilde{\Phi} \in \mathcal{X}_{\delta}$.
As before, the Lipschitz estimate \eqref{Lipschitz} together with Proposition \ref{projection} imply
\begin{align*}
\left \| \mathbf{P} \left( \mathbf{K} ( \Phi, \mathbf{u})(\tau) - \mathbf{K} ( \widetilde{ \Phi }, \mathbf{u})(\tau) \right) \right \| &\lesssim \delta e^{-\epsilon \tau} \left \| \Phi - \widetilde{\Phi} \right \|_{\mathcal X}, \\
\left \| \left(1- \mathbf{P}\right) \left( \mathbf{K} ( \Phi, \mathbf{u})(\tau) - \mathbf{K} ( \widetilde{ \Phi }, \mathbf{u})(\tau) \right) \right \| &\lesssim \delta e^{-\epsilon\tau} \left \| \Phi - \widetilde{\Phi} \right \|_{\mathcal X}
\end{align*}
and by choosing $\delta$ sufficiently small we conclude
\begin{align*}
\left \| \mathbf{K} ( \Phi, \mathbf{u}) - \mathbf{K} ( \widetilde{ \Phi }, \mathbf{u}) \right \|_{\mathcal{X}} \leq \frac{1}{2} \left \| \Phi - \widetilde{\Phi} \right \|_{\mathcal{X}}.
\end{align*}
Consequently, the claim follows by the contraction mapping principle. Uniqueness in the whole space $\mathcal X$ and the Lipschitz continuity of the solution map are routine and we omit the details.
\end{proof}
Now we turn to the particular initial data we prescribe. To this end, we define the space
\begin{align*}
\mathcal{H}^{R} := H_{\text{rad}}^{m} (\mathbb{B}_{R}^{d+2} ) \times H_{\text{rad}}^{m-1} (\mathbb{B}_{R}^{d+2}), \quad m\equiv m_{d} =\frac{d+3}{2}
\end{align*}
for $R>0$, endowed with the induced norm
\begin{align*}
\left \| \mathbf{w} \right \|_{\mathcal{H}^{R}}^{2} = \left \| (w_{1},w_{2}) \right \|_{\mathcal{H}^{R}}^{2} =\left \| w_{1}\right \|_{ H_{\text{rad}}^{m} \left( \mathbb{B}^{d+2}_R \right) } +
\left \| w_{2} \right \|_{ H_{\text{rad}}^{m-1} \left( \mathbb{B}^{d+2}_R \right) }.
\end{align*}
Recall the definition of the initial data operator $\mathbf U(\mathbf v, T)$ from Eq.~\eqref{5}.
\begin{lemma} \label{th1}
Fix $T_0>0$.
Let $\delta>0$ be sufficiently small and $\mathbf{v}$ with $|\cdot|^{-1} \mathbf{v} \in \mathcal{H}^{T_{0} + \delta}$. Then, the map
\begin{align*}
\mathbf{U} (\mathbf{v},\cdot):[T_{0}-\delta,T_{0}+\delta] \longrightarrow \mathcal{H},\quad T \longmapsto \mathbf{U} (\mathbf{v},T)
\end{align*}
is continuous. Furthermore, for all $T \in [T_{0} -\delta,T_{0}+\delta]$,
\begin{align*}
\big \| |\cdot|^{-1} \mathbf{v}\big \|_{\mathcal{H}^{T_{0}+\delta} } \leq \delta \Longrightarrow \big \| \mathbf{U} (\mathbf{v},T)\big \| \lesssim \delta.
\end{align*}
\end{lemma}
\begin{proof}
The statements are straightforward consequences of the very definition of $\mathbf U(\mathbf v,T)$, the smoothness of $\frac{f_0(\rho)}{\rho}$, and the continuity of rescaling in Sobolev spaces. We omit the details.
\end{proof}
Finally, given $T_{0}>0$ and $\mathbf{v} \in \mathcal{H}^{T_{0} + \delta}$ with $\| |\cdot|^{-1} \mathbf{v} \|_{\mathcal{H}^{T_{0}+\delta} } \leq \frac{\delta}{M}$ for $\delta>0$ sufficiently small and $M>0$ sufficiently large, we apply Lemma \ref{th1} to see that $\mathbf{u}:=\mathbf{U} (\mathbf{v},T)$ satisfies the assumptions of Theorem \ref{th2} for all $T\in [T_0-\delta,T_0+\delta]$. Hence, for all $T \in [T_{0}-\delta,T_{0}+\delta]$, the map $\mathbf K(\cdot,\mathbf U(\mathbf v,T))$ has a fixed point $\Phi_{T}:= \mathbf{\Phi} (\mathbf U(\mathbf v,T)) \in \mathcal{X}_{\delta}$. In the last step we now argue that for each $\mathbf v$, there exists a particular $T_{\mathbf{v}} \in [T_{0}-\delta,T_{0}+\delta]$ that makes the correction term vanish, i.e., $\mathbf C(\Phi_{T_{\mathbf v}},\mathbf U(\mathbf v,T_{\mathbf v}))=0$. Since $\mathbf C$ has values in $\rg \mathbf P=\langle\mathbf g\rangle$, the latter is equivalent to
\begin{align}
\label{eq:Tv}
\exists T_{\mathbf{v}} \in [T_{0}-\delta,T_{0}+\delta]: \quad \Big< \mathbf{C} \left( \Phi _{T_{\mathbf{v}}},\mathbf{U} \left( \mathbf{v},T_{\mathbf{v}} \right) \right),\mathbf{g} \Big>_{\mathcal H} = 0.
\end{align}
The key observation now is that
\[ \partial_T \left . \left ( \begin{array}{c}
\frac{1}{\rho}f_0(\frac{T}{T_0}\rho) \\
\frac{T^2}{T_0^2}f_0'(\frac{T}{T_0}\rho) \end{array} \right )\right |_{T=T_0}=\frac{2\sqrt{d-2}}{T_0}\,\mathbf g(\rho) \]
and thus, we have the expansion
\[ \Big< \mathbf{C} \left( \Phi_T,\mathbf U (\mathbf{v},T ) \right),\mathbf{g} \Big>_{\mathcal H}=\frac{2\sqrt{d-2}}{T_0}\|\mathbf g\|^2(T-T_0)
+O((T-T_0)^2)+O(\tfrac{\delta}{M}T^0)+O(\delta^2T^0).
\]
Consequently, a simple fixed point argument proves \eqref{eq:Tv}, see \cite{DonSch16}, Theorem 4.15 for full details.
In summary, we arrive at the following result.
\begin{theorem} \label{correction}
Fix $T_0>0$. Then there exist $\delta,M >0$ such that for any $\mathbf{v}$ with
\[ \| |\cdot|^{-1} \mathbf{v} \|_{\mathcal{H}^{T_{0}+\delta} } \leq \frac{\delta}{M} \] there exists a $T \in [T_{0}-\delta,T_{0}+\delta]$ and a function $\Phi \in \mathcal{X}_{\delta}$ which satisfies
\begin{align}
\label{eq:int}
\Phi (\tau) = \mathbf{S} (\tau) \mathbf{U} (\mathbf{v},T) + \int_{0}^{\tau} \mathbf{S} (\tau-s) \mathbf{N} \big( \Phi (s) \big) ds
\end{align}
for all $\tau \geq 0$. Furthermore, $\Phi$ is unique in $C \big( [0,\infty);\mathcal{H} \big)$.
\end{theorem}
\subsection{Proof of the main theorem} With the results of the previous section at hand, we can now prove the main theorem. Fix $T_0>0$ and suppose the radial initial data
$\psi [0]$ satisfy
\begin{align*}
\left \| |\cdot|^{-1} \Big( \psi [0] -\psi^{T_{0}}[0] \Big) \right \|_{ H^{\frac{d+3}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2}) \times H^{\frac{d+1}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2} )} \leq \frac{\delta}{M}
\end{align*}
with $\delta,M>0$ from Theorem \ref{correction}.
We set $\mathbf v:=\psi[0]-\psi^{T_0}[0]$, cf.~Section \ref{sec:sim}.
Then we have
\begin{align*}
\left \| |\cdot|^{-1} \mathbf{v} \right \|_{\mathcal{H}^{T_{0}+\delta}} = \left \| |\cdot|^{-1} \Big( \psi [0] -\psi^{T_{0}}[0] \Big)
\right \|_{\mathcal{H}^{T_{0}+\delta}} \leq \frac{\delta}{M}
\end{align*}
and Theorem $\ref{correction}$ yields the existence of
$T \in [T_{0}-\delta,T_{0}+\delta]$ such that Eq.~\eqref{eq:int} has a unique solution $\Phi \in \mathcal{X}$ that satisfies $\| \Phi (\tau) \| \leq \delta e^{-\epsilon \tau}$ for all $\tau\geq 0$.
By construction,
\[ \psi(t,r)=\psi^T(t,r)+\frac{r}{T-t}\phi_1\left (\log\frac{T}{T-t},\frac{r}{T-t}\right ) \]
is a solution to the original wave maps problem \eqref{cauchy}. Furthermore,
\[ \partial_t \psi(t,r)=\partial_t \psi^T(t,r)+\frac{r}{(T-t)^2}\phi_2 \left (\log\frac{T}{T-t},\frac{r}{T-t}\right ). \]
Consequently,
\begin{align*}
(T-t)^{k-\frac{d}{2}}&\left \||\cdot|^{-1}\left (\psi(t,\cdot)-\psi^T(t,\cdot)\right )
\right \|_{\dot H^k(\mathbb B^{d+2}_{T-t})} \\
&=(T-t)^{k-\frac{d}{2}-1}\left \|\phi_1\left (\log\frac{T}{T-t},\frac{|\cdot|}{T-t}\right ) \right \|_{\dot H^k(\mathbb B^{d+2}_{T-t})} \\
&=\left \|\phi_1\left (\log\frac{T}{T-t},\cdot\right ) \right \|_{\dot H^k(\mathbb B^{d+2})}
\leq \left \|\Phi\left (\log\frac{T}{T-t}\right )\right \| \\
&\leq \delta (T-t)^\epsilon
\end{align*}
for all $t\in [0,T)$ and $k=0,1,2,\dots,\frac{d+3}{2}$.
Analogously,
\begin{align*}
(T-t)^{\ell-\frac{d}{2}+1}&\left \||\cdot|^{-1}\left (\partial_t \psi(t,\cdot)-\partial_t \psi^T(t,\cdot)\right )\right \|_{\dot H^\ell(\mathbb B^{d+2}_{T-t})} \\
&=(T-t)^{\ell-\frac{d}{2}-1}\left \|\phi_2\left (\log\frac{T}{T-t},\frac{|\cdot|}{T-t}\right ) \right \|_{\dot H^\ell(\mathbb B^{d+2}_{T-t})} \\
&=\left \|\phi_2\left (\log\frac{T}{T-t},\cdot\right ) \right \|_{\dot H^\ell(\mathbb B^{d+2})}
\leq \left \|\Phi\left (\log\frac{T}{T-t}\right )\right \| \\
&\leq \delta (T-t)^\epsilon
\end{align*}
for all $\ell=0,1,2,\dots,\frac{d+1}{2}$.
| {'timestamp': '2017-06-26T02:04:02', 'yymm': '1701', 'arxiv_id': '1701.05082', 'language': 'en', 'url': 'https://arxiv.org/abs/1701.05082'} |
\section{Introduction}
In the standard flare scenario \citepads[e.g.,][]{1996ApJ...466.1054M} the
energy release of the primary flare (primary magnetic reconnection) takes place in the
current sheet below a rising magnetic rope. Here, the plasmoids (the secondary
magnetic ropes in 3D), which are a natural outcome of the reconnection process,
are formed and ejected. The ejection of plasmoids can be traced observationally
via soft X-ray and radio waves, which map the magnetic-field reconnection
\citepads{1998ApJ...499..934O, 2000A&A...360..715K, 2002A&A...395..677K, 2004A&A...417..325K}.
With increasing spatial resolution of the solar photosphere and chromosphere,
flares, jets, and plasmoids on different scales are observed
\citepads[e.g.,][]{2013Natur.493..485C, 2013Natur.493..501C}. This means that
the solar atmosphere is highly structured, and magnetic reconnection processes
are ubiquitous. As such, the current sheets initiating reconnection processes
cannot be smooth but must contain some internal fragmented structure.
Consequently, magnetic reconnection itself must be fractal. As an efficient
mechanism to cascade down to smaller scales, instabilities have proven to be an
ideal trigger.
Many nonlinear, time-dependent magnetohydrodynamic (MHD) simulations focus on linear
and nonlinear instabilities, which are initiated via arbitrarily prescribed small
perturbations
of an initially smooth, static equilibrium. These instabilities typically result in
reconnection, and in the following in fragmentation of the magnetic field and hence
the current density \citepads[e.g.,][]{2010AdSpR..45...10B, 2011ApJ...733..107K,
2011ApJ...737...24B}, forming chains of plasmoids \citepads{2007PhPl...14j0703L,
2010PhRvL.105w5002U}, coalescence and further fragmentation of plasmoids
\citepads{2010AIPC.1242...89P, 2011ApJ...733..107K}, and plasmoids on progressively smaller
scales \citepads{2001EP&S...53..473S}.
The process of cascading can also be initiated by stochastic velocity
fluctuations, generating small-scale structures of the large-scale magnetic
field \citepads{1999ApJ...517..700L, 2009ApJ...700...63K, 2011PhRvE..83e6405E}.
This turbulent approach, however, originates from external perturbations
impressed on initial background (magnetic and velocity) fields, requiring the
prescription of initial noise, e.g., in the form of power-law spectra of
perturbations. On the other hand, the turbulent reconnection can result
from a successive coalescence and fragmentation of plasmoids, their fast
heating, and an increase of the plasma beta parameter at some locations, where
the flow instabilities become important as well \citepads{2012A&A...541A..86K}.
In contrast to studies using instabilities or turbulence as the initial trigger for
fragmentation, the MHD theory itself inherently provides the cradles for fractal
structures, because the MHD is scale-free and therefore applies to large as well as to
small scales \citepads{2012EGUGA..14.3226S, 2012cosp...39.1785S}. In their studies
of the Earth's magnetotail using a quasi-static adiabatic MHD approach,
\citetads{1995GeoRL..22.2057W} previously noticed the fragmentation of a
thin current sheet. In their investigations, they found the formation of
double-structures of the current density when using nonsimilarity solutions of the
quasi-static equations. Similarly, the numerical investigations of
\citetads{2001JGR...106.3811B} revealed the formation of thin current sheets from a
sequence of static equilibria. Thus, instead of using perturbations of a smooth,
static equilibrium, one might start directly from already structured, fragmented MHD
equilibrium states. For this, one needs to construct a selfconsistent analytical
description of the time-independent, nonlinear dynamics
\citepads[see, e.g.,][]{2001ohnf.conf...57N, 2006A&A...454..797N, 2010AnGeo..28.1523N,
2012AnGeo..30..545N}.
Separatrices form during magnetic reconnection processes, which originate in
so-called X-points. These X-points can separate regions of closed and open field
lines. The open field-line regions can be regarded as field lines along which, e.g.,
the solar wind can flow into the interplanetary space, while the closed regions correspond
to, e.g., magnetic arcades or flux ropes (plasmoids) from which plasma cannot leave. To
stabilize such a configuration, in which strong flows occur outside and (almost) no flows
inside, shear currents have to keep the system in equilibrium. Hence the physical problem
can be approximately described with the static approach in regions of closed field lines,
while in regions with open field lines the problem is in steady-state \citepads[see,
e.g.,][]{2006A&A...454..797N, 2010AnGeo..28.1523N, 2012AnGeo..30..545N}.
\citetads{2010AnGeo..28.1523N} considered that the
Alfv\'en Mach number, $M_{A}$, determining the strength of the flow and therefore the
plasma velocity, vanishes within the plasmoid, so that the structure is basically
magneto-hydrostatic (MHS). However, not every closed
field-line region must necessarily be of MHS nature. Instead, plasmoids might contain
vortices, because slight asymmetries during the ejection event could result in a
nonzero angular momentum transfer. Hence the flow inside plasmoids can be sheared.
Shear flows were found to produce current filamentation not only in solar or magnetospheric
environments, but also in space and astrophysical
plasmas, e.g., in astrophysical jets, where shear flows also induce the filamentation of
currents \citepads{1998PhPl....5.3732W, 2000PhPl....7.5159K}.
In this paper, we investigate the role of shear flows within a configuration containing
a magnetic dome and detached plasmoids, resembling a typical solar-flare configuration after
a first reconnection process. In our investigations, we used a selfconsistent analytical
description of the time-independent, nonlinear dynamics. The paper is
structured as follows: in Sect.\,\ref{sec2} we introduce the basic equations
and the transformation method, while the results are described in
Sect.\,\ref{sec3}. The assumptions are discussed in Sect.\,\ref{sec4}, and the
conclusions are given in Sect.\,\ref{sec5}.
\section{Basic equations}
\label{sec2}
\begin{figure}
\centering
\includegraphics[width=0.6\hsize]{sketch.pdf}
\caption{Sketch of a magnetic configuration of a solar flare with plasmoids
formed via magnetic reconnection processes.}
\label{sketch}
\end{figure}
We assumed a magnetic configuration of a solar flare with a plasmoid formed via
some first magnetic reconnection event (see Fig.\,\ref{sketch}). The plasmoid is
enclosed by two X-points, while the plasmoid itself hosts a magnetic O-point in its center.
The equilibrium of such a system can be formulated using the
steady/static approach. This is, however, only strictly valid if the plasmoid
has no or only marginal motion in $y$ direction. The dynamics of plasmoids
depends on the reconnection rate at the X-points below and above the plasmoid,
and plasmoids ``in rest'' \citepads[see, e.g.,][] {2008SoPh..253..173B,
2008A&A...477..649B} and stable \citepads[i.e. without further coalescence,
see, e.g.,][]{2006PhPl...13c2307K} have been found by numerical simulations,
justifying our assumption.
The choice of a certain equilibrium defines an arbitrary, but fixed length-scale.
This does not allow one to make inferences on the properties of the plasmoid on (much)
smaller scales, on which, e.g., stationary shear flows related to vortex sheets might
exist. Such shear flows would generate additional forces on the former MHS states,
which can only be compensated for by changes in Lorentz forces and
pressure gradients. To maintain the force balance self-consistently, we applied
the transformation method developed by \citetads{1992PhFlB...4.1689G} and
advanced/progressed by
\citetads{1999GApFD..91..269P} and \citetads{2006A&A...454..797N}. In the past
decades many attempts have been made to find exact and analytical solutions of nonlinear
steady-state (= stationary) MHD equations \citepads[e.g.,][]{1981ApJ...245..764T,
1996ApJ...460..185C, 1997PhPl....4.3544G, 2005AdSpR..35.2067N, 2006AdSpR..37.1292N}.
However, the transformation is the only systematic method that physically and mathematically
relates steady-state MHD flows to MHS states. For such a transformation
to work, it is reasonable to request that in the stationary state
the velocity field and the magnetic field are parallel (field-aligned flows).
This guarantees that the electric field vanishes, according to the ideal Ohm's law
\begin{equation}
\vec E + \vec v \times \vec B = \vec 0 \quad
\Rightarrow \quad \vec E = \vec 0\, .
\end{equation}
We note that other transformations between steady MHD states exist, which lead to
configurations in which the velocity field and the magnetic field are not necessarily parallel
\citepads{2000PhRvE..62.8616B, 2001PhLA..291..256B, 2002PhRvE..66e6410B}. However, only in
the case of incompressible field-aligned flows one can always reduce the steady-state MHD
equations to the MHS equations. Another advantage of the transformation method is that it
is independent of the dimensions, i.e., it can be performed in 1, 2, and 3D.
\subsection{Transformation from MHS states to stationary MHD configurations}
\label{trafo}
In the following we restrict the analysis to sub-Alfv\'enic flows to emphasize
in particular their relationship to MHS states. In addition, we use normalized
parameters, for which we introduce normalization constants $\hat{B}, \hat{\rho}, \hat{l},
\hat{p}$ and $\hat{\varv}_{A}$, where $\hat{\varv}_{A}=\hat{B}/\sqrt{\mu_{0}\hat{\rho}}$
is the normalized Alfv\'en velocity. Let $\vec v$ be the plasma velocity normalized on
$\hat{\varv}_{A}$, $\rho$ the mass density normalized on $\hat{\rho}$,
$\vec j=\vec\nabla\times\vec B$ the current density vector normalized on
$\hat{B}/(\mu_{0}\hat{l})$ with $\hat{l}$ as the characteristic length scale, and
$p$ the scalar plasma pressure normalized on
$\hat{p}=\hat{B}^2/\mu_{0}$.
With these definitions, we can write the set of equations of stationary, field-aligned
incompressible MHD, consisting of the mass continuity equation, the Euler equation, the
definition for field-aligned flow and Alfv\'{e}n Mach number, the incompressibility
condition, and the solenoidal condition for the magnetic field, in the form
\begin{eqnarray}
\vec\nabla\cdot(\rho\vec v) & = & 0\, ,\label{konti}\\
\rho\left(\vec v\cdot\vec\nabla\right)\vec v & = & \vec j\times\vec B - \vec\nabla p\, ,
\label{euler0} \\
\vec v & = & \frac{M_{A}\vec B}{\sqrt{\rho}}\, ,\label{paral}\\
\vec\nabla\cdot\vec v & =& 0\, ,\label{konti2}\\
\vec\nabla\cdot\vec B & = & 0\label{divb} \, .
\end{eqnarray}
This set of equations can always be reduced to the set of static equations using
the transformation equations \citepads[for details see][]{2010AnGeo..28.1523N,
2012AnGeo..30..545N} of the form
\begin{eqnarray}
\vec B &&=\frac{\vec B_{S}}{\sqrt{1-M_{A}^2}}\, ,\label{magtrafo} \\
p &&=p_{S} - \frac{M_{A}^2\left|\vec B_{S}\right|^2}{1-M_{A}^2}
\, , \label{pressuretrafo}\\
\sqrt{\rho} \vec v &&=
\frac{M_{A}\vec B_{S}}{\sqrt{1-M_{A}^2}}
\equiv\ M_{A}\vec B\label{streaming2} \, , \\
\vec j &&=\frac{M_{A}\,\vec\nabla M_{A}\times\vec B_{S}}{\left(1-M_{A}^2
\right)^{\frac{3}{2}}}
+\frac{\vec j_{S}}{\left(1-M_{A}^2\right)^{\frac{1}{2}}} \, , \label{currenttrafo}\\
\vec\nabla p_{S} &&=\vec j_{S}\times\vec B_{S} \label{mhs1}
\,\, ,
\end{eqnarray}
where the subscript $S$ refers to the original MHS fields.
Here it is a necessary condition that the Alfv\'en Mach number $M_{A}$ and the
density $\rho$ are constant along fieldlines, i.e.,
\begin{eqnarray}
\vec B\cdot\vec\nabla M_{A} & = & 0 \label{cond1} \\
\vec B\cdot\vec\nabla\rho & = & 0\, . \label{conditions}
\end{eqnarray}
An important property of this type of transformation
is the fact that every transformed magnetic field strength $|\vec B|$ is stronger
than the original static magnetic field strength $|\vec B_{S}|$ (as long as $M_{A}\neq 0$).
Moreover, as $\vec j$ is directly proportional to the term $\vec\nabla M_{A}$, which
can have an arbitrarily (but not infinite) high value, basically every infinitesimale scale
$1/\nabla = l>0$ can be chosen. Therefore, we can produce a current that is higher than any
current threshold to excite anomalous resistivity, such that current-driven instabilities and
hence magnetic reconnection can be induced. This generated current can be even more
amplified when the Alfv\'en Mach number approaches the limit $M_{A}\la 1$.
We stress that the transformation method provides a nonlinear self-consistent solution
of the stationary MHD equations, and changing any of the physical variables produces a
nonlinear feed-back of all other variables. Variations of $M_{A}$ should not be
misunderstood as an explicit time-dependent change or sequence of the underlying MHS
equilibrium, like in the quasi-static sequences of \citetads{1995GeoRL..22.2057W} or
\citetads{2001JGR...106.3811B}. Instead, the transformation has to be interpreted as a
nonlinear variation or displacement of the former initial MHS equilibrium. That is, in
affinity to variational calculus, the steady-states are \lq located\rq~in the proximity
of MHS states.
The set of transformation equations (Eqs.\,(\ref{magtrafo} - \ref{mhs1})) together with the
conditions of Eqs.\,(\ref{cond1} - \ref{conditions}) provide a \lq recipe\rq~to construct
field-aligned, incompressible flows along the MHS structures. In practice, we first need to
calculate an MHS equilibrium.
In the following we assumed that the equilibrium has some
sort of symmetry (e.g. in z-direction), so that it can be reduced to
a pure 2-dimensional (2D) problem\footnote{A restriction to pure 2D is justified, because
it enhances the clarity of the representation of the fragmentation process. Our studies
are aimed at the fragmentation of the isocontours of the current density $j_{z}$.}.
In that case, the equilibrium value of the magnetic field has the form
$\vec B_{S} = \nabla A(x,y) \times \vec e_z$.
Next, we need to determine a Mach number profile, $M_{A}(A)$. This profile has to depend
locally only on the flux function, $A$, so that $\vec B_{S}\cdot \vec\nabla M_{A} = 0$, and
hence the condition Eq\,(\ref{cond1}) is automatically fulfilled.
The \lq new\rq~magnetic field, i.e., the steady-state
field, is then given by a new flux function $\alpha$, which is a function of $A$, such that
\begin{equation}
M_{A}^2=1-\displaystyle\frac{1}{\left(\frac{d\alpha}{dA}\right)^2}\quad\Leftrightarrow\quad
\left(\alpha'(A)\right)^2=\frac{1}{1-M_{A}^2} \label{Ma_alpha}
\end{equation}
and $\vec B = \vec\nabla\alpha \times \vec e_{z}$ \citepads[see][]{2010AnGeo..28.1523N}.
The prime denotes the derivative with respect to $A$. Armed with this Mach number
profile and $\vec B_{S}$, the set of transformation equations (Eq.\,(\ref{magtrafo}) to
(\ref{mhs1})) can be evaluated.
For the adopted 2D shape of the magnetic field the current-transformation equation
(Eq.\,(\ref{currenttrafo})) takes the form
\begin{eqnarray}
j = j_{z} & = & -\frac{M_{A} M_{A}'\,\left(\vec\nabla A\right)^2}{\left(1-M_{A}^2\right)^{3/2}} -
\frac{\Delta A}{\left(1-M_{A}^2\right)^{1/2}} \\
& = & -\Delta\alpha=-\alpha'' \left(\vec\nabla A\right)^2 - \alpha' \Delta A .
\label{currenttrafoexpl6}
\end{eqnarray}
The current fragmentation is strong where the magnetic field is strong, as the increase in
current and its spatial variation is mainly governed by $M_{A}'$ but amplified by
$\left(\vec\nabla A\right)^2$.
One interesting and important property of this transformed current is the fact that
it can have a zero-crossing even for an initially monopolar MHS-current distribution.
This means that any suitable choice of a transformation can make $j_{z}$ negative
(positive), although the MHS current is completely positive (negative). In particular,
the zero-crossing of the current requires that it has to vanish at some point, i.e.,
$j = j_{z} \stackrel{!}{=} 0$. This delivers a condition for the Mach number profile
of the form
\begin{equation}
M_{A}M_{A}'=- \frac{\Delta A}{\left(\vec\nabla A\right)^2}\,\left(1-M_{A}^2\right)\, ,
\label{determine}
\end{equation}
with the restriction $\left(\vec\nabla A\right)^2\neq 0$.
As an important example of a monopolar MHS-current we refer to Liouville's equation
given by $\Delta A=\exp(-2A)$ (see Eq.\,(\ref{lio}) below). Because $\Delta A$ is always
positive and $\left(\vec\nabla A\right)^2$ and $1-M_{A}^2$ are positive as well, the
left-hand side derivative of Eq.\,(\ref{determine}) must be negative, i.e.,
$d/dA\,(M_{A}^2/2)<0$. This condition can in principle be fulfilled with any suitable
Alfv\'en Mach number profile that is monotonically decreasing with $A$ (at least
locally). This demonstrates the power of the transformation method and shows that it can
be used to generate strong current fragmentation.
The zero-crossing is a definite sign that fragmentation can take place, but in many cases
it is sufficient to have a strong gradient concerning $M_{A}$ and/or a large
$(\nabla A)^{2}$.
On the other hand, this means that in the vicinity of a magnetic null point $M_{A}'$ must
be extremely large to compensate the vanishing magnetic field strength.
Nevertheless, depending on the choice of the Mach number profile, current
fragmentation can happen even without a zero-crossing of the transformed current.
\section{Results}
\label{sec3}
\subsection{Nonlinear static equilibria}
\label{nonlinstat}
As described in the previous section, the first step is to derive a
reasonable initial MHS equilibrium, which is able to reproduce a field-line scenario
with individual disconnected plasmoids, as drawn schematically in Fig.\,\ref{sketch}.
For this, we used two well-known equilibrium configurations and combined them.
Starting from the static magnetic field in 2D,
$\vec{B_{S}} = \vec\nabla A \times \vec e_{z}$, and inserting it into the MHS equilibrium
equation (Eq.\,(\ref{mhs1})) delivers the well-known Grad-Shafranov-equation, often also
called L\"{u}st-Schl\"{u}ter-equation \citepads[see, e.g.,][]{1957ZNatA..12..850L,
1958JETP....6..545S}
\begin{eqnarray}
\Delta A=-\frac{dp_{S}}{dA}.
\label{GSE}
\end{eqnarray}
Because $\vec{B_{S}}\cdot \vec\nabla A = 0$ is valid, lines of constant $A$
are field lines. This implies that the current $j= -\Delta A$ is constant
along field lines, as is the pressure $p_{S}$, because they are functions of $A$,
and consequently the isocontours of the current have the same topological and geometrical
structure as those of the field lines.
For the pressure function $p_{S}(A)$ we use
\begin{eqnarray}
p_{S}(A)=\frac{1}{2}\,\exp(-2A),
\end{eqnarray}
as derived in the frame of the Vlasov theory by, e.g., \citetads{1934PhRv...45..890B},
\citetads{1962IlNC...23..115H}, and \citetads{1973JGR....78.3773K}. The same function is
typically applied in MHS or MHD
configurations in which the pressure monotonically decreases in the direction perpendicular
to the current sheet. Examples are flare configurations, magnetotails, and helmet streamers
\citepads[see, e.g.,][]{1975Ap&SS..35..389B, 1998SoPh..180..439W, 2008SoPh..253..173B,
2010AdSpR..45...10B}.
With this pressure function, the Grad-Shafranov-equation has the form
\begin{eqnarray}
\Delta A=\exp(-2A)\, , \label{lio}
\end{eqnarray}
also known as Liouville's equation \citepads[e.g.,][]{1975ArRMA..58..219B}.
By defining $u=x+iy$, $\varv=x-iy$, and $i^2=-1$, Liouville's equation can be written as
\citepads[see][]{1975ArRMA..58..219B, 1978SoPh...57...81B}
\begin{eqnarray}
4\frac{\partial^{2} A}{\partial u\, \partial \varv}=
\exp(-2A)\, .
\label{lio1}
\end{eqnarray}
The general solution of Liouville's equation, Eq.\,(\ref{lio1}), is given by
\begin{eqnarray}
A(u,\varv)=\ln\frac{1+\frac{1}{4}|\Psi(u)|^2}{\displaystyle\left|\frac{d\,\Psi}{du}\right|}\, ,
\end{eqnarray}
implying that every holomorphic function $\Psi(u)$ gives us an exact solution of
the nonlinear Grad-Shafranov-equation, Eq.\,(\ref{lio}).
The classical ansatz is
\begin{equation}
\Psi=2\exp{u}\, ,
\end{equation}
leading to the Harris-sheet equilibrium \citepads{1962IlNC...23..115H}
\begin{equation}
A=\ln\cosh (x)\, ,
\end{equation}
which represents a bipolar magnetic field structure, i.e., a plasma or current sheet
separating magnetic fields with opposite orientations. This is a one-dimensional
structure.
\begin{figure}
\centering
\includegraphics[width=0.85\hsize]{static.pdf}
\caption{Field lines for the Kan magnetotail (top), the periodic sheet pinch or
periodic Harris sheet (middle), and the combined one (bottom) that serves as our
initial MHS equilibrium.}
\label{fieldlinesmhs}
\end{figure}
A modified magnetic-field structure including a normal component penetrating
the 2D current sheet was calculated by \citetads{1973JGR....78.3773K} for the case of the
Earth's magnetotail region. Such a configuration emulates the dipole structure that extends
into and influences the Earth's magnetotail. In addition, with respect to configurations
within the solar corona, such a scenario ideally resembles, e.g., magnetic dome structures.
\citetads{1973JGR....78.3773K} chose the function
$\Psi=2\exp{\left(u+d/u\right)}$, which drops off for $|u|\rightarrow \infty$.
Here, $d$ is a constant. This choice is a slight perturbation of the original 1D Harris sheet
toward a 2D magnetic-field configuration, converging for large distances back again to the
original Harris-sheet equilibrium. The corresponding flux function is
\begin{equation}
A= \ln \displaystyle\frac{\cosh\left[ x\left(1+\frac{d}{r^{2}}\right)\right]}
{\sqrt{\displaystyle\frac{(d^{2}-2d(x^{2}-y^{2})+r^{4})}{r^{4}}}}\, ,
\end{equation}
where $r=\sqrt{x^{2}+y^{2}}$. The resulting field lines, computed for $d=0.5$,
are shown in the top panel of Fig.\,\ref{fieldlinesmhs}, with the $y$-axis pointing in
magnetotail direction, and the Earth (in the original Kan picture) located at the origin of
the coordinate system. This scenario allows one to describe stretched, tail-like structures,
including a dipole-like configuration close to the Earth (i.e., for low values of $y$).
To describe the field lines of periodic structures, we applied the approach of
\citetads{1967ApJ...149..727S}, who developed a formalism to solve Liouville's equation
resulting in the so-called periodic, corrugated sheet-pinch. In this scenario,
the original Harris-sheet equilibrium is slightly modified to $\Psi=2\left( \sqrt{1+\delta^2}
\exp{u} + \delta\right)$ with $\delta$ as a constant, leading to the following flux function:
\begin{equation}
A= \ln \left(\sqrt{1+\delta^2} \cosh{x} + \delta \cos{y}\right)\, .
\end{equation}
The field lines evaluated for $\delta = 0.1$ are shown in the middle panel of
Fig.\,\ref{fieldlinesmhs}.
For our purposes, we used the approaches from both Kan and Schmid-Burgk,
and combined them, i.e., we applied the modification of the Harris-sheet found by
\citetads{1967ApJ...149..727S} to the Kan equilibrium.
This is necessary, because we aim at achieving a representation in which the equilibrium has a
strong $B_{x}$--component close to the lower boundary $y=0$ and a periodic-sheet pinch
for $y\rightarrow\infty$. This means that $\Psi$ is now represented by
\begin{equation}
\Psi= 2\left(\sqrt{1+\delta^2}\exp{\left(u+d/u\right)}+\delta\right)\, .
\end{equation}
With this function, we finally obtain a flux function of the form
\begin{equation}
A = \ln\displaystyle\frac{\sqrt{1+\delta^{2}} \cosh\left[ x\left(1+\frac{d}{r^{2}}\right)\right] +
\delta \cos\left[y\left(1-\frac{d}{r^{2}}\right)\right] }
{\sqrt{\displaystyle\frac{(d^{2}-2d(x^{2}-y^{2})+r^{4})}{r^{4}}}} \, ,
\label{finalflux}
\end{equation}
whose contour plot has field lines as depicted in the bottom panel of
Fig.\,\ref{fieldlinesmhs}. In our representation the $y$-axis corresponds to the height above
the solar photosphere, and the photosphere itself is located at $y=0$. The symmetry axis of
the post-flare magnetic field configuration is given by $x=0$. This flux function
(Eq.\,\ref{finalflux}) is used in the following and serves as our initial MHS equilibrium.
\subsection{Different transformation approaches}
There exist three different approaches to model field aligned shear flows. These are the
transformations via
\begin{itemize}
\item magnetic field amplification defined by $\alpha'$,
\item peaked plasma flows defined by $M_{A}$, and
\item asymptotical 1D current structures defined by $j$.
\end{itemize}
Each approach
requires the specification of either one of the finally transformed MHD values (such as
the current or the magnetic field, the latter is even identical to the transformation itself),
or the plasma flow of the stationary MHD configuration. In combination with the prescribed
intrinsic MHS values, the corresponding transformation between these two states can be evaluated.
As we have shown, Eq.\,(\ref{Ma_alpha}) describes two equivalent methods for a transformation,
i.e., via the calculation of either the transformed magnetic field, specified by $\alpha'$, or the
plasma flow in the transformed configuration, given by $M_A$. On the other hand, it is also
reasonable to compute the Mach number profile via Eq.\,(\ref{currenttrafoexpl6}) by specifying
the asymptotic behavior of the transformed current $j$. These three methods are basically
equivalent, but each of them emphasizes a different physical aspect of the fragmentation
problem, and consequently needs different constraints and boundary conditions.
More specifically, each method is based on the prescription of one physical parameter that
serves as control parameter.
\subsubsection{Transformation via magnetic-field amplification defined by $\alpha'$}
The first mapping method has been described in detail by \citetads{2006A&A...454..797N}.
It is based on the prescription of the magnetic field and is best applicable for potential
fields that are asymptotical homogeneous, i.e., their flux function is given by
$A\approx B_{S\infty} x$ for large $y$. In that case, the transformation between the new,
steady-state flux function $\alpha$ and the old, stationary flux function $A$ is given by
\begin{eqnarray}
\alpha(A) &=& C A + \sum\limits_{k}\, a_{k}\ln\cosh\left(\frac{A-A_{k}}{d_{k}}\right)\\
\alpha'(A) &=& C + \sum\limits_{k}\frac{a_{k}}{d_{k}}\tanh
\left(\frac{A-A_{k}}{d_{k}}\right)\, .
\end{eqnarray}
This transformation, which is based on the calculation of $\alpha'$, produces a series of $k$
Harris-sheets with different strengths $a_{k}/d_{k}$ and widths $d_{k}$, offset by $A_{k}$
from the MHS state. The parameters $C$, $A_{k}$, $a_{k}$, and $d_{k}$ are not completely
free. They have to be chosen such that $\left|\alpha'(A)\right|>1$ to guarantee sub-Alfv\'enic
flows and to satisfy the boundary conditions or constraints, provided, e.g., by observations.
The number of Harris-type current sheets $k$ depends on the number of separatrix lines
originating in potential X-points. This means that $k$ is determined or fixed by the number
of \lq pauses\rq, i.e., boundary layers, within the chosen domain. The location of the pauses
is marked by $A_{k}$.
Such transformations are ideal for a proper modeling of tail configurations as they appear in
the heliotail \citepads[or astrotails in
general, see][]{2001ohnf.conf...57N, 2006A&A...454..797N}, which require the maintenance of
strong current sheets that form the boundary layer in the vicinity of the seperatrix
(heliopause/astropause) in between the outer solar/stellar wind and the very local
interstellar medium.
\subsubsection{Transformation via peaked plasma flows defined by $M_{A}$}
For the second possible transformation method a Mach number profile $M_A(A)$ has to be
specified. To obtain a highly-structured current distribution, the Mach number profile
needs to contain strong gradients and must show strong spatial variation. This means that
$M_{A}$ cannot be given by a simple two-dimensional function, but has to be constructed
out of several pieces or branches, which need to be connected by continuous transitions,
meaning that each of these branches must be at least twice continuously differentiable at
the boundaries of the intervals so that no discontinuities in the current density profile
appear. Consequently, the function $M_{A}$ must be composed of a set of functions $m_{k}(A)$,
which exist only within some defined field-line interval and vanish outside. In addition,
the functions $m_{k}(A)$ must have a compact support to guarantee that both $m_{k}^{2}(A) < 1$
and $M_{A}^{2} < 1$, i.e., the Mach number and its constituents are bounded functions
(sub-Alfv\'{e}nic).
This goal can be achieved in different ways: (i) One may define piece-wise functions
that vanish at the boundary of some field-line interval; (ii) one may apply purely the
classical partition of unity; (iii) the partition of unity is used in combination with a
continuous sum, i.e., continuous distribution of flow tubes that can be written as an integral.
This means that we can define a general function for $M_{A}$ of the form
\begin{eqnarray}
M_{A}(A)=\sum\limits_{k} m_{k}(A) + \int\limits m(A)\, dA \,\, ,
\end{eqnarray}
which consists of a sum over $k$ individual peaked plasma flows, each defined within
discrete field-line intervals, and a continuous distribution $m(A)$ of plasma flow tubes.
Such an approach with a pure continuous distribution $m(A) \sim 1/\cosh(A)^{2}$ has been used,
e.g., by \citetads{2010AnGeo..28.1523N} to generate a single current-sheet along a magnetic
separatrix. In contrast, we show in Sect.\,\ref{example} an example in which piece-wise
functions are defined.
\subsubsection{Transformation via asymptotical 1D current structures defined by $j$\,: the inverse method}
\label{inversemethod}
The third method to determine the transformation is based on the prescription
of the transformed current, $j$, given by Eq.\,(\ref{currenttrafoexpl6}) \citepads[see
also][]{2006A&A...454..797N}. Typically, in magnetostatics
the magnetic field is directly calculated from Amp\`{e}re's equation $\Delta A
= -j$, where the current distribution $j$ is prescribed. In MHD, a prescription of the
current distribution or the magnetic field is not possible. Here, the values have to be
calculated self-consistently and simultaneously from the nonlinear MHD equations.
But the transformation method enables us to define the current distribution in
configurations that occur ubiquitously in space plasmas as an explicit function of the flux
function $A$. If we can find a way to prescribe the current density $j$
as a spatially, i.e., depending on $A(x,y)$, strongly variable current distribution,
we can generate self-consistently current fragmentations on small scales, emulating
scenarios that in the literature are often approached via turbulence originating
from external perturbations
\citepads{1999ApJ...517..700L, 2009ApJ...700...63K, 2011PhRvE..83e6405E}.
As was shown in Sect.\,\ref{trafo}, the current distribution, resulting from the
transformation, is given by
\begin{equation}
-j=\Delta\alpha=\alpha'' \left(\vec\nabla A\right)^2 + \alpha' \Delta A \, ,
\label{currenttrafoexpl1}
\end{equation}
with $j=j_{z}(x,y)$. The terms $\alpha'', \alpha'$, and $\Delta A$ are pure functions of the
flux function $A$. On the other hand, the quadratic expression $\left(\vec\nabla A\right)^2$
is a scalar function and generally a function of $x$ and $y$, which, in the case of 2D
equilibria, cannot be expressed as an explicit function of $A$ only. Instead,
Eq.\,(\ref{currenttrafoexpl1}) is an equation defining or rather determining
$j(x,y)\equiv j(A,y)$ from a given transformation $\alpha'$, which seems to be the most
consequent and logical method. However, Eq.\,(\ref{currenttrafoexpl1}) cannot be regarded as
just a pure ordinary differential equation for $\alpha'$. Therefore, calculating $\alpha'$
from Eq.\,(\ref{currenttrafoexpl1}) for a given or prescribed $j$, has in general no formal
solution (see Appendix\,\ref{append}). Based on this mathematical problem, it is necessary to
find a different approach.
Magnetohydrostatic equilibria in space plasmas often have regions where the fields are
extremely stretched. Such tail-like regions typically occur far away from bipolar or even
multipolar field regions, as, e.g., in our case (bottom panel of Fig.\,\ref{fieldlinesmhs})
in the regions of high $|x|$ values, or, in the case of the Kan equilibrium, also in the
regions of high $y$ values (top panel of Fig.\,\ref{fieldlinesmhs}), or in general for going
to $\infty$ along or in the direction of the tail axis. The regions of stretched field lines
can be approximated by a 1D configuration, which depends only weakly on a second coordinate.
Examples are asymptotically 1D
regions of exact and analytical tail equilibria or so-called weakly 2D or weakly 3D equilibria
\citepads[see, e.g.,][]{1972ASSL...32..200S, 1977JGR....82..147B, 1979SSRv...23..365S}. The
advantage of this asymptotically 1D approach is that the equilibrium problem can be treated
as if it depended on only one coordinate. This coordinate is, at least locally, a unique
function of the field-line label $A$, and vice versa, so that in a local coordinate system $A$
can always be chosen as coordinate. Therefore, in these stretched field line regions, the
problem can be solved. The solution found is, however, a general solution and not restricted
to the pure 1D region, because every found solution for $|\alpha'| > 1$ is an exact solution
of the sub-Alfv\'{e}nic steady-state problem (Eqs.\,\ref{magtrafo}-\ref{mhs1}).
Assuming that $\lim_{x,y\rightarrow\infty}\vec B_{S}= \vec B_{S\infty}$, we define
$|\vec B_{S\infty}|=B_{S\infty}(A)$. The limes has to be smooth. Then we can
introduce the asymptotical current via $\vec\nabla\times(\lim_{x,y\rightarrow\infty}
\vec B_{S})=\lim_{x,y\rightarrow\infty}(\vec\nabla\times\vec B_{S})=\lim_{x,y\rightarrow\infty}
\vec j_{S}$, with $\lim_{x,y\rightarrow\infty}|\vec j_{S}|=j_{S\infty}(A)= P_{S} \, '(A)$.
The last identification thereby again represents the Grad-Shafranov-equation (Eq.\,\ref{GSE}).
With these relations, Eq.\,(\ref{currenttrafoexpl1}) can be written as
\begin{eqnarray}
- j_{\infty}(A)=\alpha'' (A) B_{S\infty}^2(A) - \alpha'(A)\, j_{S\infty}(A)\, .
\label{currenttrafoexpl3}
\end{eqnarray}
This pure one-dimensional differential relation is now a linear ordinary differential
equation of first order for $\alpha'(A)$, which can be solved: We divide both sides of
Eq.\,(\ref{currenttrafoexpl3}) by $B_{S\infty}^2$ and multiply with the integrating factor
$\exp\left(\int (- j_{S\infty}/B_{S}^2)\, dA\right)$. This leads to a complete differential
that can be integrated, resulting in the following general solution
\begin{eqnarray}
\alpha'= \alpha'(A)=\displaystyle\frac{\displaystyle\int\exp{
\left(\int\,\displaystyle -\frac{j_{S\infty}}{B_{S\infty}^2} \, dA \right)}
\left(-\frac{j_{\infty}}{B_{S\infty}^2}\right) \, dA + C_{0}}
{\exp{\displaystyle\left(\int\, -\frac{j_{S\infty}}{B_{S\infty}^2} dA\right)}} \, .
\label{currenttrafoexpl4}
\end{eqnarray}
Thus for reasonable prescribed stationary asymptotic current densities $j_{\infty}(A)$, the
general solution of the transformation $\alpha'$ can be computed from the asymptotic MHS
functions for the current density $j_{S\infty}(A)$ and the magnetic field $B_{S\infty}(A)$.
The parameter $C_{0}$ is an integration constant, defining an offset and a boundary condition
for $\alpha'$, hence $M_{A}$.
To find suitable prescriptions for $j_{\infty}(A)$ that guarantee $\alpha'^{2}>1$ also in the
two-dimensional regions is a difficult practical task. However, with this approach it is
basically possible to generate current fragmentation scenarios based purely on the
self-consistent solution of the MHD equations. Hence, the transformation method provides a
self-consistent tool in which current fragmentation is a basic property of the nonlinear
MHD theory, because $j_{\infty}$ is basically not subject to any limitations.
\subsection{Example for shear-flow-induced current fragmentation}
\label{example}
\begin{figure}
\centering
\includegraphics[width=\hsize]{mazalow.pdf}
\caption{Constructed Mach number profile.}
\label{machnumber1}
\end{figure}
The aim of our analysis is to study the process of current fragmentation that takes place in the
vicinity and within an ejected plasmoid that was formed via magnetic reconnection in a
typical solar eruptive flare. Observations of such flare processes indicate that the
surrounding material on the open field-lines is moving upwards, while the plasma below the
X-point located in between the two plasmoids (see Fig.\,\ref{fieldlinesmhs}), i.e., within
the closed field-line region, can be assumed to be static\footnote{In dynamical flare
scenarios, the plasma within the closed arcade structure
tends to flow downwards, i.e. back to the surface because of plasma cooling, which forces the
arcade structure to shrink. However, our model aims at studying the post-eruptive flare
phase, in which the photospheric layers (i.e., the arcade structures at the bottom of our
configuration) are back in static equilibrium \citepads[see, e.g.,][]{2012ApJ...757L...5W}.}.
With this picture, it is more
convenient to apply the transformation method based on the prescribed nonzero
sub-Alfv\'{e}nic Mach number profile rather than based on the asymptotic current
distribution or the magnetic field amplification, because the latter two are extremely
difficult to extract from observations, in particular because of the still-lacking high
enough spatial resolution.
In a bipolar MHS structure, assuming the main direction of the magnetic field to be the
$y$--direction, i.e., $B_{y}>B_{x}$ outside the outer separatrix, the main component of the
magnetic field, $B_{y}$, changes its direction and therefore its sign. For
symmetric magnetic-field lines with respect to the $y$-axis, $A$ is a symmetric function of
$x$ (i.e., $\vec B_{S}$ is anti-symmetric and $A$ is symmetric with respect to the $y$--axis).
As $M_{A}$ is a function of $A$, and the plasma flow is required to be purely upstreaming
on both sides (boundary condition), the Mach number profile needs to change its sign.
Consequently, one needs to define a piecewise function $M_{A}(A)$ with at least two
different branches (left and right of the outer separatrix). Otherwise, $M_{A}$ cannot change
its sign.
\begin{figure*}
\centering
\includegraphics[width=\hsize]{mapping.pdf}
\caption{Static (left) versus stationary (right) current (top)
and its isocontours (bottom). For better visualization the current is plotted inversely and
cut off at the numerical value of 2. The maximum at the origin approaches a numerical
value of 6.}
\label{mappingfigs}
\end{figure*}
The Mach number profile, which we apply now to simulate such a scenario, consists of several
branches: Within the dipole-like region inside the closed field-lines, which is assumed to be
static, $M_{A} = 0$. In the regions outside the outer separatrix, we assume that the Mach
number profile is symmetric, i.e., $M_{A}(-x,y)=-M_{A}(x,y)$, although asymmetric profiles
might also be possible. We furthermore assume that for $x<0$ the plasma flow is parallel to
the magnetic field ($M_{A} > 0$), while for $x>0$ it is antiparallel ($M_{A} < 0$). To
guarantee a continuous transition between the positive and negative Mach number branches
(i.e., to have a smooth flow pattern), $M_{A}$ must vanish at the separatrix itself. To
ensure that the current $j$ is also continuous, $M_{A}$ and its derivative must be
continuously differentiable in the boundary region (separatrix). Furthermore, we assume that
the upper, disconnected plasmoid contains a vortex. This assumption is reasonable,
because any small asymmetry during the reconnection process and the disconnection of the
plasmoid will immediately result in a nonzero angular momentum and hence in a rotational
motion of the plasma. Hence, its representation in the Mach number profile is given by a
maximum in the center of the plasmoid, and strong gradients from the center to its edges.
With these specifications, our Mach number profile covering the region in $x$ and $y$ as
defined by the MHS configuration (see Fig.\,\ref{fieldlinesmhs}) is given by the following
four branches
\begin{displaymath}
M_{A} =
\left\lbrace \begin{array}{lcl}
-0.5 f \left( 1-\displaystyle\frac{1}{1+ \left(\frac{A-A_{\rm sep}}{A_{b}}\right)^{2} }\right) & \textrm{for} &
A > A_{\rm sep}\, ,\quad x > 0 \\
0.5 f \left( 1-\displaystyle\frac{1}{1+ \left(\frac{A-A_{\rm sep}}{A_{b}}\right)^{2} }\right) & \textrm{for} &
A > A_{\rm sep}\, ,\quad x < 0 \\
0.9 f_{p}\left( 1-\displaystyle\frac{1}{1+ \left(\frac{A-A_{\rm sep}}{A_{b}}\right)^{2} }\right) & \textrm{for} &
A < A_{\rm sep}\, ,\quad y > 6.5 \\
0 & & \textrm{elsewhere}\, ,
\end{array}
\right.
\end{displaymath}
it is displayed in Fig.\,\ref{machnumber1}. Hereby $A_{\rm sep}$ represents the outer
separatrix and has the numerical value $A_{\rm sep}=0.0875$,
and $A_{b}$ is a parameter influencing the steepness of the Mach number profile, and therefore
the width of the current sheets. For our model computations we choose $A_{b}=0.1$. The
parameters $f$ and $f_{p}$ are functions of $A$, simulating small wave-like spatial fluctuations.
For the example presented in Fig.\,\ref{mappingfigs}, we used $f = 1- 0.1 \sin{(1.1 A)}$ and
$f_{p}=1$.
Starting from the MHS equilibrium configuration for the flux function and its corresponding
current distribution (see Sect.\,\ref{nonlinstat}), we applied the mapping defined by the
Mach number profile. The resulting current and its isocontour lines are displayed in
the upper and lower right panels of Fig.\,\ref{mappingfigs}. Obviously, the current
distribution shows new features, which did not exist in the static case (left panels of
Fig.\,\ref{mappingfigs}). These are ring-like and crescent-shaped structures around both the
lower, static configuration and the upper disconnected
plasmoid. In both cases, these new current sheets are located outside but along the
separatrix. In addition, inside the detached plasmoid, the current appears dome-like in the
center, and two more current sheets (current \lq islands\rq) formed close to the separatrix.
These new current structures (sheets, islands, maximum) are
also visible in the isocontour plot. There, additional butterfly-like current islands
appear in the vicinity of the X-point of the outer separatrix.
The strongest currents can be recognized at the \lq Kan dipole\rq-region,
because the increase of $A$ and $|\vec B_{S}|$ in the region of the pole of $A$ results in
strong currents already in the MHS-state. Applying the shear at the outer separatrix
enforces this effect in particular at the bottom separatrix (in the vicinity of the
photosphere), generating two additional current peaks.
In the center of the static configuration (where $M_{A}$ was set to zero),
the current structure has not changed. In contrast to the Grad-Shafranov theory, the stationary
current isocontours do not (completely) resemble the field-line structure anymore.
To highlight the motion of the plasma, we display in Fig.\,\ref{flowxy} the $x$ and $y$
components of the normalized (with respect to density) plasma velocity field. The $y$
component shows that in the outside regions it is always positive, in agreement with an
upstream behavior of the flow on the open field-lines. The clockwise rotational flow of the
plasma within the upper plasmoid is obvious from the $y$ component of the flow being positive
on the left side of the plasmoid, and negative on the right side. The $x$ component of the
flow is generally very small with a wavy structure due to the curved open field-lines in the
vicinity of the outer separatrix. Only the circular flow inside the upper plasmoid has
slightly higher velocity.
\begin{figure}
\includegraphics[width=\hsize]{flowx.pdf}
\includegraphics[width=\hsize]{flowy.pdf}
\caption{$x$ and $y$ components of the plasma flow field.}
\label{flowxy}
\end{figure}
In summary, from an initially smooth current
distribution our applied mapping created a new distribution, which shows multiple current
filaments that can be regarded as current fragmentation.
\begin{figure}
\includegraphics[width=\hsize]{mazaAb001fp.pdf}
\includegraphics[width=\hsize]{currentinvAb001fp.pdf}
\includegraphics[width=\hsize]{currentisocontourAb001fp.pdf}
\caption{Mach number profile (top), transformed inverse current density (middle) and its
isocontours (bottom), for a narrower intrinsic current sheet with $A_{b} = 0.01$
and an intrinsically structured plasmoid with $f_{p} \neq 1$.}
\label{Ab001}
\end{figure}
\section{Discussion}
\label{sec4}
The results shown in the previous section serve as an illustration. There, the width
of the current sheet, prescribed by the parameter $A_{b}$, was set to a value of 0.1, which
resulted in pure, crescent-shaped current sheet structures. However, decreasing the value of
$A_{b}$, i.e., steepening the Mach number profile and shrinking the width of the current sheet
resulted in additional fragmentation of the crescent-shaped current sheet into several strong current
peaks, as shown in the middle panel of Fig.\,\ref{Ab001} for which $A_{b}$ was set to 0.01. For
better visualization the current was cut at the numerical value of ten. From these results we may
thus conclude that fragmentation is enforced when reaching smaller scales of the shear flows.
Furthermore, fragmentation of the flux rope's current density profile occurs when the parameter
$f_{p}$ is different from 1. In the example depicted in Fig.\,\ref{Ab001} we used
$f_{p} = 1-0.1\sin[1/(A^{2}+0.01)]$ and $f=1$. The corresponding Mach number profile, which now
already shows a small-scale structure imprinted on the plasmoid, is shown in the top panel of
Fig.\,\ref{Ab001}. The fine-structure obtained in the transformed current is obvious from
both the current density profile and its isocontours (bottom panel of Fig.\,\ref{Ab001}).
For better visualization we also show in Fig.\,\ref{Zoom} a high-resolution zoom of the
isocontours and the inverse current density for the same model parameters as in
Fig.\,\ref{Ab001}. The zoomed region contains the left side of the plasmoid. The plot of
the isocontours demonstrates that the topology of the current isolines is much more
complex than the one of the flux function isolines, i.e., the magnetic-field lines.
The bottom panel of Fig.\,\ref{Zoom} displays the comparison between the static
(dashed lines) and the stationary, fragmented current density (solid line) along the $y$-axis
within the plasmoid region for $x=-0.4$. This plot highlights the strong spatial variation of
the stationary current density, which shows steep gradients that imply fragmentation of the
initially smooth current sheet. This relatively simple example stresses that to achieve
fragmentation on much smaller scales, it is essential to use a Mach number profile, which is much
more complex and contains highly alternating structures, e.g., in the form of saw-tooth-like or
other oscillating functions. Furthermore, every Mach number profile could in principle successively
and infinitely be refined by, e.g., an iterative scheme of the form $f((M_{A})_{n}) =
(M_{A})_{n+1}$. Such an iterative mapping can be performed because $(M_{A})_{n}$ is constant along
the field lines, so every regular function or mapping has to be constant on the field lines as
well. These iterations define fractal structures and hence demonstrate the fractal nature of MHD.
The concept of fragmentation in the frame of ideal MHD remains valid down to length scales of
100\,m to $\sim 5$\,m for conditions typical for the solar corona. The value of $\sim 5$\,m
thereby corresponds to the ion inertial length, defined as the ratio of the speed of light
and the ion plasma frequency. On length scales similar to and shorther than the ion
inertial length, either the Hall-MHD or the two-fluid MHD needs to be applied.
\begin{figure}
\includegraphics[width=\hsize]{zoom_currentisocontour.pdf}
\includegraphics[width=\hsize]{zoom_currentinv.pdf}
\includegraphics[width=\hsize]{currentcomp.pdf}
\caption{High-resolution zoom into the plasmoid region for the same model as in
Fig.\,\ref{Ab001}. Shown are the isocontours (top), the transformed inverse current
density (middle), and a cut through the current density at $x=-0.4$ (bottom) for the
stationary ($j_{z}$, solid) and the static case ($j_{zs}$, dashed).}
\label{Zoom}
\end{figure}
In our analysis we ignored resistive or nonideal effects to guarantee the existence
of plausible stationary flows. However, the presence of nonideal terms, particularly
in the shape of a resistivity on the right-hand side of Ohm's law, does not automatically
imply the nonexistence of stationary solutions. The inclusion of a resistivity, $\eta$,
such that $\vec\nabla\times \left( \eta \vec j\right) = \vec 0$, supports stationary
nonideal MHD flows and hence the existence of ideal equilibria. The stationarity
of Maxwell equations in 2D demands that the electric-field component $E_{z} = \eta j_{z}$ is
constant. As $E_{z}$ is at the same time the reconnection rate, this implies that the
reconnection rate is independent of the resistivity \citepads[e.g.,][]{2006PhPl...13c2307K}.
Consequently, even if, as in our case, the flows are field-aligned and steady-state, these
MHD flows can be regarded as an analogy to steady-state reconnection solutions with constant
reconnection rate. The existence of resistive steady states with field-aligned flows and
reasonable resistivity profiles has been shown by \citetads{2000JPlPh..64..601T,
2003PhPl...10.2382T}. Under such conditions in our 2D case, Ohmic heating of the plasma is
directly proportional to $j_{z}$ and occurs everywhere where filamentation or fragmentation
takes place and could in principle contribute (at least partially) to the heating of the
corona.
Although we had limited our analysis to a pure 2D configuration, the transformation technique
is valid in all dimensions because it is based on vector analysis identities. Therefore, starting
from a 3D MHS equilibrium, the mapping would deliver current fragmentation also in 3D. However,
to find suitable MHS equilibria as starting configurations is a difficult task, hence
pre-computed fully 3D MHS equilibria are so far rare. MHS equilibria for laminar
flows and magnetic fields have been constructed by, e.g., \citetads{1999GApFD..91..269P}, which
might serve as starting points for a future 3D analysis.
\section{Conclusions}
\label{sec5}
Observations of the solar atmosphere with increasing spatial resolution reveal that the atmosphere
is highly structured or fragmented. Hence, the mechanisms initiating the formation of small-scale
structures, such as jets, flares, and plasmoids, which typically occur as a result of magnetic
reconnection processes of current sheets, must be inherently fractal.
Although solutions of the MHD equations pretend that physical parameters, such as the magnetic
field or the current, are smooth on large scales, they do not necessarily have to be smooth on small
scales. This is shown by our analysis, in which, starting from an MHS equilibrium with a smooth
current distribution for a stationary plasmoid configuration, we obtained a current structure
displaying steep gradients, i.e., strong spatial variations of the current density, as well as
an internally fragmented plasmoid, depending on the initially chosen Mach number profile.
Hence, pure MHD equilibria are able to display intrinsic fine structure, which
can serve as the seeds for instabilities, i.e., as ``secondary instabilities''
\citepads[see, e.g.,][]{2010AIPC.1242...89P}, and therefore as triggering mechanisms for
second-generation current fragmentation.
Because the MHD equations are scale-free, our results are valid
not only for the global flare scale, but also for scales close to dissipation
scales.
As a natural next step, our stationary equilibrium configuration should be implemented into
MHD simulations as the starting configuration, to see and test the onset of instabilities and the
time-dependent evolution of the resulting additional current fragmentation.
| {'timestamp': '2013-06-24T02:02:02', 'yymm': '1306', 'arxiv_id': '1306.5155', 'language': 'en', 'url': 'https://arxiv.org/abs/1306.5155'} |
\subsubsection*{References}}
\usepackage[utf8]{inputenc}
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\newcommand\mohamed[1]{{\color{red} #1}}
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\title{Gumbel-Softmax Selective Networks}
\author{%
Mahmoud Salem$^{1,2,3}$\thanks{This work was done during an internship at Borealis AI.} \quad Mohamed Osama Ahmed$^1$ \quad Frederick Tung$^1$ \quad Gabriel Oliveira$^1$ \\
$^1$Borealis AI \quad $^2$Vector Institute \quad $^3$University of Guelph\\
\texttt{[email protected]} \\
\texttt{\{mohamed.o.ahmed, frederick.tung, gabriel.oliveira\}@borealisai.com} \\
}
\begin{document}
\maketitle
\begin{abstract}
ML models often operate within the context of a larger system that can adapt its response when the ML model is uncertain, such as falling back on safe defaults or a human in the loop. This commonly encountered operational context calls for principled techniques for training ML models with the option to abstain from predicting when uncertain. Selective neural networks are trained with an integrated option to abstain, allowing them to learn to recognize and optimize for the subset of the data distribution for which confident predictions can be made. However, optimizing selective networks is challenging due to the non-differentiability of the binary selection function (the discrete decision of whether to predict or abstain). This paper presents a general method for training selective networks that leverages the Gumbel-softmax reparameterization trick to enable selection within an end-to-end differentiable training framework. Experiments on public datasets demonstrate the potential of Gumbel-softmax selective networks for selective regression and classification.
\end{abstract}
\section{Introduction}\label{sec:intro}
When an ML model is uncertain about its prediction, for example due to the uniqueness of the input with respect to previously observed training samples, it is often preferable for the model to abstain from making a prediction, instead of making a poor prediction that could erode user confidence or lead to harmful downstream consequences. In cases of abstention, the system may fall back on expert judgment or safe defaults. The automatic learning of an abstention policy frees ML system developers from having to hand-craft a set of selection rules based on heuristics.
\textit{Selective networks} are trained with an integrated reject option, i.e., the option to abstain from making a prediction when the model is uncertain \citep{geifman2019selectivenet}.
Optimizing selective networks is challenging because of the non-differentiability of the binary selection operation (the decision of whether to select or abstain). In the conventional formulation of selective networks, the non-differentiability of selection is handled by replacing the binary selection operation with a soft relaxation. However, this approximation means that in practice the selective network does not perform selection during training, but instead assigns a soft instance weight to each training sample.
In this paper, we present Gumbel-softmax selective networks, which enable binary selection decisions during training while preserving end-to-end differentiability using the Gumbel-softmax reparameterization trick \citep{jang2017,maddison2017}.
The proposed technique for training selective networks is general and does not assume a particular prediction task (e.g. classification). It leverages a principled tool to perform selection or abstention within an end-to-end training framework. Experiments on four public datasets demonstrate the potential of Gumbel-softmax selective networks for both selective regression and selective classification tasks.
\section{Related Work}
In practice, it is often useful for an ML system to have the option of abstaining from making a prediction when it detects a situation of high uncertainty. Given that the system has the option to abstain, an important question to ask is how we can train the model \textit{with the knowledge that it is allowed to abstain}. By integrating this option into model training, the model can learn to automatically recognize and optimize for the part of the data distribution for which confident predictions can be made, instead of attempting to fit the entire data distribution at training time and applying hand-crafted abstention rules at inference time.
How to train a neural network with the knowledge that it is allowed to abstain has received relatively little attention in the ML community.
\citet{geifman2019selectivenet} proposed the modern selective network (SelectiveNet), which adds a dedicated selection head to the base network. The network is trained to optimize the task performance criterion, such as classification accuracy, given a target level of coverage: the proportion of input samples for which the network should make predictions.
\citet{Liu2019deepgamblers} proposed to add the abstention option as a separate class that can be predicted. A threshold is applied to the score of the abstention class to achieve a desired level of coverage without re-training. However, this approach can be applied to classification networks only. We propose a general approach that can be applied to any predictive task.
\citet{huang2020selfadaptivetraining} used the selective classification task to illustrate the potential of their self-adaptive training technique, which improves generalization performance in the presence of noisy training data.
\section{Method}
\subsection{Preliminaries: Selective Networks}
\label{subsec:selectivenet}
A \textit{selective neural network} can be defined as a pair $(f,g)$, where $f$ is a prediction function and $g$ is a binary selection function, such that the output of the network is given by \citep{geifman2019selectivenet}:
\begin{equation}
(f,g)(x) =
\begin{cases}
f(x) & \text{ if } g(x)=1 \\
\text{Abstain}& \text{ if } g(x)=0.
\end{cases}
\end{equation}
Selective networks trade off prediction performance against coverage: the proportion of input samples that the network selects (i.e., makes predictions for). Given a set of $m$ training data points $\{{(x_i, y_i)}\}_{i=1}^{m}$, the empirical coverage is defined as
\begin{equation}
\widehat{\phi}(g) = \frac{1}{m} \sum_{i=1}^{m} g(x_i) \, ,
\end{equation}
and the empirical selective risk is defined as
\begin{equation}
\widehat{r}(f,g) = \frac{\frac{1}{m} \sum_{i=1}^{m} \ell(f(x_i),y_i)g(x_i)}{\widehat{\phi}(g)} \, ,
\end{equation}
where $\ell$ is a loss function such as cross-entropy for classification or mean squared error for regression.
The overall training objective is then a weighted combination of the empirical selective risk and a penalty term that penalizes differences between the empirical coverage and a pre-specified target coverage:
\begin{equation}
\label{eq:Lfg}
\mathcal{L}_{(f,g)} = \widehat{r}(f,g) + \lambda \Psi (c - \widehat{\phi}(g)) \, ,
\end{equation}
where $c$ is a pre-specified target coverage, $\Psi$ is a penalty function (e.g. $\Psi(a) = max(0,a)^2$), and $\lambda$ is a balancing hyperparameter.
Optimizing Eq.~\ref{eq:Lfg} is challenging because of the non-differentiability of the binary selection function $g$. \citet{geifman2019selectivenet} handle the non-differentiability of selection by replacing the binary function $g$ with a relaxed function $g : \mathcal{X} \to [0, 1]$. While this addresses the differentiability issue, the approximation means that in practice the selective network does not perform selection during training, but instead assigns a soft instance weight to each training sample.
This introduces a gap between training and inference.
To address this discrepancy, in the following we describe a differentiable method for enabling binary selection during training while preserving end-to-end training using the Gumbel-softmax reparameterization trick.
\subsection{Gumbel-softmax Selective Networks}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{GumbelFigure_v3.png}
\caption{Gumbel-softmax selective networks leverage the Gumbel-softmax reparameterization trick \citep{jang2017,maddison2017} to enable selection (abstention) decisions within an end-to-end differentiable training framework. The temperature parameter $\tau$ is annealed over time such that the softmax approaches the argmax.}
\label{fig:summary}
\end{figure}
The reparameterization trick \citep{kingma2014,rezende2014} in deep learning allows us to replace a stochastic computation graph by a differentiable computation graph with learnable parameters, acting on noise from a fixed base distribution. For example, suppose we want a stochastic node in a neural network that performs sampling from a normal distribution parameterized by mean $\mu$ and standard deviation $\sigma$. We cannot backpropagate through this stochastic node because of the non-differentiability of the sampling operation. However, we can replace this stochastic node with a parameterized differentiable computation that takes noise as input: the computation takes input noise sampled from the standard normal $\mathcal{N}(0, 1)$, scales it by $\sigma$, and then shifts the result by $\mu$. Since $\mu$ and $\sigma$ can be generated by deterministic neural network layers trainable by backpropagation, this reparameterization effectively enables sampling from an arbitrary, learnable normal distribution.
We now revisit the conventional selective network formulation and show how we can use the reparameterization trick to perform binary selection while preserving end-to-end training.
Let us re-define the output of $g$ as the probability of selecting the input (i.e., the probability the network should make the prediction instead of abstaining). The selection function becomes a stochastic operator that selects the input with probability $g$. Similar to the example at the beginning of this subsection, we have a stochastic node that performs a sampling operation. However, instead of sampling from a normal distribution, we want to sample from the Bernoulli distribution, $Bernoulli(g)$.
The Gumbel-softmax reparameterization trick \citep{jang2017,maddison2017} allows us to reparametrize a stochastic node that samples from a categorical distribution, again by replacing it with a differentiable function of learnable parameters, acting on noise from a base distribution. Given a categorical distribution of $k$ events with probability $\pi_1, ..., \pi_k$, we compute $\log \pi_1, ..., \log \pi_k$, and to each of these terms we add i.i.d. noise sampled from the Gumbel distribution \citep{gumbel1954}. We can then draw a stochastic sample $z$ (represented by a one-hot vector) by taking the argmax:
\begin{equation}
z = one\_hot (\argmax_i [G_i + \log \pi_i]) \, ,
\end{equation}
where $G_i \sim Gumbel(0, 1)$. To allow end-to-end training, we approximate the argmax with a softmax, which gives a softened vector $\tilde{z}$:
\begin{equation}
\tilde{z}_i = \frac{\exp \, ((\,\log\pi_i+G_i))/\tau)}{{\sum}_{j=1}^k{\exp \, ((\,\log\pi_j+G_j)/\tau)}} \, , \quad \text{for } i=1,...,k
\end{equation}
The temperature parameter $\tau > 0$ determines the sharpness of the softmax, and is annealed over time towards zero to recover the argmax. As $\tau \to \infty$, the Gumbel-softmax distribution converges to the uniform distribution, and as $\tau \to 0$, the Gumbel-softmax distribution converges to the categorical distribution. Therefore, we have moved the dependency on parameters $\pi_1, ..., \pi_k$ from the non-differentiable stochastic sampling function to a differentiable function consisting of softmax and log operations acting on base noise, which can be trained end-to-end with backpropagation.
Putting it all together, we perform binary selection by applying the Gumbel-softmax reparameterization trick with $\pi_1=g, \pi_2=1-g$. In the forward pass, we use the argmax form to perform binary selection. In the backward pass, we use the softmax form with temperature annealing to compute gradients and enable end-to-end training.
Figure \ref{fig:summary} shows a visual summary of the proposed approach.
\section{Experiments}
In this section, we demonstrate the potential of Gumbel-softmax selective networks on four public datasets. Due to space limitations, we defer dataset and implementation details to the supplementary.
Selective networks trained at the same level of target coverage may differ in the actual coverage achieved in evaluation (i.e., the number of predictions made on the test set).
For a fair comparison, we apply coverage calibration \citep{geifman2019selectivenet} to equalize the number of test predictions across all approaches. For example, when evaluating at a coverage level of 70\%, we compute the error metrics over the 70\% most confident predictions (highest $g$ values) among the test samples.
Table \ref{tab:regression_results} summarizes the experimental results for Gumbel-softmax selective networks and SelectiveNets on three public regression datasets, averaged over five trials.
We train all models from scratch, and for a fair comparison all shared hyperparameters and train budgets are the same.
On the Concrete Compressive Strength dataset, the results we obtain for SelectiveNet are better than those reported in the original paper \citep{geifman2019selectivenet} as we found that applying a learning rate decay schedule, instead of a constant learning rate as in \citet{geifman2019selectivenet}, substantially boosts performance.
Gumbel-softmax selective networks consistently outperform SelectiveNets at every coverage level on all three regression datasets.
\begin{table}
\centering
\scriptsize
\caption{Selective regression results on Concrete Compressive Strength, California Housing, and Ames Housing datasets. For the housing datasets, errors are computed in units of \$10,000. We highlight in bold the lowest error rates.}
\begin{tabular}{c|cc|cc|cc}
\toprule[1.2pt]
\multirow{3}{*}{Coverage} & \multicolumn{2}{c|}{Concrete Compressive Strength} & \multicolumn{2}{c|}{California Housing} & \multicolumn{2}{c}{Ames Housing} \\
& \multicolumn{2}{c|}{MSE ($\downarrow$)} & \multicolumn{2}{c|}{MAE (10,000's, $\downarrow$)} & \multicolumn{2}{c}{MAE (10,000's, $\downarrow$)} \\
& Gumbel-softmax & SelectiveNet & Gumbel-softmax & SelectiveNet & Gumbel-softmax & SelectiveNet \\
\midrule \midrule
100 & 32.84$\pm$2.50 & 32.82$\pm$0.67 & 4.51$\pm$0.03 & 4.55$\pm$0.05 & 1.68$\pm$0.07 & 1.64$\pm$0.04 \\
90 & \textbf{25.13$\pm$1.22} & 26.56$\pm$2.82 & \textbf{4.19$\pm$0.05} & 4.36$\pm$0.11 & \textbf{1.22$\pm$0.04} & 1.25$\pm$0.05 \\
80 & \textbf{21.15$\pm$0.83} & 21.80$\pm$3.25 & \textbf{3.92$\pm$0.07} & 4.24$\pm$0.17 & \textbf{1.10$\pm$0.05} & 1.11$\pm$0.03 \\
70 & \textbf{16.17$\pm$1.85} & 18.59$\pm$2.50 & \textbf{3.66$\pm$0.04} & 3.97$\pm$0.18 & \textbf{1.04$\pm$0.01} & 1.07$\pm$0.03 \\
60 & \textbf{13.72$\pm$2.44} & 17.59$\pm$2.23 & \textbf{3.38$\pm$0.09} & 3.99$\pm$0.23 & \textbf{0.97$\pm$0.03} & 1.00$\pm$0.04 \\
50 & \textbf{11.15$\pm$2.11} & 14.43$\pm$2.57 & \textbf{3.22$\pm$0.15} & 3.78$\pm$0.15 & \textbf{0.95$\pm$0.06} & 1.01$\pm$0.05 \\
\bottomrule[1.2pt]
\end{tabular}
\label{tab:regression_results}
\end{table}
Following \cite{feng2022}, Table \ref{tab:classification_results} summarizes the experimental results on the ImageNet-100 dataset, averaged over five trials. Gumbel-softmax selective networks modestly outperform SelectiveNets at higher coverage levels; both methods perform comparably at lower coverage levels.
\begin{table}
\centering
\footnotesize
\caption{Selective classification results on ImageNet-100. We highlight in bold the lowest error rates.}
\begin{tabular}{c|cc}
\toprule[1.2pt]
\multirow{3}{*}{Coverage} & \multicolumn{2}{c}{ImageNet-100} \\
& \multicolumn{2}{c}{Top-1 Accuracy ($\uparrow$)} \\
& Gumbel-softmax & SelectiveNet \\
\midrule \midrule
100 & 86.16$\pm$0.15 & 86.07$\pm$0.11 \\
90 & \textbf{89.76$\pm$0.64} & 88.68$\pm$0.30 \\
80 & \textbf{93.33$\pm$0.47} & 92.59$\pm$0.18 \\
70 & \textbf{96.03$\pm$0.33} & 95.86$\pm$0.45 \\
60 & 97.79$\pm$0.34 & \textbf{97.83$\pm$0.28} \\
50 & \textbf{99.12$\pm$0.49} & 99.06$\pm$0.23 \\
\bottomrule[1.2pt]
\end{tabular}
\label{tab:classification_results}
\end{table}
\section{Conclusion}
ML models are often deployed not in isolation, but as part of a larger system, with non-ML logic, legacy processes, or humans in the loop. In operational contexts where the system has the option of falling back on supporting processes when the ML model is uncertain, the option to abstain should be integrated directly in the ML model training. We hope that our ideas on how to train selective networks will reinvigorate interest in this practical problem.
| {'timestamp': '2022-11-22T02:03:44', 'yymm': '2211', 'arxiv_id': '2211.10564', 'language': 'en', 'url': 'https://arxiv.org/abs/2211.10564'} |
\section{Introduction}
Let $G$ be a compact, connected and simple Lie group with group unit $e\in G
. \textsl{The inverse involution} on $G$ is the periodic $2$ transformation
\gamma $ sending each group element $g\in G$ to its inverse $g^{-1}\in G$.
In this paper we present a general procedure to calculate the isomorphism
type of the fixed set
\begin{quote}
$Fix(\gamma )=\{g\in G\mid g=g^{-1}\}$
\end{quote}
\noindent of the involution $\gamma $.
Given a group element $x\in G$ let $M_{x}$, $C_{x}\subset G$ be the adjoint
orbit through $x$ and the centralizer of $x$ in $G$, respectively. That is
\begin{quote}
$M_{x}=\{gxg^{-1}\in G\mid g\in G\}$; $C_{x}=\{g\in G\mid gx=xg\}$.
\end{quote}
\noindent The map $G\rightarrow G$ by $g\rightarrow gxg^{-1}$ is constant
along the left cosets of $C_{x}$ in $G$, and induces a diffeomorphism from
the homogeneous space $G/C_{x}$ onto the orbit space $M_{x}$
\begin{enumerate}
\item[(1.1)] $f_{x}:G/C_{x}\overset{\cong }{\rightarrow }M_{x}$,
[g]\rightarrow gxg^{-1}$.
\end{enumerate}
\noindent In view of this identification the isomorphism type of the orbit
space $M_{x}$ is completely determined by the centralizer $C_{x}$. It is
crucial to notice that $x\in Fix(\gamma )$ implies that $M_{x}\subset
Fix(\gamma )$. Naturally, one asks for a partition of the space $Fix(\gamma
) $ by certain adjoint orbits $M_{x}$, and determine the isomorphism types
of the corresponding centralizers $C_{x}$.
Concerning the applications of our approach we assume the reader's
familiarity with the classification on Lie groups. In particular, all $1
--connected compact simple Lie groups consists of the three infinite
families $SU(n+1),Sp(n),$ $Spin(n+2)$, $n\geq 2,$ of \textsl{classical group
}, and the five \textsl{exceptional Lie groups}
G_{2},F_{4},E_{6},E_{7},E_{8}$. For a classical Lie group $G$ the fixed set
Fix(\gamma )$ can be easily calculated using linear algebra, see Frankel
\cite{[F]}. For this reason we shall restrict ourself to the simple
exceptional Lie groups. Explicitly we shall have
\begin{quote}
$G=G_{2},F_{4},E_{6},E_{7},E_{8}$ or $E_{6}^{\ast },E_{7}^{\ast }$,
\end{quote}
\noindent where $G^{\ast }=G/\mathcal{Z}(G)$ with $\mathcal{Z}(G)$ the
center of $G$.
Fix a maximal torus $T$ in $G$ and let $\exp :L(T)\rightarrow T$ be the
exponential map, where $L(T)$ is the tangent space to $T$ at the unit $e$.
In term of a set $\Omega =\{\omega _{1},\cdots ,\omega _{n}\}\subset L(T)$
of fundamental dominant weights of $G$ (see Definition 2.3), together with
the fundamental Weyl cell $\Delta $ corresponding to $\Omega $, our main
result is stated below.
Let $SO(n)$ and $Ss(n)$ be the \textsl{special orthogonal group} and the
\textsl{semispinor group} of order $n$, respectively. For a connected Lie
group $H$ write $[H]^{2}$ for the group with two components whose identity
component is $H$. For two manifolds $M$ and $N$ denote by $M\coprod N$ their
disjoint union.
\bigskip
\noindent \textbf{Theorem 1.1.} \textsl{For a simple Lie group }$G$\textsl{\
there is a subset }$\mathcal{F}_{G}\subset $\textsl{\ }$\Delta $\textsl{\ so
that}
\begin{enumerate}
\item[(1.2)] $Fix(\gamma )=\{e\}\coprod\limits_{u\in \mathcal{F}_{G}}M_{\exp
(u)}$\textsl{.}
\end{enumerate}
\noindent \textsl{Moreover, for each exceptional Lie group }$G$ \textsl{the
set }$\mathcal{F}_{G}$\textsl{, as well as the isomorphism type of the
adjoint orbit} $M_{\exp (u)}$\textsl{\ with }$u\in \mathcal{F}_{G}$\textsl
,\ is tabulated below}
\begin{center}
\begin{tabular}{l|l|l}
\hline\hline
$G$ & $\mathcal{F}_{G}$ & ${\small M}_{\exp (u)}{\small =G/C}_{\exp (u)}$,
u\in \mathcal{F}_{G}$ \\ \hline
$G_{2}$ & $\{\frac{{\small \omega }_{{\small 1}}}{2}\}$ & ${\small G}_{2
{\small /SO(4)}$ \\
$F_{4}$ & $\{\frac{{\small \omega }_{{\small k}}}{2}\}_{{\small k=1,4}}$ &
{\small F}_{4}{\small /Spin(9),}$ ${\small F}_{4}{\small /}\frac{{\small
Sp(3)\times Sp(1)}}{{\small Z}_{2}}$ \\
$E_{6}$ & $\{\frac{{\small \omega }_{{\small 2}}}{2},\frac{\omega
_{1}+\omega _{6}}{2}\}$ & ${\small E}_{6}{\small /}\frac{{\small SU(2)\times
SU(6)}}{{\small Z}_{2}}{\small ,}$ ${\small E}_{6}{\small /}\frac{{\small
Spin(10)\times S}^{{\small 1}}}{{\small Z}_{4}}$ \\
$E_{6}^{\ast }$ & ${\small \{}\frac{{\small \omega }_{{\small k}}}{2}{\small
\}}_{{\small k=1,2}}$ & ${\small E}_{6}^{\ast }{\small /}\frac{{\small
Spin(10)\times S}^{{\small 1}}}{{\small Z}_{4}}{\small ,}$ ${\small E
_{6}^{\ast }{\small /}\frac{{\small SU(2)\times }\frac{{\small SU(6)}}{Z_{3}
}{{\small Z}_{2}}$ \\
$E_{7}$ & $\{\frac{{\small \omega }_{{\small k}}}{2},\omega _{{\small 7}}\}_
{\small k=1,6}}$ & ${\small E}_{7}{\small /}\frac{{\small Spin(12)\times
SU(2)}}{{\small Z}_{2}}{\small ,}$ ${\small E}_{7}{\small /}\frac{{\small
Spin(12)\times SU(2)}}{{\small Z}_{2}}{\small ,}${\small \ }${\small \exp
(\omega _{7})}$ \\
$E_{7}^{\ast }$ & ${\small \{}\frac{{\small \omega }_{{\small k}}}{2}{\small
\}}_{{\small k=1,2,7}}$ & ${\small E}_{7}^{\ast }{\small /}\frac{{\small
Ss(12)\times SU(2)}}{{\small Z}_{2}}{\small ,}$ ${\small E}_{7}^{\ast
{\small /}[\frac{{\small SU(8)}}{Z_{4}}{\small ]}^{2}{\small ,}$ ${\small E
_{7}^{\ast }{\small /}[\frac{{\small E}_{6}{\small \times S}^{1}}{{\small Z
_{3}}]^{2}$ \\
$E_{8}$ & $\{\frac{{\small \omega }_{{\small k}}}{2}\}_{{\small k=1,8}}$ &
{\small E}_{8}{\small /Ss(16),}$ ${\small E}_{8}{\small /}\frac{{\small E}_
{\small 7}}{\small \times SU(2)}}{{\small Z}_{2}}$ \\ \hline\hline
\end{tabular}
{\small Table 1. The fixed sets of the inverse involution on exceptional Lie
groups}
\end{center}
Historically, the problem of determining the isomorphism type of the fixed
set $Fix(\gamma )$ of a simple Lie group $G$ has been studied by Frankel
\cite{[F]} for the classical Lie groups, and by Chen, Nagano \cite{[CN1],[N]
, Yokota \cite{[Y1],[Y2],[Y3]} for the exceptional Lie groups, see Remark
4.4. These works rely largely on the specialities of each individual Lie
group and the calculations were performed case by case. In comparison, our
approach is free of the types of simple Lie groups, and is ready to extend
to general cases, see Corollaries 5.1--5.2 of Section 5.
The paper is arranged as follows. Section \S 2 contains a brief introduction
to the roots and weight systems of simple Lie groups. In Section \S 3 the
set $\mathcal{F}_{G}$ specifying the partition on $Fix(\gamma )$ in formula
(1.2) is largely determined by Lemma 3.3. Combining Lemma 3.3 with the
algorithm calculating the isomorphism type of a centralizer $C_{x}$ obtained
in \cite{[DL]}, Theorem 1.1 is established in Section 4. Finally, general
structure of the fixed set $Fix(\gamma )$ of the inverse involution $\gamma $
on an arbitrary Lie group $G$ is discussed briefly in Section 5.
\section{Geometry of roots and weights}
For a simple Lie group $G$ with Lie algebra $L(G)$ and a maximal torus $T$,
the dimension $n=\dim T$ is called the \textsl{rank} of $G$, and the
subspace $L(T)$ of $L(G)$ is called the\textsl{\ Cartan subalgebra} of $G$.
Equip $L(G)$ with an inner product $(,)$ so that the adjoint representation
acts on $L(G)$ as isometries, and let
\begin{quote}
$d:G\times G\rightarrow \mathbb{R}$ (resp. $d:T\times T\rightarrow \mathbb{R}
$)
\end{quote}
\noindent be the induced metric on $G$ (resp. on $T$).
The restriction of the exponential map $\exp :L(G)\rightarrow G$ to $L(T)$
defines a set $\mathcal{S}(G)=\{L_{1},\cdots ,L_{m}\}$ of $m=\frac{1}{2
(\dim G-n)$ hyperplanes in $L(T)$, namely, the set of\textsl{\ singular
hyperplanes }through the origin in $L(T)$ \cite[p.168]{[BD]}. Let
l_{k}\subset L(T)$ be the normal line of the plane $L_{k}$ through the
origin, $1\leq k\leq m$. Then the map $\exp $ carries $l_{k}$ onto a circle
subgroup of $G$.
\bigskip
\noindent \textbf{Definition 2.1.} Let $\pm \alpha _{k}\in l_{k}$ be the
non--zero vectors with minimal length so that $\exp (\pm \alpha _{k})=e$,
1\leq k\leq m$. The subset
\begin{quote}
$\Phi =\{\pm \alpha _{k}\in L(T)\mid 1\leq k\leq m\}$
\end{quote}
\noindent of $L(T)$ is called the \textsl{root system of }$G$.
The \textsl{Weyl group} of $G$, denoted by $W$, is the subgroup of
Aut(L(T)) $ generated by the reflections $r_{k}$ in the hyperplane $L_{k}$,
1\leq k\leq m$. $\square $
\bigskip
\noindent \textbf{Remark 2.2. }We point out that\textbf{\ }the root system
\Phi $ by Definition 2.1 is \textsl{dual} to those that are commonly used in
literatures, e.g. \cite{[B],[Hu]}. In particular, the symplectic group
Sp(n) $ is of the type $B_{n}$, while the spinor group $Spin(2n+1)$ is of
the type $C_{n}$.$\square $
\bigskip
The planes in $\mathcal{S}(G)$ divide $L(T)$ into finitely many convex open
cones, called the \textsl{Weyl chambers} of $G$. Fix once and for all a
regular point $x_{0}\in L(T)\backslash \underset{1\leq k\leq m}{\cup }L_{m}
, and let $\mathcal{F}(x_{0})$ be the closure of the Weyl chamber containing
$x_{0}$. Assume that $L(x_{0})=\{L_{1},\cdots ,L_{n}\}$ is the subset of
\mathcal{S}(G)$ consisting of the walls of $\mathcal{F}(x_{0})$, and let
\alpha _{i}\in \Phi $ be the root normal to the wall $L_{i}\in $ $L(x_{0})$
and pointing toward $x_{0}$. Then the subset $\Delta (x_{0})=\{\alpha
_{1},\cdots ,\alpha _{n}\}$ of $\Phi $ is called \textsl{the} \textsl{system
of} \textsl{simple roots }of $G$ relative to $x_{0}$.
\bigskip
\noindent \textbf{Definition 2.3 }(\cite[p.67]{[Hu]})\textbf{.} Each root
\alpha \in \Phi $ gives rise to a linear map
\begin{enumerate}
\item[(2.1)] $\alpha ^{\ast }:L(T)\rightarrow \mathbb{R}$ by $\alpha ^{\ast
}(x)=2(x,\alpha )/(\alpha ,\alpha )$,
\end{enumerate}
\noindent called \textsl{the inverse root} of $\alpha $. The \textsl{weight
lattice of} $G$ is the subset of $L(T)$
\begin{quote}
$\Lambda =\{x\in L(T)\mid \alpha ^{\ast }(x)\in \mathbb{Z}$ for all $\alpha
\in \Phi \}$,
\end{quote}
\noindent whose elements are called \textsl{weights}. Elements in the subset
of $\Lambda $
\begin{enumerate}
\item[(2.2)] $\Omega =\{\omega _{i}\in L(T)\mid \alpha _{j}^{\ast }(\omega
_{i})=\delta _{i,j},$ $\alpha _{j}\in \Delta (x_{0})\}$
\end{enumerate}
\noindent are called the \textsl{fundamental dominant weights} of $G$
relative to $x_{0}$, where $\delta _{i,j}$ is the Kronecker symbol.$\square $
\bigskip
To be precise we adopt the convention that for each simple group $G$ with
rank $n$ its fundamental dominant weights $\omega _{1},\cdots ,\omega _{n}$
are ordered by the order of their corresponding simple roots pictured as the
vertices in the Dynkin diagram of $G$ in \cite[p.58]{[Hu]}. Useful
properties of the weights are:
\bigskip
\noindent \textbf{Lemma 2.4. }\textsl{Let\ }$\Omega =\{\omega _{1},\cdots
,\omega _{n}\}$\textsl{\ be the set of fundamental dominant weights relative
to the regular point }$x_{0}$\textsl{. Then}
\textsl{i) }$\Omega =\{\omega _{1},\cdots ,\omega _{n}\}$\textsl{\ is a
basis for }$\Lambda $ \textsl{over} $\mathbb{Z}$\textsl{;}
\textsl{ii) for each }$1\leq i\leq n$\textsl{\ the half line }$\{t\omega
_{i}\in L(T)\mid t\in \mathbb{R}^{+}\}$\textsl{\ is the edge of the Weyl
chamber }$\mathcal{F}(x_{0})$\textsl{\ opposite to the wall }$L_{i}$\textsl{
}
\textsl{iii) if }$G$\textsl{\ is simple, }$(\omega _{i},\omega _{j})>0$
\textsl{for all }$1\leq i,j\leq n$\textsl{.}$\square $
\noindent \textbf{Proof.} Property i) is well known. By (2.2) each weight
\omega _{i}\in \Omega $ is perpendicular to $\alpha _{j}$ (i.e. $\omega
_{i}\in L_{j}$) for all $j\neq i$, $1\leq j\leq n$. This verifies ii). For
iii) we refer to \cite[p.72, Exercise 8]{[Hu]}.$\square $
\bigskip
Let $\mathcal{Z}(G)$ be the center of the group $G$, and let $\Lambda
_{e}=\exp ^{-1}(e)\subset L(T)$ be the \textsl{unit lattice}. The set
\Delta (x_{0})=\{\alpha _{1},\cdots ,\alpha _{n}\}$ of simple roots spans
also a lattice $\Lambda _{r}$ on $L(T)$, known as the \textsl{root lattice
of $G$.
\bigskip
\noindent \textbf{Lemma 2.5 (\cite[(3.3)]{[DL]}) . }\textsl{In the Euclidean
space }$L(T)$\textsl{\ one has }
\textsl{i)} $\Lambda =\exp ^{-1}(\mathcal{Z}(G))$\textsl{; ii)} $\Lambda
_{r}\subseteq \Lambda _{e}\subseteq \Lambda $\textsl{, }
\noindent \textsl{where in ii), the first equality\ holds if and only if }$G$
\textsl{is }$1$\textsl{--connected,} \textsl{and} \textsl{the second
equality holds if and only if }$\mathcal{Z}(G)=\{0\}$.$\square $
\bigskip
For a simple Lie group $G$ the quotient group $\Lambda /\Lambda _{e}$ is
always finite (see \cite[p.68]{[Hu]}). As a result we can introduce the
\textsl{deficiency function }on the weight lattice
\begin{enumerate}
\item[(2.3)] $\kappa :\Lambda \rightarrow \mathbb{Z}$, $x\rightarrow \kappa
_{x}$,
\end{enumerate}
\noindent by letting $\kappa _{x}$ be the least positive integer so that
\kappa _{x}x\in \Lambda _{e}$, $x\in \Lambda $. This function provides us
with a partition $\Omega =\Omega _{1}\sqcup \Omega _{2}$ with
\begin{quote}
$\Omega _{1}=\{\omega \in \Omega \mid \kappa _{\omega }=1\}$, $\Omega
_{2}=\{\omega \in \Omega \mid \kappa _{\omega }\geq 2\}$.
\end{quote}
\noindent \textbf{Example 2.6.} Let $G$ be a simple Lie group.
If $\mathcal{Z}(G)=\{0\}$ we get from $\Lambda _{e}=\Lambda $ by Lemma 2.5
that $\Omega =\Omega _{1}$.
If $G$ is $1$--connected with $\mathcal{Z}(G)\neq \{0\}$, we have $\Lambda
_{e}=\Lambda _{r}$ by Lemma 2.5. From the expressions of the fundamental
dominant weights\textsl{\ }by simple roots in \cite[p.69]{[Hu]} one
determines the subset $\Omega _{1}$, consequently $\Omega _{2}$, as that
tabulated below
\begin{center}
\begin{tabular}{l|llllll}
\hline\hline
$G$ & $A_{n}$ & $Sp(n)$ & $Spin(2n+1)$ & $Spin(2n)$ & $E_{6}$ & $E_{7}$ \\
\hline
$\Omega _{1}$ & $\emptyset $ & $\{\omega _{i}\}_{i<n}$ & $\{\omega
_{i}\}_{i=2k}$ & $\{\omega _{i}\}_{i=2k\text{ }\leq n-2,}$ & $\{\omega
_{i}\}_{i=2,4}$ & $\{\omega _{i}\}_{i=1,3,4,6}$ \\ \hline\hline
\end{tabular
.$\square $
\end{center}
The set $\Delta (x_{0})$ of simple roots is a basis for both $L(T)$ and the
root lattice $\Lambda _{r}$. Using this basis a partial order $\prec $ on
L(T)$ (hence on $\Phi \subset L(T)$) can be introduced by the following rule:
\begin{center}
$v\prec u$\textsl{\ if and only if the difference }$u-v$\textsl{\ is a sum
of elements of }$\Delta (x_{0})$\textsl{.}
\end{center}
\noindent As in \cite[p.67]{[Hu]} we put $\Lambda ^{+}:=\Lambda \cap
\mathcal{F}(x_{0})$. An element $\omega \in \Lambda ^{+}$ is called \textsl
minimal }if $\omega \succ \omega ^{\prime }\in \Lambda ^{+}$ implies that
\omega =\omega ^{\prime }$.
\bigskip
\noindent \textbf{Lemma 2.7.} \textsl{Let }$G$ \textsl{be an }$1$\textsl
--connected simple Lie group, and\ let }$\Pi _{G}\subset \Lambda ^{+}$
\textsl{be the subset of all non--zero minimal weights. Then\ }$\Pi
_{G}\subset \Omega _{2}$\textsl{. Moreover,}
\textsl{i) for} \textsl{each} $\omega \in \Omega _{2}$ \textsl{there is
precisely one} \textsl{weight} $\omega ^{\prime }\in \Pi _{G}$ \textsl{so
that }$\omega \succ \omega ^{\prime }$\textsl{;}
\textsl{ii) the set of all non--trivial elements in }$\mathcal{Z}(G)$\textsl
\ are given without repetition by }$\{\exp (\omega )\in \mathcal{Z}(G)\mid
\omega \in \Pi _{G}\}$\textsl{.}
\noindent \textbf{Proof.} See \cite[P.92]{[Hu]}. $\square $
\bigskip
In view of Lemma 2.7 we can introduce a \textsl{retraction} $r:\Omega
_{2}\rightarrow \Pi _{G}$ and an \textsl{involution} $\tau :\Pi
_{G}\rightarrow \Pi _{G}$ respectively by the rules:
a) $\omega \succ r(\omega )\in \Pi _{G}$, $\omega \in \Omega _{2}$ (by i) of
Lemma 2.7);
b) $\tau (\omega )+\omega \in \Lambda _{e}$, $\omega \in \Pi _{G}$ (by ii)
of Lemma 2.7).
\noindent Alternatively, the element $\tau (\omega )$ is characterized by
the relation
\begin{quote}
$\exp (\omega )\exp (\tau (\omega ))=e$.
\end{quote}
\noindent \textbf{Example 2.8. }Assume that\textbf{\ }$G$ is simple and $1
--connected with $\mathcal{Z}(G)\neq \{0\}$.
a) The set $\Pi _{G}$ of minimal weights is given by (see \cite[P.92]{[Hu]}):
\begin{center}
\begin{tabular}{l|l|l|l|l|l|l}
\hline\hline
$G$ & $SU(n)$ & $Sp(n)$ & $Spin(2n+1)$ & $Spin(2n)$ & $E_{6}$ & $E_{7}$ \\
\hline
$\Pi _{G}$ & $\left\{ \omega _{i}\right\} _{1\leq i\leq n}$ & $\left\{
\omega _{n}\right\} $ & $\left\{ \omega _{1}\right\} $ & $\left\{ \omega
_{1},\omega _{n-1},\omega _{n}\right\} $ & $\left\{ \omega _{1},\omega
_{6}\right\} $ & $\left\{ \omega _{7}\right\} $ \\ \hline\hline
\end{tabular
;
\end{center}
b) The set $\Omega _{2}$, as well as the composition $\tau \circ r:\Omega
_{2}\rightarrow \Pi _{G}$, is given by
\begin{center}
\begin{tabular}{l|l|l|l|l|l}
\hline\hline
$G$ & $SU(n)$ & $Sp(n)$ & $Spin(2n+1)$ & $E_{6}$ & $E_{7}$ \\ \hline
$\Omega _{2}$ & ${\small \{\omega }_{{\small k}}{\small \}}_{{\small 1\leq
k\leq n}}$ & ${\small \{\omega }_{{\small n}}{\small \}}$ & ${\small
\{\omega }_{{\small 2k+1}}{\small \}}$ & ${\small \{\omega }_{{\small k}
{\small \}}_{{\small k=1,3,5,6}}$ & ${\small \{\omega }_{k}{\small \}}_
{\small k=2,5,7}}$ \\ \hline
${\small \tau \circ r(\omega }_{{\small k}}{\small )}$ & $\omega _{n+1-k}$ &
$\omega _{n}$ & $\omega _{1}$ &
\begin{tabular}{l}
${\small \omega }_{{\small 6}}\text{ }${\small for }${\small k=1,5}$ \\
${\small \omega }_{{\small 1}}$ {\small for }${\small k=3,6}
\end{tabular}
& $\omega _{7}$ \\ \hline\hline
\end{tabular}
\end{center}
\noindent and for $G=Spin(2n)$, by $\Omega _{2}=\{\omega _{k},\omega
_{n-1},\omega _{n}\mid k\leq n-2$ odd$\}$,
\begin{center}
$\tau \circ r(\omega _{k})=\left\{
\begin{tabular}{l}
$\omega _{1}\text{ if }k\leq n-2\text{;}$ \\
$\omega _{n}$ if either $n$ is odd, $k=n-1$, or $n$ is even, $k=n$; \\
$\omega _{n-1}$ if either $n$ is odd, $k=n$, or $n$ is even, $k=n-1$.
\square
\end{tabular
\right. $
\end{center}
\section{Computation in the fundamental Weyl cell}
For a simple Lie group $G$ elements in the root system\textsl{\ }$\Phi $ has
at most two lengths. Let $\beta $\textsl{\ }be the \textsl{maximal short roo
} relative to the partial order $\prec $ on the set $\Phi ^{+}$ of positive
roots \cite[p.55]{[Hu]}. The\textsl{\ fundamental Weyl cell} is the simplex
in the Weyl chamber $\mathcal{F}(x_{0})$ defined by
\begin{quote}
$\Delta =\{u\in \mathcal{F}(x_{0})\mid \beta ^{\ast }(u)\leq 1\}$.
\end{quote}
\noindent Let $d:T\times T\rightarrow \mathbb{R}$\ be the distance function
on $T$. It is well known that
\bigskip
\noindent \textbf{Lemma 3.1 (\cite{[C], [Cr]}).} \textsl{Let }$G$\textsl{\
be a simple Lie group. Then}
\textsl{i) the equation }$d(e,\exp (u))=\left\Vert u\right\Vert $\textsl{\
holds if and only if }$\left\Vert u\right\Vert \leq \left\Vert
u-v\right\Vert $ \textsl{holds} \textsl{for all }$v\in \Lambda _{e}$\textsl{
}
\textsl{ii)} \textsl{if }$G$\textsl{\ is }$1$\textsl{--connected, then }
u\in \Delta $ \textsl{implies that} $d(e,\exp (u))=\left\Vert u\right\Vert
\textsl{.}$\square $
\bigskip
It is well known that every element $x\in G$ is conjugate under $G$ to an
element of the form $\exp (u)\in G$ with $u\in \Delta $ and $d(e,\exp
(u))=\left\Vert u\right\Vert $. Moreover, if $x\in Fix(\gamma )$ then $2u\in
\Lambda _{e}$. This implies that
\bigskip
\noindent \textbf{Lemma 3.2.} \textsl{For a simple Lie group }$G$\textsl{\
with fundamental Weyl cell }$\Delta $\textsl{\ set}
\begin{enumerate}
\item[(3.1)] $\mathcal{K}_{G}=\{u\in \Delta \mid 2u\in \Lambda _{e}$,\
d(e,\exp (u))=\left\Vert u\right\Vert \}$.
\end{enumerate}
\noindent \textsl{Then}
\begin{enumerate}
\item[(3.2)] $Fix(\gamma )=\{e\}\bigcup\limits_{u\in \mathcal{K}_{G}}M_{\exp
(u)}$ \textsl{with }$d(e,x)=\left\Vert u\right\Vert $\textsl{\ for all }
x\in M_{\exp (u)}$\textsl{.}$\square $
\end{enumerate}
Comparing (3.2) with (1.2) we emphasis that the decomposition (3.2) on
Fix(\gamma )$ may not be disjoint, as overlap like $M_{\exp (u)}=M_{\exp
(v)} $ may occur for some $u,v\in \mathcal{K}_{G}$ with $u\neq v$. However,
based on the relation (3.2) our approach to $Fix(\gamma )$ consists of three
steps:
i) find a general expression for elements in $\mathcal{K}_{G}$;
ii) specify a subset $\mathcal{F}_{G}\subseteq \mathcal{K}_{G}$ so that the
relation (3.2) can be refined as $Fix(\gamma )=\{e\}\coprod\limits_{u\in
\mathcal{F}_{G}}M_{\exp (u)}$;
iii) decide the isomorphism types of $M_{\exp (u)}$ for all $u\in \mathcal{F
_{G}$.
\noindent In this section we accomplish step i) in the next result.
\bigskip
\noindent \textbf{Lemma 3.3. }\textsl{Let} $G$ \textsl{be a simple Lie group
} \textsl{Then }$u\in \mathcal{K}_{G}$ \textsl{implies that}
\begin{enumerate}
\item[(3.3)] $u=\left\{
\begin{tabular}{l}
$\frac{1}{2}\omega _{k}$ \textsl{for some }$\omega _{k}\in \Omega _{1}$, \\
$\frac{1}{2}(\omega _{k}+\tau \circ r(\omega _{k}))$ \textsl{for some}
\omega _{k}\in \Omega _{2}$\textsl{.
\end{tabular
\right. $
\end{enumerate}
\noindent \textbf{Proof.} For an $u\in \mathcal{K}_{G}$ we get from $u\in
\Delta $ and $2u\in \Lambda _{e}$ that
\begin{quote}
$u=\lambda _{k_{1}}\omega _{k_{1}}+\cdots +\lambda _{k_{t}}\omega _{k_{t}}$
with $\lambda _{k_{s}}>0$ and $2\lambda _{k_{s}}\in \mathbb{Z}$
\end{quote}
\noindent by Lemma 2.4, where $\{k_{1},\cdots ,k_{t}\}\subseteq \{1,\cdots
,n\}$. This implies that
\begin{enumerate}
\item[(3.4)] $2u-\omega _{k_{1}}=a\in \Lambda ^{+}$.
\end{enumerate}
\noindent The formula (3.3) will be deduced from the second constraint
d(e,\exp (u))=\left\Vert u\right\Vert $ on $u\in \mathcal{K}_{G}$ in (3.1).
If $\omega _{k_{1}}\in \Omega _{1}$ then $\omega _{k_{1}}\in \Lambda _{e}$
implies that $\left\Vert u\right\Vert \leq \left\Vert u-\omega
_{k_{1}}\right\Vert $ by i) of Lemma 3.1. That is
\begin{quote}
$\left\Vert \frac{1}{2}a+\frac{1}{2}\omega _{k_{1}}\right\Vert ^{2}\leq
\left\Vert \frac{1}{2}a-\frac{1}{2}\omega _{k_{1}}\right\Vert ^{2}$
\end{quote}
\noindent by (3.4). However, since $(\omega _{i},\omega _{j})>0$ by iii) of
Lemma 2.4 and since $a\in \Lambda ^{+}$, this is possible if and only if
a=0 $. That is
\begin{enumerate}
\item[(3.5)] $u=\frac{1}{2}\omega _{k_{1}}$.
\end{enumerate}
If $\omega _{k_{1}}\in \Omega _{2}$ we have $a\in \Lambda ^{+}$ but $a\notin
\Lambda _{e}$ by (3.4). According to Lemma 2.7 there is precisely one weight
$\omega _{s}\in \Pi _{G}$ so that $2u-\omega _{k_{1}}=a=\omega _{s}+b$ with
b$ a sum of elements of $\Delta (x_{0})$. From $2u,b\in \Lambda _{e}$ we
find that $\omega _{k_{1}}+\omega _{s}\in \Lambda _{e}$ and therefore
\left\Vert u\right\Vert \leq \left\Vert u-\omega _{k_{1}}-\omega
_{s}\right\Vert $ by i) of Lemma 3.1. That is
\begin{quote}
$\left\Vert \frac{1}{2}(\omega _{k_{1}}+\omega _{s})+\frac{1}{2}b\right\Vert
^{2}\leq \left\Vert \frac{1}{2}(\omega _{k_{1}}+\omega _{s})-\frac{1}{2
b\right\Vert ^{2}$.
\end{quote}
\noindent Again, since $(\omega _{i},\omega _{j})>0$ by iii) of Lemma 2.4
and since $b\in \Lambda ^{+}$, this is possible if and only if $b=0$. We
obtain from (3.4) that $u=\frac{1}{2}(\omega _{k_{1}}+\omega _{s})$, $\omega
_{s}\in \Pi _{G}$. Furthermore, from the calculation
\begin{quote}
$e=\exp (2u)=\exp (\omega _{k_{1}})\exp (\omega _{s})=\exp (r(\omega
_{k_{1}}))\exp (\omega _{s})$
\end{quote}
\noindent (since $\omega _{k_{1}}\succ r(\omega _{k_{1}})\in \Pi _{G}$) as
well as the definition of $\tau $ we get $\omega _{s}=$ $\tau \circ r(\omega
_{k_{1}})$. This shows that
\begin{enumerate}
\item[(3.6)] $u=\frac{1}{2}(\omega _{k_{1}}+\tau \circ r(\omega _{k_{1}}))$
with $\omega _{k_{1}}\in \Omega _{2}$.
\end{enumerate}
\noindent The proof of (3.3) has now been completed by (3.5) and (3.6).
\square $
\section{Proof of Theorem 1.1}
Assume that $G$ is an exceptional Lie group and the expression of its
maximal short root $\beta $ in term of the simple roots is\ $\beta
=m_{1}\alpha _{1}+\cdots +m_{n}\alpha _{n}$ (see \cite[p.66]{[Hu]}). By the
definition (2.2) of the fundamental dominant weights
\begin{enumerate}
\item[(4.1)] $\beta ^{\ast }(\frac{\omega _{i}}{2})=\frac{m_{i}\left\Vert
\alpha _{i}\right\Vert ^{2}}{2\left\Vert \beta \right\Vert ^{2}}$; $\quad
\beta ^{\ast }(\frac{\omega _{i}+\omega _{j}}{2})=\frac{m_{i}\left\Vert
\alpha _{i}\right\Vert ^{2}}{2\left\Vert \beta \right\Vert ^{2}}+\frac
m_{j}\left\Vert \alpha _{j}\right\Vert ^{2}}{2\left\Vert \beta \right\Vert
^{2}}$.
\end{enumerate}
\noindent Let $\mathcal{K}_{G}^{\prime }\subset \Delta $ be the subset of
the vectors $u$ satisfying (3.3). Combining (3.3) and (4.1), together with
computations in Examples 2.6 and 2.8, one determines the set $\mathcal{K
_{G}^{\prime }$ for each exceptional $G$, as that presented in the second
column of Tables 2 below.
In \cite{[DL]} an explicit procedure to calculate the isomorphism type of
the centralizer $C_{\exp (u)}\subset G$ in term of $u\in \Delta $ is
obtained. As applications those centralizers $C_{\exp (u)}$ with $u\in
\mathcal{K}_{G}^{\prime }$ are determined and presented in the third column
of Table 2 (see also in \cite[Theorem 4.4, Theorem 4.6]{[DL]}).
In general $\mathcal{K}_{G}\subseteq \mathcal{K}_{G}^{\prime }$ by Lemma
3.3. However, the centralizers $C_{\exp (u)}$ recorded in Table 2 are useful
for us to specify the desired subset $\mathcal{K}_{G}$ from $\mathcal{K
_{G}^{\prime }$. To explain this we observe that if $u\in \mathcal{K
_{G}^{\prime }$ is a vector with $u\notin \mathcal{K}_{G}$, then $d(e,\exp
(u))<\left\Vert u\right\Vert $ implies that there exists a vector $v\in L(T)$
satisfying
\begin{quote}
$\exp (v)=\exp (u)$ and $d(e,\exp (v))=\left\Vert v\right\Vert $.
\end{quote}
\noindent Take a Weyl group element $w\in W$ so that $v^{\prime }=w(v)\in
\mathcal{F}(x_{0})$. The relations $2v^{\prime }\in \Lambda _{e}$ and
d(e,\exp (v^{\prime }))=\left\Vert v^{\prime }\right\Vert $ indicate that
v^{\prime }\in \mathcal{K}_{G}$ by (3.1). In particular we have
\bigskip
\noindent \textbf{Lemma 4.1.} If $u\in \mathcal{K}_{G}^{\prime }\backslash
\mathcal{K}_{G}$ then there exists an element $v\prime \in \mathcal{K}_{G}$
so that
\begin{quote}
$C_{\exp (u)}\cong C_{\exp (v^{\prime })}$ and $\left\Vert v^{\prime
}\right\Vert <\left\Vert u\right\Vert $.$\square $
\end{quote}
\begin{center}
\begin{tabular}{l|l|l}
\hline\hline
$G$ & $\mathcal{K}_{G}^{\prime }$ & the centralizer $C_{\exp (u)}$ with
u\in \mathcal{K}_{G}^{\prime }$ \\ \hline
$G_{2}$ & $\{\frac{1}{2}\omega _{{\small 1}}\}$ & ${\small SO(4)}$ \\
$F_{4}$ & $\{\frac{1}{2}\omega _{{\small k}}\}_{{\small k=1,4}}$ & ${\small
Spin(9),}\frac{{\small Sp(3)\times Sp(1)}}{{\small Z}_{2}}$ \\
$E_{6}$ & $\{\frac{{\small \omega }_{{\small 2}}}{2},\frac{\omega
_{1}+\omega _{6}}{2}\}$ & $\frac{{\small SU(2)\times SU(6)}}{{\small Z}_{2}}
{\small ,} $\frac{{\small Spin(10)\times S}^{{\small 1}}}{{\small Z}_{4}}$
\\
$E_{6}^{\ast }$ & ${\small \{}\frac{1}{2}{\small \omega }_{{\small k}
{\small \}}_{{\small k=1,2,3,5},{\small 6}}$ & $\frac{{\small Spin(10)\times
S}^{{\small 1}}}{{\small Z}_{4}}$ {\small for }${\small k=1,6}${\small ; }
\frac{{\small SU(2)\times }\frac{{\small SU(6)}}{Z_{3}}}{{\small Z}_{2}}$
{\small for }${\small k=2,3,5}$ \\
$E_{7}$ & $\{\frac{{\small \omega }_{{\small k}}}{2},\omega _{{\small 7}}\}_
{\small k=1,6}}$ & $\frac{{\small Spin(12)\times SU(2)}}{{\small Z}_{2}}
{\small , }$\frac{{\small Spin(12)\times SU(2)}}{{\small Z}_{2}}${\small , }
{\small E}_{{\small 7}}$ \\
$E_{7}^{\ast }$ & ${\small \{}\frac{1}{2}{\small \omega }_{{\small k}
{\small \}}_{{\small k=1,2,6,7}}$ & $\frac{Ss(12){\small \times SU(2)}}
{\small Z}_{2}}${\small ,} $[\frac{{\small SU(8)}}{{\small Z}_{4}}]^{2}
{\small , }$\frac{Ss(12){\small \times SU(2)}}{{\small Z}_{2}}$, $[\frac
{\small E}_{6}{\small \times S}^{1}}{{\small Z}_{3}}]^{2}$ \\
$E_{8}$ & $\{\frac{1}{2}\omega _{{\small k}}\}_{{\small k=1,8}}$ & ${\small
Ss(16)}${\small , }$\frac{{\small E}_{{\small 7}}{\small \times SU(2)}}
{\small Z}_{2}}$ \\ \hline\hline
\end{tabular}
{\small Table 2. the set }$\mathcal{K}_{G}^{\prime }${\small \ as well as
the centralizers }${\small C}_{\exp (u)}${\small \ with }$u\in \mathcal{K
_{G}^{\prime }$.
\end{center}
\noindent \textbf{Proof of Theorem 1.1.} The proof will be divided into two
cases, depending on whether $G$ is $1$--connected. Concerning the use of
Lemma 4.1 in the forthcoming arguments, we note that in view of the
presentation of the fundamental dominant weights with respect to appropriate
Euclidean coordinates $\{\varepsilon _{1},\cdots ,\varepsilon _{m}\}$ on
L(T)$ in the standard reference \cite[p.265-277]{[B]}, the length
\left\Vert u\right\Vert $ for a vector $u\in \mathcal{K}_{G}^{\prime }$ can
be easily evaluated.
\textbf{Case I.} $G=G_{2},F_{4},E_{6},E_{7},E_{8}$. Since $G$ is $1
--connected, we have
\begin{enumerate}
\item[(4.2)] $\mathcal{K}_{G}=\mathcal{K}_{G}^{\prime }$
\end{enumerate}
\noindent by ii) of Lemma 3.1. Consequently, it follows from (3.2) that
\begin{enumerate}
\item[(4.3)] $Fix(\gamma )=\{e\}\bigcup\limits_{u\in \mathcal{K}_{G}}M_{\exp
(u)}$ with $M_{\exp (u)}\cong G/C_{\exp (u)}$.
\end{enumerate}
Assume in the decomposition (4.3) on $Fix(\gamma )$ that the relation
M_{\exp (u)}=M_{\exp (v)}$ holds for some $u,v\in \mathcal{K}_{G}$ with
u\neq v$. By (3.2) one has
\begin{quote}
i) $C_{\exp (u)}\cong C_{\exp (v)}$ and ii) $\left\Vert u\right\Vert
=\left\Vert v\right\Vert $.
\end{quote}
\noindent However, in view of the groups $C_{\exp (u)}$ presented in Table 2
the only possibility for i) to hold is when $G=E_{7}$ and $(u,v)=(\frac
{\small \omega }_{{\small 1}}}{2},\frac{{\small \omega }_{{\small 6}}}{2})$,
but in this case the calculation
\begin{quote}
$\left\Vert \frac{{\small \omega }_{{\small 1}}}{2}\right\Vert =\frac{1}
\sqrt{2}}<\left\Vert \frac{{\small \omega }_{{\small 6}}}{2}\right\Vert =1$
(see \cite[p.280]{[B]}).
\end{quote}
\noindent shows that the relation ii) does not hold. Summarizing, taking
\mathcal{F}_{G}=\mathcal{K}_{G}$ then the decomposition (1.2) on $Fix(\gamma
)$ is given by (4.3), and the proof of Theorem 1.1 is completed by the
corresponding items in Table 2.
\textbf{Case II.} $G=E_{6}^{\ast },E_{7}^{\ast }$. This case is slightly
delicate because, instead of the equality (4.2) one has $\mathcal{K
_{G}\subseteq \mathcal{K}_{G}^{\prime }$ by Lemma 3.3. Nevertheless, granted
with Lemma 4.1 and results in Table 2 we shall show that
\begin{enumerate}
\item[(4.4)] $\mathcal{K}_{G}=\left\{
\begin{tabular}{l}
${\small \{}\frac{{\small \omega }_{{\small k}}}{2}{\small \}}_{{\small
k=1,2,6}}$ if $G=E_{6}^{\ast }$ \\
${\small \{}\frac{{\small \omega }_{{\small k}}}{2}{\small \}}_{{\small
k=1,2,7}}$ if $G=E_{7}^{\ast }
\end{tabular
\right. ;$
\item[(4.5)] for $u,v\in \mathcal{K}_{G}$ the overlap $M_{\exp (u)}=M_{\exp
(v)}$ (see (3.2)) happens if and only if $G=E_{6}^{\ast }$ and $(u,v)=(\frac
{\small \omega }_{1}}{2},\frac{{\small \omega }_{{\small 6}}}{2})$.
\end{enumerate}
\noindent Consequently, setting
\begin{quote}
$\mathcal{F}_{G}=\left\{
\begin{tabular}{l}
${\small \{}\frac{{\small \omega }_{{\small k}}}{2}{\small \}}_{{\small k=1,
}}$ if $G=E_{6}^{\ast }$ \\
${\small \{}\frac{{\small \omega }_{{\small k}}}{2}{\small \}}_{{\small
k=1,2,7}}$ if $G=E_{7}^{\ast }
\end{tabular
\right. $
\end{quote}
\noindent the proof of Theorem 1.1 for this case is completed by (4.4) and
(4.5), and the relevant items in Table 2.
For $G=E_{7}^{\ast }$ we have $\mathcal{K}_{E_{7}^{\ast }}^{\prime }={\small
\{}\frac{1}{2}{\small \omega }_{{\small k}}{\small \}}_{{\small k=1,2,6,7}}$
by Table 2. Since $\omega _{7}\in \Lambda _{e}$ by Example 2.6 and since
\begin{quote}
$\frac{1}{\sqrt{2}}=\left\Vert \omega _{7}-\frac{1}{2}\omega _{6}\right\Vert
<\left\Vert \frac{1}{2}\omega _{6}\right\Vert =1$ (see \cite[p.280]{[B]})
\end{quote}
\noindent we have $\frac{1}{2}\omega _{6}\notin \mathcal{K}_{E_{7}^{\ast }}$
by i) of Lemma 3.1. The proof of (4.4) for $G=E_{7}^{\ast }$ is done by
Lemma 4.1, together with the groups $C_{\exp (u)}$ with $u\in {\small \{
\frac{1}{2}{\small \omega }_{{\small k}}{\small \}}_{{\small k=1,2,7}}$
given in Table 2.
Similarly, for $G=E_{6}^{\ast }$ we have $\omega _{1},\omega _{6}\in \Lambda
_{e}$ by Example 2.6, but
\begin{quote}
$\frac{1}{\sqrt{2}}=\left\Vert \omega _{1}-\frac{1}{2}\omega _{3}\right\Vert
<\left\Vert \frac{1}{2}\omega _{3}\right\Vert =\sqrt{\frac{5}{6}}$;
$\frac{1}{\sqrt{2}}=\left\Vert \omega _{6}-\frac{1}{2}\omega _{5}\right\Vert
<\left\Vert \frac{1}{2}\omega _{5}\right\Vert =\sqrt{\frac{5}{6}}$, see \cit
[p.276]{[B]}.
\end{quote}
\noindent We get $\frac{1}{2}\omega _{3},\frac{1}{2}\omega _{5}\notin
\mathcal{K}_{E_{6}^{\ast }}$ from i) of Lemma 3.3. The proof of (4.4) for
G=E_{6}^{\ast }$ is done by Lemma 4.1, together with the groups $C_{\exp
(u)} $ with $u\in {\small \{}\frac{1}{2}{\small \omega }_{{\small k}}{\small
\}}_{{\small k=1,2,6}}$ given in Table 2.
For (4.5) assume that in the decomposition (3.2) on $Fix(\gamma )$ the
relation $M_{\exp (u)}=M_{\exp (v)}$ holds for some $u,v\in \mathcal{K}_{G}$
with $u\neq v$. By (3.2) one has
\begin{quote}
i) $C_{\exp (u)}\cong C_{\exp (v)}$; ii) $\left\Vert u\right\Vert
=\left\Vert v\right\Vert $.
\end{quote}
\noindent In view of the groups $C_{\exp (u)}$ with $u\in \mathcal{K}_{G}$
presented in the last column of Table 2 the only possibility for both i) and
ii) to hold is when $G=E_{6}^{\ast }$ and $(u,v)=(\frac{{\small \omega }_
{\small 1}}}{2},\frac{{\small \omega }_{{\small 6}}}{2})$. Let $w_{0}$ be
the unique longest element of the Weyl group of $E_{6}$ \cite[p.171]{[B]}.
Then
\begin{quote}
$w_{0}(\frac{1}{2}\omega _{{\small 1}})=-\frac{1}{2}\omega _{{\small 6}}$
(see \cite[p.276]{[B]}).
\end{quote}
\noindent This implies that $M_{\exp (\frac{{\small \omega }_{{\small 1}}}{2
)}=M_{\exp (-\frac{1}{2}\omega _{{\small 6}})}$. We get (4.5) from the
general relation $\exp (-\frac{1}{2}u)=\exp (\frac{1}{2}u)$, $u\in \Lambda
, which holds in all groups $G$ with trivial center. This completes the
proof.$\square $
\bigskip
\noindent \textbf{Remark 4.2. }For a vector $u\in \Delta $ let $C_{\exp
(u)}^{0}$ be the identity component of the centralizer $C_{\exp (u)}$.
Indeed, for $G=E_{6}^{\ast }$ or $E_{7}^{\ast }$ the main result in \cite
Theorem 3.7]{[DL]} is applicable to determine the isomorphism type of
C_{\exp (u)}^{0}$ instead of the whole group $C_{\exp (u)}$. Therefore,
additional explanation for the groups $C_{\exp (u)}$ corresponding to
G=E_{6}^{\ast }$ or $E_{7}^{\ast }$ in Table 2 is requested.
In general, let $p:G^{\symbol{126}}\rightarrow G$ be the universal covering
of a simple Lie group $G$ and let $T^{\symbol{126}}\subset G^{\symbol{126}}$
be the maximal torus of $G^{\symbol{126}}$ corresponding to $T$ in $G$. With
respect to the standard identification $L(G^{\symbol{126}})=L(G)$ (resp.
L(T^{\symbol{126}})=L(T)$) the exponential map $\exp $ of $G$ factors
through that $\exp ^{\symbol{126}}$ of $G^{\symbol{126}}$ in the fashion
\begin{quote}
$\exp =p\circ \exp ^{\symbol{126}}:L(G^{\symbol{126}})\rightarrow G^{\symbol
126}}\rightarrow G$ (resp. $L(T^{\symbol{126}})\rightarrow T^{\symbol{126
}\rightarrow T$).
\end{quote}
\noindent Since for $u\in \mathcal{F}_{G}$ the subspace $M_{\exp ^{\symbol
126}}(u)}$ of $G^{\symbol{126}}$ is $1$--connected\textbf{\ }\cite[Corollary
3.4, p.101]{[Bo]}, $p$ restricts to a universal covering $p_{u}:M_{\exp ^
\symbol{126}}(u)}\rightarrow M_{\exp (u)}$.
On the other hand, as the set of all non--trivial covering transformations
of $p$ are in one to one correspondence with the set $\Pi _{G^{\symbol{126
}} $ of minimal weights (see Example 2.8) in the fashion
\begin{quote}
$\widetilde{g}\rightarrow \exp ^{\symbol{126}}(\omega _{s})\cdot \widetilde{
}$, $\widetilde{g}\in G^{\symbol{126}}$, $\omega _{s}\in \Pi _{G^{\symbol{12
}}}$,
\end{quote}
\noindent the set $\Pi _{u}$ of nontrivial covering transformations of
p_{u} $ can be shown to be
\begin{enumerate}
\item[(4.6)] $\Pi _{u}=\{\omega _{s}\in \Pi _{G^{\symbol{126}}}\mid $
\omega _{s}+u-w(u)\in \Lambda _{r}$ for some $w\in W\}$,
\end{enumerate}
\noindent where $\Lambda _{r}$ is the root lattice of $G^{\symbol{126}}$.
Based on this formula a direct calculation in the vector space space $L(T^
\symbol{126}})$ shows that
\begin{enumerate}
\item[(4.7)] $\Pi _{u}=\left\{
\begin{tabular}{l}
$\{\omega _{7}\}$ for $G=E_{7}^{\ast }$ and $u\in {\small \{}\frac{{\small
\omega }_{{\small k}}}{2}{\small \}}_{{\small k=2,7}}$ \\
$\emptyset $ otherwise
\end{tabular
\right. $.
\end{enumerate}
\noindent Consequently
\begin{quote}
$C_{\exp (u)}=\left\{
\begin{tabular}{l}
$\lbrack C_{\exp (u)}^{0}]^{2}$ for $G=E_{7}^{\ast }$ and $u\in {\small \{
\frac{{\small \omega }_{{\small k}}}{2}{\small \}}_{{\small k=2,7}}$ \\
$C_{\exp (u)}^{0}$ otherwise
\end{tabular
\right. $
\end{quote}
\noindent This justify the groups $C_{\exp (u)}$ corresponding to
G=E_{6}^{\ast }$ or $E_{7}^{\ast }$ in Table 2.$\square $
\bigskip
\noindent \textbf{Remark 4.3.} With the preliminary data for
G=SU(n+1),Sp(n),$ $Spin(n+2)$, $n\geq 2$, recorded in Example 2.8, one can
obtain the fixed set $Fix(\gamma )$ for the simple Lie groups of the
classical types (see \cite{[F]}) by the same argument as that used to
establish Theorem 1.1.$\square $
\bigskip
\noindent \textbf{Remark 4.4.} It is clear that $x\in Fix(\gamma )$\textsl{\
}implies that $x^{2}=e$. Consequently, the map $\sigma :G\rightarrow G$ by
\sigma (g)=xgx^{-1}$ is an involutive automorphism of $G$ with fixed
subgroup $C_{x}$, the centralizer at $x$. This indicates that the orbit
spaces $M_{\exp (u)}$ in the decomposition (1.2) are all\textsl{\ global
Riemannian symmetric spaces} of $G$ in the sense of E. Cartan. However, the
existing theory of symmetric spaces \cite{[He],[CN1],[N],[Y1],[Y2],[Y3]}
does not constitute a solution to our problem for the following reasons:
i) not every symmetric space of $G$ can appear as a component of $Fix(\gamma
)$;
ii) if a symmetric space of $G$ happens to be a component of $Fix(\gamma )$,
it may occur twice (see in (1.2) for the case $G=E_{7}$);
iii) in view of the relation $M_{\exp (u)}=G/C_{\exp (u)}$ a complete
characterization of the symmetric space $M_{\exp (u)}$ amounts to the
determination of the centralizer $C_{\exp (u)}$, which is a delicate issue
absent in the classical theory of Lie groups \cite{[He]}, and has recently
been made explicit in our paper \cite{[DL]}.
\noindent As a witness of i)--iii), for the exceptional Lie groups Nagano
\cite{[N]} stated the list of all the symmetric spaces which are connected
components of\noindent\ $Fix(\gamma )$ without specifying their embedding in
$G$. He did not write a proof in his other papers, although he promised to
do so in \cite{[N]}. In addition, the cases $G=E_{6}^{\ast }$ or
E_{7}^{\ast }$ were not considered in the papers \cite{[Y1],[Y2],[Y3]}.
Summarizing, without resorting to the theory of symmetric spaces and by a
unified approach, we have enumerated all the symmetric spaces of an
exceptional $G$ that are components of $Fix(\gamma )$, and presented
concrete realization of these spaces as the adjoint orbits of $G$.$\square $
\section{Generalities}
Result on $Fix(\gamma )$ for the simple Lie groups (i.e. Theorem 1.1 and
\cite{[F]}) is fundamental in understanding the general structure of the
fixed set $Fix(\gamma )$ for the inverse involution $\gamma $ on an
arbitrary Lie group $G$. Fairly transparent in our context we have the next
result which indicates how Theorem 1.1 could be extended to general settings.
\bigskip
\noindent \textbf{Corollary 5.1.} \textsl{For any semi--simple Lie group }$G
\textsl{\ with a maximal torus }$T$\textsl{, there is a finite subset }
\mathcal{F}_{G}\subset L(T)$\textsl{\ so that}
\textsl{i) }$\left\Vert u\right\Vert =d(e,\exp (u))$\textsl{\ for all }$u\in
\mathcal{F}_{G}$\textsl{;}
\textsl{ii) }$Fix(\gamma )=\{e\}\coprod\limits_{u\in \mathcal{F}_{G}}M_{\exp
(u)}$\textsl{.}
\noindent \textsl{In particular, if }$G=G_{1}\times \cdots \times G_{k}
\textsl{\ with all the factor groups }$G_{i}$\textsl{\ exceptional, one can
take }$\mathcal{F}_{G}=\mathcal{F}_{G_{1}}\times \cdots \times \mathcal{F
_{G_{k}}$\textsl{\ with }$\mathcal{F}_{G_{i}}$ \textsl{being given by the
second column of Table 1. Consequently, for an }$u=(u_{1},\cdots ,u_{k})\in
\mathcal{F}_{G}$\textsl{\ with }$u_{i}\in \mathcal{F}_{G_{i}}$\textsl{\ one
has}
\begin{quote}
$M_{\exp (u)}=M_{\exp (u_{1})}\times \cdots \times M_{\exp (u_{k})}$\textsl{
}$\square $
\end{quote}
A homomorphism $h:G\rightarrow G^{\prime }$ of two semisimple Lie groups $G$
and $G\prime $ clearly satisfies the relation $h(Fix(\gamma ))\subseteq
Fix(\gamma ^{\prime })$. This indicates that Corollary 5.1 can play a role
in the representation theory of Lie groups. More precisely
\bigskip
\noindent \textbf{Corollary 5.2.} \textsl{A group homomorphism }
h:G\rightarrow G^{\prime }$\textsl{\ determines uniquely a correspondence }
h^{\circ }:\mathcal{F}_{G}\rightarrow \mathcal{F}_{G^{\prime }}\sqcup \{0\}
\textsl{\ so that}
\begin{enumerate}
\item[(5.1)] $h(M_{\exp (u)})\subseteq M_{\exp (h^{\circ }(u))}$\textsl{,}
u\in \mathcal{F}_{G}$.
\end{enumerate}
\noindent \textsl{Moreover, if }$h:G\rightarrow G^{\prime }$\textsl{\ is the
inclusion of a totally geodesic subgroup, then}
\begin{enumerate}
\item[(5.2)] $\left\Vert h^{\circ }(u)\right\Vert =\left\Vert u\right\Vert
\textsl{\ for all }$u\in \mathcal{F}_{G}$\textsl{.}$\square $
\end{enumerate}
Specifying a subset $\mathcal{F}_{G}\subset L(T)$ with properties (1.2)
amounts to an explicit characterization of the embedding $Fix(\gamma
)\subset G$. Apart from the general fact that the choice of $\mathcal{F}_{G}$
may not be unique, our proof of Theorem 1.1 implies that, if $G$ is $1
--connected, there exists a unique set $\mathcal{F}_{G}$ satisfying the
relation $\mathcal{F}_{G}\subset \Delta $. Geometrically
\bigskip
\noindent \textbf{Corollary 5.3. }\textsl{If }$G$\textsl{\ is simple and }$1
\textsl{--connected,}\textbf{\ }\textsl{each adjoint orbit }$M_{\exp (u)}$
\textsl{in }$Fix(\gamma )$\textsl{\ meets the subspace }$\exp (\Delta )
\textsl{\ of }$G$\textsl{\ exactly at one point.}$\square $
\bigskip
\textbf{Acknowledgement.} The authors are grateful to Angela Pasquale for
valuable communications, and in particular, for informing us the works \cit
{[Y1],[Y2],[Y3]} by I. Yokota.
| {'timestamp': '2013-12-17T02:10:54', 'yymm': '1312', 'arxiv_id': '1312.4229', 'language': 'en', 'url': 'https://arxiv.org/abs/1312.4229'} |
\section{INTRODUCTION}\label{intro}
In large part, we are aware of the principal processes that create free
electrons in otherwise completely neutral parts of the interstellar medium
(ISM) in the disk of our Galaxy. However, for some of the contributing
factors, ones that either enhance or diminish the relative ionization of the
gas, there is a need to validate our understanding of their strengths. Most
of these processes are well understood qualitatively, but one major
quantitative uncertainty is the effectiveness of extreme
ultraviolet (EUV) and soft X-ray photons
in ionizing the gas. Two factors contribute to this uncertainty: one is the
difficulty in measuring the fluxes of diffuse photons with energies above the
ionization potential of hydrogen (13.6~eV) but below about 100~eV, and the
other is an overall assessment of how well these photons can penetrate the
neutral regions, which depends on the porosity of the gas structures and the
distribution of radiation sources.
Our ultimate goal is not only to understand these processes better, but also
to obtain an estimate for the fractional amount of the gas that is in an
ionized state. This information is relevant to gauging the strength of
heating of the gas due to the photoelectric effect from dust grains
irradiated by starlight, since the grain charge, which regulates its rate,
depends on the electron density (Weingartner \& Draine 2001b).
While generally considered to be less important than the photoelectric
effect, other means of creating thermal energy, such as the heating caused by
secondary electrons from cosmic ray and X-ray ionizations and the dissipation
of Alfv\'en waves and magnetosonic turbulence (Kulsrud \& Pearce 1969; McIvor
1977; Spangler 1991; Minter \& Spangler 1997; Lerche et al. 2007), depend on
the relative fractions of electrons. Cooling of the gas through the
collisional excitation of the fine-structure levels of C$^+$, Si$^+$ and
Fe$^+$ or, at temperatures approaching $10^4\,$K various metastable levels
and the L$\alpha$ transition of hydrogen, are likewise governed by the degree
of partial ionization (Dalgarno \& McCray 1972). In the following
paragraphs, we consider three different pathways for creating small amounts
of ionization in the mostly neutral ISM.
Except for the interiors of dense clouds where there is significant
extinction, all of the space in the disk of our Galaxy is exposed to
ultraviolet starlight photons that are capable of ionizing atoms that have
first ionization potentials less than that of hydrogen (13.6~eV). Since the
recombination rates of the ions are slow relative to their ionization rates,
the concentrations of the ionized states of these atoms are strongly
dominant. Thus, it is a simple matter to add together the contributions from
various elements that are able to supply free electrons. The only
uncertainty here is an accounting of the strengths of depletions from the gas
phase caused by these elements condensing into solid form onto dust grains.
These strengths vary collectively for all of the elements from one region to
the next. If we take such variations into account (Jenkins 2009), we
can state that most of the gas will have free electron contributions that
should be somewhere in the range $n({\rm M}^+)=0.8-1.7\times 10^{-4}n({\rm
H}_{\rm tot})$, where $n({\rm M}^+)$ is the number density of heavy elements
that are capable of being ionized\footnote{This range was computed from the
solar abundances and representative values of the gas fractions $[{\rm
X}_{\rm gas}/{\rm H}]_1$ and $[{\rm X}_{\rm gas}/{\rm H}]_0$ listed in
Jenkins (2009) for elements that can be ionized by
starlight, except that the gas-phase abundance of C was lowered by a factor
of 0.43, in accord with the recommendation by Sofia et al. (2011) that
earlier determinations of $N({\rm C~II})$ were systematically too high by a
factor of 1/0.43.} and $n({\rm H}_{\rm tot})$ is the total density of hydrogen
in both neutral and ionized forms.
As cosmic ray particles collide with gas atoms in the Galaxy, they heat and
ionize the ISM. We are unable to observe the flux of the lowest energy
particles because we are shielded from them by the heliospheric magnetic
field, and extrapolations from the observed higher energy flux distributions
are uncertain (Spitzer \& Tomasko 1968). Nevertheless, from
measurements of the relative abundances of trace molecular species, the
cosmic ray ionization rates $\zeta_{\rm CR}\approx 0.5-9\times 10^{-16}{\rm
s}^{-1}$ seems to be the most plausible range for the general ISM
(Wagenblast \& Williams 1996; Liszt 2003; Indriolo et al. 2007; Neufeld et
al. 2010; Indriolo \& McCall 2012), although details in the chemical models
may introduce some uncertainty [cf. Le Petit et al. (2004) and Shaw et al.
(2008)]. The chemical models of Bayet et al. (2011) suggest that in some
particularly active regions the ionization rates may increase to $\zeta_{\rm
CR} > 1\times 10^{-14}{\rm s}^{-1}$.
We now consider a third mechanism for ionizing the gas, one that is harder to
quantify than the other two. EUV and soft X-ray
radiation can ionize atoms, through both the action of primary
photoionizations and by creating a cascade of energetic, secondary
photoelectrons that can collisionally ionize other atoms. Estimates of the
effectiveness of these agents are difficult to synthesize, since there are
many complicating factors. At high energies, much of the radiation arises
from cooling, very hot ($T>10^6\,$K) gas coming from recently shocked regions
within the disk and halo of our Galaxy. For energies slightly above about
100~eV, the photons can survive a journey through the neutral medium up to
about $N({\rm H}^0)=2\times 10^{19}\,{\rm cm}^{-2}$, and this penetration
depth progressively increases with energy. Supplementing this ionizing
radiation is that coming from several kinds of sources that are embedded
within the neutral medium. These include stars over a wide range of spectral
types on the main sequence, active X-ray binaries, and the plentiful, but
faint white dwarf (WD) stars.
\section{STRATEGY OF THE INVESTIGATION}\label{strategy}
The objective of the current study is to use specialized observations to help
resolve the uncertainties that were mentioned above and gain a quantitative
insight on how effectively the neutral regions are partially photoionized by
EUV and X-ray radiation. We do this by repeating a method developed by Sofia
\& Jenkins (1998, hereafter SJ98), who examined
interstellar UV absorption features that could be seen in the spectra of
background stars so that they could compare the abundance of the neutral form
of argon, which is highly susceptible to being photoionized, to that of
hydrogen, which has an ionization potential very close to that of argon but
with a markedly lower ionization cross section. We propose that it is safe
to assume that the abundance of argon (both neutral and ionized) relative to
that of hydrogen should be equal to the solar ratio. For instance, SJ98
presented arguments that support the principle that argon in the gas phase is
not likely to be depleted by being incorporated into an atomic matrix within
interstellar dust grains. We will reinforce this idea with some indirect
observational evidence in Section~\ref{reference_abund}. Thus, we operate on
the principle that any deficiency of neutral argon (Ar~I) below our
expectation for the amount of H~I that is present may be considered to arise
from the conversion of Ar to its ionized form, which is invisible.
The investigation by SJ98 covered an extremely limited number of target stars
observed with the {\it Interstellar Medium Absorption Profile Spectrograph\/}
({\it IMAPS\/}) (Jenkins et al. 1996). Later, Jenkins et al.
(2000b) and Lehner et al. (2003)
reported on observations of absorption features of Ar~I observed by the {\it
Far Ultraviolet Spectroscopic Explorer\/} ({\it FUSE\/}) (Moos et al. 2000;
Sahnow et al. 2000) toward collections of WD stars inside and
slightly beyond the edge of the Local Bubble,\footnote{The Local Bubble is an
irregularly shaped region with an unusually low average density with a radius
of about 80~pc that is approximately centered on the Sun (Vergely et al.
2010; Welsh et al. 2010; Reis et al. 2011). It contains small, partly
ionized clouds immersed in a much lower density medium (Redfield 2006;
Redfield \& Linsky 2008).} in order to infer ionization conditions of clouds
subjected to the characteristic radiation field in our immediate
neighborhood. Our current goal is to extend our reach well beyond the stars
surveyed in these two studies, again by using {\it FUSE\/} spectra, so that
we can sample regions of more typical densities well outside the Local
Bubble. We do this by downloading from the Mikulski Archive for Space
Telescopes (MAST) at the Space Telescope Science
Institute a large collection of {\it FUSE\/} spectra
of hot subdwarf stars that are situated several hundred pc away from us, well
beyond the boundary of the Local Bubble.
Our ultimate objective is to compare the neutral fractions of Ar and H, as
had been done in past investigations. However, a conventional approach of
simply deriving the two column densities and dividing one by the other is not
easily achievable with the data in this survey for two reasons. First, it is
difficult to measure $N$(Ar~I) because the lines are saturated, but only
moderately so, and recorded at low resolution. Second, the amount of H~I on
a sight line can usually be determined from the damping wings of L$\alpha$,
but when the interstellar column density is low and there is a significant
stellar L$\alpha$ absorption, one must know the star's effective temperature
and surface gravity and then create a model for the stellar feature, against
which the interstellar feature is superimposed. Also, as emphasized by Sofia
et al. (2011) measurements of $N$(H~I) using the
damping wings of L$\alpha$ can give a misleading outcome if one does not know
about and correct for the effects of the velocity structures of the gas.
This problem is probably most severe for low column density cases in the
present study.
We can overcome the difficulties mentioned above if we replace H with O as
the comparison element. In the wavelength coverage of {\it FUSE\/} there are
a large number of O~I absorption features, and these lines cover a broad
range of transition probabilities. Most important, the strengths of the O~I
features are comparable to the one available feature of Ar~I.\footnote{There
are two transitions of Ar~I in the {\it FUSE\/} wavelength coverage. We can
use only the one at 1048.220$\,$\AA\ because the 1066.660$\,$\AA\ line has
interference from a pair of strong stellar Si~IV lines at nearly the same
wavelength.} By a simple comparison of the strength of this Ar~I line to
those of O~I, we can determine $N({\rm Ar~I})/N({\rm O~I})$. We describe
this process in more detail in Section~\ref{derivation}.
The ionization fraction of O is strongly locked to that of H through a strong
charge exchange reaction\footnote{We add a caution that deviations from a
nearly one-to-one relationship for the ionization fractions of O and H can
occur at low temperatures because the ionization potential of O is slightly
higher than that of H ($\Delta E/k = 229\,$K) for the lowest fine-structure
level in the O~I ground state. However, since the ionization fractions are
small and most of the gas we are considering is at temperatures much higher
than $\Delta E/k$, this deviation is generally small enough to ignore.
Nevertheless, we performed explicit calculations of the O and H ionization
fractions, as described later in Section~\protect\ref{charge_exchange}.}
(Field \& Steigman 1971; Chambaud et al. 1980; Stancil et al. 1999). Thus
we can use O as a proxy for H. The only shortcoming of this tactic is that O
can be depleted in the ISM, but the depletion factors are not very large in
the regimes of low densities considered here (Cartledge et al. 2004, 2008;
Jenkins 2009).
Section~\ref{obs} of this paper describes the selection of archival {\it
FUSE\/} spectra and how they were processed to yield useful presentations of
the absorption features for measurements of equivalent widths.
Our method of interpreting the spectra to yield the ionization of Ar relative
to that of O is presented in Section~\ref{analysis}. Section~\ref{beyond}
contains a short digression on how we verify that the target stars are beyond
the edge of the Local Bubble. In Section~\ref{fundamentals} we outline the
basic equations that take into account the processes that influence the
partial ionizations of Ar, relative to those of H and O. The equations
presented in this section are virtually identical to those outlined by SJ98,
but with some new refinements (i.e., a few reactions that were not included
earlier). We consider the creation of free electrons from the starlight
ionization of heavy elements and the effects from cosmic rays as processes
that are mostly understood and already accounted for, and view the actions
arising from EUV and X-ray ionizations as the principal unknowns whose
strengths are to be determined. In Section~\ref{known_photoionization} we
make a prediction for the degree of ionization produced by known sources of
radiation, but find that in order to satisfy the general outcomes for our
measured ratios of Ar~I to O~I, an extraordinarily low volume density of
hydrogen $n({\rm H}_{\rm tot})$ is required. In order to obtain the same
results for higher densities, we must propose a means of achieving higher
levels of ionization. In Section~\ref{additional_photoionization} we propose
two possibilities: (1) there is a large residual ionization left over from
effects of radiation emitted by nearby, but now extinct supernova remnants (SNRs)
over the past several Myr or (2) the neutral medium is porous enough to allow
external, low-energy photons to penetrate the gas with less than the expected
amount of attenuation. Section~\ref{discussion} presents an overview of the
implications of our results on an assortment of physical processes and
various other kinds of observations that depend on electron fractions in the
diffuse, neutral medium. The paper ends with a summary of the main
conclusions (Section~\ref{summary}).
Appendices to this paper give descriptions of various atomic processes that
were incorporated into the calculations, but at a level of detail that most
readers may wish to ignore. A general section on the ionizations arising
from secondary electrons (Appendix~\ref{gamma_s}) is broken into two
subsections: one treats the effects from electrons liberated by the
ionizations of H and He (Section~\ref{electrons_H_He}), while the second one
discusses primary and Auger electrons created by the inner shell X-ray
ionizations of heavy elements (Section~\ref{heavy_elem}).
Appendix~\ref{gamma_he} gives the equations for evaluating the effects from a
multitude of different kinds of ionizing photons that arise from the
recombination of He$^{++}$ and He$^+$ ions with free electrons. Finally,
Appendix~\ref{cr_ioniz} describes how we can estimate the rates of cosmic ray
ionization of H$^0$, He$^0$ and Ar$^0$ from the observed rates that apply to
molecular hydrogen in dense clouds.
\section{OBSERVATIONS}\label{obs}
\subsection{Target Selection}\label{tgt selection}
Our objective was to make use of target stars that represented intermediate
cases between nearby WD stars, whose sight lines are entirely or
heavily influenced by conditions in the Local Bubble, and the much more
distant hot main-sequence, giant or supergiant stars that can create their
own enhanced ionizations in atypical concentrations of gas associated with
their formation. Hot subdwarf stars represent a class of objects that fall
into this intermediate category. They have distances that are of order a few
hundred parsecs from us, which reduces the contribution from material in the
Local Bubble to a very minor level. Since they are old, they have had
adequate time to escape from their progenitorial gas clouds, and thus their
locations are essentially random and should show no preference for dense gas
complexes. They have the additional advantage that they are bright enough to
yield good quality spectra, but they are not so bright that they exceed the
maximum allowed count-rate levels for {\it FUSE}.
In an initial screening of prospective targets, we examined the quick-look
plots of all stars classified as sdO and sdB spectral types in the MAST
archive of {\it FUSE\/} data. In this step, we rejected all spectra that
either seemed to show very strong stellar spectral features (8 stars) or that
had an inadequate signal-to-noise ratio ($S/N$) at wavelengths in the
vicinity of 920$\,$\AA\ (70 stars), which is where most of the O~I lines are
situated. A few further rejections were made after the spectra were
downloaded and found to have observing anomalies (an extraordinarily large
number of missing observations caused by channel misalignments: 3 stars),
strong stellar features that were not evident in the quick-look plots (3
stars), or molecular hydrogen lines that were strong enough to seriously
compromise the lines that we wanted to measure (2 stars). The lack of stars
with exceptionally strong H$_2$ features helped to eliminate sight lines that
penetrate dense, cool gas clouds.
\subsection{Creation of the Spectra}\label{creation}
All of the downloads of the calibrated {\it FUSE\/} data from MAST occurred
well after the final pipeline reductions were performed for the archive with
CalFUSE version 3.2.3 (Dixon et al. 2007). For every target, we
accepted data from all of the available observing sessions (identified by
unique archive root names) but rejected any subexposures that had an
extraordinarily low count level caused by a channel misalignment during the
observation. We used exposures obtained during both orbital day and night.
Normally, one must be cautious about observations of features for either O~I
or N~I because they can be filled in by diffuse telluric emission lines
during daytime observing. However the O~I transition strengths considered
here are so weak that the telluric contributions are insignificant.
All subexposures and spectral channels that passed our initial screening were
coadded with weight factors based on the inverse squares of their respective
values of $S/N$ for intensities smoothed over a wavelength interval of
0.12$\,$\AA\ (or 9 independent spectral elements -- this ensures that weights
are not strongly influenced by random noise excursions). Before this
coaddition took place, we examined some strong interstellar features and
aligned the individual spectra in wavelength against a preliminary coaddition
with no wavelength shifts. This process enabled us to virtually eliminate
any degradation in resolution caused by drifts of the spectra in the
wavelength direction from one subexposure to the next. However, there can
still be overall small systematic errors in radial velocity of about
$10\,{\rm km~s}^{-1}$; see Appendix~A of Bowen et al. (2008). In a few
cases, the spectral $S/N$ values were too low to allow such shifts to be made
with much confidence, even after the intensities were smoothed with a median
filter for viewing. Such spectra were combined without any shifts. For
every target, two combined spectra were created: one was made up with shifts
appropriate to the spectral region covering the Ar~I line at 1048$\,$\AA,
while the other had differentials that were optimized for the wavelengths
that covered the weakest O~I lines near 920$\,$\AA.
\section{ANALYSIS}\label{analysis}
\subsection{Equivalent Widths and Their Errors}\label{EW}
We measured equivalent widths of the Ar~I and O~I lines by integrating
intensity deficits below best-fitting Legendre polynomials for the continua
defined from intensities at locations somewhat removed from the features.
Special precautions were made to account for various sources of error, which
are important for later analysis stages that assign relative weights to
different measurements and also for the estimates of the ultimate errors in
the results. First, we accounted for the direct effect that random
count-rate variations can have on the equivalent width outcome (Jenkins et
al. 1973). Next, the weakest lines are subject to uncertainties arising from
improper definitions of the continua. To construct the probable errors, we
evaluated the expected formal errors in the polynomial coefficients, as
described by Sembach \& Savage (1992), and then we multiplied them by 2 in
order to make an approximate allowance for additional uncertainties caused by
the arbitrariness in selecting the most appropriate polynomial order. To
find the effects of these continuum errors on our measurements, the
equivalent widths were re-evaluated using the probable excursions of the
continua on either side of the preferred ones. Errors in the background
subtraction in the {\it FUSE\/} data processing are small compared to the
other errors.
\begin{deluxetable}
{
l
c
c
c
c
c
c
c
c
c
c
c
c
c
c
}
\tabletypesize{\scriptsize}
\rotate
\tablecolumns{14}
\tablewidth{0pt}
\tablecaption{Equivalent Widths\tablenotemark{a}\label{EW_table}}
\tablehead{
\colhead{} & \colhead{Ar I\tablenotemark{b}} &
\multicolumn{12}{c}{O~I\tablenotemark{b,c}}\\
\cline{3-14}
\colhead{} & \colhead{1048.220} & \colhead{919.917} & \colhead{922.200} &
\colhead{925.446} &
\colhead{916.815} & \colhead{930.257} & \colhead{919.658} & \colhead{921.857}
&
\colhead{924.950} & \colhead{950.885} & \colhead{976.448} & \colhead{948.686}
&
\colhead{971.738}\\
\colhead{Star Name} & \colhead{2.440} & \colhead{$-0.788$} &
\colhead{$-0.645$} &
\colhead{$-0.484$} & \colhead{$-0.362$} & \colhead{$-0.301$} &
\colhead{$-0.137$} &
\colhead{$-0.001$} & \colhead{0.155} & \colhead{0.176} & \colhead{0.509} &
\colhead{0.778} & \colhead{1.052}\\
\colhead{} & \colhead{\nodata} & \colhead{11.6} & \colhead{15.3} &
\colhead{20.6} &
\colhead{25.0} & \colhead{27.8} & \colhead{34.5} & \colhead{40.4} &
\colhead{46.9} &
\colhead{49.0} & \colhead{\nodata} & \colhead{\nodata} & \colhead{\nodata}\\
\colhead{(1)}& \colhead{(2)}& \colhead{(3)}& \colhead{(4)}&
\colhead{(5)}& \colhead{(6)}& \colhead{(7)}& \colhead{(8)}& \colhead{(9)}&
\colhead{(10)}& \colhead{(11)}& \colhead{(12)}& \colhead{(13)}&
\colhead{(14)}
}
\startdata
2MASS15265306\\
~~~~~+7941307\dotfill&$ 77\pm 18$
&$( 14\pm 16)$&$( -4\pm 16)$&$ 29\pm 12$& \nodata&$ 24\pm 22$&$
41\pm 22$
&$ 39\pm 19$&$ 84\pm 11$&$ 59\pm 15$&$ 97\pm 12$&$ 111\pm 11$&$( 120\pm
12)$\\
AA Dor\dotfill&$ 60\pm 10$
&$ 30\pm 18$& \nodata&$ 56\pm 13$& \nodata&$ 75\pm 31$&$
51\pm 37$
& \nodata&$ 93\pm 13$&$ 73\pm 21$&$( 121\pm 19)$& \nodata&$(
225\pm 9)$\\
AGK+81 266\dotfill&$ 76\pm 19$
&$ 22\pm 15$&$ 44\pm 15$&$ 33\pm 15$& \nodata&$ 60\pm 19$&$ 76\pm
15$
&$ 87\pm 18$&$ 99\pm 14$&$ 122\pm 14$&$( 116\pm 18)$&$( 102\pm 15)$&$(
125\pm 14)$\\
BD+18 2647\dotfill&$ 43\pm 18$
&$ 22\pm 13$& \nodata&$ 37\pm 11$& \nodata&$ 26\pm 16$&$
39\pm 18$
& \nodata&$ 78\pm 11$&$ 68\pm 11$&$( 63\pm 13)$&$( 131\pm 12)$&$(
159\pm 11)$\\
BD+25 4655\dotfill&$ 33\pm 18$
&$ 3\pm 12$& \nodata&$ 18\pm 12$& \nodata&$ 22\pm 11$&$
21\pm 12$
& \nodata&$ 53\pm 11$&$ 41\pm 12$&$ 64\pm 13$&$ 77\pm 12$&$( 75\pm
11)$\\
BD+28 4211\dotfill&$ 31\pm 8$
&$ 14\pm 3$&$ 27\pm 5$&$ 24\pm 3$&$ 25\pm 4$&$ 37\pm 4$&$ 34\pm
4$
&$ 35\pm 4$&$ 42\pm 3$&$ 51\pm 3$&$( 42\pm 4)$&$( 83\pm 4)$&$(
62\pm 3)$\\
BD+37 442\dotfill&$ 166\pm 7$
& \nodata&$ 61\pm 28$&$ 107\pm 16$& \nodata&$ 95\pm 24$&
\nodata
&$ 166\pm 17$&$ 128\pm 16$& \nodata& \nodata& \nodata&
\nodata\\
BD+39 3226\dotfill&$ 54\pm 3$
&$( 21\pm 8)$&$( 81\pm 8)$&$ 34\pm 7$&$ 40\pm 8$&$ 48\pm 10$&$
47\pm 8$
&$ 38\pm 8$&$ 52\pm 7$&$ 59\pm 7$&$ 54\pm 7$&$ 69\pm 8$&$( 73\pm
7)$\\
CPD$-$31 1701\dotfill&$ 16\pm 8$
&$ 11\pm 5$& \nodata&$ 19\pm 4$& \nodata&$ 20\pm 11$&$
39\pm 6$
& \nodata&$ 45\pm 4$&$( 28\pm 5)$&$( 45\pm 7)$&$( 68\pm 5)$&$(
91\pm 4)$\\
CPD$-$71D172\dotfill&$ 34\pm 10$
&$ 31\pm 13$&$ 34\pm 13$&$ 52\pm 12$& \nodata&$ 57\pm 13$&$ 47\pm
15$
&$ 36\pm 18$&$ 67\pm 12$&$ 67\pm 13$&$( 61\pm 13)$&$( 93\pm 13)$&$(
83\pm 14)$\\
EC11481$-$2303\dotfill&$ 132\pm 16$
&$( 39\pm 17)$& \nodata&$ 75\pm 16$& \nodata&$ 92\pm 16$&$
99\pm 17$
&$ 80\pm 34$&$ 145\pm 15$&$ 121\pm 17$&$ 153\pm 17$&$ 146\pm 16$&$ 177\pm
16$\\
Feige 34\dotfill&$ 36\pm 17$
&$( 9\pm 13)$& \nodata&$ 13\pm 13$& \nodata&$ 28\pm 14$&$
38\pm 13$
& \nodata&$ 42\pm 13$&$ 46\pm 14$&$ 41\pm 21$&$ 86\pm 19$&
\nodata\\
HD$\,$113001\dotfill&$ 115\pm 20$
&$( 28\pm 10)$& \nodata&$( 50\pm 10)$& \nodata&$( 53\pm 15)$&$
63\pm 10$
& \nodata&$ 96\pm 10$&$ 93\pm 12$&$ 100\pm 13$&$ 125\pm 11$&$ 128\pm
10$\\
JL 119\dotfill&$ 112\pm 5$
&$ 52\pm 14$&$ 39\pm 27$&$ 89\pm 8$& \nodata&$ 103\pm 15$&$ 83\pm
25$
&$ 80\pm 23$&$ 138\pm 7$&$ 111\pm 19$&$( 169\pm 7)$&$( 180\pm 6)$&$(
174\pm 7)$\\
JL 25\dotfill&$ 77\pm 19$
&$ 29\pm 17$& \nodata&$ 43\pm 17$& \nodata&$ 72\pm 22$&$
29\pm 17$
& \nodata&$ 69\pm 17$&$ 94\pm 17$& \nodata&$ 111\pm 18$&$
106\pm 17$\\
JL 9\dotfill&$ 100\pm 10$
&$ 46\pm 12$& \nodata&$ 67\pm 9$& \nodata&$ 95\pm 16$&$
87\pm 14$
& \nodata&$ 106\pm 9$&$ 116\pm 9$& \nodata&$ 129\pm 11$&$(
142\pm 12)$\\
LB 1566\dotfill&$ 57\pm 6$
&$ 32\pm 13$& \nodata&$ 51\pm 11$& \nodata&$ 56\pm 14$&$
48\pm 15$
& \nodata&$ 81\pm 11$&$ 73\pm 19$&$( 98\pm 11)$&$( 108\pm 14)$&$(
111\pm 11)$\\
LB 1766\dotfill&$ 67\pm 16$
&$ -3\pm 13$& \nodata&$ 61\pm 12$& \nodata&$ 58\pm 14$&$
38\pm 14$
& \nodata&$ 98\pm 11$& \nodata&$ 120\pm 11$&$( 148\pm 10)$&$(
168\pm 10)$\\
LB 3241\dotfill&$ 56\pm 9$
&$ 29\pm 8$&$ 25\pm 9$&$ 54\pm 8$& \nodata&$ 53\pm 8$&$ 52\pm
8$
&$ 97\pm 8$&$ 82\pm 7$&$ 106\pm 7$&$( 112\pm 7)$&$( 123\pm 7)$&$(
128\pm 7)$\\
LS 1275\dotfill&$ 62\pm 15$
&$ 29\pm 10$&$ 42\pm 12$&$ 45\pm 9$&$ 52\pm 14$&$ 65\pm 13$&$ 46\pm
10$
&$( 99\pm 13)$&$ 58\pm 9$&$ 74\pm 10$&$ 80\pm 10$&$ 89\pm 11$&$( 89\pm
9)$\\
LSE 234\dotfill&$ 115\pm 12$
&$( 47\pm 16)$&$ 72\pm 17$&$ 76\pm 15$& \nodata&$ 94\pm 16$&$
85\pm 18$
&$ 123\pm 17$&$ 107\pm 15$&$ 122\pm 15$& \nodata&$ 154\pm 18$&$ 130\pm
15$\\
LSE 259\dotfill&$ 149\pm 6$
& \nodata& \nodata&$ 88\pm 14$& \nodata&$ 134\pm 20$&
\nodata
& \nodata&$ 123\pm 13$&$ 156\pm 13$& \nodata& \nodata&
\nodata\\
LSE 263\dotfill&$ 119\pm 17$
&$( 49\pm 14)$& \nodata&$ 73\pm 12$&$ 101\pm 19$&$ 80\pm 15$&$
76\pm 14$
& \nodata&$ 91\pm 13$&$ 76\pm 13$&$ 95\pm 17$&$ 139\pm 14$&$ 159\pm
12$\\
LSE 44\dotfill&$ 93\pm 12$
&$ 56\pm 11$&$ 68\pm 12$&$ 68\pm 10$& \nodata&$ 78\pm 15$&$ 61\pm
15$
&$ 94\pm 12$&$ 92\pm 9$&$ 132\pm 10$& \nodata&$( 140\pm 10)$&$(
107\pm 11)$\\
LSII+18 9\dotfill&$ 91\pm 15$
& \nodata& \nodata&$ 54\pm 21$&$ 66\pm 38$&$ 73\pm 23$&$
72\pm 21$
& \nodata&$ 78\pm 21$&$ 125\pm 21$& \nodata& \nodata&$(
119\pm 21)$\\
LSII+22 21\dotfill&$ 45\pm 12$
&$ 27\pm 10$& \nodata&$ 35\pm 9$&$ 40\pm 15$&$ 52\pm 13$&$ 44\pm
11$
&$( 116\pm 8)$&$ 63\pm 8$&$ 74\pm 9$&$( 91\pm 8)$&$( 100\pm 9)$&$(
119\pm 8)$\\
LSIV+10 9\dotfill&$ 179\pm 12$
&$( 58\pm 10)$& \nodata&$ 88\pm 9$& \nodata&$ 106\pm 18$&$
88\pm 12$
& \nodata&$ 152\pm 9$&$ 170\pm 10$&$ 231\pm 9$& \nodata&$(
192\pm 10)$\\
LSS 1362\dotfill&$ 88\pm 3$
&$( 39\pm 11)$&$ 52\pm 14$&$ 59\pm 8$& \nodata&$ 78\pm 18$&$
64\pm 20$
&$( 65\pm 8)$&$( 68\pm 10)$&$ 134\pm 7$& \nodata& \nodata&
\nodata\\
MCT 2005\\
~~~$-$5112\dotfill&$ 103\pm 8$
& \nodata& \nodata&$ 74\pm 11$& \nodata&$ 75\pm 18$&
\nodata
& \nodata&$ 113\pm 9$&$ 135\pm 12$&$( 168\pm 15)$&$( 150\pm 13)$&$(
124\pm 18)$\\
MCT 2048\\
~~~$-$4504\tablenotemark{d}\dotfill&$ 122\pm 12$
& \nodata&$ 86\pm 21$&$ 83\pm 15$& \nodata&$ 61\pm 59$&
\nodata
&$ 90\pm 28$&$ 118\pm 14$&$ 117\pm 17$&$ 149\pm 14$&$ 174\pm 16$&$( 172\pm
15)$\\
NGC6905\\
~~~star\tablenotemark{e}\dotfill&$ 147\pm 8$
&$( 45\pm 39)$& \nodata&$ 101\pm 39$& \nodata&$ 83\pm 43$&$
62\pm 46$
& \nodata&$ 139\pm 38$&$ 167\pm 38$&$ 205\pm 39$& \nodata&$(
154\pm 39)$\\
PG0919+272\dotfill&$ 55\pm 27$
&$ 34\pm 17$&$ 31\pm 16$&$ 43\pm 16$&$ 45\pm 27$&$ 83\pm 17$&$ 59\pm
17$
&$ 64\pm 16$&$ 81\pm 15$&$ 65\pm 17$&$ 87\pm 15$&$ 108\pm 16$&$( 106\pm
15)$\\
PG0952+519\dotfill&$ 52\pm 3$
&$ 19\pm 8$& \nodata&$ 36\pm 7$& \nodata&$ 43\pm 8$&$
53\pm 9$
& \nodata&$ 74\pm 7$& \nodata&$( 125\pm 6)$&$( 154\pm 8)$&$(
148\pm 6)$\\
PG1032+406\dotfill&$ -21\pm 27$
&$ 3\pm 23$&$ 11\pm 19$&$ 15\pm 18$&$( 95\pm 24)$&$ 20\pm 19$&$ -2\pm
36$
&$ 36\pm 18$&$ 15\pm 17$&$ 72\pm 17$&$ 46\pm 20$&$ 70\pm 18$&$( 94\pm
17)$\\
PG1051+501\dotfill&$ 165\pm 15$
& \nodata& \nodata&$ 81\pm 30$& \nodata&$ 64\pm 33$&
\nodata
& \nodata&$ 128\pm 27$&$ 142\pm 28$& \nodata& \nodata&$
185\pm 28$\\
PG1230+068\dotfill&$ 102\pm 15$
&$ 40\pm 8$&$ 45\pm 11$&$ 72\pm 7$& \nodata&$ 82\pm 16$&$ 86\pm
9$
&$ 79\pm 13$&$ 110\pm 7$&$ 88\pm 8$&$ 105\pm 18$&$ 126\pm 7$&$( 139\pm
7)$\\
PG1544+488\dotfill&$ 84\pm 3$
&$ 26\pm 11$&$ 25\pm 13$&$ 56\pm 8$&$ 76\pm 34$&$ 69\pm 10$&$ 74\pm
13$
&$ 89\pm 11$&$ 91\pm 7$&$ 97\pm 9$&$( 124\pm 11)$&$( 132\pm 7)$&$(
151\pm 8)$\\
PG1605+072\dotfill&$ 131\pm 15$
& \nodata& \nodata&$ 72\pm 21$& \nodata&$ 99\pm 25$&
\nodata
& \nodata&$ 136\pm 20$&$ 163\pm 20$&$ 229\pm 20$&$( 166\pm 22)$&$(
186\pm 20)$\\
PG1610+519\dotfill&$ 122\pm 9$
& \nodata& \nodata&$ 81\pm 12$& \nodata& \nodata&
\nodata
& \nodata&$ 150\pm 9$& \nodata&$ 256\pm 5$& \nodata&$(
228\pm 7)$\\
PG2158+082\dotfill&$ 117\pm 7$
& \nodata& \nodata&$ 100\pm 19$& \nodata&$ 97\pm 26$&
\nodata
& \nodata&$ 121\pm 19$&$ 175\pm 17$& \nodata&$( 154\pm 22)$&
\nodata\\
PG2317+046\dotfill&$ 68\pm 20$
&$ 49\pm 18$&$ 29\pm 31$&$ 61\pm 19$& \nodata&$ 50\pm 24$&$ 28\pm
21$
&$ 48\pm 33$&$ 71\pm 18$&$ 135\pm 18$& \nodata&$ 132\pm 19$&$( 81\pm
19)$\\
Ton 102\dotfill&$ 51\pm 13$
&$ 21\pm 15$&$ 22\pm 16$&$ 26\pm 16$& \nodata&$ 39\pm 16$&$ 34\pm
16$
&$ 79\pm 16$&$ 42\pm 17$&$ 63\pm 16$&$ 72\pm 15$& \nodata&
\nodata\\
Ton S227\dotfill&$ 80\pm 25$
&$ 27\pm 10$&$ 26\pm 11$&$ 43\pm 10$& \nodata&$ 21\pm 20$&$ 27\pm
11$
&$ 30\pm 12$&$ 58\pm 9$& \nodata&$ 85\pm 11$&$ 111\pm 12$&$ 121\pm
9$\\
UV0904$-$02\tablenotemark{f}\dotfill&$ 69\pm 4$
&$( 30\pm 9)$& \nodata&$ 43\pm 8$& \nodata&$ 59\pm 12$&$
52\pm 10$
& \nodata&$ 67\pm 7$&$ 65\pm 11$&$ 75\pm 7$&$ 84\pm 8$&$( 89\pm
7)$\\
\enddata
\tablenotetext{a}{Equivalent widths are given in m\AA. Values given in
parentheses indicate
lines that were not used in the linear fits.}
\tablenotetext{b}{Numbers given below indicate (1) wavelengths in \AA, (2)
line strengths in
terms of $\log (f\lambda)$ taken from Morton (2003), and (3) the
equivalent width in m\AA\
for $N({\rm O~I})=10^{16}\,{\rm cm}^{-2}$ and $b=6\,{\rm km~s}^{-1}$.}
\tablenotetext{c}{{}Lines are arranged in order of increasing strength.}
\tablenotetext{d}{Name recognized by SIMBAD: 2MASS J20515997-4042465.}
\tablenotetext{e}{Central star of the planetary nebula NGC 6905.}
\tablenotetext{f}{Name recognized by SIMBAD: 2MASS J09070812-0306139.}
\end{deluxetable}
\clearpage
The sources of error mentioned above are straightforward to evaluate and
would apply to just about any measure of an equivalent width. However, with
the subdwarf stars, we must also contend with the confusion produced by
stellar lines. We made no attempt to model such features, because in order
to do so we would need to know the details of the stellar parameters for each
star. Instead, we regarded the influence of stellar features as random
sources of error in our line measurements. In order to estimate the
amplitude of such errors, which can vary markedly from one star to the next
and can change with wavelength, we measured for each target the variance of a
large number of equivalent width measurements of imaginary, fake lines at
wavelengths similar to the ones under study, but that were displaced away
from known real interstellar lines, both atomic and molecular. This variance
was then used as a guide for estimating the errors that should arise from
stellar features.
Figure~\ref{sample_spectra} shows samples of spectra covering the relevant
wavelength regions for two stars. These two cases illustrate strong
differences in the degree of interference from stellar lines. The first
example, AGK+81~266, has many stellar lines that can either add an apparent
absorption to an interstellar line or distort the continuum level that is
measured on either side of the line. For this target, these effects dominate
over other sources of error and create $1\sigma$ uncertainties in $W_\lambda$
equal to 18.5 and 14.2$\,$m\AA\ for the Ar~I and O~I lines, respectively.
This star also exhibits molecular hydrogen features of moderate strength, but
here the lines are not strong enough to compromise the measurements of the
atomic lines. A far more favorable case for measuring interstellar features
is presented by the star UV0904$-$02. Here, uncertainties produced by random
stellar features should create errors of only 3.6$\,$m\AA\ for the Ar~I line
and 6.6$\,$m\AA\ for the O~I lines.
All of the errors discussed in this section were combined in quadrature to
synthesize the overall errors in the equivalent widths. Values for the
equivalent widths of all lines and their associated uncertainties are listed
for each of our targets in Table~\ref{EW_table}. The columns in this table
are arranged in a sequence from the weakest to the strongest lines.
\subsection{Derivations of [Ar~I/O~I] Values and Their
Uncertainties}\label{derivation}
\subsubsection{Reference Abundances}\label{reference_abund}
Detailed discussions on various methods of measuring the protosolar and
B-star abundances of Ar have been presented by Lodders (2008) and Lanz et
al. (2008). We adopt a mean value for the recommended outcomes of the two,
$\log ({\rm Ar/H})+12=6.60\pm 0.10$. (Since both determinations might be
subject to common errors, the error of the mean is not reduced below the
0.10$\,$dex errors specified by each of them.) This value is higher than the
solar photospheric value proposed by Asplund et al. (2009), and it remains so
even after one applies a correction for gravitational settling of $+0.07\,$dex
(Lodders 2003) to obtain a protosolar value of $\log ({\rm
Ar/H})+12=6.47\pm 0.13$. For O, we take the solar photospheric value given
by Asplund et al. (2009) and again apply a
$+0.07\,$dex settling correction to get $\log ({\rm O/H})+12=8.76\pm 0.05$.
This value agrees remarkably well with the measurement $\log ({\rm
O/H})+12=8.76\pm 0.03$ obtained for B-stars by Przybilla et al.
(2008).
We must now consider the prospect that some of the Ar and O atoms are
incorporated into solid form within or on the surfaces of dust grains, and
this effect might be large enough to distort our findings on the differences
in ionization. Unfortunately, we have no direct information about the
depletion of gas-phase Ar, since ionization corrections (the object of the
present study) can influence the outcome. While in principle it would be
beneficial if we could study $N({\rm Ar~I})$ along sight lines that penetrate
dense media, where depletions are likely to dominate over ionization effects,
this is not possible because the absorption lines are far too saturated (much
more so than in the current study).
\onecolumn
\begin{figure}
\vspace*{-2cm}
\epsscale{.95}
\plotone{sample_spectra.eps}
\caption{Examples of {\it FUSE\/} spectra, where signals from all detector
channels have been combined, for the two stars AGK+81~266 (top group of
panels) and UV0904$-$02 (bottom group). Various interstellar features are
indicated. The two panels showing the Ar~I feature best illustrate the large
contrasts in the strengths of unidentified stellar features that can
interfere with the interstellar ones.\label{sample_spectra}}
\end{figure}
\twocolumn
SJ98 presented a number of
theoretical arguments that suggested Ar is not appreciably depleted in
the low density ISM that we can observe. However, it would be good to
confirm this outlook by some independent, experimental means. Fortunately,
krypton is an element that can be observed in the ISM, and, like argon, is
chemically inert. It would be reasonable to expect that the capture of Kr
onto interstellar dust grains, if it happens, would be similar to that of Ar.
An advantage of studying Kr is that its interstellar features are weak
(Cartledge et al. 2008), which means that they can be used to measure
column densities over sight lines that have high values of $n({\rm H}_{\rm
tot})$ where element depletions should be generally very strong. As with Ar,
Kr has a photoionization cross section that is substantially larger than that
of H (Sterling 2011). Thus, any simple measure of the deficiencies
of this element in the low density ISM could simply be a product of it being
more easily photoionized than H.
A way to overcome the confusion from the offset produced by ionization is to
compare differential capture rates of elements onto grains as the conditions
that favor grain formation change. For instance, Jenkins (2009) has
determined that for sight lines with $N({\rm H}^0)>10^{19.5}$, where
ionization corrections should be small, when $5\times 10^5$ O atoms are
removed from the gas phase, $1(\pm 1)$ atom of Kr vanishes. Since O is more
abundant than Kr by a factor of $2.5\times 10^5$, any relative decrease in
the abundance of Kr in the gas phase would be about half that of O (but the
errors allow for this factor to range from zero to being equal to that of
oxygen). If we accept the idea that Ar depletes in the same manner as Kr,
probably to within a factor of $\sqrt{m_{\rm Kr}/m_{\rm Ar}}$, and that in low
density media O shows very low depletions ($<0.1\,$dex) (Jenkins 2009), it is
reasonable to adopt a gas-phase abundance ratio that is virtually the same as
the protosolar ones, $\log ({\rm Ar/O})_\odot=-2.16\pm 0.11$. If indeed
there is some mild depletion of O due to the formation of silicate dust
grains, we may understate the strength of the ionization of Ar.
\subsubsection{Interpretation of Line Strengths}\label{line_strengths}
We adopt the premise that the distribution of radial velocities of the
neutral argon atoms is identical to that of neutral oxygen (but this is not
exactly correct; we will revisit this issue later). Figure~\ref{oi_fits}
shows examples of some standard curve-of-growth plots for the O~I lines
appearing in the spectra of the same two targets that were featured in
Fig.~\ref{sample_spectra}, AGK+81~266 and UV0904$-$02. The former of the two
illustrates an average amount of line saturation for relevant features, while
the latter represents an extreme case of saturation caused by a low overall
dispersion of radial velocities. In principle, we could have derived values
for $N$(O~I) from the best-fit curves of growth shown by the dashed curves in
the figure panels and then assume that $N$(Ar~I) follows from the equivalent
width of the one available line at 1048.220$\,$\AA\ assuming the same
velocity dispersion parameter $b$ as that found for O~I. Instead, we used a
much simpler approach that sidesteps the goal of deriving explicit values of
$N$ and $b$ (whose errors are strongly correlated) and proceeds directly to
an answer for just the ratio of the two column densities. An advantage here
is that we can make a straightforward analytical determination of the
uncertainty of the outcome based on the errors of some linear fitting
coefficients.
The comparison of Ar~I to O~I is based on the following principle. If one
could imagine the existence of a hypothetical O~I line with a transition
strength $\log (f\lambda)_{\rm O~I}$ that is just right to produce a value of
$W_\lambda/\lambda$ that exactly matches that of the Ar~I line, one could
then derive the deficiency of Ar~I with respect to its expectation based on
O~I, which we denote in logarithmic form as [Ar~I/O~I]. This quantity yields
the logarithm of the ratio of the two neutral fractions relative to the solar
abundance ratio and is given by the relation
\begin{eqnarray}\label{ar_o_eqn}
[{\rm Ar~I/O~I}]&=&\log (f\lambda)_{\rm O~I}-\log (f\lambda)_{\rm Ar~I}-\log
({\rm Ar/O})_\odot\nonumber\\
&=&\log (f\lambda)_{\rm O~I}-0.28\pm 0.11,
\end{eqnarray}
where $\log (f\lambda)_{\rm Ar~I}=2.440\pm 0.004$ (Morton 2003).
\onecolumn
\begin{figure}
\epsscale{1.25}
\plottwo{oi_fit_AGK+81_266.eps}{oi_fit_UV0904-02.eps}
\caption{Examples of how the values of [Ar~I/O~I] are derived for the two
stars chosen for Fig.~\protect\ref{sample_spectra}. The equality shown in
each $x$-axis label applies to the horizontal projection of the Ar~I line
strength onto the O~I curve of growth, as expressed in
Eq.~\protect\ref{ar_o_eqn}. We performed weighted least-squares linear fits
for the logarithms of $W_\lambda/\lambda$ for lines of O~I that are most
influential in establishing the trends ({\it solid lines\/}) for a comparison
with the measurements of $\log W_\lambda/\lambda$ of the Ar~I line. Values
for $c_0$ and $c_1$ given in Eq.~\ref{linear_trend} that define these trends
are shown in the boxes. The best-fit curves of growth together with their
$b$ values are also shown, but they are not used in the derivations. O~I
lines that had strengths that were well above or below those in the important
portion of curve of growth were not included in the derivation of the best
fits, and they are indicated here as ``(not used).'' These unused lines are
identified with parentheses around the equivalent width values listed in
Table~\protect\ref{EW_table}.\label{oi_fits}}
\end{figure}
\twocolumn
To determine the strength of the hypothetical O~I line, we perform a weighted
least-squares linear fit for $\log (W_\lambda/\lambda)$ vs. the quantity on
the right-hand side of Eq.~\ref{ar_o_eqn} (the abscissa for each plot in
Fig.~\ref{oi_fits}) for an appropriate selection of O~I lines. [As with the
Ar~I line the $f$-values of the O~I transitions are from Morton
(2003).] The lines that are chosen for this fit are ones
that are situated not too far from the horizontal projection (shown by dotted
lines in the figure) of $\log (W_\lambda/\lambda)$ for the single line of
Ar~I onto the trend for the O~I lines. With this restricted fit, we define a
simple relationship in $\log (W_\lambda/\lambda)$ that is a good
approximation to a relevant portion of the curve of growth.
In the two panels of Fig.~\ref{oi_fits}, these best-fit linear trends are
shown by the straight solid lines. They depict the relation between $y=\log
W_\lambda/\lambda$ and $x=\log (f\lambda)-0.28$ according to the equation
\begin{equation}\label{linear_trend}
y=c_0+c_1(x-x_0),
\end{equation}
where
\begin{equation}\label{x_0}
x_0=\sum_i x_i\sigma (y)_i^{-2}\Big/ \sum_i \sigma (y)_i^{-2}
\end{equation}
represents a zero reference point that produces a vanishing covariance for
the errors in the fitting coefficients $c_0$ and $c_1$. A nominal value on
the $x$ axis for the projection of $y_{\rm Ar~I}$ onto the linear relation is
given by
\begin{equation}\label{aro_solution}
x_{\rm Ar~I}=x_0+{y_{\rm Ar~I}-c_0\over c_1}
\end{equation}
In the fraction part of this equation, the numerator and denominator have
errors $(\sigma(y_{\rm Ar~I})^2+\sigma(c_0)^2)^{0.5}$ and $\sigma(c_1)$,
respectively. A conventional approach for deriving the error of the quotient
is to add in quadrature the relative errors of the two terms, yielding the
relative error of the quotient. However, this scheme breaks down when the
error in the denominator is not very much less than the denominator itself.
A more robust way to derive the error of a quotient has been developed by
Geary (1930); for a concise description of this method see
Appendix~A of Jenkins (2009). We use this method here; it
is effective as long as there is little chance that the denominator minus its
error could become very close to zero or be negative, i.e., ${\rm
denom.}/\sigma({\rm denom.}) \gtrsim 3$.
The dotted lines in Figure~\ref{oi_fits} show schematically how the best fit
values and the error ranges for [Ar~I/O~I] are derived. Note that the
horizontal and vertical segments for the error limits do not exactly
intersect the best linear trend for the O~I lines because the error analysis
allows for the uncertainty for the location of this line. (But we point out
that the intersections for the worst possible error in one direction for
$y_{\rm Ar~I}$ do not occur at the locations for the worst possible
deviations in the opposite direction for the trend line.) Also, the final
errors for $[{\rm Ar~I}/{\rm O~I}]=x_{\rm Ar~I}$ are not symmetrical about
the best values. On average, the upward error bounds are about 75\% as large
as the negative ones.
The difference in atomic weights of Ar and O will cause the thermal
contributions to the Doppler broadenings of these two elements to differ from
each other. The impact of this effect on our results for the column density
ratios is small however. For instance, if we consider that we are viewing
absorption lines arising from the warm, neutral medium (WNM) and there were
no bulk motions of the gas, the line broadening parameters $b_{\rm therm.}$
caused by thermal Doppler broadening for $T=7000\,$K would be 2.7 and
$1.7\,{\rm km~s}^{-1}$ for O~I and Ar~I, respectively. For a typical
observation, such as the one for AGK+81~266 illustrated in the left-hand
panel of Fig.~\ref{oi_fits}, we find that the observed curve of growth for
the O~I lines, yielding an apparent $b_{\rm obs.}=9.9\,{\rm km~s}^{-1}$,
indicates that kinematic effects arising from turbulent motions (or multiple
velocity components) should be the most important contribution to the
broadening since $b_{\rm turb.}=\sqrt{b_{\rm obs.}^2-b({\rm O~I})_{\rm
therm.}^2}=9.53\,{\rm km~s}^{-1}$ is only slightly smaller than $b_{\rm
obs.}$. The curve of growth that characterizes the saturation of the Ar~I
line would conform to a slightly lower velocity dispersion parameter compared
to that observed for O~I, $b_{\rm obs.}=\sqrt{b_{\rm turb.}^2+b({\rm
Ar~I})_{\rm therm.}^2}=9.68\,{\rm km~s}^{-1}$, because the higher atomic
weight of Ar causes the thermal contribution to be smaller. For the observed
equivalent width of the Ar~I line, the error in the ratio of column densities
caused by our assumption that the values of $b_{\rm obs.}$ of the two
elements are identical will create an underestimate of $[{\rm
Ar~I/O~I}]=-0.012\,$dex. A similar calculation for the more extreme line
saturation exhibited by UV0904$-$02 shown in the right-hand panel of
Fig.~\ref{oi_fits} indicates that the perceived outcome for [Ar~I/O~I] could
be too low by $-0.12\,$dex. Cases showing this much saturation are rare for
our collection of sight lines.
If the kinematic line broadening is not an approximately Gaussian form that
one would expect from pure turbulent broadening, but instead results from
distinct and well-separated narrow components, the errors in [Ar~I/O~I] could
be larger than those evaluated above. However, the recordings of Ar~I lines
for 9 different stars made by {\it IMAPS\/} at a resolution of $4\,{\rm
km~s}^{-1}$ that were shown by SJ98 reveal profiles that, while not exactly
Gaussian, are nevertheless generally smooth and devoid of any narrow spikes
that are isolated from each other. Thus, the numerical estimates presented
in the above paragraph should be reasonably accurate.
One might question whether or not errors in the adopted $f$-values could
cause global systematic errors in the evaluations of [Ar~I/O~I]. While
Morton (2003) listed a very small uncertainty for the
$f$-value of the Ar~I line at 1048.220$\,$\AA, he did not specify errors for
the O~I lines. Nevertheless, empirical evidence from high quality {\it
FUSE\/} observations of WD stars by H\'ebrard et al. (2002),
Sonneborn et al. (2002) and Oliveira et al.
(2003) all showed curves of growth that are
remarkably well behaved for the same lines that are used in the present
study. While one could still pose the objection that all of the O~I lines
could collectively have a systematic error of a certain magnitude, and yet
could still yield acceptable curves of growth, this seems unlikely: the
values of [O~I/H~I] derived by Sonneborn et al. (2002) and Oliveira et al.
(2003) are generally consistent with those found elsewhere in the
ISM based on measurements of the
intersystem O~I line at 1355.6$\,$\AA\ (Jenkins 2009).
[H\'ebrard et al. (2002) did not attempt to measure
$N$(H~I).]
\subsubsection{Outcomes}\label{outcomes}
Table~\ref{aro_results} shows the outcomes of our analysis for all of the
targets in the survey, along with the applicable Galactic coordinates and
apparent magnitudes of the stars. The last two columns present some cautions
in numerical form. First, Column (9) lists for the fraction part of
Eq.~\ref{aro_solution} the denominator divided by its error. If this number
is less than about 3, the upper limit for [Ar~I/O~I] should not be trusted.
Second, Column (10) lists the probability of obtaining a worse fit of the O~I
lines to the linear trend (i.e., a higher value for $\chi^2$), given the
errors that we derived. A plot of the frequency of all of these numbers shows
a distribution that is consistent with a uniform distribution between 0 and
1, which indicates that our error estimates for $W_\lambda$ of the O~I lines
are neither too conservative nor too generous. Thus, any individual case
where this probability is low should not be considered as a real anomaly.
\begin{figure}[h!]
\epsscale{1.0}
\plotone{sorted_results.eps}
\caption{Outcomes for [Ar~/O~I] sorted according to their best
values.\label{sorted_results}}
\end{figure}
\begin{figure}[h!]
\vspace{-.3cm}
\plotone{sky_display.eps}
\caption{Locations of the stars and their values of [Ar~I/O~I] shown on an
Aitoff projection of Galactic coordinates. The sizes of the black circles
indicate the values of [Ar~I/O~I] according to the key shown at the
bottom.\label{sky_display}}
\end{figure}
Figure~\ref{sorted_results} shows a graphic representation of all of the
results and their uncertainties. They were ranked and then arranged in order
of the best values of [Ar~I/O~I] to make it easier to see the dispersion of
results and also to show that the more extreme deviations often represent
cases where the errors are somewhat larger than normal.
\onecolumn
{\setlength{\textheight}{9in}
\begin{deluxetable}{
l
c
c
c
c
c
c
c
c
c
}
\tabletypesize{\footnotesize}
\tablecolumns{10}
\tablewidth{0pt}
\tablecaption{Results for [Ar I/O I]\tablenotemark{a}\label{aro_results}}
\vspace{-.5cm}
\tablehead{
\colhead{} & \multicolumn{2}{c}{Galactic Coord.}\tablenotemark{b} &
\colhead{} & \colhead{} & \multicolumn{3}{c}{[Ar I/O I]}\\
\cline{2-3} \cline{6-8}
\colhead{Star} & \colhead{$\ell$} & \colhead{$b$} &
\colhead{} & \colhead{} & \colhead{Lower} & \colhead{Best} & \colhead{Upper}
&
\colhead{Denom. Val.}&\colhead{Prob. of}\\
\colhead{Name} & \colhead{(deg.)} & \colhead{(deg.)} &
\colhead{$m_B$\tablenotemark{b}} &
\colhead{$m_V$\tablenotemark{b}} & \colhead{Limit} & \colhead{Value} &
\colhead{Limit} & \colhead{/Error\tablenotemark{c}} &
\colhead{Worse Fit}\\
\colhead{(1)}& \colhead{(2)}& \colhead{(3)}& \colhead{(4)}&
\colhead{(5)}& \colhead{(6)}& \colhead{(7)}& \colhead{(8)}& \colhead{(9)}&
\colhead{(10)}
}
\startdata
2MASSJ15265306+7941307&$115.06$&$34.93$&$11.30$&$11.60$&$-0.45$&$-0.11$&$0.14
$&$4.74$&$0.26$\\
AA
Dor\dotfill&$280.48$&$-32.18$&$10.84$&$11.14$&$-1.18$&$-0.76$&$-0.51$&$2.60$&
$0.70$\\
AGK+81
266\dotfill&$130.67$&$31.95$&$11.60$&$11.94$&$-0.69$&$-0.47$&$-0.30$&$5.25$&$
0.89$\\
BD+18
2647\dotfill&$289.48$&$80.14$&$11.48$&$11.82$&$-1.14$&$-0.63$&$-0.34$&$3.76$&
$0.66$\\
BD+25
4655\dotfill&$81.67$&$-22.36$&$9.39$&$9.68$&$-1.25$&$-0.44$&$-0.02$&$4.68$&$0
.80$\\
BD+28
4211\dotfill&$81.87$&$-19.29$&$10.17$&$10.51$&$-1.03$&$-0.69$&$-0.44$&$7.42$&
$0.07$\\
BD+37
442\dotfill&$137.07$&$-22.45$&$9.69$&$9.92$&$-0.32$&$-0.20$&$-0.02$&$2.54$&$0
.04$\\
BD+39
3226\dotfill&$65.00$&$28.77$&$9.89$&$10.18$&$-0.49$&$-0.28$&$-0.10$&$3.42$&$0
.69$\\
CPD$-$31
1701\dotfill&$246.46$&$-5.51$&$10.22$&$10.56$&$-1.69$&$-1.04$&$-0.70$&$4.68$&
$0.42$\\
CPD$-$71
172\dotfill&$290.20$&$-42.61$&$10.90$&$10.68$&$-2.60$&$-1.46$&$-0.94$&$2.28$&
$0.77$\\
EC11481-2303\dotfill&$285.29$&$37.44$&$11.49$&$11.76$&$-0.53$&$-0.18$&$0.11$&
$4.81$&$0.43$\\
Feige
34\dotfill&$173.32$&$58.96$&$10.84$&$11.18$&$-1.16$&$-0.42$&$0.02$&$3.43$&$0.
82$\\
HD113001\dotfill&$110.97$&$81.16$&$9.65$&$9.65$&$-0.26$&$0.23$&$0.63$&$4.12$&
$0.42$\\
JL
119\dotfill&$314.61$&$-43.36$&$13.22$&$13.49$&$-0.63$&$-0.53$&$-0.45$&$5.68$&
$0.16$\\
JL
25\dotfill&$318.63$&$-29.17$&$13.09$&$13.28$&$-0.94$&$-0.27$&$0.20$&$3.54$&$0
.29$\\
JL
9\dotfill&$322.60$&$-27.04$&$12.96$&$13.24$&$-0.68$&$-0.43$&$-0.22$&$5.12$&$0
.24$\\
LB
1566\dotfill&$306.36$&$-62.02$&$12.81$&$13.13$&$-0.89$&$-0.65$&$-0.48$&$3.06$
&$0.82$\\
LB
1766\dotfill&$261.65$&$-37.93$&\nodata&$12.34$&$-1.03$&$-0.62$&$-0.34$&$4.71$
&$0.07$\\
LB
3241\dotfill&$273.70$&$-62.48$&$12.45$&$12.73$&$-0.86$&$-0.69$&$-0.55$&$7.98$
&$0.01$\\
LS
1275\dotfill&$268.96$&$2.95$&$10.94$&$11.40$&$-1.01$&$-0.44$&$0.00$&$4.69$&$0
.77$\\
LSE
234\dotfill&$329.44$&$-20.52$&\nodata&$12.63$&$-0.77$&$-0.34$&$0.00$&$3.70$&$
0.38$\\
LSE
259\dotfill&$332.36$&$-7.71$&\nodata&$12.54$&$-0.40$&$-0.26$&$-0.12$&$2.52$&$
0.08$\\
LSE
263\dotfill&$345.24$&$-22.51$&$11.40$&$11.30$&$-0.27$&$0.07$&$0.37$&$6.32$&$0
.24$\\
LSE
44\dotfill&$313.37$&$13.49$&$12.21$&$12.45$&$-0.73$&$-0.52$&$-0.34$&$5.53$&$0
.11$\\
LSII +18
9\dotfill&$55.20$&$-2.65$&$11.81$&$12.13$&$-0.65$&$-0.40$&$-0.21$&$2.43$&$0.6
6$\\
LSII +22
21\dotfill&$61.17$&$-4.95$&$12.23$&$12.58$&$-1.06$&$-0.67$&$-0.41$&$4.04$&$0.
91$\\
LSIV +10
9\dotfill&$56.17$&$-19.01$&$11.71$&$11.98$&$-0.18$&$-0.10$&$-0.03$&$11.01$&$0
.21$\\
LSS
1362\dotfill&$273.67$&$6.19$&$12.27$&$12.50$&$-0.60$&$-0.53$&$-0.48$&$7.05$&$
0.65$\\
MCT
2005$-$5112\dotfill&$347.71$&$-32.68$&$15.20$&\nodata&$-0.63$&$-0.47$&$-0.36$
&$3.82$&$0.46$\\
MCT
2048$-$4504\dotfill&$1.10$&$-39.51$&$14.85$&$15.20$&$-0.52$&$-0.28$&$-0.09$&$
4.97$&$0.97$\\
NGC6905
star\dotfill&$61.49$&$-9.57$&$16.30$&$14.50$&$-0.46$&$-0.26$&$-0.12$&$2.56$&$
0.75$\\
PG0919+272\dotfill&$200.46$&$43.89$&$12.27$&$12.69$&$-2.24$&$-0.89$&$-0.16$&$
3.83$&$0.64$\\
PG0952+519\dotfill&$164.07$&$49.00$&$12.47$&$12.80$&$-0.60$&$-0.51$&$-0.44$&$
5.11$&$0.90$\\
PG1032+406\dotfill&$178.88$&$59.01$&$10.80$&\nodata&$-3.79$&$-1.84$&$-0.41$&$
2.71$&$0.30$\\
PG1051+501\dotfill&$159.61$&$58.12$&$14.59$&\nodata&$-0.01$&$0.28$&$0.59$&$2.
77$&$0.61$\\
PG1230+068\dotfill&$290.42$&$68.85$&$12.25$&\nodata&$-0.57$&$-0.23$&$0.06$&$7
.58$&$0.03$\\
PG1544+488\dotfill&$77.54$&$50.13$&$12.80$&\nodata&$-0.48$&$-0.40$&$-0.34$&$5
.41$&$0.56$\\
PG1605+072\dotfill&$18.99$&$39.33$&$13.01$&$12.84$&$-0.52$&$-0.36$&$-0.24$&$5
.31$&$0.88$\\
PG1610+519\dotfill&$80.50$&$45.31$&$13.73$&\nodata&$-0.53$&$-0.44$&$-0.36$&$1
0.93$&$0.22$\\
PG2158+082\dotfill&$67.58$&$-35.48$&$12.66$&\nodata&$-0.91$&$-0.64$&$-0.49$&$
2.78$&$0.16$\\
PG2317+046\dotfill&$84.84$&$-51.10$&\nodata&$12.87$&$-1.42$&$-0.79$&$-0.37$&$
4.20$&$0.04$\\
Ton
102\dotfill&$127.05$&$65.78$&$13.29$&$13.54$&$-0.89$&$-0.45$&$-0.15$&$2.96$&$
0.31$\\
Ton
S227\dotfill&$201.39$&$-77.81$&$11.60$&$11.90$&$-0.34$&$0.12$&$0.44$&$9.06$&$
0.36$\\
UV0904$-$02\dotfill&$232.98$&$28.12$&$11.64$&$11.96$&$-0.39$&$-0.19$&$-0.03$&
$3.78$&$0.93$\\
\enddata
\tablenotetext{a}{$[{\rm Ar~I/O~I}]\equiv {N({\rm Ar~I})/N({\rm O~I})\over
({\rm Ar/O})_\odot}$. The range of possible values shown here do not include
an overall systematic error that arises from the uncertainties in the
constants that are included in Eq.~\protect\ref{ar_o_eqn}, which amounts to a
combined error of 0.11~dex.}
\tablenotetext{b}{Coordinates and apparent magnitudes were supplied by the
SIMBAD database.}
\tablenotetext{c}{The relevance of this quantity is discussed in the text
that follows Eq.~\protect\ref{aro_solution}. If the listed value is less
than about 3, the positive value of the error quotient may be misleading.}
\end{deluxetable}
}
\twocolumn
Figure~\ref{sky_display} shows the locations of the targets in the sky and
their respective values of [Ar~I/O~I]. While no obvious regional trends seem
to be evident, we can perform a test to determine whether or not the
variability of the outcomes exceeds what we would have expected from our
errors (assuming that they are correct). We do this by computing a value for
$\chi^2$, where the choice for the error of each case depends on whether a
test value is above or below the measurement outcome. This procedure
properly takes into account the asymmetries of the errors. Adopting this
method, we find that a minimum $\chi^2$ of 66.9 for the 44 measurements
occurs at a value $[{\rm Ar~I/O~I}]=-0.438$. This minimum for the $\chi^2$
with 43 degrees of freedom is greater than what we would have expected from
our errors alone (the probability of obtaining by chance a higher value for
$\chi^2$ is only 1\%). If we were to propose that the real variability
across the sky is $\sigma=0.11~{\rm dex}$, and we add this value in
quadrature to all of the experimental errors, the minimum $\chi^2$ drops to
42.4, which makes the probability of a worse fit equal to 50\%. The location
of this new minimum is at $[{\rm Ar~I/O~I}]=-0.427$.
The discussion above has considered only the random fluctuations arising from
either the measurements or the true variability in [Ar~I/O~I] in different
sight lines. We must not lose sight of the fact that there can be an overall
systematic error of 0.11~dex for the entire collection. This global error
arises from the uncertainties of the value for $\log ({\rm Ar/O})_\odot$ that
went into Eq.~\ref{ar_o_eqn}. It is much larger than the error in the
weighted mean value for all of the measurements.
\section{ARE THE STARS BEYOND THE BOUNDARY OF THE LOCAL
BUBBLE?}\label{beyond}
As discussed in Section~\ref{strategy}, our objective is to sample
interstellar material that is beyond the edge of the Local Bubble, enough so
that our measurements are not heavily influenced by the very low density gas
within this cavity. In principle, we could compare the three-dimensional locations of the
target stars with maps that outline the boundary of the Local Bubble
(Vergely et al. 2010; Welsh et al. 2010; Reis et al. 2011) to indicate
whether or not we are primarily sampling gas in the surrounding denser
medium. However, the distances to our targets are uncertain, which makes
this approach unworkable. Instead, we adopt a definition of the boundary
proposed by Sfeir et al. (1999) (and one that was also used by Lehner et al.
(2003)), who linked its location to the sudden onset of Na~I D-line
absorption that crossed a threshold $W_\lambda({\rm D2})=20\,$m\AA. This
threshold is equivalent to $N({\rm H~I})\approx 2\times 10^{19}{\rm cm}^{-2}$
(Ferlet et al. 1985), which in turn corresponds to $N({\rm O~I})$ that is
slightly greater than $10^{16}{\rm cm}^{-2}$. Lehner et al. (2003) found
that a typical velocity dispersion parameter $b({\rm O~I})=6\,{\rm km~s}^{-1}$
occurred inside the Local Bubble. Using these two parameters for O~I, $(N,
~b)=(10^{16}{\rm cm}^{-2},~6\,{\rm km~s}^{-1})$, we can compute the equivalent
widths for all of the O~I lines when the boundary is crossed. The third row
of numbers in the column headings in Table~\ref{EW_table} shows the values of
$W_\lambda$ (in m\AA) for all but the strongest three lines. By comparing
these values with the entries that show our measurements, particularly the
weaker but securely measured ones, we can ascertain that our targets are
beyond the edge of the Local Bubble.
\section{INTERPRETATION: FUNDAMENTAL PROCESSES AND
EQUATIONS}\label{fundamentals}
In this section, we address the basic physical processes that relate our
findings on [Ar~I/O~I] to the ionization balance of the gas and the resulting
degree of partial ionization. Our discussion about the means of ionizing the
atoms will focus mainly on the primary ionization by photons, along with the
effects of collisional ionizations caused by secondary electrons that
originate from these primary ionizations. These two processes are the most
important sources of ionization, and they represent one side of the balance
between recombinations with free electrons and charge exchange reactions
between various constituents of the medium. For completeness, we will also
cover other means of ionizing atoms and creating free electrons, such as
cosmic ray ionizations, the ionizations of inner shell electrons of heavy
elements by X-rays, the creation of ionizing photons when helium ions
recombine, and the nearly complete ionization of many elements that have
first ionization potentials below that of hydrogen.
\subsection{Direct and Secondary Ionizations of H and Ar}\label{ionization}
\subsubsection{Photoionization}\label{photoionization}
The primary photoionization cross sections for neutral Ar are larger than
those for H by about one order of magnitude at low energies, and the ratio
increases substantially at higher energies. Various secondary ionizing
processes initiated by the primary photoionizations of H and He likewise have
a stronger effect on Ar than on H. This contrast in ionization rates is the
fundamental tool that we use in the interpretation of the Ar data to quantify
the photoionization of H and the subsequent creation of free electrons. For
the collective effect of all of these ionization channels, we can construct a
simple formalism based on arguments created by SJ98. They defined a quantity
based on ionization rates $\Gamma$ and recombination coefficients $\alpha$
for the two elements,
\begin{equation}\label{p_ar}
P_{\rm Ar}= {\Gamma({\rm Ar}^0)\alpha({\rm H}^0,T)\over \Gamma({\rm
H}^0)\alpha({\rm Ar}^0,T)}~.
\end{equation}
SJ98 considered only primary photoionizations of these two elements. Going
beyond their development, we construct a more comprehensive picture by
considering some refinements in the calculations of $\Gamma$ for both ${\rm
H}^0$ and ${\rm Ar}^0$.
First, we start with the primary photoionization rates $\Gamma_p=\int
F(E)\sigma(E)dE$, where $F(E)$ is the ambient photon flux as a function of
energy $E$ and $\sigma(E)$ is the photoionization cross section for either
H$^0$
(Spitzer 1978, pp. 105-106) or Ar$^0$ (Marr \& West 1976).
Next, we include secondary ionizations with rates $\Gamma_s$ that are created
by the collisions from energetic electrons that are liberated by the primary
photoionizations of H and He. Added to this are the effects from photons
with energies above about 300$\,$eV, which can interact with the abundant
heavy elements in the ISM to produce additional energetic electrons that will
ionize H and Ar with a rate that we identify as $\Gamma_{s^\prime}$. These
electrons arise from the primary ionizations of the inner electronic shells,
and they are supplemented by one or more additional electrons from the Auger
process. Finally, we must acknowledge that recombinations of singly- and
doubly-ionized He ions with electrons create additional photons when
de-excitation occurs in the lower stage of ionization, either He$^0$ or
He$^+$. The importance of not overlooking secondary electrons and
recombination photons from the He ionizations is underscored by the fact that
while the abundance of He is only 1/10 that of hydrogen, its primary
ionization cross section is 6 to 100 times that of H over the energy range 25
to 4000$\,$eV. Most of the recombination photons are capable of ionizing
both H and Ar, and they supplement the other sources of ionization with rates
$\Gamma_{He^0}$ and $\Gamma_{He^+}$. In short, we consider that the total
ionization rates for the two elements in Eq.~\ref{p_ar} each consist of five
contributions,
\begin{equation}\label{Gammas}
\Gamma=\Gamma_p+\Gamma_s+\Gamma_{s^\prime}+\Gamma_{He^+}+\Gamma_{He^0}
\end{equation}
The details of how we compute $\Gamma_s$, $\Gamma_{s^\prime}$,
$\Gamma_{He^+}$ and $\Gamma_{He^0}$ are discussed in Appendices~\ref{gamma_s}
and \ref{gamma_he}.
The quantity $P_{\rm Ar}$ defined in Eq.~\ref{p_ar} provides a means for
evaluating how large the neutral fraction of Ar should be relative to that of
H according to the formula
\begin{equation}\label{ar_h}
[{\rm Ar~I/H~I}]=\log\left[ {1+n({\rm H}^+)/n({\rm H}^0)\over 1+P_{\rm Ar}
n({\rm H}^+)/n({\rm H}^0)}\right] .
\end{equation}
For a more accurate formulation of [Ar~I/H~I] that will be developed later in
Section~\ref{equilibrium}, we will introduce a more refined parameter
$P^\prime_{\rm Ar}$ that will be based on a calculation that is more
elaborate than the one shown in Eq.~\ref{p_ar}. This new parameter will be
substituted for $P_{\rm Ar}$ in Eq.~\ref{ar_h}. Under most conditions, the
differences between $P^\prime_{\rm Ar}$ and $P_{\rm Ar}$ are small.
Figure~\ref{p_ar_plot} shows for both H and Ar the monoenergetic cross
sections for primary ionization (dashed lines), the effective enhancements
arising from the secondary ionizations $\Gamma_s$ (dash-dot lines) for
$x_e=0.05$, and the additional effects from $\Gamma_{s^\prime}$,
$\Gamma_{He^+}$ and $\Gamma_{He^0}$, all of which give total ionization rates
shown by the solid lines. For H, secondary ionizations outweigh the primary
ones for photon energies above about 100~eV. By contrast, we find that for
Ar the enhancement at these energies is small compared to its much higher
primary cross section. A more narrowly defined version of $P_{\rm Ar}$,
which we denote as $P_{\rm Ar}(E)$, applies to an irradiation of the gas by
photons of a given energy $E$, rather than from a combination of fluxes over
a broad energy range. The definition of $P_{\rm Ar}(E)$ is illustrated in
the upper panel of the figure, and its dependence on $E$ is shown in the
bottom panel.
\onecolumn
\begin{figure}
\epsscale{.8}
\plotone{p_ar_plot.eps}
{\renewcommand{\baselinestretch}{.95}
\caption{{\it Upper panel:\/} Various combinations of effective
photoionization cross sections for neutral hydrogen and argon as a function
of energy $E$. The dashed lines depict the absorption cross sections for the
primary ionization rates $\Gamma_p$ that would apply to monoenergetic
radiation at the energy $E$ shown on the $x$-axis. The effective cross
sections that arise by including the additional effects from ionizations by
secondary electrons produced by H and He (related to $\Gamma_s$) are shown by
the dot-dash lines. These secondary ionization efficiencies were calculated
according to the principles outlined in Section~\protect\ref{electrons_H_He}
of the Appendix for an electron fraction $x_e=0.05$.; see
Eqs.~\protect\ref{Hphi} and \protect\ref{Hephi}. These two ionization
processes can be further enhanced by photons that come from recombinations of
He ions with free electrons (Appendix~\protect\ref{gamma_he}) and electrons
from inner shell ionizations of heavy elements
(Appendix~\protect\ref{heavy_elem}), leading to total effective cross
sections depicted by the solid lines. Across the top of the plot are markers
showing the energies at which half-intensity penetration depths for the
photons occur for various hydrogen column densities, assuming the gases have
a solar abundance ratio for various elements. The quantity $P_{\rm Ar}(E)$
defined in Eq.~\ref{p_ar} for a specific energy $E$ is approximately equal to
the ratio of the total effective photoionization cross sections that give
rise to $\Gamma({\rm H}^0,~{\rm Ar}^0)$ defined in Eq.~\protect\ref{Gammas},
since the recombination rates for the two elements are about the same for all
temperatures. {\it Bottom panel:\/} A plot of $P_{\rm Ar}(E)$ as a function
of energy.\label{p_ar_plot}}}
\end{figure}
\twocolumn
\subsubsection{Other Ionization Mechanisms}\label{other_ionization}
As discussed earlier in Section~\ref{intro}, neutral atoms in the ISM are
subjected to ionization by cosmic ray particles. These ionizations, with a
rate $\zeta_{\rm CR}$, add to the effects of primary and secondary
ionizations from photons discussed above. To obtain proper values of
$\zeta_{\rm CR}$ that apply to the diffuse medium, we must apply corrections
to the values of $\zeta_{\rm CR}$ that were measured for the more dense media
that have appreciable concentrations of molecules. The details of this
computation are discussed in Appendix~\ref{cr_ioniz}.
In the WNM, the main source of free electrons is from the
ionization of H and He. However, a small number of additional electrons
arise from other atoms that can be ionized by starlight photons less
energetic than the ionization potential of H. For the combined effect from
these elements, we adopt an estimate $n({\rm M}^+)$ equal to $1.5\times
10^{-4}$ times the density of hydrogen based on the assumption that the gas
we are viewing is in a regime where the element depletions are relatively
modest.
\subsection{Recombination}\label{recombination}
The radiative recombination coefficients for free electrons and ions to
create neutral hydrogen and singly-ionized helium, $\alpha({\rm H}^0,T)$ and
$\alpha({\rm He}^+,T)$, are taken from
Spitzer (1978, pp. 105-107). For $\alpha({\rm H}^0)$, we
excluded recombinations to the lowest electronic level, since they generate
Lyman limit photons that can re-ionize hydrogen atoms over a short distance
scale. Recombination coefficients for He$^0$ were taken from Aldrovandi \&
P\'equignot (1973) and those for Ar$^0$ from Shull \& Van Steenberg
(1982). The results for Ar$^0$ given by
Shull \& Van Steenberg (1982) agree well
with the radiative recombination coefficients listed by Aldrovandi \&
P\'equignot (1974). The minimum
temperature where dielectronic recombination for Ar$^0$ becomes important is
$2.5\times 10^4\,$K
(Aldrovandi \& P\'equignot 1974), which is above temperatures that we
will consider, hence we can ignore this process.
In addition to recombining with a free electron, an ion can also be
neutralized by colliding with a dust grain and removing an electron from it
(Snow 1975; Draine \& Sutin 1987; Lepp et al. 1988; Weingartner \& Draine
2001a). The operation of this effect on protons is important for regulating
the fraction of free electrons in the cold neutral medium (CNM), but it is of
lesser significance for the WNM [cf. Figs. 16.1 and 16.2 of
Draine (2011)], which probably dominates the sight lines in our study. The
rate constant $\alpha_g$ for this process is normalized to the local
hydrogen density $n({\rm H_{tot}})\equiv n({\rm H}^0)+n({\rm H}^+)$, and it
depends on several physical parameters that influence the charge on the
grains, such as the electron density $n(e)$, the rate of photoelectric
emission that is driven by the intensity $G$ of the local radiation field
between 6 and 13.6~eV, and the temperature $T$. For our equilibrium
equations in Section~\ref{equilibrium}, we have adopted parametric fits for
$\alpha_g({\rm H}^0,n(e),G,T)$ and $\alpha_g({\rm He}^0,n(e),G,T)$ from
Weingartner \& Draine (2001a). They do not supply fit coefficients
for $\alpha_g({\rm Ar}^0,n(e),G,T)$, but since the ionization potential of
Ar$^0$ is close to that of H$^0$, it is reasonable to adopt the hydrogen rate
coefficient and divide it by the square-root of the atomic weight (40) of Ar.
Throughout our analysis, we set $G=1.13$, which is the value recommended for
the general ISM by Weingartner \& Draine (2001a).
\subsection{Charge Exchange}\label{charge_exchange}
In addition to the recombination processes mentioned in the previous section,
charge exchange reactions with neutral hydrogen can also lower the ionization
state of an atom. With reference to such reactions for element X, ${\rm
X}^++{\rm H}^0\rightarrow {\rm X}^0+{\rm H}^+$ and ${\rm X}^{++}+{\rm
H}^0\rightarrow {\rm X}^++{\rm H}^+$, we adopt the notation $C^\prime({\rm
X}^+,T)$ and $C^\prime({\rm X}^{++},T)$ for the respective rate constants.
For our calculations of $C^\prime({\rm He}^+,T)$, $C^\prime({\rm
He}^{++},T)$, $C^\prime({\rm Ar}^+,T)$, and $C^\prime({\rm Ar}^{++},T)$, we
adopted Kingdon \& Ferland's (1996) fits to the
calculations from various sources (see their Table~1 for the coefficients and
references).
Since the ionization potential of neutral oxygen is close to that of
hydrogen, the rate constant for charge exchange of these two species is large
and reverse endothermic reaction is not negligible, except at very low
temperatures. The rate constant $C({\rm O}^+,T)$ for the reaction ${\rm
O}^0+{\rm H}^+\rightarrow {\rm O}^++{\rm H}^0$ can be obtained from $C^\prime
({\rm O}^+,T)$ by the principle of detailed balancing. However, in doing so,
one must treat the three fine-structure levels of the ground state of O$^0$
separately, since their energy separations are comparable to the differences
in the ionization potentials of H and O. The large rate constants in both
directions (Stancil et al. 1999) assures that the ionization fraction
of O is locked very close to a value 8/9 times that of H for $T\gtrsim
10^3\,$K. This is a key principle that allows us to use O as a substitute
for H in the present study or Ar vs. H fractional ionizations.
For completeness, we should also consider charge exchange reactions between
Ar and He, even though the abundance of He is much lower than that of H. The
charge transfer recombination reaction ${\rm Ar}^{++}+{\rm He}^0\rightarrow
{\rm Ar}^++{\rm He}^+$ has a rate constant $D^\prime({\rm
Ar}^{++},T)=1.3\times 10^{-10}{\rm cm}^3{\rm s}^{-1}$ according to Butler \&
Dalgarno (1980), which they claim to be
constant for $T\geq 10^3\,$K. The charge transfer ionization reaction with
He$^+$, i.e., ${\rm Ar}^0+{\rm He}^+ \rightarrow {\rm Ar}^++{\rm He}^0$, has
a rate constant $D({\rm Ar}^+,T)<10^{-13}{\rm cm}^3{\rm s}^{-1}$ according to
Albritton (1978), so we will ignore this process in the
equation for the ionization balance for Ar (Eq.~\ref{equi1} below).
\subsection{Equilibrium Equations}\label{equilibrium}
In a medium where both hydrogen and helium are partly ionized, the densities
of an element X in its 3 lowest levels of ionization X$^0$, X$^+$ and
X$^{++}$ are governed by the equilibrium equations\footnote{The
development here follows that given by Eqs. 12--19 of SJ98, except that we
have added the He charge exchange recombination reaction as an additional
channel for reducing the ionization of element X in its doubly charged form.
We have also added cosmic ray ionizations and have implicitly included the
various kinds of secondary photoionizations in the definition of $\Gamma$, as
indicated in Eq.~\protect\ref{Gammas}. We have corrected
Eqs.~\protect\ref{equi1} and \protect\ref{n0} here to include a missing
$n({\rm H^+})$ term, which was a typographical oversight in the equations of
SJ98.}
\begin{eqnarray}\label{equi1}
&&\Big[ \Gamma({\rm X}^0)+\zeta_{\rm CR}({\rm X}^0)+C({\rm X}^+,T)n({\rm
H^+})\Big] n({\rm X}^0)\nonumber\\
&=&\Big [\alpha({\rm X}^0,T)n(e)+C^\prime({\rm X}^+,T)
n({\rm H}^0)\nonumber\\
&+&\alpha_g({\rm X}^0,n(e),G,T)n({\rm H_{tot}})\Big] n({\rm X}^+)
\end{eqnarray}
and
\begin{eqnarray}\label{equi2}
&&\Gamma({\rm X}^+)n({\rm X}^+)=\Big[ \alpha({\rm X}^+,T)n(e)+C^\prime({\rm
X}^{++},T)n({\rm H}^0)\nonumber\\
&+&D^\prime({\rm X}^{++},T)n({\rm He}^0)\Big] n({\rm
X}^{++})
\end{eqnarray}
where $\Gamma({\rm X}^y)$ is the photoionization rate of element X in
its ionization state $y$ (neutral, +, or ++) and $\alpha({\rm X}^y,T)$ is the
recombination rate of the $y+1$ state with free electrons as a function of
temperature $T$. The simultaneous solution to these two equations yields the
fractional abundances in the three ionization levels
\begin{eqnarray*}\label{n0}
f_0({\rm X},T) \equiv {n({\rm X}^0)\over n({\rm X}^0)+n({\rm
X}^+)+n({\rm X}^{++})}=\Bigg( 1\, +
\end{eqnarray*}
\begin{mathletters}
\begin{equation}
{\Big[\Gamma({\rm X}^0)+\zeta_{\rm CR}({\rm X}^0)+C({\rm
X}^+,T)n({\rm H^+})\Big]
\Big[\Gamma({\rm X}^+)+Y\Big]\over \Big[\alpha({\rm X}^0,T)n(e) +
C^\prime({\rm X}^+,T)n({\rm H}^0)+\alpha_g({\rm X}^0,n(e),G,T)n({\rm
H_{tot}})\Big]Y}\Bigg)^{-1}
\end{equation}
with
\begin{equation}
Y=\alpha({\rm X}^+,T)n(e)+C^\prime({\rm X}^{++},T)n({\rm
H}^0)+D^\prime({\rm X}^{++},T)n({\rm He}^0)~,
\end{equation}
\end{mathletters}
\begin{equation}\label{n++}
f_{++}({\rm X},T) \equiv {n({\rm X}^{++})\over n({\rm X}^0)+n({\rm
X}^+)+n({\rm X}^{++})}={1-f_0({\rm X},T)\over 1+Y/\Gamma({\rm X}^+)}~,
\end{equation}
and
\begin{equation}\label{n+}
f_+({\rm X},T)\equiv {n({\rm X}^+)\over n({\rm X}^0)+n({\rm X}^+)+n({\rm
X}^{++})}=1-f_0({\rm X},T)-f_{++}({\rm X},T)~.
\end{equation}
\subsection{Electron Density and the Ionization Fractions of H and
He}\label{electron_density}
Before the ionization fractions of Ar can be derived, we must determine not
only the ionization balance of hydrogen, but also that for helium. We do
this by solving
Eqs.~\ref{n0}--\ref{n+} (substituting He for X and eliminating the $D^\prime$
term) along with the equation for the hydrogen ionization balance,
\begin{mathletters}
\begin{equation}\label{H_ioniz_balance}
{n({\rm H}^+)\over n({\rm H}^0)}={\Gamma({\rm H}^0)+\zeta_{\rm CR}({\rm
H}^0)+Z\over \alpha({\rm
H}^0,T)n(e)+\alpha_g({\rm X}^0,n(e),G,T)n({\rm H_{tot}})-Z}~,
\end{equation}
with
\begin{equation}\label{Z}
Z=0.1n({\rm H}^0)\big[C^\prime({\rm He}^+,T)f_+({\rm He},T)+C^\prime({\rm
He}^{++},T)f_{++}({\rm He},T)\big]~,
\end{equation}
\end{mathletters}
and the constraints
\begin{equation}\label{He_total}
n({\rm He}^0)+n({\rm He}^+)+n({\rm He}^{++})=0.1n({\rm H}^0)\left[1+{n({\rm
H}^+)\over n({\rm H}^0)}\right]
\end{equation}
and
\begin{equation}\label{n(e)}
n(e)=n({\rm H}^+)+n({\rm He}^+)+2n({\rm He}^{++})+n({\rm M}^+)~.
\end{equation}
Since the hydrogen ionization balance depends on $n(e)$ which in turn is
influenced by the ionization fractions of He (which are also influenced by
$n(e)$), we must solve
Eqs.~\ref{H_ioniz_balance}-\ref{n(e)} iteratively to obtain a solution. We
found that these equations converged very well if we kept $n({\rm H_{tot}})$
and the ionization rate of H pegged to a certain value.\footnote{Fixing
$n(e)$ instead of $n({\rm H_{tot}})$ produces unstable, oscillating solutions
under certain circumstances.} Starting with a zero helium ionization rate,
the iterations progressed slowly to successively higher rates until the
final, correct value was reached and the ionization fractions had stabilized.
After obtaining the final results for the coupled hydrogen and helium
ionization balances, we can solve for $f_0({\rm Ar})$ using
Eq.~\ref{n0}. This result can then be used to derive a more accurate value
for $P_{\rm Ar}$,
\begin{equation}\label{Pprime_ar}
P^\prime_{\rm Ar}={n({\rm H}^0)\over n({\rm H}^+)}\left( {1\over f_0({\rm
Ar})}-1\right)~,
\end{equation}
which can be substituted for $P_{\rm Ar}$ in Eq.~\ref{ar_h} to obtain a
solution for [Ar~I/H~I] that makes use of all of the physical processes that
were discussed in Sections~\ref{ionization}--\ref{charge_exchange}.
\section{OUTCOME FROM KNOWN SOURCES OF
PHOTOIONIZATION}\label{known_photoionization}
\subsection{External Radiation}\label{external_radiation}
Over many decades, the diffuse, soft X-ray background has been measured by a
large number of different experiments [for a review, see McCammon \& Sanders
(1990)]. Most of the emission below 1$\,$keV
arises from hot ($T>10^6\,$K) gas in the Galactic disk and halo, with
radiation from extragalactic sources dominating at higher energies
(Chen et al. 1997; Miyaji et al. 1998; Moretti et al. 2009). Much of the
literature on the diffuse radiation shows a distinction between contributions
from a local component with little foreground absorption and more distant
emissions with varying levels of absorption. The local background was once
identified as having originated from hot gas in the Local Bubble (Sanders et
al. 1977; Hayakawa et al. 1978; Fried et al. 1980), but in recent years it
has been recognized to be strongly contaminated, or completely dominated, by
X-rays arising from charge exchange produced by the interaction of the solar
wind with incoming interstellar atoms (Cravens 2000; Lallement 2004; Pepino
et al. 2004; Koutroumpa et al. 2006, 2007, 2009; Peek et al. 2011; Crowder et
al. 2012). For this reason, we ignore the weakly absorbed, nearly isotropic
portion of the X-ray background and focus our attention to the component that
exhibits a pattern in the sky that clearly shows absorption by gas in the
Galaxy.
Kuntz \& Snowden (2000) have performed a detailed
investigation of the nonlocal component, which they call the transabsorption
emission (TAE). They describe the strength and spectral character of the TAE
in terms of emissions from optically thin plasmas at two different
temperatures. They define a soft component that has a mean intensity over
the sky $I=2.6\times 10^{-8}{\rm erg~cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}$ over
the interval $0.1 < E < 2\,$keV and a spectrum consistent with the emission
from a plasma at a temperature $T=10^{6.06}\,$K, and this flux is accompanied
by a hard component with $I=8.5\times 10^{-9}{\rm erg~cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}$ over the same energy interval with $T=10^{6.46}\,$K.
To translate the sum of these two components into a distribution of the
photon flux as a function of energy, $F(E)$, we calculate synthetic flux
representations using the CHIANTI database and software (Version~6.0)
(Dere et al. 1997, 2009), after normalizing the
emission measures to give the intensities stated above (we find that ${\rm
EM}=10^{16.37}$ and $10^{15.81}{\rm cm}^{-5}$ for the soft and hard
components, respectively). We supplement the TAE result with an underlying
power-law extragalactic emission of the form $10.5\,{\rm phot~cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}{\rm keV}^{-1}E({\rm keV})^{-1.46}$ (Chen et al. 1997).
The ISM is opaque to X-rays at the lowest energies. The energies at which
half of the X-rays are absorbed for various column densities are shown in the
top portion of Figure~\ref{p_ar_plot}, which were derived from the
calculations of Wilms et al. (2000). At energies of
around 100$\,$eV where the ISM is neither completely opaque or transparent
for $N({\rm H}^0)\approx {\rm few}\,\times 10^{19}{\rm cm}^{-2}$,
uncertainties in the layout of emitting and absorbing regions make it
difficult to calculate with much precision how far the X-rays can penetrate
the typical gas volumes that were sampled in our survey of Ar~I and O~I.
Thus, rather than implement an elaborate attenuation function that would be
difficult to explain (and perhaps not especially correct at our level of
understanding), we apply a simplification that all of the X-rays are
transmitted above some threshold energy and none below it. The threshold
that we adopted was 90$\,$eV, on the assumption that in some directions the
gas can view the unattenuated X-ray sky through $N({\rm H}^0)$ slightly less
than $10^{19}{\rm cm}^{-2}$.
The upper panel of Figure~\ref{flux_plot} shows our synthesis of the sum of
the extragalactic power-law emission and TAE synthesis described above. For
the purpose of calculating $\Gamma$ for various elements, we make use of only
the flux depicted by the dark trace in the figure, i.e., that which starts at
the cutoff energy (90$\,$eV) and ends at an energy beyond which no
appreciable additional ionization occurs.
\subsection{Internal Radiation Sources}\label{internal_radiation}
Embedded within the ISM are sources of EUV and X-ray radiation that can make
additional contributions to $\Gamma$. We can make estimates for their
average space densities and the character of their emissions, but one
uncertainty that remains is how well the ensuing photoionizations are
dispersed throughout the ambient gas. At one extreme representing minimum
dispersal, we envision the classical Str\"omgren Spheres that surround
sources that are not moving rapidly and that emit most of their photons with
barely enough energy to ionize hydrogen. These photons have a short mean
free path in a neutral medium. Under these circumstances, the zone of
influence of the source is sharply bounded, and the resulting ionization is
nearly total inside the region and zero outside it. At the other extreme,
one can imagine that the photons, ones that have relatively high energy, can
travel over a significant fraction of the inter-source distances before they
are absorbed. In addition, the sources themselves could move rapidly enough
that they never have a chance to establish a stable condition of ionization
equilibrium. (This issue will be investigated quantitatively in
Section~\ref{recomb_time}.) These conditions could lead to the ionization
being more evenly distributed and not necessarily complete. If the sources
have large enough velocities, we can even imagine a picture where there is a
random network of ``fossil Str\"omgren trails'' (Dupree \& Raymond 1983) that
ultimately might blend together. More extreme manifestations of such trails
in denser media may have already been discovered by McCullough \& Benjamin
(2001) and Yagi et al. (2012), who observed faint, but straight and narrow
lines of H$\alpha$ emission in the sky (but were unable to identify the
sources that created them).
It is difficult to establish where in the continuum between the two extremes
for the dispersal of ionization the true effects of embedded EUV and X-ray
sources are to be found. For our treatment of the influence of these
production sites for ionizing radiation, we will adopt the simplified premise
that all of their photons are available to create a uniform but weak level of
ionization everywhere. This picture is not entirely correct, since one
should expect that very near the sources some fraction of the ionizing
photons are ``wasted'' by creating localized regions with much higher than
usual levels of ionization. Such regions dissipate most of their ionization
rapidly because their recombination times are short. For this reason, our
making use of a calculated average production rate of ionizing photons per
unit volume will lead to an equilibrium equation that overestimates the
space- and time-averaged level of ionization. (Later, it will be shown that
this overestimate is of no real consequence because we obtain an answer that
is still below that needed to explain the overall average ionization level.)
In the sections that follow, we consider three classes of sources that are
randomly distributed throughout the neutral ISM and can in principle help to
ionize it: main-sequence stars, active X-ray binaries, and WD stars.
Luminous, early-type stars contribute large amounts of ionizing radiation,
but most of their radiation completely ionizes the surrounding media and
makes the gas virtually invisible in the Ar~I and O~I lines. These stars
also tend to be clustered inside the dense clouds of gas that led to their
formation.
\onecolumn
\begin{figure}
\epsscale{0.7}
\plotone{flux_plot.eps}
\caption{{\it Upper panel:\/} External fluxes arising from the known X-ray
background radiation (solid line) and a hypothetical, time-averaged flux
(over a recombination time of $\approx 1\,$Myr) from 3 supernova remnants at
a distance of 100$\,$pc (dashed line). The gray portion of the solid curve
is in an energy range where the opacity to X-rays is high, and thus this
radiation is not likely to penetrate into much of the gas that we observe.
{\it Lower panel:\/} Internal average rates per unit volume for the injection
of ionizing photons by the coronae of main-sequence stars (MS), active
binaries (AB) and the photospheres of WD stars
(WD). (A machine readable table of all 5 flux distributions vs. $E$
shown in this figure is available in the online version of the
{\it Astrophysical Journal.})\label{flux_plot}}
\end{figure}
\twocolumn
The question may arise as to whether or not, by considering the embedded
sources as a separate contribution that adds to the external radiation
background discussed earlier, we are possibly ``double counting'' some of the
photons that could ionize the ISM. Kuntz \& Snowden (2001) have computed the
probable relative contribution of Galactic point sources that were not
explicitly taken out of their measurements of the diffuse X-ray background,
and they concluded that this contamination was, at most, only about $2-10\%$
of the radiation that was thought not to arise from the Local Bubble. Within
their highest energy bands ($0.73-2.01\,$keV), they state that the
contamination could be as high as 51\%.
\subsubsection{Main-sequence Stars}\label{ms}
For various kinds of point sources, the ratio of the emission of X-rays to
photons in the $V$ band is usually defined by the relation
\begin{equation}\label{f_X/f_V}
\log(f_X/f_V)=\log f_X+0.4m_V+5.37~,
\end{equation}
where $f_X$ is the apparent X-ray flux of the source over a specified energy
interval, expressed in ${\rm erg~cm}^{-2}{\rm s}^{-1}$, and $m_V$ is its
apparent visual magnitude.\footnote{Numerical
values for this formula are not universal, since the adopted X-ray energy
interval depends on which instrument was used. For instance, surveys that
used the {\it Einstein Observatory\/}, e.g. those reported by Maccacaro et
al. (1988) or Stocke et al. (1991), used the passband $0.3<E<3.5\,$keV for
$f_X$, whereas those based on the {\it ROSAT\/} All-Sky Survey (RASS)
(Krautter et al. 1999; Ag\"ueros et al. 2009) measured $f_X$ over
the interval $0.1<E<2.4\,$keV.} The total output of X-rays from a source with an
absolute magnitude $M_V=0$ should be $5.11\times 10^{34}(f_X/f_V)\,{\rm
erg~s}^{-1}$. Within each spectral class, there is a large dispersion in the
measured values of $\log(f_X/f_V)$, typically of order 1$\,$dex, which is
probably attributable to differences in stellar rotation velocities
(Audard et al. 2000; Feigelson et al. 2004), variations in
foreground absorption by the ISM, and time variability of the X-ray emission.
While the most significant X-ray flares from stars can create spectacular
increases in flux, their time-averaged effect has been estimated by Audard et
al. (2000) to amount to only about 10\% of the steady
emission. On the basis of white-light monitoring of dwarf stars, Walkowicz
et al. (2011) found that the duty cycle of
flaring events is only of order a few percent.
For any given stellar spectral class with a characteristic absolute magnitude
$M_V$ that spans a range $\Delta M_V$ along the main sequence and has
luminosity function $\phi(M_V)\,{\rm stars~mag}^{-1}{\rm pc}^{-3}$, the
energy density of X-rays per unit volume is given by
\begin{eqnarray}\label{F(X)}
&&F(E)_{\rm MS}=1.74\times 10^{-21}\phi(M_V)\Delta M_V\nonumber\\
&\times& 10^{\log (f_X/f_V)-0.4M_V}{\rm erg~cm}^{-3}{\rm s}^{-1}~.
\end{eqnarray}
If we use the mean values of $\log (f_X/f_V)$ for different spectral classes
listed by Ag\"ueros et al. (2009) for the X-ray band $0.1<E<2.4\,$keV,
obtain values of $M_V$ for these classes from Schmidt-Kaler
(1982), define a luminosity function for stars in the
disk of our Galaxy from the formula given by Bahcall \& Soneira
(1980), and then sum the results over all
spectral types A-M, we obtain an average energy density equal to $1.90\times
10^{-28}{\rm erg~cm}^{-3}{\rm s}^{-1}$. While we acknowledge that the
spectral character of coronal emissions can vary for different stars along
the main sequence, in the interest of simplicity we adopted for all cases a
spectrum based on the differential emission measure (DEM) for the coronal
emission from the quiet Sun, as defined in the CHIANTI database. To
obtain a final photon emission rate per unit volume and energy in the ISM, we
made use of this spectrum and normalized its energy output over the
$0.1-2.4\,$keV band to the energy density factor given above. The result is
shown by the spectrum labeled ``MS'' in the lower panel of
Figure~\ref{flux_plot}.
\subsubsection{Active Binaries and Cataclysmic Variables that Emit
X-rays}\label{ab}
At high Galactic latitudes, most of the X-ray radiation above several keV
originates from extragalactic sources. Near the direction toward the
Galactic center, however, Revnivtsev et al. (2009)
found that at 4$\,$keV about half of the X-ray emission arises from point
sources in the Galaxy that can be resolved by the {\it Chandra X-ray
Observatory\/}, with the remainder coming from either a diffuse Galactic (hot
gas) emission or an extragalactic contribution. To estimate the average
emission per unit volume from these sources, we integrate the $2-10\,$keV
X-ray emission over the entire luminosity function $\phi(\log L_{2-10\,{\rm
keV}})_{\rm AB}$ specified by Sazonov et al. (2006)
to obtain the total energy output
\begin{eqnarray}\label{L_AB}
&&L(2-10\,{\rm keV})_{\rm AB}\nonumber\\
&=&\int_{22.5}^{34.0} L_{2-10\,{\rm keV}}\phi(\log
L_{2-10\,{\rm keV}})_{\rm AB}\nonumber\\
&\times& d\,(\log L_{2-10\,{\rm keV}})= 2.13\times
10^{38}{\rm erg~s}^{-1}
\end{eqnarray}
for all of the sources in our Galaxy.\footnote{Note that the upper bound for
the integration is at $10^{34}{\rm erg~s}^{-1}$. While the energy output
over the whole Galaxy for neutron star and black hole X-ray binaries with
individual outputs $L_{2-10\,{\rm keV}}>10^{34}{\rm erg~s}^{-1}$ is
substantial, these objects are so few in number that they can no longer be
considered as embedded sources. The brightest such object in the sky,
Sco~X$-$1, is at a distance of 2.8$\,$kpc and creates a flux at our location of
only $2.8\times 10^{-7}{\rm erg~cm}^{-2}{\rm s}^{-1}$ over the $2-10\,$keV
band (Grimm et al. 2002), which is substantially lower than the
extragalactic background flux integrated over the whole sky.} We may convert
this value to an average volume emissivity by multiplying it by the stellar
mass density at our location ($0.04\,M_\odot~{\rm pc}^{-3}$) divided by
the total stellar mass of the Galaxy ($7\times 10^{10}M_\odot$), where
both of these numbers were those adopted by Sazonov et al. (2006), and
ultimately obtain a value $4.15\times 10^{-30}{\rm erg~cm}^{-3}{\rm s}^{-1}$.
To define the spectral shape for the radiation emitted by these sources, we
used the CHIANTI software to compute the emission from a plasma with
an average of the DEM functions expressed by Sanz-Forcada et al. (2002,
2003)\footnote{These authors describe their DEM functions in terms of
$N(e)N({\rm H}^+)dV/d(\log T$), whereas the convention in CHIANTI
assumes that the DEM is defined in terms of $N(e)N({\rm H}^+)dV/dT$.}
that were constructed from their {\it Extreme-Ultraviolet Explorer\/}
observations of various active
binary sources. This spectrum was then normalized such that the flux in the
$2-10\,$keV band matched the volume emissivity described above. The emission
from active binaries that we derived $F(E)_{\rm AB}$ is shown by the curve
labeled ``AB'' in the lower panel of Figure~\ref{flux_plot}.
\subsubsection{White Dwarf Stars}\label{wd}
WD stars that are hot enough to emit significant fluxes in the EUV
spectral range are much less numerous than the main-sequence stars considered
in Section~\ref{ms}. However their photospheres generate outputs in the EUV
region that exceed by far the coronal emissions from individual main-sequence
stars. This fact is demonstrated by the actual observations of the local EUV
sources compiled by Vallerga (1998), where he found that a
vast majority of the detected objects were nearby hot WD
stars.\footnote{The spectrum shown by Vallerga (1998)
indicates that the $m_V=1.5$ B2~II star Adhara ($\epsilon$~CMa) dominates the
local flux at energies below the He~I ionization edge at 24.6$\,$eV. This is
an atypical situation, since $N({\rm H~I})\lesssim 10^{18}{\rm cm}^{-2}$
toward this star (Gry \& Jenkins 2001).} Krzesinski et al.
(2009) have measured the luminosity function for
DA WDs $\phi(M_{\rm bol})_{\rm WD}$, expressed in terms of (${\rm
stars}~M_{\rm bol}^{-1}{\rm pc}^{-3}$), in our part of the Galaxy using
results from the {\it Sloan Digital
Sky Survey Data Release 4\/} (SDSS DR4) database. If we combine this information with the
theoretical computations of stellar atmosphere fluxes $F_\lambda$ (equal to 4
times the Eddington flux $H_\lambda$), expressed in the units ${\rm
erg~cm}^{-2}{\rm s}^{-1}{\rm cm}^{-1}$ (Rauch 2003; Rauch et al. 2010),
convert it to a physical flux at the stellar surface ${\mathcal F}(E)=
7.74\times 10^7\pi F_\lambda E({\rm eV})^{-3}{\rm phot~cm}^{-2}{\rm s}^{-
1}{\rm eV}^{-1}$, and assume each star has a radius $r_*=0.014r_\odot$
(Liebert et al. 1988), we can obtain a total emissivity per unit volume,
\begin{eqnarray}
&&F(E)_{\rm WD}=4\pi r_*^2 (1.65\times 10^{-34})\nonumber\\
&\times& \int{\mathcal F}(E)\phi(M_{\rm
bol})_{\rm WD}d\, M_{\rm bol}~{\rm phot~cm}^{-3}{\rm s}^{-1}{\rm eV}^{-1}~.
\end{eqnarray}
A conversion from $M_{\rm bol}$ to $T_{\rm eff}$ is shown in the plot of the
WD luminosity function presented by Krzesinski et al. (2009). The model
fluxes for stars with $T_{\rm eff}<40,000\,$K were assumed to arise from
stars with pure hydrogen atmospheres, but stars with temperatures above this
limit are known to have significant abundances of metals in their atmospheres
because radiative levitation can overcome diffusive settling (Dupuis et al.
1995; Marsh et al. 1997; Schuh et al. 2002). Thus, for $T_{\rm eff}=50,000\,
$K and above, we used the fluxes for model atmospheres with [$X$]~=~[$Y$]~=~0 and
$[Z]=-1$, which have significant sources of opacity that reduce the
radiation at energies $E>54\,$eV.
As a check on the calculations described above, we can compute the X-ray
energy outputs as a function of $T_{\rm eff}$ over the $0.1-0.28\,$keV range,
synthesize a luminosity function as a function of $L_X$ in this band, and
then compare the results to X-ray luminosity function derived by Fleming et
al. (1996) from a RASS survey of WDs. Our
source densities at the high end of the luminosity distribution compare
favorably with the distribution shown by Fleming et al., but we predict a
somewhat greater number of sources with $L_X\lesssim 10^{31}{\rm erg~s}^{-1}$
due to a large space density of stars with $30,000\leq T_{\rm eff}\leq
40,000\,$K.
\subsection{Interpretation of Internal
Ionizations}\label{interpretation_internal}
The treatments of the ionizations that arise from external and internal
sources differ in a fundamental way. External radiation above some energy
threshold is regarded as not being consumed by ionizing the gas, i.e., it is
assumed to be unattenuated and thus each gas constituent is ionized at an
appropriate rate $\Gamma$, as defined in Eq.~\ref{Gammas}, that is simply
proportional to $\int \sigma(E)F(E)dE$. Here, $\sigma(E)$ is the effective
cross section for the combination of the different ionization channels, as
depicted for Ar and H by the solid lines in Fig.~\ref{p_ar_plot}.
As indicated in the beginning of Section~\ref{internal_radiation}, for the
internally generated radiation we switch to a very different concept and
adopt the simplified premise that all of the photons emitted by embedded
sources are used up by ionizing the various gas constituents that surround
them. Thus the primary ionization rate for each kind of atom (or ion) $X$ is
given by
\begin{equation}\label{Gamma_pX}
\Gamma_p(X)=\int F(E)y(X,E)n(X)^{-1}dE
\end{equation}
where $F(E)$ in this equation is the sum of all of the photon generation
rates per unit volume specified in Sections~\ref{ms} through \ref{wd}. Each
of the different species represented by $X$ must compete with others for the
photons that are consumed. Thus the equation includes a term $y(X,E)$, which
is a sharing function for the ionization rate that is represented by the
relative probability that any photon with an energy $E$ will interact with a
given species $X$,
\begin{equation}\label{y(X,E)}
y(X,E)={n(X)\sigma(X,E)\over\sum\limits_{X^\prime}n(X^\prime)\sigma(X^\prime,
E)}~,
\end{equation}
where $n(X)$ is the number density of $X$, $\sigma(X,E)$ is the
photoionization cross section of $X$ at an energy $E$ (likewise for
$X^\prime$), and the sum in the denominator covers all of the major species
competing for photons, i.e., $X^\prime={\rm H}^0$, ${\rm He}^0$, ${\rm
He}^+$, and heavy elements whose inner shells respond to the more energetic
X-rays. For either the external or internal radiations, the secondary
ionization rates $\Gamma_s$ and $\Gamma_{s^\prime}$ follow in proportion to
the primary ones according to the descriptions given in Appendix~\ref{gamma_s}.
The ionizations from helium recombinations $\Gamma_{\rm He}^+$ and
$\Gamma_{\rm He}^0$ are treated as internal sources of ionization, and their
rates are driven by the local densities of He$^{++}$, He$^+$, and electrons,
as described in Appendix~\ref{gamma_he}.
\subsection{Predicted Level of Ionization}\label{predicted_level}
Given the computed rates of ionization in the previous sections, an
evaluation of the electron fraction $x_e=n(e)/n({\rm H}_{\rm tot})$ will
depend on both the temperature $T$ and density $n({\rm H}_{\rm tot})$ for the
gas. This fraction is higher than $n({\rm H}^+)/n({\rm H}^0)$ because some
electrons arise from the ionization of He. The coupling of the H and He
ionization fractions is governed by Eqs.~\ref{H_ioniz_balance} through
\ref{n(e)}. Our observational constraint, which must ultimately agree with
the ionization calculations, arises from the values for [Ar~I/O~I], which
respond to $n({\rm H}^+)/n({\rm H}^0)$ in accord with Eq.~\ref{ar_h}. While
the use of $P_{\rm Ar}$ in this equation will give an approximate value for
[Ar~I/O~I], a more accurate result emerges by replacing $P_{\rm Ar}$ by
$P^\prime_{\rm Ar}$, the derivation of which was described in
Sections~\ref{equilibrium} and \ref{electron_density}. Our goal will be to
explore parameters that will match a computed value for [Ar~I/O~I] to the
representative observed value $[{\rm Ar~I/O~I}]=-0.427\pm0.11$.
We have no direct knowledge about the local values of $n({\rm H}_{\rm tot})$
that apply to the gas in front of the stars in this survey. We must
therefore rely on general estimates that have appeared in the literature. We
can draw upon two resources. First, surveys of 21-cm emission indicate the
amounts of H~I in the disk at a Galactocentric distance of the Sun, but
difficulties in interpreting the outcomes arise from self-absorption effects
and ambiguities in distinguishing between cold, dense clouds
(CNM) and their surrounding WNM. Ferri\`ere (2001) lists values
for $n({\rm H}^0)_{\rm WNM}$ that are in the range $0.2-0.5\,{\rm cm}^{-3}$.
Dickey \& Lockman (1990) state a value of
$0.57\,{\rm cm}^{-3}$, but it is not clear whether this excludes
contributions from the CNM. Kalberla \& Kerp (2009) estimate that the
midplane density of the WNM is $0.1\,{\rm cm}^{-3}$, but this low value is
difficult to reconcile with measurements of the mean thermal pressures $nT=
3800\,{\rm cm}^{-3}$K by Jenkins \& Tripp (2011) and a general recognition
that $T_{\rm WNM}<10^4\,$K.
\begin{deluxetable}{
l
c
c
c
c
c
c
}
\tablecolumns{7}
\tablewidth{0pt}
\tablecaption{Ionization Rates from Known
Sources\tablenotemark{a}\label{known_source_rates}}
\tablehead{
\colhead{Ionization} & \colhead{Rate} & \multicolumn{5}{c}{Relative
Contribution\tablenotemark{b}}\\
\cline{3-7}
\colhead{Source\tablenotemark{c}} & \colhead{($10^{-17}\,{\rm s}^{-1}$)} &
\colhead{$\Gamma_p$} & \colhead{$\Gamma_{s,{\rm H}^0}$} &
\colhead{$\Gamma_{s,{\rm He}^0}$} & \colhead{$\Gamma_{s,{\rm He}^+}$} &
\colhead{$\Gamma_{s^\prime}$}\\
\colhead{(1)}& \colhead{(2)}& \colhead{(3)}& \colhead{(4)}&
\colhead{(5)}& \colhead{(6)}& \colhead{(7)}
}
\startdata
$\Gamma({\rm H}^0)_{\rm ext.}$& 5.26& 0.20& 0.22& 0.52& 0.013& 0.048\\
$\Gamma({\rm H}^0)_{\rm MS}$& 6.72& 0.26& 0.044& 0.57& 0.11& 0.012\\
$\Gamma({\rm H}^0)_{\rm AB}$& 0.649& 0.035& 0.015& 0.52& 0.22& 0.21\\
$\Gamma({\rm H}^0)_{\rm WD}$& 2.19& 0.92& 0.030& 0.044& 6.3e-3& 9.4e-6\\
$\Gamma_{He^+}({\rm H}^0)$& 0.166&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^0}({\rm H}^0)$& 4.94&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\zeta_{\rm CR}({\rm H}^0)$& 12.3&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm H}^0)$& {\bf
32.3}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm He}^0)_{\rm ext.}$& 31.4& 0.87& 0.035& 0.084& 2.2e-3& 7.5e-3\\
$\Gamma({\rm He}^0)_{\rm MS}$& 25.7& 0.80& 0.012& 0.15& 0.028& 3.0e-3\\
$\Gamma({\rm He}^0)_{\rm AB}$& 0.861& 0.33& 0.011& 0.36& 0.16& 0.14\\
$\Gamma({\rm He}^0)_{\rm WD}$& 17.0& 0.99& 4.5e-3& 5.9e-3& 8.6e-4& 1.2e-6\\
$\Gamma_{He^+}({\rm He}^0)$& 0.916&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^0}({\rm He}^0)$& 7.10&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\zeta_{\rm CR}({\rm He}^0)$& 14.4&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm He}^0)$& {\bf
97.3}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm He}^+)_{\rm ext.}$&
16.9&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm He}^+)_{\rm MS}$& 8.32&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm He}^+)_{\rm AB}$&
0.0978&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm He}^+)_{\rm WD}$&
0.224&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^+}({\rm He}^+)$& 0.317&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm He}^+)$\tablenotemark{d}& {\bf
25.9}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm Ar}^0)_{\rm ext.}$& 251.& 0.94& 0.018& 0.042& 1.1e-3& 3.9e-3\\
$\Gamma({\rm Ar}^0)_{\rm MS}$& 82.2& 0.77& 0.014& 0.18& 0.035& 3.9e-3\\
$\Gamma({\rm Ar}^0)_{\rm AB}$& 10.5& 0.77& 3.6e-3& 0.12& 0.053& 0.049\\
$\Gamma({\rm Ar}^0)_{\rm WD}$& 45.3& 0.99& 5.6e-3& 8.2e-3& 1.2e-3& 1.8e-6\\
$\Gamma_{He^+}({\rm Ar}^0)$& 1.35&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^0}({\rm Ar}^0)$& 93.6&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\zeta_{\rm CR}({\rm Ar}^0)$& 113.&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm Ar}^0)$& {\bf
597.}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm Ar}^+)_{\rm ext.}$&
123.&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm Ar}^+)_{\rm MS}$& 47.2&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm Ar}^+)_{\rm AB}$& 1.57&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm Ar}^+)_{\rm WD}$& 49.0&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^+}({\rm Ar}^+)$& 0.969&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm Ar}^+)$\tablenotemark{d}& {\bf
222.}&\nodata&\nodata&\nodata&\nodata&\nodata\\
\enddata
\tablenotetext{a}{The internal rates (with subscripts MS, AB and WD) are
based on a volume density $n({\rm H}_{\rm tot})=0.5\,{\rm cm}^{-3}$. Such
rates scale inversely with density (although not exactly so, because the
secondary ionization efficiencies change when $x_e$ changes).}
\tablenotetext{b}{Fractions of the values shown in column (2). See
Eqs.~\protect\ref{Gammas}, \protect\ref{Hphi}, \protect\ref{Hephi},
\protect\ref{auger_H}, and \protect\ref{auger_He}.}
\tablenotetext{c}{Meaning of the subscripts that follow the different forms
of $\Gamma$: ext.~=~external X-ray background radiation
(Section~\protect\ref{external_radiation}), MS~=~embedded main-sequence stars
(Section~\protect\ref{ms}), AB~=~embedded active binary stars
(Section~\protect\ref{ab}), and WD~=~embedded WD stars
(Section~\protect\ref{wd}).}
\tablenotetext{d}{The total ionization rates for the ions He$^+$ and Ar$^+$
do not include the ionizations from secondary electrons or cosmic rays.
Hence these totals underestimate the true rates.}
\end{deluxetable}
Heiles \& Troland (2003) estimate
that an overall average $\langle n({\rm H}^0)_{\rm WNM})\rangle=0.28\,{\rm
cm}^{-3}$ translates into local densities of $0.56\,{\rm cm}^{-3}$ if the WNM
has a volume filling factor of 0.5. A second method for estimating $n({\rm
H}^0)_{\rm WNM}$ is to note where the WNM branch of the theoretical thermal
equilibrium curve of Wolfire et al. (2003) intersects
the average thermal pressure $p/k=3800\,{\rm cm}^{-3}$K of Jenkins \& Tripp
(2011). This exercise yields a value of
$0.4\,{\rm cm}^{-3}$.
Table~\ref{known_source_rates} shows a detailed accounting of the ionization
rates from the different radiation sources, under the condition that the
value of $x_e$ corresponds to what the equilibrium equations yield for a WNM
at $T=7000\,$K. The entries in this table reveal that for neutral hydrogen
the ionization caused by all of the sources of EUV and X-ray photons result
in a combined total rate $\Gamma({\rm H}^0)$ that is about 1.6 times the rate of
ionization by cosmic rays $\zeta_{\rm CR}$. For neutral He and Ar, the
photon ionization rates are several times higher than those from cosmic rays.
The changing patterns in the distribution of different ionization mechanisms
revealed in Columns (3) to (7) in the table reflect differences in the
distribution of fluxes with energy shown in Fig.~\ref{flux_plot}. For
sources with hard spectra, about half of the hydrogen ionization arises from
the ionization of He$^0$, which produces energetic electrons that can cause
secondary collisional ionizations, i.e., the process associated with
$\Gamma_{s,{\rm He}^0}({\rm H}^0)$. It is only for the very soft spectrum of
the collective radiation from WD stars that we find that
$\Gamma_p({\rm H}^0)$ strongly dominates over the other ionization routes.
The ionization rates and concentrations of various primary constituents are
coupled to each other by the network of reactions described in
Section~\ref{fundamentals} and Appendices \ref{gamma_s} through
\ref{cr_ioniz}. The first column of Table~\ref{properties_table} lists
various quantities of interest, and their derived values under the assumption
that $n({\rm H_{tot}})=0.50\,{\rm cm}^{-3}$ and $T=7000\,$K are given in
Column (2). (The remaining two columns of this table will be discussed later
in Section~\ref{additional_photoionization}.)
Figure~\ref{sdosdb_plot} shows the predicted outcomes for [Ar~I/O~I] after we
perform the calculations, again using the equations and reaction rates given
in Section~\ref{fundamentals} and the Ar ionization rates derived from the
information in Sections~\ref{external_radiation} and
\ref{internal_radiation}. The upper curve in this figure that represents
$n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$ is clearly inconsistent with our
findings for [Ar~I/O~I]. At $T=7000\,$K, a lower value $n({\rm H}_{\rm
tot})=0.14\,{\rm cm}^{-3}$ is consistent with the upper error bound for
[Ar~I/O~I]. At lower temperatures, however, the predictions for [Ar~I/O~I]
once again are found to be considerably above the observed values. It is not
until a density of $0.09\,{\rm cm}^{-3}$ is reached, a value that is
unacceptably low, that the predicted ionization conditions for $T=7000\,$K
fit comfortably with the nominal value from the observations. Even here,
however, this result is not fully satisfactory because a good fraction of the
WNM is known to be at temperatures well below 7000$\,$K, where the gas is
thermally unstable (Heiles \& Troland 2003).
Our overall conclusion is that the ionization rates arising from what we
consider to be known sources of ionizing radiation are not able to maintain a
level of ionization in the WNM that is consistent with our low observed
values for [Ar~I/O~I]. Our quest to resolve this problem by exploring some
possible supplemental means for ionizing the medium will be addressed later
in Section~\ref{additional_photoionization}.
\begin{figure}[h!]
\epsscale{1.0}
\plotone{sdosdb_plot.eps}
\caption{Calculated values for $10^{\rm [Ar I/O I]}$ for three different
densities $n({\rm H}_{\rm tot})$ and a range of different gas temperatures
$T$ shown on the $x$ axis, computed using estimates for the ionization rates
from known sources of radiation, both external and internal. A general value
for $10^{\rm [Ar I/O I]}$ that arose from the survey is indicated by the
horizontal line, and the shaded band shows the range of possible
systematic errors.\label{sdosdb_plot}}
\end{figure}
\begin{deluxetable}{
l
c
c
c
}
\tablewidth{0pt}
\tablecaption{Gas Properties\tablenotemark{a}\label{properties_table}}
\tablehead{
\colhead{} & \colhead{Without SNR} & \colhead{With SNR} & \colhead{Greater
Low}\\
\colhead{Quantity} & \colhead{Contrib.\tablenotemark{b}} &
\colhead{Contrib.\tablenotemark{c}} & \colhead{Energy
Penetration\tablenotemark{d}}\\
\colhead{(1)} &\colhead{(2)} &\colhead{(3)} & \colhead{(4)}
}
\startdata
$P_{\rm Ar}$\tablenotemark{e}\dotfill&13.1&19.3&12.6\\
$P^\prime_{\rm Ar}$\tablenotemark{f}\dotfill&13.4&22.8&15.9\\
$n(e)\,({\rm cm}^{-3})$\dotfill&0.018&0.033&0.052\\
$n({\rm H}^0)\,({\rm cm}^{-3})$\dotfill&0.48&0.48&0.46\\
$n({\rm H}^+)\,({\rm cm}^{-3})$\dotfill&0.015&0.025&0.036\\
$n({\rm He}^0)\,({\rm cm}^{-3})$\dotfill&0.047&0.042&0.035\\
$n({\rm He}^+)\,({\rm cm}^{-3})$\dotfill&3.0e-03&8.1e-03&0.014\\
$n({\rm He}^{++})\,({\rm cm}^{-3})$\dotfill&2.1e-05&2.1e-04&1.0e-03\\
Total $\Gamma({\rm H}^0)\,({\rm s}^{-1})$\dotfill&3.2e-16&9.2e-16&2.1e-15\\
$\ell_c$ ($10^{-27}{\rm erg~s}^{-1}{\rm
H~atom}^{-1})$\tablenotemark{g}\dotfill&4.3&6.2&8.5\\
\enddata
\tablenotetext{a}{For $n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$ and
$T=7000\,$K.}
\tablenotetext{b}{Applies to the photoionization rates expressed in
Table~\protect\ref{known_source_rates} from the known sources only.}
\tablenotetext{c}{The radiation from known sources is supplemented by the
average flux over 1$\,$Myr from 3 supernova remnants created by explosions
with energies $E_{\rm SN}=3\times 10^{50}\,{\rm erg}$ in a medium with
$n({\rm H})=1.0\,{\rm cm}^{-3}$ located at a distance of 100$\,$pc. This
supplemental radiation is shown by the dashed line in the upper panel of
Fig.~\protect\ref{flux_plot}.}
\tablenotetext{d}{Allows for a low energy cutoff of 60$\,$eV instead of
90$\,$eV for the external radiation background; see
Section~\protect\ref{penetration}.}
\tablenotetext{e}{As defined in Eq.~\protect\ref{p_ar}.}
\tablenotetext{f}{As defined in Eq.~\protect\ref{Pprime_ar}.}
\tablenotetext{g}{Cooling rate from radiation by C$^+$ in its excited
fine-structure state.}
\end{deluxetable}
\subsection{Dependence of Recombination with Time}\label{recomb_time}
An argument that helps to support the concept that the internal sources
spread their ionization rather evenly throughout the medium is that they move
at a rate that makes their local residence time short compared to a
characteristic $e$-folding time $t_{\rm recomb.}$ for the decay of the proton
density from an initial high value to some end equilibrium state $n({\rm
H}^+)_{\rm eq.}$ as $t\rightarrow\infty$,
\begin{equation}\label{t_recomb}
t_{\rm recomb.}=-{n({\rm H}^+)-n({\rm H}^+)_{\rm eq.}\over dn({\rm H}^+)/dt}~,
\end{equation}
where
\begin{eqnarray}\label{dnpdt}
&&dn({\rm H}^+)/dt=-\Big[\alpha({\rm H}^0,T)n(e)\nonumber\\
&+&\alpha_g({\rm
H}^0,n(e),G,T)n({\rm H_{tot}})\nonumber\\
&+&\Gamma({\rm H^0})\Big] n({\rm
H}^+)+\Gamma({\rm H}^0)n({\rm H_{tot}})~.
\end{eqnarray}
This expression overlooks the complications arising from charge exchange
reactions, and when we evaluate the trend of $n({\rm H}^+)$ with time we
assume that $n(e)$ is always equal to $1.2n({\rm H}^+)$, as is the case when
the gas is in a steady-state ionization condition. Another simplification is
that $T$, and hence $\alpha({\rm H}^0,T)$, remains constant.\footnote{A lack
of variation in $T$ is probably a safe assumption for isobaric recombination,
but in the isochoric case $T$ may deviate to lower values at intermediate
times (Dong \& Draine 2011), and this would increase the value of
$\alpha({\rm H}^0,T)$ and make the recombination more rapid. This could be
of importance for very large regions of space ionized by the radiation from
old SNRs, which will be considered later in
Section~\protect\ref{previous_SNR}.} When the gas is highly ionized,
$\Gamma({\rm H}^0)$ is not equal to the value that we computed for $n({\rm
H}^+)_{\rm eq.}=0.015\,{\rm cm}^{-3}$ because the secondary ionization
processes change with $n(e)$ and depend on the strength of the helium
ionization. However, we can ignore this complication that occurs at early
times because the recombination terms in Eq.~\ref{dnpdt} are considerably
larger than the terms involving $\Gamma({\rm H}^0)$. The ionization rate
influences the character of the decay only when the gas is weakly ionized.
Hence there is no harm in declaring that at all times $\Gamma({\rm
H}^0)=3.23\times 10^{-16}{\rm s}^{-1}$, as stated in
Table~\ref{known_source_rates}. We can adopt for the final state
(equilibrium) densities the values $n({\rm H}^+)_{\rm eq.}=0.015\,{\rm
cm}^{-3}$ and $n(e)_{\rm eq.}=0.018\,{\rm cm}^{-3}$ listed in Column (2) of
Table~\ref{properties_table}.
The change of $t_{\rm recomb.}$ with time and the relaxation of $n({\rm
H}^+)$ from a fully ionized condition to $n({\rm H}^+)_{\rm eq.}$ for $n({\rm
H_{tot}})=0.50\,{\rm cm}^{-3}$ and $T=7000\,$K is illustrated in
Fig.~\ref{ioniz_decay}. Values for $t_{\rm recomb.}$ start at 0.13$\,$Myr
for an initial $n({\rm H}^+)=0.50\,{\rm cm}^{-3}$ and increase to 1.0$\,$Myr
when $n({\rm H}^+)$ reaches $0.036\,{\rm cm}^{-3}$ at $t=1.77\,$Myr, beyond
which $t_{\rm recomb.}$ very gradually climbs toward a steady value of
1.68$\,$Myr and the subsequent decay in $n({\rm H}^+)$ toward $n({\rm
H}^+)_{\rm eq.}=0.015\,{\rm cm}^{-3}$ is almost purely exponential.
\begin{figure}[h!]
\epsscale{1.0}
\plotone{ioniz_decay.eps}
\caption{The behavior of $t_{\rm recomb.}$ (upper solid curve) and $n({\rm
H}+)$ (lower solid curve) with time $t$, starting from a fully ionized
condition to an equilibrium state $n({\rm H}+)_{\rm eq.}$ for $n({\rm
H_{tot}})=0.50\,{\rm cm}^{-3}$ and $T=7000\,$K. Values of $t_{\rm recomb.}$
are defined by Eq.~\protect\ref{t_recomb}, and $n({\rm H}+)$ as a function of
$t$ can be found by solving the differential equation for $dn({\rm H}+)/dt$
given in Eq.~\protect\ref{dnpdt}. By solving a similar equation for Ar$^+$
and comparing it to that for H$^+$, we obtain the trend for [Ar~I/H~I] shown
by the dashed line.\label{ioniz_decay}}
\end{figure}
Of the three different classes of internal sources, WD stars emit
the softest radiation, which means the influences of their ionizations can be
more sharply bounded than those of the others. While this consideration may
present a challenge to our assumption about the uniformity of ionization in
space, we note that these stars have observed radial velocity dispersions of
about $25\,{\rm km~s}^{-1}$ (Falcon et al. 2010), or rms speeds of
$43\,{\rm km~s}^{-1}$, which are consistent with the transverse velocity
dispersion measurements reported by Wegg \& Phinney (2012). This indicates
that WDs typically move about 76$\,$pc during the time interval that
$n({\rm H}^+)$ decays from 0.50 to $0.036\,{\rm cm}^{-3}$. This amount of
travel is considerably larger than the radius of a Str\"omgren Sphere
\begin{equation}\label{r_S}
r_{\rm S}=\left( {3r^2_*\int_{13.6\, {\rm eV}}^\infty F(E)_{\rm WD}dE\over
n({\rm H})^2\alpha({\rm H}^0,T)}\right)^{1/3}
\end{equation}
that would be established if a WD star were stationary. (Recall
that $F(E)_{\rm WD}$ is the flux emitted at the surface of the star with a
radius $r_*$.) A typical value for $r_{\rm S}$ in the WNM would be about
1.8$\,$pc; this value applies to a WD star with $r_*=0.014r_\odot$
that has $T_{\rm eff}=50,000\,$K and is situated within a gas with a density
$n({\rm H_{tot}})=0.50\,{\rm cm}^{-3}$.
One can view the dispersal of ionization caused by the star's motion in the
context of the simplified description of ``cometary H~II regions'' described
by Raga (1986); the dimensionless elongation parameter in this
development $\beta=3v/[n({\rm H_{tot}})\alpha({\rm H}^0,T)r_{\rm S}]=14$ if
$r_{\rm S}=1.8\,$pc and $v=43\,{\rm km~s}^{-1}$, which would result in a tube
of ionization with a radius $\sim 0.5r_{\rm S}$ and a very long tail. The
fact that the ionizations produced by WD stars could be highly
diluted because they are distributed over large volumes could explain why a
sensitive survey of H$\alpha$ emission from regions around such stars carried
out by Reynolds (1987) yielded only one detection out of
nine targeted regions surrounding these stars.
\section{PROPOSALS FOR ADDITIONAL
PHOTOIONIZATION}\label{additional_photoionization}
Given that the outcome for a reasonable value of $n({\rm H}_{\rm tot})$ and
the ionization rate from known sources discussed in
Section~\ref{known_photoionization} do not agree with our observations, we
must explore some alternatives for boosting the average rate of
photoionization in the WNM. Two possibilities are discussed in the
following sections.
\subsection{Ionization from Previous Supernova Remnants}\label{previous_SNR}
A popular theme in the ISM literature of early 1970's was the consideration
that the low density portions of the ISM were ionized impulsively by UV and
X-ray radiation from supernovae and their remnants (SNRs), which represented
sources of ionization that had limited durations and that materialized at
random locations and times (Bottcher et al. 1970; Werner et al. 1970; Jura \&
Dalgarno 1972; Gerola et al. 1973; Schwarz 1973). A particularly instructive
example that shows the wide variations in temperature and fractional
ionization can be seen in the results from a numerical simulation by Gerola
et al. (1974). This topic has also been approached by Slavin et al. (2000),
who estimated the effects of irradiation of the neutral medium by SNRs, with
a special emphasis on its influence on gas in the Galactic halo. The spectral
character and strength of the emitted radiation from a remnant depends on a
combination of supernova energy and the density of the ambient medium
(Mansfield \& Salpeter 1974). The emission of soft X-rays as a remnant
evolves can last up to a late phase when radiative cooling causes a dense
shell to form, which could be opaque to the lowest energy X-rays. However
some of the radiation may continue to escape afterward if instabilities
cause the shell to break up and create gaps (Vishniac 1994; Blondin et al.
1998). The magnitudes of such instabilities are uncertain, since they might
be diminished if either the ambient medium has a low density and high
temperature (Mac Low \& Norman 1993) or if the shock is partly stabilized by
an embedded transverse magnetic field
(T\'oth \& Draine 1993).
Chevalier (1974) has computed the strength and
distribution over energy for the radiation emitted by various models for
SNRs. As a representative example, we can use the emission
over a time interval of $2.5\times 10^5\,{\rm yr}$ from his model that had an
energy $E_{\rm SN}=3\times 10^{50}\,{\rm erg}$ and was expanding into a
medium with an average density of $1\,{\rm cm}^{-3}$. If we assume that
there was a succession of about 3 such remnants that occurred once every Myr
(i.e., of order $t_{\rm recomb.}$ defined in Eq.~\ref{t_recomb} for $n({\rm
H}^+)=0.036\,{\rm cm}^{-3}$; see Fig.~\ref{ioniz_decay}) and they were at a
distance of 100$\,$pc from some location in the ISM, the resulting
time-averaged flux shown by the dashed line in the upper panel of
Fig.~\ref{flux_plot} would increase the total ionization of H from
$\Gamma({\rm H}^0)=3.2\times 10^{-16}{\rm s}^{-1}$ for the steady ionization
sources to the much higher value of $9.2\times 10^{-16}{\rm s}^{-1}$ and
create conditions that would yield values for [Ar~I/O~I] that are virtually
the same as those depicted by the curve labeled $n({\rm H}_{\rm
tot})=0.14\,{\rm cm}^{-3}$ in Fig.~\ref{sdosdb_plot}, but in this case for a
density $n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$. At $T=7000\,$K, the
densities of various constituents are represented by the numbers that are
shown in Column (3) of Table~\ref{properties_table}. For this temperature,
the calculation of [Ar~I/O~I] is just consistent with the upper error bound
from the observations.
A stronger flux from SNRs would be needed to make the
calculated value for [Ar~I/O~I] match nominal value for the observed ones.
To obtain a curve that matches the lowest curve (labeled $n({\rm H}_{\rm
tot})=0.09\,{\rm cm}^{-3}$) in Fig.~\ref{sdosdb_plot} and still maintain an
assumed local density $n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$, we would
require either an increase in the average energy of the supernovae (SNe), or the
rate would need to be increased from 3 to 8~SN~${\rm Myr}^{-1}$ at a distance
of 100$\,$pc. In this circumstance, $n(e)=0.047\,{\rm cm}^{-3}$.
While one might be tempted to ask whether some average SN rate in our
location within the Galaxy (van den Bergh \& McClure 1994; McKee \& Williams
1997; Ferri\`ere 2001) is consistent with at least 3 SNe Myr$^{-1}$ at
separations of about 100$\,$pc from some representative location, it is
probably more realistic to draw upon actual estimates of the recent history
of explosions in the general neighborhood of the Sun, which have been
inspired by the evidence of how the ISM has been disrupted by the expansions
of the explosion remnants. The Local Bubble (see footnote 2 in
Section~\ref{strategy}) and the neighboring radio emitting Loop~I
superbubble\footnote{Loop~I is coincident with the North Polar Spur of X-ray
radiation. Its center is estimated to be only 180 pc away from us (Bingham
1967).}, which intersect one another (Egger \& Aschenbach 1995), are both
thought to have been created by a series of SN explosions that
occurred over the past 10-15$\,$Myr
(Ma\'iz-Apell\'aniz 2001; Bergh\"ofer \& Breitschwerdt 2002). Fuchs
et al. (2006) analyzed the numbers and mass functions of stars within nearby
O-B associations and traced their motions back in time. From this
information, they concluded that the Local Bubble was created by 14-20
SNe. This estimate is consistent with the 19 SNe that
Breitschwerdt \& de~Avillez (2006) used in their hydrodynamic simulation of
the creation of the Local Bubble.
Within a shorter time frame, there is some evidence that atoms from a nearby
SN were deposited on the Earth some 2.8$\,$Myr ago, as revealed by an
enhancement of $^{56}$Fe in a thin layer within deep-sea ferromanganese crust
(Knie et al. 2004). Fitoussi et al. (2008)
attempted to replicate this result in a marine sediment, but the outcome did
not agree with the earlier result at the same level of significance.
However, there are alternative means for investigating past depositions of
SN elements in terrestrial samples (Bishop \& Egli 2011; Feige et al.
2012), and new results may emerge soon.
If we add to the Local Bubble contribution the past SNe in the Sco-Cen
association that created Loop~I (Iwan 1980; Egger 1998; Diehl et al. 2010),
one can imagine that, to within the uncertainties in the X-ray production
rates, the time-averaged level of ionizing radiation could conceivably be of
the same order of magnitude as that discussed in the preceding paragraphs.
A legitimate question to pose is whether or not, at some intermediate stage
of recombination, the value of [Ar~I/H~I] makes a brief excursion to a level
below the final equilibrium state. If this were the case, we might be able
to relax the requirement for a high frequency of SN explosions. The
recombination rates $\alpha({\rm H}^0,T)$ and $\alpha({\rm Ar}^0)$ are nearly
equal to each other, but the dust grain recombination rate $\alpha_g$ for Ar
should be much less than that for H because the thermal velocities of Ar are
lower. Thus, at early times where recombinations still dominate over the
steady-state ionization rate but the electron density has decreased to a
point that neutralization of Ar and H by collisions with dust grains become
important, one could imagine that the difference in grain recombination rates
might be influential in allowing the H$^+$ to recombine more quickly than
Ar$^+$. In order to investigate this possibility, we can evaluate the time
history of the Ar recombination using Eqs.~\ref{t_recomb} and \ref{dnpdt} but
with a substitution of ``Ar'' for each appearance of ``H.'' A resulting
comparison of the Ar recombinations to those of H shows that [Ar~I/H~I] shows
a steady decrease from an initial value slightly greater than 1.0 (when the
gas is just starting to recombine) to the final equilibrium state for this
quantity without any excursion to lower values. This behavior is illustrated
by the dashed line in Fig.~\ref{ioniz_decay}.
\subsection{Penetration of the WNM by Low-energy Photons}\label{penetration}
Until now, we have regarded the emission of X-rays from the hot ($T\sim
10^6\,$K) gas in our Galaxy as an external source of ionizing radiation that
must penetrate the bulk of the WNM regions under study. With this condition
in mind, we established a 90$\,$eV cutoff for the external radiation, below
which the X-rays were assumed to be absorbed in the outer layers of the
neutral gas. However, some of the soft X-ray emitting gas may be threaded
through the WNM, a picture that is reminiscent of the network of hot
interstellar tunnels proposed by Cox \& Smith (1974).
Hot, interspersed gas could allow some or all of the neutral medium to have
access to lower energy radiation. Observations indicate that O~VI, an
indicator of gas that is collisionally ionized at $T\sim 3\times 10^5\,$K,
can be seen in emission (Dixon et al. 2006; Otte \& Dixon 2006) and absorption
(Jenkins 1978a, b; Bowen et al. 2008) almost everywhere in the disk of the
Galaxy. The simulations of SNe exploding at random in the Galactic
disk by Ferri\`ere (1995) and de~Avillez \& Breitschwerdt (2005a, b) indicate
that this hot gas should be sufficiently pervasive and frothy that, even
though our measurements penetrate through column densities well in excess of
$10^{19}\,{\rm cm}^{-3}$, individual parcels of gas may be exposed to
adjacent sources of radiation through significantly lower column densities.
Indeed, there could be unusually strong X-ray emission at the boundaries
where the neutral regions come into contact with the hot gas, where charge
exchange reactions between neutrals and highly ionized species could produce
an enhancement of soft X-ray emission over that which is emitted by just the
hot gas (Wang \& Liu 2012).
It is clear from the gray parts of the external radiation spectrum depicted
in Fig.~\ref{flux_plot} that even a modest lowering of the cutoff below our
adopted 90$\,$eV threshold will create a significant increase in the
ionization of the WNM. For instance, if the threshold were dropped to
60$\,$eV, an energy just below the strong M-line complex of emission features
from highly ionized Fe at 70$\,$eV, the level of ionization would increase to
an extraordinarily large total hydrogen ionization rate $\Gamma({\rm
H}^0)=2.1\times 10^{-15}{\rm s}^{-1}$. For $n({\rm H}_{\rm tot})=0.50\,{\rm
cm}^{-3}$ and $T=7000\,$K, this increase would create a value for [Ar~I/O~I]
that is consistent with the upper error bound for the measured outcomes. In
this particular case, most of the ionization is caused by photons with
energies near 70$\,$eV. For this energy, $P_{\rm Ar}(E)$ is near its minimum
value (see the lower panel of Fig.~\ref{p_ar_plot} and the entry in Column
(4) of Table~\ref{properties_table}). As a consequence, in order to match
our observations, Eq.~\ref{ar_h} shows us that the required $n({\rm H}^+)$
(and thus $\Gamma({\rm H}^0)$) must be increased beyond the value needed from
the time-averaged SNR radiation with its larger value of
$P_{\rm Ar}$.
A consideration that disfavors the presence of a strong ionizing flux at
energies in the vicinity of 70$\,$eV, at least at our location, is that it
does not seem to be present at anywhere near the intensity level shown in
Fig.~\ref{flux_plot}. For instance, from observations by the {\it Cosmic Hot
Interstellar Plasma Spectrometer\/} ({\it CHIPS\/}) Hurwitz et al.
(2005) stated an upper limit for the flux emitted by
the Fe lines near 70$\,$eV. This limit was consistent with an emission
measure for a plasma with a solar abundance pattern at $T=10^6\,$K that is
less than 5\% of the value ${\rm EM}=10^{16.37}{\rm cm}^{-5}$ that we used
for reconstructing the emission spectrum from the soft component of the X-ray
background described in Section~\ref{external_radiation}. Jelinsky et al.
(1995) and Bloch et al. (2002)
similarly found flux upper limits at low energies (but not as stringent as
those from {\it CHIPS\/}) that were much lower than model predictions at this
energy that we obtained from fits to the observed radiation at higher
energies. A marginal detection of the Fe emission complex toward high
Galactic latitudes by McCammon et al. (2002),
$F(E)=100\pm 50\,{\rm phot~cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}$ is also well
below the peak at 70$\,$eV that appears in our reconstructed flux. The
weakness of these fluxes could be caused either by a deficiency of Fe below
the solar abundance ratio in the hot emitting gases or by our inability to
see hot gas regions whose low energy radiation is not absorbed by intervening
neutral material.
\section{DISCUSSION}\label{discussion}
We have now reached a point where it is appropriate to investigate the
consistency of the newly derived WNM ionization levels with other
observational and theoretical findings, along with some consequences of our
results on various relevant physical processes.
\subsection{Other UV Observations}\label{other_obs}
Along a number of sight lines, there have been detailed investigations of UV
absorption lines observed at high enough velocity resolutions to identify
which components came from either mostly neutral or mostly ionized clouds
(Spitzer \& Fitzpatrick 1993, 1995; Fitzpatrick \& Spitzer 1994, 1997; Wood
\& Linsky 1997; Holberg et al. 1999; Welty et al. 1999; Jenkins et al. 2000a;
Gry \& Jenkins 2001; Sonnentrucker et al. 2002). These investigations relied
either on the relative populations of the excited fine-structure level of
${\rm C}^+$ or the ratios
of ions to neutrals for various elements.
The ion-to-neutral ratios of different elements gave very different outcomes
for $n(e)$ (but systematically went up and down together from one velocity
component to the next); these disparities probably arose as a result of a
lack of a good understanding of the physical processes involved. Welty et
al. (2003) have presented a good summary of the results
that exhibited this problem.
Values of $n(e)$ identified with the WNM by the investigators cited above
generally ranged between 0.04 and $0.12\,{\rm cm}^{-3}$, which seem to be
higher than our values given in Table~\ref{properties_table}. A common theme
in the discussions of these results was that such high partial ionizations
could not be attributed to the known EUV, X-ray and cosmic ray ionization
rates -- a conclusion that is stated once again from our [Ar~I/O~I] results.
Electron densities for clouds embedded in the Local Bubble have an average
value of $0.11\,{\rm cm}^{-3}$ (Redfield \& Falcon 2008), a result
that once again relied on comparisons of excited C$^+$ to other species
(either unexcited C$^+$ or S$^+$). The fractional ionization of these clouds
appears to be much higher than what we found for the general WNM outside the
Local Bubble, if one assumes that the characteristic total density $n({\rm
H}_{\rm tot})\approx 0.2\,{\rm cm}^{-3}$ within the clouds (Redfield \&
Linsky 2008). By solving for the time-dependent ionization in Eq.~\ref{dnpdt}
for $n({\rm H_{tot}})=0.2\,{\rm cm}^{-3}$, we find that the decay from a
fully ionized condition to an approximately half ionized state takes only
0.40$\,$Myr. It is likely that this higher level of ionization might be
explained by supplemental radiation from evaporative boundaries that surround
the clouds (Slavin \& Frisch 2002), if indeed they are embedded in a very
hot, low density gas, or by the infiltration of some ionizing radiation from
$\epsilon$~CMa, which strongly dominates over other radiation sources within
the Local Bubble at low energies (Vallerga \& Welsh 1995).
\subsection{Pulsar Dispersion Measures}\label{pulsar_DM}
Cordes \& Lazio (2002) have derived characteristic
electron densities for three volumes within about 1$\,$kpc of the Sun.
Inside the Local Bubble, their model seems to best fit an average density of
only $\langle n(e)\rangle=0.005\,{\rm cm}^{-3}$, which is not surprising
because much of the volume has probably been cleared of material by SN
explosions, except for some isolated, warm clouds with an average filling
factor in the range $5.5-19$\% (Redfield \& Linsky 2008). Regions
outside the Local Bubble have higher densities: two volumes that they studied
yield $\langle n(e)\rangle=0.012$ and $0.016\,{\rm cm}^{-3}$. However, these
regions are identified with ellipsoidal volumes designated as either a
``local superbubble'' (LSB) or a ``low density region'' (LDR), so they too
could have significant voids that would dilute the apparent electron
densities. The values of $\langle n(e)\rangle$ given for these regions could
be consistent with our measures of $n(e)$ given in the last two columns of
Table~\ref{properties_table} for the WNM if this medium had a filling factor
of about 1/3 in these regions and there were no significant contributions
from the warm ionized medium (WIM).
An independent analysis of dispersion measures was carried out by de Avillez
et al. (2012) for 24 pulsars with known distances
between 0.2 and 8$\,$kpc from the Sun and with $\vert z\vert<0.2\,$kpc. They
found a distribution of $n(e)$ outcomes that was consistent with a log-normal
distribution centered on $\log n(e)=-1.47$ and a dispersion $\sigma[\log
n(e)]=0.17$. The fact that their representative values of $n(e)$ are higher
than the determination of Cordes \& Lazio (2002)
and close to our findings based on [Ar~I/O~I] may be accidental, since their
results may be strongly influenced by sight-line interceptions of fully
ionized regions.
\subsection{Emission Lines}\label{emission_lines}
Observations of diffuse line emissions in the sky most generally apply to
probing the physical conditions in fully ionized gases, either the bright
H~II regions around hot stars or the WIM
(Reynolds et al. 1977, 2002; Haffner et al. 1999; Madsen et al. 2006). Most
of the emissions are dominated by contributions from species that are
expected to be abundant in such highly ionized media, and they should
overwhelm any contributions from the very dilute ionization in the WNM.
Nevertheless, there are a few cases where emission line fluxes from some
neutral species in the WIM are expected to be very weak, and thus a
contribution from the WNM might in principle be identified. We need to
explore whether or not the predictions for line strengths from a medium with
our enhanced electron densities are not in serious violation of detections or
the upper limits for the fluxes. We explore three such cases in the
subsections that follow. The first test involving line emission from He
recombinations is especially important, because we predict that the fraction
of singly-ionized He could be as high as 16$-$28\% of the total amount of
helium.
\subsubsection{He $\lambda$5876 Recombination Radiation}\label{He_recomb}
Reynolds \& Tufte (1995) attempted to compare
the strength of the recombination line of He$^0$ at 5876$\,$\AA\ to that of
H$\alpha$ in parts of the sky away from well defined H~II regions. Their
motive was to determine the hardness of the radiation that maintains the
ionization in the WIM. While an explicit upper limit for the He recombination
line flux was not stated by Reynolds \& Tufte (1995), we estimate that the
flux in the two directions that they sampled was found to be less than 0.1$\,
$R (R~=~Rayleigh~=~$10^6/(4\pi) {\rm phot~cm}^{-2}{\rm sr}^{-1}{\rm s}^{-
1}$).
Using the line emission rates given by Benjamin et al. (1999) we find that
for $n(e)=0.04\,{\rm cm}^{-3}$, $n({\rm He}^+)=0.01\,{\rm cm}^{-3}$, and $T=
7000\,$K (representative values for the enhanced ionization cases presented
in the last two columns of Table~\ref{properties_table}) the emission should
be $3.0\times 10^{-17}{\rm phot~cm}^{-3}{\rm s}^{-1}$. If we were to
propose that the emission is seen over a path of 200$\,$pc with no
extinction (i.e., imagine a length of 400$\,$pc with a filling factor for the
WNM of 50\%), we could expect to find an emission of $1500\,{\rm phot~cm}^{-
2}{\rm sr}^{-1}{\rm s}^{-1} = 0.02\,$R. This expectation is well below the
sensitivity of the observations by Reynolds \& Tufte (1995), so our predicted
intensity from the enhanced electron density and ionization of He does not
violate their upper limit.
\subsubsection{Emission of [O~I] $\lambda$6300 from Electron
Collisions}\label{OI_excitation}
In order to determine the neutral fraction of H in the WIM, Reynolds et al.
(1998) measured the strength of the [O~I]
$\lambda$6300 line emitted in parts of the sky that had a uniform, moderately
strong H$\alpha$ emission (but away from obvious H~II regions excited by
stars), much as they had done for the He recombination line discussed above.
In three different directions, they detected intensities of 0.2, 0.09, and
0.11$\,$R.
According to Federman \& Shipsey (1983),
electron collisions dominate over those by hydrogen for the excitation of the
$^1D_2$ state of neutral oxygen when the electron fraction exceeds
$1.5\times 10^{-4}$. Hence, we can ignore hydrogen impacts. From the
fitting formula of P\'equignot (1990) to the collision strengths computed
by Berrington \& Burke (1981), we derive a
collisional rate constant $C_e$ for electron excitations to be $8.0\times
10^{-11}{\rm cm}^3{\rm s}^{-1}$ at $T=7000\,$K. If we again make the
conservative assumption expressed in Section~\ref{reference_abund} that at
the low densities of the WNM the depletion of O is negligible, we expect that
$n({\rm O}^0)=2.7\times 10^{-4}{\rm cm}^{-3}$ if $n({\rm H}^0)=0.47\,{\rm
cm}^{-3}$. For $n(e)=0.04\,{\rm cm}^{-3}$, we expect the emissivity to be
equal to $n({\rm O}^0)n(e)C_e$ multiplied by a branching fraction 0.76
(Froese Fischer \& Tachiev 2010) for the proportion of decays from
the $^1D_2$ level to the lower $^3P_2$ level. This product
equals $6.6\times 10^{-16}{\rm phot~cm}^{-3}{\rm s}^{-1}$, which should
produce 0.41$\,$R over a path of 200$\,$pc. This value is greater than the
three measurements by Reynolds et al. (1998). The
magnitude of this violation is not large, considering the uncertainties of
our assumptions about the lack of depletion of O and the adopted path length
estimate. Also, for temperatures less than 7000$\,$K, the expected strength
of the emission will be considerably less: for instance, at $T=5000\,$K, the
emission should be 4.5 times weaker than at 7000$\,$K.
\subsubsection{Emission of [N~I] $\lambda$5201 from Electron
Collisions}\label{NI_excitation}
An upper limit of 0.13$\,$R for the [N~I] $\lambda$5201 line was determined
for a single direction in the sky by Reynolds et al. (1977). If we adopt
the same calculations as in the previous section for O~I, but make the
substitution that $n({\rm N}^0)=8.0\times 10^{-5}n({\rm H}^0)=3.8\times
10^{-5}{\rm cm}^{-3}$ (again, assuming no depletion), and $C_e=8.6\times
10^{-11}{\rm cm}^3{\rm s}^{-1}$ at $T=7000\,$K (Tayal 2006), we obtain an
emissivity equal to $1.3\times 10^{-16}{\rm phot~cm}^{-3}{\rm s}^{-1}$,
which yields an intensity of 0.08$\,$R over a 200$\,$pc path. This value is
below the upper limit determined by Reynolds et al. (1977).
\subsection{Cooling Rates from Carbon Ions and Oxygen
Atoms}\label{cooling_rate}
A major coolant for the neutral ISM is the singly charged carbon ion
(Dalgarno \& McCray 1972; Wolfire et al. 1995, 2003), whose $^2P_{3/2}$
excited fine-structure level in the ground electronic state can be excited
by collisions with electrons and hydrogen atoms. The cross section for
excitation by electrons is substantially greater than that for neutral
hydrogen. For this reason, the level of partial ionization is an important
factor in the excitation rate. After excitation, an energy loss can occur
because the excited level can undergo a spontaneous radiative decay with a
transition probability $A_{21}=2.29\times 10^{-6}{\rm s}^{-1}$ to the lower
$^2P_{1/2}$ state (Nussbaumer \& Storey 1981), a process that liberates a
photon with a wavelength of $158\,\mu$m. If we assume that the abundance
ratio $({\rm C}^+/{\rm H}^0)=9.5\times 10^{-5}$ (see footnote~1 in
Section~\ref{intro}), we can use the densities that we derived for $n({\rm
H}^0)$ and $n(e)$ along with the excitation cross section by H$^0$ impacts
computed by Barinovs et al. (2005) and electron collision strengths of
Wilson \& Bell (2002) to compute the ${\rm C}^{+*}$ energy
loss rates $\ell_c$ per H atom. These rates are given in the last row of
Table~\ref{properties_table}. They do not change in direct proportion to
$x_e$ because about half of the excitations come from collisions with H
atoms, which are more numerous than the electrons. In addition, the overall
cooling rate for the WNM does not increase in direct proportion to $\ell_c$
because it accounts for only about one-third of the total cooling.
Most of the remaining cooling comes from the fine-structure excitation of O~I
and the subsequent emissions at 44 and $63\,\mu$m. The O~I cooling is
insensitive to changes in $x_e$ because the collision rate constants for
electrons (Bell et al. 1998) are much less than those for neutral
hydrogen. Using an extrapolation of the atomic hydrogen collisional rate
constants given by Abrahamsson et al. (2007)
above a temperature of $10^3\,$K, the spontaneous decay rates of the two
excited levels given by Galav\'is et al. (1997), and ${\rm O/H}=5.75\times
10^{-4}$, we find that for $n({\rm H}^0)=0.47\,{\rm cm}^{-3}$ and $T=7000\,$K
that the energy loss rate per H atom $\ell_o=1.13\times 10^{-26}{\rm
erg~s}^{-1}$.
The results for $\ell_c$ shown in the table are significantly lower than the
average value of $2\times 10^{-26}{\rm erg~s}^{-1}{\rm H~atom}^{-1}$ found
for low-velocity clouds by Lehner et al. (2004), who
measured the column densities of C~II$^*$ from spectra recorded by {\it
FUSE}. [Lehner et al. (2003) obtained similar results
for sight lines toward WD stars within or just outside the Local
Bubble.] Lehner et al. (2004) compared their results
with measurements of H$\alpha$ emission in the same directions, and they
concluded that about half of the C~II$^*$ that they detected came from fully
ionized gas (but with large variations from one sight line to the next).
This inference was supported by the fact that their values of $\ell_c$ were
slightly lower for sight lines with large values of $N$(H~I). In principle,
one can gain an insight on the relative importance of H~II regions by
comparing the observed abundances of C$^{+*}$ to those of N$^{+*}$, since the
former can come from both neutral and ionized regions, while the latter
arises only from ionized regions. Gry et al. (1992)
compared these two species in a comprehensive study of both the absorption
lines observed by the {\it Copernicus\/} satellite (Rogerson et al. 1973) and
the 158 and 205$\mu$m emission lines observed by the {\it COBE\/}
satellite (Wright et al. 1991). They concluded that H~II regions were
responsible for a major portion of the C$^{+*}$ that was observed, but this
result is clearly dependent on the assumed ratio of atomic C to N in the H~II
regions.
An important advantage of the $\ell_c$ determinations synthesized from our
results for [Ar~I/O~I] is that they apply {\it only\/} to the WNM; hence they
give more accurate indications of the carbon cooling rates within this medium
without any contamination from H~II regions. They do, of course, rely on the
value of assumed relative abundances of C and H in the gas phase.
\subsection{Heating Rates}\label{heating_rates}
\subsubsection{Thermal Time Constants}\label{thermal_time_constants}
The cooling time for a medium with $T=7000\,$K and thermal pressure
$nT=3800\,{\rm cm}^{-3}$K (Jenkins \& Tripp 2011) is about
4.1$\,$Myr [ (Wolfire et al. 2003); see their Eq.~(4)]. However, when
the medium is impulsively heated and ionized by the EUV and X-ray
illuminations from a SNR, the temperature can approach or exceed $10^4\,$K,
and the onset of L$\alpha$ cooling creates a dramatic increase in the overall
cooling rate (Dalgarno \& McCray 1972). When this happens, the
thermal relaxation timescale becomes much shorter than the mean interval
between the bursts of radiation, each of which last only about one to a few
times $10^5\,$yr. Thus, while we can compute a time-averaged heating rate
for the SNR illuminations that might explain our high levels of partial
ionization, we have no reason to expect that this average should be balanced
by the cooling rates that would apply for the medium at $T=7000\,$K.
\subsubsection{Secondary Electrons}\label{secondary_heating}
The same energetic electrons that are responsible for the secondary
ionizations $\Gamma_s$ and $\Gamma_{s^\prime}$ can also heat the gas through
collisions with other electrons. As with the calculations of the efficiency
of secondary ionizations described in Appendix~\ref{gamma_s}, we use the
analytic approximations of Ricotti et al. (2002) for
the heating efficiencies, based on the numerical results for various
conditions obtained by Shull \& Van Steenberg (1985). Aside from replacing
ionization efficiencies with heating efficiencies and multiplying by the
energies of the respective secondary electrons, the calculations here are
virtually the same as for the ionization rates.
Table~\ref{heating_rates_table} shows the outcomes for the evaluations of
secondary electron heating rates. It is no surprise that there is a
substantial increase in the heating rates when we advance from the ionization
created by known sources to either of the two hypothetical enhanced
ionization examples that could explain the electron fractions indicated by
the observations of [Ar~I/O~I].
Our high result ($1.2\times 10^{-25}{\rm erg~s}^{-1}{\rm H~atom}^{-1}$) for
the heating in the regime where there is increased penetration of X-rays down
to 60$\,$eV creates a serious problem for this model, since this steady-state
rate is considerably larger than the corresponding cooling rate
$\ell_c+\ell_o=1.98\times 10^{-26}\,{\rm erg~s}^{-1}{\rm H~atom}^{-1}$ given
in Section~\ref{cooling_rate}.
\subsubsection{Cosmic Ray Heating of Electrons}\label{CRheating_elec}
In addition to ionizing the gas and creating secondary electrons that can
heat the gas, cosmic rays can also interact with free electrons in the medium
and heat them. The heating rate per unit volume is approximately equal to
$A\zeta_{\rm CR}({\rm H}^0)n(e)$, where $A\approx 4.6\times 10^{-10}{\rm
erg}$
(Draine 2011, p. 338). With the electron densities listed in
Table~\ref{properties_table} and $\zeta_{\rm CR}=1.25\times 10^{-16}{\rm
s}^{-1}{\rm H~atom}^{-1}$, we find that for the three different electron
densities in this table the heating rates are $1.0\times 10^{-27}$ (for no
SNR contribution), $1.9\times 10^{-27}$ (with SNR contributions), and
$3.0\times 10^{-27}{\rm erg~cm}^{-3}{\rm s}^{-1}$ (for the lower energy
penetration example). After dividing these numbers by values of $n({\rm
H}^0)$ given in Table~\ref{properties_table}, we find that these rates are
about 0.4 to 1.2 times the rate from secondary electrons liberated by the
cosmic rays, and 0.05 to 0.5 times the respective heating rates from the
secondary electrons generated by ionizations from the X-ray backgrounds.
\begin{deluxetable}{
l
c
}
\tablewidth{0pt}
\tablecaption{Secondary Electron Heating
Rates\tablenotemark{a}\label{heating_rates_table}}
\tablehead{
\colhead{} & \colhead{Rate}\\
\colhead{Source\tablenotemark{b}} & \colhead{($10^{-27}{\rm erg~s}^{-1}\,{\rm
H~atom}^{-1}$)}
}
\startdata
X-ray background (no SNR)\dotfill&4.1\\
X-ray background (plus SNR)\dotfill&50\\
X-ray background (down to 60$\,$eV)\dotfill&110\\
Main-sequence stars\dotfill&1.5\tablenotemark{c}\\
Active binaries\dotfill&0.063\tablenotemark{c}\\
WDs\dotfill&0.84\tablenotemark{c}\\
Cosmic rays\dotfill&4.9\tablenotemark{c}\\
He$^+$ recomb.\tablenotemark{d} (background with no SNR)\dotfill&0.059\\
He$^+$ recomb.\tablenotemark{d} (background plus SNR)\dotfill&1.2\\
He$^+$ recomb.\tablenotemark{d} (background down to 60$\,$eV)\dotfill&8.8\\
\enddata
\tablenotetext{a}{Heating by secondary electrons that are liberated by the
ionizations of H$^0$, He$^0$ and He$^+$, expressed in terms of an energy
dissipation rate per neutral H atom. Direct interactions of cosmic rays with
free electrons produce an additional heating which is discussed in
Section~\protect\ref{CRheating_elec}.}
\tablenotetext{b}{The conditions for the top and bottom three rows correspond to those
for the last three columns in Table~\protect\ref{properties_table}.}
\tablenotetext{c}{To first order, the heating rates from internal sources and
cosmic rays should not change as the strength of the background radiation
increases above the basic rate from known sources. In
reality, they increase by modest amounts ($\sim$30\%) when the values of
$x_e$ (which drive the heating efficiency) increase.}
\tablenotetext{d}{From secondary electrons that are produced by
$\Gamma_{He^+}({\rm He}^0)$ and $\Gamma_{He^+}({\rm H}^0)$. Other
ionizations arising from helium recombinations do not produce electrons with
sufficient energy to cause any appreciable heating.}
\end{deluxetable}
\begin{deluxetable}{
l
c
c
c
c
c
c
c
c
c
c
c
}
\tablecolumns{12}
\tablewidth{0pt}
\tablecaption{Dust Grain Heating and Cooling
Rates\tablenotemark{a}\label{grain_rates_table}}
\tablehead{
\colhead{} & \multicolumn{3}{c}{Without SNR} & \colhead{} &
\multicolumn{3}{c}{With SNR} & \colhead{} & \multicolumn{3}{c}{Greater Low}\\
\colhead{$b_c$\tablenotemark{b}} &
\multicolumn{3}{c}{Contrib.\tablenotemark{c}} & \colhead{} &
\multicolumn{3}{c}{Contrib.\tablenotemark{c}} & \colhead{} &
\multicolumn{3}{c}{Energy Penetration\tablenotemark{c}}\\
\cline{2-4} \cline{6-8} \cline{10-12}
\colhead{$\times 10^5$} & \colhead{H} & \colhead{C} & \colhead{$\Delta$} &
\colhead{} & \colhead{H} & \colhead{C} & \colhead{$\Delta$} & \colhead{} &
\colhead{H} & \colhead{C} & \colhead{$\Delta$}\\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{} &
\colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{} & \colhead{(8)} &
\colhead{(9)} & \colhead{(10)}
}
\startdata
0.0&9.9&5.1&4.8&~&16.1&7.5&8.5&~&22.0&10.0&12.0\\
2.0&15.1&7.3&7.8&~&23.2&11.0&12.1&~&30.7&14.9&15.8\\
4.0&24.4&10.6&13.9&~&36.4&16.0&20.4&~&46.9&21.8&25.1\\
6.0&31.6&13.8&17.8&~&47.4&20.9&26.5&~&61.3&28.5&32.9\\
\enddata
\tablenotetext{a}{Expressed in units of $10^{-27}{\rm erg~s}^{-1}\,{\rm
H~atom}^{-1}$. H~=~heating, C~=~cooling, and $\Delta={\rm H-C}$ (i.e., net
heating rate). The values were computed for the average interstellar
radiation field (ISRF), as defined by Eq.~31 and Table~1 of Weingartner \&
Draine (2001b) (with a radiation intensity
$G=1.13$). The adopted value of $R_V\equiv A_V/E(B-V)=3.1$. The values for
the heating and cooling rates per H atom tabulated here were computed from
Eq.~44 (for grain heating) and Eq.~45 (for grain cooling) of Weingartner \&
Draine (2001b) (but without the $n({\rm H})$
factor) and the coefficients listed in their Tables~2 and 3.}
\tablenotetext{b}{The abundance of carbon, relative to hydrogen, in the
grains. A high value of $b_c$ implies a population of dust grains that is
rich in polycyclic aromatic hydrocarbons (PAHs).}
\tablenotetext{c}{The same conditions as for Columns (2)-(4) in
Table~\protect\ref{properties_table}.}
\end{deluxetable}
\subsection{Dust Grains: Photoelectric Heating and Recombination
Cooling}\label{photoelectric_grain_heating}
When dust grains are illuminated by starlight, they emit photoelectrons,
which can heat the medium (Watson 1972; Draine 1978; Pottasch et al. 1979;
Bakes \& Tielens 1994; Weingartner \& Draine 2001b). This heating is
partly offset by collisional cooling via grain-ion recombination (Draine \&
Sutin 1987). The efficiencies of these mechanisms are regulated by the
charge of the grains, which in turn depends on $n(e)$, $T$ and $G$ (the
density of starlight). Table~\ref{grain_rates_table} shows our
calculations of grain heating and cooling for the three cases (no SNR, with
SNR, and lower energy X-ray penetration) using the fitting formulae and
coefficients given by Weingartner \& Draine (2001b); see note~$a$ of the
table for details.
\placetable{grain_rates_table}
It is generally regarded that the photoelectric heating from grains is
greater than that from cosmic rays by about one order of magnitude
(Draine 2011, p. 339) [or considerably more than this if the assumed value
for $\zeta_{\rm CR}({\rm H}^0)$ is lower than that adopted here, e.g.,
Wolfire et al. (1995, 2003)].
Any enhancement in $x_e$ tends to increase the heating rate from grains,
since the grains will be more negatively charged. However, this increase is
not as strong as the effect of a greater electron fraction on the direct
cosmic ray heating, so the disparity between the two rates is decreased.
\section{SUMMARY AND CONCLUSIONS}\label{summary}
We have developed a means for deriving the representative rates of
ionization, and thus the resulting electron densities, along sight lines that
penetrate the WNM and that extend out to several
hundred pc from us, well beyond the edge of the Local Bubble. Our method
makes use of the fact that when a mostly neutral medium is exposed to the
ambient EUV and soft X-ray ionizing radiation, the argon atoms are far more
susceptible to being ionized than hydrogen atoms. Thus, by comparing the
abundances of Ar~I to those of H~I, we gain an understanding of the strength
of the photoionization and secondary processes related to it. For a number
of practical reasons, we find that it is desirable to use O~I as a proxy for
H~I. The partial ionization of oxygen is strongly coupled to that of
hydrogen through a rapid charge exchange process. Using this strategy, we
compare the column densities of Ar~I and O~I derived from absorption lines
seen in the {\it FUSE\/} spectra of 44 hot subdwarf stars.
Since the neutral forms of argon and oxygen are virtually absent in regions
that are fully ionized (either the prominent H~II regions around hot stars or
the much lower density but more pervasive WIM), our
probes sample only regions that have appreciable concentrations of H~I. This
sets our measurements apart from conventional determinations of average
electron densities (e.g., pulsar dispersion measures, H$\alpha$ intensities,
C~II fine-structure excitation, etc.), which are strongly influenced by
contributions from the fully ionized regions.
We find that, on average, the abundance of neutral argon, relative to that of
neutral oxygen, is $[{\rm Ar~I/O~I}]=-0.427\pm 0.11\,$dex below what we would
expect if both species had no partial ionization and the solar abundance
ratio is a proper standard of comparison. We interpret this deficiency in
terms of the greater susceptibility of argon to photoionization. After
accounting for a broad range of processes that can modify the fractional
ionizations, we conclude that with known sources of ionizing radiation, both
external and internal, the only straightforward way to reconcile the large
deficiency of Ar~I is to propose that the neutral medium has a characteristic
density of only $0.09\,{\rm cm}^{-3}$, which is far below a generally
accepted value of about $0.5\,{\rm cm}^{-3}$. At this latter, higher volume
density of hydrogen, known sources of ionization should produce an electron
density $n(e)=0.018\,{\rm cm}^{-3}$ and create a result $[{\rm
Ar~I/O~I}]=-0.14$, which is greater than all but a small fraction of the
measurements and well above the overall average value.
In order to achieve a result for [Ar~I/O~I] that is consistent with both our
observations and a density $n({\rm H~I})=0.5\,{\rm cm}^{-3}$, we must propose
nonconventional sources of additional ionization. We discuss two such
possibilities; in reality we might be witnessing the outcome of some
combination of both of them working together.
The first explanation is that the shielding of the external, low energy X-ray
flux is less effective than expected: if we lower the cutoff energy from a
nominally expected 90$\,$eV to only 60$\,$eV, we can gain sufficient
additional ionizing photons from the external background to create a level of
ionization that is consistent with an upper bound for the value implied by
our observed [Ar~I/O~I]. To explain this lower shielding, one might envision
a nearly sponge-like topology for the WNM, where the internal holes and
channels are filled with hot, X-ray emitting gas. Indeed, de Avillez et al.
(2012) have performed hydrodynamical simulations
of the effects from random SN explosions and stellar winds in the
plane of the Galaxy, and the outcome of their model exhibits a complex,
turbulent entanglement of hot, warm and cold gas complexes. Their
simulation, which includes non-equilibrium ionization calculations, yields an
average electron density $n(e)=0.04\,{\rm cm}^{-3}$, a value that is not far
removed from our determination for the WNM. This near match may be
coincidental, however. In the simulation, voids containing hot,
collisionally ionized gas with very low $n(e)$, in combination with dense
regions that are photoionized, may produce an outcome that is close to the
value for the WNM.
A serious shortcoming of the interpretation that we are viewing a
steady-state maintenance of a high level of ionization in the WNM is that the
calculated heating rate of $1.2\times 10^{-25}\,{\rm erg~s}^{-1}\,{\rm
H~atom}^{-1}$ from secondary electrons is unacceptably high -- much higher
than the cooling rate from atomic fine-structure excitations or
recombinations of ions onto dust grains.
Our second explanation is a time-dependent solution that can sidestep the
problem of overheating in the equilibrium case discussed above. Over a
period of one to a few times $10^5\,$yr, an SNR can create a
burst of ionizing radiation in the X-ray region that can briefly elevate the
ionization of the ISM to levels well above normal out to distances of order a
few hundred pc away. Initially, the recombination timescale is quite short,
but then it advances to about 1$\,$Myr when the level of ionization
approaches our observed average state. For heating by secondary
electrons, the temperature should exhibit a similar rapid adjustment that is
followed by a much slower trend with time. The radiation burst will indeed
create a very high heating rate, but as the gas temperature approaches or
exceeds $10^4\,$K, the abrupt onset of the very strong L$\alpha$ cooling will
dump most of the heat over a time scale that is considerably shorter than the
cooling time ($\sim 4\,$Myr) for the gas in its usual state. A good
exposition of this quick thermal recovery to $T<10^4\,$K and a comparatively
more gentle relaxation in the heat loss and recombinations can be seen in
plots of $T$ and $x_e$ vs. time in Fig.~2 of Gerola et al. (1974).
We propose that the ionization imprint of an SN explosion on the WNM
can last well beyond the time that the remnant is recognizable in X-rays. We
have good evidence that a number of SNe have exploded in our general
vicinity over the past $1-10\,$Myr. As a consequence, if one considers that
the activity in our neighborhood has recently been higher than normal for our
Galaxy, the WNM out to several hundred pc from the Sun may be ionized to a
level that is somewhat higher than the low-density neutral gas at similar
galactocentric distances elsewhere in the Galactic plane.
\acknowledgments
This research was supported by a NASA Astrophysics Data Processing grant No.
NNX10AD44G to Princeton University. All of the data presented in this paper
were obtained from the Mikulski Archive for Space Telescopes (MAST) at the
Space Telescope Science Institute (STScI). STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. Support for MAST for non-{\it HST\/} data is provided by the
NASA Office of Space Science via grant NNX09AF08G and by other grants and
contracts. The author acknowledges many useful discussions with B.~T.~Draine.
Drs. Draine and C.~Gry supplied suggestions for improvement after reading a
draft of this paper. Some of the conclusions presented here relied on the
use of the CHIANTI database and software, which is a collaborative project
involving the NRL (USA), the Universities of Florence (Italy) and Cambridge
(UK), and George Mason University (USA); other conclusions made use of the
T\"ubingen Model Atmosphere Fluxes within the framework of the German
Astrophysical Virtual Observatory (GAVO). The coordinates and apparent
magnitudes displayed in Table~\ref{aro_results} were provided by the
SIMBAD database, operated at CDS, Strasbourg, France.
{\it Facility:\/} \facility{FUSE}
| {'timestamp': '2013-01-16T02:00:12', 'yymm': '1301', 'arxiv_id': '1301.3144', 'language': 'en', 'url': 'https://arxiv.org/abs/1301.3144'} |
\section{Introduction}
Every quantitative analysis of the primordial black hole (PBH) number
and mass spectrum \cite{carr75}
requires knowledge of the threshold parameter, $\delta_{\rm c}$, separating
perturbations that form
black holes from those that do not, and the resulting black hole mass,
$M_{\rm bh}$, as a function of distance from the threshold. In order to
determine $\delta_{\rm c}$ and $M_{\rm bh}$ for various initial conditions,
we performed one-dimensional,
general relativistic simulations of the hydrodynamics of PBH
formation in the radiation-dominated phase of the early
universe. Three families of perturbation shapes were
chosen to represent ``generic'' classes of initial data, reflecting the lack of
information about the specific shape of primordial fluctuations. The
numerical technique is sketched
in Section \ref{numerics}. Defined as the excess mass within the
horizon sphere at the onset of the collapse, we find $\delta_{\rm c} \approx 0.7$
for all three perturbation shapes, indicating that the threshold value
may indeed be universal (Section \ref{hydro}). A numerical confirmation of the
previously suggested power-law scaling of $M_{\rm bh}$ with
$\delta - \delta_{\rm c}$ \cite{niejed97}, related to the well-known behavior of
collapsing space--times at the critical point of black hole formation
\cite{chop93}, is presented in Section \ref{scaling}. In this framework,
the PBH mass spectrum is determined by the dimensionless coefficient,
$K$, the scaling exponent, $\gamma$, and the initial horizon mass,
$M_{\rm h}$, such that
\begin{equation}
\label{scale}
M_{\rm bh} = K M_{\rm h} (\delta - \delta_{\rm c})^\gamma\,\,.
\end{equation}
We provide numerical results for $K$ and $\gamma$ for
the three perturbation families. A more detailed description of the
numerical technique and further results will be presented in a
forthcoming publication \cite{niejed98}.
\section{Numerical technique}
\label{numerics}
The dynamics of collapsing density perturbations in the Early Universe
are determined by the general relativistic
equations of motion for a perfect fluid, the field equations, the first
law of thermodynamics, and a radiation-dominated equation of state. The
assumption of spherical symmetry is well justified for large
fluctuations in a Gaussian distribution \cite{barea86}, reducing the
problem to one spatial dimension.
For our simulations, we chose the formulation of the hydrodynamical
equations by Hernandez and Misner \cite{hermis66} as implemented by
Baumgarte et al.~ \cite{baum95}. Based on the original equations by
Misner and Sharp
\cite{missha64}, Hernandez and Misner proposed to exchange the
Schwarzschild time variable, $t$, with the outgoing null coordinate,
$u$. In so doing, the hydrodynamical equations retain the Lagrangian
character of the Misner--Sharp equations but avoid crossing into the
event horizon of a black hole once it has formed. Covering the entire
space--time outside while asymptotically approaching the event horizon,
the Hernandez--Misner equations are perfectly suited to
follow the evolution of a black hole for long times after its
formation without encountering coordinate singularities. This allowed
us, in principle, to study the
accretion of material onto newly formed PBHs for arbitrarily long times (in
contrast with earlier calculations \cite{Nade}).
Since the expanding outer regions of our simulated piece of the
universe are most conveniently tracked in a comoving numerical
reference frame, the Lagrangian form of the Hernandez--Misner equations
is their second major asset. It also provides a simple prescription
for the outer boundary condition, which is defined to match the exact
solution of the Friedmann equations for a radiation
dominated flat universe. Hence, the pressure follows the
analytic solution
\begin{equation}
P = P_0 \left(\frac{\tau}{\tau_0}\right)^{-2}\,\,,
\end{equation}
where $\tau$ is the proper time of the outermost fluid element
(corresponding to the cosmological time $t$ in a
Friedmann--Robertson--Walker (FRW) universe) and
$P_0$ and $\tau_0$ are the initial values for pressure and proper
time.
\section{Threshold for black hole formation}
\label{hydro}
We studied the spherically symmetric evolution of three families of curvature
perturbations. Initial conditions were chosen to be
perturbations of the energy density, $\epsilon = \rho_0 e$, in
unperturbed Hubble flow specified
at horizon crossing. The first family of perturbations is described by
a Gaussian-shaped overdensity that asymptotically approaches the FRW
solution at large radii. The other two families of initial conditions
involve a spherical
Mexican Hat function and a fourth order polynomial. These functions
are characterized by
rarefaction regions outside of the horizon radius, $R_{\rm h}$, that
identically compensate
for the additional mass of the overdensities inside the horizon
volume, so that the mass derived from the total integrated
density profile is equal to that of an unperturbed FRW solution.
In our numerical experiments, the amplitude, $A$, of the perturbations
is used to tune the
initial conditions to sub- or supercriticality with respect to black
hole formation. The critical amplitude, $A_{\rm c}$, is
strongly shape-dependent, varying between $A_{\rm c} = 3.04$ for
Mexican-Hat-shaped perturbations and $A_{\rm c} = 2.05$ for the
Gaussian curve. If, however, we define the control
parameter $\delta$ as the additional mass
inside $R_{\rm h}$ in units of the horizon mass, we find strikingly similar
values --- $\delta_{\rm c} = 0.67$ (Mexican Hat), $\delta_{\rm c} = 0.70$ (Gaussian curve), and
$\delta_{\rm c} = 0.71$ (polynomial) --- for all three families of initial data in our
study. This number is considerably greater than the previously
employed threshold $\delta_{\rm c} = 1/3$
following from analytical estimates \cite{carr75}. Given
the qualitative difference of the functional forms of the different
perturbation families, the result that $\delta_{\rm c}$ of the Gaussian
perturbation lies in between
the critical values of the mass compensated functions is
surprising. Our results suggest that $\delta_{\rm c} \approx 0.7$ (with $\delta$
defined as above) is a universal statement, i.e., true for all
perturbation shapes in the radiation-dominated regime. This remains to
be verified by means of additional experiments.
\section{Scaling of PBH masses with distance to the threshold}
\label{scaling}
For a variety of matter models, it is well-known that the dynamics of
near-critical collapse
exhibit continuous or discrete self-similarity and power-law scaling
of the black hole mass with the offset from the critical point
\{Eq.~(\ref{scale}) \cite{chop93,critrev}\}. In particular, Evans
and Coleman \cite{evans94} found
self-similarity and mass scaling in numerical experiments of a
collapsing radiation fluid. They numerically determined the scaling exponent
$\gamma \approx 0.36$, followed by a linear perturbation analysis of
the critical solution by Koike et al.~ \cite{koike95} that yielded $\gamma \approx
0.3558$. Until recently, it was believed that entering the scaling regime
requires a degree of fine-tuning of the initial data
that is unnatural for any astrophysical application. It was noted
\cite{niejed97} that fine-tuning to criticality
occurs naturally in the case of PBHs forming from a steeply declining
distribution of primordial density fluctuations, as generically predicted by
inflationary scenarios. In the radiation-dominated
cosmological epoch, the only difference with the fluid collapse
studied by Evans and Coleman \cite{evans94} is the expanding,
finite-density-background space--time of a FRW universe. Assuming that
self-similarity and mass scaling are consequences of an intermediate
asymptotic solution that is independent of the asymptotic boundary
conditions, Eq.~(\ref{scale}) is applicable to PBH masses,
allowing the derivation of a universal PBH initial mass function
\cite{niejed97}.
\begin{figure}
\epsfysize=8cm
\epsfbox{scaling.eps}
\vspace{-13mm}
\caption{\label{f7} Black hole masses as a function of $\delta - \delta_{\rm c}$
for three different
perturbation shape families. The best fit parameters to equation
(\ref{scale}) are:
$\gamma = 0.36$, $K = 2.85$, $\delta_{\rm c} = 0.6745$ (Mexican Hat
perturbation, triangles); $\gamma = 0.37$, $K = 2.39$, $\delta_{\rm c} = 0.7122$
(polynomial perturbation, crosses); $\gamma = 0.34$, $K = 11.9$,
$\delta_{\rm c} = 0.7015$ (Gaussian curve perturbation, diamonds).}
\end{figure}
Figure (\ref{f7}) presents numerical evidence for mass scaling
according to Eq.~(\ref{scale}) in black hole collapse in an
asymptotic FRW space--time. All three perturbation families give rise
to scaling solutions with a scaling exponent $\gamma
\approx 0.36$.
\section{Conclusions}
This work discusses numerical collapse simulations of three
generic families of energy density perturbations, one with a finite
total excess mass
with respect to the unperturbed FRW solution and two mass compensated
ones. Among various possible definitions for the collapse control parameter
$\delta$, the total excess
gravitational mass of the perturbed space--time with respect to the
unperturbed FRW background enclosed in the initial horizon volume is
the only one that gives rise to a
similar threshold value for all three shape families, $\delta_{\rm c} \approx
0.7$. Whether this result is an indication for universality of $\delta_{\rm c}$
in this specific definition needs to be verified with the help of
additional simulations using a larger sample of initial perturbation
shapes.
The previously suggested \cite{niejed97} scaling relation between
$M_{\rm bh}$ and $\delta - \delta_{\rm c}$, based on the analogy with critical
phenomena observed in near-critical black hole collapse in
asymptotically non-expanding space--times \cite{chop93}, is confirmed
numerically for an asymptotic FRW background. For the smallest black
holes in this investigation, the scaling exponent is $\gamma \approx
0.36$, which is identical to the non-expanding numerical and
analytical results \cite{evans94,koike95} within our numerical accuracy. The
parameter $K$ of Eq.~(\ref{scale}), needed in addition to
$\gamma$ to evaluate the two-parameter PBH
IMF derived in \cite{niejed97}, ranges from $K \approx 2.4$ to $K
\approx 12$.
The author wishes to thank K. Jedamzik for a fruitful collaboration,
T. Baumgarte for providing the original version of
the hydrodynamical code, and Joan George for valuable stylistic corrections.
| {'timestamp': '1998-06-02T23:53:40', 'yymm': '9806', 'arxiv_id': 'astro-ph/9806043', 'language': 'en', 'url': 'https://arxiv.org/abs/astro-ph/9806043'} |
\section{Introduction}
$\quad$Cosmic muons are the
important contributors to background processes when
search for the conversion of a muon to an electron\cite{come,mu2e}. Cosmic
ray veto geometry surrounding the detectors and stopping target should
be carefully eliminated this background. Passive and active shielding
should provide background of $\sim$0.1 events in signal window for 3 year run of
experiments. Scintillator-based active shielding will consist of four layers.
The scintillator strips will read out by multiclad wavelength
shifting (WLS) fibers connected to photodetectors. Most promising devices
based on new technologies used as photodetectors are the silicon photomultipliers
(SiPM)\cite{golo}. Photon detection efficiency of SiPM is dependent on wavelength of
the photon, temperature, dark count\cite{yang} etc., and very difficult to be
measured and even more difficult to model\cite{kjha}. For this reason, we are
limited only to the study of the light propagation and collection.
Even for
this purpose, a Monte Carlo simulation can adequately predict the
experimental results only if the detector parameters are
sufficiently close to their true values. Some of parameter, e.g. surface
boundaries descriptions, can be tuned by using measurements for
particular scintillator strip configurations.
\input{texfig1.tex}
\section{ Some features of the plastic scintillator strip modeling }
$\quad$The Monte Carlo
simulation with all possible processes play a crucial
role in the feasibility study of the proposed detector module and in identifying
detector parameter values. Low-energy optical photons (photons with a
wavelength much greater than the typical atomic spacing) undergo the
following processes: bulk absorption, Rayleigh scattering, reflection
and refraction at medium boundaries, and wavelength shifting.
\input{texfig2.tex}
\quad The boundary processes on all scintillator play an important role in tracing
photons in strips. Compared to them, photon self-absorption in
scintillator is less significant\cite{knoll}. In Geant4.10.06 \cite{gea4}
simulation we combined
the polished scintillator surface finishes with the backpainted wrapping
option which represents diffuse (Lambertian) reflection. In this
simulation we use the UNIFIED model for the processes between two
dielectric materials.
\input{texfig31.tex}
\section{Simulation and results}
$\quad$ The peak of emission light of a plastic scintillator
(e.g., Saint Gobain BC400 series) does not matches the peak sensitivity of used
photodetectors. To solve this problem it is necessary to use the WLS fiber to
transfer light to the photodetector. In this simulation the fiber are multiclad
consisting of a scintillating core surrounded by an acrylic inner cladding
\input{texfig3.tex}
and an outer cladding which made of a fluor-acrylic material (similar
to the Kuraray double clad fibers of type Y11(200)\cite{kura}).
It was assumed that in a scintillator strip a mean value of 10000
scintillation photons per MeV of deposited energy were emitted.
For this scintillator, the maximum emission is at a wavelength of 431\,nm
and refractive index is 1.58.
\quad For WLS fiber attenuation length of 500\,cm for its own radiation, and for
plastic scintillator attenuation length of 300\,cm are assumed.
The total diameter
of fiber is 1.2\,mm. The total thickness of cladding structure is 6\%
of the diameter of a fiber. In this simulation the strip contains one or two
co-extruded grooves with 3\,mm depth and 1.3\,mm width
for insertion of the WLS fiber. The selected strip and fiber parameters
are close to those used in the test-beam measurements at JINR (Dubna, Russia).
\input{texfig9.tex}
\quad This simulation was performed using Geant4 for plastic
scintillator with the dimension 4*1*300\,cm${}^3$ and co-extruded TiO$_{2}$
white diffuse reflective (R=98$\%$) coating.
The strip contain one at the center or two grooves at a
distance of 2\,cm from each other along the entire length of scintillator strip.
We collect photons from a
WLS fiber at one of the strip ends (hereinafter referred to as
photodetector side). On the photodetector side at the fiber end
the photons are fully absorbed. The opposite ends of the fibers are
blackened.
\input{texfig4.tex}
\quad The $^{90}Sr$ source was simulated in the Geant4 framework.
The source provides an electron flux in a wide energy range
up to 2.3\,MeV (see Fig.1). The radiation source was enclosed in a shell with
a lead collimator. The diameter of the collimator outlet was 1\,mm.
The source was located at a distance of 2\,mm above or below the scintillator strip.
\quad Cosmic muons were generated according to\cite{volk} in the range
0.3-5000\,GeV. The zenith angle and energy distribution for simulated cosmic
muons are displayed in Figure\,2.
\input{texfig5.tex}
\input{texfig6.tex}
\quad In Figure\,3, we show the distribution of the energy deposited in the scintillation
strip when the middle of the side 4*300\,cm${}^{2}$ is irradiated with a ${}^{90}Sr$ source.
\quad Figure\,4 shows the light intensity distribution in the end of a fiber
as seen by the photodetector side. This simulation study show that
the light intensity increases towards the edge of the fiber core. The mean
wavelength of light collected by the photodetector is 535\,nm.
\quad The distribution of the photon number at the photodetector side
when the middle of the strip side 4*300\,cm${}^{2}$ is
irradiated with a ${}^{90}Sr$ is shown in Figure\,5.
\quad Figure\,6 show the light yield when strip with one fiber irradiated
with a $^{90}Sr$ source which located over or under the strip at Z=0.0\,cm.
The strip is located at X=$\pm$\,2.0\,cm, Y=$\pm$\,0.5\,cm, Z=$\pm$\,150.0\,cm.
The first point is 0.5\,mm away from the side (X=19.5\,mm) and each step is
1\,mm (X-coordinate).
Points close to X=0.0\,mm are located at a distance of 0.5\,mm.
In this and the following Figures, 500 events were simulated for each point.
\quad In Figures\,7 and 8, the same thing is shown as in Figure\,6 but for
the case with two fibers in the strip.
Note that, in both cases (strip with one and two
fibers), the behaviors of the light yield when a radiation source is above and
below the strip differ from each other. But this difference is not significant.
\quad Figure\,9 shows the relation between the mean number of optical
photons detected at the photodetector side and the distance between point
of impact of electrons from the radiation source and photodetector side.
This graph is fitted by a function
\vskip 0.5 cm
$\qquad\qquad\quad N_{phot}(z) = A*e^{-z/\lambda_{1}} + B*e^{-z/\lambda_{2}}.$
\vskip 0.5 cm
Note that this formula was proposed by Kaiser et al.\cite{kais} for the case
when light is collected from the ends of the scintillator using a photomultipliers.
The first term is the transmission behavior for photons that travel directly
to the photodetector side.
\input{texfig7.tex}
The second term is the transmission behavior for photons that hit the detector
after a series of reflection on a scintillator surface.
The first point in the Figure is 75 mm away from the photodetector side
and each step is 75\,mm. The curve in the figure corresponds to the parameters
$\lambda_{1}$=43.5\,m and $\lambda_{2}$=1.75\,m with almost 100\% errors.
$\quad$To study the light attenuation when the strip surface is irradiated
by cosmic muons we retreated on each side of the surface by 1\,mm and
divided it into 40 equal parts. Each sector has been
uniformly irradiated by 500 muons with energies, azimuth and
zenith angles modeled accordingly to\cite{volk}. In Figure 10, we demonstrate
the light attenuation for this case. The points in the Figure are located in the
center of each of the 40 sections. For the given points, the results of one
exponential and double exponential fit are the same (blue curve in the Figure),
$\lambda_{1}=\lambda_{2}$=5.88\,m. The green curve in Figure corresponds
to the fit by the formula
\vskip 0.5 cm
$\qquad\qquad\quad N_{phot}(z) = A*e^{-z/\lambda} + B,$
\vskip 0.5 cm
where $\lambda=$2.32\,m.
\input{texfig8.tex}
\section{Conclusion}
\qquad In this note, we modelled the light output and attenuation in a scintilltion
strip with dimensions of 4*1*300\,cm${}^{3}$.
The simulated radiation source ${}^{90}Sr$ and cosmic muons were used as beam particles.
The scintillation
strip was irradiated both from the side of the embedded fibers and from
the opposite side along and across the strip.
Optical photons was collected
from one and two fibers embedded in the strip along the entire length. It
was shown that the attenuation of light depending on the distance to the
photodetector is described by a double exponential function.
$\quad$ We are sincerely grateful to Z.\,Tsamalaidze and
Yu.\,Davydov for initiating this work.
}
\small{
\bibliographystyle{plain}
| {'timestamp': '2020-07-09T02:10:07', 'yymm': '2007', 'arxiv_id': '2007.03921', 'language': 'en', 'url': 'https://arxiv.org/abs/2007.03921'} |
\section{Introduction}
\label{sec:intro}
Membrane remodeling is required for critical cellular processes including endocytosis, formation of multivesicular bodies, retrograde trafficking and exosome formation. Viruses and other pathogens also reshape cellular membranes during different stages of their lifecycles including entry into the host cell, formation of replication complexes, construction of assembly factories, and exit (also called egress or budding). Understanding the mechanisms of viral budding and the forces that drive this process would advance our fundamental understanding of viral lifecycles, and shed light on analogous cellular processes in which membrane remodeling and vesicle formation are essential for function.
In parallel, understanding fundamental determinants of budding and membrane dynamics would facilitate the design of viral nanoparticles. There is keen interest in reengineering enveloped viral nanoparticles to be used as targeted transport vehicles capable of crossing cell membranes through viral fusion \cite{Lundstrom2009,Cheng2013,Rowan2010,Petry2003,Rohovie2017}.
All viruses contain a capsid protein, which primarily functions to protect the viral genome during viral transmission. In enveloped viruses, the internal capsid is surrounded by a host-derived lipid bilayer and viral glycoproteins (GPs) embedded in this membrane. Enveloped viruses can be sub-divided into two groups based on their sequence of virion assembly and budding. For the first group (e.g. alphaviruses, hepatitis B, herpes) budding requires the assembly of a preformed nucleocapsid core (NC), which may be ordered or disordered depending on the virus. The core then binds to membrane-bound GPs and initiates budding \cite{Sundquist2012,Hurley2010,Welsch2007,Garoff1998}. For the second group, (e.g. influenza, type C retroviruses (HIV)) capsid assembly occurs concomitant with budding \cite{Welsch2007,Sundquist2012,Vennema1996}. The advantage of one assembly mechanism over another is not obvious; particle infectivity, morphology, and stability may all influence the preferred budding process.
The importance of preformed capsids in Alphavirus assembly is of particular interest because the presence of a capsid in the particle is not necessary for production of infectious particles. The traditional view is that alphaviruses follow the preassembled NC budding pathway \cite{Garoff1978,Garoff2004,Strauss1994,Wilkinson2005}, based on the observation of high concentrations of NCs in the cytoplasm \cite{Acheson1967}, and evidence that GP-GP and NC-GP interactions are required for virion formation \cite{Suomalainen1992,Lopez1994}. However, several studies have challenged this conclusion. In particular, Forsell et al.\xspace \cite{Forsell2000} reported successful assembly and budding of alphavirus despite mutations which inhibited NC assembly by impairing interactions between NC proteins, while Ruiz-Guillen et al.\xspace \cite{Ruiz-Guillen2016} observed budding of infectious alphavirus particles from cells which did not express the capsid gene. In both cases infectious particles were assembled and released or budded from the cell. These observations suggest that GP interactions may be sufficient for alphavirus budding. This begs the question: Why do enveloped viruses have internal nucleocapsid cores? Is there an advantage to having a NC during budding?
Molecular dynamics (MD) simulations can be a useful tool to bridge the gaps between the different steps of assembly that cannot be experimentally characterized. Computational studies have already provided insightful information about virus NC assembly \cite{Perlmutter2015}, as well as the interactions between proteins and lipid membranes \cite{Reynwar2007,Simunovic2013,Bradley2016}. Previous simulations on budding of nanoscale particles led to important insights but did not consider the effect of GPs (\cite{Smith2007,Vacha2011,Ruiz-Herrero2012,Deserno2002,Jiang2015,Li2010a, Li2010,Yang2011}), although budding directed by GP adsorption or capsid assembly have been the subject of continuum theoretical modeling \cite{Lerner1993,Tzlil2004,Zhang2008}. The formation of clathrin cages during vesicle secretion, a process which bears similarities to viral budding, has also been the subject of modeling studies \cite{Foret2008,Cordella2014, Cordella2015,Matthews2013}. Most closely related to our work are previous simulations on the assembly and budding of 12-subunit capsids, which found that membrane adsorption can lower entropic barriers to assembly \cite{Matthews2012,Matthews2013a} and that membrane microdomains can facilitate assembly and budding \cite{Ruiz-Herrero2015}. In contrast to these earlier works, we consider the presence of a nucleocapsid, a larger shell (80 trimer subunits), and a different subunit geometry. We find that these modifications lead to qualitatively different assembly pathways and outcomes in some parameter ranges.
In this article, we perform MD simulations on a coarse-grained model for GPs, the NC, and a lipid bilayer membrane to elucidate the forces driving enveloped virus budding. Our model is motivated by the alphavirus structure and experimental observations on alphavirus budding \cite{Forsell2000,Ruiz-Guillen2016,Garoff2004}, but we consider our results in the broader context of enveloped viruses. To evaluate the relative roles of a preassembled NC compared to the assembly of transmembrane glycoproteins in driving budding, we perform two sets of simulations. The first focuses entirely on glycoprotein-directed budding (Fig. \ref{fig:budding}a) by including only the membrane and model GPs, whose geometry and interactions drive formation of an icosahedral shell with the geometry of the alphavirus envelope. This model directly applies to experiments on budding from cells in which capsid assembly was eliminated \cite{Forsell2000,Ruiz-Guillen2016}. The second set of simulations includes model GPs and a preassembled NC, thus allowing for NC-directed budding (Fig. \ref{fig:budding}b).
We present phase diagrams describing how assembly morphologies depend on the strength of GP-GP and NC-GP interactions. The results demonstrate that the competition between the elastic energy of membrane deformations and deviations from preferred protein curvature can lead to polymorphic morphologies, and that templating by the NC can significantly decrease the resulting polydispersity. In the presence of a preassembled NC, there is a threshold strength of NC-GP interactions above which pathways transition from GP-directed to NC-directed budding. Our simulations enable visualization of the intermediates along each of these pathways as well as analysis of their relative timescales. In both pathways, assembly proceeds rapidly until budding is approximately 2/3 complete, after which curvature of the membrane at the neck of the bud imposes a barrier to subunit diffusion that significantly slows subsequent assembly and budding. We discuss possible implications of this slowdown for enveloped viruses such as HIV that bud with incompletely formed capsids.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=\columnwidth]{figures/scheme_mechanisms2.pdf}
\caption{ Mechanisms of enveloped virus budding on membranes: glycoprotein(GP)-directed budding and nucleocapsid(NC)-directed budding. }
\label{fig:budding}
\end{center}
\end{figure}
\begin{figure*}[hbt]
\begin{center}
\includegraphics[width=0.9\textwidth]{figures/scheme_model_capsid3.pdf}
\caption{{\bf a)} Cryoelectron microscopy (CryoEM) density distribution of Sindbis virus. This central cross-section shows the inner structure of alphaviruses, with the RNA molecule (purple) enclosed by the NC (orange, red) and the lipid membrane (green) with the transmembrane GPs (blue). [{\bf b-g}] Computational model of the alphavirus GPs and NC. {\bf b)} Comparison between the GP trimer as revealed by CryoEM (PDB ID: 3J0C \cite{Zhang2002}) and our coarse-grained model trimer. Trimers are modeled as rigid bodies comprising three cones, with each cone formed by 6 pseudoatoms of increasing diameter. {\bf c)} Each of the 4 inner pseudoatoms of a cone (green) interacts with its counterpart in a neighboring cone through a Morse potential with well-depth $\epsilon_{\text{gg}}$. All pseaudoatoms, including the `excluders' (red), interact through a repulsive Lennard-Jones potential. {\bf d)} Model glycoproteins are trapped in the membrane by `membrane-excluders' (purple) which interact with membrane pseudoatoms through a repulsive Lennard-Jones potential. {\bf e)} Complete trimer subunit embedded in the membrane. To aid visibility, in subsequent figures only the membrane excluders are shown. {\bf f)} Snapshot of a typical capsid assembled by model glycoproteins in the absence of a membrane, consisting of 80 trimer subunits. {\bf g)} Snapshot of typical capsid assembled by glycoproteins around the model nucleocapsid (NC, blue) in the absence of a membrane. The NC is modeled as a rigid spherical particle. NC pseaudoatoms interact with the lowermost pseudoatom in each GP cone through a Morse potential with depth $\epsilon_{\text{ng}}$.}
\label{fig:scheme}
\end{center}
\end{figure*}
\begin{figure*}[hbt]
\begin{center}
\includegraphics[width=0.8\textwidth]{figures/simulation_trajectoryd2_all.pdf}
\caption{ {\it (Top)} Typical simulation trajectory of GP-directed budding, at simulation times 600, 1800, 2700 and 4200$\tau_{0}$, from left to right, with GP-GP interaction strength $\epsilon_{\text{gg}}=2.3$. {\it (Bottom)} Simulation trajectory of NC-directed budding, at 600, 1800, 2700 and 5600 $\tau_{0}$, with $\epsilon_{\text{gg}}=2.5$ and NC-GP interaction strength $\epsilon_{\text{ng}}=3.5$. The second timepoint in each row corresponds to an example of the intermediate with a constricted neck described in the text. Except where noted otherwise, the membrane bending modulus $\kappa_{\text{mem}}\approx 14.5k_{\text{B}}T$ throughout the manuscript.}
\label{fig:trajectory}
\end{center}
\end{figure*}
\begin{figure}
\caption{{\bf (A): Simulation Movie 1.} Animation of a typical simulation showing GP-directed budding, for $\epsilon_{\text{gg}}=2.5$ and $\kappa_{\text{mem}}=14.5 k_{\text{B}}T$. Colors are as follows: GPs, magenta; membrane head groups, cyan; membrane tails, yellow. To show the membrane neck geometry more clearly the inactive subunits are rendered invisible in this animation. {\bf (B): Simulation Movie 2.} The same simulation trajectory as in {\bf (A)}, but rendered to show a central cross-section of the budding shell and the membrane. Inactive subunits are rendered brown in this animation. }
\label{fig:MovieGPdirected}
\end{figure}
\begin{figure}
\caption{{\bf (C): Simulation Movie 3.} Animation of a typical simulation showing NC-directed budding, for $\epsilon_{\text{gg}}=2.5$, $\epsilon_{\text{ng}}=3.5$, and $\kappa_{\text{mem}}=14.5 k_{\text{B}}T$. Colors are as in Simulation Movie 1, and the NC is colored blue. To show the membrane neck geometry more clearly we do not show inactive subunits in this animation. {\bf (D): Simulation Movie 4.} Animation showing a central cross-section of the NC-directed budding (same simulation trajectory as {\bf (C)}). Inactive subunits are shown in brown in this animation. }
\label{fig:MovieNCdirected}
\end{figure}
\section{Results}
\label{sec:results}
Although intact viruses can be simulated at atomistic or near-atomistic resolution \cite{Freddolino2006,Zhao2013,Reddy2015,Perilla2015,Huber2016}, the time scales for alphavirus assembly (ms-minutes) are prohibitive at such resolution. We thus consider a coarse-grained description for the viral GPs and the membrane, which enables tractable simulation of a large membrane over biologically relevant timescales while retaining the essential physical features of membranes and virus capsid and transmembrane proteins (see Fig. \ref{fig:scheme} and section \ref{sec:methods}). The membrane is represented by a solvent-free model which can be tuned to match properties of biological membranes while allowing simulation of large systems \cite{Cooke2005}. Our model GPs are designed to roughly match the triangular shape, dimensions and aspect ratio of Sindbis virus GP trimers ~\cite{Mukhopadhyay2006,Zhang2002}. They experience lateral interactions, which in the absence of a membrane drive assembly into capsids containing 80 subunits, consistent with the 80 trimers in the alphavirus glycoprotein shell. In our simulations the GPs are embedded within the membrane, where they freely tilt and diffuse but cannot escape on simulation timescales.
Motivated by the recent observation that alphavirus nucleocapsids do not require icosahedral symmetry \cite{Wang2015} to be infectious, we model the nucleocapsid as a rigid isotropic sphere. To account for experimental observations of capsid and GP conformational dynamics, the model GPs interconvert between assembly-inactive and assembly-active conformations which are respectively compatible or incompatible with assembly (see section~\ref{sec:methods}).
To compare GP-directed and NC-directed budding, we performed two sets of simulations that respectively included or did not include a NC. To understand how these pathways depend on parameters which can be controlled in experiments or varied under evolutionary pressures, we simulated assembly as a function of parameters controlling the GP-GP interaction strength $\epsilon_{\text{gg}}$, the NC-GP interaction strength $\epsilon_{\text{ng}}$ (when a NC is present), and the membrane bending modulus $\kappa_{\text{mem}}$. All energies are reported in units of the thermal energy, $k_{\text{B}}T$. For notational convenience, we refer to particles assembled from GPs only as GP-particles, and GPs assembled around the NC as GPNC-particles.
\subsection*{Budding includes an intermediate with a constricted neck}
We show snapshots from typical trajectories for simulations in the presence and absence of a NC in Fig. \ref{fig:trajectory}. In both cases, assembly and membrane deformation proceed rapidly until the GP shell is approximately 2/3 complete (the second timepoint in each row in Fig. \ref{fig:trajectory}), when the budding region is connected to the rest of the membrane by a narrow neck. Subunits within the neck experience restricted configurations due to the high membrane curvature. Thus, the neck acts as an entropic barrier that impedes subunit diffusion to the growing shell, causing the assembly rate to slow dramatically as the shell nears completion (Fig. \ref{fig:timescales}). The neck continues to narrow as additional GPs assemble until it becomes a tether connecting the bud and membrane. In this article we do not consider ESCRT or related scission-inducing proteins, and thus the bud separates from the membrane only when a large thermal fluctuation induces membrane fission leading to scission of the tether.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=\columnwidth]{figures/phasediagramGP.pdf}
\caption{ Predominant end-products for assembly without a NC as a function of GP-GP interaction strength, along with simulation snapshots that exemplify each class of end-product.The distribution of end-products for several representative values of $\epsilon_{\text{gg}}$ is shown in Fig. \ref{fig:distributionoutcomes}.}
\label{fig:phasediagramGP}
\end{center}
\end{figure}
\subsection*{Glycoprotein-directed budding leads to complete but polydisperse particles}
\label{sec:gpdriven}
We first consider assembly in the absence of a NC, so that budding is necessarily GP-directed. Fig. \ref{fig:phasediagramGP} shows the most frequent end-product obtained as a function of the GP-GP interaction strength. For weak interactions ($\epsilon_{\text{gg}}<1.4$) assembly is unfavorable. In contrast, bulk simulations (model GPs in the absence of a membrane) exhibit shell assembly for $\epsilon_{\text{gg}}>0.97$ at trimer concentration $\phi_{3D}=3.4 \cdot 10^{-5}\sigma^{-3}$ (see Fig. ~\ref{fig:capsidsbulk}a in the SI), well below the effective concentration of trimers on the membrane $\phi_{2D}^{eff}=0.001\sigma^{-3}$ \footnote{To compare the subunit concentration in bulk $\phi_{3D}=N_{s}/L^{3}$, where $N_{s}$ is the number of subunits and $L$ the box size, with the subunit concentration on the membrane, we measure the mean deviation of the height fluctuations of the subunits on the membrane, $\lambda\approx 4.5$, and the effective concentration on the membrane is then given by $\phi_{2D}^{eff}=N_{s} / L^2 \lambda$.} . This result demonstrates that the membrane rigidity can introduce a substantial barrier to assembly~\cite{Ruiz-Herrero2012}.
Within a narrow range of interaction strengths $1.4< \epsilon_{\text{gg}}<1.7$, assembly and budding stalls at the constricted neck intermediate described above. For these parameters, the intermediate remains upon extending the simulation length to 10,500$\tau_{0}$, suggesting that it corresponds to a true steady state or a very long-lived kinetic trap. This configuration resembles partially assembled states that were predicted theoretically \cite{Zhang2008,Foret2014}, but arises due to different physics. We find that the range of $\epsilon_{\text{gg}}$ over which the state arises depends on the subunit geometry, but the state exists for any geometry we considered (see section \ref{sec:subunit} in the SI). A similar configuration was observed during simulations of assembly and budding of a 12-subunit capsid on a membrane \cite{Ruiz-Herrero2015}, suggesting it is a generic feature of assembly and budding. However, in that work assembly never proceeded past the partially assembled state for any parameter set on a homogeneous membrane, possibly due to the to the small size of the simulated capsid.
For stronger interactions, we observe complete budding. However, the morphologies of the resulting GP-particles depend on the interaction strength in two ways. First, overly strong interactions ($\epsilon_{\text{gg}}>9.0$) drive rapid assembly which can proceed simultaneously along multiple fronts within a shell, leading to the formation of holey GP-particles (Fig. \ref{fig:phasediagramGP}). This result is consistent with holey capsids that assemble under strong interactions in bulk simulations \cite{Hagan2006,Rapaport2012}. Second, for moderate interactions ($1.6< \epsilon_{\text{gg}}<9.0$) the shells are complete, but their size depends on the interaction strength, with typical sizes ranging over 95--140 subunits. The origin of this polymorphism is discussed later in this section.
\subsection*{Nucleocapsid-directed budding leads to more monodisperse particles}
\label{sec:ncdriven}
The predominant end-products of assembly in the presence of a NC are shown in Fig.~\ref{fig:phasediagramNC} as a function of the two interaction parameters: GP-GP ($\epsilon_{\text{gg}}$) and NC-GP ($\epsilon_{\text{ng}}$).
We observe complete assembly and budding for $\epsilon_{\text{ng}} > 0.9$ and $1\le \epsilon_{\text{gg}} \le 6$ (Fig.~\ref{fig:phasediagramNC}, blue region), low to moderate GP-GP interactions. Compared to the GP-directed pathway, the presence of a NC allows assembly to occur at a lower $\epsilon_{\text{gg}}$ as evidenced by obtaining complete shells even for $\epsilon_{\text{gg}}$ as low as 0.9.
Outside of this range, several other end-products arise. For $\epsilon_{\text{ng}}<0.4$ (Fig.~\ref{fig:phasediagramNC}, pink region), the NC interactions are sufficiently weak that budding is entirely GP-directed (\textit{i.e.}\xspace GP shells assemble and bud, but not around the NC). In the range $0.4< \epsilon_{\text{ng}}<0.9$ (Fig.~\ref{fig:phasediagramNC}, brown region) we observe an intermediate regime in which the NC promotes nucleation but fails to act as a perfect template. The GP shell initially starts assembling on the NC surface, but eventually separates from the surface to form a larger shell due to the effect of membrane rigidity (see Section \ref{sec:polymorphism}). The result is an asymmetric shell that is partially attached to the NC; with a typical size of 95 subunits it is smaller than a GP-particle but considerably larger than the intrinsic preferred shell size. These two assembly outcomes (GP shells partially attached or unattached to the NC) demonstrate that the presence of a NC does not necessarily imply NC-directed budding; there is a minimum NC-GP interaction strength required for the NC to direct the assembly and budding pathway.
\begin{figure*}[hbt]
\begin{center}
\includegraphics[width=0.8\textwidth]{figures/phasediagram_NC.pdf}
\caption{ Predominant end-products of the NC-directed budding, as a funtion of the NC-subunit interaction $\epsilon_{\text{ng}}$ and the subunit-subunit interaction $\epsilon_{\text{gg}}$, and snapshots showing representative examples of each outcome. }
\label{fig:phasediagramNC}
\end{center}
\end{figure*}
Strong GP-GP interactions ($\epsilon_{\text{gg}}>6$, Fig.~\ref{fig:phasediagramNC}, upper grey region) lead to holey particles. This result can be explained as in the case of holey GP-particles described above; however, notice that the threshold value of $\epsilon_{\text{gg}}$ is smaller than in the absence of the NC ($\epsilon_{\text{gg}}=9$). Interestingly, we also observe holey GPNC-particles when strong NC-GP interactions are combined with weak GP-GP interactions ($\epsilon_{\text{gg}}<1.0$, Fig.~\ref{fig:phasediagramNC}, lower grey region). In this regime, NC uptake proceeds rapidly, but GP subunits do not associate quickly enough form a complete shell as budding proceeds.
To further elucidate the interplay between the two interactions, Fig.~\ref{fig:energy} (SI) compares the total energetic contributions from GP-GP and NC-GP interactions for budded GPNC-particles. We see that GP-GP interactions account for the majority of the attractive energy stabilizing the shell, with the NC-GP providing as little as 10-20\% of the total energy. These results highlight the delicate balance between GP-GP and NC-GP interactions required to obtain a well-formed GPNC-particle.
\subsection*{Membrane-induced polymorphism}
\label{sec:polymorphism}
Although both GP-only budding and NC-directed budding lead to the formation of complete particles, the morphology of budded shells significantly differs between both mechanisms. Fig. \ref{fig:bending}a shows the mean shell size as a function of interaction strength (averaged over all closed particles). For GP-directed budding, we see a strong dependence of particle size on subunit interactions: weak interactions lead to ovoid particles containing up to 140 trimers. As $\epsilon_{\text{gg}}$ increases, particles become smaller and more spherical, more closely resembling the shells that assemble in bulk simulations. We show snapshots of typical GP-shells assembled at weak and strong interactions. On the contrary, the size of GPNC-particles is nearly constant with $\epsilon_{\text{gg}}$ and only slightly larger (81-83 subunits) than the preferred size in bulk simulations.
{\textbf{A theoretical model for membrane-induced polymorphism.} Although shell assembly is necessarily out-of-equilibrium in finite-length simulations, we can understand the dependence of size on interaction strength from a simple equilibrium model that uses the Helfrich model \cite{Helfrich1973} to account for the elastic energy associated with membrane deformation (which has no spontaneous curvature and thus favors flat configurations) and deviation of the GP shell from its preferred curvature. The calculation is detailed in the Appendix. Minimizing the total free energy for a system with fixed number of GP subunits obtains that the most probable number of subunits in a shell $n$ corresponds to the value which minimizes the elastic energy per subunit, given by
\begin{align}
n = n_{0} \left( 1 + \frac{\kappa_{\text{mem}}}{ \kappa_{\text{shell}}} \right)^{2},
\label{eq:nsubs}
\end{align}
where $n_{0}$ is the number of subunits in the equilibrium configuration in the absence of a membrane ($n_{0}=80$ in our model), and $\kappa_{\text{mem}}$ and $\kappa_{\text{shell}}$ as the membrane and shell bending moduli. Thus the preferred GP-particle size is determined by the ratio $\kappa_{\text{mem}} / \kappa_{\text{shell}}$, which quantifies the competition between the membrane and shell deformation energies.
Only in the limit where the shell rigidity dominates, $\kappa_{\text{mem}} / \kappa_{\text{shell}} \to 0$, will GP-particles exhibit the size observed in bulk simulations.
To compare the theoretical estimate to the shell sizes observed in simulations, we estimated the relationship between the GP interaction strength and the shell bending modulus as $\kappa_{\text{shell}} \approx 25.66 \epsilon_{\text{gg}}$ (we show the complete derivation in section \ref{sec:bendingestimation} in the SI), leading to a range of shell bending rigidities of $\approx 40-250k_{\text{B}}T$. This range coincides with bending rigidity values measured in AFM experiments on virus capsids (see the Discussion) \cite{Michel2006,Roos2007,May2011}. The prediction of Eq.~\eqref{eq:nsubs} using this estimate and the estimated membrane rigidity $\kappa_{\text{mem}}=14.5k_{\text{B}}T$ is shown in Fig \ref{fig:bending}a. The prediction is also compared against simulated particle sizes as a function of the parameter $\kappa_{\text{mem}} / \kappa_{\text{shell}}$ for different membrane bending rigidities in Fig.~\ref{fig:bending}b. For moderate values of $\kappa_{\text{mem}} / \kappa_{\text{shell}}$ we observe good agreement between the theory and simulation results, especially considering that there is no fit parameter. The agreement breaks down for $\kappa_{\text{mem}} / \kappa_{\text{shell}}\gtrsim0.3$, likely for several reasons. Firstly, our theory assumes a closed GP shell, whereas the size of the incomplete region of the GP shell increases in size with $\kappa_{\text{mem}} / \kappa_{\text{shell}}$, as illustrated by the snapshots in Fig.~\ref{fig:bending}b. Secondly, subunits within the largest GP particles are far from their preferred interaction angle, and thus their elastic response could be nonlinear. Finally, finite-size effects could become non-negligible for the largest buds.
In contrast to GP-directed assembly, Figs.~\ref{fig:phasediagramGP} and \ref{fig:phasediagramNC} demonstrate that a NC can dramatically change the morphology of a GP shell by acting as a template over a broad range of interaction strengths. The observed monodispersity in GPNC-particles can be understood from Eq.~\eqref{eq:nsubs} by noting that the NC is modeled as a perfectly rigid sphere in our simulations, and thus corresponds to the limit $\kappa_{\text{shell}}\rightarrow \infty$ if it acts as a perfect template for the GP shell. The relevance of this approximation to enveloped viruses is considered in the Discussion.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=\columnwidth]{figures/bending2.pdf}
\caption{ {\bf a)} Average number of subunits in budded particles as a function of GP-GP affinity, for GP-directed budding (\textcolor{blue}{$\blacksquare$} symbols) and NC-directed budding with $\epsilon_{\text{ng}}=4.5k_{\text{B}}T$ (\textcolor{red}{$\CIRCLE$} symbols). The solid green line gives the theoretical prediction (Eq.~\eqref{eq:nsubs}) for the estimated capsid rigidity $\kappa_{\text{shell}}=25.66 \epsilon_{\text{gg}}$ (see section \ref{sec:bendingestimation} (SI)) and $\kappa_{\text{mem}}=14.5k_{\text{B}}T$. {\bf b)} Average number of subunits in GP-shells as a function of the ratio between membrane and shell bending modulii, $\kappa_{\text{mem}} / \kappa_{\text{shell}}$. The data includes different sets of simulations in which either $\kappa_{\text{mem}} $ or $\kappa_{\text{shell}}$ is maintained constant, and we sweep over the other parameter: $\kappa_{\text{mem}}=14.5$ (\textcolor{blue}{$\blacksquare$}), $\kappa_{\text{mem}}=21.5$ (\textcolor{red}{$\blacktriangledown$}), $\kappa_{\text{shell}}=51.3$ (\textcolor{green}{$\CIRCLE$}), and $\kappa_{\text{shell}}=154.0$ (\textcolor{dpurple}{$\blacktriangle$}). The theoretical prediction (Eq.~\eqref{eq:nsubs}) is shown as a black solid line.}
\label{fig:bending}
\end{center}
\end{figure}
\subsection*{The nucleocapsid influences timescales for late stage budding}
\label{sec:timescales}
As noted above assembly can be divided into two stages. The shell grows rapidly until about 2/3 completion, after which neck curvature significantly slows subunit association (Fig. \ref{fig:timescales}). The timescale for the second stage depends on the interaction parameters and whether a NC is present --- for the small GPNC-particles assembly is completed quickly $\sim 450\tau_{0}$, whilst in the large GP-particles with the broadest necks it may require up to $3,000\tau_{0}$. In contrast, the timescale for the first stage is almost independent of interaction strengths and the NC, and depends only weakly on membrane bending modulus. Furthermore, as shown in section \ref{sec:conformational} (SI), conformational switching is not rate limiting, implying that assembly rates during the first stage are limited by subunit diffusion.
This observation parallels models for clathrin-independent receptor-mediated endocytosis, in which the endocytosis timescale is estimated from the time required for membrane receptors to diffuse to the enveloped particle \cite{Gao2005,Bao2005}. Applying the same analysis to our simulations, the timescale for GPs to diffuse to the budding site is given by $\tau\sim l^{2}/2D$, with $D_{\text{sub}}=\sigma^{2}/\tau_{0}$ the GP diffusion constant in our simulations, and $l\approx 45\sigma$ as the radius of the region around the budding site initially containing 80 trimers, enough to envelop the particle. This estimate yields $\sim1200\tau_{0}$, which is reasonably close to the typical timescale for stage 1 observed in the simulations, $\sim1000\tau_{0}$. Note that this model does not describe the timescale of the latter stage of assembly, since the curved neck region imposes a barrier to subunit diffusion which increases as the particle nears completion.
\subsection*{GP conformational changes avoid kinetic traps}
Finally, we note that when subunit conformation changes are not accounted for (\textit{i.e.}\xspace all subunits are in the active state), we observe complete assembly and budding only under a narrow range of interaction strengths for NC-directed budding and not at all for GP-directed budding (section \ref{sec:conformational}, SI). For most interaction strengths the simulated densities of GPs led to multiple small aggregates which failed to drive significant membrane deformation. This behavior is indicative of kinetic trapping, known to occur in assembly reactions at high concentrations or binding affinities \cite{Hagan2014,Whitelam2015}. The ability of an inactive conformation to avoid this trap is consistent with simulations of bulk assembly \cite{Lazaro2016,Grime2016}, and the ability of budding to proceed in the presence of high subunit concentrations (when conformational changes are accounted for) is consistent with the observation of high densities of GPs in the membranes of cells infected with Sindbis virus \cite{Bonsdorff1978}.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=\columnwidth]{figures/timescales4_outcomes.pdf}
\caption{ Number of GP trimers in budding shells as a function of time for trajectories at different parameter values: GP-directed budding with $\epsilon_{\text{gg}}=1.4$ (black) or $\epsilon_{\text{gg}}=1.75$ (blue) and NC-directed budding with $\epsilon_{\text{ng}}=4.5$ and $\epsilon_{\text{gg}}=1.75$ (red). The snapshots to the right show the assembly products. For each parameter set the thick line shows an average over three trajectories, and the two thin lines show two individual trajectories to give a sense of the size of fluctuations. The lines end when budding occurs, except for the GP-directed case with $\epsilon_{\text{gg}}=1.4$ which ended in the stalled partially budded state. }
\label{fig:timescales}
\end{center}
\end{figure}
\section{Discussion and conclusions }
\label{sec:discussion}
\subsection*{Advantages of NC-driven budding}
We have described dynamical simulations of the assembly and budding of GPs in the presence and absence of a preassembled NC, and presented phase diagrams showing how assembly pathways and products depend on the relevant interaction parameters.
The key difference between NC-directed and GP-directed budding identified by our results is variability of the budded particle size and morphology.
The presence of the core directs the morphology of the particle, which may have direct consequences on particle stability during transmission and the conformational changes that occur during particle entry into a new host cell.
The number of GPs in GP-particles (containing no NC) varies by $>$50\% over the range of interaction parameters in which we observe successful budding, in comparison to a variation of less than 5\% for GPNC-particles (containing a NC). A simple equilibrium model accounting for the competition between the energy costs associated with membrane deformation and deviation of the GP shell from its preferred curvature was qualitatively consistent with the simulation results. The membrane bending energy favors formation of larger particles, while a higher bending rigidity of the GP shell favors smaller particles. This prediction applies to any form of assembly on a fluid membrane, and thus is relevant to capsid-directed budding as well as the GP-directed budding studied here.
It is worth considering this observation in the context of recent experimental observations on budding in the absence of a NC, as well as experimental measurements on viral particle elastic properties.
Several experimental studies have reported GP-directed budding from cells in which the NC proteins are impaired \cite{Forsell2000,Ruiz-Guillen2016}. In particular, Ruiz-Guillen et al.\xspace \cite{Ruiz-Guillen2016} recently showed that cells expressing the genome and GPs, but not the capsid protein, for Sindbis and Semliki Forest virus generate infectious viral particles that can propagate in mammalian cells. For the wild-type virus, in which the GPs assemble around the NC, they estimated a virion diameter of 60-70nm, whereas the GP-only particles were typically 100-150nm. This increase in particle size is consistent with our simulations and theoretical model, suggesting that the relatively low rigidity of the GP shell leads to the formation of large particles. However there are two important caveats to this interpretation. First, the bending rigidity of the alphavirus GP shell has not been measured, so we cannot directly predict the increase in particle size. Second, our model assumes that the preferred curvature of the GP shell is commensurate with its size in a wild type virion, for which there is no direct evidence. A recent study of herpes simplex virus nuclear egress complex (NEC), which consists of two viral envelope proteins, found that NEC particles budded in the absence of capsid are smaller than native viral particles \cite{Hagen2015}. In our model this observation would require that the intrinsic spontaneous curvature radius of the NEC complex is smaller than that of the capsid, as suggested by the authors \cite{Hagen2015}.
Despite different tendencies for polymorphism, our results show that GP-directed and NC-directed budding share many similarities, presenting a challenge for distinguishing between NC-directed and assembly-directed budding.
For example, assembly timescales depend only weakly on the presence of a NC and interaction parameters, instead being limited by diffusion of GP subunits to the budding site. We anticipate that this result would be unchanged by adding additional ingredients to the model, since the conditions we simulated (high density of GPs in the surrounding membrane) correspond to the lower bound of GP diffusion timescales. The same principle would apply if GPs are directly targeted to the budding site (rather than diffusing on the membrane) as has been suggested for alphavirus budding \cite{Martinez2014}, except that the diffusive flux would be replaced by a targeting flux.
\subsection*{Requirement for scission machinery}
A second commonality between the two scenarios investigated is that assembly slows down considerably after the GP shell reaches approximately 2/3 completion, because the high curvature of the neck region imposes a barrier to subunit diffusion. For weak interactions, this leads to a long-lived partially budded state. For stronger interactions budding eventually completes, but completion of the shell can be preempted by scission.
Since spontaneous scission is a rare event, most viruses actively drive scission by either recruiting host cell machinery, such as the ESCRT protein complex in the case of HIV \cite{Baumgartel2011}, or encoding their own macinery, such as the M2 protein in Influenza \cite{Rossman2010}. In alphaviruses specifically the scission machinery has not been identified, though it is known that alphavirus budding is independent of ESCRT proteins \cite{Chen2008}. Although in the present article we do not consider the action of these scission-inducing mechanisms, our observation of a slowdown of assembly rates due to neck curvature could be relevant to HIV budding. If the association rate becomes sufficiently slow, ESCRT-directed scission will occur before assembly completes, leaving up to 40\% of the shell incomplete, as is observed for immature HIV virions \cite{Briggs2009}. In support of this possibility, we note that although scission is a rare event in our simulations because there is no ESCRT, it usually occurs before the final 1-3 subunits assemble causing the budded particle to have small hole at the scission site.
\subsection*{Virus elasticity and GP spontaneous curvature affect budding morphology}
Since Fig. \ref{fig:phasediagramGP} suggests that the elastic properties of the different viral components are key determinants of the assembly product,
it is worth considering the validity of the model parameters.
The mechanical properties of viruses have been extensively studied using atomic force microscopy (AFM) indentation \cite{Michel2006,Roos2007} and fluctuation spectrum analysis \cite{May2011}. Typical estimates lie within the range $\kappa=30-400k_{\text{B}}T$, with considerable variation depending on the specific virus and experimental technique. A recent work by \citet{Schaap2012} explicitly compared the stiffness of the capsid protein coat of influenza virus with that of a lipid membrane, by AFM indentation of similarly sized particles, with the goal of identifying the contribution of the matrix proteins to the virus stiffness. They found that matrix coats are approximately ten times stiffer than bare membranes, $\kappa_{\text{mem}} / \kappa_{\text{shell}} \sim 0.1$. This value lies within the range explored in our simulations.
Similarly, Kol et al.\xspace \cite{Kol2007} investigated the effect of HIV maturation on its mechanical properties.
The immature HIV particle consists of a gag polyprotein capsid surrounded by a lipid bilayer containing viral envelope proteins. During maturation, the NC and capsid portions of gag are cleaved, leaving only a thin matrix layer and the envelope proteins in contact with the bilayer. Kol et al.\xspace found that this cleavage softens the particles by an order of magnitude, suggesting that inter-protein contacts of the underlying capsid layer are necessary for the high rigidity of immature virions.
Finally, we note that the membrane elasticity properties also play a role in determining particle morphology. Depending on the virus family and host cell type, enveloped viral particles bud through different cellular membranes (the plasma membrane, the ER, the ERGIC, or the nuclear membrane), all of which have different lipid compositions and thus different bending properties. Moreover, many viruses create and/or exploit membrane microdomains with different compositions (such as lipid rafts) as preferential locations for budding \cite{Waheed2010,Welsch2007,Rossman2011}. The effect of inhomogeneous membrane elastic properties on particle morphology thus deserves further exploration. While these ingredients can be incorporated into the model, the results described here demonstrate that the interplay between the elastic properties of membranes and viral proteins and the presence of an interior core can shape the morphology of a budding particle.
\section{Methods}
\label{sec:methods}
\subsection{Glycoproteins and capsid}
\label{sec:capsid}
Our coarse-grained GP model is motivated by the geometry of Sindbis virions as revealed by cryoelectron microscopy ~\cite{Mukhopadhyay2006,Zhang2002}. The outer layer of Sindbis is comprised from heterodimers of the E1 and E2 GPs. Three such heterodimers form a tightly interwoven trimer-of-heterodimers, and 80 of these trimers are organized into a T=4 lattice. On the capsid surface each trimer forms a roughly equilateral triangle with edge-length $\sim 8$nm. In the radial direction, each E1-E2 heterodimer spans the entire lipid membrane and the ectodomain spike, totaling $\sim12$nm in length. In our model, we consider the GP trimer as the basic assembly subunit, assuming that the formation of trimers is fast relative to the timescale for assembly of trimers into a complete capsid.
Our subunit model aims to capture the triangular shape, aspect ratio, and preferred curvature of the GP trimers while minimizing computational detail. To this end, we employ the conical particles studied by Chen et al.\xspace \cite{Chen2007}, modified in two ways.
First, our GP trimer subunit comprises three cones, which are fused together and simulated as a rigid body. Second, the cones are truncated, so that they form a shell with an empty interior, as shown in Fig. \ref{fig:scheme}. The cone length and trimer organization within the capsid are consistent with the Sindbis structure (see section \ref{sec:subunit} (SI) for full details). Note that while the domains primarily responsible for curvature of alphavirus GPs are located to the exterior of the envelope, the conical regions which drive curvature of the model subunit oligomers are located within and below the plane of the membrane. We found that this arrangement facilitated completion of assembly (see section \ref{sec:subunit} in the SI).
Each cone consists of a linear array of six beads of increasing diameter.
Two nearby cones experience repulsive interactions,
mediated by a repulsive Lennard-Jones potential between all pairs of beads, with size parameter $\sigma$ equal to the bead diameter. In addition, each of the four inner beads experiences an attractive interaction with its counterpart (the bead with the same diameter) in the neighboring subunit, modeled by a Morse potential. The Morse potential depth $\epsilon_{\text{gg}}$ determines the subunit-subunit interaction strength, which is related to the GP-GP binding affinity. The equilibrium distance of the Morse potential $r_{\text{e}}$, and the Lennard-Jones diameter $\sigma$ for each interacting pair is chosen to drive binding towards a preferred trimer-trimer angle. We set the
preferred angle so that in bulk simulations (in the absence of membrane) the subunits predominantly assemble into aggregates with the target size, 80 subunits. However, there is a small amount of polydispersity, with some capsids having sizes between 79 and 82 subunits (Fig.~\ref{fig:capsidsbulk}). Although the subunit geometry locally favors hexagonal packing, formation of a closed capsid requires 12 five-fold defects \cite{Grason2016}. We find that the spatial distribution of these defects is typically not fully consistent with icosahedral symmetry for dynamically formed capsids. It is unclear whether this is a kinetic effect or indicates that icosahedral symmetry is not the free energy minimum at this particle size. However, the relatively high monodispersity observed suggests that the 80-subunits capsid is a free energy minimum and assembly is robust at these conditions.
\subsection{Lipid membrane}
\label{sec:membrane}
The lipid membrane is represented by the implicit solvent model from Cooke and Deserno \cite{Cooke2005}. This model enables on computationally accessible timescales the formation and reshaping of bilayers with physical properties such as rigidity, fluidity, and diffusivity that can be tuned across the range of biologically relevant values. Each lipid is modeled by a linear polymer of three beads connected by FENE bonds; one bead accounts for the lipid head and two beads for the lipid tail. An attractive potential between the tail beads represents the hydrophobic forces that drive lipid self-assembly. In section \ref{sec:bendingestimation} in the SI we estimate the bending rigidity of the membrane in our simulations by analyzing their fluctuations spectra. Unless otherwise specified, our simulations used $\kappa_{\text{mem}}\approx 14.5 k_{\text{B}}T$ as a typical rigidity of plasma membranes.
\subsection{Glycoprotein-membrane interactions}
The effect of individual GPs on the behavior of the surrounding membrane has not been well characterized. Moreover, to facilitate interpretation of our simulation results, we require a model in which we could independently vary subunit-subunit interactions and subunit-membrane interactions. Therefore, we use the following minimal model for the GP-membrane interaction.
We add six membrane excluder beads to our subunit, three at the top and three at the bottom of the subunit, with top and bottom beads separated by 7nm (Fig.~\ref{fig:scheme}c,d). These excluder beads interact through a repulsive Lennard-Jones potential with all membrane beads, whereas all the other cone beads do not interact with the membrane pseudoatoms. In a simulation, the subunits are initialized with membrane located between the top and bottom layer of excluders. The excluded volume interactions thus trap the subunits in the membrane throughout the length of the simulation, but allow them to tilt and diffuse laterally. Separating the subunit pseudoatoms that interact with the membrane from those which control the subunit-subunit potential allows us to independently vary subunit-subunit and subunit-membrane interactions. The position of the subunit-subunit interaction beads (cones) relative to the membrane excluders has little effect on the initial stages of assembly and budding, but strongly affects its completion (described in detail in section \ref{sec:subunit} in the SI).
We note that the model does not account for local distortions within the lipid hydrophobic tails in the vicinity of the GPs. Such interactions could drive local membrane curvature and membrane-mediated subunit interactions which could either enhance or inhibit assembly and budding. Understanding these interactions is an active area (e.g. Refs. \cite{Weikl1998,Semrau2009,Reynwar2011,Goulian1993,Deserno2009}) but beyond the scope of the present study.
\subsection{Nucleocapsid}
\label{sec:NC}
The NC is represented in our model by a rigid spherical particle. This minimal representation is based on two experimental observations. We model it as spherically symmetric because asymmetric reconstructions by Wang et al.\xspace \cite{Wang2015} showed that the alphavirus NC does not exhibit icosahedral symmetry in virions (assembled in host cells) or viruslike particles (assembled in vitro). Second, within the NC-directed hypothesis the NC assembles completely in the endoplasmic reticulum and is then transported by the secretory pathway to the budding site at the plasma membrane. The complete NC has been shown to have a significantly higher rigidity than lipid membrane or GP-coated vesicles \cite{Schaap2012,Kol2007}; thus, we model it as infinitely rigid.
Our model NC as constructed from 623 beads distributed on a spherical surface with radius $r_{\text{NC}}=19.0\sigma$, and subjected to a rigid body constraint. To represent the hydrophobic interactions between GPs cytoplasmic tails and the capsid proteins, NC beads and the third bead of the GP subunits (counting outwards) experience an attractive Morse potential, with well-depth $\epsilon_{\text{ng}}$. The radius of the NC sphere was tuned using bulk simulations to be commensurate with a capsid comprising 80 GPs. To minimize the number of parameters, we do not consider an attractive interaction between the NC and membrane, but the NC beads experience a repulsive Lennard-Jones potential with all membrane beads.
\subsection{Conformational change and subunit concentration}
\label{sec:conformationalmodel}
Experiments on several viral families suggest that viral proteins interconvert between `assembly-active' and `assembly-inactive' conformations, which are respectively compatible or incompatible with assembly into the virion \cite{Packianathan2010,Deshmukh2013,Zlotnick2011}. Computational modeling suggests that such conformational dynamics can suppress kinetic traps \cite{Lazaro2016,Grime2016}. Conformational changes of the alphavirus GPs E1 and E2 are required for dimerization in the cytoplasm, and it has been proposed that the GPs interconvert between assembly-inactive and assembly-active conformations \cite{Zlotnick2011}, possibly triggered by interaction with NC proteins \cite{Forsell2000}. Based on these considerations, our GP model includes interconversion between assembly-active and assembly-inactive conformations. The two conformations have identical geometries, but only assembly-active conformations experience attractive interactions to neighboring subunits. We adopt the `Induced-Fit' model of Ref.~\cite{Lazaro2016}, meaning that interaction with an assembling GP shell or the NC favors the assembly-active conformation. For simplicity, we consider the limit of infinite activation energy. In particular, with a periodicity of $\tau_\text{c}$ all the inactive subunits found within a distance 1.0$\sigma$ of the capsid are switched to the active conformation, while any active subunits further than this distance from an assembling shell convert to the inactive conformation. Results were unchanged when we performed simulations at finite activation energies larger than $4k_{\text{B}}T$.
In simulations performed at a constant total number of GPs the assembly rate progressively slows over the course of the simulation due to the depletion of unassembled subunits. This is an unphysical result arising from the necessarily finite size of our simulations. Moreover, during an infection additional GPs would be targeted to and inserted into the membrane via non-equilibrium process (powered by ATP). Therefore, our simulations are performed at constant subunit concentration within the membrane (outside of the region where an assembling shell is located). To achieve this, we include a third subunit type called `reservoir subunits', which effectively acts as a reservoir of inactive subunits. These subunits interact with membrane beads but experience no interactions with the other two types of GP subunits. With a periodicity of $\tau_\text{c}$, reservoir subunits located in a local region free of active or inactive subunits (corresponding to a circumference of radius 1.5 times the radius of the largest subunit bead) are switched to the assembly-inactive state.
\subsection{Simulations}
\label{sec:simulations}
We performed simulations in HOOMD-blue\cite{Nguyen2011}, version 1.3.1, which uses of GPUs to accelerate molecular simulations. Both the subunits and the NC were simulated using the Brownian dynamics algorithm for rigid bodies. The membrane dynamics was integrated using the NPT algorithm, a modified implementation of the Martina-Tobias-Klein thermostat-barostat. The box size changes in the membrane plane, to allow membrane relaxation and maintain a constant lateral pressure. The out-of-plane dimension was fixed at $200\sigma$.
Throughout the manuscript we report dimensions of length, mass, and energy in units of $\sigma$, $m_{0}$, and the thermal energy $k_{\text{B}}T$. We fixed temperature at $k_{\text{B}}T/ \epsilon_{0}=1.1$. Physical sizes and timescales can be estimated as follows. We set the diameter of the lipid head as $d_{\text{head}}= \sigma$, so that considering a 5nm-thick bilayer leads to $\sigma \approx 0.9$nm. The characteristic timescale of the simulation is determined by the subunit diffusion, which in our simulations is dominated by the interaction with the membrane lipids rather than with the bath. We define our unit of time $\tau_{0}$ as the characteristic time of a subunit to diffuse a distance $\sigma$ on the membrane. Comparing with a typical transmembrane protein diffusion constant $\sim 4\mu$m$^{2}$/s \cite{Ramudarai2009,Goose2013}, we obtain $\tau_{0}=250$ns.
Our simulated equations of motion do not account for hydrodynamic coupling between the membrane and the implicit solvent, which can accelerate the propagation of bilayer perturbations. To assess the significance of this effect, we performed an additional series of simulations which did account for hydrodynamic coupling, by evolving membrane dynamics according to the NPH algorithm in combination with a dissipative particle dynamics (DPD) thermostat.
As expected from Matthews and Likos \cite{Matthews2012}, we found that hydrodynamic interactions did enhance the rate of membrane deformations; however, budding proceeded only 1.1-1.2 times faster than with the NPT scheme. Moreover, the end-product distribution was the same with and without hydrodynamic interactions. Therefore, to avoid the increased computational cost associated with the DPD algorithm, we performed all subsequent simulations with the NPT method. The very limited effect of hydrodynamics can be understood from the fact that assembly timescales in our simulations are more strongly governed by subunit diffusion than by membrane dynamics (Fig. \ref{fig:timescales}).
Our system size was constrained by the capsid dimensions and the need to access long timescales. Taking the Sindbis virion as a reference structure, the bilayer neutral surface radius in the virion is $\approx 24 $nm \cite{Zhang2002}, so the surface area of the membrane envelope is $A_{0}\sim 7200$nm$^2$. We thus needed to simulate membrane patches that were significantly larger than $A_{0}$ to ensure that the membrane tension remained close to zero and that finite-size effects were negligible. Throughout this manuscript we report results from simulations on a membrane patch with size $170\times170$nm$^2$ ($A \sim 28,900$nm$^2$), which contains $51,842$ lipids. We compared membrane deformations, capsid size and organization from these simulations against a set of simulations on a larger membrane ($210\times210$nm$^2$, $A\sim44,100$nm$^2$) and observed no significant differences, suggesting that finite size effects were minimal.
Simulations were initialized with 160 subunits uniformly distributed on the membrane, including 4 active-binding subunits (located at the center of the membrane) with the remainder in the assembly-inactive conformation. In addition, there were 156 subunits in the reservoir conformation uniformly distributed. The membrane was then equilibrated to relax any unphysical effects from subunit placement by integrating the dynamics for 1,500 $\tau_{0}$ without attractive interactions between GPs. Simulations were then performed for 4,200 $\tau_{0}$ with all interactions turned on. The timestep was set to $\Delta t=0.0015$, and the thermostat and barostat coupling constants were $\tau_{T}=0.4$ and $\tau_{P}=0.5$, respectively. Since the tension within the cell membrane during alphavirus budding is unknown, we set the reference pressure to $P_{0}=0$ to simulate a tensionless membrane. The conformational switching timescale was set to $\tau_\text{c}=3\tau_0$, sufficiently frequent that the dynamics are insensitive to changes in this parameter.
Unless otherwise specified, for each parameter set we perform 8 independent simulations.
\section{Appendix}
\subsection*{Equilibrium model for the dependence of GP shell size on system parameters}
\label{sec:EquilModel}
In this section we give a detailed derivation of Eq.~\ref{eq:nsubs} of the main text. This expression explains the simulation results for GP shell size as a function of control parameters (Fig.~\ref{fig:bending}), and is obtained from a simple equilibrium model based on the thermodynamics of assembly \cite{Safran1994,Hagan2014} that accounts for the elasticity of the shell and the membrane.
The total free energy for the system of free subunits on the membrane, shell intermediates of size $n$, and complete shells of size $N$ can be expressed as
\begin{align}
F / k_{\text{B}}T = \sum_{n=1}^{N} \rho_{n} [\log{\rho_{n}v_{0}}-1] + \rho_{n} {G_n^{\text{shell}}} / k_{\text{B}}T ,
\label{eq:totalfreenergy}
\end{align}
where $\rho_{n}$ and ${G_n^{\text{shell}}}$ are respectively the concentration and interaction free energy for an intermediate with size $n$, and $v_{0}$ is a standard state volume. Minimization of Eq.~\eqref{eq:totalfreenergy} subject to the constraint of constant subunit concentration yields the well-known law of mass action for the equilibrium distribution of intermediate concentrations,
\begin{align}
\rho_{n}v_{0}=\exp\left[-\left({G_n^{\text{shell}}}-n \mu_1\right)/ k_{\text{B}}T \right] .
\label{eq:LMA}
\end{align}
with \begin{align}
\mu_{1} = \log{\rho_{1}v_{0}}
\label{eq:chempotential1}
\end{align}
the chemical potential of free subunits. Similarly we can compute the chemical potential for intermediates $\mu=\partial F / \partial \rho_{n}$ as
\begin{align}
\mu_{n} = \log{\rho_{n}v_{0}} + {G_n^{\text{shell}}} / k_{\text{B}}T .
\label{eq:chempotentialn}
\end{align}
For large shells, the first term in \eqref{eq:chempotentialn} is neglegible compared to the free energy of the shell, and the chemical pontential can be approximated as $\mu_{n}\approx {G_n^{\text{shell}}}/ k_{\text{B}}T$. In equilibrium, the chemical potential of free subunits must be equal to that of subunits in shells and intermediates, leading to $\mu_{1}=\mu_{n} / n \approx {G_n^{\text{shell}}} / nk_{\text{B}}T$.
The intermediate size with maximal concentration is determined by the condition
\begin{align}
\frac{d\rho_{n}}{dn}=\rho_{n}\frac{d}{dn}[-{G_n^{\text{shell}}} k_{\text{B}}T + n \mu_{1}] =0,
\label{eq:rhomax}
\end{align}
which using \eqref{eq:chempotential1} and \eqref{eq:chempotentialn} can be rewriten as
\begin{align}
\frac{d}{dn}[-{G_n^{\text{shell}}}/ k_{\text{B}}T + n \mu_{1}] \approx \left[- \frac{d {G_n^{\text{shell}}}}{dn} + {G_n^{\text{shell}}} / n \right]/ k_{\text{B}}T.
\label{eq:rhomax2}
\end{align}
Thus, the optimal size at equilibrium is that which minimizes the interaction free energy per subunit, ${G_n^{\text{shell}}}/n$.
The interaction free energy includes subunit-subunit interactions and the elastic energy of the shell and the membrane. Assuming that the shell can be described as a continuous, two-dimensional spherical shell, its elastic energy is given by the Helfrich bending energy, with bending modulus $\kappa_{\text{shell}}$ and spontaneous curvature $c_{0}=2/R_{0}$, where $R_{0}$ is the equilibrium radius of the shell. The membrane underneath is a symmetric bilayer with rigidity $\kappa_{\text{mem}}$. The free energy ${G_n^{\text{shell}}}$ thus reads
\begin{align}
{G_n^{\text{shell}}} = n \Delta g_{\text{g}} + \frac{\kappa_{\text{mem}}}{2} \int_{S} c^{2} dS + \frac{\kappa_{\text{shell}}}{2} \int_{S} (c-c_0)^{2} dS,
\label{eq:helfrich}
\end{align}
where $\Delta g_{\text{g}}$ is the free energy per subunit added to the shell, $c$ is the total curvature, and $S$ denotes the surface area. Assuming spherical symmetry and accounting for the fact that the subunits are rigid, the shell surface area is $S \approx n S_{0}$, with $S_{0}$ as the area per subunit. We can then express the total curvature as a function of the number of subunits in the particle, $c= 2/R \approx (16\pi / n A_{0})^{1/2}$.
Here we are assuming that the Gaussian modulus of the membrane is unchanged by the presence of the GPs, so that the integrated Gaussian curvature is constant for fixed topology by the Gauss-Bonnet theorem \cite{Deserno2015}. Under the (reasonable) assumption that the shell size is determined before scission, the Gaussian curvature energy then contributes a constant to the free energy and can be neglected. We have also neglected the energy from the 12 disclinations in the shell. Accounting for this could shift the theory curve in Fig.~\ref{fig:bending}b but would not change the slope.
Finally, recalling that the equilibrium configuration minimizes the interaction free energy per subunit, ${G_n^{\text{shell}}}/n$ results in \eqref{eq:nsubs} of the main text,
\begin{align}
n = n_{0} \left( 1 + \frac{\kappa_{\text{mem}}}{ \kappa_{\text{shell}}} \right)^{2},
\label{eq:nsubs}
\end{align}
where $n_{0}$ is the number of subunits in the equilibrium configuration in the absence of membrane, corresponding to $n_{0}=80$ in our model.
\subsection*{Acknowledgements}
This work was supported by the NIH, Award Number R01GM108021 from the National Institute Of General Medical Sciences (GRL and MFH), the Brandeis Center for Bioinspired Soft Materials, an NSF MRSEC, DMR-1420382 (GRL), and the NSF, award number MCB1157716 (SM). Computational resources were provided by the NSF through XSEDE computing resources (MCB090163) and the Brandeis HPCC which is partially supported by the Brandeis MRSEC.
| {'timestamp': '2017-07-27T02:00:38', 'yymm': '1706', 'arxiv_id': '1706.04867', 'language': 'en', 'url': 'https://arxiv.org/abs/1706.04867'} |
\section{Introduction}
Recent advances in neural Natural Language Processing (NLP) have pushed the frontiers. In particular, transformer-architecture-based models \cite{transformer} surpassed human performance in various NLP benchmarks such as SQuAD2.0 \cite{rajpurkar-etal-2018-know}, GLUE \cite{wang-etal-2018-glue}, and SuperGLUE \cite{NEURIPS2019_4496bf24}. This also opened new opportunities for the Grammatical Error Correction (GEC) task which we address in this work.
GEC is the task of correcting different kinds of errors in text such as spelling, punctuation, grammatical, and word choice errors. The abundance of such noise in the text can hinder not only the understanding by humans but also the performance of various downstream NLP systems. An error-free text is also more beautiful, clean, associated with a certain prestige. However, producing it may be problematic for non-native speakers, language learners, it requires additional time and effort.
NLP state of the art for GEC still has much room to improve. As of now, the best F$_{0.5}$ scores are only up to 0.72\footnote{\url{http://nlpprogress.com/english/grammatical_error_correction.html}}. Moreover, the research is mostly focused on English and a few other popular languages. To reduce this gap and contribute to the GEC progress, in this work, we investigate it for the Lithuanian language.
The Lithuanian language is one of the oldest living languages in the world. It has retained most of the features of the Indo-European Protolanguage, i.e., it is characterized by a very ancient linguistic structure: declensions (of nouns, adjectives, and pronouns), short and long vowels, diphthongs, etc. Lithuanian also has many similarities with Sanskrit -- the classical language of ancient India, still used today as a scholarly and liturgical language in Hinduism, Buddhism, and Jainism. Antoine Meillet (1886-1936), one of the most influential French linguists, once stated: ``Anyone wishing to hear how Indo-Europeans spoke should come and listen to a Lithuanian peasant''.
The Lithuanian language is synthetic and uses inflections to express syntactic relationships within a sentence. In other words, the relations in a sentence are expressed by word endings rather than with unbound morphemes and word order. This allows a lot of freedom in composing sentences. In contrast to agglutinative languages, which combine affixes by ``gluing'' them unchanged inside word ending, in Lithuanian inflectional categories are ``fused''. Meanwhile, prefixes, suffixes, and infixes are still used to derive words. Lithuanian verbs can be made from any onomatopoeia; phrasal verbs (e.g., go in, go out) are composed by adding the prefix to the verb. Lithuanian is unique for having 13 different participial forms of the verb \cite{uniqueLithuanian} while modern English has only 2 (present and past participles). It is estimated that 47\% of Lithuanian word forms are morphologically ambiguous \cite{rimkute2006morfologinio}, i.e., requiring context consideration to discern the meaning. All these mentioned features of the Lithuanian language make it interesting and important to analyze in the context of automatic GEC.
Our contributions are:
\begin{itemize}
\item We present the first GEC system for Lithuanian language based on deep neural networks.
\item We compare sub-word and byte-level tokenization approaches for Lithuanian grammatical error correction.
\item We share all the technical details, code, and model weights for open reuse and reproducibility.
\end {itemize}
\section{Related Work}
The simplest form of GEC is spellcheck. GNU Aspel\footnote{\url{http://aspell.net/}} and Hunspell\footnote{\url{http://hunspell.github.io/}} are two widely-used open source spellcheckers. In particular, Hunspell \cite{hunspell_lt} is the only system we found for Lithuanian GEC. Such systems work by keeping a large dictionary of possible words and detecting the non-words. During detection, the nearest alternatives from the dictionary are suggested. In Hunspell's case, the dictionary is made more compact by keeping only the main word forms with transformation rules, prefixes and suffixes. Spellchek systems are compact but limited to the correction of only non-words.
The first systems for a more complex GEC were based on Statistical Machine Translation (SMT) using a noisy channel model \cite{brockett-etal-2006-correcting}. A significant contribution to GEC was the introduction of the CoNLL-2014 shared task \cite{ng-etal-2014-conll}. Multiple systems were proposed, and among them, the phrase-based SMT setup was the most promising \cite{junczys-dowmunt-grundkiewicz-2016-phrase}. Yet neural approaches started to emerge, like \cite{xie2016neural}. As such systems advanced, hybrid statistical (SMT) and Neural Machine Translation (NMT) approaches \cite{grundkiewicz-junczys-dowmunt-2018-near} took the top. Only the introduction of the Transformer model \cite{NIPS2017_3f5ee243} enabled neural approaches to supersede the statistical ones. As of now, the latter systems are claiming state-of-the-art results in GEC \cite{rothe2021a, omelianchuk-etal-2020-gector}.
Novel less-supervised approaches are also emerging. A simple language model reaching a reasonable performance with minimal annotated training data was demonstrated in \cite{bryant-briscoe-2018-language}. The proposed system used $n$-gram language model to score variations of a sentence until incremental inflections do not improve the score anymore. Such a system was again improved using transformer-based language models instead of the $n$-grams in \cite{alikaniotis-raheja-2019-unreasonable}. It turns out that such a less-supervised approach can outperform fully-supervised systems that were claiming state-of-the-art results several years ago.
Currently, the main constraint for GEC is the lack of training data. Researchers make progress by including new data sources or using automatic grammatical error generation to synthesize them. A simple language-agnostic pre-training objective was proposed in \cite{rothe2021a}. The authors automatically corrupted sentences in character level: swapping, inserting, dropping spans; token level: swapping, dropping spans; word level: lower-casing, upper-casing the first letter. A bigger model and larger dataset allowed achieving state-of-the-art GEC results for 4 languages. Authors of \cite{shah-de-melo-2020-correcting} used a small corpus of spelling errors to derive statistics for typographical error generation and generate a large parallel synthetic corpus. Another way is to use the data that the model incorrectly predicted during the training. A fluency boost learning and inference mechanism was proposed in \cite{ge2018reaching} that reuses less fluent model predictions as new inputs during subsequent epochs. Similar trends are emerging with other languages. For example, simply adding new data improved a Transformer GEC system for the Czech language \cite{naplava2022czech}. To summarize, it is important for neural GEC systems to be trained on large and high-quality corpora.
\section{Dataset}
As mentioned, a large dataset is essential for training a neural GEC solution.
The data also has to be of the highest quality so that we can take it as a gold standard of grammatically-correct text.
The largest publicly available general-purpose dataset for the Lithuanian language is from OSCAR \cite{OrtizSuarezSagotRomary2019} at 5\,GB of deduplicated text. However, it is obtained from a general Common Crawl\footnote{\url{https://commoncrawl.org/}} and makes trusting the grammatical correctness problematic.
To make sure that the text is of good quality, we crawled various Lithuanian websites ourselves. We manually checked how the text is structured in every webpage so that only the relevant parts: title, summary (optionally), and the main text paragraphs would be scrapped. We crawled the following types of web pages: news, literature, blogs, encyclopedias, others.
We added titles and summaries to the main text paragraphs as additional paragraphs. Finally, we split the data into paragraphs. As a result, a single paragraph became a single sample of our dataset.
\subsection{Preprocessing}
Although we performed a well-curated data scraping, there were still some artifacts in our data that had to be corrected or removed.
We had to remove some websites because of relatively high rates of spelling errors. %
This left us with a total of 34 final websites.
Some common error patterns can be easily corrected automatically. We looked at common mistakes in Lithuanian web texts \cite{tamulioniene2015budingiausios} and performed the following corrections:
\begin{enumerate}
\item Incorrect quotation marks. In Lithuanian, the correct are ,,ABC``. Meanwhile, the English version ``ABC'' or others such as the universal "ABC" is often used instead.
\item The lack of space. It can happen before ``m.'' and ``d.'' abbreviations. For example, the text ``1918m. vasario 16d.'' must be corrected to ``1918 m. vasario 16 d.''. Additionally, the space is often omitted after a full stop: ``ir t.t.'' and ``A.Sabonis'' must be corrected to ``ir t. t.'' and ``A. Sabonis''.
\item An unnecessary space. The space must be omitted before most punctuation marks: ``tik darbui , visiškai pamirštant poilsį ,'' is corrected to ``tik darbui, visiškai pamirštant poilsį,''.
\end{enumerate}
We also filtered the text samples based on some statistics: %
\begin{enumerate}
\item The sample text length should be at least 20 characters.
\item The fraction of Lithuanian letters in a sample should be at least 0.98. This filters out text from other languages and with miscellaneous characters. In the end, we are solving a task for the Lithuanian language. We included the characters ``€₤\$\%wx'' as Lithuanian since they are used quite often.
\item The fraction of spaces to non-spaces should be at most 0.02. This allowed us to filter out samples dominated by URL addresses.
\end{enumerate}
Lastly, we deduplicated our text samples. We shuffled the resulting 29\,312\,785 samples and took a subset of 4\,194\,304 for this work. Some statistics for the subset are depicted in Table \ref{tab:dataset}.
\begin{table}[h]
\setlength{\tabcolsep}{1.8pt}
\caption{Dataset sizes by various tokenizations. The total dataset size is 4\,194\,304 samples.}
\label{tab:dataset}
\begin{tabular}{lcrc}
\toprule
\multirow{2}{*}{Tokenizer} & {Sample length,} &{Tokens,} & \multirow{2}{*}{Tokenization example}\\
& mean $\pm$ std & $\times 10^6$ & \\
\midrule
Characters & $226\pm194$ & 947& Lietuva – graži šalis\\
ByT5 \cite{xue2021byt5} & $243\pm194$ & 1\,017 & Lietuva \symbol{92}xe2\symbol{92}x80\symbol{92}x93 gra\symbol{92}xc5\symbol{92}xbei \symbol{92}xc5\symbol{92}xa1alis\\
T5 \cite{stankevivcius2021generating} & $48\pm43$ &201 & [\_Lietuva] [\_–] [\_graži] [\_šalis]\\
mT5 \cite{xue-etal-2021-mt5}&$71\pm61$ & 298& [\_] [Lietuva] [\_–] [\_] [graž] [i] [\_šal] [is]\\
Words & $30\pm26$ & 126 & [Lietuva] [graži] [šalis]\\
\bottomrule
\end{tabular}
\end{table}
The transformer model has a quadratic running time complexity $\mathcal{O}(n^{2})$ with respect to the sequence length $n$ (number of tokens). Usually, this is not a constraint as most text tasks are within the maximal sequence length of 512 (T5 \cite{2020t5}) sub-words or 1024 (ByT5 \cite{xue2021byt5}) bytes. Yet in our training dataset, we had longer examples that we did not wish to truncate and, hence, lose. Instead, we split these too-long sequences to the length of 2100 characters for the T5 model and 700 characters for the ByT5. After that, we proceeded with the corresponding tokenization. As a result, the exact numbers of samples and tokens differ for both models, but the initial dataset and the amount of text (see Table \ref{tab:dataset}) is the same.
For both runs, we set aside 0.05\% of the data for validation and another 0.05\% for testing.
\section{Methods}
\subsection{Generating Grammatical Errors}
We induce 3 groups of synthetic grammatical errors described below.
\subsubsection{Typographical Errors}
They are induced by modeling the way how humans mistype on the keyboard. We follow the exact same methodology as in \cite{stankevivcius2022correcting}: take mistyping statistics between each pair of characters on a QWERTY keyboard from an English dataset and apply them probabilistically to our texts. Out of the all characters considered, this way we corrupted 2\% of them; from which were: substitution, 36.1\%; deletion, 31.7\%; insertion, 17.8\%; transposition, 14.4\%.
\subsubsection{Confusing Similar Sounding Letters}
This is a very common source of spelling mistakes. We model them by defining sets of characters that sound alike, and randomly substituting a letter with one from the same set at a point of the generated error. For this, we use weighted sampling. The probabilities of the letters for substitution are proportional to how frequent they are in Lithuanian texts overall. For example, a in single set of letters ``iįy'' (where sounds differ only in their length), it is way more common to mistakenly write ``i'' instead of ``į'', rather than ``į'' instead of ``i'', as ``i'' is much more common. Only 2\% of all such found occurrences were replaced. Groups of letters and detailed probabilities for the group members are derived from the raw (no preprocessing) subset of 2\,909\,403 samples, and are presented in Appendix \ref{appendix1}.
\subsubsection{Other Errors}
We also introduce errors in the text by the four specific rules described below. We, again, thus corrupt 2\% of the matches of the rules.
\begin{enumerate}
\item Gemination are doubled consonant letters that sound like a single one and thus is prone to be typed only once. This also applies to any consecutive letters from ``cčsšzž'' group. For example, the words ``pusseserė užsimerkė'' may be mistakenly written as ``puseserė usimerkė'' as they sound similar due to the gemination.
\item Assimilation to an adjacent letter. This is specific to any letter of ``ptksš'' being before any of the ``bdgzž'' or vice versa. For example, the words ``dirbti, lipdavo'' may be mistakenly written as ``dirpti, libdavo'' as this is how they sound due to the assimilation.
\item Uppercasing or lowercasing the first letter in a word. For example, the word ``ąžuolas'' can start both with the lower or upper case depending on whether it is a tree (oak) or person's name. We exclude the first words in a sentence as these always have to start the upper case.
\item Delete and add space. We separately match all occurrences of spaces and all empty strings not at a word boundary.
\end{enumerate}
Some samples of the corrupted sentences are presented in Appendix \ref{appendix2}.
\subsection{Transformer Models}
In this work, we compare T5 \cite{stankevivcius2021generating} and ByT5 \cite{xue2021byt5} transformer models for grammatical error correction of Lithuanian. They are of sequence-to-sequence type. The encoder encodes the input sequence with attention operating on all input tokens while the decoder predicts output sequence tokens one by one, attending to tokens of both encoder (all) and decoder (only previous ones).
Below we further emphasize the properties of these models that make them appropriate for our task.
\subsubsection{T5}
The original T5 \cite{2020t5} was designed to be universal for multiple tasks. Authors showed that there is no difference whether a custom ``head'' is used (added on top of the pre-trained transformer) for fine-tuning purposes or a simple sequence-to-sequence formulation in text format is employed (no need to add additional weights to a pre-trained model). This way even tasks with outputs as float numbers can be formatted into a text-to-text format. Such generic task formulation made the T5 model very popular.
In previous work \cite{stankevivcius2021generating}, we adapted the T5 model for the generation of summaries of Lithuanian news articles. We trained a SentencePiece \cite{kudo2018sentencepiece} tokenizer on $10^{6}$ and the main model on 2\,027\,418 news articles. As a result, this model should be familiar with the Lithuanian language (both tokenizer and model weights) and we use it as the basis for our fine-tuning purposes.
\subsubsection{ByT5}
ByT5 is a follow-up model from the multilingual mT5 \cite{xue-etal-2021-mt5} and T5 \cite{2020t5}. The authors showed that adapting byte-level tokenization can lead to a much more efficient use of model parameters. As an example, the multilingual mT5 had over 66\% of its weights (for the base version) allocated to its multilingual word pieces (a total of 250\,000) related weights (input embedding matrix and decoder softmax layer) which were only sparsely updating during the training. Meanwhile, ByT5 vocabulary has only 384 items and the model reuses the saved parameters in more massive layers rather than indexing tokens. These benefits allowed ByT5 to surpass the small and the base versions of mT5 \cite{xue2021byt5}.
The introduction of finer byte-level tokenization is especially important for grammatical error correction. Typos, variants in spelling and capitalization, and morphological changes can lead to completely different sub-word tokens, while byte tokens are affected the least. The authors of ByT5 showed that their model outperforms mT5 if various types of noise are introduced. Therefore, we use this model in our study of Lithuanian grammatical error correction.
\subsection{Training Details}
To train the models, we used a GeForce RTX 2080 Ti GPU. Following the best practices with the T5 family of models \cite{2020t5, xue-etal-2021-mt5, rothe2021a, xue2021byt5}, we used the total batch size (number of samples to pass through the model before the gradient update) of 128 for fine-tuning. For ByT5 it was achieved by 128 gradient accumulation steps of batch size 1; while for T5, 64 gradient accumulation steps of batch size 2. We had to use multiple accumulation steps to process the total batch sequentially by smaller parts as the total batch did not fit into GPU memory at once. It took us approximately 100 hours for ByT5 and 30 hours for T5 fine-tuning. ByT5 took longer due to the longer sequences produced by finer byte-level tokenization.
We used the training script and Pytorch model implementation from the Hugging Face library \cite{wolf-etal-2020-transformers}. For simplicity, we employed an Adafactor optimizer \cite{pmlr-v80-shazeer18a} with a constant learning rate of 0.001. If not stated otherwise, we used all the default parameters as in the Hugging Face library version 4.12.0.
\subsection{Evaluation}
One of the most popular grammatical error correction evaluation metrics is ERRANT \cite{bryant-etal-2017-automatic}. It applies a set of rules operating over a set of linguistic annotations to construct the alignment and extract individual edits between corrupted, corrected, and gold-standard texts. This way precision, recall, and \textit{F}-score can be calculated. We customized the original ERRANT by using Hunspell dictionaries \cite{hunspell_lt}, stemmer\footnote{\url{https://pypi.org/project/PyStemmer/}}, spaCy version 3.2 pipeline \verb|lt_core_news_lg| \footnote{\url{https://spacy.io/models/lt}}, and corresponding part-of-speech tags for the Lithuanian language.
During the inference, we used simple greedy decoding with a beam size of 1. That is, we simply selected for each next token the one that the model assigned the highest probability to.
\section{Results}
Training dynamics of T5 and ByT5 models are depicted in Figure \ref{fig2}. Only after 6\% of training, the ByT5 score F$_{0.5}=0.85$ is already higher than the F$_{0.5}=0.80$ T5 managed to reach after the full epoch. We can also see that the performance is steadily increasing during the fine-tuning and is expected to continue doing so. The same results are indicated by the training loss. It is much lower for the ByT5, hence the model is better.
We divided our synthesized errors into several groups and corrupted the test set with each group separately from the others. Evaluation results of such setup are presented in Table \ref{tab:groups}. We can see that the easiest task for both models was adding or deleting spaces, while the hardest task is correcting assimilation and gemination mistakes. This group may lag in performance due to the smaller abundance (2\% of samples) in the training data.
We present some generated test samples in Appendix \ref{appendix2}.
\begin{figure}
\includegraphics[width=\textwidth]{lit_gec.pdf}
\caption{Training loss and F$_{0.5}$ score for both T5 and ByT5 runs.}
\label{fig2}
\centering
\end{figure}
\begin{table}[h]
\setlength{\tabcolsep}{13.4pt}
\caption{Evaluation for the separate error categories with models trained for one epoch. We applied synthetic corruption for the test set of ByT5 (total of 2\,155 samples) and T5 (total of 2\,099 samples) with each error group separately. We show both ERRANT F$_{0.5}$ score and number of samples (\#samples) affected and evaluated on.}
\label{tab:groups}
\begin{tabular}{lrrrr}
\toprule
\multirow{2.5}{*}{Error group} & \multicolumn{2}{c}{ByT5} & \multicolumn{2}{c}{T5} \\
\cmidrule(r){2-3} \cmidrule(l){4-5}
& F$_{0.5}$ &\#samples & F$_{0.5}$ & \#samples\\
\midrule
Typographical & 0.87 & 1\,916 &0.72 & 1\,868\\
Punctuation & 0.81 & 489 &0.36 &460\\
Similar sounding letters & 0.88 & 1\,115 &0.55 &1\,143\\
Add/delete spaces & 0.96 & 1\,873 &0.74 &1\,832\\
Assimilation/Gemination & 0.79 & 56 &0.30 & 43\\
Upper/Lower casing & 0.86 & 785 &0.47 &781\\
\bottomrule
\end{tabular}
\end{table}
\section{Discussion}
We trained the first reported deep-learning-based Lithuanian grammatical error correction system and compared two sequence-to-sequence transformer models for the task.
The ByT5 transformer model, based on byte-level tokenization, greatly outperformed the subword counterpart T5. We think that the main reason for this is that the fine-grained byte-level details allow the model to maximize acquired information about the sentence and thus calculate a more accurate representation. This way, the model sees a bigger picture and has to solve the task with less ambiguity. On the other hand, longer and more informative token sequences are slower to process and induce the slowdown of three times, compared to the T5. Yet even if we compare models trained for the same amount of time, ByT5 is still the leader. This shows that for the grammatical error correction it is crucial to have the best possible representation of the text.
We thought that during the T5 subword tokenizer training acquired common token patterns may be of great use. Yet our results show that this is not the case. On the contrary, it may make it harder for the model to ``understand'' the true representation behind the corrupted text.
In the future, we plan to train the ByT5 model even longer. It is clearly visible from our results that in the current state it is under-trained. Additional benefits could be expected from more data and more passes through the dataset.
We hope that this work will help both researchers and Lithuanian language users. We make our trained model and code available at \url{https://github.com/LukasStankevicius/Towards-Lithuanian-Grammatical-Error-Correction}.
\subsection*{Funding}
The research is partially funded by the joint Kaunas University of Technology Research and Innovation Fund and Vytautas Magnus University project ``Deep-Learning-Based Automatic Lithuanian Text Editor (Lituanistas)'', Project no.: PP34/2108.
\subsection*{Acknowledgements}
We thank our project collaborators from Vytautas Magnus University, especially Jurgita Kapočiūtė-Dzikienė, for valuable discussions on related topics.
\bibliographystyle{splncs04}
| {'timestamp': '2022-03-21T01:27:15', 'yymm': '2203', 'arxiv_id': '2203.09963', 'language': 'en', 'url': 'https://arxiv.org/abs/2203.09963'} |
\chapter*{Introduction : fonctions trigonométriques et symboles modulaires}
Comme Eisenstein l'explique lui-même dans \cite{Eisenstein}, sa méthode pour construire des fonctions elliptiques s'applique de manière élégante au cas plus simple des fonctions trigonométriques. C'est par là que débute le livre que Weil \cite{Weil} consacre à ce sujet; nous suivons son exemple.
\section{La relation d'addition pour la fonction cotangente}
La méthode d'Eisenstein se base sur la considération de la série
$$\varepsilon (x) = \frac{1}{2i\pi} \sideset{}{{}^e}\sum_{m \in \mathbf{Z}} \frac{1}{x+m} = \frac{1}{2i\pi} \lim_{M \to \infty} \sum_{m = -M}^{M} \frac{1}{x+m},$$
où le symbole $\sum^e$ désigne la sommation d'Eisenstein définie par la limite de droite.
Eisenstein démontre que\footnote{La normalisation appara\^{\i}tra plus naturellement par la suite; elle est bien évidemment liée au fait que la forme $dx/ (2i \pi x)$ possède un résidu égal à $1$ en $0$.} $\varepsilon (x) = \frac{1}{2i} \cot \pi x$ et ce faisant retrouve la \emph{formule d'addition}, originellement découverte par Euler, selon laquelle pour tous les nombres complexes $x$ et $y$ tels que ni $x$, ni $y$, ni $x+y$ ne soient entiers, on a
\begin{equation} \label{E:addition}
\varepsilon (x) \varepsilon (y) - \varepsilon (x) \varepsilon (x+y) - \varepsilon (y) \varepsilon (x+y) = - 1/4.
\end{equation}
Le point de départ de la démonstration d'Eisenstein est une identité élémentaire entre fractions rationnelles~:
\begin{equation} \label{E:addition0}
\frac{1}{xy} - \frac{1}{x(x+y)} - \frac{1}{y (x+y)} = 0.
\end{equation}
Formellement on a en effet
\begin{multline*}
\varepsilon (x) \varepsilon (y) - \varepsilon (x) \varepsilon (x+y) - \varepsilon (y) \varepsilon (x+y) \\ = \sum_{p,q,r} \left( \frac{1}{(x+p) (y+q)} - \frac{1}{(x+p)(x+y+r)} - \frac{1}{(y+q) (x+y+r)} \right)
\end{multline*}
où les entiers $p$, $q$ et $r$ varient dans $\mathbf{Z}$ tout en étant astreints à la relation $p+q-r=0$; les sommes n'étant pas absolument convergente cette décomposition n'a pas de sens, mais la relation \eqref{E:addition} se déduit d'une version régularisée de cette observation.
Sczech \cite{Sczech92} interprète la relation d'addition \eqref{E:addition} comme une ``relation de cocycle''; nous y reviendrons. On relie d'abord cette relation aux \emph{symboles modulaires} dans le demi-plan de Poincaré $\mathcal{H}$.
\section{Symboles modulaires}
Notons $\mathcal{H}^*$ l'espace obtenu en adjoignant à $\mathcal{H}$ les points rationnels $\mathbf{P}^1 (\mathbf{Q})$ de son bord à l'infini $\mathbf{P}^1 (\mathbf{R}) = \mathbf{R} \cup \{ \infty \}$. \'Etant donné deux points distincts $r$ et $s$ dans $\mathbf{P}^1 (\mathbf{Q})$, on note $\{ r,s \}$ la géodésique orientée reliant $r$ à $s$ dans $\mathcal{H}$.
Soit $\Delta$ le groupe abélien engendré par les symboles $\{ r, s \}$ et soumis aux relations engendrées par
$$\{ r , s \} + \{ s, r \} = 0 \quad \mbox{et} \quad \{r,s \} + \{ s,t \} + \{ t,r \} =0 .$$
On appelle \emph{symbole modulaire} l'image d'un symbole $\{r,s \}$ dans $\Delta$; on la note $[r,s]$. Manin \cite{Manin} observe que $\Delta$ est engendré par les symboles \emph{unimodulaires}, c'est à dire les $[r,s]$ avec $r=a/c$ et $s=b /d$ tels que $ad-bc = 1$, dont la géodésique associée $\{r,s\}$ est une arête de la triangulation de Farey représentée ci-dessous.
L'action de $\mathrm{SL}_2 (\mathbf{Z})$ sur $\mathcal{H}$ par homographies se prolonge en une action sur $\mathcal{H}^*$ et induit une action naturelle sur $\Delta$ de sorte que
$$g \cdot [\infty , 0 ] = [a/c , b/d], \quad \mbox{pour tout } g = \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \mathrm{SL}_2 (\mathbf{Z}).$$
Notons
$$\overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2 ) = \mathcal{M} (\mathbf{C}^2 / \mathbf{Z}^2) / \mathbf{C} \cdot \mathbf{1}$$
le quotient de l'espace des fonctions méromorphes et $\mathbf{Z}^2$-périodiques sur $\mathbf{C}^2$ par le sous-espace des fonctions constantes. L'action linéaire de $\mathrm{SL}_2 (\mathbf{Z})$ sur $\mathbf{C}^2$ induit une action sur $\mathbf{C}^2 / \mathbf{Z}^2$, et donc sur $\overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2 )$, et l'observation suivante est simplement une reformulation de la relation d'addition \eqref{E:addition}.
\medskip
\noindent
{\bf Observation.} {\it L'application $\mathbf{c} : \Delta \to \overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)$ définie sur les symboles \emph{unimodulaires} par
$$\mathbf{c} ([r,s]) (x, y) = \epsilon (dx-by) \epsilon (-cx + ay), \quad \mbox{pour } r= a/c, \ s = b/d, \ ad-bc=1, $$
est bien définie et $\mathrm{SL}_2 (\mathbf{Z})$-équivariante. }
\medskip
On dit alors que $\mathbf{c}$ est un \emph{symbole modulaire à valeurs dans $\overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)$}.
\begin{center}
\includegraphics[width=0.9\textwidth]{Fareybis.png}
\end{center}
Cette remarque élémentaire place la relation d'addition dans un nouveau contexte. Elle suggère d'étudier l'action des opérateurs de Hecke sur $\mathbf{c}$.
\section{Opérateurs de Hecke}
Les actions du groupe $\mathrm{SL}_2 (\mathbf{Z})$ sur $\Delta$ et sur $\mathbf{C}^2/ \mathbf{Z}^2$ s'étendent naturellement au monoïde $M_2 (\mathbf{Z})^\circ = M_2 (\mathbf{Z}) \cap \mathrm{GL}_2 (\mathbf{Q})$.
Cela munit $\mathrm{Hom} (\Delta , \overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2))$ d'une action à droite qui étend celle de $\mathrm{SL}_2 (\mathbf{Z})$~:
$$\phi_{| g} ([r,s]) (x,y) = \phi (g \cdot [r,s]) \left( g \cdot \left( \begin{smallmatrix} x \\ y \end{smallmatrix} \right) \right),$$
$( \phi \in \mathrm{Hom} (\Delta , \overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)), \ g \in M_2 (\mathbf{Z})^\circ ).$
L'espace
$$\mathrm{Hom} (\Delta , \overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2) )^{\mathrm{SL}_2 (\mathbf{Z})}$$
des symboles modulaires à valeurs dans $\overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)$ hérite alors d'une action à droite de l'algèbre de Hecke associée à la paire $(M_2 (\mathbf{Z})^\circ , \mathrm{SL}_2 (\mathbf{Z}))$~: étant donné un élément $g \in M_2 (\mathbf{Z})^\circ$ on décompose la double classe de $g$ par $\mathrm{SL}_2 (\mathbf{Z})$ en une union finie de classes à gauche
$$\mathrm{SL}_2 (\mathbf{Z} ) g \mathrm{SL}_2 (\mathbf{Z}) = \bigsqcup_j \mathrm{SL}_2 (\mathbf{Z}) g_j.$$
L'opérateur de Hecke associé à $g$ opère sur un symbole modulaire $\phi$ à valeurs dans $\overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)$ par
$$\mathbf{T}(g) \phi = \sum_j \phi_{| g_j}.$$
\medskip
\noindent
{\it Exemples.} Notons $\mathbf{T}_p$ l'opérateur de Hecke associé à la matrice $\left( \begin{smallmatrix} p & 0 \\ 0 & 1 \end{smallmatrix} \right)$.
Pour $p=2$, on a
$$
(\mathbf{T}_2 \mathbf{c}) ([\infty , 0]) (x,y) = \varepsilon (2x) \varepsilon (y) + \varepsilon (x) \varepsilon (2y) + \varepsilon (2x) \varepsilon (x+y) + \varepsilon (2y) \varepsilon (x+y).
$$
On laisse au lecteur le plaisir coupable de vérifier que, si $2x$, $2y$ et $x+y$ ne sont pas des entiers, on a
\begin{multline*}
\varepsilon (2x) \varepsilon (y) + \varepsilon (x) \varepsilon (2y) + \varepsilon (2x) \varepsilon (x+y) + \varepsilon (2y) \varepsilon (x+y) \\ - 2 \varepsilon (2x) \varepsilon (2y) - \varepsilon (x) \varepsilon (y) = 1/4
\end{multline*}
et donc que le symbole modulaire $\mathbf{c}$, à valeurs dans $\overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)$, est annulé par l'opérateur
$$\mathbf{T}_2 - 2[2]^* -1,$$
où l'on note $[m]^*$ le tiré en arrière par l'application $\mathbf{C}^2/ \mathbf{Z}^2 \to \mathbf{C}^2 / \mathbf{Z}^2$ induite par la multiplication par $m$, autrement dit $(x,y) \mapsto (mx,my)$.
Pour $p=3$ et $p=5$, on a respectivement
\begin{multline*}
(\mathbf{T}_3 \mathbf{c}) ([\infty , 0]) (x,y) = \varepsilon (3x) \varepsilon (y) + \varepsilon (x) \varepsilon (3y) + \varepsilon (3x) \varepsilon (x+y) \\ + \varepsilon (3y) \varepsilon (x+y) + \varepsilon (3x) \varepsilon (y-x) + \varepsilon (3y) \varepsilon (x-y).
\end{multline*}
et
\begin{multline*}
(\mathbf{T}_5 \mathbf{c}) ([\infty , 0]) (x,y) = \varepsilon (5x) \varepsilon (y) + \varepsilon (x) \varepsilon (5y) + \varepsilon (5x) \varepsilon (x+y) + \varepsilon (5y) \varepsilon (x+y) \\ + \varepsilon (5x) \varepsilon (y-2x) + \varepsilon (5y) \varepsilon (x+2y) - \varepsilon (y-2x) \varepsilon ( x+2y)
+ \varepsilon (5x) \varepsilon (y+2x) \\ + \varepsilon (5y) \varepsilon (x+3y) + \varepsilon (y+2x) \varepsilon ( x+3y) + \varepsilon (5x) \varepsilon (y-x) + \varepsilon (x-y) \varepsilon (5y) .
\end{multline*}
Au prix de fastidieux calculs on peut là encore vérifier que dans chacun de ces cas la relation
\begin{equation} \label{E:HeckeTrig}
(\mathbf{T}_p - p[p]^* -1) \mathbf{c} =0
\end{equation}
est satisfaite.
\section{Un théorème et quelques questions}
Il est naturel de conjecturer que les relations \eqref{E:HeckeTrig} sont vérifiées pour tout nombre $p$ premier. Un tel énoncé rappelle une conjecture de Busuioc \cite{Busuioc} et Sharifi \cite{Sharifi}; la démonstration récente que Sharifi et Venkatesh \cite{SharifiVenkatesh} en ont donnée implique aussi que les relations \eqref{E:HeckeTrig} sont vérifiées pour tout $p$ premier.\footnote{Le lien entre les travaux de Sharifi et Venkatesh et les questions abordées ici est expliqué dans le sixième paragraphe de cette introduction.}
En plus de cela on aimerait relever l'application $\mathbf{c}$ en un symbole modulaire --- nécessairement \emph{partiel}, au sens de la thèse de Dasgupta, voir \cite{Evil,DD} --- à valeurs dans $\mathcal{M} (\mathbf{C}^2 / \mathbf{Z}^2)$ plutôt que son quotient par les fonctions constantes. Ces deux desiderata font l'objet du théorème ci-dessous qui s'énonce plus naturellement en termes de cohomologie des groupes et requiert en partie d'augmenter le niveau.
L'application $\overline{\mathbf{S}} : g \mapsto \mathbf{c} ([\infty , g \cdot \infty ])$ définit en effet un $1$-cocycle\footnote{La relation de cocycle s'écrit
$$\overline{\mathbf{S}} (hg) = \overline{\mathbf{S}} (h) + h \cdot \overline{\mathbf{S}} (g).$$} de $\mathrm{SL}_2 (\mathbf{Z} )$ à valeurs dans
$\overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)$ et donc une classe de cohomologie dans
$$H^1 (\mathrm{SL}_2 (\mathbf{Z} ) , \overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2)).$$
\'Etant donné un entier $N$ strictement positif, on note comme d'habitude
$$\Gamma_0 (N) = \left\{ \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \mathrm{SL}_2 (\mathbf{Z}) \; : \; c \equiv 0 \ (\mathrm{mod} \ N) \right\}$$ le sous-groupe de $\mathrm{SL}_2 (\mathbf{Z})$ constitué des matrices qui fixent la droite $\langle e_1 \rangle$, engendrée par le premier vecteur de la base canonique de $\mathbf{Z}^2$, modulo $N$. On note enfin $\Delta^\circ_N \subset \Delta$
le sous-groupe engendré par les symboles $[r , s]$ avec $r, s \in \Gamma_0 (N) \cdot \infty \subset \mathbf{P}^1 (\mathbf{Q})$. Le sous-groupe $\Delta^\circ_N$ est donc engendré par les éléments $[a/Nc , b/Nd] \in \Delta$ avec $a$ et $b$ premiers avec $N$.
On note finalement $D_N$ le groupe abélien libre engendré par les combinaisons linéaires entières formelles de diviseurs positifs de $N$ et $D_N^\circ$ le sous-groupe constitué des éléments de degré $0$, c'est-à-dire des éléments $\delta = \sum_{d | N} n_d [d]$ tels que $\sum_{d|N} n_d =0$.
\begin{theorem*}
Il existe un morphisme $\delta \mapsto \mathbf{S}_\delta$ de $D_N$ vers le groupe des $1$-cocycles de $\Gamma_0 (N)$ à valeurs dans $\mathcal{M} (\mathbf{C}^2 / \mathbf{Z}^2 )$ vérifiant les propriétés suivantes.
1. On a
$$[\mathbf{S}_{[1]}] = [\overline{\mathbf{S}}] \neq 0 \quad \mbox{dans} \quad H^{1} (\Gamma_0 (N), \overline{\mathcal{M}} (\mathbf{C}^2 / \mathbf{Z}^2 ) ).$$
2. Pour tout entier $p$ premier ne divisant pas $N$ et pour tout $\delta \in D_N$, la classe de cohomologie de $\mathbf{S}_\delta$ dans $H^{1} (\Gamma_0 (N), \mathcal{M} (\mathbf{C}^2 / \mathbf{Z}^2 ) )$ annule l'opérateur $\mathbf{T}_{p} - p [p]^* - 1$.
3. Pour tout entier strictement positif $N$ et pour tout $\delta = \sum_{d | N } n_d [d]$ dans $D_N^\circ$, le $1$-cocycle $\mathbf{S}_\delta$ est cohomologue à un cocycle explicite $\mathbf{S}_{\delta}^*$ défini par
\begin{multline} \label{Sexplicit}
\mathbf{S}_{\delta}^*\left( \begin{array}{cc} a & * \\ Nc & * \end{array} \right) \\ = \left\{ \begin{array}{ll}
0 & \mbox{si } c=0, \\
\sum_{d | N} \frac{n_d }{d' c} \sum_{j \ {\rm mod} \ d' c} \varepsilon \left( \frac{1}{d' c} (y+j) \right) \varepsilon \left( dx - \frac{a}{d' c} (y+j) \right) & \mbox{sinon},
\end{array} \right.
\end{multline}
avec $dd'=N$.
\end{theorem*}
\medskip
\noindent
{\it Remarque.} Chaque application $\mathbf{S}_\delta^*$ définit en fait un symbole modulaire \emph{partiel} dans
$$\mathrm{Hom} (\Delta_N^\circ , \mathcal{M} (\mathbf{C}^2 / \mathbf{Z}^2 ))^{\Gamma_0 (N)}.$$
On le définit sur un symbole $[\infty , a/Nc]$ par $\mathbf{S}^*_\delta \left( \begin{smallmatrix} a & u \\ Nc & v \end{smallmatrix} \right)$ où $u$ et $v$ sont tels que $av-Ncu=1$.
\medskip
Ce texte est consacré à une vaste généralisation de ce théorème. Le but est de répondre aux questions suivantes~:
\begin{enumerate}
\item Que dire des produits de $n$ fonctions cotangentes lorsque $n \geq 3$ ?
\item Y a-t-il des résultats analogues pour des fonctions elliptiques ou, plus simplement, pour des fractions rationnelles ?
\end{enumerate}
Les réponses à ces questions sont énoncées au chapitre \ref{C:2} qui est une sorte de deuxième introduction dans laquelle les résultats généraux sont énoncés. Avant cela, dans le premier chapitre, on commence par détailler la construction de classes de cohomologie, pour des sous-groupes de $\mathrm{GL}_n$, qui généralisent la classe de $\mathbf{S}_{[1]}$ et ont toutes une origine topologique commune. Les classes que nous construisons sont à coefficients dans une partie de la cohomologie d'arrangements d'hyperplans dans des produits $A^n$ où $A$ est isomorphe au groupe multiplicatif ou à une courbe elliptique. Un point important, pour démontrer le théorème ci-dessus ou les résultats annoncés dans le chapitre \ref{C:2} est ensuite de montrer que cette partie de la cohomologie d'arrangements d'hyperplans peut être représentée par des formes méromorphes sur $A^n$. C'est l'objet du théorème \ref{P:Brieskorn} dont la démonstration occupe le chapitre \ref{S:OrlikSolomon}. Le calcul explicite des classes ainsi obtenues occupe le reste de l'ouvrage.
\medskip
Dans les deux derniers paragraphes de cette introduction on relie les cocycles $\mathbf{S}_\delta$, et leurs généralisations annoncées, à des objets plus classiques en théorie des nombres.
\section{\'Evaluation, terme constant, morphismes de Dedekind--Rademacher} Le slogan suivant, à retenir, distingue les avantages respectifs des cocycles $\mathbf{S}_\delta$ et $\mathbf{S}_\delta^*$.
\begin{quote}
{\it Les cocycles notés $\mathbf{S}$ peuvent être \emph{évalués} en des points de torsion alors que les cocycles notés $\mathbf{S}^*$ peuvent eux être \emph{calculés} en le point générique.}
\end{quote}
Nous verrons en effet que l'on peut contrôler l'ensemble des points $P \in \mathbf{C}^2 / \mathbf{Z}^2$ en lesquels les fonctions méromorphes $\mathbf{S}_\delta (g)$ sont régulières. Après évaluation en de tels points de torsion de $\mathbf{C}^2/\mathbf{Z}^2$ bien choisis, on retrouve alors des résultats plus classiques.
\medskip
\noindent
{\it Exemple.} Les fonctions dans l'image de $\mathbf{S}_{[1]}$ sont régulières en les points $(j/N,0)$, pour tous $j \in \{1 , \ldots , N-1 \}$. L'application
\begin{equation} \label{E:cocDR}
\Psi_{N} : \Gamma_0 (N ) \to \mathbf{C}; \quad g \mapsto \sum_{j=1}^{N-1} \mathbf{S}_{[1]} (g ) (j/N, 0)
\end{equation}
définit donc un morphisme de groupes. On retrouve ainsi
un multiple du morphisme de Dedekind--Rademacher \cite{Rademacher,Mazur} donné par
\begin{equation} \label{E:DR}
\Phi_N \left( \begin{array}{cc} a & b \\ N c & d \end{array} \right) = \left\{ \begin{array}{ll}
(N -1)b/d & \mbox{si } c=0 \\
\frac{(N -1)(a+d)}{Nc} + 12 \cdot \mathrm{sign}(c) \cdot D^N \left( \frac{a}{N |c|} \right) & \mbox{si } c \neq 0,
\end{array} \right.
\end{equation}
où, en notant $D : \mathbf{Q} \to \mathbf{Q}$ la \emph{somme de Dedekind} usuelle
$$D(a/c) = \frac{1}{c} \sum_{j=1}^{c-1}\varepsilon \left( \frac{j }{c} \right) \varepsilon \left( - \frac{j a}{c} \right) \quad \mbox{pour } c>0 \quad \mbox{et} \quad (a,c)=1,$$
on a $D^N (x) = D(x) - D(N x)$. De l'expression de $\mathbf{S}_{[1]}$ que l'on donnera au chapitre~\ref{C:7}, on peut plus précisément déduire --- comme dans \cite[\S 11]{Takagi} --- que
$$12 \cdot \Psi_N = \Phi_N .$$
\medskip
Plutôt que d'évaluer les fonctions méromorphes $\mathbf{S}^*_\delta (g)$ on peut considérer leur terme constant en $0$. Le théorème du paragraphe précédent implique alors le corollaire qui suit.
\begin{cor*}
Soit $\delta \in D_N^\circ$ tel que $\sum_{d | N} n_d d =0$ et $D^\delta (x) = \sum_{d | N} n_d D (dx)$. Alors l'application $\Psi_\delta : \Gamma_0 (N) \to \mathbf{C}$ donnée par
\begin{equation} \label{E:DD}
\Psi_\delta \left( \begin{array}{cc} a & * \\ N c & * \end{array} \right) = \left\{ \begin{array}{ll}
0 & \mbox{si } c=0 \\
\mathrm{sign}(c) \cdot D^\delta \left( \frac{a}{N |c|} \right) & \mbox{si } c \neq 0,
\end{array} \right.
\end{equation}
définit un morphisme de groupes.
\end{cor*}
On notera que le morphisme $12 \Psi_\delta$ est à valeurs entières et qu'il coïncide avec le \emph{morphisme de Dedekind--Rademacher modifié} $\Phi_\delta$ de Darmon et Dasgupta \cite{DD}.
\section{Relations avec les travaux de Kato et de Sharifi--Venkatesh}
Kato \cite{Kato} construit des unités de Siegel sur les courbes modulaires $Y_1 (N)$ à partir de fonctions theta sur la courbe elliptique universelle $E$ au-dessus de $Y_1 (N)$.
Il découle en effet de \cite[Proposition 1.3]{Kato} qu'étant donné un entier strictement positif $m$ premier à $6N$, il existe une fonction theta ${}_m \theta$ dans $\mathbf{Q} (E)^\times$ qui est une unité en dehors des points de $m$-torsion. La fonction ${}_m \theta$ est caractérisée par son diviseur dans $E[m]$ et des relations de distribution associées aux applications de multiplication par des entiers relativement premiers à $m$. Les unités de Siegel sont alors obtenues en tirant en arrière sur $Y_1 (N)$ ces fonctions theta par des sections de $N$-torsion.
Sharifi et Venkatesh \cite{SharifiVenkatesh} considèrent des analogues des fonctions ${}_m \theta$. Leur méthode permet en fait de construire des $1$-cocycles sur des sous-groupes $\Gamma$ de $\mathrm{GL}_2 (\mathbf{Z})$ à valeurs dans les groupes de $K$-théorie de degré $2$ des corps de fonctions de $\mathbf{C}^2 / \mathbf{Z}^2$ ou du carré $E^2$ de la courbe elliptique universelle. Les $1$-cocycles du premier type sont des applications de la forme $\Gamma \to K_2 (\mathbf{Q} (\mathbf{C}^2 / \mathbf{Z}^2))$.
En les composant avec
$$K_2 (\mathbf{Q} (\mathbf{C}^2 / \mathbf{Z}^2)) \to \mathcal{M} (\mathbf{C}^2 / \mathbf{Z}^2); \quad \{ f , g \} \mapsto \frac{d \log (f) \wedge d \log (g)}{dx \wedge dy},$$
où $x$ et $y$ sont les deux coordonnées de $\mathbf{C}^2$, on obtient des $1$-cocycles dont on peut vérifier qu'ils sont cohomologues aux cocycles du théorème énoncé ci-dessus. Le fait que les relations \eqref{E:HeckeTrig} sont bien vérifiées pour tout nombre premier $p$ découle alors de \cite[Lemma 4.2.9]{SharifiVenkatesh}.
Un aspect intéressant de notre construction est que nous obtenons ces cocycles à partir d'une classe purement topologique; l'émergence de fonctions méromorphes se déduit au final d'un théorème ``de type Brieskorn'' qui permet de représenter certaines classes de cohomologie singulière par des formes méromorphes. La construction est suffisamment maniable pour nous permettre de considérer plus généralement l'action de $\mathrm{GL}_n (\mathbf{Z})$ sur $\mathbf{C}^n / \mathbf{Z}^n$ ou sur $E^n$.
Les $1$-cocycles du second type chez Sharifi et Venkatesh sont des applications de la forme $\Gamma \to K_2 (\mathbf{Q} (E^2))$. Comme pour les unités de Siegel, on peut tirer en arrière ces cocycles par des sections de torsion. On obtient ainsi des homomorphismes du premier groupe d'homologie de $\Gamma$ vers le $K_2$ d'une courbe modulaire. Goncharov et Brunault \cite{Gon1,Bru1,Bru2} avaient déjà construit de tels homomorphismes en associant explicitement à certains symboles modulaires des symboles de Steinberg d'unités de Siegel. Obtenir ces morphismes par spécialisation d'un $1$-cocycle à valeurs dans $K_2 (\mathbf{Q} (E^2))$ permet de montrer que ces homomorphismes sont Hecke-équivariants.
En composant un $1$-cocycle $\Gamma \to K_2 (\mathbf{Q} (E^2))$ avec le symbole différentiel $\partial \log \wedge \partial \log$ puis en tirant le résultat en arrière par une section de torsion, on obtient un homomorphisme du premier groupe d'homologie de $\Gamma$ vers les formes modulaires de poids $2$ sur $Y_1(N)$. \'Etendu aux symboles modulaires (partiels), ce morphisme associe à un tel symbole une ``zeta modular form'' au sens de \cite[Section 4]{Kato}.
Cette construction s'étend là encore à l'action de $\mathrm{GL}_n (\mathbf{Z})$ sur le produit de $n$ courbes elliptiques universelles; voir chapitre \ref{C:2}. Les cocycles ainsi obtenus révèlent des relations cachées entre des produits de fonctions elliptiques classiques, relations gouvernées par l'homologie de sous-groupes de congruence dans $\mathrm{GL}_n (\mathbf{Z})$. On peut tirer de cela un certain nombres de conséquences arithmétiques \cite{Takagi,ColmezNous}; d'autres conséquences sont en préparation.
\section{Remerciements}
Ce texte est l'aboutissement d'une réflexion entamée il y a quelques années avec Akshay Venkatesh. Plusieurs idées sont issues de discussions avec lui (ainsi que l'impulsion d'écrire en français), un grand merci à lui pour sa générosité.
De toute évidence, ce travail doit beaucoup aux idées initiées et développées par Robert Sczech. Nous profitons de cette occasion pour lui exprimer notre gratitude.
P.C. remercie Samit Dasgupta pour lui avoir posé une question étincelle il y a 10 ans.
L.G. remercie l'IHES et le soutien de l'ERC de Michael Harris durant les premiers temps de ce projet.
N.B. tient à remercier Olivier Benoist pour de nombreuses discussions autour du chapitre 3.
Merci à Emma Bergeron pour les dessins.
Enfin, c'est un plaisir de remercier Henri Darmon, Clément Dupont, Javier Fresan, Peter Xu et les rapporteurs anonymes pour leurs nombreux commentaires.
\numberwithin{equation}{chapter}
\numberwithin{section}{chapter}
\resettheoremcounters
\chapter{Construction de cocycles : aspects topologiques} \label{C:1}
\section{Résumé}
Le cocycle $\mathbf{S}$ discuté en introduction est relié aux ``cocycles d'Eisenstein'' qui interviennent sous différentes formes dans la littérature, par exemple dans \cite{Sczech93,Nori,CD,CDG}. Un cocycle très proche est en fait explicitement considéré par Sczech dans \cite{Sczech92,Sczech93}.
Le premier but de ce texte est de donner une construction générale de cocycles de ``type Sczech'' et de montrer qu'ils ont une source topologique commune. La méthode utilisée consiste à relever certaines classes de cohomologie dans $H^{2n-1}(X^*)$, où $X$ est un $G$-espace et $X^*$ est égal à $X$ privé d'un nombre fini de ses points, en des classes de cohomologie {\'e}quivariante dans $H_{G}^{2n-1}(X^* )$. Elle rappelle la méthode proposée par Quillen \cite{Quillen} pour calculer la cohomologie d'un groupe linéaire sur un corps fini.
En pratique on considère essentiellement trois cas~:
\begin{itemize}
\item Le cas \emph{additif} (ou \emph{affine}). Dans ce cas $X = \mathbf{C}^n$, l'espace épointé $X^*$ est égal à $X$ privé du singleton $D=\{ 0 \}$ et $G = \mathrm{GL}_n (\mathbf{C})^\delta$, le groupe des transformations lin{\'e}aires de $\mathbf{C}^n$, consid{\'e}r{\'e} comme un groupe discret.
\item Le cas \emph{multiplicatif} (ou \emph{trigonométrique}). Dans ce cas $X = \mathbf{C}^n / \mathbf{Z}^n$, l'espace épointé $X^*$ est égal à $X$ privé d'un cycle $D$ de degré $0$ constitué de points de torsion et $G$ est un sous-groupe de $\mathrm{GL}_n (\mathbf{Z})$ qui préserve $D$.
\item Le cas \emph{elliptique}. Dans ce cas $X=E^n$, où est $E$ une courbe elliptique ou une famille de courbes elliptiques, l'espace épointé $X^*$ est égal à $X$ privé d'un cycle $D$ de degré $0$ constitué de points de torsion et $G$ est un sous-groupe d'indice fini de $\mathrm{GL}_n (\mathbf{Z})$, ou $\mathrm{GL}_n (\mathcal{O})$ si $E$ est à multiplication par $\mathcal{O}$, qui préserve $D$.
\end{itemize}
Dans chacun de ces cas l'action naturelle (linéaire) de $G$ sur $X$ donne lieu à un fibré d'espace total
$$EG \times_G X$$
au-dessus de l'espace classifiant $BG=EG/G$. L'isomorphisme de Thom associe alors à $D$ une classe dans
$H^{2n} ( EG \times_G X , EG \times_G X^* )$.
La construction de Borel identifie ce groupe au groupe de cohomologie {\it équivariante} $H_G^{2n} (X , X^*)$. On renvoie à l'annexe \ref{A:A} pour plus de détails sur la cohomologie équivariante; elle généralise à la fois la cohomologie des groupes et la cohomologie usuelle. La construction de Borel permet de retrouver les propriétés usuelles, comme par exemple associer une suite exacte longue à une paire de $G$-espaces. Dans la suite on considère
$$[D] \in H_G^{2n} (X , X^*)$$
et la suite exacte longue associée à la paire $(X,X^*)$~:
\begin{equation} \label{suiteexacte0}
H_G^{2n-1} (X ) \to H_G^{2n-1} ( X^*) \to H_G^{2n} (X , X^*) \to H_G^{2n} (X).
\end{equation}
L'origine topologique de nos cocycles repose alors sur le fait suivant~:
\medskip
\noindent
{\bf Fait.} {\it La classe $[D] \in H_G^{2n} (X , X^*)$ admet un relevé (privilégié)
$$E[D] \in H_{G}^{2n-1}(X^*).$$
}
\medskip
Nous démontrons ce fait au cas par cas dans les paragraphes qui suivent. Dans le cas affine il résulte du fait qu'un fibré vectoriel complexe plat possède une classe d'Euler rationnelle triviale, alors que dans le cas elliptique on le déduit d'un théorème de Sullivan qui affirme que la classe d'Euler rationnelle d'un fibré vectoriel à groupe structural contenu dans $\mathrm{SL}_n (\mathbf{Z})$ est nulle.
L'étape suivante part d'une remarque générale~: supposons que $Y/\mathbf{C}$ soit une vari\'et\'e {\em affine}, de dimension $n$, sur laquelle opère un groupe $G$, et supposons donnée une classe de cohomologie équivariante $\alpha \in H_G^{2n-1}(Y(\mathbf{C}))$. Puisque $H^i(Y(\mathbf{C}))$ s'annule pour
$i > n$, la suite spectrale pour la cohomologie \'equivariante donne une application
$$H_G^{2n-1}(Y(\mathbf{C})) \rightarrow H^{n-1}(G, H^{n}(Y(\mathbf{C}))).$$
Elle permet donc d'associer à $\alpha$ une classe de cohomologie du groupe $G$.
Dans les cas qui nous intéressent la variété $X^*$ n'est pas affine, mais on peut restreindre la classe $E[D]$ \`a un ouvert affine $U$. On voudrait aussi que $U$ soit invariant par $G$; mais un tel $U$ n'existe pas. Dans le cas additif où $X = \mathbf{C}^n$, on peut toutefois formellement prendre $U := `` \mathbf{C}^n - \bigcup_{\ell} \ell^{-1}(0)$''. Plus précisément, étant donné un ensemble fini $L$ de fonctionnelles affines, on pose
$U_L = \mathbf{C}^n - \cup_{\ell \in L} \ell^{-1}(0)$. En regardant $U$ comme la limite inverse des $U_L$, on associe à $E[D]$ une classe dans le groupe
$$H^{n-1}(G, \varinjlim_{L} H^j(U_L )).$$
La dernière étape de notre construction consiste à représenter $\varinjlim_{L} H^j(U_L)$ par des formes méromorphes. Dans le cas affine cela résulte d'un théorème célèbre de Brieskorn \cite{Brieskorn}~:
$$\varinjlim_{L} H^j(U_L ) = \begin{cases} 0, \ j > n, \\ \Omega^n_{\mathrm{aff}}, \ j = n, \end{cases} $$
où $\Omega^n_{\mathrm{aff}} \subset \Omega^n_{\mathrm{mer}} (\mathbf{C}^n )$ est une sous-algèbre de formes méromorphes, voir Définition~\ref{def:Omegamer}.
Le chapitre \ref{S:OrlikSolomon} est consacré à la démonstration d'un résultat de ce type dans les cas multiplicatif et elliptique.
En admettant pour l'instant ce théorème ``de type Brieskorn'', on consacre le présent chapitre à détailler la construction esquissée ci-dessus. Elle conduit à des classes
$$\mathbf{S}[D] \in H^{n-1} (G , \Omega^n_{\rm mer} (X )).$$
\section{Le cocycle additif}
\subsection{Une classe de cohomologie \'equivariante} \label{S21}
Soit $G= \mathrm{GL}_n (\mathbf{C})^\delta$, le groupe des transformations lin{\'e}aires de $\mathbf{C}^n$, consid{\'e}r{\'e} comme un groupe discret.
La repr{\'e}sentation linéaire\footnote{On identifie donc $\mathbf{C}^n$ à l'espace des vecteurs colonnes.}
$$G \rightarrow \mathrm{GL} (\mathbf{C}^n ); \quad g \mapsto ( z \in \mathbf{C}^n \mapsto gz)$$
donne lieu {\`a} un fibr{\'e} vectoriel $\mathcal{V}$, d'espace total $EG \times_G \mathbf{C}^n$, sur l'espace classifiant $BG$. On renvoie à l'annexe \ref{A:A} pour des rappels sur les espaces classifiants, les espaces simpliciaux et la cohomologie équivariante. Rappelons juste ici que si $X$ est un espace topologique muni d'une action continue de $G$, on a
$$H^*_G (X) = H^* (EG \times_G X) .$$
Lorsque $X$ est contractile, ce groupe se réduit à $H^* (BG)=H^* (G)$, la cohomologie du groupe $G$.
On peut considérer la classe de Thom du fibré $\mathcal{V}$~:
$$u \in H^{2n}_G ( \mathbf{C}^n , \mathbf{C}^n - \{ 0 \} )$$
à coefficients dans $\mathbf{C}$. Dans la suite exacte
$$\xymatrix{
H_G^{2n-1} (\mathbf{C}^n - \{ 0 \}) \ar[r] & H_G^{2n} (\mathbf{C}^n , \mathbf{C}^n - \{ 0 \} ) \ar[r]^{ \quad \quad c} & H_{G}^{2n} (\mathbf{C}^n )},$$
l'image de $u$ par l'application $c$ est la classe de Chern \'equivariante
$$c_{2n}(\mathcal{V}) \in H^{2n}(BG, \mathbf{C}),$$
qui est nulle parce que $\mathcal{V}$ est plat. On peut donc relever la classe $u$ en une classe dans $H_G^{2n-1} (\mathbf{C}^n - \{ 0 \})$.
\medskip
\noindent
{\it Remarque.} Ce relev\'e n'est pas unique, mais on peut consid\'erer la suite exacte
$$H^{2n-1} (G) \to H^{2n-1}_G (\mathbf{C}^n - \{ 0 \} ) \to H^{2n-1} (\mathbf{C}^n -\{ 0 \})$$
associée à la fibration $\mathcal{V}^* \to BG$, où $\mathcal{V}^*$ désigne le complémentaire de la section nulle dans $\mathcal{V}$,
Chaque relev\'e de $u$ dans $H_G^{2n-1} (\mathbf{C}^n - \{ 0 \})$ s'envoie sur la classe fondamentale dans $H^{2n-1} (\mathbf{C}^n -\{ 0 \})$.
\medskip
Le quotient $EG \times_G (\mathbf{C}^n - \{ 0 \})$ est une \emph{variété simpliciale}, c'est-à-dire un ensemble semi-simplicial dont les $m$-simplexes
$$(EG \times_G (\mathbf{C}^n - \{ 0 \}))_m = (EG_m \times (\mathbf{C}^n - \{ 0 \}) / G$$
sont des variétés et dont les applications de faces et de dégénérescences sont lisses. La \emph{réalisation grossière} $\| EG \times_G (\mathbf{C}^n - \{ 0 \}) \|$ de cette variété simpliciale est l'espace topologique
$$\| EG \times_G (\mathbf{C}^n - \{ 0 \}) \| = \sqcup_{m \geq 0} \Delta_m \times (EG \times_G (\mathbf{C}^n - \{ 0 \}))_m / \sim.$$
Ici $\Delta_m$ désigne le $m$-simplexe standard et les identifications sont données par
\begin{equation*}
(\sigma^i (t) , x) \sim (t , \sigma_i (x)), \quad t \in \Delta_{m-1} , \ x \in (EG \times_G (\mathbf{C}^n - \{ 0 \}))_m, \ i \in \{ 0 , \ldots , m \},
\end{equation*}
où $\sigma^i : \Delta_{m-1} \to \Delta_m$ désigne l'inclusion de la $i$-ème face et $\sigma_i : (EG \times_G (\mathbf{C}^n - \{ 0 \}))_m \to (EG \times_G (\mathbf{C}^n - \{ 0 \}))_{m-1}$ est l'application de face correspondante. On renvoie à l'annexe \ref{A:A} pour plus de détails sur ces objets.
Retenons que l'on a une projection continue naturelle de la réalisation grossière vers la réalisation géométrique de $EG \times_G (\mathbf{C}^n - \{ 0 \})$ et que cette application est une équivalence d'homotopie. En pratique nous travaillerons avec la réalisation grossière. En tirant en arrière le relevé de $u$ on obtient la proposition suivante.
\begin{proposition} \label{P4}
La classe de Thom $u$ admet un relev\'e dans
$$H^{2n-1} ( \| EG \times_G (\mathbf{C}^n - \{ 0 \}) \| ).$$
\end{proposition}
{\it Via} la théorie de Chern--Weil et les travaux de Mathai et Quillen \cite{MathaiQuillen}, nous construirons au paragraphe~\ref{S:61} du chapitre \ref{S:6} un relevé {\rm privilégié} de $u$ représenté par une forme différentielle.
\subsection{Effacer les hyperplans} \label{S:222}
\`A tout \'el\'ement $g\in G$ de premier vecteur ligne $v \in \mathbf{C}^n - \{ 0 \} $, on associe une forme linéaire
$$e_1^* \circ g : \mathbf{C}^n \to \mathbf{C}; \quad z \mapsto v z.$$
Pour tout $(k+1)$-uplet $(g_0 , \ldots , g_k ) \in G^{k+1}$, on note
$$U (g_0 , \ldots , g_k) = \{ z \in \mathbf{C}^n \; : \; \forall j \in \{ 0 , \ldots , k \}, \ e_1^* (g_j z) \neq 0 \}.$$
C'est un ouvert de $\mathbf{C}^n$ qui est égal au complémentaire d'un arrangement d'hyperplans~:
\begin{equation} \label{hypComp}
U (g_0 , \ldots , g_k) = \mathbf{C}^n - \cup_j H_j, \quad H_j = \mathrm{ker} (e_1^* \circ g_j ).
\end{equation}
L'action du groupe $G$ sur $\mathbf{C}^n$ préserve l'ensemble de ces ouverts~:
$$g \cdot U (g_0 , \ldots , g_k) = U (g_0 g^{-1} , \ldots , g_k g^{-1}).$$
Comme les variétés \eqref{hypComp} sont affines de dimension $n$, elles n'ont pas de cohomologie en degré $>n$. Nous montrons dans l'annexe \ref{A:A} qu'il correspond alors à la classe de cohomologie fournie par la proposition \ref{P4} une classe dans
$$H^{n-1} ( G , \lim_{\substack{\rightarrow \\ H_j}} H^n (\mathbf{C}^n - \cup_j H_j )).$$
\subsection{Une classe de cohomologie à valeurs dans les formes méromorphes}
\'Etant donné une forme linéaire $\ell$ sur $\mathbf{C}^n$, on définit une forme différentielle méromorphe sur $\mathbf{C}^n$ par la formule
\begin{equation}
\omega_\ell = \frac{1}{2i \pi} \frac{d\ell }{\ell }.
\end{equation}
Pour tout $g \in G$, on a
\begin{equation} \label{E:relg}
g^* \omega_{ g \cdot \ell} = \omega_{\ell}.
\end{equation}
D'après un théorème célèbre de Brieskorn \cite[Lemma 5]{Brieskorn}, confirmant une conjecture d'Arnold, l'application naturelle $\eta \mapsto [\eta]$ de la $\mathbf{Z}$-algèbre graduée engendrée par les formes $\omega_{\ell}$ et l'identité vers la cohomologie singulière à coefficients entiers de \eqref{hypComp} est un isomorphisme d'algèbre. Cela justifie d'introduire la définition suivante dans notre contexte.
\begin{definition} \label{def:Omegamer}
Soit
$$\Omega_{\rm aff}= \bigoplus_{p=0}^n \Omega_{\rm aff}^p$$
la $\mathbf{Z}$-algèbre graduée de formes différentielles méromorphes sur $\mathbf{C}^n$ engendrée par les formes $\omega_{\ell}$, avec $\ell \in (\mathbf{C}^n)^\vee - \{0 \}$, et par l'identité en degré $0$.
\end{definition}
Le th\'eor\`eme de Brieskorn implique que l'application naturelle
$$\Omega_{\rm aff} \to \lim_{\substack{\rightarrow \\ H_{j}}} H^\bullet (\mathbf{C}^n - \cup_{j} H_{j} )$$
est un isomorphisme. Finalement, on a démontré~:
\begin{proposition} \label{P:Sa}
La classe de cohomologie fournie par la proposition \ref{P4} induit une classe
\begin{equation} \label{E:Sa}
S_{\rm aff} \in H^{n-1} (G , \Omega_{\rm aff}^n ).
\end{equation}
\end{proposition}
Nous donnons deux représentants explicites de cette classe de cohomologie au chapitre suivant.
\section{Les cocycles multiplicatif et elliptique} \label{S:23}
On considère plus généralement une famille lisse $A \to S$ de groupes algébriques commutatifs dont les fibres sont connexes et de dimension $1$. Dans les cas multiplicatif et elliptique, chaque fibre est un groupe abélien isomorphe au groupe multiplicatif $\mathbf{G}_m$ dont le groupe des points complexes est isomorphe à $\mathbf{C}^\times = \mathbf{C} / \mathbf{Z}$, {\it via} l'application
$$ \mathbf{C} \to \mathbf{C}^\times ; \quad z \mapsto q_z = e(z) := e^{2i\pi z},$$
ou à une courbe elliptique. Soit $T \to S$ le produit fibré de $n$ copies de $A$ au-dessus de $S$. Le groupe $\mathrm{GL}_n (\mathbf{Z})$ opère sur $T$ par multiplication matricielle~: on voit un élément $\mathbf{a} \in T$ comme un vecteur colonne $\mathbf{a}=(a_1 , \ldots , a_n)$ où chaque $a_i \in A$, et un élément $g \in G$ envoie $\mathbf{a}$ sur $g\mathbf{a}$. Soit $G$ un sous-groupe de $\mathrm{GL}_n (\mathbf{Z})$.
\begin{definition}
Soit $c$ un entier supérieur à $1$. Un \emph{cycle invariant de $c$-torsion} sur $T$ est une combinaison linéaire formelle à coefficients entiers de sections de $c$-torsion de $T$ qui est invariante par $G$, autrement dit un élément
$$D \in H_G^0 (T[c]).$$
On dit de plus que $D$ est \emph{de degré $0$} si la somme de ses coefficients est égale à $0$.
\end{definition}
\medskip
\noindent
{\it Exemple.} Lorsque $A$ est une famille de courbes elliptiques, l'élément
$$[T[c] - c^{2n} \{ 0 \} ] \in H_G^0 (T[c])$$
est un cycle invariant de $c$-torsion de degré $0$.
\medskip
L'isomorphisme de Thom induit un isomorphisme
$$H_G^0 (T[c]) \to H_G^{2n} (T , T - T[c]);$$
on pourra se référer à \cite[Section 2]{Takagi} pour plus de détails sur cet isomorphisme et la deuxième partie du lemme ci-dessous.
Considérons maintenant la suite exacte longue de la paire $(T , T-T[c])$~:
\begin{equation} \label{suiteexacte}
H_G^{2n-1} (T ) \to H_G^{2n-1} ( T - T[c]) \to H_G^{2n} (T , T - T[c]) \stackrel{\delta}{\to} H_G^{2n} (T).
\end{equation}
\medskip
\begin{lem} \label{L:Sul}
{\rm (1)} Dans le cas multiplicatif, l'image dans $H_G^{2n} (T)$ d'un cycle invariant de $c$-torsion $D$ sur $T$ est {\rm rationnellement} nulle.
{\rm (2) (Sullivan \cite{Sullivan})} Dans le cas elliptique, un cycle invariant de $c$-torsion $D$ sur $T$ est de degré $0$ si et seulement si son image dans $H_G^{2n} (T)$, par l'application $\delta$ de la suite exacte \eqref{suiteexacte}, est {\rm rationnellement} nulle. Plus précisément, si $D$ est de degré $0$ son image est nulle dans $H_G^{2n} (T , \mathbf{Z} [1/c])$.
\end{lem}
\begin{proof} 1. Commençons par considérer le cas où $D = \{0\}$. On veut montrer que son image $[0]$ dans $H_G^{2n} (T)$ est nulle. Puisque cette image est contenue dans le noyau du morphisme
$$H_G^{2n} (T) \to H_G^{2n} (T- \{ 0 \})$$
induit par l'application de restriction, il suffit de montrer que son tiré en arrière par la section nulle $e=0^* [0]$ dans $H^{2n} (BG)$ est rationnellement nul. Par définition $e$ est la classe d'Euler du fibré normal de $\{0\}$ dans $EG \times_G T$ au-dessus de $BG$. Dans le cas multiplicatif il est isomorphe au fibré
$$EG \times_G \mathbf{C}^n \to BG$$
qui est complexe\footnote{Ce n'est plus vrai dans le cas elliptique car $E$ peut varier au-dessus de $S$. Dans ce cas on obtient un fibré en $\mathbf{R}^{2n}$ qui, même plat, peut avoir une classe d'Euler non nulle.} et plat. Les classes de Chern de ce fibré sont donc nulles et donc la classe d'Euler $e$ aussi. Ainsi l'image de $\{0\}$ dans $H_G^{2n} (T)$ est bien triviale.
Considérons maintenant le cas général où $D$ est un cycle de $c$-torsion. Son image $[c]_* (D)$, par l'application $[c]$ de multiplication dans les fibres, est égale au cycle $\{0\}$. On vient donc de montrer que l'image de la classe de $[c]_* (D)$ dans $H_G^{2n} (T)$ est nulle. L'application $[c]: T \to T$ étant un revêtement fini de degré $c^n$, le morphisme $[c]_*: H_G^{2n} (T) \to H_G^{2n} (T)$ est rationnellement injectif. L'image de $D$ dans $H_G^{2n} (T)$ est donc aussi (rationnellement) nulle.
2. Voir par exemple \cite[Lemma 9]{Takagi} pour plus de détails.
\end{proof}
Soit $D$ un cycle invariant de $c$-torsion $D$ sur $T$ que l'on supposera de plus de degré $0$ dans le cas où $A$ est une famille de courbes elliptiques. On peut alors relever $D$ en un élément de $H_G^{2n-1} (T - T[c])$. Toutefois, en général ce relevé n'est pas uniquement déterminé; l'ambiguïté est précisément $H_G^{2n-1} (T)$. On réduit cette ambiguïté en considérant la multiplication dans les fibres par un entier $s$, voir \cite{Faltings}. La multiplication dans les fibres induit une application propre $[s] : T \to T$ qui induit à son tour une application image directe $[s]_*$ en cohomologie (équivariante).
En supposant de plus $s$ premier à $c$, on a $[s]^{-1} T[c] = T[sc]$. L'immersion ouverte
$$j : T - T[sc] \to T -T[c]$$
induit un morphisme
$$j^* : H^\bullet (T-T[c]) \to H^\bullet (T-T[sc] ).$$
Avec un léger (abus de notation, on notera simplement $[s]_*$ la composition
$$\xymatrix{
H^\bullet (T-T[c]) \ar[r]^{j^*} & H^\bullet (T-T[sc] ) \ar[r]^{[s]_*} & H^\bullet (T-T[c])}$$
de l'application de restriction à $T-T[sc]$ par l'application d'image directe en cohomologie, et de même en cohomologie équivariante. On définit de même une application
$$[s]_* : H^\bullet (T , T - T[c]) \to H^\bullet (T , T - T[c])$$
en cohomologie et aussi en cohomologie équivariante.
Noter que, puisque $s$ est premier à $c$, on a
$$[s]_* ( [T[c] - c^{2n} \{ 0 \} ] ) = [T[c] - c^{2n} \{ 0 \} ].$$
En général, quitte à augmenter $s$, on peut supposer que $[s]_* (D) = D$. Cela motive la définition suivante.
\begin{definition} \label{Def1.7}
Soit
$$H_G^\bullet (T-T[c])^{(1)} \subset H_G^\bullet (T-T[c])$$
l'intersection, pour tout entier $s>1$ premier à $c$, des sous-espaces caractéristiques de $[s]_*$ associées à la valeur propre $1$, c'est-à-dire le sous-espace des classes de cohomologie complexes qui sont envoyées sur $0$ par une puissance de $[s]_*-1$.
\end{definition}
On définirait de même $H_G^\bullet (T)^{(1)}$, $H_G^\bullet ( T , T - T[c])^{(1)}$, et leurs analogues $H^\bullet (T-T[c])^{(1)}$ en cohomologie usuelle.
Comme dans le cas affine, la construction de Borel permet de calculer la cohomologie équivariante de $T$, resp. $T-T[c]$, comme cohomologie d'un espace fibré au-dessus de $BG$ de fibre $T$, resp. $T-T[c]$. On en déduit des suites spectrales compatibles à l'action de $[s]_*$~:
\begin{equation} \label{SST}
H^p (G , H^q (T )) \Longrightarrow H^{p+q}_G (T)
\end{equation}
et
\begin{equation} \label{SSTm}
H^p (G , H^q (T -T[c] )) \Longrightarrow H^{p+q}_G (T-T[c]).
\end{equation}
Dans le cas elliptique, les fibres sont compactes et $H_G^k (T)^{(1)} = \{ 0 \}$ si $k < 2n$. Les valeurs propres de $[s]_*$ sont donc des puissances $s^j$, avec $j>1$. Il en résulte que l'on peut projeter un relevé de $D$ sur le sous-espace propre associé à la valeur $1$ dans $H_G^{2n-1} (T - T[c])$; on renvoie à \cite[\S 3.2]{Takagi} pour les détails. On obtient ainsi que le cycle $D$ possède un relevé canonique dans $H_G^{2n-1} (T - T[c])^{(1)}$.
Dans le cas multiplicatif il n'est plus vrai que $D$ possède un relevé canonique, le relevé n'est défini que modulo $H_G^{2n-1} (T)^{(1)}$.
Comme expliqué en introduction, on voudrait maintenant restreindre cette classe à un ``ouvert affine $G$-invariant''. Un tel ouvert n'existant pas dans $T$, on considère là encore les réalisations géométriques des espaces simpliciaux correspondants.
Le but du prochain paragraphe est de montrer, en procédant comme dans le cas additif, les deux théorèmes qui suivent. Dans les deux cas on note $\Omega_{\rm mer} (T)$ l'algèbre graduée des formes différentielles méromorphes sur $T$ et $\Omega (T)$ la sous-algèbre constituée des formes partout holomorphes sur $T$.
\begin{theorem} \label{T:cocycleM}
Supposons que $A$ soit une famille de groupes multiplicatifs. Alors, tout cycle $G$-invariant $D$ donne lieu à une classe
$$S_{\rm mult} [D] \in H^{n-1} (G , \Omega^n_{\rm mer} (T))$$
qui est uniquement déterminée par $D$ modulo $\Omega^n (T)$.
\end{theorem}
\begin{theorem} \label{T:cocycleE}
Supposons que $A$ soit une famille de courbes elliptiques. Alors, tout cycle $G$-invariant $D$ de degré $0$ donne lieu à une classe
$$S_{\rm ell} [D] \in H^{n-1} (G , \Omega^n_{\rm mer} (T) )$$
qui est uniquement déterminée par $D$.
\end{theorem}
\medskip
\noindent
{\it Remarque.} La construction permet en outre de montrer que si $\mathbf{a} \in T-T[c]$ est $G$-invariant alors $S_{\rm mult} [D]$, resp. $S_{\rm ell} [D]$, est cohomologue à une classe de cohomologie à valeurs dans les formes régulières en $\mathbf{a}$. Le point de vue topologique décrit ci-dessus mène ainsi naturellement à la construction de classes de cohomologie ``à la Sczech'' comme celle évoquée en introduction.
\medskip
Sous certaines conditions supplémentaires sur le cycle $D$, nous décrivons des représentants explicites des classes $S_{\rm mult} [D]$ et $S_{\rm ell} [D]$ dans le chapitre suivant.
\section[Démonstration des théorèmes 1.7 et 1.8]{Démonstration des théorèmes \ref{T:cocycleM} et \ref{T:cocycleE}}
\subsection{Arrangement d'hyperplans trigonométriques ou elliptiques}
On fixe un groupe algébrique $A$, isomorphe au groupe multiplicatif ou à une courbe elliptique. Soit $n$ un entier naturel et soit $T=A^n$.
On appelle {\it fonctionnelle affine} toute application $\chi: T \rightarrow A$ de la forme
$$t_0 + \mathbf{a} \mapsto \chi_0(\mathbf{a})$$
où $t_0$ est un élément de $T_{\mathrm{tors}}$ et $\chi_0 : A^n \rightarrow A$ un morphisme de la forme
$\mathbf{a} = (a_1, \dots, a_n) \mapsto \sum r_i a_i$
où les $r_i$ sont des entiers.
On dit que $\chi$ est {\it primitif} si les coordonnées
$ (r_1, \dots, r_n) \in \mathbf{Z}^n$ de $\chi_0$ sont premières entre elles dans leur ensemble. Dans ce cas le lieu d'annulation de $\chi$ est un translaté de l'ensemble
$$\mathrm{ker}(\chi_0) := \{(a_1, \dots, a_n) \in A^n \; : \; \sum r_i a_i = 0\}.$$
Soit $\mathbf{v}_1, \dots, \mathbf{v}_{n-1}$ une base du sous-module de $\mathbf{Z}^n$ orthogonal au vecteur $\mathbf{r}$. Les
$\mathbf{v}_i$ définissent une application
$$A^{n-1} \longrightarrow A^n$$
qui est un isomorphisme sur son image $\mathrm{ker}(\chi_0)$.
(On peut en effet se ramener au cas où $(r_1, \dots, r_n) = (0,\dots, 0, 1)$.)
\begin{definition}
On appelle \emph{hyperplan} le lieu d'annulation (ou abusivement ``noyau'') d'une fonctionnelle affine primitive.
Un \emph{arrangement d'hyperplans} $\Upsilon$ est un fermé de Zariski dans $T$ réunion d'hyperplans. La taille $\# \Upsilon$ est le nombre d'hyperplans distincts de cet arrangement.
\end{definition}
Noter que de manière équivalente, un hyperplan est l'image d'une application
$A^{n-1} \rightarrow T$ linéaire relativement à un morphisme $A^{n-1} \rightarrow A^n$ induit par une matrice entière de taille $(n-1) \times n$.
\begin{lemma} \label{affine}
Si $\Upsilon$ contient $n$ fonctionnelles affines $\chi$ dont les vecteurs associés $\mathbf{r} \in \mathbf{Z}^n$ sont linéairement indépendants alors
le complémentaire $T-\Upsilon$ est affine.
Lorsque $A$ est une courbe elliptique, c'est même une équivalence.
\end{lemma}
\proof
S'il existe $n$ fonctionnelles affines dans $\Upsilon$ dont les vecteurs de $\mathbf{Z}^n$ associés sont linéairement indépendants alors ces fonctionnelles définissent une
application finie $T \rightarrow A^n$ et $\Upsilon$ est la pré-image de la réunion des axes de coordonnées dans $A^n$. Mais $(A-\{0\})^n$ est affine, et la pré-image d'une variété affine par une application finie est encore affine.
Maintenant, si $A$ est une courbe elliptique et que l'espace engendré par les vecteurs des fonctionnelles affines qui définissent $\Upsilon$ est un sous-module propre de $M \subset \mathbf{Z}^n$ alors la donnée d'un point de $T-\Upsilon$ et d'un vecteur de $\mathbf{Z}^n$ orthogonal à $M$ définit un plongement
$$A \longrightarrow T - \Upsilon.$$
Comme $A$ n'est pas affine, l'espace $T-\Upsilon$ ne l'est pas non plus.
\qed
\subsection[Cohomologie des arrangements d'hyperplans]{Opérateurs de dilatation et cohomologie des arrangements d'hyperplans} \label{S:dil}
On appelle {\em application de dilatation} toute application $[s] : T \rightarrow T$ associée à un entier $s>1$ et de la forme
$$[s] : \mathbf{a} \mapsto s \mathbf{a}.$$
L'image d'un hyperplan par une application de dilatation est encore un hyperplan.
Tout hyperplan est l'image d'un sous-groupe par un translation par point de \emph{torsion} dans $T$. \'Etant donné un arrangement d'hyperplans, on peut donc trouver une application de dilatation $[s]$ qui préserve cet arrangement, c'est-à-dire telle que $[s] \Upsilon \subset \Upsilon$.
Puisque $[s]$ préserve $\Upsilon$, on a une immersion ouverte
$$j : T - [s]^{-1} \Upsilon \to T - \Upsilon .$$
La dilatation $[s]$ induit une application de $T-[s]^{-1}\Upsilon$ vers $T-\Upsilon$ qui est à fibres finies. En abusant légèrement des notations on note $[s]_*$ la composition
$$\xymatrix{
H^\bullet (T-\Upsilon) \ar[r]^-{j^*} & H^\bullet (T-[s]^{-1} \Upsilon) \ar[r]^-{[s]_*} & H^\bullet (T-\Upsilon)}$$
de l'application de restriction à $T-[s]^{-1} \Upsilon$ par l'application d'image directe de $[s]$ en cohomologie.
On peut alors poser l'analogue suivant de la définition \ref{Def1.7}.
\begin{definition} \label{D:sep1}
Soit
$$H^*(T-\Upsilon, \mathbf{C})^{(1)} \subset H^*(T-\Upsilon, \mathbf{C})$$
l'intersection, pour tout entier $s>1$ tel que la dilatation $[s]$ préserve $\Upsilon$, des sous-espaces caractéristiques de $[s]_*$ associé à la valeur propre $1$, c'est-à-dire le sous-espace des classes de cohomologie complexes qui sont envoyées sur $0$ par une puissance de $[s]_*-1$.
\end{definition}
\subsection{Démonstration des théorèmes \ref{T:cocycleM} et \ref{T:cocycleE}}
On se place maintenant dans le cas où la famille lisse $A \to S$ de groupes algébriques commutatifs est soit simplement $A = \mathbf{G}_m$ ou une famille de courbes elliptiques au-dessus d'une courbe modulaire.
\`A tout vecteur ligne $v\in \mathbf{Z}^n$ dont les coordonnées sont premières entre elles dans leur ensemble, il correspond la fonctionnelle linéaire primitive
$$\chi_{v} : T \to A; \ \mathbf{a} \mapsto v \mathbf{a}.$$
Il découle du lemme \ref{L:Sul} que, sous les hypothèses des théorèmes \ref{T:cocycleM} et \ref{T:cocycleE}, le cycle $D$ se relève en un élément de
$$H_G^{2n-1} (T-T[c]) = H^{2n-1} (EG \times_G (T-T[c]))$$
et donc de
$$H^{2n-1} (\| EG \times_G (T-T[c]) \| ).$$
\`A tout $(k+1)$-uplet $\mathbf{g} \in (EG)_k$ on associe un ouvert
\begin{equation*} \label{U}
U (\mathbf{g}) = \left\{ \mathbf{a} \in T \; \Bigg| \; \begin{array}{l} \forall j \in \{ 0 , \ldots , k\}, \ \forall i \in \{ 0 , \ldots , n\}, \\ \chi_{e_i g_j } (\mathbf{a}) \notin A[c] \end{array} \right\} .
\end{equation*}
C'est le complémentaire d'un arrangement d'hyperplans dans $T$~:
\begin{equation} \label{E:arrgtHyp}
U (\mathbf{g}) = T - \cup_{j=0}^k \cup_{i=1}^n \cup_{a \in A[c]} \chi_{e_i g_j }^{-1} (a).
\end{equation}
L'action du groupe $G$ préserve l'ensemble de ces ouverts~:
$$h \cdot U (\mathbf{g}) = U (\mathbf{g} h^{-1}).$$
Il découle par ailleurs du lemme \ref{affine} que les variétés \eqref{E:arrgtHyp} sont affines; elles n'ont par conséquent pas de cohomologie en degré $>n$. Comme expliqué dans l'annexe~\ref{A:A}, il correspond à tout élément dans $H^{2n-1}_G (T -T[c])$ une classe de cohomologie dans
$$H^{n-1} (G , \lim_{\substack{\rightarrow \\ \Xi }} H^n (T - \cup_{\chi \in \Xi} \cup_{a \in A[c]} \chi^{-1} (a))),$$
où $\Xi$ désigne l'ensemble des translatés par $G$ des morphismes $\chi_{e_1}, \ldots , \chi_{e_n}$. Noter que toute classe dans $H^n ( U (\mathbf{g}))$ définie un élément de la limite inductive. En pratique, nos cocycles seront représentés par des formes régulières sur des $U (\mathbf{g})$.
L'élément de $H^{2n-1}_G (T -T[c])$ que nous considérons appartient à $H^{2n-1}_G (T -T[c])^{(1)}$, on obtient donc en fait une classe de cohomologie dans
$$H^{n-1} (G , \lim_{\substack{\rightarrow \\ \Xi }} H^n (T - \cup_{\chi \in \Xi} \cup_{a \in A[c]} \chi^{-1} (a))^{(1)}).$$
Il nous reste à représenter
$$\lim_{\substack{\rightarrow \\ \Xi }} H^n (T - \cup_{\chi \in \Xi} \cup_{a \in A[c]} \chi^{-1} (a))^{(1)},$$
par des formes méromorphes. C'est l'objet du chapitre \ref{S:OrlikSolomon} dans lequel nous démontrons un théorème ``à la Brieskorn'' dans ce contexte, cf. Théorème \ref{P:Brieskorn}.
Finalement le cycle $D$ donne donc lieu à un élément de $H^{n-1} (G , \Omega^n_{\rm mer} (T) )$. Dans le cas elliptique cet élément est uniquement déterminé. Ce n'est pas vrai dans le cas multiplicatif. Alors $T$ est elle-même affine et on en déduit un diagramme commutatif
$$\xymatrix{
H_G^{2n-1} (T)^{(1)} \ar[d] \ar[r] & H_G^{2n-1} (T-T[c] )^{(1)} \ar[d] \ar[r] & H_G^{2n} (T , T - T[c])^{(1)} \\
H^{n-1} (G , H^n (T)^{(1)} ) \ar[r] & H^{n-1} (G , \Omega^n_{\rm mer} (T))). &
}$$
La classe associée à $D$ dans $H^{n-1} (G , \Omega^n_{\rm mer} (T) )$ n'est donc déterminée qu'à un élément de $H^{n-1} (G , H^n (T)^{(1)} )$ près. En invoquant encore une fois le théorème~\ref{P:Brieskorn} on identifie cette indétermination à un élément de
$H^{n-1} (G , \Omega^n (T) )$.
Pour conclure, notons que la remarque qui suit les théorèmes \ref{T:cocycleM} et \ref{T:cocycleE} se démontre en partant non plus des hyperplans
$\chi_{e_j}^{-1} (A[c])$, pour $j \in \{1, \ldots , n \}$, translatés par les éléments de $G$ mais de $n$ hyperplans ne passant pas par $\mathbf{a}$, ce que l'on peut faire de manière $G$-équivariante puisque $\mathbf{a}$ est $G$-invariant.
\chapter{Énoncés des principaux résultats : cocycles explicites} \label{C:2}
\resettheoremcounters
\numberwithin{equation}{chapter}
Dans ce chapitre on décrit dans chacun des trois cas (affine, multiplicatif, elliptique) des cocycles explicites représentants les classes de cohomologie construites au chapitre précédent. Les démonstrations des résultats énoncés ici feront l'objet des chapitres suivants.
\section[Le cas affine]{Le cas affine : symboles modulaires universels et algèbre de Orlik--Solomon}
Un théorème célèbre de Orlik et Solomon \cite{OrlikSolomon} fournit une présentation, par générateurs et relations, de l'algèbre graduée $\Omega_{\rm aff}$ engendrée par les formes $\omega_{\ell}$ et l'identité. En particulier dans $\Omega^n_{\rm aff}$ l'ensemble des relations, entre les monômes de degré $n$, est engendré par
\begin{enumerate}
\item $\omega_{\ell_1} \wedge \ldots \wedge \omega_{\ell_{n}} = 0$ si $\det (\ell_1 , \ldots , \ell_{n} ) = 0$, et
\item $\sum_{i=0}^n (-1)^i \omega_{\ell_0} \wedge \ldots \wedge \widehat{\omega_{\ell_i}} \wedge \ldots \wedge \omega_{\ell_n} = 0$, pour tous $\ell_0 , \ldots , \ell_n$ dans $(\mathbf{C}^n)^\vee - \{ 0 \}$.
\end{enumerate}
Le fait que les relations ci-dessus soient effectivement vérifiées dans $\Omega^n_{\rm aff}$ n'est pas difficile, il est par contre remarquable qu'elles engendrent \emph{toutes} les relations.
Dans ce paragraphe on commence par expliquer que le fait que les relations soient vérifiées donne naturellement lieu à un cocycle de $G= \mathrm{GL}_n (\mathbf{C})^\delta$ à valeurs dans $\Omega^n_{\rm aff}$. On énonce alors un théorème qui relie ce cocycle à celui construit au chapitre \ref{C:1}. Finalement on explique que ce cocycle est la spécialisation d'un symbole modulaire universel.
\subsection{Un premier cocycle explicite}
De manière générale, si $X$ est un ensemble muni d'une action transitive de $G$, si $M$ est un $G$-module, si $F : X^n \to M$ est une fonction $G$-équivariante vérifiant
$$\sum_{i=0}^{n} (-1)^i F (x_0 , \ldots , \widehat{x}_i , \ldots , x_{n} ) = 0,$$
et si $x$ est un point dans $X$, alors
$$f_x (g_1 , \ldots , g_{n} ) := F(g_1^{-1} x , \ldots , g_{n}^{-1} x )$$
définit un $(n-1)$-cocycle du groupe $G$ à valeurs dans $M$. De plus, la classe de cohomologie représentée par ce cocycle ne dépend pas de $x$.
En appliquant ce principe général à
$$X = (\mathbf{C}^n )^\vee -\{0 \}, \quad M = \Omega^n_{\rm aff}, \quad F (\ell_1 , \ldots , \ell_{n}) = \omega_{\ell_1} \wedge \ldots \wedge \omega_{\ell_{n}} \quad \mbox{et} \quad x=e_1^*,$$
on obtient un $(n-1)$-cocycle homogène
\begin{equation} \label{cocycleSa}
\mathbf{S}_{\rm aff} : G^{n} \to \Omega_{\rm aff}^n; \quad (g_1 , \ldots , g_{n} ) \mapsto \omega_{\ell_1} \wedge \ldots \wedge \omega_{\ell_{n}},
\end{equation}
où $\ell_j (z) = e_1^*(g_j z)$.
\medskip
\noindent
{\it Remarque.} Le cocycle ainsi construit est homogène à droite, ce qui se traduit donc par la relation
\begin{equation} \label{invSa}
g^* \mathbf{S}_{\rm aff} (g_1 g^{-1} , \ldots , g_{n} g^{-1}) = \mathbf{S}_{\rm aff} (g_1 , \ldots , g_{n} ) ,
\end{equation}
qui découle de l'équation (\ref{E:relg}).
\medskip
\begin{theorem} \label{T:Sa}
Le cocycle $\mathbf{S}_{\rm aff}$ représente la classe (non nulle)
$$S_{\rm aff} \in H^{n-1} (G , \Omega_{\rm aff}^n )$$
de la proposition \ref{P:Sa}.
\end{theorem}
\subsection{Immeuble de Tits et symboles modulaires universels} \label{S:ARuniv}
Considérons maintenant l'immeuble de Tits $\mathbf{T}_n$. C'est le complexe simplicial dont les sommets sont les sous-espaces propres non-nuls de $\mathbf{C}^n$ et dont les simplexes sont les drapeaux de sous-espaces propres. Rappelons (voir \S \ref{S:Tits}) que l'immeuble de Tits s'identifie naturellement au bord à l'infini de l'espace symétrique associé à $\mathrm{GL}_n (\mathbf{C})$ dans la compactification géodésique. D'après le théorème de Solomon--Tits \cite{SolomonTits}, l'immeuble de Tits $\mathbf{T}_n$ a le type d'homotopie d'un bouquet de $(n-2)$-sphères. Son homologie réduite en degré $n-2$ est appelé module de Steinberg de $\mathbf{C}^n$; on note donc
$$\mathrm{St} (\mathbf{C}^n ) = \widetilde{H}_{n-2} (\mathbf{T}_n ).$$
Ash et Rudolph décrivent un ensemble explicite de générateurs de $\mathrm{St} (\mathbf{C}^n )$, appelés {\it symboles modulaires universels}, de la manière suivante~: soient $v_1 , \ldots , v_n$ des vecteurs non nuls de $\mathbf{C}^n$. Identifiant le bord de la première subdivision barycentrique $\Delta_{n-1} '$ du $(n-1)$-simplexe standard au complexe simplicial dont les sommets sont les sous-ensembles propres non vides de $\{ 1 , \ldots , n \}$, on associe aux vecteurs $v_1 , \ldots , v_n$ l'application simpliciale
\begin{equation} \label{app1}
\partial \Delta_{n-1} ' \to \mathbf{T}_n
\end{equation}
qui envoie chaque sommet $I \subsetneq \{ 1 , \ldots , n \}$ de $\partial \Delta_{n-1} '$ sur le sommet $\langle v_i \rangle_{i \in I}$ de $\mathbf{T}_n$. Le symbole modulaire universel $[v_1 , \ldots , v_n] \in \mathrm{St}(\mathbf{C}^n)$ est alors défini comme l'image de la classe fondamentale de $\partial \Delta_{n-1} '$ par l'application \eqref{app1}. D'après \cite[Prop. 2.2]{AshRudolph} le symbole $[v_1 , \ldots , v_n]$ vérifie les relations suivantes.
\begin{enumerate}
\item Il est anti-symétrique (la transposition de deux vecteurs change le signe du symbole).
\item Il est homogène de degré $0$~: pour tout $a \in \mathbf{C}^*$, on a $[ a v_1 , \ldots , v_n] =[v_1 , \ldots , v_n]$.
\item On a $[v_1 , \ldots , v_n]=0$ si $\det (v_1 , \ldots , v_n)=0$.
\item Si $v_0 , \ldots, v_{n}$ sont $n+1$ vecteurs de $\mathbf{C}^n$, on a
$$\sum_{j=0}^n (-1)^j [ v_0 , \ldots , \widehat{v}_j , \ldots , v_n] =0 .$$
\item Si $g \in \mathrm{GL}_n (\mathbf{C})$, alors $[g v_1 , \ldots , g v_n ] = g \cdot [v_1 , \ldots , v_n]$, où le point désigne l'action naturelle de $\mathrm{GL}_n (\mathbf{C})$ sur $\mathrm{St}(\mathbf{C}^n)$.
\end{enumerate}
D'après \cite[Prop. 2.3]{AshRudolph} les symboles modulaires universels engendrent $\mathrm{St}(\mathbf{C}^n)$. Kahn et Sun \cite[Corollary 2]{KahnSun} montrent que les relations ci-dessus fournissent en fait une {\it présentation} de $\mathrm{St}(\mathbf{C}^n)$.
Comme pour les symboles modulaires classiques discutés en introduction, étant donné un $G$-module $M$ une application $G$-équivariante
$\Phi : \mathrm{St}(\mathbf{C}^n) \to M$ induit un $(n-1)$-cocycle de $G$ à coefficients dans $M$~:
$$(g_1 , \ldots , g_n ) \mapsto \Phi (g_1^{-1} e_1 , \ldots , g_n^{-1} e_1).$$
\subsection{Un deuxième cocycle explicite}
Dans le cas affine notre principal résultat est le suivant.
\begin{theorem} \label{T:Sa}
L'application
\begin{multline} \label{E:applAff}
\mathrm{St}(\mathbf{C}^n) \to \Omega^n_{\rm aff}; \\ \quad [v_1 , \ldots , v_n] \mapsto \left\{ \begin{array}{ll}
0 & \mbox{si } \det (v_1 , \ldots , v_n)=0, \\
\omega_{v_1^*} \wedge \ldots \wedge \omega_{v_n^*} & \mbox{sinon}, \end{array} \right.
\end{multline}
où, dans le deuxième cas, $(v_1^* , \ldots , v_n^*)$ désigne la base duale à $(v_1 , \ldots , v_n)$, induit un $(n-1)$-cocycle $\mathbf{S}_{\rm aff}^*$ qui représente encore la classe (non nulle) $S_{\rm aff}$ dans $H^{n-1} (G , \Omega_{\rm aff}^n )$.
\end{theorem}
Le cocycle $\mathbf{S}_{\rm aff}^*$ est explicitement donné par
\begin{equation} \label{cocycleSa*}
\mathbf{S}_{\rm aff}^* : G^{n} \to \Omega_{\rm aff}^n; \quad (g_1 , \ldots , g_{n} ) \mapsto \omega_{\ell_1} \wedge \ldots \wedge \omega_{\ell_{n}},
\end{equation}
où cette fois $\ell_j$ est une forme linéaire sur $\mathbf{C}^n$ de noyau $\langle g_1^{-1} e_1 , \ldots , \widehat{g_j^{-1} e_1} , \ldots , g_{n}^{-1} e_1 \rangle$ (et identiquement nulle si les $g_j^{-1} e_1$ ne sont pas en position générale). Il découle encore immédiatement de l'équation \eqref{E:relg} au chapitre \ref{C:1} que $\mathbf{S}_{\rm aff}^*$ est homogène à droite, autrement dit qu'il vérifie la relation \eqref{invSa}. Il est par contre un peu moins évident qu'il définisse bien un cocycle. Les deux cocycles $\mathbf{S}_{\rm aff}$ et $\mathbf{S}_{\rm aff}^*$ sont considérés par Sczech dans une note non publiée \cite{Sczechprepub}; le cocycle $\mathbf{S}_{\rm aff}$ est le point de départ d'un article important de Sczech \cite{Sczech93} sur lequel nous reviendrons. Nous lui préférons $\mathbf{S}_{\rm aff}^*$ précisément parce qu'il provient de \eqref{E:applAff}.
Du fait que $\mathbf{S}_{\rm aff}^*$ provienne de l'application \eqref{E:applAff} on peut penser à ce cocycle comme à une classe de cohomologie \emph{relative} au bord de Tits. Nous donnerons un sens rigoureux à cela au cours de la démonstration que nous détaillons au chapitre \ref{S:6}. Notre démarche pour démontrer le théorème \ref{T:Sa} consiste à partir de la description topologique (\ref{E:Sa}) de $S_{\rm aff}$ et d'exhiber un représentant explicite grâce à la théorie de Chern--Weil. Outre qu'il permet de montrer que les cocycles $\mathbf{S}_{\rm aff}$ et $\mathbf{S}_{\rm aff}^*$ sont cohomologues et trouvent leur origine dans le relevé dans $H_G^{2n-1} (\mathbf{C}^n -\{ 0 \})$ de la classe fondamentale de $\mathbf{C}^n - \{ 0 \}$, l'avantage de ce point de vue est qu'il se généralise naturellement dans les cas multiplicatif et elliptique que nous discutons dans les paragraphes qui suivent.
\medskip
{\it Remarques.} 1. On peut reformuler le résultat de Ash--Rudolph \cite[Prop. 2.2]{AshRudolph} cité plus haut de la manière suivante~:
\begin{quote}
L'application qui à $(g_1 , \ldots , g_n ) \in G^n$ associe le symbole modulaire $[g_1^{-1} e_1 , \ldots , g_n^{-1} e_1]$ dans $\mathrm{St} (\mathbf{C}^n)$ est un $(n-1)$-cocycle homogène.
\end{quote}
Dans l'annexe \ref{A:B} on donne une démonstration topologique de cette assertion qui réalise la classe de cohomologie associée comme une classe d'obstruction; on pourra comparer cette manière de voir avec \cite{SharifiVenkatesh}.
2. L'application
$$\phi : \mathrm{St} (\mathbf{C}^n ) \to \mathrm{St}( (\mathbf{C}^n )^\vee )$$
qui à un symbole $[v_1 , \ldots , v_n]$ associe $0$ si $\det (v_1 , \ldots , v_n)= 0$, et $[v_1^* , \ldots , v_n^*]$ sinon, est un isomorphisme $G$-équivariant.
Le cocycle $\mathbf{S}_{\rm aff}$ se déduit alors de l'application $G$-équivariante
\begin{equation} \label{R:Sa}
\mathbf{S}_{\rm aff}^* \circ \phi^{-1} : \mathrm{St} ((\mathbf{C}^n )^\vee ) \to \Omega_{\rm aff}^n; \quad [\ell_1 , \ldots , \ell_n ] \mapsto \omega_{\ell_1} \wedge \ldots \wedge \omega_{\ell_n}.
\end{equation}
3. Les applications \eqref{R:Sa} et \eqref{E:applAff} sont $G$-équivariantes et surjectives. Or Andrew Putman et Andrew Snowden \cite{Putman} ont récemment démontré l'irréductibilité de la représentation de Steinberg $\mathrm{St} (\mathbf{C}^n)$. Les applications \eqref{R:Sa} et \eqref{E:applAff} sont donc des isomorphismes. Il s'en suit que les relations dans $\Omega^n_{\rm aff}$ sont engendrées par les images
des relations de Ash--Rudolph; on retrouve donc que les relations de Orlik et Solomon engendrent {\it toutes} les relations entre les formes $\omega_\ell$ dans $\Omega^n_{\rm aff}$.
\section[Le cas multiplicatif]{Le cas multiplicatif : formes différentielles trigonométriques et symboles modulaires} \label{S:2-2}
Considérons maintenant l'algèbre graduée $\Omega_{\rm mer} = \Omega_{\rm mer} ((\mathbf{C}^\times )^n)$ des formes méromorphes sur le produit de $n$ copies du groupe multiplicatif $\mathbf{C}^\times$ que l'on identifie au quotient $\mathbf{C}/ \mathbf{Z}$ {\it via} l'application
$$ \mathbf{C} / \mathbf{Z} \to \mathbf{C}^\times; \quad z \mapsto q=e^{2i\pi z}.$$
Rappelons que la fonction $\varepsilon (z) = \frac{1}{2i} \cot (\pi z)$ est égale à la somme --- régularisée au sens de Kronecker --- de la série
$$\frac{1}{2i \pi} \sum_{m \in \mathbf{Z}} \frac{1}{z+m}.$$
\'Etant $\mathbf{Z}$-périodique la forme $\varepsilon (z) dz$ définit bien une forme méromorphe sur $\mathbf{C}/ \mathbf{Z}$. {\it Via} l'identification $\mathbf{C} / \mathbf{Z} \cong \mathbf{C}^\times$ rappelée ci-dessus, on a
$$\varepsilon (z) dz = \frac{1}{2i\pi} \frac{dq}{q-1} - \frac{1}{4i \pi} \frac{dq}{q} \quad \mbox{et} \quad dz = \frac{1}{2i\pi} \frac{dq}{q}.$$
On note finalement $\overline{\Omega}_{\rm mer}$ le quotient de $\Omega_{\rm mer} ((\mathbf{C}^\times )^n)$ par la sous-algèbre engendrée par les formes régulières
$$dz_j = \frac{1}{2i\pi} \frac{dq_j}{q_j} \quad (j \in \{ 1 , \ldots , n \} ).$$
L'action (à gauche) des matrices $n \times n$ sur $\mathbf{C}^n$ (identifiées aux vecteurs colonnes) induit une action du monoïde $M_n (\mathbf{Z} )$
sur $\mathbf{C}^n / \mathbf{Z}^n$ et donc une action du groupe $\mathrm{SL}_n (\mathbf{Z})$. On commence par rappeler dans ce contexte les définitions des opérateurs de Hecke puis, comme dans le cas additif on décrit un premier cocycle explicite avant de faire le lien avec les symboles modulaires.
\subsection{Opérateurs de Hecke}
Soit $\Gamma$ un sous-groupe d'indice fini de $\mathrm{SL}_n (\mathbf{Z})$ et soit $S$ un sous-monoïde de $M_n (\mathbf{Z} )^\circ = M_n (\mathbf{Z} ) \cap \mathrm{GL}_n (\mathbf{Q})$ contenant $\Gamma$.
L'action, \emph{à droite}, de $M_n (\mathbf{Z} )^\circ$ sur $\Omega_{\rm mer}^n$, par tiré en arrière, induit une action de l'algèbre de Hecke associée au couple $(S, \Gamma)$ sur $H^{n-1} (\Gamma ,\Omega_{\rm mer}^n)$~: une double classe
$$\Gamma a \Gamma \quad \mbox{avec} \quad a \in S$$
induit un opérateur --- dit de Hecke --- sur $H^{n-1} (\Gamma ,\Omega_{\rm mer}^n)$, noté $\mathbf{T}(a)$, que l'on décrit comme suit~:
on décompose la double classe
$$\Gamma a \Gamma = \sqcup_j \Gamma a_j $$
où l'union est finie. Pour tout $g \in \Gamma$ on peut donc écrire
$$a_j g^{-1} = (g^{(j)})^{-1} a_{\sigma (j)} \quad \mbox{avec} \quad \sigma \mbox{ permutation et } g^{(j)} \in \Gamma.$$
\'Etant donné un cocycle (homogène à droite) $c : \Gamma^n \to \Omega^n_{\rm mer}$, on pose alors
$$\mathbf{T} (a) c (g_1 , \ldots , g_n) = \sum_j a_{j}^*c (g_1^{(j)} , \ldots , g_n^{(j)});$$
c'est encore un cocycle homogène (à droite) et on peut montrer que sa classe de cohomologie est indépendante du choix des $a_j$; voir \cite{RhieWhaples}.
Noter par ailleurs qu'un élément $a \in S$ induit une application $[a] : \mathbf{C}^n / \mathbf{Z}^n \to \mathbf{C}^n / \mathbf{Z}^n$. Notons $\mathrm{Div}_\Gamma$ le groupe (abélien) constitué des combinaisons linéaires entières formelles $\Gamma$-invariantes de points de torsion dans $\mathbf{C}^n / \mathbf{Z}^n$. Dans la suite on note $[\Gamma a \Gamma]$ l'application
$$[\Gamma a \Gamma] = \sum_j [a_j] : \mathrm{Div}_\Gamma \to \mathrm{Div}_\Gamma.$$
\subsection{Cocycles multiplicatifs I}
\'Etant donné un élément $D \in \mathrm{Div}_\Gamma$, la théorie de Chern--Weil nous permettra de construire, au chapitre \ref{S:chap9}, des représentants suffisamment explicites de la classe de cohomologie $S_{\rm mult} [D] \in H^{n-1} (\Gamma , \Omega_{\rm mer}^n )$ du théorème \ref{T:cocycleM} pour démontrer le théorème qui suit au \S \ref{S:3.2.2}.
\begin{theorem} \label{T:mult}
Soit $\chi_0 : \mathbf{C}^{n} / \mathbf{Z}^n \to \mathbf{C} / \mathbf{Z}$ un morphisme primitif. Il existe une application linéaire
$$\mathrm{Div}_\Gamma \to C^{n-1} (\Gamma , \Omega_{\rm mer}^n )^{\Gamma}; \quad D \mapsto \mathbf{S}_{\rm mult, \chi_0} [D]$$ telle que
\begin{enumerate}
\item chaque cocycle $\mathbf{S}_{\rm mult , \chi_0} [D]$ représente $S_{\rm mult} [D] \in H^{n-1} (\Gamma , \Omega_{\rm mer}^n )$;
\item chaque forme différentielle méromorphe $\mathbf{S}_{\rm mult , \chi_0} [D] (g_1 , \ldots , g_n)$ est régulière en dehors des hyperplans affines passant par un point du support de $D$ et dirigés par $\mathrm{ker} ( \chi_0 \circ g_j)$ pour un $j \in \{1 , \ldots , n \}$;
\item pour tout entier $s >1$, on a
$$\mathbf{S}_{\rm mult , \chi_0} [[s]^*D] = [s]^* \mathbf{S}_{\rm mult , \chi_0}[D] \quad \mbox{\emph{(relations de distribution)} et}$$
\item pour tout $a \in S$,
$$\mathbf{T} (a) \mathbf{S}_{\rm mult , \chi_0} [D] = \mathbf{S}_{\rm mult , \chi_0} [ [\Gamma a \Gamma]^* D] \quad \mbox{dans} \quad H^{n-1} (\Gamma , \Omega_{\rm mer}^n ).$$
\end{enumerate}
\end{theorem}
\medskip
\noindent
{\it Exemple.} Lorsque $\Gamma = \mathrm{SL}_n (\mathbf{Z})$ et $S = M_n (\mathbf{Z} )^\circ$, l'algèbre de Hecke est engendrée par les opérateurs
$$\mathbf{T}^{(k)}_p = \mathbf{T} (a^{(k)}_p) \quad \mbox{avec} \quad a^{(k)}_p=\mathrm{diag} (\underbrace{p, \ldots , p}_{k} , 1 , \ldots , 1 ),$$
où $p$ est un nombre premier et $k$ un élément de $\{ 1 , \ldots , n-1 \}$.
Maintenant, le tiré en arrière de $D_0 = \{ 0 \}$ par l'application $[\Gamma a^{(k)}_p \Gamma]$ est supporté sur l'ensemble de tous les points de $p$-torsion comptés avec multiplicité $\left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p$ sauf $0$ compté avec multiplicité $\left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p$.\footnote{Ici
$$ \left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p = \frac{(p^n-1) \cdots (p^{n-k+1} -1)}{(p^k-1) \cdots (p-1)} = \frac{(p^n-1)(p^n-p) \cdots (p^n-p^{k-1})}{(p^k-1) (p^k-p) \cdots (p^k - p^{k-1})}$$
est le coefficient $p$-binomial de Gauss, égal au nombre de sous-espaces vectoriels de dimension $k$ dans $\mathbf{F}_p ^n$.} On en déduit que
$$[\Gamma a^{(k)}_p \Gamma]^* D_0 = \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p [p]^*D_0 + \left( \left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p \right) D_0$$
et donc que la classe de cohomologie de $\mathbf{S}_{\rm mult, \chi_0} [D_0 ]$ annule l'opérateur
$$\mathbf{T}^{(k)}_p - \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p [p]^* - \left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p + \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p.$$
On retrouve ainsi les deux premiers points du théorème énoncé en introduction, le lien avec le cocycle $\mathbf{S}$ est plus précisément que pour $n=2$,
$$\mathbf{S}_{\rm mult , e_1^*} [D_0] (1 , g) = \mathbf{S}_{[1]} (g^{-1}) dx \wedge dy ,$$
où $x$ et $y$ sont les coordonnées, abscisse et ordonnée, dans $\mathbf{C}^2 / \mathbf{Z}^2$.
\medskip
\subsection{Symboles modulaires}
Soit
$$\Delta_n (\mathbf{Z} ) \subset \mathrm{St} (\mathbf{C}^n)$$
le sous-groupe abélien engendré par les symboles
$$[h] = [v_1 , \ldots , v_n] \quad \mbox{où} \quad h = ( v_1 | \cdots | v_n ) \in M_n (\mathbf{Z})^\circ .$$
On appelle \emph{symbole modulaire} tout élément $[h] = [v_1 , \ldots , v_n]$ de $\Delta_n (\mathbf{Z})$.
Par définition le $\mathbf{Z}$-module $\Delta_n (\mathbf{Z})$ est égal au quotient de $\mathbf{Z} [M_n (\mathbf{Z})^\circ ]$ par les relations (1), (2), (3) et (4) de Ash--Rudolph. On note $I_n \subset \mathbf{Z} [\mathrm{SL}_n (\mathbf{Z})]$ le sous-module engendré par les éléments
\begin{equation} \label{E:eltIdeal}
[h] + [hR] , \quad [h] + (-1)^n [hP] \quad \mbox{et} \quad [h] + [hU] + [hU^2] \quad (h \in \mathrm{SL}_n (\mathbf{Z}) ),
\end{equation}
avec
\begin{equation*}
R=(-e_2 |e_1 | e_3 | \cdots | e_n), \quad P = (e_2 | e_3 | \cdots | e_n | (-1)^{n+1} e_1)
\end{equation*}
et
$$U= (-e_1 - e_2 | e_1 | e_3 | \cdots | e_n ).$$
Bykovskii \cite{Bykovskii} démontre que
\begin{equation} \label{E:byk}
\Delta_n (\mathbf{Z}) = \mathbf{Z} [ \mathrm{SL}_n (\mathbf{Z}) ] / I_n .
\end{equation}
Un sous-groupe d'indice fini $\Gamma$ dans $\mathrm{SL}_n (\mathbf{Z})$ opère naturellement à gauche sur $\Delta_n (\mathbf{Z})$ et sur $\mathbf{Z}[\mathrm{SL}_n (\mathbf{Z} )]$, et \eqref{E:byk} est une identité entre $\mathbf{Z} [\Gamma]$-modules.
Dans l'introduction on a exhibé, dans le cas $n=2$, un premier lien entre les structures de $\Delta_2 (\mathbf{Z})$ et $\Omega^2_{\rm mer}$.
La situation générale, où $n$ est arbitraire, est plus subtile; elle fait l'objet du théorème qui suit. Commençons par naturellement étendre les définitions vues en introduction.
Le monoïde $M_n (\mathbf{Z} )^\circ$ opère \emph{à droite} sur $\mathrm{Hom} (\Delta_n (\mathbf{Z}) , \Omega_{\rm mer}^n )$ par
$$\phi_{|g} ([v_1 , \ldots , v_n]) = g^* \phi ([gv_1 , \ldots , gv_n ]) \quad (\phi \in \mathrm{Hom} (\Delta_n (\mathbf{Z}) , \Omega_{\rm mer}^n ), \ g \in M_n (\mathbf{Z} )^\circ).$$
Cette action induit en particulier une action de $\mathrm{SL}_n (\mathbf{Z})$ et, étant donné un sous-groupe d'indice fini $\Gamma \subset \mathrm{SL}_n (\mathbf{Z})$, on appelle \emph{symbole modulaire} pour $\Gamma$ à valeurs dans $\Omega_{\rm mer}^n$ un élément de
$$\mathrm{Hom} (\Delta_n (\mathbf{Z}) , \Omega_{\rm mer}^n )^{\Gamma }.$$
Soit maintenant $C$ un sous-ensemble $S$-invariant\footnote{Et donc aussi $\Gamma$-invariant.} de vecteurs non-nuls dans $\mathbf{Z}^n$. On note $\Delta_C \subset \Delta_n (\mathbf{Z})$ le sous-groupe engendré par les $[v_1 , \ldots , v_n]$, où chaque $v_j$ appartient à $C$.
\begin{definition}
Un \emph{symbole modulaire partiel} sur $C$ pour $\Gamma$ à valeurs dans $\Omega^n_{\rm mer}$ est un élément de
$$\mathrm{Hom} (\Delta_C , \Omega^n_{\rm mer} )^\Gamma.$$
\end{definition}
Soit $v_0 \in C$. Un symbole modulaire partiel $\phi$ donne lieu à un $(n-1)$-cocycle à valeurs dans $\Omega^n_{\rm mer}$
$$c_\phi : (g_1 , \ldots , g_n) \mapsto \phi (g_1^{-1} v_0 , \ldots , g_n^{-1} v_0 )$$
dont la classe de cohomologie ne dépend pas du choix de $v_0$ dans $C$; dans la suite on prendra toujours $v_0 = e_1$. Les opérateurs de Hecke opèrent sur les symboles modulaires partiels par
$$\mathbf{T} (a) \phi = \sum_j \phi_{|a_j} \quad \left( a \in S , \ \phi \in \mathrm{Hom} (\Delta_C , \Omega^n_{\rm mer} )^\Gamma \right),$$
de sorte que
$$\mathbf{T} (a) [c_\phi] = [c_{\mathbf{T} (a) \phi} ] \in H^{n-1} (\Gamma , \Omega^n_{\rm mer}) .$$
\subsection{Cocycles multiplicatifs II}
Un élément de $\mathrm{Div}_\Gamma$ est une fonction $\Gamma$-invariante $D : \mathbf{Q}^n / \mathbf{Z}^n \to \mathbf{Z}$ dont le support est contenu dans un réseau de $\mathbf{Q}^n$.
\begin{definition}
Soit $\mathrm{Div}_\Gamma^{\circ}$ le noyau du morphisme
$$\mathrm{Div}_\Gamma \to \mathbf{Z} [ (\mathbf{Q}/ \mathbf{Z})^{n-1}]$$
induit par la projection sur les $n-1$ dernières coordonnées.
\end{definition}
Lorsque $D \in \mathrm{Div}_\Gamma^{\circ}$ on peut représenter la classe $S_{\rm mult } [D]$ par un cocycle complètement explicite à valeurs dans $\Omega_{\rm mer}^n$, déduit d'un symbole modulaire partiel; c'est l'objet du théorème suivant que l'on démontre au \S \ref{S:8.3.3}. Pour simplifier les expressions on identifie, {\it via} la multiplication par la $n$-forme invariante $dz_1 \wedge \cdots \wedge dz_n$, l'espace $\Omega_{\rm mer}^n$ à $\mathcal{M} (\mathbf{C}^n / \mathbf{Z}^n)$, l'espace des fonctions méromorphes sur $\mathbf{C}^n / \mathbf{Z}^n$.
\begin{theorem} \label{T:mult2}
Soit $\Gamma$ un sous-groupe d'indice fini dans $\mathrm{GL}_n (\mathbf{Z})$ et $C = \Gamma \cdot \mathbf{Z} e_1 \subset \mathbf{Z}^n$. L'application linéaire
$$\mathrm{Div}_\Gamma^\circ \to \mathrm{Hom} (\Delta_C , \mathcal{M} (\mathbf{C}^n / \mathbf{Z}^n))^\Gamma ; \quad D \mapsto \mathbf{S}^*_{\rm mult} [D],$$
où
\begin{multline*}
\mathbf{S}^*_{\rm mult} [D] : [v_1 , \ldots , v_n] \mapsto \frac{1}{\det h} \sum_{w \in \mathbf{Q}^n / \mathbf{Z}^n} D(w) \cdot \\ \sum_{\substack{\xi \in \mathbf{Q}^n/\mathbf{Z}^n \\ h \xi = w \ (\mathrm{mod} \ \mathbf{Z}^n)}} \varepsilon (v_1^* + \xi_1 ) \cdots \varepsilon (v_{n}^* + \xi_n) ,
\end{multline*}
avec $h = ( v_1 | \cdots | v_n) \in M_n (\mathbf{Z})^\circ$, est bien définie et vérifie les propriétés suivantes.
\begin{enumerate}
\item Pour tout $D \in \mathrm{Div}_\Gamma^\circ$, le cocycle associé à $\mathbf{S}_{\rm mult}^* [D]$ représente la classe de cohomologie $S_{\rm mult} [D] \in H^{n-1} (\Gamma , \Omega^n_{\rm mer} )$.
\item Pour tout $a \in M_n (\mathbf{Z})^\circ$ préservant $C$, on a
$$\mathbf{T} (a) (\mathbf{S}^*_{\rm mult } [D] ) = \mathbf{S}^*_{\rm mult } [ [\Gamma a \Gamma]^* D].$$
\end{enumerate}
\end{theorem}
\medskip
\noindent
{\it Remarque.} On peut vérifier sur les formules que les relations de distributions
$$\mathbf{S}^*_{\rm mult } [[s]^*D] = [s]^* \mathbf{S}^*_{\rm mult } [D],$$
pour tout entier $s>1$ et pour tout $D \in \mathrm{Div}_\Gamma^\circ$, sont encore satisfaites. On peut de même montrer que pour tous
$g_1, \ldots , g_n \in \Gamma$, la fonction méromorphe
$$\mathbf{S}^*_{\rm mult } [D] (g_1 , \ldots , g_n)$$
est régulière en dehors des hyperplans affines passant par un point du support de $D$ et dirigés par
$$\langle g_1^{-1} e_1, \ldots , \widehat{g_j^{-1} e_1} , \ldots , g_n^{-1} e_1 \rangle \quad \mbox{pour un } j \in \{ 1 , \ldots , n \}.$$
On pourrait bien sûr ici remplacer $e_1$ par n'importe quel vecteur primitif $v_0$, à condition de prendre $C=\Gamma \cdot v_0$.
\medskip
\noindent
{\it Exemple.} Soit $N$ un entier strictement positif. Considérons le groupe $\Gamma$ constitué des matrices de $\mathrm{SL}_n (\mathbf{Z})$ qui fixent la droite $\langle e_1 \rangle$, engendrée par le premier vecteur de la base canonique de $\mathbf{Z}^n$, modulo $N$; dans la suite on note ce groupe $\Gamma_0 (N,n)$ ou simplement $\Gamma_0 (N)$ s'il n'y a pas d'ambiguïté sur la dimension. \`A toute combinaison linéaire formelle $\delta = \sum_{d | N} n_d [d]$ de diviseurs positifs de $N$ on associe
$$D_\delta = \sum_{d | N } n_d \sum_{j=0}^{d-1} \left[ \frac{j}{d} e_1 \right] ;$$
c'est un élément de $\mathrm{Div}_{\Gamma_0 (N)}$ qui appartient à $\mathrm{Div}_{\Gamma_0 (N)}^\circ$ si et seulement si on a $\sum_{d | N } n_d d =0$.
Par définition $D_\delta = \sum_{d | N } n_d \pi_d^* D_0 $, où $D_0$ désigne toujours l'élément de $\mathrm{Div}_{\Gamma_0 (N)}$ de degré $1$ supporté en $0$ et $\pi_d$ désigne la matrice diagonale $\mathrm{diag} (d, 1 , \ldots , 1)$. On a donc
$$\mathbf{S}_{\rm mult , \chi_0} [D_\delta ] = \sum_{d | N } n_d \mathbf{S}_{\rm mult , \chi_0} [\pi_d^* D_0 ] .$$
Noter que $D_0$ est invariant par le sous-groupe $\pi_d \Gamma_0 (N) \pi_d^{-1}$ de $\mathrm{SL}_n (\mathbf{Z})$.
On verra que pour $g_1 , \ldots , g_n \in \Gamma_0 (N)$ et pour tout diviseur $d$ de $N$, les cocycles
$$\mathbf{S}_{\rm mult , \chi_0} [\pi_d^* D_0 ] (g_1 , \ldots , g_n) \quad \mbox{et} \quad \pi_d^* \mathbf{S}_{\rm mult , \chi_0} [D_0 ] (\pi_d g_1 \pi_d^{-1} , \ldots , \pi_d g_n \pi_d^{-1})$$
représentent la même classe de cohomologie dans $H^{n-1} (\Gamma , \overline{\Omega}^n_{\rm mer})$
et donc qu'il en est de même pour
$$\mathbf{S}_{\rm mult , \chi_0} [D_\delta ] (g_1 , \ldots , g_n) \quad \mbox{et} \quad \sum_{d | N } n_d \pi_d^* \mathbf{S}_{\rm mult , \chi_0} [D_0 ] (\pi_d g_1 \pi_d^{-1} , \ldots , \pi_d g_n \pi_d^{-1}) .$$
\medskip
\noindent
{\it Remarque.} On retrouve le théorème énoncé en introduction en prenant $n=2$ et
$$\mathbf{S}_{\delta} (g^{-1}) = \mathbf{S}_{\rm mult , e_1^* } [D_{\delta^\vee} ] (1 , g ) \quad \mbox{et} \quad \mathbf{S}^*_{\delta} (g^{-1}) = \mathbf{S}^*_{\rm mult } [D_{\delta^\vee} ] (1 , g ),$$
avec $\delta^\vee = \frac{1}{N} \sum_{d |N } n_d d' [d]$.
\medskip
Le théorème \ref{T:mult2} implique que le cocycle $\mathbf{S}_{\rm mult , \chi_0} [D_\delta ]$, qui a l'avantage d'être régulier en la plupart des points de torsion, est cohomologue au cocycle qui se déduit du symbole modulaire partiel $\mathbf{S}_{\rm mult}^* [D_\delta ]$ et dont l'avantage est d'avoir une expression simple en le point générique. Un calcul direct permet en effet de vérifier qu'il associe à un élément $[v_1 , \ldots , v_n ] \in \Delta_C$, où les $n-1$ dernières coordonnées des vecteurs $v_j $ sont toutes divisibles par $N$, l'expression
$$\sum_{d | N } n_d d \pi_d^* \mathbf{c} (v_1^{(d)} , \ldots , v_n^{(d)} ),$$
où $v_j^{(d)}$ désigne le vecteur de $\mathbf{Z}^n$ obtenu à partir de $v_j$ en divisant par $d$ ses $n-1$ dernières coordonnées et
$$\mathbf{c} (v_1 , \ldots , v_n ) = \frac{1}{\det h} \sum_{\substack{\xi \in \mathbf{Q}^n/\mathbf{Z}^n \\ h \xi \in \mathbf{Z}^n }} \varepsilon (v^*_1 + \xi_1 ) \cdots \varepsilon (v^*_{n} + \xi_n ) , $$
avec toujours $h= ( v_1 | \cdots | v_n)$.
\subsection{Cocycles de Dedekind--Rademacher généralisés}
Soit $N$ un entier strictement positif et $\delta = \sum_{d | N} n_d [d]$ une combinaison linéaire formelle entière de diviseurs positifs de $N$ comme dans l'exemple précédent. Puisque $D_\delta$ est $\Gamma_0 (N)$-invariant, il lui correspond un cocycle (régulier) $\mathbf{S}_{\rm mult , e_1^*} [D_\delta ]$ du groupe $\Gamma_0 (N)$. On note
\begin{equation*}
\mathbf{\Psi}_\delta : \Gamma_0 (N)^n \to \mathcal{M} (\mathbf{C}^n / \mathbf{Z}^n); \quad (g_1 , \ldots , g_n ) \mapsto \mathbf{S}_{\rm mult , e_1^*} [D_\delta ] (g_1 , \ldots , g_n ).
\end{equation*}
C'est un $(n-1)$-cocycle du groupe $\Gamma_0 (N)$ à valeurs dans les fonctions méromorphes sur $\mathbf{C}^n / \mathbf{Z}^n$.
Sous l'hypothèse que $\delta$ est de degré $0$, c'est-à-dire $\sum_{d | N} n_d =0$, les points de $D_\delta$ ont tous une première coordonnée non nulle dans $\frac{1}{N} \mathbf{Z} / \mathbf{Z}$. Il découle donc du théorème \ref{T:mult} que l'image de $\mathbf{\Psi}_\delta$ est contenue dans les fonctions régulières en $0$.
\begin{proposition} \label{P:DRgen}
L'application
$$\Phi_\delta : \Gamma_0 (N )^n \to \mathbf{C}; \quad (g_1 , \ldots , g_n ) \mapsto \left[ \mathbf{\Psi}_\delta (g_1 , \ldots , g_n ) \right] (0) $$
définit un $(n-1)$-cocycle (à valeurs scalaires) qui représente une classe de cohomologie non-nulle $[\Phi_\delta] \in H^{n-1} (\Gamma_0 (N) , \mathbf{Q})$ telle que
\begin{enumerate}
\item la classe $d_n [\Phi_\delta ]$, où $2d_n$ désigne le dénominateur du $n$-ème nombre de Bernoulli, est entière, et
\item pour tout nombre premier $p$ qui ne divise pas $N$ et pour tout $k \in \{ 0 , \ldots , n-1 \}$, on a
$$\mathbf{T}_p^{(k)} [\Phi_\delta ] = \left( \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p + \left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p \right) \cdot [\Phi_\delta].$$
\end{enumerate}
\end{proposition}
Les cocycles $\Phi_\delta$ sont des généralisations à $\mathrm{SL}_n (\mathbf{Z})$ des cocycles de Dedekind--Rademacher pour $\mathrm{SL}_2 (\mathbf{Z} )$. Si $F$ est un corps de nombres totalement réel $F$ de degré $n$ au-dessus de $\mathbf{Q}$ et $L$ est un idéal fractionnaire de $F$, on peut considérer le groupe $U$ des unités totalement positives de $\mathcal{O}_F$ préservant $L$. Le choix d'une identification de $L$ avec $\mathbf{Z}^n$, et donc de $F$ avec $\mathbf{Q}^n$, permet de plonger $U$ dans $\mathrm{GL}_n (\mathbf{Z})$ {\it via} la représentation régulière de $U$ sur $L$. Dans le dernier paragraphe du chapitre \ref{S:chap9} on montre que l'évaluation de $\Psi_\delta$ sur la classe fondamentale dans $H_{n-1} (U , \mathbf{Z})$ est égale à la valeur en $0$ d'une stabilisation d'une fonction zeta partielle de $F$. Le théorème \ref{T:mult2} permet alors de retrouver les expressions de ces valeurs comme sommes de Dedekind explicites obtenues par Sczech \cite{Sczech93}; voir aussi \cite{CD,CDG}.
\subsection{Lien avec le cocycle de Sczech}
Les cocycles $\mathbf{S}^*_{\rm mult} [D]$ ($D \in \mathrm{Div}_\Gamma^\circ$) sont très proches du cocycle de Sczech \cite[Corollary p. 598]{Sczech93}. Il y a toutefois quelques différences notables~:
\begin{enumerate}
\item Par souci de simplicité nous nous sommes restreints ici\footnote{Mais pas dans \cite{ColmezNous} motivé par l'étude de toutes les valeurs critiques des fonctions $L$ correspondantes.} à ne considérer que de la cohomologie à coefficients triviaux et non tordue. De sorte qu'en prenant les notations de Sczech on a $P=1$ et $v=0$.
\item L'hypothèse que $D \in \mathrm{Div}_\Gamma^\circ$ conduit à une stabilisation du cocycle de Sczech qui fait disparaitre, au prix d'une augmentation du niveau, son paramètre $Q$ mais n'est défini qu'en un point \emph{générique} de la fibre $\mathbf{C}^n / \mathbf{Z}^n$.
\end{enumerate}
\section[Le cas elliptique]{Le cas elliptique : formes différentielles elliptiques et symboles modulaires}
Considérons maintenant une courbe elliptique $E$ au-dessus d'une base $Y$ et l'algèbre graduée $\Omega_{\rm mer} (E^n)$ des formes méromorphes sur le produit symétrique
$$E^n = E \times_Y \cdots \times_Y E$$
de $n$ copies de $E$.
Une matrice entière induit, par multiplication à gauche, une application $E^n \to E^n$. On en déduit donc là encore une action du semi-groupe $M_n (\mathbf{Z})^\circ = M_n (\mathbf{Z} ) \cap \mathrm{GL}_n (\mathbf{Q})$ sur $\Omega_{\rm mer} (E^n)$.
\subsection{Opérateurs de Hecke}
Soit $\Gamma$ un sous-groupe d'indice fini de $\mathrm{SL}_n (\mathbf{Z})$ et soit $S$ un sous-monoïde de $M_n (\mathbf{Z} )^\circ$ contenant $\Gamma$.
L'action, \emph{à droite}, de $M_n (\mathbf{Z} )^\circ$ sur $\Omega_{\rm mer} (E^n)$, par tiré en arrière, induit là encore une action de l'algèbre de Hecke associée au couple $(S, \Gamma)$ sur $H^{n-1} (\Gamma ,\Omega_{\rm mer} (E^n))$. On note encore $\mathbf{T}(a)$ l'opérateur de Hecke sur $H^{n-1} (\Gamma ,\Omega_{\rm mer} (E^n))$ associé à la double classe
$$\Gamma a \Gamma \quad \mbox{avec} \quad a \in S.$$
Noter par ailleurs qu'un élément $a \in S$ induit une application $[a] : E^n \to E^n$.
Notons $\mathrm{Div}_\Gamma(E^n)$ le groupe (abélien) constitué des combinaisons linéaires entières formelles $\Gamma$-invariantes de points de torsion dans $E^n$. Dans la suite on note $[\Gamma a \Gamma]$ l'application
$$[\Gamma a \Gamma] = \sum_j [a_j] : \mathrm{Div}_\Gamma(E^n) \to \mathrm{Div}_\Gamma(E^n).$$
Supposons maintenant que la base $Y$ soit une courbe modulaire $\Lambda \backslash \mathcal{H}$ avec $\Lambda$ sous-groupe d'indice fini dans $\mathrm{SL}_2 (\mathbf{Z})$.
\`A travers l'identification
\begin{equation} \label{E:ident}
\mathcal{H} \times \mathbf{R}^2 \stackrel{\simeq}{\longrightarrow} \mathcal{H} \times \mathbf{C}; \quad (\tau , (u,v)) \mapsto (\tau , u \tau +v )
\end{equation}
l'action de $\mathbf{Z}^2$ sur $\mathbf{R}^2$ par translation induit une action de $\mathbf{Z}^2$ sur $\mathcal{H} \times \mathbf{C}$ par
$$(\tau , z ) \stackrel{(m,n)}{\longmapsto} (\tau , z+ m \tau + n).$$
Le groupe $\mathrm{SL}_2 (\mathbf{R})$ opère à gauche sur $\mathcal{H} \times \mathbf{R}^2$ par
$$(\tau , (u,v)) \stackrel{g}{\longmapsto} (g \tau , (u,v) g^{-1} ).$$
\`A travers \eqref{E:ident} cette action devient
$$(\tau , z) \stackrel{\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)}{\longmapsto} \left( \frac{a\tau +b}{c\tau +d} , \frac{z}{c\tau + d} \right).$$
La courbe elliptique $E$, et plus généralement le produit fibré $E^n$, s'obtiennent comme double quotient
\begin{equation}
E^n= \Lambda \backslash \left[ ( \mathcal{H} \times \mathbf{C}^n) / \mathbf{Z}^{2n} \right] = \Lambda \backslash \left[ \mathcal{H} \times (\mathbf{R}^{2n} / \mathbf{Z}^{2n}) \right].
\end{equation}
Considérons maintenant un sous-monoïde $\Delta \subset M_2 (\mathbf{Z} )^\circ$ contenant $\Lambda$. L'action de $\Lambda$ sur \eqref{E:ident} s'étend au monoïde $\Delta$ par la formule
$$(\tau , (u,v)) \stackrel{g}{\longmapsto} (g \tau , \det (g) (u,v) g^{-1} ).$$
Cette action préserve le réseau $\mathbf{Z}^2$ dans $\mathbf{R}^2$ de sorte que tout élément $g \in \Delta$ induit une application
$$[g] : \mathcal{H} \times \mathbf{R}^{2n}/ \mathbf{Z}^{2n} \to \mathcal{H} \times \mathbf{R}^{2n} /\mathbf{Z}^{2n}.$$
Une double classe
$$\Lambda b \Lambda \quad \mbox{avec} \quad b \in \Delta$$
induit alors un op\'erateur
$$[\Lambda b \Lambda] : \Omega^n_\mathrm{mer}(E^n) \to \Omega^n_\mathrm{mer}(E^n)$$
lui aussi dit de Hecke, sur $\Omega_{\rm mer}^n (E^n)$ et donc aussi sur $H^{n-1} (\Gamma ,\Omega_{\rm mer} (E^n))$, que l'on notera simplement $T(b)$.
\subsection{Cocycles elliptiques I} \label{S:ellI} Soit $\mathrm{Div}_\Gamma^0 (E^n)$ le groupe des combinaisons linéaires entières formelles $\Gamma$-invariantes \emph{de degré $0$} de points de torsion dans $E^n$.
\'Etant donné un élément $D \in \mathrm{Div}_\Gamma^0 (E^n )$, la théorie de Chern--Weil nous permettra de construire, au chapitre \ref{S:chap10}, des représentants suffisamment explicites de la classe de cohomologie $S_{\rm ell} [D] \in H^{n-1} (\Gamma , \Omega_{\rm mer} (E^n) )$ du théorème \ref{T:cocycleE} pour démontrer le théorème suivant.
Dans la suite on fixe $n$-morphismes primitifs linéairement indépendants
$$\chi_1 , \ldots , \chi_n : \mathbf{Z}^n \to \mathbf{Z}.$$
On note encore $\chi_j : \mathbf{Q}^n \to \mathbf{Q}$ les formes linéaires correspondantes et $\chi_j : E^n \to E$ les morphismes primitifs qu'ils induisent. On pose
$$\chi = (\chi_1 , \ldots , \chi_n).$$
On démontre le théorème suivant au \S \ref{S:9.3.3}.
\begin{theorem} \label{T:ell}
Il existe une application linéaire
$$\mathrm{Div}_\Gamma^0 (E^n ) \to C^{n-1} (\Gamma , \Omega^n_{\rm mer}(E^n) )^{\Gamma}; \quad D \mapsto \mathbf{S}_{\rm ell , \chi} [D]$$
telle que
\begin{enumerate}
\item pour tout entier $s >1$, on a $\mathbf{S}_{\rm ell , \chi} [[s]^*D] = [s]^* \mathbf{S}_{\rm ell , \chi} [D]$ (\emph{relations de distribution});
\item chaque cocycle $\mathbf{S}_{\rm ell , \chi} [D]$ représente $S_{\rm ell} [D] \in H^{n-1} (\Gamma , \Omega_{\rm mer}^n (E^n) )$;
\item chaque forme différentielle méromorphe $\mathbf{S}_{\rm ell , \chi} [D] (g_1 , \ldots , g_n)$ est régulière en dehors des hyperplans affines passant par un point du support de $D$ et dirigés par $\mathrm{ker} ( \chi_i \circ g_j)$ pour $i,j \in \{1 , \ldots , n \}$;
\item pour tout $a \in S$,
$$\mathbf{T} (a) \left[\mathbf{S}_{\rm ell , \chi} [D] \right] = \left[\mathbf{S}_{\rm ell , \chi } [ [\Gamma a \Gamma]^* D] \right];$$
\item pour tout $b \in \Delta$,
$$T (b) \left[ \mathbf{S}_{\rm ell , \chi } [D] \right] = \left[ \mathbf{S}_{\rm ell , \chi } [ [\Lambda b \Lambda]^* D] \right].$$
\end{enumerate}
\end{theorem}
\medskip
\noindent
{\it Exemple.} Soit $c$ un entier supérieur à $2$. La combinaison linéaire de points de $c$-torsion $E^n [c] - c^{2n} \{0 \}$ dans $E^n$ est de degré $0$ et invariante par $\Gamma = \mathrm{SL}_n (\mathbf{Z})$; elle définit donc un élément $D_c \in \mathrm{Div}_\Gamma^0 (E^n)$. L'algèbre de Hecke associée au couple $(S , \Gamma)$, avec $S = M_n (\mathbf{Z} )^\circ$, est engendrée par les opérateurs
$$\mathbf{T}^{(k)}_p = \mathbf{T} (a^{(k)}_p) \quad \mbox{avec} \quad a^{(k)}_p=\mathrm{diag} (\underbrace{p, \ldots , p}_{k} , 1 , \ldots , 1 ),$$
où $p$ est premier et $k$ appartient à $\{ 1 , \ldots , n-1 \}$.
Soit $p$ un nombre premier. La base canonique de $\mathbf{C}$ au-dessus du point $\tau=i \in \mathcal{H}$ fournit une base de $E[p]$ et permet d'identifier $E^n [p]$ au groupe abélien des matrices $M_{n,2} (\mathbf{F}_p)$. Le tiré en arrière de $\{ 0 \}$ par l'application $[\Gamma a^{(k)}_p \Gamma]$ est supporté sur l'ensemble de tous les points de $p$-torsion et une matrice dans $M_{n,2} (\mathbf{F}_p)$ est comptée avec multiplicité\footnote{\'Egale au nombre de $k$-plans contenant un $2$-plan donné. Noter que ce nombre est $0$ si $k \leq 1$ il n'y a alors que des matrices de rang $\leq 1$ dans le support.} $\left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p$ si elle est de rang $2$, avec multiplicité
\footnote{\'Egale au nombre de $k$-plans contenant une droite donnée.} $\left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p$ si elle est de rang $1$ et avec multiplicité $\left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p$ si elle est nulle.
D'un autre côté on peut considérer la double classe $\Lambda \left( \begin{smallmatrix} p & 0 \\ 0 & 1 \end{smallmatrix} \right) \Lambda$ avec
$$\Lambda = \mathrm{SL}_2 (\mathbf{Z}).$$
La pré-image de $\{ 0 \}$ par l'application induite $E^n \to E^n$ est cette fois égale à l'ensemble des matrices de rang $1$ comptées avec multiplicité $1$ et la matrice nulle comptée avec multiplicité
$\left( \begin{smallmatrix} 2 \\ 1 \end{smallmatrix} \right)_p = p+1$.
On en déduit que pour $p$ premier à $c$, on a
\begin{multline*}
[\Gamma a_p^{k} \Gamma]^* D_c = \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p [p]^*D_c + \left( \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p \right) [ \Lambda \left( \begin{smallmatrix} p & 0 \\ 0 & 1 \end{smallmatrix} \right) \Lambda ]^* D_c \\ + \left( \left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p - \left( \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p \right) (p+1) - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p \right) D_c
\end{multline*}
et donc que la classe de cohomologie de $\mathbf{S}_{\rm ell , \chi} [D_c]$ annule l'opérateur
\begin{multline*}
\mathbf{T}_p^{(k)} - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p [p]^* - \left( \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p \right) T_p \\ - \left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p + \left( \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p \right) (p+1) + \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p.
\end{multline*}
\medskip
\subsection{Cocycles elliptiques II}
La $1$-forme différentielle $dz$ induit une trivialisation du fibré des $1$-formes relatives $\Omega^1_{E / Y}$. On utilise cette trivialisation pour identifier les sections holomorphes du fibré des formes méromorphes relatives $\Omega_{{\rm mer}, E^n / Y}^n$ à l'espace $\mathcal{M}_n (E^n)$ des fonctions $F$ méromorphes sur $(\mathcal{H} \times \mathbf{C}^n) / \mathbf{Z}^{2n}$ qui vérifie la propriété de modularité
$$F \left( \frac{a\tau +b }{c \tau +d } , \frac{z}{c\tau +d} \right) = (c\tau +d )^n F (z,\tau ) \quad \mbox{pour tout} \quad \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \Lambda .$$
Sous certaines hypothèses naturelles sur $D$ on peut, comme dans le cas multiplicatif, représenter la classe $S_{\rm ell} [D]$ par un cocycle complètement explicite déduit d'un symbole modulaire partiel. Le long d'une fibre $E_\tau$ au-dessus d'un point $[\tau] \in Y$ il est naturel de remplacer la fonction $\varepsilon$ par la série d'Eisenstein
\begin{equation}
E_1 (\tau , z) = \frac{1}{2i\pi} \sideset{}{{}^e}\sum_{\omega \in \mathbf{Z} + \mathbf{Z} \tau} \frac{1}{z+\omega} = \frac{1}{2i\pi} \lim_{M \to \infty} \sum_{m=-M}^M \left( \lim_{N \to \infty} \sum_{n=-N}^N \frac{1}{z+ m \tau + n } \right).
\end{equation}
Toutefois, deux problèmes se présentent:
\begin{enumerate}
\item La série d'Eisenstein n'est pas périodique en $z$ de période $\mathbf{Z}\tau + \mathbf{Z}$ mais vérifie seulement
\begin{equation} \label{E1per}
E_1 ( \tau , z + 1) = E_1 (\tau , z) \quad \mbox{et} \quad E_1 (\tau , z+ \tau) = E_1 (\tau , z ) - 1,
\end{equation}
voir \cite[III \ \S 4 \ (5)]{Weil}.
\item La série d'Eisenstein n'est pas non plus modulaire mais vérifie seulement que pour toute matrice $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \mathrm{SL}_2 (\mathbf{Z})$ on a
\begin{equation} \label{E1mod}
E_1 \left( \frac{a\tau + b}{c\tau +d} , \frac{z}{c\tau +d} \right) = (c\tau + d) E_1 (\tau , z) + cz,
\end{equation}
voir \cite[III \ \S 5 \ (7)]{Weil}.
\end{enumerate}
Pour remédier à ces deux problèmes on suppose dorénavant que $Y=Y_0 (N)$, avec $N$ entier et supérieur à $2$. La courbe $E$ est alors munie d'un sous-groupe $K \subset E[N]$ cyclique d'ordre $N$ et on peut considérer la fonction
\begin{equation*}
E_1^{(N)} (\tau , z) = \sum_{\xi \in K} E_1 \left( \tau , z + \xi \right) - N E_1 (\tau , z) = \sum_{j=0}^{N-1} E_1 \left( \tau , z + \frac{j}{N} \right) - N E_1 (\tau , z).
\end{equation*}
D'après \eqref{E1per} et \eqref{E1mod} cette fonction est en effet périodique et modulaire relativement au groupe $\Gamma_0 (N)$.
La fonction $E_1^{(N)} (\tau , z)$ est associée au cycle de torsion $K - N \{ 0 \}$ dans $E$ qui est \emph{de degré $0$}.
\medskip
Revenons maintenant au produit fibré $E^n$. On note $\mathrm{Div}_{\Gamma , K} (E^n) \subset \mathrm{Div}_\Gamma (E^n)$ le sous-groupe constitué des combinaisons linéaires entières formelles $\Gamma$-invariantes de points de torsion dans $K^n \subset E[N]^n$. Identifiant ces points de torsion au sous-groupe $\left( \frac{1}{N} \mathbf{Z} / \mathbf{Z} \right)^n \subset (\mathbf{Q} / \mathbf{Z} )^n$, on verra un élément de $\mathrm{Div}_{\Gamma , K} (E^n)$ comme une fonction $\Gamma$-invariante $D : (\mathbf{Q} / \mathbf{Z} )^n \to \mathbf{Z}$ dont le support est contenu dans le réseau $\left( \frac{1}{N} \mathbf{Z} \right)^n \subset \mathbf{Q}^n$.
\begin{definition}
Soit $\mathrm{Div}_{\Gamma , K}^{\circ} (E^n)$ l'intersection des noyaux des $n$ morphismes
$$\mathrm{Div}_{\Gamma , K} (E^n) \to \mathbf{Z} [(\mathbf{Q} /\mathbf{Z})^{n-1}]$$
induits par la projection sur les $n-1$ dernières coordonnées.
\end{definition}
Noter que les éléments de $\mathrm{Div}_{\Gamma , K}^{\circ} (E^n)$ sont de degré $0$.
\medskip
\noindent
{\it Exemple.} Supposons $\Gamma = \Gamma_0 (N,n)$. \`A toute combinaison linéaire formelle $\delta = \sum_{d | N} n_d [d]$ de diviseurs positifs de $N$ on associe
$$D_\delta = \sum_{d | N } n_d \sum_{\xi \in K[d]} (\xi , 0 , \ldots , 0),$$
où $K[d]$ désigne l'ensemble des éléments de $d$-torsion dans $K$. La combinaison formelle de points de torsion $D_\delta$ définit un élément de $\mathrm{Div}_{\Gamma , K}^{\circ}(E^n)$ si et seulement si $\sum_{d| N} n_d d =0$.
\medskip
Notre construction donne lieu à des classes de cohomologie à valeurs dans les formes méromorphes $\Omega_{\rm mer}^n (E^n)$. En restreignant ces formes aux fibres on obtient des classes de cohomologie à valeurs dans les sections holomorphes du fibré des formes méromorphes relatives $\Omega^n_{{\rm mer}, E^n /Y}$ et donc dans $\mathcal{M}_n (E^n )$. Le théorème suivant, que nous démontrons au \S \ref{S:9.4.2}, donne des représentants explicites de ces classes de cohomologie.
\begin{theorem} \label{T:ellbis}
Supposons $Y=Y_0 (N)$ avec $N$ entier supérieur à $2$ et notons $K \subset E[N]$ le sous-groupe cyclique d'ordre $N$ correspondant. Soit $\Gamma$ un sous-groupe d'indice fini dans $\mathrm{SL}_n (\mathbf{Z})$ et $C = \Gamma \cdot \mathbf{Z} e_1 \subset \mathbf{Z}^n$. L'application linéaire
$$\mathrm{Div}_{\Gamma , K}^{\circ} (E^n) \to \mathrm{Hom} (\Delta_C , \mathcal{M}_n (E^n) )^\Gamma ; \quad D \mapsto \mathbf{S}^*_{\rm ell} [D],$$
où
\begin{equation} \label{E:S*ell}
\mathbf{S}^*_{\rm ell} [D] : [v_1 , \ldots , v_n] \mapsto \frac{1}{\det h} \sum_{w \in E^n} D(w) \sum_{\substack{\xi \in E^n \\ h \xi = w}} E_1 (\tau , v_1^* + \xi_{1}) \cdots E_1 (\tau , v_n^* + \xi_{n}) ,
\end{equation}
avec $h = ( v_1 | \cdots | v_n) \in M_n (\mathbf{Z})^\circ$, est bien définie et représente la classe de cohomologie
$$S_{\rm ell} [D] \in H^{n-1} (\Gamma , \mathcal{M}_n (E^n) ).$$
\end{theorem}
Le fait que $\mathbf{S}^*_{\rm ell} [D]$ définit bien un cocycle est démontré d'une manière différente dans la thèse de Hao Zhang \cite[Theorem 4.1.6]{Zhang}. La démonstration repose sur la construction, par le second auteur, d'un $(n-1)$-cocycle du groupe $\mathrm{GL}_n (\mathbf{Z})$ à valeurs dans les fonctions méromorphes sur $\mathcal{H} \times \mathbf{C}^n \times \mathbf{C}^n$, voir \cite{CS}.
\medskip
\noindent
{\it Remarques.} 1. Le fait que les fonctions dans l'image de $\mathbf{S}^*_{\rm ell } [D]$, qui ne sont {\it a priori} définies que sur
$\mathcal{H} \times \mathbf{C}^n $, soient en fait $\mathbf{Z}^{2n}$-invariantes et modulaires de poids $n$ relativement à l'action de $\Lambda$ fait partie de l'énoncé du théorème mais peut être vérifié à la main. Comme dans le cas multiplicatif on peut aussi vérifier directement sur les formules que les relations de distribution
$$\mathbf{S}^*_{\rm ell } [[s]^*D] = [s]^* \mathbf{S}^*_{\rm ell } [D],$$
pour tout entier $s>1$ et pour tout $D \in \mathrm{Div}_{\Gamma , K}^{\circ} (E^n)$, sont encore vérifiées et que pour tous
$g_1, \ldots , g_n \in \Gamma$, la forme différentielle méromorphe $\mathbf{S}^*_{\rm ell } [D] (g_1 , \ldots , g_n)$ est régulière en dehors des hyperplans affines passant par un point du support de $D$ et dirigés par
$$\langle g_1^{-1} e_1, \ldots , \widehat{g_j^{-1} e_1} , \ldots , g_n^{-1} e_1 \rangle \quad \mbox{pour un } j \in \{ 1 , \ldots , n \}.$$
2. Pour tout entier $m$ premier à $N$, l'opérateur de Hecke $T_m$ correspondant à la double classe
$$\Lambda\left( \begin{array}{cc} m & 0 \\ 0 & 1 \end{array} \right) \Lambda = \bigsqcup_{\substack{a, d >0 \\ ad=m}} \bigsqcup_{b=0}^{d-1} \Lambda \left( \begin{array}{cc} a & b \\ 0 & d \end{array} \right)$$
opère sur une fonction méromorphe $F$ du type
$$F = E_1 (\tau , z_1 ) \ldots E_n (\tau , z_n),$$
par l'expression familière
$$T_m F = m^n \sum_{\substack{a, d >0 \\ ad=m}} \sum_{b=0}^{d-1} \frac{1}{d^n} E_1 \left( \frac{a\tau+b}{d} , a z_1 \right) \ldots E_n \left( \frac{a\tau+b}{d} , az_n \right).$$
\medskip
\medskip
\noindent
{\it Exemple.} Soit $\Gamma = \Gamma_0 (N,n)$ et soit $\delta = \sum_{d | N} n_d [d]$ une combinaison linéaire formelle de diviseurs positifs de $N$ telle que $\sum_{d| N} n_d d =0$. Après restriction aux fibres, le théorème \ref{T:ell} associe à tout $n$-uplet $\chi$ de morphismes primitifs et linéairement indépendants $\mathbf{Z}^n \to \mathbf{Z}$ un $(n-1)$-cocycle régulier $\mathbf{S}_{\rm ell , \chi } [D_\delta]$ du groupe $\Gamma$ à valeurs dans $\mathcal{M}_n (E^n )$. Le théorème \ref{T:mult2} implique que ce cocycle est cohomologue au cocycle qui se déduit du symbole modulaire partiel $\mathbf{S}_{\rm mult}^* [D_\delta ]$ et dont un calcul permet de vérifier qu'il associe à un élément $[v_1 , \ldots , v_n ] \in \Delta_C$, où les $n-1$ dernières coordonnées des vecteurs $v_j $ sont toutes divisibles par $N$, l'expression
$$\sum_{d | N } n_d d \pi_d^*\mathbf{E}( \tau ; v_1^{(d)} , \ldots , v_n^{(d)} ) ,$$
où $v_j^{(d)}$ désigne le vecteur de $\mathbf{Z}^n$ obtenu à partir de $v_j$ en divisant par $d$ ses $n-1$ dernières coordonnées et
\begin{equation*}
\mathbf{E} ( \tau ; v_1 , \ldots , v_n ) = \frac{1}{\det h} \sum_{\substack{\xi \in E^n \\ h \xi = 0 }} E_1 (\tau , v_1^* + \xi_{1}) \cdots E_1 (\tau , v_n^* + \xi_{n}) ,
\end{equation*}
avec toujours $h= ( v_1 | \cdots | v_n)$.
\subsection{Relèvement theta explicite}
Soit $N$ un entier supérieur à $4$ et soit $E$ la courbe elliptique universelle au-dessus de la courbe modulaire $Y_1 (N) = \Gamma_1 (N) \backslash \mathcal{H}$, où
$$\Gamma_1 (N) = \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in \mathrm{SL}_2 (\mathbf{Z}) \; : \; \left( \begin{array}{c} a \\ c \end{array} \right) \equiv \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \ (\mathrm{mod} \ N ) \right\};$$
c'est l'espace des modules fin qui paramètre la donnée d'une courbe elliptique $E_\tau = \mathbf{C} / (\mathbf{Z} \tau + \mathbf{Z})$ et d'un point de $N$-torsion $[1/N]$.
Soit $\Gamma$ un sous-groupe de congruence contenu dans $\Gamma_0 (N,n)$ et $\delta = \sum_{d | N} n_d [d]$ une combinaison de diviseurs positifs de $N$ telle que $\sum_{d | N} n_d d=0$. La fonction $D_\delta$ appartient alors à $\mathrm{Div}_\Gamma^0 (E^n)$. Prenons
$$\chi = (e_1^*, e_1^* + e_2^* , \ldots , e_1^* + e_n^*).$$
Sous l'hypothèse que $\delta$ est de degré $0$, c'est-à-dire $\sum_{d | N} n_d =0$, les points de $D_\delta$ ont tous une première coordonnée non nulle dans $\frac{1}{N} \mathbf{Z} / \mathbf{Z}$. Il découle donc du théorème \ref{T:ell} que l'image de $\mathbf{S}_{\rm ell , \chi} [D_\delta ]$ est contenue dans les formes régulières en $0$.
Après restriction de ces formes aux fibres et évaluation en $0$ on obtient une classe de cohomologie à valeurs dans l'espace $M_n (Y_1 (N))$ des formes modulaires de poids $n$ sur $Y_1 (N)$. On note $\Theta [D_\delta ] : \Gamma^n \to M_n (Y_1 (N))$ le cocycle correspondant.
\begin{theorem}
Le cocycle $\Theta [D_\delta ]$ représente une classe de cohomologie dans $H^{n-1} (\Gamma , M_n (Y_1 (N)))$ qui réalise le relèvement theta des formes modulaires paraboliques de poids $n$ pour $Y_1 (N)$ dans la cohomologie $H^{n-1} (\Gamma , \mathbf{C})$.
Le relèvement associe à une forme modulaire parabolique $f \in M_n (Y_1 (N))$ la classe de cohomologie du cocycle
$$\Theta_{f} [D_\delta ] = \langle \Theta [D_\delta ] , f \rangle_{\rm Petersson} : \Gamma^n \to \mathbf{C}$$
et pour tout entier $p$ premier ne divisant pas $N$ et pour tout $k\in \{ 1 , \ldots , n-1 \}$, on a
\begin{multline*}
\mathbf{T}_p^{(k)} \left[ \Theta_{ f} [D_\delta ] \right] = \left( \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p \right) \left[\Theta_{ T_p f} [D_\delta ] \right] \\ + \left( \left( \begin{smallmatrix} n \\ k \end{smallmatrix} \right)_p - (p+1) \left( \left( \begin{smallmatrix} n-1 \\ k-1 \end{smallmatrix} \right)_p - \left( \begin{smallmatrix} n-2 \\ k-2 \end{smallmatrix} \right)_p \right) \right) \left[ \Theta_{ f} [D_\delta ] \right] .
\end{multline*}
\end{theorem}
\medskip
\noindent
{\it Remarques.} 1. Si $f$ est une forme propre pour $T_p$ de valeur propre $a_p$, le relevé de $f$ est propre pour les opérateurs de Hecke $\mathbf{T}_p^{(k)}$ Lorsque $n=3$ les valeurs propres correspondantes aux opérateurs $\mathbf{T}_p^{(1)}$ et $\mathbf{T}_p^{(2)}$ sont respectivement $a_p+p^2$ et $pa_p+1$; ce relèvement est notamment étudié dans \cite{AshGraysonGreen}.
2. L'hypothèse $\sum_{d | N} n_d d =0$ faite sur $D_\delta$ implique que les formes modulaires dans l'image de $\Theta [D_\delta ]$ sont paraboliques à l'infini. Après évaluation en l'infini de $\Theta [D_\delta ]$ on retrouve le cocycle de Dedekind-Rademacher généralisé $\Psi_\delta$.
\medskip
Si $F$ est un corps de nombres totalement réel de degré $n$ au-dessus de $\mathbf{Q}$ et $L$ est un idéal fractionnaire de $F$, on peut là encore considérer le groupe $U$ des unités totalement positives de $\mathcal{O}_F$ préservant $L$ et le plonger dans $\mathrm{GL}_n (\mathbf{Z})$. On peut cette fois montrer que l'évaluation de $\Theta_\delta [D]$ sur la classe fondamentale dans $H_{n-1} (U , \mathbf{Z})$ est une forme modulaire de poids $n$ égale à la restriction à la diagonale d'une série d'Eisenstein partielle\footnote{Associée à la classe de $L$.} sur une variété modulaire de Hilbert; voir \cite[\S 13.3]{Takagi}. Le théorème \ref{T:ellbis} permet alors d'exprimer cette restriction comme une combinaison linéaire de produits de séries d'Eisenstein $E_1$.
\medskip
\noindent
{\it Remarque.} Lorsque $n=2$, le théorème \ref{T:ellbis} est à rapprocher d'un théorème de Borisov et Gunnells \cite[Theorem 3.16]{BorisovGunnells} selon lequel l'application qui à un symbole unimodulaire $[a/c : b/d]$, avec $c,d \neq 0$ modulo $N$, associe
$$E_1 \left(\tau , \frac{c}{N} \right) E_1 \left(\tau , \frac{d}{N} \right)$$
induit un symbole modulaire partiel Hecke équivariant à valeurs dans les formes modulaires de poids $2$ sur $Y_1 (N)$ \emph{modulo les séries d'Eisenstein de poids $2$}. Borisov et Gunnells considèrent en effet les séries d'Eisenstein, de poids $1$ et niveau $Y_1(N)$,
$$E_1 (\tau , a/N) \quad ( a \in \{ 1 , \ldots , N-1 \} ),$$
et remarquent que si $a+b+c = 0$ modulo $N$ et si $a,b,c \neq 0$, l'expression
$$E_1 (\tau , a/N) E_1 (\tau , b/N) + E_1 (\tau , b/N) E_1 (\tau , c/N) + E_1 (\tau , c/N) E_1 (\tau , a/N)$$
est une combinaison linéaire de séries d'Eisentein de poids $2$, voir \cite[Proposition 3.7 et 3.8]{BorisovGunnells}.\footnote{Cela se déduit d'une formule plus générale démontrée dans \cite{BorisovGunnellsInventiones} et que l'on peut également trouver chez Eisenstein \cite{Eisenstein}.}
\medskip
Notons finalement que la construction de ce chapitre s'applique de manière similaire à une courbe elliptique plutôt qu'à une famille de courbes elliptiques. Lors-\ que cette courbe elliptique est à multiplication complexe par l'anneau des entiers $\mathcal{O}$ d'un corps quadratique imaginaire $k$ on obtient des cocycles de degré $n-1$ de sous-groupes de congruence, non plus de $\mathrm{GL}_n (\mathbf{Z})$ mais, de $\mathrm{GL}_n (\mathcal{O})$. Ce sont des généralisations des cocycles considérés par Sczech \cite{SczechBianchi} et Ito \cite{Ito} lorsque $n=2$. Comme d'après un théorème classique de Damerell les évaluations des séries d'Eisenstein $E_1$ en des points CM sont, à des périodes transcendantes explicites près, des nombres algébriques, on peut montrer que les cocycles associés aux corps quadratiques imaginaires sont à valeurs algébriques. Leur étude fait l'objet de l'article \cite{ColmezNous} dans lequel nous démontrons une conjecture de Sczech et Colmez relative aux valeurs critiques des fonctions $L$ attachées aux caractères de Hecke d'extensions de $k$.
\chapter[Cohomologie d'arrangements d'hyperplans]{Cohomologie d'arrangements d'hyperplans~: représentants canoniques} \label{S:OrlikSolomon}
\resettheoremcounters
\numberwithin{equation}{chapter}
Dans ce chapitre, qui peut se lire de manière indépendante du reste de l'ouvrage, on démontre un théorème ``à la Orlik--Solomon'' pour les arrangements d'hyperplans dans des produits de $\mathbf{C}^\times$ ou des produits de courbes elliptiques. Le résultat principal que nous démontrons est le théorème \ref{P:Brieskorn}.
\section{Arrangement d'hyperplans trigonométriques ou elliptiques}
On fixe un groupe algébrique $A$, isomorphe au groupe multiplicatif ou à une courbe elliptique. Soit $n$ un entier naturel. On considère un $A^n$-torseur $T$ muni d'un sous-ensemble $T_{\mathrm{tors}} \subset T$ fixé qui est un torseur pour le groupe des points de torsions de $A^n$.
\medskip
\noindent
{\it Remarque.} Même dans le cas (qui nous intéresse principalement) où $T=A^n$, nous serons amené à considérer des hyperplans affines comme
$\{a \} \times A^{n-1}$ où $a \in A_{\rm tors}$. Ces derniers sont encore des $A^{n-1}$-torseurs.
\medskip
Comme dans le cas où $T=A^n$, on appelle {\it fonctionnelle affine} toute application $\chi: T \rightarrow A$ de la forme
$$t_0 + \mathbf{a} \mapsto \chi_0(\mathbf{a})$$
où $t_0$ est un élément de $T_{\mathrm{tors}}$ et $\chi_0 : A^n \rightarrow A$ un morphisme de la forme
$\mathbf{a} = (a_1, \dots, a_n) \mapsto \sum r_i a_i$
où les $r_i$ sont des entiers.\footnote{Autrement dit un morphisme standard, sauf dans le cas où $A$ est une courbe elliptique à multiplication complexe.} Rappelons que $\chi$ est {\it primitif} si les coordonnées
$ (r_1, \dots, r_n) \in \mathbf{Z}^n$ de $\chi_0$ sont premières entre elles dans leur ensemble. Dans ce cas le lieu d'annulation de $\chi$ est un translaté de $\mathrm{ker}(\chi_0)$ qui est isomorphe à l'image de l'application linéaire $A^{n-1} \to A^n$ associée à une base du sous-module de $\mathbf{Z}^n$ orthogonal au vecteur $\mathbf{r}$.
Le lieu d'annulation de $\chi$ a donc, comme $T$, une structure de torseur sous l'action de $A^{n-1}$ et il est muni d'une notion de points de torsion.
On appelle \emph{hyperplan} le lieu d'annulation (ou abusivement ``noyau'') d'une fonctionnelle affine primitive, ou de manière équivalente l'image d'une application
$A^{n-1} \rightarrow T$ linéaire relativement à un morphisme $A^{n-1} \rightarrow A^n$ induit par une matrice entière de taille $(n-1) \times n$.
\begin{definition}
Un \emph{arrangement d'hyperplans} $\Upsilon$ est un fermé de Zariski dans $T$ réunion d'hyperplans. La taille $\# \Upsilon$ est le nombre d'hyperplans distincts de cet arrangement.
\end{definition}
Comme dans le cas linéaire (Lemme \ref{affine}) on a le lemme suivant.
\begin{lemma} \label{affineb}
Si $\Upsilon$ contient $n$ fonctionnelles affines $\chi$ dont les vecteurs associés $\mathbf{r} \in \mathbf{Z}^n$ sont linéairement indépendants alors
le complémentaire $T-\Upsilon$ est affine.
Lorsque $A$ est une courbe elliptique, c'est même une équivalence.
\end{lemma}
\section[Cohomologie des arrangements d'hyperplans]{Opérateurs de dilatation et cohomologie des arrangements d'hyperplans} \label{S:dil}
On appelle {\em application de dilatation} toute application $[s] : T \rightarrow T$ associée à un entier $s>1$ et de la forme
$$[s] : t+ \mathbf{a} \mapsto t + s \mathbf{a}$$
pour un certain $t \in T_{\mathrm{tors}}$.
L'image d'un hyperplan par une application de dilatation est encore un hyperplan.
\'Etant donné un arrangement d'hyperplans $\Upsilon$ définis par des fonctionnelles affines $\chi_1 , \ldots , \chi_r$ on peut, comme dans le cas $T=A^n$, trouver une application de dilatation $[s]$ qui préserve $\Upsilon$. Quitte à augmenter $s$ on peut de plus supposer que si $\chi_1 , \ldots , \chi_r$ est une collection de fonctionnelles affines définissant des hyperplans dans $\Upsilon$, alors $[s]$ préserve les composantes connexes de la préimage de $0$ par
$$(\chi_1 , \ldots , \chi_r) : T \to A^r;$$
ces composantes connexes sont en effet des fermés de Zariski dans $T$.
On fixe dorénavant un tel choix de dilatation $[s]$. Pour ne pas alourdir les notations on notera simplement
$$H^*(T-\Upsilon, \mathbf{C})^{(1)} \subset H^*(T-\Upsilon, \mathbf{C})$$
le sous-espace caractéristique de $[s]_*$ associé à la valeur propre $1$. Noter que cette fois ce sous-espace dépend {\it a priori} du choix de la dilatation.
On utilisera les mêmes notations pour les espaces de formes différentielles.
\subsection{Faisceau de formes différentielles et un théorème de Clément Dupont}
Une forme méromorphe $\omega$ sur $T$ est {\it à pôles logarithmiques le long de $\Upsilon$} si au voisinage de chaque point $p \in T$ on peut décomposer $\omega$ comme combinaison linéaire sur $\mathbf{C}$ de formes du type
\begin{equation} \label{E31}
\nu \wedge \bigwedge_{J} \frac{d f_j}{f_j}
\end{equation}
où $\nu$ est une forme holomorphe au voisinage de $p$ et chaque indice $j \in J$ paramètre un hyperplan $H_j$ de $\Upsilon$ passant par $p$ défini par une équation linéaire locale $f_j = 0$. Notons que cette définition est indépendante du choix des $f_j$ puisque ceux-ci sont uniquement déterminés à une unité locale près.
Les formes méromorphes sur $T$ à pôles logarithmiques le long de $\Upsilon$ forment un complexe de faisceaux d'espaces vectoriels complexes sur $T$ que, suivant Dupont \cite{DupontC}, nous notons $\Omega^\bullet_{\langle T , \Upsilon \rangle}$. On prendra garde au fait que ce complexe est en général strictement contenu dans le complexe de Saito $\Omega^\bullet_T (\log \Upsilon)$ lorsque le diviseur $\Upsilon$ n'est pas à croisements normaux.
Dupont \cite[Theorem 1.3]{DupontC} démontre que l'inclusion entre complexes de faisceaux
\begin{equation} \label{Dqi}
\Omega^\bullet_{\langle T , \Upsilon \rangle} \hookrightarrow j_* \Omega^\bullet_{T - \Upsilon},
\end{equation}
où $j$ désigne l'inclusion de $T-\Upsilon$ dans $T$, est un quasi-isomorphisme. On rappelle ci-dessous les ingrédients principaux de sa démonstration.
\subsection{Opérateurs de dilatation} Il découle du lemme suivant que l'application de dilatation $[s]$ induit un opérateur $[s]_*$ sur $\Omega^\bullet_{\langle T , \Upsilon \rangle}$.
\begin{lemma}
Pour tout ouvert $V \subset T$, le morphisme de trace
$$\Omega^*_{\mathrm{mer}}([s]^{-1} V) \rightarrow \Omega^*_{\mathrm{mer}}(V)$$
induit par $[s]$ envoie $\Omega^\bullet_{\langle T , \Upsilon \rangle} ([s]^{-1} V)$ dans $\Omega^\bullet_{\langle T , \Upsilon \rangle} (V)$.
Autrement dit, la trace définit un morphisme de faisceaux
$$[s]_* \Omega^\bullet_{\langle T , \Upsilon \rangle} \rightarrow \Omega^\bullet_{\langle T , \Upsilon \rangle}.$$
\end{lemma}
\proof
Soient $p_1$ et $p_2$ deux points de $T$ tels que $[s] p_1 = p_2$ et soit $\omega_1$ une section locale de $\Omega^\bullet_{\langle T , \Upsilon \rangle}$ définie sur un voisinage de $p_1$. Puisque $[s]$ préserve $\Upsilon$, l'image par $[s]$ d'un hyperplan $H_1$ de $\Upsilon$ passant par $p_1$ est un hyperplan $H_2$ de $\Upsilon$ passant par $p_2$.
Soit maintenant $f_2 = 0$ une équation locale pour l'hyperplan $H_2$ dans un voisinage $W$ de $p_2$. Comme l'application $[s]$ est un biholomorphisme local, quitte à rétrécir $W$, l'application $f_1 = [s]^* f_2$ définit une équation locale pour l'hyperplan $H_1$ au voisinage de $p_1$. De plus, le morphisme trace induit par $[s]$ envoie $f_1 \in \Omega^0 ([s]^{-1} W)$ sur $\deg([s]) f_2$ et donc
$$[s]_* \frac{df_1}{f_1} = \frac{df_2}{f_2}.$$
Comme $\Omega^\bullet_{\langle T , \Upsilon \rangle}([s]^{-1} V)$ est engendré par des formes qui localement peuvent s'exprimer comme produits extérieurs de formes régulières et de formes du type $df_1/f_1$, le lemme s'en déduit.
\qed
\begin{definition}
Soit
$$H^* (T , \Omega^j_{\langle T , \Upsilon \rangle})^{(1)} \subset H^*(T , \Omega^j_{\langle T , \Upsilon \rangle})$$
le sous-espace caractéristique de $[s]_*$ associé à la valeur propre $1$, c'est-à-dire le sous-espace des classes de cohomologie qui sont envoyées sur $0$ par une puissance de $[s]_*-1$.
\end{definition}
Le but de ce chapitre est la démonstration du théorème qui, dans les contextes multiplicatifs et elliptiques, remplace le théorème de Brieskorn invoqué dans le cas affine pour construire la classe \eqref{E:Sa}.
\begin{theorem} \label{P:Brieskorn}
Supposons $T-\Upsilon$ affine. Pour tout degré $j \leq n = \dim(T)$ les formes dans $H^0(T, \Omega^j_{\langle T , \Upsilon \rangle})^{(1)}$ sont fermées et l'application naturelle
$$H^0(T, \Omega^j_{\langle T , \Upsilon \rangle})^{(1)} \rightarrow H^j(T-\Upsilon)^{(1)}$$
est un isomorphisme.
\end{theorem}
Après avoir terminé la rédaction de ce chapitre nous avons appris que, dans le cas multiplicatif, il existe en fait, comme dans le cas affine, une algèbre de formes différentielles algébriques fermées sur $T-\Upsilon$, qui est isomorphe à la cohomologie de $T-\Upsilon$. Dans le cas des arrangements d’hyperplans affines, cette algèbre est celle d’Arnol’d et Brieskorn. Dans le cas des arrangements toriques, elle est plus difficile à décrire, notamment parce que la cohomologie n’est pas engendrée en degré 1 en général. De Concini et Procesi \cite[Theorem 5.2]{DeConciniProcesi} traitent le cas unimodulaire; ce cas-là est analogue au cas affine, l’algèbre est engendrée par des représentants d’une base du $H^1$ du tore et les formes $d\log (\chi-a)$, où $\{\chi =a\}$ est une équation d’un des hyperplans de l'arrangement. Une description de l'algèbre dans le cas général est donnée par Callegaro, D'Adderio, Delucchi, Migliorini et Pagaria \cite{CDDMP}.
\subsection{Mise en place de la démonstration (par récurrence)}
On démontre le théorème \ref{P:Brieskorn} par récurrence sur la paire $(n=\dim \ T , \#(\Upsilon))$, relativement à l'ordre lexicographique. L'initialisation de la récurrence se fera en démontrant directement le théorème
\begin{itemize}
\item lorsque $\# \Upsilon = n$, dans le cas elliptique, et
\item lorsque $\Upsilon$ est vide, dans le cas multiplicatif.
\end{itemize}
\medskip
\noindent
{\it Remarque.} Lorsque $A$ est elliptique et que $\# \Upsilon = n$ --- cas d'initialisation de la récurrence --- les vecteurs de $\mathbf{Z}^n$ associés aux hyperplans de $\Upsilon$ sont nécessairement linéairement indépendants d'après le lemme \ref{affineb}. Le diviseur $\Upsilon \subset T$ est {\it à croisements normaux simples}.
\medskip
\`A chaque étape de récurrence on procède comme suit~: supposons que $\Upsilon'$ soit un ensemble non vide d'hyperplans avec $T-\Upsilon'$ affine.
Supposons de plus $\# \Upsilon' > n$ dans le cas elliptique. Considérons un hyperplan $H$ dans $\Upsilon'$ tel que $T-\Upsilon$, avec $\Upsilon = \Upsilon '-H$, soit affine.
Notons $\Upsilon \cap H$ l'ensemble des hyperplans de $H$ --- vu comme $A^{n-1}$-torseur --- obtenus par intersection de $H$ avec les hyperplans (de $T$) appartenant à $\Upsilon$. L'arrangement
$$H - (\Upsilon \cap H),$$
étant un fermé de Zariski de la variété affine $T-\Upsilon$, est affine.
On peut maintenant considérer le diagramme
\begin{equation} \label{E:diag}
T - \Upsilon ' \stackrel{j}{\hookrightarrow} T - \Upsilon
\stackrel{\iota}{\hookleftarrow} H - (\Upsilon \cap H),
\end{equation}
où $\iota$ est une immersion fermée et $j$ une injection ouverte.
Dans la suite $\iota$ désignera plus généralement l'inclusion de $H$ dans $T$. Noter que $\iota (H - (\Upsilon \cap H))$ est un diviseur lisse de $T-\Upsilon$ de complémentaire l'image de $j$.
Le point clé de la démonstration est le fait, non trivial et démontré par Clément Dupont, que la suite de faisceaux
\begin{equation} \label{dupont}
0 \rightarrow \Omega^q_{\langle T , \Upsilon \rangle} \rightarrow \Omega^{q}_{\langle T , \Upsilon ' \rangle} \stackrel{\mathrm{Res}}{\longrightarrow} \iota_* \Omega^{q-1}_{\langle H , \Upsilon \cap H \rangle} \rightarrow 0
\end{equation}
est \emph{exacte}.
Dans le paragraphe suivant on décrit les grandes lignes de la démonstration de Dupont.
\section{Travaux de Dupont} \label{travaux dupont}
Suivant Dupont \cite[Definition 3.4]{DupontC}, on munit le faisceau $\Omega^\bullet_{\langle T , \Upsilon \rangle}$ d'une filtration ascendante
$$W_0 \Omega^\bullet_{\langle T , \Upsilon \rangle} \subset W_1 \Omega^\bullet_{\langle T , \Upsilon \rangle} \subset \ldots $$
appelé {\it filtration par les poids}. Le $k$-ème terme de cette filtration $W_k \Omega^\bullet_{\langle T , \Upsilon \rangle}$ est engendré par les formes du type \eqref{E31} avec $\# J \leq k$. On a donc
$$W_0 \Omega^\bullet_{\langle T , \Upsilon \rangle} = \Omega^\bullet_T \quad \mbox{et} \quad W_q \Omega^q_{\langle T , \Upsilon \rangle} = \Omega^q_{\langle T , \Upsilon \rangle}.$$
Pour comprendre le complexe gradué associé considérons $\mathrm{gr}_{n}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}$,
c'est-à-dire le quotient
$$W_n \Omega^\bullet_{\langle T , \Upsilon \rangle} /W_{n-1} \Omega^\bullet_{\langle T , \Upsilon \rangle}.$$
Localement, la classe d'une forme de type \eqref{E31} dans $\mathrm{gr}_{n}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}$ ne dépend pas du choix des $f_j$ puisque ceux-ci sont uniquement déterminés à une unité locale près. Un choix approprié de coordonnées locales permet alors de vérifier que la fibre de $\mathrm{gr}_{n}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}$ au-dessus d'un point $p$ est nulle à moins que $p$ ne soit contenu dans $n$ hyperplans linéairement indépendants.
Maintenant, si $p$ appartient à $n$ hyperplans linéairement indépendants, Dupont \cite[Theorem 3.11]{DupontC} construit une application naturelle de l'algèbre de Orlik--Solomon locale en $p$ vers $\mathrm{gr}_{n}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}$. Les hyperplans de $\Upsilon$ passant par $p$ induisent un arrangement d'hyperplans (linéaires) $\Upsilon^{(p)}$ dans l'espace tangent en $p$ à $T$ qu'un choix de coordonnées locales identifie à $\mathbf{C}^n$. Orlik et Solomon \cite{OrlikSolomon} définissent une algèbre graduée $\mathrm{OS}_\bullet (\Upsilon^{(p)})$ par générateurs et relations. En énumérant $H_1 , \ldots , H_\ell$ les hyperplans de $\Upsilon^{(p)}$, autrement dit les hyperplans de $\Upsilon$ passant par $p$, Orlik et Solomon définissent en particulier $\mathrm{OS}_n (\Upsilon^{(p)})$
comme étant engendrée par des éléments $e_J$ pour $J = \{ j_1 , \ldots , j_n \} \subset \{1 , \ldots , \ell \}$ avec $j_1 < \ldots < j_n$, quotientée par l'espace des relations
engendré par les combinaisons linéaires
\begin{equation} \label{RelOS}
\sum_{i=0}^n (-1)^i e_{K-\{j_i \}},
\end{equation}
où $K=\{ j_0 , \ldots , j_n \} \subset \{1 , \ldots , \ell \}$ avec $j_0 < j_1 < \ldots < j_n$.
L'application
\begin{equation} \label{appDupont}
\mathrm{OS}_n (\Upsilon^{(p)}) \to \mathrm{gr}_{n}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}
\end{equation}
construite par Dupont est alors définie de la manière suivante. Pour chaque $j$ dans $\{1 , \ldots , \ell \}$ on choisit une équation locale $f_j =0$ définissant l'hyperplan $H_j$ au voisinage de $p$, et on envoie chaque générateur $e_J$ de $\mathrm{OS}_n (\Upsilon^{(p)})$ sur la forme
$$\wedge_{j \in J} \frac{df_j}{f_j} \in \mathrm{gr}_{n}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}.$$
Rappelons que ces éléments ne dépendent pas des choix faits pour les $f_j$, le fait qu'ils vérifient les relations \eqref{RelOS} s'obtient, comme dans le cas classique des arrangements d'hyperplans dans $\mathbf{C}^n$, en explicitant la relation de dépendance entre les $H_j$ $(j \in K)$. Cela montre que l'application \eqref{appDupont} est bien définie. Par définition, elle est surjective au voisinage de $p$.
Dupont définit plus généralement une application
\begin{equation} \label{appDupont2}
\bigoplus_{\mathrm{codim}(S) = k} \iota_{S*} \Omega_S^{\bullet} \otimes \mathrm{OS}_S (\Upsilon ) \rightarrow \mathrm{gr}_{k}^W \Omega^\bullet_{\langle T , \Upsilon \rangle},
\end{equation}
où la somme porte maintenant sur les {\it strates}, c'est-à-dire les composantes connexes d'intersections d'hyperplans, de codimension $k$ et $\iota_S$ désigne l'inclusion de la strate $S$ dans $T$. \`A tout point $p$ d'une strate $S$ il correspond une algèbre d'Orlik--Solomon locale $\mathrm{OS}_\bullet (\Upsilon^{(p)})$ et on note
$$\mathrm{OS}_S (\Upsilon) \subset \mathrm{OS}_\bullet (\Upsilon^{(p)})$$
le sous-espace engendré par les monômes $e_J$ où $J$ décrit les sous-ensembles d'hyperplans appartenant à $\Upsilon^{(p)}$ et dont l'intersection est exactement la trace $S^{(p)}$ de la strate $S$ dans l'espace tangent en $p$ à $T$. Noter que $\mathrm{OS}_S (\Upsilon)$ ne dépend pas du choix du point $p$ dans $S$.
Le facteur correspondant à une strate $S = \{ p \}$ de codimension $n$ dans \eqref{appDupont2} correspond à l'application \eqref{appDupont}.
Pour construire l'application \eqref{appDupont2} en général, considérons une strate $S$ de codimension $k$ et un point $p \in S$. Un choix d'équations locales
$f_j =0$ pour les hyperplans de $\Upsilon$ contenant $S$ permet --- par un argument similaire à celui détaillé ci-dessus --- de construire une application
\begin{equation} \label{E:app1}
\underline{\mathrm{OS}_S (\Upsilon)} \rightarrow \mathrm{gr}_{k}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}.
\end{equation}
(Ici $\underline{\mathrm{OS}_S (\Upsilon)}$ désigne le faisceau constant associé à $\mathrm{OS}_S (\Upsilon)$.)
On étend maintenant \eqref{E:app1} en une application
\begin{equation} \label{E:app2}
\Omega_T^{\bullet} \otimes \mathrm{OS}_S (\Upsilon) \rightarrow \mathrm{gr}_{k}^W \Omega^\bullet_{\langle T , \Upsilon \rangle}
\end{equation}
par linéarité. Finalement \eqref{E:app2} se factorise par $\iota_{S*} \Omega_S^{\bullet } \otimes \mathrm{OS}_S (\Upsilon)$~: étant donné une forme $\nu$ dans le noyau de l'application $\Omega_T^{\bullet } \to \iota_{S*} \Omega_S^{\bullet }$ et un monôme $e_J \in \mathrm{OS}_S (\Upsilon)$ il s'agit de montrer que \eqref{E:app2} envoie $\nu \otimes e_J$ sur $0$. Notons $f_j = 0$ les équations locales des hyperplans de $J$. Dans les coordonnées les $f_j$ engendrent l'idéal de $S$, on peut donc écrire $\nu$ comme une combinaison linéaire
$$\nu \sum \omega_j f_j + \sum \omega_j ' (df_j).$$
Il nous reste alors finalement à vérifier que
$$f_j \wedge \left( \frac{df_1}{f_1} \wedge \dots \wedge \frac{df_k}{f_k} \right) \in W_{k-1} \Omega^\bullet_{\langle T , \Upsilon \rangle}$$
et
$$df_j \wedge \left( \frac{df_1}{f_1} \wedge \dots \wedge \frac{df_k}{f_k} \right) \in W_{k-1} \Omega^\bullet_{\langle T , \Upsilon \rangle}$$
ce qui résulte des définitions.
L'application \eqref{appDupont2} est surjective, ce qui peut se voir en calculant les fibres. Dupont \cite[Theorem 3.6]{DupontC} montre en fait que \eqref{appDupont2} est un isomorphisme de complexes. On extrait ici de sa démonstration le lemme crucial (pour nous) suivant.
\begin{lemma} \label{L:exactdupont}
La suite courte de faisceaux \eqref{dupont} est exacte.
\end{lemma}
\begin{proof}
La suite de complexes \eqref{dupont} induit des suites courtes
\begin{equation} \label{dupontW}
\tag{$\mathbf{W}_k$} 0 \rightarrow W_k \Omega^q_{\langle T , \Upsilon \rangle} \rightarrow W_k \Omega^{q}_{\langle T , \Upsilon ' \rangle} \rightarrow W_{k-1} \Omega^{q-1}_{\langle H , \Upsilon \cap H \rangle} \rightarrow 0
\end{equation}
et
\begin{equation} \label{dupontgr}
\tag{$\mathbf{gr}^W_k$} 0 \rightarrow \mathrm{gr}^W_k \Omega^q_{\langle T , \Upsilon \rangle} \rightarrow \mathrm{gr}^W_k \Omega^{q}_{\langle T , \Upsilon' \rangle} \rightarrow \mathrm{gr}^W_{k-1} \Omega^{q-1}_{\langle H , \Upsilon \cap H \rangle} \rightarrow 0.
\end{equation}
Puisque $W_k \Omega^\bullet_{\langle T , \Upsilon \rangle} = \Omega^\bullet_{\langle T , \Upsilon \rangle}$ pour $k$ suffisamment grand, pour démontrer le lemme il suffit de montrer que les suites \eqref{dupontW} sont exactes. Et comme par ailleurs les suites courtes \eqref{dupontW} sont exactes à gauche et à droite, il nous faut seulement vérifier qu'elles sont exactes au milieu.
Remarquons maintenant que les suites \eqref{dupontW} et \eqref{dupontgr} donne lieu à une suite exacte courte de complexes
\begin{equation} \label{SEcx}
0 \rightarrow (\mathbf{W}_{k-1}) \rightarrow (\mathbf{W}_k) \rightarrow (\mathbf{gr}_k^W) \to 0 .
\end{equation}
La suite exacte longue associée en cohomologie implique que si les suites $(\mathbf{W}_{k-1})$ et \eqref{dupontgr} sont exactes au milieu alors \eqref{dupontW} l'est aussi. Puisque $(\mathbf{W}_{0})$ est évidemment exacte, une récurrence sur $k$ réduit la démonstration du lemme à la démonstration que les suites \eqref{dupontgr} sont exactes au milieu.
Pour abréger les expressions, on notera simplement $\mathrm{OS}^{(k)} (\Upsilon)$ le faisceau apparaissant à la source de l'application \eqref{appDupont2}. En tensorisant avec les faisceaux $\iota_{S*} \Omega_S^\bullet$ les suites exactes entre algèbres d'Orlik--Solomon obtenues par ``restriction et effacement'' on obtient que les suites de complexes
$$0 \to \mathrm{OS}^{(k)} (\Upsilon) \to \mathrm{OS}^{(k)} (\Upsilon' ) \to \mathrm{OS}^{(k)} (\Upsilon \cap H ) \to 0$$
sont exactes. En considérant les diagrammes commutatifs
$$\xymatrix{
0 \ar[r] & \mathrm{OS}^{(k)} (\Upsilon) \ar[r] \ar[d] & \mathrm{OS}^{(k)}_{\Upsilon'} \ar[r] \ar[d] & \mathrm{OS}^{(k-1)}_{\Upsilon \cap H} \ar[d] \ar[r] &0 \\
0 \ar[r] & \mathrm{gr}_k^W \Omega^\bullet_{\langle T ,\Upsilon \rangle} \ar[r] & \mathrm{gr}_k^W \Omega^\bullet_{\langle T , \Upsilon' \rangle} \ar[r]& \mathrm{gr}_{k-1}^W \Omega^\bullet_{\langle H , \Upsilon \cap H \rangle} \ar[r] & 0.
}
$$
on montre, dans un même élan, par récurrence sur le cardinal de l'arrangement d'hyperplans que les flèches verticales sont des isomorphismes et que la ligne du bas est exacte au milieu. Par hypothèse de récurrence on peut supposer que les flèches verticales à gauche et à droite sont des isomorphismes. Une chasse au diagramme montre alors que la ligne du bas est exacte au milieu pour tout $k$. Comme expliqué ci-dessus cela suffit à montrer que les suites courtes \eqref{dupontW} sont exactes. La suite exacte longue en cohomologie associée à la suite exacte courte de complexes \eqref{SEcx} implique alors que la suite \eqref{dupontgr} est enfin partout exacte (pas seulement au milieu). Finalement les deux lignes des diagrammes commutatifs ci-dessus sont exactes et, puisque les flèches verticales à gauche et à droite sont des isomorphismes, le lemme des cinq implique que la flèche du milieu est également un isomorphisme. Ce qui permet de continuer la récurrence.
\end{proof}
Terminons cette section par un autre résultat important (pour nous en tout cas) de Clément Dupont \cite[Theorem 3.13]{DupontC}.
\begin{lemma} \label{L:exactdupont}
L'inclusion \eqref{Dqi} est un quasi-isomorphisme.
\end{lemma}
\begin{proof}[Esquisse de démonstration]
La cohomologie de complexe $j_* \Omega_{T-\Upsilon}^\bullet$ en $p$ est précisément la cohomologie de l'arrangement local $\Upsilon^{(p)}$ d'hyperplans dans $\mathbf{C}^n$, c'est-à-dire l'algèbre de Orlik--Solomon $\mathrm{OS}_\bullet (\Upsilon^{(p)})$. D'un autre côté on peut calculer la cohomologie de $\Omega^\bullet_{\langle T , \Upsilon \rangle}$ en se servant de la filtration par les poids et de l'isomorphisme \eqref{appDupont2}, on obtient le même résultat.
\end{proof}
\section[Démonstration du théorème 3.5]{Démonstration du théorème \ref{P:Brieskorn}}
\subsection{Initialisation dans le cas multiplicatif} \label{B1}
On l'a dit, on démontre le théorème \ref{P:Brieskorn} par récurrence. Dans le cas multiplicatif où $A =\mathbf{G}_m$, l'initialisation de la récurrence correspond au cas où $\Upsilon$ est vide.
\begin{lemma} \label{L:B1}
Le sous-espace
\begin{equation} \label{E:H0(1)}
H^0(\mathbf{G}_m^n, \Omega^\bullet)^{(1)}
\end{equation}
est de dimension $1$ engendré par la forme $\wedge_{i=1}^{n} dz_i/z_i$ où $z_i$ désigne la $i$-ème coordonnée sur $\mathbf{G}_m^n$.
En particulier \eqref{E:H0(1)} est concentré en degré $n$.
\end{lemma}
\proof
Une forme différentielle holomorphe sur $\mathbf{G}_m^n$ se décompose de manière unique en série de Laurent à plusieurs variables. Il suffit donc de considérer les monômes
\begin{equation} \label{E:monome}
\left( \prod_{i=1}^n z_i^{a_i} \right) \cdot \bigwedge_{j \in J} \frac{dz_j}{z_j} \quad (J \subset \{1 , \ldots , n \} , \ a_1 , \ldots , a_n \in \mathbf{Z}).
\end{equation}
Maintenant, pour tout entier $a \in \mathbf{Z}$ on a
$$\sum_{w : w^s=z} w^a = \left\{ \begin{array}{cl}
s z^{a/s} & \mbox{si } s | a, \\
0 & \mbox{sinon}.
\end{array} \right.$$
En poussant en avant le monôme \eqref{E:monome} par $[s]_*$ on obtient donc
\begin{multline*}
\sum_{w_i^s = z_i} \left( \left( \prod_{i=1}^n w_i^{a_i} \right) \cdot \bigwedge_{j \in J} \frac{dw_j}{w_j} \right) \\
= \left\{ \begin{array}{ll}
s^{n- \# J} \left( \prod_{i=1}^n z_i^{a_i /s} \right) \cdot \bigwedge_{j \in J} \frac{dz_j}{z_j} & \mbox{si } s | a_i \mbox{ pour tout } i, \\
0 & \mbox{sinon}.
\end{array} \right.
\end{multline*}
Il s'en suit que tout élément de $H^0(\mathbf{G}_m^n , \Omega^\bullet )$ appartient à un sous-espace $[s]_*$-invariant de dimension fini et qu'un monôme \eqref{E:monome} appartient à $H^0(\mathbf{G}_m^n, \Omega^\bullet)^{(1)}$ si et seulement si $\# J = n$ et tous les $a_i$ sont nuls.
\qed
La cohomologie de $\mathbf{G}_m^n$ est égale à l'algèbre extérieure sur les formes fermées $dz_i /z_i$. Le sous-espace $H^\bullet ( \mathbf{G}_m^n )^{(1)}$ est donc de dimension $1$ engendré par la forme $\wedge_{i=1}^{n} dz_i/z_i$. Le lemme \ref{L:B1} implique donc que, dans ce cas où $A =\mathbf{G}_m$ et $\Upsilon$ est vide, le théorème \ref{P:Brieskorn} est bien vérifié.
\subsection{Initialisation dans le cas elliptique} \label{B2}
Dans le cas elliptique où $A =E$, l'initialisation de la récurrence correspond au cas où $\Upsilon$ est constitué de $n$ hyperplans linéairement indépendants. Dans ce cas, le faisceau $\Omega^\bullet_{\langle T , \Upsilon \rangle}$ coïncide avec le faisceau $\Omega^\bullet_T (\log \Upsilon)$ des formes différentielles logarithmiques. C'est un complexe de faisceaux et, d'après Atiyah--Hodge \cite{AtiyahHodge}, Griffiths \cite{Griffiths} et Deligne \cite{DeligneH2}, on a un isomorphisme canonique
\begin{equation} \label{G-D}
H^k (T-\Upsilon , \mathbf{C}) = \mathbf{H}^k (T , \Omega^\bullet_T (\log \Upsilon) );
\end{equation}
cf. \cite[Corollary 8.19]{Voisin}. De plus, la suite spectrale de Hodge--de Rham
$$E_1^{pq} = H^p (T , \Omega^q_T (\log \Upsilon)) \Longrightarrow \mathbf{H}^{p+q} (T , \Omega^\bullet_T (\log \Upsilon ) )$$
dégénère à la première page, cf. \cite[Corollaire 3.2.13, (ii)]{DeligneH2}.
Le lemme suivant implique donc que, dans ce cas, le théorème \ref{P:Brieskorn} est bien vérifié.
\begin{lemma} \label{L:affirmation}
Supposons que $\Upsilon$ soit constitué de $n$ hyperplans linéairement indépendants. Alors le sous-espace $H^p(T, \Omega^q_T (\log \Upsilon) )^{(1)}$ est nul pour tout $p > 0$.
\end{lemma}
\begin{proof}
On démontre le lemme en explicitant les groupes de cohomologie $H^p(T, \Omega^q_T (\log \Upsilon) )^{(1)}$. On commence par remarquer que l'on a un isomorphisme
\begin{equation} \label{E:isomF}
\bigoplus_{i=1}^n \mathcal{O}(H_i) \stackrel{\sim}{\longrightarrow} \Omega^1_T (\log \Upsilon) ,
\end{equation}
où $\Upsilon = \{ H_1, \dots, H_n \}$. Pour tout $i \in \{ 1 , \ldots , n \}$ soit $\omega_i$ une $1$-forme différentielle sur $T$ obtenue comme tirée en arrière par un caractère définissant $H_i$ d'une $1$-forme partout non-nulle sur $A$.
L'application
$$(f_i) \in \oplus_i \mathcal{O} (H_i) \longmapsto \sum f_i \omega_i $$
est globalement définie et induit un isomorphisme au niveau des fibres; elle réalise l'isomorphisme \eqref{E:isomF}.
Noter que $[s]^* \omega_i = s \omega_i$; pour tout $f \in \mathcal{O} (H_i )$ on a donc
$$[s]_*( f \omega_i) = (s^{-1} [s]_* f) \omega_i.$$
Comme par ailleurs
$$\Omega^q_T (\log \Upsilon) = \wedge^q \Omega^1_T (\log \Upsilon) \simeq \bigoplus_{\substack{J \subset \{1, \ldots , n \} \\ \# J = q}} \mathcal{O}(\sum_{j \in J} H_j )$$
le lemme \ref{L:affirmation} découle du lemme qui suit.
\end{proof}
\begin{lemma}
Soit $J \subset \{1, \ldots , n \}$ de cardinal $q$ et soit $p$ un entier strictement positif. Le morphisme $[s]_*$ opère sur $H^p(T, \mathcal{O}(\sum_{j \in J} H_j))$ par le scalaire $s^{2n-p}$ et l'espace $H^p(T, \mathcal{O}(\sum_{j \in J} H_j))$ est réduit à $0$ si $n\leq p+q$.
\end{lemma}
En particulier $s^q$ n'est pas un valeur propre généralisée.
\proof
Lorsque $q = n$, le faisceau $\mathcal{O}(\sum_{i=1}^n H_i)$ est ample. Or il découle du théorème d'annulation de Mumford \cite[\S III.16]{Mumford} que la cohomologie supérieure, de degré $p>0$, des fibrés amples est nulle sur une variété abélienne.
Supposons maintenant $q<n$ et supposons pour simplifier $T=A^n$. Les caractères définissant les hyperplans $H_j$ ($j \in J$) induisent un morphisme de variétés abéliennes
$$\pi: A^n \rightarrow A^q$$
et le fibré en droites $\mathcal{O}(\sum_{j \in J} H_j )$ sur $A^n$ est égal au tiré en arrière $\pi^* \mathcal{L}$ du fibré en droites
$\mathcal{L}$ sur $A^q$ associé au diviseur $z_1 \dots z_q = 0$ égal à la somme des hyperplans de coordonnées.
Commençons par remarquer qu'en général si $f : C \to B$ est un morphisme de variétés abéliennes à fibres connexes alors les $R^j f_* \mathcal{O}_C$ sont des fibrés vectoriels triviaux. Pour le voir, on note $F$ le noyau de $f$; c'est une variété abélienne et il existe une sous-variété abélienne $B' \subset C$ telle que
\begin{itemize}
\item l'application somme $F \times B' \to C$ et
\item l'application induite $B' \to B$
\end{itemize}
soient des isogénies (voir par exemple \cite[Proposition 12.1]{Milne}). Notons $C' = F \times B' $ et $f' : C' = F \times B' \to B'$. Le faisceau de cohomologie $R^j f'_* \mathcal{O}_{C'}$ est un faisceau constant de fibre $H^j (F , \mathcal{O}_F)$. Comme les translations de $F$ opèrent trivialement sur $H^j (F , \mathcal{O}_F)$, ce faisceau constant sur $B'$ descend sur $B$ en un faisceau, nécessairement égal à $R^j f_* \mathcal{O}_C$, constant de fibre $H^j (F, \mathcal{O}_F)$.
En général le morphisme $\pi$ n'est pas à fibres connexes mais, comme $\pi$ est un morphisme de groupes, les composantes connexes des fibres sont toutes de même dimension et le quotient $B$ de $A^n$ par les composantes connexes des fibres est un revêtement fini de $A^q$. On peut donc factoriser $\pi$ en la composition $\pi = g \circ \pi'$ de morphismes $\pi': A^n \to B$ et $g : B \to A^q$ avec $\pi'$ à fibres connexes et $g$ fini. Notons $F$ la fibre de $\pi'$. La suite spectrale de Leray associée au morphisme $\pi '$ s'écrit~:
\begin{equation} \label{suiteLeray}
H^r (B , R^p \pi '_*(\pi {}^* \mathcal{L} )) \Longrightarrow H^{p+r} (A^n , \pi {}^* \mathcal{L}).
\end{equation}
Par la formule de projection, on a un isomorphisme
\begin{equation} \label{ProjFormula}
R^p \pi '_*(\pi {}^* \mathcal{L} ) = R^p \pi '_*(\pi' {}^* g^* \mathcal{L} ) \cong g^*\mathcal{L} \otimes R^p \pi ' _* \mathcal{O}_{A^n} = g^*\mathcal{L} \otimes H^p (F , \mathcal{O}_F),
\end{equation}
où la dernière égalité découle du paragraphe précédent.
Puisque les composantes connexes des fibres de $\pi$ sont des copies de $A^{n-q}$, on obtient que
$$H^r (B , R^p \pi '_*(\pi^* \mathcal{L} )) \cong H^r (B , g^*\mathcal{L}) \otimes H^p (A^{n-q} , \mathcal{O}_{A^{n-q}} ).$$
Il découle à nouveau du théorème d'annulation de Mumford que ce groupe s'annule pour $r>0$ car $g^*\mathcal{L}$ est ample. La suite spectrale \eqref{suiteLeray} dégénère donc et on obtient un isomorphisme
\begin{equation} \label{HpDec}
H^p (A^n , \pi^* \mathcal{L}) \cong H^0 (B , g^*\mathcal{L} ) \otimes H^p (A^{n-q} , \mathcal{O}_{A^{n-q}} ).
\end{equation}
Notons de plus que
$$H^0 (B, g^*\mathcal{L} ) = H^0 (A^q , g_* (g^* \mathcal{L})) = H^0 (A^q , \mathcal{L} \otimes g_* \mathcal{O}_B)$$
où $g_* \mathcal{O}_B$ est une somme de fibrés en droites de torsion $\mathcal{T}_1 , \ldots ,\mathcal{T}_r$. On a donc
$$H^0 (B, g^*\mathcal{L} ) = \bigoplus_{j=1}^r H^0 (A^q , \mathcal{L}_j')$$
où $\mathcal{L}_j ' = \mathcal{L} \otimes \mathcal{T}_j$.
Par hypothèse, la multiplication par $s$ sur $A^n$ préserve les composantes connexes des fibres de $\pi$ et donc sa décomposition de Stein. L'endomorphisme induit $[s]_*$ sur
$$H^p (A^n , \mathcal{O}(\sum_{j \in J} H_j)) = H^p (A^n , \pi^* \mathcal{L})$$
préserve donc la décomposition \eqref{HpDec} et chaque terme $H^0 (A^q , \mathcal{L}_j')$ de la décomposition de $H^0 (B, g^*\mathcal{L} )$. On étudie l'action de $[s]_*$ sur chacun des facteurs du produit tensoriel~:
\begin{itemize}
\item L'endomorphisme $[s]_*$ opère sur\footnote{Ici on utilise que la suite spectrale de Hodge--de Rham pour $A^{n-q}$ dégénère en $E_1$.}
$$H^p (A^{n-q} , \mathcal{O}_{A^{n-q}} ) \hookrightarrow H^p(A^{n-q}, \mathbf{C})$$
par $s^{2n-2q-p}$. De plus, le groupe de gauche est nul si $n \leq p+q$.
\item Comme $H^r (A^q , \mathcal{L}_j ')=0$ pour $r>0$, il découle par exemple de \cite[Theorem 13.3]{Milne} que chaque groupe $H^0 (A^q , \mathcal{L}_j')$ est de rang $1$. La section canonique est une fonction theta sur laquelle l'homomorphisme $[s]_*$ opère par multiplication par $s^{2q}$; voir par exemple \cite[Eq. (3.1)]{Beauville}.
\end{itemize}
Finalement $[s]_*$ opère sur $H^p (A^n , \mathcal{O}(\sum_{j \in J} H_j))$ par le scalaire $s^{2n-p}$ et le groupe $H^p (A^n , \mathcal{O}(\sum_{j \in J} H_j))$ est nul si $n\leq p+q$. Il est donc exclu d'obtenir $s^q$ comme valeur propre.
\qed
\subsection{Pureté, par récurrence}
\begin{lemma} \label{C1}
La partie invariante de la cohomologie $H^j(T -\Upsilon )^{(1)}$ est pure (comme structure de Hodge \cite{DeligneH2}) de poids $2j$ pour $j \leq \dim(T)$. En particulier, si $T-\Upsilon$ est affine,
la cohomologie est pure en tout degré.
\end{lemma}
\proof
Pour $n=0$ le lemme est immédiat. Lorsque $n$ est strictement positif mais que $\# \Upsilon=0$,
la cohomologie invariante n'intervient qu'en degré cohomologique maximal. Dans le cas elliptique ce degré est $2 \dim (T)$ et il n'y a rien à démontrer. Dans le cas multiplicatif le lemme se déduit du fait que $H^1(\mathbf{G}_m)$ est pur de poids $2$.
Pour l'étape de récurrence (sur $(n, \# \Upsilon )$), on considère un triplet \eqref{E:diag} tel que $(T,\Upsilon )$ et $(H , (\Upsilon \cap H))$ vérifient tous les deux les conclusions du lemme \ref{C1}. La suite exacte longue de Gysin en cohomologie associée à ce triplet s'écrit
\begin{multline} \label{LESGysin}
H^{j-2}( H - (H \cap \Upsilon ), \mathbf{C}(-1)) \stackrel{\delta_1}{\rightarrow} H^j( T - \Upsilon ) \rightarrow H^j(T - \Upsilon ' ) \\ \rightarrow H^{j-1}(H - (H \cap \Upsilon), \mathbf{C}(-1))
\stackrel{\delta_2}{\rightarrow}
H^{j+1}( T - \Upsilon ).
\end{multline}
Comme $\mathbf{C} (-1)$ est de poids $2$ et que les applications de la suite exacte ci-dessus préservent la filtration par le poids et sont compatibles avec l'action de $[s]_*$, on conclut que $H^j(T - \Upsilon ' )^{(1)}$ est pure de poids $2j$ pour $j \leq \dim(T)$. Cela prouve la première assertion du lemme.
La deuxième partie du lemme s'en déduit puisque, si $T-\Upsilon$ est affine, la cohomologie de $T-\Upsilon$ s'annule pour $j > \dim(T)$.
\qed
\begin{lemma} \label{exactness in topology}
Supposons que $T - \Upsilon$ et $H - (H \cap \Upsilon)$ soient affines. Alors
$$0 \rightarrow H^j( T - \Upsilon)^{(1)} \rightarrow H^j(T - \Upsilon')^{(1)} \stackrel{\mathrm{Res}}{\longrightarrow} H^{j-1}(H - (H \cap \Upsilon), \mathbf{C}(-1))^{(1)} \rightarrow 0$$
est une suite exacte courte.
\end{lemma}
\proof
Puisque $T - \Upsilon$ et $H - (H \cap \Upsilon)$ sont affines, il découle du lemme \ref{C1} que pour tout degré $j$ les parties invariantes de la cohomologie $H^j(T -\Upsilon )^{(1)}$ et $H^j(H- (H \cap \Upsilon) )^{(1)}$ sont pures de poids $2j$.
Les membres de la suite \eqref{LESGysin} appartiennent à la catégorie des $\mathbf{C}$-espaces vectoriels de dimension finie munis d'une action linéaire de $\mathbf{Z}$ (le groupe engendré par $[s]_*$). Le foncteur qui à un tel espace $H$ associe $H^{(1)}$ est exact. On peut en effet décomposer $H$ en la somme directe $\oplus H^{(\lambda)}$ des sous-espaces caractéristiques de $[s]_*$ et l'application
$$\alpha \mapsto \frac{1}{\prod_{\lambda \neq 1} (1- \lambda)^m} \prod_{\lambda \neq 1} ([s]_* - \lambda)^m \quad \left( m = \dim \ H \right)$$
est un projecteur sur $H^{(1)}$.\footnote{C'est pour cette raison qu'on considère le sous-espace caractéristique $H^{(1)}$ plutôt que le sous-espace des vecteurs $1$-propre.}
En passant aux parties invariantes dans la suite \eqref{LESGysin}, on obtient donc une suite exacte dont le terme de gauche est pur de poids $2j-2$ alors que le terme suivant est pur de poids $2j$. L'application $\delta_1$ est donc nulle. De même le terme de droite est pur de poids $2j+2$ alors que le terme précédent $H^{j-1}(H - (H \cap \Upsilon), \mathbf{C}(-1))^{(1)}$ est pur de poids $2j$. L'application $\delta_2$ est donc nulle elle aussi.
\qed
\medskip
On compare maintenant ces suites exactes à la suite exacte longue associée à \eqref{dupont}, c'est-à-dire,
\begin{multline} \label{dupont2}
H^p(T, \Omega^q_{\langle T ,\Upsilon \rangle} )^{(1)} \rightarrow H^p(T, \Omega^q_{\langle T , \Upsilon ' \rangle })^{(1)} \rightarrow H^p(H, \Omega^{q-1}_{\langle H , \Upsilon \cap H\rangle })^{(1)} \\
\stackrel{\delta}{\rightarrow} H^{p+1}(T, \Omega^q_{\langle T ,\Upsilon \rangle})^{(1)}.
\end{multline}
Le quasi-isomorphisme \eqref{Dqi} donne plus précisément lieu à une suite spectrale qui calcule la cohomologie de $T-\Upsilon$ et le lemme suivant montre que la première page de cette suite spectrale permet de calculer la partie invariante de la cohomologie de $T-\Upsilon$.
\begin{lemma}\label{C2}
Supposons que $A$ soit une courbe elliptique et que $T-\Upsilon$ soit affine. Alors la suite spectrale de Hodge--de Rham
\begin{equation} \label{SSHdR}
H^p(T, \Omega^q_{\langle T , \Upsilon \rangle})^{(1)} \Longrightarrow H^{p+q}(T-\Upsilon)^{(1)}
\end{equation}
dégénère à la première page et tous les morphismes de connexions $\delta$ dans \eqref{dupont2} sont nuls.
\end{lemma}
\proof
On procède à nouveau par récurrence sur $(n , \# \Upsilon)$.
On initialise la récurrence avec le cas où $\Upsilon$ est constitué de $n$ hyperplans linéairement indépendants. Alors $H$ est un diviseur à croisements normaux et $\Omega^q_{\langle T , \Upsilon \rangle} = \Omega^q_T (\log \Upsilon )$. Dans ce cas le lemme est une conséquence de la dégénérescence de la suite spectrale de Hodge--de Rham à pôles logarithmiques, cf. \S \ref{B2}.
Supposons maintenant par récurrence que le lemme est démontré pour les arrangements $(H, H \cap \Upsilon)$ et $(T, \Upsilon)$; nous le vérifions alors pour l'arrangement $(T, \Upsilon ')$ avec toujours $\Upsilon '=\Upsilon \cup H$.
De la suite exacte courte \eqref{dupont} on tire
\begin{multline} \label{dps}
\dim \ H^p(T, \Omega^q_{\langle T , \Upsilon ' \rangle})^{(1)} \\ \leq \dim \ H^p(T, \Omega^q_{\langle T , \Upsilon \rangle})^{(1)} + \dim \ H^p(H, \Omega_{\langle H , H\cap \Upsilon \rangle }^{q-1})^{(1)}.
\end{multline}
En sommant sur tous les couples $(p,q)$ tels que $p+q=j$ on obtient tour à tour les inégalités suivantes~:
\begin{equation*}
\begin{split}
\dim \ H^j(T - \Upsilon ' )^{(1)} & \stackrel{(i)}{\leq} \sum_{p+q=j} \dim \ H^p(T, \Omega^q_{\langle T , \Upsilon ' \rangle} )^{(1)} \\
& \stackrel{(ii)}{ \leq } \sum_{p+q=j} \left( \dim \ H^p(T, \Omega^q_{\langle T , \Upsilon \rangle})^{(1)} + \dim \ H^p(H, \Omega_{\langle H , H\cap \Upsilon \rangle }^{q-1})^{(1)} \right) \\
& \stackrel{(iii)}{=} \dim \ H^j(T - \Upsilon)^{(1)} + \dim \ H^{j-1}(H - (\Upsilon \cap H))^{(1)},
\end{split}
\end{equation*}
où l'on explique chacune des (in)égalités ci-dessous.
\begin{itemize}
\item[(i)] Découle de l'existence de la suite spectrale \eqref{SSHdR}; on a égalité pour tous les $j$ si et seulement si la suite spectrale dégénère à la première page.
\item[(ii)] Découle de \eqref{dps}; on a égalité si et seulement les morphismes de connexions dans la suite exacte longue \eqref{dupont2} sont nuls.
\item[(iii)] C'est l'étape de récurrence.
\end{itemize}
Finalement le lemme \ref{exactness in topology} implique que toutes les inégalités ci-dessus sont en fait des égalités et donc que la suite spectrale pour $\Upsilon '=\Upsilon \cup H$ dégénère et que la suite exacte longue \eqref{dupont2} se scinde en suites exactes courtes.
\qed
\medskip
\noindent
{\it Remarque.} Comme nous l'a fait remarquer un rapporteur, le formalisme général des complexes de Hodge mixtes \cite[Scholie 8.1.9, (v)]{DeligneH3} et les travaux de Dupont impliquent en fait directement que la suite spectrale de Hodge--de Rham \eqref{SSHdR} dégénère à la première page.
\medskip
\begin{lemma}\label{C3}
Supposons $A = \mathbf{G}_m$. La suite \eqref{dupont} donne lieu à des suites exactes courtes :
\begin{equation} \label{dupont3}
0 \rightarrow H^0(T, \Omega^j_{\langle T,\Upsilon \rangle }) \rightarrow H^0(T, \Omega^{j}_{\langle T , \Upsilon ' \rangle } ) \rightarrow H^0(H , \Omega^{j-1}_{\langle H , H \cap \Upsilon \rangle}) \rightarrow 0.
\end{equation}
De plus, l'action de $[s]_*$ sur chacun des espaces impliqués est localement finie.
\end{lemma}
\proof
La première assertion découle du fait que qu'il n'y a pas de cohomologie supérieure puisque $T$ est affine. La seconde affirmation s'obtient par récurrence; l'initialisation est conséquence du lemme \ref{L:B1}.
\qed
\subsection{Fin de la démonstration du théorème \ref{P:Brieskorn}}
On peut maintenant démontrer le théorème par récurrence sur $(n, \# \Upsilon)$.
L'initialisation a été vérifiée aux \S \ref{B1} et \ref{B2}. Supposons maintenant par récurrence que le théorème est démontré pour les arrangements $(H, H \cap \Upsilon)$ et $(T, \Upsilon)$ (avec $T-\Upsilon$ et $H-(H\cap \Upsilon)$ affines donc). Vérifions alors le théorème pour l'arrangement $(T, \Upsilon ')$ avec toujours $\Upsilon '=\Upsilon \cup H$.
D'après les lemmes \ref{exactness in topology}, \ref{C1} et \ref{C2} on a des diagrammes commutatifs
$$
\xymatrix{
H^0(T, \Omega^j_{\langle T,\Upsilon \rangle })^{(1)} \ar[d] \ar@{^{(}->}[r] & H^0(T, \Omega^{j}_{\langle T , \Upsilon ' \rangle } )^{(1)} \ar[d] \ar@{->>}[r] & H^0(H , \Omega^{j-1}_{\langle H , H \cap \Upsilon \rangle})^{(1)} \ar[d] \\
H^j(T - \Upsilon )^{(1)} \ar@{^{(}->}[r] & H^j (T- \Upsilon ' )^{(1)} \ar@{->>}[r] & H^{j-1}( H - (H \cap \Upsilon ))^{(1)}
}
$$
où les suites horizontales sont exactes et, par hypothèse de récurrence, les morphismes verticaux à gauche et à droite sont des isomorphismes. Le morphisme vertical du milieu est donc lui aussi un isomorphisme, ce qui démontre le théorème.
\qed
\medskip
\noindent
{\it Remarque.} Comme nous l'a fait remarquer un rapporteur, la théorie de Hodge mixte et les travaux de Dupont permettent de déduire directement du lemme \ref{C1} de pureté que
$$H^0(T, \Omega^\bullet_{\langle T,\Upsilon \rangle })^{(1)} \to H^\bullet (T - \Upsilon )^{(1)}$$
est un isomorphisme. Nous avons préféré maintenir notre approche un peu plus pédestre. Dans tous les cas, le c{\oe}ur de l'argument repose sur \cite{DupontC}.
\medskip
\chapter{Formes différentielles sur l'espace symétrique associé à $\mathrm{SL}_n (\mathbf{C})$} \label{C:4}
\resettheoremcounters
Dans ce chapitre, on fixe un entier $n\geq 2$ et on note $V = \mathbf{C}^n$; on voit les éléments de $V$ comme des vecteurs colonnes.
Une matrice $g \in \mathrm{GL}_n (\mathbf{C})$ définit une forme hermitienne sur $\mathbf{C}^n$ de matrice hermitienne associée $g^{-*}g^{-1}$.\footnote{On note $M^\top$ la transposée d'une matrice $M$, et $M^*=\overline{M}^\top$. Lorsque $M$ est inversible on note enfin $M^{-\top}$ et $M^{-*}$ la transposée et la conjuguée de son inverse.} On en déduit une bijection
\begin{equation*}
S:=\mathrm{GL}_n (\mathbf{C} ) / \mathrm{U}_n \simeq \left\{H \; : \; H \text{ forme hermitienne définie positive sur } \mathbf{C}^n \right\}.
\end{equation*}
Dans ce chapitre on note
$$G = \mathrm{GL}_n (\mathbf{C}) \quad \mbox{et} \quad K = \mathrm{U}_n$$
de sorte que $S=G/K$. En identifiant $\mathbf{R}_{>0}$ au centre réel de $\mathrm{GL}_n (\mathbf{C})$ {\it via} l'application $s \mapsto s \cdot 1_n$, le quotient
$$X = \mathrm{GL}_n (\mathbf{C}) / \mathrm{U}_n \mathbf{R}_{>0}$$
est l'espace symétrique associé au groupe $\mathrm{SL}_n (\mathbf{C})$.
On prendra pour point base dans $S$ la métrique hermitienne $|\cdot |$. Pour $z = (z_1 , \ldots , z_n) \in \mathbf{C}^n$, vu comme vecteur colonne, on a $|z|^2 = |z_1 |^2 + \ldots + |z_n|^2$ et la forme hermitienne $H$ associée à une matrice $g \in \mathrm{GL}_n (\mathbf{C})$ est donc donnée par $H(z,z) = | g^{-1} z |^2$.
Dans ce chapitre on explique que la théorie de Mathai--Quillen permet de construire deux formes différentielles $G$-invariantes naturelles
$$\psi \in A^{2n-1}(S \times \mathbf{C}^n ) \quad \mbox{et} \quad \varphi \in A^{2n} (S \times \mathbf{C}^n )$$
qui décroissent rapidement le long des fibres de $S \times \mathbf{C}^n \to S$ et vérifient
\begin{enumerate}
\item $\varphi$ est une \emph{forme de Thom} au sens qu'elle est fermée et d'intégrale $1$ dans les fibres $\mathbf{C}^n$, et
\item la transformée de Mellin de $\psi$ en $0$ définit est une forme fermée sur $S\times (\mathbf{C}^n - \{ 0 \} )$ et sa restriction aux fibres $\mathbf{C}^n - \{0 \}$ représente la classe fondamentale.
\end{enumerate}
\section{Formes de Mathai--Quillen}
Dans la suite de ce chapitre on note $\mathfrak{g}$ et $\mathfrak{k}$ les algèbres de Lie respectives de $G$ et $K$ et on note $\mathfrak{p}$ le supplémentaire orthogonal de $\mathfrak{k}$ dans $\mathfrak{g}$ relativement à la forme de Killing.
\subsection{Fibré $K$-principal au-dessus de $S$} La projection $G \to G/K$ permet de voir le groupe $G$ comme un fibré $K$-principal au-dessus de $S$. Une connexion sur $G$ est une $1$-forme $\theta \in A^1(G) \otimes \mathfrak{k}$ telle que
\begin{equation}
\begin{split}
\mathrm{Ad}(k)(k^*\theta) &= \theta, \qquad k \in K, \\
\iota_X \theta &=X, \qquad X \in \mathfrak{k}.
\end{split}
\end{equation}
Le fibré $G \to S$ est naturellement $G$-équivariant (relativement aux actions naturelles, à gauche, de $G$ sur lui-même et sur $S$). Soit $\theta$ la connexion $G$-invariante sur $G$ définit comme suit~: {\it via} les isomorphismes
\[
(A^1(G) \otimes \mathfrak{k})^{G \times K} \simeq (\mathfrak{g}^* \otimes \mathfrak{k})^K \simeq \mathrm{Hom}_K(\mathfrak{g},\mathfrak{k}),
\]
une connexion $G$-invariante correspond à une section $K$-équivariante de l'inclusion $\mathfrak{k} \hookrightarrow \mathfrak{g}$. On définit $\theta$ comme étant la connexion associée à la projection $p : \mathfrak{g} \to \mathfrak{k}$. La forme de connexion $\theta$ est donc explicitement donnée par la formule
\[
\theta = p(g^{-1}dg) = \frac{1}{2}(g^{-1}dg - g^* d( g^{-*})).
\]
Sa forme de courbure associée
\[
\Omega = (\Omega_{ij} )_{1 \leq i, j \leq n} = d\theta+\theta^2 \in A^2(G) \otimes \mathfrak{k}
\]
est $G$-invariante et horizontale, autrement dit $\iota_X \Omega = 0$ pour tout $X \in \mathfrak{k}$ --- identifié au champs de vecteur engendré par l'action à droite de $\mathfrak{k}$ on $G$.
\subsection{Fibré vectoriel associé}
Soit
$$G \times^K V = [ G \times V ] / K,$$
où l'action (à droite) de $K$ on $G \times V$ est $(g,v)k = (gk,k^{-1}v)$. C'est un fibré vectoriel au-dessus de $S=G/K$ et l'action
$$h \cdot [g,v]=[hg,v]$$
l'équipe d'une structure de fibré $G$-équivariant. La forme hermitienne standard $v \mapsto v^*v$ sur $V$ le munit finalement d'une métrique $G$-\'equivariante.
\subsection{Une forme de Thom explicite} On rappelle ici l'expression de la forme de Thom construite par Mathai et Quillen \cite{MathaiQuillen} sur $G \times^K V$. Soient $z_1,\overline{z}_1, \ldots,z_n, \overline{z}_n$ les coordonnées standard sur $V$. On note respectivement $z$ et $dz$ les vecteurs colonnes $(z_1,\ldots,z_n)^\top$ et $(dz_1,\ldots,dz_n)^\top$.
\'Etant donné un sous-ensemble $I=\{i_1,\ldots,i_p\} \subseteq \{1 \ldots,n\}$, avec $i_1 < \cdots < i_p$, et un vecteur $\xi=(\xi_1,\ldots,\xi_n )$ on note
$$\xi^I = \xi_{i_1} \cdots \xi_{i_p} \quad \mbox{et} \quad \xi^{I*} = \xi_{i_p} \cdots \xi_{i_1}.$$
On note $I'=\{1,\ldots,n \}-I$ le complémentaire de $I$ et on définit une signe $\epsilon(I,I')$ par l'égalité $dz^I dz^{I'} = \epsilon(I,I') dz_1 \cdots dz_n$.
L'expression
\begin{equation} \label{E:UMQ}
U = \left(\frac{i}{2\pi}\right)^n e^{-|z|^2} \sum_{I, J} \epsilon(I,I') \epsilon(J,J') \det(\Omega_{IJ}) (dz+\theta z)^{I'} \overline{(dz+\theta z)}^{J' *},
\end{equation}
où la somme porte sur tous les couples $(I,J)$ de sous-ensembles de $\{1,\ldots,n\}$ avec $|I|=|J|$ et $\Omega_{IJ}=(\Omega_{ij})_{i \in I, j \in J}$, définit une forme $K$-invariante, fermée, horizontale et de degré $2n$ sur $G \times V$ et donc une forme dans $A_{d=0}^{2n}(G \times^K V)$; c'est la forme de Thom de Mathai--Quillen.
\subsection{Formes différentielles sur $S \times V$}
La représentation standard de $G$ dans $V$ fait du fibré trivial
$$E = S \times V \to S$$
un fibré $G$-équivariant --- pour tout $g \in G$, l'isomorphisme $\alpha(g):E \xrightarrow{\sim} g^*E$ est donné, dans chaque fibre, par la multiplication par $g$. Le fibré $E$ est naturellement muni d'une métrique hermitienne $G$-équivariante $|\cdot|_h^2$ donnée par
$|v|_H^2 = v^*Hv$, où $H=H(g)=g^{-*}g^{-1}$.
L'application
\[
\Phi: E \to G \times^{K} V, \qquad \Phi(gK,v) = (g,g^{-1}v)
\]
est une isométrie $G$-équivariante.
\begin{definition}
Soit
\[
\varphi = \Phi^*( U) \in A^{2n} (E)^G \quad \mbox{et} \quad \psi = \iota_X \varphi \in A^{2n-1} (E)^G,
\]
où $X = \sum_i (z_i \partial_{i} + \overline{z}_i \overline{\partial}_i )$ est le champs de vecteur radial sur $E$.
\end{definition}
La forme $\varphi$ est fermée, rapidement décroissante et d'intégrale $1$ le long des fibres de $S \times V \to V$; c'est une \emph{forme de Thom} en ce sens.
\section{Une $(2n-1)$-forme fermée sur $X \times (\mathbf{C}^n - \{ 0 \})$} \label{S:42}
La multiplication par un réel strictement positif $s$ sur $V=\mathbf{C}^n$ induit une application
$$[s] : S \times V \to S \times V.$$
Il découle des définitions que
\begin{equation} \label{E:tddt1}
d ([s]^* \psi ) = s \frac{d}{ds} ( [s]^* \varphi ).
\end{equation}
La proposition suivante est essentiellement due à Mathai et Quillen \cite[\S 7]{MathaiQuillen}.
\begin{proposition} \label{P:eta}
L'intégrale
\begin{equation} \label{E:eta}
\eta = \int_0^{+\infty} [s]^* \psi \frac{ds}{s}
\end{equation}
est convergente et définit une forme \emph{fermée} dans $A^{2n-1} (X \times (\mathbf{C}^n - \{ 0 \}))^G$ dont la restriction à chaque fibre $\mathbf{C}^n -\{0 \}$ de $X \times (\mathbf{C}^n - \{ 0 \} )$ représente la classe fondamentale.
\end{proposition}
\begin{proof} L'expression explicite \eqref{E:UMQ} de la forme de Thom de Mathai--Quillen permet de montrer que l'intégrale converge en $0$; en fait lorsque $s$ tend vers $0$ la forme $[s]^* \psi$ est un $O(s)$.
Maintenant si $v \in V$ est un vecteur non nul, l'image $s v$ tend vers l'infini avec $s$ et le fait que $\psi$ soit rapidement décroissante dans les fibres de $S \times V \to S$ implique que l'intégrale \eqref{E:eta} converge sur $S \times (\mathbf{C}^n - \{ 0 \})$. Comme $\eta$ est de plus invariante, par construction, par multiplication dans les fibres, elle définit finalement une forme sur $X \times (\mathbf{C}^n - \{ 0 \} )$.
La forme $\eta$ est de degré $2n-1$ et $G$-invariante. On calcule sa différentielle à l'aide de \eqref{E:tddt1}~:
\begin{equation*}
\begin{split}
d \eta & = d \left( \int_0^{+\infty} [s]^* \psi \frac{ds}{s} \right) \\
& = \int_0^{+\infty} d([s]^* \psi) \frac{ds}{s} \\
& = \int_0^{+\infty} \frac{d}{ds} ([s]^* \varphi ) ds .
\end{split}
\end{equation*}
La décroissance rapide de $\varphi$ le long des fibres de $E \to S$ montre comme ci-dessus que sur $S \times (\mathbf{C}^n - \{ 0 \} )$ les formes $[s]^* \varphi$ tendent vers $0$ quand $s$ tend vers l'infini. Enfin, il découle de \eqref{E:UMQ} que lorsque $s$ tend vers $0$ la forme $[s]^* \varphi$ tend vers
\begin{equation} \label{E:chern}
\omega = \left( \frac{i}{2\pi} \right)^n \Phi^* (\det \Omega).
\end{equation}
La différentielle $d\eta$ est donc égale à \eqref{E:chern}. Montrons maintenant que cette forme est identiquement nulle.
La $2n$-forme $G$-invariante \eqref{E:chern} est égale au tiré en arrière, par la projection $E \to S$, du représentant de Chern--Weil de la classe de Chern de degré maximal $c_n (E)$.\footnote{Le $G$-fibré $E$ étant plat la forme \eqref{E:chern} admet donc nécessairement une primitive $G$-invariante, ce qu'est précisément le forme $\eta$. Montrer que $\eta$ est fermée est un problème analogue à l'existence du relevé canonique dans la proposition \ref{P4}.}
Soit $B \subset \mathrm{GL}_n (\mathbf{C})$ le sous-groupe de Borel des matrices triangulaires supérieures. La décomposition $G = B K$ induit un difféomorphisme $B$-équivariant
$$f: B \times^{B \cap K} V \to G \times^K V.$$
Il suffit donc de montrer que la $2n$-forme différentielle
\begin{equation} \label{E:f*MQ}
\left( \frac{i}{2\pi} \right)^n f^* (\det \Omega)
\end{equation}
est identiquement nulle. Maintenant, comme $B \cap K = \mathrm{U}_1^n$, le fibré hermitien $B \times^{B \cap K} V$ se scinde métriquement en une somme directe de fibrés en droite au-dessus de $B/(B \cap K)$. Par fonctorialité la forme \eqref{E:f*MQ} au-dessus d'un point $[b]$ dans $B/(B \cap K)$ s'obtient comme $b$-translaté du produit de $n$ formes sur les facteurs $\mathrm{U}_1$ de $B \cap K$, chacune de ces formes égale à \eqref{E:f*MQ} avec $n=1$. On est ainsi réduit à considérer le cas $n=1$ et la nullité de \eqref{E:f*MQ} résulte du fait qu'il n'y a pas de $2$-forme non nulle sur le cercle.
\medskip
\noindent
{\it Remarque.} Une démonstration plus conceptuelle est possible : par l'astuce de Weyl il suffit en effet de démontrer que la forme invariante correspondant à \eqref{E:chern} sur le dual compact de $S$
$$S^c = \mathrm{U}_n = (\mathrm{U}_n \times \mathrm{U}_n) /\mathrm{U}_n$$
est identiquement nulle. Mais celle-ci est harmonique car fermée invariante et représente la classe de Chern maximale d'un fibré plat. Elle est donc nécessairement nulle.
\medskip
Il nous reste à voir que $\eta$ représente la classe fondamentale de $S \times (\mathbf{C}^n - \{ 0 \} )$. Il suffit pour cela de se restreindre à la fibre au-dessus du point base de $S$. Alors $\varphi$ est simplement
$$
\left( \frac{i}{2\pi} \right)^n e^{-|z|^2} dz_1 \wedge \ldots \wedge dz_n \wedge d\overline{z}_n \wedge \ldots \wedge d \overline{z}_1
= \frac{1}{\pi^n} e^{- \sum_j x_j^2} dx_1 \wedge \ldots \wedge dx_{2n}$$
où les $x_j$ sont les coordonnées dans une base orthonormée et orienté positivement du $\mathbf{R}$-espace vectoriel $\mathbf{C}^n$. Dans ces coordonnées le champ radial $X$ s'écrit $\sum_j x_j \partial_{x_j}$ et la forme $\psi$ est égale à
$$\frac{1}{\pi^n} e^{- \sum_j x_j^2} \sum_{j=1}^{2n} (-1)^{j-1} x_j dx_1 \wedge \ldots \wedge \widehat{dx_j} \wedge \ldots \wedge dx_{2n}.$$
On trouve finalement que la forme $\eta$, en restriction à la fibre $\mathbf{C}^n - \{ 0 \}$ au-dessus du point base de $S$, est égale à
$$\frac{\Gamma (n)}{2 \pi^n} \frac{ \sum_{j=1}^{2n} (-1)^{j-1} x_j dx_1 \wedge \ldots \wedge \widehat{dx_j} \wedge \ldots \wedge dx_{2n}}{ (x_1^2 + \ldots + x_{2n}^2)^n}$$
qui est la forme volume normalisée sur la sphère $\mathbf{S}^{2n-1}$.
\end{proof}
\medskip
\noindent
{\it Remarque.} On peut identifier $S$ à $X \times \mathbf{R}_{>0}$ {\it via} l'application $\mathrm{GL}_n (\mathbf{C} )$-équivariante $gU_n \mapsto ( [g] , | \det (g) |^{1/n})$. Les formes $\varphi$ et $\psi$ définissent alors deux formes invariantes sur $X \times \mathbf{R}_{>0} \times V$ et on a
$$\varphi = \alpha - \psi \wedge \frac{dr}{r},$$
où
$\alpha$ est une forme différentielle sur $X \times \mathbf{R}_{>0} \times V$ qui ne contient pas de composante $dr$ le long de $\mathbf{R}_{>0}$ et dont la restriction à chaque $X \times \{s \} \times V$ $(s \in \mathbf{R}_{>0})$ est égale à $[s]^* \varphi$. Le fait que $\varphi$ soit fermée est équivalent à
\eqref{E:tddt1} et, au-dessus de $V-\{0 \}$, la forme $\eta$ est le résultat de l'intégration partielle
$$\eta = \int_{\mathbf{R}_{>0}} \varphi.$$
\section{Calculs explicites dans le cas $n=1$}
On suppose dans ce paragraphe que $n=1$. Dans ce cas $S= \mathbf{R}_{>0}$ et $X$ est réduit à un point. On peut alors calculer explicitement les formes de Mathai--Quillen, cf. \cite{Takagi}. On trouve que pour $(r,z) \in \mathbf{R}_{>0} \times \mathbf{C}$ on a
\begin{equation} \label{E:phiN1}
\varphi = \frac{i}{2\pi} e^{- r^2 |z|^2 } \left( r^2 dz \wedge d \overline{z} -r^2 ( z d\overline{z} - \overline{z}dz ) \wedge \frac{dr}{r} \right)
\end{equation}
de sorte que
\begin{equation} \label{E:psiN1}
\psi = - \frac{i}{2\pi} \left( r^2 |z|^2 e^{- r^2 |z|^2 } \right) \left( \frac{dz}{z} - \frac{d\overline{z}}{\overline{z}} \right)
\end{equation}
et
\begin{equation} \label{E:etaN1}
\eta = - \frac{i}{4\pi} \left( \frac{dz}{z} - \frac{d\overline{z}}{\overline{z}} \right).
\end{equation}
\section{Formes de Schwartz et représentation de Weil} \label{S:44}
L'espace total du fibré $E$ est un espace homogène sous l'action du groupe affine $G \ltimes V = \mathrm{GL}_n (\mathbf{C}) \ltimes \mathbf{C}^n$, où
$$(g,v) \cdot (g' , v' ) = (gg' , g v' + v).$$
L'action à gauche du groupe affine sur $E= S \times V$ est transitive et le stabilisateur du point base $([K] , 0)$ est le groupe $K=\mathrm{U}_n$ plongé dans le groupe affine {\it via} l'application $k \mapsto (k,0)$.
Soit $\mathcal{S} (V)$ l'espace de Schwartz de $V$. On appelle {\it représentation de Weil} du groupe affine dans $\mathcal{S} (V)$ la représentation $\omega$ donnée par~:
$$\omega (g , v) : \mathcal{S} (V) \to \mathcal{S} (V) ; \quad \phi \mapsto \left( w \mapsto \phi ( g^{-1} (w-v)) \right).$$
Soit\footnote{L'isomorphisme est induit par l'évaluation en le point base $(eK, 0)$ dans $S \times V$.}
$$A^{k} (E , \mathcal{S}(V))^{G \ltimes V} := \left[ \mathcal{S} (V) \otimes A^k (E) \right]^{G \ltimes V} \cong \left[\mathcal{S}(V) \otimes \wedge^\bullet (\mathfrak{p} \oplus V)^* \right]^K$$
l'espace des $k$-formes différentielles invariantes sur $E$ à valeurs dans $\mathcal{S}(V)$.
\begin{definition}
Pour tout $w \in V$, on pose
$$\widetilde{\varphi} (w) = t_{-w}^* \varphi \in A^{2n} (E) \quad \mbox{et} \quad \widetilde{\psi}(w) = t_{-w}^* \psi \in A^{2n-1} (E),$$
où $t_{-w} : E \to E$ désigne la translation par $-w$ dans les fibres de $E$.
\end{definition}
\begin{lemma}
Les applications $w \mapsto \widetilde{\varphi} (w)$ et $w \mapsto \widetilde{\psi}(w)$ définissent des éléments
$$\widetilde{\varphi} \in A^{2n} (E , \mathcal{S}(V))^{G \ltimes V} \quad \mbox{et} \quad \widetilde{\psi} \in A^{2n-1} (E , \mathcal{S}(V))^{G \ltimes V}.$$
\end{lemma}
\begin{proof} Montrons d'abord l'invariance, c'est à dire que pour tous $(g,v)$ et $(h,w)$ dans $G \ltimes V$, on a
\begin{equation} \label{E:invariancephipsi}
(g,v)^* \widetilde{\varphi} (gw+v) = \widetilde{\varphi} (w) \quad \mbox{et} \quad (g,v)^* \widetilde{\psi} (gw+v) = \widetilde{\psi} (w).
\end{equation}
Ici $(g,v)^*$ désigne le tiré en arrière par le difféomorphisme de $E$ induit par l'élément $(g,v) \in G \ltimes V$.
L'identité \eqref{E:invariancephipsi} résulte des définitions; on la vérifie pour $\widetilde{\varphi}$, le cas de $\widetilde{\psi}$ se traitant de la même manière~:
\begin{equation*}
\begin{split}
(g,v)^* \widetilde{\varphi} (gw+v) & = \left[ (1,v) \cdot (g , 0) \right]^* \widetilde{\varphi} (gw+v) \\
& = g^* \left[ (1,v)^* \widetilde{\varphi} (gw+v) \right] \\
& = g^* \left[ (1,v)^* ((1, -gw -v)^* \varphi ) \right] \\
& = g^* \left[ (1 , -gw )^* \varphi \right] = \left[ (1, -gw) \cdot (g, 0 ) \right]^* \varphi \\
& = (g , -gw)^* \varphi = \left[ (g,0) \cdot (1 , -w) \right]^* \varphi \\
& = (1 , -w)^* (g^* \varphi ) = (1,-w)^* \varphi = \widetilde{\varphi} (w).
\end{split}
\end{equation*}
Il nous reste à vérifier que $\widetilde{\varphi}$ --- et de la même manière $\widetilde{\psi}$ --- définit bien une forme différentielle à valeurs dans $\mathcal{S}(V)$. Par $(G \ltimes V)$-invariance, il suffit pour cela de remarquer que pour tout $X$ dans $\wedge^{2n} (\mathfrak{p} \oplus V)$, la fonction
$$(\widetilde{\varphi} (w))_{(eK , 0)} (X) = (t_{-w}^* \varphi )_{(eK , 0)} (X) = [\varphi (-w)] (X) $$
est bien une fonction de Schwartz de $w$.
\end{proof}
\medskip
\noindent
{\it Remarque.} De notre point de vue, l'apport majeur du formalisme de Mathai--Quillen est de fournir ces fonctions tests ``archimédiennes'' auxquelles on peut alors appliquer le formalisme automorphe. Il est en effet notoirement délicat de construire les bonnes fonctions tests à l'infini en général.
\medskip
Par définition on a
$$\widetilde{\varphi} (0) = \varphi \quad \mbox{et} \quad \widetilde{\psi} (0) = \psi$$
et il découle en particulier de \eqref{E:invariancephipsi} que
\begin{equation} \label{E:pullback}
v^* \widetilde{\varphi} (w) = (v-w)^* \varphi \quad \mbox{et} \quad v^* \widetilde{\psi} (w) = (v-w)^* \psi \in A^\bullet (S) ,
\end{equation}
où l'on a identifié un vecteur de $V$ à la section constante $S \to E$ qu'il définit.
Finalement, les formes $[s]^* \varphi$ et $[s]^* \psi$ définissent à leur tour des formes différentielles $\widetilde{[s]^* \varphi}$ et $\widetilde{[s]^* \psi}$ dans $A^{\bullet} (E, \mathcal{S}(V))^{G \ltimes V}$ qui vérifient
\begin{equation} \label{E:invariancephipsi2}
\widetilde{[s]^* \varphi} (w) = [s]^* \widetilde{\varphi} (s w) \quad \mbox{et} \quad \widetilde{[s]^* \psi} (w) = [s]^* \widetilde{\psi} (s w).
\end{equation}
Le lemme suivant découle de la construction.
\begin{lemma} \label{L:convcourant}
1. Lorsque $s$ tend vers $+\infty$, les formes $\widetilde{[s]^*\varphi} (0) = [s]^* \varphi$ convergent uniformément sur tout compact de $S \times (\mathbf{C}^n - \{ 0 \})$ vers la forme nulle et convergent au sens des courants vers le courant d'intégration le long de $S \times \{ 0 \}$.
2. Soit $v \in \mathbf{C}^n - \{ 0 \}$. Lorsque $s$ tend vers $+\infty$, les formes $$\widetilde{[s]^*\varphi} (v) = [s]^* \widetilde{\varphi} (sv)$$ convergent uniformément exponentiellement vite sur tout compact de $S \times \mathbf{C}^n$ vers la forme nulle.
3. Lorsque $s$ tend vers $0$, les formes $\widetilde{[s]^*\varphi} (0) = [s]^* \varphi$ convergent uniformément sur tout compact de $S \times \mathbf{C}^n$ vers la forme nulle.
\end{lemma}
\medskip
\noindent
{\it Remarque.} Dans le dernier cas, les formes $\widetilde{[s]^*\varphi} (0) = [s]^* \varphi$ convergent uniformément sur tout compact de $S \times \mathbf{C}^n$ vers la forme $\omega$ définie en \eqref{E:chern} et dont on a montré qu'elle est identiquement nulle. Dans le dernier chapitre, on considère la forme $\varphi$ associée au produit
$$\mathrm{GL}_n (\mathbf{R} ) / \mathrm{SO}_n \times \mathrm{SL}_2 (\mathbf{R} ) /\mathrm{SO}_2 \times \mathbf{C}^n.$$
Dans ce cas les formes $\widetilde{[s]^*\varphi} (0) = [s]^* \varphi$ convergent uniformément sur tout compact de $S \times \mathbf{C}^n$ vers une forme invariante non nulle en générale.
\chapter{Compactifications de Satake, de Tits et symboles modulaires} \label{C:5}
\resettheoremcounters
Les espaces symétriques admettent de nombreuses compactifications équivariantes, cf. \cite{BorelJi}. On rappelle ici deux d'entre elles, la compactification (minimale) de Satake et la compactification de Tits. On étudie ensuite le comportement de la forme $\eta$, définie au chapitre précédent, lorsque l'on s'approche du bord de ces compactifications.
\section{Compactification de Satake}
L'espace $S$ est un cône ouvert dans l'espace vectoriel réel $\mathcal{H}$ des matrices hermitiennes de rang $n$. L'action de $G$ sur $S$
$$g : H \mapsto g^{-*} H g^{-1}$$
s'étend en une action sur $\mathcal{H}$ qui induit une action de $G$ sur l'espace projectif $\mathrm{P} (\mathcal{H})$. L'application
$$i : X \to \mathrm{P} (\mathcal{H})$$
est un plongement $G$-équivariant. L'adhérence de son image $i(X)$ dans l'espace compact $\mathrm{P} (\mathcal{H})$ est donc une compactification $G$-équivariante de $X$; elle est appelée {\it compactification (minimale) de Satake} et notée $\overline{X}^S$. Elle est convexe et donc contractile.
La compactification de Satake $\overline{X}^S$ se décompose en une union disjointe
\begin{equation} \label{E:Xsatake1}
\overline{X}^S = X \bigcup_{W} b(W) ,
\end{equation}
où $W$ parcourt les sous-espaces non nuls et propres de $\mathbf{C}^n$ et $b(W)$ désigne l'image dans $\mathrm{P} (\mathcal{H})$ du cône sur l'ensemble des matrices hermitiennes semi-définies positives dont le noyau est exactement $W$.
On note $P_W$ le sous-groupe parabolique de $\mathrm{SL}_n (\mathbf{C})$ qui préserve $W$. Si $g \in \mathrm{SL}_n (\mathbf{C})$ est tel que
$$W = g \langle e_1 , \ldots , e_{j} \rangle \quad (j=\dim W) ,$$
le groupe $P_W$ est obtenu en conjuguant par $g$ le groupe
$$P=P_j = P_{\langle e_1 , \ldots , e_{j} \rangle}.$$
Soient
\begin{equation*}
\begin{split}
N=N_j &= \left\{ \left. \begin{pmatrix} 1_j & x \\ 0 & 1_{n-j}\end{pmatrix} \ \right| \ x \in M_{j, (n-j)} (\mathbf{C} ) \right\} \\
M=M_j &= \left\{ \left. \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} \ \right| \ A \in \mathrm{GL}_j(\mathbf{C}), \ B \in \mathrm{GL}_{n-j}(\mathbf{C}) , \ |\det(A)| =|\det(B)|=1
\right\} \\
A=A_j &= \left\{ \left. a(t_1,t_2):=\begin{pmatrix} t_1 1_j & 0 \\ 0 & t_2 1_{n-j} \end{pmatrix} \ \right| \ t_1, t_2 \in \mathbf{R}_{>0}, \ \det a(t_1,t_2) = 1
\right\} .
\end{split}
\end{equation*}
Un élément $g \in \mathrm{SL}_n(\mathbf{C})$ peut s'écrire
\[
g = u m a k, \quad u \in N, \ m \in M, \ a \in A, \ k \in \mathrm{SU}_n
\]
Dans cette décomposition $u$ et $a$ sont uniquement déterminés par $g$, et $m$ et $k$ sont déterminés à un élément de $M \cap \mathrm{SU}_n$ près. On peut donc écrire $a=a(t_1(g),t_2(g))$ avec $t_1(g)$ et $t_2(g)$ déterminé par $g$; on a plus précisément
\[
t_1(g)^{-j} = \det (H(g)|_{ \langle e_1,\ldots,e_j \rangle }),
\]
où $H(g)|_{\langle e_1,\ldots,e_j \rangle}$ désigne la restriction de la métrique hermitienne $v \mapsto |g^{-1}v|^2$ déterminée par $g$ au sous-espace $\langle e_1,\ldots, e_j \rangle$.
\'Etant donné un réel $t \in \mathbf{R}^+$, on pose
\[
A_t = \{a(t_1,t_2) \in A \; : \; t_1 /t_2 \geq t \}.
\]
\begin{definition}
On appelle \emph{ensemble de Siegel} associé à un sous-ensemble relativement compact $\omega \subset NM$ le sous-ensemble
\[
\mathfrak{S}_j (t,\omega) := \omega A_t \cdot \mathrm{SU}_n \subset \mathrm{SL}_n(\mathbf{C});
\]
on parlera aussi d'ensemble de Siegel pour son image dans $X$.
On appelle plus généralement, \emph{ensemble de Siegel associé à $W$} tout sous-ensemble de la forme
\[
\mathfrak{S}_W (g,t,\omega) := g \omega A_t \cdot \mathrm{SU}_n
\]
où $g \in \mathrm{GL}_n(\mathbf{C})$ vérifie $g \langle e_1,\ldots,e_{\mathrm{dim} W} \rangle =W$.
\end{definition}
Dans la décomposition \eqref{E:Xsatake1}, les composantes de bord $b(W)$ appartiennent toutes à une même $G$-orbite $G\cdot X_{n-j}$ où l'on note $X_{n-j}$ l'espace symétrique associé au sous-espace $\langle e_{j +1},\ldots, e_n \rangle$ plongé dans $P (\mathcal{H})$ {\it via} l'inclusion
$$\mathrm{GL}_{n-j} (\mathbf{C}) \to \mathcal{H} ; \quad A \mapsto \begin{pmatrix} 0 & 0 \\ 0 & A \end{pmatrix}$$
de sorte que les formes hermitiennes de l'image aient pour noyau $\langle e_1,\ldots, e_j \rangle$.
Le groupe $G$ a en fait exactement $n$ orbites~:
\begin{equation} \label{E:Xsatake2}
\overline{X}^S = X \sqcup G \cdot X_{n-1} \sqcup \ldots \sqcup G \cdot X_1.
\end{equation}
Finalement la topologie sur $\overline{X}^S$ peut être comprise inductivement~: un ouvert $U$ relativement compact voisinage d'un point $p$ dans $X_{n-1}$ se relève en un ouvert relativement compact de $\mathrm{GL}_{n-1} (\mathbf{C}) \subset M_1$. Le produit de cet ouvert avec un sous-ensemble relativement compact de $N_1$ définit un sous-ensemble relativement compact $\omega \subset N_1 M_1$. On construit alors un voisinage de $p$ dans $\overline{X}^S$ en prenant la réunion de $U$ avec l'ensemble de Siegel $\mathfrak{S} (t , \omega )$.
\section{Compactification de Tits} \label{S:Tits}
L'immeuble de Tits $\mathbf{T}=\mathbf{T}_n$ associé au groupe $\mathrm{SL}_n (\mathbf{C})$ est un ensemble simplicial dont les simplexes non dégénérés sont en bijection avec les sous-groupes paraboliques propres de $\mathrm{SL}_n (\mathbf{C})$, ou de manière équivalente avec les drapeaux propres
\begin{equation} \label{E:flag}
W_\bullet: 0 \subsetneq W_1 \subsetneq \cdots \subsetneq W_k \subsetneq \mathbf{C}^n, \quad k \geq 0.
\end{equation}
Le stabilisateur $Q(W_\bullet) \subset \mathrm{SL}_n (\mathbf{C} )$ d'un tel drapeau est un sous-groupe parabolique propre qui définit un simplexe de dimension $k$ dans $\mathbf{T}$. La $i$-ème face de ce simplexe correspond au drapeau déduit de $W_\bullet$ en enlevant $W_i$.
Notons $Q= N_Q A_Q M_Q$ la décomposition de Langlands (associé au choix fixé du sous-groupe compact $\mathrm{SU}_n$) d'un sous-groupe parabolique $Q$ de $\mathrm{SL}_n (\mathbf{C})$ et $\mathfrak{q}$, $\mathfrak{n}_Q$, $\mathfrak{a}_Q$ et $\mathfrak{m}_Q$ les algèbres de Lie correspondantes.
Soit $\Phi^+(Q ,A_Q )$ l'ensemble des racines pour l'action adjointe de $\mathfrak{a}_Q$ on $\mathfrak{n}_Q$. Ces racines définissent une chambre positive
$$
\mathfrak{a}_Q^+ = \left\{ H \in \mathfrak{a}_Q \; : \; \alpha(H)>0, \quad \alpha \in \Phi^+(Q,A_Q) \right\}.
$$
En notant $\langle \cdot, \cdot \rangle$ la forme de Killing sur $\mathfrak{sl}_N (\mathbf{C})$, on définit un simplexe ouvert
$$
\mathfrak{a}_Q^+(\infty) = \left\{ H \in \mathfrak{a}_Q^+ \; : \; \langle H, H \rangle = 1 \right\} \subset \mathfrak{a}_Q^+
$$
et un simplexe fermé
$$
\overline{\mathfrak{a}_Q^+}(\infty) = \left\{ H \in \mathfrak{a}_Q \; : \; \alpha(H) \geq 0, \ \langle H, H \rangle = 1, \quad \alpha \in \Phi^+(Q,A_Q) \right\}
$$
dans $\mathfrak{a}_Q$.
\medskip
\noindent
{\it Remarque.} Lorsque $Q= P_W$ est maximal, l'algèbre de Lie $\mathfrak{a}_Q$ est de dimension $1$ et $\overline{\mathfrak{a}_Q^+}(\infty)$ se réduit à un point.
\medskip
Si $Q_1$ et $Q_2$ sont deux sous-groupes paraboliques propres, alors $\overline{\mathfrak{a}_{Q_1}^+}(\infty)$ est une face de $\overline{\mathfrak{a}_{Q_2}^+}(\infty)$ si et seulement si $Q_2 \subseteq Q_1$. L'immeuble de Tits peut donc être réalisé géométriquement comme
\begin{equation} \label{eq:Tits_building_geometric_realization}
\mathbf{T}_n \sim \coprod_{Q} \overline{\mathfrak{a}_Q^+}(\infty)/\sim,
\end{equation}
où l'union porte sur tous les sous-groupes paraboliques propres $Q$ dans $\mathrm{SL}_n (\mathbf{C})$ et $\sim$ désigne la relation d'équivalence induite par l'identification de $\overline{\mathfrak{a}_{Q_1}^+}(\infty)$ à une face de $\overline{\mathfrak{a}_{Q_2}^+}(\infty)$ dès que $Q_2 \subseteq Q_1$. Au niveau ensembliste on peut décomposer
$$
\mathbf{T}_n = \coprod_Q \mathfrak{a}_Q^+(\infty)
$$
en l'union disjointe des simplexes ouverts $\mathfrak{a}_Q^+(\infty)$.
La compactification de Tits $\overline{X}^T$ de $X$ a pour bord l'immeuble de Tits $\mathbf{T}_n$~: au niveau ensembliste on a
$$
\overline{X}^T = X \cup \coprod_Q \mathfrak{a}_Q^+(\infty).
$$
On renvoie à \cite[\S I.2]{BorelJi} pour une description détaillée de la topologie sur $\overline{X}^T$ et son identification avec la compactification géodésique de $X$, on se contente ici d'énoncer les trois propriétés suivantes qui caractérisent cette topologie~:
\begin{enumerate}
\item La topologie induite sur le bord $\mathbf{T}_n$ est la topologie quotient donnée par \eqref{eq:Tits_building_geometric_realization}.
\item Soit $x \in X$. Une suite $x_j \in \overline{X}^T$, $n \geq 1$, converge vers $x$ si et seulement si $x_j \in X$ pour $j \gg 1$ et $x_j$ converge vers $x$ dans $X$ muni de sa topologie usuelle.
\item Soit $H_\infty \in \mathfrak{a}_Q^+(\infty)$ et soit $(x_j)_{j \geq 1}$ une suite dans $X$. La décomposition de Langlands de $Q$ permet d'écrire $x_j=u_j \exp(H_j)m_j$ avec $u_j \in N_Q$ et $H_j \in \mathfrak{a}_Q$ uniquement déterminés et $m_j \in M_Q$ uniquement déterminé modulo $\mathrm{SU}_n \cap M_Q$. Alors $x_j \to H_\infty$ si et seulement si $x_j$ est non bornée et
\begin{enumerate}
\item[(i)] $H_j/||H_j|| \to H_\infty$ dans $\mathfrak{a}_Q$,
\item[(ii)] $d(u_j m_j x_0,x_0)/||H_j|| \to 0$,
\end{enumerate}
où $d$ désigne la métrique symétrique sur $X$.
\end{enumerate}
Muni de cette topologie, l'espace $\overline{X}^T$ est séparé et l'action de $\mathrm{SL}_n (\mathbf{C})$ sur $X$ s'étend naturellement en une action continue sur $\overline{X}^T$.
\begin{definition}
\'Etant donné deux points $x \in X$ et $x' \in \overline{X}^T$, on note $[x,x']$ l'unique segment géodésique orienté joignant $x$ à $x'$.
\end{definition}
Si $x' \in X$, on définit plus explicitement $[x,x']$ comme étant égal à l'image de l'application
\begin{equation} \label{E:segment}
s(x,x') : [0,1] \to \overline{X}^T; \quad t \mapsto s(t;x,x'),
\end{equation}
l'unique segment géodésique orienté, paramétré à vitesse constante par l'intervalle unité, reliant $x$ à $x'$ dans $X$ avec $s(0;x,x') = x$ et $s(1; x,x' )= x'$.
Si $x'$ appartient au bord de $\overline{X}^T$, il existe un unique sous-groupe parabolique $Q$ tel que $x'$ corresponde à $H_\infty \in \mathfrak{a}_Q^+ (\infty)$. Dans les coordonnées horocycliques associées à la décomposition de Langlands de $Q$, on a $x= u \exp (H)m$. On définit alors $[x,x']$ comme étant égal à l'image de l'application
\begin{equation*}
s(x,x') : [0,1] \to \overline{X}^T; \quad t \mapsto s(t;x,x') = \left\{ \begin{array}{ll}
u \exp \left( H + \frac{t}{1-t} H_\infty \right) m, & \mbox{si } t<1 ,\\
x' , & \mbox{si } t=1.
\end{array} \right.
\end{equation*}
\section{Ensembles de Siegel généralisés}
Soit $J = \{ j_1 < \ldots < j_r \}$ une suite strictement croissante d'entiers dans $\{1 , \ldots , n-1 \}$. On associe à $J$ le drapeau
$$W_J : 0 \subsetneq W_{j_1} \subsetneq \cdots \subsetneq W_{j_r} \subsetneq \mathbf{C}^n,$$
où $W_{j_k} = \langle e_1 , \ldots , e_{j_k} \rangle$, et on note $Q_J$ le sous-groupe parabolique qui stabilise $W_J$. On peut décrire explicitement la décomposition de Langlands $Q_J$~: soient
\begin{equation*}
\begin{split}
N &= N_{J} = \left\{ \begin{pmatrix} 1_{j_1} & * & \cdots & * \\ 0 & 1_{j_2} & \cdots & * \\ 0 & 0 & \ddots & * \\ 0 & 0 & 0 & 1_{j_{r+1}} \end{pmatrix}\right\} \\
M &= M_{J} = \left\{ \left. \begin{pmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & A_{r+1} \end{pmatrix} \ \right| \ A_k \in \mathrm{GL}_{j_k}(\mathbb{C}), \ |\det(A_k)|=1 \right\} \\
A &= A_{J} = \left\{ a(t_1,\ldots,t_{r+1}) \; : \; t_k>0, \ \det a(t_1,\ldots,t_{r+1}) = 1 \right\},
\end{split}
\end{equation*}
où
$$a(t_1,\ldots,t_{r+1}) =\begin{pmatrix} t_1 1_{j_1} & 0 & \cdots & 0 \\ 0 & t_2 1_{j_2} & \cdots & 0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & t_{r+1} 1_{j_{r+1}} \end{pmatrix}.$$
On a $Q_J = N A M$. Un élément $g \in \mathrm{SL}_n (\mathbf{C})$ peut être décomposé en un produit
\[
g = u ma k, \quad u \in N, \ m \in M, \ a \in A, \ k \in \mathrm{SU}_n,
\]
où $u$ et $a$ sont uniquement déterminés par $g$, et $m$ et $k$ sont déterminés à un élément de $M \cap \mathrm{SU}_n$ près.
\'Etant donné un nombre réel strictement strictement positif $t$, on pose
\[
A_t = \{a(t_1,\ldots,t_{r+1}) \in A \; : \; t_k/t_{k+1} \geq t \text{ for all } k \}.
\]
L'ensemble de Siegel généralisé déterminé par $t>0$ et un sous-ensemble relativement compact $\omega \subset NM$ est
$$\mathfrak{S}_J (t , \omega) = \omega A_t \cdot \mathrm{SU}_n;$$
on le verra aussi bien comme un sous-ensemble de $\mathrm{SL}_n (\mathbf{C})$ que de $X$.
Soit
$$\overline{\mathfrak{S}_{J} (t , \omega)}^T \subset \overline{X}^T$$
l'adhérence de $\mathfrak{S}_{J} (t , \omega)$.
Les deux affirmations suivantes découlent du fait que la compactification de Tits coïncide avec la compactification géodésique, cf. \cite[\S 1.2 et Proposition I.12.6]{BorelJi}.
\begin{quote}
{\it Affirmation 1 :} Soit $H \in \mathfrak{a}_{Q_J}^+(\infty)$ et soit $\omega \subset N_J M_J$ un sous-ensemble ouvert relativement compact. Alors pour tout réel $t>0$, l'ensemble des $x \in X$ tels qu'il existe $y \in [x, H[$ tel que
$$[ y , H ] \subset \overline{\mathfrak{S}_{J} (t , \omega)}^T$$
est ouvert.
\end{quote}
\medskip
\begin{quote}
{\it Affirmation 2 :}
Soit $x$ un point dans $X$ et soit $H \in \mathfrak{a}_{Q_J}^+(\infty)$. Il existe un sous-ensemble ouvert relativement compact $\omega \subset N_J M_J$ tel que pour tout $t>0$, il existe $y \in [x, H [$ tel que
$$[ y , H ] \subset \overline{\mathfrak{S}_{J} (t , \omega)}^T.$$
\end{quote}
\medskip
Il découle en particulier de ces deux affirmations que si $\kappa \subset X$ un sous-ensemble compact et si $t$ est un réel strictement positif, le cône
$$C (\kappa , [W_J]) = \bigcup_{\substack{x \in \kappa \\ x' \in [W_J]}} [x,x']$$
est asymptotiquement contenu dans une réunion finie d'ensembles de Siegel généralisés, autrement dit il existe un compact $\Omega \subset X$ et des ensembles relativement compacts $\omega_{J '} \subset N_{J'} M_{J'}$, pour tout sous-ensemble $J' \subset J$, tels que
$$C(\kappa , [W_J]) \subset \Omega \cup \bigcup_{J ' \subset J} \overline{\mathfrak{S}_{J'} (t , \omega_{J'} )}^T.$$
Le dessin ci-dessous représente schématiquement la décomposition en ensemble de Siegel lorsque $\# J=2$ (de sorte que $[W_J]$ est un simplexe de dimension $1$).
\begin{center}
\includegraphics[width=0.3\textwidth]{SiegelSet.png}
\end{center}
On pose plus généralement la définition suivante.
\begin{definition}
Soit $W_\bullet$ un drapeau propre de $\mathbf{C}^n$. On appelle \emph{ensemble de Siegel généralisé} associé au cusp $W_\bullet$ tout ensemble de la forme
$$\mathfrak{S}_{W_\bullet} (g,t , \omega) = g \omega A_t \cdot \mathrm{SU}_n $$
dans $\mathrm{SL}_n (\mathbf{C})$ (ou dans $X$), où $g \in \mathrm{SL}_n (\mathbf{C})$ est tel que $g^{-1} W_\bullet$ est un drapeau standard, c'est-à-dire de la forme $W_J$ pour un certain $J$.
\end{definition}
Après translation par $g$, l'observation ci-dessus implique~:
\begin{proposition} \label{P:reduction}
Soit $W_\bullet$ un drapeau propre de $\mathbf{C}^n$ de simplexe associé $[W_\bullet] \subset \mathbf{T}$, soit $t$ un réel strictement positif et soit $\kappa \subset X$ un sous-ensemble compact. Il existe un sous-ensemble relativement compact $\Omega \subset X$, un élément $g \in \mathrm{SL}_n (\mathbf{C})$ et des ensembles relativement compacts $\omega_{J '} \subset N_{J'} M_{J'}$, pour tout sous-ensemble $J' \subset J$, tels que
$$C(\kappa , [W_\bullet]) \subset \Omega \cup \bigcup_{W_\bullet ' \subset W_\bullet} \overline{\mathfrak{S}_{W_\bullet '} (g, t , \omega_{J'} )}^T.$$
\end{proposition}
\begin{proof}
On prend pour $g$ un élément tel que $g^{-1} W_\bullet = W_J$. Alors
$$C(\kappa , [W_\bullet]) = g C(g^{-1} \kappa , [W_J])$$
et on est ramené au cas du drapeau standard $W_J$ détaillé ci-dessus.
\end{proof}
\section{Comportement à l'infini de $\eta$}
\'Etant donné un sous-espace $W \subset V=\mathbf{C}^n$ on note $W^\perp$ l'orthogonal de $W$ relativement à la métrique hermitienne standard $| \cdot |$ sur $\mathbf{C}^n$ et $p_{W} : V \to W$ la projection orthogonale.
\begin{lem} \label{L8}
Soit $Q_J \subset \mathrm{SL}_n (\mathbf{C})$ le sous-groupe parabolique propre associé au drapeau $W_J$. Soit $\omega \subset N_J M_J$ un sous-ensemble relativement compact. Il existe alors des constantes strictement positives $C$, $\alpha$ et $\beta$ telles que pour tout réel $t >0$, on ait
\begin{equation*}
\| v^*\varphi \|_\infty \leq C e^{- \alpha t^\beta | p_{W_{j_r}^\perp }(v) |^2} t^{\beta n/2} \max (1 , |v|^n )
\end{equation*}
et
\begin{equation*}
\| v^* \psi \|_\infty \leq C e^{- \alpha t^\beta | p_{W_{j_r}^\perp }(v) |^2} t^{\beta n/2} \max ( |v| , |v|^n) \quad (v \in V)
\end{equation*}
en restriction à $\mathfrak{S}_J (t , \omega) \subset X (\subset S)$.
\end{lem}
\begin{proof}
Le carré de la métrique hermitienne sur $V$ associée à un élément $g$ dans $\omega A_t \cdot \mathrm{SU}_n$ est bi-Lipschitz à
$$v \mapsto \frac{1}{t_1^2} | p_{W_{j_1}} (v) |^2 + \ldots + \frac{1}{t_r^2} | p_{W_{j_{r-1}}^\perp \cap W_{j_r}} (v) |^2 + \frac{1}{t_{r+1}^2} | p_{W_{j_r}^\perp} (v) |^2 \geq \frac{1}{t_{r+1}^2} | p_{W_{j_r}^\perp} (v) |^2.$$
Mais puisque chaque $t_j$ est supérieur à $t^{r+1-j} t_{r+1}$ et que $t_1 \cdots t_{r+1} =1$, on a $1 \geq t^{j_1 + \ldots + j_r} t_{r+1}^n$ et donc
$$\frac{1}{t_{r+1}^2} \geq t^\beta \quad \mbox{avec} \quad \beta = \frac{2}{n} (j_1 + \ldots + j_r) >0.$$
Le lemme se déduit alors des expressions explicites de $\varphi$ et $\psi$ déduites de \eqref{E:UMQ}.\footnote{Noter que $\varphi(v)$ tend vers une constante quand $v$ tend vers $0$ alors que $\psi (v)$ tend vers $0$ linéairement.}
\end{proof}
On déduit de ce lemme la proposition suivante.
\begin{proposition} \label{P32}
Soit $Q_J \subset \mathrm{SL}_n (\mathbf{C})$ le sous-groupe parabolique propre associé au drapeau $W_J$. Soit $\omega \subset N_J M_J$ un sous-ensemble relativement compact, soit $t$ un réel strictement positif et soit $\kappa \subset V - W_{j_r}$ un sous-espace compact. La restriction de $\eta$ à $\mathfrak{S}_J (t,\omega) \times \kappa$ s'étend en une forme fermée, nulle à l'infini, à l'adhérence
$$\overline{\mathfrak{S}_J (t,\omega)}^S \times \kappa, \quad \mbox{resp. } \overline{\mathfrak{S}_J (t,\omega)}^T \times \kappa,$$ dans $\overline{X}^S \times \kappa$, resp. $\overline{X}^T \times \kappa$.
\end{proposition}
\begin{proof} Pour tout $s >0$ on a
$$v^* ([s]^* \psi ) = (sv)^* \psi .$$
Il découle donc du lemme précédent qu'il existe des constantes strictement positives $C$, $\alpha$ et $\beta$ telles que, en restriction à $\mathfrak{S}_J (t ,\omega) \times \kappa$, la forme $[s]^*\psi$ soit de norme
\begin{equation} \label{E:psis}
\| [s]^* \psi \|_\infty \leq s C e^{- s^2 \alpha t^\beta } .
\end{equation}
L'intégrale
$$\int_0^{+\infty} [s]^* \psi \frac{ds}{s}$$
est donc uniformément convergente sur $\mathfrak{S}_J (t ,\omega) \times \kappa$. Il découle enfin de la proposition \ref{P:reduction} et de \eqref{E:psis} que la forme $\eta$ tend uniformément vers $0$ lorsque l'on s'approche du bord de Tits (ou de Satake) dans $\mathfrak{S}_J (t ,\omega) \times \kappa$.
\end{proof}
\section{Symboles modulaires}
Soit $k$ un entier naturel. On note $\Delta_k '$ la première subdivision barycentrique du $k$-simplexe standard. On identifie chaque sommet $v$ de $\Delta_k '$ à un sous-ensemble non vide de $\{0,\ldots,k\}$ de sorte qu'un ensemble de sommets $\{v_0,\ldots,v_r\}$ forme un $r$-simplexe de $\Delta_k '$ si et seulement si
$$v_0 \subseteq \cdots \subseteq v_r.$$
On notera $\Delta_{v_0,\ldots,v_r}$ ce simplexe.
\`A tout $(k+1)$-uplet $\mathbf{q} = (q_0 , \ldots , q_{k})$ de vecteurs non nuls dans $V$ avec $k \leq n-1$, on associe maintenant une application continue
\begin{equation} \label{E:appDelta}
\Delta(\mathbf{q}) : \Delta_{k} ' \to \overline{X}^T.
\end{equation}
Supposons dans un premier temps que $\langle q_0 , \ldots, q_{k} \rangle$ soit un sous-espace propre de $V = \mathbf{C}^n$. Pour toute chaîne $v_0 \subseteq \cdots \subseteq v_r$ définissant un $r$-simplexe de $\Delta_{k}'$, le drapeau associé
\begin{equation} \label{E:flagq}
0 \subsetneq \langle q_i \; | \; i \in v_0 \rangle \subseteq \langle q_i \; | \; i \in v_1 \rangle \subseteq \cdots \subseteq \langle q_i \; | \; i \in v_r \rangle \subsetneq \mathbf{C}^n
\end{equation}
est propre et définit un $r$-simplexe (possiblement dégénéré) de l'immeuble de Tits $\mathbf{T}$. On définit alors $\Delta (\mathbf{q})$ comme étant l'application simpliciale $\Delta_{k}' \to \mathbf{T}$ qui envoie chaque $r$-simplexe $\Delta_{v_0,\ldots,v_r}$ sur le $r$-simplexe associé à \eqref{E:flagq} dans $\mathbf{T}$ (on laisse au lecteur le soin de vérifier que cette application est bien simpliciale, autrement dit qu'elle est compatible aux applications de faces et de dégénérescence).
Supposons maintenant que $k=n-1$ et que les vecteurs $q_0 , \ldots , q_{n-1}$ soient linéairement indépendants. Soit
\begin{equation} \label{E:g}
g = (q_0 |\cdots|q_{n-1}) \in \mathrm{GL}_n ( \mathbf{C})
\end{equation}
la matrice dont les vecteurs colonnes sont précisément les vecteurs $q_0 , \ldots , q_{n-1}$. On définit alors une sous-variété $\Delta^\circ (\mathbf{q})$ dans $X$ de la manière suivante. Soit $B$ le sous-groupe parabolique minimal de $\mathrm{SL}_n (\mathbf{C})$ associé au drapeau maximal
$$0 \subsetneq \langle e_1 \rangle \subsetneq \langle e_1 , e_2 \rangle \subsetneq \cdots \subsetneq \langle e_1 , \ldots , e_{n-1} \rangle \subsetneq \mathbf{C}^n.$$
En notant simplement $A$ le groupe
$$A_B=\{ \mathrm{diag}(t_1,\ldots,t_n) \in \mathrm{SL}_n (\mathbf{C}) \; : \; t_j \in \mathbf{R}_{>0}, \ t_1 \cdots t_n = 1 \} \cong \mathbf{R}_{>0}^{n-1},$$
on pose
\begin{equation}
\Delta^\circ (\mathbf{q}) := g A K \mathbf{R}_{>0} \subset G/K \mathbf{R}_{>0}=X,
\end{equation}
muni de l'orientation induite par les coordonnées $\mathrm{diag}(t_1,\ldots,t_n) \mapsto t_i$ identifiant $\Delta^\circ(\mathbf{q})$ à $\mathbf{R}_{> 0}^{n-1}$ (ce dernier étant muni de l'orientation standard). Son adhérence dans $\overline{X}^T$ est naturellement identifiée à la première subdivision barycentrique d'un $(n-1)$-simplexe dont le bord
est la réunion dans $\mathbf{T}$ des translatés par $g$ de tous les $\overline{\mathfrak{a}_Q^+} (\infty )$ où $Q$ est un sous-groupe parabolique propre de $\mathrm{SL}_n (\mathbf{C})$ contenant $A$. On construit ainsi une application \eqref{E:appDelta} pour $k=n-1$ dont la restriction au bord
$\partial \Delta_{n-1} '$ coïncide avec les applications construites précédemment.
\medskip
Concluons ce paragraphe en expliquant comment recouvrir l'image de $\Delta (\mathbf{q})$ par des ensembles de Siegel généralisés~: à chaque sommet $v$ de $\Delta_{n-1} '$, autrement dit un sous-ensemble propre non vide de $\{0,\ldots , n-1\}$, il correspond un sous-espace
\begin{equation}
W(\mathbf{q})_v = \langle q_k \; | \; k \in v \rangle
\end{equation}
dans $\mathbf{C}^n$. Il découle de la proposition \ref{P:reduction} que l'on peut recouvrir l'image de $\Delta (\mathbf{q})$ par des ensembles de Siegel généralisés associés aux cusps $W_\bullet$ formés de sous-espaces $W(\mathbf{q})_v$~:
\begin{proposition} \label{P33}
Soit $\mathbf{q} = (q_0 , \ldots , q_{n-1})$ un $n$-uplet de vecteurs non nuls dans $V$ et soit $t$ un réel strictement positif. Il existe alors un sous-ensemble relativement compact $\Omega \subset X$ et un nombre fini d'ensembles de Siegel généralisés
$\mathfrak{S}_{W_\bullet} (g , t , \omega)$, où chaque drapeau $W_\bullet$ est formé de sous-espaces $W(\mathbf{q})_v$ et chaque $g$ est une matrice dont les vecteurs colonnes sont des $q_j$, tels que l'image de l'application $\Delta (\mathbf{q})$ dans $\overline{X}^T$ soit contenue dans la réunion finie
$$\Omega \cup \bigcup_{W_\bullet, g , \omega} \overline{\mathfrak{S}_{W_\bullet} (g , t , \omega)}.$$
\end{proposition}
\medskip
\noindent
{\it Remarque.} L'adhérence de $\Delta^\circ (\mathbf{q})$ dans $\overline{X}^S$ est égale à l'enveloppe convexe conique
$$\mathrm{P} \left\{ \sum_{j=0}^{n-1} t_j m_j \in \mathcal{H} \; : \; \forall j \in \{0, \ldots , n-1 \}, \ t_j >0 \right\} \subset \overline{X}^S$$
des formes hermitiennes semi-définies positives $m_j = q_j^* (\overline{q_j^*} )^\top$ où\footnote{On prendra garde au fait que Ash et Rudolph \cite[p. 5]{AshRudolph} commettent une légère erreur en identifiant $\Delta^\circ (\mathbf{q})$ avec l'enveloppe convexe conique des formes
hermitiennes semi-définies positives $m_j = q_j \overline{q_j}^\top$.}
$$g^{-*} = (q_0^* | \cdots | q_{N-1}^* ), \quad
\mbox{de sorte que } (\overline{q_i^*} )^\top q_j = \delta_{ij} \quad \mbox{et} \quad q_j^* = g^{-*} e_{j+1} .$$
\medskip
\begin{proof} Pour tout $j \in \{0, \ldots , n-1 \}$, on a $ge_{j+1} = q_{j}$. On peut donc se ramener au cas où $\mathbf{q} = (e_1 , \ldots , e_n)$ et $$\Delta (\mathbf{q})^\circ = A K \mathbf{R}_{>0} \subset G/K \mathbf{R}_{>0}=X$$
ce qui prouve immédiatement la remarque. Le cas général s'en déduit en translatant par $g$. Noter qu'alors la forme $m_j$ est égale à $q_j^* (q_j^* )^\top$ qui a bien pour noyau
$$W(\mathbf{q})^{(j)} = W(\mathbf{q})_{\{0 , \ldots , \widehat{j} , \ldots , n-1 \}}. $$
\end{proof}
\section{\'Evaluation de $\eta$ sur les symboles modulaires}
Soit $\mathbf{q} = (q_0 , \ldots , q_{k})$ un $(k+1)$-uplet de vecteurs de non nuls dans $V$ avec $k \leq n-1$. Les propositions \ref{P32} et \ref{P33} impliquent que la forme différentielle fermée
\begin{equation}
\eta (\mathbf{q} ) = (\Delta (\mathbf{q}) \times \mathrm{id} )^* \eta \in A^{2n-1} \left( \Delta_{k}' \times \left( V - \bigcup_{|v| <n} W(\mathbf{q})_v \right) \right),
\end{equation}
où $\Delta_{k}'$ est identifié à
$$\mathbf{R}^{k} \cong \{ (t_0 , \ldots , t_{k} ) \in \mathbf{R}_+^{k+1} \; : \; t_0 + \ldots + t_{k} =1 \}$$
{\it via} les coordonnées barycentriques, est bien définie.
Il découle en outre des propositions \ref{P32} et \ref{P33} que l'intégrale partielle
$$\int_{\Delta_{k}'} \eta (\mathbf{q})$$
converge et définit une forme de degré $2n-k-1$ sur $V - \bigcup_{|v|<n} W(\mathbf{q})_v$.
\begin{proposition} \label{P34}
1. Si $\langle q_0 , \ldots , q_{k} \rangle$ est un sous-espace propre de $V$, alors la forme $\int_{\Delta_{k}'} \eta (\mathbf{q})$ est identiquement nulle.
2. Supposons $k=n-1$ et que les vecteurs $q_0 , \ldots , q_{n-1}$ soient linéairement indépendants. Alors la $n$-forme $\int_{\Delta_{n-1}'} \eta (\mathbf{q})$ est égale à
$$\frac{1}{(4i\pi)^n} \left( \frac{d \ell_0}{\ell_0} - \overline{\frac{d\ell_0}{\ell_0}} \right) \wedge \ldots \wedge \left( \frac{d \ell_{n-1}}{\ell_{n-1}} - \overline{\frac{d\ell_{n-1}}{\ell_{n-1}}} \right) \in A^n \left( V - \bigcup_{j} W(\mathbf{q})^{(j)} \right),$$
où $\ell_j$ est la forme linéaire sur $\mathbf{C}^n$, de noyau $W(\mathbf{q})^{(j)}$, qui à $z$ associe $z^\top q_j^*$.
\end{proposition}
\begin{proof} 1. Dans ce cas l'image de $\Delta (\mathbf{q})$ est contenue dans le bord de $\overline{X}^T$ et il résulte de la proposition \ref{P32} que $\int_{\Delta_{k}'} \eta (\mathbf{q})$ est nulle sur tout ouvert relativement compact de $V - \bigcup_{|v| <n} W(\mathbf{q})_v$.
2. Supposons donc $k=n-1$ et que les vecteurs $q_0 , \ldots , q_{n-1}$ soient linéairement indépendants.
En notant toujours $g$ l'élément \eqref{E:g}, la $G$-invariance de $\eta$ implique que
$$g^* \left( \int_{\Delta_{n-1}'} \eta (\mathbf{q}) \right) = \int_{\Delta_{n-1}'} \eta (e_1 , \ldots , e_n ).$$
Comme par ailleurs $g^*\ell_j$ est la forme linéaire $e_{j+1}^*$ de noyau
$$g^{-1} W(\mathbf{q})^{(j)} = \langle e_1 , \ldots , \widehat{e_{j+1}} , \ldots , e_n \rangle,$$
on est réduit à vérifier la proposition dans le cas où $\mathbf{q} = (e_1 , \ldots , e_n )$.
Il nous reste donc à calculer l'intégrale
$$\int_{AK\mathbf{R}_{>0}} \eta, \quad \mbox{où } A=\{ \mathrm{diag}(t_1,\ldots,t_n) \in \mathrm{SL}_n (\mathbf{C}) \; : \; t_j \in \mathbf{R}_{>0}, \ t_1 \cdots t_n = 1 \}.$$ D'après la remarque à la fin du paragraphe \ref{S:42} on a
\begin{equation} \label{E:intsymbMod}
\int_{AK\mathbf{R}_{>0}} \eta = \int_{\{ \mathrm{diag}(t_1,\ldots,t_n ) \; : \; t_j \in \mathbf{R}_{>0} \} K} \varphi.
\end{equation}
Or, en restriction à l'ensemble des matrices symétriques diagonales réelles, le fibré en $\mathbf{C}^n$ se scinde {\it métriquement} en une somme directe de $n$ fibrés en droites, correspondant aux coordonnées $(z_j )_{j=1 , \ldots , n}$ de $z$ et la forme $\varphi$ se décompose en le produit de $2$-formes associées à ces fibrés en droites~:
$$
\varphi^{(j)} = \frac{i}{2\pi} e^{- t_j^2 |z_j |^2 } \left( t_j^2 dz_j \wedge d \overline{z}_j
- t_j^2 ( z_j d\overline{z}_j - \overline{z}_j dz_j ) \wedge \frac{dt_j}{t_j} \right),
$$
d'après (\ref{E:phiN1}).
Finalement, on obtient que l'intégrale \eqref{E:intsymbMod} est égale à
\begin{multline*}
\frac{(-i)^n}{(2\pi)^n} \left( \prod_{j=1}^n \int_{\mathbf{R}_{>0}} t_j^2 |z_j|^2 e^{-t_j ^2 |z_j |^2} \frac{dt_j }{t_j} \right) \wedge_{j=1}^n \left( \frac{dz_j}{z_j} - \frac{d\overline{z}_j}{\overline{z}_j} \right) \\
= \frac{1}{(4i\pi)^n} \wedge_{j=1}^n \left( \frac{dz_j}{z_j} - \frac{d\overline{z}_j}{\overline{z}_j} \right) ,
\end{multline*}
comme attendu.
\end{proof}
\chapter{Cocycles de $\mathrm{GL}_n (\mathbf{C})$ explicites} \label{S:6}
\resettheoremcounters
Dans ce chapitre on note à nouveau $G=\mathrm{GL}_n (\mathbf{C})^{\delta}$. Dans un premier temps on explique comment associer à la forme de Mathai--Quillen $\eta \in A^{2n-1} (X \times (\mathbf{C}^n - \{ 0 \} ))^G$ un représentant explicite du relevé canonique $\Phi \in H_G^{2n-1} (\mathbf{C}^n -\{ 0 \})$ fourni par la proposition \ref{P4}. On utilise ensuite ce représentant explicite pour démontrer le théorème \ref{T:Sa}.
\section{Forme simpliciale associée à $\eta$} \label{S:61}
Soit $x_0 \in X$ le point base associé à la classe de l'identité dans $\mathrm{SL}_n (\mathbf{C})$. L'application $\exp : T_{x_0} X \to X$ étant un difféomorphisme, il existe une rétraction
\begin{equation} \label{E:R}
R : [0, 1] \times X \to X
\end{equation}
de $X$ sur $\{x_0 \} $. Elle est donnée par la formule
$$R_s (x ) = \exp (s \exp^{-1} (x) ) \quad (x \in X, \ s \in [0,1]).$$
Suivant \cite{Dupont} on déduit de $R$ une suite d'applications
\begin{equation} \label{E:mapr}
\rho_k : \Delta_k \times E_kG \times \mathbf{C}^n \longrightarrow X \times \mathbf{C}^n
\end{equation}
définies de la manière suivante~: pour $t = (t_0 , \ldots , t_k ) \in \Delta_k$ on pose $s_j = t_j + t_{j+1} + \ldots + t_k$ ($j=1, \ldots , k$). \'Etant donné un $(k+1)$-uplet
$$\mathbf{g} = (g_0 , \ldots , g_k ) \in E_kG$$
et un vecteur $z \in \mathbf{C}^n$, on a alors
\begin{multline} \label{E:mapr2}
\rho_k ( t , \mathbf{g} , z ) = (g_0^{-1} \cdot R_{s_1} ( g_0 g_1^{-1} \cdot R_{s_2 / s_1} ( g_1 g_2^{-1} \cdot \\ \ldots g_{j-1} g_j^{-1} \cdot R_{s_{j+1} / s_j} ( g_j g_{j+1}^{-1} \cdot \cdots R_{s_k / s_{k-1} } ( g_{k-1} g_k^{-1} \cdot x_0) \ldots ))) , z).
\end{multline}
La suite $(\rho_k )$ est constituée d'applications $G$-équivariantes qui font commuter le diagramme\footnote{Ici $\epsilon^k : \Delta_{k-1} \to \Delta_k$ désigne l'application d'inclusion de la $k$-ième face.}
\begin{equation} \label{diag:simpl}
\xymatrix{
\Delta_{k-1} \times E_kG \times \mathbf{C}^n \ar[d]^{\ \mathrm{id} \times \partial_k \times \mathrm{id}} \ar[r]^{\epsilon^k \times \mathrm{id}} & \Delta_k \times E_kG \times \mathbf{C}^n \ar[d]^{\rho_{k}} \\
\Delta_{k-1} \times E_{k-1} G \times \mathbf{C}^n \ar[r]^{\quad \quad \rho_{k-1}} & X\times \mathbf{C}^n,
}
\end{equation}
de sorte que $\rho$ induit une application $G$-équivariante
$$\rho^* : A^\bullet (X \times \mathbf{C}^n ) \to \mathrm{A}^\bullet (EG \times \mathbf{C}^n),$$
où l'espace à droite est celui des formes différentielles simpliciales sur la variété simpliciale $EG \times \mathbf{C}^n$, cf. annexe \ref{A:A}.
La proposition suivante précise la proposition \ref{P4}.
\begin{proposition}
La forme simpliciale
$$\rho^* \eta \in \mathrm{A}^{2n-1} (EG \times (\mathbf{C}^n - \{ 0 \}))^G$$
est fermée et représente une classe dans $H_G^{2n-1} (\mathbf{C}^n - \{ 0 \} )$ qui relève la classe fondamentale dans $H^{2n-1} (\mathbf{C}^n - \{ 0 \} )$.
\end{proposition}
\begin{proof} La forme $\rho^* \eta$ est fermée et $G$-invariante puisque $\eta$ l'est (proposition \ref{P:eta}) et, comme $\eta$ représente la classe fondamentale de $\mathbf{C}^n - \{ 0 \}$, la classe de cohomologie équivariante $[\rho^* \eta ]$ s'envoie sur la classe fondamentale dans $H^{2n-1} (\mathbf{C}^n - \{ 0 \} )$.
\end{proof}
\section{Cocycle associé} \label{S:62}
Considérons maintenant les ouverts
\begin{equation} \label{hypCU'}
\begin{split}
U(g_0 , \ldots , g_k ) & = \left\{ z \in \mathbf{C}^n \; : \; \forall j \in \{ 0 , \ldots , k \}, \ e_1^* (g_j z ) \neq 0 \right\} \\
& = \mathbf{C}^n - \cup_j H_j,
\end{split}
\end{equation}
où $H_j$ est le translaté par $g_j^{-1}$ de l'hyperplan $z_1=0$ dans $\mathbf{C}^n$. Ces sous-variétés sont toutes affines de dimension $n$ et n'ont donc pas de cohomologie en degré $>n$.
La forme simpliciale $\rho^* \eta$ représente une classe de cohomologie équivariante dans $H^{2n-1}_G (\mathbf{C}^n - \{0 \})$. On peut lui appliquer la méthode décrite dans l'annexe \ref{A:A} et lui associer un $(n-1)$-cocycle de $G$ à valeurs dans
$$\lim_{\substack{\rightarrow \\ H_j}} H^n (\mathbf{C}^n - \cup_j H_j ).$$
On calculera explicitement ce cocycle au paragraphe \S \ref{S:demTSa}.
Avant cela remarquons qu'on aurait pu également considérer les ouverts
\begin{equation} \label{hypC}
\begin{split}
U^* (g_0 , \ldots , g_k) & = \left\{ z \in \mathbf{C}^n \; \left| \; \begin{array}{l} \forall J \subset \{ 0 , \ldots , k \}, \\ |J| <\min (k,n) \Rightarrow z \notin \langle q_j \; : \; j \in J \rangle \end{array} \right. \right\} \\
& = \mathbf{C}^n - \bigcup_j W(\mathbf{q})^{(j)},
\end{split}
\end{equation}
où $\mathbf{q} = (q_0 , \ldots , q_k)=( g_0^{-1} e_1 , \ldots , g_{k}^{-1} e_1 )$ et $W(\mathbf{q})^{(j)} = \langle q_i \; : \; i \neq j \rangle$.
En général la sous-variété \eqref{hypC} n'est pas affine. C'est toutefois le cas lorsque les vecteurs $g_0^{-1} e_1, \ldots , g_{k}^{-1} e_1$ engendrent $\mathbf{C}^n$. L'argument de l'annexe \ref{A:A} implique donc encore que la forme simpliciale $\rho^* \eta$ détermine un $(n-1)$-cocycle de $G$ à valeurs dans
$$\lim_{\substack{\rightarrow \\ H_j}} H^n (\mathbf{C}^n - \cup_j H_j ).$$
On détaille particulièrement le calcul explicite de ce cocycle dans les paragraphes suivants; ce sont en effet ces calculs que nous généraliserons dans les cas multiplicatifs et elliptiques.
\section{Section simpliciale et homotopie} \label{S:63}
Pour tout entier $k \in [0 , n-1]$, on définit, par récurrence, une subdivision simpliciale $\left[ \Delta_k \times [0,1] \right]'$ de $\Delta_k \times [0,1]$ en joignant tous les simplexes dans $\Delta_k \times \{0 \} \cup \partial \Delta_k \times [0,1]$ au barycentre de $\Delta_k \times \{1\}$, comme sur la figure ci-dessous.
\begin{center}
\includegraphics[width=0.5\textwidth]{Figure1.png}
\end{center}
L'ensemble des $(k+1)$-simplexes de $\left[ \Delta_k \times [0,1] \right]'$ est constitué des joints
\begin{equation} \label{E:cone1}
\Delta_w \star \Delta_{v_0, \ldots , v_{k - |w|}} \quad (w \subset v_0)
\end{equation}
où $\Delta_w \subset \Delta_k = \Delta_k \times \{ 0 \}$ est le simplexe correspondant à un sous-ensemble non-vide $w \subset \{0, \ldots , k \}$, de cardinal $|w|$, et $\Delta_{v_0, \ldots , v_{k-|w|}}$ est le $(k-|w|)$-simplexe de la subdivision barycentrique $\Delta_k' = \Delta_k ' \times \{ 1 \}$ associé à une suite croissante $v_0 \subset \cdots \subset v_{k-|w|}$ de sous-ensembles de $\{0 , \ldots , k \}$ contenant tous $w$.
On définit maintenant une suite d'applications
\begin{equation}
\varrho_k : \left[ \Delta _k \times [0,1] \right]' \times E_k G \times \mathbf{C}^n \to \overline{X}^T \times \mathbf{C}^n \quad (k \in \{ 0 , \ldots , n-1 \} )
\end{equation}
de manière à ce que la restriction de $\varrho_k$ à
$$\Delta_k \times E_k G \times \mathbf{C}^n = (\Delta _k \times \{0 \} ) \times E_k G \times \mathbf{C}^n$$
coïncide avec $\rho_k$ et qu'en restriction à
$$\Delta_k ' \times E_k G \times \mathbf{C}^n \subset (\Delta _k ' \times \{1 \} ) \times E_k G \times \mathbf{C}^n$$
on ait
$$\varrho_k ( - , \mathbf{g} , -) = \Delta (\mathbf{q} ) \times \mathrm{Id}_{\mathbf{C}^n },$$
où si $\mathbf{g}=(g_0 , \ldots , g_{k} ) \in E_{k} G$ on note toujours $\mathbf{q} = ( g_0^{-1} e_1 , \ldots , g_k^{-1} e_1 ).$
En procédant par récurrence sur $k$ on est ramené à définir $\varrho_k$ sur chaque simplexe \eqref{E:cone1}. Considérons donc un sous-ensemble non-vide $w \subset \{0, \ldots , k \}$ et une suite croissante $v_0 \subset \cdots \subset v_{k-|w|}$ de sous-ensembles de $\{0 , \ldots , k \}$ contenant tous $w$. L'application $\varrho_k$ étant définie sur $\Delta_w \subset \Delta _k \times \{0 \}$ et sur $\Delta_{v_0, \ldots , v_{k-|w|}} \subset \Delta _k ' \times \{1 \} $, l'expression
$$\varrho_k ( t , \mathbf{g} , z ) = s \left( t ; \varrho_k ( 0 , \mathbf{g} , z ) , \varrho_k ( 1 , \mathbf{g} , z ) \right)$$
définit une application
$$|\Delta_w| \times |\Delta_{v_0, \ldots , v_{k-|w|}}| \times [0,1] \to \overline{X}^T \times \mathbf{C}^n$$
qui se factorise en une application
$$| \Delta_w \star \Delta_{v_0, \ldots , v_{k - |w|}} | \to \overline{X}^T \times \mathbf{C}^n.$$
On définit ainsi $\varrho_k$ sur les simplexes \eqref{E:cone1}; on laisse au lecteur le soin de vérifier les relations de compatibilité.
La proposition \ref{P33} implique que, pour tout $\mathbf{g} \in E_k G$, on peut recouvrir l'image
$$\varrho_k \left( \left[ \Delta _k \times [0,1] \right]' \times \{ \mathbf{g} \} \times \mathbf{C}^n \right) \subset \overline{X}^T \times \mathbf{C}^n$$
par un nombre fini de produits $\mathfrak{S} \times \mathbf{C}^n$ d'ensembles de Siegel par $\mathbf{C}^n$. Il découle donc de la proposition \ref{P32} que la forme différentielle fermée
$$\varrho_k^* \eta (\mathbf{g}) \in \mathrm{A}^{2n-1} \left( \left[ \Delta _k \times [0,1] \right]' \times U^* (g_0, \ldots , g_k) \right)$$
est bien définie.
\begin{definition}
Pour tout entier $k \in [0, n-1]$ et pour tout $(k+1)$-uplet $(g_0 , \ldots , g_{k}) \in E_k G$, on pose
$$H_k (g_0 , \ldots , g_k ) = \int_{\left[ \Delta _k \times [0,1] \right]'} \varrho_k^* \eta (g_0 , \ldots , g_k ) \in A^{2n-2-k} \left(\mathbf{C}^n - \bigcup_{j=0}^k W(\mathbf{q})^{(j)} \right),$$
où $\mathbf{q} = (q_0 , \ldots , q_k)$ avec $q_j = g_j^{-1} e_1$.
\end{definition}
\section{Calcul du cocycle}
Il résulte de la définition de l'application bord $\delta$ donnée dans l'annexe \ref{A:A} que, pour tout entier $k \in [0, n-1]$, on a
$$\delta H_{k-1} (g_0 , \ldots , g_{k}) = \sum_{j=0}^k H_{k-1} (g_0 , \ldots , \widehat{g}_j , \ldots , g_k ),$$
vue comme forme différentielle sur $\mathbf{C}^n - \bigcup_{j=0}^k W(\mathbf{q})^{(j)}$.
\begin{theorem} \label{T37}
Pour tout entier $k \in [0 , n-1]$ et pour tout $\mathbf{g} = (g_0 , \ldots , g_k)$ dans $E_k G$, l'intégrale $\int_{\Delta_k} \rho^* \eta (\mathbf{g})$ est égale à
$$\delta H_{k-1} (g_0 , \ldots , g_{k}) \pm d H_k (g_0 , \ldots , g_k), \quad \mbox{si} \quad k < n-1,$$
et
$$\int_{\Delta_{n-1}'} \eta (\mathbf{q}) + \delta H_{n-2} (g_0 , \ldots , g_{n-1}) \pm dH_{n-1} (g_0 , \ldots , g_{n-1}), \quad \mbox{si} \quad k=n-1,$$
dans $A^{2n-2-k} \left(\mathbf{C}^n - \bigcup_{j=0}^k W(\mathbf{q})^{(j)} \right),$
où $\mathbf{q} = (q_0 , \ldots , q_k)$ avec $q_j = g_j^{-1} e_1$.
\end{theorem}
\begin{proof} Puisque $\eta$ est fermée on a~:
$$(d_{\left[ \Delta _k \times [0,1] \right]'} \pm d ) \varrho_k^* \eta = 0$$
et donc
$$\int_{\left[ \Delta _k \times [0,1] \right]'} d_{\left[ \Delta _k \times [0,1] \right]'} \varrho_k^* \eta (g_0 , \ldots , g_k ) \pm d H_k (g_0 , \ldots , g_k ) =0.$$
Maintenant, d'après le théorème de Stokes on a
\begin{multline*}
\int_{\left[ \Delta _k \times [0,1] \right]'} d_{\left[ \Delta _k \times [0,1] \right]'} \varrho_k^* \eta (g_0 , \ldots , g_k ) = \int_{\Delta_k \times \{ 0 \}} \varrho_k^* \eta (g_0 , \ldots , g_k ) \\ + \int_{\left[(\partial \Delta_k) \times [0,1]\right]'} \varrho_k^* \eta (g_0 , \ldots , g_k ) - \int_{\Delta_k' \times \{ 1 \}} \varrho_k^* \eta (g_0 , \ldots , g_k ).
\end{multline*}
La dernière intégrale est égale à $\int_{\Delta_k '} \eta (\mathbf{q})$ et est donc nulle si $k < n-1$ d'après la proposition \ref{P34}.
Finalement, par définition de $\varrho_k$ on a
$$\int_{\Delta_k \times \{ 0 \}} \varrho_k^* \eta (g_0 , \ldots , g_k ) = \int_{\Delta_k} \rho_k^* \eta (g_0 , \ldots , g_k) $$
et
$$\int_{\left[(\partial \Delta_k) \times [0,1]\right]'} \varrho_k^* \eta (g_0 , \ldots , g_k ) = - \delta H_{k-1} (g_0 , \ldots , g_k ).$$
\end{proof}
Le corollaire suivant découle du théorème \ref{T37} et de la proposition \ref{P34}.
\begin{cor} \label{C38}
La forme différentielle simpliciale fermée $\rho^* \eta $ définit un $(n-1)$-cocycle de $G$ à valeurs dans
$$\lim_{\substack{\rightarrow \\ H_j}} H^n (\mathbf{C}^n - \cup_j H_j )$$
qui est cohomologue au cocycle
$$(g_0 , \ldots , g_{n-1} ) \mapsto \left[ \frac{1}{(4i\pi)^n} \left( \frac{d \ell_0}{\ell_0} - \overline{\frac{d\ell_0}{\ell_0}} \right) \wedge \ldots \wedge \left( \frac{d \ell_{n-1}}{\ell_{n-1}} - \overline{\frac{d\ell_{n-1}}{\ell_{n-1}}} \right) \right],$$
où $\ell_j$ est une forme linéaire sur $\mathbf{C}^n$, de noyau $W(\mathbf{q})^{(j)}$, qui à $z$ associe $z^\top q_j^*$.
\end{cor}
\section[Démonstration du théorème 2.2]{Démonstration du théorème \ref{T:Sa}} \label{S:demTSa}
Les identités immédiates suivantes entre formes différentielles sur $\mathbf{C}^*$
$$\frac{1}{2i\pi} \frac{dz}{z} = \frac{d\theta}{2\pi} + d \left( \frac{1}{2i\pi} \log r \right) \quad \mbox{et} \quad
\frac{1}{4i\pi} \left( \frac{dz}{z} - \overline{\frac{dz}{z}} \right) = \frac{d\theta}{2\pi},$$
avec $z = re^{i\theta}$, impliquent que si $\ell$ est une forme linéaire sur $\mathbf{C}^n$ les formes différentielles
$$\frac{1}{2i\pi} \frac{d\ell}{\ell} \quad \mbox{et} \quad \frac{1}{4i\pi} \left( \frac{d \ell}{\ell} - \overline{\frac{d\ell}{\ell}} \right)$$
sur $\mathbf{C}^n- \mathrm{ker} (\ell )$ sont cohomologues. Le théorème de Brieskorn \cite{Brieskorn} évoqué au \S \ref{S21} implique donc que l'application
$$\Omega_{\rm aff} \to \lim_{\substack{\rightarrow \\ H_j}} H^\bullet (\mathbf{C}^n - \cup_j H_j ); \quad \frac{1}{2i\pi} \frac{d\ell}{\ell} \mapsto \left[ \frac{1}{4i\pi} \left( \frac{d \ell}{\ell} - \overline{\frac{d\ell}{\ell}} \right) \right]$$
est un isomorphisme d'algèbre. Comme par ailleurs cet isomorphisme est $G$-équivariant, il résulte du corollaire \ref{C38}
que la classe de $[\rho^* \eta]$ donne lieu à un $(n-1)$-cocycle de $G$ dans $\Omega^n_{\rm aff}$ cohomologue à
$$\mathbf{S}_{\rm aff}^* : G^n \to \Omega^n_{\rm aff} ; \quad (g_0 , \ldots , g_{n-1}) \mapsto \frac{1}{(2i\pi )^n} \frac{d\ell_0}{\ell_0} \wedge \ldots \wedge \frac{d\ell_{n-1}}{\ell_{n-1}},$$
où $\ell_j$ est une forme linéaire de noyau $\langle g_0^{-1} e_1 , \ldots , \widehat{g_j^{-1} e_1} , \ldots , g_{n-1}^{-1} e_1 \rangle$.
Il nous reste à montrer que ce cocycle est cohomologue à $\mathbf{S}_{\rm aff}$ et que sa classe de cohomologie est non nulle.
Pour ce faire on applique à nouveau l'argument de l'annexe \ref{A:A} à la forme simpliciale fermée $\rho^*\eta$ mais en utilisant cette fois les ouverts \eqref{hypCU'}. Rappelons que dans ce cas
\begin{equation*}
U (g_0 , \ldots , g_k) = \mathbf{C}^n - \cup_j H_j
\end{equation*}
où $H_j$ est le translaté par $g_j^{-1}$ de l'hyperplan $z_1=0$ dans $\mathbf{C}^n$.
On procède alors de la même manière que pour obtenir $\mathbf{S}_{\rm aff}^*$ mais en remplaçant la compactification de Tits par celle de Satake $\overline{X}^S$. La convexité de l'ouvert des formes hermitiennes non-nulles semi-définies positives dans $\mathcal{H}$ permet de rétracter $\overline{X}^S$ sur le point (à l'infini) associé à la forme hermitienne $|e_1^* (\cdot )|^2$ de noyau $\langle e_2 , \ldots , e_n \rangle$; notons
$$\overline{R} : [0,1] \times \overline{X}^S \times \mathbf{C}^n \to \overline{X}^S \times \mathbf{C}^n$$
l'application correspondante.
Comme pour $R$, il correspond à $\overline{R}$ une suite d'applications $G$-équivariantes
$$\overline{\rho}_k : \Delta_k \times E_k G \times \mathbf{C}^n \to \overline{X}^S \times \mathbf{C}^n$$
qui font commuter le diagramme \eqref{diag:simpl} de sorte que $\overline{\rho} = (\overline{\rho}_k)$ induit une application $G$-équivariante
$$(\overline{\rho})^* : A^\bullet (\overline{X}^S \times \mathbf{C}^n ) \to \mathrm{A}^\bullet (EG \times \mathbf{C}^n ).$$
Mieux, il découle encore de la proposition \ref{P32} que les formes différentielles fermées
$$(\overline{\rho}_{n-1})^* \eta (\mathbf{g}) \in A^{2n-1} (\Delta_{n-1} \times U (g_0 , \ldots , g_{n-1}) )$$
sont bien définies, se recollent en une forme simpliciale fermée dans
$$\varinjlim A^{n}( \mathbf{C}^n - \cup_j H_j )$$
et définissent un $(n-1)$-cocycle
$$(g_0 , \ldots , g_{n-1} ) \mapsto \int_{\Delta_{n-1}} (\overline{\rho}_{n-1})^* \eta (g_0 , \ldots , g_{n-1} ).$$
Ce dernier étant obtenu en appliquant l'argument de l'annexe \ref{A:A}, il est cohomologue au cocycle du corollaire \ref{C38}.
Or, d'après la remarque suivant la proposition \ref{P33}, on a
$$\overline{\rho}_k (\Delta_k \times \{ \mathbf{g} \} \times \{ z \} ) = \overline{\Delta^\circ (\mathbf{\sigma}^*)} \times \{ z \}$$
où cette fois $\mathbf{\sigma} = (\sigma_0 , \ldots , \sigma_{n-1})$, avec $\sigma_j = g_j^\perp e_1 $, de sorte que
$$\mathrm{ker} \ e_1^* (g_j \cdot) = \langle \sigma_0^* , \ldots , \widehat{\sigma_j^*} , \ldots , \sigma_{n-1}^* \rangle.$$
On a donc
$$\int_{\Delta_{n-1}} (\overline{\rho})^* \eta (g_0 , \ldots , g_{n-1} ) = \int_{\Delta_{n-1} '} \eta (\mathbf{\sigma}^*)$$
et la proposition \ref{P34} implique finalement que l'intégrale $\int_{\Delta_{n-1}} (\overline{\rho})^* \eta (\mathbf{g})$ est nulle si les $n$ formes linéaires $\ell_j = e_1^* (g_j \cdot)$ ($j \in \{0, \ldots , n-1\}$) sont linéairement dépendantes et qu'elle est égale à
$$\frac{1}{(4i\pi)^n} \left( \frac{d \ell_0}{\ell_0} - \overline{\frac{d\ell_0}{\ell_0}} \right) \wedge \ldots \wedge \left( \frac{d \ell_{n-1}}{\ell_{n-1}} - \overline{\frac{d\ell_{n-1}}{\ell_{n-1}}} \right)$$
sinon.
En faisant à nouveau appel au théorème de Brieskorn on retrouve que $\mathbf{S}_{\rm aff}$ définit un cocycle mais surtout que celui-ci représente la même classe que $\mathbf{S}_{\rm aff}^*$. Le fait que la classe de cohomologie correspondante $S_{\rm aff} \in H^{n-1} (G , \Omega_{\rm aff}^n)$ soit non nulle résulte finalement de \cite[Theorem 3]{Sczech93} où $\mathbf{S}_{\rm aff}$ est évalué sur un $(n-1)$-cycle d'éléments unipotents. \qed
\medskip
\noindent
{\it Remarque.} Soit $W$ un sous-espace propre et non nul de $\mathbf{C}^n$, autrement dit un sommet de l'immeuble de Tits $\mathbf{T}$. Soit $X(W)$ le sous-espace de $\overline{X}^S$ constitué des matrices hermitiennes semi-définies positives dont le noyau contient $W$; il est homéomorphe à $\overline{X}_{n-\dim W}^S$ et donc contractile. Les sous-ensembles $X(\ell)$, avec $\ell \subset \mathbf{C}^n$ droites, forment un recouvrement acyclique du bord $\partial \overline{X}^S$ de la compactification de Satake. De plus, une intersection $X(\ell_1) \cap \ldots \cap X(\ell_k)$ est non-vide si et seulement si les droites $\ell_1, \ldots , \ell_k$ engendrent un sous-espace propre $W$ de $\mathbf{C}^n$, auquel cas
$$X(\ell_1) \cap \ldots \cap X(\ell_k) = X(W).$$
Comme ensemble simplicial, la première subdivision barycentrique du nerf du recouvrement acyclique de $\partial \overline{X}^S$ par les $X(\ell )$ est donc égal à $\mathbf{T}$. C'est la source de la dualité qui relie les deux cocycles $\mathbf{S}_{\rm aff}$ et $\mathbf{S}_{\rm aff}^*$.
On a en effet des isomorphismes
$$H_{n-1} ( \overline{X}^S , \partial \overline{X}^S ) \cong \widetilde{H}_{n-2} (\partial \overline{X}^S) \cong \underbrace{\widetilde{H}_{n-2} (\mathbf{T})}_{= \mathrm{St} (\mathbf{C}^n )} \cong H_{n-1} ( \overline{X}^T , \partial \overline{X}^T ).$$
Explicitement, l'isomorphisme
$$\mathrm{St} (\mathbf{C}^n ) \stackrel{\sim}{\longrightarrow} H_{n-1} ( \overline{X}^T , \partial \overline{X}^T )$$
associe à l'élément $[q_0 , \ldots , q_{n-1}] \in \mathrm{St} (\mathbf{C}^n )$ la classe de $\Delta (\mathbf{q})$ dans
$$H_{n-1} ( \overline{X}^T , \partial \overline{X}^T );$$
il est $G$-équivariant ce qui se traduit par le fait que $\mathbf{S}_{\rm aff}^*$ s'étende en un isomorphisme $G$-équivariant de $\mathrm{St} (\mathbf{C}^n )$ vers $\Omega^n_{\rm aff}$. L'isomorphisme
$$\mathrm{St} (\mathbf{C}^n ) \stackrel{\sim}{\longrightarrow} H_{n-1} ( \overline{X}^S , \partial \overline{X}^S )$$
associe quant à lui à un élément $[q_0 , \ldots , q_{n-1}] \in \mathrm{St} (\mathbf{C}^n )$ la classe de $[q_0^* , \ldots , q_{n-1}^*]$ dans
$H_{n-1} ( \overline{X}^S , \partial \overline{X}^S )$; la représentation de $G$ dans $H_{n-1} ( \overline{X}^S , \partial \overline{X}^S )$ est donc naturellement identifiée à la représentation $\mathrm{St} ((\mathbf{C}^n )^\vee)$.
\medskip
Dans la suite, on globalise la construction ci-dessus en remplaçant la forme $\eta$ par la valeur en $0$ d'une série d'Eisenstein construite à partir de la fonction test $\psi$ à l'infini. On ne considère que la compactification de Tits, plus naturelle pour notre propos comme le montre la remarque précédente.
\chapter{Séries d'Eisenstein associées à $\psi$} \label{C:7}
\resettheoremcounters
Dans ce chapitre on restreint les formes $\varphi$ et $\psi$ à l'espace symétrique associé à $\mathrm{SL}_n (\mathbf{R})$ que nous noterons $X$ en espérant ne pas créer de confusion. On note donc dorénavant
$$S = \mathrm{GL}_n (\mathbf{R}) / \mathrm{SO}_n, \ S^+ = \mathrm{GL}_n (\mathbf{R})^+ / \mathrm{SO}_n \ \mbox{et} \ X = S^+/ \mathbf{R}_{>0} = \mathrm{SL}_n (\mathbf{R}) /\mathrm{SO}_n.$$
On globalise les constructions précédentes en formant une série d'Eisenstein à partir de la fonction $\psi$. La préimage de cette série d'Eisenstein par une section de torsion est étudiée dans un registre plus général par Bismut et Cheeger \cite{BismutCheeger}. Nous considérons ici le fibré en tores; la valeur en $s=0$ de la série d'Eisenstein est une manière de régulariser la moyenne de $\eta$ relativement à un réseau de $\mathbf{R}^n$. Nous travaillons adéliquement.
\section{Quotients adéliques}
Soit $\mathbf{A}$ l'anneau des adèles de $\mathbf{Q}$ et soit
$$[\mathrm{GL}_n] = \mathrm{GL}_n (\mathbf{Q}) \backslash \mathrm{GL}_n (\mathbf{A} ) / \mathrm{SO}_n Z(\mathbf{R})^+.$$
Le théorème d'approximation forte pour $\mathrm{GL}_n$ implique que, pour tout sous-groupe compact ouvert $K \subset \mathrm{GL}_n (\mathbf{A}_f )$, le quotient
$$[\mathrm{GL}_n] / K = \mathrm{GL}_n (\mathbf{Q}) \backslash \left( (\mathrm{GL}_n (\mathbf{R} ) / \mathrm{SO}_n Z(\mathbf{R})^+ ) \times \mathrm{GL}_n (\mathbf{A}_f ) \right) / K$$
est une union finie de quotients de $X$ de volumes finis que l'on peut décrire de la manière suivante. \'Ecrivons
\begin{equation} \label{E:approxforte}
\mathrm{GL}_n (\mathbf{A}_f ) = \bigsqcup_{j} \mathrm{GL}_n (\mathbf{Q})^+ g_j K
\end{equation}
avec $\mathrm{GL}_n (\mathbf{Q} )^+ = \mathrm{GL}_n (\mathbf{Q} ) \cap \mathrm{GL}_n (\mathbf{R})^+$. Alors
\begin{equation} \label{E:quot0}
[\mathrm{GL}_n] / K = \bigsqcup_{j} \Gamma_j \backslash X,
\end{equation}
où $\Gamma_j$ est l'image de $\mathrm{GL}_n (\mathbf{Q} )^+ \cap g_j K g_j^{-1}$ dans $\mathrm{GL}_n (\mathbf{R})^+ / Z(\mathbf{R})^+$.
La composante connexe de la classe de l'identité dans $[\mathrm{GL}_n] / K$ est le quotient
\begin{equation} \label{E:quot3}
\Gamma \backslash X
\end{equation}
où $\Gamma = K \cap \mathrm{GL}_n (\mathbf{Q})^+$.
Soit $V = \mathbf{G}_a^n$ vu comme groupe algébrique sur $\mathbf{Q}$; on a en particulier
$$V (\mathbf{Q} ) = \mathbf{Q}^n \quad \mbox{et} \quad V (\mathbf{R} ) = \mathbf{R}^n.$$
Le groupe $\mathrm{GL}_n$, algébrique sur $\mathbf{Q}$, opère naturellement (par multiplication matricielle à gauche) sur $V(\mathbf{C}) = \mathbf{C}^n$. On note
$$\mathcal{G} = \mathrm{GL}_n \ltimes V$$
le groupe affine correspondant; on le voit comme groupe algébrique sur $\mathbf{Q}$. Soit
$$[\mathcal{G} ] = \mathcal{G} (\mathbf{Q} ) \backslash \left[ (\mathrm{GL}_n (\mathbf{R} ) \ltimes \mathbf{C}^n ) \cdot \mathcal{G} (\mathbf{A}_f ) \right] / \mathrm{SO}_n Z (\mathbf{R})^+.$$
On explique maintenant comment associer à ces données une famille de groupes abéliens isomorphes à $(\mathbf{C}^* )^n$.
Soit
$$L_f \subset V(\mathbf{A}_f ) = \{ I_n \} \ltimes V(\mathbf{A}_f ) \subset \mathcal{G} (\mathbf{A}_f)$$
un sous-groupe compact ouvert; l'intersection $L= L_f \cap V (\mathbf{Q})$ est un réseau dans $V$.
On suppose dorénavant que $K \subset \mathrm{GL}_n (\mathbf{A}_f)$ préserve $L_f$. Le sous-groupe
$$\mathcal{K} = \mathcal{K}_{L_f} = K \ltimes L_f \subset \mathcal{G} (\mathbf{A}_f)$$
préserve $L_f$; c'est un sous-groupe compact ouvert.
Les quotients
\begin{equation} \label{E:quot1}
[\mathcal{G} ] / L_f = \mathcal{G} (\mathbf{Q} ) \backslash \left[ (\mathrm{GL}_n (\mathbf{R} ) \ltimes \mathbf{C}^n ) \cdot \mathcal{G} (\mathbf{A}_f ) \right] / \mathrm{SO}_n Z (\mathbf{R})^+ L_f \quad \mbox{et} \quad [\mathcal{G} ] / \mathcal{K}
\end{equation}
sont des fibrés en quotients de $\mathbf{C}^n$ au-dessus de respectivement $[\mathrm{GL}_n]$ et $[\mathrm{GL}_n] / K$.
De \eqref{E:approxforte} on déduit que
\begin{equation} \label{E:TK}
[\mathcal{G} ] / \mathcal{K} \simeq \bigsqcup_j \Gamma_j \backslash (X \times \mathbf{C}^n/L_j ),
\end{equation}
où $L_j = g_j (L_f) \cap V(\mathbf{Q})$.
L'isomorphisme s'obtient de la manière suivante~: étant donné une double classe
$$\mathcal{G} (\mathbf{Q}) [( x , z) , ( g_f , v_f ) ] \mathcal{K},$$
avec $x \in \mathrm{GL}_n (\mathbf{R} )/\mathrm{SO}_n Z (\mathbf{R})^+$, $z \in \mathbf{C}^n$ et $(g_f , v_f) \in \mathcal{G} (\mathbf{A}_f)$,
on peut d'abord supposer que $x$ appartient à $X$ en multipliant à gauche par un élément de $\mathcal{G} (\mathbf{Q})$ si nécessaire. On écrit alors
$$(g_f , v_f) = (h , w)^{-1} (g_j , 0) k, \quad \mbox{avec } (h,w) \in \mathcal{G} (\mathbf{Q} )^+ \mbox{ et } k \in \mathcal{K}.$$
Alors
\begin{equation} \label{E:TK2}
\begin{split}
\mathcal{G} (\mathbf{Q}) [( x , z) , ( g_f , v_f ) ] \mathcal{K} & = \mathcal{G} (\mathbf{Q}) [( x , z) , (h , w)^{-1} (g_j , 0) ] \mathcal{K} \\
& = \mathcal{G} (\mathbf{Q})(h , w)^{-1} [( hx , hz+w) , (g_j,0) ] \mathcal{K}
\end{split}
\end{equation}
d'image $[hx, hz+w]$ dans $\Gamma_j \backslash (X \times \mathbf{C}^n/L_j )$.
Dans la suite, on note
$$\mathcal{T}_\mathcal{K} = \Gamma \backslash (X \times \mathbf{C}^n/L )$$
le fibré au-dessus de \eqref{E:quot3}.
\medskip
\noindent
{\it Exemple.} Soit $N$ un entier strictement supérieur à $1$. On note
$$K_{0} (N) \subset \mathrm{GL}_n (\widehat{\mathbf{Z}}) = \prod_p \mathrm{GL}_n (\mathbf{Z}_p)$$
le sous-groupe défini par les relations de congruences suivantes aux nombres premiers $p$ divisant $N$~: si $N = \prod p^{v_p (N)}$ alors la $p$-composante de $K_{0} (N)$ est constituée des matrices de $\mathrm{GL}_n (\mathbf{Z}_p )$ de la forme
$$\left( \begin{array}{cc} \mathbf{Z}_p^\times & * \\ 0_{1,n-1} & \mathrm{GL}_{n-1} (\mathbf{Z}_p ) \end{array} \right)$$
modulo $p^{v_p (N)}$. Le groupe $K_0 (N)$ est compact ouvert et
$$[\mathrm{GL}_n] / K_0 (N) = \Gamma_0 (N) \backslash X$$
avec
$$\Gamma_0 (N) = \left\{ A \in \mathrm{SL}_n (\mathbf{Z} ) \; : \; A \equiv \left( \begin{array}{ccc} * & * & * \\ 0 & * & * \\ \vdots & \vdots & \vdots \\ 0 & * & * \end{array} \right) \ (\mathrm{mod} \ N ) \right\}.$$
Le groupe $\mathcal{K}_0 (N) = K_0 (N) \ltimes V (\widehat{\mathbf{Z}} )$ est compact et ouvert dans $\mathcal{G} (\mathbf{A}_f )$ et
$$\mathcal{T}_{\mathcal{K}_0 (N)} = \Gamma_0 (N) \backslash \left[ X \times (\mathbf{C}^n / \mathbf{Z}^n) \right].$$
\medskip
\section{Fonctions de Schwartz et cycles associés} \label{S:15}
Soit $\mathcal{S} (V (\mathbf{A}_f ))$ l'espace de Schwartz de $V(\mathbf{A}_f)$ des fonctions $\varphi_f : V (\mathbf{A}_f) \to \mathbf{C}$ localement constantes et à support compact. Le groupe $\mathcal{G} (\mathbf{A}_f)$ opère sur $\mathcal{S} (V (\mathbf{A}_f ))$ par la ``représentation de Weil''
$$\omega (g , v ) : \mathcal{S} (V (\mathbf{A}_f )) \to \mathcal{S} (V (\mathbf{A}_f )); \quad \phi \mapsto \left( w \mapsto \phi (g^{-1} (w-v) \right).$$
Considérons maintenant l'espace $C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right)$ des fonctions lisses; on fait opérer le groupe $\mathcal{G} (\mathbf{A}_f)$ sur $C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right)$ par la représentation régulière droite~:
$$( (h ,w) \cdot f) (g ,v ) = f( g h , g w + v ).$$
Les fonctions lisses sont précisément celles qui sont invariantes sous l'action d'un sous-groupe ouvert de $\mathcal{G} (\mathbf{A}_f )$.
L'application
\begin{equation}
\mathcal{S} (V (\mathbf{A}_f )) \to C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right); \quad \phi \mapsto f_\phi : ( (g,v) \mapsto \phi (-g^{-1} v ))
\end{equation}
est $\mathcal{G} (\mathbf{A}_f)$-équivariante relativement aux deux actions définies ci-dessus.
\'Etant donné une fonction $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f ))$ on note $L_{\varphi_f}$ le réseau des périodes de $\varphi_f$. Si $K \subset \mathrm{GL}_n (\mathbf{A}_f )$ est un sous-groupe compact ouvert qui laisse $\varphi_f$ invariante (et préserve donc $L_{\varphi_f}$), alors la fonction $f_{\varphi_f}$ est invariante (à droite) sous l'action de $\mathcal{K} = K \ltimes L_{\varphi_f} \subset \mathcal{G} (\mathbf{A}_f)$.
\begin{definition}
Soit $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f ))$ une fonction de Schwartz invariante sous l'action d'un sous-groupe compact ouvert $K\subset \mathrm{GL}_n (\mathbf{A}_f )$.
\begin{itemize}
\item Soit $D_{\varphi_f}$, resp. $D_{\varphi_f , K}$, l'image de l'application
$$\mathcal{G} (\mathbf{Q} ) \left[ \left( \mathrm{GL}_n (\mathbf{R}) \times \{ 0 \} \right) \cdot \mathrm{supp} (f_{\varphi_f} ) \right] \to [\mathcal{G}] /L_{\varphi_f} ,$$
resp.
$$\mathcal{G} (\mathbf{Q} ) \left[ \left( \mathrm{GL}_n (\mathbf{R}) \times \{ 0 \} \right) \cdot \mathrm{supp} (f_{\varphi_f} ) \right] \to [\mathcal{G}]/ \mathcal{K},$$
induite par l'inclusion du support de $f_{\varphi_f}$ dans $\mathcal{G} (\mathbf{A}_f)$.
\item Soit
$$U_{\varphi_f} \subset [\mathcal{G}] /L_{\varphi_f} , \quad \mbox{resp.} \quad U_{\varphi_f, K} \subset [\mathcal{G}]/ \mathcal{K},$$
le complémentaire de $D_{\varphi_f}$, resp. $D_{\varphi_f , K}$.
\end{itemize}
\end{definition}
\medskip
\noindent
{\it Remarque.} {\it Via} l'isomorphisme \eqref{E:TK} la projection de $D_{\varphi_f , K} \subset [\mathcal{G}]/ \mathcal{K}$ est égale à la réunion finie
\begin{equation}
\bigsqcup_j \bigcup_\xi \Gamma_j \backslash (X \times (L_j + \xi )/L_j ),
\end{equation}
où $\xi$ parcourt les éléments de $V(\mathbf{Q}) / L_j$ tels que $\varphi_f (g_j^{-1} \xi )$ soit non nul.
\medskip
En effet, en suivant \eqref{E:TK2} on constate que
$$(g_f , v_f ) = (h^{-1} g_j , - h^{-1} w) k$$
et, la fonction $f_{\varphi_f}$ étant $\mathcal{K}$-invariante à droite, on a
$$f_{\varphi_f} (g_f , v_f ) = f_{\varphi_f} (h^{-1} g_j , -h^{-1} w ) = \varphi_f ( g_j^{-1} w ).$$
\medskip
L'espace $D_{\varphi_f}$ est donc un revêtement fini de $[\mathrm{GL}_n]$. La fonction $\varphi_f$ induit en outre une fonction localement constante sur $D_{\varphi_f}$, c'est-à-dire un élément de $H^0 (D_{\varphi_f })$.
Maintenant, l'isomorphisme de Thom implique que l'on a~:
\begin{equation} \label{E:thom}
H^0 (D_{\varphi_f }) \stackrel{\sim}{\longrightarrow} H^{2n} \left( [\mathcal{G}]/ L_{\varphi_f} , U_{\varphi_f } \right);
\end{equation}
on note
$$[\varphi_f] \in H^{2n} \left( [\mathcal{G}]/ L_{\varphi_f} , U_{\varphi_f } \right)$$
l'image de $\varphi_f$; cette classe est $K$-invariante, on désigne par $[\varphi_f]_K$ son image dans
$$H^{2n} \left( [\mathcal{G}]/ \mathcal{K}, U_{\varphi_f , K } \right).$$
\begin{lemma} \label{L:40}
Supposons $\widehat{\varphi}_f (0) =0$, autrement dit que $\int_{V(\mathbf{A}_f )} \varphi_f (v) dv=0$. Alors, l'image de $[\varphi_f]$ par l'application degré
$$H^0 (D_{\varphi_f }) \to \mathbf{Z}^{\pi_0 (D_{\varphi_f })}$$
est égale à $0$.
\end{lemma}
\begin{proof} Montrons en effet que pour tout $j$ on a
$$\sum_{\xi \in V(\mathbf{Q}) / L_j} \varphi_f ( g_j^{-1} \xi) = 0.$$
Quitte à remplacer $\varphi_{f}$ par $\omega (g_j ) \varphi_f$ on peut supposer que $g_j$ est l'identité. Mais
\begin{equation*}
\begin{split}
\sum_{\xi \in V(\mathbf{Q}) / L} \varphi_f (\xi ) & = \sum_{ v \in V (\mathbf{A}_f ) / L_{\varphi_f} } \varphi_f (v) \\
& = \frac{1}{\mathrm{vol} (L_{\varphi_f})} \sum_{v \in V (\mathbf{A}_f ) / L_{\varphi_f} } \int_{L_{\varphi_f} } \varphi_f (v+u) du \\
& = \frac{1}{\mathrm{vol} (L_{\varphi_f} )} \int_{V(\mathbf{A}_f )} \varphi_f (v) dv \\
& = \frac{1}{\mathrm{vol} (L_{\varphi_f} )} \widehat{\varphi}_f (0) =0.
\end{split}
\end{equation*}
\end{proof}
Dans la suite on désigne par $D_{\varphi_f }^0$ l'intersection de $D_{\varphi_f }$ avec la composante connexe $\mathcal{T}_{\mathcal{K}}$.
\medskip
\noindent
{\it Exemple.} Soit $N$ un entier strictement supérieur à $1$. Alors, le réseau
$$L_{\mathcal{K}_0 (N)} = (N^{-1} \widehat{\mathbf{Z}} ) \times \widehat{\mathbf{Z}} \times \ldots \times \widehat{\mathbf{Z}} \subset V (\mathbf{A}_f)$$
est $\mathcal{K}_0 (N)$-invariant. La fonction
\begin{equation} \label{E:exvarphif}
\sum_{j=0}^{N-1} \delta_{(\frac{j}{N} , 0 , \ldots , 0) + \widehat{\mathbf{Z}}^n} - N \delta_{\widehat{\mathbf{Z}}^n} \in \mathcal{S} (V (\mathbf{A}_f))
\end{equation}
est $\mathcal{K}_0 (N)$-invariante, à support dans $L_{\mathcal{K}_0 (N)}$ et de degré $0$.
En conservant les notations de l'exemple précédent, on désigne par $D_0 (N)$ le sous-ensemble de
$$\mathcal{T}_{\mathcal{K}_0 (N)} = \Gamma_0 (N) \backslash \left[ X \times (\mathbf{C}^n / \mathbf{Z}^n) \right]$$
associé à la fonction \eqref{E:exvarphif}. Il est constitué de tous les points dont la première coordonnée dans la fibre au-dessus de $\Gamma_0 (N) \backslash X$ est de $N$-torsion et dont toutes les autres coordonnées sont nulles.
\medskip
\section{Série theta adélique} \`A toute fonction $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f))$ il correspond les formes différentielles
$$\widetilde{\varphi} \otimes \varphi_f \quad \mbox{et} \quad \widetilde{\psi} \otimes \varphi_f \in A^{\bullet} \left( S \times \mathbf{C}^n , \mathcal{S} (\mathbf{C}^n ) \right) \otimes \mathcal{S} (V (\mathbf{A}_f ))$$
de degrés respectifs $2n$ et $2n-1$.
En appliquant la distribution theta dans les fibres, on obtient alors des applications
\begin{equation} \label{appl-theta}
\theta_\varphi \quad \mbox{et} \quad \theta_\psi : \mathcal{S} (V (\mathbf{A}_f)) \longrightarrow \left[ A^{\bullet} (S \times \mathbf{C}^n) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} )}.
\end{equation}
\medskip
\noindent
{\it Remarque.} Rappelons que
$$A^{\bullet} (S \times \mathbf{C}^n) \cong \left[ \wedge^\bullet (\mathfrak{p} \oplus \mathbf{C}^n)^* \otimes C^{\infty} (\mathrm{GL}_n (\mathbf{R}) \ltimes \mathbf{C}^n ) \right]^{\mathrm{SO}_n}.$$
Le produit tensoriel dans \eqref{appl-theta} est donc plus rigoureusement égal à
$$\mathrm{Hom}_{\mathrm{SO}_n} \left( \wedge^\bullet (\mathfrak{p} \oplus \mathbf{C}^n) , C^{\infty} \left( (\mathrm{GL}_n (\mathbf{R}) \ltimes \mathbf{C}^n) \times \mathcal{G} (\mathbf{A}_f ) \right) \right)^{\mathcal{G} (\mathbf{Q} )}$$
où $C^{\infty} \left( (\mathrm{GL}_n (\mathbf{R}) \ltimes \mathbf{C}^n) \times \mathcal{G} (\mathbf{A}_f ) \right) $ est l'espace des fonctions lisses sur un espace adélique.
\medskip
L'application $\theta_\varphi$ est définie par
\begin{equation} \label{appl-theta2}
\begin{split}
\theta_\varphi (g_f , v_f ; \varphi_f ) & = \sum_{\xi \in V (\mathbf{Q} ) } \widetilde{\varphi} (\xi ) (\omega (g_f , v_f ) \varphi_f ) (\xi ) \\
& = \sum_{\xi \in V(\mathbf{Q} ) } \varphi_f \left( g_f^{-1} (\xi -v_f ) \right) \widetilde{\varphi} (\xi )
\end{split}
\end{equation}
et de même pour $\theta_\psi$.
Rappelons que le groupe $\mathcal{G} (\mathbf{R})$ opère naturellement sur $S \times \mathbf{C}^n$; étant donné un élément $(g,v) \in \mathcal{G} (\mathbf{R})$ et une forme $\alpha \in \mathcal{A}^{\bullet} (S \times \mathbf{C}^n)$ on note $(g,v)^* \alpha$ le tiré en arrière de $\alpha$ par l'application
$$(g,v) : S \times \mathbf{C}^n \to S \times \mathbf{C}^n.$$
L'invariance sous le groupe $\mathcal{G}(\mathbf{Q})$ dans \eqref{appl-theta} signifie donc que pour tout $(g,v)$ dans $\mathcal{G} (\mathbf{Q})$ on a~:
\begin{equation} \label{E:invtheta}
(g,v)^*\theta_\varphi (g g_f , gv_f + v ;\varphi_f)=\theta_\varphi (g_f, v_f ; \varphi_f);
\end{equation}
ce qui découle de la $\mathcal{G} (\mathbf{R})$-invariance de $\widetilde{\varphi}$, voir \S \ref{S:44}.
Les applications $\theta_\varphi$ et $\theta_\psi$ entrelacent par ailleurs les actions naturelles de $\mathcal{G} (\mathbf{A}_f)$ des deux côtés~: pour tout $(h_f , w_f ) \in \mathcal{G} (\mathbf{A}_f )$ et pour $\theta = \theta_\varphi$ ou $\theta_\psi$, on a
\begin{equation} \label{E:entrelacetheta}
\theta ( \omega (h_f , w_f ) \varphi_f ) = (h_f , w_f ) \cdot \theta (\varphi_f ).
\end{equation}
En particulier, on a
$$\theta_\varphi (\varphi_f ) \quad \mbox{et} \quad \theta_\psi (\varphi_f ) \in \left[ A^{\bullet} (S \times \mathbf{C}^n) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} ) \times L_{\varphi_f}} = A^\bullet \left( \widehat{[\mathcal{G}]} / L_{\varphi_f} \right),$$
où
$$\widehat{[\mathcal{G}]} = \mathcal{G} (\mathbf{Q}) \backslash \left( (\mathrm{GL}_n (\mathbf{R}) \ltimes \mathbf{C}^n) \times \mathcal{G} (\mathbf{A}_f ) \right) / \mathrm{SO}_n.$$
Il découle en outre de \eqref{E:entrelacetheta} que si $\varphi_f$ est $K$-invariante alors les formes $\theta_\varphi (\varphi_f )$ et
$\theta_\psi (\varphi_f )$ sont $K$-invariantes à droite.
\medskip
\medskip
\subsection{Action de l'algèbre de Hecke} \label{algHecke}
Soit $p$ un nombre premier. On désigne par $\mathcal{H}_p$ l'algèbre de Hecke locale de $\mathcal{G} (\mathbf{Q}_p )$, c'est-à-dire les fonctions lisses et à support compact dans $\mathcal{G} (\mathbf{Q}_p )$ et le produit de convolution. La fonction caractéristique de $\mathcal{G} (\mathbf{Z}_p)$ appartient à $\mathcal{H}_p$.
L'\emph{algèbre de Hecke globale} $\mathcal{H} (\mathcal{G} (\mathbf{A}_f) )$ est le produit restreint des algèbres de Hecke locales $\mathcal{H}_p$ relativement aux fonctions caractéristiques de $\mathcal{G} (\mathbf{Z}_p)$, cf. \cite{Flath}.
Un élément de $\mathcal{H} (\mathcal{G} (\mathbf{A}_f) )$ est donc un produit tensoriel
$\phi = \otimes \phi_p$, où pour presque tout $p$ la fonction $\phi_p$ est égale à la fonction caractéristique de $\mathcal{G} (\mathbf{Z}_p)$. On désigne par $\mathbf{T}_\phi$ l'{\it opérateur de Hecke} sur $C^\infty (\mathcal{G} (\mathbf{A}_f ))$ qui lui correspond. Il associe à une fonction $f \in C^\infty (\mathcal{G} (\mathbf{A}_f ))$ la fonction
$$\mathbf{T}_\phi (f) : (g,v) \mapsto \int_{\mathcal{G} (\mathbf{A}_f)} f(gh , gw+v) \phi (h,w) d(h,w),$$
où la mesure de Haar est normalisée de sorte que pour tout $p$, le volume de $\mathcal{G} (\mathbf{Z}_p )$ soit égal à un.
On note encore
$$\mathbf{T}_\phi : A^\bullet (S \times \mathbf{C}^n) \otimes C^\infty (\mathcal{G} (\mathbf{A}_f )) \to A^\bullet (S \times \mathbf{C}^n) \otimes C^\infty (\mathcal{G} (\mathbf{A}_f ))$$
l'opérateur de Hecke induit; il commute à l'action (à gauche) de $\mathcal{G} (\mathbf{Q})$.
L'algèbre de Hecke $\mathcal{H} (\mathcal{G} (\mathbf{A}_f) )$ opère également sur $ \mathcal{S} (V (\mathbf{A}_f))$ {\it via} les opérateurs
$T_\phi : \mathcal{S} (V (\mathbf{A}_f)) \to \mathcal{S} (V (\mathbf{A}_f))$ qui à une fonction de Schwartz $\varphi_f$ associe la fonction
$$T_\phi (\varphi_f ) : v \mapsto \int_{\mathcal{G} (\mathbf{A}_f )} \phi (h,w) \varphi_f (h^{-1} (v-w)) d (h,w) .$$
La proposition suivante résulte alors des définitions.
\begin{proposition} \label{P:hecke1}
Soit $\phi \in \mathcal{H} (\mathcal{G} (\mathbf{A}_f ))$. Alors $\mathbf{T}_\phi$ préserve
$$\left[ A^\bullet (S \times \mathbf{C}^n) \otimes C^\infty (\mathcal{G} (\mathbf{A}_f )) \right]^{\mathcal{G} (\mathbf{Q})} $$
et
$$\mathbf{T}_\phi ( \theta_{\varphi} (\varphi_f) ) = \theta_{\varphi} ( T_\phi (\varphi_f )) \quad \mbox{et} \quad \mathbf{T}_\phi ( \theta_{\psi} (\varphi_f) ) = \theta_{\psi} ( T_\phi (\varphi_f )).$$
\end{proposition}
\subsection{Classes de cohomologie associées} \label{cohomClass}
Notons $\widehat{U_{\varphi_f}}$ la préimage de $U_{\varphi_f}$ dans $\widehat{[\mathcal{G}]} / L_{\varphi_f}$ et $\widehat{U_{\varphi_f , K}} $ la projection de $\widehat{U_{\varphi_f}}$ dans $ \widehat{[\mathcal{G}]} / \mathcal{K}$ de sorte que
$$\widehat{U_{\varphi_f , K}} = \widehat{[\mathcal{G}]} / \mathcal{K} - \mathcal{G} (\mathbf{Q}) \left( [S^+ \times \{ 0 \}] \cdot \mathrm{supp}(f_{\varphi_f} ) \right) \mathcal{K}.$$
Notons enfin $\widehat{[\varphi_f]}$ l'image de $[\varphi_f]$ dans
$$H^{2n} \left( \widehat{[\mathcal{G}]} / L_{\varphi_f} , \widehat{U_{\varphi_f}} \right).$$
\begin{proposition} \label{P:thetacohom}
La forme différentielle $\theta_\varphi (\varphi_f )$ est fermée et représente $\widehat{[\varphi_f]}$ dans $H^{2n} \left( \widehat{[\mathcal{G}]} / L_{\varphi_f} , \widehat{U_{\varphi_f}} \right)$.
\end{proposition}
\begin{proof} La forme $\theta_\varphi (\varphi_f )$ est fermée comme combinaison linéaire de formes fermées. Fixons un sous-groupe compact ouvert $K \subset \mathrm{GL}_n (\mathbf{A}_f)$ tel que $\varphi_f$ soit $K$-invariante. Alors $\theta_\varphi (\varphi_f )$ appartient à
$$\left[ A^{2n} (S \times \mathbf{C}^n) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} ) \times \mathcal{K}} \cong \bigoplus_j A^{2n} (S^+ \times \mathbf{C}^n )^{\Gamma_j \ltimes L_j},$$
où l'isomorphisme ci-dessus est obtenu en évaluant en $(g_j , 0)$. D'un autre côté, on a montré que l'image de $[\varphi_f ]_{K}$ dans
$$H^{2n} \left( \widehat{[\mathcal{G}]} / \mathcal{K} \right) \cong \bigoplus_j H^{2n} (\Gamma_j \backslash (S^+ \times \mathbf{C}^n/L_j ))$$
est égale à
$$\bigoplus_j \sum_{\xi \in V(\mathbf{Q})/ L_j} \varphi_f (g_j^{-1} \xi ) \left[ \Gamma_j \backslash (S^+ \times (L_j +\xi )/L_j ) \right].$$
Quitte à remplacer $\varphi_f$ par $\omega (g_j ) \varphi_f$ on peut donc se restreindre à la composante connexe associée à l'identité. On a alors simplement $\Gamma_j = \Gamma$ et $L_j = L$.
Maintenant, pour tout $\xi \in V (\mathbf{Q} )$ la forme différentielle
$$\varphi_f (\xi ) \widetilde{\varphi} (\xi ) = \varphi_f (\xi) (1,-\xi)^* \varphi \in A^{2n} ( S^+ \times \mathbf{C}^n )$$
est une forme de Thom et représente
$$\varphi_f (\xi) [S^+ \times \{ \xi \} ] \in H^{2n} (S^+ \times \mathbf{C}^n , S^+ \times (\mathbf{C}^n - \{ \xi \} )).$$
La moyenne
$$\theta_\varphi (1,0;\varphi_f ) = \sum_{\xi \in V (\mathbf{Q})} \varphi_f (\xi ) \widetilde{\varphi} (\xi) \in A^{2n} (S^+ \times \mathbf{C}^n )$$
représente donc l'image de
$$\sum_{\xi \in V(\mathbf{Q}) / L} \varphi_f (\xi) \left[ \Gamma \backslash (S^+ \times (L +\xi )/L ) \right]$$
par l'isomorphisme de Thom.
\end{proof}
La proposition \ref{P:eta} suggère de considérer les formes différentielles
$$\theta_{[r]^*\varphi} (\varphi_f ) \quad \mbox{et} \quad \theta_{[r]^*\psi} (\varphi_f ) \in \left[ A^{\bullet} (S \times \mathbf{C}^n) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} )} \quad (r >0).$$
Le lemme suivant est une version globale du lemme \ref{L:convcourant}. On le déduit de la formule sommatoire de Poisson.
\begin{lemma} \label{L:theta-asympt}
1. Lorsque $r$ tend vers $+\infty$, les formes $\theta_{[r]^* \varphi} (\varphi_f )$ convergent uniformément sur tout compact de
$\widehat{U_{\varphi_f , K}}$
exponentiellement vite vers la forme nulle.
2. Vues comme courants dans
$$ \left[\mathcal{D}^{2n} (S \times \mathbf{C}^n ) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} ) \times \mathcal{K}}$$
les formes $\theta_{[r]^* \varphi} (\varphi_f )$ convergent exponentiellement vite, lorsque $r$ tend vers $+\infty$, vers le courant $[\widehat{D_{\varphi_f , K}}]$ associé à la fonction localement constante $\varphi_f$ et de support
$$\mathcal{G} (\mathbf{Q}) \left( [S^+ \times \{ 0 \}] \times \mathrm{supp}(f_{\varphi_f}) \right) \mathcal{K}.$$
3. L'application qui à $r \in \mathbf{R}_+^*$ associe la forme différentielle $\theta_{[r]^* \varphi} ( \varphi_f )$ sur $S \times \mathbf{C}^n $ se prolonge en une fonction lisse sur $[0, +\infty)$ qui s'annule en $0$.
\end{lemma}
\begin{proof} Les deux premiers points découlent des points (1) et (2) du lemme \ref{L:convcourant}. On peut en outre remarquer que, lorsque $r$ varie, les formes $\theta_{[r] ^* \varphi} (\varphi_f )$ sont toutes cohomologues d'après la démonstration de la proposition \ref{P:thetacohom}.
Le troisième point découle lui du point (3) du lemme \ref{L:convcourant}.
\end{proof}
\section{Séries d'Eisenstein adéliques} \label{SEA7}
Soit $\varphi_f \in \mathcal{S} (V(\mathbf{A}_f))$. Les trois propositions suivantes sont des incarnations adéliques du procédé classique de régularisation de Hecke, voir par exemple \cite[Chapitre IV]{Wielonsky}. Elles se démontrent de la même manière; on ne détaille que la démonstration de la première qui est légèrement plus subtile.
\begin{proposition} \label{P:Eis1}
L'intégrale
\begin{equation} \label{Eis1}
E_\varphi ( \varphi_f , s) = \int_0^{\infty} r^{s} \theta_{[r]^*\varphi} (\varphi_f ) \frac{dr}{r},
\end{equation}
qui converge absolument, uniformément sur tout compact de $\widehat{U_{\varphi_f , K}}$ si $\mathrm{Re} (s) >0$, possède un prolongement méromorphe à $\mathbf{C}$ tout entier, à valeurs dans l'espace $A^{2n} (\widehat{U_{\varphi_f , K}})$, holomorphe en dehors de pôles au plus simples aux entiers strictement négatifs.
\end{proposition}
\begin{proof} D'après le lemme \ref{L:theta-asympt} (1), l'intégrale \eqref{Eis1}
est absolument convergente sur tout compact de $\widehat{U_{\varphi_f , K}}$ si $\mathrm{Re} (s) >0$. Elle définit donc une forme différentielle dans $A^{2n} (\widehat{U_{\varphi_f , K}})$. Pour prolonger cette fonction de $s$, il suffit de couper l'intégrale en deux morceaux, l'un allant de $0$ à $1$ où l'on utilise un développement limité de la fonction $r \mapsto \theta_{[r]^*\varphi} (\varphi_f )$ au voisinage de $0$, qui est bien défini d'après le lemme \ref{L:theta-asympt} (3), et l'autre de $1$ à $+\infty$ qui ne pose pas de problème d'après le lemme \ref{L:theta-asympt} (1). Le fait que $0$ ne soit pas un pôle découle du fait que $r \mapsto \theta_{[r]^*\varphi} (\varphi_f )$ s'annule en $0$.
\end{proof}
\begin{proposition} \label{P:Eis1bis}
L'intégrale
\begin{equation} \label{Eis1bis}
E_\varphi ( \varphi_f , s) = \int_0^{\infty} r^{s} \left( \theta_{[r]^*\varphi} (\varphi_f ) - [\widehat{D_{\varphi_f , K}}] \right) \frac{dr}{r},
\end{equation}
définit une application méromorphe en $s$ à valeurs dans l'espace des courants
$$\left[\mathcal{D}_{\bullet} (S \times \mathbf{C}^n ) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} ) \times \mathcal{K}}$$
avec un pôle simple en $s=0$ de résidu $- [\widehat{D_{\varphi_f , K}}]$.
\end{proposition}
\begin{proof} La démonstration est identique à celle de la proposition \ref{P:Eis1} à ceci près que cette fois on applique le lemme \ref{L:theta-asympt} (2) et que l'application qui à $r$ associe le courant $ \theta_{[r]^*\varphi} (\varphi_f ) - [\widehat{D_{\varphi_f , K}}]$ ne s'annule plus en $0$ mais est égale à $- [\widehat{D_{\varphi_f , K}}]$. La fonction qui à $s$ associe le courant $E_\varphi ( \varphi_f , s)$ a donc cette fois un pôle (simple) en $0$ de résidu $- [\widehat{D_{\varphi_f , K}}]$.
\end{proof}
\begin{proposition} \label{P:Eis2}
L'intégrale
\begin{equation} \label{Eis2}
E_\psi ( \varphi_f , s) = \int_0^{\infty} r^{s} \theta_{[r]^*\psi} (\varphi_f ) \frac{dr}{r}
\end{equation}
qui converge absolument, uniformément sur tout compact de $\widehat{U_{\varphi_f , K}}$ si $\mathrm{Re} (s) >0$, possède un prolongement méromorphe à $\mathbf{C}$ tout entier, à valeurs dans l'espace $A^{2n-1} (\widehat{U_{\varphi_f , K}})$, holomorphe en dehors de pôles au plus simples aux entiers strictement négatifs.
\end{proposition}
\begin{proof} La démonstration est identique à celle de la proposition \ref{P:Eis1}. Le fait que la fonction $r \mapsto \theta_{[r]^*\psi} (\varphi_f )$ soit nulle en $0$ découle cette fois du fait que $\psi$ s'annule en $0$.
\end{proof}
\medskip
Rappelons maintenant que
$$[\mathcal{G}] = \widehat{[\mathcal{G} ]} / Z (\mathbf{R})^+$$
où $Z (\mathbf{R})^+$ opère naturellement sur $S^+$ et trivialement sur la fibre $\mathbf{C}^n$. L'action d'un élément $\lambda \in Z(\mathbf{R})^+$ sur
$S^+ \times \mathbf{C}^n$ s'obtient donc en composant l'action de $(\lambda , 0)$ dans $\mathcal{G} (\mathbf{R})$ par la multiplication par $\lambda^{-1}$ dans la fibre $\mathbf{C}^n$. La $\mathcal{G} (\mathbf{R})$-équivariance de $\widetilde{\varphi}$ implique alors que pour tout $w \in \mathbf{C}^n$ on a
$$\lambda^* \widetilde{\varphi} (w) = [\lambda^{-1}]^* ((\lambda , 0 )^* \widetilde{\varphi} (w)) = [\lambda^{-1}]^* \widetilde{\varphi} (\lambda^{-1} w ) = \widetilde{[\lambda^{-1}]^* \varphi} (w)$$
et de même pour $\psi$. Il s'en suit que pour tout $\lambda \in Z(\mathbf{R})^+$ on a
\begin{equation}
\lambda^* \widetilde{[r]^*\varphi} = \widetilde{[\lambda^{-1} r]^*\varphi} \quad \mbox{et} \quad \lambda^* \widetilde{[r]^*\psi} = \widetilde{[\lambda^{-1} r]^*\psi}
\end{equation}
et donc
\begin{equation} \label{E:actZ}
\lambda^* E_\varphi (\varphi_f , s) = \lambda^{s} E_{\varphi} (\varphi_f , s) \quad \mbox{et} \quad \lambda^*E_\psi (\varphi_f , s) = \lambda^{s} E_\psi (\varphi_f , s).
\end{equation}
On pose
\begin{equation}
E_\psi (\varphi_f) = E_\psi (\varphi_f , 0) \in A^{2n-1} \left( [\mathcal{G}] / L_{\varphi_f} - D_{\varphi_f } \right) .
\end{equation}
\begin{theorem}
La forme différentielle
$$E_\psi (\varphi_f) \in A^{2n-1} \left( [\mathcal{G}] / L_{\varphi_f} - D_{\varphi_f } \right)$$
est \emph{fermée} et représente une classe de cohomologie qui relève la classe\footnote{Noter que dans le cas (multiplicatif) de ce paragraphe, l'image de $[\varphi_f]$ dans $H^{2n} \left( [\mathcal{G}] / L_{\varphi_f} \right)$ est nulle puisque les fibres sont de dimension cohomologique $n$.}
$$[\varphi_f] \in H^{2n} \left( [\mathcal{G}] / L_{\varphi_f} , [\mathcal{G}] / L_{\varphi_f} - D_{\varphi_f }\right)$$
dans la suite exacte longue
\begin{multline*}
\ldots \to H^{2n-1} \left( [\mathcal{G}] / L_{\varphi_f} - D_{\varphi_f } \right) \\ \to H^{2n} \left( [\mathcal{G}] / L_{\varphi_f} , [\mathcal{G}] / L_{\varphi_f} - D_{\varphi_f } \right) \to H^{2n} \left( [\mathcal{G}] / L_{\varphi_f} \right) \to \ldots
\end{multline*}
\end{theorem}
\begin{proof} C'est une version adélique de \cite[Theorem 19]{Takagi}. Fixons un sous-groupe compact ouvert $K \subset \mathrm{GL}_n (\mathbf{A}_f )$ tel que $\varphi_f$ soit $K$-invariante. Les intégrales \eqref{Eis1} et \eqref{Eis2} étant absolument convergentes sur tout compact de $\widehat{U_{\varphi_f , K}}$, il découle de (\ref{E:tddt1}) que
\begin{equation*}
\begin{split}
d E_\psi ( \varphi_f , s) & = \int_0^\infty r^{s} d (\theta_{[r]^*\psi}^* (\varphi_f )) \frac{dr}{r} \\
& = \int_0^\infty r^{s} \frac{d}{dr} \theta_{[r]^*\varphi} (\varphi_f ) dr.
\end{split}
\end{equation*}
Une intégration par parties donne donc
\begin{equation} \label{dEis}
d E_\psi ( \varphi_f , s) = - s \int_0^\infty r^{s} \theta_{[r]^*\varphi} (\varphi_f ) \frac{dr}{r} = -s E_{\varphi} (\varphi_f , s )
\end{equation}
sur $\widehat{U_{\varphi_f , K}}$.
En particulier la forme $E_\psi ( \varphi_f ) = E_\psi ( \varphi_f , 0)$ est fermée sur $\widehat{U_{\varphi_f , K}}$ et la première partie du théorème découle finalement du fait que, d'après \eqref{E:actZ}, la forme $E_\psi ( \varphi_f )$ est invariante sous l'action du centre $Z (\mathbf{R})^+$.
Enfin, la proposition \ref{P:Eis1bis} implique que l'identité \eqref{dEis} s'étend sur $\widehat{[\mathcal{G}]} / \mathcal{K}$ en une identité entre courants et qu'en ce sens
$$d E_\psi (\varphi_f) =[\widehat{D_{\varphi_f , K}}]$$
et que l'image de $[E_\psi (\varphi_f)] \in H^{2n-1} (\widehat{U_{\varphi_f , K}})$ dans $H^{2n} (\widehat{[\mathcal{G}] }/ \mathcal{K} , \widehat{U_{\varphi_f , K}})$ est égale à $[\varphi_f]_K$. Le théorème s'en déduit en remarquant encore que ces classes sont toutes invariantes sous l'action de $Z (\mathbf{R} )^+$.
\end{proof}
\medskip
Pour conclure ce paragraphe notons que pour tout $g \in \mathrm{GL}_n (\mathbf{Q}) $, il découle de \eqref{E:invtheta} que l'on a
\begin{equation} \label{E:glnqinv}
g^* (E_\psi ( \varphi_f ) (gg_f , gv_f )) = E_\psi (\varphi_f) (g_f , v_f).
\end{equation}
En prenant $g$ scalaire et $(g_f, v_f) = (g_j , 0)$ on obtient en particulier que pour $\alpha$ dans $\mathbf{Q}^*$, positif si $n$ est impair, on a
$$E_\psi (\varphi_f (\alpha \cdot )) (g_j , 0) = E_{[\alpha^{-1}]^*\psi} (\varphi_f ) (g_j , 0) = E_{\psi} (\varphi_f ) (g_j , 0) .$$
Ce qui implique que
$$[E_{\psi} (\varphi_f ) (g_j , 0) ] \in H_{\Gamma_j}^{2n-1} \left( ( \mathbf{C}^n - \cup_\xi (L_j +\xi) ) / L_j \right)^{(1)},$$
au sens de la définition \ref{Def1.7}.
La restriction de $E_\psi (\varphi_f)$ à la composante connexe $\mathcal{T}_{\mathcal{K}}$ définit une forme fermée $E_\psi (\varphi_f)^0$ sur
$$\Gamma \backslash (X \times \mathbf{C}^n )/ L - D_{\varphi_f }^0$$
et donc une classe de cohomologie équivariante
\begin{equation} \label{E:721}
[E_\psi (\varphi_f)^0] \in H_\Gamma^{2n-1} \left( ( \mathbf{C}^n - \cup_\xi (L+ \xi) ) / L \right)^{(1)} ,
\end{equation}
où $\xi$ parcourt des éléments de $V (\mathbf{Q}) \cap \mathrm{supp} (\varphi_f )$.
\medskip
\noindent
{\it Exemple.} Soit $N$ un entier strictement supérieur à $1$. En prenant pour $\varphi_f$ la fonction \eqref{E:exvarphif}, on obtient une classe de cohomologie équivariante
$$E_{D_0 (N)} \in H_{\Gamma_0 (N)}^{2n-1} (T - T [N])^{(1)}$$
avec $T = \mathbf{C}^n / \mathbf{Z}^n$. Cette classe est associée au cycle invariant de $N$-torsion et de degré $0$
$$D_0 (N) \in H_{\Gamma_0 (N)}^0 (T[N])$$
comme expliqué au paragraphe \ref{S:15}.
\medskip
\section{Comportement à l'infini de $E_\psi (\varphi_f)$} \label{S192}
\begin{definition}
\'Etant donné un sous-espace rationnel $W \subset V$, on note
$$\int_W : \mathcal{S} (V (\mathbf{A}_f)) \to \mathcal{S} (V (\mathbf{A}_f) / W (\mathbf{A}_f))$$
l'application naturelle d'intégration le long des fibres de la projection $V \to V/ W$.
\end{definition}
Dans ce paragraphe on fixe une fonction $\varphi_f \in \mathcal{S} (V(\mathbf{A}_f ))$ et un sous-groupe parabolique $Q = Q (W_\bullet )$ dans $\mathrm{SL}_n (\mathbf{Q})$ associé à un drapeau de sous-espaces rationnels de $\mathbf{Q}^n$; voir (\ref{E:flag}) dont on reprend les notations. On se propose d'étudier le comportement de la forme différentielle $E_\psi (\varphi_f)$ en restriction aux ensembles de Siegel associés à $Q$. On fixe également
\begin{itemize}
\item un réel strictement positif $t_0$,
\item un élément $g \in \mathrm{SL}_n (\mathbf{Q})$ tel que $g^{-1} W_\bullet$ soit un drapeau standard $W_J$,
\item un sous-ensemble relativement compact $\omega\subset N_J M_J$, et
\item un sous-ensemble compact $\kappa$ dans
\begin{equation*}
\left\{ (z , (g_f , v_f ) ) \in \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f ) \; \left| \; \begin{array}{l} \forall \xi \in V (\mathbf{Q}), \\ \varphi_f (g_f^{-1} (\xi - v_f)) \neq 0 \Rightarrow z \notin W_k (\mathbf{C}) + \xi \end{array} \right. \right\}.
\end{equation*}
\end{itemize}
\begin{lemma} \label{L:thetaSiegel}
Il existe des constantes strictement positives $C$, $\alpha$ et $\beta$ telles que pour tout $t \geq t_0$ les deux propriétés suivantes sont vérifiées.
\begin{enumerate}
\item Les formes $\theta_{[r]^* \psi} (\varphi_f )$ ($r\geq 1$) sont de norme
$$||\theta_{[r]^* \psi} (\varphi_f ) ||_\infty \leq C e^{-r^2 \alpha t^\beta}$$
en restriction à $\mathfrak{S}_{W_\bullet} (g ,t, \omega) \times \kappa$.
\item Si l'on suppose de plus que $\int_{W_1} \varphi_f$ est constante égale à $0$, alors les formes $\theta_{[r]^* \psi} (\varphi_f )$ ($r\leq 1$) sont de norme
$$||\theta_{[r]^* \psi} (\varphi_f ) ||_\infty \leq C e^{-r^{-2} \alpha t^\beta}$$
en restriction à $\mathfrak{S}_{W_\bullet} (g ,t, \omega) \times \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f )$.
\end{enumerate}
\end{lemma}
\begin{proof} La premier point découle du lemme \ref{L8} (la convergence uniforme de la somme sur $\xi$ résulte du fait qu'on somme des fonctions gaussiennes). Pour démontrer le deuxième point il suffit de majorer la norme ponctuelle de
$$(h,v)^*\theta_{[r]^* \psi} (\varphi_f ) = [r]^* \left(\sum_\xi \varphi_f (\xi) (\omega (r^{-1} h , v) \widetilde{\psi}) (\xi ) \right),$$
en le point base $([e] , 0)$ de $X \times \mathbf{C}^n$, pour $(h,v) \in \mathfrak{S}_{W_\bullet} (g ,t, \omega) \times \mathbf{C}^n$. Pour cela, on utilise à nouveau la formule de Poisson~:
\begin{equation*}
\sum_\xi \varphi_f (\xi) (\omega (r^{-1} h , v) \widetilde{\psi}) (\xi ) = \sum_\xi \widehat{\varphi}_f (\xi) \widehat{(\omega (r^{-1} h , v) \widetilde{\psi})} (\xi ) .
\end{equation*}
Maintenant, pour $(h , v) \in \mathrm{GL}_n (\mathbf{C}) \ltimes \mathbf{C}^n$ on a~:
$$| \widehat{\omega(h , v) \widetilde{\psi } } (\xi)| = | \det (h) \widehat{\widetilde{\psi}} (h^{\top} \xi )|.$$
On est donc ramené à étudier la croissance de $\widehat{\widetilde{\psi}}$ sur $\mathfrak{S}_{W_\bullet} (g ,t, \omega)^{-\top} \times \{ 0 \}$ où l'image par $h \mapsto h^{-\top}$ de $\mathfrak{S}_{W_\bullet} (g ,t, \omega)$ est un
ensemble de Siegel de la forme $\mathfrak{S}_{W_\bullet '} (g' ,t, \omega')$ associé au drapeau
$$(0) \varsubsetneq W_k ' \varsubsetneq W_{k-1} ' \varsubsetneq \cdots \varsubsetneq W_1 ' \varsubsetneq \mathbf{Q}^n,$$
avec $g' = w_J g^{-\top} w_J^{-1}$, $\omega' = w_J \omega^{-\top} w_J^{-1}$ et
$$w_J = \left(\begin{array}{ccc}
& & 1_{n-j_k} \\
& \reflectbox{$\ddots$} & \\
1_{n-j_1} & & \end{array} \right).$$
Or
$$\widehat{\widetilde{\psi}} (h^{\top} \xi ) = (\omega (h^{-\top} , 0 ) \widehat{\widetilde{\psi}} ) (\xi)= ((h^{-\top})^* \widehat{\widetilde{\psi}} ) (\xi)$$
et la démonstration du lemme \ref{L8} --- ou le fait que $\widehat{\widetilde{\psi}}$ soit une forme différentielle dans $A^{2n-1} (E , \mathcal{S}(V))$ --- implique qu'en restriction à $\mathfrak{S}_{W_\bullet '} (g' ,t, \omega')$ la forme
$$\widehat{\widetilde{\psi}} (h^\top \xi ) \quad \mbox{a pour norme} \quad O(e^{-\alpha' t^{\beta '} |p_{W_1} (\xi )|^2})$$
pour certaines constantes strictement positives $\alpha '$ et $\beta '$.
Finalement, par hypothèse $\int_{W_1} \varphi_f$ est identiquement nulle, de sorte que pour tout $\xi \in W_1 '$, on a\footnote{Ici $\chi$ est un caractère additif fixé.}
\begin{equation*}
\begin{split}
\widehat{\varphi}_f (\xi ) & = \int_{V (\mathbf{A}_f )} \varphi_f (x) \chi (\langle \xi , x \rangle) dx \\
& = \int_{W_1 '} \left( \int_{W_1} \varphi_f (w+w') dw' \right) \chi (\langle \xi , w \rangle) dw \\
& = 0.
\end{split}
\end{equation*}
La somme
$$\sum_{\xi \in V (\mathbf{Q} ) } \widehat{\varphi}_f (\xi ) \widehat{\psi} (\xi )$$
ne porte donc que sur les $\xi \notin W_1'$ c'est-à-dire ceux tels que $p_{W_1} (\xi) \neq 0$ et le théorème s'en déduit.
\end{proof}
\begin{proposition} \label{P12}
Supposons $\int_{W_1} \varphi_f$ constante égale à $0$. Il existe alors des constantes strictement positives $C$, $\alpha$ et $\beta$ telles que pour tout $t \geq t_0$, la forme différentielle $E_\psi (\varphi_f )$ soit de norme $\leq C e^{- \alpha t^\beta}$ en restriction à
$$\mathcal{G} (\mathbf{Q} ) \left[ \mathfrak{S}_{W_\bullet} (g ,t, \omega) \times \kappa \right] / \mathrm{SO}_n .$$
\end{proposition}
\begin{proof}
La forme différentielle $E_\psi (\varphi_f )$ est définie par l'intégrale absolument convergente
$$E_\psi ( \varphi_f ) = \int_0^{\infty} \theta_{[r]^*\psi} (\varphi_f ) \frac{dr}{r}.$$
La proposition se démontre en décomposant l'intégrale en une somme $\int_0^1 + \int_1^\infty$ et en appliquant à chacune de ces intégrales le lemme \ref{L:thetaSiegel}.
\end{proof}
\section{\'Evaluation de $E_\psi (\varphi_f )$ sur les symboles modulaires} \label{S:7.eval}
Soit $\mathbf{q} = (q_0 , \ldots , q_{k})$ un $(k+1)$-uplet de vecteurs non nuls dans $V(\mathbf{Q})$ avec $k \leq n-1$.
Rappelons que l'on a associé à $\mathbf{q}$ une application continue \eqref{E:appDelta} de la première subdivision barycentrique $\Delta_k ' $ du $k$-simplexe standard vers la compactification de Tits~:
$$\Delta (\mathbf{q}) : \Delta_k ' \to \overline{X}^T.$$
On considère ici l'application
$$\Delta (\mathbf{q}) \times \mathrm{id}_{\mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f)} : \Delta_k ' \times \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f) \to \overline{X}^T \times \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f).$$
Pour tout entier $j \in [0 , k]$ on désigne par $W(\mathbf{q})^{(j)}$ le sous-espace
$$W(\mathbf{q})^{(j)} = \langle q_0 , \ldots , \widehat{q}_j , \ldots , q_{k} \rangle$$
dans $V$.
Soit $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f))$ une fonction de Schwartz telle que pour tout entier $j$ dans $[0 , k]$ on ait
\begin{equation} \label{E:condphi}
\int_{W(\mathbf{q})^{(j)}} \varphi_f =0 \quad \mbox{dans} \quad \mathcal{S} (V (\mathbf{A}_f) / W(\mathbf{q})^{(j)} (\mathbf{A}_f) ).
\end{equation}
La proposition \ref{P33} et la proposition \ref{P12} impliquent que, pour tout sous-ensemble compact $\kappa$ dans
$$\left\{ (z , (g_f , v_f ) ) \in \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f ) \; \left| \; \begin{array}{l}\forall \xi \in V (\mathbf{Q}) \mbox{ tel que } \varphi_f (g_f^{-1} (\xi - v_f)) \neq 0, \\
z \notin \bigcup_{j=0}^k (W(\mathbf{q})^{(j)}_\mathbf{C} + \xi) \end{array} \right. \right\},$$
la forme différentielle fermée $E_\psi (\varphi_f )$ est intégrable sur
$$\Delta (\mathbf{q}) \times \mathrm{id}_{\mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f)} ( \Delta_k ' \times \kappa ).$$
On note
\begin{equation}
E_\psi (\varphi_f , \mathbf{q} ) = (\Delta (\mathbf{q}) \times \mathrm{id} )^* E_\psi (\varphi_f);
\end{equation}
son évaluation en $(g_f , v_f) \in \mathcal{G} (\mathbf{A}_f )$ donne une forme fermée dans
$$A^{2n-1} \left( \Delta_{k}' \times \left( \mathbf{C}^n - \bigcup_{j=0}^k \bigcup_{\xi} (W(\mathbf{q})^{(j)}_\mathbf{C} + \xi) \right) \right),$$
où $\xi$ parcourt l'ensemble des vecteurs de $V(\mathbf{Q})$ tels que $\varphi_f (g_f^{-1} (\xi - v_f )) \neq 0$. Dans la proposition suivante, on calcule son intégrale partielle sur $\Delta_k '$.
Supposons maintenant que $k=n$ et que $\mathbf{q}$ soit constitué de vecteurs linéairement indépendants. Soit $g \in \mathrm{GL}_n (\mathbf{Q})$ tel que $g \cdot \mathbf{e} = \mathbf{q}$. La considération du diagramme commutatif
$$\xymatrixcolsep{5pc}\xymatrix{
\Delta_n ' \times \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f) \ar[d]^{\mathrm{id} \times g \times \mathrm{id}} \ar[r]^{\Delta (\mathbf{e}) \times \mathrm{id}} & \overline{X}^T \times \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f) \ar[d]^{g} \\
\Delta_n ' \times \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f) \ar[r]^{\Delta (\mathbf{q}) \times \mathrm{id}} & \overline{X}^T \times \mathbf{C}^n \times \mathcal{G} (\mathbf{A}_f) }$$
et la formule d'invariance \eqref{E:glnqinv} impliquent que
\begin{equation} \label{E:invformsimpl0}
g^* (E_{\psi} (\varphi_f , \mathbf{q}) (gg_f , gv_f )) = E_\psi ( \varphi_f , \mathbf{e} ) (g_f , v_f ).
\end{equation}
Quitte à remplacer $\varphi_f$ par $\omega (g_f , v_f ) \varphi_f$, les calculs explicites se ramènent au cas où $(g_f , v_f) = (1,0)$. On ne considère dans la suite de ce paragraphe que les formes
$$E_\psi (\varphi_f , \mathbf{q} )^0 = E_\psi (\varphi_f , \mathbf{q} ) (1,0)$$
qui satisfont
\begin{equation} \label{E:invformsimpl}
g^* E_{\psi} ( \omega (g, 0) \varphi_f , \mathbf{q})^0 = E_\psi ( \varphi_f , \mathbf{e} )^0.
\end{equation}
\begin{proposition} \label{P34bis}
Soit $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f))$ une fonction vérifiant \eqref{E:condphi}.
{\rm 1.} Si $\langle q_0 , \ldots , q_{k} \rangle$ est un sous-espace propre de $V$, alors la forme $\int_{\Delta_{k}'} E_\psi (\varphi_f , \mathbf{q} )^0$ est identiquement nulle.
{\rm 2.} Supposons $k=n-1$ et que les vecteurs $q_0 , \ldots , q_{n-1}$ soient linéairement indépendants. On pose $g = (q_0 | \cdots | q_{n-1} ) \in \mathrm{GL}_n (\mathbf{Q})$ et on fixe $\lambda \in \mathbf{Q}^\times$ tel que la matrice $h=\lambda g$ envoie $V(\mathbf{Z}) = \mathbf{Z}^n$ dans $L_{\varphi_f}$. Alors la $n$-forme $\int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} )^0$ est égale à
\begin{multline*}
\sum_{v \in \mathbf{Q}^n / L_{\varphi_f} } \varphi_f (v ) \sum_{\substack{\xi \in \mathbf{Q}^n/\mathbf{Z}^n \\ h \xi = v \ (\mathrm{mod} \ L_{\varphi_f})}} \mathrm{Re} \left( \varepsilon (\ell_1 - \xi_1 ) d\ell_1 \right) \wedge \cdots \wedge \mathrm{Re} \left( \varepsilon (\ell_{n} - \xi_n) d \ell_{n} \right),
\end{multline*}
où $\ell_j$ est la forme linéaire sur $\mathbf{C}^n$, de noyau $W(\mathbf{q})^{(j-1)}$, telle que $h^* \ell_j = e_j^*$.
\end{proposition}
\medskip
\noindent
{\it Remarque.} Le fait que l'expression soit en fait indépendante du choix de $\lambda$ découle des relations de distribution
$$\sum_{j=0}^{m -1} \cot (\pi (z+ j /m )) = m \cot (\pi m z ).$$
\medskip
\begin{proof} 1. Dans ce cas l'image de $\Delta (\mathbf{q})$ est contenue dans le bord de $\overline{X}^T$ et il résulte de l'hypothèse faite sur $\varphi_f$, de la proposition \ref{P33} et de la proposition \ref{P12} que l'intégrale $\int_{\Delta_{k}'} E_\psi (\varphi_f , \mathbf{q} )^0$ est nulle sur tout ouvert relativement compact de
$$\mathbf{C}^n - \bigcup_{j=0}^k \bigcup_{\xi} (W(\mathbf{q})^{(j)}_\mathbf{C} + \xi).$$
2. Supposons donc $k=n-1$ et que les vecteurs $q_0 , \ldots , q_{n-1}$ soient linéairement indépendants.
En notant toujours $g = (q_0 | \cdots | q_{n-1} ) \in \mathrm{GL}_n (\mathbf{Q})$ l'élément (\ref{E:g}), il découle de \eqref{E:invformsimpl} que
\begin{equation*}
\int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} )^0 = (h^{-1})^* \left( \int_{\Delta_{n-1}'} E_\psi ( \varphi_f (h \cdot) , \mathbf{e})^0 \right),
\end{equation*}
où $\mathbf{e} = (e_1 , \ldots , e_n )$ et $h=\lambda g$.
On est donc ramené à
calculer l'intégrale
$$\int_{A\mathrm{SO}_n \mathbf{R}_{>0}} E_\psi (\varphi_f (h \cdot )) (1,0) ,$$
où
$$A=\{ \mathrm{diag}(t_1,\ldots,t_n ) \in \mathrm{SL}_n (\mathbf{R}) \; : \; t_j \in \mathbf{R}_{>0}, \ t_1 \cdots t_n = 1 \}.$$
D'après la remarque à la fin du paragraphe \ref{S:42}, on a
\begin{multline} \label{E:intti}
\int_{A\mathrm{SO}_n \mathbf{R}_{>0}} E_\psi (\varphi_f (h \cdot )) (1,0) \\ = \int_{\{ \mathrm{diag}(t_1,\ldots,t_n) \; : \; t_j \in \mathbf{R}_{>0} \} \mathrm{SO}_n} (t_1 \cdots t_n )^{s} \theta_\varphi (1 , 0 ;\varphi_f (h \cdot)) \ \Big|_{s=0}.
\end{multline}
Or, en restriction à l'ensemble des matrices symétriques diagonales réelles, le fibré en $\mathbf{C}^n$ se scinde {\it métriquement} en une somme directe de $n$ fibrés en droites, correspondant aux coordonnées $(z_j )_{j=1 , \ldots , n}$ de $z$ et la forme $\varphi$ se décompose en le produit de $n$ formes associées à ces fibrés en droites et égales, d'après (\ref{E:phiN1}), à
$$
\varphi^{(j)} = \frac{i}{2\pi} e^{- t_j^2 |z_j |^2 } \left( t_j^2 dz_j \wedge d \overline{z}_j
- t_j^2 (z_j d\overline{z}_j - \overline{z}_j dz_j ) \wedge \frac{dt_j}{t_j} \right), \quad j \in \{ 1 , \ldots , n \}.
$$
Comme $(1 , \xi )^* \widetilde{\varphi} (\xi ) = \varphi$ on obtient que
\begin{multline*}
\int_{\{ \mathrm{diag}(t_1,\ldots,t_n) \; : \; t_j \in \mathbf{R}_{>0} \} \mathrm{SO}_n} (t_1 \cdots t_n )^s \widetilde{\varphi} (\xi) \\ = \frac{(-i)^n}{(4\pi)^n} \Gamma (1+ \frac{s}{2})^n \wedge_{j=1}^n \left( \frac{dz_j}{(z_j - \xi_j) | z_j - \xi_j |^{s} } -\frac{d\overline{z}_j}{\overline{z_j - \xi_j} | z_j - \xi_j |^{s}} \right).
\end{multline*}
L'intégrale \eqref{E:intti} est donc égale à la valeur en $s=0$ de
\begin{equation*}
\Gamma (1+ \frac{s}{2})^n \sum_{\xi \in V(\mathbf{Q})} \varphi_f (h \xi ) \wedge_{j=1}^n \mathrm{Re} \left( \frac{1}{2i\pi} \frac{dz_j}{(z_j - \xi_j) | z_j - \xi_j |^{s} } \right)
\end{equation*}
c'est-à-dire,
\begin{equation*}
\Gamma (1+ \frac{s}{2})^n \sum_{\xi \in V(\mathbf{Q}) / V (\mathbf{Z})} \varphi_f (h \xi) \wedge_{j=1}^n \mathrm{Re} \left( \frac{1}{2i\pi} \sum_{m \in \mathbf{Z}} \frac{dz_j}{(z_j - \xi_j +m) | z_j - \xi_j +m |^{s} } \right) ,
\end{equation*}
où l'on a utilisé que la fonction $\varphi_f ( h \cdot)$ est $V(\mathbf{Z} )$-invariante. Rappelons maintenant que pour tout $z \in \mathbf{C}$, la fonction
$$s \mapsto \frac{1}{2i\pi} \sideset{}{'} \sum_{m \in \mathbf{Z}} \frac{1}{(z+m) |z +m|^{s}}$$
admet un prolongement méromorphe au plan des $s \in \mathbf{C}$, qui est égal à $\varepsilon (z) = \frac{1}{2i}\cot ( \pi z)$ en $s=0$; cf. \cite[VII, \S 8, p.56]{Weil}. On en déduit que l'intégrale \eqref{E:intti}
n'est autre que
$$\sum_{\xi \in V(\mathbf{Q}) / V (\mathbf{Z})} \varphi_f (h \xi) \wedge_{j=1}^n \mathrm{Re} \left(\varepsilon (z_j - \xi_j ) dz_j \right).$$
Finalement, comme $(h^{-1})^* e_{j}^*$ est égale à la forme linéaire $\ell_j$, on conclut que
\begin{multline*}
\begin{split}
\int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} )^0
& = \sum_{\xi \in V(\mathbf{Q}) / V (\mathbf{Z})} \varphi_f (h \xi) \cdot (h^{-1})^* \left( \wedge_{j=1}^n \mathrm{Re} (\varepsilon (z_j - \xi_j ) dz_j ) \right) \\
& = \sum_{\xi \in V(\mathbf{Q}) / V (\mathbf{Z})} \varphi_f (h \xi) \wedge_{j=1}^{n} \mathrm{Re} (\varepsilon ( \ell_j - \xi_{j} ) d\ell_j ) .
\end{split}
\end{multline*}
En rassemblant les vecteurs $\xi$ envoyés par $h$ sur un même vecteur modulo $L_{\varphi_f}$ on obtient la formule annoncée.
\end{proof}
\chapter{Cocycle multiplicatif du groupe rationnel $\mathrm{GL}_n (\mathbf{Q})^+$} \label{S:chap9}
\resettheoremcounters
Au chapitre précédent on a défini une forme $E_\psi (\varphi_f)^0$ représentant une classe de cohomologie équivariante \eqref{E:721} qui, d'après le théorème \ref{T:cocycleM}, induit une classe
\begin{equation*}
S_{\rm mult} [D_{\varphi_f}^0] \in H^{n-1} (\Gamma, \Omega^n_{\rm mer} ( \mathbf{C}^n ) ) .
\end{equation*}
Dans ce chapitre, on détermine explicitement des cocycles qui représentent ces classes de cohomologie. On suit la même démarche qu'au chapitre \ref{S:6} mais en considérant les séries d'Eisenstein $E_\psi (\varphi_f)$ plutôt que la forme $\eta$. Autrement dit plutôt que la distribution ``évaluation en zéro'' on considère cette fois la distribution theta. On travaille uniquement avec la bordification de Tits car il faut ici prendre garde au fait que la forme $E(\varphi_f)$ ne s'étend pas à tout le bord.
\section{Forme simpliciale associée à $E_\psi$} \label{S8.1}
L'application (\ref{E:R}) induit une rétraction
$$[0,1] \times X \times \mathbf{C}^n \to X \times \mathbf{C}^n ,$$
encore notée $R$, de $X \times \mathbf{C}^n$ sur $\{ x_0 \} \times \mathbf{C}^n$.
Soit $\Gamma$ un sous-groupe de congruence dans $\mathrm{SL}_n (\mathbf{Z})$. Il lui correspond un sous-groupe compact ouvert $K$ dans $\mathrm{GL}_n (\mathbf{A}_f )$ tel que
$$\Gamma = K \cap \mathrm{GL}_n (\mathbf{Q})^+ .$$
Dans ce chapitre on prend $L = \mathbf{Z}^n$; c'est un réseau $\Gamma$-invariant dans $V(\mathbf{Q})$.
De la même manière qu'au paragraphe \ref{S:61}, la rétraction $R$ induit une suite d'applications
\begin{equation}
\rho_k : \Delta_k \times E_k\Gamma \times \mathbf{C}^n/ \mathbf{Z}^n \longrightarrow X \times \mathbf{C}^n/ \mathbf{Z}^n .
\end{equation}
\'Etant donné un $(k+1)$-uplet
$$\mathbf{g} = (\gamma_0 , \ldots , \gamma_k ) \in E_k\Gamma$$
et un élément $z \in \mathbf{C}^n/ \mathbf{Z}^n$, l'application $\rho_k ( \cdot , \mathbf{g} , z)$ envoie le simplexe $\Delta_k$ sur le simplexe géodésique dans $X$ de sommets $\gamma_0^{-1} x_0$, $\ldots$, $\gamma_k^{-1} x_0$ défini, par récurrence, en prenant le cône géodésique sur le $(k-1)$-simplexe géodésique de sommets $\gamma_1^{-1} x_0 , \ldots , \gamma_k^{-1} x_0$ depuis $\gamma_0^{-1} x_0$.
La suite $\rho=(\rho_k )$ est constituée d'applications $\Gamma$-équivariantes et induit donc une application
$\Gamma$-équivariante
$$\rho^* : A^\bullet (X \times \mathbf{C}^n / \mathbf{Z}^n ) \to \mathrm{A}^\bullet (E\Gamma \times \mathbf{C}^n /\mathbf{Z}^n ),$$
où l'espace de droite est celui des formes différentielles simpliciales sur la variété simpliciale
\begin{equation} \label{VS1}
E\Gamma \times \mathbf{C}^n / \mathbf{Z}^n .
\end{equation}
Le groupe $\Gamma$ opère (diagonalement) sur \eqref{VS1} par
$$\left( \mathbf{g} , z \right) \stackrel{(h,w)}{\longmapsto} \left( \mathbf{g} h^{-1}, h z \right).$$
Un élément $D \in \mathrm{Div}_\Gamma$ peut-être vu comme une fonction $\Gamma$-invariante sur $\mathbf{C}^n / \mathbf{Z}^n$ à support dans les points de torsion; il lui correspond donc une fonction $\mathcal{K}$-invariante $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f ))$. On a
\begin{equation} \label{E:fibresj}
\mathbf{C}^n / \mathbf{Z}^n - \mathrm{supp} \ D = \mathbf{C}^n/\mathbf{Z}^n - \bigcup_\xi (\xi + \mathbf{Z}^n )/ \mathbf{Z}^n ,
\end{equation}
où $\xi$ parcourt l'ensemble des éléments de $\mathbf{Q}^n / \mathbf{Z}^n$ tels que $\varphi_f (\xi)$ soit non nul.
La proposition suivante découle des définitions.
\begin{proposition} \label{P:50}
La forme simpliciale
$$\mathcal{E}_\psi (\varphi_f ) := \rho^* E_\psi (\varphi_f )^0 \in \mathrm{A}^{2n-1} \left( E \Gamma \times ( \mathbf{C}^n / \mathbf{Z}^n - \mathrm{supp} \ D ) \right)^\Gamma$$
est fermée et représente la classe de cohomologie équivariante $[E_{\psi} (\varphi_f )^{(0)} ]$.
\end{proposition}
L'ouvert \eqref{E:fibresj} étant de dimension cohomologique $n$, il correspond à la classe de cohomologie équivariante $[E_{\psi} (\varphi_f )^{(0)} ]$ une classe de cohomologie dans
$$H^{n-1} ( \Gamma , H^n (( \mathbf{C}^n / \mathbf{Z}^n - \mathrm{supp} \ D )^{(1)}).$$
Il découle même de \cite[Theorem 2.3]{Dupont} que l'intégration sur les $(n-1)$-simplexes associe à la forme simpliciale $\mathcal{E}_\psi (\varphi_f )$ un $(n-1)$-cocycle qui à un $n$-uplet d'éléments de $\Gamma$
associe une $n$-forme fermée sur \eqref{E:fibresj}. Pour obtenir des cocycles à valeurs dans les formes méromorphes on procède comme dans la démonstration du théorème \ref{T:cocycleM}. Il faut pour cela effacer quelques hyperplans de manière à pouvoir invoquer le théorème \ref{P:Brieskorn}. C'est ce que l'on détaille dans le paragraphe qui vient.
\section[Les cocycles $\mathbf{S}_{{\rm mult}, \chi_0}$]{Les cocycles $\mathbf{S}_{{\rm mult}, \chi_0}$, démonstration du théorème \ref{T:mult}} \label{S:3.2.2}
Fixons un morphisme primitif $\chi_0 : \mathbf{C}^n /\mathbf{Z}^n \to \mathbf{C} / \mathbf{Z}$.
Pour tout $(k+1)$-uplet $\mathbf{g} = (\gamma_0 , \ldots , \gamma_k )$ d'éléments de $\Gamma$, on note
\begin{equation} \label{Vchi0S2}
U(\mathbf{g}) = \mathbf{C}^n / \mathbf{Z}^n - \bigcup_{\xi } \bigcup_{i=0}^k (\xi + \mathrm{ker} (\chi_0 \circ \gamma_i) ),
\end{equation}
où $\xi$ décrit les éléments du support de $D$.
Les ouverts $U(\mathbf{g})$ sont des variétés affines. On peut donc leur appliquer le théorème \ref{P:Brieskorn}.
On obtient un cocycle
\begin{equation*}
\mathbf{S}_{{\rm mult}, \chi_0} [D] : \Gamma^n \longrightarrow \Omega_{\rm mer}^n (\mathbf{C}^n / \mathbf{Z}^n )
\end{equation*}
qui représente la classe $S_{\rm mult} [D]$ et est à valeurs dans les formes méromorphes qui sont régulières en dehors des hyperplans affines $\xi + \mathrm{ker} (\chi_0 \circ \gamma )$, avec $\xi \in \mathrm{supp} \ D$ et $\gamma \in \Gamma$. Ce sont les cocycles annoncés dans le théorème \ref{T:mult}; il nous faut encore vérifier les propriétés attendues sous l'action des opérateurs de Hecke.
Considérons donc un sous-monoïde $S$ de $M_n (\mathbf{Z})^\circ$ contenant $\Gamma$. À toute double classe $\Gamma a \Gamma$, avec $a \in S$, il correspond la fonction caractéristique de $KaK$, elle appartient à $\mathcal{H} (\mathrm{GL}_n (\mathbf{A}_f) , K)$.
L'application $\mathrm{GL}_n (\mathbf{A}_f) \to \mathcal{G} (\mathbf{A}_f )$ qui à un élément $g_f$ associe $(g_f , 0)$ induit un plongement
$$K \backslash \mathrm{GL}_n (\mathbf{A}_f) / K \hookrightarrow \mathcal{K} \backslash \mathcal{G} (\mathbf{A}_f ) / \mathcal{K}$$
et donc une inclusion
$$\mathcal{H} (\mathrm{GL}_n (\mathbf{A}_f) , K) \hookrightarrow \mathcal{H} (\mathcal{G} (\mathbf{A}_f ) , \mathcal{K})$$
``extension par $0$''. Dans la suite on identifie une fonction $\phi$ dans $\mathcal{H} (\mathrm{GL}_n (\mathbf{A}_f) , K)$ à son image dans $\mathcal{H} (\mathcal{G} (\mathbf{A}_f ) , \mathcal{K})$.
Il découle alors du \S \ref{algHecke} que $\phi$ induit un opérateur de Hecke $\mathbf{T}_\phi$ sur
$$H^{n-1} (\Gamma , \Omega_{\rm mer}^n (\mathbf{C}^n / \mathbf{Z}^n ));$$
lorsque $\phi$ est la fonction caractéristique de $KaK$, l'opérateur $\mathbf{T}_\phi$ coïncide avec $\mathbf{T}(a)$ du \S~\ref{S:2-2}.
Une fonction $\phi \in \mathcal{H} (\mathrm{GL}_n (\mathbf{A}_f) , K)$ induit aussi un opérateur
\begin{equation} \label{E:applphi}
T_\phi : \mathcal{S} (V(\mathbf{A}_f ))^K \to \mathcal{S} (V(\mathbf{A}_f ))^K
\end{equation}
sur l'espace $\mathcal{S} (V(\mathbf{A}_f ))^K$ des fonctions de Schwartz $K$-invariantes. On a
\begin{equation*}
T_\phi ( \varphi_f) = \sum_{g \in \Gamma \backslash \mathrm{GL}_n (\mathbf{Q}) / \Gamma} \phi (g) \sum_{h \in \Gamma g \Gamma / \Gamma} \varphi_f (h^{-1} \cdot ).
\end{equation*}
La fonction $\phi$ induit finalement une application
\begin{equation} \label{E:applphi}
[\phi ] : \mathrm{Div}_\Gamma \to \mathrm{Div}_\Gamma \quad \mbox{avec} \quad [\phi] = \sum_{g \in \Gamma \backslash \mathrm{GL}_n (\mathbf{Q}) / \Gamma} \phi (g) \sum_{h \in \Gamma g \Gamma / \Gamma} h .
\end{equation}
Lorsque $\phi$ est la fonction caractéristique de $KaK$, on a $[\phi]=[\Gamma a \Gamma]$ (cf. \S~\ref{S:2-2}).
Il résulte des définitions que
\begin{equation} \label{E:DetT}
D_{T_{\phi} \varphi_f} = [\phi]^* D
\end{equation}
et la proposition \ref{P:hecke1} implique~:
\begin{proposition} \label{P:hecke2}
Soit
$\phi \in \mathcal{H} (\mathrm{GL}_n (\mathbf{A}_f) , K)$. On a
\begin{equation}
\mathbf{T}_\phi \left[ \mathbf{S}_{{\rm mult}, \chi_0}[D] \right]= \left[ \mathbf{S}_{{\rm mult}, \chi_0 }[[\phi]^* D] \right] .
\end{equation}
\end{proposition}
Ceci conclut la démonstration du théorème \ref{T:mult}.
\section{Le cocycle $\mathbf{S}^*_{{\rm mult}}$}
Comme dans le cas additif, on peut obtenir un cocycle explicite en appliquant l'argument de l'annexe \ref{A:A} à la forme simpliciale $\mathcal{E}_\psi (\varphi_f )$ et aux ouverts, indexés par les éléments $\mathbf{g} = (\gamma_0 , \ldots , \gamma_k) $ de $E_k \Gamma$ (avec $k \in \mathbf{N}$),
\begin{equation} \label{E:86}
U^* (\mathbf{g}) = \mathbf{C}^n /\mathbf{Z}^n - \bigcup_\xi \bigcup_{i=0}^k (W(\mathbf{q})_{\mathbf{C}}^{(i)} + \xi +\mathbf{Z}^n ) / \mathbf{Z}^n ,
\end{equation}
où $\mathbf{q} = ( \gamma_0^{-1} e_1 , \ldots , \gamma_{k}^{-1} e_1 )$ et $\xi$ parcourt l'ensemble des éléments de $\mathbf{Q}^n / \mathbf{Z}^n$ tels que $\varphi_f (\xi)$ soit non nul.
\subsection{Section simpliciale et homotopie}
Au paragraphe \ref{S:63}, on a défini des applications $\varrho_k$ que l'on restreint maintenant à $E\Gamma$. Pour tout entier $k \in [0 , n-1]$ on note encore
\begin{equation}
\varrho_k : \Delta _k \times [0,1] \times E_k \Gamma \times \mathbf{C}^n \to \overline{X}^T \times \mathbf{C}^n / \mathbf{Z}^n
\end{equation}
les applications induites. Les images des projections dans $\overline{X}^T$ sont cette fois contenues dans la bordification rationnelle de Tits obtenue en n'ajoutant que les sous-groupes paraboliques $Q=Q (W_\bullet )$ associés à des drapeaux de sous-espaces rationnels de $V(\mathbf{Q})= \mathbf{Q}^n$ engendrés par des vecteurs $q=\gamma^{-1} e_1$ avec $\gamma \in \Gamma$.
La proposition \ref{P33} implique que pour tout entier $k \in [0, n-1]$, pour tout $\mathbf{g} \in E_k \Gamma$, pour tout $z \in U^* (\mathbf{g})$ et pour tout réel strictement positif $t$, l'image
$$\varrho_k ( \Delta_k \times [0,1] \times \{ \mathbf{g} \} \times \{z \} ) \subset \overline{X}^T \times \mathbf{C}^n / \mathbf{Z}^n $$
est contenue dans une réunion finie
$$\left( \Omega \cup_{W_\bullet , h , \omega} \overline{\mathfrak{S}_{W_\bullet } (h , t , \omega )} \right) \times \{z \} ,$$
où $\Omega \subset X$ est relativement compact et chaque drapeau $W_\bullet$ est formé de sous-espaces engendrés par certains des vecteurs $\gamma_i^{-1} e_1$ où $\mathbf{g} = (\gamma_0 , \ldots , \gamma_k)$.
La proposition \ref{P12} motive la définition suivante.
\begin{definition} \label{def67}
Soit $\mathcal{S} (V (\mathbf{A}_f))^{\circ}$ le sous-espace de $\mathcal{S} (V(\mathbf{A}_f))$ constitué des fonctions $\varphi_f$ telles que pour tout sous-espace $W$ contenant $e_1$, l'image $\int_W \varphi_f $ de $\varphi_f$ dans $\mathcal{S} (V (\mathbf{A}_f) / W(\mathbf{A}_f))$ est constante égale à $0$.
\end{definition}
\medskip
\noindent
{\it Remarque.} On réserve la notation $\mathcal{S} (V (\mathbf{A}_f))^{0}$ pour l'espace des fonctions $\varphi_f$ dans $\mathcal{S} (V(\mathbf{A}_f))$ telles que $\widehat{\varphi}_f (0) =0$. Noter que puisque $V$ contient $e_1$ on a
$$\mathcal{S} (V (\mathbf{A}_f))^{\circ} \subset \mathcal{S} (V (\mathbf{A}_f))^{0}.$$
\medskip
La forme $E_\psi (\varphi_f )^{(0)}$ est $(\Gamma \ltimes \mathbf{Z}^n)$-invariante et la proposition \ref{P12} implique que si $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f ))^{\circ}$ alors pour tout $\mathbf{g} \in E_k \Gamma$ la restriction de la forme différentielle fermée
$\varrho_k^* E_\psi (\varphi_f )^{(0)}$ à $\Delta_k \times [0,1] \times \mathbf{g} \times U^* ( \mathbf{g})$ est bien définie et fermée.
\begin{definition}
Supposons $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f ))^{\circ}$. Pour tout entier $k \in [0, n-1]$ et pour tout $(k+1)$-uplet $\mathbf{g} = (\gamma_0 , \ldots , \gamma_k) \in E_k \Gamma$, on pose
$$\mathcal{H}_k[\varphi_f] (\gamma_0 , \ldots , \gamma_k ) = \int_{\Delta_k \times [0,1]} \varrho_k^*E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ).$$
C'est une forme différentielle de degré $2n-2-k$ sur $U^* (\mathbf{g})$.
\end{definition}
\subsection{Calcul du cocycle}
En remplaçant l'appel à la proposition \ref{P32} par l'utilisation de la proposition \ref{P12}, la démonstration du théorème \ref{T37} conduit au résultat suivant. On renvoie à l'annexe \ref{A:A} pour les définitions des opérateurs $\delta$ et $d$.
\begin{theorem} \label{T8.8}
Supposons $\varphi_f \in \mathcal{S} (V(\mathbf{A}_f))^{\circ}$. Pour tout entier $k \in [0 , n-1]$ et pour tout $\mathbf{g} = (\gamma_0 , \ldots , \gamma_k) \in E_k \Gamma$,
l'intégrale $\int_{\Delta_k } \mathcal{E}_{\psi} (\varphi_f ) (\mathbf{g})$ est égale à
$$\delta \mathcal{H}_{k-1} [\varphi_f ] (\mathbf{g}) \pm d \mathcal{H}_k [\varphi_f ] (\mathbf{g}), \quad \mbox{si } k < n-1,$$
et
$$\int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} )^{(0)}
+ \delta \mathcal{H}_{n-2} [\varphi_f ] (\mathbf{g}) \pm d \mathcal{H}_{n-1} [\varphi_f ] (\mathbf{g}), \quad \mbox{si } k=n-1,$$
dans $A^{2n-2-k} \left( U^* (\mathbf{g}) ) \right)$ avec toujours $\mathbf{q} = ( \gamma_0^{-1} e_1 , \ldots , \gamma_{k}^{-1} e_1 )$.
\end{theorem}
\begin{proof} Puisque $E_\psi (\varphi_f)^{(0)}$ est fermée on a~:
$$(d_{\Delta_k \times [0,1]} \pm d ) \varrho_k^*E_\psi (\varphi_f )^{(0)} = 0$$
et donc
$$\int_{\Delta_k \times [0,1]} d_{\Delta_k \times [0,1]} \varrho_k^* E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ) \pm d \mathcal{H}_k (\gamma_0 , \ldots , \gamma_k ) =0.$$
Maintenant, d'après le théorème de Stokes on a
\begin{multline*}
\int_{\Delta_k \times [0,1]} d_{\Delta_k \times [0,1]} \varrho_k^* E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ) = \int_{\Delta_k \times \{ 0 \}} \varrho_k^* E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ) \\ + \int_{(\partial \Delta_k) \times [0,1]} \varrho_k^* E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ) - \int_{\Delta_k \times \{ 1 \}} \varrho_k^* E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ).
\end{multline*}
La dernière intégrale est égale à $\int_{\Delta_k '} E_\psi (\varphi_f , \mathbf{q} )^{(0)} $ et est donc nulle si $k < n-1$ d'après la proposition \ref{P34bis}.
Finalement, par définition on a
$$\int_{\Delta_k \times \{ 0 \}} \varrho_k^* E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ) = \int_{\Delta_k} \rho^* E_\psi (\varphi_f ) (\gamma_0 , \ldots , \gamma_k) $$
et
$$\int_{(\partial \Delta_k ) \times [0,1]} \varrho_k^* E_\psi (\varphi_f )^{(0)} (\gamma_0 , \ldots , \gamma_k ) = - \delta \mathcal{H}_{k-1} (\gamma_0 , \ldots , \gamma_k ).$$
\end{proof}
Le théorème précédent motive les définitions suivantes.
\begin{definition}
\'Etant donné une fonction $K$-invariante $\varphi_f \in \mathcal{S} (V(\mathbf{A}_f))^{\circ}$ on désigne par
$$\mathbf{S}_{\rm mult}^* [\varphi_f] : \Gamma^{n} \longrightarrow \Omega^{n}_{\rm mer} (\mathbf{C}^n / \mathbf{Z}^n )$$
l'application qui à un $n$-uplet $(\gamma_0 , \ldots , \gamma_{n-1})$ associe $0$ si les vecteurs $\gamma_j^{-1} e_1$ sont liés et sinon la forme différentielle méromorphe dans $\Omega^n_{\rm mer} (\mathbf{C}^n / \mathbf{Z}^n )$ d'expression
\begin{equation} \label{E:87}
\sum_{v \in \mathbf{Q}^n / \mathbf{Z}^n} \varphi_f (v) \\ \sum_{\substack{\xi \in \mathbf{Q}^n/\mathbf{Z}^n \\ h \xi = v \ (\mathrm{mod} \ \mathbf{Z}^n )}} \varepsilon (\ell_1 - \xi_1 ) \cdots \varepsilon (\ell_{n} - \xi_n) \cdot d\ell_1 \wedge \cdots \wedge d \ell_{n} ,
\end{equation}
où $h = ( \gamma_0^{-1} e_1 | \cdots | \gamma_{n-1}^{-1} e_1)$ et $h^* \ell_j = e_{j}^*$.
\end{definition}
On déduit du théorème \ref{T8.8} le théorème suivant.
\begin{theorem} \label{T8.8cor}
Supposons $\varphi_f \in \mathcal{S} (V(\mathbf{A}_f))^{\circ}$. L'application
$$\mathbf{S}_{\rm mult}^*[\varphi_f] : \Gamma^{n} \longrightarrow \Omega^{n}_{\rm mer} (\mathbf{C}^n / \mathbf{Z}^n )$$
définit un $(n-1)$-cocycle homogène non nul du groupe $\Gamma$. Il représente la même classe de cohomologie que $\mathbf{S}_{\rm mult , \chi_0} [\varphi_f]$.
\end{theorem}
\begin{proof} Il découle du théorème \ref{T8.8} que l'application
$$(\gamma_0 , \ldots , \gamma_{n-1}) \mapsto \int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} )^0$$
définit un $(n-1)$-cocycle à valeurs dans $H^n (U^* (\mathbf{g}))$.
Or, d'après la proposition \ref{P34bis} la $n$-forme $\int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} )^0$ est nulle si les vecteurs $q_j=\gamma_j^{-1} e_1$ sont liés et elle est égale à
\begin{multline} \label{E:8.fd}
\sum_{v \in \mathbf{Q}^n / \mathbf{Z}^n} \varphi_f (v) \\ \sum_{\substack{\xi \in \mathbf{Q}^n/\mathbf{Z}^n \\ h \xi = v \ (\mathrm{mod} \ \mathbf{Z}^n)}} \mathrm{Re} (\varepsilon (\ell_1 - \xi_1 ) d \ell_1 ) \wedge \cdots \wedge \mathrm{Re} (\varepsilon (\ell_{n} - \xi_n) d \ell_{n}) ,
\end{multline}
sinon.
Remarquons maintenant qu'en posant $q=e^{2i\pi z}$, on a
\begin{equation} \label{E:8.cot}
\varepsilon (z) dz = \frac{1}{2i} \cot (\pi z) dz = \frac{dz}{e^{2i\pi z} -1} + \frac12 dz = \frac{1}{2i\pi} \left( \frac{dq}{q-1} - \frac{dq}{2q} \right).
\end{equation}
En particulier les $1$-formes différentielles
$$\mathrm{Re} (\varepsilon (z) dz ) \quad \mbox{ et } \quad \varepsilon (z) dz$$
sur $\mathbf{C} / \mathbf{Z}$ sont cohomologues. On en déduit que la forme \eqref{E:8.fd} est cohomologue à $\mathbf{S}_{\rm mult}^* [\varphi_f] (\gamma_0 , \ldots , \gamma_{n-1})$. On conclut alors la démonstration, en suivant celle du théorème \ref{T:Sa} au paragraphe \ref{S:demTSa}, mais en remplaçant le théorème de Brieskorn par sa version multiplicative, le théorème \ref{P:Brieskorn}. Celui-ci s'applique car les relations de distribution pour $\varepsilon$ impliquent que la forme \eqref{E:8.fd} représente bien une classe dans le sous-espace caractéristique associé à la valeur propre $1$ dans la cohomologie de la fibre.
\end{proof}
La proposition suivante donne une autre expression, parfois plus maniable, du cocycle $\mathbf{S}_{\rm mult}^*$.
\begin{proposition}
Supposons toujours $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f ))^{\circ}$. L'expression \eqref{E:87} est égale à
$$\sum_{v \in \mathbf{Q}^n / \mathbf{Z}^n} \varphi_f (v) \sum_{\substack{\xi \in \mathbf{Q}^n/\mathbf{Z}^n \\ h \xi = v \ (\mathrm{mod} \ \mathbf{Z}^n)}} \frac{d\ell_1 \wedge \cdots \wedge d \ell_{n}}{(e^{2i\pi (\ell_1 - \xi_1 )} -1) \ldots (e^{2i\pi (\ell_n - \xi_n )} -1)}.$$
\end{proposition}
\begin{proof} D'après \eqref{E:8.cot}, il suffit de montrer que pour tout sous-ensemble non vide et propre $J \subset \{ 1 , \ldots , n \}$ on a
\begin{equation} \label{E:8.smoothing}
\sum_{v \in \mathbf{Q}^n / \mathbf{Z}^n} \varphi_f (v) \sum_{\substack{\xi \in \mathbf{Q}^n/\mathbf{Z}^n \\ h \xi = v \ (\mathrm{mod} \ \mathbf{Z}^n )}} \wedge_{j \notin J} \varepsilon (\ell_j - \xi_j ) =0.
\end{equation}
Posons $L = \mathbf{Z}^n$ et fixons un sous-ensemble $J \subset \{ 1 , \ldots , n \}$ non vide et propre de cardinal $k$. Soit $\overline{V}$ le quotient de $V$ par la droite engendrée par les vecteurs $\gamma_j^{-1} e_1$ avec $j \in J$. Désignons par $\overline{L}$ l'image de $L$ dans $\overline{V}$.
En identifiant $\mathbf{Q}^{n-k}$ avec le quotient de $\mathbf{Q}^n$ par l'espace engendré par les vecteurs $e_j$ avec $j \in J$ et $\mathbf{Z}^{n-k}$ avec l'image de $\mathbf{Z}^n$ dans $\mathbf{Q}^{n-k}$, la matrice $h$ induit une application linéaire $\overline{h} : \mathbf{Q}^{n-k} \to \overline{V} $ telle que $\overline{h} (\mathbf{Z}^{n-k} )$ soit contenu dans $\overline{L}$.
Pour tout vecteur $v \in V$ d'image $\overline{v}$ dans $\overline{V}$, la projection de $\mathbf{Q}^{n}$ sur $\mathbf{Q}^{n-k}$ induit alors une application surjective
$$\{ \xi \in \mathbf{Q}^n / \mathbf{Z}^n \; : \; h \xi = v \ (\mathrm{mod} \ L) \} \to \{ \overline{\xi} \in \mathbf{Q}^{n-k}/\mathbf{Z}^{n-k} \; : \; \overline{h} \overline{\xi} = \overline{v} \ (\mathrm{mod} \ \overline{L}) \}$$
dont les fibres ont toutes le même cardinal, égal à $\frac{[L : h (\mathbf{Z}^{n})]}{[\overline{L} : \overline{h} (\mathbf{Z}^{n-k})]}$. Le membre de gauche de \eqref{E:8.smoothing} est donc égal à
\begin{equation*}
\sum_{w\in \overline{V} / \overline{L}} \Big( \sum_{\substack{v \in V/L \\ \overline{v}=w}} \varphi_f (v) \Big)
\frac{[L : h (\mathbf{Z}^{n})]}{[\overline{L} : \overline{h} (\mathbf{Z}^{n-k})]} \sum_{\substack{\overline{\xi} \in \mathbf{Q}^{n-k}/\mathbf{Z}^{n-k} \\ \overline{h} \overline{\xi} = w \ (\mathrm{mod} \ \overline{L})}} \wedge_{j \notin J} \varepsilon (\ell_j - \xi_j ),
\end{equation*}
où $\overline{\xi} = (\xi_j )_{j \notin J}$.
Pour conclure, remarquons que le noyau de la projection $V \to \overline{V}$ contient un vecteur qui est un translaté de $e_1$ par un élément de $\Gamma$. L'image de ce vecteur dans $V(\mathbf{A}_f)$ est donc égale à $e_1$ modulo $K$. Comme la fonction $K$-invariante $\varphi_f$ appartient à $\mathcal{S} (V (\mathbf{A}_f ))^{\circ}$, on en déduit que chaque somme
$$\sum_{\substack{v \in V/L \\ \overline{v}=w}} \varphi_f (v) $$
est égale à zéro et l'on obtient l'identité annoncée \eqref{E:8.smoothing}.
\end{proof}
\subsection{Démonstration du théorème \ref{T:mult2}} \label{S:8.3.3}
Rappelons qu'un élément $D \in \mathrm{Div}_\Gamma$ peut-être vu comme une fonction $\Gamma$-invariante sur $\mathbf{Q}^n / \mathbf{Z}^n$ à support dans un réseau de $\mathbf{Q}^n$. Notons $\varphi_f$ la fonction $K$-invariante correspondante. Sous l'hypothèse que $D \in \mathrm{Div}_\Gamma^{\circ}$ la fonction $\varphi_f$ appartient à $ \mathcal{S} (V (\mathbf{A}_f ))^{\circ}$.
On pose alors
$$\mathbf{S}^*_{\rm mult} [D] = \mathbf{S}^*_{{\rm mult}}[\varphi_f].$$
C'est un $(n-1)$-cocycle de $\Gamma$ à valeurs dans $\Omega^n_{\rm mer} (\mathbf{C}^n / \mathbf{Z}^n)$. Les cocycles $\mathbf{S}^*_{\rm mult} [D]$ et $\mathbf{S}_{{\rm mult} , \chi_0 }[D]$ représentent la même classe de cohomologie puisqu'ils proviennent tous les deux de la restriction de $\mathcal{E}_\psi (\varphi_f)$. Cela démontre le premier point du théorème \ref{T:mult2}.
Le deuxième point du théorème peut se vérifier par un calcul élémentaire; il résulte aussi de la proposition \ref{P:hecke1}. \qed
\medskip
On conclut ce chapitre en remarquant que la proposition \ref{P:DRgen} découle du théorème \ref{T:mult} sauf en ce qui concerne l'intégralité de la classe $d_n [\Phi_\delta ]$. Cette dernière propriété résulte de \cite{Takagi}. Le lecteur attentif aura pourtant noté que la série d'Eisenstein étudiée dans \cite{Takagi} est de degré total $n-1$ alors que la série d'Eisenstein $E_\psi (\varphi_f)$ étudiée ici est de degré $2n-1$. On passe de la deuxième à la première en remarquant que
le fibré $\mathbf{C}^n$ se scinde métriquement en $\mathbf{R}^n \oplus (i \mathbf{R}^n)$ au-dessus de l'espace symétrique associé à $\mathrm{SL}_n (\mathbf{R})$. La forme de Mathai--Quillen $\varphi$ associée au fibré $\mathbf{C}^n$ se décompose alors en le produit $\varphi_{\mathbf{R}^n} \wedge \varphi_{i\mathbf{R}^n}$ de deux formes égales à la forme de Thom (de degré $n$) de \cite[\S 5]{Takagi}. Si l'on applique à $\varphi_{\mathbf{R}^n}$ la distribution theta, qu'on évalue $\varphi_{i\mathbf{R}^n}$ en $0$ et que l'on contracte la forme obtenue à l'aide du multivecteur $\partial_{y_1} \wedge \ldots \wedge \partial_{y_n}$, on obtient la série d'Eisenstein étudiée dans \cite{Takagi}. La propriété d'intégralité annoncée résulte alors de \cite[\S 10 Remark p. 354]{Takagi}.
\medskip
\noindent
{\it Remarque.} L'énoncé de \cite[\S 10 Remark p. 354]{Takagi} comporte malheureusement une erreur. En adoptant les notations de \cite{Takagi}, nous affirmions que pour tout entier strictement positif $m$, la classe $(m^n -1) z(\mathbf{v})$ est $\mathbf{Z}_\ell$-entière pour tout $\ell$ premier à $m$. Toutefois la démonstration requiert que la multiplication par $m$ dans les fibres fixe la section de torsion $\mathbf{v}$, autrement que si $\mathbf{v}$ est associée à un vecteur $v$ dans $\mathbf{Q}^n$ alors $mv=v$ dans $\mathbf{Q}^n / \mathbf{Z}^n$. On ne peut donc pas montrer que $d_n z(\mathbf{v})$ est une classe entière comme annoncé dans la remarque. L'énoncé est d'ailleurs faux même lorsque $n=2$, comme le montre par exemple \cite[\'Equation (11.3)]{Takagi}.
D'un autre côté, la classe $[\Phi_\delta ]$, considérée ici, est obtenue en évaluant le cocycle $\mathbf{S}_{{\rm mult}, e_1^*} [D_\delta ]$ en $0$. La section nulle étant invariante par multiplication par \emph{tous} les entiers strictement positifs $m$, la démonstration de \cite[\S 10 Remark p. 354]{Takagi} implique bien que
la classe $d_n [\Phi_\delta ]$ est entière.
\medskip
\chapter{Cocycle elliptique du groupe rationnel $\mathrm{GL}_n (\mathbf{Q})^+$} \label{S:chap10}
\resettheoremcounters
Dans ce chapitre on explique comment modifier les constructions des deux chapitres précédents pour construire les cocycles elliptiques des théorèmes \ref{T:ell} et~\ref{T:ellbis}. On commence par détailler le quotient adélique avec lequel il nous faut cette fois travailler. On explique ensuite quelles sont les principales différences avec le cas multiplicatif. Elles sont au nombre de trois~:
\begin{enumerate}
\item De la même manière que le cycle $D$ doit être supposé de degré $0$ dans le théorème \ref{T:cocycleE}, on doit supposer $\widehat{\varphi}_f (0)=0$ pour construire le cocycle $\mathbf{S}_{{\rm ell}, \chi}$; voir Théorème \ref{T:9.4}.
\item Pour pouvoir appliquer le théorème \ref{P:Brieskorn} ``à la Orlik--Solomon'', on a besoin d'effacer suffisamment d'hyperplans pour que les fibres soient affines. Cela nous contraint à considérer un $n$-uplet $\chi = (\chi_1 , \ldots , \chi_n )$ de morphismes primitifs; voir \S \ref{S:9.3.3}.
\item Le fait que la série d'Eisenstein $E_1$ ne soit pas périodique nous contraint à considérer la série d'Eisenstein non holomorphe $E_1^*$ lors de la construction de $\mathbf{S}_{{\rm ell}}^*$; voir \S \ref{S:9.4.1}.
\end{enumerate}
\section{Quotients adéliques}
Les quotients adéliques que l'on considère dans ce chapitre sont associés au groupe (algébrique sur $\mathbf{Q}$)
$$\mathcal{G} = (\mathrm{GL}_n \times \mathrm{SL}_2 ) \ltimes M_{n, 2} .$$
\`A l'infini l'espace est
$$S \times \mathcal{H} \times \mathbf{C}^n \cong [(\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R})) \ltimes M_{n, 2} (\mathbf{R})] / \mathrm{SO}_n \times \mathrm{SO}_2,$$
où $\mathcal{H} = \mathrm{SL}_2 (\mathbf{R}) / \mathrm{SO}_2$
est le demi-plan de Poincaré. L'action de $\mathcal{G} (\mathbf{R})$ sur $S \times \mathcal{H} \times \mathbf{C}^n$ se déduit des actions suivantes~:
\begin{itemize}
\item le groupe $\mathrm{SL}_2 (\mathbf{R})$ opère par
$$(gK, \tau , z) \stackrel{B}{\longmapsto} \left( gK , \frac{a\tau+b}{c\tau+d} , \frac{z}{c \tau +d} \right), \quad \mbox{où } B=\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \mathrm{SL}_2 (\mathbf{R} ),$$
\item le groupe $\mathrm{GL}_n (\mathbf{R})$ opère par\footnote{Ici le $z =(z_1 , \ldots , z_n )$ est vu comme un vecteur colonne, l'action $z \mapsto hz$ correspond donc à l'action usuelle sur les vecteur colonnes. On pourrait aussi bien considérer l'action $z \mapsto h^{-\top}z$ mais celle-ci parait plus artificielle, une alternative similaire se présente dans l'étude de la représentation de Weil.}
$$(gK , \tau , z) \stackrel{h}{\longmapsto} (hgK, \tau , hz) \quad \left(h \in \mathrm{GL}_n (\mathbf{R}) \right),$$
et
\item une matrice
$$\left( \begin{array}{cc}
u_1 & v_1 \\
\vdots & \vdots \\
u_n & v_n
\end{array} \right) \in M_{n, 2} (\mathbf{R})$$
opère par translation
$$(gK , \tau , (z_1 , \ldots , z_n)) \mapsto (gK , \tau , (z_1 + u_1 \tau + v_1 , \ldots , z_n +u_n \tau + v_n)).$$
\end{itemize}
Noter que la loi de groupe dans le produit semi-direct $$(\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R})) \ltimes M_{n, 2} (\mathbf{R})$$ est donnée par
$$(g,B , x ) \cdot (g' , B' , x') = (gg' , BB' ,gx' B^{-1} + x ).$$
L'action décrite ci-dessus munit le fibré
\begin{equation} \label{E:9.fibre2}
\xymatrix{
S \times \mathcal{H} \times \mathbf{C}^n \ar[d] \\
S \times \mathcal{H}
}
\end{equation}
d'une structure de fibré $\mathcal{G} (\mathbf{R})$-équivariant.
Les structures rationnelles usuelles sur $\mathrm{GL}_n$, $\mathrm{SL}_2$ et $M_{n,2}$ munissent $\mathcal{G}$
d'une structure de groupe algébrique sur $\mathbf{Q}$. Dans la suite de ce chapitre on identifie $M_{n,2}$ au produit $V^2$ avec $V (\mathbf{Q}) = \mathbf{Q}^n$ et on fixe~:
\begin{itemize}
\item un sous-groupe compact ouvert $K \subset \mathrm{GL}_n (\widehat{\mathbf{Z}})$, et
\item un sous-groupe compact ouvert $L \subset \mathrm{SL}_2 (\widehat{\mathbf{Z}})$.
\end{itemize}
Il correspond à ces données le sous-groupe compact ouvert
$$\mathcal{K} = (K \times L) \ltimes V(\widehat{\mathbf{Z}})^2 \subset \mathcal{G} (\mathbf{A}_f)$$
qui préserve le réseau $V(\widehat{\mathbf{Z}})^2$ dans $V (\mathbf{A}_f )^2$.
Les quotients
$$[\mathcal{G} ] / (L \ltimes V(\widehat{\mathbf{Z}})^2) = \mathcal{G} (\mathbf{Q} ) \backslash \left[ \mathcal{G} (\mathbf{R}) \cdot \mathcal{G} (\mathbf{A}_f ) \right] / (\mathrm{SO}_n \times \mathrm{SO}_2) Z (\mathbf{R})^+ (L \ltimes V(\widehat{\mathbf{Z}})^2)$$
et
$$[\mathcal{G} ] / \mathcal{K}$$
sont des fibrés en groupes au-dessus de respectivement $[\mathrm{GL}_n]$ et $[\mathrm{GL}_n] / K$. Les fibres sont des produits (fibrés) $E^n$ de (familles de) courbes elliptiques
$$E (= E_{L}) = \Lambda \backslash \left[ (\mathcal{H} \times \mathbf{C}) / \mathbf{Z}^2 \right] \quad \mbox{où} \quad \Lambda = L \cap \mathrm{SL}_2 (\mathbf{Z}).$$
On notera simplement $Y$ la base de cette famille de courbes elliptiques; c'est une courbe modulaire $Y = \Lambda \backslash \mathcal{H}$.
Comme dans le cas multiplicatif, on déduit du théorème d'approximation forte que
\begin{equation} \label{E:9.TK}
Z( \mathbf{A}_f ) \backslash [\mathcal{G} ] / \mathcal{K} \simeq \Gamma \backslash (X \times E^n ),
\end{equation}
où $\Gamma = K \cap \mathrm{GL}_n (\mathbf{Q})$; c'est un fibré au-dessus de $\Gamma \backslash X$ que, comme dans le cas multiplicatif, nous noterons $\mathcal{T}_\mathcal{K}$.
\medskip
\noindent
{\it Exemple.} Soit $N$ un entier strictement supérieur à $1$. En pratique on considérera surtout le cas où
$K=K_{0} (N)$ et $L=L_0 (N)$ de sorte que $Y$ soit égale à la courbe modulaire $Y_0 (N)$ et
$$\mathcal{T}_{\mathcal{K}} = \Gamma_0 (N) \backslash \left[ X \times E^n \right].$$
\medskip
\section{Fonctions de Schwartz et cycles}
\subsection{Formes de Mathai--Quillen}
Le fibré \eqref{E:9.fibre2} est naturellement muni d'une orientation et d'une métrique hermitienne $\mathrm{GL}_n (\mathbf{R} ) \times \mathrm{SL}_2 (\mathbf{R})$-invariantes. Le formalisme de Mathai--Quillen s'applique donc encore pour fournir des fonctions de Schwartz $\varphi$ et $\psi$ à l'infini. Les formules sont identiques à ceci près qu'elles dépendent maintenant du paramètre $\tau$.
Au-dessus d'un point
$$(gK, \tau ) \in S \times \mathcal{H}$$
la métrique sur la fibre $\mathbf{C}^n$ de \eqref{E:9.fibre2} est en effet égale à
$$z \mapsto \frac{1}{\mathrm{Im} \ \tau } | g^{-1} z |^2.$$
La construction de Mathai--Quillen expliquée au chapitre \ref{C:4} appliquée au fibré hermitien équivariant \eqref{E:9.fibre2} conduit cette fois à une forme de Thom qui est $(\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R}))$-invariante
\begin{equation} \label{9.varphi}
\varphi \in \mathcal{A}^{2n} (S \times \mathcal{H} \times \mathbf{C}^n)^{\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R})}
\end{equation}
à décroissance rapide dans les fibres. On définit encore
\begin{equation} \label{9.psi}
\psi = \iota_X \varphi \in \mathcal{A}^{2n-1} (S \times \mathcal{H} \times \mathbf{C}^n)^{\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R})}
\end{equation}
mais il n'est plus vrai dans ce contexte que la forme
$$\eta = \int_0^{+\infty} [s]^* \psi \frac{ds}{s} \in A^{2n-1} (X \times \mathcal{H} \times \mathbf{C}^n)^{\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R})}$$
soit fermée; il découle par exemple de \cite[\S 7.4]{Takagi} que déjà dans le cas $n=1$ on a
\begin{equation} \label{9.eta1}
\eta = \frac{1}{8\pi} \left( \frac{d\tau}{y} + \frac{d \overline{\tau}}{y} \right) - \frac{i}{4\pi} \left( \frac{dz}{z} - \frac{d\overline{z}}{\overline{z}} \right)
\end{equation}
dont la différentielle est égale à la forme d'aire $\frac{dx \wedge dy}{4\pi y^2}$ sur $\mathcal{H}$.
On appelle maintenant {\it représentation de Weil} la représentation $\omega$ du groupe $\mathcal{G} (\mathbf{R})$ dans l'espace de Schwartz $\mathcal{S} (M_{n , 2} (\mathbf{R}))$ donnée par
$$\omega (g , B , x) : \mathcal{S} (M_{n , 2} (\mathbf{R})) \longrightarrow \mathcal{S} (M_{n , 2} (\mathbf{R})); \quad \phi \mapsto \left(y \mapsto \phi \left( g^{-1} (y -x ) B \right) \right) ,$$
avec $(g,B,x) \in (\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R})) \ltimes M_{n, 2} (\mathbf{R})$.
Il correspond encore à $\varphi$ et $\psi$ des formes
\begin{equation} \label{9.varphi2}
\widetilde{\varphi} \in \mathcal{A}^{2n} \left( S \times \mathcal{H} \times \mathbf{C}^n ; \mathcal{S} (M_{n , 2} (\mathbf{R})) \right)^{\mathcal{G}(\mathbf{R})}
\end{equation}
et
\begin{equation} \label{9.psi2}
\widetilde{\psi} \in \mathcal{A}^{2n-1} \left( S \times \mathcal{H} \times \mathbf{C}^n ; \mathcal{S} (M_{n , 2} (\mathbf{R})) \right)^{\mathcal{G}(\mathbf{R})}
\end{equation}
qui, après évaluation en la matrice nulle, sont respectivement égales à $\varphi$ et $\psi$. Le lemme \ref{L:convcourant} reste valable sauf le dernier point; les formes
$$\widetilde{[s]^*\varphi} (0) = [s]^* \varphi$$
convergent cette fois uniformément sur tout compact vers une forme non-nulle mais qui reste invariante sous l'action de $\mathcal{G} (\mathbf{R})$.
\subsection{Fonctions de Schwartz aux places finies et cycles invariants}
Soit
$$\mathcal{S} (M_{n,2} (\mathbf{A}_f)) = \mathcal{S} ( V (\mathbf{A}_f )^2 )$$
l'espace de Schwartz des fonctions $\varphi_f : M_{n,2} (\mathbf{A}_f) \to \mathbf{C}$ localement constantes et à support compact. Le groupe $\mathcal{G} (\mathbf{A}_f)$ opère sur $\mathcal{S} (M_{n,2} (\mathbf{A}_f ))$ par la représentation de Weil
$$\omega (g , B ,x) : \mathcal{S} (M_{n,2} (\mathbf{A}_f )) \to \mathcal{S} (M_{n,2} (\mathbf{A}_f )); \quad \phi \mapsto \left( w \mapsto \phi (g^{-1} (w-x) B \right).$$
Considérons maintenant l'espace $C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right)$ des fonctions continues localement constantes; on fait opérer le groupe $\mathcal{G} (\mathbf{A}_f)$ sur $C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right)$ par la représentation régulière à droite~:
$$( (h ,y , C) \cdot f) (g ,x , B ) = f( g h , g yB^{-1} + x , BC ).$$
L'application
\begin{equation}
\mathcal{S} (M_{n,2} (\mathbf{A}_f )) \to \mathcal{G} (\mathbf{A}_f); \quad \phi \mapsto f_\phi : ( (g, x ,B ) \mapsto \phi (-g^{-1} x B))
\end{equation}
est $\mathcal{G} (\mathbf{A}_f)$-équivariante relativement aux deux actions définies ci-dessus.
\begin{definition}
Soit $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f )^2)$ une fonction de Schwartz $\mathcal{K}$-invariante.
\begin{itemize}
\item Soit $D_{\varphi_f}$, resp. $D_{\varphi_f , K}$, l'image du cycle
$$\mathcal{G} (\mathbf{Q} ) \left[ (\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R} )) \cdot \mathrm{supp} (f_{\varphi_f} ) \right] \to [\mathcal{G}] /(L \ltimes V(\mathbf{Z} )^2) ,$$
resp.
$$\mathcal{G} (\mathbf{Q} ) \left[ (\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R} )) \cdot \mathrm{supp} (f_{\varphi_f} ) \right] \to [\mathcal{G}]/ \mathcal{K},$$
induite par l'inclusion du support de $f_{\varphi_f}$ dans $\mathcal{G} (\mathbf{A}_f)$.
\item Soit
$$U_{\varphi_f} \subset [\mathcal{G}] /(L \ltimes V(\mathbf{Z} )^2), \quad \mbox{resp.} \quad U_{\varphi_f, K} \subset [\mathcal{G}]/ \mathcal{K},$$
le complémentaire de $D_{\varphi_f}$, resp. $D_{\varphi_f , K}$.
\end{itemize}
\end{definition}
Comme dans le cas multiplicatif, {\it via} l'isomorphisme \eqref{E:9.TK} la projection de $D_{\varphi_f , K} \subset [\mathcal{G}]/ \mathcal{K}$ est égale à la réunion finie
\begin{equation} \label{E:9.9}
\bigcup_\xi \Gamma \backslash (X \times \{ [\tau , u \tau + v] \in E^n \; : \; \tau \in \mathcal{H} \} ),
\end{equation}
où $\xi = (u , v)$ parcourt les éléments de $V(\mathbf{Q})^2 / V (\mathbf{Z})^2$ tels que $\varphi_f ( \xi )$ soit non nul.
\medskip
L'espace $D_{\varphi_f}$ est donc un revêtement fini de $[\mathrm{GL}_n] \times Y$ et $\varphi_f$ induit une fonction localement constante sur $D_{\varphi_f}$ c'est-à-dire un élément de $H^0 (D_{\varphi_f })$.
Maintenant, l'isomorphisme de Thom implique que l'on a~:
\begin{equation} \label{E:thom}
H^0 (D_{\varphi_f }) \stackrel{\sim}{\longrightarrow} H^{2n} \left( [\mathcal{G}]/ (L \ltimes V(\mathbf{Z} )^2) , U_{\varphi_f } \right);
\end{equation}
on note
$$[\varphi_f] \in H^{2n} \left( [\mathcal{G}]/ (L \ltimes V(\mathbf{Z} )^2) , U_{\varphi_f } \right)$$
l'image de $\varphi_f$; cette classe est $K$-invariante et l'on désigne par $[\varphi_f]_K$ son image dans
$$H^{2n} \left( [\mathcal{G}]/ \mathcal{K}, U_{\varphi_f , K } \right).$$
En reprenant la démonstration du lemme \ref{L:40} on obtient le lemme analogue suivant.
\begin{lemma} \label{L:9.40}
Supposons $\widehat{\varphi}_f (0) =0$. Alors, l'application degré
$$H^0 (D_{\varphi_f }) \to \mathbf{Z}$$
est égale à $0$.
\end{lemma}
Suivant la définition \ref{def67} on isole finalement pour la suite un espace privilégié de fonctions de Schwartz sur $M_{n , 2} (\mathbf{A}_f)$.
\begin{definition}
Soit $\mathcal{S} (M_{n , 2} (\mathbf{A}_f ))^{\circ}$ le sous-espace de $\mathcal{S} (M_{n , 2} (\mathbf{A}_f))$ constitué des fonctions de la forme
$$\mathbf{1}_{V (\widehat{\mathbf{Z}})} \oplus \overline{\varphi}_f \in \mathcal{S} (V (\mathbf{A}_f) \oplus V (\mathbf{A}_f)) = \mathcal{S} (M_{n,2} (\mathbf{A}_f )),$$
où $\overline{\varphi}_f \in \mathcal{S} (V (\mathbf{A}_f))^\circ$ est $(K \ltimes V (\widehat{\mathbf{Z}}))$-invariante.
\end{definition}
\section[Séries theta et séries d'Eisenstein adéliques]{Séries theta et séries d'Eisenstein adéliques; \\ démonstration du théorème \ref{T:ell}}
\subsection{Séries theta adéliques}
Comme dans le cas multiplicatif, on associe à toute fonction $\varphi_f \in \mathcal{S} (M_{n , 2} (\mathbf{A}_f))$ des formes différentielles
$$\widetilde{\varphi} \otimes \varphi_f \quad \mbox{et} \quad \widetilde{\psi} \otimes \varphi_f \in A^{\bullet} \left( S \times \mathcal{H} \times \mathbf{C}^n , \mathcal{S} (M_{n,2} (\mathbf{A} )) \right)^{(\mathrm{GL}_n (\mathbf{R}) \times \mathrm{SL}_2 (\mathbf{R})) \ltimes \mathbf{C}^n}.$$
En appliquant la distribution theta dans les fibres, on obtient alors une application
\begin{equation} \label{9.appl-theta}
\theta_\varphi \quad \mbox{et} \quad \theta_\psi : \mathcal{S} (M_{n , 2} (\widehat{\mathbf{Z}})) \longrightarrow \left[ A^{\bullet} (S \times \mathcal{H} \times \mathbf{C}^n) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} )}
\end{equation}
définie cette fois par
\begin{equation} \label{9.appl-theta2}
\begin{split}
\theta_\varphi^* (g_f , B_f , x_f ; \varphi_f ) & = \sum_{\xi \in M_{n,2} (\mathbf{Q} )}\widetilde{\varphi} (\xi ) (\omega (g_f , B_f , x_f ) \varphi_f ) (\xi ) \\
& = \sum_{\xi \in M_{n,2} (\mathbf{Q} )} \varphi_f \left( g_f^{-1} (\xi -x_f ) B_f \right) \widetilde{\varphi} (\xi )
\end{split}
\end{equation}
et de même pour $\theta_\psi$.
Rappelons que le groupe $\mathcal{G} (\mathbf{R})$ opère naturellement sur $S \times \mathcal{H} \times \mathbf{C}^n$; étant donné un élément $(g,B,x) \in \mathcal{G} (\mathbf{R})$ et une forme $\alpha \in A^{\bullet} (S \times \mathcal{H} \times \mathbf{C}^n)$ on note $(g,B,x)^* \alpha$ le tiré en arrière de $\alpha$ par l'application
$$(g,B,x) : S \times \mathcal{H} \times \mathbf{C}^n \to S \times \mathcal{H} \times \mathbf{C}^n.$$
L'invariance sous le groupe $\mathcal{G}(\mathbf{Q})$ dans \eqref{9.appl-theta} signifie alors que pour tout élément $(g,B,x) \in \mathcal{G} (\mathbf{Q})$ on a~:
$$(g,B,x)^*\theta_\varphi^*(g g_f , BB_f , gx_f B^{-1} + x ;\varphi_f)=\theta_\varphi^*(g_f, B_f , x_f ; \varphi_f);$$
ce qui découle de la $\mathcal{G} (\mathbf{R})$-invariance of $\widetilde{\varphi}$, cf. \eqref{9.varphi2}.
L'application $\theta_\varphi$ entrelace les actions naturelles de $\mathcal{G} (\mathbf{A}_f)$ des deux côtés. En particulier, en supposant $\varphi_f$ invariante sous le compact ouvert $\mathcal{K}$ on obtient des formes différentielles
\begin{multline*}
\theta_\varphi (\varphi_f ) \quad \mbox{et} \quad \theta_\psi (\varphi_f ) \in \left[ A^{\bullet} (S \times \mathcal{H} \times \mathbf{C}^n) \otimes C^\infty \left( \mathcal{G} (\mathbf{A}_f ) \right) \right]^{\mathcal{G} (\mathbf{Q} ) \times \mathcal{K}} \\ = A^\bullet (S \times \mathcal{H} \times \mathbf{C}^n)^{(\Gamma \times \Lambda) \ltimes M_{n,2} (\mathbf{Z} )}
\end{multline*}
autrement dit des formes différentielles sur $\Gamma \backslash (S \times E^n)$.
Comme au \S \ref{algHecke}, les séries theta $\theta_\varphi$ et $\theta_\psi$ sont équivariantes relativement aux actions naturelles des opérateurs de Hecke et, comme au \S \ref{cohomClass}, la forme différentielle $\theta_\varphi (\varphi_f)$ est fermée et représente $[\varphi_f]_K$.
Finalement les propriétés de décroissance des formes $\theta_{[r]^* \varphi} (\varphi_f )$ sont analogues à celles décrites dans le lemme \ref{L:theta-asympt} à ceci près que cette fois $\theta_{[r]^* \varphi} (\varphi_f )$ ne tend pas nécessairement vers $0$ avec $r$. C'est pour garantir cela que nous supposerons dorénavant que $\widehat{\varphi}_f (0) = 0$; en procédant comme dans la démonstration du lemme \ref{L:thetaSiegel}, il découle en effet alors de la formule de Poisson que sous cette hypothèse $\theta_{[r]^* \varphi} (\varphi_f )$ tend vers $0$ avec $r$.
\subsection{Séries d'Eisenstein adéliques}
Soit toujours $\varphi_f \in \mathcal{S} (M_{n , 2} (\mathbf{A}_f ))$ une fonction $\mathcal{K}$-invariante dont on supposera de plus qu'elle vérifie $\widehat{\varphi}_f (0) = 0$. Comme au \S \ref{SEA7} on peut alors associer à $\varphi_f$ les séries d'Eisenstein
$$E_\varphi (\varphi_f , s) = \int_0^\infty r^s \theta_{[r]^* \varphi} (\varphi_f ) \frac{dr}{r} \quad \mbox{et} \quad E_\psi (\varphi_f , s) = \int_0^\infty r^s \theta_{[r]^* \psi} (\varphi_f ) \frac{dr}{r};$$
ce sont des formes différentielles sur la préimage de $U_{\varphi_f}$ dans $S \times \mathcal{H} \times \mathbf{C}^n$ qui sont bien définies, et invariantes par l'action du centre $Z(\mathbf{R})^+$, en $s=0$. On pose alors
\begin{equation}
E_\psi (\varphi_f ) = E_\psi (\varphi_f , 0) \in A^{2n-1} \left( [\mathcal{G}]/ \mathcal{K} - D_{\varphi_f , K} \right).
\end{equation}
Comme dans le cas multiplicatif, on obtient le théorème suivant.
\begin{theorem} \label{T:9.4}
Supposons $\widehat{\varphi}_f (0) =0$. Alors la forme différentielle
$$E_\psi (\varphi_f ) \in A^{2n-1} \left( [\mathcal{G}]/ \mathcal{K} - D_{\varphi_f , K} \right)$$
est \emph{fermée} et représente une classe de cohomologie qui relève la classe $[\varphi_f]_K$ dans la suite exacte longue
\begin{multline*}
\ldots \to H^{2n-1} \left( [\mathcal{G}]/ \mathcal{K} - D_{\varphi_f , K} \right) \\ \to H^{2n} \left( [\mathcal{G}]/ \mathcal{K} , [\mathcal{G}]/ \mathcal{K} - D_{\varphi_f , K} \right) \to H^{2n} \left( [\mathcal{G}]/ \mathcal{K} \right) \to \ldots
\end{multline*}
\end{theorem}
Noter que, sous l'hypothèse $\widehat{\varphi}_f (0) =0$, le degré de $D_{\varphi_f , K}$ est nul et l'image de $[\varphi_f]_K$ dans $H^{2n} \left( [\mathcal{G}]/ \mathcal{K} \right)$ est égale à $0$.
\medskip
\subsection{Démonstration du théorème \ref{T:ell}} \label{S:9.3.3}
Il correspond à un sous-groupe de congruence $\Gamma$ dans $\mathrm{SL}_n (\mathbf{Z})$ un sous-groupe compact ouvert $K$ dans $\mathrm{GL}_n (\widehat{\mathbf{Z}})$ tel que $K \cap \mathrm{SL}_n (\mathbf{Z}) = \Gamma$.
Un élément $D \in \mathrm{Div}_\Gamma$ peut-être vu comme une fonction $\Gamma$-invariante sur $E^n$ à support dans les points de torsion de $E^n$; il lui correspond donc une fonction $\mathcal{K}$-invariante $\varphi_f \in \mathcal{S} (M_{n , 2} (\mathbf{A}_f ))$. Si de plus $D$ est de degré $0$ alors $\widehat{\varphi}_f (0) = 0$.
En procédant comme au \S \ref{S8.1} on associe à $E_\psi (\varphi_f )$ une forme simpliciale
$$\mathcal{E}_\psi (\varphi_f ) \in \mathrm{A}^{2n-1} \left( E\Gamma \times (E^n - \mathrm{supp} \ D) \right)^{\Gamma}$$
qui est fermée et représente la même classe de cohomologie équivariante que $E_{\psi} (\varphi_f )$.
Fixons $n$ morphismes primitifs linéairement indépendants
$$\chi_1 , \ldots , \chi_n : \mathbf{Z}^n \to \mathbf{Z}.$$
On note encore $\chi_j : \mathbf{Q}^n \to \mathbf{Q}$ les formes linéaires correspondantes et $\chi_j : E^n \to E$ les morphismes primitifs qu'ils induisent. On pose
$$\chi = (\chi_1 , \ldots , \chi_n).$$
Suivant la démonstration du théorème \ref{T:cocycleE} on restreint maintenant la forme simpliciale $\mathcal{E}_\psi (\varphi_f )$ aux ouverts de $E^n$, indexés par les éléments $\mathbf{g} = (\gamma_0 , \ldots , \gamma_k)$ de $E_k \Gamma$,
\begin{equation} \label{E:9.860}
U (\mathbf{g}) = E^n - \bigcup_\xi \bigcup_{i=1}^n \bigcup_{j=0}^k [\mathrm{ker} (\chi_i \circ \gamma_j) +\xi ],
\end{equation}
où $\xi$ parcourt l'ensemble des éléments du support de $D$.
Il découle du lemme \ref{affineb} que les variétés \eqref{E:9.860} sont affines. On peut donc appliquer le théorème \ref{P:Brieskorn}.
Comme dans la démonstration du théorème \ref{T:cocycleE}, l'argument expliqué dans l'annexe \ref{A:A} permet alors d'associer à la forme simpliciale $\mathcal{E}_\psi (\varphi_f )$ un $(n-1)$-cocycle homogène
\begin{equation}
\mathbf{S}_{{\rm ell}, \chi} [D] \in C^{n-1} \left( \Gamma , \Omega_{\rm mer} (E^n ) \right)^{\Gamma}
\end{equation}
qui représente la classe $S_{\rm mult} [D]$ et qui est à valeurs dans les formes méromorphes régulières en dehors des hyperplans affines $\mathrm{ker} (\chi_i \circ g) +\xi$, avec $\xi$ dans le support de $D$ et $g$ dans $\Gamma$. Cela démontre les points (1), (2) et (3) du théorème \ref{T:ell}.
Comme dans le cas multiplicatif, l'application $\mathrm{GL}_n (\mathbf{A}_f) \to \mathcal{G} (\mathbf{A}_f )$ induit une inclusion entre algèbres de Hecke
$$\mathcal{H} (\mathrm{GL}_n (\mathbf{A}_f) , K) \hookrightarrow \mathcal{H} (\mathcal{G} (\mathbf{A}_f ) , \mathcal{K})$$
et à une fonction $\phi \in \mathcal{H} (\mathrm{GL}_n (\mathbf{A}_f) , K)$ on associe un opérateur de Hecke $\mathbf{T}_\phi$. L'analogue de la proposition \ref{P:hecke2} se démontre de la même manière, de sorte que
\begin{equation}
\mathbf{T}_\phi \left[ \mathbf{S}_{{\rm ell}, \chi}[\varphi_f] \right]= \left[ \mathbf{S}_{{\rm ell}, \chi }[ T_{\phi} \varphi_f] \right] .
\end{equation}
En prenant pour $\phi$ la fonction caractéristique d'une double classe $KaK$ avec $a$ dans $M_n (\mathbf{Z}) \cap \mathrm{GL}_n (\mathbf{Q})$, l'opérateur $\mathbf{T}_\phi$ est égal à l'opérateur $\mathbf{T} (a)$ du chapitre \ref{C:2}.
L'unicité dans le théorème \ref{T:cocycleE} ne nécessite plus de passer au quotient par les formes invariantes, l'analogue de la proposition \ref{P:hecke2} devient alors
\begin{equation*}
\mathbf{T} (a) \left[ \mathbf{S}_{{\rm ell}, \chi}[D] \right] = [ \mathbf{S}_{{\rm ell}, \chi}[[\Gamma a \Gamma ]^* D] ] \quad \mbox{dans} \quad H^{n-1} (\Gamma , \Omega^n_{\rm mer} (E^n)),
\end{equation*}
ce qui démontre le point (4) du théorème \ref{T:ell}.
Finalement, l'application $\mathrm{SL}_2 (\mathbf{A}_f) \to \mathcal{G} (\mathbf{A}_f )$ induit une inclusion entre algèbres de Hecke
$$\mathcal{H} (\mathrm{SL}_2 (\mathbf{A}_f) , L) \hookrightarrow \mathcal{H} (\mathcal{G} (\mathbf{A}_f ) , \mathcal{K}).$$
Il lui correspond une deuxième famille d'opérateurs de Hecke qui contiennent en particulier ceux notés $T(b)$ au chapitre \ref{C:2}. Le point (5) du théorème \ref{T:ell} se démontre alors de la même manière que le point (4). \qed
\section[\'Evaluation sur les symboles modulaires]{\'Evaluation sur les symboles modulaires et \\ démonstration du théorème \ref{T:ellbis}}
L'étude du comportement à l'infini des séries d'Eisenstein $E_\psi (\varphi_f)$ est similaire à ce que l'on a fait dans le cas multiplicatif au \S \ref{S:7.eval}. On ne détaille pas plus, on calcule par contre l'intégrale de $E_\psi (\varphi_f)$ sur les symboles modulaires.
\subsection{\'Evaluation de $E_\psi (\varphi_f)$ sur les symboles modulaires} \label{S:9.4.1}
Soit
$$\mathbf{q} = (q_0 , \ldots , q_k)$$
un $(k+1)$-uplet de vecteurs non nuls dans $V(\mathbf{Q})$ avec $k \leq n-1$.
Soit $\overline{\varphi}_f \in \mathcal{S} (V(\mathbf{A}_f ))$ une fonction de Schwartz $(K \ltimes V( \widehat{\mathbf{Z}} ))$-invariante telle que pour tout entier $j$ dans $[0 , k]$ on ait
\begin{equation} \label{E:condphi2}
\int_{W(\mathbf{q})^{(j)}} \overline{\varphi}_f =0 \quad \mbox{dans} \quad \mathcal{S} (V (\mathbf{A}_f) / W(\mathbf{q})^{(j)} (\mathbf{A}_f) ).
\end{equation}
Soit
$$\varphi_f = \mathbf{1}_{V (\widehat{\mathbf{Z}})} \oplus \overline{\varphi}_f \in \mathcal{S} (V (\mathbf{A}_f) \oplus V (\mathbf{A}_f)) = \mathcal{S} (M_{n,2} (\mathbf{A}_f )).$$
On considère cette fois l'application
$$\Delta (\mathbf{q}) \times \mathrm{id}_{\mathcal{H} \times \mathbf{C}^n} : \Delta_k ' \times \mathcal{H} \times \mathbf{C}^n \to \overline{X}^T \times \mathcal{H} \times \mathbf{C}^n$$
et l'on note
\begin{equation}
E_\psi (\varphi_f , \mathbf{q} ) = (\Delta (\mathbf{q}) \times \mathrm{id} )^* E_\psi (\varphi_f).
\end{equation}
On obtient ainsi une forme fermée dans
$$A^{2n-1} \Big( \Delta_{k}' \times \big( E^n - \bigcup_{j=0}^k \bigcup_{\xi} (H_\mathbf{q}^{(j)} + \xi) \big) \Big),$$
où $H_\mathbf{q}^{(j)}$ désigne l'image de
$$\mathcal{H} \times \langle q_0 , \ldots , \widehat{q_j} , \ldots , q_k \rangle_\mathbf{C} \subset \mathcal{H} \times \mathbf{C}^n$$
dans $E^n$, et $\xi$ est comme dans \eqref{E:9.9}
Il s'agit maintenant de calculer les intégrales $\int_{\Delta_{k}'} E_\psi (\varphi_f , \mathbf{q} )$; ce sont des formes différentielles sur des ouverts de $E^n$. Pour simplifier on se contente de calculer, dans la proposition suivante, la restriction de ces formes aux fibres de $E^n \to Y$.
\begin{proposition} \label{9.P34bis}
1. Si $\langle q_0 , \ldots , q_{k} \rangle$ est un sous-espace propre de $V$, alors la forme $\int_{\Delta_{k}'} E_\psi (\varphi_f , \mathbf{q} )$ est identiquement nulle.
2. Supposons $k=n-1$ et que les vecteurs $q_0 , \ldots , q_{n-1}$ soient linéairement indépendants. On pose $g = (q_0 | \cdots | q_{n-1} ) \in \mathrm{GL}_n (\mathbf{Q})$ et l'on fixe $\lambda \in \mathbf{Q}^\times$ tel que la matrice $h=\lambda g$ soit entière. Alors, en restriction à la fibre $E_\tau^n$ au-dessus d'un point $[\tau] \in Y$, la $n$-forme $\int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} )$ est égale à
\begin{multline*}
\frac{1}{\det h} \sum_{v \in V(\mathbf{Q}) / V(\mathbf{Z}) } \overline{\varphi}_f (v ) \cdot \\ \sum_{\substack{\xi \in E^n \\ h \xi = v }} \mathrm{Re} \left( E_1 ( \tau , \ell_1 - \xi_{1} ) dz_1 \right) \wedge \cdots \wedge \mathrm{Re} \left( E_1 (\tau , \ell_{n} - \xi_{n} ) dz_n \right) ,
\end{multline*}
où un vecteur $v \in V(\mathbf{Q}) / V(\mathbf{Z}) = \mathbf{Q}^n / \mathbf{Z}^n$ est identifié à un élément de la courbe elliptique $E_\tau^n = \mathbf{C}^n / (\tau \mathbf{Z}^n + \mathbf{Z}^n )$ et $\ell_j$ est la forme linéaire sur $\mathbf{C}^n$ caractérisée par $h^* \ell_j = e_j^*$.
\end{proposition}
\begin{proof} On explique comment modifier la démonstration de la proposition \ref{P34bis}. La première partie est identique. Supposons donc $k=n-1$ et que les vecteurs $q_0 , \ldots , q_{n-1}$ sont linéairement indépendants. La propriété d'invariance (\ref{E:invformsimpl}) s'étend naturellement et implique cette fois encore que
\begin{equation} \label{E:9.20-}
\int_{\Delta_{n-1}'} E_\psi (\varphi_f , \mathbf{q} ) = (h^{-1})^* \left( \int_{\Delta_{n-1}'} E_\psi ( \varphi_f (h \cdot) , \mathbf{e}) \right).
\end{equation}
On en est donc réduit à calculer l'intégrale
\begin{multline} \label{9.E:intti}
\int_{A\mathrm{SO}_n \mathbf{R}_{>0}} E_\psi (\varphi_f (h \cdot )) \\ = \int_{\{ \mathrm{diag}(t_1,\ldots,t_n) \; : \; t_j \in \mathbf{R}_{>0} \} \mathrm{SO}_n} (t_1 \cdots t_n )^{s} \theta_\varphi (\varphi_f (h \cdot) (1 , 0) \ \Big|_{s=0}.
\end{multline}
En restriction à l'ensemble des matrices symétriques diagonales réelles, le fibré en $\mathbf{C}^n$ se scinde encore {\it métriquement} en une somme directe de $n$ fibrés en droites, correspondant aux coordonnées $(z_j )_{j=1 , \ldots , n}$ de $z$ et la forme $\varphi$ se décompose en le produit de $n$ formes associées à ces fibrés en droites. Ces formes font cette fois intervenir la variable $\tau$ (cf. \cite[\S 6.1]{Takagi}) mais d'après \eqref{9.eta1}, en restriction à une fibre $E_\tau^n$, on a
\begin{multline*}
\int_{\{ \mathrm{diag}(t_1,\ldots,t_n) \; : \; t_j \in \mathbf{R}_{>0} \} \mathrm{SO}_n} (t_1 \cdots t_n )^s \widetilde{\varphi} (\xi) \\ = \frac{(-i)^n}{(4\pi)^n} \Gamma (1+ \frac{s}{2})^n \wedge_{j=1}^n \left( \frac{dz_j}{(z_j - \xi_j) | z_j - \xi_j |^{s} } -\frac{d\overline{z}_j}{\overline{z_j - \xi_j} | z_j - \xi_j |^{s}} \right).
\end{multline*}
où $\xi \in M_{n,2} (\mathbf{Q}) = V (\mathbf{Q})^2$ est identifié à un élément de $E_\tau^n$ {\it via} l'application
$$V(\mathbf{Q})^2 \to E_\tau^n; \quad (u,v) \mapsto \tau u + v.$$
L'intégrale \eqref{9.E:intti} est donc égale à
\begin{multline} \label{9.int}
\Gamma (1+ \frac{s}{2})^n \sum_{\xi \in M_{n,2} (\mathbf{Q})} \varphi_f (h \xi ) \wedge_{j=1}^n \mathrm{Re} \left( \frac{1}{2i\pi} \frac{dz_j}{(z_j - \xi_j) | z_j - \xi_j |^{s} } \right) \\
= \Gamma (1+ \frac{s}{2})^n \sum_{\xi \in M_{n,2} (\mathbf{Q}) / M_{n,2} (\mathbf{Z})} \varphi_f (h \xi) \wedge_{j=1}^n \mathrm{Re} \left( K_1 (z_j , 0 , 1+s/2) dz_j \right),
\end{multline}
où $K_1$ est la série de Eisenstein--Kronecker \cite[Chap. VIII, \ (27)]{Weil} définie, pour $\mathrm{Re} (s) > 3/2$, par\footnote{On prendra garde au fait qu'on a ajouté un facteur $1/2i\pi$.}
$$K_1 (z,0, s) = \frac{1}{2i \pi} \sideset{}{'} \sum_{w \in \mathbf{Z} \tau + \mathbf{Z}} (\overline{z} + \overline{w}) | z + w|^{-2s}.$$
Il découle de \cite[p. 81]{Weil} que pour tout $z \in \mathbf{C}$, cette fonction
admet un prolongement méromorphe au plan des $s \in \mathbf{C}$ qui, en $s=1$, est égal à la série d'Eisenstein {\it non-holomorphe}
\begin{equation} \label{E*E}
E_1^* (\tau , z) = E_1 (\tau , z) + \frac{1}{y} \mathrm{Im} (z),
\end{equation}
où $y= \mathrm{Im} (\tau)$. Noter que la série $E_1^* (\tau , z)$ est $(\mathbf{Z} \tau + \mathbf{Z})$-périodique en $z$ et modulaire de poids $1$.
En prenant $s=0$ dans l'expression \eqref{9.int} on obtient que
$$\int_{\Delta_{n-1}'} E_\psi ( \varphi_f (h \cdot) , \mathbf{e}) = \sum_{\xi \in M_{n,2} (\mathbf{Q}) / M_{n,2} (\mathbf{Z})} \varphi_f (h \xi) \wedge_{j=1}^n \mathrm{Re} \left( E_1^* (\tau , z_j - \xi_j ) dz_j \right) $$
et donc que l'intégrale \eqref{E:9.20-} est égale à
\begin{multline*}
\sum_{\xi \in M_{n,2} (\mathbf{Q}) / M_{n,2} (\mathbf{Z})} \varphi_f (h \xi) \cdot (h^{-1})^* \left( \wedge_{j=1}^n \mathrm{Re} \left( E_1^* (\tau , z_j - \xi_j ) dz_j \right) \right) \\
\begin{split}
& = \frac{1}{\det h} \sum_{\xi \in M_{n,2} (\mathbf{Q}) / M_{n,2} (\mathbf{Z})} \varphi_f (h \xi) \wedge_{j=1}^{n} \mathrm{Re} \left( E_1^* (\tau , \ell_j - \xi_j ) dz_j \right) \\
& = \frac{1}{\det h} \sum_{v \in V(\mathbf{Q}) / V(\mathbf{Z}) } \overline{\varphi}_f (v ) \cdot \\
& \quad \quad \sum_{\substack{\xi \in E^n \\ h \xi = v }} \mathrm{Re} \left( E_1^* ( \tau , \ell_1 - \xi_{1} ) dz_1 \right) \wedge \cdots \wedge \mathrm{Re} \left( E_1^* (\tau , \ell_{n} - \xi_{n} ) dz_n \right) ,
\end{split}
\end{multline*}
où la dernière expression, qui découle de la définition de $\varphi_f$, est obtenue en groupant les $\xi$ envoyés par $h$ sur un même élément dans $E^n$. Rappelons que l'on identifie une classe $\xi $ dans le quotient $M_{n,2} (\mathbf{Q}) / M_{n,2} (\mathbf{Z})$ à l'élément $\tau \alpha + \beta$ dans $E^n$, où $\alpha$ et $\beta$ sont les vecteurs colonnes de $\xi$. La dernière somme porte donc sur les classes $\alpha$ et $\beta$ dans $V(\mathbf{Q}) / V(\mathbf{Z})$ telles que $h \alpha$ soit nul modulo $V(\mathbf{Z})$ et $h \beta$ soit égal à $v$ modulo $V(\mathbf{Z})$.
\medskip
\noindent
{\it Remarque.} Le fait que l'expression soit indépendante des choix de $\lambda$ et $h$ découle des relations de distributions pour $E_1^*$
$$\sum_{\xi \in E_\tau [m]} E_1^* (\tau , z- \xi ) = m E_1^* (\tau , m z ).$$
\medskip
Il nous reste à expliquer que dans l'expression ci-dessus on peut partout remplacer $E_1^*$ par $E_1$. Pour cela on commence par remarquer que l'expression
$$\sum_{v \in V(\mathbf{Q}) / V(\mathbf{Z}) } \overline{\varphi}_f (v ) \sum_{\substack{\xi \in E^n \\ h \xi = v }} \mathrm{Re} \left( E_1^* ( \tau , \ell_1 - \xi_{1} ) dz_1 \right) \wedge \cdots \wedge \mathrm{Re} \left( E_1^* (\tau , \ell_{n} - \xi_{n} ) dz_n \right)$$
peut se réécrire
\begin{multline} \label{E:sans*}
\sum_{\substack{\alpha \in \mathbf{Q}^n / \mathbf{Z}^n \\ h \alpha = 0 \ (\mathrm{mod} \ \mathbf{Z}^n) }} \sum_{v \in V(\mathbf{Q}) / V(\mathbf{Z}) } \overline{\varphi}_f (v ) \\ \sum_{\substack{\beta \in \mathbf{Q}^n / \mathbf{Z}^n \\ h \beta = v \ (\mathrm{mod} \ \mathbf{Z}^n)}} \mathrm{Re} \left( E_1^* ( \tau , \ell_1 + \alpha_1 \tau + \beta_1 ) dz_1 \right) \wedge \cdots \wedge \mathrm{Re} \left( E_1^* (\tau , \ell_{n} + \alpha_n \tau + \beta_n ) dz_n \right).
\end{multline}
Puisque d'après \eqref{E*E} les différences
$$E_1^* ( \tau , \ell_j + \alpha_j \tau + \beta_j ) - E_1 (\tau , \ell_j + \alpha_j \tau + \beta_j) = \frac{1}{y} \mathrm{Im} (\ell_j ) + \alpha_j$$
sont indépendantes de $v$ et $\beta$, il résulte bien de la démonstration de (\ref{E:8.smoothing}) que l'on peut partout remplacer $E_1^*$ par $E_1$ dans \eqref{E:sans*}. Cela conclut la démonstration de la proposition \ref{9.P34bis}.
\end{proof}
\subsection{Démonstration du théorème \ref{T:ellbis}} \label{S:9.4.2}
On procède comme dans la démonstration du théorème \ref{T:mult2}. On utilise cette fois le résultat suivant.
\begin{lemma}
Les $1$-formes différentielles
$$\mathrm{Re}(E_1^*(\tau,z) dz) \quad \textrm{and} \quad E_1^*( \tau,z) dz$$
sont cohomologues sur $E_\tau - \{ 0 \}$, où $E_\tau$ est la courbe elliptique $\mathbf{C}/(\tau \mathbf{Z}+\mathbf{Z})$. Autrement dit, la forme
$\mathrm{Im} (E_1^*( \tau,z) dz)$ est exacte sur $E_\tau - \{ 0 \}$.
\end{lemma}
\begin{proof}
D'après \cite[Eq. (2) p. 57]{deShalit}, on a
\begin{equation}\label{eqE1*tolog}
2\pi i E_1^* (\tau , z) =\frac 1 {12} \partial_z \log \theta (\tau , z ) - \pi \frac{\overline{z}}{2y},
\end{equation}
où
$$\theta (\tau , z ) =e^{6\pi\frac{z(z-\overline{z})}{y}}q_\tau(q_z^{\frac 12}-q_z^{-\frac 12})^{12}\prod_{n\geq 1}(1-q_\tau^nq_z)^{12}(1-q_\tau^nq_z^{-1})^{12}.$$
La fonction réelle $z \mapsto | \theta (\tau , z)| $ est régulière sur $\mathbf{C} - (\tau \mathbf{Z} + \mathbf{Z})$ et $(\tau \mathbf{Z} + \mathbf{Z})$-périodique (cf. \cite[p. 49]{deShalit}). Calculons maintenant sa différentielle~:
\begin{equation*}
\begin{split}
d \log | \theta (\tau , z) | & = \mathrm{Re} (d \log \theta (\tau , z) ) \\
& = \mathrm{Re} \left(\partial_z \log \theta (\tau , z) dz + \partial_{\overline{z}} \log \theta (\tau , z) d\overline{z}\right)\\
& = \mathrm{Re} \left(\partial_z \log \theta (\tau , z) dz -6 \pi \frac{z}{y} d\overline{z} \right)\\
& = \mathrm{Re} \left( 12 (2\pi i) E_1^*(\tau,z) dz + 6\pi \frac{\overline{z}}{y} dz - 6\pi \frac{z}{y} d\overline{z} \right),
\end{split}
\end{equation*}
où la dernière ligne se déduit de (\ref{eqE1*tolog}). Comme
$$\mathrm{Re} \left(\overline{z}dz- zd\overline{z}\right)=0,$$
on obtient que
$$d \log | \theta (\tau , z) | = -24 \mathrm{Im} (E_1^*( \tau,z) dz).$$
\end{proof}
En particulier en restriction à $E_\tau$ les formes
\begin{multline*}
\sum_{v \in V(\mathbf{Q}) / V(\mathbf{Z}) } \overline{\varphi}_f (v ) \cdot \\ \sum_{\substack{\xi \in E^n \\ h \xi = v }} \mathrm{Re} \left( E_1^* ( \tau , \ell_1 - \xi_{1} ) dz_1 \right) \wedge \cdots \wedge \mathrm{Re} \left( E_1^* (\tau , \ell_{n} - \xi_{n} ) dz_n \right) ,
\end{multline*}
et
$$
\sum_{v \in V(\mathbf{Q}) / V(\mathbf{Z}) } \overline{\varphi}_f (v ) \sum_{\substack{\xi \in E^n \\ h \xi = v }} E_1^* ( \tau , \ell_1 - \xi_{1} ) \cdots E_1^* (\tau , \ell_{n} - \xi_{n} ) dz_1 \wedge \cdots \wedge dz_n ,
$$
sont cohomologues. La démonstration du théorème \ref{T8.8cor} implique alors que pour toute fonction $\mathcal{K}$-invariante $\varphi_f \in \mathcal{S} (V (\mathbf{A}_f ))^\circ$, l'application
$$\mathbf{S}_{\rm ell}^* [\varphi_f ] : \Gamma^{n} \longrightarrow \Omega^{n}_{\rm mer} (E^n )$$
qui à un $n$-uplet $(\gamma_0 , \ldots , \gamma_{n-1})$ associe $0$ si les vecteurs $\gamma_j^{-1} e_1$ sont liés et sinon la forme différentielle méromorphe
\begin{equation*}
\frac{1}{\det h} \sum_{v \in V(\mathbf{Q}) / V(\mathbf{Z}) } \overline{\varphi}_f (v ) \sum_{\substack{\xi \in E^n \\ h \xi = v }} E_1 ( \tau , \ell_1 - \xi_{1} ) \cdots E_1 (\tau , \ell_{n} - \xi_{n} ) dz_1 \wedge \cdots \wedge dz_n,
\end{equation*}
où $h = ( \gamma_0^{-1} e_1 | \cdots | \gamma_{n-1}^{-1} e_1) \in \mathrm{GL}_n (\mathbf{Q}) \cap M_n (\mathbf{Z} )$ et $h^* \ell_j = e_{j}^*$, définit un $(n-1)$-cocycle homogène non nul du groupe $\Gamma$ qui représente la même classe de cohomologie que $\mathbf{S}_{\rm ell , \chi}$.
Rappelons que, pour chaque élément $D \in \mathrm{Div}_{\Gamma,K}$ il correspond une fonction $\mathcal{K}$-invariante $\varphi_f$ dans $\mathcal{S} (V (\mathbf{A}_f ))$ qui, sous l'hypothèse que $D \in \mathrm{Div}_{\Gamma,K}^{\circ}$, appartient à $ \mathcal{S} (V (\mathbf{A}_f ))^{\circ}$. Il s'en suit finalement que le cocycle
$$\mathbf{S}^*_{\rm ell} [D] = \mathbf{S}^*_{{\rm ell}}[\varphi_f]$$
vérifie le théorème \ref{T:ellbis}. \qed
\medskip
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\begin{appendix}
\chapter{Cohomologie équivariante et complexe de de Rham simplicial} \label{A:A}
\resettheoremcounters
Les résultats de ce volume sont formulés dans le langage de la cohomologie équivariante. Il s'agit d'une théorie cohomologique pour les espaces topologiques munis d'une action de groupe. Comme la cohomologie habituelle, elle peut être calculée à l'aide de chaînes simpliciales ou, alternativement, à l'aide d'une version du complexe de de Rham. Dans cette annexe, nous rassemblons les principales définitions et faits concernant la cohomologie équivariante que nous utilisons. Pour plus de détails, le lecteur peut consulter \cite{DupontBook,Dupont}.
\section{Définition de la cohomologie équivariante}
Soit $G$ un groupe discret et $X$ un espace topologique raisonnable muni d'une action de $G$. Dans cette situation, on peut définir les groupes de cohomologie équivariante
$$
H^*_G(X).
$$
Ces groupes contiennent des informations à la fois sur la cohomologie usuelle de $X$ et sur l'action de $G$. L'idée derrière la construction est la suivante : si $G$ opère librement sur $X$ alors on peut définir
$$
H^*_G(X) = H^*(X/G),
$$
c'est-à-dire la cohomologie usuelle du quotient $X/G$. Pour des actions plus générales, cependant, beaucoup d'informations sont perdues en passant à l'espace quotient $X/G$. L'idée de la cohomologie équivariante est de remplacer le quotient $X/G$ par un {\it quotient d'homotopie}
$$
EG \times_G X := |EG \times X|/G.
$$
On définit $|EG \times X|$ ci-dessous, retenons pour l'instant que c'est un espace topologique obtenu comme le produit de $X$ avec un espace contractile et sur lequel $G$ opère librement. Ainsi, le remplacement de $X$ par $|EG \times X|$ ne change pas le type d'homotopie sous-jacent, mais maintenant $G$ opère librement sur $|EG \times X|$. Il est donc raisonnable de définir
$$
H^*_G(X) = H^*(|EG \times X|/G).
$$
Par exemple, quand $X$ est réduit à un point (ou contractile), on a
$$
H^*_G(X) = H^*(BG),
$$
où $BG:=|EG|/G$ est l'espace classifiant de $G$ et $H^*(BG)$ est la cohomologie du groupe $G$. D'un autre côté, quand $G$ est trivial la cohomologie équivariante se réduit à la cohomologie habituelle.
Ci-dessous, nous définissons $EG$ en utilisant des ensembles simpliciaux et le foncteur de réalisation simpliciale. Puis nous décrivons les résultats de Dupont \cite{Dupont} qui permettent de calculer $H^*_G(X)$ en utilisant des formes différentielles lorsque $X$ est une variété.
\section{La construction de Borel} Soit $G$ un groupe discret. Un espace classifiant pour $G$ est un espace topologique dont le groupe fondamental est isomorphe à $G$ et dont les groupes d'homotopie supérieurs sont triviaux. Un tel espace existe toujours. On en donne une construction concrète qui fait naturellement le lien avec la cohomologie des groupes.
Soit $EG_\bullet$ l'ensemble simplicial dont les $m$-simplexes sont les $(m+1)$-uplets ordonnés $(g_0 , \ldots , g_m )$ d'éléments de $g$. On note $EG_m = G^{m+1}$ l'ensemble de ces $m$-simplexes. Un élément de $(g_0 , \ldots , g_m ) \in EG_m$ est recollé aux $(m-1)$-simplexes $(g_0 , \ldots , g_{i-1},g_{i+1}, \ldots , g_m)$ de la même manière qu'un simplexe standard est recollé à ses faces. Les applications de faces et de dégénérescence sont donc
\begin{align}
\partial_i(g_0,\ldots,g_m) &= (g_0,\ldots,g_{i-1},g_{i+1},\ldots,g_m), \quad \mbox{et} \\
\sigma_i(g_0,\ldots,g_m) &= (g_0,\ldots,g_i,g_i,\ldots,g_m), \quad i=0,\ldots, m.
\end{align}
Le complexe $EG_\bullet$ est contractile. Le groupe $G$ opère librement sur $EG_\bullet$ par
\begin{equation}
g \cdot (g_0,\ldots,g_m) = (g_0g^{-1},\ldots,g_m g^{-1})
\end{equation}
de sorte que l'application quotient $EG_\bullet \to EG_\bullet / G$ est un revêtement universel. La base est donc un espace classifiant pour $G$.
L'application
\begin{equation}
(g_0,\ldots,g_m) \mapsto (g_1g_0^{-1},\ldots,g_m g_{m-1}^{-1})
\end{equation}
identifie $EG_\bullet / G$ avec l'ensemble simplicial $BG_\bullet$ défini par $BG_m = G^m$ et
\begin{align}
\partial_i(g_1,\ldots,g_m) &= \begin{cases} (g_2,\ldots,g_m) & i=0 \\ (g_1,\ldots,g_ig_{i+1},\ldots, g_m) & 0 < i < m \\ (g_1,\ldots,g_{m-1}) & i=m, \end{cases} \\
\sigma_i(g_1,\ldots,g_m) &= (g_1,\ldots,g_i,1,g_{i+1},\ldots,g_m), \quad i=0,\ldots, m.
\end{align}
Notons $|X_\bullet|$ la réalisation géométrique d'un ensemble simplicial $X_\bullet$ : c'est l'espace topologique défini comme
$$
|X_\bullet | = \bigsqcup_{k \geq 0}(\Delta_k \times X_k)/\sim,
$$
où $\Delta_k$ désigne le $k$-simplexe standard et $\sim$ est la relation d'équivalence donnée par
$$(t,f^*x) \sim (f_* t, x)$$
pour tous $t \in \Delta_k$, $x \in X_l$ et toute application croissante $f:[k] \to [l]$. Pour les principales propriétés du foncteur de réalisation simpliciale, nous renvoyons le lecteur à \cite{Segal}. L'espace $|BG_\bullet|$ est donc un espace classifiant pour $G$ et $|EG_\bullet| \to |BG_\bullet|$ est le $G$-fibré universel associé (cf. \cite[Prop. 2.7]{Burgos}).
\medskip
Considérons maintenant un espace topologique $M$ sur lequel le groupe $G$ opère par homéomorphismes, autrement dit un $G$-espace. Le groupe $G$ opère alors sur
$$EG_\bullet \times M$$
par
\begin{equation}
g \cdot (g_0,\ldots,g_m, x) = (g_0g^{-1},\ldots,g_mg^{-1}, gx ).
\end{equation}
La \emph{cohomologie équivariante} de $M$ est définie comme étant la cohomologie ordinaire du quotient
$$X=|EG_{\bullet} \times_G M|;$$
autrement dit
$$H_G^\bullet (M) = H^\bullet (X) = H^\bullet (|EG_{\bullet} \times_G M|).$$
Si $G$ est le groupe trivial, c'est juste la cohomologie de $M$. Si $M$ est contractile, l'espace $X$ est homotopiquement équivalent à l'espace classifiant $BG_\bullet$ et $H_G^\bullet (M) = H^\bullet (BG_\bullet)$ est la cohomologie du groupe $G$.
Noter que si $G$ opère librement sur $M$, la projection $EG_{\bullet} \times_G M \to M/G$ est une équivalence d'homotopie et on a $H_G^\bullet (M) = H^\bullet (M/G)$.
Donnons quelques exemples qui apparaissent dans le texte principal.
\begin{example}
Supposons que $G$ soit égal à $\mathbf{Z}^n$ pour un certain entier naturel $n$. Puisque $\mathbf{Z}^n$ opère librement sur $\mathbf{R}^n$, on a
$$
BG = |EG/G| \simeq |(\mathbf{R}/\mathbf{Z})^n|.
$$
\end{example}
\begin{example}
Soit $\Gamma$ un sous-groupe discret de $\mathrm{SL}_n (\mathbf{R} )$ et supposons que l'image de $\Gamma$ dans $\mathrm{PSL}_n (\mathbf{R} ) $ est sans torsion. De tels groupes $\Gamma$ abondent car on peut montrer que tout groupe arithmétique de $\mathrm{SL}_n (\mathbf{R} )$ contient un tel sous-groupe d'indice fini. Considérons maintenant l'espace symétrique
$$
X=\mathrm{SL}_n (\mathbf{R} )/\mathrm{SO}_n .
$$
Cet espace est contractile et porte une action de $\Gamma$ induite par la multiplication à gauche dans $\mathrm{SL}_n(\mathbf{R})$. Notre hypothèse sur $\Gamma$ garantit que cette action est libre. Il s'ensuit que
$$
B\Gamma = |E\Gamma /\Gamma| \simeq \Gamma \backslash \mathrm{SL}_n(\mathbf{R})/\mathrm{SO}_n ,
$$
c'est-à-dire que le quotient de Borel est un espace localement symétrique.
\end{example}
\begin{example}
Les deux exemples ci-dessus peuvent être combinés : considérons le produit semi-direct $\mathrm{SL}_n(\mathbf{Z}) \ltimes \mathbf{Z}^n$ et soit
$$
X = (\mathrm{SL}_n(\mathbf{R})/\mathrm{SO}_n ) \times \mathbf{R}^n.
$$
Supposons que $\Gamma \subset \mathrm{SL}_n(\mathbf{Z})$ opère librement sur $\mathrm{SL}_n(\mathbf{R})/\mathrm{SO}_n$ et soit $L \subset \mathbf{Z}^n$ un sous-réseau de rang $n$ préservé par $\Gamma$. Alors $\Gamma \ltimes L$ opère librement sur l'espace contractile $X$ et donc
$$
B(\Gamma \rtimes L ) \simeq \Gamma \backslash (\mathrm{SL}_n(\mathbf{R})/\mathrm{SO}_n \times \mathbf{R}^n/L)
$$
est un fibré sur l'espace localement symétrique $\Gamma \backslash\mathrm{SL}_n(\mathbf{R})/\mathrm{SO}_n$ dont les fibres sont des tores compacts de dimension $n$.
\end{example}
\section{Formes différentielles simpliciales}
Supposons maintenant que $M$ est une $G$-variété. Le quotient $X$ est alors une \emph{variété simpliciale}, c'est-à-dire un ensemble semi-simplicial dont les $m$-simplexes
\begin{equation}
X_m = \left( EG_{\bullet} \times_G M \right)_m = \left( EG_m \times M \right)/G
\end{equation}
sont des variétés (lisses) et dont les applications de faces et de dégénérescence sont lisses, voir par exemple \cite{Dupont}.
La projection canonique
\begin{equation}
EG_{\bullet} \times_G M \to BG_\bullet
\end{equation}
réalise $EG_{\bullet} \times_G M$ comme un fibré simplicial au-dessus de $BG_\bullet$ de fibre $M$.
Dans ce contexte, la cohomologie équivariante $H_G^\bullet(M)$ peut également être calculée à l'aide de formes différentielles, comme l'a montré Dupont \cite{Dupont}. Plus précisément, désignons par $\mathrm{A}^\bullet (X)$ le complexe de de Rham simplicial où, par définition, une $p$-forme simpliciale $\alpha$ sur $X$ est une collection d'applications
$$
\alpha^{(m)} : G^{m+1} \to A^p (\Delta_m \times M)
$$
définies pour $m \geq 0$ et vérifiant les relations de compatibilité
\begin{equation}\label{eq:app_simp_1}
(\partial_i \times \mathrm{id})^* \alpha^{(m)}(g) = \alpha^{(m-1)}(\partial_i g), \qquad g \in G^{m +1},
\end{equation}
dans $\Delta_{m-1} \times M$ pour tout $i \in \{ 0 , \ldots , m\}$ et pour tout $m \geq 1$, ainsi que la relation de $G$-invariance
\begin{equation} \label{eq:app_simp_2}
g^*\alpha^{(m)}(g \cdot g') = \alpha^{(m)}(g'), \quad g \in G, \quad g' \in G^{m+1}.
\end{equation}
En d'autres termes, une $p$-forme simpliciale $\alpha$ sur $X$ est une $p$-forme sur la \emph{réalisation grossière}
$\| X \|$ de $X$, c'est-à-dire le quotient
$$
\|X\| = \bigsqcup_{k \geq 0} (\Delta_k \times X_k)/\sim,
$$
où la relation d'équivalence est donnée par $(t,f^*x) \sim (f_*t, x)$ pour toutes les applications croissantes \emph{injectives} $f:[k] \to [l]$ (une telle $f$ est généralement appelée application ``de face''). Sous des hypothèses légères sur $X$ l'application canonique $\|X\| \to |X|$ est une équivalence d'homotopie ; ces hypothèses s'appliquent pour $X=EG \times_G M$ comme le montre Segal \cite[A.1]{Segal}.
Comme dans le cas des variétés usuelles, la différentielle extérieure et le cup-produit usuel font de $\mathrm{A}^\bullet (X)$ une algèbre différentielle graduée et Dupont démontre que la cohomologie de ce complexe est isomorphe à la cohomologie de $\|X \|$.
Enfin, chaque $\mathrm{A}^p (X)$ se décompose en la somme directe
$$\mathrm{A}^p (X) = \bigoplus_{k+l = p} \mathrm{A}^{k,l} (X)$$
où $\mathrm{A}^{k,l} (X)$ est constitué des formes dont la restriction à $\Delta_m \times X_m$ est localement somme de formes
$$a dt_{i_1} \wedge \ldots \wedge dt_{i_k} \wedge dx_{j_1} \wedge \ldots \wedge dx_{j_l}$$
où les $x_j$ sont des coordonnées locales de $X_m$ et $(t_0 , \ldots , t_m)$ sont les coordonnées barycentriques de $\Delta_m$. Les différentielles extérieures $d_\Delta$ et $d_X$ relativement aux variables respectives $t$ et $x$ décomposent $(\mathrm{A}^\bullet (X) , d)$ en un double complexe $(\mathrm{A}^{\bullet , \bullet} , d_\Delta , d_X )$.
On peut introduire un autre double complexe de formes différentielles
$$(\mathcal{A}^{\bullet,\bullet}(X_\bullet),\delta,d_X).$$
Ici $\mathcal{A}^{k, l}(X_\bullet)=A^l(X_k)$ est l'espace les formes différentielles de degré $l$ sur $X_k$. On pose $\delta=\sum_{i}(-1)^i \partial_i^*$. Dupont montre que pour tout $l$, l'application d'intégration sur les simplexes
$$\mathcal{I} : (\mathrm{A}^{\bullet ,l} (X) , d_\Delta) \to (A^l (X_\bullet ) , \delta)$$
définit une équivalence d'homotopie entre complexes de chaines. Ces équivalences induisent des isomorphismes entre les suites spectrales calculant la cohomologie des complexes totaux \cite[Cor. 2.8]{Dupont}.
Dans le texte principal, on utilise ces équivalences pour obtenir des cocycles du groupe $G$ à partir de certaines classes de cohomologie équivariante. Dans la suite de cette annexe nous expliquons cette construction. L'argument consiste à considérer les applications au bord d'une suite spectrale, mais notre objectif est de décrire ces applications explicitement. L'argument dans les cas affine et multiplicatif étant analogue, nous nous concentrerons sur le cas elliptique.
Soit $E$ une courbe elliptique et considérons $E^n$ pour un entier $n \geq 2$. Fixons un sous-groupe d'indice fini $G$ de $\mathrm{SL}_n(\mathbf{Z})$ et un $0$-cycle $G$-invariant $D$ dans $E^n$ de degré zéro et constitué de points de torsion. Soit $M=E^n-|D|$. Au chapitre \ref{C:1} nous expliquons comment $D$ donne naissance à une classe
$$
E[D] \in H^{2n-1}_G (M) = H^{2n-1}(\|(EG \times M)/G\|).
$$
Nous expliquons maintenant comment associer à $E[D]$ une classe de cohomologie dans
$$
H^{n-1}(G,\varinjlim_U H^n(U))
$$
où $U$ décrit l'ensemble des ouverts affines dans $E^n$ obtenus en supprimant un nombre fini d'hyperplans.
La construction est la suivante. Choisissons une $(2n-1)$-forme simpliciale fermée $\alpha \in A^\bullet(EG \times_G M)$ représentant $E[D]$. Autrement dit, $\alpha$ est une collection d'applications
$$
\alpha^{(m)}: G^{m+1} \to A^{2n-1}(\Delta_m \times M), \qquad m \geq 0,
$$
vérifiant \eqref{eq:app_simp_1}, \eqref{eq:app_simp_2} et
$$
(d_\Delta+d_M)\alpha^{(m)}=0.
$$
Considérons les formes
$$
\mathcal{I}\alpha^{(m)} : G^{m+1} \to A^{2n-1-m}(M), \qquad m \geq 0,
$$
definies par
$$
\mathcal{I}\alpha^{(m)}(g_0,\ldots,g_m) = \int_{\Delta_m} \alpha^{(m)}(g_0,\ldots,g_m).
$$
Il découle du théorème de Stokes que
\begin{equation}
d_M \mathcal{I}\alpha^{(0)}=0
\end{equation}
et
\begin{equation}\label{eq:app_simp_4}
\delta \mathcal{I}\alpha^{(m)} + d_M \mathcal{I}\alpha^{(m+1)}= 0
\end{equation}
pour tout $m \geq 0$. Considérons maintenant les formes
$$
\widetilde{\alpha}^{(m)} \in \widetilde{A}^{2n-1-m}:=\varinjlim A^{2n-1-m}(U)
$$
où $\widetilde{\alpha}^{(m)}$ désigne l'image de $\mathcal{I}\alpha^{(m)}$ par l'application naturelle
$$A^{2n-1-m}(M) \to \varinjlim A^{2n-1-m}(U).$$
On construit maintenant une suite d'applications $G$-équivariantes
\begin{equation}
\beta^{(m)}:G^{m+1} \to \widetilde{A}^{2n-2-m}, \qquad 0\leq m < n-1,
\end{equation}
vérifiant $d_M \beta^{(0)}=\widetilde{\alpha}^{(0)}$ et
\begin{equation} \label{eq:app_simp_5}
d_M \beta^{(m)} \pm \delta \beta^{(m-1)} = \widetilde{\alpha}^{(m)}
\end{equation}
pour tout $m \in \{1,\ldots,n-2\}$. Pour ce faire nous procédons comme suit :
notez que tout ouvert affine $U \subset E^n$ satisfait $H^k(U)=0$ pour tout $k>n$. Puisque $d_M \mathcal{I}\alpha^{(0)}(e)=0$, il existe $\beta^{(0)}(e) \in \widetilde{A}^{2n-2} $ tel que
$$
d_M \beta^{(0)}(e) = \widetilde{\alpha}^{(0)}(e).
$$
On pose $\beta^{(0)}(g)=g^*(\beta^{(0)}(e))$. La forme $\widetilde{\alpha}^{(0)}$ étant $G$-invariante, pour tout $g \in G$ on a $d_M \beta^{(0)}(g) = \widetilde{\alpha}^{(0)}(g)$. Il découle alors de \eqref{eq:app_simp_4} que
$$
d_M(\widetilde{\alpha}^{(1)}+\delta\beta^{(0)}) = 0,
$$
et en procédant comme ci-dessus on peut trouver une application $G$-invariante
$$
\beta^{(1)}: G^2 \to \widetilde{A}^{2n-3}
$$
telle que
$$
d_M \beta^{(1)} = \widetilde{\alpha}^{(1)}+\delta \beta^{(0)}.
$$
En itérant cet argument, on obtient des applications $G$-équivariantes $\beta^{(0)},\ldots, \beta^{(n-2)}$ vérifiant \eqref{eq:app_simp_5}. Maintenant, il découle de \eqref{eq:app_simp_4} que
$$
d_M(\widetilde{\alpha}^{(n-1)}\pm \delta\beta^{(n-2)})=0
$$
et, en passant aux classes de cohomologie, on obtient une application
$$
c: G^{n} \to \varinjlim H^n(U), \qquad c(g):=[\widetilde{\alpha}^{(n-1)}(g)\pm \delta \beta^{(n-2)}(g)].
$$
Comme $\widetilde{\alpha}$ et $\beta$
sont $G$-équivariantes, l'application $c$ est aussi $G$-équivariante. Noter aussi que $c$ est un $(n-1)$-cocycle :
\begin{equation}
\begin{split}
\delta c(g) &= [\delta(\widetilde{\alpha}^{(n-1)}(g)\pm \delta \beta^{(n-2)}(g))] \\
&= [\delta \widetilde{\alpha}^{(n-1)}(g)] \\
&= [-d_M \widetilde{\alpha}^{(n)}(g)] \\
&= 0.
\end{split}
\end{equation}
Le cocycle $c$ dépend des choix faits dans la construction des applications $\beta^{(k)}$, mais un argument standard montre que sa classe de cohomologie
$$
[c] \in H^{n-1}(G,\varinjlim H^n(U))
$$
est indépendante de ces choix. Elle est aussi indépendante de la forme $\alpha$ sélectionnée pour représenter la classe de cohomologie originale dans $H^{2n-1}_G(M)$.
\begin{remark}
En pratique on trouve souvent des $\widetilde{\alpha}^{(m)}$ telles que pour tout $k$ dans $\{0,\ldots ,n-2\}$ la forme $\widetilde{\alpha}^{(k)}$ soit identiquement nulle. Dans ce cas on peut prendre $\beta^{(0)},\ldots,\beta^{(n-2)}$ identiquement nulles et l'application
$$
g \mapsto [\widetilde{\alpha}^{(n-1)}(g)]: G^n \to \varinjlim H^n(U).
$$
est un cocycle qui représente la classe de $[c] $.
\end{remark}
\setcounter{equation}{0}
\chapter{Classe d'Eisenstein affine et théorie de l'obstruction} \label{A:B}
\resettheoremcounters
Soit $F$ un corps, par exemple $\mathbf{C}$, et soit $\mathbf{T}_n = \mathbf{T}_n (F)$ l'immeuble de Tits de $V=F^n$ avec $n \geq 2$. Le groupe $G = \mathrm{GL}_n(F)$ opère naturellement sur $\mathbf{T}_n$ et on considère le fibré
\begin{equation}
\begin{tikzcd}
EG \times_G \mathbf{T}_n \arrow["\pi" ',d] \\ BG
\end{tikzcd}
\end{equation}
de fibre $\mathbf{T}_n$. L'immeuble de Tits $\mathbf{T}_n$ étant $(n-3)$-connexe, la première obstruction à l'existence d'une section de $\pi$ est une classe
\begin{equation} \label{E:A1}
\omega_{n-2} \in H^{n-1}(BG,\pi_{n-2}(\mathbf{T}_n )),
\end{equation}
voir par exemple \cite[Obstruction Theory, p. 415]{Hatcher}. Dans \eqref{E:A1} on voit le groupe d'homotopie $\pi_{n-2}(\mathbf{T}_n)$ de la fibre de $\pi$ comme un système local. L'immeuble de Tits $\mathbf{T}_n$ étant $(n-3)$-connexe, on a
$$\pi_{n-2}(\mathbf{T}_n) = \widetilde{H}_{n-2} (\mathbf{T}_n ) = \mathrm{St} (F^n).$$
On peut donc voir $\omega_{n-2}$ comme un élément de $H^{n-1}(G,\mathrm{St} (F^n))$.
\begin{proposition}
La classe d'obstruction $\omega_{n-2}$ coincide avec la classe de cohomologie associée au symbole universel de Ash--Rudolph représenté par le cocycle
$$G^n \to \mathrm{St} (F^n ); \quad (g_0 , \ldots , g_{n-1} ) \mapsto [g_0^{-1} e_1 , \ldots , g_{n-1}^{-1} e_1 ] .$$
\end{proposition}
\begin{proof} On commence par construire une section explicite $s$ de $\pi$ sur le squelette de dimension $n-2$. On étudie ensuite l'obstruction à étendre cette section au $(n-1)$-squelette.
On définit la section $s$ en commençant par le $0$-squelette $BG_0$ puis on l'étend par récurrence de $BG_{\leq k}$ à $BG_{\leq k+1}$~:
\begin{equation}
\cdots
\begin{tikzcd}
(EG \times_G \mathbf{T}_n )_1 \arrow[d,"\pi_1"' ] \arrow[r, shift left] \arrow[r, shift right] & (EG \times_G \mathbf{T}_n )_0 \arrow[d,"\pi_0"' ] \\
BG_1 \arrow[u, dashed, bend right, "s_1"'] \arrow[r, shift left] \arrow[r, shift right] & BG_0 \arrow[u, dashed, bend right, "s_0"'].
\end{tikzcd}
\end{equation}
Les $k$-simplexes de $EG \times_G \mathbf{T}_n$ sont les couples $(\mathbf{g} , D_\bullet)$, où $\mathbf{g}$ est un $(k+1)$-uplet $(g_0 , \ldots , g_k) \in G^{k+1}$ et $D_\bullet \in (\mathbf{T}_n )_k$ est un $(k+1)$-drapeau de sous-espaces propres non nuls de $V$, modulo équivalence
\begin{equation}
((g_0,\ldots,g_k),D_\bullet) \sim ((g_0g^{-1},\ldots,g_kg^{-1}) , gD_\bullet), \quad g \in G.
\end{equation}
Tentons maintenant de définir une section $s$ de $\pi$ en commençant par définir $s$ sur le $0$-squelette $BG_0$ puis en l'étendant par récurrence de $BG_{\leq k}$ à $BG_{\leq k+1}$:
\begin{equation}
\cdots
\begin{tikzcd}
(EG \times_G \mathbf{T}_n )_1 \arrow[d,"\pi_1"' ] \arrow[r, shift left] \arrow[r, shift right] & (EG \times_G \mathbf{T}_n )_0 \arrow[d,"\pi_0"' ] \\
BG_1 \arrow[u, dashed, bend right, "s_1"'] \arrow[r, shift left] \arrow[r, shift right] & BG_0 \arrow[u, dashed, bend right, "s_0"'].
\end{tikzcd}
\end{equation}
Les $k$-simplexes dans $EG \times_G \mathbf{T}_n $ sont les couples $(\mathbf{g},F_\bullet)$ (où $\mathbf{g} \in G^{k+1}$ et $F_\bullet \in (\mathbf{T}_n )_k$) modulo l'équivalence
\begin{equation}
((g_0,\ldots,g_k),F_\bullet) \simeq ((g_0g^{-1},\ldots,g_kg^{-1}),gF_\bullet), \quad g \in G.
\end{equation}
On note dans la suite $[\mathbf{g},F_\bullet]$ la classe d'équivalence de $(\mathbf{g},F_\bullet)$. Pour définir la section $s_0$ de $\pi_0$ (l'application induite par $\pi$ sur le $0$-squelette) on choisit une droite $L=\langle v \rangle \subset F^n$ et on pose
\begin{equation}
s_0 ( g_0 ) = [g_0 , g_0^{-1} L].
\end{equation}
On pose ensuite
\begin{equation}
s_1(g_0 , g_1) = [(g_0 ,g_1), \Delta(g_0^{-1} L , g_1^{-1} L)]
\end{equation}
où
\begin{equation}
\Delta(g_0^{-1} L , g_1^{-1}L) = \begin{tikzcd} \stackrel{\langle g_0^{-1} v \rangle}{\bullet} \arrow[r, dash] & \stackrel{\langle g_0^{-1} v , g_1^{-1}v \rangle}{\bullet} & \arrow[l, dash] \stackrel{\langle g_1^{-1} v \rangle}{\bullet}, \end{tikzcd}
\end{equation}
et les deux segments correspondent aux drapeaux\footnote{Ce chemin allant de $g_0^{-1} L$ à $g_1^{-1}L$ dans $\mathbf{T}_n$ n'est pas unique mais tout autre chemin est plus long ou passe par un $0$-simplex correspondant à un sous-espace de $F^n$ de dimension strictement supérieure à $2$.}
$$\langle g_0^{-1} v \rangle \subseteq \langle g_0^{-1} v , g_1^{-1}v \rangle \quad \mbox{et} \quad \langle g_1^{-1}v \rangle \subseteq \langle g_0^{-1} v , g_1^{-1}v \rangle.$$
En général, on pose
\begin{equation}
s_k(g_0 ,\ldots , g_k) = [(g_0 , \ldots , g_k ), \Delta( g_0^{-1} L , \ldots, g_k^{-1} L)],
\end{equation}
où $\Delta( g_0^{-1} L , \ldots, g_k^{-1} L)$ est le sous-complexe de $\mathbf{T}_n$ isomorphe à la première subdivision barycentrique d'un $k$-simplexe de $( \mathbf{T}_n )_k$ de sommets $g_0^{-1} L, \ldots ,g_k^{-1}L$ et de barycentre le sous-espace $\langle g_0^{-1} v , \ldots , g_k^{-1} v \rangle$ de dimension $k+1$ dans $V$; de sorte que les $k$-simplexes de ce complexe sont associés aux drapeaux
$$\langle g_{\sigma (0)}^{-1} v \rangle \subset \langle g_{\sigma (0)}^{-1} v, g_{\sigma (1)}^{-1} v \rangle \subset \cdots \subset \langle g_{\sigma (0)}^{-1} v, g_{\sigma (1)}^{-1} v ,\ldots, g_{\sigma (k)}^{-1}v \rangle \quad (\sigma \in \mathfrak{S}_{k+1} ) .$$
Cette construction inductive est possible tant que $k \leq n-2$, c'est-à-dire tant que $k+1$ est strictement inférieur à la dimension de $V$. On définit ainsi section
\begin{equation}
s_{\leq n-2}:BG_{\leq n-2} \to (EG \times_G \mathbf{T}_n )_{\leq n-2}.
\end{equation}
L'obstruction à étendre $s$ au $(n-1)$-squelette $BG_{\leq n-1}$ est alors représentée par l'application $BG_{n-1} \to \pi_{n-2}(\mathbf{T}_n)$ définie par
\begin{equation}
(g_0 , \ldots , g_{n-1)} \mapsto \sum_{j=0}^{n-1} (-1)^j [s_{n-2}(g_0 , \ldots , \widehat{g_j} , \ldots , g_{n-1})].
\end{equation}
La théorie de l'obstruction implique que cette application définit un cocycle
$$c_{n-2} \in C^{n-1}(BG,\pi_{n-2}(\mathbf{T}_n ))$$
qui représente $\omega_{n-2}$; de sorte que $[c_{n-2}]$ est indépendant du choix de $L$. Vue comme classe dans $H^{n-1}(G,\mathrm{St}_n(F))$ la classe d'obstruction $\omega_{n-2}$ est finalement donnée par
\begin{equation}
\omega_1(g_0,\ldots,g_{n-1}) = [g_0^{-1} v ,g_1^{-1}v,\ldots ,g_{n-1}^{-1}v],
\end{equation}
où $[\cdot]$ désigne le symbole modulaire universel de Ash--Rudolph, cf. \S \ref{S:ARuniv}.
\end{proof}
\end{appendix}
\bibliographystyle {plain}
| {'timestamp': '2023-01-24T02:12:15', 'yymm': '2301', 'arxiv_id': '2301.09118', 'language': 'fr', 'url': 'https://arxiv.org/abs/2301.09118'} |
\section{Introduction}\label{sec:Intro}
\subsection{Background} The Voronoi diagram (the Voronoi tessellation, the Voronoi decomposition, the Dirichlet tessellation) is one of the basic structures
in computational geometry. Roughly speaking, it is a certain decomposition of a given space $X$ into cells, induced by a distance function and by a tuple of subsets $(P_k)_{k\in K}$, called the generators or the sites. More precisely, the Voronoi cell $R_k$ associated with the site $P_k$ is the set
of all the points in $X$ whose distance to $P_k$ is not greater than their distance to the union of the other sites $P_j$.
Voronoi diagrams appear in a huge number of fields in science and technology and have many applications. They have been the subject of research for at least 160 years, starting formally with L. Dirichlet \cite{Dirichlet} and G. Voronoi \cite{Voronoi}, and of extensive research during the last 40 years. For several well written surveys on Voronoi diagrams which contain extensive bibliographies and many applications, see \cite{Aurenhammer}, \cite{AurenhammerKlein}, \cite{OBSC}, and \cite{VoronoiWeb}.\\
\noindent Consider the following question:
\begin{question}\label{ques:main}
Does a small change of the sites, e.g., of their position or shape, yield a small change in the corresponding Voronoi cells?
\end{question}
This question is by all means natural, because in practice, no matter which algorithm is being used for the computation of the Voronoi cells, one approximates the sites either because of lack of exact information about them, because of inevitable numerical errors occurring when a site is represented in an analog or a digital manner, for simplification purposes and so on, and it is important to know whether the resulting Voronoi cells approximate well the real ones.
For instance, consider the Voronoi diagram whose sites are either
shops (or large shopping centers), antennas, or other facilities in some city/district such as post offices. See Figures \ref{fig:Shops}-\ref{fig:ShopsReality}.
\begin{figure*}
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.78]{figure1.eps}}
\end{center}
\caption{10 shopping centers (or post offices) in a flat city. Each shopping center is represented by a point. }
\label{fig:Shops}
\end{minipage
\hfill
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.78]{figure2.eps}}
\end{center}
\caption{In reality each shopping center/post office is not a point and its location is approximated. The combinatorial structure is somewhat different and the Voronoi cells are not exactly polygons, but still, their shapes are almost the same as in Figure \ref{fig:Shops}.}
\label{fig:ShopsReality}
\end{minipage
\end{figure*}
Each Voronoi cell is the domain of influence of its site and it can be used for various purposes, among them estimating the number of potential costumers \cite{EconomyFacilityPNAS} or understanding the spreading patterns of mobile phone viruses \cite{VoronoiVirus}. In reality, each site has a somewhat vague shape, and its real location is not known exactly. However, to simplify matters we regard each site as a point (or a finite collection of points if we consider firms of shops) located more or less near the real location. As a result, the resulting cells only approximate the real ones, but we hope that the approximation will be good in the geometric sense, i.e., that the shapes of the corresponding real and approximate cells will be almost the same. (See Section \ref{sec:examples} for many additional examples, including ones with infinitely many sites or in higher/infinite dimensional spaces.) As the counterexamples in Section \ref{sec:CounterExamples} show, it is definitely not obvious that this is the case.
A similar question to Question \ref{ques:main} can be asked regarding any geometric structure/algorithm, and, in our opinion, it is a fundamental question which is analogous to the question about the stability of the solution of a differential equation with respect to small changes in the initial conditions.
The traditional approach to Voronoi diagrams, and, in particular, to (variants of) Question \ref{ques:main}, is combinatorial. For instance, as already mentioned in Aurenhammer \cite[p. 366]{Aurenhammer}, the combinatorial structure of Voronoi diagrams (in the case of the Euclidean distance with point sites), i.e., the structure of vertices, edges and so
on, is not stable under continuous motion of the sites, but it is stable ``most of the time''. A more extensive discussion about this issue, still with point sites but possibly in higher dimensions, can be found in Weller \cite{Weller}, Vyalyi et al. \cite{VGT}, and Albers et al. \cite{AGMR}.
However, it seems that this question, in the geometric sense, has been raised or discussed only rarely in the context of Voronoi diagrams. In fact, after a comprehensive survey of the literature about Voronoi diagrams, we have found only very few places that have a very brief, particular, and intuitive discussion which is somewhat related to this question. The following cases were mentioned: the Euclidean plane with finitely many sites \cite{Kaplan}, the Euclidean plane with finitely many point sites
\cite[p. 366]{Aurenhammer}, and the $d$-dimensional Euclidean space with finitely many point sites \cite{AGMR} (see also some of the references therein). It was claimed there without proofs and exact definitions that the Voronoi cells have a continuity property: a small change in the position or the shape of the sites yields a small change in the corresponding Voronoi cells.
Another continuity property was discussed by Groemer \cite{Groemer} in the context of the geometry of numbers. He considered Voronoi diagrams generated by a lattice of points in a $d$-dimensional Euclidean space, and proved that if a sequence of lattices converges to a certain lattice (meaning that the basis elements which generate the lattices converge with respect to the Euclidean distance to the basis which generates the limit lattice), then the corresponding Voronoi cells of the origin converge, with respect to the Hausdorff distance, to the cell of the origin of the limit lattice. His result is, in a sense and in a very particular case, a stability result, but it definitely does not answer Question \ref{ques:main} (which, actually, was not asked at all in \cite{Groemer}) for several reasons: first, usually the sites or the perturbed ones do not form a (infinite) lattice. Second, in many cases they are not points (singletons). Third, a site is usually different from the perturbed site (in \cite{Groemer} the discussed sites equal $\{0\}$). In this connection, we also note that Groemer's proof is very restricted to the above setting and it uses arguments based on compactness and no explicit bounds are given.
It is quite common in the computational geometry literature to assume ``ideal conditions'', say infinite precision in the computation, exact and simple input, and so on. These conditions are somewhat non-realistic. Issues related to the stability of geometric structures under small perturbations of their building blocks (not necessarily the geometric stability) are not so common in the literature, but they can be found in several places, e.g., in \cite{AGGKKRS, AryaMalamatosMount, AttaliBoissonnatEdelsbrunner, BandyopadhyaySnoeyink, ChazalCohenSteinerLieutier, ChoiSeidel, SteinerEdelsbrunerHarer, FortuneStability, HarPeled, Khanban, LofflerPhD, LofflerKreveld, GuibasSalesinStolfi, SugiharaIriInagakiImai}. However, in many of the above places the discussion has combinatorial characteristics and there are several restrictive assumptions: for instance, the underlying setting is usually a finite dimensional space (in many cases only $\R^2$ or $\R^3$), with the Euclidean distance, and with finitely many objects of a specific form (merely points in many cases). In addition, the methods are restricted to this setting. In contrast, the infinite dimensional case or the case of (possibly infinitely many) general objects or general norms have never been considered.
\subsection{Contribution of this paper} We discuss the question of stability of Voronoi diagrams with respect to small changes of the corresponding sites. We first formalize this question precisely, and then show that the answer is positive in the case of $\R^d$, or, more generally, in the case of (possibly infinite dimensional) uniformly convex normed spaces, assuming there is a common positive lower bound on the distance between the sites. Explicit bounds are presented, and we allow infinitely many sites of a general form. We also present several counterexamples which show that the assumptions formulated in the main result are crucial. We illustrate the relevance of this result using several pictures and many real-world and theoretical examples and counterexamples. To the best of our knowledge, the main result and the approach used for deriving it are new. Two of our main tools are: a new representation theorem which characterizes the Voronoi cells as a collection of line segments and a new geometric lemma which provides an explicit geometric estimate.
\subsection{The structure of the paper} In Section \ref{sec:Definitions} we present the basic definitions and notations. Exact formulation of Question \ref{ques:main} and informal description of the main result are given in Section \ref{sec:FormalInformal}. The relevance of the main result is illustrated using many theoretical and real-world examples in Section \ref{sec:examples}. The main result is presented in Section \ref{sec:Outline}, and we discuss briefly some aspects related to its proof. In Section \ref{sec:CounterExamples} we present several interesting counterexamples showing that the assumptions imposed in the main result are crucial. We end the paper in Section \ref{sec:Concluding} with several concluding remarks. Since the proof of the main result is quite long and technical, and because the main goal of this paper is to introduce the issue and to discuss it in a qualitative manner, rather than going deep into technical details, proofs were omitted from the main body of the text. Full proofs can be found in the appendix (Section \ref{sec:appendix}) and a preliminary version in \cite{ReemPhD}.
\section{Notation and basic definitions}\label{sec:Definitions}
In this section we present our notation and basic definitions. In the main discussion we consider a closed and convex set $X\neq \emptyset$ in some uniformly convex normed space $(\widetilde{X},|\cdot|)$ (see Definition \ref{def:UniformlyConvex} below), real or complex, finite or infinite dimensional. The induced metric is $d(x,y)=|x-y|$. We assume that $X$ is not a singleton, for otherwise everything is trivial. We denote by $[p,x]$ and $[p,x)$ the closed and half open line segments connecting $p$ and $x$, i.e., the sets $\{p+t(x-p): t\in [0,1]\}$ and $\{p+t(x-p): t\in [0,1)\}$ respectively. The (possibly empty) boundary of $X$ with respect to the affine hull spanned by $X$ is denoted by $\partial X$. The open ball with center $x\in X$ and radius $r>0$ is denoted by $B(x,r)$.
\begin{defin}\label{def:dom}
Given two nonempty subsets $P,A\subseteq X$, the dominance region
$\dom(P,A)$ of $P$ with respect to $A$ is the set of all $x\in X$
whose distance to $P$ is not greater than their distance to $A$, i.e.,
\begin{equation*}
\dom(P,A)=\{x\in X: d(x,P)\leq d(x,A)\}.
\end{equation*}
Here $d(x,A)=\inf\{d(x,a): a\in A\}$ and in general we denote $d(A_1,A_2)=\inf\{d(a_1,a_2): a_1\in A_1,\,a_2\in A_2\}$ for any nonempty subsets $A_1,A_2$.
\end{defin}
\begin{defin}\label{def:Voronoi}
Let $K$ be a set of at least 2 elements (indices), possibly
infinite. Given a tuple $(P_k)_{k\in K}$ of nonempty subsets
$P_k\subseteq X$, called the generators or the sites, the Voronoi diagram induced by this tuple is the tuple $(R_k)_{k\in K}$ of non-empty subsets
$R_k\subseteq X$, such that for all $k\in K$,
\begin{equation*}
R_k=\dom(P_k,{\underset{j\neq k}\bigcup P_j})
=\{x\in X: d(x,P_k)\leq d(x,P_j)\,\,\forall j\in K ,\, j\neq k \}.
\end{equation*}
In other words, the Voronoi cell $R_k$ associated with the site $P_k$ is the set of all $x\in X$ whose distance to $P_k$ is not greater than their distance to the union of the other sites $P_j$.
\end{defin}
In general, the Voronoi diagram induces a decomposition of $X$ into its Voronoi cells and the rest. If $K$ is finite, then the union of the cells is the whole space. However, if $K$ is infinite, then there may be a ``neutral cell'': for example, if $X$ is the Euclidean plane, $K=\N=\{1,2,3,\ldots\}$ and $P_k=\R\times \{1/k\}$, then no point in the lower half-plane $\R\times (-\infty,0]$ belongs to any Voronoi cell. In the above definition and the rest of the paper we ignore the neutral cell.
We now recall the definition of strictly and uniformly convex spaces.
\begin{defin}\label{def:UniformlyConvex}\label{page:UniConvDef}
A normed space $(\widetilde{X},|\cdot|)$ is said to be strictly convex if for all $x,y\in \wt{X}$ satisfying $|x|=|y|=1$ and $x\neq y$, the inequality $|(x+y)/2|<1$ holds. $(\widetilde{X},|\cdot|)$ is said to be uniformly convex if for any $\epsilon\in (0,2]$ there exists $\delta\in (0,1]$ such that for all $x,y\in \wt{X}$, if $|x|=|y|=1$ and $|x-y|\geq \epsilon$, then $|(x+y)/2|\leq 1-\delta$.
\end{defin}
Roughly speaking, if the space is uniformly convex, then for any $\epsilon>0$ there exists a uniform positive lower bound on how deep the midpoint between any two unit vectors must penetrate the unit ball, assuming the distance between them is at least $\epsilon$. In general normed spaces the penetration is not necessarily positive, since the unit sphere may contain line segments. $\R^2$ with the max norm $|\cdot|_{\infty}$ is a typical example for this. A uniformly convex space is always strictly convex, and if it is also finite dimensional, then the converse is true too. The $m$-dimensional Euclidean space $\R^m$, or more generally, inner product spaces, the sequence spaces $\ell_p$, the Lebesgue spaces $L_p(\Omega)$ ($1<p<\infty$), and a uniformly convex product of a finite number of uniformly convex spaces, are all examples of uniformly convex spaces. See Clarkson \cite{Clarkson} and, for instance, Goebel-Reich \cite{GoebelReich} and Lindenstrauss-Tzafriri \cite{LindenTzafriri} for more information about uniformly convex spaces.
From the definition of uniformly convex spaces we can obtain a function which
assigns to the given $\epsilon$ a corresponding value $\delta(\epsilon)$. There are several ways to obtain such a function, but for our purpose we only need $\delta$ to be increasing, and to satisfy $\delta(0)=0$ and $\delta(\epsilon)>0$ for any $\epsilon\in (0,2]$. One choice, which is not necessarily the most convenient one, is the modulus of convexity, which is the function $\delta:[0,2]\to[0,1]$ defined by
\begin{equation*}
\displaystyle{\delta(\epsilon)=\inf\{1-|(x+y)/2|: |x-y|\geq \epsilon,\,|x|=|y|=1\}}.\label{eq:delta}
\end{equation*}
For specific spaces we can take more convenient functions. For instance, for the spaces $L_p(\Omega)$ or $\ell_p\,$, $1<p<\infty$, we can take
\begin{equation*}
\begin{array}{l}
\delta(\epsilon)=1-(1-\left(\epsilon/2)^p\right)^{1/p},\,\, \textnormal{for}\,\,p\geq 2,\\
\delta(\epsilon)=1-\left(1-(\epsilon/2)^q\right)^{1/q},\,\, \textnormal{for}\,\,1<p\leq 2\,\,
\textnormal{and}\,\,\frac{1}{p}+\frac{1}{q}=1.
\end{array}
\end{equation*}
We finish this section with the definition of the Hausdorff distance, a definition which is essential for the rest of the paper.
\begin{defin}\label{def:Hausdorff}
Let $(X,d)$ be a metric space. Given two nonempty sets $A_1,A_2\subseteq X$, the Hausdorff distance between them is defined by
\begin{equation*}
D(A_1,A_2)=\max\{\sup_{a_1\in A_1}d(a_1,A_2),\sup_{a_2\in A_2}d(a_2,A_1)\}.
\end{equation*}
\end{defin}
Note that the Hausdorff distance $D(A_1,A_2)$ is definitely different from the usual distance $d(A_1,A_2)=\inf\{d(a_1,a_2): a_1\in A_1,\,a_2\in A_2\}$. As a matter of fact, $D(A_1,A_2)\leq \epsilon$ if and only if $d(a_1,A_2)\leq \epsilon$ for any $a_1\in A_1$, and $d(a_2,A_1)\leq \epsilon$ for any $a_2\in A_2$. In addition, if $D(A_1,A_2)<\epsilon$, then for any $a_1\in A_1$ there exists $a_2\in A_2$ such that $d(a_1,a_2)<\epsilon$, and for any $b_2\in A_2$ there exists $b_1\in A_1$ such that $d(b_2,b_1)<\epsilon$. These properties explain why the Hausdorff distance is the natural distance to be used when discussing approximation and stability in the context of sets: suppose that our resolution is at most $r$, i.e., we are not able to distinguish between two points whose distance is at most some given positive number $r$. If it is known that $D(A_1,A_2)<r$, then we cannot distinguish between the sets $A_1$ and $A_2$, at least not by inspections based only on distance measurements. As a result of the above discussion, the intuitive phrase ``two sets have almost the same shape'' can be formulated precisely: the Hausdorff distance between the sets is smaller than some given positive parameter (note that a set and a rigid transformation of it usually have different shapes).
\section{Exact formulation of the main question and informal formulation of the main result}\label{sec:FormalInformal}
The exact formulation of Question \ref{ques:main} is based on the concept of Hausdorff distance for reasons which were explained at the end of the previous section.
\begin{question}
Suppose that $(P_k)_{k\in K}$ is a tuple of non-empty sets in $X$. Let $(R_k)_{k\in K}$ be the corresponding Voronoi diagram. Is it true that a small change of the sites yields a small change in the corresponding Voronoi cells, where both changes are measured with respect to the Hausdorff distance? More precisely, is it true that for any $\epsilon>0$ there exists $\Delta>0$ such that for any tuple $(P'_k)_{k\in K}$, the condition $D(P_k,P'_k)<\Delta$ for each $k\in K$ implies that $D(R_k,R'_k)<\epsilon$ for each $k\in K$, where $(R'_k)_{k\in K}$ is the Voronoi diagram of $(P'_k)_{k\in K}$?
\end{question}
The main result (Theorem \ref{thm:stabilityUC}) says that the answer is positive. Here is an informal description of it:
\begin{answer}
Suppose that the underlying subset $X$ is a closed and convex set of a (possibly infinite dimensional) uniformly convex normed space $\wt{X}$. Suppose that a certain boundedness condition on the distance between points in $X$ and the sites holds, e.g., when $X$ is bounded or when the sites form a (distorted) lattice. If there is a common positive lower bound on the distance between the sites, and the distance to each of them is attained, then indeed a small enough change of the (possibly infinitely many) sites yields a small change of the corresponding Voronoi cells, where both changes are measured with respect to the Hausdorff distance; in other words, the shapes of the real cells and the corresponding perturbed ones are almost the same. Moreover, explicit bounds on the changes can be derived and they hold simultaneously for all the cells. There are counterexamples which show that the assumptions imposed above are crucial.
\end{answer}
The condition that the distance to a site is attained holds, e.g., when the site is either a closed subset contained in a (translation of a) finite dimensional space, or a compact set, or a convex and closed subset in a uniformly convex Banach space. The sites can always be assumed to be closed, since the distance and the Hausdorff distance preserve their values when the involved subsets are replaced by their closures. The ``certain boundedness condition on the distance between points in $X$ and the sites'' is a somewhat technical condition expressed in \eqref{eq:BallRhokThm} (see also Remark \ref{rem:rho}).
\section{ The relevance of the main result}\label{sec:examples}
In Section \ref{sec:Intro} we explained why Question \ref{ques:main} is natural and fundamental, and mentioned the real-world example of a Voronoi diagram induced by shops/cellular antennas. The goal of this section is to illustrate further the relevance of the main result using a (far from being exhaustive) list of real-world and theoretical exampls. In these examples the shape or the position of the real sites are obviously approximated, and the main result (Theorem \ref{thm:stabilityUC}) ensures that the approximate Voronoi cells and the real ones have almost the same shape, and no unpleasant phenomenon such as the one described in Figures \ref{fig:InStability000}-\ref{fig:InStability1Full} can occur.
One example is in molecular biology for modeling the proteins structure
(Richards \cite{Richards}, Kim et al. \cite{KKCRCP}, Bourquard et al. \cite{VoronoiBiology2}), where the sites are either the atoms of a protein or special selected points in the amino acids and they are approximated by spheres/points. Another example is related to collision detection and robot motion (Goralski-Gold \cite{GoralskiGold}, Schwartz et al.
\cite{SchwartzSharirHopcroft}), where the sites are the (static or dynamic) obstacles located in an environment in which a vehicle/airplane/ship/robot/satellite should move. A third example is in solid state physics (Ashcroft-Mermin \cite{AshcroftMermin}; here the common terms are ``the first Brillouin zone'' or ``the Wigner-Seitz cell'' instead of ``the Voronoi cell''), where the sites are infinitely many point atoms in a (roughly) periodic structure which represents a crystal. A fourth example is in material engineering (Li-Ghosh \cite{LiGhosh}), where the sites are cracks in a material.
A fifth example is in numerical simulations of various dynamical phenomena, e.g., gas, fluid or urban dynamics (Slotterback et al. \cite{GranularMatter}, Mostafavi et al. \cite{VoronoiSpatial}). Here the sites are certain points/shapes taken from the sampled data of the simulated phenomena, and the cells help to cluster and analyze the data continuously. A sixth example is in astrophysics (Springel et al. \cite{DarkMatterGalactic}) where the (point) sites are actually huge regions in the universe (of diameter equals to hundreds of light years) used in simulations performed for understanding the behavior of (dark) matter in the universe. A seventh example is in image processing and computer graphics, where the sites are either certain important features/parts in an image (Tagare et al. \cite{TagareJaffeDuncan}, Dobashi et al. \cite{DobashiHagaJohanNishita}, Sabha-Dutr\'e \cite{SabhaDutre}) used for processing/analyzing it, or they are points form a useful configuration such as (an approximate) centroidal Voronoi diagram (CVD) which induces cells having good shapes (Du et al. \cite{VoronoiCVD_Review}, Liu et al. \cite{Graphics_CVD}, Faustino-Figueiredo \cite{FaustinoFigueiredo}).
An eighth example is in computational geometry, and it is actually a large collection of familiar problems in this field where Voronoi cells appear
and being used, possibly indirectly: (approximate) nearest neighbor searching/the post office problem, cluster analysis, (approximate) closest pairs, motion planning, finding (approximate) minimum spanning trees, finding good triangulations, and so on. See, e.g., Aurenhammer \cite{Aurenhammer}, Aurenhammer-Klein \cite{AurenhammerKlein}, Clarkson \cite{ClarksonNN}, Indyk \cite{Indyk}, and Okabe et al. \cite{OBSC}. Here the sites are either points or other shapes, and the space is usually $\R^n$ with some norm. In some of the above problems our stability result is clearly related because of the analysis being used (e.g., cluster analysis) or because the position/shapes of the sites are approximated (e.g., motion planning, the post office problem). However, it may be related also because in many of the previous problems the difficulty/complexity of finding an exact solution is too high, so one is forced to use approximate algorithms, to impose a general position assumption, and so on. Now, by perturbing slightly the sites (moving points, replacing a non-smooth curve by an approximating smooth one, etc.,) one may obtain much simpler and tractable configurations and now, using the geometric stability of the Voronoi cells, one may estimate how well the obtained solution approximates the best/real one.
As for a theoretical example of a different nature, we mention Kopeck\'a et al. \cite{KopeckaReemReich} in which the stability results described here have been used, in particular, for proving the existence of a zone diagram (a concept which was first introduced by Asano et al. \cite{AMTn} in the Euclidean plane with point sites) of finitely many compact sites which are strictly contained in a (large) compact and convex subset of a uniformly convex space, and also for proving interesting properties of Voronoi cells there.
Another example is for the infinite dimensional Hilbert space $L_2(I)$ for some $I$ (perhaps an interval or a finite dimensional space): functions in it are used in signal processing and in many other areas in science and technology. In practice the signals (functions) can be distorted, e.g., because of noise, and in addition, they are approximated by finite series (e.g., finite Fourier series) or integrals (e.g., Fourier transform). Given several signals, the (approximate) Voronoi cell of a given signal may help, at least in theory, to cluster or analyze data related to the sites. Such an analysis can be done also when the signal is considered as a point in a finite dimensional space of a high dimension. See, for instance, Conway-Sloane \cite[pp. 66-69, 451-477]{ConwaySloane} (coding) and Shannon \cite{Shannon} (communication) for a closely related discussion (in \cite{Shannon} Voronoi diagrams are definitely used in various places, but without their explicit name).
We mention several additional examples related to our stability result, sometimes in a somewhat unexpected way. For instance, Voronoi diagrams of infinitely many sites generated by a Poisson process (Okabe et al. \cite[pp. 39, 291-410]{OBSC}), Voronoi diagrams of atom nuclei used for the mathematical analysis of stability phenomena in matter (Lieb-Yau \cite{LiebYau}), Voronoi diagrams of infinitely many lattice points in a multi-dimensional Euclidean space which appear in the original works of Dirichlet \cite{Dirichlet} and Voronoi \cite{Voronoi} (see also Groemer \cite{Groemer} and Gruber-Lekkerkerker \cite{GruberLek} regarding the geometry of numbers and quadratic forms; Groemer used his stability result for deriving the Mahler compactness theorem \cite{Mahler}),
and packing problems such as the Kepler conjecture and the Dodecahedral conjecture (Hales \cite{HalesKepler},\cite{KeplerDCG}, Hales-McLaughlin \cite{HalesMcLaughlin}; because of continuity arguments needed in the proof) or those described in Conway-Sloane \cite{ConwaySloane}.
\section{The main result and some aspects related to its proof}\label{sec:Outline}
In this section we formulate the main result and discuss briefly issues related to its proof. See also the remarks after
Theorem \ref{thm:stabilityUC} for several relevant clarifications.
\begin{thm}\label{thm:stabilityUC}
Let $(\wt{X},|\cdot|)$ be a uniformly convex normed space. Let $X\subseteq \wt{X}$ be closed and convex. Let $(P_k)_{k\in K}$, $(P'_k)_{k\in K}$ be two given tuples of nonempty subsets of $X$ with the property that the distance between each $x\in X$ and each $P_k,P'_k$ is attained. For each $k\in K$ let $A_k=\bigcup_{j\neq k}P_j,\,A'_k=\bigcup_{j\neq k}P'_j$. Suppose that the following conditions hold:
\begin{equation}\label{eq:eta}
\eta:=\inf\{d(P_k,P_j): j,k\in K, j\neq k\}>0,
\end{equation}
\begin{multline}\label{eq:BallRhokThm}
\exists \rho\in (0,\infty)\,\, \textnormal{such that for all}\,\,k\in K\,\,\textnormal{and for all}\,\, x\in X\,\,\\
\textnormal{the open ball}\,\, B(x,\rho)\,\,\textnormal{intersects} \,\,A_k.
\end{multline}
For each $k\in K$ let $R_k=\dom(P_k,A_k),R'_k=\dom(P'_k,A'_k)$ be, respectively, the Voronoi cells associated with the original site $P_k$ and the perturbed one $P'_k$. Then for each $\epsilon\in (0,\eta/6)$ there exists $\Delta>0$ such that if $D(P_k,P'_k)<\Delta$ for each $k\in K$, then $D(R_k,R'_k)<\epsilon$ for each $k\in K$.
\end{thm}
See Figures \ref{fig:Stability_0005_lpi_Before}, \ref{fig:Stability_0005_lpi_After} for an illustration. The pictures were produced using the algorithm described in \cite{ReemISVD09}.
\begin{figure*}[t]
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.75]{figure3.eps}}
\end{center}
\caption{Illustration of Theorem \ref{thm:stabilityUC}: five sites in a square in $(\R^2,\ell_p)$ where the parameter is $p=3.14159$.}
\label{fig:Stability_0005_lpi_Before}
\end{minipage
\hfill
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.75]{figure4.eps}}
\end{center}
\caption{The sites have been slightly perturbed: the two points have merged, the ``sin'' has shrunk, and so on. The cells have been slightly perturbed.}
\label{fig:Stability_0005_lpi_After}
\end{minipage
\end{figure*}
\begin{remark}\label{rem:rho}
The assumption mentioned in \eqref{eq:BallRhokThm} may seem somewhat complicated at a first glance, but it actually expresses a certain uniform boundedness condition on the distance between any point in $X$ to its neighbor sites. No matter which point $x\in X$ and which site $P_k$ are chosen, the distance between $x$ and the collection of other sites $P_j,\,j\neq k$ cannot be arbitrary large. A sufficient condition for it to hold is when a uniform bound on the diameter of the cells (including the neutral one, if it is nonempty) is known in advance, e.g., when $X$ is bounded or when the sites form a (distorted) lattice. But \eqref{eq:BallRhokThm} can hold even if the cells are not bounded, e.g., when the setting is the Euclidean plane and $P_k=\R\times \{k\}$ where $k$ runs over all integers.
\end{remark}
\begin{remark}\label{rem:Delta}
In general, we have $\Delta=O(\epsilon^2)$. However, if there is a positive lower bound on the distance between the sites and the boundary of $X$ (relative to the affine hull spanned by $X$), i.e., if the sites are strictly contained in the (relative) interior of $X$, then actually the better estimate $\Delta=O(\epsilon)$ holds. The constants inside the big $O$ can be described explicitly: when $\Delta=C\epsilon^2$ we can take
\begin{equation*
C=\frac{1}{16(\rho+5\eta/12)}\cdot\delta\left(\frac{\eta}{12\rho+5\eta}
\right),
\end{equation*}
and when $\Delta=C\epsilon$ we can take
\begin{equation*
C=\min\left\{\displaystyle{ \frac{1}{16}\delta\left(\frac{\eta}{12\rho+5\eta}\right), \frac{d(\bigcup_{k\in K}P_k,\partial X)}{8(\rho+\eta/6)}}\right\}.
\end{equation*} \\
In the second case, in addition to $\epsilon <\eta/6$, the inequality
$\epsilon\leq 8\cdot d(\bigcup_{k\in K}P_k,\partial X)$ should be satisfied too.
\end{remark}
The proof of Theorem \ref{thm:stabilityUC} is quite long and technical, and hence it is given in the appendix. Despite this, we want to say a few words about the proof and some of the difficulties which arise along the way. First, as the counterexamples mentioned in Section \ref{sec:CounterExamples} show, one must take into account all the assumptions mentioned in the formulation of the theorem.
Second, in order to arrive to the generality described in the theorem,
one is forced to avoid many familiar arguments used in computational geometry and elsewhere, such as Euclidean arguments (standard angles, trigonometric functions, normals, etc.), arguments based on lower envelopes and algebraic arguments (since the intersections between the graphs generating the lower envelope may be complicated and since the boundaries of the cells may not be algebraic), arguments based on continuous motion of points, arguments based on finite dimensional properties such as compactness (since in infinite dimensional spaces closed and bounded balls are not compact), arguments based on finiteness (since we allow infinitely many sites and sites consist of infinitely many points) and so on. Our way to overcome these difficulties is to use a new representation theorem for dominance regions as a collection of line segments (Theorem \ref{thm:domInterval} below) and a new geometric lemma (Lemma \ref{lem:StrictSegment} below) which enables us to arrive to the explicit bounds mentioned in the theorem. As a matter of fact, we are not aware of any other way to obtain these explicit bounds even in a square in the Euclidean plane with point sites. These tools also allow us to overcome the difficulty of a potential infinite accumulated error due to the possibility of infinitely many sites/sites with infinitely many points/infinite dimension.
\begin{thm}\label{thm:domInterval}
Let $X$ be a closed and convex subset of a normed space $(\widetilde{X},|\cdot|)$, and
let $P,A\subseteq X$ be nonempty. Suppose that for all $x\in X$ the distance between $x$ and $P$ is attained. Then $\dom(P,A)$ is a union of line segments starting at the points of $P$. More precisely, given $p\in P$ and a unit vector $\theta$, let
\begin{multline*
T(\theta,p)=\sup\{t\in [0,\infty): p+t\theta\in X\,\,\mathrm{and}\,\
d(p+t\theta,p)\leq d(p+t\theta,A)\}.
\end{multline*}
Then
\begin{equation*}\label{eq:dom}
\dom(P,A)=\bigcup_{p\in P}\bigcup_{|\theta|=1}[p,p+T(\theta,p)\theta].
\end{equation*}
When $T(\theta,p)=\infty$, the notation $[p,p+T(\theta,p)\theta]$ means the ray $\{p+t\theta: t\in [0,\infty)\}$.
\end{thm}
\begin{lem}\label{lem:StrictSegment}
Let $(\widetilde{X},|\cdot|)$ be a uniformly convex normed space and let
$A\subseteq \widetilde{X}$ be nonempty. Suppose that $y,p\in \widetilde{X}$ satisfy $d(y,p)\leq d(y,A)$ and $d(p,A)>0$. Let $x\in [p,y)$. Let $\sigma\in (0,\infty)$ be arbitrary. Then $d(x,p)<d(x,A)-r$ for any $r>0$ satisfying
\begin{equation*
r\leq\!\min\!\left\{\sigma,\frac{4d(p,A)}{10},d(y,x)\delta\left(\frac{d(p,A)}{10(d(x,A)+\sigma+d(y,x))}\right)\!\right\}\!.
\end{equation*}
\end{lem}
The proof of Lemma \ref{lem:StrictSegment} is based on the strong triangle inequality of Clarkson
\cite[Theorem 3]{Clarkson}. It is interesting to note that although this inequality was formulated in the very famous paper \cite{Clarkson} of Clarkson, it seems that it has been almost totally forgotten, and in fact, despite a comprehensive search we have made in the literature, we have found evidences to its existence only in \cite{Clarkson} and later in \cite{Plant}.
\section{Counterexamples}\label{sec:CounterExamples}
\begin{figure*}
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.8]{figure5.eps}}
\end{center}
\caption{Four sites in a square in $(\R^2,\ell_{\infty})$. The cell of $P_1=\{(0,0)\}$ is displayed. The other sites
are $\,P_2=\{(2,0)\}$,$\,P_3=\{(-2,0)\}$, $\,P_4=\{(0,-2)\}$.}
\label{fig:InStability000} \label{page:InStability000}
\end{minipage
\hfill
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.8]{figure6.eps}}
\end{center}
\caption{Now either $P_4$ is the square $[-\beta,\beta]\times [-2-\beta,-2+\beta]$ or $P_1=\{(\beta,\beta)\}$, $\beta>0$ arbitrary small. The two lower rays have disappeared. No stability.}
\label{fig:InStability001}
\end{minipage
\hfill
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.8]{figure7.eps}
\end{center}
\caption{The full diagram of Figure \ref{fig:InStability000}. Note the large intersection between cells 1,2, and 3. For emphasizing this intersection, each cell is represented as a union of rays (see Theorem~ \ref{thm:domInterval} for more information) and some rays were emphasized.}
\label{fig:InStability0Full} \label{page:InStability000}
\end{minipage
\hfill
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
{\includegraphics[scale=0.8]{figure8.eps}}
\end{center}
\caption{The full diagram of Figure \ref{fig:InStability001} when $P_1=\{(\beta,\beta)\}$. Cells 2,3 have been (significantly) changed too.}
\label{fig:InStability1Full}
\end{minipage
\end{figure*}
In this section we mention several counterexamples which show that the assumptions in Theorem \ref{thm:stabilityUC} are essential.
If the space is not uniformly convex, then the Voronoi cells may not be stable as shown in Figures
\ref{fig:InStability000}-\ref{fig:InStability1Full}. Here the setting is point sites in a square in $\R^2$ with the max norm.
The positive common lower bound expressed in \eqref{eq:eta} is necessary even in a square in the Euclidean plane. Consider $X=[-10,10]^2$, $P_{1,\beta}=\{(0,\beta)\}$ and $P_{2,\beta}=\{(0,-\beta)\}$, where $\beta\in [0,1]$. As long as $\beta>0$, the cell $\dom(P_{1,\beta},P_{2,\beta})$ is the upper half of $X$. However, if $\beta=0$, then $\dom(P_{1,0},P_{2,0})$ is $X$. A more interesting example occurs when considering in $X$ the rectangle $P_{1,\beta}=[-a,a]\times [-10,-\beta]$ and the line segment $P_2=[-10,10]\times \{0\}$, where $a,\beta\in [0,1]$. If $\beta=0$, then $d(P_{1,0},P_2)=0$, and the cell $\dom(P_{1,0},P_2)$ contains the rectangle $[-a,a]\times [0,10]$. However, if $\beta>0$, then this cell does not contain this rectangle.
The assumption expressed in \eqref{eq:BallRhokThm} is essential even in the Euclidean plane with two points. Indeed, given $\beta\geq 0$, let $P_{1,\beta}=\{(\beta,1)\}$ and $P_2=\{(0,-1)\}$. Then $\dom(P_{1,0},P_2)$ is the upper half space. However, if $\beta>0$, then the half space $\dom(P_{1,\beta},P_2)$ contains points $(x,y)$ with $y\to -\infty$. Thus the Hausdorff distance between the original cell $\dom(P_{1,0},P_2)$ and the perturbed one $\dom(P_{1,\beta},P_2)$ is $\infty$, so there can be no stability.
\section{Concluding Remarks}\label{sec:Concluding}
We conclude the paper with the following remarks.
First, despite the counterexamples mentioned above, some of the assumptions can be weakened, with some caution. For instance, under a certain compactness assumption and a certain geometric condition that the sites should satisfy, the main result can be generalized to normed spaces which are not uniformly convex. A particular case of these assumptions is when the space is $m$-dimensional with the $|\cdot|_{\infty}$ norm, the sites are positively separated, and no two points of different sites are on a hyperplane perpendicular to one of the standard axes.
Second, an interesting (but not immediate) consequence of the main result is that it implies the stability of the (multi-dimensional) volume, namely a small change in the sites yields a small change in the volumes of the corresponding Voronoi cells.
Third, it can be shown that the function $T$ defined in Theorem~ \ref{thm:domInterval} also has a certain continuity property if the space is uniformly convex and this expresses a certain continuity property of the boundary of the cells.
Fourth, the estimate for $\Delta$ from Remark \ref{rem:Delta} is definitely not optimal and it can be improved, but, as simple examples show, $\Delta$ cannot be too much larger and its estimate should be taken into account when performing a relevant analysis. There is nothing strange in this and the situation is analogous to the familiar case of real valued functions. For instance, consider $f:\R\to\R$ defined by $f(x)=0$, $\,x<0$, $f(x)=(1/\beta)x,\,x\in [0,\beta]$, $f(x)=1,\,\,x>\beta$, where $\beta>0$ is given. Although $f$ is continuous, a ``large'' change near $x=0$ (of more than $\beta$) will cause a large change to $f$.
\section*{Acknowledgments}
I want to thank Dan Halperin, Maarten L{\"o}ffler, and Simeon Reich for helpful discussions regarding some of the references. I also thank all the reviewers for their comments.
\bibliographystyle{amsplain}
| {'timestamp': '2011-04-08T02:00:21', 'yymm': '1103', 'arxiv_id': '1103.4125', 'language': 'en', 'url': 'https://arxiv.org/abs/1103.4125'} |
\section{Introduction}
The Land\'{e} g-factor or gyromagnetic factor $g$ is described by the formula between particle's magnetic moment $\vec{\mu}$ and it's spin $\vec{s}$: $\vec{\mu}=g (\mu_{B} / \hbar)\vec{s}$
where $\mu_{B}$ is the Bohr magneton. In the Dirac equation, the value of $g$ is $2$ for a point-like particle. Deviation from this value $a=\frac{(g-2)}{2}$ is called as the anomalous magnetic moment, and without anomalous and radiative corrections $a=0$. However, the anomalous magnetic moment $a_{e}$ of the electron was firstly obtained using radiative corrections by Schwinger as $a_{e}=\frac{\alpha}{2\pi}=0.001161$ \cite{1}. So far, the accuracy of the $a_{e}$ was examined by many theoretical and experimental studies. These studies have provided the most precise determination of fine-structure constant $\alpha_{QED}$, since $a_{e}$ is quite senseless to the strong and weak interactions. On the other hand, the anomalous magnetic moment $a_{\mu}$ of the muon enables testing the Standart Model (SM) and investigating alternative theories to the SM. The $a_{e}$ and $a_{\mu}$ can be obtained with high sensitivity through spin precession experiment. Otherwise, the spin precession experiment cannot be used to measure the anomalous magnetic moment $a_{\tau}$ of the tau, because of the relatively short lifetime $2.906 \times 10^{-13}$ s of tau \cite{tau}. So the current bounds of the $a_{\tau}$ are obtained by collision experiments. The theoretical value of the $a_{\tau}$ from QED is given as $a^{SM}_{\tau}=0.001177 $ \cite{2,3}.
The experimental bounds on the the $a_{\tau}$ are provided by the L3: $-0.052<a_{\tau}<0.058$ , OPAL: $-0.068<a_{\tau}<0.065$ and DELPHI: $-0.052<a_{\tau}<0.013$ collaborations at the LEP at $95\%$ C.L. \cite{4,5,6}.
CP violation was firstly observed in a small fractions of $K_{L}^{0}$ mesons decaying to two pions in the SM \cite{7}. This phenomenology in the SM can be easily introduced by the Cabibbo-Kobayashi-Maskawa mechanism in the quark sector \cite{8}. On the other hand, there is no CP violation in the lepton sector. However, CP violation in the quark sector causes a very small electric dipole moment of the leptons. At least to three-loop are required in order to produce a nonzero contributing in the SM and it's crude estimate is obtained as $|d_{\tau}|\leq 10^{-34}\, e\,cm$ \cite{9}.
If at least two of the three neutrinos have different mass values, CP violation in the lepton sector can occur as similar to the CP violation in the quark sector \cite{10}. There are many different models beyond the SM inducing to CP violation in the lepton sector. These models are leptoquark \cite{11,12}, SUSY \cite{13}, left-right symmetric \cite{14,bk} and more Higgs multiplets \cite{15,16}.
The bounds at $95\%$ C. L. on the anomalous electric dipole moment of the tau
yield by LEP experiments L3: $|d_{\tau}|<3.1 \times 10^{-16}\, e\,cm$, OPAL: $|d_{\tau}|<3.7 \times 10^{-16}\, e\,cm$, and DELPHI: $|d_{\tau}|<3.7 \times 10^{-16}\, e\,cm$. The most restrictive experimental bounds are obtained by BELLE: $-2.2<Re(d_{\tau})<4.5 \times (10^{-17}\, e\,cm)$ and
$-2.5<Im(d_{\tau})<0.8 \times (10^{-17}\, e\,cm)$. There are model dependent and independent studies on the anomalous dipole moments of the tau lepton in the literature \cite{47,phe1,phe2,phe3,phe4,phe5,phe7,phe8,phe9}.
We consider that difference between $a_{\tau}^{SM}$ ($d_{\tau}^{SM}$) and $a_{\tau}^{exp}$ ($d_{\tau}^{exp}$) can be reduced to determine precisely a new term proportional to $F_{2}$ ($F_{3}$) to the SM $\tau\tau\gamma$ vertex.
For this reason, the electromagnetic vertex factor of the tau lepton can be parameterized
\begin{eqnarray}
\Gamma^{\nu}=F_{1}(q^{2})\gamma^{\nu}+\frac{i}{2 m_{\tau}}F_{2}(q^{2}) \sigma^{\nu\mu}q_{\mu}+\frac{1}{2 m_{\tau}}F_{3}(q^{2}) \sigma^{\nu\mu}q_{\mu}\gamma^{5}
\end{eqnarray}
where $\sigma_{\nu\mu}=\frac{i}{2}(\gamma_{\nu}\gamma_{\mu}-\gamma_{\mu}\gamma_{\nu})$, $q$ is the momentum transfer to the photon and $m_{\tau}=1.777$ GeV is the mass of tau lepton. In the SM, at tree level, $F_{1}=1$, $F_{2}=0$ and $F_{3}=0$. Besides, in the loop effects arising from the SM and the new physics, $F_{2}$ and $F_{3}$ may be not equal to zero. For example, the anomalous coupling $F_{2}$ is given by
\begin{eqnarray}
F_{2}(0)=a_{\tau}^{SM}+a_{\tau}^{NP}
\end{eqnarray}
where $a_{\tau}^{SM}$ is the contribution of the SM and $a_{\tau}^{NP}$ is the contribution of the new physics \cite{c1,c2,c3,c4}.Therefore, the $q^{2}$-dependent form factors $F_{1}(q^{2}),F_{2}(q^{2})$ and $F_{3}(q^{2})$ in limit $q^{2} \rightarrow 0$ are given by,
\begin{eqnarray}
F_{1}(0)=1,\: F_{2}(0)=a_{\tau},\: F_{3}(0)=\frac{2m_{\tau}d_{\tau}}{e}.
\end{eqnarray}
The Compact Linear Collider (CLIC) is a proposed future $e^{+}e^{-}$ collider, designed to fulfill $e^{+}e^{-}$ collisions at energies from $0.5$ to $3$ TeV \cite{17}, and it is planned to be constructed with a three main stages research region \cite{18}. The CLIC has been extensively studied for interactions beyond the SM \cite{20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,37,38}. The CLIC enables to investigate the $\gamma\gamma$ and $\gamma e$ interactions by converting the original $e^{-}$ or $e^{+}$ beam into a photon beam through the laser backscattering procedure \cite{34x,35,36}. One of the other well-known applications of the CLIC is the $\gamma ^{*} \gamma^{*}$ process, where the emitted quasireal photon $\gamma^{*}$ is scattered with small angle from the beam pipe of $e^{-}$ or $e^{+}$ \cite{39,40,41,42,pes}. Since these photons have a low virtuality ($Q_{max}^{2}=2$\,GeV$^{2}$), they are almost on mass shell. $\gamma ^{*} \gamma^{*}$ processes can be described by equivalent photon approximation, i.e. using the Weizsacker-Williams approximation \cite{43,44,45,46,47,48,49,50,51,52,53,54,app}. Such processes have experimentally observed at the LEP, Tevatron and LHC \cite{d1,d2,d3,d4,d5,d6,d7}.
There are two reasons why we have chosen the CLIC in this work:
First, the observation of the most stringent experimental bound on the anomalous magnetic dipole moment of the tau lepton by using multiperipheral collision at the LEP through the process $e^{+}e^{-} \rightarrow e^{+}\tau \bar{\tau}e^{-}$ \cite{6}. Secondly, the importace of high center-of-mass energies to examine the electromagnetic properties of tau lepton since anomalous $\tau\tau\gamma$ couplings depend on more energy than SM $\tau\tau\gamma$ couplings at the tree level. Therefore, we investigate the potential of CLIC via the process $e^{+}e^{-} \rightarrow e^{+}\tau \bar{\tau}e^{-}$ to examine the anomalous magnetic and electric dipole moments of tau lepton.
\section{Cross sections and numerical analysis}
During calculations, the CompHEP-$4.5.1$ program was used by including the new interaction vertices \cite{55}. Also, the acceptance cuts were imposed as $|\eta_{\tau}|<2.5$ for pseudorapidity, $p_{T}^{\tau}>20 \: $ GeV for transverse momentum cut of the final state particles, $\Delta R_{\tau \bar{\tau}}>0.5$ the separation of final tau leptons.
We show the integrated total cross-section of the process
$e^{+}e^{-} \rightarrow e^{+}\gamma^{*} \gamma^{*} e^{-}\rightarrow e^{+}\tau \bar{\tau}e^{-}$ as a function of the anomalous couplings $F_{2}$ and $F_{3}$ in Fig. $1$ for three different center-of-mass energies. As can be seen in Fig. $1$, while the total cross section is symmetric for anomalous coupling $F_{3}$, it is nonsymmetric for $F_{2}$.
We estimate $95\%$ C. L. bounds on anomalous coupling parameters $F_{2}$ and $F_{3}$ using $\chi^{2}$ test. The $\chi^{2}$ function is described by the following formula
\begin{eqnarray}
\chi^{2}=\left(\frac{\sigma_{SM}-\sigma(F_{2},F_{3})}{\sigma_{SM}\delta}\right)^{2},
\end{eqnarray}
where $\delta=\sqrt{(\delta_{st})^{2}+(\delta_{sys})^{2}}$; $\delta_{st}=\frac{1}{\sqrt{N_{SM}}}$ is the statistical error and $\delta_{sys}$ is the systematic error.
The number of expected events is calculated as the signal $N=L_{int}\times BR \times \sigma$ where
$L_{int}$ is the integrated luminosity. The tau lepton decays roughly $35\%$ of the time leptonically and $65\%$ of the time to one or more hadrons. So we consider one of the tau leptons decays leptonically and the other hadronically for the signal. Thereby, we assume that branching ratio of the tau pairs in the final state to be $BR=0.46$.
There are systematic uncertainties in exclusive production at the lepton and hadron colliders. For the process $e^{+}e^{-} \rightarrow e^{+}e^{-}\tau^{+}\tau^{-}$, systematic errors are experimentally studied between $4.3\%$ and $9\%$ at the LEP
\cite{6,55x}. Recently, exclusive lepton production at the LHC has been examined and its systematic uncertainty is $4.8\%$ \cite{d4}. Also, the process $p p \rightarrow p\tau^{+}\tau^{-}p$ with $2\%$ of the total systematic error at the LHC has investigated phenomenologically in Ref. [19]. Therefore, the sensitivity limits on the anomalous magnetic and electric dipole moments of the tau lepton through the process $e^{+}e^{-} \rightarrow e^{+}e^{-}\tau^{+}\tau^{-}$ have calculated by considering three systematic errors: $2\%$, $5\%$ and $10\%$. On the other hand, there may occur an uncertainty arising from virtuality of $\gamma^{*}$ used in the Weizsacker-Williams approximation.
In Figs. $2$-$4$, we have calculated the integrated cross sections as a function of $F_{2}$ and $F_{3}$ for different $Q_{max}^{2}$ values. We can see from these figures the total cross section changes slightly with the variation of the $Q_{max}^{2}$ value. The sensitivity limits on the anomalous couplings $a_{\tau}$ and $d_{\tau}$ for different values of photon virtuality, center-of-mass energy and luminosity has been given in Table I. It has shown that the bounds on the anomalous couplings do not virtually change when $Q_{max}^{2}$ increases. Therefore, we can understand that the large values of $Q_{max}^{2}$ do not bring an important contribution to obtain sensitivity limits on the anomalous couplings \cite{4,5,51}.
In Tables II-IV, we show 95\% C.L. sensitivity bounds of the coupling $a_{\tau}$ and ${d_{\tau}}$
for various systematic uncertainties, integrated CLIC luminosities and center of mass energies. While calculating the table values, we assumed that at a given time, only one of the anomalous couplings deviated from the SM.
In Fig. $5$, we demonstrate the sensitivity contour plot at $95\%$ C.L. for the anomalous couplings $F_{2}$ and $F_{3}$ at the $\sqrt{s}=0.5$, $1.5$ and $3$ TeV with corresponding maximum luminosities through process $e^{+}e^{-} \rightarrow e^{+}\gamma^{*} \gamma^{*} e^{-}\rightarrow e^{+}\tau \bar{\tau}e^{-}$.
\section{Conclusions}
The CLIC as a $\gamma^{*} \gamma^{*}$ collider using the Weizsacker-Williams virtual photon fields of the $e^{-}$ and $e^{+}$ provides an ideal venue to investigate the electromagnetic moments of the tau lepton. For this reason, we have studied the potential of $e^{+}e^{-} \rightarrow e^{+}\gamma^{*} \gamma^{*} e^{-}\rightarrow e^{+}\tau \bar{\tau}e^{-}$ at the CLIC to examine the anomalous magnetic and electric dipole moments of the tau lepton. The findings of this study show that the CLIC can improve the sensitivity bounds on anomalous couplings electromagnetic dipole moments of tau lepton with respect to the LEP bounds.
\pagebreak
| {'timestamp': '2014-02-11T02:08:52', 'yymm': '1306', 'arxiv_id': '1306.5620', 'language': 'en', 'url': 'https://arxiv.org/abs/1306.5620'} |
\section{Introduction}
In this paper I will discuss an old, well-known and apparently
\textquotedblleft insignificant\textquotedblright\ problem that lies in the
foundation of the relativistic quantum mechanics or, more precisely, in the
foundation of the quantum electrodynamics. A story starts with the
celebrated paper of P.A.M. Dirac \cite{1} that describes relativistic motion
of a free propagating electron, and had tremendous success of
natural explanation of spin, correct non-relativistic limit, correct
coupling with the external magnetic field, correct gyromagnetic ratio and,
finally, prediction of positron.
The mathematical formalism used requires presence of the negative kinetic
energy states in order to obtain the complete orthonormal set of the
linearly independent fundamental solutions of the suggested equation.
Indeed, such states do not make sense from the physical point of view.
Since the Dirac equation is written in the Hamiltonian form, it allows us to
work in the Heisenberg representation and determine directly whether or not
a given observable is a constant of motion. Then we find, for example, that
the momentum is a conserved quantity as it should be. However, the orbital
momentum, as well as the spin, are not conserved separately and only the sum
of them is a constant of motion. Conventionally, the spin is associated with
the internal degree of freedom of the electron and therefore apparently has
nothing to do with isotropy of the space-time continuum (indeed, if we
assign to the quantum mechanical space-time point internal algebraic
structure, then this will be rather naturally expected result). Even more
surprising result \cite{2} is obtained if we consider velocity $\dot{\vec{x}}
$ in the Dirac formulation. Instantaneous group velocity of the electron has
only values $\pm c$ in spite of the non-zero rest mass of electron. In
addition, velocity of a free moving electron is not a constant of motion.
An analytic solution for the coordinate operator of a free propagating
electron was found by E. Schr\"{o}dinger \cite{3}. It turns out that in
addition to the uniform rectilinear motion consistent with the classical
electrodynamics, the Dirac electron executes oscillatory motion, which E.
Schr\"{o}dinger called \textit{Zitterbewegung}. Let us recall that entire
non-relativistic quantum mechanics was raised in order to explain the
absence of radiation during the oscillatory motion of the electron bounded
by the electric potential of the nucleus. Therefore, the Dirac theory of
electron contains a definite prediction that the free moving electron will
loose all a kinetic energy through electromagnetic radiation \cite{4}.
It is rather surprising that the \textit{Zitterbewegung Problem} attracted
only sporadic \cite{5, 6} attention during years of development of the
theory of quantum fields and efforts to achieve the unification of all
fundamental interactions. It was demonstrated \cite{6, 7} that the \textit{%
Zitterbewegung} oscillations are due solely to interference between the
positive- and negative- energy components in the wave packet. The \textit{%
Zitterbewegung} is completely absent for a wave packet made up exclusively
of positive energy plane wave solutions. It is clear from the above analysis
that if one achieves the reformulation of the Dirac equation such that the
complete orthonormal set of linearly independent solutions will contain only
positive energy states then the \textit{Zitterbewegung} oscillations will
disappear. Indeed, the charge-conjugated solutions, associated with the
positron, must be retained.
It has been known for a long time that the algebraic structure of Dirac
equation is closely related to the quadratic normal division algebra of
quaternions. Here we suggest a quaternionic reformulation of the Dirac
equations \cite{8}, as well as an additional set of similar equations
suitable for description of the free propagating quark motion. The main
effort is made to obtain equations with the intrinsic $SU\left( 2\right)
\,\,\otimes U\left( 1\right) $ local gauge invariance. In contrast with the
approach of S.L. Adler and others \cite{9, 10}, we consider the possibility
that the previously obtained quaternionic extension \cite{11} of the Hilbert
space description of quantum fields represents a consistent mathematical
framework for the electroweak unification scheme (a brief summary of
relevant results is given in the Appendix). It is obvious that in order to
achieve unification of all fundamental interactions, the algebraic extension
beyond the quaternions is needed. We demonstrate that mathematical structure
of the obtained equations of motion suggests that the required extension may
proceed through wave functions which possess three and seven phases, whereas
the scalar product remains complex. In that case the examples of
nonextendability to octonionic quantum mechanics \cite{9} are not valid.
\section{Equations of motion for fundamental fermions}
Let us consider the algebraic structure of the Dirac equation. The problem
is to achieve factorization of the energy-momentum relation
\begin{equation} \label{eq1}
E^2\,\, = \,\,p^2c^2\,\, + \,\,m^2c^4
\end{equation}
\noindent in such a way, that the correspondent Hamiltonian is the generator
of Abelian translations in time, which is expressed by the Schr\"{o}dinger
equation
\begin{equation} \label{eq2}
i\hbar \,\,\frac{\partial \psi }{\partial t}\,\, = \,\,H\psi .
\end{equation}
It was demonstrated by P.A.M. Dirac \cite{1} that in terms of the
two-dimensional commutative quadratic division algebra of complex numbers no
solution can be found. The problem requires intrinsically an extension of
the algebraic basis of the theory. The Dirac's solution of the problem,
\begin{equation}
i\hbar \,\,\frac{\partial \psi }{\partial t}\,\,=\,\,\frac{\hbar c}{i}{%
\alpha _{j}\frac{\partial \psi }{\partial x_{j}}}\,\,-\,\,\beta
\,\,mc^{2}\,\,\psi \,\,\equiv \,\,H\psi ,\text{ \ }\ \ \text{\ }j=1,2,3\text{%
\ } \label{eq3}
\end{equation}
\newpage
\noindent uses the generators of the $C_{4}$ Clifford algebra:
\begin{equation}
\begin{array}{l}
{\alpha _{i}\alpha _{k}\,\,+\,\,\alpha _{k}\alpha _{i}\,\,=\,\,2\delta _{ik}}
\\
{\alpha _{i}\beta \,\,+\,\,\beta \alpha _{i}\,\,=\,\,0} \\
{\alpha _{i}^{2}\,\,=\,\,\beta ^{2}\,\,=\,\,1.}%
\end{array}%
\quad {i,k\,\,=\,\,1,2,3} \label{eq4}
\end{equation}
However, such a drastic growth in algebra is only apparent. The true
physical content of the obtained result is expressed more distinctly if (\ref%
{eq3}) is written in the following form:
\begin{equation}
\begin{array}{l}
i\dfrac{1}{c}\,\,\dfrac{\partial \psi }{\partial t}\,\,+\,i\sigma _{j}\dfrac{%
\partial \psi }{\partial x_{j}}\,=\,\,\dfrac{mc}{\hbar }\phi \\
\\
-i\dfrac{1}{c}\,\,\dfrac{\partial \phi }{\partial t}\,\,+i\sigma _{j}\dfrac{%
\partial \phi }{\partial x_{j}}\,\,\,=\,\,-\dfrac{mc}{\hbar }\psi%
\end{array}%
\quad j=1,2,3 \label{eq5}
\end{equation}%
(we choose to work in the Weyl representation \cite{12}
\begin{equation}
\begin{array}{l}
{\alpha _{i}\,\,=\,\,\left( {{\begin{array}{*{20}c} {\sigma _i } \hfill & 0
\hfill \\ 0 \hfill & { - \sigma _i } \hfill \\ \end{array}}}\right) } \\
\\
{\beta \,\,=\,\,\left( {{\begin{array}{*{20}c} 0 \hfill & { - 1} \hfill \\ {
- 1} \hfill & 0 \hfill \\ \end{array}}}\right) } \\
\\
{\Psi \,\,=\,\,\left( {{\begin{array}{*{20}c} \psi \\ \phi \\ \end{array}}}%
\right) }%
\end{array}%
\quad i=1,2,3 \label{eq6}
\end{equation}
\noindent by the reason, which will be explained below).
In a precise analogy with the physical content of Maxwell's equations, we
again have to deal with two mutually connected waves, which propagate
together in a space with constant velocity.
Now, the full basis of the certain algebra ($C_2 $ Clifford algebra) is
symmetrically used in (\ref{eq5}). However, it is assumed that $i$ in (\ref%
{eq5}) commutes with $\sigma _i $, that is the algebra is defined over the
field of complex numbers. Therefore, the algebraic foundation of this
formulation is based on an eight-dimensional non-division algebra. In
addition, it is customary in the applications to continue working with
complex numbers as an abstract algebra, but for the $C_2 $ Clifford algebra,
one makes use of a representation (Pauli matrices) introducing into the
theory an asymmetry, which has neither mathematical nor physical
justification.
Now, I will demonstrate that the algebraic foundation of (\ref{eq5}) may be
reduced to a four-dimensional real quadratic division algebra of quaternions
and that the structure of Dirac equations is intrinsically connected with
the functional-analytical structures mentioned above.
First of all, let us substitute
\begin{equation} \label{eq7}
e_j \,\, = \,\, - i\sigma _j \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,j\,\, = \,\,1,2,3
\end{equation}
\noindent into (\ref{eq5}). Then%
\begin{equation}
\begin{array}{l}
i\dfrac{1}{c}\,\dfrac{\partial \psi }{\partial t}\,\,-e_{j}\dfrac{\partial
\psi }{\partial x_{j}}\,\,\,=\,\,\dfrac{mc}{\hbar }\phi \\
\\
-i\dfrac{1}{c}\,\dfrac{\partial \phi }{\partial t}\,\,-\,e_{j}\dfrac{%
\partial \phi }{\partial x_{j}}\,\,\,=\,\,-\dfrac{mc}{\hbar }\psi%
\end{array}%
\quad j=1,2,3 \label{eq8}
\end{equation}
\noindent or, equivalently,
\begin{equation}
\begin{array}{l}
\left\{ {\left( {{\begin{array}{*{20}c} 0 \hfill & {e_0 } \hfill \\ {e_0 }
\hfill & 0 \hfill \\ \end{array}}}\right) \,\dfrac{1}{c}\,\dfrac{\partial }{%
\partial t}\,-\left( {{\begin{array}{*{20}c} {e_j } \hfill & 0 \hfill \\ 0
\hfill & { - e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial }{\partial
x_{j}}}\right\} \,\left( {{\begin{array}{*{20}c} \psi \hfill \\ {\psi i}
\hfill \\ \end{array}}}\right) \\
\\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=\dfrac{mc%
}{\hbar }\,\,\left( {{\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0
\hfill & { - e_0 } \hfill \\ \end{array}}}\right) \left( {{%
\begin{array}{*{20}c} \phi \hfill \\ {\phi i} \hfill \\ \end{array}}}\right)
\\
\\
\left\{ {-\left( {{\begin{array}{*{20}c} 0 \hfill & {e_0 } \hfill \\ {e_0 }
\hfill & 0 \hfill \\ \end{array}}}\right) \dfrac{1}{c}\dfrac{\partial }{%
\partial t}-\left( {{\begin{array}{*{20}c} {e_j } \hfill & 0 \hfill \\ 0
\hfill & { - e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial }{\partial
x_{j}}}\right\} \,\left( {{\begin{array}{*{20}c} \phi \hfill \\ {\phi i}
\hfill \\ \end{array}}}\right) \, \\
\\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=-\dfrac{mc}{%
\hbar }\,\left( {{\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0 \hfill
& { - e_0 } \hfill \\ \end{array}}}\right) \left( {{\begin{array}{*{20}c}
\psi \hfill \\ {\psi i} \hfill \\ \end{array}}}\right)%
\end{array}%
\quad j=1,2,3 \label{eq9}
\end{equation}
\noindent and
\begin{equation}
\begin{array}{l}
\left\{ {\left( {{\begin{array}{*{20}c} 0 \hfill & { - e_0 } \hfill \\ {e_0
} \hfill & 0 \hfill \\ \end{array}}}\right) \,\dfrac{1}{c}\,\dfrac{\partial
}{\partial t}\,-\left( {{\begin{array}{*{20}c} {e_j } \hfill & 0 \hfill \\ 0
\hfill & {e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial }{\partial
x_{j}}}\right\} \,\left( {{\begin{array}{*{20}c} \psi \hfill \\ { - \psi i}
\hfill \\ \end{array}}}\right) \\
\\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=\dfrac{%
mc}{\hbar }\,\,\left( {{\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0
\hfill & {e_0 } \hfill \\ \end{array}}}\right) \left( {{%
\begin{array}{*{20}c} \phi \hfill \\ { - \phi i} \hfill \\ \end{array}}}%
\right) \\
\\
\left\{ {-\left( {{\begin{array}{*{20}c} 0 \hfill & { - e_0 } \hfill \\ {e_0
} \hfill & 0 \hfill \\ \end{array}}}\right) \dfrac{1}{c}\,\dfrac{\partial }{%
\partial t}-\left( {{\begin{array}{*{20}c} {e_j } \hfill & 0 \hfill \\ 0
\hfill & {e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial }{\partial
x_{j}}}\right\} \,\left( {{\begin{array}{*{20}c} \phi \hfill \\ { - \phi i}
\hfill \\ \end{array}}}\right) \\
\, \\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=-\dfrac{mc}{%
\hbar }\,\left( {{\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0 \hfill
& {e_0 } \hfill \\ \end{array}}}\right) \left( {{\begin{array}{*{20}c} \psi
\hfill \\ { - \psi i} \hfill \\ \end{array}}}\right) .%
\end{array}%
\quad j=1,2,3 \label{eq10}
\end{equation}
Notice that the states have the form (A1) and all operators have the form
(A11) and (A12).
Besides that, an additional (and only one) mass term is allowed:
\begin{equation*}
\begin{array}{l}
\left\{ {\left( {{\begin{array}{*{20}c} 0 \hfill & {e_0 } \hfill \\ {e_0 }
\hfill & 0 \hfill \\ \end{array}}}\right) \dfrac{1}{c}\,\dfrac{\partial }{%
\partial t}-\left( {{\begin{array}{*{20}c} {e_j } \hfill & 0 \hfill \\ 0
\hfill & { - e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial }{\partial
x_{j}}}\right\} \,\left( {{\begin{array}{*{20}c} \psi \hfill \\ {\psi i}
\hfill \\ \end{array}}}\right) \\
\\
\text{ \ \ }=\left\{ {\dfrac{m_{1}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0 \hfill & { - e_0 }
\hfill \\ \end{array}}}\right) \,\,+\,\,\dfrac{m_{2}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} 0 \hfill & {e_0 } \hfill \\ {e_0 } \hfill & 0 \hfill
\\ \end{array}}}\right) }\right\} \,\left( {{\begin{array}{*{20}c} \phi
\hfill \\ {\phi i} \hfill \\ \end{array}}}\right) \\
\\
\left\{ {-\left( {{\begin{array}{*{20}c} 0 \hfill & {e_0 } \hfill \\ {e_0 }
\hfill & 0 \hfill \\ \end{array}}}\right) \,\dfrac{1}{c}\,\dfrac{\partial }{%
\partial t}\,\,-\,\left( {{\begin{array}{*{20}c} {e_j } \hfill & 0 \hfill \\
0 \hfill & { - e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial }{%
\partial x_{j}}\,}\right\} \,\,\left( {{\begin{array}{*{20}c} \phi \hfill \\
{\phi i} \hfill \\ \end{array}}}\right) \, \\
\\
\text{ \ \ }=\left\{ {-\dfrac{m_{1}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0 \hfill & { - e_0 }
\hfill \\ \end{array}}}\right) \,\,+\,\,\dfrac{m_{2}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} 0 \hfill & {e_0 } \hfill \\ {e_0 } \hfill & 0 \hfill
\\ \end{array}}}\right) }\right\} \,\,\left( {{\begin{array}{*{20}c} \psi
\hfill \\ {\psi i} \hfill \\ \end{array}}}\right) \\
\end{array}%
\text{ \ \ \ \ }j=1,2,3
\end{equation*}
\begin{equation}
\begin{array}{l}
\left\{ {\left( {{\begin{array}{*{20}c} 0 \hfill & { - e_0 } \hfill \\ {e_0
} \hfill & 0 \hfill \\ \end{array}}}\right) \dfrac{1}{c}\,\dfrac{\partial }{%
\partial t}-\left( {{\begin{array}{*{20}c} {e_j } \hfill & 0 \hfill \\ 0
\hfill & {e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial }{\partial
x_{j}}}\right\} \,\left( {{\begin{array}{*{20}c} \psi \hfill \\ { - \psi i}
\hfill \\ \end{array}}}\right) \, \\
\\
\text{ \ \ }=\left\{ {\dfrac{m_{1}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0 \hfill & {e_0 } \hfill
\\ \end{array}}}\right) \,\,+\,\,\dfrac{m_{2}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} 0 \hfill & { - e_0 } \hfill \\ {e_0 } \hfill & 0
\hfill \\ \end{array}}}\right) }\right\} \,\left( {{\begin{array}{*{20}c}
\phi \hfill \\ { - \phi i} \hfill \\ \end{array}}}\right) \, \\
\\
\left\{ {-\left( {{\begin{array}{*{20}c} 0 \hfill & { - e_0 } \hfill \\ {e_0
} \hfill & 0 \hfill \\ \end{array}}}\right) \,\,\dfrac{1}{c}\,\dfrac{%
\partial }{\partial t}\,\,-\,\left( {{\begin{array}{*{20}c} {e_j } \hfill &
0 \hfill \\ 0 \hfill & {e_j } \hfill \\ \end{array}}}\right) \dfrac{\partial
}{\partial x_{j}}\,}\right\} \,\,\left( {{\begin{array}{*{20}c} \phi \hfill
\\ { - \phi i} \hfill \\ \end{array}}}\right) \, \\
\\
\text{ \ \ }=\left\{ {\dfrac{m_{1}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} {e_0 } \hfill & 0 \hfill \\ 0 \hfill & {e_0 } \hfill
\\ \end{array}}}\right) \,\,+\,\,\dfrac{m_{2}c}{\hbar }\,\,\left( {{%
\begin{array}{*{20}c} 0 \hfill & { - e_0 } \hfill \\ {e_0 } \hfill & 0
\hfill \\ \end{array}}}\right) }\right\} \,\,\left( {{\begin{array}{*{20}c}
\psi \hfill \\ { - \psi i} \hfill \\ \end{array}}}\right) .%
\end{array}%
\text{ \ \ \ }j=1,2,3 \label{eq11}
\end{equation}
It may be verified that the energy-momentum relation is not spoiled if one
defines
\begin{equation}
M\equiv \,\,\sqrt{m_{1}^{2}\,\,+\,\,m_{2}^{2}}. \label{eq12}
\end{equation}
Here we are forced to consider masses as given phenomenological parameters.
If $m_{1}\,\,\neq \,\,0$, the presence of this additional term does not
increase the number of fundamental plane wave solutions of the equations (%
\ref{eq11}). Therefore, we will consider the equations (\ref{eq11}) with $%
m_{1}\,\,=\,\,0$ as a separate independent set and in order to maintain the
direct connection with the Dirac equations, will neglect the $m_{2}$ term in
the presence of the non-vanishing $m_{1}$ term.
Indeed, only two equations (\ref{eq11}) are independent:
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \psi }{\partial t}i\,\,-\,e_{j}\,\dfrac{%
\partial \psi }{\partial x_{j}}\,\,=\,\,\dfrac{mc}{\hbar }\phi \\
\\
-\dfrac{1}{c}\,\,\dfrac{\partial \phi }{\partial t}i\,\,-e_{j}\,\dfrac{%
\partial \phi }{\partial x_{j}}\,\,=\,-\,\dfrac{mc}{\hbar }\psi .%
\end{array}%
\quad j=1,2,3 \label{eq13}
\end{equation}
The form (\ref{eq13}) is very convenient for the investigation of gauge
invariance group of the Dirac \noindent equations. The $U\,\,\left( 1\right)
$ gauge invariance group from the right is generated by the transformations
\begin{equation}
\begin{array}{l}
\psi ^{\prime }\,\,=\,\,\psi \,z,\,\,\,\,\,\phi ^{\prime }\,\,=\,\,\phi \,z
\\
\\
z\,\,=\,\,a\,\,+\,\,bi,\,\,\,\,\,\left\vert z\right\vert \,\,=\,\,1,\text{ }%
a,\,\,b\,\,\text{are real numbers}.%
\end{array}
\label{eq14}
\end{equation}
Since for every pair of solutions $\left( {\psi ,\,\,\phi } \right)$ of the
linear differential equations, $\left( {\psi a,\,\,\phi a} \right)$ ($a$ is
a real number) is also a solution, it is always enough to show that the
particular transformation
\begin{equation}
{\begin{array}{*{20}c} {\psi '\,\, = \,\,\psi \,i} \\ \\ {\phi '\,\, =
\,\,\phi \,i} \\ \end{array}}\,\,\,{\begin{array}{*{20}c} \hfill \\ {\left(
{\rm{i.e.}\,\,\,\,\,a\,\, = \,\,0,\,\,\,\,\,\,\,\,\,b\,\, = \,\,1} \right)}
\hfill \\ \hfill \\ \end{array}} \label{eq15}
\end{equation}
\noindent leaves the equations invariant.
Invariance of the equations (\ref{eq13}) with respect to this transformation
is obvious. Let us consider what is a left gauge invariance group of the
Dirac equations. Remember that (\ref{eq13}) are the equations for free
propagating waves and thus admit solutions of the form
\begin{equation}
\begin{array}{l}
\psi \,\,=\,\,U_{1}\,\,\exp \dfrac{-i\left( {Et\,\,-\,\,\vec{p}\vec{x}}%
\right) }{\hbar } \\
\\
\phi \,\,=\,\,U_{2}\,\,\exp \dfrac{-i\left( {Et\,\,-\,\,\vec{p}\vec{x}}%
\right) }{\hbar }.%
\end{array}
\label{eq16}
\end{equation}
Therefore, the $U\,\,\left( 1 \right)$ transformations
\begin{equation}
\begin{array}{l}
\psi ^{\prime }\,\,=\,\,z_{1}\psi ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\phi
^{\prime }\,\,=\,\,z_{1}\phi \\
\\
z_{1}\,\,=\,\,a\,\,+\,\,bi_{1},\,\,\,\,\,\,\,\,\,\left\vert {z_{1}}%
\right\vert \,\,=\,\,1 \\
\\
i_{1}\,\,\equiv \,\,\dfrac{e_{1}p_{1}\,\,+\,\,e_{2}p_{2}\,\,+\,%
\,e_{1}e_{2}p_{3}}{\left\vert \vec{p}\right\vert }%
\end{array}
\label{eq17}
\end{equation}
\noindent leave the equations (\ref{eq13}) invariant and constitute the left
gauge invariance group of the Dirac equations.
In order to see that, it is sufficient to show again that
\begin{equation}
\begin{array}{l}
\psi ^{\prime }\,\,=\,\,i_{1}\psi \\
\phi ^{\prime }\,\,=\,\,i_{1}\phi%
\end{array}
\label{eq18}
\end{equation}
\noindent is a solution of the equations (\ref{eq13}):
\begin{equation}
\begin{array}{l}
\dfrac{i_{1}U_{1}}{c}\left( {-iE}\right) \,\,i-\,e_{1}i_{1}U_{1}\,\,\left( {%
\,ip_{1}}\right) -\,e_{2}i_{1}U_{1}\,\left( {\,ip_{2}}\right)
-\,e_{1}e_{2}i_{1}U_{1}\,\left( {\,ip_{3}}\right) =\,mc\,\,i_{1}U_{2} \\
\\
\dfrac{i_{1}U_{2}}{c}\left( {iE}\right) \,\,i-\,e_{1}i_{1}U_{2}\,\,\left( {%
\,ip_{1}}\right) -\,e_{2}i_{1}U_{2}\,\left( {\,ip_{2}}\right)
-\,e_{1}e_{2}i_{1}U_{2}\,\left( {\,ip_{3}}\right) =-mc\,\,i_{1}U_{1}.%
\end{array}
\label{eq19}
\end{equation}
Then
\begin{equation}
\begin{array}{l}
\dfrac{i_{1}U_{1}E}{c}-\left( {e_{1}p_{1}+\,e_{2}p_{2}\,+\,e_{1}e_{2}p_{3}}%
\right) \,\,i_{1}U_{1}\,i=mc\,\,i_{1}U_{2} \\
\\
-\dfrac{i_{1}U_{2}E}{c}-\left( {e_{1}p_{1}+\,e_{2}p_{2}+\,e_{1}e_{2}p_{3}}%
\right) \,\,i_{1}U_{2}\,i=-mc\,\,i_{1}U_{1}.%
\end{array}
\label{eq20}
\end{equation}
By definition (see (\ref{eq17})),
\begin{equation}
e_{1}p_{1}\,+\,e_{2}p_{2}\,+\,e_{1}e_{2}p_{3}\,=\,i_{1}\,\,\left\vert \vec{p}%
\right\vert . \label{eq21}
\end{equation}
Therefore,
\begin{equation}
\begin{array}{l}
i_{1}\,\,\left[ {\dfrac{U_{1}E}{c}\,-\,\left( {e_{1}p_{1}\,+\,e_{2}p_{2}\,+%
\,e_{1}e_{2}p_{3}}\right) \,U_{1}i}\right] \,=\,i_{1}\,\left( {mc\,U_{2}}%
\right) \\
\\
i_{1}\,\,\left[ {-\dfrac{U_{2}E}{c}-\,\left( {%
e_{1}p_{1}+e_{2}p_{2}+e_{1}e_{2}p_{3}}\right) \,U_{2}i}\right]
=\,\,i_{1}\,\left( {-mc\,U_{1}}\right) .%
\end{array}
\label{eq22}
\end{equation}
It is assumed in the Dirac equations \cite{ 13}, that
\begin{equation} \label{eq23}
\left[ {i,e_j } \right]\,\, = \,\,0,\,\,\,\,\,\,\,\,\,j\,\, = \,\,1,2,3
\end{equation}
\noindent and hence the obtained gauge invariance group is $U\left( 1
\right)\,\, \otimes U\left( 1 \right)$. Now it becomes clear why the Dirac
equations allow us to incorporate an additional charge \cite{ 14} and turn
out to be suitable for the realization of the electroweak unification scheme
\cite{ 15} without contradiction with the Aharonov-Bohm effect \cite{ 16}.
However, the group-theoretical content of this scheme \cite{ 15}, side by
side with the functional-analytical structures \cite{ 11}, suggests that the
left gauge invariance group should be larger $\left( {\mbox{at}\,\,%
\mbox{least}\,\,U\,\,\left( {1;q} \right)\,\, \cong \,\,SU\left( 2 \right)}
\right)$ and should not contain an Abelian invariant subgroup. The simplest
way to satisfy these requirements is to identify the Abelian groups (\ref%
{eq14}) and (\ref{eq17}) discussed above, that is to attach to the Dirac
equations the following form:
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \psi }{\partial t}i\,\,-\,\,\dfrac{\partial
\psi }{\partial x_{j}}e_{j}\,\,=\,\,\dfrac{mc}{\hbar }\phi \\
\\
-\dfrac{1}{c}\,\,\dfrac{\partial \phi }{\partial t}i\,\,-\,\dfrac{\partial
\phi }{\partial x_{j}}e_{j}\,\,=\,-\,\dfrac{mc}{\hbar }\psi%
\end{array}%
\quad j=1,2,3 \label{eq24}
\end{equation}
\noindent and to drop the assumption (\ref{eq23}). Then the algebraic
foundation of the theory is reduced to a four-dimensional real quadratic
division algebra of quaternions.
The $U\left( 1 \right)$ right gauge invariance of the equations (\ref{eq24})
may be maintained~if
\begin{equation} \label{eq25}
i\,\, = \,\,\frac{e_1 p_1 \,\, + \,\,e_2 p_2 \,\, + \,\,e_1 e_2 p_3 }{\left|
\vec {p} \right|}
\end{equation}
\noindent and may be demonstrated exactly in the same way as (\ref{eq18}) - (%
\ref{eq22}).
Consequently, we have obtained the additional meaning for $i$, which appears
originally in the Schr\"{o}dinger equation. An algebra itself forms a vector
space, and the basis of algebra constitutes a suitable set of orthogonal
axes in that space, for example, a complex algebra may be considered as a
two-dimensional plane with the orthogonal directions 1 and $i$. In that
space $i$ standing in the left-hand side of the Schr\"{o}dinger equation
define the direction of the time translations, which form an Abelian group.
Therefore, the $U\left( 1\right) $ right gauge invariance of the equations (%
\ref{eq24}) leads us to the conclusion that these equations define (the
condition (\ref{eq25})) the direction of the time translations at the
three-dimensional quaternionic surface (the space of quantum mechanical
phases). Then, a possible physical interpretation is that, compared with a
classical relativistic particle, a quantum particle has not only its proper
time but, in addition, a proper direction of time. Perhaps, this may serve
as an explanation of why quantum equations of motion contain the first time
derivative, whereas the classical equations of motion are expressed in terms
of the second derivative.
We investigate now how our manipulations have affected the corresponding
solutions. As it is well known, the general solution of Dirac equation may
be formed as a linear combination of the four independent solutions, which
are four spinors with four components. Two of them are obtained for $E\,>0$
for two spin states $\,U_{2}^{(\ref{eq1})}\,=\left( {{%
\begin{array}{*{20}c}
1 \hfill \\
0 \hfill \\
\end{array}}}\right) $ and $\,U_{2}^{(\ref{eq2})}\,=\left( {{%
\begin{array}{*{20}c}
0 \hfill \\
1 \hfill \\
\end{array}}}\right) $, respectively. The other two we are forced to obtain
using $E\,\,$\TEXTsymbol{<} 0 since there are no other possibilities. They
correspond to the arbitrary choice of $\,U_{1}^{(\ref{eq3})}\,=\left( {{%
\begin{array}{*{20}c}
1 \hfill \\
0 \hfill \\
\end{array}}}\right) $ and $\,U_{1}^{(\ref{eq4})}\,=\left( {{%
\begin{array}{*{20}c}
0 \hfill \\
1 \hfill \\
\end{array}}}\right) $.
Let us check what happens in our quaternionic version of the Dirac equation.
In order to maintain connection with the original Dirac solutions, let us
form a complete orthonormal set of it:%
\begin{eqnarray}
\psi _{D}^{(\ref{eq1})}\,\, &=&\,\,\left( {{\begin{array}{*{20}c} {U_1
\,\,\left( \vec {p} \right)} \hfill \\ {U_1 \,\,\left( \vec {p} \right)i}
\hfill \\ \end{array}}}\right) \,\,\exp \frac{-i\,\,\left( {Et\,\,-\,\,\vec{p%
}\vec{x}}\right) }{\hbar } \notag \\
&& \notag \\
\psi _{D}^{(\ref{eq2})}\,\, &=&\,\,\left( {{\begin{array}{*{20}c} {U_2
\,\,\left( \vec {p} \right)} \hfill \\ {U_2 \,\,\left( \vec {p} \right)i}
\hfill \\ \end{array}}}\right) \,\,\exp \frac{-i\,\,\left( {Et\,\,-\,\,\vec{p%
}\vec{x}}\right) }{\hbar } \notag \\
&& \notag \\
\psi _{D}^{(\ref{eq3})}\,\, &=&\,\,\left( {{\begin{array}{*{20}c} {U_1
\,\,\left( \vec {p} \right)} \hfill \\ { - U_1 \,\,\left( \vec {p} \right)i}
\hfill \\ \end{array}}}\right) \,\,\exp \frac{-i\,\,\left( {Et\,\,-\,\,\vec{p%
}\vec{x}}\right) }{\hbar } \label{eq26} \\
&& \notag \\
\psi _{D}^{(\ref{eq4})}\,\, &=&\,\,\left( {{\begin{array}{*{20}c} {U_2
\,\,\left( \vec {p} \right)} \hfill \\ { - U_2 \,\,\left( \vec {p} \right)i}
\hfill \\ \end{array}}}\right) \,\,\exp \frac{-i\,\,\left( {Et\,\,-\,\,\vec{p%
}\vec{x}}\right) }{\hbar } \notag
\end{eqnarray}
\noindent we have
\begin{equation}
\begin{array}{l}
\left( {\dfrac{E}{c}\,\,+\,\,\left\vert \vec{p}\right\vert }\right)
\,\,U_{1}\,\,-\,\,mc\,\,U_{2}\,\,=\,\,0 \\
\\
-\,\,mc\,\,U_{1}\,\,+\,\,\left( {\dfrac{E}{c}\,\,-\,\,\left\vert \vec{p}%
\right\vert }\right) \,\,U_{2}\,\,=\,\,0.%
\end{array}
\label{eq27}
\end{equation}%
The existence of non-trivial solutions is ensured by
\begin{equation*}
\frac{E^2}{c^2}\,\, - \,\,\left| \vec {p} \right|^2\,\, - \,\,m^2c^2\,\, =
\,\,0
\end{equation*}
\noindent and
\begin{equation} \label{eq28}
U_1^{(1,2)} \,\, = \,\,\frac{mc^2}{E\,\, + \,\,c\left| \vec {p} \right|}%
U_2^{(1,2)} .
\end{equation}
Let $\,$%
\begin{equation*}
U_{2}^{(\ref{eq1})}\,=\left( {{\begin{array}{*{20}c} 1 \hfill \\ 0 \hfill \\
\end{array}}}\right) \text{ and}\,\ U_{2}^{(\ref{eq2})}\,=\left( {{%
\begin{array}{*{20}c} 0 \hfill \\ 1 \hfill \\ \end{array}}}\right) .
\end{equation*}
Then%
\begin{equation}
\begin{tabular}{l}
$\psi _{D}^{(\ref{eq1})}=\dfrac{mc^{2}N_{1}}{E+c\left\vert \vec{p}%
\right\vert }\left( {{\begin{array}{*{20}c} 1 \hfill \\ 0 \hfill \\
\end{array}}}\right) \otimes \left( {{\begin{array}{*{20}c} 1 \hfill \\ i
\hfill \\ \end{array}}}\right) \,\,\exp \dfrac{-i\,\,\left( {Et\,\,-\,\,\vec{%
p}\vec{x}}\right) }{\hbar }\,\,$ \\
$\ \ \ \ \ \ \ \ \ \ \ \ \ =\dfrac{mc^{2}N_{1}}{E+c\left\vert \vec{p}%
\right\vert }\left( {{\begin{array}{*{20}c} 1 \hfill \\ i \hfill \\ 0 \hfill
\\ 0 \hfill \\ \end{array}}}\right) \,\exp \dfrac{-i\,\,\left( {Et\,\,-\,\,%
\vec{p}\vec{x}}\right) }{\hbar }\,\,$ \\
\\
$\psi _{D}^{(\ref{eq2})}=\,\,N_{1}\left( {{\begin{array}{*{20}c} 0 \hfill \\
1 \hfill \\ \end{array}}}\right) \otimes \left( {{\begin{array}{*{20}c} 1
\hfill \\ i \hfill \\ \end{array}}}\right) \,\,\exp \dfrac{-i\,\,\left( {%
Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }\,$ \\
$\ \ \ \ \ \ \ \ \ \ \ \ \ =N_{1}\left( {{\begin{array}{*{20}c} 0 \hfill \\
0 \hfill \\ 1 \hfill \\ i \hfill \\ \end{array}}}\right) \,\exp \dfrac{%
-i\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }\,.$%
\end{tabular}
\label{eq29}
\end{equation}
Now
\begin{equation}
U_{2}^{(3,4)}\,\,=\,\,\frac{(E+c\left\vert \vec{p}\right\vert )}{mc^{2}}%
U_{1}^{(3,4)}. \label{eq30}
\end{equation}
Let $\,$%
\begin{equation*}
U_{1}^{(\ref{eq3})}\,=\left( {{\begin{array}{*{20}c} 1 \hfill \\ 0 \hfill \\
\end{array}}}\right) \text{ and }\,U_{1}^{(\ref{eq4})}\,=\left( {{%
\begin{array}{*{20}c} 0 \hfill \\ 1 \hfill \\ \end{array}}}\right) .
\end{equation*}
Then
\begin{equation}
\begin{tabular}{l}
$\psi _{D}^{(\ref{eq3})}\,=N_{2}\left( {{\begin{array}{*{20}c} 1 \hfill \\ 0
\hfill \\ \end{array}}}\right) \otimes \left( {{\begin{array}{*{20}c} 1
\hfill \\ { - i} \hfill \\ \end{array}}}\right) \,\,\exp \dfrac{-i\,\,\left(
{Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }\,$ \\
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ =N_{2}\left( {{\begin{array}{*{20}c} 1 \hfill
\\ { - i} \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{array}}}\right) \,\exp
\dfrac{-i\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }\,$ \\
$\psi _{D}^{(\ref{eq4})}=\dfrac{(E+c\left\vert \vec{p}\right\vert )N_{2}}{%
mc^{2}}\left( {{\begin{array}{*{20}c} 0 \hfill \\ 1 \hfill \\ \end{array}}}%
\right) \otimes \left( {{\begin{array}{*{20}c} 1 \hfill \\ { - i} \hfill \\
\end{array}}}\right) \,\exp \dfrac{-i\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x}}%
\right) }{\hbar }\,$ \\
$\ \ \ \ \ \ \ \ \ \ \ \ \ =\dfrac{(E+c\left\vert \vec{p}\right\vert )N_{2}}{%
mc^{2}}\left( {{\begin{array}{*{20}c} 0 \hfill \\ 0 \hfill \\ 1 \hfill \\ {
- i} \hfill \\ \end{array}}}\right) \exp \dfrac{-i\,\,\left( {Et\,\,-\,\,%
\vec{p}\vec{x}}\right) }{\hbar }$%
\end{tabular}
\label{eq31}
\end{equation}
Our choice is made in order to compare with the standard set of the linearly
independent solutions \noindent for the Dirac equation. Indeed, the
alternative%
\begin{equation}
\begin{tabular}{ll}
& $\psi _{D}^{(\ref{eq3})}=\dfrac{mc^{2}N_{1}}{E+c\left\vert \vec{p}%
\right\vert }\left( {{\begin{array}{*{20}c} 1 \hfill \\ { - i} \hfill \\ 0
\hfill \\ 0 \hfill \\ \end{array}}}\right) \,\exp \dfrac{-i\,\,\left( {%
Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }$ \\
$\text{and}$ & \\
& $\psi _{D}^{(\ref{eq4})}=N_{1}\left( {{\begin{array}{*{20}c} 0 \hfill \\ 0
\hfill \\ 1 \hfill \\ { - i} \hfill \\ \end{array}}}\right) \,\exp \dfrac{%
-i\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }.$%
\end{tabular}%
\end{equation}
\noindent may serve us equally well and at the same time make things more
transparent, since we have \noindent obtained exactly the same solutions as $%
\psi _{D}^{(\ref{eq1})}$ and $\psi _{D}^{(\ref{eq2})}$, which would be
negative energy solutions in \noindent the Dirac equation with $\vec{p}%
^{\prime }$= -$\vec{p}$ if we make the substitution
\begin{equation}
i^{\prime }=-i=-\frac{e_{1}p_{1}+\,e_{2}p_{2}\,+\,e_{1}e_{2}p_{3}}{%
\left\vert \vec{p}\right\vert }. \label{eq32}
\end{equation}
Obviously, the obtained set is mutually orthogonal.
Finally, using standard normalization condition, we obtain:%
\begin{eqnarray}
\psi _{D}^{(\ref{eq1})}\,\, &=&\frac{1}{2}\sqrt{\frac{mc^{2}}{E+c\left\vert
\vec{p}\right\vert }}\left( {{\begin{array}{*{20}c} 1 \hfill \\ i \hfill \\
0 \hfill \\ 0 \hfill \\ \end{array}}}\right) \,\exp \frac{-i\,\,\left( {%
Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }\, \notag \\
&&\, \notag \\
\psi _{D}^{(\ref{eq2})}\,\, &=&\frac{1}{2}\sqrt{\frac{E+c\left\vert \vec{p}%
\right\vert }{mc^{2}}}\left( {{\begin{array}{*{20}c} 0 \hfill \\ 0 \hfill \\
1 \hfill \\ i \hfill \\ \end{array}}}\right) \,\exp \frac{-i\,\,\left( {%
Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar } \notag \\
&& \notag \\
\psi _{D}^{(\ref{eq3})}\,\, &=&\frac{1}{2}\sqrt{\frac{mc^{2}}{E+c\left\vert
\vec{p}\right\vert }}\left( {{\begin{array}{*{20}c} 1 \hfill \\ { - i}
\hfill \\ 0 \hfill \\ 0 \hfill \\ \end{array}}}\right) \exp \frac{%
-i\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar } \label{eq33} \\
&& \notag \\
\psi _{D}^{(\ref{eq4})}\,\, &=&\frac{1}{2}\sqrt{\frac{E+c\left\vert \vec{p}%
\right\vert }{mc^{2}}}\left( {{\begin{array}{*{20}c} 0 \hfill \\ 0 \hfill \\
1 \hfill \\ { - i} \hfill \\ \end{array}}}\right) \,\exp \frac{-i\,\,\left( {%
Et\,\,-\,\,\vec{p}\vec{x}}\right) }{\hbar }. \notag
\end{eqnarray}
The obtained solutions maintain symmetry with respect to space coordinates
that may be expected \noindent based on the assumption of homogeneity of the
space-time continuum. Indeed, the correctness of the \noindent suggested
equations may be verified only through careful comparison with the
experimental data.
Now let us consider similar equations
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \psi }{\partial t}\,\,i\,\,-\,\,\dfrac{%
\partial \psi }{\partial x_{j}}\,\,e_{j}\,\,=\,\,\dfrac{mc}{\hbar }\,\,\phi
\,i \\
\\
-\dfrac{1}{c}\,\,\dfrac{\partial \phi }{\partial t}\,\,i\,\,-\,\,\dfrac{%
\partial \phi }{\partial x_{j}}\,\,e_{j}\,\,=\,\,\dfrac{mc}{\hbar }\,\,\psi
\,i%
\end{array}%
\quad j=1,2,3 \label{eq34}
\end{equation}
\noindent and verify that they admit an additional set of plane wave
solutions, for example, in the following form:
\begin{equation}
\begin{array}{l}
\psi _{j}\,\,=\,\,U_{1}e_{j}\exp \dfrac{-e_{j}\,\,\left( {Et\,\,-\,\,\vec{p}%
\vec{x}}\right) }{\hbar }\, \\
\\
\phi _{j}\,\,=\,\,U_{2}\exp \dfrac{-e_{j}\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x%
}}\right) }{\hbar }\,\,.%
\end{array}%
\quad j=1,2,3 \label{eq35}
\end{equation}
Here $U_1 $ and $U_2 $ are assumed to be real numbers. Then
\begin{equation}
\begin{array}{l}
U_{1}e_{j}\,\,\left( {\ -\dfrac{e_{j}Ei}{c}\,\,-\,\,e_{j}p_{1}e_{1}\,\,-\,%
\,e_{j}p_{2}e_{2}\,\,-e_{j}p_{3}e_{1}e_{2}}\right) \,\,=\,\,mcU_{2}i \\
\\
U_{2}\,\,\left( {\dfrac{e_{j}Ei}{c}\,\,-\,\,e_{j}p_{1}e_{1}\,\,-\,%
\,e_{j}p_{2}e_{2}\,\,-e_{j}p_{3}e_{1}e_{2}}\right) \,\,=\,\,mcU_{1}\,\,e_{j}i%
\end{array}
\label{eq36}
\end{equation}
\noindent or
\begin{equation}
\begin{array}{l}
U_{1}e_{j}\,\,\left( {-\dfrac{e_{j}Ei}{c}\,\,-\,\,e_{j}\,\,\left( {%
e_{1}p_{1}\,\,+\,\,e_{2}p_{2}\,\,+\,\,e_{1}e_{2}p_{3}}\right) }\right)
\,\,=\,\,mcU_{2}i \\
\\
U_{2}\,\,\left( {\dfrac{e_{j}Ei}{c}\,\,-\,\,e_{j}\,\,\left( {%
e_{1}p_{1}\,\,+\,\,e_{2}p_{2}\,\,+\,\,e_{1}e_{2}p_{3}}\right) }\right)
\,\,=\,\,mcU_{1}\,\,e_{j}i.%
\end{array}
\label{eq37}
\end{equation}
But according to (\ref{eq25})
\begin{equation} \label{eq38}
e_1 p_1 \,\, + \,\,e_2 p_2 \,\, + \,\,e_1 e_2 p_3 \,\, = \,\,i\,\,\left|
\vec {p} \right|.
\end{equation}
\newpage
\noindent which gives
\begin{equation}
\begin{array}{l}
U_{1}\,\,\left( {\dfrac{E}{c}\,\,+\,\,\left\vert \vec{p}\right\vert }\right)
i\,\,=\,\,mcU_{2}i \\
U_{2}e_{j}\,\,\left( {\dfrac{E}{c}\,\,-\,\,\left\vert \vec{p}\right\vert }%
\right) i\,\,=\,\,mcU_{1}e_{j}i%
\end{array}
\label{eq39}
\end{equation}
\noindent or
\begin{equation}
\begin{array}{l}
U_{1}\,\,\left( {\dfrac{E}{c}\,\,+\,\,\left\vert \vec{p}\right\vert }\right)
\,\,=\,\,mcU_{2} \\
U_{2}e_{j}\,\,\left( {\dfrac{E}{c}\,\,-\,\,\left\vert \vec{p}\right\vert }%
\right) \,\,=\,\,mcU_{1}e_{j}%
\end{array}
\label{eq40}
\end{equation}
\noindent and
\begin{equation}
\begin{array}{l}
U_{1}\,\,\left( {\dfrac{E}{c}\,\,+\,\,\left\vert \vec{p}\right\vert }\right)
\,\,=\,\,mcU_{2} \\
U_{2}\,\,\left( {\dfrac{E}{c}\,\,-\,\,\left\vert \vec{p}\right\vert }\right)
\,e_{j}\,=\,\,mcU_{1}e_{j}.%
\end{array}
\label{eq41}
\end{equation}
Thus, we finally obtain
\begin{equation}
\begin{array}{l}
\left( {\dfrac{E}{c}\,\,+\,\,\left\vert \vec{p}\right\vert }\right)
\,\,U_{1}\,\,-\,\,mcU_{2}\,\,=\,\,0 \\
-mc\,\,U_{1}\,\,+\,\,\left( {\dfrac{E}{c}\,\,-\,\,\left\vert \vec{p}%
\right\vert }\right) \,\,U_{2}\,\,=\,\,0%
\end{array}
\label{eq42}
\end{equation}
\noindent which justifies the above-made assumption concerning the reality
of $U_{1}$ and $U_{2}$. The existence of non-trivial solutions is ensured by
\begin{equation*}
\frac{E^2}{c^2}\,\, - \,\,\left| \vec {p} \right|^2\,\, - m^2c^2\,\, = \,\,0
\end{equation*}
\noindent and
\begin{equation} \label{eq43}
U_1 \,\, = \,\,\frac{mc^2}{E\,\, + \,\,c\,\,\left| \vec {p} \right|}\,\,U_2
\end{equation}
Thus, we have obtained a triplet of solutions, each one associated with the
same mass, but with
\begin{equation}
\begin{array}{l}
\left[ {\psi _{j},\,\,\psi _{k}}\right] \,\,\neq \,0 \\
\left[ {\phi _{j},\,\,\phi _{k}}\right] \,\,\neq \,0\text{ \ \ \ \ \ \ }%
j,k\,\,=\,\,1,2,3 \\
\left[ {\psi _{j},\,\,\phi _{k}}\right] \,\,\neq 0\text{ \ \ \ \ \ \ }%
j\,\,\neq \,\,k%
\end{array}
\label{eq44}
\end{equation}
\newpage
In addition, in the capacity of $e_{j},\,\,j\,\,=\,\,1,2,3,$ one may choose
not only the quaternion basis itself but other sets, for example,
\begin{equation}
ie_{1}i,\text{ \ \ }ie_{2}i,\text{ \ \ }ie_{2}e_{1}i \label{eq45}
\end{equation}
\noindent or
\begin{equation}
e_{1}ie_{1},\text{ \ \ }e_{2}ie_{2},\text{ \ \ }e_{1}e_{2}ie_{1}e_{2}.
\label{eq46}
\end{equation}
Notice, however, that (\ref{eq46}) do not form a quaternion but
\begin{equation} \label{eq47}
e_1 ie_1 \,\, + e_2 ie_2 \,\, + \,\,e_1 e_2 ie_1 e_2 \,\, = \,\,i.
\end{equation}
Additional knowledge is required in order to define which set is relevant
and how it may be associated with the correspondent physical objects.
So far, our discussion has been restricted to the four-dimensional real
quadratic division algebra of quaternions. However, it is clear that the
equations (\ref{eq24}) and (\ref{eq34}) serve in an uniform manner also the
octonionic extension of the complex Hilbert space .
If the underlying algeraic foundation of the theory is extended to include
the eight-dimensional real quadratic division algebra of octonions, then the
corresponding additional set of solutions for the equations (\ref{eq34}) may
be obtained.
They may have, e.g., the following form:
\begin{eqnarray*}
\psi \,\, &=&\,\,U_{1}\,\,i\,\,\exp \frac{-i\,\,\left( {Et\,\,-\vec{p}\vec{x}%
}\right) }{\hbar }\, \\
\phi \,\, &=&\,\,U_{2}\,\,\exp \frac{-i\,\,\left( {Et\,\,-\vec{p}\vec{x}}%
\right) }{\hbar }\,
\end{eqnarray*}
($i$ is given by (\ref{eq25}))
\begin{equation}
\begin{array}{l}
{\psi _{k}\,\,=\,U_{1}e_{k}\exp }\dfrac{-e_{k}\,\,\left( {Et\,\,-\,\,\vec{p}%
\vec{x}}\right) }{\hbar }{\,} \\
{\phi _{k}\,\,=\,U_{2}\exp }\dfrac{-e_{k}\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x%
}}\right) }{\hbar } \\
{\psi _{j_{k}}\,\,=U_{1}j_{k}\exp }\dfrac{-j_{k}\,\,\left( {Et\,\,-\,\,\vec{p%
}\vec{x}}\right) }{\hbar }{\,\,} \\
{\phi _{j_{k}}\,\,=\,U_{2}\exp }\dfrac{-j_{k}\,\,\left( {Et\,\,-\,\,\vec{p}%
\vec{x}}\right) }{\hbar }{\,\,}%
\end{array}%
\text{ \ \ \ \ \ \ }{k\,\,=\,\,4,5,6,7} \label{eq48}
\end{equation}
\noindent where $U_1 ,\,\,U_2 $
\begin{equation*}
U_1 \,\, = \,\,\frac{mc^2}{E\,\, + \,\,c\left| \vec {p} \right|}\,\,U_2
\end{equation*}
\noindent are real numbers;
\begin{equation}
j_{k}\,\,=\,\,e_{k}i\,,\text{\ \ \ \ \ \ \ }k\,\,=\,\,4,5,6,7 \label{eq49}
\end{equation}
\noindent and, as before, form a quaternion. This quaternion turns out to
form an algebraic foundation of the momentum space and, therefore, the
algebraic symmetry between coordinate and momentum spaces may be broken in
this formulation.
The particularly symmetric case occurs, if $k\,\, = \,\,7$. Then
\begin{equation}
\begin{array}{l}
i\,\,=\,\,\dfrac{e_{1}p_{1}\,+\,\,\,e_{2}p_{2}\,\,+\,\,e_{3}p_{3}}{%
\left\vert \vec{p}\right\vert } \\
\\
j\,\,=\,\,\dfrac{e_{4}p_{1}\,+\,\,\,e_{5}p_{2}\,\,+\,\,e_{6}p_{3}}{%
\left\vert \vec{p}\right\vert }.%
\end{array}
\label{eq50}
\end{equation}
Indeed, in each case the set of equations (\ref{eq24}) and (\ref{eq34})
should be supplemented by corresponding leptonic equations, for example, for
(\ref{eq50})
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \chi }{\partial t}\,\,j\,\,-\,\,\dfrac{%
\partial \chi }{\partial x}e_{4}\,\,-\,\,\dfrac{\partial \chi }{\partial y}%
e_{5}\,\,-\,\,\dfrac{\partial \chi }{\partial z}e_{6}\,\,=\,\,\dfrac{mc}{%
\hbar }\xi \\
\\
-\dfrac{1}{c}\,\,\dfrac{\partial \xi }{\partial t}\,\,j\,\,-\,\,\dfrac{%
\partial \xi }{\partial x}e_{4}\,\,-\,\,\dfrac{\partial \xi }{\partial y}%
e_{5}\,\,-\,\,\dfrac{\partial \xi }{\partial z}e_{6}\,\,=\,\,-\dfrac{mc}{%
\hbar }\chi .%
\end{array}
\label{eq51}
\end{equation}
Based on the results of M. Zorn \cite{17} that each automorphism of the
octonion algebra is completely defined by the images of three
\textquotedblleft independent\textquotedblright\ basis units \cite{18}, it
was demonstrated by M. G\"{u}naydin and F. G\"{u}rsey \cite{19} that under
given automorphism $\sigma $ we have three quaternionic planes in the space
formed by octonion algebra (space of quantum mechanical phases), which
undergo rotations by the angles $\phi _{1},\,\,\phi _{2},\,\,\phi _{3}$,
respectively, such that
\begin{equation} \label{eq52}
\phi _1 \,\, + \,\,\phi _2 \,\, + \,\,\phi _3 \,\, = \,\,0\,\,\bmod \,\,2\pi
\end{equation}
\noindent remains invariant. The planes ($e_{i},\,\,e_{j})$ are determined
by the conditions $e_{i}e_{j}\,\,=\,\,e_{k}$ and $e_{k}$ is the fixed point
common to all of those planes (compare with (\ref{eq48}) and (\ref{eq49})).
These results might help to extract the set of independent solutions and to
obtain its correct classification.
It is worth mentioning that an alternative arrangement can be also possible.
We may consider a septet of solutions, each one associated with the same
mass, namely
\begin{equation}
\begin{array}{l}
\psi \,\,=\,\,U_{1}e_{k}\exp \dfrac{-e_{k}\,\,\left( {Et\,\,-\,\,\vec{p}\vec{%
x}}\right) }{\hbar }\,\, \\
\phi \,\,=\,\,U_{2}\exp \dfrac{-e_{k}\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x}}%
\right) }{\hbar }\,\,%
\end{array}%
\text{ \ \ \ \ \ \ }k\,\,=\,\,1,...,7 \label{eq53}
\end{equation}
\noindent and, perhaps, an additional one in the form
\begin{equation}
\begin{array}{l}
\psi \,\,=\,\,U_{1}e_{j}\exp \dfrac{-e_{j}\,\,\left( {Et\,\,-\,\,\vec{p}\vec{%
x}}\right) }{\hbar }\, \\
\phi \,\,=\,\,U_{2}\exp \dfrac{-e_{j}\,\,\left( {Et\,\,-\,\,\vec{p}\vec{x}}%
\right) }{\hbar }\,%
\end{array}%
\text{ \ \ \ \ \ \ }j\,=\,\,1,...,7\, \label{eq54}
\end{equation}
\noindent where
\begin{equation}
\begin{array}{l}
j_{k}\,\,=\,\,e_{k}ie_{k} \\
j_{k+3}\,\,=\,\,e_{k+3}\,\,je_{k+3} \\
j_{7}\,\,=\,\,e_{7}%
\end{array}%
\text{ \ \ \ \ \ \ }k\,\,=\,\,1,2,3 \label{eq55}
\end{equation}
\noindent or
\begin{equation}
\begin{array}{l}
j_{k}\,\,=\,\,ie_{k}i \\
j_{k+3}\,\,=\,\,je_{k+3}\,\,j \\
j_{7}\,\,=\,\,e_{7}%
\end{array}%
\text{ \ \ \ \ \ \ }k\,\,=\,\,1,2,3 \label{eq56}
\end{equation}
\noindent with the common fixed point $e_{7}$: $i$ and $j$ are given by (\ref%
{eq50}).
The number of independent solutions, which are arranged in such a way, is
sharply reduced and serves as a slight reminder of a similar possibility
discussed in the literature \cite{ 20}.
Now it may be clarified why we have chosen to discuss the Dirac equations in
the Weyl representation. The reason is merely technical. In order to perform
clean octonionic calculations, we assume that the solutions of the equations
(\ref{eq34}) have the form (\ref{eq35}), where $U_1 $ and $U_2 $ are real
numbers. Then the obtained relations (\ref{eq42}) justify the assumption.
Notice that the electroweak unification scheme \cite{ 15} is based on the
use of this representation of the Dirac equations; in addition, in that case
the solutions behave naturally with respect to the Lorentz transformations.
Let us demonstrate that the Dirac equations in the form (\ref{eq24}), as
well as the set (\ref{eq34}), permit a consistent probabilistic
interpretation (here the discussion is restricted to the case where the
underlying algebraic basis are quaternions).
\newpage
Consider formally
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \psi }{\partial t}\,\,i\,\,-\,\dfrac{%
\partial \psi }{\partial x_{j}}e_{j}\,\,=\,\,\dfrac{m_{1}c}{\hbar }\phi
\,\,+\,\,\dfrac{m_{2}c}{\hbar }\phi i \\
\\
-\dfrac{1}{c}\,\,\frac{\partial \phi }{\partial t}\,\,i\,\,-\,\dfrac{%
\partial \phi }{\partial x_{j}}e_{j}\,\,=\,\,-\dfrac{m_{1}c}{\hbar }\psi
\,\,+\,\dfrac{m_{2}c}{\hbar }\psi i.%
\end{array}%
\quad j=1,2,3 \label{eq57}
\end{equation}
Then
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \psi }{\partial t}\,\,\,\,+\,\,\dfrac{%
\partial \psi }{\partial x_{j}}e_{j}i\,\,\,=\,\,-\,\,\dfrac{m_{1}c}{\hbar }%
\phi \,i\,\,+\,\,\dfrac{m_{2}c}{\hbar }\phi \\
\\
\dfrac{1}{c}\,\,\dfrac{\partial \phi }{\partial t}\,\,\,-\,\,\dfrac{\partial
\phi }{\partial x_{j}}e_{j}i\,\,\,=\,\,-\dfrac{m_{1}c}{\hbar }\psi i\,\,-\,%
\dfrac{m_{2}c}{\hbar }\psi%
\end{array}%
\quad j=1,2,3 \label{eq58}
\end{equation}
\noindent and
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \bar{\psi}}{\partial t}\,\,\,\,+\,\,ie_{j}%
\dfrac{\partial \bar{\psi}}{\partial x_{j}}\,\,\,=\,\,\,\dfrac{m_{1}c}{\hbar
}i\bar{\phi}\,\,+\,\,\dfrac{m_{2}c}{\hbar }\bar{\phi} \\
\\
\dfrac{1}{c}\,\,\dfrac{\partial \bar{\phi}}{\partial t}\,\,\,\,-\,\,ie_{j}%
\dfrac{\partial \bar{\phi}}{\partial x_{j}}\,\,\,=\,\,\,\dfrac{m_{1}c}{\hbar
}i\bar{\psi}\,\,-\,\,\dfrac{m_{2}c}{\hbar }\bar{\psi}.%
\end{array}%
\quad j=1,2,3 \label{eq59}
\end{equation}%
Thus, we have
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial \psi }{\partial t}\bar{\psi}\,\,\,\,+\,\,%
\dfrac{\partial \psi }{\partial x_{j}}e_{j}i\bar{\psi}\,\,\,=\,\,\,-\dfrac{%
m_{1}c}{\hbar }\phi \,i\bar{\psi}\,\,+\,\,\dfrac{m_{2}c}{\hbar }\phi \bar{%
\psi} \\
\\
\dfrac{1}{c}\,\,\psi \dfrac{\partial \bar{\psi}}{\partial t}\,\,\,+\,\,\psi
ie_{j}\,\,\dfrac{\partial \bar{\psi}}{\partial x_{j}}\,\,\,\,=\,\,\dfrac{%
m_{1}c}{\hbar }\psi i\bar{\phi}\,\,+\,\,\dfrac{m_{2}c}{\hbar }\psi \bar{\phi}
\\
\\
\dfrac{1}{c}\,\,\dfrac{\partial \phi }{\partial t}\bar{\phi}\,\,\,\,-\,\,%
\dfrac{\partial \phi }{\partial x_{j}}e_{j}i\bar{\phi}\,\,\,=\,\,\,-\dfrac{%
m_{1}c}{\hbar }\psi \,i\bar{\phi}\,\,-\,\,\dfrac{m_{2}c}{\hbar }\psi \bar{%
\phi} \\
\\
\dfrac{1}{c}\,\,\phi \dfrac{\partial \bar{\phi}}{\partial t}\,\,\,-\,\,\phi
ie_{j}\,\,\dfrac{\partial \bar{\phi}}{\partial x_{j}}\,\,\,=\,\,\dfrac{m_{1}c%
}{\hbar }\phi i\bar{\psi}\,\,-\,\,\dfrac{m_{2}c}{\hbar }\phi \bar{\psi}.%
\end{array}%
\quad j=1,2,3 \label{eq60}
\end{equation}
\newpage
Adding all equations (\ref{eq60}), we find
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\left\{ {\dfrac{\partial \psi }{\partial t}\bar{\psi}%
\,\,+\,\,\psi \dfrac{\partial \bar{\psi}}{\partial t}\,\,+\,\,\dfrac{%
\partial \phi }{\partial t}\bar{\phi}\,\,+\,\,\phi \dfrac{\partial \bar{\phi}%
}{\partial t}}\right\} \\
\\
+\,\,\left\{ {\dfrac{\partial \psi }{\partial x_{j}}e_{j}i\bar{\psi}%
\,\,+\,\,\psi ie_{j}\dfrac{\partial \bar{\psi}}{\partial x_{j}}\,\,-\,\,%
\dfrac{\partial \phi }{\partial x_{j}}e_{j}i\bar{\phi}\,\,-\,\,\phi
ie_{j}\,\,\dfrac{\partial \bar{\phi}}{\partial x_{j}}}\right\} =0%
\end{array}%
\quad j=1,2,3 \label{eq61}
\end{equation}
Notice that the mass terms vanish separately and, hence, the derivation
holds separately for (\ref{eq24}) and (\ref{eq34}) and from now on it is
understood that we consider the solutions of these equations separately.
The equation (\ref{eq61}) is invariant under the gauge transformation (\ref%
{eq15}) with $i$ given by (\ref{eq25}).
Then we have
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial }{\partial t}\,\,\,\,\psi \bar{\psi}\,\,+\,\,%
\dfrac{1}{c}\,\,\dfrac{\partial }{\partial t}\,\,\,\phi \bar{\phi} \\
\\
+\,\,\left\{ {\dfrac{\partial \psi }{\partial x_{j}}ie_{j}\bar{\psi}%
\,\,+\,\,\psi e_{j}i\,\,\dfrac{\partial \bar{\psi}}{\partial x_{j}}\,\,-\,\,%
\dfrac{\partial \phi }{\partial x_{j}}ie_{j}\bar{\phi}\,\,-\,\,\phi
e_{j}i\,\,\dfrac{\partial \bar{\phi}}{\partial x_{j}}}\right\} =0.%
\end{array}%
\quad j=1,2,3 \label{eq62}
\end{equation}
Adding equations (\ref{eq61}) and (\ref{eq62}), we obtain
\begin{equation}
\begin{array}{l}
\dfrac{1}{c}\,\,\dfrac{\partial }{\partial t}\,\,\left( {\psi \bar{\psi}%
\,\,\,\,+\,\,\,\phi \bar{\phi}}\right) \\
\\
+\,\,\dfrac{1}{2}{\LARGE \{}{\dfrac{\partial \psi }{\partial x_{j}}\left( {%
e_{j}i\,\,+\,\,ie_{j}}\right) \,\,\bar{\psi}\,\,\,\,+\,\,\psi \left( {%
e_{j}i\,\,+\,\,ie_{j}}\right) \,\,\dfrac{\partial \bar{\psi}}{\partial x_{j}}%
\,\,} \\
\\
{-\,\,\dfrac{\partial \phi }{\partial x_{j}}\,\,\left( {e_{j}i\,\,+\,\,ie_{j}%
}\right) \,\,\bar{\phi}\,\,-\,\,\phi \,\,\left( {e_{j}i\,\,+\,\,ie_{j}}%
\right) \,\,\dfrac{\partial \bar{\phi}}{\partial x_{j}}}{\LARGE \}}=0%
\end{array}%
\text{ \ \ \ \ \ \ }j=1,2,3 \label{eq63}
\end{equation}
\noindent or
\begin{equation}
\frac{\partial \rho }{\partial t}\,\,+\,\,\mathrm{div}\,\,\vec{j}\,\,=\,\,0
\label{eq64}
\end{equation}
\noindent where
\begin{equation}
\begin{array}{l}
\rho \,\,=\,\,\psi \bar{\psi}\,\,+\,\,\phi \bar{\phi} \\
\\
j_{k}\,\,=\,\,\dfrac{c}{2}\left\{ {\,\psi \left( {e_{k}i\,\,+\,\,ie_{k}}%
\right) \,\,\bar{\psi}\,\,-\,\,\phi \,\,\left( {e_{k}i\,\,+\,\,ie_{k}}%
\right) \,\,\bar{\phi}}\right\} .%
\end{array}%
\quad \text{\ \ \ }k=1,2,3 \label{eq65}
\end{equation}
If $i$ is given by (\ref{eq25}), then
\begin{equation}
e_{k}i\,\,+\,\,ie_{k}\,\,=\,\,-\frac{2p_{k}}{\left\vert \vec{p}\right\vert }%
,\quad k=1,2,3. \label{eq66}
\end{equation}
Thus, in accordance with the Galileo, Maxwell and Schr\"{o}dinger theories,
the probability current $\vec{j}$ is proportional to the velocity operator,
which is a constant of motion for free particles.
\textit{Zitterbewegung }[3] phenomenon is absent in this formulation.
\section{Conclusion}
In account of the experimental information that became available during the
last century, it is desirable for the equations of motion for the
fundamental fermions to have the following properties:\smallskip
\noindent
- the equations should possess $SU\left( 2\right) \,\,\otimes U\left(
1\right) $ local gauge invariance \linebreak \hspace*{0.340in}intrinsically
since the electron is not a source of pure electromagnetic \linebreak
\hspace*{0.340in}radiation but also has the ability to participate in weak
interactions;
\noindent
- the electron is the only particular member of the entire family of fun-
\ \ damental fermions, it is desirable that all fermions are described in the
\ \ uniform manner;
\noindent
- there exist three replication of the families of the fundamental fermions;
\noindent
- leptons do not have a color;
\noindent
- quarks do have a color;
\noindent
- each quark appear in triplet associated with the same mass;
\noindent
- a color is associated with the internal degree of freedom;
\noindent
- a color symmetry can't be broken.
\smallskip
In contrast to the existing quaternionic formulations [9, 10] of the Dirac
equation, we suggested here closely related but essentially different sets
of equations that allow description of the free motion for electron and
neutrinos, as well as triplets of quarks. The suggested solution possesses $%
SU\left( 2\right) \,\,\otimes U\left( 1\right) $ local gauge invariance
intrinsically. All obtained solutions have the structure (A1) and (A2).
These solutions are substantially different from the standard ones and may
yield, therefore, different values for observable physical quantities. We
propose to compare and verify them against the existing experimental
information. The suggested reexamination may help to decide what is a
relevant mathematical framework suitable to achieve a solution of the
unification problem for the fundamental interactions.
During the last decades, an enormous progress in the understanding of
quantum theory of fields took place. It became almost apparent that we have
to deal with four essentially fundamental interactions, which have a similar
origin, namely, presence of phases in the quantum mechanical description of
the fundamental sources of these fields [21]. In addition, the investigation
of the properties of the fundamental sources (leptons, quarks) clearly
established that the quantum numbers in addition to electric charge (week
hypercharge and color) appear in amazing correspondence with the complex
variety of the radiated fields.
The formulation of classical mechanics and the classical theory of fields
have demonstrated that the presence of additional interactions requires a
suitable generalization of the mathematical language used. Application of
the analogy with the structure of classical physics in the framework of
functional analysis naturally concentrated around attempts to extent it on
all Hurwitz algebras, as the underlying algebraic foundation of the theory.
The above discussion may be considered as an additional step towards
realization of a program initiated by E. Schr\"{o}dinger [22] to treat all
of the physics as wave mechanics:
\smallskip
a) The universal mathematical architecture of the physics is given in terms
of ten functional - analytical frameworks, suitable to incorporate the
results of physical measurements.
Real, complex, quaternion and octonion states with real scalar product
should be equivalent to the theory of classical fields. Unification of
electromagnetism with gravitation should occur already in the classical
field theory.
Complex, quaternion and octonion states with complex scalar product should
allow realization of present unification schemes. Notice that pure
relativistic quantum electrodynamics does not exist because there are no
elementary sources of pure electromagnetic radiation. Neutrino is an
elementary source of pure weak radiation.
Quaternion and octonion states with quaternion scalar product should
describe wave mechanics of space-time continuum.
Octonion states with octonion scalar product should allow ultimate
realization of idea of elementary particles picture of natural phenomena.
\smallskip
b) One expects that the quantum mechanical space-time continuum should be
different from its classical counterpart. Perhaps, the spin is not a
dynamical variable, but the feature of the quantum mechanical world, namely,
the world point is described by the following expression (before inclusion
of quantum gravity):
\begin{equation*}
X\,\, = \,\,\left( {{\begin{array}{*{20}c} t \hfill & { - e_1 x\,\, -
\,\,e_2 y\,\, - \,\,e_3 z} \hfill \\ {e_1 x\,\, + \,\,e_2 y\,\, + \,\,e_3 z}
\hfill & t \hfill \\ \end{array} }} \right)
\end{equation*}
It is interesting to construct the correspondent metric space and the
application of Least Action should lead to the equations of motion for the
fundamental fermions.
\smallskip
c) In the present discussion we consider masses as external phenomenological
parameters. However, the structure of octonion quantum mechanics with
complex scalar product suggest natural mechanism to generate masses of the
fundamental fermions as energy gaps obtained after splitting states of
initially degenerated two-level physical system.
\bigskip
I am grateful to L.P. Horwitz, Y. Aharonov, S. Nussinov, and I.D. Vagner for
the stimulating discussions.
This work has been supported in part by the Binational Science Foundation,
Jerusalem.
\section*{Appendix}
This paper is concerned with the relativistic dynamics of single particle
states and for this reason we have used only the following results of the
particular realization of this program for the quaternionic and octonionic
Hilbert spaces with complex scalar products:
1) In the quaternionic extension, quantum mechanical states are represented
by
\begin{equation}
\begin{tabular}{ll}
& $\Psi \,\,_{\not\subset }^{\left( 1\right) \,}\,=\,\dfrac{1}{\sqrt{2}}%
\binom{f}{fe_{1}}$ \\
or & \\
& $\Psi \,\,_{\not\subset }^{\left( 2\right) \,}\,=\,\dfrac{1}{\sqrt{2}}%
\binom{f}{-fe_{1}}$%
\end{tabular}
\tag{A1} \label{A1}
\end{equation}
\noindent where $f\,\,=\,\,f_{0}\,\,+\,\,\sum\limits_{i\,\,=\,\,1}^{3}{%
f_{i}e_{i};\,\,\,\,f_{0},\,\,f_{i}\,\,}$ are real functions of the
space-time coordinates and $e_{i},\,\,i\,\,=\,\,1,\,\,2,\,\,3$ form a basis
for the real quadratic division algebra of quaternions;
In the octonionic extension quantum mechanical states are represented~by:
\begin{equation}
\begin{tabular}{ll}
& $\Psi \,\,_{\not\subset }^{\left( 3\right) \,}\,=\,\dfrac{1}{\sqrt{2}}%
\binom{f}{fe_{7}}$ \\
or & \\
& $\Psi \,\,_{\not\subset }^{\left( 4\right) \,}\,=\,\dfrac{1}{\sqrt{2}}%
\binom{f}{-fe_{7}}$%
\end{tabular}
\tag{A2} \label{A2}
\end{equation}
\noindent where $f\,\,=\,\,f_{0}\,\,+\,\,\sum\limits_{i\,\,=\,\,1}^{7}{%
f_{i}e_{i};\,\,\,\,f_{0},\,\,f_{i}\,\,}$are real functions of the space-time
coordinates and $e_{i}\,\,,\,\,i\,\,=\,\,1,\,\,...,\,\,7$ are a basis for
the real quadratic division algebra of octonions.
The $e_{1}$ and $e_{7}$ in the definition of the states (A1) and (A2) play
the role of a label for the generator of a complex field in the space of
one-body states. For example, any one of the quaternionic units or some
linear combination of them%
\begin{equation}
i\,\,=\,\,\frac{ae_{1}\,\,+\,\,be_{2}\,\,+\,\,ce_{3}}{\sqrt{%
a^{2}\,\,+\,\,b^{2}\,\,+\,\,c^{2}}} \tag{A3} \label{A3}
\end{equation}%
($a,b,c$ are arbitrary real numbers) may be used for this purpose. Thus, a
definition of this combination cannot be obtained kinematically and turns
out to be a matter of the dynamics of single particle states.
2) Consider the general form of operators, induced by the structure (A1) of
the vector space.
For the complex linear operators%
\begin{equation}
A_{z}\,\,=\,\,\left( {{\begin{array}{*{20}c} {a_{11} } \hfill & {a_{12} }
\hfill \\ {a_{21} } \hfill & {a_{22} } \hfill \\ \end{array}}}\right)
\tag{A4} \label{A4}
\end{equation}
\noindent where matrix elements $a_{ij}$ are real operators over quaternions
and, in turn, are assumed to be at least z-linear operators, we have
\begin{equation}
\Psi \,\,_{\not\subset }^{\left( 1\right) \,}\,^{\prime }=\,\frac{1}{\sqrt{2}%
}\,\left( {{\begin{array}{*{20}c} {f'} \hfill \\ {f'e_1 } \hfill \\
\end{array}}}\right) \,=\,\,A_{z}\Psi _{\not\subset }^{\left( 1\right) }=\,\,%
\frac{1}{\sqrt{2}\,}\left( {{\begin{array}{*{20}c} {a_{11} } \hfill &
{a_{12} } \hfill \\ {a_{21} } \hfill & {a_{22} } \hfill \\ \end{array}}}%
\right) \left( {{\begin{array}{*{20}c} f \hfill \\ {fe_1 } \hfill \\
\end{array}}}\right) \tag{A5} \label{A5}
\end{equation}
\noindent and%
\begin{equation}
a_{21}f\,+\,a_{22}f_{1}e_{1}\,=\,\,f^{\prime }e_{1}=\,\,\left( {%
a_{11}f\,\,+\,\,a_{12}fe_{1}}\right) \,\,e_{1}=\,a_{11}fe_{1}\,-\,\,a_{12}f.
\tag{A6} \label{A6}
\end{equation}
Therefore,%
\begin{equation}
a_{12}\,\,=\,\,-a_{21};\,\,\,\,\,a_{11}\,\,=\,\,a_{22}. \tag{A7} \label{A7}
\end{equation}
The restrictions (A7) on the matrix elements of the operator (A4), obtained
for the states of the form $\Psi _{\not\subset }^{(\ref{eq1})}$ , are also
valid if one considers the transformations%
\begin{equation}
\Psi _{\not\subset }^{(\ref{eq2})\,^{\prime }}=\,A_{z}\,\,\Psi _{\not\subset
}^{(\ref{eq2})}. \tag{A8} \label{A8}
\end{equation}
However, for the operators transforming the state $\Psi _{\not\subset }^{(%
\ref{eq1})}$ into the state $\Psi _{\not\subset }^{(\ref{eq2})}$ (and vice
versa),%
\begin{equation}
\Psi _{\not\subset }^{\left( 2\right) \,\,\,\,^{\prime }}\,\,=\,\,\frac{1}{%
\sqrt{2}}\,\,\left( {{\begin{array}{*{20}c} {f'} \hfill \\ { - f'e_1 }
\hfill \\ \end{array}}}\right) \,\,=\,\,\,\,\left( {{\begin{array}{*{20}c}
{a_{11} } \hfill & {a_{12} } \hfill \\ {a_{21} } \hfill & {a_{22} } \hfill
\\ \end{array}}}\right) \,\,\,\frac{1}{\sqrt{2}}\,\,\,\left( {{%
\begin{array}{*{20}c} f \hfill \\ {fe_1 } \hfill \\ \end{array}}}\right) ,
\tag{A9} \label{A9}
\end{equation}
\noindent we have%
\begin{equation}
a_{21}\,\,=\,\,a_{12};\,\,\,\,\,a_{11}\,\,=\,\,-a_{22}. \tag{A10}
\label{A10}
\end{equation}
Thus, we have obtained two possible types of complex linear operators,
either
\begin{equation}
{\begin{array}{*{20}c} \hfill & {A_z^{(\ref{eq1})} \,\, = \,\,\left(
{{\begin{array}{*{20}c} {a_{11} } \hfill & {a_{12} } \hfill \\ { - a_{12} }
\hfill & {a_{11} } \hfill \\ \end{array} }} \right)} \hfill & \hfill
\end{array}} \tag{A11} \label{A11}
\end{equation}%
or
\begin{equation}
{\begin{array}{*{20}c} \hfill & {A_z^{(\ref{eq2})} \,\, = \,\,\left(
{{\begin{array}{*{20}c} {a_{11} } \hfill & {a_{12} } \hfill \\ {a_{12} }
\hfill & { - a_{11} } \hfill \\ \end{array} }} \right)} \hfill & \hfill
\end{array}} \tag{A12} \label{A12}
\end{equation}
We remark that the matrix elements $a_{11} $ and $a_{12} $ do not commute.
| {'timestamp': '2005-04-01T17:29:51', 'yymm': '0504', 'arxiv_id': 'physics/0504008', 'language': 'en', 'url': 'https://arxiv.org/abs/physics/0504008'} |
\section{Introduction}\label{intro}
Shahriar~S.~Afshar~\cite{afshar1,afshar2,afshar3} has recently performed a variant of
Young's two-slit experiment, in which he claims to have demonstrated the
falsification of the celebrated Bohr complementarity principle, and thereby, the Copenhagen
interpretation of quantum mechanics. In Afshar's experiment, sketched in Fig.~\ref{f1},
a source emits photons towards a screen with two slits marked U and L.
\begin{figure}\label{f1}
\setlength{\unitlength}{1cm}
\begin{picture}(11,5)
\put(1.7,0.5){\line(0,1){1}}
\put(1.7,2){\line(0,1){1}}
\put(1.7,3.5){\line(0,1){1}}
\put(10.7,0.5){\line(0,1){4}}
\qbezier(6.2,0.5)(5.2,2.5)(6.2,4.5)
\qbezier(6.2,0.5)(7.2,2.5)(6.2,4.5)
\put(10.9,1.5){$\mbox{U}^{\prime}$}
\put(10.9,3.3){$\mbox{L}^{\prime}$}
\put(1.7,1.6){\line(1,0){9}}
\put(1.7,3.4){\line(1,0){9}}
\put(6.2,1.6){\line(5,2){4.5}}
\put(6.2,1.6){\line(-3,1){4.5}}
\put(6.2,3.4){\line(5,-2){4.5}}
\put(6.2,3.4){\line(-3,-1){4.5}}
\put(0,1.1){\line(0,1){2.8}}
\put(0.3,1.1){\line(0,1){2.8}}
\put(0,2.5){\vector(1,0){0.7}}
\put(5.3,1.1){\dashbox{0.2}(0.1,2.8)}
\put(1.3,1.6){L}
\put(1.3,3.1){U}
\put(1.6,4.8){$\sigma$}
\put(5.2,4.8){$\sigma_{1}$}
\put(10.55,4.8){$\sigma_{2}$}
\put(1.6,0){$t_{0}$}
\put(5.25,0){$t_{1}$}
\put(10.6,0){$t_{2}$}
\end{picture}
\caption{Sketch of Afshar's experiment}
\end{figure}
A suitably located converging lens downstream of the screen focuses the photon streams
into sharp images of the two slits U and L, marked $\mbox{U}^{\prime}$ and
$\mbox{L}^{\prime}$ respectively, at location $\sigma_{2}$ (for simplicity,
we do not distinguish between the image plane and the focal plane in Fig.~\ref{f1}). Afshar notes
that these images serve to provide sharp which-way information for the photons, thereby
confirming their particle status. A wire grid is placed at location $\sigma_{1}$
in Fig.~\ref{f1}, such that the (thin) wires occur at precisely the theoretically
predicted minima of the interference pattern due to the superposed
states of the photons. Afshar observes that the wire grid effectively performs
a nondestructive confirmation of the superposition; when both slits are open,
no distortion or reduction in intensity of the images at $\sigma_2$ is observed
as compared to the case when the wire grid is removed. The expected distortion and
reduction in intensity do occur when one of the slits is closed. This seemingly rules out
the classical particle state of the photons between the locations~$\sigma$ and $\sigma_{1}$
when both slits are open, and confirms their wave nature, since blocking of photons by the
wire grid does not occur to the classically expected extent. Afshar concludes that both
the sharp particle and wave natures of the photon are exhibited in a single experiment,
in violation of the Bohr complementarity principle. In particular, Afshar~\cite{afshar1} claims that
the Englert-Greenberger duality relation~\cite{greenberger,englert} is falsified in his experiment.
Afshar's interpretation of his experiment has many critics (see Refs.~\refcite{kastner1}--\refcite{steuernagel}),
but there is no consensus on why Afshar is supposedly wrong. The most prominent of these critics seems to be
Kastner~\cite{kastner1}, who rejects the existence of which-way information in not only the Afshar experiment,
but also in any classical two-slit experiment even when the wire grid is not present. Kastner's argument is
essentially based on a realist interpretation in which the presence of an interference pattern requires the photon
to `really' pass through both slits. Kastner concludes that post-selection of the photon in a `which-slit' basis does
not imply that the photon passed through only one slit. In a subsequent article, Kastner~\cite{kastner2} cites the two-slit
experiment of Srikanth~\cite{srikanth} as even more dramatic than the Afshar experiment, in that the existence of the
interference pattern is irreversibly recorded (rather than just inferred) and yet post-selection of the photon in a `which-slit'
basis is possible~\footnote{An anonymous referee has rejected the validity of Srikanth's experiment}. The following
passage from Kastner~\cite{kastner2} clearly reveals her realist argument:
\begin{quote}
This is even more dramatic than the Afshar result because clearly $V=1$ since a fully articulated interference pattern
has been irreversibly recorded -- not just indicated indirectly -- and yet a measurement can be performed after the fact
that seems to reveal `which slit' the photon went through. However, the point is that the detector's vibrational
mode remains in a superposition until that measurement is made, implying that each photon indeed went through both
slits. As Srikanth puts it,`...the amplitude contributions from both paths to the observation at [detector element]
$x$ results in a superposition of vibrational modes. The initial superposition leaves behind a remnant
superposition.'~\cite{srikanth}. So, just because one can `post-select' by measuring the vibrational observable and
end up with a particular corresponding slit eigenstate doesn't mean the particle actually went through that slit;
in a very concrete sense, it went through both slits.
\end{quote}
Kastner~\cite{kastner1,kastner2} concludes that the Bohr complementarity principle is not violated in the Afshar
experiment and that the Englert-Greenberger duality relation is not applicable in all similar experiments in which an interference pattern
can be inferred without `collapsing' the superposed state. Other critics~\cite{qureshi,reitzner} of the Afshar experiment agree with Kastner
that there is no which-way information and like Kastner, argue for a realist interpretation in which the photon
passed through both slits. However, a few workers~\cite{drezet,steuernagel} assert that while which-way information is
present in the Afshar experiment, visibility of the fringes is very small because the interference pattern is
not actually recorded in the Afshar experiment. This group rejects the inference of the existence of an interference
pattern and insists that only an experimental recording of the same is acceptable. Afshar's co-workers~\cite{flores}
and O'Hara~\cite{ohara} support the Afshar interpretation that both the particle and wave properties of the photon
co-exist in a real sense, thereby rejecting both the Bohr complementarity principle and the Englert-Greenberger
duality relation.
It is the purpose of this paper to analyze the status of the Bohr complementarity
principle in Afshar's experiment, particularly in the light of the non-Aristotelian
finitary logic~(NAFL) recently proposed by the author~\cite{ijqi,1166,acs,ract}.
We will not concern ourselves with the nuts and bolts of Afshar's experiment
in an attempt to evaluate his claims from the point of view of experimental physics.
Instead, we consider the experiment from a purely logical angle, in the case
when Afshar's claims are granted as experimentally sound. It will be seen that
the NAFL interpretation of the quantum superposition principle does indeed uphold
complementarity, \emph{despite} the co-existence of the interference pattern and the path information for the photons that arrive at the image plane.
Indeed, we argue that the interference pattern should, in principle, be reconstructable for these photons from just the path information, without any need
for the controversial grid in Afshar's experiment. In fact the grid is not only logically superfluous, but also does not provide a complete
reconstruction of the interference pattern, as would be possible from analysis of the path information.
Kastner's analysis~\cite{kastner1,kastner2} is critically examined and her claim that Cramer's Transactional
Interpretation of quantum mechanics~\cite{cramer} rescues the complementarity principle in Afshar's
experiment is disputed. What the experiments of Afshar and others
really establish is that any realist interpretation of quantum mechanics is highly problematic from a
physical and philosophical point of view. Both the Bohr complementarity principle
and the Englert-Greenberger duality relation (at its extreme ends) are upheld in the Afshar experiment
when they are properly interpreted to take into account the time dependence of logical truth, as embodied by NAFL. The NAFL
interpretation really vindicates the original non-realist Copenhagen interpretation championed by
Neils Bohr and others, which, contrary to Afshar's claims, remains unscathed by the Afshar experiment.
While the Copenhagen interpretation formulates Bohr complementarity as a physical principle, the NAFL
interpretation enshrines it as a sacred and inviolable logical principle which follows from basic
postulates that embody finitary reasoning.
\section{Summary of the main argument and conclusions}\label{summary}
This section highlights the main argument and conclusions of this paper; the details are presented in subsequent
sections. Consider the modified experiment of Afshar~\cite{afshar3}, which was originally proposed
by Wheeler~\cite{wheeler} and is described in Sec.~5 of Ref.~\refcite{flores}, as quoted below.
\begin{quote}
\dots~a laser beam impinges on a 50:50 beam splitter and produces two spatially separated coherent beams of equal
intensity. The beams overlap at some distance, where they form an interference pattern of bright and dark fringes.
At the center of the dark fringes we place thin wires.~\dots~Beyond the region of overlap the two beams fully
separate again. There, two detectors are positioned such that detector~$1^{\prime}$ detects only the photons originating
from mirror~1, and detector~$2^{\prime}$ detects only photons originating from the beam splitter (mirror~2).
Since the pathway of the photon is practically unobstructed, a study of the electric fields involved together with
conservation of momentum allows us to uniquely identify, with high probability, the respective mirror as the place
where that photon originated.
\end{quote}
The wire grid in the modified experiment plays the same role as in the original Afshar experiment and
Afshar~\cite{afshar3} concludes that the results of the two experiments are in agreement.
The above experiment is similar in essence to the quantum eraser, the delayed-choice version of which
is particularly interesting~\cite{scully}. When the two photon beams overlap, they contain interference information
and when they separate, they contain which-way information. As long as no measurements are made, the available
information may be reversed as many times as one chooses. The key point here is that \emph{when} the photon beams provide
interference information, there is no way to extract which-way information and vice versa; this is essentially
a formulation of the Bohr complementarity principle and is fully in agreement with the Copenhagen
interpretation.~\footnote{The author is grateful to an anonymous referee for suggesting this
formulation and the analogy of the quantum eraser for the Afshar experiment.}
The emphasis on `when' in this formulation is particularly important and contains within it a tacit
time-dependence that has not been explored in full depth. The only new element in the Afshar experiment is
the presence of the wire grid. As in Fig.~\ref{f1}, let the photon beams converge at a spatial location $\sigma_1$, where they
pass through the wire grid. At location $\sigma_2$ let the photon beams hit the detectors, after which
one is able to deduce both the existence of the interference pattern and the path information
for the photons. We argue below that these facts do not constitute a violation of the Bohr complementarity principle as claimed
by Afshar~\cite{afshar1,afshar2,afshar3}. Further, the wire grid is superfluous in the sense that it is not required
to deduce the existence of the interference pattern; the complete path information is sufficient for this purpose.
Consider a photon that passes through the wire grid at time~$t_1$ and then reaches a detector at time~$t_2$. Afshar's argument is essentially that
a large number of such photons that reach the detectors at time~$t_2$ must have formed an interference pattern at time $t_1$, because of the negligibly small
number of photons that impinged on the wire grid at time~$t_1$ (as compared to the number that reached the detectors). It is crucial to note that Afshar is able to argue
for the existence of the interference pattern for this large number of photons only \emph{after} they have reached the detectors, at which point of time complete path
information is also available for each of these photons. If these photon paths are superposed on to an imaginary screen at location~$\sigma_1$, a complete reconstruction
of the interference pattern is theoretically possible, without any need for the wire grid. Therefore if one accepts Afshar's argument that the Bohr complementarity principle
is violated in his experiment, one must also accept the same conclusion from a classical version of his experiment, in which the wire grid is not present. In fact the analysis
of the path information would provide a complete reconstruction of the interference pattern, whose existence is only nonconstructively deduced via the wire grid. Hence
the wire grid is not only logically superfluous in Afshar's argument, but it also provides a less satisfactory and controversial method for deducing the existence of an interference pattern.
It is extremely important to note that Afshar also accepts the validity of the path information for a photon in his experiment, post its detection at time~$t_2$. It follows that Afshar has also
essentially analyzed particle-like photons, each of which he accepts as having passed through only one of the slits, to arrive at his
conclusion that an interference pattern existed at a previous time~$t_1$ for these photons.
We may conclude that Afshar's method of deducing (or as he claims, measuring) the existence of an interference pattern does not provide any evidence whatsoever
of wave-like properties for a photon. Likewise, analysis of the path information in a classical version of Afshar's experiment (without the wire grid)
would also provide a reconstruction of the interference pattern without providing any evidence whatsoever of wave-like properties for a photon.
In contrast, the classical method for measuring an interference pattern, via a destruction of the photons by a screen located at $\sigma_1$, would eliminate
any possibility of deducing path information for these photons. It is the non-availability of the path information at time $t_1$ that establishes the wave nature of the photon
(via the superposed state $S$), whose logical consequence is the presence of an interference pattern. It is extremely important to note that the converse implication does not hold:
the presence of an interference pattern does not logically imply that that the photons are in a superposed state. Hence the co-existence of the interference pattern and the path
information, as deduced at time $t_2$, is not a contradiction as claimed by Afshar. Post the time $t_2$, the correct interpretation of the Afshar experiment (or a classical version
without the wire grid) is that particle-like photons have a non-classical probability distribution, resulting in an interference pattern. Here one must understand the time dependence
involved and give up any realist notions of the superposed state $S$, or the wave nature of the photon.
The state $S$ should be thought of as merely a formalism by which we may deduce probability distributions, which
are confirmed by the existence of the interference pattern. The Copenhagen interpretation championed by Bohr has
always been non-realist in this sense. As we will see in the ensuing sections, in the (non-realist) NAFL
interpretation, the state $S$ of the photons has the precise logical meaning that path information is not available;
it does \emph{not} mean that a single photon `really' took both paths, which is a logical impossibility for a particle.
Secondly, the retroactive assertion of the path information at time~$t_2$ is a logical truth that \emph{only}
applies for times $t \ge t_2$; it does \emph{not} mean that the path information was always available. Again, one needs
a non-classical, temporal logic like NAFL to formulate this time-dependence, which, however, was always tacitly
implied by the Copenhagen interpretation. Seen in this light, we know the following facts about the modified
Afshar experiment. Firstly, at time $t_1$, path information is not available for the photons that are passing
through the wire grid (and which will impinge on the detectors at time $t_2$), and hence they are in the superposed state
$S$ at that instant ($t_1$). The interference pattern is a logical consequence of this state. Secondly, at time $t_2$,
path information is retroactively available for the photons that impinged on the detectors at that instant;
however, this retroactive assertion itself only applies for times $t \ge t_2$. The inferred
presence of the interference pattern does not logically imply that the photons are in a superposed
state $S$; it is this fact which permits a retroactive assertion of the path information post the time $t_2$.
A contradiction would ensue from the above two facts if one insists on ascribing reality for the
superposed state $S$ and also insists that the retroactive assertion of the path information was always
applicable (as would be the case in classical logic). In the first case one would have the contradiction that
the photons `really' took both paths \emph{and} the photons took only one of the available paths; in the second case the
contradiction would be that path information is both available and not available for the said photons. Kastner's
realist stance~\cite{kastner1,kastner2} leads to these contradictions and hence forces her to reject
the retroactive assertion of the path information even in the case when the wire grid is not present (as noted in Sec.~\ref{intro}).
This stance contradicts well-accepted results in physics and fails to explain how a photon can `really' take both paths and yet exhibit
particle-like behaviour at the detectors (decoherence).
The Bohr complementarity principle is formulated in the NAFL interpretation as a logical principle (see
Sec.~\ref{afsh}). Namely, that at any given time, a photon can either be in a superposed state, in which
case the law of noncontradiction fails, or in a classical particle-like state, when the law of noncontradiction
applies; at any given time, it is not logically possible for the photon to be in both of these states. This
formulation reduces to that noted earlier in terms of which-way information and interference information.
The Englert-Greenberger duality relation~\cite{greenberger,englert}, when properly interpreted, is also
upheld in the Afshar experiment, contrary to Afshar's claims~\cite{afshar1,afshar2,afshar3}. This relation
may be expressed in the form
\begin{equation}
D^2 + V^2 \le 1. \label{eg}
\end{equation}
Here $V$ is the visibility of the interference fringes and $D$ is the distinguishability of the photon paths.
Afshar claims that in his experiment, both $D$ and $V$ are close to 1, and hence Eq.~(\ref{eg}) is violated.
However, if the time dependence of the parameters $D$ and $V$ is taken into account,
one may conclude that Eq.~(\ref{eg}) is indeed upheld in the Afshar experiment. At time~$t_1$, when the photons
pass through the wire grid, there is no path information and hence $D=0$ for these photons; further, the photons are in
a superposed state at this time with the interference fringes present, and hence $V=1$. Indeed, this is certainly true for the
photons that impinge on the wire grid. Post the time $t_2$, path information is available for the photons that impinge
on the detectors, and hence we take $D=1$ for these photons. As noted
earlier, we may infer that these photons were part of an interference pattern at time $t_1$, regardless of whether
the wire grid is present. Despite this inference, one concludes that $V=0$ for time $t \ge t_2$, because the presence of the
interference pattern does not imply that the photons are in a superposed state and so the
path information stands for these times. And when path information is available, the interference pattern is no longer a logical
consequence of any wave-like properties of the photon, which is why we take $V=0$ at these times. When $D$ and $V$ are interpreted in this manner, it is impossible for
Eq.~(\ref{eg}) to be violated in the Afshar experiment (with or without the wire grid) because it expresses precisely the Bohr
complementarity principle.
Finally, consider the analogy of the Schr\"odinger cat experiment (see Sec.~\ref{sch}). At time $t_1$,
let us say that the cat is in the box and is in a superposed state of `alive and dead'. What this state means
in the NAFL interpretation is that no information is available at time~$t_1$ as to the cat's classical state
(`alive' or `dead'). Subsequently, at time~$t_2$, when the box is opened, suppose the cat is found in the `alive'
state. One may retroactively infer at time~$t_2$ that the cat was alive at time~$t_1$. This retroactive inference
only applies for times~$t \ge t_2$ and does not contradict the fact that at time~$t_1$, the cat was in a superposed
state (to which no `reality' can be ascribed). Again the complementarity principle is upheld in both the
NAFL and the Copenhagen interpretations because at any given time, the cat is in only one state, despite the
retroactive inference. Whereas Kastner's interpretation~\cite{kastner1,kastner2} would
amount to barring the retroactive inference that the cat was in the `alive' state at time~$t_1$ because it `really'
was in the superposed state of `alive and dead' at that time. Any such `reality' is obviously aphysical and
inexplicable, and therefore untenable in our view.
\section{The NAFL interpretation of quantum superposition}\label{qsnafl}
At the outset, we hasten to note that the NAFL interpretation is still nascent and
incomplete in the sense that a lot of work remains to be done in demonstrating how
real analysis can be done in NAFL~\cite{ract}, and ultimately, how all of quantum mechanics can
be formalized in this logic. What has been accomplished at this stage is a completely
new and logical interpretation of some of the ``weird'' phenomena of quantum mechanics,
in particular, superposition and entanglement~\cite{ijqi,1166,acs}. In this section, we will
confine ourselves to a brief exposition of the NAFL interpretation of quantum superposition,
and refer the reader to the original references for further details. The reader who
is already familiar with these details may skip to Sec.~\ref{bcnafl}. Our purpose herein
is to provide just enough information on NAFL so as to enable an appreciation
of the delicate and subtle logical issues involved in the interpretation of the Bohr
complementarity principle in Afshar's experiment, which is discussed in Sec.~\ref{bcnafl}.
The language, well-formed formulae and rules of inference of
NAFL theories~\cite{ijqi} are formulated in exactly the same manner as in classical
first-order predicate logic with equality~(FOPL), where we shall assume, for convenience,
that natural deduction is used; however, there are key differences and restrictions
imposed by the requirements of the Main Postulate of NAFL, which is explained in this
section. In NAFL, truths for \emph{formal propositions} can exist \emph{only} with respect
to axiomatic theories. There are no absolute truths in just the \emph{language} of
a NAFL theory, unlike classical/intuitionistic/constructive logics. There do exist
absolute (metamathematical, Platonic) truths in NAFL, but these are truths
\emph{about} axiomatic theories and their models. As in FOPL, a NAFL theory T
is defined to be consistent if and only if T has a model, and a proposition
$P$ is undecidable in T if and only if neither $P$ nor its negation
$\neg P$ is provable in T.
\subsection{The Main Postulate of NAFL}\label{mp}
If a proposition $P$ is provable/refutable in a consistent NAFL theory T, then $P$
is true/false with respect to T (henceforth
abbreviated as `true/false in T'); \emph{i.e.}, a model for T will assign $P$ to be
true/false. If $P$ is undecidable in a consistent NAFL theory T, then the Main
Postulate~\cite{1166} provides the appropriate truth definition as follows: $P$ is true/false
in T if and only if $P$ is provable/refutable in an \emph{interpretation} T* of T. Here T*
is an axiomatic NAFL theory that, like T, temporarily resides in the human mind and acts as a
`truth-maker' for (a model of) T. The theorems of T* are precisely those propositions
that are assigned `true' in the NAFL model of T, which, unlike its classical counterpart,
is not `pre-existing' and is instantaneously generated by T*. It follows that T* must necessarily
prove all the theorems of T. Note that for a given consistent theory T, T* could vary in time according to the
free will of the human mind that interprets T; for example, T* could be T+$P$ or
T+$\neg P$ or just T itself at different times for a given human mind, or in the context
of quantum mechanics, for a given �observer�. Further, T* could vary from one observer
to another at any given time; each observer determines T* by his or her own free will.
The essence of the Main Postulate is that $P$ is true/false in T if and only if it has been
\emph{axiomatically declared} as true/false by virtue of its provability/refutability
in T*. In the absence of any such axiomatic declarations, \emph{i.e.}, if $P$ is
undecidable in T* (\emph{e.g.}~take T*=T), then $P$ is `neither true nor false' in T and
Proposition~\ref{p1} shows that consistency of T requires the laws of the excluded middle and
noncontradiction to fail in a nonclassical model for T in which $P \& \neg P$ is the case.
\begin{proposition}\label{p1}
Let $P$ be undecidable in a consistent \emph{NAFL} theory \emph{T}. Then $P \vee \neg P$
and $\neg (P \& \neg P)$ are not theorems of \emph{T}. There must exist a nonclassical
model $\mathcal{M}$ for \emph{T} in which $P \& \neg P$ is the case.
\end{proposition}
For a proof of Proposition~\ref{p1}, see Ref.~\refcite{ijqi} or Appendix~A of Ref.~\refcite{acs};
this proof also seriously questions the logical/philosophical basis for the law
of noncontradiction in both classical and intuitionistic logics.
The interpretation of $P \& \neg P$ in the nonclassical model will be explained
in Sec.~\ref{qs}. Proposition~\ref{p1} is a metatheorem, \emph{i.e.}, it is a theorem
\emph{about} axiomatic theories. The concepts in Proposition~\ref{p1}, namely, consistency,
undecidability (or provability) and the existence of a nonclassical model for a theory
and hence, quantum superposition and entanglement, are strictly metamathematical
(\emph{i.e.}, pertaining to semantics or model theory) and not formalizable in the
syntax of NAFL theories. A NAFL theory~T is either consistent or inconsistent,
and a proposition~$P$ is either provable or refutable or undecidable in T, \emph{i.e.},
the law of the excluded middle applies to these metamathematical truths.
Note that the existence of a nonclassical model does not make T inconsistent or even
paraconsistent in the conventional sense, because T does not
\emph{prove} $P \& \neg P$. However, one could assert that the model theory for T requires the framework
of a paraconsistent logic, so that the nonclassical models can be analyzed. NAFL is the only logic that
correctly embodies the philosophy of formalism~\cite{1166}; NAFL truths for formal propositions are
axiomatic, mental constructs with strictly no Platonic world required.
\subsection{Quantum superposition justified in NAFL}\label{qs}
The nonclassical model $\mathcal{M}$ of Proposition~\ref{p1} is a superposition of two or
more classical models for T, in at least one of which $P$ is true and $\neg P$ in another.
Here `(non-\nolinebreak)classical' is used strictly with respect to the status of $P$. In
$\mathcal{M}$, `$P$'~(`$\neg P$') denotes that `$\neg P$'~(`$P$') is not provable in T*, or
in other words, $\mathcal{M}$ expresses that neither $P$ nor $\neg P$ has been
axiomatically declared as (classically) true with respect to T; thus $P$, $\neg P$,
and hence $P \& \neg P$, are indeed (nonclassically) true \emph{in our world}, according
to their interpretation in $\mathcal{M}$. Note also that $P$ and $\neg P$ are
\emph{classically} `neither true nor false' in $\mathcal{M}$, where `true' and `false' have
the meanings given in the Main Postulate. The quantum superposition principle is justified
by identifying `axiomatic declarations' of truth/falsity of $P$ in T (via its
provability/refutability in T* as defined in the Main Postulate) with `measurement' in the
real world. NAFL is more in tune with the Copenhagen interpretation of quantum mechanics
than the many-worlds interpretation~(MWI). Nevertheless, the \emph{information content} in
$\mathcal{M}$ is that of two or more classical models (or `worlds'), and MWI is at least
partially vindicated in this sense.
\subsection{Example: Schr\"odinger's cat}\label{sch}
Consider the situation wherein the cat is put into the box at time $t=t_0$ and has a
probability $0.5$ of being in the `alive' state at $t=t_2$, when a `measurement'
is made of its state. Let $P$ be the proposition that `The cat is alive', with $\neg P$
denoting `The cat is dead'; obviously, $P$ is undecidable in a suitable formalization QM
of quantum mechanics, which may be taken to include definitions describing this experiment. For
$t_0 < t < t_2$, the observer makes no measurements, and in tune with the
identification noted in Sec.~\ref{qs}, makes no axiomatic declarations regarding $P$
in the interpretation QM* (say, let QM*=QM for this time period). In the resulting
nonclassical model $\mathcal{M}$ of QM, the superposed state $P \& \neg P$ is the case;
this means that the cat has not been declared (measured) to be either alive or dead, which
is certainly true in the real world. At $t=t_2$, if $P$~($\neg P$) is observed,
then the observer takes, say, QM*=QM+$V$~($\neg V$), where $V$~($\neg V$) is defined as
`The cat is alive~(dead) at $t=t_2$'; note that QM* will prove
$P$~(or $\neg P$) at $t=t_2$, \emph{i.e.}, when the observer measures the cat to be
alive~(dead) in the real world, he makes the appropriate axiomatic declarations in his
mind, thus setting up QM* as defined. It should be emphasized that a NAFL theory only
`sees' the observer's axiomatic declarations and does not care whether the real world
exists. The observer sees the real world and the proposed identification of his
measurements with his axiomatic declarations is only an informal convention that is outside
the purview of NAFL. The observer could also use his free will to make his axiomatic
declarations irrespective of (and possibly in contradiction to) what he measures in the
real world; of course, if $P$ is not about the real world, then he has no other choice.
NAFL correctly handles the temporal nature of truth via the time-dependence of QM*.
If $P$ is observed (and axiomatically asserted via QM*=QM+$V$) at $t=t_2$, then the
proposition $U$ that ``The cat was alive for $t_0 < t < t_2$'' can be formalized for
$t \ge t_2$ and proven in the NAFL theory QM*=QM+$V$; $U$ does not conflict temporally
or in meaning with the superposed state $P \& \neg P$, which applies for $t_0 < t < t_2$.
We will return to the Schr\"odinger cat example in Sec.~\ref{ctoss}, in order to further
elaborate upon the validity of this retroactive assertion of $U$.
\subsection{Theory syntax and proof syntax}\label{tsps}
A NAFL theory T requires two levels of syntax, namely the `theory syntax' and the
`proof syntax'. The theory syntax consists of precisely those propositions that are
legitimate, \emph{i.e.}, whose truth in T satisfies the Main Postulate; obviously,
the axioms and theorems of T are required to be in the theory syntax. Further, one can
only add as axioms to T those propositions that are in its theory syntax. In particular,
neither $P \& \neg P$ nor its negation $P \vee \neg P$ is in the theory syntax when
$P$ is undecidable in T. The proof syntax, however, is classical because NAFL has the
same rules of inference as FOPL; thus $\neg (P \& \neg P)$
is a valid deduction in the proof syntax and may be used to prove theorems of T. For
example, if one is able to deduce $A \Rightarrow P \& \neg P$ in the proof syntax of T
where $P$ is undecidable in T and $A$ is in the theory syntax, then one has proved $\neg A$
in T despite the fact that $\neg (P \& \neg P)$ is not a theorem (in fact not even a
legitimate proposition) of T. This is justified as follows: $\neg (P \& \neg P)$ may be
needed to prove theorems of T, but it does not follow in NAFL that
the theorems of T imply $\neg (P \& \neg P)$ if $P$ is undecidable in T.
Let $A$ and $B$ be undecidable propositions in the theory syntax of T.
Then $A \Rightarrow B$ (equivalently, $\neg A \vee B$) is
in the theory syntax of T if and only if $A \Rightarrow B$ is \emph{not}
(classically) deducible in the proof syntax of T. It is easy to check that if
$A \Rightarrow B$ is deducible in the proof syntax of T, then its (illegal) presence in the
theory syntax would force it to be a theorem of T, which is not permitted by
the Main Postulate. For, in a nonclassical model $\mathcal{M}$ for T in which both $A$
and $B$ are in the superposed state, $A \& \neg B$ must be nonclassically true, but
theoremhood of $A \Rightarrow B$ will prevent the existence of $\mathcal{M}$. If one
replaces $B$ by $A$ in this result, one obtains the previous conclusion that
$\neg (A \& \neg A)$ is not in the theory syntax. For example, take $\mbox{T}_0$ to be the
null set of axioms. Then nothing is provable in $\mbox{T}_0$, \emph{i.e.}, every legitimate
proposition of $\mbox{T}_0$ is undecidable in $\mbox{T}_0$. In particular, the proposition
$(A \& (A \Rightarrow B))\Rightarrow B$, which is deducible in the proof syntax of $\mbox{T}_0$
(via the \emph{modus ponens} inference rule), is not in the theory syntax; however,
if $A \Rightarrow B$ is not deducible in the proof syntax of $\mbox{T}_0$,
then it is in the theory syntax. Note also that $\neg \neg A \Leftrightarrow A$ is
not in the theory syntax of $\mbox{T}_0$; nevertheless, the `equivalence' between
$\neg \neg A$ and $A$ holds in the sense that one can be replaced by the other
in every model of $\mbox{T}_0$, and hence in all NAFL theories. Indeed, in a nonclassical
model for $\mbox{T}_0$, this equivalence holds in a nonclassical sense and must be
expressed by a different notation~\cite{1166}.
\section{Bohr Complementarity and the NAFL interpretation of Afshar's experiment}\label{bcnafl}
The Bohr complementarity principle easily follows from the NAFL interpretation of
quantum superposition discussed in Sec.~\ref{qsnafl}. As applied to Afshar's experiment,
the relevant definition is as follows.
\begin{definition}[Bohr complementarity principle~(BCP)]\label{bcp}
The particle and wave nature of the photon cannot be \emph{simultaneously}
demonstrated to hold at any given spatial location and time within a given experiment.
\end{definition}
Note carefully the location of the word ``\emph{simultaneously}'' in this definition;
if we were to change BCP to ``\dots cannot be demonstrated to hold \emph{simultaneously}
at any given spatial location and time within a given experiment'', then such a
definition would arguably fail even in NAFL. The ability of NAFL to handle the
temporal nature of mathematical truth and the distinctions NAFL makes between
syntax and semantics are demonstrated to be very important for a
correct logical explanation of the results in Afshar's experiment.
\subsection{Quantum superposition in Afshar's experiment}\label{qsae}
Let QM be the NAFL theory formalizing quantum mechanics. We assume that definitions providing
a detailed description of the single-photon version of Afshar's experiment~\cite{afshar1,afshar2,afshar3}
(which he has reportedly performed with the same results) have already been included in QM.
\begin{definition}\label{defp}
Let $P$ denote the proposition ``The photon passed through (only) slit~U
at location~$\sigma$ and time~$t_0$ in Fig.~\ref{f1}.''
\end{definition}
\begin{definition}\label{defnegp}
Let the negation $\neg P$ of $P$ denote the proposition ``The photon passed through
(only) slit~L at location~$\sigma$ and time~$t_0$ in Fig.~\ref{f1}.''
\end{definition}
Since $P$ and $\neg P$ are equally probable, it follows that $P$ is undecidable in QM.
Here it is important to understand the NAFL concept of negation (see Sec.~2.2 of
Ref.~\refcite{1166}); in particular, mutually exclusive classical possibilities (e.g.\ in the real world) are
negations of each other and $\neg \neg P$ is equivalent to $P$, just as in classical
logic. But unlike classical logic, $P \vee \neg P$ is \emph{not} a theorem of QM, and
unlike intuitionistic logic, $\neg (P \& \neg P)$ is also \emph{not} a theorem of QM;
as noted in Sec.~\ref{mp}, there must exist a nonclassical model $\mathcal{M}$ for QM
in which $P \& \neg P$ is the case. The interpretation of $P \& \neg P$ in $\mathcal{M}$
is identical to that explained in Secs.~\ref{qs} and~\ref{sch}; `$P$'~(resp.~`$\neg P$') of
$P \& \neg P$ has the nonclassical meaning that `$\neg P$'~(resp.~`$P$') is not provable
in the observer's interpretation QM* of QM. In keeping with the informal convention noted in
Secs.~\ref{qs} and~\ref{sch}, the observer agrees to keep his axiomatic assertions in tune with his
measurements in the real world, so that the superposition $P \& \neg P$ in $\mathcal{M}$ also has
an equivalent nonclassical meaning with `$P$'~(resp.~`$\neg P$') now denoting that ``The observer
has not measured the photon to pass through slit~L~(resp.~slit~U) at location~$\sigma$ and
time~$t_0$ in Fig.~\ref{f1}.'' One can see that the NAFL interpretation of $P \& \neg P$ is
meaningful in the real world because $\neg P$ is not \emph{really} the negation of $P$ in $\mathcal{M}$;
if the observer has not measured (axiomatically declared) that the photon passed through slit~L, it does not
follow that he has measured (axiomatically declared) that the photon passed through slit~U.
Consistency demands that the observer should never be able to \emph{prove}
$P \& \neg P$ in any NAFL theory, in particular, QM or QM*; for such a proof would imply
the contradiction: ``The photon \emph{really} passed through both slits''. The only reality
that exists, as far as the observer is concerned, is that he has not measured
or axiomatically asserted either $P$ or $\neg P$ in the real world. This same reality is
accurately modeled in the nonclassical model $\mathcal{M}$ where, somewhat paradoxically,
$P$, $\neg P$ and $P \& \neg P$ all hold, but with the nonclassical interpretations noted
above; the existence of such a nonclassical model for QM is a requirement of consistency
in NAFL.
NAFL requires that both $P \& \neg P$ and its negation $\neg (P \& \neg P)$
(or equivalently, $P \vee \neg P$) are not legitimate propositions in the theory syntax
of QM; see Sec.~\ref{tsps}. What this means is that \emph{formally}, the status of
the photon in the theory syntax of QM is indeterminate; it cannot be proven to be
either a particle (in which case $P \nolinebreak \vee \nolinebreak \neg P$ should be
a theorem) or a wave (in which case $P \& \neg P$ should be a theorem). Consequently, both
classical and nonclassical models, with respect to the proposition~$P$, exist for QM.
Note that the NAFL semantics for the nonclassical model $\mathcal{M}$ of QM does not
take a stand either on the status of the photon as a particle or non-particle.
Nevertheless, the NAFL theory QM tacitly supports (but not requires) the `reality' of
the particle nature of the photon in a \emph{metalogical} sense, \emph{i.e.}, outside of
both theory syntax and semantics. This will be fully explained in Sec.~\ref{afsh}. Here we
observe that the proof syntax of QM requires the deduction of $P \vee \neg P$;
see Sec.~\ref{tsps}, where it is noted that the rules of inference of NAFL theories,
which determine the proof syntax, must be classical. It is possible (but not necessary)
to interpret this deduction as a metalogical assertion of the particle nature of
the photon in the sense that the photon `really' had to pass through one and only
one of the two slits U and L at time~$t_0$ in Fig.~\ref{f1}, even though, at that
instant, the observer did not have a proof that either of these paths was traversed.
It is this lack of knowledge at time~$t_0$ that NAFL semantics expresses as a requirement
of logical consistency, via the nonclassical model $\mathcal{M}$, rather than any
perceived reality for the particle nature of the photon. But such a perceived reality became inevitable
the moment Definition~\ref{defnegp} was formulated in QM as the negation of $P$. Here it should be kept in
mind that the `photon' referred to is that which must necessarily pass through at least one of the slits.
The coin toss experiment, considered in Sec.~\ref{ctoss}, will further illustrate the metalogical `reality'
of $P \vee \neg P$ in NAFL.
\subsection{Bohr Complementarity in Afshar's experiment}\label{afsh}
Consider again the single-photon version of Afshar's experiment in Fig.~\ref{f1}, with $P$,
$\neg P$ and the theory QM as defined in Sec.~\ref{qsae}. At time $t = t_0$ a photon passes
through the slit(s); at $t = t_1 > t_0$, the said photon passes the wire grid in the
``interference plane'' at location~$\sigma_1$, and subsequently passes through the
converging lens; at $t = t_2 > t_1$, the photon ends up at one of the locations marked
$\mbox{U}^{\prime}$ or $\mbox{L}^{\prime}$ (say, $\mbox{U}^{\prime}$, for the sake of
definiteness). Note that in the single-photon version of Afshar's experiment, the
restriction is that only one photon at a time can pass through the slit(s), although
several photons actually end up at $\mbox{U}^{\prime}$ and $\mbox{L}^{\prime}$.
\begin{definition}\label{defq}
Take the proposition $Q$ to denote that ``The photon reaches the image $\mbox{U}^{\prime}$ of
slit U at location $\sigma_2$ and time $t_2$ in Fig.~\ref{f1}''.
\end{definition}
\begin{definition}\label{defr}
Let the proposition $R$ denote that ``The images $\mbox{L}^{\prime}$ and $\mbox{U}^{\prime}$
of the slits L and U respectively are undistorted and unchanged in intensity when the
wire grid is inserted at the calculated minima of the expected interference pattern at
location~$\sigma_1$ in Fig.~\ref{f1}''.
\end{definition}
Consider the following contentious inferences that are inherent in the controversy
over Afshar's experiment.
\begin{inference}\label{inf1}
The theory QM+$Q$ proves $P$.
\end{inference}
This, of course, implies the particle nature of the photon.
\begin{inference}\label{inf2}
The theory QM+$R$ proves $P \& \neg P$.
\end{inference}
In words, Inference~\ref{inf2} means ``The photon passed through both slits U and L at location~$\sigma$ and
time~$t_0$ in Fig.~\ref{f1}'', which in turn implies the wave nature of the photon. Since $Q$
and $R$ are true, in the sense that they are observed, one may conclude that if
Inferences~\ref{inf1} and~\ref{inf2} are granted, the resulting `true' (but inconsistent)
theory QM+$Q$+$R$ violates BCP; see Definition~\ref{bcp}. Kastner~\cite{kastner1,kastner2} seems to
permit Inference~\ref{inf1}, but disputes that it `really' establishes `which-way' information
for the photon; she argues that Cramer's Transactional Interpretation~\cite{cramer} supports
her stand. Kastner concludes that only the wave nature of the photon is unambiguously
exhibited at the slits in Afshar's experiment, via Inference~\ref{inf2}, which she allows
(although neither she nor Afshar interpret the wave nature of the photon as a proposition of
the form $P \& \neg P$, as NAFL requires). In what follows, we will argue that the NAFL
interpretation allows Inference~\ref{inf1}, but \emph{not} Inference~\ref{inf2};
this fact, when coupled with the temporal nature of mathematical truth in NAFL, means that
Bohr complementarity survives. The claim that a single photon `really' exhibits wave nature
is questionable from the NAFL point of view and will be critically examined in
Sec.~\ref{noinf2}. However, we emphasize at the outset that the wave nature of light
could well follow unambiguously in a new, as yet unknown, version of QM formalized in NAFL.
The present argument applies only to our current understanding of QM and the nature of light.
The NAFL justification of BCP is as follows. For $t_0 \le t < \nolinebreak t_2$, the observer
has the theory QM in mind with the interpretation QM*=QM. Hence by Proposition~\ref{p1} (see
Sec.~\ref{mp}), $P \& \neg P$ holds for the observer, in a nonclassical model $\mathcal{M}$
for QM. At $t=t_2$, upon measuring $Q$, the observer switches to the interpretation \linebreak
QM*=QM+$Q$ and concludes $P$ in a classical model for QM, via a proof in QM*
(Inference~\ref{inf1}). Note that $P$ only applies retroactively, for times $t \ge t_2$;
therefore it does not temporally conflict with the superposition $P \& \neg P$, which
applied for $t_0 \le t < t_2$. The second observation here is that the theory QM does not
\emph{prove} $P \& \neg P$; as noted earlier, $P \& \neg P$ is not even a legitimate
proposition in the theory syntax of QM. Such a proof of $P \& \neg P$ would make QM
inconsistent in NAFL, which is \emph{not} a conventional paraconsistent logic, as noted in Sec.~\ref{mp}.
In fact, the Main Postulate of NAFL and consequently, the existence of $\mathcal{M}$, are
strictly metamathematical results, \emph{i.e.}, pertaining to semantics
or model theory; these concepts are not formalizable in either the theory syntax or proof
syntax of QM. The theory QM* can therefore prove $P$ without loss of
consistency. Since the deduction of $P$ and the meta-deduction of the superposition
$P \& \neg P$ were made in non-overlapping time intervals, and, in particular,
were \emph{not} made \emph{simultaneously}, BCP survives in NAFL.
One might object that there is something unsatisfactory about this
state of affairs; if we interpret $P$ and $P \& \neg P$ as affirming the particle and the
wave natures of the photon respectively, they seem to be contradictory even if asserted at
non-overlapping time intervals, since, after all, they both apply to the same photon at the
same spatial location and time; this is in fact the essence of Afshar's argument~\cite{afshar1}.
The answer to this objection is that in the nonclassical
model $\mathcal{M}$, $P \& \neg P$ only means that the observer has not axiomatically
asserted $P$ via a proof in QM* (or measured $P$ in the real world) and he has not
axiomatically asserted $\neg P$ via a proof in QM* (or measured $\neg P$ in the real world);
as noted in Sec.~\ref{qsae}, $P \& \neg P$ does \emph{not} imply that the photon `really'
exhibited wave nature and passed through both slits. But the retroactive assertion of
$P$ (via a proof in QM*), valid for $t \ge t_2$, can be taken to have the \emph{metalogical}
(see Sec.~\ref{qsae}) meaning that the photon `really' passed through only
slit~U at time $t_0$, and therefore does not conflict in meaning with the meta-deduction of
$P \& \neg P$ made in $\mathcal{M}$ during the time interval $t_0 \le t < t_2$; both are
indeed `true' when appropriately interpreted. To summarize, for $t_0 \le t < t_2$,
NAFL semantics only recognizes the metamathematical truth of $P \& \neg P$ in $\mathcal{M}$.
For $t \ge t_2$, NAFL semantics asserts the retroactive truth of $P$ via a classical model
for QM; this truth may be taken to hold \emph{metalogically} for $t_0 \le t < t_2$,
\emph{i.e.}, outside of NAFL syntax and semantics. The temporal nature of
NAFL truth plays a vital role in removing the mystery associated with wave-particle
duality. In classical logic, unlike NAFL, the retroactive assertion of $P$ at $t=t_2$ would
necessarily mean that $P$ was \emph{always true} in the semantics of QM, including
at $t=t_0$. Therefore in the framework of classical logic, such a retroactive assertion
of $P$ will be problematic and will clash with the quantum superposition state that
actually held in the semantics of QM at $t=t_0$.
At this stage the reader will have the following obvious question; does not
$R$~(Definition~\ref{defr}) \emph{prove} the existence of the interference pattern
at location $\sigma_1$ of Fig.~\ref{f1} and consequently, the reality of the wave
nature of the photon when it enters the slits U and L? In the NAFL interpretation, the
answer to the second part of this question is in the negative; in Sec.~\ref{noinf2} we will
argue that Inference~\ref{inf2} is not permitted on logical grounds and therefore
the observation $R$ is irrelevant to the above-noted retroactive conclusion of $P$ by
the observer. A second question that arises is whether there are logical
grounds for banning Inference~\ref{inf1} as well. Afshar~\cite{afshar1} has asserted
that Inference~\ref{inf1} follows from standard optics and has subsequently pointed out
elsewhere that moving one of the slits U or L in Fig.~\ref{f1} causes the corresponding
image $\mbox{U}^{\prime}$ or $\mbox{L}^{\prime}$ to co-move with the slit. Here we do
not pass judgement on the experimental validity of Afshar's claims. We only wish to
point out that on purely logical grounds, NAFL does permit the \emph{conclusion} of
Inference~\ref{inf1}, namely, the retroactive assertion of $P$; for reasons mentioned
in Sec.~\ref{qsae} and in this subsection, NAFL does not object to the metalogical existence
of the photon as a particle. Thus one could also take the interpretation at
$t=t_2$ as QM*=QM+$P$ (instead of QM*=QM+$Q$ as noted above), if it turns out that there is indeed
something wrong with Inference~\ref{inf1}.
We note that although the NAFL interpretation does permit the retroactive
assertion of $P$, via Inference~\ref{inf1} or otherwise, there is no \emph{obligation}
on the part of the observer to make such an assertion (or measurement). The
observer can choose to live with just the \emph{metamathematical} conclusion that both
classical and nonclassical models for QM exist; he could choose to remain agnostic
(\emph{i.e.}, take no stand) on the \emph{metalogical} status of the photon as a
particle or non-particle in the nonclassical models and scrupulously avoid making any
retroactive assertions/measurements. Consequently, to facilitate such an agnostic
attitude, Inference~\ref{inf1} must either be disallowed or if permitted, must be weakened so as not to
imply any `reality' for the path of the photon at time~$t_0$. At time~$t_2$, the photon
behaves \emph{as though} it originated from slit~U (via the weakened Inference~\ref{inf1}), but need not
have `really' done so, from the agnostic's point of view. Kastner~\cite{kastner1,kastner2}
seems to prefer this latter approach to Inference~\ref{inf1}; however, rather than remain
agnostic, she asserts that the photon passed through both slits, as a wave. In Sec.~\ref{kast},
we criticize the limiting of Inference~\ref{inf1} as noted above, as well as Kastner's
reasons for doing so. Cramer's Transactional Interpretation~\cite{cramer} is also criticized from the
NAFL point of view in Sec.~\ref{cram}. An \emph{anti-realist} approach is also possible in NAFL. Anti-realism
is stronger than agnosticism, in the sense that it requires the observer to \emph{deny} any
reality for the state of the photon as particle or non-particle in the nonclassical
model~$\mathcal{M}$. In other words, no such reality exists in the absence of an axiomatic
declaration, which, by informal convention, the observer associates with `measurement'
in the real world. The anti-realist approach would presumably require the
banning of Inference~\ref{inf1} and other retroactive assertions of reality. Of course,
Inference~\ref{inf2} is already illegal in NAFL, as will be explained in Sec.~\ref{noinf2}.
In summary, there are three approaches possible in NAFL regarding the status of the
photon during $t_0 \le t < t_2$, when $\mathcal{M}$ is in force. These are, a metalogical
reality for the particle state, agnosticism and anti-realism. The author believes
that the first option is the most satisfactory from both philosophical and logical
points of view, \emph{given} Definition~\ref{defnegp}. With such a choice of negation
in NAFL, has the observer denied any possibility that the photon can `really'
pass through both slits, thereby affirming its metalogical particle status? This is a
tricky issue; the author believes that the answer is in the positive, i.e., the observer
has predetermined the particle nature of the photon via Definition~\ref{defnegp}. One may attempt to justify
the above choice of negation by arguing that $P$ and $\neg P$ cannot be `measured' together, although
$P \& \neg P$ could be `really' true in the real world (this would provide another reason for banning
Inference~\ref{inf2}, for otherwise the proposition $R$ in Definition~\ref{defr} surely amounts to a `measurement'
of $P \& \neg P$). However, such an attempt at justification fails, for the following reason.
NAFL truth for undecidable propositions of a theory is purely axiomatic in nature, by the
Main Postulate (see Sec.~\ref{mp}); the concept of `measurement' cannot be formalized
in the theory syntax of NAFL theories~\cite{ijqi}. The informal convention of associating
`axiomatic declaration' with `measurement' in the real world can, in principle, be broken
in NAFL; see Sec.~\ref{sch} as well as the final paragraph of Sec.~\ref{noinf2}. If the photon can `really'
(or classically) pass through both slits, there ought to be nothing to stop
us from axiomatically declaring, i.e., inferring via a proof in QM*, that
`The photon passed through both slits at $t=t_0$', even if \emph{measurement} of this
event is impossible. For such a proposition to be legal in the theory syntax of QM,
NAFL would require the negation of $P$ to be modified such that it includes the case of
the photon passing through both slits in disjunction with the case noted in
Definition~\ref{defnegp}. \emph{However}, as a consequence, BCP will fail by the Main
Postulate of NAFL. For if both the particle and wave natures of the photon are classically possible
phenomena, then the formal undecidability in QM of whether the photon exists as a particle
or a wave would demand (via Proposition~\ref{p1}) that there must exist a nonclassical
model $\mathcal{N}$ for QM in which the superposed state of the photon as a particle
\emph{and} a wave must hold. In $\mathcal{N}$, the photon will be neither a particle
nor a wave, in violation of BCP. As explained in Sec.~\ref{noinf2}, we do not believe
that this is the correct approach.
Reverting to the negation in Definition~\ref{defnegp}, the ``wave nature'' of the photon
is a proposition of the form $P \& \neg P$. As noted earlier, neither $P \& \neg P$
nor its negation $P \vee \neg P$ (which symbolizes the particle nature of the photon)
is a legitimate proposition in the theory syntax of QM. Consequently, BCP survives,
because the Main Postulate and Proposition~\ref{p1} apply only to formal propositions that
are in the theory syntax of NAFL theories; the nonclassical model~$\mathcal{N}$ noted
above need not (and does not) exist. Here we have put ``wave nature'' in quotes because
$P \& \neg P$ in the nonclassical model $\mathcal{M}$ of QM does not imply that the photon
is `really' a wave, as was argued above and in Sec.~\ref{qsae}. One can also state BCP, in
the context of the NAFL interpretation of Afshar's experiment, as follows:
\begin{quote}
At any given time, the observer, via the interpretation QM*, can generate either a
classical or a nonclassical model (but \emph{not} both) for QM, with respect to the
proposition~$P$; in the classical model, $P \vee \neg P$ (and either $P$ or $\neg P$) holds,
and the photon is a particle; in the nonclassical model, $P \& \neg P$ holds and the photon
may be loosely termed as a `wave', although its true status in the real world is left
ambiguous.
\end{quote}
One can see that the above formulation of BCP is not violated in the NAFL
interpretation of Afshar's experiment. The NAFL interpretation also neatly solves the
problem of the mysterious ``instantaneous collapse of the wavefunction'', which would
arise if and only if one insists that the photon `really' exhibits wave nature.
Indeed, when $Q$ is measured at time~$t_2$ in Fig.~\ref{f1}, all that happens is that
the observer switches his interpretation QM* of QM as noted previously. This amounts to a
switch from a state of ignorance regarding the path of the photon to one of knowledge.
Obviously, there is no implication here that the photon abruptly collapsed from a wave
to a particle at $t=t_2$.
\subsection{Critique of Inference~\ref{inf2}}\label{noinf2}
From the point of view of logic (and in particular, the NAFL interpretation), we wish to
establish that \emph{the presence of an interference pattern, such as, that observed in
Young's two-slit experiment, does \textbf{not prove} the `reality' of the
wave nature of a single photon}.
Let us first consider the single-photon version of Young's two-slit experiment.
Let the wire grid at location~$\sigma_1$ in Fig.~\ref{f1} be replaced by an
electronic screen, capable of registering and storing the arrival of
single photons. The photons reach the slits one at a time, with equal probability of
passing through either slit. In the standard quantum formalism, a photon is
\emph{assumed} to take all available paths to any particular spot on the screen at which
it ends up, \emph{i.e.}, the photon is assumed to pass through both slits and `interfere
with itself', in order to theoretically predict the interference fringes. Since these
predictions agree with the experimental observations, the `reality'
of the wave nature of the photon is concluded. At the outset, let us grant that the
interference fringes are indeed observed even in the single-photon case, even though
doubts have been expressed in this regard when the rate of emission of photons is
sufficiently small~(Ref.~\refcite{mardari}; see the paragraph ``If quanta are to be
treated as real particles, self-interference must be ruled out. \dots Nevertheless,
this still means that the evidence in favour of self-interference is inconclusive.'').
Firstly, note that when a single photon is fired at the slits, it also ends up as a single,
bright spot at a specific, unpredictable location on the screen; \emph{it does not exhibit
an interference pattern on the screen}. This fact conclusively establishes the grainy nature
of the photon as it interacts with the screen. The interference pattern, which is observed
to build up over time only after many photons have landed on the screen, has to be
interpreted as reflecting a probability distribution, with the probability density
function~(PDF) proportional to some power of the intensity of the fringes. In particular,
at the dark fringes of the screen, the photon has a vanishing PDF. Note that the PDF
for the photon is normalized over the entire area of the $\sigma_1$ plane in which
the screen is located.
\begin{remark}\label{r1}
A zero value of the probability density function~(PDF) for the photon, at a point
or a line located on a dark fringe (minimum) of the interference pattern on the screen
in Young's two-slit experiment, does \emph{not} imply a proof in the theory QM that a
given photon cannot land at that point or line and be detected.
\end{remark}
Remark~\ref{r1} seems to follow even in the standard formalism for QM. Note that the
probability (as opposed to the PDF) of the photon reaching \emph{any} given point or line on
the screen is \emph{exactly} zero; but obviously this does not constitute a \emph{proof}
that the photon cannot be detected at that point or line. Points/lines on which the PDF
vanishes, such as, the dark fringes in the two-slit experiment, are not special in this
regard. An \emph{arbitrarily} small area around such a point or line will still have a
non-zero probability of recording a photon, as long as the PDF does not vanish identically
in the entire area. So there does not appear to be any basis for excluding the possibility
that a photon can land at the minima of the fringes, although one could expect that such an
eventuality is unlikely in the real world. But one should not confuse
`statistical expectation' with `proof'. To be sure, the observed intensity at the
dark fringes is zero, and zero intensity means zero photon count rate.
But remember that we are asking if a \emph{single, given} photon can land at the dark
fringes; `count rate' and `intensity' and even `probability' are not really well-defined
for this process. For example, a very large number of photons~($N$) could be fired at
the slits and $N-1$ of these could conform precisely to the expected probability
distribution; if the remaining photon ends up at a dark fringe, that does not constitute
a violation of any law of QM. For as $N$ is increased and no further deviations are
recorded, the probability distribution will asymptotically conform to the theoretical
pattern.
At this stage one might advance the argument that a zero PDF at
a point $X$ on the screen means that the photon passed through both slits and
interfered destructively with itself at $X$, making it impossible for the photon to reach
$X$. This argument requires the \emph{assumption} that the photon is `really' a wave,
from the slits all the way up to the screen, with the wavefunction `collapsing' at
the screen in order to enable the detection of the photon as a particle. But such an
assumption is at best an \emph{axiom} that is \emph{not} provable in QM.
The assertion that a photon that is fired at the slits as a particle
mysteriously transforms itself into a wave, passes through both slits, and equally
mysteriously ends up as a particle at the screen, has no credibility
when interpreted as a `reality'. In any case the axiomatic nature of this
assertion with respect to QM means that Remark~\ref{r1} still holds. For there is no
proof that the same observed probability distribution cannot be arrived at by other,
less mysterious routes. In summary, if one interprets the interference pattern at
the screen strictly as reflecting a probability distribution for particle-like photons,
and if one views the quantum formalism as merely one possible algorithm for deriving
it, one cannot infer any `reality' for the wave nature of the photon, as
demonstrated by Remark~\ref{r1}. Young's two-slit experiment, by itself, does
\emph{not prove} the wave nature of the photon. One can at best assert that no photon
has so far been experimentally detected at the theoretical minima of the interference
pattern. But that argument does not constitute a proof of the theoretical
impossibility of such a detection in \emph{all} future experiments, or for all
future times in a given experiment, which would be required to establish the wave
nature of the photon. Indeed, such an infinitary proposition can never be `proven'
experimentally and \emph{must} be axiomatic with respect to QM. Of course, in the
event of such an axiomatic assertion being added to QM, the wave nature
of the photon (which can then be inferred) must be formulated as a classically possible
phenomenon rather than as a contradiction of the form $P \& \neg P$; as observed in
Sec.~\ref{afsh}, this will require a modification of Definition~\ref{defnegp}, and
consequently, NAFL will require the violation of BCP (Definition~\ref{bcp}) via a nonclassical model
$\mathcal{N}$ in which the photon is neither a particle nor a wave. We do
not believe that this is the correct approach, not only because of the mystery
associated with wavefunction collapse, but also in the light of Inference~\ref{inf1}
in Afshar's experiment. Inference~\ref{inf2} must be sacrificed, as will be argued below.
Next consider the NAFL interpretation, with Definition~\ref{defnegp} in place. Let
$X$ be a point located on a dark fringe in Young's two-slit experiment. To `prove'
that a photon leaving the slit(s) can never reach $X$, the standard QM formalism
proceeds as follows.
\begin{enumerate}
\item Assume that the photon reaches $X$ on the electronic screen. \label{a}
\item Assume $P$, assume $\neg P$; $P \& \neg P$ follows. The photon takes all available
paths to the point $X$. \label{b}
\item Associate a probability wave with the photon, in accordance with the
standard formalism. \label{c}
\item Conclude destructive interference at $X$. The photon has a zero probability density
function at $X$. \label{d}
\item Conclude from step~\ref{d} that step~\ref{a} is false, and the photon can never
reach $X$ on the electronic screen. \label{e}
\end{enumerate}
We already questioned step~\ref{e} in the preceding analysis; a vanishing PDF does not
imply proof of impossibility. Even if step~\ref{e} is granted, the above proof is
\emph{not} valid in the NAFL version of QM, whose rules of
inference are classical. For step~\ref{b} \emph{assumes} a contradiction of the
the form $P \& \neg P$. Classically, \emph{any} proposition can be inferred from a
contradiction, and a proof based on such an inference has no validity. Therefore
the conclusion (step~\ref{e}) cannot be established as a theorem of QM based on the above
`proof', either in NAFL or in classical logic. Indeed, the wave nature of the photon
is now classically impossible to prove as a theorem of QM, since it has been
formulated as a contradiction. This holds in NAFL as well; a proposition of the
form $P \& \neg P$ is not even legitimate in the theory syntax of the NAFL theory~QM
(see Sec.~\ref{tsps}), and so cannot be a theorem. Further, $P \& \neg P$ cannot
be added as an axiom to QM, say, at time~$t_1$, for the purpose of obtaining
an interpretation QM* that retroactively asserts the wave nature of the photon, in
the same manner that $Q$ was added at time~$t_2$; as noted in Sec.~\ref{tsps},
only propositions that are legitimate in the theory syntax of QM can be so added.
\emph{However}, it follows from Proposition~\ref{p1} that consistency
of QM requires the existence of a nonclassical model $\mathcal{M}$ for
QM in which $P \& \neg P$ is nonclassically true. The nonclassical interpretation of
$P \& \neg P$ in $\mathcal{M}$ was extensively discussed in Secs.~\ref{qsae}
and~\ref{afsh}. The model theory TM for $\mathcal{M}$ must be based on a paraconsistent
logic (see Sec.~\ref{mp}), for TM must prove $P \& \neg P$. In such a
paraconsistent logic, the classical result that \emph{any} proposition follows from a
contradiction does not hold. Therefore steps~\ref{a}-\ref{d} in the above proof can be validly
formulated in TM. Rather than asserting that the photon took all available paths to the dark
spot $X$ on the screen and destructively interfered with itself, we may simply conclude that
the photon did not take any path to $X$ (and indeed, is very unlikely to do so, via the
proof in TM); hence the dark spot. Thus we have a setting in which the paraconsistent
theory TM can, in principle, justify the standard quantum formalism in a nonclassical NAFL
model $\mathcal{M}$ for QM. But note that the theorems of QM, as well as those
of its interpretation QM* (which generates $\mathcal{M}$), must also be theorems
of TM; the contradictions provable in TM must involve only undecidable propositions
of QM*, such as, $P$. In particular, since infinite sets cannot exist in NAFL
theories~\cite{ract}, including QM, it follows that TM cannot permit these either.
One must develop real analysis in NAFL without infinite sets~\cite{ract}, so that one can
justify the quantum formalism in TM. One might ask, why bother with NAFL at all?
Why not just use a paraconsistent logic to begin with, and just develop
the theory TM? The problem is that in such an eventuality, we do not have any logical
principles by which we determine which are the contradictions that should be provable in TM.
For example, there would be no particular reason for BCP to hold; without
the guiding principles of NAFL, a paraconsistent logic might as well prove the
contradiction that the photon is both a wave and a particle at the same time. Thus we
would be reduced to using our arbitrary intuitions rather than the principles of logic
in determining what should and should not be provable. One may think of TM as
essentially a set of metamathematical rules that tell us how to combine various
classical models of QM so as to generate the desired nonclassical model $\mathcal{M}$
that conforms with the observer's interpretation QM* (which in turn conforms with
his observations/axiomatic declarations in the real world). Note that TM tries to
\emph{predict} the results of the `measurements' made by the observer in the real
world, via probabilistic reasoning. TM essentially tells us what QM* will probably
look like if the observer keeps his axiomatic declarations in QM* in tune with his
(statistically large number of) observations in the real world.
In summary, our thesis is that the observed interference pattern at the screen in
Young's two-slit experiment is a manifestation of a \emph{particular} nonclassical
NAFL model $\mathcal{M}$ for QM in which the count rate of the photon at each point
of the screen is in tune with the nonclassical probability distribution derived
in the theory TM of $\mathcal{M}$. There could be other nonclassical NAFL models for QM
that follow different probability distributions, but these may not be relevant to
our real world. The important achievement here is that we have conceptually justified
the use of $P \& \neg P$ in deducing the interference pattern \emph{without conceding
any physical reality} for the wave nature of the photon. As noted earlier, $P \& \neg P$
in $\mathcal{M}$ merely reflects the observer's ignorance of the path information
of the photon and is used \emph{only} for the purpose of deriving the probability
distribution; the contradictions in the paraconsistent structure $\mathcal{M}$ have no
`physical' reality. To the various bright spots on the screen, each individual photon
`really' takes only one path. The only mystery here is that this `reality' is not
revealed via a classical probability distribution when a large number of photons are fired
at the screen; instead, the resulting interference pattern revealed by Nature
seems to indirectly confirm, via a nonclassical probability distribution, the observer's
ignorance of the path information. Why does Nature act in this manner? The answer may
have something to do with the paradoxical nature of probability, which is, even
classically, a problematic notion that is dependent on the information available to
the observer (hence, `conditional' probabilities). While the photon
passes through only one slit at a time as a particle, the status of the other inactive
slit (\emph{i.e.}, whether it is open or closed) seems to impose some sort of conditionality
on the probability distribution. Perhaps formulation of real analysis in NAFL, development
of consistent postulates for QM, and then developing the paraconsistent
theory TM of $\mathcal{M}$ will shed further light on this mystery. Many of the concepts
of standard quantum theory, such as, the notion of probability, will not be formalizable
in the NAFL theory QM and will have to be dealt with in TM, and in general, at a
metamathematical level. This is understandable, since these concepts may be specific to
the real world and the model $\mathcal{M}$ which applies to it. It is also possible that
radically new concepts of space, time, and in particular, light, may be needed for a more
satisfactory description of Nature in a NAFL theory that does away with probability
altogether.
Let us now revert to Afshar's experiment. Afshar~\cite{afshar1} argues that if a classical
probability distribution had applied, the wire grid at location~$\sigma_1$ in Fig.~\ref{f1}
should have blocked about 6.6\% of the photons. Instead, the wire grid blocked fewer than
0.1\% of the photons, as expected from the probability distribution calculated using the
standard quantum formalism. Let us grant Afshar's argument and concede that the interference
pattern does indeed exist. As argued above, this does not legitimize Inference~\ref{inf2}.
In the light of the observation of $Q$ and Inference~\ref{inf1}, one concludes that the
photon still has a metalogical particle nature for $t_0 \le t < t_2$, but with a
nonclassical probability distribution corresponding to the interference pattern, as
deducible in a nonclassical model $\mathcal{M}$ for QM. In particular, there was no `destructive
interference' at the dark fringes corresponding to the locations of the wires; the photons that
ended up at the images simply missed the wires and the dark fringes.
Note that the axiomatic nature of NAFL truth allows the observer to
\emph{axiomatically declare}, via a choice of QM*=QM+$P$ at time~$t_0$ in Fig.~\ref{f1},
that a given photon passed through slit~U, even though such a \emph{measurement} was
not made at $t=t_0$. Thus the observer breaks with the informal convention of keeping
his axiomatic declarations in tune with his measurements, and generates a classical
particle model of the photon for $t_0 \le t < t_2$, instead of the nonclassical model
$\mathcal{M}$. If at $t=t_2$, the photon is found to end up at the image
$\mbox{U}^{\prime}$, this would vindicate the observer's choice made through guesswork and
free will. Thus in principle, NAFL allows the observer to bring the particle nature of the
photon within the scope of its semantics, even though in practice, continued success on
this front would require improbable guesswork on the part of the observer. Of course,
consistency of QM requires $\mathcal{M}$ to exist (see Proposition~\ref{p1}), even if
the observer does not choose it in any particular instance.
\subsection{Critique of Kastner's argument}\label{kast}
Kastner~\cite{kastner1,kastner2} has criticized the interpretation of Inference~\ref{inf1} as
retroactively asserting the particle nature of the photon. She has essentially two
reasons for her reservations, as stated below.
\begin{itemize}
\item According to Kastner, the photon exhibited wave nature at time~$t_0$ in Fig.~\ref{f1}
in the sense that it `really' passed through both slits, as confirmed by
Inference~\ref{inf2} (which she supports with the caveat that the wave
nature of the photon, being real, is not to be formulated as a contradiction
of the form $P \& \neg P$). So, as was noted in Sec.~\ref{afsh}, Kastner requires
Inference~\ref{inf1} to be limited to the assertion that the photon was post-selected
in the state of slit~U, without any retroactive implication that the photon `really'
passed through only slit~U (as a particle). In other words, Inference~\ref{inf1} does not
provide `which-way information' for the photon.
\item Kastner cites Cramer's Transactional Interpretation~(TI) of quantum
mechanics~\cite{cramer} as unambiguously showing that the photon was selected in the
superposed state of both slits at time~$t_0$, and also was post-selected
in the state of slit~U at time~$t_2$. However, there are intermediate times
in Fig.~\ref{f1}, when the photon was between the locations $\sigma_1$ and
$\sigma_2$, during which the ontological state of the photon is ambiguous
according to TI (see Fig.~3 of Ref.~\refcite{kastner1}). Kastner states that at
these locations, the `offer wave' of TI shows the photon to be in a superposition state
of both slits, while the backwards-in-time `confirmation wave' of TI shows the photon to
be in a state of slit~U. For this reason, Kastner asserts that it would be wrong to
retroactively infer at time~$t_2$ that the photon passed through only slit~U,
since such an inference would require the photon to be determinately a particle at all
intermediate locations between $\sigma$ and $\sigma_2$ in Fig.~\ref{f1}.
\end{itemize}
On the basis of the above criticisms, Kastner concludes that Afshar's experiment does
not refute BCP~(Definition~\ref{bcp}).
We have exhaustively addressed the problems with the first of Kastner's criticisms.
From the point of view of NAFL, Inference~\ref{inf1} is merely the observer's
retroactive axiomatic declaration of the particle nature of the photon. But the
concept of `axiomatic declaration' itself cannot be formalized in the theory syntax of
NAFL theories~\cite{ijqi}. So when the observer `post-selects' the photon as being in
the `state of slit~U', he can \emph{only} have in mind that the
photon `really' passed through slit~U as a particle. Therefore, in NAFL, the classical
model of QM that is generated at time~$t_2$ via the observer's interpretation QM*=QM+$Q$,
\emph{must} reflect such a retroactive implication for Inference~\ref{inf1}.
In fact, the completeness theorem of first-order logic (which is a metamathematical
principle in NAFL) \emph{requires} that there \emph{must} exist such a classical model
for QM. For, as we have already pointed out, QM does not prove either $\neg P$ or
Kastner's claim that the photon passed through both slits at time~$t_0$. This
latter claim, affirming the wave nature of the photon at time~$t_0$, is at best another
\emph{metamathematical} requirement that Kastner chooses to impose. In the conflict
between these two metamathematical requirements, the completeness theorem wins out
in NAFL; it is a sacred principle of logic that cannot be sacrificed. On the other
hand, the nonclassical NAFL model $\mathcal{M}$ of QM that affirms $P \& \neg P$ at
time~$t_0$ does not conflict with the above retroactive assertion of $P$, either
temporally or in meaning, as we have already pointed out in Secs.~\ref{qsae}~and~\ref{afsh}.
In particular, $\mathcal{M}$ only affirms the observer's lack of information on the path
of the photon at time~$t_0$, rather than Kastner's assertion that the photon `really'
passed through both slits.
The second of Kastner's objections is also problematic, in the sense that her stated
purpose of rescuing BCP is not served by her invocation of Cramer's TI. If TI requires
that the photon's ontological state is ambiguous between the locations $\sigma_1$
and $\sigma_2$ in Fig.~\ref{f1}, that amounts to a violation of BCP when the photon was
at these locations. For we have determined, using TI, that the photon was neither a
particle nor a wave when it was at these locations. So the NAFL model of QM that applies
at these times would have to be a superposition of both the particle and wave states of the
photon, \emph{i.e.}, a superposition of a classical and a nonclassical model of QM. But
such a superposition of models does not exist in NAFL and clearly violates BCP; see
Definition~\ref{bcp} and also the NAFL version of BCP as stated at the end
of Sec.~\ref{afsh}. This is not surprising, for TI requires us to ascribe
reality to the superposed state of the photon passing through both slits; this is
also Kastner's belief, as noted above. As a consequence, NAFL would \emph{require} that
Definition~\ref{defnegp} be modified to force, via Proposition~\ref{p1}, the existence
of a nonclassical model~$\mathcal{N}$ for QM that violates BCP (as was pointed out
in Sec.~\ref{afsh}).
\subsection{Critique of Cramer's TI in the delayed-choice scenario}\label{cram}
One example of a quantum mystery arises from the well-known delayed choice experiment of
Wheeler~\cite{wheeler}. In Fig.~\ref{f1}, \emph{after} a photon passes through the slit(s)
in the $\sigma$ plane at time~$t_0$, the observer could either choose to insert a screen
in the $\sigma_1$ plane, \emph{or} he could choose to allow the photon to pass through the
lens and reach the $\sigma_2$ plane. So according to Wheeler, this delayed choice
of measurement means that the observer could post-select (\emph{i.e., after} time~$t_0$)
each photon to pass through both slits, as a wave, or one slit only, as a particle.
But this amounts to choosing one's past after the event and seems paradoxical. In the
NAFL interpretation, the metalogical reality of the photons as particles
means that the delayed choice does \emph{not} influence the past; each photon always
passed through one and only one slit. The interference pattern, which one sees
on the screen after many photons are recorded on it, \emph{also} exists (but is not
measured) for the photons that end up at the $\sigma_2$ plane, as is spectacularly
confirmed in Afshar's experiment.
Cramer's Transactional Interpretation~(TI)~\cite{cramer}, on the other hand, provides
an explanation for delayed choice by positing an `atemporal' transaction that takes
place between `offer waves' and `confirmation waves', with the latter propagating
backwards in time. From the point of view of NAFL, however, which rejects the
relativistic conception of `spacetime' (see Appendix~B of Ref.~\refcite{acs}), such an atemporal
transaction is not physically possible, and neither can anything propagate backwards
in time. The temporal and axiomatic nature of NAFL truth requires absolute time, as well
as Euclidean space. Cramer's aphysical approach to delayed choice really amounts to taking
an anti-realist stand that the `past' exists if and only if, and only \emph{after},
it is clearly defined. This is confirmed by Cramer's assertion that ``No offer is a
transaction until it is a confirmed transaction'', which corresponds to Wheeler's ``No
phenomenon is a phenomenon until it is an observed phenomenon''. Such an anti-realist
stand is possible in NAFL by associating `measurement' or `observation' with `axiomatic
declaration', and denying any metalogical reality outside of NAFL syntax and semantics
(as was noted in Sec.~\ref{afsh}). But in NAFL, the superposed state is not really a
physical state of the photon passing through both slits, as is assumed by Cramer; the
anti-realist stand, when imposed upon the NAFL interpretation, would further require that
the photon has no determinate state in the real world when it is deemed to be in a quantum
superposition of passing through both slits. Hence from the point of view of NAFL, Cramer's
TI is not consistent with either realism or anti-realism. As was discussed in Sec.~\ref{afsh},
positing a metalogical reality for the particle nature of the photon is not only
compatible with NAFL, but is also philosophically a
more satisfactory resolution of the paradoxes associated with Afshar's experiment. From this
realist point of view, Cramer's TI, by positing waves propagating backwards in time, seems
to accept that one \emph{can} influence the past; this is denied in the NAFL interpretation
as aphysical. In Sec.~\ref{kast}, we have criticized Kastner's~\cite{kastner1} defence of
BCP, using Cramer's TI, as logically problematic; the said defense upholds BCP at the slits
by requiring its violation elsewhere, at least according to the NAFL interpretation.
\section{The coin toss and Schr\"{o}dinger cat experiments}\label{ctoss}
Consider the coin toss experiment described in Sec.~2.5 of Ref.~\refcite{1166}. An observer
tosses a fair coin and, as it lands at time~$t_0$, covers the coin under the palm of
his hand without seeing the outcome. Let $P$~($\neg P$) represent ``The outcome is
`heads'~(`tails')''. Let the observer have the NAFL theory T in mind, which includes definitions
describing this coin toss experiment. Further, at $t=t_2$, the observer lifts his hand and
sees the outcome, say, `heads'. For $t_0 \le t < t_2$, the observer chooses the interpretation
T*=T. Hence $P \& \neg P$ holds for the observer in a nonclassical model $\mathcal{T}$ for
T, signifying that he has not measured~(axiomatically declared) the outcome to be
either `heads' or `tails' during this time interval. Let $R$ denote the proposition that
``The outcome is `heads' for $t \ge t_2$''. For $t \ge t_2$, the observer takes T*=T+$R$,
in tune with his observation, so that $P$ (which is provable in T* for these times) holds
in a classical model for T. Let the proposition $Q$, formulated for $t \ge t_2$, denote
``The outcome was `heads' during $t_0 \le t < t_2$''. Our contention is
that the theory T*=T+$R$ proves $Q$, so that the observer has retroactively
asserted at $t=t_2$ that the outcome was always `heads' during $t_0 \le t < t_2$.
Is the inference of $Q$ in T+$R$ legal? In this case, the observer can feel confident that
the outcome `heads' must be metalogically~(`really') true for $t_0 \le t < t_2$.
The observer \emph{knew} that the coin was flat under the palm of his hands during
$t_0 \le t < t_2$, but as was noted in Sec.~2.5 of Ref.~\refcite{1166}, this fact, being
provably equivalent to $P \vee \neg P$ in the proof syntax of T, \emph{cannot} be
formalized as a legal proposition in the theory syntax of T. Such an (illegal)
formalization would force $P \vee \neg P$ to be a theorem of T, and prevent the
existence of the nonclassical model $\mathcal{T}$ required by Proposition~\ref{p1}
for consistency of T in NAFL. The reason this example is interesting is that
the observer \emph{knows} that the superposed state of `heads and tails' in $\mathcal{T}$
has no `physical' reality, but nevertheless it correctly reflects the observer's
ignorance of the outcome during $t_0 \le t < t_2$. In Afshar's experiment, the
situation is logically similar, but the observer does not have such a clear intuition
for the particle nature of the photon, and consequently, for the validity of
Inference~\ref{inf1}. But nevertheless, NAFL treats both cases similarly and in a
logically consistent manner.
Next reconsider the Schr\"odinger cat experiment described in \linebreak Sec.~\ref{sch}.
Once again the observer has the intuition that the retroactive assertion of
$U$, via an inference in the theory QM*=QM+$V$, is true in the real world. This
inference is based on the principle (deducible in the proof syntax of QM*) that if
one finds the cat to be alive at time~$t_2$ and if one knows that the cat was alive
at an earlier time~$t_0$, then the cat was alive for $t_0 < t < t_2$. Thus at
$t=t_2$, we \emph{know} that the cat was (metalogically, `really' and unambiguously)
alive during $t_0 < t < t_2$. NAFL supports such a rock-solid, unimpeachable
inference that conforms with the standard definition of `alive'.
\section{Concluding Remarks}
The NAFL interpretation upholds the Bohr complementarity principle~(BCP) in Afshar's
experiment~\cite{afshar1,afshar2,afshar3} by retroactively affirming the `real' particle status of the
photon, even while \emph{semantically} the photon was in a superposed state~($S$) in a
nonclassical model $\mathcal{M}$ of the NAFL theory~QM formalizing quantum mechanics.
However, such a `reality' for the particle state, which is outside of both the syntax and
the semantics of QM, can be said to be \emph{metalogical}. In NAFL, $S$ (or the `wave nature'
of the photon) only reflects the fact that the observer has not measured (axiomatically
asserted) the true, classical path of the photon; no \emph{physical} reality, to the effect
that the photon `really passed though both slits and interfered with itself', can be
assigned to $S$. The interference pattern in the $\sigma_1$ plane of Fig.~\ref{f1} must
be interpreted in NAFL as reflecting a nonclassical probability distribution for the photons,
still treated as particles, that is derivable within $\mathcal{M}$. We have no explanation
yet for \emph{why} the lack of which-way information influences the probability distribution
in this manner.
The still nascent NAFL interpretation of quantum mechanics has considerable potential
for future research. The subtle formulation of the syntax and semantics of NAFL theories to combine
both classical and intuitionistic principles, the ability of NAFL to handle the temporal nature of
mathematical truth, and the demonstration of the need for a paraconsistent logic to handle the model
theory of NAFL theories are features that make the NAFL interpretation highly suitable for the purpose of
providing a logical explanation for the mysteries of quantum mechanics, \emph{within the framework
of a single logic}. This, despite the limitations that NAFL imposes on classical infinitary
reasoning~\cite{ijqi,1166,acs,ract}, is a great advantage of The NAFL interpretation, which can be said to provide
a logical basis for many of Niels~Bohr's great physical ideas that were spelt out in the Copenhagen interpretation.
It is to be emphasized the NAFL interpretation is not \emph{essential} for the purpose of explaining the results
of the Afshar experiment and other seemingly paradoxical experiments of quantum mechanics; the Copenhagen
interpretation is adequate for this purpose. However, at present quantum mechanics cannot be
satisfactorily formalized within a single logic. For example, real analysis uses the framework of classical logic
while other quantum phenomena like superposition require nonclassical logics. If the NAFL interpretation
can be developed to its full potential, a formalization of all of quantum mechanics within a single logic
would hopefully be within reach.
| {'timestamp': '2010-06-08T02:02:44', 'yymm': '0504', 'arxiv_id': 'quant-ph/0504115', 'language': 'en', 'url': 'https://arxiv.org/abs/quant-ph/0504115'} |
\section{Introduction}
It is well-known that the positive energy theorems ensure the energies of
the solutions approaching AdS spacetime globally cannot be negative\cite{Yau,Witten,Gibbons}%
. However, if the considering spacetimes are locally asymptotically
AdS but not globally, the positive energy theorem may not hold. The
Horowitz-Myers AdS soliton solution is this kind of particular
solution~\cite{Horowitz}. This AdS soliton solution is important not
only for its negative energy, but also for the agreement with the
Casimir energy in the field theory viewed from the AdS/CFT
correspondence\cite{Maldacena}. Furthermore, it has also been found
that there is a similar phase transition like the Hawking-Page phase
transition between the Ricci flat black hole and the AdS soliton
solution, and it could be connected with the
confinement/deconfinement phase transition in QCD
\cite{Hawking,Witt,Cai}.
Although many properties of this AdS soliton solution have been
studied, the analogy of its energy and tension laws is absent, and
it is simply because its entropy is zero and the period of the
imaginary time is arbitrary. Recently, inspired from the interchange
symmetry between the KK bubble and the corresponding black hole
which are all asymptotically flat, D.Kastor et al obtained some
interesting results of the KK bubble after defining some new
quantities such as its surface gravity and the area of the KK
bubble~\cite{Kastor}(Note that the surface gravity here is
associated with the spacelike Killing field which translates around
the compact spatial coordinate, and more details can be found in
\cite{Gibbons2}). In our paper, viewed from the similar interchange
symmetry between the AdS soliton solution and the Ricci flat black
hole, we first reinvestigate the energy and tension laws of the
Ricci flat black hole by considering the contribution of the tension
term \cite{Tension,Kastor,Kurita,Traschen,Townsend}, then we
investigate the analogy of the laws of the AdS soliton solution. We
find the same analogy as that of the laws of the KK bubble. In
addition, we also investigate the laws of the boosted Ricci flat
black hole~\cite{Kastor2}. The boosted Ricci flat black hole can be
obtained from the static Ricci flat black hole by a boost
transformation along the compact spatial coordinate~\cite{Cai2}.
Note that, because the spatial coordinate is compact, the boosted
Ricci flat black hole is not equivalent to the static one
globally~\cite{Lemos,Awad,Cai2}. And these kind of globally
stationary but locally static spacetimes could be considered as the
gravitational analog of the Aharonow-Bohm
effect~\cite{Aharonow,Stachel}. Similarly, for the static AdS
soliton solution, we can also make a boost transformation along the
compact spatial coordinate of the static AdS soliton solution, and
then obtain the boosted AdS soliton solution. Like the AdS soliton
solution, the boosted AdS soliton solution also has the same
interchange symmetry with the above boosted Ricci flat black hole.
However, there are closed timelike curves and conical singularity in
the boosted AdS soliton solution, thus this solution is ill in
physics and the direct analogy of the energy and tension laws of the
boosted AdS soliton solution is of no sense. In spite of that, its
conserved charges such as the energy and momentum are well-defined,
and an interesting result is that the energy of the boosted AdS
soliton is lower than that of the AdS soliton. On the other hand,
note that although here we can easily find that the energy and
tension laws of the boosted Ricci flat black hole or the AdS soliton
solution are the same as those of the asymptotical flat case, the
underlying contents are not the same. First of all, the methods of
calculating the conserved charges are different. Because what they
discuss are the asymptotically flat cases, the well-known ADM
calculation can be used in their cases~\cite{Kastor,Kastor2}.
However, it is invalid and there have been several methods to
calculate the conserved charges in the asymptotically AdS
case~\cite{Counterterm,Euclidean,HH,AD,AM,Brown}. Here we just use
the surface counterterm method or Euclidean method. Second, they
obtain the laws by using the Hamiltonian perturbation theory
techniques~\cite{Traschen,Perturbation technique}, and more
expressive is that they should use the Hamiltonian formalism
presented by the ADM method~\cite{Kastor,Kastor2}. While for black
holes we obtain the laws just by applying the Euclidean
method~\cite{Euclidean}, and basing on this we obtain the laws of
AdS soliton by using the property of interchange symmetry. During
the derivation of laws, we do not need the explicit formalisms of
conserved charges. Thus, our results could also be considered as a
simple generalization of the results in asymptotically flat case to
the asymptotically AdS case~\cite{Kastor,Kastor2}.
The rest of paper is organized as follows. In section II, we
reinvestigate the energy and tension laws of the Ricci flat black
hole by considering the contribution of the tension term. In section
III, inspired from the interchange symmetry with the Ricci flat
black hole, we obtain the analogy of the laws of the AdS soliton
solution. In section IV, we generalize the above discussion in
section II to the case of the boosted Ricci flat black hole. In
section V, we consider the boosted AdS soliton solution. Finally, in
section VI, we give a brief conclusion and discussion.
\section{Reinvestigation of the energy and tension laws of the Ricci flat black hole%
} The so-called Ricci flat black hole solution considered here is
\cite{Horowitz,Maldacena}
\begin{equation}
ds^{2}=\frac{r^{2}}{l^{2}}[-(1-\frac{r_{0}^{4}}{r^{4}}%
)dt^{2}+dy^{2}+(dx^{i})^{2}]+(1-\frac{r_{0}^{4}}{r^{4}})^{-1}\frac{l^{2}}{%
r^{2}}dr^{2}.\ \ (i=1,2) \label{Ricci flat black hole}
\end{equation}%
which arises in the near-horizon geometry of p-brane and is
asymptotically the five dimensional AdS metric. It is easy to find
that its event horizon locates at $r_{+}=r_{0}$. And in order to
remove the conical singularity at the horizon, the Euclidean time
$\tau $ must have a period $\beta =\frac{\pi l^{2}}{r_{0}}$. Note
that, the coordinate $y$ is a compact spatial coordinate, and its
period is $\eta $. As the usual treatment, we can use the Euclidean
method to research the thermodynamics of the Ricci flat black
hole~\cite{Euclidean}. Choosing the pure AdS spacetime as the
reference background, we can easily obtain the Euclidean action of
the Ricci flat black hole
\begin{eqnarray}
I_{E}=-\frac{\beta r_{0}^{4}}{16\pi l^{5}}\eta V_{2}.
\label{Euclidean action1}
\end{eqnarray}
where $V_{2}$ is the coordinate volume of the surfaces parameterized by $%
x^{i} $. Thus, the free energy of the Ricci flat black hole
evaluated on the pure AdS background is~\cite{Euclidean}
\begin{eqnarray}
F &\equiv &\frac{I_{E}}{\beta }=E-TS=-\frac{ r_{0}^{4}}{16\pi
l^{5}}\eta V_{2}. \label{free energy1}
\end{eqnarray}
And the energy and entropy are
\begin{eqnarray}
E &=&\frac{\partial I_{E}}{\partial \beta }=\frac{3r_{0}^{4}}{16\pi l^{5}}%
\eta V_{2}, \\
S &=&\beta \frac{\partial I_{E}}{\partial \beta
}-I_{E}=\frac{\eta V_{2}r_{0}^{3}}{4l^{3}}. \label{energyandentropy}
\end{eqnarray}
From~(\ref{energyandentropy}), it can be seen that the entropy $S$
is exactly equal to 1/4 of the horizon area $A$, which implicates
that those thermodynamical equations hold
\begin{eqnarray}
dF=-SdT, dE=TdS. \label{masslaw1}
\end{eqnarray}
It should be emphasized that we have not considered the contribution
of tension term to the laws above, i.e gravitational tension. And it
is known that the gravitational tension term could contribute to the
first law in the case of the black p-branes or black string if the
size of the compact spatial coordinate is allowed to be changed. The
fact is that the geometry looks locally like the black string when
is far from the horizon of the Ricci flat black hole, thus the
gravitational tension term may also contribute to the
thermodynamical laws~\cite{Kurita}. And it is true that if assuming
the free energy in~(\ref{free energy1}) is also the function of
$\eta $, we can obtain not only the energy and entropy but also the
gravitational tension
\begin{eqnarray}
E &=&(\frac{\partial I_{E}}{\partial \beta })_{\eta }=\frac{3r_{0}^{4}}{%
16\pi l^{5}}\eta V_{2}, \notag \\
S &=&\beta (\frac{\partial I_{E}}{\partial \beta })_{\eta
}-I_{E}=\frac{\eta
V_{2}r_{0}^{3}}{4l^{3}}, \notag \\
\Gamma &=&\frac{1}{\beta }(\frac{\partial I_{E}}{\partial \eta })_{\beta }=-%
\frac{r_{0}^{4}}{16\pi l^{5}}V_{2}. \label{ESTension}
\end{eqnarray}
On the other hand, in a $d+1$ dimensional spacetime $\mathcal{M}$,
the conserved charge associated with the killing vector $\xi ^{\mu
}$ generating an isometry of the boundary geometry $\partial
\mathcal {M}$ defined through the quasilocal stress tensor
is~\cite{Counterterm,Brown}
\begin{equation}
Q_{\xi }=\int_{\Sigma }d^{d-1}x\sqrt{\sigma }(u^{\mu }T_{\mu \nu
}\xi ^{\nu }). \label{Conservedcharge}
\end{equation}%
where $\Sigma $ is a spacelike hypersurface in the boundary
$\partial \mathcal {M}$, and $u^{\mu }$ is the timelike unit vector
normal to it. $\sigma _{ab}$ is the metric on $\Sigma $ defined as
\begin{equation}
\gamma _{\mu \nu }dx^{\mu }dx^{\nu }=-N_{\Sigma }^{2}dt^{2}+\sigma
_{ab}(dx^{a}+N_{\Sigma }^{a}dt)(dx^{b}+N_{\Sigma }^{b}dt)
\label{BoundaryADM}
\end{equation}
and $\gamma _{\mu \nu }$ is the metric on the boundary. Thus, the
energy related with the timelike killing vector $\xi ^{\mu }$ and
the momentum could be defined respectively as
\begin{eqnarray}
E &=&\int_{\Sigma }d^{d-1}x\sqrt{\sigma }N_{\Sigma }(u^{\mu }T_{\mu
\nu
}u^{\nu }), \label{Mass}
\\P_{a} &=&\int_{\Sigma }d^{d-1}x\sqrt{\sigma }\sigma _{ab}u_{\mu
}T^{b\mu }. \label{Momentum}
\end{eqnarray}
According to the surface counterterm method, the quasilocal stress
tensor for the asymptotically $AdS_{5}$ solution
is~\cite{Counterterm}
\begin{equation}
T_{\mu \nu }=\frac{1}{8\pi }(\theta _{\mu \nu }-\theta \gamma _{\mu \nu }-%
\frac{3}{l}\gamma _{\mu \nu }-G_{\mu \nu }). \label{stresstensor}
\end{equation}%
where all the above tensors refer to the boundary metric $\gamma
_{\mu \nu }$ defined on the hypersurface $r=constant$, and $G_{\mu
\nu }=R_{\mu \nu }-\frac{1}{2}R\gamma _{\mu \nu }$ is the Einstein
tensor of $\gamma _{\mu \nu }$, $\theta _{\mu \nu
}=-\frac{1}{2}(\nabla _{\mu }n_{\nu }+\nabla _{v}n_{\mu })$ is the
extrinsic curvature of the boundary with the normal vector $n^{\mu
}$ in the spacetime. Therefore, we can easily obtain the useful
quasi-local stress tensor of the Ricci flat black hole~(\ref{Ricci
flat black hole})
\begin{eqnarray}
8\pi T_{tt} &=&\frac{3r_{0}^{4}}{2l^{3}r^{2}}+...
\label{stresstensor1}
\end{eqnarray}%
And the energy is
\begin{eqnarray}
E =\frac{3r_{0}^{4}}{%
16\pi l^{5}}\eta V_{2}. \label{Energy1}
\end{eqnarray}
which is consistent with the above result in~(\ref{ESTension}). In
addition, the general definition of gravitational tension in a given
asymptotically translationally-invariant spatial direction (i.e.
$x$) of a $D$ dimensional space-time is~\cite{Tension}
\begin{equation}
\Gamma =\frac{1}{\Delta t}\frac{1}{8\pi }\int_{S_{x}^{\infty
}}[F(K^{(D-2)}-K_{0}^{(D-2)})-F^{\upsilon }p_{\mu \nu }r^{\nu }]
\label{TensionDefinition}
\end{equation}%
here $S_{x}^{\infty }=\Sigma _{x}\cap \Sigma ^{\infty }$ and $\Sigma
_{x}$ is the hypersurface $x=const$ with unit normal vector $n^{\mu
}$, and $\Sigma ^{\infty }$ is the asymptotic boundary of the
spacetime with unit normal vector $r^{\mu }$. The space-like killing
vector $X^{\mu }$ corresponding to the translationally-invariant
spatial direction $x$ is decomposed into normal and tangential parts
to $\Sigma _{x}$ that
\begin{equation}
X^{\mu }=Fn^{\mu }+F^{\mu \label{Decomposition}}
\end{equation}
and the extrinsic curvature tensor on $\Sigma _{x}$ with respect to $n^{\mu }$ is $%
K_{\mu \nu }$, while $K^{(D-2)}$ is the extrinsic curvature of the surface $%
S_{x}^{\infty }$ in $\Sigma _{x}$, and $K_{0}^{(D-2)}$ is the
corresponding
extrinsic curvature of the surface $S_{x}^{\infty }$ in the reference space $%
(M,(g_{0})_{\mu \nu })$. The metric with respect to $n^{\mu }$ on
$\Sigma _{x}$ is
\begin{equation}
h_{\mu \nu }=g_{\mu \nu }-n_{\mu }n_{\nu \label{ReducedMetric}}
\end{equation}%
while the corresponding canonical momentum $p_{\mu \nu }$ with
respect to $h_{\mu \nu }$ is
\begin{equation}
p^{\mu \nu }=\frac{1}{\sqrt{h}}\pi ^{\mu \nu }=K^{\mu \nu }-Kh^{\mu
\nu \label{CanonicalMomentum}}
\end{equation}
Thus, from this general definition of gravitational
tension~(\ref{TensionDefinition}), we can obtain the gravitational
tension along the compact spatial direction $y$ in Ricci flat black
hole~(\ref{Ricci flat black hole})
\begin{eqnarray}
\Gamma =-\frac{r_{0}^{4}}{16\pi l^{5}}V_{2}. \label{Tension1}
\end{eqnarray}
which is also consistent with the above result in~(\ref{ESTension}).
And these consistences of energy, tension and entropy implicate that
after adding the contribution of tension term the first laws
in~(\ref{masslaw1}) are
\begin{eqnarray}
dF &=&-SdT+\Gamma d\eta =-\frac{1}{8\pi }A_{H}d\kappa _{H}+\Gamma
d\eta ,
\notag \\
dE &=&TdS+\Gamma d\eta =\frac{1}{8\pi }\kappa _{H}dA_{H}+\Gamma
d\eta . \label{Masslaw2}
\end{eqnarray}
where $T=1/\beta =\kappa _{H}/2\pi $ and $S=A_{H}/4$. Using these
conserved charges, we can also easily check that
\begin{equation}
E-TS=\Gamma \eta. \label{Smarr}
\end{equation}%
which is very similar with the Smarr relation. Thus
from~(\ref{Smarr}) and ~(\ref{Masslaw2}), we can obtain the tension
law that
\begin{equation}
\eta d\Gamma =-SdT. \label{Tensionlaw1}
\end{equation}%
which can be found to have the same formalism with the static
Kaluza-Klein black hole which is asymptotically flat in
Refs~\cite{Kastor,Kastor2}.
\section{The AdS soliton solution, interchange symmetry, and analogy of
energy and tension laws}
The AdS soliton solution is~\cite{Horowitz}
\begin{equation}
ds^{2}=\frac{r^{2}}{l^{2}}[(1-\frac{r_{0}^{4}}{r^{4}}%
)dy^{2}-dt^{2}+(dx^{i})^{2}]+(1-\frac{r_{0}^{4}}{r^{4}})^{-1}\frac{l^{2}}{%
r^{2}}dr^{2}.\ \ (i=1,2) \label{AdSSoliton}
\end{equation}%
with the coordinate $r$ restricted to $r\geq r_{0}$. Again, the
coordinate $y $ could be identified with period $\eta =\frac{\pi
l^{2}}{r_{0}}$ to avoid a conical singularity at $r=r_{0}$. Note
that this spacetime is completely nonsingular and globally static.
And it can be obtained from the Ricci flat black hole
metric~(\ref{Ricci flat black hole}) with the double analytic
continuation such that
\begin{equation}
t\rightarrow iy,y\rightarrow it. \label{Double analytic
continuation}
\end{equation}%
which arises an interesting interchange symmetry between the AdS soliton
solution and the Ricci flat black hole.
Using the same surface counterterm method, we can calculate the
useful quasilocal stress tension of AdS soliton~\cite{Counterterm}
\begin{eqnarray}
8\pi T_{tt} &=&-\frac{r_{0}^{4}}{2l^{3}r^{2}}+...
\label{Stresstension2}
\end{eqnarray}%
Thus, the energy is
\begin{equation}
E=-\frac{r_{0}^{4}}{16\pi l^{5}}\eta V_{2}. \label{Energy2}
\end{equation}%
In addition, the tension of the AdS soliton (along the direction of
the compact coordinate $y$) from the general
definition~(\ref{TensionDefinition}) is
\begin{equation}
\Gamma =\frac{3r_{0}^{4}}{16\pi l^{5}%
} V_{2}. \label{Tension2}
\end{equation}%
Eqs~(\ref{Energy2})~(\ref{Tension2}) can explicitly manifest the
interchange symmetry with the Ricci flat black hole compared with
its energy and tension.
In the above section, during deriving the laws of Ricci flat black
hole, we mainly base on an underlying assumption that the
equations~(\ref{Masslaw2}) hold, and then find the consistence with
the calculations by other methods. However, for the AdS soliton
solution, the first problem is these equations may not hold because
the period of the imaginary time which usually relates with the
temperature is arbitrary in the AdS soliton spacetime. Moreover, if
we take the entropy just as the usual Bekenstein-Hawking entropy (it
is the 1/4 of the horizon area), we could find the entropy is zero.
Thus, the direct analogy of the mass and tension laws of the AdS
soliton solution like~(\ref{Masslaw2})~(\ref{Tensionlaw1}) seems to
be absent. In spite of that, inspired from the interchange symmetry
between the black hole and AdS soliton, it may have the analogy. And
it is true that it has been found the similar analogy of the KK
bubble in Ref~\cite{Kastor} where it discusses the asymptotically
flat case. As same as that of KK bubble, we can also first define
some new quantities, such as the surface gravity and the area of the
AdS soliton. And according to these definitions, the surface gravity
and the area of the AdS soliton are~\cite{Kastor}
\begin{equation}
\kappa _{s}=\frac{2r_{0}}{l^{2}},A_{s}=\frac{V_{2}r_{0}^{3}}{l^{3}}.
\label{SurfacegravityandArea}
\end{equation}%
However, here we would not use the Hamiltonian perturbation
techniques to deduce the laws of AdS soliton until one finds its
appropriate formalisms of the conserved charges and gravitational
tension as those of KK bubble. And we just base on its interchange
symmetry with the Ricci flat black hole~(\ref{Ricci flat black
hole}). From the quantities in~(\ref{SurfacegravityandArea}) and
those in ~(\ref{Energy2})~(\ref{Tension2}), we can make an easy
displacement in~(\ref{Masslaw2})~(\ref{Smarr}) and
~(\ref{Tensionlaw1}) by using the interchange symmetry such that
\begin{equation}
E\rightarrow \Gamma \eta ,T\rightarrow T,S\rightarrow S\eta ,\Gamma
\rightarrow E/\eta. \label{displacement}
\end{equation}
Thus we can obtain the reduced relations
\begin{equation}
d\Gamma =\frac{1}{8\pi G}\kappa _{s}dA_{s}. \label{MassTesionlaw3}
\end{equation}
\begin{equation}
dE=-\frac{1}{8\pi G}\eta A_{s}d\kappa _{s}+(\Gamma -\frac{1}{8\pi
G}\kappa _{s}A_{s})d\eta . \label{12}
\end{equation}%
From which, we can easily check out that they hold by using the
quantities
in~(\ref{Energy2})~(\ref{Tension2})~(\ref{SurfacegravityandArea})
and see that they have the similar formalisms with the laws of black
hole. Thus they can be naturally considered as the analogy of the
energy and tension laws of the AdS soliton. The most interesting
thing is that they has the same formalism with the result of the K-K
bubble in Ref~\cite{Kastor} where it is deduced by using the
Hamiltonian perturbation theory
techniques~\cite{Traschen,Perturbation technique}. Thus, it is more
convincible that they could be considered as the analogy of the
energy and tension laws of the AdS soliton.
\section{The energy and tension laws for boosted Ricci flat black hole}
The boosted Ricci flat black hole can be obtained from~(\ref{Ricci
flat black hole}) by the following boost transformation~\cite{Cai2}
\begin{eqnarray}
t &\rightarrow &t\cosh \alpha -y\sinh \alpha , \nonumber \\
y &\rightarrow &-t\sinh \alpha +y\cosh \alpha . \label{Boost
transformation}
\end{eqnarray}%
where $\alpha $ is the boost parameter and the boost velocity is
$v=\tanh \alpha $. Thus, the metric of the boosted Ricci flat black
hole is
\begin{equation}
ds^{2}=\frac{r^{2}}{l^{2}}[-dt^{2}+dy^{2}+\frac{r_{0}^{4}}{r^{4}}(dt\cosh
\alpha -dy\sinh \alpha )^{2}+(dx^{i})^{2}]+(1-\frac{r_{0}^{4}}{r^{4}})^{-1}%
\frac{l^{2}}{r^{2}}dr^{2}.\ \ (i=1,2) \label{Boosted Ricci flat
black hole}
\end{equation}%
Note that, because the coordinate $y$ in~(\ref{Boost
transformation}) is periodic, the solution~(\ref{Boosted Ricci flat
black hole}) is not equivalent to the static Ricci flat black
hole~(\ref{Ricci flat black hole}) globally~\cite{Lemos,Awad,Cai2}.
And in order to remove the conical singularity at the horizon
$r=r_{0}$, the Euclidean time $\tau $ in~(\ref{Boosted Ricci flat
black hole}) could have a period $\beta =\frac{\pi l^{2}\cosh \alpha
}{r_{0}}$. Following the same procedure as section II, at first we
do not consider the contribution from the gravitational tension term
in the laws of thermodynamics. After choosing the pure AdS spacetime
as the background and using the same Euclidean method, we can obtain
the Euclidean action of the boosted Ricci black hole to
be~\cite{Euclidean}
\begin{equation}
I_{E}=-\frac{\beta r_{0}^{4}}{16\pi Gl^{5}}\eta V_{2}.
\label{Euclidean action2}
\end{equation}
Note that, although the Euclidean action is the same as that of the
static Ricci flat black hole~(\ref{Euclidean action1}), the
relationship between $\beta$ and $r_{0}$ is different. Moreover,
here the thermal function related with Euclidean action is the
Gibbons free energy~\cite{Euclidean}
\begin{eqnarray}
G &\equiv &\frac{I_{E}}{\beta }=E-TS-vP. \label{Gibbons Free
energy}
\end{eqnarray}
and $G$ is the function of not only $\beta$ but also the boost
velocity $v$. Thus, the energy, entropy and momentum are
\begin{eqnarray}
E &=&(\frac{\partial I_{E}}{\partial \beta })_{v}-\frac{v}{\beta }(\frac{%
\partial I_{E}}{\partial v})_{\beta }=\frac{(3+4a^{2})r_{0}^{4}}{16\pi l^{5}}%
\eta V_{2}, \notag \\
S &=&\beta (\frac{\partial I_{E}}{\partial \beta
})_{v}-I_{E}=\frac{\eta
V_{2}r_{0}^{3}}{4l^{3}}\sqrt{1+a^{2}}, \notag \\
P &=&-\frac{1}{\beta }(\frac{\partial I_{E}}{\partial v})_{\beta }=\frac{%
\eta V_{2}r_{0}^{4}}{4\pi l^{5}}a\sqrt{1+a^{2}}.
\label{EnergyEntropyMomentum}
\end{eqnarray}
where $a\equiv \sinh \alpha $ and it could be easily seen that the
entropy $S$ is also exactly equal to 1/4 of the horizon area $A$,
which implicates that the following relations hold
\begin{eqnarray}
dG &=&-SdT-Pdv, \notag \\
dE &=&TdS+vdP. \label{Masslaw3}
\end{eqnarray}
Again if assuming the Gibbons free energy $G$ in~(\ref{Gibbons Free
energy}) is also the function of $\eta $, we can also obtain the
gravitational tension
\begin{eqnarray}
E &=&(\frac{\partial I_{E}}{\partial \beta })_{v,\eta }-\frac{v}{\beta }(%
\frac{\partial I_{E}}{\partial v})_{\beta ,\eta }=\frac{(3+4a^{2})r_{0}^{4}}{%
16\pi l^{5}}\eta V_{2}, \notag \\
S &=&\beta (\frac{\partial I_{E}}{\partial \beta })_{v,\eta }-I_{E}=\frac{%
\eta V_{2}r_{0}^{3}}{4l^{3}}\sqrt{1+a^{2}}, \notag \\
P &=&-\frac{1}{\beta }(\frac{\partial I_{E}}{\partial v})_{\beta ,\eta }=%
\frac{\eta V_{2}r_{0}^{4}}{4\pi l^{5}}a\sqrt{1+a^{2}}, \notag \\
\Gamma &=&\frac{1}{\beta }(\frac{\partial I_{E}}{\partial \eta
})_{\beta ,v}=-\frac{r_{0}^{4}}{16\pi l^{5}}V_{2}. \label{ESP2}
\end{eqnarray}
On the other hand, according to the definition, the useful
quasi-local stress tensor of the boosted Ricci flat black
hole~(\ref{Boosted Ricci flat black hole}) is~\cite{Counterterm}
\begin{eqnarray}
8\pi T_{tt} &=&\frac{(3+4\sinh ^{2}\alpha
)r_{0}^{4}}{2l^{3}r^{2}}+...
\nonumber \\
8\pi T_{ty} &=&-\frac{2\sinh \alpha \cosh \alpha
r_{0}^{4}}{l^{3}r^{2}}+... \label{StressTensor2}
\end{eqnarray}%
From which the energy and momentum can be calculated to be
\begin{eqnarray}
E &=&\frac{(3+4\sinh ^{2}\alpha )r_{0}^{4}}{16\pi l^{5}}\eta V_{2},
\nonumber \\
P &=&\frac{\sinh \alpha \cosh \alpha r_{0}^{4}}{4\pi l^{5}}\eta
V_{2}. \label{MPTension}
\end{eqnarray}
where the energy and momentum are consistent with the above results
in~(\ref{ESP2}). And this consistence could implicate that after
adding the contribution of tension term the first laws
in~(\ref{Masslaw3}) are
\begin{eqnarray}
dG &=&-SdT-Pdv+\Gamma d\eta , \notag \\
dE &=&TdS+vdP+\Gamma d\eta . \label{Masslaw4}
\end{eqnarray}
However, if we use the general definition of gravitational
tension~(\ref{TensionDefinition}), we can obtain the tension
\begin{equation}
\Gamma^{'} =-\frac{(1+4\sinh ^{2}\alpha )r_{0}^{4}}{16\pi
l^{5}}V_{2}. \label{Tension3}
\end{equation}
which is not consistent with the result in~(\ref{ESP2}). Note that,
this difference has also been found by D. Kastor et al, and they
argued that the tension obtained in~(\ref{ESP2}) was in fact an
effective tension which was related to the general tension such
that~\cite{Kastor2}
\begin{equation}
\Gamma =\Gamma^{'} +\frac{vP}{\eta }. \label{TTRelation}
\end{equation}%
From which, we can also find that when the boosted velocity is zero,
the general tension is just equal to the effective tension.
Using these quantities in ~(\ref{ESP2})~(\ref{MPTension}), we can
also check that
\begin{equation}
E-TS-vP=\Gamma \eta \label{Smarr2}
\end{equation}
Thus, from this relation~(\ref{Smarr2}) and the first energy
law~(\ref{Masslaw4}), the first tension law of boosted Ricci flat
black hole is
\begin{equation}
SdT+Pdv+\eta d\Gamma =0 \label{Tensionlaw2}
\end{equation}
\section{The boosted AdS soliton solution}
Naturally, we can also make a boost transformation~(\ref{Boost
transformation}) along the compact coordinate $y$ in the static AdS
soliton solution~(\ref{AdSSoliton}). Thus, the boosted AdS soliton
solution is
\begin{equation}
ds^{2}=\frac{r^{2}}{l^{2}}[-dt^{2}+dy^{2}-\frac{r_{0}^{4}}{r^{4}}(dy\cosh
\alpha -dt\sinh \alpha )^{2}+(dx^{i})^{2}]+(1-\frac{r_{0}^{4}}{r^{4}})^{-1}%
\frac{l^{2}}{r^{2}}dr^{2}.\ \ (i=1,2) \label{BoostedAdSSoliton}
\end{equation}
Note that this solution is also different from the static AdS
soliton globally, and it is easy to see that the coordinate $y$ in
the boost transformation is compact. In addition, an interesting
result is that this boosted AdS soliton solution also has the same
interchange symmetry with the boosted Ricci flat black hole. That
is, it can also be obtained from the boosted Ricci flat black hole
solution by the double analytic continuation between the time and
the compact coordinate $y$ in~(\ref{Boosted Ricci flat black hole}).
In the above section we have obtained the analogy of the energy and
tension laws of the static AdS soliton solution through the
inspiration from the interchange symmetry with the Ricci flat black
hole. However, it's easily found that there are closed timelike
curves in the boosted AdS soliton
solution~(\ref{BoostedAdSSoliton}). Moreover, viewed from the
physical point, after boosting along the compact coordinate $y$ in
static AdS soliton~(\ref{AdSSoliton}), the period of $y$ would be
shrunk to $\gamma ^{-1}\eta $ where $\gamma =(1-v^{2})^{-1/2}=\cosh
\alpha $ is the shrinking factor. However, the new period could not
avoid the conical singularity. Thus, this boosted AdS soliton
solution is ill in physics and the direct analogy of laws is of no
sense. In spite of that, the conserved charges such as energy and
momentum are well defined because they just depend on the properties
of its asymptotic behavior. And the corresponding quasi-local stress
tensor of the boosted AdS soliton solution can be
obtained~\cite{Counterterm}
\begin{eqnarray}
8\pi T_{tt} &=&-\frac{(1+4\sinh ^{2}\alpha
)r_{0}^{4}}{2l^{3}r^{2}}+...
\nonumber \\
8\pi T_{ty} &=&\frac{2\sinh \alpha \cosh \alpha
r_{0}^{4}}{l^{3}r^{2}}+... \label{StressTensor4}
\end{eqnarray}%
Thus, the energy and momentum of the boosted AdS soliton solution
are
\begin{eqnarray}
E &=&-\frac{(1+4\sinh ^{2}\alpha )r_{0}^{4}}{16\pi l^{5}}\eta V_{2},
\nonumber \\
P &=&-\frac{\sinh \alpha \cosh \alpha r_{0}^{4}}{4\pi l^{5}}\eta
V_{2}. \label{EP4}
\end{eqnarray}%
In addition, the general tension can also be obtained from the
definition~(\ref{TensionDefinition})
\begin{equation}
\Gamma =\frac{(3+4\sinh ^{2}\alpha )r_{0}^{4}}{16\pi l^{5}}V_{2}.
\label{T4}
\end{equation}%
These quantities in~(\ref{EP4})~(\ref{T4}) could explicitly manifest
the interchange symmetry with the boosted Ricci flat black hole,
too.
\section{Conclusion and discussion}
One of the motivations of this paper is to obtain the analogy of the
energy and tension laws of the AdS soliton solution, which can give
more understanding of this solution. In order to obtain them, we
first reconsider the laws of the Ricci flat black hole by taking the
contribution of the tension term into account. Then, inspired from
the interchange symmetry between the Ricci flat black hole and AdS
soliton, we finally obtain the analogy. In spite of that, how to
understand the analogy of laws of the AdS soliton is an open
question. Particularly, whether there is some underlying physical
interpretations such as thermodynamical effects in it is worthy of
further discussion. In addition, as a more general asymptotically
AdS black hole solution, we also take the boosted Ricci flat black
hole for example to give a simple generalization of the works by
D.Kastor to the asymptotically AdS case. Note that, although here
our formalisms of the laws of black holes or the static soliton are
the same as those of the asymptotically flat cases, the underlying
deduced contents are different. In principle, if we find the
appropriate formalisms of conserved charges and gravitational
tension, we perhaps can also use the Hamiltonian perturbation method
to deduce these laws directly. And this possibility will be
considered in the future work. As the corresponding solution which
has the interchange symmetry with boosted Ricci flat black hole, we
also consider the boosted AdS soliton solution. However, although
there is the same interchange symmetry, this boosted AdS soliton
solution is ill in physics because of the existence of the closed
timelike curves and conical singularity. Thus, the direct analogy of
energy and tension laws are of no sense. In spite of that, an
interesting result is that the conserved charges such as the energy
and momentum are well-defined for the boosted AdS soliton solution.
Moreover, as we expected, its energy is smaller than that of the
static AdS soliton solution. Thus, whether it can be considered as a
violation case to the new positive energy conjecture proposed by G.T
Horowitz and R.C Myers and how to understand it from the viewpoint
of the AdS/CFT correspondence would also be interesting things to
give further discussions. In addition, during calculating the
conserved charges, we also find that perhaps there is a new way to
define the gravitational tension from the quasi-local stress tensor
defined in~(\ref{stresstensor}), because the gravitational tension
can be easily found to be related to the corresponding stress tensor
$T_{yy}$ such that
\begin{eqnarray}
\text{Ricci flat black hole} &\text{: }&\Gamma =-\frac{r_{0}^{4}}{16\pi l^{5}%
}V_{2},\text{ }T_{yy}=\frac{r_{0}^{4}}{16\pi l^{3}r^{2}}+....\text{\
}
\notag \\
\text{Static AdS soliton} &\text{:}&\Gamma =\frac{3r_{0}^{4}}{16\pi l^{5}}%
V_{2},\text{ }T_{yy}=-\frac{3r_{0}^{4}}{16\pi
l^{3}r^{2}}+....\text{\ }
\notag \\
\text{Boosted Ricci flat black hole} &\text{:}&\Gamma =-\frac{%
(1+4a^{2})r_{0}^{4}}{16\pi l^{5}}V_{2},\text{ }T_{yy}=\frac{%
(1+4a^{2})r_{0}^{4}}{16\pi l^{3}r^{2}}+.... \notag \\
\text{Boosted AdS soliton} &\text{:}&\Gamma =\frac{3r_{0}^{4}}{16\pi l^{5}}%
V_{2},\text{ }T_{yy}=-\frac{(3+4a^{2})r_{0}^{4}}{16\pi
l^{3}r^{2}}+.... \label{TTYYRelation}
\end{eqnarray}
On the other hand, viewed from the physical interpretation of the
stress tensor, its spatial diagonal components are related with the
pressure, thus it is more convincible that there is a new
possibility to define the gravitational tension. In fact,
considering the interchange symmetry and the formalisms
in~(\ref{Mass})~(\ref{TensionDefinition}), we can give the new
definition of the gravitational tension through the counterterm in
the asymptotical AdS case that
\begin{equation}
\Gamma =-\frac{1}{\Delta t}\int_{S_{x}^{\infty }}d^{d-1}x\sqrt{\sigma }%
F(n^{\mu }T_{\mu \nu }n^{\nu }) \label{TensionNewDefinition}
\end{equation}
which can be easily checked that this new definition is satisfied in
our cases.
Note that, after our paper appeared, Dr. Cristian Stelea showed me
that they had also already given an exact formalism of the
gravitational tension through the counterterm in their cases. Thus
giving a more general rigorous definition of the gravitational
tension through the counterterm is an open interesting question, and
perhaps some clues could be found in their works~\cite{Stelea}.
\section{Acknowledgements}
Y.P Hu thanks Professor Rong-Gen Cai and Dr.Li-Ming Cao, Jia-Rui
Sun, Xue-Fei Gong and Chang-Yong Liu for their helpful discussions.
And Y.P Hu also thanks Dr. Cristian Stelea for his useful
information. This work is supported partially by grants from NSFC,
China (No. 10325525, No. 90403029 and No. 10773002), and a grant
from the Chinese Academy of Sciences.
| {'timestamp': '2009-05-21T06:38:16', 'yymm': '0904', 'arxiv_id': '0904.1250', 'language': 'en', 'url': 'https://arxiv.org/abs/0904.1250'} |
\section{Introduction}
Modeling the folding phenomena of surfaces, as well as the study of its regular patterns such as mathematical origami, has attracted lots of interests in computer graphics, material design as well as mathematics. Recent examples include origamizing surfaces \cite{demaine2017origamizer}, material design with mathematical origami \cite{dudte2016programming}, a FoldSketch system manipulating the folding of clothes \cite{li2018foldsketch}, modeling curved folding surface used in fabrication and architectural design \cite{zhu2013soft, kilian2008curved}, and so on. In this paper we propose a new framework to model and study such phenomena from geometric partial differential equations (PDEs) point of view. Under this framework, applications such as generating new flat-foldable material prototypes, fold and cusp sculpting on surfaces, reasoning the flat-folded surface with only partial data, become possible.
The key equation is the so-called {\it alternating Beltrami equation}
\[
\frac{\partial f}{\partial\bar{z}}(z)=\mu(z)\frac{\partial f}{\partial z}(z), \; z\in \Omega
\]
where $\Omega$ is a bounded domain with regular boundary. The difference to the classical counterpart is that here the {\it Beltrami coefficient} $\mu$ is allowed to take values in the Riemann sphere except near the equator, in contrast to the ordinary Beltrami equation which requires the Beltrami coefficient to be strictly inside the unit disk.
There are many advantages to model and study the surface folding with alternating Beltrami equation. One of the most prominent reason is its simplicity. The introduction of conformal geometry makes the problem linear, while the Beltrami coefficient can be used to encode all possible {\it conformal distortion} of the mapping. In particular, it can be used to control the folding and unfolding of the mapping. But it must be noted that, depart from conventions but crucial to our approach, the solution of the alternating Beltrami equation is not orientation-preserving in general, and so various mathematical notions need to be adapted to this setting. This is done in Section \ref{sec2} below.
\newpage
\subsection{Problem description}
\subsubsection{Solving alternating Beltrami equations}
Solving Beltrami equations has been a central component in computational quasiconformal geometry, which has many successful applications to medical imaging etc \cite{lam2014landmark, lui2014teichmuller}. The reason for a specialized treatment for alternating Beltrami equation is that $|\mu|>1$ and $|\mu|<1$ introduce genuinely different properties of the equation. In fact the equation is no longer globally elliptic, in contrast to the classical case. One can observe from our Theorem \ref{def:gqc energy} that the analogous ``energy norm" of the corresponding second order equation can well be negative. This demonstrates the non-trivial task of numerically solving alternating Beltrami equation. Nevertheless the problem can be resolved elegantly.
\subsubsection{The problem of incomplete data}
In applications involving folded surface, one of the fundamental problems is that when only one perspective of the surface is given, it is often {\it self-occluded}. This corresponds to the situation where the data needed to solve for the desired solution of the alternating Beltrami equation is often incomplete by a large portion. In other words, we have an {\it inverse problem} of inferring the geometric missing data due to self-occlusion or other reasons. We make two assumptions about what kind of data are available to us:
\begin{enumerate}
\item The domain's general shape, and the topological type of the singular set configuration is assumed to be known. The precise definition of singular set configuration is given in Section \ref{sec2}.
\item The visible portion of the folded surface is assumed to be known.
\end{enumerate}
Precise definitions and notations will be provided when we discuss the problem in details.
\subsection{Contributions of this paper}
Our work can be considered as the first application of quasiconformal methods to the problem of modeling and studying the folding phenomena. The main contributions include our methods to attack the two problems described above, which we describe in more details below.
\subsubsection{Numerically solving alternating Beltrami equation}
We analyze the effect of $|\mu|>1$ and introduce the {\bf generalized quasiconformal energy} that is analogous to the energy norm in the classical case. It is no longer convex but the saddle point characterizes the solution to the alternating Beltrami equation. The numerical method is derived from the proposed energy, works for bounded domains with arbitrary topology and amounts to simply solving a sparse linear system.
\subsubsection{Parametrizing flat-foldable surfaces with incomplete data}
To tackle the challenge of incomplete data, we exploit the structure of the problem and design suitable optimization algorithms thereof. We mainly focus on the problem for {\it flat-foldable} surfaces, informally known as {\it planar origamis}. This is due to the following two reasons:
\begin{enumerate}
\item Finding a desirable parametrization can be formulated as an optimization problem of minimizing the conformal distortion of the proposal mapping for parametrization.
\item It is where our framework of using alternating Beltrami equation is most effective, since other, especially three dimensional geometric features of the folded surface, such as mean curvature, do not belong to our framework but can be tackled by pre- or post-processing.
\end{enumerate}
To solve this ill-posed inverse problem, we propose the ``{\bf Reinforcement Iteration}" algorithm. The algorithm starts with an initial domain with the {\it singular set configuration} of the same topological type with the target surface, and then iteratively find the domain that will result in smaller conformal distortion. So alternatively, one can view the problem as finding some desirable Beltrami coefficients of the mapping from the initial domain and the desired target surface, that is a coefficient problem for the alternating Beltrami equation. Empirically our algorithm converges at about the rate $O(1/N)$, see Figure \ref{fig:convergence}. We shall layout more details in Section \ref{sec4}.
It is also interesting that the reinforcement iteration proposed is useful in other applications such as generating new Miura-ori type patterns. See Section \ref{sec: miura} for details.
\subsection{Related work} Here we briefly list some important related works in this area, while they are by no means an exhaustive survey.
{\bf Computational quasiconformal geometry.} Computational technique of conformal mapping \cite{levy2002least, desbrun2002intrinsic} turned out to be very useful in computer graphics. Since the seminal work of Gu and Yau \cite{gu2004genus}, the conformal geometry framework in surface registration tasks has advanced significantly. Earlier work generalizing these ideas is already implicit in the work of Seidel \cite{zayer2005discrete}. The quasiconformal extension of this framework was proposed by Lui and his coauthors \cite{lui2012optimization, lui2014teichmuller, choi2015flash}, with successful applications to medical image registration and surface registration. The quasiconformal method is able to handle large deformations, where conformal methods typically fail. Our work is an extension along this line, allowing the manipulation of folding, which opens up a new area to be explored.
{\bf Modeling surface folding and mathematical origamis.} In computer graphics there has been a notable amount of work on modeling the folding phenomena of surfaces. Many interesting works focus on 3D interactive design. These include the method of thin plate form with explicit user control of folding angles for interactive 3D graphics design in \cite{zhu2013soft}, which can also achieve the sharp folding edges as we do here. Our framework and techniques are completely different, especially here we are taking advantage of the fact that alternating Beltrami equation can be solved effectively in 2D. On the other hand, there are studies of developable surface design with curved folding \cite{kilian2008curved}, taking the advantage of the special quad meshes, while we don't have this restriction, but the focus and techniques are rather different. We must also mention the work of Demaine and Tachi \cite{demaine2017origamizer}, who developed algorithms to fold a planar paper into arbitrary 3D shapes. The study of the folding phenomena also has industrial applications in such as the 4D printing of \cite{kwok2015four} and material design in \cite{dudte2016programming}. We expect to discover interesting connections to them in the future work.\\
\section{Computing quasiconformal folds} \label{sec2}
\subsection{Definitions of folding homeomorphism, singular set configuration and derivation of alternating Beltrami equation} \label{subsec2.2}
The notion of folding used in this paper departs from those of three
dimensional nature, as in \cite{kilian2008curved}. Instead, we model it via the the
continuous mapping from the domain surface to the target surface with
designated change of orientation. This is made precise in the following
definition.
\begin{definition}[Folding homeomorphism and its singular set configuration]
Let $X$ and $Y$ be oriented surfaces. A continuous, bijective and {\it discrete} mapping $f:X\to Y$
({\it discrete} means $f^{-1}(y)$ are isolated in $X$ for all $y\in Y$) is called a
{\it folding homeomorphism} if there is a subset $\Sigma\subset X$, of Hausdorff dimension $1$,
with $(X,\Sigma)$ forming a locally finite two color map, say white and black, such that when restricting to
the connected components of the white (or black) region, $f$ is an orientation-preserving (or -reversing)
homeomorphism. The two color map $(X,\Sigma)$ is called the {\it singular set configuration} of $f$, and is sometimes simply referred to as $\Sigma $ if it is clear from the context. The white and black regions are denoted usually by $X^+, X^-$ respectively. \end{definition}
Note that we have required the mapping to be a bijection in order to avoid the case of degeneracy. Sometimes we would also like to give names to the points in the singular set
according to the properties of $f$. The definition below will classify the usually encountered situation and suffices for our purposes in this paper. Readers can also refer to Gutlyanskii et al. \cite{gutlyanskii2012alternating} for related materials.
\begin{definition}[Folding point and cusp point] \label{def:1}
Let $f:X\to Y$ be a folding homeomorphism with singular set configuration $\Sigma$.
\begin{itemize}
\item A point
$p\in\Sigma$ is called a folding point if there is an open neighborhood $U$ of $p$ such that
$U\setminus\Sigma$ is disconnected into exactly $2$ simply connected components,
and the restriction $f\big|_{U}$ is topologically equivalent to $(x,y)\mapsto(x,|y|)$, where
$U\cap\Sigma$ plays the role of $x$-axis.
\item A point
$p\in\Sigma$ is called a cusp point if there is an open neighborhood $U$ of $p$ such that
$U\setminus\Sigma$ is disconnected into exactly $2n$ simply connected components, $n>1$, and the remaining
points $p'\in (\Sigma\cap U)\setminus\{p\}$ are all folding points.
\item The collection of paths in $\Sigma$ consisting of all folding points is called folding lines.
\end{itemize}
\end{definition}
We illustrate these concepts in Figure \ref{fig:def_fold}. A more famous example is the paper crane origami, in Figure \ref{fig:sing set}, where for better visualization we use yellow and purple instead of white and black. The paper crane origami is in fact {\it flat-foldable}, which we will define in Definition \ref{def: qc_fold}. \\
\begin{figure}
\centering
\begin{subfigure}[b]{0.35\textwidth}
\resizebox {\columnwidth} {!} {
\begin{tikzpicture}
\draw (1,0) to [out=120,in=200] (1,1.5);
\draw (1,1.5) to [out=20,in=190] (2,2) to [out=10,in=90] (3,1);
\draw (3,1) to [out=270,in=300] (1,0);
\draw (1,1.5) to [out=-90,in=190] (3,1);
\fill [color = lightgray] (1,1.5) to [out=-90,in=190] (3,1) to [out=90,in=10] (2,2) to [out=190,in=20] (1,1.5);
\node at (1.5,0) {\footnotesize $X^{+}$};
\node at (2,1.5) {\footnotesize $X^{-}$};
\node at (2,0.7) {\footnotesize $\Sigma$};
\end{tikzpicture} }
\caption{The singular configuration contains only folding points: the folding line separates $X^{-}$ and $X^{+}$.}
\label{fig:folding_point}
\end{subfigure}
\begin{subfigure}[b]{0.35\textwidth}
\resizebox {\columnwidth} {!} {
\begin{tikzpicture}
\draw (1,0) to [out=120,in=200] (1,1.5);
\draw (1,1.5) to [out=20,in=190] (2,2) to [out=10,in=90] (3,1);
\draw (3,1) to [out=270,in=300] (1,0);
\draw (1,1.5) to [out=-90,in=180] (2,0.8);
\draw (2,0.8) to [out=0,in=190] (3,1);
\draw (1,0) to [out=0,in=250] (2,0.8);
\draw (2,0.8) to [out=70,in=250] (2,2);
\fill [color = lightgray] (1,1.5) to [out=-90,in=180] (2,0.8) to [out=70,in=250] (2,2) to [out=190,in=20] (1,1.5);
\fill [color = lightgray] (1,0) to [out=0,in=250] (2,0.8) to [out=0,in=190] (3,1) to [out=270,in=300] (1,0);
\node at (1.3,0.3) {\footnotesize $X^{+}$};
\node at (1.7,1.5) {\footnotesize $X^{-}$};
\node at (2.6,1.3) {\footnotesize $X^{+}$};
\node at (2.3,0.3) {\footnotesize $X^{-}$};
\node at (2,0.8) {\footnotesize $\Sigma$};
\end{tikzpicture} }
\caption{The singular configuration contains both folding points and a single cusp point: the cusp point joins the folding lines.}
\end{subfigure}
\caption{Illustration of Definition \ref{def:1}}
\label{fig:def_fold}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=\textwidth]{crane_foldpattern.jpg}
\caption{Singular set configuration of the paper crane.}
\label{fig:crane_foldpattern}
\end{subfigure}
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=\textwidth]{crane.jpg}
\caption{The paper crane obtained by solving an alternating Beltrami equation.}
\label{fig:crane}
\end{subfigure}
\caption{The paper crane origami}
\label{fig:sing set}
\end{figure}
So far the above definitions have been topological. We can equip the surfaces with
Riemannian metrics and thus talk about the intrinsic geometry of the
folding homeomorphism $f:(X,g_{1})\to(Y,g_{2})$ between Riemannian
surfaces. To measure the distortion of $f$ is to compare the pullback
metric $f^{*}g_{2}$ with the original metric $g_{1}$. In the other
direction, the knowledge of the distortion will give rise to a PDE
that characterizes the mapping. We shall describe these points in
details below.
For our purpose, let us take a neighborhood $\Omega \subset\mathbb{R}^2 \cong \mathbb{C}$
of $X$ and put $g_{2}$ to be the Euclidean metric. If we consider
the pull back metric as given data in the form of a matrix field $H:\Omega\to\mathbf{S}_{++}$,
where $\mathbf{S}_{++}$ denotes the space of symmetric positive definite
$2\times2$ matrices, and assume $f$ is differentiable, it then satisfies
the nonlinear system \cite{astala2008elliptic}.
\[
Df(z)^{T}Df(z)=H(z),\ \forall z\in \Omega,
\]
where $f = (u,v)^T, z = (x,y)^T, Df(z) = \begin{bmatrix}u_{x} & u_{y}\\
v_{x} & v_{y}
\end{bmatrix}$.
Surprisingly enough, it is possible to reduce the above to a linear
equation if the data is given up to multiplying a everywhere positive
function. This is the essential advantage for us to introduce conformal
geometry in dimension two in our problem. To do this, denote
\[
\mathbf{S}(2)=\{M\in\mathbf{S}_{++}:\det M=1\}.
\]
And let $G:U\to S(2)$ be given. The above equation can be expressed
as
\[
Df(z)^{T}Df(z)=\phi(z)G(z),\ \phi(z)>0.
\]
By taking determinants on both sides, we get
\[
\phi(z)=|J_{f}(z)|
\]
where $J_{f}(z)=\det Df(z)$. Note that the absolute value is necessary
since $f$ may be orientation reversing. As a result, we get
\[
Df(z)^{T}Df(z)=|J_{f}(z)|Q(z).
\]
Multiplying $Df(x)^{-1}$ on the left of both sides, and write $f=(u,v)^{T}$,
$Q=\begin{bmatrix}q_{11} & q_{12}\\
q_{12} & q_{22}
\end{bmatrix},$ we obtain the system
\[
\begin{bmatrix}u_{x} & u_{y}\\
v_{x} & v_{y}
\end{bmatrix}^{T}=\text{sgn}(J_{f}(x))\cdot\begin{bmatrix}q_{11} & q_{12}\\
q_{12} & q_{22}
\end{bmatrix}\begin{bmatrix}v_{y} & -u_{y}\\
-v_{x} & u_{x}
\end{bmatrix}.
\]
It is a straightforward matter to rewrite the above system in complex
derivatives, obtaining
\[
\frac{\partial f}{\partial\bar{z}}(z)=\mu(z)\frac{\partial f}{\partial z}(z),
\]
where $\mu=\frac{q_{11}-q_{22}+2iq_{12}}{q_{11}+q_{22}+2\cdot \text{sgn}(J_f(x)}, \frac{\partial f}{\partial\bar{z}}=(u_{x}-v_{y})+i(u_{y}+v_{x})$
and $\frac{\partial f}{\partial z}=(u_{x}+v_{y})+i(-u_{y}+v_{x})$.
This is sometimes called
the {\it alternating Beltrami equation}, coined by Uri Srebro \cite{srebro1996branched}. The name refers to the fact that
\[
\begin{cases}
|\mu|<1 & \text{if }\text{sgn}(J_{f}(x))>0\\
|\mu|>1 & \text{if }\text{sgn}(J_{f}(x))<0
\end{cases}.
\]
This motivates a more analytic definition for the folding homeomorphism, which is the principal mathematical subject of this paper.
\begin{definition} [Quasiconformal mapping with folds] \label{def: qc_fold}
A folding homeomorphism $f:\Omega\subset\mathbb{C}\to\mathbb{C}$, $K\geq1$ is called a generalized $K$-quasiconformal mapping with singular configuration $\Sigma$
if it is a solution of a
alternating Beltrami equation $\frac{\partial f}{\partial\bar{z}}(z)=\mu(z)\frac{\partial f}{\partial z}(z)$, such that
in $\Omega^{+}$ and $\Omega^{-}$ it holds that $|\mu(z)|<1$ and $|\mu(z)|>1$, respectively, and moreover satisfies the bound
\[
\big|\frac{1+|\mu(z)|}{1-|\mu(z)|}\big|\leq K
\]
for all $z\in\Omega$ except for a set of Lebesgue measure zero. In particular, the case of $K=1$ will be called flat-foldable.
\end{definition}
The quotient inside the bound has the interpretation of linear distortion of the mapping, see \cite{astala2008elliptic}. The above definition of flat-foldability is adapted to our problem, in particular in a discrete, computational setting. It includes the case where a surface is rigidly flat-folded, whose folding lines of the singular set configuration are all Euclidean geodesics.
\begin{remark}
The rigidity associated to flat-foldability is also manifested via a well-known condition about how folding lines join each other at a cusp point, known as the {\it Kowasaki's condition}. In details, let $n>1$ be an integer and suppose there are $2n$ Euclidean geodesics emanating from a cusp point $p\in U\subset\mathbb{C}$.
Then the neighborhood $U$ is flat-foldable if the alternating sum of the angles $(\alpha_i)_{i=1}^{2n}$ formed by every
two neighboring Euclidean geodesics
satisfy the condition
$$
\sum_{i=1}^{2n}(-1)^{i}\alpha_{i} = 0.
$$
This condition is utilized in \cite{dudte2016programming} for constrained optimization.
However, this formalism will not play a significant role in the algorithms we propose in this paper.
\end{remark}
To get a better picture of the alternating Beltrami equations,
we illustrate it with the effect of the Beltrami coefficients on a
single triangle (i.e. the linearized effect at the tangent space level).
This should provide one with geometric intuition for the
solutions on a triangulated mesh.
Let us rewrite the Beltrami equation as a system of first-order
PDEs in the usual Cartesian coordinate. Suppose $f:(x,y)\mapsto(u,v)$
satisfies the equation $\frac{\partial f}{\partial\bar{z}}(z)=\mu(z)\frac{\partial f}{\partial z}(z)$.
If we write $\mu=\rho+i\tau$,
then it's not hard to see that
\begin{equation}\label{eq:1}
\begin{bmatrix}u_{y}\\
v_{y}
\end{bmatrix}=\frac{1}{(1+\rho){}^{2}+\tau^{2}}\begin{bmatrix}2\tau & |\mu|^2-1\\
1-|\mu|^2 & 2\tau
\end{bmatrix}\begin{bmatrix}u_{x}\\
v_{x}
\end{bmatrix}, \end{equation}
here we have assumed $\rho\neq-1$ and $\tau\neq0$.
Hence for a single triangle, on which we assume $f$ is linear, the
mapping is determined up to a similarity transform (uniform scaling and rotation) in the target
domain. Now suppose $f$ maps a domain triangle $[v_{1},v_{2},v_{3}]=[(0,0),(1,0),(x,y)]$
to the target triangle $[w_{1},w_{2},w_{3}]=[(0,0),(1,0),(u(x,y),v(x,y))]$.
Then
\begin{equation}
\begin{bmatrix}u(x,y)\\
v(x,y)
\end{bmatrix}=\begin{bmatrix}1 & \frac{2\tau}{(1+\rho){}^{2}+\tau^{2}}\\
0 & \frac{1-|\mu|^2}{(1+\rho){}^{2}+\tau^{2}}
\end{bmatrix}\begin{bmatrix}x\\
y
\end{bmatrix}.\label{eq:2}
\end{equation}
One can check that the set of points for the family of $\mu$ with
each fixed modulus $|\mu|\neq1$ form a circle, whereas in the case
$|\mu|=1$ the circle degenerates to the $x$-axis. An illustration of this fact is shown in Figure \ref{fig:mu_plot}.
As one goes beyond to the case $|\mu|>1$, where the anti-diagonal
terms experience a change of sign, this leads to a ``flipping'' of
the triangle. In fact, for a single triangle, everything remains the same after a mirror reflection about the $x$-axis,
and the case $\mu=\infty$ corresponds to the {\it anti-conformality} of the mapping. Here $\infty$ should be understood as the infinity point in the Riemann sphere. What is more,
for each fixed argument $\arg(\mu)$ and let the modulus $|\mu|$ vary, the set of solution points form an arc of
a circle, passing through the points $(x,y)$ and $(x,-y)$. Altogether, we see that the Beltrami coefficients in effect form a
bipolar coordinate in the plane containing the target triangle. Therefore,
it describes all possible angular distortion at the tangent space
level, including those having a change of orientation.\\
\begin{figure}
\centering
\includegraphics[width=\textwidth]{mu_plot.png}
\caption{The trajectory of the third vertex under different values of Beltrami coefficients. Circles represent the situation when $|\mu| = 1/5, 7/20, 9/20, 3/5, 5/3, 20/9, 20/7, 5$ respectively.}
\label{fig:mu_plot}
\end{figure}
\subsection{Energy formulation}
In this section we turn to the computational methods to solve the Beltrami equation. Previously proposed methods include the {\it Beltrami holomorphic flow method} as in \cite{lui2012optimization}, or the decoupling method as in \cite{lui2014teichmuller}. Both require entire boundary information for solving the Beltrami equation and it can be unrealistic in applications. Fortunately, it turns out to be also unnecessary once we realize the coupling of the two coordinate functions of the mapping. This coupling arises naturally in an energy functional of least squares type. For completeness, we analyze this problem below since we did not find it in the literature.
\subsection{A formulation of least squares quasiconformal energy}
The formulation here takes inspiration from the well-known {\it least square conformal energy}, studied in \cite{levy2002least, desbrun2002intrinsic}, which take into account the coupling of $u$ and $v$.
Its continuous formulation is
$$\int_{\Omega}\|\nabla u +\begin{bmatrix}0 & -1\\
1 & 0
\end{bmatrix} \nabla v\|^{2}dxdy.$$
The corresponding matrix associated to its discretization
is the well-known {\it cotangent weight} matrix minus
a certain ``area matrix" \cite{pinkall1993computing, desbrun2002intrinsic}. This area matrix in fact plays the role of certain Neumann boundary condition.
One would expect analogous results to hold in the quasiconformal setting.
But in the quasiconformal case, it is not an entirely trivial matter to formulate the correct analog. One could formulate arbitrary quadratic energies in such a way that
$\int_{\Omega}F(\frac{\partial f}{\partial\bar{z}},\mu,\frac{\partial f}{\partial z})dxdy$ where $F$ is a quadratic cost functional such that $F(\cdot)=0$ if
$\frac{\partial f}{\partial\bar{z}}=\mu\frac{\partial f}{\partial z}$.
This formulation includes for example
$\int_{\Omega}\|\nabla u +\begin{bmatrix}0 & -1\\
1 & 0
\end{bmatrix}A \nabla v\|^{2}dxdy, $
where $A$ is the same matrix in \eqref{eq:3}.
However, one problem with these energies is that it is not always balanced with respect to the two coordinates. There is essentially only one energy that will give rise to this necessary condition which we detail below.
\subsubsection{The decoupling method and the necessary condition} Perhaps the most straightforward way to
solve the Beltrami equation is to decouple the corresponding
first order system into two independent second order equations, namely,
\begin{proposition}[Necessary condition]\label{prop:1}
Suppose for $z\in \Omega \setminus \Sigma$, $f(z) = u(z)+iv(z)$ satisfies the equation
$\frac{\partial f}{\partial\bar{z}}(z)=\mu(z) \frac{\partial f}{\partial z}(z)$. Assume the domain $\Omega$ is given the usual Euclidean geometry, and $|\mu|\neq1$,
then we have
\begin{equation}
\begin{cases}
-\nabla\cdot(A\nabla u(z)) & =0\\
-\nabla\cdot(A\nabla v(z)) & =0
\end{cases}\label{eq:3}
\end{equation}
where $A= \frac{1}{1-|\mu|^2}\begin{bmatrix}(\rho-1)^{2}+\tau^{2} & -2\tau\\
-2\tau & (1+\rho)^{2}+\tau^{2}
\end{bmatrix}$, and $\mu = \rho+i\tau$.
\end{proposition}
\begin{proof}
Observe that $\frac{\partial f}{\partial\bar{z}}(z)=\mu(z) \frac{\partial f}{\partial z}(z)$ can be transformed into
$$
\begin{bmatrix} u_x \\u_y \end{bmatrix}
= \begin{bmatrix} 0 & 1 \\-1 & 0 \end{bmatrix}A\begin{bmatrix} v_x \\v_y \end{bmatrix}.
$$
Then making use of the commutativity of second order partial derivatives $u_{xy} = u_{yx}$ under the Euclidean coordinate,
we obtain $$\nabla\cdot(A\nabla u(z)) =0.$$ The other equation is obtained in a similar way.
\end{proof}
\begin{remark} \label{rem: alter}
Note that the coefficient matrix $A$ is positive (or negative) definite if $|\mu|< 1$ (or $|\mu|>1$, respectively). If $U$ is any open neighborhood, on which $A$ is either positive or negative but not both, then it is not hard to see that they are the Euler-Lagrange equations of the
Dirichlet type energies
\begin{equation}
E_{\tilde{A}}(u;U)=\frac{1}{2}\int_{U}\|\tilde{A}^{1/2}\nabla u\|^{2}dxdy,\quad E_{\tilde{A}}(v;U)=\frac{1}{2}\int_{U}\|\tilde{A}^{1/2}\nabla v\|^{2}dxdy,\label{eq:8}
\end{equation}
where $\tilde{A}$ denotes $A$ if it is positive definite, or $-A$ if negative definite.
Therefore, we see that in general the global variational problem
must be separated according to whether $|\mu|<1$ or $|\mu|>1$ in the domain $\Omega$. We shall often denote $\Omega^{+}$ (or $\Omega^{-}$)
to be the largest open subset such that $|\mu|<1$ (or $|\mu|>1$, respectively), which is consistent with the previous notation.
\end{remark}
The derived system \eqref{eq:3}
is a {\bf necessary condition} that in principle should be satisfied by any other method which solves the equation in the Euclidean domain $\Omega$. This motivates the following.
\begin{definition} \label{def:qc energy}
The {\bf least squares quasiconformal energy} of the mapping $z=(x,y)\mapsto (u,v)$ against Beltrami coefficient $\mu=\rho + i\tau$ is defined to be
\[
\begin{aligned}
E_{LSQC}(u,v;\mu)&=\frac{1}{2}\int_{\Omega}\|P\nabla u+\begin{bmatrix}0 & -1\\
1 & 0
\end{bmatrix}P\nabla v\|^{2}\,dxdy
\end{aligned}
\]
where
\begin{equation*}
P = \frac{1}{\sqrt{1-|\mu|^2}}\begin{bmatrix}1-\rho & -\tau \\ -\tau & 1+\rho\end{bmatrix}
\end{equation*}
so that $P^TP = A$.
\end{definition}
Consequently, we have the following identity
$$
E_{LSQC}(u,v;\mu) = \left(E_{\tilde{A}}(u;\Omega)+E_{\tilde{A}}(v;\Omega)\right)-\mathcal{A}(u,v),
$$
where $\tilde{A}=P^TP$ is the same matrix described previously in \eqref{eq:8}, and
$$\mathcal{A}(u,v):=\int_{\Omega}(u_{y}v_{x}-u_{x}v_{y})\,dxdy $$
is the (signed) area of the target surface.
\begin{remark} \label{rem: lowbd_dirichlet}
Observe that we have obtained the analog of the classical lower bound of the Dirichlet energy
\begin{equation} \label{eq:13}
E_{\tilde{A}}(u)+E_{\tilde{A}}(v) \geq \mathcal{A}(u,v).
\end{equation}
This simply follows from the fact that $E_{LSQC}(u,v; \mu) \geq 0$. The vanishing of this energy is equivalent to the existence of $f = u+iv$ as a solution of the Beltrami equation with coefficient $\mu$. The existence is guaranteed for measurable Beltrami coefficients $\mu$ with $\|\mu\|_{L^{\infty}(\Omega)} < 1$, known as the {\it measurable Riemann mapping theorem}. Note also that the solution of the Beltrami equation is unique up to post-composition of conformal mappings \cite{astala2008elliptic}.
\end{remark}
If we assume the domain $\partial\Omega$ has Lipschitz boundary, then the quantity $\mathcal{A}(u,v)$
is equal to the following integral on the boundary
$$
\frac{1}{2}\int_{\partial\Omega}(v\nabla u-u\nabla v)\times\nu \,d\Gamma, $$
where $\nu(z)$ is the outer unit normal vector, and $d\Gamma$ is the standard measure of $\partial\Omega$. Actually, the coupling between $u$ and $v$ is realized as certain boundary condition applied to solving \eqref{eq:3}. The following derivation of the second order equations with boundary condition is standard.
\begin{theorem}
Suppose $\mu$ is uniformly bounded away from $1$, $\Omega$ is connected with Lipschitz boundary, and suppose there exists one pair $(u,v)$,
$u,v\in W^{2,2}(\Omega)$, such that
\[
E_{LSQC}(u,v;\mu) = \arg\inf_{\tilde{u},\tilde{v}\in W^{1,2}} E_{LSQC}(\tilde{u},\tilde{v};\mu),
\]
then they satisfy the following Neumann boundary problem
\begin{equation} \label{eq: qc}
\begin{cases}
-\nabla\cdot(A\nabla u) =0 \text{ in } \Omega \\
-\nabla\cdot(A\nabla v) =0 \text{ in } \Omega \\
\partial_{A\nu}u +\nabla v \times \nu =0 \text{ on } \partial\Omega\\
\partial_{A\nu}v -\nabla u \times \nu =0 \text{ on } \partial\Omega
\end{cases},
\end{equation}
where as before $\nu(z)$ is the outer unit normal vector.
\end{theorem}
\subsection{Generalized quasiconformal energy}
Because of the change of orientation, the energy formulation and the associated system of equations has to be accordingly modified. It is then crucial to study the interaction between regions of the domain that corresponds to different orientations of $f$.
First of all, it follows from arguments in Remark \ref{rem: alter} and Remark \ref{rem: lowbd_dirichlet} that the alternating Beltrami equation, when {\it restricted to regions of constant orientation}, is equivalent to vanishing of the energies
\[
\begin{aligned}
E_{LSQC}^+(u,v;\mu) := \frac{1}{2}\int_{\Omega^+}\|P\nabla u+\begin{bmatrix}0 & -1\\
1 & 0
\end{bmatrix}P\nabla v\|^{2}\,dxdy = 0 \\
E_{LSQC}^-(u,v;\mu) : = \frac{1}{2}\int_{\Omega^-}\|P\nabla u+\begin{bmatrix}0 & -1\\
1 & 0
\end{bmatrix}P\nabla v\|^{2}\,dxdy = 0
\end{aligned}
\]
where $\Omega^+ = \text{int } \{z\in \Omega: |\mu(z)| <1 \}$, $\Omega^- = \text{int } \{z\in \Omega: |\mu(z)| >1 \}$.
To obtain the global solution, one could solve the equation individually in $\Omega^+$ and $\Omega^-$ and glue the solution along the singular set configuration. It turns out that this can be done implicitly. The problem now is how to combine the quasiconformal energies on regions with different orientations into a single ``energy", so that we can solve the alternating Beltrami equation on the entire domain in one shot.
\begin{theorem} [Generalized quasiconformal energy] \label{def:gqc energy}
Assume there are only finitely many cusp points. Define the generalized quasiconformal energy with Beltrami coefficient $\mu$ of the mapping $z=(x,y)\mapsto (u,v)$ in $W^{2,2}$ to be
\[
E_{GQC}(u,v;\mu) = E_{LSQC}^+(u,v;\mu) - E_{LSQC}^-(u,v;\mu).
\]
Then the alternating Beltrami equation with Beltrami coefficient $\mu$ is a critical point of the above energy.
\end{theorem}
\begin{proof}
By taking a test function in the interior of constant orientation or near the boundary $\partial \Omega$, the critical point property in these regions is verified no different from the classical case. It now suffices to check the critical point property for the region near the singular set configuration. Since the number of cusp points is finite, it will not contribute to the integration on the singular set. Hence it suffices to work locally in a small neighborhood $U$ that contains only folding points, like the situation in Figure \ref{fig:folding_point}. Take any smooth test function $\phi$ compactly supported in $U$. Then by setting
\[
\frac{d}{d\epsilon}\big|_{\epsilon = 0}E_{GQC}(u+\epsilon \phi,v) = 0
\]
we obtain
\[
\left (\int_{\Omega^+} - \int_{\Omega^-} \right) \langle P\nabla \phi, P\nabla u\rangle + \langle P\nabla \phi, \begin{bmatrix}0 & -1\\
1 & 0
\end{bmatrix}P\nabla v \rangle \, dxdy= 0.
\]
Integrating by parts, and repeating the same steps for $v$, we can derive the following Euler-Lagrange system
\begin{equation}
\begin{cases}
-\nabla\cdot(A\nabla u) =0 \text{ in } \Omega^+\cup\Omega^- \\
-\nabla\cdot(A\nabla v) =0 \text{ in } \Omega^+\cup\Omega^- \\
\partial_{A\nu}u +\nabla v \times \nu =0 \text{ on } \partial\Omega \cup \Sigma\\
\partial_{A\nu}v -\nabla u \times \nu =0 \text{ on } \partial\Omega \cup \Sigma
\end{cases},
\end{equation}
where
\[
A= \frac{1}{1-|\mu|^2}\begin{bmatrix}(\rho-1)^{2}+\tau^{2} & -2\tau\\
-2\tau & (1+\rho)^{2}+\tau^{2}
\end{bmatrix},
\]
and $\nu = \nu(z)$ is the outer unit normal vector when $z\in \partial\Omega^{+}$, or equivalently the inner unit normal if we regard $z\in \partial\Omega^{-}$. Note that the second order equation outside of the singular set and boundary has the same form, and corresponds to the alternating Beltrami equation with coefficient ${\mu}$. This finishes the proof.
\end{proof}
\begin{remark}
Note that the above Neumann boundary problem is somewhat different from convention since the singular set lies in the interior but is treated like boundary. But this very condition can be seen as the way to glue two pieces of solutions on $\Omega^{+}$ and $\Omega^{-}$ together along the singular set configuration.
\end{remark}
\begin{remark}
It should be noted that an unfolding mapping can be computed in essentially the same way if the folded surface is positioned in the plane, as is the case shown in Figure \ref{fig:crane}. The computation of unfolding mapping will be important in the reinforcement iteration introduced later.
\end{remark}
\subsubsection{Discretization and implementation details}
First we discuss the case of least squares quasiconformal energy and later extend it to the generalized case. We discretize the the equation \eqref{eq: qc} on a linear triangular mesh $\mathcal{T}$, which is encoded as a list of vertices $V$ and a list of triangles $\mathcal{T}$ (by a mild abuse of notation) taking indices into $V$. We denote the number of vertices by $|V|$ and number of triangles by $|\mathcal{T}|$. The second order operator $\nabla\cdot(A\nabla)$ is a variant of the Laplacian. Its discretization amounts to expressing the following sum
\[
\sum_{T\in\mathcal{T}}\langle P\nabla\varphi(T),P\nabla\phi(T)\rangle_{T}
\]
for any test functions $\varphi, \phi$ defined on the vertices $V$ into a quadratic form $\varphi^T \mathcal{L}_{\mu} \phi$. Here, $\langle \cdot, \cdot \rangle_{T}$ is the 2D Euclidean inner product scaled by the area of the triangle $T$. On an oriented triangle $T = [v_0, v_1, v_2] $, since the functions being considered are linear on triangles, the gradient of a function
$\varphi = (\varphi_0, \varphi_1, \varphi_2) $ on this triangle is given by
\[
\nabla\varphi=\frac{1}{2\text{Area}(T)}\begin{bmatrix}0 & -1\\ 1 & 0 \end{bmatrix}\sum_{i=0,1,2}\varphi_{i}(v_{2+i}-v_{1+i}).
\]
where indexing modulo $3$ as appropriate.
Observe that
\[
\begin{bmatrix}1-\rho & -\tau \\ -\tau & 1+\rho\end{bmatrix} \begin{bmatrix}0 & -1\\ 1 & 0 \end{bmatrix} = \begin{bmatrix}0 & -1\\ 1 & 0 \end{bmatrix} \begin{bmatrix}1+\rho & \tau \\ \tau & 1-\rho\end{bmatrix}
\]
Hence, denoting
\[
v'_i = P^{-1}v_i,
\]
we have
\[
P\nabla\varphi=\frac{1}{2\text{Area}(T')}\begin{bmatrix}0 & -1\\ 1 & 0 \end{bmatrix}\sum_{i=0,1,2}\varphi_{i}(v'_{2+i}-v'_{1+i}).
\]
Therefore, denoting the triangle $T' = [v'_0, v'_1, v'_2]$,
\[
\begin{array}{rcl} \langle P\nabla\varphi(T),P\nabla\phi(T)\rangle_{T} & = & -\frac{1}{4\text{Area}(T')}\sum_{i,j}\varphi_{i}\phi_{j}(v'_{2+i}-v'_{1+i})^{T}(v'_{2+j}-v'_{1+j})\\ & = & -\sum_{i,j}\omega_{ij}(T)\varphi_{i}\phi_{j} \end{array}
\]
where
\[
\omega_{ij}(T)=\begin{cases} -\frac{1}{2}\cot\theta'_{k},\,k\neq i,j & \text{if }i\neq j\\ \frac{1}{2}(\cot\theta'_{i+1}+\cot\theta'_{i+2}) & \text{if }i=j \end{cases}
\]
where $\theta'_k$ is the angle of at the vertex $v'_k$. This is noting but the {\it cotangent weight} but with angles changed by the effect of $\mu$.
The expression for the area integral $\mathcal{A}(u,v)$ is unchanged from the least square conformal case \cite{levy2002least}, and hence we have:
\begin{corollary} \label{cor:1}
The quadratic form (up to a nonzero constant scaling) associated to the triangular mesh discretization of the least squares quasiconformal energy is given by the following $2|V| \times 2|V|$ matrix
$$
M := \text{diag}(\mathcal{L}_{\mu},\mathcal{L}_{\mu}) - 2\mathcal{A},
$$
which is applied to the $2|V|$-coordinate vector ${\bf x} = (u,v)$. The discrete version of Equation \eqref{eq: qc} is then $M\bf{x}=0$. Here $\mathcal{L}_{\mu}$ is the cotangent matrix associated to the operator $\nabla \cdot A\nabla$,
and $\mathcal{A}$ is the (signed) area matrix of the target triangular mesh.
\end{corollary}
\begin{proof}
It follows from the discussion above that the $|V| \times |V|$ matrix $\mathcal{L}_{\mu}$ corresponds to the discretization of the differential operator $\nabla\cdot(A\nabla)$. The area matrix matrix has non-zero entries only corresponding to the boundary vertices. It is then immediate to check, by examining the corresponding rows of the linear system, that for interior vertices, the solution $(u,v)$ satisfies $-\nabla\cdot(A\nabla u) =0 $ and
$-\nabla\cdot(A\nabla v) =0 $, while on the boundary, it satisfies $\partial_{A\nu}u +\nabla v \times \nu =0 $ and
$\partial_{A\nu}v -\nabla u \times \nu =0 $ respectively.
\end{proof}
In order to obtain a nontrivial solution to the system $M\mathbf{x}= 0$, it turns out one need only pin at least two vertices. The precise statement is contained in the following proposition.
\begin{proposition}
Suppose $|\mu|$ is uniformly bounded away from $1$, and the triangular mesh is connected and has no dangling triangles (i.e. there are no triangles which share a common vertex but no common edge). Let $I_{\text{pin}}$ be the indices of the points to be pinned, with number $|I_{\text{pin}}|\geq 2$, $I_{\text{free}}$ be the indices for the free points, and $M$ be the matrix defined as in Corollary \ref{cor:1}. Then the $2|I_{\text{free}}| \times2|I_{\text{free}}|$ sub-matrix $M_{free}$ of $M$ indexed by the free points has full rank.
\end{proposition}
\begin{proof}
The idea is essentially identical to the proof in \cite{levy2002least} and we only sketch the main argument and the modification needed here. The key observation is that the (topological) triangular mesh satisfying our assumption can be constructed incrementally using two operations:
\begin{itemize}
\item Glue: adding a new vertex and connecting it to two neighboring vertices;
\item Join: joining two existing vertices.
\end{itemize}
The proof is based on this observation. To wit, we express $E_\text{LSQC}$ as $\|B\mathbf{x} - \mathbf{b}\|^2$ where $B$ is of size $2|\mathcal{T}|\times 2|I_{free}|$. It then suffices to prove $B$ has full rank. One then proceeds by proving the incremental construction preserves the full rank property. Since the modulus of the Beltrami coefficient associated to the new triangle is bounded away from $1$, the associated matrix coefficients are nonzero real numbers. And therefore the same argument in \cite{levy2002least} applies.
\end{proof}
\begin{remark}
We now have a geometric understanding in the discrete case. In fact, it can be regarded as a conformal mapping with the domain given a different {\it conformal structure}. This viewpoint in fact has been already demonstrated previously when we derive the Beltrami equation. In the case of a triangular mesh, this structure can be thought of the assignment the angle triples $(\alpha_1,\alpha_2,\alpha_3)$ to each triangular face of the mesh, or equivalently the associated Beltrami coefficient. Under this viewpoint we can relate many algorithms from conformal geometric processing to their quasiconformal counterparts.
\end{remark}
\begin{remark}
It is usually a preferred practice to choose these two points far apart from each other to reduce excessive local scale change, as is the case in the conformal flattening task \cite{levy2002least}. This is because the triangle angles associated to the Beltrami coefficients as given may not be realizable as a planar mesh, and the associated least square quasiconformal energy can never reach zero.
\end{remark}
The implementation for the generalized quasiconformal energy is essentially identical to the previous case, except that the signed area matrix is replaced by the {\bf unsigned} area matrix. This amounts to reversing the sign of the entries in the corresponding rows indexed by vertices of the mesh triangles $T$ whose $|\mu_T|>1$ when constructing $\mathcal{A}$. In the sequel, solving (alternating) Beltrami equations using the described method will be denoted as
\[
\mathcal{T'} = \text{LSQC}(\mathcal{T},(\mu_{T})_{T\in\mathcal{T}},\mathcal{C}),\]
where $\mathcal{T'}$, $\mathcal{T}$ are the computed target triangular mesh and the domain mesh, respectively; $(\mu_{T})_{T\in\mathcal{T}}$ is the set of Beltrami coefficients on each face, and $\mathcal{C}$ is the set of constraints.\\
\footnotetext{The source code is available at \url{https://github.com/sylqiu/Least-square-beltrami-solver}, and it supports the alternating Beltrami equation. }
\section{The reinforcement iteration algorithm}
In this section we give our proposed solution to the problem of incomplete boundary and singular set data, where alternating Beltrami equation will be central to our approach. The iteration consists of two key steps, which find improved folding and unfolding mappings given the current unfolded and folded surfaces in an alternating fashion. Intuitively, each unfolding step tries to find a better singular set configuration, while each folding step tries to conform with the given data.
\subsection{The idea of the reinforcement iteration}
Let us now consider the case where the entire boundary and singular set of the folded surface $S$ is given. In order to parametrize $S$,
one needs to start with some initial singular set configuration. We can easily construct a mapping $g:\Omega_{\Sigma_0} \to S$ by enforcing all the constraints.
In general, $g$ will not be a conformal mapping, as the initial singular set configuration may not coincide with the reality. So instead, there exists a quasiconformal
mapping
\[
\varphi:\Omega_{\Sigma_0}\to\Omega_{\Sigma}
\]
that maps the initial configuration to the correct one. Its relation with the desired generalized conformal mapping $f$ can be observed
as a commutative diagram below
\begin{equation}
\begin{tikzcd} \label{diagram:1}
\Omega_{\Sigma_0} \arrow[r,"\varphi"] \arrow[d,"g"] & \Omega_{\Sigma} \arrow[ld,"f"]\\
S
\end{tikzcd}
\end{equation}
In this case, once we obtain the folded surface $S$ from the mapping $g$, since the entire boundary and singular set data is given, the generalized conformal ``unfolding" homeomorphism $f^{-1}$ can be constructed by solving the alternating Beltrami equation with
\begin{equation} \label{eq:14}
\mu = \begin{cases}
0 &\text{ in }S^{+}\\
\infty &\text{ in }S^{-}
\end{cases}
\end{equation}
And in this way the mapping $\varphi$ is obtained by the composition $h\circ g$. \\
However, when only partial data of $S$ is provided, it is no longer possible to obtain the folded surface $S$ by constructing $g$ in the above manner. We need to find the folded surface and its parametrization simultaneously. We shall be looking for a folded surface that satisfies the following properties.
\begin{definition} \label{def:admissible}
Let $S_{\text{vis}}$ be a set of partial boundary and singular set data, and $\Omega_{\Sigma_{0}}$ be the domain with an initial singular set configuration. A folded surface $S$ is called admissible if
\begin{enumerate}
\item Topological equivalence: The singular set configuration of $S$ is of the same topological type with the target surface.
\item Data correspondence: There is a subset $C\subset S$ such that there is a isometry from $C$ to $S_{\text{vis}}$.
\item Cycle consistency: There exist mappings $g,\varphi,f$ such that $f$ is flat-foldable, and the diagram \eqref{diagram:1} commutes.
\end{enumerate}
\end{definition}
We now proceed to describe a fixed-point-like algorithm of finding some admissible folded surface $S$ and its parametrization simultaneously. Let
$
g_{n}:\Omega_{\Sigma_{n}}\to S_{n}
$
be a generalized quasiconformal folding homeomorphism that satisfies
the constraints
\[
g_{n}\big|_{\Omega_{\text{vis}}}:\Omega_{\text{vis}}\to S_{\text{vis}},
\]
where $\Omega_{\text{vis}}\subset\Omega_{\Sigma_n}$ is the corresponding subset corresponding to the partial boundary and singular set data $S_\text{vis}$. This step promotes data fidelity. Let $h_{n}:S_{n}\to\Omega_{\Sigma_{n}}$ be ``unfolding homeomorphism'', which is obtained by trying to solve the alternating Beltrami equation with
\begin{equation*}
\mu = \begin{cases}
0 &\text{ in }S_{n}^{+}\\
\infty &\text{ in }S_{n}^{-}
\end{cases}
\end{equation*}
with enforcing the shape constraints of $\Omega_{\Sigma_{n}}$. This step minimize the generalized conformal distortion based on fitted surface $S_n$. We observe that both the $S_{\text{vis}}$ and shape constraints are essential for convergence of the algorithm, in particular implicitly decreasing the area distortion of the mapping. The next mapping $g_{n+1}$ is constructed based on the
updated domain with its singular set configuration
\[
\Sigma_{n}=h_{n}\circ g_{n}(\Sigma_{n-1}).
\]
As $n\to \infty$, we want $g_{n}$ to converge to a generalized conformal mapping $f:\Omega\to S$, and the composition
\[
h_{n}\circ g_{n}:=\psi_{n}
\]
converges to $id_{\Omega}$, while
$$
\psi_{n}\circ\cdots\circ\psi_{2}\circ\psi_{1}:=\varphi_n
$$ converges to a quasiconformal mapping that transforms
the initial singular set configuration to a desirable one. This is shown
schematically in the following diagram
\begin{equation}
\begin{tikzcd} \label{diagram:2}
\Omega_{\Sigma_{0}} \arrow[r,"\psi_{1}"] \arrow[d,"g_1"]
& \Omega_{\Sigma_{1}} \arrow[r,"\psi_{2}"] \arrow[d,"g_2"]
& \Omega_{\Sigma_{2}} \arrow[r,"\psi_{3}"] \arrow[d,"g_3"]
& \cdots \Omega_{\Sigma_{n-1}} \arrow[r,"\psi_{n}"] \arrow[d,"g_n"]
& \Omega_{\Sigma_{n}} \cdots\\
S_{1} \arrow[ru,"h_1"]
& S_{2} \arrow[ru,"h_2"]
& S_{3} \arrow[ru,"h_3"]
& S_{n} \arrow[ru,"h_n"]
\end{tikzcd}
\end{equation}
In each step we keep enforcing the available data $S_{vis}$ by the mapping $g_n$, and by $h_n$ we keep enforcing the known boundary shape of $\Omega$, hence the name reinforcement iteration.\\
\subsection{The formal optimization problem and the algorithm} \label{sec4}
Following the notations as above, let $\Omega_{\Sigma}^{+},$ $\Omega_{\Sigma}^{-}$
be the two disjoint open sets in $\Omega$ specified by some singular
set configuration $\Sigma$ as the orientation-preserving part and
the reversing part, respectively. The original objective is to find
a folding homeomorphism $f$ such that
\[
\arg\min_{f,\Sigma}\int_{\Omega_{\Sigma}^{+}}\|\frac{\partial f}{\partial\bar{z}}\|^{2}+\int_{\Omega_{\Sigma}^{-}}\|\frac{\partial f}{\partial z}\|^{2},
\]
subject to the constraint
\[
f\big|_{\Omega_{\text{vis}}}:\Omega_{\text{vis}}\to S_{\text{vis}}.
\]
Note that $\Sigma$ is a variable in the minimization problem. This
formulation as it stands seems very hard to implement. The discussion in
the last section leads to the following relaxation, which depends
on the initial singular set configuration $\Sigma_{0}$:
\begin{equation}
\arg\min_{f,\varphi}\left(E(f,\varphi):=\int_{\Omega_{\Sigma_{0}}^{+}}\|\frac{\partial f}{\partial\bar{z}}\circ \varphi \|^{2}+\int_{\Omega_{\Sigma_{0}}^{-}}\|\frac{\partial f}{\partial z}\circ \varphi\|^{2}\right)\label{eq:6}
\end{equation}
subject to the constraints
\[
f \big|_{\Omega_{\text{vis}}}:\Omega_{\text{vis}}\to S_{\text{vis}},
\]
where the argument $f$, defined on $\Omega$, ranges in the set ``folded
homeomorphisms'', and $\varphi:\Omega\to\Omega$ ranges in the set
of quasiconformal homeomorphisms. Sometimes it is also convenient to express the quantities such as
$ \int_{\Omega_{\Sigma_{0}}^{+}}\|\frac{\partial f}{\partial\bar{z}}\circ \varphi \|^{2}$
to be $\int_{\Omega_{\Sigma_{\varphi}}^{+}}\|\frac{\partial f}{\partial\bar{z}}\|^{2},$
where $\Omega_{\Sigma_{\varphi}} = \varphi(\Omega_{\Sigma_0})$.
Note that if $\varphi$ gives the ``correct'' singular set configuration,
then the above energy vanishes for $f$ that is generalized conformal.\\
Our iteration algorithm in the last section is then to find such mappings
$\varphi$ and $f$. The overall algorithm is summarized as in Algorithm \ref{algo:1}. \\
\begin{algorithm}
\caption{Reinforcement Iteration}
\begin{algorithmic}
\STATE{Initialize $\Sigma_{0}$, $\epsilon > 0$, $\text{itermax}>0$,}
\STATE{Construct $g_{1}$, $h_1$; compute $\varphi_{1} = u_{1}\circ f_{1} $.}
\STATE{Evaluate $E(g_{1},\varphi_{1})$, $E(g_{0},\varphi_{0})=0$, $n=1$.}
\WHILE{$|E(g_{n},\varphi_{n})-E(g_{n-1},\varphi_{n-1})| > \epsilon$ and $n<\text{itermax}$.}
\STATE{Provided $\varphi_{n-1}$, construct $g_{n}$, $h_n$.}
\STATE{Compute $\varphi_{n} = h_{n}\circ g_{n}\circ \varphi_{n-1}$.}
\STATE{Evaluate $E(g_{n},\varphi_{n})$.}
\ENDWHILE
\end{algorithmic} \label{algo:1}
\end{algorithm}
The basic steps are constructions of the mappings
$g_{n}$ and $h_{n}$, whose implementations we now turn to.
\subsection{Implementation details}
\subsubsection{Construction of \texorpdfstring{$g_{n}$}{}}
Given the updated domain $\Omega_{\Sigma_{\varphi_{n-1}}}=\varphi_{n-1}(\Omega_{\Sigma_{0}})$,
we obtain the $S_{\text{vis}}$-enforced mapping $g$ by solving a alternating Beltrami equations subject to the constraints
\[
g\big|_{\Omega_{\text{vis}}}:\Omega_{\text{vis}}\to S_{\text{vis}}.
\]
Since we mainly care about the partial boundary and singular data enforcement here, the values of the Beltrami coefficients does not matter so much. But unless there is some good reason, it's better in practice not to introduce artificial distortion, hence we often set the Beltrami coefficients to be $0$ or $\infty$, corresponding the orientation-preserving or -reversing regions, respectively.
In terms of triangular meshes, suppose the domain triangular mesh is
$\mathcal{D}_{n-1},$ with $\mathcal{D}_{n-1}^{+}$,
$\mathcal{D}_{n-1}^{-}$ corresponds to $\Omega_{n-1}^{+}$
and $\Omega_{n-1}^{-}$, respectively; $S_{\text{vis}}$
is realized as certain constraint $C_\text{vis}$. Then the folded surfaced is obtained by
\[
\mathcal{S}_{g_{n}}=\text{LSQC}(\mathcal{D}_{n-1},\{\mu_{T}\}_{\mathcal{D}_{\varphi_{n-1}}},C_\text{vis})
\]
where
\[
\mu_{T}=\begin{cases}
0 & \text{if }T\in\mathcal{D}_{n-1}^{+}\\
\infty & \text{if }T\in\mathcal{D}_{n-1}^{-}
\end{cases}.
\]
\subsubsection{Construction of \texorpdfstring{$h_{n}$}{}}
Given the folded surface constructed from the last step $S_{n}=g_{n}(\Omega_{n-1})$,
recall that the unfolding map $h_{n}:S_{n}\to\Omega_{\Sigma_{n}}$ is found by solving the minimization problem
\[
\arg\min_{h}\left(\int_{S_n^{+}}\|\frac{\partial h}{\partial\bar{z}}\|^{2}+\int_{S_n^{-}}\|\frac{\partial h}{\partial z}\|^{2}\right)
\]
subject to the shape constraints
\[
h\big|_{\partial S_{n}}:\partial S_{n}\to \partial \Omega.
\]
$\varphi_{n}$ is then updated by $\varphi_{n} = h_{n}\circ g_{n}$.
In terms of triangular meshes, suppose the folded surface mesh is
$\mathcal{S}_{n},$ with $\mathcal{S}_{n}^{+}$,
$\mathcal{S}_{n}^{-}$ corresponds to $S_{n}^{+}$
and $S_{n}^{-}$, respectively; $\partial\Omega$
is realized as certain constraint $C_{\partial\Omega}$. Then the above minimization
can be solved by
\[
\mathcal{D}_{\Sigma_{n}}=\text{LSQC}(\mathcal{S}_{n},\{\mu_{T}\}_{\mathcal{S}_{n}},C_{\partial\Omega})
\]
where
\[
\mu_{T}=\begin{cases}
0 & \text{if }T\in\mathcal{S}_{n}^{+}\\
\infty & \text{if }T\in\mathcal{S}_{n}^{-}
\end{cases}.
\]
\section{Further discussion and experimental results}
Several remarks are in order. First, finding the critical point for the generalized quasiconformal energy is an easy saddle point problem by solving linear equations. We can reconcile it with the minimization problem in the construction $h_n$ by noting that the two are equivalent provided the folded and unfolded part of $h_n$ matched up, which is of course part of the continuity assumption about $h_n$. \\
Second, we notice that the desirable domain $x^*$ and its folded counterpart $y^*$ are fixed points of our iteration algorithm, which can be written as $x^* = \mathcal{F}(x^*)$,
where $\mathcal{F}$ is the iteration mapping in operator form. Note that $\mathcal{F}$ depends on its argument $x$ in a very non-linear way because of the auxiliary variable $y$ we introduced, whose computation requires the cotangent matrix associated to $x$. But approximately, in each iteration the folding and unfolding operations are inverse to each other and therefore $\mathcal{F}$ is close to the identity. The convergence of fixed-point iteration is well studied in the literature, see \cite{combettes2004solving} and references therein. For example, the convergence will be implied by the $\alpha$-averaged property of $\mathcal{F}$. As we can notice in Figure \ref{fig:iter_experiment}, as well as in many other experiments, the distortion of many of the interior mesh triangles can barely be noticed in the later phase of the iteration, while the meshes remain also well conditioned. As other fixed-point iterations, it is reasonable to expect that the iteration mapping under good conditioning of the mesh triangles and a good initial guess to have convergence. \\
\begin{figure}
\centering
\begin{subfigure}[b]{0.29\textwidth}
\includegraphics[width=\textwidth]{observed_fold.jpg}
\caption{Folded surface with occlusion.}
\label{fig:observed_fold}
\end{subfigure} \quad
\begin{subfigure}[b]{0.29\textwidth}
\includegraphics[width=\textwidth]{true_unfold.jpg}
\caption{True unfolded surface.}
\label{fig:true_unfold}
\end{subfigure}
\begin{subfigure}[b]{0.26\textwidth}
\includegraphics[width=\textwidth]{2unfold_init.jpg}
\caption{Initialised domain with partial data.}
\label{fig:2unfold_init}
\end{subfigure} \\
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_init_fold+.jpg}
\caption{Iter = 1: Frontside of registered fold.}
\label{fig:iter=1 reg+}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_init_fold-.jpg}
\caption{Iter = 1: Backside of registered fold.}
\label{fig:iter=1 reg-}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_init_unfold.jpg}
\caption{Iter = 1: Unfolded surface.}
\label{fig:iter=1 unfold}
\end{subfigure} \\
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_50_fold+.jpg}
\caption{Iter = 50: Frontside of registered fold.}
\label{fig:iter=5o reg+}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_50_fold-.jpg}
\caption{Iter = 50: Backside of registered fold.}
\label{fig:iter=50 reg-}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_50_unfold.jpg}
\caption{Iter = 50: Unfolded surface.}
\label{fig:iter=50 unfold}
\end{subfigure} \\
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_200_fold+.jpg}
\caption{Iter = 200: Frontside of registered fold.}
\label{fig:iter=120 reg+}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_200_fold-.jpg}
\caption{Iter = 200: Backside of registered fold.}
\label{fig:iter=120 reg-}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{iter_200_unfold.jpg}
\caption{Iter = 200: Unfolded surface.}
\label{fig:iter=120 unfold}
\end{subfigure} \\
\caption{Iteration results for the doubly folded surface: note that the folding lines gradually straighten out.}\label{fig:iter_experiment}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{1unfold_init.png}
\caption{Iter = 1: unfolded domain.}
\label{fig:1unfold_init}
\end{subfigure}
\begin{subfigure}[b]{0.285\textwidth}
\includegraphics[width=\textwidth]{1unfold_50.png}
\caption{Iter = 50: unfolded domain.}
\label{fig:1unfold_50}
\end{subfigure}
\begin{subfigure}[b]{0.295\textwidth}
\includegraphics[width=\textwidth]{1unfold_200.png}
\caption{Iter = 200: unfolded domain.}
\label{fig:1unfold_200}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{1fold_init.png}
\caption{Iter = 1: initially registered 1-fold.}
\label{fig:1fold_init}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{1fold_final.png}
\caption{Iter = 200: finally registered 1-fold.}
\label{fig:1fold_final}
\end{subfigure}
\caption{Iteration results for a once-folded surface: note the curved boundary in occlusion from the initial map is gradually straightened out and the folded domain becomes wider.}
\label{fig:1fold_iter}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{cusp_1.png}
\caption{Iter = 1: unfolded domain.}
\label{fig:cusp_init}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{cusp_50.png}
\caption{Iter = 50: unfolded domain.}
\label{fig:cusp_50}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{cusp_200.png}
\caption{Iter = 200: unfolded domain.}
\label{fig:cusp_200}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\includegraphics[width=\textwidth]{cusp_init.png}
\caption{Iter = 1: initially registered cusp.}
\label{fig:cusp_init1}
\end{subfigure}
\begin{subfigure}[b]{0.29\textwidth}
\includegraphics[width=\textwidth]{cusp_final.png}
\caption{Iter = 200: finally registered cusp.}
\label{fig:cusp_final}
\end{subfigure}
\caption{Iteration results for a cusped surface: note that the curved boundary and folding lines in occlusion from the initial map is gradually straightened out.}
\label{fig:cusp_iter}
\end{figure}
We implemented the described reinforcement iteration algorithm and demonstrate it for a doubly folded surface, as illustrated in Figure \ref{fig:iter_experiment}. The folded surface and its unfolded counterpart, as shown in Figure \ref{fig:observed_fold} and \ref{fig:true_unfold}, are generated according to a real folded paper and its unfolded counterpart. In Figure \ref{fig:2unfold_init} it is our initialized domain $\Omega_{\Sigma_0}$. In \ref{fig:observed_fold} and \ref{fig:2unfold_init}, the red circles mark the corresponding constraint points to the visible partial singular set and boundary data. Our algorithm works similarly well with other examples as well. This shows the robustness of our algorithm.\\
In the next three rows of Figure \ref{fig:iter_experiment} we show the iteration results at the first iteration, 50-th iteration and 200-th iteration. We can observe the curly folding lines in the first iteration in Figure \ref{fig:iter=1 reg-} and \ref{fig:iter=1 unfold}. This is due to the incomplete data and the incompatible initialized domain. In the subsequent iterations we saw significant improvement over the rigidity of the folding. In practice we also found that if we explicitly regularize the singular lines by, for example, projecting them onto a Euclidean geodesic, and then restart the iteration, the convergence will be improved in particular for the multiply-folded cases.
Observe also that in the limit, as in Figure \ref{fig:iter=120 unfold}, the singular set configuration is in not exactly the same as that of the true unfolded surface. This can be explained by the existence multiple admissible solutions to this problem. For example, another admissible solution may be obtained by some different initialization. This is of course expected.\\
In Figure \ref{fig:1fold_iter} and \ref{fig:cusp_iter} we illustrate the effect of reinforcement iteration algorithm applied to a once-folded surface and a cusped surface. The straightening effect can be easily seen from the comparison between the initial folding map and the final folding map. In Figure \ref{fig:convergence}, we plot a log-log diagram for the energy $E(g_k,\varphi_k)$ that we aim to minimize for the above three examples. We can observe that the convergence rate approaches $O(1/N)$ in the mid-stage of the iteration. That the energy decreases slightly slower in the later phase can be explained by our observation from the iterations that only a few points are adjusted while the singular set configuration is still away from flat-foldability. These adjusted points are mainly near the cusp points. This fact can be observed from Figure \ref{fig:iter=1 unfold} and \ref{fig:iter=50 unfold}. The convergence rate varies in the different phases of the iteration, illustrating the non-linear nature of the iteration.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{convergence.png}
\caption{Convergence plot: the loss is defined as $\sum_{T\in \Omega_{+}} |\mu|^2 +\sum_{T\in \Omega_{-}} 1/|\mu|^2 $ for scale invariant comparison.}
\label{fig:convergence}
\end{figure}
\section{Applications}
\subsection{Generating and editing generalized Miura-ori} \label{sec: miura}
The Miura-ori refers to a special type of Origami tessellation of the plane, which can be used to design flat-foldable materials aiming at achieving designed curvature properties \cite{dudte2016programming}. Previous approaches are based on analytic construction or constrained optimization, using the Kowasaki condition. Here we explore another possibility of creating such Origami models. Namely, we create more Miura-ori type domains and realize them via solving alternating Beltrami equations.
For simplicity, we consider the Miura-ori pattern in Figure \ref{fig:miura}. The yellow color on a triangle $T$ refers to the prescription of $\mu(T) = \infty$, and purple ones $\mu(T) = 0$. After solving the alternating Beltrami equation in 2D, we obtain the classical Miura-ori strip, which is the flat-folded state of the surface. Suitable $z$-coordinates are added for visualization in 3D.
\begin{figure}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{miura_ori_unfolded.png}
\caption{Classical Miura-ori pattern}
\label{fig:miura_ori_unfolded}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{miura_ori_folded.png}
\caption{Realization of the Miura-ori on the left}
\label{fig:miura_ori_folded}
\end{subfigure}
\begin{subfigure}[b]{0.39\textwidth}
\includegraphics[width=\textwidth]{miura_ori_strip.png}
\caption{Classical Miura-ori strip}
\label{fig:miura_ori_strip}
\end{subfigure}
\caption{Classical Miura-ori and its realization in 3D}
\label{fig:miura}
\end{figure}
To generate more Miura-ori type domains, ideally we can simply apply a conformal map on the domain. Notice that, in the continuous case, the domain obtained by compositing a flat-foldable configuration with a conformal map remains flat-foldable (satisfying the alternating Beltrami equation with the same coefficients as before), because of the angle preserving property. Such a composition can create triangles at different scales.
However, because of the discreteness, the angles is preserved only if the mapping is a uniform scaling plus rigid motion. Indeed, this follows from the our assertion on rank of the system matrix. Fortunately, applying a conformal mapping usually only yield a small and structured perturbation to the Beltrami equation, and the new Miura-ori domains can still be created via several iteration of the foldings and unfoldings, in light of the reinforcement iteration we proposed. For example, a new Miura-ori pattern in Figure \ref{fig:miura2} is created via this method, with the choice of (in this case we just made any convenient choice)
\[
\Phi(z) = 10 + 0.1z + 0.4z^2.
\]
\begin{figure}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{miura_ori_unfolded2.png}
\caption{Naive composition with $\Phi$}
\label{fig:miura_ori_unfolded2}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{miura_ori_unfolded2_1.png}
\caption{A new Miura-ori pattern after folding-unfolding iterations (maximal distortion $ = 3\times 10^{-4}$ compared to the folded state (c))}
\label{fig:miura_ori_unfolded22}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{miura_ori_folded2.png}
\caption{Realization of the Miura-ori in (b)}
\label{fig:miura_ori_folded2}
\end{subfigure}
\begin{subfigure}[b]{0.39\textwidth}
\includegraphics[width=\textwidth]{miura_ori_strip2.png}
\caption{New Miura-ori strips: note that there is a stack of strips of varing sizes}
\label{fig:miura_ori_strip2}
\end{subfigure}
\caption{A new Miura-ori pattern by composition with $\Phi$ and its realization in 3D. Here, maximal distortion is defined by $\max\{\max\{|\mu_T|\}_{T\in \Omega^{+}}, \max\{1/|\mu_T|\}_{T\in \Omega^{-}} \}$}
\label{fig:miura2}
\end{figure}
Different from approach of Dudte {\it et al.} \cite{dudte2016programming}, the surface we obtain is flat-foldable by design. Given the rich family of conformal mappings, it will be particularly interesting to study the new family of Miura-ori patterns with the aid of our algorithm. The study of different patterns' curvature approximation capacities is also a exciting future direction. We envisage a ``conformal geometric processing" approach to the modelling of Miura-ori. Under such an approach researchers can efficiently design the pattern with a simple set of CAD tools. Mathematical understanding of this problem will definitely benefit such a ``bottom-up" approach to material design with flat-foldable structures.
For a preliminary example, we can simulate and study the deformation of the Miura-ori in 3D with our solutions. Starting from the flat-folded state of the surface, one can apply the classical geometric editing methods such as as-rigid-as-possible \cite{sorkine2007rigid}. An example of such a deformation with user-defined position constraints is shown in Figure \ref{fig:miura2_arap}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{miura2_arap.png}
\caption{A rigid deformation of the Miura-ori in Figure \ref{fig:miura2}}
\label{fig:miura2_arap}
\end{figure}
\subsection{Almost rigid folding with application to fold-texture generation, fold sculpting and fold in-painting}
As one of the immediate applications, we can consider a folding transformation on the texture space to create synthetic fold-like textures, prior to applying the texture map. This can be cheap to do if high quality physical simulation and rendering is not available. In Figure \ref{fig: 3dfold_texture} we explore such a possibility of user-designed fold-like texture generation.
One of the fundamental steps in texturing a 3D surface is to find the parameterization (or the texture map) $f : S \to \Omega \in \mathbb{R}^2$. In particular, UV mapping is one of the major types of parameterization techniques in various software packages, which works well if the 3D model is created from polygon meshes. The above technique can be very useful in the interactive user design, where the user directly operates on the target mesh, and the input is transformed to the texture domain via the UV-map, to create desirable fold-like texture on the target mesh. It is also possible to incorporate proper shading effect on the transformed texture directly, making the texture look more realistic. We have implemented such a fold-like texturing method using a 3D T-shirt model, shown in Figure \ref{fig:3dtexture}. Note that the mesh is not deformed at all.
\begin{figure}
\centering
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{texturing1.png}
\caption{Before fold-texturing}
\label{fig:original}
\end{subfigure}
\begin{subfigure}[b]{0.36\textwidth}
\includegraphics[width=\textwidth]{texturing2.png}
\caption{After fold-texturing}
\label{fig:3dtexture}
\end{subfigure}
\caption{Folding effect texturing on a 3D model. Note that the mesh model is not deformed.}
\label{fig: 3dfold_texture}
\end{figure}
We can also apply the folding technique directly to the 3D meshes, as an application we would like to call {\it fold sculpting}. To illustrate this, we select a patch from the T-shirt model, as shown in Figure \ref{fig:tshirt}. We applied the folding operation to a suitable parametrization of the patch, which can obtained easily via, for example, projection or a least square conformal parametrization \cite{levy2002least}, and then glue it back to the T-shirt model. Note that our algorithm produces sharp edges. This can be mitigated by some standard smoothing operation in various mesh editing software. Figure \ref{fig: maya_smooth2} and \ref{fig: maya_smooth} show the results after appropriate smoothing, where we used the software Maya\footnotemark to the smoothing and rendering tasks. Note that such folding is not easily obtained by pure handcraft, since one part of the cloth actually folds over and covers some other part of the cloth.
\footnotetext{A software of the Autodesk Inc. See \url{https://www.autodesk.com.hk/products/maya/overview}. The results are generated under the student license obtained by the first author. }
\begin{figure}
\centering
\begin{subfigure}[b]{0.30\textwidth}
\includegraphics[width=\textwidth]{tshirt1.png}
\caption{The T-shirt model}
\label{fig:tshirt}
\end{subfigure}
\begin{subfigure}[b]{0.34\textwidth}
\includegraphics[width=\textwidth]{ori_patch1.png}
\caption{The original patch}
\label{fig:ori_patch}
\end{subfigure}
\begin{subfigure}[b]{0.34\textwidth}
\includegraphics[width=\textwidth]{fold_patch1.png}
\caption{The deformed patch}
\label{fig:def_patch}
\end{subfigure}
\caption{Patch-wise fold sculpting: the region inside the red contour is the patch selected, appropriate alternating Beltrami equation is then solved in the patch domain to obtain the desired folding effect.}
\label{fig: 3dfold}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.31\textwidth}
\includegraphics[width=\textwidth]{fold_tshirt11.png}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{fold_tshirt12.png}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{fold_tshirt13.png}
\end{subfigure}
\caption{Results of fold sculpting on the T-shirt model after appropriate smoothing: two short folds are sculpted on the right.}
\label{fig: maya_smooth2}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{fold_tshirt22.png}
\end{subfigure}
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{fold_tshirt23.png}
\end{subfigure}
\caption{Results of fold sculpting on the T-shirt model: two long folds are sculpted on the left and right.}
\label{fig: maya_smooth}
\end{figure}
The technique can also be applied after the acquisition of a folded surface using laser scans, where the folded part introduces self-occlusions and the folding is usually diminished or destroyed after applying the watertight operation. To preserve the folding details from the scans directly, we can mark the folding part that we want to preserve in the raw acquisition. By taking a patch like before and mapping it into the plane, we can solve a proper alternating Beltrami equation to obtain the desired folding effect. The folded patch can then be mapped back to the raw acquisition. To illustrate this, we have done a synthetic experiment using the above approach.
\begin{figure}
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{shirt_hole.png}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{shirt_hole_patch.png}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{shirt_hole_patch2.png}
\end{subfigure}
\caption{Patch-wise fold in-painting: from the raw acquisition of a self-occluded surface, we map the occluded, holed surface to the plane and fill the hole; we then apply a suitable folding operation to reproduce the fold that was not captured during acquisition.}
\label{fig: paint patch}
\end{figure}
We begin with an incomplete acquisition of a shirt model, such as the one on the left in Figure \ref{fig: paint fold}. As demonstrated in Figure \ref{fig: paint patch}, the patch with holes is first map to the plane and subsequently filled. A suitable folding operation is then applied to the patch to produce a plausible fold geometry given the acquisition data. The reconstruction is shown in Figure \ref{fig: paint fold} on the right.
\begin{figure}
\centering
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{shirt_hole2.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{shirt_hole1.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{shirt_hole_painted2.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{shirt_hole_painted3.png}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{shirt_hole11.png}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{shirt_hole12.png}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{shirt_hole_painted12.png}
\end{subfigure}
\caption{Results of fold in-painting: the first row and second row are two examples of the technique. For each row, the two on the left are the surface with holes from acquisition due to self occlusion. The results of in-painting are the two on the right. The corresponding holed and inpainted regions are highlighted inside the red boxes.}
\label{fig: paint fold}
\end{figure}
\subsection{Self-occlusion reasoning of flat-foldable surfaces and its application to restoration of folded images}
Given a single perspective of a folded surface, for example, shown in figure \ref{fig:experiemnt234-fold}, we can use the proposed reinforcement iteration to unfold the surface, thus enabling us to identify the self-occluded region in the unfolded domains, shown in Figure \ref{fig:mask_regions}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{example1_fold00.png}
\caption{"Mountain"}
\label{fig:onefold_a}
\end{subfigure}
\begin{subfigure}[b]{0.27\textwidth}
\includegraphics[width=\textwidth]{fold_image00.png}
\caption{"The Sower"}
\label{fig:twofold_a}
\end{subfigure}
\begin{subfigure}[b]{0.18\textwidth}
\includegraphics[width=\textwidth]{cusp_fold_texture2.png}
\caption{"Building"}
\label{fig:3fold-a}
\end{subfigure}
\caption{1-fold, 2-fold and cusp-fold examples}\label{fig:experiemnt234-fold}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{example1_masking.jpg}
\caption{1-fold}
\label{fig:ex1_mask}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{sower_sq_with_mask.png}
\caption{2-fold}
\label{fig:ex2_mask}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{building_masked.png}
\caption{cusp-fold}
\label{fig:ex3_mask}
\end{subfigure}
\caption{Occluded regions of various folded paper examples}\label{fig:mask_regions}
\end{figure}
Given a folded image, the task of restoring the image involves unfolding the image and in-painting the missing parts beneath the folded region. The performance of the final restoration results obviously depends on the realization of the texture synthesis. However, it is worth noting that the unfolding result may also drastically affects the in-painting result since many in-painting methods such as the diffusion-based \cite{bertalmio2000image, tschumperle2005vector}, exemplar-based \cite{daisy2013fast, barnes2009patchmatch, barnes2010generalized} assume the full knowledge of the computational domain (i.e., the image domain). To alleviate the difficulty arose from the incomplete knowledge of the image domain, we can employ the proposed unfolding algorithm to retrieve the geometric information of the folded subdomain with the given partial geometric information. Once we restore the intrinsic image domain consistent with the partial geometric information, well-developed in-painting techniques can then be employed correctly and provide satisfactory in-painting results.
\begin{figure}
\centering
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{inpaint_ex1_iter_unfold.png}
\caption{Ground truth}
\label{fig:onefoldresult_a}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{inpaint_ex1_true_unfold.png}
\caption{Unfolded mesh}
\label{fig:onefoldresult_b}
\end{subfigure}
\begin{subfigure}[b]{0.20\textwidth}
\includegraphics[width=\textwidth]{inpaint_ex1_fold.png}
\caption{Parametrized}
\label{fig:onefold_b}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{example1_result00.png}
\caption{Restored result}
\label{fig:onefoldresult_c}
\end{subfigure}
\caption{Unfolding and restoring result of a 1-folded image.}\label{fig:mountain_result}
\end{figure}
Figure \ref{fig:mountain_result} shows the result for the 1-folded example. The unfolding is trivial in a sense, but here we want to use it to illustrate the typical procedure of a folded-image restoration. The same procedure applies to all kinds of folds alike. In any case, we assume to have the partial boundary and singular set data (i.e., the folding edges and the boundaries) available from the folded images. Our goal is to recover the folded image \ref{fig:onefold_a} by using the proposed unfolding technique and some well-established in-painting algorithms. At the beginning of our algorithm, an initialization $\Sigma_0$ is constructed by simply using the partial boundary and singular set data. By the proposed Algorithm 1, we can successfully reconstruct the folding map based on the reinforcement iterations, which are shown in \ref{fig:onefoldresult_b}. With the folding map, we can obtain the occluded region and carry out the in-painting procedure. Notice that the unfolded mesh in \ref{fig:onefoldresult_b} highly resembles to the ground truth (see \ref{fig:onefoldresult_a}). With such unfolded domain, we can acurately approximate the masked region and apply the patch-matching based in-painting algorithm to recover the image. The corresponding parametrized folding surfaces are show in figure \ref{fig:onefold_b}. The image can then be mapped from the folded surface back to the domain complementary to the occluded region. Here, as is common the case, due to the size of masked region generated from the fold of the image, we choose the patch-matching based algorithm for image in-painting. In particular, we employ the algorithm\footnotemark proposed by Daisy et. al. \cite{daisy2013fast} in this example. The result is shown in \ref{fig:onefoldresult_c}. For comparison, the original image together with the overlapping mask (drawn as a half-transparent domain) is shown in Figure \ref{fig:ex1_mask}.
\footnotetext{The algorithm is available as a plugin for the open source GIMP2 software. The software is available at \url{https://www.gimp.org/}.}
To illustrate the adaptability of our proposed algorithm in some more complicated folded surface, we now consider correspondingly, a 2-, 3-folded examples, as shown in Figure \ref{fig:experiemnt234-fold}. There is a 2-folded painting ``The Sower" by Vincent van Gogh, and a cusp-folded ``building" image.
Similarly, to approximate the original images with the given folded data, we first have to unfold these surfaces using the proposed algorithm. Unlike the trivial 1-fold example illustrated above (which may be simply get unfolded even without the use of Algorithm 1), the folding order also takes part in this unfolding problem since different orders of folding produce different folded images. When the folding number is large, obtaining the ordering of the folds from the given data is difficult. However, using the Algorithm 1, this folding order can be obtained implicitly. In other words, the unfolding procedure using Algorithm 1 is fully automatic. Figure \ref{fig:234fold-result} shows the unfolding and the corresponding in-painting results. The leftmost column shows the folded meshes corresponding to some unidentified rectangular meshes. With only partial boundary conditions and singular set data, unfolding these surfaces are highly ill-posed. But using our proposed algorithm, we successfully obtained the unfolding surfaces (the middle-left column). Notice that by regularizing the generalized Beltrami coefficient, Algorithm 1 converges to unfolded regular meshes, where unnatural curvy edges are not presented. With these unfolded meshes, we can recover the occluded regions due to the foldings (See the middle-right column) and therefore in-painting algorithms can be employed as usual. The overall recovered images are shown in the rightmost column of Figure \ref{fig:234fold-result}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{two_fold_mesh.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{two_fold_unfold_mesh.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{sower_sq_with_mask.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{inpaint_result00.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{example4_foldedmesh.png}
\end{subfigure}
\begin{subfigure}[b]{0.21\textwidth}
\includegraphics[width=\textwidth]{example4_unfoldedmesh.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{building_masked.png}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{example4_result00.png}
\end{subfigure}
\caption{Unfolding and in-painting result for the 2-fold (first row) and cusp-fold (second row) examples. Leftmost column shows the folded meshes representing the domain of the images. The unfolded results are shown in the middle-left column and the recovered occluded domains are shown in the middle-right columns. The overall recovery results are shown in the rightmost column.}
\label{fig:234fold-result}
\end{figure}
\section{Discussion and Conclusion}
We have proposed a novel way of studying and modeling the folding phenomena of surfaces using alternating Beltrami equations. The numerical scheme is proposed to overcome the known issues of the previous method, by taking into account of the coupled nature of the two coordinate functions of the solution. The resulting method works for fewer constraints, and has a nice geometric interpretation. More importantly, it allows us to formulate and solve the inverse problem of inferring and parametrizing flat-foldable surfaces with observed partial data. We have proposed to use the ``reinforcement iteration" algorithm in order to solve the associated optimization problem, which has shown empirical convergence over various examples. Various applications are given, including fold sculpting, fold-like texture generation, generating and editing generalized Miura-ori patterns, as well as self-occlusion reasoning. Many more possible applications in manufacturing, animation and modeling shall be explored in the future. At the same time, the understanding of non-rigid folding is still largely incomplete and many more interesting examples and applications are waiting to be discovered.
\section*{Acknowledgments}
The first author would like to thank Mr. Leung Liu Yusan and Dr. Emil Saucan for some useful help and discussions in the early stage of this work. The examples meshes are generated by the software Triangle \cite{shewchuk1996triangle}. This work is supported by HKRGC GRF (Project ID: 2130549).
\bibliographystyle{siamplain}
| {'timestamp': '2019-01-09T02:04:49', 'yymm': '1804', 'arxiv_id': '1804.03936', 'language': 'en', 'url': 'https://arxiv.org/abs/1804.03936'} |
\section{Introduction}
Based on ideas of Misra \cite{mis74}, it was essentially the late
S. Bugajski who recognized that the statistical (probabilistic) framework
of quantum mechanics can be understood as a reduced classical
probability theory on the projective Hilbert space $ \ph $
\cite{bug91;93a-d,bel95a;b,hol82,stu01,bus04}. In a forthcoming paper
\cite{bus07} a suggestive definition of a classical extension of
quantum mechanics is given and it is proved that every such classical
extension is essentially equivalent to the Misra-Bugajski scheme. Moreover,
it is known that $ \ph $, considered as a real differentiable manifold,
carries a Riemannian as well as a symplectic structure,
the latter enabling one to reformulate quantum dynamics
on the (in general infinite-dimensional) phase space $ \ph $
\cite{gue77,kib79,cir84,cir90a;b,bro01,bje05}. Thus, quantum mechanics
can be considered as some reduced classical statistical mechanics.
In order to take the projective Hilbert space $ \ph $ as a sample space
for classical probability theory, $ \ph $ must be equipped
with a measurable structure. To make $ \ph $ a differentiable manifold,
it should be equipped with a topology first. The elements of $ \ph $ can
be interpreted as equivalence classes of vectors $ \varphi \in \hi $,
$ \varphi \neq 0 $, as equivalence classes of unit vectors, as the
one-dimensional subspaces of $ \hi $, or as the one-dimensional
orthogonal projections acting in $ \hi $. So, on the one hand,
two qoutient topologies can be defined on $ \ph $ whereas,
on the other hand, all the different operator topologies induce
topologies on $ \ph $. In Section 2 we undertake a systematic review
and comparison, already sketched out in \cite{bug94}, of the various
topologies on $ \ph $ and show that all the topologies are the same,
thus giving a natural topology $ \mathcal{T} $ on $ \ph $. Furthermore,
in Section 3 we present a simple proof that the Borel structure
generated by the topology $ \mathcal{T} $ coincides with the measurable structure
generated by the transition probability functions of the pure
quantum states; our proof simplifies a proof of Misra from 1974 for
a corresponding statement \cite{mis74}.
\section{The Topology of the Projective Hilbert Space}\label{sec:top}
Let $ \hi \neq \{ 0 \} $ be a nontrivial separable complex
Hilbert space. Call two vectors of $ \hi^* := \hi \setminus \{0\} $
equivalent if they differ by a complex factor, and define the
{\it projective Hilbert space $ \ph $} to be the set of the corresponding
equivalence classes which are often called {\it rays}. Instead of
$ \hi^* $ one can consider only the unit sphere of $ \hi $,
$ S := \{\varphi \in \hi \, | \, \no{\varphi} = 1 \} $. Then two unit vectors
are called equivalent if they differ by a phase factor, and the set of
the corresponding equivalence classes, i.e., the set of the {\it
unit rays}, is denoted by $ S/S^1 $ (in this context, $ S^1 $ is
understood as the set of all phase factors, i.e., as the set of all
complex numbers of modulus $1$). Clearly, $ S/S^1 $ can be
identified with the projective Hilbert space $ \ph $. Furthermore,
we can consider the elements of $ \ph $ also as the one-dimensional
subspaces of $ \hi $ or, equivalently, as the one-dimensional
orthogonal projections $ P = P_{\varphi} = \kb{\varphi}{\varphi} $, $
\no{\varphi} = 1 $.
The set $ \hi^* $ and the unit sphere $S$ carry the topologies induced by
the metric topology of $ \hi $. Using the canonical projections
$ \mu \! : \hi^* \to \ph $, $ \mu(\varphi) := [\varphi] $, and
$ \nu \! : S \to S/S^1 $, $ \nu(\chi) := [\chi]_S $, where
$ [\varphi] $ is a ray and $ [\chi]_S $ a unit ray, we can equip
the quotient sets $ \ph $ and $ S/S^1 $ with their quotient topologies
$ \mathcal{T}_{\mu} $ and $ \mathcal{T}_{\nu} $. Considering $ \mathcal{T}_{\nu} $, a set
$ O \subseteq S/S^1 $ is called open if $ \nu^{-1}(O) $ is open.
\begin{theorem}\label{thm:ss1}
The set $ S/S^1 $, equipped with the quotient topology $ \mathcal{T}_{\nu} $,
is a second-countable Hausdorff space, and $ \nu $ is an open continuous
mapping.
\end{theorem}
\proof{
By definition of $ \mathcal{T}_{\nu} $, $ \nu $ is continuous. To show that
$ \nu $ is open, let $U$ be an open set of $S$. From
\[
\nu^{-1}(\nu(U)) = \nu^{-1}(\{ [\chi]_S \, | \, \chi \in U \})
= \bigcup_{\lambda \in S^1} \lambda U ,
\]
$ S^1 = \{ \lambda \in \C \, | \, |\lambda| = 1 \} $, it follows that
$ \nu^{-1}(\nu(U)) \subseteq S $ is open. So $ \nu(U) \subseteq S/S^1 $
is open; hence, $ \nu $ is open.
Next consider two different unit rays $ [\varphi]_S $ and $ [\psi]_S $ where
$ \varphi,\psi \in S $ and $ |\ip{\varphi}{\psi}| = 1 - \varepsilon $,
$ 0 < \varepsilon \leq 1 $. Since the mapping
$ \chi \mapsto |\ip{\varphi}{\chi}| $, $ \chi \in S $, is continuous,
the sets
\begin{equation}\label{u1}
U_1 := \left\{ \chi \in S \left| \,
|\ip{\varphi}{\chi}| > 1 - \tfrac{\varepsilon}{2} \right.
\right\}
\end{equation}
and
\begin{equation}\label{u2}
U_2 := \left\{ \chi \in S \left| \,
|\ip{\varphi}{\chi}| < 1 - \tfrac{\varepsilon}{2} \right.
\right\}
\end{equation}
are open neighborhoods of $ \varphi $ and $ \psi $, respectively. Consequently,
the sets $ O_1 := \nu(U_1) $ and $ O_2 := \nu(U_2) $ are open
neighborhoods of $ [\varphi]_S $ and $ [\psi]_S $, respectively. Assume
$ O_1 \cap O_2 \neq \emptyset $. Let $ [\xi]_S \in O_1 \cap O_2 $, then
$ [\xi]_S = \nu(\chi_1) = \nu(\chi_2) $ where $ \chi_1 \in U_1 $ and
$ \chi_2 \in U_2 $. It follows that $ \chi_1 $ and $ \chi_2 $ are equivalent,
so $ |\ip{\varphi}{\chi_1}| = |\ip{\varphi}{\chi_2}| $, in contradiction to
$ \chi_1 \in U_1 $ and $ \chi_2 \in U_2 $. Hence, $ O_1 $ and $ O_2 $
are disjoint, and $ \mathcal{T}_{\nu} $ is separating.
Finally, let $ \mB = \{ U_n \, | \, n \in \N \} $ be a countable base of
the topology of $S$ and define the open sets $ O_n := \nu(U_n) $. We show
that $ \{ O_n \, | \, n \in \N \} $ is a base of $ \mathcal{T}_{\nu} $. For
$ O \in \mathcal{T}_{\nu} $, we have that $ \nu^{-1}(O) $ is an open set of
$S$ and consequently $ \nu^{-1}(O) = \bigcup_{n \in M} U_n $ where
$ U_n \in \mB $ and $ M \subseteq \N $. Since $ \nu $ is surjective,
it follows that
\[
O = \nu(\nu^{-1}(O)) = \nu \left( \bigcup_{n \in M} U_n \right)
= \bigcup_{n \in M} \nu(U_n)
= \bigcup_{n \in M} O_n .
\]
Hence, $ \{ O_n \, | \, n \in \N \} $ is a countable base of
$ \mathcal{T}_{\nu} $. \ $\square$}
Analogously, it can be proved that the topology $ \mathcal{T}_{\mu} $ on $ \ph $
is separating and second-countable and that the canonical projection $ \mu $
is open (and continuous by the definition of $ \mathcal{T}_{\mu} $). Moreover,
one can show that the natural bijection $ \beta \! : \ph \to S/S^1 $,
$ \beta([\varphi]) := \left[ \frac{\varphi}{\no{\varphi}} \right]_S $,
$ \beta^{-1}([\chi]_S) = [\chi] $, is a homeomorphism. Thus, identifying
$ \ph $ and $ S/S^1 $ by $ \beta $, the topologies $ \mathcal{T}_{\mu} $ and
$ \mathcal{T}_{\nu} $ are the same.
We denote the real vector space of the self-adjoint trace-class operators by
$ \tsh $ and the real vector space of all bounded self-adjoint operators by
$ \bsh $; endowed with the trace norm and the usual operator norm,
respectively, these spaces are Banach spaces. As is well known,
$ \bsh $ can be considered as the dual space $ (\tsh)' $ where the duality
is given by the trace functional. Let $ \sh $ be the convex set of
all positive trace-class operators of trace $1$; the operators of
$ \sh $ are the density operators and describe the quantum states. We
recall that the extreme points of the convex set $ \sh $, i.e.,
the pure quantum states, are the one-dimensional orthogonal projections
$ P = P_{\varphi} $, $ \no{\varphi} = 1 $. We denote the set of these extreme
points, i.e., the extreme boundary, by $ \esh $.
The above definition of $ \ph $ and $ S/S^1 $ as well as of their
quotient topologies is related to a geometrical point of view. From
an operator-theoretical point of view, it is more obvious to identify
$ \ph $ with $ \esh $ and to restrict one of the various operator topologies
to $ \esh $. A further definition of a topology on $ \esh $ is suggested by
the interpretation of the one-dimensional projections $ P \in \esh $ as
the pure quantum states and by the requirement that
the transition probabilities between two pure states are
continuous functions. Next we consider, taking account of
$ \esh \subseteq \sh \subset \tsh \subseteq \bsh $, the metric topologies on
$ \esh $ induced by the trace-norm topology of $ \tsh $, resp.,
by the norm toplogy of $ \bsh$. After that we introduce the weak topology on
$ \esh $ defined by the transition-probability functions as well as
the restrictions of several weak operator topologies to $ \esh $. Finally,
we shall prove the surprising result that all the many toplogies on
$ \ph \cong S/S^1 \cong \esh $ are equivalent.
\begin{theorem}\label{thm:esh-dist}
Let $ P_{\varphi} = \kb{\varphi}{\varphi} \in \esh $ and
$ P_{\psi} = \kb{\psi}{\psi} \in \esh $ where
$ \no{\varphi} = \no{\psi} = 1 $. Then
\begin{enumerate}
\item[(a)]
\[ \rho_n(P_{\varphi},P_{\psi}) := \no{P_{\varphi} - P_{\psi}}
= \sqrt{1 - |\ip{\varphi}{\psi}|^2}
= \sqrt{1 - \tr{P_{\varphi} P_{\psi}}}
\]
where the norm $ \no{\cdot} $ is the usual operator norm
\item[(b)]
\[ \rho_{\mathrm{tr}}(P_{\varphi},P_{\psi})
:= \no{P_{\varphi} - P_{\psi}}_{\mathrm{tr}}
= 2\no{P_{\varphi} - P_{\psi}},
\]
in particular, the metrics $ \rho_n $ and $ \rho_{\mathrm{tr}} $ on
$ \esh $ induced by the operator norm $ \no{\cdot} $ and the trace
norm $ \no{\cdot}_{\mathrm{tr}} $ are equivalent
\item[(c)]
\[
\no{P_{\varphi} - P_{\psi}} \leq \no{\varphi - \psi},
\]
in particular, the mapping $ \varphi \mapsto P_{\varphi} $ from $S$ into
$ \esh $ is continuous, $ \esh $ being equipped with $ \rho_n $ or
$ \rho_{\mathrm{tr}} $.
\end{enumerate}
\end{theorem}
\proof{
To prove (a) and (b), assume $ P_{\varphi} \neq P_{\psi} $, otherwise
the statements are trivial. Then the range of $ P_{\varphi} - P_{\psi} $
is a two-dimensional subspace of $ \hi $ and is spanned by the two linearly
independent unit vectors $ \varphi $ and $ \psi $. Since eigenvectors of
$P_{\varphi} - P_{\psi} $ belonging to eigenvalues $ \lambda \neq 0 $
must lie in the range of $ P_{\varphi} - P_{\psi} $, they can be written as
$ \chi = \alpha \varphi + \beta \psi $. Therefore, the eigenvalue problem
$ (P_{\varphi} - P_{\psi}) \chi = \lambda \chi $, $ \chi \neq 0 $,
is equivalent to the two linear equations
\begin{eqnarray*}
(1- \lambda) \alpha + \ip{\varphi}{\psi} \beta & = & 0 \\
-\ip{\psi}{\varphi} \alpha - (1 + \lambda) \beta & = & 0
\end{eqnarray*}
where $ \alpha \neq 0 $ or $ \beta \neq 0 $. It follows that
$ \lambda = \pm \sqrt{1 - |\ip{\varphi}{\psi}|^2} =: \lambda_{1,2} $. Hence,
$ P_{\varphi} - P_{\psi} $ has the eigenvalues
$ \lambda_1 $, $0$, and $ \lambda_2 $. Now, from
$ \no{P_{\varphi} - P_{\psi}} = \max \{ |\lambda_1|,|\lambda_2| \} $ and
$ \no{P_{\varphi} - P_{\psi}}_{\mathrm{tr}} = |\lambda_1| + |\lambda_2| $,
we obtain the statements (a) and (b).---From
\begin{eqnarray*}
\no{P_{\varphi} - P_{\psi}}^2
& = & 1 - |\ip{\varphi}{\psi}|^2
= \no{\varphi - \ip{\psi}{\varphi} \psi}^2
= \no{(I - P_{\psi}) \varphi}^2 \\
& \leq & \no{(I - P_{\psi}) \varphi}^2 + \no{\psi - P_{\psi} \varphi}^2 \\
& = & \no{(I - P_{\psi}) \varphi - (\psi - P_{\psi} \varphi)}^2 \\
& = & \no{\varphi - \psi}^2
\end{eqnarray*}
we conclude statement (c). \ $\square$}
According to statement (b) of Theorem \ref{thm:esh-dist}, the metrics $ \rho_n $ and
$ \rho_{\mathrm{tr}} $ give rise to the same topology
$ {\mathcal{T}}_n = {\mathcal{T}}_{\mathrm{tr}} $ as well as to the same uniform structures.
\begin{theorem}\label{thm:esh-sepc}
Equipped with either of the two metrics $ \rho_n $ and
$ \rho_{\mathrm{tr}} $, $ \esh $ is separable and complete.
\end{theorem}
\proof{
As a metric subspace of the separable Hilbert space $ \hi $, the unit sphere
$S$ is separable. Therefore, by statement (c) of Theorem \ref{thm:esh-dist}, the metric space
$ (\esh,\rho_n) $ is separable and so is $ (\esh,\rho_{\mathrm{tr}}) $
(the latter, moreover, implies the trace-norm separability of
$ \tsh $). Now let $ \{ P_n \}_{n \in \N} $ be a Cauchy sequence in
$ (\esh,\rho_{\mathrm{tr}}) $. Then there exists an operator
$ A \in \tsh $ such that $ \no{P_n - A}_{\mathrm{tr}} \to 0 $ as well as
$\no{P_n - A} \to 0 $ as $ n \to \infty $ (remember that, on $ \tsh $,
$ \no{\cdot}_{\mathrm{tr}} $ is stronger than $ \no{\cdot} $). From
\begin{eqnarray*}
\no{P_n - A^2} = \no{A^2 - P_n^2}
& \leq & \no{A^2 - AP_n} + \no{AP_n - P_n^2} \\
& \leq & \no{A} \no{A - P_n} + \no{A - P_n} \\
& \to & 0
\end{eqnarray*}
as $ n \to \infty $ we obtain $ A = \lim_{n \to \infty} P_n = A^2 $;
moreover,
\[
\tr{A} = \tr{AI} = \lim_{n \to \infty} \tr{P_nI} = 1.
\]
Hence, $A$ is a one-dimensional orthogonal projection, i.e.,
$ A \in \esh $. \ $\square$}
Next we equip $ \esh $ with the topology $ \mathcal{T}_0 $ generated by
the functions
\begin{equation}\label{p}
P \mapsto h_Q(P) := \tr{PQ} = |\ip{\varphi}{\psi}|^2
\end{equation}
where $ P = \kb{\psi}{\psi} \in \esh $, $ Q = \kb{\varphi}{\varphi} \in \esh $,
and $ \no{\psi} = \no{\varphi} = 1 $. That is, $ \mathcal{T}_0 $ is the coarsest
topology on $ \esh $ such that all the real-valued functions $ h_Q $
are continuous. Note that $ \tr{PQ} = |\ip{\varphi}{\psi}|^2 $ can be
interpreted as the transition probability between the two pure states
$P$ and $Q$.
\begin{lemma}\label{lem:esh}
The set $ \esh $, equipped with the topology $ \mathcal{T}_0 $, is
a second-countable Hausdorff space. A countable base of $ \mathcal{T}_0 $
is given by the finite intersections of the open sets
\begin{equation}\label{uklm}
\begin{array}{crl}
U_{klm} & := & h_{Q_k}^{-1} \left( \,
\left] q_l - \frac{1}{m},q_l + \frac{1}{m}
\right[ \, \right) \vspace{2mm}\\
& = & \left\{ P \in \esh \left| \,
\left| \tr{PQ_k} - q_l \right| < \frac{1}{m}
\right. \right\}
\end{array}
\end{equation}
where $ \{ Q_k \}_{k \in \N} $ is a sequence of one-dimensional orthogonal
projections being $ \rho_n $-dense in $ \esh $, $ \{ q_l \}_{l \in \N} $
is a sequence of numbers being dense in $ [0,1] \subseteq \R $, and
$ m \in \N $.
\end{lemma}
\proof{
Let $ P_1 $ and $ P_2 $ be any two different one-dimensional
projections. Choosing $ Q = P_1 $ in (\ref{p}), we obtain
$ h_{P_1}(P_1) = 1 \neq h_{P_1}(P_2) = 1 - \varepsilon $,
$ 0 < \varepsilon \leq 1 $. The sets
\[
U_1 := \left\{ P \in \esh \left| \,
h_{P_1}(P) > 1 - \tfrac{\varepsilon}{2} \right. \right\}
\]
and
\[
U_2 := \left\{ P \in \esh \left| \,
h_{P_1}(P) < 1 - \tfrac{\varepsilon}{2} \right. \right\}
\]
(cf.\ Eqs.~(\ref{u1}) and (\ref{u2})) are disjoint open neighborhoods of $ P_1 $ and
$ P_2 $, respectively. So $ \mathcal{T}_0 $ is separating.
For an open set $ O \subseteq \R $, $ h_Q^{-1}(O) $ is $ \mathcal{T}_0 $-open. We
next prove that
\begin{equation}\label{u}
U := h_Q^{-1}(O) = \bigcup_{U_{klm} \subseteq U} U_{klm}
\end{equation}
with $ U_{klm} $ according to (\ref{uklm}). Let $ P \in U $. Then
there exists an $ \varepsilon > 0 $ such that the interval
$ ]h_Q(P) - \varepsilon,h_Q(P) + \varepsilon[ $ is contained in $O$. Choose
$ m_0 \in \N $ such that $ \frac{1}{m_0} < \frac{\varepsilon}{2} $,
and choose a member $ q_{l_0} $ of the sequence $ \{ q_l \}_{l \in \N} $
and a member $ Q_{k_0} $ of $ \{ Q_k \}_{k \in \N} $ such that
$ |\tr{PQ} - q_{l_0}| < \frac{1}{2m_0} $ and
$ \no{Q_{k_0} - Q} < \frac{1}{2m_0} $. It follows that
\begin{eqnarray*}
|\tr{PQ_{k_0}} - q_{l_0}|
& \leq & |\tr{PQ_{k_0}} - \tr{PQ}| + |\tr{PQ} - q_{l_0}| \\
& \leq & \no{Q_{k_0} - Q} + |\tr{PQ} - q_{l_0}| \\
& < & \tfrac{1}{m_0}
\end{eqnarray*}
which, by (\ref{uklm}), means that $ P \in U_{k_0l_0m_0} $. We further have
to show that $ U_{k_0l_0m_0} \subseteq U $. To that end, let
$ \widetilde{P} \in U_{k_0l_0m_0} $. Then, from
\[
\bigl| \tr{\widetilde{P}Q} - \tr{PQ} \bigr| \leq
\bigl| \tr{\widetilde{P}Q} - \tr{\widetilde{P}Q_{k_0}} \bigr|
+ \bigl| \tr{\widetilde{P}Q_{k_0}} - q_{l_0} \bigr| + |q_{l_0} - \tr{PQ}|
\]
where the first term on the right-hand side is again smaller than
$ \no{Q - Q_{k_0}} $ and, by (\ref{uklm}), the second term is smaller than
$ \frac{1}{m_0} $, it follows that
\[
\bigl| h_Q(\widetilde{P}) - h_Q(P) \bigr|
= \bigl| \tr{\widetilde{P}Q} - \tr{PQ} \bigr|
\leq \tfrac{1}{2m_0} + \tfrac{1}{m_0} + \tfrac{1}{2m_0}
= \tfrac{2}{m_0}
< \varepsilon.
\]
This implies that $ h_Q(\widetilde{P}) \in
{]h_Q(P) - \varepsilon,h_Q(P) + \varepsilon[} \subseteq O $, i.e.,
$ \widetilde{P} \in h_Q^{-1}(O) = U $. Hence,
$ U_{k_0l_0m_0} \subseteq U $.
Summarizing, we have shown that, for $ P \in U $,
$ P \in U_{k_0l_0m_0} \subseteq U $. Hence,
$ U \subseteq \bigcup_{U_{klm} \subseteq U} U_{klm} \subseteq U $,
and assertion (\ref{u}) has been proved. The finite intersections of sets
of the form $ U = h_Q^{-1}(O) $ constitute a basis of the topology
$ \mathcal{T}_0 $. Since every set $ U = h_Q^{-1}(O) $ is the union of sets
$ U_{klm} $, the intersections of finitely many sets
$ U = h_Q^{-1}(O) $ is the union of finite intersections of sets
$ U_{klm} $. Thus, the finite intersections of the sets
$ U_{klm} $ constitute a countable base of $ \mathcal{T}_0 $. \ $\square$}
Later we shall see that the topological space $ (\esh,\mathcal{T}_0) $ is
homeomorphic to $ (\esh,\mathcal{T}_n) $ as well as to $ (S/S^1,\mathcal{T}_{\nu})
$. So it is also clear by Theorem \ref{thm:esh-sepc} or Theorem
\ref{thm:ss1} that $ (\esh,\mathcal{T}_0) $ is a second-countable Hausdorff
space. The reason for stating Lemma \ref{lem:esh} is that later we
shall make explicit use of the particular countable base given
there.
The weak operator topology on the space $ \bsh $ of the bounded
self-adjoint operators on $ \hi $ is the coarsest topology such that
the linear functionals
\[
A \mapsto \ip{\varphi}{A\psi}
\]
where $ A \in \bsh $ and $ \varphi,\psi \in \hi $, are continuous. It is
sufficient to consider only the functionals
\begin{equation}\label{afun}
A \mapsto \ip{\varphi}{A\varphi}
\end{equation}
where $ \varphi \in \hi $ and $ \no{\varphi} = 1 $. The topology $ \mathcal{T}_w $
induced on $ \esh \subset \bsh $ by the weak operator topology is the
coarsest topology on $ \esh $ such that the restrictions of the linear
functionals (\ref{afun}) to $ \esh $ are continuous. Since these restrictions
are given by
\[
P \mapsto \ip{\varphi}{P\varphi} = \tr{PQ} = h_Q(P)
\]
where $ P \in \esh $ and $ Q := \kb{\varphi}{\varphi} \in \esh $, the topology
$ \mathcal{T}_w $ on $ \esh $ is, according to (\ref{p}), just our topology $ \mathcal{T}_0 $.
Now we compare the weak topology $ \mathcal{T}_0 $ with the metric topology
$ \mathcal{T}_n $.
\begin{theorem}\label{thm:top-esh}
The weak topology $ \mathcal{T}_0 $ on $ \esh $ and the metric topology $ \mathcal{T}_n $ on
$ \esh $ are equal.
\end{theorem}
\proof{
According to (\ref{p}), a neighborhood base of $ P \in \esh $ w.r.t.\ $ \mathcal{T}_0 $
is given by the open sets
\begin{equation}\label{up}
\begin{array}{c}
U(P;Q_1,\ldots,Q_n;\varepsilon) \hspace{7.5cm} \vspace{2mm}\\
\hspace{0.8cm}
\begin{array}{crl}
& := & \displaystyle{\bigcap_{i=1}^n h_{Q_i}^{-1}
( \, ]h_{Q_i}(P) - \varepsilon,h_{Q_i}(P) + \varepsilon[ \, )}
\vspace{2mm}\\
& = & \bigl\{ \widetilde{P} \in \esh \, \bigl| \, \bigl|
h_{Q_i}(\widetilde{P}) - h_{Q_i}(P) \bigr| < \varepsilon \
{\rm for} \ i=1,\ldots,n \bigr\} \vspace{2mm}\\
& = & \bigl\{ \widetilde{P} \in \esh \, \bigl| \, \bigl|
\tr{\widetilde{P}Q_i} - \tr{PQ_i} \bigr| < \varepsilon \
{\rm for} \ i=1,\ldots,n \bigr\}
\end{array}
\end{array}
\end{equation}
where $ Q_1,\dots,Q_n \in \esh $ and $ \varepsilon > 0 $;
a neighborhood base of $P$ w.r.t.\ $ \mathcal{T}_n $ is given by the open balls
\begin{equation}\label{kep}
K_{\varepsilon}(P) := \bigl\{ \widetilde{P} \in \esh \, \bigl| \,
\bigl\| \widetilde{P} - P \bigr\| < \varepsilon
\bigr\}.
\end{equation}
If $ \bigl\| \widetilde{P} - P \bigl\| < \varepsilon $, then
\[
\bigl| \tr{\widetilde{P}Q_i} - \tr{PQ_i} \bigr|
= \bigl| \tr{Q_i(\widetilde{P} - P)} \bigr|
\leq \no{Q_i}_{\mathrm{tr}} \bigl\| \widetilde{P} - P \bigr\|
= \bigl\| \widetilde{P} - P \bigr\|
< \varepsilon;
\]
hence, $ K_{\varepsilon}(P) \subseteq U(P;Q_1,\ldots,Q_n;\varepsilon) $. To
show some converse inclusion, take account of Theorem \ref{thm:esh-dist}, part (a), and note
that
\[
\bigl\| \widetilde{P} - P \bigr\|^2
= 1 - \tr{\widetilde{P}P}
= \bigl| \tr{\widetilde{P}P} - \tr{PP} \bigr|.
\]
In consequence, by (\ref{up}) and (\ref{kep}),
$ U(P;P;\varepsilon^2) = K_{\varepsilon}(P) $. Hence,
$ \mathcal{T}_0 = \mathcal{T}_n $. \ $\square$}
It looks surprising that the topolgies $ \mathcal{T}_0 $ and
$ \mathcal{T}_n $ coincide. In fact, consider the sequence
$ \{ P_{\varphi_n} \}_{n \in \N} $ where the vectors
$ \varphi_n \in \hi $ constitute an orthonormal system. Then,
w.r.t.\ the weak operator topology, $ P_{\varphi_n} \to 0 $ as
$ n \to \infty $ whereas $ \no{P_{\varphi_n} - P_{\varphi_{n+1}}} =1 $
for all $ n \in \N $. However, $ 0 \not\in \esh $; so
$ \{ P_{\varphi_n} \}_{n \in \N} $ is convergent neither w.r.t.\
$ \mathcal{T}_w = \mathcal{T}_0 $ nor w.r.t.\ $ \mathcal{T}_n $. Finally, like in the case
of the weak operator topology, there is a natural uniform structure
inducing $ \mathcal{T}_0 $. The uniform structures that are canonically related to
$ \mathcal{T}_0 $ and $ \mathcal{T}_n $ are different: $ \{ P_{\varphi_n} \}_{n \in \N} $ is
a Cauchy sequence w.r.t.\ the uniform structure belonging to $ \mathcal{T}_0 $
but not w.r.t.\ that belonging to $ \mathcal{T}_n $, i.e., w.r.t.\ the metric
$ \rho_n $.
We remark that besides $ \mathcal{T}_0 $ and $ \mathcal{T}_w $ several further
weak topologies can be defined on $ \esh $. Let $ \csh $ be the Banach space of the compact
self-adjoint operators and remember that $ (\csh)' = \tsh $. So the weak
Banach-space topologies of $ \csh $, $ \tsh $, and $ \bsh $ as well as
the weak-* Banach-space topologies of $ \tsh $ and $ \bsh $ can
be restricted to $ \esh $, thus giving the topologies
$ \mathcal{T}_1 := \sigma(\csh,\tsh) \cap \esh $,
$ \mathcal{T}_2 := \sigma(\tsh,\csh) \cap \esh $,
$ \mathcal{T}_3 := \sigma(\tsh,\bsh) \cap \esh $,
$ \mathcal{T}_4 := \sigma(\bsh,\tsh) \cap \esh $, and
$ \mathcal{T}_5 := \sigma(\bsh,(\bsh)') \cap \esh $. Moreover,
the strong operator topology induces a topology $ \mathcal{T}_s $ on
$ \esh $. From the obvious inclusions
\[
\mathcal{T}_w \subseteq \mathcal{T}_1 \subseteq \mathcal{T}_2 \subseteq \mathcal{T}_3
\subseteq \mathcal{T}_{\mathrm{tr}} ,
\]
\[
\mathcal{T}_1 = \mathcal{T}_4 \subseteq \mathcal{T}_5 = \mathcal{T}_1 ,
\]
and
\[
\mathcal{T}_w \subseteq \mathcal{T}_s \subseteq \mathcal{T}_n
\]
as well as from the shown equality
\[
\mathcal{T}_0 = \mathcal{T}_w = \mathcal{T}_n = \mathcal{T}_{\mathrm{tr}}
\]
it follows that the topologies $ \mathcal{T}_1,\ldots,\mathcal{T}_5 $ and $ \mathcal{T}_s $
also coincide with $ \mathcal{T}_0 $.
Finally, we show that all the topologies on $ \esh $ are equivalent to
the quotient topologies $ \mathcal{T}_{\mu} $ and $ \mathcal{T}_{\nu} $ on $ \ph $, resp.,
$ S/S^1 $.
\begin{theorem}\label{thm:ss1-esh}
The mapping $ F \! : S/S^1 \to \esh $, $ F([\varphi]_S := P_{\varphi} $ where
$ \varphi \in S $, is a homeomorphism between the topological spaces
$ (S/S^1,\mathcal{T}_{\nu}) $ and $ (\esh,\mathcal{T}_0) $.
\end{theorem}
\proof{
The mapping $F$ is bijective. The map
$ h_Q \circ F \circ \nu \! : S \to \R $
where $ h_Q $ is any of the functions given by Eq.~(\ref{p}) and
$ \nu $ is the canonical projection from $S$ onto $ S/S^1 $,
reads explicitly
\[
(h_Q \circ F \circ \nu)(\varphi) = h_Q(F([\varphi]_S)) = h_Q(P_{\varphi})
= \tr{P_{\varphi}Q} = \ip{\varphi}{Q\varphi};
\]
therefore, $ h_Q \circ F \circ \nu $ is continuous. Consequently,
for an open set $ O \subseteq \R $,
\[
(h_Q \circ F \circ \nu)^{-1}(O) = \nu^{-1}(F^{-1}(h_Q^{-1}(O)))
\]
is an open set of $S$. By the definition of the quotient topology
$ \mathcal{T}_{\nu} $, it follows that $ F^{-1}(h_Q^{-1}(O)) $ is an open set of
$ S/S^1 $. Since the sets $ h_Q^{-1}(O) $, $ Q \in \esh $,
$ O \subseteq \R $ open, generate the weak topology $ \mathcal{T}_0 $,
$ F^{-1}(U) $ is open for any open set $ U \in \mathcal{T}_0 $. Hence,
$F$ is continuous.
To show that $F$ is an open mapping, let $ V \in \mathcal{T}_{\nu} $ be an open
subset of $ S/S^1 $ and let $ [\varphi_0]_S \in V $. Since the canonical
projection $ \nu $ is continuous, there exists an $ \varepsilon > 0 $
such that
\begin{equation}\label{nuke}
\nu(K_{\varepsilon}(\varphi_0) \cap S) \subseteq V
\end{equation}
where $ K_{\varepsilon}(\varphi_0) := \{ \varphi \in \hi \, | \,
\no{\varphi - \varphi_0} < \varepsilon \} $. Without loss of generality
we assume that $ \varepsilon < 1 $.
The topology $ \mathcal{T}_0 $ is generated by the functions $ h_Q $
according to (\ref{p}); $ \mathcal{T}_0 $ is also generated by the functions
$ P \mapsto g_Q(P) := \sqrt{h_Q(P)} = \sqrt{\tr{PQ}} $. In consequence,
the set
\[
U_{\varepsilon}
:= g_Q^{-1} \left( \, \left] 1 - \tfrac{\varepsilon}{2},
1 + \tfrac{\varepsilon}{2} \right[ \, \right)
\cap h_Q^{-1} \left( \, \left] 1 - \tfrac{\varepsilon^2}{4},
1 + \tfrac{\varepsilon^2}{4} \right[
\, \right)
\]
where $ Q := P_{\varphi_0} $ and $ \varphi_0 $ and $ \varepsilon $ are specified
in the preceding paragraph, is $ \mathcal{T}_0 $-open. Using the identity
\[
1 - |\ip{\varphi_0}{\varphi}|^2 = \no{\varphi - \ip{\varphi_0}{\varphi} \varphi_0}^2
\]
where $ \varphi \in \hi $ is also a unit vector, we obtain
\begin{eqnarray*}
U_{\varepsilon}
& = & \left\{ P_{\varphi} \in \esh \left| \,
|g_Q(P_{\varphi}) - 1| < \tfrac{\varepsilon}{2} \ {\rm and} \
|h_Q(P_{\varphi}) - 1| < \tfrac{\varepsilon^2}{4} \right. \right\} \\
& = & \left\{ P_{\varphi} \in \esh \left| \,
\bigl| |\ip{\varphi_0}{\varphi}| - 1 \bigr| < \tfrac{\varepsilon}{2} \
{\rm and} \
\bigl| |\ip{\varphi_0}{\varphi}|^2 - 1 \bigr| < \tfrac{\varepsilon^2}{4}
\right. \right\} \\
& = & \left\{ P_{\varphi} \in \esh \left| \,
\bigl| |\ip{\varphi_0}{\varphi}| - 1 \bigr| < \tfrac{\varepsilon}{2} \
{\rm and} \,
\no{\varphi - \ip{\varphi_0}{\varphi} \varphi_0} < \tfrac{\varepsilon}{2}
\right. \right\}.
\end{eqnarray*}
Now let $ P_{\varphi} \in U_{\varepsilon} $. Since $ \varepsilon < 1 $, we have
that $ \ip{\varphi}{\varphi_0} \neq 0 $. Defining the phase factor
$ \lambda := \frac{\ip{\varphi}{\varphi_0}}{|\ip{\varphi}{\varphi_0}|} $,
it follows that
\begin{eqnarray*}
\no{\lambda \varphi - \varphi_0}
& = & \no{\lambda \varphi - \lambda \ip{\varphi_0}{\varphi} \varphi_0}
+ \no{\lambda \ip{\varphi_0}{\varphi} \varphi_0 - \varphi_0} \\
& = & \no{\varphi - \ip{\varphi_0}{\varphi} \varphi_0}
+ \bigl\| |\ip{\varphi_0}{\varphi}| \varphi_0 - \varphi_0 \bigr\| \\
& < & \tfrac{\varepsilon}{2} + \tfrac{\varepsilon}{2} \\
& = & \varepsilon.
\end{eqnarray*}
That is, $ P_{\varphi} \in U_{\varepsilon} $ implies that
$ \lambda \varphi \in K_{\varepsilon}(\varphi_0) $; moreover,
$ \lambda \varphi \in K_{\varepsilon}(\varphi_0) \cap S $.
Taking the result (\ref{nuke}) into account, we conclude that,
for $ P_{\varphi} \in U_{\varepsilon} $,
$ [\varphi]_S = [\lambda \varphi]_S = \nu(\lambda \varphi) \in V $. Consequently,
$ P_{\varphi} = F([\varphi]_S) \in F(V) $. Hence,
$ U_{\varepsilon} \subseteq F(V) $. Since $ U_{\varepsilon} $
is an open neighborhood of $ P_{\varphi_0} $, $ P_{\varphi_0} $
is an interior point of $ F(V) $. So, for every $ [\varphi_0]_S \in V $,
$ F([\varphi_0]_S) = P_{\varphi_0} $ is an interior point of $ F(V) $, and
$ F(V) $ is a $ \mathcal{T}_0 $-open set. Hence, the continuous bijective map
$F$ is open and thus a homeomorphism. \ $\square$}
In the following, we identify the sets $ \ph $, $ S/S^1 $, and $ \esh $
and call the identified set the {\it projective Hilbert space
$ \ph $}. However, we preferably think about the elements of
$ \ph $ as the one-dimensional orthogonal projections $ P = P_{\varphi} $. On
$ \ph $ then the quotient topologies $ \mathcal{T}_{\mu} $, $ \mathcal{T}_{\nu} $, the
weak topologies $ \mathcal{T}_0 $, $ \mathcal{T}_w $, $ \mathcal{T}_1,\ldots,\mathcal{T}_5 $, $ \mathcal{T}_s $,
and the metric topologies $ \mathcal{T}_n $, $ \mathcal{T}_{\mathrm{tr}} $ coincide. So
we can say that $ \ph $ carries a natural topology $ \mathcal{T} $;
$ (\ph,\mathcal{T}) $ is a second-countable Hausdorff space.
For our purposes, it is suitable to represent this topology $ \mathcal{T} $ as
$ \mathcal{T}_0 $, $ \mathcal{T}_n $, or $ \mathcal{T}_{\mathrm{tr}} $. As already discussed,
the topologies $ \mathcal{T}_0 $, $ \mathcal{T}_n $, and $ \mathcal{T}_{\mathrm{tr}} $ are
canonically related to uniform structures. With respect to the uniform
structure inducing $ \mathcal{T}_0 $, $ \ph $ is not complete. The uniform
structures related to $ \mathcal{T}_n $ and $ \mathcal{T}_{\mathrm{tr}} $ are the same
since they are induced by the equivalent metrics $ \rho_n $ and
$ \rho_{\mathrm{tr}} $; $ (\ph,\rho_n) $ and $ (\ph,\rho_{\mathrm{tr}}) $
are separable complete metric spaces. So $ \mathcal{T} $ can be defined by
a complete separable metric, i.e., $ (\ph,\mathcal{T}) $ is a polish space.
\section{The Measurable Structure of $\ph$}\label{sec:meas}
It is almost natural to define a measurable structure on the
projective Hilbert space $ \ph $ by the $ \sigma $-algebra $ \Xi =
\Xi(\mathcal{T}) $ generated by the $ \mathcal{T} $-open sets, i.e., $ \Xi $ is the
smallest $ \sigma $-algebra containing the open sets of the natural
topology $ \mathcal{T} $. In this way $ (\ph,\Xi) $ becomes a measurable
space where the elements $ B \in \Xi $ are the Borel sets of $ \ph
$. However, since the topology $ \mathcal{T} $ is generated by the
transition-probability functions $ h_Q $ according to Eq.\
(\ref{p}), it is also obvious to define the measurable structure of
$ \ph $ by the $ \sigma $-algebra $ \Sigma $ generated by the
functions $ h_Q $, i.e., $ \Sigma $ is the smallest $ \sigma
$-algebra such that all the functions $ h_Q $ are measurable. A
result due to Misra (1974) \cite[Lemma 3]{mis74} clarifies the
relation between $ \Xi $ and $ \Sigma $. Before stating that result,
we recall the following simple lemma which we shall also use later.
\begin{lemma}\label{lem:sigt-sigb}
Let $ (M,\mathcal{T}) $ be any second-countable topological space,
$ \mB \subseteq \mathcal{T} $ a countable base, and $ \Xi = \Xi(\mathcal{T}) $ the
$ \sigma $-algebra of the Borel sets of $M$. Then
$ \Xi = \Xi(\mathcal{T}) = \Xi(\mB) $ where $ \Xi(\mB) $ is the
$ \sigma $-algebra generated by $ \mB $; $ \mB $ is
a countable generator of $ \Xi $.
\end{lemma}
\proof{
Clearly, $ \Xi(\mB) \subseteq \Xi(\mathcal{T}) $. Since every open set
$ U \in \mathcal{T} $ is the countable union of sets of $ \mB $, it follows that
$ U \in \Xi(\mB) $. Therefore, $ \mathcal{T} \subseteq \Xi(\mB) $ and consequently
$ \Xi(\mathcal{T}) = \Xi(\mB) $. \ $\square$}
\begin{theorem}[{\rm Misra}]\label{thm:misra}
The $ \sigma $-algebra $ \Xi = \Xi(\mathcal{T}) $ of the Borel sets of the
projective Hilbert space $ \ph $ and the $ \sigma $-algbra $ \Sigma $
generated by the transition-probability functions $ h_Q $, $ Q \in \ph $,
are equal.
\end{theorem}
\proof{
Since $ \mathcal{T} $ is generated by the functions $ h_Q $, the latter are
continuous and consequently $ \Xi $-measurable. Since $ \Sigma $ is
the smallest $ \sigma $-algebra such that the functions $ h_Q $ are
measurable, it follows that $ \Sigma \subseteq \Xi $.
Now, by Lemma \ref{lem:esh}, $ \mathcal{T} $ is second-countable, and a countable base
$ \mB $ of $ \mathcal{T} $ is given by the finite intersections of the sets
$ U_{klm} $ according to Eq.\ (\ref{uklm}). Since $ U_{klm} \in \Sigma $,
it follows that $ \mB \subseteq \Sigma $. By Lemma \ref{lem:sigt-sigb}, we conclude that
$ \Xi = \Xi(\mB) \subseteq \Sigma $. Hence, $ \Xi = \Sigma $. \ $\square$}
We remark that our proof of Misra's theorem is much easier than
Misra's proof from 1974. The reason is that we explicitly used the
countable base $ \mB $ of $ \mathcal{T} $ consisting of $ \Sigma $-measurable sets.
Finally, consider the $ \sigma $-algebra $ \Xi_0 $ in $ \ph $ that is
generated by all $ \mathcal{T} $-continuous real-valued functions on $ \ph $, i.e.,
$ \Xi_0 $ is the $ \sigma $-algebra of the Baire sets of $ \ph $. Obviously,
$ \Sigma \subseteq \Xi_0 \subseteq \Xi $; so Theorem \ref{thm:misra} implies that
$ \Xi_0 = \Xi $. This result is, according to a general theorem,
also a consequence of the fact that the topology $ \mathcal{T} $ of
$ \ph $ is metrizable.
Summarizing, our result $ \Sigma = \Xi_0 = \Xi $ manifests that
the projective Hilbert space carries, besides its natural topology
$ \mathcal{T} $, also a very natural measurable structure $ \Xi $.
| {'timestamp': '2007-08-09T05:30:14', 'yymm': '0708', 'arxiv_id': '0708.1208', 'language': 'en', 'url': 'https://arxiv.org/abs/0708.1208'} |
\section{Supplemental Material}
\setcounter{page}{1}
\subsection{Determination of State Populations}
Our photon-count histograms are well approximated as a weighted sum of three Poissonians (a two-parameter probability mass function, because the three weights $P_2$, $P_1$, and $P_0$ must add up to unity). These three Poissonians correspond to two bright ions ($\qubitdowndown$), a single bright ion ($\qubitdownup$ and $\qubitupdown$), and two dark ions ($\qubitupup$). If we define $k_2$, $k_1$, and $k_0$ as the mean number of counts in each Poissonian (determined via independent measurements and maintained at fixed values during the following fitting procedure), we can write the probability mass function (PMF) as
\begin{equation*}
\begin{split}
\mathrm{PMF}(n;P_0,P_1)&=P_0\frac{k_0^n \exp{(-k_0)}}{n!}+P_1\frac{k_1^n \exp{(-k_1)}}{n!} \\
&+(1-P_1-P_0)\frac{k_2^n \exp{(-k_2)}}{n!}.
\end{split}
\end{equation*}
For a given dataset of identical experiments, with each repetition in the dataset labeled by $i$, we have a set of measured photon counts $\{n_i\}$. With this we define the usual log-likelihood as
\begin{equation*}
\begin{split}
l(P_0,P_1)=\sum_{i} \ln(\mathrm{PMF}(n_i;P_0,P_1)).
\end{split}
\end{equation*}
We then numerically maximize this function to find the most likely values for $P_0$, $P_1$, and $P_2=1-P_1-P_0$.
\subsection{Fidelity Distribution}
\begin{figure}[htp]
\includegraphics{fidelity_histogram.pdf}
\caption{\label{fig:fidelity_histogram} Histogram of bootstrapped Bell-state fidelities.}
\end{figure}
Figure~\ref{fig:fidelity_histogram} gives a histogram of the Bell-state fidelities obtained from the bootstrapping procedure defined in the main text.
\subsection{Gate Errors}
1-qubit rotation fidelity is limited by qubit frequency noise, laser phase noise, and laser intensity fluctuations. Qubit frequency noise results from variations in the background magnetic field; our qubit is first-order sensitive to these variations which arise in our lab from changes such as the movement of a chair or the motion of a nearby elevator. Although the incorporation of a spin echo into our gate should alleviate the influence of these frequency variations, 1-qubit rotations (of finite duration) still suffer to some extent. To further reduce these errors, we interleave calibrations of the qubit frequency within our gate experiments, performing a Ramsey frequency calibration experiment between each repetition of the gate experiment. In this way the frequency is stabilized to approximately $\pm$250~Hz, a level at which it is expected to contribute an error of only $4\times 10^{-6}$ to the 1-qubit pulses. We use a NdFeB permanent magnet to minimize fast fluctuations of the $1.07~\text{mT}$ experiment bias field at the expense of slow (correctable) thermal drifts. The remaining error is likely caused by faster phase noise, intensity noise, or fast ambient magnetic field fluctuations.
Additional errors arise due to variations in the phase of the 729 nm laser between the various 1-qubit pulses. Such phase fluctuations can occur due to instability in the laser itself, vibrations of the mirrors along the beampath, and motion of the ion or vacuum chamber. Ramsey coherence measurements performed by repeating the gate experiment [pulses (1), (3), and (5) of Fig.~\ref{fig:pulse_profiles}(b)] while extinguishing the ODF beams indicate that this error contribution is $<2\times 10^{-4}$. Here a marked improvement was observed after the installation of an electro-optic modulator in the output path of the 729 nm laser for phase-noise stabilization.
Fundamentally the fidelity is limited by metastable decay from the $D_{5/2}$ level as well as by spontaneous photon scattering from the ODF beams. These contributions are discussed in detail in Ref.~\cite{sawyer_2021}. We estimate a photon scattering error of $1.1\times 10^{-5}$ at our c.m. frequency of 2.5~MHz. We estimate the metastable decay error as follows. Assuming on average that 50\% of the population lies in the $D_{5/2}$ level until the detection laser is turned on 50~$\mu$s later (this includes the time required for the parity analysis pulse as well as some additional short delays), there is a metastable decay error of $2\times 0.5\times 50\:\mu\mathrm{s}/1.174\:\mathrm{s}=4.3\times 10^{-5}$. In addition, the ions are illuminated with the detection laser for 50~$\mu$s before we begin counting photons (this allows the intensity of the laser to stabilize). However, due to the Zeno effect, the metastable decay rate during this second interval is roughly half its unperturbed value, leading to an error of $2.1\times 10^{-5}$ (we have confirmed this reduction in decay rate via a full simulation of the Schr{\"o}dinger equation including all the relevant levels in $^{40}$Ca$^{+}$). Therefore, the total metastable decay error is approximately $6\times 10^{-5}$. A closely related source of error is metastable decay during the measurement itself. In the case of 100~$\mu$s (200~$\mu$s) measurement duration used for the data in Fig.~\ref{fig:parity_amplitude} (Fig.~\ref{fig:population_scan}), we calculate a lifetime-limited detection error of $4.3\times 10^{-5}$ ($8.6\times 10^{-5}$), again considering the reduced decay rate during illumination. These are likely overestimates, because decays that occur later during the measurement interval should only minimally impact the photon count histograms. We have chosen not to SPAM-correct our reported fidelity in order to reduce statistical complexity.
To initialize to $\qubitdowndown$, we employ optical pumping with a circularly polarized 397 nm laser beam followed by seven cycles of two-step optical pumping with the 729~nm and 854~nm laser beams exciting along $S_{1/2}$~$(m_j=-1/2)\rightarrow D_{5/2}$~$(m_j= 1/2)\rightarrow P_{3/2}$. The initial 397~nm pumping step yields an error of $\sim1\times10^{-3}$, and the following steps further reduce this error to $<10^{-6}$, which we exclude from the error budget table in the Letter.
In the Lamb-Dicke limit there is no dependence of the gate on ion temperature~\cite{sorensen_2000}. A derivation of the higher-order deviations from perfect Lamb-Dicke assumptions reveals a gate error of $\epsilon_g=(\pi^2/4) \eta^4 \bar{n}(2\bar{n}+1)$, where $\bar{n}$ represents the thermal mean number of quanta in and $\eta$ is the Lamb-Dicke parameter of the mode \cite{sorensen_2000,ballance_2016}. This dependence, which applies independently for both the BM and c.m. finite-temperature contributions, is confirmed via numerical simulations. We estimate $\bar{n}_\mathrm{BM}<0.1$, $\bar{n}_\mathrm{c.m.}<0.1$, $\eta_\mathrm{BM}=0.063$, $\eta_\mathrm{c.m.}=0.083$ so that $\epsilon_g<2\times 10^{-5}$.
Heating of the gate mode during operation is unavoidable. We operate the gate near the BM mode instead of near the c.m. mode because of the former's significantly lower $<1.4$~quanta/s heating rate (the c.m. heating rate is $33(14)$~quanta/s). This translates into a gate error of $\epsilon_g=\dot{\bar{n}} \tau_g/(2 K)=7\times 10^{-6}$ ($\tau_g$ here represents the sum of the two ODF pulse durations, and $K=2$ is the number of loops in phase space) \cite{sorensen_2000}.
Variations in the trapping potential can lead to errors via two possible mechanisms. Fast electric field variations can change the position of the trap potential within the Ramsey sequence [Fig~\ref{fig:pulse_profiles}(b)]. This manifests as phase noise both on the optical-dipole force interaction and on the 1-qubit rotations \cite{pino_2020}. For the 2~Hz low-pass filters in use on our dc electrodes, we expect this error contribution is negligible. However, it may become relevant if filters with a higher cutoff frequency are used in the future.
Slower variations in the trapping potential (for example due to thermal drift in the DAC electronics or due to slow charging or discharging of the trap surfaces) can cause variations in the trap frequency from shot to shot of the Ramsey sequence. We have characterized the trap frequency by driving ion motion with a radio-frequency electric field and detecting the ions’ response. For an 8~ms pulse, the resulting frequency response is well fit to a Gaussian with a standard deviation of $\Delta \omega/2\pi=63(11)$~Hz. This predicts an error $\epsilon_g=(\pi^2/4) (\Delta \omega/\delta)^2\approx1\times 10^{-6}$ for our gate detuning $\delta/2\pi=114$~kHz. It is likely that this frequency uncertainty is dominated by Kerr cross-coupling between the (Doppler-cooled) radial rocking modes and the BM mode \cite{nie_2009}
To lowest order, fluctuations in laser intensity between the two ODF pulses lead to errors because of the associated uncompensated phase accumulation. To estimate the effect of gate laser beam power fluctuations, we measure the power stability of the two 532~nm laser beams separately with the photodetectors used for power stabilization. From this, we calculate an Allan variance of power fluctuations at $\tau_\mathrm{ODF}=12$~$\mu$s that is consistent with the photon shot noise at each detector. The resulting error contribution is bounded from above by $1\times10^{-5}$~\cite{sawyer_2021}, which is included with the first entry in Table~\ref{table:error_budget}.
In addition to power instability, beam pointing or ion position variations will also cause intensity to fluctuate. We measure this experimentally by performing pulses (1)-(5) while illuminating the ions with only one of the two ODF beams. Performing these measurements with sufficiently many repetitions to accumulate good statistics is time consuming, but we routinely observe no errors in $1\,000$ repetitions, bounding this error source to $<10^{-3}$. Extending this characterization to include a similar number of experiments ($\sim10\,000$) as are used for Bell-state verification would provide a more restrictive bound. Slow (shot-to-shot) variations in ODF laser-beam intensity will also contribute to the Bell-state infidelity. This error source is difficult to quantify independently of the entangling gate but is reflected in the first entry of Table~\ref{table:error_budget}.
\end{document}
\section{Supplemental Material}
\setcounter{page}{1}
\subsection{Determination of State Populations}
Our photon-count histograms are well approximated as a weighted sum of three Poissonians (a two-parameter probability mass function, because the three weights $P_2$, $P_1$, and $P_0$ must add up to unity). These three Poissonians correspond to two bright ions ($\qubitdowndown$), a single bright ion ($\qubitdownup$ and $\qubitupdown$), and two dark ions ($\qubitupup$). If we define $k_2$, $k_1$, and $k_0$ as the mean number of counts in each Poissonian (determined via independent measurements and maintained at fixed values during the following fitting procedure), we can write the probability mass function (PMF) as
\begin{equation*}
\begin{split}
\mathrm{PMF}(n;P_0,P_1)&=P_0\frac{k_0^n \exp{(-k_0)}}{n!}+P_1\frac{k_1^n \exp{(-k_1)}}{n!} \\
&+(1-P_1-P_0)\frac{k_2^n \exp{(-k_2)}}{n!}.
\end{split}
\end{equation*}
For a given dataset of identical experiments, with each repetition in the dataset labeled by $i$, we have a set of measured photon counts $\{n_i\}$. With this we define the usual log-likelihood as
\begin{equation*}
\begin{split}
l(P_0,P_1)=\sum_{i} \ln(\mathrm{PMF}(n_i;P_0,P_1)).
\end{split}
\end{equation*}
We then numerically maximize this function to find the most likely values for $P_0$, $P_1$, and $P_2=1-P_1-P_0$.
\subsection{Fidelity Distribution}
\begin{figure}[htp]
\includegraphics{fidelity_histogram.pdf}
\caption{\label{fig:fidelity_histogram} Histogram of bootstrapped Bell-state fidelities.}
\end{figure}
Figure~\ref{fig:fidelity_histogram} gives a histogram of the Bell-state fidelities obtained from the bootstrapping procedure defined in the main text.
\subsection{Gate Errors}
1-qubit rotation fidelity is limited by qubit frequency noise, laser phase noise, and laser intensity fluctuations. Qubit frequency noise results from variations in the background magnetic field; our qubit is first-order sensitive to these variations which arise in our lab from changes such as the movement of a chair or the motion of a nearby elevator. Although the incorporation of a spin echo into our gate should alleviate the influence of these frequency variations, 1-qubit rotations (of finite duration) still suffer to some extent. To further reduce these errors, we interleave calibrations of the qubit frequency within our gate experiments, performing a Ramsey frequency calibration experiment between each repetition of the gate experiment. In this way the frequency is stabilized to approximately $\pm$250~Hz, a level at which it is expected to contribute an error of only $4\times 10^{-6}$ to the 1-qubit pulses. We use a NdFeB permanent magnet to minimize fast fluctuations of the $1.07~\text{mT}$ experiment bias field at the expense of slow (correctable) thermal drifts. The remaining error is likely caused by faster phase noise, intensity noise, or fast ambient magnetic field fluctuations.
Additional errors arise due to variations in the phase of the 729 nm laser between the various 1-qubit pulses. Such phase fluctuations can occur due to instability in the laser itself, vibrations of the mirrors along the beampath, and motion of the ion or vacuum chamber. Ramsey coherence measurements performed by repeating the gate experiment [pulses (1), (3), and (5) of Fig.~\ref{fig:pulse_profiles}(b)] while extinguishing the ODF beams indicate that this error contribution is $<2\times 10^{-4}$. Here a marked improvement was observed after the installation of an electro-optic modulator in the output path of the 729 nm laser for phase-noise stabilization.
Fundamentally the fidelity is limited by metastable decay from the $D_{5/2}$ level as well as by spontaneous photon scattering from the ODF beams. These contributions are discussed in detail in Ref.~\cite{sawyer_2021}. We estimate a photon scattering error of $1.1\times 10^{-5}$ at our c.m. frequency of 2.5~MHz. We estimate the metastable decay error as follows. Assuming on average that 50\% of the population lies in the $D_{5/2}$ level until the detection laser is turned on 50~$\mu$s later (this includes the time required for the parity analysis pulse as well as some additional short delays), there is a metastable decay error of $2\times 0.5\times 50\:\mu\mathrm{s}/1.174\:\mathrm{s}=4.3\times 10^{-5}$. In addition, the ions are illuminated with the detection laser for 50~$\mu$s before we begin counting photons (this allows the intensity of the laser to stabilize). However, due to the Zeno effect, the metastable decay rate during this second interval is roughly half its unperturbed value, leading to an error of $2.1\times 10^{-5}$ (we have confirmed this reduction in decay rate via a full simulation of the Schr{\"o}dinger equation including all the relevant levels in $^{40}$Ca$^{+}$). Therefore, the total metastable decay error is approximately $6\times 10^{-5}$. A closely related source of error is metastable decay during the measurement itself. In the case of 100~$\mu$s (200~$\mu$s) measurement duration used for the data in Fig.~\ref{fig:parity_amplitude} (Fig.~\ref{fig:population_scan}), we calculate a lifetime-limited detection error of $4.3\times 10^{-5}$ ($8.6\times 10^{-5}$), again considering the reduced decay rate during illumination. These are likely overestimates, because decays that occur later during the measurement interval should only minimally impact the photon count histograms. We have chosen not to SPAM-correct our reported fidelity in order to reduce statistical complexity.
To initialize to $\qubitdowndown$, we employ optical pumping with a circularly polarized 397 nm laser beam followed by seven cycles of two-step optical pumping with the 729~nm and 854~nm laser beams exciting along $S_{1/2}$~$(m_j=-1/2)\rightarrow D_{5/2}$~$(m_j= 1/2)\rightarrow P_{3/2}$. The initial 397~nm pumping step yields an error of $\sim1\times10^{-3}$, and the following steps further reduce this error to $<10^{-6}$, which we exclude from the error budget table in the Letter.
In the Lamb-Dicke limit there is no dependence of the gate on ion temperature~\cite{sorensen_2000}. A derivation of the higher-order deviations from perfect Lamb-Dicke assumptions reveals a gate error of $\epsilon_g=(\pi^2/4) \eta^4 \bar{n}(2\bar{n}+1)$, where $\bar{n}$ represents the thermal mean number of quanta in and $\eta$ is the Lamb-Dicke parameter of the mode \cite{sorensen_2000,ballance_2016}. This dependence, which applies independently for both the BM and c.m. finite-temperature contributions, is confirmed via numerical simulations. We estimate $\bar{n}_\mathrm{BM}<0.1$, $\bar{n}_\mathrm{c.m.}<0.1$, $\eta_\mathrm{BM}=0.063$, $\eta_\mathrm{c.m.}=0.083$ so that $\epsilon_g<2\times 10^{-5}$.
Heating of the gate mode during operation is unavoidable. We operate the gate near the BM mode instead of near the c.m. mode because of the former's significantly lower $<1.4$~quanta/s heating rate (the c.m. heating rate is $33(14)$~quanta/s). This translates into a gate error of $\epsilon_g=\dot{\bar{n}} \tau_g/(2 K)=7\times 10^{-6}$ ($\tau_g$ here represents the sum of the two ODF pulse durations, and $K=2$ is the number of loops in phase space) \cite{sorensen_2000}.
Variations in the trapping potential can lead to errors via two possible mechanisms. Fast electric field variations can change the position of the trap potential within the Ramsey sequence [Fig~\ref{fig:pulse_profiles}(b)]. This manifests as phase noise both on the optical-dipole force interaction and on the 1-qubit rotations \cite{pino_2020}. For the 2~Hz low-pass filters in use on our dc electrodes, we expect this error contribution is negligible. However, it may become relevant if filters with a higher cutoff frequency are used in the future.
Slower variations in the trapping potential (for example due to thermal drift in the DAC electronics or due to slow charging or discharging of the trap surfaces) can cause variations in the trap frequency from shot to shot of the Ramsey sequence. We have characterized the trap frequency by driving ion motion with a radio-frequency electric field and detecting the ions’ response. For an 8~ms pulse, the resulting frequency response is well fit to a Gaussian with a standard deviation of $\Delta \omega/2\pi=63(11)$~Hz. This predicts an error $\epsilon_g=(\pi^2/4) (\Delta \omega/\delta)^2\approx1\times 10^{-6}$ for our gate detuning $\delta/2\pi=114$~kHz. It is likely that this frequency uncertainty is dominated by Kerr cross-coupling between the (Doppler-cooled) radial rocking modes and the BM mode \cite{nie_2009}
To lowest order, fluctuations in laser intensity between the two ODF pulses lead to errors because of the associated uncompensated phase accumulation. To estimate the effect of gate laser beam power fluctuations, we measure the power stability of the two 532~nm laser beams separately with the photodetectors used for power stabilization. From this, we calculate an Allan variance of power fluctuations at $\tau_\mathrm{ODF}=12$~$\mu$s that is consistent with the photon shot noise at each detector. The resulting error contribution is bounded from above by $1\times10^{-5}$~\cite{sawyer_2021}, which is included with the first entry in Table~\ref{table:error_budget}.
In addition to power instability, beam pointing or ion position variations will also cause intensity to fluctuate. We measure this experimentally by performing pulses (1)-(5) while illuminating the ions with only one of the two ODF beams. Performing these measurements with sufficiently many repetitions to accumulate good statistics is time consuming, but we routinely observe no errors in $1\,000$ repetitions, bounding this error source to $<10^{-3}$. Extending this characterization to include a similar number of experiments ($\sim10\,000$) as are used for Bell-state verification would provide a more restrictive bound. Slow (shot-to-shot) variations in ODF laser-beam intensity will also contribute to the Bell-state infidelity. This error source is difficult to quantify independently of the entangling gate but is reflected in the first entry of Table~\ref{table:error_budget}.
\end{document}
| {'timestamp': '2021-10-19T02:40:03', 'yymm': '2105', 'arxiv_id': '2105.05828', 'language': 'en', 'url': 'https://arxiv.org/abs/2105.05828'} |
\section{Introduction}
The prospects for new physics searches at the LHC and other future
colliders are already constrained by rare processes that are sensitive to
small deviations from the Standard Model. A prime example of such a
low-energy constraint is $b \to s \gamma$ decay \cite{bsg,bsgth}. This, the anomalous
magnetic moment of the muon, $g_\mu - 2$ \cite{g-2}, and the Higgs mass, $m_h$ \cite{mh},
are among the most important indirect constraints on extensions of the
Standard Model, such as the minimal supersymmetric extension of the
Standard Model (MSSM).
The decay $B_s \to \mu^+ \mu^-$ has been emerging as another interesting
potential constraint on the parameter space of models for physics beyond
the Standard Model, such as the MSSM \cite{Dedes,Arnowitt,ko,baer}.
The Fermilab Tevatron collider
already has an interesting upper limit on the branching ratio for $B_s \to
\mu^+ \mu^-$ \cite{cdf}, and is expected soon to increase significantly its
sensitivity to $B_s \to \mu^+ \mu^-$ decay, whilst the LHC sensitivity
will reach down to the Standard Model rate \cite{Buras}. Since its branching ratio is
known to increase rapidly for large values of the ratio of Higgs v.e.v.'s,
$\tan \beta$ \cite{calcs,Babu,Bobeth}, increasing like its sixth power, these present and future
sensitivities are particularly important for models with large $\tan
\beta$.
In view of its potential importance for the MSSM, it is important to treat
the $B_s \to \mu^+ \mu^-$ decay constraint with some care, as has already
been done for the $b \to s \gamma$, $g_\mu - 2$ and $m_h$ constraints. In
each of these cases, the interpretation depends on uncertainties in
theoretical calculations, which should be propagated carefully and
combined with the experimental errors if limits are to be calculated at
some well-defined confidence level, or if a global fit to MSSM parameters
is to be attempted.
Both of these issues are important also in the case of $B_s \to \mu^+
\mu^-$ decay. As concerns the theoretical uncertainties, it is important
to include consistently all the available one-loop MSSM contributions, and
avoid any approximate treatments of any individual terms, since
cancellations are potentially important, and also to include knowledge of
the higher-order QCD corrections to the lowest-order MSSM loop diagrams.
As we discuss in this paper, other sources of error and uncertainty are
also important. These include, for example, the uncertainties in the $B_s$
meson parameters, principally the decay constant $f_{B_s}$. Since the $B_s
\to \mu^+ \mu^-$ decay rate is dominated by the exchange of the
pseudoscalar Higgs boson $A$, the value of $m_A$ is also an important
uncertainty.
Bounds on supersymmetry are often explored in a constrained model with
universal soft supersymmetry-breaking scalar masses $m_0$, gaugino masses
$m_{1/2}$ and trilinear couplings $A_0$ at some GUT input scale. In this
CMSSM, the values of $m_A$ and magnitude of the Higgs mixing parameter
$\mu$ (but not its sign) are in principle fixed by the electroweak
vacuum conditions. However, these predictions are necessarily approximate.
For example, the value of $m_A$ predicted as a function of the independent
parameters $m_{1/2}, m_0, A_0$ and $\tan \beta$ has significant
uncertainties associated with the lack of precision with which the heavy
quark masses $m_t$ and $m_b$ are known, as we discuss extensively later in
this paper. Moreover, rather different values of $m_A$ would be predicted
in models where the universality assumptions of the CMSSM are relaxed. For
example, much smaller values of $m_A$ are attainable in models with
non-universal Higgs masses (NUHM).
When interpreting experimental upper bounds (or measurements) within any
specific model such as the CMSSM, care must be taken to incorporate the
uncertainties in auxiliary parameters such as $f_{B_s}$, $m_t$ and $m_b$.
These should be propagated and combined with the experimental likelihood
function when quoting sensitivities in, e.g., the $(m_{1/2}, m_0)$ plane
at some fixed level of confidence. Moreover, one must also be aware of
model dependences within the assumed framework such as the value of $A_0$
in the CMSSM, as well as the effects of possible deviations from the model
framework such as non-universal Higgs masses.
We exemplify these points in a discussion of uncertainties in the
interpretation of the present and likely future sensitivities of the
Fermilab Tevatron collider and the LHC to $B_s \to \mu^+ \mu^-$ decay,
assuming $\mu > 0$ as preferred by $g_\mu - 2$. In particular, we show
that the uncertainties in $f_{B_s}$, $m_t$ and $m_b$ each shrink
significantly the regions that might otherwise appear to be excluded by
the present limit, or might appear to be if the decay is not discovered at
the likely future sensitivity. We compare the resulting $B_s \to \mu^+
\mu^-$ constraints with other existing constraints such as $b \to s
\gamma$, discussing how they vary with $A_0$ and commenting on the
situation within the NUHM.
\section{Calculation of {\boldmath $B_s \goto \mu^+ \, \mu^-$} Decay}
The branching ratio for the decay $B_s \goto \mu^+ \, \mu^-$ is given by
\begin{eqnarray}
\mathcal{B}(B_s \goto \mu^+ \, \mu^-) &=& \frac{G_F^2 \alpha^2}{16 \pi^3} \frac{M_{B_s}^5 f_{B_s}^2
\tau_B }{4} |V_{tb}V_{ts}^*|^2 \sqrt{1-\frac{4 m_\mu^2}{M_{B_s}^2}} \nonumber \\
&\times& \left\{ \left(1-\frac{4 m_\mu^2}{M_{B_s}^2}\right) | C_S |^2
+ \left |C_P-2 \, C_A \frac{m_\mu}{M_{B_s}^2} \right |^2 \right\} \, ,
\label{eq:braratio}
\end{eqnarray}
where the one-loop corrected Wilson coefficients $C_{S,P}$ are taken
from~\cite{Bobeth} and $C_A$ is defined in terms of $Y(x_t)$,
following~\cite{Logan}, as $C_A=Y(x_t)/\sin^2 \theta_W$ where
\begin{equation}
Y(x_t)=1.033 \left( \frac{m_t(m_t)}{170 {\rm \, Ge\kern-0.125em V}} \right)^{1.55} \, .
\end{equation}
The function $Y(x_t)$ incorporates both leading \cite{Inami} and
next-to-leading order~\cite{Buras} QCD corrections, and $m_t(m_t)$ is the
running top-quark mass in the $\overline{MS}$ scheme.
The precise value of $m_t(m_t)$ depends somewhat
on the set of supersymmetric parameters and our choice of the physical top
quark mass $m_t=178 \pm 4$~GeV~\cite{D0} that we use in this paper.
The Wilson coefficients $C_{S,P}$ have been multiplied by
$1/(1+\epsilon_b)^2$, where $\epsilon_b$ incorporates the full
supersymmetric one-loop correction to the bottom-quark Yukawa
coupling~\cite{mbcor,Carena, Pierce}. It is known that, since $\epsilon_b$ is
proportional to $\tan \beta$, this correction may be significant in the
large-$\tan \beta$ regime we study here \cite{Dedes,Arnowitt}.
These corrections to $\epsilon_b$ are flavour independent.
In addition, it is important to include the flavour-violating
contributions arising from the Higgs and chargino couplings
at the one-loop level. These corrections result in effective
one-loop corrected values for the Kobayashi-Maskawa (KM)
matrix elements~\cite{Babu,Isidori}, which we include as described
in~\cite{Buras1,Tata}. In particular, these corrections modify the Wilson
coefficients involved in Eq.~(\ref{eq:braratio}), as can be seen
in Eqs. (6.35) and (6.36) in ~\cite{Buras1} or in Eq. (14) in ~\cite{Tata}.
In addition, the latter study includes the effects of squark mixing, which are included here as well.
Below, we discuss the behaviour of the dominant
contribution to the Wilson coefficients, mainly to illustrate their
dependence on the pseudo-scalar Higgs boson mass.
The Wilson coefficients $C_{S,P}$ receive four contributions in the
context of MSSM, due to Higgs bosons, counter-terms, box and penguin
diagrams. The Higgs-boson corrections were calculated in~\cite{Logan},
and the rest of the supersymmetric corrections in~\cite{calcs,Babu}.
The full one-loop corrections have been studied and presented
comprehensively in~\cite{Bobeth}.
In our numerical analysis, we implement the full
one-loop corrections taken from this work
as well as the dominant NLO QCD corrections
studied in~\cite{Buras2}, as well as
the flavour-changing gluino contribution~\cite{Tata,Bobeth2} .
The Higgs-boson, box and
penguin corrections to $C_{S,P}$ are proportional to $\tan^2\beta$, while
the counter-term corrections dominate in the large-$\tan \beta$ limit, as
they are proportional to $\tan^3\beta$.
In order to understand the behaviour of the branching ratio in the
$(m_{1/2},m_0)$ plane in the context of the CMSSM, we focus attention on
the counter-terms which are mediated by $A,H,h$ exchange as
seen in Eqs. (5.1) and (5.2) of~\cite{Bobeth}. As seen in Eq. (5.13) of~\cite{Bobeth},
the $B_s \goto \mu^+ \, \mu^-$ decay amplitude $\propto 1/m_{A}^2$ in the large-$\tan \beta$
limit, and the decay rate $\propto 1/m_{A}^4$. This underlines the
importance of knowing or calculating $m_{A}$ as accurately as
possible.
The counter-term contribution to $C_{S,P}$ is given by \cite{Bobeth}
\begin{eqnarray}\label{susy:result:count}
C_{S,P}^{\rm CT} &=&\mp
\frac{m_\mu\tan^3\beta}{\sqrt{2}M_W^2
m_{A}^2}\sum_{i=1}^{2}\sum_{a=1}^{6}\sum_{m,n=1}^{3}
[m_{\tilde{\chi}_i^{\pm}} D_3(y_{ai})U_{i2}(\Gamma^{U_L})_{am}\Gamma^a_{imn}],
\end{eqnarray}
where
\begin{equation}\label{susy:result:gamma}
\Gamma^a_{imn}= \frac{1}{2\sqrt{2}\sin^2\theta_W}
[\sqrt{2}M_W V_{i1}(\Gamma^{U_L\dagger})_{na}-(M_U)_{nn}V_{i2}
(\Gamma^{U_R\dagger})_{na}]\lambda_{mn},
\end{equation}
and $M_U \equiv {\rm diag}(m_u, m_c, m_t)$. $U$ and $V$ are the chargino
diagonalization matrices, $\Gamma^{U_L}$ and $\Gamma^{U_R}$ are $6 \times 3$
squark diagonalization matrices, and $D_3(x) \equiv x\ln x/(1-x)$.
Additionally, $y_{ai}$ is defined in
Eq. (5.10) of~\cite{Bobeth} as $y_{ai} \equiv \msup{a}^2/\mchr{i}^2$,
where $\msup{a}^2 \equiv \{ m_{{\tilde{u}_L}}^2, m_{{\tilde{c}_L}}^2,
m_{{\tilde{t}_1}}^2,
m_{{\tilde{u}_R}}^2, m_{{\tilde{c}_R}}^2, m_{{\tilde{t}_2}}^2 \}$.
Finally, $\lambda_{mn} \equiv V_{mb}V^*_{ns}/V_{tb}V^*_{ts}$.
We can split the counter-term contribution into two terms:
one that is proportional to $M_W$ and another that is
proportional to $m_t$.
Starting with the term whose numerator depends on $M_W$,
it is easy to see that the non-vanishing terms
stem from the following combinations of
indices:
$\{a,n,m\}=\{111,222,333,633\}$ and $i=1,2$. However, the first term
$\{111\}$ is
suppressed, as it is proportional to
$\lambda_{11}=V_{ub}\, V_{us}/V_{tb}\, V_{ts} \simeq - 0.022$,
whereas the
second is not suppressed, because it is
proportional to
$\lambda_{22}=V_{cb}\, V_{cs}/V_{tb}\, V_{ts} \simeq - 1$.
Nor are the third and fourth terms suppressed, as they are multiplied by
$\lambda_{33}=1$.
Thus, the part of the counter-term contribution to the Wilson coefficient
that is $\propto M_W$ is
\begin{eqnarray}
C_S^{CT,M_W}=&-& \sqrt{2} M_W \, f \left\{ \mchr{1}\, V_{11}\,U_{12}
\left[ \lambda_{22}\, D_3(y_{21})+ \lambda_{33}\, \left( \cos^2 \theta_{\tilde t} \, D_3(y_{31}) + \sin^2 \theta_{\tilde t} \,
D_3(y_{61} ) \right) \right] \right. \nonumber \\
&+& \left. \mchr{2}\, V_{21}\,U_{22}
\left[ \lambda_{22}\,D_3(y_{22})+ \lambda_{33}\, \left(\cos^2 \theta_{\tilde t} \, D_3(y_{32}) + \sin^2 \theta_{\tilde t}
\, D_3(y_{62}) \right) \right] \right\} ,
\label{eq:ctmw}
\end{eqnarray}
where
\begin{equation}
f \equiv \frac{m_\mu \tan^3\beta }{4M_W^2 \sin^2 \theta_W m_A^2} \, ,
\label{eq:ffact}
\end{equation}
and we have ignored in (\ref{eq:ctmw}) terms that are proportional
to $\lambda_{11}$. The unitarity
of the KM
matrix implies that $\lambda_{11}+\lambda_{22}+\lambda_{33}=0$, which
for small $\lambda_{11}$ means $\lambda_{22}=-\lambda_{33}$, resulting
in the suppression of $C_S^{CT,M_W}$.
Turning now to the terms that increase with the charge-2/3 quark masses,
we see that the terms with $n=3$ (the top-quark contributions) dominate
the first- and second-generation terms in $\Gamma_{imn}^a$.
Specifically, the dominant terms have $\{a,n,m\}=\{333,633\}$. In
addition, we notice that the $i=2$ part is important, since it is
proportional to $V_{22} \, U_{22}$, while the $i=1$ term is proportional
to $V_{12} \, U_{12}$. Hence it is sufficient to take
\begin{equation}
C_S^{CT,m_t} = m_t \, f \,\, \left( \frac{\sin2\theta_{\tilde t}}{2}\right)
\mchr{2}\, [ D(y_{32})-D(y_{62}) ] \, ,
\label{eq:ctmt}
\end{equation}
where we set $V_{22}\simeq U_{22} \simeq 1$.
The expression
(\ref{eq:ctmt}) is the approximation derived in~\cite{Arnowitt}.
The GIM cancellation in (\ref{eq:ctmw}) means that the counterterm
contribution to the Wilson coefficients, which is the dominant one, can be
approximated relatively well by (\ref{eq:ctmt}). However,
in our analysis we use the full expressions given in~\cite{Bobeth}, as
well as the gluino correction and the flavour-violating corrections to
Kobayashi-Maskawa matrix elements as described above.
\section{Dependence of {\boldmath $m_A$} on {\boldmath $m_t$} and on
{\boldmath $m_b$}}
It is clear from the discussion above that the mass of the
pseudoscalar Higgs boson, $m_{A}$, is an important ingredient in
calculating the branching ratio for the decay $B_s \goto \mu^+ \, \mu^-$, since it enters in
the fourth power in the denominators of the Wilson coefficients $C_S$ and
$C_P$ in (\ref{eq:braratio}). Therefore, to further our discussion of the
uncertainties in the $B_s \goto \mu^+ \, \mu^-$ branching ratio, we now discuss the
uncertainties in the calculated value of $m_A$.
The electroweak symmetry breaking conditions may be written in the form:
\begin{equation}
m_A^2 = m_{H_1}^2 + m_{H_2}^2 + 2 \mu^2 + \Delta_A
\label{eq:mamass}
\end{equation}
and
\begin{equation}
\mu^2 = \frac{m_{H_1}^2 - m_{H_2}^2 \tan^2 \beta + \frac{1}{2} m_{\ss Z}^2 (1 - \tan^2 \beta)
+ \Delta_\mu^{(1)}}{\tan^2 \beta - 1 + \Delta_\mu^{(2)}},
\label{eq:minmu}
\end{equation}
where $\Delta_A$ and $\Delta_\mu^{(1,2)}$ are loop
corrections~\cite{Pierce,Barger:1993gh,deBoer:1994he,Carena:2001fw,erz}.
The exact forms of the radiative corrections to $\mu$ and $m_A$
are not needed here, but it is important to note that, at large $\tan
\beta$, the dominant contribution to $\Delta_\mu^{(1)}$ contains a term
which is proportional to $h_t^2 \tan \beta^2$, whereas the dominant
contribution to $m_A^2$ contains terms proportional to
$h_t^2 \tan \beta$ and $h_b^2 \tan \beta$.
Therefore, the $m_{H_2}^2$ term along with a piece proportional
to $h_t^2$ in $\Delta_\mu^{(1)}$ are dominant in (\ref{eq:minmu}) in the
large-$\tan \beta$ regime, so $\mu$ depends rather mildly on $m_b$.
We illustrate in Fig.~\ref{fig:errors} the logarithmic sensitivity of
$m_A$, namely $\Delta m_A/m_A$, to $m_b$ and $m_t$ along slices through
the $(m_{1/2}, m_0)$ plane for $\tan \beta = 57$, $A_0 = 0$ and $\mu > 0$.
We use as representative errors $\Delta m_t = 1$~GeV and $\Delta
m_b = 0.1$~GeV.
Panel (a) shows the effect of varying $m_0$ for fixed $m_{1/2} =
300$~GeV, and panel (b) shows the effect of varying $m_{1/2}$ for fixed
$m_0 = 400$~GeV.
\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=error2As.eps,height=3in}
\hspace*{-0.17in}
\epsfig{file=error1As.eps,height=3in}
\hfill
\end{minipage}
\caption{
{\it
The sensitivities of $m_A$ to $m_t$ and $m_b$, assuming $\Delta m_t =
1$~GeV
and $\Delta m_b = 0.1$~GeV, along slices through the $(m_{1/2}, m_0)$
CMSSM plane for $A_0 = 0$ and $\tan \beta = 57$. Panel (a) fixes $m_{1/2}
= 300$~GeV and varies $m_0$, while panel (b) fixes $m_0
= 400$~GeV and varies $m_{1/2}$.}
}
\label{fig:errors} \end{figure}
One can understand the behaviour depicted by employing (\ref{eq:mamass}),
(\ref{eq:minmu}) and the renormalization-group equations (RGEs) that
govern the evolution of the $m_{H_i}$ from $M_{GUT}$ to $M_Z$. The
one-loop RGE for $m_{H_1}^2$ depends on $h_b$:
\begin{equation}
\frac{\partial m_{H_1}^2}{\partial \ln Q}=\frac{1}{16 \pi^2}
\left\{ (-6g_2^2 \, M_2^2-\frac{6}{3}g_1^2 \,M_1^2)\,
+\,6|h_b|^2(m_{Q_3}^2 + m_{D_3}^2 + m_{H_1}^2 +A_b^2 ) \right\} \, ,
\label{eq:rgeh1}
\end{equation}
where we ignore, for simplicity, the $h_\tau$ contribution, while the RGE
for $m_{H_2}^2$ depends on $h_t$:
\begin{equation}
\frac{\partial m_{H_2}^2}{\partial \ln Q}=\frac{1}{16 \pi^2}
\left\{ (-6g_2^2 \, M_2^2-\frac{6}{3}g_1^2 \,M_1^2)\,
+\,6|h_t|^2(m_{Q_3}^2 + m_{U_3}^2 + m_{H_2}^2 +A_t^2 ) \right\} \, .
\label{eq:rgeh2}
\end{equation}
We see from (\ref{eq:rgeh1}) that, as $m_b$ increases, the value of
$m_{H_1}^2$ at $M_Z$ decreases, tending to decrease $m_A$
(\ref{eq:mamass}). More importantly, the term proportional to $h_b^2$ in
$\Delta_A$ in (\ref{eq:mamass}) is proportional to $m_{{\tilde
b}_1}^2/(m_{{\tilde b}_1}^2 - m_{{\tilde b}_2}^2)\log(m_{{\tilde b}_1}^2/
m_{{\tilde b}_2}^2) - 1$, which is negative across the $(m_{1/2}, m_0)$
plane for $\tan \beta = 57$ and $A_0 = 0$, with a magnitude that decreases
with $m_{1/2}$. Thus, as $m_b$ increases, we obtain a further decrease in
$m_A$. This term in fact provides most of the numerical dependence of
$m_A$ on $m_b$. Since it is enhanced by $\tan\beta$, the $m_b$ dependence
becomes milder for smaller values of $\tan\beta$.
In addition, the effect on $m_A$ is augmented if the role of the bottom
Yukawa coupling in the RGE (\ref{eq:rgeh1}) is enhanced, which occurs at
large $m_0$. This increases the sensitivity
of $m_A$ to $m_b$, as seen in Fig. \ref{fig:errors}(a). In contrast,
when $m_{1/2}$ is increased, the sensitivity to $\Delta_A$ is diminished
and, at the same time, the gaugino part of the RGE is enhanced. Both
effects lead to a reduced change in $m_A$ at large $m_{1/2}$, as can be
seen in Fig. \ref{fig:errors}(b).
Turning now to the dependence of $m_A$ on $m_t$, we see that the evolution
of $m_{H_2}^2$ shown in (\ref{eq:rgeh2}) is similar to (\ref{eq:rgeh1}),
apart from the substitution of $h_t$ and analogous mass substitutions. As
$m_t$ increases, the value of $m_{H_2}^2$ at $M_Z$ is driven to larger
negative values. However, the change in $m_A$ is dominated by the change
in $\mu$, which grows with an increase in $m_t$. The net effect is an
increase in $m_A$, as seen in Fig. \ref{fig:errors}, which increases with
$m_0$ and decreases with $m_{1/2}$, as seen in panels (a) and (b),
respectively.
Since the $B_s \goto \mu^+ \, \mu^-$ decay rate depends on the fourth power of $m_A$, the
sensitivity of $m_A$ to both $m_b$ and $m_t$ displayed in
Fig.~\ref{fig:errors} can lead to a large uncertainty in $B_s \goto \mu^+ \, \mu^-$ for $\tan
\beta = 57$. We have also evaluated the sensitivities to $m_t$ and $m_b$
for $\tan \beta = 40$. These sensitivities do not vary significantly with
$m_{1/2}$ nor with $m_0$. They are always smaller
than for $\tan \beta = 57$, and the difference is rather substantial for
large $m_0$ and small $m_{1/2}$.
Numerically, in the following analysis we assume
\begin{equation}
\Delta m_t \, = \, 4~{\rm GeV}, \; \Delta m_b \, = \, 0.11~{\rm GeV},
\label{deltamtmb}
\end{equation}
with central values for the physical pole mass $m_t = 178$~GeV and the
running mass $m_b^{\overline {MS}}(m_b) = 4.25$~GeV.
The first of the uncertainties in (\ref{deltamtmb}) is taken directly from
measurements at
the Fermilab Tevatron collider, and may be reduced by a factor of 2 to 4
by future measurements there and at the LHC. The following analysis shows
that such reductions would be most welcome also in the analysis of $B_s \goto \mu^+ \, \mu^-$
decay. The uncertainty in $m_b$ is taken from a recent review~\cite{E-KL}
that combines determinations from ${\bar b}b$ systems, $b$-flavoured
hadrons and high-energy processes. Our 1-$\sigma$ range given
in~\cite{E-KL} is contained
within the preferred range quoted by the Particle Data Group~\cite{pdg},
and is very
similar to the ranges quoted recently by the UKQCD group~\cite{UKQCD}
in the unquenched approximation and
in the review
given by Rakow at the Lattice 2004 conference~\cite{Rakow}.
It is easy to see how important these uncertainties could be. For example,
when $\Delta m_A/m_A = 0.05$ for $\Delta m_t = 1$~GeV, which occurs when
$\tan \beta = 57$ for $(m_{1/2}, m_0) = $ (300, 900) or (100, 400)~GeV,
the change in $m_A$ for $|\Delta m_t| = 4$~GeV is $\pm 0.2 m_A$,
corresponding to a change in the $A$ contribution to the $B_s \goto \mu^+ \, \mu^-$ decay rate
by a factor 2.07 or 0.41, depending on the sign of $\Delta m_t$.
We note in passing that both the CDF~\cite{CDFH} and D0~\cite{D0H}
collaborations have recently published new upper limits on Higgs
production at the Fermilab Tevatron collider. In particular, the D0
limit~\cite{D0H} is relevant to the MSSM at very large $\tan \beta$ and
small $m_A$. However, such small values of $m_A$ are already excluded at
large $\tan \beta$ in the CMSSM by other constraints such as $b \to s
\gamma$ and the lower limit on $m_h$, so the present D0 limit does not
further restrict the part of the CMSSM parameter space of interest here.
\section{The Effects of Uncertainties on {\boldmath $B_s \goto \mu^+ \, \mu^-$} Limits in the
CMSSM}
In order to assess how important these auxiliary uncertainties may be in
the interpretation of $B_s \goto \mu^+ \, \mu^-$ experiments, we display in
Fig.~\ref{fig:CMSSMerrors} their individual effects on the present $B_s \goto \mu^+ \, \mu^-$
constraint in the $(m_{1/2}, m_0)$ plane of the CMSSM for $A_0 = 0$, $\mu
> 0$ and $\tan \beta = 57$. This is close to the largest value of $\tan
\beta$ for which we find suitable electroweak vacua in generic domains of
the $(m_{1/2}, m_0)$ plane, and so maximizes the potential impact of the
$B_s \goto \mu^+ \, \mu^-$ constraint, which increases asymptotically as the sixth power of
$\tan \beta$. The dark (brick) shaded regions in the bottom-right corners
of each panel are excluded because there the lightest supersymmetric
particle (LSP) would be the charged ${\tilde \tau_1}$. The pale (blue)
shaded strips are those favoured by WMAP, if all the cold dark matter is
composed of LSPs. The supersymmetric spectrum
and relic density calculations have been descibed elsewhere
(see e.g. \cite{else}). The near-vertical dashed (black) lines at small
$m_{1/2}$ are the constraint imposed by the non-observation of charginos
at LEP, and the near-vertical dash-dotted (red) lines are those imposed by
the non-observation of the lightest MSSM Higgs boson, as calculated using
the {\tt FeynHiggs} code \cite{FeynHiggs}.
The medium (green) shaded regions are excluded
by the rare decay $b \to s \gamma$.
The branching ratio for this has been measured
by the CLEO, BELLE and BaBar collaborations~\cite{bsg}.
The theoretical prediction of $b \to s
\gamma$~\cite{bsgth,Ambrosio,Buras1} contains uncertainties which stem from
the uncertainties
in $m_b$, $\alpha_s$, the measurement of the semileptonic branching ratio
of the $B$ meson, and the effect of the scale dependence.
In particular, the scale dependence of the theoretical prediction arises from
the dependence on three scales: the scale where the QCD corrections to the
semileptonic decay are calculated and the high and low energy scales
relevant to $b \to s \gamma$ decay~\cite{scale}.
These sources of uncertainty can be
combined to determine a total
theoretical uncertainty~\footnote{According to a recent
analysis~\cite{Neubert},
these theoretical uncertainties may be significantly larger, resulting to
a weaker bound on the masses of supersymmetric particles.}.
The experimental measurement is converted into a Gaussian likelihood and
convolved with a theoretical likelihood to determine the total
likelihood~\cite{efgo},
which is used to calculate the excluded region at 95\% CL.
It is important to note that the dependence of this excluded
region on $m_A, m_b,$ and $m_t$ is quite weak in comparison,
as we have checked numerically.
Finally, the ellipsoidal contours represent the nominal $B_s \goto \mu^+ \, \mu^-$ branching
ratio, calculated (like all the others) using the current central values
$m_t = 178$~GeV and $m_b^{\overline {MS}}(m_b) = 4.25$~GeV. The numerical
labels for the two outer solid lines are exponents in the branching ratio:
$10^{-7}, 10^{-8}$, the thinner-dashed line is for $2 \times 10^{-8}$, the
thicker-dashed line for $5 \times 10^{-8}$.
The most stringent experimental upper bound on the $B_s \goto \mu^+ \, \mu^-$ branching
ratio is that given by an updated CDF measurement:
$1.5 \times 10^{-7}$ ($2.0 \times 10^{-7})$ at the 90\% (95\%)
CL~\cite{cdf}.
The innermost thick solid line of Fig.~\ref{fig:CMSSMerrors} is
the contour for the present nominal 95\% CL experimental upper limit of
$2.0 \times
10^{-7}$.
\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=currnotnobfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=currnobfs.eps,height=3.3in}
\hfill
\end{minipage}
\begin{minipage}{8in}
\epsfig{file=currnotfs.eps,height=3.3in}
\hspace*{-0.2in}
\epsfig{file=currfs.eps,height=3.3in} \hfill
\end{minipage}
\caption{
{\it
The effects of auxiliary uncertainties on the region of the $(m_{1/2},
m_0)$ plane for $A_0 = 0$, $\mu > 0$ and $\tan \beta = 57$ currently
excluded by the
Fermilab Tevatron collider. (a) The effect of $B_s$ meson uncertainties
alone, principally that in $f_{B_s}$. (b) These uncertainties combined
with the uncertainty $\Delta m_t = 4$~GeV. (c) The $B_s$ meson
uncertainties combined with the uncertainty $\Delta m_b = 0.11$~GeV. (d)
All uncertainties combined. The various contours and shadings in the
$(m_{1/2}, m_0)$ plane are explained in the text.}
}
\label{fig:CMSSMerrors}
\end{figure}
Panel (a) of Fig.~\ref{fig:CMSSMerrors} displays as a blue shaded region
the effect on the interpretation of the present experimental limit of
including the uncertainties in the $B_s$ meson parameters. The most
important uncertainty is that in the decay constant $f_{B_s}$, for which
we assume~\cite{bernard}
\begin{equation}
f_{B_s} = 230 \pm 30~{\rm MeV}.
\end{equation}
In addition, we use~\cite{pdg}
\begin{equation}
m_{B_s}=5369.6 \pm 2.4~{\rm MeV}, \qquad \tau_{B_s}=(1.461 \pm 0.057) \times
10^{-12}~{\rm s}.
\end{equation}
In calculating the uncertainty we add
quadratically the uncertainties that result from these errors as well as
those in the KM elements. We see that incorporating these
uncertainties does not change the overall shape of the excluded region,
but does shrink it slightly. There may be some possibility to reduce
the uncertainty in $f_{B_s}$ in the foreseeable future, but in this
analysis we retain it fixed in the following panels and other figures.
The blue shaded region in panel (b) of Fig.~\ref{fig:CMSSMerrors}
incorporates the present uncertainty in $m_t$, assumed to be $\Delta m_t =
4$~GeV, which is propagated through the CMSSM calculation of $m_A$ as
discussed in the previous Section. We see that this uncertainty is more
important for larger $m_0$, truncating the upper part of the exclusion
domain. This effect can readily be understood from panel (a) of
Fig.~\ref{fig:errors}, where we saw that $\Delta m_t$ has a particularly
important effect on $m_A$ at large $m_0$.
The blue shaded region in panel (c) of Fig.~\ref{fig:CMSSMerrors} shows
the parallel effect of the uncertainty in $m_b$, assumed to be $\Delta m_b
= 0.11$~GeV, which is also propagated through the CMSSM calculation of
$m_A$ as discussed in the previous Section. This uncertainty is also more
important for larger $m_0$, providing an independent mechanism for
truncating the upper part of the exclusion domain. This can also readily
be understood from panel (a) of Fig.~\ref{fig:errors}, where we saw that
$\Delta m_b$ also has a particularly important effect on $m_A$ at large
$m_0$.
There is also some tendency in both panels (b) and (c) of
Fig.~\ref{fig:CMSSMerrors} for the exclusion domain to separate from the
axis $m_{1/2} \sim 100$ GeV, particularly at large $m_0$. This can be understood
from panel (b) of Fig.~\ref{fig:errors}, where we see that the effects of
both $\Delta m_t$ and $\Delta m_b$ on $m_A$ are enhanced when $m_{1/2} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}
200$~GeV. The uncertainties in each of $m_t$ and $m_b$ become
particularly important when $m_{1/2}$ is small and $m_0$ large, as seen
separately in panels (b) and (c) of Fig.~\ref{fig:CMSSMerrors}.
The similar tendencies in panels (b) and (c) of Fig.~\ref{fig:CMSSMerrors}
are reinforced when we combine the uncertainties in $m_t$ and $m_b$, as
shown by the blue shaded region in panel (d). We find that $B_s \goto \mu^+ \, \mu^-$ decay
is currently unable to exclude any value of $m_0$ above about 350~GeV,
whereas the exclusion region would have extended up to $m_0 \sim 450$~GeV
if the auxiliary uncertainties had not been taken into account, and $\sim
400$~GeV if either of the $m_t$ or $m_b$ uncertainties had been ignored.
On the other hand, the reduction in the excluded range of $m_{1/2}$ at
lower $m_0$ is less important, typically $\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 30$~GeV.
We observe that the region currently excluded by $B_s \goto \mu^+ \, \mu^-$ is always included
within the region already excluded by $b \to s \gamma$ and/or $m_h$, even
without including the auxiliary uncertainties. The same is even more true
for smaller values of $\tan \beta$: in the case $\tan \beta = 40$ (not
shown), the region currently excluded by $B_s \goto \mu^+ \, \mu^-$ has $m_{1/2} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 180$~GeV
and $m_0 \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 170$~GeV, within the strips excluded by $m_h$ and
$m_{\chi^\pm}$ but allowed by $b \to s \gamma$. We recall that the $b \to
s \gamma$ limit is very dependent on the assumption of universal scalar
masses for the squarks, which does not play a role elsewhere in the
analysis of constraints on CMSSM parameters, and is of course untested.
Clearly $B_s \goto \mu^+ \, \mu^-$ has the potential to complement or even, in the future,
supplant the $b \to s \gamma$ constraint, though it also relies on
squark-mass universality.
The input value of the trilinear soft supersymmetry-breaking parameter
$A_0$ has a significant effect on the allowed CMSSM parameter space, and
is also important for $m_h$ as well as the $b \to s \gamma$ and $B_s \goto \mu^+ \, \mu^-$
decays. Therefore, we display in Fig.~\ref{fig:varyA} the interplays of
these constraints for $\tan \beta = 57$, $\mu > 0$, and (a) $A_0 = 2
m_{1/2}$ and (b) $A_0 = - 2 m_{1/2}$. The qualitative conclusions are
similar to the $A_0 = 0$ case discussed previously: the region currently
disallowed by $B_s \goto \mu^+ \, \mu^-$ decay largely overlaps with the regions previously
disfavoured by $m_h$ and $b \to s \gamma$, and decreases in extent as
$A_0$ is reduced from positive to negative values.
\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=currApfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=currAnfs.eps,height=3.3in}
\hfill
\end{minipage}
\caption{
{\it
The disallowed regions in the $(m_{1/2}, m_0)$ plane for
$\mu > 0$ and $\tan \beta = 57$, for (a) $A_0 = 2 m_{1/2}$ and (b) $A_0 =
- 2 m_{1/2}$. The various contours and shadings in the $(m_{1/2}, m_0)$
plane are as explained in the text describing Fig.~\ref{fig:CMSSMerrors}.
}}
\label{fig:varyA}
\end{figure}
\section{Treatment of Errors}
There are several ways to treat the auxiliary errors in the $B_s \goto \mu^+ \, \mu^-$
analysis. In the above, we have implicitly assumed one of the most
conservative treatments, in the sense that it excludes the smallest region
of the $(m_{1/2}, m_0)$ plane for given fixed values of the uncertainties.
However, other treatments are possible, and here we compare their results.
For this comparison, we include all the uncertainties in $f_{B_s}$, $m_t$
and $m_b$ discussed in the previous Section.
In our previous treatment, see panel (d) of Fig.~\ref{fig:CMSSMerrors}, we
assumed that all the uncertainties have Gaussian error distributions, and
defined the allowed region by discarding the upper 2.5\% tail of the
likelihood distribution obtained by combining the experimental and
auxiliary errors. This would be correct if the central value of the
experimental measurement, after subtracting any backgrounds, was strictly
zero. Alternatively, one might discard the upper 5 \% of the combined
likelihood distribution.
This would give the correct experimental upper limit if the central
experimental value were far enough above zero that no significant part of
the lower tail of the likelihood distribution extends below zero $B_s \goto \mu^+ \, \mu^-$
decay rate, but is otherwise clearly more conservative than the previous
treatment. Finally, experimentalists sometimes subtract one theoretical
(systematic) error from the measured result and then plot the 95 \%
confidence-level contour given by the experimental error.
The two alternative prescriptions
yield similar upper limits on $m_0$, though the shapes of the allowed regions
are different. They both reach up to $m_0 \sim 400$~GeV,
as compared the range $m_0 \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 350$~GeV found in the previous analysis,
shown in panel (d) of Fig.~\ref{fig:CMSSMerrors}. We prefer to use the
more conservative prescription used in drawing the previous figures, also
because it is demonstrably appropriate if the central experimental value
is negligible, as is currently the case.
\section{Possible Future Developments}
It is expected that the Fermilab Tevatron collider experiments will continue
to improve the present sensitivity to $B_s \goto \mu^+ \, \mu^-$ decay.
To assess the likely impact of this improved sensitivity, we
exhibit in Fig.~\ref{fig:future} the potential $(m_{1/2}, m_0)$ planes for
$A_0 = 0$ and $\tan \beta = 57, 40$, obtained using the conservative error
prescription and neglecting possible improvements in the determinations of
$f_{B_s}$, $m_t$ and $m_b$, in the pessimistic case that no signal is
seen.
\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=futfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=fut40fs.eps,height=3.3in}
\hfill
\end{minipage}
\begin{minipage}{8in}
\epsfig{file=lhcfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=lhc40fs.eps,height=3.3in}
\hfill
\end{minipage}
\caption{
{\it
The potential disallowed regions in the $(m_{1/2}, m_0)$ plane for $A_0 =
0$, $\mu > 0$ and (a) $\tan \beta = 57$, (b) $\tan \beta = 40$, obtained
assuming a Fermilab Tevatron upper limit on $B_s \goto \mu^+ \, \mu^-$ that is improved
to a 95\% CL upper limit of $5 \times 10^{-8}$,
with no parallel reductions in the uncertainties in $f_{B_s}$,
$m_t$ and $m_b$. Panels (c) and (d) show the corresponding disallowed
domains assuming a conjectural LHC measurement $BR(B_s \goto \mu^+ \, \mu^-) = (3.9 \pm
1.3) \times 10^{-9}$. The various contours and shadings in the $(m_{1/2},
m_0)$
plane are as explained in the text describing Fig.~\ref{fig:CMSSMerrors}.
}}
\label{fig:future}
\end{figure}
In panel (a) of Fig.~\ref{fig:future} for $\tan \beta = 57$, we show that
the $B_s \goto \mu^+ \, \mu^-$ constraint would, under these pessimistic assumptions, begin to
exclude a region in the neighbourhood of $(m_{1/2}, m_0) = (400, 400)$~GeV
that is favoured by WMAP and allowed by all the other present constraints,
assuming a future 95\% CL upper limit of $5 \times 10^{-8}$.
The region at small $m_{1/2}$ is allowed because of the high sensitivity to
$m_t$ and $m_b$ seen in panel (b) of Fig.~\ref{fig:errors}. For $\tan
\beta = 40$, as seen in panel (b) of Fig.~\ref{fig:future}, the region
disallowed by $B_s \goto \mu^+ \, \mu^-$ would still lie within the region already disallowed
by $b \to s \gamma$. The much reduced sensitivity of $B_s \goto \mu^+ \, \mu^-$ decay to $m_t$
and $m_b$ at small $m_{1/2}$ for $\tan \beta = 40$ implies that there is
no allowed `gap' at small $m_{1/2}$.
Panels (c) and (d) of Fig.~\ref{fig:future} show the corresponding
sensitivities at the LHC, assuming a conjectural 3-$\sigma$ measurement
whose central value coincides with the central value predicted by the
Standard Model, i.e., $BR(B_s \goto \mu^+ \, \mu^-) = (3.9 \pm 1.3) \times 10^{-9}$. In
panel (c) for $\tan \beta = 57$ we see that such a measurement would cover
a very large fraction (but not all) of the CMSSM parameter space allowed
by WMAP. On the other hand, the fraction covered in panel (d) for $\tan
\beta = 40$ is somewhat smaller. Of course, it is quite likely that the
present errors in $f_{B_s}$, $m_t$ and $m_b$ will be substantially reduced
by the time of this ultimate LHC measurement. Reducing the $f_{B_s}$
error, in particular, would reduce the scope available for a CMSSM
contribution, and extend the $B_s \goto \mu^+ \, \mu^-$ exclusion region to larger $m_{1/2}$.
\section{Conclusions}
We have seen that the interpretation of the present and prospective
experimental limits on $B_s \goto \mu^+ \, \mu^-$ decay are very sensitive to auxiliary
uncertainties, principally those in $f_{B_s}$, $m_t$ and $m_b$. At the
present time, these restrict significantly the regions of the CMSSM
parameter space that can be excluded by the current upper limit on this
decay, and their uncertainties may not be reduced significantly during the
remaining operation of the Fermilab Tevatron collider, with the likely
exception of $m_t$. However, one might hope that the uncertainties in each
of $f_{B_s}$, $m_t$ and $m_b$ could be reduced by the time the LHC
achieves its ultimate sensitivity to $B_s \goto \mu^+ \, \mu^-$ decay. As an exercise, we have
considered the possibility that their uncertainties might be reduced to
$\Delta f_{B_s} = 10$~MeV, $\Delta m_t = 1$~GeV and $\Delta m_b =
0.05$~GeV. In this case, the LHC reach in the $(m_{1/2}, m_0)$ plane for
$\tan \beta = 57$ would extend to $m_{1/2} \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 1400$~GeV along the WMAP
strip.
Beyond the framework of the CMSSM, it would be interesting to study the
interpretation of the $B_s \goto \mu^+ \, \mu^-$ constraint also in the frameworks of more
general models, such as those with non-universal Higgs masses
(NUHM)~\footnote{We expect the situation in the general low-energy
effective supersymmetric theory (LEEST) to be similar.}~\cite{nuhm}. A
complete study of the situation within the NUHM would take us beyond the
scope of this paper, so we restrict ourselves to a few remarks. The models
likely to be disfavoured by $B_s \goto \mu^+ \, \mu^-$ are those with a low value of $m_A$ and
large $\tan \beta$. Such models also tend to predict large
neutralino-nucleus scattering cross sections~\cite{ko}. We have examined a
sample of NUHM scenarios that are apparently excluded by the recent
CDMS~II upper limit on the direct scattering of supersymmetric dark
matter, and found that about half of them are excluded by $B_s \goto \mu^+ \, \mu^-$. The next
step would be to examine models apparently allowed by CDMS~II, to see how
many of them are also excluded by $B_s \goto \mu^+ \, \mu^-$. However, for consistency, one
should also study the effects on the dark matter scattering cross section
of auxiliary uncertainties such as those in $m_t$ and $m_b$, which has not
yet been done in the manner described here for $B_s \goto \mu^+ \, \mu^-$. We plan to present
elsewhere such a unified treatment of the uncertainties.
\section*{Acknowledgments}
\noindent
The work of K.A.O. and V.C.S. was supported in part
by DOE grant DE--FG02--94ER--40823. We would like to thank
D. Cronin-Hennessy, C.~Sachrajda
and M.~Voloshin for helpful discussions.
| {'timestamp': '2005-11-01T23:56:33', 'yymm': '0504', 'arxiv_id': 'hep-ph/0504196', 'language': 'en', 'url': 'https://arxiv.org/abs/hep-ph/0504196'} |
\section{Introduction}
The first report of a detection of \igr\ in the hard X-ray band is found in
the {\it INTEGRAL}/IBIS all-sky survey catalogues, which are based on data
taken before the end of 2006 \citep{krivonos07,bird07}. At that time, its
nature was unknown, other than it was a transient source, detected during a
series of observations between December 2002 and February 2004 but which
was below the threshold of the {\it INTEGRAL} detectors between 2004-2007
\citep{bikmaev08}. A {\it Chandra} observation was performed on 18 December
2006, that is, during the off state of the {\it INTEGRAL} instruments.
However, a weak source consistent with the position of \igr\ was detected.
The {\it Chandra} observation allowed the refinement of its X-ray position
and the suggestion of an optical counterpart \citep{sazonov08}.
Low-resolution ($FWHM \sim 15$ \AA) optical spectroscopic observations of
the likely counterpart indicated a B3 star. Although the
H$\alpha$ line was found in absorption, \igr\ was proposed to be a
high-mass X-ray binary with a Be star companion. It was argued that the
star was going through a disc-loss episode at the time of the observations
\citep{bikmaev08}.
Be/X-ray binaries are a class of high-mass X-ray binaries that consist of a
Be star and a neutron star \citep{reig11}. The mass donor in these systems
is a relatively massive ($\simmore 10 ~{\rm M}_\odot$) and fast-rotating
($\simmore$80\% of break-up velocity) star, whose equator is surrounded by
a disc formed from photospheric plasma ejected by the star. \ha\ in
emission is typically the dominant feature in the spectra of such stars.
In fact, the strength of the Balmer lines in general and of \ha\ in
particular (whether it has ever been in emission) together with a
luminosity class III-V constitute the defining properties of this class of
objects. The equatorial discs are believed to be quasi-Keplerian and
supported by viscosity \citep{okazaki01}. The shape and strength of the
spectral emission lines are useful indicators of the state of the disc.
Global disc variations include the transition from a Be phase, i.e., when
the disc is present, to a normal B star phase, i.e., when the disc is
absent and also cyclic V/R changes, i.e., variation in the ratio of the
blue to red peaks of a split profile that are attributed to the precession
of a density perturbation inside the disc \citep{okazaki91,papaloizou06}.
In this work we present the first long-term study of the optical
counterpart to the X-ray source \igr\ and report a disc-loss episode. The
absence of the disc allows us to derive some of the fundamental physical
parameters such as reddening, distance, and rotation velocity, without
contamination from the disc. We also confirm that \igr\ is a Be/X-ray
binary, although with an earlier spectral type than the one suggested by
\citet{bikmaev08}.
\begin{table}
\caption{Log of the spectroscopic observations in the blue region.}
\label{blue}
\centering
\begin{tabular}{@{~~}l@{~~}c@{~~}l@{~~}l@{~~}c}
\noalign{\smallskip} \hline \noalign{\smallskip}
Date &JD &Telescope &Wavelength &num. of \\
&(2,400,000+) & &coverage (\AA) &spectra \\
\noalign{\smallskip}\hline\noalign{\smallskip}
13-09-2012 &56184.34 &SKO &3810--5165 &3 \\
14-09-2012 &56185.31 &SKO &3800--5170 &5 \\
15-10-2012 &56216.74 &FLWO &3890--4900 &5 \\
26-12-2012 &56288.36 &WHT &3870--4670 &3 \\
15-07-2013 &56488.87 &FLWO &3900--4900 &3 \\
23-08-2013 &56528.58 &WHT &3840--4670 &3 \\
\noalign{\smallskip} \hline
\end{tabular}
\end{table}
\begin{figure*}
\resizebox{\hsize}{!}{\includegraphics{./fig1.eps}}
\caption[]{Evolution of the \ha\ and He I 6678 \AA\ lines. Absorption by
the disc along the line of sight produces very narrow lines (shell
profiles). }
\label{haprof}
\end{figure*}
\section{Observations}
Optical spectroscopic and photometric observations of the optical
counterpart to the INTEGRAL source \igr\ were obtained from the 1.3m
telescope of the Skinakas observatory (SKO) in Crete (Greece) and from the
Fred Lawrence Whipple Observatory (FLWO) at Mt. Hopkins (Arizona). In
addition, \igr\ was observed in service time with the 4.2-m William
Herschel Telescope telescope (WHT) of El Roque de los Muchachos observatory
in La Palma (Spain).
The 1.3\,m telescope of the Skinakas Observatory was equipped with a
2000$\times$800 ISA SITe CCD and a 1302 l~mm$^{-1}$ grating, giving a
nominal dispersion of $\sim$1.04 \AA/pixel. On the nights 29 September 2009
and 6 September 2011, a 2400 l~mm$^{-1}$ grating with a dispersion of
$\sim$0.46 \AA/pixel was used. We also observed \igr\ in queue mode with
the 1.5-m telescope at Mt. Hopkins (Arizona), and the FAST-II spectrograph
\citep{fabricant98} plus FAST3 CCD, a backside-illuminated 2688x512 UA
STA520A chip with 15$\mu$m pixels and a 1200 l~mm$^{-1}$ grating (0.38
\AA/pixel). The WHT spectra were obtained in service mode on the nights 26
December 2012 and 23 August 2013 with the ISIS spectrograph and the R1200B
grating plus the EEV12 4096$\times$2048 13.5-$\mu$m pixel CCD (0.22
\AA/pixel) for the blue arm and the R1200R grating and REDPLUS
4096$\times$2048 15-$\mu$m pixel CCD (0.25 \AA/pixel) for the red arm. The
spectra were reduced with the dedicated packages for spectroscopy of the
{\tt STARLINK} and {\tt IRAF} projects following the standard procedure.
In particular, the FAST spectra were reduced with the FAST pipeline
\citep{tokarz97}. The images were bias subtracted and flat-field corrected.
Spectra of comparison lamps were taken before each exposure in order to
account for small variations of the wavelength calibration during the
night. Finally, the spectra were extracted from an aperture encompassing
more than 90\% of the flux of the object. Sky subtraction was performed by
measuring the sky spectrum from an adjacent object-free region. To ensure
the homogeneous processing of the spectra, they were normalized with
respect to the local continuum, which was rectified to unity by employing
a spline fit.
The photometric observations were made from the 1.3-m telescope of the
Skinakas Observatory. \igr\ was observed through the Johnson/Bessel $B$,
$V$, $R$, and $I$ filters \citep{bessel90}. For the photometric
observations the telescope was equipped with a 2048$\times$2048 ANDOR CCD
with a 13.5 $\mu$m pixel size (corresponding to 0.28 arcsec on the sky) and
thus provides a field of view of 9.5 arcmin $\times$ 9.5 arcmin. The gain
and read out noise of the CCD camera at a read-out velocity of 2
$\mu$s/pixel are 2.7 $e^{-}$/ADU and 8 $e^{-}$, respectively. The FWHM
(seeing estimate) of the point sources in the images varied from 4 to 6
pixels (1.1''--1.7'') during the different campaigns. Reduction of the
data was carried out in the standard way using the IRAF tools for aperture
photometry. First, all images were bias-frame subtracted and flat-field
corrected using twilight sky flats to correct for pixel-to-pixel variations
on the chip. The resulting images are therefore free from the instrumental
effects. All the light inside an aperture with radius 4.5'' was summed up
to produce the instrumental magnitudes. The sky background was determined
as the statistical mode of the counts inside an annulus 5 pixels wide and
20 pixels from the center of the object. The absorption caused by the
Earth's atmosphere was taken into account by nightly extinction corrections
determined from measurements of selected stars that also served as
standards. Finally, the photometry was accurately corrected for colour
equations and transformed to the standard system using nightly observations
of standard stars from Landolt's catalogue \citep{landolt92,landolt09}. The
error of the photometry was calculated as the root-mean-square of the
difference between the derived final calibrated magnitudes of the standard
stars and the magnitudes of the catalogue.
The photometric magnitudes are given in Table~\ref{phot}, while information
about the spectroscopic observations can be found in Tables~\ref{red} and
\ref{blue}.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{./fig2.eps}}
\caption[]{From top to bottom, evolution of the \ha\ equivalent width,
V/R ratio, $V$ magnitude, $(B-V)$ colour, and velocity shift with time.}
\label{specpar}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=16cm,height=10cm]{./fig3.eps}
\caption[]{WHT spectrum of \igr\ and identified lines used for spectral
classification. The spectrum was smoothed with a Gaussian filter (FWHM=1).}
\label{wht}
\end{center}
\end{figure*}
\section{Results}
\subsection{The \ha\ line: evolution of spectral parameters}
\label{haevol}
The \ha\ line is the prime indicator of the circumstellar disc state. In
particular, its equivalent width (\ew) provides a good measure of the size
of the circumstellar disc \citep{quirrenbach97,tycner05,grundstrom06}. \ha\
emission results from recombination of photoionised electrons by the
optical and UV radiation from the central star. Thus, in the absence of the
disc, no emission should be observed and the line should display an
absorption profile. The \ha\ line of the massive companion in \igr\ is
highly variable, both in strength and shape. When the line appears in
emission it always shows a double-peaked profile, but the relative
intensity of the blue (V) over the red (R) peaks varies. The central
absorption that separates the two peaks goes beyond the continuum, placing
\igr\ in the group of the so-called {\em shell} stars
\citep{hanuschik95,hanuschik96a,hummel00,rivinius06}. Figure \ref{haprof}
displays the evolution of the line profiles. V/R variability is clearly
seen, indicating a distorted disc \citep{hummel97}. Significant changes in
the structure of the equatorial disc on timescales of months are observed.
In addition, a long-term growth/dissipation of the disc is suggested by the
increase of the equivalent width and subsequent decrease.
Table~\ref{red} gives the log of the spectroscopic observations and some
important parameters that resulted from fitting Gaussian functions to the
\ha\ line profile. Due to the deep central absorption, three Gaussian
components (two in emission and one in absorption) were generally needed to
obtain good fits. Column 5 gives the equivalent width of the entire (all
components) \ha\ line. The main source of uncertainty in
the equivalent width stems from the always difficult definition of the
continuum. The \ew\ given in Table~\ref{red} correspond to the average of
twelve measurements, each one from a different definition of the continuum
and the quoted error is the scatter (standard deviation) present in those
twelve measurements.
Column 6 shows the ratio between the core intensity of the blue and red
humps. The V/R ratio is computed as the logarithm of the ratio of the
relative fluxes at the blue and red emission peak maxima. Thus negative
values indicate a red-dominated peak, that is, $V<R$, and positive values a
blue-dominated line, $V>R$ .
Columns 7 and 8 in Table~\ref{red} give the ratio of the peak flux
of each component over the minimum flux of the deep absorption core. This
ratio simply serves to confirm the shell nature of \igr\ in a more
quantitative way. \citet{hanuschik96a} established an empirical {\em shell
criterion} based on the H$\alpha$ line , whereby shell stars are those with
$F_{\rm p}/F_{\rm cd}\simmore 1.5$, where $F_{\rm p}$ and $F_{\rm cd}$ are
the mean peak and trough flux, respectively.
Column 9 is the velocity shift of the central narrow absorption shell
feature when the disc is present or of the absorption profile of the \ha\
line in the absence of the disc. Prior to the measurement of the velocity
shift, all the spectra were aligned taking the value of the insterstellar
line at 6612.8 \AA\ as reference. Note that these shifts do not
necessarily represent the radial velocity of the binary, as the \ha\ line
is strongly affected by circumstellar matter \citep[see e.g.,][for a
discussion of the various effects when circumstellar matter is present in
the system]{harmanec03}. Typically, the He I lines in the blue end part of
the spectrum are used for radial velocity studies. Note, however, that in
\igr, even these lines are affected by disc emission (see
Sect.~\ref{diskc}). Nevertheless, we measured the radial velocity of the
binary by cross-correlating the higher resolution blue-end spectra obtained
from the FLWO and WHT (Table~\ref{blue}) with a template using the {\em
fxcor} task in the {\it IRAF} package. This template was generated from the
BSTAR2006 grid of synthetic spectra \citep{lanz07} and correspond to a
model atmosphere with $T_{\rm eff}=25000$ K, $log g=3.75$ convolved by a
rotational profile with $v \sin i=380$ km s$^{-1}$. The results for the
July and August 2013 observations, when the contribution of the disc is
expected to be minimum, are $v_r=-127\pm30$ km s$^{-1}$ (HJD 2,456,488.863)
and $v_r=-120\pm10$ km s$^{-1}$ (HJD 2,456,528.582), respectively.
Figure \ref{specpar} shows the evolution of \ew, the V/R ratio, the V
magnitude, the $(B-V)$ colour, and the velocity shift with time. In the
top panel of this figure, different symbols represent the equivalent width
of the different components of the line. Open circles give the sum of the
equivalent widths of the individual V and R peaks, while the squares are
the equivalent width of the deep central absorption. These values were
obtained from the Gaussian fits. The overall equivalent width (filled
circles) was measured directly from the spectra. In the bottom panel
of Fig.~\ref{specpar}, open symbols correspond to the velocity shifts
measured from the \ha\ line, while filled symbols are radial velocities
obtained by crossc-orrelating the 3950--4500 \AA\ spectra with the
template. Black circles denote SKO spectra, blue triangles correspond to
data taken from the FLWO, and red diamonds spectra obtained with the
WHT.
\subsection{Spectral classification}
\label{specl}
Figure~\ref{wht} shows the average blue spectrum of \igr\ obtained with the
4.2-m WHT on the night 23 August 2013. The main spectral features have been
identified. The 3900--4600 \AA\ spectrum is dominated by hydrogen and
neutral helium absorption lines, clearly indicating an early-type B star.
The earliest classes (B0 and B0.5) can be ruled out because no ionised
helium is present. However, \ion{Si}{III} 4552-68-75 is crearly detected,
favouring a spectral type B1-B1.5. The relatively weakness of of
\ion{Mg}{II} at 4481 \AA\ also indicates a spectral type earlier than B2.
The strength of the \ion{C}{III}+\ion{O}{II} blend at 4070 \AA\ and 4650
\AA\ agrees with this range (B1--B2) and points toward an evolved star. A
subgiant or giant star, i.e., luminosity class IV or III, is also favoured
by the presence of \ion{O}{II} at 4415-17 \AA. However, in this case, the
triplet {\ion{Si}{III} 4552-68-75 \AA\ and \ion{Si}{IV} 4089 \AA\ should be
stronger than observed. We conclude that the spectral type of the optical
counterpart to \igr\ is in the range B1--B1.5 V--III, with a prefered
classification of B1IV.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{./fig4.eps}}
\caption[]{Comparison of two spectra of \igr\ at different epochs, when the
disc was present (solid black line) and when the disc presumably had
vanished (dashed red line). The spectra were taken from the FLWO on 15
October 2012 and 15 July 2013, respectively.}
\label{widthcomp}
\end{figure}
\subsection{Contribution of the disc to the spectral lines and colours}
\label{diskc}
Figure~\ref{widthcomp} shows a comparison of two blue-end spectra of \igr\
taken from the FLWO at two different epochs. The October 2012 spectrum
corresponds to a Be phase when the \ha\ line was strongly in emission,
while the July 2013 spectrum corresponds to a B phase when this line
displayed an absorption profile. One of the most striking results that can
be directly derived from the visual comparison of the spectra is the
significantly narrower width of the spectral lines, particularly those of
the Balmer series and the He I lines, when the disc is present. To estimate
the contribution of the disc to the width of the lines we measured the FWHM
of the \ion{He}{I} at 6678 \AA\ (Fig.~\ref{haprof}) as a function of time.
The result can be seen in Fig.~\ref{hei}, where the evolution of \ew\ with
time is also plotted. The width of the helium line was significantly
narrower and the core deeper during the strong shell phase (spectra taken
during 2012, MJD $\sim$56100--56200) than at instances where the disc was
weak, in 2009 and 2013).
The disc emission also affects the photometric magnitudes and colours. The
observed $(B-V)$ colour when the disc was present (observation taken in
2011) is 0.05 mag larger than during 2013 when the disc disappeared. That
is, the disc introduces and extra reddening component.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{./fig5.eps}}
\caption[]{Evolution of the full width half maximum of the He I
$\lambda$6678 line. Note the progressive narrowing of the line as the shell
phase develops and the large values when \ew\ is positive (interpreted as
the absence of the disc).}
\label{hei}
\end{figure}
\subsection{The He I lines: rotational velocity}
\label{rotvel}
Shell stars are Be stars with strongly rotationally broadened photospheric
emission lines with a deep absorption core \citep{rivinius06}. The
rotational velocity is believed to be a crucial parameter in the formation
of the circumstellar disc. A rotational velocity close to the break-up or
critical velocity (i.e. the velocity at which centrifugal forces balance
Newtonian gravity) reduces the effective equatorial gravity to the extent
that weak processes such as gas pressure and/or non-radial pulsations may
trigger the ejection of photospheric matter with sufficient energy and
angular momentum to make it spin up into a Keplerian disc. Because stellar
absorption lines in Be stars are rotationally broadened, their widths can
be used to estimate the projected rotational velocity, $v \sin i$, where
$v$ is the equatorial rotational velocity and $i$ the inclination angle
toward the observer. However, to obtain a reliable measurement of the
rotational velocity of the Be star companion the He I lines have to be free
of disc emission. As shown in the previous section, the width of the
line becomes narrower as the disc grows, underestimating the value of the
rotational velocity.
We estimated the projected rotational velocity of \igr\ by measuring
the full width at half maximum (FWHM) of He I lines, following the
calibration by \citet{steele99}. These authors used four neutral helium
lines, namely 4026 \AA, 4143 \AA, 4387 \AA, and 4471 \AA, to derive
rotational velocities. We measured the width of these lines from the August
2013 WHT spectrum as it provides the highest resolution in our sample and
correspond to a disc-loss phase, as indicated by the absorption profile of
the \ha\ line. We made five different selections of the continuum and
fitted Gaussian profiles to these lines. We also corrected the lines for
instrumental broadening by subtracting in quadrature the FWHM of a nearby
line from the calibrated spectra. The projected rotational velocity
obtained as the average of the values from the four He I lines was
$v \sin i=365\pm15$ km s$^{-1}$. The quoted errors are the standard
deviation of all the measurements.
The rotational velocity can also be estimated by comparing the
high-resolution August 2013 WHT spectrum with a grid of synthetic spectra
broadened at various values of the rotational velocity. We employed the
BSTAR2006 grid \citep{lanz07}, which uses the code TLUSTY
\citep{hubeny88,hubeny92,hubeny94} to create the model atmosphere and
SYNSPEC\footnote{http://nova.astro.umd.edu} to calculate the emergent
spectrum. We assumed a model atmosphere with solar composition, $T_{\rm
eff}=25000$ K and $\log g=3.50$ and a microturbulent velocity of 2 km
s$^{-1}$. This spectrum was convolved with rotational and intrumental
(gaussian) profiles using ROTIN3. Thirteen rotational velocities from 300
km s$^{-1}$ to 420 km s$^{-1}$ with steps of 10 km s$^{-1}$ were
considered. The rotational velocity that minimises the sum of the squares
of the difference between data and model corresponded to $v \sin i=380$ km
s$^{-1}$, consistent with the previous value.
\subsection{Reddening and distance}
To estimate the distance, the amount of reddening to the source has to be
determined. In a Be star, the total measured reddening is made up of two
components: one produced mainly by dust in the interstellar space through
the line of sight and another produced by the circumstellar gas around the
Be star \citep{dachs88,fabregat90}. Although the physical origin and
wavelength dependence of these two reddenings is completely different,
their final effect upon the colours is very difficult to disentangle
\citep{torrejon07}. In fact, interstellar reddening is caused by {\em
absorption} and {\em scattering} processes, while circumstellar reddening
is due to extra {\em emission} at longer wavelenths. The disc-loss episode
observed in \igr\ allows us to derive the true magnitudes and colours of
the underlying Be star, without the contribution of the disc. Thus the
total reddening measured during a disc-loss episode corresponds entirely to
interstellar extinction.
The observed colour of \igr\ in the absence of the disc is
$(B-V)=0.50\pm0.02$ (Table~\ref{phot}), while the expected one for a
B1--1.5V--IV star is $(B-V)_0=-0.26$
\citep{johnson66,fitzgerald70,gutierrez-moreno79,wegner94}. Thus we derive
a colour excess of $E(B-V)=0.76\pm0.02$ or visual extinction $A_{\rm V}=R
\times E(B-V)= 2.4\pm0.1$, where the standard extinction law $R=3.1$ was
assumed. Taking an average absolute magnitude of $M_V=-3.0$, typical of a
star of this spectral type \citep{humphreys84,wegner06}, the distance to
\igr\ is estimated to be 8.7$\pm$1.3 kpc. The final error was obtained by
propagating the errors of $B-V$ (0.02 mag), $A_V$ (0.1 mag) and $M_V$ (0.3
mag).
\section{Discussion}
\label{discussion}
We have investigated the long-term variability of the optical counterpart
to the X-ray source \igr. \citet{bikmaev08} suggested that \igr\ is a
high-mass X-ray binary with a B3e companion, even though their
spectroscopic observations showed \ha\ in absorption. They argued that the
system could be in a disc-loss phase. We confirm the Be nature of \igr, but
suggest an earlier type companion, in agreement with the spectral type
distribution of Be/X-ray binaries in the Milky Way. All spectroscopically
identified optical companions of Be/X-ray binaries in the Galaxy do not
have spectral type later than B2 \citep{negueruela98b}.
\begin{table*}
\caption{Comparison of the characteristic time scales of \igr\ with other
Be/X-ray binaries. $P_{\rm orb}$ is the orbital period, $T_{\rm V/R}$is
the time needed to complete a V/R cycle, and $T_{\rm disc}$ is the time
for the formation and dissipation of the disc. }
\label{comp}
\centering
\begin{tabular}{@{~~}l@{~~}l@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}l}
\noalign{\smallskip} \hline \noalign{\smallskip}
X-ray &Spectral &Disc-loss &P$_{\rm orb}$ &$T_{\rm V/R}$ &$T_{\rm disc}$ &Reference \\
source &type &episode\tablefootmark{$^\dag$}&(days) &(year) &(year) & \\
\noalign{\smallskip} \hline \noalign{\smallskip}
\igr &B1V-IV &yes &-- &0.5-0.8 &5--6 &This work \\
4U 0115+634 &B0.2V &yes &24.3 &0.5--1 &3--5 &1,2 \\
RX J0146.9+6121 &B1III-V &no &-- &3.4 &-- &3 \\
V 0332+53 &O8-9V &no &34.2 &1 &-- &4 \\
X-Per &O9.5III &yes &250 &0.6--2 &7 &5,6,7 \\
RX J0440.9+4431 &B1III-V &yes &-- &1.5--2 &$>10$ &8 \\
1A 0535+262 &O9.7III &yes &111 &1--1.5 &4--5 &9,10,11 \\
IGR J06074+2205 &B0.5IV &yes &-- &-- &4--5 &12 \\
RX J0812.4-3114 &B0.5III-V &yes &81.3 &-- &3--4 &13 \\
4U 1145-619 &B0.2III &no &187 &3 &-- &14 \\
4U 1258-61 &B2V &yes &132 &0.36 &-- &15 \\
SAX J2103.5+4545&B0V &yes &12.7 &-- &1.5--2 &16 \\
\noalign{\smallskip} \hline
\end{tabular}
\tablefoot{
\tablefoottext{$^\dag$}{By disc-loss episodes we mean periods when the \ew\ was seen to be positive.}
}
\tablebib{
(1) \citet{negueruela01}; (2) \citet{reig07b}; (3) \citet{reig00} ;
(4) \citet{negueruela98a}; (5) \citet{lyubimkov97} ;
(6) \citet{delgado01}; (7) \citet{clark01}; (8) \citet{reig05b} ;
(9) \citet{clark98}; (10) \citet{haigh04}; (11) \citet{grundstrom07} ;
(12) \citet{reig10b}; (13) \citet{reig01};
(14) \citet{stevens97} ; (15) \citet{corbetgx86} ; (16) \citet{reig10a}
}
\end{table*}
\subsection{Spectral evolution and variability time scales}
Our monitoring of \igr\ reveals large amplitude changes in the shape and
strength of the spectral lines and two different time scales
associated with the variability of the disc: disc formation/dissipation is
estimated to occur on time scales of years, while V/R variability is seen
on time scales of months.
Our first observation was performed in July 2009 and shows the contribution
of a weak disc. Although the \ew\ is positive, indicating that absorption
dominates over emission, its value is smaller than that expected from a
pure photospheric line, which according to \citet{jaschek87} should be
$\sim$3.5--4 \AA. Also, the two peaks, V and R, separated by the central
depression can already be distinguished in our first spectrum. The
strength of the \ha\ line increased and its shape changed from an
absorption dominated profile into an emission dominated one during the
period July 2009--September 2012. As the intensity increased, the line
became progressively more asymmetric. After September 2012, \ew\ began to
decrease with a faster rate than the rise. By Summer 2013, the \ha\ line
profile had turned into absorption, that is, the system entered a disc-loss
episode.
Due to the observational gaps, it is difficult to determine the overall
time scale for the formation and dissipation of the circumstellar disc.
The observations of \igr\ by \citet{bikmaev08} were made in Spring 2007
(low resolution) and Autumn 2007 (high-resolution) and show the \ha\ line
in absorption, although in the high-resolution spectrum the shape of the
line is reminiscent of a shell profile, i.e., with the contribution of a
small disc. If the Spring 2007 spectrum really showed an absorption profile
similar to our latest observations, then we can estimate the
formation/dissipation cycle to be about six years.
In addition to large amplitude changes in the strength of the \ha\ line,
\igr\ also display marked variations in the shape of the spectral lines.
The most prominent spectroscopic evidence of disc activity is the long-term
V/R variability, that is, the cyclic variation of the relative intensity of
the blue ($V$) and red ($R$) peaks in the split profile of the line. The V/R
variability is believed to be caused by the gradual change of the amount of
the emitting gas approaching the observer and that receding from the
observer due to the precession of a density perturbation in the disc
\citep{kato83,okazaki91,okazaki97,papaloizou06}. Double-peak symmetric
profiles are expected when the high-density part is behind or in front of
the star, while asymmetric profiles are seen when the high-density
perturbation is on one side of the disc \citep{telting94}.
In principle, it is possible from the data themselves to find out whether
the perturbation travels in the same direction as Keplerian orbits of the
material in the disc (prograde precession) or in opposite direction
(retrograde precession). If the motion of the density perturbation is
prograde and the disc is viewed at a high inclination angle (as in \igr),
then the $V>R$ phase should be followed by a $V=R$ phase with a strong
shell profile corresponding to the case where the perturbation lies between
the star and the observer \citep[see][for a sketch of prograde
motion]{telting94}.
In \igr, as the disc grew, the \ha\ line changed from a symmetric to an
asymmetric profile. When the disc was weak, that is, when the equivalent
width of the \ha\ line was a few Angstrom (before 2012), the intensity of
the blue and red peaks was roughly equal, $V=R$. The data reveal that the
perturbation developed between the end of 2011 and beginning of 2012. From
August 2012, a blue dominated profile is clearly present ($V>>R$). However,
the V/R ratio showed a fast decrease through the 2012 observations
(Fig.~\ref{specpar}), indicating that the V$>$R phase was coming to an end.
This was confirmed by the January 2013 observations, where an extreme
red-dominated profile is seen, $V/R\approx -0.9$.
It is worth noticing the extremely fast V/R time scales. Although it is
not possible to pin down the exact moment of the onset of the V/R cycle due
to an observational gap of about a year (September 2011-August 2012), the
changes occurred very rapidly once the cycle started. During the $V>R$
phase, the V/R ratio changed from $\sim+0.4$ to $\sim+0.1$ in less than one
month, September-October 2012 (see Fig.~\ref{haprof} and Table~\ref{red}).
Likewise, the change from a blue-dominated $V>R$ to a red-dominated $V<R$
profile occurred in just two months (October-December 2012). If the motion
is prograde, a $V=R$ phase with a strong shell profile should be observed
in between the blue-dominated and red-dominate phases. We seem to have
missed most of this phase, which would have occurred between October and
December 2012. That is, in just two months the density perturbation must
have gone through the shell $V=R$ phase and most of the $V<R$ phase.
Extrapolating this behaviour, we estimate the duration of a whole
revolution to be of 6--9 months. These changes are among the fastest in a
BeXB. The very short time scale of the observed V/R variations
rises the question of whether these spectral changes are modulated by the
orbital period. Phase-locked V/R variations have been observed for various
Be binaries with hot companions \citep[][and references
therein]{harmanec01}, but possibly only on one BeXB (4U\,1258--61).
\citet{corbetgx86} found that the probability that the V/R variability
observed in 4U\,1258--61 was modulated by the X-ray flare period of $\sim
132$ d, which was proposed to be the orbital period of the system, was
$\sim$87\%. Table~\ref{comp} gives a few well-studied characteristic time
scales of BeXBs: the orbital period, $P_{\rm orb}$, the V/R quasi-periods,
$T_{\rm V/R}$, and the approximate duration of the formation/dissipation of
the disc, $T_{\rm disc}$.
In common with other BeXBs \citep{reig05b,reig10b},
asymmetric profiles are not seen until the disc reaches certain size and
density. During the initial stages of disc growth, the shape of the \ha\
line is always symmetric. Only when \ew\ $\simmore -6$ \AA\ the \ha\ line
displays an asymmetric profile. \igr\ also agrees with this
result. As can be seen in Fig.~\ref{haprof}, all the spectra during the
period 2009--2011 show a symmetric profile and a small \ew. The star took
all this time to build the disc. Once a critical size and density was
reached (some time at the beginning of 2012), the density perturbation
developed and started to travel around in the disc. This result is also
apparent from a comparison of the two panels in Fig.~\ref{specpar}. Before
MJD 55800, $V\approx R$ and \ew\ low. After MJD 56000, a strong asymmetric
emission profile is seen.
Although the maximum \ew\ measured is relatively small, \ew $\approx -8$
\AA, compared to other BeXB, it is consistent with a well developed disc.
In systems viewed edge-on, the maximum \ew\ is much smaller than in the
face-on case, because the projected area of the optically thick disc on the
sky is much smaller \citep{hummel94,sigut13}. On the other hand, although
a fully developed disc must have been formed, it may not extend too far
away from the star. The fast V/R changes favoured a relatively compact
disc, where the density perturbation is capable to achieve a complete
revolution in a few months.
\subsection{Shell lines and disc contribution}
The Be star companion in \igr\ is a shell star as implied by the deep
central absorption between the two peaks of the \ha. This central
depression clearly goes beyond the continuum. The shell profiles are
thought to arise when the observer's line of sight toward the central star
intersects parts of the disc, which is cooler than the stellar photosphere
\citep{hanuschik95,rivinius06}. Statistical studies on the distribution
of rotational velocities of Be stars are consistent with the idea that Be
shell stars are simply normal Be stars seen near edge-on, that is, seen at
a large inclination angle \citep{porter96}. Our observations agree with
this idea. Figure~\ref{widthcomp} clearly shows that the spectral lines are
significantly narrower and deeper when the disc is present. The narrower
lines would result from the fact that the disc conceals the equator of the
star, where the contribution to the rotational velocity is largest. The
deeper lines would result from absorption of the photospheric emission by
the disc. Both circumstances require a high inclination angle. Further
evidence for a high inclination angle is provided by the photometric
observations. Both, positive and negative correlations between the
emission-line strength and light variations have been observed and
attributed to geometrical effects \citep{harmanec83,harmanec00}. Stars
viewed at very high inclination angle show the inverse correlation because
the inner parts of the Be envelope block partly the stellar photosphere,
while the small projected area of the disc on the sky keeps the disc
emission to a minimum. In stars seen at certain inclination angle,
$i\simmore i_{\rm crit}$, the effect of the disc is to increase the
effective radius of the star, that is, as the disc grows an overall (star
plus disc) increase in brightness is expected. The value of the critical
inclination angle is not known but a rough estimate based on available data
suggest $i_{\rm crit}\sim 75^{\circ}$ \citep{sigut13}. \igr\ exhibits the
inverse correlation, that is, it becomes fainter at the beginning
of a new emission episode ($\sim$ JD 2,455,800, see Fig.~\ref{specpar}).
We have assesed the contribution of the disc on the width of helium lines,
which is the main parameter to estimate the star's rotational velocity, by
measuring the FWHM of the \ion{He}{I} 6678 \AA\ over time and by
determining the rotational velocity when the disc was present. A difference
of up to 7 \AA\ was measured between the witdh of the \ion{He}{I} 6678 \AA\
line with and without disc (see Fig.\ref{hei}). Repeating the calculation
performed in Sect.~\ref{rotvel} on the \ion{He}{I} 4026 \AA, 4387 \AA, and
4471 \AA\ when the disc was present and using the WHT December 2012
spectra, we obtain $v\sin i=170\pm20$ km s$^{-1}$. Thus, the contribution
of the disc clearly underestimates the true rotational velocity.
In \igr, the shell profile seems to be a permanent feature. We do not
observe any transition from a shell absorption profile to a Be "ordinary"
(i.e., pure emission) profile. In models that favour geometrically thin
discs with small opening angles, this result implies that the inclination
angle must be well above $70^{\circ}$ \citep{hanuschik96a}. Thus, with
such a large inclination angle, the true rotational velocity is estimated
to be $v_{\rm rot}\sim$380-400 km s$^{-1}$, and the ratio of the equatorial
rotational velocity over the critical break-up velocity, $w=v_{\rm
rot}/v_{\rm crit}\sim0.8$\footnote{The break-up velocity of a B1Ve star
is $\sim 500$ km s$^{-1}$ \citep{porter96,townsend04,cranmer05}.}
If gravity darkening is taken into account, then the fractional rotational
velocity would be even larger. Gravity darkening results from fast
rotation. Rapidly rotating B stars have centrifugally distorted shapes with
the equatorial radius larger than the polar radius. As a result, the poles
have a higher surface gravity, and thus higher temperature. Gravity
darkening breaks the linear relationship between the line width and the
projected rotational velocity and makes fast rotators display narrower
profiles, hence underestimate the true rotational velocity. The reduction
of the measured rotational velocities with respect to the true critical
velocity amounts to 10--30\%, with the larger values corresponding to the
later spectral subtypes \citep{townsend04}. Correcting for gravity
darkening, the rotational velocity would be $v_{\rm rot} \sim \approx 450$
km s$^{-1}$ (assuming $i=80^{\circ}$) and the fractional rotational
velocity of the Be companion of \igr\ $w \approx0.9$, confirming the idea
that shell stars are Be stars rotating at near-critical rotation limit. We
caution the reader that the values of the break-up velocity assume that it
is possible to assign to each Be star a mass and radius equal to that of a
much less rapidly rotating B star, e.g., from well studied eclipsing
binaries. Given that there is not a single direct measurement of the mass
and radius for any known Be star, the break-up velocity should be taken as
an approximation \citep[see e.g][]{harmanec00}.
\section{Conclusion}
We have performed optical photometric and spectroscopic observations of the
optical counterpart to \igr. Our observations show that \igr\ is a
high-mass X-ray binary with a Be shell type companion. Its long-term
optical spectroscopic variability is characterised by global changes in the
structure of the equatorial disc. These global changes manifest
observationally as asymmetric profiles and significant intensity
variability of the \ha\ line. The changes in the strength of the line are
associated with the formation and dissipation of the circumstellar disc. At
least since 2009, \igr\ has been in an active Be phase that ended in mid
2013, when a disc-loss episode was observed. The entire
formation/dissipation cycle is estimated to be six years, although given
the lack of data before 2009, this figure needs to be confirmed by future
observations. In contrast, the V/R variability is among the fastest
observed in Be/X-ray binaries with characteristic timescales of the order
of few weeks for each V/R phase.
The absence of the disc left the underlying B star exposed, allowing us to
derive its astrophysical parameters. From the ratios of various metallic
lines we have derived a spectral type B1IVe. The width of He I lines imply
a rotational velocity of $\sim370$ km s$^{-1}$. Using the photometric
magnitudes and colours we have estimated the interstellar colour excess
$E(B-V)\sim0.76$ mag, and the distance $d\sim$ 8.5 kpc.
The presence of shell absorption lines indicate that the line of sight to
the star lies nearly perpendicular to its rotation axis. Although the
Balmer lines show the most clearly marked shell variability, the helium lines
are also strongly affected by disc emission, making them narrower than in
the absence of the disc.
\begin{acknowledgements}
We thank the referee P. Harmanec for his useful comments and suggestions
which has improved the clarity of this paper. We also thank observers P.
Berlind and M. Calkins for performing the FLWO observations and I.
Psaridaki for helping with the Skinakas observations. Skinakas Observatory
is a collaborative project of the University of Crete, the Foundation for
Research and Technology-Hellas and the Max-Planck-Institut f\"ur
Extraterrestrische Physik. The WHT and its service programme (service
proposal references SW2012b14 and SW2013a19) are operated on the island of
La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de
los Muchachos of the Instituto de Astrof\'{\i}sica de Canarias. This paper
uses data products produced by the OIR Telescope Data Center, supported by
the Smithsonian Astrophysical Observatory. This work has made use of NASA's
Astrophysics Data System Bibliographic Services and of the SIMBAD database,
operated at the CDS, Strasbourg, France.
\end{acknowledgements}
\bibliographystyle{aa}
| {'timestamp': '2013-11-14T02:06:54', 'yymm': '1311', 'arxiv_id': '1311.3093', 'language': 'en', 'url': 'https://arxiv.org/abs/1311.3093'} |
\subsection{Band structure and Fermi surface}
The evolution of the band structure as a function of pressure is presented in
FIG.~\ref{bands}, showing no qualitative change in the
electronic states at the Fermi surface up to about 38.8 GPa.
From 56.7~GPa to 78.4~GPa, we observe some changes along the $\Gamma$-H
and $\Gamma$-N lines.
A 3D picture of the Fermi surface at ambient pressure and 56.7~GPa is drawn
in FIG.~\ref{Fermi}. The lower energy band
forms the octahedron centered at the $\Gamma$-point. With increasing pressure,
this octahedron shrinks and it becomes surrounded
by six little ellipsoids when the previously described band-structure changes
on the $\Gamma$-H line appear.
Around N-point, the Fermi surface forms ellipsoids that are disconnected at
lower pressure and becomes connected by necks to the four neighboring
ellipsoids
above 56.7~GPa. In addition, a complicated open sheet structure, often
referred as "jungle gym", extends from $\Gamma$ to the H points in the BZ.
Our results are in good agreement with
the detailed studies of Anderson {\em et al.}~\cite{Anderson-1}
for the band structure and of Ref.~[\onlinecite{Anderson-2}] for the
Fermi surface.
\begin{figure} \epsfxsize=8cm \centerline{
\includegraphics[scale=0.33,angle=-90]{bands.ps}} \caption{\label{bands}
The band structures of niobium at several pressures (in GPa).}
\end{figure}
\begin{figure}
\caption{\label{Fermi} Fermi surfaces from two bands (the lower energy
band in the left panels and the higher energy band in the right panels)
at ambient pressure (top panels) and at 56.7
GPa (bottom panels). Pictures obtained with
the XCrySDen package.\cite{xcrysden}} \end{figure}
\section{Results for niobium under pressure}
\subsection{Technical details}
The calculations of the ground-state electronic and vibrational
properties of Nb were performed using the Local-Density Approximation
(LDA) and an ultrasoft pseudopotential.
A kinetic energy cut-off of 45~Ry (270~Ry) was chosen for the expansion
into plane waves of the wavefunctions (density). Such high cut-offs were
necessary to obtain accurate values for some "strategic" low-frequency
phonons, located mostly near the $\Gamma$-point. In fact, even small
errors in this region of the spectrum lead to large relative errors in
the estimate of the $\alpha^{2}F$ function and of $\lambda$.
The integration over the Brillouin zone (BZ) requires special techniques
to account for the Fermi surface. We used the broadening technique proposed
in Ref.~[\onlinecite{Paxton}] with a smearing parameter of 0.03 Ry
(which was tested\cite{SdG-broad} to reproduce well
the experimental spectra).
The grids for the electronic BZ integration ({\bf k}-grid)
and for the phononic BZ integration ({\bf q}-grid)
have been chosen according to the Monkhorst-Pack scheme.\cite{mesh}
Details of the numerical quadrature used to evaluate the double-delta
term appearing in Eq.(2), together with convergence tests,
are given in the Appendix.
All calculations were performed using the
{\sc quantum-espresso}\cite{espresso} suite of codes.
\subsection{Structural properties}
Niobium in the body-centered cubic structure was studied at eight values
of the lattice parameter from 6.34~a.u. to
5.64~a.u in steps of 0.1 a.u. These lattice parameters correspond to
pressures ranging from -16.6~GPa to 78.4~GPa. The results are reported
in TABLE~\ref{units}. The calculated static equilibrium lattice constant
(zero-point motion and thermal effects not included) is about 6.14~a.u.,
slightly underestimating (as it usually happens within LDA)
the experimental value\cite{Ashcroft} of 6.24~a.u. The calculated bulk
modulus at the theoretical equilibrium lattice is 192~GPa, versus a
room-temperature
experimental value\cite{bulkmod} of 170~GPa and a calculated value of
162~GPa at the experimental lattice parameter of 6.24 a.u.
The calculated bulk modulus is very sensitive to the volume: it varies
by more than a factor three in the considered range of pressures.
\begin{table}
\begin{tabular}{cccc}
\hline \hline
\\[0.01mm]
$\;\;\;$ $a$ $\;\;\;$ & $\;\;\;$ V/V$_{0}$ $\;\;\;$ &
$\;\;\;$ $P$ $\;\;\;$ & $\;\;\;$ $B$ $\;\;\;$ \\[0.2cm]
\hline \\[0.01mm]
6.34 & 1.10 & -16.6 & 134 \\
6.24 & 1.05 & -9.5 & 162 \\
6.14 & 1.00 & -0.6 & 192 \\
6.04 & 0.95 & 10.0 & 220 \\
5.94 & 0.91 & 22.9 & 249 \\
5.84 & 0.86 & 38.8 & 308 \\
5.74 & 0.82 & 56.7 & 354 \\
5.64 & 0.78 & 78.4 & 424 \\[0.05cm]
\hline \hline
\end{tabular}
\caption{\label{units} The lattice constant $a$ (in a.u.), the corresponding
volume ratio V/V$_{0}$,
the pressure $P$ (in GPa), and bulk modulus $B$ (in GPa) calculated for BCC
niobium crystal at several pressures.
The experimental lattice constant\cite{Ashcroft} is 6.24 a.u.
and the bulk modulus\cite{bulkmod} is 170 GPa.}
\end{table}
\section{Electron-phonon coupling}
\subsection{Definitions}
The Hamiltonian for the electron-phonon interaction is given
in second quantization by
\begin{equation}
H_{el-ph} = \sum_{{\bf k}{\bf q }\nu}
g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu ,mn} \;
c_{{\bf k}+{\bf q}}^{\dagger m}c_{{\bf k}}^{n} \; (b_{-{\bf q}\nu}^{\dagger} + b_{{\bf q}\nu})
\end{equation}
\noindent
where $c_{{\bf k}+{\bf q}}^{\dagger m}$ and $c_{{\bf k}}^{n}$ are the creation and
the annihilation operators for the quasiparticles with energies
$\varepsilon_{{\bf k}+{\bf q},m}$ and $\varepsilon_{{\bf k},n}$
in bands $m$ and $n$ with wavevectors ${\bf k}+{\bf q}$ and ${\bf k}$,
respectively;
$b_{{\bf q}\nu}^{\dagger}$ and $b_{{\bf q}\nu}$ are the creation
and the annihilation operators for
phonons with energy $\omega_{{\bf q}\nu}$ and wavevector ${\bf q}$;
the matrix element $g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}$
describes the electron-phonon coupling.
The coupling constants $g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}$
define the spectral function, $\alpha^{2}F( \omega )$, its first reciprocal
momentum, $\lambda$, and the superconducting electron-phonon coupling constant,
$\lambda_{{\bf q}\nu}$, by the following set of equations:
\begin{eqnarray}
\alpha^{2}F( \omega) &=& \frac{1}{N(\varepsilon_{F})} \sum_{mn}\sum_{{\bf q}\nu}
\delta (\omega - \omega_{{\bf q}\nu})\sum_{\bf k}
|g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}|^{2} \nonumber \\
&& \times \delta (\varepsilon_{{\bf k}+{\bf q},m}-\varepsilon_{F})
\delta (\varepsilon_{{\bf k},n}-\varepsilon_{F}), \label{a2F} \\
\lambda & = &
2 \int \frac{\alpha^{2}F( \omega)}{\omega} d\omega = \sum_{{\bf q}\nu} \lambda_{{\bf q}\nu}, \label{lambda} \\
\lambda_{{\bf q}\nu} & = & \frac{2}{N(\varepsilon_{F})\omega_{{\bf q}\nu}} \sum_{mn} \sum_{\bf k}
|g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu,mn} |^{2} \nonumber \\
&& \times \delta ( \varepsilon_{{\bf k}+{\bf q},m} - \varepsilon_{F})
\delta ( \varepsilon_{{\bf k},n} - \varepsilon_{F} ). \label{dirlam}
\end{eqnarray}
\noindent
The quantity $N(\varepsilon_{F})$ is the density of states at the Fermi energy,
$\varepsilon_{F}$, per both spins.
We introduce, after Allen,\cite{AllenGamma} the phonon linewidth
$\gamma_{{\bf q}\nu}$:
\begin{eqnarray}
\gamma_{{\bf q}\nu} & = & 2 \pi \omega_{{\bf q}\nu} \sum_{mn}\sum_{{\bf k}}
|g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}|^{2} \; \nonumber \\
&& \times \delta (\varepsilon_{{\bf k}+{\bf q},m}-\varepsilon_{F})
\delta (\varepsilon_{{\bf k},n}-\varepsilon_{F}) \label{eq5}
\end{eqnarray}
\noindent
which enters the Eliashberg function, $\alpha^{2}F$, and the
electron-phonon coupling constant, $\lambda_{{\bf q}\nu}$, as follows:
\begin{eqnarray}
\alpha^{2}F( \omega) & = & \frac{1}{2 \pi \; N(\varepsilon_{F})} \sum_{{\bf q}\nu}
\frac{\gamma_{{\bf q}\nu}}{\omega_{{\bf q}\nu}} \delta (\omega - \omega_{{\bf q}\nu}),
\label{a2Feq} \\
\lambda_{{\bf q}\nu} & = & \frac{\gamma_{{\bf q}\nu}}
{\pi \; N(\varepsilon_{F}) \; \omega_{{\bf q}\nu}^{2}}. \label{lambdaeq}
\end{eqnarray}
\subsection{Matrix elements of the electron-phonon interactions}
Within DFPT\cite{DFPT,Review} the electron-phonon matrix elements
can be obtained from the first-order derivative of the self-consistent
Kohn-Sham\cite{DFT} (KS) potential, $V_{KS}$,
with respect to atomic displacements, $\vec{u}_{s{\bf R}}$ for the
$s-$th atom in lattice position ${\bf R}$, as:
\begin{equation}
g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn} = \left( \frac{\hbar}{2\omega_{{\bf q}\nu}} \right)^{1/2}
\langle \psi_{{\bf k}+{\bf q},m} | \Delta V_{KS}^{{\bf q}\nu} | \psi_{{\bf k},n} \rangle ,
\label{gg}
\end{equation}
where $\psi_{{\bf k},n}$ is the $n-$th valence KS orbital of wavevector
${\bf k}$ and
\begin{equation}
\Delta V_{KS}^{{\bf q}\nu} = \sum_{\bf R} \sum_{s} \frac{\partial V_{KS}}{\partial \vec{u}_{s{\bf R}}}
\cdot \vec{u}_{s}^{{\bf q}\nu} \frac{e^{i{\bf qR}}}{\sqrt{N}}
\label{dV}
\end{equation}
is the self-consistent first order variation of the KS potential,
$N$ is the number of cells in the crystal, and
$\vec{u}_{s}^{{\bf q}\nu}$ is the displacement pattern for phonon mode
$\vec{v}_{s}^{{\bf q}\nu}$:
\begin{equation}
\vec{u}_{s}^{{\bf q}\nu} = \frac{\vec{v}_s^{{\bf q}\nu}}{\sqrt{M_{s}}}
\end{equation}
The latter is obtained from the diagonalization
of the dynamical matrix, $\varPhi^{\alpha\beta}_{ss'}({\bf q})$:
\begin{equation}
\sum_{s'\beta} \frac{\varPhi^{\alpha\beta}_{ss'}({\bf q})}{\sqrt{M_{s}M_{s'}}}
v_{s'\beta}^{{\bf q}\nu}
= \omega_{{\bf q}\nu}^{2} v_{s\alpha}^{{\bf q}\nu}. \label{eq11}
\end{equation}
$M_{s}$ is the mass of atom $s$, $\alpha, \beta$ denote cartesian coordinates.
\section{Discussion}
After examination of the phonons and the electron-phonon coupling,
we notice that the decrease of $\lambda$ at high pressure
can be easily related to the decrease of the $\alpha^{2}F(\omega)$ peak
and to its shift toward higher frequencies for all modes.
The origin of the increase of $\lambda$ at low pressure between
$\approx$0 GPa and $\approx$10 GPa is instead more difficult to trace.
We found out that it is mainly determined by the anomalous dispersion
of the T1 and T2 modes close to $\Gamma$-point (see FIG.~\ref{ph-p})
in the frequency region below 1~THz,
that determines a low-frequency peak in the Eliashberg function of
the T1 mode above 10 GPa. This peak is instead absent at ambient pressure
and reappears for an expanded lattice only at about -16 GPa.
It has been shown \cite{Vonsovsky} for the Eliashberg model
that the contribution to $T_{c}$ from acoustic modes close to the $\Gamma$-point
vanishes. For the low-frequency modes which are associated to Kohn anomalies,
however, we can expect important contribution to $T_{c}$ because phonon
softening may occur with a finite phonon linewidth.
Therefore, the region near $\Gamma$ can give a very large contribution
to the electron-phonon coupling in niobium for all studied pressures
(see TABLE~\ref{DOS-lambda}).
Let us now consider the band structure.
In order to explain low-pressure anomalies in $T_c$,
Struzhkin {\em et al.}\cite{Struzhkin} proposed
the existence of necks between the
ellipsoids around N and the "jungle gym" open sheet extending from
$\Gamma$ to H along the $\Gamma$-$\Sigma$ line at a pressure below 5~GPa,
and the disappearance of these features at the higher pressures.
Our calculations do not support this suggestion.
The detailed analysis of the Fermi surface reported by Ostanin
{\em et al.} \cite{Ostanin} gives results close to ours.
Previous theoretical
\cite{Solanki,Anderson-2,Neve,lambda-1.52,fermi-1,fermi-2} and
experimental \cite{Anderson-2,fermi-3,fermi-4} investigations of the
Fermi surface also did not detect any changes at low pressure.
One can, therefore, connect the high-pressure decrease
in electron-phonon coupling to changes in the band structure, while
the origin of the low-pressure anomaly remains unclear.
Therefore, we need a different tool in order to detect tiny features
of the Fermi surface.
\begin{figure}
\epsfxsize=8cm
\includegraphics[scale=0.25,angle=-90.0]{nest-a.ps}
\includegraphics[scale=0.25,angle=-90.0]{nest-b.ps}
\vspace{-0.4cm}
\caption{\label{nest} The nesting factor, $X_{\bf q}$,
of the Fermi surface in niobium at eight studied pressures (in GPa),
for selected high-symmetry lines (top panel) and
high-symmetry points (bottom panel).}
\end{figure}
\begin{figure}
\caption{\label{nest-BZ} The isosurface of the nesting factor in the whole BZ,
$X_{\bf q}=1.0$ (arbitrary units), for niobium at pressures of -0.6 GPa (left panel)
and 10.0 GPa (right panel).
Pictures obtained with the XCrySDen package.\cite{xcrysden}} \end{figure}
We propose to look closer at Fermi surface nesting by plotting the
dispersion of the nesting factor:
\begin{equation}
X_{\bf q}=\sum_{\bf k} \delta (\varepsilon_{{\bf k}}-\varepsilon_{F})
\delta (\varepsilon_{{\bf k}+{\bf q}}-\varepsilon_{F}).
\end{equation}
Large nesting factors correspond to large regions of the Fermi surface
being connected by the nesting vector, $\bf q$, and are expected to correspond
to large electron-phonon couplings.
In FIG.~\ref{nest}, the factor $X_{\bf q}$ is reported as a function of pressure.
This quantity has been computed numerically using the smearing
technique with the broadening of 0.03 Ry. The maximal nesting takes place at
the $\Gamma$-point with much smaller maxima at the high-symmetry points
H, P and N, and in the middle of the lines: $\Gamma$-H, H-P, P-$\Gamma$
and $\Gamma$-N. The nesting factor decreases monotonically with
increasing pressure in the whole BZ except around ambient pressure
(0\--10~GPa).
At the equilibrium lattice constant, a large damping
of the nesting factor moves the whole nesting curve below its value
at 38.8~GPa and makes it even similar to the
curve drawn for the pressure of 56.7~GPa.
The damping of the nesting factor close to ambient pressure
occurs in the whole BZ, as we can see in FIG.~\ref{nest-BZ} in comparison
to nesting for a pressure of 10.0 GPa.
This damping explains the jump in the total electron-phonon coupling constant
which happens below and above the ambient pressure (see TABLE~\ref{DOS-lambda}).
\section{Introduction}
Niobium is a superconductor with a quite high critical temperature,
$T_{c}$=~9.25~K, for a simple metal. The experiments under pressure
by Struzhkin {\em et al.} \cite{Struzhkin} show discontinuities of $T_{c}$
at about 5~GPa and at 50-60~GPa.
The low-pressure discontinuity manifests itself as an increase of $T_{c}$
by about 1~K. The high-pressure anomaly is associated with a decrease
of the critical temperature. To date, the nature of these pressure-induced
discontinuities is not clear. Previous theoretical
studies\cite{Ostanin,Ma,Anderson-1,Anderson-2} agree in attributing the
high-pressure anomalies to some visible change in the band structure.
The low-pressure discontinuity of $T_{c}$, however, remains mysterious.
The goal of this work is to give more information about the nature and
the origin of the anomalous behavior of $T_{c}$ in niobium under pressure.
In particular, we want to understand the role of Kohn
anomalies\cite{anomalies} of the phonon spectra. Kohn anomalies are known
to drastically change the critical temperature in superconductors and are
believed to play a role in all body-centered cubic (bcc) metals
Nb, Mo, V, Ta.\cite{anomalies-exp}
Therefore, we look to the details of the electronic structure and dynamical
properties of Nb at eight pressures in the range from -16 GPa to 78 GPa.
We show that Kohn anomalies are responsible for both discontinuities,
but the origin of the low-pressure Kohn anomaly is different
from that of the high-pressure one.
Both can be identified by a closer study of the Fermi-surface
nesting and of the band structure.
The accurate calculation of the electron-phonon coupling $\lambda$ and of the
spectral function $\alpha^{2}F$ is crucial for our problem. To this end
we use density-functional perturbation theory\cite{DFPT,Review} (DFPT)
in a pseudopotential plane-wave approach. The Eliashberg function and the nesting
factor require an integration of the double delta over the Fermi surface, which
needs to be done with a high numerical accuracy.
We give a few advices for an efficient calculation of
the electron-phonon coupling. Some of the technical details, however,
can be used in general for calculations of other properties which
require an accurate numerical integration with the delta function.
This paper is organized as follows: In the next Section we remind the
physical definitions and give some details of the calculation of
electron-phonon interaction coefficients using Vanderbilt's ultrasoft
pseudopotentials.\cite{Vanderbilt}
In Sec. III, we give the technical details (Subsec. A) and present results
for several properties under pressure:
the lattice constant and bulk modulus (Subsec. B),
the band structure and Fermi surface (Subsec. C),
the phonon frequencies and linewidths (Subsec. D), and
the Eliashberg function and electron-phonon coupling constant (Subsec. E).
In Sec. IV, we discuss the origin of the anomalies, and we summarize in Sec. V.
In the Appendix, we give numerical details for the
calculation of the Eliashberg function.
\subsection{Matrix elements with ultrasoft pseudopotentials}
The use of ultrasoft (US) pseudopotentials (PPs)\cite{Vanderbilt} allows
in many cases a significant reduction of the needed plane-wave kinetic
energy cutoff, as compared with standard norm-conserving pseudopotentials.
\cite{HBS1,KB,MT} This enables a more efficient calculation, at the price of
introducing additional terms originating from the augmentation charges
employed in this scheme.\cite{Vanderbilt}
A detailed description of DFPT with US PPs has been given
elsewhere by Dal Corso.\cite{AndreaUS} Here, we only briefly describe
terms appearing in the electron-phonon coupling.
With US PPs the KS orbitals, $\psi_{{\bf k},n}$, satisfy
a generalized eigenvalue problem
\begin{equation}
\left( -\frac{\nabla^{2}}{2} + V_{KS} - \varepsilon_{{\bf k},n}
S \right) \psi_{{\bf k},n}=0
\end{equation}
where the overlap matrix, S, is given by
\begin{eqnarray}
S({\bf r}_{1},{\bf r}_{2}) &=& \delta ({\bf r}_{1}-{\bf r}_{2}) + \sum_{snm} q_{nm} \nonumber \\
&& \times \beta_{n}({\bf r}_{1}-{\bf R}_{s})
\beta_{m}^{\ast}({\bf r}_{2}-{\bf R}_{s}).
\label{SS}
\end{eqnarray}
The charge correction $q_{nm}$, in the above formula, is defined with the
augmentation functions,
$Q_{nm}({\bf r}-{\bf R}_{s})$, as follows
\begin{equation}
q_{nm} = \int d^{3}r \; Q_{nm}({\bf r}-{\bf R}_{s}),
\label{charge}
\end{equation}
and the projector functions $\beta_{n}({\bf r}-{\bf R}_{s})$ are specific for the
type of atom at the position ${\bf R}_{s}$, and are obtained from the atomic calculations.
The valence charge density is then computed as
\begin{eqnarray}
\rho({\bf r}) & = & \sum_{{\bf k},i} (|\psi_{i}({\bf r})|^{2} + \sum_{smn} Q_{mn}({\bf r}-{\bf R}_{s})
\langle \psi_{{\bf k},i} | \beta_{m} \rangle \langle \beta_{n}| \psi_{{\bf k},i} \rangle ) \nonumber \\
& = & \sum_{{\bf k},i} \int \int d^{3}r_{1}d^{3}r_{2}
\psi_{{\bf k},i}^{\ast} ({\bf r}_{1}) K({\bf r}; {\bf r}_{1},{\bf r}_{2})
\psi_{{\bf k},i} ({\bf r}_{2}),
\end{eqnarray}
where the sum over ${\bf k}$ and $i$ runs on occupied KS orbitals,
the kernel
\begin{eqnarray}
K({\bf r}; {\bf r}_{1},{\bf r}_{2}) & = & \delta ({\bf r}-{\bf r}_{1})
\delta ({\bf r}-{\bf r}_{2}) + \sum_{snm} Q_{nm}({\bf r}-{\bf R}_{s}) \nonumber \\
&& \times \beta_{n}({\bf r}_{1}-{\bf R}_{s})
\beta_{m}^{\ast}({\bf r}_{2}-{\bf R}_{s})
\end{eqnarray}
has been introduced for later convenience.
The KS selfconsistent potential in the US-PP scheme reads
\begin{eqnarray}
V_{KS} ({\bf r}_{1},{\bf r}_{2}) & = & V_{NL}({\bf r}_{1},{\bf r}_{2}) \nonumber \\
&& + \int d^{3}r \; V_{eff}({\bf r})\; K({\bf r}; {\bf r}_{1},{\bf r}_{2}) .
\label{VNL}
\end{eqnarray}
The effective potential, $V_{eff}$, contains the local,
the Hartree and the exchange-correlation (xc) terms
\begin{equation}
V_{eff} ({\bf r}) = V_{loc} ({\bf r}) + \int d^{3}r_{1} \;
\frac{\rho({\bf r}_{1})}{|{\bf r}_{1}-{\bf r}|} + V_{xc} ({\bf r}),
\end{equation}
while the nonlocal term generalizes the usual Kleinman-Bylander\cite{KB}
form allowing several projectors for a given angular momentum component
\begin{equation}
V_{NL} ({\bf r}_{1},{\bf r}_{2}) = \sum_{snm} D_{nm}^{0} \;
\beta_{n}({\bf r}_{1}-{\bf R}_{s}) \beta_{m}^{\ast}({\bf r}_{2}-{\bf R}_{s}).
\end{equation}
When augmentation charges vanish ($Q_{nm}=0$) the above formulas
reduce to the standard norm-conserving formulation.
In order to generalize Eq.~(\ref{gg}) to the case of US PPs, one needs to
compute
first order perturbation theory in presence of overlap matrix, $S$, as below
\begin{equation}
g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn} =
\langle \psi_{{\bf k}+{\bf q},m} | \Delta V_{KS}^{{\bf q}\nu}
- \varepsilon_{{\bf k},n} \Delta S | \psi_{{\bf k},n} \rangle ,
\end{equation}
where
\begin{equation}
\Delta S = \sum_{\bf R} \sum_{s} \frac{\partial S}{\partial \vec{u}_{s{\bf R}}}
\cdot \vec{u}_{s}^{{\bf q}\nu} \frac{e^{i{\bf qR}}}{\sqrt{N}},
\end{equation}
and $\Delta V_{KS}^{{\bf q}\nu}$ is given by Eq.~(\ref{dV}).
The derivative of the Kohn-Sham potential is given in Ref.~[\onlinecite{AndreaUS}]
and for gradient-corrected functionals in Ref.~[\onlinecite{GGA}].
\subsection{Phonon frequencies and linewidths}
\begin{figure*}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.45,angle=-90]{ph-6.ps}}
\caption{Phonon dispersions in niobium at several pressures (in GPa),
for the two transverse modes (left and medium panels, notation "T1" and "T2")
and the longitudinal mode (right panels, notation "L");
the scale on the vertical axis is the same for all three modes.}
\label{ph-p}
\end{figure*}
\begin{figure*}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.45,angle=-90]{gam-6.ps}}
\caption{Phonon linewidths (in THz) in niobium
at several pressures (in GPa).
The three modes of $\gamma_{{\bf q}\nu}$
correspond to the three phonon modes shown in FIG.~\ref{ph-p};
the scale on the vertical axis is the same for all three modes.}
\label{gam-p}
\end{figure*}
\begin{figure*}
\epsfxsize=8cm
\leftline{
\includegraphics[scale=0.45,angle=-90]{a2F-6.ps}}
\caption{Eliashberg function, $\alpha^{2}F$, in niobium at several pressures (in GPa);
the scale on the horizontal axis is the same,
within a given mode, for all pressures.}
\label{a2F-p}
\end{figure*}
The phonon spectra are presented in FIG.~\ref{ph-p}. We observe
an overall increase in phonon frequencies with pressure,
especially at H, P and N high-symmetry points.
Close to the $\Gamma$-point along the $\Gamma$-H symmetry
line, very low frequency phonon modes with an anomalous
pressure dependence can be observed. At both the experimental and
the calculated equilibrium lattice constants, the two
transverse modes (T1 and T2 in the following) display a
very flat dispersion close to $\Gamma$-point.
At variance with all other modes in the BZ, the T1 and T2 modes along $\Gamma$-H
line soften with pressure (between $\approx$10 to $\approx$55 GPa in
our calculations).
This anomalous, non-monotonic, dispersion relation
eventually disappears and become a smooth curve
at the highest pressure we have considered.
The phonon linewidths $\gamma_{{\bf q}\nu}$ have a weak dependence upon
pressure (see FIG.~\ref{gam-p}), with two important exceptions:
i) at low pressure,
large variations in phonon linewidth occur for the T2 and L modes near the N
high-symmetry point, and ii) at high pressure (above $\approx$55 GPa),
a significant reduction in linewidth is observed in many parts of the BZ,
especially along the $\Gamma$-H direction.
\subsection{Eliashberg function and electron-phonon coupling constant}
\label{sec-lambda}
By integrating the calculated phonon linewidths and frequencies we can
obtain Eliashberg $\alpha^{2}F(\omega)$ function, Eq.\ (\ref{a2Feq}), that
we present in FIG.~\ref{a2F-p} for the three phonon branches separately.
As a general feature, all peaks in $\alpha^{2}F(\omega)$ move to higher
frequency with pressure, as expected from the global positive frequency shift
with pressure visible in FIG.~\ref{ph-p}.
For the T1 and T2 modes the height of the main peak decreases with pressure,
while for the L mode the maximal height of the peak appears at the equilibrium
lattice constant.
Our calculated $\lambda$ and the density of states at the studied pressures
are presented in TABLE~\ref{DOS-lambda}.
The pressure behavior of the electron-phonon coupling constant
is similar to what could be expected from the experimental pressure
dependence for $T_{c}$: $\lambda$ shows a positive jump at low pressure
and decreases significantly at high pressure.
\begin{table}
\begin{tabular}{ccccccc}
\hline \hline
\\[0.01mm]
$\;\;\;$ $P$ $\;\;\;$ & & $\;\;\;$ $N(\varepsilon_F)$ $\;\;\;$
&& $\;\;\;$ $\lambda$ $\;\;\;$ &&
$\;\;\;$ $\sim T_{c}^{exp}$ $\;\;\;$ \\[0.2cm]
\hline\\[0.01mm]
-16.6 & & 11.41 && 1.91 && \-- \\
-9.5 & & 10.80 && 1.60 && \-- \\
-0.6 & & 10.12 && 1.41 && 9.2 \\
10.0 & & 9.69 && 1.65 && 10.0 \\
22.9 & & 9.16 && 1.47 && 9.8 \\
38.8 & & 8.55 && 1.29 && 9.7 \\
56.7 & & 7.71 && 1.10 && 9.5 \\
78.4 & & 6.55 && 0.86 && 8.8 \\[0.05cm]
\hline \hline
\end{tabular}
\caption{The parameters of niobium under pressure, $P$ (in GPa):
the electronic density of states at the Fermi surface
$N(\varepsilon_F)$ (states per spin and per Ry),
the electron-phonon coupling constant $\lambda$,
and the experimental critical temperature $T_{c}^{exp}$
from Ref.~[\onlinecite{Struzhkin}].}
\label{DOS-lambda}
\end{table}
A good review of theoretical works on the electron-phonon coupling in Nb
at ambient pressure is given by Solanki {\it et al.} in Ref.~[\onlinecite{Solanki}],
where the reported values of the electron-phonon coupling constant range
from 0.59, obtained\cite{Anderson-1} from augmented plane-wave method
(APW), to 1.52, obtained\cite{lambda-1.52} later from the same method.
More recently, Savrasov\cite{Sav-lambda} obtained a value of 1.26,
Bauer {\it et al.} \cite{Bauer} obtained a value of 1.33,
while our results for $\lambda$ at ambient pressure is 1.41.
We notice that the main reason for the spread in the
reported values of $\lambda$ is the difference in calculated values
for $N(\varepsilon_F)$, entering the definition of $\lambda$ in the
denominator (Eqs.~(\ref{a2F})-(\ref{dirlam})).
In fact a large value of 14.1 (states per spin and per Ry) for the DOS and a
small value of 0.59 for $\lambda$ have been
reported in Ref.~[\onlinecite{Anderson-1}], while
a value of 8.89 for the DOS and $\lambda$=1.52 have been reported in
Ref.~[\onlinecite{lambda-1.52}].
Consistent with this trend, our calculated DOS of 10.12 at ambient pressure
is somewhat lower than the DOS of 10.21 obtained by Savrasov,\cite{Sav-lambda},
while other authors obtained
$\sim 11$ (Ref.~[\onlinecite{Ostanin}]), 11.77 (Ref.~[\onlinecite{Singalas}]),
and 9.84 (Ref.~[\onlinecite{Crabtree}]).
The experimental values of $\lambda$ obtained from the electronic
tunneling spectroscopy\cite{lambda-exp1,lambda-exp2} are 1.04 and 1.22,
while Haas-van Alphen data\cite{Crabtree} yield $\lambda=1.33$.
\section{Summary}
We investigated the origin of the two discontinuities of the superconducting
critical temperature, observed in Niobum at low pressure, about 5~GPa,
and at high pressure, about 60~GPa. \cite{Struzhkin}
For this purpose, we developed computational tools for the accurate
calculation of the electron-phonon coupling.
We find that the anomalous behavior of $T_{c}$ in Nb
under pressure originates in Kohn anomalies close to the $\Gamma$-point
in the BZ, so the measured discontinuities are caused by low-frequency phonons.
In agreement with previous authors,\cite{Ostanin,Ma,Anderson-1,Anderson-2}
we find that the high-pressure discontinuity of $T_{c}$ is associated to
a visible change in the band structure.
As for the low-pressure discontinuity, we notice that such anomaly
shows up as a general decrease of the nesting factor without any visible change
in the shape of the Fermi surface. Such a decrease is uniform in the whole BZ,
explaining why previous calculations, Refs. [\onlinecite{Ostanin,Ma}], did not detect any
anomaly in the electronic structure of Nb near ambient pressure.
The total electron-phonon coupling constant varies with pressure
as expected from the measured critical temperatures.
In conclusion, both discontinuities of $T_{c}$ in niobium, at low and high pressures,
can be reproduced when the electron-phonon spectral function is calculated accurately,
and the anomalies can be explained by the closer look into the details of
the Fermi-surface nesting and the band structure.
\section{Implementation details and test calculations}
The electron-phonon coupling constant $\lambda$ and the spectral
function $\alpha^{2}F(\omega)$ are defined by a double-delta
integration on the Fermi surface (Eqs.~(\ref{a2F}) and (\ref{dirlam})).
The accurate calculation of these integrands requires a very dense
sampling in both the electronic ({\bf k}) and the phononic
({\bf q}) grids. One can use either the broadening technique
\cite{broadening} or the tetrahedron method.\cite{tetra} We choose
the former to perform the quadrature on the Fermi surface, and
the latter to evaluate the electron-phonon and phonon densities of
states as functions of the vibrational frequencies. In the broadening
scheme, a finite energy width is attributed to each state. For any
function $f$ which has to be integrated with the double-delta,
one can use the formula
\begin{eqnarray}
I &= &\int d{\bf k} \int d{\bf q} \; f({\bf k},{\bf q}) \;
\delta(\epsilon_{\bf k}-\epsilon_{F}) \delta(\epsilon_{{\bf k}+{\bf q}}-\epsilon_{F})
\nonumber \\
& \simeq & {\Omega_{BZ}^2\over N_k N_q}
\sum_{\bf k} \sum_{\bf q} f({\bf k},{\bf q}) \;
\frac{1}{\sqrt{2\pi}\sigma}
exp \left( - \frac{(\epsilon_{\bf k}-\epsilon_{F})^{2}}{\sigma^{2}} \right) \nonumber \\
&& \times \frac{1}{\sqrt{2\pi}\sigma}
exp \left( - \frac{(\epsilon_{{\bf k}+{\bf q}}-\epsilon_{F})^{2}}{\sigma^{2}} \right),
\label{integral}
\end{eqnarray}
where $\sigma$ is the broadening, $N_k$ and $N_q$ the number of {\bf k}-
and {\bf q}-points, $\Omega_{BZ}$ the volume of the BZ.
For infinitely dense grids
of {\bf k}- and {\bf q}-points,
convergence of the integrand is achieved when $\sigma$ approaches zero.
For finite grids, however, one has to find a range of $\sigma$ values
yielding close results for the integrands at different grids.
In metals like Nb the presence of Kohn anomalies in the phonon
spectra sets additional requirements for the accuracy. Thus, it is expected
that the aforementioned integrands have to be calculated at very dense
{\bf k}-point and {\bf q}-point grids.
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.34,angle=-90]{fig-1.ps}}
\caption{The density of states and the double-delta integrand on the Fermi surface of Nb.}
\label{DOS}
\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.34,angle=-90]{fig-2.ps}}
\caption{The phonon linewidth $\gamma_{{\bf q}\nu}$ for two selected {\bf q}-vectors.}
\label{q2q4}
\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.40,angle=0]{a2F.ps}}
\vspace{-3mm}
\caption{The Eliashberg function of Nb for fixed broadening $\sigma$=0.03~Ry
and different q-grid (top panel) and the Eliashberg function for
fixed q-grid and different broadening $\sigma$ (bottom panel).}
\label{spectra}
\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.30,angle=-90]{lambda.ps}}
\caption{The total electron-phonon coupling $\lambda$ of Nb for different
q-grids as a function of
broadening $\sigma$.}
\label{lambda-tot}
\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.25,angle=-90.0]{a2F-0.ps}}
\caption{The Eliashberg function for Nb at calculated equilibrium lattice constant (6.14~a.u.).}
\label{a2F-0}
\end{figure}
Since the electron-phonon coupling matrix elements are smooth functions
of {\bf k} and {\bf q}, we resort to an interpolation procedure.
For a chosen {\bf k}- and {\bf q}-vector grid, we calculate matrix
elements $g_{\bf k+q, k}^{{\bf q}s\alpha,mn}$, defined as in Eq. (\ref{gg})
but with respect to the displacement of a single atom $s$ along
cartesian component $\alpha$. For each {\bf q}-vector we make a
linear interpolation in {\bf k}-space of the matrix elements to a
denser {\bf k}-vector grid. We then perform the integration, using
the denser grid and gaussian broadening as in Eq. (\ref{integral}),
of auxiliary phonon linewidths
$\widetilde\gamma^{\alpha\beta}_{ss'}({\bf q})$, defined as:
\begin{eqnarray}
\widetilde\gamma^{\alpha\beta}_{ss'}({\bf q}) & = & \sum_{mn}\sum_{{\bf k}}
(g_{{\bf k+q, k}}^{{\bf q}s\alpha, mn})^*
g_{{\bf k+q, k}}^{{\bf q}s'\beta, mn} \; \nonumber \\
&& \times \delta (\varepsilon_{{\bf k}+{\bf q},m}-\varepsilon_{F})
\delta (\varepsilon_{{\bf k},n}-\varepsilon_{F}).
\label{auxgamma}
\end{eqnarray}
These are related to the $\gamma_{{\bf q}\nu}$ of Eq. (\ref{eq5})
through the relation
\begin{equation}
\gamma_{{\bf q}\nu} = 2\pi\omega_{{\bf q}\nu} \sum_{s\alpha}\sum_{s'\beta}
(u^{{\bf q}\nu}_{s\alpha})^*\widetilde \gamma^{\alpha\beta}_{ss'}({\bf q})
u^{{\bf q}\nu}_{s'\beta}.
\end{equation}
Symmetry is exploited to reduce the number of {\bf k}- and
{\bf q}-points used in the calculation. A convenient way to achieve
such a goal is to perform {\em symmetrization}. Let us denote
with $T^{\alpha \beta}$ the symmetry operators of the small group
of {\bf q} (i.e. the subgroup of crystal symmetry that leaves {\bf q}
unchanged). We restrict summation on {\bf k}-points to the irreducible
BZ calculated with respect to the small group of {\bf q}. Then
symmetrization is performed, separately for each {\bf q}-vector,
as follows:
\begin{eqnarray}
\widetilde \gamma^{\alpha \beta}_{ss'}({\bf q}) & = &
\sum_{\alpha' \beta'} \; T^{\alpha \alpha'} T^{\beta \beta'}
\widetilde\gamma^{\alpha' \beta'}_{T(s)T(s')}({\bf q}) \nonumber \\
& & \times exp \left( i \vec{\bf q}\cdot
( \vec{\tau}_{s}-\vec{\tau}_{T(s)}-\vec{\tau}_{s'}+\vec{\tau}_{T(s')}) \right).
\label{symmetrization}
\end{eqnarray}
In the above formula, atom $s$ with atomic position $\vec{\tau}_s$
transform into atom $T(s)$ with atomic position $\vec{\tau}_{T(s)}$
after application of operation $T$.
The symmetrized matrix at each {\bf q} is subsequently rotated,
using the remaining crystal symmetries that are not in the small
group of {\bf q}, and the symmetrized matrices at all {\bf q}-vectors
in the star of {\bf q} are thus obtained with minimal computational
effort. The same procedure can be applied to dynamical matrices,
Eq. (\ref{eq11}).
Once the electron-phonon coupling matrix of Eq. (\ref{auxgamma})
are calculated on a {\bf q}-vector grid, it is possible to perform
Fourier interpolation and to interpolate to a finer grid. In this way,
the integration in the {\bf q} space needed to calculate
$\alpha^{2}F(\omega)$, Eq. (\ref{a2F}), and $\lambda$, Eq. (\ref{lambda}),
can be accurately performed with a reasonable computational effort.
Let us turn now to some numerical experiments.
FIG.~\ref{DOS} shows the density of states for Nb (left panel)
and the double-delta integrand of a constant function (right panel)
at the Fermi surface for a selected {\bf q}-vector, as a function
of the broadening $\sigma$ and of the Monkhorst-Pack\cite{mesh}
{\bf k}-point grid. For large enough $\sigma$ the results
for different grids -- except the (8,8,8) grid which is too coarse --
converge to the same values, which however depend on $\sigma$.
For small enough $\sigma$ different grids yield different results.
Since we are interested in the $\sigma\rightarrow 0$ limit, we have
to choose a grid of an affordable size that yields converged results
for a $\sigma$ as small as possible. A reasonable choice is the
(64,64,64) grid with $\sigma$=0.02~Ry, yielding a DOS at the Fermi
energy about 10.1 states per spin and per Ry.
The convergence of the double-delta function is a little bit slower
than that of a single-delta.
The phonon linewidths, $\gamma_{{\bf q}\nu}$, for two selected phonons
{\bf q}$\nu$ are displayed in FIG.~\ref{q2q4}. The self-consistent
calculations were performed i) at the {\bf k}-grids of (16,16,16) and
(32,32,32), interpolated to a denser (64,64,64) grid; ii)
at the (24,24,24) {\bf k}-grid, interpolated to (72,72,72); iii)
at the (48,48,48) {\bf k}-grid, interpolated to (96,96,96).
The integration weights for the {\bf k}-space quadrature,
i.e. the gaussians centered around the single-particle energies, were
obtained from the accurate self-consistent calculation at the
corresponding dense grids. As one can see in FIG.~\ref{q2q4}, the
convergence in {\bf k}-points is obtained quite easily even for the
SCF calculation at the grid (16,16,16).
In order to obtain the spectral function $\alpha^{2}F(\omega)$ one
needs to perform the {\bf q}-space quadrature of the phonon
linewidths; Eq. (\ref{a2Feq}). For the phonon and the electron-phonon
densities of states at given frequency $\omega$, we employ the
tetrahedron method within the scheme proposed by
Bl\"ochl.\cite{Bloechl} FIG.~\ref{spectra} (upper panel) shows the
Eliashberg function, $\alpha^{2}F$, of Nb for {\bf q}-grid (8,8,8),
without interpolation and interpolated into denser grids.
This has been done for a fixed broadening, $\sigma$=0.03~Ry. The
lower panel of the same figure shows the $\alpha^{2}F$ function at the
{\bf q}-grid of (8,8,8) interpolated to (20,20,20)-point grid for
several broadenings $\sigma$.
In FIG.~\ref{lambda-tot}, we report the variation of total
electron-phonon coupling constant $\lambda$ with the broadening
$\sigma$ for three {\bf q}-meshes.
In FIG.~\ref{a2F-0}, we present the electron-phonon density of states
for niobium under ambient pressure. We fit the curve of the spectral
function with cubic splines \cite{NR} for a finer plot.
| {'timestamp': '2006-02-07T13:25:56', 'yymm': '0504', 'arxiv_id': 'cond-mat/0504077', 'language': 'en', 'url': 'https://arxiv.org/abs/cond-mat/0504077'} |
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\begin{document}
\begin{center}
{\Large\sc Brown-York Energy and Radial Geodesics}
\end{center}
\vspace{0.2cm}
\begin{center}
{\large\sc Matthias Blau} and {\large\sc Blaise Rollier}
\end{center}
\vskip 0.2 cm
\centerline{\it Institut de Physique, Universit\'e de Neuch\^atel}
\centerline{\it Rue Breguet 1, CH-2000 Neuch\^atel, Switzerland}
\vspace{1cm}
We compare the Brown-York (BY) and the standard Misner-Sharp (MS)
quasilocal energies for round spheres in spherically symmetric space-times
from the point of view of radial geodesics. In particular, we show that
the relation between the BY and MS energies is precisely analogous to that
between the (relativistic) energy $E$ of a geodesic and the effective
(Newtonian) energy $E_{\text{eff}}$
appearing in the geodesic equation, thus
shedding some light on the relation between the two. Moreover, for
Schwarzschild-like metrics we establish a general relationship between
the BY energy and the geodesic effective potential which explains and
generalises the recently observed connection between negative BY energy
and the repulsive behaviour of geodesics in the Reissner-Nordstr\o m
metric. We also comment on the extension of this connection between
geodesics and the quasilocal BY energy to regions inside a horizon.
\section{Introduction}
It is a consequence of the fundamental general covariance of general
relativity that there is no well-defined covariant notion of the local
energy density of the gravitational field. The next best thing is perhaps
the notion of a quasilocal energy (QLE), i.e.\ the energy contained
in a two-dimensional surface. Numerous definitions of QLE have been
proposed in the literature (for a detailed and up-to-date review with many
references see \cite{szab}), and these tend to be mutually inequivalent
even in simple cases such as the Kerr metric \cite{berg}.
There is at least one case, however, in which there appears to be
\textit{almost} universal agreement as to what the QLE should be,
namely for round spheres (i.e.\ orbits of the rotational isometry
group) in spherically symmetric space-times. In that case, the
classical Misner-Sharp (MS) energy \cite{ms} (see e.g.\ \cite{szab} or
\cite{hayward} for recent discussions) is widely considered to be the
``standard'' definition of the energy for round spheres.
One serious contender to this definition is based on the Brown-York (BY)
QLE \cite{by1}. The definition of the BY energy is based on the covariant
Hamilton-Jacobi formulation of general relativity, and this makes it a
natural object to consider in a variety of contexts, with numerous
attractive features. However, the standard
BY energy for round spheres does not
agree with the standard MS energy (even for the Schwarzschild metric), and
this fact has occasionally been used as an argument against the BY energy
as a ``good'' definition of a QLE (see e.g.\ the discussions in
\cite{szab,hayward}).
In this article we will look at the relationship and differences between
the MS and BY energies for round spheres from the point of view of
geodesics and
their associated energy concepts like the relativistic geodesic energy
and the effective Newtonian potential. In general, one would not expect
point-like objects to be able to probe something not quite local like a
QLE. However, the situation is different for round spheres for which the
QLE is independent of the angular coordinates. In such a situation it is
fair to ask whether there is a relation between the gravitational energy
as felt by a point-like observer (geodesic) and that defined according
to some QLE prescription.
Originally, our investigation of these issues was prompted by an
observation and a remark in \cite{lsy}. There it was observed that
for the Reissner-Nordstr\o m metric the BY energy becomes negative
for sufficiently small radius. In \cite{lsy} it was suggested that
this negative energy is strictly related to the well-known repulsive
behaviour exhibited by the geodesics of massive neutral particles in
the Reissner-Nordstr\o m metric.
What supports this point of
view is the fact that the energy indeed becomes negative at precisely
the radius where radial geodesics begin to experience the repulsive
behaviour of the Reissner-Nordstr\o m core. This clearly hints at a
deeper connection between geodesic and quasilocal energy, or, in the words of
\cite{lsy}:``\textit{The turnaround radius agrees with the radius where
the quasilocal energy becomes negative, so it seems that the two effects
are very likely connected.}''
We will indeed be able to establish a general relationship between the BY
energy and the geodesic effective potential (for radial geodesics) and,
in particular, a relation between negative BY energy and a repulsive
behaviour for geodesics.
Our results also shed some light on the difference between the MS and
BY energies for round spheres. In particular, for Schwarzschild-like
metrics $ds^2 = -f(r)^2 dt^2 + f(r)^{-2} dr^2 + r^2 d\Omega^2$, for
which geodesics are conveniently described in terms of an effective
Newtonian potential, we observe that the MS energy is directly related
to the effective potential $V_{\text{eff}}(r)$ for radial goedesics, and that
the relation between the MS and BY energies is strictly analogous to the
relation $E_{\text{eff}} = \trac{1}{2}(E^2 -1)$ between the energy appearing in
the effective potential equation and the relativistic geodesic energy $E$
of the particle. Therefore, inasmuch as $E$ is a relativistic energy and
$E_{\text{eff}}$ an effective Newtonian energy, perhaps one interpretation of
the difference between the BY and MS energies for round spheres is to say
that the former provides one with a relativistic notion of gravitational
energy while the MS energy is more like an effective Newtonian quantity.
We also briefly discuss the extension of these results to regions inside
a horizon. An extension of the BY energy to this case was proposed in
\cite{lsy}. However, it has been remarked\footnote{e.g.\ by one of the
referees, and by Ruth Durrer (private communication)} that the proposal of
\cite{lsy} should perhaps better be thought of as a quasi-local momentum.
Our geodesic perspective is compatible with this
point of view since, as we will show, the relation between the BY
``energy'' of \cite{lsy} and the effective potential inside the horizon
is identical to that between $E$ and $E_{\text{eff}}$ provided that one
considers geodesics that are \textit{spacelike} (outside the horizon).
We believe that the message of this work is two-fold: First of all,
it shows that there are situations where geodesic test particles can be
useful to probe candidate definitions of QLE. Moreover, these results
also illuminate the difference between the BY and MS energies
and provide further evidence that the BY definition of a QLE provides
a good (relativistic) measure of the gravitational energy even though
(or even precisely because) it does not agree with the standard (and
perhaps somewhat more Newtonian) MS energy for round spheres.
\section{Brown-York and Misner-Sharp Energy for Spherical Symmetry}
We briefly recall the definition of the BY and MS energies
for round spheres, referring to the original literature (e.g.\
\cite{by1,by2,lsy} and \cite{ms})
and the review article \cite{szab} for details.
We will consider a general
spherically symmetric metric written in the form
\be
ds^2 = -N(t,r)^2 dt^2 + f(t,r)^{-2}dr^2 + r^2 d\Omega^2
\label{met2}
\ee
with $d\Omega^2= d\theta^2 + \sin^2\theta d\phi^2$
the standard line-element on the unit 2-sphere.
Even though we will
only consider 4-dimensional space-times in this article, the extension to
higher dimensions is rather straightforward.
In contrast to \cite{by2} we prefer to work directly
with the area radius $r$ as the radial coordinate.
The \textit{round spheres} in this space-time (the orbits of the
rotational isometry group) are the 2-spheres $t=\text{const.},
r=\text{const}$.
It was shown in \cite{by1,by2} that,
in any region in which $\del_t$ is timelike and $\del_r$ is
spacelike, the standard
BY quasilocal energy $E_{BY}(t,r)$ associated to a round sphere
of radius $r$, and calculated with respect to the standard static
observers associated to the spatial slicing $t=\text{const.}$
is given by
\be
E_{BY}(t,r) = \frac{r}{G_N}(1-f(t,r))\;\;,
\label{eby2}
\ee
where $G_N$ is Newton's constant.
This BY energy differs from the ``standard''
Misner-Sharp (MS) energy \cite{ms} for round spheres which, for a metric of
the type (\ref{met2}) and for any $(t,r)$, is given by
\be
E_{MS}(t,r) = \frac{r}{2G_N}\left(1-f(t,r)^2\right)\;\;.
\label{ems1}
\ee
For example, for the Reissner-Nordstr\o m metric
$N(r)^2 = f(r)^2 = 1 - \frac{2m}{r} + \frac{e^2}{r^2}$
one has
\be
\begin{aligned}
E_{BY}(r) &= \frac{r}{G_N}
\left(1-\sqrt{1 - \frac{2m}{r}+\frac{e^2}{r^2}}\right)\\
E_{MS}(r) &= \frac{1}{G_N}\left(m-\frac{e^2}{2r}\right)\;\;.
\end{aligned}
\ee
Both reduce to the ADM mass $M=m/G_N$ asymptotically,
$\lim_{r\ra\infty} E_{BY}(r) =
\lim_{r\ra\infty} E_{MS}(r) = M$
(and for the Schwarzschild metric one evidently
has $E_{MS}(r)=M$ for all $r$).
Moreover, for sufficiently small values of $r$, $r<r_0 = e^2/2m$
both the MS and the BY energy are negative (note that the expression for the
BY energy is also valid inside the inner horizon $r_-$ and that $r_0 < r_-$).
The qualitative (and not just quantitative) difference bewteen the
BY and MS energies is e.g.\ illustrated by the fact that,
unlike the MS energy, the BY energy is finite at $r=0$,
$E_{BY}(0)=-|e|/G_N$ \cite{lsy}.
\section{Brown-York and geodesic energy for Schwarzschild-like metrics}
In order to analyse the BY energy (and its relationship with the MS energy)
from the point of view of geodesics, we now specialise to Schwarzschild-like
metrics, i.e.\ static spherically symmetric metrics with $N(r)=f(r)$ (the
extension to time-dependent Schwarzschild-like metrics with $f=f(t,r)$ is
straightforward),
\be
ds^2 = -f(r)^2 dt^2 + f(r)^{-2} dr^2 + r^2 d\Omega^2\;\;.
\label{met4}
\ee
In this case the behaviour of timelike radial
geodesics is governed by the effective potential equation
\be
\trac{1}{2}\dot{r}^2 + V_{\text{eff}}(r) = E_{\text{eff}}\;\;,
\label{veff2}
\ee
where the effective (and effectively Newtonian) potential
$V_{\text{eff}}(r)$ is related to $f(r)^2$ by
\be
f(r)^2 = 1 + 2 V_{\text{eff}}(r)\;\;,
\label{fv}
\ee
and the effective energy $E_{\text{eff}}$ is given in terms of the
relativistic geodesic energy per unit rest mass
$E=f(r)^2 \dot{t}$ of the particle by
\be
E_{\text{eff}} = \trac{1}{2}(E^2 - 1)\;\;.
\label{eeff}
\ee
For later we note that $E=f(r_m)$ where $r_m$ (the index could indicate
a minimum or maximum) is a turning point, $\dot{r}_m=0$, of the trajectory,
and that
in the asymptotically flat case (which we take here to simply mean
$\lim_{r\ra\infty} f(r)=1$) for scattering trajectories that reach
(or start out at) $r\ra\infty$ one also has the relation
$E^2 = 1 + \dot{r}_{\infty}^2\geq 1$ between $E$ and
the velocity at infinity.
In particular, for scattering trajectories in
the Reissner-Nordstr\o m metric, for the minimal radius $r_m=r_m(E)$ one has
$r_m(E) \leq r_m(E=1) = e^2/2m$,
which, as noted in \cite{lsy}, agrees with the radius
$r_0$ where the BY (and MS) energy becomes negative.
In order to now
study the relations among the Brown-York energy, the Misner-Sharp
energy, and the geodesic effective potential, it turns out to be convenient to
introduce the corresponding potentials
\be
V_{BY}(r) := -G_{N}\frac{E_{BY}(r)}{r}\qquad\qquad
V_{MS}(r) := -G_{N}\frac{E_{MS}(r)}{r}\;\;.
\ee
Using the definition (\ref{ems1}) of the MS energy and (\ref{fv}),
one immediately sees that
\be
V_{MS}(r) = -\trac{1}{2}(1-f(r)^2) = V_{\text{eff}}(r)\;\;.
\label{mseff}
\ee
Thus the MS potential agrees on the nose with the effective potential
and has a clear physical interpretation in the present context. In
particular, negative MS energy is strictly correlated with a repulsive
behaviour of the effective potential for radial geodesics.
What about the BY potential? Given the above relation between the
MS energy and radial geodesics, one's first thought may perhaps
be\footnote{This was not our first thought, but we are grateful to one
of the referees for reminding us that it should perhaps have been.}
that to establish a link with the BY energy one should calculate the
latter for freely falling (geodesic) rather than static observers,
or for static observers in comoving (Novikov) coordinates.
The Schwarzschild BY energy for geodesic obervers was first determined
in \cite{bm} and more recently, also motivated by the appearance of
the first version of the present article on the arXiv, in \cite{yc}
(with a slightly different prescription).
For example, the result of \cite{bm} (for an observer initially at rest at
infinity) is
\be
E_{BY}^{\text{freefall}}(r) = \frac{r}{G_N}
\left(\sqrt{1+ \frac{2m}{r}}-1\right)\;\;.
\ee
In Novikov coordinates $(\tau,R)$, where $\tau$ is the proper time of a
radially infalling observer and $R$ is related to the maximal radius
$r_m$ of the geodesic by $R= \sqrt{\frac{r_m}{2m}-1}$,
the Schwarzschild metric reads (see e.g.\ \cite[\S 31.4]{mtw})
\be
ds^2 = -d\tau^2 + \frac{R^2+1}{R^2} \left(\frac{\del r}{\del R}\right)^2 dR^2
+ r(\tau,R)^2 d\Omega^2\;\;.
\ee
Calculating the BY energy for ``static'' observers in this space-time, one
finds
\be
\label{enov}
E_{BY}^{\text{Novikov}}(r) = \frac{r}{G_N}
\left(1 - \frac{R}{\sqrt{R^2 +1}}\right)
= \frac{r}{G_N}
\left(1 - \sqrt{1-\frac{2m}{r_m}}\right)\;\;.
\ee
This can e.g.\ be seen to agree with the result of
\cite{yc}, based on the calculation of the freefall BY energy in Kruskal
coordinates. Thus, neither do the above results reproduce the MS
energy, nor do they appear to be related to the effective geodesic potential
in any other particularly useful or illuminating way.
In this context it is perhaps also worth pointing out that in
\cite{by2} a change of coordinates (foliation) $t\ra T(t,r)$ for the
Schwarzschild metric was exhibited with respect to which the standard
BY energy takes the MS value $E_{BY}(r) = M$. Such a foliation, giving
$E_{BY}(r)=E_{MS}(r)$, can also readily be constructed for the general
Schwarzschild-like metric \eqref{met4}. It suffices to choose $T(t,r)$ such
that
\be
\label{T}
dT = dt + \frac{1-f^2}{f^2(1+f^2)} dr
\;\;.
\ee
To see this note that for a general spherically symmetric metric of the
form
\be
ds^2 = -N(t,r)^2 dt^2 + F(t,r)^{-2}(dr + A(t,r)dt)^2 + r^2 d\Omega^2
\ee
the BY energy is still given by \eqref{eby2} (with $f\ra F$),
and that with the choice \eqref{T} one has
$1-F = \trac{1}{2}(1-f^2)$, so that indeed $E_{BY}(r) = E_{MS}(r)$.
However, the physical significance of this choice of foliation escapes us, and
this construction does not appear to shed any light on the relationship
between the BY energy and geodesic notions of energy.
Thus we now return to the task of relating the standard BY energy
\eqref{eby2} to the geodesic effective potential.
Substituting (\ref{fv}) in (\ref{eby2}), one finds
\be
E_{BY}(r) = \frac{r}{G_N}(1-\sqrt{1+2 V_{\text{eff}}(r)})\;\;,
\ee
or
\be
1+ V_{BY}(r) = \sqrt{1+2V_{\text{eff}}(r)}\;\;.
\label{vbyveff}
\ee
While this relation, which we may also read as the relation between the MS
energy and the BY energy, may appear to be somewhat obscure, it reveals several
interesting features of the BY potential $V_{BY}(r)$
and its relation to $V_{\text{eff}}(r)=V_{MS}(r)$:
\begin{enumerate}
\item First of all
we observe that (\ref{vbyveff}) and (\ref{veff2})
allow us to express the BY potential in terms of geodesic quantities as
\be
\label{vbyr}
1+ V_{BY}(r) = \sqrt{E^2 -\dot{r}^2}\leq E\;\;.
\ee
In other words, the relation between the Brown-York potential and the
relativistic energy $E$
(per unit rest mass) of the particle can be phrased as
\textit{The energy $E$ of the geodesic particle is greater or equal to
the sum of its rest mass and the gravitational potential energy (as measured
by $V_{BY}(r)$), with equality at points where $\dot{r}=0$.}
Thus the BY potential appears to provides a reasonable
measure of the energy of the gravitational field in this context.
The inequality $E\geq 1+ V_{BY}(r)$ should be compared and
contrasted with the analogous equation $E_{\text{eff}} \geq V_{MS}(r)$
for the MS (or effective) potential that follows from (\ref{veff2}).
This suggests a certain
analogy $E_{BY} \lra E$ and $E_{MS} \lra E_{\text{eff}}$.
\item This analogy is strengthened by the observation that \eqref{vbyveff}
implies
\be
V_{\text{eff}}(r) = \trac{1}{2}\left((1+V_{BY}(r))^2-1\right),
\label{vv2}
\ee
which shows that the relation between $V_{\text{eff}}$ and
$1+V_{BY}$ is identical to the relation $E_{\text{eff}} = \frac{1}{2}(E^2-1)$
\eqref{eeff}
between the effective energy $E_{\text{eff}}$ and the geodesic particle
energy $E$.
Thus, since $E$ is a relativistic energy and $E_{\text{eff}}$ an effective
Newtonian quantity, it is tempting to say that the BY energy provides one
with a relativistic notion of gravitational energy while the MS energy
is really more like an effective Newtonian quantity. So far, however,
this is only a suggestion, based on the geodesic analogy that we have
developed here, and further analysis of this issue, in other settings,
will be required to substantiate (or disprove) this interpretation of
the difference between $E_{MS}$ and $E_{BY}$.
\item
Finally, \eqref{vbyveff} implies that
$V_{\text{eff}}(r)$ and
$V_{BY}(r)$ have the same zeros
and that the BY potential is repulsive/positive whenever (and
whereever) the effective potential is repulsive.
Thus the BY energy is negative if and only if the effective
potential is repulsive. In particular, \eqref{vbyr} leads
to a simple expression for the BY energy at any
turning point $r_m$ ($\dot{r}_m=0$) of the potential, namely
$1+ V_{BY}(r_m) = E$ or
\be
E_{BY}(r_m) = \frac{r_m}{G_N}(1-E)\;\;.
\ee
This also follows directly from the definition \eqref{eby2} and the
previously noted $E=f(r_m)$.
In particular, $E_{BY}(r_m)$ is negative for scattering trajectories with
$E> 1$. Thus non-positive BY energy is necessary for a repulsive
behaviour of radial geodesics. This provides a simple explanation and
proof of a generalisation of the
observation made in \cite{lsy} in the context of the Reissner-Nordstr\o m
metric. Note also that, for the Schwarzschild metric, at $r=r_m$ one has
$E_{BY}(r_m)=E_{BY}^{\text{Novikov}}(r_m)$,
so that static and freely falling
observers can agree on the energy at a turning point of the freely falling
observer, as they should.
\end{enumerate}
All in all this provides us with a coherent picture of the relation
between geodesic notions of energy on the one hand, and the quasilocal
gravitational MS and BY energies for round spheres on the other.
Finally we comment briefly, from the present geodesic point of view,
on the extension $E_{LSY}(r)$
of the BY energy $E_{BY}(r)$ to the interior of a horizon proposed in
\cite{lsy}. Writing the Schwarzschild-like metric as
\be
ds^2 = -\epsilon f(r)^2 dt^2 + \epsilon f(r)^{-2} dr^2 + r^2 d\Omega^2\;\;,
\label{met5}
\ee
with $\epsilon=\pm 1$ corresponding to the exterior (interior) region,
the definition of \cite{lsy} is
\be
E_{LSY}(r) = \frac{r}{G_N}(1-\epsilon f(r))
\ee
($E_{LSY}(r) = E_{BY}(r)$ in the region $\epsilon=+1$).
We now write the effective potential equation for radial timelike
($\lambda=+1$) or spacelike ($\lambda=-1$) geodesics as
\be
\trac{1}{2}\dot{r}^2 + V_{\text{eff}}^{\lambda}(r) = E_{\text{eff}}^{\lambda}
\ee
where
$E_{\text{eff}}^{\lambda} = \trac{1}{2}(E^2-\lambda)$.
Then one easily finds
\be
\epsilon\lambda V_{\text{eff}}^{\lambda}(r) =
\trac{1}{2}\left((1+V_{LSY}(r))^2-\epsilon\right)\;\;.
\label{vv3}
\ee
This relation between the effective potential
$V_{\text{eff}}^{\lambda}(r)$ and
$1+V_{LSY}(r)$ is identical to the relation
between the effective Newtonian energy $E_{\text{eff}}^\lambda$
and the relativistic geodesic
energy $E$ for any $\epsilon$ provided that one
correlates the region of interest (specified by
$\epsilon$) with the character of the geodesic (indicated by $\lambda$)
by making the choice
$\epsilon = \lambda$.
Thus for $\epsilon=-1$
$E_{LSY}(r)$ appears to be naturally associated with spacelike
geodesics.
\subsection*{Acknowledgements}
We are grateful to the referees for their useful comments and suggestions.
This work forms part of a Master thesis project of B.R.\ performed jointly
at the Universit\'e de Gen\`eve and the Universit\'e de Neuch\^atel. M.B.\
acknowledges financial support from the Swiss National Science Foundation
and the EU under contract MRTN-CT-2004-005104.
\renewcommand{\Large}{\normalsize}
| {'timestamp': '2008-04-04T17:24:07', 'yymm': '0708', 'arxiv_id': '0708.0321', 'language': 'en', 'url': 'https://arxiv.org/abs/0708.0321'} |
\section*{Dynamical Behaviour of $O$ in Lattice Gases}
The dynamical behaviour of the anisotropic order parameter $m$ [see Eq.~\eqref{eq:def-m} in the Letter] following a quench to the critical point is well described by
the Gaussian theory for all the three lattice gas models studied, $i.e.,$ driven lattice gas with either constant (IDLG) or random (RDLG) infinite drive and equilibrium lattice gas (LG). In other words, in the short-time regime, $m \sim t^{1/2}$ [see Eq. \eqref{eq:mt}] and the Binder cumulant $g$ of the lowest transverse mode [defined in Eq. \eqref{eq:binder}] is zero in this regime. The alternative order parameter $O,$ however, distinguishes between the driven (IDLG, RDLG) and the equilibrium (LG) lattice gases.
In order to understand this, we first write the phenomenological scaling form for $O$, analogous to Eq. \eqref{eq:scalingass} in the Letter,
\begin{eqnarray}
O (t, L_{\parallel} ; S_\Delta) = L_{\parallel}^{-\beta/[\nu(1+\Delta)]} \tilde f_O (t/L_{\parallel}^{z/(1+\Delta)} ; S_\Delta).\quad
\label{eq:Oscalingass}
\end{eqnarray}
We already remarked that, in the LG, this scaling form is not compatible with the prediction $O \sim t^{1/8} L_{\parallel}^{-1/2}$ of the Gaussian theory. However, following Ref. \cite{AS2002}, it can be argued that, at short times, the only dependence of $O$ on the system size $L_{\parallel}$ is of the form $O \sim L_\parallel^{-1/2}$ which is very well confirmed by numerical simulations. Accordingly, the generic behaviour of $O$ can be assumed to be
\begin{eqnarray}
O \sim t^{\alpha} L_\parallel^{-1/2}, \label{eq:O}
\end{eqnarray}
where $\alpha$ is a phenomenological exponent to be determined. This, along with Eq. \eqref{eq:Oscalingass}, implies $\tilde f_O(x) \sim x^{\alpha}.$ Comparing the finite-size behaviour in Eq.~\eqref{eq:O} with Eq.~\eqref{eq:Oscalingass} one actually infers,
\begin{eqnarray}
\alpha &=& \frac{1+ \Delta -2 \beta/\nu}{2 \, (4- \eta)}. \label{eq:alpha}
\end{eqnarray}
This equation, together with the hyperscaling relation $\Delta - 2 \beta/\nu= - \eta$ in two spatial dimensions, shows that the prediction $\alpha = 1/8$ of the Gaussian theory [see Eq. \eqref{eq:Ot}] can be obtained only when $\eta=0,$ which is the case for the IDLG (exactly) and the RDLG (approximately) but not for the LG.
On the other hand, Eq.~\eqref{eq:alpha} predicts $\alpha = 1/10$ upon substituting the values of the critical exponents corresponding to the Ising universality class (LG). This is consistent with the numerical simulation results presented in the main text, see Fig. \ref{fig:ising}(b) therein.
\begin{figure}[th]
\vspace*{0.2 cm}
\centering
\includegraphics[width=10 cm]{./compare_binder.pdf}
\caption{Comparison between the temporal evolution of the Binder cumulants $g$ corresponding to the $12^{th}$ transverse mode, $i.e.,$ with $n_\perp =12,$ in the LG (lowest curve), IDLG and RDLG (two upper curves) on a $32 \times 32$ lattice. \label{fig:b}}
\label{fig:binder}
\end{figure}
The emergence of this new value $1/10$ of the exponent $\alpha$ must be traced back to the non-Gaussian nature of higher fluctuating modes in the LG. In fact, even though the lowest mode behaves identically in all the three models we considered, characterized by the same behaviour of $m$, higher modes show a significant difference in the non-driven case.
To illustrate this, we measured the Binder cumulants of higher modes which is defined analogously to Eq.~(11), using transverse modes other than the first, i.e., with $\mu=\tilde \sigma(0,2 \pi n_\bot/L_\bot)$ and $n_\bot>1.$
Figure \ref{fig:b} compares the same for all the three lattice gases for the mode with $n_\perp =12$ on a $32 \times 32$ lattice. Clearly, the curve corresponding to the LG (lowest, blue) departs from Gaussian behaviour $g=0$ (in practice, $e.g.,$ $|g| \lesssim 0.005,$ corresponding to the shaded gray area) much earlier than it does for the IDLG or RDLG (two upper curves, red and green respectively).
Accordingly, the different dynamical behaviour of $O$, which involves a sum over all modes, can be attributed to the non-Gaussian nature of the higher modes in the LG.
Such a departure is not entirely surprising. In fact, for higher modes, mesoscopic descriptions such as the ones in Eqs. \eqref{eq:L-DLG} or \eqref{eq:g_evol} are not expected to hold, while the anisotropy at the microscopic level could be the mechanism leading to the Gaussianity of higher modes in the driven models.
| {'timestamp': '2017-02-16T02:07:38', 'yymm': '1607', 'arxiv_id': '1607.01689', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.01689'} |
\section{Introduction}\label{sec:intro}
The ATLAS and CMS collaborations at Large Hadron Collider (LHC) have
recently discovered a new bosonic resonance of mass around 125 GeV
\cite{:2012gk, ATLAS:science,:2012gu,CMS:science, :2012br}. Measuring its coupling
to different Standard Model (SM) particles and establishing its nature are going to
be leading aims of future LHC runs. Although it is yet to be confirmed
as SM Higgs, in this paper we specify this resonance as Higgs and denote it by $H$.
Any deviations from its SM nature should exhibit in its coupling to
different particles. Anomalous couplings of Higgs may come in both gauge and Yukawa
sectors. Establishing the nature of the Higgs will require a precise
measurement of its gauge as well as Yukawa couplings. In future LHC runs the coupling of Higgs to $W, Z$ bosons
will be measured in several different channels such as $H \to Z Z^{*}\to 4 \ell$. However, measuring
its coupling to fermions as well as loop induced couplings like $HZ\gamma$ are going to be
relatively more challenging.
In this regard, several studies
\cite{Isidori:2013cla,Gonzalez-Alonso:2014rla,Buchalla:2013mpa,Gao:2014xlv,Bhattacharya:2014rra,Keung:1983ac,Hagiwara:1993sw,Curtin:2013fra,Bodwin:2014bpa,Korchin:2014kha,Delaunay:2013pja,Giudice:2008uua,Kagan:2014ila,Chen:2014gka,
Beneke:2014sba,Manohar:2000dt,Isidori:2013cga,Brod:2013cka,Falkowski:2014ffa,Bergstrom:1985hp,Grinstein:2013vsa,Bodwin:2013nua}
have been
directed towards rare Higgs decays such as $H \to Z V$; $V$ being a vector quarkonium ($J^{\rm{PC}} = 1^{--}$).
Although the branching ratios are small, rare Higgs decays offer
complimentary information about Higgs couplings~\cite{Isidori:2013cla} and can serve as important probe of ``New Physics'' (NP).
Besides, subsequent decays of $Z$ and $V$ into pair of leptons make them experimentally clean channels.
Moreover, the decay rates are further enhanced by resonant production of $V$ and could be seen in high
luminosity LHC runs or in future colliders. Among rare Higgs decays, the decay to a vector quarkonium
($ J/\psi, \Upsilon$) received considerable attention in recent times.
Refs.~\cite{Gonzalez-Alonso:2014rla,Gao:2014xlv,Bhattacharya:2014rra} have studied $H \to Z V$ process
with the aim to probe Higgs couplings and new physics.
There exist three different processes that contribute to the $H \to Z V$
decay. Refs.~\cite{Isidori:2013cla,Gonzalez-Alonso:2014rla} calculate
the decay rates for $H \to Z V$ via $H \to Z^* Z$ with $Z^* \to V$.
Although in SM the process $H \to Z \gamma^{*} \to Z V$ is loop suppressed,
Ref.~\cite{Gao:2014xlv} shows that it can provide a
significant contribution depending on the nature of the vector boson $V$.
There exist a third contribution where $H \to Z V$ is produced via $H \to q \bar{q}\to Z V$
and Ref.~\cite{Bhattacharya:2014rra} studies this channel assuming anomalous coupling
in Yukawa sector. As any of the above three processes could be anomalous, in this paper we
perform a model independent analysis of $H\to ZV$ decay without making any assumption on its
origin.
We first calculate the SM contribution of the three processes and their respective interferences
to the $H\to ZV$ decay.
We also write down the most general $H Z V$ vertex and derive corresponding angular asymmetries from
it. These asymmetries have been discussed in Ref.~\cite{Modak:2013sb,Modak:2014zca} in the context of
$H \to Z Z^*\to 4 \ell$ and also in Ref~.\cite{Buchalla:2013mpa} to probe non standard Higss coupling
via angular analysis. They provide powerful tools which can probe SM as well as any anomalous contributions to
the decay. Similar asymmetry has also been discussed in Ref.~\cite{Bhattacharya:2014rra}
to measure $CP$ odd properties of Yukawa sector in $H \to Z V$ decays.
In our work we construct all possible asymmetries and perform a case by case analysis discussing relative contributions
of different diagrams and their consequences on respective asymmetries.
The plan of the paper goes as follows. In section~\ref{sec:direct} we compute the SM contributions
of the three processes and compare their relative strengths.
Section~\ref{sec:anghvz} is devoted to formalize the angular analysis and construction of angular
asymmetries for $H \to Z V$ with further decays of $V$ and $Z$ into pair of leptons. We also
discuss how to probe different Higgs coupling using these angular asymmetries. In Section~\ref{sec:conclusion} we conclude
our results.
\section{Standard Model contribution of different channels to $H \to Z V $ process}
\label{sec:direct}
\begin{figure*}[hbtp!]
\centering
\includegraphics[width=.9\textwidth]{hvz.eps}
\caption{Feynman diagrams contributing to $H \to Z V$, $V$ being a vector quarkonium resonance.
The diagrams originate from three different couplings: (a) tree level $HZZ$ coupling,
(b) loop induced $HZ\gamma$ coupling, (c) $H q \bar{q}$ Yukawa coupling.}
\label{fig1}
\end{figure*}
We start our discussion by first estimating the relative strength of SM contribution of different channels to
the process $H \to Z V$, where $V$ is a vector quarkonium ($J^{PC}=1^{--}$). In particular we will focus on $J/\psi(1S)$ and $\Upsilon(1S)$ but our analysis is general and can be used for any vector quarkonium. These decays receive contributions from three
different diagrams as shown in Fig.\ref{fig1}. Some of these contributions have been individually studied in previous
works \cite{Gonzalez-Alonso:2014rla, Gao:2014xlv, Bhattacharya:2014rra} but a combined analysis is still lacking.
The relative strengths of the diagrams and their interference terms
vary depending on the final vector quarkonium resonance. Because of quite different masses of
$J/\psi$ and $\Upsilon$ resonances the relative
strengths of these diagrams differ appreciably in the two cases.
We explicitly calculate the individual contributions for $J/\psi (1S)$ and $\Upsilon(1S)$
to demonstrate this fact.
In SM the first diagram Fig.\ref{fig1}(a), originates from tree level $HZZ$ gauge coupling.
The matrix element for it is given by
\begin{align}
\mathcal{M}_1 & = -\mathcal{K}_1\left(a^{ZZ}_1 g_{\mu \nu}\right)\epsilon_1^{*\mu}\epsilon_2^{*\nu}
\end{align}
where
\begin{align}
\mathcal{K}_1=\frac{2 g~g^q_V f_V}{\cos\theta_W}\frac{M_V M_Z^2}{M_Z^2-M_V^2}\label{k1},
\end{align}
with $\theta_W$ as Weinberg angle, $g^q_V=(\frac{1}{4}-\frac{2}{3}\sin^2\theta_W)$ for Charm($c$) quark and
$g^q_V=(-\frac{1}{4}+\frac{1}{3}\sin^2\theta_W)$ for Bottom($b$) quark. Also, $\epsilon_1^{\mu}(q_1)$ and $\epsilon_2^{\nu}(q_2)$
are the polarization vectors for $Z$ and $V$ having momenta $q_1$ and $q_2$ respectively. Moreover,
$f_V$ is defined by the matrix element $\bracket{0}{\bar{q}\gamma^{\mu}q}{V(q_2,\epsilon_2)}=f_V M_V \epsilon_2^{\mu}$.
Since in SM the $HZ\gamma$ coupling is forbidden at tree level, the second diagram Fig.\ref{fig1}(b),
can only arise via loop processes. One can compute this process by writing down an effective lagrangian
for the $HZ\gamma$ coupling \cite{Korchin:2013ifa,Gao:2014xlv,Bergstrom:1985hp,Hagiwara:1993sw}
The matrix element for this diagram is given by
\begin{align}
\mathcal{M}_2 &=-\mathcal{K}_2 \left(a^{Z\gamma}_1 ~q_1.q_2 ~g_{\mu \nu}- a^{Z\gamma}_2 q_{1\nu}q_{2\mu}\right)
\epsilon_1^{*\mu}\epsilon_2^{*\nu}
\end{align}
where
\begin{align}
\mathcal{K}_2 = \frac{g\,\alpha\, Q^f f_V}{2 \pi v} \frac{C_{Z\gamma}}{M_V}.
\end{align}
$C_{Z\gamma}$ is the dimensionless effective
coupling constant for the $HZ\gamma$ vertex \cite{Bergstrom:1985hp,Hagiwara:1993sw,Korchin:2013ifa}, $\alpha = \frac{e^2}{4 \pi}$ and
$Q^f = \frac{2}{3}, \frac{-1}{3}$ for $V = J/\psi, \Upsilon$ respectively.
The third contribution Fig.\ref{fig1}(c) comes from $Hq\bar{q}$ Yukawa coupling and is given by
\begin{align}
\mathcal{M}_3 &=-\mathcal{K}_3 \left(a^{Zq\bar{q}}_1 ~q_1.q_2 ~g_{\mu \nu}- a^{Zq\bar{q}}_2 q_{2\mu} q_{1\nu}\right)
\epsilon_1^{*\mu}\epsilon_2^{*\nu}
\end{align}
where
\begin{align}
\mathcal{K}_3 = \frac{4\sqrt{3} g g^q_V \phi_0}{\cos \theta_W\left(M_H^2-M_Z^2-M_V^2\right)}\left(\frac{M_V G_F}{2\sqrt{2}}\right)^{\frac{1}{2}},
\end{align}
and $\phi_0$ is the wave function of the
vector quarkonium resonance evaluated at zero three momentum \cite{Bodwin:2013gca,Bodwin:1994jh,Bodwin:2014bpa}.
The total decay width for $H \to Z V$ process is combination of all three contributions
given by
\begin{align}
\Gamma_{total}& =\Gamma_1+\Gamma_{2}+\Gamma_{3}+\Gamma_{12}+\Gamma_{13}+\Gamma_{23}.
\end{align}
where $\Gamma_i$ are obtained from $|\mathcal{M}_i|^2$ and $\Gamma_{ij}$
are interference terms between $\mathcal{M}_i$ and $\mathcal{M}_j$ with $i,j=1,2,3$. The individual
contributions for both $ J/\psi(1S) $ and $\Upsilon(1S)$ are listed in Table~\ref{tab}.
\begin{table}[htb]
\caption{Contributions to the branching fraction from the three contributing diagrams and their interferences for $J/\psi(1S)$ and $\Upsilon(1S)$ resonances. The total decay width of Higgs is assumed to be $4.07$ MeV. Values of $f_V=0.405(0.680)$ GeV
\cite{Gonzalez-Alonso:2014rla} and $\phi^2_0=0.073(0.512)$ $\rm{GeV}^3$\cite{Bodwin:2013gca} for $J/\psi(\Upsilon)$.}
\begin{tabular}{c c c}\hline\hline
$\mathcal{B}r(H \to Z V)$ \hspace{1cm} & $ J/\psi(1S) $ \hspace{1cm} & $\Upsilon(1S)$ \\ \hline
$\mathcal{B}r_{\Gamma_1}$ \hspace{1cm} & $1.75 \times 10^{-6}$ \hspace{1cm} & $1.68 \times 10^{-5}$ \\
$\mathcal{B}r_{\Gamma_2}$ \hspace{1cm} & $1.14 \times 10^{-6}$ \hspace{1cm} & $8.33 \times 10^{-8}$ \\
$\mathcal{B}r_{\Gamma_3}$ \hspace{1cm} & $8.52 \times 10^{-9}$ \hspace{1cm} & $5.80 \times 10^{-7}$ \\
$\mathcal{B}r_{\Gamma_{12}}$ \hspace{1cm} & $4.50 \times 10^{-7}$ \hspace{1cm} & $1.10\times 10^{-6}$ \\
$\mathcal{B}r_{\Gamma_{13}}$ \hspace{1cm} & $3.89 \times 10^{-8}$ \hspace{1cm} & $2.89 \times 10^{-6}$ \\
$\mathcal{B}r_{\Gamma_{23}}$ \hspace{1cm} & $1.97\times 10^{-7}$ \hspace{1cm} & $4.40\times 10^{-7}$ \\ \hline \hline
\end{tabular}
\label{tab}
\end{table}
From Table\,\ref{tab} it is clear that the relative contributions of the three channels is different
for $ J/\psi $ and $\Upsilon$ resonances. In case of $ J/\psi$ the dominant contributions
come from $\Gamma_{1}$ and $\Gamma_{2}$ corresponding to $HZZ$ and $HZ\gamma$ couplings respectively.
The subleading contributions come from the interference terms $\Gamma_{12}$ and $\Gamma_{23}$. The
contribution $\Gamma_{3}$ coming from $Hq\bar{q}$ coupling is negligibly small. The
major contribution from Yukawa sector will come from the interference term $\Gamma_{23}$.
Therefore while probing the anomalous Yukawa couplings one should not neglect the contribution of
the interference terms over $\Gamma_{3}$.
However, in case of $\Upsilon$ the situation is quite different. The leading contribution comes only from the $\Gamma_{1}$ term
whereas $\Gamma_{12}$ and $\Gamma_{13}$ provide the subleading contributions.
The contribution of $\Gamma_{3}$ is now larger than $\Gamma_{2}$ but still negligibly small
compared to $\Gamma_{1}$. Again as before while probing anomalous Yukawa coupling the
effect of interference terms can not be neglected.
As discussed above the rare Higgs decays $H \to Z V$ are sensitive not only
to $HZZ$ coupling but also to $HZ\gamma$ and $Hq\bar{q}$ couplings. Moreover,
depending on nature of $V$, the contribution of various Higgs couplings to the decay
widths vary widely. Hence these decay modes have potential to provide information
complimentary to $H \to Z Z^{*} \to 4 \ell$ ``golden channel''.
Also, $H \to Z V$ decays followed by a subsequent
decay of $Z$ and $V$ into pair of leptons will provide a experimentally clean channel
that can be used to probe them in future colliders or high luminosity LHC runs.
In next section we will discuss the angular analysis technique which provide a powerful tool
for probing such couplings.
\section{Angular analysis and Observables for $H \to Z V \to 4 \ell$ process}
\label{sec:anghvz}
In this section we formalize the necessary technique to probe $HZV$ vertex.
We start with writing down general structure of the vertex and the different helicity amplitudes for
$H \to Z V$ process.
The general Lorentz structure of the $HZV$ vertex can be written as
\begin{equation}\label{eq:VHZZ0}
V^{\alpha \beta}_{HZV}=
\bigg( a_1 \, g^{\alpha \beta} + a_2 \, P^{\alpha}P^{\beta}
+ i a_3 \,\epsilon^{\alpha \beta \mu \nu} \; q_{1\mu}\,q_{2\nu} \bigg),
\end{equation}
where $a_1$, $a_2$ and $a_3$ are vertex factors, $P$ is the momentum of
Higgs boson and $q_1$ and $q_2$ are momenta of $Z$ and $V$ respectively. In SM
\begin{align}
a^{\rm{SM}}_1 & = -(\mathcal{K}_1\,a^{ZZ}_1+\mathcal{K}_2\,a^{Z\gamma}_1 q_1.q_2+\mathcal{K}_3\,a^{q\bar{q}}_1 q_1.q_2),
\label{a1} \\
a^{\rm{SM}}_2 & = \left(\mathcal{K}_2~a^{Z\gamma}_2+\mathcal{K}_3 a^{q\bar{q}}_2\right), \label{a2} \\
a^{\rm{SM}}_3 & = 0. \label{a3}
\end{align}
The couplings $a_1$, $a_2$ and $a_3$ can be extracted via angular asymmetries discussed below.
Any deviation from SM values will indicate anomalous nature of $H \to Z V$ decay.
The decay under consideration can be expressed in terms of three helicity
amplitudes $\mathcal{A}_{L}$, $\mathcal{A}_{\parallel}$ and $\mathcal{A}_{\perp}$ defined
in the transversity basis as
\begin{align}
\mathcal{A}_{L} &= (M_H^2 - M_Z^2 - M_V^2)\, a_1 + M_H^2\, X^2 \, a_2, \label{eq:AL}\\%[2ex]
\mathcal{A}_{\parallel} &= \sqrt{2 }M_H M_V \, a_1, \label{eq:AA}\\%[2ex]
\mathcal{A}_{\perp} &= \sqrt{2}M_H^2 M_V\, X \, a_3, \label{eq:AP}
\end{align}
where $M_H$, $M_Z$ and $M_V$ are masses of $H$, $Z$ and $V$ respectively with
\begin{eqnarray}\label{eq:X}
X&=&\displaystyle\frac{\sqrt{\displaystyle\lambda(M_H^2,M_Z^2,M_V^2)}}{\displaystyle 2M_H}
\end{eqnarray}
where
$\lambda(x,y,z)= x^2+y^2+z^2-2\,x\,y-2\,x\,z-2\,y\,z$.
The helicity amplitudes $\mathcal{A}_{L}$, $\mathcal{A}_{\parallel}$ and $\mathcal{A}_{\perp}$ have
definite parity properties. $\mathcal{A}_{L}$, $\mathcal{A}_{\parallel}$ are $CP$ even in nature
where as $\mathcal{A}_{\perp}$ is $CP$ odd.
The full angular distribution for $H \to Z_{(\ell^+ \ell^-)} V_{(\ell^+ \ell^-)}$ is given
by following expression
\begin{widetext}
\begin{align}\label{angdist}
& \frac{8\pi}{\Gamma}\frac{d^3\Gamma}{ d\cos{\theta_1} \;
d\cos{\theta_2} \; d\phi} = 1 +\frac{|f_{\parallel}|^2-|f_{\perp}|^2}{4}
\cos\,2\phi\big(1-P_2(\cos\theta_1)\big)
\big(1-P_2(\cos\theta_2)\big)
+\frac{1}{2} \text{Im}(f_{\parallel} f_{\perp}^*)\,\sin\,2\phi \nonumber \\
& \times \big(1-P_2(\cos\theta_1)\big) \big(1-P_2(\cos\theta_2)\big)
+\frac{1}{2}(1-3
\modulus{f_L}^2)\,\big(P_2(\cos\theta_1)+P_2(\cos\theta_2)\big)
+\frac{1}{4}(1+3\modulus{f_L}^2)\,P_2(\cos\theta_1)P_2(\cos\theta_2)\nonumber\\
& +\frac{9}{8\sqrt{2}} \left( \text{Re}(f_L
f_{\parallel}^*)\,\cos\phi + \text{Im}(f_L f_{\perp}^*)\,\sin\phi
\right)
\sin\,2\theta_1\,\sin\,2\theta_2
+\eta \Bigg(\frac{3}{2} \text{Re}(f_{\parallel}
f_{\perp}^*)\big(\cos\theta_2 (2 + P_2(\cos\theta_1)) \nonumber\\
& - \cos\theta_1
(2 + P_2(\cos\theta_2))\big)
-\frac{9}{2\sqrt{2}}\text{Re}(f_L f_{\perp}^*)
\cos\theta_2\cos\phi \sin\theta_1\sin\theta_2
+\frac{9}{2\sqrt{2}}\text{Im}(f_L f_{\parallel}^*)
\cos\theta_2\sin\phi
\sin\theta_1\sin\theta_2\Bigg),
\end{align}
\end{widetext}
where the angle $\theta_1$($\theta_2$) is the angle between three momenta of $\ell^+$ in $Z$($V$) rest frame
and the direction of three momenta of $Z$($V$) in $H$ rest frame. The angle $\phi$ is defined as the angle between
the normals to the planes defined by $Z\to \ell^+ \ell^-$ and $V\to \ell^+ \ell^-$ in $H$ rest frame.
The expressions for {\it{helicity fractions}} $f_L$, $f_{\parallel}$ and $f_{\perp}$ are given in the appendix.
Integrating Eq.\eqref{angdist} with respect to the angles
$\cos{\theta_1}$ or $\cos{\theta_2}$ or $\phi$, one can obtain following
uniangular distributions:
\begin{align}
\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta_1} &=
\frac{1}{2} + t_2\,P_2(\cos\theta_1) - t_1
\cos\theta_1, \label{eq:t1dist} \\
\frac{1}{\Gamma}\frac{d\Gamma}{ d\cos\theta_2} &=
\frac{1}{2} + t_2\,P_2(\cos\theta_2) , \label{eq:t2dist} \\
\frac{2\pi}{\Gamma}\frac{d\Gamma}{d\phi} &= 1 +
u_2\,\cos\,2\phi + v_2\,\sin\,2\phi \label{eq:phidist}
\end{align}
where $P_2(\cos\theta_{1,2})$ are second degree Legendre Polynomial and
\begin{align}
\label{eq:t1}
t_1 &=\frac{3}{2} \, \eta \text{Re}(f_{\parallel} f_{\perp}^*),\\
\label{eq:t2}
t_2 &=\frac{1}{4} (1-3\modulus{f_L}^2),\\
\label{eq:v2}
v_2 &=\frac{1}{2}\,\mathrm{Im}(f_{\parallel}f_{\perp}^*),\\
\label{eq:u2}
u_2 &=\frac{1}{4} (|f_{\parallel}|^2-|f_{\perp}|^2).
\end{align}
The uniangular distributions in Eq.\eqref{eq:t1dist}, Eq.\eqref{eq:t2dist} and Eq.\eqref{eq:phidist}
will give us arsenal to probe the $H\to Z V$ coupling.
The observables $t_1$, $t_2$, $u_2$ and $v_2$ can be extracted using following
asymmetries:
\begin{align}
& t_1 = \frac{1}{\Gamma} \left( \int_{-1}^{0} - \int_{0}^{+1} \right)
\frac{d\Gamma}{ d\cos\theta_1} \, d\cos\theta_1,
\label{eq:T10asym}\\%
& t_2 = \!\frac{4}{3 \Gamma}\! \left( \int_{-1}^{-\frac{1}{2}} -
\int_{-\frac{1}{2}}^{+\frac{1}{2}} + \int_{+\frac{1}{2}}^{+1} \right)
\frac{d\Gamma}{d\cos\theta_{1,2}} d\cos\theta_{1,2}
\label{eq:T20asym}\\%
& v_2 = \frac{\pi}{2\Gamma} \left( \int_{-\pi}^{-\frac{\pi}{2}} -
\int_{-\frac{\pi}{2}}^{0} + \int_{0}^{+\frac{\pi}{2}} -
\int_{+\frac{\pi}{2}}^{+\pi} \right) \frac{d\Gamma}{d\phi}d\phi,
\label{eq:V20asym}\\%
& u_2 = \frac{\pi}{2 \Gamma} \left( \int_{-\pi}^{-\frac{3\pi}{4}} -
\int_{-\frac{3\pi}{4}}^{-\frac{\pi}{4}} +
\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}
-\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} + \int_{\frac{3\pi}{4}}^{\pi}
\right)\frac{d\Gamma}{d\phi} d\phi.
\label{eq:U20asym}
\end{align}
The observables $t_1$, $t_2$, $u_2$, $v_2$ are functions of $a_1$, $a_2$,
$a_3$ and their measurements will allow us to probe $H\to Z V$ coupling.
In SM $t_1$, $t_2$, $u_2$, $v_2$ have unique values which can be computed using
the SM values of the couplings $a_1$, $a_2$, $a_3$ given in Eq.\eqref{a1},
Eq.\eqref{a2} and Eq.\eqref{a3}. The anomalous nature, if any, of $a_1$, $a_2$ , $a_3$ will
show up in the observables as deviation from their SM values.
As discussed in Section \ref{sec:direct}, the rare Higgs decays are sensitive to $HZZ$, $HZ\gamma$
and $Hq\bar{q}$ couplings. Therefore, any deviation of the observables $t_1$, $t_2$, $u_2$, $v_2$ from their SM values
can not a priori be attributed to anomalous nature of any one sector. However, when taken in conjugation with
other decays like $H\to Z Z^{*}\to 4\ell$ they can provide complimentary information about $HZ\gamma$
and $Hq\bar{q}$ couplings. For example, if any hint of anomalous nature is found in $H\to Z Z^{*}\to
4\ell$ decay, one expects to see corresponding deviations in the observables of $H \to ZV$ for both $J/\psi$ and $\Upsilon$.
On the other hand if $H\to Z Z^{*}\to 4\ell$ decay observables turn out to be consistent with the SM
values then $HZZ$ contribution in rare decays should also be SM like. In such a scenario any observed anomaly in $H \to Z V$
can only come from either $HZ\gamma$ or $H q\bar{q}$ couplings. As the magnitude of their contributions in $H \to J/\psi$ and $H \to \Upsilon$ are quite different, this fact can be exploited to further narrow down the origin of the anomalous behaviour.
Moreover, for Higgs decay to both $J/\psi$ and $\Upsilon$, the effect of any anomaly in Yukawa sector will manifest itself predominantly
through the interference terms.
In principle Higgs can have anomalous couplings in more than one sector. If so, it will be relatively more difficult to make any definite conclusions about the relative contributions of the three sectors to the anomalous couplings of Higgs in
$H \to Z V$ decays.
\section{Conclusion}\label{sec:conclusion}
We discuss the feasibility of probing the rare Higgs decays $H\to Z V$
followed by further decay of $V$ and $Z$ into pair of leptons. We have shown that the $H\to Z V$ decays
receive contributions from gauge as well as Yukawa sector and depending on the nature of $V$, can be sensitive to anomalous couplings in more than one sector. They can provide complimentary information about the gauge and Yukawa couplings of Higgs.
We then showed how observables extracted as angular asymmetries will allow us to extract information about $HZV$ coupling.
Such asymmetries have the potential to infer the nature of gauge and Yukawa couplings of Higgs in future colliders or
high luminosity LHC runs.
\acknowledgments
We thank Dibyakrupa Sahoo and Rahul Sinha for many fruitful discussions.
| {'timestamp': '2014-11-11T02:10:24', 'yymm': '1411', 'arxiv_id': '1411.2210', 'language': 'en', 'url': 'https://arxiv.org/abs/1411.2210'} |
\section{Introduction}\label{sec:intro}
Let ${\mathbf{A}}$ be an $m$-by-$n$ real-valued matrix with singular value decomposition (SVD)
\begin{equation*}
{\mathbf{A}}=[{\mathbf{U}} \,\, {\mathbf{U}}_{\perp} ] \left[ \begin{array}{c c}
\boldsymbol{\Sigma}_1 & 0 \\
0 & \boldsymbol{\Sigma}_2
\end{array} \right] \left[ \begin{array}{c}
{\mathbf{V}}^\top \\
{\mathbf{V}}_{\perp}^\top
\end{array} \right],
\end{equation*}
where ${\mathbf{U}} \in \mathbb{R}^{m\times r}$, ${\mathbf{V}} \in \mathbb{R}^{n\times r}$, $[{\mathbf{U}} \,\, {\mathbf{U}}_{\perp}], [{\mathbf{V}} \,\, {\mathbf{V}}_{\perp}]$ are orthogonal and $\boldsymbol{\Sigma}_1,\boldsymbol{\Sigma}_2$ are (pseudo) diagonal matrices with decreasing singular values of ${\mathbf{A}}$. Suppose ${\mathbf{B}} = {\mathbf{A}} + {\mathbf{Z}} \in \mathbb{R}^{m \times n}$, where ${\mathbf{Z}}$ is some perturbation matrix. We similarly write down the SVD of ${\mathbf{B}}$ as
\begin{equation*}
\begin{split}
{\mathbf{B}} =& [\widehat{{\mathbf{U}}} \,\, \widehat{{\mathbf{U}}}_{\perp} ] \left[ \begin{array}{c c}
\widehat{\boldsymbol{\Sigma}}_1 & 0 \\
0 & \widehat{\boldsymbol{\Sigma}}_2
\end{array} \right] \left[ \begin{array}{c}
\widehat{{\mathbf{V}}}^\top \\
\widehat{{\mathbf{V}}}_{\perp}^\top
\end{array} \right]
\end{split}
\end{equation*}
such that $\widehat{{\mathbf{U}}}$ and $\widehat{{\mathbf{V}}}$ share the same dimensions as ${\mathbf{U}}$ and ${\mathbf{V}}$, respectively. The relationship between the singular structures of ${\mathbf{A}}$ and ${\mathbf{B}}$ is a central topic in matrix perturbation theory. Since the seminal work by Weyl \cite{weyl1912asymptotische}, Davis-Kahan \cite{davis1970rotation}, Wedin \cite{wedin1972perturbation}, the perturbation analysis for singular values (i.e., $\boldsymbol{\Sigma}_1,\boldsymbol{\Sigma}_2$ versus $\widehat{\boldsymbol{\Sigma}}_1, \widehat{\boldsymbol{\Sigma}}_2$) and the leading singular vectors (i.e., ${\mathbf{U}},{\mathbf{V}}$ versus $\widehat{\mathbf{U}},\widehat{\mathbf{V}}$) have attracted enormous attentions. For example, \cite{vaccaro1994second,xu2002perturbation,liu2008first} studied perturbation expansion for singular value decomposition; \cite{li1998relative,li1998relative2,londre2000note,stewart2006perturbation} established the relative perturbation theory for eigenvectors of Hermitian matrices and singular vectors of general matrices; \cite{demmel1990accurate,barlow1990computing,demmel1992jacobi,drmavc2008new} studied the numeric computation accuracy for singular values and vectors; more recently, \cite{yu2014useful,cai2018rate,cape2019two} developed several new perturbation results under specific structural assumptions motivated by emerging applications in statistics and data science. The readers are referred to \cite{stewart1990matrix,ipsen2000overview,bhatia2013matrix} for overviews of the historical development of matrix perturbation theory.
While most of the existing works focused on ${\mathbf{U}}$ and ${\mathbf{V}}$ or $\boldsymbol{\Sigma}_1$ and $\boldsymbol{\Sigma}_2$, there are fewer studies on the perturbation analysis of the true matrix ${\mathbf{A}}$ itself. In this paper, we consider the estimation of rank-$r$ matrix ${\mathbf{A}}$ (i.e., $\boldsymbol{\Sigma}_2 = 0$) via rank-$r$ truncated SVD (i.e., best rank-$r$ approximation) of ${\mathbf{B}}$: $\widehat {\mathbf{A}} := \widehat{\mathbf{U}} \widehat\boldsymbol{\Sigma}_1\widehat{\mathbf{V}}^\top$. Such a low-rank assumption and estimation method are widely used in many applications including matrix denoising \cite{gavish2014optimal,donoho2014minimax}, signal processing \cite{tufts1993estimation,jolliffe2002principal} and multivariate statistical analysis \cite{morrison1976multivariate}, etc. We focus on the estimation error in matrix Schatten-$q$ norm: $\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_q$. A tight upper bound on $\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_q$ can provide an important benchmark for both algorithmic and statistical analysis in the applications mentioned above; moreover, it can be used to study some other basic perturbation quantities, such as the pseudo-inverse perturbation $\|\widehat{{\mathbf{A}}}^\dagger - {\mathbf{A}}^\dagger\|_q$ \cite{wedin1973perturbation,stewart1977perturbation}.
As a starting point, it is straightforward to apply the classical perturbation bounds for singular values and vectors to obtain an upper bound on $\|\widehat{\mathbf{A}}-{\mathbf{A}}\|_q$. For example, Wedin \cite{wedin1972perturbation} proved via $\sin\Theta$ Theorem that
\begin{equation}\label{ineq: wedin matrix reconstruction}
\| \widehat{{\mathbf{A}}} - {\mathbf{A}} \|_q \leq \|{\mathbf{Z}}\|_q \left(3 + \| {\mathbf{B}} - \widehat{\mathbf{A}}\|_q/\sigma_r({\mathbf{B}}) \right).
\end{equation}
Another way is utilizing the optimality of SVD (Eckart-Young-Mirsky Theorem) and some basic norm inequalities to obtain:
\begin{equation}\label{ineq: matrix reconstruction via q-norm}
\| \widehat{{\mathbf{A}}} - {\mathbf{A}} \|_q \leq \| \widehat{{\mathbf{A}}} - {\mathbf{B}} \|_q + \|{\mathbf{A}} - {\mathbf{B}}\|_q \leq 2 \| {\mathbf{A}} - {\mathbf{B}} \|_q \leq 2 \| {\mathbf{Z}} \|_q,
\end{equation}
\begin{equation} \label{ineq: matrix reconstruction via spectral norm}
\| \widehat{{\mathbf{A}}} - {\mathbf{A}} \|_q \leq r^{1/q} \| \widehat{{\mathbf{A}}} - {\mathbf{A}} \| \overset{\eqref{ineq: matrix reconstruction via q-norm}}\leq 2 r^{1/q} \|{\mathbf{Z}}\|.
\end{equation}
In contrast, we establish the following result in this paper:
\begin{Theorem}\label{th: low rank matrix reconstruction}
Suppose ${\mathbf{B}} = {\mathbf{A}} + {\mathbf{Z}}$, where ${\mathbf{A}}$ is an unknown rank-$r$ matrix, ${\mathbf{B}}$ is the observation, and ${\mathbf{Z}}$ is the perturbation. Let $\widehat{{\mathbf{A}}} = \widehat{\mathbf{U}} \widehat\boldsymbol{\Sigma}_1\widehat{\mathbf{V}}^\top$ be the best rank-$r$ approximation of ${\mathbf{B}}$. Then,
\begin{equation}\label{ineq:hat-A-A}
\begin{split}
\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_q & \leq \left\{ \begin{array}{ll}
(2^q + 1)^{1/q} \left\|{\mathbf{Z}}_{\max(r)}\right\|_q, & 1\leq q \leq 2;\\
\sqrt{5} \left\|{\mathbf{Z}}_{\max(r)}\right\|_q, & 2\leq q < \infty;\\
2\|{\mathbf{Z}}_{\max(r)}\|, & q=\infty.
\end{array} \right.
\end{split}
\end{equation}
Here ${\mathbf{Z}}_{\max(r)}$ is defined as the best rank-$r$ approximation of ${\mathbf{Z}}$.
\end{Theorem}
The proof of Theorem \ref{th: low rank matrix reconstruction} relies on a careful characterization of $\|P_{\widehat{\mathbf{U}}_\perp}{\mathbf{A}}\|_q$ (where $P_{\widehat {\mathbf{U}}_\perp}$ is the projection onto the subspace spanned by $\widehat{{\mathbf{U}}}_\perp$) in Theorem \ref{th:SVD-projection}, which we refer as the \emph{perturbation projection error bound}. The details will be presented in Section 2.
The established bound \eqref{ineq:hat-A-A} is sharper than the classic results \eqref{ineq: wedin matrix reconstruction}, \eqref{ineq: matrix reconstruction via q-norm} and \eqref{ineq: matrix reconstruction via spectral norm} since $\|{\mathbf{Z}}_{\max(r)}\|_q\leq \|{\mathbf{Z}}\|_q, r^{1/q}\|{\mathbf{Z}}\|$ for any ${\mathbf{Z}}$. When $m, n \gg r$ and the first $r$ singular values of ${\mathbf{Z}}$ decay fast, which commonly happens in many large-scale matrix datasets \cite{udell2019big}, $\|{\mathbf{Z}}_{\max(r)}\|_q$ can be much smaller than $\|{\mathbf{Z}}\|_q, r^{1/q}\|{\mathbf{Z}}\|$ (see an example in Section \ref{sec:main-result}) so that the upper bound of \eqref{ineq:hat-A-A} can be much smaller than \eqref{ineq: wedin matrix reconstruction}, \eqref{ineq: matrix reconstruction via q-norm} and \eqref{ineq: matrix reconstruction via spectral norm}.
Then, we further introduce two lower bounds to justify the tightness of the upper bound in Theorem \ref{th: low rank matrix reconstruction}. Specifically for any $\epsilon > 0$, $1\leq q \leq \infty$, we construct a triplet of matrices $({\mathbf{A}},{\mathbf{Z}},{\mathbf{B}})$ such that
\begin{equation}\label{ineq:general-lower-bound}
\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_q \geq ((2^q + 1)^{1/q} - \epsilon)\|{\mathbf{Z}}_{\max(r)}\|_q >0,
\end{equation}
which suggests that the constant in \eqref{ineq:hat-A-A} cannot be further improved for $q\in [1, 2]\cup\{\infty\}$. In addition, we introduce an estimation error lower bound to show that the rank-$r$ truncated SVD estimator (i.e., $\widehat {\mathbf{A}}$) is minimax rate-optimal over the class of all rank-$r$ matrices.
As a byproduct of the theory in this paper, we derive a subspace (singular vectors) sin$\Theta$ perturbation bound (definition of Schatten-$q$ sin$\Theta$ distance is in Section \ref{sec: notation}):
$$\max\left\{\|\sin \Theta(\widehat{{\mathbf{U}}}, {\mathbf{U}})\|_q, \|\sin \Theta(\widehat{{\mathbf{V}}}, {\mathbf{V}})\|_q\right\} \leq \frac{2\| {\mathbf{Z}}_{\max(r)} \|_q }{\sigma_r({\mathbf{A}})}.$$
This bound is ``user-friendly" as it is free of ${\mathbf{B}}$, $\widehat{\mathbf{U}}$, and $\widehat{\mathbf{V}}$, which are often perturbed and uncontrolled quantities in practice (see more discussions in Section \ref{sec:wedin}).
The rest of this paper is organized as follows. After a brief introduction on notation and preliminaries in Section \ref{sec: notation}, we present the proof of Theorem \ref{th: low rank matrix reconstruction} in Section \ref{sec:main-result} and develop the corresponding lower bounds in Section \ref{sec:lower-bound}. The new $\sin\Theta$ perturbation analysis is done in Section \ref{sec:wedin}. We provide numerical studies to corroborate our theoretical findings in Section \ref{sec: numerical study}. Conclusion and discussions are made in Section \ref{sec: conclusion}. Additional technical results and proofs are collected in Section \ref{sec:additional-proof}.
\subsection{Notation and Preliminaries} \label{sec: notation}
The following notation will be used throughout this paper. The lowercase letters (e.g., $a, b$), lowercase boldface letters (e.g., $\u, \v$), uppercase boldface letters (e.g., ${\mathbf{U}}, {\mathbf{V}}$) are used to denote scalars, vectors, matrices, respectively. For any two numbers $a,b$, let $a \wedge b = \min\{a,b\}$, $a \vee b = \max\{a,b\}$. For any matrix ${\mathbf{A}} \in \mathbb{R}^{m\times n}$ with singular value decomposition $\sum_{i=1}^{m \land n} \sigma_i({\mathbf{A}})\u_i \v_i^\top$, let ${\mathbf{A}}_{\max(r)}= \sum_{i=1}^{r} \sigma_i({\mathbf{A}})\u_i \v_i^\top$ be the best rank-$r$ approximation of ${\mathbf{A}}$, and ${\mathbf{A}}_{\max(-r)} = \sum_{i=r+1}^{m \land n} \sigma_i({\mathbf{A}})\u_i \v_i^\top$ be the remainder. For $q \in [1, \infty]$, the Schatten-$q$ norm of matrix ${\mathbf{A}}$ is defined as $\|{\mathbf{A}}\|_q := \left( \sum_{i=1}^{m \land n} \sigma^q_i({\mathbf{A}}) \right)^{1/q}$. Especially, Frobenius norm $\|\cdot\|_F$ and spectral norm $\|\cdot\|$ are Schatten-$2$ norm and -$\infty$ norm, respectively. In addition, let ${\mathbf{I}}_r$ be the $r$-by-$r$ identity matrix. Let $\mathbb{O}_{r}$ be the set of $r$-by-$r$ orthogonal matrices, $\mathbb{O}_{p, r} = \{{\mathbf{U}} \in \mathbb R^{p\times r}: {\mathbf{U}}^\top {\mathbf{U}}={\mathbf{I}}_r\}$ be the set of all $p$-by-$r$ matrices with orthonormal columns. For any ${\mathbf{U}}\in \mathbb{O}_{p, r}$, $P_{{\mathbf{U}}} = {\mathbf{U}}\U^\top$ is the projection matrix onto the column span of ${\mathbf{U}}$. We also use ${\mathbf{U}}_\perp\in \mathbb{O}_{p, p-r}$ to represent the orthonormal complement of ${\mathbf{U}}$. We use bracket subscripts to denote sub-matrices. For example, ${\mathbf{A}}_{[i_1,i_2]}$ is the entry of ${\mathbf{A}}$ on the $i_1$-th row and $i_2$-th column; ${\mathbf{A}}_{[(r+1):m, :]}$ contains the $(r+1)$-th to the $m$-th rows of ${\mathbf{A}}$.
We use the $\sin \Theta$ norm to quantify the distance between singular subspaces. Suppose ${\mathbf{U}}_1$ and ${\mathbf{U}}_2$ are two $p$-by-$r$ matrices with orthonormal columns. Let the singular values of ${\mathbf{U}}_1^\top {\mathbf{U}}_2$ be $\sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_r \geq 0$. Then $\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)$ is defined as a diagonal matrix with principal angles between ${\mathbf{U}}_1$ and ${\mathbf{U}}_2$:
$$\Theta({\mathbf{U}}_1, {\mathbf{U}}_2) = {\rm diag}\left( \cos^{-1} (\sigma_1), \ldots, \cos^{-1} (\sigma_r) \right).$$
Then the Schatten-$q$ $\sin\Theta$ distance is defined as
\begin{equation}\label{label:Schatten-q-distance}
\|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q = \left\|{\rm diag}(\sin \cos^{-1}(\sigma_1), \ldots, \sin \cos^{-1}(\sigma_r) )\right\|_q = \left(\sum_{i=1}^r (1-\sigma_r^2)^{q/2}\right)^{1/q}.
\end{equation}
Importantly, $\| {\mathbf{U}}_{1 \perp}^\top {\mathbf{U}}_2\|_q = \left\| \sin \Theta({\mathbf{U}}_1, {\mathbf{U}}_2) \right\|_q$ for any $q \in [1,\infty]$ \cite[Lemma 2.1]{li1998relative2}. Several basic properties of the Schatten-$q$ sin$\Theta$ distance are established in Lemma \ref{lm: spectral of sin theta} in Section \ref{sec:additional-proof}.
\section{Proof of Theorem \ref{th: low rank matrix reconstruction}}\label{sec:main-result}
We first introduce the following Theorem \ref{th:SVD-projection}, which quantifies the projection error $\|P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}}\|_q$ (or $\|{\mathbf{A}} P_{\widehat{{\mathbf{V}}}_\perp}\|_q$) under the perturbation model. This result plays a crucial role in the proof of Theorem \ref{th: low rank matrix reconstruction} and may also be of independent interest.
\begin{Theorem}[A perturbation projection error bound]\label{th:SVD-projection}
Suppose ${\mathbf{B}} = {\mathbf{A}} + {\mathbf{Z}}$ for some rank-$r$ matrix ${\mathbf{A}}$ and perturbation matrix ${\mathbf{Z}}$. Then for any $q \in [1, \infty]$,
\begin{equation}\label{ineq:P_U-P_V}
\max\left\{\|P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}}\|_q,\|{\mathbf{A}} P_{\widehat{{\mathbf{V}}}_\perp}\|_q \right\} \leq 2 \|{\mathbf{Z}}_{\max(r)}\|_q.
\end{equation}
\end{Theorem}
Next, the following Lemma \ref{lm:partition-l_q-norm} characterizes the Schatten-$q$ norm of matrix orthogonal projections.
\begin{Lemma}\label{lm:partition-l_q-norm}
Suppose ${\mathbf{A}},{\mathbf{B}}\in \mathbb{R}^{m\times n}$, ${\mathbf{U}}\in \mathbb{O}_{m, r}$, $q\geq 1$. Then,
$$\|P_{\mathbf{U}}{\mathbf{A}} + P_{{\mathbf{U}}_\perp} {\mathbf{B}}\|_q \leq \left\{
\begin{array}{ll}
\left(\|P_{\mathbf{U}}{\mathbf{A}}\|_q^2 + \|P_{{\mathbf{U}}_\perp}{\mathbf{B}}\|_q^2\right)^{1/2}, & 2\leq q\leq \infty;\\
\left(\|P_{\mathbf{U}}{\mathbf{A}}\|_q^q + \|P_{{\mathbf{U}}_\perp}{\mathbf{B}}\|_q^q\right)^{1/q}, & 1\leq q \leq 2.
\end{array}
\right.$$
\end{Lemma}
\begin{proof}[Proof of Lemma \ref{lm:partition-l_q-norm}]
Let ${\mathbf{T}} = P_{\mathbf{U}}{\mathbf{A}} + P_{{\mathbf{U}}_\perp} {\mathbf{B}}$. We construct ${\mathbf{T}}_1 = P_{{\mathbf{U}}} {\mathbf{T}} = P_{\mathbf{U}}{\mathbf{A}}$, ${\mathbf{T}}_2 = P_{{\mathbf{U}}_\perp} {\mathbf{T}} = P_{{\mathbf{U}}_\perp} {\mathbf{B}}$. First we have ${\mathbf{T}}^\top {\mathbf{T}} = {\mathbf{T}}_1^\top {\mathbf{T}}_1 + {\mathbf{T}}_2^\top {\mathbf{T}}_2$. So for $p \geq 1$
\begin{equation*}
\|{\mathbf{T}}\|_{2p}^2 = \|{\mathbf{T}}^\top {\mathbf{T}}\|_p = \|{\mathbf{T}}_1^\top {\mathbf{T}}_1 + {\mathbf{T}}_2^\top {\mathbf{T}}_2\|_p \leq \|{\mathbf{T}}_1^\top {\mathbf{T}}_1\|_p + \|{\mathbf{T}}_2^\top {\mathbf{T}}_2\|_p = \|{\mathbf{T}}_1\|_{2p}^2 + \|{\mathbf{T}}_2\|_{2p}^2,
\end{equation*} and this proves the first part.
For the second part, note that when $q = 1$, the inequality holds by triangle inequality. Next we show the inequality holds when $1 < q \leq 2$. Let ${\mathbf{X}} = \begin{bmatrix} ({\mathbf{T}}_1^\top {\mathbf{T}}_1)^{1/2} & ({\mathbf{T}}_2^\top {\mathbf{T}}_2)^{1/2}
\end{bmatrix}$. Note that for any $1\leq p < \infty$,
\begin{equation} \label{ineq: whole big than block}
\|{\mathbf{T}}_1^\top {\mathbf{T}} _1 + {\mathbf{T}}_2^\top {\mathbf{T}}_2\|_p^p = \| {\mathbf{X}} {\mathbf{X}}^\top \|_p^p = \|{\mathbf{X}}^\top {\mathbf{X}}\|_p^p \overset{(a)}\geq \|{\mathbf{T}}_1^\top {\mathbf{T}}_1\|_p^p + \| {\mathbf{T}}_2^\top {\mathbf{T}}_2\|_p^p,
\end{equation} where (a) is because the norm of the diagonal part of a matrix is always smaller than the norm of the whole matrix \cite{bhatia1988clarkson}. So we have
\begin{equation} \label{ineq: contraction of A}
\|{\mathbf{T}}\|_{2p}^{2p} = \|{\mathbf{T}}^\top {\mathbf{T}}\|_{p}^{p} = \|{\mathbf{T}}_1^\top {\mathbf{T}} _1 + {\mathbf{T}}_2^\top {\mathbf{T}}_2\|_{p}^{p} \overset{\eqref{ineq: whole big than block}} \geq \|{\mathbf{T}}_1^\top {\mathbf{T}}_1\|_p^p + \| {\mathbf{T}}_2^\top {\mathbf{T}}_2\|_p^p = \|{\mathbf{T}}_1\|_{2p}^{2p} + \|{\mathbf{T}}_2\|_{2p}^{2p}.
\end{equation}
Since $2 \leq 2p < \infty$ and Schatten-$q$ norm is the dual of Schatten-$2p$ norm with $1 < q \leq 2$, the second part follows by the duality argument. Specifically, we consider the linear mapping $\mathcal{A}$ such that $\mathcal{A}({\mathbf{T}}) = \begin{bmatrix} {\mathbf{T}}_1 & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{T}}_2
\end{bmatrix}$. It is easy to verify its adjoint $\mathcal{A}^*$ satisfies $\mathcal{A}^*\left(\begin{bmatrix} {\mathbf{T}}_1 & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{T}}_2
\end{bmatrix}\right) = {\mathbf{T}}$. From \eqref{ineq: contraction of A} we have shown for any $2\leq p < \infty$, $\|\mathcal{A}({\mathbf{T}})\|_p \leq \|{\mathbf{T}}\|_p $, i.e., $\mathcal{A}$ is contractive with respect to $\|\cdot\|_{p}$. By duality its adjoint is also a contractive map with respect to $\|\cdot\|_q$ for $1/p + 1/q = 1$ with $1 < q \leq 2$, i.e., $\|{\mathbf{T}}\|_{q}^{q} \leq \|{\mathbf{T}}_1\|_{q}^{q} + \|{\mathbf{T}}_2\|_{q}^{q}$. This finishes the proof.
\end{proof}
Next, we prove Theorem \ref{th: low rank matrix reconstruction} based on Theorem \ref{th:SVD-projection} and Lemma \ref{th: low rank matrix reconstruction}.
\begin{proof}[Proof of Theorem \ref{th: low rank matrix reconstruction}]
For $1 \leq q < \infty$, since $\widehat{{\mathbf{A}}} = {\mathbf{B}}_{\max(r)}$ and $\widehat{\mathbf{U}}$ is composed of the first $r$ left singular vectors of ${\mathbf{B}}$, we have $\widehat{\mathbf{A}} = P_{\widehat{\mathbf{U}}}{\mathbf{B}}$ and
\begin{equation*}
\begin{split}
\left\| \widehat{{\mathbf{A}}} - {\mathbf{A}} \right\|_q & = \left\|P_{\widehat{{\mathbf{U}}}}{\mathbf{B}} - P_{\widehat{{\mathbf{U}}}} {\mathbf{A}} - P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}} \right\|_q = \left\|P_{\widehat{{\mathbf{U}}}}{\mathbf{Z}} - P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}} \right\|_q\\
& \overset{(a)} \leq \left\{ \begin{array}{ll}
\left(\left\|P_{\widehat{{\mathbf{U}}}}{\mathbf{Z}} \right\|^q_q + \left\| P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{A}} \right\|^q_q \right)^{1/q}, & 1\leq q \leq 2;\\
\left(\left\|P_{\widehat{{\mathbf{U}}}}{\mathbf{Z}} \right\|^2_q + \left\| P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{A}} \right\|^2_q \right)^{1/2}, & 2\leq q < \infty
\end{array}\right. \\
& \overset{(b)}\leq \left\{ \begin{array}{ll}
(2^q + 1)^{1/q} \left\|{\mathbf{Z}}_{\max(r)}\right\|_q, & 1\leq q \leq 2;\\
\sqrt{5} \left\|{\mathbf{Z}}_{\max(r)}\right\|_q, & 2\leq q < \infty
\end{array} \right.
\end{split}
\end{equation*}
Here, (a) is due to Lemma \ref{lm:partition-l_q-norm} and (b) is due to Theorem \ref{th:SVD-projection}. For $q = \infty$,
\begin{equation*}
\begin{split}
\| \widehat{{\mathbf{A}}} - {\mathbf{A}} \| \leq \| \widehat{{\mathbf{A}}} - {\mathbf{B}} \| + \|{\mathbf{A}} - {\mathbf{B}}\| \overset{(a)}\leq 2 \| {\mathbf{A}} - {\mathbf{B}} \| \leq 2 \| {\mathbf{Z}} \| = 2\|{\mathbf{Z}}_{\max(r)}\|.
\end{split}
\end{equation*}
Here $(a)$ comes from the fact that $\widehat{\mathbf{A}}$ is the best rank-$r$ approximation of ${\mathbf{B}}$.
\end{proof}
In the rest of this section, we focus on the proof of Theorem \ref{th:SVD-projection}. To this end, we introduce several additional lemmas on the properties of matrix singular values and norms. First, the following lemma introduces a dual characterization of the truncated matrix Schatten-$q$ norm.
\begin{Lemma}[Dual representation of Truncated Schatten-$q$ norm]\label{lm: charac of Schatten-q norm}
Let ${\mathbf{X}} \in \mathbb{R}^{m \times n} (m \leq n)$ be a matrix with full singular value decomposition ${\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top$ with ${\mathbf{W}} \in \mathbb{R}^{m \times m}, \boldsymbol{\Lambda} \in \mathbb{R}^{m \times n}, {\mathbf{M}} \in \mathbb{R}^{n \times n}$ and singular values $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_m$, then for any ${\mathbf{B}} \in \mathbb{R}^{m \times n}$ such that ${\rm rank}({\mathbf{B}}) \leq r \leq m$, we have
\begin{equation}\label{ineq: inner product upper bound via q norm}
\left| \langle {\mathbf{B}}, {\mathbf{X}} \rangle \right| \leq \|{\mathbf{B}} \|_q \left\| {\mathbf{X}}_{\max (r)}\right\|_p
\end{equation}
for any $q \geq 1$ and $1/p + 1/q = 1$. The equality is achieved if ${\mathbf{B}} = {\mathbf{W}}_{[:,1:r']} \boldsymbol{\Sigma} {\mathbf{M}}_{[:,1:r']}^\top$, where $r' = r \wedge {\rm rank}({\mathbf{X}})$ and $\boldsymbol{\Sigma} \in \mathbb{R}^{r' \times r'}$ is a non-zero diagonal matrix satisfying
\begin{equation*}
\boldsymbol{\Sigma}^q_{[1,1]}/\lambda_1^p = \cdots = \boldsymbol{\Sigma}_{[r',r']}^q/\lambda_{r'}^{p}.
\end{equation*}
Alternatively,
\begin{equation}\label{eq: truncated schatten q norm}
\|{\mathbf{X}}_{\max(r)}\|_p = \sup_{\|{\mathbf{B}}\|_q \leq 1, {\rm rank}({\mathbf{B}}) \leq r} \langle {\mathbf{B}}, {\mathbf{X}} \rangle.
\end{equation}
If ${\rm rank}({\mathbf{X}}) \leq r$, then
\begin{equation}\label{eq: schatten q norm of rank r matrix}
\|{\mathbf{X}}\|_p = \sup_{\|{\mathbf{B}}\|_q \leq 1, {\rm rank}({\mathbf{B}}) \leq r} \langle {\mathbf{B}}, {\mathbf{X}} \rangle.
\end{equation}
\end{Lemma}
\begin{proof}
First, \eqref{eq: truncated schatten q norm} and \eqref{eq: schatten q norm of rank r matrix} follow from \eqref{ineq: inner product upper bound via q norm}. So we only need to prove \eqref{ineq: inner product upper bound via q norm}. Denote ${\mathbf{U}} {\mathbf{K}} {\mathbf{V}}^\top$ as a full singular value decomposition of ${\mathbf{B}}$, where ${\mathbf{U}} \in \mathbb{R}^{m \times m}, {\mathbf{K}} \in \mathbb{R}^{m \times n}, {\mathbf{V}} \in \mathbb{R}^{n \times n}$. Then
\begin{equation} \label{ineq: diag inequality}
\begin{split}
|\langle{\mathbf{B}}, {\mathbf{X}} \rangle| = |{\rm tr}({\mathbf{B}}^\top {\mathbf{X}})| = |{\rm tr}({\mathbf{V}} {\mathbf{K}}^\top {\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top)| =& |{\rm tr}({\mathbf{K}}^\top {\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}})|\\
\leq & {\rm diag}({\mathbf{K}})^\top |{\rm diag}({\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}})|.
\end{split}
\end{equation}
Here ${\rm diag}({\mathbf{K}})$ is a vector consisting diagonal entries of ${\mathbf{K}}$ and the $|\cdot|$ is taken entrywise for a given vector.
Since ${\rm rank}({\mathbf{B}}) \leq r$, by H\"older's inequality, we have
\begin{equation}\label{ineq: holder for inner product}
{\rm diag}({\mathbf{K}})^\top |{\rm diag}({\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}})|\leq \left( \sum_{i=1}^r {\mathbf{K}}^q_{[i,i]} \right)^{1/q} \left( \sum_{i=1}^r \left|\left( {\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}} \right)_{[i,i]}\right|^p \right)^{1/p}
\end{equation}
for any $q \geq 1$, $1/p + 1/q = 1$. To finish the proof, we only need to show
\begin{equation}\label{ineq: diag value and singular value}
\left( \sum_{i=1}^r \left|\left( {\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}} \right)_{[i,i]}\right|^p \right)^{1/p} \leq \left( \sum_{i=1}^r \lambda_i^p \right)^{1/p}.
\end{equation}
To show \eqref{ineq: diag value and singular value}, we first introduce the following property of Ky Fan norm \cite{fan1949theorem} of matrix ${\mathbf{A}} \in \mathbb{R}^{m\times n}$: for any $1 \leq s \leq n \land m$
\begin{equation}\label{eq: ky norm property}
K_s({\mathbf{A}}) := \sum_{i=1}^s \sigma_i({\mathbf{A}}) = \sup_{{\mathbf{U}} \in \mathbb{O}_{m,s}, {\mathbf{V}} \in \mathbb{O}_{n, s}} {\rm tr}({\mathbf{U}}^\top {\mathbf{A}} {\mathbf{V}}).
\end{equation}
Denote $a_1 \geq a_2 \geq \ldots \geq a_r \geq 0$ as the values of $\left\{ \left|\left( {\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}} \right)_{[i,i]}\right| \right\}_{i=1}^r$ in descending order. By \eqref{eq: ky norm property}, we have
\begin{equation*}
\sum_{i=1}^s a_i \leq K_s( {\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}} ) = \sum_{i=1}^s \sigma_i\left( {\mathbf{U}}^\top {\mathbf{W}} \boldsymbol{\Lambda} {\mathbf{M}}^\top {\mathbf{V}} \right) = \sum_{i=1}^s \lambda_i, \quad \text{for }s = 1, \ldots, r.
\end{equation*} The last equality is due to the fact that ${\mathbf{U}},{\mathbf{V}},{\mathbf{W}},{\mathbf{M}}$ are all orthogonal matrices. Then equation \eqref{ineq: diag value and singular value} follows from the following Lemma \ref{lm:sequence}.
\begin{Lemma}[Karamata's inequality]\label{lm:sequence}
Suppose $x_1\geq x_2 \geq \cdots \geq x_k \geq 0$ and $y_1\geq y_2 \geq \cdots \geq y_k \geq 0$. For any $1\leq j \leq k$, $\sum_{i=1}^j x_i \leq \sum_{i=1}^j y_i$. Then for any $p\geq 1$,
$$\sum_{i=1}^k x_i^p \leq \sum_{i=1}^k y_i^p.$$
The equality holds if and only if $(x_1,\ldots, x_k) = (y_1,\ldots, y_k)$.
\end{Lemma}
\begin{proof}
See \cite[Theorem 1]{kadelburg2005inequalities}.
\end{proof}
By Lemma \ref{lm:sequence}, the equality in \eqref{ineq: diag value and singular value} holds if ${\mathbf{U}}_{[:,1:r']} = {\mathbf{W}}_{[:,1:r']}, {\mathbf{V}}_{[:,1:r']} = {\mathbf{M}}_{[:,1:r']}$; in the meantime, the equalities in \eqref{ineq: diag inequality} and \eqref{ineq: holder for inner product} hold if we further have
\begin{equation*}
{\mathbf{K}}_{[1,1]}^q/\lambda_1^p = \cdots = {\mathbf{K}}_{[r',r']}^q/\lambda_{r'}^p
\end{equation*}
for non-zero singular values in ${\mathbf{K}}$ and ${\mathbf{K}}_{[j,j]} = 0$ for $j > r'$. This has finished the proof.
\end{proof}
Recall that a matrix norm $\|\cdot\|$ is unitarily invariant if $\|{\mathbf{A}}\| = \|{\mathbf{U}}{\mathbf{A}} {\mathbf{V}}\|$ for any matrix ${\mathbf{A}}$ and orthogonal matrices ${\mathbf{U}}, {\mathbf{V}}$. We have the following Lemmas for $\|(\cdot)_{\max(r)}\|_q$.
\begin{Lemma}\label{lem: triangle of trun schatten q}
For any $q \geq 1$, $\|(\cdot)_{\max(r)}\|_q$ is a unitarily invariant matrix norm, i.e., for any ${\mathbf{A}},{\mathbf{B}} \in \mathbb{R}^{m\times n}$ and $\lambda \in \mathbb{R}$,
\begin{itemize}
\item $\|{\mathbf{A}}_{\max(r)}\|_q \geq 0$; $\|{\mathbf{A}}_{\max(r)}\|_q=0$ if and only if ${\mathbf{A}}=0$;
\item $\|(\lambda {\mathbf{A}})_{\max(r)}\|_q = |\lambda|\cdot \|{\mathbf{A}}_{\max(r)}\|_q$, $\forall \lambda \in \mathbb{R}$;
\item $ \|\left({\mathbf{A}} + {\mathbf{B}}\right)_{\max(r)}\|_q \leq \| {\mathbf{A}}_{\max(r)} \|_q + \|{\mathbf{B}}_{\max(r)}\|_q;$
\item $\|{\mathbf{A}}_{\max(r)}\|_q = \|{\mathbf{U}}{\mathbf{A}}_{\max(r)}{\mathbf{V}}\|_q$ for any orthogonal matrices ${\mathbf{U}}$ and ${\mathbf{V}}$.
\end{itemize}
\end{Lemma}
\begin{Lemma}\label{lm: optimality of SVD in truncated Schatten-q norm}
Given matrix ${\mathbf{A}} \in \mathbb{R}^{m \times n}$ and any non-negative integer $k \leq m \wedge n$, for any matrix ${\mathbf{M}}$ with ${\rm rank}({\mathbf{M}}) \leq r$, we have
\begin{equation*}
\left\|\left({\mathbf{A}}_{-\max(r)}\right)_{\max(k)} \right\|_q \leq \| \left({\mathbf{A}} - {\mathbf{M}}\right)_{\max(k)} \|_q.
\end{equation*}
The equality is achieved when ${\mathbf{M}} = {\mathbf{A}}_{\max(r)}$.
\end{Lemma}
The proofs for Lemmas \ref{lem: triangle of trun schatten q} and \ref{lm: optimality of SVD in truncated Schatten-q norm} are deferred to Section \ref{sec:additional-proof}. Now we are in position to prove Theorem \ref{th:SVD-projection}.
\begin{proof}[Proof of Theorem \ref{th:SVD-projection}]
We only study $\|P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}}\|_q$ since the proof of the upper bound of $\|{\mathbf{A}} P_{\widehat{{\mathbf{V}}}_\perp}\|_q$ follows by symmetry. Denote $\sum_{k=1}^r \sigma_{k}({\mathbf{A}})\u_k \v_k^\top$ as a singular value decomposition of ${\mathbf{A}}$. Since ${\rm rank}(P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}}) \leq {\rm rank}({\mathbf{A}})= r$, for $p \geq 1$ satisfying $1/p + 1/q = 1$, we have
\begin{align
\left\|P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}} \right\|_q \overset{(a)}= & \sup_{\|{\mathbf{X}}\|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r } \langle P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{A}}, {\mathbf{X}} \rangle \nonumber\\
= & \sup_{\|{\mathbf{X}}\|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r } \langle P_{\widehat{{\mathbf{U}}}_\perp} \left({\mathbf{A}} +{\mathbf{Z}} \right) - P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{Z}} , {\mathbf{X}} \rangle \nonumber\\
\leq & \sup_{\|{\mathbf{X}}\|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r } \langle P_{\widehat{{\mathbf{U}}}_\perp} ({\mathbf{A}} +{\mathbf{Z}}), {\mathbf{X}} \rangle + \sup_{\|{\mathbf{X}}\|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r } \langle P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{Z}}, {\mathbf{X}} \rangle \nonumber\\
\overset{(b)}\leq & \sup_{\|{\mathbf{X}}\|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r } \|{\mathbf{X}}\|_p \left\| \left(P_{\widehat{{\mathbf{U}}}_\perp} \left({\mathbf{A}} +{\mathbf{Z}} \right)\right)_{\max (r)} \right\|_q \nonumber\\
& + \sup_{\|{\mathbf{X}}\|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r } \|{\mathbf{X}}\|_p \left\| \left(P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{Z}} \right)_{\max (r)} \right\|_q\nonumber\\
\overset{(c)}\leq & \min_{{\rm rank}({\mathbf{M}}) \leq r} \left\| \left({\mathbf{A}} +{\mathbf{Z}} -{\mathbf{M}} \right)_{\max (r)} \right\|_q + \left\| \left(P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{Z}} \right)_{\max (r)} \right\|_q \nonumber\\
\leq & \left\| \left({\mathbf{A}} +{\mathbf{Z}} - P_{{\mathbf{U}}}({\mathbf{A}} + {\mathbf{Z}}) \right)_{\max (r)} \right\|_q + \left\| \left(P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{Z}} \right)_{\max (r)} \right\|_q \nonumber\\
\leq & \left\| \left(P_{{\mathbf{U}}_\perp} {\mathbf{Z}} \right)_{\max (r)} \right\|_q + \| (P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{Z}} )_{\max (r)} \|_q \label{ineq:projection}\\
\leq & \|{\mathbf{Z}}_{\max (r)}\|_q + \|{\mathbf{Z}}_{\max (r)}\|_q \leq 2\|{\mathbf{Z}}_{\max (r)}\|_q.\nonumber
\end{align}
Here (a) (b) are due to Lemma \ref{lm: charac of Schatten-q norm} and (c) is due to Lemma \ref{lm: optimality of SVD in truncated Schatten-q norm}.
\end{proof}
We make several remarks on Theorems \ref{th: low rank matrix reconstruction} and \ref{th:SVD-projection}.
First, as discussed in Section \ref{sec:intro}, one can derive the matrix estimation error bounds relying on $\|{\mathbf{Z}}\|_q$ or $r^{1/q}\|{\mathbf{Z}}\|$ via the existing perturbation theory in the literature. The following example illustrates that our result can be much sharper when the singular values of ${\mathbf{Z}}$ has some polynomial decay.
\begin{Example}
Suppose ${\mathbf{Z}}$ satisfies that $\sigma_k({\mathbf{Z}}) = k^{-1/q}$ for $q > 1$. Then
\begin{equation*}
\|{\mathbf{Z}}_{\max(r)}\|_q = \left(\sum_{k=1}^r k^{-1}\right)^{1/q} \approx (1+\log r)^{1/q},
\end{equation*}
which can be much smaller than
\begin{equation*}
\begin{split}
\|{\mathbf{Z}}\|_q & = \left(\sum_{k=1}^{m \wedge n} k^{-1}\right)^{1/q} \approx (1+\log (m \wedge n))^{1/q},\qquad r^{1/q}\|{\mathbf{Z}}\| = r^{1/q}.
\end{split}
\end{equation*}
\end{Example}
Second, Theorem \ref{th:SVD-projection} may not be simply implied by the classic results. For example, the classic Wedin's $\sin\Theta$ Theorem \cite{wedin1972perturbation},
\begin{equation} \label{ineq:wedin-perturbation}
\max\left\{\|\sin\Theta({\mathbf{U}}, \widehat{{\mathbf{U}}})\|_q, \sin \Theta({\mathbf{V}}, \widehat{{\mathbf{V}}})\|_q\right\} \leq \frac{\max \{ \|{\mathbf{Z}}\widehat{{\mathbf{V}}}\|_q, \|\widehat{{\mathbf{U}}}^\top {\mathbf{Z}}\|_q \} }{\sigma_r({\mathbf{B}})},
\end{equation}
yields
\begin{equation} \label{ineq: matrix projection bound via sin Theta}
\begin{split}
\|P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{A}}\|_q & = \| \widehat{{\mathbf{U}}}_\perp {\mathbf{U}} \boldsymbol{\Sigma}_1 {\mathbf{V}}^\top \|_q \overset{\text{Lemma } \ref{lm: singular value characterization}}\leq \| \widehat{{\mathbf{U}}}_\perp^\top {\mathbf{U}} \|_q \sigma_1({\mathbf{A}}) = \|\sin\Theta({\mathbf{U}}, \widehat{{\mathbf{U}}})\|_q\sigma_1({\mathbf{A}}) \\
& \leq \max \left\{ \| {\mathbf{Z}} \widehat{{\mathbf{V}}} \|_q, \|\widehat{{\mathbf{U}}}^\top {\mathbf{Z}}\|_q \right\} \frac{\sigma_1({\mathbf{A}}) }{\sigma_r({\mathbf{B}})}.
\end{split}
\end{equation}
This bound \eqref{ineq: matrix projection bound via sin Theta} can be less sharp or practical for its dependency on $\sigma_1({\mathbf{A}})/\sigma_r({\mathbf{B}})$. As pointed out by \cite{udell2019big}, the spectrum of large matrix datasets arising from applications often delay fast. If the singular values of ${\mathbf{A}},{\mathbf{B}}$ decay fast, $\sigma_1({\mathbf{A}})/\sigma_r({\mathbf{B}})\gg 1$ and \eqref{ineq: matrix projection bound via sin Theta} can be loose. In contrast, our bound \eqref{ineq:P_U-P_V} in Theorem \ref{th:SVD-projection} is free of any ratio of singular values, which can be a significant advantage in practice. We will further illustrate the difference between \eqref{ineq:P_U-P_V} and \eqref{ineq: matrix projection bound via sin Theta} by simulation in Section \ref{sec: numerical matrix perturbation projection}.
Third, it is noteworthy by \eqref{ineq:projection} in the proof of Theorem \ref{th:SVD-projection}, we have actually proved
\begin{equation}\label{ineq:projection-error}
\begin{split}
&\|P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}}\|_q \leq \left\|\left(P_{\widehat{\mathbf{U}}_{\perp}}{\mathbf{Z}}\right)_{\max(r)}\right\|_q + \left\|\left(P_{{\mathbf{U}}_{\perp}}{\mathbf{Z}}\right)_{\max(r)}\right\|_q \\
& \|{\mathbf{A}} P_{\widehat{{\mathbf{V}}}_\perp}\|_q \leq \left\|\left({\mathbf{Z}} P_{\widehat{\mathbf{V}}_{\perp}}\right)_{\max(r)}\right\|_q + \left\|\left({\mathbf{Z}} P_{{\mathbf{V}}_{\perp}}\right)_{\max(r)}\right\|_q
\end{split}
\end{equation}
under the setting of Theorem \ref{th:SVD-projection}. The bound \eqref{ineq:projection-error} can be better than the one in Theorem \ref{th:SVD-projection} in some scenarios. For example, when ${\mathbf{Z}}$ is (or is close to) ${\mathbf{U}} \Sigma_{\mathbf{Z}} {\mathbf{V}}^\top$ for some $r$-by-$r$ matrix $\Sigma_{\mathbf{Z}}$, the bound in \eqref{ineq:projection-error} is smaller than $\|{\mathbf{Z}}_{\max(r)}\|_q$. On the other hand, the proposed bound in Theorem \ref{th:SVD-projection} is strong enough for proving Theorem \ref{th: low rank matrix reconstruction}, does not involve $P_{\widehat{\mathbf{U}}_{\perp}}$ or $P_{\widehat{\mathbf{V}}_{\perp}}$, and can be more convenient to use.
\section{Lower bounds}\label{sec:lower-bound}
The following Theorem \ref{th:lower bound} shows that the error upper bound for the rank-$r$ truncated SVD estimator $\widehat {\mathbf{A}}$ in Theorem \ref{th: low rank matrix reconstruction} is sharp.
\begin{Theorem}\label{th:lower bound}
For any $\varepsilon>0$ and $q \geq 1$, there exist ${\mathbf{A}}$, ${\mathbf{B}}$, and ${\mathbf{Z}}\neq 0$ such that ${\rm rank}({\mathbf{A}}) = r$, ${\mathbf{B}}={\mathbf{A}}+{\mathbf{Z}}$, and
$$\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_q > ((2^q+1)^{1/q}-\varepsilon)\|{\mathbf{Z}}_{\max(r)}\|_q.$$
\end{Theorem}
\begin{proof}
Without loss of generality we assume $0<\varepsilon<1$. We choose a value $\eta \in (0, \frac{(2^q+1)^{1/q}}{(2^q+1)^{1/q} - \varepsilon} - 1)$. Define
\begin{equation*}
{\mathbf{A}} =
\begin{bmatrix}
2{\mathbf{I}}_r & \mathbf{0}_{r\times r} & \mathbf{0}\\
\mathbf{0}_{r\times r} & \mathbf{0}_{r\times r} & \mathbf{0}\\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{bmatrix}, \quad {\mathbf{Z}} = \begin{bmatrix}
-(1+\eta){\mathbf{I}}_r & \mathbf{0}_{r\times r} & \mathbf{0}\\
\mathbf{0}_{r\times r} & {\mathbf{I}}_r & \mathbf{0}\\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{bmatrix},
\end{equation*} and
\begin{equation*}
{\mathbf{B}} = \begin{bmatrix}
(1-\eta){\mathbf{I}}_r & \mathbf{0}_{r\times r} & \mathbf{0}\\
\mathbf{0}_{r\times r} & {\mathbf{I}}_r & \mathbf{0}\\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{bmatrix}.
\end{equation*}
Then,
\begin{equation*}
\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_q = \left\|\begin{bmatrix}
-2{\mathbf{I}}_r & \mathbf{0}_{r\times r}\\
\mathbf{0}_{r\times r} & {\mathbf{I}}_r
\end{bmatrix}\right\|_q = (2^q r + r)^{1/q},\quad \|{\mathbf{Z}}_{\max(r)}\|_q = (1+\eta) r^{1/q}.
\end{equation*}
We thus have
\begin{equation*}
\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_q > ((2^q + 1)^{1/q} - \varepsilon)\|{\mathbf{Z}}_{\max(r)}\|_q.
\end{equation*}
\end{proof}
Theorem \ref{th: low rank matrix reconstruction} and Theorem \ref{th:lower bound} together imply that the constants in \eqref{ineq:hat-A-A} are not improvable when $1\leq q\leq 2$ and $q = \infty$. For $2 < p < \infty$, it would be an interesting future work to close the gap between the upper bound ($\sqrt{5}\|{\mathbf{Z}}_{\max(r)}\|_q$) and the lower bound ($(2^q+1)^{1/q}\|{\mathbf{Z}}_{\max(r)}\|_q$).
Apart from checking the sharpness of the upper bound \eqref{ineq:hat-A-A}, another nature question is, whether the rank-$r$ truncated SVD estimator is an optimal estimator in estimating ${\mathbf{A}}$. To answer this question, we consider the minimax estimation error lower bound among all possible data-dependent procedures $\widecheck{{\mathbf{A}}} = \widecheck{{\mathbf{A}}}({\mathbf{B}})$ (i.e., $\widecheck{{\mathbf{A}}}$ is a deterministic or random function of matrix ${\mathbf{B}}$). We specifically focus on the following class of $(\widetilde{{\mathbf{A}}}, \widetilde{{\mathbf{Z}}}, \widetilde{{\mathbf{B}}})$ triplets:
\begin{equation*}
\mathcal{F}_r(\xi) = \left\{ (\widetilde{{\mathbf{A}}}, \widetilde{{\mathbf{Z}}},\widetilde{{\mathbf{B}}}): \widetilde{\mathbf{B}} = \widetilde{\mathbf{A}} + \widetilde{\mathbf{Z}},{\rm rank}(\widetilde{{\mathbf{A}}}) = r, \left\| \widetilde{{\mathbf{Z}}}_{\max(r)} \right\|_q \leq \xi \right\}.
\end{equation*}
Here, $\xi$ corresponds to $\left\|{\mathbf{Z}}_{\max(r)}\right\|_q$ in the context of Theorem \ref{th: low rank matrix reconstruction}.
\begin{Theorem}[Schatten-$q$ minimax lower bound] \label{th: matrix reconstruction lower bound}
For the low-rank perturbation model, if $m \wedge n \geq 2r$, then, for any $q \geq 1$, we have
\begin{equation*}
\inf_{\widecheck{{\mathbf{A}}}} \sup_{(\widetilde{{\mathbf{A}}}, \widetilde{{\mathbf{Z}}},\widetilde{{\mathbf{B}}}) \in \mathcal{F}_r(\xi)} \left\| \widecheck{{\mathbf{A}}} - \widetilde{{\mathbf{A}}} \right\|_q \geq 2^{1/q-1} \xi.
\end{equation*}
Here the infimum is taken over all the estimation procedures.
\end{Theorem}
\begin{proof}
The proof is done by construction. We construct
\begin{equation*}
{\mathbf{Z}}_1 = \left(\begin{array}{ccc}
{\mathbf{0}}_{r \times r} & {\mathbf{0}} & {\mathbf{0}}\\
{\mathbf{0}} & \frac{\xi}{r^{1/q}} {\mathbf{I}}_r & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}}
\end{array} \right),\quad \widebar{{\mathbf{A}}}_1 = \left( \begin{array}{c c c}
\frac{\xi }{r^{1/q}} {\mathbf{I}}_r & {\mathbf{0}} & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}}_{r \times r} & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}}
\end{array} \right),
\end{equation*} and
\begin{equation*}
{\mathbf{Z}}_2 = \left( \begin{array}{c c c}
\frac{\xi}{ r^{1/q}}{\mathbf{I}}_r & {\mathbf{0}} & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}}_{r \times r} & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}}
\end{array} \right),\quad \widebar{{\mathbf{A}}}_2 = \left( \begin{array}{c c c}
{\mathbf{0}}_{r \times r} & {\mathbf{0}} & {\mathbf{0}}\\
{\mathbf{0}} & \frac{\xi }{r^{1/q}} {\mathbf{I}}_r & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}}
\end{array} \right).
\end{equation*}
By the construction above, we have $\| ({\mathbf{Z}}_1)_{\max(r)} \|_q = \xi$, $\|({\mathbf{Z}}_2)_{\max(r)} \|_q = \xi$, and $\widebar{{\mathbf{A}}}_1 + {\mathbf{Z}}_1 = \widebar{{\mathbf{A}}}_2 + {\mathbf{Z}}_2$. So
\begin{equation*}
\begin{split}
& \inf_{\widecheck{{\mathbf{A}}}} \sup_{(\widetilde{{\mathbf{A}}}, \widetilde{{\mathbf{Z}}}, \widetilde{{\mathbf{B}}}) \in \mathcal{F}_r(\xi)} \left\| \widecheck{{\mathbf{A}}} - \widetilde{{\mathbf{A}}} \right\|_q \geq \inf_{\widecheck{{\mathbf{A}}}} \left(\max \left\{ \| \widecheck{{\mathbf{A}}} - \widebar{{\mathbf{A}}}_1 \|_q, \| \widecheck{{\mathbf{A}}} - \widebar{{\mathbf{A}}}_2 \|_q \right\} \right)\\
\geq & \frac{1}{2} \inf_{\widecheck{{\mathbf{A}}}} \left( \| \widecheck{{\mathbf{A}}} - \widebar{{\mathbf{A}}}_1 \|_q + \| \widecheck{{\mathbf{A}}} - \widecheck{{\mathbf{A}}}_1 \|_q \right) \geq \frac{1}{2} \|\widebar{{\mathbf{A}}}_1 - \widebar{{\mathbf{A}}}_2\|_q = 2^{1/q-1}\xi.
\end{split}
\end{equation*}
\end{proof}
Combining Theorems \ref{th: low rank matrix reconstruction} and \ref{th: matrix reconstruction lower bound}, we conclude that the truncated SVD $\widehat{\mathbf{A}}$ achieves the optimal rate of low-rank matrix estimation error among all possible procedures $\widecheck{{\mathbf{A}}}$ in the class of $\mathcal{F}_r(\xi)$.
\section{Subspace perturbation bounds}\label{sec:wedin}
In this section, we apply the perturbation projection error bound established in Theorem \ref{th:SVD-projection} to derive a user-friendly subspace (singular vectors) perturbation bound.
\begin{Theorem}\label{th: variant 2 of Wedin} Consider the same perturbation setting as in Theorem \ref{th: low rank matrix reconstruction}. For any $q \geq 1$, we have
\begin{equation*}
\max\left\{\|\sin \Theta(\widehat{{\mathbf{U}}}, {\mathbf{U}})\|_q, \|\sin \Theta(\widehat{{\mathbf{V}}}, {\mathbf{V}})\|_q\right\} \leq \frac{2\| {\mathbf{Z}}_{\max(r)} \|_q }{\sigma_r({\mathbf{A}})}.
\end{equation*}
\end{Theorem}
\begin{proof}
By Theorem \ref{th:SVD-projection}, we have
\begin{equation*}
\| P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{A}} \|_q \leq 2 \| {\mathbf{Z}}_{\max(r)} \|_q.
\end{equation*}
Since the left singular subspace of ${\mathbf{A}}$ is ${\mathbf{U}}$, we have ${\mathbf{U}}\U^\top{\mathbf{A}} = P_{{\mathbf{U}}} {\mathbf{A}} = {\mathbf{A}}$. Then
\begin{equation*}
\|\sin \Theta(\widehat{{\mathbf{U}}}, {\mathbf{U}})\|_q = \| \widehat{{\mathbf{U}}}_\perp^\top {\mathbf{U}} \|_q \overset{\text{Lemma } \ref{lm: singular value characterization} }\leq \frac{\| \widehat{{\mathbf{U}}}_\perp^\top{\mathbf{U}}\U^\top{\mathbf{A}} \|_q}{\sigma_r({\mathbf{U}}^\top{\mathbf{A}})} = \frac{\|P_{\widehat{\mathbf{U}}_\perp}{\mathbf{A}}\|_q}{\sigma_r({\mathbf{A}})} \leq \frac{2 \| {\mathbf{Z}}_{\max(r)} \|_q}{\sigma_r({\mathbf{A}})}.
\end{equation*}
\end{proof}
We note that several similar bounds are developed towards the applications in statistics and machine learning in the past few years, for example, \cite[Corollary 4.1]{vu2013minimax}, \cite[Theorem 2]{yu2014useful}, and \cite[Lemma 5.1]{lei2015consistency}. When the matrix is positive semidefinite, these results yield
\begin{equation}\label{ineq:vu}
\begin{split}
& \left\|\sin\Theta(\widehat{\mathbf{U}}, {\mathbf{U}})\right\|_F \leq \frac{\sqrt{2}\|{\mathbf{Z}}\|_F}{\sigma_r({\mathbf{A}})}, \quad \text{(\cite[Corollary 4.1]{vu2013minimax})},
\end{split}
\end{equation}
\begin{equation}\label{ineq:yu-lei}
\begin{split}
& \left\|\sin\Theta(\widehat{\mathbf{U}}, {\mathbf{U}})\right\|_F \leq \frac{2\min\{r^{1/2}\|{\mathbf{Z}}\|, \|{\mathbf{Z}}\|_F\}}{\sigma_r({\mathbf{A}})} \quad \text{\cite[Theorem 2]{yu2014useful}, \cite[Lemma 5.1]{lei2015consistency}}.
\end{split}
\end{equation}
When ${\mathbf{A}}, {\mathbf{Z}}, {\mathbf{B}}$ are asymmetric, \cite{yu2014useful} also proved
\begin{equation}\label{ineq:yu}
\begin{split}
\left\|\sin\Theta(\widehat{\mathbf{U}}, {\mathbf{U}})\right\|_F \leq \frac{2(2\|{\mathbf{A}}\| + \|{\mathbf{Z}}\|)\min\{r^{1/2}\|{\mathbf{Z}}\|, \|{\mathbf{Z}}\|_F\}}{\sigma_r^2({\mathbf{A}})} \quad \text{\cite[Theorem 3]{yu2014useful}}.
\end{split}
\end{equation}
The perturbation bounds \eqref{ineq:vu}\eqref{ineq:yu-lei}\eqref{ineq:yu}, along with Theorem \ref{th: variant 2 of Wedin} in this paper, are ``user friendly" as they do not involve $\widehat{\mathbf{U}}$, $\widehat{\mathbf{V}}$ or ${\mathbf{B}}$ in contrast to the classical Wedin's $\sin\Theta$ bound \eqref{ineq:wedin-perturbation}. This advantage facilitates the application of these perturbations to many settings when ${\mathbf{A}}$ and ${\mathbf{Z}}$ are the given arguments: one no longer needs to further bound $\|{\mathbf{Z}}\widehat{{\mathbf{V}}}\|_q$, $\|\widehat{\mathbf{U}}^\top{\mathbf{Z}}\|_q$. The ``user friendly" advantage is also important in many settings as the denominator of \eqref{ineq:wedin-perturbation}, $\sigma_r({\mathbf{B}})$, depends highly on the perturbation ${\mathbf{Z}}$ and can be rather small due to perturbation \cite{yu2014useful}. In addition, our new result in Theorem \ref{th: variant 2 of Wedin} has a better dependence on both ${\mathbf{Z}}$ and $\sigma_r({\mathbf{A}})$ than \eqref{ineq:vu}\eqref{ineq:yu-lei}\eqref{ineq:yu} because
$$\|{\mathbf{Z}}_{\max(r)}\|_F \leq \min\left\{r^{1/2} \|{\mathbf{Z}}\|, \|{\mathbf{Z}}\|_F\right\},$$
while the opposite side of this inequality does not hold. Moreover, Theorem \ref{th: variant 2 of Wedin} covers the more general asymmetric matrices in Schatten-$q$ sin$\Theta$ norms for any $q \in [1, \infty]$.
\section{Simulations} \label{sec: numerical study}
In this section, we provide numerical studies to support our theoretical results. We specifically compare the low-rank matrix estimation error bound (Theorem \ref{th: low rank matrix reconstruction}) and the matrix perturbation projection error bound (Theorem \ref{th:SVD-projection}) in Section \ref{sec:main-result} with the results in previous literature. In each setting, we randomly generate a perturbation ${\mathbf{Z}} = \u \v^\top + \widetilde{{\mathbf{Z}}}$, draw ${\mathbf{A}}$ by a to-be-specified scheme, and construct ${\mathbf{B}} = {\mathbf{A}} + {\mathbf{Z}}$. Here $\u, \v$ are randomly generated unit vectors and $\widetilde{{\mathbf{Z}}}$ has i.i.d. $N(0, \sigma^2)$ entries. Throughout the simulation studies, we consider the Schatten-$2$ norm (i.e., Frobenius norm) as the error metric. Each simulation setting is repeated for 100 times and the average values are reported.
\subsection{Numerical Comparison of Low-Rank Matrix Estimation Error Bounds} \label{sec: numerical matrix reconstruction}
We first compare the low-rank matrix estimation error bound $\|\widehat{\mathbf{A}} - {\mathbf{A}}\|_q$ in Theorem \ref{th: low rank matrix reconstruction} and the bounds in \eqref{ineq: matrix reconstruction via q-norm} and \eqref{ineq: matrix reconstruction via spectral norm}. We set $n \in \{100,300\}, r \in \{4,6,\ldots, 16\}$, $\sigma = 0.02$, and generate ${\mathbf{A}} = {\mathbf{U}} \boldsymbol{\Sigma}_1 {\mathbf{V}}^\top$, where ${\mathbf{U}} \in \mathbb{R}^{n \times r}, {\mathbf{V}} \in \mathbb{R}^{n \times r}$ are independently drawn from $\mathbb{O}_{n,r}$ uniformly at random; $\boldsymbol{\Sigma}_1$ is a diagonal matrix with singular values decaying polynomially as: $(\boldsymbol{\Sigma}_1)_{[i,i]} = \frac{10}{i}$, $1 \leq i \leq r$.
The evaluations of the upper bounds in Theorem \ref{th: low rank matrix reconstruction}, \eqref{ineq: matrix reconstruction via q-norm}, \eqref{ineq: matrix reconstruction via spectral norm}, and the true value of $\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_F$ are given in Figure \ref{fig: simulation2}. It shows that the upper bound in Theorem \ref{th: low rank matrix reconstruction} is tighter than the upper bounds in \eqref{ineq: matrix reconstruction via q-norm}, \eqref{ineq: matrix reconstruction via spectral norm} in all settings. In addition, when $n$ increases from $100$ to $300$, the upper bound of \eqref{ineq: matrix reconstruction via q-norm} significantly increases while the upper bound of Theorem \ref{th: low rank matrix reconstruction} remains steady. This is because the upper bounds of \eqref{ineq: matrix reconstruction via q-norm} and Theorem \ref{th: low rank matrix reconstruction} rely on $\|{\mathbf{Z}}\|_F$ and $\|{\mathbf{Z}}_{\max(r)}\|_F$, respectively.
\begin{figure}
\centering
\subfigure[$n = 100$]{\includegraphics[height = 0.3\textwidth]{matrix_reconstruction_poly_decay_p100.pdf}}
\hskip1cm
\subfigure[$n = 300$]{\includegraphics[height = 0.3\textwidth]{matrix_reconstruction_poly_decay_p300.pdf}}
\caption{Low-rank matrix estimation error bound (Theorem \ref{th: low rank matrix reconstruction}), upper bounds \eqref{ineq: matrix reconstruction via q-norm}, \eqref{ineq: matrix reconstruction via spectral norm} and the true value of $\|\widehat{{\mathbf{A}}} - {\mathbf{A}}\|_F$
} \label{fig: simulation2}
\end{figure}
\subsection{Numerical Comparison of Matrix Perturbation Projection Error Bounds} \label{sec: numerical matrix perturbation projection}
Next, we compare the matrix perturbation projection error bound in Theorem \ref{th:SVD-projection} with the upper bound \eqref{ineq: matrix projection bound via sin Theta} derived from Wedin's sin$\Theta$ Theorem. We generate ${\mathbf{B}}, {\mathbf{Z}}$ in the same way as the previous simulation setting. When generating $\boldsymbol{\Sigma}_1$ in ${\mathbf{A}}$, apart from the polynomial singular value decaying pattern considered in the last setting, we also consider the following exponential singular value decaying pattern: $(\boldsymbol{\Sigma}_1)_{[i,i]} = 2^{5-i}$, $1 \leq i \leq r$.
The values of the upper bounds in Theorem \ref{th:SVD-projection} and \eqref{ineq: matrix projection bound via sin Theta}, along with the true value of $\|P_{\widehat{{\mathbf{U}}}_\perp}{\mathbf{A}}\|_q$, are presented in Figure \ref{fig: simulation1}. We find the bound of Theorem \ref{th:SVD-projection} is much tighter than the bound in \eqref{ineq: matrix projection bound via sin Theta}. As $r$ increases or singular value decaying pattern becomes exponential, i.e., ${\mathbf{A}}$ becomes ill-conditioned, \eqref{ineq: matrix projection bound via sin Theta} becomes loose while Theorem \ref{th:SVD-projection} can still be sharp.
\begin{figure}
\centering
\subfigure[Singular values of ${\mathbf{A}}$ decay polynomially]{\includegraphics[height = 0.3\textwidth]{matrix_projection_poly_decay.pdf}}
\hskip1cm
\subfigure[Singular values of ${\mathbf{A}}$ decay exponentially]{\includegraphics[height = 0.3\textwidth]{matrix_projection_expdecay.pdf}}
\caption{Matrix perturbation projection error upper bound (Theorem \ref{th:SVD-projection}), upper bound via Wedin's sin$\Theta$ Theorem \eqref{ineq: matrix projection bound via sin Theta}, and the true value of $\|P_{\widehat{\mathbf{U}}_\perp}{\mathbf{A}}\|_F$.
} \label{fig: simulation1}
\end{figure}
\section{Discussions} \label{sec: conclusion}
In this paper, we prove a sharp upper bound for estimation error of rank-$r$ truncated SVD ($\|\widehat{\mathbf{A}}- {\mathbf{A}}\|_q$) under perturbation, and show its optimality in low-rank matrix estimation. The key technical tool we use is a novel matrix perturbation projection error bound for $\|P_{\widehat{{\mathbf{U}}}_\perp} {\mathbf{A}}\|_q$. As a byproduct, we also provide a sharper user-friendly sin$\Theta$ perturbation bound. The numerical studies demonstrate the advantages of these new results over the ones in the literature.
Throughout the paper, we study the additive perturbations and it is a future work to extend the results to multiplicative perturbations \cite{li1998relative,li1998relative2}. Also for convenience of presentation, we focus on the real number field in this paper. It is interesting to extend the developed results to the field of complex numbers. The main technical work for such an extension includes a dual representation of the truncated Schatten-$q$ norm in the field of complex numbers, i.e., a complex version of Lemma \ref{lm: charac of Schatten-q norm}.
Apart from the widely studied perturbation theory on singular value decomposition, the perturbation theory for other problems, such as pseudo-inverses \cite{wedin1973perturbation,stewart1977perturbation}, least squares problems \cite{stewart1977perturbation}, orthogonal projection \cite{stewart1977perturbation,xu2020perturbation,fierro1996perturbation,chen2016perturbation}, rank-one perturbation \cite{zhu2019rank}, are also important topics. It would be interesting to explore whether the tools developed in this paper is useful in studying the perturbation theory for these problems.
\section{Additional Lemmas and Proofs}\label{sec:additional-proof}
\begin{proof}[Proof of Lemma \ref{lem: triangle of trun schatten q}]
Since $\|{\mathbf{A}}_{\max(r)}\|_q = (\sum_{i=1}^r \sigma_i^q({\mathbf{A}}))^{1/q}$, we have $\|{\mathbf{A}}_{\max(r)}\|_q \geq 0$,
\begin{equation*}
\|{\mathbf{A}}_{\max(r)}\|_q = 0 \text{ if and only if } \sigma_1({\mathbf{A}}) = 0 \text{ if and only if }{\mathbf{A}} = 0.
\end{equation*}
Since $\sigma_i(\lambda{\mathbf{A}}) = |\lambda|\sigma_i({\mathbf{A}})$, we have $\|(\lambda{\mathbf{A}})_{\max(r)}\|_q = |\lambda|\cdot \|{\mathbf{A}}_{\max(r)}\|_q$.
Next, we apply Lemma \ref{lm: charac of Schatten-q norm} to prove the triangle inequality:
\begin{equation*}
\begin{split}
& \|({\mathbf{A}} + {\mathbf{B}})_{\max(r)}\|_q \overset{\text{Lemma \ref{lm: charac of Schatten-q norm}}}{=} \sup_{\|{\mathbf{X}} \|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r} \left\langle {\mathbf{A}} + {\mathbf{B}}, {\mathbf{X}} \right \rangle\\
\leq & \sup_{\|{\mathbf{X}} \|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r} \left\langle {\mathbf{A}} , {\mathbf{X}} \right \rangle + \sup_{\|{\mathbf{X}} \|_p \leq 1, {\rm rank}({\mathbf{X}}) \leq r} \left\langle {\mathbf{B}}, {\mathbf{X}} \right \rangle
\overset{\text{Lemma \ref{lm: charac of Schatten-q norm}}}{=} \|{\mathbf{A}}_{\max(r)}\|_q + \|{\mathbf{B}}_{\max(r)}\|_q.
\end{split}
\end{equation*}
Finally for any orthogonal matrices ${\mathbf{U}}$ and ${\mathbf{V}}$, since $\sigma_i({\mathbf{A}}) = \sigma_i({\mathbf{U}}{\mathbf{A}}{\mathbf{V}})$, we have $\|{\mathbf{A}}_{\max(r)}\|_q = \|{\mathbf{U}}{\mathbf{A}}_{\max(r)}{\mathbf{V}}\|_q$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lm: optimality of SVD in truncated Schatten-q norm}]
By the well-known Eckart-Young-Mirsky Theorem \cite{eckart1936approximation,mirsky1960symmetric,golub1987generalization}, the truncated SVD achieves the best low-rank matrix approximation in any unitarily invariant norm. This lemma follows from the Eckart-Young-Mirsky Theorem and the fact that $\|(\cdot)_{\max(k)}\|_q$ is a unitarily invariant matrix norm (Lemma \ref{lem: triangle of trun schatten q}).
\end{proof}
\begin{Lemma}[Properties of $\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)$]\label{lm: spectral of sin theta}
Suppose ${\mathbf{U}}_1,{\mathbf{U}}_2,{\mathbf{U}}_3 \in \mathbb{O}_{p,r}$ are $p \times r\, (r \leq p)$ matrices with orthonormal columns.
\begin{itemize}[leftmargin=*]
\item (Spectrum of $\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)$) ${\mathbf{U}}_{1 \perp}^\top {\mathbf{U}}_2$ and $\sin \Theta({\mathbf{U}}_1, {\mathbf{U}}_2)$ share the same singular values, i.e.,
\begin{equation} \label{eq: spectral equality}
\sigma_i({\mathbf{U}}_{1 \perp}^\top {\mathbf{U}}_2) = \sigma_i(\sin \Theta({\mathbf{U}}_1, {\mathbf{U}}_2)), \quad i = 1, \ldots, r.
\end{equation}
In particular, $\| {\mathbf{U}}_{1 \perp}^\top {\mathbf{U}}_2\|_q = \left\| \sin \Theta({\mathbf{U}}_1, {\mathbf{U}}_2) \right\|_q$ for any $q \in [1,\infty]$.
\item (Triangle Inequality)
$$\|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q \leq \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_3)\|_q + \|\sin\Theta({\mathbf{U}}_2, {\mathbf{U}}_3)\|_q $$
\item (Equivalence to other distances) The Schatten-$q$ $\sin\Theta$ distance defined as \eqref{label:Schatten-q-distance} is equivalent to other metrics, as the following inequality holds,
\begin{equation*}
\|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q \leq \inf_{\mathbf{O}\in \mathbb{O}_r} \|{\mathbf{U}}_1 - {\mathbf{U}}_2\mathbf{O}\|_q \leq 2 \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q;
\end{equation*}
\begin{equation*}
\|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q \leq \|{\mathbf{U}}_1{\mathbf{U}}_1^\top - {\mathbf{U}}_2{\mathbf{U}}_2^\top\|_q \leq 4 \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q.
\end{equation*}
\end{itemize}
\end{Lemma}
\begin{proof}
\begin{itemize}[leftmargin=*]
\item {\bf (Spectrum of $\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)$)}. Suppose ${\mathbf{U}}_{1 \perp}^\top {\mathbf{U}}_2$ has singular value decomposition ${\mathbf{W}}_1 \boldsymbol{\Sigma} {\mathbf{V}}_1^\top$ and ${\mathbf{U}}_1^\top {\mathbf{U}}_2$ has singular value decomposition ${\mathbf{W}}_2 \boldsymbol{\Lambda} {\mathbf{V}}_2^\top$, where ${\mathbf{W}}_1 \in \mathbb{R}^{(p-r) \times r}; {\mathbf{W}}_2, \boldsymbol{\Sigma}, \boldsymbol{\Lambda}, {\mathbf{V}}_1, {\mathbf{V}}_2 \in \mathbb{R}^{r \times r}$. By the definition of $\sin \Theta$ distance, $\sigma_i \left(\sin \Theta({\mathbf{U}}_1, {\mathbf{U}}_2) \right) = \sqrt{ 1- \boldsymbol{\Lambda}^2_{[r-i,r-i]}}$. So to show the result of \eqref{eq: spectral equality}, we only need to show
\begin{equation} \label{eq: relationship of bLemaba and bSigma}
\sqrt{1- \boldsymbol{\Lambda}^2_{[r-i,r-i]}} = \boldsymbol{\Sigma}_{[i, i]}.
\end{equation}
Since ${\mathbf{V}}_1, {\mathbf{V}}_2$ are both orthogonal matrices, suppose ${\mathbf{V}}_1 {\mathbf{R}} = {\mathbf{V}}_2$ where ${\mathbf{R}}$ is a $r \times r$ orthogonal matrix. Then
\begin{equation*}
\begin{split}
{\mathbf{I}} & = {\mathbf{U}}_2^\top {\mathbf{U}}_{1\perp}{\mathbf{U}}_{1\perp}^\top {\mathbf{U}}_2 + {\mathbf{U}}_2^\top {\mathbf{U}}_{1}{\mathbf{U}}_{1}^\top {\mathbf{U}}_2 \\
& = {\mathbf{V}}_1 \boldsymbol{\Sigma}^2 {\mathbf{V}}_1^\top + {\mathbf{V}}_2 \boldsymbol{\Lambda}^2 {\mathbf{V}}_2^\top = {\mathbf{V}}_1 \left( \boldsymbol{\Sigma}^2 + {\mathbf{R}} \boldsymbol{\Lambda}^2 {\mathbf{R}}^\top\right) {\mathbf{V}}_1^\top.
\end{split}
\end{equation*}
So $\boldsymbol{\Sigma}^2 + {\mathbf{R}} \boldsymbol{\Lambda}^2 {\mathbf{R}}^\top = {\mathbf{I}}$, and this means that ${\mathbf{R}}$ could only be a permutation matrix. And since $\boldsymbol{\Sigma}_{[1,1]} \geq \ldots \geq \boldsymbol{\Sigma}_{[r,r]}$ and $\boldsymbol{\Lambda}_{[1,1]} \geq \ldots \geq \boldsymbol{\Lambda}_{[r,r]}$, the only way that can make $\boldsymbol{\Sigma}^2 + {\mathbf{R}} \boldsymbol{\Lambda}^2 {\mathbf{R}}^\top = {\mathbf{I}}$ to be true for a permutation matrix ${\mathbf{R}}$ is
\begin{equation*}
\boldsymbol{\Lambda}^2_{[r-i,r-i]} + \boldsymbol{\Sigma}^2_{[i, i]} = 1.
\end{equation*}
This has finished the proof for $\sigma_i({\mathbf{U}}_{1\perp}^\top {\mathbf{U}}_2) = \sigma_i(\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2))$ for any $i=1,\ldots, r$. Thus, $\|{\mathbf{U}}_{1\perp}^\top{\mathbf{U}}_2\|_q = \|\sin\Theta({\mathbf{U}}_1,{\mathbf{U}}_2)\|_q$ for any $q\in [1,\infty]$.
\item {\bf (Triangle Inequality)}.
\begin{equation*}
\begin{split}
\|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q & \overset{(a)}= \| {\mathbf{U}}_{1\perp}^\top {\mathbf{U}}_2 \|_q = \| {\mathbf{U}}_{1\perp}^\top (P_{{\mathbf{U}}_3} + P_{{\mathbf{U}}_{3\perp}}) {\mathbf{U}}_2 \|_q \\
& \overset{(b)}\leq \| {\mathbf{U}}_{1\perp}^\top P_{{\mathbf{U}}_3} {\mathbf{U}}_2 \|_q + \| {\mathbf{U}}_{1\perp}^\top P_{{\mathbf{U}}_{3\perp}} {\mathbf{U}}_2 \|_q \\
& \leq \| {\mathbf{U}}_{1\perp} {\mathbf{U}}_3 \|_q \| {\mathbf{U}}_3^\top {\mathbf{U}}_2 \| + \| {\mathbf{U}}_{1\perp}^\top {\mathbf{U}}_{3\perp} \| \| {\mathbf{U}}_{3\perp}^\top {\mathbf{U}}_2 \|_q \\
& \overset{(c)} \leq \| {\mathbf{U}}_{1\perp} {\mathbf{U}}_3 \|_q + \| {\mathbf{U}}_{3\perp}^\top {\mathbf{U}}_2 \|_q\\
& = \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_3)\|_q + \|\sin\Theta({\mathbf{U}}_3, {\mathbf{U}}_2)\|_q.
\end{split}
\end{equation*}
Here, (a) is due to \eqref{eq: spectral equality}, (b) is a triangle inequality, (c) is due to $ \| {\mathbf{U}}_{1\perp}^\top {\mathbf{U}}_{3\perp}\| \leq 1, \| {\mathbf{U}}_3^\top {\mathbf{U}}_2 \| \leq 1$.
\item {\bf (Equivalence to other distances)}.
Since all metrics mentioned in Lemma \ref{lm: spectral of sin theta} is rotation invariant, i.e. for any $\mathbf{O} \in \mathbb{O}_{p}$, $\sin \Theta ({\mathbf{U}}_1, {\mathbf{U}}_2) = \sin \Theta(\mathbf{O} {\mathbf{U}}_1, \mathbf{O}{\mathbf{U}}_2)$, without loss of generality, we can assume
\begin{equation*}
{\mathbf{U}}_2 = \left[ \begin{array}{c}
{\mathbf{I}}_r \\
{\mathbf{0}}_{(p-r) \times r}
\end{array} \right].
\end{equation*} In this case
\begin{equation*}
\begin{split}
\inf_{\mathbf{O}\in \mathbb{O}_r} \|{\mathbf{U}}_1 - {\mathbf{U}}_2\mathbf{O}\|_q & = \inf_{\mathbf{O}\in \mathbb{O}_r} \left\| \begin{array}{c}
({\mathbf{U}}_1)_{[1:r,:]} - \mathbf{O} \\
({\mathbf{U}}_1)_{[(r+1):p,:]}
\end{array} \right\|_q \\
& \overset{(a)}= \inf_{\mathbf{O}\in \mathbb{O}_r} \sup_{\|{\mathbf{X}}\|_p \leq 1} \left\langle\begin{array}{c}
({\mathbf{U}}_1)_{[1:r,:]} -\mathbf{O} \\
({\mathbf{U}}_1)_{[(r+1):p,:]}
\end{array}, {\mathbf{X}} \right\rangle \\
& \geq \sup_{\|{\mathbf{X}}\|_p \leq 1} \left\langle
({\mathbf{U}}_1)_{[(r+1):p,:]} , {\mathbf{X}}_{[(r+1):p,:]} \right\rangle\\
& \overset{(b)}= \|({\mathbf{U}}_1)_{[(r+1):p,:]} \|_q = \|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1 \|_q = \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q.
\end{split}
\end{equation*}
Here, (a)(b) are due to Lemma \ref{lm: charac of Schatten-q norm}.
Now we prove the upper bound of $\inf_{\mathbf{O}\in \mathbb{O}_r} \|{\mathbf{U}}_1 - {\mathbf{U}}_2\mathbf{O}\|_q$. Recall ${\mathbf{U}}_1^\top {\mathbf{U}}_2$ has singular value decomposition ${\mathbf{W}}_2 \boldsymbol{\Lambda} {\mathbf{V}}_2^\top$. Then
\begin{equation*}
\begin{split}
\inf_{\mathbf{O}\in \mathbb{O}_r} \|{\mathbf{U}}_1 - {\mathbf{U}}_2\mathbf{O}\|_q & \leq \|{\mathbf{U}}_1 - {\mathbf{U}}_2 {\mathbf{W}}_2 {\mathbf{V}}_2^\top \|_q \\
& \leq \| P_{{\mathbf{U}}_2} ({\mathbf{U}}_1 - {\mathbf{U}}_2 {\mathbf{W}}_2 {\mathbf{V}}_2^\top) \|_q + \| P_{{\mathbf{U}}_{2\perp}} ({\mathbf{U}}_1 - {\mathbf{U}}_2 {\mathbf{W}}_2 {\mathbf{V}}_2^\top) \|_q \\
& \leq \| {\mathbf{W}}_2 ({\mathbf{I}}_r - \boldsymbol{\Lambda}) {\mathbf{V}}_2^\top \|_q + \|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1\|_q \\
& = \left( \sum_{i=1}^r (1 - \boldsymbol{\Lambda}_{[i,i]})^q \right)^{1/q} + \|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1\|_q\\
& \leq \left( \sum_{i=1}^r \left(\sqrt{1 - \boldsymbol{\Lambda}^2_{[i,i]}}\right)^q \right)^{1/q} + \|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1\|_q \\
& \overset{(a)}\leq 2 \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q.
\end{split}
\end{equation*}
Here, (a) is due the relationship between the singular values of ${\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1$ and ${\mathbf{U}}_2^\top {\mathbf{U}}_1$ established in \eqref{eq: relationship of bLemaba and bSigma}.
Finally, we prove the equivalency of $\|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q$ and $\|{\mathbf{U}}_1{\mathbf{U}}_1^\top - {\mathbf{U}}_2{\mathbf{U}}_2^\top\|_q$. First
\begin{equation*}
\|{\mathbf{U}}_1{\mathbf{U}}_1^\top - {\mathbf{U}}_2{\mathbf{U}}_2^\top\|_q \geq \| {\mathbf{U}}_{2\perp} ( {\mathbf{U}}_1{\mathbf{U}}_1^\top - {\mathbf{U}}_2{\mathbf{U}}_2^\top ) \|_q = \|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1\|_q = \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q.
\end{equation*}
On the other hand, notice the decomposition
\begin{equation*}
\begin{split}
{\mathbf{U}}_1{\mathbf{U}}_1^\top - {\mathbf{U}}_2{\mathbf{U}}_2^\top & = (P_{{\mathbf{U}}_2} + P_{{\mathbf{U}}_{2\perp}}) {\mathbf{U}}_1{\mathbf{U}}_1^\top (P_{{\mathbf{U}}_2} + P_{{\mathbf{U}}_{2\perp}})- {\mathbf{U}}_2{\mathbf{U}}_2^\top\\
& = {\mathbf{U}}_2 \left( ({\mathbf{U}}_2^\top {\mathbf{U}}_1 ) ({\mathbf{U}}_2^\top {\mathbf{U}}_1 )^\top - {\mathbf{I}}_r \right) {\mathbf{U}}_2^\top + P_{{\mathbf{U}}_{2\perp}} {\mathbf{U}}_1{\mathbf{U}}_1^\top P_{{\mathbf{U}}_2} \\
& \quad + P_{{\mathbf{U}}_2} {\mathbf{U}}_1{\mathbf{U}}_1^\top P_{{\mathbf{U}}_{2\perp}} + P_{{\mathbf{U}}_{2\perp}} {\mathbf{U}}_1{\mathbf{U}}_1^\top P_{{\mathbf{U}}_{2\perp}}.
\end{split}
\end{equation*}
So,
\begin{equation*}
\begin{split}
\|{\mathbf{U}}_1{\mathbf{U}}_1^\top - {\mathbf{U}}_2{\mathbf{U}}_2^\top\|_q & \overset{(a)}\leq \|({\mathbf{U}}_2^\top {\mathbf{U}}_1 ) ({\mathbf{U}}_2^\top {\mathbf{U}}_1 )^\top - {\mathbf{I}}_r \|_q + 3\|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1 \|_q \\
& \leq \|{\mathbf{I}}_r - \boldsymbol{\Lambda}^2 \|_q + 3\|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1 \|_q\\
& = \left( \sum_{i=1}^r \left(1 - \boldsymbol{\Lambda}^2_{[i,i]}\right)^q \right)^{1/q} + 3\|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1\|_q\\
& \leq \left( \sum_{i=1}^r \left(\sqrt{1 - \boldsymbol{\Lambda}^2_{[i,i]}}\right)^q \right)^{1/q} + 3\|{\mathbf{U}}_{2\perp}^\top {\mathbf{U}}_1\|_q \\
& \overset{(b)}= 4 \|\sin\Theta({\mathbf{U}}_1, {\mathbf{U}}_2)\|_q.
\end{split}
\end{equation*}
Here, (a) is due to triangle inequality and $\|{\mathbf{U}}_2^\top {\mathbf{U}}_1\| \leq 1$, (b) is due to \eqref{eq: spectral equality} and \eqref{eq: relationship of bLemaba and bSigma}.
\end{itemize}
\end{proof}
The following Lemma \ref{lm: singular value characterization} characterizes the singular values of the product of matrices.
\begin{Lemma}[Singular values of the product of two matrices] \label{lm: singular value characterization}
Suppose ${\mathbf{A}}\in \mathbb{R}^{m \times n}$, ${\mathbf{B}} \in \mathbb{R}^{n \times b}$. Then
\begin{equation}\label{ineq: singular value bound}
\sigma_i({\mathbf{A}} {\mathbf{B}}) \leq \sigma_i({\mathbf{A}}) \|{\mathbf{B}}\|, \quad \sigma_i({\mathbf{A}} {\mathbf{B}}) \geq \sigma_i({\mathbf{A}}) \sigma_n({\mathbf{B}}),
\end{equation}
\begin{equation} \label{ineq: shatten q norm bound}
\|{\mathbf{A}}{\mathbf{B}}\|_q \leq \|{\mathbf{A}}\|_q \|{\mathbf{B}}\|, \quad \|{\mathbf{A}}{\mathbf{B}}\|_q \geq \|{\mathbf{A}}\|_q \sigma_n({\mathbf{B}})
\end{equation}
for any $1 \leq i \leq m \wedge n$ and $q \geq 1$.
\end{Lemma}
\begin{proof} First
\begin{equation*}
\begin{split}
\sigma_i({\mathbf{A}} {\mathbf{B}}) = \lambda_i^{1/2}({\mathbf{A}} {\mathbf{B}} {\mathbf{B}}^\top {\mathbf{A}}^\top) = \lambda_i^{1/2}({\mathbf{B}} {\mathbf{B}}^\top {\mathbf{A}}^\top {\mathbf{A}}) &\overset{(a)} \geq (\lambda_i({\mathbf{A}}^\top {\mathbf{A}}) \lambda_{n-i+1}({\mathbf{B}} {\mathbf{B}}^\top) )^{1/2} \\
& \geq \sigma_i({\mathbf{A}}) \sigma_n({\mathbf{B}}).
\end{split}
\end{equation*}
Here (a) is due to \cite[P.371 Theorem H.1.d.]{marshall1979inequalities}.
Next we show $\sigma_i({\mathbf{A}}{\mathbf{B}})\leq \sigma_i({\mathbf{A}})\|{\mathbf{B}}\|$. Recall the best low-rank approximation property of SVD \cite{mirsky1960symmetric,golub1987generalization}, we have
\begin{equation*}
\sigma_i({\mathbf{A}}) = \min_{{\mathbf{X}} \in \mathbb{R}^{m \times n}, {\rm rank}({\mathbf{X}}) \leq i-1} \| {\mathbf{A}} - {\mathbf{X}} \|.
\end{equation*}
So
\begin{equation*}
\begin{split}
\sigma_i({\mathbf{A}} {\mathbf{B}}) & = \min_{{\mathbf{X}} \in \mathbb{R}^{m \times n}, {\rm rank}({\mathbf{X}}) \leq i-1} \| {\mathbf{A}}{\mathbf{B}} - {\mathbf{X}} \| \leq \left\| {\mathbf{A}}{\mathbf{B}} - \sum_{k=1}^{i-1} \sigma_k({\mathbf{A}}) \u_k \v_k^\top {\mathbf{B}} \right\| \\
&= \left\| {\mathbf{A}}_{\max(-(i-1))} {\mathbf{B}}\right\| \leq \left\|{\mathbf{A}}_{\max(-(i-1))} \right\|\cdot \|{\mathbf{B}} \| = \sigma_i({\mathbf{A}}) \|{\mathbf{B}}\|.
\end{split}
\end{equation*}
Next,
$$\|{\mathbf{A}}{\mathbf{B}}\|_q=(\sum_i \sigma_i^q({\mathbf{A}}{\mathbf{B}}))^{1/q} \leq (\sum_i \sigma_i^q({\mathbf{A}})\|{\mathbf{B}}\|^q)^{1/q} = \|{\mathbf{A}}\|_q \|{\mathbf{B}}\|,$$
$$\|{\mathbf{A}}{\mathbf{B}}\|_q=(\sum_i \sigma_i^q({\mathbf{A}}{\mathbf{B}}))^{1/q} \geq (\sum_i \sigma_i^q({\mathbf{A}})\sigma^q_n({\mathbf{B}}))^{1/q} = \|{\mathbf{A}}\|_q \sigma_n({\mathbf{B}}).$$
\end{proof}
\bibliographystyle{alpha}
| {'timestamp': '2020-11-23T02:07:14', 'yymm': '2008', 'arxiv_id': '2008.01312', 'language': 'en', 'url': 'https://arxiv.org/abs/2008.01312'} |
\section{Introduction}\vspace*{-2.8mm}
\label{sec:intro}
As the number of skilled maintenance workers decreases worldwide, the demand for automatic
sound-monitoring system has been increasing.
Such systems aim to detect anomalous states of a machine from its sound.
Because anomalous sound data can rarely be obtained in practice, unsupervised anomaly detection
methods are often adopted for these systems \cite{Suefusa2020,Koizumi2020}.
Neural generative models such as a variational autoencoder (VAE) \cite{Kingma2014a} and a normalizing flows (NF) \cite{Tabak2013,Dinh2015}
are the most commonly used methods for unsupervised anomaly detection
because of their high detection performance.
These methods try to detect data with different distributions from normal data, without
using anomalous data.
However, not only a machine's anomalous states but also changes in its physical parameters
(domain shifts) or aging can affect the distributions of the normal data, which induces
false positives when using unsupervised methods.
Aging causes changes in data distributions over a long period of time,
and these changes can be handled by continual learning or incremental learning \cite{Wiewel2019}.
On the other hand, domain shifts, which are the focus in this paper,
can induce sudden, huge differences in data
distributions, because physical parameters can change within a short period of time.
Moreover, because these physical parameters are often numerical values, it is impossible
to collect a sufficient amount of data for all possible parameters.
Accordingly, we need an unsupervised method that can handle domain shifts while requiring
sound data with only a few sets of physical parameters.
In this paper, we develop an unsupervised anomalous sound detection method
that can handle domain shifts due to changes in physical parameters.
Our idea is to disentangle the factors of domain shifts
and perform anomaly detection by using a space that is invariant with respect to these shifts.
Specifically, we propose to constrain a neural generative model so that
some of the latent variables represent factors of domain shifts and
other latent variables represent components that are invariant with respect to domain shifts.
As a result, the anomaly scores calculated using the latter latent variables are not affected by
domain shifts but only
by other variation factors such as a machine's anomalous state,
which can lead to fewer false positives.\vspace*{-2.8mm}
\section{Problem Statement}\vspace*{-2.8mm}
\label{sec:statement}
Anomalous sound detection is the task of identifying whether a machine
is normal or anomalous according to an anomaly score calculated by a trained model from the machine's sound.
Each piece of input sound data is determined as anomalous if its anomaly score
exceeds a threshold value.
We consider unsupervised anomalous sound detection, in which only normal sound is
available for training.
We also assume that a machine's physical parameters are only available during training.
This assumption is realistic because sensors to measure the physical parameters may not be available
in real-world operation, depending on environmental conditions.
This problem setting is similar to that of DCASE 2021 Challenge Task 2 \cite{Kawaguchi2021},
in which machines have up to three different numerical and physical parameters that cause domain shifts.\vspace*{-2.8mm}
\section{Relation to prior work}\vspace*{-2.8mm}
\subsection{Semi-supervised disentanglement learning}\vspace*{-2.8mm}
Learning disentangled representations has been at the core of representation learning research \cite{Bengio2014}.
Unsupervised disentanglement learning methods, in which
a VAE with regularizers is commonly used to encourage disentanglement, have mainly been investigated \cite{Higgins2017}.
However, it has been pointed out that unsupervised disentanglement is impossible without
inductive biases \cite{Locatello2019a}.
Semi-supervised disentanglement learning methods, on the other hand, explicitly use
a few labeled pieces of data to disentangle factors of variation \cite{Locattello2020, Nie2020}.
Locatello et al. \cite{Locattello2020} trained a VAE with an added loss term to incorporate
label information during training.
Esling et al. \cite{Esling2020} used an NF to disentangle categorical tag information from a latent space.
Our proposed method is a semi-supervised disentanglement learning method that uses an
NF to disentangle numerical and physical parameters without an additional loss term. \vspace*{-2.8mm}
\subsection{Anomalous sound detection under domain shifts}\vspace*{-2.8mm}
For DCASE2021 Challenge Task 2, we published the MIMII DUE dataset \cite{Tanabe2021}, the first dataset
for anomalous sound detection under domain shifts.
In this dataset, we changed physical parameters between the source and target domain
to induce domain shifts. The source domain data and a few samples from the target domain data were available during training.
The top-ranked approaches in the challenge first used autoencoder-based methods or classifiers to extract embeddings
from the data and then used likelihood-based methods like a Gaussian mixture model (GMM) to
calculate anomaly scores from the embeddings.
Kuroyanagi et al. \cite{Kuroyanagi2021} proposed to calculate anomaly scores by training a GMM for each domain
on the autoencoder's reconstruction errors.
Sakamoto et al. \cite{Sakamoto2021} attained the highest scores for the target data by
estimating the mean of the target data under an assumption that the
mean changes between the source and target domains.
Wilkinghoff \cite{Wilkinghoff2021} trained a classifier that discriminates each set of physical parameters to obtain embeddings.
Our proposed method does not require the target domain data for training,
as it explicitly uses numerical and physical parameters to obtain a domain-shift-invariant
latent space that does not change between the source and target domains.\vspace*{-2.8mm}
\section{conventional approach}\vspace*{-2.8mm}
\label{sec:conventional}
\subsection{Semi-supervised disentanglement learning using VAE}\vspace*{-2.8mm}
Locatello et al. \cite{Locattello2020}
proposed a semi-supervised disentanglement learning method to
disentangle the factors of variations represented by a few labeled data instances, denoted as $\mathbf{y}$.
They modified the loss function of VAE to incorporate supervision:
\begin{equation}
\begin{split}
L = -&\E_{q_\phi (\mathbf{z}|\mathbf{x})} [\log p_\theta (\mathbf{x}|\mathbf{z})] + \beta (D_{KL} (q_\phi (\mathbf{z}|\mathbf{x}))) \\
&+ \gamma \E_{\mathbf{x}, \mathbf{y}} [R(q_\phi (\mathbf{y}|\mathbf{x}))],
\end{split}
\label{Locatello}
\end{equation}
where $\mathbf{x}$ denotes the input data,
$\mathbf{z}$ is the latent variables,
$\beta$ is a hyperparameter introduced in \cite{Higgins2017},
$\gamma$ is another hyperparameter,
and $R(\cdot)$ is a function to induce supervised disentanglement.\vspace*{-2.8mm}
\subsection{Unsupervised anomaly detection using NF}\vspace*{-2.8mm}
Among neural generative models for unsupervised anomalous sound detection,
an NF evaluates the exact likelihood of the input data and
has shown better detection performance than
other models, including a VAE \cite{Dohi2021}.
The NF models a series of invertible transformations
$f=f_1 \circ f_2 \circ \cdots \circ f_K$
between an input data distribution $p(\mathbf{x})$ and
a known distribution $p(\mathbf{z})$.
The log likelihood of the input data can be calculated as
\begin{equation}
\log p(\mathbf{x}) = \log p(\mathbf{z}_{0}) + \sum_{i=1}^K \log |\det (\frac{d\mathbf{z}_{i}}{d\mathbf{z}_{i-1}})|,
\label{likelihood}
\end{equation}
\begin{equation}
\mathbf{z_i} = f_i (\mathbf{z}_{i-1}),
\end{equation}
where $\mathbf{z_0}$ is a latent vector that follows a known distribution such as the standard isotropic gaussian $N(0, 1)$,
and $z_i$ $(i=1,2,\cdots,K)$ is an intermediate latent vector.
The anomaly score can be calculated as the negative
log likelihood (NLL) of the input data \cite{Schmidt2019, Ryzhikov2019, Dias2020},
\begin{equation}
a(\mathbf{x}) = -\log p(\mathbf{x}).
\label{conv_score}
\end{equation}
The NF has mainly been used as an unsupervised method, which fails
to handle distribution changes due to domain shifts.
Specifically, when a machine's physical parameters change and the distribution of its
normal sound data changes, unsupervised methods can output high anomaly scores, leading to false positives.
\vspace*{-2.8mm}
\section{Proposed Approach}\vspace*{-2.8mm}
\label{sec:proposed}
\subsection{Learning of domain-shift-invariant latent space for anomaly detection using NF}\vspace*{-2.8mm}
To handle distribution changes due to domain shifts, we propose to disentangle the factors
of domain shifts and construct a domain-shift-invariant latent space
for anomaly score calculation.
For this purpose, we use an NF because of its high expressive power.
First, we train an NF model to obtain domain-shift-invariant representations.
As long as the latent variables follow an isotropic Gaussian, if we let some latent variables
$\mathbf{z}_{d}$ represent a factor of domain shifts, then
the latent space constructed by other latent variables $\mathbf{z}_{c}$
should be invariant with respect to that factor.
To make $\mathbf{z}_{d}$ represent a numerical and physical parameter $v$
that causes domain shifts,
we constrain $\mathbf{z}_{d}$ to follow a Gaussian distribution $N(kv, 1)$, where
$k$ is a hyperparameter to induce stable training of the model.
If a set of sound data with different values of the parameter $v$ is available, then the model can learn
to map input data into a latent space that shifts linearly with $v$.
This forces some latent variables $\mathbf{z}_{d}$ to represent the physical parameter $v$
while making other latent variables $\mathbf{z}_{c}$ invariant to that parameter.
Using the obtained domain shift-invariant latent space, we then calculate anomaly scores
that are unaffected by domain shifts.
The likelihood of the latent variables, $p(\mathbf{z}_{0})$, can be factorized as
\begin{equation}
\log p(\mathbf{z}_{0}) = \log p(\mathbf{z}_{c}) + \log p(\mathbf{z}_{d}). \label{factor}
\end{equation}
From (\ref{likelihood}) and (\ref{factor}), the likelihood of input data $\mathbf{x}$ can be written as:
\begin{equation}
\log p(\mathbf{x})
= \log p(\mathbf{z}_{c}) + \log p(\mathbf{z}_{d}) + \sum_{i=1}^K \log |\det (\frac{d\mathbf{z}_{i}}{d\mathbf{z}_{i-1}})|, \label{data}
\end{equation}
Because the latent variables $\mathbf{z}_{d}$ are constrained to represent
a factor $v$ of domain shifts, they cannot be used for calculating domain shift-invariant
anomaly scores.
Also, the third term of (\ref{data}) can remain constant if
it is calculated using the same learned parameters.
Accordingly, the anomaly score in (\ref{conv_score}) can be calculated
using only the first term of (\ref{data}):
\begin{equation}
a(\mathbf{z}_{c}) = -\log p(\mathbf{z}_{c}).
\label{score}
\end{equation}\vspace*{-2.8mm}
\subsection{Multi-scale architecture in NF for learning disentangled reprens\-\-\-entations}\vspace*{-2.8mm}
\label{ssec:multi-scale}
A multi-scale architecture in an NF was first introduced in \cite{Dinh2017} and has since been commonly
used in other NF models \cite{Kingma2018}.
To apply our method in a multi-scale architecture, we modify only the last block of the
architecture.
Let $c, h, w$ denote the channel size, height, and width of a feature vector, respectively, and
assume that the architecture consists of $N$ blocks.
In the $i$th block $(i=1,2,\cdots,N-1)$, an input $\mathbf{x}_{i}$ with dimensions $(c, h, w)$ is squeezed to give
a feature with dimensions $(4c, h/2, w/2)$.
After some flow transformations, half the channels
are used as the input $\mathbf{x}_{i+1}$ to the next block, while the other channels, $\mathbf{z}_{ic}$, are factored out; here,
$\mathbf{z}_{ic}$ is constrained to follow $N(0, 1)$.
In the last block, we constrain half the channels, $\mathbf{z}_{d}$,
to follow $N(0, 1)$ and the others, $\mathbf{z}_{Nc}$, to follow $N(kv, 1)$.
Because only the latent variables in the last block are constrained to represent a factor of domain shifts,
the model has to propagate domain-shift-dependent components to the last block.
As a result, the latent variables factored out at each block can represent domain-shift-invariant components
at each different scale.
The shift-invariant latent space $\mathbf{z}_{c}$ is obtained by concatenating the $\mathbf{z}_{ic}$
from all the blocks:
\begin{equation}
\mathbf{z}_{c} = (\mathbf{z}_{1c}, ..., \mathbf{z}_{Nc}).
\end{equation}
The anomaly score is then calculated using (\ref{score}).\vspace*{-2.8mm}
\section{Experiments}\vspace*{-2.8mm}
\label{sec:typestyle}
\subsection{Dataset}\vspace*{-2.8mm}
We prepared two slide rails with the same machine ID, which operated with a
designated velocity of 50--750 mm/s and a distance of 100--500 mm.
Figure \ref{fig:spec} shows examples of log-mel spectrograms of the sound with different
operation velocities.
We first recorded normal sound data with 15 different velocities (50, 100, ..., and 750 mm/s)
and three different distances (100, 250, and 500 mm), giving 45 different
physical parameter sets in total. Each parameter set had 10 sound clips, with each clip consisting of
a 10-second single-channel 16-kHz recording.
We used data with different velocities for the training and test data:
seven of the 15 velocities
(100, 200, 300, 400, 500, 600, and 700 mm/s) were used for the
training data, while other velocities were used for evaluating the model's ability to disentangle
velocities that were not included in the training data.
To evaluate the anomaly detection performance, we also r\-e\-c\-o\-r\-d\-e\-d pairs of normal and anomalous sound data.
Both the normal and anomalous sound data had
15 different velocities
and a fixed distance of 500 mm, with 10 sound clips for each velocity.
The normal sound data was recorded using the same slide rail as the one used for the training data.
The anomalous sound data was recorded using the other slide rail
\begin{figure}[t]
\begin{center}
\begin{minipage}{0.49\hsize}
\begin{center}
\includegraphics[width=1.0\hsize,height=1.8cm,clip]{vel100_dis500.png}\\
(a) 100 mm/s\\
\end{center}
\end{minipage}
\begin{minipage}{0.49\hsize}
\begin{center}
\includegraphics[width=1.0\hsize,height=1.8cm,clip]{vel300_dis500.png}\\
(b) 300 mm/s\\
\end{center}
\end{minipage}
\begin{minipage}{0.49\hsize}
\begin{center}
\includegraphics[width=1.0\hsize,height=1.8cm,clip]{vel500_dis500.png}\\
(c) 500 mm/s\\
\end{center}
\end{minipage}
\begin{minipage}{0.49\hsize}
\begin{center}
\includegraphics[width=1.0\hsize,height=1.8cm,clip]{vel700_dis500.png}\\
(d) 700 mm/s\\
\end{center}
\end{minipage}
\caption{Examples of log-mel spectrograms with different operation velocities (mm/s) and
an operation distance of 500 mm.}
\label{fig:spec}
\end{center}
\vspace*{-3.5mm}
\end{figure}
\vspace*{-2.8mm}
\subsection{Experimental conditions}\vspace*{-2.8mm}
We used Glow \cite{Kingma2018} for the NF model because it is often chosen
in out-of-distribution detection and anomaly detection tasks \cite{Dohi2021, Haunschmid2020}.
We also used a VAE with the loss function given in (\ref{Locatello}) to compare the
disentanglement performance with that of our proposed method.
To obtain input features, we applied the same procedure for Glow and the VAE.
Each frame of the log-mel spectrograms was first computed
with a length of 1024, a hop size of 512, and 128 mel bins.
At least 313 frames were generated for each sound clip, and 64 consecutive frames
with 48 overlappings were concatenated to generate the input features.
We prepared two Glow models with and without the multi-scale architecture.
The Glow model with the multi-scale architecture (multi-scale Glow) had three blocks with
five flow steps in each block, and each flow step had
three CNN layers with 32 hidden layers.
For the Glow model without the multi-scale architecture (single-scale Glow), the only
difference from the multi-scale Glow was it had just one block.
When using the proposed method, we constrained half the channels of the last block
to follow $N(kv, 1)$, as described in Sec. \ref{ssec:multi-scale}.
In contrast, the single-scale Glow and the multi-scale Glow without the proposed method were trained by
constraining all the latent variables to follow $N(0, 1)$.
The VAE model had 10 linear layers with 128 dimensions, except for the fifth
layer, which had eight dimensions.
The first dimension of the latent variables was constrained to represent
the velocity $v$, and $R(\cdot)$ in (\ref{Locatello}) was calculated by taking the mean squared error between
the value of the first latent variable and the actual velocity.
For the hyperparameters we set $\beta=1$ and $\gamma=0.01$.
All models were trained for 1000 epochs by using the
Adam optimizer \cite{Kingma2014} with a learning rate of
0.0001 and a batch size of 128
\begin{figure}[t]
\centering
\includegraphics[width=1.0\hsize,height=0.20\vsize,clip]{total.png}
\caption{Means of the latent variables for unseen velocities.}
\label{fig:estimation}
\vspace*{-5.0mm}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=0.95\hsize,height=0.25\vsize,clip]{vel0_spell.png}
\caption{Anomaly scores of sound data with the single-scale Glow and the conventional method.}
\label{fig:vel0}
\vspace*{-3.0mm}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.95\hsize,height=0.25\vsize,clip]{vel1_spell.png}
\caption{Anomaly scores of sound data with the single-scale Glow and the proposed method.}
\label{fig:vel1}
\vspace*{-5.0mm}
\end{figure*}
\vspace*{-2.8mm}
\subsection{Results}\vspace*{-2.8mm}
We first estimated the unseen velocities from the trained models to investigate whether these
velocities could be disentangled.
For both the single-scale and multi-scale Glow,
the estimated velocity was calculated by taking the mean of $\mathbf{z}_{d}$ in the last block.
For the VAE, the first dimension of the latent variables was used.
The velocities estimated from input features in the same sound clip were averaged to give
one estimation for each clip.
Figure \ref{fig:estimation} shows the estimation for each unseen velocity.
The multi-scale Glow showed the best performance, especially for the
lower velocities of 50--250 mm/s and the higher velocities of 650 and 750 mm/s.
Though the estimates tended to be larger for the medium velocities of 350--550 mm/s,
they still showed positive correlations with the actual velocities.
On the other hand, the VAE failed to estimate the higher velocities.
This result shows that the Glow models were better at disentangling the factors of variations
with their higher expressive power.
The multi-scale Glow gave better estimation results than the single-scale Glow, especially for
higher velocities.
This may be because the multi-scale Glow has more learnable parameters and the
multi-scale architecture enables extraction of representations at different scales.
\begin{table}[]
\begin{center}
\caption{AUCs (in \%) of sound data with the seen velocities (velocities in the training
data), unseen velocities (velocities only in the test data), and
all velocities.}
\begin{tabular}{lccc}
\bhline{1.5pt}
Method&
\begin{tabular}[c]{@{}c@{}}
Seen\\
velocities
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
Unseen\\
velocities
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
All\\
velocities
\end{tabular}\\ \hline
\begin{tabular}[c]{@{}l@{}}
VAE using \\
reconstruction error \\
\end{tabular} & 52.9 & 57.5 & 55.2\\\hline
\begin{tabular}[c]{@{}l@{}}
VAE using KLD \\
\end{tabular} & 47.0 & 52.4 & 49.9\\\hline
\begin{tabular}[c]{@{}l@{}}
VAE with loss in (\ref{Locatello}) \\
\end{tabular} & 64.8 & 60.3 & 62.3\\\hline
\begin{tabular}[c]{@{}l@{}}
Single-scale Glow \\
\end{tabular} & 89.1 & 72.7 & 78.6\\\hline
\begin{tabular}[c]{@{}l@{}}
Single-scale Glow using \\
proposed method \\
\end{tabular} & 97.7 & 87.3 & 91.8\\\hline
\begin{tabular}[c]{@{}l@{}}
Multi-scale Glow\\
\end{tabular} & \bf{100.0} & 86.0 & 91.9\\\hline
\begin{tabular}[c]{@{}l@{}}
Multi-scale Glow using \\
proposed method \\
\end{tabular} & \bf{100.0} & \bf{89.9} & \bf{94.5}\\\bhline{1.5pt}
\end{tabular}
\label{auc}
\end{center}
\vspace*{-7.0mm}
\end{table}
We then calculated the anomaly scores
for each sound clip.
To evaluate the detection performance of each model,
we calculated the
area under the receiver operating characteristic curve (AUC) for the seen velocities,
unseen velocities,
and all velocities.
For the VAE, we calculated three different scores:
the reconstruction error (the first term in (\ref{Locatello})) from a model trained using a conventional loss (first and second terms in (\ref{Locatello})),
the Kullback-Leibler divergence (KLD, second term in (\ref{Locatello})) from the same model,
and the KLD from a model trained using the loss in (\ref{Locatello}).
For Glow, we used single-scale and multi-scale versions with and without
the proposed method.
Table \ref{auc} lists the results, which indicate that the proposed method improved the AUC by
13.2\% for the single-scale Glow and 2.6\% for the multi-scale Glow.
In addition, the Glow models outperformed all the VAE models,
even though VAE with the loss in (\ref{Locatello}) outperformed
the other VAEs with the conventional loss.
The proposed method showed greater improvement in the single-scale Glow
than in the multi-scale Glow. This may be because the number of learnable
parameters in the single-scale Glow was not enough to learn the
distribution of the normal data, which made the effect of using the
domain-shift-invariant latent space more evident.
Figures \ref{fig:vel0} and \ref{fig:vel1} show the anomaly scores
for data using the single-scale Glow with the conventional method and the proposed method,
respectively.
In Fig. \ref{fig:vel0}, the anomaly scores of the normal sound data with unseen velocities, especially for
50, 250, 450, and 750 mm/s, were higher than those of the normal and even the anomalous sound data
with seen velocities.
Because the normal data with unseen velocities cannot be used for training,
the anomaly detector may detect normal samples with these velocities as anomalous samples.
On the other hand, in Fig. \ref{fig:vel1}, the normal sound data with unseen velocities
showed about the same scores as with seen velocities, except for 50 mm/s.
This result indicates that the proposed methods can lower the anomaly scores of normal sound
data with unseen velocities by disentangling the operation velocity.
For 50 mm/s,
because the distribution of the normal data can significantly change around this lower
velocity, the anomaly score for the normal data was higher with this velocity than
with the other velocities.
The anomaly scores of the anomalous sound data became higher
for several velocities with the proposed method. As the model is
trained to disentangle velocities by using the normal data, it may have disentangled components that
that did not represent the velocity of the anomalous sound data, thus raising the anomaly scores.
\vspace*{-2.8mm}
\section{Conclusion}\vspace*{-3.0mm}
We proposed an anomalous sound detection method that can handle domain shifts.
The proposed method uses an NF model to
disentangle the numerical and physical parameters of a machine, which gives
a domain-shift-invariant latent space.
Experimental results demonstrated that the proposed method disentangles the
factors of domain shifts better than a VAE does, thus enabling improvement in
the anomaly detection performance
for data with unseen physical parameters.
\clearpage
\ninept
\bibliographystyle{IEEEbib}
| {'timestamp': '2021-11-15T02:06:23', 'yymm': '2111', 'arxiv_id': '2111.06539', 'language': 'en', 'url': 'https://arxiv.org/abs/2111.06539'} |