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+arXiv:1001.0001v1  [cs.IT]  30 Dec 2009On the structure of non-full-rank perfect codes
+Olof Heden and Denis S. Krotov∗
+Abstract
+The Krotov combining construction of perfect 1-error-corr ecting binary codes
+from 2000 and a theorem of Heden saying that every non-full-r ank perfect 1-error-
+correcting binary code can be constructed by this combining construction is gener-
+alized to the q-ary case. Simply, every non-full-rank perfect code Cis the union of
+a well-defined family of ¯ µ-components K¯µ, where ¯µbelongs to an “outer” perfect
+codeC⋆, and these components are at distance three from each other. Compo-
+nents from distinct codes can thus freely be combined to obta in new perfect codes.
+The Phelps general product construction of perfect binary c ode from 1984 is gen-
+eralized to obtain ¯ µ-components, and new lower bounds on the number of perfect
+1-error-correcting q-ary codes are presented.
+1. Introduction
+LetFqdenote the finite field with qelements. A perfect1-error-correcting q-ary code of
+lengthn, for short here a perfect code , is a subset Cof the direct product Fn
+q, ofncopies of
+Fq, having the property that any element of Fn
+qdiffers in at most one coordinate position
+from a unique element of C.
+The family of all perfect codes is far from classified or enumerated. We will in this
+short note say something about the structure of these codes. W e need the concept of
+rank.
+We consider Fn
+qas a vector space of dimension nover the finite field Fq. Therank
+of aq-ary codeC, here denoted rank( C), is the dimension of the linear span < C >of
+the elements of C. Trivial, and well known, counting arguments give that if there exist s
+a perfect code in Fn
+qthenn= (qm−1)/(q−1), for some integer m, and|C|=qn−m. So,
+for every perfect code C,
+n−m≤rank(C)≤n.
+If rank(C) =nwe will say that Chasfull rank.
+∗This research collaboration was partially supported by a grant from Swedish Institute; the work of
+the second author was partially supported by the Federal Target Program “Scientific and Educational
+PersonnelofInnovation Russia”for 2009-2013(governmentco ntract No. 02.740.11.0429)and the Russian
+Foundation for Basic Research (grant 08-01-00673).
+1We will show thatevery non-full-rankperfect code isa unionofso ca lled ¯µ-components
+K¯µ, and that these components may be enumerated by some other pe rfect codeC⋆, i.e,
+¯µ∈C⋆. Further, the distance between any two such components will be a t least three.
+This implies that we will be completely free to combine ¯ µ-components from different
+perfect codesofsamelength, toobtainotherperfect codes. Ge neralizing aconstruction by
+Phelps of perfect 1-error correcting binary codes [8], we will obtain further ¯µ-components.
+As an application of our results we will be able to slightly improve the lowe r bound on
+the number of perfect codes given in [6].
+Our results generalize corresponding results for the binary case. In [3] it was shown
+that a binary perfect code can be constructed as the union of diffe rent subcodes (¯ µ-
+components) satisfying some generalized parity-check property , each of them being con-
+structed independently or taken from another perfect code. In [2] it was shown that every
+non-full-rank perfect binary code can be obtained by this combining construction.
+2. Every non-full-rank perfect code is the union of ¯µ-
+components
+We start with some notation. Assume we have positive integers n,t,n1, ...,ntsuch that
+n1+...+nt≤n. Anyq-aryword ¯xwill berepresented intheblockform ¯ x= (¯x1|¯x2|...|
+¯xt|¯x0) = (¯x∗|¯x0), where ¯xi= (xi1,xi2,...,x ini),i= 0,1,...,t,n0=n−n1−...−nt,
+¯x∗= (¯x1|¯x2|...|¯xt). For every block ¯ xi,i= 1,2,...,t, we define σi(¯xi) by
+σi(¯xi) =ni/summationdisplay
+j=1xij,
+and, for ¯x,
+¯σ(¯x) = ¯σ(¯x∗) = (σ1(¯x1),σ2(¯x2),...,σ t(¯xt))
+Recall that the Hamming distance d(¯x,¯y) between two words ¯ x, ¯yof the same length
+means the number of positions in which they differ.
+Amonomial transformation is a map of the space Fn
+qthat can be composed by a
+permutation of the set of coordinate positions and the multiplication in each coordinate
+position with some non-zero element of the finite field Fq.
+Aq-ary codeCislinearifCis a subspace of Fn
+q. A linear perfect code is called a
+Hamming code .
+Theorem 1. LetCbe any non-full-rank perfect code Cof lengthn= (qm−1)/(q−1).
+To any integer r<m, satisfying
+1≤r≤n−rank(C),
+there is aq-ary Hamming code C⋆of lengtht= (qr−1)/(q−1), such that for some
+monomial transformation ψ
+ψ(C) =/uniondisplay
+¯µ∈C⋆K¯µ,
+2where
+K¯µ={(¯x1|¯x2|...|¯xt|¯x0) : ¯σ(¯x) = ¯µ,¯x1,¯x2,...,¯xt∈Fqs
+q,¯x0∈C¯µ(¯x∗)}(1)
+for some family of perfect codes C¯µ(¯x), of length 1+q+q2+...+qs−1, wheres=m−r,
+and satisfying, for each ¯µ∈C⋆,
+d(¯x∗,¯x′
+∗)≤2 =⇒C¯µ(¯x∗)∩C¯µ(¯x′
+∗) =∅. (2)
+The codeC⋆will be called an outercode toψ(C). The subcodes K¯µwill be called
+¯µ-components ofψ(C). As the minimum distance of Cis three, the distance between any
+two distinct ¯ µ-components will be at least three.
+Proof. LetDbe any subspace of Fn
+qcontaining<C >, and of dimension n−r. By
+using a monomial transformation ψof space we may achieve that the dual space of ψ(D)
+is the nullspace of a r×n-matrix
+H=
+| | | | | | | |
+¯α11···¯α1n1¯α21···¯α2n2···¯αt1···¯αtnt¯0···¯0
+| | | | | | | |
+
+where ¯αij= ¯αi, fori= 1,2,...,t, the first non-zero coordinate in each vector ¯ αiequals
+1, ¯αi/ne}ationslash= ¯αi′, fori/ne}ationslash=i′, and where the columns of Hare in lexicographic order, according
+to some given ordering of Fq.
+To avoid too much notation we assume that Cwas such that ψ= id.
+LetC⋆be the null space of the matrix
+H⋆=
+| | |
+¯α1¯α2···¯αt
+| | |
+
+Define, for ¯ µ∈C⋆,
+K¯µ={(¯x1|¯x2|...|¯xt|¯x0)∈C: (σ1(¯x1),σ2(¯x2),...,σ(¯xt)) = ¯µ}.
+Then,
+C=/uniondisplay
+¯µ∈C⋆K¯µ.
+Further, since any two columns of H⋆are linearly independent, for any two distinct words
+¯µand ¯µ′ofC⋆
+d(K¯µ,K¯µ′)≥3. (3)
+We will show that K¯µhas the properties given in Equation (1).
+Any word ¯x= (¯x1|¯x2|...|¯xt|¯x0) must be at distance at most one from a word of
+C, and hence, the word ( σ1(¯x1),σ2(¯x2),...,σ t(¯xt)) is at distance at most one from some
+word ofC⋆. It follows that C⋆is a perfect code, and as a consequence, as C⋆is linear, it
+is a Hamming code with parity-check matrix H⋆. As the number of rows of H⋆isr, we
+then get that the number tof columns of H⋆is equal to
+t=qr−1
+q−1= 1+q+q2+...+qr−1.
+3For any word ¯ x∗ofFn1+n2+...+ntq with ¯σ(¯x∗) = ¯µ∈C⋆, we now define the code C¯µ(¯x∗)
+of lengthn0by
+C¯µ(¯x∗) ={¯c∈Fn0
+q: (¯x∗|¯c)∈C}.
+Again, using the fact that Cis a perfect code, we may deduce that for any ¯ x∗such
+that the set C¯µ(¯x∗) is non empty, the set C¯µ(¯x∗) must be a perfect code of length n0=
+(qs−1)/(q−1), for some integer s.
+From the fact that the minimum distance of Cequals three, we get the property in
+Equation (2).
+Let ¯eidenote a word of weight one with the entry 1 in the coordinate positio ni. It
+then follows that the two perfect codes C¯µ(¯x∗) andC¯µ(¯x∗+ ¯e1−¯ei), fori= 2,3,...,n 1,
+must be mutually disjoint. Hence, n1is at most equal to the number of perfect codes in
+a partition of Fn0qinto perfect codes, i.e.,
+n1≤(q−1)n0+1 =qs.
+Similarly,ni≤qs, fori= 2,3,...,t.
+Reversing these arguments, using Equation (3) and the fact that Cis a perfect code,
+we find that ni, for eachi= 1,2,...,t, is at least equal to the number of words in an
+1-ball ofFn0q.
+We conclude that ni=qs, fori= 1,2,...,t, and finally
+n=qs(1+q+q2+...+qr−1)+1+q+q2+...+qs−1= 1+q+q2+...+qr+s−1.
+Givenr, we can then find sfrom the equality
+n= 1+q+q2+...+qm−1.
+△
+3. Combining construction of perfect codes
+In the previous section, it was shown that a perfect code, depend ing on its rank, can
+be divided onto small or large number of so-called ¯ µ-components, which satisfy some
+equation with ¯ σ. The construction described in the following theorem realizes the ide a
+of combining independent ¯ µ-components, differently constructed or taken from different
+perfect codes, in one perfect code.
+A functionf: Σn→Σ, where Σ is some set, is called an n-ary(ormultary)quasigroup
+of order |Σ|if in the equality z0=f(z1,...,z n) knowledge of any nelements of z0,z1,
+...,znuniquely specifies the remaining one.
+Theorem 2. Letmandrbe integers, m>r,qbe a prime power, n= (qm−1)/(q−1)
+andt= (qr−1)/(q−1). Assume that C∗is a perfect code in Ft
+qand for every ¯µ∈C∗
+we have a distance- 3codeK¯µ⊂Fn
+qof cardinality qn−m−(t−r)that satisfies the following
+generalized parity-check law:
+¯σ(¯x) = (σ1(x1,...,x l),...,σ t(xlt−l+1,...,x lt)) = ¯µ
+4for every ¯x= (x1,...,x n)∈K¯µ, wherel=qm−rand¯σ= (σ1,...,σ t)is a collections of
+l-ary quasigroups of order q. Then the union
+C=/uniondisplay
+¯µ∈C∗K¯µ
+is a perfect code in Fn
+q.
+Proof. It is easy to check that Chas the cardinality of a perfect code. The distance
+at least 3 between different words ¯ x, ¯yfromCfollows from the code distances of K¯µ(if
+¯x, ¯ybelong to the same K¯µ) andC∗(if ¯x, ¯ybelong to different K¯µ′,K¯µ′′, ¯µ′,¯µ′′∈C∗).△
+The ¯µ-components K¯µcanbeconstructedindependentlyortakenfromdifferentperfec t
+codes. In the important case when all σiare linear quasigroups (e.g., σi(y1,...,y l) =
+y1+...+yl) the components can be taken from any perfect code of rank at m ostn−r, as
+followsfromtheprevioussection(itshouldbenotedthatif ¯ σislinear, thena ¯ µ-component
+can be obtained from any ¯ µ′-component by adding a vector ¯ zsuch that ¯σ(¯z) = ¯µ−¯µ′).
+In general, the existence of ¯ µ-components that satisfy the generalized parity-check law
+for arbitrary ¯ σis questionable. But for some class of ¯ σsuch components exist, as we will
+see from the following two subsections.
+Remark. It is worth mentioning that ¯ µ-components can exist for arbitrary length tof
+¯µ(for example, in the next two subsections there are no restriction s ont), if we do not
+require the possibility to combine them into a perfect code. This is esp ecially important
+for the study of perfect codes of small ranks (close to the rank o f a linear perfect code):
+once we realize that the code is the union of ¯ µ-components of some special form, we may
+forget about the code length and consider ¯ µ-components for arbitrary length of ¯ µ, which
+allows to use recursive approaches.
+3.1. Mollard-Phelps construction
+Here we describe the way to construct ¯ µ-components derived from the product construc-
+tion discovered independently in [7] and [9]. In terms of ¯ µ-components, the construction
+in [9] is more general; it allows substitution of arbitrary multary quasig roups, and we will
+use this possibility in Section 4.
+Lemma 1. Let¯µ∈Ft
+qand letC#be a perfect code in Fk
+q. Letvandhbe(q−1)-ary
+quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) : ¯y∈Fq−1
+q}is perfect. Let
+V1, ...,VtandH1, ...,Hkbe respectively (k+1)-ary and (t+1)-ary quasigroups of order
+q. Then the set
+K¯µ=/braceleftBig
+(¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :
+¯xij∈Fq−1
+q,
+(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
+(H1(h(¯x11),...,h(¯xt1),z1),...,H k(h(¯x1k),...,h(¯xtk),zk))∈C#/bracerightBig
+is a¯µ-component that satisfies the generalized parity-check law with
+σi(·,...,·,·) =Vi(v(·),...,v(·),·).
+5(The elements of F(q−1)kt+k+t
+q in this construction may be thought of as three-dimensional
+arrays where the elements of ¯xijare z-lined, every underlined block is y-lined, and the
+tuple of blocks is x-lined. Naturally, the multary quasigroups Vimay be named “vertical”
+andHi, “horizontal”.)
+The proof of the code distance is similar to that in [9], and the other pr operties of a
+¯µ-component are straightforward. The existence of admissible ( q−1)-ary quasigroups v
+andhis the only restriction on the q(this concerns the next subsection as well). If Fqis
+a finite field, there are linear examples: v(y1,...,y q−1) =y1+...+yq−1,v(y1,...,y q−1) =
+α1y1+...+αq−1yq−1whereα1, ...,αq−1are all the non-zero elements of Fq. Ifqis not
+a prime power, the existence of a q-ary perfect code of length q+1 is an open problem
+(with the only exception q= 6, when the nonexistence follows from the nonexistence of
+two orthogonal 6 ×6 Latin squares [1, Th.6]).
+3.2. Generalized Phelps construction
+Here we describe another way to construct ¯ µ-components, which generalizes the construc-
+tion of binary perfect codes from [8].
+Lemma 2. Let¯µ∈Ft
+q. Let for every ifrom1tot+1the codesCi,j,j= 0,1,...,qk−k
+form a partition of Fk
+qinto perfect codes and γi:Fk
+q→ {0,1,...,qk−k}be the corre-
+sponding partition function:
+γi(¯y) =j⇐⇒¯y∈Ci,j.
+Letvandhbe(q−1)-ary quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) :
+¯y∈Fq−1
+q}is perfect. Let V1, ...,Vtbe(k+ 1)-ary quasigroups of order qandQbe a
+t-ary quasigroup of order qk−k+1.
+K¯µ=/braceleftBig
+(¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :
+¯xij∈Fq−1
+q,
+(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
+Q(γ1(h(¯x11),...,h(¯x1k)),...,γ t(h(¯xt1),...,h(¯xtk))) =γt+1(z1,...,zk)/bracerightBig
+is a¯µ-component that satisfies the generalized parity-check law with
+σi(·,...,·,·) =Vi(v(·),...,v(·),·).
+The proof consists of trivial verifications.
+4. On the number of perfect codes
+In this section we discuss some observations, which result in the bes t known lower bound
+on the number of q-ary perfect codes, q≥3. The basic facts are already contained in
+other known results: lower bounds on the number of multary quasig roups of order q, the
+6construction [9] of perfect codes from multary quasigroups of or derq, and the possibility
+to choose the quasigroup independently for every vector of the o uter code (this possibility
+was not explicitly mentioned in [9], but used in the previous paper [8]).
+A general lower bound, in terms of the number of multary quasigrou ps, is given by
+Lemma 3. In combination with Lemma 4, it gives explicit numbers.
+Lemma 3. The number of q-ary perfect codes of length nis not less than
+Q/parenleftBiggn−1
+q,q/parenrightBiggRn−1
+q
+whereQ(m,q)is the number of m-ary quasigroups of order qand whereRn′=qn′/(n′q−
+q+1)is the cardinality of a perfect code of length n′.
+Proof. Constructing a perfect code like in Theorem 2 with t=n−1
+q, we combine
+Rn−1
+qdifferent ¯µ-components.
+Constructing every such a component as in Lemma 2, k= 1,t=n−1
+q, we are free
+to choose the t-ary quasigroup Qof orderqinQ(t,q) ways. Clearly, different t-ary
+quasigroups give different components. (Equivalently, we can use L emma 1 and choose
+the (t+1)-ary quasigroup H1, but should note that the value of H1in the construction is
+always fixed when k= 1, because C#consists of only one vertex; so we again have Q(t,q)
+different choices, not Q(t+1,q)). △
+Lemma 4. The number Q(m,q)ofm-ary quasigroups of order qsatisfies:
+(a) [5]Q(m,3) = 3·2m;
+(b) [11]Q(m,4) = 3m+1·22m+1(1+o(1));
+(c) [4]Q(m,5)≥23n/3−0.072;
+(d) [10]Q(m,q)≥2((q2−4q+3)/4)n/2for oddq(the previous bound [4]wasQ(m,q)≥
+2⌊q/3⌋n);
+(e) [4]Q(m,q1q2)≥Q(m,q1)·Q(m,q2)qm
+1.
+For oddq≥5, the number of codes given by Lemmas 3 and 4(c,d) improves the
+constantcin the lower estimation of form eecn(1+o(1))for the number of perfect codes, in
+comparison with the last known lower bound [6]. Informally, this can be explained in the
+following way: the construction in [6] can be described in terms of mu tually independent
+small modifications of the linear multary quasigroup of order q, while the lower bounds
+in Lemma 4(c,d) are based on a specially-constructed nonlinear multa ry quasigroup that
+allows a lager number of independent modifications. For q= 3 andq= 2s, the number
+of codes given by Lemmas 3 and 4(a,b,e) also slightly improves the boun d in [6], but do
+not affect on the constant c.
+7References
+1. S. W. Golomb and E. C. Posner. Rook domains, latin squares, and e rror-distributing
+codes.IEEE Trans. Inf. Theory , 10(3):196–208, 1964.
+2. O. Heden. On the classification of perfect binary 1-error corre cting codes. Preprint
+TRITA-MAT-2002-01, KTH, Stockholm, 2002.
+3. D. S. Krotov. Combining construction of perfect binary codes. Probl. Inf. Transm. ,
+36(4):349–353, 2000. translated from Probl. Peredachi Inf. 36 (4) (2000), 74-79.
+4. D. S. Krotov, V. N. Potapov, and P. V. Sokolova. On reconstru cting reducible n-ary
+quasigroups and switching subquasigroups. Quasigroups Relat. Syst. , 16(1):55–67,
+2008. ArXiv:math/0608269
+5. C. F. Laywine and G. L. Mullen. Discrete Mathematics Using Latin Squares . Wiley,
+New York, 1998.
+6. A. V. Los’. Construction of perfect q-ary codes by switchings o f simple components.
+Probl. Inf. Transm. , 42(1):30–37, 2006. DOI: 10.1134/S0032946006010030 transla ted
+from Probl. Peredachi Inf. 42(1) (2006), 34-42.
+7. M. Mollard. A generalized parity function and its use in the constru ction of perfect
+codes.SIAM J. Algebraic Discrete Methods , 7(1):113–115, 1986.
+8. K. T. Phelps. A general product construction for error corre cting codes. SIAM J.
+Algebraic Discrete Methods , 5(2):224–228, 1984.
+9. K. T. Phelps. A product construction for perfect codes over a rbitrary alphabets.
+IEEE Trans. Inf. Theory , 30(5):769–771, 1984.
+10. V. N. Potapov and D. S. Krotov. On the number of n-ary quasigroups of finite order.
+Submitted. ArXiv:0912.5453
+11. V.N.PotapovandD.S.Krotov. Asymptoticsforthenumbero fn-quasigroupsoforder
+4.Sib. Math. J. , 47(4):720–731, 2006. DOI: 10.1007/s11202-006-0083-9 tran slated
+from Sib. Mat. Zh. 47(4) (2006), 873-887. ArXiv:math/0605104
+O. Heden
+Department of Mathematics, KTH
+S-100 44 Stockholm, Sweden
+email:[email protected]
+D. Krotov
+Sobolev Institute of Mathematics
+and
+Mechanics and Mathematics Department, Novosibirsk State Univer sity
+Novosibirsk, Russia
+email:[email protected]
+8

1001.0002.txt

@@ -0,0 +1,771 @@
+arXiv:1001.0002v2  [hep-th]  9 Mar 2010Gravity duals for logarithmic conformal field theories
+Daniel Grumiller and Niklas Johansson
+Institute for Theoretical Physics, Vienna University of Te chnology
+Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria
+E-mail:[email protected], [email protected]. ac.at
+Abstract. Logarithmic conformal fieldtheories with vanishingcentra l charge describe systems
+withquencheddisorder, percolation ordiluteself-avoidi ngpolymers. Inthesetheories theenergy
+momentum tensor acquires a logarithmic partner. In this tal k we address the construction of
+possible gravity duals for these logarithmic conformal fiel d theories and present two viable
+candidates for such duals, namely theories of massive gravi ty in three dimensions at a chiral
+point.
+Outline
+Thistalk isorganized asfollows. Insection 1werecall sali ent featuresof2-dimensionalconformal
+field theories. In section 2 we review a specific class of logar ithmic conformal field theories where
+the energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-list
+for gravity duals to logarithmic conformal field theories. I n section 4 we discuss two examples
+of massive gravity theories that comply with all the items on that list. In section 5 we address
+possible applications of an Anti-deSitter/logarithmic co nformal field theory correspondence in
+condensed matter physics.
+1. Conformal field theory distillate
+Conformal field theories (CFTs) are quantum field theories th at exhibit invariance under angle
+preserving transformations: translations, rotations, bo osts, dilatations and special conformal
+transformations. In two dimensions the conformal algebra i s infinite dimensional, and thus
+two-dimensional CFTs exhibit a particularly rich structur e. They arise in various contexts in
+physics, including string theory, statistical mechanics a nd condensed matter physics, see e.g. [1].
+The main observables in any field theory are correlation func tions between gauge invariant
+operators. There exist powerful tools to calculate these co rrelators in a CFT. The operator
+content of various CFTs may differ, but all CFTs contain at leas t an energy momentum tensor
+Tµν. Conformal invariance requires the energy momentum tensor to be traceless, Tµ
+µ= 0,
+in addition to its conservation, ∂µTµν= 0. In lightcone gauge for the Minkowski metric,
+ds2= 2dzd¯z, these equations take a particularly simple form: Tz¯z= 0,Tzz=Tzz(z) :=OL(z)
+andT¯z¯z=T¯z¯z(¯z) :=OR(¯z). Conformal Ward identities determine essentially unique ly the form
+of 2- and3-point correlators between thefluxcomponents OL/Rof theenergy momentum tensor:∝an}b∇acketle{tOR(¯z)OR(0)∝an}b∇acket∇i}ht=cR
+2¯z4(1a)
+∝an}b∇acketle{tOL(z)OL(0)∝an}b∇acket∇i}ht=cL
+2z4(1b)
+∝an}b∇acketle{tOL(z)OR(0)∝an}b∇acket∇i}ht= 0 (1c)
+∝an}b∇acketle{tOR(¯z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht=cR
+¯z2¯z′2(¯z−¯z′)2(1d)
+∝an}b∇acketle{tOL(z)OL(z′)OL(0)∝an}b∇acket∇i}ht=cL
+z2z′2(z−z′)2(1e)
+∝an}b∇acketle{tOL(z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht= 0 (1f)
+∝an}b∇acketle{tOL(z)OL(z′)OR(0)∝an}b∇acket∇i}ht= 0 (1g)
+The real numbers cL,cRare the left and right central charges, which determine key p roperties of
+the CFT. We have omitted terms that are less divergent in the n ear coincidence limit z,¯z→0 as
+well as contact terms, i.e., contributions that are localiz ed (δ-functions and derivatives thereof).
+If someone provides us with a traceless energy momentum tens or and gives us a prescription
+how to calculate correlators,1but does not reveal whether the underlying field theory is a CF T,
+thenwecanperformthefollowing check. Wecalculate all 2- a nd3-point correlators of theenergy
+momentum tensor with itself, and if at least one of the correl ators does not match precisely with
+the corresponding correlator in (1) then we know that the fiel d theory in question cannot be a
+CFT. On the other hand, if all the correlators match with corr esponding ones in (1) we have
+non-trivial evidence that the field theory in question might be a CFT. Let us keep this stringent
+check in mind for later purposes, but switch gears now and con sider a specific class of CFTs,
+namely logarithmic CFTs (LCFTs).
+2. Logarithmic CFTs with an energetic partner
+LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTs
+and postpone a physics discussion until the end of the talk, s ee [3,4] for reviews. There are two
+conceptually different, but mathematically equivalent, way s to define LCFTs. In both versions
+there exists at least one operator that acquires a logarithm ic partner, which we denote by Olog.
+We focus in this talk exclusively on theories where one (or bo th) of the energy momentum
+tensor flux components is the operator acquiring such a partn er, for instance OL. We discuss
+now briefly both ways of defining LCFTs.
+According to the first definition “acquiring a logarithmic pa rtner” means that the
+Hamiltonian Hcannot be diagonalized. For example
+H/parenleftbigg
+Olog
+OL/parenrightbigg
+=/parenleftbigg
+2 1
+0 2/parenrightbigg/parenleftbigg
+Olog
+OL/parenrightbigg
+(2)
+Theangularmomentum operator Jmay ormay not bediagonalizable. Weconsider onlytheories
+whereJis diagonalizable:
+J/parenleftbigg
+Olog
+OL/parenrightbigg
+=/parenleftbigg
+2 0
+0 2/parenrightbigg/parenleftbigg
+Olog
+OL/parenrightbigg
+(3)
+The eigenvalues 2 arise because the energy momentum tensor a nd its logarithmic partner both
+correspond to spin-2 excitations.
+1This is exactly what the AdS/CFT correspondence does: given a gravity dual we can calculate the energy
+momentum tensor and correlators.The second definition makes it more transparent why these CFT s are called “logarithmic”
+in the first place. Suppose that in addition to OL/Rwe have an operator OMwith conformal
+weightsh= 2+ε,¯h=ε, meaning that its 2-point correlator with itself is given by
+∝an}b∇acketle{tOM(z,¯z)OM(0,0)∝an}b∇acket∇i}ht=ˆB
+z4+2ε¯z2ε(4)
+The correlator of OMwithOLvanishes since the latter has conformal weights h= 2,¯h= 0, and
+operators whose conformal weights do not match lead to vanis hing correlators. Suppose now
+that we send the central charge cLand the parameter εto zero, and simultaneously send ˆBto
+infinity, such that the following limits exist:
+bL:= lim
+cL→0−cL
+ε∝ne}ationslash= 0B:= lim
+cL→0/parenleftbigˆB+2
+cL/parenrightbig
+(5)
+Then we can define a new operator Ologthat linearly combines OL/M.
+Olog=bLOL
+cL+bL
+2OM(6)
+Taking the limit cL→0 leads to the following 2-point correlators:
+∝an}b∇acketle{tOL(z)OL(0,0)∝an}b∇acket∇i}ht= 0 (7a)
+∝an}b∇acketle{tOL(z)Olog(0,0)∝an}b∇acket∇i}ht=bL
+2z4(7b)
+∝an}b∇acketle{tOlog(z,¯z)Olog(0,0)∝an}b∇acket∇i}ht=−bLln(m2
+L|z|2)
+z4(7c)
+These 2-point correlators exhibit several remarkable feat ures. The flux component OLof the
+energy momentum tensor becomes a zero norm state (7a). Never theless, the theory does not
+become chiral, because the left-moving sector is not trivia l:OLhas a non-vanishing correlator
+(7b) with its logarithmic partner Olog. The 2-point correlator (7c) between two logarithmic
+operators Ologmakes it clear why such CFTs have the attribute “logarithmic ”. The constant
+bL, sometimes called “new anomaly”, defines crucial propertie s of the LCFT, much like the
+central charges do in ordinary CFTs. The mass scale mLappearing in the last correlator above
+has no significance, and is determined by the value of Bin (5). It can be changed to any finite
+value by the redefinition Olog→ Olog+γOLwith some finite γ. We setmL= 1 for convenience.
+Conformal Ward identities determine again essentially uni quely the form of 2- and 3-point
+correlators in a LCFT. For the specific case where the energy m omentum tensor acquires a
+logarithmic partner the 3-point correlators were calculat ed in [5]. The non-vanishing ones are
+given by
+∝an}b∇acketle{tOL(z,¯z)OL(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=bL
+z2z′2(z−z′)2(8a)
+∝an}b∇acketle{tOL(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=−2bLln|z′|2+bL
+2
+z2z′2(z−z′)2(8b)
+∝an}b∇acketle{tOlog(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=lengthy
+z2z′2(z−z′)2(8c)
+If alsoORacquires a logarithmic partner O/tildewiderlogthen the construction above can be repeated,
+changing everywhere L→R,z→¯zetc. In that case we have a LCFT with cL=cR= 0 andbL,bR∝ne}ationslash= 0. Alternatively, it may happen that only OLhas a logarithmic partner Olog. In that
+case we have a LCFT with cL=bR= 0 andbL,cR∝ne}ationslash= 0. This concludes our brief excursion into
+the realm of LCFTs.
+Given that LCFTs are interesting in physics (see section 5) a nd that a powerful way to
+describe strongly coupled CFTs is to exploit the AdS/CFT cor respondence [6] it is natural to
+inquire whether there are any gravity duals to LCFTs.
+3. Wish-list for gravity duals to LCFTs
+In this section we establish necessary properties required for gravity duals to LCFTs. We
+formulate them as a wish-list and explain afterwards each it em on this list.
+(i) We wishfora 3-dimensional action Sthat dependsonthemetric gµνandpossiblyonfurther
+fields that we summarily denote by φ.
+(ii) We wish for the existence of AdS 3vacua with finite AdS radius ℓ.
+(iii) We wish for a finite, conserved and traceless Brown–Yor k stress tensor, given by the first
+variation of the full on-shell action (including boundary t erms) with respect to the metric.
+(iv) We wish that the 2- and 3-point correlators of the Brown– York stress tensor with itself are
+given by (1).
+(v) We wish for central charges (a la Brown–Henneaux [7]) tha t can be tuned to zero, without
+requiring a singular limit of the AdS radius or of Newton’s co nstant. For concreteness we
+assumecL= 0 (in addition cRmay also vanish, but it need not).
+(vi) We wish for a logarithmic partner to the Brown–York stre ss tensor, so that we obtain a
+Jordan-block structure like in (2) and (3).
+(vii) We wish that the 2- and non-vanishing 3-point correlat ors of the Brown–York stress tensor
+with its logarithmic partner are given by (7) and (8) (and the right-handed analog thereof).
+We explain now why each of these items is necessary. (i) is req uired since the AdS/CFT
+correspondence relates a gravity theory in d+1 dimensions to a CFT in ddimensions, and we
+chosed= 2 on the CFT side. (ii) is required since we are not merely loo king for a gauge/gravity
+duality, butreallyforanAdS/CFTcorrespondence,whichre quirestheexistenceofAdSsolutions
+on the gravity side. (iii) is required since we desire consis tency with the AdS dictionary, which
+relates the vacuum expectation value of the renormalized en ergy momentum tensor in the CFT
+∝an}b∇acketle{tTij∝an}b∇acket∇i}htto the Brown–York stress tensor TBY
+ij:
+∝an}b∇acketle{tTij∝an}b∇acket∇i}ht=TBY
+ij=2√−gδS
+δgij/vextendsingle/vextendsingle/vextendsingle
+EOM(9)
+The right hand side of this equation contains the first variat ion of the full on-shell action with
+respect to the metric, which by definition yields the Brown–Y ork stress tensor. (iv) is required
+since the 2- and 3-point correlators of a CFT are fixed by confo rmal Ward identities to take
+the form (1). (v) is required because of the construction pre sented in section 2, where a LCFT
+emerges from taking an appropriate limit of vanishing centr al charge, so we need to be able
+to tune the central charge without generating parametric si ngularities. Actually, there are
+two cases: either left and right central charge vanish and bo th energy momentum tensor flux
+components acquire a logarithmic partner, or only one of the m acquires a logarithmic partner,
+which for sake of specificity we always choose to be left. (vi) is required, since we consider
+exclusively LCFTs where the energy momentum tensor acquire s a logarithmic partner. (vii) is
+required since the 2- and 3-point correlators of a LCFT are fix ed by conformal Ward identities to
+taketheform(7), (8). Ifanyoftheitemsonthewish-listabo veisnotfulfilleditisimpossiblethat
+the gravitational theory under consideration is a gravity d ual to a LCFT of the type discussedin section 2.2On the other hand, if all the wishes are granted by a given grav itational theory
+there are excellent chances that this theory is dual to a LCFT . Until recently no good gravity
+duals for LCFTs were known [8–12].
+Before addressing candidate theories that may comply with a ll wishes we review briefly how
+to calculate correlators on the gravity side [6], since we sh all need such calculations for checking
+several items on the wish-list. The basic identity of the AdS /CFT dictionary is
+∝an}b∇acketle{tO1(z1)O2(z2)...On(zn)∝an}b∇acket∇i}ht=δ(n)S
+δj1(z1)δj2(z2)...δjn(zn)/vextendsingle/vextendsingle/vextendsingle
+ji=0(10)
+The left hand side is the CFT correlator between noperators Oi, whereOiin our case comprise
+theleft-andright-moving fluxcomponentsoftheenergymome ntumtensor andtheirlogarithmic
+partners. The right hand side contains the gravitational ac tionSdifferentiated with respect to
+appropriate sources jifor the corresponding operators. According to the AdS/CFT d ictionary
+“appropriate sources” refers to non-normalizable solutio ns of the linearized equations of motion.
+We shall be more concrete about the operators, actions, sour ces and non-normalizable solutions
+to the linearized equations of motion in the next section. Fo r now we address possible candidate
+theories of gravity duals to LCFTs.
+The simplest candidate, pure 3-dimensional Einstein gravi ty with a cosmological constant
+described by the action
+SEH=−1
+8πGN/integraldisplay
+Md3x√−g/bracketleftig
+R+2
+ℓ2/bracketrightig
+−1
+4πGN/integraldisplay
+∂Md2x√−γ/bracketleftig
+K−1
+ℓ/bracketrightig
+(11)
+does not comply with the whole wish list. Only the first four wi shes are granted: The 3-
+dimensional action (12) depends on the metric. The equation s of motion are solved by AdS 3.
+ds2
+AdS3=gAdS3µνdxµdxν=ℓ2/parenleftbig
+dρ2−1
+4cosh2ρ(du+dv)2+1
+4sinh2ρ(du−dv)2/parenrightbig
+(12)
+The Brown–York stress tensor (9) is finite, conserved and tra celess. The 2- and 3-point
+correlators on the gravity side match precisely with (1). Ho wever, the central charges are given
+by [7]
+cL=cR=3ℓ
+2GN(13)
+and therefore allow no tuning to cL= 0 without taking a singular limit. Moreover, there is no
+candidate for a logarithmic partner to the Brown–York stres s tensor. Thus, pure 3-dimensional
+Einstein gravity cannot be dual to a LCFT.
+Adding matter fields to Einstein gravity does not help neithe r. While this may lead to other
+kinds of LCFTs, it cannot produce a logarithmic partner for t he energy momentum tensor. This
+is so, because the energy momentum tensor corresponds to gra viton (spin-2) excitations in the
+bulk, and the only field producing such excitations is the met ric.
+Therefore, what we need is a way to provide additional degree s of freedom in the gravity
+sector. The most natural way to do this is by considering high er derivative interactions of the
+metric. Thefirstgravity modelofthistypewas constructedb yDeser, Jackiw andTempleton [13]
+who introduced a Chern–Simons term for the Christoffel connec tion.
+SCS=−1
+16πGNµ/integraldisplay
+d3xǫλµνΓρσλ/bracketleftig
+∂µΓσρν+2
+3ΓσκµΓκσν/bracketrightig
+(14)
+2Other types of LCFTs exist, e.g. with non-vanishing central charge or with logarithmic partners to operators
+other than the energy momentum tensor. The gravity duals for such LCFTs need not comply with all the items
+on our wish list.Hereµis a real coupling constant. Adding this action to the Einste in–Hilbert action (11)
+generates massive graviton excitations in the bulk, which i s encouraging for our wish list since
+we need these extra degrees of freedom. The model that arises when summing the actions (11)
+and (14),
+SCTMG=SEH+SCS (15)
+is known as “cosmological topologically massive gravity” ( CTMG) [14]. It was demonstrated by
+KrausandLarsen[15]that thecentral charges inCTMG areshi ftedfromtheir Brown–Henneaux
+values:
+cL=3ℓ
+2GN/parenleftbig
+1−1
+µℓ/parenrightbig
+cR=3ℓ
+2GN/parenleftbig
+1+1
+µℓ/parenrightbig
+(16)
+This is again good news concerning our wish list, since cLcan be made vanishing by a (non-
+singular) tuning of parameters in the action.
+µℓ= 1 (17)
+CTMG (15) with the tuning above (17) is known as “cosmologica l topologically massive gravity
+at the chiral point” (CCTMG). It complies with the first five it ems on our wish list, but we still
+have to prove that also the last two wishes are granted. To thi s end we need to find a suitable
+partner for the graviton.
+4. Keeping logs in massive gravity
+4.1. Login
+In this section we discuss the evidence for the existence of s pecific gravity duals to LCFTs that
+has accumulated over the past two years. We start with the the ory introduced above, CCTMG,
+and we end with a relatively new theory, new massive gravity [ 16].
+4.2. Seeds of logs
+Given that we want a partner for the graviton we consider now g raviton excitations ψaround
+the AdS background (12) in CCTMG.
+gµν=gAdS3µν+ψµν (18)
+Li,SongandStrominger[17]foundanicewaytoconstructthe m,andwefollowtheirconstruction
+here. Imposing transverse gauge ∇µψµν= 0 and defining the mutually commuting first order
+operators
+/parenleftbig
+DM/parenrightbigβ
+µ=δβ
+µ+1
+µεµαβ∇α/parenleftbig
+DL/R/parenrightbigβ
+µ=δβ
+µ±ℓεµαβ∇α (19)
+allows to write the linearized equations of motion around th e AdS background (12) as follows.
+(DMDLDRψ)µν= 0 (20)
+A mode annihilated by DM(DL) [DR]{(DL)2but not by DL}is called massive (left-moving)
+[right-moving] {logarithmic }and is denoted by ψM(ψL) [ψR]{ψlog}. Away from the chiral
+point,µℓ∝ne}ationslash= 1, the general solution to the linearized equations of moti on (20) is obtained from
+linearly combining left, right and massive modes [17]. At th e chiral point DMdegenerates with
+DLand the general solution to the linearized equations of moti on (20) is obtained from linearly
+combining left, right and logarithmic modes [18]. Interest ingly, we discovered in [18] that the
+modesψlogandψLbehave as follows:
+(L0+¯L0)/parenleftbigg
+ψlog
+ψL/parenrightbigg
+=/parenleftbigg
+2 1
+0 2/parenrightbigg/parenleftbigg
+ψlog
+ψL/parenrightbigg
+(21)whereL0=i∂u,¯L0=i∂vand
+(L0−¯L0)/parenleftbigg
+ψlog
+ψL/parenrightbigg
+=/parenleftbigg
+2 0
+0 2/parenrightbigg/parenleftbigg
+ψlog
+ψL/parenrightbigg
+(22)
+If we define naturally the Hamiltonian by H=L0+¯L0and the angular momentum by
+J=L0−¯L0we recover exactly (2) and (3), which suggests that the CFT du al to CCTMG
+(if it exists) is logarithmic, as conjectured in [18]. It was further shown with Jackiw that the
+existence of the logarithmic excitations ψlogis not an artifact of the linearized approach, but
+persists in the full theory [19].
+Thus, also the sixth wish is granted in CCTMG. The rest of this section discusses the last
+wish.
+4.3. Growing logs
+We assume now that there is a standard AdS/CFT dictionary [6] available for LCFTs and check
+if CCTMG indeed leads to the correct 2- and 3-point correlato rs. To this end we have to identify
+the sources jithat appear on the right hand side of the correlator equation (10). Following the
+standard AdS/CFT prescription the sources for the operator sOL(OR) [Olog] are given by left
+(right) [logarithmic] non-normalizablesolutions tothel inearized equations of motion (20). Thus,
+our first task is to find all solutions of the linearized equati ons of motion and to classify them
+into normalizable and non-normalizable ones, where “norma lizable” refers to asymptotic (large
+ρ) behavior that is exponentially suppressed as compared to t he AdS background (12).
+A construction of all normalizable left and right solutions was provided in [17], and the
+normalizable logarithmic solutions were constructed in [1 8].3The non-normalizable solutions
+were constructed in [25]. It turned out to be convenient to wo rk in momentum space
+ψL/R/log
+µν(h,¯h) =e−ih(t+φ)−i¯h(t−φ)FL/R/log
+µν(ρ) (23)
+The momenta h,¯hare called “weights”. All components of the tensor Fµνare determined
+algebraically, except for one that is determined from a seco nd order (hypergeometric) differential
+equation. Ingeneral oneofthelinearcombinations of theso lutionsis singularattheorigin ρ= 0,
+whiletheother isregular there. We keep onlyregular soluti ons. For each given set ofweights h,¯h
+the regular solution is either normalizable or non-normali zable. It turns out that normalizable
+solutions exist for integer weights h≥2,¯h≥0 (orh≤ −2,¯h≤0). All other solutions are
+non-normalizable.
+An example for a normalizable left mode is given by the primar y with weights h= 2,¯h= 0
+ψL
+µν(2,0) =e−2iu
+cosh4ρ
+1
+4sinh2(2ρ) 0i
+2sinh(2ρ)
+0 0 0
+i
+2sinh(2ρ) 0 −1
+
+µν(24)
+Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant.
+The corresponding logarithmic mode is given by
+ψlog
+µν(2,0) =−1
+2(i(u+v)+lncosh2ρ)ψL
+µν(2,0) (25)
+Evidently, it behaves asymptotically like its left partner (24), except for overall linear growth in
+ρ. It is also worthwhile emphasizing that the logarithmic mod e (25) depends linearly on time
+3All these modes are compatible with asymptotic AdS behavior [20,21], and they appear in vacuum expectation
+values of 1-point functions. Indeed, the 1-point function /angbracketleftTij/angbracketrightinvolves both ψlogandψR[21–24].t= (u+v)/2. Both features are inherent to all logarithmic modes. All o ther normalizable
+modes can be constructed from the primaries (24), (25) algeb raically.
+An example for a non-normalizable left mode is given by the mo de with weights h= 1,
+¯h=−1
+ψL
+µν(1,−1) =1
+4e−iu+iv
+0 0 0
+0 cosh(2 ρ)−1−2i/radicalig
+cosh(2ρ)−1
+cosh(2ρ)+1
+0−2i/radicalig
+cosh(2ρ)−1
+cosh(2ρ)+1−4
+cosh(2ρ)+1
+
+µν(26)
+Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant,
+except for the vv-component, which grows like e2ρ. The corresponding logarithmic mode grows
+again faster than its left partner (26) by a factor of ρand depends again linearly on time.
+Given a non-normalizable solution ψLobviously also αψLis a non-normalizable solution,
+with some constant α. To fix this normalization ambiguity we demand standard coup ling of the
+metric to the stress tensor:
+S(ψuL
+v,Tv
+u) =1
+2/integraldisplay
+dtdφ/radicalig
+−g(0)ψuu
+LTuu=/integraldisplay
+dtdφe−ihu−i¯hvTuu (27)
+HereSis either someCFT action withbackgroundmetric g(0)or adualgravitational action with
+boundary metric g(0). The non-normalizable mode ψLis the source for the energy-momentum
+flux component Tuu. The requirement (27) fixes the normalization. The discussi on above
+focussed on left modes. For the right modes essentially the s ame discussion applies, but with
+the substitutions L↔R,h↔¯handu↔v.
+4.4. Logging correlators
+Generically the 2-point correlators on the gravity side bet ween two modes ψ1(h,¯h) andψ2(h′,¯h′)
+in momentum space are determined by
+∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)∝an}b∇acket∇i}ht=1
+2/parenleftbig
+δ(2)SCCTMG(ψ1,ψ2)+δ(2)SCCTMG(ψ2,ψ1)/parenrightbig
+(28)
+where∝an}b∇acketle{tψ1ψ2∝an}b∇acket∇i}htstands for the correlation function of the CFT operators dua l to the graviton
+modesψ1andψ2. On the right hand side one has to plug the non-normalizable m odesψ1
+andψ2into the second variation of the on-shell action and symmetr ize with respect to the two
+modes. The second variation of the on-shell action of CCTMG
+δ(2)SCCTMG=−1
+16πGN/integraldisplay
+d3x√−g/parenleftbig
+DLψ1∗/parenrightbigµνδGµν(ψ2)+boundary terms (29)
+turns out to be very similar to the second variation of the on- shell Einstein–Hilbert action
+δ(2)SEH=−1
+16πGN/integraldisplay
+d3x√−gψ1µν∗δGµν(ψ2)+boundary terms (30)
+Thissimilarity allows ustoexploitresultsfromEinsteing ravity forCCTMG,aswenowexplain.4
+The bulk term in CCTMG (29) has the same form as in Einstein the ory (30) with ψ1replaced
+byDLψ1. Now, consider boundary terms. Possible obstructions to a w ell-defined Dirichlet
+boundary value problem can come only from the variation δGµν(ψ2), sinceDLis a first order
+operator. Thus any boundary terms appearing in (29) contain ing normal derivatives must be
+4Alternatively, one can follow the program of holographic re normalization, as it was done by Skenderis, Taylor
+and van Rees [23]. Their results for 2-point correlators agr ee with the results presented here.identical with those in Einstein gravity upon substituting ψ1→ DLψ1. In addition there can be
+boundary terms which do not contain normal derivatives of th e metric. However, it turns out
+that such terms can at most lead to contact terms in the hologr aphic computation of 2-point
+functions. The upshot of this discussion is that we can reduc e the calculation of all possible 2-
+point functions in CCTMG to the equivalent calculation in Ei nstein gravity with suitable source
+terms. To continue we go on-shell.5
+DLψL= 0 DLψR= 2ψRDLψlog=−2ψL(31)
+These relations together with the comparison between CCTMG (29) and Einstein gravity (30)
+then establish
+∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htEH (32a)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32b)
+∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32c)
+∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32d)
+∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH (32e)
+Here the sign ∼means equality up to contact terms. Evaluating the right han d sides in Einstein
+gravity yields
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH=δh,h′δ¯h,¯h′cBH
+24h
+¯h(h2−1)t1/integraldisplay
+t0dt (33)
+and similarly for the right modes, with h↔¯h. The quantity cBHis the Brown–Henneaux
+central charge (13). The calculation of the 2-point correla tor between two logarithmic modes
+cannot be reduced to a correlator known from Einstein gravit y. The result is given by [25]
+∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −δh,h′δ¯h,¯h′ℓ
+4GNh
+¯h(h2−1)/parenleftbig
+ψ(h−1)+ψ(−¯h)/parenrightbigt1/integraldisplay
+t0dt(34)
+whereψis the digamma function. An ambiguity in defining ψlog, viz.,ψlog→ψlog+γψL, was
+fixed conveniently in the result (34). This ambiguity corres ponds precisely to the ambiguity of
+the LCFT mass scale mLin (7c) (see also the discussion below that equation).
+To compare the results (32)-(34) with the Euclidean 2-point correlators in the short-
+distance limit (1), (7) we take the limit of large weights h,−¯h→ ∞(e.g. lim h→∞ψ(h) =
+lnh+O(1/h)) and Fourier-transform back to coordinate space (e.g. h3/¯his Fourier-transformed
+into∂4
+z/(∂z∂¯z)δ(2)(z,¯z)∝∂4
+zln|z| ∝1/z4). Straightforward calculation establishes perfect
+agreement with the LCFT correlators (1), (7), provided we us e the values
+cL= 0 cR=3ℓ
+GNbL=−3ℓ
+GN(35)
+These are exactly the values for central charges cL,cR[15] and new anomaly bL[23,25] found
+before. Thus, at the level of 2-point correlators CCTMG is in deed a gravity dual for a LCFT.
+5Above by “on-shell” we meant that the background metric is Ad S3(12) and therefore a solution of the classical
+equations of motion. Here by “on-shell” we mean additionall y that the linearized equations of motion (20) hold.Ψ1
+Ψ3Ψ2
+Figure 1. Witten diagram for three graviton correlator
+We evaluate now the Witten diagram in Fig. 1, which yields the 3-point correlator on the
+gravity side between three modes ψ1(h,¯h),ψ2(h′,¯h′) andψ3(h′′,¯h′′) in momentum space.
+∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)ψ3(h′′,¯h′′)∝an}b∇acket∇i}ht=1
+6/parenleftbig
+δ(3)SCCTMG(ψ1,ψ2,ψ3)+5 permutations/parenrightbig
+(36)
+On the right hand side one has to plug the non-normalizable mo desψ1,ψ2andψ3into the third
+variation of the on-shell action and symmetrize with respec t to all three modes.
+δ(3)SCCTMG∼ −1
+16πGN/integraldisplay
+d3x√−g/bracketleftig/parenleftbig
+DLψ1/parenrightbigµνδ(2)Rµν(ψ2,ψ3)+ψ1µν∆µν(ψ2,ψ3)/bracketrightig
+(37)
+The quantity δ(2)Rµν(ψ2,ψ3) denotes the second variation of the Ricci-tensor and the te nsor
+∆µν(ψ2,ψ3) vanishes if evaluated on left- and/or right-moving soluti ons. All boundary terms
+turn out to be contact terms, which is why only bulk terms are p resent in the result (37) for the
+third variation of the on-shell action. We compare again wit h Einstein gravity.
+δ(3)SEH∼ −1
+16πGN/integraldisplay
+d3x√−gψ1µνδ(2)Rµν(ψ2,ψ3) (38)
+Once more we can exploit some results from Einstein gravity f or CCTMG, and we find the
+following results [25] for 3-point correlators without log -insertions:
+∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htEH (39a)
+∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39b)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39c)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39d)
+with one log-insertion:
+∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40a)
+∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40b)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htEH (40c)and with two or more log-insertions:
+lim
+|weights|→∞∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (41a)
+lim
+|weights|→∞∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′Plog(h,h′,¯h,¯h′)
+¯h¯h′(¯h+¯h′)(41b)
+lim
+|weights|→∞∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′lengthy
+¯h¯h′(¯h+¯h′)(41c)
+Thelast two correlators so far could becalculated qualitat ively only (Plogis a known polynomial
+in the weights and also contains logarithms in the weights, a s expected on general grounds),
+and it would be interesting to calculate them exactly. They a re in qualitative agreement with
+corresponding LCFT correlators. All other correlators hav e been calculated exactly [25], and
+they are in precise agreement with the LCFT correlators (1), (8), provided we use again the
+values (35) for central charges and new anomaly.
+Inconclusion, also theseventh wishisgranted forCCTMG.6Thus, thereareexcellent chances
+that CCTMG is dual to a LCFT with values for central charges an d new anomaly given by (35).
+4.5. Logs don’t grow on trees
+From the discussion above it is clear that possible gravity d uals for LCFTs are sparse in theory
+space: Einstein gravity (11) does not provide a gravity dual for any tuning of parameters and
+CTMG (15) does potentially provide a gravity dual only for a s pecific tuning of parameters (17).
+Any candidate for a novel gravity dual to a LCFT is therefore w elcomed as a rare entity.
+Very recently another plausible candidate for such a gravit ational theory was found [26].
+That theory is known as “new massive gravity” [16].
+SNMG=1
+16πGN/integraldisplay
+d3x√−g/bracketleftig
+σR+1
+m2/parenleftbig
+RµνRµν−3
+8R2/parenrightbig
+−2λm2/bracketrightig
+(42)
+Heremis a mass parameter, λa dimensionless cosmological parameter and σ=±1 the sign of
+the Einstein-Hilbert term. If they are tuned as follows
+λ= 3 ⇒m2=−σ
+2ℓ2(43)
+then essentially the same story unfolds as for CTMG at the chi ral point. The main difference
+to CCTMG is that both central charges vanish in new massive gr avity at the chiral point
+(CNMG) [27,28].
+cL=cR=3ℓ
+2GN/parenleftbigg
+σ+1
+2ℓ2m2/parenrightbigg
+= 0 (44)
+Therefore, both left and right flux component of the energy mo mentum tensor acquire a
+logarithmic partner. It is easy to check that CNMG grants us t he first six wishes from section
+3. The seventh wish requires again the calculation of correl ators. The 3-point correlators have
+not been calculated so far, but at the level of 2-point correl ators again perfect agreement with
+a LCFT was found, provided we use the values [26]
+cL=cR= 0bL=bR=−σ12ℓ
+GN(45)
+6The sole caveat is that two of the ten 3-point correlators wer e calculated only qualitatively. It would be
+particularly interesting to calculate the correlator betw een three logarithmic modes (41c), since it contains an
+additional parameter independent from the central charges and new anomaly that determines LCFT properties.Itislikely thatasimilarstorycanberepeatedforgeneralm assivegravity [16], whichcombines
+new massive gravity (42) with a gravitational Chern–Simons term (14). Thus, even though they
+are sparse in theory space we have found a few good candidates for gravity duals to LCFTs:
+cosmological topologically massive gravity, new massive g ravity and general massive gravity. In
+all cases we have to tune parameters in such a way that a “chira l point” emerges where at least
+one of the central charges vanishes.
+4.6. Chopping logs?
+Sofarwe were exclusively concerned with findinggravitatio nal theories wherelogarithmic modes
+can arise. In this subsection we try to get rid of them. The rat ionale behind the desire to
+eliminate the logarithmic modes is unitarity of quantum gra vity. Gravity in 2+1 dimensions is
+simple and yet relevant, as it contains black holes [29], pos sibly gravity waves [13] and solutions
+that are asymptotically AdS. Thus, it could provide an excel lent arena to study quantum gravity
+in depth provided one is able to come up with a consistent (uni tary) theory of quantum gravity,
+for instance by constructing its dual (unitary) CFT. Indeed , two years ago Witten suggested a
+specific CFT dual to 3-dimensional quantum gravity in AdS [30 ]. This proposal engendered a
+lot of further research (see [31–37] for some early referenc es), including the suggestion by Li,
+Song and Strominger [17] to construct a quantum theory of gra vity that is purely right-moving,
+dubbed“chiral gravity”. To make a long story [18,19,24,38– 81] short, “chiral gravity” is nothing
+but CCTMG with the logarithmic modes truncated in some consi stent way.
+We discuss now two conceptually different possibilities of im plementing such a truncation.
+The first option was proposed in [18]. If one imposes periodic ity in time for all modes, t→t+β,
+then only the left- and right-moving modes are allowed, whil e the logarithmic modes are
+eliminated since they grow linearly in time, see e.g. (25). T he other possibility was pursued
+in [22]. It is based upon the observation that logarithmic mo des grow logarithmically faster in
+e2ρthan their left partners, see e.g. (25). Thus, imposing boun dary conditions that prohibit this
+logarithmic growth eliminates all logarithmic modes.
+Currently it is not known whether chiral gravity has its own d ual CFT or if it exists merely
+as a zero-charge superselection sector of the logarithmic C FT. In the latter case it is unclear
+whether or not the zero-charge superselection sector is a fu lly-fledged CFT. Another alternative
+is that neither the LCFT nor its chiral truncation dual to chi ral gravity exists. In that case
+CTMG is unlikely to exist as a consistent quantum theory on it s own. Rather, it would require
+a UV completion, such as string theory.
+4.7. Logout
+We summarize now the key results reviewed in this section as w ell as some open issues.
+Cosmological topologically massive gravity (15) at the chi ral point (17) is likely to be dual
+to a LCFT with a logarithmic partner for one flux component of t he energy momentum tensor
+since 2- [23] and 3-point correlators [25] match. The values of central charges and new anomaly
+are given by (35). The detailed calculation of the correlato r with three log-insertions (41c)
+still needs to be performed and will determine another param eter of the LCFT. New massive
+gravity (42) at the chiral point (43) is likely to be dual to a L CFT with a logarithmic partner
+for both flux components of the energy momentum tensor since 2 -point correlators match [26].
+The central charges vanish and the new anomalies are given by (45). The calculation of 3-
+point correlators still needs to be performed and will provi de a more stringent test of the
+conjectured duality to a LCFT. A similar story is likely to re peat for general massive gravity
+(the combination of topologically and new massive gravity) at a chiral point, and it could be
+rewardingtoinvestigate thisissue. Finallyweaddressedp ossibilitiestoeliminatethelogarithmic
+modes and their partners, since such an elimination might le ad to a chiral theory of quantum
+gravity [17], called “chiral gravity”. The issue of whether chiral gravity exists still remains open.5. Towards condensed matter applications
+In this final section we review briefly some condensed matter s ystems where LCFTs do arise,
+see [3,4] for more comprehensive reviews. We focus on LCFTs w here the energy-momentum
+tensor acquires a logarithmic partner, i.e., the class of LC FTs for which we have found possible
+gravity duals.7Condensed matter systems described by such LCFTs are for ins tance systems
+at (or near) a critical point with quenched disorder, like sp in glasses [83]/quenched random
+magnets [84,85], dilute self-avoiding polymers or percola tion [86]. “Quenched disorder” arises
+in a condensed matter system with random variables that do no t evolve with time. If the
+amount of disorder is sufficiently large one cannot study the e ffects of disorder by perturbing
+around a critical point without disorder — standard mean fiel d methods break down. The
+system is then driven towards a random critical point, and it is a challenge to understand its
+precise nature. Mathematically, the essence of the problem lies in the infamous denominator
+arising in correlation functions of some operator Oaveraged over disordered configurations (see
+e.g. chapter VI.7 in [87])
+∝an}b∇acketle{tO(z)O(0)∝an}b∇acket∇i}ht=/integraldisplay
+DVP[V]/integraltext
+Dφexp/parenleftbig
+−S[φ]−/integraltext
+d2z′V(z′)O(z′)/parenrightbig
+O(z)O(0)/integraltext
+Dφexp/parenleftbig
+−S[φ]−/integraltext
+d2z′V(z′)O(z′)/parenrightbig (46)
+HereS[φ] is some 2-dimensional8quantum field theory action for some field(s) φandV(z) is a
+random potential with some probability distribution. For w hite noise one takes the Gaussian
+probability distribution P[V]∝exp/parenleftbig
+−/integraltext
+d2zV2(z)/(2g2)/parenrightbig
+, wheregis a coupling constant that
+measuresthestrengthoftheimpurities. Ifit werenot forth edenominatorappearingontheright
+hand side of the averaged correlator (46) we could simply per form the Gaussian integral over
+the impurities encoded in the random potential V(z). This denominator is therefore the source
+of all complications and to deal with it requires suitable me thods, see e.g. [88]. One possibility is
+to eliminate the denominator by introducing ghosts. This so -called “supersymmetric method”
+works well if the original quantum field theory described by t he actionS[φ] is very simple, like a
+free field theory. Another option is the so-called replica tr ick, where one introduces ncopies of
+the original quantum field theory, calculates correlators i n this setup and takes the limit n→0
+in the end, which formally reproduces the denominator in (46 ). Recently, Fujita, Hikida, Ryu
+and Takayanagi combined the replica method with the AdS/CFT correspondence to describe
+disordered systems [89] (see [90,91] for related work), ess entially by taking ncopies of the CFT,
+exploiting AdS/CFT to calculate correlators and taking for mally the limit n→0 in the end.
+Like other replica tricks their approach relies on the exist ence of the limit n→0.
+One of the results obtained by the supersymmetric method or r eplica trick is that correlators
+like the one in (46) develop a logarithmic behavior, exactly as in a LCFT [84]. In fact, in
+then→0 limit prescribed by the replica trick, the conformal dimen sions of certain operators
+degenerate. This produces a Jordan block structure for the H amiltonian in precise parallel to
+theµℓ→1 limit of CTMG. More concretely, LCFTs can be used to compute correlators of
+quenched random systems!
+This suggests yet-another route to describe systems with qu enched disorder, and our present
+results add to this toolbox. Namely, instead of taking ncopies of an ordinary CFT we may
+start directly with a LCFT. If this LCFT is weakly coupled we c an work on the LCFT side
+perturbatively, using the results mentioned above [3,4,84 –86]. On the other hand, if the LCFT
+becomes strongly coupled, perturbative methods fail. To ge t a handle on these situations we
+can exploit the AdS/LCFT correspondence and work on the grav ity side. Of course, to this end
+7A well-studied alternative case is a LCFT with c=−2 [2,82]. There is no obvious way to construct a gravity
+dual for such LCFTs, even when considering CTMG or new massiv e gravity away from the chiral point. We thank
+Ivo Sachs for discussions on this issue.
+8Analog constructions work in higher dimensions, but we focu s here on two dimensions.one needs to construct gravity duals for LCFTs. The models re viewed in this talk are simple
+and natural examples of such constructions.
+Acknowledgments
+We thank Matthias Gaberdiel, Gaston Giribet, Olaf Hohm, Rom an Jackiw, David Lowe, Hong
+Liu, Alex Maloney, John McGreevy, Ivo Sachs, Kostas Skender is, Wei Song, Andy Strominger
+and Marika Taylor for discussions. DG thanks the organizers of the “First Mediterranean
+Conference on Classical and Quantum Gravity” for the kind in vitation and for all their efforts to
+make the meeting very enjoyable. DG and NJ are supported by th e START project Y435-N16
+of the Austrian Science Foundation (FWF). During the final st age NJ has been supported by
+project P21927-N16 of FWF. NJ acknowledges financial suppor t from the Erwin-Schr¨ odinger-
+Institute (ESI) during the workshop “Gravity in three dimen sions”.
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1001.0003.txt

@@ -0,0 +1,843 @@
+arXiv:1001.0003v3  [hep-th]  10 May 2010Preprint typeset in JHEP style - HYPER VERSION KUL-TF-09/28
+HD-THEP-09-31
+A landscape of non-supersymmetric AdS vacua on
+coset manifolds
+Paul Koerber∗
+Instituut voor Theoretische Fysica, Katholieke Universit eit Leuven, Celestijnenlaan
+200D, B-3001 Leuven, Belgium
+Email:koerber atitf.fys.kuleuven.be
+Simon K¨ ors
+Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg, Philosophenweg 16-19, D-69120
+Heidelberg, Germany
+Email:s.koers atthphys.uni-heidelberg.de
+Abstract: We construct new families of non-supersymmetric sourceles s type IIA AdS 4
+vacua on those coset manifolds that also admit supersymmetr ic solutions. We investigate
+the spectrum of left-invariant modes and find that most, but n ot all, of the vacua are stable
+under these fluctuations. Generically, there are also no mas sless moduli.
+∗Postdoctoral Fellow FWO – Vlaanderen.Contents
+1. Introduction 1
+2. Ansatz 3
+3. Solutions 6
+4. Stability analysis 11
+5. Conclusions 15
+A. SU(3)-structure 15
+B. Type II supergravity 16
+1. Introduction
+The reasons for studying AdS 4vacua of type IIA supergravity are twofold: first they are
+examples of flux compactifications away from the Calabi-Yau r egime, where all the moduli
+can be stabilized at the classical level. Secondly, they can serve as a gravity dual in the
+AdS4/CFT3-correspondence, which became the focus of attention due to recent progress
+in the understanding of the CFT-side as a Chern-Simons-matt er theory describing the
+world-volume of coinciding M2-branes [1].
+Itismucheasiertofindsupersymmetricsolutionsofsupergr avityasthesupersymmetry
+conditions are simpler than the full equations of motion, wh ile at the same time there
+are general theorems stating that the former – supplemented with the Bianchi identities
+of the form fields – imply the latter [2, 3, 4, 5]. Although spec ial type IIA solutions
+that came from the reduction of supersymmetric M-theory vac ua were already known (see
+e.g. [6, 7, 8]), it was only in [3] that the supersymmetry cond itions for type IIA vacua with
+SU(3)-structure were first worked out in general. It was disc overed that there are natural
+solutions to these equations on the four coset manifolds G/Hthat have a nearly-K¨ ahler
+limit [9, 10, 11, 12, 13, 14] (solutions on other manifolds ca n be found in e.g. [3, 15, 16]).1
+To be precise these are the manifolds SU(2) ×SU(2),G2
+SU(3),Sp(2)
+S(U(2)×U(1))andSU(3)
+U(1)×U(1).2
+These solutions are particularly simple in the sense that bo th the SU(3)-structure, which
+determines the metric, as well as all the form fluxes can be exp anded in terms of forms
+which are left-invariant under the action of the group G. The supersymmetry equations
+1For an early appearance of these coset manifolds in the strin g literature see e.g. [17].
+2See [18] for a review and a proof that these are the only homoge neous manifolds admitting a nearly-
+K¨ ahler geometry.
+– 1 –of [3] then reduce to purely algebraic equations and can be ex plicitly solved. Nevertheless,
+these solutions still have non-trivial geometric fluxes as o pposed to the Calabi-Yau or torus
+orientifolds of [15, 16]. Similarly to those papers it is pos sible to classically stabilize all
+left-invariant moduli [14]. Inspired by the AdS 4/CFT3correspondence more complicated
+type IIA solutions have in the meantime been proposed. The so lutions have a more generic
+form for the supersymmetry generators, called SU(3) ×SU(3)-structure [19], and are not
+left-invariant anymore [20, 21, 22, 23] (see also [24]). Sup ersymmetric AdS 4vacua in type
+IIB with SU(2)-structure have also been studied in [25, 26, 2 7, 28] and in particular it has
+been shown in [28] that also in this setup classical moduli st abilization is possible.
+At some point, however, supersymmetry has to be broken and we have to leave
+the safe haven of the supersymmetry conditions. In this pape r we construct new non-
+supersymmetric AdS 4vacua without source terms. This means that the more complic ated
+equations of motion of supergravity should be tackled direc tly3. In order to simplify the
+equations we use a specific ansatz: we start from a supersymme tric AdS 4solution and scan
+for non-supersymmetric solutions with the samegeometry (and thus SU(3)-structure), but
+withdifferent NSNS- and RR-fluxes. Moreover, we expand these form fields in t erms of the
+SU(3)-structure and its torsion classes. This may seem rest rictive at first, but it works for
+11D supergravity, where solutions like this have been found and are known as Englert-type
+solutions [31, 32, 33] (see [34] for a review). To be specific, for each supersymmetric M-
+theory solution of Freund-Rubin type (which means the M-the ory four-form flux has only
+legs along the external AdS 4space, i.e.F4=fvol4wherefis called the Freund-Rubin
+parameter) it is possible to construct a non-supersymmetri c solution with the same inter-
+nal geometry but with a different four-form flux. The modified fo ur-form of the Englert
+solution has then a non-zero internal part: ˆF4∝η†γm1m2m3m4ηdxm1m2m3m4, whereηis
+the 7D supersymmetry generator, and a different Freund-Rubin parameterfE=−(2/3)f.
+Also the Ricci scalar of the AdS 4space, and thus the effective 4D cosmological constant,
+differs:R4D,E= (5/6)R4D. In type IIA with non-zero Romans mass (so that there is no lif t
+to M-theory) non-supersymmetric solutions of this form hav e been found as well: for the
+nearly-K¨ ahler geometry in [35, 29, 36] and for the K¨ ahler- Einstein geometry in [35, 20, 37].
+In this paper we show that this type of solutions is not restri cted to these limits and sys-
+tematically scan for them. Applying our ansatz to the coset m anifolds with nearly-K¨ ahler
+limit, mentioned above, we find that the most interesting man ifolds areSp(2)
+S(U(2)×U(1))and
+SU(3)
+U(1)×U(1), on which we find several families of non-supersymmetric AdS 4solutions. We
+also find some non-supersymmetric solutions in regimes of th e geometry that do not allow
+for a supersymmetric solution.
+These non-supersymmetric solutions are not necessarily st able. For instance, it is
+known that if there is more than one Killing spinor on the inte rnal manifold (which holds
+in particular for S7, the M-theory lift of CP3=Sp(2)
+S(U(2)×U(1))), the Englert-type solution is
+unstable [38]. We investigate stability of our solutions ag ainst left-invariant fluctuations.
+This means we calculate the spectrum of left-invariant mode s, and check for each mode
+3Anotherroute would be tofindsome alternative first-ordereq uations, which extendthe supersymmetry
+conditions in that they still automatically imply the full e quations of motion in certain non-supersymmetric
+cases, see e.g. [29, 30].
+– 2 –whether the mass-squared is above the Breitenlohner-Freed man bound [39, 40]. This is not
+a complete stability analysis in that there could still be no n-left-invariant modes that are
+unstable. We do believe it provides a good first indication. I n particular, we find for the
+type IIA reduction of the Englert solution on S7that the unstable mode of [38] is among
+our left-invariant fluctuations and we find the exact same mas s-squared.
+These non-supersymmetric AdS 4vacua are interesting, because, provided they are
+stable, they should have a CFT-dual. For instance in [20] the CFT-dual for a non-
+supersymmetric K¨ ahler-Einstein solution on CP3was proposed. Furthermore, for phe-
+nomenologically more realistic vacua, supersymmetry-bre aking is essential. Really, one
+would like to construct classical solutions with a dS 4-factor, which are necessarily non-
+supersymmetric. Because of a series of no-go theorems – from very general to more specific:
+[41, 42, 43, 44, 45] – this is a very non-trivial task. For pape rs nevertheless addressing this
+problemsee[46,47,45,48,49,28]. Inthiscontext thelands capeofthenon-supersymmetric
+AdS4vacua of this paper can be considered as a playground to gain e xperience before try-
+ing to construct dS 4-vacua. In fact, in [48] an ansatz very similar to the one used in this
+paper was proposed in order to construct dS 4-vacua. Applied to the coset manifolds above,
+it did however not yield any solutions, in agreement with the no-go theorem of [45].
+In section 2 we explain our ansatz in full detail, while in sec tion 3 we present the
+explicit solutions we found on the coset manifolds. In secti on 4 we analyse the stability
+against left-invariant fluctuations before ending with som e short conclusions. We provide
+an appendix with some useful formulae involving SU(3)-stru ctures and an appendix on our
+supergravity conventions.
+Thenon-supersymmetricsolutions of this paperappearedbe forein thesecond author’s
+PhD thesis [50].
+2. Ansatz
+In this section we explain the ansatz for our non-supersymme tric solutions. The reader
+interested in the details might want to check out our SU(3)-s tructure conventions in ap-
+pendix A, while towards the end of the section we need the type II supergravity equations
+of motion outlined in appendix B.
+We start with a supersymmetric SU(3)-structure solution of type IIA supergravity.
+The SU(3)-structure is defined by a real two-form Jand a complex decomposable three-
+form Ω satisfying (A.1). Moreover, Jand Ω together determine the metric as in (A.2). In
+order for the solution to preserve at least one supersymmetr y (N= 1) [3] one finds that
+the warp factor Aand the dilaton Φ should be constant, the torsion classes W1,W2purely
+imaginary and all other torsion classes zero (for the definit ion of the torsion classes see
+(A.3)). This implies
+dJ=3
+2W1ReΩ, (2.1a)
+dReΩ = 0, (2.1b)
+dImΩ =W1J∧J+W2∧J, (2.1c)
+– 3 –where we defined W1≡ −iW1andW2≡ −iW2. The fluxes can then be expressed in terms
+of Ω,Jand the torsion classes and are given by
+eΦˆF0=f1, (2.2a)
+eΦˆF2=f2J+f3ˆW2, (2.2b)
+eΦˆF4=f4J∧J+f5ˆW2∧J, (2.2c)
+eΦˆF6=f6vol6, (2.2d)
+H=f7ReΩ, (2.2e)
+where for the supersymmetric solution
+f1=eΦm, f 2=−W1
+4, f3=−w2, f4=3eΦm
+10,
+f5= 0, f 6=9W1
+4, f7=2eΦm
+5.(2.3)
+Using the duality relation f=˜F0=−⋆6ˆF6=−e−Φf6(see (B.6)) we find that f6is
+proportional to the Freund-Rubin parameter f, whilef1is proportional to the Romans
+massm. Furthermore, we introduced here a normalized version of W2, enabling us later
+on to use (2.2) as an ansatz for the fluxes also in the limit W2→0:
+ˆW2=W2
+w2,withw2=±/radicalbig
+(W2)2, (2.4)
+where one can choose a convenient sign in the last expression .
+The Bianchi identity for ˆF2imposes dW2∝ReΩ. Working out the proportionality
+constant [3] we find
+dW2=−1
+4(W2)2ReΩ. (2.5)
+Furthermore, using the values for the fluxes (2.3) it fixes the Romans mass:
+e2Φm2=5
+16/parenleftbig
+3(W1)2−2(W2)2/parenrightbig
+. (2.6)
+We now want to construct non-supersymmetric AdS solutions o n the manifolds men-
+tioned in the introduction with the samegeometry as in the supersymmetric solution, and
+thus the same SU(3)-structure ( J,Ω), but with different fluxes. We make the ansatz that
+the fluxes can still be expanded in terms of J,Ω and the torsion class ˆW2as in (2.2), but
+with different values for the coefficients fi. To this end we plug the ansatz for the geometry
+(J,Ω) — eqs. (2.1) — and the ansatz for the fluxes — eqs. (2.2) — into the equations of
+motion (B.7) and solve for the fi. We will make one more assumption, namely that
+ˆW2∧ˆW2=cJ∧J+pˆW2∧J, (2.7)
+withc,psome parameters. This is an extra constraint only for theSU(3)
+U(1)×U(1)coset and
+we will discuss its relaxation later.4Wedging with Jwe find then immediately c=−1/6.
+4With the ansatz (2.2) the constraint is forced upon us. Indee d, suppose that instead ˆW2∧ˆW2=
+−1/6J∧J+pˆW2∧J+P∧J, wherePis a non-zero simple (1,1)-form independent of ˆW2. We find then
+from the equation of motion for Hand the internal part of the Einstein equation respectively f5f3= 0 and
+(f3)2−(f5)2−(w2)2= 0. So the only possibility is then f5= 0 and f3=±w2, which leads in the end to
+the supersymmetric solution. They way out is to also include Pas an expansion form in (2.2).
+– 4 –Furthermore we need expressions for the Ricci scalar and ten sor, which for a manifold with
+SU(3)-structure can be expressed in terms of the torsion cla sses [51]. Taking into account
+that onlyW1,2are non-zero we find:
+R6D=15(W1)2
+2−(W2)2
+2, (2.8a)
+Rmn=1
+6gmnR6D+W1
+4W2(m·Jn)+1
+2[W2m·W2n]0+1
+2Re/bracketleftbig
+dW2|(2,1)m·¯Ωn/bracketrightbig
+,(2.8b)
+where (P)2andPm·Pnfor a form Pare defined in (B.2) and |0indicates taking the
+traceless part. From eq. (2.5) follows that for our purposes dW2|2,1= 0 so that the last
+term in (2.8b) vanishes. Moreover, using (2.7) [ W2m·W2n]0can be expressed in terms of
+W2(m·Jn).
+Plugging the ansatz for the fluxes (2.2) into the equations of motion (B.7) and using
+eqs. (2.1), (2.5), (A.5), (2.7), (B.5), (B.6) and (2.8b) we fi nd:
+BianchiF2: 0 =3
+2W1f2−1
+4w2f3+f1f7,
+eomF4: 0 = 3W1f4+1
+4w2f5−f6f7,
+eomH: 0 = 6W1f7−3f1f2−12f4f2−6f4f6−f3f5,
+0 =w2f7+f1f3+f2f5−2f3f4−f5f6+pf3f5, (2.9)
+dilaton eom : 0 = R4D+R6D−2f2
+7,
+Einstein ext. : 0 = R4D+(f1)2+3(f2)2+12(f4)2+(f6)2+(f3)2+(f5)2,
+Einstein int. : 0 = R6D−6(f7)2+1
+2/bracketleftbig
+3(f1)2+3(f2)2−12(f4)2−3(f6)2+(f3)2−(f5)2/bracketrightbig
+,
+0 = 4(f2f3+2f4f5)−w2W1−p/bracketleftbig
+(f3)2−(f5)2−(w2)2/bracketrightbig
+.
+In the equation of motion for Hwe get separate conditions from the coefficients of J∧J
+andˆW2∧Jrespectively. In the internal Einstein equation we find like wise a separate
+condition from the trace and the coefficient of W2(m·Jn). In the next section we find
+explicit solutions to these equations for the coset manifol ds with nearly-K¨ ahler limit, the
+stability of which we investigate in section 4.
+Flipping signs
+The Einstein and dilaton equation are quadratic in the form fl uxes and thus insensitive to
+flipping the signs of these fluxes. Taking into account also th e flux equations of motion
+and Bianchi identities, we find that for each solution to the s upergravity equations, we
+automatically obtain new ones by making the following sign fl ips:
+H→ −H,ˆF0→ −ˆF0,ˆF2→ˆF2,ˆF4→ −ˆF4,ˆF6→ˆF6,
+H→ −H,ˆF0→ˆF0,ˆF2→ −ˆF2,ˆF4→ˆF4,ˆF6→ −ˆF6,
+H→H,ˆF0→ −ˆF0,ˆF2→ −ˆF2,ˆF4→ −ˆF4,ˆF6→ −ˆF6.(2.10)
+In particular, these sign flips will transform a supersymmet ric solution into another super-
+symmetric solution (as can be verified using the conditions ( 2.1),(2.3) allowing for suitable
+– 5 –sign flips of J, ReΩ and ImΩ compatible with the metric). If some fluxes are ze ro, more
+sign flips are possible. For instance for ˆF0=ˆF4= 0 we find the following extra sign-flip,
+known as skew-whiffing in the M-theory compactification literature [52] (see also t he review
+[34])
+H→ ±H,ˆF2→ˆF2,ˆF6→ −ˆF6, (2.11)
+which transforms a supersymmetric solution into a non-supersymmetric one. When dis-
+cussing different solutions, we will from now on implicitly co nsider each solution together
+with its signed-flipped counterparts.
+3. Solutions
+Let us now solve the equations obtained in the previous secti on for the coset manifolds that
+admit sourceless supersymmetric solutions, namelyG2
+SU(3), SU(2)×SU(2),Sp(2)
+S(U(2)×U(1))and
+SU(3)
+U(1)×U(1). For the supersymmetricsolutions on these manifolds we wil l use the conventions
+and presentation of [13, 14]. For moredetails, includingin particular ourchoice of structure
+constants for the relevant algebras, we refer to these paper s.
+On a coset manifold G/Hone can define a coframe emthrough the decomposition of
+the Lie-valued one-form L−1dL=emKm+ωaHain terms of the algebras of GandH. Here
+Lis a coset representative, the Haspan the algebra of Hand theKmspan the complement
+of this algebra within the algebra of G. The exterior derivative on the emis then given
+in terms of the structure constants through the Maurer-Cart an relation. Furthermore,
+the forms that are left-invariant under the action of Gare precisely those forms that are
+constant in the basis spanned by emand for which the exterior derivative is also constant
+in this basis. For these forms the exterior derivative can th en be expressed solely in terms
+of the structure constants only involving the Km. We refer to [53, 54] for a review on coset
+technology or to the above papers for a quick explanation.
+G2
+SU(3)and SU(2) ×SU(2)
+We start from the supersymmetric nearly-K¨ ahler solution o nG2
+SU(3). The SU(3)-structure
+is given by
+J=a(e12−e34+e56),
+Ω =a3/2/bracketleftbig
+(e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
+,(3.1)
+whereais the overall scale.
+Since this SU(3)-structure corresponds to a nearly-K¨ ahle r geometry the torsion class
+W2is zero. Furthermore we find
+W1=−2√
+3a−1/2, w2=p= 0. (3.2)
+– 6 –Plugging this into the equations (2.9) we find exactly three s olutions for ( f1,...,f7) (up
+to the sign flips (2.10)):
+a−1/2(√
+5
+2,1
+2√
+3,0,3
+4√
+5,0,−9
+2√
+3,1√
+5),
+a−1/2(/radicalbigg
+5
+3,0,0,0,0,5√
+3,0),
+a−1/2(1,1√
+3,0,−1
+2,0,√
+3,1).(3.3)
+The first is the supersymmetric solution, while the last two a re non-supersymmetric solu-
+tions, which were already found in [35, 29, 36]. Truncating t o the 4D effective theory it
+was shown in [30] that a generalization of this family of solu tions is quite universal as it
+appears in a large class of N= 2 gauged supergravities.
+On the SU(2) ×SU(2) manifold, requiring the same geometry as the supersym metric
+solution and not allowing for source terms will restrict us t o the nearly-K¨ ahler point. The
+analysis is then basically the same as forG2
+SU(3)above.
+Sp(2)
+S(U(2)×U(1))
+The family of supersymmetric solutions on this manifold has , next to the overall scale,
+an extra parameter determining the shape of the solutions. I t is then possible to turn on
+the torsion class W2and venture away from the nearly-K¨ ahler geometry. This mak es this
+class much richer and enables us this time to find new non-supe rsymmetric solutions. The
+SU(3)-structure is given by [12, 13, 14]
+J=a(e12+e34−σe56),
+Ω =a3/2σ1/2/bracketleftbig
+(e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
+,(3.4)
+whereais the overall scale and σis the shape parameter. We find for the torsion classes
+and the parameter p:
+W1= (aσ)−1/22+σ
+3,
+(W2)2= (aσ)−18(1−σ)2
+3⇒w2= (aσ)−1/22√
+2(1−σ)√
+3,
+ˆW2=−1√
+3/parenleftbig
+e12+e34+2σe56/parenrightbig
+,
+p=−/radicalbig
+2/3.(3.5)
+We easily read off that σ= 1 corresponds to the nearly-K¨ ahler geometry. Note that ev en
+thoughW2→0 forσ→1,ˆW2is well-defined and non-zero in this limit so that we can
+still use it as an expansion form for the fluxes. The points σ= 2 andσ= 2/5 are also
+special, since eq. (2.6) then implies that the supersymmetr ic solution has zero Romans
+mass and, in particular, can be lifted to M-theory. Moreover , these are the endpoints of
+the interval where supersymmetric solutions exist (since o utside this interval we would find
+from eq. (2.6) that m2<0). They are indicated as vertical dashed lines in the plots.
+– 7 –Figure 1:Sp(2)
+S(U(2)×U(1))-model: plot of aR4Dfor the supersymmetric solutions (light green) and
+the new non-supersymmetric solutions (other colors) in terms of t he shape parameter σ. Unstable
+solutions are indicated in red.
+Pluggingeqs.(3.5) intothesupergravityequationsofmoti on(2.9)wefindnumericallya
+rich spectrumofsolutions, whicharedisplayed infigures1a nd2. Note that thedependence
+on the overall scale can be easily extracted from all plotted quantities by multiplying by
+ato a suitable power. We plotted the value of the 4D Ricci scala rR4Dof the AdS-space
+against the shape parameter σin figure 1. Note that R4Dis inversely proportional to the
+AdS-radius squared and related to the effective 4D cosmologic al constant and the vev of
+the 4D scalar potential Vas follows
+Λ =∝angb∇acketleftV∝angb∇acket∇ight=R4D/4. (3.6)
+The supersymmetric solutions are plotted in light green, wh ile red is used for the non-
+supersymmetric solutions found to be unstable in section 4. For completeness of the pre-
+sentation of our numeric results, we provide the values of ea ch of the coefficients fiof the
+ansatz (2.2) in figure 2.
+The first point to note is that where the supersymmetric solut ions are restricted to
+the interval σ∈[2/5,2], there exist non-supersymmetric solutions in the somewh at larger
+intervalσ∈[0.39958,2.13327]. Furthermore, there are up to five non-supersymmetri c
+solutions for each supersymmetric solution.
+We remark that the parameters σand the overall scale are not continuous moduli since
+they are determined by the vevs of the fluxes, which in a proper string theory treatment
+shouldbequantized. Indeed, inthenextsection wewill show that generically all moduliare
+stabilized. We leave the analysis of flux quantization, whic h is complicated by the fact that
+– 8 –(a) Plot of a1/2f1(Romans mass)
+ (b) Plot of a1/2f2(J-part of ˆF2)
+(c) Plot of a1/2f3(ˆW2-part of ˆF2)
+ (d) Plot of a1/2f4(J∧J-part of ˆF4)
+(e) Plot of a1/2f5(J∧ˆW2-part of ˆF4)
+ (f) Plot of a1/2f6(Freund-Rubin parameter)
+(g) Plot of a1/2f7(ReΩ part of H)
+Figure 2: Plots of the solutions on the cosetSp(2)
+S(U(2)×U(1)). Different colors indicate different
+solutions. Unstable solutions are indicated in red (see section 4) and the supersymmetric solutions
+in light green. By a suitable rescaling of the coefficients the dependen ce on the overall scale ais
+taken out.
+– 9 –there is non-trivial H-flux (twisting the RR-charges), to further work. The expect ation is
+that the continuous line of supergravity solutions is repla ced by discrete solutions.
+Let us now take a look at some special values of σ. Forσ= 1 we find five solutions
+of which three (including the supersymmetric one) are up to s caling equivalent to the
+solutions (3.3) onG2
+SU(3)of the previous section [35, 29, 36, 30]. They have f3=f5= 0 and
+so the fluxes are completely expressed in terms of J. However, there are also two new non-
+supersymmetric solutions (the dark green and the purple one ) which have f3∝negationslash= 0,f5∝negationslash= 0.
+Next we turn to the case σ= 2. This point is special in that the metric becomes
+the Fubini-Study metric on CP3and the bosonic symmetry of the geometry enhances
+from Sp(2) to SU(4). In fact, since the RR-forms of the supers ymmetric solution can be
+expanded in terms of the closed K¨ ahler form ˜J= (1/3)J+(2a)1/2W2of the Fubini-Study
+metric, the symmetry group of the whole supersymmetric solu tion is SU(4). One can also
+show that the supersymmetry enhances from the generic N= 1 toN= 6 [6]. In [37] it
+was found that there is an infinite continuous family of non-s upersymmetric solutions and
+two discrete separate solutions (see also [35] for an incomp lete early discussion), which all
+have SU(4)-symmetry. They are notdisplayed in the plot since they can not be found by
+taking a continuous limit σ→2. For these solutions H= 0 (f7= 0) and ˆF2andˆF4are
+expanded in terms of ˜J(for more details see [37]).
+Instead, in the plot we find apart from the supersymmetric sol ution (which merges
+with the dark green solution at σ= 2) two more discrete non-supersymmetric solutions,
+which have only Sp(2)-symmetry (since the fluxes cannot be ex pressed in terms of ˜Jonly).
+The blue one is new, while the red one turns out to be the reduct ion of the Englert-type
+solution. Indeed for the Englert-type solution we expect
+f1= 0, no Romans mass ,(3.7a)
+f2=f2,susy, f3=f3,susy, same geometry in M ⇒sameˆF2as susy,(3.7b)
+f7=−2f4=−(1/3)f6,susy, f5= 0, fromˆF4in M-theory ,(3.7c)
+f6= (−2/3)f6,susy, Freund-Rubin parameter changes ,(3.7d)
+R4D= (5/6)R4D,susy, 4D Λ changes ,(3.7e)
+which agrees with the values displayed in the figures for the r ed curve at σ= 2.
+Also forσ= 2/5 we find apart from the supersymmetric solution, the Englert solution
+(the purplecurve) andoneextra non-supersymmetricsoluti on (the darkgreen curve). Note
+that while the supersymmetric curve joins the olive green cu rve atσ= 2/5, the purple
+curve only joins the dark green curve at σ= 0.39958.
+SU(3)
+U(1)×U(1)
+For this manifold the SU(3)-structure is given by [13, 14]:
+J=a(−e12+ρe34−σe56),
+Ω =a3/2(ρσ)1/2/bracketleftbig
+(e245+e135+e146−e236)+i(e235+e136+e246−e145)/bracketrightbig
+,(3.8)
+– 10 –whereρandσare the shape parameters of the model. Furthermore we find for the torsion
+classes:
+W1=−(aρσ)−1/21+ρ+σ
+3,
+W2=−(2/3)a1/2(ρσ)−1/2/bracketleftbig
+(2−ρ−σ)e12+ρ(1−2ρ+σ)e34−σ(1+ρ−2σ)e56/bracketrightbig
+.(3.9)
+It turns out that the ansatz (2.7) is only satisfied for
+ρ= 1, σ= 1 orρ=σ. (3.10)
+In all three of these cases the equations (2.9) forSU(3)
+U(1)×U(1)reduce to exactly the same
+equations as forSp(2)
+S(U(2)×U(1))so that we obtain the same solution space. However, as we
+will see in the next section, the stability analysis will be d ifferent since the model on
+SU(3)
+U(1)×U(1)has two extra left-invariant modes.
+In order to find further non-supersymmetric solutions, we sh ould go beyond the ansatz
+(2.7). Let us put
+ˆW2∧ˆW2= (−1/6)J∧J+p1ˆW2∧J+p2ˆP∧J, (3.11)
+whereˆPis a primitive normalized (1,1)-form (so that it is orthogon al toJandˆP2= 1).
+Furthermore, we also choose it orthogonal to ˆW2i.e.
+ˆW2·ˆP= 0 or equivalently J∧ˆW2∧ˆP= 0. (3.12)
+From the last equation one finds, using (2.1c), that d ˆP∧ImΩ = 0, which implies on
+SU(3)
+U(1)×U(1)that
+dˆP= 0. (3.13)
+One can now allow the RR-fluxes ˆF2andˆF4to have pieces proportional to ˆPandˆP∧
+Jrespectively and adapt the equations (2.9) accordingly to a ccommodate for the new
+contributions. Now it is possible to numerically find non-su persymmetric solutions for ρ
+andσnot satisfying (3.10). In particular, there are Englert-ty pe solutions on the ellipse of
+values for (ρ,σ) where the supersymmetric solution has zero Romans mass. Fr om eq. (2.6)
+we find that this ellipse is described by
+m2=5
+16ρσ/bracketleftbig
+−5(ρ2+σ2)+6(ρ+σ+ρσ)−5/bracketrightbig
+= 0. (3.14)
+We will not go into more detail on these solutions in this pape r.
+4. Stability analysis
+Inthissectionweinvestigate whetherthenewnon-supersym metricsolutionsonSp(2)
+S(U(2)×U(1))
+andSU(3)
+U(1)×U(1)are stable5. To this end we calculate the spectrum of scalar fluctuations . We
+5In [36] it was found that the non-supersymmetric solutions o nG2
+SU(3)and the similar solutions on the
+nearly-K¨ ahler limits of the other two coset manifolds unde r study are stable. We find exactly the same
+spectrum as the authors of that paper, which provides a consi stency check on our approach. We thank
+Davide Cassani for providing us with these numbers, which ar e not explicitly given in their paper. We did
+not investigate the spectrum of the similar solution on SU(2 )×SU(2), which is more complicated as there
+are more modes.
+– 11 –use the well-known result of [39, 40] that in an AdS 4vacuum a tachyonic mode does not yet
+signal an instability. Only a mode with a mass-squared below the Breitenlohner-Freedman
+bound,
+M2<−3|Λ|
+4, (4.1)
+where Λ<0 is the 4D effective cosmological constant, leads to an instab ility. We restrict
+ourselves to left-invariant fluctuations, which implies th at even if we do not find any modes
+below the Breitenlohner-Freedman bound, the vacuum might s till be unstable, since there
+might be fluctuations with sufficiently negative mass-square d that are not left-invariant.
+This analysis can however pinpoint many unstable vacua and w e do believe it gives a
+valuable first indication for the stability of the others.
+Truncatingtotheleft-invariant modesonthecoset manifol dsunderstudyleads toa4D
+N= 2 gauged supergravity6. It has been shown in [36] that this truncation is consistent .
+The spectrum of the scalar fields can then be obtained from the 4D scalar potential. In
+fact, this computation is analogous to the one performed in [ 14] for the supersymmetric
+N= 1 vacua on the coset spaces. As opposed to the models here, th e models in that
+paper included orientifolds, which broke the supersymmetr y of the 4D effective theory
+fromN= 2 toN= 1. However, also in the present case the N= 1 approach is applicable
+and effectively we have used exactly the same procedure, i.e. u sing theN= 1 scalar
+fluctuations and obtaining the scalar potential from the N= 1 superpotential and K¨ ahler
+potential (see [55, 56, 57, 58]).7The reason is the following. The N= 2 scalar fluctuations
+in the vector multiplets are
+Jc=J−iB= (ki−ibi)ωi=tiωi, (4.2)
+whereωispan the left-invariant two-forms of the coset manifold. Th e orientifold projection
+of theN= 1 theory would then project out the scalar fluctuations comi ng from expanding
+oneventwo-forms, which are absent for the N= 1 theory on the coset manifolds under
+study. The scalar fluctuations in the N= 2 vector multiplets are thus exactly the same as
+the scalars in the chiral multiplets of the K¨ ahler moduli se ctor of the N= 1 theory. The
+6It is important to make the distinction between the number of supersymmetries of respectively the
+4D effective theory, the 10D compactifications, and their 4D t runcation (which are the solutions of the
+4D effective theory [36]). In the presence of one left-invari ant internal spinor, the effective theory will be
+N= 2 since this same spinor can be used in the 4+ 6 decomposition of both ten-dimensional Majorana-
+Weyl supersymmetry generators, but multiplied with indepe ndent four-dimensional spinors. On the other
+hand, for a certain compactification to preserve the supersy mmetry, certain differential conditions, which
+follow from putting the variations of the fermions to zero mu st be satisfied. In the presence of RR-fluxes,
+these conditions mix both ten-dimensional Majorana-Weyl s pinors, putting the four-dimensional spinors in
+both decompositions equal. A generic supersymmetric compa ctification therefore only preserves N= 1.
+Theσ= 2 supersymmetric K¨ ahler-Einstein solution on CP3on the other hand is non-generic in that it
+preserves N= 6, of which only one internal spinor is left-invariant unde r the action of Sp(2) and remains
+after truncation to 4D.
+7It is interesting to note that (in N= 1 language) all the D-terms vanish, so that the supersymmet ry
+breaking is purely due to F-terms. Indeed, in [58] it is shown thatD= 0 is equivalent to d H(e2A−ΦReΨ1) =
+0 in the generalized geometry formalism. For SU(3)-structu re this translates to d( e2A−ΦReΩ) = 0 and
+H∧ReΩ = 0, which is satisfied for our ansatz, eq. (2.1) and (2.2).
+– 12 –(a) Spectrum ofSp(2)
+S(U(2)×U(1))
+(b) Two extra modes of theSU(3)
+U(1)×U(1)-model
+Figure 3: Spectrum of left-invariant modes of the solutions onSp(2)
+S(U(2)×U(1))andSU(3)
+U(1)×U(1).
+expansion forms can then be chosen to bethe same as the Y(2−)
+iof [14]. Furthermore, there
+is one tensor multiplet, which contains the dilaton Φ, the tw o-formBµνand two axions ξ
+and˜ξcoming from the expansion of the RR-potential C3:
+C3=ξα+˜ξβ, (4.3)
+where a choice for αandβspanning the left-invariant three-forms would be Y(3−)and
+Y(3+)of [14] respectively. In the presence of Romans mass or ˆF2-flux the two-form Bµν
+becomes massive and cannot be dualized to a scalar. The dilat on and˜ξappear in a chiral
+multiplet of the complex moduli sector of the N= 1 theory, while Bµνandξare projected
+out by the orientifold. By using the N= 1 approach we thus loose the information on just
+one scalarξ. A proper N= 2 analysis would however learn that ξdoes not appear in the
+scalarpotential (seee.g.[36]), implyingthatitismassle ssandthusabovetheBreitenlohner-
+Freedman bound. Moreover, the scalar potential should be th e same whether it is obtained
+directly from reducing the 10D supergravity action (as in [5 9]) or whether it is obtained
+usingN= 2 orN= 1 technology8. Furthermore we note that the massless scalar field ξ
+not appearing in the potential is not a modulus, since it is ch arged [60, 61], and therefore
+eaten by a vector field becoming massive.
+Thespectraof left-invariant modesforSp(2)
+S(U(2)×U(1))andSU(3)
+U(1)×U(1)aredisplayed infigure
+3. The Breitenlohner-Freedman bound is indicated as a horiz ontal dashed line. The Sp(2)-
+model has six scalar fluctuations entering the potential: ki,biwithi= 1,2 from the two
+vector multiplets, and Φ ,˜ξfrom the universal hypermultiplet, while the SU(3)-model h as
+two more fluctuations from the extra vector multiplet. These two extra modes make a big
+difference for the stability analysis since one of them tends t o be below the Breitenlohner-
+Freedman bound for the purple and dark green solution. As a re sult, even though the
+solutions for the Sp(2)- and SU(3)-model take the same form, the SU(3)-model has more
+unstable solutions: compare figure 1 and 4.
+8The only potential difference between the latter two would be the contribution from the orientifold.
+We have checked that this contribution vanishes in the scala r potentials of [14] in the limit of the orientifold
+chargeµ→0.
+– 13 –Figure 4:SU(3)
+U(1)×U(1)-model: plot of aR4Din terms of the shape parameter σ. Unstable solutions
+are indicated in red.
+Inparticular, wenotethatthereductionoftheEnglert-typ esolutionisunstablefor σ=
+2 in the Sp(2)-model, in agreement with [38], since the M-the ory lift of the corresponding
+supersymmetric solution has eight Killing spinors. We inde ed find the same negative mass-
+squaredM2=−(4/5)|Λ|for the unstable mode as in that paper. On the other hand,
+forσ= 2/5 the Englert-type solution is stable against left-invaria nt fluctuations. This is
+still in agreement with [38] which relied on the existence of at least two Killing spinors,
+while the M-theory lift of the N= 1 supersymmetric solution at σ= 2/5 has only one
+Killing-spinor. For the SU(3)-model, all Englert-type sol utions turn out to be unstable
+(including the ones outside the condition (3.10)).
+We also investigated the stability of the additional soluti ons at the special point σ= 2
+found in [37]. We found that for the Sp(2)-model all these sol utions are stable against left-
+invariant fluctuations. For the SU(3)-model on theother han dit turnsout that thediscrete
+solutions ineqs.(3.16) and(3.17) ofthatreferenceareuns table, whilethecontinuous family
+of eq. (3.18) becomes unstable for
+γ2
+β2>5(75∓16√
+21)
+8217, (4.4)
+for the±sign choice in front of the square root in eq. (3.18) of that pa per respectively
+(note that the supersymmetric solution corresponds to the p ointγ2/β2= 0 in this family).
+Finally, we note that generically (i.e. unless an eigenvalu e is crossing zero at a special
+value forσ) all the plotted modes are massive. For a range of values for σone of the
+eigenvalues for the dark green and purple solution takes a sm all, but still non-zero value.
+– 14 –5. Conclusions
+In this paper we presented new families of non-supersymmetr ic AdS 4vacua. In fact,
+extrapolating from our analysis on these specific coset mani folds and under the assumption
+that a proper treatment of flux quantization does not kill muc h more vacua than in the
+supersymmetric case, it would seem that there are more of the se non-supersymmetric
+vacua than supersymmetric ones. This would imply that such v acua cannot be ignored
+in landscape studies. We have moreover shown that many of the m are stable against a
+specific set of fluctuations, namely the ones that can be expan ded in terms of left-invariant
+forms. If these vacua turn out to be stable against all fluctua tions they should also have
+a CFT-dual, which could be studied along the lines of [20], wh ere the three-dimensional
+Chern-Simons-matter theory dual to a particular highly sym metric non-supersymmetric
+vacuum was proposed. Furthermore, the nice property of some IIA vacua that all moduli
+enter the superpotential and thus can be stabilized at a clas sical level [15] also extends to
+our non-supersymmetric vacua.
+A next step would be to relax the constraint that the solution s should have the same
+geometry as the supersymmetric solution. It is also interes ting to investigate whether a
+similar ansatz and techniques can be used to look for tree-le vel dS-vacua [62].
+Acknowledgments
+We thank Davide Cassani for useful email correspondence and proofreading, and further-
+more Claudio Caviezel for active discussions and initial co llaboration. We would further
+like to thank the Max-Planck-Institut f¨ ur Physik in Munich , where both of the authors
+were affiliated during the bulk of the work on this paper. P.K. i s a Postdoctoral Fellow
+of the FWO – Vlaanderen. The work of P.K. is further supported in part by the FWO –
+Vlaanderen project G.0235.05 and in part by the Federal Office for Scientific, Technical and
+Cultural Affairs through the ’Interuniversity Attraction Po les Programme Belgian Science
+Policy’ P6/11-P. S.K. is supported by the SFB – Transregio 33 “The Dark Universe” by
+the DFG.
+A. SU(3)-structure
+A real non-degenerate two-form Jand a complex decomposable three-form Ω define an
+SU(3)-structure on the 6D manifold M6iff:
+Ω∧J= 0, (A.1a)
+Ω∧¯Ω =8i
+3!J∧J∧J∝negationslash= 0, (A.1b)
+and the associated metric is positive-definite. This metric is determined by Jand Ω as
+follows:
+gmn=−JmpIpn, (A.2)
+withIthe complex structure associated (in the way of [63]) to Ω. Th e volume-form is
+given by vol 6=1
+3!J3=−(i/8)Ω∧¯Ω.
+– 15 –Theintrinsictorsionofthemanifold M6decomposesintofivetorsionclasses W1,...,W5.
+Alternatively they correspond to the SU(3)-decomposition of the exterior derivatives of J
+and Ω [64]. Intuitively, they parameterize the failure of th e manifold to be of special
+holonomy, which can also be thought of as the deviation from c losure ofJand Ω. More
+specifically we have:
+dJ=3
+2Im(W1¯Ω)+W4∧J+W3,
+dΩ =W1J∧J+W2∧J+¯W5∧Ω,(A.3)
+whereW1is a scalar, W2is a primitive (1,1)-form, W3is a real primitive (1 ,2)+(2,1)-form,
+W4is a real one-form and W5a complex (1,0)-form. In this paper only the torsion classes
+W1,W2are non-vanishing and they are purely imaginary, so it will b e convenient to define
+W1,2so thatW1,2=iW1,2. A primitive (1,1)-form P(such asW2) transforms under the 8
+of SU(3) and satisfies
+P∧J∧J= 0. (A.4)
+The Hodge dual is given by
+⋆6P=−P∧J. (A.5)
+A primitive (1 ,2)(or (2,1))-formQon the other hand transforms as a 6(or¯6) under SU(3)
+and satisfies
+Q∧J= 0. (A.6)
+B. Type II supergravity
+The bosonic content of type II supergravity consists of a met ricG, a dilaton Φ, an NSNS
+three-form Hand RR-fields Fn. We use the democratic formalism of [65], in which the
+number of RR-fields is doubled, so that nruns over 0 ,2,4,6,8,10 in type IIA and over
+1,3,5,7,9 in IIB. We will often collectively denote the RR-fields with the polyform F=/summationtext
+nFn. We have also doubled the RR-potentials, collectively deno ted byC=/summationtext
+nC(n−1).
+These potentials satisfy F= dHC+me−B= (d +H∧)C+me−B. In type IIB there is
+of course no Romans mass m, so that the second term vanishes. In type IIA we find in
+particularF0=m.
+The bosonic part of the pseudo-action of the democratic form alism then simply reads
+S=1
+2κ2
+10/integraldisplay
+d10X√
+−G/braceleftbigg
+e−2Φ/bracketleftbigg
+R+4(dΦ)2−1
+2H2/bracketrightbigg
+−1
+4F2/bracerightbigg
+, (B.1)
+where we defined F2=/summationtext
+nF2
+nand the square of an l-formPas follows
+P2=P·P=1
+l!Pm1...mlPm1...ml, (B.2a)
+where the indices are raised with the inverse of the metric Gmnor the internal metric gmn
+(defined later on), depending on the context. In the followin g it will also be convenient to
+define:
+Pm·Pn=ιmP·ιnP=1
+(l−1)!Pmm2...mlPnm2...ml. (B.2b)
+– 16 –The extra degrees of freedom for the RR-fields in the democrat ic formalism have to be
+removed by hand by imposing the following duality condition at the level of the equations
+of motion after deriving them from the action (B.1):
+Fn= (−1)(n−1)(n−2)
+2⋆10F10−n. (B.3)
+That is why (B.1) is only a pseudo-action.
+The fermionic content consists of a doublet of gravitinos ψMand a doublet of dilatinos
+λ. The components of the doublets are of different chirality in t ype IIA and of the same
+chirality in type IIB.
+In this paper we look for vacuum solutions that take the form A dS4×M6. In principle
+there could also be a warp factor A, but it will always be constant for the solutions in this
+paper. We can choose it to be zero. The compactification ansat z for the metric then reads
+ds2
+10=GmndXmdXn= ds2
+4+gmndxmdxn, (B.4)
+where ds2
+4is the line-element for AdS 4andgmnis the metric on the internal space M6. For
+the RR-fluxes the ansatz becomes
+F=ˆF+vol4∧˜F, (B.5)
+whereˆFand˜Fonly have internal indices. The duality constraint (B.3) im plies that ˜Fis
+not independent of ˆF, and given by
+˜Fn= (−1)(n−1)(n−2)
+2⋆6ˆF6−n. (B.6)
+What we need in this paper are the type II equations of motion, which can be found
+from the pseudo-action (B.1). We use them as they are written down in [5] (originally they
+were obtained for massive type IIA in [35]), but take some lin ear combinations in order
+to further simplify then. Without source terms (i.e. we put jtotal= 0 in the equations of
+motion of [5]), they then read:
+dHF= 0 (Bianchi RR fields) , (B.7a)
+d−H⋆10F= 0 (eom RR fields) , (B.7b)
+dH= 0 (BianchiH), (B.7c)
+d/parenleftbig
+e−2Φ⋆10H/parenrightbig
+−1
+2/summationdisplay
+n⋆10Fn∧Fn−2= 0 (eom H), (B.7d)
+2R−H2+8/parenleftbig
+∇2Φ−(∂Φ)2/parenrightbig
+= 0 (dilaton eom) , (B.7e)
+2(∂Φ)2−∇2Φ−1
+2H2−e2Φ
+8/summationdisplay
+nnF2
+n= 0 (trace Einstein/dilaton eom) ,(B.7f)
+RMN+2∇M∂NΦ−1
+2HM·HN−e2Φ
+4/summationdisplay
+nFnM·FnN= 0 (B.7g)
+(Einstein eq./dilaton/trace) .
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1001.0005.txt

@@ -0,0 +1,441 @@
+arXiv:1001.0005v1  [astro-ph.CO]  30 Dec 2009Astronomy& Astrophysics manuscriptno.akari˙RXJ1716˙v5 c∝circlecopyrtESO 2018
+October30,2018
+Environmentaldependenceof 8 µmluminosityfunctionsof
+galaxiesatz ∼0.8
+Comparison between RXJ1716.4 +6708 andthe AKARI NEP deep field.⋆,⋆⋆
+Tomotsugu Goto1,2,⋆⋆⋆, Yusei Koyama3,T.Wada4,C.Pearson5,6,7,H.Matsuhara4,T.Takagi4, H.Shim8, M.Im8,
+M.G.Lee8, H.Inami4,9,10,M.Malkan11, S.Okamura3,T.T.Takeuchi12, S.Serjeant7, T.Kodama2, T.Nakagawa4,
+S.Oyabu4,Y.Ohyama13, H.M.Lee8, N.Hwang2, H.Hanami14, K.Imai15,and T.Ishigaki16
+1Institute for Astronomy, University of Hawaii,2680 Woodla wnDrive, Honolulu, HI,96822, USA
+e-mail:[email protected]
+2National Astronomical Observatory, 2-21-1 Osawa,Mitaka, Tokyo, 181-8588,Japan
+3Department of Astronomy, School of Science,The University of Tokyo, Tokyo113-0033, Japan
+4Institute of Space and Astronautical Science, JapanAerosp ace Exploration Agency, Sagamihara,Kanagawa 229-8510
+5Rutherford Appleton Laboratory, Chilton, Didcot,Oxfords hire OX110QX, UK
+6Department of Physics,Universityof Lethbridge, 4401 Univ ersity Drive,Lethbridge, AlbertaT1J 1B1, Canada
+7Astrophysics Group, Department of Physics, The OpenUniver sity, MiltonKeynes, MK76AA, UK
+8Department of Physics& Astronomy, FPRD,Seoul National Uni versity, Shillim-Dong,Kwanak-Gu, Seoul 151-742, Korea
+9Spitzer Science Center,California Institute ofTechnolog y, Pasadena, CA91125
+10Department of Astronomical Science,The Graduate Universi tyfor Advanced Studies
+11Department of Physicsand Astronomy, UCLA,Los Angeles, CA, 90095-1547 USA
+12Institute for Advanced Research, Nagoya University, Furo- cho, Chikusa-ku, Nagoya 464-8601
+13Academia Sinica,Institute of Astronomyand Astrophysics, Taiwan
+14Physics Section,Facultyof Humanities and SocialSciences , Iwate University, Morioka, 020-8550
+15TOMER&D Inc. Kawasaki, Kanagawa 2130012, Japan
+16Asahikawa National College of Technology, 2-1-6 2-joShunk ohdai, Asahikawa-shi, Hokkaido 071-8142
+Received September 15, 2009; accepted December 16, 2009
+ABSTRACT
+Aims.Weaim to reveal environmental dependence of infraredlumin osity functions (IR LFs)of galaxies at z ∼0.8 using the AKARI
+satellite. AKARI’s wide field of view and unique mid-IR filter s help us to construct restframe 8 µm LFs directly without relying on
+SEDmodels.
+Methods. We construct restframe 8 µm IR LFs in the cluster region RXJ1716.4 +6708 at z=0.81, and compare them with a blank
+field using the AKARI North Ecliptic Pole deep field data at the same redshift. AKARI’s wide field of view (10’ ×10’) is suitable to
+investigate wide range of galaxy environments. AKARI’s 15 µm filter is advantageous here since it directly probes restfr ame 8µm at
+z∼0.8, without relyingona large extrapolation based ona SEDfi t,which was the largestuncertainty inprevious work.
+Results. We have found that cluster IR LFsat restframe 8 µm have a factor of 2.4smaller L∗and a steeper faint-end slope than that
+of the field. Confirming this trend, we also found that faint-e nd slopes of the cluster LFs becomes flatter and flatter with de creasing
+local galaxy density. These changes in LFs cannot be explain ed by a simple infall of field galaxy population into a cluster . Physics
+that canpreferentiallysuppress IR luminous galaxies inhi gh density regions is requiredtoexplain the observed resul ts.
+Keywords. galaxies: evolution, galaxies:interactions, galaxies:s tarburst, galaxies:peculiar, galaxies:formation
+1. Introduction
+It hasbeenobservedthat galaxypropertieschangeas a funct ion
+of galaxyenvironment;the morphology-densityrelation re ports
+that fractionof elliptical galaxiesis largerat highergal axyden-
+sity(Gotoetal.,2003);thestarformationrate(SFR)ishig herin
+lower galaxy density (G´ omezet al., 2003; Tanakaet al., 200 4)
+. However, despite accumulating observational evidence, w e
+⋆This research is based on the observations with AKARI, a JAXA
+project withthe participationof ESA.
+⋆⋆Based on data collected at Subaru Telescope, which is operat ed by
+the National Astronomical Observatory ofJapan.
+⋆⋆⋆JSPSSPDfellowstill do not fully understand the underlying physics govern ing
+environmental-dependentevolutionofgalaxies.
+Infrared (IR) emission of galaxies is an important
+probe of galaxy activity since at higher redshift, a sig-
+nificant fraction of star formation is obscured by dust
+(Takeuchi,Buat,&Burgarella, 2005; Gotoetal., 2010).
+Although there exist low-z cluster studies (Baiet al., 2006 ;
+Shimet al., 2010; Tranetal., 2010), not much attention has
+been paid to the infrared properties of high redshift cluste r
+galaxies, mainly due to the lack of sensitivity in previous I R
+satellites such as ISO and IRAS. Superb sensitivity of recen tly
+launched Spitzer and AKARI satellites can revolutionize th e
+infraredviewofenvironmentaldependenceofgalaxyevolut ion.2 Gotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8
+Fig.1.Restframe 8 µm LFs of cluster RXJ1716.4 +6708 at
+z=0.81 in the squares, and those of the AKARI NEP deep
+field in the triangles. For RXJ1716.4 +6708, only photometric
+and spectroscopic cluster member galaxies are used. For the
+NEP deep field, galaxies with photo-z/specz in the range of
+0.65< z <0.9are used. The dot-dashed lines are 8 µm LFs
+of RXJ1716.4 +6708, but scaled down for easier comparison.
+Thethindottedlinesarethebest-fitdoublepowerlaws.Vert ical
+arrows show the 5 σflux limits of deep/shallow regions of the
+cluster (red) and the NEP deep field (blue) in terms of L8µmat
+z=0.81.
+In this work, we comparerestframe8 µm LFs between clus-
+ter and field regions at z=0.8 using data from the AKARI.
+Monochromaticrestframe 8 µm luminosity ( L8µm) is important
+since it is known to correlate well with the total IR luminosi ty
+(Babbedgeet al., 2006; Huanget al., 2007), andhence,with t he
+SFR of galaxies (Kennicutt, 1998). This is especially true f or
+star-forminggalaxiesbecausethe rest-frame8 µm fluxare dom-
+inatedbyprominentPAHfeaturessuchasat6.2,7.7and8.6 µm
+(Desert,Boulanger,&Puget, 1990).
+ImportantadvantagesbroughtbytheAKARIareasfollows:
+(i) At z=0.8, AKARI’s 15 µm filter (L15) covers the redshifted
+restframe 8 µm, thus we can estimate 8 µm LFs without using
+a large extrapolation based on SED models, which were the
+largest uncertainty in previous work. (ii) Large field of vie w of
+the AKARI’smid-IRcamera(IRC, 10’ ×10’)allowsustostudy
+wider area including cluster outskirts, where important ev olu-
+tionary mechanisms are suggested to be at work (Gotoet al.,
+2004; Kodamaet al., 2004). For example, passive spiral gala x-
+ies have been observed in such an environment (Gotoet al.,
+2003). Unless otherwise stated, we adopt a cosmology with
+(h,Ωm,ΩΛ) = (0.7,0.3,0.7)(Komatsuet al., 2009).
+2. Data & Analysis
+2.1. LFs ofclusterRXJ1716.4 +6708
+The AKARI is a Japanese infrared satellite (Murakamiet al.,
+2007), which has continuous filter coverage in the mid
+IR wavelengths ( N2,N3,N4,S7,S9W,S11,L15,L18Wand
+L24). The AKARI has observed a massive galaxy cluster,Fig.2.Restframe 8 µm LFs of cluster RXJ1716.4 +6708 at
+z=0.81, divided according to the local galaxy density ( Σ5th).
+Thestars,circlesandsquaresareforgalaxieswith logΣ5th≥2,
+1.6≤logΣ5th<2,andlogΣ5th<1.6,respectively.
+RXJ1716.4 +6708, in N3,S7andL15(Koyamaetal., 2008).
+RXJ1716.4 +6708 is at z=0.81 and has σ= 1522+215
+−150km s−1,
+LXbol= 13.86±1.04×1044ergs−1,kT= 6.8+1.0
+−0.6keV.Mass
+estimate from weak lensing and X-ray are 3.7 ±1.3×1014M⊙
+and 4.35 ±0.83×1014M⊙, respectively (see Koyamaet al.,
+2007, forreferences).
+An important advantage of the AKARI observation is L15
+filter, which corresponds to the restframe 8 µm at z=0.81. With
+15 (3) pointings, L15reaches 66.5 (96.5) µJy in deep (shal-
+low) regions at 5 σ. Here flux is measured in 11” aperture,
+and coverted to total flux using AKARI’s IRC correction table
+(2009.5.1)1.ClusterstudieswiththeSpitzerareoftenperformed
+in 24µm and thus needed a large extrapolation to estimate ei-
+therL8µmor total infrared luminosity ( LTIR,8−1000µm).
+Note that we do not claim the L8µmis a better indicator of
+thetotalIRluminositythanotherindicators(Brandlet al. ,2006;
+Calzetti et al., 2007; Riekeet al., 2009), but it is importan t that
+theAKARIcanmeausureredshifted 8µmfluxdirectlyinoneof
+thefilters.
+Thanks to the AKARI’s wide field of view (10’ ×10’), the
+total area coverage around the cluster is 200 arcmin2, which
+cover larger area than previous cluster studies with the Spi tzer,
+allowingustostudyIRsourcesintheoutskirts,whereimpor tant
+galaxyevolutiontakesplace(e.g.,Gotoet al.,2003).Prev iously,
+Koyamaet al. (2008) reporteda high fractionof L15sourcesin
+the intermediatedensity regionin the cluster,suggesting a pres-
+enceofenvironmentaleffectintheintermediatedensityen viron-
+ment.
+Thissameregionwasimagedwith Suprime-Camin VRi′z′
+and has a good photometric redshift estimate (Koyamaet al.,
+2007).Usedinthisworkare54 L15-detectedgalaxieswhichare
+well identifiedwithopticalsourceswith 0.76≤zphoto≤0.83.
+With the L15filter covering the restframe 8 µm, we simply
+convert the observed flux to 8 µm monochromatic luminosity
+1http://www.ir.isas.jaxa.jp/ASTRO-
+F/Observation/DataReduction/IRC/ApertureCorrection 090501.htmlGotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8 3
+Table 1.Best doublepower-lawfit parametersforLFs
+Sample L∗
+8µm(L⊙)φ∗(Mpc−3dex−1)α β
+NEPDeepfield 6.1 ±0.5×10100.0010±0.0003 1.1 ±0.3 5.7 ±1.2
+RXJ1716.4 +67082.5±0.1×10100.74±0.04 2.6 ±0.1 5.5 ±0.4
+(L8µm) using a standard cosmology. Completeness was mea-
+sured by distributing artificial point sources with varying flux
+withinthe field andby examiningwhat fractionofthem wasre-
+coveredasafunctionofinputflux.Sincewehavedeepercover -
+age at the center of the cluster, the completeness was measur ed
+separately in the central deep region and the outer regions o f
+the field. More detail of the method is described in Wada et al.
+(2008).
+Oncethefluxisconvertedtoluminosityandcompletenessis
+takenintoaccount,it is straightforwardto construct L8µmLFs,
+which we show in the squares in Fig.1. Errors of the LFs are
+assumedtofollowPoissondistribution.Here,wetakeanang ular
+distance of the most distant source from the cluster center a s
+a cluster radius ( Rmax= 6.2Mpc). We assumed4
+3πR3
+maxas
+the volume of the cluster to obtain galaxy density ( φ). This is
+only one of many ways to define a cluster volume, and thus, a
+cautionmustbetakentocompare absolute valuesofourLFsto
+other work such as Shimet al. (2010). This cluster is elongat ed
+inangulardirection(Koyamaet al.,2007),andthus,thevol ume
+mightnotbespherical.Yet,comparisonofthe shapeoftheLFs
+isvalid.
+2.2. LFs inthe AKARI NEP Deepfield
+Our field LFs are based on the AKARI NEP Deep field
+data. The AKARI performed deep imaging in the North
+Ecliptic Pole Region (NEP) from 2-24 µm, with 4 pointings
+in each field over 0.4 deg2(Matsuharaet al., 2006, 2007;
+Wada etal., 2008). The 5 σsensitivity in the AKARI IR filters
+(N2,N3,N4,S7,S9W,S11,L15,L18WandL24) are 14.2,
+11.0, 8.0, 48, 58, 71, 117, 121 and 275 µJy (Wada etal., 2008).
+Flux is measured in 3 pix radius aperture (=7”), then correct ed
+tototal flux.
+AsubregionoftheNEP-Deepfield(0.25deg2)hasancillary
+datafromSubaru BVRi′z′(Imaiet al.,2007;Wada etal.,2008),
+CFHTu′(Serjeant et al. in prep.), KPNO2m/FLAMINGOs J
+andKs(Imaietal., 2007), GALEX FUVandNUV(Malkan
+et al. in prep.). For the optical identification of MIR source s,
+we adopt the likelihood ratio method (Sutherland&Saunders ,
+1992).Usingthesedata,weestimatephotometricredshifto fL15
+detectedsourcesintheregionwiththe LePhare (Ilbertet al.,
+2006; Arnoutset al., 2007).Themeasurederrorsonthephoto -z
+against 293 spec-z galaxies from Keck/DEIMOS (Takagi et al.
+in prep.) are∆z
+1+z=0.036 at z≤0.8. We have excluded those
+sourcesbetterfit with QSO templatesfromtheLFs.
+To construct field LFs, we have selected L15sources at
+0.65< zphotoz<0.9. There remained 289 IR galaxies with
+a median redshift of 0.76. L15flux is converted to L8µmus-
+ing the photometric redshift of each galaxy. LFs are com-
+puted using the 1/ Vmaxmethod. We used the SED templates
+(Lagache,Dole,&Puget, 2003) for k-corrections to obtain the
+maximumobservableredshiftfromthefluxlimit.Completene ss
+of theL15detection is corrected using Pearsonet al. (2009b).
+Thiscorrectionis25%atmaximum,sincewe onlyusethesam-
+plewherethecompletenessisgreaterthan80%.
+The resulting field LFs are shown in the dotted line and tri-
+angles in Fig.1. Errors of the LFs are computed using a 1000Monte Caro simulation with varying zand flux within their er-
+rors. These estimated errors are added to the Poisson errors in
+eachLFbinin quadrature.
+We performed a detaild comparison of restframe 8 µm
+LFs to those in the literature in Gotoetal. (2010). Briefly,
+there is an oder of difference between Caputiet al. (2007) an d
+Babbedgeetal. (2006), reflecting difficulty in estimating L8µm
+dominatedbyPAHemissionsusingSpitzer24 µmflux.Ourfield
+8µm LF lies between Caputi etal. (2007) and Babbedgeet al.
+(2006). Compared with these work, we have directly observed
+restframe 8 µm using the AKARI L15filter, eliminating the un-
+certaintlyinfluxconversionbasedonSEDmodels.Moredetai ls
+andevolutionoffieldIRLFsaredescribedinGotoet al.(2010 ).
+3. Results& Discussion
+3.1. 8µmIRLFs
+In Fig.1, we show restframe 8 µm LFs of cluster
+RXJ1716.4 +6708 in the squares, and LFs of the field re-
+gion in the triangles. First of all, cluster LFs have by a fact or
+of∼700 higher density than the field LFs, reflecting the fact
+the galaxy clusters is indeed high density regions in terms o f
+infraredsources.
+Next, to compare the shape of the LFs, we normalized the
+cluster LF to match the field LFs at the faintest end, and show
+in the dash-dottedline. In contrast to the field LFs, which sh ow
+flattening of the slope at log L8µm<10.8L⊙, the cluster LF
+maintainsthesteepslopeintherangeof 10.0L⊙<logL8µm<
+10.6L⊙.Thedifferenceissignificant,consideringthesizeofer-
+rorsoneachLF.
+Wefitadouble-powerlawtobothclusterandfieldLFsusing
+thefollowingformulae.
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−α
+dL/L∗,(L < L∗) (1)
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−β
+dL/L∗,(L > L∗) (2)
+Free parameters are: L∗(characteristic luminosity, L⊙),φ∗
+(normalization, Mpc−3),αandβ(faint and bright end slopes),
+respectively.ThebestfitvaluesforfieldandclusterLFsare sum-
+marisedinTable1andshowninthedottedlinesinFig.1.
+The bright-end slopes are not very different, but L∗of the
+cluster LF is smaller than the field by a factor of 2.4, and the
+faint-endtailofclusterLF issteeperthanthatoffieldLF.
+To further examine the difference at the faint end of the
+LFs, we divide the cluster LF using the local galaxy density
+(Σ5th) measuredbyKoyamaet al. (2008). Thisdensityis based
+on the distance to the 5th nearest neighbor in the transverse di-
+rection using all the optical photo-z members, and thus, is a
+surface galaxy density. We separate LFs using similar crite ria,
+logΣ5th≥2(dense),1.6≤logΣ5th<2(intermediate), and
+logΣ5th<1.6(sparse), then plot LFs of each region in the4 Gotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8
+stars, circles, and squares in Fig.2. A fraction of the total vol-
+umeofthe clusteris assignedto eachdensitygroupin invers ely
+proportionaltothe sumof Σ3/2
+5thofeachgroup.
+Interestingly, the faint-end slope becomes flatter and flatt er
+with decreasing local galaxy density. This result is consis tent
+with our comparison with the field in Fig.1. In fact, the lowes t
+densityLF(squares)hasaflatfaint-endtailsimilartothat ofthe
+fieldLF.SincetheseLFsarebasedonthesamedata,changesin
+the faint-end slope are not likely due to the errors in comple te-
+ness correction nor calibration problems. The completenes s of
+the deep and shallow regions of the cluster are measured sep-
+arately. The changes in the slope is much larger than the maxi -
+mumcompletenesscorrectionof25%.Wealsocheckedtheclus -
+ter LFs as a function of cluster centric radius, to find no sign ifi-
+cantdifference,perhapsduetotheelongatedmorphologyof this
+cluster. At the same time, assuming the same cluster volume,
+Fig.2 shows that a possible contamination from the field gala x-
+ies to cluster LFs is only ∼0.1% in the dense region and ∼1%
+eveninthe sparseregion.
+It is interesting that not just the change in the scale of the
+LFs, but there is a change in the L∗and the faint-end slope ( α)
+of the LFs, resulting in the deficit in the 10.2L⊙<logL8µm<
+10.8L⊙for cluster LFs. One might imaginea change just in L∗
+might explain the difference in Fig.1. However, in Fig.2, th ere
+clearlyisachangein theslopeasafunctionof Σ5th.
+However,interpretationis rathercomplicated;a shapeofL F
+would not change if field galaxies infall into cluster unifor mly
+withoutchangingtheirstar-formationactivity.Although inclus-
+ter environment,a fractionof MIR luminousgalaxiesis smal ler
+than field (Koyamaet al., 2008), uniformand instant quenchi ng
+of star-formation activity of field galaxies can only shift a LF,
+butcannotaccountforachangein L∗andαoftheLFs.
+Two important findings in this work are; (i) L∗is smaller
+in the cluster. (ii) the faint-end slopes become steeper tow ard
+higher-density regions. To explain these changes in LFs, IR -
+luminousgalaxiesneedtobepreferentiallyreduced,witha rela-
+tive increase of IR-faint galaxies. However, an environmen tal-
+driven physical process such as the ram-pressure stripping or
+galaxy-merging would quench star-formation not only in mas -
+sivegalaxiesbutinlessmassivegalaxiesaswell,andthusi snot
+abletoexplaintheobservedchangesinLFs.
+Ontheotherhand,ithasbeenfrequentlyobservedthatmore
+massive galaxies formed earlier in the Universe. This downs iz-
+ing scenario also depends on the environment,in the sense th at
+galaxieswith same mass are moreevolvedin higherdensityen -
+vironmentsthangalaxisin less denseenvironments(Gotoet al.,
+2005; Tanakaet al., 2005, 2008). Statistically, a good corr ela-
+tionhasbeenfoundbetween LTIRandstellarmass(Elbazet al.,
+2007). Our finding of the relative lack of IR-luminous galaxi es
+in the cluster environmentmay be consistent with the downsi z-
+ing scenario, where higher density regions have more evolve d
+galaxies and lacks massive star-forming galaxies. In contr ast,
+in lower density regions more massive galaxies are still sta r-
+forming. However, since the data we have shown is in IR lumi-
+nosity, to conclude on this, we need good stellar mass estima te
+basedondeepernear-IRdata.
+Although a specific mechanism is unclear, the steep faint-
+end could also result from the enhanced star-formation in le ss
+massive galaxies. In the above scenario, massive galaxies h ave
+already ceased their star-formation in the cluster, but les s mas-
+sive galaxiesare still formingstars. These less massive ga laxies
+may stop star-formation soon to join the faint-end of the red -
+sequence(Koyamaet al., 2007).Fig.3.TotalinfraredLFsofclusterRXJ1716.4 +6708atz=0.81
+in the solid line, and those of the AKARI NEP deep field in the
+dashed line. Overplottedare the LFs of MS1054 from Bai et al.
+(2007).
+3.2. Total IRLFs
+To compare the L8µmLF in Fig.1 to those in the literature, we
+needtoconvert L8µmtoLTIR.Weusethethefollowingrelation
+byCaputiet al.(2007);
+LTIR= 1.91×(νLν8µm)1.06(±55%) (3)
+Thisis better tunedfor a similar luminosityrange used here
+than the originalrelationby Bavouzetetal. (2008). The con ver-
+sion, however, has been the largest source of errors in estim at-
+ingLTIRfromL8µm.Caputi etal.(2007)report55%ofdisper-
+sion around the relation. It should be kept in mind that the re st-
+frame8µm is sensitive to the star-formation activity, but at the
+same time, it is where the SED models have strongest discrep-
+anciesduetothecomplicatedPAHemissionlines(seeFig.12 of
+Caputiet al.,2007; Gotoetal., 2010).
+Theestimated LTIRcanbe,then,convertedtoSFRusingthe
+followingrelationfor a Salpeter IMF, φ(m)∝m−2.35between
+0.1−100M⊙(Kennicutt, 1998).
+SFR(M⊙yr−1) = 1.72×10−10LTIR(L⊙) (4)
+In Fig.3, we show the LTIRLFs. Symbols are the same as
+Fig.1. Inthe topaxis,we showcorrespondingSFR. Overplott ed
+asterisks are cluster LF of MS1054 at z=0.83 with ×2 larger
+mass by Bai et al. (2007), which show good agreement with
+ourLFsofRXJ1716.4 +6708in thesquares.Notethat Bai et al.
+(2007) covered only the central region of MS1054 due to the
+smaller field of view of the Spitzer. The shape of their LF look s
+more similar to our LFs in the highest density bin in Fig.2. A
+shift in scale is perhaps due to difference in esimating clus ter
+volumes.
+Amajordifferenceofourworktothat ofBai etal. (2007)is
+that they were not able to compare in detail on the shape of the
+LFs between field and cluster regions, due mainly to a smaller
+fieldcoverageandlargererrorsonLFs.Theyhadtofixthefain t-
+end slope with a local value. The largest source of errors is i nGotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8 5
+converting Spitzer 24 µm flux into 8 µm. Both cluster and field
+LFs of this work use L15filter, which measures restframe 8 µm
+fluxdirectly,eliminatingthelargestsourceoferrors.Ina ddition,
+bothclusterandfiledLFsaremeasuredwithanessentiallysa me
+methodology,allowingusa faircomparisonofLFs.
+4. Summary
+We constructed restframe 8 µm LFs of a massive galaxy cluster
+(RXJ1716.4 +6708) and a rarefied field region (the NEP deep
+field)at z ∼0.8usingessentially thesame methodanddata from
+the AKARI telescope. AKARI’s 15 µm filter nicely covers rest-
+frame8µm at z∼0.8,and thuswe do not needa large interpola-
+tion based on SED models. AKARI’s wide field of view allows
+ustoinvestigatevarietyofclusterenvironmentswith2ord ersof
+differenceinlocal galaxydensity.
+We found that L∗of the cluster 8 µm LF is smaller than the
+field by a factor of 2.4, and the faint-end tail of cluster IR LF s
+becomesteeperandsteeperwithincreasinglocalgalaxyden sity.
+This difference cannot be explained by a simple infall of fiel d
+galaxies into a cluster. Physics that preferentially supre sses IR
+luminous galaxes in higer density regions is needed to expla in
+theobservedresults.
+Acknowledgments
+Wethanktheanonymousrefereeformanyinsightfulcomments ,
+which significantly improved the paper. We are greateful to
+MasayukiTanakaforusefuldiscussion.WethankL.Baiforpr o-
+vidingdataforcomparison.
+T.G.,Y.K. and H.I. acknowledgesfinancial support from the
+JapanSocietyforthePromotionofScience(JSPS)throughJS PS
+ResearchFellowshipsforYoungScientists.MIwassupporte dby
+the Korea Science and Engineering Foundation(KOSEF) grant
+No. 2009-0063616, funded by the Korea government(MEST)”
+HML acknowledgesthe supportfrom KASI throughits cooper-
+ativefundin2008.
+This research is based on the observations with AKARI, a
+JAXA projectwiththe participationofESA.
+Theauthorswishtorecognizeandacknowledgetheverysig-
+nificant cultural role and reverence that the summit of Mauna
+Kea has always had within the indigenous Hawaiian commu-
+nity. We are most fortunate to have the opportunity to conduc t
+observationsfromthissacredmountain.
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1001.0006.txt

@@ -0,0 +1,447 @@
+arXiv:1001.0006v2  [astro-ph.CO]  4 May 2010Draft version November 2, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+COMPARISON OF HECTOSPEC VIRIAL MASSES WITH SZE MEASUREMENT S
+Kenneth Rines1,2, Margaret J. Geller2, and Antonaldo Diaferio3,4
+Draft version November 2, 2018
+ABSTRACT
+We present the first comparison of virial masses of galaxy clusters with their Sunyaev-Zel’dovich
+Effect (SZE) signals. We study 15 clusters from the Hectospec Clus ter Survey (HeCS) with
+MMT/Hectospec spectroscopy and published SZE signals. We measu re virial masses of these clusters
+from an average of 90 member redshifts inside the radius r100. The virial masses of the clusters are
+strongly correlated with their SZE signals (at the 99% confidence lev el using a Spearman rank-sum
+test). This correlation suggests that YSZcan be used as a measure of virial mass. Simulations predict
+a powerlaw scaling of YSZ∝Mα
+200withα≈1.6. Observationally, we find α=1.11±0.16, significantly
+shallower (given the formal uncertainty) than the theoretical pr ediction. However, the selection func-
+tion of our sample is unknown and a bias against less massive clusters c annot be excluded (such a
+selection bias could artificially flatten the slope). Moreover, our sam ple indicates that the relation
+between velocity dispersion (or virial mass estimate) and SZE signal has significant intrinsic scatter,
+comparable to the range of our current sample. More detailed stud ies of scaling relations are therefore
+needed to derive a robust determination of the relation between clu ster mass and SZE.
+Subject headings: galaxies: clusters: individual — galaxies: kinematics and dynamics — co smology:
+observations
+1.INTRODUCTION
+Clusters of galaxies are the most massive virialized
+systems in the universe. The normalization and evo-
+lution of the cluster mass function is therefore a sen-
+sitive probe of the growth of structure and thus cos-
+mology (e.g., Rines et al. 2007, 2008; Vikhlinin et al.
+2009; Henry et al. 2009; Mantz et al. 2008; Rozo et al.
+2008, and references therein). Many methods exist
+to estimate cluster masses, including dynamical masses
+from either galaxies (Zwicky 1937) or intracluster gas
+(e.g., Fabricant et al. 1980), gravitational lensing (e.g.,
+Smith et al.2005;Richard et al.2010), andthe Sunyaev-
+Zel’dovich effect (SZE Sunyaev & Zeldovich 1972). In
+practice, these estimates are often made using simple
+observables, such as velocity dispersion for galaxy dy-
+namics or X-ray temperature for the intracluster gas.
+If one of these observable properties of clusters has a
+well-defined relation to the cluster mass, a large survey
+can yield tight constraints on cosmological parameters
+(e.g., Majumdar & Mohr 2004). There is thus much
+interest in identifying cluster observables that exhibit
+tight scaling relations with mass (Kravtsov et al. 2006;
+Rozo et al. 2008). Numerical simulations indicate that
+X-ray gas observables (Nagai et al. 2007) and SZE sig-
+nals (Motl et al. 2005) are both candidates for tight scal-
+ing relations. Both methods are beginning to gain ob-
+servational support (e.g., Henry et al. 2009; Lopes et al.
+2009; Mantz et al. 2009; Locutus Huang et al. 2009).
+Dynamical masses from galaxy velocities are unbiased
+[email protected]
+1Department of Physics & Astronomy, Western Washington
+University, Bellingham, WA 98225; [email protected]
+2Smithsonian Astrophysical Observatory, 60 Garden St, Cam-
+bridge, MA 02138
+3Universit` a degli Studi di Torino, Dipartimento di Fisica G en-
+erale “Amedeo Avogadro”, Torino, Italy
+4Istituto Nazionale di Fisica Nucleare (INFN), Sezione di
+Torino, Torino, Italyin numerical simulations (Diaferio 1999; Evrard et al.
+2008), and recent results from hydrodynamical simula-
+tions indicate that virial masses may have scatter as
+small as ∼5% (Lau et al. 2010).
+Previous studies have compared SZE signals to hydro-
+staticX-raymasses(Bonamente et al.2008;Plagge et al.
+2010) and gravitational lensing masses (Marrone et al.
+2009, hereafter M09). Here, we make the first compar-
+ison between virial masses of galaxy clusters and their
+SZE signals. We use SZE measurements from the lit-
+erature and newly-measured virial masses of 15 clus-
+ters from extensive MMT/Hectospec spectroscopy. This
+comparison tests the robustness of the SZE as a proxy
+for cluster mass and the physical relationship between
+the SZE signal and cluster mass. Large SZ cluster sur-
+veys are underway and are beginning to yield cosmologi-
+calconstraints(Carlstrom et al.2010;Hincks et al.2010;
+Staniszewski et al. 2009).
+We assume a cosmology of Ω m=0.3, Ω Λ=0.7, and
+H0=70 km s−1Mpc−1for all calculations.
+2.OBSERVATIONS
+2.1.Optical Photometry and Spectroscopy
+We are completing the Hectospec Cluster Survey
+(HeCS), a study of an X-ray flux-limited sample of 53
+galaxy clusters at moderate redshift with extensive spec-
+troscopy from MMT/Hectospec. HeCS includes all clus-
+ters with ROSAT X-ray fluxes of fX>5×10−12erg
+s−1at [0.5-2.0]keVfrom the Bright Cluster Survey (BCS
+Ebeling et al.1998)orREFLEXsurvey(B¨ ohringer et al.
+2004) with optical imaging in the Sixth Data Release
+(DR6) of SDSS (Adelman-McCarthy et al. 2008). We
+use DR6 photometry to select Hectospec targets. The
+HeCS targets are all brighter than r=20.8 (SDSS cata-
+logs are 95% complete for point sources to r≈22.2). Out
+of the HeCS sample, 15 clusters have published SZ mea-
+surements.2 Rines, Geller, & Diaferio
+2.1.1.Spectroscopy: MMT/Hectospec and SDSS
+HeCS is a spectroscopic survey of clusters in the red-
+shift range 0.10 ≤z≤0.30. We measure spectra with
+the Hectospec instrument (Fabricant et al. 2005) on the
+MMT 6.5m telescope. Hectospec provides simultaneous
+spectroscopy of up to 300 objects across a diameter of
+1◦. This telescope and instrument combination is ideal
+for studying the virial regions and outskirts of clusters
+at these redshifts. We use the red sequence to preselect
+likely cluster members as primary targets, and we fill
+fibers with bluer targets (Rines et al. in prep. describes
+the details of target selection). We eliminate all targets
+withexistingSDSSspectroscopyfromourtargetlistsbut
+include these in our final redshift catalogs.
+Ofthe15clustersstudiedhere,onewasobservedwitha
+single Hectospec pointing and the remaining 14 were ob-
+served with two pointings. Using multiple pointings and
+incorporatingSDSS redshifts of brighterobjectsmitigate
+fiber collision issues. Because the galaxy targets are rel-
+atively bright ( r≤20.8), the spectra were obtained with
+relativelyshortexposuretimes of3x600sto 4x900sunder
+a variety of observing conditions.
+Figure 1 shows the redshifts of galaxies versus their
+projected clustrocentric radii for the 15 clusters stud-
+ied here. The infall patterns are clearly present in all
+clusters. We use the caustic technique (Diaferio 1999)
+to determine cluster membership. Briefly, the caustic
+technique uses a redshift-radius diagram to isolate clus-
+ter members in phase space by using an adaptive ker-
+nel estimator to smooth out the galaxies in phase space,
+and then determining the edges of this distribution (see
+Diaferio 2009, for a recent review). This technique has
+been successfully applied to optical studies of X-ray clus-
+ters, and yields cluster mass estimates in agreement
+with estimatesfromX-rayobservationsandgravitational
+lensing (e.g., Rines et al. 2003; Biviano & Girardi 2003;
+Diaferio et al. 2005; Rines & Diaferio 2006; Rines et al.
+2007, and references therein).
+We apply the prescription of Danese et al. (1980) to
+determine the mean redshift cz⊙and projected velocity
+dispersion σpof each cluster from all galaxies within the
+caustics. We calculate σpusing only the cluster members
+projected within r100estimated from the caustic mass
+profile.
+2.2.SZE Measurements
+The SZE detections are primarily from
+Bonamente et al. (2008, hereafter B08), supplemented
+by three measurements from Marrone et al. (2009,
+hereafter M09). Most of the SZ data were obtained with
+the OVRO/BIMA arrays; the additional clusters from
+M09 were observed with the Sunyaev-Zel’dovich Array
+(SZA; e.g., Muchovej et al. 2007).
+Numerical simulations indicate that the integrated
+Compton y-parameter YSZhas smaller scatter than the
+peak y-decrement ypeak(Motl et al. 2005), so B08 and
+M09 report only YSZ. Although ypeakshould be nearly
+independent of redshift, YSZdepends on the angular size
+of the cluster. The quantity YSZD2
+Aremoves this depen-
+dence. Thus, we compare our dynamical mass estimates
+to this quantity rather than ypeakorYSZ. Table 1 sum-
+marizes the SZ data and optical spectroscopy.
+It is also critical to determine the radius within whichYSZis determined. B08 use r2500, the radius that en-
+closes an average density of 2500 times the critical den-
+sity at the cluster’s redshift; r2500has physical values of
+300-700kpc forthe massiveclustersstudied by B08(470-
+670kpcforthesubsamplestudiedhere). M09useaphys-
+ical radius of 350 kpc because this radius best matches
+their lensing data.
+To use both sets of data, we must estimate the con-
+version between YSZ(r2500) measured within r2500and
+YSZ(r= 350 kpc) measured within the smaller radius
+r=350 kpc. There are 8 clusters analyzed in both B08
+and M09 (5 of which are in HeCS). We perform a least-
+squaresfit to YSZ(r2500)−YSZ(r= 350kpc) to determine
+an approximate aperture correction for the M09 clusters.
+We list both quantities in Table 1.
+3.RESULTS
+We examine two issues: (1) the strength of the corre-
+lation between SZE signal and the dynamical mass and
+(2) the slope of the relationship between them. Figure 2
+shows the YSZ−σprelation. Here, we compute σpfor all
+galaxies inside both the caustics and the radius r100,cde-
+fined by the caustic mass profile [ rδis the radius within
+which the enclosed density is δtimes the critical density
+ρc(z)].
+Because we make the first comparison of dynami-
+cal properties and SZE signals, we first confirm that
+these two variables are well correlated. A nonparametric
+Spearman rank-sum test (one-tailed) rejects the hypoth-
+esis of uncorrelated data at the 98.4% confidence level.
+The strong correlation in the data suggests that both σp
+andYSZD2
+Aincrease with increasing cluster mass.
+Hydrodynamic numerical simulations indicate that
+YSZ(integrated to r500) scales with cluster mass as
+YSZ∝Mα
+500, whereα=1.60 with radiative cooling and
+star formation, and 1.61 for simulations with radiative
+cooling, star formation, and AGN feedback ( α=1.70 for
+non-radiative simulations, Motl et al. 2005). Combin-
+ing this result with the virial scaling relation of dark
+matter particles, σp∝M0.336±0.003
+200 (Evrard et al. 2008),
+the expected scaling is YSZ∝σ4.76(we assume that
+M100∝M500). The right panels of Figure 2 shows this
+predicted slope (dashed lines).
+The bisector of the least-squares fits to the data has
+a slope of 2 .94±0.74, significantly shallower than the
+predicted slope of 4.8.
+We recompute the velocity dispersions σp,Afor all
+galaxies within one Abell radius (2.14 Mpc) and in-
+side the caustics. Surprisingly, the correlation is slightly
+stronger (99.4% confidence level). This result supports
+the idea that velocity dispersions computed within a
+fixedphysicalradiusretainstrongcorrelationswith other
+cluster observables, even though we measure the velocity
+dispersion inside different fractions of the virial radius
+for clusters of different masses. Because cluster veloc-
+ity dispersions decline with radius (e.g. Rines et al. 2003;
+Rines & Diaferio 2006), σp,Amay be smaller than σp,100
+(measured within r100,c) for low-mass clusters, perhaps
+exaggerating the difference in measured velocity disper-
+sionsrelativeto the differences in virialmass(i.e., σp,Aof
+a low-mass cluster may be measured within 2 r100while
+σp,Aof a high-mass cluster may be measured within r100;
+the ratio σp,Aof these clusters would be exaggerated rel-Hectospec Virial Masses and SZE 3
+Fig. 1.— Redshift versus projected clustrocentric radius for the 15 HeCS clusters studied here. Clusters are ordered left-to-r ight and
+top-to-bottom by decreasing values of YSZD2
+A(r2500). The solid lines show the locations of the caustics, which w e use to identify cluster
+members. The Hectospec data extend out to ∼8 Mpc; the figure shows only the inner 4 Mpc to focus on the viria l regions.
+ative to the ratio σp,100). Future cluster surveys with
+enough redshifts to estimate velocity dispersions but too
+few to perform a caustic analysis should still be sufficient
+for analyzing scaling relations.
+Because of random errors in the mass estimation, the
+virial mass and the caustic mass within a given radius
+do not necessarily coincide. Therefore, the radius r100
+depends on the mass estimator used. Figure 2 shows
+the scaling relationsfor two estimated masses M100,cand
+M100,v;M100,cis the mass estimated within r100,c(where
+bothquantitiesaredefinedfromthecausticmassprofile),
+andM100,vis the mass estimated within r100,v(both
+quantities are estimated with the virial theorem, e.g.,
+Rines & Diaferio 2006). including galaxies projected in-
+sider100,v. Similar to σp, there is a clear correlation
+between M100,vandYSZD2
+A(99.0% confidence with a
+Spearman test). The strong correlation of dynamical
+mass with SZE also holds for M100,cestimated directlyfrom the caustic technique (99.8% confidence).
+The bisector of the least-squares fits has a slope of
+1.11±0.16, again significantly shallower than the pre-
+dicted slope of 1.6. This discrepancy has two distinct
+origins. By looking at the distribution of the SZE sig-
+nals in Figure 2, we see that, at a given velocity disper-
+sion or mass, the SZE signals have a scatter which is a
+factor of ∼2. Alternatively, at fixed SZE signal, there
+is a scatter of a factor of ∼2 in estimated virial mass.
+Unless the observational uncertainties are significantly
+underestimated, the data show substantial intrinsic scat-
+ter. Moreover, this scatter is comparable to the range of
+our sample and, therefore, the error on the slope derived
+from our least-squares fit to the data is likely to be un-
+derestimated (see Andreon & Hurn 2010, for a detailed
+discussionofaBayesianapproachtofittingrelationswith
+measurement uncertainties and intrinsic scatter in both
+quantities).4 Rines, Geller, & Diaferio
+TABLE 1
+HeCS Dynamical Masses and SZE Signals
+Cluster z σ p M100,vM100,c YSZD2
+AYSZD2
+ASZE
+(350 kpc) ( r2500)
+km s−11014M⊙1014M⊙10−5Mpc−210−4Mpc2Ref.
+A267 0.2288 743+81
+−616.86±0.82 4.26 ±0.14 3.08 ±0.34 0.42 ±0.06 1
+A697 0.2812 784+77
+−596.11±0.69 5.96 ±3.51 – 1.29 ±0.15 1
+A773 0.2174 1066+77
+−6318.4±1.7 16.3 ±0.7 5.40 ±0.57 0.90 ±0.10 1
+Zw2701 0.2160 564+63
+−473.47±0.42 2.69 ±0.30 1.46 ±0.016 0.17 ±0.02a2
+Zw3146 0.2895 752+92
+−676.87±0.89 4.96 ±0.91 – 0.71 ±0.09 1
+A1413 0.1419 674+81
+−606.60±0.85 3.49 ±0.15 3.47 ±0.24 0.81 ±0.12 1
+A1689 0.1844 886+63
+−5215.3±1.4 9.44 ±5.66 7.51 ±0.60 1.50 ±0.14 1
+A1763 0.2315 1042+79
+−6416.9±1.6 12.6 ±1.5 3.10 ±0.32 0.46 ±0.05a2
+A1835 0.2507 1046+66
+−5519.6±1.6 20.6 ±0.3 6.82 ±0.48 1.37 ±0.11 1
+A1914 0.1659 698+46
+−386.70±0.57 6.21 ±0.21 – 1.08 ±0.09 1
+A2111 0.2290 661+57
+−454.01±0.41 4.77 ±1.23 – 0.55 ±0.12 1
+A2219 0.2256 915+53
+−4512.8±1.0 12.0 ±4.7 6.27 ±0.26 1.19 ±0.05a2
+A2259 0.1606 735+67
+−535.59±0.60 4.90 ±1.69 – 0.27 ±0.10 1
+A2261 0.2249 725+75
+−577.13±0.83 5.10 ±2.07 – 0.71 ±0.09 1
+RXJ2129 0.2338 684+88
+−644.31±0.57 2.94 ±0.13 – 0.40 ±0.07 1
+Note. —aExtrapolated to r2500using the best-fit relation between YSZD2
+A(350kpc) and YSZD2
+A(r2500) for eight clusters in common
+between B08 and M09.
+Note. — Redshift zand velocity dispersion σpare computed for galaxies defined as members using the causti cs. Masses M100,vand
+M100,care evaluated using the virial mass profile and caustic mass p rofile respectively.
+Note. — REFERENCES: SZE data are from (1) Bonamente et al. 2008 and (2) Marrone et al. 2009.
+Our shallow slopes may also arise in part from the fact
+that our sample, which has been assembled from the lit-
+erature and whose selection function is difficult to deter-
+mine, is likely to be biased against clusters with small
+mass and low SZE signal. Larger samples should deter-
+mine whether unknown observational biases or issues in
+the physical understanding of the relation account for
+this discrepancy.
+4.DISCUSSION
+Thestrongcorrelationbetweenmassesfromgalaxydy-
+namics and SZE signals indicates that the SZE is a rea-
+sonableproxyforcluster mass. B08compareSZEsignals
+toX-rayobservables,inparticularthetemperature TXof
+the intracluster medium and YX=MgasTX, whereMgas
+isthemassoftheICM(seealsoPlagge et al.2010). Both
+of these quantities are measured within r500, a signifi-
+cantly smaller radius than r100where we measure virial
+mass. M09 compare SZE signals to masses estimated
+from gravitational lensing measurements. The lensing
+masses are measured within a radius of 350 kpc. For the
+clusters studied here, this radius is smaller than r2500
+and much smaller than r100. Numerical simulations indi-
+cate that the scatter in masses measured within an over-
+densityδdecreases as δdecreases (White 2002), largely
+because variations in cluster cores are averaged out at
+larger radii. Thus, the dynamical measurement reaching
+to larger radius may provide a more robust indication
+of the relationship between the SZE measurements and
+cluster mass.
+TheYSZD2
+A−Mlensdata presented in M09 show a
+weakercorrelationthanouropticaldynamicalproperties.
+A Spearman test rejects the hypothesis of uncorrelated
+data for the M09 data at only the 94.8% confidence level,
+compared to the 98.4-99.8% confidence levels for our op-
+tical dynamical properties. One possibility is that Mlensis more strongly affected by substructure in cluster cores
+and by line-of-sight structures than are the virial masses
+and velocity dispersions we derive.
+Few measurements of SZE at large radii ( > r500) are
+currently available. Hopefully, future SZ data will allow
+a comparisonbetween virialmass and YSZwithin similar
+apertures.
+5.CONCLUSIONS
+Our first direct comparison of virial masses, velocity
+dispersions, and SZ measurements for a sizable clus-
+ter sample demonstrates a strong correlation between
+these observables (98.4-99.8% confidence). The SZE sig-
+nal increases with cluster mass. However, the slopes of
+both the YSZ−σrelation ( YSZ∝σ2.94±0.74
+p) and the
+YSZ−M100relation ( YSZ∝M1.11±0.16
+100) are significantly
+shallower(giventheformaluncertainties)thantheslopes
+predictedbynumericalsimulations(4.76and1.60respec-
+tively).
+This result may be partly explained by a bias against
+less massive clusters that could artificially flatten our
+measured slopes. Unfortunately, the selection function
+of our sample is unknown and we are unable to quan-
+tify the size of this effect. More importantly, our sample
+indicates that the relation between SZE and virial mass
+estimates (or velocity dispersion) has a non-negligible in-
+trinsicscatter. Acomplete, representativeclustersample
+is required to robustly determine the size of this scatter,
+its origin, and its possible effect on the SZE as a mass
+proxy.
+Curiously, YSZis more strongly correlated with both
+σpandM100than with Mlens(M09). Comparison of
+lensingmassesandclustervelocitydispersions(andvirial
+masses)forlarger,complete, objectivelyselected samples
+of clusters may resolve these differences.
+Thefull HeCS sampleof53clusterswill providealargeHectospec Virial Masses and SZE 5
+Fig. 2.— Integrated S-Z Compton parameter YSZD2
+Aversus dynamical properties for 15 clusters from HeCS. Left panels: SZE data
+versus virial mass M100estimated from the virial mass profile (top) and the caustic m ass profile (bottom). Solid and open points indicate
+SZ measurements from B08 and M09 respectively. The dashed li ne shows the slope of the scaling predicted from numerical si mulations:
+YSZ∝M1.6(Motl et al. 2005), while the solid line shows the ordinary le ast-squares bisector. Arrows show the aperture correction s to
+the SZE measurements (see text). Right panels: SZE data versus projected velocity dispersions measured fo r galaxies inside the caustics
+and (top) inside r100,cestimated from the caustic mass profile and (bottom) inside t he Abell radius 2.14 Mpc. The dashed line shows the
+scaling predicted from simulations: YSZ∝M1.6(Motl et al. 2005) and σ∝M0.33(Evrard et al. 2008). The solid line shows the ordinary
+least-squares bisector. Data points and arrows are defined a s in the left panels.
+sample of clusters with robustly measured velocity dis-
+persions and virial masses as a partial foundation for
+these comparisons.
+We thank Stefano Andreon for fruitful discussions
+about fitting scaling relations with measurement errorsand intrinsic scatter in both quantities. AD gratefully
+acknowledges partial support from INFN grant PD51.
+We thank Susan Tokarz for reducing the spectroscopic
+data and Perry Berlind and Mike Calkins for assisting
+with the observations.
+Facilities: MMT (Hectospec)
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1001.0007.txt

@@ -0,0 +1,124 @@
+arXiv:1001.0007v1  [astro-ph.CO]  30 Dec 2009Cosmicstarformation history
+revealedby the AKARI
+& Spatially-resolvedspectroscopyofan E+A(Post-starbur st)system
+Tomotsugu GOTO∗, the AKARINEPDteam†,M.Yagi∗∗andC.Yamauchi†
+∗InstituteforAstronomy,Universityof Hawaii,2680Woodla wnDrive, Honolulu,HI,96822,USA
+†JapanAerospaceExplorationAgency,Sagamihara,Kanagawa 229-8510,Japan
+∗∗NationalAstronomicalObservatory,2-21-1Osawa,Mitaka, Tokyo,181-8588,Japan
+Abstract. We reveal cosmic star-formation history obscured by dust us ing deep infrared observa-
+tionwiththeAKARI.Acontinuousfiltercoverageinthemid-I Rwavelength(2.4,3.2,4.1,7,9,11,
+15, 18, and 24 µm) by the AKARI satellite allows us to estimate restframe 8 µm and 12 µm lumi-
+nositieswithoutusingalargeextrapolationbasedonaSEDfi t,whichwasthelargestuncertaintyin
+previouswork. We found that restframe 8 µm (0.38<z<2.2), 12µm (0.15<z<1.16), and total
+infrared (TIR) luminosity functions (LFs) (0 .2<z<1.6) constructed from the AKARI NEP deep
+data, show a continuous and strong evolution toward higher r edshift. In terms of cosmic infrared
+luminosity density ( ΩIR), which was obtained by integrating analytic fits to the LFs, we found a
+goodagreementwithpreviousworkat z<1.2,withΩIR∝(1+z)4.4±1.0.Whenweseparatecontri-
+butionsto ΩIRby LIRGs and ULIRGs, we foundmore IR luminoussourcesare inc reasinglymore
+importantathigherredshift.WefoundthattheULIRG(LIRG) contributionincreasesbyafactorof
+10(1.8)from z=0.35toz=1.4.
+Keywords: galaxies:evolution,galaxies:starburst
+PACS:98.70.Lt
+Introduction .Revealingthecosmicstarformationhistoryisoneofthemaj orgoals
+of the observational astronomy. However, UV/optical estim ation only provides us with
+alowerlimitofthestarformationrate(SFR) duetotheobscu rationbydust.Astraight-
+forward way to overcome this problem is to observe in infrare d, which can capture the
+starformation activityinvisiblein the UV. The superb sens itivitiesofrecently launched
+SpitzerandAKARI satellitescan revolutionizethefield.
+However,most of theSpitzer work relied on a large extrapola tionfrom 24 µm flux to
+estimate the 8, 12 µm or total infrared (TIR) luminosity, due to the limited numb er of
+mid-IR filters. AKARI has continuous filter coverage across t he mid-IR wavelengths,
+thus, allows us to estimate mid-IR luminosity without using a largek-correction based
+on the SED models, eliminating the largest uncertainty in pr evious work. By taking
+advantage of this, we present the restframe 8, 12 µm TIR LFs, and thereby the cosmic
+starformationhistoryderivedfrom theseusingtheAKARINE P-Deep data.
+Data&Analysis .TheAKARIhasobservedtheNEPdeepfield(0.4deg2)in9filters
+(N2,N3,N4,S7,S9W,S11,L15,L18WandL24) to the depths of 14.2, 11.0, 8.0, 48, 58,
+71, 117, 121 and 275 µJy (5σ)[14]. This region is also observed in BVRi′z′(Subaru),
+u′(CFHT), FUV,NUV(GALEX), and J,Ks(KPNO2m), with which we computed
+photo-zwithΔz
+1+z=0.043. Objects which are better fit with a QSO template are re movedFIGURE 1. (left) Restframe 8 µm LFs. The blue diamonds, purple triangles, red squares, and orange
+crosses show the 8 µm LFs at 0 .38<z<0.58,0.65<z<0.90,1.1<z<1.4, and 1.8<z<2.2,
+respectively. The dotted lines show analytical fits with a do uble-power law. Vertical arrows show the
+8µm luminosity corresponding to the flux limit at the central re dshift in each redshift bin. Overplotted
+are Babbedge et al. [1] in the pink dash-dotted lines, Caputi et al. [2] in the cyan dash-dotted lines,
+and Huang et al. [6] in the green dash-dotted lines. AGNs are e xcluded from the sample. (middle)
+Restframe 12 µm LFs. The blue diamonds, purple triangles, and red squares s how the 12 µm LFs at
+0.15<z<0.35,0.38<z<0.62, and 0 .84<z<1.16, respectively. Overplotted are Pérez-González
+et al. [11] at z=0.3,0.5and 0.9 in the cyan dash-dottedlines, and Rush, Mal kan, & Spinoglio [12] at z=0
+inthegreendash-dottedlines. (right)TIRLFs.
+from the analysis. We compute LFs using the 1/ Vmaxmethod. Data are used to 5 σwith
+completeness correction. Errors of the LFs are from 1000 rea lization of Monte Carlo
+simulation.
+8µm LF.Monochromatic 8 µm luminosity ( L8µm) is known to correlate well with
+the TIR luminosity [1, 6], especially for star-forming gala xies because the rest-frame
+8µmfluxaredominatedbyprominentPAHfeaturessuchasat6.2,7 .7and8.6 µm.The
+leftpanelofFig.1showsastrongevoltuionof8 µmLFs.Overplottedpreviousworkhad
+torelyonSEDmodelstoestimate L8µmfromtheSpitzer S24µmintheMIRwavelengths
+whereSEDmodelingisdifficultduetothecomplicatedPAHemi ssions.Here,AKARI’s
+mid-IR bands are advantageous in directly observing redshi fted restframe 8 µm flux in
+one of the AKARI’s filters, leading to more reliable measurem ent of 8µm LFs without
+uncertaintyfromtheSED modeling.
+12µm LF.12µm luminosity ( L12µm) represents mid-IR continuum, and known to
+correlate closely with TIR luminosity [11]. The middle pane l of Fig.1 shows a strong
+evoltuion of 12 µm LFs. Here the agreement with previous work is better becaus e (i)
+12µm continuum is easier to be modeled, and (ii) the Spitzer also captures restframe
+12µm inS24µmat z=1.
+TIRLF.Lastly,weshowtheTIRLFsintherightpanelofFig.1.Weused Lagache,
+Dole, & Puget [8]’s SED templates to fit the photometry using t he AKARI bands
+at>6µm (S7,S9W,S11,L15,L18WandL24). The TIR LFs show a strong evolution
+comparedto localLFs. At 0 .25<z<1.3,L∗
+TIRevolvesas ∝(1+z)4.1±0.4.FIGURE2. Evolutionof TIRluminositydensitybasedon TIRLFs (redcir cles),8µmLFs (stars), and
+12µm LFs (filled triangles). The blue open squares and orange fill ed squares are for LIRG and ULIRGs
+only,alsobasedonour LTIRLFs.Overplotteddot-dashedlinesareestimatesfromtheli terature:LeFloc’h
+et al. [9], Magnelli et al. [10] , Pérez-González et al. [11], Caputi et al. [2], and Babbedge et al. [1] are
+in cyan, yellow, green, navy, and pink, respectively. The pu rple dash-dotted line shows UV estimate by
+Schiminovichet al.[13].Thepinkdashedlineshowsthe tota lestimateofIR(TIRLF)andUV [13].
+Cosmic star formation history .We fit LFs in Fig.1 with a double-power law, then
+integrate to estimate total infrared luminosity density at various z. The restframe 8
+and 12µm LFs are converted to LTIRusing [11, 2] before integration. The resulting
+evolution of the TIR density is shown in Fig.2. The right axis shows the star formation
+densityassumingKennicutt[7].We obtain ΩIR(z)∝(1+z)4.4±1.0. Comparisonto ΩUV
+[13] suggests that ΩTIRexplains 70% of Ωtotalatz=0.25, and that by z=1.3, 90% of
+the cosmic SFD is explained by the infrared. This implies tha tΩTIRprovides good
+approximationofthe Ωtotalatz>1.
+In Fig.2, we also show the contributions to ΩTIRfrom LIRGs and ULIRGs. From
+z=0.35 to z=1.4,ΩIRby LIRGs increases by a factor of ∼1.6, andΩIRby ULIRGs
+increases byafactorof ∼10. Moredetailsarein Gotoet al. [3].
+Spatially-Resolved Spectroscopy of an E+A (post-starburs t) System .We per-
+formed a spatially-resolved medium resolution long-slit s pectroscopy of a nearby E+A
+(post-starburst) galaxy system with FOCAS/Subaru [4]. Thi s E+A galaxy has an obvi-
+ous companion galaxy 14kpc in front (Fig.3, left) with the ve locity difference of 61.8
+km/s.
+WefoundthatH δequivalentwidth(EW)oftheE+Agalaxyisgreaterthan7Å gal axy
+wide (8.5 kpc) with no significant spatial variation. We dete cted a rotational velocity in
+the companion galaxy of >175km/s. The progenitor of the companion may have beenFIGURE 3. (left) The SDSS g,r,i-composite image of the J1613+5103. The long-slit position s are
+overlayed.The E+A galaxy is to the right (west), with bluer c olour. The companion galaxy is to the left
+(east). (right) H δEW is plotted against D4000. The diamonds and triangles are f or the E+A core/north
+spectra, respectively. The squares and crosses are for the c ompanion galaxy’s core/north spectra. Gray
+lines are population synthesis models with 5-100% delta bur st population added to the 10G-year-old
+exponentially-decaying( τ=1Gyr)underlyingstellarpopulation.SalpeterIMFandmet allicityof Z=0.008
+areassumed.Onthe models,burstagesof0.1,0.25,0.5and2 G yraremarkedwiththefilled circles.
+a rotationally-supported, but yet passive S0 galaxy. The ag e of the E+A galaxy after
+quenching the star formation is estimated to be 100-500Myr, with its centre having
+slightly younger stellar population. The companion galaxy is estimated to have older
+stellarpopulationof >2 Gyrs ofagewithnosignificantspatialvariation(Fig.3, ri ght).
+Thesefindingsareinconsistentwithasimplepicturewheret hedynamicalinteraction
+createsinfallofthegasreservoirthatcausesthecentrals tarburst/post-starburst.Instead,
+ourresultspresentanimportantexamplewherethegalaxy-g alaxyinteractioncantrigger
+agalaxy-widepost-starburstphenomena.
+REFERENCES
+1. BabbedgeT.S.R., et al.,2006,MNRAS, 370,1159
+2. CaputiK.I.,et al.,2007,ApJ,660,97
+3. GotoT.,et al. 2010,A&AAKARI specialissue
+4. GotoT.,YagiM.,YamauchiC., 2008,MNRAS, 391,700
+5. HopkinsA.M.,ConnollyA. J.,HaarsmaD. B.,CramL. E.,200 1,AJ, 122,288
+6. HuangJ.-S.,et al.,2007,ApJ, 664,840
+7. KennicuttR. C.,Jr., 1998,ARA&A,36,189
+8. LagacheG., DoleH.,PugetJ.-L.,2003,MNRAS, 338,555
+9. LeFloc’hE.,etal., 2005,ApJ,632,169
+10. MagnelliB., et al.2009,A&A,496,57
+11. Pérez-GonzálezP. G.,etal., 2005,ApJ,630,82
+12. RushB., MalkanM. A.,SpinoglioL.,1993,ApJS,89,1
+13. SchiminovichD.,et al.,2005,ApJ, 619,L47
+14. Wada T.,et al.,2008,PASJ, 60,517

1001.0008.txt

@@ -0,0 +1,322 @@
+arXiv:1001.0008v2  [hep-th]  6 Jan 2010Multi-Stream Inflation: Bifurcations and Recombinations i n the Multiverse
+Yi Wang∗
+Physics Department, McGill University, Montreal, H3A2T8, Canada
+In this Letter, we briefly review the multi-stream inflation s cenario, and discuss its implications in
+the string theory landscape and the inflationary multiverse . In multi-stream inflation, the inflation
+trajectory encounters bifurcations. If these bifurcation s are in the observable stage of inflation, then
+interesting observational effects can take place, such as do main fences, non-Gaussianities, features
+and asymmetries in the CMB. On the other hand, if the bifurcat ion takes place in the eternal stage
+of inflation, it provides an alternative creation mechanism of bubbles universes in eternal inflation,
+as well as a mechanism to locally terminate eternal inflation , which reduces the measure of eternal
+inflation.
+I. INTRODUCTION
+Inflation [1] has become the leading paradigm for the
+very early universe. However, the detailed mechanism
+for inflation still remains unknown. Inspired by the pic-
+ture of string theory landscape [2], one could expect that
+the inflationary potential has very complicated structure
+[3]. Inflation in the string theory landscape has impor-
+tantimplicationsinbothobservablestageofinflationand
+eternal inflation.
+The complicated inflationary potentials in the string
+theory landscape open up a great number of interest-
+ing observational effects during observable inflation. Re-
+searchesinvestigatingthecomplicatedstructureofthein-
+flationary potential include multi-stream inflation [4, 5],
+quasi-single field inflation [6], meandering inflation [7],
+old curvaton [8], etc.
+Thestringtheorylandscapealsoprovidesaplayground
+for eternal inflation. Eternal inflation is an very early
+stage of inflation, during which the universe reproduces
+itself, so that inflation becomes eternal to the future.
+Eternal inflation, if indeed happened (for counter ar-
+guments see, for example [9]), can populate the string
+theory landscape, providing an explanation for the cos-
+mological constant problem in our bubble universe by
+anthropic arguments.
+In this Letter, we shall focus on the multi-stream infla-
+tion scenario. Multi-stream inflation is proposed in [4].
+And in [5], it is pointed out that the bifurcations can
+lead to multiverse. Multi-stream inflation assumes that
+during inflation there exist bifurcation(s) in the inflation
+trajectory. For example, the bifurcations take place nat-
+urally in a random potential, as illustrated in Fig. 1. We
+briefly review multi-stream inflation in Section II. The
+details of some contents in Section II can be found in
+[4]. We discuss some new implications of multi-stream
+inflation for the inflationary multiverse in Section III.
+∗[email protected]
+FIG. 1. In this figure, we use a tilted random potential to
+mimic a inflationary potential in the string theory landscap e.
+One can expect that in such a random potential, bifurcation
+effects happens generically, as illustrated in the trajecto ries
+in the figure.
+FIG. 2. One sample bifurcation in multi-stream inflation.
+The inflation trajectory bifurcates into AandBwhen the
+comoving scale k1exits the horizon, and recombines when
+the comoving scale k2exits the horizon.
+II. OBSERVABLE BIFURCATIONS
+In this section, we discuss the possibility that the bi-
+furcation of multi-stream inflation happens during the
+observable stage of inflation. We review the production
+of non-Gaussianities, features and asymmetries [4] in the2
+FIG. 3. In multi-stream inflation, the universe breaks up
+into patches with comoving scale k1. Each patch experienced
+inflation either along trajectories AorB. These different
+patches can be responsible for the asymmetries in the CMB.
+CMB, and investigate some other possible observational
+effects.
+To be explicit, we focus on one single bifurcation, as
+illustrated in Fig. 2. We denote the initial (before bifur-
+cation) inflationary direction by ϕ, and the initial isocur-
+vature direction by χ. For simplicity, we let χ= 0 before
+bifurcation. When comoving wave number k1exits the
+horizon, the inflation trajectory bifurcates into Aand
+B. When comoving wave number k2exits the horizon,
+the trajectories recombines into a single trajectory. The
+universe breaks into of order k1/k0patches (where k0de-
+notes the comoving scale of the current observable uni-
+verse), each patch experienced inflation either along tra-
+jectories AorB. The choice of the trajectories is made
+by the isocurvature perturbation δχat scale k1. This
+picture is illustrated in Fig. 3.
+We shall classify the bifurcation into three cases:
+Symmetric bifurcation . If the bifurcation is symmetric,
+in other words, V(ϕ,χ) =V(ϕ,−χ), then there are two
+potentially observable effects, namely, quasi-single field
+inflation, and a effect from a domain-wall-like objects,
+which we call domain fences.
+As discussed in [4], the discussion of the bifurcation
+effect becomes simpler when the isocurvature direction
+has mass of order the Hubble parameter. In this case,
+except for the bifurcation and recombination points, tra-
+jectoryAand trajectory Bexperience quasi-single field
+inflation respectively. As there are turnings of these tra-
+jectories, the analysis in [6] can be applied here. The
+perturbations, especially non-Gaussianities in the isocur-
+vature directions are projected onto the curvature direc-
+tion, resultingin a correctionto the powerspectrum, and
+potentially large non-Gaussianities. As shown in [6], the
+amount of non-Gaussianity is of order
+fNL∼P−1/2
+ζ/parenleftbigg1
+H∂3V
+∂χ3/parenrightbigg/parenleftBigg˙θ
+H/parenrightBigg3
+, (1)
+whereθdenotes the angle between the true inflation di-
+rection and the ϕdirection.
+As shown in Fig. 3, the universe is broken into patches
+during multi-stream inflation. There arewall-likebound-
+aries between these patches. During inflation, theseboundaries are initially domain walls. However, after
+the recombination of the trajectories, the tensions of
+these domain walls vanish. We call these objects domain
+fences. As is well known, domain wall causes disasters
+in cosmology because of its tension. However, without
+tension, domain fence does not necessarily cause such
+disasters. It is interesting to investigate whether there
+are observational sequences of these domain fences.
+Nearly symmetric bifurcation If the bifurcation is
+nearly symmetric, in other words, V(ϕ,χ)≃V(ϕ,−χ),
+but not equal exactly, which can be achieved by a spon-
+taneous breaking and restoring of an approximate sym-
+metry, then besides the quasi-single field effect and the
+domain fence effect, there will be four more potentially
+observable effects in multi-stream inflation, namely, the
+features and asymmetries in CMB, non-Gaussianity at
+scalek1and squeezed non-Gaussianity correlating scale
+k1and scale kwithk1< k < k 2.
+The CMB power asymmetries are produced because,
+as in Fig. 3, patches coming from trajectory AorBcan
+have different power spectra PA
+ζandPB
+ζ, which are de-
+termined by their local potentials. If the scale k1is near
+to the scale of the observational universe k0, then multi-
+stream inflation provides an explanation of the hemi-
+spherical asymmetry problem [10].
+The features in the CMB (here feature denotes extra
+large perturbation at a single scale k1) are produced as
+a result of the e-folding number difference δNbetween
+two trajectories. From the δNformalism, the curvature
+perturbation in the uniform density slice at scale k1has
+an additional contribution
+δζk1∼δN≡ |NA−NB|. (2)
+These features in the CMB are potentially observable
+in the future precise CMB measurements. As the addi-
+tional fluctuation δζk1does not obey Gaussian distribu-
+tion, there will be non-Gaussianity at scale k1.
+Finally, there are also correlations between scale k1
+and scale kwithk1< k < k 2. This is because the ad-
+ditional fluctuation δζk1and the asymmetry at scale k
+are both controlled by the isocurvature perturbation at
+scalek1. Thus the fluctuations at these two scales are
+correlated. As estimated in [4], this correlation results in
+a non-Gaussianity of order
+fNL∼δζk1
+ζk1PA
+ζ−PB
+ζ
+PA
+ζP−1/2
+ζ. (3)
+Non-symmetric bifurcation If the bifurcation is not
+symmetric at all, especially with large e-folding number
+differences (of order O(1) or greater) along different tra-
+jectories, the anisotropy in the CMB and the large scale
+structure becomes too large at scale k1. However, in
+this case, regions with smaller e-folding number will have
+exponentially small volume compared with regions with
+larger e-folding number. Thus the anisotropy can behave
+in the form of great voids. We shall address this issue in
+more detail in [11]. Trajectories with e-folding number3
+difference from O(10−5) toO(1) in the observable stage
+of inflation are ruled out by the large scale isotropy of
+the observable universe.
+At the remainderof this section, we would like to make
+several additional comments for multi-stream inflation:
+The possibility that the bifurcated trajectories never re-
+combine. In this case, one needs to worry about the do-
+main walls, which do not become domain fence during
+inflation. These domain walls may eventually become
+domain fence after reheating anyway. Another prob-
+lem is that the e-folding numbers along different tra-
+jectories may differ too much, which produce too much
+anisotropies in the CMB and the large scale structure.
+However, similar to the discussion in the case of non-
+symmetric bifurcation, in this case, the observable effect
+could become great voids due to a large e-folding number
+difference. The case without recombination of trajectory
+also has applications in eternal inflation, as we shall dis-
+cuss in the next section.
+Probabilities for different trajectories . In [4], we con-
+sidered the simple example that during the bifurcation,
+the inflaton will run into trajectories AandBwith equal
+probabilities. Actually, this assumption does not need to
+be satisfied for more general cases. The probability to
+run into different trajectories can be of the same order
+of magnitude, or different exponentially. In the latter
+case, there is a potential barrier in front of one trajec-
+tory, which can be leaped over by a large fluctuation of
+theisocurvaturefield. Alargefluctuationoftheisocurva-
+ture field is exponentially rare, resulting in exponentially
+different probabilities for different trajectories. The bi-
+furcation of this kind is typically non-symmetric.
+Bifurcation point itself does not result in eternal infla-
+tion. As is well known, in single field inflation, if the
+inflaton releases at a local maxima on a “top of the hill”,
+a stage of eternal inflation is usually obtained. However,
+at the bifurcation point, it is not the case. Because al-
+though the χdirection releases at a local maxima, the ϕ
+direction keeps on rolling at the same time. The infla-
+tiondirectionisacombinationofthesetwodirections. So
+multi-stream inflation can coexist with eternal inflation,
+but itself is not necessarily eternal.
+III. ETERNAL BIFURCATIONS
+In multi-stream inflation, the bifurcation effect may ei-
+ther take place at an eternal stage of inflation. In this
+case, it provides interesting ingredients to eternal infla-
+tion. These ingredients include alternative mechanism to
+producedifferentbubble universesandlocalterminations
+for eternal inflation, as we shall discuss separately.
+Multi-stream bubble universes . The most discussed
+mechanisms to produce bubble universes are tunneling
+processes, such as Coleman de Luccia instantons [12] and
+Hawking Moss instantons [13]. In these processes, the
+tunneling events, which are usually exponentially sup-
+pressed, create new bubble universes, while most parts
+FIG. 4. Cascade creation of bubble universes. In this figure,
+we assume trajectory Ais the eternal inflation trajectory, and
+trajectory Bis the non-eternal inflation trajectory.
+of the spatial volume remain in the old bubble universe
+at the instant of tunneling.
+If bifurcations of multi-stream inflation happen dur-
+ing eternal inflation, two kinds of new bubble universes
+can be created with similar probabilities. In this case,
+at the instant of bifurcation, both kinds of bubble uni-
+verseshavenearlyequalspatialvolume. Withachangeof
+probabilities, the measures for eternal inflation should be
+reconsideredformulti-streamtype bubble creationmech-
+anism.
+If the inflation trajectories recombine after a period of
+inflation, the different bubble universes will eventually
+have the same physical laws and constants of nature. On
+the other hand, if the different inflation trajectories do
+not recombine, then the different bubble universes cre-
+ated by the bifurcation will have different vacuum ex-
+pectation values of the scalar fields, resulting to different
+physical laws or constants of nature. It is interesting
+to investigate whether the bifurcation effect is more ef-
+fective than the tunneling effect to populate the string
+theory landscape.
+Note that in multi-stream inflation, it is still possi-
+ble that different trajectorieshaveexponentiallydifferent
+probabilities, as discussed in the previous section. In this
+case, multi-stream inflation behaves similar to Hawking
+Moss instantons during eternal inflation.
+Local terminations for eternal inflation . It is possible
+that during multi-stream inflation, a inflation trajectory
+bifurcates in to one eternal inflation trajectory and one
+non-eternal inflation trajectory with similar probability.
+Inthiscase,theinflatonintheeternalinflationtrajectory
+frequently jumps back to the bifurcation point, resulting
+in a cascade creation of bubble universes, as illustrated
+in Fig. 4. This cascade creation of bubble universes, if4
+realized, is more efficient in producing reheating bubbles
+than tunneling effects. Thus it reduces the measure for
+eternal inflation.
+There are some other interesting issues for bifurcation
+in the multiverse. For example, the bubble walls may
+be observable in the present observable universe, and the
+bifurcations can lead to multiverse without eternal infla-
+tion. These possibilities are discussed in [5].
+IV. CONCLUSION AND DISCUSSION
+To conclude, webriefly reviewedmulti-stream inflation
+during observable inflation. Some new issues such as do-main fences and connection with quasi-single field infla-
+tion are discussed. We also discussed multi-stream infla-
+tion in the context of eternal inflation. The bifurcation
+effect in multi-stream inflation provides an alternative
+mechanism for creating bubble universes and populating
+the string theory landscape. The bifurcation effect also
+provides a very efficient mechanism to locally terminate
+eternal inflation.
+ACKNOWLEDGMENT
+We thank Yifu Cai for discussion. This work was sup-
+ported by NSERC and an IPP postdoctoral fellowship.
+[1] A. H. Guth, Phys. Rev. D 23, 347 (1981). A. D. Linde,
+Phys. Lett. B 108, 389 (1982). A. J. Albrecht and
+P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
+[2] R. Bousso and J. Polchinski, JHEP 0006, 006 (2000)
+[arXiv:hep-th/0004134]. S. B. Giddings, S. Kachru
+and J. Polchinski, Phys. Rev. D 66, 106006 (2002)
+[arXiv:hep-th/0105097]. S. Kachru, R. Kallosh, A. Linde
+and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003)
+[arXiv:hep-th/0301240]. M. R. Douglas, JHEP 0305, 046
+(2003) [arXiv:hep-th/0303194].
+[3] Q. G. Huang and S. H. Tye, arXiv:0803.0663 [hep-th].
+[4] M. Li and Y. Wang, JCAP 0907, 033 (2009)
+[arXiv:0903.2123 [hep-th]].
+[5] S. Li, Y. Liu and Y. S. Piao, arXiv:0906.3608 [hep-th].
+[6] X. Chen and Y. Wang, arXiv:0909.0496 [astro-ph.CO].
+X. Chen and Y. Wang, arXiv:0911.3380 [hep-th].
+[7] S. H. Tye and J. Xu, arXiv:0910.0849 [hep-th].
+[8] J. O. Gong and M. Sasaki, JCAP 0901, 001 (2009)
+[arXiv:0804.4488 [astro-ph]]. J. O. Gong, C. Lin andY. Wang, arXiv:0912.2796 [astro-ph.CO].
+[9] V. F. Mukhanov, L. R. W. Abramo and R. H. Bran-
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+(1982).

1001.0009.txt

@@ -0,0 +1,389 @@
+arXiv:1001.0009v1  [q-bio.BM]  30 Dec 2009Jamming proteins with slipknots and their free energy lands cape
+Joanna I. Su/suppress lkowska1, Piotr Su/suppress lkowski2,3,4and Jos´ e N. Onuchic1
+1Center for Theoretical Biological Physics,
+University of California San Diego,
+Gilman Drive 9500, La Jolla 92037,
+2Physikalisches Institute and Bethe Center for Theoretical Physics,
+Universit¨ at Bonn, Nussallee 12, 53115 Bonn, Germany
+3California Institute of Technology, Pasadena, CA 92215,
+4Institute for Nuclear Studies,
+Ho˙ za 69, 00-681 Warsaw, Poland
+Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their un-
+folding times when the stretching force crosses an intermed iate threshold. This behavior arises as
+a consequence of the existence of alternative unfolding rou tes that are dominant at different force
+ranges. Responsible for longer unfolding times at higher fo rces is the existence of an intermediate,
+metastable configuration where the slipknot is jammed. Simu lations are performed with a coarsed
+grained model with further quantification using a refined des cription of the geometry of the slip-
+knots. The simulation data is used to determine the free ener gy landscape (FEL) of the protein,
+which supports recent analytical predictions.
+PACS numbers: 87.15.ap, 87.14.E-, 87.15.La, 82.37.Gk, 87. 10.+e
+The large increase in determining new protein struc-
+tures has led to the discovery of several proteins with
+complicated topology. This new fact has arised the ques-
+tion if their energy landscape and the folding mechanism
+is similar to typical proteins. One class of such proteins
+includes knotted proteins which comprise around 1% of
+all structures deposited in the PDB database [1, 2]. A
+related class of proteins contains more subtle geometric
+configurations called slipknots [3, 4]. Recent theoretical
+studies using structure-based models (where native con-
+tacts are dominant) suggest that slipknot-like conforma-
+tions act like intermediates during the folding of knotted
+proteins [5]. This entire new mechanism is consistent
+with energy landscape theory (FEL) and the funnel con-
+cept [7, 8]. It was shown that the slipknot formation
+reduces the topological barrier. Complementing regular
+folding studies, additional information about the land-
+scape was obtained by mechanical manipulation of the
+knotted protein with atomic force microscopy [9] both
+experimentally in [10, 11] and theoretically in [12, 13, 14].
+For example, [12] it has been showen that unfolding pro-
+ceeds via a series of jumps between various metastable
+conformations, a mechanism opposite to the smooth un-
+folding in knotted homopolymers.
+Motivated by these early results, we now propose a uni-
+fied picture for the mechanical unfolding of proteins with
+slipknots. In this Letter this question is addressed by
+explaining the role of topological barriers along their me-
+chanical unfolding pathways. Supported by our previous
+results that knotted proteins can still have a minimally
+frustrated funnel-like energy landscape, structure-based
+theoretical coarse-grained models are used [15] to ana-
+lyze the behavior of a slipknot protein under stretching.
+Studies are performed for the α/β class protein thymi-dine kinase (PDB code: 1e2i [17]).
+2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket17.27.57.88.1logΤ
+FIG. 1: Dependence of the unfolding times τon the stretch-
+ing force Ffor 1e2i (solid line, in red). In this Letter we
+describe this mechanism as a superposition of two unfolding
+pathways: I for small forces (dashed (lower) line, in blue),
+and II for intermidiate and large forces (dashed-dotted (up -
+per) line, green).
+Most of our analysis is based on stretching simulations
+under constant force [16]. The crucial signature for this
+process is the overall unfolding time from the beginning
+of the stretching until the protein fully unfolds. Normally
+one expects that the transition between the native and
+the unfolded basins to be limited by overcoming the free
+energy barrier, which gets effectively reduced upon an
+application of a stretching force. The rate by which this
+barrier is reduced depends on the distance between the
+unfolded basin and the top of the barrier measured along
+the stretching coordinate x. This idea was first devel-
+oped in the phenomenological model of Bell [18], which
+states that the unfolding time τdecreases exponentially
+with applied stretching force Fasτ(F) =τ0e−Fx
+kBT. A2
+refined analysis performed in ref. [19] revealed that this
+dependence is more complicated but still monotonically
+decreasing.
+The unfolding times for 1e2i measured in our simula-
+tions are shown as the red curve in Fig. 1. In contrast to
+the above expectations, increasing the force in the range
+3-3.5ǫ/˚A surprisingly results in a larger stability of the
+protein. ǫis the typical effective energy of tertiary na-
+tive contacts that is consistent with the value ǫ/˚A≃71
+pN derived in [15]. A solution for this paradox is accom-
+plished by realizing that unfolding is dominated by two
+distinct, alternative routes that are dominant at different
+force regimes. A routing switch occurs when threshold is
+crossed between weak and intermediate forces. At higher
+forces, mechanical unfolding is dominated by a route that
+involves a jammed slipknot. This jamming gives rise to
+the unexpected dependence of unfolding time on applied
+force. Characterizing this mechanism is the central goal
+of this Letter.
+FIG. 2: A slipknot (left) consists of a threaded loop (k1−k2,
+in red) which is partialy threaded through a knotting loop
+(k2−k3, in blue). An example of a protein configuration with
+a tightened slipknot is shown in the right panel.
+To describe the evolution of a slipknot quantitatively
+requires a refined description. A slipknot is character-
+ized by the three points shown in Fig. 2. The first
+pointk1is determined by eliminating amino acids con-
+secutively from one terminus until the knot configura-
+tion is reached (which can be detected e.g. by applying
+the KMT algorithm [20]). The two additional points,
+k2andk3, correspond to the ends of this knot. In the
+native state the protein 1e2i contains a slipknot with
+k1= 10,k2= 128,k3= 298. These three points divide
+the slipknot into two loops, which are called the knotting
+loop and thethreaded loop . The former one is the loop of
+the trefoil knot and the latter one is threaded through the
+knotting loop. Unfolding of the slipknot upon stretch-
+ing depends on the relative shrinking velocity of these
+two loops (see Fig. 3). When the threaded loop shrinks
+faster than the knotting loop, the slipknot unties. In the
+opposite case the slipknot gets (temporarily) tightened
+or jammed, resulting in a metastable state associated
+to a local minimum in the protein’s FEL. Upon further
+stretching, this configuration eventually also unties. The
+evolution of both loops of the slipknot is encoded in thetime dependence of the points k1,k2,k3, see Fig. 3.
+pathway I pathway II
+catch−bonds slip−bondspathway II
+catch−bonds slip−bondspathway I
+FIG. 3: The behavior of the slipknot during stretching (top)
+is determined by the relative behavior of its two loops, en-
+coded in the time dependence of k1,k2andk3(bottom). If the
+threaded loop shrinks faster than the knotting loop, k1merges
+withk2(bottom left) and the slipknot untightens (pathway I,
+top left). If the knotting loop shrinks faster, k2approaches k3
+(bottom right, ≃14000τ) and the slipknot gets temporarily
+tightened (pathway II, top right). This is a metastable stat e
+which can eventually untie further stretching , with k1finally
+merging with k2(bottom right, ≃19000τ). Kinetic stud-
+ies were performed slightly above folding temperature usin g
+overdamped Langevin dynamics with typical folding times of
+10000τ.
+Before discussing the stretching of 1e2i, we explain why
+a slipknot formed by a uniformly elastic polymer should
+smoothly unfold under stretching. To simplify the discus-
+sion we approximate the threaded and knotting loops by
+circles of size RtandRk. These two loops shrink during
+stretching and, when the threaded one eventually van-
+ishes, the slipknot gets untied. If both loops have similar
+sizes, the slipknot is very unstable and unties immedi-
+ately. When the threaded loop is much larger than the
+knotting one, Rt>> R k, untightening can be explained
+as follows. The elastic energy associated to local bend-
+ing is proportional to the square of the curvature. If the
+loop is approximated by a circle of radius R, then its local
+curvature is constant and equals R−1. The total elastic
+energy is/contintegraltext
+dsR−2∼R−1[21]. From the assumption
+Rt>> R kwe conclude that upon stretching it is ener-
+getically favorable to decrease Rtrather than Rk. This
+happens until both radii become equal and then, just as
+above, the slipknot gets very unstable and untightens. In
+this discussion we have not yet taken into account that
+when a slipknot is stretched some parts of a chain slide
+along each other. This effect could be incorporated by in-
+cluding the friction generated by the sliding [22]. But in
+the slipknot the sliding region associated with the knot-
+ting loop is much longer than the region associated to3
+the threaded loop. Thus this effect results in a faster
+tightening of the threaded rather than the knotting loop,
+facilitating even more the untightening of the slipknot.
+The above argument should apply to slipknots in
+biomolecules because they are characterized by a per-
+sistence length that in principle is simply related to their
+elasticity [23]. For DNA this effect is described by worm-
+like-chain models (WLC) [24] and it has been confirmed
+experimentally. Although WLC models are too simple
+to describe the protein general behavior, they are use-
+ful in some limited applications. Thus at first sight one
+might expect that slipknots in proteins should smoothly
+untie upon stretching. Proteins, however, are much more
+complicated than DNA or uniformly elastic polymers.
+The presence of stabilizing native tertiary contacts leads
+to a jumping character during stretching [12]. In addi-
+tion their bending energy is not uniform along the chain
+due to the heterogeneity of the amino-acid sequence. As
+a consequence it turns out that the intuition obtained
+through the above analysis of polymers or WLC models
+is misleading.
+2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket10.51Prob/LParen1pathway I/RParen1
+FIG. 4: Dependence on the applied stretching force of the
+probability of choosing pathway I rather than II (see Fig. 3) .
+This varying probability leads to the complicated dependen ce
+of the total unfolding time on the stretching force observed in
+Fig. 1.
+Our analysis of the evolution of the endpoints k1,k2,k3
+(Fig. 3, bottom) reveals that for various stretching forces
+unfolding proceeds along two distinct pathways (Fig. 3,
+top). In pathway I the slipknot smoothly unties, which
+is observed for relatively weak forces. At intermediate
+forces pathway II starts to dominate and the knotting
+loop can shrink tightly before the threaded one vanishes.
+In this regime the protein gets temporarily jammed (Fig.
+3, right), leading to much longer unfolding times (catch
+pathway). The probability of choosing pathway I at dif-
+ferent forces is shown in Fig. 4. This pathway competi-
+tion explains the nontrivial total unfolding time depen-
+dence observed in Fig. 1.
+The two different pathways I and II arise from com-
+pletely different unfolding mechanisms. Pathway I starts
+and continues mostly from the C-terminal side, along
+16α, 15β, 14α, 13β, 12(helices bundle), 11 α(here the
+number denotes a consecutive secondary structure ascounted from N-terminal, and αorβspecifies whether
+this is a helix or a β-sheet; for more details about the
+structure of 1e2i see the PDB). This is followed by unfold-
+ing of helices 11 α, 10αthat allows breaking of the con-
+tacts inside the β-sheet created by the N-terminal, with
+unfolding proceeding also from the N-terminal. Pathway
+II also starts from the C-terminal but rapidly (as soon
+as helix 15 is unfolded) switches to the N-terminal. In
+this case, differently from pathway I, the β-sheet from
+the N-terminal unfolds even before 13 β. These scenarios
+indicate that the pathway I should be dominant at weak
+forces since they are not sufficient to break the β-sheet
+during first steps of unfolding. The jammed pathway is
+typical only if stretching forces are sufficiently strong for
+unfolding to proceed from the two terminals of the pro-
+tein.
+A similar phenomenon was firstly proposed in ref. [25]
+and referred to as catch-bonds. Experimental evidence
+suggesting this mechanism was first observed for adhe-
+sion complexes [26, 27]. Using AFM, at large forces the
+ligand-receptor pair becomes entangled and therefore ex-
+pands the unfolding time. A theoretical description of
+this mechanism was given in ref. [28, 29, 30].
+The kinetic data can also be used to determine the as-
+sociated free energy landscape (FEL) [7]. In an initial
+simplification we associate the barriers along the stretch-
+ing coordinate as the the kinetic bottlenecks during the
+mechanical unfolding event. Generalizing Bell’s model,
+a recent description of two-state mechanical unfolding in
+the presence of a single transition barrier has been devel-
+oped in [19], with the rate equation
+τ(F) =τ0/parenleftBig
+1−νFx†
+∆G/parenrightBig1−1/ν
+e−∆G
+kBT/parenleftbig
+1−(1−νFx†/∆G)1/ν/parenrightbig
+,
+(1)
+whereνencodes the shape of the barrier. Here x†denotes
+the distance between the barrier and the unfolded basin
+(in a first approximation it can be regarded as Findepen-
+dent) and lies on the reaction coordinate along the AFM
+pulling direction. It can be experimentally determined
+by measuring how the stretching force modulates the un-
+folding times τ. The height of the barrier is denoted by
+∆G. Fig. 1 (unfolding times are given by solid red line)
+shows that this single barrier theory is not sufficient for
+the full range of forces. As described before, in the higher
+force regime, additional basins have to be included in the
+energy landscape. Models with several metastable basins
+have been called multi-state FEL models [31]. Evidence
+supporting the need of multi-states FEL was confirmed
+by AFM experiments in different systems [32, 33].
+To construct a multi-state FEL that incorporates two
+unfolding pathways I and II we use a linear combina-
+tion of eq. (1)-like expressions with different shapes and
+barrier heights. Each one of them essentially accounts
+for the distinct barrier along a relevant unfolding route.
+Fitting the stretching data to eq. (1) with a cusp-like4
+2.5 33.5 4F/LBracket1Ε/Slash1/Angstrom/RBracket16.67.6logΤ
+N U I
+FIG. 5: Pathway II with two barriers. Left: dependence of the
+unfolding time on the applied force with the data and the fit
+to the formula (1) for the first maximum (lower, in green) and
+for the second maximum (upper, in blue). Right: schematic
+free energy landscape for this pathway, with jammed slipkno t
+in a minimum between two barriers.
+ν= 1/2 approximation (another possibility ν= 2/3 for
+the cubic potential in general leads to similar results [19])
+determines accurately the location and the height of the
+potential barriers. Pathway II involves two barriers: first
+until the moment of creation of the intermediate which
+is followed the untieing event. They are characterized by
+(x1,∆G1) and (x2,∆G2) arising respectively from the
+lower and upper fits in Fig. 5 (left). The superposition
+of these two fits gives the overall mean unfolding time for
+pathway II (dotted-dashed curve in green in Fig. 1). For
+the ordinary slipknot unfolding (pathway I), the results
+xIandGIarise from the dashed blue curve in Fig. 1.
+This analysis leads to the results
+x1= 2.3kBT˚A
+ǫ, x2= 0.7kBT˚A
+ǫ, xI= 1.4kBT˚A
+ǫ,
+∆G1= 8.0kBT, ∆G2= 4.2kBT, ∆GI= 4.7kBT.
+We conclude that the free energy landscape consists of
+two “valleys”. The force-dependent probability of choos-
+ing one of the valleys during stretching depends on the
+details of the protein structure. It is determined from our
+simulations as shown in Fig. 4. Using these probability
+values and the parameters above for xand ∆G, we can
+accurately represent the simulation data using a linear
+combination of equations of the form (1). This agreement
+supports our analytical analysis and generalizes eq. (1)
+for the full of range forces. In addition it demonstrates
+that structure-based models sufficiently capture the ma-
+jor geometrical properties of a slipknotted protein. A
+schematic representation of the free energy landscape for
+pathway II is shown in Fig. 5 (right).
+Summarizing, we have analyzed the process of tighten-
+ing of the slipknot in protein 1e2i and determined the cor-
+responding free energy landscape. Its main feature is the
+presence of a metastable configuration with a tightened
+slipknot, which is observed for sufficiently large pulling
+forces. This phenomenon does not exist for uniformly
+elastic polymers. In this Letter we concentrated on pro-
+tein 1e2i but similar behavior has also been observed forother proteins with slipknots, e.g. 1p6x. Our results
+provide testable predictions that can now be verified by
+AFM stretching experiments.
+We appreciate useful comments of O. Dudko. The
+work of J.S. was supported by the Center for Theo-
+retical Biological Physics sponsored by the NSF (Grant
+PHY-0822283) with additional support from NSF-MCB-
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1001.0011.txt

@@ -0,0 +1,463 @@
+arXiv:1001.0011v2  [cond-mat.mes-hall]  16 Apr 2010Guided plasmons in graphene p-njunctions
+E. G. Mishchenko,1A. V. Shytov∗,1and P. G. Silvestrov2
+1Department of Physics and Astronomy, University of Utah, Sa lt Lake City, Utah 84112, USA
+2Theoretische Physik III, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany
+Spatial separation of electrons and holes in graphene gives rise to existence of plasmon waves
+confined to the boundary region. Theory of such guided plasmo n modes within hydrodynamics of
+electron-hole liquid is developed. For plasmon wavelength s smaller than the size of charged domains
+plasmon dispersion is found to be ω∝q1/4. Frequency, velocity and direction of propagation of
+guided plasmon modes can be easily controlled by external el ectric field. In the presence of magnetic
+field spectrum of additional gapless magnetoplasmon excita tions is obtained. Our findings indicate
+that graphene is a promising material for nanoplasmonics.
+PACS numbers: 73.23.-b, 72.30.+q
+Introduction . Breakthrough progress in synthesis and
+characterization has made graphene [2] a promising ob-
+ject for nanoelectronics. Operation of graphene-based
+transistors [3] and other components would rely on the
+propertiesofits single-particle excitations–electronsand
+holes. However, one can also envisage a completely dif-
+ferent set of applications which employ collective excita-
+tions, such as plasmons. Currently, plasmon excitations
+in metallic structures are a subject of nanoplasmonics, a
+new field which has emerged at the confluence of optics
+and condensed matter physics with one of the aims be-
+ing the developing of plasmon-enhanced high resolution
+near-field imaging methods [4, 5]. Another objective is
+possible utilization of plasmons in integrated optical cir-
+cuits. However, perspectives of graphene for nanoplas-
+monics are largely unexplored since plasmon modes of
+graphene flakes have not been addressed so far. As our
+results indicate a great amount of control over graphene
+plasmon properties makes it a very promising material
+for applications.
+Fundamentally, the spectrum of collective chargeoscil-
+lations reflects the long-rangenature of Coulomb interac-
+tion. In conventional two dimensional systems, such as
+those created in semiconducting heterostructures, plas-
+mons are gapless, ω2(q) = 2πe2nq/m∗, withnandm∗
+being electron density and effective mass, respectively
+[6]. Such oscillations can be treated hydrodynamically.
+In clean graphene at zero temperature the plasmon fre-
+quency,ω2∝ |EF|, vanishes with decreasing the doping
+levelEF. It has been argued [7] that the interaction be-
+tweenelectronsandholesinthefinalstatecanmodifythe
+response functions of Dirac fermions and open up a pos-
+sibility for the propagation of charge oscillations at low
+frequencies ω < qv, wherevis electron velocity. Still, hy-
+drodynamic( ω > qv)analogofconventionalplasmonsre-
+mains absent unless either temperature is non-zero [8] or
+graphene is driven away from the charge neutrality point
+by doping or gating [9]. Expectedly, in both cases plas-
+mon spectrum has the conventional form, ω(q)∝q1/2.
+In the present paper we investigate spectra of hydro-
+dynamic plasmons in spatially inhomogeneous grapheneflakes. Realistic graphene samples are typically subject
+to disorder potential and mechanical strain [10] that lead
+totheformationofchargedelectronandholepuddles[11]
+with boundaries between nandpregions being the lines
+ofzerochemicalpotential. Moreover,controlled p-njunc-
+tions can be made with the help of metallic gates [12].
+Alsop-njunctions can be created by applying electric
+field within the plane of a graphene flake, see Fig. 1a.
+The field separates electrons and holes spatially in a way
+that allows control of both the amount of induced charge
+(and thus plasmon frequency) and spatial orientation of
+the junction (the direction of plasmon propagation).
+b)2d 2d
+Ea)
+0n n
+p p
+FIG.1: Twotypesofgraphene p-njunctions: a)field-induced,
+b) gate-induced. Dot-dashed line indicates boundary betwe en
+electron and hole regions and, correspondingly, the direct ion
+of plasmon propagation. In case of field-induced junction it
+is controlled by the direction of external electric field E0.
+Below, we demonstrate that such p-njunctions can
+guide plasmons. We show the existence of charge oscil-
+lations which are localized at the junction and have the
+amplitude decaying with the distance to the junction.
+For wavelengths shorter than the width of the charged
+domains, we find the plasmon spectrum of the form,
+ω2
+n(q) =αne2v
+¯h/radicalbigg
+q|ρ′
+0|
+e, (1)
+whereρ′
+0is the gradient of equilibrium charge density
+at the junction, vis electron velocity, and n= 0,1,2,...2
+enumerates the solutions. The lowest mode has α0=
+4√
+2πΓ(3/4)/Γ(1/4)≈3.39.
+Below we derive this result and discuss plasmon prop-
+erties for the two types of p-njunctions: electric field
+controlled and gate controlled, as shown in Fig. 1.
+Hydrodynamics of charge density oscillations. We uti-
+lize the hydrodynamic approach to describe the motion
+of charged Dirac fermions. The rate of change of electric
+current density Jdue to dynamic electric field Efollows
+from the usual intra-band Drude conductivity with the
+corresponding density of states [13],
+˙J(r,t) =e2
+π¯h2|µ(r)|E(r,t), (2)
+determined by the local value of chemical potential µ(r)
+as measured from the Dirac point (positive for electrons
+and negative for holes). Electric current is related to the
+variation of charge density δρby means of the continuity
+equation,
+δ˙ρ(r,t)+∇·J(r,t) = 0. (3)
+Finally, the variation of charge density produces electric
+field according to the Coulomb law [14],
+E(r,t) =−∇/integraldisplay
+d2r′δρ(r′,t)
+|r−r′|. (4)
+Equations (2)-(4) give a closed system for plasmon exci-
+tations in graphene flakes. We apply it to a p-njunction
+created in a strip infinite along the y-axis (direction of
+plasmon propagation). Using the Fourier representation,
+δρ(r,t) =δρ(x)exp(iqy−iωt), and eliminating Eand
+Jwe arrive at the equation for the oscillating part of
+electron density,
+ω2δρ(x)+2e2v√π¯h/braceleftBigg
+d
+dx/radicalbigg
+|ρ0(x)|
+ed
+dx−q2/radicalbigg
+|ρ0(x)|
+e/bracerightBigg
+×/integraldisplayd
+−ddx′δρ(x′)K0(|q||x−x′|) = 0,(5)
+HereK0is the modified Bessel function and 2 dis
+the width of graphene flake. Within the Thomas-
+Fermi approximation equilibrium charge density ρ0(x)
+is related to the chemical potential via ρ0(x) =
+−sgn(µ)eµ2(x)/π¯h2v2(electron charge is taken to be
+−e). This follows from the condition that the electro-
+chemical potential µ(x)−eφ(x) is constant throughout
+the system. The solutions of Eq. (5) will now be consid-
+ered for large and small plasmon momenta separately.
+Short wavelength, q≫1/d. In this case the decay
+of plasmon density δρ(x) occurs over a distance much
+smaller than the width of the system and the limits
+of integration in Eq. (5) can be extended to infinity.
+Assuming (cf. Eq. (11) below) the linear dependence,
+ρ0(x) =ρ′
+0x, we observe that the integro-differentialequation (5) acquires obvious scaling property. Intro-
+ducing the variable ξ=qxwe arrive at the plasmon
+spectrum in the form (1), with dimensionless constants
+αndetermined from the eigenvalue problem:
+−2√π/parenleftbiggd
+dξ/radicalbig
+|ξ|d
+dξ−/radicalbig
+|ξ|/parenrightbigg
+×/integraldisplay∞
+−∞dξ′δρ(n)(ξ′)K0(|ξ−ξ′|) =αnδρ(n)(ξ).(6)
+Interestingly, this integro-differential equation allows a
+complete analytic solution, though the detailed analysis
+is beyond the scope of this paper. Our main findings
+are as follows. Solutions are enumerated by n= 0,1,2...
+with even/odd numbers corresponding to even/odd den-
+sity profile, δρ(n)(−ξ) = (−1)nδρ(n)(ξ). Surprisingly,
+eigenvalues are doubly-degenerate and given by
+α2n=α02n+1
+4n+1·3·7··(4n−1)
+1·5··(4n−3), α2n+1=α2n.(7)
+At large distances all modes have exponential depen-
+dence,δρ(n)(ξ)∼e−|ξ|, while at |ξ| ≪1 even and
+odd solutions exhibit different behavior, δρ(even)∼1−
+const/radicalbig
+|ξ|andδρ(odd)∼sign(ξ)//radicalbig
+|ξ|. The first pair
+of solutions (belonging to the lowest eigenvalue α0) in
+the Fourier representation δρ(n)(k) =/integraltext
+dξδρ(n)(ξ)eikξ
+acquires a simple form:
+δρ(0)(k)∝1
+(1+k2)3/4, δρ(1)(k)∝k
+(1+k2)3/4.(8)
+Long wavelength, q≪1/d. In contrast to the above
+result (1) plasmon spectrum at small qis sensitive to a
+specific realization of the p-njunction. We address the
+long-wavelength behavior of plasmons in field controlled
+junctions. We expect this case to be of more interest,
+in addition it allows a more complete description. Be-
+fore analyzing plasmons in this structure, we discuss the
+equilibrium density profile. As shown in Fig. 1a the flake
+of width 2 dis placed in external electric field E0applied
+along the x-direction. The equilibrium density distribu-
+tionρ(x) is found from,
+E0x+sgn(x)¯hv
+e/radicalbiggπ
+e|ρ0(x)|+2/integraldisplayd
+0dx′ρ0(x′)lnx+x′
+|x−x′|= 0,
+(9)
+where it is used that ρ0(x) =−ρ0(−x). Prior to solv-
+ing Eq. (9) it is instructive to analyze validity of the
+semiclassical approach. The first condition implies that
+the change of the electron wavelength is smooth on the
+scale of itself, d/dx(¯hv/µ)≪1. Estimating µ(x)∼eE0x
+we obtain that the distance to the p-njunction line
+(x= 0) should exceed the characteristic electric field
+lengthlE=/radicalbig
+e/E0≪x. The second condition requires
+that the electron wavelength is small compared with the
+width of the system, d≫¯hv/µ. Noting that in graphene3
+¯hv∼e2we can rewrite this second condition simply as
+lE≪d. Thus, the Thomas-Fermi equation (9) for the
+equilibrium charge density and the hydrodynamic equa-
+tion (5) for its variation are applicable as long as
+lE≪d, q≪1/lE. (10)
+However, the ratio of qand 1/dcan be arbitrary. For a
+moderate external electric field ∼104V/m the value of
+electric length lE∼0.4µm and the first of the conditions
+(10) is satisfied easily for micron-sized samples.
+AnalyticsolutionofEq.(9)ispossiblewhenthe second
+term is small, in which case the charge density is [15]
+ρ0(x) =E0x√
+d2−x2. (11)
+Substituting this expression back into Eq. (9) we ob-
+serve that the second term is indeed negligible as long
+asx≫l2
+E/d. This is assured whenever the condi-
+tions (10) are satisfied. It is also worth pointing out
+that Eq. (11) justifies the linear approximation for the
+charge density used in deriving Eq. (1) for q≫1/d, with
+ρ′
+0/e= 1/(l2
+Ed).
+We now turn to the analysis of plasma oscillations
+propagating on top of the density distribution, Eq. (11).
+For small plasmon momenta, q≪1/d, electric field ex-
+tends beyond the width of the flake and the equation (5)
+needs to be supplemented with the boundary condition,
+which ensures that electric field (and thus the current)
+vanishes at the edges, x=±d:
+P/integraldisplayd
+−ddxδρ(x)
+x±d= 0. (12)
+The spectrum of the lowest symmetric mode can be most
+easily found by integrating Eq. (5) across the width of
+the flake. The first term in the brackets will then van-
+ish exactly due to the boundary condition (12). The
+remaining integral can now be calculated to the log-
+arithmic accuracy with the help of the approximation
+K0(q|x−x′|) =−lnq|x−x′|:
+/integraldisplayd
+−ddx/radicalbigg
+|ρ0(x)|
+eln(q|x−x′|)≈2dΓ2(3/4)
+lE√πln(qd).
+(13)
+Eqs. (5) and (13) combine to give the equation, [ ω2−
+ω2
+0(q)]/integraltextd
+−ddxδρ(x) = 0, that yields the dispersion of the
+gapless symmetric plasmon,
+ω2
+0(q) = Γ2(3/4)4e2vd
+π¯hlEq2ln(1/qd),(14)
+reminiscent of the plasmon spectrum in quasi-one-
+dimensional wires, The remaining modes, n≥1, are
+gapped. For these modes/integraltextd
+−ddxδρ(x) = 0 and simple
+procedure of integrating Eq. (5) over the width of theflake is not useful. Instead, the equation for the n-th fre-
+quency gap can be obtained by setting q= 0 in Eq. (5).
+We observe that
+ω2
+n(0) =βne2v
+¯hlEd, (15)
+whereβnare the eigenvalues of the equation,
+2√πd
+dξ/radicalbig
+|ξ|
+(1−ξ2)1/4/integraldisplay1
+−1dξ′δρ(n)(ξ′)
+ξ−ξ′=βnδρ(n)(ξ).(16)
+The zeroth mode β0= 0, see Eq. (14), is found ana-
+lytically: δρ(0)∝1//radicalbig
+1−ξ2. It describes charge dis-
+tribution in the strip in response to a (uniform along
+xdirection and smooth along y-direction) change of its
+chemical potential [16]. Other solutions of Eq. (16) are
+found numerically:
+β1= 1.41, β2= 6.49, β3= 6.75,... (17)
+With increasing nthe eigenmodes of integro-differential
+equation (16) oscillate faster, but in generaldo not follow
+the oscillation theorem familiar from quantum mechan-
+ics. In particular, the solutions with n= 0 andn= 3 are
+even while n= 1,n= 2 are odd [17].
+Finally, we mention the case of a gate-controlled p-n
+junction, Fig.1b. Theequilibriumdensityprofileislinear
+nearx= 0 and saturates for large |x|[18]. Eq. (1) is still
+applicable for q >1/d. In the limit q <1/done should
+take into account the screening of long-range Coulomb
+interaction by metallic gates. In this case the logarithm
+in the spectrum of the gapless plasmon disappears, and
+the lowest mode Eq. (14) becomes sound-like.
+Magnetoplasmons. If external magnetic field Bis ap-
+plied perpendicularly to the plane of graphene the plas-
+mon spectra acquire new modes. The equation of motion
+(2) should now be modified to include the Lorentz force,
+˙J(r,t) =e2
+π¯h2|µ(x)|E(r,t)−ev2
+cµ(x)J×B.(18)
+The relative coefficient between electric and magnetic
+terms in this equation follows from the expression for
+the Lorentz force acting on a single particle. The last
+term has opposite sign for electrons and holes. Note that
+the frequency of cyclotron motion ωB(x) =ev2B/cµ(x)
+in graphene p-njunctions is position-dependent. The
+remaining equations (3)-(4) are intact in the presence of
+magnetic field. The boundary condition requires now the
+vanishing of the normal component of electric current at
+the boundary, rather than simply vanishing of the elec-
+tric field, as in Eq. (12). Eliminating JandEwe arrive
+at the generalization of equation (5),
+δρ(x)+2e2
+π/braceleftbigg
+q2Z −q
+ω(ωBZ)′−d
+dxZd
+dx/bracerightbigg
+×/integraldisplayd
+−ddx′δρ(x′)K0(|q||x−x′|) = 0,(19)4
+whereZ(x) =|µ(x)|/(ω2
+B(x)−ω2).
+The most interesting effect described by Eq. (19) is
+the appearance of a set of new modes, chiral magne-
+toplasmons, similar to those considered in Ref. [19] for
+conventional 2D electron systems with smooth bound-
+aries. To find their dispersion in strong magnetic fields,
+whenω≪ωB(x) (the exact condition is given below),
+one should retain only the second term in Eq. (19).
+Noticing that ( ωBZ)′=πl2
+Bρ′
+0(x)/e=πl2
+B/(l2
+Ed), where
+lB=/radicalbig
+¯hc/eBis the magnetic length, we arrive at the
+integral equation
+−2c
+Bq
+ωdρ0(x)
+dx/integraldisplayd
+−ddx′δρ(x′)K0(|q||x−x′|) =δρ(x).(20)
+SinceK0is positive, propagation of magnetoplasmons
+withq >0is quenched, indicative oftheir chiral property
+[20]. As seen from Eq. (20), the plasmon density δρ(x) is
+concentratedwhere ρ′
+0(x) isthestrongest. Thederivative
+of the charge density in field-induced junctions (11) fea-
+tures strong singularitynearthe edges of the flake. Thus,
+low-frequency magnetoplasmon spectrum is strongly de-
+pendent on the microscopic regularization of this singu-
+lar behavior and is, therefore, beyond the scope of the
+Thomas-Fermi approximation used throughout this pa-
+per.
+Thegate-induced junctions, however, allow a rather
+simple analytical description of these modes if we ap-
+proximate that ρ′
+0(x) =e/l2
+Edfor|x| ≤dandρ′
+0(x) = 0
+for|x|> d. The oscillating density δρ(x) then vanishes
+for|x|> d. The solution inside the strip, |x| ≤d, can
+be easily found for q≫1/d, where one can assume the
+range of integration in Eq. (20) to be infinite. The eigen-
+functions of Eq. (20) are simply given by sin[ q⊥(x+d)],
+with the values of q⊥=πn/2ddetermined from the con-
+dition,δρq(±d) = 0. The spectrum of magnetoplasmons
+is then found to be,
+ωn(q) =−2πe2l2
+B
+¯hl2
+Edq/radicalbig
+q2+π2n2/4d2, n= 1,2...(21)
+The magnetoplasmon spectrum (21) is derived under
+the assumption that magnetic field is strong, ωB(d)≫ω,
+which implies that lB≪lE. In order to neglect the first
+and third terms in the brackets in Eq. (19) one has to
+ensure that q≪(lE/lB)4/d. This condition might turn
+out to be more orless restrictivethan the hydrodynamics
+condition q≪1/lE, depending on the particular value of
+the ratio lB/lE. Note that the smallness of this ratio is
+not in contradiction to the non-quantized description of
+electron motion in magnetic filed. The latter is valid as
+long as the filling factor is large, eEd≫ωB(d), which
+means that lB≫l2
+E/d. For magnetic field ∼1T, and
+lB∼25nm, using the estimate below Eq. (10) that lE∼
+400nm we conclude that the width of the flake should
+exceedd >10µm. The magnetoplasmon modes (21) are
+∼(lB/lE)2slowerthan electrons. Note that these modesare undamped since single-particle excitations cannot be
+induced at frequencies below cyclotron frequency ωB.
+Conclusions . Graphene p-njunctions are among the
+most simple and promising applications of this material.
+Single-electron properties of p-njunctions have been ex-
+tensively studied. In the present paper we investigated
+their collective excitations both with and without mag-
+netic field. We anticipate that plasmon modes will be
+crucial for the optical response of graphene nanostruc-
+tures and realistic samples containing electron-hole pud-
+dles. High degree of experimental control should make
+them of special interest to nanoplasmonics and electron-
+ics. Among the most promising applications of plasmons
+inp-njunctions we envisage a possibility of a “plasmon
+transistor” [4]. In particular, by simply switching the
+direction of electric field from across the flake to along
+it (and back) the propagation of plasmons can be facil-
+itated (or prevented). In addition, as follows from the
+above Eqs. (1), (11), the plasmon velocity can be con-
+trolled with simple change in the magnitude of electric
+field. This is in a sharp contrast to plasmons in metal-
+lic nanostructures, whose spectra are typically fixed once
+devices are fabricated.
+Acknowledgments. Useful discussions with M. Raikh
+and O. Starykh are gratefully acknowledged. This
+work was supported by DOE, Grant No. DE-FG02-
+06ER46313. P.G.S. was supported by the SFB TR 12.
+[*] Present address: School of Physics, University of Exete r,
+EX4 4QL, U.K.
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+[5] S.A. Maier, Plasmonics: Fundamentals and Applications
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1001.0012.txt

@@ -0,0 +1,1189 @@
+arXiv:1001.0012v2  [astro-ph.EP]  20 Dec 2010Draft version May 20, 2018
+Preprint typeset using L ATEX style emulateapj v. 8/13/10
+THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPL ANETS
+Nicolas B. Cowan1,2, Eric Agol2,
+Draft version May 20, 2018
+ABSTRACT
+If both the day-side and night-side effective temperatures of a pla net can be measured, it is possible
+to estimate its Bond albedo, 0 < AB<1, as well as its day–night heat redistribution efficiency,
+0< ε <1. We attempt a statistical analysis of the albedo and redistribution efficiency for 24
+transiting exoplanets that have at least one published secondary e clipse. For each planet, we show
+how to calculate a sub-stellar equilibrium temperature, T0, and associated uncertainty. We then use
+a simple model-independent technique to estimate a planet’s effective temperature from planet/star
+flux ratios. We use thermal secondary eclipse measurements —tho se obtained at λ >0.8 micron—
+to estimate day-side effective temperatures, Td, and thermal phase variations —when available— to
+estimatenight-sideeffectivetemperature. Westronglyruleoutth e“nullhypothesis”ofasingle ABand
+εforall 24planets. If wealloweachplanet to havedifferent paramete rs,we find that lowBond albedos
+are favored ( AB<0.35 at 1σconfidence), which is an independent confirmation of the low albedos
+inferred from non-detection of reflected light. Our sample exhibits a wide variety of redistribution
+efficiencies. When normalized by T0, the day-side effective temperatures of the 24 planets describe
+a uni-modal distribution. The two biggest outliers are GJ 436b (abno rmally hot) and HD 80606b
+(abnormally cool), and these are the only eccentric planets in our sa mple. The dimensionless quantity
+Td/T0exhibits no trend with the presence or absence of stratospheric in versions. There is also no
+clear trend between Td/T0andT0. That said, the 6 planets with the greatest sub-stellar equilibrium
+temperatures ( T >2400 K) have low ε, as opposed to the 18 cooler planets, which show a variety
+of recirculation efficiencies. This hints that the very hottest trans iting giant planets are qualitatively
+different from the merely hot Jupiters. We propose an explanation o f this trend based on how a
+planet’s radiative and advective times scale with temperature: both timescales are expected to be
+shorter for hotter planets, but the temperature-dependance of the radiative timescale is stronger,
+leading to decreased heat recirculation efficiency.
+Subject headings: methods: data analysis — (stars:) planetary systems —
+1.INTRODUCTION
+Short-period exoplanets are expected to have atmo-
+spheric compositions and dynamics that differ signifi-
+cantly from Solar System giant planets3. These planets
+orbit∼100×closer to their host stars than Jupiter does
+from the Sun. As a result, they receive ∼104×more flux
+andexperiencetidalforces ∼106×strongerthanJupiter.
+In contrast to Jupiter, which releases roughly as much
+power in its interior as it receives from the Sun, short-
+period exoplanets have power budgets dictated by the
+flux they receive from their host stars. Roughly speak-
+ing, the stellar flux incident on a planet does one of two
+things: it is reflected back into space, or advected else-
+where on the planet and re-radiated at different wave-
+lengths. The physical parameters that describe these
+processes are the planet’s Bond albedo and redistribu-
+tion efficiency.
+1.1.Albedo
+1CIERA Fellow, Northwestern University, 2131 Tech Dr,
+Evanston, IL 60208
+email: [email protected]
+2Astronomy Department, University of Washington, Box
+351580, Seattle, WA 98195
+3For our purposes a “short period” exoplanet is one where the
+periastron distance is less than 0 .1 AU, regardless of its actual
+period, and regardless of mass, which may range from Neptune -
+sized to Brown Dwarf. They are all Class IV and V extrasolar
+giant planets in the scheme of Sudarsky et al. (2003).Giant planets in the Solar System have albedos greater
+than 50%because ofthe presenceofcondensedmolecules
+(H2O, CH 4, NH3, etc.) in their atmospheres. Planets
+with effective temperatures exceeding ∼400 K should be
+cloud free, leading to albedos of 0.05–0.4 (Marley et al.
+1999). If pressure-broadenedNa and K opacity is impor-
+tant at optical wavelengths (as it is for brown dwarfs,
+Burrows et al. 2000), then the Bond albedos of hot
+Jupiters may be less than 10% (Sudarsky et al. 2000).
+But the very hottest planets, the so-called class V extra-
+solar giant planets ( Teff>1500 K), might have very high
+albedosdue to a high silicate cloud layer(Sudarsky et al.
+2000). For a planet whose albedo is dominated by
+clouds (as opposed to Rayleigh scattering) the albedo
+depends on the composition and size of cloud particles
+(Seager et al. 2000).
+Earlyattempts to observe reflected light from exoplan-
+ets (Charbonneau et al. 1999; Collier Cameron et al.
+2002a; Leigh et al. 2003a,b; Rodler et al. 2008, 2010) in-
+dicated that they might not be as reflective as Solar Sys-
+tem gas giants (for a review, see Langford et al. 2010).
+Measurements of HD 209458b taken with the Cana-
+dian MOST satellite revealed a very low albedo ( <8%,
+Rowe et al.2008), andit hassincebeentakenforgranted
+that all short-period planets have albedos on par with
+that of charcoal.
+From the standpoint of the planet’s climate, the im-
+portant factor is not the albedo at any one wavelength,2 Cowan & Agol
+Aλ, but rather the integrated albedo, weighted by the in-
+cident stellar spectrum, known as the Bond albedo and
+denoted in this paper as AB. The relation between Aλ
+and the planet’s Bond albedo is not trivial. If the albedo
+is dominated by gray clouds, then the albedo at a sin-
+gle wavelength can indeed be extrapolated to obtain AB.
+For non-grayreflectance spectra, however, it is critical to
+measureAλat the peak emitting wavelength of the host
+startoobtainagoodestimateofthe planet’senergybud-
+get. For example, as pointed out in Marley et al. (1999),
+planets with identical albedo spectra, Aλ, mayhaveradi-
+cally different ABdepending on the spectraltype oftheir
+host stars.
+1.2.Redistribution Efficiency
+The first few measurements of hot Jupiter phase vari-
+ations showed signs that these planets are not all cut
+from the same cloth. Harrington et al. (2006) and
+Knutson et al. (2007a) quoted very different phase func-
+tion amplitudes for the υAndromeda and HD 189733
+systems. It was not clear whether the differences were
+intrinsic to the planets, however, because the data
+were taken with different instruments, at different wave-
+lengths, and with very different observation schemes (in
+any case, subsequent re-analysis of the original data and
+newly aquired Spitzerobservations of υAndromeda b
+paint a completely different picture of that system:
+Crossfield et al. 2010).
+The uniform study presented in Cowan et al. (2007),
+on the other hand, showed that HD 179949b and
+HD209458bexhibit significantlydifferentdegreesofheat
+recirculation, confirming suspicions. But it was not clear
+whether hot exoplanets were uni-modal or bi-modal in
+redistribution: are HD 179949b and HD 209458b end-
+members of a single distribution, or prototypes for two
+fundamentally different sorts of exoplanets?
+The presence or lack of a stratospheric tempera-
+ture inversion (Hubeny et al. 2003; Fortney et al. 2006;
+Burrows et al. 2007, 2008; Zahnle et al. 2009) on the
+day-sides of exoplanets has been invoked to explain
+a purported bi-modality in recirculation efficiency on
+hot Jupiters (Fortney et al. 2008). The argument, sim-
+ply put, is that optical absorbers high in the atmo-
+sphere of extremely hot Jupiters (equilibrium temper-
+atures greater than ∼1700 K) would absorb incident
+photons where the radiative timescales are short, mak-
+ingit difficult forthese planets torecirculateenergy. The
+most robust detection of this temperature inversionis for
+HD 209458b (Knutson et al. 2008), but this planet does
+not exhibit a large day-night brightness contrast at 8 µm
+(Cowan et al. 2007). So while temperature inversions
+seem to exist in the majority of hot Jupiter atmospheres
+(Knutson et al. 2010), their connection to circulation ef-
+ficiency —if any— is not clear.
+1.3.Outline of Paper
+It has been suggested (e.g., Harrington et al. 2006;
+Cowan et al. 2007) that observations of secondary
+eclipses and phase variations each constrain a combina-
+tion of a planet’s Bond albedo and circulation efficiency.
+But observations —even phase variations— at a single
+waveband do little to constrain a planet’s energy bud-
+get. In this work we show how observations in differentwavebands and for different planets can be meaningfully
+combined to estimate these planetary parameters.
+In§2 we introduce a simple model to quantify the
+day-side and night-side energy budget of a short-period
+planet, and show how a planet’s Bond albedo, AB, and
+redistribution efficiency, ε, can be constrained by ob-
+servations. In §3 we use published observations of
+24 transiting planets to estimate day-side and —where
+appropriate—night-sideeffective temperatures. We con-
+struct a two-dimensionaldistribution function in ABand
+εin§4. We state our conclusions in §5.
+2.PARAMETERIZING THE ENERGY BUDGET
+2.1.Incident Flux
+Short-period planets have a power budget entirely dic-
+tated by the flux they receive from their host star,
+which dwarfs tidal heating or remnant heat of forma-
+tion. Following Hansen (2008), we define the equi-
+librium temperature at the planet’s sub-stellar point:
+T0(t) =Teff(R∗/r(t))1/2, whereTeffandR∗are the star’s
+effective temperature and radius, and r(t) is the planet–
+star distance (for a circular orbit ris simply equal to the
+semi-major axis, a). For shorthand, we define the geo-
+metrical factor a∗=a/R∗, which is directly constrained
+by transit lightcurves (Seager & Mall´ en-Ornelas 2003).
+The incident flux on the planet is given by Finc=
+1
+2σBT4
+0, and it is significant that this quantity has some
+associated uncertainty. For a planet on a circular orbit,
+the uncertainty in T0=Teff/√a∗is related —to first
+order— to the uncertainties in the host star’s effective
+temperature, and the geometrical factor:
+σ2
+T0
+T2
+0=σ2
+Teff
+T2
+eff+σ2
+a∗
+4a2∗. (1)
+For a planet with non-zero eccentricity, T0varies with
+time, but we are only concerned with its value at su-
+perior conjunction: secondary eclipse occurs at superior
+conjunction, when we are seeing the planet’s day-side.
+At that point in the orbit, the planet–star distance is
+rsc=a(1−e2)/(1−esinω), whereeandωare the
+planet’s orbital eccentricity and argument of periastron,
+respectively.
+For planets with non-zero eccentricity, the uncertainty
+inT0is given by
+σ2
+T0
+T2
+0=σ2
+Teff
+T2
+eff+σ2
+a∗
+4a2∗+/parenleftBig
+e2cos2ω
+1−e2/parenrightBig
+σ2
+ecosω
++/parenleftBig
+esinω
+1−e2−1
+2(1−esinω)/parenrightBig
+σ2
+esinω,(2)
+whereσecosωandσesinωarethe observationaluncertain-
+ties in the two components of the planet’s eccentricity4.
+2.2.Emergent Flux
+At secondary eclipse, and in the absence of albedo or
+energy circulation, the equilibrium temperature of a re-
+gion on the planet depends on the normalized projected
+4This formulation is preferable to an error estimate based on σe
+andσω, because the eccentricity and argument of periastron are
+highlycorrelated inorbitalfits. Thatsaid, the uncertaint iesσecosω
+andσesinωare often not included in the literature, in which case
+we use a slightly different —and more conservative— formulat ion
+of the error budget using σeandσω.Albedo and Heat Recirculation on Hot Exoplanets 3
+distance,γ, from the center of the planetary disc as
+T(γ) =T0(1−γ2)1/8. The thermal secondary eclipse
+depth in this limit is given by:
+Fday
+F∗=/parenleftbiggRp
+R∗/parenrightbigg2/parenleftbigghc
+λkT0/parenrightbigg8/parenleftBig
+ehc/λkT∗
+b−1/parenrightBig
+×/integraldisplay(λkT0/hc)8
+0dx
+exp(x−1/8)−1, (3)
+whereT∗
+bis the brightness temperature of the star at
+wavelength λ.
+In the no-circulation limit, then, the day-side emer-
+gent spectrum is not exactly that of a blackbody, even
+if each annulus has a blackbody spectrum. But these
+differences are not important for the present study, since
+we are concerned with bolometric flux. By integrating
+Equation 3 over λ, one obtains the effective tempera-
+tureoftheday-sideintheno-albedo,no-circulationlimit:
+Tε=0= (2/3)1/4T0(see also Burrows et al. 2008; Hansen
+2008). Indeed, treatingtheplanet’sday-sideasauniform
+hemisphere emitting at this temperature gives nearly the
+same wavelength dependence as the more complex Equa-
+tion 3. The Tε=0temperatures for our sample of 24 tran-
+siting planets are shown in Table 1. These set the max-
+imum possible day-side effective temperature we should
+expect to measure.
+The integrated day-side flux in the general —non-zero
+circulation— case is more subtle: heat may be trans-
+ported to the planet’s night-side, and/or to its poles. In
+this paper we neglect the E-W asymetry in the planet’s
+temperature map due to zonal flows and hence phase
+offsets in the thermal phase variations. Under this as-
+sumption, the day-night temperature contrast can more
+directly be extracted from the observed thermal phase
+variations.
+In practice, manystudies haveadopted asingle param-
+eter to represent bothzonal and meridional transport. It
+is instructive to consider the apparent day-side effective
+temperatures in variouslimits: uniform day-sidetemper-
+ature andT= 0 on the night-side (this is often referred
+to as the planet’s “equilibrium temperature”): Tequ=
+(1/2)1/4T0; in the case of perfect longitudinal transport
+but no latitudinal transport: Tlong= (8/(3π2))1/4T0;
+and in the limit of a uniform temperature everywhere
+on the planet: Tuni= (1/4)1/4T0.
+Comparing the apparent day-side temperatures in the
+three limits of circulation above leads to the following
+simple parametrization of the day-side effective temper-
+ature in terms of the planetary albedo, AB, and circula-
+tion efficiency, ε:
+Td=T0(1−AB)1/4/parenleftbigg2
+3−5
+12ε/parenrightbigg1/4
+,(4)
+where 0< ε <1. Note that εis related to —but dif-
+ferent from— the ǫused in (Cowan & Agol 2010). The
+former is merely a parametrization of the observed disk-
+integrated effective temperature, while the latter, which
+can take values from 0 to ∞, is a precisely defined ratio
+of radiative and advective timescales. The ǫ= 0 case is
+precisely equal to the ε= 0 case, while the ǫ→ ∞limit
+is equivalent to ǫ≈0.95.
+Our definition of εis similar to the Burrows et al.(2006) definition of Pnandyieldsthe sameno-circulation
+limit. But our ε= 1 limit produces a lower day-side
+brightness than the Pn= 0.5 limit, because we as-
+sume that the planet’s day-side has a uniform tempera-
+ture distribution in that limit (for a discussion of differ-
+ent redistribution parameterizations, see the appendix of
+Spiegel & Burrows 2010).
+In reality, efficient longitudinal transport (read: fast
+zonalwinds) mayleadtomorebandingandthereforeless
+efficient latitudinal transport. So one could argue that
+in the limit of perfect day-night temperature homoge-
+nization, both the day and night apparent temperatures
+should beTd= (8/(3π2))1/4T0, in between the Burrows
+et al. value of Td= (1/3)1/4T0and that suggested by
+our parameterization, Td= (1/4)1/4T0. At moderate
+day-night recirculation efficiencies, however, there is a
+good deal of latitudinal transport (I. Dobbs-Dixon, priv.
+comm.), so implicitly assuming a constant T∝cos1/4
+latitudinal dependence —as done by Burrows et al.— is
+not founded, either. The bottom line is that any single-
+parameter implementation of advection is incapable of
+capturing the real complexities involved, but longitudi-
+nal transport is the dominant factor in determining day
+and night effective temperatures.
+Not withstanding the subtleties discussed above and
+noting that cooling tends to latitudinaly homogenize
+night-side temperatures (Cowan & Agol 2010), we get a
+night-side temperature of:
+Tn=T0(1−AB)1/4/parenleftBigε
+4/parenrightBig1/4
+. (5)
+Note thatTdandTnare the equator-weighted tempera-
+tures of their respective hemispheres (ie, as seen by an
+edge-on viewer). As such, they will tend to be slightly
+higher than the hemisphere-averaged temperature, ex-
+cept in the ε= 1 limit. This is also why the quantity
+T4
+d+T4
+nis still a weak function of ε.
+Fig. 1.— Different kinds of idealized observations constrain the
+Bond albedo, ABand circulation efficiency, ε, differently. A mea-
+surement of the secondary eclipse depth at optical waveleng ths is
+a measure of albedo (solid line). A secondary eclipse depth a t
+thermal wavelengths gives a joint constraint on albedo and r ecir-
+culation (dotted line). A measurement of the night-side effe ctive
+temperature from thermal phase variations yields a constra int (the
+dashed line) nearly orthogonal to the day-side measurement .
+In Figure 1 we show how different kinds of observa-4 Cowan & Agol
+tions constrain ABandε. For this example, we chose
+constraints consistent with AB= 0.2 andε= 0.3. The
+solid line is a locus of constant AB; the dotted line is
+the locus of constant Td/T0; the dashed line is a lo-
+cus of constant Tn/T0. From this figure it is clear that
+the measurements complement each other: measuring
+two of the three quantities (Bond albedo, effective day-
+side or night-side temperatures) uniquely determines the
+planet’s albedo and circulation efficiency. When obser-
+vations have some associated uncertainty, they define a
+swath through the AB–εplane.
+3.ANALYSIS
+3.1.Planetary & Stellar Data
+We begin by considering all the photometric obser-
+vations of short-period exoplanets published through
+November 2010, summarized in Table 1. We have dis-
+carded photometric observations of non-transiting plan-
+ets because of their unknown radius and orbital inclina-
+tion5. This leaves us with 24 transiting exoplanets for
+which there are observations in at least one waveband
+at superior conjunction, and in some cases in multiple
+wavebands and at multiple planetary phases.
+Stellar and planetary data are taken from the Ex-
+oplanet Encyclopedia (exoplanet.eu), and references
+therein. We repeated parts of the analysis with the
+Exoplanet Data Explorer database (exoplanets.org) and
+found identical results, within the uncertainties. When
+the stellar data are not available, we have assumed typi-
+cal parameters for the appropriate spectral class, and so-
+lar metallicity. Insofar as we are only concerned with the
+broadband brightnesses of the stars, our results should
+not depend sensitively on the input stellar parameters.
+Knowing the stars’ Teff, loggand [Fe/H], we
+use the PHOENIX/NextGen stellar spectrum grids
+(Hauschildt et al. 1999) to determine their brightness
+temperatures at the observed bandpasses. At each wave-
+band for which eclipse or phase observations have been
+obtained, we determine the ratio of the stellar flux to the
+blackbodyfluxatthatgridstar’s Teff. Wethenapplythis
+factor to the Teffof the observed star.
+It is worth noting that the choice of stellar model leads
+to systematic uncertainties in the planetary brightness
+that are of order the photometric uncertainties. For
+example, Christiansen et al. (2010) use stellar models
+for HAT-P-7 from Kurucz (2005), while we use those
+of Hauschildt et al. (1999). The resulting 8 µm bright-
+ness temperatures for HAT-P-7b differ by as much as
+600 K, or slightly more than 1 σ. Our uniform use
+of Hauschildt et al. (1999) models should alleviate this
+problem, however.
+3.2.From Flux Ratios to Effective Temperature
+The planet’s albedo and recirculation efficiency gov-
+ern its effective day-side and night-side temperatures, Td
+andTn, respectively. Observationally, we can only mea-
+sure the brightness temperature, ideally at a number of
+different wavelengths: Tb(λ). If one knew, a priori, the
+5For completeness, these are: τ-Bootis b, υ-Andromeda b,
+51 Peg b, Gl 876d, HD 75289b, HD 179949b and HD 46375b
+(Charbonneau et al. 1999; Collier Cameron et al. 2002b;
+Leigh et al. 2003a,b; Harrington et al. 2006; Cowan et al. 200 7;
+Seager & Deming 2009; Crossfield et al. 2010; Gaulme et al. 201 0)emergent spectrum of a planet, one could trivially con-
+vert a single brightness temperature to an effective tem-
+perature. Alternatively, if observations were obtained at
+a number of wavelengths bracketing the planet’s black-
+body peak, it would be possible to estimate the planet’s
+bolometric flux and hence its effective temperature in a
+model-independent way (e.g., Barman 2008).
+We adopt the latter empirical approach of converting
+observed flux ratios into brightness temperatures, then
+using these to estimate the planet’s effective tempera-
+ture. The secondary eclipse depth in some waveband di-
+vided by the transit depth is a direct measureofthe ratio
+of the planet’s day-side intensity to the star’s intensity
+at that wavelength, ψ(λ). Knowing the star’s brightness
+temperature at a given wavelength, it is possible to com-
+pute the apparent brightness temperature of the planet’s
+day side:
+Tb(λ) =hc
+λk/bracketleftbigg
+log/parenleftbigg
+1+ehc/λkT∗
+b(λ)−1
+ψ(λ)/parenrightbigg/bracketrightbigg−1
+.(6)
+On the Rayleigh-Jeans tail, the fractional uncertainty
+in the brightness temperature is roughly equal to the
+fractional uncertainty in the eclipse depth; on the Wien
+tail, the fractional error on brightness temperature can
+be smaller because the flux is very sensitive to tempera-
+ture.
+By the same token, a secondary eclipse depth and
+phase variation amplitude at a given wavelength can be
+combined to obtain a measure of the planet’s night-side
+brightness temperature at that waveband.
+Since the albedo and recirculation efficiency of the
+planet are not known ahead of time, it is not immedi-
+atelyobviouswhich wavelengthsaresensitiveto reflected
+light and which are dominated by thermal emission. For
+each planet, we compute the expected blackbody peak if
+the planet has no albedo and no recirculation of energy,
+λε=0= 2898/Tε=0µm. Insofar as real planets will have
+non-zero albedo and non-zero recirculation, the day side
+should never reach Tε=0, and the actual spectral energy
+distributionwillpeakatslightlylongerwavelengths. The
+coolest planet in our sample, Gl 436b, would exhibit a
+blackbody peak at λε=0= 3.1µm, while the hottest
+planet we consider, WASP-12b, has λε=0= 0.9µm.
+In practice this means that ground-based near-IR and
+space-based mid-IR (e.g., Spitzer) observations are as-
+sumed to measure thermal emission, while space-based
+optical observations (MOST, CoRoT, Kepler) may be
+contaminated by reflected starlight.
+In Figure2, wedemonstratetwo alternativetechniques
+to convert an array of brightness temperatures, Tb(λ),
+into an estimate of a planet’s effective temperature, Teff.
+The solid black line shows a model spectrum of ther-
+mal emission from Fortney et al. (2008), with an ef-
+fective temperature of Teff= 1941 K shown with the
+black dashed line. The expected blackbody peak of
+the planet is marked with a vertical dotted line. The
+red points are the expected brightness temperatures in
+the J, H, and K sbands (crosses), as well as the IRAC
+(asterisks) and MIPS (diamond) instruments on Spitzer
+(Fazio et al. 2004; Rieke et al. 2004; Werner et al. 2004).
+Since the majority of the observations of exoplanets have
+been obtained with SpitzerIRAC, we focus on estimat-
+ingTeffbasedonlyon brightness temperatures in thoseAlbedo and Heat Recirculation on Hot Exoplanets 5
+Fig. 2.— The solid black line shows a model spectrum from
+Fortney et al. (2008) including only thermal emission (ie: n o re-
+flected light). The planet’s effective temperature is shown w ith the
+black dashed line, while the expected wavelength of the blac kbody
+peak of the planet is marked with a black dotted line. The red
+points show the expected brightness temperatures in the J, H , and
+Ksbands (crosses), as well as the IRAC (asterisks) and MIPS (di a-
+mond) instruments on Spitzer. The linear interpolation technique
+described in the text is shown with the red line.
+four bandpasses.
+Wien Displacement: The first approach is to simply
+adopt the brightness temperature of the bandpass clos-
+est to the planet’s blackbody peak (the black dotted
+line). If only the four IRAC channels are available, the
+best one can do is the 3.6 µm measurement, yielding
+Teff= 1925 K. There is —however— some subtlety in
+estimating the peak wavelength, as this is dependent on
+knowing the planet’s temperature (and hence ABandε)
+a priori.
+Linear Interpolation: The linear interpolation tech-
+nique, shown with the red line in Figure 2, obviates the
+need for an estimate of the planet’s temperature. The
+brightness temperature is assumed to be constant short-
+ward of the shortest- λobservation, and longward of the
+longest-λobservation. Between bandpasses, the bright-
+ness temperature changes linearly with λ. As long as
+the various brightness temperatures do not differ grossly
+from one another, this technique implicitly gives more
+weight to observations near the hypothetical blackbody
+peak. The bolometric flux of this “model” spectrum is
+then computed, and admits a single effective tempera-
+ture, which is Teff= 1927 K for the current example.
+Since we hope to apply our routine to planets with well
+sampled blackbody peaks, we adopt the linear interpola-
+tion technique, as it can make use of multiple brightness
+temperature estimates near the peak.
+Thetwotechniquesdescribedaboveproducesimilaref-
+fective temperatures, though —unsurprisingly— neither
+gives precisely the correct answer. But these system-
+atic errors are comparable or smaller than the photo-
+metric uncertainty in observations of individual bright-
+ness temperatures (see Table 1). The best IR observa-
+tions for the nearest, brightest planetary systems (e.g.,
+HD 189733b and HD 209458b) lead to observational un-
+certainties of approximately 50 K in brightness temper-
+ature. For many planets, the uncertainty is 100–200 K.
+By that metric, either the Wien displacement or the lin-
+ear interpolation routines give adequate estimates of the
+effective temperature, with errors of 16 K and 14 K, re-spectively.
+Wemakeamorequantitativeanalysisofthesystematic
+uncertainties involved in the Linear Interpolation tem-
+perature estimates as follows. We produce 8800 mock
+data sets: 100 realizations for 11 models and data in
+up to 8 wavebands (J, H, K, IRAC, MIPS; Since this nu-
+mericalexperiment choosesrandom bands from the eight
+available, the results should not be very different if ad-
+ditional wavebands are considered). We run our Linear
+Interpolation technique on each of these and plot in Fig-
+ure 3 the estimated day-side temperature normalized by
+the actual model effective temperature versus the num-
+ber of wavebands used in the estimate. The temperature
+estimates cluster near Test/Teff= 1, indicating that the
+technique is not significantly biased. The scatter in es-
+timates decreases as more wavebands are used, from a
+standard deviation of 7.6% if only a single brightness
+temperature is used, down to 2.4% if photometry is ac-
+quired in eight bands. We incorporate this systematic
+error into our analysis by adding it in quadrature to
+the observational uncertainties described in the follow-
+ing paragraph. This has the desirable effect that planets
+with fewer observations have a larger systematic uncer-
+tainty on their effective temperature.
+Fig. 3.— The Linear Interpolation technique for estimating day-
+side effective as tested on a suite of eleven hot Jupiter spect ral
+models provided by J.J. Fortney. The y-axis shows the estima ted
+day-side effective temperature normalized by the actual mod el ef-
+fective temperature. The x-axis represents the number of br ight-
+ness temperatures used in the estimate. Each color correspo nds to
+one of the eleven models used in the comparison. The black err or
+bars represent the standard deviation in the normalized tem pera-
+ture estimates.
+Inpractice,wewouldliketopropagatethephotometric
+uncertainties to the estimate of Teff. For the Wien Dis-
+placement technique, this uncertainty propagates triv-
+ially to the effective temperature. For the linear inter-
+polation technique, a Monte Carlo can be used to esti-
+mate the uncertainty in Teff: the input eclipse depths
+are randomly shifted 1000 times in a manner consistent
+with their photometric uncertainties —assuming Gaus-
+sianerrors—andtheeffectivetemperatureisrecomputed
+repeatedly. Thescatterintheresultingvaluesof Teffpro-
+vides an estimate of the observational uncertainty in the
+parameter, to which we add in quadrature the estimate
+ofsystematicerrordescribedabove. The resultinguncer-
+tainties are listed in Table 1. These uncertainties should6 Cowan & Agol
+be compared to the uncertainties in Tε=0(also listed in
+Table 1), which are computed using the uncertainty in
+the star’s properties and the planet’s orbit.
+There are two practical issues with the linear interpo-
+lation temperature estimation technique. In some cases,
+onlyupperlimitshavebeenobtained, thereforeonecould
+setψ= 0, with the appropriate1-sigmauncertainty. But
+this approach leads to huge uncertainties in Tefffor plan-
+ets with a secondary eclipse upper-limit near their black-
+body peak. Instead of “punishing” these planets, we opt
+to not use upper-limits (though for completeness we in-
+clude them in Table 1). Secondly, when multiple mea-
+surements of an eclipse depth have been published for
+a given waveband, we use the most recent observation,
+indicated with a superscript “ e” in Table 1. In all cases
+these observations either explicitly agree with their older
+counterpart, or agree with the re-analyzed older data.
+4.RESULTS
+4.1.Looking for Reflected Light
+For each planet, we use thermal observations (essen-
+tially those in the J, H, K s, andSpitzerbands) to es-
+timate the planet’s effective day-side temperature, Td,
+and —when phase variations are available— Tn. These
+values are listed in Table 1. In five cases (CoRoT-
+1b, CoRoT-2b, HAT-P-7b, HD 209458b, TrES-2b), sec-
+ondary eclipses and/or phase variations have been ob-
+tained at optical wavelengths. Such observations have
+the potential to directly constrain the albedo of these
+planets. One approach is to adopt the Tdfrom thermal
+observations and calculate the expected contrast ratio at
+optical wavelengths, under the assumption of blackbody
+emission (see also Kipping & Bakos 2010). Insofar as
+the observed eclipse depths are deeper than this calcu-
+lated depth, one can invoke the contribution of reflected
+light and compute a geometric albedo, Ag. If one treats
+the planet as a uniform Lambert sphere, the geometric
+albedo is related to the spherical albedo at that wave-
+length byAλ=3
+2Ag. These values are listed in Table 1.
+But reflected light is not the only explanation for an
+unexpectedly deep optical eclipse. Alternatively, the
+emissivity of the planets may simply be greater at op-
+tical wavelengths than at mid-IR wavelengths, in agree-
+mentwith realisticspectralmodelsofhotJupiters, which
+predict brightness temperatures greater than Teffon the
+Wien tail (see, for example, the Fortney et al. model
+showninFigure2, whichdoesnotincludereflectedlight).
+Note that this increasein emissivityshould occurregard-
+less of whether or not the planet has a stratosphere: by
+definition, the depth at which the optical thermal emis-
+sion is emitted is the depth at which incident starlight
+is absorbed, which will necessarily be a hot layer —
+assuming the incident stellar spectrum peaks in the op-
+tical.
+Determining the albedo directly (ie: by observing re-
+flected light) can be difficult for short period planets,
+because there is no way to distinguish between reflected
+and re-radiated photons. The blackbody peaks of the
+star and planet often differ by less than a micron. There-
+fore, unlike Solar System planets, these worlds do not
+exhibit a minimum in their spectral energy distribution
+between the reflected and thermal peaks. The hottest
+—and therefore most ambiguous case— of the five tran-siting planets with optical constraints is HAT-P-7b. If
+one takes the mid-IR eclipse depths at face value, the
+planet has a day-side effective temperature of ∼2000 K.
+When combined with the Kepler observations, one com-
+putesanalbedoofgreaterthan50%. Thelargeday-night
+amplitude seen in the Kepler bandpass is then simply
+due to the fact that the planet’s night-side reflects no
+starlight, and the cool day-side can be attributed to high
+ABand/orε. If, on the other hand, one takes the op-
+tical flux to be entirely thermal in origin ( Aλ= 0), the
+day-side effective temperature is ∼2800 K. This is very
+close to that planet’s Tε=0, leaving very little power left
+for the night-side, again explaining the large day-night
+contrast observed by Kepler. The truth probably lies
+somewhere between these two extremes, but in any case
+this degeneracy will be neatly broken with Warm Spitzer
+observations: the two scenarios outlined above will lead
+to small and large thermal phase variations, respectively.
+It is telling that the only optical measurement in Table 1
+that is unanimously considered to constrain albedo —
+and not thermal emission— is the MOST observations
+of HD 209458b (Rowe et al. 2008), the coolest of the five
+transiting planets with optical photometric constraints.
+The bottom line is that extracting a constraint on re-
+flected light from optical measurements of hot Jupiters is
+best done with a detailed spectral model. But even when
+reflectedlightcanbedirectlyconstrained,convertingthis
+constraint on Aλinto a constraint on ABalso requires
+detailedknowledgeofboththestarandtheplanet’sspec-
+tral energy distributions, making for a model-dependent
+exercise.
+4.2.Populating the AB-εPlane
+Setting aside optical eclipses and direct measurements
+of albedo, we may use the rich near- and mid-IR data to
+constrain the Bond albedo and redistribution efficiency
+of short-period giant planets. We define a 20 ×20 grid in
+ABandεand use Equations 4 & 5 to calculate the nor-
+malized day-side and night-side effective temperatures,
+Td/T0andTn/T0, at each grid point, ( i,j). For each
+planet, we have an observational estimate of the day-side
+effective temperature, and in three cases we also have an
+estimate of the night-side effective temperature (as well
+as associated uncertainties).
+We first verifywhether ornot the observationsarecon-
+sistent with a single ABandε. To evaluate this “null
+hypothesis”, we compute the usual χ2=/summationtext24
+i=1(model−
+data)2/error2at each grid point. We use only the esti-
+mates of day-side and (when available) night-side effec-
+tive temperatures to calculate the χ2, giving us 27-2=25
+degreesoffreedom. The“best-fit”has χ2= 132(reduced
+χ2= 5.3), so the current observations strongly rule out
+a single Bond albedo and redistribution efficiency for all
+24 planets.
+For 21 of the 24 planets considered here, we construct
+a two-dimensional distribution function for each planet
+as follows:
+PDF(i,j) =1/radicalbig
+2πσ2
+de−(Td−Td(i,j))2/(2σd)2.(7)
+This defines a swath through parameter space with the
+same shape as the dotted line in Figure 1.
+For the three remaining planets (HD 149026b,Albedo and Heat Recirculation on Hot Exoplanets 7
+HD 189733b, HD 209458b), phase variation measure-
+ments help break the degeneracy:
+PDF(i,j) =1√
+2πσ2
+de−(Td−Td(i,j))2/(2σd)2
+×1√
+2πσ2ne−(Tn−Tn(i,j))2/(2σn)2.(8)
+Fig. 4.— The global distribution function for short-period exo-
+planets in the AB–εplane. The gray-scale shows the sum of the
+normalized probability distribution function for the 24 pl anets in
+our sample. The data mostly consist of infrared day-side flux es,
+leading to the dominant degeneracy (see first the dotted line in
+Figure 1).
+We create a two-dimensional normalized probability
+distribution function (PDF) for each planet, then add
+these together to create the global PDF shown in Fig-
+ure 4. This is a democratic way of representing the data,
+since each planet’s distribution contributes equally.
+In Figures 5 and 6 we show the distribution functions
+for the albedo and circulation of the 24 planets in our
+sample,obtainedbymarginalizingtheglobalPDFofFig-
+ure 4 over either ABorε.
+Fig. 5.— The solid black line shows the projection of the 2-
+dimensional probability distribution function (the gray- scale of
+Figure 4) projected onto the ε-axis. The dashed line shows the
+ε-distribution if one requires that all planets have Bond alb edos
+less than 0.1; under this assumption, we see hints of a bimoda l
+distribution in heat circulation efficiency.Fig. 6.— The solid black line shows the projection of the 2-
+dimensionalprobabilitydistributionfunction (the gray- scale ofFig-
+ure 4) projected onto the AB-axis. The cumulative distribution
+function (not shown) yields a 1 σupper limit of AB<0.35.
+The solid line in Figure 5 shows no evidence of bi-
+modality in heat redistribution efficiency, although there
+is a wide range of behaviors. The dashed line in Figure 5
+shows theε-distribution if one requires the albedo to be
+low,AB<0.1. There are then many high-recirculation
+planets, since advection is the only way to depress the
+day-side temperature in the absence of albedo. Inter-
+estingly, the dashed line doesshow tentative evidence of
+two separate peaks in ε: if short-period giant planets
+have uniformly low albedos, then there appear to be two
+modes of heat recirculation efficiency. We revisit this
+idea below.
+Figure 6 shows that planets in this sample are consis-
+tent with a low Bond albedo. Note that this constraint
+is based entirely on near- and mid-infrared observations,
+and is thus independent from the claims of low albedo
+based on searches for reflected light (Rowe et al. 2008,
+and references therein). Furthermore, this is a constraint
+on the Bond albedo, rather than the albedo in any lim-
+ited wavelength range.
+In Figure 7 we plot the dimensionless day-side effec-
+tive temperature, Td/T0, against the maximum expected
+day-side temperature, Tε=0. The cyan asterisks in Fig-
+ure 7 show the four hot Jupiters without temperature
+inversions, while most of the remaining planets have in-
+versions (Knutson et al. 2010). The presence or absence
+of an inversion does not appear to affect the efficiency of
+day–night heat recirculation.
+Planets should lie below the solid red line in Figure 7,
+which denotes Tε=0= (2/3)1/4T0. Of the 24 planets in
+our sample, only one (Gl 436b) has a day-side effective
+temperature significantly above the Tε=0limit6. This
+planet is by far the coolest in our sample, it is on an ec-
+centric orbit, and observations indicate that it may have
+a non-equilibrium atmosphere (Stevenson et al. 2010).
+There is no reason, on the other hand, that planets
+shouldn’t lie below the red dotted line in Figure 7:
+all it would take is non-zero Bond albedo. That said,
+only 3 of the 24 planets we consider are in this region,
+6This is driven by the abnormally high 3.6 micron brightness
+temperature; including the 4.5 micron eclipse upper limit d oes not
+significantly change our estimate of this planet’s effective temper-
+ature.8 Cowan & Agol
+Fig. 7.— The dimensionless day-side effective temperature,
+Td/T0, plotted against the maximum expected day-side temper-
+ature,Tε=0. The red lines correspond to three fiducial limits of
+recirculation, assuming AB= 0: no recirculation (solid), uniform
+day-hemisphere (dashed), and uniform planet (dotted). The gray
+points indicate the default values (using only observation s with
+λ >0.8 micron) for the four planets whose optical eclipse depths
+may be probing thermal emission rather than just reflected li ght
+(from left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-P-7b ).
+For these planets we have here elected to include optical mea sure-
+ments in our estimate of the day-side bolometric flux and effec tive
+temperature, shown in black. The cyan asterisks denote thos e hot
+Jupiters known notto have a stratospheric inversion according
+to (Knutson et al. 2010). They are, from left to right: TrES-1 b,
+HD 189733b, TrES-3b, WASP-4b. The two red x’s denote the ec-
+centric planets in our sample, which are also the two worst ou tliers.
+with the greatest outlier being HD 80606b, a planet on
+an extremely eccentric orbit with superior conjunction
+nearly coinciding with periastron. As such, it is likely
+that much of the energy absorbed by the planet at that
+point in its orbit performs mechanical work (speeding up
+winds, puffingupthe planet, etc. SeealsoCowan & Agol
+2010) rather than merely warming the gas. Gl 436b and
+HD 80606b are denoted by red x’s in Figure 7.
+The gray points in Figure 7 indicate the default val-
+ues (using only observationswith λ>0.8 micron) for the
+four planets whose optical eclipse depths may be probing
+thermal emission rather than just reflected light (from
+left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-
+P-7b). For these planets we have here elected to use
+all available flux ratios (including optical observations
+potentially contaminated by reflected light) to estimate
+the day-side bolometric flux and effective temperature,
+shown as black points in Figure 7.
+If one takes these day-side effective temperature es-
+timates at face value, it appears that the planets with
+Tε=0<2400 K exhibit a wide-variety of redistribution
+efficiencies and/or Bond albedos, but are consistent with
+AB= 0. It is worth noting that many of the best char-
+acterized planets in this region have Td/T0≈0.75, and
+this accounts for the sharp peak in the dotted line of Fig-
+ure 5 atε= 0.75. The hottest 6 planets, on the other
+hand, have uniformly high Td/T0, indicating that they
+have both low Bond albedo andlow redistribution effi-
+ciency. These planets must not have the high-altitude,
+reflective silicate clouds hypothesized in Sudarsky et al.
+(2000). But this conclusion is dependent on how one
+interprets the Keplerobservations of HAT-P-7b: if the
+large optical flux ratio is due to reflected light, then this
+planet is cooler than we think, and even the hottest tran-siting planets exhibit a variety of behaviors.
+5.SUMMARY & CONCLUSIONS
+We have described how to estimate a planet’s incident
+power budget ( T0), where the uncertainties are driven by
+the uncertainties in the host star’s effective temperature
+and size, as well as the planet’s orbit. We then described
+a model-independent technique to estimate the effective
+temperature of a planet based on planet/star flux ra-
+tiosobtained at variouswavelengths. When the observed
+day-side and night-side effective temperatures are com-
+pared, one can constrain a combination of the planet’s
+Bond albedo, AB, and its recirculation efficiency, ε. We
+applied this analysis on 24 known transiting planets with
+measured infrared eclipse depths.
+Our principal results are:
+1. Essentially all of the planets are consistent with low
+Bond albedo.
+2. We firmly rule out the “null hypothesis”, whereby all
+transiting planets can be fit by a single ABandε. It
+is not immediately clear whether this stems from differ-
+ences in Bond albedo, recirculation efficiency, or both.
+3. In the few cases where it is possible to unambiguously
+infer an albedo based on optical eclipse depths, they are
+extremely low, implying correspondingly low Bond albe-
+dos (<10%). If one adopts such low albedos for all
+the planets in our sample, the discrepancies in day-side
+effective temperature must be due to differences in recir-
+culation efficiency.
+4. These differences in recirculation efficiency do not
+appear to be correlated with the presence or absence of
+a stratospheric inversion.
+5. Planets cooler than Tε=0= 2400 K exhibit a wide va-
+riety of circulation efficiencies that do not appear to be
+correlated with equilibrium temperature. Alternatively,
+theseplanetsmayhavedifferent (but generallylow)albe-
+dos. Planets hotter than Tε=0= 2400 K have uniformly
+low redistribution efficiencies and albedos.
+The apparent decrease in advective efficiency with
+increasing planetary temperature remains unexplained.
+One hypothesis, mentioned earlier, is that TiO and VO
+would provide additional optical opacity in atmospheres
+hotter than T∼1700 K, leading to temperature in-
+versions and reduced heat recirculation on these plan-
+ets (Fortney et al. 2008). But if our sample shows any
+sharp change it behavior it occurs near 2400 K, rather
+than 1700K. One couldinvokeanotheroptical absorber,
+but in any case the lack of correlation —pointed out in
+thisworkandelsewhere—betweenthepresenceofatem-
+perature inversionand the efficiency of heat recirculation
+makes this explanation suspect. Another possible expla-
+nation for the observed trend is that the hottest planets
+have the most ionized atmospheres and may suffer the
+most severe magnetic drag (Perna et al. 2010).
+The simplest explanation for this trend is simply that
+the radiative time is a steeper function of temperature
+than the advective time: advective efficiency is given
+roughly by the ratio of the radiative and advective times
+(eg: Cowan & Agol 2010). In the limit of Newtonian
+cooling, the radiative time scales as τrad∝T−3. If one
+assumes the wind speed to be of order the local sound
+speed, then the advective time scales as τadv∝T−0.5.
+One might therefore naively expect the advective effi-
+ciency to scale as T−2.5. Such an explanation would notAlbedo and Heat Recirculation on Hot Exoplanets 9
+explain the apparent sharp transition seen at 2400 K,
+however.
+The combination of optical observations of secondary
+eclipses and thermal observations of phase variations is
+the best way to constrain planetary albedo and circu-
+lation. The optical observations should be taken near
+the star’s blackbody peak, both to maximize signal-to-
+noise, and to avoidcontaminationfrom the planet’s ther-
+mal emission, but this separationmay not be possible for
+the hottest transiting planets. The thermal observations,
+likewise, should be near the planet’s blackbody peak to
+better constrain its bolometric flux. Note that this wave-
+length is shortwardof the ideal contrastratio, which typ-
+ically falls on the planet’s Rayleigh-Jeans tail. Further-
+more, the thermal phase observations should span a full
+planetaryorbit: thelightcurveminimumisthemostsen-
+sitive measure of ε, and should occur nearly half an orbit
+apart from the light curve maximum, despite skewed di-
+urnal heatingpatterns (Cowan & Agol 2008, 2010). This
+means that observing campaigns that only cover a little
+more than half an orbit (transit →eclipse) are probably
+underestimating the real peak-trough phase amplitude.A possible improvement to this study would be to per-
+form a uniform data reduction for all the Spitzerexo-
+planet observations of hot Jupiters. These data make up
+the majority of the constraints presented in our study
+and most are publicly available. And while the pub-
+lished observations were analyzed in disparate ways, a
+consensus approach to correcting detector systematics is
+beginning to emerge.
+N.B.C. acknowledges useful discussions of aspects of
+this work with T. Robinson, M.S. Marley, J.J. Fort-
+ney, T.S. Barman and D.S. Spiegel. Thanks to our
+referee B.M.S. Hansen for insightful feedback, and to
+E.D. Feigelson for suggestions about statistical methods.
+N.B.C. was supported by the Natural Sciences and Engi-
+neering Research Council of Canada. E.A. is supported
+by a National Science Foundation Career Grant. Sup-
+port for this work was provided by NASA through an
+award issued by JPL/Caltech. This research has made
+use of the Exoplanet Orbit Database and the Exoplanet
+Data Explorer at exoplanets.org.
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+TABLE 1
+Secondary Eclipses & Phase Variations of Exoplanets
+Planet Tε=0[K]aλ[µm]bEclipse DepthcTbright[K] Phase AmplitudecDerived Quantitiesd
+CoRoT-1b12424(84) 0.60(0.42) 1 .6(6)×10−42726(141) Td=2674(144) K
+0.71(0.25) 1 .26(33)×10−42409(75) 1 .0(3)×10−4Aλ<0.1
+2.10(0.02) 2 .8(5)×10−32741(125) Td(A= 0)=2515(84) K
+2.15(0.32) 3 .36(42)×10−32490(157)
+3.6(0.75) 4 .15(42)×10−32098(116)
+4.5(1.0) 4 .82(42)×10−32084(106)
+CoRoT-2b21964(42) 0.60(0.42) 6(2) ×10−52315(85) Td=1864(233) K
+0.71(0.25) 1 .02(20)×10−42215(49) Aλ= 0.16(7)
+1.65(0.25) <1.7×10−3(3σ) Td(A= 0)=2010(144) K
+2.15(0.32) 1 .6(9)×10−31914(292)
+3.6(0.75) 3 .55(20)×10−31798(40)
+4.5(1.0)e4.75(19)×10−31791(33)
+4.5(1.0) 5 .10(42)×10−3
+8.0(2.9) 4 .1(1.1)×10−3
+8.0(2.9)e4.09(80)×10−31318(143)
+Gl 436b3934(41) 3.6(0.75) 4 .1(3)×10−41145(23) Td=1082(38) K
+4.5(1.0) <1.0×10−4(3σ)
+5.8(1.4) 3 .3(1.4)×10−4797(106)
+8.0(2.9)e4.52(27)×10−4737(17)
+8.0(2.9) 5 .7(8)×10−4
+8.0(2.9) 5 .4(7)×10−4
+16(5) 1 .40(27)×10−3963(126)
+24(9) 1 .75(41)×10−31016(182)
+HAT-P-1b41666(38) 3.6(0.75) 8 .0(8)×10−41420(47) Td=1439(59) K
+4.5(1.0) 1 .35(22)×10−31507(100)
+5.8(1.4) 2 .03(31)×10−31626(128)
+8.0(2.9) 2 .38(40)×10−31564(151)
+HAT-P-7b52943(95) 0.65(0.4) 1 .30(11)×10−43037(35) 1 .22(16)×10−4Td=2086(156) K
+3.6(0.75) 9 .8(1.7)×10−42063(152) Aλ= 0.58(5)
+4.5(1.0) 1 .59(22)×10−32378(179) Td(A= 0)=2830(86) K
+5.8(1.4) 2 .45(31)×10−32851(235)
+8.0(2.9) 2 .25(52)×10−32512(403)
+HD 80606b61799(50) 8.0(2.9) 1 .36(18)×10−31137(73) Td=1137(113) K
+HD 149026b71871(17) 8.0(2.9)e3.7(0.8)×10−4976(276) 2 .3(7)×10−4Td=1571(231) K
+8.0(2.9) 8 .4(1.1)×10−4Tn=976(286) K
+HD 189733b81537(16) 2.15(32) <4.0×10−4(1σ) Td=1605(52) K
+3.6(0.75) 2 .56(14)×10−31639(34) Tn=1107(132) K
+4.5(1.0) 2 .14(20)×10−31318(45)
+5.8(1.4) 3 .10(34)×10−31368(69)
+8.0(2.9) 3 .381(55)×10−3
+8.0(2.9) 3 .91(22)×10−31.2(2)×10−3
+8.0(2.9)e3.440(36)×10−31259(7) 1 .2(4)×10−3
+16(5) 5 .51(30)×10−31338(52)
+24(9) 5 .98(38)×10−3
+24(9)e5.36(27)×10−31202(46) 1 .3(3)×10−3
+HD 209458b91754(15) 0.5(0.3) 7(9) ×10−62368(156) Td=1486(53) K
+2.15(0.32) <3×10−4(1σ) Aλ= 0.09(7)
+3.6(0.75) 9 .4(9)×10−41446(45) Td(A= 0)=2031(128) K
+4.5(1.0) 2 .13(15)×10−31757(57) Tn=1476(304) K
+5.8(1.4) 3 .01(43)×10−31890(149)
+8.0(2.9) 2 .40(26)×10−31480(94) <1.5×10−3(2σ)
+24(9) 2 .60(44)×10−31131(143)
+OGLE-TR-56b102874(84) 0.90(0.15) 3 .63(91)×10−42696(116) Td=2696(236) K
+OGLE-TR-113b111716(33) 2.15(0.32) 1 .7(5)×10−31918(164) Td=1918(219) K
+TrES-1b121464(16) 3.6(0.75) <1.5×10−3(1σ) Td=998(67) K
+4.5(1.0) 6 .6(1.3)×10−4972(56)
+8.0(2.9) 2 .25(36)×10−31152(94)
+TrES-2b131917(21) 0.65(0.4) 1 .14(78)×10−52020(132) Td=1623(76) K
+2.15(0.32) 6 .2(1.2)×10−41655(80) Aλ= 0.06(3)
+3.6(0.75) 1 .27(21)×10−31490(84) Td(A= 0) = 1751(80) K
+4.5(1.0) 2 .30(24)×10−31652(74)
+5.8(1.4) 1 .99(54)×10−31373(177)
+8.0(2.9) 3 .59(60)×10−31659(163)
+TrES-3b142093(32) 0.7(0.3) <6.2×10−4(1σ) Td=1761(66) K
+1.25(0.16) <5.1×10−4(3σ)
+2.15(0.32) 2 .41(43)×10−3
+2.15(0.32)e1.33(17)×10−31770(58)
+3.6(0.75) 3 .46(35)×10−31818(73)12 Cowan & Agol
+TABLE 1
+Secondary Eclipses & Phase Variations of Exoplanets
+4.5(1.0) 3 .72(54)×10−31649(107)
+5.8(1.4) 4 .49(97)×10−31621(173)
+8.0(2.9) 4 .75(46)×10−31480(82)
+TrES-4b152250(37) 3.6(0.75) 1 .37(11)×10−31889(63) Td=1891(81) K
+4.5(1.0) 1 .48(16)×10−31727(83)
+5.8(1.4) 2 .61(59)×10−32112(283)
+8.0(2.9) 3 .18(44)×10−32168(197)
+WASP-1b162347(35) 3.6(0.75) 1 .17(16)×10−31678(87) Td=1719(89) K
+4.5(1.0) 2 .12(21)×10−31923(91)
+5.8(1.4) 2 .82(60)×10−32042(253)
+8.0(2.9) 4 .70(46)×10−32587(176)
+WASP-2b171661(69) 3.6(0.75) 8 .3(3.5)×10−41264(164) Td=1280(121) K
+4.5(1.0) 1 .69(17)×10−31380(53)
+5.8(1.4) 1 .92(77)×10−31299(232)
+8.0(2.9) 2 .85(59)×10−31372(154)
+WASP-4b182163(60) 3.6(0.75) 3 .19(31)×10−32156(97) Td=2146(140) K
+4.5(1.0) 3 .43(27)×10−31971(75)
+WASP-12b193213(119) 0.9(0.15) 8 .2(1.5)×10−43002(104) Td=2939(98) K
+1.25(0.16) 1 .31(28)×10−32894(149)
+1.65(0.25) 1 .76(18)×10−32823(88)
+2.15(0.32) 3 .09(13)×10−33018(51)
+3.6(0.75) 3 .79(13)×10−32704(49)
+4.5(1.0) 3 .82(19)×10−32486(68)
+5.8(1.4) 6 .29(52)×10−33167(179)
+8.0(2.9) 6 .36(67)×10−32996(229)
+WASP-18b203070(50) 3.6(0.75) 3 .1(2)×10−33000(107) Td=2998(138) K
+4.5(1.0) 3 .8(3)×10−33128(150)
+5.8(1.4) 4 .1(2)×10−33095(103)
+8.0(2.9) 4 .3(3)×10−32991(153)
+WASP-19b212581(49) 1.65(0.25) 2 .59(45)×10−32677(135) Td=2677(244) K
+XO-1b221526(24) 3.6(0.75) 8 .6(7)×10−41300(32) Td=1306(47) K
+4.5(1.0) 1 .22(9)×10−31265(34)
+5.8(1.4) 2 .61(31)×10−31546(89)
+8.0(2.9) 2 .10(29)×10−31211(87)
+XO-2231685(33) 3.6(0.75) 8 .1(1.7)×10−41447(102) Td=1431(98) K
+4.5(1.0) 9 .8(2.0)×10−41341(105)
+5.8(1.4) 1 .67(36)×10−31497(155)
+8.0(2.9) 1 .33(49)×10−31179(219)
+XO-3241982(82) 3.6(0.75) 1 .01(4)×10−31875(30) Td=1871(63) K
+4.5(1.0) 1 .43(6)×10−31965(40)
+5.8(1.4) 1 .34(49)×10−31716(330)
+8.0(2.9) 1 .50(36)×10−31625(236)
+aThe planet’s expected day-side effective temperature in the absence of reflection or recirculation ( AB= 0,ε= 0). The 1 σuncertainty is shown
+in parenthese.
+bThe bandwidth is shown in parenthese.
+cEclipse depths and phase amplitudes are unitless, since the y are measured relative to stellar flux.
+dTdandTndenote the day-side and night-side effective temperatures o f the planet, as estimated from thermal secondary eclipse de pths and
+thermal phase variations, respectively. The estimated 1 σuncertainties are shown in parentheses. The default day-si de temperature is computed
+using only observations at λ >0.8µm. Eclipse measurements at shorter wavelengths may then be u sed to estimate the planet’s albedo at those
+wavelengths, Aλ. Note that this is a spherical albedo; the geometric albedo i s given by Ag=2
+3Aλ. If —on the other hand— AB= 0 is assumed,
+then all the day-side flux is thermal, regardless of waveband , yielding the second Tdestimate.
+eWhen multiple measurements of an eclipse depth have been pub lished in a given waveband, we use the most recent observatio n. In all cases
+these observations are either explicitly agree with their o lder counterpart, or agree with the re-analyzed older data.
+1Snellen et al. (2009); Alonso et al. (2009b); Gillon et al. (2 009); Rogers et al. (2009); Deming et al. (2010),2Alonso et al. (2009a); Snellen et al.
+(2010); Gillon et al. (2010); Alonso et al. (2010); Deming et al. (2010),3Deming et al. (2007); Demory et al. (2007); Stevenson et al. ( 2010);
+Knutson et al. in prep.,4Todorov et al. (2010),5Borucki et al. (2009); Christiansen et al. (2010),6Laughlin et al. (2009),7Knutson et al.
+(2009b),8Deming et al. (2006); Knutson et al. (2007a); Barnes et al. (2 007); Charbonneau et al. (2008); Knutson et al. (2009c); Ago l et al.
+(2010),9Richardson et al. (2003); Deming et al. (2005); Cowan et al. ( 2007); Rowe et al. (2008); Knutson et al. (2008),10Sing & L´ opez-Morales
+(2009),11Snellen & Covino (2007),12Charbonneau et al. (2005); Knutson et al. (2007b),13O’Donovan et al. (2010); Croll et al. (2010a);
+Kipping & Bakos (2010b),14Fressin et al. (2010); Croll et al. (2010b); Christiansen et al. (2010b),15Knutson et al. (2009a),16,17Wheatley et al.
+(2010),18Beerer et al. (2010),19L´ opez-Morales et al. (2010); Campo et al. (2010); Croll et a l. (2010c),20Nymeyer et al. (2010),21Anderson et al.
+(2010),22Machalek et al. (2008),23Machalek et al. (2009),24Machalek et al. (2010)

1001.0013.txt

@@ -0,0 +1,1084 @@
+arXiv:1001.0013v2  [astro-ph.CO]  8 Jan 2010Astronomy& Astrophysics manuscriptno.akari˙LF˙aa˙v7 c∝circlecopyrtESO 2018
+October30,2018
+EvolutionofInfraredLuminosityfunctionsofGalaxiesint he
+AKARINEP-Deepfield
+Revealing thecosmic star formationhistory hidden by dust⋆,⋆⋆
+Tomotsugu Goto1,2,⋆⋆⋆,T.Takagi3,H.Matsuhara3,T.T.Takeuchi4,C.Pearson5,6,7, T.Wada3,T.Nakagawa3,O.Ilbert8,
+E.LeFloc’h9,S.Oyabu3, Y.Ohyama10,M.Malkan11, H.M.Lee12, M.G.Lee12,H.Inami3,13,14, N.Hwang2, H.Hanami15,
+M.Im12, K.Imai16,T.Ishigaki17,S.Serjeant7,and H.Shim12
+1Institute for Astronomy, University of Hawaii,2680 Woodla wnDrive, Honolulu, HI,96822, USA
+e-mail:[email protected]
+2National Astronomical Observatory, 2-21-1 Osawa,Mitaka, Tokyo, 181-8588,Japan
+3Institute of Space and Astronautical Science, JapanAerosp ace Exploration Agency, Sagamihara,Kanagawa 229-8510
+4Institute for Advanced Research, Nagoya University, Furo- cho, Chikusa-ku, Nagoya 464-8601
+5Rutherford Appleton Laboratory, Chilton, Didcot,Oxfords hire OX110QX, UK
+6Department of Physics,Universityof Lethbridge, 4401 Univ ersity Drive,Lethbridge, AlbertaT1J 1B1, Canada
+7Astrophysics Group, Department of Physics, The OpenUniver sity, MiltonKeynes, MK76AA, UK
+8Laboratoire d’Astrophysique de Marseille, BP8,Traverse d u Siphon, 13376 Marseille Cedex 12, France
+9CEA-Saclay,Service d’Astrophysique, France
+10Academia Sinica,Institute of Astronomyand Astrophysics, Taiwan
+11Department of Physicsand Astronomy, UCLA,Los Angeles, CA, 90095-1547 USA
+12Department of Physics& Astronomy, FPRD,Seoul National Uni versity, Shillim-Dong,Kwanak-Gu, Seoul 151-742, Korea
+13Spitzer Science Center,California Institute ofTechnolog y, Pasadena, CA91125
+14Department of Astronomical Science,The Graduate Universi tyfor Advanced Studies
+15Physics Section,Facultyof Humanities and SocialSciences , Iwate University, Morioka, 020-8550
+16TOMER&D Inc. Kawasaki, Kanagawa 2130012, Japan
+17Asahikawa National College of Technology, 2-1-6 2-joShunk ohdai, Asahikawa-shi, Hokkaido 071-8142
+Received September 15, 2009; accepted December 16, 2009
+ABSTRACT
+Aims.Dust-obscured star-formation becomes much more important with increasing intensity, and increasing redshift. We aim to
+reveal cosmic star-formationhistoryobscured bydust usin g deep infraredobservation withthe AKARI.
+Methods. We construct restframe 8 µm, 12µm, and total infrared (TIR) luminosity functions (LFs) at 0.15< z <2.2using 4128
+infraredsources intheAKARINEP-Deepfield.Acontinuous fil tercoverage inthemid-IRwavelength(2.4,3.2,4.1,7,9,11 , 15,18,
+and 24µm) by the AKARI satellite allows us to estimate restframe 8 µm and 12 µm luminosities without using a large extrapolation
+based ona SEDfit,which was the largestuncertainty inprevio us work.
+Results. Wehavefoundthatall8 µm(0.38< z <2.2),12µm(0.15< z <1.16),andTIRLFs( 0.2< z <1.6),showacontinuous
+andstrongevolutiontowardhigher redshift.Intermsofcos micinfraredluminositydensity( ΩIR),whichwasobtainedbyintegrating
+analytic fits to the LFs,we found a good agreement withprevio us work at z <1.2. We found the ΩIRevolves as ∝(1+z)4.4±1.0.
+Whenweseparatecontributionsto ΩIRbyLIRGsandULIRGs,wefoundmoreIRluminoussourcesareinc reasinglymoreimportant
+at higher redshift. Wefound that the ULIRG(LIRG)contribut ionincreases bya factor of 10(1.8) from z=0.35 toz=1.4.
+Keywords. galaxies: evolution, galaxies:interactions, galaxies:s tarburst, galaxies:peculiar, galaxies:formation
+1. Introduction
+Studies of the extragalactic background suggest at least ha lf
+the luminous energy generated by stars has been reprocessed
+into the infrared(IR) by dust (Lagacheetal., 1999; Pugetet al.,
+1996; Franceschini,Rodighiero,&Vaccari, 2008), suggest ing
+that dust-obscured star formation was much more important a t
+higherredshiftsthantoday.
+⋆This research is based on the observations with AKARI, a JAXA
+project withthe participationof ESA.
+⋆⋆Based on data collected at Subaru Telescope, which is operat ed by
+the National Astronomical Observatory ofJapan.
+⋆⋆⋆JSPSSPDfellowBell etal. (2005) estimate that IR luminosity density is 7
+times higher than the UV luminosity density at z ∼0.7 than lo-
+cally. Takeuchi,Buat, &Burgarella (2005) reported that UV -to-
+IRluminositydensityratio, ρL(UV)/ρL(dust),evolvesfrom3.75
+(z=0) to 15.1 by z=1.0 with a careful treatment of the sample
+selection effect, and that 70% of star formation activity is ob-
+scured by dust at 0.5 < z <1.2. Both works highlight the im-
+portance of probing cosmic star formation activity at high r ed-
+shift in the infrared bands. Several works found that most ex -
+tremestar-forming(SF) galaxies,whichareincreasinglyi mpor-
+tant at higher redshifts, are also more heavily obscured by d ust
+(Hopkinsetal., 2001; Sullivanet al., 2001; Buatet al.,200 7).2 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Despite the value of infrared observations, studies of
+infrared galaxies by the IRAS and the ISO were re-
+stricted to bright sources due to the limited sensitiv-
+ities (Saundersetal., 1990; Rowan-Robinsonet al., 1997;
+Floreset al., 1999; Serjeantet al., 2004; Takeuchiet al., 2 006;
+Takeuchi,Yoshikawa,&Ishii, 2003), until the recent launc h of
+theSpitzer andtheAKARI satellites. Theirenormousimprov ed
+sensitivitieshaverevolutionizedthefield.Forexample:
+Le Floc’het al. (2005) analyzed the evolution of the total
+and 15µm IR luminosity functions (LFs) at 0< z <1based
+on the the Spitzer MIPS 24 µm data (>83µJy andR <24) in
+the CDF-S, and found a positive evolution in both luminosity
+and density, suggesting increasing importance of the LIRG a nd
+ULIRGpopulationsathigherredshifts.
+P´ erez-Gonz´ alezetal. (2005) used MIPS 24 µm observations
+oftheCDF-SandHDF-N( >83µJy)tofindthatthat L∗steadily
+increasesbyanorderofmagnitudeto z∼2,suggestingthatthe
+luminosity evolution is stronger than the density evolutio n. The
+ΩTIRscalesas(1+z)4.0±0.2fromz=0to0.8.
+Babbedgeet al. (2006) constructed LFs at 3.6, 4.5, 5.8, 8
+and 24µm over0< z < 2using the data from the Spitzer
+Wide-areaInfraredExtragalactic(SWIRE)Surveyin a 6.5de g2
+(S24µm>230µJy). They found a clear luminosity evolu-
+tion in all the bands, but the evolution is more pronounced at
+longer wavelength; extrapolatingfrom 24 µm, they inferred that
+ΩTIR∝(1+z)4.5. They constructed separate LFs for three dif-
+ferentgalaxySED (spectral energydistribution)typesand Type
+1 AGN, finding that starburst and late-type galaxies showed
+strongerevolution.Comparisonof3.6and4.5 µmLFswithsemi-
+analytic and spectrophotometricmodelssuggested that the IMF
+is skewed towards higher mass star formation in more intense
+starbursts.
+Caputi etal.(2007)estimatedrestframe8 µmLFsofgalaxies
+over 0.08deg2in the GOODS fields based on Spitzer 24 µm (>
+80µJy) atz=1 and 2. They found a continuousand strong posi-
+tiveluminosityevolutionfrom z=0toz=1,andto z=2.However,
+theyalsofoundthatthenumberdensityofstar-forminggala xies
+withνL8µm
+ν>1010.5L⊙(AGNs are excluded.) increases by a
+factor of 20 from z=0 to 1, but decreases by half from z=1 to 2
+mainlyduetothe decreaseofLIRGs.
+Magnelliet al. (2009) investigated restframe 15 µm, 35µm
+and total infrared (TIR) LFs using deep 70 µm observations
+(∼300µJy) in the Spitzer GOODS and FIDEL (Far Infrared
+Deep Extragalactic Legacy Survey) fields (0.22 deg2in total)
+atz <1.3. They stacked 70 µm flux at the positions of 24 µm
+sources when sources are not detected in 70 µm. They found no
+changeintheshapeoftheLFs,butfoundapureluminosityevo -
+lutionproportionalto(1+z)3.6±0.5,andthatLIRGsandULIRGs
+have increased by a factor of 40 and 100 in number density by
+z∼1.
+Also, see Daiet al. (2009) for 3.6-8.0 µm LFs based on the
+IRACphotometryintheNOAODeepWide-FieldSurveyBootes
+field.
+However, most of the Spitzer work relied on a large
+extrapolation from 24 µm flux to estimate the 8, 12 µm or
+TIR luminosity. Consequently, Spitzer results heavily de-
+pended on the assumed IR SED library (Dale&Helou, 2002;
+Lagache,Dole,&Puget, 2003; Chary& Elbaz, 2001). Indeed
+many authors pointed out that the largest uncertainty in the se
+previous IR LFs came from SED models, especially when one
+computesTIRluminositysolelyfromobserved24 µmflux(e.g.,
+see Fig.5ofCaputiet al.,2007).
+AKARI, the first Japanese IR dedicated satellite, has con-
+tinuous filter coverage across the mid-IR wavelengths, thus , al-Fig.1. Photometric redshift estimates with LePhare
+(Ilbertet al., 2006; Arnoutset al., 2007; Ilbertet al., 200 9)
+for spectroscopically observed galaxies with Keck/DEIMOS
+(Takagi et al. in prep.). Red squares show objects where AGN
+templates were better fit. Errors of the photoz is∆z
+1+z=0.036 for
+z≤0.8, but becomes worse at z >0.8to be∆z
+1+z=0.10 due
+mainlyto therelativelyshallownear-IRdata.
+lows us to estimate MIR (mid-infrared)-luminositywithout us-
+ing a large k-correction based on the SED models, eliminating
+thelargestuncertaintyinpreviouswork.Bytakingadvanta geof
+this, we present the restframe 8, 12 µm and TIR LFs using the
+AKARI NEP-Deepdatainthiswork.
+Restframe 8 µm luminosity in particular is of primary rele-
+vance for star-forming galaxies, as it includes polycyclic aro-
+matic hydrocarbon (PAH) emission. PAH molecules charac-
+terize star-forming regions (Desert,Boulanger,&Puget, 1 990),
+and the associated emission lines between 3.3 and 17 µm dom-
+inate the SED of star-forming galaxies with a main bump lo-
+cated around 7.7 µm. Restframe 8 µm luminosities have been
+confirmed to be good indicators of knots of star formation
+(Calzetti etal., 2005) and of the overall star formation act ivity
+of star forming galaxies (Wuet al., 2005). At z=0.375, 0.875,
+1.25 and 2, the restframe 8 µm is covered by the AKARI S11,
+L15,L18WandL24filters. We present the restframe 8 µm LFs
+at theseredshiftsatSection3.1.
+Restframe 12 µm luminosity functions have also been
+studied extensively (Rush,Malkan,& Spinoglio, 1993;
+P´ erez-Gonz´ alezet al., 2005). At z=0.25, 0.5 and 1, the
+restframe12 µmiscoveredbytheAKARI L15,L18WandL24
+filters. We present the restframe 12 µm LFs at these redshifts in
+Section3.3.
+We also estimate TIR LFs through the SED fit using all
+the mid-IR bands of the AKARI. The results are presented in
+Section3.5.
+Unless otherwise stated, we adopt a cosmology with
+(h,Ωm,ΩΛ) = (0.7,0.3,0.7)(Komatsuet al., 2008).
+2. Data & Analysis
+2.1. Multi-wavelength data inthe AKARI NEP Deepfield
+AKARI, the Japanese infraredsatellite (Murakamiet al., 20 07),
+performed deep imaging in the North Ecliptic Region (NEP)
+from 2-24 µm, with 14 pointings in each field over 0.4
+deg2(Matsuharaet al., 2006, 2007; Wada et al., 2008). DueGotoet al.:InfraredLuminosityfunctions withthe AKARI 3
+Fig.2.Photometricredshiftdistribution.
+Fig.3.8µmluminositydistributionsofsamplesusedtocompute
+restframe 8 µm LFs. From low redshift, 533, 466, 236 and 59
+galaxiesarein eachredshiftbin.
+to the solar synchronous orbit of the AKARI, the NEP
+is the only AKARI field with very deep imaging at these
+wavelengths. The 5 σsensitivity in the AKARI IR filters
+(N2,N3,N4,S7,S9W,S11,L15,L18WandL24) are 14.2,
+11.0, 8.0, 48, 58, 71, 117, 121 and 275 µJy (Wada etal., 2008).
+These filters provide us with a unique continuous wavelength
+coverage at 2-24 µm, where there is a gap between the Spitzer
+IRAC and MIPS, and the ISO LW2andLW3. Please consult
+Wada etal. (2007, 2008); Pearsonet al. (2009a,b) for data ve ri-
+ficationandcompletenessestimateatthesefluxes.ThePSFsi zes
+are 4.4, 5.1, and 5.4” in 2−4,7−11,15−24µm bands. The
+depths of near-IR bands are limited by source confusion, but
+thoseofmid-IRbandsarebyskynoise.In analyzingthese observations,we first combinedthe three
+images of the MIR channels, i.e. MIR-S( S7,S9W, andS11)
+and MIR-L( L15,L18WandL24), in order to obtain two high-
+quality images. In the resulting MIR-S and MIR-L images, the
+residual sky has been reduced significantly, which helps to o b-
+tain more reliable source catalogues. For both the MIR-S and
+MIR-Lchannels,we use SExtractorforthecombinedimagesto
+determineinitialsourcepositions.
+We follow Takagietal. (2007) procedures for photometry
+and band-merging of IRC sources. But this time, to maximize
+the number of MIR sources, we made two IRC band-merged
+catalogues based on the combined MIR-S and MIR-L images,
+andthenconcatenatedthese catalogues,eliminatingdupli cates.
+Intheband-mergingprocess,thesourcecentroidineachIRC
+image has beendetermined,starting fromthe sourcepositio n in
+the combined images as the initial guess. If the centroid det er-
+mined in this way is shifted from the original position by >3′′,
+we reject such a source as the counterpart. We note that this
+band-mergingmethodisusedonlyforIRCbands.
+We comparedraw numbercountswith previouswork based
+on the same data but with different source extraction method s
+(Wadaet al., 2008; Pearsonet al., 2009a,b) and found a good
+agreement.
+A subregion of the NEP-Deep field was observed in the
+BVRi′z′-bands with the Subaru telescope (Imaiet al., 2007;
+Wada etal., 2008), reaching limiting magnitudes of zAB=26
+in one field of view of the Suprime-Cam.We restrict our analy-
+sis to the data in this Suprime-Cam field (0.25 deg2), where we
+have enough UV-opical-NIR coverage to estimate good photo-
+metricredshifts.The u′-bandphotometryinthisareaisprovided
+by the CFHT (Serjeant et al. in prep.). The same field was also
+observed with the KPNO2m/FLAMINGOs in JandKsto the
+depth ofKsVega<20(Imaiet al., 2007). GALEX coveredthe
+entirefieldtodepthsof FUV <25andNUV < 25(Malkanet
+al.in prep.).
+In the Suprime-Cam field of the AKARI NEP-Deep field,
+there are a total of 4128 infrared sources down to ∼100µJy in
+theL18Wfilter. All magnitudesare given in AB system in this
+paper.
+For the optical identification of MIR sources, we adopt the
+likelihood ratio (LR) method (Sutherland&Saunders, 1992) .
+For the probability distribution functions of magnitude an d an-
+gular separation based on correct optical counterparts (an d for
+this purpose only), we use a subset of IRC sources, which are
+detected in all IRC bands. For this subset of 1100 all-band–
+detected sources, the optical counterparts are all visuall y in-
+spected and ambiguous cases are excluded. There are multipl e
+opticalcounterpartsfor35%ofMIRsourceswithin <3′′. Ifwe
+adoptedthenearestneighborapproachfortheopticalident ifica-
+tion,theopticalcounterpartsdiffersfromthat oftheLRme thod
+for20%ofthesourceswith multipleopticalcounterparts.T hus,
+in total we estimate that less than 15% of MIR sources suffer
+fromseriousproblemsofopticalidentification.
+2.2. Photometric redshift estimation
+For these infrared sources, we have computed photomet-
+ric redshift using a publicly available code, LePhare1
+(Ilbertet al., 2006; Arnoutsetal., 2007; Ilbertet al., 200 9).
+The input magnitudes are FUV,NUV (GALEX), u(CFHT),
+B,V,R,i′,z′(Subaru), J,andK(KPNO2m).Wesummarizethe
+filtersusedinTable1.
+1http://www.cfht.hawaii.edu/∼arnouts/lephare.html4 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Table 1.Summaryoffiltersused.
+Estimate Redshift Filter
+Photoz0.15<z<2.2FUV,NUV ,u,B,V,R,i′,z,J, andK
+8µm LF 0.38 <z<0.58 S11(11 µm)
+8µm LF 0.65 <z<0.90 L15(15 µm)
+8µm LF 1.1 <z<1.4 L18W (18 µm)
+8µm LF 1.8 <z<2.2 L24(24 µm)
+12µm LF 0.15 <z<0.35 L15(15 µm)
+12µm LF 0.38 <z<0.62 L18W (18 µm)
+12µm LF 0.84 <z<1.16 L24(24 µm)
+TIRLF 0.2 <z<0.5S7,S9W,S11,L15,L18WandL24
+TIRLF 0.5 <z<0.8S7,S9W,S11,L15,L18WandL24
+TIRLF 0.8 <z<1.2S7,S9W,S11,L15,L18WandL24
+TIRLF 1.2 <z<1.6S7,S9W,S11,L15,L18WandL24
+Among various templates and fitting parameters we tried,
+we found the best results were obtained with the following: w e
+used modified CWW (Coleman,Wu,& Weedman, 1980) and
+QSO templates.TheseCWW templatesareinterpolatedandad-
+justed to better match VVDS spectra (Arnoutsetal., 2007). W e
+included strong emission lines in computing colors. We used
+the Calzetti extinction law. More details in training LePhare
+isgiveninIlbertet al.(2006).
+The resulting photometric redshift estimates agree reason -
+ably well with 293 galaxies ( R <24) with spectroscopic red-
+shifts taken with Keck/DEIMOS in the NEP field (Takagi et al.
+inprep.).Themeasurederrorsonthephoto- zare∆z
+1+z=0.036for
+z≤0.8and∆z
+1+z=0.10 for z >0.8. The∆z
+1+zbecomes signifi-
+cantly larger at z >0.8, where we suffer from relative shallow-
+ness of our near-IR data. The rate of catastrophic failures i s 4%
+(∆z
+1+z>0.2)amongthespectroscopicsample.
+In Fig.1, we compare spectroscopic redshifts from
+Keck/DEIMOS (Takagi et al.) and our photometric red-
+shift estimation. Those SEDs which are better fit with a QSO
+template are shown as red triangles. We remove those red
+triangle objects ( ∼2% of the sample) from the LFs presented
+below. We caution that this can only remove extreme type-1
+AGNs, and thus, fainter, type-2 AGN that could be removedby
+X-raysoropticalspectroscopystill remainin thesample.
+Fig.2showsthedistributionofphotometricredshift.Thed is-
+tributionhasseveralpeaks,whichcorrespondstogalaxycl usters
+in the field (Gotoetal., 2008). We have 12% of sources that do
+nothaveagoodSEDfit toobtainareliablephotometricredshi ft
+estimation.Weapplythisphoto- zcompletenesscorrectiontothe
+LFs we obtain.Readers are referredto Negrelloet atal. (200 9),
+who estimated photometricredshifts using only the AKARI fil -
+terstoobtain10%accuracy.
+2.3. The1/ Vmaxmethod
+WecomputeLFsusingthe1/ Vmaxmethod(Schmidt,1968).The
+advantage of the 1/ Vmaxmethod is that it allows us to compute
+a LF directly from data, with no parameter dependence or an
+assumed model. A drawback is that it assumes a homogeneous
+galaxy distribution, and is thus vulnerable to local over-/ under-
+densities(Takeuchi,Yoshikawa,&Ishii,2000).
+A comoving volume associated with any source of a given
+luminosity is defined as Vmax=Vzmax−Vzmin, wherezmin
+is the lower limit of the redshift bin and zmaxis the maximum
+redshiftat whichthe objectcouldbe seen giventhe fluxlimit of
+the survey, with a maximum value corresponding to the upperredshiftoftheredshiftbin.Moreprecisely,
+zmax= min(z maxof the bin ,zmaxfromthe flux limit) (1)
+We usedtheSED templates(Lagache,Dole,&Puget, 2003) for
+k-corrections to obtain the maximum observable redshift fro m
+thefluxlimit.
+Foreachluminositybinthen,theLFisderivedas
+φ=1
+∆L/summationdisplay
+i1
+Vmax,iwi, (2)
+whereVmaxis a comoving volume over which the ith galaxy
+couldbeobserved, ∆Listhesizeoftheluminositybin(0.2dex),
+andwiis the completeness correction factor of the ith galaxy.
+WeusecompletenesscorrectionmeasuredbyWadaet al.(2008 )
+for11and24 µmandPearsonet al.(2009a,b)for15and18 µm.
+Thiscorrectionis25%atmaximum,sincewe onlyusethesam-
+plewherethecompletenessisgreaterthan80%.
+2.4. Monte Carlo simulation
+Uncertainties of the LF values stem from various factors suc h
+as fluctuations in the numberof sources in each luminosity bi n,
+the photometric redshift uncertainties, the k-correction uncer-
+tainties, and the flux errors. To compute these errors we per-
+formedMonteCarlosimulationsbycreating1000simulatedc at-
+alogs,whereeach catalogcontainsthesame numberof source s,
+but we assign each source a new redshift following a Gaussian
+distribution centered at the photometric redshift with the mea-
+sured dispersion of ∆z/(1 +z) =0.036 for z≤0.8and
+∆z/(1+z) =0.10forz >0.8(Fig.1). The flux of each source
+is also allowed to vary accordingto the measuredflux error fo l-
+lowingaGaussiandistribution.For8 µmand12µmLFs,wecan
+ignore the errors due to the k-correction thanks to the AKARI
+MIR filter coverage. For TIR LFs, we have added 0.05 dex of
+error for uncertaintyin the SED fitting following the discus sion
+in Magnelliet al. (2009). We did not consider the uncertaint y
+on the cosmic variance here since the AKARI NEP field cov-
+ers a large volume and has comparable number counts to other
+generalfields(Imaiet al.,2007,2008).Eachredshiftbinwe use
+covers∼106Mpc3of volume. See Matsuharaetal. (2006) for
+morediscussion on the cosmic variancein the NEP field. These
+estimated errors are added to the Poisson errors in each LF bi n
+inquadrature.
+3. Results
+3.1. 8µm LF
+Monochromatic 8 µm luminosity ( L8µm) is known to cor-
+relate well with the TIR luminosity (Babbedgeet al., 2006;
+Huanget al.,2007),especiallyforstar-forminggalaxiesb ecause
+the rest-frame 8 µm flux are dominated by prominent PAH fea-
+turessuchasat 6.2,7.7and8.6 µm.
+Since the AKARI has continuous coverage in the mid-IR
+wavelengthrange,therestframe8 µmluminositycanbeobtained
+without a large uncertainty in k-correction at a corresponding
+redshift and filter. For example, at z=0.375, restframe 8 µm is
+redshiftedinto S11filter. Similarly, L15,L18WandL24cover
+restframe 8 µm atz=0.875, 1.25 and 2. This continuous filter
+coverageisanadvantagetoAKARIdata.OftenSEDmodelsare
+used to extrapolate from Spitzer 24 µm flux in previous work,Gotoet al.:InfraredLuminosityfunctions withthe AKARI 5
+producingasourceofthe largestuncertainty.We summarise fil-
+tersusedinTable1.
+To obtain restframe 8 µm LF, we applied a flux limit
+of F(S11) <70.9, F(L15) <117, F(L18W) <121.4, and
+F(L24)<275.8µJy atz=0.38-0.58, z=0.65-0.90, z=1.1-1.4
+andz=1.8-2.2,respectively.Thesearethe5 σlimitsmeasuredin
+Wada etal. (2008). We exclude those galaxies whose SEDs are
+betterfit withQSO templates( §2).
+We use the completeness curve presented in Wada et al.
+(2008) and Pearsonet al. (2009a,b) to correct for the incom-
+pleteness of the detection. However, this correction is 25% at
+maximumsincethesampleis80%completeatthe5 σlimit.Our
+mainconclusionsarenotaffectedbythisincompletenessco rrec-
+tion. To compensatefor the increasing uncertaintyin incre asing
+z, we use redshift binsize of 0.38 < z <0.58, 0.65 < z <0.90,
+1.1< z <1.4,and 1.8 < z <2.2.We show the L8µmdistribution
+in each redshift rangein Fig.3. Within each redshift bin, we use
+1/Vmaxmethodto compensateforthefluxlimit ineachfilter.
+We show the computed restframe 8 µm LF in Fig.4. Arrows
+show the 8 µm luminosity correspondingto the flux limit at the
+central redshift in each redshift bin. Errorbarson each poi nt are
+basedontheMonteCarlosimulation( §2.3).
+For a comparison, as the green dot-dashed line, we also
+show the 8 µm LF of star-forming galaxies at 0< z < 0.3
+by Huanget al. (2007), using the 1/ Vmaxmethod applied to the
+IRAC 8µm GTO data. Compared to the local LF, our 8 µm LFs
+showstrongevolutionin luminosity.Intherangeof 0.48< z <
+2,L∗
+8µmevolvesas ∝(1+z)1.6±0.2. Detailedcomparisonwith
+theliteraturewill bepresentedin §4.
+3.2. Bolometric IR luminosity density basedonthe 8 µm
+LF
+Constraining the star formation history of galaxies as a fun c-
+tion of redshift is a key to understanding galaxy formation i n
+the Universe. One of the primary purposes in computing IR
+LFs is to estimate the IR luminosity density, which in turn is a
+goodestimatorof thedust hiddencosmic star formationdens ity
+(Kennicutt, 1998). Since dust obscurationis more importan t for
+more actively star forming galaxies at higher redshift, and such
+star formationcannotbeobservedinUV light,it is importan tto
+obtainIR-basedestimateinordertofullyunderstandtheco smic
+star formationhistoryoftheUniverse.
+Weestimatethetotalinfraredluminositydensitybyintegr at-
+ingtheLFweightedbytheluminosity.First, weneedtoconve rt
+L8µmto the bolometric infrared luminosity. The bolometric IR
+luminosity of a galaxy is produced by the thermal emission of
+its interstellarmatter. Instar-forminggalaxies,the UV r adiation
+producedbyyoungstarsheatstheinterstellardust,andthe repro-
+cessed lightisemittedin theIR. Forthisreason,in star-fo rming
+galaxies,thebolometricIRluminosityisagoodestimatoro fthe
+current SFR (star formation rate) of the galaxy. Bavouzetet al.
+(2008) showed a strong correlation between L8µmand total in-
+frared luminosity ( LTIR) for 372 local star-forming galaxies.
+TheconversiongivenbyBavouzetet al.(2008)is:
+LTIR= 377.9×(νLν)0.83
+rest8µm(±37%) (3)
+Caputi etal. (2007) further constrained the sample to lumi-
+nous, high S/N galaxies ( νL8µm
+ν>1010L⊙and S/N>3in all
+MIPS bands) in order to better match their sample, and derive d
+thefollowingequation.Fig.4.Restframe 8 µm LFs based on the AKARI NEP-Deep
+field. The blue diamons, purple triangles, red squares, and o r-
+ange crosses show the 8 µm LFs at 0.38< z <0.58,0.65<
+z <0.90,1.1< z <1.4, and1.8< z <2.2, respectively.
+AKARI’s MIR filters can observe restframe 8 µm at these red-
+shifts in a corresponding filter. Errorbars are from the Mont e
+Caro simulations ( §2.4). The dotted lines show analytical fits
+with a double-power law. Vertical arrows show the 8 µm lumi-
+nosity corresponding to the flux limit at the central redshif t in
+each redshift bin. Overplotted are Babbedgeet al. (2006) in the
+pink dash-dotted lines, Caputiet al. (2007) in the cyan dash -
+dotted lines, and Huanget al. (2007) in the green dash-dotte d
+lines.AGNsareexcludedfromthe sample( §2.2).
+LTIR= 1.91×(νLν)1.06
+rest8µm(±55%) (4)
+Since ours is also a sample of bright galaxies, we use this
+equation to convert L8µmtoLTIR. Because the conversion is
+based on local star-forming galaxies, it is a concern if it ho lds
+at higher redshift or not. Bavouzetet al. (2008) checked thi s by
+stacking 24 µm sources at 1.3< z <2.3in the GOODS fields
+to find the stacked sources are consistent with the local rela -
+tion. They concluded that equation (3) is valid to link L8µm
+andLTIRat1.3< z <2.3. Takagiet al. (2010) also show
+that local L7.7µmvsLTIRrelation holds true for IR galaxies
+at z∼1 (see their Fig.10). Popeetal. (2008) showed that z∼2
+sub-millimeter galaxies lie on the relation between LTIRand
+LPAH,7.7that has been established for local starburst galaxies.
+S70/S24ratios of 70 µm sources in Papovichet al. (2007) are
+also consistent with local SED templates. These results sug gest
+it isreasonabletouse equation(4) foroursample.
+The conversion, however, has been the largest source of er-
+rorinestimating LTIRfromL8µm.Bavouzetet al.(2008)them-
+selvesquote37%ofuncertainty,andthatCaputietal.(2007 )re-
+port 55% of dispersion around the relation. It should be kept in
+mind that the restframe 8µm is sensitive to the star-formation
+activity, but at the same time, it is where the SED models have
+strongest discrepancies due to the complicated PAH emissio n
+lines. A detailed comparison of different conversions is pr e-
+sented in Fig.12 of Caputiet al. (2007), who reported factor of
+∼5ofdifferencesamongvariousmodels.6 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Then the 8 µm LF is weighted by the LTIRand integrated
+to obtain TIR density. For integration, we first fit an ana-
+lytical function to the LFs. In the literature, IR LFs were
+fit better by a double-power law (Babbedgeet al., 2006) or
+a double-exponential (Saunderset al., 1990; Pozziet al., 2 004;
+Takeuchiet al., 2006; Le Floc’het al., 2005) than a Schechte r
+function, which declines too suddenlly at the high luminosi ty,
+underestimating the number of bright galaxies. In this work ,
+we fit the 8 µm LFs using a double-powerlaw (Babbedgeet al.,
+2006)asshownbelow.
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−α
+dL/L∗,(L < L∗) (5)
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−β
+dL/L∗,(L > L∗) (6)
+First, the double-powerlaw is fitted to the lowest redshift L F at
+0.38< z <0.58 to determine the normalization( Φ∗) and slopes
+(α,β).Forhigherredshiftswedonothaveenoughstatisticstosi -
+multaneouslyfit 4parameters( Φ∗,L∗,α,andβ).Therefore,we
+fixedtheslopesandnormalizationat the localvaluesandvar ied
+onlyL∗atforthehigher-redshiftLFs.Fixingthefaint-endslope
+isacommonprocedurewiththedepthofcurrentIRsatellites ur-
+veys (Babbedgeet al., 2006; Caputi etal., 2007). The strong er
+evolution in luminosity than in density found by previous wo rk
+(P´ erez-Gonz´ alezet al., 2005; LeFloc’het al., 2005) also justi-
+fies this parametrization. Best fit parameters are presented in
+Table2.Oncethebest-fitparametersarefound,weintegrate the
+doublepowerlawoutsidetheluminosityrangeinwhichwehav e
+data to obtain estimate of the total infrared luminosity den sity,
+ΩTIR.
+The resulting total luminosity density ( ΩIR) is shown in
+Fig.5 as a function of redshift. Errors are estimated by vary ing
+thefit within1 σofuncertaintyin LFs, thenerrorsin conversion
+fromL8µmtoLTIRare added. The latter is by far the larger
+source of uncertainty. Simply switching from equation (3) ( or-
+ange dashed line) to (4) (red solid line) produces a ∼50% dif-
+ference. Cyan dashed lines show results from LeFloc’het al.
+(2005) for a comparision. The lowest redshift point was cor-
+rectedfollowingMagnellietal. (2009).
+We also show the evolution of monochromatic 8 µm lumi-
+nosity (L8µm), which is obtained by integrating the fits, but
+without converting to LTIRin Fig.6. The Ω8µmevolves as
+∝(1+z)1.9±0.7.
+The SFR and LTIRare related by the following equation
+for a Salpeter IMF, φ(m)∝m−2.35between0.1−100M⊙
+(Kennicutt,1998).
+SFR(M⊙yr−1) = 1.72×10−10LTIR(L⊙) (7)
+The right ticks of Fig.5 shows the star formation density
+scale,convertedfrom ΩIRusingtheaboveequation.
+In Fig.5, ΩIRmonotonically increases toward higher z.
+Comparedwith z=0,ΩIRis∼10timeslargerat z=1.Theevolu-
+tionbetween z=0.5andz=1.2isalittleflatter,butthisisperhaps
+duetoamoreirregularshapeofLFsat0.65 < z <0.90,andthus,
+wedonotconsideritsignificant.Theresultsobtainedherea gree
+with previous work (e.g., Le Floc’het al., 2005) within the e r-
+rors. We compare the results with previous work in more detai l
+in§4.Fig.5.Evolution of TIR luminosity density computed by inte-
+grating the 8 µm LFs in Fig.4.The red solid lines use the con-
+version in equation (4). The orange dashed lines use equatio n
+(3).ResultsfromLeFloc’hetal.(2005)areshownwiththecy an
+dottedlines.
+Fig.6.Evolution of 8 µm IR luminosity density computed by
+integrating the 8 µm LFs in Fig.4. The lowest redshift point is
+fromHuanget al.(2007).
+3.3. 12µm LF
+In this subsection we estimate restframe 12 µm LFs based
+on the AKARI NEP-Deep data. 12 µm luminosity ( L12µm)
+has been well studied through ISO and IRAS, and known to
+correlate closely with TIR luminosity (Spinoglioetal., 19 95;
+P´ erez-Gonz´ alezet al.,2005).
+As was the case for the 8 µm LF, it is advantageous that
+AKARI’s continuous filters in the mid-IR allow us to estimate
+restframe 12 µm luminosity without much extrapolation based
+onSEDmodels.Gotoet al.:InfraredLuminosityfunctions withthe AKARI 7
+Table 2.Best fit parametersfor8,12 µmandTIRLFs
+Redshift λ L∗(L⊙)Φ∗(Mpc−3dex−1)α β
+0.38<z<0.58 8 µm (2.2+0.3
+−0.1)×1010(2.1+0.3
+−0.4)×10−31.75+0.01
+−0.013.5+0.2
+−0.4
+0.65<z<0.90 8 µm (2.8+0.1
+−0.1)×10102.1×10−31.75 3.5
+1.1<z<1.4 8 µm (3.3+0.2
+−0.2)×10102.1×10−31.75 3.5
+1.8<z<2.2 8 µm (8.2+1.2
+−1.8)×10102.1×10−31.75 3.5
+0.15<z<0.35 12 µm (6.8+0.1
+−0.1)×109(4.2+0.7
+−0.6)×10−31.20+0.01
+−0.022.9+0.4
+−0.2
+0.38<z<0.62 12 µm (11.7+0.3
+−0.5)×1094.2×10−31.20 2.9
+0.84<z<1.16 12 µm (14+2
+−3)×1094.2×10−31.20 2.9
+0.2<z<0.5 Total (1.2+0.1
+−0.2)×1011(5.6+1.5
+−0.2)×10−41.8+0.1
+−0.43.0+1.0
+−1.0
+0.5<z<0.8 Total (2.4+1.8
+−1.6)×10115.6×10−41.8 3.0
+0.8<z<1.2 Total (3.9+2.3
+−2.2)×10115.6×10−41.8 3.0
+1.2<z<1.6 Total (14+1
+−2)×10115.6×10−41.8 3.0
+Fig.7.12µm luminosity distributions of samples used to com-
+puterestframe12 µmLFs. Fromlowredshift,335,573,and213
+galaxiesarein eachredshiftbin.
+Targeted redshifts are z=0.25, 0.5 and 1 where L15,L18W
+andL24filterscovertherestframe12 µm,respectively.Wesum-
+marise the filters used in Table 1. Methodology is the same as
+for the 8µm LF; we used the sample to the 5 σlimit, corrected
+for the completeness, then used the 1/ Vmaxmethod to com-
+pute LF in each redshift bin. The histogram of L12µmdistri-
+bution is presented in Fig.7. The resulting 12 µm LF is shown
+in Fig.8. Compared with Rush,Malkan,& Spinoglio (1993)’s
+z=0 LF based on IRAS Faint Source Catalog, the 12 µm LFs
+show steady evolution with increasing redshift. In the rang e of
+0.25< z <1,L∗
+12µmevolvesas ∝(1+z)1.5±0.4.
+3.4. Bolometric IR luminosity density basedonthe 12 µm
+LF
+12µm is one of the most frequentlyused monochromaticfluxes
+to estimate LTIR. The total infrared luminosity is computed
+from theL12µmusing the conversionin Chary& Elbaz (2001);
+P´ erez-Gonz´ alezet al.(2005).
+logLTIR= log(0.89+0.38
+−0.27)+1.094logL12µm (8)Fig.8.Restframe 12 µm LFs based on the AKARI NEP-Deep
+field.Thebluediamonds,purpletriangles,andredsquaress how
+the 12µm LFs at 0.15< z <0.35,0.38< z <0.62, and
+0.84< z <1.16, respectively. Vertical arrows show the 12 µm
+luminosity corresponding to the flux limit at the central red -
+shift in each redshift bin. Overplotted are P´ erez-Gonz´ al ezet al.
+(2005) at z=0.3,0.5 and 0.9 in the cyan dash-dotted lines, and
+Rush,Malkan,& Spinoglio (1993) at z=0 in the green dash-
+dottedlines. AGNsareexcludedfromthesample( §2.2).
+Takeuchietal. (2005) independently estimated the relatio n
+tobe
+logLTIR= 1.02+0.972logL12µm, (9)
+which we also use to check our conversion. As both au-
+thors state, these conversions contain an error of factor of 2-3.
+Therefore, we should avoid conclusions that could be affect ed
+bysucherrors.
+Then the 12 µm LF is weighted by the LTIRand integrated
+to obtain TIR density. Errors are estimated by varying the fit
+within 1σof uncertainty in LFs, and errors in converting from
+L12µmtoLTIRareadded.Thelatter isbyfarthe largestsource
+of uncertainty. Best fit parameters are presented in Table 2. In
+Fig.10,we showtotal luminositydensitybasedonthe12 µmLF8 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Fig.9.Evolution of 12 µm IR luminosity density computed by
+integratingthe12 µmLFsinFig.8.
+Fig.10. TIR luminosity density computed by integrating the
+12µmLFsin Fig.8.
+presented in Fig.8. The results show a rapid increase of ΩIR,
+agreeing with previous work (LeFloc’hetal., 2005) within t he
+errors.
+We also integrate monochromatic L12µmover the LFs
+(without converting to LTIR) to derive the evolution of to-
+tal12µmmonochromatic luminosity density, Ω12µm. The re-
+sults are shown in Fig.9, which shows a strong evolution of
+Ω12µm∝(1 +z)1.4±1.4. It is interesting to compare this to
+Ω8µm∝(1 +z)1.9±0.7obtained in §3.2. Although errors are
+significantonbothestimates, Ω12µmandΩ8µmshowa possibly
+differentevolution,suggestingthatthecosmicinfrareds pectrum
+changesits SED shape.Whetherthisisdueto evolutionindus t,
+or dusty AGN contribution is an interesting subject for futu re
+work.Fig.11.An example of the SED fit. The red dashed line shows
+thebest-fitSEDfortheUV-optical-NIRSED,mainlytoestima te
+photometricredshift.Thebluesolidlineshowsthebest-fit model
+fortheIRSEDat λ >6µm,toestimate LTIR.
+3.5. TIRLF
+AKARI’scontinuousmid-IRcoverageisalsosuperiorforSED -
+fitting to estimate LTIR, since for star-forming galaxies, the
+mid-IR part of the IR SED is dominated by the PAH emissions
+whichreflectthe SFR ofgalaxies,andthus,correlateswell w ith
+LTIR, which is also a good indicator of the galaxy SFR. The
+AKARI’scontinuousMIRcoveragehelpsustoestimate LTIR.
+After photometric redshifts are estimated using the UV-
+optical-NIRphotometry,we fix the redshift at the photo- z,then
+use the same LePhare code to fit the infrared part of the SED
+to estimate TIR luminosity. We used Lagache,Dole,&Puget
+(2003)’s SED templates to fit the photometryusing the AKARI
+bands at >6µm (S7,S9W,S11,L15,L18WandL24). We
+showanexampleoftheSEDfitinFig.11,wherethereddashed
+and blue solid lines show the best-fit SEDs for the UV-optical -
+NIR and IR SED at λ >6µm, respectively. The obtained total
+infraredluminosity( LTIR) is shown as a functionofredshift in
+Fig.12,withspectroscopicgalaxiesinlargetriangles.Th efigure
+shows that the AKARI can detect LIRGs ( LTIR>1011L⊙)
+up toz=1 and ULIRGs ( LTIR>1012L⊙) toz=2. We also
+checkedthatusingdifferentSEDmodels(Chary& Elbaz,2001 ;
+Dale& Helou,2002) doesnotchangeouressentialresults.
+Galaxies in the targeted redshift range are best sampled in
+the 18µm band due to the wide bandpass of the L18Wfilter
+(Matsuharaet al., 2006). In fact, in a single-band detectio n, the
+18µm image returns the largest number of sources. Therefore,
+we applied the 1/ Vmaxmethod using the detection limit at
+L18W. We also checked that using the L15flux limit does
+not change our main results. The same Lagache,Dole,&Puget
+(2003)’s models are also used for k-corrections necessary to
+compute VmaxandVmin. The redshift bins used are 0.2 <
+z <0.5,0.5< z <0.8,0.8< z <1.2,and 1.2 < z <1.6. A distri-
+butionof LTIRineachredshiftbinis showninFig.13.
+Theobtained LTIRLFsareshowninFig.14.Theuncertain-
+ties are esimated through the Monte Carlo simulations ( §2.4).
+For a local benchmark, we overplot Sanderset al. (2003) who
+derived LFs from the analytical fit to the IRAS Revised Bright
+Galaxy Sample, i.e., φ∝L−0.6forL < L∗andφ∝L−2.2for
+L > L∗withL∗= 1010.5L⊙. The TIR LFs show a strong evo-
+lutioncomparedtolocalLFs.At 0.25< z <1.3,L∗
+TIRevolvesGotoet al.:InfraredLuminosityfunctions withthe AKARI 9
+Fig.12.TIR luminosity is shown as a function of photometric
+redshift. The photo- zis estimated using UV-optical-NIR pho-
+tometry.LTIRisobtainedthroughSED fit in7-24 µm.
+Fig.13.AhistogramofTIRluminosity.Fromlow-redshift,144,
+192, 394, and 222 galaxies are in 0.2 < z <0.5, 0.5< z <0.8,
+0.8< z <1.2,and1.2 < z <1.6,respectively.
+as∝(1 +z)4.1±0.4. We further compare LFs to the previous
+workin§4.
+3.6. Bolometric IR luminosity density basedonthe TIRLF
+Using the same methodology as in previous sections, we inte-
+grateLTIRLFs in Fig.14 through a double-power law fit (eq.
+5 and 6). The resulting evolution of the TIR density is shown
+with red diamonds in Fig.15, which in in good agreement with
+LeFloc’hetal.(2005)withintheerrors.Errorsareestimat edby
+varying the fit within 1 σof uncertainty in LFs. For uncertainty
+intheSEDfit,weadded0.15dexoferror.Bestfitparametersar e
+presented in Table 2. In Fig.15, we also show the contributio ns
+toΩTIRfromLIRGsandULIRGswiththebluesquaresandor-
+ange triangles, respectively. We further discuss the evolu tion of
+ΩTIRin§4.Fig.14.TIRLFs.Verticallinesshowtheluminositycorrespond-
+ing to the flux limit at the central redshift in each redshift b in.
+AGNsareexcludedfromthesample( §2.2).
+Fig.15. TIR luminosity density (red diamonds) computed by
+integrating the total LF in Fig.14. The blue squares and oran ge
+trianglesareforLIRG andULIRGsonly.
+4. Discussion
+4.1. Comparison with previouswork
+In this section, we compare our results to previous work, esp e-
+ciallythosebasedontheSpitzerdata.Comparisonsarebest done
+inthesamewavelengths,sincetheconversionfromeither L8µm
+orL12µmtoLTIRinvolves the largest uncertainty. Hubble pa-
+rametersinthepreviousworkareconvertedto h= 0.7forcom-
+parison.10 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+4.1.1. 8µm LFs
+Recently, using the Spitzer space telescope, restframe 8 µm LFs
+ofz∼1 galaxies have been computed in detail by Caputiet al.
+(2007) in the GOODS fields and by Babbedgeetal. (2006) in
+theSWIREfield.Inthissection,wecompareourrestframe8 µm
+LFs(Fig.4)tothese anddiscusspossibledifferences.
+In Fig.4, we overplot Caputi etal. (2007)’s LFs at z=1 and
+z=2inthecyandash-dottedlines.Their z=2LFisingoodagree-
+ment with our LF at 1.8 < z <2.2. However, their z=1 LF is
+larger than ours by a factor of 3-5 at logL >11.2. Note that
+the brightest ends( logL∼11.4)are consistent with each other
+to within 1 σ. They have excluded AGN using optical-to-X-ray
+flux ratio, and we also have excluded AGN through the optical
+SED fit. Therefore, especially at the faint-end, the contami na-
+tionfromAGN isnot likelyto be the maincauseof differences .
+Since Caputiet al. (2007) uses GOODS fields, cosmic variance
+may play a role here. The exact reason for the difference is un -
+known, but we point out that their ΩIRestimate at z=1 is also
+higherthanotherestimatesbyafactorofafew(seetheirFig .15).
+Once converted into LTIR, Magnelliet al. (2009) also reported
+Caputiet al.(2007)’s z=1LF ishigherthantheirestimatebased
+on 70µm by a factor of several (see their Fig.12). They con-
+cluded the difference is from different SED models used, sin ce
+their LF matched with that of Caputi etal. (2007)’s once the
+same SED models were used. We will compare our total LFs
+tothosein theliteraturebelow.
+Babbedgeet al. (2006) also computed restframe 8 µm LFs
+using the Spitzer/SWIRE data. We overplot their results at
+0.25< z <0.5and0.5< z <1in Fig.4 with the pink dot-
+dashedlines.Inbothredshiftranges,goodagreementisfou ndat
+higherluminositybins( L8µm>1010.5L⊙).However,atallred-
+shift ranges including the ones not shown here, Babbedgeet a l.
+(2006) tends to show a flatter faint-end tail than ours, and a
+smallerφby a factor of ∼3. Although the exact reason is un-
+known, the deviation starts toward the fainter end, where bo th
+works approach the flux limits of the surveys. Therefore,pos si-
+blyincompletesamplingmaybeoneofthereasons.Itisalsor e-
+portedthat thefaint-endof IRLFsdependson theenvironmen t,
+in the sense that higher-density environment has steeper fa int-
+end tail (Gotoet al., 2010). Note that at z=1, Babbedgeet al.
+(2006)’s LF (pink) deviates from that by Caputiet al. (2007)
+(cyan) by almost a magnitude. Our 8 µm LFs are between these
+works.
+These comparisons suggest that even with the current gen-
+eration of satellites and state-of-the-art SED models, fac tor-of-
+several uncertainties still remain in estimating the 8 µm LFs
+at z∼1. More accurate determination has to await a larger
+and deeper survey by the next generation IR satellites such a s
+HerschelandWISE.
+To summarise, our 8 µm LFs are between those by
+Babbedgeetal.(2006)andCaputiet al.(2007),anddiscrepa ncy
+is by a factor of several at most. We note that both of the previ -
+ous works had to rely on SED models to estimate L8µmfrom
+the Spitzer S24µmin the MIR wavelengths where SED model-
+ing is difficult. Here, AKARI’s mid-IR bands are advantageou s
+indirectlyobservingredshiftedrestframe8 µmfluxinoneofthe
+AKARI’s filters, leading to more reliable measurement of 8 µm
+LFswithoutuncertaintyfromtheSED modeling.
+4.1.2. 12 µm LFs
+P´ erez-Gonz´ alezet al. (2005) investigated the evolution of rest-
+frame12µmLFsusingthe SpitzerCDF-S andHDF-N data.Weoverplot their results in similar redshift ranges as the cya n dot-
+dashed lines in Fig.8. Consideringboth LFs have significant er-
+ror bars, these LFs are in good agreement with our LFs, and
+show significant evolution in the 12 µm LFs compared with the
+z=012µmLFbyRush,Malkan,&Spinoglio(1993).Theagree-
+ment is in a stark contrast to the comparison in 8 µm LFs in
+§4.1.1, wherewe sufferedfromdifferncesof a factor of sever al.
+Apossiblereasonforthisisthat12 µmissufficientlyredderthan
+8µm, that it is easier to be extrapolated from S24µmin case of
+the Spitzer work. In fact, at z=1, both the Spitzer 24 µm band
+and AKARI L24observe the restframe 12 µm directly. In addi-
+ton, mid-IR SEDs around 12 µm are flatter than at 8 µm, where
+PAH emissions are prominent.Therefore,SED modelscan pre-
+dict the flux more accurately. In fact, this is part of the rea-
+sonwhyP´ erez-Gonz´ alezet al.(2005)chosetoinvestigate 12µm
+LFs. P´ erez-Gonz´ alezetal. (2005) used Chary&Elbaz (2001 )’s
+SEDtoextrapolate S24µm,andyet,theyagreewellwithAKARI
+results, which are derived from filters that cover the restfr ame
+12µm. However, in other words, the discrepancy in 8 µm LFs
+highlights the fact that the SED models are perhaps still imp er-
+fect in the 8 µm wavelengthrange, and thus, MIR-spectroscopic
+data that covers wider luminosity and redshift ranges will b e
+necessary to refine SED models in the mid-IR. AKARI’s mid-
+IR slitless spectroscopy survey (Wada, 2008) may help in thi s
+regard.
+4.1.3. TIRLFs
+Lastly,we compareourTIRLFs(Fig.14) withthoseinthelite r-
+ature.AlthoughtheTIRLFs canalso be obtainedbyconvertin g
+8µmLFsor12 µmLFs,wealreadycomparedourresultsinthese
+wavelengths in the last subsections. Here, we compare our TI R
+LFstoLe Floc’het al.(2005)andMagnellietal. (2009).
+LeFloc’het al. (2005) obtained TIR LFs using the Spitzer
+CDF-S data. They have used the best-fit SED among various
+templatestoestimate LTIR.WeoverplottheirtotalLFsinFig.14
+with the cyan dash-dotted lines. Only LFs that overlapwith o ur
+redshit ranges are shown. The agreement at 0.3< z <0.45
+and0.6< z <0.8is reasonable, considering the error bars on
+bothsides.However,inallthreeredshiftranges,LeFloc’h et al.
+(2005)’sLFsare higherthanours,especiallyfor 1.0< z <1.2.
+We also overplot TIR LFs by Magnellietal. (2009), who
+used Spitzer 70 µm flux and Chary& Elbaz (2001)’s model to
+estimateLTIR.Inthetwobins(centeredon z=0.55and z=0.85;
+pink dash-dotted lines) which closely overlap with our reds hift
+bins, excellent agreement is found. We also plot Huynhet al.
+(2007)’s LF at 0.6< z <0.9in the navy dash-dotted lines,
+whichis computedfromSpitzer 70µmimagingin the GOODS-
+N, and this also shows very good agreement with ours. These
+LFs are on top of each other within the error bars, despite the
+fact that these measurements are from different data sets us ing
+differentanalyses.
+This means that LeFloc’hetal. (2005)’s LFs is also higher
+thanthatofMagnelliet al.(2009),inadditiontoours.Apos sible
+reasonis that both Magnelliet al. (2009) and we removedAGN
+(at least bright ones), whereas Le Floc’het al. (2005) inclu ded
+them. This also is consistent with the fact that the differen ce
+is larger at 1.0< z <1.2where both surveys are only sen-
+sitive to luminous IR galaxies, which are dominated by AGN.
+Another possible source of uncertainty is that Magnelliet a l.
+(2009) and we used a single SED library, while LeFloc’het al.
+(2005)pickedthebestSEDtemplateamongseverallibraries for
+eachgalaxy.Gotoet al.:InfraredLuminosityfunctions withthe AKARI 11
+Fig.16.EvolutionofTIRluminositydensitybasedonTIRLFs(redcir cles),8µmLFs(stars),and12 µmLFs(filledtriangles).The
+blue open squaresand orangefilled squaresare for LIRG and UL IRGs only, also based on our LTIRLFs. Overplotteddot-dashed
+lines are estimates from the literature: LeFloc’het al. (20 05), Magnelliet al. (2009) , P´ erez-Gonz´ alezet al. (2005) , Caputiet al.
+(2007), and Babbedgeet al. (2006) are in cyan, yellow, green , navy,and pink, respectively.The purple dash-dottedline shows UV
+estimatebySchiminovichetal. (2005).Thepinkdashedline showsthetotalestimateofIR(TIRLF)andUV (Schiminoviche t al.,
+2005).
+4.2. Evolution of ΩIR
+In this section, we compare the evolution of ΩIRas a function
+ofredshift.InFig.16, weplot ΩIRestimatedfromTIRLFs(red
+circles), 8 µm LFs (brown stars), and 12 µm LFs (pink filled tri-
+angles),as a functionof redshift.Estimatesbased on12 µmLFs
+and TIR LFs agree each other very well, while those from 8 µm
+LFs show a slightly higher value by a factor of a few than oth-
+ers. This perhaps reflects the fact that 8 µm is a more difficult
+part of the SED to be modeled, as we had a poorer agreement
+amongpapersintheliteraturein8 µmLFs.Thebright-endslope
+of the double-power law was 3.5+0.2
+−0.4in Table 2. This is flat-
+ter than a Schechter fit by Babbedgeet al. (2006) and a double-
+exponential fit by Caputiet al. (2007). This is perhaps why we
+obtainedhigher ΩIRin8µm.
+We overplot estimates from various papers in the litera-
+ture(LeFloc’hetal.,2005; Babbedgeet al.,2006;Caputiet al.,
+2007; P´ erez-Gonz´ alezet al., 2005; Magnelliet al., 2009) in the
+dash-dottedlines. Our ΩIRhasverygoodagreementwith these
+at0< z <1.2,withalmostallthedash-dottedlineslyingwithin
+ourerrorbarsof ΩIRfromLTIRand12µmLFs.Thisisperhaps
+because an estimate of an integrated value such as ΩIRis more
+reliablethanthat ofLFs.
+Atz >1.2, ourΩIRshows a hint of continuous increase,
+while Caputiet al. (2007) and Babbedgeetal. (2006) observe da slight decline at z >1. However,as both authorsalso pointed
+out, at this high-redshift range, both the AKARI and Spitzer
+satellites are sensitive to onlyLIRGs and ULIRGs, and thust he
+extrapolationto fainterluminositiesassumesthefaint-e ndslope
+of the LFs, which couldbe uncertain.In addition,this work h as
+a poorerphoto-zestimate at z >0.8(∆z
+1+z=0.10)due to the rel-
+atively shallow near-IR data. Several authors tried to over come
+thisproblembystackingundetectedsources.However,ifan un-
+detectedsourceisalsonotdetectedatshorterwavelengths where
+positions for stacking are obtained, it would not be include d in
+the stacking either. Next generation satellite such as Hers chel,
+WISE, and SPICA (Nakagawa, 2008) will determine the faint-
+endslopeat z >1moreprecisely.
+We parameterize the evolution of ΩIRusing a following
+function.
+ΩIR(z)∝(1+z)γ(10)
+By fitting this to the ΩIRfrom TIR LFs, we obtained γ=
+4.4±1.0. This is consistent with most previous works.
+For example, LeFloc’hetal. (2005) obtained γ= 3.9±
+0.4, P´ erez-Gonz´ alezet al. (2005) obtained γ= 4.0±0.2,
+Babbedgeetal. (2006) obtained γ= 4.5+0.7
+−0.6, Magnelliet al.
+(2009) obtained γ= 3.6±0.4. The agreement was expected
+fromFig.16,butconfirmsastrongevolutionof ΩIR.12 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Fig.17. Contribution of ΩTIRtoΩtotal= ΩUV+ ΩTIRis
+shownasa functionofredshift.
+4.3. Differential evolution among ULIRG,LIRG,normal
+galaxies
+In Fig. 15, we also plot the contributions to ΩIRfrom LIRGs
+and ULIRGs (measured from TIR LFs) with the blue open
+squares and orange filled squares, respectively. Both LIRGs
+and ULIRGs show strong evolution, as has been seen for to-
+talΩIRin the red filled circles. Normal galaxies ( LTIR<
+1011L⊙) are still dominant, but decrease their contribution to-
+ward higher redshifts. In contrast, ULIRGs continueto incr ease
+their contribution. From z=0.35 to z=1.4,ΩIRby LIRGs in-
+creases by a factor of ∼1.6, andΩIRby ULIRGs increases by
+a factor of ∼10. The physical origin of ULIRGs in the local
+Universe is often merger/interaction(Sanders& Mirabel, 1 996;
+Taniguchi&Shioya, 1998; Goto, 2005). It would be interesti ng
+to investigate whether the merger rate also increases in pro por-
+tion to the ULIRG fraction, or if different mechanisms can al so
+produceULIRGsathigherredshift.
+4.4. Comparison tothe UVestimate
+We have been emphasizing the importance of IR probes of the
+total SFRD of the Universe. However, the IR estimates do not
+take into account the contribution of the unabsorbed UV ligh t
+produced by the young stars. Therefore, it is important to es ti-
+matehowsignificantthisUV contributionis.
+Schiminovichet al. (2005) found that the energy density
+measured at 1500 ˚A evolves as ∝(1+z)2.5±0.7at0< z <1
+and∝(1 +z)0.5±0.4atz >1. using the GALEX data sup-
+plemented by the VVDS spectroscopic redshifts. We overplot
+their UV estimate of ρSFRwith the purple dot-dashed line in
+Fig.16. The UV estimate is almost a factor of 10 smaller than
+the IR estimate at most of the redshifts, confirming the impor -
+tanceofIRprobeswheninvestingtheevolutionofthetotalc os-
+mic star formation density. In Fig.16 we also plot total SFD ( or
+Ωtotal)byadding ΩUVandΩTIR,withthemagentadashedline.
+In Fig.17, we show the ratio of the IR contribution to the to-
+tal SFRD of the Universe ( ΩTIR/ΩTIR+ ΩUV) as a function
+of redshift. Although the errors are large, Fig.17 agrees wi thTakeuchi,Buat,& Burgarella (2005), and suggests that ΩTIR
+explains 70% of Ωtotalatz=0.25, and that by z=1.3, 90% of
+the cosmic SFD is explained by the infrared. This implies tha t
+ΩTIRprovidesgoodapproximationofthe Ωtotalatz >1.
+5. Summary
+We have estimated restframe 8 µm, 12µm, and total infrared lu-
+minosity functions using the AKARI NEP-Deep data. Our ad-
+vantage over previous work is AKARI’s continuous filter cov-
+erage in the mid-IR wavelengths (2.4, 3.2, 4.1, 7, 9, 11, 15, 1 8,
+and24µm),whichallowustoestimate mid-IRluminositywith-
+out a large extrapolationbased on SED models, which were the
+largest uncertainty in previous work. Even for LTIR, the SED
+fitting is much more reliable due to this continuouscoverage of
+mid-IRfilters.
+Ourfindingsareasfollows:
+–8µm LFs show a strong and continuous evolution from
+z=0.35 to z=2.2. Our LFs are larger than Babbedgeet al.
+(2006), but smaller than Caputi etal. (2007). The differenc e
+perhaps stems from the different SED models, highlighting
+a difficulty in SED modeling at wavelengths crowded by
+strong PAH emissions. L∗
+8µmshows a continuous evolution
+asL∗
+8µm∝(1+z)1.6±0.2in therangeof 0.48< z <2.
+–12µm LFs show a strong and continuous evolution from
+z=0.15toz=1.16with L∗
+12µm∝(1+z)1.5±0.4. Thisagrees
+well with P´ erez-Gonz´ alezet al. (2005), including a flatte r
+faint-endslope. A better agreementthan with 8 µm LFs was
+obtained, perhaps because of smaller uncertainty in model-
+ing the 12 µm SED, and less extrapolationneededin Spitzer
+24µmobservations.
+–The TIR LFs show good agreement with Magnelliet al.
+(2009), but are smaller than Le Floc’het al. (2005). At
+0.25< z <1.3,L∗
+TIRevolvesas ∝(1+z)4.1±0.4.Possible
+causes of the disagreement include different treatment of
+SEDmodelsinestimating LTIR,andAGNcontamination.
+–TIR densities estimated from 12 µm and TIR LFs show a
+strong evolution as a function of redshift, with ΩIR∝
+(1 +z)4.4±1.0.ΩIR(z)also show an excellent agreement
+withpreviousworkat z <1.2.
+–We investigated the differential contribution to ΩIRby
+ULIRGsandLIRGs.WefoundthattheULIRG(LIRG)con-
+tribution increases by a factor of 10 (1.8) from z=0.35 to
+z=1.4, suggesting IR galaxies are more dominant source of
+ΩIRathigherredshift.
+–We estimated that ΩIRcaptures80% of the cosmic star for-
+mationatredshiftslessthan1,andvirtuallyallofitathig her
+redshift.Thusaddingtheunobscuredstarformationdetect ed
+at UV wavelengths would not change SFRD estimates sig-
+nificantly.
+Acknowledgments
+We are grateful to S.Arnouts for providing the LePhare code,
+and kindly helping us in using the code. We thank the anony-
+mousrefereeformanyinsightfulcomments,whichsignifican tly
+improvedthe paper.
+T.G. and H.I. acknowledgefinancial supportfrom the Japan
+Society for the Promotion of Science (JSPS) through JSPS
+Research Fellowships for Young Scientists. HML acknowl-
+edges the support from KASI through its cooperative fund in
+2008. TTT has been supported by Program for Improvement
+of Research Environment for Young Researchers from SpecialGotoet al.:InfraredLuminosityfunctions withthe AKARI 13
+CoordinationFundsforPromotingScienceandTechnology,a nd
+the Grant-in-Aid for the Scientific Research Fund (20740105 )
+commissioned by the Ministry of Education, Culture, Sports ,
+Science and Technology (MEXT) of Japan. TTT has been also
+partially supported from the Grand-in-Aid for the Global CO E
+Program “Quest for Fundamental Principles in the Universe:
+from Particles to the Solar System and the Cosmos” from the
+MEXT.
+This research is based on the observations with AKARI, a
+JAXA projectwiththe participationofESA.
+Theauthorswishtorecognizeandacknowledgetheverysig-
+nificant cultural role and reverence that the summit of Mauna
+Kea has always had within the indigenous Hawaiian commu-
+nity. We are most fortunate to have the opportunity to conduc t
+observationsfromthissacredmountain.
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1001.0018.txt

@@ -0,0 +1,354 @@
+arXiv:1001.0018v2  [quant-ph]  28 Jan 2010Nonadaptive quantum query complexity
+Ashley Montanaro∗
+October 1, 2018
+Abstract
+We studythe powerofnonadaptivequantum queryalgorithms,whic h arealgorithms
+whose queries to the input do not depend on the result of previous q ueries. First, we
+show that any bounded-error nonadaptive quantum query algorit hm that computes
+some total boolean function depending on nvariables must make Ω( n) queries to the
+input in total. Second, we show that, if there exists a quantum algor ithm that uses k
+nonadaptive oracle queries to learn which one of a set of mboolean functions it has
+been given, there exists a nonadaptive classical algorithm using O(klogm) queries to
+solve the same problem. Thus, in the nonadaptive setting, quantum algorithms can
+achieve at most a very limited speed-up over classical query algorith ms.
+1 Introduction
+Many of the best-known results showing that quantum compute rs outperform their classical
+counterparts are proven in the query complexity model. This model studies the number of
+queries to the input xwhich are required to compute some function f(x). In this work, we
+study two broad classes of problem that fit into this model.
+In the first class of problems, computational problems, one wishes to compute some
+boolean function f(x1,...,x n) using a small number of queries to the bits of the input
+x∈ {0,1}n. The query complexity of fis the minimum number of queries required for any
+algorithm to compute f, with some requirement on the success probability. The dete rmin-
+istic query complexity of f,D(f), is the minimum number of queries that a deterministic
+classical algorithm requires to compute fwith certainty. D(f) is also known as the decision
+tree complexity of f. Similarly, the randomised query complexity R2(f) is the minimum
+number of queries required for a randomised classical algor ithm to compute fwith success
+probability at least 2 /3. The choice of 2 /3 is arbitrary; any constant strictly between 1 /2
+and 1 would give the same complexity, up to constant factors.
+There is a natural generalisation of the query complexity mo del to quantum computa-
+tion, which gives rise to the exact and bounded-error quantu m query complexities QE(f),
+Q2(f) (respectively). In this generalisation, the quantum algo rithm is given access to the
+∗Department of Computer Science, University of Bristol, Woo dland Road, Bristol, BS8 1UB, UK;
+[email protected] .
+1inputxthrough a unitary oracle operator Ox. Many of the best-known quantum speed-ups
+can be understood in the query complexity model. Indeed, it i s known that, for certain
+partial functions f(i.e. functions where there is a promise on the input), Q2(f) may be ex-
+ponentially smaller than R2(f)[14]. However, if fis atotal function, D(f) =O(Q2(f)6) [4].
+See [6, 10] for good reviews of quantum and classical query co mplexity.
+In the second class of problems, learning problems, one is given as an oracle an unknown
+functionf?(x1,...,x n), which is picked from a known set Cofmboolean functions f:
+{0,1}n→ {0,1}. These functions can be identified with n-bit strings or subsets of [ n], the
+integers between 1 and n. The goal is to determine which of the functions in Cthe oraclef?
+is, with some requirement on the success probability, using the minimum number of queries
+tof?. Note that the success probability required should be stric tly greater than 1 /2 for
+this model to make sense.
+Borrowing terminology fromthe machinelearning literatur e, each function in Cis known
+as aconcept, andCis known as a concept class [13]. We say that an algorithm that can
+identify any f∈ Cwith worst-case success probability plearnsCwith success probability
+p. This problem is known classically as exact learning from me mbership queries [3, 13],
+and also in the literature on quantum computation as the orac le identification problem [2].
+Many interesting results in quantum algorithmics fit into th is framework, a straightforward
+example being Grover’s quantum search algorithm [9]. It has been shown by Servedio and
+Gortler that the speed-up that may be obtained by quantum que ry algorithms in this model
+is at most polynomial [13].
+1.1 Nonadaptive query algorithms
+This paper considers query algorithms of a highly restricti ve form, where oracle queries are
+not allowed to depend on previous queries. In other words, th e queries must all be made
+at the start of the algorithm. We call such algorithms nonadaptive , but one could also call
+themparallel, in contrast to the usual serial model of query complexity, w here one query
+follows another. It is easy to see that, classically, a deter ministic nonadaptive algorithm
+that computes a function f:{0,1}n→ {0,1}which depends on all ninput variables must
+query allnvariables (x1,...,x n). Indeed, for any 1 ≤i≤n, consider an input xfor which
+f(x) = 0, butf(x⊕ei) = 1, where eiis the bit string which has a 1 at position i, and is 0
+elsewhere. Then, if the i’th variable were not queried, changing the input from xtox��ei
+would change the output of the function, but the algorithm wo uld not notice.
+In the case of learning, the exact number of queries required by a nonadaptive determin-
+istic classical algorithm to learn any concept class Ccan also be calculated. Identify each
+concept in Cwith ann-bit string, and imagine an algorithm Athat queries some subset
+S⊆[n] of the input bits. If there are two or more concepts in Cthat do not differ on any of
+the bits inS, thenAcannot distinguish between these two concepts, and so canno t succeed
+with certainty. On the other hand, if every concept x∈ Cis unique when restricted to S,
+thenxcan be identified exactly by A. Thus the number of queries required is the minimum
+size of a subset S⊆[n] such that every pair of concepts in Cdiffers on at least one bit in S.
+We will be concerned with the speed-up over classical query a lgorithms that can be
+2achieved by nonadaptive quantum query algorithms. Interes tingly, it is known that speed-
+ups can indeed be found in this model. In the case of computing partial functions, the
+speed-up can be dramatic; Simon’s algorithm for the hidden s ubgroup problem over Zn
+2, for
+example, is nonadaptive and gives an exponential speed-up o ver the best possible classical
+algorithm [14]. Thereare also known speed-upsfor computin g total functions. For example,
+the parity of nbits can be computed exactly using only ⌈n/2⌉nonadaptive quantum queries
+[8]. More generally, anyfunction of nbits can be computed with bounded error using only
+n/2+O(√n)nonadaptivequeries, byaremarkablealgorithmofvanDam[ 7]. Thisalgorithm
+in fact retrieves allthe bits of the input xsuccessfully with constant probability, so can also
+be seen as an algorithm that learns the concept class consist ing of all boolean functions on
+nbits usingn/2+O(√n) nonadaptive queries.
+Finally, one of the earliest results in quantum computation can be understood as a
+nonadaptive learning algorithm. The quantum algorithm sol ving the Bernstein-Vazirani
+parity problem [5] uses one query to learn a concept class of s ize 2n, for which any classical
+learning algorithm requires nqueries, showing that there can be an asymptotic quantum-
+classical separation for learning problems.
+1.2 New results
+We show here that these results are essentially the best poss ible. First, any nonadap-
+tive quantum query algorithm that computes a total boolean f unction with a constant
+probability of success greater than 1 /2 can only obtain a constant factor reduction in the
+number of queries used. In particular, if we restrict to nona daptive query algorithms, then
+Q2(f) = Θ(D(f)). In the case of exact nonadaptive algorithms, we show that the factor of
+2 speed-up obtained for computing parity is tight. More form ally, our result is the following
+theorem.
+Theorem 1. Letf:{0,1}n→ {0,1}be a total function that depends on all nvariables,
+and letAbe a nonadaptive quantum query algorithm that uses kqueries to the input to
+computef, and succeeds with probability at least 1−ǫon every input. Then
+k≥n
+2/parenleftBig
+1−2/radicalbig
+ǫ(1−ǫ)/parenrightBig
+.
+In the case of learning, we show that the speed-up obtained by the Bernstein-Vazirani
+algorithm [5] is asymptotically tight. That is, the query co mplexities of quantum and
+classical nonadaptive learning are equivalent, up to a loga rithmic term. This is formalised
+as the following theorem.
+Theorem 2. LetCbe a concept class containing mconcepts, and let Abe a nonadaptive
+quantum query algorithm that uses kqueries to the input to learn C, and succeeds with
+probability at least 1−ǫon every input, for some ǫ <1/2. Then there exists a classical
+nonadaptive query algorithm that learns Cwith certainty using at most
+4klog2m
+1−2/radicalbig
+ǫ(1−ǫ)
+queries to the input.
+31.3 Related work
+We note that the question of putting lower bounds on nonadapt ive quantum query algo-
+rithms has been studied previously. First, Zalka has obtain ed a tight lower bound on the
+nonadaptive quantum query complexity of the unordered sear ch problem, which is a par-
+ticular learning problem [15]. Second, in [12], Nishimura a nd Yamakami give lower bounds
+on the nonadaptive quantum query complexity of a multiple-b lock variant of the ordered
+search problem. Finally, Koiran et al [11] develop the weigh ted adversary argument of Am-
+bainis [1] to obtain lower bounds that are specific to the nona daptive setting. Unlike the
+situation considered here, their bounds also apply to quant um algorithms for computing
+partial functions.
+We now turn to proving the new results: nonadaptive computat ion in Section 2, and
+nonadaptive learning in Section 3.
+2 Nonadaptive quantum query complexity of computation
+LetAbe a nonadaptive quantum query algorithm. We will use what is essentially the
+standard model of quantum query complexity [10]. Ais given access to the input x=
+x1...xnvia an oracle Oxwhich acts on an n+1 dimensional space indexed by basis states
+|0/an}brack⌉tri}ht,...,|n/an}brack⌉tri}ht, and performs the operation Ox|i/an}brack⌉tri}ht= (−1)xi|i/an}brack⌉tri}ht. We define Ox|0/an}brack⌉tri}ht=|0/an}brack⌉tri}htfor
+technical reasons (otherwise, Acould not distinguish between xand ¯x). Assume that A
+makeskqueries toOx. As the queries are nonadaptive, we may assume they are made i n
+parallel. Therefore, the existence of a nonadaptive quantu m query algorithm that computes
+fand fails with probability ǫis equivalent to the existence of an input state |ψ/an}brack⌉tri}htand a
+measurement specified by positive operators {M0,I−M0}, such that /an}brack⌉tl⌉{tψ|O⊗k
+xM0O⊗k
+x|ψ/an}brack⌉tri}ht ≥
+1−ǫfor all inputs xwheref(x) = 0, and /an}brack⌉tl⌉{tψ|O⊗k
+xM0O⊗k
+x|ψ/an}brack⌉tri}ht ≤ǫfor all inputs xwhere
+f(x) = 1.
+The intuition behind the proof of Theorem 1 is much the same as that behind “adver-
+sary” arguments lower bounding quantum query complexity [1 0]. As in Section 1.1, let ej
+denote the n-bit string which contains a single 1, at position j. In order to distinguish two
+inputsx,x⊕ejwheref(x)/n⌉}ationslash=f(x⊕ej), the algorithm must invest amplitude of |ψ/an}brack⌉tri}htin
+components where the oracle gives information about j. But, unless kis large, it is not
+possible to invest in many variables simultaneously.
+We will use the following well-known fact from [5].
+Fact 3(Bernstein and Vazirani [5]) .Imagine there exists a positive operator M≤Iand
+states|ψ1/an}brack⌉tri}ht,|ψ2/an}brack⌉tri}htsuch that /an}brack⌉tl⌉{tψ1|M|ψ1/an}brack⌉tri}ht ≤ǫ, but/an}brack⌉tl⌉{tψ2|M|ψ2/an}brack⌉tri}ht ≥1−ǫ. Then |/an}brack⌉tl⌉{tψ1|ψ2/an}brack⌉tri}ht|2≤
+4ǫ(1−ǫ).
+We now turn to the proof itself. Write the input state |ψ/an}brack⌉tri}htas
+|ψ/an}brack⌉tri}ht=/summationdisplay
+i1,...,ikαi1,...,ik|i1,...,ik/an}brack⌉tri}ht,
+4where, for each m, 0≤im≤n. It is straightforward to compute that
+O⊗k
+x|i1,...,ik/an}brack⌉tri}ht= (−1)xi1+···+xik|i1,...,ik/an}brack⌉tri}ht.
+Asfdepends on all ninputs, for any j, there exists a bit string xjsuch thatf(xj)/n⌉}ationslash=
+f(xj⊕ej). Then
+(OxjOxj⊕ej)⊗k|i1,...,ik/an}brack⌉tri}ht= (−1)|{m:im=j}||i1,...,ik/an}brack⌉tri}ht;
+in other words ( OxjOxj⊕ej)⊗knegates those basis states that correspond to bit strings
+i1,...,ikwherejoccurs an odd number of times in the string. Therefore, we hav e
+|/an}brack⌉tl⌉{tψ|(OxjOxj⊕ej)⊗k|ψ/an}brack⌉tri}ht|2=
+/summationdisplay
+i1,...,ik|αi1,...,ik|2(−1)|{m:im=j}|
+2
+=
+1−2/summationdisplay
+i1,...,ik|αi1,...,ik|2[|{m:im=j}|odd]
+2
+=: (1−2Wj)2.
+Now, by Fact 3, (1 −2Wj)2≤4ǫ(1−ǫ) for allj, so
+Wj≥1
+2/parenleftBig
+1−2/radicalbig
+ǫ(1−ǫ)/parenrightBig
+.
+On the other hand,
+n/summationdisplay
+j=1Wj=n/summationdisplay
+j=1/summationdisplay
+i1,...,ik|αi1,...,ik|2[|{m:im=j}|odd]
+=/summationdisplay
+i1,...,ik|αi1,...,ik|2n/summationdisplay
+j=1[|{m:im=j}|odd]
+≤/summationdisplay
+i1,...,ik|αi1,...,ik|2k=k.
+Combining these two inequalities, we have
+k≥n
+2/parenleftBig
+1−2/radicalbig
+ǫ(1−ǫ)/parenrightBig
+.
+3 Nonadaptive quantum query complexity of learning
+In the case of learning, we use a very similar model to the prev ious section. Let Abe a
+nonadaptivequantumqueryalgorithm. Aisgiven access toanoracle Ox, whichcorresponds
+toabit-string xpickedfromaconcept class C.Oxactsonann+1dimensionalspaceindexed
+by basis states |0/an}brack⌉tri}ht,...,|n/an}brack⌉tri}ht, and performs the operation Ox|i/an}brack⌉tri}ht= (−1)xi|i/an}brack⌉tri}ht, withOx|0/an}brack⌉tri}ht=|0/an}brack⌉tri}ht.
+5Assume that Amakeskqueries toOxand outputs xwith probability strictly greater than
+1/2 for allx∈ C.
+We will prove limitations on nonadaptive quantum algorithm s in this model as follows.
+First, we show that a nonadaptive quantum query algorithm th at useskqueries to learn C
+is equivalent to an algorithm using one query to learn a relat ed concept class C′. We then
+show that existence of a quantum algorithm using one query th at learns C′with constant
+success probability greater than 1 /2 implies existence of a deterministic classical algorithm
+usingO(log|C′|) queries. Combining these two results gives Theorem 2.
+Lemma 4. LetCbe a concept class over n-bit strings, and let C⊗kbe the concept class
+defined by
+C⊗k={x⊗k:x∈ C},
+wherex⊗kdenotes the (n+ 1)k-bit string indexed by 0≤i1,...,ik≤n, withx⊗k
+i1,...,ik=
+xi1⊕ ··· ⊕xik, and we define x0= 0. Then, if there exists a classical nonadaptive query
+algorithm that learns C⊗kwith success probability pand usesqqueries, there exists a classical
+nonadaptive query algorithm that learns Cwith success probability pand uses at most kq
+queries.
+Proof.Given access to x, an algorithm Acan simulate a query of index ( x1,...,x k) ofx⊗k
+by using at most kqueries to compute x1⊕··· ⊕xk. Hence, by simulating the algorithm
+for learning C⊗k,Acan learn C⊗kwith success probability pusing at most kqnonadaptive
+queries. Learning C⊗ksuffices to learn C, because each concept in C⊗kuniquely corresponds
+to a concept in C(to see this, note that the first nbits ofx⊗kare equal to x).
+Lemma 5. LetCbe a concept class containing mconcepts. Assume that Ccan be learned
+using one quantum query by an algorithm that fails with proba bility at most ǫ, for some
+ǫ<1/2. Then there exists a classical algorithm that uses at most (4log2m)/(1−2/radicalbig
+ǫ(1−ǫ))
+queries and learns Cwith certainty.
+Proof.Associate each concept with an n-bit string, for some n, and suppose there exists a
+quantum algorithm that uses one query to learn Cand fails with probability ǫ<1/2. Then
+by Fact 3 there exists an input state |ψ/an}brack⌉tri}ht=/summationtextn
+i=0αi|i/an}brack⌉tri}htsuch that, for all x/n⌉}ationslash=y∈ C,
+|/an}brack⌉tl⌉{tψ|OxOy|ψ/an}brack⌉tri}ht|2≤4ǫ(1−ǫ),
+or in other words/parenleftBiggn/summationdisplay
+i=0|αi|2(−1)xi+yi/parenrightBigg2
+≤4ǫ(1−ǫ). (1)
+We now show that, if this constraint holds, there must exist a subset of the inputs S⊆[n]
+such that every pair of concepts in Cdiffers on at least one input in S, and|S|=O(logm).
+By the argument of Section 1.1, this implies that there is a no nadaptive classical algorithm
+that learns Mwith certainty using O(logm) queries.
+We will use the probabilistic method to show the existence of S. For anyk, form a
+subsetSof at most kinputs between 1 and nby a process of krandom, independent
+6choices of input, where at each stage input iis picked to add to Swith probability |αi|2.
+Now consider an arbitrary pair of concepts x/n⌉}ationslash=y, and letS+,S−be the set of inputs on
+which the concepts are equal and differ, respectively. By the c onstraint (1), we have
+4ǫ(1−ǫ)≥/parenleftBiggn/summationdisplay
+i=0|αi|2(−1)xi+yi/parenrightBigg2
+=
+/summationdisplay
+i∈S+|αi|2−/summationdisplay
+i∈S−|αi|2
+2
+=
+1−2/summationdisplay
+i∈S−|αi|2
+2
+,
+so/summationdisplay
+i∈S−|αi|2≥1
+2−/radicalbig
+ǫ(1−ǫ).
+Therefore, at each stage of adding an input to S, the probability that an input in S−is
+added is at least1
+2−/radicalbig
+ǫ(1−ǫ). So, after kstages of doing so, the probability that none
+of these inputs has been added is at most/parenleftBig
+1
+2+/radicalbig
+ǫ(1−ǫ)/parenrightBigk
+. As there are/parenleftbigm
+2/parenrightbig
+pairs of
+conceptsx/n⌉}ationslash=y, by a union bound the probability that none of the pairs of con cepts differs
+on any of the inputs in Sis upper bounded by
+/parenleftbiggm
+2/parenrightbigg/parenleftbigg1
+2+/radicalbig
+ǫ(1−ǫ)/parenrightbiggk
+≤m2/parenleftbigg1
+2+/radicalbig
+ǫ(1−ǫ)/parenrightbiggk
+.
+For anykgreater than
+2log2m
+log22/(1+2/radicalbig
+ǫ(1−ǫ))<4log2m
+1−2/radicalbig
+ǫ(1−ǫ)
+this probability is strictly less than 1, implying that ther e exists some choice of S⊆[n]
+with|S| ≤ksuch that every pair of concepts differs on at least one of the in puts inS. This
+completes the proof.
+We are finally ready to prove Theorem 2, which we restate for cl arity.
+Theorem. LetCbe a concept class containing mconcepts, and let Abe a nonadaptive
+quantum query algorithm that uses kqueries to the input to learn C, and succeeds with
+probability at least 1−ǫon every input, for some ǫ <1/2. Then there exists a classical
+nonadaptive query algorithm that learns Cwith certainty using at most
+4klog2m
+1−2/radicalbig
+ǫ(1−ǫ)
+queries to the input.
+Proof.LetOxbe the oracle operator corresponding to the concept x. Then a nonadaptive
+quantum algorithm Athat learns xusingkqueries to Oxis equivalent to a quantum
+algorithm that uses one query to O⊗k
+xto learnx. It is easy to see that this is equivalent to
+Ain fact using one query to learn the concept class C⊗k. By Lemma 5, this implies that
+there exists a classical algorithm that uses at most (4 klog2m)/(1−2/radicalbig
+ǫ(1−ǫ)) queries
+to learn C⊗kwith certainty. Finally, by Lemma 4, this implies in turn tha t there exists a
+classical algorithm that uses the same number of queries and learnsCwith certainty.
+7Acknowledgements
+I would like to thank Aram Harrow and Dan Shepherdfor helpful discussions and comments
+on a previous version. This work was supported by the EC-FP6- STREP network QICS and
+an EPSRC Postdoctoral Research Fellowship.
+References
+[1] A. Ambainis. Polynomial degree vs. quantum query comple xity.J. Comput. Syst. Sci. ,
+72(2):220–238, 2006. quant-ph/0305028 .
+[2] A. Ambainis, K. Iwama, A. Kawachi, H. Masuda, R. Putra, an d S. Yamashita. Quan-
+tum identification of Boolean oracles. In Proc. STACS 2004 , pages 93–104. Springer,
+2004.quant-ph/0403056 .
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+quantum computation in determining parity. Phys. Rev. Lett. , 81:5442–5444, 1998.
+quant-ph/9802045 .
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+[10] P. Høyer and R. ˇSpalek. Lower bounds on quantum query complexity. Bulletin
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+quant-ph/0509153 .
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+9

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@@ -0,0 +1,378 @@
+arXiv:1001.0019v1  [gr-qc]  30 Dec 2009On the instability of Reissner-Nordstr¨ om black holes in de Sitter backgrounds
+Vitor Cardoso∗
+CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico,
+Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal &
+Department of Physics and Astronomy, The University of Miss issippi, University, MS 38677-1848, USA
+Madalena Lemos†and Miguel Marques‡
+CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico,
+Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
+(Dated: November 3, 2018)
+Recent numerical investigations have uncovered a surprisi ng result: Reissner-Nordstr¨ om-de Sitter
+black holes are unstable for spacetime dimensions larger th an 6. Here we prove the existence of
+such instability analytically, and we compute the timescal e in the near-extremal limit. We find very
+good agreement with the previous numerical results. Our res ults may me helpful in shedding some
+light on the nature of the instability.
+PACS numbers: 04.50.Gh,04.70.-s
+I. INTRODUCTION
+In physics, stability of a given configuration (solution
+of some set of equations), is a useful criterium for rele-
+vance of that solution. Unstable configurations are likely
+not tobe realizablein practice, and representaninterme-
+diate stage in the evolution of the system. Nevertheless,
+the instability itself is of great interest, since an under-
+standing of the mechanism behind it may help one to
+better grasp the physics involved. In particular, it is of
+interest to be able to predict which other systems display
+similar instabilities, or even have a deeper understanding
+of the physics behind the instability (why is the system
+unstable? is there some fundamental principle behind
+the instability?).
+In General Relativity, the Kerr family exhausts the
+blackhole solutionsto the electro-vacEinstein equations.
+Kerr black holes are stable, and can therefore describe
+astrophysicalobjects. However,there aremanyinstances
+of instabilities afflicting objects with an event horizon,
+such as the Gregory-Laflamme [1], the ultra-spinning [2]
+or superradiant instabilities [3] and other instabilities of
+higher-dimensional black holes in alternative theories [4,
+5](for a review see Ref. [6]).
+Konoplya and Zhidenko (hereafter KZ) recently stud-
+ied small perturbations in the vicinity of a charged black
+hole in de Sitter background, a Reissner-Nordstr¨ om de
+Sitter black hole (RNdS) [7]. Their (numerical) results
+show that when the spacetime dimensionality D >6, the
+spacetime is unstable, provided the charge is larger than
+agiventhreshold, determined byKZforeach D. Because
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+‡Electronic address: [email protected] results are so surprising (the mechanism behind it is
+not yet understood), we set out to to investigate this in-
+stability and hopefully understand it better. Our results
+can be summarized as follows: (i) we can prove analyti-
+cally the existence of unstable modes for charge Qhigher
+thanacertainthreshold. (ii)inthenear-extremalregime,
+we are able to find an explicit solution for the unstable
+modes, determining the instability timescale analytically.
+We hope that our incursion in this topic helps to better
+understand the physics at work.
+II. EQUATIONS
+This work focuses on the higher dimensional RNdS ge-
+ometry, described by the line element
+ds2=−f dt2+f−1dr2+r2dΩ2
+n, (1)
+wheredΩ2
+nis the line element of the nsphere and
+f= 1−λr2−2M
+rn−1+Q2
+r2n−2. (2)
+the background electric field is E0=q/rn, withqthe
+electric charge. The quantities MandQare related to
+the physical mass M and charge qof the black hole [8],
+andλto the cosmological constant. The spacetime di-
+mensionality is D=n+2.
+The above geometry possesses three horizons: the
+black-holeCauchyhorizonat r=ra, the black hole event
+horizon is at r=rband the cosmological horizon is at
+r=rc, whererc> rb> ra, the only real, positive zeroes
+off. For convenience, we set rb= 1, i.e., we measure all
+quantities in terms of the event horizon rb. We thus get
+2M= 1+Q2−λ, (3)2
+Furthermore, we can also write
+λ=r−4−n
+c(rn+2
+c−r3
+c)(rn+2
+c−Q2r3
+c)
+rn+2c−rc.(4)
+For a fixed rcand spacetime dimension D, the existence
+ofaregulareventhorizonimposesthatthecharge Qmust
+be smaller than a certain value Qext. With our units this
+maximum charge is
+Q2
+ext=rn
+c/parenleftbig
+−2rc+(n+1)rn
+c−(n−1)rn+2
+c/parenrightbig
+−rc/parenleftbig
+rc(n+1)−2nrnc+(n−1)r2n+1c/parenrightbig.(5)
+Gravitational perturbations of this spacetime couple to
+the electromagnetic field, and were completely character-
+ized by Kodama and Ishibashi [8]. They can be reduced
+to a set of two second order ordinary differential equa-
+tions of the form,
+d2
+dr2∗Φ±+/parenleftbig
+ω2−VS±/parenrightbig
+Φ±= 0, (6)where the tortoise coordinate r∗and the potentials VS±
+are defined through
+r∗≡/integraldisplay
+f−1dr, V S±=fU±
+64r2H2
+±.(7)
+We have
+H+= 1−n(n+1)
+2δx, (8)
+H−=m+n(n+1)
+2(1+mδ)x, (9)
+and the quantities U±are given by
+U+=/bracketleftbig
+−4n3(n+2)(n+1)2δ2x2−48n2(n+1)(n−2)δx
+−16(n−2)(n−4)]y−δ3n3(3n−2)(n+1)4(1+mδ)x4
++4δ2n2(n+1)2/braceleftbig
+(n+1)(3n−2)mδ+4n2+n−2/bracerightbig
+x3
++4δ(n+1)/braceleftbig
+(n−2)(n−4)(n+1)(m+n2K)δ−7n3+7n2−14n+8/bracerightbig
+x2
++/braceleftbig
+16(n+1)/parenleftbig
+−4m+3n2(n−2)K/parenrightbig
+δ−16(3n−2)(n−2)/bracerightbig
+x
++64m+16n(n+2)K, (10)
+U−=/bracketleftbig
+−4n3(n+2)(n+1)2(1+mδ)2x2+48n2(n+1)(n−2)m(1+mδ)x
+−16(n−2)(n−4)m2/bracketrightbig
+y−n3(3n−2)(n+1)4δ(1+mδ)3x4
+−4n2(n+1)2(1+mδ)2/braceleftbig
+(n+1)(3n−2)mδ−n2/bracerightbig
+x3
++4(n+1)(1+ mδ)/braceleftbig
+m(n−2)(n−4)(n+1)(m+n2K)δ
++4n(2n2−3n+4)m+n2(n−2)(n−4)(n+1)K/bracerightbig
+x2
+−16m/braceleftbig
+(n+1)m/parenleftbig
+−4m+3n2(n−2)K/parenrightbig
+δ
++3n(n−4)m+3n2(n+1)(n−2)K/bracerightbig
+x
++64m3+16n(n+2)m2K. (11)
+The variables x,yand parameters µ,mare defined
+through
+x≡2M
+rn−1, y≡λr2, (12)
+µ2≡M2+4mQ2
+(n+1)2, m≡k2−nK,(13)
+andthe quantity δis implicitly givenby µ= (1+2mδ)M.
+Note that the following relations holds Q2= (n+
+1)2M2δ(1+mδ).
+Note also that for the spacetime considered in this pa-
+perK= 1, whichmeansthatthe eigenvalues k2aregivenbyk2=l(l+n−1), where lis the angular quantum
+number, that gives the multipolarity of the field. The
+behavior of the potentials varies considerably over the
+range of parameters. In Fig. 1 we show V−forD= 8,
+rc= 1/0.95,l= 2andthreedifferentvaluesofthecharge,
+Q= 0.2,0.35,0.44.
+III. A CRITERIUM FOR INSTABILITY
+A sufficient (but not necessary) condition for the exis-
+tence of an unstable mode has been proven by Buell and3
+/s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51 /s48/s44/s48/s52 /s48/s44/s48/s53/s45/s50/s48/s50/s52/s54
+/s49/s48/s52
+/s32/s86
+/s45/s49/s48/s52
+/s32/s86
+/s45
+/s32/s32/s86
+/s45
+/s114/s45/s49/s32/s81/s61/s48/s46/s50/s48
+/s32/s81/s61/s48/s46/s51/s53
+/s32/s81/s61/s48/s46/s52/s52/s49/s48/s51
+/s32/s86
+/s45
+FIG. 1: Behavior of V−for different parameters, for D= 8.
+Here we fix the event horizon at rb= 1, and the cosmological
+horizon at rc= 1/0.95. We consider l= 2 modes and three
+different charges, Q= 0.2,0.35,0.44.
+Shadwick [9] and is the following,
+/integraldisplayrc
+rbV
+fdr <0. (14)
+The instability region is depicted in figure 2 for several
+/s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s32
+/s32/s81/s47/s81
+/s101/s120/s116
+/s114
+/s98/s47/s114
+/s99
+FIG. 2: The parametric region of instability in Q/Qext−rb/rc
+coordinates, according to criterim (14), for l= 2. Top to
+bottom, D= 7,8,9,10,11.
+spacetime-dimension D, which can be compared with the
+numerical results by KZ, their figure 4. It is apparent
+that condition (14) very accurately describes the numer-
+ical results for rb/rc∼1, a regime we explore below in
+Section IV. As one moves away from extremality cri-
+terium (14) is just too restrictive. An improved analysis
+and refined criterium would be necessary to describe the
+whole rangeofthe numericalresults. Nevertheless, figure2 is very clear: higher-dimensional ( D >6) RNdS black
+holes are unstable for a wide range of parameters.
+IV. AN EXACT SOLUTION IN THE NEAR
+EXTREMAL RNDS BLACK HOLE
+Let us now specialize to the near extremal RNdS black
+hole, which we define as the spacetime for which the cos-
+mological horizon rcis very close (in the rcoordinate)
+to the black hole horizon rb, i.e.rc−rb
+rb≪1. The wave
+equationin this spacetime can be solvedexactly, in terms
+of hypergeometric functions [10]. The key point is that
+the physical region of interest (where the boundary con-
+ditions are imposed), lies between rbandrc. Thus,
+f∼2κb(r−rb)(rc−r)
+rc−rb, (15)
+where we have introduced the surface gravity κbassoci-
+ated with the event horizon at r=rb, as defined by the
+relationκb=1
+2df/drr=rb. For near-extremal black holes,
+it is approximately
+κb∼(rc−rb)(n−1)
+2r2
+b/parenleftbig
+1−nQ2/parenrightbig
+.(16)
+In this limit, one can invert the relation r∗(r) of (7) to
+get
+r=rce2κbr∗+rb
+1+e2κbr∗. (17)
+Substituting this on the expression (15) for fwe find
+f=(rc−rb)κb
+2cosh(κbr∗)2. (18)
+As such, and taking into account the functional form of
+the potentials for wave propagation, we see that for the
+near extremal RNdS black hole the wave equation (6) is
+of the form
+d2Φ(ω,r)
+dr2∗+/bracketleftBigg
+ω2−V0
+cosh(κbr∗)2/bracketrightBigg
+Φ(ω,r) = 0,(19)
+with
+V0=(rc−rb)κb
+2VS±(rb)
+f(20)
+The potential in (19) is the well known P¨ oshl-Teller po-
+tential [11]. The solutions to (19) were studied and they
+are of the hypergeometric type, (for details see Refs.
+[12, 13]). Itshouldbesolvedunderappropriateboundary
+conditions:
+Φ∼e−iωr∗, r∗→ −∞ (21)
+Φ∼eiωr∗, r∗→ ∞. (22)4
+These boundary conditions impose a non-trivial condi-
+tion onω[12, 13], and those that satisfy both simultane-
+ously are called quasinormal frequencies. For the P¨ oshl-
+Teller potential one can show [12, 13] that they are given
+by
+ω=κb/bracketleftBigg
+−/parenleftbigg
+j+1
+2/parenrightbigg
+i+/radicalBigg
+V0
+κ2
+b−1
+4/bracketrightBigg
+, j= 0,1,....
+(23)
+We conclude therefore that an instability is present
+TABLE I: The threshold of instability for near-extremal
+RNdS black holes (i.e., black holes for which the cosmologic al
+and event horizon almost coincide) for l= 2 modes. We show
+the prediction from the exact, analytic expression obtaine d
+in the near extremal limit (24), which we label Q/QN
+extand
+the one from criterium (14) which we label as Q/QV
+ext. Both
+these results are compared to the numerical results by KZ.
+D
+7 8 9 10 11 D→ ∞
+Q/QN
+ext0.913 0.774 0.683 0.617 0.567p
+2/D
+Q/QV
+ext0.913 0.775 0.684 0.618 0.568p
+2/D
+Q/QNum
+ext0.94 0.78 0.68 0.61 0.55 —
+whenever V0is negative. The threshold of stability in
+the near-extremal regime is therefore given by
+VS±(rb)
+f= 0, (24)
+The expression for VS±(rb)/fis lengthy, and we won’t
+presentit here. Thevaluesofthe charge Q/Qextthat sat-
+isfy the condition above are given in Table I (for l= 2),
+and compared to the prediction from the analysis in Sec-
+tion III, criterium (14). The agreement is excellent. Fur-thermore, we compare these predictions against the nu-
+merical results by KZ, extrapolated to the extremal limit
+(ρ= 1 in KZ notation). The agreement is remarkable.
+V. CONCLUSIONS
+We have shown analytically that charged black holes
+in de Sitter backgrounds are unstable for a wide range of
+charge and mass of the black hole, confirming previous
+numerical studies [7]. The stability properties of the ex-
+tremalD= 6 black hole remain unknown. Our methods
+and results and inconclusive at this precise point, further
+dedicated investigations would be necessary.
+Ouranalyticalresultinthenear-extremalregimecould
+be used to investigate further the nature of this instabil-
+ity, something we have not attempted to do here. A
+possible refinement concerns the large- Dlimit of the in-
+stability, where it couldbe possible to find an analytical
+expression throughout all range of parameters. We have
+inmind resultsandtechniquessimilartothoseofKoland
+Sorkin [14]. It would also be interesting to investigate
+the stability properties, using this or other techniques, of
+near-extremal Kerr-dS black holes, which have recently
+been conjectured to have an holographic description [15].
+Acknowledgements
+We warmly thank Roman Konoplya and Alexander
+Zhidenko for useful correspondence and for sharing their
+numerical results with us. This work was partially
+funded by Funda¸ c˜ ao para a Ciˆ encia e Tecnologia (FCT)-
+Portugal through projects PTDC/FIS/64175/2006,
+PTDC/ FIS/098025/2008,PTDC/FIS/098032/2008and
+CERN/FP/109290/2009.
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1001.0020.txt

@@ -0,0 +1,756 @@
+arXiv:1001.0020v2  [nlin.SI]  3 Mar 2010Classification of integrable hydrodynamic chains
+A.V. Odesskii1,2, V.V. Sokolov1
+1L.D. Landau Institute for Theoretical Physics (Russia)
+2Brock University (Canada)
+Abstract Using the method of hydrodynamic reductions, we find all inte-
+grable infinite (1+1)-dimensional hydrodynamic-type chains of shif t one. A
+class of integrable infinite (2+1)-dimensional hydrodynamic-type c hains is
+constructed.
+MSC numbers: 17B80, 17B63, 32L81, 14H70
+Address : L.D. Landau Institute for Theoretical Physics of Russian Academ y of Sciences,
+Kosygina 2, 119334, Moscow, Russia
+E-mail: [email protected], [email protected]
+1Contents
+1 Introduction 3
+2 Integrable chains and hydrodynamic reductions 4
+3 GT-systems 5
+4 Canonical forms of GT-systems associated
+with integrable chains 7
+5 Generic case 12
+6 Trivial GT-system and 2+1-dimensional integrable hydrodynamic chains 14
+7 Infinitesimal symmetries of triangular GT-systems 17
+21 Introduction
+We consider integrable infinite quasilinear chains of the form
+uα,t=φα,1u1,x+···+φα,α+1uα+1,x, α= 1,2,..., φ α,α+1/negationslash= 0, (1.1)
+whereφα,j=φα,j(u1,...,uα+1).Two chains are called equivalent if they are related by a trans-
+formation of the form
+uα→Ψα(u1,...,uα),∂Ψα
+∂uα/negationslash= 0, α= 1,2,... (1.2)
+By integrability we mean the existence of an infinite set of hydrodyna mic reductions [1, 2,
+3, 4, 5, 6].
+Example 1. The Benney equations [7, 8, 9]
+u1,t=u2,x, u 2,t=u1u1,x+u3,x,... u αt= (α−1)uα−1u1,x+uα+1,x,... (1.3)
+provide the most known example of integrable chain (1.1). The hydro dynamic reductions for
+the Benney chain were investigated in [10]. /square
+In [4, 5, 6] integrable divergent chains of the form
+u1t=F1(u1,u2)x, u2t=F2(u1,u2,u3)x,···, uit=Fi(u1,u2,...,ui+1)x,··· (1.4)
+were considered. In [6] some necessary integrability conditions we re obtained. Namely, a non-
+linear overdetermined system of PDEs for functions F1,F2was presented. The general solution
+of the system was not found. Another open problem was to prove t hat the conditions are
+sufficient. In other words, for any solution F1,F2of the system one should find functions
+Fi,i>2 such that the resulting chain is integrable.
+Probably any integrable chain (1.1) is equivalent to a divergent chain. However, the diver-
+gent coordinates are not suitable for explicit formulas. Our main obs ervation is that a conve-
+nient coordinates are those, in which the so-called Gibbons-Tsarev type system (GT-system)
+related to integrable chain is in a canonical form.
+Using our version (see [11, 12]) of the hydrodynamic reduction meth od, we describe all
+integrable chains (1.1). We establish an one-to-one corresponden ce between integrable chains
+(1.1) and infinite triangular GT-systems of the form
+∂ipj=P(pi,pj)
+pi−pj∂iu1, i/negationslash=j, (1.5)
+∂i∂ju1=Q(pi,pj)
+(pi−pj)2∂iu1∂ju1, i/negationslash=j, (1.6)
+∂ium= (gm,0+gm,1pi+···+gm,m−1pm−1
+i)∂iu1, g m,j=gm,j(u1,...,um), gm,m−1/negationslash= 0,
+3wherem= 2,3,...andi,j= 1,2,3.The functions P,Qare polynomials quadratic in each of
+variablespiandpj,with coefficients being functions of u1,u2.The functions p1,p2,p3,u1,u2,...
+in (3.11) depend on r1,r2,r3,and∂i=∂
+∂ri.
+Example 1-1 (continuation of Example 1.) The system (1.5),(1.6) corresponding t o the
+Benney chain has the following form
+∂ipj=∂iu1
+pi−pj, ∂ i∂ju1=2∂iu1∂ju1
+(pi−pj)2, (1.7)
+∂ium= (−(m−2)um−2−···−2u2pm−2
+i−u1pm−3
+i+pm−1
+i)∂iu1. (1.8)
+Equations (1.7) were firstly obtained in [10]. /square
+Given GT-system (1.5), (1.6) the coefficients of (1.1) are uniquely de fined by the following
+relations
+pi∂ium=φm,1∂iu1+···+φm,m+1∂ium+1, m= 2,3,... (1.9)
+Namely, equating the coefficients at different powers of piin (1.9), we get a triangular system
+of linear algebraic equations for φi,j. Thus, the classification problem for chains (1.1) is reduced
+to a description of all GT-systems (1.5), (1.6) . The latter problem is solved in Section 4-6.
+The paper is organized as follows. Following [11, 12], we recall main defin itions in Section
+2 (see [1, 2, 3, 11] for details). We consider only 3-component hyd rodynamic reductions since
+the existence of reductions with N >3 gives nothing new [1]. In Section 3 we formulate
+our previous results that are needed in the paper. Section 4 is devo ted to a classification of
+admissible polynomials PandQin (1.5), (1.6). In Sections 5,6 we construct integrable chains
+for the generic case and for some degenerations. Section 6 also co ntains examples of (2+1)-
+dimensional infinite hydrodynamic-type chains integrable from the v iewpoint of the method
+of hydrodynamic reductions. Infinitesimal symmetries of GT-syst ems are studied in Section 7.
+These symmetries seem to be important basic objects in the hydrod ynamic reduction approach.
+Acknowledgments. Authors thank M.V. Pavlov for fruitful discussions. V.S. is gratefu l to
+Brock University for hospitality. He was partially supported by the R FBR grants 08-01-464,
+09-01-22442-KE, and NS 3472.2008.2.
+2 Integrable chains and hydrodynamic reductions
+According to [1, 2, 3, 4, 5, 6] a chain (1.1) is called integrable if it admits sufficiently many
+so-called hydrodynamic reductions.
+Definition. A hydrodynamic (1+1)-dimensional N-component reduction of a chain (1.1)
+is a semi-Hamiltonian (see formula (3.18) ) system of the form
+ri
+t=pi(r1,...,rN)ri
+x, i= 1,..,N (2.10)
+4and functions uj(r1,...,rN), j= 1,2,...such that for each solution of (2.10) functions uj=
+uj(r1,...,rN), i= 1,...satisfy (1.1).
+Substituting ui=ui(r1,...,rN), i= 1,...into (1.1), calculating tandx-derivatives by virtue
+of (2.10) and equating coefficients at rs
+xto zero, we obtain
+∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
+It is clear from this system that
+∂suk=gk(ps,u1,...,uk)∂su1, k= 2,3,...
+wheregk(p,u1,...,uk) is a polynomial of degree k−1 inpfor eachk= 2,3,...Compatibility
+conditions∂i∂juk=∂j∂iukgive us a system of linear equations for ∂ipj, ∂jpi, ∂i∂ju1, i/negationslash=j.
+This system should have a solution (otherwise we would not have suffic iently many reductions).
+Moreover, expressions for ∂suk, k= 2,3,..., ∂jpi, ∂i∂ju1, i/negationslash=jshould be compatible and form
+a so-called GT-system.
+Remark. In the sequel we assume N= 3 because the case N >3 gives nothing new [1].
+3 GT-systems
+Definition. A compatible system of PDEs of the form
+∂ipj=f(pi,pj,u1,...,un), ∂iu1j/negationslash=i,
+∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, j/negationslash=i, (3.11)
+∂iuk=gk(pi,u1,...,un)∂iu1, k= 1,...,n−1,
+wherei,j= 1,2,3 is called n-fields GT-system . Herep1,p2,p3,u1,...,unare functions of
+r1,r2,r3and∂i=∂
+∂ri.
+Definition. Two GT-systems are called equivalent if they are related by a transformation
+of the form
+pi→λ(pi,u1,...,un), (3.12)
+uk→µk(u1,...,un), k= 1,...,n. (3.13)
+Example 2 [13]. Leta0,a1,a2be arbitrary constants, R(x) =a2x2+a1x+a0. Then the
+system
+∂ipj=a2p2
+j+a1pj+a0
+pi−pj∂iu1, ∂ i∂ju1=2a2pipj+a1(pi+pj)+2a0
+(pi−pj)2∂iu1∂ju1(3.14)
+is an one-field GT-system. The original Gibbons-Tsarev system (1.7 ) corresponds to a2=a1=
+0,a0= 1.The polynomial R(x) can be reduced to one of the following canonical forms: R= 1,
+5R=x,R=x2, orR=x(x−1) by a linear transformation (3.12). A wide class of integrable
+3D-systems of hydrodynamic type related to (3.14) is described in [1 3]. An elliptic version of
+this GT-system and the corresponding integrable 3D-systems wer e constructed in [15]. /square
+Definition. An additional system
+∂iuk=gk(pi,u1,...,un+m)∂iun, k=n+1,...,n+m (3.15)
+suchthat(3.11)and(3.15)arecompatibleiscalled an extension of(3.11)byfields un+1,...,un+m.
+It turns our that
+∂iun+1=f(pi,un+1,u1,...,un)∂iu1
+is an extension for GT-system (3.11). Stress that here fis the same function as in (3.11). We
+call this extension the regular extension byun+1.
+Example 2-1. The generic case of Example 2 corresponds to R=x(x−1). The regular
+extension by u2is given by
+∂iu2=u2(u2−1)
+pi−u2∂iu1.
+If we express u1from this formula and substitute it to (3.14), we get the following one -field
+GT-system
+∂ipj=pj(pj−1)(pi−u1)
+u1(u1−1)(pi−pj)∂iu1,
+∂i∂ju1=pipj(pi+pj)−p2
+i−p2
+j+(p2
+i+p2
+j−4pipj+pi+pj)u1
+u1(u1−1)(pi−pj)2∂iu1∂ju1./square(3.16)
+The second basic notion of the hydrodynamic reduction method is so -called GT-family of
+(1+1)-dimensional hydrodynamic-type systems.
+Definition. An (1+1)-dimensional 3-component hydrodynamic-type system o f the form
+ri
+t=vi(r1,...,rN)ri
+x, i= 1,2,3, (3.17)
+is called semi-Hamiltonian if the following relation holds
+∂j∂ivk
+vi−vk=∂i∂jvk
+vj−vk, i/negationslash=j/negationslash=k. (3.18)
+Definition. A Gibbons-Tsarev family associated with the Gibbons-Tsarev type s ystem
+(4.25) is a (1+1)-dimensional hydrodynamic-type system of the fo rm
+ri
+t=F(pi,u1,...,um)ri
+x, i= 1,2,3, (3.19)
+semi-Hamiltonian by virtue of (3.11).
+6Example 2-2 [13]. Applying the regular extension to the generic GT-system (3.14) two
+times, we get the following GT-system:
+∂ipj=pj(pj−1)
+pi−pj∂iw, ∂ ijw=2pipj−pi−pj
+(pi−pj)2∂iw∂jw, i/negationslash=j, (3.20)
+∂iuj=uj(uj−1)∂iw
+pi−uj, j= 1,2. (3.21)
+Consider the generalized hypergeometric [14] linear system of the f orm
+∂2h
+∂uj∂uk=sj
+uj−uk·∂h
+∂uk+sk
+uk−uj·∂h
+∂uj, j/negationslash=k, (3.22)
+∂2h
+∂uj∂uj=−/parenleftBigg
+1+n+2/summationdisplay
+k=1sk/parenrightBigg
+sj
+uj(uj−1)·h+sj
+uj(uj−1)n/summationdisplay
+k/negationslash=juk(uk−1)
+uk−uj·∂h
+∂uk+
+/parenleftBiggn/summationdisplay
+k/negationslash=jsk
+uj−uk+sj+sn+1
+uj+sj+sn+2
+uj−1/parenrightBigg
+·∂h
+∂uj.(3.23)
+Herei,j= 1,2 ands1,...,s4are arbitrary parameters. It easy to verify that this system is in
+involution and therefore the solution space is 3-dimensional. Let h1,h2,h3be a basis of this
+space. For any hwe put
+S(p,h) =u1(u1−1)(p−u2)hh1,u1−hu1h1
+h1+u2(u2−1)(p��u1)hh1,u2−hu2h1
+h1.
+Then the formula
+F=S(p,h3)
+S(p,h2)(3.24)
+defines the generic linear fractional GT-family for (3.20). /square
+4 Canonical forms of GT-systems associated
+with integrable chains
+For integrable chains the corresponding GT-systems involve infinite number of fields ui, i=
+1,2,...(see Example 1-1). In this Section we show that these GT-systems are equivalent to
+infinite triangular extensions of one-field GT-systems from Example s 2,3.
+A compatible system of PDEs of the form
+∂ipj=f(pi,pj,u1,...,un)∂iu1, i/negationslash=j,
+∂iuk=gk(pi,u1,...,uk)∂iu1, k= 1,2,...,, (4.25)
+7∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, i/negationslash=j,
+wherei,j= 1,2,3 is called triangular GT-system . Herep1,p2,p3,u1,u2,...are functions of
+r1,r2,r3,and∂i=∂
+∂ri.
+Definition. A chain (1.1) is called integrable if there exists a Gibbons-Tsarev type system
+of the form (4.25) and a Gibbons-Tsarev family
+ri
+t=F(pi,u1,...,um)ri
+x, i= 1,2,3, (4.26)
+such that (1.1) holds by virtue of (4.25), (4.26).
+Due to the equivalence transformations (3.12) we can assume witho ut loss of generality that
+F(p,u1,...,um) =p. (4.27)
+Under this assumption we have
+uj,t=/summationdisplay
+s∂sujrs
+t=/summationdisplay
+s∂sujpsrs
+x.
+and similar
+uj,x=/summationdisplay
+s∂sujrs
+x.
+Substituting these expressions into (1.1) and equating coefficients atrs
+xto zero, we obtain
+∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
+Using (4.25) and replacing psbyp, we get
+p=φ1,1+φ1,2g2, pg2=φ2,1+φ2,2g2+φ2,3g3, pg3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...
+Solving this system with respect to g2, g3,..., we obtain
+gi(p) =ψi,0+ψi,1p+...+ψi,i−1pi−1.
+Hereψi,jare functions of u1,...,ui. For example,
+g2=−p
+φ1,2−φ1,1
+φ1,2. (4.28)
+Remark. Since we assume that φi,i−1/negationslash= 0,we haveψi,i−1/negationslash= 0 for all i. Therefore g1=
+1,g2,...is a basis in the linear space of all polynomials in p. The coefficients φi,jof our chain
+are just entries of the matrix of multiplication by pin this basis. More generally, if we don’t
+normalizeF=p, then the coefficients φi,jcan be found from the equations
+F(p) =φ1,1+φ1,2g2, F(p)g2=φ2,1+φ2,2g2+φ2,3g3,
+F(p)g3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...(4.29)
+8Compatibility conditions ∂i∂juα=∂j∂iuα, α= 2,3,4 give a system of linear equations for
+∂ipj, ∂jpi, ∂i∂ju1. Solving this system, we obtain formulas (1.5),(1.6), where in principa lP, Q
+coulddependon u1,u2,u3,u4. However, itfollowsfromcompatibility conditions ∂i∂jpk=∂j∂ipk
+thatP, Qdepend onu1, u2only.
+Written (1.5) in the form
+∂ipj=/parenleftbiggR(pj)
+pi−pj+(z4p2
+j+z5pj+z6)pi+z4p3
+j+z3p2
+j+z7pj+z8/parenrightbigg
+∂iu1, (4.30)
+whereR(x) =z4x4+z3x3+z2x2+z1x+z0,one can derive from the compatibility conditions
+∂i∂jpk=∂j∂ipk,∂i∂ju1=∂j∂iu1that the equation (1.6) has the following form
+∂i∂ju1=/parenleftbigg2z4p2
+ip2
+j+z3pipj(pi+pj)+z2(p2
+i+p2
+j)+z1(pi+pj)+2z0
+(pi−pj)2+z9/parenrightbigg
+∂iu1∂ju1.(4.31)
+It is easy to verify that we can normalize z9=z6−z7, g2=pby a transformation (1.2).
+Then the coefficients zi(x,y),i= 0,...,8 satisfy the following pair of compatible dynamical
+systems with respect to yandx:
+z0,y= 2z0z5−z1z6, z 1,y= 4z0z4+z1z5−2z2z6, z 2,y= 3z1z4−3z3z6,
+z3,y= 2z2z4−z3z5−4z4z6, z 4,y=z3z4−2z4z5, z 5,y=z4z7−z4z6−z2
+5,
+z6,y=z4z8−z5z6, z 7,y= 2z1z4−2z3z6−z5z6+z4z8, z 8,y= 2z0z4−z2
+6−z6z7+z5z8,
+and
+z0,x=−z0z2−z0z6+3z0z7−z1z8, z 1,x=−z1z2+3z0z3−z1z6+2z1z7−2z2z8,
+z2,x=−z2
+2+2z1z3+4z0z4−z2z6+z2z7−3z3z8, z 3,x= 3z1z4−z3z6−4z4z8,
+z4,x=z2z4−z4z6−z4z7, z 5,x=z1z4−z5z6−z4z8, z 6,x=z0z4−z2
+6,
+z7,x=z1z3+3z0z4+z1z5−z2z6−z2z7+z2
+7−z3z8−2z5z8,
+z8,x=z0z3+z0z5−z2z8−2z6z8+z7z8.
+These is a complete description of the GT-systems related to integr able chains (1.1).
+To solve the dynamical systems we bring the polynomial Rto a canonical form sacrificing
+to the normalization (4.27).
+It is obvious that linear transformations pi→api+b, wherea,bare functions of u1,u2,
+preserve the form of GT-system (4.30),(4.31). Moreover, there exist transformations of the
+form
+pi=a¯pi+b
+¯pi−ψ, i= 1,2,3 (4.32)
+9preserving the form of GT-system (4.30),(4.31). Such admissible tr ansformations are described
+by the following conditions:
+au2=z4(b+aψ), b u2=z4bψ+z5b−z6a, ψ u2=z4ψ2+z5ψ+z6.
+Under transformations (4.32) the polynomial Ris transformed by the following simple way:
+R(pi)→(pi−ψ)4R/parenleftBigapi+b
+pi−ψ/parenrightBig
+.
+Suppose that Rhas distinct roots. It is possible to verify that by an admissible trans formation
+(4.32) we can move three of the four roots to 0 ,1 and∞. It follows from compatibility
+conditionsfortheGT-system thatthenthefourthroot λ(u1,u2)doesnotdependon u2. Making
+transformation of the form u1→q(u1) we arrive at the canonical forms λ=u1orλ=const. It
+is straightforwardly verified that in the first case equations (4.30) , (4.31) coincides with (3.16).
+In the second case the GT-system does not exist.
+In the case of multiple roots the polynomial R(x) can be reduced to one of the following
+forms:R= 0,R= 1,R=x,R=x2, orR=x(x−1).In all these cases equations (4.30),
+(4.31) coincides with the corresponding equations from Example 2.
+Thus, the following statement is valid:
+Proposition 1. There are 6 non-equivalent cases of GT-systems (4.30), (4.31). T he canon-
+ical forms are:
+Case 1: (3.16) (generic case);
+Case 2: (3.14) with R(x) =x(x−1);
+Case 3: (3.14) with R(x) =x2;
+Case 4: (3.14) with R(x) =x;
+Case 5: (3.14) with R(x) = 1.
+Case 6: (3.14) with R(x) = 0./square
+Remark. Cases 2-6 can be obtained from Case 1 by appropriate limit procedur es. For
+example, Case 2 corresponds to the limit u1→u1
+ε, ε→0.
+It follows from (4.27), (4.28) that for any canonical form the func tionsFandg2have the
+following structure:
+g2(pi) =k1pi+k2
+k3pi+k4, F(pi) =f1pi+f2
+k3pi+k4, (4.33)
+where the coefficients are functions of u1,u2.
+Lemma 1. For theCase 1 any function g2can bereduced by anappropriatetransformation
+10¯u2=σ(u1,u2) to one of the following canonical forms:
+a1:g2(p) =u2(u2−1)(p−u1)
+u1(u1−1)(p−u2)(regular extension);
+b1:g2(p) =1
+p−u1;
+c1:g2(p) =u−λ
+1(u1−1)λ−1
+p−λλ= 1,0;
+d1:g2(p) =u1−u2
+u1(u1−1)p+u2−1
+u1−1./square
+The GT-system from the Case 1 possesses a discrete automorphis m groupS4interchanging
+the points 0 ,1,∞,u1. The group is defined by generators
+σ1:u1→1−u1, pi→1−pi, σ 2:u1→u1
+u1−1, pi→pi
+pi−1,
+and
+σ3:u1→1−u1, pi→(1−u1)pi
+pi−u1.
+Up to this group the cases b1,c1,d1are equivalent and one can take say the case d1for further
+consideration. The case a1is invariant with respect to the group.
+Remark. The casesb1, c1, d1are degenerations of the case a1. Namely, they can be
+obtained as appropriate limit u2→u1,u2→λ, u2→ ∞correspondingly.
+All possible functions g2for Cases 2-5 are described in the following
+Lemma 2. For the GT-system (3.14) (excluding Case 6) any function g2can be reduced
+by an appropriate transformation ¯ u2=σ(u1,u2) to one of the following canonical forms:
+a2:g2(p) =R(u2)
+p−u2(regular extension);
+b2:g2(p) =1
+p−λ,whereR(λ) = 0;
+c2:g2(p) =p−a2u2.
+The discrete automorphism of the GT-system interchanges the ro ots ofRin the case b2./square
+Lemma 3. For the GT-system (3.14) with R(x) = 0 (Case 6) any function g2can be
+reduced to g2(p) =pby an appropriate transformation ¯ u2=σ(u1,u2). Furthermore, the
+corresponding triangular GT-system has the form
+∂ipj= 0, ∂ i∂ju1= 0, ∂ iuk=pk−1
+iu1, k= 2,3,.../square (4.34)
+115 Generic case
+The next step in the classification is to find all functions Fof the form (4.28) for each pair
+consisting of a GT-system from Proposition 1 and the correspondin gg2from Lemmas 1-3.
+The semi-Hamiltonian condition (3.18) yields a non-linear system of PDE s for the functions
+f1(u1,u2),f2(u1,u2).For each case this system can be reduced to the linear generalized h yper-
+geometric system (3.22), (3.23) with a special set of parameters s1,s2,s3,s4or to a degeneration
+of this system.
+The general linear fractional GT-family for the generic case 1, a1is given by (3.24). Ac-
+cording to (4.33), the additional restriction is that the root of the denominator has to be equal
+u2.It is easy to verify that this is equivalent to s2= 0,h1,u2=h2,u2= 0. The latter means that
+h1(u1),h2(u1) are linear independent solutions of the standard hypergeometric equation
+u(u−1)h(u)′′+[s1+s3−(s3+s4+2s1)u]h(u)′+s1(s1+s3+s4+1)h(u) = 0.(5.35)
+The function h3(u1,u2) is arbitrary solution of (3.22), (3.23) with s2= 0 linearly independent
+ofh1(u1),h2(u1). Without loss of generality we can choose
+h3(u1,u2) =/integraldisplayu2
+0(t−u1)s1ts3(t−1)s4dt.
+Formula (3.24) gives
+F(p,u1,u2) =f1(u1,u2)p−f2(u1,u2)
+p−u2, (5.36)
+where
+f1=u2(u2−1)h1h3,u2+u1(u1−1)(h1h3,u1−h3h′
+1)
+u1(u1−1)(h1h′
+2−h2h′
+1),
+f2=u1u2(u2−1)h1h3,u2+u2u1(u1−1)(h1h3,u1−h3h′
+1)
+u1(u1−1)(h1h′
+2−h2h′
+1).
+Notice that h1h′
+2−h2h′
+1=const(u1−1)s1+s4us1+s3
+1.
+For integer values of s1,s3,s4the hypergeometric system can be solved explicitly. For
+example, if s1=s3=s4= 0, the above formulas give rise to F=g2.Ifs4=−2−s1−s3then
+F=(u2−u1)s1+1us3+1
+2(u2−1)−1−s1−s3
+p−u2;
+ifs4= 0,then
+F=(p−1)(u2−u1)s1+1us3+1
+2(u1−1)−1−s1
+p−u2.
+Nowwearetofindthefunctions g3,g4,...in(4.25). Thesefunctionsaredefineuptoarbitrary
+transformation (1.2), where α= 3,4,.... In practice, one can look for functions g3,g4,...linear
+inui,i>2 (cf. (1.8)). An extension linear in ui,i>2 is given by
+g3(p) =−(u1−u2)(u2−1)p
+u1(u1−1)(p−u2)2,
+12gi(p) =(i−3)(u1−u2)(u2−1)pui
+u1(u1−1)(p−u2)2−(u1−u2)i−3(u2−1)2p(p−u1)(p−1)i−4
+u1(u1−1)i−2(p−u2)i−1−
+i−4/summationdisplay
+s=1(i−s−2)(u1−u2)s(u2−1)2p(p−u1)(p−1)s−1ui−s
+u1(u1−1)s+1(p−u2)s+2.
+The coefficients of the chain (1.1) corresponding to Case 1, a1are determined from (4.29),
+whereFis given by (5.36). Relations (4.29) are equivalent to a triangular syst em of linear
+algebraic equations. Solving this system, we find that for i>4 coefficients of the chain read:
+φi,i+1=(u1−1)(f1u2−f2)
+(u2−1)(u1−u2)def=Q1, φ i,i=f2−f1
+u2−1def=Q2,
+φi,4=−uiQ1, φ i,3=−/parenleftBig
+(u4+i−3)ui+(2−i)ui+1/parenrightBig
+Q1def=Ai,
+andφi,j= 0 for all remaining i,j.Fori≤4 we have
+φ1,1=f1u1−f2
+u1−u2, φ 1,2=−u1
+u2Q1,
+φ2,1=(u2−1)(f1u2−f2)
+(u1−1)(u1−u2), φ 2,2=f2u1−f1u2
+2
+u2(u1−u2), φ 2,3=f1u2−f2,
+φ3,1=φ3,2= 0, φ 3,3=Q2−(u4−1)Q1, φ 3,4=−Q1,
+φ4,1=φ4,2= 0, φ 4,3=A4, φ 4,4=Q2−u4Q1, φ 4,5=Q1.(5.37)
+The explicit formulas for other cases of Proposition 1 can be obtaine d by limits from the
+above formulas. We outline the limit procedures for the case 1, d1. In this case the limit is
+given byu2→u1+εu2, ε→0.It is easy to check that under this limit the extension a1
+turns tod1. The limit of the system (3.22), (3.23) with s2= 0 can be easily found. The general
+solution of the system thus obtained is given by h=c1(u2−u1)1+s1+s3+s4+h1,whereh1is the
+general solution of (5.35). Let h1,h2be solutions of (5.35), and h3= (u2−u1)1+s1+s3+s4. Then
+the limit procedure in (5.36) gives rise to
+F(p,u1,u2) =Q×/parenleftBig
+(1+s1+s3+s4)h1(p−u1)+u1(u1−1)h′
+1/parenrightBig
+,
+where
+Q= (u2−u1)1+s1+s3+s4(u1−1)−1−s1−s4u−1−s1−s3
+1.
+As usual, the most degenerate cases in classification of integrable P DEs could be interesting
+for applications. In our classification they are Case 5, c2and Case 6. The Benney chain
+(see Examples 1 and 1-1) belongs to Case 5, case c2(i.eg2=p). Any GT-family has the form
+F=f1(u1,u2)p+f2(u1,u2). Iff1= 1 thenF=p+k2u2+k1u1.The Benney case corresponds to
+13k1=k2= 0. For arbitrary kiwe get the Kupershmidt chain [16]. In the case f1=A(u1),A′/negationslash= 0
+we obtain:
+f1=k2exp(λu1)+k1, f 2=k2k3exp(λu1)+λk1(k3u1−u2).
+In the generic case
+F= exp(λu2)(S1(u1)p+S2(u1)),
+where the functions Sican be expressed in terms of the Airy functions.
+6 Trivial GT-system and 2+1-dimensional integrable hy-
+drodynamic chains
+It was observed in [11] that (2+1)-dimensional systems of hydro dynamic type with the trivial
+GT-system usually admit some integrable multi-dimensional generaliza tions. For the chains
+such GT-system is defined by (4.34). That is why the Case 6 is of a gre at importance in our
+classification. The automorphisms of (4.34) are given by
+pj→pj, j= 1,...,N, u i→νui+γi, i= 1,2,...; (6.38)
+pj→apj+b, j= 1,...,N, u i→ai−1ui+(i−1)ai−2bui−2+...+bi−1u1, i= 1,2,...
+The corresponding GT-families are of the form F(p) =A(u1,u2)p+B(u1,u2), where
+A(x,y),B(x,y) satisfies the following system of PDEs:
+AByy=AyBy, AB xy=AyBx, AB xx=AxBx,
+AAyy=A2
+y, AA xy=AxAy, AA xx=A2
+x+AxBy−AyBx.(6.39)
+This system can be easily solved in elementary functions. For each so lution formula (4.29)
+defines the corresponding integrable chain (1.1).
+It follows from (6.39) that there are two types of u2-dependence:
+1(generic case). F(p) = exp(λu2)/parenleftBig
+a(u1)p+b(u1)/parenrightBig
+,
+2. F(p) =a(u1)p+λu2+b(u1).
+In the first case there are two subcases: b′/negationslash= 0 andb′= 0.The first subcase gives rise to
+a=σ′, b=k1σ σ(x) =c1exp(µ1x)+c2exp(µ2x),wherec1c2(λk1−µ1µ2) = 0.
+The second subcase leads to
+b=c1, a(x) =c2exp(µx)+c3,wherec2(c1λ−c3µ) = 0.
+The same subcases for the case 2 yield
+a=σ′, b=k1σ σ(x) =c1+c2x+c3exp(µx),wherec3(λ−c2µ) = 0,
+14and
+b=c1, a(x) =c2exp(µx)+c3,wherec2(λ−c3µ) = 0.
+It is easy to verify that in the generic case the function Fcan be reduced by (6.38) to the
+form
+F(p) =eu2+u1(p−1)+eu2−u1(p+1).
+In this case the corresponding chain reads as
+uk,t= (eu2+u1+eu2−u1)uk+1,x+(eu2−u1−eu2+u1)uk,x, k= 1,2,3,... (6.40)
+Asusual, thischainisthefirstmember ofaninfinitehierarchy. These condflowofthishierarchy
+is given by
+uk,τ= (eu2+u1+eu2−u1)uk+2,x+(u3−u1)(eu2+u1+eu2−u1)uk+1,x+
+(eu2+u1(u1−u3−1)+eu2−u1(u3−u1−1))uk,x, k= 1,2,3,...
+In the case 2 with c3=λ= 0,k1= 1 we get the chain
+uk,t=uk+1,x+u1uk,x, k= 1,2,3,... (6.41)
+This chain is equivalent to the chain of the so-called universal hierarc hy [17]. The chain (6.41)
+is a degeneration of the chain
+uk,t=uk+1,x+u2uk,x, k= 1,2,3,... (6.42)
+Following the line of [3, 11] it is not difficult to find (2+1)-dimensional inte grable generaliza-
+tions for all (1+1)-dimensional integrable chains constructed abo ve. Some families of functions
+Fdescribed above linearly depend on two parameters. Denote these parameters by γ1,γ2.The
+corresponding integrable chain
+uk,t=γ1(φk,1u1,x+···+φk,k+1uk+1,x)+γ2(ψk,1u1,x+···+ψk,k+1uk+1,x)
+is also linear in γ1,γ2.We claim that the following (2+1)-dimensional chain
+uk,t= (φk,1u1,x+···+φk,k+1uk+1,x)+(ψk,1u1,y+···+ψk,k+1uk+1,y) (6.43)
+is integrable from the viewpoint of the method of hydrodynamic redu ctions. For each case the
+reductions can be easily described.
+For example, in the generic case
+F(p) =γ1eu2+u1(p−1)+γ2eu2−u1(p+1)
+formula (6.43) yields (2+1)-dimensional chain
+uk,t=eu2+u1(uk+1,x−uk,x)+eu2−u1(uk+1,y+uk,y), k= 1,2,3,... (6.44)
+15After a change of variables of the form
+x→ −1
+2x, y→1
+2y, u 1→1
+2u0, u2→u1+1
+2u0, u3→ −2u2+1
+2u0,...
+(6.44) can be written as
+u0,t=eu1u0,y+eu1(u1,y−eu0u1,x), u i,t=eu0+u1ui,x+eu1(eu0ui+1,x−ui+1,y),(6.45)
+wherei= 1,2,.... Probably (6.45) is a first example of a (2+1)-dimensional chain integ rable
+from the viewpoint of the hydrodynamic reduction approach.
+TriangularGT-systemsrelatedtointegrable(2+1)-dimensionalch ainswithfields u0,u1,u2,...
+have the form
+∂ipj=f1(pi,qi,pj,qj,u0,...,un)∂iu0, ∂ iqj=f2(pi,qi,pj,qj,u0,...,un)∂iu0,
+∂i∂ju0=h(pi,qi,pj,qj,u0,...,un)∂iu0∂ju0, (6.46)
+∂iuk=gk(pi,qi,u0,...,uk+1)∂iu0, k= 0,1,2,...
+Herei/negationslash=j, i,j= 1,...,3,p1,...,p3, q1,...,q3,u0,u1,u2,...,arefunctionsof r1,r2,r3.Inparticular,
+the GT-system associated with (6.45) has the form:
+∂ipj=∂i∂ju0= 0, ∂ iqj=/parenleftBigpiqi−pjqj
+pi−pj−qiqj/parenrightBig
+∂iu0, ∂ iuk=−pi
+(pi−1)k∂iu0.
+Thehydrodynamicreductionsof(6.45)isgivenbythepairofsemi-ha miltonian(1+1)-dimensional
+systems
+ri
+y=eu0/parenleftBig
+1−1
+qi/parenrightBig
+ri
+x, ri
+t=eu0+u1/parenleftBig1
+(pi−1)qi+1/parenrightBig
+ri
+x.
+Chain (6.45) is the first member of an infinite hierarchy of pairwise com muting flows where
+the corresponding ”times” are t1=t, t2, t3,.... These flows and their hydrodynamic reductions
+can be described in terms of the generating function U(z) =u1+u2z+u3z2+...The hierarchy
+is given by
+D(z)u0=eU(z)/parenleftBig
+u0,y+U(z)y−eu0U(z)x/parenrightBig
+,
+D(z1)U(z2) =eu0+U(z1)U(z2)x+(1+z1)eU(z1)/parenleftBig
+eu0U(z1)x−U(z2)x
+z1−z2−U(z1)y−U(z2)y
+z1−z2/parenrightBig
+,
+whereD(z) =∂
+∂t1+z∂
+∂t2+z2∂
+∂t3+...The reductions can be written as
+D(z)ri=eu0+U(z)/parenleftBig
+1+1+z
+(pi−1−z)qi/parenrightBig
+ri
+x.
+Other (2+1)-dimensional integrable chains related to 2-dimensiona l vector spaces of solu-
+tions for system (6.39) are degenarations of (6.45). In particular F=γ1eu1p+γ2(p+u2) leads
+to the following (2+1)-dimensional integrable generalization of (6.44 ):
+uk,t=eu1uk+1,x+uk+1,y+u2uk,y, k= 1,2,3,....
+16Conjecture. Any chain of the form (6.43) integrable by the hydrodynamic reduct ion
+method is a degeneration of (6.45).
+We are planning to consider the problem of classification of integrable chains (6.43) in a
+separate paper.
+7 Infinitesimal symmetries of triangular GT-systems
+A scientific way to construct the functions g3,g4,...for different cases from Proposition 1 is
+related to infinitesimal symmetries of the corresponding GT-syste m1. The whole Lie algebra
+of symmetries is one the most important algebraic structures relat ed to any triangular GT-
+system (4.25). In particular, this algebra acts on the hierarchy of the commuting flows for the
+corresponding chain (1.1).
+A vector field
+S=N/summationdisplay
+j=1X(pj,u1,...,us)∂
+∂pj+∞/summationdisplay
+m=1Ym(u1,...,ukm)∂
+∂um,∂Ym
+∂ukm/negationslash= 0 (7.47)
+is called a symmetry of the triangular GT-system (4.25) if it commutes with all ∂i.Notice that
+it follows from the definition that
+S(∂iu1) =∂i(Y1).
+We call (7.47) a symmetry of shift difkm=m+dform>>0.LetMbe the minimal integer
+such thatkm=m+d,m>M. If the functions gi,i= 1,...,M+dfrom (4.25) are known, then
+the functions X,Y1,...YMcan be found from the compatibility conditions
+S(∂ipj) =∂iS(pj), S(∂iuk) =∂iS(uk), k= 1,...,M.
+The functions YM+1,YM+2,...can be chosen arbitrarily. After that gM+d+1,gM+d+2,...are
+uniquely defined by the remaining compatibility conditions.
+The generic case 1, a 1. Looking for symmetries of shift one, we find X=Y1= 0 and
+M= 1. Hence without loss of generality we can take
+S=∞/summationdisplay
+m=2um+1∂
+∂um
+for the symmetry. This fact gives us a way to construct all functio nsgi,i >3 in the infinite
+triangular extension for the case 1, a1.Indeed, it follows from the commutativity conditions
+S(∂iuk) =∂iS(uk) thatgk+1=S(gk),wherek= 2,3,.... In particular,
+g3=(pj−u1)(2pju2−pj−u2
+2)u3
+u1(u1−1)(pj−u2)2.
+1Note that these functions are not unique because of the triangula r group of symmetries (1.2) acting on the
+fieldsu3,u4,...
+17The functions githus constructed are not linear in u3.The corresponding chain (1.1) is equiv-
+alent to the chain constructed in Section 5 but not so simple.
+It would be interesting to describe the Lie algebra of all symmetries in this case. Here we
+present the essential part for symmetry of shift 2:
+X=pj(pj−1)u2
+3
+(pj−u2)u2(u2−1), Y 1=u1(u1−1)u2
+3
+(u1−u2)u2(u2−1),
+Y2=−3
+2u4+(2u1−1)u2
+3
+u2(u2−1)+u3./square
+The case 1, d 1. One can add fields u3,...in such a way that the whole triangular GT-
+system admits the following symmetry of shift 1:
+S=u2
+u1(u1−1)N/summationdisplay
+i=1pi(pi−1)∂
+∂pi+∞/summationdisplay
+i=1ui+1∂
+∂ui.
+As in the previous example, one can easily recover the whole GT-syst em. For example,
+∂iu3=/parenleftbiggu3(pi+u1−1)
+u1(u1−1)+2u2
+2pi(pi−1)
+u2
+1(u1−1)2/parenrightbigg
+∂iu1./square
+Below we describe the symmetry algebra for the case 5, c2(in particular, for the Benney
+chain).
+The case 5, c 2. For the triangular GT-system (1.7), (1.8) there exists an infinite L ie
+algebra of symmetries Si,i∈Z,whereSiis a symmetry of shift i. The simplest symmetries
+are the following:
+S−2=∂
+∂u1+∞/summationdisplay
+i=3/parenleftBig
+−ui−2+/summationdisplay
+k+m=i−3ukum−/summationdisplay
+k+m+l=i−4ukumul+···/parenrightBig∂
+∂ui,
+S−1=N/summationdisplay
+j=1∂
+∂pj+∞/summationdisplay
+i=1(i−1)ui−1∂
+∂ui,
+S0=N/summationdisplay
+j=1pj∂
+∂pj+∞/summationdisplay
+i=1(i+1)ui∂
+∂ui,
+S1=N/summationdisplay
+j=1(p2
+j+3u1)∂
+∂pj+∞/summationdisplay
+i=1(i+3)ui+1∂
+∂ui+∞/summationdisplay
+i=2/summationdisplay
+k+m=iukum∂
+∂ui+∞/summationdisplay
+i=23(i−1)u1ui−1∂
+∂ui,
+S2=N/summationdisplay
+j=1(p3
+j+4u1pj+5u2)∂
+∂pj+∞/summationdisplay
+i=1(i+5)ui+2∂
+∂ui+∞/summationdisplay
+i=14iu1ui∂
+∂ui+∞/summationdisplay
+i=25(i−1)u2ui−1∂
+∂ui+
+18∞/summationdisplay
+i=1/summationdisplay
+k+m=i+13ukum∂
+∂ui+∞/summationdisplay
+i=3/summationdisplay
+k+m+l=iukumul∂
+∂ui.
+The whole algebra is generated by S1,S2,S−1,S−2.It is isomorphic to the Virasoro algebra with
+zero central charge.
+LetDtibe the vector fields corresponding to commuting flows for the Benn ey chain. Here
+Dt1=Dx, Dt2=Dt. Then the commutator relations
+[S1,Dti] = (i+1)Dti+1
+hold. Thus the vector field S1plays the role of a master-symmetry for the Benney hierarchy.
+/square
+The case 6 . In this case there exist infinitesimal symmetries of form
+Ti=ui+1∂
+∂u1+ui+2∂
+∂u2+..., i= 0,1,2,...
+Si=N/summationdisplay
+j=1pi+1
+j∂
+∂pj+ui+2∂
+∂u2+2ui+3∂
+∂u3+3ui+4∂
+∂u4+..., i=−1,0,1,2,...
+Note that [Si,Sj] = (j−i)Si+j,[Ti,Tj] = 0,[Si,Tj] =jTi+j./square
+References
+[1]E.V. Ferapontov, K.R. Khusnutdinova , On integrability of (2+1)-dimensional quasilinear
+systems, Comm. Math. Phys. 248(2004) 187-206,
+[2]E.V. Ferapontov, K.R. Khusnutdinova , The characterization of 2-component (2+1)-
+dimensional integrablesystemsofhydrodynamic type, J.Phys. A: Math.Gen. 37(8)(2004)
+2949 - 2963.
+[3]E.V. Ferapontov, K.R. Khusnutdinova , Hydrodynamic reductions of multidimensional
+dispersionless PDEs: the test for integrability, J. Math. Phys. 45(6) (2004) 2365 - 2377.
+[4]M.V. Pavlov , Classification of the Egorov hydrodynamic chains. Theor. Math. P hys.138
+No. 1 (2004) 55-71.
+[5]M.V. Pavlov , Classification of integrable hydrodynamic chains and generating fu nctions
+of conservation laws, J. Phys. A: Math. Gen. 39(34) (2006) 10803–10819.
+[6]E.V. Ferapontov, D.G. Marshal , Differential-geometric approach to the integrability of
+hydrodynamic chains: the Haanties tensor, Math. Ann. 339(1), (2007) 61–99.
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+20

1001.0021.txt

@@ -0,0 +1,884 @@
+arXiv:1001.0021v3  [cond-mat.quant-gas]  8 Oct 2010Strong-coupling expansionforthe two-species Bose-Hubba rd model
+M. Iskin
+Department of Physics, Koc ¸ University, Rumelifeneri Yolu , 34450 Sariyer, Istanbul, Turkey
+(Dated: August 28, 2018)
+Toanalyze the ground-state phase diagram ofBose-Bose mixt ures loadedinto d-dimensional hypercubic op-
+tical lattices, we perform a strong-coupling power-series expansion in the kinetic energy term (plus a scaling
+analysis) for the two-species Bose-Hubbard model with onsi te boson-boson interactions. We consider both
+repulsive and attractive interspecies interaction, and ob tain an analytical expression for the phase boundary be-
+tweentheincompressibleMottinsulatorandthecompressib lesuperfluidphaseuptothirdorderinthehoppings.
+In particular, we find a re-entrant quantum phase transition from paired superfluid (superfluidity of composite
+bosons, i.e. Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two and three di-
+mensions, whentheinterspecies interactionissufficientl ylargeandattractive. Wehope thatsome ofourresults
+couldbe testedwithultracoldatomic systems.
+PACS numbers: 03.75.-b, 37.10.Jk,67.85.-d
+I. INTRODUCTION
+Single-species Bose-Hubbard (BH) model is the bosonic
+generalization of the Hubbard model, and was introduced
+originallytodescribe4Heinporousmediaordisorderedgran-
+ular superconductors [1]. For hypercubic lattices in all di -
+mensions d, there are only two phases in this model: an in-
+compressible Mott insulator at commensurate (integer) fill -
+ings and a compressible superfluid phase otherwise. The su-
+perfluid phase is well described by weak-coupling theories,
+buttheinsulatingphaseisastrong-couplingphenomenonth at
+only appearswhen the system is on a lattice. Transition from
+the Mott insulator to the superfluid phase occurs as the hop-
+ping, particle-particleinteraction,or the chemical pote ntial is
+varied[1].
+It is the recent observation of this transition in effective ly
+three- [2], one- [3], and two-dimensional [4, 5] optical lat -
+tices, which has been considered one of the most remarkable
+achievements in the field of ultracold atomic gases, since it
+paved the way for studying other strongly correlated phases
+in similar setups. Such lattices are created by the intersec tion
+of laser fields, and they are nondissipative periodic potent ial
+energy surfaces for the atoms. Motivated by this success in
+experimentally simulating the single-species BH model wit h
+ultracoldatomic Bose gasesloaded into optical lattices, t here
+has been recently an intense theoretical activity in analyz ing
+BH aswell asFermi-Hubbardtypemodels[6].
+For instance, in addition to the Mott insulator and single-
+species superfluid phases, it has been predicted that the two -
+species BH model has at least two additional phases: an in-
+compressible super-counter flow and a compressible paired
+superfluidphase[7–16]. Ourmaininteresthereisinthelatt er
+phase,wherea directtransitionfromtheMott insulatorto t he
+paired superfluid phase (superfluidity of composite bosons,
+i.e. Bose-Bose pairs) has been predicted, when both species
+have integer fillings and the interspecies interaction is su ffi-
+ciently large and attractive. Given that the interspecies i nter-
+actions can be fine tuned in ongoing experiments, e.g. with
+41K-87Rb with mixtures [17, 18], via using Feshbach reso-
+nances,we hopethat someof ourresults couldbe tested with
+ultracoldatomicsystems.Inthispaper,weexaminetheground-statephasediagramof
+the two-species BH model with on-site boson-boson interac-
+tionsind-dimensionalhypercubiclattices, includingboth the
+repulsive and attractive interspecies interaction, via a s trong-
+coupling perturbation theory in the hopping. We carry the
+expansion out to third-order in the hopping, and perform a
+scaling analysis using the known critical behavior at the ti p
+of the insulating lobes, which allows us to accurately predi ct
+the critical point, and the shape of the insulating lobes in t he
+plane of the chemical potential and the hopping. This tech-
+niquewaspreviouslyusedtodiscussthephasediagramofthe
+single-species BH model [19–23], extended BH model [24],
+and of the hardcore BH model with a superlattice [25], and
+its results showed an excellent agreement with Monte Carlo
+simulations [23, 25]. Motivated by the success of this tech-
+nique with these models, here we apply it to the two-species
+BH model, hoping to develop an analytical approach which
+couldbeasaccurateasthenumericalones.
+The remaining paper is organized as follows. After in-
+troducing the model Hamiltonian in Sec. II, we develop the
+strong-coupling expansion in Sec. III, where we derive an
+analytical expression for the phase boundary between the in -
+compressible Mott insulator and the compressible superflui d
+phase. Then, in Sec. IV, we proposea chemical-potentialex-
+trapolation technique based on scaling theory to extrapola te
+ourthird-orderpower-seriesexpansioninto a functionalf orm
+thatisappropriatefortheMottlobes,anduse ittoobtainty p-
+ical ground-state phase diagrams. A brief summary of our
+conclusionsisgiveninSec.V.
+II. TWO-SPECIESBOSE-HUBBARDMODEL
+TodescribeBose-Bosemixturesloadedintoopticallattice s,
+weconsiderthe followingtwo-speciesBH Hamiltonian,
+H=−/summationdisplay
+i,j,σtij,σb†
+i,σbj,σ+/summationdisplay
+i,σUσσ
+2/hatwideni,σ(/hatwideni,σ−1)
++U↑↓/summationdisplay
+i/hatwideni,↑/hatwideni,↓−/summationdisplay
+i,σµσ/hatwideni,σ, (1)2
+where the pseudo-spin σ≡ {↑,↓}labels the trapped hyper-
+fine states of a given species of bosons, or labels different
+types of bosons in a two-species mixture, tij,σis the tun-
+neling (or hopping) matrix between sites iandj,b†
+i,σ(bi,σ)
+is the boson creation (annihilation) and /hatwideni,σ=b†
+i,σbi,σis
+the boson number operator at site i,Uσσ′is the strength of
+the onsite boson-bosoninteraction between σandσ′compo-
+nents, and µσis the chemical potential. In this manuscript,
+we considera d-dimensionalhypercubiclattice with Msites,
+forwhich we assume tij,σis a real symmetricmatrix with el-
+ementstij,σ=tσ≥0foriandjnearest neighbors and 0
+otherwise. Thelattice coordinationnumber(orthe numbero f
+nearestneighbors)forsuchlatticesis z= 2d.
+We take the intraspecies interactions to be repulsive
+({U↑↑,U↓↓}>0), but discuss both repulsive and attractive
+interspecies interaction U↑↓as long as U↑↑U↓↓> U2
+↑↓. This
+guarantees the stability of the mixture against collapse wh en
+U↑↓≪0,andagainstphaseseparationwhen U↑↓≫0. How-
+ever,whentheinterspeciesinteractionissufficientlylar geand
+attractive, we note that instead of a direct transition from the
+Mottinsulatortoasingleparticlesuperfluidphase,itispo ssi-
+bletohaveatransitionfromtheMottinsulatortoa pairedsu -
+perfluid phase (superfluidity of composite bosons, i.e. Bose -
+Bose pairs) [7–16]. Therefore, one needs to consider both
+possibilities,asdiscussednext.
+III. STRONG-COUPLINGEXPANSION
+We use the many-body version of Rayleigh-Schr¨ odinger
+perturbation theory in the kinetic energy term to perform th e
+expansion (in powers of t↑andt↓) for the different energies
+needed to carryout our analysis. The strong-couplingexpan -
+sion technique was previously used to discuss the phase di-
+agram of the single-species BH model [19–21, 23], extended
+BHmodel[24],andofthehardcoreBHmodelwithasuperlat-
+tice [25], and its results showed an excellent agreement wit h
+Monte Carlo simulations [23, 25]. Motivated by the success
+of this technique with these models, here we apply it to the
+two-speciesBH model.
+To determine the phase boundary separating the incom-
+pressible Mott phase from the compressible superfluid phase
+within the strong-coupling expansion method, one needs the
+energyoftheMottphaseandofits‘defect’states(thosesta tes
+whichhaveexactlyoneextraelementaryparticleorholeabo ut
+the ground state) as a function of t↑andt↓. At the point
+where the energy of the incompressible state becomes equal
+to its defect state, the system becomes compressible, assum -
+ing that the compressibility approaches zero continuously at
+the phaseboundary. Here,we remarkthat thistechniquecan-
+notbeusedtocalculatethephaseboundarybetweentwocom-
+pressiblephases.A. Ground-StateWave Functions
+The perturbation theory is performed with respect to the
+ground state of the system when t↑=t↓= 0, and therefore
+we first need zeroth order wave functions of the Mott phase
+and of its defect states. To zerothorderin t↑andt↓, the Mott
+insulatorwavefunctioncanbewrittenas,
+|Ψins(0)
+Mott/an}bracketri}ht=1/radicalbig
+n↑!n↓!/productdisplay
+i(b†
+i,↑)n↑(b†
+i,↓)n↓|0/an}bracketri}ht,(2)
+where/an}bracketle{t/hatwideni,σ/an}bracketri}ht=nσis anintegernumbercorrespondingto the
+ground-stateoccupancyofthe pseudo-spin σbosons,/an}bracketle{t···/an}bracketri}htis
+thethermalaverage,and |0/an}bracketri}htisthevacuumstate. Ontheother
+hand, the wave functions of the defect states are determined
+by degenerate perturbation theory. The reason for that lies
+in the fact that when exactly one extra elementary particle o r
+hole is added to the Mott phase, it could go to any of the M
+lattice sites, since all of those states share the same energ y
+whent↑=t↓= 0. Therefore, the initial degeneracy of the
+defectstates isoforder M.
+Whentheelementaryexcitationsinvolveasingle- σ-particle
+(exactly one extra pseudo-spin σboson) or a single- σ-hole
+(exactly one less pseudo-spin σboson), this degeneracy is
+lifted at first order in t↑andt↓. The treatment for this case is
+very similar to the single-species BH model [19, 24], and the
+wave functions(to zerothorderin t↑andt↓) forthe single- σ-
+particleandsingle- σ-holedefectstates turnouttobe
+|Ψsσp(0)
+def/an}bracketri}ht=1√nσ+1/summationdisplay
+ifsσp
+ib†
+i,σ|Ψins(0)
+Mott/an}bracketri}ht,(3)
+|Ψsσh(0)
+def/an}bracketri}ht=1√nσ/summationdisplay
+ifsσh
+ibi,σ|Ψins(0)
+Mott/an}bracketri}ht, (4)
+wherefsσp
+i=fsσh
+iis the eigenvector of the hopping matrix
+tij,σwith the highest eigenvalue (which is ztσwithz= 2d)
+such that/summationtext
+jtij,σfsσp
+j=ztσfsσp
+i.The normalizationcondi-
+tion requires that/summationtext
+i|fsσp
+i|2= 1. Notice that we choose the
+highest eigenvalue of tij,σbecause the hoppingmatrix enters
+theHamiltonianas −tij,σ,andweultimatelywantthelowest-
+energystates.
+However,whentheelementaryexcitationsinvolvetwopar-
+ticles (exactly one extra boson of each species) or two holes
+(exactly one less boson of each species), the degeneracy is
+lifted at second order in t↑andt↓. Such elementary excita-
+tions occur when U↑↓is sufficiently large and attractive [26],
+and the wave functions (to zeroth order in t↑andt↓) for the
+two-particleandtwo-holedefectstatescanbewrittenas
+|Ψtp(0)
+def/an}bracketri}ht=1/radicalbig
+(n↑+1)(n↓+1)/summationdisplay
+iftp
+ib†
+i,↑b†
+i,↓|Ψins(0)
+Mott/an}bracketri}ht,(5)
+|Ψth(0)
+def/an}bracketri}ht=1√n↑n↓/summationdisplay
+ifth
+ibi,↑bi,↓|Ψins(0)
+Mott/an}bracketri}ht, (6)
+whereftp
+i=fth
+iturns out to be the eigenvector of the
+tij,↑tij,↓matrix with the highest eigenvalue (which is zt↑t↓
+withz= 2d)suchthat/summationtext
+jtij,↑tij,↓ftp
+j=zt↑t↓ftp
+i.Sincethe
+elementaryexcitationsinvolvetwo particlesor two holes, the3
+degeneratedefectstatescannotbeconnectedbyonehopping ,
+but rather require two hoppings to be connected. Therefore,
+oneexpectsthedegeneracytobeliftedatleastatsecondord er
+int↑andt↓, asdiscussednext.
+B. Ground-StateEnergies
+Next, we employ the many-body version of Rayleigh-
+Schr¨ odinger perturbation theory in t↑andt↓with respect to
+the ground state of the system when t↑=t↓= 0, and cal-
+culate the energy of the Mott phase and of its defect states.
+The energy of the Mott state is obtained via nondegenerate
+perturbation theory, and to third order in t↑andt↓it is given
+by
+Eins
+Mott
+M=/summationdisplay
+σUσσ
+2nσ(nσ−1)+U↑↓n↑n↓−/summationdisplay
+σµσnσ
+−/summationdisplay
+σnσ(nσ+1)zt2
+σ
+Uσσ+O(t4). (7)Thisis anextensivequantity,i.e. Eins
+Mottis proportionalto the
+number of lattice sites M. The odd-order terms in t↑andt↓
+vanishforthe d-dimensionalhypercubiclatticesconsideredin
+thismanuscript,whichissimplybecausetheMott state give n
+in Eq. (2) cannot be connected to itself by only one hopping,
+but ratherrequirestwo hoppingsto be connected. Notice tha t
+Eq. (7) recovers the known result for the single-species BH
+modelwhenoneofthepseudo-spincomponentshavevanish-
+ingfilling,e.g. n↓= 0[19,24].
+Thecalculationofthedefect-stateenergiesismoreinvolv ed
+since it requires using degenerate perturbation theory. As
+mentioned above, when the elementary excitations involve a
+single-σ-particleorasingle- σ-hole,thedegeneracyisliftedat
+firstorderin t↑andt↓. Alengthybutstraightforwardcalcula-
+tionleadstotheenergyofthesingle- σ-particledefectstateup
+tothirdorderin t↑andt↓as
+Esσp
+def=Eins
+Mott+U↑↓n−σ+Uσσnσ−µσ−(nσ+1)ztσ
+−nσ/bracketleftbiggnσ+2
+2+(nσ+1)(z−3)/bracketrightbiggzt2
+σ
+Uσσ−2n−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓zt2
+−σ
+U−σ−σ
+−nσ(nσ+1)/bracketleftbig
+nσ(z−1)2+(nσ+1)(z−1)(z−4)+(nσ+2)(3z/4−1)/bracketrightbigzt3
+σ
+U2σσ
+−4(nσ+1)n−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓/parenleftBigg
+z−1−U2
+−σ−σ
+U2
+−σ−σ−U2
+↑↓/parenrightBigg
+ztσt2
+−σ
+U2
+−σ−σ+O(t4), (8)
+where(− ↑)≡↓and vice versa. Here, we assume Uσσ≫tσand{U−σ−σ,|U−σ−σ±U↑↓|} ≫t−σ. Equation(8) is valid for
+alld-dimensionalhypercubiclattices,andit recoversthe know nresult forthesinglespeciesBH modelwhen n−σ= 0[19, 24].
+Note that this expression also recovers the known result for the single species BH model when U↑↓= 0, which provides an
+independentcheckofthe algebra. To thirdorderin t↑andt↓, we obtaina similarexpressionfortheenergyofthe single- σ-hole
+defectstate givenby
+Esσh
+def=Eins
+Mott−U↑↓n−σ−Uσσ(nσ−1)+µσ−nσztσ
+−(nσ+1)/bracketleftbiggnσ−1
+2+nσ(z−3)/bracketrightbiggzt2
+σ
+Uσσ−2n−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓zt2
+−σ
+U−σ−σ
+−nσ(nσ+1)/bracketleftbig
+(nσ+1)(z−1)2+nσ(z−1)(z−4)+(nσ−1)(3z/4−1)/bracketrightbigzt3
+σ
+U2σσ
+−4nσn−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓/parenleftBigg
+z−1−U2
+−σ−σ
+U2
+−σ−σ−U2
+↑↓/parenrightBigg
+ztσt2
+−σ
+U2
+−σ−σ+O(t4), (9)
+which is also valid for all d-dimensional hypercubic lattices, and it also recovers the known result for the single-species BH
+modelwhen n−σ= 0orU↑↓= 0[19, 24]. Here, we againassume Uσσ≫tσand{U−σ−σ,|U−σ−σ±U↑↓|} ≫t−σ. We also
+checkedtheaccuracyofthesecond-ordertermsinEqs.(8)an d(9)viaexactsmall-cluster(two-site)calculationswith oneσand
+two−σparticles.
+We note that the mean-field phase boundarybetween the Mott ph ase and its single- σ-particle and single- σ-holedefect states
+canbecalculatedas
+µpar,hol
+σ=Uσσ(nσ−1/2)+U↑↓n−σ−ztσ/2±/radicalbig
+U2σσ/4−Uσσ(nσ+1/2)ztσ+z2t2σ/4. (10)4
+This expression is exact for infinite-dimensionalhypercub iclattices, and it recoversthe knownresult for the single s pecies BH
+model when n−σ= 0orU↑↓= 0[1]. In the d→ ∞limit (while keeping dtσconstant), we checked that our strong-coupling
+perturbationresultsgiveninEqs.(8)and(9)agreewiththi sexactsolutionwhenthelatterisexpandedouttothirdorde rint↑and
+t↓,providinganindependentcheckofthealgebra. Equation(1 0)alsoshowsthat,forinfinite-dimensionallattices,theM ottlobes
+are separatedby U↑↓n−σ, but theirshapesandcritical points(thelatter are obtain edbysetting µpar
+σ=µhol
+σ) are independentof
+U↑↓. This is not the case for finite-dimensional lattices as can b e clearly seen from our results. It is also important to menti on
+herethat boththe shapesandcritical pointsare independen tofthe sign of U↑↓in finite dimensions(at the third-orderpresented
+here)ascanbeseen inEqs.(8) and(9).
+However, when the elementary excitations involve two parti cles or two holes (which occurs when U↑↓is sufficiently large
+and attractive [26]), the degeneracyis lifted at second ord erint↑andt↓. A lengthybut straightforwardcalculationleads to the
+energyofthetwo-particledefectstate uptothirdorderin t↑andt↓as
+Etp
+def=Eins
+Mott+U↑↓(n↑+n↓+1)+/summationdisplay
+σ(Uσσnσ−µσ)+2(n↑+1)(n↓+1)
+U↑↓zt↑t↓
++/summationdisplay
+σ/bracketleftbigg(nσ+1)2
+U↑↓−nσ(nσ+2)
+2Uσσ+U↑↓+2nσ(nσ+1)
+Uσσ/bracketrightbigg
+zt2
+σ+O(t4). (11)
+Here, we assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Equation (11) is valid for all d-dimensional hypercubiclattices, where the
+odd-ordertermsin t↑andt↓vanish[27]. Tothirdorderin t↑andt↓,weobtainasimilarexpressionfortheenergyofthetwo-hol e
+defectstate givenby
+Eth
+def=Eins
+Mott−U↑↓(n↑+n↓−1)−/summationdisplay
+σ[Uσσ(nσ−1)−µσ]+2n↑n↓
+U↑↓zt↑t↓
++/summationdisplay
+σ/bracketleftbiggn2
+σ
+U↑↓−(n2
+σ−1)
+2Uσσ+U↑↓+2nσ(nσ+1)
+Uσσ/bracketrightbigg
+zt2
+σ+O(t4), (12)
+which is also valid for all d-dimensional hypercubic lattices,
+where the odd-order terms in t↑andt↓vanish [27]. Here,
+we again assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Since
+the single- σ-particleandsingle- σ-holedefectstateshavecor-
+rections to first order in the hopping, while the two-particl e
+and two-hole defect states have corrections to second order
+in the hopping, the slopes of the Mott lobes are finite as
+{t↑,t↓} →0in the former case, but they vanish in the lat-
+tercase. Hence,theshapeoftheinsulatinglobesareexpect ed
+to be very different for two-particle or two-hole excitatio ns.
+In addition, the chemical-potential widths ( µσ) of all Mott
+lobes are Uσσin the former case, but they [ (µ↑+µ↓)/2] are
+U↑↓+(U↑↑+U↓↓)/2inthelatter.
+We note that in the limit when t↑=t↓=t,U↑↑=U↓↓=
+U0,U↑↓=U′,n↑=n↓=n0,µ↑=µ↓=µ, andz= 2
+(ord= 1), Eq. (12) is in complete agreementwith Eq. (3) of
+Ref. [11], providing an independent check of the algebra. In
+addition, in the limit when t↑=t↓=J,U↑↑=U↓↓=U,
+U↑↓=W≈ −U,n↑=n↓=m, andµ↑=µ↓=µ,
+Eqs. (11) and (12) reduce to those given in Ref. [12] (after
+settingUNN= 0there). However, the terms that are propor-
+tional tot↑t↓are not included in their definitions of the two-
+particle and two-hole excitation gaps. We also checked the
+accuracy of Eqs. (11) and (12) via exact small-cluster (two-
+site) calculationswithoneparticleofeachspecies.
+Wewouldalsoliketoremarkinpassingthattheenergydif-
+ferencebetweentheMottphaseanditsdefectstatesdetermi ne
+the phase boundaryof the particle and hole branches. This is
+because at the point where the energy of the incompressiblestate becomes equal to its defect state, the system becomes
+compressible, assuming that the compressibility approach es
+zero continuously at the phase boundary. While Eins
+Mottand
+its defects Esσp
+def,Esσh
+def,Etp
+defandEth
+defdepend on the lattice
+sizeM, their difference do not. Therefore, the chemical po-
+tentialsthatdeterminetheparticleandholebranchesarei nde-
+pendentof Mat thephaseboundaries. Thisindicatesthat the
+numerical Monte Carlo simulations should not have a strong
+dependenceon M.
+It is known that the third-order strong-coupling expansion
+isnotveryaccuratenearthetipoftheMottlobes,as t↑andt↓
+arenotverysmallthere[19,24]. Forthisreason,anextrapo la-
+tion technique is highly desirable to determine more accura te
+phase diagrams. Therefore, having discussed the third-ord er
+strong-coupling expansion for a general two-species Bose-
+Bose mixtures with arbitary hoppings tσ, interactions Uσσ′,
+densities nσ, and chemical potentials µσ, next we show how
+todevelopa scalingtheory.
+IV. EXTRAPOLATIONTECHNIQUE
+In this section, we propose a chemical potential extrapo-
+lation technique based on scaling theory to extrapolate our
+third-orderpower-seriesexpansionintoafunctionalform that
+is appropriate for the entire Mott lobes. It is known that the
+criticalpointatthetipofthelobeshasthescalingbehavio rof
+a(d+1)-dimensional XYmodel,andthereforethelobeshave5
+Kosterlitz-Thouless shapes for d= 1and power-law shapes
+ford >1. For illustration purposes, here we analyze only
+the latter case, but this techniquecan be easily adapted to t he
+d= 1case [19].
+A. ScalingAnsatz
+Fromnowonwe considera two-speciesmixturewith t↑=
+t↓=t,U↑↑=U↓↓=U,U↑↓=V,n↑=n↓=n, and
+µ↑=µ↓=µ. Whend >1, we proposethe followingansatz
+which includes the known power-law critical behavior of the
+tipofthe lobes
+µ±
+U=A(x)±B(x)(xc−x)zν, (13)
+whereA(x) =a+bx+cx2+dx3+···andB(x) =α+βx+
+γx2+δx3+···areregularfunctionsof x= 2dt/U,xcisthe
+critical point which determines the location of the lobes, a nd
+zνis the critical exponent for the ( d+ 1)-dimensional XY
+model which determines the shape of the lobes near xc=
+2dtc/U. In Eq. (13), the plus sign correspondsto the particle
+branch, and the minus sign corresponds to the hole branch.
+Theformoftheansatzistakentobethesameforbothsingle-
+and two-partice (or single- and two-hole) excitations, but the
+parametersareverydifferent.
+The parameters a,b,candddepend on U,Vandn, and
+they are determined by matching them with the coefficients
+givenbyourthird-orderexpansionsuchthat A(x) = (µpar+
+µhol)/(2U).Here,µparandµholare our strong-couplingex-
+pansion results determined from Eqs. (8) and (9) for the
+single-particle and single-hole excitations, or from Eqs. (11)
+and(12)forthetwo-particleandtwo-holeexcitations,res pec-
+tively. Writing our strong-coupling expansion results for the
+particleandhole branchesin the form µpar=U/summationtext3
+n=0e+
+nxn
+andµhol=U/summationtext3
+n=0e−
+nxn, leads to a= (e+
+0+e−
+0)/2,
+b= (e+
+1+e−
+1)/2,c= (e+
+2+e−
+2)/2, andd= (e+
+3+e−
+3)/2.
+To determine the U,Vandndependence of the parameters
+α,β,γ,δ,xcandzν, we first expand the left hand side of
+B(x)(xc−x)zν= (µpar−µhol)/(2U)in powers of x, and
+matchthecoefficientswiththecoefficientsgivenbyourthir d-
+orderexpansion,leadingto
+α=e+
+0−e−
+0
+2xzνc, (14)
+β
+α=zν
+xc+e+
+1−e−
+1
+e+
+0−e−
+0, (15)
+γ
+α=zν(zν+1)
+2x2c+zν
+xce+
+1−e−
+1
+e+
+0−e−
+0+e+
+2−e−
+2
+e+
+0−e−
+0,(16)
+δ
+α=zν(zν+1)(zν+2)
+6x3c+zν(zν+1)
+2x2ce+
+1−e−
+1
+e+
+0−e−
+0
++zν
+xce+
+2−e−
+2
+e+
+0−e−
+0+e+
+3−e−
+3
+e+
+0−e−
+0. (17)
+We fixzνat its well-known values such that zν≈2/3for
+d= 2andzν= 1/2ford >2. If the exact value of xcis known via other means, e.g. numerical simulations, α,β,
+γandδcan be calculated accordingly, for which the extrap-
+olation technique gives very accurate results [23, 25]. If t he
+exact value of xcis not known, then we set δ= 0, and solve
+Eqs. (14), (15), (16) and the δ= 0equation to determine
+α,β,γandxcself-consistently, which also leads to accurate
+results [19, 24]. Next we present typical ground-state phas e
+diagrams for (d= 2)- and (d= 3)-dimensional hypercubic
+latticesobtainedfromthisextrapolationtechnique.
+B. Numerical Results
+In Figs. 1 and 2, the results of the third-order strong-
+couplingexpansion(dottedlines)arecomparedtothoseoft he
+extrapolationtechnique(hollowpink-squaresandsolidbl ack-
+circles) when V= 0.5UandV=−0.85U, respectively, in
+two (d= 2orz= 4) andthree ( d= 3orz= 6) dimensions.
+We recall here that t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=V,
+n↑=n↓=n, andµ↑=µ↓=µ.
+In Fig. 1, we show the chemical potential µ(in units of U)
+versusx= 2dt/Uphasediagramfor(a)two-dimensionaland
+(b) three-dimensional hypercubic lattices, where we choos e
+the interspecies interaction to be repulsive V= 0.5U. Com-
+paring Eqs. (8) and (9) with Eqs. (11) and (12), we expect
+that the excited state of the system to be the usual superfluid
+for allV >0for allt. The dotted lines correspond to phase
+boundary for the Mott insulator to superfluid state as deter-
+mined from the third-order strong-coupling expansion, and
+the hollow pink-squares correspond to the extrapolation fit s
+forthesingle-particleandsingle-holeexcitationsdiscu ssedin
+the text. We recall here that an incompressible super-count er
+flow phase [7–9, 13] also exists outside of the Mott insulator
+lobes, but our current formalism cannot be used to locate its
+phaseboundary.
+TABLE I. List of the critical points (location of the tips) xc=
+2dtc/Ufor the first two Mott insulator lobes that are found from
+the chemical potential extrapolation technique described in the text.
+Here,t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=V,n↑=n↓=n, and
+µ↑=µ↓=µ. These critical points for the single-particle or single-
+hole excitations are determined from Eqs. (8) and (9), and th ey tend
+tomove inas Vincreases, andare independent of the signof V.
+d= 2 d= 3
+V/Un= 1n= 2n= 1n= 2
+0.00.234 0.138 0.196 0.116
+0.10.234 0.138 0.196 0.115
+0.20.233 0.137 0.195 0.115
+0.30.230 0.136 0.194 0.114
+0.40.227 0.134 0.193 0.113
+0.50.223 0.131 0.190 0.112
+0.60.217 0.128 0.187 0.110
+0.70.208 0.123 0.182 0.107
+0.80.197 0.116 0.174 0.102
+0.90.193 0.113 0.163 0.0956
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(a) Two dimensions (V=0.5U)
+n=1n=2n=3sp/sh ext
+third order
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(a) Two dimensions (V=0.5U)
+sp/sh ext
+third order
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(b) Three dimensions (V=0.5U)
+n=1n=2n=3sp/sh ext
+third order
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(b) Three dimensions (V=0.5U)
+sp/sh ext
+third order
+FIG. 1. (Color online) Chemical potential µ(in units of U) versus
+x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
+hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
+V= 0.5U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
+correspond to phase boundary for the Mott insulator to super fluid
+state as determined from the third-order strong-coupling e xpansion,
+and the hollow pink-squares to the extrapolation fit for the s ingle-
+particle or single-hole excitations discussed in the text. Recall that
+anincompressiblesuper-counterflowphasealsoexistsouts ideofthe
+Mott insulator lobes.
+Att= 0, the chemical potential width of all Mott lobes
+areU(similar to the single-species BH model), but they are
+separated from each other by Vas a function of µ. Astin-
+creasesfromzero,therangeof µaboutwhichthegroundstate
+is a Mott insulator decreases, and the Mott insulator phasedisappears at a critical value of t, beyond which the system
+becomes a superfluid. In addition, similar to what was found
+forthesingle-speciesBH model[19,24],thestrong-coupli ng
+expansionoverestimatesthe phase boundaries,and it leads to
+unphysical pointed tips for all Mott lobes, which is expecte d
+since a finite-order expansion cannot describe the physics o f
+thecriticalpointcorrectly. Ashortlistof V/Uversusthecrit-
+ical points xc= 2dtc/Uis presented for the first two Mott
+insulator lobes in Table I, where it is shown that the criti-
+cal points tend to move in as Vincreases. This is because
+presence of a second species (say −σones) screens the on-
+site intraspeciesrepulsion Uσσbetweenσ-species, and hence
+increasesthesuperfluidregion.
+In Fig. 2, we show the chemical potential µ(in units of
+U)versusx= 2dt/Uphasediagramfor(a) two-dimensional
+and (b) three-dimensionalhypercubiclattices, where in th ese
+figures we choose the interspecies interaction to be attract ive
+V=−0.85U. Comparing Eqs. (8) and (9) with Eqs. (11)
+and (12), we expect that the excited state of the system to
+be a paired superfluid for all V <0whent→0. This is
+clearlyseen inthefigurewherethedottedlinescorrespondt o
+phaseboundaryfortheMottinsulatortosuperfluidstateasd e-
+termined from the third-orderstrong-couplingexpansion, the
+hollow pink-squares correspond to the extrapolation fits fo r
+thesingle-particleandsingle-holeexcitations(shownon lyfor
+illustration purposes), and the solid black-circles corre spond
+to the extrapolation fits for the two-particle and two-hole e x-
+citations(thisisthe expectedtransition)discussedin th etext.
+Att= 0, the chemical potential width of all Mott lobes
+areV+U= 0.15U, which is in contrast with the single-
+species BH model. As tincreases from zero, the range of µ
+aboutwhichthegroundstateisaMottinsulatordecreaseshe re
+as well, and the Mott insulator phase disappears at a critica l
+value oft, beyondwhich the system becomesa paired super-
+fluid. The strong-couplingexpansionagain overestimatest he
+phaseboundaries,anditagainleadstounphysicalpointedt ips
+for all Mott lobes. In addition, a short list of V/Uversus the
+critical points xc= 2dtc/Uare presented for the first two
+MottinsulatorlobesinTableI. Ourresultsareconsistentw ith
+the expectation that, for small V, the locations of the tips in-
+crease as a function of V, because the presence of a nonzero
+Viswhatallowedthesestatestoforminthefirstplace. How-
+ever, when Vis largerthan some critical value ( ∼0.6U), the
+locationsofthetipsdecrease,andtheyeventuallyvanishw hen
+V=−U. Thismay indicatean instabilitytowardsa collapse
+sinceat thispoint U↑↑U↓↓is exactlyequalto U2
+↑↓.
+Compared to the V >0case shown in Fig. 1, note that
+shape of the Mott insulator to paired superfluidphase bound-
+ary is very different, showing a re-entrant behavior in all d i-
+mensions from paired superfluid to Mott insulator and again
+to a paired superfluid phase, as a function of t. Our results
+are consistent with an early numerical time-evolving block
+decimation (TEBD) calculation [11], where such a re-entran t
+quantumphasetransitionin onedimensionwaspredicted.
+The re-entrant quantum phase transition occurs when co-
+efficient of the hopping term in Eq. (12) is negative [so7
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(a) Two dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(a) Two dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(b) Three dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(b) Three dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+FIG. 2. (Color online) Chemical potential µ(in units of U) versus
+x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
+hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
+V=−0.85U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
+correspond to phase boundary for the Mott insulator to super fluid
+statedeterminedfromthethird-order strong-coupling exp ansion, the
+hollow pink-squares to the extrapolation fit for the single- particle or
+single-hole excitations (shown only for illustration purp oses), and
+the solid black-circles to the extrapolation fit for the two- particle or
+two-hole excitations (the expected transition) discussed inthe text.
+that the two-hole excitation branch has a negative slope in
+(µ↑+µ↓)/2versustσphase diagram when tσ→0], i.e.
+−(2n↑n↓/U↑↓)zt↑t↓−/summationtext
+σ[n2
+σ/U↑↓−(n2
+σ−1)/(2Uσσ+
+U↑↓)+2nσ(nσ+1)/Uσσ]zt2
+σterm,whichoccursforthefirst
+few Mott lobes beyond a critical U↑↓. When this coefficient
+is negative, its value is most negative for the first Mott lobe ,TABLE II. List of the critical points (location of the tips) xc=
+2dtc/Uthat are found from the chemical potential extrapolation
+techniquedescribedinthetext. Here, t↑=t↓=t,U↑↑=U↓↓=U,
+U↑↓=V,n↑=n↓=n, andµ↑=µ↓=µ. These critical
+points for the two-particle or two-hole excitations are det ermined
+from Eqs. (11) and (12) when V <0. Note that, for small V,xc’s
+tend to increase as a function of V, since the presence of a nonzero
+Vis what allowed these states to form in the first place. Howeve r,
+xc’s decrease beyond a critical V, and they eventually vanish when
+V=−U,which mayindicate an instabilitytowards a collapse.
+d= 2 d= 3
+V/Un= 1n= 2n= 1n= 2
+-0.010.0543 0.0337 0.0611 0.0379
+-0.030.0937 0.0582 0.105 0.0655
+-0.050.121 0.0749 0.136 0.0843
+-0.070.142 0.0883 0.160 0.0994
+-0.10.169 0.105 0.190 0.118
+-0.20.233 0.145 0.262 0.164
+-0.30.277 0.173 0.311 0.195
+-0.40.307 0.193 0.345 0.217
+-0.50.325 0.205 0.366 0.230
+-0.60.331 0.209 0.372 0.235
+-0.70.321 0.203 0.362 0.228
+-0.80.291 0.183 0.327 0.206
+-0.90.225 0.141 0.253 0.159
+-0.930.193 0.121 0.217 0.136
+-0.950.166 0.103 0.187 0.116
+-0.970.1304 0.0812 0.147 0.0913
+-0.990.0764 0.0474 0.0860 0.0534
+and thereforethe effect is strongest there. However,the co ef-
+ficientincreasesandeventuallybecomespositiveasafunct ion
+offilling,andthusthere-entrantbehaviorbecomesweakera s
+fillingincreases,anditeventuallydisappearsbeyondacri tical
+filling. For the parametersused in Fig. 2, this occursonlyfo r
+the first lobe, as can be seen in the figures. We also note that
+the sign of this coefficientis independentof the dimensiona l-
+ity of the lattice, since z= 2dentersinto the coefficient only
+asanoverallfactor.
+What happenswhen t↑/ne}ationslash=t↓and/orU↑↑/ne}ationslash=U↓↓? We donot
+expectany qualitativechangefor attractiveinterspecies inter-
+actions. However, for repulsive interspecies interaction s, this
+lifts the degeneracyof the single-particle or single-hole exci-
+tation energies. While the transition is from a double Mott
+insulator to a double superfluid of both species in the degen-
+erate case, it is from a double-Mott insulator of both specie s
+toaMottinsulatorofonespeciesandasuperfluidoftheother
+inthenondegeneratecase.
+V. CONCLUSIONS
+We analyzed the zero temperature phase diagram of the
+two-species Bose-Hubbard (BH) model with on-site boson-
+boson interactions in d-dimensional hypercubic lattices, in-8
+cluding both the repulsive and attractive interspecies in-
+teraction. We used the many-body version of Rayleigh-
+Schr¨ odinger perturbation theory in the kinetic energy ter m
+with respect to the ground state of the system when the ki-
+netic energy term is absent, and calculate ground state ener -
+gies needed to carry out our analysis. This technique was
+previously used to discuss the phase diagram of the single-
+speciesBH model[19–21, 23], extendedBH model[24],and
+of the hardcore BH model with a superlattice [25], and its
+resultsshowedanexcellentagreementwithMonteCarlosim-
+ulations [23, 25]. Motivated by the success of this techniqu e
+with these models, here we generalized it to the two-species
+BH model, hoping to develop an analytical approach which
+couldbeasaccurateasthe numericalones.
+We derived analytical expressions for the phase boundary
+betweentheincompressibleMottinsulatorandthecompress -
+iblesuperfluidphaseuptothirdorderinthehoppings. Weals o
+proposed a chemical potential extrapolation technique bas ed
+on the scaling theory to extrapolateour third-orderpower s e-
+riesexpansionintoafunctionalformthatisappropriatefo rthe
+Mott lobes. In particular, when the interspecies interacti on is
+sufficiently large and attractive, we found a re-entrant qua n-
+tum phase transition from paired superfluid (superfluidity o f
+compositebosons,i.e. Bose-Bosepairs)toMottinsulatora nd
+again to a paired superfluid in all one, two and three dimen-sions. SincetheavailableMonteCarlocalculations[9,10] do
+not provide the Mott insulator to superfluid transition phas e
+boundary in the experimentally more relevant chemical po-
+tentialversushoppingplane,wecouldnotcompareourresul ts
+with them. This comparison is highly desirable to judge the
+accuracyofourstrong-couplingexpansionresults.
+A possible direction to extend this work is to consider the
+limit where hopping of one-species is much larger than the
+other. In this limit, the two-species BH model reduces to
+theBose-BoseversionoftheFalicov-Kimballmodel[28],th e
+Fermi-Fermi version of which has been widely discussed in
+the condensed-matter literature and the Fermi-Bose versio n
+has just been studied [29]. It is known for such models that
+thereisa tendencytowardsbothphaseseparationanddensit y
+wave order [30], which requires a new calculation partially
+similar to that of Ref. [24]. One can also examine how the
+momentumdistributionchangeswiththehoppingintheinsu-
+latingphases[23, 31], whichhasdirect relevanceto ultrac old
+atomicexperiments.
+VI. ACKNOWLEDGMENTS
+The author thanks Anzi Hu, L. Mathey and J. K. Freer-
+icksfordiscussions,andTheScientificandTechnologicalR e-
+searchCouncilofTurkey(T ¨UB˙ITAK)forfinancialsupport.
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1001.0022.txt

@@ -0,0 +1,651 @@
+arXiv:1001.0022v2  [hep-ph]  17 Mar 2010Preprint typeset in JHEP style - HYPER VERSION MADPH–09-1552
+µτProduction at Hadron Colliders
+Tao Han∗, Ian Lewis†
+Department of Physics, University of Wisconsin, Madison, W I 53706, U.S.A.
+Marc Sher‡
+Particle Theory Group, College of William and Mary, William sburg, Virginia 23187
+Abstract: Motivated by large νµ−ντflavor mixing, we consider µτproduction at hadron
+colliders via dimension-6 effective operators, which can be a ttributed to new physics in the
+flavor sector at a higher scale Λ. Current bounds on many of the se operators from low energy
+experiments are very weak or nonexistent, and they may lead t o cleanµ+τ−andµ−τ+signals
+at hadron colliders. At the Tevatron with 8 fb−1, one can exceed current bounds for most
+operators, with most 2 σsensitivities being in the 6 −24 TeV range. We find that at the LHC
+with 1 fb−1(100 fb−1) integrated luminosity, one can reach a2 σsensitivity for Λ ∼3−10 TeV
+(Λ∼6−21 TeV), depending on the Lorentz structure of the operator. For some operators,
+an improvement of several orders of magnitude in sensitivit y can be obtained with only a few
+tens of pb−1at the LHC.
+Keywords: Lepton flavor physics; Hadron collider phenomenology..
+∗[email protected]
+†[email protected]
+‡[email protected]
+1. Introduction 1
+2.µτProduction at Hadron Colliders 3
+3. Signal Identification and Backgrounds 4
+3.1τDecay to Electrons 5
+3.1.1 Signal Reconstruction 5
+3.1.2 Backgrounds and their Suppression 6
+3.2τDecay to Hadrons 9
+3.3 Sensitivity Reach at the Tevatron 10
+3.4 Sensitivity Reach at the LHC 10
+4. Discussions and Conclusions 13
+A. New Physics Bounds 14
+B. Partial Wave Unitarity Bounds 14
+1. Introduction
+The most important discovery in particle physics in the past decade has only deepened the
+mystery of “flavor” of quarks and leptons. The fact that the mi xing angles in the leptonic
+sector are large [1, 2] stands in sharp contrast with the obse rved small mixing angles in the
+quarksector. Inparticular, mixingbetweenthesecondandt hirdgeneration neutrinosappears
+to be maximal. Of course, this large mixing could occur from d iagonalizing the neutrino mass
+matrix, the charged lepton mass matrix, or both. At present, the source of this large mixing
+is a mystery.
+In view of this, it is tempting to explore other interactions which change lepton flavor
+between the second and third generations. Several years ago , two of us (TH, MS), along with
+Black and He (BHHS) [3], performed a comprehensive analysis of constraints on these inter-
+actions based on low energy meson physics. BHHS chose an effect ive field theory approach,
+in which all dimension-6 operators of the form
+(¯µΓτ)(¯qαΓqβ), (1.1)
+– 1 –were studied, where Γ contains possible Dirac γ-matrices. With six flavors of quarks, there
+were 12 possible combinations of qaandqb(assuming Hermiticity), six diagonal and six off-
+diagonal, and four choices S,P,V,A of the gamma matrices were considered. All of these
+operators were considered, and most were bounded by conside ringτ,K,Bandtdecays.
+In particular, BHHS considered operators of the form
+∆L= ∆L(6)
+τµ=/summationdisplay
+j,α,βCj
+αβ
+Λ2(µΓjτ)/parenleftBig
+qαΓjqβ/parenrightBig
++ H.c., (1.2)
+where Γ j∈(1, γ5, γσ, γσγ5) denotes relevant Dirac matrices, specifying scalar, pseu doscalar,
+vector and axial vector couplings, respectively. They did n ot consider tensor operators since
+the hadronic matrix elements were not known and the bounds we re expected to be weak in
+any event. They chose a value of
+Cj
+αβ= 4πO(1) (default) , (1.3)
+which corresponds to an underlying theory with a strong gaug e coupling of αS=O(1).
+Arguments can be made for multiplying or dividing this by 4 π, for naive dimensional analysis
+or for weakly coupled theories, respectively. A discussion is found in BHHS; we simply choose
+the above definition of Λ and other choices can be made by simpl e rescaling.
+Besides the four fermion operators in Eq. (1.2), there may be other induced operators
+involving the SM gauge bosons, such as the electroweak trans ition operator
+∆L=κv
+Λ2¯µσµντFµν, (1.4)
+wherevis the vacuum expectation value of the Standard Model Higgs fi eld andFµνis the
+electroweak field tensor. However, when these operators are compared to the underlying
+new strong dynamics of the four fermion interaction in Eq. (1 .2), it is found that they are
+suppressed by O(MW/Λ), where MWis the mass of the electroweak gauge boson. For new
+physics scales of order 1 TeV or greater, this is at least an or der of magnitude suppression.
+Thus, we ignore these operators.
+BHHS found that operators involving the three lightest quar ks were strongly bounded,
+with bounds ranging from 3 to 13 TeV on the related value of Λ. T hese bounds can be found
+in Appendix A. Not surprisingly, operators involving the to p quark were either unbounded or
+very weakly bounded, with only the tuoperator for vector and axial vector couplings being
+bounded by Λ <650 GeV (the bound arises through a loop in B→µτdecay). Operators
+involving the b-quark and a light quark also have bounds on Λ which were gener ally in
+the several TeV range. However, there were some surprises. T he scalar and pseudoscalar
+operators involving cuandccwere completely unbounded, and the bboperator was essentially
+unbounded for all S,P,V,A operators. And, as noted above, noneof the tensor operators
+were considered at all, for all quark combinations.
+In this note, we point out that the operators in Eq. (1.1) (wit hout involving top quarks)
+will contribute to µ−τproduction at hadron colliders. Given that many of the possi ble
+– 2 –operators, as noted above, are completely unbounded or weak ly bounded from the current
+low energy data, study of pp→µτat the LHC or pp→µτat the Tevatron will probe
+unexplored territory.
+There have been some previous discussions of µ−τproduction at hadron colliders. Han
+and Marfatia [4] looked at the lepton-violating decay h→µτat hadron colliders, and a very
+detailed analysis of signals and backgrounds was carried ou t by Assamagan et al. [5] after-
+wards. Other work looking at Higgs decays focused on mirror f ermions [6], supersymmetric
+models [7], seesaw neutrino models [8], and Randall-Sundru m models [9]. In addition to
+Higgs decays, others have considered lepton-flavor violati on in the decays of supersymmetric
+particles [10] and in horizontal gauge boson models [11]. Th ese analyses, however, were done
+in the context of very specific models (often relying on the as sumption that the µandτare
+emitted in the decay of a single particle). Here, we will use a much more general effective
+field theory approach.
+This paper is organized as follows. In the next section, we di scuss the cross sections
+forµτproduction via the various operators. A detailed analysis o f the signal identification
+and background subtraction is in Section 3, and Section 4 con tains some discussions and our
+conclusions. Appendix A reiterates the bounds from BHHS for comparison, and Appendix B
+outlines the calculation of partial-wave unitarity bounds .
+2.µτProduction at Hadron Colliders
+Dueto the absenceof appreciable µτproductionin theSM, their production can beestimated
+via the effective operators in Eq. (1.1). On dimensional groun ds, the cross section for ¯ qiqj→
+µτgrows with center of mass energy, i.e.,
+σ(¯qiqj→µτ)∝s
+Λ4, (2.1)
+where√sis the center of mass energy for the partonic system. This gro wth of cross section
+with energy will eventually violate unitarity bounds. Expa nding the scattering amplitudes in
+partial waves, we find the unitarity bounds to be (see Appendi x B)
+s≤/braceleftBigg
+2Λ2for scalar ,pseudoscalar ,and tensor;
+3Λ2vector and axial vector case .(2.2)
+The total cross sections for µτproduction at the hadronic level after convoluting with
+the parton distribution functions (pdfs) are
+σScalar=π
+3S
+Λ4/integraldisplayτmax
+τ0dτ(q⊗q)(τ)/parenleftbigg
+1−τ0
+τ/parenrightbigg2
+τ (2.3)
+σVector=4π
+9S
+Λ4/integraldisplayτmax
+τ0dτ(q⊗q)(τ)/parenleftbigg
+1−τ0
+τ/parenrightbigg2/parenleftbigg
+1+τ0
+2τ/parenrightbigg
+τ (2.4)
+σTensor=8π
+9S
+Λ4/integraldisplayτmax
+τ0dτ(q⊗q)(τ)/parenleftbigg
+1−τ0
+τ/parenrightbigg2/parenleftbigg
+1+2τ0
+τ/parenrightbigg
+τ, (2.5)
+– 3 –whereτ=s/S,τ0=m2
+τ/S,mτis the tau mass, and√
+Sis the center of mass energy in the
+lab frame. The pseudoscalar cross section is of the same form as the scalar cross section, and
+the axial vector cross section is of the same form as the vecto r cross section. Our perturbative
+calculation will become invalid at the unitarity bound, hen ce there is a maximum on the τ
+integration. It is given by τmax= 2Λ2/Sfor the scalar, pseudoscalar, and tensor cases, and
+τmax= 3Λ2/Sfor the vector and axial-vector cases. Also, q(x) is the quark distribution
+function with flavor sum suppressed, and ⊗denotes the convolution defined as
+(g1⊗g2)(y) =/integraldisplay1
+0dx1/integraldisplay1
+0dx2g1(x1)g2(x2)δ(x1x2−y). (2.6)
+The CTEQ6L parton distribution function set is used for all o f the results [12].
+Results for the cross sections for the scalar, pseudoscalar , vector, axial vector, and tensor
+structures at the Tevatron, LHC at 10 TeV and 14 TeV are given i n Table 1. Thecross section
+for the pseudoscalar (axial vector) current is the same as fo r the scalar (vector) current. For
+all cases, Λ is set equal to 2 TeV and the unitarity bounds are t aken into consideration. At
+this rather high scale, the production rates are dominated b y the valence quark contributions.
+The cross sections at the LHC are larger than those at the Teva tron by roughly an order of
+magnitude, reaching about 100 pb.
+For some cases the bounds from BHHS are greater than 2 TeV, hen ce the cross section
+needs to be scaled to determine a realistic cross section at h adron colliders. The partonic
+cross sections scale at Λ−4, but at the hadronic level a complication arises since the un itarity
+bounds introduce a dependence on the new physics scale in the integration over pdfs. If
+the unitarity bounds are ignored ( τmax= 1), one finds that with Λ = 2 TeV neglecting the
+unitarity bounds has at most a 10% effect on the cross sections a t the LHC for both 10 TeV
+and 14 TeV and no effect at the Tevatron since the unitarity boun ds are greater than the lab
+frame energy. Hence, if Λ is increased from 2 TeV, at the LHC it is a good approximation to
+assume the cross section scales as Λ−4and at the Tevatron the cross section scales exactly as
+Λ−4. For example, the lower bound on Λ for the vector u¯ucoupling from BHHS is 12 TeV, so
+the maximum cross section at the 14 TeV LHC from this operator would be approximately
+160×(2/12)4pb = 120 fb. On the other hand, there is no bound whatsoever for the vector
+u¯ccoupling, and thus a cross section limit of 110 pb would yield a new limit of 2 TeV on the
+scale of this operator. This would constitute an improvemen t of many orders of magnitude.
+3. Signal Identification and Backgrounds
+Upon production at hadron colliders, τ’s will promptly decay and are detected via their decay
+products. About 35% of the time the τdecays to two neutrinos and an electron or muon, the
+other 65% of the time the τdecays to a few hadrons plus a neutrino. We will consider the τ
+decay to an electron as well as hadronic decays in this work. T he decay to a muon will result
+in aµ+µ−final state that has a large Drell-Yan background. We will stu dy the signal reach
+at the Tevatron and at the 14 TeV LHC.
+– 4 –Table 1: Cross sections for all the scalar, pseudoscalar, vector, axial ve ctor, and tensor structures at
+the Tevatron at 2 TeV, the LHC at 10 TeV, and the LHC at 14 TeV. Th e pseudoscalar (axial vector)
+cross section is the same as the scalar (vector) cross section. All cross sections were evaluated with
+the new physics scale Λ = 2 TeV and the unitarity bounds are taken int o consideration.
+Tevatron 2 TeV ( p¯p)LHC 10 TeV ( pp) LHC 14 TeV ( pp)
+σ(pb)1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u8.4 11 22 63 85 170 120 160 310
+d¯d2.5 3.3 6.7 38 51 100 72 98 190
+s¯s0.18 0.24 0.49 5.5 7.4 15 11 15 30
+d¯s1.3 1.7 3.4 34 45 91 66 89 180
+d¯b0.50 0.67 1.3 17 22 45 34 46 90
+s¯b0.13 0.17 0.34 5.0 6.7 13 11 14 28
+u¯c1.5 2.0 3.9 41 55 110 80 110 210
+c¯c0.070 0.094 0.19 2.6 3.5 7.0 5.5 7.3 15
+b¯b0.021 0.028 0.056 1.1 1.5 2.9 2.4 3.2 6.4
+3.1τDecay to Electrons
+3.1.1 Signal Reconstruction
+Theτdecays to an electron plus two neutrinos about 18% of the time . We thus search for a
+final state of an electron and muon
+e+µ. (3.1)
+The electromagnetic calorimeter resolution is simulated b y smearing the electron energies
+according to a Gaussian distribution with a resolution para meterized by
+σ(E)
+E=a/radicalbig
+E/GeV⊕b, (3.2)
+where the constants are a= 10% and b= 0% at the Tevatron [13], a= 5% and b= 0.55% at
+the LHC [14], and ⊕indicates addition in quadrature. For simplicity, we have u sed the same
+form of smearing for the muons.
+The decay of the τleaves us with some missing energy and we need to consider how to
+effectively reconstructthe τmomentum. Forourprocessallthemissingtransversemoment um
+is coming from the τ, hence
+pτ
+T=pe
+T+pmiss
+T. (3.3)
+At hadron colliders, we have no information on the longitudi nal component of the missing
+momentum on an event-by-event basis. However, the τwill be highly boosted and its decay
+– 5 –products will be collimated. Hence, the missing momentum sh ould be aligned with the
+electron momentum and the ratio pe
+z/pmiss
+zshould be the same as the ratio of the magnitudes
+of the transverse momenta, pe
+T/pmiss
+T. Therefore, the longitudinal component of the τcan be
+reconstructed as [4]
+pτ
+z=pe
+z/parenleftbigg
+1+pmiss
+T
+pe
+T/parenrightbigg
+. (3.4)
+Once the three-momentum is reconstructed, we can solve for t heτenergy,E2
+τ=p2
+τ+m2
+τ.
+Figure 1 illustrates the effectiveness of this method at the Te vatron. Figure 1(a) (Figure 1(b))
+shows the transverse momentum (longitudinal momentum) dis tribution for the theoretically
+generated (solid) and kinematically reconstructed (dashe d)τmomenta. As can be seen, the
+τmomentum is reconstructed effectively.
+We first apply some basic cuts on the transverse momentum and t he pseudo rapidity
+to simulate the detector acceptance and triggering, as well as to isolate the signal from the
+background,
+pµ
+T>20 GeV,|ηµ|<2.5,
+pe
+T>20 GeV,|ηe|<2.5. (3.5)
+Since the signal does not contain any jets, we also require a j et veto such that there are no
+jets with pT>50 GeV and |η|<2.5.
+There are several distinctive kinematic features of our sig nal. The decay products of the
+τwill be highly collimated, and the electron transverse mome ntum will be traveling in the
+same direction as the missing transverse momentum. Also, in the transverse plane the muon
+and tau should be back to back. Since the electron will mostly be in the direction of the τ,
+it will also be nearly back to back with the muon. Finally, the τandµhave equal transverse
+momenta; hence, the decay products of the τhave less transverse momentum than the µ. We
+can measure this discrepancy using the momentum imbalance
+∆pT=pµ
+T−pe
+T. (3.6)
+For the signal, this observable should be positive. Based on the kinematics of our signal, we
+apply the further cuts [5]
+δφ(pµ
+T,pe
+T)>2.75 rad, δφ(pmiss
+T,pe
+T)<0.6 rad, (3.7)
+∆pT>0.
+3.1.2 Backgrounds and their Suppression
+Theleadingbackgrounds are W+W−pair production, Z0/γ⋆→τ+τ−, andt¯tpair production
+[5]. The total rates for these backgrounds at the Tevatron an d the LHC are given in Table 2
+with consecutive cuts. We consider both of the final states wi thµ+andµ−.
+– 6 –0 100 200 300
+pτ
+T (GeV)10-410-310-210-1dσ/dpT (pb/GeV)Generated
+Reconstructed
+(a)-300 -200 -100 0 100 200 300
+pτ
+z (GeV)10-310-210-1dσ/dpz (pb/GeV)Generated
+Reconstructed
+(b)
+Figure 1: Distributions of the theoretically generated (solid line) and kinematic ally reconstructed
+(dashed line) τmomentum at the Tevatron at 2 TeV with a u¯cinitial state, scalar coupling, and new
+physics scale of 1 TeV. Fig. (a) is the τtransverse momentum distribution, and Fig. (b) is the τ
+longitidunal momentum distribution.
+Table 2: Leading backgrounds to the τ’s electronic decay before and after consecutive kinematic and
+invariant mass cuts for (a) the Tevatron at 2 TeV and (b) the LHC a t 14 TeV.
+Backgrounds (pb) No Cuts Cuts Eq. (3.5) + Eq. (3.7) + Eq. (3.8)
+(a) Tevatron 2 TeV
+W+W−→µ±νµτ∓ντ0.032 0.0046 0.0012 2.6×10−4
+W+W−→µ±νµe∓νe0.18 0.13 0.0060 9.8×10−4
+Z0/γ⋆→τ+τ−→µ±νµτ∓610 0.21 0.091 1.4×10−4
+t¯t→µ±νµbτ∓ντ¯b 0.020 6.5×10−47.4×10−54.4×10−5
+t¯t→µ±νµbe∓νe¯b 0.11 0.0099 7.3×10−42.7×10−4
+(b) LHC 14 TeV
+W+W−→µ±νµτ∓ντ0.34 0.030 0.0088 0.0031
+W+W−→µ±νµe∓νe 1.9 0.99 0.051 0.014
+Z0/γ⋆→τ+τ−→µ±νµτ∓2300 1.1 0.49 0.0014
+t¯t→µ±νµbτ∓ντ¯b 1.9 0.070 0.010 0.0077
+t¯t→µ±νµbe∓νe¯b 11 1.5 0.10 0.050
+The partonic cross section of our signal increases with ener gy while the cross sections
+of the backgrounds will decrease with energy. Hence, the inv ariant mass distribution of our
+signal does not fall off as quickly as the backgrounds.
+Figure 2(a) shows the invariant mass distributions of backg rounds and our signal at the
+Tevatron with initial states c¯candu¯cwith various couplings and a new physics scale of
+1 TeV after applying the cuts in Eqs. (3.5) and (3.7). The cros s section for the pseudoscalar
+– 7 –0 200 400 600800 1000
+Mµτ (GeV)10-510-410-310-2dσ/dMµτ (pb/GeV)eµνν
+τ+τ−
+Tensor
+Vector
+Scalaru c-bar
+c c-barTevatron
+(a)0 200 400 600800 1000
+Mµτ (GeV)10-510-410-310-2dσ/dMµτ (pb/GeV)eµνν
+τ+τ−1 TeV
+2 TeV
+3 TeVTevatron
+(b)
+Figure 2: The invariant mass distributions of the reconstructed τ−µsystem at the Tevatron at 2
+TeV. Fig. (a) shows the distributions of the leading backgrounds (d otted and dot-dot-dash) and of
+our signal for the u¯candc¯cinitial states with coupling of various Lorentz structures and a new physics
+scale of 1 TeV. Fig. (b) shows the distributions of the leading backgr ounds (dotted and dashed) and
+of our signal (solid) for the u¯cinitial state with scalar coupling and various new physics scales. The
+cuts in Eqs. (3.5) and (3.7) have been applied.
+(axial-vector) couplings are the same as those for the scala r (vector) couplings. The decline
+in the signal rates is due to a suppression of the pdfs at large x. Although the signal rates
+steeply decline with invariant mass the background falls off faster. The u¯csignal is still clearly
+above background due to a valence quark in the initial state, but thec¯csignal distribution is
+much closer to the background distribution due to the steep f all with invariant mass and a
+lack of an initial state valence quark. Figure 2(b) shows the invariant mass distributions of
+backgroundsandoursignalattheTevatronwithinitial stat eu¯candscalarcouplingforvarious
+new physics scales. The 3 TeV new physics scale invariant mas s distribution is approaching
+the background distribution. A higher cutoff on the invarian t mass will be needed to separate
+the weak signal from the backgrounds. Based on Fig. 2, we prop ose a selection cut on
+Mµτ>250 GeV . (3.8)
+Table 2 shows the effects of the invariant mass cut on the backgr ounds in the last column.
+Similar analyses can be carried out for the LHC. Figure 3(a) s hows the invariant mass
+distribution for our signal with the u¯candc¯cinitial states and various Lorentz structures, as
+well as the backgroundsafter thecuts in Eqs. (3.5) and(3.7) . Thenewphysics scale was set to
+1 TeV and the unitarity bound is imposed. Figure 3(b) shows th e invariant mass distribution
+of theu¯cinitial state with various new physics scales. The cutoff on t he invariant mass
+corresponds to the unitarity bound, the scale at which the pe rturbative calculation becomes
+untrustworthy. In the lack of the knowledge for the new physi cs to show up at the scale Λ,
+we simply impose a sharp cutoff at the unitarity bound. As comp ared with the Tevatron,
+the LHC signal rates fall off much less quickly with invariant mass since the Tevatron’s lower
+– 8 –0 500 1000 1500
+Mµτ (GeV)10-410-310-210-1dσ/dMµτ (pb/GeV)Tensor
+Vector
+Scalar
+bbeµνντ+τ−u c-bar
+c c-barLHC
+(a)0 1 2 3 4
+Mµτ (TeV)10-510-410-310-210-1dσ/dMµτ (pb/GeV)bbeµνν
+τ+τ−1 TeV
+2 TeV
+3 TeV
+4 TeV
+5 TeVLHC
+(b)
+Figure 3: The invariant mass distributions of the reconstructed τ−µsystem at the LHC at 14 TeV.
+Fig. (a) shows the distributions of the leading backgrounds (dotte d and dot-dot-dash) and of our
+signal for the u¯candc¯cinitial states with coupling of various Lorentz structures and a new physics
+scale of 1 TeV. Fig. (b) shows the distributions of the leading backgr ounds (dotted and dashed) and
+of our signal (solid) for the u¯cinitial state with scalar coupling and various new physics scales. The
+cut offs in the distributions at high invariant mass are due to the unita rity bounds. The cuts in Eqs.
+(3.5) and (3.7) have been applied.
+energy leads to a suppression from the pdfs at large x. As can be seen, as the new physics
+scale increases the cross section decreases and the backgro und becomes more problematic at
+lower invariant mass. Also, as the new physics scale increas es the unitarity bound becomes
+less strict. Hence, although the backgrounds at the LHC are c onsiderably larger than at the
+Tevatron, for large new physics scales the LHC has an enhance ment in the signal cross section
+from the large invariant mass region.
+3.2τDecay to Hadrons
+Although with significantly larger backgrounds, the signal fromτhadronic decays can be
+very distinctive as well. We limit the hadronic τdecays to 1-prong decays to pions, i.e.,
+τ±→π±ντ,τ±→π±π0ντ, andτ±→π±2π0ντ. Theτ’s have 1-prong decays to these final
+states about 50% of the time. We thus search for a final state of aτjet and a muon
+jτ+µ. (3.9)
+To simulate detector resolution effects, the energy is smeare d according to Eq. (3.2) with
+a= 80% and b= 0% for the jet at the Tevatron [13] and a= 100% and b= 5% at the LHC
+[14]. As in the electronic decay, the τis highly boosted and its decay products are collimated.
+Hence, all the missing energy in the event should be aligned w ith theτ. The signal is then
+reconstructed as described in Eqs. (3.3) and (3.4) with the e lectron momentum replaced by
+the momentum of the τ-jet.
+– 9 –The hadronic decay of the τalso has the backgrounds W+W−pair production, Z0/γ⋆→
+τ+τ−, andt¯tpair production plus an additional background of W+jet, where the jet is
+misidentified as a τ-jet. At the Tevatron, we assume a τ-jet tagging efficiency of 67% and
+that a light jet is mistagged as a τ-jet 1.1% of the time [15] and at the LHC we assume a τ-jet
+tagging efficiency of 40% and a light jet misidentification rat e of 1% [14]. Even with a low
+rate of misidentification, the W+jet background is large. To suppress this background, we
+note that for hadronic decays most of the τtransverse momentum will be carried by the jet.
+Hence the τ-jet should be traveling in the same direction as the reconst ructedτmomentum.
+Motivated by this observation, we apply the same cuts as Eqs. (3.5), (3.7), and (3.8) with the
+electron momentum replaced by the τ-jet momentum and the additional cuts
+pτ−jet
+T
+pτ
+T>0.6 ∆ R(pτ−jet
+T,pτ
+T)<0.2 rad. (3.10)
+3.3 Sensitivity Reach at the Tevatron
+One can determine the sensitivity of the Tevatron to the new p hysics scale with 8 fb−1of
+data. Table 3 shows the sensitivity of the Tevatron for (a) el ectronic and (b) hadronic τ
+decays. The tables list the maximum new physics scale sensit ivity at 2 σand 5σlevel at the
+Tevatron. The reaches for scalar (vector) and pseudoscalar (axial-vector) are the same at the
+Tevatron, although the previous bounds from BHHS for the sca lar (vector) and pseudoscalar
+(axial-vector) couplings may not be the same. The bounds fro m BHHS can be found in
+Appendix A. If only one of the bounds for scalar (vector) or ps eudoscalar (axial-vector)
+coupling from BHHS is greater than the Tevatron reach one sta r is placed next to the new
+physics scale, if both bounds are greater than the Tevatron r each two stars are placed next
+to the new physics scale. Due to the larger backgrounds from W+jet, the Tevatron is much
+less sensitive to the τhadronic decays than the τelectronic decays.
+There were no bounds from BHHS for the tensor couplings, so th e Tevatron will be
+able to exlude some of the parameter space. Since the tensor c ross sections are generally at
+least twice as large as the scalar cross sections, the Tevatr on is more sensitive to the tensor
+couplings than it is to scalar couplings. Also, in general, t he Tevatron is more sensitive to
+processes with initial state valence quarks than those with out initial state quarks. With 8
+fb−1of data most of the bounds can be increased, some quite string ently.
+Somewhat similar leptonic final states have been searched fo r in a model-independent
+way at the Tevatron [16], although these included substanti al missing energy and possible
+jets. We encourage the Tevatron experimenters to carry out t he analyses as suggested in this
+article.
+3.4 Sensitivity Reach at the LHC
+The LHC is also sensitive to flavor changing operators. For th e signal and background anal-
+ysis, we used the same kinematical cuts as we used at the Tevat ron, see Eqs. (3.5), (3.7),
+and (3.8). Table 4 shows the sensitivity of the LHC to all poss ible initial states and the
+– 10 –Table 3: Maximum new physics scales the Tevatron is sensitive to with 8 fb−1of data at the 2 σ
+and 5σlevels. The sensitivities are presented for both (a) electronic and ( b) hadronic τdecays with
+various initial states. One star indicates that the Tevatron reach is less than only one of the scalar
+(vector) or pseudoscalar (axial-vector) bounds from BHHS, and two stars indicates that the Tevatron
+reach is less than both bounds from BHHS. BHHS does not contain bo unds on the tensor coupling.
+(a)τ→e
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u20 21 24 14 15 17
+d¯d17 18 21 12 13 15
+s¯s9.9 10 12 7.2* 7.7** 8.7
+d¯s15 16 18 10 11* 13
+d¯b13 14 16 9.8 10* 11
+s¯b9.5 10 11 6.9 7.3 8.3
+u¯c17 18 20 12 13 14
+c¯c7.9 8.3 9.5 5.7 6.0 6.9
+b¯b6.4 6.8 7.7 4.6 4.9 5.6
+(b)τ→h±
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u8.6** 9.2** 10 6.5** 6.9** 8.1
+d¯d5.7** 6.1** 7.1 4.3** 4.6** 5.4
+s¯s1.8* 1.9** 2.3 1.4** 1.4** 1.7
+d¯s3.7 4.0* 4.6 2.8* 3.0** 3.5
+d¯b2.7* 2.9* 3.4 2.0** 2.2 2.5
+s¯b1.5** 1.6** 1.9 1.1** 1.2** 1.4
+u¯c3.9 4.1 4.8 2.9 3.1 3.6
+c¯c1.1 1.2 1.4 0.89 0.95** 1.1
+b¯b0.91 0.97 1.1 0.68 0.73 0.86
+couplings under consideration with 100 fb−1of data. The table contains the maximum new
+physics scales the LHC is sensitive to at the 2 σand 5σlevels. As with the Tevatron, the
+LHC reach for scalar (vector) couplings is the same as that fo r pseudoscalar (axial-vector)
+couplings, although the bounds from BHHS may be different. If o nly one of the bounds for
+scalar (vector) or pseudoscalar (axial-vector) coupling f rom BHHS is greater than the LHC
+reach one star is placed next to the new physics scale, if both bounds are greater than the
+LHC reach two stars are placed next to the new physics scale. D espite the larger backrounds
+for the hadronic τdecays, at the LHC the reaches for the hadronic and electroni cτdecays are
+– 11 –Table 4: Maximum new physics scales the LHC is sensitive to at 14 TeV with 100 fb−1of data at the
+2σand 5σlevels. The sensitivities are presented for both (a) electronic and ( b) hadronic τdecays with
+various initial states. One star indicates that the LHC reach is less t han only one of the scalar (vector)
+or pseudoscalar (axial-vector) bounds from BHHS, and two stars indicates that the LHC reach is less
+than both bounds from BHHS. BHHS does not contain bounds on the tensor coupling.
+(a)τ→e
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u18 19 21 14 15 17
+d¯d16 17 19 12 13 15
+s¯s9.0* 9.6* 11 7.1* 7.6** 8.6
+d¯s13 14 16 10 11* 13
+d¯b12 13 14 9.7 10 11
+s¯b8.7 9.2 10 6.8 7.3 8.2
+u¯c15 16 18 12 13 14
+c¯c7.2 7.6 8.6 5.7 6.0 6.8
+b¯b5.8 6.2 7.0 4.6 4.9 5.5
+(b)τ→h±
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+u¯u15 16 18 12 13 14
+d¯d13 14 16 10* 11* 13
+s¯s7.9* 8.4** 9.7 6.2* 6.7** 7.7
+d¯s11 12* 14 9.3 9.9* 11
+d¯b10 11 13 8.4* 8.9 10
+s¯b7.6 8.1 9.3 6.0 6.4 7.4
+u¯c13 14 16 10 11 12
+c¯c6.3 6.7 7.8 5.0 5.3 6.2
+b¯b5.1 5.5 6.3 4.1 4.3 5.0
+much more similar than at the Tevatron since the LHC cross sec tion receives an enhancement
+from the large invariant mass region. For electronic (hadro nic)τdecays the LHC with 100
+fb−1of data is less (more) sensitive than the Tevatron with 8 fb−1of data.
+Figure 4 shows the integrated luminosities needed for 2 σand 5σobservation at the LHC
+with various initial states and τdecay to electrons as a function of the new physics scale.
+For some initial states and Lorentz structures BHHS had a bou nd on the new physics scale
+larger than 1 TeV. In those cases the distribution does not be gin until the BHHS bound on
+the new physics scale. The sensitivity for the pseudoscalar (axial-vector) is the same as the
+scalar (vector) state, although the bounds from BHHS are diffe rent. Note that extraordinary
+– 12 –1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)Scalar
+Vector
+Tensor
+2σu c-bar Initial State
+5σLHC 14 TeV
+(a)1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)
+Scalar
+Vector
+Tensor
+2σc c-bar Initial State
+5σLHC 14 TeV
+(b)
+1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)Scalar
+Vector
+Tensor
+2σd b-bar Initial State
+5σLHC 14 TeV
+(c)1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)
+Scalar
+Vector
+Tensor
+2σs b-bar Initial State
+5σLHC 14 TeV
+(d)
+Figure 4: The luminosity at the 14 TeV LHC needed for 2 σand 5σobservation as a function of the
+new physics scales with couplings of various Lorentz structures an d electronic τdecay. The sensitivity
+for theu¯cinitial state is shown in (a), for the c¯cinitial state in (b), for the d¯binitial state in (c), and
+for thes¯binitial state in (d). The lower bounds on the new physics scale were ta ken from BHHS.
+improvementintheboundscouldbefound(oradiscoverymade )withrelatively lowintegrated
+luminosity. Consider, for example, the u¯cinitial state. There is currently no bound at all;
+in principle, Λ could be tens of GeV. The figure shows that a tot al integrated luminosity of
+an inverse picobarn would give a 5 σsensitivity for a Λ of 1 TeV. An integrated luminosity of
+an inverse femtobarn would give substantial improvements f or all of the operators shown in
+Fig. 4.
+4. Discussions and Conclusions
+In a previous article, motivated by discovery of large νµ−ντmixing in charged current
+interactions, bounds on the analogous mixing in neutral cur rent interactions were explored.
+A general formalism for dimension-6 fermionic effective oper ators involving τ−µmixing with
+– 13 –typical Lorentz structure ( µΓτ)(qαΓqβ) was presented, and the low-energy constraints on
+the new physics scale associated with each operator were der ived, mostly from experimental
+bounds on rare decays of τ, hadrons or heavy quarks. Tensor operators were not conside red,
+and some of the operators, such as cuµτ, were completely unbounded.
+Inthis article, weconsider µτproductionat hadroncolliders viatheseoperators. Tables 3
+and4 list thenewphysics scales that are accessible at the Te vatron and theLHC, respectively.
+Duetomuchsmallerbackgrounds, boththeLHCandTevatronar emoresensitivetoelectronic
+τdecays than hadronic τdecays. For hadronic τdecays, the LHC receives an enhancement
+from the large invariant mass region and is more sensitive th an the Tevatron. Since the
+backgrounds to electronic τdecays at the Tevatron are much smaller than those at the LHC,
+the Tevatron is more sensitive than the LHC to electronic τdecays. We found that at the
+Tevatron with 8 fb−1, one can exceed current bounds for most operators, with most 2σ
+sensitivities being in the 6 −24 TeV range. We find that at the LHC with 1 fb−1(100 fb−1)
+integrated luminosity, one can reach a 2 σsensitivity for Λ ∼3−10 TeV (Λ ∼6−21 TeV),
+depending on the Lorentz structure of the operator.
+Acknowledgments
+We would like to thank Vernon Barger and Xerxes Tata for discu ssions. MS would like to
+thank the Wisconsin Phenomenology Institute, in particula r Linda Dolan, for hospitality
+during his visit. The work of TH and IL was supported by the US D OE under contract
+No. DE-FG02-95ER40896, and that of MS was supported in part b y the National Science
+Foundation PHY-0755262.
+A. New Physics Bounds
+The bounds from BHHS in units of TeV are presented in Table 5. T he *s indicate there are
+no bounds on the new physics scale. Also, there are no bounds f rom BHHS for the tensor
+coupling.
+B. Partial Wave Unitarity Bounds
+Since the cross section from our higher-dimensional operat ors increases as s, it is necessary
+to determine the unitarity bound for q¯q→µτ. The partial wave expansion for a+b→1+2
+can be written as
+M(s,t) = 16π∞/summationdisplay
+J=M(2J+1)aJ(s)dJ
+µµ′(cosθ)
+where
+aJ(s) =1
+32π/integraldisplay1
+−1M(s,t)dJ
+µµ′(cosθ)dcosθ,
+µ=sa−sb,µ′=s1−s2andJ≤max(|µ|,|µ′|). The condition for unitarity is |ℜ(aJ)| ≤1/2.
+– 14 –Coupling type 1 γ5 γµ γµγ5
+u¯u 2.6 12 12 11
+d¯d 2.6 12 12 11
+s¯s 1.5 9.9 14 9.5
+d¯s 2.3 3.7 13 3.6
+d¯b 2.2 9.3 2.2 8.2
+s¯b 2.6 2.8 2.6 2.5
+u¯c * * 0.55 0.55
+c¯c * * 1.1 1.1
+b¯b * * 0.18 *
+Table 5: Bounds on the new physics scales from BHHS in units of TeV for variou s operators and the
+scalar, pseudoscalar, vector, and axial-vector couplings. The *s indicate there were no bounds.
+It is straightforward to calculate the coefficients for the S, V,T operators. For example,
+for the scalar operator
+M=4π
+Λ2¯vλ1(p1)uλ2(p2)¯uλ3(p3)vλ4(p4)
+one can just plug in the explicit expressions:
+uλ(p)≡/parenleftBigg/radicalbig
+E−λ|p|χλ(ˆp)/radicalbig
+E+λ|p|χλ(ˆp)/parenrightBigg
+vλ(p)≡/parenleftBigg
+−/radicalbig
+E+λ|p|χ−λ(ˆp)/radicalbig
+E−λ|p|χ−λ(ˆp)/parenrightBigg
+whereχ+(ˆz) =/parenleftbig1
+0/parenrightbig
+,χ−(ˆz) =/parenleftbig0
+1/parenrightbig
+. In the massless limit, this simply gives a0=s/(4Λ2) and
+so the unitarity bound gives s≤2Λ2. For the vector case, a0= 0 and a1=s/(6Λ2) giving
+the unitarity bound s≤3Λ2. The tensor case gets contributions from both a0anda1, and
+the stronger bound then applies.
+References
+[1] S. Fukuda, et al., [Super-Kamiokande Collaboration], Phys. Rev. Lett. 85, 3999 (2000); 86,
+5656 (2001); 82, 1810 (1999); 81, 1562 (1998); 81, 1158 (1998); and T. Toshito,
+[Super-Kamiokande Collaboration], hep-ex/0105023 .
+[2] Q. R. Ahmad, et al.,[SNO collaboration], Phys. Rev. Lett. 87, 071301 (2001). Q. R. Ahmad, et
+al.,[SNO Collaboration], Phys. Rev. Lett. (2002), nucl-ex/0204008 andnucl-ex/0204009 .
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+– 15 –[4] T. Han and D. Marfatia, Phys. Rev. Lett. 86, 1442 (2001) [arXiv:hep-ph/0008141].
+[5] K. A. Assamagan, A. Deandrea and P. A. Delsart, Phys. Rev. D 67, 035001 (2003)
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+[10] F. Deppisch, J. Kalinowski, H. Pas, A. Redelbach and R. Ruckl, ar Xiv:hep-ph/0401243.
+[11] H. U. Bengtsson, W. S. Hou, A. Soni and D. H. Stork, Phys. Re v. Lett.55, 2762 (1985).
+[12] D. Stump, J. Huston, J. Pumplin, W. K. Tung, H. L. Lai, S. Kuhlma nn and J. F. Owens, JHEP
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+– 16 –

1001.0024.txt

@@ -0,0 +1,659 @@
+arXiv:1001.0024v1  [q-fin.CP]  30 Dec 2009November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Journal of Circuits, Systems, and Computers
+c/circlecopyrtWorld Scientific Publishing Company
+BAYESIAN INFERENCE OF STOCHASTIC VOLATILITY MODEL
+BY HYBRID MONTE CARLO
+Tetsuya Takaishi†
+Hiroshima University of Economics,
+Hiroshima 731-0192 JAPAN
+†[email protected]
+Received (Day Month Year)
+Revised (Day Month Year)
+Accepted (Day Month Year)
+The hybrid Monte Carlo (HMC) algorithm is applied for the Bay esian inference of the
+stochastic volatility (SV) model. We use the HMC algorithm f or the Markov chain Monte
+Carloupdates of volatility variables of the SV model. First we compute parameters of the
+SV model by using the artificial financial data and compare the results from the HMC
+algorithm with those from the Metropolis algorithm. We find t hat the HMC algorithm
+decorrelates the volatility variables faster than the Metr opolis algorithm. Second we
+make an empirical study for the time series of the Nikkei 225 s tock index by the HMC
+algorithm. We find the similar correlation behavior for the s ampled data to the results
+from the artificial financial data and obtain a φvalue close to one ( φ≈0.977), which
+means that the time series has the strong persistency of the v olatility shock.
+Keywords : Hybrid Monte Carlo Algorithm, Stochastic Volatility Mode l, Markov Chain
+Monte Carlo, Bayesian Inference, Financial Data Analysis
+1. Introduction
+Many empirical studies of financial prices such as stock indexes, ex change rates
+have confirmed that financial time series of price returns shows va rious interesting
+properties which can not be derived from a simple assumption that th e price re-
+turns follow the geometric Brownian motion. Those properties are n ow classified
+as stylized facts1,2. Some examples of the stylized facts are (i) fat-tailed distribu-
+tion of return (ii) volatility clustering (iii) slow decay of the autocorre lation time
+of the absolute returns. The true dynamics behind the stylized fac ts is not fully
+understood. In order to imitate the real financial markets and to understand the
+origins of the stylized facts, a variety of models have been propose d and examined.
+Actually many models are able to capture some of the stylized facts3-14.
+In empirical finance the volatilityis an important value to measurethe risk. One
+of the stylized facts of the volatility is that the volatility of price retu rns changes
+in time and shows clustering, so called ”volatility clustering”. Then the histogram
+of the resulting price returns shows a fat-tailed distribution which in dicates that
+1November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+2Authors’ Names
+the probability of having a large price change is higher than that of th e Gaussian
+distribution. In order to mimic these empirical properties of the vola tility and to
+forecast the future volatility values, Engle advocated the autore gressive conditional
+hetroskedasticity (ARCH) model15where the volatility variable changes determin-
+istically depending on the past squared value of the return. Later t he ARCH model
+is generalized by adding also the past volatility dependence to the vola tility change.
+This model is known asthe generalizedARCH(GARCH) model16. The parameters
+of the GARCH model applied to financial time series are conventionally determined
+by the maximum likelihood method. There are many extended versions of GARCH
+models, such as EGARCH17, GJR18, QGARCH19,20models etc., which are de-
+signed to increase the ability to forecast the volatility value.
+The stochastic volatility (SV) model21,22is another model which captures the
+propertiesofthevolatility.IncontrasttotheGARCHmodel,thevo latilityoftheSV
+model changes stochastically in time. As a result the likelihood functio n of the SV
+model is given as a multiple integral of the volatility variables. Such an in tegral in
+general is not analytically calculable and thus the determination of th e parameters
+of the SV model by the maximum likelihood method becomes difficult. To o vercome
+this difficulty in the maximum likelihood method the Markov Chain Monte Ca rlo
+(MCMC) method based on the Bayesian approach is proposed and de veloped21. In
+the MCMC of the SV model one has to update not only the parameter variables
+but also the volatility ones from a joint probability distribution of the p arameters
+and the volatility variables. The number of the volatility variables to be updated
+increases with the data size of time series. The first proposed upda te scheme of
+the volatility variables is based on the local update such as the Metro polis-type
+algorithm21. It is however known that when the local update scheme is used for
+the volatility variables having interactions to their neighbor variables in time, the
+autocorrelationtime ofsampledvolatilityvariablesbecomeslargeand thusthe local
+update scheme becomes ineffective23. In order to improve the efficiency of the local
+update method the blocked scheme which updates several variable s at once is also
+proposed23,24. A recent survey on the MCMC studies of the SV model is seen in
+Ref.25.
+In our study we use the HMC algorithm26which had not been considered
+seriously for the MCMC simulation of the SV model. In finance there ex ists an
+application of the HMC algorithm to the GARCH model27where three GARCH
+parameters are updated by the HMC scheme. It is more interesting to apply the
+HMC for updates of the volatility variables because the HMC algorithm is a global
+update scheme which can update all variables at once. This feature of the HMC
+algorithm can be used for the global update of the volatility variables which can not
+be achieved by the standard Metropolis algorithm. A preliminary stud y28shows
+that the HMC algorithmsamplesthe volatilityvariableseffectively.In t his paperwe
+give a detailed description of the HMC algorithm and examine the HMC alg orithm
+with artificial financial data up to the data size of T=5000. We also ma ke an
+empirical analysis of the Nikkei 225 stock index by the HMC algorithm.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 3
+2. Stochastic Volatility Model
+The standard version of the SV model21,22is given by
+yt=σtǫt= exp(ht/2)ǫt, (1)
+ht=µ+φ(ht−1−µ)+ηt, (2)
+whereyt= (y1,y2,...,yn) represents the time series data, htis defined by ht= lnσ2
+t
+andσtiscalledvolatility.Wealsocall htvolatilityvariable.Theerrorterms ǫtandηt
+are taken from independent normal distributions N(0,1) andN(0,σ2
+η) respectively.
+We assume that |φ|<1. When φis close to one, the model exhibits the strong
+persistency of the volatility shock.
+For this model the parameters to be determined are µ,φandσ2
+η. Let us use θ
+asθ= (µ,φ,σ2
+η). Then the likelihood function L(θ) for the SV model is written as
+L(θ) =/integraldisplayn/productdisplay
+t=1f(ǫt|σ2
+t)f(ht|θ)dh1dh2...dhn, (3)
+where
+f(ǫt|σ2
+t) =/parenleftbig
+2πσ2
+t/parenrightbig−1
+2exp/parenleftbigg
+−y2
+t
+2σ2
+t/parenrightbigg
+, (4)
+f(h1|θ) =/parenleftBigg
+2πσ2
+η
+1−φ2/parenrightBigg−1
+2
+exp/parenleftbigg
+−[h1−µ]2
+2σ2η/(1−φ2)/parenrightbigg
+, (5)
+f(ht|θ) =/parenleftbig
+2πσ2
+η/parenrightbig−1
+2exp/parenleftbigg
+−[ht−µ−φ(ht−1−µ)]2
+2σ2η/parenrightbigg
+. (6)
+As seen in Eq.(3), L(θ) is constructed as a multiple integral of the volatility vari-
+ables. For such an integral it is difficult to apply the maximum likelihood me thod
+which estimates values of θby maximizing the likelihood function. Instead of using
+the maximum likelihood method we perform the MCMC simulations based o n the
+Bayesian inference as explained in the next section.
+3. Bayesian inference for the SV model
+From the Bayes’ rule, the probability distribution of the parameter sθis given by
+f(θ|y) =1
+ZL(θ)π(θ), (7)
+whereZis the normalization constant Z=/integraltext
+L(θ)π(θ)dθandπ(θ) is a prior disti-
+bution of θfor which we make a certian assumption. The values of the paramete rs
+are inferred as the expectation values of θgiven by
+/an}bracketle{tθ/an}bracketri}ht=/integraldisplay
+θf(θ|y)dθ. (8)
+In general this integral can not be performed analytically. For tha t case, one can
+use the MCMC method to estimate the expectation values numerically .November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+4Authors’ Names
+In the MCMC method, we first generate a series of θwith a probability of
+P(θ) =f(θ|y). Letθ(i)= (θ(1),θ(2),...,θ(k)) be values of θgenerated by the MCMC
+sampling. Then using these kvalues the expectation value of θis estimated by an
+average as
+/an}bracketle{tθ/an}bracketri}ht=1
+kk/summationdisplay
+i=1θ(i). (9)
+The statistical error for kindependent samples is proportional to1√
+k. When the
+sampled data are correlated the statistical error will be proportio nal to/radicalbigg
+2τ
+kwhere
+τis the autocorrelation time between the sampled data. The value of τdepends
+on the MCMC sampling scheme we take. In order to reduce the statis tical error
+within limited sampled data it is better to choose an MCMC method which is able
+to generate data with a small τ.
+3.1.MCMC Sampling of θ
+For the SV model, in addition to θ, volatility variables htalso have to be updated
+sincetheyshouldbeintegratedoutasinEq.(3).Let P(θ,ht)be thejointprobability
+distribution of θandht. ThenP(θ,ht) is given by
+P(θ,ht)∼¯L(θ,ht)π(θ), (10)
+where
+¯L(θ,ht) =n/productdisplay
+t=1f(ǫt|ht)f(ht|θ). (11)
+For the prior π(θ) we assume that π(σ2
+η)∼(σ2
+η)−1and for others π(µ) =π(φ) =
+constant.
+The MCMC sampling methods for θare given in the following21,22. The prob-
+ability distribution for each parameter can be derived from Eq.(10) b y extracting
+the part including the corresponding parameter.
+•σ2
+ηupdate scheme.
+The probability distribution of σ2
+ηis given by
+P(σ2
+η)∼(σ2
+η)−n
+2−1exp/parenleftbigg
+−A
+σ2η/parenrightbigg
+, (12)
+where
+A=1
+2{(1−φ2)(h1−µ)2+n/summationdisplay
+t=2[ht−µ−φ(ht−1−µ)]2}.(13)
+Since Eq.(12) is an inverse gamma distribution we can easily draw a value
+ofσ2
+ηby using an appropriate statistical library in the computer.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 5
+•µupdate scheme.
+The probability distribution of µis given by
+P(µ)∼exp/braceleftbigg
+−B
+2σ2η(µ−C
+B)2/bracerightbigg
+, (14)
+where
+B= (1−φ2)+(n−1)(1−φ)2, (15)
+and
+C= (1−φ2)h1+(1−φ)n/summationdisplay
+t=2(ht−φht−1). (16)
+µis drawn from a Gaussian distribution of Eq.(14).
+•φupdate scheme.
+The probability distribution of φis given by
+P(φ)∼(1−φ2)1/2exp{−D
+2σ2η(φ−E
+D)2}, (17)
+where
+D=−(h1−µ)2+n/summationdisplay
+t=2(ht−1−µ)2, andE=/summationtextn
+t=1(ht−µ)(ht−1−µ).(18)
+In order to update φwith Eq.(17), we use the Metropolis-Hastings
+algorithm30,31. Let us write Eq.(17) as P(φ)∼P1(φ)P2(φ) where
+P1(φ) = (1−φ2)1/2, (19)
+P2(φ)∼exp{−D
+2σ2η(φ−E
+D)2}. (20)
+SinceP2(φ) is a Gaussian distribution we can easily draw φfrom Eq.(20).
+Letφnewbe a candidate given from Eq.(20). Then in order to obtain the
+correct distribution, φnewis accepted with the following probability PMH.
+PMH= min/braceleftbiggP(φnew)P2(φ)
+P(φ)P2(φnew),1/bracerightbigg
+= min/braceleftBigg/radicalBigg
+(1−φ2new)
+(1−φ2),1/bracerightBigg
+.(21)
+In addition to the abovestep we restrict φwithin [−1,1]to avoida negative
+value in the calculation of square root.
+3.2.Probability distribution for ht
+The probability distribution of the volatility variables htis given by
+P(ht)≡P(h1,h2,...,hn)∼ (22)
+exp/parenleftBig
+−/summationtextn
+i=1{ht
+2+ǫ2
+t
+2e−ht}−[h1−µ]2
+2σ2
+η/(1−φ2)−/summationtextn
+i=2[ht−µ−φ(ht−1−µ)]2
+2σ2
+η/parenrightBig
+.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+6Authors’ Names
+Thisprobabilitydistributionisnotasimplefunction todrawvaluesof ht.Aconven-
+tional method is the Metropolis method30,31which updates the variables locally.
+There are several methods21,22,23,24developed to update htfrom Eq.(22). Here
+we use the HMC algorithm to update htglobally. The HMC algorithm is described
+in the next section.
+4. Hybrid Monte Carlo Algorithm
+Originallythe HMCalgorithmis developedforthe MCMCsimulationsofthe lattice
+QuantumChromoDynamics(QCD) calculations26. Amajordifficultyofthe lattice
+QCDcalculationsistheinclusionofdynamicalfermions.Theeffectoft hedynamical
+fermions is incorporated by the determinant of the fermion matrix. The computa-
+tional work of the determinant calculation requires O(V3) arithmetic operations29,
+whereVis the volume of a 4-dimensional lattice. A typical size of the volume is
+V >104. The standard Metropolis algorithm which locally updates variables do es
+not work since each local update requires O(V3) arithmetic operations for a deter-
+minant calculation,which results in unacceptable computational cos t in total. Since
+the HMC algorithm is a global update method, the computational cos t remains in
+the acceptable region.
+The basic idea of the HMC algorithm is a combination of molecular dynamic s
+(MD) simulation and Metropolis accept/reject step. Let us conside r to evaluate the
+following expectation value /an}bracketle{tO(x)/an}bracketri}htby the HMC algorithm.
+/an}bracketle{tO(x)/an}bracketri}ht=/integraldisplay
+O(x)f(x)dx=/integraldisplay
+O(x)elnf(x)dx, (23)
+wherex= (x1,x2,...,xn),f(x) is a probability density and O(x) stands for an
+function of x. First we introduce momentum variables p= (p1,p2,...,pn) conjugate
+to the variables xand then rewrite Eq.(23) as
+/an}bracketle{tO(x)/an}bracketri}ht=1
+Z/integraldisplay
+O(x)e−1
+2p2+lnf(x)dxdp=1
+Z/integraldisplay
+O(x)e−H(p,x)dxdp. (24)
+whereZis a normalization constant given by
+Z=/integraldisplay
+exp/parenleftbigg
+−1
+2p2/parenrightbigg
+dp, (25)
+andp2stands for/summationtextn
+i=1p2
+i.H(p,x) is the Hamiltonian defined by
+H(p,x) =1
+2p2−lnf(x). (26)
+Note that the introduction of pdoes not change the value of /an}bracketle{tO(x)/an}bracketri}ht.
+In the HMC algorithm, new candidates of the variables ( p,x) are drawn by
+integrating the Hamilton’s equations of motion,
+dxi
+dt=∂H
+∂pi, (27)
+dpi
+dt=−∂H
+∂xi. (28)November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 7
+In general the Hamilton’s equations of motion arenot solved analytic ally. Therefore
+wesolvethemnumericallybydoingthe MDsimulation.Let TMD(∆t) beanelemen-
+tary MD step with a step size ∆ t, which evolves ( p(t),x(t)) to (p(t+∆t),x(t+∆t)):
+TMD(∆t) : (p(t),x(t))→(p(t+∆t),x(t+∆t)). (29)
+Any integrator can be used for the MD simulation provided that the f ollowing
+conditions are satisfied26
+•area preserving
+dp(t)dx(t)dx=dp(t+∆t)dx(t+∆t). (30)
+•time reversibility
+TMD(−∆t) : (p(t+∆t),x(t+∆t))→(p(t),x(t)). (31)
+The simplest and often used integrator satisfying the above two co nditions is
+the 2nd order leapfrog integrator given by
+xi(t+∆t/2) =xi(t)+∆t
+2pi(t)
+pi(t+∆t) =p(t)i−∆t∂H
+∂xi
+xi(t+∆t) =xi(t+∆t/2)+∆t
+2pi(t+∆t). (32)
+In this study we use this integrator.The numericalintegration is pe rformedNsteps
+repeatedly by Eq.(32) and in this case the total trajectory length λof the MD is
+λ=N×∆t.
+At the end of the trajectory we obtain new candidates ( p′,x′). These candidates
+are accepted with the Metropolis test, i.e. ( p′,x′) are globally accepted with the
+following probability,
+P= min{1,exp(−H(p′,x′))
+exp(−H(p,x))}= min{1,exp(−∆H)}, (33)
+where∆Histhe energydifferencegivenby∆ H=H(p′,x′)−H(p,x). Sinceweinte-
+grate the Hamilton’s equations of motion approximately by an integra tor, the total
+Hamiltonianisnotconserved,i.e.∆ H/ne}ationslash= 0.Theacceptanceorthe magnitudeof∆ H
+is tuned by the step size ∆ tto obtain a reasonable acceptance. Actually there ex-
+ists the optimal acceptance which is about 60 −70%for 2nd order integrators32,33.
+Surprisingly the optimal acceptance is not dependent of the model we consider. For
+the n-th order integrator the optimal acceptance is expected to be32∼exp/parenleftbigg
+−1
+n/parenrightbigg
+.
+We could also use higher order integrators which give us a smaller ener gy dif-
+ference ∆ H. However the higher order integrators are not always effective sin ce
+they need more arithmetic operations than the lower order integra tors32,33. The
+efficiency of the higher order integrators depends on the model we consider. ThereNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+8Authors’ Names
+also exist improved integrators which have less arithmetic operation s than the con-
+ventional integrators34.
+For the volatility variables ht, from Eq.(22), the Hamiltonian can be defined by
+H(pt,ht) =n/summationdisplay
+i=11
+2p2
+i+n/summationdisplay
+i=1{hi
+2+ǫ2
+i
+2e−hi}+[h1−µ]2
+2σ2η/(1−φ2)+n/summationdisplay
+i=2[hi−µ−φ(hi−1−µ)]2
+2σ2η,(34)
+wherepiis defined as a conjugate momentum to hi. Using this Hamiltonian we
+perform the HMC algorithm for updates of ht.
+5. Numerical Studies
+In order to test the HMC algorithm we use artificial financial time ser ies data
+generatedbythe SVmodel with a setofknownparametersand per formthe MCMC
+simulations to the artificial financial data by the HMC algorithm. We als o perform
+the MCMC simulations by the Metropolis algorithm to the same artificial data and
+compare the results with those from the HMC algorithm.
+Using Eq.(1) with φ= 0.97,σ2
+η= 0.05 andµ=−1 we have generated 5000
+time series data. The time series generated by Eq.(1) is shown in Fig.1. From those
+data we prepared 3 data sets: (1)T=1000 data (the first 1000 of the time series),
+(2)T=2000data (the first 2000ofthe time series)and (3) T=5000 (the whole data).
+To these data sets we made the Bayesian inference by the HMC and M etropolis
+algorithms.Preciselyspeakingboth algorithmsareusedonlyfor the MCMC update
+of the volatility variables. For the update of the SV parameters we u sed the update
+schemes in Sec.3.1.
+For the volatility update in the Metropolis algorithm, we draw a new can didate
+of the volatility variables randomly, i.e. a new volatility hnew
+tis given from the
+previous value hold
+tby
+hnew
+t=hold
+t+δ(r−0.5), (35)
+whereris a uniform random number in [0 ,1) andδis a parameter to tune the
+acceptance. The new volatility hnew
+tis accepted with the acceptance Pmetro
+Pmetro= min/braceleftbigg
+1,P(hnew
+t)
+P(hold
+t)/bracerightbigg
+, (36)
+whereP(ht) is given by Eq.(22).
+The initial parameters for the MCMC simulations are set to φ= 0.5,σ2
+η= 1.0
+andµ= 0. The first 10000 samples are discarded as thermalization or burn -in
+process. Then 200000samples are recorded for analysis. The tot al trajectory length
+λof the HMC algorithm is set to λ= 1 and the step size ∆ tis tuned so that the
+acceptance of the volatility variables becomes more than 50%.
+First we analyze the sampled volatility variables. Fig.2 shows the Mont e Carlo
+(MC) history of the volatility variable h100fromT= 2000 data set. We take h100
+as the representative one of the volatility variables since we have ob served theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 9
+0 1000 2000 3000 4000 5000t-6-4-20246yt
+Fig. 1. The artificial SV time series used for this study.
+50000 55000 60000
+Monte Carlo history-2-10123h100HMC
+50000 55000 60000
+Monte Carlo history-2-10123h100Metropolis
+Fig. 2. Monte Carlo histories of h100generated by HMC (left) and Metropolis (right) with
+T= 2000 data set. The Monte Carlo histories in the window from 5 0000 to 60000 are shown.
+similar behavior for other volatility variables. See also Fig.3 for the sim ilarity of the
+autocorrelation functions of the volatility variables.
+AcomparisonofthevolatilityhistoriesinFig.2clearlyindicatesthatth ecorrela-
+tion of the volatility variable sampled from the HMC algorithm is smaller th an that
+from the Metropolis algorithm. To quantify this we calculate the auto correlation
+function (ACF) of the volatility variable. The ACF is defined as
+ACF(t) =1
+N/summationtextN
+j=1(x(j)−/an}bracketle{tx/an}bracketri}ht)(x(j+t)−/an}bracketle{tx/an}bracketri}ht)
+σ2x, (37)
+where/an}bracketle{tx/an}bracketri}htandσ2
+xare the average value and the variance of xrespectively.
+Fig.3 shows the ACF for three volatility variables, h10,h20andh100sampled
+by the HMC. It is seen that those volatility variables have the similar co rrelation
+behavior. Other volatility variables also show the similar behavior. Thu s hereafter
+we only focus on the volatility variable h100as the representative one.
+Fig.4 compares the ACF of h100by the HMC and Metropolis algorithms. It
+is obvious that the ACF by the HMC decreases more rapidly than that by theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+10Authors’ Names
+0 20 40 60 80t0.010.11ACFh10
+h20
+h100
+Fig. 3. Autocorrelation functions of three volatility vari ablesh10,h20andh100sampled by the
+HMC algorithm for T= 2000 data set. These autocorrelation functions show the si milar behavior.
+0 100 200 300 400 500t0.010.11ACFHMC
+Metropolis
+Fig. 4. Autocorrelation function of the volatility variabl eh100by the HMC and Metropolis
+algorithms for T= 2000 data set.
+Metropolis algorithm. We also calculate the autocorrelation time τintdefined by
+τint=1
+2+∞/summationdisplay
+t=1ACF(t). (38)
+The results of τintof the volatility variables are given in Table 1. The values in
+the parentheses represent the statistical errors estimated by the jackknife method.
+We find that the HMC algorithm gives a smaller autocorrelation time tha n the
+Metropolis algorithm, which means that the HMC algorithm samples the volatility
+variables more effectively than the Metropolis algorithm.
+Next we analyze the sampled SV parameters. Fig.5 shows MC histories of the
+φparameter sampled by the HMC and Metropolis algorithms. It seems t hat both
+algorithms have the similar correlationfor φ. This similarity is also seen in the ACF
+in Fig.6(left), i.e. both autocorrelation functions decrease in the sim ilar rate with
+timet. The autocorrelation times of φare very large as seen in Table 1. We also
+find the similar behavior for σ2
+η, i.e. both autocorrelation times of σ2
+ηare large.
+On the other hand we see small autocorrelations for µas seen in Fig.6(right).
+Furthermore we observe that the HMC algorithm gives a smaller τintforµthanNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 11
+φ µ σ2
+η h100
+true 0.97 -1 0.05
+T=1000 HMC 0.973 -1.13 0.053
+SD 0.010 0.51 0.017
+SE 0.0004 0.003 0.001
+2τint 360(80) 3.1(5) 820(200) 12(1)
+Metropolis 0.973 -1.14 0.053
+SD 0.011 0.40 0.017
+SE 0.0005 0.003 0.0013
+2τint 320(60) 10.1(8) 720(160) 190(20)
+T=2000 HMC 0.978 -0.92 0.053
+SD 0.007 0.26 0.012
+SE 0.0003 0.001 0.0009
+2τint 540(60) 3(1) 1200(150) 18(1)
+Metropolis 0.978 -0.92 0.052
+SD 0.007 0.26 0.011
+SE 0.0003 0.003 0.0009
+2τint 400(100) 13(2) 1000(270) 210(50)
+T=5000 HMC 0.969 -1.00 0.056
+SD 0.005 0.11 0.009
+SE 0.0003 0.0004 0.0007
+2τint 670(100) 4.2(7) 1250(170) 10(1)
+Metropolis 0.970 -1.00 0.054
+SD 0.005 0.12 0.008
+SE 0.00023 0.0011 0.0005
+2τint 510(90) 30(10) 960(180) 230(28)
+Table 1. Results estimated by the HMC and Metropolis algorit hms.SDstands for Standard
+Deviation and SEstands for Statistical Error. The statistical errors are es timated by the jackknife
+method. We observe no significant differences on the autocorr elation times among three data sets.
+that of the Metropolis algorithm, which means that HMC algorithm sam plesµ
+more effectively than the Metropolis algorithm although the values of τintforµ
+take already very small even for the Metropolis algorithm.
+The values of the SV parameters estimated by the HMC and the Metr opolis
+algorithms are listed in Table 1. The results from both algorithms well r eproduce
+the true values used for the generation of the artificial financial d ata. Furthermore
+for each parameter and each data set, the estimated parameter s by the HMC and
+the Metropolis algorithms agree well. And their standard deviations a lso agree
+well. This is not surprising because the same artificial financial data, thus the same
+likelihood function is usedfor both MCMC simulationsby the HMC and Met ropolis
+algorithms. Therefore they should agree each other.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+12Authors’ Names
+40000 45000 50000
+MC history0.940.950.960.970.980.991φ
+HMC
+40000 45000 50000
+MC history0.940.950.960.970.980.991φ
+Metropolis
+Fig. 5. Monte Carlo histories of φgenerated by HMC (left) and Metropolis (right) for T= 2000
+data set.
+0 1000t0.010.11ACFHMC
+Metropolis
+0 100 200 300t0.0010.010.1 ACFHMC
+Metropolis
+Fig. 6. Autocorrelation functions of φ(left) and µ(right) by the HMC and Metropolis algorithm
+forT= 2000 data set.
+6. Empirical Analysis
+In this section we make an empirical study of the SV model by the HMC algorithm.
+The empirical study is based on daily data of the Nikkei 225 stock inde x. The
+sampling period is 4 January 1995 to 30 December 2005 and the numbe r of the
+observations is 2706. Fig.7(left) shows the time series of the data. Letpibe the
+Nikkei 225 index at time i. The Nikkei 225 index piare transformed to returns as
+ri= 100ln( pi/pi−1−¯s), (39)
+where ¯sis the average value of ln( pi/pi−1). Fig.7(right) shows the time series of
+returns calculated by Eq.(39). We perform the same MCMC sampling b y the HMC
+algorithm as in the previous section. The first 10000 MC samples are d iscarded and
+then 20000 samples are recorded for the analysis. The ACF of samp ledh100and
+sampled parameters are shown in Fig.8. Qualitatively the results of t he ACF are
+similar to those from the artificial financial data, i.e. the ACF of the v olatility and
+µdecrease quickly although the ACF of φandσ2
+ηdecrease slowly. The estimated
+values of the parameters are summarized in Table 2. The value of φis estimated to
+beφ≈0.977. This value is very close to one, which means the time series has th e
+strong persistency of the volatility shock. The similar values are also seen in theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 13
+HMC φ µ σ2
+η h100
+0.977 0.52 0.020
+SD 0.006 0.13 0.005
+SE 0.001 0.0016 0.001
+2τint560(190) 4(1) 1120(360) 21(5)
+Table 2. Results estimated by the HMC for the Nikkei 225 index data.
+050010001500200025003000t10000150002000025000
+Nikkei 225 Index
+050010001500200025003000t-505rt
+Fig. 7. Nikkei 225 stock index from 4 January 1995 to 30 Decemb er 2005(left) and returns(right).
+0 20 40 60t0.010.11ACFh100
+0 200 400 600800 1000t0.010.11ACFφ
+ση2
+µ
+Fig. 8. Autocorrelation functions of the volatility variab leh100(left) and the sampled parameters
+(right).
+previous studies21,22.
+7. Conclusions
+We applied the HMC algorithm to the Bayesian inference of the SV mode l and
+examined the property of the HMC algorithm in terms of the autocor relation times
+of the sampled data. We observed that the autocorrelation times o f the volatility
+variables and µparameter are small. On the other hand large autocorrelation times
+are observed for the sampled data of φandσ2
+ηparameters. The similar behavior
+for the autocorrelation times are also seen in the literature22.
+From comparison of the HMC and Metropolis algorithms we find that th e HMCNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+14Authors’ Names
+algorithmsamplesthevolatilityvariablesand µmoreeffectivelythantheMetropolis
+algorithm. However there is no significant difference for φandσ2
+ηsampling. Since
+the autocorrelation times of µfor both algorithms are estimated to be rather small
+the improvement of sampling µby the HMC algorithm is limited. Therefore the
+overall efficiency is considered to be similar to that of the Metropolis a lgorithm.
+By using the artificial financial data we confirmed that the HMC algor ithm cor-
+rectly reproduces the true parameter values used to generate t he artificial financial
+data. Thus it is concluded that the HMC algorithm can be used as an alt ernative
+algorithm for the Bayesian inference of the SV model.
+If we are only interested in parameter estimations of the SV model, t he HMC
+algorithm may not be a superior algorithm. However the HMC algorithm samples
+thevolatilityvariableseffectively.ThustheHMC algorithmmayservea sanefficient
+algorithm for calculating a certain quantity including the volatility varia bles.
+Acknowledgments.
+The numerical calculations were carried out on SX8 at the Yukawa In stitute for
+Theoretical Physics in Kyoto University and on Altix at the Institute of Statistical
+Mathematics.
+Note added in proof. After this work was completed the author noticed a sim-
+ilar approach by Liu35. The author is grateful to M.A. Girolami for drawing his
+attention to this.
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1001.0025.txt

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+arXiv:1001.0025v1  [cs.CR]  30 Dec 2009GNSS-based Positioning: Attacks and Countermeasures
+Panos Papadimitratos and Aleksandar Jovanovic
+EPFL
+Switzerland
+Email: firstname.lastname@epfl.ch
+Abstract
+Increasing numbers of mobile computing devices, user-
+portable, or embedded in vehicles, cargo containers, or the
+physical space, need to be aware of their location in order
+to provide a wide range of commercial services. Most often,
+mobile devices obtain their own location with the help of
+Global Navigation Satellite Systems (GNSS), integrating,
+for example, a Global Positioning System (GPS) receiver.
+Nonetheless, an adversary can compromise location-aware
+applications by attacking the GNSS-based positioning: It
+can forge navigation messages and mislead the receiver into
+calculating a fake location. In this paper, we analyze this
+vulnerability and propose and evaluate the effectiveness of
+countermeasures. First, we consider replay attacks, which
+can be effective even in the presence of future cryptographic
+GNSS protection mechanisms. Then, we propose and an-
+alyze methods that allow GNSS receivers to detect the re-
+ception of signals generated by an adversary, and then re-
+ject fake locations calculated because of the attack. We
+consider three diverse defense mechanisms, all based on
+knowledge, in particular, own location ,time, andDoppler
+shift, receivers can obtain prior to the onset of an attack.
+We find that inertial mechanisms that estimate location
+can be defeated relatively easy. This is equally true for the
+mechanism that relies on clock readings from off-the-shelf
+devices; as a result, highly stable clocks could be needed.
+On the other hand, our Doppler Shift Test can be effective
+without any specialized hardware, and it can be applied to
+existing devices.
+1 Introduction
+As wireless communications enable an ever-broadening
+spectrum of mobile computing applications, location or
+position information becomes increasingly important for
+those systems. Devices need to determine their own posi-
+tion,1to enable location-based or location-aware function-
+ality and services. Examples of such systems include: sen-
+sors reporting environmental measurements; cellular tele -
+phones or portable digital assistants (PDAs) and comput-
+ers offering users information and services related to their
+1In this paper, we are not concerned with the related but ortho g-
+onal localization problem of allowing a specific entity to de termine
+and ascertain the location of other devices.surroundings; mobile embedded units, such as those for
+Vehicular Communication (VC) systems seeking to pro-
+vide transportation safety and efficiency; or, merchandize
+(container) and fleet (truck) management systems.
+Global navigation satellite systems (GNSS), such as the
+Global Positioning System (GPS), its Russian counter-
+part (GLONAS), and the upcoming European GALILEO
+system, are the most widely used positioning technology.
+GNSS transmit signals bearing reference information from
+a constellation of satellites; computing platforms nodes),
+equipped with the appropriate receiver, can decode them
+and determine their own location.
+However, commercial instantiations of GNSS systems,
+which are within the scope of this paper, are open to
+abuse: An adversary can influence the location informa-
+tion,loc(V), a node Vcalculates, and compromise the node
+operation. For example, in the case of a fleet management
+system, an adversary can target a specific truck. First, the
+adversary can use a transmitter of forged GNSS signals
+that overwrite the legitimate GNSS signals to be received
+by the victim node (truck) V. This would cause a false
+loc(V) to be calculated and then reported to the fleet cen-
+ter, essentially concealing the actual location of Vfrom the
+fleet management system. Once this is achieved, physical
+compromise of the truck (e.g., breaking into the cargo or
+hijacking the vehicle) is possible, as the fleet management
+system would have limited or no ability to protect its as-
+sets.
+This is an important problem, given the consequences
+such attacks can have. In this paper, we are concerned
+with methods to mitigate such a vulnerability. In partic-
+ular, we propose mechanisms to detect and reject forged
+GNSS messages, and thus avoid manipulation of GNSS-
+based positioning. Our investigation is complementary
+to cryptographic protection, which commercial GNSS sys-
+tems do not currently provide but are expected to do so
+in the future (e.g., authentication services by the upcom-
+ing GALILEO system [5]). Our approach is motivated by
+the fundamental vulnerability of GNSS-based positioning
+toreplay attacks [9], which can be mounted even against
+cryptographically protected GNSS.
+The contribution of this paper consists of three mecha-
+nisms that allow receivers to detect forged GNSS messages
+and fake GNSS signals. Our countermeasures rely on in-
+formation the receiver obtained before the onset of an at-
+1tack, or more precisely, before the suspected onset of an
+attack. We investigate mechanisms that rely on own (i)
+location information, calculated by GNSS navigation mes-
+sages, (ii) clock readings, without any re-synchronization
+with the help of the GNSS or any other system, and (iii)
+received GNSS signal Doppler shift measurements. Based
+on those different types of information, our mechanisms
+can detect if the received GNSS signals and messages orig-
+inate from adversarial devices. If so, location informatio n
+induced by the attack can be rejected and manipulation
+of the location-aware functionality be avoided. We clarify
+that the reaction to the detection of an attack, and mecha-
+nisms that mitigate unavailability of legitimate GNSS sig-
+nals is out of the scope of this paper.
+We briefly introduce the GNSS operation and related
+work in Sec. 2. We discuss the adversary model and specific
+attack methods in Sec. 3.2. We then present and analyze
+the three defensive mechanisms in Sec. 4. Our findings
+support that highly accurate clocks can be very effective
+at the expense of appropriate clock hardware; but they
+can otherwise be susceptible, when off-the-shelf hardware
+is used. Location-based mechanisms can also be defeated
+relatively easily. On the contrary, our Doppler Shift Test
+(DST) provides accurate detection of attacks, even against
+a sophisticated adversary.
+2 GNSS Overview
+2.1 Basic Operation
+Each GNSS-equipped node Vcan receive simultaneously
+a set of navigation messages NAV ifrom each satellite Si
+in the visible constellation . Satellite transmitters utilize a
+spread-spectrum technique and each satellite is assigned a
+unique spreading code Ci. These codes are a priori pub-
+licly known. Navigation messages allow Vto determine its
+position, loc(V) = (XV, YV, ZV), in a Cartesian system, as
+well global time, by obtaining a clock correction or time
+offset,tV, also called the synchronization error . At least
+four satellites should be visible in order for a receiver to
+compute position and exact time, the so-called PVT (Po-
+sition, Velocity and Time) or navigation solution [6]. This
+computation relies on the pseudo-range measurements per-
+formed by V, one pseudo-range per visible satellite, that is,
+estimating the satellite-receiver distance based on the es ti-
+mated signal propagation delay, ρi. For each pseudo-range
+ρiestimated at V, the following equation is formed:
+ρi=|si−loc(V)|+c·tV (1)
+The satellite Siposition is si, the receiver position is
+loc(V),cis the speed of light, and tVis the synchronization
+error for V.2.2 Future Cryptographic GNSS Protec-
+tion
+Cryptographic protection ensures the authenticity and in-
+tegrity of GNSS messages, i.e., ensures that NAV messages
+generated solely by GNSS entities, with no modification,
+are accepted and used by nodes. Currently, cryptography is
+used in military systems, but it is not available for commer-
+cial systems to provide authenticity and integrity. Public
+or asymmetric key cryptography is a flexible and scalable
+approach that does not require tamper-resistant receivers .2
+Independently of the number of receivers present in the sys-
+tem (possibly, millions or eventually hundreds of millions ),
+a pair of private/public keys ki, Kican be assigned to each
+satellite Si, with the public key bound to the satellite iden-
+tity via a certificate provided by a Certification Authority.
+Each receiver obtains the certified public keys of all satel-
+lites in order to be able to validate NAV messages digitally
+signed with the corresponding ki.Navigation Message Au-
+thentication (NMA) [5] will be available as a GALILEO
+service.
+To further enhance protection, a different public-key
+NMA approach was proposed in [7]. Each Sichooses a
+secret spreading code for each NAV message but discloses
+this, along with a hidden timing marker , in a delayed and
+authenticated manner to the receiving nodes. If nodes can
+maintain accurate clocks by means other than the GNSS
+system alone, they can then safely detect messages that are
+forged or replayed between the time of their creation and
+the code disclosure. A similar idea using Secret Spreading
+Codes (SSC) was presented in [11].
+3 Attacking GNSS
+3.1 Adversary model
+The location (position) GNSS-equipped nodes obtain can
+be manipulated by an external adversary, without any ad-
+versarial control on the GNSS entities (the system ground
+stations, the satellites, the ground-to-satellite commun ica-
+tion, and the receiver). If any cryptographic protection
+is present, we assume that cryptographic primitives are
+not breakable and that the private keys of satellites can-
+not be compromised. The adversary can receive signals
+from all available satellites (depending on the locations o f
+the adversary-controlled receivers). It is also fully awar e
+of the GNSS implementation specifics and thus can pro-
+duce fully compliant signals, i.e., with the same modula-
+tion, transmission frequency equal to the nominal one, ft,
+or any frequency in the range of received ones, fr; similarly,
+transmitted and received signal powers, as well as message
+preambles and body format (header, content).
+We classify adversaries based on their ability to re-
+produce GNSS messages and signals, considering ones
+equipped with:
+2To prevent the compromise of a single, system-wide symmetri c
+key, shared among the GNSS and all nodes.
+21. Single or multiple radios, each transmitting at the
+same constant power, Pc
+t, and frequency fc
+t.
+2. Single or multiple radios, each being ability to adapt
+its transmission frequency, fj
+t, over time; jis an index
+of adversarial radios.
+3. Multiple radios with adaptive transmission capabili-
+ties as above, and additionally the ability to estab-
+lish fast communication among any of the adversarial
+nodes equipped with those radios.
+Adversarial radios in all above cases can record GNSS
+signals and navigation messages for long periods. For all
+adversaries above, we consider a nominal range R, within
+which adversarial transmissions can be received, with this
+value varying for different adversarial radios. We denote
+this as the area under attack . Clearly, the more powerful
+and the more numerous radios an adversary has, the higher
+its potential impact can be. In the sense, it can influence a
+larger system area and potentially mislead more receivers.
+We assume that the area under attack does not coin-
+cide with the wireless system area. In other words, the
+adversary has limited physical presence and communica-
+tion capabilities. This implies that nodes can lock on ac-
+tual GNSS signals for a period of time before entering an
+area under attack. We do not dwell on how frequently and
+under what circumstances nodes are under attack. Rather,
+we investigate the strength of different defense mechanisms
+given that a node is under attack. We abstract the phys-
+ical properties of the adversarial equipment and consider
+the periods of time it can cause unavailability and maintain
+the receiver locked on the spoofed signal.
+We emphasize that our attack model is notthe worst
+case; this would be a receiver under attack during its cold
+start, that is, the first time it is turned on and searches for
+GNSS signals to lock on. However, our adversary model
+corresponds to a broad range of realistic cases and it is a
+powerful one. For example, returning to the cargo example
+of the introduction: It will be hard for an adversary to
+control a receiver from its installation, e.g., on a contain er,
+and then throughout a trip. But it would be rather easy to
+select a location and time to mount its attack. Regarding
+the strength of the attacker, it is noteworthy that attacks
+are possible without any physical access to and without
+tampering with the victim node(s) software and hardware.
+3.2 Mounting Attacks against GNSS Re-
+ceivers
+The adversary can construct a transmitter that emits sig-
+nals identical to those sent by a satellite, and mislead the
+receiver that signals originate from a visible satellite. H ow-
+ever, the attacker has to first force the receiver to lose
+its “lock” on the satellite signals. This can be achieved
+byjamming legitimate GNSS signals, by transmitting a
+sufficiently powerful signal that interferes with and ob-
+scures the GNSS signals [12]. Jammers are simple to con-
+struct with low cost and very effective: for example, withReceived GNSS signal delayed 
+Transmit after  treplay NAV message buffering 
+Preamble 
+detection 
+Victim receiver 
+V
+Total 
+delay treplay Adversary 
+Figure 1: Illustration of the replay attack: the adversary
+captures and replays the signal after some time treplay =
+tmin
+replay +τ, with the τ≥0 chosen by the adversary, and
+tmin
+replay >0 imposed by the specifics of the attack configu-
+ration and the adversary capabilities.
+1 Watt of transmission power, the reception of GNSS sig-
+nals is stopped within a radius of approximately 35 km
+radius [6,12].
+Then, the adversary can spoof GNSS signals, i.e., forge
+and transmit signals at the same frequency and with power
+thatexceeds that of the legitimate GNSS signal at the re-
+ceiver’s antenna. Satellite simulators are capable of broa d-
+casting simultaneously signals carrying counterfeit navi ga-
+tion data from ten satellites.3The spoofed signal can also
+be generated by manipulating and rebroadcasting actual
+signals ( meaconing ). As long as the lock of the victim re-
+ceiver Von the spoofed signal persists, loc(V) is under the
+influence or full control of the adversary.
+Apart from jamming, the adversary could take advan-
+tage of gaps in coverage , i.e., areas and periods of time for
+which Vcannot lock on to more than three satellite sig-
+nals. Clearly, this can be often possible in urban areas or
+because of the terrain, such as tunnels or obstructions from
+high-rise buildings. We do not consider further this case,
+as such loss of satellite signals is not under the control of
+the attacker. Nonetheless, the tests we propose here are ef-
+fective independently of what causes receivers to loose loc k
+on GNSS signals.
+3.3 Replay attack
+Thereplay attack can be viewed as a part of a more general
+class of relay attacks : the attacker receives at one location
+legitimate GNSS signals, relays those to another location
+3The adversary can deceive the receiver after down-conversi on
+of the satellite signal, with one component in-phase and one in-
+quadrature:
+I(t) =aiCa(t)M(t)cos(ft) (2)
+Q(t) =aqCa(t)M(t)sin(ft) (3)
+Cais the C/A (Course/Aquisition) code, M(t) is the NAV message,
+and coefficients aiandaqrepresent the signal attenuation. The at-
+tacker could pick the amplifying coefficients aiandaqsuch that the
+received signal power exceeds the nominal power od a GPS sign al [13].
+3where it retransmits them without any modification. This
+way the adversary can avoid detection if cryptography is
+employed, while it can “present” a victim with GNSS sig-
+nals that are not normally visible at the victim’s location.
+In this paper, we abstract away the placement of adversar-
+ial nodes, and we characterize the replay attack by two fea-
+tures: (i) the adversarial node capability to receive, reco rd
+and replay GNSS signals, and (ii) the delay treplay between
+reception and re-transmission of a signal.
+The GNSS signal reception and replay can be done
+at the message or symbol level, or it can be done by
+recording the entire frequency band and replaying it with-
+out de-spreading signals. The latter, more involved and
+thus costly, would enable the attacker to mount an at-
+tack against the delayed-disclosure secret spreading code
+approach, as pointed out in [7], not only for long replay-
+ing delays but also for very short ones. Clearly, such an
+instantiation of the replaying attack implies a more sophis -
+ticated adversary than one replaying symbols or messages.
+For example, the adversary would need to infer, possibly by
+possessing a legitimate receiver, the start of NAV messages
+to replay signals accordingly
+Thetreplay delay between reception and re-transmission
+depends on the attack configuration (e.g., the distance be-
+tween the receiving and re-transmitting adversarial radio s,
+the physics of the signal propagation, and, when applica-
+ble, the delay for the adversary to decode the GNSS signal).
+We capture such factors by considering tmin
+replay >0, a min-
+imum delay that the adversary cannot avoid. Beyond this,
+the attacker can choose some additional delay τ≥0, such
+that it replays the signal after treplay =tmin
+replay +τ. We
+illustrate a replay attack in Fig. 1: The recording of the
+NAV message starts after its beginning is detected, due to
+the preamble 10001011, with length of eight chips, and the
+decoding of the NAV message first bit. This corresponds
+totmin
+replay = 20ms: the transmission rate of 50 bit/s implies
+that 20ms are needed for the first bit to be received by an
+adversarial radio.
+The adversary can choose different treplay values for sig-
+nals from different satellites, even though “blind” replayi ng
+of all NAV signals with the same delay can be effective. The
+selection of which signals (from which satellites) to relay of-
+fer flexibility. But even the “blind” replaying of all NAV
+signals (the entire band) can be effective: treplay controls
+the “shift” in the PVT solution. Essentially, treplay con-
+trols the “shift” in the PVT solution the adversary induces
+to the victim node(s).
+Fig. 2 shows the impact of a replay attack as a function
+of the spoofing stage of the attack: (i) the location offset
+or error, i.e., the distance between the attack-induced and
+the actual victim receiver position, and (ii) the time offset
+or error, that is, the time difference between the attack-
+induced clock value and the actual time. We consider for
+this example trelay= 20ms, as the first bit decoding de-
+lay dwarfs the preamble detection and propagation delays.
+This is indeed a very subtle attack we refer to [9] for a range
+oftreplay values, which shows that the larger the treplay, as0 50 100 150 200 250 300010002000300040005000600070008000900010000
+Attack duration  [s]Distance offset  [m]
+(a)
+0 50 100 150 200 250 300050100150200250300350
+Attack duration [s]Time offset   [ms]
+  
+(b)
+Figure 2: Impact of the replay attack, as a function of
+thespoofing attack duration. (a) Location offset or er-
+ror: Distance between the attack-induced and the actual
+victim receiver position. (b) Time offset or error: Time
+difference between the attack-induced clock value and the
+actual time.
+the adversary tunes its τvalue, the higher the location and
+time offsets.
+Even for a very low treplay, while the mobile node re-
+ceiver is still locked on the attacker-transmitted signals , the
+location error increases, with the victim receiver “dragge d”
+away from its actual position. Each millisecond of trelay
+translates approximately into 300m of location offset for
+each pseudorange (as the speed of light, c, is taken into
+account), with the actual “displacement” of the victim de-
+pending on the geometry (e.g., position of the satellite
+whose signals were replayed).
+As for the time offset, which can be viewed as a side-
+effect of the attack: it is in the order of less than one mil-
+lisecond per second, and it can very well go easily unnoticed
+by the user. With a given trelay, every time the victim re-
+ceiver re-synchronizes, typically at the end of a NAV mes-
+sage that lasts 30 sec, treplay will emerge as tVfrom the
+PVT solution and thus will be accumulated as part of the
+time offset shown in Fig. 2.
+4 Defense mechanisms
+We investigate three defense mechanisms that rely on a
+common underlying three-step idea. First, the receiver col -
+lects data for a given parameter during periods of time it
+deems it is not under attack; we term this the normal mode .
+4Second, based on the normal mode data, the receiver pre-
+dicts the value of the parameter in the future. When it
+suspects it is under attack, it enters what we term alert
+mode. In this mode, the receiver compares the predicted
+values with the ones it obtains from the GNSS functional-
+ity. If the GNSS-obtained values differ, beyond a protocol-
+selectable threshold, from the predicted ones, the receive r
+deems it is under attack . In that case, all PVT solutions
+obtained in alert mode are discarded. Otherwise, the sus-
+pected PVT solutions are accepted and the receiver reverts
+to the normal mode.
+In this work, we consider three parameters: location ,
+time, andDoppler Shift , and we present the corresponding
+detection mechanisms, Location Inertial Test ,Clock Offset
+Test, andDoppler Shift Test . We emphasize again that all
+three mechanisms rely on the availability of prior informa-
+tion collected in normal mode. But they are irrelevant if
+the receiver starts its operation without any such informa-
+tion (i.e., a cold start ).
+To evaluate the proposed schemes, we use GPS traces
+collected by an ASHTECH Z-XII3T receiver that out-
+puts observation and navigation (.obs and .nav) data into
+RINEX ( Receiver Independent Exchange Format ) [8]. We
+implement the PVT solution functionality in Matlab, ac-
+cording to the receiver interface specification [8]. Our im-
+plementation operates on the RINEX data, which include
+pseudoranges and Doppler frequency shift and phase mea-
+surements. We simulate the movement of receivers over a
+period of T= 300 s, with their position updated at steps of
+Tstep= 1sec.
+4.1 Location Inertial Test
+At the transition to alert mode, the node utilizes own lo-
+cation information obtained from the PVT solution, to
+predict positions while in attack mode. If those positions
+match the suspected as fraudulent PVT ones, the receiver
+returns to normal mode. We consider two approaches for
+the location prediction: (i) inertial sensors and (ii) Kalm an
+filtering.
+Inertial sensors , i.e., altimeters, speedometers, odome-
+ters, can calculate the node (receiver) location indepen-
+dently of the GNSS functionality.4However, the accuracy
+of such (electro-mechanical) sensors degrades with time.
+One example is the low-cost inertial MEMS Crista IMU-15
+sensor (Inertial Measurement Unit).
+Fig. 3 shows the position error as a function of time [4],
+which is in our context corresponds to the period the re-
+ceiver is in the alert mode. As the inertial sensor inaccurac y
+increases, the node has to accept as normal attack-induced
+locations. Fig. 4 shows a two-dimensional projection of
+two trajectories, the actual one and the estimated and er-
+roneously accepted one. We see that over a short period
+4They have already been used to provide continuous navigatio n
+between the update periods for GNSS receivers, which essent ially are
+discrete-time position/time sensors with sampling interv al of approx-
+imately one second0102030405060708090100050100150200250300
+GNSS unavailability period [s]Inertial navigation error [m]
+  
+Figure 3: Location error of Crista IMU-15 inertial sensor,
+as a function of the GNSS unavailability period.
+3.456 3.458 3.46 3.462 3.464 3.466 3.468
+x 1065.295.35.315.325.335.345.355.365.375.38x 105
+ 
+X coordinate [m] Y coordinate [m]
+Attacker−induced trajectory
+Actual trajectory
+Figure 4: Illustration of location error using inertial sen -
+sors: Actual vs. estimated when under attack trajectory.
+of time, a significant difference is created because of the
+attack.
+A more effective approach is to rely on Kalman filtering
+of location information obtained during normal mode. Pre-
+dicted locations can be obtained by the following system
+model:
+Sk+1= Φ kSk+Wk (4)
+withSkbeing the system state, i.e., location ( Xk, Yk, Zk)
+and velocity ( V xk, V yk, V zk) vectors, Φ kthe transition
+matrix, and Wkthe noise. Fig. 5 illustrates the location
+offset for a set of various trajectories. Unlike the case that
+only inertial sensors are used, with measurements of iner-
+tial sensors (with the error characteristics of Fig. 3 used
+as data when GNSS signals are unavailable, filtering pro-
+vides a linearly increasing error with the period of GNSS
+unavailability.
+Overall, for short unavailability periods, inertial mech-
+anisms can be effective. As long as the error (Y axes of
+Figs. 4, 5) does not grow significantly, the replay attack
+can be detected. But for sufficiently high errors, the re-
+play attack impact can remain undetected. We remind the
+reader that the x-axes in Fig. 2 provide the duration of the
+spoofing attack - the transmission (replay) of GNSS signals
+- and they are not to be confused with the duration of the
+GNSS period of unavailability in the x-axis of Figs. 4, 5.
+50 50 100 150 200 250 300020040060080010001200
+Time [s]Distance offset [m]
+Figure 5: Distance error of inertial mechanisms with
+Kalman filtering, as a function of the GNSS unavailabil-
+ity period.
+0 5 10 15 20 25 30−9−8.5−8−7.5−7−6.5−6x 10−3
+Time  [30s step]Time offset  [s]
+  
+Figure 6: Clock offset for the ASHTECH Z-XII3T receiver,
+during a 900 sec period with no re-synchronization.
+4.2 Clock Offset Test
+Each receiver has a clock that is in general imprecise, due
+to the drift errors of the quartz crystal. If the reception
+of GNSS signals is disrupted, the oscillator switches from
+normal to holdover mode. Then, the time accuracy de-
+pends only on the stability of the local oscillator [2,6]. Th e
+quartz crystals of different clocks run at slightly different
+frequencies, causing the clock values to gradually diverge
+from each other (skew error).
+A simulation based study [2] of quartz clocks claims that
+coarse time synchronization can be maintained at microsec-
+ond accuracy without GPS reception for 350 sec in 95%
+cases. This means that quartz oscillators can maintain
+millisecond synchronization for few hours, including ran-
+dom errors and temperature change inaccuracies. Indeed,
+in such a case, the adversary would need to cause GNSS
+availability for long periods of time, for example, tens of
+hours, before being able to mount a relay attack that causes
+a time offset in the order of tens of milliseconds.
+However, without highly stable clocks, mounting attacks
+against the Clock Offset Test can be significantly easier.
+This can be the case for a ASHTECH receiver, for which
+time offset values are shown at successive points in time,
+each 30 seconds apart, in Fig. 6. We clarify this is notto be perceived as criticism for a given receiver or to be
+the basis for the suitability of the Clock Offset Test. As
+explained above, the stability of the receiver clock deter-
+mines the strength of this test. But the data in Fig. 6,
+over a period of 900 seconds, exactly demonstrates that
+for commodity receivers significant instability is observe d;
+time offset values are in the order of ten milliseconds (or
+slightly less). Consequently, the adversary would need to
+jam for roughly a couple of minutes, force the receiver to
+consider as acceptable a time offset of 20 to 32 millisec-
+onds, and thus be mislead by a replay attack as detailed in
+Sec. 3.
+Finally, we note that we do not consider here the case
+of synchronization by means external to the GNSS system.
+For example, if the receiver could connect to the Internet
+and run NTP, it could obtain accurate time. But this would
+be an infrequent operation (in the order of magnitude of
+days), thus useful only if highly stable clock hardware were
+available.
+4.3 Doppler Shift Test (DST)
+Based on the received GNSS signal Doppler shift, with
+respect to the nominal transmitter frequency ( ft=
+1.575GHz), the receiver can predict future Doppler Shift
+values. Once lock to GNSS signals is obtained again, pre-
+dicted Doppler shift values are compared to the ones cal-
+culated due to the received GNSS signal. If the latter are
+different than the predicted ones beyond a threshold, the
+GNSS signal is deemed adversarial and rejected. What
+makes this approach attractive is the smooth changes of
+Doppler shift and the ability to predict it with low, es-
+sentially constant errors over long periods of time. This
+in dire in contrast to the inertial test based on location,
+whose error grows exponentially with time.
+The Doppler shift is produced due to the relative motion
+of the satellite with respect to the receiver. The satellite
+velocity is computed using ephemeris information and an
+orbital model available at the receiver. The received fre-
+quency, fr, increases as the satellite approaches and de-
+creases as it recedes from the receiver; it can be approxi-
+mated by the classical Doppler equation:
+fr=ft·(1−vr·a
+c) (5)
+where ftis nominal (transmitted) frequency, frreceived
+frequency, vris the satellite-to-user relative velocity vector
+andcspeed of radio signal propagation. The product vr·
+arepresents the radial component of the relative velocity
+vector along the line-of-sight to the satellite.
+If the frequency shift differs from the predicted shift for
+each visible satellite Siin the area depending on the data
+obtained from the almanac (in the case when the naviga-
+tion history is available), for more than defined thresholds
+(∆fmin,∆fmax) or estimated Doppler shift from naviga-
+tion history differs for more than the estimated shift, know-
+ing the rate ( r), the receiver can deem the received signal
+as product of attack.
+650 100 150 200 250 3002300235024002450250025502600265027002750
+Time [s]Frequency offset [Hz]
+  
+Measured Doppler shift [Hz ] 
+Linear approximation
+Prediction bounds
+Figure 7: Measured and approximated Doppler frequency
+shift.
+TheAlmanac contains approximate position of the satel-
+lites, ( Xsi, Y si, Zsi), time and the week number ( WN, t ),
+and the corrections, such that the receiver is aware of the
+expected satellites, their position, and the Doppler offset .
+Because of the high carrier frequencies and large satel-
+lite velocities, large Doppler shifts are produced ( ±5kHz),
+and vary rapidly (1 Hz/s). The oscillator of the receiver
+has frequency shift of ±3KHz, thus the resultant frequency
+shift goes therefore up to ±9KHz. Without the knowledge
+of the shift, the receiver has to perform a search in this
+range of frequencies in order to acquire the signal. The
+rate of Doppler shift receiving frequency caused by the rel-
+ative movement between GPS satellite and vehicles approx-
+imately 40 Hz per minute to the maximum. These varia-
+tions are linear for every satellite. If the receiver is mobi le,
+the Doppler shift variation can be estimated knowing the
+velocity of the receiver( [3]).
+In our simulations, Doppler shift is analyzed for each
+available satellite (number of available satellites varie s). To
+be consistent with results shown for other mechanisms, we
+present results for DST for the 300sec period.
+We observe in Fig. 7 the Doppler shift variation based
+on data collected by an ASHTECH receiver: the maximum
+change in rate is within + /−20Hz around a linear curve
+fitted to the data. This clues that with sufficient samples,
+the future Doppler Shift rate, and thus the shift per se,
+values can be predicted. In practice, we observe that 50
+sec of samples, with one sample per second, appear to be
+sufficient.
+More precisely, the rate of change of the frequency shift,
+Di(t), is computed for each satellite, Si, as:
+ri=dDi(t)
+dt(6)
+which can be approximated by numerical methods. Based
+on prior samples for each Di, available for some time win-
+dow the frequency shift can be predicted based those sam-
+ples and the estimate rate of change of the Doppler shift.
+Based on prior measured statistics of the signal at the re-
+ceiver, the variance σ2of a random component, assumed
+to beN(0, σ2), can be estimated. This random component0 50 100 150 200 250 300−10000100020003000
+Time [s]Frequency offset [Hz]SV−1
+0 50 100 150 200 250 300−10000−50000
+Time [s]Frequency offset [Hz]SV−4
+  
+0 50 100 150 200 250 3000200040006000
+Time [s]Frequency offset [Hz]SV−7
+0 50 100 150 200 250 3000100020003000
+Time [s]Frequency offset [Hz]SV−13
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz]SV−20
+0 50 100 150 200 250 300−10000100020003000
+Time [s]Frequency offset [Hz] SV−24
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz] SV−25
+Figure 8: Doppler shift attack; unsophisticated adversary .
+The dotted line represents the predicted and the solid line
+the measured frequency offset.
+is due to signal variation (including receiver mobility, RF
+multipath, scattering). Its estimation can serve to deter-
+mine an acceptable interval around the predicted values.
+The adversary is mostly at the ground and static or mov-
+ing with speed that is much smaller than the satellite ve-
+locity, which is in a range around 3km/s. Thus, the adver-
+sary will not be able to produce the same Doppler shift as
+the satellites, unless it changes its transmission frequen cy
+to match the one receivers would obtain from GNSS sig-
+nals due to the Doppler shift. An unsophisticated attacker
+would then be easily detected. This is illustrated in Fig. 8:
+After a “gap” corresponding to jamming, there is a striking
+difference, between 100 and 150 seconds, when comparing
+the Doppler shift due to the attack to the predicted one.
+The case of A sophisticated adversary that controls its
+transmission frequency (the attack starts at 160 s)is shown
+in the Fig. 9. The adversary has multiple adaptive ra-
+dios and it operates according to the following principle: i t
+predicts the Doppler frequency shift at the location of the
+receiver, and it then changes its transmission frequency
+accordingly. If the attacker is not precisely aware of the
+actual location and motion dynamics of the victim node
+(receiver), there is still a significant difference between t he
+predicted and the adversary-caused Doppler shift. This
+is shown, with a magnitude of approximately 300 Hz, in
+Fig. 9; a difference that allows detection of the attack.
+5 Conclusion
+Existing GNSS receivers are vulnerable to a number of
+attacks that manipulate the location and time the re-
+ceivers compute. We qualitatively and quantitatively ana-
+lyze those in this paper, and identify memory-based mech-
+anisms that can help in securing GNNS signals. In particu-
+lar, we realize that location-based inertial mechanisms an d
+a clock offset test can be relatively easily defeated, with th e
+adversary causing (through jamming) a sufficiently long
+period of unavailability. In the latter case, only special-
+ized highly stable clock hardware could enable detection of
+fraudulent GNSS signals. Our Doppler Shift Test provides
+70 50 100 150 200 250 300020004000
+Time [s]Frequency offset [Hz]SV−1
+  
+0 50 100 150 200 250 300−10000−50000
+Time [s]Frequency offset [Hz]SV−21
+  
+0 50 100 150 200 250 3000500010000
+Time [s]Frequency offset [Hz]SV−7
+  
+0 50 100 150 200 250 300020004000
+Time [s]Frequency offset [Hz]SV−25
+  
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz]SV−9
+  
+0 50 100 150 200 250 3000100020003000
+Time [s]Frequency offset [Hz]SV−29
+  
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz]SV−13
+  
+Figure 9: Doppler shift attack; sophisticated adversary.
+The dotted line represents the predicted and the solid line
+the measured frequency offset.
+resilience to long unavailability periods without special ized
+equipment.
+Our results are the first, to the best of our knowledge,
+to provide tangible demonstration of effective mechanisms
+to secure mobile systems from location information manip-
+ulation via attacks against the GNSS systems.
+As part of on-going and future work, we intent to further
+refine and generalize the simulation framework we utilized
+here, to consider precisely the effect of counter-measures
+that only partially limit the attack impact. Moreover, we
+will consider more closely the cost of mounting attacks of
+differing sophistication levels, especially through proof -of-
+concept implementations.
+References
+[1] N. Bertelsen, K. Borre, The GPS Code Software Re-
+ceiver , Aalborg University, Birkhauser, 2007
+[2] W. Franz and H. Hartenstein, Inter-Vehicle Communi-
+cations, FleetNet project , University Karlruhe, 2005
+[3]http://www.freepatentsonline.com/5036329.html
+[4] S. Godha, Performance Evaluation of Low Cost
+MEMS-Based IMU Integrated with GPS for Land Ve-
+hicle Navigation Appplication , University of Calgary,
+2006
+[5] G.W. Hein and F. Kneissl, Authenticating GNSS Proofs
+Against Spoofs , InsideGNSS, September/October 2007
+[6] E.D. Kaplan, Understanding GPS - Principles and Ap-
+plications , Artech House, 2006
+[7] M. Kuhn, An asymetric Security Mechanism for Nav-
+igation Signals , Sixth Information Hiding Workshop,
+Toronto, Canada, 2004
+[8] NAVSTAR GPS Joint Program Office, NAVSTAR
+Global Positioning System - Interface Specification IS-
+GPS 200 Space Segment/Navigation User Interfaces ,
+SMC/GP, CA, USA, 2004[9] P. Papadimitratos and A. Jovanovic, Protection and
+Fundamental Vulnerability of GNSS , IWSSC, Toulouse,
+2008
+[10] A.D. Rabbany, Introduction to GPS , Artech House,
+2002
+[11] L. Scott, Anti-Spoofing and Authenticated Signal Ar-
+chitectures for Civil Navigation Signals , ION-GNNS,
+Portand, Oregon, 2003
+[12] J.A. Volpe, Vulnearability Assesment of the Trans-
+portation Infrastructure Relying on GPS , NTSC, NAV-
+CEN draft report, 2001
+[13] H. Wen, P. Huang, and J. Fagan, Countermeasures for
+GPS signal spoofing , The University of Oklahoma, 2004
+[14] J. Zogg, GPS Basics - Introduction to the System , U-
+blox AG, 2002
+8

1001.0026.txt

@@ -0,0 +1,318 @@
+arXiv:1001.0026v1  [astro-ph.SR]  30 Dec 2009Detectionof solar-likeoscillations from Keplerphotometry ofthe open
+cluster NGC 6819
+DennisStello,1Sarbani Basu,2HansBruntt,3Benoˆ ıt Mosser,3Ian R. Stevens,4
+TimothyM.Brown,5Jørgen Christensen-Dalsgaard,6Ronald L. Gilliland,7Hans Kjeldsen,6
+Torben Arentoft,6J´ erˆ omeBallot,8CarolineBarban,3TimothyR. Bedding,1WilliamJ. Chaplin,4
+YvonneP. Elsworth,4Rafael A.Garc´ ıa,9Marie-Jo Goupil,3SaskiaHekker,4Daniel Huber,1
+SavitaMathur,10Søren Meibom,11Reza Samadi,3VinothiniSangaralingam,4
+Charles S. Baldner,2KevinBelkacem,12KatiaBiazzo,13Karsten Brogaard,6
+Juan Carlos Su´ arez,14Francesca D’Antona,15Pierre Demarque,2LisaEsch,2NingGai,2,16
+Frank Grundahl,6YvelineLebreton,17Biwei Jiang,16NadaJevtic,18ChristofferKaroff,4
+AndreaMiglio,12JoannaMolenda- ˙Zakowicz,19JosefinaMontalb´ an,12ArletteNoels,12
+Teodoro RocaCort´ es,20,21Ian W. Roxburgh,22AldoM. Serenelli,23VictorSilvaAguirre,23
+ChristiaanSterken,24Peter Stine,18Robert Szab´ o,25AchimWeiss,23WilliamJ. Borucki,26
+DavidKoch,26JonM. Jenkins27– 2 –
+1SydneyInstituteforAstronomy(SIfA),SchoolofPhysics,U niversityofSydney,NSW2006,Australia
+2DepartmentofAstronomy,YaleUniversity,P.O.Box 208101, New Haven,CT 06520-8101
+3LESIA,CNRS,Universit´ ePierreetMarieCurie,Universit´ eDenisDiderot,ObservatoiredeParis,92195Meudon,
+France
+4SchoolofPhysicsandAstronomy,UniversityofBirmingham, Edgbaston,BirminghamB152TT,UK
+5LasCumbresObservatoryGlobalTelescope,Goleta,CA 93117 ,USA
+6DepartmentofPhysicsandAstronomy,AarhusUniversity,80 00AarhusC,Denmark
+7SpaceTelescopeScienceInstitute,3700San MartinDrive,B altimore,Maryland21218,USA
+8Laboratoired’AstrophysiquedeToulouse-Tarbes,Univers it´ edeToulouse,CNRS,14avE.Belin,31400Toulouse,
+France
+9Laboratoire AIM, CEA/DSM-CNRS, Universit´ e Paris 7 Didero t, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-
+Yvette,France
+10IndianInstituteofAstrophysics,Koramangala,Bangalore 560034,India
+11Harvard-SmithsonianCenterforAstrophysics,60GardenSt reet,Cambridge,MA,02138,USA
+12Institutd’AstrophysiqueetdeG´ eophysiquedel’Universi t´ edeLi` ege,17All´ eedu6Aoˆ ut,B-4000Li` ege,Belgium
+13ArcetriAstrophysicalObservatory,LargoE.Fermi5,50125 ,Firenze,Italy
+14InstitutodeAstrof´ ısicadeAndaluc´ ıa(CSIC),Dept. Stel larPhysics,C.P. 3004,Granada,Spain
+15INAF -Osservatoriodi Roma,via diFrascati 33,I-00040,Mon teporzio,Italy
+16DepartmentofAstronomy,BeijingNormalUniversity,Beiji ng100875,China
+17GEPI,ObservatoiredeParis,CNRS, Universit´ eParisDider ot,5Place JulesJanssen,92195Meudon,France
+18Departmentof Physics& EngineeringTechnology,Bloomsbur gUniversity,400East SecondSt, BloomsburgPA
+17815,USA
+19AstronomicalInstitute,UniversityofWrocław,ul.Kopern ika11,51-622Wrocław,Poland
+20DepartmentodeAstrof´ ıca,Universidadde LaLaguna,38207 LaLaguna,Tenerife,Spain
+21InstitutodeAstrof´ ıcadeCanarias,38205La Laguna,Tener ife,Spain
+22QueenMaryUniversityofLondon,Mile EndRoad,LondonE14NS ,UK
+23MaxPlanckInstituteforAstrophysics,KarlSchwarzschild Str. 1,GarchingbeiM¨ unchen,D-85741,Germany
+24Vrije UniversiteitBrussel, Pleinlaan2,B-1050Brussels, Belgium
+25KonkolyObservatory,H-1525Budapest,P.O. Box67,Hungary
+26NASA AmesResearchCenter,MS 244-30,MoffatField,CA 94035 ,USA
+27SETIInstitute/NASA AmesResearchCenter,MS244-30,Moffa tField, CA 94035,USA– 3 –
+ABSTRACT
+Asteroseismology of stars in clusters has been a long-sough t goal because the as-
+sumption of a common age, distance and initial chemical comp osition allows strong
+tests of the theory of stellar evolution. We report results f rom the first 34 days of sci-
+encedatafromthe KeplerMission fortheopenclusterNGC6819—oneoffourclus-
+ters in the field of view. We obtain the first clear detections o f solar-like oscillations
+in the cluster red giants and are able to measure the large fre quency separation, ∆ν,
+andthefrequencyofmaximumoscillationpower, νmax. Wefindthattheasteroseismic
+parameters allow us to test cluster-membership of the stars , and even with the limited
+seismicdatainhand,wecan alreadyidentifyfourpossiblen on-membersdespitetheir
+havinga betterthan 80% membershipprobabilityfrom radial velocitymeasurements.
+We are also able to determine the oscillation amplitudes for stars that span about two
+orders of magnitude in luminosity and find good agreement wit h the prediction that
+oscillation amplitudesscale as the luminosityto the power of 0.7. These early results
+demonstrate the unique potential of asteroseismology of th e stellar clusters observed
+byKepler.
+Subjectheadings: stars: fundamentalparameters—stars: oscillations—star s: interi-
+ors—techniques: photometric—openclustersandassociati ons: individual(NGC6819)
+1. Introduction
+Openclustersprovideuniqueopportunitiesinastrophysic s. Starsinopenclustersarebelieved
+to be formed from the same cloud of gas at roughly the same time . The fewer free parameters
+available to model cluster stars make them interesting targ ets to analyze as a uniform ensemble,
+especiallyforasteroseismicstudies.
+Asteroseismology is an elegant tool based on the simple prin ciple that the frequency of a
+standing acoustic wave inside a star depends on the sound spe ed, which in turn depends on
+the physical properties of the interior. This technique app lied to the Sun (helioseismology) has
+provided extremely detailed knowledge about the physics th at governs the solar interior, (e.g.,
+Christensen-Dalsgaard2002). Allcoolstarsareexpectedt oexhibitsolar-likeoscillationsofstand-
+ing acoustic waves – called p modes – that are stochastically driven by surface convection. Using
+asteroseismology to probe the interiors of cool stars in clu sters, therefore, holds promise of re-
+warding scientific return (Gough& Novotny 1993; Brown& Gill iland 1994). This potential has
+resulted in several attempts to detect solar-like oscillat ions in clusters using time-series photome-
+try. These attempts were often aimed at red giants, since the iroscillation amplitudesare expected– 4 –
+tobelargerthanthoseofmain-sequenceorsubgiantstarsdu etomorevigoroussurfaceconvection.
+Despite these attempts, only marginal detections have been attained so far, limited either by the
+lengthofthetimeseriesusuallyachievablethroughobserv ationswiththe HubbleSpaceTelescope
+(Edmonds& Gilliland 1996; Stello&Gilliland 2009) or by the difficulty in attaining high preci-
+sion from ground-based campaigns (e.g., Gillilandetal. 19 93; Stelloet al. 2007; Frandsen et al.
+2007).
+InthisLetterwereportcleardetectionsofsolar-likeosci llationsinred-giantstarsintheopen
+cluster NGC 6819 using photometry from NASA’s Kepler Mission (Borucki et al. 2009). This
+cluster,oneoffourinthe Keplerfield, isabout2.5Gyrold. Itisatadistanceof2.3kpc, andha sa
+metallicityof[Fe/H] ∼ −0.05(see Holeet al. 2009, and references herein).
+2. Observations anddata reduction
+The data were obtained between 2009 May 12 and June 14, i.e., t he first 34 days of con-
+tinuous science observations by Kepler(Q1 phase). The spacecraft’s long-cadence mode ( ∆t≃
+30minutes) used in this investigation provided a total of 1639 data points in the time series of
+each observed star. For this Letter we selected 47 stars in th e field of the open cluster NGC 6819
+with membership probability PRV>80% from radial velocity measurements (Holeet al. 2009).
+Figure1showsthecolor-magnitudediagram(CMD)oftheclus terwiththeselectedstarsindicated
+by green symbols. The eleven annotated stars form a represen tative subset, which we will use to
+illustrate our analyses in Sections 3 and 4. We selected the s tars in this subset to cover the same
+brightnessrangeasourfullsample,whilegivinghighweigh ttostarsthatappeartobephotometric
+non-members (i.e., stars located far from the isochrone in t he CMD). Data for each target were
+checked carefully to ensure that the time-series photometr y was not contaminated significantly
+by other stars in the field, which could otherwise complicate the interpretation of the oscillation
+signal.
+Fourteen data points affected by the momentum dumping of the spacecraft were removed
+from the time series of each star. In addition, we removed poi nts that showed a point-to-point
+deviation greater than 4σ, whereσis the local rms of the point-to-point scatter within a 24 hou r
+window. This process removed on average one data-point per t ime series. Finally, we removed a
+linear trend from each time series and then calculated the di screte Fourier transform. The Fourier
+spectraathighfrequencyhavemeanlevelsbelow5partsperm illion(ppm)inamplitude,allowing
+usto search forlow-amplitudesolar-likeoscillations.– 5 –
+3. Extractionofasteroseismicparameters
+Figure 2 shows the Fourier spectra (in power) of 9 stars from o ur subset. These range from
+thelowerred-giant branch to thetip ofthe branch (see Figur e1). The stars are sorted by apparent
+magnitude, which for a cluster is indicative of luminosity, with brightest at the top. Note that the
+redgiantsinNGC6819aresignificantlyfainter( 12/lessorsimilarV/lessorsimilar14)thanthesampleof Keplerfieldred
+giants (8/lessorsimilarV/lessorsimilar12) studied by Beddinget al. (2010). Nevertheless, it is clear from Figure 2 that
+we can detect oscillations for stars that span about two orde rs of magnitude in luminosity along
+theclustersequence.
+Weusedfourdifferentpipelines(Hekkeret al.2009a;Huber et al.2009a;Mathuret al.2009;
+Mosser& Appourchaux 2009) to extract the average frequency separation between modes of the
+same degree (the so-called large frequency separation, ∆ν). We have also obtained the frequency
+of maximum oscillation power, νmax, and the oscillation amplitude. The measured values of ∆ν
+are indicated by vertical dotted lines in Figure 2 centered o n the highest oscillation peaks near
+νmax. While the stars in Figure 2, particularly in the lower panel s, show the regular series of
+peaks expected for solar-likeoscillations,the limitedle ngth of the time-series datadoes not allow
+such structureto be clearly resolved for the mostluminouss tars in our sample— thosewith νmax
+/lessorsimilar20µHz. We do, however, see humps of excess power in the Fourier sp ectra (see Figure 2 star
+no. 2 and 8) with νmaxand amplitude in mutual agreement with oscillations. With l onger time
+series weexpectmorefirm resultsforthesehigh-luminosity giants.
+4. Cluster membership from asteroseismology
+It isimmediatelyclear fromFigure2thatnotallstars follo wtheexpected trendofincreasing
+νmaxwith decreasing apparent magnitude, suggesting that some o f the stars might be intrinsically
+brighterorfainterthanexpected. Sinceoscillationsinas taronlydependonthephysicalproperties
+of the star, we can use asteroseismology to judge whether or n ot a star is likely to be a cluster
+member independentlyof its distanceand of interstellarab sorption and reddening. For cool stars,
+νmaxscaleswiththeacousticcut-offfrequency,anditiswelles tablishedthatwecanestimate νmax
+by scalingfromthesolarvalue(Brownet al. 1991; Kjeldsen& Bedding 1995):
+νmax
+νmax,⊙=M/M⊙(Teff/Teff,⊙)3.5
+L/L⊙, (1)
+whereνmax,⊙= 3100µHz. The accuracy of such estimates is good to within 5% (Stell oet al.
+2009)assumingwehavegoodestimatesofthestellarparamet ersM,L, andTeff.
+In thefollowingweassumetheidealisticscenario whereall clustermembersfollowstandard
+stellar evolutiondescribed by the isochrone. Stellar mass along the red giant branch of thecluster– 6 –
+isochrone varies by less than 1%. The variation is less than 5 % even if we also consider the
+asymptoticgiant branch. For simplicity,we therefore adop t a mass of 1.55M⊙for all stars, which
+is representativefortheisochronefrom Marigoet al. (2008 )(Figure 1) and a similarisochroneby
+VandenBerg etal. (2006). Neglectingbinarity (see Table 1) , we derivethe luminosityof each star
+in our subset from its V-band apparent magnitude, adopting reddening and distance modulus of
+E(B−V) = 0.1and(M−m)V= 12.3,respectively(obtainedfromsimpleisochronefitting,see
+Holeetal.2009). WeusedthecalibrationofFlower(1996)to convertthestellar (B−V)0colorto
+Teff. BolometriccorrectionswerealsotakenfromFlower(1996) . Thederivedquantitieswerethen
+used toestimate νmaxfor each star(Eq.1), and compared withtheobservedvalue(s eeFigure3).
+Figure 3 shows four obvious outliers (no. 1, 3, 8 and 11), thre e of which are also outliers in
+theCMD (no. 1, 3, and11). Fortherest ofthestars weseegood a greement between theexpected
+andobservedvalue,indicatingthattheuncertaintyonthe νmaxestimatesarerelativelysmall. Since
+thevariationsinmassandeffectivetemperatureamongthec lustergiantstarsaresmall,deviations
+fromthedottedlinemustbecausedbyanincorrectestimateo ftheluminosity. Thisimpliesthatthe
+luminositiesofstarsfallingsignificantlyaboveorbelowt helinehavebeenover-orunderestimated,
+respectively. The simplest interpretation is that these ou tliers are fore- or background stars, and
+hence not members of the cluster. To explain the differences between the observed and expected
+value ofνmaxwould require the deviant stars to have Verrors of more than 1 magnitude, and in
+some cases B−Verrors of about 0.2 magnitude if they were cluster members. B inarity may
+explain deviations above the dotted line, but only by up to a f actor of two in L(and hence, in the
+ratio of the observed to expected νmax). The deviation of only one star (no.1) could potentially
+be explained this way. However, that would be in disagreemen t with its single-star classification
+from multi-epoch radial velocity measurements, assuming i t is not a binary viewed pole-on (see
+Table 1). Hence, under the assumptionof a standard stellar e volution, the most likely explanation
+forallfouroutliersinFigure3isthereforethatthesestar sarenotclustermembers. Thisconclusion
+is, however,in disagreementwith theirhighmembershippro babilityfrom measurementsofradial
+velocity (Holeet al. 2009) and proper motion (Sanders 1972) (see Table 1). Another interesting
+possibility is that the anomalous pulsation properties mig ht be explained by more exotic stellar
+evolutionscenariosthan isgenerally anticipatedforopen -clusterstars.
+5. Asteroseismic“color-magnitude diagrams”
+ItisclearfromFigure2thattheamplitudesoftheoscillati onsincreasewithluminosityforthe
+seismicallydeterminedclustermembers. Basedoncalculat ionsbyChristensen-Dalsgaard& Frandsen
+(1983), Kjeldsen& Bedding (1995) have suggested that the ph otometric oscillation amplitude of
+p modes scale as (L/M)sTeff−2, withs= 1(the velocity amplitudes, meanwhile, would scale as– 7 –
+(L/M)s). This was revised by Samadi etal. (2007) to s= 0.7based on models of main sequence
+stars. Takingadvantageofthefewerfreeparameterswithin thisensembleofstars,ourobservations
+allow us to make some progress towards extrapolating this sc aling to red giants and determining
+thevalueof s.
+In Figure4 weintroduceanewtypeofdiagramthatissimilart oaCMD, butwithmagnitude
+replaced by an asteroseismicparameter – in thiscase, theme asured oscillationamplitude. Ampli-
+tudeswereestimatedforallstarsinoursample(exceptfort hefouroutliers)usingmethodssimilar
+tothatofKjeldsenet al.(2008)(seealsoMichelet al.2008) ,whichassumethattherelativepower
+betweenradialandnon-radialmodesisthesameasintheSun. Thisdiagramconfirmstherelation-
+ship between amplitude and luminosity. Despite a large scat ter, which is not surprising from this
+relatively short timeseries, we see that s= 0.7provides a much better match than s= 1.0. Once
+verifiedwithmoredata,thisrelationwillallowtheuseofth emeasuredamplitudeasanadditional
+asteroseismic diagnostic for testing cluster membership a nd for isochrone fitting in general. We
+notethat theother clusters observed by Keplerhave different metallicitiesthan NGC 6819, which
+willallowfutureinvestigationon themetallicitydepende nce oftheoscillationamplitudes.
+We expect to obtain less scatter in the asteroseismic measur ements when longer time series
+become available. That will enable us to expand classical is ochrone fitting techniques to include
+diagramslikethis,whereamplitudecouldalsobereplacedb yνmaxor∆ν. Inparticular,weshould
+beabletodeterminetheabsoluteradiiaidedby ∆νoftheredgiantbranchstars,whichwouldbean
+importantcalibratorfor theoretical isochrones. Additio nally,thedistributionsoftheasteroseismic
+parameters – such as νmax– can potentially be used to test stellar population synthes is models
+(Hekkeret al.2009b;Miglioet al.2009b). Applyingthisapp roachtoclusterscouldleadtofurther
+progress in understanding of physical processes such as mas s loss during the red-giant phase (see
+e.g.,Miglioet al.2009a). Notethatafewclearoutliersare indicativeofnon-membershiporexotic
+stellarevolution,asaresultoffactorssuchasstellarcol lisionsorheavymassloss,whileageneral
+deviationfromthetheoreticalpredictionsbyalargegroup ofstarswouldsuggestthatthestandard
+theorymay need revision.
+Finally, we note that NGC 6819 and another Keplercluster, NGC 6791, contain detached
+eclipsingbinaries(Talamantes& Sandquist2009;Street et al.2005;deMarchi et al.2007;Mochejskaetal.
+2005). For these stars masses and radii can be determined ind ependently (Grundahl et al. 2008),
+whichwillfurtherstrengthenresultsofasteroseismicana lyses.– 8 –
+6. Discussion& Conclusions
+PhotometricdataofredgiantsinNGC6819obtainedbyNASA’s KeplerMission haveenabled
+ustomakethefirst cleardetectionofsolar-likeoscillatio nsin clusterstars. Thegeneral properties
+of the oscillations ( ∆ν,νmax, and amplitudes) agree well with results of field red giants m ade by
+Kepler(Bedding etal.2010)andCoRoT(deRidderet al.2009;Hekker et al.2009b). Wefindthat
+the oscillation amplitudes of the observed stars scale as (L/M)0.7Teff−2, suggesting that previous
+attemptstodetect oscillationsinclustersfrom groundwer eat thelimitofdetection.
+We find that the oscillation properties provide additional t ests for cluster membership, al-
+lowing us to identify four stars that are either non-members or exotic stars. All four stars have
+membership probability higher than 80% from radial-veloci ty measurements, but three of them
+appear to be photometric non-members. We further point out t hat deviations from the theoretical
+predictionsoftheasteroseismicparametersamongalarges ampleofclusterstarshavethepotential
+ofbeingusedasadditionalconstraintsintheisochronefitt ingprocess,whichcanleadtoimproved
+stellarmodels.
+Our results, based on limited data of about one month, highli ght the unique potential of as-
+teroseismologyon the brighteststars in thestellarcluste rs observed by Kepler. With longerseries
+sampled at the spacecraft’s short cadence ( ≃1 minute), we expect to detect oscillations in the
+subgiantsand turn-offstars, as wellas inthebluestraggle rsinthiscluster.
+FundingforthisDiscoverymissionisprovidedbyNASA’sSci enceMissionDirectorate. The
+authorswouldliketothanktheentire Keplerteamwithoutwhomthisinvestigationwouldnothave
+been possible. The authors also thank all funding councils a nd agencies that have supported the
+activitiesofWorkingGroup 2ofthe KeplerAsteroseismicScience Consortium(KASC).
+Facilities: Kepler.
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+Samadi, R., Georgobiani, D., Trampedach, R., Goupil, M. J., Stein, R. F., & Nordlund, ˚A. 2007,
+A&A,463,297
+Sanders, W. L. 1972,A&A,19,155
+Stello,D., Chaplin,W. J.,Basu, S., Elsworth,Y., &Bedding , T.R. 2009, MNRAS, 400,80
+Stello,D., &Gilliland,R. L.2009,ApJ, 700,949
+Stello,D., et al. 2007,MNRAS, 377,584
+Street, R. A.,et al. 2005,MNRAS, 358,795
+Talamantes, A., & Sandquist, E. L. 2009, in Bulletin of the Am erican Astronomical Society,
+Vol.41,320
+VandenBerg, D. A., Bergbusch, P. A., &Dowler,P. D. 2006,ApJ S, 162,375
+ThispreprintwaspreparedwiththeAAS L ATEXmacrosv5.2.– 11 –
+Table1:Cross identificationsandmembership.
+ID ID WOCS ID ID Mem.ship Mem.ship Mem.ship
+Thiswork KICaHoleet al. Sanders Holeet al.bSanderscThiswork
+1 5024272 003003 SM95% no
+2 5024750 001004 141 SM93% 83% yes
+3 5023889 004014 42 SM95% 90% no
+4 5023732 005014 27 SM94% 90% yes
+5 5112950 003005 148 SM95% 92% yes
+6 5112387 003007 73 SM95% 88% yes
+7 5024512 003001 116 SM93% 90% yes
+8 4936335 007021 9 SM95% 68% no
+9 5024405 004001 100 SM93% 91% yes
+10 5112072 009010 39 SM95% 91% yes
+11 4937257 009015 144 SM88% 80% no
+aIDfromthe KeplerInputCatalogue (Lathamet al. 2005).
+bClassification (SM:singlemember)andmembershipprobabil ityfromradialvelocity(Holeetal. 2009).
+cMembershipprobabilityfrompropermotion(Sanders1972).– 12 –
+Fig. 1.— Color-magnitude diagram of NGC 6819. Plotted stars have membership probability
+PRV>80% as determined by Holeet al. (2009). Photometric indices ar e from the same source.
+Theisochroneis from Marigoet al. (2008)(Age=2.4 Gyr, Z=0. 019,modified for theadopted red-
+dening of 0.1mag). Color-coded stars have been analyzed, an d the annotated numbers refer to the
+legend in panels of Figure 2 and star numbers in Figure 3 (see a lso Table 1). Insets show light
+curves in parts per thousand of two red giants oscillating on different timescales. The variations
+ofthelightcurves inPanelA and Baredominatedby thestella roscillationswithperiodsofafew
+days andofaboutsix hours,respectively.– 13 –
+Fig. 2.— Fourierspectraofa representativeset ofred giant salongtheclustersequence sortedby
+apparent magnitude. Annotated numbers in each panel refer t o the star identification (see Fig. 1
+and Table 1). ‘AM’ indicates that the star is an asteroseismi c member. Red solid curves show the
+smoothed spectrum for stars with νmax<20µHz. To guide the eye, we have plotted dotted lines
+toindicatethemeasuredaveragelargefrequencyseparatio n. Thecentraldottedlineiscenteredon
+thehighestoscillationpeaksnear νmax. Notethatsince ∆νisgenerallyfrequencydependent,only
+thecentraldottedlineisexpectedtolineupwithapeakinth eoscillationspectrum. Theredarrows
+indicate the position of the expected νmax(see Eq. 1) for stars where the observed value does not
+agree withtheexpectationsforthiscluster(seeSection 4) .– 14 –
+Fig. 3.— Ratioofobservedandexpected νmax. 1-σerrorbarsindicatetheuncertaintyon νmax(obs).
+Stars clearly above or below the dotted line are either not cl uster members or members whose
+evolutionhavenot followedthestandardscenario.– 15 –
+Fig. 4.— Amplitude color diagram of red giant stars in NGC 681 9 with the Marigoet al. (2008)
+isochrone overlaid with three values of sin the amplitude scaling relation: (L/M)sTeff−2. The
+solarvalueusedin thisscalingis 4.7ppm(Kjeldsen &Bedding 1995).

1001.0027.txt

@@ -0,0 +1,389 @@
+arXiv:1001.0027v1  [astro-ph.GA]  30 Dec 2009New candidate Planetary Nebulae in the IPHAS survey: the cas e of
+PNe with ISM interaction.
+Laurence SabinA, Albert A. ZijlstraA, Christopher WareingB, Romano L.M.
+CorradiC, Antonio MampasoC, Kerttu ViironenC, Nicholas J. WrightDand
+Quentin A. ParkerE
+AJodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester,
+Manchester M13 9PL, UK
+BDepartment of Applied Mathematics, University of Leeds, Le eds, LS2 9JT, UK
+CInstituto de Astrofisica de Canarias, Tenerife, Spain
+DHarvard-Smithsonian Center for Astrophysics, 60 Garden St reet, Cambridge, MA, 02138, USA
+EMacquarie University/Anglo-Australian Observatory, Dep artment of Physics, North Ryde, Sydney
+NSW 2190, AUSTRALIA
+AEmail: [email protected]
+Abstract: We present the results of the search for candidate Planetary Nebulae interacting with
+the interstellar medium (PN-ISM) in the framework of the INT Photometric H αSurvey (IPHAS)
+and located in the right ascension range 18h-20h. The detect ion capability of this new Northern
+survey, in terms of depth and imaging resolution, has allowe d us to overcome the detection problem
+generally associated to the low surface brightness inheren t to PNe-ISM. We discuss the detection of
+21 IPHAS PN-ISM candidates. Thus, different stages of intera ction were observed, implying various
+morphologies i.e. from the unaffected to totally disrupted s hapes. The majority of the sources belong
+to the so-called WZO2 stage which main characteristic is a br ightening of the nebula’s shell in the
+direction of motion. The new findings are encouraging as they would be a first step into the reduction
+of the scarcity of observational data and they would provide new insights into the physical processes
+occurring in the rather evolved PNe.
+Keywords: Planetary nebulae, ISM interaction, survey.
+1 Introduction
+Large Hαsurveys have so far allowed the detection of
+∼3000 planetary nebulae (PNe) in the Galaxy. The
+data can be principally found in the Strasbourg-ESO
+Catalogue (Acker et al.1992)andtherecentMacquarie-
+AAO-StrasbourgH αPlanetaryNebulaCatalogues: MASH
+IandII(Parker et al.(2006)andMiszalski et al(2008)).
+Unfortunatelyalimitation inour understandingofthis
+short and rather complex phase of stellar evolution lies
+either in the deepness of the detections realised or the
+type of PNe investigated. Indeed, although enormous
+progress has been made over the years in terms of ob-
+servations, the well-studied PNe are generally bright
+and often young. This hampers the study of:
+•PNe hidden by the interstellar medium, partic-
+ularly those located at low galactic height.
+•PNe with (very)low surface brightness where we
+find the group of old PNe.
+•Very distant PNe which appear as unresolved
+and not recognisable as nebulae.•PNe located in crowded areas such as the galac-
+tic plane.
+Moreover, excluding these objects from global studies
+(morphology, abundances,luminosityfunction...etc)may
+bias our understanding of planetary nebulae. As an il-
+lustration, few PNe are described in the literature as
+“PNe with ISM interaction”, which is the step before
+the complete dilution of the nebulae in the interstel-
+lar medium (Borkowski et al. (1990), Ali et al. (2000),
+Xilouris et al. (1996) and Tweedy et al. (1996)). The
+study of the interaction process would give new in-
+sights intoseveral aspects of the PNevolution. Indeed,
+the density difference between ISM and PNe will affect
+their shape. This is expected to be observable in old
+objects where the nebular density declines sufficiently
+to be overcome by the ISM density. Other phenom-
+ena like the flux and brightness enhancement following
+the compression of the external shell, the increase of
+the recombination rate in the PN Rauch et al. (2000),
+the occurrence of turbulent Rayleigh-Taylor instabili-
+ties and the implication of magnetic fields Dgani et al.
+(1998) are among the physical processes which need
+to be addressed not only from a theoretical but also
+observational point of view.
+12 Publications of the Astronomical Society of Australia
+The low surface brightness generally associated to
+PNe-ISM has for a long time prevented any deeper ob-
+servation and good statistical study of these interac-
+tions, where only the interacting rim is well seen. New
+generations of H αsurveys have overcome this prob-
+lem. A perfect example is the discovery of PFP 1 by
+Pierce et al. (2004)intheframeworkoftheAAO/UKST
+SuperCOSMOS H αsurvey (SHS) (Parker et al. 2005).
+This PN, starting to interact with the ISM at the
+rim, is very large (radius = 1.5 ±0.6 pc) and very
+faint (logarithm of the H αsurface brightness equal
+to -6.05 ergcm−2.s−1.sr−1). In order to unveil and
+study this “missing PN population” in the Northern
+hemisphere we need surveys providing the necessary
+observing depth: the Isaac Newton Telescope (INT)
+Photometric H αSurvey (IPHAS) is one of them and
+will complete the work done in the South by the SHS.
+2 IPHAS contribution
+IPHAS is a new fully photometric CCD survey of the
+Northern Galactic Plane, started in 2003 (Drew et al.
+(2005), Gonzalez-Solares et al (2008)) and which has
+now been completed1. Using the 2.5m Isaac Newton
+Telescope (INT)in LaPalma (Canary Islands, SPAIN)
+and the Wide Field Camera (WFC) offering a field of
+view of 34.2 ×34.2 arcmin2, IPHAS targets the Galac-
+tic plane in the Northern hemisphere, at a latitude
+range of -5◦<b<5◦and covers 1800 deg2. This
+international survey is conducted not only in H αbut
+also makes use of two continuum filters, respectively
+the Sloan r’ and i’. IPHAS is viewed as an enhance-
+ment to former narrow-band surveys, first due to the
+use of CCD and the particularly small pixel scale al-
+lowed bytheWFCwith0.33 arcsec pix−1butalso (and
+mainly) due to the depth reached for point sources de-
+tection. Thus sources with a r’ magnitude between 13
+and 19.5-20 could be detected with a very good pho-
+tometric accuracy. The most interesting characteristic
+for our purpose is the ability to detect resolved ex-
+tended emissions with an H αsurface brightness down
+to 2×10−17erg cm−2s−1arcsec−2.
+In this paper we will focus on extended (candi-
+date) PNe (i.e. objects with a size greater than 5 arc-
+sec). They were searched for via a visual inspection of
+2 deg2Hα-r(continuum removal) mosaics made from
+the different IPHAS observations. And in order to al-
+low the detection of objects of multiple size and bright-
+ness level, the mosaics were binned at respectively 15
+pixels×15 pixels (5 arcsec) and 5 pixels ×5 pixels (1.7
+arcsec). The first binning level, which is of particu-
+lar interest to us, helps to detect resolved, low surface
+brightness objects (down to the IPHAS limit) and to
+accentuate the contours/shape of the nebulae (this is
+particularly useful to see, for example, the full extent
+of an outflow or a tail). The second set, is used to de-
+tect intermediate size nebulae i.e. smaller than ∼15-20
+arcsec in diameter.
+1http://www.iphas.orgThe first area that has been fully investigated is the re-
+gion between RA=18h and RA=20h. We detected 233
+candidate PNe among which other nebulosities may be
+found e.g. small HII regions (Sabin, PhD thesis, to be
+published). Around 20% of this sample have been so
+far spectroscopically confirmed as PNe (Sabin et al.,
+in preparation). If we look at the particularities of the
+PNe and candidate PNe uncovered, we observe that
+from thepointofviewofthesize, large objects (greater
+than 20 arcsec) constitute the main new group (Fig.
+1). As large objects are generally considered as more
+evolved, we are confident in finding in this group new
+old PNe and byextension new cases of PNe interacting
+with the surrounding ISM (PNe/ISM).
+Figure 1: Galactic distribution of the IPHAS neb-
+ulae according to their size.
+3 Candidate PNewith ISMin-
+teraction
+Fundamental in PN development, the interaction with
+the ISM does not only concern old PNe, as may be
+commonly thought. Indeed, the PN-ISM interaction
+has mainly been detected in a rather small number of
+nebulae, which are generally bright objects (“young”
+and“mid-age”PNe). Rauch et al.(2000)andWareing et al.
+(2007) showed that different stages of interaction are
+exhibited during the PNe life. The low surface bright-
+ness, generally associated with nebulae mixing with
+the ISM and “old” PNe, has for a long time prevented
+any deeper observation and good statistical study of
+these interactions. Although faint objects will still re-
+main difficult to detect, the IPHAS survey provides
+a noticeable improvement. Nevertheless, a caveat is
+the difficulty to visually separate PNe-ISM from other
+faint and extended structures like old HII regions, Su-
+pernovae (SNRs) or diffuse H αstructures. As an ex-
+ample, faint bow shocks generally characteristics ofwww.publish.csiro.au/journals/pasa 3
+PNemixingwiththeISMcanalsobefilamentarystruc-
+tures from old SNRs. A spectroscopic analysis is the
+only way to have a clear identification.
+The work presented here is based on the classification
+from Wareing et al. (2007) (WZO 1-4 called after the
+authors’ names) and will allow us to establish the de-
+gree of interaction for each nebula. Their classifica-
+tion is the result of the first extensive investigation
+of the applicable parameter space, varying stellar pa-
+rameters, relative velocities through the ISM and ISM
+densities.
+The depth reached by the IPHAS survey combined
+with the binning detection method allowed us to iden-
+tify 21 cases of interacting candidate PNe.
+3.1 WZO1 type
+The first group of PNe/ISM concerns those where the
+main PN is still unaffected and which may display a
+distant bow shock. In our area of study (18h-20h), the
+majority of candidates answer the first condition, but
+none show the outer bow shock. Outside this area, the
+nicknamed“EarNebula”orIPHASXJ205013.7+465518
+with a 6 arcmin size may be coincident with a WZO1
+description as this object is a confirmed bipolar PN
+(Fig. 3) surrounded by a shell which may be an AGB
+remnant shell or would indicate a multiple shell nebula
+(Fig. 2).
+3.2 WZO2 type
+This category concerns PNe showing a bright rim in
+the direction of motion. This is the most common fea-
+ture found in our sample and 17 objects out of 21 fall
+under this classification. Fig. 4 presents 3 examples
+with different angular sizes, although they all display
+a diameter on the order of a few arcmin (we consid-
+ered the assumed full extent of the round nebulae).
+We point out in Fig. 4-Top the difficulty to determine
+the true direction of motion regarding the CS position
+and off-axis bow shock. Such a geometry could be ex-
+plained by an ISM gradient from high on the left to
+low on the right. We also notice a particularly low
+observed surface brightness (SB) which may explain
+previous non detections.
+3.3 WZO3 type
+This type is exemplified by PNe whose geometric cen-
+tres are shifted away from the central star (CS): both
+are no longer coincident. An example, is the ancient
+PN Sh 2-188 around which IPHAS has uncovered an
+extended structure (Wareing et al. 2006). We identi-
+fied 3 candidate PNe coincident with this description.
+The most probing WZO3 type in our sample is pre-
+sented in figure 5 and corresponds, according to hy-
+drodynamical models, to a PN with a CS velocity of
+about 100 km/s.3.4 WZO4 type
+The WZO4 corresponds to the most difficult types of
+PN to be detected: the CS has left the vicinity of
+the now totally disrupted PN, leaving an amorphous
+structure. The challenge does not lie in the detection
+ability (it enters in the IPHAS range of detection) but
+more intheselection oftheobjects as possible PNedue
+to the total lack of symmetry or axi-symmetry. This
+type of interaction is also discussed in more detail by
+Wareing et al. in these proceedings.
+We identified 1 candidate PN which could fit the
+given description. Fig. 6 presents the selected can-
+didate in the top panel. We suggest the the nebular
+material has been moved from the front to the rear
+leaving a remnant “wall of material”. We also notice
+that some features may be linked to turbulence effects.
+The comparison with the hydrodynamical model (bot-
+tom panel) seems to support this hypothesis. Nev-
+ertheless a spectroscopic confirmation of the nebula’s
+nature will be needed. The model implies a velocity
+relative to the ISM of 100 Km/s and an evolution in
+the post-AGB phase of 10 000 years.
+3.5 Distribution of the candidates
+Fig. 7-top shows that the majority of the WZO2 nebu-
+lae typesare locatedinzones ofrelatively lowISMden-
+sity (compared to the Galactic Centre). The low stress
+exerted on the nebulae may explain why they still keep
+their quasi circular shape. The ISM is more dense in
+the Galactic Plane than in the zone towards the anti-
+centreor thezoneaboveaheightof100pc(from obser-
+vation of neutral hydrogen gas, Dickey et al. (1990)).
+We therefore expected a greater influence of the in-
+teraction process in this area. Indeed, we observed
+that the most advanced stages of interaction, namely
+WZO3 and WZO4, are detected in areas of high ISM
+density, where PN are more likely to be affected by
+such densities.
+Thesizedistribution, Fig. 7-bottom, indicatesthat
+althoughmostofthedetectedcandidatePNearelarge2,
+i.e with a size greater than 100 arcsec, or of medium
+size i.e. between 20 and 100 arcsec, small nebulae
+also show signs of interaction. This confirms that the
+ISM interaction process does not “a priori” only im-
+ply “old” nebulae. We also observe that large objects
+mainly lie at higher latitudes than smaller nebulae but
+it is also interesting to notice that we detect large ob-
+jects inzones ofhighextinction; large PNeseemtosur-
+vive at relatively low latitudes. They would undergo
+strong alteration by the ISM and would display more
+advanced stages of interaction. Those disruptions tend
+to affect them more than smaller size nebulae at the
+same latitude range.
+4 Conclusion and Perspectives
+In the first fully analysed area of the Galactic plane,
+RA=18h to RA=20h, the new H αphotometric survey
+2The sizes here are defined in terms of angular sizes, so
+the physical correspondence will depend on the distance.4 Publications of the Astronomical Society of Australia
+IPHASappearstobeanexcellenttooltostudyPNein-
+teracting with the ISM. Indeed the survey contributes
+to the detection of nebulae so far hidden mainly due to
+their faintness. Thus, 21 objects have been identified
+aspossible planetarynebulaeinteractingwiththeISM.
+They show diverse sizes (although the majority display
+a diameter greater than 100 arcsec) and morphologies
+corresponding to the four different cases of interaction
+commonly defined going from the unaffected to the to-
+tally disrupted nebula. The most common stage is the
+WZO2correspondingtonebulaeshowingabrightening
+of their rim in the direction of motion. This is coinci-
+dent with the observations made by Wareing and al (in
+these proceedings) crossing different H αsurveys. We
+were also able to reach those targets at low latitudes
+and found that some could survive in those environ-
+ments although they would be strongly affected by the
+ISM. The total lack of PNe/ISM at the highest point
+of ISM density (b= ±0.5 deg and 30 deg <l<50 deg)
+can either be due to the limitation of IPHAS or be-
+cause they have been totally destroyed by the effects
+of ISM interaction.
+The next logical step is the spectroscopic identification
+of these sources, their central star study and physical
+size determination. The low surface brightness implies
+the use of particular means such as integral field spec-
+troscopy to be able to retrieve the maximum infor-
+mation. Therefore a new programme of IPHAS PN
+candidate follow-up spectroscopy led by Q. Parker, A.
+Zijlstra and R. Corradi is now underway.
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+Direction of motionThick outer
+Filamentsshell: rim
++ Bipolar outflowBright edges of the bipolar
+Sharp structures
+Faint opposite edge
+Figure 2: An example of WZO1 type: The “Ear Nebula” IPHAS PN. Nort h on the top and East on the
+left.
+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1erg/cm2/s/A erg/cm2/s/A
+Figure 3: WHT spectra of the “EarNebula” using the R300Band R158 R gratings. This nebula, for which
+we show some of the “strongest” emission lines useful for an identifi cation, presents a clear [NII] over-
+intensity and it has been confirmed as true PN using the revised diagn ostic diagram from Riesgo et al.
+(2006) (particularly the log [H α/[SII]]vslog [Hα/[NII]] diagram).6 Publications of the Astronomical Society of Australia
+Interacting rimDirection of motion
+CS candidate
+Direction of motion
+Geometric centerBow shock CS candidates
+Direction of motionCS candidateBright rim
+Figure 4: Examples of WZO2 types. Top: Size=1.2 arcmin and SB=3.4e−17erg cm−2s−1arcsec−2.
+Middle: Size= 8.5 arcmin and SB=1.1e−16erg cm−2s−1arcsec−2, Bottom: 4.3 arcmin and SB=2.7e−16
+erg cm−2s−1arcsec−2. North on the top and East on the left.
+ Direction of motion
+Most probable CS
+Bright rim
+Figure 5: WZO3 type of ISM interaction in a IPHAS candidate PNe. Nor th on the top and East on the
+left.www.publish.csiro.au/journals/pasa 7
+ 
+Direction of motion
+Dense "wall" of nebular material gas and dust
+or
+turbulences Traces ofFaint frontal bow shock
+Figure 6: A possible example of WZO4 ISM interaction in one IPHAS PN ca ndidate (top: North on the
+top and East on the left) with the corresponding hydrodynamical m odel (bottom) [reproduction of figure
+5(d) from Wareing et al. (2007)].8 Publications of the Astronomical Society of Australia
+Figure 7: Galactic distribution of the candidate PNe/ISM according t o their stage of interaction and
+their size.
+Figure 8: Example of candidates with sizes greater than 100 arcsec (respectively 7.7 and 2.9 arcmin) and
+located at b= ±1 deg. These objects present a WZO2 stage of interaction and only their (very) faint
+interacting rim are seen.

1001.0028.txt

@@ -0,0 +1,1777 @@
+arXiv:1001.0028v2  [math.CO]  28 Feb 2012CYCLIC SIEVING FOR GENERALISED NON-CROSSING
+PARTITIONS ASSOCIATED WITH COMPLEX REFLECTION
+GROUPS OF EXCEPTIONAL TYPE
+Christian Krattenthaler†andThomas W. M ¨uller‡
+†Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien,
+Nordbergstraße 15, A-1090 Vienna, Austria.
+WWW:http://www.mat.univie.ac.at/ ~kratt
+‡School of Mathematical Sciences,
+Queen Mary & Westfield College, University of London,
+Mile End Road, London E1 4NS, United Kingdom.
+WWW:http://www.maths.qmw.ac.uk/ ~twm/
+Dedicated to the memory of Herb Wilf
+Abstract. We prove that the generalised non-crossing partitions associated with
+well-generated complex reflection groups of exceptional type obe y two different cyclic
+sieving phenomena, as conjectured by Armstrong, and by Bessis a nd Reiner. The
+computational details are provided in the manuscript “Cyclic sieving for generalised
+non-crossing partitions associated with complex reflectio n groups of exceptional type
+— the details” [arχiv:1001.0030 ].
+1.Introduction
+In his memoir [2], Armstrong introduced generalised non-crossing partitions asso-
+ciated with finite (real) reflection groups, thereby embedding Krew eras’ non-crossing
+partitions [22], Edelman’s m-divisible non-crossing partitions [12], thenon-crossing par-
+titions associated with reflection groups due to Bessis [6] and Brady and Watt [10] into
+one uniform framework. Bessis and Reiner [9] observed that Arms trong’s definition can
+be straightforwardly extended to well-generated complex reflection groups (see Section 2
+for the precise definition). These generalised non-crossing partit ions possess a wealth
+of beautiful properties, and they display deep and surprising relat ions to other combi-
+natorial objects defined for reflection groups (such as the gene ralised cluster complex
+2000Mathematics Subject Classification. Primary 05E15; Secondary 05A10 05A15 05A18 06A07
+20F55.
+Key words and phrases. complex reflection groups, unitary reflection groups, m-divisible non-
+crossing partitions, generalised non-crossing partitions, Fuß–Ca talan numbers, cyclic sieving.
+†Research partially supported by the Austrian Science Foundation F WF, grants Z130-N13 and
+S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics
+and Probabilistic Number Theory.”
+‡Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13.
+12 C. KRATTENTHALER AND T. W. M ¨ULLER
+of Fomin and Reading [13], or the extended Shi arrangement and the geometric multi-
+chains of filters of Athanasiadis [4, 5]); see Armstrong’s memoir [2] and the references
+given therein.
+Ontheotherhand, cyclic sieving isaphenomenonbroughttolightbyReiner, Stanton
+and White [30]. It extends the so-called “( −1)-phenomenon” of Stembridge [34, 35].
+Cyclic sieving can be defined in three equivalent ways (cf. [30, Prop. 2.1]). The one
+which gives the name can be described as follows: given a set Sof combinatorial
+objects, an action on Sof a cyclic group G=/an}bracketle{tg/an}bracketri}htwith generator gof ordern, and
+a polynomial P(q) inqwith non-negative integer coefficients, we say that the triple
+(S,P,G)exhibits the cyclic sieving phenomenon , if the number of elements of Sfixed
+bygkequalsP(e2πik/n). In [30] it is shown that this phenomenon occurs in surprisingly
+many contexts, and several further instances have been discov ered since then.
+In [2, Conj. 5.4.7] (also appearing in [9, Conj. 6.4]) and [9, Conj. 6.5], Ar mstrong,
+respectively Bessis and Reiner, conjecture that generalised non- crossing partitions for
+irreducible well-generated complex reflection groups exhibit two diffe rent cyclic sieving
+phenomena (see Sections 3 and 7 for the precise statements).
+According to the classification of these groups due to Shephard an d Todd [32], there
+are two infinite families of irreducible well-generated complex reflectio n groups, namely
+the groups G(d,1,n) andG(e,e,n), wheren,d,eare positive integers, and there are 26
+exceptional groups. For the infinite families of types G(d,1,n) andG(e,e,n), the two
+cyclic sieving conjectures follow from the results in [19].
+Thepurposeofthepresent articleistopresent aproofofthecyc licsieving conjectures
+of Armstrong, and of Bessis and Reiner, for the 26 exceptional ty pes, thus completing
+the proof of these conjectures. Since the generalised non-cros sing partitions feature a
+parameterm, from the outset this is nota finite problem. Consequently, we first need
+several auxiliary results to reduce the conjectures for each of t he 26 exceptional types
+to afiniteproblem. Subsequently, we use Stembridge’s Maplepackagecoxeter [36]
+and theGAPpackageCHEVIE[14, 28] to carry out the remaining finitecomputations.
+The details of these computations are provided in [21]. In the presen t paper, we con-
+tent ourselves with exemplifying the necessary computations by go ing through some
+representative cases. It is interesting to observe that, for the verification of the type
+E8case, it is essential to use the decomposition numbers in the sense o f [17, 18, 20] be-
+cause, otherwise, the necessary computations would not be feas ible in reasonable time
+with the currently available computer facilities. We point out that, fo r the special case
+where the aforementioned parameter mis equal to 1, the first cyclic sieving conjecture
+has been proved in a uniform fashion by Bessis and Reiner in [9]. (See [3 ] for a —
+non-uniform — proof of cyclic sieving for non-crossing partitions as sociated with real
+reflection groups under the action of the so-called Kreweras map, a special case of the
+second cyclic sieving phenomenon discussed in the present paper.) T he crucial result on
+which the proof of Bessis and Reiner is based is (5.5) below, and it plays an important
+rolein our reduction of the conjectures forthe 26 exceptional gr oupsto a finite problem.
+Our paper is organised as follows. In the next section, we recall the definition of
+generalised non-crossing partitions for well-generated complex re flection groups and of
+decomposition numbers in the sense of [17, 18, 20], and we review so me basic facts.
+The first cyclic sieving conjecture is subsequently stated in Section 3. In Section 4, we
+outline an elementary proof that the q-Fuß–Catalan number, which is the polynomial
+Pin the cyclic sieving phenomena concerning the generalised non-cros sing partitionsCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 3
+for well-generated complex reflection groups, is always a polynomial with non-negative
+integer coefficients, as required by the definition of cyclic sieving. (F ull details can be
+found in [21, Sec. 4]. The reader is referred to the first paragraph of Section 4 for
+comments on other approaches for establishing polynomiality with no n-negative coeffi-
+cients.) Section 5 contains the announced auxiliary results which, fo r the 26 exceptional
+types, allow a reduction of the conjecture to a finite problem. In Se ction 6, we discuss
+a few cases which, in a representative manner, demonstrate how t o perform the re-
+maining case-by-case verification of the conjecture. For full det ails, we refer the reader
+to [21, Sec. 6]. The second cyclic sieving conjecture is stated in Sect ion 7. Section 8
+contains the auxiliary results which, for the 26 exceptional types, allow a reduction of
+the conjecture to a finite problem, while in Section 9 we discuss some r epresentative
+cases of the remaining case-by-case verification of the conjectu re. Again, for full details
+we refer the reader to [21, Sec. 9].
+2.Preliminaries
+Acomplex reflection group isa groupgeneratedby(complex) reflections in Cn. (Here,
+a reflection is a non-trivial element of GLn(C) which fixes a hyperplane pointwise and
+which hasfiniteorder.) Wereferto[24]foranin-depthexpositionof thetheorycomplex
+reflection groups.
+Shephard and Todd provided a complete classification of all finitecomplex reflection
+groups in [32] (see also [24, Ch. 8]). According to this classification, a n arbitrary
+complex reflection group Wdecomposes into a direct product of irreducible complex
+reflection groups, acting on mutually orthogonal subspaces of th e complex vector space
+onwhichWisacting. Moreover, thelistofirreduciblecomplexreflectiongroups consists
+of the infinite family of groups G(m,p,n), wherem,p,nare positive integers, and 34
+exceptional groups, denoted G4,G5,...,G 37by Shephard and Todd.
+In this paper, we are only interested in finite complex reflection grou ps which are
+well-generated . A complex reflection group of rank nis called well-generated if it is
+generated by nreflections.1Well-generation can be equivalently characterised by a
+duality property due to Orlik and Solomon [29]. Namely, a complex reflec tion group of
+ranknhastwo sets ofdistinguished integers d1≤d2≤ ··· ≤dnandd∗
+1≥d∗
+2≥ ··· ≥d∗
+n,
+called its degreesandcodegrees , respectively (see [24, p. 51 and Def. 10.27]). Orlik and
+Solomon observed, using case-by-case checking, that an irreduc ible complex reflection
+groupWof ranknis well-generated if and only if its degrees and codegrees satisfy
+di+d∗
+i=dn
+for alli= 1,2,...,n. The reader is referred to [24, App. D.2] for a table of the degree s
+and codegrees of all irreducible complex reflection groups. Togeth er with the classi-
+fication of Shephard and Todd [32], this constitutes a classification o f well-generated
+complex reflection groups: the irreducible well-generated complex r eflection groups are
+— the two infinite families G(d,1,n) andG(e,e,n), whered,e,nare positive inte-
+gers,
+— the exceptional groups G4,G5,G6,G8,G9,G10,G14,G16,G17,G18,G20,G21of
+rank 2,
+1We refer to [24, Def. 1.29] for the precise definition of “rank.” Roug hly speaking, the rank of a
+complex reflection group Wis the minimal nsuch that Wcan be realized as reflection group on Cn.4 C. KRATTENTHALER AND T. W. M ¨ULLER
+— the exceptional groups G23=H3,G24,G25,G26,G27of rank 3,
+— the exceptional groups G28=F4,G29,G30=H4,G32of rank 4,
+— the exceptional group G33of rank 5,
+— the exceptional groups G34,G35=E6of rank 6,
+— the exceptional group G36=E7of rank 7,
+— and the exceptional group G37=E8of rank 8.
+In this list, we have made visible the groups H3,F4,H4,E6,E7,E8which appear as
+exceptional groups in the classification of all irreducible realreflection groups (cf. [16]).
+LetWbe a well-generated complex reflection group of rank n, and letT⊆Wdenote
+theset of all(complex) reflections inthegroup. Let ℓT:W→Zdenotethewordlength
+in terms of the generators T. This word length is called absolute length orreflection
+length. Furthermore, we define a partial order ≤TonWby
+u≤Twif and only if ℓT(w) =ℓT(u)+ℓT(u−1w). (2.1)
+This partial order is called absolute order orreflection order . As is well-known and
+easy to see, the equation in (2.1) is equivalent to the statement tha t every shortest
+representation of uby reflections occurs as an initial segment in some shortest produc t
+representation of wby reflections.
+Now fix a (generalised) Coxeter element2c∈Wand a positive integer m. The
+m-divisible non-crossing partitions NCm(W) are defined as the set
+NCm(W) =/braceleftbig
+(w0;w1,...,w m) :w0w1···wm=cand
+ℓT(w0)+ℓT(w1)+···+ℓT(wm) =ℓT(c)/bracerightbig
+.
+A partial order is defined on this set by
+(w0;w1,...,w m)≤(u0;u1,...,u m) if and only if ui≤Twifor 1≤i≤m.
+We have suppressed the dependence on c, since we understand this definition up to
+isomorphism of posets. To be more precise, it can be shown that any two Coxeter
+elements are related to each other by conjugation and (possibly) a n automorphism on
+the field of complex numbers (see [33, Theorem 4.2] or [24, Cor. 11.2 5]), and hence the
+resulting posets NCm(W) are isomorphic to each other. If m= 1, thenNC1(W) can
+be identified with the set NC(W) of non-crossing partitions for the (complex) reflection
+groupWasdefined byBessis andCorran(cf.[8]and[7, Sec.13]; theirdefinit ionextends
+the earlier definition by Bessis [6] and Brady and Watt [10] for real r eflection groups).
+The following result has been proved by a collaborative effort of seve ral authors (see
+[7, Prop. 13.1]).
+2An element of an irreducible well-generated complex reflection group Wof ranknis called a
+Coxeter element if it isregularin the sense of Springer [33] (see also [24, Def. 11.21]) and of order dn.
+An element of Wis called regular if it has an eigenvector which lies in no reflecting hyperp lane of a
+reflection of W. It follows from an observation of Lehrer and Springer, proved un iformly by Lehrer
+and Michel [23] (see [24, Theorem 11.28]), that there is always a regu lar element of order dnin an
+irreducible well-generated complex reflection group Wof rankn. More generally, if a well-generated
+complex reflection group Wdecomposes as W∼=W1×W2×···×Wk, where the Wi’s are irreducible,
+then a Coxeter element of Wis an element of the form c=c1c2···ck, whereciis a Coxeter element of
+Wi,i= 1,2,...,k. IfWis arealreflection group, that is, if all generators in Thave order 2, then the
+notion of generalised Coxeter element given above reduces to that of a Coxeter element in the classical
+sense (cf. [16, Sec. 3.16]).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 5
+Theorem 1. LetWbe an irreducible well-generated complex reflection group, and let
+d1≤d2≤ ··· ≤dnbe its degrees and h:=dnits Coxeter number. Then
+|NCm(W)|=n/productdisplay
+i=1mh+di
+di. (2.2)
+Remark1.(1) The number in (2.2) is called the Fuß–Catalan number for the reflection
+groupW.
+(2) Ifcis a Coxeter element of a well-generated complex reflection group Wof rank
+n, thenℓT(c) =n. (This follows from [7, Sec. 7].)
+We conclude this section by recalling the definition of decomposition nu mbers from
+[17, 18, 20]. Although we need them here only for (very small) real re flection groups,
+and although, strictly speaking, they have been only defined for re al reflection groups in
+[17, 18, 20], this definition can be extended to well-generated comple x reflection groups
+without any extra effort, which we do now.
+Given a well-generated complex reflection group Wof rankn, typesT1,T2,...,T d(in
+the sense of the classification of well-generated complex reflection groups) such that the
+sumoftheranksofthe Ti’sequalsn, andaCoxeter element c, thedecompositionnumber
+NW(T1,T2,...,T d) is defined as the number of “minimal” factorisations c=c1c2···cd,
+“minimal” meaning that ℓT(c1) +ℓT(c2) +···+ℓT(cd) =ℓT(c) =n, such that, for
+i= 1,2,...,d, the type of cias a parabolic Coxeter element is Ti. (Here, the term
+“parabolic Coxeter element” means a Coxeter element in some parab olic subgroup. It
+follows from [31, Prop.6.3] that any element ciis indeed a Coxeter element in a unique
+parabolic subgroup of W.3By definition, the type of ciis the type of this parabolic
+subgroup.) Since any two Coxeter elements are related to each oth er by conjugation
+plus field automorphism, the decomposition numbers are independen t of the choice of
+the Coxeter element c.
+The decomposition numbers for real reflection groups have been c omputed in [17,
+18, 20]. To compute the decomposition numbers for well-generated complex reflection
+groups is a task that remains to be done.
+3.Cyclic sieving I
+In this section we present the first cyclic sieving conjecture due to Armstrong [2,
+Conj. 5.4.7], and to Bessis and Reiner [9, Conj. 6.4].
+Letφ:NCm(W)→NCm(W) be the map defined by
+(w0;w1,...,w m)/mapsto→/parenleftbig
+(cwmc−1)w0(cwmc−1)−1;cwmc−1,w1,w2,...,w m−1/parenrightbig
+.(3.1)
+It is indeed not difficult to see that, if the ( m+ 1)-tuple on the left-hand side is an
+element ofNCm(W), then so is the ( m+1)-tuple on the right-hand side. For m= 1,
+this action reduces to conjugation by the Coxeter element c(applied to w1). Cyclic
+sieving arising from conjugation by chas been the subject of [9].
+3The uniqueness can be argued as follows: suppose that ciwere a Coxeter element in two parabolic
+subgroups of W, sayU1andU2. Then it must also be a Coxeter element in the intersection U1∩U2.
+On the other hand, the absolute length of a Coxeter element of a co mplex reflection group Uis always
+equal to rk( U), the rank of U. (This follows from the fact that, for each element uofU, we have
+ℓT(u) = codim/parenleftbig
+ker(u−id)/parenrightbig
+, with id denoting the identity element in U; see e.g. [31, Prop. 1.3]). We
+conclude that ℓT(ci) = rk(U1) = rk(U2) = rk(U1∩U2), This implies that U1=U2.6 C. KRATTENTHALER AND T. W. M ¨ULLER
+It is easy to see that φmhacts as the identity, where his the Coxeter number of W
+(see (5.1) and Lemma 29 below). By slight abuse of notation, let C1be the cyclic group
+of ordermhgenerated by φ. (The slight abuse consists in the fact that we insist on C1
+to be a cyclic group of order mh, while it may happen that the order of the action of
+φgiven in (3.1) is actually a proper divisor of mh.)
+Given these definitions, we are now in the position to state the first c yclic sieving
+conjecture of Armstrong, respectively of Bessis and Reiner. By t he results of [19] and
+of this paper, it becomes the following theorem.
+Theorem 2. For an irreducible well-generated complex reflection group Wand any
+m≥1, the triple (NCm(W),Catm(W;q),C1), whereCatm(W;q)is theq-analogue of
+the Fuß–Catalan number defined by
+Catm(W;q) :=n/productdisplay
+i=1[mh+di]q
+[di]q, (3.2)
+exhibits the cyclic sieving phenomenon in the sense of Reine r, Stanton and White [30].
+Here,nis the rank of W,d1,d2,...,d nare the degrees of W,his the Coxeter number
+ofW, and[α]q:= (1−qα)/(1−q).
+Remark2.We write Catm(W) for Catm(W;1).
+By definition of the cyclic sieving phenomenon, we have to prove that Catm(W;q) is
+a polynomial in qwith non-negative integer coefficients, and that
+|FixNCm(W)(φp)|= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh, (3.3)
+for allpin the range 0 ≤p<mh. The first fact is established in the next section, while
+the proof of the second is achieved by making use of several auxiliar y results, given
+in Section 5, to reduce the proof to a finite problem, and a subseque nt case-by-case
+analysis. Alldetails ofthisanalysiscanbefoundin[21, Sec. 6]. Inthe present paper, we
+content ourselves with discussing the cases where W=G24and whereW=G37=E8,
+since these suffice to convey the flavour of the necessary comput ations.
+4.Theq-Fusz–Catalan numbers Catm(W;q)
+The purpose of this section is to provide an elementary, self-conta ined proof of the
+fact that, for all irreducible complex reflection groups W, theq-Fuß–Catalan number
+Catm(W;q) is a polynomial in qwith non-negative integer coefficients. For most of
+the groups, this is a known property. However, aside from the fac t that, for many of
+the known cases, the proof is very indirect and uses deep algebraic results on rational
+Cherednik algebras, there still remained some cases where this pro perty had not been
+formally established. The reader is referred to the “Theorem” in Se ction 1.6 of [15],
+whichsaysthat, undertheassumptionofacertainrankcondition( [15, Hypothesis2.4]),
+theq-Fuß–Catalan number Catm(W;q) is a Hilbert series of a finite-dimensional quo-
+tient of the ring of invariants of Wand also the graded character of a finite-dimensional
+irreducible representation of a spherical rational Cherednik algeb ra associated with
+W. At present, this rank condition has been proven for all irreducible well-generated
+complex reflection groups apart from G17,G18,G29,G33,G34; see [26, Tables 8 and 9,
+column “rank”], and the recent paper [27], which establishes the res ult in the case of
+G32.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 7
+In the sequel, aside from the standard notation [ α]q= (1−qα)/(1−q) forq-integers,
+we shall also use the q-binomial coefficient, which is defined by
+/bracketleftbigg
+n
+k/bracketrightbigg
+q:=/braceleftBigg
+1, ifk= 0,
+[n]q[n−1]q···[n−k+1]q
+[k]q[k−1]q···[1]q,ifk>0.
+We begin with several auxiliary results.
+Proposition 3. For all non-negative integers nandk, theq-binomial coefficient [n
+k]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.This is a well-known fact, which can be derived either from the recurr ence rela-
+tion(s) satisfied by the q-binomial coefficients (generalising Pascal’s recurrence relation
+for binomial coefficients; cf. [1, eqs. (3.3.3) and (3.3.4)]), or from th e fact that the q-
+binomial coefficient [n
+k]qis the generating function for (integer) partitions with at most
+kparts all of which are at most n−k(cf. [1, Theorem 3.1]). /square
+Proposition 4. For all non-negative integers mandn, theq-Fuß–Catalan number of
+typeAn,
+1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q,
+is a polynomial in qwith non-negative integer coefficients.
+Proof.In [25, Sec. 3.3], Loehr proves that
+1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q
+=/summationdisplay
+v∈V(m)
+nqm(n
+2)+/summationtext
+i≥0(m(vi
+2)−ivi)/productdisplay
+i≥1qvi/summationtextm
+j=1(m−j)vi−j/bracketleftbigg
+vi+vi−1+···+vi−m−1
+vi/bracketrightbigg
+q,(4.1)
+whereV(m)
+ndenotes the set of all sequences v= (v0,v1,...,v s) (for some s) of non-
+negative integers with v0>0,vs>0, andv0+v1+···+vs=n, and such that there
+is never a string of mor more consecutive zeroes in v. By convention, vi= 0 for all
+negativei. His proof works by showing that the expressions on both sides of ( 4.1)
+satisfy the same recurrence relation and initial conditions, using cla ssicalq-binomial
+identities. We refer the reader to [25] for details. By Proposition 3, the expression on
+the right-hand side of (4.1) is manifestly a polynomial in qwith non-negative integer
+coefficients. /square
+Lemma 5. Ifaandbare coprime positive integers, then
+[ab]q
+[a]q[b]q(4.2)
+is a polynomial in qof degree (a−1)(b−1), all of whose coefficients are in {0,1,−1}.
+Moreover, if one disregards the coefficients which are 0, then+1’s and(−1)’s alternate,
+and the constant coefficient as well as the leading coefficient o f the polynomial equal +1.
+Proof.LetΦn(q)denotethe n-thcyclotomicpolynomialin q. Usingtheclassicalformula
+1−qn=/productdisplay
+d|nΦd(q),8 C. KRATTENTHALER AND T. W. M ¨ULLER
+we see that
+(1−q)(1−qab)
+(1−qa)(1−qb)=/productdisplay
+d1|a,d1/ne}ationslash=1
+d2|a,d2/ne}ationslash=1Φd1d2(q),
+so that, manifestly, the expression in (4.2) is a polynomial in q. The claim concerning
+the degree of this polynomial is obvious.
+In order to establish the claim on the coefficients, we start with a sub -expression of
+(4.2),
+(1−qab)
+(1−qa)(1−qb)=/parenleftbiggb−1/summationdisplay
+i=0qia/parenrightbigg/parenleftbigg∞/summationdisplay
+j=0qjb/parenrightbigg
+=∞/summationdisplay
+k=0Ckqk, (4.3)
+say. The assumption that aandbare coprime implies that 0 ≤Ck≤1 fork≤
+(a−1)(b−1). Multiplying both sides of (4.3) by 1 −q, we obtain the equation
+[ab]q
+[a]q[b]q= (1−q)(a−1)(b−1)/summationdisplay
+k=0Ckqk+(1−q)∞/summationdisplay
+k=(a−1)(b−1)+1Ckqk. (4.4)
+By our previous observation on the coefficients Ckwithk≤(a−1)(b−1), it is obvious
+that the coefficients of the first expression on the right-hand side of (4.4) are alternately
++1 and−1, when 0’s are disregarded. Since we already know that the left-ha nd side is
+a polynomial in qof degree (a−1)(b−1), we may ignore the second expression.
+The proof is concluded by observing that the claims on the constant and leading
+coefficients are obvious. /square
+Corollary 6. Letaandbbe coprime positive integers, and let γbe an integer with
+γ≥(a−1)(b−1). Then the expression
+[γ]q[ab]q
+[a]q[b]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.Let
+[ab]q
+[a]q[b]q=(a−1)(b−1)/summationdisplay
+k=0Dkqk.
+We then have
+[γ]q[ab]q
+[a]q[b]q=(a−1)(b−1)+γ−1/summationdisplay
+N=0qNN/summationdisplay
+k=max{0,N−γ+1}Dk. (4.5)
+IfN≤γ−1, then, by Lemma 5, the sum over kon the right-hand side of (4.5) equals
+1−1+1−1+···, which is manifestly non-negative. On the other hand, if N >γ−1,
+then we may rewrite the sum over kon the right-hand side of (4.5) as
+N/summationdisplay
+k=max{0,N−γ+1}Dk=(a−1)(b−1)/summationdisplay
+k=N−γ+1Dk=(a−1)(b−1)+γ−1−N/summationdisplay
+k=0D(a−1)(b−1)−k.
+Again, by Lemma 5, this sum equals 1 −1 + 1−1 +···, which is manifestly non-
+negative. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 9
+The next lemmas all have a very similar flavour, and so do their proofs . In order to
+avoid repetition, proof details are only provided for Lemmas 7 and 16 ; the proofs of
+Lemmas 9–15, 22–24 follow the pattern exhibited in the proof of Lem ma 7, while the
+proofs of Lemmas 17–21 follow that of the proof of Lemma 15. Full d etails are found
+in [21, Sec. 4].
+Lemma 7. Letαandβbe positive integers with α≥6andβ≥8. Then the expression
+[α]q3[β]q4[72]q[3]q[4]q
+[8]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[72]q[3]q[4]q
+[8]q[9]q[12]q
+= (1−q3+q9−q15+q18)(1−q4+q8−q12+q16−q20+q24−q28+q32).
+It should be observed that both factors on the right-hand side ha ve the property that
+coefficients are in {0,1,−1}and that (+1)’s and ( −1)’s alternate, if one disregards the
+coefficients which are 0. If we now apply the same idea as in the proof o f Corollary 6,
+then we see that [ α]q3times the first factor is a polynomial in qwith non-negative
+integer coefficients, as is [ β]q4times the second factor. Taken together, this establishes
+the claim. /square
+Lemma 8. Letαandβbe positive integers with α≥26andβ≥8. Then the expression
+[α]q[β]q4[15]q
+[3]q[5]q[72]q[3]q[4]q
+[8]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 9. Letαandβbe positive integers with α≥18andβ≥3. Then the expression
+[α]q3[β]q4[90]q[3]q[4]q
+[5]q[6]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 10. Letαandβbe positive integers with α≥20andβ≥18. Then the
+expression
+[α]q[β]q3[90]q[3]q
+[5]q[6]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 11. Letαbe a positive integer with α≥26. Then the expression
+[α]q[15]q
+[3]q[5]q[12]q3
+[3]q3[4]q3
+is a polynomial in qwith non-negative integer coefficients.10 C. KRATTENTHALER AND T. W. M ¨ULLER
+Lemma 12. Letαbe a positive integer with α≥14. Then the expression
+[α]q[15]q
+[3]q[5]q[6]q3
+[2]q3[3]q3
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 13. Letαandβbe positive integers with α≥30andβ≥20. Then the
+expression
+[α]q[β]q2[84]q[2]q
+[4]q[6]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 14. Letαandβbe positive integers with α≥24andβ≥68. Then the
+expression
+[α]q[β]q[105]q
+[3]q[5]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 15. Letαandβbe positive integers with α≥24andβ≥34. Then the
+expression
+[α]q[β]q[70]q
+[2]q[5]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 16. Letαandβbe positive integers with α≥4andβ≥2. Then the expression
+[α]q2[β]q5[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q= 1+q−q3−q4−q5+q7+q8.
+If we multiply this expression by [ α]q2, then, forα= 4 we obtain
+1+q+q2−q5−q9+q12+q13+q14,
+forα= 5 we obtain
+1+q+q2−q5+q8−q11+q14+q15+q16,
+and, forα≥6, we obtain
+1+q+q2−q5+q8+q10+p1(q)+q2α−4+q2α−2−q2α+1+q2α+4+q2α+5+q2α+6,
+wherep1(q) is a polynomial in qwith non-negative coefficients of order at least 11 and
+degree at most 2 α−5. In all cases it is obvious that the product of the result and [ β]q5,
+withβ≥2, is a polynomial in qwith non-negative coefficients. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11
+Lemma 17. Letαandβbe positive integers with α≥14andβ≥2. Then the
+expression
+[α]q[β]q5[14]q
+[2]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 18. Letαandβbe positive integers with α≥32andβ≥12. Then the
+expression
+[α]q[β]q2[35]q
+[5]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 19. Letαandβbe positive integers with α≥16andβ≥2. Then the
+expression
+[α]q2[β]q5[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 20. Letαandβbe positive integers with α≥56andβ≥4. Then the
+expression
+[α]q[β]q2[35]q
+[5]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 21. Letαandβbe positive integers with α≥38andβ≥2. Then the
+expression
+[α]q[β]q5[14]q
+[2]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 22. Letαandβbe positive integers with α≥30andβ≥26. Then the
+expression
+[α]q[β]q3[126]q[3]q
+[6]q[7]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 23. Letαandβbe positive integers with α≥66andβ≥54. Then the
+expression
+[α]q[β]q3[252]q[3]q
+[7]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 24. Letαandβbe positive integers with α≥54andβ≥34. Then the
+expression
+[α]q[β]q2[140]q[2]q
+[4]q[7]q[10]q
+is a polynomial in qwith non-negative integer coefficients.12 C. KRATTENTHALER AND T. W. M ¨ULLER
+We are now ready for the proof of the main result of this section.
+Theorem 25. For all irreducible well-generated complex reflection grou ps and posi-
+tive integers m, theq-Fuß–Catalan number Catm(W;q)is a polynomial in qwith non-
+negative integer coefficients.
+Proof.First, letW=An. In this case, the degrees are 2 ,3,...,n+1, and hence
+Catm(An;q) =1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.
+Next, letW=G(d,1,n). In this case, the degrees are d,2d,...,nd , and hence
+Catm(G(d,1,n);q) =/bracketleftbigg
+(m+1)n
+n/bracketrightbigg
+qd,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+Now, letW=G(e,e,n). In this case, the degrees are e,2e,...,(n−1)e,n, and hence
+Catm(G(e,e,n);q) =[m(n−1)e+n]q
+[n]qn−1/productdisplay
+i=1[m(n−1)e+ie]q
+[ie]q
+=/bracketleftbigg
+(m+1)(n−1)
+n−1/bracketrightbigg
+qe+qn[e]qn/bracketleftbigg
+(m+1)(n−1)
+n/bracketrightbigg
+qe,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+It remains to verify the claim for the exceptional groups.
+For the groups W=G6,G9,G14,G17,G21,and partially for the groups W=G20,G23,
+G28,G30,G33,G35,G36,G37(depending on congruence properties of the parameter m),
+polynomiality and non-negativity of coefficients of the correspondin gq-Fuß–Catalan
+number can be directly read off by a proper rearrangement of the t erms in the defining
+expression; for example, for W=G21(with degrees given by 12 ,60) we have
+Catm(G21;q) =[60m+12]q[60m+60]q
+[12]q[60]q= [5m+1]q12[m+1]q60,
+which is manifestly a polynomial in qwith non-negative integer coefficients.
+For the groups G5,G10,G18,G26,G27,G29,G34, the terms in the defining expres-
+sion of the corresponding q-Fuß–Catalan number can be arranged in a manner so
+that aq-binomial coefficient appears; polynomiality and non-negativity of co efficients
+then follow from Proposition 3. For example, for W=G34(with degrees given by
+6,12,18,24,30,42) we have
+Catm(G34;q) =[42m+6]q[42m+12]q[42m+18]q[42m+24]q[42m+30]q[42m+42]q
+[6]q[12]q[18]q[24]q[30]q[42]q
+= [m+1]q42/bracketleftbigg
+7m+5
+5/bracketrightbigg
+q6,
+which, written in this form, is obviously a polynomial in qwith non-negative integer
+coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13
+On the other hand, for the groups G4,G8,G16,G25,G32, the terms in the defining
+expression of the corresponding q-Fuß–Catalan number can be arranged in a manner so
+that aq-Fuß–Catalannumber of type Aappears andProposition 4 applies; for example,
+forW=G32(with degrees given by 12 ,18,24,30) we have
+Catm(G32;q) =[30m+12]q[30m+18]q[30m+24]q[30m+30]q
+[12]q[18]q[24]q[30]q
+=1
+[5m+6]q6/bracketleftbigg
+5m+6
+5/bracketrightbigg
+q6,
+which indeed fits into the framework of Proposition 4 and, hence, is a polynomial in q
+with non-negative integer coefficients.
+In the other cases, the more “specialised” auxiliary results given in C orollary 6 and
+Lemmas7–24havetobeapplied. Forthesakeofillustration, weexhib it oneexample for
+each of them below, with full details being provided in [21, Sec. 4]. In ge neral, the idea
+is that, given a rational expression consisting of cyclotomic factor s, as in the definition
+oftheq-Fuß–Catalannumbers, onetriestoplacedenominator factorsbe lowappropriate
+numerator factors so that one can divide out the denominator fac tor completely. For
+example, if we were to encounter the expression
+[30m+12]q·(other terms)
+[12]q·(other terms)
+and know that mis even, then we would try to simplify this to
+/bracketleftbig5m+2
+2/bracketrightbig
+q12·(other terms)
+(other terms),
+where [5m+2
+2]q12is manifestly a polynomial in qwith non-negative integer coefficients.
+On the other hand, in a situation where twodenominator factors “want” to divide a
+singlenumerator factor, we “extract” as much as we can from the nume rator factor and
+compensate by additional “fudge” factors. To be more concrete , if we encounter the
+expression
+[14m+14]q·(other terms)
+[6]q[14]q·(other terms)
+and we know that m≡0 (mod 3), then we would try the rewriting
+/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2[2]q·(other terms)
+(other terms),
+with the idea that we might find somewhere else a term [2 α]q, which could be combined
+with the term[2] qin the denominator into [2 α]q/[2]q= [α]q2, andthen apply Corollary6
+to see that
+[α]q2[21]q2
+[3]q2[7]q2
+is a polynomial in qwith non-negative integer coefficients (provided αis at least 12),
+with/bracketleftbigm+1
+3/bracketrightbig
+q42being such a polynomial in any case.
+In situations where threedenominator factors “want” to divide a singlenumerator
+factor, one has to perform more complicated rearrangements, in order to be able to
+apply one of the Lemmas 7–24.14 C. KRATTENTHALER AND T. W. M ¨ULLER
+For example, for W=G24, the degrees are 4 ,6,14, and hence
+Catm(G24;q) =[14m+4]q[14m+6]q[14m+14]q
+[4]q[6]q[14]q.
+We have
+Catm(G24;q) =
+
+/bracketleftbig7m
+2+1/bracketrightbig
+q4/bracketleftbig14m
+6+1/bracketrightbig
+q6[m+1]q14,ifm≡0 (mod 6),/bracketleftbig7m+2
+3/bracketrightbig
+q6/bracketleftbig7m+3
+2/bracketrightbig
+q4[m+1]q14, ifm≡1 (mod 6),
+/bracketleftbig7m
+2+1/bracketrightbig
+q4[7m+3]q2/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2,ifm≡2 (mod 6),
+[7m+2]q2/bracketleftbig7m
+3+1/bracketrightbig
+q6/bracketleftbigm+1
+2/bracketrightbig
+q28[14]q2
+[2]q2[7]q2,ifm≡3 (mod 6),
+/bracketleftbig7m+2
+6/bracketrightbig
+q12[6]q2
+[2]q2[3]q2[7m+3]q2[m+1]q14,ifm≡4 (mod 6),
+[7m+2]q2/bracketleftbig7m+3
+2/bracketrightbig
+q4/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2,ifm≡5 (mod 6),
+which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all
+cases.
+ForW=G30=H4, the degrees are 2 ,12,20,30, and hence
+Catm(H4;q) =[30m+2]q[30m+12]q[30m+20]q[30m+30]q
+[2]q[12]q[20]q[30]q.
+Ifmis odd, then we may write
+Catm(H4;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[5m+2]q6[3m+2]q10/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q6[10]q2[15]q2,
+which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients.
+ForW=G35=E6, the degrees are 2 ,5,6,8,9,12, and hence
+Catm(E6;q) =[12m+2]q[12m+5]q[12m+6]q[12m+8]q[12m+9]q[12m+12]q
+[2]q[5]q[6]q[8]q[9]q[12]q.
+Ifm≡5 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5[2m+1]q6
+×[3m+2]q4[4m+3]q3/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,
+which, by Lemma 7, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡7 (mod 30),then we have
+Catm(E6;q) =/bracketleftbig6m+1
+2/bracketrightbig
+q4[12m+5]q/bracketleftbig2m+1
+15/bracketrightbig
+q90
+×[90]q[3]q[4]q
+[5]q[6]q[9]q[3m+2]q4[4m+3]q3/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6 and Lemma 9, is a polynomial in qwith non-negative integer
+coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15
+Ifm≡8 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8
+×/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+3/bracketrightbig
+q36[12]q3
+[3]q3[4]q3,
+which, by Lemma 11, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡13 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×[3m+2]q4/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡22 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+15/bracketrightbig
+q90[90]q[3]q
+[5]q[6]q[9]q
+×/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3[m+1]q12,
+which, by Lemma 10, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡23 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
+×[3m+2]q4/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,
+which, by Lemma 8, is a polynomial in qwith non-negative integer coefficients.
+ForW=G36=E7, the degrees are 2 ,6,8,10,12,14,18, and hence
+Catm(E7;q) =[18m+2]q[18m+6]q[18m+8]q[18m+10]q
+[2]q[6]q[8]q[10]q
+×[18m+12]q[18m+14]q[18m+18]q
+[12]q[14]q[18]q.
+Ifm≡18 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+28/bracketrightbig
+q168[84]q2[2]q2
+[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6 and Lemma 13, is a polynomial in qwith non-negative integer
+coefficients.16 C. KRATTENTHALER AND T. W. M ¨ULLER
+Ifm≡23 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+35/bracketrightbig
+q210[105]q2
+[3]q2[5]q2[7]q2[9m+4]q2[9m+5]q2
+×[3m+2]q6[9m+7]q2/bracketleftbigm+1
+2/bracketrightbig
+q36[6]q6
+[2]q6[3]q6,
+which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡54 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+70/bracketrightbig
+q140[70]q2
+[2]q2[5]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat, byCorollary6
+and Lemma 15, this is a polynomial in qwith non-negative integer coefficients.
+ForW=G37=E8, the degrees are 2 ,8,12,14,18,20,24,30, and hence
+Catm(E7;q) =[30m+2]q[30m+8]q[30m+12]q[30m+14]q
+[2]q[8]q[12]q[14]q
+×[30m+18]q[30m+20]q[30m+24]q[30m+30]q
+[18]q[20]q[24]q[30]q.
+Ifm≡3 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4/bracketleftbig15m+4
+7/bracketrightbig
+q14[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡8 (mod 84),then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+42/bracketrightbig
+q252[126]q2[3]q2
+[6]q2[7]q2[9]q2[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡11 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 17
+which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡16 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+84/bracketrightbig
+q504[252]q2[3]q2
+[7]q2[9]q2[12]q2[m+1]q30,
+which, by Lemma 23, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡18 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18
+/bracketleftbig3m+2
+28/bracketrightbig
+q280[140]q2[2]q2
+[4]q2[7]q2[10]q2/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Lemma 24, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡21 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 17, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡25 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡27 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer
+coefficients.
+All other cases are disposed of in a similar fashion. /square
+5.Auxiliary results I
+This section collects several auxiliary results which allow us to reduce the problem
+of proving Theorem 2, or the equivalent statement (3.3), for the 2 6 exceptional groups
+listed in Section 2 to a finite problem. While Lemmas 27 and 28 cover spec ial choices
+of the parameters, Lemmas 26 and 30 afford an inductive procedur e. More precisely,18 C. KRATTENTHALER AND T. W. M ¨ULLER
+if we assume that we have already verified Theorem 2 for all groups o f smaller rank,
+then Lemmas 26 and 30, together with Lemmas 27 and 31, reduce th e verification of
+Theorem 2 for the group that we are currently considering to a finit e problem; see
+Remark 3. The final lemma of this section, Lemma 32, disposes of com plex reflection
+groups with a special property satisfied by their degrees.
+Letp=am+b, 0≤b<m. We have
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;ca+1wm−b+1c−a−1,ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
+caw1c−a,...,cawm−bc−a/parenrightbig
+,(5.1)
+where∗stands for the element of Wwhich is needed to complete the product of the
+components to c.
+Lemma 26. It suffices to check (3.3)forpa divisor of mh. More precisely, let pbe
+a divisor of mh, and letkbe another positive integer with gcd(k,mh/p) = 1, then we
+have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πikp/mh (5.2)
+and
+|FixNCm(W)(φp)|=|FixNCm(W)(φkp)|. (5.3)
+Proof.For (5.2), this follows immediately from
+lim
+q→ζ[α]q
+[β]q=/braceleftBigg
+α
+βifα≡β≡0 (modd),
+1 otherwise ,(5.4)
+whereζis ad-th root of unity and α,βare non-negative integers such that α≡β
+(modd).
+In order to establish (5.3), suppose that x∈FixNCm(W)(φp), that is,x∈NCm(W)
+andφp(x) =x. It obviously follows that φkp(x) =x, so thatx∈FixNCm(W)(φkp).
+To establish the converse, note that, if gcd( k,mh/p) = 1, then there exists k′with
+k′k≡1 (modmh
+p). It follows that, if x∈FixNCm(W)(φkp), that is, if x∈NCm(W) and
+φkp(x) =x, thenx=φk′kp(x) =φp(x), whencex∈FixNCm(W)(φp). /square
+Lemma 27. Letpbe a divisor of mh. Ifpis divisible by m, then(3.3)is true.
+Proof.According to (5.1), the action of φponNCm(W) is described by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;cp/mw1c−p/m,...,cp/mwmc−p/m/parenrightbig
+.
+Hence, if (w0;w1,...,w m) is fixed by φp, then each individual wimust be fixed under
+conjugation by cp/m.
+Using the notation W′= Cent W(cp/m), theprevious observationmeans that wi∈W���,
+i= 1,2,...,m. Springer [33, Theorem 4.2] (see also [24, Theorem 11.24(iii)]) prove d
+thatW′is a well-generated complex reflection group whose degrees coincide with those
+degrees ofWthat are divisible by mh/p. It was furthermore shown in [9, Lemma 3.3]
+that
+NC(W)∩W′=NC(W′). (5.5)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 19
+Hence, the tuples ( w0;w1,...,w m) fixed byφpare in fact identical with the elements of
+NCm(W′), which implies that
+|FixNCm(W)(φp)|=|NCm(W′)|. (5.6)
+Application of Theorem 1 with Wreplaced by W′and of the “limit rule” (5.4) then
+yields that
+|NCm(W′)|=/productdisplay
+1≤i≤n
+mh
+p|dimh+di
+di= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh. (5.7)
+Combining (5.6) and (5.7), we obtain (3.3). This finishes the proof of t he lemma. /square
+Lemma 28. Equation (3.3)holds for all divisors pofm.
+Proof.Using (5.4) and the fact that the degrees of irreducible well-genera ted complex
+reflection groups satisfy di<hfor alli<n, we see that
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh=/braceleftBigg
+m+1 ifm=p,
+1 ifm/ne}ationslash=p.
+On the other hand, if ( w0;w1,...,w m) is fixed by φp, then, because of the action (5.1),
+we must have w1=wp+1=···=wm−p+1andw1=cwm−p+1c−1. In particular,
+w1∈CentW(c). By the theorem of Springer cited in the proof of Lemma 27, the
+subgroup Cent W(c) is itself a complex reflection group whose degrees are those degre es
+ofWthat are divisible by h. The only such degree is hitself, hence Cent W(c) is the
+cyclic group generated by c. Moreover, by (5.5), we obtain that w1=ε, the identity
+element of W, orw1=c. Therefore, for m=pthe set Fix NCm(W)(φp) consists of the
+m+1 elements ( w0;w1,...,w m) obtained by choosing wi=cfor a particular ibetween
+0 andm, all otherwj’s being equal to ε, while, for m/ne}ationslash=p, we have
+FixNCm(W)(φp) =/braceleftbig
+(c;ε,...,ε)/bracerightbig
+,
+whence the result. /square
+Lemma 29. LetWbe an irreducible well-generated complex reflection group a ll of
+whose degrees are divisible by d. Then each element of Wis fixed under conjugation by
+ch/d.
+Proof.By the theorem of Springer cited in the proof of Lemma 27, the subg roupW′=
+CentW(ch/d) is itself a complex reflection group whose degrees are those degre es ofW
+that are divisible by d. Thus, by our assumption, the degrees of W′coincide with the
+degrees ofW, and hence W′must be equal to W. Phrased differently, each element of
+Wis fixed under conjugation by ch/d, as claimed. /square
+Lemma 30. LetWbe an irreducible well-generated complex reflection group o f rankn,
+and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. Without loss of
+generality, we assume that gcd(h1,m2) = 1. Suppose that Theorem 2has already been
+verified for all irreducible well-generated complex reflect ion groups with rank <n. Ifh2
+does not divide all degrees di, then Equation (3.3)is satisfied.20 C. KRATTENTHALER AND T. W. M ¨ULLER
+Proof.Let us write h1=am2+b, with 0 ≤b < m 2. The condition gcd( h1,m2) = 1
+translates into gcd( b,m2) = 1. From (5.1), we infer that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;ca+1wm−m1b+1c−a−1,ca+1wm−m1b+2c−a−1,...,ca+1wmc−a−1,
+caw1c−a,...,cawm−m1bc−a/parenrightbig
+.(5.8)
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=ca+1wi+m−m1bc−a−1, i= 1,2,...,m 1b,
+wi=cawi−m1bc−a, i=m1b+1,m1b+2,...,m,
+which, after iteration, implies in particular that
+wi=cb(a+1)+(m2−b)awic−b(a+1)−(m2−b)a=ch1wic−h1, i= 1,2,...,m.
+It is at this point where we need gcd( b,m2) = 1. The last equation shows that each wi,
+i= 1,2,...,m, and thus also w0, lies in Cent W(ch1). By the theorem of Springer cited
+in the proof of Lemma 27, this centraliser subgroup is itself a complex reflection group,
+W′say, whose degrees are those degrees of Wthat are divisible by h/h1=h2. Since,
+by assumption, h2does not divide alldegrees,W′has rank strictly less than n. Again
+by assumption, we know that Theorem 2 is true for W′, so that in particular,
+|FixNCm(W′)(φp)|= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/mh.
+The arguments above together with (5.5) show that Fix NCm(W)(φp) = Fix NCm(W′)(φp).
+On the other hand, using (5.4) it is straightforward to see that
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/mh.
+This proves (3.3) for our particular p, as required. /square
+Lemma 31. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. We assume
+thatgcd(h1,m2) = 1. Ifm2>nthen
+FixNCm(W)(φp) =/braceleftbig
+(c;ε,...,ε)/bracerightbig
+.
+Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(φp) and that there exists a
+j≥1 such that wj/ne}ationslash=ε. By (5.8), it then follows for such a jthat alsowk/ne}ationslash=εfor
+allk≡j−lm1b(modm), where, as before, bis defined as the unique integer with
+h1=am2+band 0≤b < m 2. Since, by assumption, gcd( b,m2) = 1, there are
+exactlym2suchk’s which are distinct mod m. However, this implies that the sum of
+the absolute lengths of the wi’s, 0≤i≤m, is at least m2> n, a contradiction to
+Remark 1.(2). /square
+Remark 3.(1) If we put ourselves in the situation of the assumptions of Lemma 30,
+then we may conclude that equation (3.3) only needs to be checked f or pairs (m2,h2)
+subject to the following restrictions:
+m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (5.9)
+Indeed, Lemmas 27 and 30 together imply that equation (3.3) is alway s satisfied in all
+other cases.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 21
+(2) Still putting ourselves in the situation of Lemma 30, if m2>nandm2h2does not
+divide any of the degrees of W, then equation (3.3) is satisfied. Indeed, Lemma 31 says
+thatinthiscasetheleft-handsideof (3.3)equals1,whileastraightf orwardcomputation
+using (5.4) shows that in this case the right-hand side of (3.3) equals 1 as well.
+(3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider,
+whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite
+number of choices for p=h1m1to be checked.
+Lemma 32. LetWbe an irreducible well-generated complex reflection group o f rankn
+with the property that di|hfori= 1,2,...,n. Then Theorem 2is true for this group
+W.
+Proof.By Lemma 26, we may restrict ourselves to divisors pofmh.
+Suppose that e2πip/mhis adi-th rootof unity for some i. In other words, mh/pdivides
+di. Sincediis a divisor of hby assumption, the integer mh/palso divides h. But this
+is equivalent to saying that mdividesp, and equation (3.3) holds by Lemma 27.
+Now assume that mh/pdoes not divide any of the di’s. Then, by (5.4), the right-
+hand side of (3.3) equals 1. On the other hand, ( c;ε,...,ε) is always an element of
+FixNCm(W)(φp). To see that there are no others, we make appeal to the classific a-
+tion of all irreducible well-generated complex reflection groups, whic h we recalled in
+Section 2. Inspection reveals that all groups satisfying the hypot heses of the lemma
+have rank n≤2. Except for the groups contained in the infinite series G(d,1,n)
+andG(e,e,n) for which Theorem 2 has been established in [19], these are the grou ps
+G5,G6,G9,G10,G14,G17,G18,G21. We now discuss these groups case by case, keeping
+the notation of Lemma 30. In order to simplify the argument, we not e that Lemma 31
+implies that equation (3.3) holds if m2>2, so that in the following arguments we
+always may assume that m2= 2.
+CaseG5. The degrees are 6 ,12, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG6. The degrees are 4 ,12, and therefore, according to Remark 3.(1), we need
+only consider the casewhere h2= 4andm2= 2, that is, p= 3m/2. Then (5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+(5.10)
+If (w0;w1,...,w m) isfixed by φpandnot equal to ( c;ε,...,ε), there must exist an iwith
+1≤i≤m
+2such thatℓT(wi) =ℓT(wm
+2+i) = 1,wm
+2+i=cwic−1,wiwm
+2+i=wicwic−1=c,
+and allwj, withj/ne}ationslash=i,m
+2+i, equalε. However, with the help of the GAPpackage
+CHEVIE[14, 28], one verifies that there is no wiinG6such that
+ℓT(wi) = 1 and wicwic−1=c
+are simultaneously satisfied. Hence, the left-hand side of (3.3) is eq ual to 1, as required.
+CaseG9. The degrees are 8 ,24, and therefore, according to Remark 3.(1), we need
+only consider the case where h2= 8 andm2= 2, that is, p= 3m/2. This is the same p
+as forG6. Again, CHEVIEfinds no solution. Hence, the left-hand side of (3.3) is equal
+to 1, as required.
+CaseG10. The degrees are 12 ,24, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.22 C. KRATTENTHALER AND T. W. M ¨ULLER
+CaseG14. The degrees are 6 ,24, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG17. The degrees are 20 ,60, and therefore, according to Remark 3.(1), we need
+only consider the cases where h2= 20 orh2= 4. In the first case, p= 3m/2, which is
+the samepas forG6. Again,CHEVIEfinds no solution. In the second case, p= 15m/2.
+Then (5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.(5.11)
+By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus, on
+elements fixed by φp, the above action of φpreduces to the one in (5.10). This action
+was already discussed in the first case. Hence, in both cases, the le ft-hand side of (3.3)
+is equal to 1, as required.
+CaseG18. The degrees are 30 ,60, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG21. The degrees are 12 ,60, and therefore, according to Remark 3.(1), we need
+only consider the cases where h2= 12 orh2= 4. In the first case, p= 5m/2, so that
+(5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.(5.12)
+If (w0;w1,...,w m) is fixed by φpand not equal to ( c;ε,...,ε), there must exist an i
+with 1≤i≤m
+2such thatℓT(wi) = 1 andwic2wic−2=c. However, with the help of
+theGAPpackageCHEVIE[14, 28], one verifies that there is no such solution to this
+equation. In the second case, p= 15m/2. Then (5.8) becomes the action in (5.11).
+By Lemma 29, every element of NC(W) is fixed under conjugation by c5, and, thus,
+on elements fixed by φp, the action of φpin (5.11) reduces to the one in the first case.
+Hence, in both cases, the left-hand side of (3.3) is equal to 1, as re quired.
+This completes the proof of the lemma. /square
+6.Exemplification of case-by-case verification of Theorem 2
+It remains to verify Theorem 2 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
+G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
+details can be found in [21, Sec. 6]. We content ourselves with illustra ting the type of
+computation that is needed here by going through the case of the g roupG24, and by
+discussing some of the arguments needed for the group G37=E8.
+In the sequel we write ζdfor a primitive d-th root of unity.
+CaseG24.The degrees are 4 ,6,14, and hence we have
+Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
+[14]q[6]q[4]q.
+Letζbe a 14m-th root of unity. In what follows, we abbreviate the assertion tha t “ζis
+a primitive d-th root of unity” as “ ζ=ζd.” The following cases on the right-hand sideCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 23
+of (3.3) occur:
+lim
+q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (6.1a)
+lim
+q→ζCatm(G24;q) =7m+3
+3,ifζ=ζ6,ζ3,3|m, (6.1b)
+lim
+q→ζCatm(G24;q) =7m+2
+2,ifζ=ζ4,2|m, (6.1c)
+lim
+q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (6.1d)
+lim
+q→ζCatm(G24;q) = 1,otherwise. (6.1e)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.1). The only cases not covered by Lemma 27 are the on es in (6.1b),
+(6.1c), and (6.1e). (In both (6.1a) and (6.1d) we have d|h.)
+We first consider (6.1b). By Lemma 26, we are free to choose p= 7m/3 ifζ=ζ6,
+respectively p= 14m/3 ifζ=ζ3. In both cases, mmust be divisible by 3.
+We start with the case that p= 7m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w2m
+3+1c−3,c3w2m
+3+2c−3,...,c3wmc−3,c2w1c−2,...,c2w2m
+3c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w2m
+3+ic−3, i= 1,2,...,m
+3, (6.2a)
+wi=c2wi−m
+3c−2, i=m
+3+1,m
+3+2,...,m. (6.2b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.2) reduce to
+t1=c3t3c−3, (6.3a)
+t2=c2t1c−2, (6.3b)
+t3=c2t2c−2. (6.3c)
+One of these equations is in fact superfluous: if we substitute (6.3b ) and (6.3c) in
+(6.3a), then we obtain t1=c7t1c−7which is automatically satisfied due to Lemma 29
+withd= 2.
+Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with
+(6.3), we infer that
+t1(c2t1c−2)(c4t1c−4) =c. (6.4)
+With the help of CHEVIE, one obtains 7 solutions for t1in this equation, each of them
+giving rise to m/3 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/3.
+In total, we obtain 1 + 7m
+3=7m+3
+3elements in Fix NCm(G24)(φp), which agrees with
+the limit in (6.1b).
+The case where p= 14m/3 can be treated in a similar fashion. In the end, it
+turns out that we have to solve the same enumeration problem as fo rp= 7m/3, and,24 C. KRATTENTHALER AND T. W. M ¨ULLER
+consequently, the number of elements of Fix NCm(G24)(φp) is the same, namely7m+3
+3, as
+required.
+Our next case is (6.1c). Proceeding in a similar manner as before, we s ee that there is
+againthe trivial possibility ( c;ε,...,ε), and otherwise we have to find t1withℓT(t1) = 1
+satisfying the inequality
+t1(c3t1c−3)≤Tc. (6.5)
+With the help of CHEVIE, one obtains 7 solutions for t1in this relation, each of them
+giving rise to m/2 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/2.
+In total, we obtain 1 + 7m
+2=7m+2
+2elements in Fix NCm(G24)(φp), which agrees with
+the limit in (6.1c).
+Finally, we turn to (6.1e). By Remark 3, the only choices for h2andm2to be consid-
+ered areh2= 1 andm2= 3,h2=m2= 2, andh2= 2 andm2= 3. These correspond
+to the choices p= 14m/3,p= 7m/2, respectively p= 7m/3, all of which have already
+been discussed as they do not belong to (6.1e). Hence, (3.3) must n ecessarily hold, as
+required.
+CaseG37=E8.The degrees are 2 ,8,12,14,18,20,24,30, and hence we have
+Catm(E8;q) =[30m+30]q[30m+24]q[30m+20]q[30m+18]q
+[30]q[24]q[20]q[18]q
+×[30m+14]q[30m+12]q[30m+8]q[30m+2]q
+[14]q[12]q[8]q[2]q.
+Letζbe a 30m-th root of unity. The cases occurring on the right-hand side of (3 .3)
+not covered by Lemma 27 are:
+lim
+q→ζCatm(E8;q) =5m+4
+4,ifζ=ζ24,4|m, (6.6a)
+lim
+q→ζCatm(E8;q) =3m+2
+2,ifζ=ζ20,2|m, (6.6b)
+lim
+q→ζCatm(E8;q) =5m+3
+3,ifζ=ζ18,ζ9,3|m, (6.6c)
+lim
+q→ζCatm(E8;q) =15m+7
+7,ifζ=ζ14,ζ7,7|m, (6.6d)
+lim
+q→ζCatm(E8;q) =(5m+4)(5m+2)
+8,ifζ=ζ12,2|m, (6.6e)
+lim
+q→ζCatm(E8;q) =(5m+4)(15m+4)
+16,ifζ=ζ8,4|m, (6.6f)
+lim
+q→ζCatm(E8;q) =(5m+4)(3m+2)(5m+2)(15m+4)
+64,ifζ=ζ4,2|m,(6.6g)
+lim
+q→ζCatm(E8;q) = Catm(E8),ifζ=−1 orζ= 1, (6.6h)
+lim
+q→ζCatm(E8;q) = 1,otherwise. (6.6i)
+We now have to prove that the left-hand side of (3.3) in each case ag rees with the
+values exhibited in (6.6). Since the corresponding computations in th e various cases are
+very similar, we concentrate here only on the cases (6.6f) and (6.6g ), these two being
+representative of the types of arguments arising. As before, we refer the reader to [21,
+Sec. 6] for full details.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 25
+Letusconsiderthecasein(6.6f)first. ByLemma26, wearefreeto choosep= 15m/4.
+In particular, mmust be divisible by 4. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4wm
+4+1c−4,c4wm
+4+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
+4c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4wm
+4+ic−4, i= 1,2,...,3m
+4, (6.7a)
+wi=c3wi−3m
+4c−3, i=3m
+4+1,3m
+4+2,...,m. (6.7b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+4such that
+1≤ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4)≤2, (6.8a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there are i1andi2with 1≤i1<i2≤m
+4such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+4) =ℓT(wi2+m
+4)
+=ℓT(wi1+2m
+4) =ℓT(wi2+2m
+4) =ℓT(wi1+3m
+4) =ℓT(wi2+3m
+4) = 1,(6.8b)
+and all other wjare equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+4wi+2m
+4wi+3m
+4≤Tc,
+or
+wi1wi2wi1+m
+4wi2+m
+4wi1+2m
+4wi2+2m
+4wi1+3m
+4wi2+3m
+4=c.
+Together with equations (6.7)–(6.8), this implies that
+wi=c15wic−15andwi(c11wic−11)(c7wic−7)(c3wic−3)≤Tc, (6.9)
+or that
+wi1=c15wi1c−15, wi1=c15wi2c−15,
+andwi1wi2(c11wi1c−11)(c11wi2c−11)(c7wi1c−7)(c7wi2c−7)(c3wi1c−3)(c3wi2c−3) =c.
+(6.10)
+Here, the first equation in (6.9) and the first two equations in (6.10) are automatically
+satisfied due to Lemma 29 with d= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 30 solutions
+forwiin (6.9) with ℓT(wi) = 1, 45 solutions for wiwithℓT(wi) = 2 and wiof type
+A2
+1(as a parabolic Coxeter element; see the end of Section 2), and 20 s olutions for
+wiwithℓT(wi) = 2 and wiof typeA2. Each of them gives rise to m/4 elements of
+FixNCm(E8)(φp) sinceiranges from 1 to m/4.
+The number of solutions in Case (iii) can be computed from our knowled ge of the
+solutions in Case (ii) according to type, using some elementary count ing arguments.
+Namely, the number of solutions of (6.10) is equal to
+45·2+20·3 = 150,26 C. KRATTENTHALER AND T. W. M ¨ULLER
+since an element of type A2
+1can be decomposed in two ways into a product of two
+elements of absolute length 1, while for an element of type A2this can be done in 3
+ways.
+In total, we obtain 1 + (30 + 45 + 20)m
+4+ 150/parenleftbigm/4
+2/parenrightbig
+=(5m+4)(15m+4)
+16elements in
+FixNCm(E8)(φp), which agrees with the limit in (6.6f).
+Next, we discuss the case in (6.6g). By Lemma 26, we are free to cho osep= 15m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c8wm
+2+ic−8, i= 1,2,...,m
+2, (6.11a)
+wi=c7wi−m
+2c−7, i=m
+2+1,m
+2+2,...,m. (6.11b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+1≤ℓT(wi) =ℓT(wi+m
+2)≤4, (6.12a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,andℓ1+ℓ2≤4,
+(6.12b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1,i2,i3with 1≤i1<i2<i3≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,
+ℓ3:=ℓT(wi3) =ℓT(wi3+m
+2)≥1,andℓ1+ℓ2+ℓ3≤4,(6.12c)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(v) there are i1,i2,i3,i4with 1≤i1<i2<i3<i4≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi4)
+=ℓT(wi1+m
+2) =ℓT(wi2+m
+2) =ℓT(wi3+m
+2) =ℓT(wi4+m
+2) = 1,(6.12d)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc, respectively
+wi1wi2wi3wi1+m
+2wi2+m
+2wi3+m
+2≤Tc,
+respectively
+wi1wi2wi3wi4wi1+m
+2wi2+m
+2wi3+m
+2wi4+m
+2=c.
+Together with equations (6.11)–(6.12), this implies that
+wi=c15wic−15andwi(c7wic−7)≤Tc, (6.13)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15,andwi1wi2(c7wi1c−7)(c7wi2c−7)≤Tc,(6.14)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 27
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15,
+andwi1wi2wi3(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)≤Tc,(6.15)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15, wi4=c15wi4c−15,
+andwi1wi2wi3wi4(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)(c7wi4c−7) =c.(6.16)
+Here, the first equation in (6.13), the first two in (6.14), the first t hree in (6.15), and
+the first four in (6.16), are all automatically satisfied due to Lemma 2 9 withd= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains
+— 45 solutions for wiin (6.13) with ℓT(wi) = 1,
+— 150 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2
+1,
+— 100 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2,
+— 75 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3
+1,
+— 165 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA1∗A2,
+— 90 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3,
+— 15 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
+1∗A2,
+— 45 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA1∗A3;
+— 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
+2,
+— 18 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA4,
+— 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeD4.
+Each of them gives rise to m/2 elements of Fix NCm(E8)(φp) sinceiranges from 1 to m/2.
+There are no solutions for wiin (6.13) with wiof typeA4
+1.
+Letting the computer find all solutions in cases (iii)–(v) would take ye ars. However,
+the number of these solutions can be computed from our knowledge of the solutions
+in Case (ii) according to type, if this information is combined with the de composition
+numbers in the sense of [17, 18, 20] (see the end of Section 2) and some elementary
+(multiset) permutation counting. The decomposition numbers for A2,A3,A4, andD4
+of which we make use can be found in the appendix of [18].
+To begin with, the number of solutions of (6.14) with ℓ1=ℓ2= 1 is equal to
+n1,1:= 150·2+100·NA2(A1,A1) = 600,
+since an element of type A2
+1can be decomposed in two ways into a product of two
+elements of absolute length 1, while for an element of type A2this can be done in
+NA2(A1,A1) = 3 ways. Similarly, the number of solutions of (6.14) with ℓ1= 2 and
+ℓ2= 1 is equal to
+n2,1:= 75·3+165·(1+NA2(A1,A1))+90·NA3(A2,A1) = 1425,
+the number of solutions of (6.14) with ℓ1= 3 andℓ2= 1 is equal to
+n3,1:= 15·(2+NA2(A1,A1))+45·(1+NA3(A2,A1))+5·(2NA2(A1,A1))
++18·(NA4(A3,A1)+NA4(A1∗A2,A1))+5·(ND4(A3,A1)+ND4(A3
+1,A1)) = 660,28 C. KRATTENTHALER AND T. W. M ¨ULLER
+the number of solutions of (6.14) with ℓ1=ℓ2= 2 is equal to
+n2,2:= 15·(2+2NA2(A1,A1))+45·(2NA3(A2,A1))+5·(2+NA2(A1,A1)2)
++18·(NA4(A2,A2)+NA4(A2
+1,A2
+1)+2NA4(A2,A2
+1))
++5·(ND4(A2,A2)+2ND4(A2,A2
+1)) = 1195,
+the number of solutions of (6.15) with ℓ1=ℓ2=ℓ3= 1 is equal to
+n1,1,1:= 75·3!+165·(3NA2(A1,A1))+90NA3(A1,A1,A1) = 3375,
+the number of solutions of (6.15) with ℓ1= 2 andℓ2=ℓ3= 1 is equal to
+n2,1,1:= 15·(2+NA2(A1,A1)+2·2·NA2(A1,A1))+45·(2NA3(A2,A1)+NA3(A1,A1,A1))
++5·(2NA2(A1,A1)+2NA2(A1,A1)2)+18·(NA4(A2,A1,A1)+NA4(A2
+1,A1,A1))
++5·(ND4(A2,A1,A1)+ND4(A2
+1,A1,A1)) = 2850,
+and the number of solutions of (6.16) is equal to
+n1,1,1,1:= 15·(12NA2(A1,A1))+45·(4NA3(A1,A1,A1))+5·(6NA2(A1,A1)2)
++18·NA4(A1,A1,A1,A1)+5·ND4(A1,A1,A1,A1) = 6750.
+In total, we obtain
+1+(45+150+100+75+165+90+15+45+5+18+5)m
+2+(n1,1+2n2,1+2n3,1+n2,2)/parenleftbiggm/2
+2/parenrightbigg
++(n1,1,1+3n2,1,1)/parenleftbiggm/2
+3/parenrightbigg
++n1,1,1,1/parenleftbiggm/2
+4/parenrightbigg
+=(5m+4)(3m+2)(5m+2)(15m+4)
+64
+elements in Fix NCm(E8)(φp), which agrees with the limit in (6.6g).
+7.Cyclic sieving II
+In this section we present the second cyclic sieving conjecture due to Bessis and
+Reiner [9, Conj. 6.5].
+Letψ:NCm(W)→NCm(W) be the map defined by
+(w0;w1,...,w m)/mapsto→/parenleftbig
+cwmc−1;w0,w1,...,w m−1/parenrightbig
+. (7.1)
+Form= 1, we have w0=cw−1
+1, so that this action reduces to the inverse of the
+Kreweras complement Kc
+idas defined by Armstrong [2, Def. 2.5.3].
+It is easy to see that ψ(m+1)hacts as the identity, where his the Coxeter number of
+W(see (8.1) below). By slight abuse of notation as before, let C2be the cyclic group
+of order (m+1)hgenerated by ψ.
+Given these definitions, we are now in the position to state the secon d cyclic sieving
+conjecture of Bessis and Reiner. By the results of [19] and of this p aper, it becomes the
+following theorem.
+Theorem 33. For an irreducible well-generated complex reflection group Wand any
+m≥1, the triple (NCm(W),Catm(W;q),C2), whereCatm(W;q)is theq-analogue of
+the Fuß–Catalan number defined in (3.2), exhibits the cyclic sieving phenomenon.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 29
+By definition of the cyclic sieving phenomenon, we have to prove that
+|FixNCm(W)(ψp)|= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h, (7.2)
+for allpin the range 0 ≤p<(m+1)h.
+8.Auxiliary results II
+This section collects several auxiliary results which allow us to reduce the problem of
+proving Theorem 33, respectively the equivalent statement (7.2), for the 26 exceptional
+groups listed in Section 2 to a finite problem. The corresponding lemma s, Lemmas 34–
+39, are analogues of Lemmas 26–28 and 30–32 in Section 5.
+Letp=a(m+1)+b, 0≤b<m+1. We have
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (ca+1wm−b+1c−a−1;ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
+caw0c−a,...,cawm−bc−a/parenrightbig
+.(8.1)
+Lemma 34. It suffices to check (7.2)forpa divisor of (m+1)h. More precisely, let pbe
+a divisor of (m+1)h, and letkbe another positive integer with gcd(k,(m+1)h/p) = 1,
+then we have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πikp/(m+1)h (8.2)
+and
+|FixNCm(W)(ψp)|=|FixNCm(W)(ψkp)|. (8.3)
+Proof.For (8.3), this follows in the same way as (5.3) in Lemma 26.
+For (8.2), we must argue differently than in Lemma 26. Let us write ζ=e2πip/(m+1)h.
+For a given group W, we writeS1(W) for the set of all indices isuch thatζdi−h= 1,
+and we write S2(W) for the set of all indices isuch thatζdi= 1. By the rule of de
+l’Hospital, we have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h=
+
+0 if |S1(W)|>|S2(W)|,/producttext
+i∈S1(W)(mh+di)/producttext
+i∈S2(W)di/producttext
+i/∈S1(W)(1−ζdi−h)
+/producttext
+i/∈S2(W)(1−ζdi),if|S1(W)|=|S2(W)|.
+(8.4)
+Since, by Theorem 25, Catm(W;q) is a polynomial in q, the case |S1(W)|<|S2(W)|
+cannot occur.
+We claim that, for the case where |S1(W)|=|S2(W)|, the factors in the quotient of
+products/producttext
+i/∈S1(W)(1−ζdi−h)/producttext
+i/∈S2(W)(1−ζdi)
+cancel pairwise. If we assume the correctness of the claim, it is obv ious that we get
+the same result if we replace ζbyζk, where gcd( k,(m+1)h/p) = 1, hence establishing
+(8.2).
+In order to see that our claim is indeed valid, we proceed in a case-by- case fash-
+ion, making appeal to the classification of irreducible well-generated complex reflection
+groups, which werecalled inSection2. Firstofall, since dn=h, thesetS1(W)isalways
+non-empty as it contains the element n. Hence, if we want to have |S1(W)|=|S2(W)|,30 C. KRATTENTHALER AND T. W. M ¨ULLER
+the setS2(W) must be non-empty as well. In other words, the integer ( m+ 1)h/p
+must divide at least one of the degrees d1,d2,...,d n. In particular, this implies that,
+for each fixed reflection group Wof exceptional type, only a finite number of values of
+(m+1)h/phas to be checked. Writing Mfor (m+1)h/p, what needs to be checked is
+whether the multisets (that is, multiplicities of elements must be taken into account)
+{(di−h) modM:i /∈S1(W)}and{dimodM:i /∈S2(W)}
+are the same. Since, for a fixed irreducible well-generated complex r eflection group,
+thereisonlyafinitenumber ofpossibilities for M, thisamountstoaroutineverification.
+/square
+Lemma 35. Letpbe a divisor of (m+ 1)h. Ifpis divisible by m+ 1, then(7.2)is
+true.
+We leave the proof to the reader as it is completely analogous to the p roof of
+Lemma 27.
+Lemma 36. Equation (7.2)holds for all divisors pofm+1.
+Proof.We have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h=/braceleftBigg
+0 ifp<m+1,
+m+1 ifp=m+1.
+Here, the first case follows from (8.4) and the fact that we have S1(W)⊇ {n}and
+S2(W) =∅ifp|(m+1) andp<m+1.
+Ontheother hand, if ( w0;w1,...,w m) is fixed by ψp, then onecanapply anargument
+similar to that in Lemma 28 with any witaking the role of w1, 0≤i≤m. It follows
+that ifp=m+1, the set Fix NCm(W)(ψp) consists of the m+1 elements ( w0;w1,...,w m)
+obtained by choosing wi=cfor a particular ibetween 0 and m, all otherwj’s being
+equal toε. Ifp<m+1, then there is no element in Fix NCm(W)(ψp). /square
+Lemma 37. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
+assume that gcd(h1,m2) = 1. Suppose that Theorem 33has already been verified for
+all irreducible well-generated complex reflection groups w ith rank< n. Ifh2does not
+divide all degrees di, then equation (7.2)is satisfied.
+We leave the proof to the reader as it is completely analogous to the p roof of
+Lemma 30.
+Lemma 38. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
+assume that gcd(h1,m2) = 1. Ifm2>nthen
+FixNCm(W)(ψp) =∅.
+We leave the proof to the reader as it is analogous to the proof of Le mma 31.
+Remark 4.By applying the same reasoning as in Remark 3 with Lemmas 30 and 31
+replaced by Lemmas 37 and 38, respectively, it follows that we only ne ed to check (7.2)
+for pairs (m2,h2) satisfying (5.9) and m2≤n. This reduces the problem to a finite
+number of choices.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 31
+Lemma 39. LetWbe an irreducible well-generated complex reflection group o f rankn
+with the property that di|hfori= 1,2,...,n. Then Theorem 33is true for this group
+W.
+Proof.Proceeding in a fashion analogous to the beginning of the proof of Le mma 32, we
+mayrestricttothecasewhere p|(m+1)hand(m+1)h/pdoesnotdivideanyofthe di’s.
+Inthiscase, itfollowsfrom(8.4)andthefactthatwehave S1(W)⊇ {n}andS2(W) =∅
+that the right-hand side of (7.2) equals 0. Inspection of the classifi cation of all irre-
+ducible well-generated complex reflection groups, which we recalled in Section 2, reveals
+that all groups satisfying the hypotheses of the lemma have rank n≤2. Except for the
+groups contained in the infinite series G(d,1,n) andG(e,e,n) for which Theorem 2 has
+been established in [19], these are the groups G5,G6,G9,G10,G14,G17,G18,G21. The
+verification of (7.2) can be done in a similar fashion as in the proof of Le mma 32. We
+illustrate this by going through the case of the group G6. In analogy with the earlier
+situation, we note that Lemma 38 implies that equation (7.2) holds if m2>2, so that
+in the following arguments we may assume that m2= 2.
+CaseG6. The degrees are 4 ,12, and therefore, according to Remark 4, we need only
+consider the case where h2= 4 andm2= 2, that is, p= 3(m+1)/2. Then the action
+ofψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c2wm+1
+2c−2;c2wm+3
+2c−2,...,c2wmc−2,cw0c−1,...,cw m−1
+2c−1/parenrightbig
+.
+(8.5)
+If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1
+2such that
+ℓT(wi) = 1,wicwic−1=c, and allwj,j/ne}ationslash=i,m+1
+2+i, equalε. However, with the help of
+CHEVIE, one verifies that there is no such solution to this equation. Hence, the left-hand
+side of (7.2) is equal to 0, as required.
+This completes the proof of the lemma. /square
+9.Exemplification of case-by-case verification of Theorem 3 3
+It remains to verify Theorem 33 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
+G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
+details can be found in [21, Sec. 9]. We content ourselves with discuss ing the case of
+the groupG24, as this suffices to convey the flavour of the necessary computat ions.
+In order to simplify our considerations, it should be observed that t he action of ψ
+(given in(7.1)) is exactly the same as the actionof φ(given in (3.1)) with mreplaced by
+m+1on the components w1,w2,...,w m+1, that is, if we disregard the 0-th component
+of the elements of the generalised non-crossing partitions involved . The only difference
+which arises is that, while the ( m+ 1)-tuples ( w0;w1,...,w m) in (7.1) must satisfy
+w0w1···wm=c, forw1,w2,...,w m+1in (3.1) we only must have w1w2···wm+1≤Tc.
+Consequently, we may use the counting results from Section 6, exc ept that we have to
+restrict our attention to those elements ( w0;w1,...,w m,wm+1)∈NCm+1(W) for which
+w1w2···wm+1=c, or, equivalently, w0=ε.
+CaseG24.The degrees are 4 ,6,14, and hence we have
+Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
+[14]q[6]q[4]q.32 C. KRATTENTHALER AND T. W. M ¨ULLER
+Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (9.1a)
+lim
+q→ζCatm(G24;q) =7m+7
+3,ifζ=ζ6,ζ3,3|(m+1), (9.1b)
+lim
+q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (9.1c)
+lim
+q→ζCatm(G24;q) = 0,otherwise. (9.1d)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.1). The only cases not covered by Lemma 35 are the on es in (9.1b) and
+(9.1d). On the other hand, the only cases left to consider accordin g to Remark 4 are
+the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, andh2=m2= 2. These
+correspond to the choices p= 14(m+1)/3,p= 7(m+1)/3, respectively p= 7(m+1)/2.
+The first two cases belong to (9.1b), while p= 7(m+1)/2 belongs to (9.1d).
+In the case that p= 7(m+1)/3, the action of ψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c3w2m+2
+3c−3;c3w2m+5
+3c−3,...,c3wmc−3,c2w0c−2,...,c2w2m−1
+3c−2/parenrightbig
+.
+Hence, for an iwith 0≤i≤m−2
+3, we must find an element wi=t1, wheret1satisfies
+(6.4), so that we can set wi+m+1
+3=c2t1c−2,wi+2m+2
+3=c4t1c−4, and all other wj’s equal
+toε. We have found seven solutions to the counting problem (6.4), and e ach of them
+gives rise to ( m+1)/3 elements in Fix NCm(G24)(ψp) since the index iranges from 0 to
+(m−2)/3.
+On the other hand, if p= 14(m+1)/3, then the action of ψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c5wm+1
+3c−5;c5wm+4
+3c−5,...,c5wmc−5,c4w0c−4,...,c4wm−2
+3c−4/parenrightbig
+.
+By Lemma 29, every element of NC(W) is fixed under conjugation by c7, and, thus, the
+equations for t1in this case are the same as in the previous one where p= 7(m+1)/3.
+Hence, in either case, we obtain 7m+1
+3=7m+7
+3elements in Fix NCm(G24)(ψp), which
+agrees with the limit in (9.1b).
+Ifp= 7(m+ 1)/2, the relevant counting problem is (6.5). However, no element
+(w0;w1,...,w m)∈FixNCm(G24)(ψp) can be produced in this way since the counting
+problem imposes the restriction that ℓT(w0) +ℓT(w1) +···+ℓT(wm) be even, which
+contradicts the fact that ℓT(c) =n= 3. This is in agreement with the limit in (9.1d).
+Acknowledgements
+The authors thank an anonymous referee for a very careful rea ding of the original
+manuscript, and for numerous pertinent suggestions which have h elped to considerably
+improve the original manuscript.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 33
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1001.0032.txt

@@ -0,0 +1,527 @@
+arXiv:1001.0032v1  [astro-ph.SR]  30 Dec 2009Draft version November 15, 2018
+Preprint typeset using L ATEX style emulateapj v. 08/22/09
+ASTEROSEISMIC INVESTIGATION OF KNOWN PLANET HOSTS IN THE KEPLER FIELD
+J. Christensen-Dalsgaard1,2, H. Kjeldsen1,2, T. M. Brown3, R. L. Gilliland4, T. Arentoft1,2, S. Frandsen1,2,
+P.-O. Quirion1,2,5, W. J. Borucki6, D. Koch6, and J. M. Jenkins7
+Draft version November 15, 2018
+ABSTRACT
+In addition to its great potential for characterizing extra-solar p lanetary systems the Kepler mis-
+sionis providing unique data on stellar oscillations. A key aspect of Keplerasteroseismology is the
+application to solar-like oscillations of main-sequence stars. As an ex ample we here consider an ini-
+tial analysis of data for three stars in the Keplerfield for which planetary transits were known from
+ground-based observations. For one of these, HAT-P-7, we obt ain a detailed frequency spectrum and
+hence strong constraints on the stellar properties. The remaining two stars show definite evidence for
+solar-like oscillations, yielding a preliminary estimate of their mean dens ities.
+Subject headings: stars: fundamental parameters — stars: oscillations — planetary systems
+1.INTRODUCTION
+The main goal of the Kepler mission is to character-
+ize extra-solar planetary systems, particularly Earth-like
+planets in the habitable zone (e.g., Borucki et al. 2009).
+The mission detects the presence of planets through the
+minute reduction of the light from a star as a planet
+crosses the line of sight. Several observations of such
+reductions at fixed time intervals for a given star, and
+extensive follow-up observations, are used to verify that
+the effect results from planet transits and to characterize
+the planet. To ensure a reasonable chance of detection
+Keplerobserves more than 100,000 stars simultaneously,
+in a fixed field in the Cygnus-Lyra region. Most stars
+are observed at a cadence of 29.4 min, but a subset of
+up to 512 stars can be observed at a short cadence (SC)
+of 58.85s. Keplerwas launched on 6 March 2009 and
+data from the commissioning period and the first month
+of regular observations are now available.
+The very high photometric accuracy required to detect
+planet transits (Borucki et al. 2010; Koch et al. 2010)
+also makes the Keplerobservations of great interest for
+asteroseismic studies of stellar interiors. In particular,
+the SC data allow investigations of solar-like oscillations
+in main-sequence stars. Apart from the great astrophys-
+ical interest of such investigations they also provide pow-
+erful tools to characterize stars that host planetary sys-
+tems (Kjeldsen et al. 2009).
+In stars with effective temperature Teff<∼7000K we
+expect to see oscillations similar to those observed in the
+Sun (e.g., Christensen-Dalsgaard 2002), excited stochas-
+ticallybythe near-surfaceconvection. Theseareacoustic
+modes of high radial order; in main-sequence stars such
+1Department of Physics and Astronomy, Aarhus University,
+DK-8000 Aarhus C, Denmark: e-mail [email protected]
+2Danish AsteroSeismology Centre
+3Las Cumbres Observatory Global Telescope, Goleta, CA 93117
+4Space Telescope Science Institute, 3700 San Martin Drive, B al-
+timore, MD 21218
+5Canadian Space Agency, 6767 Route de l’A´ eroport, Saint-
+Hubert, QC, J3Y 8Y9 Canada (present address)
+6NASA Ames Research Center, MS 244-30, Moffett Field, CA
+94035, USA
+7SETI Institute/NASA Ames Research Center, MS244-30, Mof-
+fett Field, CA 94035, USAmodes approximately satisfy the asymptotic relation
+νnl≃∆ν0(n+l/2+ǫ)−l(l+1)D0 (1)
+(Vandakurov 1967; Tassoul 1980). Here νnlis the cyclic
+frequency, nis the radial order of the mode and lis
+the degree, l= 0 corresponding to radial (i.e., spher-
+ically symmetric) oscillations. Also, ∆ ν0is essentially
+the inverse sound travel time across the stellar diameter;
+this is closely related to the mean stellar density ∝angbracketleftρ∗∝angbracketright:
+∆ν0∝ ∝angbracketleftρ∗∝angbracketright1/2.D0depends sensitively on conditions
+near the center of the star; for stars during the central
+hydrogenburningphasethisprovidesameasureofstellar
+age. Finally, ǫis determined by conditions near the stel-
+lar surface. This regular form of the frequency spectrum
+simplifies the analysis of the observations, and the close
+relation between the stellar properties and the param-
+eters characterizing the frequencies make them efficient
+diagnostics of the properties of the star. This has been
+demonstrated in the last few years through observations
+of solar-like oscillations from the ground and from space
+(for reviews, see Bedding & Kjeldsen 2008; Aerts et al.
+2009; Gilliland et al. 2010a).
+Even observations allowing a determination of ∆ ν0
+provide useful constraints on ∝angbracketleftρ∗∝angbracketright. With a reliable de-
+termination of individual frequencies ∝angbracketleftρ∗∝angbracketrightis tightly con-
+strained and an estimate of the stellar age can be ob-
+tained. This can greatly aid the interpretation of obser-
+vations of planetary transits (e.g., Gilliland et al. 2010b;
+Nutzman et al. 2010). We note that photometric obser-
+vations such as those carried out by Keplerare predom-
+inantly sensitive to modes of degree l= 0−2. As indi-
+catedbyEq.(1)thesearesufficienttoobtaininformation
+about the core properties of the star.
+Ground-based transit observations have identified
+three planetary systems in the Keplerfield: TrES-2
+(O’Donovan et al. 2006; Sozzetti et al. 2007), HAT-P-7
+(P´ al et al. 2008), and HAT-P-11 (Dittmann et al. 2009;
+Bakos et al. 2010). These systems have been observed
+byKeplerin SC mode. Their properties (cf. Table 1)
+indicate that they should display solar-like oscillations
+at observable amplitudes, and hence they are obvious
+targets for Keplerasteroseismology. Here we report the
+results of a preliminary asteroseismic characterization of2 Christensen-Dalsgaard et al.
+TABLE 1
+Properties of transiting systems.
+Name KIC No Teff(K) [Fe/H] L/L⊙log(g) (cgs) vsiniSource
+(kms−1)
+HAT-P-7 10666592 6350 ±80 0.26±0.08 4 .9±1.1 4.07±0.06 3.8±0.5 (a)
+6525±61 0.31±0.07 4 .09±0.08 (b)
+HAT-P-11 10748390 4780 ±50 0.31±0.05 0.26±0.02 4.59±0.03 1.5±1.5 (c)
+TrES-2 11446443 5850 ±50−0.15±0.10 1.17±0.10 4.4±0.1 2 ±1 (d)
+5795±73 0.06±0.08 4 .30±0.13 (b)
+Note. — Sources: (a): P´ al et al. (2008); (b): Ammler-von Eif et al . (2009); (c): Bakos et al. (2010); (d): Sozzetti et al. (2007 ). In some
+cases asymmetric error bars have been symmetrized.
+the central stars in the systems, based on the early Ke-
+plerdata.
+2.OBSERVATIONS AND DATA ANALYSIS
+We have analyzed data from Kepler for three
+planet-hosting stars using a pipeline developed for
+fast and robust analysis of all Keplerp-mode data
+(Christensen-Dalsgaard et al. 2008; Huber et al. 2009).
+Each time series contains 63324 data points. SC data
+characteristics and minor post-pipeline processing are
+discussed in Gilliland et al. (2010c). In addition a limb-
+darkened transit light curve model fit has been removed
+and 5-σclipping applied to remove outlying data points
+from each of the time series. The frequency analysis con-
+tains four main steps:
+1. We calculate an oversampled (factor of four) ver-
+sion ofthe power spectrum by using a least-squares
+fitting. We smoothed the spectrum to 3 µHz reso-
+lution to remove the fine structure caused by the
+finite mode lifetime.
+2. We correlated the smoothed power spectrum with
+an equally spaced comb of delta functions, sepa-
+ratedby∆ ν0/2,andconfinedtoaGaussian-shaped
+band with a full width at half maximum of 5∆ ν0.
+We adopted the maximum of this convolution over
+lags between 0 and 0.5 ∆ ν0as the filter output for
+each ∆ν0.
+3. After identifying the peak correlation for the best
+matched model filter and extracting the large sep-
+aration corresponding to this peak we calculate the
+folded spectrum (see Fig. 1b), i.e., the sum of the
+power as a function of frequency modulo the opti-
+mumlargeseparation(theonecorrespondingtothe
+peak correlation). The summed power is used to
+locate the p-mode structure and identify the ridges
+corresponding to the different mode degrees (based
+on the asymptotic relation).
+4. From the asymptoticrelationandthe identification
+of mode degrees we finally identify the position of
+the individual p-mode frequencies in the smoothed
+version of the power spectrum; when more than
+one mode is seen near the expected frequency we
+use the power-weighted average of the two peaks.
+Those extracted frequencies and the mode identifi-
+cations are used in the modeling.
+For observations with low signal-to-noise ratio it may
+not be possible to identify the individual frequencies. In       0        1        2
+Fig. 1.— (a)PowerspectrumofHAT-P-7forfrequencies between
+300 and 3000 µHz. The spectrum is smoothed with a gaussian filter
+with a FWHM of 3 µHz. The noise level at high frequencies corre-
+sponds to 1.1 ppm in amplitude. The white curve is a smoothed
+power spectrum with a gaussian filter (150 µHz FWHM). A fit to
+the background (dashed white curve) is also shown. The exces s
+power and the individual p-modes are evident. (b) Folded pow er
+spectrum, between 750 and 1500 µHz, for HAT-P-7 for a large sep-
+aration of 59 .22µHz. Indicated are the positions corresponding to
+radial modes ( l= 0) and non-radial modes with l= 1 and 2. The
+measured positions are used to identify the individual osci llation
+modes in panel (a). (c) ´Echelle diagram (see text) for frequencies
+of degree l= 0, 1, and 2 in HAT-P-7; a frequency separation of
+59.36µHz and a starting frequency of 10 .8µHz were used. The
+filled symbols, coded for degree as indicated, show the obser ved
+frequencies, while the open symbols are for Model 3 in Table 2 ,
+minimizing χ2
+ν.3
+such cases the analysis is carried through step 2, to de-
+termine the maximum response and hence an estimate of
+the large separation.
+Results on the three individual cases are presented in
+§4.
+3.MODEL FITTING
+Stellar evolution models and adiabatic oscillation
+frequencies were computed using the Aarhus codes
+(Christensen-Dalsgaard 2008a,b), with the OPAL
+equation of state (Rogers et al. 1996) and opacity
+(Iglesias & Rogers 1996) and the NACRE nuclear reac-
+tion parameters (Angulo et al. 1999). In some cases (see
+below) diffusion and settling of helium were included,
+using the simplified formulation of Michaud & Proffitt
+(1993). Convection was treated with the B¨ ohm-Vitense
+(1958) mixing-length formulation, with a mixing length
+αML= 2.00 in units of the pressure scale height roughly
+corresponding to a solar calibration. In some models
+with convective cores, overshoot was included over a dis-
+tance of αovpressure scale heights. Evolution started
+from chemically homogeneous zero-age models. The ini-
+tial abundances by mass X0andZ0of hydrogen and
+heavy elements were characterized by the assumed value
+of [Fe/H], using as reference a present solar surface com-
+position with Zs/Xs= 0.0245 (Grevesse & Noels 1993)
+and assuming, from galactic chemical evolution, that
+X0= 0.7679−3Z0.
+From the observed ∆ ν0, effective temperature and
+composition an initial estimate of the stellar parame-
+ters was obtained using the grid-based SEEK pipeline
+(Quirion et al., in preparation). Smaller grids were then
+computed in the vicinity of these initial parameters, to
+obtaintighterconstraintsonstellarproperties. ForHAT-
+P-7 the analysis of the observations yielded frequencies
+of individually identified modes; here the analysis was
+based on
+χ2
+ν=1
+N−1/summationdisplay
+nl/parenleftBigg
+ν(obs)
+nl−ν(mod)
+nl
+σν/parenrightBigg2
+,(2)
+whereν(obs)
+nlandν(mod)
+nlare the observed and model fre-
+quencies, σνis the standard error in the observed fre-
+quencies (assumed to be constant) and Nis the num-
+ber of observed frequencies. In addition, we considered
+χ2=χ2
+ν+χ2
+T, whereχ2
+Tis the corresponding normalized
+square difference between the observed and model effec-
+tive temperature. When χ2
+νwas available we minimized
+it along each evolution track and considered the result-
+ing minimum values, and the corresponding value of χ2,
+as a function of the parameters characterizing the mod-
+els (see Gilliland et al. 2010b, for details). When only
+the large separation ∆ ν0could be determined from the
+observations, we identified the model along each track
+which matched ∆ ν0and considered the resulting χ2
+Tas
+a function of the model parameters.
+4.RESULTS
+4.1.HAT-P-7
+The observed power spectrum for HAT-P-7 is shown
+in Fig. 1a. The presence of solar-like p-mode peaks, with
+a maximum power around 1.1mHz, is evident. At high
+frequency the noise level in the amplitude spectrum is1.1 parts per million (ppm), with some increase at lower
+frequency, likely due to the effects of stellar granulation.
+Carrying out the correlation analysis described in §2
+we determined the large separation as ∆ ν0= 59.22µHz.
+Figure 1b shows the resulting folded spectrum. This
+clearly shows two closely spaced peaks, identified as cor-
+responding to modes of degree l= 0 and 2, and single
+peak separated from these two by approximately ∆ ν0/2,
+corresponding to l= 1. On this basis we finally deter-
+mined the individual frequencies, identifying the modes
+from the asymptotic relation; the final set includes 33
+p-mode frequencies, determined with a standard error
+σν= 1.4µHz. These frequencies, corresponding to ra-
+dial orders between 11 and 24, are illustrated in Fig. 1c
+in an ´ echelle diagram (see below).
+A grid of models was computed for masses between
+1.41 and 1 .61M⊙, [Fe/H] between 0.17 and 0.38, and
+αov= 0,0.1 and 0.2, extending well beyond the end of
+central hydrogen burning. The modeling did not include
+diffusion and settling. At the mass of this star the outer
+convection zone is quite thin, and as a result the set-
+tling timescale is much shorter than the age of the star.
+Including settling, without compensating effects such as
+partial mixing in the radiative region or mass loss, leads
+to a rapid change in the surface composition which is
+inconsistent with the observed [Fe/H]; for simplicity we
+therefore neglected these effects for HAT-P-7.8
+The computed frequencies were corrected according to
+the procedure of Kjeldsen et al. (2008) for errors in the
+modeling of the near-surface layers, by adding a(ν/ν0)b
+wherea= 0.1158µHz,ν0= 1000µHz andb= 4.9. As
+discussed in §3, for each evolution track, characterized
+by a set of model parameters, we minimized the depar-
+tureχ2
+νof the model frequencies from the observations,
+defining the best model for this set.
+We first consider χ2
+νas a function of the effective
+temperature of the models (Fig. 2a). It is evident
+that there is a clear minimum in χ2
+ν; this is consistent
+with the determination of Teffby P´ al et al. (2008) but
+not with the somewhat higher temperature obtained by
+Ammler-von Eif et al. (2009) (see also Table 1). Thus
+in the following we use the observed quantities from
+P´ al et al. (2008).
+Since the frequencies to leading order are determined
+by the mean stellar density ∝angbracketleftρ∗∝angbracketright, Fig. 2b,c show χ2
+νand
+χ2as functions of ∝angbracketleftρ∗∝angbracketright. It is evident that the best-fitting
+modelsoccupyanarrowrangeof ∝angbracketleftρ∗∝angbracketright, withawell-defined
+minimum. Fittingaparabolato χ2inpanel(c)weobtain
+the estimate ∝angbracketleftρ∗∝angbracketright= 0.2712±0.0032gcm−1. In Fig. 2d
+χ2is shown against model age. Here the variation with
+model parameters is substantially stronger, resulting in
+a greater spread in the inferred age; in particular, it is
+evident, not surprisingly, that the results depend on the
+extent of convective overshoot. From the figure we esti-
+matethattheageofHAT-P-7isbetween1.4and2.3Gyr.
+Examples of evolution tracks are shown in Fig. 3; pa-
+rameters for these models are provided in Table 2. They
+were chosen to give the smallest χ2
+νfor each of the three
+values of αovconsidered. Also shown are the locations
+8Artificially suppressing settling in the outer layers, whil e in-
+cluding diffusion and settling in the core, leads to results t hat are
+very similar to those presented here.4 Christensen-Dalsgaard et al.
+TABLE 2
+Stellar evolution models fitting the observed frequencies for HAT-P-7.
+No M ∗/M⊙Age Z0X0αovR∗/R⊙/angbracketleftρ∗/angbracketrightTeffL∗/L⊙χ2
+νχ2
+(Gyr) (gcm−3) (K)
+1 1.53 1.758 0.0270 0.6870 0.0 1.994 0.2718 6379 5.91 1.08 1.2 1
+2 1.52 1.875 0.0290 0.6809 0.1 1.992 0.2708 6355 5.81 1.04 1.0 4
+3 1.50 2.009 0.0270 0.6870 0.2 1.981 0.2718 6389 5.87 1.00 1.2 4
+Note. — Models minimizing χ2
+ν(cf. Eq. 2) along the evolution tracks, illustrated in Fig. 3 . The models have been selected as providing
+the smallest χ2
+νfor each of the three values of the overshoot parameter αov. The smallest value of χ2
+νis obtained for Model 3.
+Fig. 2.— Results of fitting the observed frequencies to a grid of
+stellar models (see text for details). Plusses, stars and di amonds
+correspond to modelswith αov= 0 (no overshoot), 0.1, and 0.2. (a)
+Minimum mean square deviation χ2
+νof the frequencies (cf. Eq. 2)
+along each evolution track, against the effective temperatu reTeff
+of the corresponding models. The vertical dashed and dotted lines
+indicate the effective temperatures found by P´ al et al. (200 8) and
+Ammler-von Eif et al. (2009). (b) Minimum mean square devia-
+tionχ2
+νagainst the mean density /angbracketleftρ∗/angbracketrightof the corresponding models.
+(c) As (b), but showing the combined χ2. (d)χ2against the age
+for the models that minimize χ2
+ν; the different ridges correspond to
+the different masses in the grid, the more massive models resu lting
+in a lower estimate of the age.Fig. 3.— Theoretical HR diagram with selected evolutionary
+tracks, corresponding to the models defined in Table 2. The ’+ ’ in-
+dicate the models along the full set of evolutionary sequenc es mini-
+mizing the difference between the computed and observed freq uen-
+cies. The box is centered on the LandTeffas given by P´ al et al.
+(2008), with a size matching the errors on these quantities.
+of the models minimizing χ2
+νalong each of the computed
+tracks; these evidently fall close to a line in the HR di-
+agram, corresponding to the small range in ∝angbracketleftρ∗∝angbracketright. The
+range of luminosities, from P´ al et al. (2008), is based on
+modeling and hence has not been used in our fit; even
+so, it is gratifying that the present models are essentially
+consistentwiththesevalues. Also,asindicatedbyFig.2a
+andTable 2, the best-fitting models areclose to the value
+ofTeffobtained by P´ al et al. (2008).
+The match of the best-fitting model (Model 3 of Ta-
+ble 2) to the observed frequencies is illustrated in a so-
+called´ echelle diagram (Grec et al. 1983) in Fig. 1c. In
+accordance with Eq. (1) the frequency spectrum is di-
+vided into slices of length ∆ ν, starting at a frequency of
+10.8µHz; the figure shows the location of the observed
+(filled symbols) and computed (open symbols) frequen-
+cies within each slice, against the starting frequency of
+the slice; the model results extend to the acoustical cut-
+off frequency, 1930 µHz, of the model. There is clearly a
+very good overall agreement between model and obser-
+vations, including the detailed variation with frequency
+which reflects the frequency dependence of the large sep-
+aration, as a possible diagnostics of the outer layers of
+the star (e.g., Houdek & Gough 2007).
+We have finally made a fit of the inferred ∝angbracketleftρ∗∝angbracketright, as
+well asTeffand [Fe/H] from P´ al et al. (2008), to com-
+puted evolutionary tracks from the Yonsei-Yale compi-
+lation (Yi et al. 2001). This was based on a Markov
+Chain Monte Carlo analysis to obtain the statistical
+properties of the inferred quantities (see Brown 2010,
+for details). This resulted in M= 1.520±0.036M⊙,
+R= 1.991±0.018R⊙and an age of 2 .14±0.26Gyr.
+We note that the age estimate reflects the specific as-5
+sumptions in the Yonsei-Yale evolution calculations; as
+indicated by Fig. 2d the true uncertainty in the age de-
+termination is likely somewhat larger.
+4.2.HAT-P-11
+For HAT-P-11 the oscillation amplitudes were much
+smaller than in HAT-P-7, as expected from the general
+scaling of amplitudes with stellar mass and luminosity
+(e.g., Kjeldsen & Bedding 1995). Thus with the present
+short run of data it has only been possible to determine
+thelargeseparation∆ ν0= 180.1µHzfromthemaximum
+in the correlation analysis. We have matched this to a
+grid of models, including diffusion and settling of helium,
+with masses between 0.7 and 0 .9M⊙and [Fe/H] between
+0.21 and 0.41. These models provide a good fit to the
+observed TeffandL/L⊙; note that in the presentcase the
+luminosity is based on a reasonably well-determined par-
+allax. We havedetermined an estimateof ∝angbracketleftρ∗∝angbracketrightbyaverag-
+ing the results of those models which match the observed
+∆ν0and lie within 2 standard deviations ( ±100K) from
+the value of Teffprovided by Bakos et al. (2010); the re-
+sult is∝angbracketleftρ∗∝angbracketright= 2.5127±0.0009gcm−3. Although the for-
+mal error is extremely small, owing to a tight relation
+between the large separation and the mean density for
+stars in this region in the HR diagram, the true error is
+undoubtedly substantially larger. In particular, we ne-
+glected the error in the determination of ∆ ν0and these
+data have not allowed a correction for the systematic
+errors in the modeling of the near-surface layers of the
+star.
+4.3.TrES-2
+Here also we were unable to determine individual fre-
+quencies from the present set of data. The expected am-
+plitudes are smaller than for HAT-P-7, and the noise
+level higher due to the fainter magnitude of TrES-2.
+The correlation analysis yielded two possible values of
+∆ν0: 97.7µHz and 130 .7µHz. For this star ∝angbracketleftρ∗∝angbracketrighthas
+been determined from the analysis of the transit light
+curve. Sozzetti et al. (2007) obtained ∝angbracketleftρ∗∝angbracketright= 1.375±
+0.065gcm−3, while Southworth (2009) found ∝angbracketleftρ∗∝angbracketright=
+1.42±0.13gcm−3. From the scaling with ∝angbracketleftρ∗∝angbracketright1/2thesmaller of the two possible values of ∆ ν0is clearly incon-
+sistent with these values of ∝angbracketleftρ∗∝angbracketright, while ∆ ν0= 130.7µHz
+yields models that are consistent with the observed Teff
+and log(g) of Sozzetti et al. (2007) as well as with these
+values of the mean density. Here we considered a grid
+of models with helium diffusion and settling, masses be-
+tween 0.85 and 1 .1M⊙and [Fe/H] between −0.25 and
+−0.05. Determining again the mean value of ∝angbracketleftρ∗∝angbracketrightfor
+those models that matched ∆ ν0and had Teffwithin two
+standard deviations of the value of Sozzetti et al. (2007)
+we obtained ∝angbracketleftρ∗∝angbracketright= 1.3233±0.0027gcm−3. As in the
+case of HAT-P-11 the true error is likely substantially
+higher.
+5.DISCUSSION AND CONCLUSION
+The present preliminary analysis provides a striking
+demonstration of the potential of Keplerasteroseismol-
+ogyanditssupportingroleintheanalysisofplanethosts.
+Thesestarswill undoubtedly be observedthroughout the
+mission and hence the quality of the data will increase
+substantially. For HAT-P-7 the detected frequencies are
+already close to what will be required for a detailed anal-
+ysis of the stellar interior, beyond the determination of
+the basic parameters of the star. Thus here we can look
+forward to a test of the assumptions of the stellar mod-
+eling; the resulting improvements will further constrain
+the overall properties of the star, in particular its age.
+Also, given the observed vsiniwe expect a rotational
+splitting comparable to that observed in the Sun, and
+hence likely detectable with a few months of observa-
+tions. For the other two stars there is strong evidence
+for the presence of solar-like oscillations; thus continued
+observations will very likely result in the determination
+of individual frequencies and hence further constraints
+on the properties of the stars.
+Funding for this Discovery mission is provided by
+NASA’s Science Mission Directorate. We are very grate-
+ful to the entire Keplerteam, whose efforts have led to
+this exceptional mission. The present work was sup-
+ported by the Danish Natural Science Research Council.
+Facilities: The Kepler Mission
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1001.0033.txt

@@ -0,0 +1,1148 @@
+arXiv:1001.0033v1  [astro-ph.SR]  30 Dec 2009WIYN OPEN CLUSTER STUDY. XXXVIII. STELLAR RADIAL VELOCITIE S IN THE YOUNG OPEN
+CLUSTER M35 (NGC 2168)
+Aaron M. Geller∗, Robert D. Mathieu∗, Ella K. Braden∗, Søren Meibom∗,†
+Department of Astronomy, University of Wisconsin - Madison , WI 53706, USA
+and
+Imants Platais
+Department of Physics and Astronomy, The Johns Hopkins Univ ersity, Baltimore, MD 21218, USA
+and
+Christopher J. Dolan∗
+Department of Astronomy, University of Wisconsin - Madison , WI 53706, USA
+ABSTRACT
+We present 5201 radial-velocity measurements of 1144 stars, as p art of an ongoing study of the
+young (150 Myr) open cluster M35 (NGC 2168). We have observed M 35 since 1997, using the Hydra
+Multi-Object Spectrograph on the WIYN 3.5m telescope. Our stellar sample covers main-sequence
+stars over a magnitude range of 13.0 ≤V≤16.5 (1.6 - 0.8 M ⊙) and extends spatially to a radius of
+30 arcminutes (7 pc in projection at a distance of 805 pc or ∼4 core radii). Due to its youth, M35
+provides a sample of late-type stars with a range of rotation period s. Therefore, we analyze the radial-
+velocity measurement precision as a function of the projected rot ational velocity. For narrow-lined
+stars (vsini≤10 km s−1), the radial velocities have a precision of 0.5 km s−1, which degrades to 1.0
+km s−1for stars with vsini= 50 km s−1. The radial-velocitydistribution shows a well-defined cluster
+peak with a central velocity of -8.16 ±0.05 km s−1, permitting a clean separation of the cluster and
+field stars. For stars with ≥3 measurements, we derive radial-velocity membership probabilities a nd
+identify radial-velocity variables, finding 360 cluster members, 55 of which show significant radial-
+velocity variability. Using these cluster members, we construct a co lor-magnitude diagram for our
+stellar sample cleaned of field star contamination. We also compare th e spatial distribution of the
+single and binary cluster members, finding no evidence for mass segr egation in our stellar sample.
+Accounting for measurement precision, we place an upper limit on the radial-velocity dispersion of
+the cluster of 0 .81±0.08 km s−1. After correction for undetected binaries, we derive a true radia l-
+velocity dispersion of 0 .65±0.10 km s−1.
+(galaxy:) open clusters and associations: individual (NGC 2168) - (s tars:) binaries: spectroscopic
+1.INTRODUCTION
+Young open clusters are laboratories for the direct
+study of the near-primordial characteristics of stellar
+populations. Their properties, and particularly those of
+the binary systems, offer unique insights into how stars
+arebornand provideessentialguidancefor N-bodystud-
+ies of star clusters. Indeed, with sophisticated N-body
+simulations now able to model real open clusters (e.g.,
+Hurley et al. 2005), knowledge of the correct initial con-
+ditions are all the more important. In particular, the ini-
+tial binary population has a vast impact on the dynami-
+cal evolution of the cluster, and the characteristics of the
+initial binary population will affect the overallfrequency,
+formation rate and formation mechanisms of anomalous
+stars, like blue stragglers, as interactions with binaries
+are thought to be catalysts for the formation of these ex-
+otic objects (Hurley et al. 2005; Knigge et al. 2009). As
+a rich open cluster with an age of ∼150 Myr, M35 is a
+prime cluster to define these hitherto poorly known ini-
+tial conditions for the binary population required for any
+∗Visiting Astronomer, Kitt Peak National Observatory, Nati onal
+Optical Astronomy Observatory, which is operated by the Ass o-
+ciation of Universities for Research in Astronomy (AURA) un der
+cooperative agreement with the National Science Foundatio n.
+†Current address: Harvard-Smithsonian Center for Astrophy sics,
+60 Garden Street, Cambridge, MA 02138, USAopen cluster simulation.
+M35 is a fundamental cluster in the WIYN Open Clus-
+ter Study (WOCS; Mathieu 2000), and as such has a
+strong base of astrometric and photometric observations
+fromboththeWOCScollaborationandothers. Theclus-
+ter is centered at α= 6h09m07.s5 andδ= +24◦20′28′′
+(J2000),towardstheGalacticanticenter. Numerouspho-
+tometric studies have identified the rich main-sequence
+population (e.g., Kalirai et al. 2003; von Hippel et al.
+2002; Sung & Bessell 1999). WOCS CCD photometry
+places the cluster at a distance of 805 ±40 pc, with an
+ageof150 ±25Myr, ametallicityof[Fe/H]=-0.18 ±0.05
+and a reddening of E(B−V)=0.20±0.01 (C. Deliyan-
+nis, private communication). The most recent published
+parameters, from Kalirai et al. (2003), place the cluster
+at a distance of 912+70
+−65pc ((m−M)0= 9.80±0.16)
+with an age of 180 Myr, adopting a E(B−V)=0.20 and
+[Fe/H] = -0.21. (See Kalirai et al. (2003) for a thorough
+review of previous photometry references and their de-
+rived cluster parameters). We note that these two recent
+studies used different isochrone families.
+There have been multiple proper-motion studies
+of the cluster (Ebbighausen 1942; Cudworth 1971;
+McNamara & Sekiguchi 1986a), although none deter-
+mine clustermembership for individual starsfainter than2 Geller et al.
+V≈15.0. Using proper motions, Leonard & Merritt
+(1989) derive a cluster mass from 1600-3200 M ⊙within
+3.75 pc. Detailed observations have also been made
+in M35 to study tidal evolution in binary stars
+(Meibom & Mathieu 2005; Meibom et al. 2006, 2007),
+lithium abundances (Steinhauer & Deliyannis 2004;
+Barrado y Navascu´ es et al. 2001), and white dwarfs
+(Reimers & Koester 1988; Williams et al. 2004, 2006,
+2009).
+This is the first paper in a series studying the dy-
+namical state of M35 through the use of radial-velocity
+(RV) measurements. The data and results presented
+in this series will form the largest database of spec-
+troscopic cluster membership and variability in M35 to
+date. In this paper, we present results from our ongo-
+ing radial-velocity study of the cluster, which we began
+in September 1997. Our stellar sample includes solar-
+type main-sequence stars within the magnitude range of
+13.0≤V≤16.5, which corresponds to a mass range1of 1.6
+- 0.8 M ⊙. The main-sequence turnoff is at V∼9.5,∼4
+M⊙. In Section 2, we describe this stellar sample, ob-
+servations and data reduction in detail. We thoroughly
+investigate our RV measurement precision and the effect
+of stellar rotation in Section 3. Then in Section 4 we de-
+rive RV membership probabilities, and use our study of
+the RV precision to identify RV variables, which we as-
+sume to be binaries or higher-order systems. Within this
+mass range, we identify 360 solar-type main-sequence
+members; 305 are single2(non-RV-variable) stars while
+55 show significant RV variability. We then use these re-
+sults to plot a color-magnitude diagram (CMD) cleaned
+offield starcontamination, tosearchfor evidenceofmass
+segregation and to study the cluster RV dispersion (Sec-
+tion5). Finally, inSection6, weprovideabriefsummary.
+In future papers, we will study the binary population of
+M35in detail, providingobservationsthat will be used to
+directly constrain the initial binary population of open
+cluster simulations.
+2.OBSERVATIONS AND DATA REDUCTION
+In the following section, we define our stellar sample,
+provide a detailed description of our observations and
+data reduction process, and discuss the completeness of
+our spectroscopic observations.
+2.1.Photometric Target Selection
+Initially, we created our M35 target list from the stars
+in three wide-field CCD images centered on M35, taken
+by T. von Hippel with the Kitt Peak National Observa-
+tory (KPNO) Burrell Schmidt telescope on November 18
+and 19, 1993. These images have Vexposures of 4 s, 20 s
+and 180 s and Bexposures of 4 s, 25 s and 240s covering
+a 70′×70′field. We obtained BandVphotometry with
+a limiting magnitude of V= 17, denoted as source 1 in
+1This mass range is derived from a 180 Myr Padova isochrone
+(Marigo et al. 2008) using the distance, reddening and metal licity
+from Kalirai et al. (2003).
+2In the following, we use the term “single” to identify stars
+with no significant RV variation. Certainly, many of these st ars
+are also binaries, although generally with longer periods a nd/or
+lower mass ratios ( q=m2/m1) than the binaries identified in this
+study. When applicable, we have attempted to reduce this bin ary
+contamination amongst the single star sample by photometri cally
+identifying objects as binaries that lie well above the sing le-star
+main-sequence (see Section 4.2).Fig. 1.— Color-magnitude diagram for stars in the field
+of M35 highlighting the selected region used in this survey.
+We plot all stars in the field with the gray points to show
+the location of our selected sample relative to the full clus -
+ter. Our stellar sample is bounded by the solid black lines.
+Within this region, we plot observed stars in the solid black
+points. Additionally, for reference we plot a 180 Myr Padova
+isochrone (Marigo et al. 2008) using the distance, reddenin g
+and metallicity from Kalirai et al. (2003)
+Table 3. Additionally, we derive astrometry from these
+plates, tied to the Tycho catalogue.
+More recently, we added to our database the BVpho-
+tometry of Deliyannis (private communication), taken
+on the WIYN30.9m telescope with the S2KB 2K by
+2K CCD. This photometry derives from a mosaic of five
+fields. Each field has a 20′×20′field-of-view, with one
+central field and four tiled around the center, for a total
+field-of-view of of 40′×40′. This photometry is denoted
+as source 2 in Table 3, and covers 74% of the objects
+we have observed in this study. We note that this pho-
+tometry is more precise than that of source 1. The star-
+by-star difference in Vmagnitudes for the two sources is
+roughly Gaussian with σ= 0.06 mag. However there is
+a tail that extends beyond three times this sigma value.
+Therefore we caution the reader when using magnitudes
+from source 1.
+We selected stars for the RV master list based on three
+constraints. The faintest sources that can be observed
+efficiently at echelle resolution using the Hydra Multi-
+Object Spectrograph (MOS) on the WIYN 3.5m have
+V=16.5; this therefore sets our faint limit for observa-
+tions. Stars bluer than ( B−V)∼0.6 ((B−V)0∼0.4)
+do not provide precise RV measurementsdue to rapid ro-
+tationandpaucityofspectrallines; thisthereforesetsour
+blue limit for observations. Finally, we perform a photo-
+metric selection of cluster member candidates, shown as
+the outlined region4in Figure 1. This region includes a
+wide swath above the main sequence so as to not select
+against binary stars (e.g. Dabrowski & Beardsley 1977),
+yet also removes stars that are very likely cluster non-
+members. This photometric selection allows for an effi-
+cient survey of the cluster. Our sample extends radially
+to 30 arcminutes from the cluster center. At a distance
+of 805 pc, this corresponds to the inner ∼7 pc of the
+3The WIYN Observatory is a joint facility of the University of
+Wisconsin-Madison, Indiana University, Yale University, and the
+National Optical Astronomy Observatories.
+4Specifically, we select stars with 0 .6<(B−V)<1.5, 13.0<
+V <16.5 and between the lines defined by 5 .7(B−V)+8.6< V <
+5.7(B−V)+11.0.WOCS. RV Measurements in M35 3
+Fig. 2.— Completeness of our observations as a function of Vmagnitude (left) and projected radius (right). We plot the
+completeness in stars observed ≥3 times with the dashed line, and stars observed ≥1 time with the solid line.
+cluster in projection. Given the core radius derived by
+Mathieu (1983) of1.9 ±0.1pc, oursample isdrawn from
+the inner ∼4 core radii.
+Note that we lack BVphotometry for ∼11% of the
+point sources found within 30 arcminutes from the clus-
+tercenterin2MASS.Formostoftheseobjects, itislikely
+that there is a nearby or overlapping additional object
+which has prevented accurate BVphotometric measure-
+ments from either of our sources. These objects, by de-
+fault, are not included in our stellar sample. In total, our
+stellar sample contains 1344 stars.
+2.2.Spectroscopic Observations
+Since September 1997, we have collected 5201 spec-
+tra of 1144 stars within this stellar sample as part of
+an ongoing observing program using the WIYN Hydra
+MOS. For the majority of these observations, we use Hy-
+dra’s blue-sensitive 300 µm fibers, which project to a 3.1′
+aperture on the sky. We use the 316 lines mm−1echelle
+grating, isolating the 11th order with the X14 filter. The
+resulting spectra span a wavelength range of ∼25 nm,
+with a dispersion of 0.015 nm pixel−1, centered on 512.5
+nm. We have also occasionally centered our observation
+on 637.5 nm using a very similar setup. In this region,
+we use the same grating, but isolate the 9th order with
+the X18 filter. These observations span a slightly larger
+wavelength range of ∼30 nm, and have a dispersion of
+0.017 nm pixel−1. Due to a broken filter, observations
+taken after the spring of 2008 use different observing se-
+tups than discussed above; most are centered on 560 nm
+and all use the echelle grating. We have not noticed any
+decrease in performance from the new wavelength range,
+but we caution the reader that we lack sufficient obser-
+vations in these setups to reliably determine our RV pre-
+cision for these measurements. During this same period
+certain upgrades were made to the spectrograph collima-
+tor5. All observed regions are rich in metal lines. The
+typical velocity resolution is 15 km s−1. In a two-hour
+integration, the spectra have signal-to-noise (S/N) ra-
+tios ranging from ∼18 per resolution element for V=16.5
+stars to∼100 per resolution element for V=13 stars.
+We create fiber configurations ( pointings ) for our ob-
+servations using a similar method as Geller et al. (2008).
+Monte Carlo simulations show that we require at least
+5http://www.astro.wisc.edu/ ∼mab/research/bench upgrade/threeobservationsoverthecourseofayearinordertoen-
+sure 90% confidence that a star is either constant or vari-
+able in RV out to binary periods of 1000 days (Mathieu
+1983, Geller & Mathieu, in preparation). Given three
+observations with consistent RV measurements over a
+timespan of at least a year and typically longer, we clas-
+sify a given star as single (strictly, non-RV variable) and
+finished, andmoveittothelowestpriority. Ifagivenstar
+has three RV measurements with a standard deviation
+>2.0 km s−1(four times our precision for narrow-lined
+stars; see Section 4.2), we classify the star as RV vari-
+able and give it the highest priority for observation on a
+schedule appropriate to its timescale of variability. This
+prioritization allows us to most efficiently derive orbital
+solutions for our detected binaries.
+We place our shortest-period binaries at the highest
+priority for observations each night, followed by longer-
+period binaries to obtain 1-2 observations per run. Be-
+low the confirmed binaries we place, in the following or-
+der, “candidate binaries” (once-observedstars with a RV
+measurement outside the cluster RV distribution or stars
+with a few measurements that span only 1.5 - 2.5 km
+s−1), once observed and then twice observed non-RV-
+variable likely members, twice observed non-RV-variable
+likely non-members, unobserved stars, and finally, “fin-
+ished” stars. Within each group, we prioritize by dis-
+tance from the cluster center, giving those stars nearest
+to the center the highest priority. A typical pointing
+will contain ∼70 fibers placed on individual stars in our
+sample and ∼10 sky fibers.
+For a given pointing we obtain three consecutive ex-
+posures, each of 40 minutes. In poor transparency or
+with a particularly bright sky, we restrict the targets
+toV <15.0 and shorten the integration time, gener-
+ally to 20 minute exposures. We obtain Thorium-Argon
+(ThAr), oroccasionallyCopper-Argon(CuAr), emission-
+lamp comparison spectra (300 s integrations) before and
+after each set of science integrations for wavelength cal-
+ibration and to check for wavelength shifts during the
+observing sequence. For each set of integrations we also
+obtain one flat-field image (200 s) of a white spot on
+the dome illuminated by incandescent lights. Associat-
+ing the flat-field images with the science integrations is
+particularly critical for calibrating throughput variations
+between the fibers in order to apply sky subtractions. In
+total, we have observed 106 distinct pointings in M354 Geller et al.
+over the roughly 11 years since our survey began.
+2.3.Data Reduction
+For a thorough description of our data reduction pro-
+cess, seeGeller et al.(2008). Inshort, weperformastan-
+dard bias and flat-field correction to the images using
+the overscan strip and the flat-field images, respectively.
+The flat-field spectra are used to trace each aperture in
+a given pointing and thereby extract the science spec-
+tra. Wavelength solutions derived from the emission-
+lamp spectra are applied, followed by sky-subtraction
+using sky fibers from each pointing. The three sets of
+spectra (one set from each integration in a given con-
+figuration) are then combined via a median filter to re-
+movecosmicraysignalsandimproveS/N.These reduced
+spectra are cross-correlated with a high S/N solar spec-
+trum, obtained using a dusk sky exposure taken on the
+WIYN 3.5m with the same instrument setup as the given
+pointing. A Gaussian fit to the cross-correlation func-
+tion (CCF) yields a RV and a full width at half maxi-
+mum (FWHM, in km s−1) for each stellar observation.
+The mean UT time is used to find and correct each RV
+measurement for the Earth’s heliocentric velocity. Fi-
+nally we apply the unique fiber-to-fiber RV offsets de-
+rived by Geller et al. (2008) for the WIYN-Hydra data
+to these RVs. As in Geller et al. (2008), to ensure a
+sufficient quality of measurement, we incorporate into
+our database only those spectra with a CCF peak height
+higher than 0.4. Additionally, we examine the distri-
+bution of RVs for each individual star and visually in-
+spect any measurements that are outliers in the distri-
+bution. Occasionally we remove a measurement whose
+CCF, though having a peak height above 0.4, clearly
+provides a spurious measurement (e.g., inadequate sky
+subtraction).
+2.4.Completeness of Spectroscopic Observations
+We have at least one observation for 1144 of the
+1344 stars in our stellar sample, for a completeness of
+85% across our entire sample. 60% of the stars in our
+stellar sample have sufficient observations for their RVs
+to be considered final (813/1344). For these stars, we ei-
+ther have ≥3 RV measurements that show no variation,
+or, if we do see RV variability, we have found a binary
+orbital solution. (These 813 stars comprise the SM, SN,
+BM and BN classes; see Section 4.1). Of those stars not
+finalized, 231 have only one or two observations, and an-
+other 100stars arevariable but do not yet havedefinitive
+orbital solutions.
+In Figure 2, we show the completeness of our observa-
+tions as functions of Vmagnitude (left) and projected
+radius (right). We plot the completeness in stars ob-
+served≥3 times with the dashed line and stars observed
+≥1 time with the solid line. Our prioritization of stars
+by distance from the cluster center is evident by our de-
+creasing completeness with cluster radius. The decreas-
+ing completeness towards fainter stars reflects the need
+for dark skies with minimal sky contamination in order
+to obtain sufficient S/N in our spectra to derive reliable
+RVsforfaint stars. Thereare37starswith V <15in our
+stellar sample that do not have RV measurements, one of
+which is a proper-motion member. 15 were observed but
+did not yield reliable RVs, mostly due to rapid rotation.22 were not observed, 15 of which are farther than 20
+arcminutes in radius from the cluster center.
+Thedifferenceincompletenessbetweenbrightstarsob-
+served≥1 and≥3 times is also a result of an increas-
+ing population of rapidly rotating stars towards bluer
+(B−V) color. For many of these stars, we have multi-
+ple observations of which only a few, and sometimes one,
+exceed this cutoff value of CCF peak height >0.4 and
+therefore are included in our database. For purposes of
+future research we also include seven rapid rotators in
+Table 3 for which we have been unable to derive RVs
+from our spectra.
+3.EFFECTS OF STELLAR ROTATION ON
+MEASUREMENT PRECISION
+3.1.Observed Rotation
+Because of its youth, M35 provides a sample of
+late-type stars with a range of rotational periods
+(Meibom et al. 2009); some of these stars have projected
+rotational velocities that exceed our spectral resolution.
+As such, the cluster presents an opportunity to explore
+empiricallythedependence ofourmeasurementprecision
+on increasing vsini, whereiis the inclination angle of
+the stellar rotation axis to our line of sight.
+Fig. 3.— FWHM as a function of vsinifor observations
+in the 512.5 nm region. FWHM values are measured from
+the CCF peaks derived from a series of artificially broadened
+templates, of known vsini, correlated against the original
+narrow-lined spectrum. We also show a polynomial fit to the
+data, which we then use to derive vsinivalues for observed
+stars in M35. Additionally we plot a dashed line at vsini
+= 10 km s−1, below which the curve flattens out due to our
+spectral resolution. We impose a floor in vsiniat this value
+as we are unable to reliably measure slower rotation.
+The measured FWHM of the CCF for a given star is
+directly related to the vsini(Rhode et al. 2001). Thus
+in order to derive a vsinivalue, we first measure the
+FWHM of the CCF peak. To do so, we fit a Gaussian
+function to the peak, forcing the baseline of the Gaus-
+sian to start at the background level of the CCF. Specif-
+ically, we subtract from the CCF a polynomial fit to this
+backgroundlevel, and then fit the Gaussian to the subse-
+quent “continuum subtracted” CCF. We only use spec-
+tra from the 512.5 nm region to measure the FWHM, as
+the FWHM is dependent on the setup (i.e., the disper-
+sion, etc.), and most of our observations were taken in
+the 512.5 nm region. We then use a similar technique as
+Rhode et al. (2001), to convert this FWHM to a vsini.WOCS. RV Measurements in M35 5
+Fig. 4.— Histogram of vsinimeasurements (left) and vsinias a function of ( B−V)0(right) for the cluster members of M35.
+We have removed double-lined binaries and any binaries with known periods less than 10.2 days, the circularization peri od in
+M35 (Meibom & Mathieu 2005). We only show stars with mean vsinivalues derived from ≥3 observations within the 512.5
+nm region. Notice that the stars with the largest rotation ar e generally also the bluest stars in our sample.
+We create a series of artificially broadened templates by
+convolving our standard solar template with a series of
+theoretical rotation profiles of specific vsinivalues. We
+then cross correlate this series of broadened templates
+with the original narrow-lined template and measure the
+FWHM of the CCF peak as described above. In Fig-
+ure 3 we show the results of this analysis along with a
+polynomial fit to the data. We use this curve to derive
+vsinivalues for all observations of stars in M35 in the
+512.5 nm region. We then take the mean vsinifor each
+star, using only our highest quality (CCF peak height
+>0.4) spectra, and provide these values in Table 3. We
+are unable to reliably measure vsinivalues below 10 km
+s−1, due to the spectral resolution; we therefore impose
+a floor to the vsiniat this value.
+The median FWHM value that we observe is 46.1 km
+s−1which corresponds to vsini= 10.3 km s−1. Exclud-
+ing stars rotating slower than 10 km s−1, we find a preci-
+sion of 1.4 km s−1for individual vsinivalues of ≤25 km
+s−1, which increasesto 1.6km s−1forvsini >25km s−1.
+These precision values were derived in the same manner
+as for our RV precision, with a fit to a χ2function; see
+Section 3.2 and Geller et al. (2008). Where possible, we
+derive a mean vsinifor a given star from multiple, gen-
+erally≥3, observations within the 512.5 nm region. We
+have compared our vsinimeasurements to the rotation
+periods from Meibom et al. (2009) for stars observed in
+both studies, and find the vsiniand rotation periods to
+be consistent.
+In the left panel of Figure 4, we plot a histogram of
+the mean vsinimeasurements for M35 cluster members.
+(See Section 4.1 for our membership criteria.) In this
+and the other panel, we have excluded any binaries with
+periods known to be less than the circularization period
+in M35 of 10.2 days (Meibom & Mathieu 2005), as the
+rotation of the stars in these binaries have likely been af-
+fectedbytidalprocesses. Wehavealsoremovedanystars
+that appearto be indouble-lined binaries, asthe spectral
+lines in many of these observations are broadened due to
+the secondary spectrum at similar, though slightly off-
+set, RV. In the right panel of Figure 4, we plot the mean
+vsinias a function of ( B−V)0for M35 cluster mem-
+bers. We see a clear trend of increasing rotation towards
+bluer stars, as has also been observed in other young
+open clusters and the field (e.g., field, Hyades, Pleiades,Kraft 1967; Pleiades, Soderblom et al. 1993; Blanco 1,
+Mermilliod et al. 2008; IC 2391, Platais et al. 2007).
+3.2.Radial-Velocity Precision
+We determine the RV measurement precision following
+Geller et al. (2008), where a χ2distribution is fit to the
+distribution of the standard deviations of the first three
+RV measurements for each star in an ensemble of stars.
+Here we do this operation on samples of stars with dif-
+feringvsini. Specifically, we consider stars with vsini
+of≤10 km s−1, 10 - 20 km s−1and 20 - 80 km s−1.
+The bin sizes were chosen arbitrarily in order to pro-
+vide sufficiently large samples. The first bin contains all
+narrow-lined stars for which we have imposed a floor to
+thevsini(see Section 3.1); these stars have line widths
+characteristicof the auto-correlationof our spectralreso-
+lution. The remainingbins containstarswith line widths
+increased by stellar rotation.
+A detailed study of the RV measurement precision of
+our observation and data-reduction pipeline has been
+done by Geller et al. (2008) for late-type stars in the
+old open cluster NGC 188. For the narrow-lined stars
+in NGC 188 they find a single-measurement precision of
+0.4 km s−1. This precision is also a function of the S/N
+of the spectrum, as shown in Geller et al. (2008) by the
+degrading precision with increasing Vmagnitude as well
+as decreasing CCF peak height. The largest S/N effect
+seen for narrow-lined stars in NGC 188 is to degrade the
+precision by 0.25 km s−1. The effect of rotation is larger
+than this amount. Here, we derive a relationship be-
+tween the measurement precision and vsiniand use this
+relationship in our analysis throughout this paper.
+In Figure 5 we show the RV precision as a function
+ofvsiniin M35 for observations taken in the 512.5 nm
+region. The narrow-lined stars have a RV precision of
+0.5 km s−1, similar to that found for the narrow-lined
+stars in NGC 188 observed with this same setup. As ex-
+pected, the value of the measurement precision increases
+with increasing line width. For the most rapidly rotating
+stars (vsini >50 km s−1), the measurement precision
+degrades to ∼1.0 km s−1. We fit a linear relationship to
+the points in Figure 5, shown as the dashed line:
+σi= 0.38+0.012(vsini) km s−1,(1)
+whereσiis our precision. We use this equation with
+the mean measured vsinifor a given star to calculate6 Geller et al.
+the single-measurement RV precision for that star. We
+adopt a floor to our precision at 0.5 km s−1, as found for
+our narrow-lined stars, and shown by the break in the
+dashed line in Figure 5.
+Fig. 5.— RV measurement precision as a function of the
+averagevsini(in km s−1) for single lined stars with ≥3 ob-
+servations. The bins are vsiniof≤10 km s−1, 10 - 20 km
+s−1and 20 - 80 km s−1, chosen to provide sufficiently large
+samples. The gray horizontal bars indicate the bin sizes for
+each point. The black vertical error bars show the one sigma
+errors on the precision fit values. The dotted line shows the
+fit to these data, and provided in Equation 1; we impose a
+floor to our precision at 0.5 km s−1.
+We lack sufficient observations to perform this same
+analysis using observations in the 637.5 nm region or
+for observations taken after the spring of 2008 (see Sec-
+tion 2.2). Therefore, for the 129 stars that do not have
+anyobservationsinthe512.5nmregion( ∼11%ofourob-
+served stars), we visually inspect the spectra and CCFs.
+For narrow-lined stars, we set the precision to 0.5 km
+s−1, and for rotating stars we set the precision to 1.0 km
+s−1. We can then use this RV precision value for a given
+star to determine whether our observations for this star
+are constant or variable in velocity (see Section 4.2). We
+note that only 13 of these stars have sufficient observa-
+tions for their RVs to be considered final, and only 2 are
+probable members.
+4.RESULTS
+The full M35 database is available with the electronic
+version of this paper; here we show a sample of our re-
+sults in Table 3. The first column in Table 3 contains
+the WOCS identification number ( IDW). These num-
+bers are defined in the same manner as in Hole et al.
+(2009), with the cluster center set at α= 6h9m7.s5 and
+δ= +24◦20′28′′(J2000). Nextwe givethe corresponding
+IDs from Meibom et al. (2009), McNamara & Sekiguchi
+(1986a) and Cudworth (1971) ( IDM,IDMcandIDC).
+The next few columns provide the right ascension ( RA),
+declination ( DEC), theBVphotometry and the source
+number ( S) for this photometry (see Section 2.1). Next,
+we show the number of RV measurements ( N) and the
+mean and standard error of the RV measurements. For
+stars with only one RV measurements, we show the
+single-measurementRV precision instead of the standard
+error. Next we provide this single-measurement RV pre-
+cision (σi, derived using equation 1), the mean and stan-TABLE 1
+Gaussian Fit Parameters For Cluster
+and Field RV Distributions
+Cluster Field
+Ampl. (Number) 69.0 ±2.0 2.4 ±0.4
+RV(km s��1) -8.17 ±0.05 13 ±4
+σ(km s−1) 0.92 ±0.08 34 ±4
+dard error of the vsinimeasurements6, thee/ivalue
+(see Section 4.2), the calculated RV membership proba-
+bility(P RV, seeSection4.1), theproper-motionmember-
+ship probability from McNamara & Sekiguchi (1986a)
+(PPM1) and Cudworth (1971) ( PPM2), where available,
+andthen, theclassificationoftheobject(seeSection4.3).
+For RV-variable stars with orbital solutions, we present
+the center-of-mass ( γ) RV with the derived error in place
+of the mean RV and its standard error, and add the com-
+ment SB1 or SB2 for single- and double-lined binaries,
+respectively. Additionally, for binaries without orbital
+solutionsthatappeartobedouble-lined, weaddthecom-
+ment of SB2. Finally, for purposes of future research we
+include seven rapid rotators for which we have been un-
+able to derive RVs from our spectra, and label them with
+the comment RR.
+4.1.Membership
+The RV distribution of M35 is clearly distinguished
+from that of the field when we plot a histogram of the
+mean RVs for the observedstars in our stellar sample. In
+Figure 6, we show a histogram of the mean RVs for stars
+with≥3 RV measurements whose standard deviations
+are<2 km s−1, as well as the γ-RVs for binary stars
+with orbitalsolutions, thus excludingfrom the fit anyRV
+variables whose γ-RVs are unknown. The cluster shows
+a well-defined peak rising above the broad distribution
+of the field stars. We simultaneously fit one-dimensional
+Gaussian functions, Fc(v) andFf(v), to represent the
+cluster and field RV distributions, respectively, and then
+use these fits to calculate RV membership probabilities
+for each individual star. We compute the membership
+probability PRV(v) with the usual formula:
+PRV(v) =Fc(v)
+Ff(v)+Fc(v)(2)
+(Vasilevskis et al. 1958). We plot these Gaussian fits in
+Figure 6 with the dashed lines, and show the fit param-
+eters in Table 1.
+For a given single star, we use the mean RV to com-
+pute the RV membership probability. For a given binary
+star with an orbital solution, we compute the RV mem-
+bership probability from the γ-RV. For RV-variable stars
+without orbital solutions, the γ-RVs are not known, and
+therefore we cannot calculate RV membership probabil-
+ities. For these stars, we provide a preliminary member-
+ship classification, described in Section 4.3.
+6For double-lined binaries and stars with no observation in t he
+512.5 nmregion, wedo notderive a vsinivalue. For starswith only
+one measurement in the 512.5 nm region, we convert the 1-sigm a
+error on the FWHM (derived from the Gaussian fit to the CCF
+peak) to an error on the vsiniusing the fit shown in Figure 3. As
+this relationship is not linear, we provide the mean of the de rived
+upper and lower errors on vsini.WOCS. RV Measurements in M35 7
+Fig. 6.— RV histogram for stars in the field of M35. We
+include the mean RVs for stars observed ≥3 times with RV
+standarddeviations <2kms−1andtheγ-RVsfor binarystars
+with orbital solutions, excluding RV variables whose γ-RVs
+are unknown. The bin sizes are 0.5 km s−1, equal to our
+RV precision for narrow-lined stars, as found in Section 3.
+The dashed lines show the simultaneous Gaussian fits to the
+cluster and field RV distributions.
+Fig. 7.— Histogram of membership probabilities, P RV, for
+stars observed ≥3 times with RV standard deviations <2 km
+s−1and for binaries whose γ-RVs are known. For the single
+stars, we compute P RVusing the mean observed RV; for bi-
+naries with orbital solutions, P RVis based on the γ-RV. We
+show our membership cutoff of P RV=50% with the dashed
+line, above which we classify a star as a cluster member. Note
+that we do not show the full height of the bin at lowest mem-
+bership probability for clarity.
+In Figure 7, we show the distribution of RV member-
+ship probabilities, displaying a clean separation between
+the cluster members and field stars. In the following
+analysis, we use a probability cutoff of P RV≥50 % to
+define our cluster member sample. Using the 344 single
+clustermembersandbinaryclustermemberswithorbital
+solutions, we find a mean cluster RV of -8.16 ±0.05 km
+s−1. From the area under the fit to the cluster and field
+distributions, we estimate a field contamination of 6%
+within our cluster member sample (P RV≥50%). Though
+this estimate is derived excluding the RV variables that
+do not have orbital solutions, the percent contamination
+should be valid for the cluster as a whole.
+Our RV membership probabilities agree well with the
+proper-motion memberships of Cudworth (1971) and
+McNamara & Sekiguchi (1986a). We note that our stel-
+lar sample covers only the faintest portion of either
+proper-motion study. There are 24 Cudworth (1971)proper-motionmembers within our observed stellar sam-
+ple, of which we find 14 (58%) to also have ≥50% RV
+membership probabilities. Cudworth (1971) note that
+forV >13 they begin to find significant errors in
+their photometry and expect many field stars to con-
+taminate their proper-motion member sample; this can
+likely explain the 10 discrepant stars. There are 70
+McNamara & Sekiguchi(1986a)proper-motionmembers
+within our observed stellar sample, of which we find 64
+(91%) to also have ≥50% RV membership probabilities.
+McNamara & Sekiguchi (1986a) expects up to 15 field
+stars contaminating their cluster member sample from
+13< V <15, which can easily account for the 6 dis-
+crepant stars.
+We also note that NGC 2158 is only ∼28 arcminutes
+away from the center of M35, at α= 6h07m25sandδ=
++24◦05′48′′(J2000), and thus is within the spatialregion
+that we have surveyed. Scott et al. (1995) find a mean
+RV for NGC 2158 of 28 ±4 km s−1. There are five stars
+within our sample that lie within the cluster radius of 2.5
+arcminutes Carraro et al. (2002) from the center of NGC
+2158 and have RVs within three times the standard error
+(12 km s−1) of the mean RV : 125044, 39017, 111050,
+57037, 54048. Two of these stars (125044 and 57037)
+have less than three observations; the remaining three
+have≥3observationsandappeartobenon-RV-variables.
+4.2.Radial-Velocity Variability
+RV-variable stars are distinguishable by the larger
+standard deviations of their RV measurements. Here,
+we assume that such velocity variability is the result of
+a binary companion, or perhaps multiple companions.
+Specifically, we consider a star to be a RV variable if the
+ratio of the standard deviation of its RV measurements
+to the single-measurement RV precision7(e/i) for that
+star is greater than four (Geller et al. 2008). We provide
+thee/ivalue for each single-lined star in Table 3; we
+label double-lined systems as RV variables directly, and
+include the comment of SB2 in Table 3.
+MonteCarloanalysishasshownthat, forsimilarobser-
+vations ofsolar-typestars in NGC 188, Geller & Mathieu
+(in preparation) can detect the majority of binaries with
+periods less than 104days and a negligible fraction of
+longer-period binaries. Though the slightly poorer preci-
+sionforthe M35datawill effect the specific completeness
+numbers, we can assume a similarly high completeness in
+detected binaries with periods less than 104days and a
+corresponding drop in completeness for longer-period bi-
+naries. Some of the undetected systems are evident from
+their separation from the main sequence (see Figure 8).
+We have currently identified 55 RV-variable members
+of M35, and have derived orbital solutions for 71%
+(39/55) of this sample. In following papers we will pro-
+vide the orbital solutions for these systems, including
+all derived parameters. We will then perform a detailed
+analysis of the distributions of these orbital parameters
+as well as the binary frequency of the cluster.
+7We use the same nomenclature of “ e/i” as in Geller et al.
+(2008), though in other sections, for clarity, we have label ed the
+precision as σi, so as not to confuse the precision with an inclina-
+tion angle.8 Geller et al.
+TABLE 2
+Number of Stars
+or Star Systems
+Within Each
+Membership
+Class
+Class Number
+SM 305
+SN 452
+BM 39
+BN 17
+BLM 16
+BU 16
+BLN 68
+U 231
+4.3.Membership Classification of Radial-Velocity
+Variable Stars
+We follow the same classification system as
+Geller et al. (2008) and Hole et al. (2009) in order
+to provide a qualitative guide to a given star’s mem-
+bership and variability, in addition to the calculated
+RV memberships and e/ivalues. We provide these
+classifications for all observed stars, while the member-
+ships and e/ivalues are only provided for a subset of
+appropriate stars.
+For stars with e/i<4, we classify those with P RV≥50%
+as single members (SM), and those with P RV<50% as
+singlenon-members(SN). Ifa starhas e/i≥4and enough
+measurements from which we are able to derive an or-
+bital solution, we use the γ-RV to compute a secure
+RV membership. For these binaries, we classify those
+with P RV≥50% as binary members (BM) and those with
+PRV<50% as binary non-members (BN). For RV vari-
+ables without orbital solutions, we split our classifica-
+tions into three categories. If the mean RV results in
+PRV≥50%,weclassifythesystemasabinarylikelymem-
+ber (BLM). If the mean RV results in P RV<50% but the
+range of measured RVs includes the cluster mean RV, we
+classify the system as a binary with unknown member-
+ship (BU). Finally, if the RV measurements for a given
+star all lie either at a lower or higher RV than the clus-
+ter distribution, we classify the system as a binary likely
+non-member (BLN), since it is unlikely that any orbital
+solution could place the binary within the cluster distri-
+bution. We classify stars with <3 RV measurements as
+unknown (U), as these stars do not meet our minimum
+criterion for deriving RV memberships or e/imeasure-
+ments. In the following analysis, we include the SM,
+BM and BLM stars as cluster members. Including these
+stars, we find 360 total cluster members in our sample.
+We list the number of stars within each class in Table 2.
+5.DISCUSSION
+In the following section, we present a CMD for M35
+cleaned of field star contamination (Section 5.1), com-
+pare the spatial distribution of the single and binary
+members (Section 5.2), and analyze the RV dispersion
+of the cluster (Section 5.3).
+5.1.Color-Magnitude Diagram
+In Figure 8, we show the CMD for all RV cluster mem-
+bers in M35 from this study for which we have pho-
+tometry from the WIYN 0.9m, as this set of photom-Fig. 8.— Color-magnitude diagram of M35 including only
+cluster members (P RV≥50%) with photometry from WIYN
+0.9m (source 2). We plot the RV variables with orbital so-
+lutions with circles and without orbital solutions with dia -
+monds. We show the 180 Myr Padova isochrone as the black
+line. The solid gray line shows where binaries with mass ra-
+tios of 1.0 lie on the CMD, and the dashed gray line shows the
+deviation from the isochrone of twice the photometric error .
+etry is of higher precision than that taken on the Bur-
+rell Schmidt (see Section 2.1). We also plot a 180 Myr
+Padova isochrone using the cluster parameters derived
+by Kalirai et al. (2003) in the black curve. Binaries with
+orbital solutions are circled and RV variables without or-
+bital solutions are marked by diamonds.
+Additionally, we use the Padova isochrone to plot the
+location on the CMD of binaries with mass ratios q= 1,
+shown as the gray line. We note that there are a number
+ofstarsobservedbrighterandtothe redofthis line, some
+that we have not identified as RV variables. In this loca-
+tion on the CMD one would expect to find either higher-
+order systems or field stars. There are 33 RV members
+that lie above the q= 1 line; 22 are single and 11 show
+RV variability. We expect a 6% field star contamination
+within the cluster members sample (Section 4.1). If we
+include only the 309 cluster members that have photom-
+etry from the WIYN 0.9m (and are therefore shown in
+Figure 8), this results in 19 possible field stars; including
+our entire cluster member sample results in 22 possible
+field stars. Therefore field star contamination cannot ac-
+count for all of these sources, suggesting that a subset
+of these stars are indeed higher-order systems. We also
+notethatthereareanadditional12clustermemberswith
+photometry from the Burrell Schmidt that lie above the
+q= 1 line, but recall that this source of photometry is of
+poorer precision.
+Finally, for use in Sections 5.2, we follow a similar pro-
+cedure as Montgomery et al. (1993) to attempt to photo-
+metricallyidentify binariesthatlie farfromtheisochrone
+ontheCMD. Wederivethedistanceofeachstarfromthe
+main-sequenceisochroneand fit a Gaussianfunction rep-
+resenting the photometric error distribution to the dis-
+tribution of these distances. We notice a clear excess in
+the observed distribution from the Gaussian fit at 2 σ,
+shown as the dashed gray line in Figure 8. We attribute
+this excess to photometric binaries. A 1 M ⊙star in M35
+with the additional light from a companion of mass-ratio
+q= 0.78 would lie on this line. Therefore sources ob-
+served above this line are likely binaries with larger mass
+ratios (q >0.78), or very infrequently, field stars. We
+observe 42 cluster members above this line that showWOCS. RV Measurements in M35 9
+no significant RV variation (and therefore fall into the
+SM class). Many of these are likely long-period binaries
+that are outside of our detection limits, as the hard-soft
+boundary for solar-type stars in M35 is ∼105−106days,
+and we only detect binaries with P/lessorsimilar104days (Geller
+& Mathieu, in preparation).
+5.2.Spatial Distribution and Mass Segregation
+In Figure 9 we compare the cumulative projected ra-
+dial distributions of the single and binary members of
+M35. We have attempted to reduce the contamination
+from undetected binaries within our single-star sample
+by only including stars with no detectable RV variation
+(SM) that arefainter and bluer than the dashed grayline
+in Figure 8. This conservative cut removes large- q(i.e.,
+high total mass) binaries that have periods longer than
+our detection limit. We have not applied any correc-
+tion for the spatial bias found in our observations (Sec-
+tion 2.4), because this bias will be present in both the
+single- and binary-star samples and should therefore not
+effect this analysis. A Kolmogorov-Smirnov test shows
+no significant difference between these two populations
+with a value of 60%. We therefore conclude that the
+solar-type main-sequence binaries in M35 show no evi-
+dence for central concentration as compared to the single
+stars.
+Mathieu (1983) finds a half-mass relaxation time for
+the cluster of 150 Myr, comparable to the cluster age.
+This study of the radial spatial distributions for proper-
+motion-selected member stars in the 8 .0< V < 14.5
+(∼4.4 - 1.2 M ⊙) range revealed mass segregation only
+amongstarsmoremassivethan 2M ⊙. The degreeofseg-
+regationlessenswith decreasingmass, and is largelynon-
+existent among solar-like stars. McNamara & Sekiguchi
+(1986b) found similar results in their proper-motion se-
+lected sample, which covered stars down to V= 14.5 (∼
+1.2 M⊙). We only include primary stars with masses of
+1.6 - 0.8 M ⊙, and therefore most of our binaries have
+total masses that are lower than the higher-mass stars
+that have been shown to be mass segregated. Mathieu
+(1983) found that M35 is fit well with a multi-mass King
+model. In such models the reduction in mass segregation
+for lower-mass systems derives from more severe tidal
+truncation of higher-dispersion velocity distributions in
+a cluster potential dominated by the solar-like stars.
+5.3.Cluster Radial-Velocity Dispersion
+To determine the true RV dispersion of the cluster, we
+followtheprocedureofGeller et al.(2008). Wefirstlimit
+oursampletoonlyincludeSMstarsthathave vsini≤10
+km s−1. We limit the vsinivalue to ensure that we only
+use the highest precision RV measurements for this anal-
+ysis. These narrow-lined stars have a precision σi= 0.5
+km s−1. We will discuss the effect of undetected binaries
+that likely remain within this sample in Section 5.3.1.
+Usingthissampleof67SMstars,wefirstderivetheob-
+serveddispersion σobsbytakingthestandarddeviationof
+themeanRVsforeachstar,andwefind σobs= 0.86±0.07
+km s−1. This observed dispersion is a function of our
+measurement precision and is also inflated by undetected
+binaries. Therefore, in order to derive the true RV dis-
+persion, we must first account for the precision on theseFig. 9.— Cumulative projected radial spatial distributions
+of the M35 single and binary cluster members. We have ex-
+cluded any stars from the single-star sample that are bright er
+and redder than the dashed grey line in Figure 8, as these
+stars are likely long-period binaries that are outside of ou r de-
+tection limits. We plot the single stars with the black point s
+andtheRVvariables with theopen diamonds. We findnosig-
+nificant evidence for central concentration of the RV-varia ble
+population.
+RV measurements. We derive8the “combined RV dis-
+persion” σcbfrom :
+σ2
+cb=σ2
+obs−1
+nn/summationdisplay
+i=1ξ2
+i. (3)
+Here,n= 67 is the number of stars used in this analysis,
+andξiis the mean errorof the RV for the ith star defined
+as,
+ξi=
+m/summationdisplay
+j=1/parenleftbig
+RVj−RVi/parenrightbig2
+m(m−1)
+1/2
+(4)
+whereRVjis one of the mnumber of RV measurements
+for a given star i, andRViis the mean RV for that star.
+WenotethatthesecondterminEquation3isverynearly
+equal to σ2
+i/3, as we have 3 RVs for most of the stars in
+this sample, and these narrow-lined stars all have the
+same precision of σi= 0.5 km s−1. Following this proce-
+dure, wederiveacombineddispersionof σcb= 0.81±0.08
+km s−1. The error on this combined dispersion is almost
+entirely due to the statistical error on σobs.
+We find no significant difference in the combined RV
+dispersion of the SM or BM stars. For the BM stars,
+we use the γ-RVs in place of the mean RVs, and substi-
+tute the measurement precision for the standard devia-
+tion portion in Equation 4. There is also no significant
+variation in the combined RV dispersion as a function
+of radius, although due to the small sample sizes our
+binned RV dispersion values have large uncertainties (of
+0.1 - 0.15 km s−1for bins of 10 arcmin).
+This combined RV dispersion is inflated by undetected
+binaries. In the following section, we quantify this effect
+and apply the correctionto derivethe true RV dispersion
+ofM35, an improvementon the procedureofGeller et al.
+(2008).
+8The use of Equations 3 and 4 is an improvement over the pro-
+cedure of Geller et al. (2008) adapted from McNamara & Sander s
+(1977). The uncertainty on σcbalso follows McNamara & Sanders
+(1977).10 Geller et al.
+5.3.1.Contribution from Undetected Binaries
+The combined RV dispersion defined in Equation 3 is
+also described by,
+σcb=σc+β (5)
+whereσcis the true RV dispersion of the cluster and
+βrepresents the contribution from undetected binaries
+within our sample. Therefore, in order to derive the true
+RV dispersion of the cluster we have performed a Monte
+Carlo analysis to determine this contribution from unde-
+tected binaries.
+We first create a set of simulated binaries with or-
+bital parameters distributed according to the Galactic
+field solar-type binaries studied by Duquennoy & Mayor
+(1991). Specifically, these binaries have a log-normal pe-
+riod distribution centered on log( P[days] ) = 4.8 with
+σ= 2.3, and a Gaussian eccentricity distribution cen-
+tered on e= 0.3. For binaries with periods below the
+circularization period of 10.2 days (Meibom & Mathieu
+2005), we set the eccentricity to zero. We use only solar-
+massprimary stars, and a distribution in secondarymass
+between0.08-1M ⊙describedby aGaussiancenteredon
+M2= 0.23 M⊙withσ= 0.42 M⊙(Kroupa et al. 1990).
+Duquennoy & Mayor (1991) found this Gaussian to be
+the best fit to their solar-type field binaries, and this dis-
+tribution is also consistent with that of Goldberg et al.
+(2003) for their field binaries with primary masses >0.67
+M⊙. The orbital inclinations and phases of the binaries
+are chosen randomly. We then generate three RVs for
+these simulated binaries distributed in time according to
+the actual distribution of our first three observations for
+starsin M35. The majorityofthe SM starsin oursample
+haveonlythreeobservations. TotheseRVs, wealsoadda
+randomerrorgeneratedfromaGaussiancenteredonzero
+and with σ= 0.5 km s−1, the RV precision for narrow-
+lined stars in M35. We also add a random velocity offset
+generated from a Gaussian centered on zero with a stan-
+dard deviation equal to an adopted one-dimensional RV
+dispersion.
+To this sample, we add a number of simulated single
+stars to produce a desired binary frequency. We generate
+three RVs for each single star from a Gaussian described
+byourprecision. To themean RVforeverysinglestarwe
+also add a random offset described by the assumed RV
+dispersion in the same manner as for the simulated bi-
+naries. We then keep only those simulated binaries (and
+single stars) whose first three RVs result in an e/i <4,
+and whose mean RVs are within three standard devia-
+tions of the mean RV from a Gaussian fit to the simu-
+lated RV distribution. This cutoff in standard deviation
+reflects our membership criterion of P RV≥50% for the
+M35 observations. These binaries would be undetected
+within the SM sample.
+We then follow the equations given above to derive β
+fora rangeofbinaryfrequenciesand velocitydispersions,
+σc. In Figure 10, we plot the true cluster RV dispersion
+(σc) as a function of the combined RV dispersion ( σcb)
+for a range of total binary frequencies, where each line
+corresponds to a different binary frequency between 0%
+(far left) to 100% (far right) in steps of 10%. We can
+then use the results shown in Figure 10 to derive the
+true RV dispersion for M35. Furthermore, the results
+shown in this figure are also applicable to RV dispersionFig. 10.— The true cluster RV dispersion ( σc) plotted
+against the combined RV dispersion ( σcb) for a range of total
+binaryfrequencies. Eachlinecorrespondstoadifferentbin ary
+frequency in steps of 10%, with 0% at the far left and 100%
+at the far right. With the vertical gray rectangle, we plot th e
+region included in the combined RV dispersion for M35 of
+0.81±0.08 km s−1. The diagonal gray region covers the pos-
+sible lines within our extrapolated true binary frequency i n
+M35 of 66% ±8% (derived assuming the M35 binaries follow
+a Duquennoy & Mayor (1991) period distribution). Finally,
+we plot the resulting true RV dispersion in M35 of 0.65 ±
+0.10 km s−1with the black point at the intersection of these
+two shaded regions.
+analyses for other star clusters, provided that the binary
+population is consistent with the Duquennoy & Mayor
+(1991) field binaries.
+5.3.2.True Radial-Velocity Dispersion
+To date, we have detected 55 binaries in M35 out
+of 360 cluster members. If we assume a similar com-
+pleteness as in Geller & Mathieu (in preparation), for
+NGC 188, then we can assume that we have detected
+63% of the binaries with periods less than 104days
+(and a negligible fraction of binaries with longer peri-
+ods). This correction results in a binary frequency of
+24%±3% forP <104days. This binary frequency
+is consistent with that of solar-type stars in the Galac-
+tic field Duquennoy & Mayor (1991) out to the same
+period limit. If we assume the M35 binaries follow
+a Duquennoy & Mayor (1991) period distribution, then
+our binary frequency for P <104days implies a total bi-
+nary frequency of 66% ±8%, with the inclusion of wider
+binaries currently beyond our detection limits. We then
+take this value for the total binary frequency and correct
+our combined RV dispersion for undetected binaries.
+In the filled gray areas in Figure 10 we show the re-
+gions defined by our M35 combined RV dispersion and
+the total binary frequency. At the intersection, we plot
+the derivedtrue RV dispersionin M35 of σc= 0.65±0.10
+km s−1. Using a flat distribution in secondary mass (and
+mass ratio), as has been suggested by some studies (e.g.,
+Mazeh et al. 1992, 2003), has a negligible effect on the
+derived true RV dispersion. This true RV dispersion is
+consistent with the projected velocity dispersion of 1.0
+±0.15 km s−1, derived by Leonard & Merritt (1989) us-
+ingtheproper-motiondatafromMcNamara & Sekiguchi
+(1986a).
+6.SUMMARY
+This is the first paper in a series studying the dynam-
+ical state of the young ( ∼150 Myr) open cluster M35WOCS. RV Measurements in M35 11
+(NGC 2168). In this first paper, we present our RV ob-
+servations and provide initial results from this survey.
+Our stellar sample extends to 30 arcminutes in radius
+from the cluster center (7 pc in projection at a distance
+of 805 pc or ∼4 core radii), and we have selected a region
+from aV, (B−V) CMD (Figure 1) which covers a mass
+range of 1.6 - 0.8 M ⊙. We have used the WIYN 3.5m
+telescope with the Hydra MOS to obtain 5201 spectra of
+1144 stars within this stellar sample. From these spec-
+tra, we derive RV measurements with a precision of 0.5
+km s−1for narrow-lined stars. The vast majority of the
+observed stars have multiple measurements, allowing de-
+termination of cluster membership and identification of
+spectroscopic binary stars. We detect 360 cluster mem-
+bers, 55 of which show significant variability in their RV
+measurements. Binary orbital solutions have been ob-
+tained for 39 of these RV variables, which we will present
+in detail in the next paper in this series. Observations
+of the rest of the RV variables and the remainder of our
+stellar sample are ongoing. Table 3 provides the first RV
+membership database for M35 and extends ∼1.5 magni-
+tudes deeper than any previous membership catalogue.
+Using the RV cluster members, we study the spa-
+tial distribution and velocity dispersion of the single
+and binary stars. We find their spatial distributions
+to be indistinguishable. This lack of central concentra-
+tion for the binaries is consistent with earlier observa-
+tional studies of stars in M35 as well as with a fully re-
+laxed dynamical model for the cluster (Mathieu 1983;
+McNamara & Sekiguchi 1986b). In these studies, mass
+segregationisseeninhigher-massstars,butdiminishesto
+being undetectable for stars in our observed mass range.After correcting for measurement precision, but not for
+binaries, we place an upper limit on the RV dispersion of
+the cluster of 0 .81±0.08 km s−1. When we also correct
+for undetected binaries, we derive a true RV dispersion
+of 0.65±0.10 km s−1.
+The WOCS group will continue our survey of M35 in
+ordertoderiveRVmembershipsforallstarsin ourstellar
+sample and obtain orbital solutions for all binaries with
+periods less than a few thousand days, as well as some
+with longer periods. In future papers, we will study the
+binary population of M35 in detail, providing all orbital
+solutions and analyzing the binary frequency and dis-
+tributions of orbital parameters. These data will form
+essential constraints on the hitherto poorly known initial
+binary populations used in sophisticated N-body models
+of open clusters.
+The authors would like to express their gratitude to
+the staff of the WIYN Observatory for their skillful and
+dedicated work that have allowed us to obtain these ex-
+cellent spectra. We thank Ata Sarajedini and Ted von
+Hippel for the acquisition of the Schmidt images, Vera
+Platais for work on the astrometry and photometry as
+well as John Bjorkman for early photometry work. We
+also thank the many undergraduate and graduate stu-
+dents who have contributed late nights to obtain the
+spectra for this project. This work was supported by
+NSF grant AST 0406615and the Wisconsin Space Grant
+Consortium.
+Facilities: WIYN 3.5m
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+Radial-Velocity Data Table
+IDWIDMIDMcIDCRA DEC V (B−V)S N RV RV eσivsini(vsini)ePRVPPM1PPM2e/i Class Comment
+96041 410 ··· ··· 6:10:28.65 24:11:52.0 16.416 0.960 1 4 51.89 1.55 0.52 11.8 1 .1 ··· ·· · · ·· 5.93 BLN ···
+36042 209 ··· ··· 6:10:34.30 24:14:07.8 14.835 0.859 1 3 -8.74 0.55 0.64 21.6 0 .8 96 ·· · · ·· 1.49 SM ···
+36045 209 ··· ··· 6:10:43.69 24:16:08.9 14.497 0.824 1 17 -9.58 0.19 0.50 10.3 0.2 91 ·· · · ·· 77.46 BM SB1
+138057 366 ··· ··· 6:10:50.20 24:04:50.7 16.368 1.165 1 1 -25.13 0.50 0.50 ··· ··· ··· ·· · · ·· · ·· U ···
+64052 312 ··· ··· 6:10:43.70 24:07:00.8 15.884 1.023 1 4 -8.85 0.64 0.55 14.2 4 .2 96 ·· · · ·· 2.34 SM ···
+15036 180 ··· 731 6:10:15.70 24:11:31.7 13.450 0.690 2 1 57.98 0.50 0.50 ··· ··· ··· ·· · 0 · ·· U ···
+49051 227 ··· ··· 6:10:51.32 24:11:10.6 15.086 0.890 1 4 88.83 0.34 0.50 10.0 ··· 0 ·· · · ·· 1.35 SN ···
+29047 87 ··· ··· 6:10:44.59 24:13:44.3 14.948 0.895 1 4 56.95 0.28 0.50 10.0 ··· 0 ·· · · ·· 1.10 SN ···
+40032 ··· ··· ··· 6:10:11.15 24:14:01.5 15.280 0.820 2 3 -7.72 0.32 0.55 14.3 0 .8 96 ·· · · ·· 1.02 SM ···
+20037 193 ··· 758 6:10:22.30 24:14:39.3 14.270 0.650 2 1 3.88 0.50 0.50 ··· ··· ··· ·· · 7 · ·· U ···
+The contents of each column are defined in Section 4.

1001.0034.txt

@@ -0,0 +1,594 @@
+arXiv:1001.0034v1  [math.NT]  4 Jan 2010NEW IDENTITIES INVOLVING q-EULER
+POLYNOMIALS OF HIGHER ORDER
+T. Kim AND Y. H. Kim
+Abstract. In this paper, we present new generating functions which are relat ed to
+q-Euler numbers and polynomials of higher order. From these genera ting functions, we
+give new identities involving q-Euler numbers and polynomials of higher order.
+§1. Introduction/ Preliminaries
+LetCbe the complex number field. We assume that q∈Cwith|q|<1 and
+theq-number is defined by [ x]q=1−qx
+1−qin this paper. The q-factorial is given by
+[n]q! = [n]q[n−1]q···[2]q[1]qand theq-binomial formulae are known that
+(x:q)n=n/productdisplay
+i=1(1−xqi−1) =n/summationdisplay
+i=0/parenleftbiggn
+i/parenrightbigg
+qq(i
+2)(−x)i,(see [3, 14, 15]) ,
+and
+1
+(x:q)n=n/productdisplay
+i=1/parenleftbigg1
+1−xqi−1/parenrightbigg
+=∞/summationdisplay
+i=0/parenleftbiggn+i−1
+i/parenrightbigg
+qxi,(see [3, 5, 14, 15]) ,
+where/parenleftbign
+i/parenrightbig
+q=[n]q!
+[n−i]q![i]q!=[n]q[n−1]q···[n−i+1]q
+[i]q!.
+The Euler polynomials are defined by2
+et+1ext=/summationtext∞
+n=0En(x)tn
+n!, for|t|< π. In the
+special case x= 0,En(=En(0)) are called the n-th Euler numbers. In this paper, we
+consider the q-extensions of Euler numbers and polynomials of higher orde r. Barnes’
+multiple Bernoulli polynomials are also defined by
+(1)
+tr
+/producttextr
+j=1(eajt−1)ext=∞/summationdisplay
+n=0Bn(x,r|a1,···,ar)tn
+n!,where|t|<max
+1≤i≤r2π
+|ai|, (see [1, 14]).
+Key words and phrases. : multiple q-zeta function, q-Euler numbers and polynomials, higher
+order q-Euler numbers, Laurent series, Cauchy integral.
+2000 AMS Subject Classification: 11B68, 11S80
+The present Research has been conducted by the research Grant of Kw angwoon University in 2010
+Typeset by AMS-TEX
+1In one of an impressive series of papers (see [1, 6, 14]), Barn es developed the so-called
+multiple zeta and multiple gamma function. Let a1,···,aNbe positive parameters.
+Then Barnes’ multiple zeta function is defined by
+ζN(s,w|a1,···,aN) =/summationdisplay
+m1,···,mN=0(w+m1a1+···+mNaN)−s,(see [1]),
+whereℜ(s)> N,ℜ(w)>0. Form∈Z+, we have
+ζN(−m,w|a1,···,aN) =(−1)mm!
+(N+m)!BN+m(w,N|a1,···,aN).
+In this paper, we consider Barnes’ type multiple q-Euler numbers and polynomials.
+The purpose of this paper is to present new generating functi ons which are related
+toq-Euler numbers and polynomials of higher order. From the Mel lin transformation
+of these generating functions, we derive the q-extensions o f Barnes’ type multiple
+zeta functions, which interpolate the q-Euler polynomials of higher order at negative
+integer. Finally, we give new identities involving q-Euler numbers and polynomials of
+higher order.
+§2.q-Euler numbers and polynomials of higher order
+In this section, we assume that q∈Cwith|q|<1. Letx,a1,... ,a rbe complex
+numbers with positive real parts. Barnes’ type multiple Eul er polynomialsare defined
+by
+(2)2r
+/producttextr
+j=1(eajt+1)ext=∞/summationdisplay
+n=0E(r)
+n(x|a1,... ,a r)tn
+n!,for|t|<max
+1≤i≤rπ
+|wi|,(see [6]),
+andE(r)
+n(a1,... ,a r)(=E(r)
+n(0|a1,... ,a r)) are called the n-th Barnes’ type multiple
+Euler numbers. First, we consider the q-extension of Euler polynomials. The q-Euler
+polynomials are defined by
+(3)Fq(t,x) =∞/summationdisplay
+n=0En,q(x)tn
+n!= [2]q∞/summationdisplay
+m=0(−q)me[m+x]qt,(see [8, 11, 13, 14, 15]) .
+From (3), we have
+En,q(x) =[2]q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+ql+1).
+In the special case x= 0,En,q(=En,q(0)) are called the n-thq-Euler numbers. From
+(3), we can easily derive the following relation.
+E0,q= 1,andq(qE+1)n+En,q= 0 ifn≥1,(see [8, 16, 17]) ,
+2where we use the standard convention about replacing EkbyEk,q.It is easy to show
+that
+lim
+q→1Fq(t,x) =2
+et+1ext=∞/summationdisplay
+n=0En(x)tn
+n!,(see [2, 3, 19-23]) ,
+whereEn(x) are the n-th Euler polynomials. For r∈N, the Euler polynomials of
+orderris defined by
+(4)/parenleftbigg2
+et+1/parenrightbiggr
+ext=∞/summationdisplay
+n=0E(r)
+n(x)tn
+n!,for|t|< π.
+Now we consider the q-extension of (4).
+(5)F(r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
+n=0E(r)
+n,q(x)tn
+n!,
+whereE(r)
+n,q(x) are called the n-thq-Euler polynomials of order r(see [10-15]). From
+(5), we can derive
+(6) E(r)
+n,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+ql+1)r.
+By (5) and (6), we see that
+(7) F(r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)me[m+x]qt.
+Thus, we note that lim q→1F(r)
+q(t,x) =/parenleftBig
+2
+et+1/parenrightBigr
+ext=/summationtext∞
+n=0E(r)
+n(x)tn
+n!.In the special
+casex= 0,E(r)
+n,q(=E(r)
+n,q(0)) are called the n-thq-Euler numbers of order r. By (5),
+(6) and (7), we obtain the following proposition.
+Proposition 1. Forr∈N, let
+F(r)
+q(t,x) = [2]r
+q/summationdisplay
+m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
+n=0E(r)
+n,q(x)tn
+n!.
+Then we have
+E(r)
+n,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+ql+1)r= [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)m[m+x]n
+q.
+3From the Mellin transformation of F(r)
+q(t,x), we can derive the following equation.
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q(−t,x)ts−1dt= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr
+[m1+···+mr+x]sq
+= [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)m1
+[m+x]sq, (8)
+wheres∈C,x/negationslash= 0,−1,−2,.... By (8), we can define the multiple q-zeta function
+related to q-Euler polynomials.
+Definition 2. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the multiple q-zeta
+function related to q-Euler polynomials as
+ζq,r(s,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr
+[m1+···+mr+x]sq.
+Note that ζq,r(s,x) is a meromorphic function in whole complex s-plane. From (8),
+we also note that
+ζq,r(s,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)m1
+[m+x]sq.
+By Laurent series and the Cauchy residue theorem in (5) and (8 ), we see that
+ζq(−n,x) =E(n)
+n,q(x),forn∈Z+.
+Therefore, we obtain the following theorem.
+Theorem 3. Forr∈N,n∈Z+, andx∈Rwithx/negationslash= 0,−1,−2,..., we have
+ζq(−n,x) =E(r)
+n,q(x).
+Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
+the generalized q-Euler polynomial attached to χare considered by
+Fq,χ(x) =∞/summationdisplay
+n=0En,χ,q(x)tn
+n!= [2]q∞/summationdisplay
+m=0(−q)mχ(m)e[m+x]qt.
+From (3) and (9), we have
+En,χ,q(x) =[2]q
+[2]qff−1/summationdisplay
+a=0(−q)aχ(a)En,qf(x+a
+f).
+4In the special case x= 0,En,χ,q=En,χ,q(0) are called the n-th generated q-Euler
+number attached to χ.
+It is known that the generalized Euler polynomials of order rare defined by
+(10) (2/summationtextf−1
+a=0(−1)aχ(a)eat
+eft+1)rext=∞/summationdisplay
+n=0E(r)
+n,χ(x)tn
+n!,
+for|t|<π
+f.
+We consider the q-extension of (10). The generalized q-Euler polynomials of order
+rattached to χare defined by
+F(r)
+q,χ(t,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(r/productdisplay
+i=1χ(mi))e[m1+···+mr+x]qt
+=∞/summationdisplay
+n=0E(r)
+n,χ,q(x)tn
+n!,(see [14, 15]) . (11)
+Note that
+lim
+q→1F(r)
+q,χ(t,x) = (2/summationtextf−1
+a=0(−1)aχ(a)eat
+eft+1)r.
+By (11), we easily see that
+E(r)
+n,χ,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg
+(−qx)lf−1/summationdisplay
+a1,...,ar=0(r/productdisplay
+j=1χ(aj))(−ql+1)/summationtextr
+i=1ai
+(1+q(l+1)f)r
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(r/productdisplay
+i=1χ(mi))[m1+···+mr+x]n
+q.
+Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we have
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q,χ(−t,x)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(/producttextr
+i=1χ(mi))
+[m1+···+mr+x]sq,(see [15]) . (12)
+From (12), we can consider the Dirichlet’s type multiple q-l-function as follows :
+Definition 4. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
+type multiple q-l-function as
+lq(s,x|χ) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(/producttextr
+i=1χ(mi))
+[m1+···+mr+x]sq,(see [15]) .
+By Laurent series and the Cauchy residue theorem in (11) and ( 12), we obtain the
+following theorem.
+5Theorem 5. Forn∈Z+, we have
+lq(−n,x|χ) =E(r)
+n,χ,q(x).
+Forh∈Zandr∈N, we consider the extended r-pleq-Euler polynomials.
+F(h,r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(−1)/summationtextr
+j=1mje[m1+···+mr+x]qt
+=∞/summationdisplay
+n=0E(h,r)
+n,q(x)tn
+n!. (13)
+Note that
+lim
+q→1F(h,r)
+q(t,x) = (2
+et+1)rext=∞/summationdisplay
+n=0E(r)
+n(x)tn
+n!.
+From (13), we note that
+E(h,r)
+n,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−qx)l
+(−qh−r+l+1:q)r
+= [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+q(−qh−r+1)m[m+x]n
+q. (14)
+By (14), we easily see that
+(15)F(h,r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+q(−qh−r+1)me[m+x]qt,(see [11, 13, 14]) .
+Using the Mellin transform for F(h,r)
+q(t,x), we have
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q(−t,x)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq/summationtextr
+j=1(h−j+1)mj
+[m1+···+mr+x]sq,(see [13, 14, 15]) ,(16)
+fors∈C,x∈Rwithx/negationslash= 0,−1,−2,.... Now we can define the extended q-zeta
+function associated with E(h,r)
+n,q(x).
+6Definition 6. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the (h, q)-zeta
+function as
+ζ(h)
+q,r(s,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq/summationtextr
+j=1(h−j+1)mj
+[m1+···+mr+x]sq.
+Notethat ζ(h)
+q,r(s,x)isalsoa meromorphic function inwholecomplex s-plane. From
+(16) and (15), we note that
+(17) ζ(h)
+q,r(s,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+q(−qh−j+1)m1
+[m+x]sq.
+Using the Cauchy residue theorem and Laurent series in (16), we obtain the following
+theorem.
+Theorem 7. Forn∈Z+, we have
+ζ(h)
+q,r(−n,x) =E(h,r)
+n,q(x).
+We consider the extended r-ple generalized q-Euler polynomials as follows :
+F(h,r)
+q,χ(t,x)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(−1)/summationtextr
+j=1mj(r/productdisplay
+j=1χ(mj))e[m1+···+mr+x]qt(18)
+=∞/summationdisplay
+n=0E(h,r)
+n,χ,q(x)tn
+n!.
+By (18), we see that
+E(h,r)
+n,χ,q(x) =[2]r
+q
+(1−q)nf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))n/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlxq(h−j+l+1)aj
+(−q(h−r+l+1)f:qf)r
+=[2]r
+q
+[2]r
+qf[f]n
+qf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))q/summationtextr
+j=1(h−j+1)ajζ(h)
+qf,r(−n,x+/summationtextr
+j=1aj
+f).(19)
+Therefore, we obtain the following theorem.
+7Theorem 8. Forn∈Z+, we have
+E(h,r)
+n,χ,q(x)
+=[2]r
+q
+[2]r
+qf[f]n
+qf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))q/summationtextr
+j=1(h−j+1)ajζ(h)
+qf,r(−n,x+/summationtextr
+j=1aj
+f).
+From (18), we note that
+1
+Γ(s)/integraldisplay∞
+0F(h,r)
+q,χ(−t,x)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(/producttextr
+j=1χ(mj))(−1)m1+···+mr
+[m1+···+mr+x]sq, (20)
+wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,....
+From (20), we define the Dirichlet’s type multiple ( h,q)-l-function associated with
+the generalized multiple q-Euler polynomials attached to χ.
+Definition 9. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
+type multiple q-l-function as follows :
+l(h)
+q(s,x|χ) = [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(/producttextr
+i=1χ(mi))(−1)m1+···+mr
+[m1+···+mr+x]sq.
+Note that l(h)
+q(s,x|χ) is a meromorphic function in whole complex plane. It is easy
+to show that
+l(h)
+q(s,x|χ)
+=[2]r
+q
+[2]r
+qf1
+[f]sqf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))q/summationtextr
+j=1(h−j+1)ajζ(h)
+qf,r(s,x+/summationtextr
+j=1aj
+f).
+By (19) and (20), we obtain the following theorem.
+Theorem 10. Forn∈Z+, we have
+l(h)
+q(−n,x|χ) =E(h,r)
+n,χ,q(x).
+Finally, we give the q-extension of Barnes’ type multiple Euler polynomials in (2 ).
+Forx,a1,... ,a r∈Cwith positive real part, let us define the Barnes’ type mutipl e
+8q-Euler polynomials in Cas follows :
+F(r)
+q(t,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mre[a1m1+···+armr+x]t(21)
+=∞/summationdisplay
+n=0E(r)
+n,q(x|a1,... ,a r;b1,... ,b r)tn
+n!,
+whereb1,... ,b r∈Z. By (21), we see that
+E(r)
+n,q(x|a1,... ,a r;b1,... ,b r)
+=[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+qla1+b1+1)···(1+qlar+br+1)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mr[a1m1+···+armr+x]n
+q.
+From (21), we note that
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
+[a1m1+···+armr+x]sq. (22)
+By (22), we define the Barnes’ type multiple q-zeta function as follows :
+ζq,r(s,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
+[a1m1+···+armr+x]sq,
+wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,.... By (21), (22) and (23), we obtain the
+following theorem.
+Theorem 11. Forn∈Z+, we have
+ζq,r(s,x|a1,... ,a r;b1,... ,b r) =E(r)
+n,q(x|a1,... ,a r;b1,... ,b r).
+Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
+the generalized Barnes’ type multiple q-Euler polynomials attached to χare defined
+9by
+F(r)
+q,χ(t,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(r/productdisplay
+i=1χ(mi))e[a1m1+···+armr+x]qt(24)
+=∞/summationdisplay
+n=0E(r)
+n,χ,q(x|a1,... ,a r;b1,... ,b r)tn
+n!,
+From (24), we note that
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q,χ(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
+i=1χ(mi))
+[a1m1+···+armr+x]sq. (25)
+By (25), we can define Barnes’ type multiple q-l-function in C. Fors∈C,x∈Rwith
+x/negationslash= 0,−1,−2,..., let us define the Barnes’ type multiple q-l-function as follows :
+l(r)
+q(s,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
+i=1χ(mi))
+[a1m1+···+armr+x]sq. (26)
+Note that l(r)
+q(s,x|a1,... ,a r;b1,... ,b r) is a meromorphic function in whole complex
+s-plane. By (24), (25) and (26), we easily see that
+l(r)
+q(−n,x|a1,... ,a r;b1,... ,b r) =E(r)
+n,χ,q(x|a1,... ,a r;b1,... ,b r)
+forn∈Z+, (see [1-18]).
+References
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+[18] T. Kim, Y. Simsek, Analytic continuation of the multiple Daehee q-l-functions associated
+with Daehee numbers , Russ. J. Math. Phys. 15(2008), 58–65.
+[19] Y. H. Kim, W. Kim, C. S. Ryoo, On the twisted q-Euler zeta function associated with
+twistedq-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009), 93-100.
+[20] H.Ozden, I.N.Cangul, Y.Simsek, Remarks on q-Bernoulli numbers associated with Daehee
+numbers , Adv. Stud. Contemp. Math. 18(2009), 41-48.
+[21] K. Shiratani, S. Yamamoto, On ap-adic interpolation function for the Euler numbers and
+its derivatives , Mem. Fac. Sci., Kyushu University Ser. A 39(1985), 113-125.
+[22] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers , Advan. Stud.
+Contemp. Math. 11(2005), 205–218.
+[23] Z. Zhang, Y. Zhang, Summation formulas of q-series by modified Abel’s lemma , Adv. Stud.
+Contemp. Math. 17(2008), 119–129.
+Taekyun Kim
+Division of General Education-Mathematics, Kwangwoon Uni versity,
+Seoul 139-701, S. Korea e-mail: [email protected]
+Young-Hee Kim
+Division of General Education-Mathematics,
+Kwangwoon University,
+Seoul 139-701, S. Korea e-mail: [email protected]
+11

1001.0035.txt

@@ -0,0 +1,934 @@
+arXiv:1001.0035v2  [hep-th]  25 Mar 2010Reconstruction of Baxter Q-operator fromSklyanin SOV
+for cyclic representations ofintegrable quantum models
+G. Niccoli
+DESY,Notkestr. 85, 22603 Hamburg, GermanyDESY 09-227
+Abstract
+In [1], the spectrum (eigenvaluesand eigenstates) of a latt ice regularizationsof the Sine-
+Gordon model has been completely characterized in terms of p olynomial solutions with
+certain propertiesof the Baxter equation. This characteri zation for cyclic representations
+hasbeenderivedbytheuseofthe SeparationofVariables(SO V)methodofSklyaninand
+bythedirectconstructionoftheBaxter Q-operatorfamily. Here,wereconstructtheBaxter
+Q-operatorandthesamecharacterizationofthespectrumbyo nlyusingtheSOVmethod.
+This analysis allows us to deduce the main features required for the extension to cyclic
+representationsofotherintegrablequantummodelsofthis kindofspectrumcharacteriza-
+tion.
+Keywords: Integrable Quantum Systems;Separation of Variables;Baxt erQ-operator; PACScode 02.30.IK2
+1. Introduction
+Theintegrabilityofa quantummodelisbydefinitionrelated to theexistenceofamutuallycommu-
+tativefamily Qofself-adjointoperators Tsuchthat
+(A) [T,T′] = 0,
+(B) [T,U] = 0,
+(C) if [ T,O] = 0,∀T,T′∈ Q,
+∀T∈ Q,
+∀T∈ Q,thenO=O(Q),(1.1)
+whereUis the unitary operatordefining the time-evolutionin the mo del; note that the property(C)
+stays for the completeness of the family Q. In the framework of the quantum inverse scattering
+method [2, 3, 4] the Lax operator L(λ)is the mathematical tool which allows to define the transfer
+matrix:
+T(λ) = trC2M(λ),M(λ)≡/parenleftbiggA(λ)B(λ)
+C(λ)D(λ)/parenrightbigg
+≡LN(λ)...L1(λ),(1.2)
+aoneparameterfamilyofmutualcommutativeself-adjointo perators. Theintegrabilityofthemodel
+follows from T(λ)if the properties (B) and (C) of definition (1.1) can be proven for it. In some
+quantummodeltheintegrabilityisderivedbyprovingtheex istenceofafurtherone-parameterfamily
+ofself-adjointoperatorsthe Q-operatorwhichbydefinitionsatisfiesthefollowingproper ties:
+[Q(λ),Q(µ)] = 0,[T(λ),Q(µ)] = 0,∀λ,µ∈C, (1.3)
+plusthe Baxterequationwith thetransfermatrix:
+T(λ)Q(λ) =a(λ)Q(q−1λ)+d(λ)Q(qλ). (1.4)
+This is in particular the case for those models (like Sine-Go rdon [1]) for which the time-evolution
+operatorUis expressedin terms of Q. A naturalquestionarises: Is the integrablestructure of t hese
+quantummodelscompletelycharacterizedbythetransferma trixT(λ)?
+Note that a standard procedure1to prove the existence of Q(λ)is by a direct construction of an
+operatorsolutionoftheBaxterequation(1.4). Moreover,t hecoefficients a(λ)andd(λ)aswellasthe
+analytic and asymptotics properties of Q(λ)are some model dependent features which are derived
+bythe construction. Let usrecall thatthe generalstrategy [11, 12,13,14, 15] ofthisconstructionis
+to findagaugetransformation2such that the action of each gaugetransformedLax matrixon Q(λ)
+becomesupper-triangular. Thenthe Q-operatorassumesa factorized localformandthe problemof
+its existence in such a form is reduced to the problem of the ex istence of some model dependent
+specialfunction3.
+1Itis worth recalling that there are also others constructio ns of theQ-operator. An interesting example is presented in the
+series of works [5,6,7] by V.V. Bazhanov, S.L.Lukyanov and A .B.Zamolodchikov on the integrable structure of conformal
+field theories. In [6,7] the Q-operator is obtained asatransfer-matrix byatrace proced ure ofafundamental L-operator with
+q-oscillator representation for the auxiliary space (see al so [8, 9]). This construction can be extended to massive inte grable
+quantum field theories as itwas argued by thesame authors in [ 10].
+2Itleaves unchanged thetransfer matrix while modifies the mo nodromy matrix M(λ)defined in (1.2) .
+3Thequantumdilogarithm functions [16,17,18,19,20,21,22,23,24,25]forexample a ppear intheSinh-Gordon model
+[26],in their non-compact form,and in the Sine-Gordon model[1],in their cyclicform.3
+It is worth pointing out that on the one hand the construction of these special functionsfor general
+modelscanrepresenta concretetechnicalproblem4andthat ontheotherhandtheexistenceofsuch
+functionsis onlya sufficientcriterionforthe existence of Q(λ). It is then a relevantquestionif it is
+possibletobypassthiskindofconstructionprovidingadif ferentproofofthe existenceof Q(λ).
+Given an integrable quantum model the first fundamental task to solve is the exact solution of its
+spectral problem , i.e. the determination of the eigenvalues and the simultan eous eigenstates of the
+operator family Q, defined in (1.1). There are several methods to analyze this s pectral problem as
+thecoordinate Bethe ansatz [27, 28, 29], the TQmethod [28], the algebraic Bethe ansatz (ABA)
+[2, 3, 4], the analyticBethe ansatz [30] and the separation of variables (SOV) meth od of Sklyanin
+[31, 32, 33]; this last one seems to be more promising. Indeed , on the one hand it resolves the
+problems related to the reduced applicability of other meth ods (like ABA) and on the other hand
+it directly implies the completeness of the characterizati on of the spectrum which instead for other
+methodshastobeproven.
+For cyclic representations [34] of integrable quantum mode ls the SOV method should lead to the
+characterizationoftheeigenvaluesandthesimultaneouse igenstatesofthetransfermatrix T(λ)bya
+finite5systemofBaxter-likeequations. However,itisworthpoint ingoutthatsuchacharacterization
+of the spectrum is not the most efficient; this is in particula r true in view of the analysis of the
+continuum limit. Here the main question reads: Is it possibl e to define a set of conditions under
+whichtheSOVcharacterizationofthespectrumcanbereform ulatedintermsofafunctionalBaxter
+equation? In fact, this is equivalent to ask if we can reconst ruct theQ-operator from the finite
+system of Baxter-like equations. In this case the solution o f the spectral problem is reduced to the
+classification of the solutions of the Baxter equation which satisfy some analytic and asymptotic
+propertiesfixedbythe operators TandQ.
+The lattice Sine-Gordonmodelis used asa concreteexamplew herethese questionsaboutquantum
+integrability find a complete and affirmative answer. Indeed , in section 3, we show that the SOV
+characterization of the transfer matrix spectrum is exactl y equivalent to a functional equation of
+the form detD(Λ) = 0, whereD(λ)(see (3.21)) is a one-parameter family of quasi-tridiagonal
+matrices. In section 4, we show that this functional equatio n is indeed equivalent to the Baxter
+functional equation and, in section 5, we use these results t o reconstruct the Baxter Q-operator
+with the same level of accuracy obtained by the direct constr uction presented in [1]. It is worth
+pointingoutthat these resultsallowusto provethat thetra nsfermatrix T(λ)(plustheΘ-chargefor
+even chain) describes the family Qof complete commuting self-adjoint charges which implies t he
+quantum integrabilityof the model accordingto definition ( 1.1). So that in the Sine-Gordonmodel
+theBaxter Q-operatorplaysonlytheroleofa usefulauxiliaryobject.
+Let us point out that one of the main advantages of the spectru m characterization derived for the
+Sine-Gordonmodelisthe possibilityto proveanexactrefor mulationin termsof non-linearintegral
+4TheSine-Gordon model at irrational values of the coupling β2is asimple case where this kind of problem emerges.
+5Thenumber of equations in thesystem is finite and related to t hedimension of thecyclic representation.4
+equations6(NLIE).Thiswill bethe subjectof a futurepublicationwher ethe NLIEcharacterization
+will lead us by the implementation of the continuum limit to t he description of the Sine-Gordon
+spectrum in all the interesting regimes. These results will be shown to be consistent with those
+obtained previously in the literature7[37, 38, 39, 40, 41, 42] (see [43, 44] for reviews). Note that
+the methodbasedon thereformulationofthe spectralproble min termsofNLIEhasbeenalso used
+recently [49] to derive the Sinh-Gordonspectrum in finite vo lume and to characterize the spectrum
+in theinfraredandultravioletlimits.
+The analysis of the Sine-Gordon model allows us to infer the m ain features required to extend this
+kindofspectrumcharacterizationtocyclicrepresentatio nsofotherintegrablequantummodels. This
+is particularly relevant for those models for which a direct construction of the Baxter Q-operator
+encounterstechnicaldifficulties.
+Acknowledgments. I would like to thank J. Teschner for stimulating discussion s and suggestions on a prelimi-
+nary versionof this workand J.-M.Maillet for the interests hown.
+I gratefullyacknowledge support from the ECbythe Marie Cur ie Excellence GrantMEXT-CT-2006-042695.
+2. The Sine-Gordon model
+Weusethissectiontorecallthemainresultsderivedin[1]o nthedescriptionintermsofSOVofthe
+lattice Sine-Gordonmodel. This will be used as the starting point to introducea characterizationof
+the spectrumof the transfermatrix T(λ)which will lead to the constructionof the Q-operatorfrom
+SOV.
+2.1 Definitions
+Thelattice Sine-Gordonmodelcanbecharacterizedbythefo llowingLaxmatrix8:
+LSG
+n(λ) =κn
+i/parenleftigg
+iun(q−1
+2κnvn+q+1
+2κ−1
+nv−1
+n)λnvn−λ−1
+nv−1
+n
+λnv−1
+n−λ−1
+nvniu−1
+n(q+1
+2κ−1
+nvn+q−1
+2κnv−1
+n)/parenrightigg
+,(2.5)
+whereλn≡λ/ξnfor anyn∈ {1,...,N}withξnandκnparameters of the model. For any n∈
+{1,...,N}thecoupleofoperators( un,vn)defineaWeyl algebra Wn:
+unvm=qδnmvmun,whereq=e−πiβ2. (2.6)
+We will restrictourattentiontothecase inwhich qisarootofunity,
+β2=p′
+p, p,p′∈Z>0, (2.7)
+6Thistype of equations werebefore introduced in adifferent framework in [35,36]
+7See [45, 46] for a related model analyzed in the framework of A BA and [47, 48] for the corresponding finite volume
+continuum limit.
+8Thelattice regularization of the Sine-Gordon model that we consider here goes back to [4,50] and is related to formula-
+tions which have morerecently been studied in [51,52,53].5
+withp≡2l+ 1odd andp′even so that qp= 1. In this case each Weyl algebra Wnadmits a
+finite-dimensional representation of dimension p. In fact, we can represent the operators un,vnon
+thespace ofcomplex-valuedfunctions ψ:SN
+p→Cas
+un·ψ(z1,...,z N) =unznψ(z1,...,z n,...,z N),
+vn·ψ(z1,...,z N) =vnψ(z1,...,q−1zn,...,z N).(2.8)
+whereSp={q2n;n= 0,...,2l}is a subset of the unit circle; note that Sp={qn;n= 0,...,2l}
+sinceq2l+2=q.
+Themonodromymatrix M(λ)definedin(1.2)intermsoftheLax-matrix(2.5)satisfiesthe quadratic
+relations:
+R(λ/µ)(M(λ)⊗1)(1⊗M(µ)) = (1⊗M(µ))(M(λ)⊗1)R(λ/µ), (2.9)
+wheretheauxiliary R-matrixisgivenby
+R(λ) =
+qλ−q−1λ−1
+λ−λ−1q−q−1
+q−q−1λ−λ−1
+qλ−q−1λ−1
+. (2.10)
+The elements of M(λ)generate a representation RNof the so-called Yang-Baxter algebra char-
+acterized by the 4Nparametersκ= (κ1,...,κ N),ξ= (ξ1,...,ξ N),u= (u1,...,u N)and
+v= (v1,...,v N); in the present paper we will restrict to the case un= 1,vn= 1,n= 1,...,N.
+The commutation relations (2.9) are at the basis of the proof of the mutual commutativity of the
+T-operators.
+Inthecase ofa latticewith Nevenquantumsites, we havealso tointroducetheoperator:
+Θ =N/productdisplay
+n=1v(−1)1+n
+n, (2.11)
+whichplaystheroleofa gradingoperator inthe Yang-Baxteralgebra:
+Proposition 6 of [1] Θcommuteswiththetransfermatrixandsatisfiesthefollowin gcommutation
+relationswith theentriesofthemonodromymatrix:
+ΘC(λ) =qC(λ)Θ,[A(λ),Θ] = 0, (2.12)
+B(λ)Θ =qΘB(λ),[D(λ),Θ] = 0. (2.13)
+Moreover,the Θ-chargeallowstoexpressthe asymptoticsofthetransferma trixas:
+lim
+logλ→∓∞λ±NT(λ) =/parenleftiggN/productdisplay
+a=1κaξ±1
+a
+i/parenrightigg
+/parenleftbig
+Θ+Θ−1/parenrightbig
+. (2.14)6
+Let us denotewith ΣTthe spectrum(the set of the eigenvaluefunctions t(λ)) of the transfer matrix
+T(λ). By the definitions(1.2) and (2.5), then ΣTis contained9inC[λ2,λ−2](N+eN−1)/2, where we
+haveusedthenotatione N= 0forNoddand1forNeven.
+Notethat inthecase of Neven,the Θ-chargenaturallyinducesthegrading ΣT=/uniontextl
+k=0Σk
+T,where:
+Σk
+T≡/braceleftigg
+t(λ)∈ΣT: lim
+logλ→∓∞λ±Nt(λ) =/parenleftiggN/productdisplay
+a=1κaξ±1
+a
+i/parenrightigg
+(qk+q−k)/bracerightigg
+.(2.15)
+This simply follows by the asymptotics of T(λ)and by its commutativity with Θ. In particular,
+anyt(λ)∈Σk
+Tis aT-eigenvalue corresponding to simultaneous eigenstates of T(λ)andΘwith
+Θ-eigenvalues q±k.
+2.2 CyclicSOVrepresentations
+TheseparationofvariablesmethodofSklyaninisbasedonth eobservationthatthespectralproblem
+forT(λ)simplifies considerablyif one worksin an auxiliaryreprese ntationwherethe commutative
+familyofoperators B(λ)isdiagonal.
+InthecaseoftheSine-Gordonmodelthevectorspace10CpNunderlyingtheSOVrepresentationcan
+beidentifiedwiththespaceoffunctions Ψ(η)definedforηtakenfromthediscreteset
+BN≡/braceleftbig
+(qk1ζ1,...,qkNζN); (k1,...,k N)∈ZN
+p/bracerightbig
+, (2.16)
+onthesefunctions B(λ)actsasa multiplicationoperator,
+BN(λ)Ψ(η) =ηeN
+Nbη(λ)Ψ(η), b η(λ)≡N/productdisplay
+n=1κn
+i[N]/productdisplay
+a=1(λ/ηa−ηa/λ) ; (2.17)
+where[N]≡N−eNandη1,...,η[N]are the zerosof bη(λ). In the case of even Nit turns out that
+we needa supplementaryvariable ηNinordertobeable toparameterizethe spectrumof B(λ).
+In[1]wehaveproventhatforgeneralvaluesoftheparameter sκandξoftheoriginalrepresentation
+it is possible to construct these SOV representationsand mo reoverwe have defined the map which
+fixestheSOVparameter ηintermsoftheparameters κandξ.
+In these SOV representations the spectral problem for T(λ)is reduced to the following discrete
+system ofBaxter-likeequationsin thewave-function Ψt(η) =/a\}bracketle{tη|t/a\}bracketri}htofaT-eigenstate |t/a\}bracketri}ht:
+t(ηr)Ψ(η) =a(ηr)T−
+rΨ(η)+d(ηr)T+
+rΨ(η)∀r∈ {1,...,[N]},(2.18)
+9Herewith C[x,x−1]Mwearedenoting the linear space ofthe Laurentpolynomials o f degreeMin thevariable x∈C.
+10It is always possible to provide the structure of Hilbert spa ce to this finite-dimensional linear space. In particular, t he
+scalar product in the SOVspace is naturally introduced by th e requirement that the transfer matrix is self-adjoint in th e SOV
+representation. Appendix B addresses this issue.7
+whereT±
+raretheoperatorsdefinedby
+T±
+rΨ(η1,...,η N) = Ψ(η1,...,q±1ηr,...,η N),
+whilethe coefficients a(λ)andd(λ)are definedby:
+a(λ) =N/productdisplay
+n=1κn
+iλn(1−iq−1/2λnκn)(1−iq−1/2λn
+κn),d(λ) =qNa(−λq).(2.19)
+Inthecase of Nevenwe haveto addto thesystem(2.18) thefollowingequatio ninthevariable ηN:
+T+
+NΨ±k(η) =q±kΨ±k(η), (2.20)
+fort(λ)∈Σk
+Twithk∈ {0,...,l}.NotethatthecyclicityoftheseSOVrepresentationsisexpr essed
+bytheidentificationof (T±
+j)pwith theidentityforany j∈ {1,...,N}.
+3. SOV characterization of T-eigenvalues
+Let usintroducetheoneparameterfamily D(λ)ofp×pmatrix:
+D(λ)≡
+t(λ)−d(λ) 0 ··· 0 −a(λ)
+−a(qλ)t(qλ)−d(qλ) 0 ··· 0
+0......
+... ···...
+... ···...
+...... 0
+0... 0−a(q2l−1λ)t(q2l−1λ)−d(q2l−1λ)
+−d(q2lλ) 0 ... 0−a(q2lλ)t(q2lλ)
+(3.21)
+wherefornow t(λ)isjust anevenLaurentpolynomialofdegree N+eN−1inλ.
+Lemma 1. Thedeterminant detpDisanevenLaurentpolynomialofmaximaldegree N+eN−1in
+Λ≡λp.
+Proof.Let us start observingthat D(λq)is obtainedby D(λ)exchangingthe first and p-th column
+andafterthefirst and p-throw,so that
+det
+pD(λq) = det
+pD(λ)∀λ∈C, (3.22)
+whichimpliesthat detpDisfunctionof Λ. Let usdevelopthedeterminant:
+det
+pD(Λ) =p/productdisplay
+h=1a(λqh)+p/productdisplay
+h=1a(−λqh)−qNa(λ)a(−λ) det
+2l−1D(1,2l+1),(1,2l+1)(λ)
+−qNa(λq)a(−λq) det
+2l−1D(1,2),(1,2)(λ)+t(λ)det
+2lD1,1(λ), (3.23)8
+whereD(h,k),(h,k)(λ)denotes the (2l−1)×(2l−1)sub-matrix of D(λ)obtained removing the
+rowsandcolumns handkwhileDh,k(λ)denotesthe 2l×2lsub-matrixof D(λ)obtainedremoving
+therowhandcolumn k. Theinteresttowardthisdecompositionof detpD(Λ)isduetothefact that
+the matrices D(1,2),(1,2)(λ),D(1,2l+1),(1,2l+1)(λ)andD1,1(λ)aretridiagonal matrices. Following
+thesamereasoningusedinLemma4toprovethat det2lD1,1(λ)isanevenfunctionof λwecanalso
+showthatthisistruefor det2l−1D(1,2),(1,2)(λ)anddet2l−1D(1,2l+1),(1,2l+1)(λ). Fromtheparityof
+these functionsthe parityof detpD(Λ)followsbyusing(3.23).
+Beinga(λ),d(λ)andt(λ)Laurentpolynomialofdegree Ninλ,inthecaseof Neventhestatement
+ofthelemmaisalreadyproven;so wehavejust toshowthat:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) = 0 (3.24)
+forNoddwhichfollowsobservingthat:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) =i±pNN/productdisplay
+n=1κp
+nξ±p
+ndet
+p/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δh,k+1−q(1∓1)N/2δh,k−1/vextenddouble/vextenddouble/vextenddouble.
+(3.25)
+The interesttowardthe function detpD(Λ)isdueto the fact thatit allowsthefollowingcharacteri-
+zationofthe T-spectrum:
+Lemma 2. ΣTis the set of all the functions t(λ)∈C[λ2,λ−2](N+eN−1)/2which satisfy the system
+of equations:
+det
+pD(ηp
+a) = 0∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN, (3.26)
+plusin thecaseof Neven:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) = 0. (3.27)
+Proof.The requirement that the system of equations (2.18) admits a non-zerosolution leads to the
+equations(3.26),while theequation(3.27) foreven Nsimplyfollowsbyobservingthat:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) = det
+p/vextenddouble/vextenddouble/vextenddoubleq(1∓1)N/2δi,j−1+q−(1∓1)N/2δi,j+1−(qk+q−k)δi,j/vextenddouble/vextenddouble/vextenddouble
+×(−1)N/productdisplay
+n=1/parenleftbig
+iκnξ±
+n/parenrightbigp= 0. (3.28)
+Note that the above characterization of the T-spectrum ΣTrequires as input the knowledge of BN,
+i.e. the lattice of zeros of the operator B(λ). It is so interesting to notice that this characterization9
+has in fact a reformulation which is independent from the kno wledge of BN. To explain this let us
+notethatLemma1allowsto introducethefollowingmap:
+Dp,N:t(λ)∈C[λ2,λ−2](N+eN−1)/2→ Dp,N(t(λ))≡det
+pD(Λ)∈C[Λ2,Λ−2](N+eN−1)/2.
+(3.29)
+Intermsofthismapwecanintroduceafurthercharacterizat ionofthespectrumofthetransfermatrix
+T(λ).
+Theorem 1. The spectrum ΣTof the transfer matrix T(λ)coincides with the kernel NDp,N⊂
+C[λ2,λ−2](N+eN−1)/2ofthe map Dp,N.
+Proof.The inclusion NDp,N⊂ΣTis trivial thanks to Lemma 2, vice-versa if t(λ)∈ΣTthen
+the function detpD(Λ)is zero in N+eNdifferent values of Λ2which thanks to Lemma 1 implies
+detpD(Λ)≡0,i.e.ΣT⊂ NDp,N.
+That is the set of eigenvalues of the transfer matrix T(λ)is exactly characterized as the subset of
+C[λ2,λ−2](N+eN−1)/2whichcontainsallthesolutionsofthefunctionalequation detpD(Λ) = 0. In
+thenextsectionwewill showthat thisfunctionalequationi s nothingelse thattheBaxterequation.
+Remark 1. Let us note that the same kind of functional equation detD(Λ) = 0 also appears
+in [54, 55, 56]. There it recasts, in a compact form, the funct ional relations which result from the
+truncatedfusionsoftransfermatrixeigenvalues. Itissor elevanttopointoutthatforthe BBS-model11
+in the SOV representation the non-triviality condition of t he solutions of the system of Baxter-like
+equations has been shown [60] to be equivalent to the truncat ion identity in the fusion of transfer
+matrixeigenvalues.
+4. Baxterfunctional equation
+The main consequence of the previous analysis is that it natu rally leads to the complete character-
+ization of the transfer matrix spectrum in terms of polynomi al solutions of the Baxter functional
+equation.
+Theorem2. Lett(λ)∈ΣTthent(λ)definesuniquelyuptonormalizationapolynomial Qt(λ)that
+satisfiestheBaxterfunctionalequation:
+t(λ)Qt(λ) =a(λ)Qt(λq−1)+d(λ)Qt(λq)∀λ∈C. (4.30)
+Proof.The fact that given a t(λ)∈C[λ2,λ−2](N+eN−1)/2there exists up to normalizationat most
+one polynomial Qt(λ)that satisfies the Baxter functional equationhas been prove nin Lemma 2 of
+[1]. So we have to prove only the existence of Qt(λ)∈C[λ]. An interesting point about the proof
+givenhereisthatit isa constructiveproof.
+11TheBBS-model [12, 57,58,59] has been analyzed in the SOVapp roach in aseries of works [60,61,62].10
+Let usnoticethatthe condition t(λ)∈ΣT≡ NDp,Nimpliesthatthe p×pmatrixD(λ)hasrank2l
+foranyλ∈C\{0}. Letusdenotewith
+Ci,j(λ) = (−1)i+jdet
+2lDi,j(λ) (4.31)
+the(i,j)cofactorof the matrix D(λ); then the matrix formedout of these cofactorshasrank 1, i.e.
+all thevectors:
+Vi(λ)≡(Ci,1(λ),Ci,2(λ),...,Ci,2l+1(λ))T∈Cp∀i∈ {1,...,2l+1}(4.32)
+areproportional:
+Vi(λ)/Ci,1(λ) =Vj(λ)/Cj,1(λ)∀i,j∈ {1,...,2l+1},∀λ∈C. (4.33)
+Theproportionality(4.33)oftheeigenvectorsV i(λ)implies:
+C2,2(λ)/C2,1(λ) =C1,2(λ)/C1,1(λ) (4.34)
+which,byusingtheproperty(A.69),canberewrittenas:
+C1,1(λq)/C1,2l+1(λq) =C1,2(λ)/C1,1(λ). (4.35)
+Moreover,thefirst elementinthe vectorialcondition D(λ)V1(λ) =0¯reads:
+t(λ)C1,1(λ) =a(λ)C1,2l+1(λ)+d(λ)C1,2(λ). (4.36)
+Let us note that from the form of a(λ),d(λ)andt(λ)∈ΣTit follows that all the cofactors are
+Laurentpolynomialofmaximaldegree122lNinλ:
+Ci,j(λ) = Ci,jλ−2lN+ai,j4lN−(ai,j+bi,j)/productdisplay
+h=1(λ(i,j)
+h−λ). (4.37)
+In Lemma 5, we show that the equations (4.35) and (4.36) imply that if C 1,1(λ)has a common
+zero with C 1,2(λ)then this is also a zero of C 1,2l+1(λ)and that the same statement holds ex-
+changing C 1,2(λ)with C 1,2l+1(λ). So we can denote with C1,1C1,1(λ),C1,2l+1C1,2l+1(λ)and
+C1,2C1,2(λ)the polynomials of maximal degree 4lNobtained simplifying the common factors in
+C1,1(λ), C1,2l+1(λ)and C1,2(λ). Then,byequation(4.35),theyhavetosatisfythe relation s:
+C1,2l+1(λ) =q¯N1,1C1,1(λq−1),C1,2(λ) =q−¯N1,1C1,1(λq)andC1,2l+1=ϕC1,1,(4.38)
+whereϕ≡C1,1/C1,2and¯N1,1is the degree of the polynomial C1,1(λ). So that equation (4.36)
+assumestheformofa Baxterequationin thepolynomial C1,1(λ):
+t(λ)C1,1(λ) = ¯a(λ)C1,1(λq−1)+¯d(λ)C1,1(λq), (4.39)
+12Theai,jandbi,jare non-negative integers and λ(i,j)
+h/ne}ationslash= 0for anyh∈ {1,...,4lN−(ai,j+bi,j)}.11
+with coefficients ¯a(λ)≡q¯N1,1ϕa(λ)and¯d(λ)≡q−¯N1,1ϕ−1d(λ). Note that the consistence of the
+aboveequationimpliesthat ϕisap-rootoftheunity. Indeed,denotingwith ¯D(Λ)thematrixdefined
+asin(3.21) butwithcoefficients ¯a(λ)and¯d(λ), equation(4.39) implies:
+0 = det
+p¯D(Λ)≡(ϕp−1)/parenleftiggp/productdisplay
+h=1a(λqh)−ϕ−pp/productdisplay
+h=1a(−λqh)/parenrightigg
+. (4.40)
+The expansionfor detp¯D(Λ)in (4.40) is derivedby using the expansion(3.23) for detp¯D(Λ), the
+formulae13:
+det
+2lD1,1(λ) = det
+2lD1,1(λ), (4.41)
+det
+2l−1D(1,2),(1,2)(λ) = det
+2l−1D(1,2),(1,2)(λ), (4.42)
+det
+2l−1D(1,2l+1),(1,2l+1)(λ) = det
+2l−1D(1,2l+1),(1,2l+1)(λ), (4.43)
+andthecondition t(λ)∈ΣT. Finally,if wedefine:
+Qt(λ)≡λaC1,1(λ), (4.44)
+whereq−a=q¯N1,1ϕwitha∈ {0,..,2l},we getthestatementofthetheorem.
+Remark2. Theprevioustheoremimpliesthatforany t(λ)∈ΣTthepolynomialsolution Qt(λ)of
+theBaxterequationcanberelatedtothedeterminantofatri diagonalmatrixoffinitesize p−1. Note
+thatthe spectrumoftheSine-Gordonmodelinthecase ofirra tionalcoupling ¯β2shouldbededuced
+fromβ2=p′/prational in the limit β2→¯β2. In particular, this implies that underthis limit ( p→
++∞)thedimensionoftherepresentationdivergesaswellasthe sizeofthetridiagonalmatrixwhose
+determinant is associated to the solution Qt(λ)of the Baxter equation. It is then relevant to point
+out that in the case of the quantum periodic Toda chain the sol utions of the corresponding Baxter
+equationareexpressedintermsofdeterminantsofsemi-infi nitetridiagonalmatrices[63,13, 64].
+It is worth noticing that the set of polynomials Qt(λ), introducedin the previoustheorem,admitsa
+moreprecisecharacterization:
+Theorem 3. Lett(λ)∈ΣTthent(λ)defines uniquely up to normalization a polynomial solution
+Qt(λ)oftheBaxterfunctionalequation(4.30) ofmaximaldegree 2lN.
+Inthecase Nodd,it results:
+Qt(0)≡Q0/\e}atio\slash= 0,andlim
+λ→∞λ−2lNQt(λ)≡Q2lN/\e}atio\slash= 0. (4.45)
+In the case Neven, the condition (4.45) selects t(λ)∈Σ0
+Twhile fort(λ)∈Σk
+Twithk∈ {1,...,l}
+we havethecharacterization Q0=Q2lN= 0and:
+lim
+λ→0Qt(λq)
+Qt(λ)=q±k,lim
+λ→∞Qt(λq)
+Qt(λ)=q−(N±k). (4.46)
+13They follow from the tridiagonality of these matrices and by using Lemma3.12
+Proof.Thankstoformula(A.74),thecofactor C 1,1(λ)∈C[λ,λ−1]2lNiseveninλandso it admits
+theexpansions:
+C1,1(λ) = C1,1λ−2lN+2˜a1,12lN−(˜a1,1+˜b1,1)/productdisplay
+i=1(λ(1,1)
+i−λ)(λ(1,1)
+i+λ).(4.47)
+Let us note now that by using the properties(A.69) and (A.74) , the relation (4.34) can be rewritten
+as:
+C1,1(λq)C1,1(λ) =qNC1,2(λ)C1,2(−λ). (4.48)
+Usingthat andthegeneralrepresentation(4.37)forthe cof actor C 1,2(λ), weget:
+a1,2= 2˜a1,1≡2a,b1,2= 2˜b1,1≡2b,C2
+1,2=C2
+1,1q−2(N+b)(4.49)
+and:/parenleftig
+λ(1,1)
+i/parenrightig2
+=/parenleftig
+λ(1,2)
+i/parenrightig2
+≡¯λ2
+i,/parenleftig
+λ(1,2)
+i+2lN−(a+b)/parenrightig2
+=/parenleftbig¯λi/q/parenrightbig2(4.50)
+with¯λi/\e}atio\slash= 0for anyi∈ {1,...,2lN−(a+b)}withaandb∈Z≥0. Note that the equation (4.49)
+andthefactthat ϕ≡C1,1/C1,2isap-rootofthe unityimply ϕ=qb+N. Thenwecanwrite:
+C1,1(λ) = Cλ−2lN+2a2lN−(a+b)/productdisplay
+i=1(¯λi+λ)(¯λi−λ), (4.51)
+C1,2(λ) =qaCλ−2lN+2a2lN−(a+b)/productdisplay
+i=1(¯λi+λ)((−1)H(x−i)¯λi−λq), (4.52)
+whereC≡C1,1andH(n)≡ {0forn <0,1forn≥0}is the Heaviside step function. Here, x
+isanon-negativeintegerwhichisfixedtozerothankstoform ula(4.38). Thenthesolution Qt(λ)of
+theBaxter equation(4.30) belongsto C[λ]2lNandhastheform:
+Qt(λ)≡λa2lN−(a+b)/productdisplay
+i=1(¯λi−λ). (4.53)
+Let usshow nowthe remainingstatementsof thetheoremconce rningthe asymptoticsof Qt(λ). To
+thisaimwe computethe limits:
+lim
+logλ→∓∞λ±2lNC1,1(λ) = det
+2l/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δi,j+1+q(1∓1)N/2δi,j−1−(qk+q−k)δeN,1δi,j/vextenddouble/vextenddouble/vextenddouble
+i/ne}ationslash=1,j/ne}ationslash=1
+×N/productdisplay
+h=1(κhξ±1
+h
+i)2l= (δeN,1(1+(2l+1)δk,0)−1)N/productdisplay
+h=1(κhξ∓1
+h
+i)2l,(4.54)
+whichimply:
+a=b= 0, (4.55)13
+forNoddandNevenwitht(λ)∈Σ0
+T,i.e. thecondition(4.45). Inthe remainingcases, Nevenand
+t(λ)/∈Σ0
+T,the sameformulaimplies:
+a/\e}atio\slash= 0,b/\e}atio\slash= 0, (4.56)
+sothatQ0=Q2lN= 0,whiletheasymptoticsbehaviors(4.46)simplyfollowtaki ngtheasymptotics
+oftheBaxterequationsatisfied by Qt(λ).
+5.Q-operator: Existence andcharacterization
+Let us denote with Σtthe eigenspace of the transfer matrix T(λ)corresponding to the eigenvalue
+t(λ)∈ΣT,then:
+Definition 1. LetQ(λ)betheoperatorfamily definedby:
+Q(λ)|t/a\}bracketri}ht ≡Qt(λ)|t/a\}bracketri}ht ∀|t/a\}bracketri}ht ∈Σtand∀t(λ)∈ΣT, (5.57)
+withQt(λ)the element of C[λ]2lNcorresponding to t(λ)∈ΣTby the injection defined in the
+previoustheorem.
+Under the assumptions ξandκreal or imaginarynumbers, which assure the self-adjointne ssof the
+transfermatrix T(λ)forλ∈R,thefollowingtheoremholds:
+Theorem4. Theoperatorfamily Q(λ)isaBaxter Q-operator:
+(A)Q(λ)satisfieswith T(λ)thecommutationrelations:
+[Q(λ),T(µ)] = [Q(λ),Q(µ)] = 0∀λ,µ∈C, (5.58)
+plusthe Baxterequation:
+T(λ)Q(λ) =a(λ)Q(λq−1)+d(λ)Q(λq)∀λ∈C. (5.59)
+(B)Q(λ)isa polynomialofdegree 2lNinλ:
+Q(λ)≡2lN/summationdisplay
+n=0Qnλn,
+with coefficients Qnself-adjointoperators.
+(C)Inthecase Nodd,the operator Q2lN=idandQ0isaninvertibleoperator.
+(D)Inthecase Neven,Q(λ)commuteswiththe Θ-chargeandtheoperator Q2lNistheorthogonal
+projectionontothe Θ-eigenspacewith eigenvalue1. Q0hasnon-trivialkernel coincidingwith
+theorthogonalcomplementto the Θ-eigenspacewith eigenvalue1.14
+Proof.Note that the self-adjointness of the transfer matrix T(λ)implies that Q(λ)is well defined,
+indeed its action is defined on a basis. The property (A) is a tr ivial consequence of Definition 1.
+Notethat theinjectivityofthemap t(λ)∈ΣT→Qt(λ)∈C[λ]2lNimplies:
+(Qt(λ))∗=Qt(λ∗)∀λ∈C (5.60)
+being(a(λ))∗=d(λ∗)and(t(λ))∗=t(λ∗). So we get the Hermitian conjugation property
+(Q(λ))†=Q(λ∗), i.e. the self-adjointness of the operators Qn. The properties (C) and (D) of
+the operators Q0andQ2lNdirectly follow from the asymptotics of the eigenfunction Qt(λ)while
+thecommutativityof Q(λ)andΘisa directconsequenceofthecommutativityof T(λ)andΘ.
+6. Conclusion
+Intheprevioussectionwehaveshownthatbyonlyusingthech aracterizationofthespectrumofthe
+transfer matrix obtained by the SOV method we were able to rec onstruct the Q-operator. It is also
+interestingto pointoutastheresultsderivedin [1]togeth erwiththoseofthepresentarticleyield:
+Theorem5. Thefamily Qwhichcharacterizesthequantumintegrabilityofthelatti ceSine-Gordon
+model(see definition(1.1)) isdescribedby thetransfermat rixT(λ)fora chainwith Noddnumber
+of siteswhile by T(λ)plustheΘ-chargefora chainwith Nevennumberof sites.
+Proof.LetusstartnoticingthatProposition3andTheorem4of[1]a rederivedonlyusingtheSOV
+method (i.e. without any assumption about the existence of t heQ-operator). So only using SOV
+analysis we have derived that for Nodd the transfer matrix T(λ)has simple spectrum while for
+Neven this is true for T(λ)plus theΘ-charge; i.e. they define a complete family of commuting
+observables and so satisfy the properties(A) and (C) of the d efinition (1.1). In this article we have
+moreover shown that the Q-operator is defined as a function of the transfer matrix whic h implies
+the property(B) of (1.1) recalling that in [1] the time-evol utionoperator Uhas been expressed as a
+functionofthe Q-operator.
+Let us shortly point out the main features required in abstra ct to extend to cyclic representationsof
+other integrable quantum models the same kind of spectrum ch aracterization derived here for the
+lattice Sine-Gordonmodel.
+R1.The model admits an SOV description and the spectrum of the tr ansfer matrix can be charac-
+terizedbyasystem ofBaxter-likeequationsin the T-wave-function Ψ(η) =/a\}bracketle{tη|t/a\}bracketri}ht:
+t(ηr)Ψ(η) =a(ηr)Ψ(η1,...,q−1ηr,...,η N)+d(ηr)Ψ(η1,...,qη r,...,η N),(6.61)
+where(η1,...,ηN)∈BNwithBNtheset ofzerosofthe B-operatorintheSOV representation.
+Here,theparameter qisa rootofunitydefinedasin (2.6) and(2.7).15
+Note that for cyclic representationsof an integrable quant um model the set BNis a finite subset of
+CN. So the coefficients a(ηr)andd(ηr)are specified only in a finite number of points where they
+satisfy thefollowingaveragevaluerelations14:
+A(ηp
+r) =p/productdisplay
+k=1a(qkηr),D(ηp
+r) =p/productdisplay
+k=1d(qkηr). (6.62)
+HereA(Λ)andD(Λ)are the average values of the operator entries A(λ)andD(λ)of the mon-
+odromy matrix. Let us recall that the operator entries of the monodromymatrix are expected to be
+polynomials(orLaurentpolynomials)inthespectralparam eterλsothecorrespondingaverageval-
+uesarepolynomials(orLaurentpolynomials)in Λ≡λp. Itisthennaturaltointroducethefunctions
+a(λ)andd(λ)aspolynomial(orLaurentpolynomial)solutionsofthefoll owingaveragerelations:
+A(Λ)+γB(Λ) =p/productdisplay
+k=1a(qkλ),D(Λ)+δB(Λ) =p/productdisplay
+k=1d(qkλ), (6.63)
+whereB(Λ)istheaveragevalueoftheoperator B(λ)andγandδare constanttobe fixed.
+R2.Let usdenotewith Zf(λ)the set ofthezerosofthefunctions f(λ), then:
+∃λ0∈Za(λ):λ0/∈ ∪2l−1
+h=0Zd(λqh). (6.64)
+R3.Theaveragevaluesofthefunctions aanddarenotcoincidinginallthezerosofthe B-operator:
+A(ηp
+a)/\e}atio\slash=D(ηp
+a)∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN. (6.65)
+The requirement R1yields the introduction of the p×pmatrixD(λ), defined as in (3.21), by
+the functions a(λ)andd(λ)solutions of (6.63). This should allow us to reformulate the spectral
+problem for the transfer matrix as the problem to classify al l the solutions t(λ)to the functional
+equationdetpD(Λ) = 0ina modeldependentclassoffunctions.
+The requirement R2implies that the rank of the matrix D(λ)is almost everywhere 2l. Indeed, the
+condition (6.64) implies C 1,p(λ0)/\e}atio\slash= 0, independently from the function t(λ). Being the cofactor
+C1,p(λ)acontinuousfunctionofthespectralparametertheabovest atementontherankofthematrix
+D(λ)follows. Underthisconditionwecanfollowtheprocedurepr esentedinTheorem2toconstruct
+the solutionsof the Baxter equation. Then the self-adjoint nessof the transfer matrix Tallows us to
+proceedasinsection5to showthe existenceofthe Q-operatorasa functionof T.
+The requirement R3is a sufficient criterion15to show the simplicity of the spectrum of Twhich
+should imply that the full integrable structure of the quant um model should be described by the
+14Theequations in (6.62) are trivial consequences of the SOVr epresentation and of the cyclicity.
+15It is worth noticing that in the case of the Sine-Gordon model the criterion R3does not apply to the representations
+withun=vn= 1. Nevertheless, we have shown the simplicity of Tby using some model dependent properties of the
+coefficients a(λ)andd(λ), see section 5of [1].16
+transfermatrixassoonastheproperty(B)indefinition(1.1 )isshownforthemodelunderconsider-
+ation.
+Following the schema here presented, in a future publicatio n we will address the analysis of the
+spectrumfortheso-called α-sectorsoftheSine-Gordonmodel(see[1]). Theuseofthisapproachi s
+in particularrelevantin these sectorsof theSine-Gordonm odelbecausea direct constructionof the
+Q-operatorleadstosometechnicaldifficulty.
+A. Properties ofthecofactors C i,j(λ)
+Let usconsideran M×Mtridiagonal matrix16O:
+O≡
+z1y10··· 0 0
+x1z2y20··· 0
+0x2z3y3...
+.........
+...... 0
+0...0xM−2zM−1yM−1
+0 0...0xM−1zM
+(A.66)
+i.e. a matrix with non-zero entries only along the principal diagonal and the next upper and lower
+diagonals.
+Lemma 3. The determinantof atridiagonalmatrix is invariantundert he transformation ̺αwhich
+multiplies for αthe entries above the diagonal and for α−1the entries below the diagonal leaving
+theentriesonthediagonalunchanged.
+Proof.Letusnotethatthedeterminantofa tridiagonalmatrixadmi tsthefollowingexpansion:
+det
+MO=z1det
+M−1O1,1+x1y1det
+M−2O(1,2),(1,2), (A.67)
+wherewe haveused thesame notationsintroducedafterformu la(3.23). By usingit, we getthat the
+actionof̺αreads:
+det
+M̺α(O) =z1det
+M−1̺α(O)1,1+x1y1det
+M−2̺α(O)(1,2),(1,2). (A.68)
+Then the statement follows by induction noticing that the tr ansformation ̺αleaves always un-
+changedthedeterminantofa 2×2matrix.
+16An interesting analysis of the eigenvalue problem for tridi agonal matrices is presented in [65].17
+Lemma 4. Thefollowingpropertieshold:
+Ch+i,k+i(λ) =Ch,k(λqi)∀i,h,k∈ {1,...,2l+1}, (A.69)
+and:
+C1,1(λ) =C1,1(−λ)andC2,1(λ) =qNC1,2(−λ). (A.70)
+Proof.Note that by the definition (4.31) of the cofactors C i,j(λ)the equations (A.69) are simple
+consequencesof qp= 1andareprovenexchangingrowsandcolumnsin thedeterminan ts.
+Let us provenow that the cofactor C 1,1(λ) = det 2lD1,1(λ)is an even function of ��. The tridiago-
+nalityofthematrix D1,1(λ)allowsusto usethepreviouslemma:
+C1,1(λ)≡det
+2l/vextenddouble/vextenddoublet(λqh)δh,k−a(λqh)δh,k+1−qNa(−λqh+1)δh,k−1/vextenddouble/vextenddouble
+h>1,k>1
+= det
+2l/vextenddouble/vextenddoublet(λqh)δh,k−qNa(λqh)δh,k+1−a(−λqh+1)δh,k−1/vextenddouble/vextenddouble
+h>1,k>1
+= det
+2l/vextenddouble/vextenddoublet(λqh)δh,k−d(−λqk)δk,h−1−a(−λqk)δk,h+1/vextenddouble/vextenddouble
+h>1,k>1
+≡det
+2l(D1,1(−λ))T=C1,1(−λ). (A.71)
+To provenowthesecondrelationin (A.70) weexpandthe cofac tors:
+C2,1(λ) =2l+1/productdisplay
+h=2a(λqh)+d(λ) det
+2l−1D(1,2),(1,2)(λ), (A.72)
+C1,2(λ) =2l/productdisplay
+h=1d(λqh)+a(λq) det
+2l−1D(1,2),(1,2)(λ). (A.73)
+Byusingthesamestepsshownin(A.71),thetridiagonalityo fthematrixD (1,2),(1,2)(λ)impliesthat
+its determinant is an even function of λfrom which the statement C 2,1(λ) =qNC1,2(−λ)follows
+recallingthat d(λ) =qNa(−λq).
+Remark 3. Inthisarticlewe needonlytheproperties(A.70);however, it isworthpointingoutthat
+theyarespecialcasesofthefollowingpropertiesofthe cof actors:
+Ci,j(λ) =qN(i−j)Cj,i(−λ)∀i,j∈ {1,...,2l+1}. (A.74)
+Theproofof(A.74)canbedonesimilarlytothatof(A.70) but we omitit forsimplicity.
+Let ususe onceagainthenotation Zfforthe set ofthezerosofafunction f(λ), then:
+Lemma 5. Theequations(4.35)and(4.36) imply:
+ZC1,1∩ZC1,2≡ZC1,1∩ZC1,2l+1. (A.75)18
+Proof.The inclusions/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+\Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+\Zd⊂
+ZC1,1∩ZC1,2triviallyfollowbyequation(4.36).
+Let us observe now that C 1,2(λq−1)has no common zero with a(λ)and that C 1,2l+1(λq)has no
+common zero with d(λ). These statements simply follow from (A.73), (A.69)and(A. 72) when we
+recall that a(λ)has no common zero with/producttext2l−1
+h=0d(λqh)and thatd(λ)has no common zero with/producttext2l+1
+h=2a(λqh). So,if/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+∩Zaisnotemptyand λ0∈/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+∩Za,theequation
+(4.35) computed in λ=q−1λ0implies C 1,2l+1(λ0) = 0being C 1,2(λ0q−1)/\e}atio\slash= 0, i.e.λ0∈
+ZC1,1∩ZC1,2l+1. Similarly,if/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+∩Zdisnotemptyand λ0∈/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+∩Zd,
+the equation (4.35) computed in λ=λ0implies C 1,2(λ0) = 0being C 1,2l+1(λ0q)/\e}atio\slash= 0, i.e.λ0∈
+ZC1,1∩ZC1,2. So that (4.35) implies the inclusions/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+∩Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+∩Zd⊂ZC1,1∩ZC1,2inthiswaycompletingthe proofofthelemma.
+B. Scalarproduct inthe SOV space
+Here is described as a natural structure of Hilbert space can be provided to the linear space of the
+SOV representationbypreservingtheself-adjointnessoft hetransfermatrix.
+B.1 CyclicrepresentationsoftheWeylalgebra
+Here,we considerthecyclicrepresentationsoftheWeyl alg ebraW(n)
+qinthecase:
+up
+n=vp
+n= 1forβ2=p′/pwithp′evenandp= 2l+1odd. (B.76)
+At anysitenofthechain,weintroducethe quantumspace Rnwithvn-eigenbasis:
+vn|k,n/a\}bracketri}ht=qk|k,n/a\}bracketri}ht ∀|k,n/a\}bracketri}ht ∈Bn={|k,n/a\}bracketri}ht,∀k∈ {−l,...,l}}. (B.77)
+Note that the eigenvaluesof vndescribe the unit circle Sp={qk:k∈ {−l,...,l}},indeedql+1=
+q−l. OnRnisdefinedap-dimensionalrepresentationoftheWeyl algebrabysetting :
+un|k,n/a\}bracketri}ht=|k+1,n/a\}bracketri}ht ∀k∈ {−l,...,l} (B.78)
+with thecyclicitycondition:
+|k+p,n/a\}bracketri}ht=|k,n/a\}bracketri}ht. (B.79)
+B.2 Representationin the SOVbasis
+The analysis developed in [1] define recursively the eigenba sis{|¯η1qh1,...,¯ηNqhN/a\}bracketri}ht}of theB-
+operator in the original representation, i.e. as linear com binations of the elements of the basis
+{|h1,...,hN/a\}bracketri}ht ≡/circlemultiplytextN
+n=1|hn,n/a\}bracketri}ht}, where|hn,n/a\}bracketri}htare the elements of the vn-eigenbasis defined in
+(B.77). To writethischangeofbasisin amatrixformlet usin troducethe followingnotations:
+|yj/a\}bracketri}ht ≡ |¯η1qh1,...,¯ηNqhN/a\}bracketri}htand|xj/a\}bracketri}ht ≡ |h1,...,hN/a\}bracketri}ht (B.80)19
+where:
+j:=h1+N/summationdisplay
+a=2(2l+1)(a−1)(ha−1)∈ {1,...,(2l+1)N}, (B.81)
+notethat thisdefinesa oneto onecorrespondencebetween N-tuples(h1,...,hN)∈ {1,...,2l+1}N
+and integers j∈ {1,...,(2l+1)N}, which just amountsto chose an orderingin the elementsof th e
+two basis. Underthisnotation,wehave:
+|yj/a\}bracketri}ht=W|xj/a\}bracketri}ht=(2l+1)N/summationdisplay
+i=1Wi,j|xi/a\}bracketri}ht, (B.82)
+where we are representing |xj/a\}bracketri}htas the vector |j/a\}bracketri}htin the natural basis in C(2l+1)NandW=||Wi,j||
+is a(2l+1)N×(2l+1)Nmatrix. The matrix Wis defined by recursion in terms of the kernel K
+constructedinappendixCof[1], letususethenotation:
+K({h1,...,hN},k1,{k2,...,kN})≡KN(η|χ2;χ1), (B.83)
+whereweareconsideringthecase N−M = 1. Thenthe recursionreads:
+W(N)
+i,j=2l+1/summationdisplay
+k2,...,kN=1K({h1(j),...,hN(j)},h1(i),{k2,...,kN})W(N−1)
+¯h(i),a(k2,...,kN),(B.84)
+where we have introduced the index (N)and(N−1)in the matrices Wto make clear the step
+of the recursion. Here, (h1(j),...,hN(j))is the unique N-tuples corresponding to the integer j∈
+{1,...,(2l+ 1)N}andh1(i)is the first entry in the unique N-tuples corresponding to the integer
+i∈ {1,...,(2l+1)N}. Moreover,wehavedefined:
+¯h(i) := 1+i−h1(i)
+2l+1∈ {1,...,(2l+1)(N−1)}anda(k2,...,kN) =k2+N/summationdisplay
+a=3(2l+1)(a−2)(ka−1),
+(B.85)
+Remarks:
+a)Underthechangeofbasis {|xj/a\}bracketri}ht} → {|yj/a\}bracketri}ht}thegenericoperatorX transformsforsimilarity:
+XSOV≡W−1XW, (B.86)
+so fromtheactionofthezerooperators ηaandtheshift operators T±
+aontheB-eigenbasis |yj/a\}bracketri}ht:
+ηa|yj/a\}bracketri}ht= ¯ηaqha(j)|yj/a\}bracketri}htandT±
+a|yj/a\}bracketri}ht=|yj±(2l+1)(a−1)/a\}bracketri}ht (B.87)
+we havethat:
+(ηa)SOV= ¯ηa||qha(j)δi,j||and/parenleftbig
+T±
+a/parenrightbig
+SOV=||δi,j±(2l+1)(a−1)||. (B.88)20
+Fromtheaboveexpressionwe have17:
+(ηa)†
+SOV= (ηa)∗
+SOVand/parenleftbig
+T±
+a/parenrightbig†
+SOV=/parenleftbig
+T∓
+a/parenrightbig
+SOV. (B.89)
+b) The known transformation properties of the entries of the monodromy matrix in the original
+representationimply:
+/parenleftbiggDSOV(λ)CSOV(λ)
+BSOV(λ)ASOV(λ)/parenrightbigg
+=/parenleftigg
+G−1(ASOV(λ∗))†G−G−1(BSOV(λ∗))†G
+−G−1(CSOV(λ∗))†G G−1(DSOV(λ∗))†G/parenrightigg
+,(B.90)
+withGisapositiveself-adjointmatrixdefinedby G:=W†W.
+c)Thequantumdeterminantrelationis invariantundersimi laritytransformationsandso we have:
+a(λ)d(λq−1) =ASOV(λ)DSOV(λq−1)−BSOV(λ)CSOV(λq−1), (B.91)
+Lemma 6. Thebasis {|yj/a\}bracketri}ht}isnotanorthogonalbasisw.r.t. thenaturalscalarproduct on{|xj/a\}bracketri}ht}.
+Proof.Note that the condition {|yj/a\}bracketri}ht}is an orthogonal basis is equivalent to the statement Gis
+a diagonal matrix (with positive diagonal entries). Let us r ecall that the Hermitian conjugation
+propertyofB(λ)togetherwiththeYang-Baxtercommutationrelationsimply :
+[B†(λ),B(µ)] = [B(µ),C(λ∗)] =q−q−1
+λ∗/µ−µ/λ∗(A(λ∗)D(µ)−A(µ)D(λ∗))/\e}atio\slash= 0 (B.92)
+that is the operator B(λ)is not a normal operator. Now let us show that the non-normali tyofB(λ)
+impliesthat Gisnotdiagonal. Indeed,wecanwrite:
+[B†(λ),B(µ)] =/parenleftbig
+W†/parenrightbig−1(BSOV(λ))†GBSOV(µ)W−1−WBSOV(µ)G−1(BSOV(λ))†W†
+=W(G−1(BSOV(λ))†GBSOV(µ)−BSOV(µ)G−1(BSOV(λ))†G)W−1.(B.93)
+Notenowthatifweassume Gdiagonal,then Gcommutesbothwith BSOV(λ)andwith(BSOV(λ))†,
+being all diagonal matrices in the SOV representation, whic h implies the absurd [B†(λ),B(µ)] =
+0.
+B.3 Scalarproductin theSOVspace
+The self-adjointness of the family T(λ)implies that the transfer matrix eigenstates are orthogona l
+undertheoriginalscalar product:
+δi,j= (|ti/a\}bracketri}ht,|tj/a\}bracketri}ht), (B.94)
+we have chosen the orthonormal ones. Note that the above equa tion naturally induces a scalar
+productintheSOV representationobtainedunderchangeofb asis:
+(|b/a\}bracketri}ht,|a/a\}bracketri}ht)SOV≡(G|b/a\}bracketri}ht,|a/a\}bracketri}ht) (B.95)
+17Here, weare using the standard notation for the adjoint X†≡(X∗)t.21
+thatisascalarproductforwhichtheadjointofavector |a/a\}bracketri}htisthenaturaladjointtimesthematrix G:
+|b/a\}bracketri}ht†SOV≡ /a\}bracketle{tb|Gwith/a\}bracketle{tb|=/parenleftig
+(|b/a\}bracketri}ht)t/parenrightig∗
+, (B.96)
+andsoforthegenericoperator Xwehave:
+X†SOV≡G−1X†G. (B.97)
+It istrivialtonoticethat:
+Lemma 7. The family of operators TSOV(λ)is self-adjoint w.r.t. †SOVand the eigenstates
+|tj/a\}bracketri}htSOV≡W−1|tj/a\}bracketri}htare orthonormal w.r.t. the scalar product defined in (B.95). Moreover, it
+results:
+/parenleftigg
+(ASOV(λ∗))†SOV(BSOV(λ∗))†SOV
+(CSOV(λ∗))†SOV(DSOV(λ∗))†SOV/parenrightigg
+=/parenleftbiggDSOV(λ)−CSOV(λ)
+−BSOV(λ)ASOV(λ)/parenrightbigg
+.(B.98)
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1001.0036.txt

@@ -0,0 +1,1085 @@
+The Computational Structure of Spike Trains
+Robert Haslinger,1, 2Kristina Lisa Klinkner,3and Cosma Rohilla Shalizi3, 4
+1Martinos Center for Biomedical Imaging,Massachusetts General Hospital, Charlestown MA
+2Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge MA
+3Department of Statistics, Carnegie Mellon University, Pittsburgh PA
+4Santa Fe Institute, Santa Fe NM
+(Dated: September 2008; January 2009)
+Neurons perform computations, and convey the results of those computations
+through the statistical structure of their output spike trains. Here we present a
+practical method, grounded in the information-theoretic analysis of prediction, for
+inferring a minimal representation of that structure and for characterizing its com-
+plexity. Starting from spike trains, our approach nds their causal state models
+(CSMs), the minimal hidden Markov models or stochastic automata capable of
+generating statistically-identical time series. We then use these CSMs to objec-
+tively quantify both the generalizable structure and the idiosyncratic randomness
+of the spike train. Specically, we show that the expected algorithmic informa-
+tion content (the information needed to describe the spike train exactly) can be
+split into three parts describing (1) the time-invariant structure (complexity) of
+the minimal spike-generating process, which describes the spike train statistically ,
+(2) the randomness (internal entropy rate) of the minimal spike-generating process,
+and (3) a residual pure noise term not described by the minimal spike generating
+process. We use CSMs to approximate each of these quantities. The CSMs are in-
+ferred non-parametrically from the data, making only mild regularity assumptions,
+via the Causal State Splitting Reconstruction (CSSR) algorithm. The methods
+presented here complement more traditional spike train analyses by describing not
+only spiking probability, and spike train entropy, but also the complexity of a spike
+train's structure. We demonstrate our approach using both simulated spike trains
+and experimental data recorded in rat barrel cortex during vibrissa stimulation.
+I. INTRODUCTION
+The recognition that neurons are computational devices is one of the foundations of modern neuroscience (McCulloch
+& Pitts, 1943). However, determining the functional form of such computation is extremely dicult, if only because
+while one often knows the output (the spikes) the input (synaptic activity) is almost always unknown. Often, therefore,
+scientists must draw inferences about the computation from its results, namely the output spike trains and their
+statistics. In this vein, many researchers have used information theory to determine, via calculation of the entropy
+rate, a neuron's channel capacity, i.e., how much information the neuron could conceivably transmit, given the
+distribution of observed spikes (Rieke et al., 1997). However, entropy quanties randomness, and says little about
+how much structure a spike train has, or the amount and type of computation which must have, at a minimum, taken
+place to produce this structure. Here, and throughout this paper, we mean \computational structure" information-
+theoretically, i.e., the most compact eective description of a process capable of statistically reproducing the observed
+spike trains. The complexity of this structure is the number of bits needed to describe it. This is dierent from the
+algorithmic information content of a spike train, which is the number of bits needed to reproduce the latter exactly ,
+describing not only its regularities, but also its accidental, noisy details.
+Our goal is to develop rigorous yet practical methods for determining the minimal computational structure necessary
+and sucient to generate neural spike trains. We are able to do this through non-parametric analysis of the directly-
+observable spike trains, without resorting to a priori assumptions about what kind of structure they have. We do this
+by identifying the minimal hidden Markov model (HMM) which can statistically predict the future of the spike train
+without loss of information. This HMM also generates spike trains with the same statistics as the observed train.
+It thus denes a program which describes the spike train's computational structure, letting us quantify, in bits, the
+structure's complexity.
+From multiple directions, several groups, including our own, have shown that minimal generative models of time
+series can be discovered by clustering histories into \states", based on their conditional distributions over future events
+(Crutcheld & Young, 1989; Grassberger, 1986; Jaeger, 2000; Knight, 1975; Littman et al., 2002; Shalizi & Crutcheld,
+2001). The observed time series need notbe Markovian (few spike trains are), but the construction always yieldsarXiv:1001.0036v1  [q-bio.NC]  30 Dec 20092
+the minimal HMM capable of generating and predicting the original process. Following Shalizi (2001); Shalizi &
+Crutcheld (2001), we will call such a HMM a \Causal State Model" (CSM). Within this framework, the model
+discovery algorithm called Causal State Splitting Reconstruction , or CSSR (Shalizi & Klinkner, 2004) is an adaptive
+non-parametric method which consistently estimates a system's CSM from time-series data. In this paper we adapt
+CSSR for use in spike train analysis.
+CSSR provides us with non-parametric estimates of the time- and history- dependent spiking probabilities found by
+more familiar parametric analyses. Unlike those analyses, it is also capable, in the limit of innite data, of capturing all
+the information about the computational structure of the spike-generating process contained in the spikes themselves.
+In particular, the CSM quanties the complexity of the spike-generating process by showing how much information
+about the history of the spikes is relevant to their future, i.e., how much information is needed to reproduce the
+spike train statistically. This is equivalent to the log of the eective number of statistically-distinct states of the
+process (Crutcheld & Young, 1989; Grassberger, 1986; Shalizi & Crutcheld, 2001). While this is not the same as
+the algorithmic information content, we show that CSMs can also approximate the average algorithmic information
+content, splitting it into three parts: (1) The generative process's complexity in our sense; (2) the internal entropy
+rateof the generative process, the extra information needed to describe the exact state transitions the undergone while
+generating the spike train; and (3) the residual randomness in the spikes, unconstrained by the generative process.
+The rst of these quanties the spike train's structure, the last two its randomness.
+Below, we give precise denitions of these quantities, both their ensemble averages ( xII.C) and their functional
+dependence on time ( xII.D). The time-dependent versions allow us to determine when the neuron is traversing states
+requiring complex descriptions. Our methods put hard numerical lower bounds on the amount of computational
+structure which must be present to generate the observed spikes. They also quantify, in bits, the extent to which the
+neuron is driven by external forces. We demonstrate our approach using both simulated and experimentally recorded
+single-neuron spike trains. We discuss the interpretation of our measures, and how they add to our understanding of
+neuronal computation.
+II. THEORY AND METHODS
+Throughout this paper we treat spike trains as stochastic binary time series, with time divided into discrete, equal-
+duration bins steps (typically at one millisecond resolution); \1" corresponds to a spike and \0" to no spike. Our aim is
+to nd a minimal description of the computational structure present in such a time series. Heuristically, the structure
+present in a spike train can be described by a \program" which can reproduce the spikes statistically. The information
+needed to describe this program (loosely speaking the program length) quanties the structure's complexity. Our
+approach uses minimal, optimally predictive HMMs, or Causal State Models (CSMs), reconstructed from the data, to
+describe the program. (We clarify our use of \minimal" below.) The CSMs are then used to calculate various measures
+of the computational structure, such as its complexity.
+The states are chosen so that they are optimal predictors of the spike train's future, using only the information
+available from the train's history. (We discuss the limitations of this below.) Specically the states Stare dened
+by grouping the histories of past spiking activity Xt
+1which occur in the spike train into equivalence classes, where
+all members of a given equivalence class are statistically equivalent in terms of predicting the future spiking X1
+t+1.
+(Xt
+t0denotes the sequence of random observables, i.e., spikes or their absence, between t0andt>t0whileXtdenotes
+the random observable at time t. The notation is similar for the states.) This construction ensures that the causal
+states are Markovian, even if the spike train is not (Shalizi & Crutcheld, 2001, Lemma 6, p. 839). Therefore, at all
+timestthe system and its possible future evolution(s) can be specied by the state St. Like all HMMs, a CSM can
+be represented pictorially by a directed graph, with nodes standing for the process's hidden states and directed edges
+the possible transitions between these states. Each edge is labeled with the observable/symbol emitted during the
+corresponding transition (\1" for a spike and \0" for no spike), and the probability of traversing that edge given that
+the system started in that state. The CSM also species the time-averaged probability of occupying any state (via
+the ergodic theorem for Markov chains).
+The theory is described in more detail below, but at this point examples may clarify the ideas. Figures 1 A and B
+show two simple CSMs. Both are built from simulated 40 Hz spike trains 200 seconds in length (1 msec time bins,
+p= 0:04 IID at each time when spiking is possible). However, spike trains generated from the CSM in Figure 1 B
+have a 5 msec refractory period after each spike (when p= 0), while the spiking rate in non-refractory periods is still
+40 Hz (p= 0:04). The refractory period is additional structure, represented by the extra states. State Arepresents
+the status of the neuron during 40 Hz spiking, outside of the refractory periods. While in this state, the neuron either
+emits no spike ( Xt+1= 0), staying in state A, or emits a spike ( Xt+1= 1) with probability p= 0:04 and moves to
+stateB. The equivalence class of past spiking histories dening state Atherefore includes all past spiking histories
+for which the most recent ve symbols are 0, symbolically f00000g. StateBis the neuron's state during the rst3
+msec of the refractory period. It is dened by the set of spiking histories f1g. No spike can be emitted during a
+refractory period so the transition to state Cis certain and the symbol emitted is always '0'. In this manner the
+neuron proceeds through states CtoFand back to state Awhereupon it is possible to spike again.
+The rest of this section is divided into four subsections. First, we brie
+(for details, see Shalizi (2001); Shalizi & Crutcheld (2001)) and discuss why they can be considered a good choice for
+understanding the structural content of spike trains. Second, we describe the Causal State Splitting Reconstruction
+(CSSR) algorithm used to reconstruct CSMs from observed spike trains (Shalizi & Klinkner, 2004). We emphasize
+that CSSR requires no a priori knowledge of the structure of the CSM which is discovered from the spike train. Third,
+we discuss two dierent notions of spike train structure, namely statistical complexity and algorithmic information
+content. These two measures can be interpreted as dierent aspects of a spike train's computational structure, and
+each can be related to the reconstructed CSM. Fourth and nally, we show how the reconstructed CSM can be used
+to predict spiking, measure the neural response and detect the in
+A. Causal State Models
+The foundation of the theory of causal states is the concept of a predictively sucient statistics . A statistic, ,
+on one random variable, X, is sucient for predicting another random variable, Y, when(X) andXhave the
+same information1aboutY,I[X;Y] =I[(X);Y]. This holds if and only if XandYare conditionally independent
+given(X):P(YjX;(X)) = P(Yj(X)). This is a close relative of the familiar idea of parametric suciency;
+in Bayesian statistics, where parameters are random variables, parametric suciency is a special case of predictive
+suciency (Bernardo & Smith, 1994). Predictive suciency shares all of parametric suciency's optimality properties
+(Bernardo & Smith, 1994). However, a statistic's predictive suciency depends only on the actual joint distribution of
+XandY, not on any parametric model of that distribution. Again as in the parametric case, a minimal predictively
+sucient statistic is one which is a function of every other sucient statistic , i.e.,(X) =h((X)) for some h.
+Minimal sucient statistics are the most compact summaries of the data which retain all the predictively-relevant
+information. A basic result is that a minimal sucient statistic always exists and is (essentially) unique, up to
+isomorphism (Bernardo & Smith, 1994; Shalizi & Crutcheld, 2001).
+In the context of stochastic processes, such as spike trains, is the minimal sucient statistic of the history Xt
+1
+for predicting future of the process, X1
+t+1. This statistic is the optimal predictor of the observations. The sequence
+of values of the minimal sucient statistic, St=(Xt
+1), is another stochastic process. This process is always a
+homogeneous Markov chain, whether or not the Xtprocess is (Knight, 1975; Shalizi & Crutcheld, 2001). Turned
+around, this means that the original Xtprocess is always a random function of a homogeneous Markov chain, whose
+latent states, named the causal states by Crutcheld & Young (1989), are optimal, minimal predictors of the future
+of the time series.
+Acausal state model orcausal state machine is a stochastic automaton or HMM constructed so that its Markov
+states are minimal sucient statistics for predicting the future of the spike train, and consequently can generate spike
+trains statistically identical to those observed.2Causal state reconstruction means inferring the causal states from
+the observed spike train. Following Crutcheld & Young (1989); Shalizi & Crutcheld (2001), the causal states can be
+seen as equivalence classes of spike-train histories Xt
+1which maximize the mutual information between the state(s)
+and the future of the spike train X1
+t+1. Because they are sucient, they predict the future of the spike train as well
+as it can be predicted from its history alone. Because they are minimal, the number of states or equivalence classes
+is as small as it can be without discarding predictive power.3
+Formally, two histories, xandy, are equivalent when P(X1
+t+1jXt
+1=x) =P(X1
+t+1jXt
+1=y). The equiva-
+lence class of xis [x]. Dene the function which maps histories to their equivalence classes:
+(x)[x]
+=
+y:P(X1
+t+1jXt
+1=y) =P(X1
+t+1jXt
+1=x)	
+1See Cover & Thomas (1991) for information-theoretic denitions and notation.
+2Some authors use \hidden Markov Model" only for models where the current observation is independent of all other
+variables given the current state, and call the broader class which includes CSMs \partially observable Markov
+model".
+3There may exist more compact representations, but then the states, or their equivalents, can never be empirically
+identied | see Shalizi & Crutcheld (2001, thm. 3, p. 846), or L ohr & Ay (2009).4
+The causal states are the possible values of , i.e., the equivalence classes; each corresponds to a distinct distribution
+for the future. The state at time tisSt=(Xt
+1). Clearly,(x) is a sucient statistic. It is also minimal, since if 
+is sucient, then (x) =(y) implies(x) =(y). One can further show (Shalizi & Crutcheld, 2001, Theorem
+3) thatis the unique minimal sucient statistic, meaning that any other must be isomorphic to it.
+In addition to being minimal sucient statistics, the causal states have some other important properties which
+make them ideal for quantifying structure (Shalizi & Crutcheld, 2001). (1) As mentioned, fStgis a Markov process,
+and one can write the observed process Xas a random function of the causal state process, i.e., Xhas a natural
+hidden-Markov-model representation. (2) The causal states are recursively calculable; there is a function Tsuch
+thatSt+1=T(St;Xt+1) | see Appendix A. (3) CSMs are closely related to the \predictive state representations" of
+controlled dynamical systems (Littman et al., 2002); see Appendix C.
+B. Causal State Splitting Reconstruction
+Our goal is to nd a minimal sucient statistic for the spike train, which will form a hidden Markov model. As
+stated previously, the states of this model are equivalence classes of spiking histories Xt
+1. In practice, we need an
+algorithm which can both cluster histories into groups which preserve their conditional distribution of futures, and
+nd the history length  at which the past may be truncated while preserving the computational structure of the
+spike train. The former is accomplished by the CSSR algorithm (Shalizi & Klinkner, 2004) for inferring causal states
+from data by building a recursive next-step-sucient statistic.4We do the latter by minimizing Schwartz's Bayesian
+Information Criterion (BIC) over .
+To save space, we just sketch the CSSR algorithm here.5CSSR starts by treating the process as an independent,
+identically-distributed sequence, with one causal state. It adds states when statistical tests show that the current set
+of states is not sucient. Suppose we have a sequence xN
+1=x1;x2;:::xNof lengthNfrom a nite alphabet Aof
+sizek. We wish to derive from this an estimate ^ of the minimal sucient statistic . We do this by nding a set  of
+states, each of which will be a set of strings, or nite-length histories. The function ^ will then map a history xto
+whichever state contains a sux of x(taking \sux" in the usual string-manipulation sense). Although each state
+can contain multiple suxes, one can check (Shalizi & Klinkner, 2004) that the mapping ^ will never be ambiguous.
+The null hypothesis is that the process is Markovian on the basis of the states in ,
+P(XtjXt1
+tL=axt1
+tL+1) =P(Xtj^S= ^(xt1
+tL+1)) (1)
+for alla2A. In words, adding an extra piece of history does not change the conditional distribution for the next
+observation. We can check this with standard statistical tests, such as 2or Kolmogorov-Smirnov. In this paper, we
+used a KS test of size = 0:01.6If we reject this hypothesis, we fall back on a restricted alternative hypothesis , that
+we have the right set of conditional distributions, but have matched them with the wrong histories. That is,
+P(XtjXt1
+tL=axt1
+tL+1) =P(Xtj^S=s) (2)
+for somes2, buts6= ^(xt1
+tL+1). If this hypothesis passes a test of size , thensis the state to which we assign
+the history7. Only if the (2) is itself rejected do we create a new state, with the sux axt1
+tL+1.8
+4A next-step-sucient statistic contains all the information needed for optimal one-step-ahead prediction,
+I[Xt+1;(Xt
+1)] =I[Xt+1;Xt
+1], but not necessarily for longer predictions. CSSR relies on the theorem that
+ifis next-step sucient, and it is recursively calculable, then is sucient for the whole of the future (Shalizi &
+Crutcheld, 2001, pp. 842{843). CSSR rst nds a next-step sucient statistic, and then renes it to be recursive.
+5In addition to Shalizi & Klinkner (2004), which gives pseudocode, some details of convergence, and applications to
+process classication, are treated in Klinkner & Shalizi (2009); Shalizi et al. (2009). An open-source C++ imple-
+mentation is available at http://bactra.org/CSSR/ . The CSMs generated by CSSR can be displayed graphically,
+as we do in this paper, with the open-source program dot(http://www.graphviz.org/
+6For niteN, decreasing tends to yield simpler CSMs with fewer states. In a sense, it is a sort of regularization
+coecient. The in
+of this paper, varying in the range 0 :001<< 0:1 made little dierence.
+7If more than one such state sexists, we chose the one for which bP(Xtj^S=s) diers least, in total variation
+distance, from bP(Xtjt1
+tL=axt1
+tL+1), which is plausible and convenient. However, which state we chose is irrelevant
+in the limit N!1 , so long as the dierence between the distributions is not statistically signicant.
+8The conceptually similar algorithm of Kennel & Mees (2002) in eect always creates a new state, which leads to
+more complex models, sometimes innitely more complex ones; see Shalizi & Klinkner (2004).5
+The algorithm itself has three phases. Phase I initializes  to a single state, which contains only the null sux ;.
+(That is,;is a sux of any string.) The length of the longest sux in  is L; this starts at 0. Phase II iteratively
+tests the successive versions of the null hypothesis, Eq. 1, and Lincreases by one each iteration, until we reach some
+maximum length . At the end of II, ^ is (approximately) next-step sucient. Phase III makes ^ recursively calculable,
+by splitting the states until they have deterministic transitions. Under mild technical conditions (a nite true number
+of states, etc.), CSSR converges in probability on the correct CSM as N!1 , provided only that  is long enough
+to discriminate all of the states. The error of the predicted distributions of futures P(X1
+t+1jXt
+1), measured by total
+variation distance, decays as N1=2. Section 4 of Shalizi & Klinkner (2004) details CSSR's convergence properties.
+Comparisons of CSSR's performance with that of more traditional expectation maximization based approaches can
+also be found in Shalizi & Klinkner (2004) as can time complexity bounds for the algorithm. Depending upon the
+machine used, CSSR can process an N= 106time series in under a minute.
+1. Choosing 
+CSSR requires no a priori knowledge of the CSM's structure, but does need a choice of of ; here pick it by
+minimizing the BIC of the reconstructed models over , i.e.,
+BIC2 logL+dlogN (3)
+whereLis the likelihood, Nis the data length and dis the number of model parameters, in our case the number
+of predictive states9BIC's logarithmic-with- Npenalty term helps keep the number of causal states from growing
+too quickly with increased data size, which is why we use it instead of the Akaike Information Criterion (AIC). Also,
+BIC is known to be consistent for selecting the order of Markov chains and variable-length Markov models (Csisz ar
+& Talata, 2006), both of which are sub-classes of CSMs.
+Writing the observed spike train as xN
+1, and the state sequence as sN
+0, the total likelihood of the spike train is
+L=X
+sN
+02N+1P(XN
+1=xN
+1jSN
+0=sN
+0)P(SN
+0=sN
+0); (4)
+the sum over all possible causal state sequences of the joint probability of the spike train and the state sequence.
+Since the states update recursively, st+1=T(st;xt+1), the starting state s0and the spike train xN
+1x the entire state
+sequencesN
+0. Thus the sum over state sequences can be replaced by a sum over initial states
+L=X
+si2P(XN
+1=xN
+1jS0=si)P(S0=si) (5)
+with the state probabilities P(S0=si) coming from the CSM. By the Markov property,
+P(XN
+1=xN
+1jS0=si) =NY
+j=1P(Xj=xjjSj1=sj1) (6)
+Selecting  is now straightfoward: for each value of , we build the CSM from the spike train, calculate the
+likelihood using Eq. 5 and 6, and pick the value, and CSM, minimizing Eq. 3. We try all values of  up to a model-
+independent upper bound. For a wide range of stochastic processes, Marton & Shields (1994) showed that the length
+mof subsequences for which probabilities can be consistently and non-parametrically estimated can grow as fast as
+logN=h, wherehis the entropy rate, but no faster. CSSR estimates the distribution of the next symbol given the
+previous  symbols, which is equivalent to estimating joint probabilities of blocks of length m=  + 1. Thus Marton
+and Shield's result limits the usable values of :
+logN
+h1 (7)
+9The number of independent parameters dinvolved in describing the CSM will be (number of states)*(number of
+symbols - 1) since the sum of the outgoing probabilities for each state is constrained to be 1. Thus, for a binary
+alphabet,d= number of states.6
+Using Eq. 7 requires the entropy rate h. The latter can either be upper bounded as the log of the alphabet size (here,
+log 2 = 1), or by some other, less pessimistic, estimator of the entropy rate (such as the output of CSSR with  = 1).
+Use of an upper bound on hresults in a conservative maximum value for . For example, a 30 minute experiment
+with 1 msec time bins lets us use at least 20 by the most pessimistic estimate of h= 1; the actual maximum
+value of  may be much larger. We use  25 in this paper but see no indication that this can't be extended further,
+if need be.
+2. Condensing the CSM
+For real neural data, the number of causal states can be very large | hundreds or more. This creates an interpreta-
+tion problem, if only because it is hard to t such a CSM on a single page for inspection. We thus developed a way to
+reduce the full CSM while still accounting for most of the spike train's structure. Our \state culling" technique found
+the least-probable states and selectively removed them, appropriately redirecting state transitions and reassigning
+state occupation probabilities. By keeping the most probable states, we focus on the ones which contribute the most
+to the spike train's structure and complexity. Again, we used BIC as our model selection criterion.
+First, we sorted the states by probability, nding the least probable state (\remove" state) with a single incoming
+edge from a state (its \ancestor") with outgoing transitions to two dierent states, the remove state and a second,
+\keep" state. We redirected both of the ancestor's outgoing edges to the keep state. Second, we reassigned the remove
+state's outgoing transitions to the keep state. If the outgoing transitions from the keep state were still deterministic (at
+most a single 0 emitting edge and a single 1 emitting edge), we stopped. If the transitions were non-deterministic, we
+merged states reached by emitting 0s with each other (likewise those reached by 1s), repeating this until termination.
+Third, we checked that there existed a state sequence of the new model which could generate the observed spikes. If
+there was, we accepted the new CSM. If not, we rejected the new CSM and chose the next lowest probability state
+from the original CSM to remove.
+This culling was iterated until removing any state made it impossible for the CSM to generate the spike train. At
+each iteration, we calculated BIC (as described in the previous section), and ultimate chose the culled CSM with the
+minimum BIC. This gave a culled CSM for each value of ; the nal one we used was chosen after also minimizing
+BIC over . The CSMs shown below in the results section paper result from this minimizing of BIC over  and state
+culling.
+3. ISI Bootstrapping
+While we do model selection with BIC, we also want to do model checking or adequacy-testing. For the most part,
+we do this by using the CSM to bootstrap point-wise condence bounds on the interspike interval (ISI) distribution,
+and checking their coverage of the empirical ISI distribution. Because this distribution is not used by CSSR in
+reconstructing the CSM, it provides a check on the latter's ability to accurately describe the spike-train's statistics.
+Specically, we generated condence bounds as follows. To simulate one spike train, we picked a random starting
+state according to the CSM's inferred state-occupation probabilities, and then ran the CSM forward for Ntime-steps,
+Nbeing the length of the original spike train. This gives a binary time-series, where a \1" stands for a spike and
+a \0" for no-spike, and gave us a sample of inter-spike intervals from the CSM. This in turn gave an \empirical"
+ISI distribution. Repeated over 104independent runs of the CSM, and taking the 0 :005 and 0:995 quantiles of the
+distributions at each ISI length, gives 99% pointwise condence bounds. (Pointwise bounds are necessary because of
+the ISI distribution often modulates rapidly with ISI length.) If the CSM is correct, the empirical ISI will, by chance,
+lie outside the bounds at 1% of the ISI lengths.
+If we split the data into training and validation sets, a CSM reconstructed from the training set can be used to
+bootstrap ISI condence bounds, which can be compared to the ISI distribution of the test set. We discuss this sort
+of of cross validation, as well as an additional test based on the time rescaling theorem, in Appendix B.7
+C. Complexity and Algorithmic Information Content
+The algorithmic information content K(xn
+1) of a sequence xn
+1is the length of the shortest complete (input-free)
+computer program which will output xn
+1exactly and then halt (Cover & Thomas, 1991)10. In general, K(xn
+1) is
+strictly uncomputable, but when xn
+1is the realization of a stochastic process Xn
+1, the ensemble-averaged algorithmic
+information essentially coincides with the Shannon entropy (\Brudno's theorem"; see Badii & Politi (1997)), re
+the fact that both are maximized for completely random sequences (Cover & Thomas, 1991). Both the algorithmic
+information and the Shannon entropy can be conveniently written in terms of a minimal sucient statistic Q:
+E[K(Xn
+1)] =H[Xn
+1] +o(n)
+=H[Q] +H[Xn
+1jQ] +o(n) (8)
+The equality H[Xn
+1] =H[Q] +H[Xn
+1jQ] holds because Qis a function of Xn
+1, soH[QjXn
+1] = 0.
+The key to determining a spike train's expected algorithmic information is thus to nd a minimal sucient statistic.
+By construction, causal state models provide exactly this; a minimal sucient statistic for xn
+1is the state sequence
+sn
+0=s0;s1;:::sn(Shalizi & Crutcheld, 2001). Thus the ensemble-averaged algorithmic information content, dropping
+termso(n) and smaller, is
+E[K(Xn
+1)] =H[Sn
+0] +H[Xn
+1jSn
+0]
+=H[S0] +nX
+i=1H[SijSi1] +nX
+i=1H[XijSi;Si1] (9)
+Going from the rst to the second line uses the causal states' Markov property. Assuming stationarity, Eq. 9 becomes
+E[K(Xn
+1)] =H[St] +n(H[StjSt1] +H[XtjSt;St1])
+=C+n(J+R) (10)
+This separates terms representing structure from those representing randomness.
+The rst term in Eq. 10 is the complexity ,C, of the spike-generating process (Crutcheld & Young, 1989; Grass-
+berger, 1986; Shalizi et al., 2004).
+C=H[St] =E[logP(St)] (11)
+Cis the entropy of the causal states, quantifying the structure present in the observed spikes. This is distinct from
+the entropy of the spikes themselves, which quanties not their structure but their randomness (and is approximated
+by the other two terms). Intuitively, Cis the (time-averaged) amount of information about the past of the system
+which is relevant to predicting its future. For example, consider again the IID 40 Hz Bernoulli process of Figure
+1 A. With p= 0:04, this has an entropy of 0 :24 bits/msec, but because it can be described by a single state, the
+complexity is zero. (That state emits either a \0" or a \1", with respective probabilities 0 :96 and 0:04, but either way
+the state transitions back to itself.) In contrast, adding a 5 ms refractory period to the process means six states are
+needed to describe the spike trains (Figure 1 B). The new structure of the refractory period is quantied by the higher
+complexity, C= 1:05 bits.
+The second and third terms in Eq. 10 both describe randomness, but of distinct kinds. The second term, the
+internal entropy rate J, quanties the randomness in the state transitions; it is the entropy of the next state given
+the current state.
+J=H[St+1jSt] =E[logP(St+1jSt)] (12)
+This is the average number of bits per time-step needed to describe the sequence of states the process moved through
+(beyond those given by C). The last term in Eq. 10 accounts for any residual randomness in the spikes which is not
+captured by the state transitions.
+R=H[Xt+1jSt;St+1] =E[logP(Xt+1jSt;St+1)] (13)
+10The algorithmic information content is also called the Kolmogorov complexity. We do not use this term, to avoid
+confusion with our \complexity" Cthe information needed to reproduce the spike train statistically rather than
+exactly (Eq. 11). See Badii & Politi (1997) for a detailed comparison of complexity measures.8
+For long trains, the entropy of the spikes, H[Xn
+1], is approximately the sum of these two terms, H[Xn
+1]n(J+R).
+Computationally, Crepresents the xed generating structure of the process, which needs to be described once, at
+the beginning of the time series, and n(J+R) represents the growing list of details which pick out a particular time
+series from the ensemble which could be generated; this needs, on average, J+Rextra bits per time-step. (Cf. the
+\sophistication" of G acs et al. (2001).)
+Consider again the 40 Hz Bernoulli process. As there is only one state, the process always stays in that state. Thus
+the entropy of the next state J= 0. However, the state sequence yields no information about the emitted symbols
+(the process is IID), so the residual randomness R= 0:24 bits/msec | as it must be, since the total entropy rate is
+0:24 bits/msec. In contrast, the states of the 5 msec refractory process are informative about the process's future. The
+internal entropy rate J= 0:20 bits/msec and the residual randomness R= 0. All of the randomness is in the state
+transitions, because they uniquely dene the output spike train. The randomness in the state transition is conned
+to stateA, where the process \decides" whether it will stay in A, emitting no spike, or emit a spike and go to B. The
+decision needs, or gives, 0 :24 bits of information. The transitions from BthroughFand back to Aare xed and
+contribute 0 bits, reducing the expected J.
+The important point is that the structure present in the refractory period makes the spike train less random,
+lowering its entropy. Averaged over time, the mean ring rate of the process is p= 0:0333. Were the spikes IID, the
+entropy rate would be 0 :21 bits/msec, but in fact J+R= 0:20 bits/msec. This is because a minimal description
+of a long sequence Xt1:::XtN=XtN
+t1, the generating process only needs to be described once (C), while the internal
+entropy rate and randomness need to be updated at each time step ( n(J+R)). Simply put, a complex, structured
+spike train can be exactly described in fewer bits than one which is entirely random. The CSM lets us calculate this
+reduction in algorithmic information, and quantify the structure by means of the complexity.
+D. Time-Varying Complexity and Entropies
+The complexity and entropy are ensemble-averaged quantities. In the previous section the ensemble was the entire
+time series, and the averaged complexity and entropies were analogous to a mean ring rate. The time-varying
+complexity and entropies are also of interest, for example their variation after stimuli. A peri-stimulus time histogram
+(PSTH) shows how the ring probability varies with time; the same idea works for the complexity and entropy.
+Since the states form a Markov chain, and any one spike train stays within a single ergodic component, we can
+invoke the ergodic theorem (Gray, 1988), and (almost surely) assert that
+X
+St;St+1P(St;St+1;Xt+1)f(St;St+1;Xt+1) = lim
+N!11
+NNX
+t=1f(St;St+1;Xt+1)
+= lim
+N!1hf(St;St+1;Xt+1)iN (14)
+for arbitrary integrable functions f(St;St+1;Xt+1).
+In the case of the mean ring rate, the function to time-average is l(t)Xt+1. For the time averaged-complexity,
+internal entropy and residual randomness, the functions (respectively c,jandr) are
+c(t) =logP(St)
+j(t) =logP(St+1jSt)
+r(t) =logP(Xt+1jSt;St+1) (15)
+and time-varying entropy h(t) =j(t) +r(t).
+The PSTH averages over an ensemble of stimulus presentations, rather than time:
+PSTH (t) =1
+MMX
+i=1li(t) =1
+MMX
+i=1Xt+1;i (16)
+withMbeing number of stimulus presentations, and tre-set to zero at each presentation. Analogously, the \PSTH"
+of the complexity is
+CPSTH (t) =1
+MMX
+i=1ci(t) =1
+MMX
+i=1logP(St;i) (17)9
+For the entropies, replace cwithj,rorhas appropriate. Similar calculations can be made with any well-dened
+ensemble of reference times, not just stimulus presentations; we will also calculate cand the entropies as functions of
+the time since the latest spike.
+We can estimate the error these time-dependent quantities as the standard error of the mean as a function of time,
+SEt=st=p
+Mwherestis the sample standard deviation in each time bin tandMis the number of trials. The
+probabilities appearing in the denitions of c(t),j(t),r(t) also have some estimation errors, either because of sampling
+noise or, more interestingly, because the ensemble is being distorted by outside in
+between their averages (over time or stimuli) and what the CSM predicts for those averages. In the next section, we
+explain how to use this to measure the in
+E. The In
+If we know that St=s, the CSM predicts that ring probability is (t) =P(Xt+1= 1jSt=s). By means of the
+CSM's recursive ltering property (Appendix A), once a transient regime has passed, the state is always known with
+certainty. Thereafter, the CSM predicts what the ring probability should be at all times, incorporating the eects of
+the spike train's history. As we show in the next section, these predictions give good matches to the actual response
+function in simulations where the spiking probability depends only on the spike history. But real neurons' spiking
+rates generally also depend on external processes, e.g., stimuli. As currently formulated, the CSM is (or, rather,
+converges on) the optimal predictor of the future of the process given its own past. Such an \output only" model
+does not represent the (possible) eects of other processes, and so ignores external covariates and stimuli. Presently,
+determining the precise form of spike trains' responses to external forces is best left to parametric models.
+However, we can use output-only CSMs to learn something about the computation: the PSTH-calculated entropy
+rateHPSTH (t) =JPSTH (t) +RPSTH (t) quanties the extent to which external processes drive the neuron. (The
+PSTH subscript is henceforce supressed.) Suppose we know the true ring probability true(t). At each time step,
+the CSM predicts the ring probability CSM(t). IfCSM(t) =true(t), then the CSM correctly describes the spiking
+and the PSTH entropy rate is
+HCSM(t) =CSM(t) log [CSM(t)](1CSM(t)) log [1CSM(t)] (18)
+However, if CSM(t)6=true(t), then the CSM mis-describes the spiking, because it neglects the in
+processes. Simply put, the CSM has no way of knowing when the stimuli happen. The PSTH entropy rate calculated
+using the CSM becomes
+HCSM(t) =true(t) log [CSM(t)](1true(t)) log [1CSM(t)] (19)
+Solvingtrue(t),
+true(t) =HCSM(t) + log [1CSM(t)]
+log [1CSM(t)]log [CSM(t)](20)
+The discrepancy between CSM(t) andtrue(t) indicates how much of the apparent randomness in the entropy rate
+is actually due to external driving. The true PSTH entropy rate Htrue(t) is
+Htrue(t) =true(t) log [true(t)](1true(t)) log [1true(t)] (21)
+The dierence between HCSM(t) andHtrue(t) quanties, in bits, the driving by external forces as a function of the
+time since stimulus presentation.
+
H=HCSM(t)Htrue(t)
+=true(t) logtrue(t)
+CSM(t)
++ (1true(t)) log1true(t)
+1CSM(t)
+(22)
+This stimulus-driven entropy 
His the relative entropy or Kullback-Leibler divergence D(XtruekXCSM) between the
+true distribution of symbol emissions and that predicted by the CSM. Information-theoretically, this relative entropy
+is the error in our prediction of the next state due to assuming the neuron is running autonomously when it's actually
+externally driven. Since every state corresponds to a distinct distribution over future behavior, this is our error in
+predicting the future due to ignorance of the stimulus.11
+11Cf. the informational coherence introduced by Klinkner et al. (2006) to measure of information-sharing between10
+III. RESULTS
+We now present a few examples. (All of them use a time-step of 1 millisecond.) We begin with idealized model
+neurons to illustrate our technique. We recover CSMs for the model neurons using only the simulated spike trains
+as input to our algorithms. From the CSM we calculate the complexity, entropies, and, when appropriate, stimulus-
+driven entropy (Kullback-Leibler divergence between the true and CSM predicted ring probabilities) of each model
+neuron. We then analyze spikes recorded in vivo from a neuron in layer II/III of rat SI (barrel) cortex. We use spike
+trains recorded both with and without external stimulation of the rat's whiskers. See Andermann & Moore (2006)
+for experimental details.
+A. Model neuron with a \soft" refractory period and bursting
+We begin with a refractory, bursting model neuron, whose spiking rate depends only on the time since the last
+spike. The baseline rate is 40 Hz. Every spike is followed by a 2 msec \hard" refractory period, during which spikes
+never occur. The spiking rate then rebounds to twice its baseline, to which it slowly decays. (See dashed line in the
+rst panel of Figure 3 B.) This history dependence mimics that of a bursting neuron, and is, intuitively, more complex
+than the simple refractory period of the model in Figure 1.
+Figure 2 shows the 17-state CSM reconstructed from a 200 second spike train (at 1 msec resolution) generated by
+this model. It has a complexity of C= 3:16 bits (higher than that of the model in Figure 1, as anticipated), an
+internal entropy rate of J= 0:25 bits/msec and a residual randomness of R= 0 bits/msec. The CSM was obtained
+with  = 17 (selected by BIC). Figure 3 A shows how the 99% ISI bounds bootstrapped from the CSM enclose the
+empirical ISI distribution, with the exception of one short segment.
+The CSM is easily interpreted. State Ais the baseline state. When it emits a spike, the CSM moves to state
+B. There are then two deterministic transitions, to Cand thenD, which never emit spikes; this is the hard 2 msec
+refractory period. Once in Dit is possible to spike again, and if that happens, the transition is back to state B.
+However, if no spike is emitted, the transition is to state E. This is repeated, with varying ring probabilities, as
+statesEthroughQare traversed. Eventually, the process returns to Aand so to baseline.
+Figure 3 B plots the ring rate, complexity, and internal entropies as functions of the time since the last spike
+conditional on no subsequent spike emission . This lets us compare the ring rate predicted by the CSM (solid line
+squares) to the specication of the model which generated the spike train (dashed line) and a PSTH calculated by
+triggering on the last spike (solid line). Except at 16 and 17 msec post spike, the CSM-predicted ring rate agrees
+with both the generating model and the PSTH. The discrepancy arises because the CSM only discerns the structure
+in the data, and most of the ISIs are shorter than 16 msec. There is much closer agreement between the CSM and
+the PSTH if ring rates are plotted as a function of time since a spike without conditioning on no subsequent spike
+emission (not shown).
+The second and third panels of Figure 3 plot the time-dependent complexity and entropies. The complexity is
+much higher after the emission of a spike than during baseline, because the states traversed (B-Q) are less probable,
+and represent the additional structures of refractoriness and bursting. The time-dependent entropies (third panel)
+show that just after a spike, the refractory period imposes temporary determinism on the spike train, but burstiness
+increases the randomness before the dynamics return to the baseline state.
+B. Model neuron under periodic stimulation
+Figure 4 shows the CSM for a periodically-stimulated model neuron . This CSM was reconstructed from 200 seconds
+of spikes with a baseline ring rate of 40 Hz ( p= 0:04). Each second, the ring rate rose over the course of 5 msec to
+p= 0:54 spikes/msec, falling slowly back to baseline over the next 50 msec. This mimics the periodic presentation of
+a strong external stimulus. (The exact inhomogeneous ring rate used was (t) = 0:93[et=10et=2] + 0:04 witht
+in msec. See Figure 5 B, rst panel, dashed line.) In this model, the ring rate does not directly depend on the spike
+train's history, but there is a sort of history dependence in the stimulus time-course, and this is what CSSR discovers.
+BIC selected  = 7, giving a 16 state CSM with C= 0:89 bits,J= 0:27 bits/msec and R= 0:0007 bits/msec. The
+baseline is again state Aand if no spike is emitted then the process stays in A. Spikes are either spontaneous and
+neurons, by quantifying the error in predicting the distribution of the future of one neuron due to ignoring its
+coupling with another.11
+random, or stimulus-driven. Because the stimulus is external, it is not immediately clear which of these two causes
+produced a given spike. Thus, if a spike is emitted, the CSM traverses states BthroughF, deciding, so to speak,
+whether or not the spike is due to a stimulus. If two spikes happen within 3 msec of each other, then the CSM decides
+that it is being stimulated and goes to one of states G,HorM. StatesGthroughPrepresent the response to the
+stimulus. The CSM moves between these states until no spike is emitted for 3 msec, when it returns to the baseline,
+A.
+The ISI distribution from the CSM matches that from the model (Figure 5 A). However, because the stimulus
+doesn't depend on spike train's history, the CSM makes inaccurate predictions during stimulation. The rst panel of
+Figure 5 B plots the ring rate as a function of time since stimulus presentation, comparing the model (dashed line)
+and the PSTH (solid line) with the CSM's prediction (line with squares). The discrepancy between these is due to
+the CSM having no way of knowing that an external stimulus has been applied until several spikes in a row have been
+emitted (represented, as we just say, by states B{F)12. Despite this, c(t) shows that something more complex than
+simple random ring is happening (second panel of Figure 5 B), as do j(t) andr(t) (third panel). Further, something
+is clearly wrong with the entropy rate, because it should be upper-bounded by h= 1 bit/msec (when p= 0:5). The
+fact thath(t) exceeds this bound indicates an external force, not fully captured by the CSM, is at work.
+As discussed in Methods ( xII.E), drive from the stimulus can be quantied with a relative entropy (Figure 5 C).
+Stimuli are presented at t= 1 msec, where 
 H(t)>1 bit. It is not until 25 msec post-stimulus that 
 H(t)0 and
+the CSM once again correctly describes the internal entropy rate. Thus, as expected, the stimulus strongly in
+neuronal dynamics immediately after its presentation. The true internal entropy rate Htrue(t) is slightly less than 1
+bit/msec shortly after stimulation, when the true spiking rate has a maximum of pmax= 0:54. The fact that the CSM
+gives an inaccurate value for Jactually lets us nd the number of bits of information gain supplied by the stimulus,
+e.g., 
H > 1 bit immediately after the stimulus is presented.
+C. Spontaneously Spiking Barrel Cortex Neuron
+We reconstructed a CSM from 90 seconds of spontaneous (no vibrissa de
+II/III FSU barrel cortex neuron. CSSR, using  = 21, discovered a CSM with 315 states, a complexity of C= 1:78
+bits, and internal entropy rate of J= 0:013 bits/msec. After state culling ( xII.B.2), the reduced CSM, plotted in
+Figure 6, has 14 states, C= 1:02,J= 0:10 bits/msec, and residual randomness of R= 0:005 bits/msec. We focus on
+the reduced CSM from this point onwards.
+This CSM resembles that of the spontaneously-ring model neuron of xIII.A and Fig. 2. The complexity and
+entropies are lower than those of our model neuron because the mean spike rate is much lower, and so simple
+descriptions suce most of the time. (Barrel cortex neurons exhibit notoriously low spike rates, especially during
+anesthesia.) There is a baseline state Awhich emits a spike with probability p= 0:01, i.e., 10 Hz. When a spike is
+emitted, the CSM moves to state Band then on through the chain of states CthroughN, return to Aif no spike is
+subsequently emitted. However, the CSM can emit a second or even third spike after the rst, and indeed this neuron
+displays spike doublets and triplets. In general, emitting a spike moves the CSM to B, with some exceptions that
+show the structure to be more intricate than the model neuron's.
+Figure 7 A shows the CSM's 99% condence bounds almost completely enclosing the empirical ISI distribution.
+The rst panel of Figure 7 B plots the history-dependent ring probability predicted by the CSM as a function of the
+time since the latest spike, according to both the PSTH and the CSM's prediction. They are highly similar in the
+rst 13 msec post-spike, indicating that the CSM gets the spiking statistics right in this epoch. The CSM and PSTH
+the diverge after this, for two reasons. First, as with the model neuron, there are few ISIs of this length. Most of the
+ISIs are either shorter, due to the nueron's burstiness, or much longer, due to the low baseline ring rate. Secondly,
+90 seconds is not very much data. We show in Figure 10 that a CSM reconstructed from a longer spike train does
+capture all of the structure. We present the results of this shorter spike train to emphasize that, as a non-parametric
+method, CSSR only uncovers the statistical structure in the data , no more, no less.
+Finally, the second and third panels of Figure 6 B show, respectively, the complexity and entropies as functions of
+the time since the latest spike. As with the model of xIII.A, the structure in the process occurs after spiking, during
+the refractory and bursting periods. This is when the complexity is largest, and also when the entropies vary most.
+12In eect, this part of the CSM implements Bayes's rule, balancing the increased likelihood of a spike after a stimulus
+against the low a priori probability or base-rate of stimulation.12
+D. Periodically Stimulated Barrel Cortex Neuron
+We reconstructed CSMs from 335 seconds of spike trains taken from the same neuron used above, but recorded
+while it was being periodically stimulated by vibrissa de
+in Figure 8. (Before state culling, the original CSM had 1916 states, C= 2:55 andJ= 0:11.) The reduced CSM has a
+complexity of C= 1:97 bits, an internal entropy rate of J= 0:10 bits/msec, and a residual randomness of R= 0:005
+bits/msec. Note that Cis higher when the neuron is being stimulated as opposed to when it is spontaneously ring,
+indicating more structure in the spike train.
+While at rst the CSM may seem to only represents history-dependent refractoriness and bursting, ignoring the
+external stimulus, this is not quite true. Once again, there is a baseline state A, and most of the other states ( B{X)
+comprise a refractory/bursting chain, like this neuron has during spontaneous ring. However, the transition upon A
+emitting a spike is not back to Band then down the chain again, but to either state C1, and subsequently C2, or more
+often to state ZZ. These three states represent the structure induced by the external stimulus, as we saw with the
+model stimulated neuron of xIII.B and Figure 4. (The state ZZis comparable to the state Mof the model stimulated
+neuron: both loop back to themselves if they emit a spike.) Three states are enough because, in this experiment,
+barrel cortex neurons spike extremely sparsely, 0 :1{0:2 spikes per stimulus presentation.
+Figure 9 A plots the ISI distribution, nicely enclosed by the bootstrapped condence bounds. Figure 9 B shows
+the ring rate, complexity and entropies as functions of the time since stimulus presentation (averaged over all
+presentations). These plots look much like those in Figure 7 B. However, there is a clear indication that something
+more complex takes place after stimulation: the CSM's ring-rate predictions are wrong. The stimulus-driven entropy
+
Hturns out to be as large as 0 :02 bits within 5{15 msec post-stimulus. This agrees with the known 5{10 msec
+stimulus propagation time between vibrissae and barrel cortex (Andermann & Moore, 2006). The reason that 
 H
+is so much smaller for the real neuron than the stimulated model neuron of xIII.B is that the former's ring rate is
+much lower. Although the ring rate post-stimulus can be almost twice as large as the CSM's prediction, the actual
+rate is still low, max (t)0:04 spikes/msec. Most of the time the neuron does not spike, even when stimulated, so
+on average, the stimulus provides little information per presentation. For completeness, Figure 10 shows the spike
+probability, complexity and entropies as functions of the time since the latest spike. Averaged over this ensemble, the
+CSM's predictions are highly accurate.
+IV. DISCUSSION
+The goal of this paper was to present methods for determining the structural content of spike trains while making
+minimal a priori assumptions as to the form which that structure takes. We use the CSSR algorithm to build
+minimal, optimally predictive hidden Markov models (CSMs) from spike trains, Schwartz's Bayesian Information
+Criterion to nd the optimal history length  of the CSSR algorithm, and bootstrapped condence bounds on the
+ISI distribution from the CSM to check goodness-of-t. We demonstrated how CSMs can estimate a spike train's
+complexity, thus quantifying its structure, and its mean algorithmic information content, quantifying the minimal
+computation necessary to generate the spike train. Finally we showed how to quantify, in bits, the in
+external stimuli upon the spike-generating process. We applied these methods both to simulated spike trains, for
+which the resulting CSMs agreed with intuition, and to real spike trains recorded from a layer II/II rat barrel cortex
+neuron, demonstrating increased structure, as measured by the complexity, when the neuron was being stimulated.
+We are unaware of any other practical techniques for quantifying the complexity and computational structure of
+a spike train as we dene them. Intuitively, neither random (Poisson), nor highly ordered (e.g., strictly periodic, as
+in Olufsen et al. (2003)) spike trains should be thought of as complex since they do not possess structure requiring a
+sophisticated program to generate. Instead, complexity lies between order and disorder (Badii & Politi, 1997), in the
+non-random variation of the spikes. Higher complexity means a greater degree of organization in neural activity than
+would be implied by random spiking. It is the reconstruction of the CSM through CSSR which allows us to calculate
+the complexity.
+Our denition of complexity stands in stark contrast to other complexity measures which assign high values to
+highly disordered systems. Some of these, such as Lempel Ziv complexity (Amigo et al., 2002, 2004; Jimenez-Montano
+et al., 2002; Szczepanski et al., 2004) and context free grammar complexity (Rapp et al., 1994) have been applied to
+spike trains. However, both of these are measures of the amount of information required to reproduce the spike train
+exactly , and take on very high values for completely random sequences. These \complexity" measures are therefore
+much more similar to total algorithmic information content and even to the entropy rate than to our sort of complexity.
+Our measure of complexity is the entropy of the distribution of causal states. This has the desired property of
+being maximized for structured, rather than ordered or disordered systems, because the causal states are dened
+statistically, as equivalence classes of histories conditioned on future events. Other researchers have also calculated13
+complexity measures which are entropies of state distributions, but have dened their states dierently. Amigo et al.
+(2002) uses the observables (symbol strings) present in the spike train to dene a k-th order Markov process and calls
+each individual length k string which appears in the spike train a state. Gorse and Taylor (1990) similarly use single
+sux symbol strings to dene the states of a Markov process. In both cases, IID Bernoulli sequences could exhibit
+up to 2kstates (in long enough sequences), and possess an extremely high \complexity". However, all of these states
+make the same prediction for the future of the process. The minimal representation is a single causal state, a CSM
+with a complexity of zero.
+It should be noted that there are also many works which model spike trains using HMMs, but in which the hidden
+states represent macro -states of the system (awake/asleep, Up/Down, etc.), and spiking rates are modeled separately
+in each macro-state (Abeles et al., 1995; Achtman et al., 2007; Chen et al., 2008; Danoczy & Hahnloser, 2005; Jones
+et al., 2007). Although the graphical representation of such HMMs may look like those of CSMs, the two kinds of
+states have very dierent meanings. Finally, there are also state space methods which model the dynamical state of
+the system as a continuous hidden variable, the most well known of which is the linear Gaussian model with Kalman
+ltering. These have been extensively applied to neural encoding and decoding problems (Eden et al., 2007; Smith
+et al., 2004; Srinivasan et al., 2007). Interestingly, for a univariate Gaussian ARMA model in state-space form, the
+Kalman lter's one-step-ahead prediction and mean-squared prediction error are, jointly, minimal-sucient for next-
+step prediction, and since they can be updated recursively they in fact constitute the minimal sucient statistic, and
+hence the causal state in this special case.
+Neurons are driven by their aerent synapses. Although as discussed in Appendix C, there is a parallel \transducer"
+formalism for generating CSMs which take external in
+mented, and our current approach reconstructs CSMs only from the spike train. Since the history of the neuron under
+study is typically connected with the history of the network in which it is located, this CSM will, in general, re
+more than a neuron's internal biophysical properties. Nonetheless, in both our model neurons and in the real barrel
+cortex neuron, states not interpretable as simple refractoriness or bursting appeared when a stimulus was present,
+proving we can detect stimulus-driven complexity. Further, we showed that the CSM can be used to determine the
+extent (in bits) to which a neuron is driven by external stimuli.
+The methods presented here complement more established modes of spike-train analysis, which have dierent goals.
+Parametric methods, such as PSTHs or maximum likelihood estimation (Brown et al., 2004; Truccolo et al., 2005)
+generally focus on determining a neuron's ring rate (mean, instantaneous or history-dependent), and on how known
+external covariates modulate that rate. They have the advantage of requiring less data than non-parametric methods
+such as CSSR, but the disadvantage, for our purposes, of imposing the structure of the model at the outset. When
+the experimenter wants to know how a neuron encodes a particular aspect of a covariate, e.g., how neurons in the
+sensory periphery or primary sensory cortices encode stimuli, parametric methods have proved highly illuminating.
+However, in many cases the identity or even existence of relevant external covariates is uncertain. For example, one
+could envision using CSMs to analyze recordings in pre-frontal cortex during dierent cognitive tasks, or to perhaps
+compare spiking structure during dierent attentional states. In both cases, the relevant external covariates are not
+at all clear, but CSMs could still be used to quantify changes in computational structure, for single neurons or for
+groups of them. For neural populations one can envision generating distributions (over the population) of complexities
+and examining how these distributions change in dierent cortical macro-states. This would be entirely analagous to
+analyzing distributions of ring rates or tuning curves.
+In addition to calculations of the complexity, the whole array of mutual-information analyses can be applied to
+CSMs, but instead of calculating mutual information between the spikes and the covariates (which could include other
+spike trains), one can calculate the mutual information between the covariates and the causal states . The advantage
+is that the causal states represent the behavioral patterns of the spike-generating process, and so are closer to the
+actual state of the system than the spikes (output observables) are themselves. Results on calculating the mutual
+information between the causal states of dierent neurons (informational coherence) in a large simulated network
+show that synchronous neuronal dynamics are more eectively revealed than when calculated directly from the spikes
+(Klinkner et al., 2006).
+In closing, our methods provide a way to understand structure in spike trains, and should be considered as comple-
+ments to traditional analysis methods. We rigorously dene structure, and show how to discover it from the data itself.
+Our methods go beyond those which seek to describe the observed variation in the spiking rates by also describing
+the underlying computational process (in the form of a CSM) needed to generate that variation. A CSM can show
+not only that the spike rate has changed, but also show exactly howit has changed.
+Acknowledgments The authors thank Mark Andermann and Christopher Moore for the use of their data. RH thanks
+Emery Brown, Anna Dreyer and Christopher Moore for valuable discussions. CRS thanks Anthony Brockwell, Dave
+Feldman, Chris Genovese, Rob Kass and Alessandro Rinaldo for valuable discussions.14
+APPENDIX A: Filtering with CSMs
+A common diculty with hidden Markov models is that predictions can only be made from a knowledge of the state,
+which must itself be guessed at from the time series, since it is, after all, hidden. This creates the state estimation
+orltering problem. Under strong assumptions (linear Gaussian stochastic dynamics, linearly observed through IID
+additive Gaussian noise) the Kalman lter is an optimal yet tractable solution. For non-linear processes, however,
+optimal ltering essentially amounts to maintaining a posterior distribution over the states and updating it via Bayes's
+rule (Ahmed, 1998). (This distribution is sometimes called the process's \information state".)
+One convenient and important feature of CSMs is that this whole machinery of ltering is unnecessary, because of
+their recursive-updating property. Given the state at time t,St, and the observation at time t+ 1,Xt+1, the state at
+timet+ 1 is xed, St+1=T(St;Xt+1) for some transition function T. Clearly, if the state is known with certainty
+at any time, it will remain known. However, the same recursive updating property also allows us to show that the
+state does become certain, i.e., that after some nite (but possibly random) time ,P(S=sjX
+1) is either 0 or 1 for
+all statess. For Markov chains of order k, clearlyk; under more general circumstances P(t) goes to zero
+exponentially or faster.
+Thus, after a transient period, the state is completely unambiguous. This will be useful to us in multiple places,
+including understanding the computational structure of the process and predicting the ring rate of the neuron. It also
+leads to considerable numerical simplications, compared to approaches which demand conventional ltering. Further,
+recursive ltering is easily applied to a new spike train, not merely the one from which the CSM was reconstructed.
+This helps in cross-validating CSMs, as discussed in the next appendix.
+APPENDIX B: Cross-Validation
+It is often desirable to cross-validate a statistical model by spliting one's data set in two, using one part (generally
+the larger) as a training set for the model and the other part to validate the model by some statistical test. In the
+case of CSMs it is particularly important to check the validity of the BIC used to regularize the  control-setting.
+One possible test is the ISI bootstrapping of xII.B.3. A second, somewhat stronger, goodness-of-t test is based
+on the time rescaling theorem of Brown et al. (2002). This test rescales the interspike intervals as a function of the
+integrated history-dependent spiking rate over the ISI:
+k= 1eRtk+1
+tk(t)dt(B1)
+where theftkgare the spike times and (t) is the history-dependent spiking rate from the CSM. If the CSM describes
+the data well, then rescaled ISI's fkgshould follow a uniform distribution. This can be tested using either a
+Kolmogorov Smirnov test or by plotting the empirical CDF of the rescaled times against the CDF of the uniform
+distribution (Kolmogorov Smirnov or \KS" plot) (Brown et al., 2002).
+Figure 11 gives cross-validation results for the rat barrel cortex neuron, during both spontaneous ring and periodic
+vibrissae de
+validation set. The 335 seconds of stimulus-evoked ring were split into a 270 second training set and a 65 second
+validation set. Panels A and B show the ISI bootstrapping results for the spontaneous and stimulus evoked ring
+respectively. The dashed lines are 99% condence bounds from a CSM reconstructed from the training set and the
+solid line is the ISI distribution of the validation set. The ISI distribution largely falls within these bounds for both
+the spontaneous and stimulus evoked data.
+Panels C-F display the time rescaling test. Panels C and D show the time rescaling plots for the spontaneous and
+stimulus evoked training data respectively. The dashed lines are 95% condence bounds. The spontaneous KS plot
+largely falls within the bounds. The stimulus-evoked does not, but this is expected because, as discussed, the CSM
+does not completely capture the imposition of the external stimulus. (The jagged \steps" in both plots result from
+the 1 msec temporal discretization.) Panels E and F show the time rescaling plots for, respectively, the spontaneous
+and stimulus evoked validation data. The ts here are somewhat worse. In the stimulated case, this is not surprising.
+In the spontaneous case the cause is likely non-stationarity in the data, a problem shared with other spike train
+analysis techniques, such as the Generalized Linear Model approaches described in the next Appendix. It should be
+emphasized that the point of reconstructing CSMs is not to obtain perfect ts to the data, but instead to estimate
+the structure inherent in the spike train, and the cross-validation results should be viewed in this light.15
+APPENDIX C: Causal State Transducers and Predictive State Representations
+Mathematically, CSMs can be expanded to include the in
+state transducers , which are optimal representations of the history-dependent mapping from inputs to outputs (Shalizi,
+2001, ch. 7). Such causal state transducers are a type of partially-observable Markov decision process, closely related
+to predictive state representations (PSRs) (Littman et al., 2002). In both formalisms, the right notion of \state"
+is a statistic, a measurable function of the observable past of the process. Causal states represent this through an
+equivalence relation on the space of observable histories. For PSRs, the representation is through \tests", i.e., a dis-
+tinguished set input/output sequence pairs; the idea is that states can be uniquely characterized by their probabilities
+of producing the output sequences conditional on the input sequences.
+An algorithm for reconstructing causal state transducers would begin by estimating probability distributions of
+future histories conditioned on both the history of the spikes and the history of an external covariate Y, e.g.
+P(X1
+t+1jXt
+1;Yt
+1), and otherwise be entirely parallel to CSSR. This has not yet implemented.
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+A 0 | 0.96 1 | 0.04
+A0 | 0.96 B 1 | 0.04 C0 | 1 D0 | 1
+E0 | 1
+F0 | 1
+0 | 1A
+B
+FIG. 1 Two simple CSMs reconstructed from 200 sec of simulated spikes using CSSR. States are represented as the nodes
+of a directed graph. The transitions between states are labeled with the symbol emitted during the transition (1 = spike, 0
+= no spike) and the probability of the transition given the origin state. (A) The CSM for a 40 Hz Bernoulli spiking process
+consists of a single state Awhich always transitions back to itself, emitting a spike with probability p= 0:04 per msec. (B)
+CSM for 40 Hz Bernoulli spiking process with a 5 msec refractory period imposed after each spike. State Aagain spikes with
+probability p= 0:04. Upon spiking the CSM transitions through a deterministic chain of states B{F(squares) which represent
+the refractory period. The increased structure of the refractory period requires a more complex representation.18
+A0 | 0.957
+B1 | 0.043
+C0 | 1.000
+D0 | 1.000
+H1 | 0.053
+I0 | 0.9471 | 0.069
+J0 | 0.931F1 | 0.024
+G0 | 0.9761 | 0.039
+0 | 0.9611 | 0.076
+K0 | 0.924E1 | 0.012
+0 | 0.9881 | 0.003
+0 | 0.997
+P1 | 0.069
+Q0 | 0.9310 | 0.933
+1 | 0.067
+O1 | 0.074
+0 | 0.926N1 | 0.078
+0 | 0.922M1 | 0.080
+0 | 0.920L1 | 0.075
+0 | 0.9251 | 0.079
+0 | 0.921
+FIG. 2 CSM reconstructed from a 200 sec simulated spike train with a \soft" refractory/bursting structure. C= 3:16,J= 0:25,
+R= 0. State A(circle) is the baseline 40 Hz spiking state. Upon emitting a spike the transition is to state B. States Band
+C(squares) are \hard" refractory states from which no spike may be emitted. States Dthrough Q(hexagons) compromise a
+refractory/bursting chain from which if a spike is emitted the transition is back to state B. Upon exiting the chain the CSM
+returns to the baseline state A.19
+0 5 10 15 20 25 3000.20.4Entropies0 5 10 15 20 25 300246Complexity (C(t))0 5 10 15 20 25 3000.050.1History Dependent Firing ProbabilityISI Distribution
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.08
+time since most recent spike (msec)msecspikes/msec bits bitsA
+B
+λCSM(t)
+λPSTH (t)
+J(t)
+R(t)H(t)λmodel (t)
+FIG. 3 \Soft" refractory and bursting model ISI distribution and time dependent ring probability, complexity and entropies.
+(A) ISI distribution and 99% condence bounds bootstrapped from the CSM. (B) First panel: Firing probability as a function
+of time since the most recent spike. Line with squares = ring probability predicted by CSM. Solid line = ring probability
+deduced from PSTH. Dashed line = model ring rate used to generate spikes. Second panel: Complexity as a function of time
+since most recent spike. Third panel: Entropies as a function of time since most recent spike. Squares = internal entropy rate,
+circles = residual randomness, solid line = entropy rate. (overlaps squares)20
+A0 | 0.957
+B1 | 0.043
+C0 | 0.944
+H1 | 0.056D1 | 0.055
+F0 | 0.945
+E0 | 0.885
+M1 | 0.1150 | 0.882
+1 | 0.1180 | 0.948
+G1 | 0.0520 | 0.821
+1 | 0.179
+I0 | 0.791
+1 | 0.209 J0 | 0.839
+K1 | 0.1610 | 0.878
+1 | 0.122
+L0 | 0.677 1 | 0.323
+1 | 0.205
+N0 | 0.795
+1 | 0.392 0 | 0.6081 | 0.366
+O0 | 0.6341 | 0.265
+P0 | 0.7350 | 0.816
+1 | 0.184
+FIG. 4 16-state CSM reconstructed from 200 sec of simulation of periodically-stimulated spiking. C= 0:89,J= 0:27,
+R= 0:0007. State Ais the baseline state. States Bthrough F(octagons) are \decision" states in which the CSM evaluates
+whether a spike indicates a stimulus or was spontaneous. Two spikes within 3 msec cause the CSM to transition to states G
+through P, which represent the structure imposed by the stimulus. If no spikes are emitted within 5 (often fewer) sequential
+msec, the CSM goes back to the baseline state A.21
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.080.10.12
+0 5 10 15 20 25 30 35 40 45 5000.20.40.6Time Dependent Firing Probability
+0 5 10 15 20 25 30 35 40 45 500510Complexity C(t)
+0 5 10 15 20 25 30 35 40 45 500123EntropiesISI Distribution
+msec
+Time since stimulus (msec)
+Time since stimulus (msec)bits bitsspikes/msec
+Stimulus Driven Entropy  (ΔH(t))A
+B
+CλCSM(t)
+λPSTH (t)
+J(t)
+R(t)
+H(t)λmodel (t)
+0 5 10 15 20 25 30 35 40 45 5000.511.5 bits
+FIG. 5 Stimulus model ISI distribution and time-dependent complexity and entropies. (A) ISI distribution and 99% condence
+bounds. (B) First panel: Firing probability as a function of time since stimulus presentation. Second panel: Time dependent
+complexity. Third panel: time-dependent entropies. (C) The stimulus-driven entropy is >1 bit, indicating strong external
+drive. See text for discussion.22
+A0 | 0.990
+B1 | 0.010
+D
+E0 | 0.9281 | 0.072
+1 | 0.065
+F0 | 0.935C
+0 | 0.9611 | 0.039 0 | 0.9911 | 0.009
+H1 | 0.046 G0 | 0.954
+M1 | 0.042
+N0 | 0.9580 | 0.959
+1 | 0.041
+L1 | 0.036
+0 | 0.964K1 | 0.041
+0 | 0.959J
+1 | 0.0320 | 0.968I1 | 0.047
+0 | 0.9531 | 0.052
+0 | 0.9481 | 0.053
+0 | 0.947
+FIG. 6 14-state CSM reconstructed from 90 sec of spiking recorded from a spontaneously spiking (no stimulus) neuron located
+in layer II/III of rat barrel cortex. C= 1:02,J= 0:10,R= 0:005. State A(circle) is baseline 10 Hz spiking. States Bthrough
+Ncomprise a refractory/bursting chain similar to, but with a somewhat more intricate structure than, that of the model neuron
+in Figure 223
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.080.1
+0 5 10 15 20 25 3000.020.040.06History Dependent Firing Probability
+0 5 10 15 20 25 3002468Complexity C(t)
+0 5 10 15 20 25 3000.20.4EntropiesISI Distribution
+msecspikes/msecbits bits
+time since most recent spike (msec)BA
+J(t)
+R(t)
+H(t)λCSM(t)
+λPSTH (t)
+FIG. 7 Spontaneously spiking barrel cortex neuron. (A) ISI distribution and 99% bootstrapped condence bounds. (B) First
+panel: Time dependent ring probability as a function of time since most recent spike. See text for explanation of discrepancy
+between CSM and PSTH spike probabilities. Second Panel: Complexity as a function of time since most recent spike. Third
+Panel: Entropy rates as a function of time since most recent spike.24
+A0 | 0.99
+B1 | 0.01
+ZZ
+1 | 0.02
+C2 0 | 0.99C1 | 0.010 | 0.99
+X0 | 0.98
+1 | 0.02L
+1 | 0.03M0 | 0.97
+1 | 0.04N0 | 0.96D0 | 0.95
+C11 | 0.05H2
+1 | 0.050 | 0.95
+1 | 0.06E0 | 0.94
+H1
+1 | 0.01 0 | 0.99
+1 | 0.040 | 0.96
+O1 | 0.030 | 0.97
+1 | 0.03P0 | 0.977J
+1 | 0.04K0 | 0.96
+1 | 0.040 | 0.97H
+1 | 0.04
+I0 | 0.96
+1 | 0.040 | 0.96G
+1 | 0.050 | 0.95F
+1 | 0.050 | 0.95
+1 | 0.060 | 0.94
+0 | 0.991 | 0.01 W
+0 | 0.981 | 0.02V
+1 | 0.020 | 0.98S
+1 | 0.03T0 | 0.97
+1 | 0.03U0 | 0.98Q
+1 | 0.03R0 | 0.97
+1 | 0.030 | 0.97
+1 | 0.030 | 0.97
+1 | 0.020 | 0.98
+FIG. 8 29-state CSM reconstructed from 335 seconds of spikes recorded from a layer II/III barrel cortex neuron undergoing
+periodic (125 msec inter-stimulus interval) stimulation via vibrissa de
+states are devoted to refractory/bursting behavior, however states \C1", \C2" and \ZZ" represent the structure imposed by the
+external stimulus. See text for discussion.25
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.08
+0 5 10 15 20 25 30 35 40 45 5000.020.04Time Dependent Firing Probability
+0 5 10 15 20 25 30 35 40 45 501.522.53Complexity C(t)
+0 5 10 15 20 25 30 35 40 45 5000.20.4EntropiesISI Distribution A
+msecspikes/msecbits bits
+time since stimulus presentation (msec)λCSM(t)
+λPSTH (t)
+J(t)
+R(t)
+H(t)B
+C
+time since stimulus presentation (msec)0 5 10 15 20 25 30 35 40 45 5000.0050.010.0150.020.025Stimulus Driven Entropy  (ΔH(t))bits
+FIG. 9 Stimulated barrel cortex neuron ISI distribution and time-dependent complexity and entropies. (A) ISI distribution
+and 99% condence bounds. (B) First panel: Firing probability as a function of time since stimulus presentation. Second
+panel: Time-dependent complexity. Third panel: time-dependent entropies. (C) The stimulus driven entropy (maximum of
+0:02 bits/msec) is low because the number of spikes per stimulus ( 0:10:2) is very low and hence the stimulus does not
+supply much information.26
+0 5 10 15 20 25 3000.020.040.06History Dependent Firing Probability
+0 5 10 15 20 25 30 35 40 45 500510Complexity C(t)
+0 5 10 15 20 25 3000.20.4Entropiesspikes/msecbits bits
+time since most recent spike (msec)J(t)
+R(t)
+H(t)λCSM(t)
+λPSTH (t)
+FIG. 10 Firing probability, complexity and entropies of the stimulated barrel cortex neuron as a function of time since the
+most recent spike.27
+0 10 20 30 40 5000.020.040.060.080.10.12
+0 10 20 30 40 5000.020.040.060.080.1
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+0 0.2 0.4 0.6 0.8 100.20.40.60.81ISI Distrib. Spont. Spiking
+      Validation DataISI Distrib. Stimulated Spiking           Validation Data
+KS Plot, Spont. Spiking      Training Data
+Validation DataKS Plot, Stimulated Spiking          Training Data
+Validation Datamsec msecA B
+C D
+E F
+FIG. 11 Cross validation of CSMs reconstructed from spontaneously ring and stimulus evoked rat barrel cortex on an inde-
+pendent validation training set. (A,B) ISI distribution of spontaneously and stimulus evoked ring validation sets and 99%
+condence bounds bootstrapped from CSM. (C-D) Time rescaling plots of training data sets for spontaneously ring and stim-
+ulus evoked ring respectively. Dashed lines are 95% condence bounds and the solid line is the rescaled ISIs. The solid line
+along the digagonal is for visual comparison to an ideal t. (E-F) Similar time rescaling plots for the validation data sets.

1001.0038.txt

@@ -0,0 +1,1123 @@
+arXiv:1001.0038v1  [math.CA]  30 Dec 2009BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC
+SPACES
+ALEXANDER VOLBERG†AND BRETT D. WICK‡
+Abstract. In this paper we study “Bergman-type” singular integral ope rators on Ahlfors regular
+metric spaces. The main result of the paper demonstrates tha t if a singular integral operator on a
+Ahlfors regular metric space satisfies an additional estima te, then knowing the “T(1)” conditions
+for the operator imply that the operator is bounded on L2. The method of proof of the main result
+is an extension and another application of the work originat ed by Nazarov, Treil and the first author
+on non-homogeneous harmonic analysis.
+1.Introduction and Statements of results
+We are interested in Calder´ on-Zygmund operators living on metric spaces. In particular, these
+kernels will live on a metric space of homogeneous type. We br iefly recall these types of metric
+spaces. A metric space of homogeneous type is a space X, a quasi-metric ρ, and a non-negative
+Borel measure νon the space X. The key property that defines these spaces is that all balls B(x,r)
+defined byρare open, and the measure νsatisfies the doubling condition
+ν(B(x,2r))≤Cdoubν(B(x,r))∀x∈X, r∈R+.
+We also require that ν(B(x,r))<∞for allx∈Xandr∈R+. The main example of the metric
+spaces that the reader should keep in mind is the case of Rnwith the standard metric and Lebesgue
+measure. Instead of the standard doubling condition, we wil l impose a slightly stronger condition.
+Let(X,ν,ρ) beaAhlforsregularmetricmeasurespace. By thiswemeanth at (X,ρ) isacomplete
+metric space, ν≥0 is a Borel measure on X, and there exist constants 0 < c1< c2,n >0, such
+that, for all r≥0 andx∈X:
+c1rn≤ν(B(x,r))≤c2rn. (1.1)
+It is easy to see that condition 1.1implies the doubling condition on νwithCdoub=c2
+c12n.
+We next recall the definition of Calder´ on-Zygmund operator s on metric spaces as introduced by
+Christ, [1]. For anyx,y∈X, we set
+λ(x,y) =ν(B(x,ρ(x,y)))≈ρ(x,y)n.
+A simple calculation shows that λ(x,y)≈λ(y,x) because of the doubling condition on ν. Then
+astandard kernel is a function k:X×X\ {x=y} →Csuch that there exists constants CCZ,
+τ,δ>0
+|k(x,y)| ≤CCZ
+λ(x,y)=CCZ
+ρ(x,y)n∀x/\e}atio\slash=y∈X;
+and
+|k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
+ρ(x,y)τ1
+λ(x,y)=CCZρ(x,x′)τ
+ρ(x,y)τ+n
+†Research supported in part by a National Science Foundation DMS grant.
+‡The second author is supported by National Science Foundati on CAREER Award DMS# 0955432 and an
+Alexander von Humboldt Fellowship.
+12 A. VOLBERG AND B. D. WICK
+provided that ρ(x,x′)≤δρ(x,y). In this situation, we say that the kernel ksatisfies the standard
+estimates. Again, the canonical examples to keep in mind are the usual Calder´ on-Zygmund kernels
+onRn.
+However, we will be interested in kernels that satisfy estim ates as if they lived on a “smaller
+space”. First, suppose that we have another measure µon the metric space X(which need not be
+doubling), but satisfies the following relationship, for so me 0≤m<n
+µ(B(x,r))/lessorsimilarrm∀x∈X,∀r. (H)
+Then, we define a standard kernel of order 0<m≤nas a function k:X×X\{x=y} →C
+such that there exists constants CCZ,τ,δ >0
+|k(x,y)| ≤CCZ
+ρ(x,y)m∀x/\e}atio\slash=y∈X;
+and
+|k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
+ρ(x,y)τ+m
+provided that ρ(x,x′)≤δρ(x,y). In this situation, we say that the kernel ksatisfies the standard
+estimates. In this case, we then define the Calder´ on-Zygmun d operator associated to µas
+Tµ(f)(x) :=/integraldisplay
+Xk(x,y)f(y)dµ(y).
+For “nice” functions f, this integral is well defined and
+These definitions are motivated by the Calder´ on-Zygmund ke rnels that live in Rn, but satisfy
+estimates as if they lived in Rmwithm≤n. One should think of the measure µas given by the
+m-dimensional Lebesgue measure after restricting to a m-dimensional hyperplane.
+The constants CCZ,τ,δandmwill be referred to as the Calder´ on-Zygmund constants of th e
+kernelk(x,y).
+We will also be interested in the kernels that have the additi onal property that satisfy
+|k(x,y)| ≤1
+max(dm(x),dm(y)),
+whered(x) := dist(x,X\Ω) = inf{ρ(x,y) :y∈X\Ω}and Ω being an open set in X.
+Our main result is the following theorem:
+Theorem 1. Let(X,ρ,ν)be a Ahlfors regular metric space. Let k(x,y)be a Calder´ on-Zygmund
+kernel of order mon(X,ρ,ν), with Calder´ on-Zygmund constants CCZandτ, that satisfies
+|k(x,y)| ≤1
+max(d(x)m,d(y)m),
+whered(x) := dist(x,X\Ω). Letµbe a probability measure with compact support in Xand all
+balls such that µ(B(x,r))>rmlie in an open set Ω. Finally, suppose also that a “ T1Condition”
+holds for the operator Tµ,mwith kernel kand for the operator T∗
+µ,mwith kernel k(y,x):
+/bardblTµ,mχQ/bardbl2
+L2(X;µ)≤Aµ(Q),/bardblT∗
+µ,mχQ/bardbl2
+L2(X;µ)≤Aµ(Q). (1.2)
+Then/bardblTµ,m/bardblL2(X;µ)→L2(X;µ)≤C(A,m,d,τ ).
+The balls for which we have µ(B(x,r))>rmwill be called “non-Ahlfors balls”. The key hypoth-
+esis is that we can capture all the non-Ahlfors balls in some o pen set Ω. To mitigate against this
+difficulty, we will have to suppose that our Calder´ on-Zygmun d kernels have an additional estimate
+in terms of the behavior in terms of the distance to the comple ment of Ω.
+An immediate application of Theorem 1is a new proof of results by the authors in [ 9]. In
+[9] a variant of Theorem 1was obtained in the Euclidean setting, and then is further ex tended
+to Calder´ on-Zygmund kernels in the natural metric associa ted to the Heisenberg group on theBERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 3
+unit ball. This was then used to characterize the Carleson me asures for the analytic Besov–Sobolev
+spaces on the unit ball in Cn. The connection between Carleson measures and a variant of T heorem
+1is provided since a measure is Carleson if and only if a certai n naturally occurring Calder´ on-
+Zygmund operator is bounded on L2. The operator to be studied is amenable to the methods of
+non-homogeneous harmonic analysis.
+Themethod of proof of Theorem 1will beto usethe tools of non-homogeneous harmonicanalysi s
+as developed by F. Nazarov, S. Treil, and the first author in th e series of papers [ 3–6] and further
+explained in the book by the first author [ 8]. We essentially adapt the proof given by the authors in
+[9] to the case of metric spaces considered in this paper. Const ants will bedenoted by Cthroughout
+the paper.
+2.Proof of Theorem 1
+The proof of this theorem will be divided in several parts. We first recall the construction of M.
+Christ of “dyadic cubes” on a metric space of homogeneous typ e, see [1]. The interested reader
+can also consult the paper by E. Sawyer and R. Wheeden, [ 7], where a similar construction is
+performed.
+Theorem 2 (M. Christ, [ 1]).There exists a collection of open sets {Qk
+α⊂X:k∈Z,α∈Ik}and
+constantsκ∈(0,1),a0>0, andη>0andC1,C2<∞such that
+(i)ν(X\/uniontext
+αQk
+α) = 0∀k∈Z;
+(ii)Ifl≥k, then either Ql
+β⊂Qk
+αorQl
+β⊂Qk
+α=∅;
+(iii)For each (k,α)and eachl<kthere is a unique βsuch thatQk
+α⊂Ql
+β;
+(iv)The diameter of Qk
+αis an absolute constant multiple of κk;
+(v)EachQk
+αcontains some ball B(zα,a0κk);
+(vi)ν{x∈Qk
+α: dist(x,X\Qk
+α)≤tκk}/lessorsimilartην(Qk
+α)
+HereIkis a (possibly finite) index set, depending only on k∈Z.
+The construction of these cubes uses only the properties of t he homogeneous space ( X,ρ,ν).
+One can think of the cubes Qk
+αas being cubes or balls of diameter κkand centerzk
+α. We will let D
+denote the collection of dyadic cubes on Xthat exists by the above Theorem.
+We further remark that it is possible to “randomize” this con struction. In a recent paper by
+Hyt¨ onen and Martikainen, [ 2], they studied this construction in and showed that it is pos sible to
+construct several random dyadic grids of the type above. The details of this construction aren’t
+immediately important for the proof of the main results in th is paper, only the existence of these
+randomgrids. We recommendthat thereaderconsultthewell- written paper[ 2] fortheconstruction
+of these grids. In particular, Section 10 of that paper conta ins the necessary modifications of
+Theorem 2to construct the random dyadic lattices in a metric space.
+We also define the dilation of a set E⊂Xby a parameter λ≥1 by
+λE:={x∈X:ρ(x,E)≤(λ−1)diam(E)}.
+2.1.Terminal and transit cubes. We will call the cube Q∈ Daterminal cube if the parent of
+Q(which exists and is unique by (iii) of Theorem 2) is contained in our open set Ω or µ(Q) = 0.
+All other cubes are called transitcubes. Then, denote by DtermandDtranas the terminal and
+transit cubes from D. We first state two obvious Lemmas.
+Lemma 3. IfQbelongs to Dterm, then
+|k(x,y)| ≤1
+κm.
+This follows since Qbelongs to its parent which is a subset of Ω and so for x,y∈Qwe have
+thatd(x)≥κand similarly for y. Another obvious lemma:4 A. VOLBERG AND B. D. WICK
+Lemma 4. IfQbelongs to Dtran, then
+µ(B(x,r))/lessorsimilarrm.
+We assume that F= suppµlies in a grand child cube of Qwhere, this Qis a certain (fixed) unit
+cube. We then take two “random” lattices as constructed by Hy t¨ onen and Martikainen in [ 2]. Now,
+letD1andD2be two such dyadic lattices, that have the property that the u nit cube contains the
+support ofµdeep inside a unit cube of the corresponding lattice. We will decompose our functions
+fandgwith respect to the lattices D1andD2.
+We would like to denote Qjas a dyadic cube belonging to the dyadic lattice Dj. Unfortunately,
+this makes the notation later very cumbersome. So, we will us e the letter Qto denote a dyadic
+cube belonging to the lattice D1and the letter Rto denote a dyadic cube belonging to the lattice
+D2. We will also let s(Q) denote the “size” or “scale” of the cube, namely, what gener ation of the
+construction from Theorem 2the cube belongs to.
+From now on, we will always denote by Qjthe dyadic subcubes of a cube Qenumerated in some
+“natural order”. Similarly, we will always denote by Rjthe dyadic subcubes of a cube RfromD2.
+Next, notice that there are special unit cubes Q0andR0of the dyadic lattices D1andD2
+respectively. They have the property that they are both tran sit cubes and contain Fdeep inside
+them.
+2.2.Projections Λand∆Q.LetDbe one of the dyadic lattices above. For a function ψ∈
+L1(X;µ) and for a cube Q⊂X, denote by /a\}bracketle{tψ/a\}bracketri}htQthe average value of ψoverQwith respect to the
+measureµ, i.e.,
+/a\}bracketle{tψ/a\}bracketri}htQ:=1
+µ(Q)/integraldisplay
+Qψdµ
+(of course, /a\}bracketle{tψ/a\}bracketri}htQmakes sense only for cubes Qwithµ(Q)>0). Put
+Λϕ:=/a\}bracketle{tϕ/a\}bracketri}htQ0.
+Clearly, Λϕ∈L2(X;µ) for allϕ∈L2(X;µ), and Λ2= Λ, i.e., Λ is a projection. Note also, that
+actually Λ does not depend on the lattice Dbecause the average is taken over the whole support
+of the measure µregardless of the position of the cube Q0(orR0).
+Below we will start almost every claim by “Assume (for definit eness) that s(Q)≤s(R)...”.
+Below, for ease of notation, we will write that a cube Q∈ X ∩Y to mean that the dyadic cube Q
+has both property XandYsimultaneously.
+For every transit cube Q∈ D1, define ∆Qϕby
+∆Qϕ/vextendsingle/vextendsingle
+X\Q:= 0, ,∆Qϕ/vextendsingle/vextendsingle
+Qj:=
+
+/bracketleftBig
+/a\}bracketle{tϕ/a\}bracketri}htQj−/a\}bracketle{tϕ/a\}bracketri}htQ/bracketrightBig
+ifQjis transit;
+ϕ−/a\}bracketle{tϕ/a\}bracketri}htQifQjis terminal.
+Observe that for every transit cube Q, we haveµ(Q)>0, so our definition makes sense since no
+zero can appear in the denominator. We repeat the same definit ion forR∈ D2.
+We then have have following Lemma that collects several easy properties of ∆Qϕ. To check these
+properties is left to the reader as an exercise.
+Lemma 5. For everyϕ∈L2(X;µ)and every transit cube Q,
+(1) ∆Qϕ∈L2(X;µ);
+(2)/integraltext
+X∆Qϕdµ= 0;
+(3) ∆Qis a projection, i.e., ∆2
+Q= ∆Q;
+(4) ∆QΛ = Λ∆Q= 0;BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 5
+(5) IfQ,/tildewideQare transit,/tildewideQ/\e}atio\slash=Q, then ∆Q∆eQ= 0.
+We next note that it is possible to decompose functions ϕinto the corresponding projections Λ
+and ∆Q.
+Lemma 6. LetQ0be a transit cube. For every ϕ∈L2(X;µ)we have
+ϕ= Λϕ+/summationdisplay
+Qtransit∆Qϕ,
+the series converges in L2(X;µ)and, moreover,
+/bardblϕ/bardbl2
+L2(µ)=/bardblΛϕ/bardbl2
+L2(µ)+/summationdisplay
+Qtransit/bardbl∆Qϕ/bardbl2
+L2(µ).
+Proof.Note first of all that if one understands the sum
+/summationdisplay
+Qtransit
+as limk→∞/summationtext
+Qtransit:s(Q)>δk, then forµ-almost every x∈X, one has
+ϕ(x) = Λϕ(x)+/summationdisplay
+Qtransit∆Qϕ(x).
+Indeed, the claim is obvious if the point xlies in some terminal cube. Suppose now that this is not
+the case. Observe that
+Λϕ(x)+/summationdisplay
+Qtransit:s(Q)>κk∆Qϕ(x) =/a\}bracketle{tϕ/a\}bracketri}htQk,
+whereQkis the dyadic cube of size κk, containing x. Therefore, the claim is true if
+/a\}bracketle{tϕ/a\}bracketri}htQk→ϕ(x).
+But, the exceptional set for this condition has µ-measure 0. Now the orthogonality of all ∆ Qϕ
+between themselves, and their orthogonality to Λ ϕproves the lemma. /square
+3.Good and bad functions
+We consider the functions fandg∈L2(X;µ). We fix two dyadic lattices D1andD2as before
+and define decompositions of fandgvia Lemma 6,
+f= Λf+/summationdisplay
+Q∈Dtran
+1∆Qf, g= Λg+/summationdisplay
+R∈Dtran
+2∆Rg.
+For a dyadic cube Rwe denote ∪i∈Ik∂RibyskR, called the skeleton ofR. Here theRiare the
+dyadic children of R.
+Letτ,mbe parameters of the Calder´ on-Zygmundkernel k. We fixα=τ
+2τ+2m.
+Definition 7. Fix a small number δ >0 andS≥2 to be chosen later. Choose an integer rsuch
+that
+κ−r≤δS<κ−r+1. (3.1)
+A cubeQ∈ D1is called bad(δ-bad) if there exists R∈ D2such that
+(1)s(R)≥κrs(Q);
+(2) dist(Q,skR)<s(Q)αs(R)1−α.6 A. VOLBERG AND B. D. WICK
+LetB1denote the collection of all bad cubes and correspondingly l etG1denote the collection of
+good cubes. The symmetric definition gives the collection of badcubesR∈ D2, denotes as B2.
+We say, that ϕ=/summationtext
+Q∈Dtran
+1∆Qϕisbadif in the sum only bad Q’s participate in this decompo-
+sition with the same appling to ψ=/summationtext
+Q∈Dtran
+2∆Qψ. In particular, given two distinct lattices D1
+andD2we fix the decomposition of fandginto good and bad parts:
+f=fgood+fbad,wherefgood= Λf+/summationdisplay
+Q∈Dtran
+1∩G1∆Qf.
+The same applies to g= Λg+/summationtext
+R∈Dtran
+2∆Rg=ggood+gbad.
+Theorem 8. One can choose S=S(α)in such a way that for any fixed Q∈ D1,
+P{Qis bad} ≤δ2. (3.2)
+By symmetry P{Ris bad} ≤δ2for any fixed R∈ D2.
+Theproof of this Theorem can befoundin thepaper [ 2]. Theuseof Theorem 8gives usS=S(α)
+in such a way that for any fixed Q∈ D1,
+P{Qis bad} ≤δ2. (3.3)
+We are now ready to prove
+Theorem 9. Consider the decomposition of ffrom Lemma 6. Then one can choose S=S(α)in
+such a way that
+E(/bardblfbad/bardblL2(X;µ))≤δ/bardblf/bardblL2(X;µ). (3.4)
+The proof depends only on the property ( 3.3) and not on a particular definition of what it means
+to be a bad or good function.
+Proof.By Lemma 6(its left inequality),
+E(/bardblfbad/bardblL2(X;µ))≤E/parenleftBig/summationdisplay
+Q∈Dtran
+1∩B1/bardbl∆Qf/bardbl2
+L2(X;µ)/parenrightBig1/2
+.
+Then
+E(/bardblfbad/bardblL2(X;µ))≤/parenleftBig
+E/summationdisplay
+Q∈Dtran
+1∩B1/bardbl∆Qf/bardbl2
+L2(X;µ)/parenrightBig1/2
+.
+LetQbe a fixed cube in D1; then, using ( 3.3), we conclude:
+E/bardbl∆Qf/bardbl2
+L2(X;µ)=P{Qis bad}/bardbl∆Qf/bardbl2
+L2(X;µ)≤δ2/bardbl∆Qf/bardbl2
+L2(X;µ).
+Therefore, we can continue as follows:
+E(/bardblfbad/bardblL2(X;µ))≤δ/parenleftBig/summationdisplay
+Q∈Dtran
+1∩B1/bardbl∆Qf/bardbl2
+L2(X;µ)/parenrightBig1/2
+≤δ/bardblf/bardblL2(X;µ).
+The last inequality uses Lemma 6again (its right inequality). /square
+This theorem can also be found in the paper [ 2].BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 7
+3.1.Reduction to Estimates on Good Functions. We consider two random dyadic lattices
+D1andD2as constructed in [ 2]. Take now two functions fandg∈L2(X;µ) decomposed according
+to Lemma 6
+f= Λf+/summationdisplay
+Q∈Dtran
+1∆Qf, g= Λg+/summationdisplay
+R∈Dtran
+2∆Rg.
+Recall that we can now write f=fgood+fbad,g=ggood+gbad. Then
+(Tf,g) = (Tfgood,ggood)+R(f,g),whereR(f,g) = (Tfbad,g)+(Tfgood,gbad).
+Theorem 10. LetTbe any operator with bounded kernel. Then
+E|R(f,g)| ≤2δ/bardblT/bardblL2(X;µ)→L2(X;µ)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+Remark 11.Notice that the estimate depends on the norm of Tnot on the bound on its kernel.
+Proof.The procedure of taking the good and bad part of a function are projections in L2(X;µ)
+and so they do not increase the norm. Since we have that the ope ratorTis bounded, then
+|R(f,g)| ≤ /bardblT/bardblL2(X;µ)→L2(X;µ)/parenleftbig
+/bardblg/bardblL2(X;µ)/bardblfbad/bardblL2(X;µ)+/bardblf/bardblL2(X;µ)/bardblgbad/bardblL2(X;µ)/parenrightbig
+Therefore, upon taking expectations we find
+E|R(f,g)| ≤ /bardblT/bardblL2(X;µ)→L2(X;µ)/parenleftbig
+/bardblg/bardblL2(X;µ)E(/bardblfbad/bardblL2(X;µ))+/bardblf/bardblL2(X;µ)E(/bardblgbad/bardblL2(X;µ))/parenrightbig
+.
+Using Theorem 9we finish the proof.
+/square
+We see that we need now only to estimate
+|(Tfgood,ggood)| ≤C(τ,m,d,T 1)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ). (3.5)
+In fact, considering any operator Twith bounded kernel we conclude
+(Tf,g) =E(Tf,g) =E(Tfgood,ggood)+ER(f,g).
+Using Theorem 10and (3.5) we have
+|(Tf,g)| ≤C/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ)+2δ/bardblT/bardblL2(X;µ)→L2(X;µ)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+From here, taking the supremum over fandgin the unit ball of L2(X;µ), and choosing δ=1
+4we
+get
+/bardblT/bardblL2(X;µ)→L2(X;µ)≤2C.
+3.2.Splitting (Tfgood,ggood)into Three Sums. First let us get rid of the projection Λ. We fix
+two corresponding dyadic lattices D1andD2. Recall that F= suppµis deep inside a unit cube Q
+of the standard dyadic lattice Das well as inside the shifted unit cubes Q0∈ D1andR0∈ D2. If
+f∈L2(X;µ), we have
+/bardblTΛf/bardblL2(X;µ)=/a\}bracketle{tf/a\}bracketri}htQ0/bardblTχQ0/bardblL2(X;µ)
+≤A1.2/bardblf/bardblL2(X;µ)µ(Q0)1/2
+µ(Q0)µ(Q0)1/2
+=A1.2/bardblf/bardblL2(X;µ).
+So we can replace fbyf−Λfand identically we can repeat this argument with gand from now
+on we may assume further that/integraldisplay
+Xf(x)dµ(x) = 0 and/integraldisplay
+Xg(x)dµ(x) = 0.
+Based on the reductions above, we can now think that fandgare good functions with zero
+averages. We skip mentioning below that Q∈ Dtran
+1andR∈ Dtran
+2, since this will always be the
+case by the convention established above.8 A. VOLBERG AND B. D. WICK
+To study the action of the Calder´ on-Zygmund operator Tonfandg, we split the pairing in
+the following manner,
+(Tf,g) =/summationdisplay
+Q∈G1,R∈G2,s(Q)≤s(R)(∆Qf,∆Rg)+/summationdisplay
+Q∈G1,R∈G2,s(Q)>s(R)(∆Qf,∆Rg).
+The question of convergence of the infinite sum can be avoided here, as we can think that the
+functionsfandgare only finite sums. This removes the question of convergenc e and allows us to
+rearrange and group the terms in the sum in any way we want.
+We need to estimate only the first sum, as the second will follo w by symmetry. For the sake of
+notational simplicity we will skip mentioning that the cube sQandRare good and we will skip
+mentioning s(Q)≤s(R). So, for now on,
+/summationdisplay
+Q,R:other conditionsmeans/summationdisplay
+Q,R:s(Q)≤s(R),Q∈G1,R∈G2,other conditions.
+Remark 12.It is convenient sometimes to think that the summation/summationdisplay
+Q,R:other conditions
+goes over good QandallR. Formally, this does not matter, since the functions fandgare good
+functions, and so this merely reduces to adding or omitting s everal zeros to the sum. For the
+symmetric sum over Q,R:s(Q)> s(R) the roles of QandRin this remark must of course be
+interchanged.
+The definition of δ-badness involved a large integer r, see (3.1). Use this notation to write our
+sum overs(Q)≤s(R) as follows
+/summationdisplay
+Q,R(∆Qf,∆Rg) =/summationdisplay
+Q,R:s(Q)≥κ−rs(R)+/summationdisplay
+Q,R:s(Q)<κ−rs(R)=/summationdisplay
+Q,R:s(Q)≥κ−rs(R),dist(Q,R)≤s(R)+
+/bracketleftbigg/summationdisplay
+Q,R:s(Q)≥κ−rs(R),dist(Q,R)>s(R)+/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q∩R=∅/bracketrightbigg
++/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q∩R/\egatio\slash=∅
+=:σ1+σ2+σ3.
+3.3.Three Potential Estimates of/integraltext
+X/integraltext
+Xk(x,y)f(x)g(y)dµ(x)dµ(y).Recall that the kernel
+k(x,y) ofTsatisfies the estimate
+|k(x,y)| ≤1
+max(d(x)m,d(y)m), d(x) = dist(x,X\Ω),
+Ω being an open set in X, and
+|k(x,y)| ≤CCZ
+ρ(x,y)m∀x/\e}atio\slash=y∈X;
+and
+|k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
+ρ(x,y)τ+m
+provided that ρ(x,x′)≤δρ(x,y), with some fixed constants numbers CCZ,τ,m.
+First, we will sometimes write
+/integraldisplay
+X/integraldisplay
+Xk(x,y)f(x)g(y)dµ(x)dµ(y) =/integraldisplay
+X/integraldisplay
+X[k(x,y)−k(x0,y)]f(x)g(y)dµ(x)dµ(y)
+using the fact that our fandgwill actually be ∆ Qfand ∆ Rgand so their integrals are zero.
+Temporarily write K(x,y) for either k(x,y) ork(x,y)−k(x0,y).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 9
+After that we have three logical possibilities to estimate
+/integraldisplay
+X/integraldisplay
+XK(x,y)f(x)g(y)dµ(x)dµ(y).
+(1) Estimate |K|inL∞, andf,ginL1norms;
+(2) Estimate |K|inL∞L1norm, and finL1norm,ginL∞norm (or maybe, do this
+symmetrically);
+(3) Estimate |K|inL1norm, andf,ginL∞norms.
+The third method is widely used for Calder´ on–Zygmund estim ates on homogeneous spaces (say
+with respect to Lebesgue measure), but it is very dangerous t o use in the case of a nonhomogeneous
+measure. Here is the reason. After fandgare estimated in the L∞norm, one needs to continue
+these estimates to have L2norms. There is nothing strange in that as usually fandgare almost
+proportional to characteristic functions. But for fliving onQsuch thatf=cQχQ(cQis a
+constant),
+/bardblf/bardblL∞(X:µ)≤1
+µ(Q)1/2/bardblf/bardblL2(X;µ).
+The same reasoning applies for gonR. Then
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+XK(x,y)f(x)g(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle≤1
+(µ(Q)µ(R))1/2/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+And the nonhomogeneous measure has no estimate from below. H aving two uncontrollable almost
+zeroes in the denominator is a very bad idea. We will never use the estimate of type (3).
+On the other hand, estimates of type (2) are much less dangero us (although requires the care as
+well). This is because, in this case one applies
+/bardblf/bardblL1(X;µ)≤µ(Q)1/2/bardblf/bardblL2(X;µ)and/bardblg/bardblL∞(X;µ)≤1
+µ(R)1/2/bardblg/bardblL2(µ),
+and gets
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+XK(x,y)f(x)g(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle≤/parenleftbiggµ(Q)
+µ(R)/parenrightbigg1/2
+/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+If we choose to use estimate of the type (2) only for pairs Q,Rsuch thatQ⊂Rwe are in good
+shape. This approach is what we will end up going when estimat ingσ3.
+Plan. The first sum is the “diagonal” part of the operator, σ1. The second sum, σ2is the “long
+range interaction”. The final sum, σ3, is the “short range interaction”. The diagonal part will be
+estimated using our T1 assumption of Theorem 1.2, for the long range interaction we will use the
+first type of estimates described above, for the short range i nteraction we will use estimates of types
+(1) and (2) above. But all this will be done carefully!
+4.The Long Range Interaction: Controlling Term σ2
+We first prove a lemma that demonstrates that for functions wi th supports that are far apart,
+we have some good control on the bilinear form induced by our C alder´ on-Zygmund operator T.
+For two dyadic cubes QandR, we set
+D(Q,R) :=s(Q)+s(R)+dist(Q,R).10 A. VOLBERG AND B. D. WICK
+Lemma 13. Suppose that QandRare two cubes in X, such that s(Q)≤s(R). LetϕQ,ψR∈
+L2(X;µ). Assume that ϕQvanishes outside Q, andψRvanishes outside R;/integraltext
+XϕQdµ= 0and, at
+last,dist(Q,suppψR)≥s(Q)αs(R)1−α. Then
+|(ϕQ,TψR)| ≤ACs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)µ(R)/bardblϕQ/bardblL2(X;µ)/bardblψR/bardblL2(X;µ).
+Remark 14.Note that we require only that the support of the function ψRlies far from the cube
+Q; the cubes QandRthemselves may intersect! Such situations will arise when e stimating the
+termσ2.
+Proof.LetxQbe the center of the cube Q. Note that for all x∈Q,y∈suppψR, we have
+ρ(xQ,y)≥s(Q)
+2+dist(Q,suppψR)≥s(Q)
+2+2r(1−α)s(Q)/greaterorsimilars(Q)/greaterorsimilarρ(x,xQ).
+Therefore,
+|(ϕQ,TψR)|=/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+Xk(x,y)ϕQ(x)ψR(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+X[k(x,y)−k(xQ,y)]ϕQ(x)ψR(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle
+/lessorsimilars(Q)τ
+dist(Q,suppψR)m+τ/bardblϕQ/bardblL1(X;µ)/bardblψR/bardblL1(X;µ).
+There are two possible cases.
+Case 1: dist(Q,suppψR)≥s(R).Then
+D(Q,R) :=s(Q)+s(R)+dist(Q,R)≤3dist(Q,suppψR)
+and therefore
+s(Q)τ
+dist(Q,suppψR)m+τ/lessorsimilars(Q)τ
+D(Q,R)m+τ/lessorsimilars(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ.
+Case 2:s(Q)αs(R)1−α≤dist(Q,suppψR)≤s(R).ThenD(Q,R)≤3s(R) and we get
+s(Q)τ
+dist(Q,suppψR)m+τ≤s(Q)τ
+[s(Q)αs(R)1−α]m+τ=s(Q)τ
+2s(R)τ
+2
+s(R)m+τ/lessorsimilars(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ.
+Here, key to the proof was the choice of α=τ
+2(τ+m). Now, to finish the proof of the lemma, it
+remains only to note that
+/bardblϕQ/bardblL1(X;µ)≤/radicalbig
+µ(Q)/bardblϕQ/bardblL2(X;µ)and/bardblψR/bardblL1(X;µ)≤/radicalbig
+µ(R)/bardblψR/bardblL2(X;µ).
+/square
+Applying this lemma to ϕQ= ∆QfandψR= ∆Rg, we obtain
+|σ2|/lessorsimilar/summationdisplay
+Q,Rs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)/radicalbig
+µ(R)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ). (4.1)
+To control term σ2the computations above suggest that we will define a matrix op erator, de-
+pending on the cubes QandRand show that it is a bounded operator on ℓ2.
+Lemma 15. Define
+TQ,R:=s(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)/radicalbig
+µ(R) (Q∈ Dtr
+1, R∈ Dtr
+2, s(Q)≤s(R)).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 11
+Then, for any two families {aQ}Q∈Dtr
+1and{bR}R∈Dtr
+2of nonnegative numbers, one has
+/summationdisplay
+Q,RTQ,RaQbR≤AC/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+Remark 16.Note thatTQ,Ris defined for all QandRwiths(Q)≤s(R) and that the conditions
+dist(Q,R)≥s(Q)αs(R)1−α(or even the condition Q∩R=∅) no longer appears as a condition in
+the summation!
+Assuming Lemma 15for the moment, the estimate of σ2then proceeds in an obvious fashion.
+|σ2|/lessorsimilar/summationdisplay
+Q,Rs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)/radicalbig
+µ(R)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ)
+/lessorsimilar
+/summationdisplay
+Q/bardbl∆Qf/bardbl2
+L2(X;µ)
+1/2/parenleftBigg/summationdisplay
+R/bardbl∆Rg/bardbl2
+L2(X;µ)/parenrightBigg1/2
+/lessorsimilar/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+Here the first line follows by ( 4.1), the second by Lemma 15, and finally the last by Lemma 6. We
+now turn to the proof of Lemma 15.
+Proof.Let us “slice” the matrix TQ,Raccording to the ratios(Q)
+s(R). Namely, let
+T(k)
+Q,R=/braceleftBigg
+TQ,Rifs(Q) =κ−ks(R);
+0 otherwise ,
+(k= 0,1,2,...). To prove the lemma, it is enough to show that for every k≥0,
+/summationdisplay
+Q,RT(k)
+Q,RaQbR≤C2−τ
+2k/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+The matrix {T(k)
+Q,R}has a “block” structure since the variables bRcorresponding to the cubes
+R∈ Dtr
+2for whichs(R) =κjcan only interact with the variables aQcorresponding to the cubes
+Q∈ Dtr
+1, for which s(Q) =κj−k. Thus, to get the desired inequality, it is enough to estimat e each
+block separately, i.e., to demonstrate that
+/summationdisplay
+Q,R:s(Q)=κj−k,s(R)=κjT(k)
+Q,RaQbR≤C/bracketleftBig/summationdisplay
+Q:s(Q)=κj−ka2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+R:s(R)=κjb2
+R/bracketrightBig1
+2.
+Let us introduce the functions
+F(x) :=/summationdisplay
+Q:s(Q)=κj−kaQ/radicalbig
+µ(Q)χQ(x) and G(x) :=/summationdisplay
+R:ℓ(R)=κjbR/radicalbig
+µ(R)χR(x).
+Note that the cubes of a given size in one dyadic lattice do not intersect (Property (ii) of Theorem
+2), and therefore at each point x∈X, at most one term in the sum can be non-zero. Also observe
+that
+/bardblF/bardblL2(X;µ)=/bracketleftBig/summationdisplay
+Q:s(Q)=κj−ka2
+Q/bracketrightBig1
+2and/bardblG/bardblL2(X;µ)=/bracketleftBig/summationdisplay
+R:s(R)=κjb2
+R/bracketrightBig1
+2.
+Then the estimate we need can be rewritten as/integraldisplay
+X/integraldisplay
+XKj,k(x,y)F(x)G(y)dµ(x)dµ(y)≤C/bardblF/bardblL2(X;µ)/bardblG/bardblL2(X;µ),12 A. VOLBERG AND B. D. WICK
+where
+Kj,k(x,y) =/summationdisplay
+Q,R:s(Q)=κj−k,s(R)=κjs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τχQ(x)χR(y).
+Again, for every pair of points x,y∈X, only one term in the sum can be nonzero. Since ρ(x,y)+
+s(R)≤3D(Q,R) for anyx∈Qandy∈R, we obtain
+Kj,k(x,y) =Cκ−τ
+2ks(R)τ
+D(Q,R)m+τ
+/lessorsimilarκ−τ
+2kκjτ
+[κj+ρ(x,y)]m+τ=:κ−τ
+2kkj(x,y).
+So, it is enough to check that/integraldisplay
+X/integraldisplay
+Xkj(x,y)F(x)G(y)dµ(x)dµ(y)/lessorsimilar/bardblF/bardblL2(X;µ)/bardblG/bardblL2(X;µ).
+We remind the reader that we called the balls “non-Ahlfors ba lls” if
+µ(B(x,r))>rm.
+According to the Schur test, it would suffice to prove that for e veryy∈X, one has the estimate/integraltext
+Xkj(x,y)dµ(x)/lessorsimilar1 and vice versa (i.e., for every x∈X, one has/integraltext
+Xkj(x,y)dµ(y)/lessorsimilar1). Then
+the norm of the integral operator with kernel kjinL2(X;µ) would be bounded by a constant and
+the proof of Lemma 15would be over. If we assumed a priori that the supremum of radi i of all
+non-Ahlfors balls centered at y∈Rwiths(R) =κj,were less than κj+1, then the needed estimate
+would be immediate. In fact, we can write/integraldisplay
+Xkj(x,y)dµ(x) =/integraldisplay
+B(y,κj+1)kj(x,y)dµ(x)+/integraldisplay
+X\B(y,κj+1)kj(x,y)dµ(x)
+/lessorsimilarκ−jmµ(B(y,κj+1))+/integraldisplay
+X\B(y,κj+1)κjτ
+ρ(x,y)m+τdµ(x)
+/lessorsimilarκ−jmµ(B(y,κj+1))+∞/summationdisplay
+k=0κjτ
+(κkκj+1)m+τµ(B(y;κkκj+1))
+/lessorsimilar/parenleftBig
+1+∞/summationdisplay
+k=01
+κkτ/parenrightBig
+≈1.
+The passage from the second to the third line follows by exhau sting the the space X\B(y,κj+1)
+by “annular regions” and making obvious estimates using con dition (H).
+The difficulty with this approach is that we cannot guarantee t he supremum of the radii of all
+non-Ahlfors balls centered at ybe less than κj+1for everyy∈X. Our measure may not have this
+uniform property.
+So, generally speaking, we are unable to show that the integr al operator with kernel kj(x,y) acts
+inL2(X;µ); but we do not need that much! We only need to check that the corresponding bilin ear
+form is bounded on two givenfunctionsFandG. So, we are not interested in the points y∈X
+for whichG(y) = 0 (or in the points x∈X, for which F(x) = 0). But, by definition, Gcan be
+non-zero only on the transit cubes in D2. Here we used our convention that we omit in all sums
+the fact that QandRare transit cubes, however they are!
+Now let us notice that if (and this is the case for all Rin the sum we estimate in our lemma)
+R∈ Dtran
+2, then the supremum of radii of all non-Ahlfors balls centere d aty∈Ris bounded by
+s(R) for every y∈R. Indeed, this is just Lemma 4. The same reasoning shows that if Q∈ Dtran
+1,
+then the supremum of radii of all non-Ahlfors balls centered atx∈Qis bounded by κj−k+1≤κj+1
+wheneverF(x)/\e}atio\slash= 0, and we are done with Lemma 15. /squareBERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 13
+Now, wehope, thereader will agree that thedecision to decla re thecubescontained inΩterminal
+was a good one. As a result, the fact that the measure µis not Ahlfors did not put us in any real
+trouble – we barely had a chance to notice this fact at all. But , it still remains to explain why we
+were so eager to have the extra condition
+|k(x,y)| ≤1
+max(dm(x),dm(y)), d(x) := dist(x,X\Ω)
+on our Calder´ on–Zygmund kernel. The answer is found in the n ext two sections.
+5.Short Range Interaction and Nonhomogeneous Paraproducts: Controlling
+Termσ3.
+Recall that the sum σ3is taken over the pairs Q,R, for which s(Q)<κ−rs(R) andQ∩R/\e}atio\slash=∅.
+We would like to improve this condition to the demand that Qlie “deep inside” one of the subcubes
+Rj. Recall also that we defined the skeletonskRof the cube Rby
+skR:=/uniondisplay
+j∂Rj.
+We have declared a cube Q∈ D1bad if there exists a cube R∈ D2such thats(R)>κrs(Q) and
+dist(Q,skR)≤s(Q)αs(R)1−α. Now, for every good cube Q∈ D1, the conditions s(Q)<κ−rs(R)
+andQ∩R/\e}atio\slash=∅together imply that Qlies inside one of the children RjofR. We will denote this
+subcube by RQ. The sum σ3can now be split into
+σterm
+3:=/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis terminal(∆Qf,T∆Rg)
+and
+σtran
+3:=/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∆Rg).
+5.1.Estimation of σterm
+3.First of all, write (recall that Rjdenote the children of R):
+σterm
+3=/summationdisplay
+j/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q⊂Rj∈Dterm
+2(∆Qf,T∆Rg).
+Clearly, it is enough to estimate the inner sum for every fixed , and so let us do this for j= 1. We
+have/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q⊂R1∈Dterm
+2(∆Qf,T∆Rg) =/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:s(Q)<κ−rs(R),Q⊂R1(∆Qf,T∆Rg).
+Recall that the kernel kof our operator Tsatisfies the estimate of Lemma 3
+|k(x,y)|/lessorsimilar1
+s(R)mfor allx∈R1,y∈X. (5.1)
+Hence,
+|T∆Rg(x)|/lessorsimilar/bardbl∆Rg/bardblL1(X;µ)
+s(R)mfor allx∈R1, (5.2)
+and therefore
+/bardblχR1·T∆Rg/bardblL2(X;µ)/lessorsimilar/bardbl∆Rg/bardblL1(X;µ)/radicalbig
+µ(R1)
+s(R)m
+/lessorsimilarµ(R)
+s(R)m/bardbl∆Rg/bardblL2(X;µ)≤AB/bardbl∆Rg/bardblL2(X;µ).14 A. VOLBERG AND B. D. WICK
+This follows because /bardbl∆Rg/bardblL1(X:µ)≤/radicalbig
+µ(R)/bardbl∆Rg/bardblL2(X;µ)andµ(R1)≤µ(R) hold trivially. Ad-
+ditionally, by Lemma 4we have
+µ(R)/lessorsimilars(R)m(5.3)
+becauseR(the father of the cube R1) is a transit cube if R1is terminal.
+Now, recalling Lemma 6, and taking into account that ∆Qf≡0 outsideQ, we get
+/summationdisplay
+Q:Q⊂R1|(∆Qf,T∆Rg)|=/summationdisplay
+Q:Q⊂R1|(∆Qf,χR1·T∆Rg)|
+/lessorsimilar/bardblχR1·T∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2
+/lessorsimilar/bardbl∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2.
+So, we obtain/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:Q⊂R1|(∆Qf,T∆Rg)|
+/lessorsimilar/summationdisplay
+R:R1∈Dterm
+2/bardbl∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2
+/lessorsimilar/bracketleftBig/summationdisplay
+R:R1∈Dterm
+2/bardbl∆Rg/bardbl2
+L2(X;µ)/bracketrightBig1
+2/bracketleftBig/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2.
+But the terminal cubes in D2do not intersect! Therefore every ∆Qfcan appear at most once in
+the last double sum, and we get the bound
+/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:Q⊂R1|(∆Qf,T∗∆Rψ)|
+/lessorsimilar/bracketleftBig/summationdisplay
+R/bardbl∆Rg/bardbl2
+L2(X;µ)/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Q/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2/lessorsimilar/bardblf/bardblL2(X;µ)/bardblψ/bardblL2(X;µ).
+Lemma6has been used again in the last inequality.
+5.2.Estimation of σtran
+3.Recall that
+σtran
+3=/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∗∆Rg).
+Split every term in the sum as
+(∆Qf,T∆Rψ) = (∆Qf,T(χRQ∆Rg))+(∆Qf,T∗(χR\RQ∆Rg)).
+Observe that since Qis good,Q⊂R, ands(Q)<κ−rs(R), we have
+dist(Q,suppχR\RQ∆Rg)≥dist(Q,skR)≥s(Q)αs(R)1−α.
+Using Lemma 13and taking into account that the norm /bardblχR\RQ∆Rψ/bardblL2(X;µ)does not exceed
+/bardbl∆Rψ/bardblL2(X;µ), we conclude that the sum
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit|(∆Qf,T∗(χR\RQ∆Rg))|BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 15
+can be estimated by the sum ( 4.1). Thus, our task is to find a good bound for the sum
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∗(χRQ∆Rg)).
+Recalling the definition of ∆Rψand recalling that RQis atransitcube, we get
+χRQ∆Rg=cR,QχRQ,
+where
+cRQ=/a\}bracketle{tψ/a\}bracketri}htRQ−/a\}bracketle{tg/a\}bracketri}htR
+is aconstant. So, our sum can be rewritten as
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transitcRQ(∆Qf,T∗(χRQ)).
+Our next goal will be to extend the function χRQto the function 1 in every term.
+Let us observe that
+(∆Qf,T∗(χX\RQ)) =/integraldisplay
+X/integraldisplay
+X\RQk(x,y)∆Qf(x)dµ(x)dµ(y)
+=/integraldisplay
+X/integraldisplay
+X\RQ[k(x,y)−k(xQ,y)]∆Qf(x)dµ(x)dµ(y).
+Note again that for every x∈Q,y∈X\RQ, we have
+ρ(xQ,y)≥s(Q)
+2+dist(Q,X\RQ)/greaterorsimilars(Q)/greaterorsimilarρ(x,xQ).
+Therefore,
+|k(x,y)−k(xQ,y)|/lessorsimilar/parenleftBiggρ(x,xQ)
+ρ(xQ,y)/parenrightBiggτ1
+ρ(x,y)m/lessorsimilars(Q)τ
+ρ(xQ,y)m+τ,
+and
+|(∆Qf,T(χX\RQb))|/lessorsimilars(Q)τ/bardbl∆Qf/bardblL1(X;µ)/integraldisplay
+X\RQdµ(y)
+ρ(xQ,y)m+τ.
+Now let us consider the sequence of cubes R(j)∈ D2, beginning with R(0)=RQand gradually
+ascending ( R(j)⊂R(j+1),s(R(j+1)) =κs(R(j))) to the starting cube R0=R(N)of the lattice D2.
+Clearly, all these cubes R(j)are transit cubes.
+We have
+/integraldisplay
+X\RQdµ(y)
+ρ(xQ,y)m+τ=/integraldisplay
+R0\RQdµ(y)
+ρ(xQ,y)m+τ=N/summationdisplay
+j=1/integraldisplay
+R(j)\R(j−1)dµ(y)
+ρ(xQ,y)m+τ.
+We call the j-th term of this sum Ij. Note now that, since Qis good and s(Q)< κ−rs(R)≤
+κ−rs(R(j)) for allj, we have
+dist(Q,R(j)\R(j−1))≥dist(Q,skR(j))≥s(Q)αs(R(j))1−α.
+Hence
+Ij≤1
+[s(Q)αs(R(j))1−α]m+τ/integraldisplay
+R(j)dµ.16 A. VOLBERG AND B. D. WICK
+Recalling that α=τ
+2(m+τ), we see that the first factor equals
+1
+s(Q)τ
+2s(R(j))m+τ
+2.
+SinceR(j)is transit, we have /integraldisplay
+R(j)dµ/lessorsimilarµ(R(j))/lessorsimilars(R(j))m.
+Thus,
+Ij/lessorsimilars(Q)τ
+2s(R(j))τ
+2=κ−(j−1)ε
+2s(Q)τ
+2s(R)τ
+2.
+Summing over j≥1, we get
+/integraldisplay
+X\RQ|b(y)|dµ(y)
+ρ(xQ,y)m+τ=N/summationdisplay
+j=1Ij/lessorsimilar1−κ−τ
+21
+s(Q)τ
+2s(R)τ
+2.
+Now let us note that
+|cRQ| ≤/bardbl∆Rg/bardblL1(RQ,µ)
+µ(RQ)≤/bardbl∆Rg/bardblL2(RQ,µ)/radicalbig
+µ(RQ). (5.4)
+We finally obtain
+|(∆Qf,T∗(χX\RQ))|
+/lessorsimilar1
+η(1−κ−τ
+2)/bracketleftbiggs(Q)
+s(R)/bracketrightbiggτ
+2/radicalBigg
+µ(Q)
+µ(RQ)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ)
+and
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit|cR,Q|·|(∆Qf,T∗(χX\RQ))|
+/lessorsimilar1
+η(1−κ−τ
+2)/summationdisplay
+j/summationdisplay
+Q,R:Q⊂Rj/bracketleftbiggs(Q)
+s(R)/bracketrightbiggτ
+2/radicalBigg
+µ(Q)
+µ(Rj)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
+Lemma 17. For every two families {aQ}Q∈Dtr
+1and{bR}R∈Dtr
+2of nonnegative numbers, one has
+/summationdisplay
+Q,R:Q⊂R1TQ,RaQbR≤1
+1−κ−τ
+2/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+Proof.Let us “slice” the matrix TQ,Raccording to the ratios(Q)
+s(R). Namely, let
+T(k)
+Q,R=/braceleftBigg
+TQ,R,ifQ⊂R1, s(Q) =κ−ks(R);
+0,otherwise
+(k= 1,2,...). It is enough to show that for every k≥0,
+/summationdisplay
+Q,RT(k)
+Q,RaQbR≤κ−τ
+2k/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+The matrix {T(k)
+Q,R}has a very good “block” structure: every aQcan interact with only onebR. So,
+it is enough to estimate each block separately, i.e., to show that for every fixed R∈ Dtran
+2,
+/summationdisplay
+Q:Q⊂R1,ℓ(Q)=κ−kℓ(R)κ−τ
+2k/radicalBigg
+µ(Q)
+µ(R1)aQbR≤κ−τ
+2k/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2bR.BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 17
+But, reducing both parts by the non-essential factor κ−τ
+2kbR, we see that this estimate is equivalent
+to the trivial estimate
+/summationdisplay
+Q:Q⊂R1,s(Q)=κ−ks(R)/radicalBigg
+µ(Q)
+µ(R1)aQ
+≤/bracketleftBig/summationdisplay
+Q:Q⊂R1,s(Q)=κ−ks(R)µ(Q)
+µ(R1)/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2≤/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2,
+(since cubes Q∈ D1of fixed size do not intersect,/summationtext
+Q:Q⊂R1,s(Q)=κ−ks(R)µ(Q)≤µ(R1)).
+/square
+Remark 18.We did not use here the fact that {aQ},{bR}are supported on transit cubes. We
+actually proved
+Lemma19. The matrix {TQ,R}defined by
+TQ,R:=/bracketleftbiggs(Q)
+s(R)/bracketrightbiggτ
+2/radicalBigg
+µ(Q)
+µ(R1)(Q⊂R1),
+generates a bounded operator in l2.
+We just finished estimating an extra term which appeared when we extendχRQto the whole
+1. So, the extension of χRQto the function 1 does not cause much harm, and we are left with
+estimating the sum/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transitcRQ(∆Qf,T∗1).
+Note that the inner product (∆Qf,T∗1)does not depend onRat all, so it seems to be a good idea
+to sum over Rfor fixedQfirst.
+Recalling that
+cRQ=/a\}bracketle{tg/a\}bracketri}htRQ−/a\}bracketle{tg/a\}bracketri}htR
+and that Λψ= 0⇐⇒ /a\}bracketle{tψ/a\}bracketri}htR0= 0, we conclude that for every Q∈ Dtran
+1that really appears in the
+above sum,/summationdisplay
+R:R⊃Q,s(R)>κms(Q),RQis transitcRQ=/a\}bracketle{tg/a\}bracketri}htRQ.
+Definition. LetR(Q) be the smallest transitcubeR∈ D2containing Qand such that s(R)≥
+κrs(Q).
+So, we obtain the sum
+/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(∆Qf,T∗1)
+to take care of.
+Remark. Let us recall that we had the convention that says that the cub esQconsidered are
+only good ones (and of course they are only transit cubes). Th e range of summation should be
+Q∈ Dtran
+1,Qis good (default); there exists a cube R∈ Dtran
+2such thats(Q)<κ−rs(R),Q⊂R
+and the child RQ(the one containing Q) ofRis transit. In other words, in fact, the sum is written18 A. VOLBERG AND B. D. WICK
+formally incorrectly. We have to replace R(Q) byRQin the summation. However, the smallest
+transit cube containing Q(this isR(Q)) and the smallest transit child (containing Q) of a certain
+subcubeRofR0(this child is RQ) are of course the same cube, unless R(Q) =R0. Thus the sum
+formally has some extra terms corresponding to R(Q) =R0. But, they all are zeros! In one of the
+first reductions, we were allowed to work only with with gsuch that Λ g= 0 (recall that Λ gmeans
+the average of gwith respect to µ), so/a\}bracketle{tg/a\}bracketri}htR(Q)= 0 ifR(Q) =R0.
+5.3.Pseudo-BMOand special paraproduct. Tointroducetheparaproductoperator, werewrite
+our sum as follows/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(∆Qf,T∗1) =/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(f,∆∗
+QT∗1)
+=
+f,/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)∆QT∗1
+.
+We use the fact that ∆∗
+Q= ∆Q. We now introduce the paraproduct operator, which will allo w
+us to control term σtran
+3.
+Definition 20. Given a function F, theparaproduct with symbol Fis the function
+ΠFg(x) :=/summationdisplay
+R∈D2,R⊂R0/a\}bracketle{tg/a\}bracketri}htR/summationdisplay
+Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)∆QF(x).
+As in the case when the metric space is Rd, the behavior of the paraproduct operators will be
+governed by “BMO” conditions on the symbol F. In the case of a metric space though, we face an
+additional wrinkle since we have to overcome the challenge o f dealing with the dyadic cubes, and
+we need an appropriate notion of “dilation” in the metric spa ce.
+Recall that we defined the dilation by the parameter λ≥1 of a setS⊂Xby
+λ·S:={x∈X: dist(x,S)≤(λ−1)diamS}
+Note thatS⊂λ·S.
+Definition 21. A function F∈L2(X;µ) will be called a “pseudo- BMOfunction” if there exists
+Λ>1 such that for any cube Qwithµ(sQ)≤Ksmdiam(Q)m,s≥1, we have
+/integraldisplay
+Q|F(x)−/a\}bracketle{tF/a\}bracketri}htQ|2dµ(x)≤Cµ(ΛQ).
+Lemma 22. Letµ,Tsatisfy the assumptins of Theorem 1. Then
+T∗1∈pseudo-BMO . (5.5)
+HereCdepends only on the constants of Theorem 1.
+Proof.Forx∈Qwe writeT∗1(x) = (T∗χΛQ)(x)+(T∗χX\ΛQ)(x) =:ϕ(x)+ψ(x). First, we notice
+that
+x,y∈Q⇒ |ψ(x)−ψ(y)| ≤C(K,Λ),
+whereKis the constant form our definition above. This is easy:
+|ψ(x)−ψ(y)| ≤/integraldisplay
+X\ΛQ|k(x,t)−k(y,t)|dµ(t) =∞/summationdisplay
+j=1/integraldisplay
+Λj+1Q\ΛjQ|k(x,t)−k(y,t)|dµ(t)≤
+∞/summationdisplay
+j=1diam(Q)τ
+(Λjdiam(Q))m+τK(Λjdiam(Q))m=∞/summationdisplay
+j=1K
+Λjτ≤C(K,Λ,τ).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 19
+Therefore,/integraldisplay
+Q|ψ(x)−/a\}bracketle{tψ/a\}bracketri}htQ|2dµ(x)/lessorsimilarµ(Q)≤µ(ΛQ).
+But,/integraldisplay
+Q|ϕ(x)−/a\}bracketle{tϕ/a\}bracketri}htQ|2dµ(x)/lessorsimilar/integraldisplay
+Q|T∗χΛQ|2dµ≤Aµ(ΛQ)
+by theT1 assumption of Theorem 1.
+/square
+Lemma 23. Letµ,Tsatisfy the assumptins of Theorem 1. Then
+/bardblΠT∗1/bardblL2(X;µ)→L2(X;µ)≤C. (5.6)
+HereCdepends only on the constants of Theorem 1.
+Proof.LetF=T∗1. In the definition of Π Fall ∆Qare mutually orthogonal. So it is easy to see
+that
+/bardblΠFg/bardbl2
+L2(X;µ)=/summationdisplay
+R∈D2,R⊂R0|/a\}bracketle{tg/a\}bracketri}htR|2/summationdisplay
+Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)/bardbl∆QF/bardblL2(X;µ).
+Put
+aR:=/summationdisplay
+Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)/bardbl∆QF/bardblL2(X;µ).
+By Carleson Embedding Theorem, it is enough to prove that for everyS∈ D2
+/summationdisplay
+R∈D2,R⊂SaR≤Cµ(S). (5.7)
+This is the same as
+/summationdisplay
+Q∈D1,Qtransit,s(Q)≤κ−rs(R),dist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ)≤Cµ(R). (5.8)
+Let us consider a Whitney decomposition of Rinto disjoint cubes P, such that 1 .5P⊂R, 1.4P
+have only bounded multiplicity C(d) of intersection. This can be accomplished by modifying the
+arguments found in Section 7 of [ 2].
+Consider the sums
+sP:=/summationdisplay
+Q∈D1,Qtransit,s(Q)≤κ−rs(R),Q∪P/\egatio\slash=∅,dist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ).(5.9)
+ThissPcan be zero if there is no transit cubes as above intersecting it. But ifsP/\e}atio\slash= 0 then
+necessarily
+µ(P)≤A(d)s(P)m,
+and moreover
+µ(sP)≤A(d)sms(P)m,∀s≥1.
+In fact, in this case Pintersects a transit cube Q, which by elementary geometry is “smaller”’
+thanP:s(Q)≤c(r,d)s(P). But then the above inequalities follow from the definition oftransit.
+It is also clear that for large rand forQ,Pas above
+Q∩P/\e}atio\slash=∅ ⇒Q⊂1.2P .
+Therefore,
+sP/\e}atio\slash= 0⇒sP≤/summationdisplay
+Q∈D1,Qtransit,s(Q)≤κ−rs(R),Q⊂1.2Pdist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ).20 A. VOLBERG AND B. D. WICK
+So
+sP/\e}atio\slash= 0⇒sP≤/integraldisplay
+1.2P|F−/a\}bracketle{tF/a\}bracketri}ht1.2P|2dµ≤Cµ(1.4P).
+The last inequality follows from Lemma 22.
+Now we add all sP’s. We get ≤C/summationtextµ(1.4P). This is smaller than C1µ(R) as 1.4P’s have
+multiplicity C(d)<∞. /square
+6.The Diagonal Sum: Controlling Term σ1.
+To complete the estimate of |(Tfgood,ggood)|in only remains to estimate σ1. But notice that
+/bardbl∆Qf/bardblL1(X;µ)≤ /bardbl∆Qf/bardblL2(X;µ)/radicalbig
+µ(Q) and/bardbl∆Rg/bardblL1(X;µ)≤ /bardbl∆Rg/bardblL2(X;µ)/radicalbig
+µ(R).
+Remember that all cubes QandRin the sums considered at this point are transit cubes. In
+particular, in σ1we have that QandRare close and of the almost same size. If a son of Q,S(Q),
+is terminal, then by Lemma 3
+|(TχS(Q)∆Qf,∆Rg)| ≤/radicalbig
+µ(Q)/radicalbig
+µ(R)
+s(Q)m/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
+The sons are terminal, but QandRare transit, so µ(Q)/lessorsimilars(Q)m≈s(R)m. Summing such pairs
+(and symmetric ones, where a son of Ris terminal) we get C(r)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+We are left with the part of σ1, where we sum over QandRsuch that their sons are transit.
+Then we use pairing
+|(TχS(Q)∆Qf,χS(R)∆Rg)| ≤ |cS(Q)||cS(R)|/radicalbig
+µ(S(Q))µ(S(R)).
+The estimate above follows from our T1 assumption in Theorem 1. Now using ( 5.4), again we
+obtain
+|(TχS(Q)∆Qf,χS(R)∆Rg)| ≤C/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
+This completes the proof of Theorem 1.
+References
+[1] Michael Christ, AT(b)theorem with remarks on analytic capacity and the Cauchy int egral, Colloq. Math. 60/61
+(1990), no. 2, 601–628. ↑1,3
+[2] T. Hyt¨ onen and H. Martikainen, Non-Homogeneous Tb Theorem on Metric Spaces , preprint. ↑3,4,6,7,19
+[3] F. Nazarov, S. Treil, and A. Volberg, TheTb-theorem on non-homogeneous spaces , Acta Math. 190(2003), no. 2,
+151–239. ↑3
+[4] ,Accretive system Tb-theorems on nonhomogeneous spaces , Duke Math. J. 113(2002), no. 2, 259–312. ↑3
+[5] ,Weak type estimates and Cotlar inequalities for Calder´ on- Zygmund operators on nonhomogeneous spaces ,
+Internat. Math. Res. Notices (1998), no. 9, 463–487. ↑3
+[6] ,Cauchy integral and Calder´ on-Zygmund operators on nonhomo geneous spaces , Internat. Math. Res. No-
+tices (1997), no. 15, 703–726. ↑3
+[7] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclide an and homogeneous spaces ,
+Amer. J. Math. 114(1992), no. 4, 813–874. ↑3
+[8] A. Volberg, Calder´ on-Zygmund capacities and operators on nonhomogene ous spaces , CBMS Regional Conference
+Series in Mathematics, vol. 100, Published for the Conferen ce Board of the Mathematical Sciences, Washington,
+DC, 2003. ↑3
+[9] A. Volberg and B. D. Wick, Bergman-type Singular Operators and the Characterization of Carleson Measures for
+Besov–Sobolev Spaces on the Complex Ball (2009), preprint. ↑2,3
+Alexander Volberg, Department of Mathematics, Michigan St ate University, East Lansing, MI
+USA 48824
+E-mail address :[email protected] SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 21
+Alexander Volberg, Department of Mathematics, University of Edinburgh, James Clerk Maxwell
+Building, The King’s Buildings, Mayfield Road, Edinburgh S cotland EH9 3JZ
+E-mail address :[email protected]
+Brett D. Wick, School of Mathematics, Georgia Institute of T echnology, 686 Cherry Street,
+Atlanta, GA 30332-1060 USA
+E-mail address :[email protected]
+URL:http://people.math.gatech.edu/~bwick6/

1001.0039.txt

@@ -0,0 +1,166 @@
+arXiv:1001.0039v1  [astro-ph.IM]  30 Dec 2009Astronomical Data Analysis Software and Systems XIX P45
+ASP Conference Series, Vol. XXX, 2009
+Y. Mizumoto, K.-I. Morita, and M. Ohishi, eds.
+TGCat, The Chandra Transmission Grating Catalog and
+Archive: Systems, Design and Accessibility
+Arik W. Mitschang1, David P. Huenemoerder2, Joy S. Nichols1
+Abstract. The recently released Chandra Transmission Grating Catalog and
+Archive, TGCat, presents a fully dynamic on-line catalog allowing users to
+browse and categorize Chandra gratings observations quickly and easily, gen-
+erate custom plots of resulting response corrected spectra on- line without the
+need for special software and to download analysis ready product s from multi-
+ple observations in one convenient operation. TGCathas been registered as
+a VO resource with the NVO providing direct access to the catalogs in terface.
+The catalog is supported by a back-end designed to automatically fe tch newly
+public data, process , archive and catalog them, At the same time ut ilizing an
+advanced queue system integrated into the archive’s MySQL datab ase allowing
+large processing projects to take advantage of an unlimited numbe r of CPUs
+across a network for rapid completion. A unique feature of the cat alog is that
+all of the high level functions used to retrieve inputs from the Chan dra archive
+and to generate the final data products are available to the user in an ISIS writ-
+ten library with detailed documentation. Here we present a structu ral overview
+of the Systems, Design, and Accessibility features of the catalog a nd archive.
+1. Introduction
+TGCataims to be the definitive end-user source for all Chandra HETG S and
+LETGS observations. In order to achieve this goal the catalo g must both have
+the absolute best processed and calibrated data, as well as h ave an interface
+which makes it easy for users to find, review and download thei r observations
+of choice. The science requirements for accurately process ing gratings data are
+discussed elsewhere ( Huenemoerder et. al. 2010 ). This writ ing will focus on
+the automated system that collects and processes new public data, the interfaces
+used for administrative review, and the interfaces and syst ems provided for user
+access.
+2. Database, Archive, and Subsystems
+TheTGCatprocessing system is comprised of three major components:
+•MySQL database storing meta-data
+•File archive
+•Processing Software
+1Smithsonian Astrophysical Observatory, Cambridge, MA, US A
+2MIT Kavli Institute for Space Research, Cambridge, MA
+12 MITSCHANG, HUENEMOERDER, NICHOLS
+Figure 1. TGCatdata flow
+These components are described in the context of TGCatprocessing in this
+section.
+Tables: Thetables relevant toprocessingincludetheextractions, source, spec-
+tral properties, files, and queue tables. The extractions ta ble has one entry per
+processed extraction, where extraction is taken to mean a si ngle source in a
+uniqueChandra observation ID (ObsID). Any one ObsIDcan have many sources
+and any one source can be in many ObsIDs, the extractions tabl e will store one
+entry for each combination thereof. In order to consolidate all extractions of a
+single source, there is a source table indexed on SIMBAD1identifier, a TGCat
+identifier, and coordinates, that associates entries in the extractions table. The
+files table tracks processing output products and summary im ages allowing the
+webpagestoeasily displayinformation onfilesavailable fo rdownload. Thespec-
+tral properties table stores and indexes spectral properti es in several different
+wavebands, these data are also available in a fits table for do wnload.
+Data Flow: The flow of data through the system is rather simple, as illus-
+trated in Figure 1. A bash script run via cronat regular intervals downloads a
+list of gratings observations from the public chandra archi ve2and compares with
+the list of ObsIDs that have been submitted to TGCat. Any not in TGCat
+will be added to the queue table to be processed in line. Daemo n processes,
+written in python, run on any number of network connected hos ts continuously
+requesting entries from the queue in a FIFOmanner, the first process to ask
+entries from the queue table will retrieve the first entry ( by time of creation
+). They then parse the queue entry which can contain a number o f custom
+processing parameters, setup work spaces and logging, fork off the processing
+which is implemented in ISIS ( Houck & Denicola 2000 ) interac tive library
+functions, which are available for download3and custom use, and finally ingest
+1http://simbad.u-strasbg.fr/simbad
+2http://cda.harvard.edu/chaser/
+3http://space.mit.edu/cxc/analysis/tgcatTGCat 3
+data into the database and archive. During the ingest step, s everal checks are
+done to evaluate whether or not the resultant processed data are a complete
+set and worth being manually reviewed by TGCatscientists. If so, meta data
+are added to the database’s main extractions table which ret urns a unique iden-
+tifier that is then used to tag the file based products. At this t ime linking is
+done between the source table and extractions table. If not, the queue entry is
+marked as an error and notification sent. TGCatoperators have the choice to
+investigate the existing processing workspace or simply re -evaluate parameters
+and re-queue the extraction as new.
+Validation and Verification: Each extraction available for browsing has
+been reviewed by a member of the TGCatscience team in order to confirm
+zeroth order placement, proper masking, etc. This is done vi a an internal web-
+site nearly mimicking the public interface for reviewing ex tractions but with
+the addition of forms for queueing processing and for markin g the extraction as
+“good”, “bad”, or “warning” and optional comment fields whic h are available
+for review by the end user. “bad” extractions are never shown on the public
+site and administrators have the option of rejecting any oth er extraction for
+whatever reason. Because each extraction is tagged with the date of processing,
+the version of TGCatused for processing, and a so called group ID which is
+unique per group of extractions intended to be of the same obj ect for a single
+ObsID, keeping track of accepted sources and avoiding dupli cates is made easy.
+3. Access
+TGCathas several interfaces for data access. Currently three are operational
+includingthewebbrowserinterface, andtwoVirtualObserv atoryaccessprotocol
+interfaces. Each oftheseaccess interfaces areimplemente dasa“plug-in”written
+in php. Plug-in functions to create output are called by a cat alog independent
+generalized query library written in php under TGCatdevelopment dubbed
+“queryLib”. This modular approach makes it trivial to add ac cess interfaces to
+TGCat.
+queryLib: queryLib stores state and type information of an individual query
+in a database table. Each time a query is performed, fields in t his table such as
+awhereclause, sort fields, indexed IDs and columns are populated wi th infor-
+mation specific to that query. As a user interacts with the par ticular interface,
+information in this table entry are updated to reflect the new state of the query
+through object method calls. queryLib initialized a query b y creating a new en-
+try in the table with two unique IDs: one is the primary key ide ntifier assigned
+at creation and can be used to reference the query, the other i s created on the fly
+using a combination of the primary key and a random hash strin g that is very
+unlikely to be guessed. This is done so that a malicious user c annot change the
+state of anyone’s query by simply guessing an ID. The queryLi b speeds up the
+process of redrawing a table of results by storing the indexe d primary keys of
+all entries selected rather than having to rerun a potential ly complicated where
+clause.4 MITSCHANG, HUENEMOERDER, NICHOLS
+Web Catalog Interface: The web interface4toTGCatallows users browse
+extraction and object entries in a way that is most suitable t o the catalog data
+structure. Each extraction has an associated preview page w here a set of stan-
+dard plots produced at the time of processing is available fo r viewing. A table
+of spectral properties providing quick general categoriza tion, and a table of each
+individual file available for the extraction along with a lin k to download that file
+are provided. Perhaps one of the most important features of t he web interface
+is the interactive plotting of individual or combined spect ra directly on the site.
+This is implemented server side by taking the POST request fo r plotting param-
+eters along with a unique file name, creating a small script of the commands for
+loading data and plotting, and piping this script into an ISI S process. Malicious
+useisprevented by checking all parameter inputsfor approp riatevalues, running
+the ISIS process as an unprivileged user, and checking the te mp file name for
+validity before piping. Since the file name is known at the tim e of the request,
+the page simply needs to reload to show the new plot. The comma nds, any error
+output, and an ASCII dump of the plot data are available for do wnload as well.
+Virtual Observatory: TGCatis a registered VO service providing both
+the Simple Cone Search5and Simple Image Access6Protocols. These are both
+implemented as an XML output plug-ins taking input from a php script that
+provides appropriate GET parameters, then creates a query o bject after parsing
+the input data, finally running the query in exactly the same w ay as for the web
+and ASCII interfaces. Error handling is done by the calling s cript and meta
+data is provided by running a query with parameters known to h ave no catalog
+entries ( indexed ID=0 ). VO queries, and any other type, can b e tracked in
+the query table using the type column. In this way we can track statistics on
+requests coming from services such as datascope7.
+Data Package Downloads: Generally, users will want to download more
+than one file at a time for analysis, such as such as spectra and responses or
+productsfrom multiple extractions of the same target. To th is endTGCatruns
+apackager processthatparsesaqueuetablemuchlikethepro cessingqueuetable
+described above. Requested packages are added to the packag e queue table,
+validated and read by the packager which then fetches data to a temp space
+placing them in a directory hierarchy tagged with ObsID and TGCatID. The
+entire hierarchy is then tarred, compressed and provided to the user along with
+file checksums via HTTP.
+Acknowledgements. We would like to thank Dan Dewey for extensive input
+on data organization and interface layout, and Mike Nowak fo r ISIS plotting
+routines and advice on the interactive plotting interface. This work is supported
+4http://tgcat.mit.edu
+5http://tgcat.mit.edu/tgCli.php?OUTPUT=V
+6http://tgcat.mit.edu/tgSia.php
+7http://heasarc.gsfc.nasa.gov/cgi-bin/vo/datascope/i nit.plTGCat 5
+by the Chandra X-ray Center (CXC) NASA contract NAS8-03060. DPH was
+supportedby NASA through the Smithsonian Astrophysical Ob servatory (SAO)
+contract SV3-73016 for the Chandra X-Ray Center and Science Instruments.
+References
+Huenemoerder, et. al. 2010, ( in preparation )
+Houck, J. C., Denicola, L. A. 2000, in ASP Conf. Ser. 216, ADASS IX, ed. N. Manset,
+C. Veillet, & D. Crabtree (San Francisco: ASP), 591
+Williams, Roy, Hanisch, Robert, Szalay, Alex, Plante, Ray 2009, Simple Cone Search
+Version1.03,http://www.ivoa.net/Documents/REC/DAL/ConeSea rch-20080222.html
+Tody, Doug & Plante, Ray 2009, Simple Image Access Specification Ve rsion 1.0,
+http://www.ivoa.net/Documents/SIA/20091008/PR-SIA-1.0-20 091008.html

1001.0040.txt

@@ -0,0 +1,725 @@
+arXiv:1001.0040v2  [math-ph]  16 Sep 2010COURANT ALGEBROIDS FROM CATEGORIFIED
+SYMPLECTIC GEOMETRY
+CHRISTOPHER L. ROGERS
+Abstract. In categorified symplectic geometry, one studies the cate-
+gorified algebraic and geometric structures that naturally arise on man-
+ifolds equipped with a closed nondegenerate ( n+ 1)-form. The case
+relevant to classical string theory is when n= 2 and is called ‘2-plectic
+geometry’. Just as the Poisson bracket makes the smooth func tions on
+a symplectic manifold into a Lie algebra, there is a Lie 2-alg ebra of
+observables associated to any 2-plectic manifold. String t heory, closed
+3-forms and Lie 2-algebras also play important roles in the t heory of
+Courant algebroids. Courant algebroids are vector bundles which gen-
+eralize the structures found in tangent bundles and quadrat ic Lie alge-
+bras. It is known that a particular kind of Courant algebroid (called an
+exact Courant algebroid) naturally arises in string theory , and that such
+an algebroid is classified up to isomorphism by a closed 3-for m on the
+base space, which then induces a Lie 2-algebra structure on t he space of
+global sections. In this paper we begin to establish precise connections
+between 2-plectic manifolds and Courant algebroids. We pro ve that any
+manifold Mequipped with a 2-plectic form ωgives an exact Courant
+algebroid EωoverMwithˇSevera class [ ω], and we construct an embed-
+ding of the Lie 2-algebra of observables into the Lie 2-algeb ra of sections
+ofEω. We then show that this embedding identifies the observables as
+particular infinitesimal symmetries of Eωwhich preserve the 2-plectic
+structure on M.
+1.Introduction
+The underlying geometric structures of interest in categor ified symplectic
+geometry are multisymplectic manifolds: manifolds equipp ed with a closed,
+nondegenerate form of degree ≥2 [8]. This kind of geometry originated in
+the work of DeDonder [10] and Weyl [24] on the calculus of vari ations, and
+more recently has been used as a formalism to investigate cla ssical field the-
+ories [11, 12, 13]. In this paper, we call a manifold ‘ n-plectic’ if it is equipped
+with a closed nondegenerate ( n+ 1)-form. Hence ordinary symplectic ge-
+ometry corresponds to the n= 1 case, and the corresponding 1-dimensional
+field theory is just the classical mechanics of point particl es. In general,
+examples of n-plectic manifolds include phase spaces suitable for descr ibing
+n-dimensional classical field theories. We will be primarily concerned with
+Date: October 29, 2018.
+This work was partially supported by a grant from The Foundat ional Questions
+Institute.
+12 CHRISTOPHER L. ROGERS
+then= 2case. Thisis the firstreally new case of n-plectic geometry andthe
+corresponding 2-dimensional field theories of interest inc lude bosonic string
+theory. Indeed, just as the phase space of the classical part icle is a mani-
+fold equipped with a closed, nondegenerate 2-form, the phas e space of the
+classical string is a finite-dimensional manifold equipped with a closed non-
+degenerate 3-form. This phase space is often called the ‘mul tiphase space’
+of the string [11] in order to distinguish it from the infinite -dimensional
+symplectic manifolds that are used as phase spaces in string field theory [6].
+In classical mechanics, the relevant mathematical structu res are not just
+geometric, butalsoalgebraic. Thesymplecticformgives th espaceofsmooth
+functions the structure of Poisson algebra. Analogously, i n classical string
+theory, the 2-plectic form induces a bilinear skew-symmetr ic bracket on
+a particular subspace of differential 1-forms, which we call H amiltonian.
+The Hamiltonian 1-forms and smooth functions form the under lying chain
+complex of an algebraic structure known as a semistrict Lie 2 -algebra. A
+semistrict Lie 2-algebra can be viewed as a categorified Lie a lgebra in which
+the Jacobi identity is weakened and is required to hold only u p to isomor-
+phism. Equivalently, it can be described as a 2-term L∞-algebra, i.e. a
+generalization of a 2-term differential graded Lie algebra in which the Ja-
+cobi identity is only satisfied up to chain homotopy [1, 14]. J ust as the
+Poisson algebra of smooth functions represents the observa bles of a system
+of particles, it has been shown that the Lie 2-algebra of Hami ltonian 1-forms
+contains the observables of the classical string [2]. In gen eral, ann-plectic
+structure will give rise to a L∞-algebra on an n-term chain complex of dif-
+ferential forms in which the ( n−1)-forms correspond to the observables of
+ann-dimensional classical field theory [15].
+Many of the ingredients found in 2-plectic geometry are also found in
+the theory of Courant algebroids, which was also developed b y generalizing
+structures found in symplectic geometry. Courant algebroi ds were first used
+by Courant [9] to study generalizations of pre-symplectic a nd Poisson struc-
+tures in the theory of constrained mechanical systems. Roug hly, a Courant
+algebroid is a vector bundle that generalizes the structure of a tangent bun-
+dle equipped with a symmetric nondegenerate bilinear form o n the fibers.
+In particular, the underlying vector bundle of a Courant alg ebroid comes
+equipped with a skew-symmetric bracket on the space of globa l sections.
+However, unlike the Lie bracket of vector fields, the bracket need not satisfy
+the Jacobi identity.
+In a letter to Weinstein, ˇSevera [20] described how a certain type of
+Courant algebroid, known as an exact Courant algebroid, app ears naturally
+when studying 2-dimensional variational problems. In clas sical string the-
+ory, the string can be represented as a map φ: Σ→Mfrom a 2-dimensional
+parameter space Σ into a manifold Mcorresponding to space-time. The
+imageφ(Σ) is called the string world-sheet. The map φextremizes the inte-
+gral of a 2-form θ∈Ω2(M) over its world-sheet. Hence the classical string
+is a solution to a 2-dimensional variational problem. The 2- formθis calledCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 3
+the Lagrangian and depends on elements of the first jet bundle of the trivial
+bundleΣ ×M. TheLagrangian isnotunique. A solution φremainsinvariant
+if an exact 1-form or ‘divergence’ is added to θ. It is, in fact, the 3-form dθ
+that is relevant. Inthiscontext, ˇSeveraobserved that the3-form dθuniquely
+specifies (up to isomorphism) the structure of an exact Coura nt algebroid
+overM. The general correspondence between exact Courant algebro ids and
+closed 3-forms on the base space was further developed by ˇSevera, and also
+by Bressler and Chervov [4], to give a complete classificatio n. An exact
+Courant algebroid over Mis determined up to isomorphism by its ˇSevera
+class: an element [ ω] in the third de Rham cohomology of M.
+Just as in 2-plectic geometry, the underlying geometric str ucture of a
+Courant algebroid has an algebraic manifestation. Roytenb erg and Wein-
+stein [16] showed that the bracket on the space of global sect ions induces
+anL∞structure. If we are considering an exact Courant algebroid , then
+the global sections can be identified with ordered pairs of ve ctor fields and
+1-forms on the base space. Roytenberg and Weinstein’s resul ts imply that
+these sections, when combined with the smooth functions on t he base space,
+form a semistrict Lie 2-algebra [23]. Moreover, the bracket of the Lie 2-
+algebra is determined by a closed 3-form corresponding to a r epresentative
+of theˇSevera class [21].
+Thus there are striking similarities between 2-plectic man ifolds and ex-
+act Courant algebroids. Both originate from attempts to gen eralize certain
+aspects of symplectic geometry. Both come equipped with a cl osed 3-form
+that gives rise to a Lie 2-algebra structure on a chain comple x consisting
+of smooth functions and differential 1-forms. In this paper, w e prove that
+there is indeed a connection between the two. We show that any manifold
+Mequipped with a 2-plectic form ωgives an exact Courant algebroid Eω
+withˇSevera class [ ω], and that there is an embedding of the Lie 2-algebra
+of observables into the Lie 2-algebra corresponding to Eω. Moreover, this
+embedding allows us to characterize the Hamiltonian 1-form s as particular
+infinitesimal symmetries of Eωwhich preserve the 2-plectic structure on M.
+2.Courant algebroids
+Here we recall some basic facts and examples of Courant algeb roids and
+then we proceed to describe ˇSevera’s classification of exact Courant alge-
+broids. There are several equivalent definitions of a Couran t algebroid found
+in the literature. In this paper we use the definition given by Roytenberg
+[17].
+Definition 2.1. ACourant algebroid is a vector bundle E→Mequipped
+with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a skew-
+symmetric bracket /llbracket·,·/rrbracketonΓ(E), and a bundle map (called the anchor)
+ρ:E→TMsuch that for all e1,e2,e3∈Γ(E)and for all f,g∈C∞(M)the
+following properties hold:
+(1)/llbrackete1,/llbrackete2,e3/rrbracket/rrbracket−/llbracket/llbrackete1,e2/rrbracket,e3/rrbracket−/llbrackete2,/llbrackete1,e3/rrbracket/rrbracket=−DT(e1,e2,e3),4 CHRISTOPHER L. ROGERS
+(2)ρ([e1,e2]) = [ρ(e1),ρ(e2)],
+(3) [e1,fe2] =f[e1,e2]+ρ(e1)(f)e2−1
+2/an}bracketle{te1,e2/an}bracketri}htDf,
+(4)/an}bracketle{tDf,Dg/an}bracketri}ht= 0,
+(5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{t[e1,e2]+1
+2D/an}bracketle{te1,e2/an}bracketri}ht,e3/an}bracketri}ht+/an}bracketle{te2,[e1,e3]+1
+2D/an}bracketle{te1,e3/an}bracketri}ht/an}bracketri}ht,
+where[·,·]is the Lie bracket of vector fields, D:C∞(M)→Γ(E)is the map
+defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f, and
+T(e1,e2,e3) =1
+6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
+The bracket in Definition 2.1 is skew-symmetric, but the first property
+implies that it needs only to satisfy the Jacobi identity “up toDT”. (The
+notation suggests we think of this as a boundary.) The functi onTis often
+referred to as the Jacobiator . (When there is no risk of confusion, we shall
+refer to the Courant algebroid with underlying vector bundl eE→MasE.)
+Note that the vector bundle Emay be identified with E∗via the bilinear
+form/an}bracketle{t·,·/an}bracketri}htand therefore we have the dual map
+ρ∗:T∗M→E.
+Hence the map Dis simply the pullback of the de Rham differential by ρ∗.
+Thereisanalternatedefinitiongiven by ˇSevera[20]forCourantalgebroids
+which uses a bilinear operation on sections that satisfies a J acobi identity
+but is not skew-symmetric.
+Definition 2.2. ACourant algebroid is a vector bundle E→Mtogether
+with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a bilinear
+operation ◦onΓ(E), and a bundle map ρ:E→TMsuch that for all
+e1,e2,e3∈Γ(E)and for all f∈C∞(M)the following properties hold:
+(1)e1◦(e2◦e3) = (e1◦e2)◦e3+e2◦(e1◦e3),
+(2)ρ(e1◦e2) = [ρ(e1),ρ(e2)],
+(3)e1◦fe2=f(e1◦e2)+ρ(e1)(f)e2,
+(4)e1◦e1=1
+2D/an}bracketle{te1,e1/an}bracketri}ht,
+(5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{te1◦e2,e3/an}bracketri}ht+/an}bracketle{te2,e1◦e3/an}bracketri}ht,
+where[·,·]is the Lie bracket of vector fields, and D:C∞(M)→Γ(E)is the
+map defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f.
+The “bracket” ◦is related to the bracket given in Definition 2.1 by:
+x◦y=/llbracketx,y/rrbracket+1
+2D/an}bracketle{tx,y/an}bracketri}ht. (1)
+Roytenberg [17] showed that if Eis a Courant algebroid in the sense of
+Definition 2.1 with bracket /llbracket·,·/rrbracket, bilinear form /an}bracketle{t·,·/an}bracketri}htand anchor ρ, thenEis
+a Courant algebroid in the sense of Definition 2.2 with the sam e anchor and
+bilinear form but with bracket ◦given by Eq. 1. Unless otherwise stated, all
+Courant algebroids mentioned in this paper are Courant alge broids in the
+sense of Definition 2.1. We introduced Definition 2.2 mainly t o connect our
+results here with previous results in the literature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 5
+Example 1.An important example of a Courant algebroid is the standard
+Courant algebroid E0=TM⊕T∗Mover any manifold Mwith bracket
+/llbracket(v1,α1),(v2,α2)/rrbracket0=/parenleftbigg
+[v1,v2],Lv1α2−Lv2α1−1
+2d(ιv1α2−ιv2α1)/parenrightbigg
+,(2)
+and bilinear form
+/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. (3)
+In this case the anchor ρ:E0→TMis the projection map, and for a
+function f∈C∞(M),Df= (0,df).
+The standard Courant algebroid is the prototypical example of anexact
+Courant algebroid [4].
+Definition 2.3. A Courant algebroid E→Mwith anchor map ρ:E→TM
+isexactiff
+0→T∗Mρ∗
+→Eρ→TM→0
+is an exact sequence of vector bundles.
+2.1.TheˇSevera class of an exact Courant algebroid. ˇSevera’s clas-
+sification originates in the idea that choosing a splitting o f the above short
+exact sequence corresponds to defining a kind of connection.
+Definition 2.4. Aconnection on an exact Courant algebroid Eover a
+manifold Mis a map of vector bundles A:TM→Esuch that
+(1)ρ◦A= idTM,
+(2)/an}bracketle{tA(v1),A(v2)/an}bracketri}ht= 0for allv1,v2∈TM,
+whereρ:E→TMand/an}bracketle{t·,·/an}bracketri}htare the anchor and bilinear form, respectively.
+IfAis a connection and θ∈Ω2(M) is a 2-form then one can construct a
+new connection:
+(A+θ)(v) =A(v)+ρ∗θ(v,·). (4)
+(A+θ) satisfies the first condition of Definition 2.4 since ker ρ= imρ∗. The
+second condition follows from the fact that we have by definit ion ofρ∗:
+/an}bracketle{tρ∗(α),e/an}bracketri}ht=α(ρ(e)) (5)
+for alle∈Γ(E) andα∈Ω1(M). Furthermore, one can show that any two
+connections on an exact Courant algebroid must differ (as in Eq . 4) by a
+2-form on M. Hence the space of connections on an exact Courant algebroi d
+is an affine space modeled on the vector space of 2-forms Ω2(M) [4].
+The failure of a connection to preserve the bracket gives a su itable notion
+of curvature:
+Definition 2.5. IfEis an exact Courant algebroid over Mwith bracket /llbracket·,·/rrbracket
+andA:TM→Eis a connection then the curvature is a map F:TM×
+TM→Edefined by
+F(v1,v2) =/llbracketA(v1),A(v2)/rrbracket−A([v1,v2]).6 CHRISTOPHER L. ROGERS
+IfFis the curvature of a connection Athen given v1,v2∈TM, it follows
+from exactness and axiom 2 in Definition 2.1 that there exists a 1-form
+αv1,v2∈Ω1(M) such that F(v1,v2) =ρ∗(αv1,v2). Since Ais a connection,
+its image is isotropic in E. Therefore for any v3∈TMwe have:
+/an}bracketle{tF(v1,v2),A(v3)/an}bracketri}ht=/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht.
+The above formula allows one to associate the curvature Fto a 3-form on
+M:
+Proposition 2.6. LetEbe an exact Courant algebroid over a manifold M
+with bracket /llbracket·,·/rrbracketand bilinear form /an}bracketle{t·,·/an}bracketri}ht. LetA:TM→Ebe a connection
+onE. Then given vector fields v1,v2,v3onM:
+(1)The function
+ω(v1,v2,v3) =/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht
+defines a closed 3-form on M.
+(2)Ifθ∈Ω2(M)is a 2-form and ˜A=A+θthen
+˜ω(v1,v2,v3) =/an}bracketle{t/llbracket˜A(v1),˜A(v2)/rrbracket,˜A(v3)/an}bracketri}ht
+=ω(v1,v2,v3)+dθ(v1,v2,v3).
+Proof.The statements in the proposition are proven in Lemmas 4.2.6 , 4.2.7,
+and 4.3.4 in the paper by Bressler and Chervov [4]. In their wo rk they
+define a Courant algebroid using Definition 2.2, and therefor e their bracket
+satisfies the Jacobi identity, but is not skew-symmetric. In our notation,
+their definition of the curvature 3-form is:
+ω′(v1,v2,v3) =/an}bracketle{tA(v1)◦A(v2),A(v3)/an}bracketri}ht.
+In particular they show that ◦satisfying the Jacobi identity implies ω′is
+closed. The Jacobiator corresponding to the Courant bracke t is non-trivial
+in general. However the isotropicity of the connection and E q. 1 imply
+A(v1)◦A(v2) =/llbracketA(v1),A(v2)/rrbracket∀v1,v2∈TM.
+Henceω′=ω, so all the needed results in [4] apply here. /square
+Thus the above proposition implies that the curvature 3-for m of an exact
+Courantalgebroid over Mgives awell-defined cohomology class in H3
+DR(M),
+independent of the choice of connection.
+2.2.Twisting the Courant bracket. The previous section describes how
+to go from exact Courant algebroids to closed 3-forms. Now we describe the
+reverse process. In Example 1 we showed that one can define the standard
+Courant algebroid E0over any manifold M. The total space is the direct
+sumTM⊕T∗M, the bracket and bilinear form are given in Eqs. 2 and 3,
+and the anchor is simply the projection. The inclusion A(v) = (v,0) of the
+tangent bundle into the direct sum is obviously a connection onE0and it
+is easy to see that the standard Courant algebroid has zero cu rvature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 7
+ˇSevera and Weinstein [20, 21] observed that the bracket on E0could be
+twisted by a closed 3-form ω∈Ω3(M) on the base:
+/llbracket(v1,α1),(v2,α2)/rrbracketω=/llbracket(v1,α1),(v2,α2)/rrbracket0+ω(v1,v2,·).
+This gives a new Courant algebroid Eωwith the same anchor and bilinear
+form. Using Eqs. 2 and 3 we can compute the curvature 3-form of this new
+Courant algebroid:
+/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht=/an}bracketle{t/llbracket(v1,0),(v2,0)/rrbracket,(v3,0)/an}bracketri}ht
+=/an}bracketle{t([v1,v2],ω(v1,v2,·)),(v3,0)/an}bracketri}ht
+=ω(v1,v2,v3),
+and we see that Eωis an exact Courant algebroid over MwithˇSevera class
+[ω].
+3. 2-plectic geometry
+We nowgive abriefoverview of 2-plectic geometry. Moredeta ils including
+motivation for several of the definitions presented here can be found in our
+previous work with Baez and Hoffnung [2, 3].
+Definition 3.1. A3-formωon aC∞manifold Mis2-plectic , or more
+specifically a 2-plectic structure , if it is both closed:
+dω= 0,
+and nondegenerate:
+∀v∈TxM, ιvω= 0⇒v= 0
+Ifωis a2-plectic form on Mwe call the pair (M,ω)a2-plectic manifold .
+The 2-plectic structure induces an injective map from the sp ace of vector
+fields on Mto the space of 2-forms on M. This leads us to the following
+definition:
+Definition 3.2. Let(M,ω)be a2-plectic manifold. A 1-form αonMis
+Hamiltonian if there exists a vector field vαonMsuch that
+dα=−ιvαω.
+We sayvαis theHamiltonian vector field corresponding to α. The set
+of Hamiltonian 1-forms and the set of Hamiltonian vector fiel ds on a 2-
+plectic manifold are both vector spaces and are denoted as Ham(M)and
+VectH(M), respectively.
+The Hamiltonian vector field vαis unique if it exists, but note there may
+be 1-forms αhaving no Hamiltonian vector field. Furthermore, two distin ct
+Hamiltonian 1-forms may differ by a closed 1-form and therefor e share the
+same Hamiltonian vector field.
+We can generalize thePoisson bracket of functionsin symple ctic geometry
+by defining a bracket of Hamiltonian 1-forms.8 CHRISTOPHER L. ROGERS
+Definition 3.3. Givenα,β∈Ham(M), thebracket {α,β}is the
+1-form given by
+{α,β}=ιvβιvαω.
+Proposition 3.4. Letα,β,γ∈Ham(M)and letvα,vβ,vγbe the respective
+Hamiltonian vector fields. The bracket {·,·}has the following properties:
+(1)The bracket of Hamiltonian forms is Hamiltonian:
+d{α,β}=−ι[vα,vβ]ω, (6)
+so in particular we have
+v{α,β}= [vα,vβ].
+(2)The bracket is skew-symmetric:
+{α,β}=−{β,α} (7)
+(3)The bracket satisfies the Jacobi identity up to an exact 1-form :
+{α,{β,γ}}−{{α,β},γ}−{β,{α,γ}}=dJα,β,γ (8)
+withJα,β,γ=ιvαιvβιvγω.
+Proof.See Proposition 3.7 in [2]. /square
+4.Lie2-algebras
+Both the Courant bracket and the bracket on Hamiltonian 1-fo rms are,
+roughly, Lie brackets which satisfy the Jacobi identity up t o an exact 1-
+form. This leads us to the notion of a Lie 2-algebra: a categor y equipped
+with structures analogous to those of a Lie algebra, for whic h the usual laws
+involving skew-symmetry and the Jacobi identity hold up to i somorphism
+[1, 19]. A Lie 2-algebra in which the isomorphisms are actual equalities
+is called a strict Lie 2-algebra. A Lie 2-algebra in which the laws govern-
+ing skew-symmetry are equalities but the Jacobi identity ho lds only up to
+isomorphism is called a semistrict Lie 2-algebra.
+Here we define a semistrict Lie 2-algebra to be a 2-term chain c omplex
+of vector spaces equipped with structures analogous to thos e of a Lie al-
+gebra, for which the usual laws hold up to chain homotopy. In t his guise,
+a semistrict Lie 2-algebra is nothing more than a 2-term L∞-algebra. For
+more details, we refer the reader to the work of Lada and Stash eff [14], and
+the work of Baez and Crans [1].
+Definition 4.1. Asemistrict Lie 2-algebra is a2-term chain complex
+of vector spaces L= (L1d→L0)equipped with:
+•a chain map [·,·]:L⊗L→Lcalled the bracket;
+•an antisymmetric chain homotopy J:L⊗L⊗L→Lfrom the chain
+map
+L⊗L⊗L→L
+x⊗y⊗z/mapsto−→[x,[y,z]],COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 9
+to the chain map
+L⊗L⊗L→L
+x⊗y⊗z/mapsto−→[[x,y],z]+[y,[x,z]]
+called the Jacobiator ,
+such that the following equation holds:
+[x,J(y,z,w)] +J(x,[y,z],w) +J(x,z,[y,w]) +[J(x,y,z),w]
++[z,J(x,y,w)] =J(x,y,[z,w]) +J([x,y],z,w)
++[y,J(x,z,w)] +J(y,[x,z],w) +J(y,z,[x,w]).(9)
+We will also need a suitable notion of morphism:
+Definition 4.2. Given semistrict Lie 2-algebras LandL′with bracket and
+Jacobiator [·,·],Jand[·,·]′,J′respectively, a homomorphism fromLto
+L′consists of:
+•a chain map φ= (φ0,φ1) :L→L′, and
+•a chain homotopy φ2:L⊗L→Lfrom the chain map
+L⊗L→L
+x⊗y/mapsto−→[φ(x),φ(y)]′,
+to the chain map
+L⊗L→L
+x⊗y/mapsto−→φ([x,y])
+such that the following equation holds:
+J′(φ0(x),φ0(y),φ0(z))−φ1(J(x,y,z)) =
+φ2(x,[y,z])−φ2([x,y],z)−φ2(y,[x,z])−[φ2(x,y),φ0(z)]′
++[φ0(x),φ2(y,z)]′−[φ0(y),φ2(x,z)]′.(10)
+This definition is equivalent to the definition of a morphism b etween 2-
+termL∞-algebras. (The same definition is given in [1], but it contai ns a
+typographical error.)
+4.1.The Lie 2-algebra from a 2-plectic manifold. Given a 2-plectic
+manifold( M,ω), wecanconstructasemistrictLie2-algebra. Theunderlyi ng
+2-term chain complex is namely:
+L=C∞(M)d→Ham(M)
+wheredis the usual exterior derivative of functions. This chain co mplex is
+well-defined, since any exact form is Hamiltonian, with 0 as i ts Hamiltonian
+vector field. We can construct a chain map
+{·,·}:L⊗L→L,
+by extending the bracket on Ham( M) trivially to L. In other words, in
+degree 0, the chain map is given as in Definition 3.3:
+{α,β}=ιvβιvαω,10 CHRISTOPHER L. ROGERS
+and in degrees 1 and 2, we set it equal to zero:
+{α,f}= 0,{f,α}= 0,{f,g}= 0.
+The precise construction of this Lie 2-algebra is given in th e following the-
+orem:
+Theorem 4.3. If(M,ω)is a2-plectic manifold, there is a semistrict Lie
+2-algebraL(M,ω)where:
+•the space of 0-chains is Ham(M),
+•the space of 1-chains is C∞,
+•the differential is the exterior derivative d:C∞→Ham(M),
+•the bracket is {·,·},
+•the Jacobiator is the linear map JL: Ham(M)⊗Ham(M)⊗Ham(M)→
+C∞defined by JL(α,β,γ) =ιvαιvβιvγω.
+Proof.See Theorem 4.4 in [2]. /square
+4.2.The Lie 2-algebra from a Courant algebroid. Given any Courant
+algebroid E→Mwith bilinear form /an}bracketle{t·,·/an}bracketri}ht, bracket /llbracket·,·/rrbracket, and anchor ρ:E→
+TM, we can construct a 2-term chain complex
+C=C∞(M)D→Γ(E),
+with differential D=ρ∗d. The bracket /llbracket·,·/rrbracketon global sections can be ex-
+tended to a chain map /llbracket·,·/rrbracket:C⊗C→C. Ife1,e2are degree 0 chains then
+/llbrackete1,e2/rrbracketis the original bracket. If eis a degree 0 chain and f,gare degree 1
+chains, then we define:
+/llbrackete,f/rrbracket=−/llbracketf,e/rrbracket=1
+2/an}bracketle{te,Df/an}bracketri}ht
+/llbracketf,g/rrbracket= 0.
+This extended bracket gives a semistrict Lie 2-algebra on th e complex C:
+Theorem 4.4. IfEis a Courant algebroid, there is a semistrict Lie 2-
+algebraC(E)where:
+•the space of 0-chains is Γ(E),
+•the space of 1-chains is C∞(M),
+•the differential the map D:C∞(M)→Γ(M),
+•the bracket is /llbracket·,·/rrbracket,
+•the Jacobiator is the linear map JC: Γ(M)⊗Γ(M)⊗Γ(M)→C∞(M)
+defined by
+JC(e1,e2,e3) =−T(e1,e2,e3)
+=−1
+6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
+Proof.Theproofthat aCourantalgebroid inthesenseofDefinition 2 .1gives
+rise to a semistrict Lie 2-algebra follows from the work done by Roytenberg
+on graded symplectic supermanifolds [18] and Lie 2-algebra s [19]. In partic-
+ular we refer the reader to Example 5.4 of [19] and Section 4 of [18].COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 11
+Onthe other hand, theoriginal construction of Roytenberg a nd Weinstein
+[16] gives a L∞-algebra on the complex:
+0→kerDι→C∞(M)D→Γ(E),
+with trivial structure maps lnforn≥3. Moreover, the map l2(correspond-
+ing to the bracket /llbracket·,·/rrbracketgiven above) is trivial in degree >1 and the map
+l3(corresponding to the Jacobiator JC) is trivial in degree >0. Hence we
+can restrict this L∞-algebra to our complex Cand use the results in [1] that
+relateL∞-algebras with semistrict Lie 2-algebras. /square
+5.The Courant algebroid associated to a 2-plectic manifold
+Now we have the necessary machinery in place to describe how C ourant
+algebroids connect with 2-plectic geometry. First, recall the discussion in
+Section 2.2 on twisting the bracket of the standard Courant a lgebroid E0by
+a closed 3-form. From Definition 3.1, we immediately have the following:
+Proposition 5.1. Let(M,ω)be a2-plectic manifold. There exists an exact
+Courant algebroid EωwithˇSevera class [ω]overMwith underlying vector
+bundleTM⊕T∗M→M, anchor ρ(v,α) =v, and bracket and bilinear form
+given by:
+/llbracket(v1,α1),(v2,α2)/rrbracketω=/parenleftbigg
+[v1,v2],Lv1α2−Lv2α1−1
+2d(ιv1α2−ιv2α1)+ιv2ιv1ω/parenrightbigg
+,
+/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1.
+More importantly, the Courant algebroid constructed in Pro position 5.1
+not only encodes the 2-plectic structure ω, but also the corresponding Lie
+2-algebra constructed in Theorem 4.3:
+Theorem 5.2. Let(M,ω)be a2-plectic manifold and let Eωbe its corre-
+sponding Courant algebroid. Let L(M,ω)andC(Eω)be the semistrict Lie
+2-algebras corresponding to (M,ω)andEω, respectively. Then there exists
+a homomorphism embedding L(M,ω)intoC(Eω).
+Before we prove the theorem, we introduce some lemmas to ease the
+calculations. In the notation that follows, if α,βare Hamiltonian 1-forms
+with corresponding vector fields vα,vβ, then
+B(α,β) =1
+2(ιvαβ−ιvβα). (11)
+Also by the symbol /anticlockwisewe mean cyclic permutations of the symbols α,β,γ.
+Lemma 5.3. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
+fieldsvα,vβ, thenLvαβ={α,β}+dιvαβ.
+Proof.SinceLv=ιvd+dιv,
+Lvαβ=ιvαdβ+dιvαβ=−ιvαιvβω+dιvαβ={α,β}+dιvαβ.
+/square12 CHRISTOPHER L. ROGERS
+Lemma 5.4. Ifα,β,γ∈Ham(M)with corresponding Hamiltonian vector
+fieldsvα,vβ,vγ, then
+ι[vα,vβ]γ+/anticlockwise=−3ιvαιvβιvγω+2/parenleftbig
+ιvαdB(β,γ)+ιvγdB(α,β)+ιvβdB(γ,α)/parenrightbig
+.
+Proof.The identity ι[v1,v2]=Lv1ιv2−ιv2Lv1and Lemma 5.3 imply:
+ι[vα,vβ]γ=Lvαιvβγ−ιvβLvαγ
+=Lvαιvβγ−ιvβ({α,γ}+dιvαγ)
+=ιvαdιvβγ−ιvβιvγιvαω−ιvβdιvαγ,
+where the last equality follows from the definition of the bra cket.
+Hence it follows that:
+ι[vγ,vα]β=ιvγdιvαβ−ιvαιvβιvγω−ιvαdιvγβ,
+ι[vβ,vγ]α=ιvβdιvγα−ιvγιvαιvβω−ιvγdιvβα.
+Finally, note 2 ιvαdB(β,γ) =ιvαdιvβγ−ιvαdιvγβ. /square
+Lemma 5.5. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
+fieldsvα,vβ, then
+Lvβα−Lvαβ=−2({α,β}+dB(α,β)).
+Proof.Follows immediately from Lemma 5.3 and the definition of B(α,β).
+/square
+Proof of Theorem 5.2. We will construct a homomorphism from L(M,ω) to
+C(Eω). LetLbe the underlying chain complex of L(M,ω) consisting of
+Hamiltonian 1-forms in degree 0 and smooth functions in degr ee 1. Let
+Cbe the underlying chain of C(Eω) consisting of global sections of Eωin
+degree 0 and smooth functions in degree 1. The bracket /llbracket·,·/rrbracketωdenotes the
+extension of the bracket on Γ( Eω) to the complex Cin the sense of Theorem
+4.4. Let φ0:L0→C0be given by
+φ0(α) = (vα,−α),
+wherevαis the Hamiltonian vector field corresponding to α. Letφ1:L1→
+C1be given by
+φ1(f) =−f.
+Finally let φ2:L0⊗L0→C1be given by
+φ2(α,β) =−B(α,β) =−1
+2(ιvαβ−ιvβα).
+Nowweshow φ2isawell-definedchainhomotopyinthesenseofDefinition
+4.2. For degree 0 we have:
+/llbracketφ0(α),φ0(β)/rrbracketω=/parenleftbigg
+[vα,vβ],Lvα(−β)−Lvβ(−α)+1
+2d/parenleftbig
+ιvαβ−ιvβα/parenrightbig
++ιvβιvαω/parenrightbigg
+= ([vα,vβ],−{α,β}+dφ2(α,β)),COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 13
+where the last equality above follows from Lemma 5.5. By Prop osition 3.4,
+the Hamiltonian vector field of {α,β}is [vα,vβ]. Hence we have:
+/llbracketφ0(α),φ0(β)/rrbracketω−φ0({α,β}) =dφ2(α,β).
+Indegree1, thebracket {·,·}istrivial. Henceitfollows fromthedefinition
+of/llbracket·,·/rrbracketωand the bilinear form on Eω(given in Proposition 5.1 ) that
+/llbracketφ0(α),φ1(f)/rrbracketω=−/llbracketφ1(f),φ0(α)/rrbracketω=1
+2/an}bracketle{t(vα,−α),(0,−df)/an}bracketri}ht=φ2(α,df).
+Therefore φ2is a chain homotopy.
+It remains to show the coherence condition (Eq. 10 in Definiti on 4.2) is
+satisfied. We rewrite the Jacobiator JCas:
+JC(φ0(α),φ0(β),φ0(γ)) =−1
+6/an}bracketle{t/llbracketφ0(α),φ0(β)/rrbracket,φ0(γ)/an}bracketri}ht+/anticlockwise
+=−1
+6/an}bracketle{t([vα,vβ],−{α,β}−dB(α,β)),(vγ,−γ)/an}bracketri}ht+/anticlockwise
+=1
+6/parenleftBig
+ι[vα,vβ]γ+ιvγιvβιvαω+ιvγdB(α,β)/parenrightBig
++/anticlockwise
+=−JL(α,β,γ)+1
+2/parenleftbig
+ιvγdB(α,β)+/anticlockwise/parenrightbig
+.
+The last equality above follows from Lemma 5.4 and the definit ion of the
+Jacobiator JL. Therefore the left-hand side of Eq. 10 is
+JC(φ0(α),φ0(β),φ0(γ))−φ1(JL(α,β,γ)) =1
+2/parenleftbig
+ιvγdB(α,β)+/anticlockwise/parenrightbig
+.
+By the skew-symmetry of the brackets, the right-hand side of Eq. 10 can
+be rewritten as:
+(/llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise)−(φ2({α,β},γ)+/anticlockwise).
+From the definitions of the bracket, bilinear form and φ2we have:
+/llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise=1
+2/an}bracketle{t(vα,−α),(0,dφ2(β,γ)/an}bracketri}ht+/anticlockwise
+=−1
+2ιvαdB(β,γ)+/anticlockwise,
+and:
+φ2({α,β},γ)+/anticlockwise=−1
+2/parenleftBig
+ι[vα,vβ]γ−ιvγιvβιvαω/parenrightBig
+=−(ιvαdB(β,γ)+/anticlockwise).
+The last equality above follows again from Lemma 5.4. Theref ore the right-
+hand side of Eq. 10 is
+1
+2/parenleftbig
+ιvγdB(α,β)+/anticlockwise/parenrightbig
+.
+Hencethemaps φ0,φ1,φ2give ahomomorphismofsemistrictLie2-algebras.
+/square14 CHRISTOPHER L. ROGERS
+We note that Roytenberg [19] has shown that a Courant algebro id defined
+using Definition 2.2 with the bilinear operation ◦induces a hemistrict Lie
+2-algebra on the complex Cdescribed in Theorem 4.4 above. A hemistrict
+Lie 2-algebra is a Lie 2-algebra in which the skew-symmetry h olds up to
+isomorphism, while the Jacobi identity holds as an equality . We have proven
+in previous work [2] that a 2-plectic structure also gives ri se to a hemistrict
+Lie 2-algebra on the complex described in Theorem 4.3. One ca n show that
+all results presented above, in particular Theorem 5.2, car ry over to the
+hemistrict case.
+6.Hamiltonian 1-forms as infinitesimal symmetries of the
+Courant algebroid
+Givena2-plecticmanifold( M,ω), theLie2-algebraofobservables L(M,ω)
+identifies particular infinitesimal symmetries of the corre sponding Courant
+algebroid Eωvia the embedding described in the proof of Theorem 5.2. To
+see this, we first recall some basic facts concerning automor phisms of exact
+Courant algebroids. The presentation here follows the work of Bursztyn,
+Cavalcanti, and Gualtieri [7].
+Definition 6.1. LetE→Mbe a Courant algebroid with bilinear form /an}bracketle{t·,·/an}bracketri}ht,
+bracket /llbracket·,·/rrbracket, and anchor ρ:E→TM. Anautomorphism is a bundle
+isomorphism F:E→Ecovering a diffeomorphism ϕ:M→Msuch that
+(1)ϕ∗/an}bracketle{tF(e1),F(e2)/an}bracketri}ht=/an}bracketle{te1,e2/an}bracketri}ht,
+(2)F(/llbrackete1,e2/rrbracket) =/llbracketF(e1),F(e2)/rrbracket,
+(3)ρ(F(e1)) =ϕ∗(ρ(e1)).
+Consider the exact Courant algebroid Eωdescribed in Section 2.2 with
+underlyingvector bundle TM⊕T∗M→MandˇSevera class [ ω]∈H3
+DR(M).
+Given a 2-form B∈Ω2(M), one can define a bundle isomorphism
+expB:TM⊕T∗M→TM⊕T∗M
+by
+expB(v,α) = (v,α+ιvB).
+The map exp Bis known as a ‘gauge transformation’. It covers the identity
+id:M→Mand therefore is compatible (in the sense of Definition 6.1) w ith
+the anchor ρ(v,α) =v. Since Bis skew-symmetric, exp Bpreserves the
+bilinear form /an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. However a simple compu-
+tation shows that exp Bpreserves the bracket /llbracket·,·/rrbracketω(defined in Eq. 2.2) if
+and only if Bis a closed 2-form:
+/llbracketexpB(v1,α1),expB(v2,α2)/rrbracketω= expB/parenleftbig
+/llbracket(v1,α1),(v2,α2)/rrbracketω+dB/parenrightbig
+.
+Given a diffeomorphism ϕ:M→Mof the base space, one can define a
+bundle isomorphism Φ: TM⊕T∗M→TM⊕T∗Mby
+Φ(v,α) =/parenleftBig
+ϕ∗v,(ϕ∗)−1α/parenrightBig
+.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 15
+ThemapΦsatisfiesconditions1and3ofDefinition6.1butdoes notpreserve
+the bracket in general:
+/llbracketΦ(v1,α1),Φ(v2,α2)/rrbracketω= Φ/parenleftBig
+/llbracket(v1,α1),(v2,α2)/rrbracketϕ∗ω/parenrightBig
+.
+Bursztyn, Cavalcanti, and Gualtieri [7] showed that any aut omorphism F
+of the exact Courant algebroid Eωmust be of the form
+F= ΦexpB, (12)
+where Φ is constructed from a diffeomorphism ϕ:M→Msuch that
+ω−ϕ∗ω=dB. (13)
+This classification of automorphisms allows one to classify the infinitesimal
+symmetries as well. Let
+Ft= ΦtexptB=/parenleftBig
+ϕt∗exptB,(ϕ∗
+t)−1exptB/parenrightBig
+bea 1-parameter family of automorphismsof the Courant alge broidEωwith
+F0= idEω. Letu∈Vect(M) be the vector field that generates the flow ϕ−t.
+Then differentiation gives:
+dFt
+dt(v,α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+t=0= ([u,v],Luα+ιvB).
+Sinceω−ϕ∗
+tω=tdB, it follows that uandBmust satisfy the equality:
+Luω=dB. (14)
+Theseinfinitesimaltransformationsarecalled derivations [7]oftheCourant
+algebroid Eω, sincetheycorrespondtolinearfirstorderdifferential oper ators
+which act as derivations of the non-skew-symmetric bracket :
+(v1,α1)◦ω(v2,α2) =/llbracket(v1,α1),(v2,α2)/rrbracketω+1
+2d/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht.(15)
+= ([v1,v2],Lv1α2−ιv2dα1+ιv2ιv1ω). (16)
+In general, derivations are pairs ( u,B)∈Vect(M)⊕Ω2(M) satisfying Eq.
+14. They act on global sections ( v,α)∈Γ(Eω) by:
+(u,B)·(v,α) = ([u,v],Luα+ιvB).
+Global sections themselves naturally act as derivations vi a anadjoint
+action[7]. Given ( u,β)∈Γ(Eω) letBbe the 2-form
+B=−dβ+ιuω. (17)
+Define ad (u,β): Γ(Eω)→Γ(Eω) by
+ad(u,β)(v,α) = (u,B)·(v,α) = ([u,v],Luα+ιv(−dβ+ιuω)).(18)
+One can see this is indeed the adjoint action in the usual sens e if one con-
+siders the non-skew-symmetric bracket given in Eq. 15:
+ad(u,β)(v,α) = (u,β)◦ω(v,α).16 CHRISTOPHER L. ROGERS
+Recall that in the proof of Theorem 5.2 we constructed a homom orphism
+of Lie 2-algebras using the map φ0: Ham(M)→Γ(Eω) defined by
+φ0(α) = (vα,−α),
+wherevαis the Hamiltonian vector field correspondingto α. Comparing Eq.
+17 to Definition 3.2 of a Hamiltonian 1-form, we see that a sect ion (u,β)∈
+Γ(Eω) is in the image of the map φ0if and only if its adjoint action ad (u,β)
+corresponds to the pair ( u,0)∈Vect(M)⊕Ω2(M). This implies that ad (u,β)
+preserves the 2-plectic structure on Mand that −βis a Hamiltonian 1-form
+with Hamiltonian vector field u. Also if uis complete, then Eqs. 12 and 13
+imply that the 1-parameter family Ftof Courant algebroid automorphisms
+generated by ad (u,β)correspondsto a1-parameter family of diffeomorphisms
+ϕt:M→Mwhich preserve the 2-plectic structure:
+ϕ∗
+tω=ω.
+In analogy with symplectic geometry, we call such automorph ismsHamil-
+tonian 2-plectomorphisms .
+We provide the following proposition as a summary of the disc ussion
+presented in this section:
+Proposition 6.2. Let(M,ω)be a 2-plectic manifold and let Eωbe its cor-
+responding Courant algebroid. There is a one-to-one correspo ndence be-
+tween the Hamiltonian 1-forms Ham(M)on(M,ω)and sections (u,β)of
+Eωwhose adjoint action satisfies the equality
+ad(u,β)(v,α) = (Luv,Luα).
+7.Conclusions
+We suspect that the results presented here are preliminary a nd indicate
+a deeper relationship between 2-plectic geometry and the th eory of Courant
+algebroids. For example, the discussion of connections and curvature in
+Section 2.1 is reminiscent of the theory of gerbes with conne ction [5], whose
+relationship with Courant algebroids has been already stud ied [4, 20]. In 2-
+plecticgeometry, gerbeshavebeenconjecturedtoplayarol einthegeometric
+quantization ofa2-plecticmanifold[2]. Itwillbeinteres tingtoseehowthese
+different points of view complement each other.
+In general, much work has been done on studying the geometric struc-
+tures induced by Courant algebroids (e.g. Dirac structures , twisted Dirac
+structures). Perhaps this work can aid 2-plectic geometry s ince many geo-
+metric structures in this context are somewhat less underst ood or remain
+ill-defined (e.g. the notion of a 2-Lagrangian submanifold o r 2-polarization).
+On the other hand, n-plectic manifolds are well understood in the role
+they play in classical field theory [11], and are also underst ood algebraically
+in the sense that an n-plectic structure gives an n-termL∞-algebra on a
+chain complex of differential forms [15]. Perhaps these insig hts can aid inCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 17
+understanding ‘higher’ analogs of Courant algebroids (e.g . Lien-algebroids)
+and complement similar ideas discussed by ˇSevera in [22].
+8.Acknowledgments
+We thank John Baez, Yael Fregier, Dmitry Roytenberg, Urs Sch rieber,
+James Stasheff and Marco Zambon for helpful comments, questi ons, and
+conversations.
+References
+[1] J. Baez and A. Crans, Higher-dimensional algebra VI: Lie 2-algebras, TAC12(2004),
+492–528. Also available as arXiv:math/0307263.
+[2] J. Baez, A. Hoffnung, and C. Rogers, Categorified symplect ic geometry and
+the classical string, Comm. Math. Phys. 293(2010), 701–715. Also available as
+arXiv:0808.0246.
+[3] J. Baez and C. Rogers, Categorified symplectic geometry a nd the string Lie 2-algebra,
+available as arXiv:0901.4721.
+[4] P. Bressler and A. Chervov, Courant algebroids, J. Math. Sci. (N.Y.) 128(2005),
+3030–3053. Also available as arXiv:hep-th/0212195.
+[5] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantiz ation,
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+and generalized complex structures, Adv. Math. 211(2007), 726–765. Also available
+as arXiv:math/0509640.
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+J. Austral. Math. Soc. (Series A) 66(1999), 303–330.
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+[10] T. DeDonder, Theorie Invariantive du Calcul des Variations , Gauthier–Villars, Paris,
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+Theor. Phys. 32(7) (1993), 1087–1103. Also available as hep-th/9209099.
+[15] C. Rogers, L∞-algebras from multisymplectic geometry, in preparation.
+[16] D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie
+algebras, Lett. Math. Phys. 46(1998), 81–93. Also available as arXiv:math/9802118.
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+manifolds , Ph.D. thesis, UC Berkeley, 1999. Also available as arXiv:m ath/9910078.
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+temp. Math. ,315, AMS, Providence, RI, 2002, pp. 169–185. Also available as
+arXiv:math/0203110.18 CHRISTOPHER L. ROGERS
+[19] D. Roytenberg, On weak Lie 2-algebras, in: P. Kielanows kiet al(eds.) XXVI Work-
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+Theor. Phys. Suppl. 144(2001), 145–154. Also available as arXiv:math/0107133.
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+E-mail address :[email protected]
+Department of Mathematics, University of California, Rive rside, Califor-
+nia 92521, USA

1001.0041.txt

@@ -0,0 +1,497 @@
+arXiv:1001.0041v2  [math.MG]  2 Jul 2010Almost-Euclidean subspaces of ℓN
+1via tensor
+products: a simple approach to randomness
+reduction
+Piotr Indyk1⋆and Stanislaw Szarek2⋆⋆
+[email protected]
+2CWRU & Paris 6 [email protected]
+Abstract. It has been known since 1970’s that the N-dimensional ℓ1-
+space contains nearly Euclidean subspaces whose dimension isΩ(N).
+However, proofs of existence of such subspaces were probabi listic, hence
+non-constructive, which made the results not-quite-suita ble for subse-
+quently discovered applications to high-dimensional near est neighbor
+search, error-correcting codes over the reals, compressiv e sensing and
+other computational problems. In this paper we present a “lo w-tech”
+scheme which, for any γ >0, allows us toexhibitnearly Euclidean Ω(N)-
+dimensional subspaces of ℓN
+1while using only Nγrandom bits. Our re-
+sults extend and complement (particularly) recent work by G uruswami-
+Lee-Wigderson. Characteristic features of our approach in clude (1) sim-
+plicity (we use only tensor products) and (2) yielding almos t Euclidean
+subspaces with arbitrarily small distortions.
+1 Introduction
+It is a well-known fact that for any vector x∈RN, itsℓ2andℓ1norms are
+related by the (optimal) inequality /⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤√
+N/⌊ar⌈⌊lx/⌊ar⌈⌊l2. However, classical
+results in geometric functional analysis show that for a “substant ial fraction” of
+vectors , the relation between its 1-norm and 2-norm can be made m uch tighter.
+Specifically, [FLM77,Kas77,GG84] show that there exists a subspace E⊂RNof
+dimensionm=αN, and a scaling constant Ssuch that for all x∈E
+1/D·√
+N/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤S/⌊ar⌈⌊lx/⌊ar⌈⌊l1≤√
+N/⌊ar⌈⌊lx/⌊ar⌈⌊l2 (1)
+whereα∈(0,1) andD=D(α), called the distortion ofE, are absolute (notably
+dimension-free) constants.Overthe last few years,such “almos t-Euclidean”sub-
+spaces ofℓN
+1have found numerous applications, to high-dimensional nearest
+neighbor search [Ind00], error-correcting codes over reals and c ompressive sens-
+ing [KT07,GLR08,GLW08], vector quantization [LV06], oblivious dimension ality
+⋆This research has been supported in part by David and Lucille Packard Fellowship,
+MADALGO (Center for Massive Data Algorithmics, funded by th e Danish National
+Research Association) and NSF grant CCF-0728645.
+⋆⋆Supported in part by grants from the National Science Founda tion (U.S.A.) and the
+U.S.-Israel BSF.reduction and ǫ-samples for high-dimensional half-spaces [KRS09], and to other
+problems.
+For the above applications, it is convenient and sometimes crucial th at the
+subspaceEis defined in an explicit manner3. However, the aforementioned re-
+sults do not providemuch guidance in this regard,since they use the probabilistic
+method. Specifically, either the vectors spanning E, or the vectors spanning the
+space dual to E, are i.i.d. random variables from some distribution. As a result,
+the constructionsrequire Ω(N2)independent randomvariablesasstartingpoint.
+Until recently, the largest explicitly constructible almost-Euclidean subspace of
+ℓN
+1, due to Rudin [Rud60] (cf. [LLR94]), had only a dimension of Θ(√
+N).
+During the last few years, there has been a renewed interest in the prob-
+lem[AM06,Sza06,Ind07,LS07,GLR08,GLW08],withresearchersusingide asgained
+from the study of expanders, extractorsand error-correctin gcodes to obtain sev-
+eral explicit constructions. The work progressed on two fronts, focusing on (a)
+fully explicit constructions of subspaces attempting to maximize the dimension
+and minimize the distortion [Ind07,GLR08], as well as (b) construction s using
+limited randomness, with dimension and distortion matching (at least q ualita-
+tively)theexistentialdimensionanddistortionbounds[Ind00,AM06,L S07,GLW08].
+The parameters of the constructions are depicted in Figure 1. Qua litatively,
+they show that in the fully explicit case, one can achieve either arbitr arily low
+distortion or arbitrarily high subspace dimension, but not (yet?) bo th. In the
+low-randomness case, one can achieve arbitrarily high subspace dim ension and
+constant distortion while using randomness that is sub-linear in N; achieving
+arbitrarily low distortion was possible as well, albeit at a price of (super )-linear
+randomness.
+Reference Distortion Subspace dimension Randomness
+[Ind07] 1+ǫ N1−oǫ(1)explicit
+[GLR08] (logN)Oη(logloglog N)(1−η)N explicit
+[Ind00] 1+ǫ Ω(ǫ2/log(1/ǫ))NO(Nlog2N)
+[AM06,LS07] Oη(1) (1−η)N O(N)
+[GLW08] 2Oη(1/γ)(1−η)N O(Nγ)
+This paper 1+ǫ (γǫ)O(1/γ)N O(Nγ)
+Fig.1.The best known results for constructing almost-Euclidean s ubspaces of ℓN
+1. The
+parameters ǫ,η,γ∈(0,1) are assumed to be constants, although we explicitly point
+out when the dependence on them is subsumed by the big-Oh nota tion.
+3For the purpose of this paper “explicit” means “the basis of Ecan be generated
+by a deterministic algorithm with running time polynomial i nN.” However, the
+individual constructions can be even “more explicit” than t hat.Our result In this paper we show that, using sub-linear randomness, one can
+constructasubspacewitharbitrarilysmalldistortionwhilekeepingit sdimension
+proportional to N. More precisely, we have:
+Theorem 1 Letǫ,γ∈(0,1). GivenN∈N, assume that we have at our dis-
+posal a sequence of random bits of length max{Nγ,C(ǫ,γ)}log(N/(ǫγ)). Then,
+in deterministic polynomial (in N) time, we can generate numbers M >0,
+m≥c(ǫ,γ)Nand anm-dimensional subspace of ℓN
+1E, for which we have
+∀x∈E,(1−ǫ)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤(1+ǫ)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2
+with probability greater than 98%.
+In a sense, this complements the result of [GLW08], optimizing the dist ortion
+of the subspace at the expense of its dimension. Our approach also allows to
+retrieve – using a simpler and low-tech approach – the results of [GLW 08] (see
+the comments at the end of the Introduction).
+Overview of techniques The ideas behind many of the prior constructions as
+well as this work can be viewed as variants of the related developmen ts in the
+context of error-correcting codes. Specifically, the construct ion of [Ind07] resem-
+bles the approach of amplifying minimum distance of a code using expan ders
+developed in [ABN+92], while the constructions of [GLR08,GLW08] were in-
+spired by low-density parity check codes. The reason for this stat e of affairs is
+that a vector whose ℓ1norms and ℓ2norms are very different must be “well-
+spread”, i.e., a small subset of its coordinates cannot contain most of itsℓ2
+mass (cf. [Ind07,GLR08]). This is akin to a property required from a g ood error-
+correcting code, where the weight (a.k.a. the ℓ0norm) of each codeword cannot
+be concentrated on a small subset of its coordinates.
+In this vein, our construction utilizes a tool frequently used for (lin ear) error-
+correcting codes, namely the tensor product . Recall that, for two linear codes
+C1⊂ {0,1}n1andC2⊂ {0,1}n2, their tensor product is a code C⊂ {0,1}n1n2,
+such that for any codeword c∈C(viewed as an n1×n2matrix), each column of
+cbelongs toC1and each row of cbelongs toC2. It is known that the dimension
+ofCis a product of the dimensions of C1andC2, and that the same holds
+for the minimum distance. This enables constructing a code of “large ” block-
+lengthNkby starting from a code of “small” block-length Nand tensoring it k
+times. Here, we roughly show that the tensor product of two subs paces yields a
+subspace whose distortion is a product of the distortions of the su bspaces. Thus,
+we can randomly choose an initial small low-distortion subspace, and tensor it
+with itself to yield the desired dimension.
+However, tensoring alone does not seem sufficient to give a subspac e with
+distortionarbitrarilycloseto1.Thisisbecausewecanonlyanalyzeth edistortion
+of the product space for the case when the scaling factor Sin Equation 1 is
+equal to 1 (technically, we only prove the left inequality, and rely on t he general
+relation between the ℓ2andℓ1for the upper bound). For S= 1, however, the
+best achievable distortion is strictly greater than 1, and tensoring can make itonly larger. To avoid this problem, instead of the ℓN
+1norm we use the ℓN/B
+1(ℓB
+2)
+norm, for a “small” value of B. The latter norm (say, denoted by /⌊ar⌈⌊l · /⌊ar⌈⌊l) treats
+the vector as a sequence of N/B“blocks” of length B, and returns the sum of
+theℓ2norms of the blocks. We show that there exist subspaces E⊂ℓN/B
+1(ℓB
+2)
+such that for any x∈Ewe have
+1/D·/radicalbig
+N/B/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤/radicalbig
+N/B/⌊ar⌈⌊lx/⌊ar⌈⌊l2
+forDthat is arbitrarily close to 1. Thus, we can construct almost-Euclide an
+subspaces of ℓ1(ℓ2) of desired dimensions using tensoring, and get rid of the
+“inner”ℓ2norm at the end of the process.
+We point out that if we do not insist on distortion arbitrarily close to 1,
+the “blocks” are not needed and the argument simplifies substantia lly. In par-
+ticular, to retrieve the results of [GLW08], it is enough to combine the scalar-
+valued version of Proposition 1 below with “off-the-shelf” random co nstructions
+[Kas77,GG84] yielding – in the notation of Equation 1 – a subspace E, for which
+the parameter αis close to 1.
+2 Tensoring subspaces of L1
+We start by defining some basic notions and notation used in this sect ion.
+Norms and distortion In this section we adopt the “continuous” notation for
+vectorsand norms. Specifically, considera real Hilbert space Hand a probability
+measureµover [0,1]. Forp∈[1,∞] consider the space Lp(H) ofH-valuedp-
+integrable functions fendowed with the norm
+/⌊ar⌈⌊lf/⌊ar⌈⌊lp=/⌊ar⌈⌊lf/⌊ar⌈⌊lLp(H)=/parenleftbigg/integraldisplay
+/⌊ar⌈⌊lf(x)/⌊ar⌈⌊lp
+Hdµ(x)/parenrightbigg1/p
+In what follows we will omit µfrom the formulae since the measure will be
+clear from the context (and largely irrelevant). As our main result c oncerns
+finite dimensional spaces, it suffices to focus on the case where µis simply the
+normalizedcountingmeasureoverthe discreteset {0,1/n,...(n−1)/n}for some
+fixedn∈N(although the statements hold in full generality). In this setting, t he
+functionsffromLp(H) areequivalent to n-dimensional vectorswith coordinates
+inH.4The advantage of using the Lpnorms as opposed to the ℓpnorms that
+the relation between the 1-norm and the 2-norm does not involve sc aling factors
+that depend on dimension, i.e., we have /⌊ar⌈⌊lf/⌊ar⌈⌊l2≥ /⌊ar⌈⌊lf/⌊ar⌈⌊l1for allf∈L2(H) (note
+that, for the Lpnorms, the “trivial” inequality goes in the other direction than
+for theℓpnorms). This simplifies the notation considerably.
+4The values from Hroughly correspond to the finite-dimensional “blocks” in th e
+construction sketched in the introduction. Note that Hcan be discretized similarly
+as theLp-spaces; alternatively, functions that are constant on int ervals of the type/parenleftBig
+(k−1)/N,k/N/parenrightBig
+can be considered in lieu of discrete measures.We will be interested in lialmost subspaces E⊂L2(H) on which the 1-norm
+and 2-norm uniformly agree, i.e., for some c∈(0,1],
+/⌊ar⌈⌊lf/⌊ar⌈⌊l2≥ /⌊ar⌈⌊lf/⌊ar⌈⌊l1≥c/⌊ar⌈⌊lf/⌊ar⌈⌊l2 (2)
+for allf∈E. The best (the largest) constant cthat works in (2) will be denoted
+Λ1(E). For completeness, we also define Λ1(E) = 0 if noc>0 works.
+Tensor products IfH,Kare Hilbert spaces, H⊗2Kis their (Hilbertian) tensor
+product, which may be (for example) described by the following prop erty: if (ej)
+is an orthonormal sequence in Hand (fk) is an orthonormal sequence in K,
+then (ej⊗fk) is an orthonormal sequence in H ⊗2K(a basis if ( ej) and (fk)
+were bases). Next, any element of L2(H)⊗ Kis canonically identified with a
+function in the space L2(H ⊗2K); note that such functions are H ⊗K-valued,
+but are defined on the same probability space as their counterpart s fromL2(H).
+IfE⊂L2(H) is a linear subspace, E⊗Kis – under this identification – a linear
+subspace of L2(H⊗2K).
+As hinted in the Introduction, our argument depends (roughly) on the fact
+that the property expressed by (1) or (2) “passes” to tensor p roducts of sub-
+spaces, and that it “survives” replacing scalar-valued functions b y ones that
+have values in a Hilbert space. Statements to similar effect of various degrees
+of generality and precision are widely available in the mathematical liter ature,
+see for example [MZ39,Bec75,And80,FJ80]. However, we are not awar e of a ref-
+erence that subsumes all the facts needed here and so we presen t an elementary
+self-contained proof.
+We start with two preliminary lemmas.
+Lemma 1 Ifg1,g2,...∈E⊂L2(H), then
+/integraldisplay/parenleftbig/summationdisplay
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+H/parenrightbig1/2dx≥Λ1(E)/parenleftBig/integraldisplay/summationdisplay
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+Hdx/parenrightBig1/2
+.
+ProofLetKbe an auxiliary Hilbert space and ( ek) an orthonormal sequence
+(O.N.S.) in K. We will apply Minkowski inequality – a continuous version of
+the triangle inequality, which says that for vector valued functions /⌊ar⌈⌊l/integraltext
+h/⌊ar⌈⌊l ≤/integraltext/⌊ar⌈⌊lh/⌊ar⌈⌊l– to the K-valued function h(x) =/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊lHek. As is easily seen,
+/⌊ar⌈⌊l/integraltext
+h/⌊ar⌈⌊lK=/⌊ar⌈⌊l/summationtext
+k/parenleftbig/integraltext
+/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊lHdx/parenrightbig
+ek/⌊ar⌈⌊lK=/parenleftbig/summationtext
+k/⌊ar⌈⌊lgk/⌊ar⌈⌊l2
+L1(H)/parenrightbig1/2. Given that gk∈
+E,/⌊ar⌈⌊lgk/⌊ar⌈⌊lL1(H)≥Λ1(E)/⌊ar⌈⌊lgk/⌊ar⌈⌊lL2(H)and so
+/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+h/vextenddouble/vextenddouble/vextenddouble
+K≥Λ1(E)/parenleftBig/integraldisplay/summationdisplay
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+Hdx/parenrightBig1/2
+On the other hand, the left hand side of the inequality in Lemma 1 is exa ctly/integraltext
+/⌊ar⌈⌊lh/⌊ar⌈⌊lK, so the Minkowski inequality yields the required estimate.
+We are now ready to state the next lemma. Recall that Eis a linear subspace
+ofL2(H), andKis a Hilbert space.Lemma 2 Λ1(E⊗K) =Λ1(E)
+IfE⊂L2=L2(R), the lemma says that any estimate of type (2) for scalar
+functionsf∈Ecarries overto their linear combinations with vector coefficients,
+namely to functions of the type/summationtext
+jvjfj,fj∈E,vj∈ K. In the general case,
+any estimate for H-valued functions f∈E⊂L2(H) carries over to functions of
+the form/summationtext
+jfj⊗vj∈L2(H⊗2K), withfj∈E,vj∈ K.
+Proof of Lemma 2 Let (ek) be an orthonormalbasis of K. In fact w.l.o.g. we may
+assume that K=ℓ2and that (ek) is the canonical orthonormal basis. Consider
+g=/summationtext
+jfj⊗vj, wherefj∈Eandvj∈ K. Then also g=/summationtext
+kgk⊗ekfor some
+gk∈Eand hence (pointwise) /⌊ar⌈⌊lg(x)/⌊ar⌈⌊lH⊗2K=/parenleftbig/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+H/parenrightbig1/2. Accordingly,
+/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K)=/parenleftbig/integraltext/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+Hdx/parenrightbig1/2,while/⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)=/integraltext/parenleftbig/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+H/parenrightbig1/2dx.
+Comparing such quantities is exactly the object of Lemma 1, which imp lies that
+/⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)≥Λ1(E)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K).Sinceg∈E⊗Kwasarbitrary,it follows that
+Λ1(E⊗K)≥Λ1(E). The reverse inequality is automatic (except in the trivial
+case dim K= 0, which we will ignore).
+IfE⊂L2(H) andF⊂L2(K) are subspaces, E⊗Fis the subspace of
+L2(H ⊗2K) spanned by f⊗gwithf∈E,g∈F. (For clarity, f⊗gis a
+function on the product of the underlying probability spaces and is defined by
+(x,y)→f(x)⊗g(y)∈ H⊗K .)
+The next proposition shows the key property of tensoring almost- Euclidean
+spaces.
+Proposition 1. Λ1(E⊗F)≥Λ1(E)Λ1(F)
+ProofLet (ϕj) and (ψk) be orthonormal bases of respectively EandFand let
+g=/summationtext
+j,ktjkϕj⊗ψk.Weneedtoshowthat /⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)≥Λ1(E)Λ1(F)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K),
+where thep-norms refer to the product probability space, for example
+/⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)=/integraldisplay /integraldisplay/vextenddouble/vextenddouble/summationdisplay
+j,ktjkϕj(x)⊗ψk(y)/vextenddouble/vextenddouble
+H⊗2Kdxdy.
+Rewriting the expression under the sum and subsequently applying L emma 2 to
+the inner integral for fixed ygives
+/integraldisplay/vextenddouble/vextenddouble/summationdisplay
+j,ktjkϕj(x)⊗ψk(y)/vextenddouble/vextenddouble
+H⊗2Kdx=/integraldisplay/vextenddouble/vextenddouble/summationdisplay
+jϕj(x)⊗/parenleftBig/summationdisplay
+ktjkψk(y)/parenrightBig/vextenddouble/vextenddouble
+H⊗2Kdx
+≥Λ1(E)/parenleftBig/integraldisplay/vextenddouble/vextenddouble/summationdisplay
+jϕj(x)⊗/parenleftBig/summationdisplay
+ktjkψk(y)/parenrightBig/vextenddouble/vextenddouble2
+H⊗2Kdx/parenrightBig1/2
+=Λ1(E)/parenleftBig/summationdisplay
+j/vextenddouble/vextenddouble/summationdisplay
+ktjkψk(y)/vextenddouble/vextenddouble2
+K/parenrightBig1/2In turn,/summationtext
+ktjkψk∈F(for allj) and so, by Lemma 1,
+/integraldisplay/parenleftBig/summationdisplay
+j/vextenddouble/vextenddouble/summationdisplay
+ktjkψk(y)/vextenddouble/vextenddouble2
+K/parenrightBig1/2
+dy≥Λ1(F)/parenleftBig/integraldisplay/summationdisplay
+j/vextenddouble/vextenddouble/summationdisplay
+ktjkψk(y)/vextenddouble/vextenddouble2
+Kdy/parenrightBig1/2
+=Λ1(F)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K).
+Combining the above formulae yields the conclusion of the Proposition .
+3 The construction
+In this section we describe our low-randomness construction. We s tart from a
+recap of the probabilistic construction, since we use it as a building blo ck.
+3.1 Dvoretzky’s theorem, and its “tangible” version
+For general normed spaces, the following is one possible statement of the well-
+known Dvoretzky’s theorem [Dvo61]:
+Givenm∈Nandε>0there isN=N(m,ε)such that, for any norm on RN
+there is an m-dimensional subspace on which the ratio of ℓ1andℓ2norms is
+(approximately) constant, up to a multiplicative factor 1+ε.
+For specific norms this statement can be made more precise, both in describing
+the dependence N=N(m,ε) and in identifying the constant of (approximate)
+proportionality of norms. The following version is (essentially) due to Milman
+[Mil71].
+Dvoretzky’s theorem (Tangible version) Consider the N-dimensional Eu-
+clidean space (real orcomplex) endowed with the Euclidean norm /⌊ar⌈⌊l·/⌊ar⌈⌊l2and some
+other norm /⌊ar⌈⌊l·/⌊ar⌈⌊lsuch that, for some b>0,/⌊ar⌈⌊l·/⌊ar⌈⌊l ≤b/⌊ar⌈⌊l·/⌊ar⌈⌊l2. LetM=E/⌊ar⌈⌊lX/⌊ar⌈⌊l, whereX
+is a random variable uniformly distributed on the unit Euclidean sphere . Then
+there exists a computable universal constant c >0, so that if 0< ε <1and
+m≤cε2(M/b)2N, then for more than 99% (with respect to the Haar measure)
+m-dimensional subspaces Ewe have
+∀x∈E,(1−ε)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤(1+ε)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2. (3)
+Alternative good expositions of the theorem are in, e.g., [FLM77], [MS8 6] and
+[Pis89]. We point out that standard and most elementary proofs yield m≤
+cε2/log(1/ε)(M/b)2N; the dependence on εof orderε2was obtained in the
+important papers [Gor85,Sch89], see also [ASW10].
+3.2 The case of ℓn
+1(ℓB
+2)
+OurobjectivenowistoapplyDvoretzky’stheoremandsubsequent lyProposition
+1 to spaces of the form ℓn
+1(ℓB
+2) for some n,B∈N, so from now on we set/⌊ar⌈⌊l·/⌊ar⌈⌊l:=/⌊ar⌈⌊l·/⌊ar⌈⌊lℓn
+1(ℓB
+2)To that end, we need to determine the values of the parameter
+Mthat appears in the theorem. (The optimal value of bis clearly√n, as in
+the scalar case, i.e., when B= 1.) We have the following standard (cf. [Bal97],
+Lecture 9)
+Lemma 3
+M(n,B) :=Ex∈SnB−1/⌊ar⌈⌊lx/⌊ar⌈⌊l=Γ(B+1
+2)
+Γ(B
+2)Γ(nB
+2)
+Γ(nB+1
+2)n.
+In particular,/radicalBig
+1+1
+n−1/radicalBig
+2
+π√n>M(n,1)>/radicalBig
+2
+π√nfor alln∈N(the scalar
+case) andM(n,B)>/radicalBig
+1−1
+B√nfor alln,B∈N.
+The equality is shown by relating (via passing to polar coordinates) sp heri-
+cal averages of norms to Gaussian means: if Xis a random variable uniformly
+distributed on the Euclidean sphere SN−1andYhas the standard Gaussian
+distribution on RN, then, for any norm /⌊ar⌈⌊l·/⌊ar⌈⌊l,
+E/⌊ar⌈⌊lY/⌊ar⌈⌊l=√
+2Γ(N+1
+2)
+Γ(N
+2)E/⌊ar⌈⌊lX/⌊ar⌈⌊l
+The inequalities follow from the estimates/radicalBig
+x−1
+2<Γ(x+1
+2)
+Γ(x)<√x(forx≥1
+2),
+which in turn are consequences of log-convexity of Γand its functional equation
+Γ(y+1) =yΓ(y). (Alternatively, Stirling’s formula may be used to arrive at a
+similar conclusion.)
+Combining Dvoretzky’s theorem with Lemma 3 yields
+Corollary 1 If0< ε <1andm≤c1ε2n, then for more than 99%of the
+m-dimensional subspaces E⊂ℓn
+1we have
+∀x∈E(1−ε)/radicalbigg
+2
+π√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤(1+ε)/radicalbigg
+1+1
+n−1/radicalbigg
+2
+π√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2(4)
+Similarly, if B >1andm≤c2ε2nB, then for more than 99%of them-
+dimensional subspaces E⊂ℓn
+1(ℓB
+2)we have
+∀x∈E(1−ε)/radicalbigg
+1−1
+B√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2 (5)
+We point out that the upper estimate on /⌊ar⌈⌊lx/⌊ar⌈⌊lin the second inequality is valid
+for allx∈ℓn
+1(ℓB
+2) and, like the estimate M(n,B)≤√n, follows just from the
+Cauchy-Schwarz inequality.
+Since a random subspace chosen uniformly according to the Haar me asure
+on the manifold of m-dimensional subspaces of RN(orCN) can be constructed
+from anN×mrandom Gaussian matrix, we may apply standard discretization
+techniques to obtain the followingCorollary 2 There is a deterministic algorithm that, given ε,B,m,n as in
+Corollary 1 and a sequence of O(mnlog(mn/ǫ))random bits, generates sub-
+spacesEas in Corollary 1 with probability greater than 98%, in time polynomial
+in1/ε+B+m+n.
+We point out that in the literature on the “randomness-reduction” , one typ-
+ically uses Bernoulli matrices in lieu of Gaussian ones. This enables avoid ing the
+discretization issue, since the problem is phrased directly in terms of random
+bits. Still, since proofs of Dvoretzky type theorems for Bernoulli m atrices are
+often much harder than for their Gaussian counterparts, we pre fer to appeal in-
+stead to a simple discretization ofGaussian random variables.We not e, however,
+that the early approach of [Kas77] was based on Bernoulli matrices .
+We are now ready to conclude the proof of Theorem 1. Given ε∈(0,1)
+andn∈N, chooseB=⌈ε−1⌉andm=⌊cε2(1−1
+B)nB⌋ ≥c0ε2nB. Corollary
+2 (Equation 5) and repeated application of Proposition 1 give us a sub space
+F⊂ℓν
+1(ℓβ
+2) (whereν=nkandβ=Bk) of dimension mk≥(c0ε2)kνβsuch that
+∀x∈F(1−ε)3k/2nk/2/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤nk/2/⌊ar⌈⌊lx/⌊ar⌈⌊l2.
+Moreover,F=E⊗E⊗...⊗E, whereE⊂ℓn
+1(ℓB
+2) is a typical m-dimensional
+subspace.Thusin ordertoproduce E, henceF,weonlyneed togeneratea“typi-
+cal”m≈c0ε2(νβ))1/ksubspace of the nB= (νβ))1/k-dimensional space ℓn
+1(ℓB
+2).
+Note that for fixed εandk >1,nBandmare asymptotically (substantially)
+smaller than dim F. Further, in order to efficiently represent Fas a subspace of
+anℓ1-space, we only need to find a good embedding of ℓβ
+2intoℓ1. This can be
+done using Corollary 2 (Equation 4); note that βdepends only on εandk. Thus
+we reduced the problem of finding “large” almost Euclidean subspace s ofℓN
+1to
+similar problems for much smaller dimensions.
+Theorem 1 now follows from the above discussion. The argument give s, e.g.,
+c(ε,γ) = (cεγ)3/γandC(ε,γ) =c(ε,γ)−1.
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+317–366; (North-Holland, Amsterdam, 2003), Vol. 2, pp. 181 9–1820.
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+Internat. Sympos. Linear Spaces (Jerusalem, 1960) . Jerusalem Academic Press,
+Jerusalem; Pergamon, Oxford,1961, pp. 123–160.
+[FJ80] T. Figiel and W. B. Johnson. Large subspaces of ℓn
+∞and estimates of the
+Gordon-Lewis constant. Israel J. Math. 37 (1980), 92–112.
+[FLM77] T. Figiel, J. Lindenstrauss, and V. D. Milman. The di mension of almost
+spherical sections of convex bodies. Acta Math. 139 (1977), no. 1-2, 53–94.
+[GG84] A. Garnaev and E. Gluskin. The widths of a Euclidean ba ll.Soviet Math.
+Dokl.30 (1984), 200–204 (English translation)
+[Gor85] Y. Gordon. Some inequalities for Gaussian processe s and applications. Israel
+J. Math. 50 (1985), 265–289.
+[GLR08] V. Guruswami, J. Lee, and A. Razborov. Almost euclid ean subspaces of l1
+via expander codes. SODA, 2008.
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+randomness and error-correction over the reals. RANDOM , 2008.
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+ceedings of the Ninth ACM-SIAM Symposium on Discrete Algori thms, 2000.
+[Ind07] P. Indyk. Uncertainty principles, extractors and e xplicit embedding of l2 into
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+Euclidean embeddings in spaces of finite volume ratio via ran dom matrices. J.
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1001.0042.txt

@@ -0,0 +1,629 @@
+arXiv:1001.0042v2  [hep-th]  16 Apr 2010Topological Gravity in Seven Dimensions
+H. L¨ u†‡and Yi Pang⋆
+†China Economics and Management Academy
+Central University of Finance and Economics, Beijing 100081
+‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
+⋆Key Laboratory of Frontiers in Theoretical Physics
+Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
+ABSTRACT
+We obtain new topological gravity in seven dimensions by add ing two topological terms
+to the Einstein-Hilbert action. For certain choice of the co upling constants, these terms
+may have an origin as the R4correction to the 3-form field equation of eleven-dimension al
+supergravity. We derive the full set of the equations of moti on, and obtain large classes of
+solutions including static AdS black holes, squashed seven spheres and Q111spaces.1 Introduction
+There has been considerable interest in topological gauge t heories [1] because of their wide
+application in physics. The most studied example is the thre e-dimensional one. In addition
+to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
+S=1
+µ/integraldisplay
+d3xTr(dω∧ω−2
+3ω∧ω∧ω), (1)
+whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
+logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
+also generate Lorentz violation dynamically [3]. Topologi cal gravity [4] becomes dynamical
+with a propagating massive particle, with the mass proporti onal to the coupling constant
+µ. Recently, a cosmological constant is added and the corresp onding boundary conformal
+field theory (CFT) is discussed [5]. The three-dimensional m assive topological gravity is
+conjectured to be unitary for certain parameter region even though the theory has higher
+derivatives in time [6].
+The attention on higher dimensional generalizations is con siderably less. The five di-
+mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
+terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
+tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
+supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
+ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
+gauge potential Ato the connection Γ. Moreover, as we shall see later, seven-d imensional
+topological gravity has a direct origin in eleven-dimensio nal supergravity, while any higher-
+dimensional origin of the three-dimensional theory remain s unknown.
+In section 2, we present the two topological terms in seven di mensions, and discuss their
+properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
+we find it is more convenient to lift the system to eight dimens ions in order to derive the
+equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large
+classes of solutions including static Anti-de Sitter (AdS) black holes, squashed S7andQ111.
+We emphasize that all the previously-known static (AdS) bla ck holes remain to be solutions
+whenthetopological termsare addedinto theaction. Thisis analogous to threedimensions,
+where the BTZ black hole remains to be a solution in topologic al massive gravity. We
+conclude in section 4.
+22 The theory
+In seven dimensions, there are two topological terms; they a re given by
+S1=µ/integraldisplay
+Ω(7)
+1=µ/integraldisplay
+Tr(Γ∧Θ+1
+3Γ3)∧Tr(Θ2) =µ/integraldisplay
+Ω(3)∧dΩ(3), (2)
+S2=ν/integraldisplay
+Ω(7)
+2=ν/integraldisplay
+Tr(Θ3∧Γ+2
+5Θ2∧Γ3+1
+5Θ∧Γ2∧Θ∧Γ+1
+5Θ∧Γ5+1
+35Γ7),
+with Ω(3)= Tr(dΓ∧Γ−2
+3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ−Γ∧Γ,
+andµ,νare two parameters of length dimension 5. (We rescale the tot al action by the
+seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
+Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
+1and Ω(7)
+2are
+topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
+D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
+dΩ(7)
+1=Y(8)
+1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
+2=Y(8)
+2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
+As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
+Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
+Pontryagin term is proportional to Y(8)
+1−2Y(8)
+2, corresponding to ν=−2µ. In eleven-
+dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
+1
+2F(4)∧F(4)+X(8), whereX(8)is given by
+X(8)∝Y(8)
+1−4Y(8)
+2. (4)
+Thus forν=−4µ, the topological terms can be obtained from the S4reduction of super-
+gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
+For large fluxes, this topological term dominates the higher -order corrections.
+To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
+to perform their variation with respect to the metric. These topological terms are not
+manifestly invariant under the general coordinate transfo rmation, but Y(8)
+1andY(8)
+2are.
+Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
+Let us first consider the variation of S1. In terms of coordinate components, we have
+/integraldisplay
+dΩ(7)
+1=1
+16/integraldisplay
+d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
+1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
+rescaling the field B→B/gandF→F/gand setting g=−2, one can obtain the same expressions as the
+ones given here.
+3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
+represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
+For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
+lowing relation
+δRµ
+ναβ=δΓµ
+νβ;α−δΓµ
+να;β, (6)
+we find that
+/integraldisplay
+dδΩ(7)
+1=−1
+2/integraldisplay
+d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
+;ν8
+≡1
+2/integraldisplay
+d∗J, (7)
+where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
+introduced a 1-form current J=Jαdxα. Its components are given by
+Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
+Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
+δΩ(7)
+1=1
+2∗J, (9)
+up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
+only, we have
+δS1= 4µ/integraldisplay
+Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
+The variation of S2can be obtained in the same manner, given by
+δS2= 4ν/integraldisplay
+Tr(Θ∧Θ∧Θ∧δΓ). (11)
+Finally, we make use of the variation of the connection
+δΓi
+mj=1
+2gin(δgnm;j+δgnj;m−δgml;n), (12)
+and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
+terms, given by
+Cij
+1=δS1√gδgij=µ
+4√g[ǫij1j2j3j4j5j6(Ri1
+i2j1j2Ri2
+i1j3j4Rjk
+j5j6);k+i↔j],
+Cij
+2=δS2√gδgij=ν
+4√g[ǫij1j2j3j4j5j6(Rk
+i1j1j2Ri1
+i2j3j4Rji2
+j5j6);k+i↔j].(13)
+For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
+constant Λ and S1+S2, the corresponding full set of EOMs is given by
+Rij−1
+2gijR+Λgij+Cij
+1+Cij
+2= 0. (14)
+4It should be remarked that under a large gauge transformatio n Γ→ O−1ΓO−O−1dO,
+the action transforms as S→S+µv(O)+νw(O), where
+v(O) =/integraldisplay
+−1
+3d/parenleftBig
+Tr(O−1dO)3∧Ω(3)/parenrightBig
+;w(O) =−1
+35/integraldisplay
+Tr(O−1dO)7.(15)
+Thevterm is trivial and gives no restriction to the parameter µ, while the wterm should
+be classified by the seventh homotopy group of SO(1,6)
+π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
+The invariance of eiSrequires that
+ν= 2πn, n = 0,±1,±2.... (17)
+This quantization condition is clearly consistent with the M5-brane quantization, since
+it has a direct origin in D= 11. This result is completely different from that in three
+dimensions, where the SO(1,2) is homotoplically trivial and the mass parameter is not
+quantized. Moreover, since νis quantized, S2will not be renormalized in the quantum
+theory. This suggests some intriguing properties in the cor responding CFT dual.
+3 Solutions
+Spherically-symmetric solutions:
+Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
+in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
+is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
+the spherically-symmetric ansatz, given by
+ds2=−F(r)dt2+dr2
+G(r)+r2dΩ2
+5. (18)
+We find that for this ansatz, the contributions from the topol ogical terms Cij
+1andCij
+2
+vanish identically. This implies that the previously-know n static (AdS) black holes, charged
+or neutral, are still solutions when the topological terms a re added to the action. This is
+analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
+topological gravity. However the thermodynamic quantitie s such as the mass and entropy
+will acquire modifications [9, 10].
+S3bundle over S4:
+5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
+large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
+loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
+the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
+by
+ds2=α3/summationdisplay
+i=1(σi−cos2(1
+2θ)˜σi)2+β/parenleftBig
+dθ2+1
+4sin2θ3/summationdisplay
+i=1˜σ2
+i/parenrightBig
+. (19)
+whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
+2ǫijkσj∧σkand
+d˜σi=1
+2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
+4orα=1
+5β=9
+100.
+The first case corresponds to the round S7and the second is a squashed S7that is also
+Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
+to
+2α2+4αβ(7β−2)−β2= 0, (20)
+together with
+√α(α−β)3(4(10α+β)µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
+It is clear from (20) that there exists one and only one positi veαfor any positive β. The
+squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
+2. Note that when 2 µ= 3ν,
+the squashed S7that is Eisntein remains Einstein.
+S1bundle over CP3:
+There is another way of squashing an S7, which can be viewed as an S1bundle over
+CP3. The metric ansatz is given by
+ds2=α(dτ+sin2θ(dψ+B))2+βds2
+CP3,
+ds2
+CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
+d˜θ2+1
+4sin2˜θcos2˜θσ2
+3
++1
+4sin2˜θ(σ2
+1+σ2
+2)/parenrightBig
+,
+B=1
+2sin2˜θσ3. (22)
+It is of a round S7whenα=β= 1. In general, the EOMs imply that
+α=β(8−7β),8µ+ν+β3
+10976(β−1)2√α= 0. (23)
+The squashing parameter γ≡α/βlies in the range (0 ,8).
+Squashed Q111spaces:
+6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
+consider the following ansatz
+ds2=α/parenleftBig
+dψ+3/summationdisplay
+i=1cosθidφi/parenrightBig2
++β3/summationdisplay
+i=1(dθ2
+i+sinθ2
+idφ2
+i). (24)
+It is ofQ111provided that α=1
+2β= 1/16, and it remains so for ν= 0. In general, we have
+α= 4β(1−7β),8(α−β)(2α−β)µ+α(2α−3β)ν+β5(α−8β+60β2)
+4α3/2= 0.(25)
+Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
+the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
+theory, and we shall not enumerate them further.
+4 Conclusions
+This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
+with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
+Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
+symmetric solutions are unmodified by the inclusion of these topological terms. We also
+obtain squashed S7andQ111spaces, where the squashing parameter is related to the cou-
+pling constants of the topological terms. It is intriguing t o see that these known squashed
+homogeneous spaces which appear to have no connection can no w be unified under our new
+gravity theory.
+As in three dimensions, our topological gravity should play an important role in explor-
+ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
+multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
+backgrounds. The quantization condition for one of the coup ling constant suggests an un-
+usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
+include a classification of all topological gravities in (4 k+3) dimensions, investigating the
+linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
+Acknowledgement
+We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
+grant No.1053060/A050207 and the NSFC group grant No.10821 504.
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+[6] W. Li, W. Song and A. Strominger, “Chiral gravity in three dimensions,” JHEP 0804,
+082 (2008) [arXiv:0801.4566 [hep-th]]. E.A. Bergshoeff, O. H ohm and P.K. Townsend,
+“Massive gravity in three dimensions,” Phys. Rev. Lett. 102, 201301 (2009)
+[arXiv:0901.1766 [hep-th]].
+[7] M. Gunaydin, G. Sierra and P.K. Townsend, “Quantization of the gauge coupling
+constant in a five-dimensional Yang-Mills/Einstein superg ravity theory,” Phys. Rev.
+Lett.53, 322 (1984).
+[8] M. Pernici, K. Pilch and P. van Nieuwenhuizen, “Gauged ma ximally extended super-
+gravity in seven-dimensions,” Phys. Lett. B 143, 103 (1984).
+[9] S. Deser and B. Tekin, “Energy in topologically massive g ravity,” Class. Quant. Grav.
+20, L259 (2003) [arXiv:gr-qc/0307073].
+[10] Y. Tachikawa, “Black hole entropy in the presence of Che rn-Simons terms,” Class.
+Quant. Grav. 24, 737 (2007) [arXiv:hep-th/0611141].
+[11] D.D.K. Chow, C.N. Pope and E. Sezgin, “Exact solutions o f topologically massive
+gravity,” arXiv:0906.3559 [hep-th].
+8arXiv:1001.0042v2  [hep-th]  16 Apr 2010Seven-Dimensional Gravity with Topological Terms
+H. L¨ u†‡and Yi Pang⋆
+†China Economics and Management Academy
+Central University of Finance and Economics, Beijing 100081
+‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
+⋆Key Laboratory of Frontiers in Theoretical Physics
+Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
+ABSTRACT
+We construct new seven-dimensional gravity by adding two to pological terms to the
+Einstein-Hilbert action. For certain choice of the couplin g constants, these terms may be
+related to the R4correction to the 3-form field equation of eleven-dimension al supergravity.
+We derive the full set of the equations of motion. We find that t he static spherically-
+symmetric black holes are unmodified by the topological term s. We obtain squashed AdS 7,
+and also squashed seven spheres and Q111spaces in Euclidean signature.1 Introduction
+There has been considerable interest in topological gauge t heories [1] because of their wide
+application in physics. The most studied example is the thre e-dimensional one. In addition
+to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
+S=1
+µ/integraldisplay
+d3xTr(dω∧ω+2
+3ω∧ω∧ω), (1)
+whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
+logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
+also generate Lorentz violation dynamically [3]. Topologi cally massive gravity [4] becomes
+dynamical with a propagating massive particle, with the mas s proportional to the coupling
+constantµ. Recently, a cosmological constant is added and the corresp onding boundary
+conformal field theory (CFT) is discussed [5]. The three-dim ensional massive topological
+gravity is conjectured to be unitary for certain parameter r egion even though the theory
+has higher derivatives in time [6].
+The attention on higher dimensional generalizations is con siderably less. The five di-
+mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
+terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
+tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
+supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
+ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
+gauge potential Ato the connection Γ. As we shall see later, these topological terms in
+seven dimensions may be related to the anomaly cancelation t erms in eleven-dimensional
+supergravity.
+In section 2, we present the two topological terms in seven di mensions, and discuss their
+properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
+we find it is more convenient to lift the system to eight dimens ions in order to derive the
+equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large classes
+of solutions. We find that the static spherically-symmetric black holes are unmodified by
+the topological terms. This is analogous to three dimension s, where the BTZ black hole
+remains to be a solution in topologically massive gravity. I n Euclidean signature, we obtain
+squashedS7andQ111spaces. In particular, one of the squashed seven sphere can b e Wick
+rotated to become squashed AdS 7. We conclude in section 4.
+22 The theory
+In seven dimensions, there are two topological terms; they a re given by
+S1= ˜µ/integraldisplay
+Ω(7)
+1= ˜µ/integraldisplay
+Tr(Γ∧Θ−1
+3Γ3)∧Tr(Θ2) = ˜µ/integraldisplay
+Ω(3)∧dΩ(3), (2)
+S2= ˜ν/integraldisplay
+Ω(7)
+2= ˜ν/integraldisplay
+Tr(Θ3∧Γ−2
+5Θ2∧Γ3−1
+5Θ∧Γ2∧Θ∧Γ+1
+5Θ∧Γ5−1
+35Γ7),
+with Ω(3)= Tr(dΓ∧Γ+2
+3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ+Γ∧Γ,
+and ˜µ,˜νare two parameters of length dimension 5. (We rescale the tot al action by the
+seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
+Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
+1and Ω(7)
+2are
+topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
+D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
+dΩ(7)
+1=Y(8)
+1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
+2=Y(8)
+2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
+As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
+Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
+Pontryagin term is proportional to Y(8)
+1−2Y(8)
+2, corresponding to ˜ ν=−2˜µ. In eleven-
+dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
+1
+2F(4)∧F(4)+X(8), whereX(8)is given by
+X(8)∝Y(8)
+1−4Y(8)
+2. (4)
+Thus for ˜ν=−4˜µ, the topological terms can be obtained from the S4reduction of super-
+gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
+For large fluxes, this topological term dominates the higher -order corrections.
+To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
+to perform their variation with respect to the metric. These topological terms are not
+manifestly invariant under the general coordinate transfo rmation, but Y(8)
+1andY(8)
+2are.
+Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
+Let us first consider the variation of S1. In terms of coordinate components, we have
+/integraldisplay
+dΩ(7)
+1=1
+16/integraldisplay
+d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
+1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
+rescaling the field B→B/gandF→F/gand setting g= 2, one can obtain the same expressions as the
+ones given here.
+3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
+represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
+For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
+lowing relation
+δRµ
+ναβ=δΓµ
+νβ;α−δΓµ
+να;β, (6)
+we find that
+/integraldisplay
+dδΩ(7)
+1=−1
+2/integraldisplay
+d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
+;ν8
+≡1
+2/integraldisplay
+d∗J, (7)
+where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
+introduced a 1-form current J=Jαdxα. Its components are given by
+Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
+Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
+δΩ(7)
+1=1
+2∗J, (9)
+up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
+only, we have
+δS1= 4˜µ/integraldisplay
+Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
+The variation of S2can be obtained in the same manner, given by
+δS2= 4˜ν/integraldisplay
+Tr(Θ∧Θ∧Θ∧δΓ). (11)
+Finally, we make use of the variation of the connection
+δΓi
+mj=1
+2gin(δgnm;j+δgnj;m−δgml;n), (12)
+and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
+terms, given by
+Cij
+1=δS1√gδgij=µ
+4√g[ǫij1j2j3j4j5j6(Ri1
+i2j1j2Ri2
+i1j3j4Rjk
+j5j6);k+i↔j],
+Cij
+2=δS2√gδgij=ν
+4√g[ǫij1j2j3j4j5j6(Rk
+i1j1j2Ri1
+i2j3j4Rji2
+j5j6);k+i↔j].(13)
+For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
+constant Λ and S1+S2, the corresponding full set of EOMs is given by
+Rij−1
+2gijR+Λgij+Cij
+1+Cij
+2= 0. (14)
+4It should be remarked that under a large gauge transformatio n Γ→ OΓO−1−dOO−1,
+the action transforms as S→S+ ˜µv(O)+ ˜νw(O), where
+v(O) =/integraldisplay
+1
+3d/parenleftBig
+Tr(dOO−1)3∧Ω(3)/parenrightBig
+;w(O) =1
+35/integraldisplay
+Tr(dOO−1)7.(15)
+Thevterm is trivial and gives no restriction to the parameter ˜ µ, while the wterm should
+be classified by the seventh homotopy group of SO(1,6)
+π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
+The invariance of eiSrequires that
+64π4˜ν= 2πn, n = 0,±1,±2.... (17)
+This result is completely different from that in three dimensi ons, where the SO(1,2) is
+homotoplically trivial and the mass parameter is not quanti zed. Moreover, since ˜ νis quan-
+tized,S2will not be renormalized in the quantum theory. This suggest s some intriguing
+properties in the corresponding CFT dual.
+3 Solutions
+Spherically-symmetric solutions:
+Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
+in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
+is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
+the spherically-symmetric ansatz, given by
+ds2=−F(r)dt2+dr2
+G(r)+r2dΩ2
+5. (18)
+We find that for this ansatz, the contributions from the topol ogical terms Cij
+1andCij
+2
+vanish identically. This implies that the previously-know n static (AdS) black holes, charged
+or neutral, are still solutions when the topological terms a re added to the action. This is
+analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
+topological gravity. However the thermodynamic quantitie s such as the mass and entropy
+will acquire modifications [9, 10].
+As we shall discuss presently, there also exist squashed AdS 7solutions.
+S3bundle over S4:
+5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
+large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
+loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
+the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
+by
+ds2=α3/summationdisplay
+i=1(σi−cos2(1
+2θ)˜σi)2+β/parenleftBig
+dθ2+1
+4sin2θ3/summationdisplay
+i=1˜σ2
+i/parenrightBig
+. (19)
+whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
+2ǫijkσj∧σkand
+d˜σi=1
+2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
+4orα=1
+5β=9
+100.
+The first case corresponds to the round S7and the second is a squashed S7that is also
+Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
+to
+2α2+4αβ(7β−2)−β2= 0, (20)
+together with
+√α(α−β)3(4(10α+β)˜µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
+It is clear from (20) that there exists one and only one positi veαfor any positive β. The
+squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
+2. Note that when 2˜ µ= 3˜ν,
+the squashed S7that is Eisntein remains Einstein.
+S1bundle over CP3:
+There is another way of squashing an S7, which can be viewed as an S1bundle over
+CP3. This example can be generalized to Minkowskian signature t o give rise to squashed
+AdS7[12]. The metric ansatz is given by
+ds2=α(dτ+sin2θ(dψ+B))2+βds2
+CP3,
+ds2
+CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
+d˜θ2+1
+4sin2˜θcos2˜θσ2
+3
++1
+4sin2˜θ(σ2
+1+σ2
+2)/parenrightBig
+,
+B=1
+2sin2˜θσ3. (22)
+It is of a round S7whenα=β= 1. In general, the EOMs imply that
+α=β(8−7β),8˜µ+ ˜ν+β3
+10976(β−1)2√α= 0. (23)
+The squashing parameter γ≡α/βlies in the range (0 ,8).
+Squashed Q111spaces:
+6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
+consider the following ansatz
+ds2=α/parenleftBig
+dψ+3/summationdisplay
+i=1cosθidφi/parenrightBig2
++β3/summationdisplay
+i=1(dθ2
+i+sinθ2
+idφ2
+i). (24)
+It is ofQ111provided that α=1
+2β= 1/16, and it remains so for ˜ ν= 0. In general, we have
+α= 4β(1−7β),8(α−β)(2α−β)˜µ+α(2α−3β)˜ν+β5(α−8β+60β2)
+4α3/2= 0.(25)
+Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
+the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
+theory, and we shall not enumerate them further.
+4 Conclusions
+This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
+with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
+Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
+symmetric solutions are unmodified by the inclusion of these topological terms. We also
+obtain squashed AdS 7, and squashed S7andQ111spaces in Euclidean signature, where
+the squashing parameter is related to the coupling constant s of the topological terms. It is
+intriguing to see that these known squashed homogeneous spa ces which appear to have no
+connection can now be unified under our new gravity theory.
+As in three dimensions, our topological gravity should play an important role in explor-
+ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
+multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
+backgrounds. The quantization condition for one of the coup ling constant suggests an un-
+usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
+include a classification of all topological gravities in (4 k+3) dimensions, investigating the
+linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
+Acknowledgement
+We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
+grant No.1053060/A050207 and the NSFC group grant No.10821 504.
+7References
+[1] S. Deser, R. Jackiw and S. Templeton, “Topologically mas sive gauge theories,” An-
+nals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988
+APNYA,281,409-449.2000)].
+[2] F. Wilczek and A. Zee, “Linking numbers, spin, and statis tics of solitons,” Phys. Rev.
+Lett.51, 2250 (1983).
+[3] S.M. Carroll, G.B. Field and R. Jackiw, “Limits on a Loren tz and parity violating
+modification of electrodynamics,” Phys. Rev. D 41, 1231 (1990).
+[4] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional E instein gravity: dynamics
+of flat space,” Annals Phys. 152, 220 (1984).
+[5] E. Witten, “Three-dimensional gravity revisited,” arX iv:0706.3359 [hep-th].
+[6] W. Li, W. Song and A. Strominger, “Chiral gravity in three dimensions,” JHEP 0804,
+082 (2008) [arXiv:0801.4566 [hep-th]]. E.A. Bergshoeff, O. H ohm and P.K. Townsend,
+“Massive gravity in three dimensions,” Phys. Rev. Lett. 102, 201301 (2009) arXiv:
+0901.1766 [hep-th].
+[7] M. Gunaydin, G. Sierra and P.K. Townsend, “Quantization of the gauge coupling
+constant in a five-dimensional Yang-Mills/Einstein superg ravity theory,” Phys. Rev.
+Lett.53, 322 (1984).
+[8] M. Pernici, K. Pilch and P. van Nieuwenhuizen, “Gauged ma ximally extended super-
+gravity in seven-dimensions,” Phys. Lett. B 143, 103 (1984).
+[9] S. Deser and B. Tekin, “Energy in topologically massive g ravity,” Class. Quant. Grav.
+20, L259 (2003), gr-qc/0307073.
+[10] Y. Tachikawa, “Black hole entropy in the presence of Che rn-Simons terms,” Class.
+Quant. Grav. 24, 737 (2007), hep-th/0611141.
+[11] D.D.K. Chow, C.N. Pope and E. Sezgin, “Exact solutions o f topologically massive
+gravity,” arXiv:0906.3559 [hep-th].
+[12] P. Hoxha, R.R. Martinez-Acosta and C.N. Pope, “Kaluza- Klein consistency, Killing
+vectors, and Kaehler spaces,” Class. Quant. Grav. 17, 4207 (2000), hep-th/0005172.
+8

1001.0043.txt

@@ -0,0 +1,399 @@
+arXiv:1001.0043v2  [astro-ph.EP]  13 Jan 2010Strong Constraints to the Putative Planet Candidate around VB 10 using
+Doppler spectroscopy1
+Guillem Anglada-Escud´ e
+Department of Terrestrial Magnetism, Carnegie Institution o f Washington
+5241 Broad Branch Road, NW, Washington, DC 20015 USA
+[email protected]
+Evgenya Shkolnik
+Department of Terrestrial Magnetism, Carnegie Institution o f Washington
+5241 Broad Branch Road, NW, Washington, DC 20015 USA
+[email protected]
+Alycia J. Weinberger
+Department of Terrestrial Magnetism, Carnegie Institution o f Washington
+5241 Broad Branch Road, NW, Washington, DC 20015 USA
+[email protected]
+Ian B. Thompson
+The Observatories of the Carnegie Institution of Washington
+813 Santa Barbara Street, Pasadena, CA 91101 USA
+[email protected]
+David J. Osip
+Las Campanas Observatory, Carnegie Institution of Washington
+Colina El Pino Casilla 601, La Serena, Chile
+[email protected]
+John H. Debes
+Goddard Space Flight Center, NASA Postdoctoral Program
+8463 Greenbelt Rd, Greenbelt, MD 20770, USA
+[email protected]
+ABSTRACT– 2 –
+We present new radial velocity measurements of the ultra-co ol dwarf VB 10,
+which was recently announced to host a giant planet detected with astrometry. The
+new observations were obtained using optical spectrograph s(MIKE/Magellan and ES-
+PaDOnS/CHFT) and cover a 63% of the reported period of 270 day s. We apply Least-
+squares periodograms to identify the most significant signa ls and evaluate their corre-
+spondingFalse Alarm Probabilities. We show that this metho d is the proper generaliza-
+tion to astrometric data because (1) it mitigates the coupli ng of the orbital parameters
+with the parallax and proper motion, and (2) it permits a dire ct generalization to
+include non-linear Keplerian parameters in a combined fit to astrometry and radial ve-
+locity data. In fact, our analysis of the astrometry alone un covers the reported 270 d
+period and an even stronger signal at ∼50 days. We estimate the uncertainties in the
+parameters using a Markov Chain Monte Carlo approach. The no minal precision of the
+new Doppler measurements is about 150 s−1while their standard deviation is 250 ms−1.
+However, the best fit solutions still have RMS of 200 ms−1indicating that the excess
+in variability is due to uncontrolled systematic errors rat her than the candidate com-
+panions detected in the astrometry. Although the new data al one cannot rule-out the
+presence of a candidate, when combined with published radia l velocity measurements,
+the False Alarm Probabilities of the best solutions grow to u nacceptable levels strongly
+suggesting that the observed astrometric wobble is not due t o an unseen companion.
+Subject headings: astrometry, methods: statistical, stars: individual (VB 1 0), tech-
+niques: radial velocities
+1. Introduction
+Pravdo & Shaklan (2009) recently announced the discovery of an astrometric companion to
+VB 10, an ultra-cool dwarf with a mass of ≈0.08 M ⊙. From a Keplerian fit to the motion, they
+determined a mass of 6 M Jand a period of 270 d. Thus VB 10 became the lowest mass star
+known to harbor a planetary companion. The mass ratio betwee n VB 10 and its companion, ∼13,
+also is intriguing. A similar mass ratio for a Solar-type sta r would make the companion a brown
+dwarf, but brown dwarfs as small separation companions to st ars are quite rare. VB 10 is itself the
+secondary in a wide binary with V1428 Aql, a M2.5 star (van Bie sbroeck 1961). At a distance of
+1Based on observations collected with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory,
+Chile, at the W. M. Keck Observatory and the Canada-France-H awaii Telescope (CFHT). The Keck Observatory is
+operated as a scientific partnership between the California Institute of Technology, the University of California, and
+NASA, and was made possible by the generous financial support of the W. M. Keck Foundation. CFHT is operated
+by the National Research Council of Canada, the Institut Nat ional des Sciences de l’Univers of the Centre National
+de la Recherche Scientique of France, and the University of H awaii.– 3 –
+5.8 pc from the Sun, the 74′′separation of this proper motion binary corresponds to a pro jected
+separation of 430 AU.
+Such low-mass stars have not been the target of intensive pre cision radial velocity (PRV)
+monitoring because they have low visual fluxes and high stell ar activity. For example, the dedicated
+HARPS M-dwarf planet search observes stars2only brighter than V=14 and of moderate to low
+activity levels (Bonfils et al. 2007). VB 10 has V mag = 17.3 and is known to be a flare star
+(Berger et al. 2008). PRV and lensing planet searches have so far found only 13 stars under 0.5
+M⊙hosting 18 planets, and of these, more than half have masses b elow 0.1 M J.
+Despite the challenges, searches for planetary companions to low mass stars are of continuing
+interest. Low-mass stars appear less likely to have lower ma ss stellar companions and less likely to
+harbor planets than Solar-mass stars (Cumming et al. 2008). When they do have companions, they
+tend to be stars of nearly equal mass to the primary (Burgasse r et al. 2007). The mass function of
+planets orbiting M dwarfs, and how it differs from the planet ma ss function for higher-mass stars,
+provides a constraint on the planet formation mechanism(s) in general. Disks sufficiently massive
+to form Jupiter-mass planets appear to be rare around brown d warfs, whose disks generally look
+like lower mass versions of T Tauri disks (Scholz et al. 2006) . High mass companions would have
+to form via a binary-like fragmentation mechanism (e.g. Fon t-Ribera et al. 2009). Thus how a 6
+MJplanet could form around an ∼80 MJstar and how common such high-mass ratio companions
+remain important questions (Boss et al. 2009).
+The reported planet’s astrometric orbit predicts a radial v elocity (RV) amplitude of at least 1
+km s−1for a circular orbit and up to several km s−1for an eccentric orbit. This magnitude signal is
+detectable with ordinary RV measurements without requirin g the adoption of precision techniques
+such as an iodine cell or simultaneous thorium reference.
+Although several RV measurements of VB 10 exist in the litera ture before 2009, it is difficult
+to combine the historical RVs (see list in Table 4 of Pravdo & S haklan (2009)), as each observation
+used different calibration techniques and/or RV standards th at introduce zero-point offsets and the
+typical uncertainties are also large ( ∼1.5 km s−1). The most precise measurements in the literature
+were recently published by Zapatero Osorio et al. (2009) (he reafter Z09), but provide only a “hint
+of variability.” These data did little to constrain the orbi tal parameters of the planet beyond what
+the astrometry had already done (Anglada-Escud´ e et al. 200 9).
+Here, we present a more precise set of RV observations over 17 5 days (or 65% of the reported
+orbital period). We also present general techniques for joi nt fitting of astrometric and RV data and
+show how they can be used to constrain the orbit of the candida te planet.
+2http://www.eso.org/sci/observing/proposals/77/gto/h arps/3.txt– 4 –
+2. New Data
+We acquired spectra at 7 epochs in 2009 with the MIKE spectrog raph at the Magellan Clay
+telescope at Las CampanasObservatory(Chile). We usedthe0 .35′′andthe0.5′′slits whichproduce
+a spectral resolution of ≈45,000 and 35,000, respectively, across the 4900 – 10000 ˚A range of the
+red chip. The seeing was in the range from 0 .5 to 1.1′′. These data were reduced using the facility
+pipeline (Kelson 2003).
+We also have in hand a single spectrum of VB 10 taken in 2006 usi ng the HIRES (Vogt et al.
+1994)) on the Keck I 10-m telescope. We used the 0.861′′slit to obtain a spectral resolution of
+λ/∆λ≈58,000 at λ∼7000˚A. We used the GG475 order-blocking filter and the red cross-di sperser
+to maximize throughput in the red orders.
+To increase the phase coverage, an additional spectrum was o btained using the ESPaDOnS on
+the CFHT 3.6-m telescope. ESPaDOnS is fiber fed from the Casse grain to Coud´ e focus where the
+fiber image is projected onto a Bowen-Walraven slicer at the s pectrograph entrance. ESPaDOnS’
+‘star+sky’ mode records the full spectrum over 40 grating or ders covering 3700 to 10400 ˚A at
+a spectral resolution of λ/∆λ≈68,000. The data were reduced using Libre Esprit described in
+Donati et al. (1997, 2007).
+Each stellar exposure is bias-subtracted and flat-fielded fo r pixel-to-pixel sensitivity variations.
+After optimal extraction, the 1-D spectra are wavelength ca librated with a thorium-argon arc. To
+correct for instrumental drifts, we used the telluric molec ular oxygen A band (from 7620 – 7660
+˚A) which aligns the MIKE spectra to 40 m s−1, after which we corrected for the heliocentric
+velocity. Consistency tests with the bluer Oxygen band show s comparable values but with larger
+measurement error.
+The final spectra are of moderate S/N reaching ≈25 per pixel at 8000 ˚A. Each night, spectra
+were also taken of a M-dwarf RV standard, namely GJ 699 (Barna rd’s star; SpT = M4V) and/or
+GJ 908 (SpT = M1V).
+To measure VB 10’s RV, we cross-correlated each of 9 orders be tween 7000 and 9000 ˚A (ex-
+cluding those with strong telluric absorption) where VB 10 e mits most of its optical light, with the
+spectrumofGJ699 and/orGJ908taken onthesamenight usingI RAF’s1fxcorroutine(Fitzpatrick
+1993). Both GJ 699 and GJ 908 have been monitored for planets f or years and none has been found
+within the RV stability level of 0.1 km s−1. Here we use the systemic RVs published by a planet-
+search team (Nidever et al. 2002): RV(GJ 699) = –110.506 km s−1and RV(GJ 908) = –71.147
+km s−1. The zero-point of the absolute RVs is uncertain at the 0.4 km s−1level. We measured
+the RVs from the gaussian peak fitted to the cross-correlatio n function (CCF) of each order and
+adopt the average RV of all orders with a mean standard deviat ion of the individual measurements
+of 0.150 km s−1. The average of all our measurements is 36.02 km s−1with a standard deviation
+1IRAF (Image Reduction and Analysis Facility), http://iraf .noao.edu/– 5 –
+of 0.25 km s−1. An observing log with the measured RVs and uncertainties fo r VB 10 is shown in
+Table 1.
+3. Data Analysis: Combining Astrometry and Radial Velocities
+In this section, we reanalyze the original astrometric data to calculate the likelihood of astro-
+metrically allowed solutions, and then combine the astrome try and RV data sets in a consistent
+framework to quantify how the new RV measurements constrain the possibleorbits of the candidate
+signals observed in the astrometry of VB 10b.
+3.1. Least-squares periodograms
+The most popular method to look for periodicities in data is t he so-called Lomb-Scargle
+periodogram. A version adapted to deal with astrometric two -dimensional data developed by
+Catanzarite et al. (2006) (Joint Lomb Scargle periodogram) was implemented in the discovery pa-
+per of VB 10b (Pravdo & Shaklan 2009). Any method based on the L omb-Scargle periodogram
+performs optimally only under an important implicit assump tion: all other signals (e.g. linear
+trend, an average offset, etc.) can be subtracted from the data without affecting the significance of
+the signal under investigation. This assumption does not ho ld for astrometry because the proper
+motion and the parallax are also a significant part of the sign al and they typically correlate with
+the periodic motion of a planet (see Black & Scargle 1982).
+We use instead a Least-squares periodogram. The weighted Le ast-squares solution is obtained
+by fitting all the free parameters in the model for a given peri od. The sum of the weighted residuals
+divided by Nis the so-called χ2statistic. Then, each χ2
+Pof a given model with kPparameters, can
+be compared to the χ2
+0of the null hypothesis with k0free parameters by computing the power, z,
+as
+z(P) =(χ2
+0−χ2
+P)/(kP−k0)
+χ2
+P/(Nobs−kP)(1)
+where a large zis interpreted as a very significant solution. The values of zfollow a Fisher F-
+distribution with kP−k0andNobs−kPdegrees of freedom (Scargle 1982; Cumming 2004). Even
+if only noise is present, a periodogram will contain several peaks (see Scargle 1982, as an example)
+whoseexistencehavetobeconsideredinobtainingtheproba bilityofaspuriousdetection. Assuming
+Gaussian noise, the probability that a peak in the periodogr am has a power higher than z(P) by
+chance is the so-called False Alarm Probability (FAP) :
+FAP = 1 −(1−Prob[z > z(P)])M(2)
+whereMis the number of independent frequencies. In the case of unev en sampling, Mcan be quite
+large and is roughly the number of periodogram peaks one coul d expect from a data set with only– 6 –
+Gaussian noise and thesame cadence as thereal observations . We adopt the recipe M≈2∆T/Pmin
+given in Cumming (2004, Sec 2.2), where ∆ Tis the time-span of the observations and Pminis the
+minimum period searched. One still has to select Pminarbitrarily. Assuming a Pmin= 20 days, the
+astrometric data alone has M∼300, and the combination of astrometry and RVs has M∼360.
+In our particular problem, the null hypothesis is the basic k inematic model with k0= 6 param-
+eters: 2 coordinates, 2 proper motions, parallax and system ic RV. As a first approach, our simplest
+non-null hypothesis considers circular orbits, astrometr ic data only and one RV measurement. For
+a given period, the number of free parameters is then kP= 10: the 6 kinematic ones plus the four
+Thiele Innes elements A,B,FandG(e.g. Wright & Howard 2009). Since the model is linear in all
+10 parameters, the power can be efficiently computed for many p eriods between 20 days and 4000
+days to obtain a familiar representation of the periodogram that we call a Circular Least-squares
+Periodogram (CLP). The CLP of the astrometric data, shown at top in Figure 1, displays two
+obvious peaks: the reported one at 270 days (Pravdo & Shaklan 2009) and a more significant one
+at 49.9 days, both with high power and very low FAPs.
+To find the full Keplerian solution for both periods and estim ate their FAPs, we perform a
+Least-squares periodogram sampling a grid of fixed eccentri city-period (eP) pairs and fitting all
+other parameters. For each eP pair kPis 11: the null-hypothesis ( X0,Y0,µX,µY,πandv0)
+plus all the other Keplerian elements: Mass of the planet, Ω, ω,i, and the Mean anomaly at the
+initial epoch M0(see Wright & Howard 2009, for a recent review). We analyze bo thastrometry
+onlyandastrometry+RVs . Theχ2of the best fit solution is then used to obtain each FAP as
+previously described. Figure 1 shows the resulting color-c oded FAPs for each eccentricity–period
+pair (eP-map).
+3.1.1. Astrometry only
+A value of M= 300 has been used to obtain the FAP, and our result at 270 d qua litatively
+agrees with Pravdo & Shaklan (2009), however the more signifi cant period is at ∼50 d. For both
+periods, there are regions with FAP <1% spanning all possible eccentricities (second row in Fig-
+ure 1). The best fits and their χ2per degree of freedom (¯ χ2) are summarized in Table 2. The
+obtained results for the 270 d period are in agreement with th ose reported in the discovery paper
+by Pravdo & Shaklan (2009). The best fit solution for the 50 d pe riod has mass ∼15 MJ, which
+would be a very low mass brown dwarf. It is important to point o ut that the best fit inclination is
+close to 90 (edge on) for both solutions. The uncertainties o n the orbital parameters are quantified
+in Sec 3.2.– 7 –
+3.1.2. Astrometry+RVs
+We now fit jointly for the best orbital solution to the astrome try and RVs. Our campaign
+covered about 65% of the 270 d orbit. The standard deviation o f all our RVs measurements is
+250 m s−1(null hypothesis) which is larger than the individual uncer tainties in Table 4. When we
+cross-correlate our standards, we measure a similar RMS of 2 00 m s−1, which indicates that the
+difference is due to an uncontrolled or unmeasured systematic . The RMS of the RVs for the best
+fit solution is 200 m s−1, which is not statistically different from the RMS of the null h ypothesis.
+This is another indication that our measurements contain sy stematic errors at the level of 100 −200
+m/s. Despite of that, we use the nominal errors in the Least-s quares solution as the best estimates
+for the individual uncertainties we can provide. In Figure 2 , we show the best solutions to both
+signals including all the data.
+For the 270 d period, our RV non detection cannot exclude a small region of orbital solutions
+arounde∼0.8 with a FAP between 1%–5% – see Figure 1, third row right panel . We now add
+the RVs measurements by Z09 and solve for a joint solution. A z ero-point offset between datasets
+is added as an additional free parameter. The combined RV mea surements force the eccentricity
+to large values which apparently still provides a reasonabl e fit to the astrometry (see top panels in
+Figure 2). However, the FAPs are now all higher than 10% (Figu re 1, bottom right panel), which
+indicates that the signal can be barely distinguished from t he noise fluctuations. The “hint” of
+detection in Z09 based on one discrepant value at 3 .1−σout of five can be due to random errors
+with a non negligible probability.
+For the 50 d period, there are still several orbits that provi de a decent fit to the combined
+astrometry and the new RV data with a FAP lower than 1%. These o ccupy a small space around
+the best joint solution, with e= 0.90 (see Figure 1, 3rd row, left panel) and an inclination clos e
+to 0. Large eccentricity causes the duration of fast RV varia tion to be very short (and difficult to
+catch); an inclination close to 0 tends to suppress any RVs si gnal. Such inclination is in apparent
+contradiction with the one obtained using the astrometry al one (∼90 deg). The reason is the
+following: while the new fit to the astrometry forced by the RV s is much worse than the one
+obtained from the astrometry alone, such a solution still re presents an improvement compared to
+the null hypothesis. Adding Z09 data to the fit increases the F AP of the most likely solution to 2%,
+an eccentricity of 0 .91 and the inclination close to 0 (see Table 2). This suggests that the signal at
+50 d is also spurious, even though it has slightly better chan ces of survival than the one at 270 d.
+3.2.A Posteriori Probability Distributions
+We adapt the method developed by Ford (2005, 2006) to assess u ncertainties in orbit determi-
+nations by obtaining the a posteriori probability distribution for the parameters using a Markov
+Chain with a Gibbs sampler strategy. Our problem is identica l to the one described by Ford (2005),
+where now the χ2contains both RV and astrometric observations and the model has a few more– 8 –
+free parameters. Several properly adjusted MCMC with 106steps have been computed obtaining
+compatible results. The step sizes of the Gibbs sampler are i nitialized with the formal errors from
+the best fit Least-squares solution, and adjusted to obtain a transition probability between 10%
+and 20%. The first 105steps of each chain are rejected. The final distributions mat ch very well the
+areas of low FAPs in the eP-maps (see Figure 3 as an example) gi ving further proof that the chains
+have converged to the equilibrium distributions. The MCMC c ontains 13 free parameters – the 11
+from the Least-squares periodogram plus eccentricity and p eriod. When the RVs measurements
+from Z09 are included, and additional offset parameter is incl uded.
+Table 2 presents the standard deviations obtained via the MC MC for both the 50 d and 270 d
+periods using astrometry alone and astrometry + all RV data. As an example, we show the two
+dimensional density of states in period-eccentricity spac e in Figure 3 (left) obtained in both cases
+aroundthe 270 d signal. Themarginalized distributionsfor ein theform of histograms are shownin
+Figure 3 (right). For the astrometry-alone case, the distri bution of eis almost uniform. It becomes
+strongly peaked towards high eccentricities when all the RV data are included. Since the best fit
+solution at 270 d is poor (¯ χ2= 1.76), the corresponding χ2minimum is not very deep which is
+reflected in a significant increase in the derived uncertaint ies (See Table 2). The same happens to
+the signal at 50 d with the exception of the inclination that h as a small uncertainty (4 deg) close
+to 0. Even though this solution has a low FAP, the inclination has to be coincidentally very small
+to suppress any RV signal and very different from using astrome try only (94 deg), rasing serious
+doubts of its reality.
+4. Discussion and Conclusions
+The non-detection of a significant RV variation in our data se t already discards most orbital
+configurations allowed by the astrometry. When combined wit h Z09 RVs measurements, there are
+no remaining solutions with a FAP lower than 10% around the 27 0 d period, so the presence of
+a planet candidate at that period is not supported by the obse rvations. For the 50 d period, the
+constraints arealso strongandbecome almost definitive whe ntheZ09data is included. Even highly
+eccentric solutions have a relatively large FAP ( >2%). We find that particular combinations of
+eccentricity, inclination and ωcan fit an almost flat RV curve indicating that the analytic met hods
+applied to estimate FAPs for high eccentricities tend to giv e over optimistic results and that this
+issue should be studied in more detail.
+We have developed and implemented useful tools for detailed analysis of combined astrometric
+and RV data: Circular Least-squares periodogram as the prop er generalization of the classic Lomb-
+Scargle periodogram to deal with astrometric data, eP-maps to visualize the most likely period–
+eccentricity combinations and a Bayesian characterizatio n of the parameter uncertainties based on
+a MCMC approach.
+VB 10 is also part of the Carnegie Astrometric Planet Search p rogram (Boss et al. 2009). RV– 9 –
+measurements with precision techniques in the near-infrar ed (Bean et al. 2009) may provide the
+required accuracy to put even stronger limits to the existen ce of VB10b or find other planets in the
+system. VB 10 will certainly be observed by the space astrome try mission Gaia (Perryman et al.
+2001), which would be capable of finding a planet with a period of 270 d and as small as 0 .2 MJ.
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+This preprint was prepared with the AAS L ATEX macros v5.2.– 11 –
+Table 1. Log of RVs
+Telescope UT Date HJD Slit Width RV (w/ GJ 699)aRV (w/ GJ 908)a
++Instrument –2450000′′km s−1km s−1
+Keck I + HIRES 2006 Aug 12 3959.57 0.86 – 35.59 ±0.15
+Clay+MIKE 2009 Jun 06 4988.74 0.35 36.23 ±0.13 35.99 ±0.15
+Clay+MIKE 2009 Jun 07 4989.82 0.50 36.22 ±0.13 36.09 ±0.20
+Clay+MIKE 2009 Jun 08 4990.75 0.50 36.15 ±0.12 36.10 ±0.22
+Clay+MIKE 2009 Jun 30 5012.72 0.35 35.72 ±0.11 –
+Clay+MIKE 2009 Jul 25 5037.66 0.50 35.96 ±0.11 36.03 ±0.11
+Clay+MIKE 2009 Sep 04 5078.58 0.50 35.96 ±0.09 36.37 ±0.24
+Clay+MIKE 2009 Oct 15 5119.54 0.50 36.30 ±0.14 36.41 ±0.13
+Clay+MIKE 2009 Oct 26 5130.51 0.50 36.41 ±0.16 36.27 ±0.18
+CFHT+ESPaDOnS 2009 Nov 29 5164.69 –b– 35.74 ±0.20
+aUncertainties are the standard deviation of the 9 orders of t he cross correlation and do not include the 40
+m s−1systematic uncertainty from the telluric wavelength corre ction. Absolute radial velocity determination
+has an uncertainty of 0 .4 km/s but it is not relevant for orbital fitting purposes.
+bESPaDOnS is a fiber fed spectrograph with an effective resolut ion of R ∼68000 in the wavelength range
+of interest
+Table 2. Best fitting valuesa. Uncertainties obtained from a MCMC with 106steps.
+Parameter Astrometry 50 d Astrometry 270 d Astro+ all RV 50 d A stro+ all RV 270 d
+X0(mas) -16.6 ±1.6 -14.1d±3.2 -21.15±2.3 -17.9 ±4.7
+Y0(mas) -408.0 ±1.9 -406.1d±3.5 409.52±2.8 -410.5 ±5.51
+µR.A.(mas/yr) -588.98 ±0.25 -589.08 ±0.25 -588.66±0.29 -589.21 ±0.26
+µDec(mas/yr) -1360.95 ±0.25 -1361.08 ±0.24 -1361.02 ±0.25 -1361.36 ±0.20
+π(mas) 168.3 ±1.51 169.5 ±1.4 169.95±1.37 169.24 ±1.30
+v0(km/s) 35.2 ±1.4 35.4d±1.050 36.06±0.11 36.05 ±0.08
+voffset(km/s) - - 1.5±0.42 1.5 ±0.36
+P(days) 49.7 ±0.5 272.1 ±4.1 49.84±0.11 278.5 ±2.7
+Mass(MJ) 17.5 ±4.4 7.1 ±2.7 13.7±6.4 5.0 ±2.9
+e 0.22c±0.30 0.48c±0.31 0.91±0.13 0.90 ±0.16
+i(deg) 93 ±5 90 ±15 4±5 110c±50
+Ω(deg) 40 ±20 220 ±25 13c±100 40c±66
+ω(deg) 20 ±40 30c±80 122c±60 17c±90
+M0(deg) 270 ±0 170c±108 340c±70 156c±80
+a(AU)d0.12 0.36 0.12 0.36
+¯χ2
+02.28 2.28 2.75 2.75
+¯χ20.87 0.93 1.62 1.76
+aThe mass of VB 10 is assumed to be 0 .078 M ⊙according to Pravdo & Shaklan (2009)
+bLarge uncertainty due to correlation with the eccentricity
+cUnconstrained or poorly constrained
+dDerived quantity using Kepler equations– 12 –
+Fig. 1.— Top panel. Circular Least-squares periodogramshowingthe two most si gnificant periods
+with their corresponding False Alarm Probabilities (FAP). Second row. FAPs obtained for a grid
+of Eccentricity–Period pairs around the 50 d (left) and the 2 70 d (right) when only astrometry is
+considered. Third row. FAPs obtained when our new RV are included to the fit. Bottom row.
+Final FAPs obtained when all published RV data are combined i n a joint fit.– 13 –
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810R.A.(mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810 Dec (mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810R.A.(mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810 Dec (mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase32333435363738RV (km/s)
+0 0.2 0.4 0.6 0.8 132333435363738
+0 0.2 0.4 0.6 0.8 1
+Phase32333435363738
+MIKE
+HIRES
+CFHT
+NIRSPEC Z09P = 49.84 days P = 278.5 days
+Fig. 2.— The best fit (lowest χ2) joint solutions to the Pravdo & Shaklan (2009) astrometry a nd
+used RVs for the two signals. Top panels contain the astromet ric offsets after the removal of the
+corresponding parallax and the proper motion. The lower pan els contain all RVs used. Each RV
+point represents the weighted average of the values obtaine d using both reference stars if available.
+The best fit offset has been applied to Z09 data (Green triangles ). Phase 0 corresponds to the first
+astrometric epoch at JD 2451438 .64 and the corresponding folding periods are given on the top .
+Fig. 3.— Left: Steps in period-eccentricity space of a Marko v Chain of 106elements applied to the
+astrometry only (black) and to the astrometry+all RV data (b rown). The distribution resembles
+the FAP contours on the eP-maps around 270 days indicating th at the chain has successfully
+converged to the equilibrium distribution. Right: Histogr am reproducing the marginalized density
+distributions in efor the astrometry only and astrometry+RV around the 270 d so lution.

1001.0044.txt

@@ -0,0 +1,1486 @@
+arXiv:1001.0044v3  [math.PR]  31 Mar 2011A law of large numbers approximation for
+Markov population processes with countably
+many types
+A. D. Barbour∗and M. J. Luczak†
+Universit¨ at Z¨ urich and London School of Economics
+Abstract
+When modelling metapopulation dynamics, the influence of a s in-
+gle patch on the metapopulation depends on the number of indi vidu-
+als in the patch. Since the population size has no natural upp er limit,
+this leads to systems in which there are countably infinitely many
+possible types of individual. Analogous considerations ap ply in the
+transmission of parasitic diseases. In this paper, we prove a law of
+large numbers for quite general systems of this kind, togeth er with
+a rather sharp bound on the rate of convergence in an appropri ately
+chosen weighted ℓ1norm.
+Keywords: Epidemic models, metapopulation processes, countably many
+types, quantitative law of large numbers, Markov population proce sses
+AMS subject classification: 92D30, 60J27, 60B12
+Running head: A law of large numbers approximation
+1 Introduction
+There are many biological systems that consist of entities that diffe r in their
+influence according to the number of active elements associated wit h them,
+∗Angewandte Mathematik, Universit¨ at Z¨ urich, Winterthurertra sse 190, CH-8057
+Z¨URICH; ADB was supported in part by Schweizerischer Nationalfond s Projekt Nr. 20–
+107935/1.
+†London School of Economics; MJL was supported in part by a STICE RD grant.
+1and can be divided into types accordingly. In parasitic diseases (Bar bour &
+Kafetzaki 1993, Luchsinger 2001a,b, Kretzschmar 1993), the in fectivity of a
+host depends on the number of parasites that it carries; in metapo pulations,
+the migration pressure exerted by a patch is related to the number of its
+inhabitants (Arrigoni 2003); the behaviour of a cell may depend on the num-
+ber of copies of a particular gene that it contains (Kimmel & Axelrod 2 002,
+Chapter 7); and so on. In none of these examples is there a natura l upper
+limit to the number of associated elements, so that the natural set ting for
+a mathematical model is one in which there are countably infinitely man y
+possible types of individual. In addition, transition rates typically incr ease
+with the number of associated elements in the system — for instance , each
+parasite has an individual death rate, so that the overall death ra te of par-
+asites grows at least as fast as the number of parasites — and this le ads
+to processes with unbounded transition rates. This paper is conce rned with
+approximations to density dependent Markov models of this kind, wh en the
+typical population size Nbecomes large.
+In density dependent Markov population processes with only finitely
+many types of individual, a law of large numbers approximation, in the f orm
+ofasystemofordinarydifferentialequations, wasestablishedbyK urtz(1970),
+together with a diffusion approximation (Kurtz, 1971). In the infinit e di-
+mensional case, the law of large numbers was proved for some spec ific mod-
+els (Barbour & Kafetzaki 1993, Luchsinger 2001b, Arrigoni 2003 , see also
+L´ eonard 1990), using individually tailored methods. A more general result
+was then given by Eibeck & Wagner (2003). In Barbour & Luczak (20 08),
+the law of large numbers was strengthened by the addition of an err or bound
+inℓ1that is close to optimal order in N. Their argument makes use of an
+intermediate approximation involving an independent particles proce ss, for
+which the law of large numbers is relatively easy to analyse. This proce ss is
+then shown to be sufficiently close to the interacting process of act ual inter-
+est, by means of a coupling argument. However, the generality of t he results
+obtained is limited by the simple structure of the intermediate proces s, and
+the model of Arrigoni (2003), for instance, lies outside their scop e.
+In this paper, we develop an entirely different approach, which circu m-
+vents the need for an intermediate approximation, enabling a much w ider
+class of models to be addressed. The setting is that of families of Mar kov
+population processes XN:= (XN(t), t≥0),N≥1, taking values in the
+countable space X+:={X∈ZZ+
++;/summationtext
+m≥0Xm<∞}. Each component repre-
+2sents the number of individuals of a particular type, and there are c ountably
+many types possible; however, at any given time, there are only finit ely
+many individuals in the system. The process evolves as a Markov proc ess
+with state-dependent transitions
+X→X+Jat rate NαJ(N−1X), X∈ X+, J∈ J,(1.1)
+where each jump is of bounded influence, in the sense that
+J ⊂ {X∈ZZ+;/summationdisplay
+m≥0|Xm| ≤J∗<∞},for some fixed J∗<∞,(1.2)
+so that the number of individuals affected is uniformly bounded. Dens ity
+dependence is reflected in the fact that the arguments of the fun ctionsαJ
+are counts normalised by the ‘typical size’ N. Writing R:=RZ+
++, the func-
+tionsαJ:R →R+are assumed to satisfy
+/summationdisplay
+J∈JαJ(ξ)<∞, ξ∈ R0, (1.3)
+whereR0:={ξ∈ R:ξi= 0 for all but finitely many i}; this assumption
+implies that the processes XNare pure jump processes, at least for some
+non-zero length of time. To prevent the paths leaving X+, we also assume
+thatJl≥ −1 for each l, and that αJ(ξ) = 0 ifξl= 0 for any J∈ Jsuch
+thatJl=−1. Some remarks on the consequences of allowing transitions J
+withJl≤ −2 for some lare made at the end of Section 4.
+Thelawoflargenumbersisthenformallyexpressed intermsofthesy stem
+ofdeterministic equations
+dξ
+dt=/summationdisplay
+J∈JJαJ(ξ) =:F0(ξ), (1.4)
+to be understood componentwise for those ξ∈ Rsuch that
+/summationdisplay
+J∈J|Jl|αJ(ξ)<∞,for alll≥0,
+thus by assumption including R0. Here, the quantity F0represents the in-
+finitesimal average drift of the components of the random proces s. However,
+in this generality, it is not even immediately clear that equations (1.4) h ave
+a solution.
+3In order to make progress, it is assumed that the unbounded comp onents
+in the transition rates can be assimilated into a linear part, in the sens e
+thatF0can be written in the form
+F0(ξ) =Aξ+F(ξ), (1.5)
+again to be understood componentwise, where Ais a constant Z+×Z+
+matrix. These equations are then treated as a perturbed linear sy stem
+(Pazy 1983, Chapter 6). Under suitable assumptions on A, there exists a
+measure µonZ+, defining a weighted ℓ1norm/⌊ard⌊l · /⌊ard⌊lµonR, and a strongly
+/⌊ard⌊l·/⌊ard⌊lµ–continuoussemigroup {R(t), t≥0}oftransitionmatriceshaving point-
+wise derivative R′(0) =A. IfFis locally /⌊ard⌊l·/⌊ard⌊lµ–Lipschitz and /⌊ard⌊lx(0)/⌊ard⌊lµ<∞,
+this suggests using the solution xof the integral equation
+x(t) =R(t)x(0)+/integraldisplayt
+0R(t−s)F(x(s))ds (1.6)
+as an approximation to xN:=N−1XN, instead of solving the deterministic
+equations (1.4) directly. We go on to show that the solution XNof the
+stochastic system can be expressed using a formula similar to (1.6), which
+has an additional stochastic component in the perturbation:
+xN(t) =R(t)xN(0)+/integraldisplayt
+0R(t−s)F(xN(s))ds+/tildewidemN(t),(1.7)
+where
+/tildewidemN(t) :=/integraldisplayt
+0R(t−s)dmN(s), (1.8)
+andmNis the local martingale given by
+mN(t) :=xN(t)−xN(0)−/integraldisplayt
+0F0(xN(s))ds. (1.9)
+The quantity mNcanbe expected to be small, at least componentwise, under
+reasonable conditions.
+To obtain tight control over /tildewidemNin all components simultaneously, suf-
+ficient to ensure that sup0≤s≤t/⌊ard⌊l/tildewidemN(s)/⌊ard⌊lµis small, we derive Chernoff–like
+boundsonthedeviations ofthemost significant components, witht hehelpof
+a family of exponential martingales. The remaining components are t reated
+usingsomegeneral a prioriboundsonthebehaviourofthestochasticsystem.
+4This allows us to take the difference between the stochastic and det erministic
+equations (1.7) and (1.6), after which a Gronwall argument can be c arried
+through, leading to the desired approximation.
+The main result, Theorem 4.7, guarantees an approximation error o f or-
+derO(N−1/2√logN) in the weighted ℓ1metric/⌊ard⌊l·/⌊ard⌊lµ, except on an event of
+probability of order O(N−1logN). More precisely, for each T >0, there
+exist constants K(1)
+T,K(2)
+T,K(3)
+Tsuch that, for Nlarge enough, if
+/⌊ard⌊lN−1XN(0)−x(0)/⌊ard⌊lµ≤K(1)
+T/radicalbigg
+logN
+N,
+then
+P/parenleftBig
+sup
+0≤t≤T/⌊ard⌊lN−1XN(t)−x(t)/⌊ard⌊lµ> K(2)
+T/radicalbigg
+logN
+N/parenrightBig
+≤K(3)
+TlogN
+N.(1.10)
+Theerrorboundissharper, byafactoroflog N, thanthatgiveninBarbour&
+Luczak(2008),andthetheoremisapplicabletoamuch widerclassof models.
+However, the method of proof involves moment arguments, which r equire
+somewhat stronger assumptions on the initial state of the system , and, in
+models such as that of Barbour & Kafetzaki (1993), onthe choice of infection
+distributions allowed. The conditions under which the theorem holds c an be
+divided into three categories: growth conditions on the transition r ates, so
+that the a prioribounds, which have the character of moment bounds, can
+be established; conditions on the matrix A, sufficient to limit the growth of
+the semigroup R, and (together with the properties of F) to determine the
+weights defining the metric in which the approximation is to be carried o ut;
+and conditions on the initial state of the system. The a priori bounds are
+derived in Section 2, the semigroup analysis is conducted in Section 3, and
+the approximation proper is carried out in Section 4. The paper conc ludes
+in Section 5 with some examples.
+The form (1.8) of the stochastic component /tildewidemN(t) in (1.7) is very simi-
+lar to that of a key element in the analysis of stochastic partial differ ential
+equations; see, for example, Chow (2007, Section 6.6). The SPDE a rguments
+used for its control are however typically conducted in a Hilbert spa ce con-
+text. Our setting is quite different in nature, and it does not seem cle ar how
+to translate the SPDE methods into our context.
+52 A priori bounds
+We begin by imposing further conditions on the transition rates of th e pro-
+cessXN, sufficient to constrain its paths to bounded subsets of X+dur-
+ing finite time intervals, and in particular to ensure that only finitely ma ny
+jumps can occur in finite time. The conditions that follow have the flav our
+of moment conditions on the jump distributions. Since the index j∈Z+is
+symbolic in nature, we start by fixing an ν∈ R, such that ν(j) reflects in
+some sense the ‘size’ of j, with most indices being ‘large’:
+ν(j)≥1 for allj≥0 and lim
+j→∞ν(j) =∞. (2.1)
+We then define the analogues of higher empirical moments using the q uanti-
+tiesνr∈ R, defined by νr(j) :=ν(j)r,r≥0, setting
+Sr(x) :=/summationdisplay
+j≥0νr(j)xj=xTνr, x∈ R0, (2.2)
+where, for x∈ R0andy∈ R,xTy:=/summationtext
+l≥0xlyl. In particular, for X∈ X+,
+S0(X) =/⌊ard⌊lX/⌊ard⌊l1. Note that, because of (2.1), for any r≥1,
+#{X∈ X+:Sr(X)≤K}<∞for allK >0. (2.3)
+To formulate the conditions that limit the growth of the empirical mom ents
+ofXN(t) witht, we also define
+Ur(x) :=/summationdisplay
+J∈JαJ(x)JTνr;Vr(x) :=/summationdisplay
+J∈JαJ(x)(JTνr)2, x∈ R.(2.4)
+The assumptions that we shall need are then as follows.
+Assumption 2.1 There exists a νsatisfying (2.1)andr(1)
+max,r(2)
+max≥1such
+that, for all X∈ X+,
+/summationdisplay
+J∈JαJ(N−1X)|JTνr|<∞,0≤r≤r(1)
+max,(2.5)
+the case r= 0following from (1.2)and(1.3); furthermore, for some non-
+negative constants krl, the inequalities
+U0(x)≤k01S0(x)+k04,
+U1(x)≤k11S1(x)+k14, (2.6)
+Ur(x)≤ {kr1+kr2S0(x)}Sr(x)+kr4,2≤r≤r(1)
+max;
+6and
+V0(x)≤k03S1(x)+k05,
+Vr(x)≤kr3Sp(r)(x)+kr5,1≤r≤r(2)
+max, (2.7)
+are satisfied, where 1≤p(r)≤r(1)
+maxfor1≤r≤r(2)
+max.
+The quantities r(1)
+maxandr(2)
+maxusually need to be reasonably large, if Assump-
+tion 4.2 below is to be satisfied.
+Now, for XNas in the introduction, we let tXNndenote the time of its
+n-th jump, with tXN
+0= 0, and set tXN∞:= lim n→∞tXNn, possibly infinite. For
+0≤t < tXN∞, we define
+S(N)
+r(t) :=Sr(XN(t));U(N)
+r(t) :=Ur(xN(t));V(N)
+r(t) :=Vr(xN(t)),
+(2.8)
+once again with xN(t) :=N−1XN(t), and also
+τ(N)
+r(C) := inf {t < tXN
+∞:S(N)
+r(t)≥NC}, r≥0,(2.9)
+where the infimum of the empty set is taken to be ∞. Our first result shows
+thattXN∞=∞a.s., and limits the expectations of S(N)
+0(t) andS(N)
+1(t) for any
+fixedt.
+In what follows, we shall write F(N)
+s=σ(XN(u),0≤u≤s), so that
+(F(N)
+s:s≥0) is the natural filtration of the process XN.
+Lemma 2.2 Under Assumptions 2.1, tXN∞=∞a.s. Furthermore, for any
+t≥0,
+E{S(N)
+0(t)} ≤(S(N)
+0(0)+Nk04t)ek01t;
+E{S(N)
+1(t)} ≤(S(N)
+1(0)+Nk14t)ek11t.
+Proof. Introducing the formal generator ANassociated with (1.1),
+ANf(X) :=N/summationdisplay
+J∈JαJ(N−1X){f(X+J)−f(X)}, X∈ X+,(2.10)
+we note that NUl(x) =ANSl(Nx). Hence, if we define M(N)
+lby
+M(N)
+l(t) :=S(N)
+l(t)−S(N)
+l(0)−N/integraldisplayt
+0U(N)
+l(u)du, t ≥0,(2.11)
+7for 0≤l≤r(1)
+max, it is immediate from (2.3), (2.5) and (2.6) that the process
+(M(N)
+l(t∧τ(N)
+1(C)), t≥0) is a zero mean F(N)–martingale for each C >0.
+In particular, considering M(N)
+1(t∧τ(N)
+1(C)), it follows in view of (2.6) that
+E{S(N)
+1(t∧τ(N)
+1(C))} ≤S(N)
+1(0)+E/braceleftBigg/integraldisplayt∧τ(N)
+1(C)
+0{k11S(N)
+1(u)+Nk14}du/bracerightBigg
+≤S(N)
+1(0)+/integraldisplayt
+0(k11E{S(N)
+1(u∧τ(N)
+1(C))}+Nk14)du.
+Using Gronwall’s inequality, we deduce that
+E{S(N)
+1(t∧τ(N)
+1(C))} ≤(S(N)
+1(0)+Nk14t)ek11t,(2.12)
+uniformly in C >0, and hence that
+P/bracketleftBig
+sup
+0≤s≤tS1(XN(s))≥NC/bracketrightBig
+≤C−1(S1(xN(0))+k14t)ek11t(2.13)
+also. Hence sup0≤s≤tS1(XN(s))<∞a.s. for any t, limC→∞τ(N)
+1(C) =∞
+a.s., and, from (2.3) and (1.3), it thus follows that tXN∞=∞a.s. The bound
+onE{S(N)
+1(t)}is now immediate, and that on E{S(N)
+0(t)}follows by applying
+the same Gronwall argument to M(N)
+0(t∧τ(N)
+1(C)).
+The next lemma shows that, if any T >0 is fixed and Cis chosen large
+enough, then, with high probability, N−1S(N)
+0(t)≤Cholds for all 0 ≤t≤T.
+Lemma 2.3 Assume that Assumptions 2.1 are satisfied, and that S(N)
+0(0)≤
+NC0andS(N)
+1(0)≤NC1. Then, for any C≥2(C0+k04T)ek01T, we have
+P[{τ(N)
+0(C)≤T}]≤(C1∨1)K00/(NC2),
+whereK00depends on Tand the parameters of the model.
+Proof. It is immediate from (2.11) and (2.6) that
+S(N)
+0(t) =S(N)
+0(0)+N/integraldisplayt
+0U(N)
+0(u)du+M(N)
+0(t)
+≤S(N)
+0(0)+/integraldisplayt
+0(k01S(N)
+0(u)+Nk04)du+ sup
+0≤u≤tM(N)
+0(u).(2.14)
+8Hence, from Gronwall’s inequality, if S(N)
+0(0)≤NC0, then
+S(N)
+0(t)≤/braceleftbigg
+N(C0+k04T)+ sup
+0≤u≤tM(N)
+0(u)/bracerightbigg
+ek01t.(2.15)
+Now, considering the quadratic variation of M(N)
+0, we have
+E/braceleftBigg
+{M(N)
+0(t∧τ(N)
+1(C′))}2−N/integraldisplayt∧τ(N)
+1(C′)
+0V(N)
+0(u)du/bracerightBigg
+= 0 (2.16)
+for anyC′>0, from which it follows, much as above, that
+E/parenleftBig
+{M(N)
+0(t∧τ(N)
+1(C′))}2/parenrightBig
+≤E/braceleftbigg
+N/integraldisplayt
+0V(N)
+0(u∧τ(N)
+1(C′))du/bracerightbigg
+≤/integraldisplayt
+0{k03ES(N)
+1(u∧τ(N)
+1(C′))+Nk05}du.
+Using (2.12), we thus find that
+E/parenleftBig
+{M(N)
+0(t∧τ(N)
+1(C′))}2/parenrightBig
+≤k03
+k11N(C1+k14T)(ek11t−1)+Nk05t,(2.17)
+uniformlyforall C′. Doob’smaximal inequality appliedto M(N)
+0(t∧τ(N)
+1(C′))
+now allows us to deduce that, for any C′,a >0,
+P/bracketleftBig
+sup
+0≤u≤TM(N)
+0(u∧τ(N)
+1(C′))> aN/bracketrightBig
+≤1
+Na2/braceleftbiggk03
+k11(C1+k14T){ek11T−1}+k05T/bracerightbigg
+=:C1K01+K02
+Na2,
+say, so that, letting C′→ ∞,
+P/bracketleftBig
+sup
+0≤u≤TM(N)
+0(u)> aN/bracketrightBig
+≤C1K01+K02
+Na2
+also. Taking a=1
+2Ce−k01Tand putting the result into (2.15), the lemma
+follows.
+In the next theorem, we control the ‘higher ν-moments’ S(N)
+r(t) ofXN(t).
+9Theorem 2.4 Assume thatAssumptions 2.1are satisfied, andthat S(N)
+1(0)≤
+NC1andS(N)
+p(1)(0)≤NC′
+1. Then, for 2≤r≤r(1)
+maxand for any C >0, we
+have
+E{S(N)
+r(t∧τ(N)
+0(C))} ≤(S(N)
+r(0)+Nkr4t)e(kr1+Ckr2)t,0≤t≤T.(2.18)
+Furthermore, if for 1≤r≤r(2)
+max,S(N)
+r(0)≤NCrandS(N)
+p(r)(0)≤NC′
+r,
+then, for any γ≥1,
+P[ sup
+0≤t≤TS(N)
+r(t∧τ(N)
+0(C))≥NγC′′
+rT]≤Kr0γ−2N−1, (2.19)
+where
+C′′
+rT:= (Cr+kr4T+/radicalbig
+(C′
+r∨1))e(kr1+Ckr2)T
+andKr0depends on C,Tand the parameters of the model.
+Proof. Recalling (2.11), use the argument leading to (2.12) with the martin-
+galesM(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C)), for any C′>0, to deduce that
+ES(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))
+≤S(N)
+r(0)+/integraldisplayt
+0/parenleftBig
+{kr1+Ckr2}E/braceleftBig
+S(N)
+r(u∧τ(N)
+1(C′)∧τ(N)
+0(C))/bracerightBig
++Nkr4/parenrightBig
+du,
+for 1≤r≤r(1)
+max, sinceN−1S(N)
+0(u)≤Cwhenu≤τ(N)
+0(C): define k12= 0.
+Gronwall’s inequality now implies that
+ES(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))≤(S(N)
+r(0)+Nkr4t)e(kr1+Ckr2)t,(2.20)
+for 1≤r≤r(1)
+max, and (2.18) follows by Fatou’s lemma, on letting C′→ ∞.
+Now, also from (2.11) and (2.6), we have, for t≥0 and each r≤r(1)
+max,
+S(N)
+r(t∧τ(N)
+0(C))
+=S(N)
+r(0)+N/integraldisplayt∧τ(N)
+0(C)
+0U(N)
+r(u)du+M(N)
+r(t∧τ(N)
+0(C))
+≤S(N)
+r(0)+/integraldisplayt
+0/parenleftBig
+{kr1+Ckr2}S(N)
+r(u∧τ(N)
+0(C))+Nkr4/parenrightBig
+du
++ sup
+0≤u≤tM(N)
+r(u∧τ(N)
+0(C)).
+10Hence, from Gronwall’s inequality, for all t≥0 andr≤r(1)
+max,
+S(N)
+r(t∧τ(N)
+0(C))≤/braceleftBig
+N(Cr+kr4t)+ sup
+0≤u≤tM(N)
+r(u∧τ(N)
+0(C))/bracerightBig
+e(kr1+Ckr2)t.
+(2.21)
+Now, as in (2.16), we have
+E/braceleftBigg
+{M(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))}2−N/integraldisplayt∧τ(N)
+1(C′)∧τ(N)
+0(C)
+0V(N)
+r(u)du/bracerightBigg
+= 0,
+(2.22)
+from which it follows, using (2.7), that, for 1 ≤r≤r(2)
+max,
+E/parenleftBig
+{M(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))}2/parenrightBig
+≤E/braceleftBigg
+N/integraldisplayt∧τ(N)
+1(C′)∧τ(N)
+0(C))
+0V(N)
+r(u)du/bracerightBigg
+≤/integraldisplayt
+0{kr3ES(N)
+p(r)(u∧τ(N)
+1(C′)∧τ(N)
+0(C))+Nkr5}du
+≤N(C′
+r+kp(r),4T)kr3
+kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2t)−1)+Nkr5T,
+this last by (2.20), since p(r)≤r(1)
+maxfor 1≤r≤r(2)
+max. Using Doob’s
+inequality, it follows that, for any a >0,
+P/bracketleftBig
+sup
+0≤u≤TM(N)
+r(u∧τ(N)
+0(C))> aN/bracketrightBig
+≤1
+Na2/braceleftbiggkr3(C′
+r+kp(r),4T)
+kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2T)−1)+kr5T/bracerightbigg
+=:C′
+rKr1+Kr2
+Na2.
+Takinga=γ/radicalbig
+(C′
+r∨1) and putting the result into (2.21) gives (2.19), with
+Kr0= (C′
+rKr1+Kr2)/(C′
+r∨1).
+Note also that sup0≤t≤TS(N)
+r(t)<∞a.s. for all 0 ≤r≤r(2)
+max, in view of
+Lemma 2.3 and Theorem 2.4.
+In what follows, we shall particularly need to control quantities of t he
+form/summationtext
+J∈JαJ(xN(s))d(J,ζ), where xN:=N−1XNand
+d(J,ζ) :=/summationdisplay
+j≥0|Jj|ζ(j), (2.23)
+11forζ∈ Rchosen such that ζ(j)≥1 grows fast enough with j: see (4.12).
+Defining
+τ(N)(a,ζ) := inf/braceleftBigg
+s:/summationdisplay
+J∈JαJ(xN(s))d(J,ζ)≥a/bracerightBigg
+,(2.24)
+infinite if there is no such s, we show in the following corollary that, under
+suitable assumptions, τ(N)(a,ζ) is rarely less than T.
+Corollary 2.5 Assume that Assumptions 2.1 hold, and that ζis such that
+/summationdisplay
+J∈JαJ(N−1X)d(J,ζ)≤ {k1N−1Sr(X)+k2}b(2.25)
+for some 1≤r:=r(ζ)≤r(2)
+maxand some b=b(ζ)≥1. For this value
+ofr, assume that S(N)
+r(0)≤NCrandS(N)
+p(r)(0)≤NC′
+rfor some constants
+CrandC′
+r. Assume further that S(N)
+0(0)≤NC0,S(N)
+1(0)≤NC1for some
+constants C0,C1, and define C:= 2(C0+k04T)ek01T. Then
+P[τ(N)(a,ζ)≤T]≤N−1{Kr0γ−2
+a+K00(C1∨1)C−2},
+for anya≥ {k2+k1C′′
+rT}b, whereγa:= (a1/b−k2)/{k1C′′
+rT},Kr0andC′′
+rT
+are as in Theorem 2.4, and K00is as in Lemma 2.3.
+Proof. In view of (2.25), it is enough to bound the probability
+P[ sup
+0≤t≤TS(N)
+r(t)≥N(a1/b−k2)/k1].
+However, Lemma 2.3 and Theorem 2.4 together bound this probability by
+N−1/braceleftbig
+Kr0γ−2
+a+K00(C1∨1)C−2/bracerightbig
+,
+whereγais as defined above, as long as a1/b−k2≥k1C′′
+rT.
+If (2.25) is satisfied,/summationtext
+J∈JαJ(xN(s))d(J,ζ)isa.s. bounded on0 ≤s≤T,
+becauseS(N)
+r(s) is. The corollary shows that the sum is then bounded by
+{k2+k1C′′
+r,T}b, except on an event of probability of order O(N−1). Usually,
+one can choose b= 1.
+123 Semigroup properties
+We make the following initial assumptions about the matrix A: first, that
+Aij≥0 for alli/ne}ationslash=j≥0;/summationdisplay
+j/negationslash=iAji<∞for alli≥0,(3.1)
+and then that, for some µ∈RZ+
++such that µ(m)≥1 for each m≥0, and
+for some w≥0,
+ATµ≤wµ. (3.2)
+We then use µto define the µ-norm
+/⌊ard⌊lξ/⌊ard⌊lµ:=/summationdisplay
+m≥0µ(m)|ξm|onRµ:={ξ∈ R:/⌊ard⌊lξ/⌊ard⌊lµ<∞}.(3.3)
+Note that there may be many possible choices for µ. In what follows, it is
+important that Fbe a Lipschitz operator with respect to the µ-norm, and
+this has to be borne in mind when choosing µ.
+Setting
+Qij:=AT
+ijµ(j)/µ(i)−wδij, (3.4)
+whereδis the Kronecker delta, we note that Qij≥0 fori/ne}ationslash=j, and that
+0≤/summationdisplay
+j/negationslash=iQij=/summationdisplay
+j/negationslash=iAT
+ijµ(j)/µ(i)≤w−Aii=−Qii,
+using (3.2) for the inequality, so that Qii≤0. Hence Qcan be augmented to
+a conservative Q–matrix, in the sense of Markov jump processes, by adding a
+coffin state ∂, and setting Qi∂:=−/summationtext
+j≥0Qij≥0. LetP(·) denote the semi-
+group of Markov transition matrices corresponding to the minimal p rocess
+associated with Q; then, in particular,
+Q=P′(0) and P′(t) =QP(t) for all t≥0 (3.5)
+(Reuter 1957, Theorem 3). Set
+RT
+ij(t) :=ewtµ(i)Pij(t)/µ(j). (3.6)
+13Theorem 3.1 LetAsatisfy Assumptions (3.1)and(3.2). Then, with the
+above definitions, Ris a strongly continuous semigroup on Rµ, and
+/summationdisplay
+i≥0µ(i)Rij(t)≤µ(j)ewtfor alljandt. (3.7)
+Furthermore, the sums/summationtext
+j≥0Rij(t)Ajk= (R(t)A)ikare well defined for all
+i,k, and
+A=R′(0)andR′(t) =R(t)Afor allt≥0.(3.8)
+Proof. We note first that, for x∈ Rµ,
+/⌊ard⌊lR(t)x/⌊ard⌊lµ≤/summationdisplay
+i≥0µ(i)/summationdisplay
+j≥0Rij(t)|xj|=ewt/summationdisplay
+i≥0/summationdisplay
+j≥0µ(j)Pji(t)|xj|
+≤ewt/summationdisplay
+j≥0µ(j)|xj|=ewt/⌊ard⌊lx/⌊ard⌊lµ, (3.9)
+sinceP(t) is substochastic on Z+; henceR:Rµ→ Rµ. To show strong
+continuity, we take x∈ Rµ, and consider
+/⌊ard⌊lR(t)x−x/⌊ard⌊lµ=/summationdisplay
+i≥0µ(i)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+j≥0Rij(t)xj−xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/summationdisplay
+i≥0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleewt/summationdisplay
+j≥0µ(j)Pji(t)xj−µ(i)xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤(ewt−1)/summationdisplay
+i≥0/summationdisplay
+j≥0µ(j)Pji(t)xj+/summationdisplay
+i≥0/summationdisplay
+j/negationslash=iµ(j)Pji(t)xj+/summationdisplay
+i≥0µ(i)xi(1−Pii(t))
+≤(ewt−1)/summationdisplay
+j≥0µ(j)xj+2/summationdisplay
+i≥0µ(i)xi(1−Pii(t)),
+from which it follows that lim t→0/⌊ard⌊lR(t)x−x/⌊ard⌊lµ= 0, by dominated conver-
+gence, since lim t→0Pii(t) = 1 for each i≥0.
+The inequality (3.7) follows from the definition of Rand the fact that P
+is substochastic on Z+. Then
+(ATRT(t))ij=/summationdisplay
+k/negationslash=iQikµ(i)
+µ(k)ewtµ(k)
+µ(j)Pkj(t)+(Qii+w)ewtµ(i)
+µ(j)Pij(t)
+=µ(i)
+µ(j)[(QP(t))ij+wPij(t)]ewt,
+14with (QP(t))ij=/summationtext
+k≥0QikPkj(t) well defined because P(t) is sub-stochastic
+andQis conservative. Using (3.5), this gives
+(ATRT(t))ij=µ(i)
+µ(j)d
+dt[Pij(t)ewt] =d
+dtRT
+ij(t),
+and this establishes (3.8).
+4 Main approximation
+LetXN,N≥1, beasequence ofpure jumpMarkov processes asinSection 1,
+withAandFdefined as in (1.4) and (1.5), and suppose that F:Rµ→ Rµ,
+withRµas defined in (3.3), for some µsuch that Assumption (3.2) holds.
+Suppose also that Fis locally Lipschitz in the µ-norm: for any z >0,
+sup
+x/negationslash=y:/bardblx/bardblµ,/bardbly/bardblµ≤z/⌊ard⌊lF(x)−F(y)/⌊ard⌊lµ//⌊ard⌊lx−y/⌊ard⌊lµ≤K(µ,F;z)<∞.(4.1)
+Then, for x(0)∈ RµandRas in (3.6), the integral equation
+x(t) =R(t)x(0)+/integraldisplayt
+0R(t−s)F(x(s))ds. (4.2)
+has a unique continuous solution xinRµon some non-empty time interval
+[0,tmax), such that, if tmax<∞, then/⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞ast→tmax(Pazy 1983,
+Theorem 1.4, Chapter 6). Thus, if Awere the generator of R, the function x
+would be a mild solution of the deterministic equations (1.4). We now wish
+to show that the process xN:=N−1XNis close to x. To do so, we need a
+corresponding representation for XN.
+To find such a representation, let W(t),t≥0, be a pure jump path on X+
+that has only finitely many jumps up to time T. Then we can write
+W(t) =W(0)+/summationdisplay
+j:σj≤t∆W(σj),0≤t≤T, (4.3)
+where ∆W(s) :=W(s)−W(s−)andσj,j≥1, denote thetimes when Whas
+its jumps. Now let Asatisfy (3.1) and (3.2), and let R(·) be the associated
+semigroup, as defined in (3.6). Define the path W∗(t), 0≤t≤T, from the
+equation
+W∗(t) :=R(t)W(0)+/summationtext
+j:σj≤tR(t−σj)∆j−/integraltextt
+0R(t−s)AW(s)ds,
+(4.4)
+15where ∆ j:= ∆W(σj). Note that the latter integral makes sense, because
+each of the sums/summationtext
+j≥0Rij(t)Ajkis well defined, from Theorem 3.1, and
+because only finitely many of the coordinates of Ware non-zero.
+Lemma 4.1 W∗(t) =W(t)for all0≤t≤T.
+Proof. Fix any t, and suppose that W∗(s) =W(s) for alls≤t. This is
+clearly the case for t= 0. Let σ(t)> tdenote the time of the first jump
+ofWaftert. Then, for any 0 < h < σ(t)−t, using the semigroup property
+forRand (4.4),
+W∗(t+h)−W∗(t)
+= (R(h)−I)R(t)W(0)+/summationdisplay
+j:σj≤t(R(h)−I)R(t−σj)∆j (4.5)
+−/integraldisplayt
+0(R(h)−I)R(t−s)AW(s)ds−/integraldisplayt+h
+tR(t+h−s)AW(t)ds,
+where, in the last integral, we use the fact that there are no jumps ofW
+between tandt+h. Thus we have
+W∗(t+h)−W∗(t)
+= (R(h)−I)
+
+R(t)W(0)+/summationdisplay
+j:σj≤tR(t−σj)∆j−/integraldisplayt
+0R(t−s)AW(s)ds
+
+
+−/integraldisplayt+h
+tR(t+h−s)AW(t)ds
+= (R(h)−I)W(t)−/integraldisplayt+h
+tR(t+h−s)AW(t)ds. (4.6)
+But now, for x∈ X+,
+/integraldisplayt+h
+tR(t+h−s)Axds= (R(h)−I)x,
+from (3.8), so that W∗(t+h) =W∗(t) for all t+h < σ(t), implying that
+W∗(s) =W(s) for all s < σ(t). On the other hand, from (4.4), we have
+W∗(σ(t))−W∗(σ(t)−) = ∆W(σ(t)), so that W∗(s) =W(s) for alls≤σ(t).
+Thus we can prove equality over the interval [0 ,σ1], and then successively
+over the intervals [ σj,σj+1], until [0 ,T] is covered.
+16Now suppose that Warises as a realization of XN. ThenXNhas transi-
+tion rates such that
+MN(t) :=/summationdisplay
+j:σj≤t∆XN(σj)−/integraldisplayt
+0AXN(s)ds−/integraldisplayt
+0NF(xN(s))ds(4.7)
+is a zero mean local martingale. In view of Lemma 4.1, we can use (4.4) t o
+write
+XN(t) =R(t)XN(0)+/tildewiderMN(t)+N/integraldisplayt
+0R(t−s)F(xN(s))ds,(4.8)
+where
+/tildewiderMN(t) :=/summationdisplay
+j:σj≤tR(t−σj)∆XN(σj)
+−/integraldisplayt
+0R(t−s)AXN(s)ds−/integraldisplayt
+0R(t−s)NF(xN(s))ds.(4.9)
+Thus, comparing (4.8) and (4.2), we expect xNandxto be close, for
+0≤t≤T < tmax, provided that we can show that supt≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis small,
+where/tildewidemN(t) :=N−1/tildewiderMN(t). Indeed, if xN(0) andx(0) are close, then
+/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ
+≤ /⌊ard⌊lR(t)(xN(0)−x(0))/⌊ard⌊lµ
++/integraldisplayt
+0/⌊ard⌊lR(t−s)[F(xN(s))−F(x(s))]/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ
+≤ewt/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ
++/integraldisplayt
+0ew(t−s)K(µ,F;2ΞT)/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ,(4.10)
+by (3.9), with the stage apparently set for Gronwall’s inequality, ass uming
+that/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµand sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµare small enough that then
+/⌊ard⌊lxN(t)/⌊ard⌊lµ≤2ΞTfor 0≤t≤T, where Ξ T:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ.
+Bounding sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis, however, not so easy. Since /tildewiderMNis not
+itselfamartingale, wecannotdirectlyapplymartingaleinequalitiestoc ontrol
+its fluctuations. However, since
+/tildewiderMN(t) =/integraldisplayt
+0R(t−s)dMN(s), (4.11)
+17we can hope to use control over the local martingale MNinstead. For this
+and the subsequent argument, we introduce some further assum ptions.
+Assumption 4.2
+1. There exists r=rµ≤r(2)
+maxsuch that supj≥0{µ(j)/νr(j)}<∞.
+2. There exists ζ∈ Rwithζ(j)≥1for alljsuch that (2.25)is satisfied
+for some b=b(ζ)≥1andr=r(ζ)such that 1≤r(ζ)≤r(2)
+max, and that
+Z:=/summationdisplay
+k≥0µ(k)(|Akk|+1)/radicalbig
+ζ(k)<∞. (4.12)
+The requirement that ζsatisfies (4.12) as well as satisfying (2.25) for some
+r≤r(2)
+maximplies in practice that it must be possible to take r(1)
+maxandr(2)
+max
+to be quite large in Assumption 2.1; see the examples in Section 5.
+Note that part 1 of Assumption 4.2 implies that lim j→∞{µ(j)/νr(j)}= 0
+for some r= ˜rµ≤rµ+1. We define
+ρ(ζ,µ) := max {r(ζ),p(r(ζ)),˜rµ}, (4.13)
+wherep(·) is as in Assumptions 2.1. We can now prove the following lemma,
+which enables us to control the paths of /tildewiderMNby using fluctuation bounds for
+the martingale MN.
+Lemma 4.3 Under Assumption 4.2,
+/tildewiderMN(t) =MN(t)+/integraldisplayt
+0R(t−s)AMN(s)ds.
+Proof. From (3.8), we have
+R(t−s) =I+/integraldisplayt−s
+0R(v)Adv.
+Substituting this into (4.11), we obtain
+/tildewiderMN(t) =/integraldisplayt
+0R(t���s)dMN(s)
+18=MN(t)+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+dMN(s)
+=MN(t)+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+dXN(s)
+−/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+F0(xN(s))ds.
+It remains to change the order of integration in the double integrals , for
+which we use Fubini’s theorem.
+In the first, the outer integral is almost surely a finite sum, and at e ach
+jump time tXN
+lwe havedXN(tXN
+l)∈ J. Hence it is enough that, for each i,
+mandt,/summationtext
+j≥0Rij(t)Ajmis absolutely summable, which follows from Theo-
+rem 3.1. Thus we have
+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+dXN(s) =/integraldisplayt
+0R(v)A{XN(t−v)−XN(0)}dv.
+(4.14)
+For the second, the k-th component of R(v)AF0(xN(s)) is just
+/summationdisplay
+j≥0Rkj(v)/summationdisplay
+l≥0Ajl/summationdisplay
+J∈JJlαJ(xN(s)). (4.15)
+Now, from (3.7), we have 0 ≤Rkj(v)≤µ(j)ewv/µ(k), and
+/summationdisplay
+j≥0µ(j)|Ajl| ≤µ(l)(2|All|+w), (4.16)
+becauseATµ≤wµ. Hence, puttingabsolutevaluesinthesummandsin(4.15)
+yields at most
+ewv
+µ(k)/summationdisplay
+J∈JαJ(xN(s))/summationdisplay
+l≥0|Jl|µ(l)(2|All|+w).
+Now, in view of (4.12) and since ζ(j)≥1 for allj, there is a constant K <∞
+such that µ(l)(2|All|+w)≤Kζ(l). Furthermore, ζsatisfies (2.25), so that,
+by Corollary 2.5,/summationtext
+J∈JαJ(xN(s))/summationtext
+l≥0|Jl|ζ(l) is a.s. uniformly bounded in
+0≤s≤T. Hence we can apply Fubini’s theorem, obtaining
+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+F0(xN(s))ds=/integraldisplayt
+0R(v)A/braceleftbigg/integraldisplayt−v
+0F0(xN(s))ds/bracerightbigg
+dv,
+19and combining this with (4.14) proves the lemma.
+We now introduce the exponential martingales that we use to bound the
+fluctuations of MN. Forθ∈RZ+bounded and x∈ Rµ,
+ZN,θ(t) :=eθTxN(t)exp/braceleftBig
+−/integraltextt
+0gNθ(xN(s−))ds/bracerightBig
+, t≥0,
+is a non-negative finite variation local martingale, where
+gNθ(ξ) :=/summationdisplay
+J∈JNαJ(ξ)/parenleftBig
+eN−1θTJ−1/parenrightBig
+.
+Fort≥0, we have
+logZN,θ(t) =θTxN(t)−/integraldisplayt
+0gNθ(xN(s−))ds
+=θTmN(t)−/integraldisplayt
+0ϕN,θ(xN(s−),s)ds, (4.17)
+where
+ϕN,θ(ξ) :=/summationdisplay
+J∈JNαJ(ξ)/parenleftBig
+eN−1θTJ−1−N−1θTJ/parenrightBig
+,(4.18)
+andmN(t) :=N−1MN(t). Note also that we can write
+ϕN,θ(ξ) =N/integraldisplay1
+0(1−r)D2vN(ξ,rθ)[θ,θ]dr, (4.19)
+where
+vN(ξ,θ′) :=/summationdisplay
+J∈JαJ(ξ)eN−1(θ′)TJ,
+andD2vNdenotes thematrixofsecond derivatives withrespect totheseco nd
+argument:
+D2vN(ξ,θ′)[ζ1,ζ2] :=N−2/summationdisplay
+J∈JαJ(ξ)eN−1(θ′)TJζT
+1JJTζ2(4.20)
+for anyζ1,ζ2∈ Rµ.
+Now choose any B:= (Bk, k≥0)∈ R, and define ˜ τ(N)
+k(B) by
+˜τ(N)
+k(B) := inf/braceleftBigg
+t≥0:/summationdisplay
+J:Jk/negationslash=0αJ(xN(t−))> Bk/bracerightBigg
+.
+Our exponential bound is as follows.
+20Lemma 4.4 For anyk≥0,
+P
+sup
+0≤t≤T∧˜τ(N)
+k(B)|mk
+N(t)| ≥δ
+≤2exp(−δ2N/2BkK∗T).
+for all0< δ≤BkK∗T, whereK∗:=J2
+∗eJ∗, andJ∗is as in(1.2).
+Proof. Takeθ=e(k)β, forβto be chosen later. We shall argue by stopping
+the local martingale ZN,θat timeσ(N)(k,δ), where
+σ(N)(k,δ) :=T∧˜τ(N)
+k(B)∧inf{t:mk
+N(t)≥δ}.
+Note that eN−1θTJ≤eJ∗, so long as |β| ≤N, so that
+D2vN(ξ,rθ)[θ,θ]≤N−2/parenleftBigg/summationdisplay
+J:Jk/negationslash=0αJ(ξ)/parenrightBigg
+β2K∗.
+Thus, from (4.19), we have
+ϕN,θ(xN(u−))≤1
+2N−1Bkβ2K∗, u≤˜τ(N)
+k(B),
+and hence, on the event that σ(N)(k,δ) = inf{t:mk
+N(t)≥δ} ≤(T∧˜τ(N)
+k(B)),
+we have
+ZN,θ(σ(k,δ))≥exp{βδ−1
+2N−1Bkβ2K∗T}.
+But since ZN,θ(0) = 1, it now follows from the optional stopping theorem
+and Fatou’s lemma that
+1≥E{ZN,θ(σ(N)(k,δ))}
+≥P/bracketleftBig
+sup
+0≤t≤T∧˜τ(N)
+k(B)mk
+N(t)≥δ/bracketrightBig
+exp{βδ−1
+2N−1Bkβ2K∗T}.
+We can choose β=δN/B kK∗T, as long as δ/BkK∗T≤1, obtaining
+P
+sup
+0≤t≤T∧˜τ(N)
+k(B)mk
+N(t)≥δ
+≤exp(−δ2N/2BkK∗T).
+Repeating with
+˜σ(N)(k,δ) :=T∧˜τ(N)
+k(B)∧inf{t:−mk
+N(t)≥δ},
+21and choosing β=δN/B kK∗T, gives the lemma.
+Theprecedinglemmagivesaboundforeachindividualcomponentof MN.
+We need first to translate this into a statement for all components simulta-
+neously. For ζas in Assumption 4.2, we start by writing
+Z(1)
+∗:= max
+k≥1k−1#{m:ζ(m)≤k};Z(2)
+∗:= sup
+k≥0µ(k)(|Akk|+1)/radicalbig
+ζ(k).(4.21)
+Z(2)
+∗is clearly finite, because of Assumption 4.2, and the same is true for Z(1)
+∗
+also, since Zof Assumption 4.2 is at least # {m:ζ(m)≤k}/√
+k, for each k.
+Then, using the definition (2.24) of τ(N)(a,ζ), note that, for every k,
+/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
+|Jk|ζ(k)≤ah(k)
+ζ(k),(4.22)
+for anyt < τ(N)(a,ζ) and any h∈ R, and that, for any K ⊆Z+,
+/summationdisplay
+k∈K/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
+k∈K/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
+|Jk|ζ(k)
+≤a
+mink∈K(ζ(k)/h(k)). (4.23)
+From (4.22) with h(k) = 1 for all k, if we choose B:= (a/ζ(k), k≥0), then
+τ(N)(a,ζ)≤˜τ(N)
+k(B) for allk. For this choice of B, we can take
+δ2
+k:=δ2
+k(a) :=4aK∗TlogN
+Nζ(k)=4BkK∗TlogN
+N(4.24)
+in Lemma 4.4 for k∈κN(a), where
+κN(a) :=/braceleftbig
+k:ζ(k)≤1
+4aK∗TN/logN/bracerightbig
+={k:Bk≥4logN/K∗TN},
+(4.25)
+since then δk(a)≤BkK∗T. Note that then, from (4.12),
+/summationdisplay
+k∈κN(a)µ(k)δk(a)≤2Z/radicalbig
+aK∗TN−1logN, (4.26)
+withZas defined in Assumption 4.2, and that
+|κN(a)| ≤1
+4aZ(1)
+∗K∗TN/logN. (4.27)
+22Lemma 4.5 If Assumptions 4.2 are satisfied, taking δk(a)andκN(a)as
+defined in (4.24)and(4.25), and for any η∈ R, we have
+1.P
+/uniondisplay
+k∈κN(a)/braceleftBig
+sup
+0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≥δk(a)/bracerightBig
+≤aZ(1)
+∗K∗T
+2NlogN;
+2.P
+/summationdisplay
+k/∈κN(a)Xk
+N(t) = 0for all0≤t≤T∧τ(N)(a,ζ)
+≥1−4logN
+K∗N;
+3. sup
+0≤t≤T∧τ(N)(a,ζ)
+
+/summationdisplay
+k/∈κN(a)η(k)|Fk(xN(t))|
+
+≤aJ∗
+mink/∈κN(a)(ζ(k)/η(k)).
+Proof. For part 1, use Lemma 4.4 together with (4.24) and (4.27) to give
+the bound. For part 2, the total rate of jumps into coordinates w ith indices
+k /∈κN(a) is
+/summationdisplay
+k/∈κN(a)/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))≤a
+mink/∈κN(a)ζ(k),
+ift≤τ(N)(a,ζ),using(4.23)with K= (κN(a))c,which, combinedwith(4.25),
+proves the claim. For the final part, if t≤τ(N)(a,ζ),
+/summationdisplay
+k/∈κN(a)η(k)|Fk(xN(t))| ≤/summationdisplay
+k/∈κN(a)η(k)/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))J∗,
+and the inequality follows once more from (4.23).
+LetB(1)
+N(a) andB(2)
+N(a) denote the events
+B(1)
+N(a) :=
+
+/summationdisplay
+k/∈κN(a)Xk
+N(t) = 0 for all 0 ≤t≤T∧τ(N)(a,ζ)
+
+;
+B(2)
+N(a) :=
+/intersectiondisplay
+k∈κN(a)/braceleftBig
+sup
+0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≤δk(a)/bracerightBig
+,(4.28)
+and setBN(a) :=B(1)
+N(a)∩B(2)
+N(a). Then, by Lemma 4.5, we deduce that
+P[BN(a)c]≤aZ(1)
+∗K∗T
+2NlogN+4logN
+K∗N, (4.29)
+23of order O(N−1logN) for each fixed a. Thus we have all the components
+ofMNsimultaneously controlled, except on a set of small probability. We
+now translate this into the desired assertion about the fluctuation s of/tildewidemN.
+Lemma 4.6 If Assumptions 4.2 are satisfied, then, on the event BN(a),
+sup
+0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ≤√aK4.6/radicalbigg
+logN
+N,
+where the constant K4.6depends on Tand the parameters of the process.
+Proof. From Lemma 4.3, it follows that
+sup
+0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ (4.30)
+≤sup
+0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊lmN(t)/⌊ard⌊lµ+ sup
+0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
+0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds.
+For the first term, on BN(a) and for 0 ≤t≤T∧τ(N)(a,ζ), we have
+/⌊ard⌊lmN(t)/⌊ard⌊lµ≤/summationdisplay
+k∈κN(a)µ(k)δk(a)+/integraldisplayt
+0/summationdisplay
+k/∈κN(a)µ(k)|Fk(xN(u))|du.
+The first sum is bounded using (4.26) by 2 Z√aK∗T N−1/2√logN, the sec-
+ond, from Lemma 4.5 and (4.25), by
+TaJ∗
+mink/∈κN(a)(ζ(k)/µ(k))≤Z(2)
+∗2J∗/radicalbigg
+Ta
+K∗/radicalbigg
+logN
+N.
+For the second term in (4.30), from (3.7) and (4.16), we note that
+/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµ≤/summationdisplay
+k≥0µ(k)/summationdisplay
+l≥0Rkl(t−s)/summationdisplay
+r≥0|Alr||mr
+N(s)|
+≤ew(t−s)/summationdisplay
+l≥0µ(l)/summationdisplay
+r≥0|Alr||mr
+N(s)|
+≤ew(t−s)/summationdisplay
+r≥0µ(r){2|Arr|+w}|mr
+N(s)|.
+24OnBN(a) and for 0 ≤s≤T∧τ(N)(a,ζ), from (4.12), the sum for r∈κN(a)
+is bounded using
+/summationdisplay
+r∈κN(a)µ(r){2|Arr|+w}|mr
+N(s)|
+≤/summationdisplay
+r∈κN(a)µ(r){2|Arr|+w}δr(a)
+≤/summationdisplay
+r∈κN(a)µ(r){2|Arr|+w}/radicalBigg
+4aK∗TlogN
+Nζ(r)
+≤(2∨w)Z/radicalbig
+4aK∗T/radicalbigg
+logN
+N.
+The remaining sum is then bounded by Lemma 4.5, on the set BN(a) and
+for 0≤s≤T∧τ(N)(a,ζ), giving at most
+/summationdisplay
+r/∈κN(a)µ(r){2|Arr|+w}|mr
+N(s)|
+≤/summationdisplay
+r/∈κN(a)µ(r){2|Arr|+w}/integraldisplays
+0|Fr(xN(t))|dt
+≤(2∨w)saJ∗
+mink/∈κN(a)(ζ(k)/µ(k){|Akk|+1})
+≤(2∨w)Z(2)
+∗2J∗/radicalbigg
+Ta
+K∗/radicalbigg
+logN
+N.
+Integrating, it follows that
+sup
+0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
+0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds
+≤(2T∨1)ewT/braceleftBigg/radicalbig
+4aK∗TZ+Z(2)
+∗J2J∗/radicalbigg
+Ta
+K∗/bracerightBigg/radicalbigg
+logN
+N,
+and the lemma follows.
+This has now established the control on sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµthat we need,
+in order to translate (4.10) into a proof of the main theorem.
+25Theorem 4.7 Suppose that (1.2),(1.3),(3.1),(3.2)and(4.1)are all satis-
+fied, and that Assumptions 2.1 and 4.2 hold. Recalling the defi nition(4.13)
+ofρ(ζ,µ), forζas given in Assumption 4.2, suppose that S(N)
+ρ(ζ,µ)(0)≤NC∗
+for some C∗<∞.
+Letxdenote the solution to (4.2)with initial condition x(0)satisfying
+Sρ(ζ,µ)(x(0))<∞. Thentmax=∞.
+Fix any T, and define ΞT:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ. If/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤
+1
+2ΞTe−(w+k∗)T, wherek∗:=ewTK(µ,F;2ΞT), then there exist constants c1,c2
+depending on C∗,Tand the parameters of the process, such that for all N
+large enough
+P/parenleftBigg
+sup
+0≤t≤T/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ>/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+c1/radicalbigg
+logN
+N/parenrightBigg
+ek∗T/parenrightBigg
+≤c2logN
+N. (4.31)
+Proof. AsS(N)
+ρ(ζ,µ)(0)≤NC∗, it follows also that S(N)
+r(0)≤NC∗for all
+0≤r≤ρ(ζ,µ). Fix any T < tmax, takeC:= 2(C∗+k04T)ek01T, and observe
+that, for r≤ρ(ζ,µ)∧r(2)
+max, and such that p(r)≤ρ(ζ,µ), we can take
+C′′
+rT≤/tildewideCrT:={2(C∗∨1)+kr4T}e(kr1+Ckr2)T, (4.32)
+in Theorem 2.4, since we can take C∗to bound CrandC′
+r. In particular,
+r=r(ζ) as defined in Assumption 4.2 satisfies both the conditions on r
+for (4.32) to hold. Then, taking a:={k2+k1/tildewideCr(ζ)T}b(ζ)in Corollary 2.5, it
+follows that for some constant c3>0, on the event BN(a),
+P[τ(N)(a,ζ)≤T]≤c3N−1.
+Then, from (4.29), for some constant c4,P[BN(a)c]≤c4N−1logN. Here,
+the constants c3,c4depend on C∗,Tand the parameters of the process.
+We now use Lemma 4.6 to bound the martingale term in (4.10). It fol-
+lows that, on the event BN(a)∩ {τ(N)(a,ζ)> T}and on the event that
+/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµ≤ΞTfor all 0≤s≤t,
+/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
+logN
+N/parenrightBigg
++k∗/integraldisplayt
+0/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds,
+26wherek∗:=ewTK(µ,F;2ΞT). Then from Gronwall’s inequality, on the
+eventBN(a)∩{τ(N)(a,ζ)> T},
+/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
+logN
+N/parenrightBigg
+ek∗t,
+(4.33)
+for all 0≤t≤T, provided that
+/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
+logN
+N/parenrightBigg
+≤ΞTe−k∗T.
+This is true for all Nsufficiently large, if /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤1
+2ΞTe−(w+k∗)T,
+which we have assumed. We have thus proved (4.31), since, as show n above,
+P(BN(a)c∪{τ(N)(a,ζ)> T}c) =O(N−1logN).
+We now use this to show that in fact tmax=∞. Forx(0) as above, we
+can take xj
+N(0) :=N−1⌊Nxj(0)⌋ ≤xj(0), so that S(N)
+ρ(ζ,µ)(0)≤NC∗forC∗:=
+Sρ(ζ,µ)(x(0))<∞. Then, by (4.13), lim j→∞{µ(j)/νρ(ζ,µ)(j)}= 0, so it fol-
+lowseasilyusing boundedconvergence that /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ→0asN→ ∞.
+Hence, for any T < t max, it follows from (4.31) that /⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ→D0
+asN→ ∞, fort≤T, with uniform bounds over the interval, where ‘ →D’
+denotes convergence in distribution. Also, by Assumption 4.2, ther e is a con-
+stantc5such that /⌊ard⌊lxN(t)/⌊ard⌊lµ≤c5N−1S(N)
+rµ(t) for each t, whererµ≤r(2)
+maxand
+rµ≤ρ(ζ,µ). Hence, using Lemma 2.3 and Theorem 2.4, sup0≤t≤2T/⌊ard⌊lxN(t)/⌊ard⌊lµ
+remains bounded in probability as N→ ∞. Hence it is impossible that
+/⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞asT→tmax<∞,implyingthatinfact tmax=∞forsuchx(0).
+Remark . The dependence on the initial conditions is considerably compli-
+cated by the way the constant Cappears in the exponent, for instance in the
+expression for /tildewideCrTin the proof of Theorem 4.7. However, if kr2in Assump-
+tions 2.1 can be chosen to be zero, as for instance in the examples be low, the
+dependence simplifies correspondingly.
+Therearebiologicallyplausiblemodelsinwhichtherestrictionto Jl≥ −1
+is irksome. In populations in which members of a given type lcan fight one
+another, a natural possibility is to have a transition J=−2e(l)at a rate
+proportional to Xl(Xl−1), which translates to αJ=α(N)
+J=γxl(xl−N−1),
+a function depending on N. Replacing this with αJ=γ(xl)2removes the
+27N-dependence, but yields a process that can jump to negative value s ofXl.
+For this reason, it is useful to be able to allow the transition rates αJto
+depend on N.
+Since the arguments inthis paper are not limiting arguments for N→ ∞,
+it does not require many changes to derive the corresponding resu lts. Quan-
+tities such as A,F,Ur(x) andVr(x) now depend on N; however, Theorem 4.7
+continues toholdwithconstants c1andc2thatdo notdepend on N, provided
+thatµ,w,ν, theklmfrom Assumption 2.1 and ζfrom Assumption 4.2 can
+be chosen to be independent of N, and that the quantities Z(l)
+���from (4.21)
+can be bounded uniformly in N. On the other hand, the solution x=x(N)
+of (4.2) that acts as approximation to xNin Theorem 4.7 now itself depends
+onN, through R=R(N)andF=F(N). IfA(and hence R) can be taken
+to be independent of N, and lim N→∞/⌊ard⌊lF(N)−F/⌊ard⌊lµ= 0 for some fixed µ–
+Lipschitz function F, a Gronwall argument can be used to derive a bound
+for the difference between x(N)and the (fixed) solution xto equation (4.2)
+withN-independent RandF. IfAhas to depend on N, the situation is
+more delicate.
+5 Examples
+We begin with some general remarks, to show that the assumptions are sat-
+isfied in many practical contexts. We then discuss two particular ex amples,
+those of Kretzschmar (1993) and of Arrigoni (2003), that fitte d poorly or
+not at all into the general setting of Barbour & Luczak (2008), th ough the
+other systems referred to in the introduction could also be treate d similarly.
+In both of our chosen examples, the index jrepresents a number of individ-
+uals — parasites in a host in the first, animals in a patch in the second —
+and we shall for now use the former terminology for the preliminary, general
+discussion.
+Transitions that can typically be envisaged are: births of a few para sites,
+which may occur either in the same host, or in another, if infection is b eing
+represented; births and immigration of hosts, with or without para sites; mi-
+gration of parasites between hosts; deaths of parasites; death s of hosts; and
+treatment of hosts, leading to the deaths of many of the host’s pa rasites. For
+births of parasites, there is a transition X→X+J, whereJtakes the form
+Jl= 1;Jm=−1;Jj= 0, j/ne}ationslash=l,m, (5.1)
+28indicating that one m-host has become an l-host. For births of parasites
+within a host, a transition rate of the form bl−mmXmcould be envisaged,
+withl > m, the interpretation being that there are Xmhosts with parasite
+burdenm, each of which gives birth to soffspring at rate bs, for some small
+values of s. For infection of an m-host, a possible transition rate would be
+of the form
+Xm/summationdisplay
+j≥0N−1Xjλpj,l−m,
+since an m-host comes into contact with j-hosts at a rate proportional to
+their density in the host population, and pjrrepresents the probability of a
+j-host transferring rparasites to the infected host during the contact. The
+probability distributions pj·can be expected to be stochastically increasing
+inj. Deaths of parasites also give rise to transitions of the form (5.1),
+but now with l < m, the simplest form of rate being just dmXmforl=
+m−1, though d=dmcould also be chosen to increase with parasite burden.
+Treatment of a host would lead to values of lmuch smaller than m, and
+a rate of the form κXmfor the transition with l= 0 would represent fully
+successful treatment of randomly chosen individuals. Births and d eaths of
+hosts and immigration all lead to transitions of the form
+Jl=±1;Jj= 0, j/ne}ationslash=l. (5.2)
+Fordeaths, Jl=−1, anda typical ratewould be d′Xl. Forbirths, Jl= 1, and
+a possible rate would be/summationtext
+j≥0Xjb′
+jl(withl= 0 only, if new-born individuals
+are free of parasites). For immigration, constant rates λlcould be supposed.
+Finally, for migration of individual parasites between hosts, transit ions are
+of the form
+Jl=Jm=−1;Jl+1= 1;Jm−1= 1;Jj= 0, j/ne}ationslash=l,m,l+1,m−1,
+(5.3)
+a possible rate being γmXmN−1Xl.
+For all the above transitions, we can take J∗= 2 in (1.2), and (1.3) is
+satisfied in biologically sensible models. (3.1) and (3.2) depend on the wa y in
+which the matrix Acan be defined, which is more model specific; in practice,
+(3.1) is very simple to check. The choice of µin (3.2) is influenced by the
+need to have (4.1) satisfied. For Assumptions 2.1, a possible choice o fνis to
+takeν(j) = (j+1) for each j≥0, withS1(X) then representing the num-
+ber of hosts plus the number of parasites. Satisfying (2.5) is then e asy for
+29transitions only involving the movement of a single parasite, but in gen eral
+requires assumptions as to the existence of the r-th moments of the distri-
+butions of the numbers of parasites introduced at birth, immigratio n and
+infection events. For (2.6), in which transitions involving a net reduc tion
+in the total number of parasites and hosts can be disregarded, th e parasite
+birth events are those in which the rates typically have a factor mXmfor
+transitions with Jm=−1, withmin principle unbounded. However, at such
+events, an m-individual changes to an m+sindividual, with the number s
+of offspring of the parasite being typically small, so that the value of JTνr
+associated with this rate has magnitude mr−1; the product mXmmr−1, when
+summed over m, then yields a contribution of magnitude Sr(X), which is al-
+lowable in(2.6). Similar considerations showthat theterms N−1S0(X)Sr(X)
+accommodate the migration rates suggested above. Finally, in orde r to have
+Assumptions 4.2 satisfied, it is in practice necessary that Assumptio ns 2.1
+are satisfied for large values of r, thereby imposing restrictions on the dis-
+tributions of the numbers of parasites introduced at birth, immigra tion and
+infection events, as above.
+5.1 Kretzschmar’s model
+Kretzschmar (1993) introduced a model of a parasitic infection, in which the
+transitions from state Xare as follows:
+J=e(i−1)−e(i)at rate Niµxi, i ≥1;
+J=−e(i)at rate N(κ+iα)xi, i≥0;
+J=e(0)at rate Nβ/summationtext
+i≥0xiθi;
+J=e(i+1)−e(i)at rate Nλxiϕ(x), i ≥0,
+wherex:=N−1X,ϕ(x) :=/⌊ard⌊lx/⌊ard⌊l11{c+/⌊ard⌊lx/⌊ard⌊l1}−1withc >0, and/⌊ard⌊lx/⌊ard⌊l11:=/summationtext
+j≥1j|x|j; here, 0≤θ≤1, andθidenotes its i-th power (our θcorresponds
+to the constant ξin [7]). Both (1.2) and (1.3) are obviously satisfied. For
+Assumptions (3.1), (3.2) and (4.1), we note that equation corresp onding
+to (1.5) has
+Aii=−{κ+i(α+µ)};AT
+i,i−1=iµandAT
+i0=βθi, i≥2;
+A11=−{κ+α+µ};AT
+10=µ+βθ;
+A00=−κ+β, i≥1,
+30with all other elements of the matrix equal to zero, and
+Fi(x) =λ(xi−1−xi)ϕ(x), i≥1;F0(x) =−λx0ϕ(x).
+Hence Assumption (3.1) isimmediate, andAssumption (3.2)holds for µ(j) =
+(j+1)s, for any s≥0, withw= (β−κ)+. For the choice µ(j) =j+1,F
+maps elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
+K(µ,F;Ξ) =c−2λΞ(2c+Ξ).
+For Assumptions 2.1, choose ν=µ; then (2.5) is a finite sum for each
+r≥0. Turning to (2.6), it is immediate that U0(x)≤βS0(x). Then, for
+r≥1,
+/summationdisplay
+i≥0λϕ(N−1X)Xi{(i+2)r−(i+1)r} ≤λS1(X)
+S0(X)/summationdisplay
+i≥0rXi(i+2)r−1
+≤r2r−1λSr(X),
+since, by Jensen’s inequality, S1(X)Sr−1(X)≤S0(X)Sr(X). Hence we can
+takekr2=kr4= 0 and kr1=β+r2r−1λin (2.6), for any r≥1, so that
+r(1)
+max=∞. Finally, for (2.7),
+V0(x)≤(κ+β)S0(x)+αS1(x),
+so thatk03=κ+β+αandk05= 0, and
+Vr(x)≤r2(κS2r(x)+αS2r+1(x)+µS2r−1(x)+22(r−1)λS2r−1(x))+βS0(x),
+so that we can take p(r) = 2r+1,kr3=β+r2{κ+α+µ+22(r−1)λ}, and
+kr5= 0 for any r≥1, and so r(2)
+max=∞. In Assumptions 4.2, we can clearly
+takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 17.
+5.2 Arrigoni’s model
+Inthemetapopulation model ofArrigoni (2003), thetransitions f romstate X
+are as follows:
+J=e(i−1)−e(i)at rateNixi(di+γ(1−ρ)), i ≥2;
+J=e(0)−e(1)at rateNx1(d1+γ(1−ρ)+κ);
+J=e(i+1)−e(i)at rateNibixi, i ≥1;
+J=e(0)−e(i)at rateNxiκ, i ≥2;
+J=e(k+1)−e(k)+e(i−1)−e(i)at rateNixixkργ, k ≥0, i≥1;
+31as before, x:=N−1X. Here, the total number N=/summationtext
+j≥0Xj=S0(X) of
+patches remains constant throughout, and the number of animals in any one
+patch changes by at most one at each transition; in the final (migra tion)
+transition, however, the numbers in two patches change simultane ously. In
+the above transitions, γ,ρ,κare non-negative, and ( di),(bi) are sequences of
+non-negative numbers.
+Once again, both (1.2) and (1.3) are obviously satisfied. The equatio n
+corresponding to (1.4) can now be expressed by taking
+Aii=−{κ+i(bi+di+γ)};AT
+i,i−1=i(di+γ);AT
+i,i+1=ibi, i≥1;
+A00=−κ,
+with all other elements of Aequal to zero, and
+Fi(x) =ργ/⌊ard⌊lx/⌊ard⌊l11(xi−1−xi), i≥1;F0(x) =−ργx0/⌊ard⌊lx/⌊ard⌊l11+κ,
+where we have used the fact that N−1/summationtext
+j≥0Xj= 1. Hence Assumption (3.1)
+is again immediate, and Assumption (3.2) holds for µ(j) = 1 with w= 0,
+forµ(j) =j+ 1 with w= max i(bi−di−γ−κ)+(assuming ( bi) and (di)
+to be such that this is finite), or indeed for µ(j) = (j+1)swith any s≥2,
+with appropriate choice of w. With the choice µ(j) =j+1,Fagain maps
+elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
+K(µ,F;Ξ) = 3ργΞ.
+To check Assumptions 2.1, take ν=µ; once again, (2.5) is a finite sum
+for each r. Then, for (2.6), it is immediate that U0(x) = 0. For any r≥1,
+using arguments from the previous example,
+Ur(x)≤r2r−1/braceleftBigg/summationdisplay
+i≥1ibixi(i+1)r−1+/summationdisplay
+i≥1/summationdisplay
+k≥0iργxixk(k+1)r−1/bracerightBigg
+≤r2r−1{max
+ibiSr(x)+ργS1(x)Sr−1(x)}
+≤r2r−1{max
+ibiSr(x)+ργS0(x)Sr(x)},
+so that, since S0(x) = 1, we can take kr1=r2r−1(maxibi+ργ) andkr2=
+kr4= 0 in (2.6), and r(1)
+max=∞. Finally, for (2.7), V0(x) = 0 and, for r≥1,
+Vr(x)
+≤r2/braceleftBig
+22(r−1)max
+ibiS2r−1(x)+max
+i(i−1di)S2r(x)+γ(1−ρ)S2r−1(x)
++ργ(22(r−1)S1(x)S2r−2(x)+S0(x)S2r−1(x))/bracerightBig
++κS2r(x),
+32so that we can take p(r) = 2r, and (assuming i−1dito be finite)
+kr3=κ+r2{22(r−1)(max
+ibi+ργ)+max
+i(i−1di)+γ},
+andkr5= 0 for any r≥1, andr(2)
+max=∞. In Assumptions 4.2, we can again
+takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 16.
+Acknowledgement
+We wish to thank a referee for recommendations that have substa ntially
+streamlined our arguments. ADB wishes to thank both the Institut e for
+MathematicalSciencesoftheNationalUniversityofSingaporeand theMittag–
+Leffler Institute for providing a welcoming environment while part of t his
+work was accomplished. MJL thanks the University of Z¨ urich for th eir hos-
+pitality on a number of visits.
+References
+[1]Arrigoni, F. (2003). Deterministic approximation of a stochastic
+metapopulation model. Adv. Appl. Prob. 35691–720.
+[2]Barbour, A. D. andKafetzaki, M. (1993). A host–parasite model
+yielding heterogeneous parasite loads. J. Math. Biology 31157–176.
+[3]Barbour, A. D. andLuczak, M. J. (2008). Laws of large numbers
+for epidemic models with countably many types. Ann. Appl. Probab. 18
+2208–2238.
+[4]Chow, P.-L. (2007).Stochastic partial differential equations. Chapman
+and Hall, Boca Raton.
+[5]Eibeck, A. andWagner, W. (2003). Stochastic interacting particle
+systems and non-linear kinetic equations. Ann. Appl. Probab. 13845–
+889.
+[6]Kimmel, M. andAxelrod, D. E. (2002).Branching processes in biol-
+ogy.Springer, Berlin.
+33[7]Kretzschmar, M. (1993).Comparison ofaninfinite dimensional model
+for parasitic diseases with a related 2-dimensional system. J. Math. Anal-
+ysis Applics 176235–260.
+[8]Kurtz, T. G. (1970). Solutions of ordinary differential equations as
+limits of pure jump Markov processes. J. Appl. Probab. 749–58.
+[9]Kurtz, T. G. (1971).Limit theorems forsequences ofjumpMarkov pro-
+cesses approximating ordinary differential processes. J. Appl. Probab. 8
+344–356.
+[10]L´eonard, C. (1990).Some epidemic systems are long range interacting
+particle systems. In: Stochastic Processes in Epidemic Theory , Eds J.-P.
+Gabriel, C. Lef` evre & P. Picard, Lecture Notes in Biomathematics 86
+170–183: Springer, New York.
+[11]Luchsinger, C. J. (1999).MathematicalModelsofaParasiticDisease,
+Ph.D. thesis, University of Z¨ urich.
+[12]Luchsinger, C. J. (2001a). Stochastic models of a parasitic infection,
+exhibiting three basic reproduction ratios. J. Math. Biol. 42, 532–554.
+[13]Luchsinger, C. J. (2001b). Approximating the long term behaviour
+of a model for parasitic infection. J. Math. Biol. 42, 555–581.
+[14]Pazy, A. (1983).Semigroups of Linear Operators and Applications to
+Partial Differential Equations. Springer, Berlin.
+[15]Reuter, G. E. H. (1957). Denumerable Markov processes and the
+associated contraction semigroups on l.Acta Math. 97, 1–46.
+34

1001.0045.txt

@@ -0,0 +1,454 @@
+arXiv:1001.0045v1  [hep-ph]  31 Dec 2009EPJ manuscript No.
+(will be inserted by the editor)
+Axial Anomaly and Mixing Parameters of
+Pseudoscalar Mesons
+Yaroslav N. Klopot1,a, Armen G. Oganesian2,b, and Oleg V. Teryaev1,c
+1Joint Institute for Nuclear Research, Bogoliubov Laborato ry of Theoretical Physics, Dubna 141980,
+Russia
+2Institute of Theoretical and Experimental Physics, B.Cher emushkinskaya 25, Moscow 117218, Russia
+Abstract. In this work the analysis of mixing parameters of the system i nvolving
+η,η′mesons and some third massive state Gis carried out. We use the generalized
+mixing scheme with three angles. The framework of the disper sive approach to
+Abelian axial anomaly of isoscalar non-singlet current and the analysis of exper-
+imental data of charmonium radiative decays ratio allow us t o get a number of
+quite strict constraints for the mixing parameters. The ana lysis shows that the
+equal values of axial current coupling constants f8andf0are preferable which
+may be considered as a manifestation of SU(3) and chiral symmetry.
+1 Introduction
+This work is developing the approach of the papers [1,2] and is devot ed to the significant
+problem of mixing of pseudoscalar mesons. It is especially important w ith a number of current
+and planned experiments.
+The problem of η-η′mixing has been studied for many years. The usual approach with on e
+mixingangledominatedfordecades,butintherecentyearsthemor eelaboratedschemesappear
+to be unavoidable [3–8]. In particular, the theoretical ground of th is was based on the recent
+progress in the ChPT [9–11]. On the other hand, it was shown, that t he current experimental
+data cannot satisfactory describe the whole set of experiments w ithin the one-angle mixing
+scheme.
+The mixing schemes are usually enunciated either in terms of SU(3) or quark basis. In
+our paper [2] we construct and use the generalization of SU(3) basis similar to the mixing of
+massive neutrinos. This is because we use the dispersive approach t o axial anomaly ( [12], [13]
+for a review) to find some model-independent and precise restrictio n on the mixing parameters.
+It was shown that any scheme with more than one angle unavoidably d emands an additional
+admixture of higher mass state. If we restrict ourselves to only on e additional state G(denoted
+as a glueball without really specifying its nature) then the general m ixing scheme can be
+described in terms of 3 angles. In particular cases the number of an gles can be reduced to two.
+In the paper [2] the analysis of different conventional (and most ph ysically interesting)
+particular cases was performed (including two-angle mixing schemes ) basing on the dispersive
+representation of axial anomaly from one side and charmonium deca ys ratio from the other
+side.
+ae-mail:[email protected]
+be-mail:[email protected]
+ce-mail:[email protected] Will be inserted by the editor
+The main conclusion of the paper [2] is that in all considered cases the only reasonable
+solutions appear at f8=f0≃fπ. The main aim of this work is to check whether this relation
+remains valid in the most general case with some specific constraints imposed.
+This paper is organized as follows. In the Sec. 2 we introduce our not ation and the general
+approach to the mixing. In Sec. 3 we derive the basic equations relyin g on the dispersive
+approach to Abelian axial anomaly of isoscalar non-singlet current J8
+µ5and the charmonium
+radiativedecayratio RJ/Ψ, while in Sec. 4 we performthe numerical analysisofthese equations .
+Finally, in Sec. 5 we present the conclusion.
+2 Mixing scheme
+We start with a ( N-component) vector of physical pseudoscalar fields consisting of the fields of
+the lightest pseudoscalar mesons and other fields:
+/tildewideΦ≡
+π0
+η
+η′
+G
+...
+. (1)
+We are not able to specify the physical nature of the other compon ents with higher masses,
+the lowest of which Gcan be either a glueball or some excited state1. Let us also introduce,
+following [16,17], a set of SU(3) fields ϕ3,ϕ8,ϕ0(Φ1,Φ2,Φ3) and complement them with other
+(sterile) fields gi(Φi,i= 4..N)
+Φ=
+ϕ3
+ϕ8
+ϕ0
+g
+...
+. (2)
+The three upper fields ϕ3,ϕ8,ϕ0are the only ones which define the generalized PCAC
+relationforaxialcurrent Ja
+µ5=qγµγ5λa
+√
+2q(nosummationover acontraryto jandkisassumed):
+∂µJa
+µ5=faδ∆L
+δΦa=FajMjkΦk, a= 3,8,0, j,k= 1..N, (3)
+where∆Lis the mass term in the effective Lagrangian with a non-diagonal mass matrixM(as
+fieldsΦkare not orthogonal to each other):
+∆L=1
+2ΦTMΦ, (4)
+andFis a matrix of decay constants2:
+F≡
+f30 0 0...0
+0f80 0...0
+0 0f00...0
+. (5)
+In order to proceed from initial SU(3) fields Φto physical mass fields /tildewideΦthe unitary (real,
+as the CP-violating effects are negligible) matrix Uis introduced
+1Note, that the mixing with the excited states is usually(e.g . [14,15]) supposed to be suppressed.
+2Note, that matrix of decay constants Fis non-square expressing the fact that generally the number
+ofSU(3) currents is less then the number of all possible states in volved in mixing. The similar situation
+takes place (see e.g. [18]) in one of the extensions of the Sta ndard Model – neutrino mixing scenario
+involving sterile neutrinos.Will be inserted by the editor 3
+/tildewideΦ=UΦ (6)
+that diagonalizes the mass matrix
+UMUT=/tildewiderM≡diag(m2
+π0,m2
+η,m2
+η′,m2
+G,...), (7)
+wheremπ,mη,mη′andmGare the masses of the π,η,η′mesons and glueball state G,
+respectively.
+Simple transformations of Eq.(3) read:
+∂µJµ5=FUT/tildewiderM/tildewideΦ (8)
+This formula is close to those obtained in [16,17] (in the limit of small mix ing). When the decay
+constants are equal, it is reduced to formula (3.40) in [19].
+The matrix elements of ∂µJµ5between vacuum state and physical states |/tildewiderΦk∝angbracketright
+∝angbracketleft0|∂µJa
+µ5|/tildewiderΦk∝angbracketright=Fa
+i(UT/tildewiderM)i
+k (9)
+can be compared to the standard definition of the ”physical” couplin g constants of axial cur-
+rents:
+∝angbracketleft0|Ja
+µ5|/tildewiderΦk∝angbracketright=ifa
+kqµ. (10)
+From (9) and (10) follows the relation
+fa
+k=Fa
+i(UT)i
+k=fa(UT)a
+k. (11)
+This expression (recall, that there is no summation over a) clearly shows that fa
+kare obtained
+bymultiplicationofeachlineof UTbyrespectivecoupling faandformanon-diagonal(contrary
+toF) matrix.
+Taking into account the well-known smallness of π0mixing with the η,η′sector [16,17,20]
+and neglecting all higher contributions we restrict our consideratio n to three physical states
+η,η′,Gand two currents J8
+µ5,J0
+µ5. Then the divergencies of the axial currents (recall, that Gis
+a first mass state heavier than η′):
+/parenleftbigg∂µJ8
+µ5
+∂µJ0
+µ5/parenrightbigg
+=/parenleftbigg
+f80 0
+0f00/parenrightbigg
+UT
+m2
+η0 0
+0m2
+η′0
+0 0m2
+G
+
+η
+η′
+G
+. (12)
+Exploring the mentioned similarity of the meson and lepton mixing, we us e the Euler
+parametrization for the mixing matrix U(we use notation ci≡cosθi,si≡sinθi):
+U=
+c8c3−c0s3s8−c3s8−c8c0s3s3s0
+s3c8+c3c0s8−s3s8+c3c8c0−c3s0
+s8s0 c8s0 c0
+. (13)
+In the following consideration we will need the divergency of the octe t current ∂µJ8
+µ5, so let
+us write it out explicitly:
+∂µJ8
+µ5=f8(m2
+ηη(c8c3−c0s3s8)+m2
+η′η′(s3c8+c3c0s8)+m2
+GG(s8s0)). (14)
+As soon as in the chiral limit J8
+µ5should be conserved, from Eq.(14) follows that coefficients
+of the terms m2
+η′,m2
+Gmust decrease at least as ( mη/mη′,G)2. More specifically, we expect the
+following limits for the terms of Eq.(14):
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
+mG/parenrightbigg2
+. (15)4 Will be inserted by the editor
+3 Abelian axial anomaly and charmonium decays ratio
+In our paper the dispersive form of the anomaly sum rule will be exten sively used, so we remind
+briefly the main points of this approach (see e.g. review [13] for det ails).
+Consider a matrix element of a transition of the axial current to two photons with momenta
+pandp′
+Tµαβ(p,p′) =∝angbracketleftp,p′|Jµ5|0∝angbracketright. (16)
+The general form of Tµαβfor a case p2=p′2can be represented in terms of structure
+functions (form factors):
+Tµαβ(p,p′) =F1(q2)qµǫαβρσpρp′
+σ+
+1
+2F2(q2)[pα
+p2ǫµβρσpρp′
+σ−p′
+β
+p2ǫµαρσpρp′
+σ−ǫµαβσ(p−p′)σ],(17)
+whereq=p+p′. The functions F1(q2),F2(q2) can be described by dispersion relations with
+no subtractions and anomaly condition in QCD results in the sum rule:
+∞/integraldisplay
+0Im F1(q2)dq2= 2αNc/summationdisplay
+e2
+q, (18)
+whereeqare quark electric charges and Ncis the number of colors. This sum rule [21] was
+developed by Jiˇ r´ ı Hoˇ rejˇ s´ ı [22], and later generalized [23]. Not ice that in QCD this equation
+does not have any perturbative corrections [24], and it is expected that it does not have any
+non-perturbative corrections as well due to the ’t Hooft’s consist ency principle [25]. It will be
+important for us that as q2→ ∞the function ImF1(q2) decreases as 1 /q4(see discussion
+in Ref. [2]). Note also that the relation (18) contains only mass-indep endent terms, which is
+especially important for the 8th component of the axial current J8
+µ5containing strange quarks:
+J8
+µ5=1√
+6(¯uγµγ5u+¯dγµγ5d−2¯sγµγ5s). (19)
+The general sum rule (18) takes the form:
+∞/integraldisplay
+0Im F1(q2)dq2=2√
+6α(e2
+u+e2
+d−2e2
+s)Nc=/radicalbigg
+2
+3α , (20)
+whereeu= 2/3,ed=es=−1/3,Nc= 3.
+In order to separate the form factor F1(q2), multiply Tµαβ(p,p′) byqµ/q2. Then, taking the
+imaginary part of F1(q2), using the expression for ∂µJ8
+µ5from Eq.(12) and unitarity we get:
+ImF1(q2) =Im qµ1
+q2∝angbracketleft2γ|J(8)
+µ5|0∝angbracketright=
+−f8
+q2∝angbracketleft2γ|[m2
+ηη(c8c3−c0s3s8)+m2
+η′η′(s3c8+c3c0s8)+m2
+GGs8s0]|0∝angbracketright=
+πf8[Aηδ(q2−m2
+η)(c8c3−c0s3s8)+Aη′δ(q2−m2
+η′)(s3c8+c3c0s8)+AGδ(q2−m2
+G)(s8s0)].
+(21)
+If we employ the sum rule (20), we obtain a simple equation:
+(c8c3−c0s3s8)+β(s3c8+c3c0s8)+γ(s8s0) =ξ, (22)
+whereWill be inserted by the editor 5
+β≡Aη′
+Aη=/radicaligg
+Γη′→2γ
+Γη→2γm3η
+m3
+η′, γ≡AG
+Aη=/radicaligg
+ΓG→2γ
+Γη→2γm3η
+m3
+G, (23)
+ξ≡/radicaligg
+α2m3η
+96π3Γη→2γ1
+f2
+8, Γη→2γ=m3
+η
+64πA2
+η. (24)
+Note that if we include higher resonancesin this equation, they will be suppressed as 1 /m2
+res
+by virtue of the mentioned above asymptotic behavior of F1(q2)∝1/q4. For the last two terms
+in (22) we can specify this constraint as follows:
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilarβ
+γ/parenleftbiggmη′
+mG/parenrightbigg2
+. (25)
+As an additional experimental constraint we use, following [26,27], t he data of the decay
+ratioRJ/Ψ= (Γ(J/Ψ)→η′γ)/(Γ(J/Ψ)→ηγ).
+As it was pointed out in [28], the radiative decays J/Ψ→η(η′)γare dominated by non-
+perturbative gluonic matrix elements, and the ratio of the decay ra tesRJ/Ψ= (Γ(J/Ψ)→
+η′γ)/(Γ(J/Ψ)→ηγ) can be expressed as follows:
+RJ/Ψ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∝angbracketleft0|G/tildewideG|η′∝angbracketright
+∝angbracketleft0|G/tildewideG|η∝angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenleftbiggpη′
+pη/parenrightbigg3
+, (26)
+wherepη(η′)=MJ/Ψ(1−m2
+η(η′)/M2
+J/Ψ)/2. The advantage of this ratio is expected smallness of
+perturbative and non-perturbative corrections.
+The divergencies of singlet and octet components of the axial curr ent in terms of quark
+fields can be written as:
+∂µJ8
+µ5=1√
+6(muuγ5u+mddγ5d−2mssγ5s), (27)
+∂µJ0
+µ5=1√
+3(muuγ5u+mddγ5d+mssγ5s)+1
+2√
+33αs
+4πG/tildewideG. (28)
+Following [26], neglect the contribution of u- and d- quark masses, th en the matrix elements
+of the anomaly term between the vacuum and η,η′states are:
+√
+3αs
+8π∝angbracketleft0|G/tildewideG|η∝angbracketright=∝angbracketleft0|∂µJ(0)
+µ5|η∝angbracketright+1√
+2∝angbracketleft0|∂µJ(8)
+µ5|η∝angbracketright, (29)
+√
+3αs
+8π∝angbracketleft0|G/tildewideG|η′∝angbracketright=∝angbracketleft0|∂µJ(0)
+µ5|η′∝angbracketright+1√
+2∝angbracketleft0|∂µJ(8)
+µ5|η′∝angbracketright. (30)
+Using Eq. (12), (26), (29), (30) we deduce:
+RJ/Ψ=/bracketleftiggf0(−s3s8+c3c8c0)+1√
+2f8(s3c8+c3c0s8)
+f0(−c3s8−c8c0s3)+1√
+2f8(c8c3−c0s3s8)/bracketrightigg2
+×/parenleftbiggmη′
+mη/parenrightbigg4/parenleftbiggpη′
+pη/parenrightbigg3
+.(31)
+4 Analysis
+For further analysis it is convenient to rewrite the equations (22), (31) in terms of angles
+θ1≡θ8+θ3,θ8andθ0:
+1
+2(c1+c2−c0(c2−c1))+β
+2(s1−s2+c0(s1+s2))+γ(s8s0) =ξ. (32)6 Will be inserted by the editor
+RJ/Ψ=/bracketleftiggf0(c1−c2+c0(c1+c2))+1√
+2f8(s1−s2+c0(s1+s2))
+f0(−s1−s2−c0(s1−s2))+1√
+2f8(c1+c2−c0(c2−c1))/bracketrightigg2/parenleftbiggmη′
+mη/parenrightbigg4/parenleftbiggpη′
+pη/parenrightbigg3
+,(33)
+whereθ2≡2θ8−θ1.
+The angles θ1,θ8,θ0have the explicit physical meaning. From the definition (13) of the
+mixing matrix Uone can see that the angle θ1describes the overlap in the η−η′system with
+an accuracy ∼θ2
+0/2 and coincides with their mixing angle as θ0→0. At the same time θ0is
+responsible for the glueball admixture to η−η′system, and s8s0describes the contribution of
+the glueball state Gto the octet component of axial current ∂J8
+µ5only.
+In the further analysis we will use the following assumptions:
+I) As we discussed in Sec. 2, the last term in (14) should be suppress ed as (mη/mG)2. So
+we impose the following constraint:
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
+mG/parenrightbigg2
+. (34)
+II) In sec 3 we found another constraint, which follows from the as ymptotic behavior of
+ImF1(see 25):
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilarβ
+γ/parenleftbiggmη′
+mG/parenrightbigg2
+. (35)
+III) In our numerical analysis we suppose that γcannot exceed 1 (i.e. ΓG→2γ/m3
+G/lessorsimilar
+Γη→2γ/m3
+η). This restriction corresponds to the assumption that 2-photon decay widths of
+pseudoscalar mesons grow like the third power of their masses, or in other words, the glueball
+coupling to quarks is of the same order as for the meson octet stat es.
+IV) We accept that the decay constants obey the relation f8/greaterorsimilarf0/greaterorsimilarfπ(for various kinds
+of justification see, e.g., [3,9]).
+For the purposes of numerical analysis, the values of RJ/Ψ(RJ/Ψ= 4.8±0.6), masses
+and two-photon decay widths of η,η′mesons are taken from PDG [29]. Using the values
+mη,mη′,Γη→2γ,Γη′→2γ, we see that the relation for the constraint (34) is more strict tha n the
+constraint (35). Supposing the minimal mass of the glueball to be of ordermG≃3mη≃1.5
+GeV, we get the estimation:
+|s8s0|/|s3c8+c3c0s8|/lessorsimilar0.1. (36)
+On Fig. 1 the plots of the equations (32) and (33) in the parameter s pace (θ8,θ1) are shown
+for different values of decay constants f8,f0and mixing angle θ0. The dashed curves denote
+experimental uncertainties. The intersection points of the curve s represent the solutions of both
+equations (32),(33). The filled area indicates the region, where the constraint (36) is valid. The
+plotted range of angle θ1is limited to the physically interesting region, where the solution for
+relatively small angles θ0exists. Let us note for completeness, that there is another solut ion for
+θ1∼90◦,θ0/greaterorsimilar50◦which does not seem to have a physical sense.
+The numerical analysis shows, that the solution of the equations (3 2) and (33) satisfying
+the mentioned above constraint is possible only for rather small mixin g angleθ0and for decay
+constants f8,f0closetoeachotherandcloseto fπ:forf8/fπ=f0/fπ= 1.0the possiblerangeof
+mixing angle θ0isθ0= (0÷25)◦(see Fig. 1(a)-1(c) for demonstration), for f8/fπ=f0/fπ= 1.1
+the possible range of mixing angle θ0isθ0= (0÷20)◦.
+There is no solutions for decay constant values f8/fπ= 1.1,f0/fπ= 1.0 for any θ0(see Fig.
+1(d)-1(f) for demonstration), and for any f0/lessorsimilarf8in case of f8/fπ≥1.2. The obtained results
+are quite stable: even if we relax the constraint (36) making its r.h.s. several times larger, all
+the conclusions are preserved.
+Note finally, that this result is in contradiction with the prediction for the decay constant
+f8/fπ= 1.34 [10] obtained in the Large NcChPT.Will be inserted by the editor 7
+/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(a) (f8,f0) = (1.0,1.0)fπ,θ0=
+0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(b) (f8,f0) = (1.0,1.0)fπ,θ0=
+5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(c) (f8,f0) = (1.0,1.0)fπ,θ0=
+30◦
+/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(d) (f8,f0) = (1.1,1.0)fπ,θ0=
+0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(e) (f8,f0) = (1.1,1.0)fπ,θ0=
+5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(f) (f8,f0) = (1.1,1.0)fπ,θ0=
+30◦
+Fig. 1.The solutions of the Eq. (32) (thin curves, blue online) and ( 33)(thick curves, red online) with
+the experimental uncertainties (dashed curves) for differe nt values of the parameters f8,f0andθ0. The
+shaded area indicates the region, where the relation (36) is valid.
+5 Conclusion
+In this paper we studied what can be learnt about the mixing in the pse udoscalar sector from
+the dispersive approach to axial anomaly.
+Our analysis shows that the equal values of axial current coupling c onstants f8andf0are
+favorablewhich may be considered as a manifestation of SU(3) and chiral symmetry. Moreover,
+with a less definiteness the relation fπ≈f8≈f0[2] is also supported.
+Theanalysisdemands f8<1.2fπwhich deviatesat 10%levelfrom theresultsofcalculations
+within the chiral perturbation theory ( f8= 1.34fπ) [10].
+Thevalueofthe mixingangle θ0,whichisresponsibleforthe glueballadmixturetothe η−η′,
+is limited to θ0<25◦for (f8,f0) = (1.0,1.0)fπand toθ0<20◦for (f8,f0) = (1.0,1.0)fπ.
+The improvement of the experimental data of RJ/Ψcan significantly limit the constraints
+for the parameters θ0,θ8andf8,f0.
+We thank J. Hoˇ rejˇ s´ ı, B. L. Ioffe and M. A. Ivanov for useful co mments and discussions.
+Y. K. and O. T. gratefully acknowledge the organizers of the works hop for hospitality and
+support. This work was supported in part by RFBR (Grants 09-02- 00732,09-02-01149),by the
+funds from EC to the project ”Study of the Strong Interacting M atter” under contract N0.
+R113-CT-2004-506078 and by CRDF Project RUP2-2961-MO-09.
+References
+1. Y. N. Klopot, A. G. Oganesian, O. V. Teryaev, arXiv:0810.1 217 [hep-ph] (2008).8 Will be inserted by the editor
+2. Y. N. Klopot, A. G. Oganesian, O. V. Teryaev, arXiv:0911.0 180 [hep-ph] (2009).
+3. T. Feldmann, P. Kroll, B. Stech, Phys. Lett. B449, 339 (1999).
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+5. F. De Fazio, M. R. Pennington, JHEP07, 051 (2000).
+6. P. Kroll, Mod. Phys. Lett. A20, 2667 (2005).
+7. R. Escribano, J.-M. Frere, JHEP06, 029 (2005).
+8. V. Mathieu, V. Vento arXiv:0910.0212 (2009).
+9. H. Leutwyler, Nucl. Phys. Proc. Suppl. 64, 223 (1998).
+10. R. Kaiser, H. Leutwyler, hep-ph/9806336 (1998).
+11. R. Kaiser, H. Leutwyler, Eur. Phys. J. C17, 623 (2000).
+12. A. D. Dolgov, V. I. Zakharov, Nucl. Phys. B27, 525 (1971).
+13. B. L. Ioffe, Int. J. Mod. Phys. A21, 6249 (2006).
+14. R. Escribano, arXiv:0712.1814 [hep-ph] (2007).
+15. H.-Y. Cheng, H.-n. Li, K.-F. Liu, Phys. Rev. D79, 014024 (2009).
+16. B. L. Ioffe, Yad. Fiz. 29, 1611 (1979).
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+18. S. M. Bilenky, C. Giunti, W. Grimus, Prog. Part. Nucl. Phys. 43, 1 (1999).
+19. D. Diakonov, M. V. Polyakov, C. Weiss, Nucl. Phys. B461, 539 (1996).
+20. B. L. Ioffe, A. G. Oganesian, Phys. Lett. B647, 389 (2007).
+21. Y. Frishman, A. Schwimmer, T. Banks, S. Yankielowicz, Nucl. Phys. B177, 157 (1981).
+22. J. Horejsi, Phys. Rev. D32, 1029 (1985).
+23. O. L. Veretin, O. V. Teryaev, Phys. Atom. Nucl. 58, 2150 (1995).
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+25. J. Horejsi, O. Teryaev, Z. Phys. C65, 691 (1995).
+26. R. Akhoury, J. M. Frere, Phys. Lett. B220, 258 (1989).
+27. P. Ball, J. M. Frere, M. Tytgat, Phys. Lett. B365, 367 (1996).
+28. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, V. I. Zakha rov,Nucl. Phys. B165, 55 (1980).
+29. C. Amsler, et al.,Phys. Lett. B667, 1 (2008).

1001.0046.txt

@@ -0,0 +1,262 @@
+arXiv:1001.0046v1  [math.CO]  31 Dec 2009THE CAUCHY-SCHWARZ INEQUALITY IN CAYLEY GRAPH
+AND TOURNAMENT STRUCTURES ON FINITE FIELDS
+STEPHAN FOLDES AND L ´ASZL´O MAJOR
+Abstract. The Cayley graph construction provides a natural grid struc ture
+on a finite vector space over a field of prime or prime square car dinality, where
+the characteristic is congruent to 3 modulo 4, in addition to the quadratic
+residue tournament structure on the prime subfield. Distanc e from the null
+vector in the grid graph defines a Manhattan norm. The Hermiti an inner prod-
+uct on these spaces over finite fields behaves in some respects similarly to the
+real and complex case. An analogue of the Cauchy-Schwarz ine quality is valid
+with respect to the Manhattan norm. With respect to the non-t ransitive order
+provided by the quadratic residue tournament, an analogue o f the Cauchy-
+Schwarz inequality holds in arbitrarily large neighborhoo ds of the null vector,
+when the characteristic is an appropriate large prime.
+1.Manhattan norms and grid graphs
+We consider the finite fields FpandFp2of prime and prime square cardinality,
+wherep≡3 mod 4. The field Fp2has a natural graph structure with the field
+elements as vertices, two distinct vertices u,zbeing adjacent if ( z−u)4= 1. The
+subfieldFpofFp2then induces a subgraph in which xandyare adjacent if and only
+if (y−x)2= 1.The graph Fp2is isomorphic to the Cartesian square C2
+p=Cp/squareCp,
+whereCpis ap-cycle and within Fp2the induced subgraph Fpis itself a p-cycle.
+Clearly the graph Fp2is not planar, but can be drawn as a grid on the torus.
+For any connected graph whose vertex set is a group, the distanc e of any vertex
+zfrom the identity element of the group is called the normofz, denoted N(z).
+In general, distances and norms measured in connected subgraph s induced by sub-
+groups can be larger than distances and norms measured with refe rence to the
+whole graph. However, with respect to the distance-preserving s ubgraph induced
+byFpinFp2, the norm of any z∈Fpis the same as its norm with respect to the
+whole graph Fp2: this is simply the length of the shortest path from 0 to zin the
+cycle induced by Fp.
+Forq=porq=p2, then-dimensional vector space Fn
+qis also endowed with
+the Cartesian product graph structure Fq/square···/squareFqisomorphic to Cn
+porC2n
+p. The
+norm of a vector v= (v1,...,v n) inFn
+qis then equal to the sum N(v1)+···+N(vn)
+and we also write N(v) for this vector norm.
+The Gaussian integers Z[i] also constitute a graph in which uandzare adjacent
+if and only if ( z−u)4= 1.
+Date: Dec 24, 2009.
+1991Mathematics Subject Classification. Primary 05C12, 05C20, 05C25; Secondary 06F99,
+11T99.
+Key words and phrases. Cauchy-Schwarz inequality, triangle inequality, submult iplicativity,
+finite field, quadratic field extension, quadratic residue to urnament, grid graph, Manhattan dis-
+tance, discrete norm, Gaussian integers, graph product, gr aph quotient, Cayley graph.
+12 STEPHAN FOLDES AND L ´ASZL´O MAJOR
+It iseasytoseethatthenorminthis infinite Manhattan grid satisfiesthetriangle
+and submultiplicative inequalities
+N(u+z)≤N(u)+N(z)
+N(uz)≤N(u)N(z)
+To emphasize that the norms on Fp2,Fn
+p2andZ[i] are understood with reference
+to the specific grid graphs defined above, we call these norms Manhattan norms .
+Throughout this paper we think of Fp2as the ring quotient Z[i]/(p).
+2.Graph quotients and Cayley graphs
+Given a graph G(undirected, with possible loops) on vertex set Vand an equi-
+valence relation ≡onV, thequotient graph G/≡is defined as follows: the vertices
+ofG/≡are the equivalence classes of ≡, and classes A,Bare adjacent if for some
+a∈A,b∈B, the elements a,bare adjacent in G. Note that the distance of Ato
+Bin the quotient graph is at most equal to, but possibly less than the m inimum of
+the distances atobfor alla∈A,b∈B. Note also that G/≡can have loops even
+ifGhas not.
+Given a group Gwith identity element eand a set Γ of group elements that
+generates G, the(left) Cayley graph C(G,Γ) ofGwith respect to Γ has vertex set
+G, elements a,b∈Gbeing considered adjacent if ab−1orba−1belongs to Γ. For
+each congruence ≡of the group G, corresponding to some normal subgroup H, Γ
+yields a generating set Γ ≡ofG/≡consisting with those classes of ≡that intersect
+Γ. The graph quotient of C(G,Γ) by the equivalence ≡coincides with the Cayley
+graph of the quotient graph G/≡with respect to Γ ≡. ForR⊆Ginducing a
+connected subgraph [ R] inC(G,Γ), denote by dR(x,y) the distance function of the
+subgraph [ R]. Denoting by xHtheH-coset of any x∈G, this relates to norms in
+C(G,Γ) andC(G,Γ)/≡as follows: for all x∈R,
+dR(x,e)≥N(x)≥N(xH)
+Both inequalities can be strict. However, we have:
+Cayley Graph Quotient Lemma. Let a group Gwith identity ebe generated
+byΓ⊆G, and consider any normal subgroup Hwith corresponding congruence ≡.
+There is a set R⊆Ghaving exactly one element in common with each congruence
+class modulo H, and such that for every x∈R
+dR(x,e) =N(x) =N(xH)
+Proof.We can define the unique (representative) element r(A)∈R∩Afor each
+cosetAby induction on the distance d(H,A) ofAfromHinC(G,Γ)/≡. Let
+r(H) =e. Assuming r(A) defined for all Awithd(H,A)≤m, let a coset Bhave
+distance m+1 from H. Choose any coset Aadjacent to Bwithd(H,A) =mand
+elements a∈A,b∈Bthat are adjacent in C(G,Γ). Letr(B) =ba−1r(A)./square
+We can apply the above lemma in the case where G=Z[i], Γ ={1,i}and
+H=pZ[i] ={pa+pbi:a,b∈Z}foraprimeinteger p≡3 mod 4. Now C(G,Γ)and
+C(G,Γ)/≡are the Manhattan grid graphs on Z[i] andZ[i]/H=Fp2, respectively.
+Referringtothe set Rofrepresentativesinthe lemma, forany H-cosetsX,Yletx,y
+be the unique elements in X∩R,Y∩R. Asxy∈XY, we have N(XY)≤N(xy).
+By the submultiplicative inequality in Z[i] we have N(xy)≤N(x)N(y). Using the3
+lemmawehave N(x)N(y) =N(X)N(Y). Thisyieldsasubmultiplicativeinequality
+inFp2and a similar reasoning on the coset X+Yyields a triangle inequality:
+Triangle and Submultiplicative Inequalities in Fp2.For allu,zinFp2
+N(u+z)≤N(u)+N(z)
+N(uz)≤N(u)N(z)
+/square
+This indicates that Manhattan distance provides a well-behaved not ion of neigh-
+borhood of 0 in the finite fields Fp2.
+3.Squares in Fpand non-transitive order
+For each prime p≡3 mod 4 the quadratic residue tournament onFpis the
+directed graph with vertex set Fpin which there is an arrowfrom vertex xto vertex
+yify−xis a non-zero square in Fp, in which case we write x <py. We write x≤py
+ifx <pyorx=y. The relation ≤pis reflexive, anti-symmetric but not transitive,
+and for every x/ne}ationslash=yexactly one of x≤pyory≤pxholds. Using Dirichlet’s
+theorem on primes in arithmetic progressions, Kustaanheimo showe d [4] that for
+every positive integer k, there is a prime p≡3 mod 4, such that ≤pis a transitive
+(and linear) order relation on {0,1,...,k} ⊆Fp, that is, all positive integers up to
+kare quadratic residues mod p. Obviously kcannot exceed ( p−1)/2. Implications
+of [4] and related questions were investigated by J¨ arnefelt, Kust aanheimo, Quist
+[3, 5], in particular with a view to discrete models in physics, also in subse quent
+application-oriented work between the 1950’s (Coish [1]) and the 198 0’s (Nambu
+[6]). For further references see [2]. In particular [4] implies that for every positive
+integerk, there is a prime p≡3 mod 4, such that all z∈Fp2withN(z)≤kare
+squares in Fp2. (Note that all elements of the prime subfield Fpare squares in Fp2.)
+To emphasise the analogy of the relation ≤pwith the ordinary inequality relation
+≤among numbers, we say that a non-zero z∈Fp2ispositiveifz∈Fpand 0≤pz.
+4.Inner products compared in non-transitive order
+The only non-trivial automorphism of the field Fp2associates to each z∈Fp2
+itsconjugate z. Theinner product v·wof vectors v= (v1,...,v n) andw=
+(w1,...,w n) inFn
+p2is defined as the scalar v1w1+···+vnwn∈Fp2. This inner
+product is left and right distributive over vector addition, satisfies v·w=w·v,
+c(v·w) = (cv)·w=v·(cw) for allc∈Fp2. However, while v·vbelongs to the
+prime subfield Fp,v·vis not necessarily positive, and can be 0 even if v/ne}ationslash=0. Still,
+a conditional version of positive definiteness holds locally:
+Conditional Positive Definiteness. For every k≥1there is a prime p≡3
+mod 4, such that for all n≥1and for all vectors v∈Fn
+p2of Manhattan norm
+N(v)≤k,we have 0≤pv·vwith equality if and only if v=0.
+Proof.By Kustaanheimo’s result in [4] there is a prime integer p≡3 mod 4 such
+that 0,1,...,2k3are all quadratic residues mod p. Forv= (v1,...,v n) inFn
+p2, let
+vj=aj+bji, wherei2=−1. IfN(v)≤kthen for all j,N(aj)≤kandN(bj)≤k,
+vjvj=a2
+j+b2
+jbelongs to the set of squares {0,...,2k2}. Sincevjcan be non-zero
+for at most kindices 1 ≤j≤nonly, the sum of the corresponding terms a2
+j+b2
+j
+belongs to the set of squares {0,1,...,2k3}. /square4 STEPHAN FOLDES AND L ´ASZL´O MAJOR
+Note that for all vectors v,w∈Fn
+p2
+(v·w)(w·v) = (v·w)(v·w)∈Fpand
+(v·v)(w·w)∈Fp
+Ifvandwareproportional , i.e. if there exists a scalar cinFp2such that v=cw
+orw=cv, then the above two products are equal. Generally, they are relat ed in
+the quadratic residue tournament of Fpas follows.
+Cauchy-Schwarz Inequality for Quadratic Residue Tourname nts.For eve-
+ryk≥1there is a prime p≡3 mod 4 , such that for all n≥1and for all vectors
+v,w∈Fn
+p2of Manhattan norm at most k,
+(v·w)(w·v)≤p(v·v)(w·w)
+Proof.Forn= 1 the inequality holds trivially as the two sides are equal. Assume
+n≥2,v= (v1,...,v n),w= (w1,...,w n). Forall1 ≤i≤n,N(vi)≤k,N(wi)≤k.
+By Kustaanheimo’s result [4] there is a prime p≡3 mod 4 such that all positive
+integers up to 4 k6are quadratic residues modulo p. For each of the/parenleftbign
+2/parenrightbig
+pairs
+{i,j} ⊆ {1,...,n},i/ne}ationslash=j, by the triangle and submultiplicative inequalities in Fp2
+N[(viwj−vjwi)(viwj−vjwj)]≤(k2+k2)2= 4k4
+Thus the element
+(viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj) = (viwj−vjwi)(viwj−vjwj)
+is a square of Manhattan norm at most 4 k4inFp, and it is non-zero for at most/parenleftbigk
+2/parenrightbig
+≤k2pairs{i,j}. Summing over all pairs {i,j}, all but at most/parenleftbigk
+2/parenrightbig
+≤k2terms
+vanish in the sum/summationdisplay
+[(viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj)]
+which therefore has Manhattan norm at most 4 k6and it must also be a square in
+Fp. But this sum is equal to the difference of products
+n/summationdisplay
+i=1vivin/summationdisplay
+j=1wjwj−n/summationdisplay
+i=1viwin/summationdisplay
+j=1vjwj= (v·v)(w·w)−(v·w)(w·v)
+which is consequently a square in Fp. /square
+Remark. From the proof it is clear that, in analogy with the classical Cauchy-
+Schwarz inequality, for vectors v,wof norm not exceeding kinFn
+p2, wherepis
+related to kas stipulated above, the Cauchy-Schwarz inequality with respect t o≤p
+holds with equality if and only if viwj−vjwi= 0 for all i,j, i.e. if and only if v,w
+are proportional.
+We note that the inequality established above is conditional, it holds on ly in a
+specified Manhattan neighborhood of the null vector. Every non- zero element of
+Fpcan be written as a sum of two squares, in particular there are a,b∈Fp, such
+thata2+b2=−1. Forz=a+biwe have zz=−1. As soon as n≥2, inFn
+p2let
+v= (a,b,0,...,0) and w= (bz,−az,0,...,0)
+The inequality ( v·w)(w·v)≤p(v·v)(w·w) fails because the left-hand side is 0
+and the right-hand side is −1. In fact if n≥3, the inequality can be invalidated
+with vectors v,winFn
+pas follows. Taking again a,b∈Fpwitha2+b2=−1, let
+v= (1,a,b,0,...,0) and w= (1,0,0,0,...,0)5
+However,the Cauchy-Schwarzinequalityholdsunconditionallyinthe 2-dimensional
+case for vectors with components in Fp:
+Special case of F2
+p.Letpbe a prime congruent 3modulo4. For all vectors v,w
+inF2
+p
+(v·w)(w·v)≤p(v·v)(w·w)
+Proof.Nowtheconjugationappearinginthe innerproductsisthe identity. Written
+in components,
+(v·v)(w·w)−(v·w)(w·v) = (v2
+1+v2
+2)(w2
+1+w2
+2)−(v1w1+v2w2)2=
+=v2
+1w2
+2+v2
+2w2
+1−2v1w1v2w2= (v1w2−v2w1)2
+/square
+5.Manhattan norm of inner product
+The Manhattan norm can be seen to be submultiplicative not only on th e ring
+Z[i] and its quotient field Fp2, but on all vector spaces Fn
+p2, with respect to the
+inner product:
+Cauchy-Schwarz Inequality for Manhattan Norm on Fn
+p2.Consider any
+primep≡3 mod 4 and letn≥1. For all v,w∈Fn
+p2
+N(v·w)≤N(v)N(w)
+Proof.Letv= (v1,...,v n),w= (w1,...,w n)∈Fn
+p2. Thenv·w=/summationtextvjwj. Clearly
+N(z) =N(z) for any z∈Fp2. By the triangle and submultiplicative inequalities in
+Fp2we have
+N(v·w) =N/parenleftbig/summationtextvjwj/parenrightbig
+≤/summationtextN/parenleftbig
+vjwj/parenrightbig
+≤/summationtextN(vj)N(wj)≤
+≤/summationtextN(vj)/summationtextN(wj) =N(v)N(w)
+/square
+Remark. The inequality N(v·w)≤N(v)N(w) is easily interpreted and continues
+to hold for v,win the module ( Z[i]/mZ[i])nfor any positive integer m. As soon as
+mis composite, or a prime not congruent to 3 modulo 4, the ring Z[i]/mZ[i] fails
+to be an integral domain.
+References
+[1] H.R. Coish, Elementary particles in a finite world geomet ry, Phys. Rev. 114 - 1 (1959) 383-388
+[2] S. Foldes, The Lorentz group and its finite field analogues : local isomorphism and approxima-
+tion, J. Math. Phys. 49, 093512 (2008)
+[3] G. J¨ arnefelt, P. Kustaanheimo, An observation on finite geometries, in Proc. Skandinaviske
+Matematikerkongress i Trondheim 1949, 166-182
+[4] P. Kustaanheimo, A note on a finite approximation of the eu clidean plane geometry, Comment.
+Phys.-Math. Soc. Sc. Fenn. XV. 19 (1950) 1-11
+[5] P. Kustaanheimo, B. Qvist, On differentiation in Galois fi elds. Ann. Acad. Sci. Fennicae. Ser.
+A. I. Math.-Phys. 1952, (1952). no. 137, 12 pp.
+[6] Y. Nambu, Field theory of Galois fields, in Field Theory an d Quantum Statistics, eds. J.A.
+Batalin et.al., Institute of Physics Publishing 1987, pp. 6 25-6366 STEPHAN FOLDES AND L ´ASZL´O MAJOR
+Stephan Foldes
+Institute of Mathematics,
+Tampere University of Technology,
+PL 553, 33101 Tampere, Finland
+E-mail address :[email protected]
+L´aszl´o Major
+Institute of Mathematics,
+Tampere University of Technology,
+PL 553, 33101 Tampere, Finland
+E-mail address :[email protected]

1001.0047.txt

@@ -0,0 +1,953 @@
+Coulomb interaction and transient charging of excited states in open nanosystems
+Valeriu Moldoveanu,1Andrei Manolescu,2Chi-Shung Tang,3and Vidar Gudmundsson4
+1National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Magurele, Romania
+2Reykjavik University, School of Science and Engineering, Kringlan 1, IS-103 Reykjavik, Iceland
+3Department of Mechanical Engineering, National United University, Lienda, Miaoli 36003, Taiwan
+4Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
+We obtain and analyze the eect of electron-electron Coulomb interaction on the time dependent
+current 
+the contact is gradually switched on in time and we calculate the time dependent reduced density
+operator of the sample using the generalized master equation. The many-electron states (MES) of
+the isolated sample are derived with the exact diagonalization method. The chemical potentials of
+the two leads create a bias window which determines which MES are relevant to the charging and
+discharging of the sample and to the currents, during the transient or steady states. We discuss the
+contribution of the MES with xed number of electrons Nand we nd that in the transient regime
+there are excited states more active than the ground state even for N= 1. This is a dynamical
+signature of the Coulomb blockade phenomenon. We discuss numerical results for three sample
+models: short 1D chain, 2D lattice, and 2D parabolic quantum wire.
+PACS numbers: 73.23.Hk, 85.35.Ds, 85.35.Be, 73.21.La
+I. INTRODUCTION
+Due to the increasing interest in ultra-fast electron
+dynamics considerable progress occurred recently in the
+theoretical description of time dependent mesoscopic
+transport. New methods and numerical implementations
+are rapidly evolving. Transient currents in open nanos-
+tructures are studied with Green-Keldysh formalism,1,2,3
+scattering theory,4and quantum master equation.5,6,7,8
+Most of the results were obtained for noninteracting elec-
+trons due to the well known computational diculties to
+include time-dependent Coulomb eects.
+It is nevertheless clear that the electron-electron inter-
+action is important in such problems. An eort to incor-
+porate it has been recently done by Kurth et al.9followed
+by My oh anen et al.10who have described correlated time-
+dependent transport in a short 1D chain dened by a
+lattice Hamiltonian. The 1D sample was connected to
+external leads and the current was driven by a time-
+dependent bias. Those authors used a method based
+on the Kadano-Baym equation for the non-equilibrium
+Green's function combined with the time-dependent den-
+sity functional theory to include the Coulomb interac-
+tion in the sample. Once the Green's functions were
+calculated total average quantities of interest could be
+obtained, like charge density or current, both in the tran-
+sitory and in the steady state. However this method does
+not say much about the dynamics of specic internal
+states of the sample system. In view of the spectroscopy
+of excited states11it is important to have a theoretical
+tool for understanding separately the charging and re-
+laxation of the ground states and excited states in meso-
+scopic systems in time-dependent conditions.
+Our alternative is to use the statistical, or density op-
+erator. The complete information about the time evo-
+lution of each quantum state of the sample is captured
+in the reduced density operator (RDO), which is the so-lution of the generalized master equation (GME). Once
+the RDO is dened in the Fock space the inclusion of
+the Coulomb interaction becomes a known computational
+problem: obtaining the many-electron states (MES) of
+the sample. The RDO matrix is then calculated in the
+basis of the interacting MES.
+Let us enumerate some of the previous theoretical
+schemes to treat transport and electron-electron inter-
+action with the master equation. One of the rst at-
+tempts to derive a master equation for an interacting sys-
+tem with time-dependent perturbations belongs to Lan-
+greth and Nordlander for the Anderson model.12Gurvitz
+and Prager started from the time-dependent Shr odinger
+equation for the MES wave functions and ended up with
+Bloch-like rate equations for the density matrix of a quan-
+tum dot.13The electronic currents were calculated in the
+steady state and it was shown that the Coulomb interac-
+tion renormalizes the tunneling rates between the leads
+and the system. In the same context K onig et al.14de-
+veloped a powerful diagrammatic technique by expand-
+ing the RDO of a mesoscopic system in powers of the
+tunneling Hamiltonian. The time-dependence of the sta-
+tistical operator of the coupled and interacting system
+implies a quantum master equation for the so called pop-
+ulations. In this method the Coulomb interactions are
+treated exactly, which makes it appealing for studying
+various correlation eects like cotunneling.15The con-
+nection between the real-time diagrammatic approach of
+K onig et al.14and the Nakajima-Zwanzig approach16,17
+to the generalized master equation (GME) approach was
+made transparent by Timm.18
+More recently Li and Yan19combined the n-resolved
+master equation and the time dependent density-
+functional method to write down a Kohn-Sham master
+equation for the reduced single-particle density matrix.
+Also, Esposito and Galperin,20using the equation of mo-
+tion for the Hubbard operators, have obtained a many-arXiv:1001.0047v1  [cond-mat.mes-hall]  30 Dec 20092
+body description of quantum transport in an open sys-
+tem and established a connection between the GME and
+non-equilibrium Greeen's functions. They studied simple
+systems in the steady state regime: a resonant level cou-
+pled to a a single vibration mode, an interacting dot with
+two spins, and a two-level bridge. Another recent work
+by Darau et al.21implemented the GME for a benzene
+single-electron transistor and used exact MES to compute
+steady state currents within the Markov approximation.21
+The stability diagram and the conductance peaks were
+obtained and a current blocking due to interferences be-
+tween degenerated orbitals was noticed.
+In our previous papers7,8we considered the GME
+method for the RDO of independent electrons in the Fock
+space. We discussed the transient transport through
+quantum dots and quantum wires. The contact between
+the leads and the sample was switched on at a certain ini-
+tial moment t0. We discussed extensively the occupation
+of the states within the bias window and the geometrical
+eects on the transient currents. We described the cou-
+pling between the sample and the leads via a tunneling
+Hamiltonian in which we took into account the spatial
+extension of the wave functions of both subsystems in
+the contact region.
+In spite of earlier or more recent attempts a complete
+description of the Coulomb eects in the time-dependent
+transport is still missing, especially in sample models
+larger than a few sites. In the present work we com-
+bine the GME method with the Coulomb interaction in
+the sample and we analyze the dynamics of the electrons
+starting with the moment when the leads are coupled to
+the sample until a steady state is reached. The Coulomb
+interaction is included in the Hamiltonian of the isolated
+sample and the interacting MES are calculated with the
+exact diagonalization method. This means the Coulomb
+interaction is fully included with no mean eld assump-
+tion or density-functional model. The number of single-
+electron states (SES) used to dene the matrix elements
+of the Hamiltonian of interacting electrons is suciently
+large such that the MES of interest are convergent. Due
+to the nite bias window only a limited number of MES
+participate to the charge transport through the sample,
+i. e.only those energetically compatible with the elec-
+trons in the leads. Hence the MES of interest are selected
+by the chemical potentials in the leads. We calculate the
+RDO matrix elements in the subspace of these MES using
+the GME. The electron-electron interaction in the leads
+is neglected.
+It is well known that the Fock space increases expo-
+nentially with the number of SES. In addition the time
+dependent numerical solution of the GME is also com-
+putational expensive. So at this stage we are limited to
+describe only few electrons in the system: up to ve in a
+small system, but only up to three in a larger one.
+The paper is organized as follows. In Section 2 we
+brie
+interaction, and the selection of the MES. Next, in Sec-
+tion 3, we show results for three models: a short 1Dchain, a 2D lattice of 12 10 sites, and a nite quantum
+wire with parabolic lateral connement. Conclusions and
+discussions are presented in Section 4.
+II. GME METHOD AND COULOMB
+INTERACTION
+In this section we summarize the main lines of our
+method. The equations apply both to the lattice and
+continuous models. The time-dependent transport prob-
+lem is considered within the partitioning approach which
+is known both from the pioneering work of Caroli22and
+from the derivation of the GME. Prior to an initial time
+t0the left lead (L) having a \source" role, and the right
+lead (R) having a \drain" role, are not connected to the
+sample and therefore can be characterized by equilibrium
+states with chemical potentials LandRrespectively.
+Our aim is to compute the time dependent currents 
+ing through the sample and leads starting at moment t0,
+when the three subsystems are connected, until a station-
+ary state is reached.
+The generic Hamiltonian of the total system consisting
+of the sample plus the leads is:
+H(t) =HL+HR+HS+HT(t): (1)
+Hlwithl=L;R are the Hamiltonians of the leads. We
+denote by "qland qlthe single-particle energies and
+wave functions respectively, for each lead. Using the cre-
+ation and annihilation operators associated to the single-
+particle states, cy
+qlandcql, we can write
+Hl=Z
+dq"qlcy
+qlcql: (2)
+HSis the Hamiltonian of the sample. In the absence
+of the interaction the SES have discrete energies denoted
+asEnand corresponding one-body wave functions n(r).
+Using now the creation and annihilation operators for the
+sample SES, dy
+nanddn, we can write
+HS=X
+nEndy
+ndn+1
+2X
+nm
+n0m0Vnm;n0m0dy
+ndy
+mdm0dn0:(3)
+The second term in Eq. (3) is the Coulomb interaction.
+In the SES basis the two-body matrix elements are given
+by:
+Vnm;n0m0=Z
+drdr0
+n(r)
+m(r0)u(rr0)n0(r)m0(r0);
+(4)
+whereu(rr0) is the Coulomb potential.
+The third term of Eq. (1) is the so-called tunneling
+Hamiltonian describing the transfer of particles between
+the leads and the sample:
+HT(t) =X
+l=L;RX
+nZ
+dql(t)(Tl
+qncy
+qldn+h:c:):(5)3
+HTcontains two important elements: (1) The time de-
+pendent switching functions l(t) which open the contact
+between the leads and the sample; these functions mimic
+the presence of a time dependent potential barrier. (2)
+The coupling Tl
+qnbetween a state with momentum qof
+the leadland the state nof the isolated sample, with
+wave function n. The coupling coecients Tl
+qndepend
+on the energies of the coupled states and, maybe more
+important, on the amplitude of the wave functions in the
+contact region. As we have shown in our previous work7,8
+this construction allows us to capture geometrical eects
+in the electronic transfer. A precise denition of the cou-
+pling coecients is however model specic, and will be
+mentioned in the next section.
+The evolution of our system is completely determined
+by the statistical operator W(t) associated to the total
+Hamiltonian H(t) dened in Eq.(1). W(t) is the solution
+of the quantum Liouville equation with a known initial
+value, prior to the coupling of the sample and leads:
+i~_W(t) = [H(t);W(t)]; W (tt0) =LRS;(6)
+The isolated leads are described by equilibrium distribu-
+tions,
+l=e(HllNl)
+Trlfe(HllNl)g; l=L;R; (7)
+and the isolated sample by the density operator S. Af-
+ter the coupling moment the dynamics of the sample is
+conveniently described by the RDO which is dened by
+averaging the total statistical operator over those degrees
+of freedom belonging to the leads:
+(t) = TrLTrRW(t); (t0) =S: (8)
+In the absence of the electron-electron interaction the
+MES eigenvectors of HSare bit-strings of the form ji=
+ji
+1;i
+2;::;i
+n:::i, wherei
+n= 0;1 is the occupation number
+of then-th SES. The set fgis a basis in the Fock space
+of the isolated sample and the RDO can be seen as a
+matrix in this basis. From Eqs. (6)-(8) we obtain in the
+lowest (2-nd) order in the coupling parameters Tl
+qnthe
+GME (see Ref. 7 for details):
+_(t) =i
+~[HS;(t)]
+1
+~2X
+l=L;RZ
+dql(t)([Tql;
+ql(t)] +h:c:);(9)
+where the coupling operator Tqlhas matrix elements
+(Tql)=X
+nTl
+qnhjdyji: (10)
+The operators 
+ qland qlare dened as
+
+ql(t) =eitHSZt
+t0dsl(s)ql(s)ei(st)"qleitHS;
+ql(s) =eisHS
+Ty
+ql(s)(1fl)(s)Ty
+qlfl
+eisHSandflis the Fermi function of the lead l.
+In the presence of the electron-electron interaction in
+the sample the MES which are eigenstates of HSare lin-
+ear combinations of bit-strings: HSj) =Ej), where
+j) =P
+Cji,Cbeing the mixing coecients
+which can be found together with the energies Eby
+diagonalizing HS. (To distinguish better between the
+noninteracting and the interacting MES we use the right
+angular bracket for the former and the regular curved
+bracket for the later.) Using now the set fgas a basis,
+i. e.theinteracting MES, the GME has the same form
+as Eq. (9), where the matrix elements of all operators
+are now dened in the interacting basis and the matrix
+elements of the coupling operators are
+(Tql)=X
+nTl
+qn(jdyj): (11)
+Because the sample is open the number of electrons N
+contained in the sample is not xed. The Hamiltonian
+HSgiven in Eq. (3) commutes with the total \number"
+operatorP
+ndy
+ndn. ThusNis a \good quantum number"
+such that any state j) has a xed number of electrons.
+So the MES can also be labeled as j) =jN;i) with
+i= 0;1;2;:::an index for the ground and excited states
+of the MES subset with Nelectrons. The many-body
+energies can also be written as E=E(i)
+N. In the practical
+calculations Nvaries between 0 (the vacuum state) and
+Nmaxwhich is the total number of SES considered in the
+numerical diagonalization of HS. The total number of
+MES is thus 2Nmax.
+If the coupling between the leads and the sample is
+not too strong we expect that only a limited number of
+MES participate eectively to the electronic transport.
+These states are naturally selected by the bias window
+[R;L]. In the following examples, by selecting suitable
+values of the chemical potentials in the leads, we will
+truncate the basis of interacting MES to a reasonably
+small subset such that we can solve numerically Eq. (9)
+with our available computing resources. To relate the
+bias window with the eective MES we need to consider
+the chemical potential of the isolated sample containing
+Nelectrons,
+(i)
+N=E(i)
+NE(0)
+N1; (12)
+which is the energy required to add the N-th electron on
+top of the ground state with N1 to obtain the i-th MES
+withNparticles.23We expect the current associated to
+the MESjN;i) to depend on the location of the chemical
+potential(i)
+Nrelatively to the bias window. In particular
+it is clear that if at the coupling moment t0the sample is
+empty all MES with (i)
+NLwill remain empty both
+during the transient and the steady states, so they can
+be safely ignored when solving the GME.4
+III. MODELS AND RESULTS
+We have numerically implemented the GME method
+both for lattice and continuous models. The sample mod-
+els are: a short 1D chain with 5 sites, a 2D rectangular
+lattice with 1210 = 120 sites, and a short quantum
+wire withe parabolic lateral connement. In all cases the
+coupling functions have the form
+l(t) = 12
+e
+with
+ment, which is t0= 0, we have l(0) = 0 (no coupling),
+and in the steady state, for t!1 ,l= 1 (full coupling).
+A. A toy model: short 1D chain
+In this model the two semi-innite leads are attached
+to the ends of a 1D chain with 5 sites. The coupling
+between a lead state with wave function  qland a sam-
+ple state with wave function nis given by the product
+between the wave functions at the contact site:
+Tl
+qn=Vl 
+ql(0)n(il); (14)
+where 0 is the contact site of the lead l=L;R, the end
+sites of the sample being iL= 1 andiR= 5.
+FIG. 1: (Color online) The equilibrium chemical potentials
+(0)
+Nfor 1N5 as a function of the interaction strength
+U. The dotted lines mark the chemical potentials of the leads
+selected in the transport simulations shown in the next gure,
+i. e.L= 5:25 andR= 4:75.
+The reason to call this a toy model is that we can ob-
+tain the complete set of 25= 32 MES, i. e.we do not
+need to cut the basis of the 5 SES. We also do not need
+to cut the MES basis, all matrix elements of the statisti-
+cal operator can be numerically calculated, even if not all
+of them might be important for the currents. In addition
+we will consider the strength of the Coulomb interaction
+as a free parameter U, whereas in a realistic systems this
+is xed by the electron charge and the dielectric con-
+stant of the material. Our goal is to have a qualitativeunderstanding of the underlying physics, and in particu-
+lar to show the presence of the Coulomb blocking eects
+at certain values of Uor of the chemical potentials of the
+leads. The Coulomb matrix elements dened in Eq. (4)
+are calculated as
+Vnm;n0m0=X
+i6=i0
+n(i)
+m(i0)U
+jii0jn0(i)m0(i0):(15)
+In Fig.1 we show the equilibrium chemical potentials
+(0)
+Ncorresponding to ground states with 1 N5 par-
+ticles against the interaction strength U. One observes
+a linear dependence of (0)
+NonU, with slope increasing
+withN. Obviously the total Coulomb energy increases
+both withUandN.
+Let us now brie
+scenario.24Suppose the isolated sample contains Nelec-
+trons and the chemical potentials of the leads are cho-
+sen such that (0)
+N< R< L< (0)
+N+1. Then the bias
+window [R;L] may include one or more of the excited
+congurations with Nparticles. In general some states
+withNelectrons may have excitation energies exceed-
+ingLor even(0)
+N+1. This situation corresponds to the
+Coulomb blockade phenomenon. Indeed, the addition of
+the (N+ 1)-th electron is energetically forbidden. Con-
+sequently the current in the steady state should vanish.
+However, shorter or longer transient currents are gener-
+ated by all many-body congurations in the vicinity of
+the bias window.
+Fig. 2(a) and 2(b) show the total currents in the left
+lead and the total charge residing in the sample for sev-
+eral values of the interaction strength. Uis measured
+in units of tS, the hopping parameter in the sample,7
+and the time is expressed in units of ~=tSwhile the cur-
+rent is in units of etS=~. The coupling constant in Eq.
+(13) is
+(0) =j00000ih00000j.
+The chemical potentials of the leads, L= 5:25 and
+R= 4:75, are chosen such that in the absence of
+Coulomb interaction, i. e. forU= 0,(0)
+4is located
+within the bias window. In this case we obtain in the
+steady state the mean number of electrons about 3.6 and
+a non-vanishing current in the leads. This is understand-
+able, since (0)
+4=E4= 5, which is the 4-th level of
+the isolated sample. The occupation of this level in the
+steady state is about 0.6, the other states being either full
+or empty. Also in this case, the excited states have small
+contributions to the steady state current as the system
+tends to be in the ground state with N= 3 electrons.
+Those contributions may also depend on the coupling
+strength of individual states with the leads, but in gen-
+eral remain small.25
+The situation may change for U6= 0. For the inter-
+acting system, e. g. forU= 0:3, the system settles down
+in the Coulomb blockade regime, the total current be-
+ing almost suppressed in the steady state. This happens
+because the interaction pushes the chemical potentials
+upwards such that for U= 0:3 both ground states with5
+FIG. 2: (Color online) The total current entering the 5 1
+sample from the left lead as a function of time for the dierent
+values of the interaction strength U. The chemical potentials
+of the leads L= 5:25 andR= 4:75.
+N= 3 andN= 4 electrons are outside the bias win-
+dow and cannot produce steady currents. When the in-
+teraction strength is further increased to U= 0:5 and
+U= 1 the steady state currents are gradually restored.
+This could look surprising, but one can see in Fig.1 that
+by increasing Uthe ground state conguration with 3
+electrons approaches and enters the bias window. Con-
+sequently the transport becomes again possible. Note
+that while the steady state currents are not monotonous
+w.r.t.Uthe charge absorbed in the system continuously
+decreases, Fig. 2(b).
+In transport experiments the strength of the electron-
+electron interaction is indeed xed. The usual way to
+obtain the Coulomb blockade is to vary the chemical po-
+tentials of the leads relatively to the energy levels of the
+sample, or vice versa. In Fig. 3 we show the currents
+in both leads for dierent values of the chemical poten-
+tialR, while keeping xed L= 7. The strength of
+the Coulomb interaction is U= 1 and(0)
+4almost equals
+L. The steady state value of the current decreases as
+Rincreases, because fewer states are included in the
+bias window. The Coulomb blockade onset occurs for
+R>5, when(0)
+3drops below R. We observe that the
+FIG. 3: The time-dependent total currents in the left and
+right leads at dierent values of the chemical potential R.
+The current in the right lead starts at negative values. Other
+parameters: VL=VR= 0:750,U= 1:0.
+maximum value of the total current in the left lead does
+not change much when Rvaries. In contrast, the tran-
+sient current in the right lead is negative and increases
+in magnitude as Rincreases. This means that the right
+lead feeds the many-body congurations that fall below
+R.
+The contribution of the excited states to the transient
+and steady state currents depends strongly on the bias
+window. In Fig. 4 we show the currents entering the sam-
+ple from the left lead, carried by the states with N= 2
+andN= 3 electrons, for R= 3;4;5 (the cases with non-
+vanishing current in the steady state). We also show sep-
+arately the contribution to the currents associated to the
+ground state congurations, related to (0)
+2and(0)
+3, and
+the complementary contribution of all the excited states
+with 2 and 3 particles. In this case the wave vectors of
+the ground states are mostly given by the non-interacting
+wave vectors:j11000iwith weight 97% and j11100iwith
+98% forN= 2 andN= 3 respectively.
+ForR= 3 the steady state current of the ground
+state conguration is vanishingly small and so the total
+negative current associated to two-particle states comes
+mostly from the excited states. In the many-body energy
+spectrum of the isolated sample we obtain 5 excited con-
+gurations with (i)
+22[R;L] = [3;7]. AsRmoves up
+the steady state current of the ground state with N= 2
+becomes also negative. The combined contributions of
+the excited states vanishes at R= 5. As can be seen
+from Fig. 1 R= 5 is well above (0)
+2, but very close to
+(0)
+3. Consequently, the ground conguration with N= 2
+is heavily populated in the steady state, whereas the ex-
+cited states have low probability and thus weak current.
+Actually, as we have checked, all the currents associated
+to each excited state with N= 2 vanish individually. In
+the transient regime however the N= 2 currents in all
+three cases are dominated by the excites states.
+The currents of the excited states having N= 3 elec-6
+FIG. 4: The separate contributions to the current of the
+ground state with Nparticles and of allexcited states with
+Nparticles, for dierent values of R. For completeness we
+also include the total currents JLfor the same congura-
+tions. The discussion is made in the text. Other parameters:
+VL=VR= 0:750,U= 1:0.
+trons are positive at R= 3, but change sign at R= 4.
+ForR= 5 their magnitude exceeds the contribution of
+the ground state which is always positive. A more de-
+tailed analysis of the currents carried by specic excited
+states will be given for the 2D model.
+Finally, both in the transient and in the steady states
+the currents have small periodic 
+by the permanent transitions of electrons between the
+states in the sample and the states in the leads and
+back.25They are best seen in Fig. 2(a). Such 
+have also been obtained very recently by Kurth et al. us-
+ing combination of the non-equilibrium Green's functions
+and the time dependent density-functional theory of the
+Coulomb interaction.26
+B. 2D lattice
+We show now results for a 2D rectangular lattice with
+1210 sites. For a lattice constant of a= 5 nm this
+sample can be seen as a discrete version of a quantum
+dot of 60 nm50 nm. We used the lowest 10 SES of the
+non-interacting sample in the numerical diagonalization
+of the interacting Hamiltonian. This number is sucient
+to produce convergent results for the rst 50 MES for
+an interaction strength U= 0:8. The Coulomb matrix
+elements are calculated in the same way as for the 1Dcase, Eq. (15), except that now the site indices are two-
+dimensional, i. e.i= (ix;iy) andi0= (i0
+x;i0
+y).
+The two contact sites are chosen at diagonally opposite
+corners of the sample. The coupling coecients are cal-
+culated with Eq. (14), like for the 1D chain, and depend
+on the wave function of the particular SES at the con-
+tact sites. These coecients are illustrated in Fig. 5(a).
+The reduced density matrix is calculated using the rst
+50 MES. This allowed us to take into account many-body
+congurations with up to 3 electrons.
+In Fig. 5(b) we show the chemical potentials (i)
+Nfor
+the ground and excited states with N= 1;2;and 3 par-
+ticles. At the initial moment t0= 0 the system is empty.
+Based on the previous example, the main contribution to
+the currents in the steady states is expected from those
+MES with ground state chemical potentials located inside
+the bias window [ R;L]. One also observes excited con-
+gurations with Nparticles having chemical potentials
+larger than (0)
+N+1.
+FIG. 5: (Color online) (a) The coupling amplitudes jTqnj2
+forn= 1;::;5 between single-particle states in the leads with
+momentum qand the lowest 5 single-particle states of the
+isolated dot. (b) The generalized chemical potentials for N-
+particle interacting congurations. The red crosses mark (0)
+N
+while the other ones correspond to generalized potentials (i)
+N
+related to the i-th excited state of the Nparticle system.
+In the following we discuss the currents carried by the7
+various many-body states involved in transport. In a rst
+series of calculations we selected the chemical potential
+R= 0:2 and used two values of the chemical potential
+of the left lead L= 0:4 andL= 0:6. ForR= 0:2 and
+L= 0:4 the bias window contains only the 1-st and the
+2-nd excited congurations with N= 1, Fig. 5(b). The
+ground states for N= 1 andN= 2 are instead located
+below and above the bias window, respectively. Conse-
+quently the steady state current is very small. When
+Lincreases to 0.6 the ground state conguration with
+N= 2 enters the bias window and the current increases,
+Fig. 6(a).
+To analyze the transient regime we split the current
+into contributions given by the ground state and excited
+states with 1 electron (see Fig. 6(b)). When L= 0:4 the
+1-st and 2-nd excited state carry currents exceeding the
+current associated to the ground state, which survive all
+the way to the steady state. The current corresponding
+to the 2-nd excited state is smaller than the current of the
+1-st excited state, but comparable to that of the ground
+state. This is explained by the strength of the coupling
+coecients shown in Fig. 5(a), the 2-nd single-particle
+state being stronger coupled to the leads. The remaining
+higher excited states give oscillating and fast decaying
+transient currents. In Fig. 6(c) L= 0:6 and therefore
+higher excited states enter the bias window; their tran-
+sient currents are still decaying but at a smaller rate.
+Comparing with Fig. 6(a) it in clear that the transient
+regime is dominated by excited states.
+Next we discuss currents associated with states having
+2 and 3 electrons. We keep now xed R= 0:35 and
+again increase Lstarting with 0.6. Fig. 7(a) shows the
+total currents in the left lead for N= 2 andN= 3. As
+the bias increases the transient currents are enhanced,
+but they become comparable as the system approaches
+the steady state. In Fig. 7(b) the total current on three
+particle states shows a dierent behavior: the steady
+states value increases drastically when Lmoves up. To
+explain this one can look again at the diagram of the
+chemical potentials, Fig. 5(b). At L= 0:6 the 3-particle
+congurations are above the bias window and as such
+they contribute less to the current. In contrast, as L
+increases the ground state conguration with N= 3 en-
+ters the bias window, the window is closer to the excited
+states, and thus the total current increases. Actually, for
+L= 0:8 and 0:9 the current for N= 3 does not reach
+the steady state in the time interval considered.
+Now we look at the contribution of the excited states
+withN= 2 for two cases, L= 0:6 andL= 0:9. Again,
+the inspection of the diagram in Fig. 5(b) predicts the
+results of Fig. 8. When L= 0:6 there is just one excited
+conguration within the bias window, in addition to the
+ground state. In Fig. 8(a) we see that in the steady state
+these two congurations give signicant contributions to
+the current, whereas the higher excited states play a role
+only in the transient regime. Fig. 8(b) shows that at
+L= 0:9 the currents of the excited states and of the
+ground state are decreasing, some of them reaching even
+FIG. 6: (a) The total currents in the left and right leads for
+L= 0:6 andL= 0:4, while keeping R= 0:2. (b) The
+partial currents in the left lead for single-particle states when
+L= 0:4 andR= 0:2. (c) The partial currents in the left
+lead for single-particle states when L= 0:6 andR= 0:2
+negative values towards the steady state. This happens
+because the bias window includes now the ground state
+withN= 3 and the excited states with N= 2 deplete in
+the favor of the ground state.
+The sign of the current carried by states with Npar-
+ticles depends on the placement of the corresponding
+ground state chemical potential relatively to the bias win-
+dow. For example if we x L= 1:5 andR= 0:65 we8
+FIG. 7: (a) The total current in the left lead carried by all
+many-body congurations with N= 2, for increasing values
+ofL(i. e.0.6,0.7,0.8 and 0.9) and R= 0:2. (b) The same
+forN= 3.
+obtain(0)
+2< L. Fig. 9(a) shows the N-particle cur-
+rents when the sample initially contains two electrons in
+the ground state. This initial state evolves faster to the
+steady state than the empty system. While for N= 3
+the current in the left lead is positive, for both N= 2
+andN= 1 the currents are negative. The charge re-
+siding on each N-particle state and the total charge are
+shown in Fig. 9(b). Since single-particle congurations
+are unlikely their occupation vanishes. The total charge
+accumulated on the N= 3 states increases up to 2, while
+the total charge on the N= 2 states decreases from 2 to
+0.75. The sign of the current for N= 2 becomes pos-
+itive when Ris lowered to 0.2, Fig. 9(c), and exceeds
+the current carried by the states with N= 3. This is
+because the 1-st and the 2-nd SES practically determine
+the ground state with two electrons and thus (0)
+2, and
+also because the 1-st SES is strongly coupled to the leads.
+However, the current with N= 1 is still negative.
+FIG. 8: (a) The total current in the left lead carried by all
+many-body congurations with N= 2 atL= 0:6. (b) The
+same forL= 0:9. Other parameters R= 0:35.
+C. Parabolic quantum wire
+In this subsection we apply the GME with Coulomb
+interaction to describe the transport through a short
+quantum wire of length Lx= 300 nm with a parabolic
+connement in the y-direction perpendicular to the di-
+rection of transport. The contact ends of the isolated
+wire atLx=2 are described by hard walls. This is
+now a continuous model, where a large functional ba-
+sis is used to expand the eigenfunctions of the system
+in. In a similar manner we use a functional basis with
+complete truncated sets of continuous and discrete func-
+tions to expand the eigenfunctions of the semi-innite
+parabolic leads in. To show that we can describe the com-
+bined geometrical eects imposed on the system by it's
+geometry and an external perpendicular magnetic eld
+we place the quantum wire is in an external magnetic
+eld of strength 1 :0 T. The characteristic connement
+energy is given by ~
+0= 1:0 meV. We assume GaAs
+parameters with m= 0:067me,= 12:4 meV. The
+magnetic length modied by the parabolic connement
+isaw=p
+~=(m
+w), with 
+2
+w= 
+2
+0+!2
+c. and the
+cyclotron frequency !c=eB=(mc). AtB= 1:0 T,
+aw= 23:87 nm. The semi-innite leads having the same9
+FIG. 9: (a) The total current in the left lead carried by N-
+particle states and the total charge. for L= 1:5 and for
+R= 0:65. (b) The occupation number of the N-particle
+states. (c) The total current in the left lead carried by N-
+particle states for L= 1:2 andR= 0:2. (d) The occupation
+number of the N-particle states and the total charge.parabolic connement and being subject to the same ex-
+ternal perpendicular magnetic eld have a continuous en-
+ergy spectrum with discrete Landau sub-bands.
+The Coulomb potential in Eq. (4) in the 2D wire is
+described by
+u(rr0) =e2
+p
+(xx0)2+ (yy0)2+2; (16)
+with the small convergence parameter ( =aw) = 0:01
+to facilitate the two-dimensional numerical integration
+needed for the matrix elements (4).
+After the GME (9) has been transformed to the in-
+teracting many-electron basis by the unitary transfor-
+mation obtained by the diagonalization of HS(3) we
+truncate the RDO (8) to 32 MES. For the bias range
+0:0
=LR1:7 meV used here 10 SES are
+sucient to obtain these lowest 32 states with good accu-
+racy. We will be omitting singly occupied states of high
+energy that should not be relavant for the parameters
+here. The natural strength of the Coulomb interaction
+will only give us MES that are occupied by one or two
+electrons in the energy range 0 to 6 meV covered by the
+32 MES.
+Since in the partitioning approach [ HS;HL] = 0 we
+have to construct Tl
+qnas a non-local overlap ofnand
+ L;R
+qon the contact regions Cl; l=L;R:8
+Tl
+qn=Z
+Cldrdr0
+ 
+ql(r)n(r)gl
+qn(r;r0) +h:c:

+:(17)
+gl
+qn(r;r0) =gl
+0exp
+l
+1(xx0)2l
+2(yy0)2
+expjEn"qlj
+
l
+E
+: (18)
+As before"qlis the energy spectrum of lead l, andEn
+is the energy of the SES numbered by nin the quan-
+tum wire. The quantum number qfor the states in leads
+represents both the discrete Landau band number and a
+continuous quantum number that can be related to the
+momentum of a particular state. Here we use the pa-
+rameters1a2
+w=2a2
+w= 0:25, 
LR
+E= 0:25 meV, and
+gLR
+0= 40 meV for B= 1:0 T. The domain of the over-
+lap integral for the leads is 2awinto the lead or the
+system forxandx0from each end of the wire at Lx=2
+and between4awforyandy0, see Ref. (8) for an exact
+denition. All the SES will be coupled to the leads, but
+the coupling strength will depend on the character of the
+SES, whether it is an edge- or bulk state and other ner
+geometrical details that is brought about by the magnetic
+eld.
+The right chemical potential Ris held at 1 :4 meV
+and the transport properties are calculated for dierent
+values of the bias 
 by varyingL. Figure 10 compares
+the total occupation of all one- electron and two-electron
+MES for the interacting system at two dierent values of
+the bias. At, 
 = 0:2 meV we see that almost solely10
+FIG. 10: (Color online) The total charge residing in one- and
+two-electron states as a function of time for two dierent val-
+ues of the bias 
 .B= 1:0 T,Lx= 300 nm, ~
+0= 1:0
+meV.
+one-electron states are occupied, while for 
 = 1:2 meV
+initially it is likely to have one-electron states occupied,
+but very soon the occupation of the two-electron states
+becomes as probable with the likelihood of the occupa-
+tion of the one-electron states fast reducing with time.
+We also have to admit here that even though the steady
+state value of the total current through the system can
+be deduced by the values of the current at 270 ps, the
+charging of the system takes much longer time, since we
+are using here a very weak coupling to the leads that
+mimics a tunneling regime.
+If we now use the average value of the current in the left
+and right leads at t= 270 ps as a measure of the steady
+state current we get the information displayed in Fig.
+11, where the steady state value of the current is shown
+for the interacting system as a function of the bias and
+compared to the charge in the system. We have a clear
+Coulomb blocking in the interacting system. In the case
+of a non-interacting system the lack of a gap between the
+one- and two electron MES and a strong mixing of the
+energy regimes of two- and three-electron states the two-
+electron plateau only appears as a small shoulder. The
+FIG. 11: The total steady state current for interacting 10
+SES, and the total charge at t= 270 ps. for dierent values
+of the bias 
 .B= 1:0 T,Lx= 300 nm, ~
+0= 1:0 meV.
+32 MES selected here include no three-electron or MES
+with higher number of electrons. It should be mentioned
+here that a dierent choice of the right bias Rcan result
+in the system charging faster and thus at the same time
+the total current through it being smaller. This comes
+from the fact that the states have a dierent coupling to
+the leads and the time range shown here is very much in
+the transient- or it's long exponential decay regime.
+Figure 12 displaying the current in the right lead gives
+an idea how the Coulomb blocking plateau appears after
+the transition regime. The transition regime where the
+FIG. 12: The total current in the right lead for interacting
+and non-interacting 10 SES as a function of the bias 
 and
+time.B= 1:0 T,Lx= 300 nm, ~
+0= 1:0 meV.
+right current goes negative, i. e.where it supplies charge
+to the system is partially truncated from the gure.11
+IV. SUMMARY AND CONCLUSIONS
+We calculated time-dependent currents in open meso-
+scopic systems composed by a sample attached to two
+semi-innite leads, by solving the generalized master
+equation for the reduced density operator acting in the
+Fock space of the sample. This is the natural frame-
+work for including the Coulomb electron-electron inter-
+action in the sample, which is the main achievement of
+this work. The Coulomb interaction is treated in the
+spirit of the exact diagonalization method, i. e.in a pure
+many-body manner. The interacting many-body states
+of the sample are expanded in the basis of non-interacting
+\bit-string" states with unspecied number of electrons.
+We believe our method is a viable alternative to a recent
+approach based on a time-dependent density-functional
+model.9,10,26We used three sample models, a short 1D
+wire with 5 sites, but also a larger 2D lattice with 120
+sites and a continuous model, whereas the cited group
+used much smaller samples even with no structure.26
+Indeed, due to computational limitations we could use
+only a restricted, eective number of many-body states in
+the GME, between 30-50 depending on the model, from
+the bottom of the energy spectrum. We chose the bias
+window [R;L] and the strength of the sample-leads
+coupling parameters VR;Lsuch that only the eective
+states contribute to the transport of electrons through
+the sample, whereas the states with higher energy are
+unreachable by the electrons. Consequently the number
+of electrons in the sample can be only up to 3 or 4.
+We could calculate the contribution to the charge and
+currents in the sample and in the leads respectively, cor-
+responding to any particular many-body state. We use
+the 1D chain as a toy model to emphasize the dominant
+role of the excited states in the transient regime and the
+onset of the Coulomb blockade in the steady state. A
+similar 1D model with 4 sites 1D has been considered
+recently by My oh anen et al.10
+As shown also in our previous works on time-dependenttransport in non-interacting systems the GME method
+includes information on the energy structure of the sam-
+ple, but also on the geometrical properties re
+the wave functions and sample-lead contacts.7,8,25Here
+we illustrate these aspects, in the interacting case, for two
+nanosystems: a two-dimensional quantum dot described
+by a lattice Hamiltonian and a short parabolic quantum
+wire. The time-dependent occupation of specic many-
+body states was thoroughly analyzed, for dierent values
+of the chemical potentials of the leads. It turned out
+that the excited states with Nelectrons contribute to
+the steady state currents if the ground state conguration
+withN+ 1 particles is not available for transport. How-
+ever, if(0)
+N<Rand at the same time (0)
+N+1lies within
+the bias window the excited states with Nparticles
+are active only in the transient regime and become de-
+populated in the steady state regime. This behavior is of
+interest in the excited-state spectroscopy experiments.11
+To our knowledge the time-dependent currents associated
+to excited states have not been discussed theoretically so
+far.
+Acknowledgments
+The authors acknowledge nancial support from the
+Development Fund of the Reykjavik University Grant
+No. T09001, the Research and Instruments Funds of the
+Icelandic State, the Research Fund of the University of
+Iceland, the Icelandic Science and Technology Research
+Programme for Postgenomic Biomedicine, Nanoscience
+and Nanotechnology, the National Science Council of Tai-
+wan under contract No. NSC97-2112-M-239-003-MY3.
+V.M. also acknowledges the hospitality of the Reykjavik
+University, Science Institute and the partial nancial sup-
+port from PNCDI2 program (grant No. 515/2009) and
+grant No. 45N/2009.
+1G. Stefanucci, S. Kurth, A. Rubio and E. K. U. Gross,
+Phys. Rev. B 77, 075339 (2008).
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+Rev. B 76, 085330 (2007)
+3X. Zheng, F. Wang, C. Y. Yam, Y. Mo, and G.H. Chen,
+Phys. Rev. B 75, 195127 (2007).
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+Moldoveanu, Phys. Rev. B 77, 035329 (2008).
+5U. Harbola, M. Esposito, and S. Mukamel, Phys. Rev. B
+74, 235309 (2006).
+6S. Welack, M. Schreiber, and U. Kleinekath ofer, J. Chem.
+Phys. 124, 044712 (2006)
+7V. Moldoveanu, A. Manolescu and V. Gudmundsson, New
+J. Phys. 11, 073019 (2009).
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+A. Manolescu, to appear in New J. Phys.
+9S. Kurth, G. Stefanucci, C.-O. Almbladh, A. Rubio, andE. K. U. Gross, Phys. Rev. B 72, 035308 (2005).
+10P. My oh anen, A. Stan, G. Stefanucci, and R. van Leeuwen,
+Phys. Rev. B 80, 115107 (2009), Europhys. Lett. 84, 67001
+(2008).
+11T. Fujisawa, D. G. Austing, Y. Tokura, Y. Hirayama, and
+S. Tarucha, J. Phys.: Condens. Matter 15, R1395 (2003).
+12D. C. Langreth and P. Nordlander, Phys. Rev. B 43, 2541
+(1991)
+13S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15932
+(1996).
+14J. K onig, H. Schoeller, and G. Sch on, Phys. Rev. Lett. 76,
+1715 (1996); J. K onig, J. Schmid, H. Schoeller, and G.
+Sch on, Phys. Rev. B 54, 16 820 (1996).
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+16S. Nakajima, Prog. Theor. Phys. 20, 948 (1958).
+17R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).12
+18C. Timm, Phys. Rev. B 77, 195416 (2008).
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+20M.Esposito and M. Galperin, Phys. Rev. B 79, 205303
+(2009).
+21D. Darau, G. Begemann, A. Donarini, and M. Grifoni,
+Physical Review B 79235404 (2009).
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+J. Phys. C 4, 916 (1971).
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+24L.P. Kouwenhouven et al., in Mesoscopic Electron Trans-port, edited by L.L. Sohn, L.P. Kouwenhouven, and G.
+Schn, NATO Advanced Study Institute, Series E, Vol. 345
+(Kluwer, Dordrecht, 1997).
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+Rev. B 80, 205325 (2009).
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+(2009).

1001.0048.txt

@@ -0,0 +1,1038 @@
+arXiv:1001.0048v2  [math.AP]  7 Jan 2010Nonlinear stability of periodic traveling wave solutions o f
+viscous conservation laws in dimensions one and two
+Mathew A. Johnson∗Kevin Zumbrun†
+November 12, 2018
+Keywords : Periodic traveling waves; Bloch decomposition; modulate d waves.
+2000 MR Subject Classification : 35B35.
+Abstract
+Extending results of Oh and Zumbrun in dimensions d≥3, we establish nonlin-
+ear stability and asymptotic behavior of spatially-periodic traveling- wave solutions of
+viscous systems of conservation laws in critical dimensions d= 1,2, under a natural
+set of spectral stability assumptions introduced by Schneider in th e setting of reaction
+diffusion equations. The key new steps in the analysis beyond that in d imensionsd≥3
+are a refined Green function estimate separating off translation as the slowest decaying
+linear mode and a novel scheme for detecting cancellation at the leve l of the nonlinear
+iteration in the Duhamel representation of a modulated periodic wav e.
+1 Introduction
+Nonclassical viscous conservation laws arising in multiph ase fluid and solid mechanics ex-
+hibit a rich variety of traveling wave phenomena, including homoclinic (pulse-type) and
+periodic solutions along with the standard heteroclinic (s hock, or front-type) solutions
+[GZ, Z6, OZ1, OZ2]. Here, we investigate stability of period ic traveling waves: specifi-
+cally, sufficient conditions for stability of the wave. Our ma in result, generalizing results of
+Oh and Zumbrun [OZ4] in dimensions d≥3, is to show that strong spectral stability in the
+sense of Schneider [S1, S2, S3] implies linearized and nonli nearL1∩HK→L∞bounded
+stability, for all dimensions d≥1, andasymptotic stability for dimensions d≥2.
+∗Indiana University, Bloomington, IN 47405; matjohn@india na.edu: Research of M.J. was partially sup-
+ported by an NSF Postdoctoral Fellowship under NSF grant DMS -0902192.
+†Indiana University, Bloomington, IN 47405; kzumbrun@indi ana.edu: Research of K.Z. was partially
+supported under NSF grants no. DMS-0300487 and DMS-0801745 .
+11 INTRODUCTION 2
+More precisely, we show that small L1∩Hsperturbations of a planar periodic solution
+u(x,t)≡¯u(x1) (without loss of generality taken stationary) converge at Gaussian rate in
+Lp,p≥2 to a modulation
+(1.1) ¯ u(x1−ψ(x,t)),
+of the unperturbed wave, where x= (x1,˜x), ˜x= (x2,...,xd), andψis a scalar function
+whosex- andt-gradients likewise decay at least at Gaussian rate in all Lp,p≥2, but which
+itself decays more slowly by a factor t1/2; in particular, ψis merely bounded in L∞for
+dimensiond= 1.
+The one-dimensional study of spectral stability of spatial ly periodic traveling waves of
+systems of viscous conservation laws was initiated by Oh and Zumbrun [OZ1] in the “quasi-
+Hamiltonian” case that the traveling-wave equation posses ses an integral of motion, and in
+the general case by Serre[Se1]. An important contribution o f Serre was to point out a larger
+connection between the linearized dispersion relation (th e functionλ(ξ) relating spectra to
+wave number of the linearized operator about the wave) near z ero and the formal Whitham
+averaged system obtained by slow modulation, or WKB, approx imation.
+In [OZ3], this was extended to multi-dimensions, relating t he linearized dispersion rela-
+tion near zero to
+(1.2)∂tM+/summationdisplay
+j∂xjFj= 0,
+∂t(ΩN)+∇x(ΩS) = 0,
+whereM∈Rndenotes the average over one period, Fjthe average of an associated flux,
+Ω =|∇xΨ| ∈R1the frequency, S=−Ψt/|∇xΨ| ∈R1the speeds, andN=∇xΨ/|∇xΨ| ∈
+Rdthe normal νassociated with nearby periodic waves, with an additional c onstraint
+(1.3) curl (Ω N) = curl ∇xΨ≡0.
+As an immediate corollary, similarly as in [OZ1], [Se1] in th e one-dimensional case, this
+yieldedas anecessary condition formulti-dimensional sta bility hyperbolicityoftheaveraged
+system (1.2)–(1.3).
+The present study is informed by but does not directly rely on this observation relating
+Whitham averaging and spectral stability properties. Like wise, the Evans function tech-
+niquesusedin[Se1,OZ3]toestablishthisconnection play n oroleinouranalysis; indeed, the
+Evans function makes no appearance here. Rather, we rely on a direct Bloch-decomposition
+argument in the spirit of Schneider [S1, S2, S3], combining s harp linearized estimates with
+subtle cancellation in nonlinear source terms arising from the modulated wave approxima-
+tion. The analytical techniques used to realize this progra m are somewhat different from
+those of [S1, S2, S3], however, coming instead from the theor y of stability of viscous shock
+fronts through a line of investigation carried out in [OZ1, O Z2, OZ3, OZ4, HoZ]. In partic-
+ular, the nonsmooth dispersion relation at ξ= 0 typical for convection-diffusion equations1 INTRODUCTION 3
+requires different treatment from that of [S1, S2, S3] in the re action diffusion case; see Re-
+mark 2.4. Moreover, we detect nonlinear cancellation in the physicalx-tdomain rather than
+the frequency domain as in [S1, S2, S3]. The main difference bet ween the present analysis
+and that of [OZ4] is the systematic incorporation of modulat ion approximation (1.1).
+1.1 Equations and assumptions
+Consider a parabolic system of conservation laws
+(1.4) ut+/summationdisplay
+jfj(u)xj= ∆xu,
+u∈ U(open)∈Rn,fj∈Rn,x∈Rd,d≥1,t∈R+, and a periodic traveling wave solution
+(1.5) u= ¯u(x·ν−st),
+of periodX, satisfying the traveling-wave ODE ¯ u′′= (/summationtext
+jνjfj(¯u))′−s¯u′with boundary
+conditions ¯u(0) = ¯u(X) =:u0.Integrating, we obtain a first-order profile equation
+(1.6) ¯ u′=/summationdisplay
+jνjfj(¯u)−s¯u−q,
+where (u0,q,s,ν,X )≡constant. Without loss of generality take ν=e1,s= 0, so that
+¯u= ¯u(x1) represents a stationary solution depending only on x1.
+Following [Se1, OZ3, OZ4], we assume:
+(H1)fj∈CK+1,K≥[d/2]+4.
+(H2) Themap H:R×U×R×Sd−1×Rn→Rntaking (X;a,s,ν,q)/mapsto→u(X;a,s,ν,q)−a
+is a submersion at point ( ¯X;¯u(0),0,e1,¯q), whereu(·;·) is the solution operator for (1.6).
+Conditions (H1)–(H2) imply that the set of periodic solutio ns in the vicinity of ¯ uform
+a smooth (n+d+1)-dimensional manifold {¯ua(x·ν(a)−α−s(a)t)}, withα∈R,a∈Rn+d.
+1.1.1 Linearized equations
+Linearizing (1.4) about ¯ u(·), we obtain
+(1.7) vt=Lv:= ∆xv−/summationdisplay
+(Ajv)xj,
+where coefficients Aj:=Dfj(¯u) are now periodic functions of x1. Taking the Fourier
+transform in the transverse coordinate ˜ x= (x2,···,xd), we obtain
+(1.8)ˆvt=L˜ξˆv= ˆvx1,x1−(A1ˆv)x1−i/summationdisplay
+j/negationslash=1Ajξjˆv−/summationdisplay
+j/negationslash=1ξ2
+jˆv,
+where˜ξ= (ξ2,···,ξd) is the transverse frequency vector.1 INTRODUCTION 4
+1.1.2 Bloch–Fourier decomposition and stability conditions
+Following [G, S1, S2, S3], we define the family of operators
+(1.9) Lξ=e−iξ1x1L˜ξeiξ1x1
+operating on the class of L2periodic functions on [0 ,X]; the (L2) spectrum of L˜ξis equal
+to the union of the spectra of all Lξwithξ1real with associated eigenfunctions
+(1.10) w(x1,˜ξ,λ) :=eiξ1x1q(x1,ξ1,˜ξ,λ),
+whereq, periodic, is an eigenfunction of Lξ. By continuity of spectrum, and discreteness of
+the spectrum of the elliptic operators Lξon the compact domain [0 ,X], we have that the
+spectra ofLξmay be described as the union of countably many continuous su rfacesλj(ξ).
+Without loss of generality taking X= 1, recall now the Bloch–Fourier representation
+(1.11) u(x) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xˆu(ξ,x1)dξ1d˜ξ
+of anL2functionu, where ˆu(ξ,x1) :=/summationtext
+ke2πikx1ˆu(ξ1+ 2πk,˜ξ) are periodic functions of
+periodX= 1, ˆu(˜ξ) denoting with slight abuse of notation the Fourier transfo rm ofuin the
+full variable x. By Parseval’s identity, the Bloch–Fourier transform u(x)→ˆu(ξ,x1) is an
+isometry in L2:
+(1.12) /ba∇dblu/ba∇dblL2(x)=/ba∇dblˆu/ba∇dblL2(ξ;L2(x1)),
+whereL2(x1) is taken on [0 ,1] andL2(ξ) on [−π,π]×Rd−1. Moreover, it diagonalizes the
+periodic-coefficient operator L, yielding the inverse Bloch–Fourier transform representation
+(1.13) eLtu0=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xeLξtˆu0(ξ,x1)dξ1d˜ξ
+relating behavior of the linearized system to that of the dia gonal operators Lξ.
+Following [OZ4], weassumealongwith(H1)–(H2) the strong spectral stability conditions:
+(D1)σ(Lξ)⊂ {Reλ<0}forξ/ne}ationslash= 0.
+(D2) Reσ(Lξ)≤ −θ|ξ|2,θ>0, forξ∈Rdand|ξ|sufficiently small.
+(D3)λ= 0 is a semisimple eigenvalue of L0of multiplicity exactly n+1.1
+For each fixed angle ˆξ:=ξ/|ξ|, expandLξ=L0+|ξ|L1+|ξ|2L2. By assumption (D3)
+and standard spectral perturbation theory, there exist n+1 smooth eigenvalues
+(1.14) λj(ξ) =−iaj(ξ)+o(|ξ|)
+1The zero eigenspace of L0is at least ( n+1)-dimensional by the linearized existence theory and (H2 ),
+and hence n+ 1 is the minimal multiplicity; see [Se1, OZ3]. As noted in [O Z1, OZ3], minimal dimension
+of this zero eigenspace implies that ( M,NΩ) of (1.2) gives a nonsingular coordinatization of the fami ly of
+periodic traveling-wave solutions near ¯ u.1 INTRODUCTION 5
+ofLξbifurcating from λ= 0 atξ= 0, where −iajare homogeneous degree one functions
+given by |ξ|times the eigenvalues of Π 0L1|KerL0, with Π 0the zero eigenprojection of L0.
+Conditions(D1)–(D3) areexactly thespectralassumptions of[S1,S2,S3], corresponding
+to “dissipativity” of the large-time behavior of the linear ized system. As in [OZ4], we make
+the further nondegeneracy hypothesis:
+(H3) The eigenvalues λ=−iaj(ξ)/|ξ|of Π0L1
+KerL0are simple.
+The functions ajmay be seen to be the characteristics associated with the Whi tham av-
+eraged system (1.2)–(1.3) linearized about the values of M,S,N, Ω associated with the
+background wave ¯ u; see [OZ3, OZ4]. Thus, (D1) implies weak hyperbolicity of (1 .2)–(1.3)
+(reality ofaj), while (H1) corresponds to strict hyperbolicity.
+1.2 Main results
+With these preliminaries, we can now state our main results.
+Theorem 1.1. Assuming (H1)–(H3) and (D1)–(D3), for some C >0andψ∈WK,∞(x,t),
+(1.15)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0,
+|˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
+4|˜u−¯u|L1∩HK|t=0,
+|(ψt,ψx)|WK+1,p≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0,
+and
+(1.16) |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
+2(1−1
+p)+1
+2|˜u−¯u|L1∩HK|t=0
+for allt≥0,p≥2,d= 1, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
+In particular, ¯uis nonlinearly bounded L1∩HK→L∞stable for dimension d= 1.
+Theorem 1.2. Assuming (H1)–(H3) and (D1)–(D3), for any ε >0, someC >0and
+ψ∈WK,∞(x,t),
+(1.17)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0,
+|˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
+4|˜u−¯u|L1∩HK|t=0,
+|(ψt,ψx)|WK+1,p≤C(1+t)−d
+2(1−1/p)+ε−1
+2|˜u−¯u|L1∩HK|t=0,
+and
+(1.18)|˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
+2(1−1
+p)+ε|˜u−¯u|L1∩HK|t=0,
+|˜u−¯u|HK(t),|ψ(t)|HK≤C(1+t)−d
+4+ε|˜u−¯u|L1∩HK|t=0,
+for allt≥0,p≥2,d= 2, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
+In particular, ¯uis nonlinearly asymptotically L1∩HK→HKstable for dimension d= 2.1 INTRODUCTION 6
+Remark 1.1. In Theorem 1.2, derivatives in x∈R2refer to total derivatives. Moreover,
+unless specified by an appropriate index, throughout this pa per derivatives in spatial variable
+xwill always refer to the total derivative of the function.
+In dimension one, Theorem 1.1 asserts only bounded L1∩HK→L∞stability, a very
+weak notion of stability. The absence of decay in perturbati on ˜u−¯uindicates the delicacy
+of the nonlinear analysis in this case. In particular, it is c rucial to separate off the slower-
+decaying modulated behavior (1.1) in order to close the nonl inear iteration argument.
+Remark 1.2. In dimension d= 1, it is straightforward to show that the results of Theorem
+1.1 extend to all 1 ≤p≤ ∞using the pointwise techniques of [OZ2]; see Remark 3.3.
+Remark 1.3. The slow decay of |˜u−¯u|Lp(t)∼ |ψ(t)|Lpin (1.16) is due to nonlinear
+interactions; as shown in [OZ2, OZ4], the linearized decay r ate is faster by factor (1+ t)−1/2
+(Proposition 2.1). In [OZ4], it was shown that for d≥3, where linear effects dominate
+behavior, (1.16) may be replaced by the stronger estimate |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+
+t)−d
+2(1−1
+p)|˜u−¯u|L1∩HK|t=0.These distinctions reflect fine details of both linearized es timates
+(Section 3) and nonlinear structure (Sections 4.1–4.2) tha t are not immediately apparent
+from the formal Whitham approximation (1.2)–(1.3).
+1.3 Discussion and open problems
+Linearized stability under the same assumptions, with shar p rates of decay, was established
+ford= 1 [OZ2] and for d≥1 in [OZ4], along with nonlinear stability for d≥3. Theorem
+1.1 completes this line of investigation by establishing no nlinear stability in the critical
+dimensions d= 1,2, a fundamental open problem cited in [OZ1, OZ4].
+This gives a generalization of the work of [S1, S2, S3] for rea ction diffusion equations
+to the case of viscous conservation laws. Recall that the ana lysis of [S1, S2, S3] concerns
+also multiply periodic waves, i.e., waves that are either pe riodic or else constant in each
+coordinate direction. It is straightforward to verify that the methods of this paper apply
+essentially unchanged to this case, to give a corresponding stability result under the analog
+of (H1)–(H3), (D1)–(D3), as we intend to report further in a f uture work. Likewise, the
+extension from the semilinear parabolic case treated here t o the general quasilinear case is
+straightforward, following the treatment of [OZ4].
+On the other hand, as noted in [OZ2], condition (D3) is in the c onservation law setting
+nongeneric, corresponding to the special “quasi-Hamilton ian” situation studied there; in
+particular, it implies that speed is to first order constant a mong the family of spatially pe-
+riodic traveling-wave solutions nearby ¯ u. In the generic case that (D3) is violated, behavior
+is essentially different [OZ1, OZ2], and perturbations decay more slowly at the linearized
+level. Nonlinear stability remains an interesting open pro blem in this setting.
+Our approach to stability in the critical dimensions d= 1,2, as suggested in [OZ4], is,
+loosely following the approach of [S1, S2, S3], to subtract o ut a slower-decaying part of the
+solution describedby anappropriatemodulation equation a ndshowthat theresidualdecays2 BASIC LINEARIZED STABILITY ESTIMATES 7
+sufficiently rapidly to close a nonlinear iteration. It is wor th noting that the modulated
+approximation ¯ u(x1−ψ(x,t)) of (1.1) is not the full Ansatz
+(1.19) ¯ ua(Ψ(x,t)),
+Ψ(x,t) :=x1−ψ(x,t), associated with the Whitham averaged system (1.2)–(1.3) , where ¯ua
+isthemanifoldofperiodicsolutions near ¯ uintroducedbelow(H2), butonlythetranslational
+part not involving perturbations ain the profile. (See [OZ3] for the derivation of (1.2)–
+(1.3) and (1.19).) That is, we don’t need to separate out all v ariations along the manifold
+of periodic solutions, but only the special variations conn ected with translation invariance.
+The technical reason is an asymmetry in y-derivative estimates in the parts of the Green
+function associated with these various modes, something th at is not apparent without a
+detailed study of linearized behavior as carried out here. T his also makes sense formally,
+if one considers that (1.2) indicates that variables a,∇xΨ are roughly comparable, which
+would suggest, by the diffusive behavior Ψ >>∇xΨ, thatais neglible with respect to Ψ.
+However, note that in the case that (D3) holds, hence wave spe ed is stationary along the
+manifold of periodic solutions, the final equation of (1.2) d ecouples to (Ψ x)t= (ΩN)t= 0,
+and could be written as Ψ t= 0 in terms of Ψ alone. Hence, there is some ambiguity in this
+degenerate case which of Ψ, Ψ xis the primary variable, and in terms of linear behavior, the
+decay of variations aand Ψ are in fact comparable [OZ4]; in the generic case, aand Ψxare
+comparable at the linearized level [OZ2]. It would be very in teresting to better understand
+the connection between the Whitham averaged system (or suit able higher-order correction)
+and behavior at the nonlinear level, as explored at the linea r level in [OZ3, OZ4, JZ1, JZB].
+2 Basic linearized stability estimates
+We begin by recalling the basic linearized stability estima tes derived in [OZ4]. We will
+sharpen these afterward in Section 3. By standard spectral p erturbation theory [K], the
+total eigenprojection P(ξ) onto the eigenspace of Lξassociated with the eigenvalues λj(ξ),
+j= 1,...,n+1describedintheintroductioniswell-definedandanalyti cinξforξsufficiently
+small, since these (by discreteness of the spectra of Lξ) are separated at ξ= 0 from the rest
+of the spectrum of L0. Introducing a smooth cutoff function φ(ξ) that is identically one
+for|ξ| ≤εand identically zero for |ξ| ≥2ε,ε >0 sufficiently small, we split the solution
+operatorS(t) :=eLtinto low- and high-frequency parts
+(2.1) SI(t)u0:=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆu0(ξ,x1)dξ1d˜ξ
+and
+(2.2) SII(t)u0:=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·x/parenleftbig
+I−φP(ξ)/parenrightbig
+eLξtˆu0(ξ,x1)dξ1d˜ξ.2 BASIC LINEARIZED STABILITY ESTIMATES 8
+2.1 High-frequency bounds
+By standard sectorial bounds [He, Pa] and spectral separati on ofλj(ξ) from the remaining
+spectra ofLξ, we have trivially the exponential decay bounds
+(2.3)/ba∇dbleLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ce−θt/ba∇dblf/ba∇dblL2([0,X]),
+/ba∇dbleLξt(I−φP(ξ))∂l
+x1f/ba∇dblL2([0,X])≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+/ba∇dbl∂l
+x1eLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+forθ,C >0, and 0 ≤m≤K(Kas in (H1)). Together with (1.12), these give immediately
+the following estimates.
+Proposition 2.1 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D2), for some θ,
+C >0, and allt>0,2≤p≤ ∞,0≤l≤K+1,0≤m≤K,
+(2.4)/ba∇dbl∂l
+xSII(t)f/ba∇dblL2(x),/ba∇dblSII(t)∂l
+xf/ba∇dblL2(x)≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2(x),
+/ba∇dbl∂m
+xSII(t)f/ba∇dblLp(x),/ba∇dblSII(t)∂m
+xf/ba∇dblLp(x)≤Ct−d
+2(1
+2−1
+p)−m
+2e−θt/ba∇dblf/ba∇dblL2(x),
+where, again, derivatives in the variable x∈Rdrefer to total derivatives.
+Proof.The first inequalities follow immediately by (1.12) and (2.3 ). The second follows for
+x1derivatives in the case p=∞,m= 0 by Sobolev embedding from
+/ba∇dblSII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
+4e−θt/ba∇dblf/ba∇dblL2([0,X])
+and
+/ba∇dbl∂x1SII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
+4−1
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+which follow by an application of (1.12) in the x1variable and the Hausdorff–Young in-
+equality /ba∇dblf/ba∇dblL∞(˜x)≤ /ba∇dblˆf/ba∇dblL1(˜ξ)in the variable ˜ x. The result for derivatives in x1and general
+2≤p≤ ∞then follows by Lpinterpolation. Finally, the result for derivatives in ˜ xfollows
+from the inverse Fourier transform, equation (2.2), and the large|ξ|bound
+|eLtf|L2(x1)≤e−θ|˜ξ|2t|f|L2(x1),|ξ|sufficiently large ,
+which easily follows from Parseval and the fact that Lξis a relatively compact perturbation
+of∂2
+x−|ξ|2. Thus, by the above estimate we have
+/ba∇dbleLt∂˜xf/ba∇dblL2(x)≤C/ba∇dbleLξt|˜ξ|ˆf/ba∇dblL2(x1,ξ)
+≤Csup/parenleftBig
+e−θ|˜ξ|2t|ξ|/parenrightBig
+/ba∇dblˆf/ba∇dblL2(x1,ξ)
+≤Ct−1/2/ba∇dblf/ba∇dblL2(x).
+A similar argument applies for 1 ≤m≤K.2 BASIC LINEARIZED STABILITY ESTIMATES 9
+2.2 Low-frequency bounds
+Denote by
+(2.5) GI(x,t;y) :=SI(t)δy(x)
+the Green kernel associated with SI, and
+(2.6) [ GI
+ξ(x1,t;y1)] :=φ(ξ)P(ξ)eLξt[δy1(x1)]
+the corresponding kernel appearing within the Bloch–Fouri er representation of GI, where
+the brackets on [ Gξ] and [δy] denote the periodic extensions of these functions onto the
+whole line. Then, we have the following descriptions of GI, [GI
+ξ], deriving from the spectral
+expansion (1.14) of Lξnearξ= 0.
+Proposition 2.2 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3),
+(2.7)[GI
+ξ(x1,t;y1)] =φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗,
+GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)[GI
+ξ(x1,t;y1)]dξ
+=/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
+where∗denotes matrix adjoint, or complex conjugate transpose, qj(ξ,·)and˜qj(ξ,·)are right
+and left eigenfunctions of Lξassociated with eigenvalues λj(ξ)defined in (1.14), normalized
+so that/an}b∇acketle{t˜qj,qj/an}b∇acket∇i}ht ≡1, whereλj/|ξ|is a smooth function of |ξ|andˆξ:=ξ/|ξ|andqjand˜qj
+are smooth functions of |ξ|,ˆξ:=ξ/|ξ|, andx1ory1, withℜλj(ξ)≤ −θ|ξ|2.
+Proof.Smooth dependence of λjand ofq, ˜qas functions in L2[0,X] follow from standard
+spectral perturbation theory [K] using the fact that λjsplit to first order in |ξ|asξis varied
+along rays through the origin, and that Lξvaries smoothly with angle ˆξ. Smoothness of
+qj, ˜qjinx1,y1then follow from the fact that they satisfy the eigenvalue eq uation forLξ,
+which has smooth, periodic coefficients. Likewise, (2.7)(i) is immediate from the spectral
+decomposition of elliptic operators on finite domains. Subs tituting (2.5) into (2.1) and
+computing
+(2.8) /hatwideδy(ξ,x1) =/summationdisplay
+ke2πikx1/hatwideδy(ξ+2πke1) =/summationdisplay
+ke2πikx1e−iξ·y−2πiky1=e−iξ·y[δy1(x1)],
+where the second and third equalities follow from the fact th at the Fourier transform either
+continuous or discrete of the delta-function is unity, we ob tain
+GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xφP(ξ)eLξt/hatwideδy(ξ,x1)dξ
+=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)φP(ξ)eLξt[δy1(x1)]dξ,2 BASIC LINEARIZED STABILITY ESTIMATES 10
+yielding (2.7)(ii) by (2.6)(i) and the fact that φis supported on [ −π,π].
+Proposition 2.3 ([OZ4]).Under assumptions (H1)-(H3) and (D1)-(D3),
+(2.9) sup
+y/ba∇dblGI(·,t,;y)/ba∇dblLp(x),sup
+y/ba∇dbl∂x,yGI(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
+2(1−1
+p)
+for all2≤p≤ ∞,t≥0, whereC >0is independent of p.
+Proof.From representation (2.7)(ii) and ℜλj(ξ)≤ −θ|ξ|2, we obtain by the triangle in-
+equality
+(2.10) /ba∇dblGI/ba∇dblL∞(x,y)≤C/ba∇dble−θ|ξ|2tφ(ξ)/ba∇dblL1(ξ)≤C(1+t)−d
+2,
+verifying the bounds for p=∞. Derivative bounds follow similarly, since derivatives fa lling
+onqjor ˜qjare harmless, whereas derivatives falling on eiξ·(x−y)bring down a factor of ξ,
+again harmless because of the cutoff function φ.
+To obtain bounds for p= 2, we note that (2.7)(ii) may be viewed itself as a Bloch–
+Fourier decomposition with respect to variable z:=x−y, withyappearing as a parameter.
+Recalling (1.12), we may thus estimate
+(2.11)sup
+y/ba∇dblGI(x,t;y)/ba∇dblL2(x)=/summationdisplay
+jsup
+y/ba∇dblφ(ξ)eλj(ξ)tqj(·,z1)˜q∗
+j(·,y1)/ba∇dblL2(ξ;L2(z1∈[0,X]))
+≤C/summationdisplay
+jsup
+y/ba∇dblφ(ξ)e−θ|ξ|2t/ba∇dblL2(ξ)/ba∇dblqj/ba∇dblL2(0,X)/ba∇dbl˜qj/ba∇dblL∞(0,X)
+≤C(1+t)−d
+4,
+where we have used in a crucial way the boundedness of ˜ qj; derivative bounds follow simi-
+larly. Finally, bounds for 2 ≤p≤ ∞follow byLp-interpolation.
+Remark 2.4. In obtaining the key L2-estimate, we have used in an essential way the
+periodic structure of qj, ˜qj. For, viewing GIas a general pseudodifferential expression
+rather than a Bloch–Fourier decomposition, we find that the s moothness of qj, ˜qjis not
+sufficient to apply standard L2→L2bounds of H¨ ormander, which require blowup in ξ
+derivatives at less than the critical rate |ξ|−1found here; see, e.g., [H] for further discussion.
+Nor do the weighted energy estimate techniques used in [S1, S 2, S3] apply here, as these also
+rely on the property of smoothness of λj,qj, ˜qjwith respect to ξat the origin ξ= 0. The
+lack of smoothness of the linearized dispersion relation at the origin is an essential technical
+difference separating the conservation law from the reaction diffusion case; see [OZ4] for
+further discussion.
+Remark 2.5. Underlying the above analysis, and also the technically rat her different
+approach of [OZ2], is the fundamental relation
+(2.12) G(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)[Gξ(x1,t;y1)]dξ2 BASIC LINEARIZED STABILITY ESTIMATES 11
+which, provided σ(Lξ) is semisimple, yields the simple formula
+G(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)/summationdisplay
+jeλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ
+resembling that of the constant-coefficient case, where λjruns through the spectrum of Lξ.
+The basic idea in both cases is to separate off the principal pa rt of the series involving small
+λj(ξ) and estimate the remainder as a faster-decaying residual.
+Corollary 2.6 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3), for all p≥2,t≥0,
+(2.13) /ba∇dblSI(t)f/ba∇dblLp,/ba∇dbl∂xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂xf/ba∇dblLp≤C(1+t)−d
+2(1−1
+p)/ba∇dblf/ba∇dblL1.
+Proof.Immediate, from (2.9) and the triangle inequality, as, for e xample,
+/ba∇dblSI(t)f(·)/ba∇dblLp=/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+RdGI(x,t;y)f(y)dy/vextenddouble/vextenddouble/vextenddouble
+Lp(x)≤/integraldisplay
+Rdsup
+y/ba∇dblGI(·,t;y)/ba∇dblLp|f(y)|dy.
+Proposition 2.1 ([OZ4]).Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,
+p≥2,0≤l≤K,
+(2.14) /ba∇dblS(t)∂l
+xu0/ba∇dblLp≤Ct−l
+2(1+t)−d
+2(1
+2−1
+p)+l
+2t−d
+4−l
+2/ba∇dblu0/ba∇dblL1∩L2.
+Proof.Immediate, from (2.4) and (2.13).
+2.3 Additional estimates
+Lemma 2.7. Assuming (H1)–(H3), (D1)–(D3), for all t≥0,0≤l≤K,
+(2.15) /ba∇dbl∂l
+xSI(t)f/ba∇dblLp(x),/ba∇dblSI(t)∂l
+xf/ba∇dblLp(x)≤C(1+t)−d
+2(1/2−1/p)/ba∇dblf/ba∇dblL2(x).
+Proof.From boundedness of the spectral projections Pj(ξ) =qj/an}b∇acketle{t˜qj,·/an}b∇acket∇i}htinL2[0,X] and their
+derivatives, another consequence of first-order splitting of eigenvalues λj(ξ) at the origin,
+we obtain boundedness of φ(ξ)P(ξ)eLξtand thus, by (1.12), the global bounds
+(2.16) /ba∇dbl∂l
+xSI(t)f/ba∇dblL2(x),/ba∇dblSI(t)∂l
+xf/ba∇dblL2(x)≤C/ba∇dblf/ba∇dblL2(x),
+for allt≥0, yielding the result for p= 2. Moreover, by boundedness of ˜ q,qin allLp(x1),
+we have
+|φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|P(ξ)ˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|ˆf(ξ,·)|L2(x1),3 REFINED LINEARIZED ESTIMATES 12
+C, θ>0, yielding by SIf=/parenleftBig
+1
+2π/parenrightBigd/integraltextπ
+−π/integraltext
+Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆf(ξ,x1)dξ1d˜ξthe bound
+(2.17)/ba∇dblSI(t)f/ba∇dblL∞(x)≤/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1|φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)dξ1d˜ξ
+≤/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1Cφ(ξ)e−θ|ξ|2t|ˆf(ξ,·)|L2(x1)dξ1d˜ξ
+≤C|φ(ξ)e−θ|ξ|2t|L2(ξ)|ˆf|L2(ξ,x1)
+=C(1+t)−d
+4/ba∇dblf/ba∇dblL2([0,X]),
+yielding the result for p=∞,l= 0. The result for p=∞, 1≤l≤Kfollows by a similar
+argument. The result for general 2 ≤p≤ ∞then follows by Lpinterpolation between p= 2
+andp=∞.
+By Riesz–Thorin interpolation between (2.15) and (2.13), w e obtain the following, ap-
+parently sharp bounds between various LqandLp.2
+Corollary 2.8. Assuming (H0)–(H3) and (D1)–(D3), for all 1≤q≤2≤p,t≥0,
+0≤l≤K,
+(2.18) /ba∇dbl∂l
+xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂l
+xf/ba∇dblLp≤C(1+t)−d
+2(1
+q−1
+p)/ba∇dblf/ba∇dblLq.
+Proposition 2.2. Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,1≤q≤
+2≤p, and0≤l≤K,
+(2.19) /ba∇dblS(t)∂l
+xu0/ba∇dblLp≤C(1+t)−d
+2(1
+2−1
+p)+l
+2t−d
+2(1
+q−1
+2)−l
+2/ba∇dblu0/ba∇dblLq∩L2.
+Proof.Immediate, from (2.4) and (2.8).
+3 Refined linearized estimates
+The bounds of Proposition 2.1 are sufficient to establish nonl inear stability and asymptotic
+behavior in dimensions d≥3, as shown in [OZ4]. However, they are not sufficient in the
+critical dimensions d= 1,2; see Remark 1, Section 7 of [OZ4]. Comparison with standard
+diffusive stability arguments as in [Z7] show that this is due t o the fact that the full solution
+operator |S(t)∂x|decays no faster than S(t), or, equivalently, Gyno faster than G.
+Following the basic strategy introduced in [ZH, Z1, MaZ2, Ma Z4] in the context of vis-
+cous shock waves, we now perform a refined linearized estimat e separating slower-decaying
+translational modes from a faster-decaying “good” part of t he solution operator. This will
+be used in Section 4 in combination with certain nonlinear ca ncellation estimates to show
+convergence to the modulated approximation (1.1) at a faste r rate sufficient to close the
+nonlinear iteration.
+The key to this decomposition is the following observation.
+2The inclusion of general p≥2 in Lemma 2.7 repairs an omission in [OZ4], where the bounds ( 2.8) were
+stated but not used.3 REFINED LINEARIZED ESTIMATES 13
+Lemma 3.1. Assuming (H1)–(H3), (D1)–(D3), let λj(ξ/|ξ|,ξ),qj(ξ/|ξ|,ξ,·),˜qj(ξ/|ξ|,ξ,·)
+denote the eigenvalues and associated right and left eigenf unctions of Lξ, withqj,˜qjsmooth
+functions of ξ/|ξ|and|ξ|as noted in Prop. 2.2. Then, without loss of generality, q1(ω,0,·)≡
+¯u′, while˜qj(ω,0,·)forj/ne}ationslash= 1are constant functions depending only on angle ω=ξ/|ξ|.
+Proof.Expanding Lξ=L0+|ξ|L1
+ξ/|ξ|+|ξ|2L2
+ξ/|ξ|as in the introduction, consider the con-
+tinuous family of spectral perturbation problems in |ξ|indexed by angle ω=ξ/|ξ|. Then,
+both facts follow by standard perturbation theory [K] using the observations that ¯ u′is in
+the right kernel of L0and constant functions care in the left kernel of L0, with
+/an}b∇acketle{tc,L1¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,(ω1(2∂x1−A1)−/summationdisplay
+j/negationslash=1ωjAj))¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,ω1∂2
+x1¯u−/summationdisplay
+j/negationslash=1ωj∂x1fj(¯u)/an}b∇acket∇i}ht ≡0,
+where/an}b∇acketle{t·,·/an}b∇acket∇i}htdenotesL2(x1) inner product on the interval x1∈[0,X], that the dimension
+of kerL0by assumption is ( n+ 1), so that the orthogonal complement of ¯ u′in KerL0
+is dimension nso exactly the set of constant functions, and that by (H3) the functions
+qj(ω,0,·) and ˜qj(ω,0) are right and left eigenfunctions of Π 0L1|kerL0(Π0as earlier denoting
+the zero eigenprojection associated with L0).
+Remark 3.2. The key observation of Lemma 3.1 can be motivated by the form o f the
+Whitham averaged system (1.2). For, recalling (Section 1.3 ) that (D3) implies that speed
+sis stationary to first order at ¯ ualong the manifold of nearby periodic solutions, we find
+that the last equation of (1.2) reduces to ( ∇xΨ)t= 0, i.e., the equation for the translational
+variation Ψ decouples from the equations for variations in o ther modes. This corresponds
+heuristically to the fact derived above that the translatio nal mode ¯u′(x1) decouples in the
+first-order eigenfunction expansion.
+Corollary 3.1. Under assumptions (H1)–(H3), (D1)–(D3), the Green functio nG(x,t;y)
+of(1.7)decomposes as G=E+˜G,
+(3.1) E= ¯u′(x)e(x,t;y),
+where, for some C >0, allt>0,1≤q≤2≤p≤ ∞,0≤j,k,l,j+l≤K,1≤r≤2,
+(3.2)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)t−1
+2(1/q−1/2)|f|Lq∩L2,
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂r
+y˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)−1
+2+r
+2
+×t−d
+2(1/q−1/2)−r
+2|f|Lq∩L2,
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂r
+t˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)−1
+2+r
+×t−d
+2(1/q−1/2)−r|f|Lq∩L2.3 REFINED LINEARIZED ESTIMATES 14
+(3.3)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂j
+x∂k
+t∂l
+ye(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp≤(1+t)−d
+2(1/q−1/p)−(j+k)
+2|f|Lq.
+Moreover,e(x,t;y)≡0fort≤1.
+Proof.We first treat the simpler case q= 1. Recalling that
+(3.4) GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
+define
+(3.5) ˜e(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)eλ1(ξ)t˜q1(ξ,y1)∗dξ,
+so that
+(3.6)
+GI(x,t;y)−¯u′(x1)˜e(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=2eλj(ξ)tqj(ξ/|ξ|,0,x1)˜qj(ξ,y1)∗dξ
++/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdn+1/summationdisplay
+j=1eiξ·(x−y)φ(ξ)eλj(ξ)tO(|ξ|)dξ.
+Noting, by Lemma 3.1, that ∂y˜q(ω,0,y)≡constant for j/ne}ationslash= 1, we have therefore
+(3.7)∂r
+y(GI(x,t;y)−¯u′(x1)˜e(x,t;y)) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tO(|ξ|)dξ,
+which readily gives
+(3.8) |∂r
+y(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤C(1+t)−d
+2(1−1/p)−1
+2,
+p≥2, by the same argument used to prove (2.9), and similarly
+(3.9) |∂r
+t(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤c(1+t)−d
+2(1−1/p)−1
+2.
+These yield (3.2) by the triangle inequality.
+Defininge(x,t;y) :=χ(t)˜e(x,t;y), whereχisasmoothcutofffunctionsuchthat χ(t)≡1
+fort≥2 andχ(t)≡0 fort≤1, and setting ˜G:=G−¯u′(x1)e(x,t;y), we readily obtain the
+estimates (3.2) by combining (3.9) with bound (2.4) on GII. Bounds (3.3) follow from (3.5)
+by the argument used to prove (2.9), together with the observ ation thatx- ort-derivatives
+bring down factors of |ξ|, followed again by an application of the triangle inequalit y.
+Thecases1 ≤q≤2followsimilarly, bytheargumentsusedtoprove(2.15)and (2.8).4 NONLINEAR STABILITY IN DIMENSION ONE 15
+Remark 3.3. Despite their apparent complexity, the above bounds may be r ecognized
+as essentially just the standard diffusive bounds satisfied fo r the heat equation [Z7]. For
+dimensiond= 1, it may be shown using pointwise techniques as in [OZ2] tha t the bounds
+of Corollary 3.1 extend to all 1 ≤q≤p≤ ∞.
+Note the strong analogy between the Green function decompos ition of Corollary 3.1
+and that of [MaZ3, Z4] in the viscous shock case. We pursue thi s analogy further in the
+nonlinearanalysisofthefollowingsections, combiningth e“instantaneous tracking” strategy
+of [ZH, Z1, Z4, Z7, MaZ2, MaZ4] with a type of cancellation est imate introduced in [HoZ].
+4 Nonlinear stability in dimension one
+For clarity, we carry out the nonlinear stability analysis i n detail in the most difficult,
+one-dimensional, case, indicating afterward by a few brief remarks the extension to d= 2.
+Hereafter, take x∈R1, dropping the indices on fjandxjand writing ut+f(u)x=uxx.
+4.1 Nonlinear perturbation equations
+Given a solution ˜ u(x,t) of (1.4), define the nonlinear perturbation variable
+(4.1) v=u−¯u= ˜u(x+ψ(x,t))−¯u(x),
+where
+(4.2) u(x,t) := ˜u(x+ψ(x,t))
+andψ:R×R→Ris to be chosen later.
+Lemma 4.1. Forv,uas in(4.1),(4.2),
+(4.3) ut+f(u)x−uxx= (∂t−L)¯u′(x1)ψ(x,t)+∂xR+(∂t+∂2
+x)S,
+where
+R:=vψt+vψxx+(¯ux+vx)ψ2
+x
+1+ψx=O(|v|(|ψt|+|ψxx|)+/parenleftBig|¯ux|+|vx|
+1−|ψx|/parenrightBig
+|ψx|2)
+and
+S:=−vψx=O(|v|(|ψx|).
+Proof.To begin, notice from the definition of uin (4.2) we have by a straightforward
+computation
+ut(x,t) = ˜ux(x+ψ(x,t),t)ψt(x,t)+ ˜ut(x+ψ,t)
+f(u(x,t))x=df(˜u(x+ψ(x,t),t))˜ux(x+ψ,t)·(1+ψx(x,t))4 NONLINEAR STABILITY IN DIMENSION ONE 16
+and
+uxx(x,t) = (˜ux(x+ψ(x,t),t)·(1+ψx(x,t)))x
+= ˜uxx(x+ψ(x,t),t)·(1+ψx(x,t))+(˜ux(x+ψ(x,t),t)·ψx(x,t))x.
+Using the fact that ˜ ut+df(˜u)˜ux−˜uxx= 0, it follows that
+(4.4)ut+f(u)x−uxx= ˜uxψt+df(˜u)˜uxψx−˜uxxψx−(˜uxψx)x
+= ˜uxψt−˜utψx−(˜uxψx)x
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x)) = 0
+by translation invariance, we have
+(∂t−L)¯u′(x)ψ= ¯uxψt−¯utψx−(¯uxψx)x.
+Subtracting, and using the facts that, by differentiation of ( ¯u+v)(x,t) = ˜u(x+ψ,t),
+(4.5)¯ux+vx= ˜ux(1+ψx),
+¯ut+vt= ˜ut+ ˜uxψt,
+so that
+(4.6)˜ux−¯ux−vx=−(¯ux+vx)ψx
+1+ψx,
+˜ut−¯ut−vt=−(¯ux+vx)ψt
+1+ψx,
+we obtain
+ut+f(u)x−uxx= (∂t−L)¯u′(x)ψ+vxψt−vtψx−(vxψx)x+/parenleftBig
+(¯ux+vx)ψ2
+x
+1+ψx/parenrightBig
+x,
+yielding (4.3) by vxψt−vtψx= (vψt)x−(vψx)tand (vxψx)x= (vψx)xx−(vψxx)x.
+Corollary 4.2. The nonlinear residual vdefined in (4.1)satisfies
+(4.7) vt−Lv= (∂t−L)¯u′(x1)ψ−Qx+Rx+(∂t+∂2
+x)S,
+where
+(4.8) Q:=f(˜u(x+ψ(x,t),t))−f(¯u(x))−df(¯u(x))v=O(|v|2),
+(4.9) R:=vψt+vψxx+(¯ux+vx)ψ2
+x
+1+ψx,
+and
+(4.10) S:=−vψx=O(|v|(|ψx|).
+Proof.Taylor expansion comparing (4.3) and ¯ ut+f(¯u)x−¯uxx= 0.4 NONLINEAR STABILITY IN DIMENSION ONE 17
+4.2 Cancellation estimate
+Our strategy in writing (4.7) is motivated by the following b asic cancellation principle.
+Proposition 4.3 ([HoZ]).For anyf(y,s)∈Lp∩C2withf(y,0)≡0, there holds
+(4.11)/integraldisplayt
+0/integraldisplay
+G(x,t−s;y)(∂s−Ly)f(y,s)dyds=f(x,t).
+Proof.Integrating the left hand side by parts, we obtain
+(4.12)/integraldisplay
+G(x,0;y)f(y,t)dy−/integraldisplay
+G(x,t;y)f(y,0)dy+/integraldisplayt
+0/integraldisplay
+(∂t−Ly)∗G(x,t−s;y)f(y,s)dyds.
+Noting that, by duality,
+(∂t−Ly)∗G(x,t−s;y) =δ(x−y)δ(t−s),
+δ(·) here denoting the Dirac delta-distribution, we find that th e third term on the righthand
+side vanishes in (4.12), while, because G(x,0;y) =δ(x−y), the first term is simply f(x,t).
+The second term vanishes by f(y,0)≡0.
+Remark 4.1. Forψ=ψ(t), term (∂t−L)¯u′ψin (4.7) reduces to the term ˙ψ(t)¯u′(x)
+appearing in the shock wave case [ZH, Z1, Z4, Z7, MaZ2, MaZ4].
+4.3 Nonlinear damping estimate
+Proposition 4.2. Letv0∈HK(Kas in (H1)), and suppose that for 0≤t≤T, theHK
+norm ofvand theHK(x,t)norms ofψtandψxremain bounded by a sufficiently small
+constant. There are then constants θ1,2>0so that, for all 0≤t≤T,
+(4.13) |v(t)|2
+HK≤Ce−θ1t|v(0)|2
+HK+C/integraldisplayt
+0e−θ2(t−s)/parenleftBig
+|v|2
+L2+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+(s)ds.
+Proof.Subtracting from the equation (4.4) for uthe equation for ¯ u, we may write the
+nonlinear perturbation equation as
+(4.14) vt+(df(¯u)v)x−vxx=Q(v)x+ ˜uxψt−˜utψx−(˜uxψx)x,
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Using (4.6) to replace ˜ uxand ˜utrespectively by ¯ ux+vx−(¯ux+vx)ψx
+1+ψx
+and ¯ut+vt−(¯ux+vx)ψt
+1+ψx, and moving the resulting vtψxterm to the lefthand side of
+(4.14), we obtain
+(4.15)(1+ψx)vt−vxx=−(df(¯u)v)x+Q(v)x+ ¯uxψt
+−((¯ux+vx)ψx)x+/parenleftBig
+(¯ux+vx)ψ2
+x
+1+ψx/parenrightBig
+x.4 NONLINEAR STABILITY IN DIMENSION ONE 18
+Taking the L2inner product in xof/summationtextK
+j=0∂2j
+xv
+1+ψxagainst (4.15), integrating by parts, and
+rearranging the resulting terms, we arrive at the inequalit y
+∂t|v|2
+HK(t)≤ −θ|∂K+1
+xv|2
+L2+C/parenleftBig
+|v|2
+HK+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+,
+for someθ >0,C >0, so long as |˜u|HKremains bounded, and |v|HKand|(ψt,ψx)|HK(x,t)
+remain sufficiently small. Using the Sobolev interpolation |v|2
+HK≤ |∂K+1
+xv|2
+L2+˜C|v|2
+L2for
+˜C >0 sufficiently large, we obtain ∂t|v|2
+HK(t)≤ −˜θ|v|2
+HK+C/parenleftBig
+|v|2
+L2+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+from which (4.13) follows by Gronwall’s inequality.
+4.4 Integral representation/ ψ-evolution scheme
+By Proposition 4.3, we have, applying Duhamel’s principle t o (4.7),
+(4.16)v(x,t) =/integraldisplay∞
+−∞G(x,t;y)v0(y)dy
++/integraldisplayt
+0/integraldisplay∞
+−∞G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds+ψ(t)¯u′(x).
+Definingψimplicitly as
+(4.17)ψ(x,t) =−/integraldisplay∞
+−∞e(x,t;y)u0(y)dy
+−/integraldisplayt
+0/integraldisplay+∞
+−∞e(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
+following [ZH, Z4, MaZ2, MaZ3], where eis defined as in (3.1), and substituting in (4.16)
+the decomposition G= ¯u′(x)e+˜Gof Corollary 3.1, we obtain the integral representation
+(4.18)v(x,t) =/integraldisplay∞
+−∞˜G(x,t;y)v0(y)dy
++/integraldisplayt
+0/integraldisplay∞
+−∞˜G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
+and, differentiating (4.17) with respect to t, and recalling that e(x,s;y)≡0 fors≤1,
+(4.19)∂j
+t∂k
+xψ(x,t) =−/integraldisplay∞
+−∞∂j
+t∂k
+xe(x,t;y)u0(y)dy
+−/integraldisplayt
+0/integraldisplay+∞
+−∞∂j
+t∂k
+xe(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds.
+Equations (4.18), (4.19) together form a complete system in the variables ( v,∂j
+tψ,∂k
+xψ),
+0≤j≤1, 0≤k≤K, from the solution of which we may afterward recover the shif tψvia
+(4.17). From the original differential equation (4.7) togeth er with (4.19), we readily obtain
+short-time existence and continuity with respect to tof solutions ( v,ψt,ψx)∈HKby a
+standard contraction-mapping argument based on (4.13), (4 .17), and and (3.3).4 NONLINEAR STABILITY IN DIMENSION ONE 19
+4.5 Nonlinear iteration
+Associated with the solution ( u,ψt,ψx) of integral system (4.18)–(4.19), define
+(4.20)ζ(t) := sup
+0≤s≤t|(v,ψt,ψx)|HK(s)(1+s)1/4.
+Lemma 4.3. For allt≥0for whichζ(t)is finite, some C >0, andE0:=|u0|L1∩HK,
+(4.21) ζ(t)≤C(E0+ζ(t)2).
+Proof.By (4.9)–(4.10) and definition (4.20),
+(4.22) |(Q,R,S)|L1∩L∞≤ |(v,vx,ψt,ψx)|2
+L2+|(v,vx,ψt,ψx)|2
+L∞≤Cζ(t)2(1+t)−1
+2,
+so long as |ψx| ≤ |ψx|HK≤ζ(t) remains small, and likewise (using the equation to bound t
+derivatives in terms of x-derivatives of up to two orders)
+(4.23) |(∂t+∂2
+x)S|L1∩L∞≤ |(v,ψx)|2
+H2+|(v,ψx)|2
+W2,∞≤Cζ(t)2(1+t)−1
+2.
+Applying Corollary 3.1 with q= 1,d= 1 to representations (4.18)–(4.19), we obtain for
+any 2≤p<∞
+(4.24)|v(·,t)|Lp(x)≤C(1+t)−1
+2(1−1/p)E0
++Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1/2−1/p)(t−s)−3
+4(1+s)−1
+2ds
+≤C(E0+ζ(t)2)(1+t)−1
+2(1−1/p)
+and
+(4.25)
+|(ψt,ψx)(·,t)|WK,p≤C(1+t)−1
+2E0+Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1−1/p)−1/2(1+s)−1
+2ds
+≤C(E0+ζ(t)2)(1+t)−1
+2(1−1/p).
+Using (4.13) and(4.24)–(4.25), we obtain |v(·,t)|HK(x)≤C(E0+ζ(t)2)(1+t)−1
+4. Combining
+this with (4.25), p= 2, rearranging, and recalling definition (4.20), we obtain (4.3).
+Proof of Theorem 1.1. By short-time HKexistence theory, /ba∇dbl(v,ψt,ψx)/ba∇dblHKis continuous
+so long as it remains small, hence ηremains continuous so long as it remains small. By
+(4.3), therefore, it follows by continuous induction that η(t)≤2Cη0fort≥0, ifη0<1/4C,
+yielding by (4.20) the result (1.15) for p= 2. Applying (4.24)–(4.25), we obtain (1.15) for
+2≤p≤p∗for anyp∗<∞, with uniform constant C. Takingp∗>4 and estimating
+|Q|L2,|R|L2,|S|L2(t)≤ |(v,ψt,ψx)|2
+L4≤CE0(1+t)−3
+45 NONLINEAR STABILITY IN DIMENSION TWO 20
+in place of the weaker (4.22), then applying Corollary 3.1 wi thq= 2,d= 1, we obtain
+finally (1.15) for 2 ≤p≤ ∞, by a computation similar (4.24)–(4.25); we omit the detail s of
+this final bootstrap argument. Estimate (1.16) then follows using (3.3) with q=d= 1, by
+(4.26)
+|ψ(t)|Lp≤CE0+Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1−1/p)(1+s)−1
+2ds≤C(1+t)1
+2p(E0+ζ(t)2),
+together with the fact that ˜ u(x,t)−¯u(x) =v(x−ψ,t)+(¯u(x)−¯u(x−ψ),so that|˜u(·,t)−¯u|
+is controlled by the sum of |v|and|¯u(x)−¯u(x−ψ)| ∼ |ψ|. This yields stability for
+|u−¯u|L1∩HK|t=0sufficiently small, as described in the final line of the theore m.
+5 Nonlinear stability in dimension two
+We now briefly sketch the extension to dimension d= 2. Given a solution ˜ u(x,t) of (1.4),
+define the nonlinear perturbation variable
+(5.1) v=u−¯u= ˜u(x1+ψ(x,t),x2,t)−¯u(x1),
+where
+(5.2) u(x,t) := ˜u(x1+ψ(x,t),t)
+andψ:Rd×R→Ris to be chosen later.
+Lemma 5.1. Forv,uas in(5.2),
+(5.3)ut+d/summationdisplay
+j=1fj(u)xj−d/summationdisplay
+j=1uxjxj= (∂t−L)¯u′(x1)ψ(x,t)+d/summationdisplay
+j=1∂xjRj+∂tS+T,
+where
+Rj=O((|v,ψt,ψx)||(v,vx,ψt,ψx)|), S:=−vψx1= (|v|(|ψx|), T:=O(|ψx|3+|(v,ψx)||ψxx|).
+Proof.Similarly as in the proof of Lemma 4.1, it follows by a straigh tforward computation
+Using the fact that ˜ ut+/summationtext
+jdfj(˜u)˜uxj−/summationtext
+j˜uxjxj= 0, it follows that
+(5.4)ut+/summationdisplay
+jdfj(u)uxj−/summationdisplay
+juxjxj= ˜ux1ψt−˜utψx1+/summationdisplay
+j/negationslash=1dfj(˜u)˜ux1ψxj
+−/summationdisplay
+j/negationslash=1˜uxjx1ψxj−/summationdisplay
+j(˜ux1ψxj)xj,
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x1)) = 0
+by translation invariance, we have
+(∂t−L)¯u′(x1)ψ= ¯ux1ψt−¯utψx1+/summationdisplay
+j/negationslash=1dfj(¯u)¯ux1ψxj−/summationdisplay
+j/negationslash=1¯uxjx1ψxj−/summationdisplay
+j(¯ux1ψxj)xj.5 NONLINEAR STABILITY IN DIMENSION TWO 21
+Subtracting, and using (4.5) and
+(5.5)¯uxj+vxj= ˜uxj+ ˜ux1ψxj,
+¯ut+vt= ˜ut+ ˜ux1ψt,
+so that
+(5.6)˜uxj−¯uxj−vxj=−(¯ux1+vx1)ψxj
+1+ψx1,
+˜ut−¯ut−vt=−(¯ux1+vx1)ψt
+1+ψx1,
+we obtain
+ut+/summationdisplay
+jdfj(u)uxj−/summationdisplay
+juxjxj= (∂t−L)¯u′(x1)ψ+vx1ψt−vtψx1
++/summationdisplay
+j/negationslash=1(dfj(˜u)˜ux1−dfj(¯u)¯ux1)ψxj
+−/summationdisplay
+j/negationslash=1(˜uxjx1−¯uxjx1)ψxj−/summationdisplay
+j((˜ux1−¯ux1)ψxj)xj.
+Usingvx1ψt−vtψx1= (vψt)x1−(vψx1)t,
+dfj(˜u)˜ux1=f(u)x1−dfj(˜u)˜ux1ψx1=f(u)x1(1−ψx)−dfj(˜u)˜ux1ψ2
+x1,
+and ˜uxjx1= (˜uxj)x1−˜uxjx1ψx1= (˜uxj)x1(1−ψx1)+ ˜uxjx1ψ2
+x1,and rearranging, we obtain
+ut+/summationdisplay
+jdfj(u)uxj−/summationdisplay
+juxjxj= (∂t−L)¯u′(x1)ψ+(vψt)x1−(vψx1)t
++/summationdisplay
+j/negationslash=1(fj(u)−fj(¯u))x1)ψxj
+−/summationdisplay
+j/negationslash=1f(u)x1ψx1ψxj−/summationdisplay
+j/negationslash=1dfj(˜u)˜ux1ψ2
+x1ψxj
+−/summationdisplay
+j/negationslash=1(˜uxj−¯uxj)x1ψxj+/summationdisplay
+j/negationslash=1(˜uxj)x1ψx1ψxj
++/summationdisplay
+j/negationslash=1˜uxjx1ψ2
+x1ψxj
+−/summationdisplay
+j(vx1ψx1)xj−/summationdisplay
+j/parenleftBig
+(¯ux1+vx1)ψxjψx1
+1+ψx1/parenrightBig
+xj.
+Noting that
+(fj(u)−fj(¯u))x1)ψxj= ((fj(u)−fj(¯u)ψxj)x1−(fj(u)−fj(¯u))ψxjx1,5 NONLINEAR STABILITY IN DIMENSION TWO 22
+f(u)x1ψx1ψxj= (f(u)ψx1ψxj)x1−f(u)(ψx1ψxj)x1,
+and
+(˜uxj−¯uxj)x1ψxj= ((˜uxj−¯uxj)ψxj)x1−(˜uxj−¯uxj)ψxjx1,
+with|fj(u)−fj(¯u)|=O(|v|) and|˜uxj−¯uxj|=O(|v|),we obtain the result
+Proof of Theorem 1.2. The result of Lemma 5.1 is the only part of the analysis that di ffers
+essentially from that of the one-dimensional case. The canc ellation and nonlinear damping
+arguments go through exactly as before to yield the analogs o f Propositions 4.3 and (4.2).
+Likewise, we obtain a Duhamel representation analogous to ( 4.18)–(4.19), forming a closed
+system in variables ( v,ψx,ψt).
+To obtain the analog of Lemma 4.3, completing the proof of non linear stability, we can
+carry out a somewhat simpler argument than in the one-dimens ional case, using Corollary
+3.1 withd= 2,q= 2 for all estimates, not only the final bootstrap argument, g iving in
+place of (4.24) the estimate
+(5.7)
+|v(·,t)|Lp(x)≤C(1+t)−(1−1/p)E0+Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(t−s)−1
+2(1+s)−1ds
+≤C(E0+ζ(t)2)(1+t)−(1−1/p),
+(5.8)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
+2E0
++Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(t−s)−1
+2(1+s)−1ds
+≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
+2
+for divergence-form source terms, and
+(5.9)|v(·,t)|Lp(x)≤Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(1+s)−3
+2ds
+≤C(E0+ζ(t)2)(1+t)−(1−1/p),
+(5.10)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
+2E0
++Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(t−s)−1
+2(1+s)−3
+2ds
+≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
+2
+for faster-decaying nondivergence-form source terms.
+We omit the details, which are entirely similar to, but subst antially simpler than, those
+of the one-dimensional case.REFERENCES 23
+References
+[G] R. Gardner, On the structure of the spectra of periodic traveling waves , J. Math.
+Pures Appl. 72 (1993), 415-439.
+[GZ] R. Gardner and K. Zumbrun, The Gap Lemma and geometric criteria for in-
+stability of viscous shock profiles , Comm. Pure Appl. Math. 51 (1998), no. 7,
+797–85.
+[He] D. Henry, Geometric theory of semilinear parabolic equations , Lecture Notes in
+Mathematics, Springer–Verlag, Berlin (1981).
+[HoZ] D. Hoff and K. Zumbrun Asymptotic behavior of multidimensional scalar viscous
+shock fronts , Indiana Univ. Math. Journal, Vol. 49, No. 2 (2000).
+[H] I.L. Hwang, TheL2-boundedness of pseudodifferential operators, Trans. Amer.
+Math. Soc. 302 (1987) 55–76.
+[JZ1] M. Johnson and K. Zumbrun, Rigorous Justification of the Whitham Modulation
+Equations for the Generalized Korteweg-de Vries Equation, preprint (2009).
+[JZB] M. Johnson, K. Zumbrun, and J. Bronski, Bloch wave expansion vs. Whitham
+Modulation Equations for the Generalized Korteweg-de Vrie s Equation, inprepa-
+ration.
+[K] T. Kato, Perturbation theory for linear operators , Springer–Verlag, Berlin Hei-
+delberg (1985).
+[MaZ2] C. Mascia and K. Zumbrun, Stability of small-amplitude shock profiles of sym-
+metric hyperbolic-parabolic systems, Comm. Pure Appl. Math. 57 (2004), no. 7,
+841–876.
+[MaZ3] C. Mascia and K. Zumbrun, Pointwise Green function bounds for shock profiles
+of systems with real viscosity. Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177–
+263.
+[MaZ4] C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles
+of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (2004), no. 1,
+93–131.
+[OZ1] M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation
+laws with viscosity- 1. Analysis of the Evans function , Arch. Ration. Mech. Anal.
+166 (2003), no. 2, 99–166.
+[OZ2] M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation
+laws with viscosity- Pointwise bounds on the Green function , Arch.Ration. Mech.
+Anal. 166 (2003), no. 2, 167–196.REFERENCES 24
+[OZ3] M. Oh, and K. Zumbrun, Low-frequency stability analysis of periodic traveling-
+wave solutions of viscous conservation laws in several dime nsions, Journal for
+Analysis and its Applications, 25 (2006), 1–21.
+[OZ4] M. Oh, and K. Zumbrun, Stability and asymptotic behavior of traveling-wave
+solutions of viscous conservation laws in several dimensio ns, to appear, Arch.
+Ration. Mech. Anal.
+[Pa] A. Pazy, Semigroups of linear operators and applications to partial differen-
+tial equations, Applied Mathematical Sciences, 44, Springer-Verlag, New Y ork-
+Berlin, (1983) viii+279 pp. ISBN: 0-387-90845-5.
+[S1] G.Schneider, Nonlinear diffusive stability of spatially periodic soluti ons– abstract
+theorem and higher space dimensions , Proceedings of the International Confer-
+ence on Asymptotics in Nonlinear Diffusive Systems (Sendai, 1 997), 159–167,
+Tohoku Math. Publ., 8, Tohoku Univ., Sendai, 1998.
+[S2] G. Schneider, Diffusive stability of spatial periodic solutions of the Swi ft-
+Hohenberg equation, (English. English summary) Comm. Math. Phys. 178
+(1996), no. 3, 679–702.
+[S3] G. Schneider, Nonlinear stability of Taylor vortices in infinite cylinder s,Arch.
+Rat. Mech. Anal. 144 (1998) no. 2, 121–200.
+[Se1] D. Serre, Spectral stability of periodic solutions of viscous conser vation laws:
+Large wavelength analysis , Comm. Partial Differential Equations 30 (2005), no.
+1-3, 259–282.
+[Z1] K. Zumbrun, Refined wave–tracking and stability of viscous Lax shocks , Methods
+Appl. Anal. 7 (2000) 747–768.
+[Z4] K. Zumbrun, Stability of large-amplitude shock waves of compressible N avier–
+Stokes equations, withanappendixbyHelge KristianJenssenandGregoryLyng,
+in Handbook of mathematical fluid dynamics. Vol. III, 311–53 3, North-Holland,
+Amsterdam, (2004).
+[Z6] K. Zumbrun, Dynamical stability of phase transitions in the p-system wi th
+viscosity-capillarity , SIAM J. Appl. Math. 60 (2000), 1913-1929.
+[Z7] K. Zumbrun, Instantaneous shock location and one-dimensional nonlinea r sta-
+bility of viscous shock waves, preprint (2009).
+[ZH] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of vis-
+cous shock waves . Indiana Mathematics Journal V47 (1998), 741–871; Errata,
+Indiana Univ. Math. J. 51 (2002), no. 4, 1017–1021.

1001.0050.txt

@@ -0,0 +1,509 @@
+arXiv:1001.0050v1  [physics.hist-ph]  30 Dec 2009Fundamental times, lengths and physical constants:
+some unknown contributions by Ettore Majorana
+S. Esposito
+Dipartimento di Scienze Fisiche, Universit` a di Napoli “Fe derico II” and I.N.F.N. Sezione di Napoli,
+Complesso Universitario di Monte S. Angelo, Via Cinthia, 80 126 Naples, Italy∗
+G. Salesi
+Facolt` a di Ingegneria, Universit` a di Bergamo, viale Marc oni 5, 24044 Dalmine (BG) Italy
+and I.N.F.N. Sezione di Milano, via Celoria 16, 20133 Milan, Italy†
+We review the introduction in physics of the concepts of an el ementary space length and of a
+fundamental time scale, analyzing some related unknown con tributions by Ettore Majorana. In
+particular, we discuss the quasi-Coulombian scattering in presence of a finite length scale, as well
+as the introduction of an intrinsic (universal) time delay i n the expressions for the retarded elec-
+tromagnetic potentials. Finally, we also review a special m odel considered by Majorana in order to
+deduce the value of the elementary charge, in such a way antic ipating key ideas later introduced in
+quantum electrodynamics.
+INTRODUCTION: THE KNOWN STORY
+The idea of elementary space or time intervals has been quite a recur ring one in the philosophical and scientific
+literature of every time and, as almost all the fundamental physica l ideas, from time to time it has been recovered
+from oblivion. Typical examples range from the ancient Greek period to our days, namely from Zeno’s paradox on
+Achilles and the tortoise to Poincar´ e and Mach, who regarded the c oncept of continuum as a consequence of our
+physiological limitations.1
+As often shown in the widespread literature on the structure of pa rticles (e.g., see below, the “chronon theory”),
+a time-discretization for an elementary object involves an “interna l” (that is, referred to the center-of-mass frame)
+motion associated with microscopic space distances. For such an ele mentary particle, an “extended-like” structure,
+ratherthanapointlikeone,isexpectedandthishasimportantcons equencesevenforthefundamentaltheorydescribing
+thatparticlessince, indeed, theclassicaltheoryofapointlikechar gedparticleleadstowell-knowndivergencyproblems,
+which are only apparently solved by the renormalization group theor y.
+In the realm of particle physics, the concepts of an elementary time and length are substantially equivalent for
+several reasons: in most cases one can pass from time to space ju st by introducing the speed of light. Already in
+1930’s (see below), it was early realized that a fundamental length is directly related to the existence of a “cut-off” in
+the momentum transferred during the particle-detector interac tion, which is required in order to avoid the emergence
+of the so-called ultraviolet catastrophe in quantum field theories. A nother way to provide a small distance cut-off in
+field theory is to formulate it on a discrete lattice: this approach was introduced in 1940 by Wentzel [2], but only
+later it went into detail, becoming a standard technique in modern num erical approach to high energy interactions,
+as e.g. in quantum chromodynamics.
+Space-time discretizations, or fundamental scales, are introduc ed in classical and quantum theories mainly for two
+reasons: to prevent undesired infinities in some physical quantities , or (though less known) to explain the emergence
+of an intrinsic angular momentum for spinning particles.
+The first appearance in physics of a fundamental length was sugge sted by H.A. Lorentz, who introduced the so-called
+classical electron radius Rcl≡e2/mc2≃2.82·10−13cm in his famous classical relativistic model of the electron;2it is
+equivalently known as “Lorentz radius”or “Thomsonscattering len gth”. Roughly speaking, such a radius corresponds
+to the electron size required for obtaining its mass from the electro static potential energy, assuming that quantum
+mechanicaleffects are irrelevant. Actually, such effects playa non -negligiblerolein the behaviorofchargedparticlesat
+very short distances, and the classical electron radius may be con sidered as the length scale at which renormalization
+effects become relevant in quantum electrodynamics. In other wor ds,Rcldenotes the watershed between classical and
+1“...le temps et l’espace ne repr` esentent, au point de vue ph ysiologique, qu’un continue apparent, qu’ils se composent tr` es vraisemblable-
+ment d’elements discontinus, mais qu’on ne peut distinguer nettement les uns des autres.” [1]
+2Lorentz considered electrons just as rigid spheres, albeit contracting when moving in the ether [3].2
+quantum electrodynamics; it appears in the semiclassical theory fo r the non-relativistic Thomson scattering, as well
+as in the relativistic Klein-Nishina formula [4].
+In early semiclassical models, the electron was substantially regard ed as a charged sphere but, unfortunately, the
+electromagnetic self-energy of such a sphere diverges in the limit of pointlike charge. This was, then, the basic
+motivation that led M.Abraham, Lorentz and, later, P.A.M.Dirac, to derive an equation of motion for the electron
+where a finite, non-vanishing size is fully taken into account [5]. In suc h a way a fundamental length was effectively
+introduced in order to properly describe the motion of charged par ticles. Particularly extensive work on extended,
+not pointlike, charges was carried out by A.Sommerfeld [6]. In stud ying the motion of a sphere of radius Rwith
+a uniform surface charge, he obtained the following “retarded-like ” first-order differential-finite difference equation
+(T≡2R/c):
+ma=e(E+v×B)+2
+3e2
+Rc2v(t−T)−v(t)
+T, (1)
+where the electromagnetic field of the charge itself causes a self-f orce, which has a delayed effect on its motion. The
+“delay time” 2 R/cappearing in the Sommerfeld equation, corresponding to the time th e light takes to go across
+the sphere, was one of the earlier elementary times introduced in mic rophysics. Later in the 1920s, J.J.Thomson [7]
+suggested that the electric force may act in a discontinuous way, p roducing finite increments of momentum separated
+by finite intervals of time.
+The relativistic version of Sommerfeld’s non-relativistic approach wa s developed by P.Caldirola [8]. His theory of
+the electron is one of the first and simplest theories which assumes a prioria minimum time interval: it is based on
+the existence of an elementary proper-time interval, the so-called chronon. When applied to electrons in an external
+electromagnetic field, such a finite-difference theory succeeds (a lready at the classical level) in overcoming all the
+known difficulties met by the Abraham-Lorentz-Dirac (ALD) theory , like the so-called pre-acceleration problem and
+the emergence of run-away solutions [9, 10]. It is also able to give a c lear answer to the problem related to some
+ambiguities associated with the hyperbolic motion of the electron [11], as well as to the question of whether a free
+falling charged particle does or does not emit radiation [12]. Furtherm ore, at the quantum level, Caldirola’s chronon
+theory is seemingly able to explain the origin of the “classical (Schwing er’s) part”, e/planckover2pi1/2mc·α/2π=e3/4πmc2, of
+the anomalous magnetic momentum of the electron as well as the mas s spectrum of the charged leptons [8]. In this
+theory, time flows continuously, but, when an external force act s on the electron, the reaction of the particle to the
+applied force is not continuous: the value of the electron velocity uµis supposed to jump from uµ(τ−τ0) touµ(τ)
+only at certain positions snalong its world line. These “discrete positions” are such that the elec tron takes a time τ0
+for traveling from one position sn−1to the next one sn. In principle, the electron is still considered as pointlike, but
+the ALD equation for the relativistic radiating electron is replaced: ( i) by a corresponding finite–difference (retarded)
+equation in the velocity uµ(τ)
+m
+τ0/braceleftbigg
+uµ(τ)−uµ(τ−τ0)+uµ(τ)uν(τ)
+c2[uν(τ)−uν(τ−τ0)]/bracerightbigg
+=e
+cFµν(τ)uν(τ) (2)
+(which reduces to the ALD equation when τ0≪τ) and (ii) by a second equation (the so-called transmission law ),
+connecting the discrete positions xµ(τ) along the world line of the particle among themselves:
+xµ(nτ0)−xµ((n−1)τ0) =τ0
+2{uµ(nτ0)−uµ[(n−1)τ0]}, (3)
+which is valid inside each discrete interval τ0and describes the “internal” motion of the electron. In such equat ions
+the chronon associated with the electron results to beτ0
+2≡θ0=2
+3ke2
+mc3≃6.266·10−24s (k≡1/4πε0), depending,
+therefore, on the particle properties (namely, on its charge eand on its rest mass m).
+A different elementary length appearing in modern physics is the Compton wavelength , introduced in 1923 by
+A.H.Compton [13] in his explanation of the scattering of photons by e lectrons. It is defined as λC≡h/mc, and
+corresponds to the wavelength of a photon whose energy equals t he particle rest mass; for the electron, the Compton
+wavelength is about 2 .42·10−10cm. The meaning of this length is quite clear: it indicates the watershe d between
+classical mechanics and quantum wave-mechanics, since for spatia l distances below h/mca microsystem behaves as
+a very quantum object. This was promptly realized by E.Schr¨ oding er in 1930-31 [14] in his study of the so-called3
+Zitterbewegung , which emerges in the Dirac theory by means of the Darwin term appe aring in the (decomposed)
+Hamiltonian of the Dirac equation, as found by C.G.Darwin [15] already in 1928. Indeed, because of the Zitterbewe-
+gung (see also below) the electron undergoes extremely rapid fluct uations on scales just of the order of the Compton
+wavelength, causing, for example, the electrons moving inside an at om to experience a smeared nuclear Coulomb
+potential (this peculiar form of microscopic kinetic energy is presen tly better known as “quantum potential”of the
+Schr¨ odinger equation [16]).
+The Compton wavelength plays also the role of the spatial scale which naturally arise in any semi-classical model of
+spinning particles, when attempting to formulate a kinematical pictu re of the intrinsic angular momentum. Already in
+the 1920s, another extended-like structure of particles (in addit ion to the electrodynamical one mentioned above) was
+discovered, namely the mechanical (or, rather, kinematical) configuration in the particle’s center-of -mass frame due
+to the spin. Indeed, in 1925 R. Kronig, G. Uhlenbeck and S. Goudsmit [17] introduced the celebrated self-rotating
+electron hypothesis, suggesting a physical interpretation for particles sp inning around their own axis. Now, it is
+interesting to notice that any mechanical model of a self-rotating electron fatally leads to a fundamental length for
+this very peculiar rotator, resulting to be just the Compton wavele ngthλC. Anyway, models with a self-rotating
+spinning charge are very naive and problematic.3In the subsequent literature, these problems were overcome by
+considering kinematical theories of the spin based on the Zitterbew egung [19, 20], where spinning particles, though
+not effectively extended, can be considered as something which is ha lf-way between a point and a rotating body (as,
+e.g., a top). In these models the center of the (even pointlike) char ge is spatially distinct from the center-of-mass, so
+thatvelocityandmomentumarenolongerparallelvectors,andanin ternal(microscopic), rotationalspin motionofthe
+particle arises in addition to the external (macroscopic) translatio nal motion of the center-of-mass. As a consequence,
+in the classical limit the global inertial motion of a free spinning particle turns out to be no longer uniform and
+rectilinear, but rather accelerated and helical, with a radius once ag ain equal to the Compton wavelength.
+A different insight was provided by quantum mechanics with its discret e energy levels and the uncertainty principle.
+This led physicists to speculate about space-time discreteness as e arly as the 1930’s. W. Heisenberg himself [21], for
+example, in 1938 noted that physics musthave a fundamental length scale which, together with handc, might allow
+the derivation of the mass of particles. In Heisenberg’s view, this len gth scale would have been around 10−13cm,
+thus corresponding to the classical electron radius. Later, H.S.S nyder [22] in 1947 introduced a minimum length by
+“quantizing” the spacetime, thus anticipating of more than half a ce ntury the so-called “non-commutative geome-
+try” models (see below). Spacetime coordinates are represented by quantum operators and, in Snyder’s approach,
+these operators have a discrete spectrum, thus entailing a discre te interpretation of spacetime. Subsequently, also
+T.D.Lee [23] introduced an effective time discretization on the basis o f the necessarily finite number of measurements
+performable in any finite interval of time.
+The concept of non-continuous spacetime has recently returned into fashion in GUT’s [24], in string theories [25], in
+quantum gravity [26] and in the approaches regarding spacetime eit her as a sort of quantum ether or as a spacetime
+foam [27], or even as endowed with a non-commutative geometry (like in Deformed or Double Special Relativity
+[28]4). In particular, M-Theory, Loop Quantum Gravity [29, 30] and Ca usal Dynamical Triangulation [31] lead
+to postulate an essentially discrete and quantum spacetime, where (as expected from the uncertainty relations)
+fundamental momentum and mass-energy scales naturally arise, in addition to /planckover2pi1andc. In contemporary physics the
+most important case of an elementary space scale is doubtless the “ Planck length” ℓP=/radicalbigg
+/planckover2pi1G
+c3≃1.62·10−33cm; for
+example, it plays a fundamental role in string theory, where it is defin ed as the minimum length of a typical string.
+As a consequence, any space distance smaller than ℓPis deprived of any physical meaning or possibility of being
+measured. M.Planck himself, early in 1899 [32] proposed a system of units to be used in fundamental physics (a
+“System of Natural Units”), where fundamental units are the Pla nck length, the Planck time, and the Planck mass.
+Although Planck did not know yet quantum mechanics and general re lativity, such scales mark out the regions where
+these theories become, in a sense, reciprocally in contrast, so tha t a proper quantum gravity theory is required. At
+the present the Planck length is the minimum spacetime scale present in physics, at which we expect a substantial
+3It is quite famous a statement by A.O. Barut: “If a spinning pa rticle is not quite a point particle, nor a solid three dimens ional top,
+what can it be?” [18]
+4It is interesting to point out that in the first of Refs.[28] it is stated that “the special role of the time coordinate in the structure of
+k-Minkowski spacetime forces one to introduce an element of d iscretization in the time direction”.4
+“unification” of high energy microphysics and early cosmology.
+This is, in short terms and until now, the known story of the introdu ction in physics of “fundamental” constants
+with the meaning of elementary lengths or times. However, with the t horough study, started in recent years, of the
+copious set of unpublished notes left by another protagonist of th eoretical physics [33], Ettore Majorana [34], we
+have been acquainted with some unknown contributions pointing out his own elaboration about the arguments just
+described above. In the remaining part of this paper, we will then de scribe and comment such contributions, present
+in the so-called Volumetti [35] and Quaderni [36], containing study and research notes which were not published b y
+Majorana.
+In these booklets, among many different topics the author studied extensively several arguments of electrodynamics
+(both at a classical and at a quantum level). Many of these regarde d topics and methods “usually” discussed in
+the scientific workplaces of 1930s, although they were dealt with in a very original manner and, sometimes, with
+extraordinary results (see, for example, Ref. [37]). Here, howev er, we will focus just on three of these notes, which
+result to be particularly relevant for the topics considered here. A characteristic of the Majorana notes is that they
+always concern specific examples rather than generic theoretical issues. In the following section, then, we report
+the study performed by Majorana about the scattering of partic les by a quasi-Coulombian potential, where a non
+vanishing radius for the scatterer is considered. In Sect.3 we inst ead discuss the introduction of an intrinsic time delay
+in the propagation of the (classical) electromagnetic field, consider ed by Majorana in order to take into account the
+possible effect on the electromagnetic potentials of a fundamental constant length. Finally, in Sect.4 we will describe
+an interesting (though leading to incorrect results) attempt to fin d a relation between the fundamental physical
+constants e,/planckover2pi1andc. Our conclusions and outlook then follow.
+QUASI-COULOMBIAN SCATTERING
+The set of Majorana’s Quaderni opens with the study of the problem of the scattering of particles f rom a quasi-
+Coulombian potential of the form
+V(r) =k√
+r2+a2, (4)
+wherekis a positive constant (related to the electric charge of the potent ial source) and ais, according to Majorana,
+the “magnitude of the radius of the scatterer”. We do not know pr ecisely the motivations for such a study (likely,
+for applications to atomic and nuclear physics problems; see below), but, as a matter of fact, the original intention
+of the author was to extend the well-known Rutherford formula fo r the scattering of a beam of particles (with charge
+Z′eend mass m) from a given body (of charge Ze). In this case we have a pure Coulomb scattering, and the cross
+section (number of scattering particles at an angle θper unit time and solid angle) is given by the classical formula
+f(θ) =Z2Z′2e4
+4m2v4sin4θ/2=Z2Z′2e4
+16T2sin4θ/2, (5)
+wherevis the velocity of the incident particles and Tis their kinetic energy. Majorana deduced the above Rutherford
+formulain his Volumetti , usingboth classicalmechanicsargumentsandthe quantum Borna pproximationmethod [35].
+As well-known, Eq.(5) was employed for the first time by E. Rutherf ord in his experiments on αparticles impinging
+on a gold target, aimed at explaining the structure of the atom (with or without a compact nucleus at its center).
+The modification of the Coulomb scattering potential, considered by Majorana in Eq.(4), certainly introduces an
+improvement in the description of the physical phenomenon, with th e introduction of the dimensions of the scattering
+center (in the Rutherford formula assumed to be pointlike, a= 0), but the story does not end here. In fact, the most
+simple approximation in this line of thinking is to consider the scattering center as a uniformly charged sphere of
+radiusabut, in such a case, the potential outside the sphere is strictly Cou lombian (thanks to the Gauss law), and
+Eq.(4) would not apply. Of course, at the time when Majorana perf ormed his calculations, it was known that the
+nucleus of a given atom is not uniformly charged, being formed by indiv idual particles, but, again, such a situation is
+not described by the simple formula in Eq.(4). An example is the Yukaw a potential [38], where the Coulomb potential
+acquires a screening factor with an exponential form ruled by one m ore parameter, giving the range of nuclear forces.
+In any case, it is striking that the only application of Eq.(4) reporte d in the Quaderni [36] was to the hydrogen
+atom, where just one proton forms its nucleus, thus being conside red effectively as a uniformly charged particle. Thus
+the reasoning of Majorana behind Eq.(4) has a different starting p oint, upon which we will briefly speculate below.5
+For the moment, we only note that, from a strictly mathematical po int of view, the introduction of a non-vanishing
+radius for the scattering center has the effect of regularizing the Coulomb potential which, otherwise, would diverge
+forr→0 (however, by contrast to other cases, Majorana did not resto re the full Coulomb potential by taking the
+limita→0 at the end of his calculations, but always maintained in the present c ase a finite value for a).
+In the problem proposed, Majorana studied the deviations from pu re Coulomb scattering by parameterizing it with
+the ratio i/iRof the effective scattering intensity under an angle θ(i.e. the flux of scattered particles per unit surface
+(normal to the incident direction) and per unit time) with respect to that deduced in the Rutherford approximation.
+Besides the radius a, this ratio depends also on the energy and momentum of the incident particles, but Majorana
+preferred to parameterize these by means of the scattering par ameter (or, as denoted by himself, the “minimum
+approach distance”) bin the Coulomb limit, defined by K/b=T, and from the wavelength λof the free particle. By
+settingα=a/(λ/2π),β=b/(λ/2π), he obtained:
+i=f(α,β,θ)iR. (6)
+Of course, in the pure Coulomb limit, the Rutherford formula should b e recovered, so that f(0,β,θ) = 1. The
+calculations then proceed at evaluating the function f(α,β,θ) by keeping αfixed (that is, for fixed scatterer size)
+and considering the limit β→0 (for scattering particles with increasing momentum, approaching nearer and nearer
+the center). The wavefunction of the system is, then, evaluated perturbatively by expanding it in a series ruled by
+the experimental parameter β, at zeroth order ( β= 0) being used the WKB method [39]. After some passages5. He
+obtains the following analytic result in the approximation considered ( ρ=r/(λ/2π))
+f(α,β,θ) = 2sinθ
+2/integraldisplay∞
+0ρ/radicalbig
+ρ2+α2sin/bracketleftbigg
+2ρsinθ
+2/bracketrightbigg
+dρ, (7)
+which can be expressed in term of the Bessel kfunction k(αsinθ/2) [40]. In this approximation Majorana then finds
+that the actual scattering intensity may be substantially different from that of the Rutherford formula, for backward
+scattering and provided that the radius ais appreciably different from zero (the ratio i/iRapproaching zero for θ=π
+and increasing α).
+This remarkable result is not explicitly reported in the Quaderni , but Majorana gives an attempt to tabulate the
+radial wavefunction uℓ, satisfying the equation
+u′
+ℓ+/parenleftBigg
+1−β/radicalbig
+ρ2+α2−ℓ(ℓ+1)
+r2/parenrightBigg
+ue= 0, (8)
+obtained numerically with the method of the particular solutions. Alth ough numerical solutions are searched only for
+the particular case of ℓ= 0 andβ= 0.4 (and not β= 0), here the interesting point is that Majorana explicitly reports
+that “for the hydrogen atom we consider the values β= 0.4,0.5,0.6,0.7andα= 0,0.2,0.4,0.6,0.8,1”. Unfortunately,
+no further discussion is given in the Quaderni ; the key point is, however, the fact that Majorana uses λ/2πas
+the length scale for boththe lengths aandb. Now, it is very natural to measure the scattering parameter (or “the
+minimum approachdistance”) in terms ofthe free particlewavelengt hor, equivalently, (the inverseof) the free particle
+momentum ( λ=h/p), since it is obvious that increasing the momentum (or decreasing th e probe wavelength), the
+incident particle approaches nearer and nearer the scattering ce nter, thus decreasing b. The same would not apply,
+instead, to the radius of the scattering center which should be inde pendent of the incident particle properties, unless
+Majoranaconsiders effective, momentum-dependent, dimensionsforthe scatteringcenter. I n suchacase, theattention
+is evidently shifted from the investigation of an actual, particular ph ysical system considered (particles in a scattering
+potential) to the study of more general properties of the backgr ound field or space. We will come back later on this
+point.
+Finally, we conclude this section by mentioning also another modificatio n of the Coulomb potential considered by
+5In order to avoid convergence problems in the usage of the Gre en method, Majorana assumes that the scattering force acts o nly for
+distances less then a quantity R, and than let R→ ∞at the end of calculations. The potential in Eq.(4) is thus re placed during
+calculations, by k„1√
+r2+a2−1
+R«6
+Majorana [36], reminiscent of the Gamow potential for the descript ion ofα-decay process, namely
+V=
+
+V0,forr < R,
+k
+r,forr > R.(9)
+Here the role of the scattering center radius is played by the range R,α=R/(λ/2π), and one more parameter enters,
+that is the depth V0of the potential, measured with respect to the kinetic energy of th e incident particles, A=V0/T.
+Although, in this case, the calculations are only sketched, the key p oints envisaged above are as well present. Now
+the correction function in Eq.(6) is replaced by
+i
+iR=f/parenleftbiggV0
+T,R
+λ/2π,b
+λ/2π,θ/parenrightbigg
+(10)
+where “for the hydrogen” Majorana considers the same values as before for the α,βparameters (except, obviously,
+α= 0,0.2), while A=2,1.5,1,0.5,0,-0.5,-1,-1.5,-2,-2.5,-3,...,-8. Interestingly, besides negative values for V0(as in the
+Gamow model), Majorana also considers few positive values for the d epth of the potential. As above, however, the
+very notable point is that both the potential parameters V0andRappear to be strictly related to the energy and
+momentum of the probe particles.
+For both the cases studied, in the presence of continuum states ( Eq.(4)) or bound states (Eq.(9)), the reasoning
+is thus the same. This is even more intriguing if compared to the compr ehensive study of the scattering from a pure
+Coulomb potential, performed by Majorana [36] some pages after those discussed here up to now. Indeed, in this case
+the length unit is no more the free particle wavelength, the Bohr rad ius being introduced (and the energy unit is the
+Rydberg), evidently denoting possible applications to atomic problem s. From what discussed above, it is thus clear
+that Majorana’s original motivations for the studies of quasi-Coulo mbian scattering are different from those standard
+related to atomic and nuclear physics.
+RETARDED ELECTROMAGNETIC FIELDS AND THE INTRODUCTION OF A F UNDAMENTAL
+LENGTH
+Some light on this issue may come from other pages of Majorana’s Quaderni [36], where the possibility is considered
+of introducing an intrinsic constant time delay (or, equivalently, an in trinsic space constant) in the expression for the
+retarded electromagnetic fields. Here the treatment is fully classic al, and goes as follows.
+The starting point is the wave equation satisfied by any component o f the electromagnetic potentials, denoted
+generically with f(x,y,z,t), and then the evaluation of the D’Alembert operator for the stan dard retarded field
+denoted by
+ϕ(x,y,x,t) =f/parenleftBig
+x,y,z,t−r
+c/parenrightBig
+≡f(x,y,z,t). (11)
+The known result is, of course, the following:
+/squaref=∇2ϕ+2
+rc˙ϕ+2
+c∂2ϕ
+∂r∂t, (12)
+where a dot indicates time differentiation. Majorana then introduce s an intrinsic space constant ε, so that Eq.(11) is
+replaced by
+ϕ(x,y,z,t) =f/parenleftBigg
+x,y,z,t−√
+r2+ε2
+c/parenrightBigg
+=˜f(x,y,z,t). (13)
+The D’Alembertian term entering into the wave equation is thus
+/tildewider/squaref=∇2ϕ−ε2
+c2(r2+ε2)¨ϕ+2r2+3ε2
+c(r2+ε2)3/2˙ϕ+2r
+c√
+r2+ε2∂2ϕ
+∂r∂t, (14)
+which replacesEq.(12). Majoranaalsointroducesexplicitly atime de layτwhich, in his view, is a“universalconstant”
+taking the value τ= 0 classically; in his calculations, however, he always uses the length
+ε=τc (15)7
+(already introduced), so that we here follow this line of reasoning.
+The wave equation is apparently solved by using the Green method; d enoting with S′the generic source function
+(charge density ρor current density J), the modified generic potential Φ, solution of the wave equation, a ssumes now
+the following from:
+Φ =/integraldisplay1√
+R2+ε2S/parenleftBigg
+x′,y′,z′,t−√
+R2+ε2
+c/parenrightBigg
+dx′dy′dz′(16)
+(withR=|r−r′|, as usual). Majorana then ends his explicit calculations with the writin g of Eq.(16) expanded up
+to second order in εforε→0 (thus approaching the classical limit, as of course coming from exp eriments):
+Φ =/integraldisplay1
+RS/parenleftbigg
+x′,y′,z′,t−R
+c/parenrightbigg
+dx′dy′dz′
+−ε2/braceleftbigg/integraldisplay1
+2R3S/parenleftbigg
+x′,y′,z′,t−R
+c/parenrightbigg
+dx′dy′dz′
++/integraldisplay1
+2R2cS′/parenleftbigg
+x′,y′,z′,t−R
+c/parenrightbigg
+dx′dy′dz′/bracerightbigg
++... . (17)
+Some comments are now in order. Unfortunately, again we do not kn ow the motivations for such a study, but
+it is intriguing the following observation. By taking the ordinary electr ic monopole limit [41] into Eq.(16), with
+S(r,t−r
+c) =ρ(r),|r−r′|∼=|r|=rand/integraldisplay
+ρ(r′)dx′dy′dz′=Qbeing the electric charge, one can easily obtain
+Φ =Q√
+r2+ε2, (18)
+that is exactly the potential considered in Eq.(4).
+Although Eq.(18) is not explicitly reported in the Quaderni , it seems very likely that the topics discussed here
+and in the previous section (and considered in no other place in the Quaderni ) are related. If so, the “magnitude of
+the radius of the scatterer” in the scattering problem considered above, whose interpretation in terms of the physical
+dimensions of the scattering center led to difficulties, should be inter preted instead as the “universal” space constant
+introduced here. In this case Majorana’s original intention for the study of quasi-Coulombian scattering was not that
+of exploring the properties of the scattering particle (an atom or it s nucleus or even other elementary particles) but
+rather that of investigating the underlying properties of the phys ical space. The intrinsic space constant then plays
+the role of a fundamental length, like the one introduced by Planck in 1899 or that conjectured by Heisenberg in
+1938 (see the the Introduction). We do not know if Majorana effec tively thought to a given particular value but, in
+any case, to the best of our knowledge, with Majorana it is the first time that such a detailed study was effectively
+undertaken, as early as in the 1930s.
+A RELATION AMONG THE FUNDAMENTAL CONSTANTS
+The great interest of Majorana for electrodynamics [37], in its class ical or (especially) quantum form, is very well
+documented in his writings (see mainly Refs.[35] and [36]), so that ev en the particular studies described above do
+not seem at all surprising. Nevertheless, the line of thinking of such studies goes well beyond mere applications or
+generalizations of standard electrodynamics, as shown in the prev ious pages, involving questions about fundamental
+constants. It is intriguing that yet another time Majorana reason ed about such fundamental questions, though in an
+(apparently) different context, when he was still a student, as ea rly as in 1928. Indeed, in a page of his Volumetti (see
+section 31 of Volumetto II, in Ref.[35]), Majorana attempted to fin d a relation between the fundamental constants
+e,handc, or, more explicitly, an expression for the elementary charge eappearing in the Coulomb law, describing,
+for example, the electrostatic force between two electrons. The reasoning of Majorana goes as follows. In the region
+of space surrounding the two electrons, which are at a distance ℓapart, an electromagnetic field is present that, “in
+some sense” (according to Majorana’s wording), is quantized. Her e we should point out the first interesting thing
+since, in what follows (see below), it is evident that what is considered “quantized” by Majorana is undoubtedly the
+electromagnetic field acting among the two electrons. Nevertheles s, differently from what appearing elsewhere in his
+notes, Majorana does not refer explicitly to an electromagnetic field, but rather to the ’ether’ surrounding the two8
+electrons, that is (apparently) the physical space itself (in the imp roper language of that time). The two electrons are
+assumed to interact amongthem by means of a point particle (what w e could term the quantum of the electromagnetic
+field) moving with a group velocity equal to the speed of light c. Such particle is further assumed to propagate freely
+from one electron to the other, whereas it inverts its motion when “ colliding” with the electrons. The momentum of
+the quantum particle is deduced by a sort of Sommerfeld quantizatio n rule
+|p|=nh
+2ℓ,
+withn= 1. The “kick” received by the electron is therefore 2 |p|=h/ℓ, and since the number of collisions per unit
+time is 1 /T=c/2ℓ, the magnitude of the force acting on each electron is then F= 2|p|/T=hc/2ℓ2. By equating
+this expression to the Coulomb law, Majorana finally deduces the exp ression he searched for the elementary charge
+e=/radicalbig
+hc/2 which, however, as pointed out explicitly by himself, gives a value “21 times greater than the real one”.
+Though leading to results far from the truth, what discussed by Ma jorana is interesting for many reasons, one of
+which already mentioned above. Indeed, first of all, the mechanical model considered here is the first one to have
+been proposed as an attempt to deduce the value of the electric ch arge (the famous paper by Dirac in Ref.[42] was
+published few years after). However, as it should be evident from w hat reported above, the interpretation given by
+Majorana of the quantized electromagnetic field substantially coincides with that introduced mo re than a decade later
+by R.P. Feynman in quantum electrodynamics [43], the point particles e xchanged by electrons being assumed to be
+the photons. It is then remarkable that fundamental questions r egarding the constants of Nature discussed properly
+only in more recent times, were addressed in pioneering works by Maj orana as early as at the end of 1920s, where,
+on the other hand, the key ideas of quantum field theory were alrea dy present.
+CONCLUSIONS
+The problem of the existence and of the effects of a fundamental le ngth dates back to the early years of the last
+century. As reviewed in the introduction, this concept was introdu ced and rediscovered in physics several times since
+then, in order to solve difficulties of different kind as, e.g., in regularizin g some divergent results. It is, however, quite
+notable the presence of “side” effects associated to (or, rather , postulated about) the existence of such elementary
+space intervals, which could allow, in Heisenberg’s view for example, th e derivation of the mass of the particles, or
+even play a key role in the physical interpretation of the spin variable .
+Strictly related to the introduction of a fundamental length is, on t he other hand, the assumption of the existence
+of a fundamental time, although such hypothesis is less fashioned a mong physicists than the first one.
+In more recent times, the present subject is embedded into the mo re general framework of the quantization of the
+spacetime, where attempts to provide a well-defined quantum theo ry of gravity manifest into different theoretical
+approaches, with the common denominator of a fundamental lengt h scale.
+In this paper we have shown that the known picture about this intrig uing concept has been further enriched by
+several, yet unknown, contributions by Ettore Majorana. As ear ly as the beginning of 1930s, indeed, the Italian
+physicist not only conjectured (as already Planck and, later, othe rs did) the existence of both an elementary length
+and of a fundamental time scale, but also investigated in some detail the immediate physical consequences of such
+hypothesis. In this respect, his contributions once again [33] reve al a farsighted intuition, and result to be quite
+precious pieces of research both on the historical point of view and for present day theoretical investigations.
+Majorana considered, for example, the scattering from electrica lly charged particles when the underlying properties
+of the space are such that an intrinsic length exists, resulting in the modification of the pure Coulomb potential as
+in Eq.(4) or (18). Even just at the classical level, he obtained the s ame effect by introducing an intrinsic time delay,
+which is considered a “universal constant”, in the expression for t he retarded electromagnetic fields.
+At the quantum level, instead, it is quite interesting still another con tribution by Majorana about fundamental
+constants, aimed at finding a relation between e,/planckover2pi1,cor, rather, aimed at deducing the value of the elementary
+charge. Such a scope was pursued, few years later, in a famous pa per by Dirac [42], although based on a different
+theoretical scheme and with different results. Indeed, the partic ularnumerical prediction by Majorana (as opposed to
+the particular theoretical prediction of the existence of magnetic monopoles by Dirac) is far fr om the truth, but the key
+ideabehind it anticipatedofmorethan adecadethe assumption, intr oducedbyFeynmanin quantumelectrodynamics,
+of photon exchange during electromagnetic processes.
+We expect that Majorana’s important ideas and applications recove red from the oblivion through this paper will
+bring some benefit also to the present day abstract theoretical r esearch on the fundamental properties of the physical
+spacetime.9
+Acknowledgments
+The authors warmly thank E.Recami for interesting discussions an d suggestions, while acknowledge the very kind
+collaboration by M.G.Cammarota and M.Longo.
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
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