Tautological relations in Hodge field theory
TAUTOLOGICAL RELATIONS IN HODGE FIELD
THEORY
A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
Abstract. We propose a Hodge field theory construction that
captures algebraic properties of the reduction of Zwiebach invari-
ants to Gromov-Witten invariants. It generalizes the Barannikov-
Kontsevich construction to the case of higher genera correlators
with gravitational descendants.
We prove the main theorem stating that algebraically defined
Hodge field theory correlators satisfy all tautological relations.
From this perspective the statement that Barannikov-Kontsevich
construction provides a solution of the WDVV equation looks as
the simplest particular case of our theorem. Also it generalizes the
particular cases of other low-genera tautological relations proven
in our earlier works; we replace the old technical proofs by a novel
conceptual proof.
Contents
1. Introduction 1
2. Gromov-Witten theory 5
3. Zwiebach invariants 9
4. Construction of correlators in Hodge field theory 12
5. String, dilaton, and tautological relations 17
6. Vanishing of the BV structure 21
7. Main Lemma 24
8. Proof of Theorem 3 27
References 33
1. Introduction
In this paper we present an attempt to formalize what may be called
a string field theory (SFT) for (closed) topological strings with Hodge
property.
From the very first days of string theory it was considered as a kind
of generalization of the perturbative expansion of the quantum field
theory in the (functional) integral representation. The space of graphs
with g loops with metrics on edges (Schwinger proper times) was gen-
eralized to moduli space of Riemann surfaces. Indeed, the latter space
really looks like a principle U(1)n bundle over the former space near the
http://arxiv.org/abs/0704.1001v1
2 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
points of maximal degeneracy (i.e., where maximal number of handles
are pinched).
A natural question is whether there are special string theories that
degenerate exactly to quantum field theories (may be, of the special
kind). Would it happen such theories should enjoy both finiteness
of string theory and (functional) integral description of quantum field
theory.
One of the first attempts to construct a theory of this type was
done by Zwiebach in [37]. He divided the moduli space into two re-
gions: the internal piece and the boundary. He observed that surfaces
representing the boundary region may be constructed from those rep-
resenting the internal piece by gluing them with the help of cylinders
(with flat metric). Therefore, he proposed to take integrals over the
moduli spaces in two steps: first, to take an integral over the internal
pieces, such that this would produce vertices, and then take an integral
along metrics on cylinders, that would exactly reproduce integral along
the Schwinger parameters on graphs in QFT prescription.
In this approach, he came with the infinite number of vertices of
different internal genera and with different number of external legs.
However, he observed that such vertices satisfy quadratic relations that
where a quantum version of some infinity-structure. At that time com-
munity of theoretical physicists seemed not to be impressed by the
Lagrangian with infinite number of (almost uncomputable1) vertices.
The next attempt was done by Witten [36]. He assumed that in
topological string theories there may be a limit in the space of two-
dimensional theories such that the measure of integration goes to the
vicinity of the points of maximal degeneration. In the type B theories
such limit seems to be the large volume limit of the target space; this
motivated Witten’s Chern-Simons-like representation for the topologi-
cal string theory. This approach was further developed by Bershadsky,
Cecotti, Ooguri, and Vafa in [3]. We note that the tropical limit of
Gromov-Witten theory [26] (type A topological strings) seems to real-
ize the same QFT degeneration of string theory. Indeed, the tropical
limit of a Riemann surface mapped to a toric variety is represented by
the graph mapped to the moment map domain.
In the development of topological string theory it became clear that
the proper object is not just a measure on the moduli space of complex
structures of Riemann surfaces, but rather a differential form on this
space. In original formulation these differential forms were assigned
to the tensor algebra of cohomology of some complex; such objects
are called Gromov-Witten invariants. We say that Gromov-Witten
invariants are QFT-like if the differential forms of non-zero degree have
support only in a vicinity of the points of maximal degeneration.
1 Note, that computation of an integral over a subspace with a boundary is
harder than that one over a compact space.
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 3
We generalized the definition of Gromov-Witten invariants in [22] by
lifting it from the cohomology of a complex to the full complex. Such
generalization involved enlargement of the moduli space from Deligne-
Mumford space to Kimura-Stasheff-Voronov space [15], and we called it
Zwiebach invariants (in fact, some pieces of this construction appeared
earlier in [37] and [8]). The complex of states involved in the definition
of Zwiebach invariants is a bicomplex due to the action of the second
differential. The second differential represents the substitution of a
special vector field corresponding to the constant rotation of the phase
of the local coordinate at a marked point into differential forms on the
Kimura-Stasheff-Voronov space.
Once we have some Zwiebach invariants, it is possible to produce new
Zwiebach invariants by contraction of an acyclic Hodge sub-bicomplex.
In fact, it is one of the main properties of Zwiebach invariants. Con-
sider a sub-bicomplex, where these two differentials act freely. We call
it Hodge contractible bicomplex. The operation of contraction of a
Hodge contractible bicomplex turns Zwiebach invariants into induced
Zwiebach invariants on the coset with respect to contactible bicomplex.
Induced Zwiebach invariants are differential forms whose support is a
union of the support of the initial Zwiebach invariants and small neigh-
bourhoods of the points of maximal degeneration. This procedure is a
generalization from intervals to cylinders of the procedure of induction
of L∞-structures, see e. g. [30, 28].
This way we can obtain QFT-like Gromov-Witten invariants. We
just should start with Zwiebach invariants that have (in some suit-
able sense) no support inside the Kimura-Stasheff-Voronov spaces. In
fact, it is even enough to consider a weaker condition, motivated by
applications. That is, usually people consider the integrals of Gromov-
Witten invarians only over the tautological classes in the moduli space
of curves. So, we call a set of Zwiebach invariants vertex-like if the
integral over the Kimura-Stasheff-Voronov spaces of any their non-
zero component multipled by the pullback (from the Deligne-Mumford
space) of any tautological class vanishes.
Consider vertex-like Zwiebach invariants. Assume we contract a
Hodge contractible bicomplex down to cohomology. We obtain differ-
ential forms on the Deligne-Mumford space, such that the integral of
the product of any such form of non-zero degree with any tautological
class vanishes the interior of the moduli space. Integrals of such forms
over the moduli spaces turn out to be sums over graphs (corresponding
to degenerate Riemann surfaces). They resemble Feynman diagramms,
and generation function for the integral over moduli spaces resemble
diagrammatic expansion of perturbative quantum field theory.
In this paper, we don’t construct examples of vertex-like Zwiebach
invariants (we are going to do this explicitly in a future publication, as
well as the corresponding theory for the spaces introduced in [20, 21]).
4 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
Rather we conjecture that they exist and study the consequences of
this assumption. We call the emerging construction the Hodge field
theory, and now we will explain it in some detail.
First of all, degree zero parts of vertex-like Zwiebach invariants in-
duce the structure of homotopy cyclic Hodge algebra on the target
complex [22]. We remind that a cyclic Hodge algebra is just a Hodge
dGBV-algebra with one additional axiom (1/12-axiom, see below).
In fact, this structure is interesting by itself, without any reference to
Zwiebach invariants. It has first appeared in the paper of Barannikov
and Kontsevich [1]; it captures the properties of polyvector fields on
Calabi-Yau manifolds. More examples of dGBV algebras are studied
in [25] and [23]. It is possible to understand the structure of dGBV-
algebra as a natural generalization of the algebraic structure studied
in [19].
In the Hodge field theory construction we consider only a particular
case, where we obtain axioms of of a cyclic Hodge algebra itself, not
up to homotopy. We are aware of the fact that demanding existance of
vertex-like Zwiebach invariants simultaneously with vanishing homo-
topy piece of cyclic Hodge algebra conditions may be too restrictive,
and while considering only those relations that lead to axioms of cyclic
Hodge algebra may be too weak, however we proceed.
In the Hodge field theory construction we define graph expressions for
the analogues of Gromov-Witten invariants multiplied by tautological
classes using only cyclic Hodge algebra data. We call them Hodge field
theory correlators. The corresponding action of the Hodge field theory
is written down explicitely in Section 6.2.
Our main result is the proof that the Hodge field theory correla-
tors satisfy all universal equations that follow from relations among
tautological classes in cohomology.
The first result of this kind is due to Barannikov and Kontsevich.
They have noticed that there is a solution of the WDVV equation that
is associated to a dGBV-algebra (this solution is the critical value of
the BCOV action [3], see [1, Appendix] and [22, Appendix]). Later, we
reproved this in [22]. Then, in [22, 31, 32, 33] we proved some other
low-genera universal equations. Here we generalize all these result and
put all calculations done before in a proper framework.
In particular, the main problem for us was to define a graph expres-
sion in tensors of a cyclic Hodge algebra that corresponds to the full
Gromov-Witten potential with descendants. The first steps were done
in [31, 32], where we introduced the definition of descendants at one
point in Hodge field theory (mostly for combinatorial reasons). But
then we observed that it is a part of a natural definition of potential
with descendants in cyclic Hodge algebras that appears as a special
case of degeneration of vertex-like Zwiebach invariants multiplied by
tautological classes.
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 5
In this paper, we present and study this construction. We prove in
a completely algebraic way that Hodge field theory correlators satisfy
the same equations as a Gromov-Witten potential: string, dilaton, and
the whole system of PDEs coming from tautological relations in the
cohomology of the moduli space of curves (see also [33] for some pre-
liminary results). In what follows we will not only present the proof
but also will do our best relating algebraic definitions and statements
on Hodge field theory to analoguous constructions and theorems in the
theory of Zwiebach invariants.
1.1. Organization of the paper. In Section 2 we remind all neces-
sary facts about the axiomatic Gromov-Witten theory. In Section 3
we define Zwiebach invariants and explain the motivation to consider
the sums over graphs in cyclic Hodge algebras. In Section 4 we de-
fine cyclic Hodge algebras and the corresponding descendant potential.
In Section 5 we state the main properties of the descendant potential
in cyclic Hodge algebras, and the rest of the paper is devoted to the
proofs.
1.2. Acknowledgments. A.L. was supported by the Russian Fed-
eral Agency of Atomic Energy and by the grants INTAS-03-51-6346,
NSh-8065.2006.2, NWO-RFBR-047.011.2004.026 (RFBR-05-02-89000-
NWO-a), and RFBR-07-02-01161-a.
S.S. was supported by the grant SNSF-200021-115907/1. S.S. is
grateful to the participants of the Moduli Spaces program at the Mittag-
Leffler Institute (Djursholm, Sweden) for the fruitful discussions of
the preliminary versions of the results of this paper. The remarks
of C. Faber, O. Tommasi, and D. Zvonkine were especially helpful.
I.S. was supported by the grant RFBR-06-01-00037.
2. Gromov-Witten theory
In this section we remind what is Gromov-Witten theory and explain
its basic properties that we are going to reproduce in Hodge field theory
construction.
2.1. Gromov-Witten invariants. Let us fix a finite dimensional vec-
tor space H0 over C together with the choice of a homogeneous basis
H0 = 〈e1, . . . , es〉 and a non-degenerate scalar product ηij = (·, ·) on it.
Let e1 be a distinguished even element of the basis.
Consider the moduli spaces of curves Mg,n. On each Mg,n we take
a differential form Ωg,n of mixed degree with values in H
0 . The whole
system of forms {Ωg,n} is called Gromov-Witten invariants, if it satisfies
the axioms [18, 24]:
(1) There are two actions of the symmetric group Sn on Ωg,n. First,
we can relabel the marked points on curves in Mg,n; second,
we can interchange the factors in the tensor product H⊗n0 . We
6 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
require that Ωg,n is equivariant with respect to these two actions
of Sn. In other words, one can think that each copy of H0 in the
tensor product is assigned to a specific marked point on curves
in Mg,n.
(2) The forms must be closed, dΩg,n = 0.
(3) Consider the mapping π : Mg,n+1 → Mg,n forgetting the last
marked point. Then the correspondence between Ωg,n and Ωg,n+1
is given by the formula
(1) π∗Ωg,n = (Ωg,n+1, e1) .
The meaning of the right hand side is the following. We want
to turn a H⊗n+10 -valued form into a H
0 -valued one. So, we
take the copy of H0 corresponding to the last marked point and
contract it with the vector e1 using the scalar product.
(4) Consider an irreducible boundary divisor inMg,n, whose generic
point is represented by a two-component curve. It is the image
of a natural mapping σ : Mg1,n1+1 ×Mg1,n2+1 → Mg,n, where
g = g1 + g1 and n = n1 + n2. We require that
(2) σ∗Ωg,n =
Ωg1,n1+1 ∧ Ωg2,n2+1, η
Here on the right hand side we contract with a scalar product
the two copies of H0 that correspond to the node.
In the same way, consider the divisor of genus g − 1 curves
with one self-intersection. It is the image of a natural mapping
σ : Mg−1,n+2 → Mg,n. In this case, we require that
(3) σ∗Ωg,n =
Ωg−1,n+2, η
As before, we contract two copies of H0 corresponding to the
node.
(5) We also assume that (Ω0,3, e1 ⊗ ei ⊗ ej) = (ei, ej) = ηij .
2.2. Gromov-Witten potential. Let us associate to each ei the set
of formal variables Tn,i, n = 0, 1, 2 . . . . By Fg denote the formal power
series in these variables defined as
(4) Fg :=
a1,...,an≥0
ψaii ,
ejTai,j
The first sum is taken over n ≥ 3 for g = 0, n ≥ 1 for g = 1, and n ≥ 0
for g ≥ 2. On the right hand side, we contract each copy of H0 with
the factor of the tensor product associated to the same marked point.
The formal power series F := exp(
g≥0 ~
g−1Fg) is called Gromov-
Witten potential associated to the system of Gromov-Witten invariants
{Ωg,n}. The coefficients of Fg, g ≥ 0, are called correlators and denoted
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 7
(5) 〈τa1,i1 . . . τan,in〉g :=
Vectors ei1 , . . . , ein are called primary fields.
The main properties of GW potentials come from geometry of the
moduli space of curves. First, one can prove that coefficients of F
satisfy string and dilaton equations:
〈τ0,1
τaj ,ij〉g =
〈τaj−1,ij
k 6=j
τak ,ik〉g;(6)
〈τ1,1
τaj ,ij〉g = (2g − 2 + n)〈
τaj ,ij〉g.(7)
The string equation is a corollary of the fact that π∗ψj = ψj − Dj ;
here π : Mg,n+1 → Mg,n is the projection forgetting the last marked
point andDj is the divisor inMg,n+1 whose generic point is represented
by a two-component curve with one node such that one component
has genus 0 and contains exactly two marked points, the i-th and the
(n+ 1)-th ones. It is assumed that
j=1 aj > 0.
The dilaton equation is a corollary of the fact that, in the same
notations, π∗ψn+1 = 2g−2+n. Of course, we assume that 2g−2+n > 0.
Second, any relation in the cohomology of Mg,n among natural ψ-κ-
strata gives a relation for the correlators. Let us explain this in more
detail.
2.3. Tautological relations.
2.3.1. Stable dual graphs. The moduli space of curves Mg,n [12] has a
natural stratification by the topological type of stable curves. We can
combine natural strata with ψ-classes at marked points and at nodes
and κ-classes on the moduli spaces of irreducible components. These
objects are called ψ-κ-strata.
A convenient way to describe a ψ-κ-stratum in Mg,n is the language
of stable dual graphs. Take a generic curve in the stratum. To each
irreducible component we associate a vertex marked by its genus. To
each node we associate an edge connecting the corresponding vertices
(or a loop, if it is a double point of an irreducible curve). If there is a
marked point on a component, then we add a leaf at the corresponding
vertex, and we label leaves in the same way as marked points. If we
multiply a stratum by some ψ-classes, then we just mark the corre-
sponding leaves or half-edges (in the case when we add ψ-classes at
nodes) by the corresponding powers of ψ. Also we mark each vertex
by the κ-class associated to it.
8 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
Let us remark that by κ-classes on Mg,n we mean the classes
(8) κk1,...,kl := π∗
where π : Mg,n+l → Mg,n is the projections forgetting the last l marked
points. It is just another additive basis in the ring generated by the
ordinary κ-classes (κk, k ≥ 1, in our notations). The basic properties
of these classes are stated in [6].
2.3.2. Integrals over ψ-κ-strata. Using the properties of GW invariants,
one can express the integral of Ωg,n over a ψ-κ-stratum S in terms of
correlators.
Consider a special case, when S is represented by a two-vertex graph
with no ψ- and κ-classes. Then, according to axiom 4, the integral of
Ωg,n is the product of integrals of Ωg1,n1+1 and Ωg2,n2+1 over the moduli
spaces corresponding to the vertices, contracted by the scalar product:
Ωg,n,
Mg1,n1+1
Ωg1,n1+1,
eij ⊗ ei′
Mg2,n2+1
Ωg2,n2+1,
eij ⊗ ei′′
Here we assume that the genus of one component of a generic curve
in S is g1 and n1 marked points with labels j ∈ J1, |J1| = n1, are on
this component. The other component has genus g2 and n2 marked
points with labels j ∈ J2, |J2| = n2, lie on it. Of course, g1 + g2 = g,
n1 + n2 = n.
Now consider a special case, when S is represented by a one-vertex
graph with ψ- and κ-classes. Let us assign a vector in the basis of H0
to each leaf (to each marked point). Then, according to axiom 3 the
integral
j κb1,...,bk ,
is equal to
Mg,n+k
Ωg,n+k
n+j ,
eij ⊗ e
Combining these two special cases one can obtain an expression in
correlators that corresponds to an arbitrary ψ-κ-stratum.
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 9
2.3.3. Relations for correlators. Suppose that we have a linear combi-
nation L of ψ-κ-strata that is equal to 0 in the cohomology of Mg,n (a
tautological relation). Since dΩg,n = 0, the integral of
Ωg,n,
j=1 eij
over L is equal to zero, for an arbitrary choice of primary fields. This
gives an equation for correlators.
Usually, one consider also the pull-backs of L to Mg,n+n′, n
′ ≥ 0,
multiplied by arbitrary monomials of ψ-classes. Of course, they are also
represented as vanishing linear combinations of ψ-κ-strata. This gives a
system of PDEs for the formal power series Fg, g ≥ 0. For the detailed
description of the correspondence between tautological relations and
universal PDEs for GW potentials see, e. g., [10] or [6].
There are 8 basic tautological relations known at the moment: WDVV,
Getzler, Belorousski-Pandharipande, and topological recursion rela-
tions in M0,4, M1,1, M2,1, M2,2, M3,1 [10, 9, 2, 16].
3. Zwiebach invariants
In Gromov-Witten theory (and also in topological string theory) the
Gromov-Witten invariants is usually a structure on the cohomology
of a target manifold (the space H0) of on the cohomology of a com-
plex of some other gometric origin. We have introduced the notion of
Zwiebach invariants in [22] in order to formalize in a convenient way
what physicists mean by topological conformal quantum field theory
at the level of a complex rather than at the level of the cohomology.
The very general principles of homological algebra imply that al-
gebraic stuctures on the cohomology are often induced by some fun-
damental structures on a full complex (the standard example is the
induction of the infinity-structures from differential graded algebraic
structures). Such induction usually can be represented as a sum over
trees with vertices corresponding to fundamental operations and edges
corresponding to the homotopy that contracts the complex to its co-
homology.
We would like to stress that Gromov-Witten invariants also can be
considered as an induced structure on the cohomology of a complex.
In this case, the fundamental structure on the whole complex is deter-
mined by Zwiebach invariants.
We are able to associate some structure on a bicomplex with a special
compactification of the moduli space curves (Kimura-Stasheff-Voronov
compactification). So, complexes are replaced by bicomplexes, where
the second differential reflects the rotation of attached cylinders (or
circles). This is an appearance of the string nature of the problem.
As an induced structure we indeed obtain a Gromov-Witten-type
theory that, under some additional assumptions, can be presented in
terms of a sum over graphs. Below we explain the whole construction,
following [22] and with some additional details.
10 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
3.1. Kimura-Stasheff-Voronov spaces. We remind the construc-
tion of the Kimura-Stasheff-Voronov compactification Kg,n of the mod-
uli space of curves of genus g with n marked point. It is a real blow-up
of Mg,n; we just remember the relative angles at double points. We
can also choose an angle of the tangent vector at each marked point;
this way we get the principal U(1)n-bundle over Kg,n. We denote the
total space of this bundle by Sg,n.
There are also the standard mappings between different spaces Sg,n.
First, one can consider the projection π : Sg,n+1 → Sg,n forgetting the
last marked point. Suppose that under the projection we have to con-
tract a sphere that contains the points xi, xn+1, and a node. Denote
the natural coordinates on the circles corresponding to xi and a node
on a curve in Sg,n+1 by φi and θ. Let φ̃i be a coordinate on the circle
corresponding to xi in Sg,n. Then φ̃i = φi + θ under the projection
π. In the same way, if we contract a sphere that contains two nodes
and xn+1, then θ̃ = θ1 + θ2, where θ1 and θ2 are the coordinates on the
circles corresponding to the two nodes of a curve in Sg,n+1 and θ̃ is a
coordinate on the circle at the resulting node in Sg,n
In the same way, when we consider the mappings σ : Sg1,n1+1 ×
Sg1,n2+1 → Sg,n representing the natural boundary components of Sg,n,
we have θ = φn1+1 + φn2+1, where φn1+1 and φn2+1 are the coordinates
on the circles corresponding the points that are glued by σ into the
node and θ is the coordinate on the circle at the node. For the map-
ping σ : Sg−1,n+2 → Sg,n we also have θ = φn+1 + φn+2 with the same
notations.
3.2. Zwiebach invariants. Let us fix a Hodge bicomplex H with two
differentials denoted by Q and G− and with an even scalar product
η = (·, ·) invariant under the differentials:
(12) (Qv,w) = ±(v,Qw), (G−v, w) = ±(v,G−w).
The Hodge property means that
(13) H = H0 ⊕
〈eα, Qeα, G−eα, QG−eα〉,
where QH0 = G−H0 = 0 and H0 is orthogonal to H4.
Below we consider the action of Q and G− on H
⊗n. We denote
by Q(k) and G
− the action of Q and G− respectively on the k-th
component of the tensor product.
On each Sg,n we take a differential form Cg,n of the mixed degree with
values in H⊗n0 . The whole system of forms {Cg,n} is called Zwiebach
invariants, if it satisfies the axioms:
(1) Ωg,n is Sn-equivariant.
(2) (d+Q)Ωg,n = 0, Q =
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 11
(3) (G
− + ık)Cg,n = 0 for all 1 ≤ k ≤ n (we denote by ık the
substitution of the vector field generating the action on Sg,n of
the k-th copy of U(1)); Cg,n is invariant under the action of
U(1)n;
(4) π∗Cg,n = (Cg,n+1, e1), where π : Sg,n+1 → Sg,n is the mapping
forgetting the last marked point.
(5) σ∗Cg,n = (Cg1,n1+1 ∧ Cg2,n2+1, η
−1), where σ : Sg1,n1+1×Sg1,n2+1 →
Sg,n represents the boundary component. In the same way,
σ∗Cg,n = (Cg−1,n+2, η
−1) for the mapping σ : Sg−1,n+2 → Sg,n.
(6) (C0,3, e1 ⊗ vα ⊗ vβ) = ((Id+ dφ2G−)vα, (Id+ dφ3G−)vβ), φ2 and
φ3 are the coordinates on the circles at the corresponding points.
Zwiebach invariants on the bicomplex with zero differentials deter-
mine Gromov-Witten invariants. Indeed, in this case the factorization
property implies that {Cg,n} is lifted from the blowdown of Kimura-
Stasheff-Voronov spaces, i.e. it is determined by a set of forms on
Deligne-Mumford spaces. Then it is easy to check that this system of
forms satisfied all axioms of Gromov-Witten invariants.
3.3. Induced Zwiebach invariants. Induced Zwiebach invariants are
obtained by the contraction of H4. We denote by G+ the contraction
operator. This means that G+H0 = 0, Π = {Q,G+} is the projection
to H4 along H0, {G+, G−} = 0, and (G+v, w) = ±(v,G+w).
We construct an induced Zwiebach form C indg,n on a homotopy equiv-
alent modification S̃g,n of the space Sg,n. At each boundary component
γ we glue the cylinder γ× [0,+∞] such that γ in Sg,n is identified with
γ × {0} in the cylinder.
So, we have the mappings σ̃ : S̃g1,n1+1×S̃g1,n2+1× [0,+∞] → S̃g,n and
σ̃ : S̃g−1,n+2 × [0,+∞] → S̃g,n representing the boundary components
with glued cylinders. We take a form Cg,n, restrict it to H
0 , and
extend it to the glued cylinder by the rule that
(14) σ̃∗C indg,n =
C indg1,n1+1 ∧ C
g2,n2+1
, [e−tΠ−dt·G+ ]
in the first case and
(15) σ̃∗C indg,n =
C indg−1,n+2, [e
−tΠ−dt·G+ ]
in the second case, where [e−tΠ−dt·G+ ] is the bivector obtained from the
operator e−tΠ−dt·G+ , t is the coordinate on [0,+∞]. This determines
C indg,n completely.
Now it is a straightforward calculation to check that the forms C indg,n
are (d+Q)-closed and satisfy the factorization property when restricted
to the strata γ × {+∞}.
3.4. Induced Gromov-Witten theory. The induced Zwiebach in-
variants determine Gromov-Witten invariants. The correlators of the
corresponding Gromov-Witten potential are given by the integrals over
12 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
the fundamental cycles of K̃g,n (we just forget the circles at marked
points in S̃g,n) of the forms C
ψaii .
In fact, the fundamental class of K̃g,n is represented as a sum over all
irreducible boundary strata in Mg,n. Indeed, a boundary stratum γ in
Mg,n has real codimension equal to the doubled number of the nodes
of its generic curve. But then we add in K̃g,n a real two-dimensional
cylinder for each node. A simple explicit calculation allows to ex-
press the integral over the component of the fundamental cycle of K̃g,n
corresponding to γ. It splits into the integrals of the initial Zwiebach
invariants (multiplied by ψ-classes) over the moduli spaces correspond-
ing to the irreducible components of curves in γ; they are contracted
with the bivectors [G−G+] (obtained from the operator G−G+ via the
scalar product) corresponding to the nodes according to the topology
of curves in γ.
So, we represent the correlators of the induced Gromov-Witten the-
ory as sums over graphs. Then one can observe that C0,3 determines
a multiplication on H . Topology of the spaces S0,4 and S1,1 implies
that the whole algebraic structure that we obtain on H is the structure
of cyclic Hodge algebra up to Q-homotopy, see [22]. Let us assume
that the initial system of Zwiebach invariants is simple enough, i. e., it
induces the explicit structure of cyclic Hodge algebra on H and only
the integrals of the zero-degree parts of the initial Zwiebach invariants
(multiplied by ψ-classes) are non-vanishing on fundamental cycles. In
this case, the induced Gromov-Witten potential can be described in
very simple algebraic terms. It is the motivation of the definition of
the Hodge field theory construction given in the next section.
4. Construction of correlators in Hodge field theory
In this section, we describe in a very formal algebraic way the sum
over graphs obtained as an expression for the Gromov-Witten potential
induced from Zwiebach invariants in the previous section.
4.1. Cyclic Hodge algebras. In this section, we recall the definition
of cyclic Hodge dGBV-algebras [22, 31, 32, 24] (cyclic Hodge algebras,
for short). A supercommutative associative C-algebra H with unit
is called cyclic Hodge algebra, if there are two odd linear operators
Q,G− : H → H and an even linear function
: H → C called integral.
They must satisfy the following axioms:
(1) (H,Q,G−) is a bicomplex:
(16) Q2 = G2− = QG− +G−Q = 0;
(2) H = H0 ⊕ H4, where QH0 = G−H0 = 0 and H4 is repre-
sented as a direct sum of subspaces of dimension 4 generated
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 13
by eα, Qeα, G−eα, QG−eα for some vectors e ∈ H4, i. e.
(17) H = H0 ⊕
〈eα, Qeα, G−eα, QG−eα〉
(Hodge decomposition);
(3) Q is an operator of the first order, it satisfies the Leibniz rule:
(18) Q(ab) = Q(a)b+ (−1)ãaQ(b)
(here and below we denote by ã the parity of a ∈ H);
(4) G− is an operator of the second order, it satisfies the 7-term
relation:
G−(abc) = G−(ab)c+ (−1)
b̃(ã+1)bG−(ac) + (−1)
ãaG−(bc)(19)
−G−(a)bc− (−1)
ãaG−(b)c− (−1)
ã+b̃abG−(c).
(5) G− satisfies the property called 1/12-axiom:
(20) str(G− ◦ a·) = (1/12)str(G−(a)·)
(here a· and G−(a)· are the operators of multiplication by a and
G−(a) respectively, str means supertrace).
Define an operator G+ : H → H related to the particular choice
of Hodge decomposition. We put G+H0 = 0, and on each subspace
〈eα, Qeα, G−eα, QG−eα〉 we define G+ as
G+eα = G+G−eα = 0,(21)
G+Qeα = eα,
G+QG−eα = G−eα.
We see that [G−, G+] = 0; Π4 = [Q,G+] is the projection to H4 along
H0; Π0 = Id− Π4 is the projection to H0 along H4.
Consider the integral
: H → C. We require that
Q(a)b = (−1)ã+1
aQ(b),(22)
G−(a)b = (−1)
aG−(b),
G+(a)b = (−1)
aG+(b).
These properties imply that
G−G+(a)b =
aG−G+(b),
Π4(a)b =
aΠ4(b), and
Π0(a)b =
aΠ0(b).
We can define a scalar product on H as (a, b) =
ab. We suppose
that this scalar product is non-degenerate. Using the scalar product
we may turn any operator A : H → H into the bivector that we denote
by [A].
14 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
4.2. Tensor expressions in terms of graphs. Here we explain a
way to encode some tensor expressions over an arbitrary vector space
in terms of graphs.
Consider an arbitrary graph (we allow graphs to have leaves and we
require vertices to be at least of degree 3, the definition of graph that
we use can be found in [24]). We associate a symmetric n-form to each
internal vertex of degree n, a symmetric bivector to each egde, and a
vector to each leaf. Then we can substitute the tensor product of all
vectors in leaves and bivectors in edges into the product of n-forms in
vertices, distributing the components of tensors in the same way as the
corresponding edges and leaves are attached to vertices in the graph.
This way we get a number.
Let us study an example:
v ⊗ v
w ⊗ w
We assign a 5-form x to the left vertex of this graph and a 3-form y
to the right vertex. Then the number that we get from this graph is
x(a, b, c, v, w) · y(v, w, d).
Note that vectors, bivectors and n-forms used in this construction
can depend on some variables. Then what we get is not a number, but
a function.
4.3. Usage of graphs in cyclic Hodge algebras. Consider a cyclic
Hodge algebra H . There are some standard tensors over H , which we
associate to elements of graphs below. Here we introduce the notations
for these tensors.
We always assign the form
(24) (a1, . . . , an) 7→
a1 · · · · · an
to a vertex of degree n.
There is a collection of bivectors that will be assigned below to edges:
[G−G+], [Π0], [Id], [QG+], [G+Q], [G+], and [G−]. In pictures, edges
with these bivectors will be denoted by
, , ,
QG+ ,
G+Q ,
respectively. Note that an empty edge corresponding to the bivector
[Id] can usually be contracted (if it is not a loop).
The vectors that we will put at leaves depend on some variables. Let
{e1, . . . , es} be a homogeneous basis of H0. In particular, we assume
that e1 is the unit of H . To each vector ei we associate formal variables
Tn,i, n ≥ 0, of the same parity as ei. Then we will put at a leaf one of
the vectors En =
i=1 eiTn,i, n ≥ 0, and we will mark such leaf by the
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 15
number n. In our picture, an empty leaf is the same as the leaf marked
by 0.
4.3.1. Remark. There is a subtlety related to the fact that H is a Z2-
graded space. In order to give an honest definition we must do the
following. Suppose we consider a graph of genus g. We can choose
g edges in such a way that the graph being cut at these edge turns
into a tree. To each of these edges we have already assigned a bivector
[A] for some operator A : H → H . Now we have to put the bivector
[JA] instead of the bivector [A], where J is an operator defined by the
formula J : a 7→ (−1)ãa.
In particular, consider the following graph (this is also an example
to the notations given above):
An empty loop corresponds to the bivector [Id]. An empty leaf corre-
sponds to the vector E0. A trivalent vertex corresponds to the 3-form
given by the formula (a, b, c) 7→
If we ignore this remark, then what we get is just the trace of the
operator a 7→ E0 · a. But using this remark we get the supertrace of
this operator.
In fact, this subtlety will play no role in this paper. It affects only
some signs in calculations and all these signs will be hidden in lemmas
shared from [22, 31]. So, one can just ignore this remark.
4.4. Correlators. We are going to define the potential using correla-
tors. Let
(27) 〈τk1(V1) . . . τkn(Vn)〉g
be the sum over graphs of genus g with n leaves marked by τki(Vi),
i = 1, . . . , n, where V1, . . . , Vn are vectors in H , and τki are just formal
symbols. The index of each internal vertex of these graphs is ≥ 3; we
associate to it the symmetric form (24). There are two possible types of
edges: edges marked by [G−G+] (thick black dots in pictures, “heavy
edges” in the text) and edges marked by [Id] (empty edges). Since
an empty edge connecting two different vertices can be contracted, we
assume that all empty edges are loops.
Consider a vertex of such graph. Let us describe all possible half-
edges adjusted to this vertex. There are 2g, g ≥ 0, half-edges coming
from g empty loops; m half-edges coming from heavy edges of graph,
and l leaves marked τka1 (Va1), . . . , τkal (Val). Then we say that the type
of this vertex is (g,m; ka1, . . . , kal). We denote the type of a vertex v
by (g(v), m(v); ka1(v), . . . , kal(v)(v)).
Consider a graph Γ in the sum determining the correlator
(28) 〈τk1(V1) . . . τkn(Vn)〉g
16 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
We associate to Γ a number: we contract according to the graph struc-
ture all tensors corresponding to its vertices, edges, and leaves (for
leaves, we take vectors V1, . . . , Vn). Let us denote this number by T (Γ).
Also we weight each graph by a coefficient which is the product of
two combinatorial constants. The first factor is equal to
(29) V (Γ) =
v∈V ert(Γ) 2
g(v)g(v)!
|aut(Γ)|
Here |aut(Γ)| is the order of the automorphism group of the labeled
graph Γ, V ert(Γ) is the set of internal vertices of Γ. In other words, we
can label each vertex v by g(v), delete all empty loops, and then we get
a graph with the order of the automorphism group equal to 1/V (Γ).
The second factor is equal to
(30) P (Γ) =
v∈V ert(Γ)
Mg(v),m(v)+l(v)
a1(v)
1 . . . ψ
al(v)(v)
The integrals used in this formula can be calculated with the help of
the Witten-Kontsevich theorem [35, 17, 29, 27, 13, 14, 4].
So, the whole contribution of the graph Γ to the correlator is equal
to V (Γ)P (Γ)T (Γ). One can check that the non-trivial contribution to
the correlator 〈τk1(V1) . . . τkn(Vn)〉g is given only by graphs that have
exactly 3g − 3 + n−
i=1 ki heavy edges.
The geometric meaning here is very clear. The number T (Γ) comes
from the integral of the induced Gromov-Witten invariants of degree
zero, while the coefficient V (Γ)P (Γ) is exactly the combinatorial in-
terpretation of the intersection number of ψk11 . . . ψ
n with the stratum
whose dual graph is obtained from Γ by the procedure described after
the definition of V (Γ).
4.5. Potential. We fix a cyclic Hodge algebra and consider the formal
power series F = F(Tn,i) defined as
(31) F = exp
g−1Fg
= exp
a1,...,an∈Z≥0
〈τa1(Ea1) . . . τan(Ean)〉g
Abusing notations, we allow to mark the leaves by τa(Ea), Ea, or a;
all this variants are possible and denote the same.
4.6. Trivial example. For example, consider the trivial cyclic Hodge
algebra: H = H0 = 〈e1〉, Q = G− = 0,
e1 = 1. Then Ea = e1 · ta, and
the correlator 〈τa1(Ea1) . . . τan(Ean)〉g consists just of one graph with
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 17
one vertex, g empty loops, and n leaves marked by a1, . . . , an. The
explicit value of the coefficient of this graph is, by definition,
(32) 〈τa1 . . . τan〉g :=
ψa11 . . . ψ
So, in the case of trivial cyclic Hodge algebra we obtain exactly the
Gromov-Witten potential of the point (i. e.,
Ωg,n, e
≡ 1)) that we
denote below by F pt.
4.6.1. Remark about notations. Abusing notation, we use the same
symbol 〈 〉g for the correlators in GW theory and in Hodge field the-
ory. We hope that it does not lead to a confusion. For instance,
〈τa1(Ea1) . . . τan(Ean)〉g in the trivial example above is the correlator of
the trivial Hodge field theory, while 〈τa1 . . . τan〉g is the correlator of the
trivial GW theory.
5. String, dilaton, and tautological relations
In this section, we prove that the potential (31) satisfies the same
string and dilaton equations as GW potentials.
5.1. String equation.
Theorem 1. If
j=1 aj > 0, we have:
(33) 〈τ0(e1)
τaj (eij )〉g =
〈τaj−1(eij )
k 6=j
τak(eik)〉g;
Proof. Consider a graph Γ contributing to the correlator on the left
hand side of the string equation. The special leaf that we are going
to remove is marked by τ0(e1) and is attached to a vertex v of genus
gv (i. e., with gv attached light loops) with lv more attached leaves
labeled by indices in Iv, |Iv| = lv, and mv attached half-edges coming
from heavy edges and loops.
Let us remove the leaf τ0(e1) and change the label of one of the leaves
attached to the same vertex from τaj (eij ) to τaj−1(eij ). This way we
obtain a graph Γj contributing to the j-th summand of the right hand
side of (33). We take the sum of these graphs over j ∈ Iv. Of course,
we skip the summands where aj = 0.
Note that this sum is not empty (if Γ gives a non-zero contribution
to the left hand side of (33)). Indeed, if it is empty, this means that
aj = 0 for all j ∈ Iv. Therefore, since we expect that the contribution
to P (Γ) of the vertex v on the left hand side is nonzero, it follows that
gv = 0 and mv + lv = 2. So, there are three possible local pictures:
τ0(ej)
τ0(e1)
, and
τ0(ej1)
τ0(ej2)
τ0(e1) .
18 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
The first picture can be replaced with the bivector [G−G+G−G+],
which is equal to zero. Therefore, T (Γ) is also equal to zero. In the
second case, we also get 0 since G−G+(e1ej) = G−G+(E0) = 0. The
third picture is possible only when it is the whole graph, and this is in
contradiction with the assumption that
j=1 aj > 0.
Note also that T (Γ) = T (Γj) and V (Γ) = V (Γj) for all j. Indeed,
we have just removed the leaf with the unit of the algebra, so this
can’t change anything in the contraction of tensors. Therefore, T (Γ) =
T (Γj). Also both the leaf τ0(e1) and the vertex v are the fixed points
of any automorphism of Γ. The same is for the vertex corresponding
to v in Γj . Therefore, the automorphism groups are isomorphic for
both graphs. Since we make no changes for empty loops, it follows
that V (Γ) = V (Γj).
Let us prove that P (Γ) =
P (Γj). Indeed, the vertices of Γ
and Γj are in a natural one-to-one correspondence. Moreover, the lo-
cal pictures for all of them except for v and its image in Γj are the
same. Therefore, the corresponding intersection numbers contributing
to P (Γ) and P (Γj) are the same. The unique difference appeares when
we take the intersection numbers corresponding to v and its images in
Γj, j ∈ Iv. But then we can apply the string equation (6) of the GW
theory of the point (32), and we see that
Mgv,kv+lv+1
Mgv,kv+lv
k 6=j
(here o : Iv → {1, . . . , l} is an arbitrary on-to-one mapping). This
implies that P (Γ) =
P (Γj).
So, we have V (Γ)P (Γ)T (Γ) =
V (Γj)P (Γj)T (Γj). In order to
complete the proof of (33), we should just notice that when we write
down this expression for all graphs contributing to the left hand side
of (33), we use each graph contributing to the right hand side of (33)
exactly once.
5.2. Dilaton equation.
Theorem 2. If 2g − 2 + n > 0, we have:
(36) 〈τ1(e1)
τaj (eij )〉g = (2g − 2 + n)〈
τaj (eij )〉g;
Proof. Consider a graph Γ contributing to the correlator on the left
hand side of (36). The special leaf that we are going to remove is
marked by τ1(e1) and is attached to a vertex v of genus gv (i. e., with
gv attached light loops) with lv more attached leaves labeled by indices
in Iv, |Iv| = lv, and mv attached half-edges coming from heavy edges
and loops.
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 19
Let us remove the leaf τ1(e1). We obtain a graph Γ
′ contributing to
the right hand side of (33). Let us prove this. Indeed, if we remove a
leaf and don’t get a proper graph, it follows that we have a trivalent
vertex. Since the contribution of this vertex to P (Γ) should be non-
zero, it follows that the unique possible local picture is
(37) τ1(e1) .
But this picture is the whole graph, and it is in contradiction with the
condition 2g − 2 + n > 0.
The same argument as in the proof of the string equation shows that
T (Γ) = T (Γ′) and V (Γ) = V (Γ′). Also, the contribution to P (Γ) and
P (Γ′) of all vertices except for the changed one is the same. The change
of the intersection number corresponding to the vertex v is captured
by the dilaton equation (6) of the trivial GW theory (32):
Mgv,kv+lv+1
= (2gv−2+kv+ lv)
Mgv,kv+lv
(again, o : Iv → {1, . . . , l} is an arbitrary on-to-one mapping). This
implies that P (Γ) = (2gv − 2 + kv + lv)P (Γ
′), and, therefore,
(39) V (Γ)P (Γ)T (Γ) = (2gv − 2 + kv + lv)V (Γ
′)P (Γ′)T (Γ′).
Let us write down the last equation for all graphs Γ contributing to
the left hand side (36). Observe that any graph Γ′ contributing to the
right hand side occurs |V ert(Γ′)| times, since the leaf τ1(e1) could be
attached to any its vertex. Therefore, any graph Γ′ contributing to the
right hand side of (36) appears in these equations with the coefficient
v∈V ert(Γ′)
(2gv − 2 + kv + lv) = 2g − 2 + n.
This completes the proof.
5.3. Tautological equations. As we have explained in Section 2.3.3,
any linear relation L among ψ-κ-strata in the cohomology of the mod-
uli space of curves gives rise to a family of universal relations for the
correlators of a Gromov-Witten theory.
Theorem 3 (Main Theorem). The system of universal relations com-
ing from a tautological relation in the cohomology of the moduli space of
curves holds for the correlators 〈
j=1 τaj (eij )〉g of cyclic Hodge algebra.
Note that some special cases of this theorem were proved in [22, 31,
32]. Our argument below is a natural generalization of the technique
introduced in these papers. Also we are able now to give an explana-
tion why we have managed to perform all our calculations there, see
Remark 8.6.1.
20 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
Let us give here a brief account of the proof of this theorem. First,
the definition of correlators of the Hodge field theory can be extended
to the intersection with an arbitrary tautological class α of degree K
in the space Mg,n, not only the a monomial in ψ-classes. In that case,
we do the following. Again, we consider the sum over all graphs Γ
with 3g − 3 + n − K heavy edges, and the number T (Γ) is defined
as above. Instead of the coefficient V (Γ)P (Γ) we use the intersection
number of α with the stratum whose dual graph is obtained from Γ by
the procedure described right after the definition of V (Γ). Namely, a
vertex with g loops is replaced by a vertex marked by g.
This definition is very natural from the point of view of Zwiebach
invariants. However, we know from Gromov-Witten theory that this
extension of the notion of correlator is unnecessary. Indeed, all integrals
with arbitrary tautological classes can be expressed in terms of the
integrals with only ψ-classes via some universal formulas.
The main question is whether these universal formulas also work
in Hodge field theory. Actually, the main result that more or less
immediately proves the theorem is the positive answer to this question.
5.3.1. Organization of the proof. The rest of the paper is devoted to
the proof of Main Theorem, and here we would like to overview it here.
In Section 6, we study the structure of graphs that can appear in
formulas for the correlators of Hodge field theory. We prove that if
T (Γ) 6= 0 and there is at least one heavy edge in Γ, then all vertices
have genus ≤ 1, i. e., there is at most one empty loop at any vertex.
This basically means that in calculations we’ll have to deal only with
genera 0 and 1. Also this allows us to write down the action of a Hodge
field theory.
In Section 7, we prove the main technical result (Main Lemma). In-
formally, it states that Q = −G−ψ when we apply these two operators
to the correlators of a Hodge field theory. In order to prove it, we
look at a small piece (consisting just of one heavy edge and one or two
vertices that are attached to it) in one of the graphs of a correlator. Of
course, in the correlator we can vary this small piece in an arbitrary
way, such that the rest of the graph remains the same. So, when we
consider the sum of all these small pieces, it is also a correlator of the
Hodge field theory. Thus we reduce the proof to a special case of the
whole statement. But since the genus of a vertex is ≤ 1, it appear now
to be a low-genera statement that can be done by a straightforward
calculation.
In Section 8, we present the proof of the Main Theorem. Consider
a ψ-κ-stratum α whose stable dual graph has k ≥ 1 edges. There is a
universal expression of the integral over α coming from Gromov-Witten
theory. It includes k entries of the scalar product restricted to H0. In
terms of graphs, it means that we are to introduce new edges with the
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 21
bivector [Π0] on them, and there are k such edges in our expression. A
direct corollary of the Main Lemma is that we can always replace [Π0]
by [Id]− ψ[G−G+]− [G−G+]ψ.
In Sections 8.2, 8.3, and 8.4, we show that when we replace [Π0] by
[Id] − ψ[G−G+] − [G−G+]ψ at all edges corresponding to the scalar
product restricted to H0, we obtain a new expression for the integral
over α that again contains only heavy edges and empty loops, as any
ordinary correlator. The main problem now is to understand the com-
binatorial coefficient of a graph Γ obtained this way.
Since we have a sum over graphs with heavy edges and empty loops,
it is natural to identify again these graphs with the corresponding strata
in the moduli space of curves. Then we can calculate the intersection
index of the stratum corresponding to a graph Γ and the initial class
α. Roughly speaking, the main thing that we have to do is to decide
about each node in α (represented initially by [Π0]), whether we have
this node in stratum corresponding to Γ. If yes, then we have an
excessive intersection (so, we must put −ψ on one of the half-edges of
the corresponding edge), and we keep this edge in Γ (so, we replace
[Π0] with −ψ[G−G+] or −[G−G+]ψ). If no, then we don’t have this
edge in Γ, so we contract [Π0], i. e., replace it with [Id].
So, the procedure that we used to get rid of the scalar product is
the same as the procedure of the intersection of α with strata of the
complementary dimension. This means (Section 8.5) that the universal
formula coming from Gromov-Witten theory is equivalent to the natu-
ral formula for the “correlator with α” coming from Zwiebach theory.
The tautological relation is a sum of classes equal to zero. So, while
the universal formula coming from Gromov-Witten theory gives (in the
case of a vanishing class) a non-trivial expression in correlators, the nat-
ural formula coming from Zwiebach theory gives identically zero. This
proves our theorem, see Section 8.6.
6. Vanishing of the BV structure
In this section, we recall several useful lemmas shared in [34, 31]. In
particular, these lemmas give some strong restrictions on graphs that
can give a non-zero contirbution to the correlators defined above.
6.1. Lemmas.
Lemma 1. [34, 31] The following vectors and bivectors are equal to
zero:
, G− .
Also let us remind another lemma in [31] that is very useful in cal-
culations.
22 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
Lemma 2. [31] For any vectors V0, V1, . . . , Vk, k ≥ 2,
...A1 A2 Ak
...G−A1 A2 Ak
+ · · ·+
...A1 Ak−1 G−Ak
,(42)
...A1 A2 Ak
...G−A1 A2 Ak
+ · · ·+
...A1 Ak−1 G−Ak
.(43)
Both lemmas are just simple corollaries of the axioms of cyclic Hodge
algebra.
6.2. Structure of graphs. Consider a graph studied in Section 4.4.
It can have leaves, empty and heavy loops, and heavy edges. Consider
a vertex of such graph. Let us assume that there are A empty loops,
B heavy loops, C heavy edges going to the other vertices of the graph,
and D leaves attached to this vertex:
(44) D C
This picture can be considered as an C +D form. Let us denote it by
Φ(A,B,C,D).
Lemma 3. If A ≥ 2 and B + C ≥ 1, then Φ(A,B,C,D) = 0.
In other words, if there are at least two empty loops at a vertex, then
there should not be any heavy loops or edges attached to this vertex.
Otherwise the contribution of the whole graph vanishes. This implies
Corollary 1. In the definition of correlators one should consider only
graphs of one of the following two types:
(1) One-vertex graphs with no heavy edges (loops).
(2) Arbitrary graphs with at most one empty loop at each vertex.
The contribution of all other graphs vanishes.
This corollary dramatically simplifies all our calculations with graphs
given below. Also we can write down now the action of the Hodge field
theory.
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 23
Let F 0g (v0, v1, v2, . . . ), vi ∈ H ⊗ C[[{Tn,i}]] be the “dimension zero”
part of the potential of the Hodge field theory, namely,
(45) F 0g :=
a1+···+an=3g−3+n
〈τa1(va1) . . . τan(van)〉g.
The first sum is taken over n ≥ 0 such that 2g − 2 + n > 0. So, it is
exactly the generating function for the vertices of our graph expressions.
Then the action of the Hodge field theory is equal to
(46) A(v) := F 00 (E0 +G−v, E1, E2, . . . ) + ~F
1 (E0 +G−v, E1, E2, . . . )
gF 0g (E0, E1, E2, . . . )−
Qv ·G−v.
If we put Tn,i = 0 for n ≥ 1, then we immediately obtain the BCOV-
type action discussed in [1, Appendix] and [22, Appendix]. The similar
actions were also studied in [5] and [7].
6.3. Proof of Lemma 3. We consider the form Φ(A,B,C,D) and we
assume that A ≥ 2.
First, let us study the case when C ≥ 1. In this case, our C+D-form
can be represented as a contraction via the bivector [Id] of two forms,
Φ(A−2, B, C−1, D+1) and Φ(2, 0, 1, 1). Let us prove that the last one
is equal to zero. Indeed, this two-form can be represented as Φ̃(·, G+·),
where the two-form Φ̃ is represented by the picture
According to Lemma 1, Φ̃ = 0. Therefore, Φ(2, 0, 1, 1) = 0 and the
whole form Φ(A,B,C,D) is also equal to zero.
Now consider the case when B ≥ 1. In this case, our C + D-form
can be represented as a contraction via the bivector [Id] of two forms,
Φ(A − 2, B − 1, C,D + 1) and Φ(2, 1, 0, 1). Let us prove that the last
one is equal to zero. Indeed,
(48) Φ(2, 1, 0, 1) = =
Here the first equality is definition of Φ(2, 1, 0, 1), the second one is just
an equivalent redrawing, the third equality is application of Lemma 2,
the fourth one is again an equivalent redrawing.
24 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
The last picture contains the bivector (47) which is equal to zero
according to Lemma 1. Therefore, the whole picture is equal to zero,
and Φ(2, 1, 0, 1) = 0. So, the whole form Φ(A,B,C,D) is equal to zero
also in this case. This proves the lemma.
7. Main Lemma
7.1. Statement. The main technical tool that we use in the proof of
Theorem 3 is the lemma that we prove in this section.
Lemma 4 (Main Lemma). For any v1, . . . , vn ∈ H, a1, . . . , an ≥ 0,
〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉g =
〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉g.
A simple corollary of this lemma is the following:
Lemma 5. For any w ∈ H, v1, . . . , vn ∈ H0,
(50) 〈τa0(Qw)τa1(v1) . . . τan(vn)〉g
+ 〈τa0+1(G−w)τa1(v1) . . . τan(vn)〉g = 0.
In other words, we can state informally that Q+ ψG− = 0.
7.2. Special cases. The proof of the lemma can be reduced to a small
number of special cases. We consider correlators whose graphs have
only one heavy edge.
7.2.1. The first case is the following: Let
i=1 ai = n− 4. We prove
that for any v1, . . . , vn ∈ H ,
〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉0 =
〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉0.
First, we see that according to the definition of the correlator, the
left hand side of Equation (51) is the sum over graphs with two vertices
and with [G−G+] on the unique edge that connects the vertices. For
each I ⊔J = {1, . . . , n} we can consider the corresponding distribution
of leaves between the vertices (to be precise, let us assume that 1 ∈ I).
Then the coefficient of such graph is 〈τ0
i∈I τai〉0〈τ0
j∈J τaj〉0, and
we take the sum over all possible positions of Q at the leaves.
Using the Leibniz rule for Q and the property that [Q,G−G+] =
−G−, we see that this sum is equal to the sum over graphs with two
vertices and with [−G−] on the unique edge that connects the vertices.
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 25
For each I ⊔J = {1, . . . , n}, |I|, |J | ≥ 2, we consider the corresponding
distribution of leaves between the vertices. Then the coefficient of
such graph is still 〈τ0
i∈I τai〉0〈τ0
j∈J τaj〉0, and the underlying tensor
expression can be written (after we multiply the whole sum by −1) as
i∈I\{1}
vi ·G−(
Let us recall that the 7-term relation for G− implies that
vj) =
i,j∈J, i<j
G−(vivj)
k∈J\{i,j}
vk(53)
− (|J | − 2)
G−(vj)
i∈J\{i,j}
Using this, we can rewrite the whole sum over graphs as
1<i<j
k 6=1,i,j
vk ·G−(vivj)
I′⊔J ′⊔{i,j}={2,...,n}
〈τa1τ0
τak〉0〈τaiτajτ0
k∈J ′
τak〉0
i 6=1
j 6=1,i
vj ·G−(vi)
I′⊔J ′⊔{i}={2,...,n}
(|J ′| − 1) 〈τa1τ0
τaj〉0〈τaiτ0
j∈J ′
τaj〉0
Using that
I′⊔J ′⊔{i,j}={2,...,n}
〈τa1τ0
τak〉0〈τaiτajτ0
k∈J ′
τak〉0 =
〈τa1+1
k 6=1
τak〉0,
Equation (53), and the fact that
(56) (n− 3)〈τa1+1
j 6=1
τaj 〉0
I′⊔J ′⊔{i}={2,...,n}
(|J ′| − 1) 〈τa1τ0
τaj〉0〈τaiτ0
j∈J ′
τaj〉0
= 〈τai+1
j 6=i
τaj〉0,
26 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
we can rewrite Expression (54) as
G−(vi),
j 6=i
〈τai+1
j 6=i
τaj 〉0.
The last formula coincides by definition with the right hand side of
Equation (51) multiplied by −1. This proves the first special case.
7.2.2. The second case is in genus 1. Let
i=1 ai = n− 1. We prove
that for any v1, . . . , vn ∈ H ,
〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉1 =
〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉1.
According to the definition of the correlator, the left hand side of
Equation (58) is the sum over graphs of two possible types. The first
type include graphs with two vertices and two edges. The first edge
is heavy and connects the vertices; the second edge is an empty loop
attached to the first vertex. For each I ⊔ J = {1, . . . , n}, |J | ≥ 2,
we can consider the corresponding distribution of leaves between the
vertices (we assume that leaves with indices in I are at the first edge).
Then the coefficient of such graph is 〈τ0
i∈I τai〉1〈τ0
j∈J τaj 〉0. The
second type include graphs with one vertex and one heavy loop. All
leaves are attached to this vertex, and the coefficient of such graph
is 〈τ 20
i=1 τai〉0. For both types of graphs, we take the sum over all
possible positions of Q at the leaves.
Using the Leibniz rule for Q and the property that [Q,G−G+] =
−G−, we get the same graphs as before, but there is no Q, and instead
of [G−G+] we have −[G−] on the corresponding edge. Using Lemma 2,
we move −G− in graphs of the first type to the leaves marked by indices
in J . Using the 1/12-axiom and Lemma 2, we move −G− in graphs of
the second type to all leaves.
This way we get graphs of the same type in both cases. We get
graphs with one vertex, one empty loop attached to it, all leaves are
also attached to this vertex, and there is −G− on one of the leaves.
One can easily check that the coefficient of the graphs with −G− at
the i-th leaf is equal to
I⊔J⊔{i}={1,...,n}
τak〉1〈τ0τai
τak〉0 +
〈τ 20
τak〉0
= 〈τai+1
k 6=i
τak〉1
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 27
It is exactly the unique graph contributing to the i-th summand of
the right hand side of Equation (58), and the coefficient is right. This
proves that special case.
7.2.3. Consider g ≥ 2. Let
i=1 ai = 3g + n − 4. In this case, the
statement that for any v1, . . . , vn ∈ H ,
〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉g =
〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉g,
is immediately reduced to 0 = 0; it is a simple corollary of Lemma 1.
7.3. Proof of Main Lemma. Consider the left hand side of Equa-
tion (49). As usual, using the Leibniz rule for Q and the property that
[Q,G−G+] = −G−, we can remove all Q, but then we must change one
of [G−G+] on edges to −[G−]. Let us cut out the peaces of graphs that
includes this edges with −[G−], all empty loops, leaves and halves of
heavy edges attached to the ends of this special edge.
Since we consider the sum over all possible graphs contributing to
correlators, these small pieces can be gathered into groups according
to the type of the rest of the initial graph. Each group forms exactly
one of the special cases studied above. So, we know that −G− should
jump either to one of the leaves or to one of the heavy edges attached
to the ends of its edge. In the first case, we get exactly the graphs in
the right hand side of Equation (49); in the second case, we get zero.
One can easily check that we get the right coefficients for the graphs
in the right hand side of Equation (49). This proves the lemma.
8. Proof of Theorem 3
8.1. Equivalence of expression in graphs. Consider the expres-
sion in correlators corresponding to a ψ-κ-stratum as it is described in
Section 2.3.2. To each vertex of the corresponding stable dual graph
we assign the sum of graphs that forms correlator in the sense of Sec-
tion 4.4. The leaves of these graphs corresponding to the edges of the
stable dual graph (nodes) are connected in these pictures by edges with
[Π0] (the restriction of the scalar product toH0). We call the edges with
[Π0] “white edges” and mark them in pictures by thick white points,
see (25).
The axioms of cyclic Hodge algebra imply a system of linear equa-
tions for the graphs of this type. In particular, it has appeared that
playing with this linear equations we can always get rid of white edges
in the sum of pictures corresponding to a stable dual graph, see [22,
31, 32]. However, previously it was just an experimental fact. Now we
can show how it works in general.
28 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
The numerous examples of the correspondence between stable dual
graphs and graphs expressions in cyclic Hodge algebras and also of the
linear relations implied by the axioms of cyclic Hodge algebra are given
in [22, 31, 32].
Below, we explain how one can represent the expression in correlators
corresponding to a ψ-κ-stratum in terms of graphs with only empty and
heavy edges and with no white edges. The unique tool that we need is
Lemmas 4 and 5 proved above.
8.2. The simplest example. Consider a stable dual graph with two
vertices and one edge connecting them:
g1 g2
ψa1 . . . ψan1 ψb1 . . . ψbn2
κu1,...,ul1 κv1,...,vl2
ψa0 ψb0
The corresponding expression in correlators is
(62) 〈τa0(ej1)
τai(e•)
τui(e1)〉g1·
ηj1j2 · 〈τb0(ej2)
τbi(e•)
τvi(e1)〉g1
(here we denote by e• an arbitrary choice of ei ∈ H0). It is convenient
for us to rewrite this expression as
(63) 〈τa0(xα1)
τai(e•)
τui(e1)〉g1·
α1α2 · 〈τb0(xα2)
τbi(e•)
τvi(e1)〉g1,
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 29
where {xα} is the basis of the whole H . Using the fact that Π0 =
Id−QG+ −G+Q and applying Lemma 5, we obtain
(64) 〈τa0(xα1)
τai(e•)
τui(e1)〉g1·
α1α2 · 〈τb0(xα2)
τbi(e•)
τvi(e1)〉g1
= 〈τa0(xα1)
τai(e•)
τui(e1)〉g1·
[Id]α1α2 · 〈τb0(xα2)
τbi(e•)
τvi(e1)〉g1
− 〈τa0+1(xα1)
τai(e•)
τui(e1)〉g1·
[G−G+]
α1α2 · 〈τb0(xα2)
τbi(e•)
τvi(e1)〉g1
− 〈τa0(xα1)
τai(e•)
τui(e1)〉g1·
[G−G+]
α1α2 · 〈τb0+1(xα2)
τbi(e•)
τvi(e1)〉g1
In all three summands of the right hand side we still have two correla-
tors, whose leaves corresponding to the nodes are connected by some
special edges. But now the connecting edge is either marked by [Id]
(an empty edge) or by [G−G+] (an ordinary heavy edge). So, this way
we get rid of the white edge in this case.
Informally, in terms of pictures, we can describe Equation (64) as
When we put ψ, we mean that we add one more ψ-class at the node
at the corresponding branch of the curve. Dashed circles denote corre-
lators.
8.3. Example with two nodes. Now we consider an example of stra-
tum, whose generic point is represented by a three-component curve.
Again, we allow arbitrary ψ-classes at marked points and two branches
at nodes.
30 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
We perform the same calculation as above, but now we explain it in
terms of informal pictures from the very beginning. So, the first step
is the same as above:
=(66)
Then we apply Lemma 4 to each of the summands in the right hand
side:
=(67)
We take the sum of these three expressions, and we see that all pictures
where we have edges with [G−] and [G+] are cancelled. So, we get an
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 31
expression for the sum of graphs representing the initial stratum in
terms of graphs with only empty and heavy edges.
8.4. General case. The general argument is exactly the same as in
the second example. In fact, this gives a procedure how to write an
expression in graphs with only empty and heavy edges (and no white
edges) starting from a stable dual graph. Let us describe this proce-
dure.
Take a stable dual graph corresponding to a ψ-κ-stratum of dimen-
sion k in Mg,n. First, we are to decorate it a little bit. For each
edge, we either leave it untouched, or substitute it with an arrow (in
two possible ways). At the pointing end of the arrow, we increase the
number of ψ-classes by 1. Each of these graphs we weight with the
inversed order of its automorphism group (automorphisms must pre-
serve all decorations) multiplied by (−1)arr, where arr is the number
of arrows.
Consider a decorated dual graph. To each its vertex we associate the
corresponding correlator of cyclic Hodge algebra (we add new leaves in
order to represent κ-classes). Then we connect the leaves corresponding
to the nodes either by empty edges (if the corresponding edge of the
decorated graph is untouched) or by heavy edges (if the corresponding
edge of the dual graph is decorated by an arrow).
It is obvious that the number of heavy edges in the final graphs is
equal to k.
8.5. Coefficients. We can simplify the resulting graphs obtained in
the previous subsection. First, we can contract empty edges (as much
as it is possible; it is forbidden to contract loops). Second, we can
remove leaves added for the needs of κ-classes. Indeed, each such leaf
is equipped with a unit of H , so it doesn’t affect the contraction of
tensors corresponding to a graph. Moreover, when we remove all leaves
corresponding to κ-classes, we still have graph with at least trivalent
vertices. Otherwise, this graph is equal to zero, c. f. arguments in the
proofs of string and dilaton equations.
So, we obtain final graphs that have the same number of heavy edges
as the dimension of the initial ψ-κ-stratum, the same number of leaves
as the initial dual graph, and some number of empty loops, at most
one at each vertex. The exceptional case is when k = 0; in this case we
obtain only one graph, with one vertex, n leaves, and g empty loops.
In the first case, let us turn a graph like this into a stable dual
graph. Just replace its vertices with no empty loops by vertices of
genus zero, vertices with empty loops by vertices of genus one, heavy
edges are edges, and leaves are leaves. There are no ψ- or κ- classes. It
is obvious that the codimension of the stratum corresponding to this
dual graph is k. Indeed, in this case it is just the number of nodes.
32 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG
So, to each ψ-κ-stratum X of dimension k > 0 we associate a linear
combination
ciYi of strata of codimension k with no ψ- or κ-classes,
whose curves have irreducible components of genus 0 and 1 only.
Proposition 1. We have ci = X · Yi.
Proof. We prove it two steps. First, consider a one-vertex stable dual
graph with no edges (just a correlator). In this case, the intersection
numberX ·Yi is just by definition ci = V (Γi)P (Γi), where Γi is the cyclic
Hodge algebra graph that turns into Yi via the procedure described
above.
Then, consider a stable dual graph with one edge. It is the intersec-
tion of the one-vertex stable dual graph with an irreducible component
of the boundary. For a given Yi, this component of the boundary either
intersects it transversaly, or we have an excessive intersection. In the
first case, the corresponding node is not represented in Yi. This means
that in Γi it should be an empty edge. In the second case, this node
is one of the nodes of Yi, so it should be a heavy edge of Γi. Also, it
is an excessive intersection, so we are to add the sum of ψ-classes with
the negative sign at the marked points (half-edges) corresponding to
the node, see [11, Appendix].
Exactly the same argument works for an arbitrary number of nodes,
we just extend it by induction. �
In the case of k = 0, we get just one final graph with coefficient c.
Proposition 2. If k = 0, the coefficient of the final graph is equal to
the number of points in the initial ψ-κ-stratum.
Proof. If k = 0, this means that each of the vertices of the initial stable
dual graph also has dimension 0, and the corresponding correlator of
cyclic Hodge algebra is represented by one one-vertex graph with no
heavy edges. Also this means that each edge of the initial stable dual
graph is replaced in the algorithm above by an empty edge. So, we can
think that we just work with the correlators of the Gromov-Witten
theory of the point. In this case Proposition becomes obvious. �
Now we deduce Theorem 3 from these propositions.
8.6. Proof of Theorem 3. Consider the system of subalgebras
(70) RH∗1 (Mg,n) ⊂ RH
∗(Mg,n)
of the cohomological tautological algebras of Mg,n generated by strata
with no ψ- or κ-classes and with irreducible curves of genus 0 and 1
only.
Let L be a linear combination of ψ-κ-strata of dimension k in Mg,n.
Then the expression in correlators of cyclic Hodge algebras correspond-
ing to L is equaivalent to a sum of some graphs with coefficients equal
to the intersection of L with classes in RHk1 (Mg,n).
TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 33
So, if the class of L is equal to zero, then the corresponding equation
(and also the whole system of equations that we described in Sec-
tion 2.3.3) for correlators of cyclic Hodge algebra is valid. Theorem is
proved.
8.6.1. Remark. Evidently, RH∗1 (Mg,n) is a module over RH
∗(Mg,n).
Also it is obvious that RH∗1 (Mg,n) is closed under pull-backs and push-
forwards via the forgetful morphisms. This explains why it was enough
to make only one check in the simplest case in order to get the system
of equations in [22, 32] (cf. an argument in the last section in [22]).
8.7. An interpretation of Propositions 1 and 2. From the point
of view of the theory of Zwiebach invariants, both propositions look
very natural. Indeed, we try to give a graph expression for the integral
of an induced Gromov-Witten form multiplied by a tautological class
X . Since we know that we are able to integrate only degree zero parts
of induced Gromov-Witten invariants, we should just take the sum over
all graphs that correspond to the strata of complimentary dimension
in RH∗1 (Mg,n). The coefficients are to be the intersection numbers of
these strata with X .
On the other hand, we know that in any Gromov-Witten theory it is
enough to fix the integrals of Gromov-Witten invariants multiplied by
ψ-classes. Then the integrals of Gromov-Witten invariants multiplied
by arbitrary tautological classes are expressed by universal formulas.
We can try to use these universal formulas also in Hodge field theory.
They are exactly our expressions with white edges.
So, we have two different natural ways to express in terms of graphs
the integrals of induced Gromov-Witten invariants multiplied by tau-
tological classes. Propositions 1 and 2 state that these two different
expressions coinside.
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TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 35
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Institute for Theoretical and Experimental Physics, Bolshaya Che-
remushkinskaya 25, Moscow, 117218, Russia.
E-mail address : losev@itep.ru
Department of Mathematics, University of Zurich, Winterthurer-
strasse 190, CH-8057 Zurich, Switzerland.
E-mail address : sergey.shadrin@math.unizh.ch
Department of Algebra, Faculty of Mechanics and Mathematics,
Moscow State University, Leninskie Gory, GSP, Moscow, 119899, Rus-
E-mail address : shneiberg@mtu-net.ru
	1. Introduction
	2. Gromov-Witten theory
	3. Zwiebach invariants
	4. Construction of correlators in Hodge field theory
	5. String, dilaton, and tautological relations
	6. Vanishing of the BV structure
	7. Main Lemma
	8. Proof of Theorem 3
	References
