INTERSECTION NUMBERS AND RANK ONE COHOMOLOGICAL FIELD THEORIES IN GENUS ONE

ALEXANDRE KABANOV AND TAKASHI KIMURA

Abstract. We obtain a simple, recursive presentation of the tautological (κ, ψ, and λ) classes on the moduli space of curves in genus 0 and 1 in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, restricted, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann, Manin, and Zagier.

Recently, there has been a great deal of interest in the topology of the moduli space of curves. Much of this interest has been due to the important role that these spaces (and their cousins, the moduli space of stable maps) play in the theory of Gromov-Witten invariants and quantum cohomology [22, 32, 30] whose origins in the physical literature are called a topological gravity [32]. They furnish nontrivial examples of cohomological field theories (CohFTs) [22, 25], in genus zero (and conjecturally for higher genera). Often, this structure is enough to completely determine the Gromov-Witten invariants themselves. The moduli spaces of curves are endowed with tautological classes whose generating functions for their associated intersection numbers obey a system of differential equations which often possess remarkable properties [32, 21]. In this paper, we apply a mixture of algebraic geometry and combinatorics to find a simple presentation of these classes in genus 0 and 1 to obtain a generalization of some equations due to Witten and Dijkgraaf [32, 5]. These generating functions parameterize the potentials associated to the space of all normalized, restricted, rank one cohomological field theories in genus one and endow this space with coordinates which are additive with respect to tensor product in the category of CohFTs. This paper is motivated by the work of Kaufmann, Manin, and Zagier [19].

The moduli space of genus g curves with n marked points, M_{g,n} := { [C; x_{1}, x_{2}, …, x_{n}] }, is the moduli space of configurations of n marked points, on a smooth, complex curve (Riemann surface) C of genus g. We assume throughout that the stability condition 2 − 2g − n < 0 is satisfied. This moduli space has a compactification \overline{M}_{g,n} (due to Deligne-Knudsen-Mumford) which is the moduli space of stable curves of genus g with n marked points where the boundary divisor \overline{M}_{g,n} − M_{g,n} is the locus of degenerate curves. The space \overline{M}_{g,n} is a stratified, complex orbifold (stack) of complex dimension 3g − 3 + n where each stratum is indexed by a decorated graph (stable graph) which denotes the type of degeneration that curves in that stratum have. A stable graph represents a cohomology class on \overline{M}_{g,n} by taking the closure of the corresponding stratum and applying Poincaré duality to the associated (rational) homology class.

Date: June 18, 1997.

The space \overline{M}_{g,n} is endowed with tautological cohomology classes whose study was initiated by Mumford [28]. Let L_{i} be the line bundle over \overline{M}_{g,n} whose fiber over a point [C; x_{1}, x_{2}, …, x_{n}] is T_{x_{i}}^{∗}C. Then ψ_{(g,n),i} = c_{1}(L_{i}), the first Chern class of L_{i}. The classes κ_{(g,n),i} in H^{2i}(\overline{M}_{g,n}, ℚ)^{S_{n}} are defined by κ_{(g,n),i} := π_{∗}(c_{1}(ω_{g,n}(D))^{i+1}) where ω_{g,n} is the cotangent bundle to the fibers of the universal curve π : \overline{M}_{g,n+1} → \overline{M}_{g,n}, D is the sum of the images of the canonical sections, and π_{∗} is fiber integration [1]. Integrals of products of these classes (intersection numbers) are of great geometric interest and are the main object of study in this paper. In particular, \frac{1}{2π^{2}} κ_{(g,n),1} is the class of the Weil-Petersson symplectic form [1]. Zograf in [33] obtained a recursion formula for the classical Weil-Petersson volumes in genus zero. The intersection numbers of the κ classes studied in [19] are called higher Weil-Petersson volumes.

In the first part of this paper we study a generating function H(t; s) ∈ ℂ[[t, s]] which incorporates all intersection numbers of the ψ and κ classes. Here t = (t_{0}, t_{1}, …) and s = (s_{1}, s_{2}, …) are formal variables. The function H_{g}(t; s) denotes the summand of H(t; s) corresponding to genus g. This function has the property that H(t, 0) = F (t), the generating function for the ψ intersection numbers defined by Witten in [32, 31]. On the other hand, setting t_{i} = 0 for i ≥ 1 gives a generating function for the intersection numbers of the κ classes. This function is closely related to that considered in [19].

We give a simple, recursive presentation of powers of the κ and ψ classes in terms of boundary strata in genus zero and genus one and derive a simple system of differential equations for H in genus zero and one which completely determine those intersection numbers. Taking appropriate limits in genus zero, we obtain equations due to Witten [32] for the ψ classes and equations for the κ classes which are equivalent of those in [19] and much simpler. (A differential equation for the classical Weil-Petersson volumes in genus zero was first obtained in [27].) This simplification arises because our presentation of the κ classes in terms of boundary strata is simpler. Furthermore, the genus one equation can be solved to obtain the relation

H_{1} = \frac{1}{24} log H′′′_{0}, (1)

where ′ denotes the partial derivative with respect to t_{0}, for all values of s and t generalizing the result of Dijkgraaf and Witten saying that F_{1} = \frac{1}{24} log F′′′_{0} [5]. We were informed that (1) was known to Zograf [26] in the special case where t_{j} = 0 for all j ≥ 1 and s_{i} = 0 for all i ≥ 2.

Witten [32] conjectured (and Kontsevich [21] proved) that F (s) was the logarithm of a tau function of the KdV hierarchy after rescaling the variables. This tau function was completely characterized [5, 17, 24] by being annihilated by a sequence of differential operators L_{n} for n ≥ −1 satisfying [L_{m}, L_{n}] = (m − n)L_{m+n}, the relations of the Virasoro algebra.^{1} The equations corresponding to L_{−1} and L_{0} were proven by Witten in [32, 31] and are essentially the so-called puncture and dilaton equations. We derive the analog of the puncture and dilaton equations for intersection numbers of κ and ψ classes. These equations are valid for all genera and do not use the presentation of the tautological classes in terms of boundary strata. Solving these equations provides another proof of (1). It is not clear which one of these approaches will prove to be most useful in higher genera.

In the second part of our paper, we apply the previous to describe the moduli space of normalized, restricted, rank one cohomological field theories (CohFTs) in genus one generalizing the results of Kaufmann, Manin, and Zagier in genus zero [19].

A (complete) CohFT of rank r [22, 25] is an r dimensional vector space with metric (V, h) together with a collection of linear maps H_{•}(\overline{M}_{g,n}) → T^{n}V which are equivariant under the action of the permutation groups and which satisfy some compatibility conditions arising from inclusion of strata on \overline{M}_{g,n} In the language of Getzler and Kapranov [13], the maps form a morphism of modular operads. Restricting to g = 0, a CohFT (V, h) is equivalent to endowing (V, h) with the structure of a (formal) Frobenius manifold [6, 16, 25]. The most spectacular examples of such theories arise when (V, h) is the cohomology ring of certain smooth, projective varieties with its intersection pairing and the morphisms come from the Gromov-Witten invariants associated to the manifold thereby endowing the cohomology ring with a deformed cup product giving it the structure of quantum cohomology. In many cases, e.g. ℂℙ^{n} or Grassmannians, the structure of a CohFT is strong enough to completely determine, recursively, the number of rational curves in the manifold counted with multiplicity (see [22, 10]).

1This result is all the more interesting because of the recent conjecture in [7] which predicts a Virasoro algebra playing a similar role in the the case where the target manifold is nontrivial.

A CohFT in genus zero can be described in terms of a certain generating function (potential) associated to the structure morphisms which must satisfy the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations (see [10, 22]). These equations encode the relations between the boundary strata in \overline{M}_{0,n} due to Keel in genus zero. Recently, Getzler [11] derived equations which are the analogs of WDVV in genus one by proving new relations which plays an analogous role in genus one to those of Keel in genus zero. His equation allowed him to predict the elliptic Gromov-Witten invariants of ℂℙ^{2} and ℂℙ^{3} (see also [2]).

Kaufmann, Manin, and Zagier proved [19] that the moduli space of normalized, rank one, CohFTs in genus zero has coordinates s such that the tensor product in the category of CohFTs is additive in these coordinates. We prove that the moduli space of normalized, rank one, restricted, CohFTs in genus one has similar coordinates (s, u). Here the variable u arises because in genus one we need to introduce another tautological cohomology class (called λ) due to Mumford. Our proof involves a mixture of techniques from Kaufmann, Manin, and Zagier [19], Getzler [11], and results from the first part of this paper.

In section 1, we review the geometry of the moduli space of stable curves, its stratification in terms of stable graphs, tautological cohomology classes, their intersection numbers, and associated generating functions. In section 2, we obtain a simple presentation for these classes in terms of stable graphs in genus 0 and 1 and derive differential equations satisfied by the generating functions associated to their intersection numbers. In section 3, we derive the analogues of the puncture and dilaton equations. In section 4, we write closed form expressions for these intersection numbers. In section 5, we use analytic properties of the generating function to prove an asymptotic formula for the Weil-Petersson volumes of the moduli space of genus 1 curves as the number of punctures becomes very large. Finally in section 6, we describe the moduli space of rank 1, restricted, CohFTs in genus one.

Acknowledgment . We are greatful to the Max Planck Institut für Mathematik for their financial support and for providing a wonderfully stimulating atmosphere. We would like to thank R. Dijkgraaf, E. Getzler, and Yu. Manin for useful conversations. We are grateful to J. Stasheff for his comments on an earlier version of this paper. We would also like to thank K. Belabas for his TEXnical assistance and for providing the music.

1. Moduli Space of Curves

Notation . In this paper we always consider cohomology with the rational coefficients: H^{•}(X) stands for H^{•}(X; ℚ). We denote the set {1, …, n} by [n]. If I is a finite set we denote its cardinality by |I|.

1.1. Basic Definitions. Let M_{g,n} be the moduli space of smooth curves of genus g with n marked points, where 2g − 2 + n > 0, i.e. M_{g,n} = { [Σ; x_{1}, x_{2}, …, x_{n}] } where Σ is a genus g Riemann surface and x_{1}, x_{2}, …, x_{n} are distinct marked points on Σ. Two such configurations are equivalent if they are related by a biholomorphic map. The moduli space M_{g,n} has a natural compactification due to Deligne, Knudsen, and Mumford denoted by \overline{M}_{g,n} = \{ [C; x_{1}, x_{2}, …, x_{n}] \} which is the moduli space of stable curves of genus g with n punctures in which M_{g,n} sits as a dense open subset. The spaces \overline{M}_{g,n} are connected, compact, complex orbifolds (in fact, stacks) with complex dimensions 3g − 3 + n. The complement of M_{g,n} in \overline{M}_{g,n} is a divisor with normal crossings and consists of those stable curves which have double points.

The moduli space \overline{M}_{g,n} forms the base of a universal family. Let π : C_{g,n} → \overline{M}_{g,n} be the universal curve which can be identified with C_{g,n} = \overline{M}_{g,n+1} where π is the projection obtained by forgetting the (n + 1)^{st} puncture and followed by collapsing any resulting unstable irreducible components of the curve, if any, to a point. The universal curve C_{g,n} → \overline{M}_{g,n} is furthermore endowed with canonical sections σ_{1}, σ_{2}, …, σ_{n} such that σ_{i} maps [C; x_{1}, x_{2}, …, x_{n}] ↦ [C′; x′_{1}, x′_{2}, …, x′_{n+1}] where C′ is obtained from C by attaching a three punctured sphere to x_{i} at one of its punctures to create a double point, then labeling the remaining two punctures on the sphere x′_{i} and x′_{n+1}, and finally setting all other x′_{j} = x_{j}. The sections σ_{i} are well-defined since \overline{M}_{0,3} is a point. The image of σ_{i} : \overline{M}_{g,n} → C_{g,n} gives rise to a divisor D_{i} in C_{g,n} for all i = 1, 2, …, n.

1.2. Natural Stratification. In the sequel it will be convenient to consider markings by arbitrary finite sets rather than by just [n]. If I is a finite set we denote by \overline{M}_{g,I} ≅ \overline{M}_{g,|I|} the corresponding moduli space.

The natural stratification of \overline{M}_{g,n} is best described in terms of graphs, and therefore we start with fixing the notation concerning graphs. We will consider only connected graphs. Each graph Γ can be described in terms of its set of vertices V (Γ), set of edges E(Γ), and set of tails S(Γ). Each edge has two endpoints belonging to V (Γ) which are allowed to be the same. Each tail has only one endpoint. If v ∈ V (Γ),we denote by n(v) the number of half-edges emanating from v, where each edge gives rise to two half-edges, and each tail to one half-edge.

The natural stratification of \overline{M}_{g,n} is determined by the type of the degeneration of the curve representing a point in the moduli space, and its strata can be labeled by stable graphs. A stable graph consists of a triple (Γ, g, μ), where Γ is a connected graph as above, g : V (Γ) → ℤ_{≥0}, and μ is a bijection between S(Γ) and a given set I. Moreover, one requires that for each vertex v, the stability condition 2g(v)−2+n(v) > 0 is satisfied. If [C; x_{1}, x_{2}, …, x_{n}] is a stable, n-pointed curve one obtains the corresponding stable graph, called its dual graph, by collapsing each irreducible component to a point (vertex), connecting any two vertices if their corresponding components share a double point and attaching a tail to a vertex for each marked point on that component.

We define the genus g(Γ) of Γ to be b_{1}(Γ) + ∑_{v∈V (Γ)} g(v), where b_{1}(Γ) is the first Betti number of Γ. We denote by G_{g,n} the set of the equivalence classes of stable graphs of genus g with n tails labeled by [n]. There is a natural action of the symmetric group S_{n} on G_{g,n}. Associating a stable curve to its dual graph provides an S_{n}-equivariant bijection between the strata of the natural stratification of \overline{M}_{g,n} and the elements of G_{g,n}.

Let \overline{M}_{Γ} be (the closure of) the moduli space of stable curves whose dual graph is Γ. It is a closed irreducible subvariety of codimension |E(Γ)| of \overline{M}_{g(Γ),S(Γ)}. Moreover, it is isomorphic to a quotient of the cartesian product

∏_{v∈V (Γ)} \overline{M}_{g(v),n(v)}

by Aut(Γ), where the automorphisms of a stable graph (Γ, g, μ) are required to preserve g and μ. This quotient morphism can be made canonical if one creates a pair of labels for each edge of Γ and labels the n(v) half-edges emanating from v by the corresponding elements of S(Γ) with the labels corresponding to the edges.

Each \overline{M}_{Γ} determines the fundamental class, in the sense of orbifolds, lying in H^{•}(\overline{M}_{g(Γ),n(Γ)}). The pull back of this class under the morphism π : \overline{M}_{g,n+1} → \overline{M}_{g,n} is represented by a subvariety of \overline{M}_{g,n+1} corresponding to the |V (Γ)| graphs each of which is obtained by attaching a tail numbered n + 1 to a vertex of Γ. It is also easy to push down these fundamental classes. Let \overline{M}_{Γ′} represents an element of H^{•}(\overline{M}_{g,n+1}). The image of this element under π_{∗} : H^{•+2}(C_{g,n}) → H^{•}(\overline{M}_{g,n}), induced by the fiber integration, is zero if after removing the (n + 1)^{st} tail from Γ′ the graph remains stable. In the other case, when the removal of the (n + 1)^{st} tail destabilizes Γ′, the image is obtained by stabilization, i.e., contracting the edge connecting the unstable vertex with the rest of the graph.

1.3. Tautological Classes. We will now describe three types of tautological cohomology classes (ψ, κ, and λ) associated to the universal curve. Consider the universal curve C_{g,n} → \overline{M}_{g,n}. The cotangent bundle to its fibers (in the orbifold sense) forms the holomorphic line bundle ω_{g,n}. Let L_{(g,n),i} → \overline{M}_{g,n} be given by the pullback L_{(g,n),i} = σ_{i}^{∗}ω_{(g,n)}. The tautological classes ψ_{(g,n),i} in H^{2}(\overline{M}_{g,n}) are defined by

ψ_{(g,n),i} := c_{1}(L_{(g,n),i})

where c_{1} denotes the first Chern class.^{2}

The tautological classes κ_{(g,n),i} in H^{2i}(\overline{M}_{g,n}) for i = 0, 1, …, (3g − 3 + n) are defined as follows. Consider the bundle ω_{g,n}(D) → C_{g,n} consisting of ω_{g,n} twisted by the divisor D = ∑_{i=1}^{n} D_{i}, then

κ_{(g,n),i} := π_{∗}(c_{1}(ω_{g,n}(D))^{i+1}).

In particular, κ_{(g,n),0} = 2g − 2 + n ∈ H^{0}(\overline{M}_{g,n}) is the negative of the Euler characteristic of a smooth curve of genus g with n points removed. We also have the equality ω_{g,n}(D) = L_{(g,n+1),n+1} [1, 15]. Therefore

κ_{(g,n),i} = π_{∗}(ψ_{(g,n+1),n+1}^{i+1}). (2)

The tautological λ classes are defined to be

λ_{(g,n),i} := c_{l}(π_{∗} ω_{g,n}) ∈ H^{2i}(\overline{M}_{g,n}),

where l = 1, …, g because π_{∗} ω_{g,n} is an orbifold bundle of rank g. (There are no λ classes in genus 0 and we define λ_{(0,n),i} := 0.) One can easily see that λ_{(g,n+1),i} = π^{∗}λ_{(g,n),i}. Therefore, all of the λ classes are pull backs of the λ classes on \overline{M}_{1,1} and \overline{M}_{g,0}, g ≥ 2. They can be expressed in terms of the κ classes, the ψ classes, and the cohomology classes lying at the boundary [28]. In particular,

2The Chern classes are in the sense of orbifolds and are therefore rational.

κ_{(g,n),1} = 12λ_{(g,n),1} − δ_{(g,n)} + ∑_{i=1}^{n} ψ_{(g,n),i}, (3)

where δ_{(g,n)} is the fundamental class of \overline{M}_{g,n} − M_{g,n} [3, 9, 28]. (This formula was brought to our attention by E. Getzler.)

We will drop subscripts associated to the genus and the number of punctures if there is no ambiguity.

Notation . Let S_{k} be the set of infinite sequences of non-negative integers m = (m_{k}, m_{k+1}, m_{k+2}, …) such that m_{i} = 0 for all i sufficiently large. We denote by δ_{a} the infinite sequence which has only one nonzero entry 1 at the a^{th} place. For m = (m_{0}, m_{1}, m_{2}, …) ∈ S_{0} and t = (t_{0}, t_{1}, t_{2}, …), a family of independent formal variables, we will use notation of the type

|m| := ∑_{i≥0} i m_{i}, ||m|| := ∑_{i≥0} m_{i}, m! := ∏_{i≥0} m_{i}!, t^{m} := ∏_{i≥0} t_{i}^{m_{i}}.

We say that l ≤ m if l_{i} ≤ m_{i} for all i. If l ≤ m we let

\binom{m}{l} := ∏_{i≥0} \binom{m_{i}}{l_{i}}

We will use the same notation when m ∈ S_{1}.

1.4. Generating Functions. Witten [31] defined a generating function which incorporates all of the information about the integrals of products of the ψ classes. In order to describe this function we need to introduce the following notation. Let

⟨τ_{d_{1}}τ_{d_{2}} … τ_{d_{n}}⟩ := ∫_{\overline{M}_{g,n}} ψ_{1}^{d_{1}}ψ_{2}^{d_{2}} … ψ_{n}^{d_{n}},

where g is determined by the equation 3g − 3 + n = d_{1} + d_{2} + ⋯ + d_{n}. If there exists no such g, then the left hand side is by definition zero. In case we want to mention the genus explicitly we will write ⟨τ_{d_{1}}τ_{d_{2}} … τ_{d_{n}}⟩_{g}. Note that this expression is symmetric with respect to d_{1}, d_{2}, …, d_{n} since ψ_{i}’s are interchanged under the action of the symmetric group S_{n}. Therefore one can write it as ⟨τ_{0}^{m_{0}}τ_{1}^{m_{1}}τ_{2}^{m_{2}} …⟩, where the set {d_{1}, …, d_{n}} contains m_{0} zeros, m_{1} ones, etc. The generating function is defined by

F (t_{0}, t_{1}, t_{2}, …) := ⟨exp ∑_{j=0}^{∞} t_{j}τ_{j}⟩ = ∑_{\{m_{i}\}∈S_{0}} ∏_{i=0}^{∞}⟨τ_{0}^{m_{0}}τ_{1}^{m_{1}}τ_{2}^{m_{2}} …⟩ \frac{t_{i}^{m_{i}}}{m_{i}!}.

We will also use the notation

∑_{m∈S_{0}}⟨τ^{m}⟩ \frac{t^{m}}{m!}

for the last expression. Note that one can also write F (t) = ∑_{g=0}^{∞} F_{g}(t), where F_{g}(t) := ∑_{m∈S_{0}}⟨τ^{m}⟩_{g} \frac{t^{m}}{m!}.

Witten conjectured in [32] and Kontsevich proved in [21] that F is the logarithm of a τ-function in the KdV-hierarchy.

In [19] Kaufmann, Manin, and Zagier considered a similar generating function for κ classes. If one defines

⟨κ^{p}⟩_{g} = ⟨κ_{1}^{p_{1}}κ_{2}^{p_{2}} …⟩_{g} := ∫_{\overline{M}_{g,n}} κ_{1}^{p_{1}}κ_{2}^{p_{2}} …,

then their generating function is

K_{g}(x; s) = K_{g}(x; s_{1}, s_{2}, …) := ∑_{p∈S_{1}}⟨κ^{p}⟩_{g} \frac{x^{|p|}}{|p|!} \frac{s^{p}}{p!}.

Note that here it is important to indicate the genus. The number of punctures n is then determined from 3g − 3 + n = |p|.

We introduce the generating function H which incorporates both of the ψ and κ classes. We shall see that F and K enjoy similar properties which arise because H obeys those properties.

First we introduce the following notation. Let m ∈ S_{0} and p ∈ S_{1}. Define

⟨τ^{m}κ^{p}⟩ := ∫_{\overline{M}_{g,n}} ψ_{1}^{d_{1}}ψ_{2}^{d_{2}} … ψ_{n}^{d_{n}}κ_{1}^{p_{1}}κ_{2}^{p_{2}} …,

where the set {d_{1}, …, d_{n}} contains m_{0} zeros, m_{1} ones, etc., and (g, n) is determined by the equations n = ||m||, 3g − 3 + n = |m| + |p|. If no such g exists we set the expression above to zero. As before, we write ⟨τ^{m}κ^{p}⟩_{g} when we want to fix g.

Definition 1.1. Let t = (t_{0}, t_{1}, …) and s = (s_{1}, s_{2}, …) be independent families of independent formal variables. We define

H(t; s) := ∑_{m∈S_{0},p∈S_{1}}⟨τ^{m}κ^{p}⟩ \frac{t^{m}}{m!} \frac{s^{p}}{p!}.

One can split H into the sum of H_{g}, g = 0, 1, . . .. Each H_{g} lies in a kernel of a certain scaling differential operator, i.e., it satisfies the charge conservation equation. The multiplication of H by ||m|| is equivalent to applying the operator E := ∑ t_{i}∂_{i}, and by |m| is equivalent to applying ∑ i t_{i}∂_{i}. Therefore one has

[3(1 − g) + ∑_{i=0}^{∞}(i − 1) t_{i}∂_{i} + ∑_{i=1}^{∞} i s_{i}d_{i}] H_{g} = 0, (4)

where ∂_{i} := ∂/∂t_{i} and d_{i} := ∂/∂s_{i}.

Clearly H(t; 0) = F (t). In order to relate H to K one fixes a genus g, sets t_{1} = t_{2} = ⋯ = 0, and t_{0} = x. The infinite sequence m reduces to m_{0} = n = |p| + 3 − 3g. It follows that

H_{g}(x, 0; s) = ∑_{p∈S_{1}}⟨κ^{p}⟩_{g} \frac{x^{|p|+3−3g}}{(|p| + 3 − 3g)!} \frac{s^{p}}{p!}.

In this paper we are primarily interested in genus 0 and 1. Then K_{0}(x; s) = H′′′_{0}(x, 0; s) and K_{1}(x; s) = H_{1}(x, 0; s), where the prime denotes the partial derivative with respect to x.

2. Presentation of the Tautological Classes via Graphs

In this section we will mainly focus on H_{0} and H_{1}, the generating functions for the intersection numbers of the κ and ψ classes in genus 0 and genus 1. First we show that H_{0} satisfies a system of nonlinear differential equations. These equations when, restricted to the ψ classes, were first obtained by Witten [31, 32]. When restricted to the κ classes in genus zero, equivalent but much more complicated equations were obtained by Kaufmann, Manin, and Zagier [19], the difference being accounted for by our simple, recursive presentation of powers of κ and ψ classes in genus zero and one in terms of boundary divisors. We prove that H_{0} satisfies differential equations of the same form as those obtained by Witten for just the ψ classes. We then obtain a system of differential equations relating H_{0} and H_{1}, and the explicit formula (1).

Here we present a geometric approach using the explicit presentation of the ψ and κ classes in terms of the boundary strata.

2.1. Basic Relation. In the beginning we want to state some general facts which hold for all genera. Let π : C_{g,n} = \overline{M}_{g,n+1} → \overline{M}_{g,n} be the universal curve. To simplify the notation in this subsection, we denote ψ_{(g,n),i} by ψ_{i}, ψ_{(g,n+1),i} by ψ̂_{i}, and we use the same convention for the κ classes. Recall from Sec. 1 that D_{i} denotes the image of the i^{th} canonical section σ_{i} : \overline{M}_{g,n} → \overline{M}_{g,n+1}. We will denote by the same letter its dual cohomology class in H^{2}(\overline{M}_{g,n+1}).

Lemma 2.1. For each k ≥ 1

ψ̂_{i}^{a} = π^{∗}ψ_{i}^{a} + σ_{i∗}ψ_{i}^{a−1}, k ≥ 1. (5)

Proof. It is easy to show that the equation ψ̂_{i}^{a} = π^{∗}ψ_{i}^{a} +π^{∗}ψ_{i}^{a−1}D_{i} from [31] can be rewritten as

ψ̂_{i}^{a} = π^{∗}ψ_{i}^{a} + (−1)^{a−1}D_{i}^{a}, k ≥ 1.

Applying the functor σ_{i∗}σ_{i}^{∗}, which is the multiplication by D_{i}, to the above equation, and using that σ_{i}^{∗}ψ̂_{i} = 0 one gets

(−1)^{a}D_{i}^{a+1} = σ_{i∗}ψ_{i}^{a}, k ≥ 0.

These two equalities imply the lemma.

2.2. Explicit Presentation in Genus 0, 1. In genus 0 and genus 1 equation (5) allows us to express inductively all powers of ψ classes, and therefore κ classes, in terms of the boundary strata. Because the ψ classes are interchanged under the action of the symmetric group this is enough to compute ψ_{(g,n),1}, g = 0, 1, and all its powers. In the calculations below we will use the properties stated in Sec. 1 regarding the pull backs and push forwards of the cohomology classes represented by graphs.

Let us introduce the following notation for the rest of this section. On graphs we denote by • vertices of genus 0, and by ∘ vertices of genus 1. We always assume that the ψ classes are associated to the marked point labeled 1, and subsequently omit it from the notation. We denote ψ_{(1,n),1} by ψ_{(n)}, and we denote ψ_{(0,J),1} by ϕ_{(J)}, where J is a finite set, 1 ∈ J. Similarly, we denote κ_{(1,n),a} by κ_{(n),a}, and κ_{(0,J),a} by ω_{(J),a}.

We also adopt the following convention. Let Γ be a stable graph. According to Sec. 1, Γ determines a canonical finite quotient map from a product of moduli spaces to \overline{M}_{Γ} provided certain choices have been made. We denote by ρ_{Γ} the composition

ρ_{Γ} : ∏_{v∈V (Γ)} \overline{M}_{g(v),n(v)} ⟶ \overline{M}_{Γ} ⟶ \overline{M}_{g(Γ),S(Γ)}, (6)

where the first arrow is the quotient morphism, and the second arrow is the inclusion.

Let γ_{v} ∈ H^{•}(\overline{M}_{g(v),n(v)}). We denote \frac{1}{| Aut(Γ)|}ρ_{Γ∗}(⊗_{v}γ_{v}) by the picture of Γ where each vertex v is in addition labeled by the cohomology class γ_{v}. We may omit the label of v if γ(v) is the fundamental class of \overline{M}_{g(v),n(v)}. In particular, the fundamental class of \overline{M}_{Γ} (in the orbifold sense) is represented by Γ with all additional labels omitted. The only possible ambiguity would arise if ⊗_{v}γ_{v} were not invariant under Aut(Γ), but this situation will not arise in this paper.

In the pictures the dashed line with two arrows indicates the (sub)set of the tails emanating from a particular vertex. If ϕ or ω labels a vertex of a graph we will omit the subscript from the notation because it is determined by the graph. (We also assume that ϕ is associated to the marked point labeled by 1.) The power ϕ^{0} represents the fundamental class.

Proposition 2.2. If n ≥ 4, a ≥ 1, then the following holds in H^{•}(\overline{M}_{0,n})

ϕ_{(n)}^{a} = ∑_{\begin{matrix}I⊔J=[n−2] \\ 1∈J \end{matrix}} ^{I n−1 n ϕ^{a−1} 1 J} (7)

Remark . The class ϕ_{(n)}^{a} is invariant under the subgroup S_{n−1} ⊂ S_{n} whose elements fix 1. Therefore instead of n − 1, n we can choose any two labels a, b, 2 ≤ a < b ≤ n to be distinguished.

Proof. The statement is true when n = 4 and a = 1 because ∫_{\overline{M}_{0,4}} ψ_{1} = 1. We shall first prove by induction that the statement is true for all n ≥ 4 and a = 1. Let us consider the projection π : \overline{M}_{0,n+1} → \overline{M}_{0,n} which “forgets” the n + 1^{st} marked point. By the induction hypothesis we assume that the statement is true for some n and a = 1. Applying π^{∗} one gets:

ϕ_{(n+1)} − ^{1 n+1 [n−2]  n n−1}

= ∑_{\begin{matrix}I⊔J=[n−2] \\ 1∈J \end{matrix}}( ^{I n−1 n n+1  1 J} + ^{I n−1 n n+1 1 J} )

The left hand side is equal to ϕ_{(n+1)} − D_{1} = π^{∗}ϕ_{(n)}. Moving D_{1} to the right hand side and relabeling the tails marked n − 1, n by n, n + 1 respectively one gets the statement in case of n + 1, a = 1.

Now assume that a ≥ 2. If n = 4, then the statement of the proposition is trivially satisfied because all terms vanish by dimensional considerations. We assume that the statement is true for a pair (n, a) and all pairs (n′, a′), where a′ < a, and prove it for (n + 1, a). Applying π^{∗} one gets

ϕ_{(n+1)}^{a} − σ_{1∗}ϕ_{(n)}^{a−1} = ∑_{\begin{matrix}I⊔J=[n−2] \\ 1∈J \end{matrix}}( ^{J I  1 n+1 n n−1 ϕ^{a−1}}

+ ^{J I  1 n+1 n n−1 ϕ^{a−1}} − σ_{1∗}( ^{I n−1 n ϕ^{a−2}  1 J} ))

According to the induction hypothesis the terms with σ_{1∗} on the left hand side and the right hand side cancel each other. Thus, we get the statement of the proposition for the pair (n + 1, a).

Let π_{1} be the morphism \overline{M}_{0,n+1} → \overline{M}_{0,n} forgetting the first marked point. Applying it to (7) with a + 1, using (2), and renumbering the labels {2, …, n + 1} by the elements of [n] we get the following

Corollary 2.3. If n ≥ 4, a ≥ 1, then the following holds in H^{•}(\overline{M}_{0,n})

ω_{(n),a} = ∑_{I⊔J=[n−2]} ^{I n−1 n ω_{a−1} J} (8)

In (8) if a = 1 one should use that ω_{(J⊔∗),0} associated to the right vertex is equal to |J| − 1 times the fundamental class (cf. Sec. 1). This agrees with π_{1∗} of (7) when a = 1.

Now we establish a relation between the ψ and κ classes in genus 0 and 1. The proof is virtually identical to that of Prop. 2.2, and we will not reproduce it.

It is shown in [4, VI.4] that

ψ_{(1)} = \frac{1}{12}  ^{1}

Note that we take the coefficient is \frac{1}{12} rather than \frac{1}{24} due to the nontrivial automorphism of the graph. Using this and (5) one obtains the following

Proposition 2.4. If n ≥ 1, a ≥ 1, then the following holds in H^{•}(\overline{M}_{1,n})

ψ_{(n)}^{a} = \frac{1}{12} ^{ϕ^{a−1}  1 [n]} + ∑_{\begin{matrix}I⊔J=[n] \\ 1∈J \end{matrix}} ^{I  ϕ^{a−1}  1 J} (9)

Pushing down the above along π_{1} : \overline{M}_{1,n+1} → \overline{M}_{1,n}, and renumbering the labels one gets Corollary 2.5. If n ≥ 1, a ≥ 1, then the following holds in H^{•}(\overline{M}_{1,n})

κ_{(n),a} = \frac{1}{12}  ^{\begin{aligned}& [n] \\ ω_{a−1} \end{aligned}} + ∑_{I⊔J=[n]} ^{I  ω_{a−1}  J} (10)

2.3. Recursion Relations and Differential Equations. Now we derive the corresponding recursion relations and differential equations for the intersection numbers of the products of the ψ and κ classes using the explicit graph presentations above. In order to obtain the recursion relations we use a method from [19] to integrate the product of the ψ and κ classes over the Poincaré dual of a chosen ψ or κ class. In order to do this we need to know how the tautological classes restrict to the strata of the natural stratification. The restriction of a ψ class to a boundary stratum is obvious.

In order to restrict products of the κ classes we use Lemma 1.3 from [19] where the authors show the following restriction property for the κ classes. (They show it in the case of genus 0, but their proof is in fact valid for all genera.) Let (Γ, g, μ) be a stable graph, ρ_{Γ} is the corresponding morphism defined by (6), and κ^{p} is a product of the κ classes on \overline{M}_{g(Γ),S(Γ)}. Then

∫_{∏_{v∈V (Γ)} \overline{M}_{g(v),n(v)}} \frac{ρ_{Γ}^{∗}(κ^{p})}{p!} = ∑_{\begin{matrix}p^{v}: v∈V (Γ) \\ ∑ p^{v}=p \end{matrix}} ∏_{v∈V (Γ)} \frac{⟨κ^{p^{v}}⟩_{g(v)}}{p^{v}!}

The argument uses a fact proved in [1] that the collection κ_{(g,n),a} for each fixed a forms a logarithmic cohomological field theory (cf. Sec. 6), i.e., the κ classes satisfy the relation

ρ_{Γ}^{∗}(κ_{a}) = ∑_{v∈V (Γ)} κ_{g(v),n(v)}. (11)

We start with genus 0. Recall that H(t; s) is the generating function incorporating the intersection numbers for the products of the ψ and κ classes defined in Sec. 1, ∂_{a}, d_{a} are partial derivative with respect to t_{a}, a ≥ 0, s_{a}, a ≥ 1, and E = ∑_{i=0}^{∞} t_{i}∂_{i}.

Theorem 2.6. For each m ∈ S_{1}, p ∈ S, k, l ≥ 0, and a ≥ 1 one has

⟨τ^{m+δ_{k}+δ_{l}+δ_{a}}κ^{p}⟩_{0}

= ∑_{\begin{matrix}m′+m′′=m \\ p′+p′′=p \end{matrix}}\binom{m}{m′}\binom{p}{p′}⟨τ^{m′+δ_{k}+δ_{l}+δ_{0}}κ^{p′}⟩_{0}⟨τ^{m′′+δ_{a−1}+δ_{0}}κ^{p′′}⟩_{0},

⟨τ^{m+δ_{k}+δ_{l}}κ^{p+δ_{a}}⟩_{0}

= ∑_{\begin{matrix}m′+m′′=m \\ p′+p′′=p \end{matrix}}\binom{m}{m′}\binom{p}{p′}⟨τ^{m′+δ_{k}+δ_{l}+δ_{0}}κ^{p′}⟩_{0} ⟨τ^{m′′+δ_{0}}κ^{p′′+δ_{a−1}}⟩_{0}.

Equivalently, for each k, l ≥ 0 the function H_{0}(t; s) satisfies

∂_{a}∂_{k}∂_{l}H_{0} = (∂_{k}∂_{l}∂_{0}H_{0})(∂_{a−1}∂_{0}H_{0}) when a ≥ 1,
d_{1}∂_{k}∂_{l}H_{0} = (∂_{k}∂_{l}∂_{0}H_{0})((E − 1)∂_{0}H_{0}),
d_{a}∂_{k}∂_{l}H_{0} = (∂_{k}∂_{l}∂_{0}H_{0})(d_{a−1}∂_{0}H_{0}) when a ≥ 2.

This system together with H_{0}(t_{0}, 0; 0) = \frac{t_{0}^{3}}{6} uniquely determines H_{0}.

Proof. The recursion relations are a direct consequence of (7), (8), and the restriction properties of the ψ and κ classes described above.

In order to derive the differential equations from the recursion relations one notices that the increment of m_{a} or p_{a} by one in a recursion relation corresponds to taking the partial derivative with respect to t_{a} or s_{a}.

The operator E appears because the second recursion relation when a = 1 produces ω_{0}. The corresponding moduli space is \overline{M}_{0,J⊔∗}. As |J ⊔ ∗| = ||m′′|| + 1, it follows that ω_{0} = |J| − 1 = ||m′′|| − 1, and we use that the multiplication by m_{i} can be expressed by t_{i}∂_{i}.

Remark . Setting s = 0 in the first equation one gets differential equations satisfied by F_{0} (cf. [32]).

Remark . Setting k = l = 0, t_{0} = x, and t_{1} = t_{2} = … = 0 in the second equation one gets differential equations satisfied by H(x, 0; s) whose third derivative with respect to x is K_{0}(x; s). These equations are a simple consequence of the results in [19, Sec. 1].

Now we turn to genus 1. We use the explicit presentations (9) and (10) and take into account the automorphism groups of the graphs to obtain the following Theorem 2.7. For each m ∈ S_{0}, p ∈ S_{1}, and a ≥ 1 one has

⟨τ^{m+δ_{a}}κ^{p}⟩_{1} = \frac{1}{24}⟨τ^{m+2δ_{0}+δ_{a−1}}κ^{p}⟩_{0}
+ ∑_{\begin{matrix}m′+m′′=m \\ p′+p′′=p \end{matrix}}\binom{m}{m′}\binom{p}{p′}⟨τ^{m′+δ_{0}}κ^{p′}⟩_{1}⟨τ^{m′′+δ_{0}+δ_{a−1}}κ^{p′′}⟩_{0},

⟨τ^{m}κ^{p+δ_{a}}⟩_{1} = \frac{1}{24}⟨τ^{m+2δ_{0}}κ^{p+δ_{a−1}}⟩_{0}
+ ∑_{\begin{matrix}m′+m′′=m \\ p′+p′′=p \end{matrix}}\binom{m}{m′}\binom{p}{p′}⟨τ^{m′+δ_{0}}κ^{p′}⟩_{1}⟨τ^{m′′+δ_{0}}κ^{p′′+δ_{a−1}}⟩_{0}.

Equivalently, the functions H_{1}(t; s) and H_{0}(t; s) satisfy

∂_{a}H_{1} = \frac{1}{24}∂_{a−1}∂_{0}∂_{0}H_{0} + (∂_{0}H_{1})(∂_{a−1}∂_{0}H_{0}) when a ≥ 1,

d_{1}H_{1} = \frac{1}{24}E∂_{0}∂_{0}H_{0} + (∂_{0}H_{1})((E − 1)∂_{0}H_{0}),

d_{a}H_{1} = \frac{1}{24}d_{a−1}∂_{0}∂_{0}H_{0} + (∂_{0}H_{1})(d_{a−1}∂_{0}H_{0}) when a ≥ 2.

This system together with the system and the initial conditions from Thm. 2.6 uniquely determines the pair H_{0}, H_{1}.

Remark . The first recursion relation when p = 0 was obtained by Witten in [32].

The system of differential equations above can be solved explicitly for H_{1} in terms of H_{0} to derive (1).

Corollary 2.8. The functions H_{1} and H_{0} are related by

H_{1} = \frac{1}{24} log ∂_{0}^{3}H_{0}.

Proof. Because of uniqueness it is enough to check that \frac{1}{24} log ∂_{0}^{3}H_{0} satisfies the differential equation in Thm. 2.7. This is a straight forward calculation which makes use of the differential equations from Thm. 2.6.

Remark . Setting s = 0 we recover a result from [5, Sec. 2.2]. Setting t_{0} = x, t_{1} = t_{2} = … = 0 we get K_{1} = \frac{1}{24} log K_{0}.

3. Puncture and Dilaton Equations

In this section we introduce an approach which does not use explicit presentations of ψ and κ classes in terms of graphs. Instead we introduce the analogues of the puncture and dilaton equations. These equations generalize the classical puncture and dilaton equations obtained by Witten [32, 31]. (We shall explain these equations below.) This will allow us to write differential equations for H, and then, using these differential equations, prove that H_{0} and H_{1} satisfy (1).

3.1. Recursion Relations. In Sec. 1 we introduce the notation incorporating the intersection numbers of both of the ψ and κ classes. Now we shall to prove certain recursion relations for these numbers.

Lemma 3.1. The following recursion relations are satisfied:

(12) ⟨τ^{m+δ_{0}}κ^{p}⟩ = ∑_{i=1}^{∞} m_{i}⟨τ^{m+δ_{i−1}−δ_{i}}κ^{p}⟩ + ∑_{\begin{matrix}j=0 \\ |j|>0 \end{matrix}}^{p} \binom{p}{j}⟨τ^{m}κ^{p−j+δ_{|j|−1}}⟩,

and for each a ≥ 1

(13) ⟨τ^{m+δ_{a}}κ^{p}⟩ = ∑_{j=0}^{p} \binom{p}{j}⟨τ^{m}κ^{p−j+δ_{|j|+a−1}}⟩.

Proof. We continue to use the notation from 2.1, i.e., π is the universal curve over \overline{M}_{g,n}, (ψ̂_{i}, κ̂_{i}) and (ψ_{i}, κ_{i}) are classes upstairs and downstairs respectively, σ_{i} is the i^{th} canonical section of π, and D_{i} is its image.

It was shown in [31] that ψ̂_{i}^{a} = π^{∗}ψ_{i}^{a} + π^{∗}ψ_{i}^{a−1}D_{i} and in [1] that κ̂_{i} = π^{∗}κ_{i}+ψ̂_{n+1}^{i}. Note also that ψ̂_{i}D_{i} = 0, ψ̂_{n+1}D_{i} = 0 for i = 1, …, n, and D_{i}D_{j} = 0 when i ≠ j. Using this one derives that

π_{∗}(ψ̂_{1}^{d_{1}} …ψ̂_{n}^{d_{n}} κ̂_{1}^{p_{1}} κ̂_{2}^{p_{2}} …)

= π_{∗}( (π^{∗}ψ_{1}^{d_{1}} + π^{∗}ψ_{1}^{d_{1}−1}D_{1}) … (π^{∗}ψ_{n}^{d_{n}} + π^{∗}ψ_{n}^{d_{n}−1}D_{n})

× (π^{∗}κ_{1} + ψ̂_{n+1})^{p_{1}}(π^{∗}κ_{2} + ψ̂_{n+1}^{2})^{p_{2}} …)

= ∑_{i:d_{i}≠0} ψ_{1}^{d_{1}} … ψ_{i}^{d_{i}−1} … ψ_{n}^{d_{n}}κ_{1}^{p_{1}}κ_{2}^{p_{2}} …

+ ∑_{\begin{matrix}j=0 \\ |j|>0 \end{matrix}}^{p} \binom{p}{j}ψ_{1}^{d_{1}} … ψ_{n}^{d_{n}}κ_{|j|−1}κ_{1}^{p_{1}−j_{1}}κ_{2}^{p_{2}−j_{2}} …

Similarly one can show that

π_{∗}(ψ̂_{1}^{d_{1}} …ψ̂_{n}^{d_{n}} ψ̂_{n+1}^{a}κ̂_{1}^{p_{1}} κ̂_{2}^{p_{2}} …)

= ∑_{j=0}^{p} \binom{p}{j}ψ_{1}^{d_{1}} … ψ_{n}^{d_{n}}κ_{|j|+a−1}κ_{1}^{p_{1}−j_{1}}κ_{2}^{p_{2}−j_{2}} …

Recall that κ_{0} = 2g − 2 + n.

One can further integrate the push forward formulas above to obtain the statement of the lemma.

Remark . Recursion relations (12) and (13) do not mix intersection numbers in different genera.

Remark . If p = 0, then the second sum in the first relation vanishes, and we obtain the classical puncture equation. If p = 0 and a = 1 in the second relation, then we get the classical dilaton equation. Note that both classical equations involve only ψ classes.

Remark . This is clear recursion relations (12) and (13) allow to eliminate τ from the intersection number, i.e., to express all mixed intersection numbers through the intersection numbers on \overline{M}_{0,3}, \overline{M}_{1,1}, and the intersection numbers of the κ classes on \overline{M}_{g,0}, g ≥ 2.

In [19, Cor. 2.3] the authors obtained an explicit expression for the intersection numbers of the κ classes through the intersection numbers of the ψ classes. This should be related to (13), but we do not know how to derive their formula from it.

3.2. Differential Operators. Now we derive differential equations for H using recursions (12) and (13). Recall that ∂_{i}, d_{i} denote the partial derivatives with respect to t_{i}, s_{i}.

Theorem 3.2. The function exp(H(t; s)) is annihilated by the following differential operators:

−∂_{0} + ∑_{j: |j|≥2} \frac{s^{j}}{j!} d_{|j|−1} + ∑_{i=0}^{∞} t_{i}∂_{i−1}

+ s_{1}(∑_{i=0}^{∞} \frac{2i + 1}{3}t_{i}∂_{i} + ∑_{i=1}^{∞} \frac{2}{3}i s_{i}d_{i}) + \frac{1}{2}t_{0}^{2}δ_{g,0} + \frac{1}{24}s_{1}δ_{g,1},

−∂_{1} + ∑_{j: |j|≥1} \frac{s^{j}}{j!} d_{|j|} + (∑_{i=0}^{∞} \frac{2i + 1}{3}t_{i}∂_{i} + ∑_{i=1}^{∞} \frac{2}{3}i s_{i}d_{i}) + \frac{1}{24}δ_{g,1},

−∂_{a} + ∑_{j} \frac{s^{j}}{j!} d_{|j|+a−1} when a ≥ 2.

Remark . The differential operators above do not mix genus, and therefore they annihilate each exp(H_{g}(t; s)) separately.

Remark . The function exp(F (t)) is annihilated by differential operators L_{i}, i ≥ −1, which, after a rescaling of variables, satisfy the Virasoro relations [5]. The first two differential operators in the statement of the theorem are analogues of L_{−1} and L_{0} respectively, which encode the puncture and dilaton equations, respectively.

Proof. The differential operators above are the direct translation of the recursion relations (12) and (13). The addition of δ_{i} to m or p translates into taking the corresponding partial derivative. The subtraction of δ_{i} from m, and multiplying the term by m_{i} translates into the multiplication by t_{i}. One should also change the summation index to obtain the second summand of each differential operator.

The terms in parentheses in the first two equations come from the value of κ_{0} = 2g − 2 + n. We use (4) in order to express this number in terms of differential operators. Finally, the last terms in the first two equations correspond to the initial conditions ⟨τ_{0}^{3}⟩_{0} = 1, ⟨κ_{1}⟩_{1} = \frac{1}{24}, and ⟨τ_{1}⟩_{1} = \frac{1}{24}.

The theorem above leads to another proof of Cor. 1. First one notes that H_{0} and H_{1} are uniquely determined by the differential operators above. Therefore it suffices to check that \frac{1}{24}∂_{0}^{3}H_{0} satisfies the genus 1 equations. This is a direct calculation.

4. Explicit Expressions

In this section we write the closed form expressions for the intersection numbers in genus 1 as sums of the multinomial coefficients.

Notation . If b = (b_{1}, …, b_{k}) is a vector with integer entries we denote by [b] the multinomial coefficient \frac{(b_{1}+···+b_{k})!}{b_{1}!...b_{k}!}, and we set it to zero if at least one entry is negative. Recall also that ||b|| denotes the sum b_{1} + … + b_{k}.

In genus 0 the intersection numbers of the ψ classes are very simple: ⟨τ_{b_{1}} … τ_{b_{k}}⟩_{0} = [b]. In order to state our result in genus 1 we define for each k ≥ 1 a function f_{k} : ℤ_{≥1}^{k} → ℤ_{≥1} by

f_{k}(b) = f_{k}(b_{1}, …, b_{k}) := ⟨τ_{0}^{||b||−k}τ_{b_{1}} … τ_{b_{k}}⟩_{1}.

Clearly each f_{k} is invariant under the permutations of its arguments.

Proposition 4.1. For each k ≥ 1

f_{k}(b) = \frac{1}{24}[b] − \frac{1}{24} ∑_{\begin{matrix}ε∈\{0,1\}^{k} \\ ||ε||≥2 \end{matrix}}(||ε|| − 2)! [b − ε]. (14)

Proof. The intersection numbers of the ψ classes in genus 1 are determined by the classical puncture equation and the classical dilaton equation. (See the second remark after Lemma 3.1.) Reformulated in terms of the collection {f_{k}} these equations say that this collection is uniquely determined by the following properties:

• f_{1}(b_{1}) ≡ \frac{1}{24},

• f_{k} is invariant under the permutation of the arguments,

• f_{k}(b) = ∑_{i=1}^{k} f_{k}(b − δ_{i}) when b_{i} ≥ 2 for all i,

• f_{k}(b_{1}, …, b_{k−1}, 1) = (b_{1} + ⋯ + b_{k−1})f_{k−1}(b_{1}, …, b_{k−1}). The first three properties are obviously satisfied by the expression given in the statement of the proposition. A direct computation verifies the last property.

An explicit expression for the intersection numbers of the κ classes in genus 1 can be obtained by substitution of (14) into Cor. 2.3 from [19]. This expression is quite complicated. We do not know how to simplify this expression, and therefore we do not present the resulting formula for the κ classes here.

5. Asymptotic Formulas for Volumes of \overline{M}_{1,n}

In this section, we derive an asymptotic formula for the Weil-Petersson volumes of \overline{M}_{1,n} in the limit that n becomes very large extending the proof of a similar result for genus zero in [19]. We do so by using analytic properties of the generating function K(x; s) in the case where all s_{i} = 0 for all i ≥ 2. In some sense, these results are complementary to those of Penner [29] who obtains similar formulas for the case where the genus becomes very large.

The class of the Weil-Petersson symplectic form on \overline{M}_{g,n} is precisely \frac{1}{2π^{2}} κ_{(g,n),1}. The symplectic volume of \overline{M}_{g,n} is called the Weil-Petersson volume of \overline{M}_{g,n}. For this reason, the intersection numbers associated to the κ classes are sometimes called higher Weil-Petersson volumes. To avoid unnecessary factors, we shall work instead with the quantity

Definition 5.1. w_{g,n} := ∫_{\overline{M}_{g,n}} κ_{1}^{3g−3+n}.

Theorem 5.2 ([19]). The genus zero Weil-Petersson volumes satisfy the asymptotic relation as n → ∞

w_{0,n+3} ∼ \frac{γ_{0} 2^{\frac{3}{2}}}{C\sqrt{π}} \frac{2^{2n}n^{2n+ \frac{1}{2}}}{C^{n}e^{2n}} ,

where γ_{0} ≈ 2.40482555777 … is the smallest zero of the Bessel function J_{0} and C = −2γ_{0}J′_{0}(γ_{0}) ≈ 2.496918339 . . ..

They proved this by noticing that the function H′′_{0}(x, 0; s) is invertible, and when all s_{i} = 0 except for s_{1} the inverse function satisfies Bessel’s equation, after a change of variables. Combining their results with ours for genus one, we obtain the following.

Theorem 5.3. The genus one Weil-Petersson volumes satisfy the asymptotic relation as n → ∞

w_{1,n} ∼ \frac{π}{24} \frac{(2n)^{2n}}{C^{n}e^{2n}} ,

where C is same constant as in the above.

Proof. One uses the asymptotic formulas for the genus zero case and our result that the generating functions are related by (1) .

Remark . The theorem above supports a conjecture of Itzykson regarding the existence of such an asymptotic formula for all genera with a constant C independent of the genus (cf. [19, p. 765]).

6. The Moduli Space of Cohomological Field Theories

The moduli space of normalized, rank one cohomological field theories of genus zero was described in Kontsevich, Manin, and Zagier [19]. The generating function associated to the κ classes endows this moduli space with coordinates which behave nicely with respect to taking tensor products of cohomological field theories, a notion introduced in [23] (see also [18]).

In this section, we introduce the notion of a restricted, normalized, cohomological field theory in genus one and describe the moduli space of rank one theories of this kind. Such CohFTs turn out to be almost completely determined by their genus zero part using the relations between the boundary strata of \overline{M}_{1,n} recently obtained by Getzler [11]. The analogous set of coordinates are constructed for this moduli space but to do so, we must introduce the λ classes, as well.

6.1. Cohomological Field Theories. Consider G_{g,n}, the set of stable graphs of genus g and n tails labeled with the set [n] . Each G_{g,n} is acted upon by the permutation group S_{n} which permutes the labels on the tails. There are composition maps

G_{g_{1},n_{1}} × G_{g_{2},n_{2}} → G_{g_{1}+g_{2},n_{1}+n_{2}−2}

taking (Γ, Γ′) ↦ Γ ∘_{(i_{1},i_{2})} Γ′ for all i_{1} in [n_{1}] and i_{2} in [n_{2}] given by grafting the tail i_{1} of Γ with tail i_{2} of Γ′ and then relabeling the remaining tails with elements of the set [n_{1} + n_{2} − 2] by inserting orders. There are another set of composition maps G_{g,n} → G_{g+1,n−2} taking Γ ↦ tr_{(i_{1},i_{2})}Γ for all distinct pairs i_{1} and i_{2} in [n] in which the tails i_{1} and i_{2} of Γ are grafted together. These composition maps are equivariant with respect to the action of the permutation groups. Let ℂ[G_{g,n}] be the vector space over ℂ with a basis G_{g,n} then the compositions and permutations group actions can be extended ℂ-linearly. Let ℂ[G] denote the direct sum of ℂ[G_{g,n}] for all stable pairs g, n.

The collection { ℂ[G_{g,n}] } (or, for that matter, { G_{g,n} }) together with the composition maps and actions of the permutation groups described above forms an example of a modular operad, a notion due to by Getzler and Kapranov [13]. By restricting to just the genus zero subcollection { ℂ[G_{0,n}] } and forgetting about the composition maps tr, we obtain an example of a cyclic operad [14].

Similarly, the homology groups H_{•}(\overline{M}_{g,n}) are endowed with an action of S_{n} which relabels the punctures on the stable curve and there are composition maps

∘_{i_{1},i_{2}} : H_{p_{1}}(\overline{M}_{g_{1},n_{1}}) ⊗ H_{p_{2}}(\overline{M}_{g_{2},n_{2}}) → H_{p_{1}+p_{2}}(\overline{M}_{g_{1}+g_{2},n_{1}+n_{2}−2})

for all i_{1} in [n_{1}], i_{2} in [n_{2}] and tr_{(i_{1},i_{2})} : H_{p}(\overline{M}_{g,n}) → H_{p}(\overline{M}_{g+1,n−2}) for all distinct i_{1} and i_{2} in [n], both of which are induced from the inclusion of strata. These composition maps are equivariant under the action of the permutation groups.

The natural maps α_{g,n} : ℂ[G_{g,n}] → H_{•}(\overline{M}_{g,n}) mapping Γ ↦ [\overline{M}(Γ)] where \overline{M}(Γ) := ∏_{v∈V (Γ)} \overline{M}_{g(v),n(v)} preserves the above structures and gives rise to the sequence of morphisms

0 ⟶ ⟨R_{g,n}⟩ ⟶ ℂ[G_{g,n}] ⟶^{α_{g,n}} H_{•}(\overline{M}_{g,n}) (15)

where the kernel of α_{g,n} is denoted by ⟨R_{g,n}⟩, the ideal in ℂ[G] generated by some space of relations R_{g,n}.

Definition 6.1. The modular operad H := { H_{g,n} } is the collection of

H_{g,n} := \frac{ℂ[G_{g,n}]}{⟨R_{g,n}⟩}

The canonical diagonal maps \overline{M}_{g,n} → \overline{M}_{g,n} × \overline{M}_{g,n} induce maps H_{•}(\overline{M}_{g,n}) → H_{•}(\overline{M}_{g,n}) ⊗ H_{•}(\overline{M}_{g,n}) making H_{•}(\overline{M}_{g,n}) into a Hopf modular operad in the natural way [13]. This endows H_{g,n} with the structure of a Hopf modular operad, as well.

In the case of g = 0, the results of [22] and [20] implies that α_{0,n} is surjective and H_{0,n} is isomorphic to H_{•}(\overline{M}_{0,n}). Furthermore, the relations R_{0,n} are those due to Keel [20] which come from a lift of the basic codimension one relations R_{0,4} on \overline{M}_{0,4} via the canonical forgetful map \overline{M}_{0,n} → \overline{M}_{0,4}.

In the case of g = 1, α_{1,n} is known not to be surjective since \overline{M}_{1,n} has odd dimensional homology classes. However, Getzler [11] has shown that the space of relations R_{1,n}, in addition to those coming from Keel’s relations, contains the lifts of two other relations. The first is the lift of the basic codimension one relation on \overline{M}_{1,2} which contains no genus one vertices – this may be regarded as the image of Keel’s relations under the self-sewing morphism tr_{(3,4)} : ℂ[G_{0,4}] → ℂ[G_{1,2}]. The second relation, which contains genus one vertices, is between codimension two strata and is of the form

12δ_{2,2} − 4δ_{2,3} − 2δ_{2,4} + 6δ_{3,4} + δ_{0,3} + δ_{0,4} − 2δ_{β} = 0 (16)

where each term is an S_{4}-invariant combination of graphs of a given topological type and each graph Γ represents the homology class [\overline{M}_{Γ}]. (See [11] for details.) Getzler also states [11] that he has shown [12] that α_{1,n} maps surjectively onto the even dimensional homology of \overline{M}_{1,n} and that the relations mentioned above do in fact generate all of R_{1,n}.

A cohomological field theory is essentially a representation, in the sense of operads, of H_{•}(\overline{M}_{g,n}). In order to define such an object, we need to define the appropriate notion of the endomorphisms of a vector space is in this context. Let V be a vector space over ℂ with a symmetric, nondegenerate bilinear form h of degree zero. Let End(V)_{g,n} := T^{n}V be the n^{th} tensor power of V for all nonnegative integers g, n such that 2g − 2 + n > 0 where T^{0}V is understood to be ℂ. S_{n} acts upon End(V)_{g,n} by permuting the tensor factors and the composition maps End(V)_{g_{1},n_{1}} ⊗ End(V)_{g_{2},n_{2}} → End(V)_{g_{1}+g_{2},n_{1}+n_{2}−2} taking (μ, μ′) ↦ μ ∘_{(i_{1},i_{2})} μ′ for all i_{1} in [n_{1}] and i_{2} in [n_{2}] given by applying the inverse of h to the the corresponding tensor factors of μ and μ′, and inserting the remaining factors in the usual way. Similarly, the composition End(V)_{g,n} → End(V)_{g+1,n−2} taking μ ↦ tr_{(i_{1},i_{2})}μ for all distinct pairs i_{1} and i_{2} in [n] corresponds to applying the inverse of h to the appropriate pair of tensor factors of μ.

Definition 6.2 (Cohomological Field Theory). A (complete) cohomological field theory (CohFT) of rank r, (V, h), is a morphism of modular operads μ_{g,n} : H_{•}(\overline{M}_{g,n}) → End(V)_{g,n} where (V, h) is an rdimensional vector space with an invariant, symmetric bilinear form h. A CohFT of genus g are maps μ_{g′,n} : H_{•}(\overline{M}_{g′,n}) → End(V)_{g′,n} which are defined only for g′ ≤ g which satisfy all the axioms of a CohFT in which no higher genus maps appear. A restricted CohFT is a morphism μ_{g,n} : H_{g,n} → End(V)_{g,n}.

A CohFT can also be described described dually in terms of maps End(V)_{g,n} → H^{•}(\overline{M}_{g,n}).

Notice that a restricted CohFT of genus zero is the same as a CohFT of genus zero since H_{•}(\overline{M}_{0,n}) = H_{0,n}.

Remark . In the language of [31, 32], a topological gravity (coupled to topological matter) is a CohFT and the morphisms μ_{g,n} are the correlation functions of the theory. The genus zero CohFT is said to be tree level while a genus one CohFT is said to be one loop.

Remark . The natural Hopf structure on H_{•}(\overline{M}_{g,n}) endows the category of CohFTs with a tensor product as is usual in representation theory.

An restricted CohFT is completely determined by a generating function called its potential. If the CohFT is not restricted then one can still define the notion of a potential (essentially since the modular operad H_{•}(\overline{M}_{g,n}) is the quotient of some free modular operad) but we will not need to work in such generality.

Definition 6.3. The potential Φ = ∑_{g=0}^{∞} Φ_{g} of a restricted CohFT μ : H → End(V) of rank r is defined by choosing a basis { e_{1}, …, e_{r} } for V where I_{g,n}(e_{a_{1}}, e_{a_{2}}, …, e_{a_{n}}) is the number obtained by using h to pair μ_{g,n}([\overline{M}_{g,n}]) with e_{a_{1}} ⊗ e_{a_{2}} ⊗ … ⊗ e_{a_{n}} and

Φ_{g}(x) := ∑_{n=0}^{∞} I_{g,n}(e_{a_{1}}, e_{a_{2}}, …, e_{a_{n}}) \frac{x^{a_{1}}x^{a_{2}} … x^{a_{n}}}{n!} .

(where the summation convention has been used) which is regarded as an element in ℂ[[x^{1}, …, x^{r}]].

Theorem 6.4. A element Φ_{0} in ℂ[[x^{1}, …, x^{r}]] is the potential of a rank r, genus zero CohFT (V, h) if and only if [22, 25] it satisfies the WDVV equation

(∂_{a}∂_{b}∂_{e}Φ_{0}) h^{ef} (∂_{f}∂_{c}∂_{d}Φ_{0}) = (−1)^{|x_{a}|(|x_{b}|+|x_{c}|)} (∂_{b}∂_{c}∂_{e}Φ_{0}) h^{ef} (∂_{f}∂_{a}∂_{d}Φ_{0}),

where h_{a,b} := h(e_{a}, e_{b}), h^{ab} is in inverse matrix to h_{ab}, ∂_{a} is derivative with respect to x^{a}, and the summation convention has been used.

If (Φ_{0}, Φ_{1}) is the potential associated to a restricted, rank r CohFT of genus one then Φ_{0} must satisfy the WDVV equation and (Φ_{0}, Φ_{1}) must satisfy Getzler’s equation from proposition (3.14) in [11].

The WDVV equation can be read off from the basic codimension one relation on \overline{M}_{0,4}. Similarly, Getzler’s equation can be seen from his relation (equation 16). The second statement will become an if and only if after the proof in [12] appears.

6.2. Rank One Cohomological Field Theories. Let (V, h) be a rank one CohFT with a fixed unit vector e. The morphisms H_{g,n} → End(V)_{g,n} are completely determined by the collection of numbers \{ I_{g,n} \} where I_{g,n} := μ_{g,n}([\overline{M}_{g,n}])(\underbrace{e, e, …, e}_{n}) which must satisfy re-

lations between themselves reflecting the way that the boundary strata in \overline{M}_{g,n} fit together. The potential in this case is

Φ_{g} = ∑_{n=0}^{∞} I_{g,n} \frac{x^{n}}{n!} ,

where I_{g,n} is defined to vanish for pairs (g, n) which are not stable.

We will see that tautological classes on the moduli space of curves give rise to complete rank one CohFTs. In order to describe the moduli space of restricted, rank one CohFTs of genus one, we need to introduce a combination of the λ classes which behave nicely with respect to restriction.

Definition 6.5. For all stable pairs, (g, n), let Λ_{g,n} be an element in H^{•}(\overline{M}_{g,n})[s, u] (where s = (s_{1}, s_{2}, …) and u = (u_{1}, u_{2}, u_{3}, …)) then let

Λ_{g,n} := exp( ∑_{i=1}^{∞}( s_{i} κ_{(g,n),i} + u_{i} γ_{(g,n),i}) )

where γ_{(g,n),i} := ch_{2i−1}(π_{∗} ω_{g,n}). Here ch_{i} is the i^{th} Chern character, and π_{∗} ω_{g,n} is the pushforward of the relative dualizing sheaf. (Notice that ch_{2i}(π_{∗} ω_{g,n}) vanishes for all i [25, 8].)

The classes γ_{(g,n),i} are polynomials in the λ classes. In particular, γ_{1} = λ_{1}.

Theorem 6.6. The collection Λ := { Λ_{g,n} } gives rise to a complete, rank one CohFT for all values of u and s by integrating the cohomology classes Λ_{g,n} over the homology classes on \overline{M}_{g,n}. Furthermore, the tensor product of the CohFT associated to parameter values (s_{1}, u_{1}) and (s_{2}, u_{2}) is the CohFT associated to (s_{1} + s_{2}, u_{1} + u_{2}). Similarly,

Proof. In the case where u_{i} vanishes for all i, this was proven in [23] where it was realized that the κ classes form a logarithmic CohFT following the work of Arbarello and Cornalba [1] (see equation 11).

The proof for the case where all the s_{i} vanish is as follows. Consider the bundles E_{g,n} := π_{∗} ω_{g,n} on \overline{M}_{g,n} from section 1.3. If Γ is a graph of genus g with n tails, then it determines the morphism ρ_{Γ} : ∏_{v∈V (Γ)} \overline{M}_{g(v),n(v)} → \overline{M}_{g(Γ),S(Γ)}. The pull back of E_{g,n} under ρ_{Γ} differs from ⊕_{v∈V (Γ)}E_{g(v),n(v)} by a trivial bundle. It follows that for each k ≥ 1 the collection of the Chern characters γ_{k} = ch_{2k−1}E_{g,n} forms a logarithmic CohFT (see [8, 9]).

The first part of the theorem follows by combining these two results. The proof that the coordinates (s, u) are additive with respect to taking tensor products follows from the definition of coproduct which is induced from the diagonal map.

Corollary 6.7. The potential of the rank one CohFT associated to Λ (for given values of s and u) is precisely the generating function χ_{g} for the intersection numbers of κ_{i} and γ_{i} classes

χ_{g}(x; s, u) := ⟨exp(x τ_{0} + ∑_{i=1}^{∞} ( s_{i} κ_{i} + u_{i} γ_{i}))⟩_{g} = ∑_{n=0}^{∞} I_{g,n} \frac{x^{n}}{n!}

where

I_{g,n} = ∑_{r,m} \frac{s^{m}}{m!} \frac{u^{r}}{r!} ⟨ κ^{m} τ_{0}^{n} γ^{r} ⟩_{g} .

It is understood that I_{g,n} := 0 for unstable pairs (g, n),

Notice that the CohFT arising Λ have the property that I_{0,3} = 1 for all values of u and s. This motivates the following definition which will play an important role in what follows.

Definition 6.8. A rank one, CohFT of genus g is said to be invertible if I_{0,3} is nonzero and normalized if I_{0,3} = 1.

6.3. Cohomological Field Theories in Genus Zero and One. Let us recall the results of Kaufmann, Manin, and Zagier for rank one CohFTs of genus zero [19]. A rank one CohFT in genus zero is uniquely determined by its potential Φ_{0}(x) = ∑_{n=3}^{∞} I_{0,n} \frac{x^{n}}{n!} . Furthermore, any function Φ_{0}(x) in x^{3} ℂ[[x]] arises from some rank one, CohFT of genus zero since the WDVV equation is trivially satisfied for rank one theories. Therefore, the moduli space of CohFTs of genus zero are parameterized by the independent variables I_{0,n} for n ≥ 3. What is nontrivial, however, is the behavior of these potentials under tensor product. In particular, the coordinates I_{0,n} do not behave nicely under tensor product. However, the generating function associated to the genus zero κ classes H_{0}(x, 0; s) (which is equal to χ_{0} with u = 0) allows them to introduce coordinates on the space of normalized, rank one CohFTs which behave nicely under tensor products.

Theorem 6.9 ([19]). The moduli space of normalized, rank one CohFTs in genus zero are parameterized by s with potential Φ_{0}(x; s) = H_{0}(x, 0; s) in ℂ[s][[x]], our generating function for the intersection numbers of κ classes in genus zero. Furthermore, taking tensor products is additive with respect to the coordinates s.

We now treat the case of genus one and discover that the κ classes are not sufficient to describe the entire moduli space of normalized, rank one CohFTs. We will see that one needs to introduce the first λ classes.

Theorem 6.10. If the pair (Φ_{0}, Φ_{1}) is a potential associated to a restricted, rank one CohFT of genus one then the following equation holds in ℂ[[x]]

−(Φ_{0}^{(3)})^{2}Φ_{1}^{(2)} + Φ_{0}^{(3)}Φ_{0}^{(4)}Φ_{1}^{(1)} − \frac{1}{12}(Φ_{0}^{(4)})^{2} + \frac{1}{24}Φ_{0}^{(3)}Φ_{0}^{(5)} = 0

where Φ_{g}^{(l)} is the l^{th} derivative of Φ_{g}.

Proof. Equation 6.10 above is nothing more than the equation due to Getzler in theorem 6.4 for the case of rank one theories. Our equation can be seen from equation 16 directly by associating to each graph

Γ ↦ \frac{1}{|Aut(Γ)|} ∏_{v∈V (Γ)} \frac{∂^{n(v)}Φ_{g(v)}}{∂x^{n(v)}}

and then extending linearly to linear combinations of graphs. One will obtain −36 times the equation above.

Unlike the case of genus zero where the WDVV equation is trivially satisfied, solutions to this equation fall into two classes depending upon whether Φ_{0}^{(3)} is invertible in the ring of formal power series ℂ[[x]].

Theorem 6.11. The pair (Φ_{0}, Φ_{1}) is a potential associated to an invertible, restricted, rank one CohFT of genus one if and only if Φ_{0}(x) is of the form I_{0,3} \frac{x^{3}}{6} + x^{4} ℂ[[x]] for I_{0,3} nonzero and

Φ_{1} = \frac{1}{24} log Φ′′′_{0} + BΦ′′_{0}

where B is an arbitrary constant. Therefore, an invertible, restricted CohFT of genus one is uniquely determined by arbitrary values of I_{0,n} for all n ≥ 4, I_{0,3} ≠ 0, and I_{1,1}.

If the restricted, rank one, CohFT is not invertible then Φ_{0} = 0 and Φ_{1} obeys no constraints. Therefore, the space of such theories is parameterized by all values of I_{1,n} for all n ≥ 1.

Proof. If the pair (Φ_{0}, Φ_{1}) is a potential associated to an invertible restricted, rank one CohFT then since Φ′′′_{0} has an inverse in ℂ[[x]], one can solve equation 6.10 explicitly.

The converse is more difficult in the absence of the proof that the lifts of the relations described above genus 1 span the entire space of relations R_{1,n}. However, we will not need this statement but will explicitly construct restricted, normalized, rank one CohFTs in genus one which realize all solutions to equation 6.11 above. This we do in the next subsection.

Since I_{1,1} = I_{0,4} + BI_{0,3}, when I_{0,3} nonzero varying B is the same as varying I_{1,1} and leaving all of the I_{0,n} unchanged.

In the case that the restricted, rank one CohFT is not invertible then our result follows from the equation.

We shall not discuss noninvertible CohFTs any further in this paper. From now on, we shall restrict ourselves to normalized CohFTs.

It is worth observing that by incorporating the ψ classes, one can use the previous result to obtain yet another proof of the formula H_{1} = \frac{1}{24} log H0^{′′′}. It is not clear which of these approaches will prove most useful in higher genera.

6.4. Potentials in Genus Zero and One. In this subsection, we construct potentials for a class of normalized, restricted, rank one CohFTs in genus one explicitly and show that they span the entire space of solutions to equation 6.11 completing the proof of that theorem. These potentials are generating functions associated to the κ classes and λ_{1}. This will give rise to coordinates which are additive under tensor product in analogy with the case of genus zero in [19].

We begin with a useful lemma.

Lemma 6.12. The tautological class λ_{1} on \overline{M}_{1,n} can be written in terms of boundary classes as follows:

λ_{1} = \frac{1}{12}  ^{[n]}

Proof. The proof follows from the fact that λ_{(1,n),1} = π^{∗}λ_{(1,1),1} via the forgetful map π : \overline{M}_{1,n} → \overline{M}_{1,1}. One uses (3) to express λ_{(1,1),1} in terms of boundary classes.

In the sequel, let χ̃_{g}(s, u) be equal to the generating function χ_{g}(s, u) where all values of u_{i} are set to zero except for u := u_{1}.

Theorem 6.13. The intersection numbers above satisfy the following:

χ̃_{0}(x; s, u) = H_{0}(x, 0; s)

and

χ̃_{1}(x; s, u) = \frac{u}{24}H′′_{0}(x, 0; s) + \frac{1}{24} log H′′′_{0}(x, 0; s)

where ′ denotes differentiation with respect to x.

Proof. Using that λ_{1} vanishes on \overline{M}_{0,n}, the presentation of λ_{1} on \overline{M}_{1,n} in terms of boundary strata above, and the fact that the κ classes and λ_{1} forms a logarithmic CohFT, we obtain the equations

⟨κ^{m}λ_{1}^{r}τ_{0}^{n}⟩_{0} = 0.

and

⟨κ^{m}λ_{1}^{r}τ_{0}^{n}⟩_{1} = \begin{cases}\frac{1}{24} ⟨κ^{m}τ_{0}^{n+2}⟩_{0} & if r = 1, \\ 0 & if r ≥ 2. \end{cases}

Rewriting these identities in terms of χ̃_{g}, using the fact that χ̃_{g}(s, x, 0) = H_{g}(s, x) and theorem 1, we obtain the desired result.

Proof. (completion of theorem 6.11) By setting B := \frac{u}{24} in the previous theorem and Φ_{g} = χ̃_{g} for g = 0, 1, we conclude the proof of theorem 6.11 since theorem 6.9 implies that by forgetting Φ_{1}, we obtain all possible CohFTs in genus zero. Furthermore, by varying u, one obtains all possible values of I_{1,1} without changing the values of I_{0,n}.

Remark . The relations between the intersection numbers obtained in the previous proof can be encoded in the differential equations

\frac{∂}{∂u}χ̃_{0} = 0 and \frac{∂}{∂u}χ̃_{1} = \frac{1}{24} \frac{∂^{2}}{∂x^{2}} χ̃_{0} (17)

Putting everything together, we arrive at the following theorem.

Theorem 6.14. The moduli space of normalized, restricted, rank one CohFTs of genus one is parameterized by coordinates (s, u) via potentials (χ̃_{0}, χ̃_{1}) where χ̃_{0}(x) belongs to \frac{x^{3}}{6} +x^{4} ℂ[s, u][[x]] and χ̃_{1}(x) belongs to x ℂ[s, u][[x]] satisfying theorem 6.13. The tensor product is additive in the coordinates (s, u).

Given two rank one, normalized CohFTs in genus zero, it is not obvious how to write down the potential of the tensor product CohFT explicitly in terms of the potentials of the tensor factors. In [19], the authors show that the operation of tensor product corresponds to multiplication of the formal Laplace transforms of the two potentials associated to the tensor factors. Because a rank one, normalized, restricted CohFTs in genus one is determined by its genus zero potential and the value of u, an explicit expression for the potential associated to the tensor product of two such theories follows from the genus zero result of [19] and theorem 6.13.

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Max Planck Institut für Mathematik, Gottfried Claren Str. 26, 53225 Bonn, GERMANY and Department of Mathematics, Michigan State University, Wells Hall, East Lansing, MI 48824-1027, USA

  E-mail address: kabanov@math.msu.edu

Max Planck Institut für Mathematik, Gottfried Claren Str. 26, 53225 Bonn, GERMANY and Department of Mathematics, Boston University, Boston, MA 02215, USA

  E-mail address: kimura@math.bu.edu