A new cohomology class on the moduli space of curves

We define a collection $\Theta_{g,n}\in H^{4g-4+2n}(\overline{\cal M}_{g,n},\mathbb{Q})$ for $2g-2+n>0$ of cohomology classes that restrict naturally to boundary divisors. We prove that the intersection numbers $\int_{\overline{\cal M}_{g,n}}\Theta_{g,n}\prod_{i=1}^n\psi_i^{m_i}$ can be recursively calculated. We conjecture that a generating function for these intersection numbers is a tau function of the KdV hierarchy. This is analogous to the conjecture of Witten proven by Kontsevich that a generating function for the intersection numbers $\int_{\overline{\cal M}_{g,n}}\prod_{i=1}^n\psi_i^{m_i}$ is a tau function of the KdV hierarchy.


INTRODUCTION
Let M g,n be the moduli space of genus g stable curves-curves with only nodal singularities and finite automorphism group-with n labeled points disjoint from nodes. Define ψ i = c 1 (L i ) ∈ H 2 (M g,n , Q) to be the first Chern class of the line bundle L i → M g,n with fibre above [(C, p 1 , . . . , p n )] given by T * p i C. Consider the natural maps given by the forgetful map which forgets the last point (1) M g,n+1 π −→ M g,n and the gluing maps which glue the last two points In this paper we construct cohomology classes Θ g,n ∈ H * (M g,n , Q) for g ≥ 0, n ≥ 0 and 2g − 2 + n > 0 satisfying the following four properties: (i) Θ g,n ∈ H * (M g,n , Q) is of pure degree, (ii) φ * irr Θ g,n = Θ g−1,n+2 , φ * h,I Θ g,n = π * 1 Θ h,|I|+1 · π * 2 Θ g−h,|J|+1 , (iii) Θ g,n+1 = ψ n+1 · π * Θ g,n , (iv) Θ 1,1 = 0 where π i is projection onto the ith factor of M h,|I|+1 × M g−h,|J|+1 . We prove below that the properties (i)-(iv) uniquely define intersection numbers of the classes Θ g,n with the classes ψ i and more generally with classes in the tautological ring RH * (M g,n ) ⊂ H 2 * (M g,n , Q). Remark 1.1. One can replace (ii) by the equivalent property φ * Γ Θ g,n = Θ Γ . for any stable graph Γ, defined in Section 3, of genus g with n external edges. Here where π v is projection onto the factor M g(v),n (v) . This generalises (ii) from 1-edge stable graphs given by φ Γ irr = φ irr and φ Γ h,I = φ h,I .

Remark 1.2.
The sequence of classes Θ g,n satisfies many properties of a cohomological field theory (CohFT). It is essentially a 1-dimensional CohFT with vanishing genus zero classes, not to be confused with Hodge classes which are trivial in genus zero but do not vanish there. The trivial cohomology class 1 ∈ H 0 (M g,n , Q), which is a trivial example of a CohFT known as a topological field theory, satisfies conditions (i) and (ii), while the forgetful map property (iii) is replaced by Θ g,n+1 = π * Θ g,n .
Theorem 1. There exists a class Θ g,n satisfying (i) -(iv) and furthermore any such class satisfies the following properties.
The main content of Theorem 1 is the existence of Θ g,n , the rigidity property (IV) and the uniqueness property (V). The existence of Θ g,n is constructed via the push-forward of a class over the moduli space of spin curves in Section 2. The rigidity property (IV) is proven in Section 3 by starting with Θ 1,1 = λψ 1 and determining constraints on λ to arrive at λ = 3 which does occur due to the construction of Θ g,n . The uniqueness result (V) involving classes in the tautological ring RH * (M g,n ) is non-constructive since it relies on the existence of non-explicit tautological relations. The proofs of properties (I) -(III) are straightforward and presented in Section 3. Section 4 describes how the classes Θ g,n naturally combine with any cohomological field theory. Remark 1.3. Properties (i) -(iv) uniquely define the classes Θ g,n for g ≤ 4 and all n, but it is not known if they uniquely define the classes Θ g,n in general. Uniqueness would follow from injectivity of the pull-back map to the boundary RH 2g−2 (M g ) → RH 2g−2 (∂M g ) which holds for g = 2, 3 and 4. It would show that Θ g ∈ RH 2g−2 (M g ) is uniquely determined from its restriction, and consequently Θ g,n would coincide with the classes constructed in Section 2 for all n ≥ 0.
The following conjecture allows one to recursively calculate all intersection numbers M g,n Θ g,n ∏ n i=1 ψ m i i via relations coming out of the KdV hierarchy. Such a recursive calculation would strengthen property (V) since intersections of Θ g,n with ψ classes determine all tautological intersections with Θ g,n algorithmically. The Brézin-Gross-Witten KdV tau function Z BGW was defined in [6,30]. Conjecture 1 has been verified up to g = 7, i.e. the coefficients of the expansion of the logarithm of the Brézin-Gross-Witten tau function are given by intersection numbers of the classes Θ g,n for g ≤ 7 and all n. Progress towards Conjecture 1, including a purely combinatorial formulation that can be stated without reference to the moduli space of stable curves or the KdV hierarchy is discussed in Section 6.
Acknowledgements. I would like to thank Dimitri Zvonkine for his ongoing interest in this work which benefited immensely from many conversations together. I would also like to thank Vincent Bouchard, Alessandro Chiodo, Alessandro Giacchetto, Oliver Leigh, Danilo Lewanksi, Rahul Pandharipande, Johannes Schmitt, Mehdi Tavakol, Ran Tessler, Ravi Vakil and Edward Witten for useful conversations, the anonymous referee for comments which improved the paper, and the Institut Henri Poincaré where part of this work was carried out.

EXISTENCE
The existence of a cohomology class Θ g,n ∈ H * (M g,n , Q) satisfying (i) -(iv) is proven here using the moduli space of stable twisted spin curves M spin g,n which consist of pairs (Σ, θ) given by a twisted stable curve Σ equipped with an orbifold line bundle θ together with an isomorphism θ ⊗2 ∼ = ω log Σ . See precise definitions below. We first construct a cohomology class on M spin g,n and then push it forward to a cohomology class on M g,n .
A stable twisted curve, with group Z 2 , is a 1-dimensional orbifold, or stack, C such that generic points of C have trivial isotropy group and non-trivial orbifold points have isotropy group Z 2 . A stable twisted curve is equipped with a map which forgets the orbifold structure ρ : C → C where C is a stable curve known as the coarse curve of C. We say that C is smooth if its coarse curve C is smooth. Each nodal point of C (corresponding to a nodal point of C) has non-trivial isotropy group, the local picture at each node is {xy = 0}/Z 2 with Z 2 -action given by (−1) · (x, y) = (−x, −y), and all other points of C with non-trivial isotropy group are labeled points of C.
A line bundle L over C is a locally equivariant bundle over the local charts, such that at each nodal point there is an equivariant isomorphism of fibres. Hence each orbifold point p associates a representation of Z 2 on L| p acting by multiplication by exp(2πiλ p ) for λ p = 0 or 1 2 . One says L is banded at p by λ p . The equivariant isomorphism at nodes guarantees that the representations agree on each local irreducible component at the node.
The canonical bundle ω C of C is generated by dz for any local coordinate z. At an orbifold point x = z 2 the canonical bundle ω C is generated by dz hence it is banded by 1 2 i.e. dz → −dz under z → −z. Over the coarse curve ω C is generated by dx = 2zdz. In other words ρ * ω C ∼ = ω C however ω C ∼ = ρ * ω C . Moreover, deg ω C = 2g − 2 and For ω log C = ω C (p 1 , ..., p n ), locally dx x = 2 dz z so ρ * ω log C ∼ = ω log C and deg ω log C = 2g − 2 + n = deg ω log C . Following [1], define the moduli space of stable twisted spin curves by M spin g,n = {(C, θ, p 1 , ..., p n , φ) | φ : Here ω log C and θ are line bundles over the stable twisted curve C with labeled orbifold points p j and deg θ = g − 1 + 1 2 n. The pair (θ, φ) is a spin structure on C. The relation θ 2 ∼ = −→ ω log C is possible because the representation associated to ω log C at p i is trivial-dz/z z →−z −→ dz/z. The equivariant isomorphism of fibres over nodal points forces the balanced condition λ p + = λ p − for p ± corresponding to p on each irreducible component.
We can now define a vector bundle over M spin g,n using the dual bundle θ ∨ on each stable twisted curve. Denote by E the universal spin structure on the universal stable twisted spin curve over M spin g,n . Given a map S → M spin g,n , E pulls back to θ giving a family (C, θ, p 1 , ..., p n , φ) where π : C → S has stable twisted curve fibres, p i : S → C are sections with orbifold isotropy Z 2 and φ : θ 2 ∼ = −→ ω log C/S = ω C/S (p 1 , .., p n ). Consider the push-forward sheaf π * E ∨ over M spin g,n . We have Furthermore, for any irreducible component C ′ i → C, the pole structure on sections of the log canonical bundle at nodes yields Since the irreducible component C ′ is stable its log canonical bundle has negative degree and deg θ ∨ | C ′ < 0.
Negative degree of θ ∨ restricted to any irreducible component implies R 0 π * E ∨ = 0 and the following definition makes sense. Definition 2.1. Define a bundle E g,n = −Rπ * E ∨ over M spin g,n with fibre H 1 (θ ∨ ).
Represent the band of θ at the labeled points by σ = (σ 1 , ..., σ n ) ∈ {0, 1} n so that at each labeled point p i the representation of Z 2 on θ| p i is given by multiplication by exp(2πiλ p i ) for λ p i = 1 2 σ i ∈ {0, 1 2 }. The number of p i with λ p i = 0 is even due to evenness of the degree of the push-forward sheaf |θ| := ρ * O C (θ) on the coarse curve C, [31]. In the smooth case, the boundary type of a spin structure is determined by an associated quadratic form applied to each of the n boundary classes which vanishes since it is a homological invariant, again implying that the number of p i with λ p i = 0 is even. The moduli space of stable twisted spin curves decomposes into components determined by the band σ: where M spin g,n, σ consists of those spin curves with θ banded by σ, and the union is σ is connected except when | σ| = n, in which case there are two connected components determined by their Arf invariant, and known as even and odd spin structures. This follows from the case of smooth spin curves proven in [43].
Restricted to M spin g,n, σ , the bundle E g,n has rank (3) rank E g,n = 2g − 2 + 1 2 (n + | σ|) by the following Riemann-Roch calculation. Orbifold Riemann-Roch takes into account the representation information . Alternatively, one can use the usual Riemann-Roch calculation on the push-forward of θ to the underlying coarse curve C as follows. The sheaf of local sections O C (L) of any line bundle L on C pushes forward to a sheaf |L| := ρ * O C (L) on C which can be identified with the local sections of L invariant under the Z 2 action. Away from nodal points |L| is locally free, hence a line bundle. At nodal points, the pushforward |L| is locally free when L is banded by the trivial representation, and |L| is a torsion-free sheaf that is not locally free when L is banded by the non-trivial representation-see [25]. The pull-back bundle is given by since locally invariant sections must vanish when the representation is non-trivial. Hence deg |θ ∨ | = deg θ ∨ − 1 2 | σ|. Hence Riemann-Roch on the coarse curve yields the same result as above: It is proven in [25] that H i (θ ∨ ) = H i (|θ ∨ |) so the calculations agree.
We have h 0 (θ ∨ ) = 0 since deg θ ∨ = 1 − g − 1 2 n < 0, and the restriction of θ ∨ to any irreducible component C ′ , say of type (g ′ , n ′ ), also has negative degree, Thus H 1 (θ ∨ ) gives fibres of a rank 2g − 2 + 1 2 (n + | σ|) vector bundle. The analogue of the boundary maps φ irr and φ h,I defined in (2) are multivalued maps defined as follows. Consider a node p ∈ C for (C, θ, p 1 , ..., p n , φ) ∈ M spin g,n . Denote the normalisation by ν :C → C with points p ± ∈C that map to the node p = ν(p ± ). WhenC is not connected, the spin structure ν * θ decomposes into two spin structures θ 1 and θ 2 . Any two spin structures θ 1 and θ 2 with bands at p + and p − that agree can glue, but not uniquely, to give a spin structure on C. This gives rise to a multivalued map, as described in [26, p.27], which uses the fibre product: and is given by M spin g,n, σ where σ and I uniquely determine σ 1 and σ 2 since θ must be banded by λ p = 0 at an even number of orbifold points, which uniquely determines the band λ p + = λ p − at the separating node.
WhenC is connected, a spin structure θ on C pulls back to a spin structurẽ θ = ν * θ onC. As above, any spin structureθ with bands at p + and p − that agree glues non-uniquely, to give a spin structure on C, and defines a multiply-defined map which uses the fibre product: and is given by Again, φ irr •ν −1 naturally restricts to components M spin g−1,n+2, σ ′ M spin g,n, σ but unlike the case of φ h,I •ν −1 above, σ does not uniquely determine σ ′ . The mapν now depends on θ and there are two cases corresponding to the decomposition of the fibre product M g−1,n+2 × M g,n M spin g,n, σ into two components which depend on Proof. When the bundle θ is banded by λ p ± = 0, the map between sheaves of local holomorphic sections is not surjective whenever U ∋ p. The image consists of local sections that agree, under an identification of fibres, at p + and p − . Hence the dual bundle θ ∨ on C is a quotient sheaf where OC(I, U) is generated by the element of the dual that sends a local section s ∈ OC(ν * θ, ν −1 U) to s(p + ) − s(p − ). Note that evaluation s(p ± ) only makes sense after a choice of trivialisation of ν * θ at p + and p − , but the ideal I is independent of this choice. The complex (6) splits as follows. We can choose a representative φ upstairs of any element from the quotient space so that φ(p + ) = 0, i.e. O C (θ ∨ , U) corresponds to elements of OC(ν * θ ∨ , ν −1 U) that vanish at p + . This is achieved by adding the appropriate multiple of In other words we can identify θ ∨ with ν * θ ∨ (−p + ) in the complex: In a family π : and it has negative degree on all irreducible components. Also R 1 π * (ν * θ ∨ | p + ) = 0 since p + has relative dimension 0. Thus We can identify the dual of the sequence (8) with the sequences (4) and (5) as follows. For the first term of (8), we have ν * θ ∨ | p + ∼ = C canonically via evaluation, The second and third terms of (8) are identified with the corresponding terms of (4), respectively the second and third terms of (5), bŷ Remark 2.4. In Lemma 2.2, the nodal band is λ p ± = 1 2 and we have λ p + + λ p − = 1. We see from Lemma 2.3 that λ p ± = 0 really wants one of λ p ± to be 1 to preserve λ p + + λ p − = 1. Definition 2.5. For 2g − 2 + n > 0 define the Chern class On the component M spin g,n, σ of M spin g,n for | σ| = n this defines the top Chern class, or Euler class. The Chern class vanishes on all other components because by (3) the rank of E g,n = 2g − 2 + 1 2 (| σ| + n) < 2g − 2 + n when | σ| < n. Note that Ω 0,n = 0 for n ≥ 3 because rank(E 0,n ) = n − 2 is greater than dim M spin 0,n = n − 3 so its top Chern class vanishes.
The cohomology classes Ω g,n behave well with respect to inclusion of strata.
Proof. When | σ| = n and θ is banded by 1 2 at the nodal point then this is an immediate application of Lemma 2.2: and the naturality of c 2g−2+n = c top . We have ) . When | σ| = n and θ is banded by 0 at the nodal point then the nodal point is necessarily non-separating and we must consider the restriction of Ω g,n to the This vanishing result is a special case of the pull-back by φ * irr since Ω g−1,n+2 vanishes on M spin g−1,n+2, σ ′ for | σ ′ | = n. Finally, when | σ| < n this is simply the pull-back of the trivial class being trivial, since in each case the restriction to an irreducible component has at least one labeled point with band = 0 so that the right hand side vanishes.
The cohomology classes Ω g,n also behave well with respect to the forgetful map π : M spin g,n+1 → M spin g,n which is defined on components with θ banded by 1 2 at p n+1 as follows. Define π(C, θ, p 1 , ..., p n+1 , φ) = (ρ(C), ρ * θ, p 1 , ..., p n , ρ * φ) where ρ(C) forgets the orbifold structure at p n+1 . The push-forward sheaf ρ * θ consists of local sections invariant under the Z 2 action. Since the representation at p n+1 is given by multiplication by −1, any invariant local section must vanish at p n+1 . In terms of a local orbifold coordinate x = z 2 , an invariant section is of the form z f (x)s for s a generator of θ and its square In other words its square is a section of ω log C with no pole at p n+1 and hence a section of ω log The forgetful map π is used to denote any family π : C → S since M spin g,n+1 is essentially the universal curve of M spin g,n . Tautological line bundles L p i → M spin g,n , i = 1, ..., n are defined analogously to those defined over M g,n as follows. Consider a family π : C → S with sections p i : S → C, i = 1, ..., n, and define Proof. Over a family π : C → S where S → M spin g,n+1 and θ → C is the universal spin structure (also denoted by E ), tensor the exact sequence of sheaves This induces a long exact sequence which simplifies to the following short exact sequence: The first of these vanishing results uses the identification θ ∨ (p n+1 ) = π * θ ∨ described below together with the vanishing R 0 π * θ ∨ = 0 due to negative degree on each irreducible component described earlier. The second of these vanishing results uses the simple dimension argument that R 1 π * vanishes on the image of p n+1 which has relative dimension 0.
Property (iv) of Θ g,n is given by the following calculation.
Proof. A one-pointed twisted elliptic curve (E , p) is a one-pointed elliptic curve (E, p) such that p has isotropy Z 2 . The degree of the divisor p in E is 1 2 and the degree of every other point in E is 1. If dz is a holomorphic differential on E (where E = C/Λ and z is the identity function on the universal cover C) then locally near p we have z = t 2 so dz = 2tdt vanishes at p. In particular, the canonical divisor (ω E ) = p has degree 1 2 and (ω log E ) = (ω E (p)) = 2p has degree 1. A spin structure on E is a degree 1 2 line bundle L satisfying L 2 = ω log E . Line bundles on E correspond to divisors on E up to linear equivalence. Note that meromorphic functions on E are exactly the meromorphic functions on E. The four spin structures on E are given by the divisors θ 0 = p and θ i = q i − p, i = 1, 2, 3, where q i is a non-trivial order 2 element in the group E with identity p. Clearly since there is a meromorphic function ℘(z) − ℘(q i ) on E with a double pole at p and a double zero at q i . Its divisor on E is 2q i − 4p, since p has isotropy Z 2 , hence 2q i − 2p ∼ 2p.

It leads leads to uniqueness of intersection numbers
a reduction argument, and consequently property (V) of Theorem 1. The proofs in this section of properties (II), (III) and (V) apply for any λ = 0. We finish the section with a rigidity result given by Theorem 2 proving that necessarily λ = 3.
We first prove the following lemma which will be needed later.
Proof of (I). Write d(g, n) = degree(Θ g,n ) which exists by (i). Note that the degree here is half the cohomological degree so Θ g,n ∈ H 2d(g,n) (M g,n , Q).
Proof of (II). This is an immediate consequence of (I) since ,n(v) = 0 since the genus 0 vertex contributes a factor of 0 to the product.
Proof of (III). Property (iii) implies that The proof of (V) follows from the special case of the intersection of Θ g,n with a polynomial in κ and ψ classes.

Proposition 3.2.
For any Θ g,n satisfying properties (i) -(iii), the intersection numbers Proof. For n > 0, we will push forward the integral (9) via the forgetful map π : M g,n → M g,n−1 as follows. Consider first the case when there are no κ classes. The presence of ψ n in Θ g,n = ψ n · π * Θ g,n−1 gives so we have reduced an intersection number over M g,n to an intersection number over M g,n−1 . In the presence of κ classes, replace κ ℓ j by κ ℓ j = π * κ ℓ j + ψ ℓ j n and repeat the push-forward as above on all summands. By induction, we see that for i.e. the intersection number (9) reduces to an intersection number over M g of Θ g times a polynomial in the κ classes. When g = 1, the right hand side is instead For g > 1, by (I) deg Θ g = 2g − 2, so we may assume the polynomial p consists only of terms of homogeneous degree g − 1 (where deg κ r = r). But by a result of Faber and Pandharipande [24,Proposition 2], which strengthens Looijenga's theorem [39], a homogeneous degree g − 1 monomial in the κ classes is equal in the tautological ring to the sum of boundary terms, i.e. the sum of push-forwards of polynomials in ψ and κ classes by the the maps (φ Γ ) * . Such relations arise from Pixton's relations and are described algorithmically in [11]. Now property (ii) of Θ g , shows that the pull-back of Θ g to these boundary terms is Θ g ′ ,n ′ for g ′ < g so we have expressed (9) as a sum of integrals of Θ g ′ ,n ′ against ψ and κ classes. By induction, one can reduce to the integral M 1,1 Θ 1,1 = λ 24 and the proposition is proven.
A consequence of Proposition 3.2 is property (V) of Theorem 1 stated as Corollary 3.3 below. Let us first recall the definition of tautological classes in H * (M g,n , Q). Dual to any point (C, p 1 , ..., p n ) ∈ M g,n is its stable graph Γ with vertices V(Γ) representing irreducible components of C, internal edges representing nodal singularities and a (labeled) external edge for each p i . Each vertex is labeled by a genus g(v) and has valency n(v). The genus of a stable graph is g( The strata algebra S g,n is a finite-dimensional vector space over Q with basis given by isomorphism classes of pairs (Γ, ω), for Γ a stable graph of genus g with n external edges and ω ∈ H * (M Γ , Q) a product of κ and ψ classes in The map q allows one to define a multiplication on S g,n , essentially coming from intersection theory in M g,n , which can be described purely graphically. The image q(S g,n ) ⊂ H * (M g,n , Q) is the tautological ring RH * (M g,n ) and an element of the kernel of q is a tautological relation. See [48, Section 0.3] for a detailed description of S g,n .
Proof. The tautological ring RH * (M g,n ) consists of polynomials in the classes κ i , ψ i and boundary classes, which are pushforwards under (φ Γ ) * of polynomials in κ i and ψ i . By the natural restriction property (ii) satisfied by Θ g,n , given a monomial in κ and ψ classes ω ∈ H * (M Γ , Q), The final term is a product over the vertices of Γ of intersections Θ classes with

Remark 3.4. The intersection numbers
i with no κ classes. This essentially reverses the reduction shown in the proof of Proposition 3.2. Explicitly, for π : M g,n+N → M g,n and m = (m 1 , ..., m N ) define a polynomial in κ classes by The polynomials R m (κ 1 , κ 2 , ...) generate all polynomials in the κ i so (10) can be used to remove any κ class.

Example 3.5. A genus two relation proven by Mumford
which induces the relation

the relation on the level of intersection numbers is given by
Until now Θ 1,1 = λψ 1 for any non-zero λ ∈ Q. The following theorem proves the rigidity condition (IV) that λ = 3. The proof of the theorem relies on the fact that for low genus and small n, the cohomology is tautological. This allows us to work in the tautological ring in order to construct Θ g,n from properties (i) -(iv).
and set the initial condition to be Proof. The existence proof in Section 2 shows that λ = 3 is possible but it does not exclude other values. The strategy of proof of this theorem is to attempt to construct classes, beginning with the initial condition Θ 1,1 = λψ 1 . Importantly, condition (iii) determines Θ g,n for all n > 0 uniquely from Θ g so the main calculation occurs over M g . We consider classes in RH 2g−2 (M g ) since for small values of g it is known that H 2 * (M) g = RH * (M g ). The essential idea is as follows. A class Θ g ∈ H 2g−2 (M g , Q) pulls back under boundary maps to Θ g−1,2 and Θ g−1,1 ⊗ Θ 1,1 . The relationship Θ g−1,2 = ψ 2 π * Θ g−1,1 constrains the class Θ g . We find that Θ 2 exists (and hence also Θ 2,n exists for all n) for all λ ∈ Q, but that Θ 3 (and Θ 3,n ) exists only for λ = 3 or λ = −11/15. The existence of Θ 4 constrains λ further, allowing only λ = 3.
In genus 3, H 2 * (M 3 , Q) = RH * (M 3 ) due to the calculation of the cohomology H * (M 3 , Q), for example by using the calculation of H * (M 3,1 , Q) in [28], together with the calculation of the tautological ring RH * (M 3 , Q) via Pixton's relations [48] implemented using the Sage package admcycles [12]. We have dim RH 4 (M 3 , Q) = 7 and we write Θ 3 as a general linear combination of basis vectors in RH 4 (M 3 ): The following pull-back map is injective has 2-dimensional kernel and is surjective onto the S 2 -invariant part of RH 4 (M 2,2 ). Hence the condition φ * irr Θ 3 = Θ 2,2 = ψ 1 ψ 2 π * Θ 2 determines Θ 3 up to parameters s, t ∈ Q: has three dimensional image, and the condition ) cannot be satisfied with only two parameters in Θ 3 for general λ, forcing λ to satisfy a polynomial relation. We find In genus 4, H 2 * (M 4 ) = RH * (M 4 ) is due to the calculation by Bergström and Tommasi [4] of the Hodge polynomial of M 4 together with the calculation of the tautological ring RH * (M 4 ) via Pixton's relations using admcycles [12]. We choose a general element θ 4 ∈ RH 6 (M 4 ) which is a linear combination of basis vectors for the 32 dimensional space RH 6 (M 4 ). The pull-back map of RH 6 (M 4 ) to the boundary can be shown to be injective using admcycles.
The main purpose of the g = 4 calculation is to prove that λ = − 11 15 is impossible, so we substitute λ = − 11 15 into Θ 3 above to get Θ 3 = 2783 81000 κ 4 1 − 11011 13500 κ 2 1 κ 2 + 59939 10125 κ 1 κ 3 + 16093 9000 κ 2 2 − 474287 13500 κ 4 − 1232 1125 B 2 As in the g = 3 case above we consider the pull-back map which has a six dimensional kernel. The S 2 -invariant part of H 12 (M 3,2 ) is proven in [3] to be 31 dimensional, and using admcycles it can be shown to be tautological. The condition φ * irr Θ 4 = Θ 3,2 = ψ 1 ψ 2 π * Θ 3 produces a system of 31 equations in 32 unknowns. Using admcycles, we find that Θ 3,2 lies in the image of the pull-back map, and constrains Θ 4 to depend linearly on 6 parameters. The pull-back map composed with projection uniquely determines the 6 parameters and finally the resulting class Θ 4 is shown under pull-back map composed with projection

COHOMOLOGICAL FIELD THEORIES
The class Θ g,n combines with known enumerative invariants, such as Gromov-Witten invariants, to give rise to new invariants. More generally, Θ g,n pairs with any cohomological field theory, which is fundamentally related to the moduli space of curves M g,n , retaining many of the properties of the cohomological field theory, and is in particular often calculable.
A cohomological field theory is a pair (H, η) composed of a finite-dimensional complex vector space H equipped with a symmetric, bilinear, nondegenerate form, or metric, η and a sequence of S n -equivariant maps. Many CohFTs are naturally defined on H defined over Q, nevertheless we use C in order to relate them to Frobenius manifolds, and to use normalised canonical coordinates, defined later.
that satisfy compatibility conditions from inclusion of strata: The CohFT has flat identity if there exists a vector 1 1 ∈ H compatible with the forgetful map π : M g,n+1 → M g,n by and For a one-dimensional CohFT, i.e. dim H = 1, identify Ω g,n with the image Ω g,n (1 1 ⊗n ), so we write Ω g,n ∈ H * (M g,n , C). A trivial example of a CohFT is Ω g,n = 1 ∈ H 0 (M g,n , C) which is a topological field theory as we now describe.
A two-dimensional topological field theory (TFT) is a vector space H and a sequence of symmetric linear maps Ω 0 g,n : H ⊗n → C for integers g ≥ 0 and n > 0 satisfying the following conditions. The map Ω 0 0,2 = η defines a symmetric, bilinear, nondegenerate form η, and together with Ω 0 0,3 it defines a product · on H via Consider the natural isomorphism H 0 (M g,n ) ∼ = C. The degree zero part of a CohFT Ω g,n is a TFT: We often write Ω 0,3 = Ω 0 0,3 interchangeably. Associated to Ω g,n is the product (14) built from η and Ω 0,3 .
Remark 4.1. The classes Θ g,n satisfy properties (11) and (12) of a one-dimensional CohFT. In place of property (13), they satisfy Θ g,n+1 ( The product defined in (14) there is a canonical basis {u 1 , ..., u N } ⊂ H such that u i · u j = δ ij u i . The metric is then necessarily diagonal with respect to the same basis, η(u i , u j ) = δ ij η i for some η i ∈ C \ {0}, i = 1, ..., N. The Givental-Teleman theorem described in Section 5 gives a construction of semisimple CohFTs. 4.1. Cohomological field theories coupled to Θ g,n .

Definition 4.2. For any
CohFT Ω defined on (H, η) define Ω Θ = {Ω Θ g,n } to be the sequence of S n -equivariant maps Ω Θ g,n : H ⊗n → H * (M g,n , C) given by This is essentially the tensor products of CohFTs, albeit involving Θ g,n . The tensor products of CohFTs is obtained as above by cup product on H * (M g,n , C), generalising Gromov-Witten invariants of target products and the Künneth formula Generalising Remark 4.1, Ω Θ g,n satisfies properties (11) and (12) of a CohFT on (H, η). In place of property (13), it satisfies and Ω Θ 0,3 = 0.
Given a CohFT Ω = {Ω g,n }, or a more general collection of classes such as Ω = {Ω Θ g,n }, and a basis {e 1 , ..., e N } of H, the partition function of Ω is defined by: .., N} and k j ∈ N. For dim H = 1 and Ω g,n = 1 ∈ H * (M g,n , C), its partition function is gives its partition function. Property (iii) is realised by the following homogeneity property: and for Z Θ (h, t 0 , t 1 , ...) in the following proposition.

Gromov-Witten invariants.
Let X be a projective algebraic variety and consider (C, x 1 , . . . , x n ) a connected smooth curve of genus g with n distinct marked points. For β ∈ H 2 (X, Z) the moduli space of stable maps M g,n (X, β) is defined by is a morphism from a connected nodal curve C containing distinct points {x 1 , . . . , x n } that avoid the nodes. Any genus zero irreducible component of C with fewer than three distinguished points (nodal or marked), or genus one irreducible component of C with no distinguished point, must not be collapsed to a point. We quotient by isomorphisms of the domain C that fix each x i . The moduli space of stable maps has irreducible components of different dimensions but it has a virtual class of dimension (17) dim For i = 1, . . . , n there exist evaluation maps: (18) ev i : M g,n (X, β) −→ X, ev i (π) = π(x i ) and classes γ ∈ H * (X, Z) pull back to classes in H * (M g,n (X, β), C) n (X, β), C). The forgetful map p : M g,n (X, β) → M g,n maps a stable map to its domain curve followed by contraction of unstable components. The push-forward map p * on cohomology defines a CohFT Ω X on the even part of the cohomology H = H even (X, C) (and a generalisation of a CohFT on H * (X, C)) equipped with the symmetric, bilinear, nondegenerate form We have (Ω X ) g,n : H even (X, C) ⊗n → H * (M g,n , C) defined by Note that it is the dependence of p = p(g, n, β) on β (which is suppressed) that allows (Ω X ) g,n (α 1 , ...α n ) to be composed of different degree terms. The partition function of the CohFT Ω X with respect to a chosen basis e α of H even (X; C) is It stores ancestor invariants. These are different to descendant invariants which use in place of ψ j = c 1 (L j ), Ψ j = c 1 (L j ) for line bundles L j → M g,n (X, β) defined as the cotangent bundle over the ith marked point. Following Definition 4.2, we define Ω Θ X by Let Θ PD g,n ⊂ A g−1 (M g,n , C) be the (g − 1)-dimensional Chow class given by the push-forward of the top Chern class of the bundle E g,n defined in Definition 2.1. The virtual dimension of the pull-back of Θ PD g,n is: Comparing the dimension formulae (17) and (20) we see that elliptic curves now take the place of Calabi-Yau 3-folds to give virtual dimension zero moduli spaces, independent of genus and degree. The invariants of a target curve X are trivial when the genus of X is greater than 1 and computable when X = P 1 , [45], producing results analogous to the usual Gromov-Witten invariants in [46]. For c 1 (X) = 0 and dim X > 1, the invariants vanish for g > 1, while for g = 1 it seems to predict an invariant associated to maps of elliptic curves to X.

Its partition function stores Weil-Petersson volumes
and deformed Weil-Petersson volumes studied by Mirzakhani [40]. Weil-Petersson volumes of the subvariety of M g,n dual to Θ g,n make sense even before we find such a subvariety. They are given by which are calculable since they are given by a translation of Z BGW . If we include ψ classes, we get polynomials V Θ g,n (L 1 , ..., L n ) which give the deformed volumes analogous to Mirzakhani's volumes. In [44,52] the polynomials V Θ g,n (L 1 , ..., L n ) are related to the volume of the moduli space of Super Riemann surfaces.

ELSV formula. Another example of a 1-dimensional CohFT is given by
where λ i = c i (E) is the ith Chern class of the Hodge bundle E → M g,n defined to have fibres H 0 (ω C ) over a nodal curve C.

Definition 4.4.
Define the simple Hurwitz number H g,µ to be the weighted count of genus g connected covers of P 1 with ramification µ = (µ 1 , ..., µ n ) over ∞ and simple ramification elsewhere. Each cover π is counted with weight 1/|Aut(π)|.
Using Ω Θ g,n = Θ · c(E ∨ ) we can define an analogue of the ELSV formula: It may be that H Θ g,µ has an interpretation of enumerating simple Hurwitz covers. Note that it makes sense to set all µ i = 0, and in particular there are non-trivial primary invariants over M g , unlike for simple Hurwitz numbers. An example calculation: The versal deformation space of the A 2 singularity. The A 2 singularity has a two-dimensional versal deformation space M ∼ = C 2 = {(t 1 , t 2 )} parametrising the family that admits a semisimple Frobenius manifold structure. Dubrovin [15] associated a family of linear systems (24) depending on the canonical coordinates (u 1 , ..., u N ) of any semisimple Frobenius manifold M. This produces a CohFT Ω A 2 defined on C 2 from the A 2 singularity using Definition 5.2 in Section 5. More generally, to any point of a Frobenius manifold one can associate a cohomological field theory and conversely the genus zero part of a cohomological field theory defines a Frobenius manifold [15].
Recall that a Frobenius manifold is a complex manifold M equipped with an associative product on its tangent bundle compatible with a flat metric-a nondegenerate symmetric bilinear form-on the manifold. It is encoded by a single function F(t 1 , ..., t N ), known as the prepotential, that satisfies a nonlinear partial differential equation known as the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation: For a semisimple conformal Frobenius manifold, multiplication by the Euler vector field E produces an endomorphism U with eigenvalues {u 1 , ..., u N } known as canonical coordinates on M. They give rise to vector fields ∂/∂u i with respect to which the metric η, product · and Euler vector field E are diagonal: The differential equation (24) in z is defined at any point of the Frobenius manifold using U, the endomorphism defined by multiplication by the Euler vector field E, and the endomorphism which is a direct consequence of substitution of Y = R(z −1 )e zU into (24) with

N )
hence it uniquely determines the topological field theory. We find the normalised canonical basis most useful for comparisons with topological recursion-see Section 5.2.1.
The Frobenius manifold structure on the versal deformation space M of the A 2 singularity was constructed in [15,49]. The product on tangent spaces of the family W t (z) = z 3 − t 2 z + t 1 is induced from the isomorphism The metric is given by , which correspond respectively to the images of 1 and W t (z) in C[z]/W ′ t (z).

Remark 4.5.
The matrix R(z) defined in (23)-which uses the normalised canonical basis for H so that η is the dot product-is related to the matrix R(z) in [48] by conjugation by the transition matrix Ψ from flat coordinates to normalised canonical coordinates (23) produces the cohomological field theory Ω A 2 associated to the A 2 singularity at the point (t 1 , t 2 ) = (0, 3) via Definition 5.2 below.

GIVENTAL CONSTRUCTION OF COHOMOLOGICAL FIELD THEORIES.
Givental produced a construction of partition functions of cohomological field theories in [29]. He defined an action of the twisted loop group, and elements of z 2 C N [[z]] known as translations, on partition functions of cohomological field theories and used this to build partition functions of semisimple cohomological field theories out of the basic building block Z KW (h, t 0 , t 1 , ...) combined with the vector 1 1 ∈ C N which represents the topological field theory. This action was interpreted as an action on the actual cohomology classes in H * (M g,n , C) independently, by Katzarkov-Kontsevich-Pantev, Kazarian and Teleman-see [48,50].
The Givental action is defined on more general sequences of cohomology classes in H * (M g,n , C) such as the collection of classes Θ g,n or Ω Θ g,n defined from any Co-hFT Ω g,n in Definition 4.2. If Ω g,n is semisimple the classes Ω Θ g,n can be obtained by applying Givental's action to the collection Θ g,n .
5.0.1. The twisted loop group action. The loop group LGL (N, C) is the group of formal series (N, C). Define the twisted loop group L (2) GL(N, C) ⊂ LGL(N, C) to be the subgroup of elements satisfying R 0 = I and

R(z)R(−z) T = I.
Elements of L (2) GL(N, C) naturally arise out of solutions to the linear system where Y(z) ∈ C N , U = diag(u 1 , ..., u N ) for u i distinct and V is skew symmetric. One can choose a solution of (24) which behaves asymptotically for z → ∞ as This defines a power series R(z) with coefficients given by N × N matrices which is easily shown to satisfy R(z)R T (−z) = I, hence R(z) ∈ L (2) GL(N, C). Givental [29] constructed an action on CohFTs using a triple as follows. For a given stable graph Γ of genus g and with n external edges we have Given (R(z), T(z), 1 1) ∈ L (2) Givental's action is defined via weighted sums over stable graphs. For R(z) ∈ L (2) GL(N, C), define which has the power series expansion on the right since R −1 (z) is also an element of the twisted loop group so the numerator I − R −1 (z)R −1 (w) T vanishes at w = −z.
. We say Γ is stable if any vertex labeled by g = 0 is of valency ≥ 3 and there are no isolated vertices labeled by g = 1. We write n v for the valency of the vertex v. Define G g,n to be the finite set of all stable, connected, genus g, decorated graphs with n ordinary leaves and at most 3g − 3 + n dilaton leaves.
We consider only the even part of H * (M g,n , C) so (25) is independent of the order in which we take the product of cohomology classes. If {Ω g,n } is a CohFT defined on (C, η) for H ∼ = C N , then the classes {Ω g,n } in (25) satisfy the same restriction conditions and hence define a CohFT on (C, η) with the same degree zero, or topological field theory, terms as those of Ω ′ . If we choose T(z) ≡ 0, then the sum in (25), which is over stable graphs without dilaton leaves, defines the action of the twisted loop group on CohFTs. If we choose R(z) ≡ I, then (25) where π : M g,n+m → M g,n is the forgetful map. The sum over m ∈ N defining (T · Ω ′ ) g,n is finite since T(z) ∈ z 2 H[[z]], so dim M g,n+m = 3g − 3 + n + m grows more slowly in m than the degree 2m coming from T resulting in at most 3g − 3 + n terms. We can relax this condition and allow T(z) ∈ zH [[z]] if we control the growth of the degrees of all terms of Ω ′ g,n in n to ensure T(z) produces a finite sum. In particular, Θ g,n , and more generally Ω ′Θ g,n for any CohFT Ω ′ g,n , is annihilated by terms of degree > g − 1 hence the sum defining (TΩ ′ ) g,n consists of at most g − 1 terms when T(z) ∈ zH [[z]]. The tensor product Ω → Ω Θ given in Definition 4.2 commutes with the action of R and commutes with the action of T up to rescaling. For a CohFT Ω, and R(z) ∈ L (2) GL(N, C) and The first relation in (26) uses the restriction properties (ii) of Θ g,n and the second of these uses the forgetful property (iii) of Θ g,n as follows: and sum over m to get T · Ω Θ = (zT) · Ω Θ . The Givental-Teleman theorem [29,53] proves that the action defined in Definition 5.2 is transitive on semisimple CohFTs. In particular, a semisimple CohFT defined on a vector space of dimension N can be constructed via the Givental action on N copies of the trivial CohFT. Given a semisimple CohFT Ω, there exists (R(z), T(z), 1 1) ∈ L (2) such that Ω g,n is defined by the weighted sum over graphs (25)

using R(z), T(z)
and Ω ′ g,n given by the topological field theory underlying Ω g,n . Note that a semisimple topological field theory of dimension N is equivalent to 1 1 ∈ C N which gives the unit vector in terms of a basis in which the product is diagonal and the metric η is the dot product, known as a normalised canonical basis.
On the level of partition functions, the construction of a semisimple CohFT from the trivial CohFT is realised via an action of quantised differential operatorsR and T on products of Z KW (h, t 0 , t 1 , ...), a KdV tau function defined in the next section.
The partition function of (25) is given in [19,29,50] by the following formula: . Vertex weightsŵ(v) store products of Z KW corresponding to the partition function of a topological field theory, edge weightsŵ(e) store coefficients of the series E (w, z), and leaf weightsŵ(ℓ) store the variables t α k in a series weighted by coefficients of the series R −1 (−z). We do not give explicit formulae for the weights, see [19,29,50], and instead use an equivalent elegant formulation given by topological recursion, defined in Section 5.2.
A consequence of the relations (26) is the following proposition which modifies the construction of a semisimple CohFT Ω to produce Ω Θ .

Proposition 5.4. Given a semisimple
CohFT Ω defined via (25) using g,n the degree 0 part of Ω g,n determined by the vector 1 1 ∈ C N . Its partition function (27) by a copy of Z Θ (h, {t k }) and shifting the operatorT.

KdV tau functions.
The KdV hierarchy is a sequence of partial differential equations beginning with the KdV equation.  (28), and later equations U t k = P k (U, U t 0 , U t 0 t 0 , ...) for k > 1 determine U uniquely from U(t 0 , 0, 0, ...). See [41] for the full definition. The Kontsevich-Witten tau function Z KW is defined by the initial condition U KW (t 0 , 0, 0, ...) = t 0 for U KW =h ∂ 2 ∂t 2 0 log Z KW . The low genus terms of log Z KW are  log Z BGW of the KdV hierarchy arises out of a unitary matrix model studied in [6,30]. It is defined by the initial condition The low genus g terms (= coefficient ofh g−1 ) of log Z BGW are (29) +h 2 63 1024 It satisfies the homogeneity property which coincides with (16) satisfied by Z Θ (h, t 0 , t 1 , ...). A proof of this homogeneity property for Z BGW can be found in [2,14]. The tau function Z BGW (h, t 0 , t 1 , ...) shares many properties of the famous Kontsevich-Witten tau function Z KW (h, t 0 , t 1 , ...) introduced in [55]. An analogue of Theorem 3 is given by Conjecture 1 which postulates that the function The tau function Z BGW appears in a generalisation of Givental's decomposition of CohFTs in [9].
The same shift T 0 = 1 z T(z) is used by Z BGW (h, {t k }) and Z Θ (h, {t k }) due to their common homogeneity property (16). One can also replace only some copies of Z KW (h, {t k }) in (27) by copies of Z BGW (h, {t k }) and shift components ofT. For example, in [13] the enumeration of bipartite dessins d'enfant is shown to have partition function for R and T determined by the curve xy 2 + xy + 1 = 0 as described in Section 5.2. Figure 1 summarises the contents of this section.

FIGURE 1. Constructions of CohFT partition functions
The upper horizontal arrow in the figure represents Givental's construction of a partition function defined in (27) and Definition 5.2. Topological recursion is defined in Section 5.2-it produces a partition function from a spectral curve S = (C, x, y, B) consisting of a Riemann surface C equipped with meromorphic functions x and y and a bidifferential B. We begin with a description of the left vertical arrow which represents the construction of an element R(z) ∈ L (2) (31) and T(z) and 1 1 from (C, x, y) in (37) and (36). We then define topological recursion in 5.2.1 and state the result of [18], that topological recursion encodes the graphical construction in (27) and gives equality of partition functions, represented by the right vertical arrow.
An element of the twisted loop group R(z) ∈ L (2) GL(N, C) can be naturally constructed from a Riemann surface Σ equipped with a bidifferential B(p 1 , p 2 ) on Σ × Σ and a meromorphic function x : Σ → C, for N = the number of zeros of dx. A basic example is the function x = z 2 on Σ = C which gives rise to the constant element R(z) = 1 ∈ GL(1, C). More generally, any function x that looks like this example locally, i.e. x = s 2 + c for s a local coordinate around a zero of dx and c ∈ C, gives R(z) = I + R 1 z + ... ∈ L (2) GL(N, C) which is in some sense a deformation of I ∈ GL (N, C), or N copies of the basic example. Definition 5.6. On any compact Riemann surface (Σ, {A i } i=1,...,g ) with a choice of A-cycles, define a fundamental normalised bidifferential of the second kind B(p, p ′ ) to be a symmetric tensor product of differentials on Σ × Σ, uniquely defined by the properties that it has a double pole on the diagonal of zero residue, double residue equal to 1, no further singularities and normalised by p∈A i B(p, p ′ ) = 0, i = 1, ..., g, [27]. On a rational curve, which is sufficient for the examples in this paper, B is the Cauchy kernel The bidifferential B(p, p ′ ) acts as a kernel for producing meromorphic differentials on the Riemann surface Σ via ω(p) = Λ λ(p ′ )B(p, p ′ ) where λ is a function defined along the contour Λ ⊂ Σ. Depending on the choice of (Λ, λ), ω can be a differential of the 1st kind (holomorphic), 2nd kind (zero residues) or 3rd kind (simple poles). For (Σ, x), a Riemann surface equipped with a meromorphic function, define evaluation of any meromorphic differential ω at a simple zero P of dx by

Definition 5.7.
where we choose a branch of x(p) − x(P ) once and for all at P to remove the ±1 ambiguity.
A fundamental example of Definition 5.7 required here is B(P, p) which is a normalised (trivial A-periods) differential of the second kind holomorphic on Σ\P with a double pole at the simple zero P of dx.
In order to produce an element of the twisted loop group, Shramchenko [51] constructed a solution Y(z) of the linear system (24) The proof in [51] is indirect, showing that Y(z) i j satisfies an associated set of PDEs in u i , and using the Rauch variational formula to calculate ∂ u k B(P α , p). Instead, here we work directly with the associated element R(z) of the twisted loop group. Definition 5.8. Define the asymptotic series in the limit z → 0 by where Γ β is a path of steepest descent for −x(p)/z containing x(P β ).
Note that the asymptotic expansion of the contour integral (31) for z → 0 depends only the intersection of Γ β with a neighbourhood of p = P β . When α = β, the integrand has zero residue at p = P β so we deform Γ β to go around P β to get a well-defined integral. Locally, this is the same as defining R s −2 exp(−s 2 )ds = −2 √ π by integrating the analytic function z −2 exp(−z 2 ) along the real line in C deformed to avoid 0.

Lemma 5.9 ([51]). The asymptotic series R(z) defined in (31) satisfies the twisted loop group condition
Proof. The proof here is taken from [17]. We have where the first equality uses the fact that the only poles of the integrand occur at Putting the two sides together yields the following result due to Eynard [21] (34)B α,β (z 1 , Equation (32) is an immediate consequence of (34) and the finiteness ofB α,β (z 1 , z 2 ) at z 2 = −z 1 .

Topological recursion.
Topological recursion is a procedure which takes as input a spectral curve, defined below, and produces a collection of symmetric tensor products of meromorphic 1-forms ω g,n on C n . The correlators store enumerative information in different ways. Periods of the correlators store top intersection numbers of tautological classes in the moduli space of stable curves M g,n and local expansions of the correlators can serve as generating functions for enumerative problems.
A spectral curve S = (C, x, y, B) is a Riemann surface C equipped with two meromorphic functions x, y : C → C and a bidifferential B(p 1 , p 2 ) defined in (5.6), which is the Cauchy kernel in this paper. Topological recursion, as developed by Chekhov, Eynard, Orantin [8,22], is a procedure that produces from a spectral curve S = (C, x, y, B) a symmetric tensor product of meromorphic 1-forms ω g,n on C n for integers g ≥ 0 and n ≥ 1, which we refer to as correlation differentials or correlators. The correlation differentials ω g,n are defined by and ω 0,2 (p 1 , p 2 ) = B(p 1 , p 2 ) and for 2g − 2 + n > 0 they are defined recursively via the following equation.
Here, we use the notation L = {2, 3, . . . , n} and p I = {p i 1 , p i 2 , . . . , p i k } for I = {i 1 , i 2 , . . . , i k }. The outer summation is over the zeroes α of dx and p →p is the involution defined locally near α satisfying x(p) = x(p) andp = p. The symbol • over the inner summation means that we exclude any term that involves ω 0,1 . Finally, the recursion kernel is given by .
which is well-defined in the vicinity of each zero of dx. It acts on differentials in p and produces differentials in p 1 since the quotient of a differential in p by the differential dx(p) is a meromorphic function. For 2g − 2 + n > 0, each ω g,n is a symmetric tensor product of meromorphic 1-forms on C n with residueless poles at the zeros of dx and holomorphic elsewhere. A zero α of dx is regular, respectively irregular, if y is regular, respectively has a simple pole, at α. A spectral curve is irregular if it contains an irregular zero of dx. The order of the pole in each variable of ω g,n at a regular, respectively irregular, zero of dx is 6g − 4 + 2n, respectively 2g. Define Φ(p) up to an additive constant by dΦ(p) = y(p)dx(p). For 2g − 2 + n > 0, the invariants satisfy the dilaton equation [22] ∑ α Res p=α Φ(p) ω g,n+1 (p, p 1 , . . . , p n ) = (2g − 2 + n) ω g,n (p 1 , . . . , p n ), where the sum is over the zeros α of dx. This enables the definition of the so-called symplectic invariants Res p=α Φ(p)ω g,1 (p).
The correlators ω g,n are normalised differentials of the second kind in each variable since they have zero A-periods, and poles only at the zeros P α of dx of zero residue. Their principal parts are skew-invariant under the local involution p →p.
A basis of such normalised differentials of the second kind is constructed from x and B in the following definition.
where evaluation B(P α , p) at P α is given in Definition 5.7.
The correlators ω g,n are polynomials in the auxiliary differentials V α k (p). To any spectral curve S, one can define a partition function Z S by assembling the polynomials built out of the correlators ω g,n [18,21,47].
As usual define F g to be the contribution from ω g,n : log

From topological recursion to Givental's construction.
The input data for Givental's construction is a triple (R(z), T(z), 1 1) ∈ L (2) Its output is a CohFT Ω, and its partition function Z Ω (h, {t α k }). The input data for topological recursion is a spectral curve S = (C, x, y, B). Its output is the correlators ω g,n which can be assembled into a partition function Z S (h, {t α k }). From a compact spectral curve define a triple 1 1 i = dy(P α ), P α regular (ydx)(P α ), P α irregular which is the unit in normalised canonical coordinates, and , P α regular (ydx)(P α ), P α irregular which defines 1 1, hence the right hand side of (37) lives in ], when P α is regular, respectively irregular. When Ω is a CohFT with flat identity-see (13) in Section 4-given by 1 1 ∈ C N , then 1 1 determines the transla- ]. In this special case y satisfies (38) ( which uniquely determines y from its first order data {dy(P α )} at each P α .
The map (C, x, y, B) → (R(z), T(z), 1 1) produces the left vertical arrow in Figure 1 and its generalisation to irregular spectral curves, i.e. a correspondence between the input data, and via the graphical construction (27) this produces the same output Z Ω (h, {t α k }) = Z S (h, {t α k }) which is the main result of [18] stated in the following theorem. on which x and B correspond to R(z) via Definition 5.8 and y corresponds to T(z) and 1 1 via (37) and (36), giving the partition function of the CohFT

Theorem 4 ([18]). Given a CohFT Ω built from
. In general, the spectral curve S in Theorem 4 is a local spectral curve which is a collection of disks neighbourhoods of zeros of dx on which B and y are define locally, although we only consider compact spectral curves S in this paper. Theorem 4 was proven only in the case T(z) = z 1 1 − R −1 (z)1 1 in [18] but it has been generalised to allow any T(z) ∈ z 2 C N [[z]]-see [9,38]. We will use the converse of Theorem 4 proven in [17], beginning instead from S. Theorem 4 was also generalised in [9] to show that the operatorsΨ,R andT acting on copies of Z BGW analogous to (27) arises by applying topological recursion to an irregular spectral curve. Equivalently, periods of the correlators of an irregular spectral curve store linear combinations of coefficients of log Z BGW . The appearance of Z BGW is due to its relationship with topological recursion applied to the curve x = 1 2 z 2 , y = 1 z [14].

Spectral curve examples.
We demonstrate Theorem 4 with four key examples of rational spectral curves equipped with the bidifferential B(p 1 , p 2 ) given by the Cauchy kernel. The spectral curves in the Examples 5.12 and 5.13, denoted S Airy and S Bes , have partition functions Z KW and Z BGW respectively. Any spectral curve at regular, respectively irregular, zeros of dx is locally isomorphic to S Airy , respectively S Bes . A consequence is that the tau functions Z KW and Z BGW are fundamental to the correlators produced from topological recursion. Moreover, the topological recursion partition function Z S is constructed via (27), using a product of copies of Z KW and copies of Z BGW , as in (30), where R and T are obtained from the spectral curve as described in Section 5.2.2. The third example, given by Theorem 5, brings together Z KW and Z Θ and conjecturally Z BGW in the limit. Proposition 5.4, which gives the relationship between the Givental construction of a semisimple CohFT Ω and its associated Ω BGW , has an elegant consequence for spectral curves. This is demonstrated explicitly in the fourth example which shows the relationship between the spectral curves of a CohFT Ω A 2 associated to the A 2 singularity and (Ω A 2 ) BGW . Examples 5.12 and 5.13 below use the differentials ξ m (z) = (2m + 1)!!z −(2m+2) dz defined by (35) for x = 1 2 z 2 with respect to a global rational parameter z for the curve C ∼ = C.

Example 5.12. Topological recursion applied to the Airy curve
produces correlators which are proven in [23] to store intersection numbers Example 5.13. Topological recursion applied to the Bessel curve It is proven in [14] that For the next example define the following collection of differentials ξ α m (z, t) us- , σ = 0, 1, m = −1, 0, 1, 2, ...
For m ≥ 0, these are linear combinations of the V i m (p) defined in (35). The following theorem uses the Chern polynomial

Theorem 5 ([38]). Topological recursion applied to the spectral curve
produces correlators ω g,n satisfying Proof of Theorem 5. Theorem 5 is a specialisation of a theorem in [38] which applies to a generalisation of the moduli space of spin curves to the moduli space of r-spin curves M 1/r g,n = {(C, θ, p 1 , ..., p n , φ) | φ : For any s ∈ Z, there is a line bundle E on the universal r-spin curve over M 1/r g,n with fibres given by the universal rth root of ω log C s . Its derived push-forward R * π * E defines a virtual bundle over M 1/r g,n . For example, when s = 1 and r = 1, −R * π * E is the Hodge bundle and when s = −1 and r = 2, −R * π * E = E g,n coincides with Definition 2.1 (where E ∨ has now become E due to s = −1.) Note that [38] considers rth roots of ω log C s − n ∑ i=1 σ i p i for C the underlying coarse curve of C with forgetful map ρ : C → C. The rth roots in [38] coincide with the push-forward |θ| = ρ * θ which is the locally free sheaf of Z 2 -invariant sections of the push-forward sheaf of θ, and the isotropy representation at p i determines σ i as described in Section 2. For r = 2, i.e. θ 2 ∼ = ω log C , at any point p i banded by 1/2 the push-forward locally satisfies |θ| 2 At any point p i banded by 0 the push-forward does not change local degree and corresponds to σ i = 0.
The Chern character of the virtual bundle −R * π * E is given by Chiodo's generalisation of Mumford's formula for the Chern character of the Hodge bundle. For σ ∈ {0, 1, ..., r − 1}, let j σ : Sing σ → M 1/r g,n be the map from the singular set of the universal spin curve banded by σ/r where now the local isotropy is Z n . Let B m (x) be the mth Bernoulli polynomial. Chiodo proved the following formula in [10]: The total Chern class of a virtual bundle c(E − F) := c(E)/c(F) can be calculated from its Chern character and in this case is given by The components of M 1/r g,n are given by M 1/r g,n, σ for σ ∈ Z n r . The push-forward of the restriction of c(−R * π * E ) to a component is known as the Chiodo class The sum of this push-forward over all components of M 1/r g,n is expressed as a weighted sum over stable graphs in [33] which encodes a twisted loop group action as described in Section 5, with edge and vertex weights proven in [38, Theorem 4.5] to exactly match the edge and vertex weights arising from the following spectral curve:x where ∼ means the asymptotic expansion in the limith → 0. Hence topological recursion applied to this spectral curve produces correlators with expansion in terms of the local coordinate e −x i = e −x(z i ) = z i e −z r i around z i = 0, where ∼ means expansion in a local coordinate, (− k) r ∈ {0, ..., r − 1} n the residue class of − k modulo r, and We have usedx = z r − log z and y = r 1+ s r s z s here, rather thanx = −z r + log z and y = z s used in [38], because the convention for the kernel K(p 1 , p) used here differs by sign from [38], and also to remove a factor of r 1+ s r s 2−2g−n from the correlators. Chiodo's formula and the asymptotic expansion (42) are true for any s ∈ Z, hence (43) holds for any s ∈ Z, although it is stated only for s ≥ 0 in [38].
In [38] (− k) r ∈ {1, ..., r} n , however replacing k i = r by k i = 0 leaves the Chiodo class invariant since it does not change the component, rather it twists the universal bundle E over the component resulting in adding a direct summand of a trivial bundle to the virtual bundle −R * π * E which does not affect the total Chern class. The invariance of the total Chern class, or equivalently the positive degree terms of the Chern character, can also be seen in Chiodo's formula via properties of the Bernoulli polynomials.
We will use (43) in the case r = 2. Definê which have local expansion at z = 0 given bŷ Each ψ i in the denominator of the right hand side of (43) produces monomials ( 1 2 k i ψ i ) m i , hence (43) with r = 2 becomeŝ ω g,n (z 1 , ..., z n ) = ∑ σ, m M g,n C g,n (2, s; σ) Change (x,ŷ) → (x, y) by The differentials defined in (39) using x are given by Hence where the last equality uses (t/2) ∑ m i = (t/2) 3g−3+n−deg for the degree operator deg c k (E σ g,n ) = k then (t/2) − deg is absorbed into the Chern polynomial. Set s = −1 to get the desired result.

A 2 singularity.
In this section we calculate the spectral curves of the CohFT Ω A 2 and (Ω A 2 ) Θ . We begin with a general result relating the spectral curve of any semisimple CohFT Ω with the spectral curve of Ω BGW .  Given an irregular spectral curve, it is proven in [9] that its partition function is obtained from (27) with translation operator given by (37) Given a semisimple CohFT Ω encoded by the regular spectral curve S = (C, x, y, B), defineŷ = dy dx . Then we see that since dy =ŷdx the translation operator shifts by T(z) α → 1 z T(z) α which proves that Ω BGW is encoded by the spectral curvê S = (C, x,ŷ = dy dx , B). Define the spectral curves (44) The partition functions associated to S = S A 2 defined in (4.1.4) and S = S Θ A 2 are built out of correlators ω S g,n by Z S (h, {t α k }) = exp ∑ g,nh g−1 n! ω S g,n ξ α k (z i )=t α k using the differentials ξ α k (z) defined on C by (45) , α ∈ {1, 2}, k ∈ N.
Proof. By Theorem 5 which is regular in t since rank E σ g,n = 2g − 2 + 1 2 (n + | σ|) so the Chern polynomial has degree at most 2g − 2 + n in t −1 . Hence for | σ| = n lim t→0 (−1) n t 2g−2+n 2 1−g p * c E σ g,n , 2 t = (−1) n 2 g−1+n p * c 2g−2+n E σ g,n = Θ g,n while for | σ| < n, rank E σ g,n < 2g − 2 + n so lim t→0 (−1) n t 2g−2+n 2 1−g p * c E σ g,n , Givental's construction produces Ω A 2 g,n , although it does not know about the degree bound and produces classes in the degrees where Ω A 2 g,n vanishes. This leads to sums of tautological classes representing the zero class, i.e. relations given by the degree d > 1 3 (g − 1 + n) part of the sum over stable graphs in (25)  Since Ω A 2 has flat identity, the push-forward classes in (25) produce κ polynomials, hence only graphs without dilaton leaves in the sum are required and the classes ω R,T,1 1 Γ consist of products of ψ and κ classes associated to each vertex of Γ. The main result of [48] is the construction of elements R d g,A ∈ S g,n for A = (a 1 , ..., a n ), a α ∈ {0, 1} satisfying q(R d g,A ) = 0 which push forward to tautological relations in H 2d (M g,n , Q). They are defined by R d g,A = degree d part of Ω A 2 g,n (v A ) for a basis {v 0 , v 1 }. The element R 1 2 ∈ H 2 (M 2 , Q) is given in Example 3.5. When n ≤ g − 1 and g > 1 we have d = g − 1 > 1 3 (g − 1 + n) hence there exist non-trivial relations R g−1 g,A . This produces the following sum over graphs Θ g,n · R g−1 g,A = 0 which defines a relation, for each A, between intersection numbers of ψ classes with Θ g,n , i.e. coefficients of Z Θ (h, {t k }). This uses Θ g,n · (φ Γ ) * = (φ Γ ) * Θ Γ together with Remark 3.4 to replace κ classes by ψ classes. We saw this in Example 3.5 arising from the genus two Pixton relation (49)  Θ 3,2 · ψ 2 1 and M 3,2 Θ 3,2 · ψ 1 ψ 2 from lower genus coefficients of Z Θ (h, t 0 , t 1 , ...).
Proof. For each g > 1, n and ⌊ n+1 2 ⌋ possible A ∈ {0, 1} n (due to symmetry and vanishing of half for parity reasons), R g−1 g,A = 0 defines a non-trivial Pixton relation. For each of these choices of g, n and A, due to restriction and pull-back properties of Θ g,n as explained above, Θ g,n · R g−1 g,A = 0 defines a relation between coefficients of Z Θ (h, {t k }), such as (49).
The main aim of the proof is to prove that the corresponding coefficients of Z BGW (h, {t k }) also satisfy this infinite set of relations. To do this, we study the partition function Z BGW Ω A 2 , defined in Definition 5.5 via the spectral curve S BGW A 2 defined in (44). The relations between coefficients of Z BGW (h, {t k }) will be stored in the spectral curve. This will produce identical relations satisfied by both the coefficients of Z BGW and Z Θ . To summarise, we have vanishing of certain coefficients of Z Θ A 2 (h, {t α k }) due to the cohomological viewpoint shown in the upper row in Figure 1, and vanishing of corresponding coefficients of Z BGW Ω A 2 (h, {t α k }) due to Givental's construction neatly encoded by topological recursion shown in the lower row in Figure 1.
Pixton relations induce relations between intersection numbers of ψ and κ classes or ψ classes alone, i.e. coefficients of Z KW (h, {t k }). These relations are realised by unexpected vanishing of coefficients of the partition function Z A 2 (h, {t α k }). Similarly, unexpected vanishing of coefficients of the partition function Z BGW A 2 (h, {t α k }) correspond to relations between coefficients of Z BGW (h, {t k }).
The coefficients of log Z BGW A 2 (h, {t α k }) are obtained from the correlators ω BGW,A 2 homogeneous by applying topological recursion to x(z) = z 3 − 3Q 2 z and y = √ −3/x ′ (z) which are homogeneous in z and Q. Then ω A 2 g,n (Q, z 1 , ..., z n ) is homogeneous in z and Q of degree 2 − 2g − n: ω The intersection numbers of Θ g,n stored in log Z Θ (h, t 0 , t 1 , ...) = ∑ g,n, kh g−1 n! M g,n Θ g,n · n ∏ j=1 ψ k j j ∏ t k j are calculated recursively via relations among tautological classes in H * (M g,n , Q). The calculation of these intersection numbers up to genus 2 can be found throughout the text. We assemble them here for convenience, then present the genus 3 calculations. g = 0 Theorem 1 property (II) gives Θ 0,n = 0 which agrees with the vanishing of all genus 0 terms in Z BGW . g = 1 Proposition 2.9 gives Θ 1,1 = 3ψ 1 hence M 1,1 Θ 1,1 = 1 8 . We use this together with the dilaton equation to get M 1,n Θ 1,n = (n−1)! 8 . This agrees with − 1 8 log(1 − t 0 ) in log Z BGW . g = 2 Using Mumford's relation [42] κ 1 = sum of boundary terms in M 2 which coincides with a genus 2 Pixton relation, Example 3.5 produced the genus 2 intersection numbers from the genus 1 intersection numbers.