1 Introduction↩︎

1.1 Structures of topological recursion relations↩︎

Let \(\overline{\mathcal{M}}_{g,n}\) be the moduli space of stable genus-\(g\) curves with \(n\) marked points. The tautological rings \(R^*(\overline{\mathcal{M}}_{g,n})\) may be defined simultaneously for all \(g,n\geq0\) with \(2g-2+n>0\), as the smallest system of subrings of the Chow ring \(A^*(\overline{\mathcal{M}}_{g,n})\) of the moduli spaces of curves closed under pullback via the forgetful morphisms \(\pi\colon \overline{\mathcal{M}}_{g,n+1} \rightarrow \overline{\mathcal{M}}_{g,n}\) and the gluing morphisms \(\iota_{g_1,g_2}\colon \overline{\mathcal{M}}_{g_1,n_1+1} \times \overline{\mathcal{M}}_{g_2,n_2+1} \rightarrow \overline{\mathcal{M}}_{g_1+g_2, n_1+n_2},\) and \(\iota_{g-1}\colon \overline{\mathcal{M}}_{g,n+2} \rightarrow \overline{\mathcal{M}}_{g+1,n}\). This elegant definition is due to Faber and Pandharipande [2], who also proved that \(R^*(\overline{\mathcal{M}}_{g,n})\) admits an explicit set of additive generators which are the images of strata classes in \(\overline{\mathcal{M} }_{g,n}\). These strata classes are formed from three ingredients: \(\psi\) classes, \(\kappa\) classes, and generalized gluing maps corresponding to stable graphs. The \(\psi\) classes are \(\psi_i=c_1(s_i^*(\omega_{\pi})), \; i=1, \ldots, n\) and the \(\kappa\) classes are \(\kappa_d = \pi_*(\psi_{n+1}^{d+1}), \; d \geq 0\), where \(\pi\colon \overline{\mathcal{M}}_{g,n+1} \rightarrow \overline{\mathcal{M}}_{g,n}\) forgets the last marked point, \(s_i\) is the section of \(\pi\) given by the \(i\)-th marked point and \(\omega_{\pi}\) denotes the relative dualizing sheaf.

In the study of the tautological ring, one of the most important problems is to determine the tautological relations: the linear relations among the strata classes. In this paper, we focus on linear equations only involving \(\psi\) classes and boundary classes, which we refer to as topological recursion relations.

It is well known that such topological recursion relations produce universal equations for Gromov–Witten invariants. Such equations play important role in studying properties for Gromov–Witten invariants and its associated integrable systems. For example they play a crucial role in the study of the low genus Virasoro conjecture (cf.[3]) and can be used to analyze the second Poisson bracket of Dubrovin-Zhang hierarchy (cf. [4], [5]).

The lower the degree of topological recusion relations, the more powerful are the corresponding universal equations. From the results in [6], [2], and [7], we know that for every \(g\), there exist tautological relations that express any degree \(g\) polynomial in \(\psi\) classes on \(\overline{\mathcal{M}}_{g,n}\) in terms of tautological classes supported on the boundary \(\overline{\mathcal{M}}_{g,n} \setminus \mathcal{M}_{g,n}\). Recently, in [8], with E. Clader and D. Zakharov, we significantly strengthened this result, and proved the existence of topological recursion relations expressing any degree \(g\) polynomial in \(\psi\) classes on \(\overline{\mathcal{M}}_{g,n}\) in terms of boundary classes. The proof is constructive, and gives an algorithm to compute its coefficients. Degree \(g\) is the smallest for which such a set of relations can exist.

At this point, all explicit topological recursion relations have genus \(g \leq 4\). Moreover while the genus \(0\) WDVV equations are simple, starting from Getzler’s equation in genus \(1\), the known topological recursion relations are very complicated (cf.[9], [10], [11], [1], [12] and [13]).

In this paper, we study basic structures of degree-\(g\) topological recursion relations in arbitrary genus.

Our first result concerns the coefficient of the bouquet class on \(\overline{\mathcal{M}}_{g,n}\) in topological recursion relations, that is the class \(\left(\xi_{\Gamma^\mathrm{loop}_{0,g,[n]}}\right)_*(1)\), where \(\Gamma^\mathrm{loop}_{0,g,[n]}\) is the stable graph with only one genus-0 vertex equipped with \(g\) loops and all the markings.

Theorem 1 gives an affirmative answer to a conjecture about the coefficients of the bouquet class in topological recursion relations proposed by T. Kimura and X. Liu in [1]. In fact, the conjecture about the coefficients of the bouquet class for \(n=1\) was proposed in [1] and later generalized to arbitrary \(n\) in [14].

It is also very interesting to compute the coefficients of topological recursion relations corresponding to strata of curves with rational tails. In general, this is complicated since there are many such coefficients. However, we do have the following linear relations among these coefficients:

1.2 Uniqueness of the coefficients in topological recursion relations↩︎

Theorems 1 and 2 prove the existence of topological recursion relations of specific shape. It is an interesting question how unique such relations are. This would give much further constraints on the structures of topological recursion relations.

While in general, we expect there to be many topological recursion relations of the shape of Theorems 1 and 2, we have the following expectations and results about uniqueness.

In Theorem 13, we give some evidence to this conjecture.

The coefficients corresponding to strata of curves with rational tails have a very interesting close relationship to Faber’s intersection number conjecture (cf. [15]), which was proved in [16], [17], [18]. For \(n\geq3\), the coefficients corresponding to strata of curves with rational tails are in general not unique. This can be seen via Belorousski-Pandharipande relation in \(R^2(\overline{M}_{2,3})\) (cf. [11]). For \(n\geq3\), motivated by Faber’s intersection number conjecture, we expect, in the expression of topological recursion relation \(\prod_{i=1}^{n}{\psi_{i}}^{k_i}=\text{boundary classes}\), there should be a common factor \(\frac{1}{\prod_{i=1}^{n}(2k_i-1)!!}\) in the coefficients of all rational tails loci. As we at the moment lack explicit formulae of these coefficients, we postpone the study of this interesting problem.

1.3 Universal equations for Gromov–Witten invariants↩︎

Let \(X\) be a smooth projective variety and \(\{\phi_\alpha: \alpha=1,\dots,N\}\) be a basis of its cohomology ring \(H^*(X;\mathbb{C})\) with \(\phi_1=\mathbf{1}\) the identity. Recall that the big phase space for Gromov–Witten invariants of \(X\) is defined to be \(\prod_{n=0}^{\infty}H^*(X;\mathbb{C})\) with standard basis \(\{\tau_{n}(\phi_\alpha): \alpha=1,\dots,N, n\geq0\}\). Let \(\{t_n^\alpha\}\) be the coordinates on the big phase space with respect to the standard basis. Let \(F_g\) be the genus-\(g\) generating function, which is a formal power series of \(\mathbf{t}=(t_n^\alpha)\) with coefficients being the genus-\(g\) Gromov–Witten invariants (see Section 5 for details). Denote \(\langle\langle\tau_{n_1}(\phi_{\alpha_1}),\dots,\tau_{n_k}(\phi_{\alpha_k})\rangle\rangle_g\) for the derivatives of \(F_g\) with respect to the variables \(t_{n_1}^{\alpha_1},\dots,t_{n_k}^{\alpha_k}\). More generally, the \(k\)-th covariant derivative of the generating functions \(F_g\) with respect to the trivial connection on the big phase space is denoted by tensors \(\langle\langle W_1\ldots W_k\rangle\rangle_{g}\), where \(W_1, \dots, W_k\) are vector fields. For convenience, we identify \(\tau_n(\phi_\alpha)\) with the coordinate vector field \(\frac{\partial}{\partial t_n^\alpha}\) on \(\prod_{n=0}^{\infty}H^*(X;\mathbb{C})\) for \(n\geq0\). If \(n<0\), \(\tau_n(\phi_\alpha)\) is understood to be the \(0\) vector field. We also abbreviate \(\tau_0(\phi_\alpha)\) by \(\phi_\alpha\). Define \(\eta=(\eta_{\alpha\beta})\) to be the matrix of the intersection pairing on \(H^{*}(X;\mathbb{C})\) in the basis \(\{\phi_{1},\dots,\phi_{N}\}\). We will use \(\eta=(\eta_{\alpha\beta})\) and \(\eta^{-1}=(\eta^{\alpha\beta})\) to lower and raise indices, for example \(\phi^{\alpha}:=\eta^{\alpha\beta}\phi_{\beta}\) for any \(\alpha\). Here we use the summation convention that repeated indices should be summed over their entire ranges.

For any vector fields \(W_{1}\) and \(W_{2}\) on the big phase space, the quantum product of \(W_{1}\) and \(W_{2}\) is defined by \[W_{1}\bullet W_{2}:=\sum_{\alpha}\langle\langle{W_{1}W_{2} \phi^{\alpha}}\rangle\rangle_0\phi_{\alpha}.\] Define the operator \(T\) on the space of vector fields by \[T(W)=\tau_{+}(W)-\sum_{\alpha}\langle\langle W\phi^\alpha\rangle\rangle_0\phi_\alpha\] where \(\tau_{+}\) is the operator which increases the level of descendants by one, i.e.\(\tau_+(\tau_k(\phi)) = \tau_{k + 1}(\phi)\). The operator \(T\) is very useful for the translating topological recursion relations into universal equations (cf.[19]). It is well known that such universal equations play an important role in computing higher genus Gromov–Witten invariants and studying the famous Virasoro conjecture (cf.[3]). It is also conjectured that all such equations can determine all higher genus Gromov–Witten invariants in terms of genus-0 Gromov–Witten invariants in the semisimple case (cf.[19]), which differs from the approach of Givental (cf.[20]) or Dubrovin-Zhang (cf.[21]). Thus finding as explicit as possible formulas for such universal equations is very important, even though they are very complicated in general.

As a byproduct of Theorem 1, we have

For a function \(f\) on the big phase space, we say \(f\overset{g}{\approx}0\) if \(f\) can be expressed as a polynomial function of \(F_0,F_1,\dots,F_{g-1}\) and their derivatives. Then Theorem 2 implies:

1.4 Coefficient of the bouquet class in the DR formula of the Hodge class \(\lambda_g\)↩︎

In [22], a nice formula for the top Chern class of Hodge bundle \(\lambda_g\) was found, which is supported on the divisor of curves with a nonseparating node: \[\begin{align} \lambda_g=\frac{(-1)^g}{2^g} \mathcal{D}_{g,n}^{g}(0,\dots,0) \in R^{g}(\overline{\mathcal{M}}_{g,n}) \end{align}\] where \(\mathcal{D}_{g,n}(a_1,\dots,a_n)\) is the Pixton’s formula for double ramification cycles defined in Section 2.2. As an application of Theorem 1, we may compute the coefficient of the bouquet class in this formula for \(\lambda_g\).

1.5 Intersection numbers on moduli space of curves↩︎

The Gromov–Witten theory of a point, or equivalently, the intersection numbers of the moduli space of stable curves has been known since Kontsevich’s famous proof of Witten’s conjecture ([23]). There are two parts in Kontsevich’s original proof. First, he reduced the geometric problem to a combinatorial problem, using a topological cell decomposition of the moduli space of curves to derive formulae for integrals of tautological \(\psi\) classes. Then he derived a matrix integral formula for these expressions, and used this to prove Witten’s conjecture.

The degree \(3g-3+n\) topological recursion relations on \(\overline{\mathcal{M}}_{g,n}\), applied to a point, give a new way to do the first part of this procedure; that is we find a new combinatorial recursive formula for integrals of tautological \(\psi\) classes on the moduli space of curves.

In principle, by induction on \(g\) and \(n\), this theorem provides us a combinatorial recursion formula for computing the intersection numbers \(\int_{\overline{\mathcal{M}}_{g,n}} \prod_{i=1}^{n}{\psi_i}^{k_i}\) for any \(2g-2+n>0\) from initial condition \(\int_{\overline{\mathcal{M}}_{0,3}}1=1\) and \(\int_{\overline{\mathcal{M}}_{1,1}}\psi_1=\frac{1}{24}\). The techniques we use are purely combinatorial from Pixton’s double ramification relation, and thus we obtain a very different approach from Kontsevich’s topological model.

It would be very interesting to try to recover Witten’s conjecture from the algorithm in Theorem 16.

This paper is organised as follows: In Section 2, we review the tautological ring and the algorithm from [8] for computing topological recursion relations. In Section 3, we study the coefficients of the bouquet class in topological recursion relations and prove Theorem 1. In Section 4, we study the locus of curves with rational tails in topological recursion relations and prove Theorem 2. In Section 5, we give some direct applications in Gromov–Witten theory. In Section 6, we compute the coefficient of the bouquet class in the DR formula for the top Hodge class \(\lambda_g\). In Section 7, we propose a recursive algorithm for computing intersection numbers on the moduli space of stable curves.

Acknowledgements↩︎

The authors would like to thank E. Clader and D. Zakharov for the collaboration [8] and for discussions related to this work. They also thank the anonymous referee for carefully reading the manuscript, and for making helpful suggestions to improve the clarity of this paper. They are especially grateful to X. Liu for sharing his generalized conjecture on the coefficient of the bouquet class. The first author was partially supported by NSF grants DMS-2054830 and DMS-2239320. The second author was partially supported by National Science Foundation of China (Grant No. 12071255) and Shandong Provincial Natural Science Foundation (Grant No. ZR2021MA101).

2 The tautological ring and topological recursion relations↩︎

2.1 The strata algebra and the tautological ring of \(\overline{\mathcal{M}}_{g,n}\)↩︎

The additive generators of the tautological ring of \(\overline{\mathcal{M}}_{g,n}\) are described by the strata algebra. In this section, we review basic definitions following [8], and refer the reader to [24] and [25] for more details.

A stable graph \(\Gamma=(V,H,E,L,g,p,\iota,m)\) of genus \(g\) with \(n\) legs consists of the following data:

1. a finite set of vertices \(V\) with a genus function \(g\colon V\to \mathbb{Z}_{\geq 0}\);

2. a finite set of half-edges \(H\) with a vertex assignment \(p\colon H\to V\) and an involution \(\iota\colon H\to H\);

3. a set of edges \(E\), which is the set of two-point orbits of \(\iota\);

4. a set of legs \(L\), which is the set of fixed points of \(\iota\), and which is marked by a bijection \(m\colon\{1,\ldots,n\}\to L\).

5. The graph \((V,E)\) is connected.

6. For every vertex \(v\in V\), we have \[2g(v)-2+n(v) > 0,\] where \(n(v)=| p^{-1}(v)|\) is the valence of the vertex \(v\)

7. The genus of the graph is \(g\), in the sense that \[\label{eq:genus} g=h^1(\Gamma)+\sum_{v\in V}g(v),\tag{1}\] where \(h^1(\Gamma)= |E|- |V|+1\).

An automorphism of a stable graph \(\Gamma\) consists of permutations of the sets \(V\) and \(H\) which leave invariant the structures \(g\), \(p\), \(\iota\) and \(m\) (and hence preserve \(L\) and \(E\)). We denote by \(\mathop{\mathrm{Aut}}(\Gamma)\) the automorphism group of \(\Gamma\).

To each stable graph \(\Gamma\), let \[\overline{\mathcal{M}}_{\Gamma}:=\prod_{v\in V}\overline{\mathcal{M}}_{g(v),n(v)}.\] There is a generalized gluing map \[\label{eq:xi} \xi_{\Gamma}\colon\overline{\mathcal{M}}_{\Gamma}\to \overline{\mathcal{M}}_{g,n},\tag{2}\] whose image is the locus in \(\overline{\mathcal{M}}_{g,n}\) having generic point corresponding to a curve with stable graph \(\Gamma\). To construct \(\xi_{\Gamma}\), a family of stable pointed curves over \(\overline{\mathcal{M}}_{\Gamma}\) is required. Such a family is easily defined by attaching the pull-backs of the universal families over each of the \(\overline{\mathcal{M}}_{g(v),n(v)}\) along the sections corresponding to half-edges. The degree of \(\xi_{\Gamma}\), as a map of Deligne–Mumford stacks, is equal to \(|\mathop{\mathrm{Aut}}(\Gamma)|\).

Additive generators of the tautological ring can be described in terms of certain decorations on stable graphs \(\Gamma\). Namely, let \[\gamma=(x_i:V\to \mathbb{Z}_{\geq 0},y:H\to \mathbb{Z}_{\geq 0})\] be a collection of functions such that \[d(\gamma_v)=\sum_{i>0}ix_i[v]+\sum_{h\in p^{-1}(v)} y[h]\leq 3g(v)-3+n(v)\] for all \(v\in V\). Then, for each \(v\), define \[\gamma_v=\prod_{i>0}\kappa_i^{x_i[v]}\prod_{h\in p^{-1}(v)} \psi_h^{y[h]}\in A^{d(\gamma_v)}(\overline{\mathcal{M}}_{g(v),n(v)}).\] Associated to any such choice of decorations \(\gamma\), there is a basic class on \(\overline{\mathcal{M}}_{\Gamma}\), also denoted by \(\gamma\), defined by \[\gamma=\prod_{v\in V}\gamma_v \in A^{d(\gamma)}(\overline{\mathcal{M}}_{\Gamma}).\] Here, the degree is \(d(\gamma):=\sum_{v\in V}d(\gamma_v)\), and we abuse notation slightly by using a product over classes on the vertex moduli spaces to denote a class on \(\overline{\mathcal{M}}_{\Gamma}\).

The strata algebra \(\mathcal{S}_{g,n}\) is defined to be the finite dimensional \(\mathbb{Q}\)-vector space spanned by isomorphism classes of pairs \([\Gamma,\gamma]\), where \(\Gamma\) is a stable graph of genus \(g\) with \(n\) legs and \(\gamma\) is a basic class on \(\overline{\mathcal{M}}_{\Gamma}\). The product is defined by intersection theory (see [26]). More precisely, for \([\Gamma_1,\gamma_1], [\Gamma_2,\gamma_2]\in \mathcal{S}_{g,n}\), the fiber product of \(\xi_{\Gamma_1}\) and \(\xi_{\Gamma_2}\) over \(\overline{\mathcal{M}}_{g,n}\) is a disjoint union of \(\xi_{\Gamma}\) over all graphs \(\Gamma\) having edge set \(E=E_1\cup E_2\), such that \(\Gamma_1\) is obtained by contracting all edges outside of \(E_1\) and \(\Gamma_2\) is obtained by contracting all edges outside of \(E_2\). We then define the product by \[\begin{align} \label{eqn:stra-prod} [\Gamma_1,\gamma_1]\cdot [\Gamma_2,\gamma_2]=\sum_{\Gamma}\left[\Gamma,\gamma_1\gamma_2\prod_{(h,h')\in E_1\cap E_2}(-\psi_h-\psi_{h'})\right]. \end{align}\tag{3}\]

Pushing forward elements of the strata algebra along the gluing maps 2 defines a ring homomorphism \[\begin{align} &q\colon\mathcal{S}_{g,n}\to A^*(\overline{\mathcal{M}}_{g,n})\\ &q([\Gamma,\gamma])=\xi_{\Gamma*}(\gamma), \end{align}\] and the image of \(q\) is precisely the tautological ring \(R^*(\overline{\mathcal{M}}_{g,n})\). Elements of \(\mathcal{S}_{g,n}\) in the kernel of \(q\) are referred to as tautological relations. Let \(\mathcal{S}^{trr}_{g,n}\subset \mathcal{S}_{g,n}\) denote the subalgebra spanned by classes which do not have any \(\kappa\) classes. Then the elements of \(\mathcal{S}^{trr}_{g,n}\) in the kernel of \(q\) are referred to as topological recursion relations.

For each basis element \([\Gamma,\gamma]\), we define its degree as \[\deg[\Gamma,\gamma]=|E|+d(\gamma).\] Since product 3 preserves the degree, then \(\mathcal{S}_{g,n}\) is a graded ring: \[\mathcal{S}_{g,n}=\bigoplus_{d=0}^{3g-3+n}\mathcal{S}_{g,n}^d.\]

2.2 Pixton’s DR formula↩︎

Fix \(g\) and \(n\), and fix a collection of integers \(A = (a_1, \ldots, a_n)\) such that \(\sum_j a_j =0\). In this subsection, we recall the definition of Pixton’s class, which is an inhomogeneous element of \(\mathcal{S}_{g,n}\) depending on \(A\).

For our purpose, we first define auxiliary classes \(\widetilde{\mathcal{D}}_{g,n}^r\), for an additional positive integer parameter \(r\). For a stable graph \(\Gamma=(V,H,g,p,\iota)\) of genus \(g\) with \(n\) legs, a weighting modulo \(r\) on \(\Gamma\) is defined to be a map \[w\colon H \to \{0, \dotsc, r - 1\}\] satisfying three properties:

1. For any \(i \in \{1, \dotsc, n\}\) corresponding to a leg \(\ell_i\) of \(\Gamma\), we have \(w(\ell_i) \equiv a_i \pmod{r}\).

2. For any edge \(e\in E\) corresponding to two half-edges \(h, h'\in H\), we have \(w(h) + w(h') \equiv 0 \pmod{r}\).

3. For any vertex \(v \in V\), we have \(\sum_{h\in p^{-1}(v)} w(h) \equiv 0 \pmod{r}\).

Define \(\widetilde{\mathcal{D}}_{g,n}^r\) to be the class \[\sum_{\Gamma\in G_{g,n}} \frac{1}{|\mathop{\mathrm{Aut}}(\Gamma)|} \frac{1}{r^{h^1(\Gamma)}} \sum_{\substack{w \text{ weighting }\\ \text{mod } r \text{ on }\Gamma}} \left[\Gamma, \prod_{i=1}^n e^{\frac{1}{2} a_i^2\psi_i} \prod_{(h,h')\in E} \frac{1 - e^{-\frac{1}{2} w(h)w(h')(\psi_h + \psi_{h'})}}{\psi_h + \psi_{h'}}\right] \in \mathcal{S}_{g,n}.\]

Pixton proved the class \(\widetilde{\mathcal{D}}_{g,n}^r\) is a polynomial in \(r\) for \(r\) sufficiently large (see [27]). Then Pixton’s class is defined to be the constant term of this polynomial. Furthermore, by the fact that \[a_1 = -(a_2 + \cdots +a_n),\] Pixton’s class can be expressed in terms of the variables \(\{a_i\}_{i=2}^{n}\), then we denote it by \[\mathcal{D}_{g,n}(a_2, \ldots, a_n),\] and its degree \(d\) component is denoted by \(\mathcal{D}^d_{g,n}(a_2, \ldots, a_n)\). The following statement was conjectured by Pixton and proved in [28]:

2.3 Polynomiality properties↩︎

In fact, Pixton proved the following result about the polynomiality of \(\mathcal{D}_{g,n}(a_2, \ldots, a_n)\), which Spelier provided an alternative proof:

Combining Theorems 6 and 7, the coefficient of any monomial \(a_2^{b_2} \cdots a_n^{b_n}\) in the class \(\mathcal{D}_{g,n}^d(a_2, \ldots, a_n)\) yields a tautological relation in \(R^d(\overline{\mathcal{M}}_{g,n})\), for each \(d > g\): \[q\left( \bigg[\mathcal{D}_{g,n}^d(a_2, \ldots, a_n)\bigg]_{a_2^{b_2} \cdots a_n^{b_n}} \right) = 0.\]

2.4 An algorithm for deriving topological recursion relations on \(\overline{\mathcal{M}}_{g,n}\)↩︎

First, we recall the algorithm for computing general degree-\(g\) topological recursion relations on \(\overline{\mathcal{M}}_{g,n}\) from [8].

Fix a genus \(g\) and a number of marked points \(n\). Let \(M\) be a monomial of degree \(D\leq 2g+1\) in the variables \(a_2, \ldots, a_n\), and let \[\begin{align} N:= n+2g+2-D. \end{align}\] Define \[\Omega_{g,M}^{\text{pre}}=\left[\mathcal{D}^{g+1}_{g,N}(a_2, \ldots, a_N)\right]_{M\cdot a_{n+1}\dots a_{N}} \in \mathcal{S}^{g+1}_{g,N}\] to be the coefficient of the monomial \(M \cdot a_{n+1} \cdots a_N\) in Pixton’s class \(\mathcal{D}^{g+1}_{g,N}(a_2, \ldots, a_N)\). Note that we use here the polynomiality discussed in Section 2.3. Let \[\Pi\colon \overline{\mathcal{M}}_{g,N}=\overline{\mathcal{M}}_{g,n+1+(2g+1-D)} \rightarrow \overline{\mathcal{M}}_{g,n+1}\] be the forgetful map, and define \[\Omega_{g,M}:= \Pi_*(\Omega_{g,M}^{\text{pre}} \cdot \psi_{n+2} \cdots \psi_{N}) \in \mathcal{S}_{g,n+1}^{g+1},\] where we use \(\Pi_*\) to denote the induced map on strata algebras. Via the dilaton equation and string equation, pushing-forward \(\Omega_{g,M}\) along the forgetful map \(\pi\colon \overline{\mathcal{M}}_{g,n+1} \rightarrow \overline{\mathcal{M}}_{g,n}\), we obtain a degree-\(g\) topological recursion relation.

Topological recursion relations which express any specific degree \(g\) monomial \(\psi_1^k \prod_{j=2}^n\psi_j^{k_j}\) of \(\psi\) classes in terms of boundary classes, are obtained recursively in terms of linear combinations of the above topological recursion relations. To describe them explicitly, we define an order on the set of monomials \(\{\psi_1^k \prod_{j=2}^n \psi_i^{l_j}: k+\sum_{j=2}^{n}l_j=g\}\): for any monomials \[\label{eq:psimon} M=\psi_1^k \prod_{j=2}^n \psi_i^{l_j},\quad M'=\psi_1^{k'} \prod_{j=2}^{n} \psi_i^{l_j'}\tag{4}\] of degree \(g\) in the \(\psi\) classes on \(\overline{\mathcal{M}}_{g,n}\), we say that the \(\psi\)-monomial \(M'\) is lower than \(M\) if \(l_j' \le l_j\) for \(j \ge 2\) with at least one of this inequalities being strict.

We consider the linear combination on \(\overline{\mathcal{M}}_{g,n}\) \[\label{eq:cancellation} \sum_{d_2, \dotsc, d_n = 0}^1 \frac{\prod_{j = 2}^n (-2l_j)^{1 - d_j}}{(2g + 1 - \sum_{j=2}^{n} (2l_j - d_j))!} \cdot \pi_* \Omega_{g, a_2^{2l_2 - d_2} \cdot \dotsb \cdot a_n^{2l_n - d_n}}=0\tag{5}\] defined when \(l_j \ge 1\) for all \(j\). By [8], up to a constant multiple, the combination 5 gives a topological recursion relation of the form \[\label{eqn:want} \psi_1^{g-\sum_{j=2}^{n} l_j} \prod_{j=2}^n \psi_j^{l_j} +\sum_{(k_2,\dots,k_n)<(l_2,\dots,l_n)}c_{k_2,k_3,\dots,k_n}\psi_1^{g-\sum_{j=2}^{n}k_j} \prod_{j=2}^n \psi_j^{k_j}=\text{boundary terms}\tag{6}\] where \((k_2,\dots,k_n)<(l_2,\dots,l_n)\) refers to the monomials of \(\psi\) classes \(\psi_1^{g-\sum_{j=2}^{n}k_j}\prod_{j=2}^{n}\psi_j^{k_j}\) strictly lower than \(\psi_1^{g-\sum_{j=2}^{n}l_j}\prod_{j=2}^{n}\psi_j^{l_j}\), and all \(c_{k_2,k_3,\dots,k_n}\) are certain rational numbers determined from Equation 5 . Also note that being topological recursion relations, the boundary terms in 6 do not involve any \(\kappa\)-classes. Using 6 repeatedly yields:

2.5 Generating series formula for \(\Omega_{g, M}\)↩︎

In this section, we prove a simplified formula for the coefficients of \(\Omega_{g, M}\).

First, fix integers \((a_2, \dotsc, a_n)\), an integer \(r > 0\), a stable graph \(\Gamma=(V,H,g,p,\iota)\) of genus \(g\) with \(n + 1\) legs, and an additional assignment \(x_v \in \mathbb{Z}\) for every \(v \in V\). Given this, we set \[a_1 = -(a_2 + \dotsb + a_n + \sum_{v \in V} x_v).\] Then, we define a weighting modulo \(r\) on \((\Gamma, x)\) to be a map \[w\colon H \to \{0, \dotsc, r - 1\}\] satisfying three properties:

1. For any \(i \in \{1, \dotsc, n\}\) corresponding to a leg \(\ell_i\) of \(\Gamma\), we have \(w(\ell_i) \equiv a_i \pmod{r}\). We set \(w(\ell_{n + 1}) = 0\).

2. For any edge \(e\in E\) corresponding to two half-edges \(h, h'\in H\), we have \(w(h) + w(h') \equiv 0 \pmod{r}\).

3. For any vertex \(v \in V\), we have \(\sum_{h\in p^{-1}(v)} w(h) \equiv -x_v \pmod{r}\).

Define \(\widetilde{\mathcal{D}}_{\Gamma}^r\) to be the class \[\frac{1}{r^{h^1(\Gamma)}} \sum_{\substack{w \text{ weighting }\\ \text{mod } r \text{ on }\Gamma}} [\Gamma, \gamma_w] \in \mathcal{S}_{g,n + 1},\] where \[\gamma_w = \prod_{i=1}^n e^{\frac{1}{2} a_i^2\psi_i} \prod_{(h,h')\in E} \frac{1 - e^{-\frac{1}{2} w(h)w(h')(\psi_h + \psi_{h'})}}{\psi_h + \psi_{h'}}.\] This class is a polynomial for \(r \gg 0\), and we will use \(\mathcal{D}_{\Gamma}\) to denote the constant part of the resulting polynomial. The class \(\mathcal{D}_{\Gamma}\) is a polynomial in the variables \(a_2, \dotsc, a_n\) and \(x_v\) for \(v \in V\), and so we will denote it by \[\mathcal{D}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}).\]

2.6 The push-forward formula of \(\pi_*(\Omega_{g,M})\)↩︎

We now compute the \(\pi\)-push-forward of the formula \(\Omega_{g,M}\) in Proposition 9 from the previous section.

First, note that given a dual graph \(\Gamma\) on \(\overline{\mathcal{M}}_{g,n}\), there are three types of dual graphs \(\Gamma'\) of \(\overline{\mathcal{M}}_{g,n + 1}\) mapping to it:

1. \(\Gamma'\) is obtained from \(\Gamma\) by adding an extra \((n+1)\)-th leg to one of the vertices of \(\Gamma\).

2. \(\Gamma'\) is obtained from \(\Gamma\) by replacing the \(i\)-th leg by an edge connected a new genus zero vertex with the \(i\)-th and \((n+1)\)-st leg.

3. \(\Gamma'\) is obtained from \(\Gamma\) by dividing an edge into two edges, and putting the \(i\)-th leg on the resulting new genus zero vertex.

Fix integers \((a_2, \dotsc, a_n)\), an integer \(r > 0\), a stable graph \(\Gamma=(V,H,g,p,\iota)\) of genus \(g\) with \(n\) legs and an assignment \(\{x_v \in \mathbb{Z}: v \in V\}\). Given this, we set \[a_1 = -(a_2 + \dotsb + a_n + \sum_{v \in V} x_v).\] Then, we define a weighting modulo \(r\) on \((\Gamma, x)\) to be a map \[w\colon H \to \{0, \dotsc, r - 1\}\] satisfying three properties:

1. For any \(i \in \{1, \dotsc, n\}\) corresponding to a leg \(\ell_i\) of \(\Gamma\), we have \(w(\ell_i) \equiv a_i \pmod{r}\).

2. For any edge \(e\in E\) corresponding to two half-edges \(h, h'\in H\), we have \(w(h) + w(h') \equiv 0 \pmod{r}\).

3. For any vertex \(v \in V\), we have \(\sum_{h\in p^{-1}(v)} w(h) \equiv -x_v \pmod{r}\).

Motivated by three operations from \(\Gamma'\) to \(\Gamma\) as above mentioned , we define three different tautological classes as follows:

1. \(\widetilde{\mathcal{D}}_{\Gamma}^r\{1,v^*\}\) to be the class \[\frac{1}{r^{h^1(\Gamma)}} \sum_{\substack{w \text{ weighting }\\ \text{mod } r \text{ on }(\Gamma,x)}} [\Gamma, \gamma_w\{1,v^*\}] \in \mathcal{S}_{g,n},\] where \[\gamma_w\{1,v^*\} = \left(\sum_{h\in H:p(h)=v^*}\frac{e^{\frac{1}{2} a_h^2\psi_h}-1}{\psi_{h}}\prod_{i=1:i\neq h}^{n} e^{\frac{1}{2} a_i^2\psi_i}\right) \prod_{(h,h')\in E} \frac{1 - e^{-\frac{1}{2} w(h)w(h')(\psi_h + \psi_{h'})}}{\psi_h + \psi_{h'}}.\]

2. \(\widetilde{\mathcal{D}}_{\Gamma}^r\{2,i\}\) to be the class \[\frac{1}{r^{h^1(\Gamma)}} \sum_{\substack{w \text{ weighting }\\ \text{mod } r \text{ on }(\Gamma,x)}} [\Gamma, \gamma_w\{2,i\}] \in \mathcal{S}_{g,n},\] where \[\gamma_w\{2,i\} =\frac{1-e^{\frac{1}{2}(a_i+x_{v^*})^2\psi_i}}{\psi_i}\prod_{j=1: j\neq i}^n e^{\frac{1}{2} a_j^2\psi_j} \prod_{(h,h')\in E} \frac{1 - e^{-\frac{1}{2} w(h)w(h')(\psi_h + \psi_{h'})}}{\psi_h + \psi_{h'}}.\]

3. \(\widetilde{\mathcal{D}}_{\Gamma}^r\{3,e\}\) to be the class \[\frac{1}{r^{h^1(\Gamma)}} \sum_{\substack{w \text{ weighting }\\ \text{mod } r \text{ on }(\Gamma,x)}} [\Gamma, \gamma_w\{3,e\}] \in \mathcal{S}_{g,n},\] where \[\begin{align} \gamma_w\{3,e\}=&\prod_{j=1}^n e^{\frac{1}{2} a_j^2\psi_j} \cdot \frac{1 - e^{-\frac{1}{2} w(h(e))(r-w(h(e)))(\psi_{h(e)} )}}{\psi_{h(e)}} \frac{1 - e^{-\frac{1}{2}(r-w(h'(e)))w(h'(e)) ( \psi_{h'(e)})}}{ \psi_{h'(e)}} \\&\cdot\prod_{(h,h')\in E-\{e\}} \frac{1 - e^{-\frac{1}{2} w(h)w(h')(\psi_h + \psi_{h'})}}{\psi_h + \psi_{h'}} \end{align}\] where on the special edge \(e\), we modify condition (2) to be \[w(h(e))+w(h'(e))\equiv x_{v^*}\mod r.\] The condition (2) is unchanged on the other edges \(E-\{e\}\).

These three kinds of classes are all polynomials for \(r \gg 0\), and we will use \(\mathcal{D}_{\Gamma}^{\{1,v^*\}}\), \(\mathcal{D}_{\Gamma}^{\{2,i\}}\) and \(\mathcal{D}_{\Gamma}^{\{3,e\}}\) to denote the constant part of the resulting polynomials respectively. These classes \(\mathcal{D}^{\{1,v^*\}}_{\Gamma}\), \(\mathcal{D}^{\{2,i\}}_{\Gamma}\) and \(\mathcal{D}^{\{3,e\}}_{\Gamma}\) are all polynomials in the variables \(a_2, \dotsc, a_n\) and \(x_v\) for \(v \in V\cup\{v^*\}\), and so we will denote them respectively by \[\mathcal{D}^{\{1,v^*\}}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}),\,\, \mathcal{D}^{\{2,i\}}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}),\,\, \mathcal{D}^{\{3,e\}}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}).\] Furthermore, let \[\mathcal{D}^{\{1,v^*\}, d}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}),\,\, \mathcal{D}^{\{2,i\}, d}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}),\,\, \mathcal{D}^{\{3,e\}, d}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}),\] be the corresponding degree \(d\) components.

3 The coefficient of the bouquet class in topological recursion relations↩︎

In this section, we compute the coefficient of the bouquet class in the degree-\(g\) topological recursion relations on \(\overline{\mathcal{M}}_{g,n}\). First, we compute contributions of stable graphs to \(\pi_{*}\Omega_{g,n}\), in particular, the trivial graph \(\Gamma_{g,[n]}\) of genus \(g\) with \(n\) markings, and the dual graph \(\Gamma^\mathrm{loop}_{g-1, 1, [n]}\) with a single genus \(g-1\) vertex, a loop and all of the markings. Second, we establish a crucial identity between these two contributions. From there, the algorithm 5 determines the coefficient of the bouquet class inductively, and the identities will lead the proof of Theorem 1.

3.1 Contributions of stable graphs to \(\pi_* \Omega_{g, M}\)↩︎

Consider the trivial graph \(\Gamma_{g,[n]}\) of genus \(g\) with \(n\) markings, and the dual graph \(\Gamma^\mathrm{loop}_{g-1, 1, [n]}\) with a single genus \(g-1\) vertex, a loop and all of the markings. Furthermore, fix a monomial \(M = \psi_2^{b_2} \dots \psi_n^{b_n}\) of degree \(D\). For each of these graphs, we will consider the corresponding contribution to \(\pi_* \Omega_{g, M}\) in Proposition 11, and simplify it.

The contribution of the trivial graph \(\Gamma_{g,[n]}\) is \[\begin{align} &\mathop{\mathrm{Cont}}_{\Gamma_{g,[n]}}\pi_*\Omega_{g,M}\\= & \frac{(2g + 2 - D)!(4g - 1 + n - D)!}{(2g - 2 + n)!} \cdot [\mathcal{D}^{\{1,v^*\}, g+1}_{\Gamma_{g, [n]}}(a_2, \dotsc, a_n, x_v)]_{M \cdot x_v^{2g + 2 - D}} \\ &+\sum_{i=1}^{n}\sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \\&\cdot [\mathcal{D}^{\{2,i\}, g+1}_{\Gamma_{g, [n]}}(a_2, \dotsc, a_n, x_v, x_{v^*})]_{M \cdot x_v^m x_{v^*}^{m^* + 1}} \end{align}\] where \(v\) is the single vertex of \(\Gamma_{g,[n]}\). Note that \(a_1 + x_{v^*} = -\sum_{i = 2}^n a_i - x_v\), and hence does not depend on \(x_{v^*}\). Therefore, the \(i = 1\)-summand in the above expression vanishes. Unfolding the definition of \(\mathcal{D}^{\{2,i\}, g+1}_{\Gamma_{g, [n]}}\), we expand the contribution to \[\begin{align} \label{eqn:cont-g-[n]} \mathop{\mathrm{Cont}}_{\Gamma_{g,[n]}}\pi_*\Omega_{g,M}=&\frac{(2g + 2 - D)!(4g - 1 + n - D)!}{(2g - 2 + n)!} \cdot \left[\Gamma_{g, [n]}, \pi_* \prod_{i=1}^{n} e^{\frac{1}{2} a_i^2\psi_i}\right]^{g + 1}_{M \cdot x_v^{2g + 2 - D}} \nonumber \\&+ \sum_{i=2}^{n}\sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \nonumber\\&\cdot\left[\Gamma_{g, [n]}, \frac{1-e^{\frac{1}{2}(a_i+x_{v^*})^2\psi_i}}{\psi_i}\prod_{j=1: j\neq i}^n e^{\frac{1}{2} a_j^2\psi_j}\right]^{g + 1}_{M \cdot x_v^m x_{v^*}^{m^* + 1}}. \end{align}\tag{10}\]

We now pass to the contribution of the dual graph \(\Gamma^\mathrm{loop}_{g-1, 1, [n]}\). It is given by \[\begin{align} &\mathop{\mathrm{Cont}}_{\Gamma^\mathrm{loop}_{g-1,1,[n]}}\pi_*\Omega_{g,M}\\=& \frac{(2g + 2 - D)!(4g - 1 + n - D)!}{2 \cdot (2g - 2 + n)!} \cdot [\mathcal{D}^{\{1,v^*\}, g+1}_{\Gamma^\mathrm{loop}_{g-1, 1, [n]}}(a_2, \dotsc, a_n, x_v)]_{M \cdot x_v^{2g + 2 - D}} \\ &+\sum_{i=1}^{n}\sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{2 \cdot (2g - 3 + n)!} \\&\cdot [\mathcal{D}^{\{2,i\}, g+1}_{\Gamma^\mathrm{loop}_{g-1, 1, [n]}}(a_2, \dotsc, a_n, x_v, x_{v^*})]_{M \cdot x_v^m x_{v^*}^{m^* + 1}} \\ &+ \sum_{m + m^* = 2g + 1 - D} \frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{2 \cdot (2g - 3 + n)!} \\&\cdot[\mathcal{D}^{\{3,e\}, g+1}_{\Gamma^\mathrm{loop}_{g-1, 1, [n]}}(a_2, \dotsc, a_n, x_v, x_{v^*})]_{M \cdot x_{v^*}^{m^* + 1} x_v^{m}}. \end{align}\] Note that the first two lines vanish by Remark 10. Unfolding the definition of \(\mathcal{D}^{\{3,e\}, g+1}_{\Gamma^\mathrm{loop}_{g-1, 1, [n]}}\), we expand the remaining contribution to \[\begin{align} \label{eqn:cont-g-1-[n]-0} &\mathop{\mathrm{Cont}}_{\Gamma^\mathrm{loop}_{g-1,1,[n]}}\pi_*\Omega_{g,M} \nonumber\\=&\sum_{m+m^*=2g+1-D} \frac{(m^*+1)!(2g+1-D)!(2g-3+n+m)!}{2(2g-3+n)!} \nonumber\\&\cdot\Bigg[\xi_{\Gamma^\mathrm{loop}_{g-1,1,[n]}},\mathop{\mathrm{Coeff}}_{r^0}\frac{1}{r}\sum_{\substack{w_1,w_2=0\\w_1+w_2\equiv x_{v^*} \mod r}}^{r-1}\Bigg[\exp\Big(\frac{1}{2}(\sum_{j=2}^{n}a_j+x_v+x_{v^*})^2\psi_1+\frac{1}{2}\sum_{j=2}^{n}a_j^2\psi_j\Big) \nonumber\\&\cdot\frac{1-e^{\frac{1}{2}w_1^2\psi_{h(e)}}}{\psi_{h(e)}} \frac{1-e^{\frac{1}{2}w_2^2\psi_{h'(e)}}}{\psi_{h'(e)}}\Bigg]_{M\cdot x_v^{m}x_{v^*}^{m^*+1}}\Bigg]. \end{align}\tag{11}\] To simplify further, we use the following:

Then by using Lemma 1 and degree considerations, equation 11 becomes: \[\begin{align} \label{eqn:cont-ga-g-1[n]0} &\mathop{\mathrm{Cont}}_{\Gamma^\mathrm{loop}_{g-1,1,[n]}}\pi_*\Omega_{g,M} \nonumber\\=&\sum_{m+m^*=2g+1-D} \frac{(m^*+1)!(2g+1-D)!(2g-3+n+m)!}{2(2g-3+n)!} \nonumber\\&\cdot\Bigg[\xi_{\Gamma^\mathrm{loop}_{g-1,1,[n]}},\mathop{\mathrm{Coeff}}_{r^0}\frac{1}{r}\sum_{\substack{w_1,w_2=0\\w_1+w_2\equiv x_{v^*} \mod r}}^{r-1}\Bigg[\exp\Big(\frac{1}{2}(\sum_{j=2}^{n}a_j+x_v+x_{v^*})^2\psi_1+\frac{1}{2}\sum_{j=2}^{n}a_j^2\psi_j\Big) \nonumber\\&\cdot\sum_{d_1,d_2=0}^{\infty}\frac{w_1^{2d_1+2}w_2^{2d_2+2}\psi^{d_1}(\psi')^{d_2}}{2^{d_1+d_2+2}(d_1+1)!(d_2+1)!}\Bigg]_{M\cdot x_v^{m}x_{v^*}^{m^*+1}}\Bigg] \nonumber\\=&-\sum_{m+m^*=2g+1-D} \frac{(m^*+1)!(2g+1-D)!(2g-3+n+m)!}{2(2g-3+n)!} \nonumber\\&\cdot\Bigg[\xi_{\Gamma^\mathrm{loop}_{g-1,1,[n]}},\Bigg[\exp\Big(\frac{1}{2}(\sum_{j=2}^{n}a_j+x_v+x_{v^*})^2\psi_1+\frac{1}{2}\sum_{j=2}^{n}a_j^2\psi_j\Big) \nonumber\\&\cdot\sum_{d_1,d_2=0}^{\infty}\frac{(2d_1+1)!!(2d_2+1)!!(x_{v^*})^{2d_1+2d_2+4}\psi^{d_1}(\psi')^{d_2}}{(2d_1+2d_2+4)!}\Bigg]_{M\cdot x_v^{m}x_{v^*}^{m^*+1}}\Bigg]. \end{align}\tag{12}\]

3.2 Useful identities↩︎

We collect several useful lemmas, which will be used repeatedly in the proof of Theorem 1.

We will use the following to simplify some of the sums over \(m\) and \(m^*\) appearing in the contributions.

3.3 A strengthening of Theorem 1↩︎

We formulate a strengthening of both Conjecture 3 and Theorem 1 using two new ingredients:

• Define a linear form \(\mathcal{F}\) on the space of polynomials of \(\psi\) classes via \[\mathcal{F}(\prod_{i=1}^{n}\psi_i^{k_i}) = \prod_{i=1}^{n}\frac{1}{(2k_i + 1)!!}.\]

• For any \(h \le g\), we let \(\Gamma^\mathrm{loop}_{h, g-h, [n]}\) be the graph with a single vertex of genus \(h\) containing all markings and \(g-h\) loops. We call these bouquet type graphs. As a special case \(\Gamma^\mathrm{loop}_{g, 0, [n]} = \Gamma_{g, [n]}\)

Below in Section 3.5, we will discuss the proof of this theorem.

We now note that Theorem 1 is a consequence of Theorem 13: Corollary 3 yields topological recursion relations of the same shape as in Theorem 1, and from there Theorem 13 implies the correct coefficient of the bouquet class.

3.4 Applying \(\mathcal{F}\)↩︎

In this subsection, we collect useful identities related to applying the operator \(\mathcal{F}\). To start, we note the identities: \[\mathcal{F} e^{\frac{1}{2} a^2\psi} = \sum_{d = 0}^\infty \frac{a^{2d}}{(d)! 2^d (2d + 1)!!} = \sum_{d = 0}^\infty \frac{a^{2d}}{(2d + 1)!} = \frac{\sinh(a)}{a},\] \[\mathcal{F} \frac{e^{\frac{1}{2} a^2\psi} - 1}{\psi} = \sum_{d = 1}^\infty \frac{a^{2d}}{(d)! 2^d (2d - 1)!!} = \sum_{d = 1}^\infty \frac{a^{2d}}{(2d)!} = \cosh(a) - 1.\]

We now apply \(\mathcal{F}\) to the contributions to \(\pi_* \Omega_{g, M}\). For the trivial graph \(\Gamma_{g,[n]}\), using 10 , we obtain \[\begin{align} &\mathcal{F}\mathop{\mathrm{Cont}}_{\Gamma_{g,[n]}}\pi_*\Omega_{g,M}\\=&\frac{(2g + 2 - D)!(4g - 1 + n - D)!}{(2g - 2 + n)!} \cdot \left[\sum_{i = 1}^n \left(\sum_{d = 1}^\infty \frac{a_i^{2d}}{(2d)!}\right) \prod_{j = 1: j \neq i}^n \left(\sum_{d = 0}^\infty \frac{a_j^{2d}}{(2d + 1)!}\right) \right]_{M \cdot x_v^{2g + 2 - D}} \\ -& \sum_{i=2}^{n}\sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{(a_i + x_{v^*})^{2d}}{(2d)!}\right) \prod_{j=1: j\neq i}^n \left(\sum_{d = 0}^\infty \frac{a_j^{2d}}{(2d + 1)!}\right)\right]_{M \cdot x_v^m x_{v^*}^{m^* + 1}} \\ &:= A + \sum_{i = 2}^n B_i - \sum_{i = 2}^n C_i, \end{align}\] where \(A\) and \(B_i\) are the \(i = 1\) and \(i \ge 2\) summands in the first term, and the \(C_i\) are the summands in the second term. We note that there is a part of \(C_i\) which cancels with \(B_i\): \[\begin{align} &\sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{a_i^{2d}}{(2d)!}\right) \prod_{j=1: j\neq i}^n \left(\sum_{d = 0}^\infty \frac{a_j^{2d}}{(2d + 1)!}\right)\right]_{M \cdot x_v^m x_{v^*}^{m^* + 1}} \\ = &\sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \binom{2g + 2 - D}{m^* + 1} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{a_i^{2d}}{(2d)!}\right) \left(\sum_{d = 0}^\infty \frac{(a + x_v)^{2d}}{(2d + 1)!}\right) \prod_{j=2: j\neq i}^n \left(\sum_{d = 0}^\infty \frac{a_j^{2d}}{(2d + 1)!}\right)\right]_{M \cdot x_v^{2g + 2 - D}} \\ = &\frac{(2g + 2 - D)!(4g - 1 + n - D)!}{(2g - 2 + n)!} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{a_i^{2d}}{(2d)!}\right) \left(\sum_{d = 0}^\infty \frac{(a + x_v)^{2d}}{(2d + 1)!}\right) \prod_{j=2: j\neq i}^n \left(\sum_{d = 0}^\infty \frac{a_j^{2d}}{(2d + 1)!}\right)\right]_{M \cdot x_v^{2g + 2 - D}} \\ = & B_i, \end{align}\] where \(a:=\sum_{j=2}^{n}a_j\) and we used Lemma 4 in the second equality.

Using Lemma 4, we may also simplify the remaining terms of \(C_i\): \[\begin{align} &\sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{(a_i + x_{v^*})^{2d} - a_i^{2d}}{(2d)!}\right) \prod_{j=1: j\neq i}^n \frac{\sinh(a_j)}{a_j}\right]_{M \cdot x_v^m x_{v^*}^{m^* + 1}} \\ =&\sum_{e = 1}^\infty \sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{a_i^{2d - e}}{(2d - e)! e!}\right) \prod_{j=1: j\neq i}^n \frac{\sinh(a_j)}{a_j}\right]_{M \cdot x_v^m x_{v^*}^{m^* + 1 - e}} \\ =&\sum_{e = 1}^\infty \sum_{m + m^* = 2g + 1 - D}\frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{(2g - 3 + n)!} \binom{2g + 2 - D - e}{m^* + 1 - e} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{a_i^{2d - e}}{(2d - e)! e!}\right) \frac{\sinh(a + x_v)}{a + x_v} \prod_{j=2: j\neq i}^n \frac{\sinh(a_j)}{a_j}\right]_{M \cdot x_v^{2g + 2 - D - e}} \\ =&\sum_{e = 1}^\infty \frac{(2g + 1 - D)! (4g + n - D)!}{(2g - 2 + n + e)!} \\ & \qquad \qquad \qquad \cdot \left[\left(\sum_{d = 1}^\infty \frac{a_i^{2d - e}}{(2d - e)!}\right) \frac{\sinh(a + x_v)}{a + x_v} \prod_{j=2: j\neq i}^n \frac{\sinh(a_j)}{a_j}\right]_{M \cdot x_v^{2g + 2 - D - e}} \\ =&\sum_{e = 0}^\infty \frac{(2g + 1 - D)! (4g + n - D)!}{(2g - 1 + n + 2e)!} \cdot \left[a_i \frac{\sinh(a + x_v)}{a + x_v} \prod_{j=2}^n \frac{\sinh(a_j)}{a_j}\right]_{M \cdot x_v^{2g + 1 - D - 2e}} \\&+\sum_{e = 1}^\infty \frac{(2g + 1 - D)! (4g + n - D)!}{(2g - 2 + n + 2e)!} \cdot \left[\cosh(a_i) \frac{\sinh(a + x_v)}{a + x_v} \prod_{j=2: j\neq i}^n \frac{\sinh(a_j)}{a_j}\right]_{M \cdot x_v^{2g + 2 - D - 2e}}, \end{align}\] where in the last equality we divide the sum into two parts according to whether \(e\) is odd or even.

Note that \[\begin{align} &\mathcal{F} \sum_{d_1,d_2=0}^{\infty}\frac{(2d_1+1)!!(2d_2+1)!!(x_{v^*})^{2d_1+2d_2+4}\psi^{d_1}(\psi')^{d_2}}{(2d_1+2d_2+4)!} \\=&\sum_{d_1,d_2=0}^{\infty}\frac{(x_{v^*})^{2d_1+2d_2+4}}{(2d_1+2d_2+4)!} =\sum_{d=1}^{\infty}(d-1)\frac{(x_{v^*})^{2d}}{(2d)!}. \end{align}\] Then applying \(\mathcal{F}\) to the contribution 12 of the graph \(\Gamma^\mathrm{loop}_{g-1, 1, [n]}\) to \(\pi_* \Omega_{g, M}\), we have \[\begin{align} \label{eqn:F-cont-ga-g-1[n]0} &\mathcal{F}\mathop{\mathrm{Cont}}_{\Gamma^\mathrm{loop}_{g-1,1,[n]}}\pi_*\Omega_{g,M} \nonumber\\= &- \sum_{m + m^* = 2g + 1 - D} \frac{(m^* + 1)!(2g + 1 - D)!(2g - 3 + n + m)!}{2 \cdot (2g - 3 + n)!} \nonumber\\&\cdot\left[\prod_{j = 1}^n \left(\sum_{d = 0}^\infty \frac{a_i^{2d}}{(2d + 1)!}\right) \sum_{d = 1}^\infty \frac{d - 1}{(2d)!} x_{v^*}^{2d}\right]_{M \cdot x_{v^*}^{m^* + 1} x_v^{m}}. \end{align}\tag{13}\]

3.5 Proof of Theorem 1↩︎

We now discuss how to use the identities collected in this section to prove Theorem 13, which implies Theorem 1.

Because ?? is linear in the coefficients of the topological recursion relation, it suffices to consider relations of the shape \(\pi_{*}\Omega_{g,M}\) for arbitrary monomials \(M =\prod_{i=2}^{n}a_i^{b_i}\) of degree \(D\leq 2g+1\), as well as for gluing map push-forwards of those relations of lower genus. We note that the only bouquet type graphs appearing in the relations \(\pi_{*}\Omega_{g,M}\) are \(\Gamma_{g, [n]}\) and \(\Gamma^\mathrm{loop}_{g-1, 1, [n]}\). Thus, ?? holds for such relations by the following identity, whose proof will be discussed below:

It is further not hard to verify that Proposition 14 also implies that the gluing-map push-forwards of the relations \(\pi_{*}\Omega_{g,M}\) must satisfy ?? . We conclude that given Proposition 14, all TRRs as in Corollary 3 satisfy ?? . This completes the proof of Theorem 13, and thus Theorem 1.

4 The coefficients of rational tail classes in topological recursion relations↩︎

In this section, we study applications of Faber’s intersection number conjecture to topological recursion relations to yield linear relations satisfied by the rational tail coefficients of topological recursion relations. We will assume \(g \ge 2\) throughout this section.

4.1 Faber’s intersection number conjecture↩︎

In 1993, C. Faber proposed remarkable conjectures about the structure of tautological ring \(R^*(\mathcal{M}_{g})\) (cf. [15]). An important part of Faber’s conjectures is the famous Faber intersection number conjecture, which is the following relation in \(R^{g-2}(\mathcal{M}_{g})\), if \(\sum_{i=1}^{n}k_i=g-2\), \[\begin{align} \pi_*(\psi_1^{k_1+1}\dots \psi_{n}^{k_n+1})=\frac{(2g-3+n)!(2g-1)!!}{(2g-1)!\prod_{i=1}^{n}(2k_i+1)!!} \kappa_{g-2} \end{align}\] where \(\pi: \mathcal{M}^{rt}_{g,n}\rightarrow\mathcal{M}_{g}\) is the forgetful morphism and \(\mathcal{M}^{rt}_{g,n}\subset\overline{\mathcal{M}}_{g,n}\) is the partial compactification of \(\mathcal{M}_{g,n}\) by stable nodal curves with rational tails. By now, there are many proofs of Faber’s intersection number conjecture (cf.[16], [17], [18]). It was pointed out in [16] that Faber’s intersection number conjecture is equivalent to the following identity of Hodge integrals \[\begin{align} \label{eqn:faber-int-conj-hodge} \int_{\overline{\mathcal{M}}_{g,n}}\psi_1^{k_1}\dots \psi_{n}^{k_n}\lambda_g\lambda_{g-1}=\frac{(2g-3+n)! |B_{2g}|}{2^{2g-1}(2g)!\prod_{i=1}^{n}(2k_i-1)!!} \end{align}\tag{14}\] for any \(\sum_{i=1}^{n}k_i=g-2+n\).

4.2 Proof of Theorem 2↩︎

For \(g=1\), the genus 1 topological recursion relation on \(\overline{\mathcal{M}}_{1,n}\) is obtained by pullback from the fundamental topological recursion relation \(\psi_1=\frac{1}{12}\delta_{irr}\in R^1(\overline{\mathcal{M}}_{1,1})\), where \(\delta_{irr}\) is the irreducible boundary divisor. For \(g\geq2, n=2\), assume that the genus \(g\) topological recursion relation has the form \[\begin{align} \label{eqn:thm-trr-g-2-assume} \psi_1^{k_1}\psi_2^{k_2}=c_{g,2}\cdot\left(\xi_{\Gamma_{g;0,\{1,2\}}}\right)_*\psi_{h(e)}^{g-1}+\cdots \end{align}\tag{15}\] where \(\Gamma_{g;0,\{1,2\}}\) is the stable graph having two vertices connected by one edge, with one genus-\(g\) vertex and one genus-0 vertex with markings \(\{1,2\}\), and \(h(e)\) is the half edge attached to the genus-\(g\) vertex. Multiplying by \(\lambda_{g}\lambda_{g-1}\) on both sides of 15 , and using properties of Hodge classes, we obtain \[\begin{align} \int_{\overline{\mathcal{M}}_{g,2}}\psi_1^{k_1}\psi_{2}^{k_2}\lambda_g\lambda_{g-1}=c_{g,2}\int_{\overline{\mathcal{M}}_{g,1}}\psi_1^{g-1}\lambda_g\lambda_{g-1}. \end{align}\] By 14 , we have \[\begin{align} \frac{(2g-1)! |B_{2g}|}{2^{2g-1}(2g)!\prod_{i=1}^{2}(2k_i-1)!!}=\frac{(2g-2)! |B_{2g}|}{2^{2g-1}(2g)!(2g-3)!!} \cdot c_{g,2}, \end{align}\] thus \[c_{g,2}=\frac{(2g-1)!!}{(2k_1-1)!!(2k_2-1)!!},\] as desired. For \(g\geq2, n\geq3\), by Corollary 3, there exists a degree \(g\) topological recursion relation of the form \[\begin{align} \label{eqn:deg61g-rational} \prod_{i=1}^{n}\psi_i^{k_i}=a_0\xi_{*}(\psi_{\bullet}^{g-1})+\sum_{i<j}a_{ij}\xi_{ij*}(\psi^{g-2}_{\bullet})+\cdots, \end{align}\tag{16}\] where \(\xi:\overline{\mathcal{M}}_{g,\{\bullet\}}\times\overline{\mathcal{M}}_{0,\bar{\bullet}\cup\{1,\dots,n\}}\rightarrow \overline{\mathcal{M}}_{g,n}\) and \(\xi_{ij}:\overline{\mathcal{M}}_{g,\{\bullet\}}\times\overline{\mathcal{M}}_{0,\bar{\bullet}\cup\{1,\dots,\hat{i},\dots,\hat{j},\dots,n\}\cup\circ}\times\overline{\mathcal{M}}_{0,\{\bar{\circ},i,j\}}\rightarrow \overline{\mathcal{M}}_{g,n}\) are the gluing maps. The omitted terms consist of other boundary classes with no \(\kappa\) classes and no genus-0 vertex decorated by \(\psi\) classes. Multiplying by \(\lambda_g\lambda_{g-1}\cdot\xi_{*}({\psi_{\bar{\bullet}}}^{n-3})\) on both sides of equation 16 , note that by product formula 3 and degree reason, \[\begin{align} \xi_{*}(\psi_{\bullet}^{g-1})\cdot\xi_{*}(\psi_{\bar{\bullet}}^{n-3})=\xi_{*}(\psi_{\bullet}^{g-1}\psi_{\bar{\bullet}}^{n-3}(-\psi_{\bullet}-\psi_{\bar{\bullet}})) \end{align}\] and \[\begin{align} \xi_{ij*}(\psi_{\bullet}^{g-2})\cdot\xi_{*}(\psi_{\bar{\bullet}}^{n-3})=\xi_{ij*}(\psi_{\bullet}^{g-2}\psi_{\bar{\bullet}}^{n-3}(-\psi_{\bullet}-\psi_{\bar{\bullet}})). \end{align}\] Integrating 16 over \(\overline{\mathcal{M}}_{g,n}\) yields \[\begin{align} 0=a_0+\sum_{1\leq i<j\leq n}a_{ij}. \end{align}\] In more detail, by a direct computation, we have \[\begin{align} \int_{\overline{\mathcal{M}}_{g,n}}\prod_{i=1}^{n}\psi_{i}^{k_i} \cdot \xi_{*}(\psi_{\bar{\bullet}}^{n-3})\lambda_{g}\lambda_{g-1}=\int_{\overline{\mathcal{M}}_{g,\{\bullet\}}}\lambda_{g}\lambda_{g-1} \int_{\overline{\mathcal{M}}_{0,\bar{\bullet}\cup\{1,\dots,n\}}}\psi_{\bar{\bullet}}^{n-3}\prod_{i=1}^{n}\psi_{i}^{k_i}=0, \end{align}\] \[\begin{align} &\int_{\overline{\mathcal{M}}_{g,n}} \xi_{*}(\psi_{\bullet}^{g-1})\xi_{*}(\psi_{\bar{\bullet}}^{n-3})\lambda_{g}\lambda_{g-1}=\int_{\overline{\mathcal{M}}_{g,n}} \xi_{*}(\psi_{\bullet}^{g-1}\psi_{\bar{\bullet}}^{n-3}(-\psi_{\bullet}-\psi_{\bar{\bullet}}))\lambda_{g}\lambda_{g-1} \\=&-\int_{\overline{\mathcal{M}}_{g,\{\bullet\}}}\psi_{\bullet}^{g-1}\lambda_{g}\lambda_{g-1}\int_{\overline{\mathcal{M}}_{0,\bar{\bullet}\cup\{1,\dots,n\}}}\psi_{\bar{\bullet}}^{n-2} \\=&-\frac{(2g-2)! |B_{2g}|}{2^{2g-1}(2g)!(2g-3)!!}, \end{align}\] and for every \(1 \le i < j \le n\), we have \[\begin{align} &\int_{\overline{\mathcal{M}}_{g,n}}\xi_{ij*}(\psi_{\bullet}^{g-2})\xi_{*}(\psi_{\bar{\bullet}}^{n-3})\lambda_{g}\lambda_{g-1} =\int_{\overline{\mathcal{M}}_{g,n}}\xi_{ij*}(\psi_{\bullet}^{g-2}\psi_{\bar{\bullet}}^{n-3}(-\psi_{\bullet}-\psi_{\bar{\bullet}}))\lambda_{g}\lambda_{g-1} \\=&-\int_{\overline{\mathcal{M}}_{g,\{\bullet\}}}\psi_{\bullet}^{g-1}\lambda_{g}\lambda_{g-1}\int_{\overline{\mathcal{M}}_{0,\bar{\bullet}}\cup\{1,\dots,\hat{i},\dots,\hat{j},\dots,n\}\cup\circ}\psi_{\bullet}^{n-3} \\=&-\frac{(2g-2)!|B_{2g}|}{2^{2g-1}(2g)!(2g-3)!!}. \end{align}\] In addition, it is not hard to see the intersection of all omitted terms and \(\lambda_g\lambda_{g-1}\cdot\xi_{*}({\psi_{\bar{\bullet}}}^{n-3})\) vanishes. This finishes the proof of Theorem 2.

5 Applications to Gromov–Witten theory↩︎

In this section, we disscus some applications of topological recursion relations in Gromov–Witten theory. Let \(X\) be a smooth projective variety. By definition, the small phase space is the vector space \(H^{*}(X,\mathbb{C})\) and the big phase space is defined to be \(\mathcal{P}:=\prod_{n=0}^{\infty} H^{*}(X,\mathbb{C})\). For simplicity, we assume \(H^{\text{odd}}(X,\mathbb{C})=0\). Let \(\{\phi_{1},\dots,\phi_{N}\}\) be a fixed basis of \(H^{*}(X,\mathbb{C})\), where \(\phi_{1}\) is the identity element of the cohomology ring of \(M\). The corresponding basis for the \(n\)-th copy of \(H^{*}(X,\mathbb{C})\) in this product is denoted by \(\{ \tau_{n}(\phi_\alpha) \mid \alpha=1, \ldots, N\}\) for \(n \geq 0\). Let \(\{t_{n}^{\alpha}\}\) be the coordinates on \(\mathcal{P}\) with respect to the standard basis \(\{ \tau_{n}(\phi_\alpha) \mid \alpha=1, \ldots, N, \,\,\, n \geq 0\}\). We will also identify the small phase space with the subspace of \(\mathcal{P}\) defined by \(t_n^{\alpha}=0\) for \(n>0\). We will identify \(\tau_{n}(\phi_{\alpha})\) with the vector field \(\frac{\partial}{\partial t_{n}^{\alpha}}\) on the big phase space. If \(n<0\), \(\tau_{n}(\phi_{\alpha})\) is understood to be the zero vector field. We will also write \(\tau_{0}(\phi_{\alpha})\) simply as \(\phi_{\alpha}\). We use \(\tau_{+}\) and \(\tau_{-}\) to denote the operators which shift the level of descendants by \(1\), i.e. \[\tau_{\pm}\Big(\sum_{n,\alpha}f_{n,\alpha}\tau_{n}(\phi_{\alpha})\Big) =\sum_{n,\alpha}f_{n,\alpha}\tau_{n\pm1}(\phi_{\alpha})\] where \(f_{n,\alpha}\) are functions on the big phase space.

Let \(\eta=(\eta_{\alpha\beta})\) be the matrix of the intersection pairing on \(H^{*}(X;\mathbb{C})\) in the basis \(\{\phi_{1},\dots,\phi_{N}\}\). We will use \(\eta=(\eta_{\alpha\beta})\) and \(\eta^{-1}=(\eta^{\alpha\beta})\) to lower and raise indices, respectively, for example \(\phi^{\alpha}:=\eta^{\alpha\beta}\phi_{\beta}\) for any \(\alpha\). Here we use the Einstein summation convention that repeated indices should be summed over their entire ranges.

Let \[\langle\tau_{n_{1}}(\phi_{\alpha_{1}})\cdot\cdot\cdot \tau_{n_{k}}(\phi_{\alpha_{k}})\rangle_{g,k,\beta} :=\int_{[\overline{\mathcal{M}}_{g,k}(X,\beta)]^{vir}}\bigcup_{i=1}^{k}(\Psi_{i}\cup ev_{i}^{*}(\phi_{\alpha_{i}}))\] be the genus-\(g\), degree-\(\beta\), descendant Gromov–Witten invariant associated to \(\phi_{\alpha_{1}},\dots,\phi_{\alpha_{k}}\) and nonnegative integers \(n_{1},\dots,n_{k}\) (cf. [31]). Here \(\overline{\mathcal{M}}_{g,n}(X,\beta)\) is the moduli space of stable maps from genus-\(g\), \(k\)-marked curves to \(X\) of degree \(\beta\in H_{2}(X;\mathbb{Z})\) and \([\overline{\mathcal{M}}_{g,n}(X,\beta)]^{vir}\) is its virtual fundamental class. \(\Psi_{i}\) is the first Chern class of the tautological line bundle over \(\overline{\mathcal{M}}_{g,n}(X,\beta)\) whose geometric fiber is the cotangent space of the domain curve at the \(i\)-th marked point and \(ev_{i}\colon \overline{\mathcal{M}}_{g,n}(X,\beta)\rightarrow X\) is the \(i\)-th evaluation map for all \(i=1,\dots,k\). The genus-\(g\) generating function is defined to be \[\begin{align} F_{g}:=\sum_{k\geq0}\sum_{\alpha_{1},\dots,\alpha_{k}} \sum_{n_{1},\dots,n_{k}} \frac{1}{k!}t_{n_{1}}^{\alpha_{1}}\cdot\cdot\cdot t_{n_{k}}^{\alpha_{k}}\sum_{\beta}q^{\beta} \left<\tau_{n_{1}}(\phi_{\alpha_{1}})\cdot\cdot\cdot \tau_{n_{k}}(\phi_{\alpha_{k}})\right>_{g,k,\beta} \end{align}\] where \(q^{\beta}\) belongs to the Novikov ring. Here we set all the undefined terms in \(F_g\) to be zero. This function is understood as a formal power series in the variables \(\{t_{n}^{\alpha}\}\) with coefficients in the Novikov ring.

Define a \(k\)-tensor \(\langle\langle\cdot\cdot\cdot\rangle\rangle_g\) by \[\begin{align} \langle\langle{W_{1}W_{2}\cdot\cdot\cdot W_{k}}\rangle\rangle_g :=\sum_{m_{1},\alpha_{1},\dots,m_{k},\alpha_{k}}f_{m_{1},\alpha_{1}}^{1} \cdot\cdot\cdot f_{m_{k},\alpha_{k}}^{k}\frac{\partial^{k}}{\partial t_{m_{1}}^{\alpha_{1}}\cdot\cdot\cdot \partial t_{m_{k}}^{\alpha_{k}}}F_{g} \end{align}\] for vector fields \(W_{i}=\sum_{m,\alpha}f_{m,\alpha}^{i}\frac{\partial}{\partial t_{m}^{\alpha}}\) where \(f_{m,\alpha}^{i}\) are functions on the big phase space.

For any vector fields \(W_{1}\) and \(W_{2}\) on the big phase space, the quantum product of \(W_{1}\) and \(W_{2}\) is defined by \[W_{1}\bullet W_{2}:=\sum_{\alpha}\langle\langle{W_{1}W_{2} \phi^{\alpha}}\rangle\rangle_0\phi_{\alpha}.\]

It is well known that topological recursion relations in \(R^*(\overline{\mathcal{M}}_{g,n})\) can be translated into universal equations for Gromov–Witten invariants via the splitting axiom and cotangent line comparison equations. Define the operator \(T\) on the space of vector fields by \[\begin{align} \label{eqn:T-operator} T(W)=\tau_{+}(W)-\sum_{\alpha}\langle\langle W\phi^\alpha\rangle\rangle_0\phi_\alpha. \end{align}\tag{17}\] The operator is very useful for translating topological recursion relations into universal equations (cf.[19]). In the process, each marked point corresponds to a vector field, and the cotangent line class corresponds to the operator \(T\). Each node is translated into a pair of primary vector fields \(\phi_\alpha\) and \(\phi^\alpha\). For the convenience of the reader, we give a proof of the translation rule in Appendix 8.2.

It is well known that such universal equations play an important role in computing higher genus Gromov–Witten invariants and the study of the famous Virasoro conjecture (cf.[3]). It is also conjectured the collection of all universal equations determines all higher genus Gromov–Witten invariants in terms of genus-0 Gromov–Witten invariants in the semisimple case (cf.[19]), which differs from the approach of Givental (cf.[20]) or Dubrovin-Zhang (cf.[21]). Thus it is important to find as explicit as possible formulas for such universal equations, even though they tend to be complicated in general.

Theorem 1 implies:

For a function \(f\) on the big phase space, we say \(f\overset{g}{\approx}0\) if \(f\) can be expressed as a polynomial of \(F_0,F_1,\dots,F_{g-1}\) and their derivatives. Theorem 2 implies the following:

For \(n\geq3\), the number of rational type strata grows rapidly, and it becomes very complicated to compute the coefficients in the degree-\(g\) topological recursion relations explicitly. We do still have the following weaker result on topological recursion relations, which is a corollary from the existence of general degree-\(g\) topological recursion relations 6 on \(\overline{\mathcal{M}}_{g,n}\).

6 Application to the Hodge class \(\lambda_g\)↩︎

In [22], a nice formula for the top Chern class \(\lambda_g\) of the Hodge bundle was found, which is supported on the divisor of curves with a nonseparating node. More explicitly, we have \[\begin{align} \lambda_g=\frac{(-1)^g}{2^g} \mathcal{D}_{g,n}^{g}(0,\dots,0) \in R^{g}(\overline{\mathcal{M}}_{g,n}) \end{align}\] where \(\mathcal{D}_{g,n}(a_1,\dots,a_n)\) is Pixton’s formula for double ramification cycles defined in Section 2.2. As an application of Theorem 1, we may compute the coefficient of the bouquet class in this formula for \(\lambda_g\):

We state a simple lemma used in the proof of the proposition.

7 Intersection numbers on the moduli space of stable curves↩︎

In this section, we modify the algorithm from Section 2.4 to derive a recursive algorithm for the intersection numbers \(\int_{\overline{\mathcal{M}}_{g,n}}\prod_{i=1}^{n}{\psi_i}^{k_i}\) with any \(2g-2+n>0\). We will show that all intersection numbers are uniquely determined by the recursion with initial conditions \(\int_{\overline{\mathcal{M}}_{0,3}}1=1\) and \(\int_{\overline{\mathcal{M}}_{1,1}}\psi_1=\frac{1}{24}\).

Fix a genus \(g\) and a number of marked points \(n\) satisfying \(2g-3+n>0\). Let \(M\) be a monomial of degree \(D\leq 2(3g-3+n)\) in the variables \(a_2,\dots,a_n\), and let \(N:=n+2(3g-3+n)-D\). Define \[\widehat{\Omega}^{pre}_{g,M}=[\mathcal{D}_{g,N}^{3g-3+n}(a_2,\dots,a_N)]_{M\cdot a_{n+1}\dots a_{N}}\in \mathcal{S}_{g,N}^{3g-3+n}\] to be the coefficient of the monomial \(M\cdot a_{n+1}\dots a_N\) in Pixton’s class \(\mathcal{D}_{g,N}^{3g-3+n}(a_2,\dots,a_N)\). Let \(\pi\colon \overline{\mathcal{M}}_{g,N}\rightarrow \overline{\mathcal{M}}_{g,n}\) be the forgetful map. Define \[\widehat{\Omega}_{g,M}=\pi_*\left(\widehat{\Omega}_{g,M}^{pre}\cdot \psi_{n+1}\dots\psi_{N}\right)\in \mathcal{S}^{3g-3+n}_{g,n}.\] In the range \(2g-3+n>0\), integrating the class \(\widehat{\Omega}_{g,M}\) gives \[\begin{align} \label{eqn:int-trr-3g-343n} \int_{\overline{\mathcal{M}}_{g,n}}\widehat{\Omega}_{g,M} =0. \end{align}\tag{18}\] The following Theorem 16 shows that 18 yields a new combinatorial recursive formula for integrals of \(\psi\) classes on the moduli space of curves.

To state the theorem, first, fix integers \((a_2, \dotsc, a_n)\), an integer \(r > 0\), a stable graph \(\Gamma=(V,H,g,p,\iota)\) of genus \(g\) with \(n\) legs, and an assignment \(x_v \in \mathbb{Z}\) for every \(v \in V\). Given this, we set \[a_1 = -(a_2 + \dotsb + a_n + \sum_{v \in V} x_v).\] With this, we define a weighting modulo \(r\) on \((\Gamma, x)\) to be a map \[w\colon H \to \{0, \dotsc, r - 1\}\] satisfying three properties:

1. For any \(i \in \{1, \dotsc, n\}\) corresponding to a leg \(\ell_i\) of \(\Gamma\), we have \(w(\ell_i) \equiv a_i \pmod{r}\).

2. For any edge \(e\in E\) corresponding to two half-edges \(h, h'\in H\), we have \(w(h) + w(h') \equiv 0 \pmod{r}\).

3. For any vertex \(v \in V\), we have \(\sum_{h\in p^{-1}(v)} w(h) \equiv -x_v \pmod{r}\).

Define \(\widehat{\mathcal{D}}_{(\Gamma,x)}^r\) to be the class \[\frac{1}{r^{h^1(\Gamma)}} \sum_{\substack{w \text{ weighting }\\ \text{mod } r \text{ on }(\Gamma,x)}} \left[\Gamma, \prod_{i=1}^n e^{\frac{1}{2} a_i^2\psi_i} \prod_{(h,h')\in E} \frac{1 - e^{-\frac{1}{2} w(h)w(h')(\psi_h + \psi_{h'})}}{\psi_h + \psi_{h'}}\right] \in \mathcal{S}_{g,n}.\] This class is a polynomial for \(r \gg 0\), and we will use \(\widehat{\mathcal{D}}_{(\Gamma,x)}\) to denote the constant part of the resulting polynomial. The class \(\widehat{\mathcal{D}}_{(\Gamma,x)}\) is a polynomial in the variables \(a_2, \dotsc, a_n\) and \(x_v\) for \(v \in V\), and so we will denote it by \[\widehat{\mathcal{D}}_{\Gamma}(a_2, \dotsc, a_n, \{x_v\}).\]

Now we give an example to show how to use Theorem 16 to do explicit calculations.

References↩︎

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1. We use the standard convention \((-1)!!=1\).↩︎

2. Here we set \(a_0\) or \(a_{ij}\) to be zero if \(\xi\) or \(\xi_{ij}\) is not defined.↩︎