August 25, 2025
We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov-Witten theory of the projective line and is dictated by \(\mathbb{Z}_2\)-equivariant topological recursion.
We prove a relation among the \(\kappa\)-classes on the moduli spaces of curves \(\overline{\mathcal{M}}_{g,n}\), conjectured in [1]. Let \(\overline{\mathcal{M}}_{g,n}\) be the moduli space of stable curves of genus \(g\) with \(n\) marked points, and let \(R^*(\overline{\mathcal{M}}_{g,n})\) denote its tautological ring. Define the constants \(s_i\), \(i=1,2,\dots\) by the formula \[\begin{align} \label{eq:J-first} \exp(-\textstyle \sum_{i=1}^\infty s_i t^i) = \sum_{i=0}^\infty (-1)^i i! t^i, \end{align}\tag{1}\] where \(t\) is a formal variable. Consider the class \[\begin{align} \label{eq:J-second} \mathbb{J}=1+\sum_{i=1}^{\infty} J_i = \exp(\textstyle \sum_{i=1}^\infty s_i \kappa_i), \end{align}\tag{2}\] where \(J_i\in R^i(\overline{\mathcal{M}}_{g,n})\) and \(\kappa_i\), \(i=1,2,\dots\), are the Miller-Morita-Mumford \(\kappa\)-classes. Equivalently, this class can be defined as follows: \[\begin{align} \label{eq:J-class} \mathbb{J}=1+\sum_{m=1}^{\infty} \frac{1}{m!} \sum_{a_1,\dots,a_m=1}^\infty \kappa_{a_1,\dots,a_m} \prod_{i=1}^m (-1)^{a_i-1} a_i!, \end{align}\tag{3}\] where \(\kappa_{a_1,\dots,a_m}\) are the multi-index \(\kappa\)-classes obtained from the push-forwards of the \(\psi\)-classes as \(\pi_*(\prod_{i=1}^m \psi_{n+i}^{a_i+1})\), where \(\pi \colon \overline{\mathcal{M}}_{g,n+m}\to \overline{\mathcal{M}}_{g,n}\) is the forgetful map.
The main result of this paper is the following
Theorem 1. We have \(J_p=0\) in \(R^*(\overline{\mathcal{M}}_{g,n})\) for \(p>2g-2+n\). Moreover, \(J_{2g-2+n}=0\) for \(n\geq 2\).
This statement was conjectured in [1]. Note that the first part of this conjecture concerns different polynomial relations in \(\kappa\)-classes and is proven in [2].
Furthermore, let \(\lambda_1,\dots,\lambda_g\) be the Chern classes of the Hodge bundle on \(\overline{\mathcal{M}}_{g,n}\). We prove the following expression for \(J_{2g-2+n}\) in the exceptional cases \(n\le1\):
Theorem 2. We have \(J_{2g-1}=(-1)^{g-1}\lambda_g\lambda_{g-1}\) in \(R^*(\overline{\mathcal{M}}_{g,1})\) and \(J_{2g-2}=(-1)^{g-2}\lambda_g\lambda_{g-2}\) in \(R^*(\overline{\mathcal{M}}_{g,0})\).
This statement was conjectured in [1].
The methods of the proof come from the analysis of the spin Gromov-Witten theory of \((\mathbb{P}^1,\mathcal{O}(-1))\) along the lines of [3] via localization and materialization, and the arguments that we use come from a statement that relates the stationary sector of the spin Gromov-Witten theory of \((\mathbb{P}^1,\mathcal{O}(-1))\) to topological recursion [4], more precisely, a \(\mathbb{Z}_2\)-equivariant version of topological recursion in the sense of [5]. To this end, we have the following statements.
Firstly, we connect the classes \(J_p\) to the stationary sector of the spin Gromov-Witten theory of \((\mathbb{P}^1,\mathcal{O}(-1))\). We refer the reader to the basic setup of this theory to [3] that matches the notation that we use here, as well as [6]. Let \([\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)]^{\mathrm{loc},\mathcal{O}(-1)}\in A_{g-1+d+n}(\overline{\mathcal{M}}_{g,n}(\mathbb{P},d))\), \(d\geq 0\), be the corresponding localized virtual cycle and consider the class \(C_{g,n,d}\) defined as \[\begin{align} C_{g,n,d} \mathrel{\vcenter{:}}= [\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)]^{\mathrm{loc},\mathcal{O}(-1)} \cdot \prod_{i=1}^n ev_i^*([\mathrm{pt}]). \end{align}\] Let \(p\colon \overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d) \to \overline{\mathcal{M}}_{g,n}\), \(2g-2+n>0\), be the natural forgetful morphism. We have:
Theorem 3. For any \(g\geq 0\), \(n\geq 0\), \(2g-2+n>0\), \(d\geq 0\) the Poincaré dual of \((-1)^{g+d+1} p_*C_{g,n,d}\) is equal to \(J_{2g-2+n-d}\) (in \(R^*(\overline{\mathcal{M}}_{g,n})\)).
Secondly, the stationary sector of the descendant spin Gromov-Witten theory of \((\mathbb{P}^1,\mathcal{O}(-1))\) whose correlators are defined as \[\begin{align} \label{eq:stationarysector} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),{\rm des}}_{g} \mathrel{\vcenter{:}}= \sum_{d=0}^\infty (-1)^{g+1}Q^d\int_{C_{g,n,d}} \prod_{i=1}^n \psi_i^{k_i} \end{align}\tag{4}\] can be computed by \(\mathbb{Z}_2\)-equivariant topological recursion:
Theorem 4. Let \(\omega^{(g)}_n\) be the system of differentials produced via \(\mathbb{Z}_2\)-equivariant topological recursion on the curve \(\mathbb{P}^1\) with the following input data \[\begin{align} x& =\sqrt{-Q/2} \Bigl(z+\frac{1}{z}\Bigr); & y&= 2 \log z; & B & =\frac{dz_1dz_2}{(z_1-z_2)^2} + \frac{d\iota(z_1)dz_2}{(\iota(z_1)-z_2)^2}; & \iota\colon z\to -\frac{1}{z}, \end{align}\] where \(z\) is a global rational coordinate, and \(\iota\) is the involution generating the \(\mathbb{Z}_2\)-action. Then for \(2g-2+n\geq 0\) near \(z=0\) in the local coordinate \(x^{-1}\) we have: \[\begin{align} \omega^{(g)}_n = 2^{2g-2+n} \sum_{k_1,\dots,k_n} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),{\rm des}}_{g} \prod_{i=1}^n d \frac{(2k_i-1)!!}{x_i^{2k_i+1}} +\delta_{g,0}\delta_{n,2} \frac{dx_1dx_2}{(x_1-x_2)^2}. \end{align}\]
This theorem might be considered as a spin version of the Norbury-Scott conjecture [7] proved in [8], see also [9], [10].
In Section 2 we recall a particular localization formula for the \(\mathbb{C}^*\)-equivariant spin Gromov-Witten theory of \(\mathbb{P}^1\). Then we first analyze its numerical consequences that give an expression for the stationary sector of the ancestor potential of the \(\mathbb{C}^*\)-equivariant spin Gromov-Witten theory of \(\mathbb{P}^1\) in terms of the Kontsevich-Witten tau function with shifted times. Repeating this analysis on the level of classes (which is a version of Givental’s materialization) we derive tautological relations that prove Theorems 1 and 2, as well as Theorem 3.
In Section 3 we explain the origin of this computation: we recall a topological recursion reformulation of the numerical version of [1] given in [1] and reformulate it in terms of \(\mathbb{Z}_2\)-equivariant topological recursion. The latter recursion appears to be the one given in Theorem 4, and we conclude by proving this theorem.
Throughout the text we use the following notation:
\(\llbracket {n} \rrbracket\) denotes \(\{1,\dots,n\}\).
\(z_I\) denotes \(\{z_i\}_{i\in I}\) for \(I\subseteq \llbracket {n} \rrbracket\).
\([u^d]\) denotes the operator that extracts the corresponding coefficient from the whole expression to the right of it, that is, \([u^d]\sum_{i=-\infty}^\infty a_iu^i \mathrel{\vcenter{:}}= a_d\).
\(\mathop{\big\lfloor_{{u}\to {v}}}\) denotes the operator of substitution (or restriction), that is, \(\mathop{\big\lfloor_{{u}\to {v}}} f(u) \mathrel{\vcenter{:}}= f(v)\).
The authors thank Xavier Blot and Reinier Kramer for very useful discussions.
A. A. was supported by the Institute for Basic Science (IBS-R003-D1). B. B. was supported by the ISF Grant 876/20. B. B., P. D.-B., and M. K. were supported by the Russian Science Foundation (grant No. 24-11-00366). S. S. was supported by the Dutch Research Council (OCENW.M.21.233).
The Center of Geometry and Physics of the Institute of Basic Studies, located in Pohang, is acknowledged for hospitality.
Consider the \(\mathbb{C}^*\) action on \(\mathbb{P}^1\) with two fixed points, \(0\) and \(\infty\), with the tangent weights \(-1\) and \(1\), respectively. Consider the moduli space of stable maps \(\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)\), \(g,n\geq 0\), \(d\geq 1\). Let \(\pi\colon\overline{\mathcal{C}}_{g,n}(\mathbb{P}^1,d) \to \overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)\) be the universal curve and \(f\colon\overline{\mathcal{C}}_{g,n}(\mathbb{P}^1,d) \to\mathbb{P}^1\) be the universal map. Our main tool is a formula for the \(\mathbb{C}^*\)-equivariant class \[\label{eq:MainClass} C_{g,n,d}(T)\mathrel{\vcenter{:}}= \Big(\prod_{i=1}^n ev_i^*(\mathbf{0})\Big) \cdot c_{g-1+d} (R^1\pi_*f^*\mathcal{O}(-1)) \cdot [\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)]^{\mathrm{vir}},\tag{5}\] where \(\mathbf{0}\) stands for the equivariant Poincaré dual of \(0\). This formula is derived in [3], [11] and we recall it in the next subsection. Note that here we have to choose a lift of the \(\mathbb{C}^*\) action to \(\mathcal{O}(-1)\). We do it by requiring that the fiber weights over \(0\) and \(\infty\) are \(1\) and \(0\), respectively. This way the localization formula will produce only graphs of star tree type — this idea goes back to [12] and is efficiently used in [11], [13].
Remark 5. By construction, \(C_{g,n,d}(T)\in A_{g-1+d}^{\mathbb{C}^*} (\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d))\) over the ring \(H^*_{\mathbb{C}^*}(\mathrm{pt})\cong \mathbb{Q}[T]\). It is a polynomial in \(T\), though the localization formula presents it as a Laurent polynomial. The vanishing of the coefficients of the negative powers of \(T\) gives us relations between classes in \(A_*(\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d))\).
Remark 6. The non-equivariant limit of the class \(c_{g-1+d} (R^1\pi_*f^*\mathcal{O}(-1)) \cdot [\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)]^{\mathrm{vir}}\) is equal to the localized virtual cycle \([\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)]^{\mathrm{loc},\mathcal{O}(-1)}\) of the spin curve \((\mathbb{P}^1,\mathcal{O}(-1))\). Thus, \(\mathop{\big\lfloor_{{T}\to {0}}}C_{g,n,d}(T) = C_{g,n,d}\) and the intersection numbers \(\int_{C_{g,n,d}(T)} \prod_{i=1}^n \psi_i^{k_i}\) compute at \(T=0\) the stationary sector of the descendant spin Gromov-Witten theory of \((\mathbb{P}^1,\mathcal{O}(-1))\) defined in 4 .
We fix integers \(g,n\geq 0\) and \(d\geq 1\). We denote by \(\mathrm{SRT}_{g,n}(\mathbb{P}^1,d)\) the set of star rooted trees of genus \(g\) with \(n\) legs, with edges of total degree \(d\), which are the graphs \[\begin{align} \Gamma=(V,H,\iota \colon H\to H, H^{\iota}\simeq \{\sigma_1,\ldots, \sigma_{n}\}, v_0\in V, d\colon E\to \mathbb{Z}_{\geq 1}, o\colon E\to \llbracket {E} \rrbracket), \end{align}\] where \(V=V(\Gamma)\) is the set of vertices, \(H=H(\Gamma)\) is the set of half-edges, \(\iota\) is the involution on \(H\) whose cycles of length two form the set of edges \(E=E(\Gamma)\) and whose fixed points are the \(n\) legs labeled from \(1\) to \(n\). Furthermore, \(g\colon V\to \mathbb{Z}_{\geq 0}\) is the genus function such that the total arithmetic genus of the graph is \(g\), \(\sum_{e\in E} d(e) = d\), and we demand that
the graph \((V,E)\) must be a rooted tree, and all edges are between the root \(v_0\in V\) and another vertex (hence the term “star rooted tree”);
the legs \(\sigma_{1},\dots,\sigma_{n}\) are attached to the root vertex \(v_0\in V\).
Furthermore, for each vertex \(v\) let \(n(v)\) denote the valence of \(v\). In particular, \(n(v_0) = n+|E|\) and \(n(v) = 1\) if \(v\not=v_0\). We say that a vertex is stable if \(2g(v)-2+n(v)>0\), and unstable otherwise.
Note that since \(\Gamma\) is a tree we have \(\sum_{v\in V} g(v)=g\) and since \(\Gamma\) is a star rooted tree we have a natural isomorphism \(V\setminus \{v_0\} \cong E\). It is convenient to order the edges in \(E\) by choosing an isomorphism \(o\colon E\to \{1,\dots,|E|\}\), and consider the ordering as a part of the data of a star rooted tree. It implies an ordering on \(V\setminus \{v_0\}\), and we denote by \(e_i\) the \(i\)-th edge, by \(v_i\) the \(i\)-th non-root vertex (attached to \(e_i\)) , and let \(d_i\) stand for \(d(e_i)\) and \(g_i\) for \(g(v_i)\), respectively, \(i=1,\dots,|E|\). Additionally, let \(g_0=g(v_0)\).
We want to assign to \(\Gamma\) a class in \(A_*( \overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)) \otimes_{\mathbb{Q}} \mathbb{Q}[T,T^{-1}]\). First, we assign to \(\Gamma\) a stratum in \(\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)\) that consists of stable maps with \(1+2|E|\) components, where we associate to \(v_0\) a constant map to \(0\in \mathbb{P}^1\) of a curve in \(\overline{\mathcal{M}}_{g_0,n+|E|}\) and to each \(v_i\in V\setminus \{v_0\}\) a constant map to \(\infty\) of a curve in \(\overline{\mathcal{M}}_{g_i,1}\) (the unstable vertices are kept for the bookkeeping reasons, we discuss their contraction below). To each edge \(e_i\) we associate a Galois covering of \(\mathbb{P}^1\) of degree \(d_i\). Let \(B_\Gamma\) be the boundary map from \(\overline{\mathcal{M}}_{g_0,n+|E|}\times \prod_{i=1}^{|E|} \overline{\mathcal{M}}_{g_i,1}\) to the corresponding stratum in \(\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d)\)
The \(n(v_0)=n+|E|\) marked points on a curve assigned to the root vertex match the \(n\) legs and \(E\) edges attached to \(v_0\) such that the first \(n\) marked points correspond to the legs and the last \(|E|\) to the edges, respectively, preserving their given ordering.
Proposition 7 (See [3], [11]). For \(d\geq 1\) we have: \[\begin{gather} \label{eq:C40T41-formula} C_{g,n,d}(T) = (-1)^{g+1}\; T^{2g-2+n-d} \\ \times\sum_{\Gamma\in \mathrm{SRT}_{g,n,d}} \frac{(-1)^{|E|}}{|E|!} (B_\Gamma)_* \left( \prod_{i=1}^{|E|} \frac{d_i^{d_i+1}}{d_i! T} \frac{1}{1-d_i \frac{\psi_{n+i}}{T}} \otimes \bigotimes_{i=1}^{|E|} \frac{\frac{(-1)^{g_i}\lambda_{g_i}}{T^{g_i}} \Lambda(\frac{1}{T})}{1+d_i \frac{\psi_1}{T}} \right). \end{gather}\qquad{(1)}\]
Here \(\Lambda(u)=\sum_{k=1} u^k \lambda_k\).
Remark 8. This formula involves in some cases the non-existing moduli spaces \(\overline{\mathcal{M}}_{0,2}\) (for the root vertex in the case \(n=1\), \(g_0=0\), \(|E|=1\) or \(n=0\), \(g_0=0\), \(|E|=2\)) and \(\overline{\mathcal{M}}_{0,1}\) (for the non-root vertices \(v\) such that \(g(v)=0\) and \(n(v)=1\) or the root vertex in the case \(n=0\), \(g_0=0\), \(|E|=1\)). These cases should be considered as the bookkeeping devices with the following conventions: \[\begin{align} \int_{\overline{\mathcal{M}}_{0,1}} \frac{1}{1-a\psi_1} & = a^{-2}; & \int_{\overline{\mathcal{M}}_{0,2}} \frac{1}{1-a\psi_1} \frac{1}{1-b\psi_2} &= (a+b)^{-1}. \end{align}\]
Let \(p\colon \overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,d) \to \overline{\mathcal{M}}_{g,n}\). Denote by \(C_{g,n}(T,Q)\) the (Poincaré dual of the) class \[\label{eq:Cgn40T44Q41} C_{g,n}(T,Q)\mathrel{\vcenter{:}}= \sum_{d=1}^\infty (-1)^{g+1}Q^d p_* C_{g,n,d}(T)\in R^*(\overline{\mathcal{M}}_{g,n}) \otimes_{\mathbb{Q}} \mathbb{Q}[T,Q].\tag{6}\] In this section, we compute the correlators of the ancestors of \(\mathbf{0}\) in the \(\mathbb{C}^*\)-equivariant extension of the spin Gromov-Witten theory of \((\mathbb{P}^1,\mathcal{O}(-1))\) given by 6 : \[\begin{align} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm anc}}_{g} \mathrel{\vcenter{:}}= \int_{\overline{\mathcal{M}}_{g,n}}C_{g,n}(T,Q)\;\psi_1^{k_1}\dots\psi_n^{k_n}. \end{align}\] The general theory implies that this quantity is a polynomial in \(T\) and \(Q\) of homogeneous degree \(d_{\rm tot}=\sum k_i-g+1\). In particular, it is identically equal to zero if \(d_{\rm tot}<0\). The regular dependence of this expression in \(Q\) is seen from the very its form. On the other hand, its regular dependence in \(T\) (that is, the fact that the contributions of all summands with \(d>d_{\rm tot}\) cancel out), is a manifestation of the localization formula. Below, we compute these correlators in a closed form that allows one to take a limit as \(T\to 0\). In the next section, we reproduce the same computations directly for the classes \(C_{g,n}(T,Q)\) themselves. When we pass to \(T\to 0\) limit, we routinely add the \(d=0\) terms to the definition of the ancestor correlators, that is, we define \[\begin{align} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),{\rm anc}}_{g} \mathrel{\vcenter{:}}= (-1)^{g+1}\int_{\overline{\mathcal{M}}_{g,n}} \sum_{d=0}^\infty Q^d p_*C_{g,n,d}\; \psi_1^{k_1}\dots\psi_n^{k_n}. \end{align}\]
Let us collect these correlators to the potentials \[\begin{align} F^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm anc}}(t_0,t_1,\dots;\hbar)& \mathrel{\vcenter{:}}=\sum_{g,n}\frac{\hbar^{2g-2+n}}{n!} \sum_{k_1,\dots,k_n\ge0} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm anc}}_{g}t_{k_1}\dots t_{k_n}; \\ \notag F^{\mathbb{P}^1,\mathcal{O}(-1),{\rm anc}}(t_0,t_1,\dots;\hbar)& \mathrel{\vcenter{:}}=\sum_{g,n}\frac{\hbar^{2g-2+n}}{n!} \sum_{k_1,\dots,k_n\ge0} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),{\rm anc}}_{g}t_{k_1}\dots t_{k_n}. \end{align}\]
Introduce the rational functions \[\label{eq:Vk} \begin{gather} V_k=\Bigl(\frac{Q}{T-Q}\frac{d}{dQ}\Bigr)^k\frac{Q}{T},\quad k\ge0, \\V_0=\frac{Q}{T},\quad V_1=\frac{Q}{T(T-Q)}=\frac{1}{T-Q}-\frac{1}{T}, \quad V_2=\frac{Q}{(T-Q)^3},\quad\dots. \end{gather}\tag{7}\] Note that \(V_k\in\mathbb{Q}[T,(T-Q)^{-1}]\) for \(k\ge2\). Define \[\begin{align} \label{eq:hatPk} \widehat P_{k}&= \delta_{k\ge2}V_{k}+\sum_{g_1=1}^\infty\hbar^{2g_1}\sum_{\substack{a+k=2g_1-2\\a,k\ge0}} (-1)^{g_1+k}T^{g_1-a} V_{k}\int\limits_{\overline{\mathcal{M}}_{g_1,1}}\!\!\!\lambda_{g_1}\lambda_a\psi_1^{k}. \end{align}\tag{8}\]
By construction, the coefficient of any power of \(\hbar\) in \(\widehat P_{k}\) is an element of \(\mathbb{Q}[T,(T-Q)^{-1}]\). The following Proposition expresses the ancestor potential in terms of the Kontsevich-Witten potential \[F^{\rm KW}(t_0,t_1,\dots;\hbar)=\sum_{g,n}\frac{\hbar^{2g-2+n}}{n!} \sum_{k_1,\dots,k_n\ge0} \int_{\overline{\mathcal{M}}_{g,n}}\psi_1^{k_1}\dots\psi_n^{k_n}\;t_{k_1}\dots t_{k_n}.\]
Proposition 9. The following formula holds true \[\begin{gather} \label{eq:TQshifts} F^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm anc}}(t_0,t_1,\dots;\hbar)\approx F^{\rm KW}\bigl(t_0-\tfrac{\widehat P_{0}}{\hbar},t_1-\tfrac{\widehat P_{1}}{\hbar}, t_2-\tfrac{\widehat P_{2}}{\hbar},\dots;(T-Q)\hbar), \end{gather}\qquad{(2)}\] with the following reservation: the symbol \(\approx\) here and below means that the equality holds up to unstable terms with \(2g-2+n\le0\), and also the terms with \(d=0\) or \(n\le1\) and arbitrary \(g\).
The exceptional terms not covered by Proposition 9 are also known; they are treated separately, see below. Note that the shifts of times \(t_k\) to finite values in the Kontsevich-Witten potential are well defined for \(k\ge2\) only. In our case the shifts involve \(\hbar\), and since the expansions of \(\widehat P_0/\hbar\) and \(\widehat P_1/\hbar\) contain strictly positive powers of \(\hbar\) only, the shifts of \(t_0\) and \(t_1\) are also well defined if treated in the expansion in \(\hbar\). This formula expresses each ancestor correlator as a finite combination of intersection numbers of \(\psi\) classes with coefficients in \(\mathbb{Q}[T,(T-Q)^{\pm1}]\). The geometrical origin of the potential implies that the poles at \(T=Q\) cancel out and the coefficients of the potential are actually polynomial in \(T\) and \(Q\). Taking the limit at \(T\to 0\) we get (after including the \(d=0\) terms): \[\begin{align} \widehat P_0\bigm|_{T=0}=\widehat P_1\bigm|_{T=0}&=0;\quad \widehat P_k\bigm|_{T=0}=(-\partial_Q)^{k-2}\bigl(-\tfrac1{Q^2}\bigr)=-\tfrac{(k-1)!}{Q^k}, \quad k \geq 2; \\ \label{eq:Anc-WK} F^{\mathbb{P}^1,\mathcal{O}(-1),{\rm anc}} &= F^{\rm KW}\bigl(t_0,t_1, t_2+\tfrac{1!}{\hbar Q^2},t_3+\tfrac{2!}{\hbar Q^3},t_4+\tfrac{3!}{\hbar Q^4},\dots;-Q\hbar). \end{align}\tag{9}\] As a corollary, we conclude that the right hand side depends regularly on \(Q\). This is one of reformulations of the conjecture in [1], see [1], where we identify \(-Q\) with the parameter \(\epsilon\) there. We prove, thereby, (the numerical part of) this conjecture and ?? provides its one-parameter extension.
Remark 10. Note that formally Equation 9 holds exactly as it is presented, unlike the statement of Proposition 9, where we use the symbol \(\approx\). So it is formally not a direct corollary and some additional analysis of the \(d=0\) and/or \(n\leq1\), and unstable terms is required. We refer for that to Section 2.3 and 2.4.
Remark 11. The equality of Proposition holds, following 6 , in the series expansion in the positive powers of the variable \(Q\). On the other hand, the coefficients on the right hand side in ?? are represented as rational functions in \(Q\) and \(T\). The specializations \(T=0\) for individual terms entering the right hand side of ?? have no cohomological meaning. However, since the coefficients of this function are actually polynomial in \(Q\) and \(T\), we can compute the non-equivariant limit \(T=0\) for the whole function by taking this specialization in the rational coefficients of its particular terms.
Proof of Proposition 9. We start with the descendant correlators \[\begin{align} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm des}}_{g} \mathrel{\vcenter{:}}= \sum_{d=1}^\infty (-1)^{g+1} Q^d\int_{C_{g,n,d}(T)}\psi_1^{k_1}\dots\psi_n^{k_n} \end{align}\] (the nonequivariant limit \(\mathop{\big\lfloor_{{T}\to {0}}}\) of this formula is given by 4 up to \(d=0\) terms).
It will be more convenient for future computations to rewrite ?? in the form \[\begin{gather} \label{eq:C40T41-formula-T} (-1)^{g+1}C_{g,n,d}(T) = T^{-d} \sum_{\Gamma\in \mathrm{SRT}_{g,n,d}} \frac{T^{2g_0-2+n+|E|}}{|E|!}(-1)^{|E|}\times \\ (B_\Gamma)_* \left( \prod_{i=1}^{|E|} \frac{\frac{d_i^{d_i-1}}{d_i!}}{1-\frac{d_i}{T} \psi_{n+i}} \otimes \bigotimes_{i=1}^{|E|} \frac{\bigl(\frac{d_i}{T}\bigr)^2(-T)^{g_i}\lambda_{g_i}\Lambda(\frac{1}{T})}{1+\frac{d_i}{T}\psi_1} \right). \end{gather}\tag{10}\]
Substituting this to the correlators we express them as \[\begin{gather} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm des}}_{g}\\ \approx \sum_{g_0=0}^gT^{2g_0-2+n}\langle \tau_{k_1}\dots\tau_{k_n} [\hbar^{2(g-g_0)}]e^{-T\sum_{k=0}^\infty P_k\tau_k} \rangle^{\rm pt}_{g_0}-T^{2g-2+n}\langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\rm pt}_{g}, \end{gather}\] \[\\P_k=\sum_{d=1}^\infty \Bigl(\frac{Q}{T}\Bigr)^d\;\frac{d^{d-1}}{d!}\;\Bigl(\frac{d}{T}\Bigr)^k\biggl(1+ \sum_{g_1=1}^\infty\hbar^{2g_1}\!\!\!\int\limits_{\overline{\mathcal{M}}_{g_1,1}}\!\!\! \frac{\bigl(\frac{d}{T}\bigr)^2(-T)^{g_1}\lambda_{g_1}\Lambda(\frac{1}{T})}{1+\frac{d}{T}\psi_1} \biggr).\] Here by \(\langle\cdot\rangle^{\rm pt}\) we denote the Kontsevich-Witten correlators, that is, those enumerating intersection numbers of just \(\psi\) classes over moduli spaces of curves. Again, the symbol \(\approx\) means that the equality is not applicable in the cases \(n\le1\); these cases should be treated separately. Then, the obtained expression for the correlators leads to the corresponding formula for the potential: \[\begin{align} F^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm des}}(t_0,t_1,\dots;\hbar)&= \sum_{g,n}\frac{\hbar^{2g-2+n}}{n!} \sum_{k_1,\dots,k_n\ge0} \langle \tau_{k_1}\dots\tau_{k_n}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm des}}_{g}t_{k_1}\dots t_{k_n} \\\notag&\approx e^{-\hbar^{-1}\sum_{k=0}^\infty P_k\partial_{t_k}} F^{\rm KW}\bigl(t_0,t_1,\dots;T\hbar) \\\notag&= F^{\rm KW}\bigl(t_0-\tfrac{P_0}{\hbar},t_1-\tfrac{P_1}{\hbar},\dots;T\hbar). \end{align}\] Next, passing from the descendant to the ancestor potential [14], [15] results in just a linear change of coordinates (with a special treatment of the unstable terms): \[\begin{align} \label{eq:ansc-desc} F&{}^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm anc}}\approx F^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm des}}\bigm|_{t_k\to\sum_{m=0}^\infty \frac{Q^m}{m!}t_{k+m}} \\\notag &\approx F^{\rm KW}\left(\sum_{m=0}^\infty\tfrac{Q^m}{m!} t_m-\tfrac{P_0}{\hbar},\sum_{m=0}^\infty \tfrac{Q^m}{m!}t_{1+m}-\tfrac{P_1}{\hbar},\dots;T\hbar\right) \\\notag &= F^{\rm KW}\left(\sum_{m=0}^\infty \tfrac{Q^m}{m!}\bigl(t_m-\tfrac{\widetilde{P}_m}{\hbar}\bigr),\sum_{m=0}^\infty \tfrac{Q^m}{m!}\bigl(t_{1+m}-\tfrac{\widetilde{P}_{1+m}}{\hbar}\bigr),\dots;T\hbar\right), \end{align}\tag{11}\] where \[\begin{align} \widetilde{P}_k&=\sum_{m=0}^\infty \frac{(-Q)^m}{m!}P_{k+m} \\\notag &=\sum_{m=0}^\infty\sum_{d=1}^\infty \Bigl(\frac{Q}{T}\Bigr)^d\;\frac{d^{d-1}}{d!}\frac{(-Q)^m\bigl(\frac{d}{T}\bigr)^m}{m!}\Bigl(\frac{d}{T}\Bigr)^k\biggl(1+ \sum_{g_1=1}^\infty\hbar^{2g_1}\!\!\!\int\limits_{\overline{\mathcal{M}}_{g_1,1}}\!\!\! \frac{\bigl(\frac{d}{T}\bigr)^2(-T)^{g_1}\lambda_{g_1}\Lambda(\frac{1}{T})}{1+\frac{d}{T}\psi_1} \biggr) \\\notag &=\sum_{d=1}^\infty X^d\;\frac{d^{d-1}}{d!}\Bigl(\frac{d}{T}\Bigr)^k\biggl(1+ \sum_{g_1=1}^\infty\hbar^{2g_1}\!\!\!\int\limits_{\overline{\mathcal{M}}_{g_1,1}}\!\!\! \frac{\bigl(\frac{d}{T}\bigr)^2(-T)^{g_1}\lambda_{g_1}\Lambda(\frac{1}{T})}{1+\frac{d}{T}\psi_1} \biggr), \\\notag X&=\tfrac{Q}{T}e^{-\frac{Q}{T}}. \end{align}\] Using known properties of the Lambert function given in our case by the equation \(X=\tfrac{Q}{T}e^{-\frac{Q}{T}}\), including the identity \(T^{-1}X\frac{d}{dX}=\frac{Q}{T-Q}\frac{d}{dQ}\), we compute \[\begin{align} \sum_{d=1}^\infty X^d\frac{d^{d-1}}{d!}&=\frac{Q}{T}=V_0, \\\sum_{d=1}^\infty X^d\frac{d^{d-1}}{d!}\left(\frac{d}{T}\right)^k&=\left(T^{-1}X\frac{d}{dX}\right)^k\sum_{d=1}^\infty X^d\frac{d^{d-1}}{d!}= \left(\frac{Q}{T-Q}\frac{d}{dQ}\right)^k\frac{Q}{T}=V_k. \end{align}\] This gives \[\widetilde{P}_k=\widehat P_k+\delta_{k,0}\frac{Q}{T}+\delta_{k,1}\Bigl(\frac{1}{T-Q}-\frac{1}{T}\Bigr),\] where \(\widehat P_k\) is given by 8 . Applying the string and dilaton equations to the Kontsevich-Witten potential we compute finally: \[\begin{align} & F^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm anc}}(t_0,t_1,\dots;\hbar) \\\notag & \qquad \textstyle \approx F^{\rm KW}\bigl(\sum_{m=0}^\infty \tfrac{Q^m}{m!}\bigl(t_m-\tfrac{\widetilde{P}_m}{\hbar}\bigr),\sum_{m=0}^\infty \tfrac{Q^m}{m!}\bigl(t_{1+m}-\tfrac{\widetilde{P}_{1+m}}{\hbar}\bigr),\dots;T\hbar) \\\notag& \qquad \approx F^{\rm KW}\bigl(t_0-\tfrac{\widetilde{P}_0}{\hbar}+\tfrac{Q}{T\hbar},t_{1}-\tfrac{\widetilde{P}_1}{\hbar}, t_{2}-\tfrac{\widetilde{P}_2}{\hbar},\dots;T\hbar) \\\notag& \qquad \approx F^{\rm KW}\bigl(t_0-\tfrac{\widetilde{P}_0}{\hbar}+\tfrac{Q}{T\hbar}, t_{1}-\tfrac{\widetilde{P}_1}{\hbar}+\tfrac{1}{\hbar}\bigl(\tfrac{1}{T-Q}-\tfrac1{T}\bigr), t_{2}-\tfrac{\widetilde{P}_2}{\hbar},\dots;(T-Q)\hbar) \\\notag& \qquad \approx F^{\rm KW}\bigl(t_0-\tfrac{\widehat P_0}{\hbar},t_{1}-\tfrac{\widehat P_1}{\hbar}, t_{2}-\tfrac{\widehat P_2}{\hbar},\dots;(T-Q)\hbar). \end{align}\] ◻
Remark 12. Note that both \(F^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm des}}\) and \(F^{\mathbb{P}^1,\mathcal{O}(-1),\mathbb{C}^*,{\rm anc}}\), up to controllable corrections, are solutions of the KdV integrable hierarchy, therefore can be considered in the framework of reductions of both KP and BKP hierarchies.
In this section we compute the class \(C_{g,n}(T,Q)\in R^*(\overline{\mathcal{M}}_{g,n})\otimes \mathbb{Q}[T,T^{-1}][[Q]]\) itself. This computation involves the string and the dilaton equation and is known in the realm of localization formulas under the name “materialization” [14], [16]. In order to describe the answer, we need an “\(\overline{\mathcal{M}}_{g,n}\) version” of star rooted trees \(\mathrm{SRT}_{g,n}\). For completeness, we give a full definition that largely repeats the one given above for \(\mathrm{SRT}_{g,n}(\mathbb{P}^1,d)\).
Let \(\mathrm{SRT}_{g,n}\) denote the set of stable graphs of genus \(g\) with \(n\) legs, which are the graphs \[\begin{align} \Gamma=(V,H,\iota \colon H\to H, H^{\iota}\simeq \{\sigma_1,\ldots, \sigma_{n}\}, v_0\in V, o\colon E\to \llbracket {|E|} \rrbracket), \end{align}\] where \(V\), \(H\), \(\iota\), and \(E\) are the vertices, half-edges, involution, and edges, as above, and \(g\colon V\to \mathbb{Z}_{\geq 0}\) is again the genus function. As before, the graph \((V,E)\) must be a star rooted tree with all legs \(\sigma_{1},\dots,\sigma_{n}\) attached to the root vertex \(v_0\in V\). The valence of a vertex \(v\) is denoted by \(n(v)\), and we demand that all vertices are stable, that is, \(2g(v)-2+n(v)>0\).
Note that we still have \(\sum_{v\in V} g(v)=g\) and the edges \(E\) and the non-root vertices \(V\setminus \{v_0\}\) are ordered by \(o\colon E\to \llbracket {|E|} \rrbracket\); \(e_i\), resp. \(v_i\) denotes the \(i\)-th edge, resp., the \(i\)-th non-root vertex attached to \(e_i\). Let \(g_i=g(v_i)\), \(i=0,\dots,|E|\). Note that the stability condition implies \(g_i>0\) for \(i\ne0\).
We want to assign to \(\Gamma\) a class in \(R^*( \overline{\mathcal{M}}_{g,n}) \otimes_{\mathbb{Q}} \mathbb{Q}[T,T^{-1}][[Q]]\). Let \(b_\Gamma\) be the boundary map \(\overline{\mathcal{M}}_{g_0,n+|E|}\times \prod_{i=1}^{|E|} \overline{\mathcal{M}}_{g_i,1}\to \overline{\mathcal{M}}_{g,n}\). As before, the \(n+|E|\) marked points on the first irreducible component are ordered such that the first \(n\) marked points correspond to the legs and the last \(|E|\) to the edges/nodes, respectively, preserving their given ordering. We decorate the vertices and half-edges of \(\Gamma\) by \(\psi\)-, \(\kappa\)-, and \(\lambda\)-classes in the following way:
The component corresponding to the root vertex is endowed with the class \[\label{eq:root-contribution} \mathcal{V}\mathrel{\vcenter{:}}= (T-Q)^{2g_0-2+n+|E|} \Big( 1+\sum_{m=1}^\infty\frac{(-1)^m(T-Q)^m}{m!}\sum_{i_1,\dots,i_m\ge1}V_{i_1+1}\dots V_{i_m+1}\,\kappa_{i_1,\dots,i_m}\Big),\tag{12}\] which can alternatively be written as \[\begin{gather} (T-Q)^{2g_0-2+n+|E|} \exp\Big({\sum_{i=1}^\infty s_i\kappa_i}\Big) \end{gather}\] for the coefficients \(s_i\), \(i\geq 1\) determined by \(e^{-\sum_{i=i}^\infty s_it^i}=1+\sum_{k=1}^\infty (T-Q)V_{k+1}t^k\).
For each edge \(e_i\), \(i\in \llbracket {|E|} \rrbracket\), we combine the \(\psi\)-classes \(\psi'\) and \(\psi''\) assigned to its half-edges attached to \(v_0\) and \(v_i\), respectively, as well as the \(\lambda\)-classes assigned to \(v_i\), into the following single expression: \[\label{eq:edge-contribution} \mathcal{E}_i\mathrel{\vcenter{:}}= \sum_{k,l=0}^\infty\sum_{a=0}^{g_i-1} (-1)^{g_i+l+1} T^{g_i-a} V_{k+l+2}\; (\psi')^{k}\otimes(\psi'')^{l} \lambda_{g_i}\lambda_{a}.\tag{13}\] (here \(\psi'\) turns into \(\psi_{n+i}\) on the component corresponding to \(v_0\) and \(\psi''\) is the \(\psi_1\) on the component corresponding to \(v_i\), \(i=1,\dots,|E|\)).
The coefficients \(V_k\) are given by 7 which we remind here for convenience: \[\label{eq:Vk-reminder} V_k=\sum_{d=1}^\infty X^d \frac{d^{d-1}}{d!}\Bigl(\frac{d}{T}\Bigr)^k =\Bigl(\frac{Q}{T-Q}\frac{d}{dQ}\Bigr)^k\frac{Q}{T}, \quad X=\frac{Q}{T}e^{-\frac{Q}{T}}.\tag{14}\] This formula implies that \(V_k\) is expressed as a rational function in \(T\) and \(Q\), moreover, \(V_k\in \mathbb{Q}[T,(T-Q)^{-1}]\) for \(k\ge2\) and \(V_{1} = \frac{Q}{T(T-Q)}\). In an exceptional case \(n=0\) of the formula below we will need also the functions \[\label{eq:Vkl-reminder} V_{k,l}=\sum_{d_1,d_2=1}^\infty X^{d_1+d_2} \frac{T^{-1}d_1^{d_1}d_2^{d_2}}{(d_1+d_2)d_1!d_2!} \Bigl(\frac{d_1}{T}\Bigr)^k\Bigl(\frac{d_2}{T}\Bigr)^l =\Bigl(\frac{Q}{T-Q}\frac{d}{dQ}\Bigr)^{-1}V_{k+1}V_{l+1},\tag{15}\] where by \((Q/(T-Q)\partial_Q)^{-1}\) we mean formal integration without the constant term, that is, the operator \(\int_0^Q \frac{T-Q}{Q}(\cdot)dQ\).
Proposition 13. We have: \[\begin{align} \label{eq:MainFormula-Cgn} \notag C_{g,n}(T,Q) & = \sum_{\Gamma\in\mathrm{SRT}_{g,n}} \frac{1}{|E|!} (b_\Gamma)_* \Big(\mathcal{V}\prod_{i=1}^{|E|}\mathcal{E}_i\Big) + \delta_{n,1}\sum_{a=0}^{g-1} (-1)^{g} T^{g-a}{\lambda_g \lambda_a} \sum_{l=0}^\infty (-1)^{l+1}V_{l+1}\, {\psi_1^{l}} \\ & \qquad +\delta_{n,0}\sum_{a=0}^{g-1} (-1)^{g} T^{g-a}{\lambda_g \lambda_a} \\\notag &\qquad\qquad \times\biggl(\sum_{l=0}^\infty (-1)^{l} \bigl(\tfrac{V_{l+1}}{T-Q}-V_{1,l}\bigr)\, \kappa_{l} +\frac{1}{2} \sum_{k,l=0}^\infty (-1)^{k+l}V_{k+1,l+1} b_* \bigl((\psi')^{k}\otimes (\psi'')^{l}\bigr)\biggr). \end{align}\qquad{(3)}\] For the exceptional terms we use here \(\kappa_0|_{\overline{\mathcal{M}}_{g,0}} = 2g-2\); \(b\) is the boundary morphism providing a \(2:1\) parametrization of the boundary divisor, and \(\psi',\psi''\) are the \(\psi\) classes associated with the two branches of a curve at its singular point.
Proof. It is convenient to abuse the notation and to think of \(p_* \sum_{d=1}^\infty (-1)^{g+1}Q^d C_{g,n,d}(T)\) rather than of \(\sum_{d=1}^\infty (-1)^{g+1} Q^d p_* C_{g,n,d}(T)\), though it is not a single push-forward but rather a system of push-forwards that depends on \(d\). This allows us to take the sum over all labels \(d\colon E\to \mathbb{Z}_{\geq 1}\) for each given pre-stable star rooted tree, with no condition on the sum of \(d(e)\) over \(e\in E\). For the class ?? or 10 rewritten in this way we assemble the dependence on \(Q\) for each edge \(e\) of genus \(g(e)=h\) into the series \[\label{eq:edge} \begin{gather} \sum_{d=1}^\infty \frac{d^{d-1}}{d!} \Big(\frac{Q}{T}\Big)^d \sum_{k,l=0}^\infty \sum_{a=0}^{h-1} (-1)^{h+l+1} T^{h-a}\Bigl(\frac{d}{T}\Bigr)^{k+l+2} (\psi')^{k} \otimes (\psi'')^l\lambda_h \lambda_a |_{\overline{\mathcal{M}}_{h,1}},\quad h\ge1, \\ -\sum_{d=1}^\infty \frac{d^{d-1}}{d!} \Big(\frac{Q}{T}\Big)^d \sum_{k=0}^\infty \Bigl(\frac{d}{T}\Bigr)^{k} (\psi')^{k},\quad h=0. \end{gather}\tag{16}\] The remaining dependence on \(T\) is the overall factor \(T^{2g_0-2+n+|E|}\) assigned to the root vertex.
Under the push-forward to \(\overline{\mathcal{M}}_{g,n}\) we firstly have to use the string and the dilaton equations to eliminate all contributions as in the summands with \(k=0\) and \(k=1\) on the second line, cf. [16]. By the computations similar to those of the previous section, the string equation amounts to the formal substitution \(Q/T \mapsto Q/Te^{-Q/T}\). Note that if \(X=Q/Te^{-Q/T}\), then \(\sum_{d=1}^\infty \frac{d^{d-1}}{d!} X^d=Q/T\) and \(T^{-1}X\partial_X=Q/(T-Q)\partial_Q\). The dilaton equation multiplies the root vertex by \((1-Q/T)^{2g_0-2+n+|E|}\). Thus the combined effect of the string and the dilaton equation implies the following intermediate form of the edge contributions under the push-forward: \[\label{eq:edge-string} \begin{gather} \sum_{k,l=0}^\infty \sum_{a=0}^{h-1} (-1)^{h+l+1} T^{h-a} V_{k+l+2}~ (\psi')^{k} \otimes (\psi'')^l\lambda_h \lambda_a |_{\overline{\mathcal{M}}_{h,1}},\quad h\ge1, \\ - \sum_{k=2}^\infty V_k~ (\psi')^{k},\quad h=0, \end{gather}\tag{17}\] where \(V_k\) is the series in \(X=Q/Te^{-Q/T}\) given by 14 , while the root vertex is further equipped with the coefficient \((T-Q)^{2g_0-2+n+|E|}\).
At the next step we forget all edges with unstable vertices, which does not change the first line in 17 but converts the second line into the following contribution to the root vertex expressed in terms of the kappa classes on the moduli space corresponding to the root vertex: \[\begin{align} 1+\sum_{m=1}^\infty \frac{(-1)^m (T-Q)^m}{m!} \sum_{a_1,\dots,a_m\geq 1} \kappa_{a_1,\dots,a_m} \prod_{i=1}^m V_{a_i+1}. \end{align}\]
Note that there is an exceptional case when within the push-forward the moduli space corresponding to the root vertex becomes unstable (or it was unstable initially). In this case, either \(n=0\) or \(n=1\). If \(n=1\), the above computation gives the following class on \(\overline{\mathcal{M}}_{g,1}\): \[\begin{align} \label{eq:exceptional} & \sum_{l=0}^\infty \sum_{a=0}^{g-1} (-1)^{g+l+1} T^{g-a} V_{l+1}~ \psi_1^{l} \lambda_g \lambda_a . \end{align}\tag{18}\]
If \(n=0\), then we have to handle the cases of graphs in \(\mathrm{SRT}_{g,0}(\mathbb{P}^1,d)\) where \(g_0=0\) and at most two non-root vertices have positive genus. In this case, we can use directly the materialization equations for the unstable cases, see [16], which give the exceptional \(\delta_{n,0}\) terms in the form \[\begin{gather} \sum_{l=0}^\infty \sum_{a=0}^{g-1} (-1)^{g+l} T^{g-a} \bigl(\tfrac{V_{l+1}}{T-Q}-V_{1,l}\bigr)\; \kappa_{l} {\lambda_g \lambda_a} + \frac{1}{2} \sum_{\substack{g_1+g_2=g\\g_1,g_2\geq 1}} \sum_{l_1,l_2=0}^\infty \sum_{a_1=0}^{g_1-1}\sum_{a_2=0}^{g_2-1} \\ (-1)^{g+l_1+l_2} T^{g-a_1-a_2}V_{l_1+1,l_2+1}~ (b_{g_1,g_2})_* \Big(\psi_1^{l_1} \lambda_{g_1}\lambda_{a_1}\otimes \psi_1^{l_2}\lambda_{g_2}\lambda_{a_2}\Big). \end{gather}\] where \(b_{g_1,g_2}:\overline{\mathcal{M}}_{g_1,1}\times\overline{\mathcal{M}}_{g_2,1}\to\overline{\mathcal{M}}_{g,0}\) is the boundary morphism and \(V_{k,l}\) is a series in \(X=Q/Te^{-Q/T}\) given by 15 . This expression is equivalent to that one of ?? by the behavior of the \(\lambda\) classes under restriction to the boundary strata.
Alternatively, one can use the following trick: exactly the same graphs but with one extra leg attached are also present in \(\mathrm{SRT}_{g,0}(\mathbb{P}^1,d)\), and the coefficients are proportional with the factor of \(d\) (the latter observation is a corollary of the string equation, \(\int_{\overline{\mathcal{M}}_{0,|E|}} \prod_{i=1}^{|E|} (1-d_i\psi_i)^{-1} / \int_{\overline{\mathcal{M}}_{0,1+|E|}} \prod_{i=1}^{|E|} (1-d_i\psi_{1+i})^{-1} = 1/d\)). Thus, we can obtain the coefficients for these exceptional cases by applying \((Q\partial_Q)^{-1}\), which gives the same answer as a direct application of the materialization equations for the unstable cases.
An observation on the equivalence of the functions \(V_k\) and \(V_{k,l}\) represented as series in \(X=Q/Te^{-Q/T}\) to their representations as rational functions in \(T\) and \(Q\) completes the proof. ◻
Denote by \(C^{(p)}_{g,n}(T,Q)\) the homogeneous component of (cohomological) degree \(p\) in the class \(C_{g,n}(T,Q)\), that is, \(C^{(p)}_{g,n}(T,Q)\in R^{p}(\overline{\mathcal{M}}_{g,n})[T,Q]\). By construction, it is a polynomial in \(T\) and \(Q\) of homogeneous degree \(2g-2+n-p\). In particular, it identically vanishes if \(p > 2g-2+n\). The goal of this section is to analyze the tautological relations that these vanishings generate.
We use notation \(S\mathrel{\vcenter{:}}= T-Q\). The dependence of the terms entering the main formula ?? on \(T\) and \(Q\) is through the functions \(V_k\) and \(V_{k,l}\) whose definitions in terms of \(T\) and \(S\) reads \[\begin{align} V_k&=\Bigl(\frac{S-T}{S}\partial_S\Bigr)^k\frac{T-S}{T},\\ V_{k,l}&=\Bigl(\frac{S-T}{S}\partial_S\Bigr)^{-1}V_{k+1}V_{l+1}, \end{align}\] where the integration constant in the last formula is chosen such that \(V_{k,l}|_{S=T}=0\).
Lemma 1. All the functions \(V_k\), \(k\ge1\), and \(V_{k,l}\), \(k,l\ge1\) are Laurent polynomials in \(T\) and \(S\) of homogeneous degrees \(-k\) and \(-k-l-1\), respectively.
Corollary 1. The contribution of each graph in \(\mathrm{SRT}_{g,n}\) and each exceptional summand in ?? to \(C_{g,n}^{(p)}(T,Q)\) is a Laurent polynomial in \(T\) and \(S\) of homogeneous degree \(2g-2+n-p\).
The Laurent polynomials in \(T\) and \(S=T-Q\) should be thought of as infinite power series in \(Q\). However, since the total polynomial \(C_{g,n}^{(p)}(T,Q)\) is represented as a finite sum of Laurent polynomials, the total contribution of each monomial in \(T\) and \(S\) which is of negative degree in either \(T\) or \(S\) should vanish, hence, it provides a tautological relation in \(R^{p}(\mathcal{M}_{g,n})\). In particular, if \(p>2g-2+n\), then the vanishing of the total contribution of each monomial gives a tautological relation.
In fact, with certain exception for \(n=0\), the monomials with negative exponent of \(T\) never appear in computations, as the following lemma shows.
Lemma 2. We have:
\(V_1=\frac{1}{S}-\frac{1}{T}\) and for \(k\ge2\) the Laurent polynomial \(V_k\) is regular in \(T\), that is, \[V_k\in\mathbb{Q}[S^{-1},T],\quad k\ge2.\] Moreover, \[\label{eq:Vk-at-T610} V_k\bigm|_{T=0}=(-1)^{k-1}\frac{(k-1)!}{S^k},\quad k\ge2.\tag{19}\]
\(V_{1,0}=\frac{1}{2T^2}-\frac{1}{ST}+\frac{1}{2S^2}\), and for \(k,l\ge1\), up to one explicitly given monomial in \(T\), the function \(V_{k,l}\) is also regular in \(T\): \[\label{eq:Vkl-at-T610} V_{k,l}-\frac{(-1)^{k+1}B_{k+l}}{(k+l)T^{k+l+1}}\in\mathbb{Q}[S^{\pm1},T],\quad k,l\ge1.\tag{20}\]
Here \(B_l\), \(l\geq 0\), are the Bernoulli numbers given by \[\begin{align} \sum_{l=0}^\infty B_l \frac{t^l }{l!}= \frac{te^t}{e^t-1}. \end{align}\]
The proof of both lemmas are straightforward elementary computations. 0◻
Using the last lemma, one can extract some particular coefficients of monomials in \(T\) and \(S=T-Q\) in ?? . Our special interest is the coefficient of \(T^0S^d\) for different \(d\) (both positive and negative).
Lemma 3. For any \(d\in\mathbb{Z}\), the coefficient of the monomial \(T^0S^d\) in the expression for \(C_{g,n}(T,Q)\) suggested by ?? is equal to \[\label{eq:T610contribution} J_{2g-2+n-d}+\delta_{n,1}\delta_{d,0}(-1)^{g}\lambda_g\lambda_{g-1} +\delta_{n,0}\delta_{d,0}(-1)^{g-1}\lambda_g\lambda_{g-2}\in R^{2g-2+n-d}(\overline{\mathcal{M}}_{g,n}),\tag{21}\] where the \(J_p\) are the polynomials in \(\kappa\) classes defined by 1 –3 .
Proof. Since \(V_k\) is regular in \(T\) for \(k\ge2\), all summands in the main term in ?? are also regular in \(T\). Moreover, since the exponent \(g_i-a\) in 13 is strictly positive, all graphs in \(\mathrm{SRT}_{g,n}\) with a nonempty edge set provide trivial contributions to the coefficient of \(T^0\). The only graph that provides a nontrivial contribution is the graph with a single vertex of genus \(g\) and no edges. By 12 and 19 , the contribution of this graph is equal exactly to the corresponding \(J\)-class.
By the same reason, a nontrivial contribution of the exceptional terms for \(n=1\) in ?? to the coefficient of \(T^0\) is possible for \(l=0\) and \(a=g-1\) only, and this contribution is equal to \((-1)^{g}\lambda_g\lambda_{g-1}\).
In opposite to the cases considered above, the exceptional summand with \(n=0\) in ?? is not regular in \(T\), so we need a more careful computation here. Using 20 and ignoring the terms regular in \(T\) in the third line in ?? , we compute the contribution of the exceptional summand with \(n=0\) to the coefficient of \(T^0\) equal to \[\begin{gather} \label{eq:n610proof1} [T^0]\sum_{a=0}^{g-1} (-1)^{g} T^{g-a}{\lambda_g \lambda_a} \biggl(\frac{\kappa_0}{2T^2}+\sum_{l=1}^\infty \frac{B_{l+1}}{(l+1)T^{l+2}}\, \kappa_{l} \biggr. \\ +\biggl.\frac{1}{2} \sum_{k,l=0}^\infty (-1)^{k}\frac{B_{k+l+2}}{(k+l+2)T^{k+l+3}} b_* \bigl((\psi')^{k}\otimes (\psi'')^{l}\bigr)\biggr) \\=(1-g)(-1)^g\lambda_g\lambda_{g-2}+[T^0]\sum_{a=0}^{g-1} (-1)^{g} T^{g-a-2}{\lambda_g \lambda_a} \sum_{k=1}^\infty\frac{B_{2k}}{2k}\frac{\mu_{2k-1}}{T^{2k-1}}, \end{gather}\tag{22}\] where the classes \(\mu_l\) for odd \(l\) are defined by \[\mu_l=\kappa_l+\frac{1}{2}b_*\frac{(\psi')^l+(\psi'')^l}{\psi'+\psi''}.\] Recall that by Mumford’s formula [17], we have \[e^{\sum_{k=1}^\infty\frac{B_{2k}}{(2k-1)2k}\mu_{2k-1}t^{2k-1}}=\sum_{a=0}^g\lambda_at^a.\] Applying \(t\partial_t\) we get as a corollary the identity \[\biggl(\sum_{k=1}^\infty\frac{B_{2k}}{2k}\mu_{2k-1}t^{2k-1}\biggr) \sum_{a=0}^g\lambda_gt^g=\sum_{a=0}^ga\,\lambda_at^a.\] By this identity, 22 can be rewritten as \[\begin{gather} (1-g)(-1)^g\lambda_g\lambda_{g-2}+[T^0](-1)^{g} T^{g-2}\lambda_g \sum_{a=0}^ga\,\frac{\lambda_a}{T^a}\\ =(1-g)(-1)^g\lambda_g\lambda_{g-2}+(g-2)(-1)^g\lambda_g\lambda_{g-2}=-(-1)^{g}\lambda_g\lambda_{g-2}, \end{gather}\] which agrees with 21 . ◻
For \(d<0\) the equality of Lemma 3 is a tautological relation: the total coefficient of \(T^0S^d\) in the expression for the class \(C_{g,n}(T,Q)\) for negative \(d\) must be equal to zero.
If \(d=0\), then the monomial \(T^0S^d\) is a constant. Hence, the total contribution to this monomial should also vanish since the class \(C_{g,n}(T,Q)\) does not contain the constant terms in the expansion in \(Q\), by definition.
This gives immediately the proofs of Theorems 1 and 2.
In the case \(d>0\), Lemma 3 is also applicable. In this case, it provides not a tautological relation any more but rather an explicit formula for the corresponding term of \(C_{g,n}\) immediately implying the \(d>0\) part of the statement of Theorem 3.
Finally, for \(d=0\) case of Theorem 3, we compute the classes explicitly without applying localization, just directly from its definition (see [6], [18] and [3]). To this end, we use the following trick: we consider \((\mathbb{P},\mathcal{O}(-1))\) as the normal bundle of the exceptional divisor \(\tilde{X}\) in \(\tilde{Y}\) obtained as a blow-up at one point of a K3 surface \(Y\). Then we can just apply the computation of [19]. We have \(c_1(\tilde{Y}) = -[\tilde{X}]\), and \([\overline{\mathcal{M}}_{g,n}(\tilde{Y},0)]^{\mathrm{vir}} = [\overline{\mathcal{M}}_{g,n} \times \tilde{Y}] \cap (\lambda_g\lambda_{g-1} [\tilde{X}] + \lambda_{g}\lambda_{g-2} [\tilde{X}]^2)\). Then for \(p\colon \overline{\mathcal{M}}_{g,n} \times \tilde{Y} \to \overline{\mathcal{M}}_{g,n}\) we have \[\begin{align} \label{eq:d0-comuptations-v1} (-1)^{g+1} p_* \Big( [\overline{\mathcal{M}}_{g,1} \times \tilde{Y}] \cap (\lambda_g\lambda_{g-1} [\tilde{X}] + \lambda_{g}\lambda_{g-2} [\tilde{X}]^2) \cap ev_1^*(-[\tilde{X}]) \Big) & = (-1)^{g-1} \lambda_{g} \lambda_{g-1}; \\ \notag (-1)^{g+1} p_* \Big( [\overline{\mathcal{M}}_{g,0} \times \tilde{Y}] \cap (\lambda_g\lambda_{g-1} [\tilde{X}] + \lambda_{g}\lambda_{g-2} [\tilde{X}]^2) \Big) & = (-1)^{g} \lambda_g \lambda_{g-2}. \end{align}\tag{23}\] 0◻
Remark 14. Alternatively, one can use [6] in combination with [6]. This theorem states that in the case of \(d=0\) we have to replace the class \(c_{g-1+d} (R^1\pi_*f^*\mathcal{O}(-1))\) in Equation 5 by the class \(-c_{g-1} (R^1\pi_*f^*\mathcal{O}(-1) - R^0\pi_* f^*\mathcal{O}(1))\). On \(\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,0) = \overline{\mathcal{M}}_{g,n}\times \mathbb{P}^1\) the latter class is equal to \((-1)^{g-1}(\lambda_{g-1}\otimes 1 + 3 \lambda_{g-2} \otimes [\mathrm{pt}])\). Recall that \([ \overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,0)]^{\mathrm{vir}} = (-1)^{g-1} (\lambda_{g}\otimes 1 - 2 \lambda_{g-1}\otimes [\mathrm{pt}])\) [19] and in the case \(n=1\) we have \(ev_1^*([\mathrm{pt}])= 1\otimes [\mathrm{pt}]\). Then for \(p\colon \overline{\mathcal{M}}_{g,n}\times \mathbb{P}^1\to \overline{\mathcal{M}}_{g,n}\), for \(n=0,1\), we have: \[\begin{align} (-1)^{g+1} p_* \Big([\overline{\mathcal{M}}_{g,1}\times \mathbb{P}^1]\cap (1\otimes [\mathrm{pt}]) \cdot (-1)^{g-1} (\lambda_{g}\otimes 1 - 2 \lambda_{g-1}\otimes [\mathrm{pt}]) \cdot \quad & \\ \notag (-1)^{g-1}(\lambda_{g-1}\otimes 1 + 3 \lambda_{g-2} \otimes [\mathrm{pt}]) \Big) & = (-1)^{g-1} \lambda_g\lambda_{g-1}; \\ \notag (-1)^{g+1} p_* \Big([\overline{\mathcal{M}}_{g,0}\times \mathbb{P}^1]\cap (-1)^{g-1} (\lambda_{g}\otimes 1 - 2 \lambda_{g-1}\otimes [\mathrm{pt}]) \cdot \quad & \\ \notag (-1)^{g-1}(\lambda_{g-1}\otimes 1 + 3 \lambda_{g-2} \otimes [\mathrm{pt}]) & = (-1)^g \lambda_g\lambda_{g-2} \end{align}\] as we have already checked in Equation 23 .
The purpose of this section is to explain where the computations of the previous section are originated from. The original conjecture on kappa classes in [1] is linked in op. cit. to a regularity question of a system of differentials obtained by topological recursion of Chekhov-Eynard-Orantin [4]. In this section we give a reformulation of the setup for topological recursion proposed in [1] in terms of the so-called \(\mathbb{Z}_2\)-equivariant topological recursion introduced in [5].
We prove that this \(\mathbb{Z}_2\)-equivariant topological recursion describes the stationary sector of the spin Gromov-Witten theory of \(\mathbb{P}^1\), thus establishing the regularity proposed in [1]. This proof is in fact a numerical avatar of the argument in the previous section that we presented in Section 2.2.
First, let us recall the definition of the original topological recursion of Chekhov-Eynard-Orantin [4], [20]. It associates a system of meromorphic differentials \(\omega^{(g)}_n\) (also known as \(n\)-point differentials), \(g\geq 0\), \(n\geq 1\), \(2g-2+n\geq 0\), to an input data (the spectral curve data) that consists of a Riemann surface \(\Sigma\) and a finite set of points \(\mathcal{P}\subset \Sigma\), two functions \(x\) and \(y\) on \(\Sigma\) such that \(dx\) and \(dy\) are meromorphic differentials on \(\Sigma\) with \(q\in\mathcal{P}\) being simple critical points of \(x\), and \(\mathop{\big\lfloor_{{}\to {q}}} {dy}\not= 0\) for each \(q\in \mathcal{P}\), and a bi-differential \(B\) with a double pole on the diagonal with biresidue \(1\) and no other poles.
The symmetric differentials \(\omega^{(g)}_{n}\), \(g\geq 0\), \(n\geq 1\), are constructed as follows: \[\begin{align} \label{eq:om0102} \omega_{0,1}(z_1) = y(z_1) dx(z_1) ; \qquad \omega_{0,2}(z_1,z_2) = B(z_1,z_2); \end{align}\tag{24}\] and for \(2g-2+n>0\) the following recursion is used: \[\begin{align} \label{eq:TopologicalRecursion} \omega_{g,n} (z_1,\dots,z_n) \mathrel{\vcenter{:}}= \;& \frac{1}{2} \sum_{\xi\in\mathcal{P}} \mathop{\rm res}_{z\to \xi} \frac{\int_z^{\sigma_\xi(z)} B(z_1,\cdot)}{(y(\sigma_\xi(z)) - y(z))\,dx(z) }\Bigg( \omega_{g-1,n+1}(z,\sigma_\xi(z),z_{\llbracket n \rrbracket \setminus \{1\}}) \\ \notag & + \sum_{\substack{g_1+g_2 = g, I_1\sqcup I_2 = {\llbracket n \rrbracket \setminus \{1\}} \\ (g_1,|I_1|),(g_2,|I_2|) \not= (0,0) }} \omega_{g_1,1+|I_1|}(z,z_{I_1})\omega_{g_2,1+|I_2|}(\sigma_\xi(z), z_{I_2})\Bigg), \end{align}\tag{25}\] where \(\sigma_\xi\) is the deck transformation of \(x\) near \(\xi\) (that is, it is a local involution such that \(x(\sigma_\xi(z))=x(z)\)).
Now recall the definition of equivariant topological recursion introduced by Giacchetto-Kramer-Lewański [5]:
Definition 1. Let \(G\) be a finite group. A \(G\)-equivariant spectral curve data \[\begin{align} (\Sigma,\phi, x,y,B, \chi, \upsilon, \beta) \end{align}\] consists of
a Riemann surface \(\Sigma\) (not necessarily compact or connected) with a free action \(\phi \colon G \times \Sigma \to \Sigma\), for which the following notation is used: \(\phi (\gamma,z) = \phi_\gamma z = \gamma z\), \(\gamma\in G\), \(z\in \Sigma\);
a function \(x \colon \Sigma \to \mathbb{C}\) such that its differential \(dx\) is meromorphic and has finitely many zeros \(a_1,\dots,a_r\) that are simple;
a meromorphic function \(y \colon \Sigma \to \mathbb{C}\) such that \(d y\) does not vanish at the zeros of \(d x\);
a symmetric bidifferential \(B\) on \(\Sigma \times \Sigma\);
three one-dimensional representations \(\chi, \upsilon, \beta \colon G \to \mathbb{C}^*\), such that for any \(\gamma \in G\) \[\label{Eqdefs} dx (\gamma z) = \chi_\gamma \, dx (z) \,, \qquad dy (\gamma z) = \upsilon_\gamma \, dy(z)\,, \qquad B(\gamma z_1,z_2) = \beta_\gamma \, B(z_1,z_2)\,,\tag{26}\] and \(B(z_1,z_2) - B^{G,\beta}(z_1,z_2)\) is regular as \(z_1 \to \gamma z_2\), where \[\label{eq:Bgb} B^{G,\beta}(z_1,z_2) \mathrel{\vcenter{:}}= \sum_{\eta \in G} \beta_\eta^{-1} \frac{d (\eta z_1) dz_2}{(\eta z_1 - z_2)^2}\,.\tag{27}\]
This version of topological recursion constructs the \(n\)-point differentials via the usual formula 25 , whereas for the unstable cases it is natural to define them as \[\begin{align} \omega^{(0)}_1(z_1)& = \frac{1}{|G|} y(z_1) dx(z_1); & \omega^{(0)}_2(z_1,z_2) & = B(z_1,z_2). \end{align}\]
Remark 15. Definition 1 as given here differs from the one given in [5] by the absence of the factor \(|G|^{-1}\) in formula 27 as compared to the respective formula in [5]. This factor amounts to a simple rescaling of \(n\)-point differentials by some factors of \(2\) and its absence or presence does not affect any properties; for our purposes it is more convenient not to have it.
We also slightly modified 26 , namely, we formulate it as a condition on the differential \(dy\) comparing to the condition on the function \(y\) in [5]. This definition is slightly more general, but due to the way \(y\) enters 25 all statements regarding equivariant topological recursion hold for a curve with such a relaxed assumption. We need this relaxed condition in the example below.
An example of the spectral curve data for the original topological recursion that is of primary interest for this paper is given in [1] (and has its roots in [9]): \[\begin{align} \label{eq:SigmaKN} \Sigma&=\mathbb{P}^1; & x&=\frac{w^2}{2}; & y&=\frac{2\mathop{\mathrm{arcsinh}}(w/\sqrt{2\epsilon})}{\sqrt{w^2+2\epsilon}}; & B &=\frac{dw_1dw_2}{(w_1-w_2)^2}. \end{align}\tag{28}\]
An example of a \(G\)-equivariant spectral curve data is given in Theorem 4. It is a \(\mathbb{Z}_2\)-equivariant spectral curve data according to Definition 1. Namely, we have \[\begin{align} \label{eq:GZ2} G&=\mathbb{Z}_2; \quad \Sigma=\mathbb{P}^1; \quad \iota(z)=-\frac{1}{z};\\ x&=\sqrt{\epsilon/2} \left(z+\frac{1}{z}\right); \quad y=2\log z; \quad B =\frac{dz_1dz_2}{(z_1-z_2)^2} + \frac{d\iota(z_1)dz_2}{(\iota(z_1)-z_2)^2}, \end{align}\tag{29}\] where \(\iota\) denotes the action of the generator of \(\mathbb{Z}_2\) (that is \(\iota (z) = \phi(1,z)\), where \(\mathbb{Z}_2\) is \(\{0,1\}\) with 0 being the group unit); and we have replaced \(Q\) with \(-\epsilon\). For this \(\mathbb{Z}_2\)-equivariant spectral curve \(\chi=\upsilon\) is the sign representation and \(\beta\) is the trivial representation of \(\mathbb{Z}_2\).
Now we want to compare the \(n\)-point differentials produced by \(\mathbb{Z}_2\)-equivariant topological recursion on the curve 29 with the ones produced by topological recursion on the curve 28 . In order not to get confused, let us mark all objects associated with 28 with tilde (that is, for that curve we will write \(\tilde{\Sigma}\), \(\tilde{x}\), \(\tilde{y}\), \(\tilde{B}\), \(\tilde{\omega}^{(g)}_n\)), while keeping the notation for 29 (with the \(n\)-point differentials produced by \(\mathbb{Z}_2\)-equivariant topological recursion on 29 denoted by \(\omega^{(g)}_n\)).
The following proposition implements this comparison:
Proposition 16. The systems of differentials \(\{\tilde{\omega}^{(g)}_n\}\) and \(\{\omega^{(g)}_n\}\) are related by the following formula: \[\begin{align} \prod_{i=1}^n \mathop{\big\lfloor_{{w_i}\to {\sqrt{\epsilon/2}(z_i-z_i^{-1})}}} 2^{2g-2+n} \tilde{\omega}^{(g)}_n(w_1,\dots,w_n) = \omega^{(g)}_n(z_1,\dots,z_n). \end{align}\]
Proof. Consider \(\tilde{y}d\tilde{x}\) (that is, \(ydx\) for the spectral curve 28 ): \[\tilde{y} d \tilde{x} = \frac{2\mathop{\mathrm{arcsinh}}(w/\sqrt{2\epsilon})}{\sqrt{w^2+2\epsilon}} d\frac{w^2}{2},\] where \(w\) is a global rational coordinate on \(\tilde{\Sigma}=\mathbb{P}^1\).
Introduce a new variable \(z\) related to \(w\) by \(w=\sqrt{\epsilon/2}(z-z^{-1})\); \(z\) is a global rational coordinate on the double cover of \(\tilde{\Sigma}=\mathbb{P}^1\) (which is \(\mathbb{P}^1\) itself; this will be \(\Sigma\) of 29 ). In the new coordinate we have locally near the point \(w=0\), \(z=1\): \[\frac{2\mathop{\mathrm{arcsinh}}(w/\sqrt{2\epsilon})}{\sqrt{w^2+2\epsilon}} d\frac{w^2}{2} = 2\log(z)\, d \left(\sqrt{{\epsilon/2}}\left(z+\frac{1}{z}\right) \right),\] where the latter expression is equal to \(ydx\), with \(x\) and \(y\) of 29 . Now notice that for \(\iota:z \mapsto -\frac{1}{z}\) we have \[\tilde{B}=\frac{dw_1dw_2}{(w_1-w_2)^2} = \frac{dz_1dz_2}{(z_1-z_2)^2} + \frac{d(\iota(z_1))dp_2}{(\iota(z_1)-z_2)^2}=B,\] hence the recursion on 28 coincides locally with the recursion on 29 whose data is taken locally near \(z=1\). The latter recursion is called (according to the terminology of [5]) the reduced version of the full recursion 29 and its differentials are tautologically equal to \(\tilde{\omega}^{(g)}_n\) . The full equivariant version of the latter reduced recursion has an extra critical point at \(z=-1\) and by [5] we have \(\omega^{(g)}_n = 2^{2g-2+n} \tilde{\omega}^{(g)}_n\). ◻
In this Section, we give a proof of Theorem 4.
Proof of Theorem 4. Note that by Eynard’s formula [9], as presented in [1], we have for \(2g-2+n>0\) \[\begin{align} \tilde{\omega}^{(g)}_n (w_1,\dots,w_n) = (-1)^n \sum_{k_1,\dots,k_n=0}^\infty \int_{\overline{\mathcal{M}}_{g,n}} \sum_{d=0}^\infty \epsilon^d J_{2g-2+n-d} \prod_{i=1}^n \psi_i^{k_i} \prod_{i=1}^n \frac{(2k_i+1)!!}{w_i^{2k_i+2}} dw_i. \end{align}\] By Theorems 1 and 3, we have \[\begin{align} \tilde{\omega}^{(g)}_n (w_1,\dots,w_n) & = (-1)^{g+d+n+1} \sum_{k_1,\dots,k_n=0}^\infty \sum_{d=0}^\infty \epsilon^d \int_{p_*C_{g,n,d}} \prod_{i=1}^n \psi_i^{k_i} \prod_{i=1}^n \frac{(2k_i+1)!!}{w_i^{2k_i+2}} dw_i. \end{align}\] This gives the stationary sector of the ancestor spin Gromov-Witten theory of \(\mathbb{P}^1\). Passing to descendants in the stable range \(2g-2+n>0\) is given by a linear change of variables as in 11 , and it amounts to the re-expansion of \(\tilde{\omega}^{(g)}_n (w_1,\dots,w_n)\) in \(\sqrt{w^2+2\epsilon} = x(z)\) at the point \(z=0\), due to the following easily verified identity: \[\frac{(2k+1)!!}{x^{2k+2}}dx=\sum_{m=0}^\infty\frac{ Q^m}{m!}\frac{(2(k+m)+1)!!}{w^{2(k+m)+2}}dw,\] where we identify \(Q=-\epsilon\). We obtain: \[\begin{align} \tilde{\omega}^{(g)}_n (w_1,\dots,w_n) & = (-1)^{g+n+1} \sum_{k_1,\dots,k_n=0}^\infty \sum_{d=0}^\infty Q^d \int_{C_{g,n,d}} \prod_{i=1}^n \psi_i^{k_i} \prod_{i=1}^n \frac{(2k_i+1)!!}{x_i^{2k_i+2}} dx_i. \end{align}\] Note also that \(\omega^{(g)}_n = 2^{2g-2+n} \tilde{\omega}^{(g)}_n\). Hence, \[\begin{align} \omega^{(g)}_n (w_1,\dots,w_n) & = (-1)^{n} \sum_{k_1,\dots,k_n=0}^\infty \sum_{d=0}^\infty Q^d 2^{2g-2+n} (-1)^{g+1} \int_{C_{g,n,d}} \prod_{i=1}^n \psi_i^{k_i} \prod_{i=1}^n \frac{(2k_i+1)!!}{x_i^{2k_i+2}} dx_i. \end{align}\] Taking into account the extra sign \((-1)^{g+1}\) that was natural for us to include in the definition of the descendant invariants, we obtain the statement of Theorem 4 in the stable range \(2g-2+n>0\). It extends further to the unstable case \((0,2)\) by a straightforward computation that essentially repeats the proof of Proposition 13. Indeed, applying [16] in our case, we get for \((g,n)=(0,2)\) \[\begin{align} \sum_{k_1,k_2=0}^\infty \frac{(-1)^{k_1+k_2+1}}{\zeta_1^{k_1+1}\zeta_2^{k_2+1}}\langle \tau_{k_1}\tau_{k_2}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),{\rm dec}}_{0} = \frac{e^{\frac{Q}{\zeta_1} + \frac{Q}{\zeta_2}} -1}{\zeta_1+\zeta_2}, \end{align}\] and then one can use this formula to check directly that \[\begin{align} \frac{dw_1dw_2}{(w_1-w_2)^2} - \frac{dx_1dx_1}{(x_1-x_2)^2} = \sum_{k_1,k_2=0}^\infty \langle \tau_{k_1}\tau_{k_2}\rangle^{\mathbb{P}^1,\mathcal{O}(-1),{\rm dec}}_{0} \frac{(2k_1+1)!!}{x_1^{2k_1+2}}\frac{(2k_2+1)!!}{x_2^{2k_2+2}} dx_1 dx_2 \end{align}\] in expansion near \(x=\infty\). ◻