Recursion between Mumford volumes of moduli spaces
\textit{M. Mirzakhani} has described in [Invent. Math. 167, No. 1, 179--222 (2007; Zbl 1125.30039)] and [J. Am. Math. Soc. 20, No. 1, 1--23 (2007; Zbl 1120.32008)] recursive relations between the Weil-Petersson volumes of \(\mathcal{M}_{g,n}(L_1,\dots, L_n)\), the moduli spaces of curves of genus \(g\) with \(n\) geodesic boundaries of lengths \(L_1,\dots, L_n\). In the present article the author presents both a new proof and a generalisation of Mirzakhani's relations. The main result of the paper is Theorem 1.1, which states that the Mumford volumes associated with fixed (conjugated) Kontsevich times satisfy the recursive relations for the correlation functions from the solutions to the loop equations for the matrix model. The recursion relations are interpreted in terms of expectation values in Kontsevich's integral, i.e., are related to ribbon graph decompositions of Riemann surfaces. The reviewed article makes heavy use of previous results of Eynard and coauthors, in particular [\textit{L. Chekhov} and \textit{B. Eynard}, J. High Energy Phys. No. 3, 014, 18 p. (2006; Zbl 1226.81137)] and [\textit{B. Eynard} and \textit{N. Orantin}, Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)]. Let \(\overline{\mathcal{M}}_{g,n}\) be the Deligne-Mumford compactification of the moduli (orbi)space of curves of genus \(g\) with \(n\) distinct marked points. It carries tautological line bundles \(\mathcal{L}_i\), \(i=1,\dots, n\). \textit{E. Witten} [Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243--310 (1991; Zbl 0757.53049)] introduced a generating function \(F(t_0,t_1,\dots)\) encoding the intersection numbers of the various \(\mathcal{L}_i\), for different values of \(g\) and \(n\). This generating function is a formal series in infinitely many indeterminates. Witten conjectured a relation between \(F(t_0,t_1,\dots)\) and the asymptotics of certain matrix integrals \(\int_{\mathrm{Her}_N}\exp (\mathrm{tr} P(X))dX\) as \(N\to\infty\), where \(\mathrm{Her}_N\) is the vector space of hermitian \(N\times N\) matrices and \(P\) is a polynomial. Witten's conjecture was based on the premise that since ``gravity is unique'', partition functions arising from different approaches to quantisation of two-dimensional gravity should coincide. \textit{M. Kontsevich} in [Commun. Math. Phys. 147, 1--23 (1992; Zbl 0756.35081)] studied formal matrix integrals \[ Z(\Lambda)= \int_{\mathrm{Her}_N}\exp(\mathrm{tr} P(X))d\mu_{\Lambda}(X), \] where \(P\) is a cubic polynomial and \(d\mu_\Lambda(X)\) is a certain gaussian measure, depending on a choice of a positive-definite matrix \(\Lambda\in \mathrm{Her}_N\). Kontsevich related the asymptotics of \(Z(\Lambda)\) to \(F(t_0,t_1,\dots)\). He proved that \(F(t_0(\Lambda), t_1(\Lambda),\dots)\) is an assymptotic expansion of \(\log Z(\Lambda)\) as \(\Lambda^{-1}\to 0\), where \(t_i(\Lambda)= -(2i-1)!! \mathrm{tr} \;\Lambda^{-(2i+1)}\). Kontsevich's proof involves the construction of a cell structure \(\mathcal{M}_{g,n}^{\mathrm{comb}}\) on the moduli space, with cells corresponding to different classes of Feynman ribbon graphs (fatgraphs). Kontsevich proved that \(Z(\Lambda)\) is a tau-functions for the KdV hierarchy, thus relating it to Witten's matrix integrals and proving Witten's conjecture. ``Matrix models'' have appeared in the 1980-ies as a discrete version of 2-dimensional quantum gravity, and their partition functions are formal matrix integrals. These have been of interest for giving rise to various combinatorial generating functions related to enumeration of triangulations of Riemann surfaces of fixed genus, or to various spaces of maps between these, etc. Kontsevich's integral is a particular example of a formal matrix integral. Matrix integrals provide a link between algebraic geometry and integrable systems: they are tau--functions of integrable hierarchies and possess a ``topological expansion'': the logarithm of the partition function can be written as \[ \sum_{g=0}^\infty N^{2-2g}F^{(g)}, \] where the quantities \(F^{(g)}\) can be interpreted in algebro-geometric terms. Numerous techniques have been developed for computing formal matrix integrals and their large \(N\) expansion. Among the most successful ones is the so-called ``loop equations method''. In [Chekhov and Eynard, Zbl 1226.81137] a new method for computing large \(N\) expansion of matrix integrals is proposed, extending the work in [\textit{J. Ambørn, L. Chekhov, C. Kristjansen} and \textit{Yu. Makeenko}, Nucl. Phys. B 404, No. 1--2, 127--172 (1993; Zbl 1043.81636)]. The authors solve the loop equations recursively in \(1/N^2\), and to leading order they obtain the equation of a plane algebraic curve, ``the classical spectral curve''. They notice that various quantities related to the matrix model -- e.g., free energy and correlation functions -- can be expressed entirely in terms of the geometry of the spectral curve. Motivated by this, in [Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)], \textit{B. Eynard} and \textit{N. Orantin} define, for each plane curve \(\mathcal{E}\subset\mathbb{C}^2\), and points \(z_i\in\mathcal{E}\), \(i=1\dots n\), an infinite sequence of meromorphic forms \(W_{g,n}(z_1,\dots,z_n)\), by imposing the recursive relations motivated by the loop equations from the matrix model. In [``Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models'', \url{arXiv:0705.3600}], \textit{B. Eynard} and \textit{N. Orantin} observe that the Laplace transform of Mirzakhani's relations coincides with the solutions to the loop equations for Kontsevich's integral. These results allow the author of the reviewed article to prove Theorem 1.1. Given a set of conjugated Kontsevich times \(\widetilde{t}_0, \widetilde{t}_1,\dots\) the ``Mumford volumes'' \[ W_{g,n}(z_1,\dots,z_n)=2^{-d_{g,n}}(t_3-2)^{2-2g-n}\sum_{d_0+\ldots+ d_n =d_{g,n}}\sum_{k=1}^{d_0}\frac{1}{k!}\times \] \[ \sum_{b_1+\dots+b_k=d_0,b_i>0}\prod_{i=1}^n\frac{2d_i+1!}{d_i!}\frac{dz_i}{z_i^{2d_i+2}}\prod_{l=1}^k\widetilde{t}_{b_l} \left\langle\prod_{l=1}^k\kappa_{b_l}\prod_{i=1}^n\psi_i^{d_i}\right\rangle_{g,n}, \] where \(d_{g,n}=\dim\mathcal{M}_{g,n}=3g-3+n\), satisfy the following recursion relations: \(W_{0,1}=0\), \(W_{0,2}=\frac{dz_1dz_2}{(z_1-z_2)^2}\), \[ W_{g,z_{n+1}}= \frac{1}{2}Res _{z\to 0}\frac{dz_{n+1}}{(z_{n+1}^2-z^2)(y(z)-y(-z))dz}\times \] \[ \left[W_{g-1,n+2}(z,-z,K)+ \sum_{h=0}^g\sum_{J\subset K}W_{h,1+|J|}(z,J)W_{g-h,1+n-|J|}(-z,K/J) \right] \] where \(K=\{z_1,\dots z_n\}\) and \(y(z) = z-\frac{1}{2}\sum_{k=0}^\infty t_{2k+3}z^{2k+1}\). Here the relation between the Kontsevich KdV times \(t_{2d+3}\) and their conjugate times \(\widetilde{t}_k\), is given by \[ f(z)=\sum_{a=1}^\infty\frac{(2a+1)!}{a!}\frac{t_{2a+3}}{2-t_3}z^a\to \widetilde{f}(z) = -\ln(1-f(z))=\sum_{b=1}^\infty \widetilde{t}_b z^b. \] Mirzakhani's relations for the Weil-Petersson volumes (Corollary 1.1) are obtained by setting \(\widetilde{t}_1=4\pi\), \(\widetilde{t}_k=0\) (\(k>1\)), and performing Laplace transform. The strategy of proof is as follows: The author identifies \(W_{g,n}(z_1,\dots,z_n)\) with certain expectation values in Kontsevich's integral, and expands these into Feynman ribbon graphs. The value assigned to each ribbon graph is the Laplace transform of the volume of a certain cell in the cell decomposition \(\mathcal{M}_{g,n}^{\mathrm{comb}}\). The inverse Laplace transform of the sum over all graphs gives the Weil-Petersson volumes, and these -- by construction -- satisfy recursive relations. The key observation from Eynard and Orantin [\url{arXiv:0705.3600}] completes the proof of Theorem 1.1.
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