Recursion between Mumford volumes of moduli spaces
type: Item
instance of: scholarly article
title: Recursion between Mumford volumes of moduli spaces
zbMATH Open document ID: 1245.14013
author: Bertrand Eynard
published in: Annales Henri Poincaré
publication date: 2012-02-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0706.4403
review text: \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 \(\int_{\mathrm{Her}_N}\exp (\mathrm{tr} P(X))dX\) …
reviewed by: Peter Dalakov
Mathematics Subject Classification ID: 14D21
zbMATH DE Number: 6005007
zbMATH Keywords: Mirzakhani relations
zbMATH Keywords: Weil-Petersson volumes
zbMATH Keywords: Kontsevich integral
zbMATH Keywords: matrix model
zbMATH Keywords: topological expansion
zbMATH Keywords: loop equations
MaRDI profile type: Publication
OpenAlex ID: W2964099357
arXiv ID: 0706.4403
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DOI: 10.1007/S00023-011-0113-4
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description: scientific article