Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

Bertrand Eynard, Nicolas Orantin

TL;DR

The paper addresses the problem of relating Mirzakhani's recursion for Weil-Petersson volumes to random matrix loop equations. By applying a Laplace transform, the authors recast the geometric recursion as a residue-based topological recursion for the Kontsevich curve, showing that the transformed volumes become EOFg invariants with Kontsevich times. A key result is the identification of Kontsevich's integral as the generating function for these volumes and the derivation of a concrete relation for $Vol({\cal M}_{g,0})$, including explicit values such as $V_{2,0}={43\pi^6\over 2160}$. This unifies moduli-space geometry with matrix-model recursion and suggests avenues to study invariants for more general spectral curves and times.

Abstract

We show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(M_{g,0}).

Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

TL;DR

The paper addresses the problem of relating Mirzakhani's recursion for Weil-Petersson volumes to random matrix loop equations. By applying a Laplace transform, the authors recast the geometric recursion as a residue-based topological recursion for the Kontsevich curve, showing that the transformed volumes become EOFg invariants with Kontsevich times. A key result is the identification of Kontsevich's integral as the generating function for these volumes and the derivation of a concrete relation for , including explicit values such as . This unifies moduli-space geometry with matrix-model recursion and suggests avenues to study invariants for more general spectral curves and times.

Abstract

We show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(M_{g,0}).

Paper Structure

This paper contains 4 sections, 3 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Laplace transform
  3. Examples
  4. Conclusion

Key Result

Theorem 2.1

For any $2g-2+n+1>0$, the $W_{n+1}^g$ satisfy the recursion relation where the RHS includes all possible $W_k^{h}$, including $W_1^0=0$ and $W_2^0$, and where is a set of $n$ variables.

Theorems & Definitions (3)

  • Theorem 2.1
  • Corollary 2.1
  • Theorem 2.2