Recursion between Mumford volumes of moduli spaces

TL;DR

This paper introduces a new proof and generalization of Mirzakhani's recursion for moduli space volumes, connecting it to Kontsevich's integral and extending it to higher Mumford kappa classes.

Contribution

It provides a novel proof and extends Mirzakhani's recursion to include all higher Mumford's kappa classes, broadening the understanding of moduli space volumes.

Findings

Generalization of Mirzakhani's recursion to higher kappa classes

Interpretation of recursion via Kontsevich's integral and Ribbon graph decomposition

New proof of the recursion relations

Abstract

We propose a new proof, as well as a generalization of Mirzakhani's recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich's integral, i.e. we relate them to a Ribbon graph decomposition of Riemann surfaces. We find a generalization of Mirzakhani's recursions to measures containing all higher Mumford's kappa classes, and not only kappa1 as in the Weil-Petersson case.