Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy

TL;DR

This paper reformulates Mirzakhani's recursion for Weil-Petersson volumes as a Virasoro constraint, linking it to the KdV hierarchy and extending the Witten-Kontsevich generating function with a new parameter.

Contribution

It introduces a differential form of Mirzakhani's recursion, connecting it to Virasoro constraints and the KdV hierarchy, and generalizes the Witten-Kontsevich generating function.

Findings

Differential version of Mirzakhani's recursion relation.

Identification of Virasoro constraints on the generating function.

The generating function is a 1-parameter solution to the KdV hierarchy.

Abstract

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter is set to be 0.