A Simple Recursion for the Mirzakhani Volume and its Super Extension

TL;DR

This paper introduces a straightforward recursion formula for Weil-Petersson volumes of hyperbolic surface moduli spaces, demonstrating polynomiality and equivalence with existing formulas, and extends these results to super-analogues.

Contribution

It presents a new simple recursion for Weil-Petersson volumes and establishes their polynomiality and equivalence with known formulas, including super-extensions.

Findings

Derived a simple recursion formula for Weil-Petersson volumes

Proved polynomiality of the volume functions

Extended results to super-analogues

Abstract

In this paper, we derive a simple recursion formula for the Weil-Petersson volumes of moduli spaces of hyperbolic surfaces with boundaries. This formula demonstrates the polynomiality of the volume functions. By constructing the Laplace transform for both the original and our alternative formulas, we show that they are directly equivalent. By considering the top and lowest degree terms of these formulas, we recover the DVV identity and cohomology class identities for $\mathcal{M}_{g,n}$. Similar conclusions are drawn for the super-analog of these results.