A Simple Recursion for the Mirzakhani Volume and its Super Extension

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.