# Topological recursion for Masur-Veech volumes

## TL;DR

This paper establishes a new approach using topological recursion to compute Masur-Veech volumes of moduli spaces of quadratic differentials, connecting geometric, combinatorial, and hyperbolic methods.

## Contribution

It introduces a geometric recursion framework to derive formulas for Masur-Veech volumes, linking them to stable graphs and providing a new computational method.

## Key findings

- Volumes are constant terms of polynomials governed by topological recursion.
- Derived explicit formulas relating volumes to hyperbolic geometry and stable graphs.
- Proposed conjectural formulas for volumes at low genus and all puncture counts.

## Abstract

We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of \cite{Delecroix} proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain an expression of the area Siegel--Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.