A flat perspective on moduli spaces of hyperbolic surfaces

TL;DR

This paper introduces a new, general way to define volumes of moduli spaces of hyperbolic cone surfaces without angle restrictions, connecting flat geometry and quantum gravity insights, and offers a novel proof of Mirzakhani's recursion and Witten-Kontsevich's theorem.

Contribution

It proposes a unified volume definition for hyperbolic cone surfaces using flat geometry limits, extending previous restrictions and linking to quantum gravity models.

Findings

Generalized volume definition for hyperbolic cone surfaces.

Derived Mirzakhani's recursion formula from new volume properties.

Provided a new proof of Witten-Kontsevich's theorem.

Abstract

Volumes of moduli spaces of hyperbolic cone surfaces were previously defined and computed when the angles of the cone singularities are at most 2pi. We propose a general definition of these volumes without restriction on the angles. This construction is based on flat geometry as our proposed volume is a limit of Masur-Veech volumes of moduli spaces of multi-differentials. This idea generalizes the observation in quantum gravity that the Jackiw-Teitelboim partition function is a limit of minimal string partition functions from Liouville gravity. Finally, we use the properties of these volumes to recover Mirzakhani's recursion formula for Weil-Petersson polynomials. This provides a new proof of Witten-Kontsevich's theorem.