A flat perspective on moduli spaces of hyperbolic surfaces

Adrien Sauvaget

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arXiv · 2405.10869

A flat perspective on moduli spaces of hyperbolic surfaces

Adrien Sauvaget

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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.

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BibTeX RIS

Adrien Sauvaget. 2024-05-17. A flat perspective on moduli spaces of hyperbolic surfaces. https://arxiv.org/abs/2405.10869

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