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A tale of two volumes of moduli spaces: Weil-Petersson and Masur-Veech

Dawei Chen, Scott Mullane

Abstract

Weil-Petersson and Masur-Veech volumes measure the sizes of moduli spaces of Riemann surfaces equipped with hyperbolic and flat metrics, respectively. Over the past several decades, the computation of these volumes has inspired remarkable developments in combinatorial enumeration, intersection theory, and recursion relations. In this survey, we review key results, methods, open problems, as well as interesting parallels that emerge in the approaches to computing both types of volumes.

A tale of two volumes of moduli spaces: Weil-Petersson and Masur-Veech

Abstract

Weil-Petersson and Masur-Veech volumes measure the sizes of moduli spaces of Riemann surfaces equipped with hyperbolic and flat metrics, respectively. Over the past several decades, the computation of these volumes has inspired remarkable developments in combinatorial enumeration, intersection theory, and recursion relations. In this survey, we review key results, methods, open problems, as well as interesting parallels that emerge in the approaches to computing both types of volumes.
Paper Structure (51 sections, 7 theorems, 67 equations)

This paper contains 51 sections, 7 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Weil--Petersson volumes
  3. Moduli of hyperbolic surfaces
  4. The Weil--Petersson metric and volume
  5. Fenchel--Nielsen coordinates
  6. The Weil--Petersson metric
  7. Wolpert: The volume for $\mathbf{L}=\mathbf{0}$ is finite
  8. Mirzakhani: An innovative proof of Witten's conjecture
  9. Moduli of hyperbolic cone surfaces
  10. Compactifications via metric completions: nodal hyperbolic surfaces
  11. Algebraic compactifications: Hassett stability
  12. Wall-crossing volume polynomials
  13. Recursion and generalized string and dilaton equations
  14. Higher cone angles
  15. Large genus asymptotics
  16. ...and 36 more sections

Key Result

Theorem 3.1

The strata $\mathcal{H}(\mu)$ of holomorphic differentials of type $\mu$ have up to three connected components, where additional components arise due to hyperelliptic and spin structures.

Theorems & Definitions (8)

  • Theorem 3.1: A condensed version of KZ03
  • Theorem 3.2: M82V82
  • Lemma 3.3: EO01
  • proof : Sketch of proof
  • Corollary 3.4: EO01
  • Proposition 3.5: S18
  • Theorem 3.6: CMSZ20
  • Theorem 3.8: A condensed version of EZ15, A20, CMSZ20