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Weil-Petersson volumes and cone surfaces

Norman Do, Paul Norbury

TL;DR

This work develops new recursion relations for Weil-Petersson volume polynomials $V_{g,n}(\\mathbf{L})$ of moduli spaces of hyperbolic surfaces by introducing intermediary cone-point moduli spaces and leveraging a generalized McShane formula. It connects volume coefficients to intersection numbers on the Deligne–Mumford moduli spaces via ψ and κ classes, deriving string- and dilaton-type equations that relate $V_{g,n+1}$ to $V_{g,n}$ and to lower-genus data. The authors show that their recursions yield efficient genus-0 algorithms for computing volumes and discuss higher derivatives at $L_{n+1}=2\\pi i$, highlighting links to Virasoro constraints and recent extensions to full volume polynomials. The framework provides a bridge between hyperbolic geometry, intersection theory, and topological recursion, with potential implications for understanding cone-point moduli and their role in volume polynomials.

Abstract

The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the boundaries by computing the volumes recursively. In this paper we give new recursion relations between the volume polynomials.

Weil-Petersson volumes and cone surfaces

TL;DR

This work develops new recursion relations for Weil-Petersson volume polynomials of moduli spaces of hyperbolic surfaces by introducing intermediary cone-point moduli spaces and leveraging a generalized McShane formula. It connects volume coefficients to intersection numbers on the Deligne–Mumford moduli spaces via ψ and κ classes, deriving string- and dilaton-type equations that relate to and to lower-genus data. The authors show that their recursions yield efficient genus-0 algorithms for computing volumes and discuss higher derivatives at , highlighting links to Virasoro constraints and recent extensions to full volume polynomials. The framework provides a bridge between hyperbolic geometry, intersection theory, and topological recursion, with potential implications for understanding cone-point moduli and their role in volume polynomials.

Abstract

The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the boundaries by computing the volumes recursively. In this paper we give new recursion relations between the volume polynomials.

Paper Structure

This paper contains 10 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Volume of the moduli space
  3. Fenchel-Nielsen coordinates.
  4. Coordinates on a hyperbolic surface
  5. Characteristic classes of surface bundles
  6. Surface bundles.
  7. Intersection numbers
  8. Use of recursion relations
  9. Classical volumes and low genus algorithms.
  10. Higher derivatives

Key Result

Theorem 1

$V_{g,n}({\bf L})$ is a polynomial in ${\bf L}=(L_1,...,L_n)$. The coefficient of $L^{\alpha}=L^{\alpha_1}..L^{\alpha_n}$ lies in $\pi^{6g-6+2n-|\alpha|}\mathbb{Q}$, $|\alpha|=\alpha_1+...+\alpha_n$.

Theorems & Definitions (9)

  • Theorem 1: Mirzakhani MirSim
  • Theorem 2
  • Theorem 3: Mirzakhani
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Completion of the proof of Theorem \ref{['th:string']}
  • proof : Proof of (iii).