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Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes

Norman Do

TL;DR

This survey explains how moduli spaces of hyperbolic surfaces carry Weil–Petersson volumes that are polynomials in boundary lengths with coefficients given by psi- and kappa-class intersections on $\overline{\mathcal{M}}_{g,n}$. It outlines Mirzakhani’s geometric approach using symplectic reduction and generalized McShane identities to derive a computable recursion for $V_{g,n}({\mathbf L})$, connect these volumes to intersection theory, and recover the Witten–Kontsevich theorem. The work also surveys limits and asymptotics, including cone-point degenerations, large-genus behavior, and the asymptotic link to Kontsevich’s combinatorial model via ribbon graphs and Pfaffians. Altogether, the framework provides a deep bridge between hyperbolic geometry, moduli of curves, and matrix-model–style combinatorics, with both exact recursion machinery and rich asymptotic structure. The results have broad implications for intersection theory, quantum geometry, and the enumeration of geometric structures on surfaces.

Abstract

Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.

Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes

TL;DR

This survey explains how moduli spaces of hyperbolic surfaces carry Weil–Petersson volumes that are polynomials in boundary lengths with coefficients given by psi- and kappa-class intersections on . It outlines Mirzakhani’s geometric approach using symplectic reduction and generalized McShane identities to derive a computable recursion for , connect these volumes to intersection theory, and recover the Witten–Kontsevich theorem. The work also surveys limits and asymptotics, including cone-point degenerations, large-genus behavior, and the asymptotic link to Kontsevich’s combinatorial model via ribbon graphs and Pfaffians. Altogether, the framework provides a deep bridge between hyperbolic geometry, moduli of curves, and matrix-model–style combinatorics, with both exact recursion machinery and rich asymptotic structure. The results have broad implications for intersection theory, quantum geometry, and the enumeration of geometric structures on surfaces.

Abstract

Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.

Paper Structure

This paper contains 21 sections, 32 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Moduli spaces of hyperbolic surfaces
  3. Hyperbolic surfaces
  4. Teichmüller theory
  5. Symplectification and compactification
  6. Combinatorial moduli space
  7. Weil--Petersson volumes
  8. Early results
  9. Symplectic reduction
  10. Polynomiality of Weil--Petersson volumes
  11. A recursion for Weil--Petersson volumes
  12. The volume of ${\mathcal{M}}_{1,1}(0)$
  13. McShane identities
  14. Mirzakhani's recursion
  15. Applications of Mirzakhani's recursion
  16. ...and 6 more sections

Key Result

Proposition 1

On a hyperbolic surface, non-trivial homotopy classes of closed curves have unique geodesic representatives. Furthermore, such geodesic representatives realise minimal intersection and self-intersection numbers. In particular, every simple closed curve is homotopic to a simple closed geodesic.

Theorems & Definitions (35)

  • Proposition 1
  • Theorem 2: Uniformisation theorem
  • Lemma 3
  • Theorem 4
  • Proposition 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9: Marsden--Weinstein theorem
  • Theorem 10
  • ...and 25 more