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Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy

Motohico Mulase, Brad Safnuk

TL;DR

This work uncovers a deep Virasoro and KdV structure underlying Mirzakhani's recursion for Weil-Petersson volumes of moduli spaces. By developing a differential version of Mirzakhani's recursion and formulating a generating function $G(s,t_0,t_1,\ldots)$, the authors show a Virasoro constraint $V_k e^{G}=0$ with $[V_n,V_m]=(n-m)V_{n+m}$, and demonstrate that the mixed $\kappa_1$ and $\psi$-class intersections yield a 1-parameter KdV tau-function related to the Witten-Kontsevich function $F$ via $G=F$ under a shift of higher times: $G(s,t_0,t_1,t_2+\gamma_2,t_3+\gamma_3,\ldots)=F(t_0,t_1,t_2,t_3,\ldots)$ with $\gamma_i=\frac{(-1)^i}{(2i+1)i!}s^{i-1}$. The differential recursion and the associated matrix of Virasoro constraints reveal that Mirzakhani’s geometric recursion encodes the same intersection-theoretic data as Witten-Kontsevich, now interpreted through a hyperbolic-geometric, symplectic-reduction lens. Consequently, $e^{G}$ is a $\tau$-function for KdV for every fixed $s$, and the relationship to $F$ provides a principled bridge between hyperbolic geometry and the algebraic-geometric framework of tautological intersection numbers. The results suggest that Virasoro and KdV structures are intrinsic to the domain-assembly processes for surfaces and may point toward a matrix-model interpretation of Mirzakhani's formulas.

Abstract

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter is set to be 0.

Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy

TL;DR

This work uncovers a deep Virasoro and KdV structure underlying Mirzakhani's recursion for Weil-Petersson volumes of moduli spaces. By developing a differential version of Mirzakhani's recursion and formulating a generating function , the authors show a Virasoro constraint with , and demonstrate that the mixed and -class intersections yield a 1-parameter KdV tau-function related to the Witten-Kontsevich function via under a shift of higher times: with . The differential recursion and the associated matrix of Virasoro constraints reveal that Mirzakhani’s geometric recursion encodes the same intersection-theoretic data as Witten-Kontsevich, now interpreted through a hyperbolic-geometric, symplectic-reduction lens. Consequently, is a -function for KdV for every fixed , and the relationship to provides a principled bridge between hyperbolic geometry and the algebraic-geometric framework of tautological intersection numbers. The results suggest that Virasoro and KdV structures are intrinsic to the domain-assembly processes for surfaces and may point toward a matrix-model interpretation of Mirzakhani's formulas.

Abstract

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter is set to be 0.

Paper Structure

This paper contains 10 sections, 7 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mirzakhani's Recursion Relation
  3. Notations
  4. McShane's identity
  5. Integration over the moduli spaces
  6. Volume calculation
  7. Relation to intersection numbers
  8. A rational recursion relation
  9. From Mirzakhani's Recursion Relation to the Virasoro Algebra
  10. Relationship to KdV hierarchy

Key Result

Theorem 1.1

For every $k\ge -1$, let us define where $\alpha_{i} = \frac{(-2)^i }{(2i+1)! }$. Then we have: Moreover, these properties uniquely determine $G$ and enable one to calculate all coefficients of the expansion. Since $G$ contains all information of the rational volumes $v_{g,n}(\boldsymbol{L})$, we conclude that the Virasoro constraint condition is indeed equivalent to the Mirzakhani recursion r

Figures (3)

  • Figure 1: Decomposing a surface
  • Figure 2: Removing a pair of pants from a surface
  • Figure 3: Capping off a bordered surface

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Mirzakhani art:MirzakhaniWP
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Theorem 4.1