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

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.

21 pages, 3 figures; v3. new introduction, minor revisions

References in corpus (6)

• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• Quantum Cohomology and Virasoro Algebra
• An algebro-geometric proof of Witten's conjecture
• Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves
• A simple proof of Witten conjecture through localization
• Combinatorial classes on the moduli space of curves are tautological

Cited by in corpus (33)

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• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• Hurwitz numbers, matrix models and enumerative geometry
• The Weil-Petersson metric geometry
• A short overview of the "Topological recursion"
• The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers
• Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus
• Topological recursion for Masur-Veech volumes
• Nonlinear PDEs for Fredholm determinants arising from string equations
• Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes
• Page curve from dynamical branes in JT gravity
• Super Quantum Airy Structures
• Towards large genus asymtotics of intersection numbers on moduli spaces of curves
• Deformations of JT Gravity via Topological Gravity and Applications
• Mirzakharni's recursion formula is equivalent to the Witten-Kontsevich theorem
• Descendent integrals and tautological rings of moduli spaces of curves
• Lectures and notes: Mirzakhani's volume recursion and approach for the Witten-Kontsevich theorem on moduli tautological intersection numbers
• Virasoro constraints and polynomial recursion for the linear Hodge integrals
• Algebro-Geometric Solutions of the Generalized Virasoro Constraints
• Emergent Geometry of Matrix Models with Even Couplings
• An effective recursion formula for computing intersection numbers
• Polynomial recursion formula for linear Hodge integrals
• Multi-boundary correlators in JT gravity
• Integration on moduli spaces of stable curves through localization
• Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursions
• A proof of loop equations in 2d topological gravity
• Cusps and the family hyperbolic metric
• Topological Recursion for Generalized $bc$-Motzkin Numbers
• A remark on Mirzakhani's asymptotic formulae
• The Laplace transform, mirror symmetry, and the topological recursion of Eynard-Orantin
• Topological recursion, topological quantum field theory and Gromov-Witten invariants of BG