Topological recursion for Masur-Veech volumes

Abstract

We study the Masur-Veech volumes $M{V}_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $M{V}_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of \cite{Delecroix} proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain an expression of the area Siegel--Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

75 pages, v2: added a section on enumeration of square-tiled surfaces

References in corpus (5)

• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves
• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves
• Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials
• On the Kontsevich geometry of the combinatorial Teichmüller space

Cited by in corpus (17)

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• Large Genus Asymptotics for Intersection Numbers and Principal Strata Volumes of Quadratic Differentials
• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves
• Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials
• Volumes of moduli spaces of flat surfaces
• An intersection-theoretic proof of the Harer-Zagier formula
• Masur-Veech volume of the gothic locus
• Taking limits in topological recursion
• On the Kontsevich geometry of the combinatorial Teichmüller space
• Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursions
• Masur--Veech volumes of quadratic differentials and their asymptotics
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• The factorial growth of topological recursion
• Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers
• Higher genus meanders and Masur-Veech volumes
• Around the combinatorial unit ball of measured foliations on bordered surfaces
• Local topological recursion governs the enumeration of lattice points in ${\overline{\mathcal{M}}}_{g,n}$