Topological recursion for Masur–Veech volumes

Jørgen Ellegaard Andersen, Gaëtan Borot, Séverin Charbonnier, Vincent Delecroix*, Alessandro Giacchetto, Danilo Lewański, Campbell Wheeler

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Abstract

We study the Masur–Veech volumes (Formula presented.) of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus (Formula presented.) with (Formula presented.) punctures. We show that the volumes (Formula presented.) are the constant terms of a family of polynomials in (Formula presented.) 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 [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] 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 [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. 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 (Formula presented.) and (Formula presented.), which leads us to propose conjectural formulae for low (Formula presented.) but all (Formula presented.). We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

OriginalsprogEngelsk
TidsskriftJournal of the London Mathematical Society
Vol/bind107
Udgave nummer1
Sider (fra-til)254-332
ISSN0024-6107
DOI
StatusUdgivet - jan. 2023

Bibliografisk note

Funding Information:
We thank Don Zagier for suggesting a more compact formula in Conjecture 5.4 , and Martin Möller for discussions related the intersection theory aspects of the paper. J.E.A. was supported in part by the Danish National Sciences Foundation Centre of Excellence grant ‘Quantum Geometry of Moduli Spaces’ and is supported by the ERC Synergy grant ‘Recursive and Exact New Quantum Theory’ (ReNewQuantum) which receive funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 810573. G.B., S.C., V.D., A.G., D.L. and C.W. benefited from the support of the Max‐Planck‐Gesellschaft.

Publisher Copyright:
© 2022 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

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