Topological recursion for Masur-Veech volumes

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

Research output: Working paperResearch

10 Downloads (Pure)


We study the Masur-Veech volumes $MV_{g,n}$ 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 $MV_{g,n}$ 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.
Original languageEnglish
Number of pages75
Publication statusPublished - 24. May 2019

Bibliographical note

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


Dive into the research topics of 'Topological recursion for Masur-Veech volumes'. Together they form a unique fingerprint.

Cite this