Masur-Veech volumes, Siegel-Veech constants and intersection numbers of moduli spaces

The cotangent space to the moduli space of complex curves of genus g with n marked points can be identified with the moduli space of pairs (C,q), where C is a complex curve of genus g, and q is a meromorphic quadratic differential on C with n simple poles and no other poles. This cotangent space comes with a natural symplectic form and the associated volume form is called the Masur-Veech volume form. We provide a formula for the volume of the level hypersurface of quadratic differentials of area 1/2. We also provide a formula of similar nature for the Siegel-Veech constant. As a concrete application, we get a large table of exact numerical values of the volumes and Siegel-Veech constants for all small g and n extending previously known data. This data was obtained by Goujard by completely different approach designed by Eskin and Okounkov. Both the volume and the Siegel-Veech constant are expressed as polynomials in the intersection numbers of psi-classes supported on the boundary components of the Deligne-Mumford compactification of the moduli space of curves. The formulae are derived from lattice point counting involving the Kontsevich volume polynomials that also appear in Mirzakhani’s topological recursion for the Weil-Petersson volumes of the moduli space of curves. Analogous formula for the Masur-Veech volume of the moduli space of holomorphic quadratic differentials was first obtained by Mirzakhani by different method (after a joint work with V.Delecroix, E.Goujard and P.Zograf)