Seminar

Let M_{g,n} be the moduli space of Riemann surfaces of genus g with n punctures. Its cotangent space identifies to the bundle Q_{g,n} of quadratic differentials. Q_{g,n} is endowed with two measures: the Weil-Petersson and the Masur-Veech volume forms. We call Weil-Petersson and Masur-Veech volumes the total mass of M_{g,n} for these volume forms. M. Mirzakhani proved two important results about them.



1) The Weil-Petersson volumes WP_{g,n}(L1, ..., Ln) of surfaces with fixed boundary length L1, ..., Ln is a polynomial. These polynomials admit a "topological recursion" that allows to compute all of them from WP_{0,3} and WP_{1,1}.



2) There is a geometric function on M_{g,n} whose integral against the Weil-Petersson measure is the Masur-Veech volume.



As explained in the talk of E. Goujard (http://www.msri.org/workshops/894/schedules/27300) I will recall a combinatorial formula expressing the Masur-Veech volumes of Q_{g,n} as a sum over stable graphs of some quantities involving psi-classes appearing in Witten's conjecture. I will explain how this formula can be turned into polynomials MV_{g,n}(L1, ..., Ln) that correspond to Masur-Veech volumes of quadratic differentials with double poles and real residues (or equivalently with "horizontal" boundaries). The polynomials MV_{g,n} satisfy a topological recursion in a flavor similar to 1).



Finally, I hope to discuss an intriguing fact about a generalization of 2) that allows to express the Masur-Veech volumes as an integral of a rational function over the combinatorial version of M_{g,n} given by Strebel differentials.