Weil-petersson Volumes of Moduli Spaces of Curves and the Genus Expansion in Two Dimensional Gravity

The aim of this note is to prove a rather explicit formula for the generating function of the Weil-Petersson volumes of the moduli spaces Mg,n of smooth npointed curves of genus g that was conjectured in [12]. E. Getzler noticed that the conecture in [12] is identical to the formulas in [5], Sect. 5, and [2], Sect. 3 and Appendix D, for the genus expansion of the free energy in Witten’s two dimensional gravity [4]. C. Faber numerically checked this conjecture for the moduli spaces M3,n with n ≤ 12 [3]. All that evidence convinced the author that the conjecture must actually be true not only for the Weil-Petersson volumes (see Theorem 1 below), but also for their higher analogues in the sense of [7] (i.e., intesection numbers of Mumford’s tautological classes κ1, ..., κ3g−3+n on Mg,n). We start by introducing the necessary notation. It is well known that the Kähler form ωWP of the Weil-Petersson metric on Mg,n extends as a closed current to the Deligne-Mumford compactification Mg,n and represents a real cohomology class [ωWP ] ∈ H(Mg,n,R) which coincides (up to a factor of π) with Mumford’s first tautological class κ1 on Mg,n [9, 10]. It means that the Weil-Petersson volume of the moduli space Mg,n is finite and is given by V olWP (Mg,n) = π 〈κ 1 〉 n!(3g − 3 + n)! .