Topological Relations on Witten–Kontsevich and Hodge Potentials

Abstract

Let Mg;n denote the moduli space of genus g stable alge- braic curves with n marked points. It carries the Mumford cohomology classesi. A homology class in H�(Mg;n) is said to be �-zero if the integral of any monomial in the �-classes vanishes on it. We show that any �-zero class implies a partial differential equation for generating se- ries for certain intersection indices on the moduli spaces. The genus homogeneous components of the Witten-Kontsevich potential, as well as of the more general Hodge potential, which include, in addition to -classes, intersection indices for �-classes, are special cases of these generating series, and the well-known partial differential equations for them are instances of our general construction. 2000 Math. Subj. Class. 14H10, 14H70.