Topological relations on Witten–Kontsevich and Hodge potentials

Let $\overline{\mathcal{M}}_{g;n}$ denote the moduli space of genus $g$ stable algebraic curves with $n$ marked points. It carries the Mumford cohomology classes $\kappa_i$. A homology class in $H_*(\overline{\mathcal{M}}_{g;n})$ is said to be $\kappa$-zero if the integral of any monomial in the $\kappa$-classes vanishes on it. We show that any $\kappa$-zero class implies a partial differential equation for generating series 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 $\psi$-classes, intersection indices for $\lambda$-classes, are special cases of these generating series, and the well-known partial differential equations for them are instances of our general construction.