Topological relations on Witten–Kontsevich and Hodge potentials

M. E. Kazarian^abc, S. K. Lando<sup>cb

a</sup> Steklov Mathematical Institute RAS, 8 Gubkina Str., 119991, Moscow, Russia
^b Poncelet Laboratory, Independent University of Moscow
^c National Research University Higher School of Economics, 7 Vavilova Str., 117312, Moscow, Russia

Abstract: 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.

Key words and phrases: Moduli spaces, Deligne–Mumford compactification, Witten–Kontsevich potential, Hodge integrals.

MSC: 14H10, 14H70

Received: February 23, 2011

Language: English

DOI: 10.17323/1609-4514-2012-12-2-397-411

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