Topological relations on Witten-Kontsevich and Hodge potentials

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Abstract

Let Mg;n denote the moduli space of genus g stable algebraic curves with n marked points. It carries the Mumford cohomology classes κi. A homology class in H*(Mg;n) is said to be κ-zero if the integral of any monomial in the κ-classes vanishes on it. We show that any κ-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 Ψ-classes, intersection indices for λ-classes, are special cases of these generating series, and the well-known partial differential equations for them are instances of our general construction.

Original languageEnglish
Pages (from-to)397-411
Number of pages15
JournalMoscow Mathematical Journal
Volume12
Issue number2
DOIs
Publication statusPublished - 2012
Externally publishedYes

Keywords

  • Deligne-Mumford compactification
  • Hodge integrals
  • Moduli spaces
  • Witten-Kontsevich potential

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