Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves

TL;DR

This paper explores the relationship between combinatorial and algebro-geometric cohomology classes on moduli spaces of curves, proposing a link via a theorem connecting derivatives of the Witten-Kontsevich partition function.

Contribution

It demonstrates that the conjectured relationship holds in complex codimension 1 and provides explicit identities relating combinatorial classes to polynomials in algebro-geometric classes.

Findings

The link works in complex codimension 1.

Explicit identities relate combinatorial classes to algebro-geometric polynomials.

The approach connects derivatives of the partition function to cohomology class identities.

Abstract

Based on the combinatorial description of the moduli spaces of curves provided by Strebel differentials, Witten and Kontsevich have introduced combinatorial cohomology classes $W_{(m_0,m_1,m_2,\dots),n}$, and conjectured that these can be expressed in terms of Mumford-Morita-Miller classes. It is argued that this link should be provided by a theorem of Di Francesco, Itzykson and Zuber which relates the derivatives of the Witten-Kontsevich partition function with respect to one set of variables to the derivatives with respect to the other set of variables. Two things are shown. First of all that this works in complex codimension 1. Secondly that in all the cases when it has been possible to make the Di Francesco, Itzykson and Zuber correpondence explicit this translates into identities of the type $$ \int_{W_{(m_0,m_1,m_2,\dots),n}}\prod\psi_i^{d_i} =\int_{\overline{\cal{M}}_{g,n}} X_{(m_0,m_1,m_2,\dots),n}\prod\psi_i^{d_i} $$ where the $X_{(m_0,m_1,m_2,\dots),n}$ are explicit polynomials in the algebro-geometric classes and the $\psi_i$ are the Chern classes of the point bundles, for any choice of $d_1,\dots,d_n$.