Intersection numbers with Witten’s top Chern class

Shadrin, Sergey¹ ; Zvonkine, Dimitri²

¹ Korteweg–de Vries Institute for Mathematics, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, and, Institute of System Research, Nakhimovskii Prospekt 36-1, 117218 Moscow, Russia
² Poncelet Laboratory, CNRS, and, Independent University of Moscow, Bolshoi Vlassievsky per, 11, 119002 Moscow, Russia

Geometry & topology, Tome 12 (2008) no. 2, pp. 713-745.

Résumé

Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with $r$–spin structures. It plays a key role in Witten’s conjecture relating to the intersection theory on these moduli spaces.

Our first goal is to compute the integral of Witten’s class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals.

This allows us, using the methods of the first author [Int. Math. Res. Not. 38 (2003) 2051-2094], to find an algorithm for computing the intersection numbers of the Witten class with powers of the $\psi$–classes over any moduli space of $r$–spin structures, in short, all numbers involved in Witten’s conjecture.

Détail

DOI : 10.2140/gt.2008.12.713

Keywords: moduli space of curves, intersection theory, Witten top Chern class

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