#### Volume 12, issue 2 (2008)

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Intersection numbers with Witten's top Chern class

### Sergey Shadrin and Dimitri Zvonkine

Geometry & Topology 12 (2008) 713–745
##### Abstract

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.

##### Keywords
moduli space of curves, intersection theory, Witten top Chern class
Primary: 14H10
Secondary: 14H70
##### Publication
Received: 5 January 2006
Revised: 22 January 2008
Accepted: 5 October 2007
Published: 12 May 2008
Proposed: Eleny Ionel
Seconded: Jim Bryan, Yasha Eliashberg
##### Authors
 Sergey Shadrin 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 Dimitri Zvonkine Poncelet Laboratory, CNRS and Independent University of Moscow Bolshoi Vlassievsky per, 11 119002 Moscow Russia