Volume 12, issue 2 (2008)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 ψ–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
Mathematical Subject Classification 2000
Primary: 14H10
Secondary: 14H70
References
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