Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces
endowed with –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
–classes over any
moduli space of –spin
structures, in short, all numbers involved in Witten’s conjecture.
Keywords
moduli space of curves, intersection theory, Witten top
Chern class
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