Intersection numbers with Witten's top Chern class

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 [15], to find an algorithm for computing the intersection numbers of the Witten class with powers of the \psi-classes (or tautological classes) over any moduli space of r-spin structures, in short, all numbers involved in Witten's conjecture.