Integrals of $ψ$-classes on twisted double ramification cycles and spaces of differentials

Abstract

We prove a closed formula for the integral of a power of a single ψ-class on strata of k-differentials. In many cases, these integrals correspond to intersection numbers on twisted double ramification cycles.Then we conjecture an expression of a refinement of double ramification cycles according to the parity of spin structures. Assuming that this conjecture is valid, we also compute the integral of a single ψ-class on the even and odd components of strata of k-differentials.As an application of these results we give a closed formula for the Euler characteristic of components of minimal strata of abelian differentials. Contents 1. Introduction 1 2. Double ramification cycles and moduli spaces of multi-scale differentials 9 3. Splitting formulas for ψ-classes on double ramification cycles 17 4. Identities satisfied by A g 24 5. Top-ψ for spin components 31 6. Euler characteristic of the minimal strata 41 Appendix A. Polynomiality properties in the splitting formula 54 References 58