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Integrals of psi-classes over double ramification cycles

A. Buryak, S. Shadrin, L. Spitz, D. Zvonkine

Abstract

DR-cycles are certain cycles on the moduli space of curves. Intuitively, they parametrize curves that allow a map to \mathbb{P}^1 with some specified ramification profile over two points. They are known to be tautological classes, but in general there is no known expression in terms of standard tautological classes. In this paper, we compute the intersection numbers of those DR-cycles with any monomials in psi-classes when this intersection is zero-dimensional.

Integrals of psi-classes over double ramification cycles

Abstract

DR-cycles are certain cycles on the moduli space of curves. Intuitively, they parametrize curves that allow a map to \mathbb{P}^1 with some specified ramification profile over two points. They are known to be tautological classes, but in general there is no known expression in terms of standard tautological classes. In this paper, we compute the intersection numbers of those DR-cycles with any monomials in psi-classes when this intersection is zero-dimensional.

Paper Structure

This paper contains 59 sections, 34 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Relative stable maps and double ramification cycles
  3. Problem.
  4. Several $\psi$-classes
  5. Completed cycles as a particular case of Theorem \ref{['Thm:1']}
  6. Theorem
  7. Theorem
  8. A conjecture on $\psi^m$
  9. Acknowledgements
  10. DR-cycle times a $\psi$-class
  11. The splitting formulas - formulations
  12. Proof.
  13. Proof.
  14. The splitting formulas - proofs
  15. Plan of proof
  16. ...and 44 more sections

Key Result

Theorem \oldthetheorem

We have where $[z^{2g}]$ denotes the coefficient of $z^{2g}$.

Theorems & Definitions (66)

  • Definition 1.2
  • Conjecture 1.3
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark 1.5
  • Remark 1.6
  • Example 1.7
  • Example 1.8
  • Conjecture 1.10
  • Theorem \oldthetheorem
  • ...and 56 more