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

Matteo Costantini, Adrien Sauvaget, Johannes Schmitt

TL;DR

This work delivers a closed formula for the top power of a single $\psi$-class on strata of $k$-differentials, packaging the intersection numbers as $\mathcal{A}_g(a)$ and expressing them via a coefficient-extracted generating series. It connects these integrals to twisted double ramification cycles and develops a comprehensive DR/multi-scale framework, including the proposed spin refinement $\mathrm{DR}_g^{\rm spin}$ and corresponding $\mathcal{A}_g^{\rm spin}(a)$, which yield spin-sensitive psi-intersections under suitable assumptions. The authors derive splitting formulas for $\psi$-classes on both DR cycles and strata of $k$-differentials, enabling a decomposition of top intersections along two-level graphs and their vertex-branch data. They further establish identities constraining $\mathcal{A}_g$ that characterize it uniquely, and extend the technology to spin components, deriving parity-sensitive boundary behavior and spin-twist constructions. As applications, they obtain closed formulas for the orbifold Euler characteristics of minimal strata and lay out conjectures for the asymptotics and spin refinements, linking to volumes and Lyapunov-type data in flat surface theory. Overall, the paper provides a coherent framework linking DR cycles, strata of differentials, and spin structures to computable intersection-theoretic invariants with implications for volumes, Euler characteristics, and asymptotics in the moduli of curves and 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.

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

TL;DR

This work delivers a closed formula for the top power of a single -class on strata of -differentials, packaging the intersection numbers as and expressing them via a coefficient-extracted generating series. It connects these integrals to twisted double ramification cycles and develops a comprehensive DR/multi-scale framework, including the proposed spin refinement and corresponding , which yield spin-sensitive psi-intersections under suitable assumptions. The authors derive splitting formulas for -classes on both DR cycles and strata of -differentials, enabling a decomposition of top intersections along two-level graphs and their vertex-branch data. They further establish identities constraining that characterize it uniquely, and extend the technology to spin components, deriving parity-sensitive boundary behavior and spin-twist constructions. As applications, they obtain closed formulas for the orbifold Euler characteristics of minimal strata and lay out conjectures for the asymptotics and spin refinements, linking to volumes and Lyapunov-type data in flat surface theory. Overall, the paper provides a coherent framework linking DR cycles, strata of differentials, and spin structures to computable intersection-theoretic invariants with implications for volumes, Euler characteristics, and asymptotics in the moduli of curves and differentials.

Abstract

We prove a closed formula for the integral of a power of a single -class on strata of -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 -class on the even and odd components of strata of -differentials. As an application of these results we give a closed formula for the Euler characteristic of components of minimal strata of abelian differentials.
Paper Structure (29 sections, 36 theorems, 218 equations, 2 figures, 1 table)

This paper contains 29 sections, 36 theorems, 218 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Strata of $k$-differentials
  3. Double ramification cycles
  4. Spin refinement
  5. Applications
  6. Outlook and open questions
  7. Organization of the paper
  8. Double ramification cycles and moduli spaces of multi-scale differentials
  9. Twisted and level graphs
  10. A formula for twisted double ramification cycles
  11. The spin double ramification cycle
  12. The moduli space of multi-scale differentials.
  13. The incidence variety compactification
  14. The moduli space of multi-scale differentials
  15. Intersection theory on the space of multi-scale differentials
  16. ...and 14 more sections

Key Result

Theorem 1.1

For all $g, n \geq 0$ satisfying $2g-2+n>0$ and $a \in \mathbb{Z}^n$, we have where $k=|a|/(2g-2+n)$, $\mathcal{S}(z)=\frac{{\rm sinh}(z/2)}{z/2},$ and $[z^{2g}](\cdot)$ stands for the coefficient of $z^{2g}$ in the formal series.

Figures (2)

  • Figure 1: Euler characteristics of even and odd components of some minimal holomorphic strata.
  • Figure 1: The local picture of the gluing construction at one node associated to a preimage $\widehat{e}$ of the edge $e$.

Theorems & Definitions (80)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.8
  • Conjecture 1.11
  • Definition 2.1: FP
  • Definition 2.2
  • Definition 2.3
  • ...and 70 more