Euler characteristics of strata of $k$-differentials

Abstract

We compute the orbifold Euler characteristics of strata of $k$-differentials on the moduli space of smooth pointed curves. Our main result is a closed coefficient-extraction formula in terms of hyperbolic functions, valid for arbitrary strata and generalizing the Harer--Zagier formula for the Euler characteristic of the moduli space of curves. In particular, these Euler characteristics are polynomial in the prescribed orders at the marked points. The proof first uses Riemann--Roch and Serre duality to reduce the computation to the Hurwitz case $k=0$, which is then evaluated using the representation theory of the symmetric group and Fock-space techniques. We also give an intersection-theoretic expression for the same invariants as tautological integrals over double ramification cycles and derive a recursion which uniquely determines them from genus zero. For odd $k$ and odd orders, we introduce a spin-weighted refinement and formulate corresponding closed and intersection-theoretic conjectural expressions. Finally, we determine the large-genus asymptotics of the Euler characteristics.