Double ramification cycles on the moduli spaces of curves

Abstract

Curves of genus $g$ which admit a map to $\mathbf {P}^{1}$ with specified ramification profile $\mu$ over $0\in \mathbf {P}^{1}$ and $\nu$ over $\infty\in \mathbf {P}^{1}$ define a double ramification cycle $\mathsf{DR}_{g}(\mu,\nu)$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle $\mathsf{DR}_{g}(\mu,\nu)$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for $\mathsf{DR}_{g}(\mu,\nu)$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When $\mu=\nu=\emptyset$ , the formula for double ramification cycles expresses the top Chern class $\lambda_{g}$ of the Hodge bundle of $\overline {\mathcal{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.