Double ramification cycles on the moduli spaces of curves

Abstract

Curves of genus g which admit a map to P1 with specified ramification profile μ over 0∈P1 and ν over ∞∈P1 define a double ramification cycle DRg(μ,ν) 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 DRg(μ,ν) 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 DRg(μ,ν) 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 μ=ν=∅, the formula for double ramification cycles expresses the top Chern class λg of the Hodge bundle of M¯¯¯¯¯¯g as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.