Double ramification cycles on the moduli spaces of curves

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 DR_g(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 DR_g(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 DR_g(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 λ_g 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.