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Curves of genus g which admit a map to CP1 with specified ramificationprofile mu over 0 and nu over infinity define a double ramification cycleDR_g(mu,nu) on the moduli space of curves. The study of the restrictions ofthese 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 doubleramification cycles on the moduli space of Deligne-Mumford stable curves.  The cycle DR_g(mu,nu) for stable curves is defined via the virtualfundamental class of the moduli of stable maps to rubber. Our main result isthe proof of an explicit formula for DR_g(mu,nu) in the tautological ringconjectured by Pixton in 2014. The formula expresses the double ramificationcycle as a sum over stable graphs (corresponding to strata classes) withsummand equal to a product over markings and edges. The result answers aquestion of Eliashberg from 2001 and specializes to Hain's formula in thecompact type case.  When mu and nu are both empty, the formula for double ramification cyclesexpresses the top Chern class lambda_g of the Hodge bundle of the moduli spaceof stable genus g curves as a push-forward of tautological classes supported onthe divisor of nonseparating nodes. Applications to Hodge integral calculationsare given.

Double ramification cycles on the moduli spaces of curves