Double ramification cycles with target varieties

Let (Formula presented.) be a nonsingular projective algebraic variety over (Formula presented.), and let (Formula presented.) be the moduli space of stable maps (Formula presented.) from genus (Formula presented.), (Formula presented.) -pointed curves (Formula presented.) to (Formula presented.) of degree (Formula presented.). Let (Formula presented.) be a line bundle on (Formula presented.). Let (Formula presented.) be a vector of integers which satisfy (Formula presented.) Consider the following condition: the line bundle (Formula presented.) has a meromorphic section with zeros and poles exactly at the marked points (Formula presented.) with orders prescribed by the integers (Formula presented.). In other words, we require (Formula presented.) to be the trivial line bundle on (Formula presented.). A compactification of the space of maps based on the above condition is given by the moduli space of stable maps to rubber over (Formula presented.) and is denoted by (Formula presented.). The moduli space carries a virtual fundamental class (Formula presented.) in Gromov–Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push-forward via the forgetful morphism of (Formula presented.) to (Formula presented.). In case (Formula presented.) is a point, the result here specializes to Pixton's formula for the double ramification cycle proven in (Janda, Pandharipande, Pixton and Zvonkine, Publ. Math. Inst. Hautes Études Sci. 125 (2017) 221–266). Several applications of the new formula are given.