Curves of genus g which admit a map to P 1 with specified ramification profile µ over 0 ∈ P 1 and ν over ∞ ∈ P 1 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 nonseparating nodes. Applications to Hodge integral calculations are given.

Witten's class on the moduli space of 3-spin curves defines a (nonsemisimple) cohomological field theory. After a canonical modification, we construct an associated semisimple CohFT with a non-trivial vanishing property obtained from the homogeneity of Witten's class. Using the classification of semisimple CohFTs by Givental-Teleman, we derive two main results. The first is an explicit formula in the tautological ring of M g,n for Witten's class. The second, using the vanishing property, is the construction of relations in the tautological ring of M g,n .Pixton has previously conjectured a system of tautological relations on M g,n (which extends the established Faber-Zagier relations on M g ). Our 3-spin construction exactly yields Pixton's conjectured relations. As the classification of CohFTs is a topological result depending upon the Madsen-Weiss theorem (Mumford's conjecture), our construction proves relations in cohomology. The study of Witten's class and the associated tautological relations for r-spin curves via a parallel strategy will be taken up in a following paper.

We use the Gromov-Witten/Pairs descendent correspondence for toric 3-folds and degeneration arguments to establish the GW/P correspondence for several compact Calabi-Yau 3-folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role.The GW/P correspondence for Calabi-Yau complete intersections provides a structure result for the Gromov-Witten invariants in a fixed curve class. After change of variables, the Gromov-Witten series is a rational function in the variable −q = e iu invariant under q ↔ q −1 .

We construct a fully equivariant correspondence between Gromov-Witten and stable pairs descendent theories for toric 3-folds X. Our method uses geometric constraints on descendents, A n surfaces, and the topological vertex. The rationality of the stable pairs descendent theory plays a crucial role in the definition of the correspondence. We prove our correspondence has a non-equivariant limit.As a result of the construction, we prove an explicit non-equivariant stationary descendent correspondence for X (conjectured by MNOP). Using descendent methods, we establish the relative GW/Pairs correspondence for X/D in several basic new log Calabi-Yau geometries. Among the consequences is a rationality constraint for non-equivariant descendent Gromov-Witten series for P 3 .