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.
Here we give a simpler proof of this result. In particular, it implies that in any semi-simple GromovWitten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the geometric Gromov-Witten potential coincides with the potential constructed via Givental's group action.As the most important application we show that our results suffice to deduce the statement of a 1991 Witten conjecture relating the r-KdV hierarchy to the intersection theory on the space of r-spin structures on stable curves. We use the fact that Givental's construction is, in this case, compatible with Witten's conjecture, as Givental himself showed in [10].R. -Dans [11], A. Givental a introduit une action de groupe sur l'espace des potentiels de Gromov-Witten et a prouvé sa transitivité sur les potentiels semi-simples. Dans [24,25], Y.-P. Lee a montré, modulo certains résultats annoncés par C. Teleman, que cette action préserve les relations tautologiques dans l'anneau de cohomologie de l'espace des modules Mg,n des courbes stables époin-tées.Ici nous donnons une démonstration plus simple de ce résultat. Il en découle, entre autres, que si dans une théorie de Gromov-Witten semi-simple on peut exprimer n'importe quel corrélateur en fonction des corrélateurs de genre 0 en utilisant uniquement des relations tautologiques, alors le potentiel de Gromov-Witten géométrique coïncide avec le potentiel construit via l'action du groupe de Givental.Ces résultats suffisent pour démontrer une conjecture de Witten de 1991 qui relie la hiérarchie r-KdV à la théorie de l'intersection sur l'espace des structures r-spin sur les courbes stables. Nous utilisons pour cela la compatibilité entre la construction de Givental dans ce cas et la conjecture de Witten, compatibilité établie dans [10] par Givental lui-même.ANNALES SCIENTIFIQUES DE L'ÉCOLE NORMALE SUPÉRIEURE 0012-9593/04
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