Tautological relations and the $r$-spin Witten conjecture

Faber, Carel; Shadrin, Sergey; Zvonkine, Dimitri

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 4, pp. 621-658.

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 ${\overline{ℳ}}_{g,n}$ des courbes stables épointées. 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.

In [11], A. Givental introduced a group action on the space of Gromov-Witten potentials and proved its transitivity on the semi-simple potentials. In [24, 25], Y.-P. Lee showed, modulo certain results announced by C. Teleman, that this action respects the tautological relations in the cohomology ring of the moduli space ${\overline{ℳ}}_{g,n}$ of stable pointed curves. Here we give a simpler proof of this result. In particular, it implies that in any semi-simple Gromov-Witten 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].

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