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 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 -KdV hierarchy to the intersection theory on the space of -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].
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 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 -KdV à la théorie de l’intersection sur l’espace des structures -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.
Keywords: quantization of Frobenius manifolds, Gromov-Witten potential, moduli of curves, $r$-spin structures, Witten’s conjecture
Mot clés : quantification des variétés de Frobenius, potentiel de Gromov-Witten, modules des courbes, structures $r$-spin, conjecture de Witten
@article{ASENS_2010_4_43_4_621_0, author = {Faber, Carel and Shadrin, Sergey and Zvonkine, Dimitri}, title = {Tautological relations and the $r$-spin {Witten} conjecture}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {621--658}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 43}, number = {4}, year = {2010}, doi = {10.24033/asens.2130}, mrnumber = {2722511}, zbl = {1203.53090}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2130/} }
TY - JOUR AU - Faber, Carel AU - Shadrin, Sergey AU - Zvonkine, Dimitri TI - Tautological relations and the $r$-spin Witten conjecture JO - Annales scientifiques de l'École Normale Supérieure PY - 2010 SP - 621 EP - 658 VL - 43 IS - 4 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2130/ DO - 10.24033/asens.2130 LA - en ID - ASENS_2010_4_43_4_621_0 ER -
%0 Journal Article %A Faber, Carel %A Shadrin, Sergey %A Zvonkine, Dimitri %T Tautological relations and the $r$-spin Witten conjecture %J Annales scientifiques de l'École Normale Supérieure %D 2010 %P 621-658 %V 43 %N 4 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2130/ %R 10.24033/asens.2130 %G en %F ASENS_2010_4_43_4_621_0
Faber, Carel; Shadrin, Sergey; Zvonkine, Dimitri. Tautological relations and the $r$-spin Witten conjecture. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 4, pp. 621-658. doi : 10.24033/asens.2130. http://www.numdam.org/articles/10.24033/asens.2130/
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