Tautological relations and the r -spin Witten conjecture

Carel Faber; Sergey Shadrin; Dimitri Zvonkine

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 4, page 621-658
  • ISSN: 0012-9593

Abstract

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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 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].

How to cite

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Faber, Carel, Shadrin, Sergey, and Zvonkine, Dimitri. "Tautological relations and the $r$-spin Witten conjecture." Annales scientifiques de l'École Normale Supérieure 43.4 (2010): 621-658. <http://eudml.org/doc/272233>.

@article{Faber2010,
abstract = {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 $\{\overline\{\mathcal \{M\}\}\}_\{g,n\}$ 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].},
author = {Faber, Carel, Shadrin, Sergey, Zvonkine, Dimitri},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {quantization of Frobenius manifolds; Gromov–Witten potential; moduli of curves; $r$-spin structures; Witten’s conjecture},
language = {eng},
number = {4},
pages = {621-658},
publisher = {Société mathématique de France},
title = {Tautological relations and the $r$-spin Witten conjecture},
url = {http://eudml.org/doc/272233},
volume = {43},
year = {2010},
}

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
PB - Société mathématique de France
VL - 43
IS - 4
SP - 621
EP - 658
AB - 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 ${\overline{\mathcal {M}}}_{g,n}$ 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].
LA - eng
KW - quantization of Frobenius manifolds; Gromov–Witten potential; moduli of curves; $r$-spin structures; Witten’s conjecture
UR - http://eudml.org/doc/272233
ER -

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