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].

Classification : 14H10, 14N35, 53D45, 53D50
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

[1] D. Abramovich & T. J. Jarvis, Moduli of twisted spin curves, Proc. Amer. Math. Soc. 131 (2003), 685-699 (electronic). | MR | Zbl

[2] L. Caporaso, C. Casagrande & M. Cornalba, Moduli of roots of line bundles on curves, Trans. Amer. Math. Soc. 359 (2007), 3733-3768 (electronic). | MR | Zbl

[3] A. Chiodo, The Witten top Chern class via $K$-theory, J. Algebraic Geom. 15 (2006), 681-707. | MR | Zbl

[4] A. Chiodo, Stable twisted curves and their $r$-spin structures, preprint arXiv:math.AG/0603687. | Numdam | MR | Zbl

[5] B. Dubrovin & Y. Zhang, Bi-Hamiltonian hierarchies in $2$D topological field theory at one-loop approximation, Comm. Math. Phys. 198 (1998), 311-361. | MR | Zbl

[6] C. Faber & R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215-252. | MR | Zbl

[7] C. Faber & R. Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), 13-49. | MR | Zbl

[8] A. B. Givental, Semisimple Frobenius structures at higher genus, Int. Math. Res. Not. 2001 (2001), 1265-1286. | MR | Zbl

[9] A. B. Givental, Symplectic geometry of Frobenius structures, Mosc. Math. J. 1 (2001), 551-568. | Zbl

[10] A. B. Givental, $A_{n-1}$ singularities and $n$KdV hierarchies, Mosc. Math. J. 3 (2003), 475-505, 743. | MR | Zbl

[11] A. B. Givental, Gromov-Witten invariants and quantization of quadratic hamiltonians, in Frobenius manifolds, Aspects Math. E36, Vieweg, 2004, 91-112. | Zbl

[12] T. Graber & R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), 93-109. | MR | Zbl

[13] T. Graber & R. Vakil, On the tautological ring of ${\overline{ℳ}}_{g,n}$, Turkish J. Math. 25 (2001), 237-243. | MR | Zbl

[14] R. Hain & E. Looijenga, Mapping class groups and moduli spaces of curves, in Algebraic geometry-Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., 1997, 97-142. | MR | Zbl

[15] E.-N. Ionel, Topological recursive relations in $H^{2g}\left({ℳ}_{g,n}\right)$, Invent. Math. 148 (2002), 627-658. | MR | Zbl

[16] T. J. Jarvis, Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000), 637-663. | MR | Zbl

[17] T. J. Jarvis, T. Kimura & A. Vaintrob, Tensor products of Frobenius manifolds and moduli spaces of higher spin curves, in Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud. 22, Kluwer Acad. Publ., 2000, 145-166. | MR | Zbl

[18] T. J. Jarvis, T. Kimura & A. Vaintrob, Gravitational descendants and the moduli space of higher spin curves, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math. 276, Amer. Math. Soc., 2001, 167-177. | MR | Zbl

[19] T. J. Jarvis, T. Kimura & A. Vaintrob, Moduli spaces of higher spin curves and integrable hierarchies, Compositio Math. 126 (2001), 157-212. | MR | Zbl

[20] T. J. Jarvis, T. Kimura & A. Vaintrob, Spin Gromov-Witten invariants, Comm. Math. Phys. 259 (2005), 511-543. | MR | Zbl

[21] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1-23. | MR | Zbl

[22] M. Kontsevich & Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525-562. | MR | Zbl

[23] Y.-P. Lee, Witten's conjecture and the Virasoro conjecture for genus up to two, in Gromov-Witten theory of spin curves and orbifolds, Contemp. Math. 403, Amer. Math. Soc., 2006, 31-42. | MR | Zbl

[24] Y.-P. Lee, Invariance of tautological equations. I. Conjectures and applications, J. Eur. Math. Soc. (JEMS) 10 (2008), 399-413. | MR | Zbl

[25] Y.-P. Lee, Invariance of tautological equations. II. Gromov-Witten theory, J. Amer. Math. Soc. 22 (2009), 331-352. | MR | Zbl

[26] Y.-P. Lee, Witten's conjecture, Virasoro conjecture, and invariance of tautological equations, preprint arXiv:math.AG/0311100. | MR | Zbl

[27] Y.-P. Lee & R. Pandharipande, Frobenius manifolds, Gromov-Witten theory, and Virasoro constraints, preprint http://www.math.princeton.edu/~rahulp/Part1.ps and http://www.math.princeton.edu/~rahulp/Part2.ps.

[28] T. Mochizuki, The virtual class of the moduli stack of stable $r$-spin curves, Comm. Math. Phys. 264 (2006), 1-40. | MR | Zbl

[29] D. Mumford, Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II, Progr. Math. 36, Birkhäuser, 1983, 271-328. | MR | Zbl

[30] A. Polishchuk, Witten's top Chern class on the moduli space of higher spin curves, in Frobenius manifolds, Aspects Math., E36, Vieweg, 2004, 253-264. | MR | Zbl

[31] A. Polishchuk & A. Vaintrob, Algebraic construction of Witten's top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math. 276, Amer. Math. Soc., 2001, 229-249. | MR | Zbl

[32] S. Shadrin, Geometry of meromorphic functions and intersections on moduli spaces of curves, Int. Math. Res. Not. 2003 (2003), 2051-2094. | MR | Zbl

[33] S. Shadrin, Intersections in genus 3 and the Boussinesq hierarchy, Lett. Math. Phys. 65 (2003), 125-131. | MR | Zbl

[34] S. Shadrin & D. Zvonkine, Intersection numbers with Witten's top Chern class, Geom. Topol. 12 (2008), 713-745. | MR | Zbl

[35] E. Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, in Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, 1993, 235-269. | MR | Zbl