Intersection numbers with Witten’s top Chern class.

*(English)*Zbl 1141.14012If \(C\) is a smooth projective curve of genus \(g\) with \(n\geq 1\) distinct marked points \(x_1,\dots,x_n\), an \(r\)-spin curve of type \((a_1,\dots,a_n)\) on \(C\) is a line bundle \(\mathcal T\) on \(C\) together with an identification between \(\mathcal T^{\otimes r}\) and \(\omega_C(-\sum a_ix_i)\). This definition can be extended to all Deligne–Mumford stable curves of genus \(g\) with \(n\) marked points, giving rise to the moduli space of stable curves with \(r\)-spin structure of type \((a_i)\), introduced by T. J. Jarvis [Int. J. Math. 11, No. 5, 637–663 (2000; Zbl 1094.14504)] and D. Abramovich and T. J. Jarvis [Proc. Am. Math. Soc. 131, No. 3, 685–699 (2003; Zbl 1037.14008)]. The structure of the moduli space of \(r\)-spin curves as a finite covering of the moduli space \(\overline{\mathcal M}_{g,n}\) of stable curves allows to define on the latter space a special cohomology class, called Witten’s top Chern class, which satisfies simple factorization rules.

The article under review deals with the computation of the intersection numbers of Witten’s class with powers of the \(\psi\)-classes. The main result is that these intersection numbers are entirely determined by genus \(0\) intersection numbers involving no \(\psi\)-classes, and the factorization rules for Witten’s class. This is proven by giving an algorithm for computing all these intersection numbers, completing the approach of S. V. Shadrin [Int. Math. Res. Not. 2003, No. 38, 2051–2094 (2003; Zbl 1070.14030)]. In particular, this involves giving a closed formula for the integrals of Witten’s class over certain divisors of \(\overline{\mathcal M}_{1,n}\), the so-called double ramification divisors.

The interest in the intersection numbers of Witten’s top Chern class and \(\psi\)-classes is motivated by the conjecture of E. Witten [in: Topological methods in modern mathematics, 235–269 (1993; Zbl 0812.14017)] that they can be arranged into a generating series which is a solution of the \(r\)-th Gelfand–Dikii hierarchy. This conjecture has been recently proved by the authors and C. Faber [Tautological relations and the \(r\)-spin Witten conjecture. Preprint 2006, arxiv:math/0612510].

The article under review deals with the computation of the intersection numbers of Witten’s class with powers of the \(\psi\)-classes. The main result is that these intersection numbers are entirely determined by genus \(0\) intersection numbers involving no \(\psi\)-classes, and the factorization rules for Witten’s class. This is proven by giving an algorithm for computing all these intersection numbers, completing the approach of S. V. Shadrin [Int. Math. Res. Not. 2003, No. 38, 2051–2094 (2003; Zbl 1070.14030)]. In particular, this involves giving a closed formula for the integrals of Witten’s class over certain divisors of \(\overline{\mathcal M}_{1,n}\), the so-called double ramification divisors.

The interest in the intersection numbers of Witten’s top Chern class and \(\psi\)-classes is motivated by the conjecture of E. Witten [in: Topological methods in modern mathematics, 235–269 (1993; Zbl 0812.14017)] that they can be arranged into a generating series which is a solution of the \(r\)-th Gelfand–Dikii hierarchy. This conjecture has been recently proved by the authors and C. Faber [Tautological relations and the \(r\)-spin Witten conjecture. Preprint 2006, arxiv:math/0612510].

Reviewer: Orsola Tommasi (Hannover)

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14H70 | Relationships between algebraic curves and integrable systems |

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\textit{S. Shadrin} and \textit{D. Zvonkine}, Geom. Topol. 12, No. 2, 713--745 (2008; Zbl 1141.14012)

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##### References:

[1] | D Abramovich, T J Jarvis, Moduli of twisted spin curves, Proc. Amer. Math. Soc. 131 (2003) 685 · Zbl 1037.14008 |

[2] | L Chen, Y Li, K Liu, Localization, Hurwitz numbers and the Witten conjecture · Zbl 1208.14053 |

[3] | A Chiodo, Stable twisted curves and their \(r\)-spin structures, to appear in Annales de l’Institut Fourier · Zbl 1179.14028 |

[4] | A Chiodo, The Witten top Chern class via \(K\)-theory, J. Algebraic Geom. 15 (2006) 681 · Zbl 1117.14008 |

[5] | C Faber, R Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. \((\)JEMS\()\) 7 (2005) 13 · Zbl 1084.14054 |

[6] | C Faber, S Shadrin, D Zvonkine, Tautological relations and the \(r\)-spin Witten conjecture · Zbl 1203.53090 |

[7] | E N Ionel, Topological recursive relations in \(H^{2g}(\mathscr M_{g,n})\), Invent. Math. 148 (2002) 627 · Zbl 1056.14076 |

[8] | T J Jarvis, Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000) 637 · Zbl 1094.14504 |

[9] | M E Kazarian, KP hierarchy and Hodge integrals, preprint (2007) · Zbl 1168.14006 |

[10] | M E Kazarian, S K Lando, An algebro-geometric proof of Witten’s conjecture, J. Amer. Math. Soc. 20 (2007) 1079 · Zbl 1155.14004 |

[11] | Y S Kim, K Liu, A simple proof of Witten conjecture through localization |

[12] | M Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992) 1 · Zbl 0756.35081 |

[13] | M Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007) 1 · Zbl 1120.32008 |

[14] | D Mumford, Towards an enumerative geometry of the moduli space of curves, Progr. Math. 36, BirkhĂ¤user (1983) 271 · Zbl 0554.14008 |

[15] | A Okounkov, R Pandharipande, Gromov-Witten theory, Hurwitz numbers, and Matrix models, I · Zbl 1205.14072 |

[16] | A Polishchuk, Witten’s top Chern class on the moduli space of higher spin curves, Aspects Math. E36, Vieweg (2004) 253 · Zbl 1105.14010 |

[17] | A Polishchuk, A Vaintrob, Algebraic construction of Witten’s top Chern class, Contemp. Math. 276, Amer. Math. Soc. (2001) 229 · Zbl 1051.14007 |

[18] | S V Shadrin, Geometry of meromorphic functions and intersections on moduli spaces of curves, Int. Math. Res. Not. (2003) 2051 · Zbl 1070.14030 |

[19] | E Witten, Two-dimensional gravity and intersection theory on moduli space, Lehigh Univ. (1991) 243 · Zbl 0757.53049 |

[20] | E Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, Publish or Perish (1993) 235 · Zbl 0812.14017 |

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