Recursion formulae of higher Weil-Petersson volumes. (English) Zbl 1186.14059

Liu, Kefeng; Xu, Hao

Int. Math. Res. Not. 2009, No. 5, 835-859 (2009).

Let \(\overline{M}_{g,n}\) be the moduli space of stable curves of genus \(g\) with \(n\) marked points. In this paper the authors study intersection numbers \[ \int_{\overline{M}_{g,n}}\kappa_{b_1}\cdots \kappa_{b_k}\psi_1^{d_1}\cdots \psi_n^{d_n}\tag{1} \] of \(\kappa\) and \(\psi\) classes. When \(d_1 = \dots= d_n = 0\), these intersection numbers are called the higher Weil-Petersson volumes. The authors find recursion relations for the integrals, generalizing the ones due to M. Mulase and B. Safnuk which in turn generalize Mirzakhani’s recursion. The proof uses a result of which converts integrals with \(\kappa\) classes into integrals without \(\kappa\) classes, and also the Witten-Kontsevich theorem on descendant integrals. They also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves.

Reviewer: David Chataur (Villeneuve d’Ascq)

MSC:

14N35

Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)