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

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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