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Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy

Motohico Mulase·Brad Safnuk
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Abstract

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter is set to be 0.

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Motohico Mulase, Brad Safnuk. 2006-03-04. Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy. https://arxiv.org/abs/math/0601194

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