Weil-Petersson volumes and intersection theory on the moduli space of curves

Mirzakhani, Maryam

doi:10.1090/S0894-0347-06-00526-1

Weil-Petersson volumes and intersection theory on the moduli space of curves
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by Maryam Mirzakhani

J. Amer. Math. Soc. 20 (2007), 1-23

DOI: https://doi.org/10.1090/S0894-0347-06-00526-1

Published electronically: March 8, 2006

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Abstract:

In this paper, we establish a relationship between the Weil- Petersson volume $V_{g,n}(b)$ of the moduli space $\mathcal {M}_{g,n}(b)$ of hyperbolic Riemann surfaces with geodesic boundary components of lengths $b_{1}$, …, $b_{n}$, and the intersection numbers of tautological classes on the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. As a result, by using the recursive formula for $V_{g,n}(b)$ obtained in the author’s Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, preprint, 2003, we derive a new proof of the Virasoro constraints for a point. This result is equivalent to the Witten-Kontsevich formula.

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