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

Author: Maryam Mirzakhani
Journal: J. Amer. Math. Soc. 20 (2007), 1-23
MSC (2000): Primary 32G15, 14H15
DOI: https://doi.org/10.1090/S0894-0347-06-00526-1
Published electronically: March 8, 2006
MathSciNet review: 2257394
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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 $b_{1},\ldots, b_{n}$ , 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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