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

Abstract

In this paper, we establish a relationship between the Weil- Petersson volume V g , n ( b ) V_{g,n}(b) of the moduli space M g , n ( b ) \mathcal {M}_{g,n}(b) of hyperbolic Riemann surfaces with geodesic boundary components of lengths b 1 b_{1} , …, b n b_{n} , and the intersection numbers of tautological classes on the moduli space M ¯ g , n \overline {\mathcal {M}}_{g,n} of stable curves. As a result, by using the recursive formula for V g , n ( b ) 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.