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Weil-Petersson volumes and intersection theory on the moduli space of curves. (English) Zbl 1120.32008
Let \(M_{g,n}\) be the moduli space of genus \(g\) curves with \(n\) marked points and \(\overline M_{g, n}\) its Deligne-Mumford compactification, and let \(M_{g,n}(b)=M_{g,n}(b_1,\dots,b_n)\) be the moduli space of hyperbolic Riemann surfaces with genus \(g\), \(n\) marked points and \(n\) geodesic boundary components of lengths \(b_1,\dots,b_n\).
In the paper under review, the author establishes a relation between the Weil-Petersson volume \(V_{g,n}(b)\) of \(M_{g,n}(b)\) and the intersection numbers of tautological classes on the moduli space \(\overline M_{g, n}\) of stable curves. By using a previous recursive formula for the volume \(V_{g,n}(b)\), the author derives a new proof of the Virasoro constraints of a point, which is equivalent to the Witten-Kontsevich formula [M. Kontsevich, Commun. Math. Phys. 147, No. 1, 1–23 (1992; Zbl 0756.35081)].

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
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