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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)
##### Keywords:
Weil-Petersson volume; moduli space
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##### References:
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