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Higher Weil-Petersson volumes of moduli spaces of stable $$n$$-pointed curves. (English) Zbl 0890.14011
The moduli spaces $$\bar{M}_{g,n}$$ of stable $$n$$-pointed complex curves of genus $$g$$ carry natural rational cohomology classes $$\omega_{g,n}(a)$$ of degree $$2a$$, which were introduced by Mumford for $$n=0$$ and subsequently by E. Arbarello and M. Cornalba [J. Algebr. Geom. 5, No. 4, 705-749 (1996; Zbl 0886.14007)] for all $$n$$. Integrals of products of these classes over $$\bar{M}_{g,n}$$ are called higher Weil-Petersson volumes; if only $$\omega_{g,n}(1)$$ is involved they reduce to classical WP volumes.
P. Zograf [in: Mapping class groups and moduli spaces of Riemann surfaces, Proc. Workshops Göttingen 1991, Seattle 1991, Contemp. Math. 150, 367-372 (1993; Zbl 0792.32016)] obtained recursive formulas for the classical WP volumes involving binomial coefficients. The authors generalise them in several ways: first they give both recursive formulas and closed formulas involving multinomial coefficients for higher WP volumes in genus 0, secondly they obtain a closed formula for higher WP volumes in arbitrary genus, where the multinomial coefficients get replaced by the less well known correlation numbers $$\langle \tau_{d_1} \cdots \tau_{d_n}\rangle$$.
Finally the authors describe the 1-dimensional cohomological field theories occurring in an article by M. Kontsevich and Yu. Manin with an appendix by R. Kaufmann [Invent. Math. 124, No. 1-3, 313-339 (1996; Zbl 0853.14021)] explicitly using the generating function they found for the higher WP volumes in genus 0. This last description has been generalised by A. Kabanov and T. Kimura [“Intersection numbers and rank one cohomological field theories in genus one”, preprint 97-61, MPI Bonn] to the genus one case.

MSC:
 14H10 Families, moduli of curves (algebraic) 14H20 Singularities of curves, local rings
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References:
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