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#### Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves.

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##### Citation

Kaufmann, R., Manin, Y., & Zagier, D. (1996). Higher Weil-Petersson volumes of
moduli spaces of stable n-pointed curves.* Communications in Mathematical Physics,* *181*(3), 763-787.

Cite as: http://hdl.handle.net/21.11116/0000-0004-38EF-1

##### Abstract

The moduli spaces \barMg,n of stable n-pointed complex curves of genus g carry natural rational cohomology classes ωg,n(a) of degree 2a, which were introduced by Mumford for n=0 and subsequently by \it E. Arbarello and \it M. Cornalba [J. Algebr. Geom. 5, No. 4, 705-749 (1996; Zbl 0886.14007)] for all n. Integrals of products of these classes over \barMg,n are called higher Weil-Petersson volumes; if only ωg,n(1) is involved they reduce to classical WP volumes. \par \it 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 τd1 ⋅s τdn\rangle.\par Finally the authors describe the 1-dimensional cohomological field theories occurring in an article by \it M. Kontsevich and \it Yu. Manin with an appendix by \it 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 \it A. Kabanov and \it T. Kimura [``Intersection numbers and rank one cohomological field theories in genus one'', preprint 97-61, MPI Bonn] to the genus one case.