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

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.

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Genre: Journal Article

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Kaufmann, R.1, Author
Manin, Yu.1, Author
Zagier, D.1, Author
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201

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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.

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Dates: 1996
Publication Status: Published in print
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Identifiers: eDoc: 744904
Other: 111
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Title: Communications in Mathematical Physics
Source Genre: Journal
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Publ. Info: Springer-Verlag, Berlin
Pages: - Volume / Issue: 181 (3) Sequence Number: - Start / End Page: 763 - 787 Identifier: ISSN: 0010-3616