an:01002494
Zbl 0890.14011
Kaufmann, R.; Manin, Yu.; Zagier, D.
Higher Weil-Petersson volumes of moduli spaces of stable \(n\)-pointed curves
EN
Commun. Math. Phys. 181, No. 3, 763-787 (1996).
00036911
1996
j
14H10 14H20
moduli spaces of stable pointed curves; Weil-Petersson volumes; 1-dimensional cohomological field theories; rational cohomology classes
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 \textit{E. Arbarello} and \textit{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.
\textit{P. Zograf} [in: Mapping class groups and moduli spaces of Riemann surfaces, Proc. Workshops G??ttingen 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 \textit{M. Kontsevich} and \textit{Yu. Manin} with an appendix by \textit{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 \textit{A. Kabanov} and \textit{T. Kimura} [``Intersection numbers and rank one cohomological field theories in genus one'', preprint 97-61, MPI Bonn] to the genus one case.
Martin Pikaart (Essen)
Zbl 0792.32016; Zbl 0853.14021; Zbl 0886.14007