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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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[1] [AC] Arbarello, E., Cornalba, M.: Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. Preprint, 1995 (to appear in J. of Alg. Geom). · Zbl 0886.14007
[2] [G] Getzler, E.: Operads and moduli spaces of genus zero Riemann surfaces In: The Moduli Space of Curves, ed. by R. Dijkgraaf, C. Faber, G. van der Geer, Progress in Math. vol.129, Basel-Boston; Birkhäuser, 1995, pp. 199–230 · Zbl 0851.18005
[3] [Ke] Keel, S.: Intersection theory of moduli spaces of stablen-pointed curves of genus zero. Trans. AMS330, 545–574 (1992) · Zbl 0768.14002
[4] [K1] Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys.147, 1–23 (1992) · Zbl 0756.35081
[5] [K2] Kontsevich, M.: Enumeration of rational curves via torus actions. In: The Moduli Space of Curves, ed. by R. Dijkgraaf, C. Faber, G. van der Geer, Progress in Math. vol.129, Basel-Boston; Birkhäuser, 1995, pp. 335–368 · Zbl 0885.14028
[6] [KM] Kontsevich, M., Yu. Manin.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys.164:3, 525–562 (1994) · Zbl 0853.14020
[7] [KMK] Kontsevich, M., Yu. Manin (with Appendix by R. Kaufmann): Quantum cohomology of a product. Inv. Math.124, f. 1–3, 313–340 (1996) · Zbl 0853.14021
[8] [LS] Logan, B., Shepp, L.: A variational problem for random Young tableaux. Adv. in Math.26, (1977) 206–222 · Zbl 0363.62068
[9] [M] Yu. Manin.: Generating functions in algebraic geometry and sums over trees. In: The Moduli Space of Curves, ed. by R. Dijkgraaf, C. Faber, G. van der Geer, Progress in Math., vol.129, Basel-Boston; Birkhäuser, 1995, pp. 401–418. · Zbl 0871.14022
[10] [Ma] Matone, M.: Nonperturbative model of Liouville gravity. Preprint hep-th/9402081
[11] [W] Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv. in Diff. Geo.1, 243–310 (1991) · Zbl 0757.53049
[12] [VK] A. Vershik, S. Kerov: Asymptotic theory of the characters of the symmetric group. Func. An. Appl.15:4, 15–27 (1981)
[13] [Z] Zograf, P.: The Weil-Petersson volume of the moduli spaces of punctured spheres. In: Cont. Math. 150 (1993), ed. by R.M. Hain and C.F. Bödigheimer, pp. 267–372 · Zbl 0792.32016
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