×

zbMATH — the first resource for mathematics

Invertible cohomological field theories and Weil-Petersson volumes. (English) Zbl 1001.14008
This paper is concerned with intersection theory on moduli spaces \(M_{g,n}\) of pointed curves. Viewing the homology ring \(H_\ast M_{g,n}=H_\ast (\overline{M}_{g,n+1},K)\) (where \(K\) is a field of characteristic zero) as an operad, cohomological field theories are cyclic algebras over it, and the invertible objects of this category form its Picard group. This group, graded by codimension, is largely unknown, except for elements \(\kappa_a\) and \(\mu_a\) in every odd codimension \(a\geq 1\) and one even element \(\kappa_a\) in every even codimension \(a\geq 2\). The construction of these classes, which uses the relative dualizing sheaf of the universal curve, is recalled, and the Weil-Petersson volume, as well as the more general potential (formal) function, are constructed in terms of these classes. The authors then relate the total free energy (formal) series and the Weil-Petersson volume by means of Schur polynomials; this allows them to produce a genus expansion for the volume, using the one given by C. Itzykson and J.-B. Zuber [Int. J. Mod. Phys. A 7, No. 23, 5661-5705 (1992; Zbl 0972.14500)] for the potential. The authors also provide a topological interpretation of the coefficients of the expansion; and, through a very impressive tour de force, which includes a version of the stationary phase method and a proof-by-picture created by Don Zagier (a computer-generated graph that provides a deformation of a contour of integration), they calculate the coefficients of the asymptotic expansion of the Weil-Petersson volumes, confirming a conjecture by Itzykson [cf. P. Zograf, “Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity”, preprint math.AG/9811026; http://front.math.ucdavis.edu)].
The paper also contains a number of suggested directions for extending the theory: proving Virasoro-type constraints to the potential function; defining a generalized quantum cohomology for any projective smooth algebraic variety \(V\); defining a quantum motivic fundamental group of \(V\), derived from the bialgebra structure of the operad \(H^\ast (\overline{M}_{g,2},K)\).

MSC:
14H10 Families, moduli of curves (algebraic)
18D50 Operads (MSC2010)
58D29 Moduli problems for topological structures
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
81T70 Quantization in field theory; cohomological methods
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] E. ARBARELLO, M. CORNALBA, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, Journ. Alg. Geom., 5 (1996), 705-749. · Zbl 0886.14007
[2] T. EGUCHI, Y. YAMADA, S.-K. YANG, On the genus expansion in the topological string theory, Rev. Mod. Phys., 7 (1995), 279. · Zbl 0837.58043
[3] C. FABER, R. PANDHARIPANDE, Hodge integrals and Gromov-Witten theory, Preprint math.AG/9810173. · Zbl 0960.14031
[4] E. GETZLER, M. M. KAPRANOV, Modular operads, Comp. Math., 110 (1998), 65-126. · Zbl 0894.18005
[5] V. GORBOUNOV, D. ORLOV, P. ZOGRAF (in preparation).
[6] C. ITZYKSON, J.-B. ZUBER, Combinatorics of the modular group II: the kont-sevich integrals, Int. J. Mod. Phys., A7 (1992), 5661. · Zbl 0972.14500
[7] A. KABANOV, T. KIMURA, Intersection numbers and rank one cohomological field theories in genus one, Comm. Math. Phys., 194 (1998), 651-674. · Zbl 0974.14018
[8] R. KAUFMANN, Yu. MANIN, D. ZAGIER, Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves, Comm. Math. Phys., 181 (1996), 763-787. · Zbl 0890.14011
[9] M. KONTSEVICH, Yu. MANIN, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525-562. · Zbl 0853.14020
[10] M. KONTSEVICH, Yu. MANIN, (with Appendix by R. Kaufmann), Quantum cohomology of a product, Inv. Math., 124 (1996), 313-340. · Zbl 0853.14021
[11] J. MORAVA, Schur Q-functions and a Kontsevich-Witten genus, Contemp. Math., 220 (1998), 255-266. · Zbl 0937.55003
[12] D. MUMFORD, Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry (M. Artin and J. Tate, eds.), Part II, Birkhäuser, 1983, 271-328. · Zbl 0554.14008
[13] F. W. J. OLVER, Introduction to asymptotics and special functions, Academic Press, 1974. · Zbl 0308.41023
[14] E. WITTEN, Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom., 1 (1991), 243-310. · Zbl 0808.32023
[15] S. WOLPERT, The hyperbolic metric and the geometry of the universal curve, J. Diff. Geo., 31 (1990), 417-472. · Zbl 0698.53002
[16] P. ZOGRAF, Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity, Preprint math.AG/9811026.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.