Invertible cohomological field theories and Weil-Petersson volumes.

*(English)*Zbl 1001.14008This 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 \(H^\ast (\overline{M}_{g,2},K)\).

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 \(H^\ast (\overline{M}_{g,2},K)\).

Reviewer: Emma Previato (Boston)

##### 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 |

##### Keywords:

moduli spaces; operads; Weil-Petersson volumes; cohomological field theories; Schur polynomials; stationary phase method; Virasoro constraints; potential; quantum cohomology; KdV equation
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\textit{Y. I. Manin} and \textit{P. Zograf}, Ann. Inst. Fourier 50, No. 2, 519--535 (2000; Zbl 1001.14008)

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