Quantum cohomology of a product (with appendix by R. Kaufmann).

*(English)*Zbl 0853.14021In a foregoing paper [Commun. Math. Phys. 164, No. 3, 525-562 (1994; see the preceding review)], the authors introduced an axiomatic approach to the Gromov-Witten classes of a complex projective algebraic manifold. They showed that these objects can be used to give an appropriate definition of the potential function \(\Phi^V\) for the related quantum cohomology associated with the manifold \(V\), and that in certain cases (Fano varieties) this construction of \(\Phi^V\) is geometrically meaningful and practically manageable. In this context, it was already sketched how the potential \(\Phi^{V \times W}\) of a product \(V \times W\) could be calculated in terms of \(\Phi^V\) and \(\Phi^W\), and that this link should correspond to the tensor multiplication of cohomological field theories or, equivalently, to that of algebras over the moduli operad of the moduli space of stable rational curves with marked points.

In the present paper, this result (together with the detailed, rather involved conceptual tools and calculations) is rigorously proved. The formalism used in the course of the proof, which by itself represents the main body of the paper, is then applied to the twisting operation in rank-one cohomological field theories. The potential of such a theory turns out to be a characteristic function involving Weil-Petersson volume forms, and a suitable generalization of Weil-Petersson forms leads to the construction of a canonical coordinate system on the group of invertible cohomological field theories of rank one. The authors’ method also gives a generalization of Zograf’s recursive Weil-Petersson volume formulas for moduli spaces of punctured spheres [cf. P. Zograf, Contemp. Math. 150, 367-372 (1993; Zbl 0792.32016)].

The appendix by R. Kaufmann (pp. 337–338) provides an important topological intersection formula for the boundary strata in the involved moduli spaces of pointed rational curves.

Although the authors amply review their framework developed in the foregoing paper reviewed above, it is recommended to study that work in the first place.

In the present paper, this result (together with the detailed, rather involved conceptual tools and calculations) is rigorously proved. The formalism used in the course of the proof, which by itself represents the main body of the paper, is then applied to the twisting operation in rank-one cohomological field theories. The potential of such a theory turns out to be a characteristic function involving Weil-Petersson volume forms, and a suitable generalization of Weil-Petersson forms leads to the construction of a canonical coordinate system on the group of invertible cohomological field theories of rank one. The authors’ method also gives a generalization of Zograf’s recursive Weil-Petersson volume formulas for moduli spaces of punctured spheres [cf. P. Zograf, Contemp. Math. 150, 367-372 (1993; Zbl 0792.32016)].

The appendix by R. Kaufmann (pp. 337–338) provides an important topological intersection formula for the boundary strata in the involved moduli spaces of pointed rational curves.

Although the authors amply review their framework developed in the foregoing paper reviewed above, it is recommended to study that work in the first place.

Reviewer: W.Kleinert (Berlin)

##### MSC:

14J45 | Fano varieties |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

14H10 | Families, moduli of curves (algebraic) |

58D30 | Applications of manifolds of mappings to the sciences |