Quantum Cohomology of a Product

0. Introduction 0.1. Quantum cohomology. Quantum cohomology of a projective algebraic manifold V is a formal deformation of its cohomology ring. Parameters of this deformation are coordinates on the space H * (V), and the structure constants are the third derivatives of a formal series Φ V (potential, or free energy) whose coefficients count the number of parametrized rational curves on V with appropriate incidence conditions (see [KM] for details). A natural problem arises how to calculate Φ V ×W in terms of Φ V and Φ W. In [KM] it was suggested that this operation corresponds to that of tensor multiplication of Cohomological Field Theories, or equivalently, algebras over the moduli operad {H * (M 0,n+1)}. The definition of the tensor product depends on a theorem on the structure of H * (M 0,n+1) whose proof was only sketched in [KM] (Theorem 7.3). One of the main goals of this note is to present this proof and related calculations in full detail. (For another proof, see [G]). We also discuss the rank one CohFT's and the respective twisting operation. An interesting geometric example of such a theory is furnished by Weil–Petersson forms. Potential of this theory is a characteristic function involving Weil–Petersson volumes calculated in [Z]. We show that a generalization of WP–forms allows one to construct a canonical coordinate system on the group of invertible CohFT's. We start with a brief review of the relevant structures from [KM]. Let H be a finite-dimensional Z 2 –graded linear space over a field K of characteristic zero, endowed with a non–degenerate even symmetric scalar product g. The main fact is the equivalence of two notions: i) A formal solution Φ of associativity, or WDVV, equations on (H, g). ii) A structure of CohFT on (H, g).