Quantum cohomology and free-field representation Original Research Article

In our previous article we have proposed that the Virasoro algebra controls the quantum cohomology of Fano varieties at all genera. In this paper we construct a free-field description of Virasoro operators and quantum cohomology. We shall show that to each even (odd) homology class of a Kähler manifold we have a free bosonic (fermionic) field and Visasoro operators are given by a simple bilinear form of these fields. We shall show that the Virasoro condition correctly reproduces the Gromov-Witten invariants also in the case of manifolds with non-vanishing non-analytic classes (hp,q ≠ 0, p ≠ q) and suggest that the Virasoro condition holds universally for all compact smooth Kähler manifolds.