Quantum Cohomology and Free Field Representation

TL;DR

This paper develops a free field representation of Virasoro operators to describe quantum cohomology of Kähler manifolds, extending previous work and supporting the universality of Virasoro constraints in Gromov-Witten theory.

Contribution

It constructs a free field framework for Virasoro operators in quantum cohomology, applicable to all compact Kähler manifolds, including those with non-analytic classes.

Findings

Virasoro operators expressed via free bosonic and fermionic fields.

Virasoro conditions reproduce Gromov-Witten invariants for various manifolds.

Universal applicability of Virasoro constraints in quantum cohomology suggested.

Abstract

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 K\"{a}hler manifold we have a free bosonic (fermionic) field and Virasoro 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 ($h^{p,q}\not=0,p\not=q$) and suggest that the Virasoro condition holds universally for all compact smooth K\"{a}hler manifolds.