Gromov-Witten Invariants and Quantization of Quadratic Hamiltonians

Abstract. We describe a formalism based on quantizationof quadratic hamil-tonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about Gromov-Witten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gro-mov – Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere. 1. Gromov – Witten invariants. Let X be a compact almost Kähler manifold of complex dimension D. Denote by Xg,m,d the moduli (orbi)space of degree d stable holomorphic maps to X of genus g curves with m marked points 27, 3. The degree d takes values in the lattice H2(X). The moduli space is compact and can be equipped 2, 29, 36 with a rational coefficient virtual fundamental cycle Xg,m,d of complex dimension m+ (1 − g)(D − 3) + ∫ d c1(TX). The total descendent potential of X is defined as DX: = exp g−1FgX, where FgX is the genus g descendent potential m,d