Gromov - Witten invariants and quantization of quadratic hamiltonians

TL;DR

This paper introduces a formalism based on quantization and symplectic actions to unify and extend results and conjectures related to Gromov-Witten invariants and Frobenius structures, including Virasoro constraints.

Contribution

It develops a new formalism for Gromov-Witten invariants using quantization of quadratic Hamiltonians and symplectic loop group actions, enabling proofs of key conjectures.

Findings

Established Virasoro constraints for semisimple Frobenius structures

Outlined a proof of the Virasoro conjecture for projective spaces

Provided a framework connecting Gromov-Witten invariants with symplectic geometry

Abstract

We describe a formalism based on quantization of quadratic hamiltonians 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 Gromov -- Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.