Topological conformal field theories and Calabi-Yau categories

TL;DR

This paper develops an algebraic framework for Gromov-Witten invariants using Calabi-Yau A-infinity categories, connecting algebraic and symplectic geometry in the context of topological conformal field theories.

Contribution

It constructs a purely algebraic approach to Gromov-Witten invariants at all genera via Calabi-Yau categories, unifying B-model and A-model perspectives.

Findings

Algebraic construction of Gromov-Witten invariants at all genera.

Recovery of classical Gromov-Witten invariants from Fukaya categories.

Establishment of conditions linking Hochschild homology to homology of the manifold.

Abstract

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A-infinity category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A-infinity version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that a good theory of open-closed Gromov-Witten invariants exists for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.