Topological strings on noncommutative manifolds

TL;DR

This paper explores a deformation of the N=2 supersymmetric sigma model on Calabi-Yau manifolds that interpolates between A- and B-models using noncommutative geometry and generalized complex structures, revealing new insights into topological strings and D-branes.

Contribution

It introduces a noncommutative deformation of the sigma model that unifies A- and B-models via generalized complex structures, expanding the understanding of topological string theory.

Findings

Deformation interpolates between A- and B-models on hyperkahler manifolds.

Path integral localizes on generalized holomorphic maps under deformation.

Derived a Chern character constraint for topological D-branes.

Abstract

We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases.