A-branes and Noncommutative Geometry

TL;DR

This paper proposes an equivalence between the category of A-branes and a noncommutative deformation of B-branes on certain symplectic manifolds, using the Seiberg-Witten transform, with tests on symplectic tori.

Contribution

It introduces a novel equivalence between A-branes and noncommutative B-branes, extending the understanding of brane categories beyond Mirror Symmetry.

Findings

A-brane category is equivalent to a noncommutative deformation of B-branes.

The equivalence is demonstrated on symplectic tori.

The approach involves the Seiberg-Witten transform.

Abstract

We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to the derived category of coherent sheaves) on the same manifold. This equivalence is different from Mirror Symmetry and arises from the Seiberg-Witten transform which relates gauge theories on commutative and noncommutative spaces. More generally, we argue that for certain generalized complex manifolds the category of generalized complex branes is equivalent to a noncommutative deformation of the derived category of coherent sheaves on the same manifold. We perform a simple test of our proposal in the case when the manifold in question is a symplectic torus.