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Published May 11, 2016 | Submitted
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A-branes and Noncommutative Geometry

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.

Additional Information

(Submitted on 23 Feb 2005) February 1, 2008. I would like to thank Dima Orlov, Oren Ben-Bassat, Jonathan Block, Tony Pantev, and Marco Gualtieri for helpful discussions. I am also grateful to the organizers of the Workshop on Mirror Symmetry at the University of Miami for providing a stimulating atmosphere. This work was supported in part by the DOE grant DE-FG03-92-ER40701.

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August 19, 2023
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