The derived category of a generalized complex torus

Abstract

For any translation invariant generalized complex structure J on a torus we construct the derived category D(J) generalizing the derived category of coherent sheaves on a complex torus and the affine/convergent Fukaya category of a symplectic torus. The category is defined in terms of a non-commutative curved deformation of the Dolbeault algebra of a complex torus. We show that this category is independent of the chosen deformation path, which we prove using non-commutative Fourier-Mukai duality. We also prove that the action of O(2n,2n; Z) on the space of generalized complex structures leaves the category D(J) invariant. In particular, T-duality acts trivially on D(J). We construct concrete objects associated to affine Lagrangians in the derived category associated to a symplectic torus and show that their spaces of morphisms are given by Floer cohomology. We show that our construction is consistent with the homological mirror symmetry conjecture (including its non-commutative version).