Categories of holomorphic vector bundles on noncommutative two-tori

Abstract

In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus.

AMSLatex, 23 pages, references added

References in corpus (5)

• Dirichlet branes, homological mirror symmetry, and stability
• Real Multiplication and noncommutative geometry
• Differential and complex geometry of two-dimensional noncommutative tori
• Non-commutative compactifications and elliptic curves
• Theta-functions on noncommutative tori

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• Noncommutative complex geometry of the quantum projective space
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• Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
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• Sklyanin algebras revisited
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• Noncommutative Riemann Conditions
• Abelian and derived deformations in the presence of Z-generating geometric helices
• Noncommutative Geometry in the Framework of Differential Graded Categories
• Some remarks on mirror symmmetry and noncommutative elliptic curves
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