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Anton Kapustin : Mirror Symmetry and Noncommutative Geometry

I review the homological mirror symmetry conjecture of Kontsevich and its relation with topological D-branes. This conjecture is nontrivial even for tori, which allows one to perform some interesting checks. It turns out that in the presence of a general B-field Kontsevich's conjecture must be modified. In particular, on the B-side, coherent sheaves on a Calabi-Yau must be replaced by coherent sheaves on a certain `noncommutative deformation' of the Calabi-Yau. By contrast, an algebraic description of the A-model branes, if it exists, should involve a noncommutative algebra even for a vanishing B-field.

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