Topological strings on noncommutative manifolds
arXiv:hep-th/0310057 · doi:10.1142/S0219887804000034
Abstract
We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases.
36 pages, AMS latex. v2: a reference to a related work has been added. v3: An error in the discussion of the Fourier-Mukai transform for twisted coherent sheaves has been fixed, resulting in several changes in Section 2. The rest of the paper is unaffected. v4: an incorrect statement concerning Lie algebroid cohomology has been fixed
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