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Extending mirror conjecture to Calabi-Yau with bundles. (English) Zbl 0986.14500
The author extends the usual understanding of mirror symmetry to the case of Calabi-Yau base manifolds with a stable bundle as additional data. The corresponding mirror objects are supersymmetric cycles on the mirror manifold. For a Calabi-Yau $$n$$-fold $$N$$ copies of $$D$$-branes wrapped over it will yield a $$U(N)$$-bundle which will be stable by the required supersymmetry. The motivation originates from type II string theory. It should be expected that the deformation of the bundle corresponds to the deformation of the supersymmetric cycle inside the mirror. This relation is studied for Calabi-Yau threefolds (e.g. for Chern-Simons theory) in more detail.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J81 Relationships with physics 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 14J30 $$3$$-folds
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