Fourier-Mukai Transform and Mirror Symmetry for D-Branes on Elliptic Calabi-Yau

الأصل · EN

Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau three-fold X is seen to have an expected adiabatic form for its induced cohomology operation only when an appropriately twisted operation resp. twisted charge is defined. Some differences with the case of K3 as well as connections with the spectral cover construction for bundles on X are pointed out. In the context of mirror symmetry Kontsevich's association of line bundle twists (resp. a certain 'diagonal' operation) with monodromies (esp. the conifold monodromy) is made explicit and checked for two example models. Interpreting this association as a relation between FM transforms and monodromies, we express the fibrewise FM transform through known monodromies. The operation of fibrewise duality as well as the notion of a certain index relevant to the computation of the moduli space of the bundle is transported to the sLag side. Finally the moduli space for D4-branes and its behaviour under the FM transform is considered with an application to the spectral cover.