Branes on Generalized Calibrated Submanifolds

We extend previous results on generalized calibrations to describe supersymmetric branes in supergravity backgrounds with diverse fields turned on, and provide several new classes of examples. As an important application, we show that supersymmetric D-branes in compactifications with field strength fluxes, and on SU(3)-structure spaces, wrap generalized calibrated submanifolds, defined by simple conditions in terms of the underlying globally defined, but non-closed, 2- and 3-forms. We provide examples where the geometric moduli of D-branes (for instance D7-branes in 3-form flux configurations) are lifted by the generalized calibration condition. In addition, we describe supersymmetric D6-branes on generalized calibrated 3-submanifolds of half-flat manifolds, which provide the mirror of B-type D-branes in IIB CY compactifications with 3-form fluxes. Supersymmetric sets of such D-branes carrying no homology charges are mirror to supersymmetric sets of D-branes which are homologically non-trivial, but trivial in K-theory. As an additional application, we describe models with chiral gauge sectors, realized in terms of generalized calibrated brane box configurations of NS- and D5-branes, which are supersymmetric but carry no charges, so that no orientifold planes are required in the compactification.