Stable D-branes, calibrations and generalized Calabi-Yau geometry

We introduce generalized calibrations that take into account the gauge field on the D-brane so that calibrated submanifolds minimize the Dirac-Born-Infeld energy. We establish the calibration bound and show that the calibration form is closed in a supersymmetric background with non-vanishing NS-NS 3-form H and dilaton. We show that the calibration conditions are equivalent to the existence of unbroken supersymmetry on the D-brane. We study the problem of supersymmetric D-branes in the presence of non-vanishing H also from the world-sheet approach and find exactly the same conditions. Finally, we show that our notion of generalized calibrations is equivalent to the calibrations introduced in the context of generalized Calabi-Yau geometry in math.DG/0401221.