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

TL;DR

The paper develops generalized calibrations for D-branes in flux backgrounds, incorporating the world-volume gauge field and linking calibration bounds to unbroken supersymmetry. It shows that the calibration form derived from SUSY is closed and saturates the Dirac-Born-Infeld energy exactly when the gluino variation vanishes, establishing a precise equivalence between generalized calibrations and SUSY D-branes. The world-sheet approach yields the same SUSY conditions, and the results are embedded in the framework of generalized Calabi–Yau geometry, connecting B- and A-branes to generalized complex structures and pure spinors. The work also demonstrates how these calibrations align with Hitchin/Gualtieri notions of generalized Calabi–Yau geometry and highlights mirror symmetry as an exchange of structures between the two generalized complex structures.

Abstract

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.