D-branes, categories and N=1 supersymmetry

We show that boundary conditions in topological open string theory on Calabi-Yau (CY) manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise criterion determining the supersymmetry preserving branes at any point in CY moduli space, completing the proposal of II-stability. (C) 2001 American Institute of Physics.