D-brane stability, geometric engineering, and monodromy in the Derived Category

We discuss aspects of topological B-type D-branes in the framework of the derived category of coherent sheaves D(X) on a Calabi–Yau 3-fold X. We analyze the link between massless D-branes and monodromies in the CFT moduli space. A classification of all massless D-branes at any point in the moduli space is conjectured, together with an associated monodromy. We test the conjectures in two independent ways. First we establish a composition formula for certain Fourier-Mukai functors, which is a consequence of the triangulated structure of D(X). Secondly, using π-stability we rederive the stable soliton spectrum of the pure N = 2 supersymmetric SU(2) SeibergWitten theory. In this approach, the simplicity of the spectrum rests on Grothendieck’s theorem concerning vector bundles over P1. ∗Email: [email protected]