Phases of supersymmetric D-branes on Kähler manifolds and the McKay correspondence

We study the topological zero mode sector of type II strings on a Kahler manifold X in the presence of boundaries. We construct two nite bases, in a sense bosonic and fermionic, that generate the the topo- logical sector of the Hilbert space with boundaries. The fermionic basis localizes on compact submanifolds in X.A variation of the FI terms interpolates between the description of these ground states in terms of the ring of chiral elds at the boundary at small volume and helices of exceptional bundles at large volume, respectively. The iden- tication of the bosonic/fermionic basis with the dual bases for the non-compact/compact K-theory group on X gives a natural explana- tion of the McKay correspondence in terms of a linear sigma model and suggests a simple generalization of McKay to singular resolutions. The construction provides also a very eective way to describe D-brane states on generic, compact Calabi{Yau manifolds and allows to recover detailed information on the moduli space, such as monodromies and analytic continuation matrices, from the group theoretical data of a simple orbifold.