Resolution of stringy singularities by non-commutative algebras

In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploi ts the non-commutative structure of D-branes, so the space is described by an algebraic geometry of non-commutative rings. The paper is devoted to the study of examples of these algebras. In our study there is an auxiliary co mmutative algebraic geometry of the center of the (local) algebras which pl ays an important role as the target space geometry where closed strings pro pagate. The singularities that are resolved will be the singularities of th is auxiliary geometry. The singularities are resolved by the non-commutativ e algebra if the local non-commutative rings are regular. This definition g uarantees that D-branes have a well defined K-theory class. Homological fun ctors also play an important role. They describe the intersection theory of D-branes and lead to a formal definition of local quivers at singularities , which can be computed explicitly for many types of singularities. These r esults can be interpreted in terms of the derived category of coherent shea ves over the non-commutative rings, giving a non-commutative version of rec ent work by M. Douglas. We also describe global features like the Betti num bers of compact singular Calabi-Yau threefolds via global holomorphic secti ons of cyclic homology classes.