Resolution of stringy singularities by non-commutative algebras

Abstract

In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploits 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 commutative algebraic geometry of the center of the (local) algebras which plays an important role as the target space geometry where closed strings propagate. The singularities that are resolved will be the singularities of this auxiliary geometry. The singularities are resolved by the non-commutative algebra if the local non-commutative rings are regular. This definition guarantees that D-branes have a well defined K-theory class. Homological functors 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 results can be interpreted in terms of the derived category of coherent sheaves over the non-commutative rings, giving a non-commutative version of recent work by M. Douglas. We also describe global features like the Betti numbers of compact singular Calabi-Yau threefolds via global holomorphic sections of cyclic homology classes.