D-branes, derived categories, and Grothendieck groups

In this paper we describe how Grothendieck groups of coherent sheaves and l ocally free sheaves can be used to describe type II D-branes, in the case t hat all Pbranes are wrapped on complex varieties and all connections are ho lomorphic. Our proposal is in the same spirit as recent discussions of K-th eory and D-branes; within the restricted class mentioned, Grothendieck grou ps encode a choice of connection on each D-brane world-volume, in addition to information about the C-infinity bundles. We also point out that derived categories can also be used to give insight into D-brane constructions, an d analyze how a Z(2) subset of the T-duality group acting on D-branes on to ri can be understood in terms of a Fourier-Mukai transformation. (C) 1999 E lsevier Science B.V. All rights reserved.