D-branes, B fields, and Ext groups

In this paper we extend previous work on calculating massless boundary Ramond sector spectra of open strings to include cases with nonzero flat B fields. In such cases, D-branes are no longer well-modeled precisely by sheaves, but rather they are replaced by 'twisted' sheaves, reflecting the fact that gauge transformations of the B field act as affine translations of the Chan-Paton factors. As in previous work, we find that the massless boundary Ramond sector states are counted by Ext groups - this time, Ext groups of twisted sheaves. As before, the computation of BRST cohomology relies on physically realizing some spectral sequences. Subtleties that cropped up in previous work also appear here.