D-branes, B fields, and Ext groups
A. Caldararu, S. Katz, E. Sharpe
TL;DR
The paper addresses open-string massless boundary Ramond spectra for D-branes on Calabi–Yau manifolds in the presence of a flat B-field. It develops a framework in which D-branes are modeled by twisted sheaves and shows that spectra are counted by twisted Ext groups $Ext^*_{X, \omega}(i_*\mathcal{E}, j_*\mathcal{F})$, with spectral sequences between twisted sheaf cohomology and Ext groups realized in BRST cohomology. The work extends prior untwisted results, provides concrete examples and general intersection results, and argues that twisted derived categories (Brauer-twisted) are the natural mathematical setting for D-branes with flat B-fields. This advances the program of understanding D-branes via derived categories by incorporating background $B$-fields and points toward a broader twisted-geometric formulation of brane physics.
Abstract
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-modelled 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.