Classifying A-field and B-field configurations in the presence of D-branes

We "solve" the Freed-Witten anomaly equation, i.e., we find a geometrical classification of the B-field and A-field configurations in the presence of D-branes that are anomaly-free. The mathematical setting being provided by the geometry of gerbes, we find that the allowed configurations are jointly described by a coset of a certain hypercohomology group. We then describe in detail various cases that arise according to such classification. As is well-known, only under suitable hypotheses the A-field turns out to be a connection on a canonical gauge bundle. However, even in these cases, there is a residual freedom in the choice of the bundle, naturally arising from the hypercohomological description. For a B-field which is flat on a D-brane, fractional or irrational charges of subbranes naturally appear; for a suitable gauge choice, they can be seen as arising from "gauge bundles with not integral Chern class": we give a precise geometric interpretation of these objects.