D-branes in a topologically nontrivial B-field

TL;DR

The authors address global worldsheet anomalies for open strings in a topologically nontrivial B-field, showing that anomaly cancellation dictates a precise relation between t'Hooft flux on D-branes and the B-field class [H]. They develop a framework where D-brane gauge fields are connections on modules over Azumaya algebras, linking this to twisted K-theory K_H(M) in the torsion case. Their key mechanism relies on interpreting the trace of holonomy and the B-field phase as line-bundle data whose pullbacks to worldsheet mapping spaces become trivial under the anomaly-canceling conditions, thereby yielding a well-defined path integral. The results extend to superstrings and Type I theories, establishing β'(y)=[H] (and W_3(N) terms) as central cancellation conditions and positioning twisted K-theory as the natural D-brane charge classification in these backgrounds. Overall, the work connects global anomaly cancellation with noncommutative geometry and twisted K-theory in string theory, clarifying the geometric meaning of D-brane gauge fields and the role of torsion B-fields.

Abstract

We study global worldsheet anomalies for open strings ending on several coincident D-branes in the presence of a B-field. We show that cancellation of anomalies is made possible by a correlation between the t'Hooft magnetic flux on the D-branes and the topological class of the B-field. One application of our results is a proper understanding of the geometric nature of the gauge field living on D-branes: rather than being a connection on a vector bundle, it is a connection on a module over a certain noncommutative algebra. Our argument works for a general closed string background. We also explain why in the presence of a topologically nontrivial B-field whose curvature is pure torsion D-branes represent classes in a twisted K-theory, as conjectured by E. Witten.