D-branes, B-fields and twisted K-theory

TL;DR

This work addresses the classification of D-brane charges in backgrounds with a nontrivial NSNS B-field by proposing twisted K-theory built from sections of an infinite-dimensional algebra bundle $\mathcal{E}_{[H]}$ with fibre $\mathcal{K}$ and Dixmier-Douady invariant $\delta(\mathcal{E}_{[H]})=[H]$. It proves that the twisted K-theory groups $K^j(X,[H])$ are given by $K_j(C_0(X,\mathcal{E}_{[H]}))$, recovering Witten's Azumaya/Morita picture in the torsion case and generalizing to non-torsion B-fields. The paper also connects Rosenberg's twisted K-theory with Donovan–Karoubi's approach, provides a local gauge-bundle interpretation of $K^0(X,[H])$, and discusses extensions to noncompact spaces and potential links to cyclic homology and Connes-Cern–type maps. Overall, it offers a natural, noncommutative-geometric framework for D-brane charge classification across broad NS backgrounds, tying together finite- and infinite-dimensional bundle descriptions.

Abstract

In this note we propose that D-brane charges, in the presence of a topologically non-trivial B-field, are classified by the K-theory of an infinite dimensional C^*-algebra. In the case of B-fields whose curvature is pure torsion our description is shown to coincide with that of Witten.