D-brane charge, flux quantisation and relative (co)homology

TL;DR

This paper explores the relationship between D-brane charge, flux quantization, and relative (co)homology in the WZW model, revealing topological obstructions and clarifying the role of gerbes in flux ambiguity.

Contribution

It clarifies the connection between D0-charge and relative cohomology, identifying obstructions due to NS 3-form cohomology and explaining flux ambiguity via gerbes.

Findings

D0-charge is a reduction of a class in relative cohomology.

Obstruction to line bundle existence is given by NS 3-form cohomology.

Flux ambiguity is not detected by D0-charge, related to gerbes.

Abstract

We reconsider the problem of U(1) flux and D0-charge for D-branes in the WZW model and investigate the relationship between the different definitions that have been proposed recently. We identify the D0-charge as a particular reduction of a class in the relative cohomology of the group modulo the D-submanifold. We investigate under which conditions this class is equivalent to the first Chern class of a line bundle on the D-submanifold and we find that in general there is an obstruction given by the cohomology class of the NS 3-form. Therefore we conclude that for topologically nontrivial B-fields, there is strictly speaking no U(1) gauge field on the D-submanifold. Nevertheless the ambiguity in the flux is not detected by the D0-charge. This has a natural interpretation in terms of gerbes.