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

TL;DR

This work reframes D0-charge and flux quantisation for D-branes in WZW models using relative cohomology on the group $G$ modulo the D-submanifold $Q$. It shows that the physically meaningful D0-charge is the reduction modulo $k$ of the relative class $[(H,oldsymbol{})]/2oldsymbol{}$ in $H^3(G,Q;oldsymbol{})$, with an obstruction given by the NS 3-form $H$ to the existence of a global line bundle on $Q$. When $H$ is nontrivial, no canonical line bundle exists; instead, a gerbe provides the natural global structure, and locally defined line bundles or gauge fields reproduce the D0-charge modulo $k$. The analysis connects worldsheet anomaly cancellation, relative cohomology, and gerbe language, offering a coherent picture applicable to general backgrounds with a nonzero NS 3-form.

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.

Figures (3)

• Figure 1: The relation $\partial M = g(\Sigma) + D$. In the figure $M$ is the solid object whose boundary is $g(\Sigma) + D$, where $D$ is contained in the D-submanifold $Q$.
• Figure 2: Gluing $M_1$ and $M_2$ along $g(\Sigma)$ to obtain the relative cycle $Z$, and gluing $D_1$ and $D_2$ along $g(\partial\Sigma)$ to obtain its boundary $\partial Z= S$.
• Figure 3: Two ways of computing the D0-charge of a spherical D2-brane $Q$ in $\text{SU}(2)$.