A note on D-branes in group manifolds: flux quantisation and D0-charge

TL;DR

This work shows that D-branes in group manifolds, described by (twisted) conjugacy classes, are intrinsically tied to a gauge-invariant two-form $\omega$ on their worldvolume, fixed by the gluing data $R$ and identified with $B+2\pi\alpha'F$. Quantum consistency of the boundary WZW action imposes two integrality conditions, yielding a discrete set of allowed branes and, specifically for $\mathrm{SU}(2)$ at level $k$, $k+1$ branes whose masses match CFT results. The authors propose a gauge-invariant, quantised D0-charge given by the relative-homology expression $\frac{1}{2\pi}[\int_{\partial\mathcal{B}} \tilde g^*\omega - \int_{\mathcal{B}} \tilde g^*H] \bmod k$, which encapsulates both bulk and boundary contributions and aligns with Page-like charge logic. They also find no evidence for a quantised $U(1)$ flux on this brane class, suggesting $\omega$ fully encodes the brane's gauge data in backgrounds with nontrivial $H$. These results illuminate the interplay between boundary state and sigma-model viewpoints and unify geometric, topological, and quantum aspects of D-branes in WZW backgrounds.

Abstract

We show that a D-brane in a group manifold given by a (twisted) conjugacy class is characterised by a gauge invariant two-form field determined in terms of the matrix of gluing conditions. Using a quantisation argument based on the path integral one obtains the known quantisation condition for the corresponding D-branes. We find no evidence for the existence of a quantised U(1) gauge field flux. We propose an expression for the D0 charge of such D-branes.