D-branes in the WZW model

It is stated in the literature that $D$-branes in the Wess-Zumino-Witten (WZW) model associated with the gluing condition $J=-\overline{J}$ along the boundary correspond to branes filling out the whole group volume. We show instead that the end points of open strings are rather bound to stay on “integer” conjugacy classes. In the case of the $\mathrm{SU}\left(2\right)\mathrm{}$ level $k$ WZW model we obtain $k-1$ two-dimensional Euclidean $D$-branes and two $D$ particles sitting at the points $e$ and $-e.\mathrm{}$