Conformal Toda theory with a boundary

We investigate $s\ell_n$ conformal Toda theory with maximally symmetric boundaries, using classical and conformal bootstrap methods. We compare the results with the well-known case of Liouville theory ($n=2$). For all values of $n\ge 2$, we find that there exist D-branes of all possible dimensions $0\le d\le n-1$. For $n\ge 3$ however, there appear infinite multiplicities in the boundary spectrum, in sharp contrast with the Liouville case.