Constraining boundary conditions in non-rational CFTs

We revisit the question of conformal boundary conditions in the compact free boson CFT in two dimensions. Besides the well-known Neumann and Dirichlet cases, there is an additional proposed one-parameter family of boundary states when the radius is an irrational multiple of the self-dual radius. These additional states have a continuous open string spectrum, and we give an explicit formula for the density of states. We also discuss several pathologies of these states, including the possible violation of the cluster condition, and that they have a divergent g-function.