On the Absence of Symmetric Simple Conformal Boundary Conditions

Non-trivial 't Hooft anomaly obstructs the existence of a simple symmetric conformal boundary condition in a CFT. Conversely, there is a common piece of lore that trivial 't Hooft anomaly promises the existence of a simple symmetry conformal boundary condition in a given CFT. Recently, counter examples to this lore was realized in tetracritical Ising CFT [1] and compact boson [2] -- the simple conformal boundary conditions preserving certain anomaly-free subsymmetry are absent in these CFTs. In this work, we uncover the underlying reason for the absence of these boundary conditions in counter examples, and propose a criterion diagnosing when the lore fails for any given 2d CFT. The Symmetry TFT description for boundary conditions plays a crucial role.