Boundary conditions and anomalies of conformal field theories in 1+1 dimensions

We study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in 1+1 dimensions.For a given CFT with a global symmetry, we consider symmetric "gapping potentials" which are relevant perturbations to the CFT.If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT.From this equivalence, if there exists a Cardy boundary state which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential.This means that the symmetry of the CFT is anomaly free.Using this approach, we systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries.They include the free compact boson theory, Wess-Zumino-Witten models, and unitary minimal models.When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system.Our results are consistent with, where available, known results in the literature.Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries.