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

Abstract

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.

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Cited by in corpus (7)

• Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries
• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Boundary condition and reflection anomaly in $2+1$ dimensions
• Boundary critical phenomena in the quantum Ashkin-Teller model