Symmetric boundary conditions in boundary critical phenomena

TL;DR

This paper classifies conformally invariant boundary conditions in minimal models, analyzes their symmetry properties, and identifies unique and problematic boundary conditions in various models, revealing insights into their physical consistency.

Contribution

It provides a classification of boundary conditions using Lie algebra automorphisms and characterizes invariant solutions in unitary minimal models, highlighting differences across series.

Findings

Invariant boundary conditions correspond to fixed points in the product graph $A imes G$.

Unique invariant boundary condition exists in $(A,A)$ models.

Some invariant boundary conditions in $(A,D)$ and $(A,E_6)$ models are unphysical.

Abstract

Conformally invariant boundary conditions for minimal models on a cylinder are classified by pairs of Lie algebras $(A,G)$ of ADE type. For each model, we consider the action of its (discrete) symmetry group on the boundary conditions. We find that the invariant ones correspond to the nodes in the product graph $A \otimes G$ that are fixed by some automorphism. We proceed to determine the charges of the fields in the various Hilbert spaces, but, in a general minimal model, many consistent solutions occur. In the unitary models $(A,A)$, we show that there is a unique solution with the property that the ground state in each sector of boundary conditions is invariant under the symmetry group. In contrast, a solution with this property does not exist in the unitary models of the series $(A,D)$ and $(A,E_6)$. A possible interpretation of this fact is that a certain (large) number of invariant boundary conditions have unphysical (negative) classical boundary Boltzmann weights. We give a tentative characterization of the problematic boundary conditions.