Symmetry breaking boundary conditions and WZW orbifolds

TL;DR

The paper develops a comprehensive framework for symmetry-breaking boundary conditions in WZW theories via Z2 orbifolds, deriving explicit reflection coefficients and constructing the full orbifold modular data, including S^O and T^O, for both inner and outer automorphisms. It systematically builds twisted-sector characters through twining characters and orbit Lie algebras, and then identifies the corresponding boundary conditions using a classifying algebra, clarifying how untwisted and twisted sectors encode preserved versus broken bulk symmetries. The results unify the treatment of inner and outer automorphisms, connect to simple currents, and provide explicit fusion rules in the orbifold theory, enabling practical analysis of D-brane-like boundary conditions in group-manifold backgrounds. The approach is applicable to abelian orbifolds in general and offers concrete tools to relate geometric and algebraic descriptions of boundary phenomena in rational CFTs.

Abstract

Symmetry breaking boundary conditions for WZW theories are discussed. We derive explicit formulae for the reflection coefficients in the presence of boundary conditions that preserve only an orbifold subalgebra with respect to an involutive automorphism of the chiral algebra. The characters and modular transformations of the corresponding orbifold theories are computed. Both inner and outer automorphisms are treated.