Symmetry breaking boundaries I. General theory

TL;DR

The paper provides a general framework for conformally invariant boundary conditions that partially break bulk symmetries, restricting to subalgebras fixed by a finite abelian orbifold group. By recasting the bulk theory as a simple-current extension and carefully treating untwisted stabilizers, the authors construct boundary blocks, boundary states, reflection coefficients, and annulus amplitudes, proving the integrality of the open-string channel data. Central to the construction is the classifying algebra ${\cal C}({\bar{\mathfrak A}})$ with a diagonalizing matrix $\tilde{S}$, which organizes boundary conditions into irreducible one-dimensional representations and connects boundary data to chiral blocks via Verlinde-like relations. The work also develops the ${\cal G}'$-extension to establish integrality of the annulus coefficients and demonstrates consistency checks that align open/closed string perspectives, enriching the understanding of symmetry-breaking boundaries with a robust algebraic structure. These results generalize known boundary-condition architectures (e.g., charge conjugation, simple-current automorphisms) and provide tools applicable to open-string sectors and statistical-model boundary phenomena.

Abstract

We study conformally invariant boundary conditions that break part of the bulk symmetries. A general theory is developped for those boundary conditions for which the preserved subalgebra is the fixed algebra under an abelian orbifold group. We explicitly construct the boundary states and reflection coefficients as well as the annulus amplitudes. Integrality of the annulus coefficients is proven in full generality.