Symmetry breaking boundaries I. General theory

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.Comment: 60 pages, LaTeX2e; typos fixed and other minor correction

Symmetry breaking boundaries

“…The results established in [1] clearly demonstrate an unexpectedly nice behavior of the space of conformally invariant boundary conditions. In the present paper we show that indeed this space is endowed with even more structure.…”

“…In [1] we have studied conformally invariant boundary conditions for an arbitrary conformal field theory that preserve a (consistent) subalgebraĀ of A, such that A = A G is the subalgebra that is fixed under a finite abelian group G of automorphisms of A. We have shown that such boundary conditions are governed by a classifying algebra C(Ā), in the sense that the reflection coefficients [2,3] -the data that characterize the boundary condition -are precisely the one-dimensional irreducible representations of C(Ā).…”

“…Our numbering of sections will be consecutive, i.e. by sections 1 to 6 we refer to those of the preceding paper, while here we will start with section 7; accordingly equation numbers with section label ≤ 6 refer to formulas in [1]. 1 A very brief summary of the pertinent results of [1] is as follows.…”

“…Here the stabilizer S λ and the untwisted stabilizer U λ are subgroups of the simple current group G = G * ; S λ consists of all simple currents in G that leaveλ fixed, and U λ is the subgroup of S λ on which a certain alternating bi-homomorphism F λ , which can be defined through the modular properties of one-point chiral blocks on the torus, is trivial. For the precise meaning of these terms we refer to [1] (in particular appendix A) and to [4]. Quite generally, the number of basis elements of C(Ā) -or equivalently, the number of independent boundary blocks, i.e.…”

“…Afterwards, in section 16 we provide several classes of examples with more complicated orbifold groups, in which for instance untwisted stabilizer subgroups occur that are proper subgroups of the full stabilizers. Finally, some of the pertinent formulae from [1] that will be needed in the sequel are collected in appendix A.…”