Completeness conditions for boundary operators in 2D conformal field theory

Abstract

In non-diagonal conformal models, the boundary fields are not directly related to the bulk spectrum. We illustrate some of their features by completing previous work of Lewellen on sewing constraints for conformal theories in the presence of boundaries. As a result, we include additional open sectors in the descendants of D odd SU (2) WZW models. A new phenomenon emerges, the appearance of multiplicities and fixed-point ambiguities in the boundary algebra not inherited from the closed sector. We conclude by deriving a set of polynomial equations, similar to those satisfied by the fusion-rule coefficients N k ij , for a new tensor A i ab that determines the open spectrum.

Symmetry breaking boundaries

“…In the special case of trivial automorphism type, g = 1, we can use the result [1] that S (λ,ψ),ρ = S λ,ρ for all ρ with g ρ = 1 to see that the ideal C (1) (Ā) is nothing but the fusion rule algebra of the A-theory, so that we recover the known results [15,3] for boundary conditions that do not break any of the bulk symmetries. It should also be noticed that the precise form of the structure constants of the ideals C (g) (Ā) does depend on the choice of representativesλ • and ψ • (except when g = 1, where independence of this choice follows as a consequence of the simple current relation (8.4)).…”

“…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).…”

The conformal boundary states for SU(2) at level 1

“…On the other hand, as far as we are aware, a general proof that the algebra is associative is still missing. In simple cases such as rational conformal field theories with charge-conjugate partition function and with standard gluing conditions imposed (which, in particular, preserve the full symmetry algebra), one can show that the classifying algebra is nothing but the fusion ring, M ab c = N ab c , see [10,7,2]. Under the same assumptions, solutions to all sewing relations were found in [11], expressed in terms of the representation category of the chiral algebra.…”

D-branes and BCFT in Hpp-wave backgrounds

“…These structure constants satisfy a set of factorization constraints first derived by Cardy and Lewellen [75,76] (see also [77,78,79]). For the sake of clarity, we will briefly review the sewing constraints for a generic CFT and in the next section we will write them explicitly for the Nappi-Witten model.…”