Symmetry breaking boundaries

Abstract

Various structural properties of the space of symmetry breaking boundary conditions that preserve an orbifold subalgebra are established. To each such boundary condition we associate its automorphism type. We show that correlation functions in the presence of such boundary conditions are expressible in terms of twisted boundary blocks which obey twisted Ward identities. The subset of boundary conditions that share the same automorphism type is controlled by a classifying algebra, whose structure constants are shown to be traces on spaces of chiral blocks. T-duality on boundary conditions is not a one-to-one map in general. These structures are illustrated in a number of examples. Several applications, including the construction of non-BPS boundary conditions in string theory, are exhibited.

Monstrous Branes

“…One of these conditions is the factorisation (or cluster) condition that requires that certain bulk-boundary structure constants satisfy a set of non-linear equations (sometimes also referred to as the classifying algebra in this context). It was shown in [38] that a W-preserving boundary state ||B satisfies this factorisation constraint provided that it preserves the full symmetry algebra 27) for all modes of fields in H 0 . (Here g ∈ G depends on the boundary condition ||B .)…”

Conformal Field Theory, Boundary Conditions and Applications to String Theory

“…This analysis has been generalized beyond the Cardy case [21]. We do not describe those results here, but indicate the main idea: The fusion rules, which are the structure constants of the classifying algebra in the Cardy case, are the dimensions of the spaces of three-point blocks.…”

Orientation matters for NIMreps

“…Since the boundaries are in one-to-one correspondence with simple current orbits, we can assign the corresponding charge to each boundary. As is well known [8], zero charge boundaries are symmetry preserving, the others symmetry breaking. Now consider the matrix elements A i0 b0 a0 , A i0 b1 a1 , A i0 b2 a2 , where the subscript on the labels denotes three times the charge.…”

“…The relation to completeness of boundaries was understood in [1], after which many papers appeared giving solutions in various cases with various degrees of explicitness, see e.g. [2] and [5][6][7][8][9][10][11][12][13][14][15].…”