Boundary conditions in rational conformal field theories

Abstract

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for s (2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for WZW s (3) theories. Finally, a systematic generalization of the formalism of Cardy-Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.

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.…”

“…Under the same assumptions, solutions to all sewing relations were found in [11], expressed in terms of the representation category of the chiral algebra. The structure of (2.23) was further clarified and extended to certain non-diagonal modular invariants of SU (2) and Virasoro minimal models in [2]; for these cases, the M ab c are structure constants of a Pasquier algebra, which in turn opens up interesting connections to quantum symmetries, see [12].…”

“…the Peter-Weyl Theorem (see for example [19]), the matrix elements D j m,n (g) define an orthogonal basis of L 2 functions on the group manifold SU (2). Since the product of two matrix elements D j m,n (g) is again an L 2 function on SU(2), it can be expanded uniquely in terms of these matrix elements; the expansion coefficients are precisely the structure constants M , that are thus uniquely determined by (3.6).…”

“…In particular, if the conformal field theory is rational with respect to some chiral algebra, the boundary states that preserve this symmetry algebra have been classified at least for a large class of cases [1,2]. However, much less is known about boundary conditions that only preserve a symmetry algebra with respect to which the original theory is not rational.…”

Twisted brane charges for non-simply connected groups

“…Here λ ∈ P + k (ḡ) is an arbitrary highest-weight representation of the affine Lie algebra g at level k, dim(λ) is the Weyl-dimension of the corresponding representation of the horizontal subalgebraḡ, and N λa b are the NIM-rep coefficients appearing in the Cardy analysis (for an introduction to these matters see [13,14]). …”