Valentina B. Petkova

Boundary conditions in rational conformal field theories

We develop further the theory of RationalConformalFieldTheories (RCFTs) on a cylinder with specified boundaryconditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardys 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 conformalboundaryconditions are labelled by the nodes of G . This approach is carried to completion for sl(2) theories leading to complete sets of conformalboundaryconditions, their associated cylinder partition functions and the A -D -E classification. We also review the current status for WZW sl(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.

On the classification of bulk and boundary conformal field theories

The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardys equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties.

FROM CFT TO GRAPHS

In this paper, we pursue the discussion of the connections between rational conformal field theories (CFT) and graphs. We generalise our recent work on the relations of operator product algebra (OPA) structure constants of sl(2) theories with the Pasquier algebra attached to the graph. We show that in a variety of CFTs built on sl(n) (typically conformal embeddings and orbifolds), similar considerations enable one to write a linear system satisfied by the matrix elements of the Pasquier algebra in terms of conformal data (quantum dimensions and fusion coefficients). In some cases this provides sufficient information for the determination of all the eigenvectors of an adjacency matrix, and hence of a graph.

Uq(s1(2)) invariant operators and minimal theories fusion matrices

Abstract The existence of U q (s1(2)) invariant operators for q p =1 leads to relation for the quantum Clebsch-Gordan kernels and for the quantum 6 j -symbols (=fusion matrices). These relations effectively reduce some equalities, inherited from the generic q case, and imply, in particular, that the polynomial identities for the quantum 6 j -symbols are consistent with the minimal theories chiral fusion rules.

FUSION MATRICES AND C<1 (QUASI) LOCAL CONFORMAL THEORIES

General properties of the fusion matrices and their explicit expression given by the Uq(sl(2)) quantum 6j-symbols are exploited to analyze some two dimensional c<1 conformal theories. The primary fields structure constants of the local theory, corresponding to the (A10, E6) modular invariant, and of the Z2 quasilocal analogs of the (A, D) and (A10, E6) series, are computed. The list of the Γ0(2) submodular invariant partition functions on the torus is extended.

A note on decoupling conditions for generic level and fusion rules

Abstract We find the solution of the sl (3) k singular vector decoupling equations on 3-point functions for the particular case when one of the fields is of weight w0·kΛ0. The result is a function with non-trivial singularities in the flag variables, namely a linear combination of 2F1 hypergeometric functions. This calculation fills in a gap in [Nucl. Phys. B 518 (1998) 645; Commun. Math. Phys. 202 (1999) 701]. and confirms the sl (3) k fusion rules determined there both for generic κ∉ Q and fractional levels. We have also analysed the fusion in sl (3) k using algebraic methods generalising those of Feigin and Fuchs and again find agreement with [Nucl. Phys. B 518 (1998) 645; Commun. Math. Phys. 202 (1999) 701]. In the process we clarify some details of previous treatments of the fusion of sl (2) k fractional level admissible representations.

On sl(3) reduction, quantum gauge transformations, and W-algebra singular vectors☆

Abstract The problem of describing the singular vectors of W 3 and W 3(2) Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of A2(1). Singular vectors of an A2(1) Verma module are mapped into W algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of the highest weight DS gauge transformations.

Verlinde Nim-Reps for Charge Conjugate SL ( N ) WZW Theory

We compute the representations (“NIM-reps”) of the fusion algebra of \( s{\overset{\lower0.5em\hbox{

Uq(sl(2)) Invariant operators and reduced polynomial identities

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Conformal boundary conditions and what they teach us

}}{l}} \)(N) which determine the boundary conditions of \( s{\overset{\lower0.5em\hbox{