Jean-Bernard Zuber

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.

Generalised twisted partition functions

Abstract We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D conformal field theory (CFT). Axa0consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanus algebraic construction, and might offer a new route to the study of properties of CFT.

The many faces of Ocneanu cells

We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra

On the classification of bulk and boundary conformal field theories

{cal A},

Conformal boundary conditions and what they teach us

a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the

Discrete Symmetries of Conformal Theories

3j

Minimal Model Correlation Functions on the Torus

-symbols of the weak Hopf algebra

INTEGRABLE BOUNDARIES, CONFORMAL BOUNDARY CONDITIONS AND A-D-E FUSION RULES

{cal A}

Conformal Field Theories, Graphs and Quantum Algebras

, reappear in several guises. % With

Large q expansions for q-state gauge-matter Potts models in lagrangian form

{cal A}