Conformal Field Theories, Graphs and Quantum Algebras

Abstract

This article reviews some recent progress in our understanding of the structure of rational conformal field theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions—open, twisted—are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract “quantum” algebra, whose 6j-and 3j-symbols contain essential information on the operator product algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of “Weak C*- Hopf algebras”.KeywordsPartition FunctionDefect LineConformal Field TheoryVertex Operator AlgebraModular InvariantThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.