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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''.
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Submitted on : Friday, February 16, 2007 - 9:52:38 PM
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Valentina Petkova, Jean-Bernard Zuber. Conformal Field Theories, Graphs and Quantum Algebras. 2001. ⟨hal-00131589⟩



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