Conformal Field Theories, Graphs and Quantum Algebras

TL;DR

This paper reviews recent advances in Rational Conformal Field Theories (RCFTs), focusing on algebraic structures, boundary conditions, and their connection to quantum algebras and Hopf algebras.

Contribution

It introduces a framework linking RCFT consistency conditions with quantum algebra representations and suggests classifying RCFTs via Weak C*-Hopf algebras.

Findings

Matrix representations encode boundary condition multiplicities.

Construction of quantum algebras with 6j- and 3j-symbols.

Potential classification of RCFTs through Weak C*-Hopf 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''.