6 O ct 1 99 8 ALGEBRAIC COSET CONFORMAL FIELD THEORIES

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary Coset Conformal Field Theories, i.e., gauged Wess-ZuminoWitten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic Quantum Field Theory (QFT) to approach Coset Conformal Field Theories. Two main conjectures are formulated and their consequences are discussed. Some results are presented which prove the main conjectures in special cases. In particular, one of the results states that an interesting class of representations of Coset WN (N ≥ 3) algebras with critical parameters are irreducible, and under the natural fusion product (Connes’ fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a longstanding conjecture about the representations of these algebras . §