1 6 M ar 1 99 9 ALGEBRAIC COSET CONFORMAL FIELD THEORIES II

Some mathematical questions relating to Coset Conformal field theories (CFT) are considered in the framework of Algebraic Quantum Field Theory as developed previously by us. We consider the issue of fix point resolution in the diagonal coset of type A. We show how to decompose certain reducible representations into irreducibles, and prove that the coset CFT gives rise to a unitary Modular Tensor Category in the sense of Turaev, and therefore may be used to construct 3-manifold invariants. We prove that if the coset inclusion satisfies certain conditions which can be checked in examples, the Kac-Wakimoto Hypothesis (KWH) is equivalent to the the Kac-Wakimoto Conjecture (KWC), a result which seems to be hard to prove by purely representation considerations. Examples are also presented. §