  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 fixed point resolution in the
diagonal coset of type A, and show how to decompose reducible representations
into irreducibles. We show the corresponding coset CFT gives rise to a unitary
tensor modular category in the sense of Turaev, and therefore may be used to
construct 3-manifold invariants. We also show that Kac-Wakimoto Hypothesis
(KWH) and Kac-Wakimoto Conjecture (KWC) are equivalent under general conditions
which can be checked in examples, a result which seems to be hard to prove by
purely representation considerations. Examples are also presented.
