THE QUASI-RATIONAL FUSION STRUCTURE OF SU(M/N) CHERN-SIMONS AND WZW THEORIES

The SU(m\n)K fusion ring is found. The existence of a positive trace on this ring, rather than integrability, is the common principle underlying the fusion rings of WZW models, ordinary and super. The argument proceeds by means of a combined examination of the ring of positive q-superdimensions, associated Chern - Simons observables, and several WZW correlation functions. The tensor product ring of U(q)(su(m\n)) for q a root of unity admits a doubly truncated tensor product structure isomorphic to the SU(m\n)K fusion ring in the cases we examine. The SU(m\n)K WZW models are found to be quasi-rational (non-unitary) conformal field theories. Many important quantities in SU(m\n)K theories are given exactly by analogous SU(m - n)K quantities, including the conformal weights, fusion coefficients, Chern - Simons observables, and the matrix elements of modular transformations of the relevant affine supercharacters. These matrix elements are related to the fusion ring via a (slightly modified) Verlinde formula. The SU(n + N\n)K and SU(k + K\k)N Chern - Simons and WZW theories are related by group-level duality.