Free fermions and WZW theories on nonsimple groups

We consider conformally and Kac-Moody invariant theories based on the groups G = G(N) x G(N tilde) where G(N) is any of the classical groups. For the values k = N tilde, k tilde = N of the Kac-Moody central charges, the monodromy problem involved in the computation of the four point function for primary fields in the defining representation of G possesses two distinct solutions. As a consequence, the WZW theory on G (with an addition U(1) factor if G(N) = SU(N)) cannot be equivalent to a theory of free fermions. (orig.)