Free fermions and WZW theories on nonsimple groups

We consider conformally and Kac-Moody invariant theories based on the groupsG=G(N)×G(Ñ) whereG(N) is any of the classical groups. For the valuesk=Ñ, (tilde k = 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 ofG possesses two distinct solutions. As a consequence, the WZW theory onG (with an additionalU(1) factor ifG(N)=SU (N)) cannot be equivalent to a theory of free fermions.