The inequivalence of fermionic and bosonic theories sharing the same energy-momentum tensor

The conjecture that an isomorphism between the current (Kac-Moody) algebras and conformal (Virasoro) algebras of a sigma model with critically coupled Wess-Zumino term and a free fermion theory in two dimensions is sufficient to ensure full equivalence between the theories is tested for the groups Sp(N) . Sp(1) by examining four-point correlation functions of bosonic fields and their fermionic counterparts. After demonstrating the equivalence for N=1 (a result confirming the equivalence for one of the orthogonal groups, SO(4)) the application of the same method for N>1 reveals the equivalence does not hold in these cases. (orig.)