Local quantum field theories involving the U(1) current algebra on the circle

The OPE algebra Q=Q(g²) generated by a pair of oppositely charged currents (z,±g)(|z|=1) of spin is specified by the leading terms in the small distance expansions of (z₁,g)(z₂, -g) and (z₁,g)(z₂,g). The current (z,g) splits into a product of a U(1)-Thirring field and a Zamolodchikov-Fattev parafermionic current. The quasilocal(i.e.single-or double-valued) representations of Q are classified. The level k states involve 2(k+1) (ks–k+1) lowest weights (dimensions). The results can be viewed as an extension of the (known) representation theory of the SU(2) current algebra in the bosonic case corresponding to even values of g² and of the N=2 extended superconformal algebra in the fermionic case corresponding to odd g².