A Uni ed Conformal Field Theory Descriptionof Paired Quantum

The wave functions of the Haldane-Rezayi paired Hall state have been previously described by a non-unitary conformal eld theory with central charge c = ?2. Moreover, a relation with the c = 1 unitary Weyl fermion has been suggested. We construct the complete unitary theory and show that it consistently describes the edge excitations of the Haldane-Rezayi state. Actually, we show that the unitary (c = 1) and non-unitary (c = ?2) theories are related by a local map between the two sets of elds and by a suitable change of conjugation. The unitary theory of the Haldane-Rezayi state is found to be the same as that of the 331 paired Hall state. Furthermore, the analysis of modular invariant partition functions shows that no alternative uni-tary descriptions are possible for the Haldane-Rezayi state within the class of rational conformal eld theories with abelian current algebra. Finally, the known c = 3=2 conformal theory of the Pfaaan state is also obtained from the 331 theory by a reduction of degrees of freedom which can be physically realized in the double-layer Hall systems.