Conformal field theories with ZN and Lie algebra symmetries

We construct two-dimensional conformal field theories with a ZN symmetry, based on the second solution of Fateev-Zamolodchikov for the parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group (ZNxZ2, where Z2 stands for the ZN charge conjugation), as singlets, [(N-1)/2] different doublets, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra B(N-1)/2 when N is odd, and DN/2 when N is even. The unitary theories are representations of the coset SOn(N)xSO2(N)/SOn+2(N), with n=1,2,.... We suggest that physically they realize the series of multicritical points in statistical systems having a ZN symmetry