Conformal field theories with ZN and Lie algebra symmetries

We construct two-dimensional conformal field theories with a Z(N) 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 (Z(N) x Z(2), where Z(2) stands for the Z(N) 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 D-N/2 when N is even. The unitary theories are representations of the coset SOn(N) x SO2(N)/SOn+2(N), with n = 1, 2,.... We suggest that physically they realize the series of multicritical points in statistical systems having a Z(N) symmetry. (C) 2004 Elsevier B.V. All rights reserved.