Parafermionic theory with the symmetry Z_N, for N even

TL;DR

This paper completes the construction of Z_N parafermionic theories for even N, classifies primary operators, and relates the theories to multicritical points in statistical systems with Z_N symmetry.

Contribution

It introduces the second solution for Z_N parafermionic theories with N even, classifies operators, and links the theories to specific coset models and multicritical points.

Findings

Constructed Z_N parafermionic theories for even N.

Classified primary operators under dihedral group symmetries.

Connected theories to multicritical points in statistical models.

Abstract

Following our previous papers (hep-th/0212158 and hep-th/0303126) we complete the construction of the parafermionic theory with the symmetry Z_N based on the second solution of Fateev-Zamolodchikov for the corresponding parafermionic chiral algebra. In the present paper we construct the Z_N parafermionic theory for N even. 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 two singlets, doublet 1,2,...,N/2-1, 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 D_{N/2}. The unitary theories are representations of the coset SO_n(N) x SO_2(N) / SO_{n+2}(N), with n=1,2,.... We suggest that physically they realise the series of multicritical points in statistical systems having a Z_N symmetry.