Conformal field theories with Z_N and Lie algebra symmetries

TL;DR

This paper constructs two-dimensional conformal field theories with Z_N symmetry using parafermionic algebras, classifies primary operators, and relates them to Lie algebra structures, proposing their physical realization in multicritical statistical systems.

Contribution

It introduces a new class of Z_N symmetric conformal field theories based on parafermionic algebras and Lie algebra classifications, extending previous models.

Findings

Classified primary operators under dihedral group symmetries.

Connected vertex operators to Lie algebra weight lattices.

Proposed physical realization in multicritical statistical models.

Abstract

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 SO_n(N) x SO_2(N) / SO_{n+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.