ExtendedU(1) Conformal Field Theories and Zk-Parafermions

Abstract

A constructive approach is developed for studying local chiral algebras generated by a pair of oppositely charged fields Ψ (z, ±g) such that the operator product expansion (OPE) of Ψ(z1, g) Ψ(z2, −g) involves a U(1) current. The main tool in the study is the factorization property of the charged fields (exhibited in [PT 2, 3]) for Virasoro central charge c < 1 into U(1)-vertex operators tensored with ZAMOLODCHIKOV-FATEEV [ZF1] (generalized) Zk-parafermions. The case Δ2 = 4 (Δ1 − 1), where Δv = Δk−v(Δ0 = 0) are the conformaldimensions of the parafermionic currents, is studied in detail. For Δv = 2v(1 − v/k) the theory is related to GEPNER'S [Ge] Z2 [so (k)] parafermions and the corresponding quantum field theoretic (QFT) representations of the chiral algebra are displayed. The Coulomb gas method of [CR] is further developed to include an explicit construction of the basic parafermionic current Ψ of weight Δ = Δ1. The characters of the positive energy representations of the local chiral algebra are written as sums of products of Kac's string functions and classical θ-functions.