Discrete Holomorphicity in the Chiral Potts Model

TL;DR

This paper constructs lattice parafermions for the $Z(N)$ chiral Potts model, revealing twisted discrete-holomorphicity conditions linked to quantum group currents, with implications for critical and perturbed conformal field theories.

Contribution

It introduces a novel construction of lattice parafermions tied to quantum group currents, establishing twisted DH conditions that extend previous results to non-critical regimes.

Findings

At criticality, parafermions match known holomorphic ones.

Twisted DH conditions correspond to deformed current conservation.

Results apply to both $N eq 2$ and $N=2$ Ising cases.

Abstract

We construct lattice parafermions for the $Z(N)$ chiral Potts model in terms of quasi-local currents of the underlying quantum group. We show that the conservation of the quantum group currents leads to twisted discrete-holomorphicity (DH) conditions for the parafermions. At the critical Fateev-Zamolodchikov point the parafermions are the usual ones, and the DH conditions coincide with those found previously by Rajabpour and Cardy. Away from the critical point, we show that our twisted DH conditions can be understood as deformed lattice current conservation conditions for an underlying perturbed conformal field theory in both the general $N\geq 3$ and $N=2$ Ising cases.