Composing parafermions: a construction of $Z_{N}$ fractional quantum Hall systems and a modern understanding of confinement and duality

TL;DR

The paper introduces anomaly-free $Z_{N}$ simple currents as composite parafermions to construct cylinder partition functions for $Z_{N}$ fractional quantum Hall states, linking discrete torsion, anyon condensation, and bulk-edge correspondence. It analyzes untwisted/twisted sectors, modular invariance, and applies the framework to explicit models such as a $Z_{N}$ FQHE, a coupled three-state Potts system, and $SU(N)_{K}$ WZW theories, while also extending to nonunitary CFTs via Galois shuffle cancellation. The authors propose a $Z_{N}$-style duality (composite parafermionic T duality) and confinement-like behavior, and connect the construction to the SU$(N)$ Haldane conjecture through anomaly matching and level-rank duality. They argue that their operator-formalism partition functions provide a universal lens on topological order, edge modes, and their RG flows, with deep connections to 2d gravity coupled to matter and potential experimental relevance.

Abstract

In this work, we propose a modern view of the integer spin simple currents which have played a central role in discrete torsion. We reintroduce them as nonanomalous composite particles constructed from $Z_{N}$ parafermionic field theories. These composite particles have an analogy with the Cooper pair in the Bardeen-Cooper-Schrieffer theory and can be interpreted as a typical example of anyon condensation. Based on these $Z_{N}$ anomaly free composite particles, we propose a systematic construction of the cylinder partition function of $Z_{N}$ fractional quantum Hall effects (FQHEs). One can expect realizations of a class of general topological ordered systems by breaking the bulk-edge correspondence of the bosonic parts of these FQH models. We also give a brief overview of various phenomena in contemporary condensed matter physics, such as $SU(N)$ Haldane conjecture, general gapless and gapped topological order with respect to the quantum anomaly defined by charges of these simple currents and bulk and boundary renormalization group flow. Moreover, we point out an analogy between these FQHEs and 2d quantum gravities coupled to matter, and propose a $Z_{N}$ generalization of supersymmetry known as "fractional supersymmetry" in the composite parafermionic theory and study its analogy with quark confinement. Our analysis gives a simple but general understanding of the contemporary physics of topological phases in the form of the partition functions derived from the operator formalism.

Figures (6)

• Figure 1: Analogy between quark confinement and FQHE
• Figure 2: Bulk and boundary RG interpretation of CFT/BQFT correspondence and its ambiguities. In a general topological order, one can attach different CFTs connected under RGs at the edge theoretically. This nonuniqueness of edge theory may result in difficulty in observing edge CFT in experimental settings.
• Figure 3: Chirality from Fradkin-Kadanoff transformations
• Figure 4: RG flows of $SU(N)$ models. Assocativity of red arrows and blue arrows corresponds to the Haldane conjecture.
• Figure 5: Summary of matching and cancellation flows. Similar to anomaly analysis by considering the simple charges, one may be possible to analyze the Galois shuffle property by considering the monodromy charges.