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

Yoshiki Fukusumi
Dec 2022
摘要
In this work, we propose a modern view of the integer spin simple currentswhich have played a central role in discrete torsion. We reintroduce them asnonanomalous composite particles constructed from $Z_{N}$ parafermionic fieldtheories. These composite particles have an analogy with the Cooper pair in theBardeen-Cooper-Schrieffer theory and can be interpreted as a typical example ofanyon condensation. Based on these $Z_{N}$ anomaly free composite particles, wepropose a systematic construction of the cylinder partition function of $Z_{N}$fractional quantum Hall effects (FQHEs). One can expect realizations of a classof general topological ordered systems by breaking the bulk-edge correspondenceof the bosonic parts of these FQH models. We also give a brief overview ofvarious phenomena in contemporary condensed matter physics, such as $SU(N)$Haldane conjecture, general gapless and gapped topological order with respectto the quantum anomaly of simple currents and bulk and boundary renormalizationgroup flow. Moreover, we point out an analogy with the FQHE and the 2d quantumgravity coupled to matter, and propose a $Z_{N}$ generalization ofsupersymmetry known as "fractional supersymmetry" in the compositeparafermionic theory and study its analogy with quark confinement. Our analysisgives a simple but general understanding of the contemporary physics oftopological phases in the form of the partition functions derived from theoperator formalism.
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