Non-minimal quasi-holes and topological degeneracy of FQHE

TL;DR

This paper analyzes the topological order in fractional quantum Hall effect (FQHE) by extending the Vafa SUSY model, focusing on quasi-hole braiding, topological degeneracy, and higher genus generalizations.

Contribution

It introduces a detailed analysis of quasi-hole braiding and topological degeneracy in FQHE using an extended Vafa SUSY framework, including higher genus cases.

Findings

Quasi-holes correspond to irreducible SU(2) representations.

Braiding statistics are described by the Knizhnik-Zamolodchikov connection.

Higher genus generalizations involve tt* geometry.

Abstract

We extend the analysis of the Vafa $\mathcal{N}=4$ SUSY model of FQHE and discuss other observables which characterize the FQHE topological order. We consider in particular the braiding properties of quasi-holes with generic charge. As one naturally expects, any quasi-hole is associated with an irreducible representation of $SU(2)$ and the statistics of a bunch of $d$ punctures is described by the Knizhnik-Zamolodchikov connection specialized to the corresponding representation. We also discuss the higher genus generalization of the Vafa model of FQHE and the corresponding $tt^{*}$ geometry.