Quantum Hall effect on $S^3$, edge states and fuzzy $S^3/{\bf Z}_2$

TL;DR

This paper explores the quantum Hall effect on the three-sphere, analyzing edge states and fuzzy geometry, revealing connections to higher-dimensional spaces and providing matrix representations of fuzzy spheres.

Contribution

It introduces a detailed analysis of the Landau problem on $S^3$, derives the effective edge action, and presents a method to represent fuzzy $S^3/{f Z}_2$ algebra with finite matrices.

Findings

Effective edge action for quantum Hall on $S^3$ derived.

Space $S^2 imes S^2$ identified as relevant for many considerations.

Matrix representation of fuzzy $S^3/{f Z}_2$ algebra provided.

Abstract

We analyze the Landau problem and quantum Hall effect on $S^3$ taking a constant background field proportional to the spin connection on $S^3$. The effective strength of the field can be tuned by changing the dimension of the representation to which the fermions belong. The effective action for the edge excitations of a quantum Hall droplet in the limit of a large number of fermions is obtained. We find that the appropriate space for many of these considerations is $S^2 \times S^2$, which plays a role similar to that of ${\bf CP}^3$ vis-a-vis $S^4$. We also give a method of representing the algebra of functions on fuzzy $S^3/{\bf Z}_2$ in terms of finite dimensional matrices.