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

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 ${\mathbf{CP}}^3$ vis-a-vis $S^4$. We also give a method of representing the algebra of functions on fuzzy $S^3/{\mathbf{Z}}_2$ in terms of finite dimensional matrices.

LaTex, 18 pages

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