Dimensional Hierarchy in Quantum Hall Effects on Fuzzy Spheres

TL;DR

This paper constructs higher-dimensional quantum Hall systems using fuzzy spheres, revealing a dimensional hierarchy where lower-dimensional branes condense into higher-dimensional incompressible liquids, with fractional charges and topological excitations.

Contribution

It introduces a novel framework for higher-dimensional quantum Hall effects based on fuzzy spheres and elucidates their topological and hierarchical properties.

Findings

Fuzzy spheres realized as spheres in colored monopole backgrounds.

Fractionally charged excitations in higher-dimensional systems.

Higher dimensional systems exhibit a hierarchy of branes condensing into incompressible liquids.

Abstract

We construct higher dimensional quantum Hall systems based on fuzzy spheres. It is shown that fuzzy spheres are realized as spheres in colored monopole backgrounds. The space noncommutativity is related to higher spins which is originated from the internal structure of fuzzy spheres. In $2k$-dimensional quantum Hall systems, Laughlin-like wave function supports fractionally charged excitations, $q=m^{-{1/2}k(k+1)}$ (m is odd). Topological objects are ($2k-2$)-branes whose statistics are determined by the linking number related to the general Hopf map. Higher dimensional quantum Hall systems exhibit a dimensional hierarchy, where lower dimensional branes condense to make higher dimensional incompressible liquid.