Non-commutative geometry of 4-dimensional quantum Hall droplet

TL;DR

This paper develops the non-commutative geometric framework for the 4D quantum Hall effect, explicitly constructing fuzzy monopole harmonics and their algebra, with potential applications to M-theory and quantum Hall systems.

Contribution

It introduces explicit constructions of fuzzy monopole harmonics and their algebra, and proposes a fusion scheme, advancing the understanding of 4D quantum Hall geometry.

Findings

Explicit matrix algebra for fuzzy $S^{4}$ is established.

Fusion rules for monopole harmonics are derived.

Applications to M-theory and 4D quantum Hall systems are discussed.

Abstract

We develop the description of non-commutative geometry of the 4-dimensional quantum Hall fluid's theory proposed recently by Zhang and Hu. The non-commutative structure of fuzzy $S^{4}$ appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in this non-commutative geometry, are explicitly constructed and their obeying the matrix algebra is obtained. This matrix algebra is associative. We also propose a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and determine the fusion rule in such fusion scheme. By products, we provide some essential ingredients of the theory of SO(5) angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of SO(5) angular momentum, are given. It is discussed that some possible applications of our results to the 4-dimensional quantum Hall system and the matrix brane construction in M-theory.