The Quantum Hall Fluid and Non-Commutative Chern Simons Theory

TL;DR

This paper reviews the Chern-Simons approach to the Quantum Hall effect, proposes an equivalence with non-commutative Chern-Simons theory at certain levels, and explores its formulation as a matrix and mapping theory.

Contribution

It demonstrates the exact equivalence between abelian non-commutative Chern-Simons theory at level n and Laughlin's theory at filling fraction 1/n, and connects it to matrix and non-commutative space theories.

Findings

Non-commutative Chern-Simons theory is equivalent to Laughlin theory at 1/n.

The theory can be formulated as a matrix model similar to D0-branes.

It can be viewed as a quantum mapping between non-commutative spaces.

Abstract

The first part of this paper is a review of the author's work with S. Bahcall which gave an elementary derivation of the Chern Simons description of the Quantum Hall effect for filling fraction $1/n$. The notation has been modernized to conform with standard gauge theory conventions. In the second part arguments are given to support the claim that abelian non-commutative Chern Simons theory at level $n$ is exactly equivalent to the Laughlin theory at filling fraction $1/n$. The theory may also be formulated as a matrix theory similar to that describing D0-branes in string theory. Finally it can also be thought of as the quantum theory of mappings between two non-commutative spaces, the first being the target space and the second being the base space.