The Quantum Hall Fluid and Non-Commutative Chern Simons Theory

TL;DR

The paper addresses the fractional quantum Hall fluid by building a field-theoretic description that begins with a charged-fluid in a strong magnetic field and culminates in a non-commutative Chern-Simons theory. The approach connects area-preserving diffeomorphism symmetry, a Maxwell–like regime, and a long-distance CS description, then elevates it to a non-commutative setting that reproduces Laughlin filling fractions via exact matrix-model realizations. It shows that quasiparticle charge is $e\nu$ and exchange statistics follow from a Berry-phase calculation, with fermionic and bosonic cases corresponding to $\nu=1/(2n+1)$ and $\nu=1/(2n)$, respectively, and general $\nu$ yielding anyons; a Seiberg–Witten map links the non-commutative theory to gauge-invariant observables. The work predicts a phase transition to a Wigner crystal at low filling, explores non-abelian NC-CS generalizations for multi-layer states, and highlights deep connections to D-brane matrix models, suggesting a rich, unified non-commutative geometric framework for quantum Hall physics with implications for both theory and potential realizations.

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.