Non-commutative Chern-Simons for the Quantum Hall System and Duality

TL;DR

This paper investigates the dual non-commutative Chern-Simons descriptions of the quantum Hall system, demonstrating that the effective action becomes non-commutative at higher orders and exploring the duality between these formulations.

Contribution

It shows that the hydrodynamic Chern-Simons theory has a non-commutative gauge symmetry emerging at all perturbation orders, clarifying the duality between two descriptions.

Findings

Effective action is non-commutative beyond leading order.

Non-commutative gauge symmetry is a quantum symmetry.

Duality between hydrodynamic and statistical descriptions is established.

Abstract

The quantum Hall system is known to have two mutually dual Chern-Simons descriptions, one associated with the hydrodynamics of the electron fluid, and another associated with the statistics. Recently, Susskind has made the claim that the hydrodynamic Chern-Simons theory should be considered to have a non-commutative gauge symmetry. The statistical Chern-Simons theory has a perturbative momentum expansion. In this paper, we study this perturbation theory and show that the effective action, although commutative at leading order, is non-commutative. This conclusion is arrived at through a careful study of the three-point function of Chern-Simons gauge fields. The non-commutative gauge symmetry of this system is thus a quantum symmetry, which we show can only be fully realized only through the inclusion of all orders in perturbation theory. We discuss the duality between the two non-commutative descriptions.