Non-commutative Chern–Simons for the quantum Hall system and duality

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 hydrodynamical 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 through the inclusion of all orders in perturbation theory. We discuss the duality between the two non-commutative descriptions.