Dualities in Quantum Hall System and Noncommutative Chern-Simons Theory

TL;DR

This paper explores various dualities in the Quantum Hall Effect using noncommutative Chern-Simons theory, linking different descriptions through dualities, brane models, and integrable systems, and proposing new interpretations of phase transitions.

Contribution

It introduces a brane-based framework connecting dualities, integrable models, and phase transitions in Quantum Hall systems, expanding the theoretical understanding of QHE.

Findings

Morita duality maps abelian noncommutative to nonabelian commutative descriptions.

Ruijsenaars system describes finite-electron QHE on the torus.

Brane models relate 2D sigma models, 3D Chern-Simons theory, and phase transitions.

Abstract

We discuss different dualities of QHE in the framework of the noncommutative Chern-Simons theory. First, we consider the Morita or T-duality transformation on the torus which maps the abelian noncommutative CS description of QHE on the torus into the nonabelian commutative description on the dual torus. It is argued that the Ruijsenaars integrable many-body system provides the description of the QHE with finite amount of electrons on the torus. The new IIB brane picture for the QHE is suggested and applied to Jain and generalized hierarchies. This picture naturally links 2d $\sigma$-model and 3d CS description of the QHE. All duality transformations are identified in the brane setup and can be related with the mirror symmetry and S duality. We suggest a brane interpretation of the plateu transition in IQHE in which a critical point is naturally described by $SL(2,R)$ WZW model.