Dualities in Quantum Hall System and Noncommutative Chern-Simons Theory

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.

31 pages, 4 figures

References in corpus (6)

• Quantum Hall states as matrix Chern-Simons theory
• Quantum Hall Physics = Noncommutative Field Theory
• Quantum Hall states on the cylinder as unitary matrix Chern-Simons theory
• D-brane Dynamics and the Quantum Hall Effect
• D-branes in Massive IIA and Solitons in Chern-Simons Theory
• Aspects of Gauge Theory on Commutative and Noncommutative Tori

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• Generalized Devil's staircase and RG flows
• Finite Temperature effects on the Induced Chern-Simons term in noncommutative geometry
• 5d-4d Correspondence in Twisted M-theory on a Conifold