SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry

TL;DR

This paper reviews the development of the supersymmetric quantum Hall effect, introducing a SUSY framework on supermanifolds, exploring Landau problems, wavefunctions, and topological states, with a focus on non-anti-commutative geometry and effective field theories.

Contribution

It presents a SUSY formulation of the quantum Hall effect on supermanifolds, explicitly constructs SUSY wavefunctions, and develops a Chern-Simons theory, unifying various quantum Hall states.

Findings

Non-anti-commutative geometry emerges in the lowest Landau level.

SUSY provides a unified picture of Laughlin and Moore-Read states.

A SUSY Chern-Simons effective field theory is developed.

Abstract

We review the recent developments of the SUSY quantum Hall effect [hep-th/0409230, hep-th/0411137, hep-th/0503162, hep-th/0606007, arXiv:0705.4527]. We introduce a SUSY formulation of the quantum Hall effect on supermanifolds. On each of supersphere and superplane, we investigate SUSY Landau problem and explicitly construct SUSY extensions of Laughlin wavefunction and topological excitations. The non-anti-commutative geometry naturally emerges in the lowest Landau level and brings particular physics to the SUSY quantum Hall effect. It is shown that SUSY provides a unified picture of the original Laughlin and Moore-Read states. Based on the charge-flux duality, we also develop a Chern-Simons effective field theory for the SUSY quantum Hall effect.