SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry

Abstract

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-anticommutative 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.

A Relation Between Supertriality and Supertwistor

“…(1) For each x, y, and z we have, x, yz = y, z, x = (−1)ỹz z, y, x = (−1)ỹz x, zy , so that µ is symmetric.…”

Divergence theorems and the supersphere

“…The supersphere occurs naturally in the theory of the supersymmetric quantum hall effect ( [Has08], [HT13]) and underlies certain field theories, see [SW05]. In a series of papers ( [DBS07], [CDS09], [Cou12]), an integration over the supersphere was introduced, first by an extension of Pizzetti's formula to super-polynomials, then by a formula for general superfunctions later expressed in terms of an embedding, while a particular case of this integral was already studied in [Jar88].…”

Hyperbolic supersymmetric quantum Hall effect

“…Many-body problems on supermanifolds, which we call the SUSY QHE, have been also explored in Refs. [18,[23][24][25][26]. The SUSY QHE was first formulated on a supersphere [23], and next on a superplane [18,24].…”