W-Infinity Algebras from Noncommutative Chern-Simons Theory

TL;DR

This paper derives a nonlinear deformation of the $w_$ algebra from noncommutative Chern-Simons theory on a plane with a hole, linking it to quantum Hall effect parameters and discussing quantization issues.

Contribution

It introduces a new nonlinear deformation of the $w_$ algebra arising from noncommutative Chern-Simons theory, connecting algebraic structures with quantum Hall parameters.

Findings

Deformation depends on level and noncommutativity parameter.

Algebra relates to fractional quantum Hall effect.

Quantization depends on ordering choices.

Abstract

We examine Chern-Simons theory written on a noncommutative plane with a `hole', and show that the algebra of observables is a nonlinear deformation of the $w_\infty$ algebra. The deformation depends on the level (the coefficient in the Chern-Simons action), and the noncommutativity parameter, which were identified, respectively, with the inverse filling fraction and the inverse density in a recent description of the fractional quantum Hall effect. We remark on the quantization of our algebra. The results are sensitive to the choice of ordering in the Gauss law.