A Note on Noncommutative Chern-Simons Theories

TL;DR

This paper investigates the quantization of the level in noncommutative Chern-Simons theories on ^2_{ heta} imes r, demonstrating that gauge invariance requires the level to be quantized, with extensions to noncommutative tori discussed.

Contribution

It establishes the quantization condition of the level in noncommutative Chern-Simons theories under finite gauge transformations.

Findings

Level quantization depends on gauge transformations at infinity.

Action changes by integer multiples of 2 under gauge transformations.

Brief discussion of noncommutative torus case.

Abstract

The three dimensional Chern-Simons theory on $\rr^2_{\theta}\times \rr$ is studied. Considering the gauge transformations under the group elements which are going to one at infinity, we show that under arbitrary (finite) gauge transformations action changes with an integer multiple of $2\pi$ {\it if}, the level of noncommutaitive Chern-Simons is {\it quantized}. We also briefly discuss the case of the noncommutaitve torus and some other possible extensions.