On Consistency Of Noncommutative Chern-Simons Theory

TL;DR

This paper analyzes the noncommutative extension of Chern-Simons theory, demonstrating its formal expansion, stability under certain corrections, anomaly freedom, and the vanishing of its beta function, despite interpretational challenges.

Contribution

It provides a detailed analysis of the quantum properties of noncommutative Chern-Simons theory, including its expansion, stability, anomaly status, and renormalization behavior.

Findings

The theory can be expanded in power series of the noncommutative parameter theta.

The theory is free of anomalies.

The beta function of the Chern-Simons coupling vanishes.

Abstract

We consider the noncommutative extension of Chern-Simons theory. We show the the theory can be fully expanded in power series of the noncommutative parameter theta and that no non-analytical sector exists. The theory appears to be unstable under radiative corrections, but we show that the infinite set of instabilities, to all orders in \hbar and in theta, is confined to a BRS exact cocycle. We show also that the theory is anomaly free. The quantum theory cannot be written in terms of the Groenewald-Moyal star product, and hence doubts arise on the interpretation of the noncommutative nature of the underlying spacetime. Nonetheless, the deformed theory is well defined as a quantum field theory, and the beta function of the Chern-Simons coupling constant vanishes, as in the ordinary Chern-Simons theory.