On the Non-Renormalization Properties of Gauge Theories with Chern-Simons Terms

TL;DR

This paper proves that three-dimensional Chern-Simons gauge theories, with or without matter, are finite and have vanishing beta functions due to their gauge invariance properties, valid in all orders of perturbation theory.

Contribution

It establishes the all-order validity of a trace identity leading to non-renormalization and finiteness of Chern-Simons gauge theories in three dimensions.

Findings

Beta function for Chern-Simons coupling vanishes.

Chern-Simons theories are finite in the presence of Yang-Mills terms.

Trace identity holds at all perturbative orders.

Abstract

Considering three-dimensional Chern-Simons theory, either coupled to matter or with a Yang-Mills term, we show the validity of a trace identity, playing the role of a local form of the Callan-Symanzik equation, in all orders of perturbation theory. From this we deduce the vanishing of the $\beta$-function associated to the Chern-Simons coupling constant and the full finiteness in the case of the Yang-Mills Chern-Simons theory. The main ingredient in the proof of the latter property is the noninvariance of the Chern-Simons form under the gauge transformations. Our results hold for the three-dimensional Chern-Simons model in a general Riemannian manifold.