The Chern-Simons Coefficient in Supersymmetric Yang-Mills Chern-Simons Theories

TL;DR

This paper analyzes one-loop quantum corrections to the Chern-Simons coefficient in supersymmetric Yang-Mills theories, revealing how supersymmetry affects the shift in the coefficient and its implications for gauge theories.

Contribution

It provides a detailed calculation of the one-loop correction to the Chern-Simons coefficient in various supersymmetric Yang-Mills theories, highlighting the role of supersymmetry in canceling quantum shifts.

Findings

In pure bosonic theories, the shift is $k ightarrow k + c_v$.

In $N=1$ supersymmetry, the shift is halved to $k ightarrow k + c_v/2$.

In $N=2,3$ supersymmetry, the correction vanishes due to complete cancellation.

Abstract

We study one-loop correction to the Chern-Simons coefficient $\kappa=k/4\pi$ in $N=1,2,3$ supersymmetric Yang-Mills Chern-Simons systems. In the pure bosonic case, the shift of the parameter $k$ is known to be $k\rightarrow k + c_v$, where $c_v$ is the quadratic Casimir of the gauge group. In the $N=1$ case, the fermionic contribution cancels the bosonic contribution by half and the shift is $k \rightarrow k+ c_v/2$, making the theory anomalous if $c_v$ is odd. In the $N=2,3$ cases, the fermionic contribution cancels the bosonic contribution completely and there is no correction. We also calculate the mass corrections, showing the supersymmetry is preserved. As the matter fields decouple from the gauge field in the pure Chern-Simons limit, this work sheds some light on the regularization dependency of the correction in pure Chern-Simons systems. We also discuss the implication to the case when the gauge symmetry is spontaneously broken by the Higgs mechanism.