Gauge Dependence in Chern-Simons Theory

TL;DR

This paper investigates how the one-loop effective action in non-Abelian Chern-Simons theory depends on gauge choices, revealing a novel gauge-dependent term and confirming consistency with Nielsen identities, with implications for Wilson loop expectations.

Contribution

It introduces a detailed analysis of gauge parameter dependence in Chern-Simons theory, identifying a new gauge-dependent term and validating results with Nielsen identities.

Findings

Discovery of a new gauge-dependent term involving $rac{ ext{alpha}}{ ext{sqrt}(p^2)}$

Confirmation that gauge dependence aligns with Nielsen identities

Potential impact on vacuum expectation values of Wilson loops

Abstract

We compute the contribution to the modulus of the one-loop effective action in pure non-Abelian Chern-Simons theory in an arbitrary covariant gauge. We find that the results are dependent on both the gauge parameter ($\alpha$) and the metric required in the gauge fixing. A contribution arises that has not been previously encountered; it is of the form $(\alpha / \sqrt{p^2}) \epsilon _{\mu \lambda \nu} p^\lambda$. This is possible as in three dimensions $\alpha$ is dimensionful. A variant of proper time regularization is used to render these integrals well behaved (although no divergences occur when the regularization is turned off at the end of the calculation). Since the original Lagrangian is unaltered in this approach, no symmetries of the classical theory are explicitly broken and $\epsilon_{\mu \lambda \nu}$ is handled unambiguously since the system is three dimensional at all stages of the calculation. The results are shown to be consistent with the so-called Nielsen identities which predict the explicit gauge parameter dependence using an extension of BRS symmetry. We demonstrate that this $\alpha$ dependence may potentially contribute to the vacuum expectation values of products of Wilson loops.