A note on the Faddeev-Popov determinant and Chern-Simons perturbation theory

TL;DR

This paper refines the Faddeev-Popov determinant expression for gauge theories around reducible solutions and applies it to Chern-Simons perturbation theory, confirming the finiteness and metric-independence of the partition function.

Contribution

It provides a new refined expression for the Faddeev-Popov determinant applicable to reducible solutions in gauge theories and demonstrates its consistency within Chern-Simons perturbation theory.

Findings

Faddeev-Popov determinant expression is refined for reducible solutions.

Perturbative expansions remain finite and metric-independent.

Results extend previous findings to more general flat gauge fields.

Abstract

A refined expression for the Faddeev-Popov determinant is derived for gauge theories quantised around a reducible classical solution. We apply this result to Chern-Simons perturbation theory on compact spacetime 3-manifolds with quantisation around an arbitrary flat gauge field isolated up to gauge transformations, pointing out that previous results on the finiteness and formal metric-independence of perturbative expansions of the partition function continue to hold.