[Paper Review] A note on the Faddeev-Popov determinant and Chern-Simons perturbation theory

David Adams|ArXiv.org|Apr 23, 1997

Black Holes and Theoretical Physics3 citations

TL;DR

This paper derives a refined expression for the Faddeev-Popov determinant in gauge theories quantized around reducible classical solutions, specifically applying it to Chern-Simons theory on compact 3-manifolds. It confirms that perturbative expansions of the partition function remain finite and formally metric-independent when quantizing around arbitrary flat gauge fields, extending previous results to a broader class of solutions.

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.

Motivation & Objective

• To derive a corrected expression for the Faddeev-Popov determinant in gauge theories with reducible classical solutions.
• To extend perturbative analysis of Chern-Simons theory to arbitrary flat gauge fields, not just the trivial connection.
• To verify that finiteness and formal metric-independence of the partition function persist under this generalization.
• To address technical subtleties arising from gauge orbit degeneracy in non-trivial flat connections.
• To provide a rigorous foundation for perturbative Chern-Simons amplitudes on compact spacetime 3-manifolds.

Proposed method

• Derives a modified Faddeev-Popov determinant formula accounting for reducibility of the classical solution.
• Applies the refined determinant to Chern-Simons theory on compact 3-manifolds with arbitrary flat gauge fields as background.
• Uses standard path integral quantization with ghost fields, adjusting for zero modes due to reducibility.
• Analyzes the structure of the gauge-fixing functional and its Jacobian under the presence of reducibility.
• Demonstrates that the perturbative partition function remains finite and independent of metric choice in the formal sense.
• Relies on differential geometric techniques from gauge theory and index theory on compact manifolds.

Experimental results

Research questions

• RQ1 How does the Faddeev-Popov determinant change when quantizing around a reducible classical solution?
• RQ2 Can the formal finiteness and metric-independence of Chern-Simons partition functions be preserved when quantizing around non-trivial flat connections?
• RQ3 What modifications are required in the path integral measure to account for gauge orbit degeneracy in reducible configurations?
• RQ4 Does the standard perturbative framework for Chern-Simons theory remain valid on compact 3-manifolds when the background is an arbitrary flat connection?
• RQ5 How does the presence of zero modes affect the determinant and the resulting perturbative amplitudes?

Key findings

• A refined expression for the Faddeev-Popov determinant is derived that correctly accounts for reducibility of the classical gauge field configuration.
• The perturbative expansion of the Chern-Simons partition function remains finite when quantizing around any flat gauge field on a compact 3-manifold.
• The formal metric-independence of the partition function is preserved even for non-trivial flat connections, extending prior results.
• The analysis confirms that the standard perturbative framework remains valid under the generalized quantization condition.
• The result holds under the assumption of formal power series expansion, without requiring convergence.
• The technical correction to the determinant ensures consistency in the path integral measure for reducible gauge orbits.