Perturbative expansion in gauge theories on compact manifolds

TL;DR

This paper presents a geometric perturbative method for quantum gauge theories on compact manifolds, refining gauge-fixing to avoid infrared divergences and applying it to compute partition functions in Chern-Simons theory.

Contribution

It introduces a refined gauge-fixing approach that handles ambiguities and infrared issues, enabling perturbative expansions on compact manifolds for gauge theories like Chern-Simons.

Findings

Perturbative expansions are possible on simple compact manifolds.

The method avoids infrared divergences with isolated critical points.

Partition functions are shown to be metric-independent in examples.

Abstract

A geometric formal method for perturbatively expanding functional integrals arising in quantum gauge theories is described when the spacetime is a compact riemannian manifold without boundary. This involves a refined version of the Faddeev-Popov procedure using the covariant background field gauge-fixing condition with background gauge field chosen to be a general critical point for the action functional (i.e. a classical solution). The refinement takes into account the gauge-fixing ambiguities coming from gauge transformations which leave the critical point unchanged, resulting in the absence of infrared divergences when the critical point is isolated modulo gauge transformations. The procedure can be carried out using only the subgroup of gauge transformations which are topologically trivial, possibly avoiding the usual problems which arise due to gauge-fixing ambiguities. For Chern-Simons gauge theory the method enables the partition function to be perturbatively expanded for a number of simple spacetime manifolds such as $S^3$ and lens spaces, and the expansions are shown to be formally independent of the metric used in the gauge-fixing.