Gribov pendulum in the Coulomb gauge on curved spaces

TL;DR

This paper extends the Gribov pendulum equation to curved spacetimes in the Coulomb gauge, analyzing solution existence on static spherically symmetric backgrounds, with implications for gauge fixing ambiguities in different asymptotic geometries.

Contribution

It provides a rigorous analysis of the Gribov pendulum in curved spaces, establishing conditions for solutions and exploring differences between asymptotically flat and AdS backgrounds.

Findings

Asymptotically flat backgrounds do not admit Gribov copies satisfying strong boundary conditions.

Asymptotically AdS backgrounds can support such Gribov copies.

Existence and uniqueness of solutions are proven for certain curved spacetimes.

Abstract

In this paper the generalization of the Gribov pendulum equation in the Coulomb gauge for curved spacetimes is analyzed on static spherically symmetric backgrounds. A rigorous argument for the existence and uniqueness of solution is provided in the asymptotically AdS case. The analysis of the strong and weak boundary conditions is equivalent to analyzing an effective one-dimensional Schrodinger equation. Necessary conditions in order for spherically symmetric backgrounds to admit solutions of the Gribov pendulum equation representing copies of the vacuum satisfying the strong boundary conditions are given. It is shown that asymptotically flat backgrounds do not support solutions of the Gribov pendulum equation of this type, while on asymptotically AdS backgrounds such ambiguities can appear. Some physical consequences are discussed.