Absence of the Gribov ambiguity in a quadratic gauge

TL;DR

This paper introduces a Lorentz invariant quadratic gauge fixing that removes the Gribov ambiguity on a compact manifold, addressing limitations of algebraic gauges and preserving BRST invariance.

Contribution

The paper demonstrates a specific quadratic gauge that is Lorentz invariant and eliminates the Gribov ambiguity on a compact manifold, with preserved BRST symmetry.

Findings

The quadratic gauge removes the Gribov ambiguity on .

The gauge preserves BRST invariance.

The ambiguity persists in algebraic gauges but not in this quadratic gauge.

Abstract

The Gribov ambiguity exists in various gauges except algebraic gauges. However, algebraic gauges are not Lorentz invariant, which is their fundamental flaw. In addition, they are not generally compatible with the boundary conditions on the gauge fields, which are needed to compactify the space i.e., the ambiguity continues to exist on a compact manifold. Here we discuss a quadratic gauge fixing, which is Lorentz invariant. We consider an example of a spherically symmetric gauge field configuration in which we prove that this Lorentz invariant gauge removes the ambiguity on a compact manifold $\mathbb{S}^3$, when a proper boundary condition on the gauge configuration is taken into account. Thus, providing one example where the ambiguity is absent on a compact manifold in the algebraic gauge. We also show that the \tmem{BRST} invariance is preserved in this gauge.