Consistent axial--like gauge fixing on hypertori

TL;DR

This paper investigates the Gribov problem in Yang-Mills theories on tori, proposing an improved axial gauge fixing that explicitly characterizes residual gauge symmetries and the space of gauge orbits, impacting the understanding of physical states.

Contribution

It introduces an improved axial gauge fixing on tori, explicitly determines the residual gauge group, and describes the gauge orbit space as an orbifold, advancing the understanding of gauge fixing ambiguities.

Findings

Residual gauge group is explicitly characterized as a discrete group.

The space of gauge orbits is described as an orbifold.

Implications for the structure of physical states in Yang-Mills theories.

Abstract

We analyze the Gribov problem for $\SU(N)$ and $\U(N)$ Yang-Mills fields on $d$-dimensional tori, $d=2,3,\ldots$. We give an improved version of the axial gauge condition and find an infinite, discrete group $\cG'=\Z^{dr}\rtimes({\Z_2}^{N-1}\rtimes\Z_2)$, where $r=N-1$ for $\GG=\SU(N)$ and $r=N$ for $\GG=\U(N)$, containing all gauge transformations compatible with that condition. This residual gauge group $\cG'$ provides (generically) all Gribov copies and allows to explicitly determine the space of gauge orbits which is an orbifold. Our results apply to Yang-Mills gauge theories either in the Lagrangian approach on $d$-dimensional space-time $T^d$, or in the Hamiltonian approach on $(d+1)$-dimensional space-time $T^d\times \R$. Using the latter, we argue that our results imply a non-trivial structure of all physical states in any Yang-Mills theory, especially if also matter fields are present.