In gauge theory, especially in non-abelian gauge theories, global problems at gauge fixing are often encountered. Gauge fixing means choosing a representative from each gauge orbit, that is, choosing a section of a fiber bundle. The space of representatives is a submanifold (of the bundle as a whole) and represents the gauge fixing condition. Ideally, every gauge orbit will intersect this submanifold once and only once. Unfortunately, this is often impossible globally for non-abelian gauge theories because of topological obstructions and the best that can be done is make this condition true locally. A gauge fixing submanifold may not intersect a gauge orbit at all or it may intersect it more than once. The difficulty arises because the gauge fixing condition is usually specified as a differential equation of some sort, e.g. that a divergence vanish (as in the Landau or Lorenz gauge). The solutions to this equation may end up specifying multiple sections, or perhaps none at all. This is called a Gribov ambiguity (named after Vladimir Gribov). Gribov ambiguities lead to a nonperturbative failure of the BRST symmetry, among other things. A way to resolve the problem of Gribov ambiguity is to restrict the relevant functional integrals to a single Gribov region whose boundary is called a Gribov horizon. Still one can show that this problem is not resolved even when reducing the region to the first Gribov region. The only region for which this ambiguity is resolved is the fundamental modular region (FMR).
Background When doing computations in gauge theories, one usually needs to choose a gauge. Gauge degrees of freedom do not have any direct physical meaning, but they are an artifact of the mathematical description we use to handle the theory in question. In order to obtain physical results, these redundant degrees of freedom need to be discarded in a suitable way In Abelian gauge theory (i.e. in quantum electrodynamics) it suffices to simply choose a gauge. A popular one is the Lorenz gauge ∂ μ A μ = 0 {\displaystyle \partial ^{\mu }A_{\mu }=0} , which has the advantage of being Lorentz invariant. In non-Abelian gauge theories (such as quantum chromodynamics) the situation is more complicated due to the more complex structure of the non-Abelian gauge group. The Faddeev–Popov formalism, developed by Ludvig Faddeev and Victor Popov, provides a way to deal with the gauge choice in non-Abelian theories. This formalism introduces the Faddeev–Popov operator, which is essentially the Jacobian determinant of the transformation necessary to bring the gauge field into the desired gauge. In the so-called Landau gauge ∂ μ A μ a = 0 {\displaystyle \partial _{\mu }A_{\mu }^{a}=0} , this operator has the form
∂ μ D μ a b , {\displaystyle \partial _{\mu }{\mathcal {D}}_{\mu }^{ab}\;,}
where D μ a b {\displaystyle {\mathcal {D}}_{\mu }^{ab}} is the covariant derivative in the adjoint representation. The determinant of this Faddeev–Popov operator is then introduced into the path integral using ghost fields. This formalism, however, assumes that the gauge choice (like ∂ μ A μ a = 0 {\displaystyle \partial _{\mu }A_{\mu }^{a}=0} ) is unique — i.e. for each physical configuration there exists exactly one A μ a {\displaystyle A_{\mu }^{a}} that corresponds to it and that obeys the gauge condition. In non-Abelian gauge theories of Yang–Mills type, this is not the case for a large class of gauges, though, as was first pointed out by Gribov in 1978.
Gribov's construction Gribov considered the question of, given a certain physical configuration, how many different gauge copies of this configuration obey the Landau gauge condition ∂ μ A μ a = 0 {\displaystyle \partial _{\mu }A_{\mu }^{a}=0} . No configurations without any representatives are known. It is perfectly possible, though, for there to be more than one. Consider two gauge fields A μ a {\displaystyle A_{\mu }^{a}} and A μ a ′ {\displaystyle {A_{\mu }^{a}}'} , and assume they both obey the Landau gauge condition. If A μ a ′ {\displaystyle {A_{\mu }^{a}}'} is a gauge copy of A μ a {\displaystyle A_{\mu }^{a}} , we would have (assuming they are infinitesimally close to each other):
… excerpt ends here. Continue reading the full article.
