Common-boundary conjecture for the first Gribov region and the fundamental modular region in the infinite-volume limit

Determine whether, in the infinite-volume limit of Landau-gauge nonabelian gauge theory, the physically relevant gauge-field configurations lie on the common boundary of the first Gribov region and the fundamental modular region, thereby validating that random selection of Gribov copies within the first Gribov region suffices for calculating Green’s functions.

Background

In nonabelian gauge theories, gauge fixing is not unique and leads to Gribov copies—multiple configurations along the same gauge orbit that satisfy the gauge condition. The first Gribov region consists of local minima of the Landau gauge-fixing functional (equivalently, configurations with positive Faddeev–Popov operator), while the fundamental modular region (FMR) comprises absolute minima.

Because numerical and functional approaches sample Gribov copies in different, uncontrolled ways, a conjecture proposes that in the infinite-volume limit the important configurations lie on the common boundary of the first Gribov region and the FMR. If true, random sampling within the first Gribov region would be adequate, simplifying comparisons across methods.