Enumerating Gribov copies on the lattice

TL;DR

This paper investigates the number of Gribov copies in lattice gauge fixing, using analytical and numerical methods, revealing orbit dependence in some gauges and challenging previous claims about suppression in stereographic gauges.

Contribution

It introduces three methods to count Gribov copies on the lattice and compares their effectiveness, providing new insights into orbit dependence and gauge choices.

Findings

n is orbit-dependent for minimal lattice Landau gauge

n is orbit-independent for absolute lattice Landau gauge

n is not exponentially suppressed in stereographic gauge in multiple dimensions

Abstract

In the modern formulation of lattice gauge-fixing, the gauge fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions called Gribov copies. The dependence of the number of Gribov copies, n[U] on the different gauge orbits plays an important role in constructing the Faddeev-Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n[U] for different orbits using three complimentary methods: 1. analytical results in lower dimensions, and some lower bounds on n[U] in higher dimensions, 2. the numerical polynomial homotopy continuation method, which numerically finds all Gribov copies for a given orbit for small lattices, and 3. numerical minimisation ("brute force"), which finds many distinct Gribov copies, but not necessarily all. Because n for the coset SU(N_c)/U(1) of an SU(N_c) theory is orbit-independent, we concentrate on the residual compact U(1) case in this article and establish that n is orbit-dependent for the minimal lattice Landau gauge and orbit-independent for the absolute lattice Landau gauge. We also observe that contrary to a previous claim, n is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension.