Comments on the Gribov Ambiguity

TL;DR

This paper investigates the Gribov ambiguity in $SU(m)\times U(1)$ gauge theories on $n$-spheres, showing the uniqueness of gauge transformation classes and analyzing their homotopy properties.

Contribution

It demonstrates the conjugacy of gauge transformation groups for these theories and determines their homotopy type using properties of mapping spaces.

Findings

Unique conjugacy class of gauge transformations established

Homotopy type of gauge groups characterized in terms of $G$

Non-trivial homotopy groups imply the presence of Gribov ambiguities

Abstract

We discuss the existence of Gribov ambiguities in $SU(m)\times U(1)$ gauge theories over the $n-$spheres. We achieve our goal by showing that there is exactly one conjugacy class of groups of gauge transformations for the theories given above. This implies that these transformation groups are conjugate to the ones of the trivial $SU(m)\times U(1)$ fiber bundles over the $n-$spheres. By using properties of the space of maps $Map_{\ast}(S^n,G)$ where $G$ is one of $U(1)$, $SU(m)$ we are able to determine the homotopy type of the groups of gauge transformations in terms of the homotopy groups of $G$. The non-triviality of these homotopy groups gives the desired result.