Comments on the Gribov Ambiguity

We discuss the existence of Gribov ambiguities in SU(m)×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)×U(1) fiber bundles over the n−spheres. By using properties of the space of maps Map∗(S , 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. † Research supported in part by the DOE grant DE-FG02-85ER40231 ⋆ e-mail [email protected]