The Gribov ambiguity in gauge theories on the four-torus

It is shown that it is impossible to fix the gauge (i.e., there is a Gribov ambiguity) in a non-abelian gauge theory, with any gauge group, defined on the four-torus (which corresponds to a four-dimensional euclidean gauge theory with periodic boundary conditions). It is also shown that a Gribov ambiguity exists in SU(2) gauge theories defined on S2 X S2 and the existence of the Gribov phenomenon is related to the existence of inequivalent quantizations of these theories. (orig.)