The Gribov phenomenon in flat and curved spaces

The quantization of Yang-Mills theories relies on the gauge-fixing procedure. However, in the non-Abelian case this procedure leads to the well known Gribov ambiguity. In order to solve the ambiguity a modification of the functional integral formula must be introduced. As a consequence of this, the Green functions get deep modifications in the infrared. We consider in particular the SU (N) case and show that in the pure gauge case the ghost propagator is enhanced, while the gluon propagator is suppressed in this limit, therefore the study of the Gribov ambiguity may shed some light on the mass gap problem and on colour confinement. We discuss some recent developments on the subject in the case of a curved background. We argue that the concurrent presence of a spacetime curvature and the Gribov ambiguity may introduce further modifications to the Green functions in the infrared.