Gribov copies, BRST-exactness and Chern-Simons theory

The path integral approach for a 3D Chern-Simons theory is discussed with a focus on the question of metric independence and BRST-exactness in the light of Gribov ambiguity. Copies of the vacuum satisfying the strong boundary conditions and with trivial winding number are shown to exist. This problem is relevant for Yang-Mills theory in 2+1 dimensions as well. Furthermore, it is also shown that there are regular background metrics supporting an infinite number of zero modes of the Abelian Faddeev-Popov operator in the Coulomb gauge. In order to avoid overcounting one could restrict the possible metrics which enter in the gauge fixing but this would imply that, at a quantum level, three-dimensional Chern-Simons theories do depend on the metric.