Gauge fixing, families index theory, and topological features of the space of lattice gauge fields

The recently developed families index theory for the overlap lattice Dirac operator is applied to demonstrate an interplay between topological features of the space of SU(N) lattice gauge fields on the 4-torus and the existence question for G0 gauge fixings which do not have the Gribov problem. The continuum version of our considerations leads to topological obstructions to the existence of such gauge fixings. These are found to be absent in the lattice setting though, thus providing an indication that such gauge fixings may exist on the lattice (which is already known to be the case in the U(1) theory when an admissibility condition is imposed). Instead, the lattice version of our considerations leads to the existence of noncontractible even-dimensional spheres in the topological sectors of the space of SU(N) lattice gauge fields when N ≥ 3, and noncontractible circles when N=2, which are constructed quite explicitly.