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

Abstract

The families index theory for the overlap lattice Dirac operator is applied to derive topological features of the space of SU(N) lattice gauge fields on the 4-torus: The topological sectors, specified by the fermionic topological charge, are shown to contain noncontractible even-dimensional spheres when $N\ge 3$, and noncontractible circles in the N=2 case. We describe how certain obstructions to the existence of gauge fixings without the Gribov problem in the continuum setting correspond on the lattice to obstructions to the contractibility of these spheres and circles. We also point out a canonical connection on the space of lattice gauge fields with monopole-like singularities associated with the spheres.

Comments: Latex, 26 pages. v2: Substantially revised to take account of the fact that ${\mathcal{G}}_0$ gauge fixings without the Gribov problem exist on the lattice (e.g. maximal tree gauges); main results unchanged; to appear in Nucl. Phys. B. (This version of the material is the one that I presented at Lattice 2002.)