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

TL;DR

This paper explores the topological structure of the space of lattice gauge fields using families index theory, revealing noncontractible spheres and circles in different gauge groups and relating continuum obstructions to lattice gauge fixing issues.

Contribution

It applies families index theory to lattice gauge fields, uncovering topological features and obstructions related to gauge fixing and Gribov problems on the lattice.

Findings

Topological sectors contain noncontractible spheres for N≥3

Noncontractible circles in the N=2 case

Obstructions to gauge fixing correspond to topological features

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 $N\ge3$, 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.