Topological aspects of lattice gauge theories

Abstract

This review discusses topological aspects of nonabelian gauge theories defined on the lattice. The pertinent topological object is the principal bundle, so, following Lüscher, it is shown how to reconstruct the bundle from the data of the lattice gauge field. This culminates in the algorithm of Phillips and Stone for computing the topological charge, or second Chern number, of an SU(N) lattice gauge field. For SU(2) this algorithm is very fast. The relation of the topological susceptibility χ t to the bare (i.e. Monte Carlo) susceptibility at nonzero lattice spacing is clarified. (The details of this discussion have a somewhat different tone than at the conference, but the conclusions are the same.) Other methods for computing the topological charge of a lattice gauge field are summarized, and the strengths and weakness are compared to the fiberbundle method. The numerical results of moderate to high statistics Monte Carlo determinations of χ t are presented with remarks on scaling behavior. The review concludes with some comments on the progress, some hints of what can be done next, and some vague hope of a hidden utility for the topology of lattice gauge fields.