Topological Charge in Lattice Gauge Theory.

The mathematical and physical aspects of some of the topological excitations which occur in statistical mechanical and field theoretical models are reviewed. The problems involved in translating these concepts to a lattice version of a theory are discussed and then illustrated in detail for the case of the O(3) non-linear sigma model. The nature of the topological excitations of gauge fields in the continuum is reviewed and a general method is given for sensibly defining the topology of a lattice gauge field. This method is described explicitly for the cases of the two dimensional U(1) theory and the four dimensional SU(2) Yang-Mills theory. A simple technique for computing the topological charge of sufficiently smooth lattice gauge field configurations is given and the results of Monte -Carlo calculations of the topological susceptibility and the (THETA) dependence of the vacuum energy for the four dimensional SU(2) theory are reported.