Small instantons in CP1 and CP2 sigma models

The anomalous scaling behavior of the topological susceptibility chi(t) in two-dimensional CPN-1 sigma models for N <= 3 is studied using the overlap Dirac operator construction of the lattice topological charge density. The divergence of chi(t) in these models is traced to the presence of small instantons with a radius of order a(=lattice spacing), which are directly observed on the lattice. The observation of these small instantons provides detailed confirmation of Luscher's argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in CP1 and CP2, leading to a divergent topological susceptibility in the continuum limit. For the CPN-1 models with N > 3 the topological susceptibility is observed to scale properly with the mass gap. These larger N models are not dominated by instantons, but rather by coherent, one-dimensional regions of topological charge which can be interpreted as domain wall or Wilson line excitations and are analogous to D-brane or "Wilson bag" excitations in QCD. In Lorentz gauge, the small instantons and Wilson line excitations can be described, respectively, in terms of poles and cuts of an analytic gauge potential.