Small Instantons in $CP^1$ and $CP^2$ Sigma Models.pdf

a r X i v : h e p - l a t / 0 6 0 7 0 2 6 v 1 2 0 J u l 2 0 0 6 Small Instantons in CP 1 and CP 2 Sigma Models Yaogang Lian and H. B. Thacker Department of Physics, University of Virginia, P.O. Box 400714 Charlottesville, VA 22901-4714 (Dated: February 1, 2008) The anomalous scaling behavior of the topological susceptibility χt 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 χt 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 confirma- tion of Lu?scher’s argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in CP 1 and CP 2, leading to a diver- gent 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. I. INTRODUCTION Recent lattice studies of topological charge structure in pure-glue QCD have provided evidence that the topological charge density, as constructed from the overlap Dirac operator, organizes into low-dimensional long-range structure [1], consisting of 3-dimensional coherent sheets in 4- dimensional spacetime. A further study demonstrated that this structure is inherently global and that topological charge fluctuations in QCD should be understood not in terms of individual localized lum