Anomalies, gauge field topology, and the lattice

Abstract

Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I use the fourth power of the naive Dirac operator to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally "fall through the lattice." Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.

arXiv comment22 pages, 12 figures, final version to appear in Annals of Physics

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• More about exactly massless quarks on the lattice
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Cited by in corpus (7)

• Confinement, chiral symmetry, and the lattice
• Correlation and localization properties of topological charge density and the pseudoscalar glueball mass in SU(3) lattice Yang-Mills theory
• Instanton liquid properties from lattice QCD
• Influence of topology on the scale setting
• Topological susceptibility from the twisted mass Dirac operator spectrum
• Quark masses, the Dashen phase, and gauge field topology
• Chiral anomalies and Wilson fermions