Anomalies, gauge field topology, and the lattice

Understand

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.