Axial anomaly and Ginsparg-Wilson fermions in the Lattice Dirac Sea picture

TL;DR

This paper derives the axial anomaly in 1+1D lattice QED using Ginsparg-Wilson fermions, showing the conditions under which the anomaly is correctly reproduced and discussing extensions to higher dimensions.

Contribution

It demonstrates how the axial anomaly emerges on the lattice with Ginsparg-Wilson fermions and clarifies the role of the Wilson parameter and adiabatic conditions.

Findings

Correct anomaly reproduced for slow gauge field variations

Non-zero Wilson r parameter necessary for anomaly

Anomaly independent of r for r>0.5

Abstract

The axial anomaly equation in 1+1 dimensional QED is obtained on the lattice for fermions obeying the Ginsparg-Wilson relation. We make use of the properties of the Lattice Dirac sea to investigate the connection between the anomaly and the Ginsparg-Wilson operator in the Hamiltonian picture. The correct anomaly is reproduced for gauge fields whose characteristic time is much larger than the lattice spacing, which is the regime where the adiabatic approximation applies. A non-zero Wilson $r$ parameter is necessary to get the correct anomaly. The anomaly is shown to be independent of $r$ for $r>0.5$. The generalization to 3+1 dimensions is also discussed.