The Index and Axial Anomaly of a lattice Dirac operator

TL;DR

This paper discusses a Ginsparg-Wilson lattice Dirac operator that, unlike its continuum counterpart, lacks topological zero modes in nontrivial gauge backgrounds despite correctly reproducing axial anomalies.

Contribution

It introduces and analyzes a lattice Dirac operator with unique topological properties differing from continuum operators.

Findings

Does not possess topological zero modes in nontrivial backgrounds

Reproduces correct axial anomaly for trivial gauge fields

Is exponentially-local and doublers-free

Abstract

A remarkable feature of a lattice Dirac operator is discussed. Unlike the Dirac operator for massless fermions in the continuum, this Ginsparg-Wilson lattice Dirac operator does not possess topological zero modes for any topologically-nontrivial background gauge fields, even though it is exponentially-local, doublers-free, and reproduces correct axial anomaly for topologically-trivial gauge configurations.