The Index of a Ginsparg-Wilson Dirac operator

TL;DR

This paper discusses a new Ginsparg-Wilson lattice Dirac operator that, unlike its continuum counterpart, lacks topological zero modes in nontrivial gauge backgrounds, while maintaining key properties like locality and correct anomaly reproduction.

Contribution

It introduces a Ginsparg-Wilson Dirac operator with unique topological properties, expanding understanding of lattice fermion formulations.

Findings

Does not possess topological zero modes in nontrivial backgrounds

Maintains exponential locality and doublers-free nature

Reproduces correct axial anomaly in trivial backgrounds

Abstract

A novel feature of a Ginsparg-Wilson lattice Dirac operator is discussed. Unlike the Dirac operator for massless fermions in the continuum, this 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.