Generalized Ginsparg-Wilson algebra and index theorem on the lattice

TL;DR

This paper explores a generalized algebraic form of the Ginsparg-Wilson relation on the lattice, demonstrating that various operators within this class share topological and anomaly properties, and establishing their locality in certain conditions.

Contribution

It introduces a generalized Ginsparg-Wilson algebra, proves the locality of associated Dirac operators for zero gauge fields, and links their topological properties across different parameter choices.

Findings

Local chiral anomaly is identical across operators

Index related to instantons is consistent for all operators

Locality is proven for zero gauge fields

Abstract

Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of the Ginsparg-Wilson relation in the form $\gamma_{5}(\gamma_{5}D)+(\gamma_{5}D)\gamma_{5} = 2a^{2k+1}(\gamma_{5}D)^{2k+2}$ where $k$ stands for a non-negative integer. The choice $k=0$ corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. It is shown that local chiral anomaly and the instanton-related index of all these operators are identical. The locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been established yet. We suggest that the Wilsonian effective action is essential to avoid infrared singularities encountered in general perturbative analyses.