A continuum limit of the chiral Jacobian in lattice gauge theory

TL;DR

This paper presents a non-perturbative method to derive the continuum limit of the chiral Jacobian in lattice gauge theory, connecting lattice formulations with continuum anomalies using the Ginsparg-Wilson relation.

Contribution

It introduces an alternative, rule-based approach to obtain the continuum chiral Jacobian from lattice gauge theory without perturbation, utilizing the Ginsparg-Wilson relation.

Findings

Successfully recovers continuum anomaly results from lattice formulations.

Provides a non-perturbative derivation of the continuum limit of the chiral Jacobian.

Illustrates the method explicitly with Neuberger's lattice Dirac operator.

Abstract

We study the implications of the index theorem and chiral Jacobian in lattice gauge theory, which have been formulated by Hasenfratz, Laliena and Niedermayer and by L\"{u}scher, on the continuum formulation of the chiral Jacobian and anomaly. We take a continuum limit of the lattice Jacobian factor without referring to perturbative expansion and recover the result of continuum theory by using only the general properties of the lattice Dirac operator. This procedure is based on a set of well-defined rules and thus provides an alternative approach to the conventional analysis of the chiral Jacobian and related anomaly in continuum theory. By using an explicit form of the lattice Dirac operator introduced by Neuberger, which satisfies the Ginsparg-Wilson relation, we illustrate our calculation in some detail. We also briefly comment on the index theorem with a finite cut-off from the present viewpoint.