A continuum limit of the chiral Jacobian in lattice gauge theory

TL;DR

This work connects the lattice formulation of chiral symmetry, via the Ginsparg-Wilson relation and the overlap (Neuberger) Dirac operator, to the continuum Atiyah-Singer index theorem and chiral anomaly without relying on perturbation theory. It shows that the lattice Jacobian reproduces the standard continuum anomaly in the continuum limit, provided doublers are eliminated, and it clarifies how the finite-cutoff lattice index theorem aligns with continuum topological results. A two-step continuum-limit procedure isolates the anomaly contribution and demonstrates regulator independence of the global and local index in this framework. The findings support the potential of lattice regularization to provide a nonperturbative foundation for chiral symmetry, with implications for chiral gauge theories and nonperturbative gauge dynamics.

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 Lüscher, 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.