Exact chiral symmetry, topological charge and related topics

TL;DR

This work surveys how Dirac operators satisfying the Ginsparg–Wilson relation resolve the lattice chirality problem by preserving chiral Ward identities and enabling an exact lattice index theorem, while avoiding additive mass renormalization and $O(a)$ artifacts. It connects the fixed-point and overlap (Neuberger) constructions to the GW framework, clarifying locality, spectrum, and topological charge definitions. The presentation details the chiral decomposition on the lattice, the role of the theta parameter, and practical considerations for measuring topological charge, with extensions to chiral gauge theories via Lüscher’s approach. The results collectively establish a principled path toward nonperturbative, chirally symmetric lattice QCD and point to practical avenues for implementing chiral gauge theories in a fully discretized setting.

Abstract

It has been shown recently that Dirac operators satisfying the Ginsparg-Wilson relation provide a solution of the chirality problem in QCD at finite lattice spacing. We discuss different ways to construct these operators and their properties. The possibility to define lattice chiral gauge theories is briefly discussed as well.

Figures (3)

• Figure 1: Spectrum of the Wilson-Dirac operator (W), and of Neuberger's operator (N).
• Figure 2: Locality of different GW Dirac operators for the free fermions.
• Figure 3: Locality of Neuberger's Dirac operator in gauge field background.