The Fuzzy Ginsparg-Wilson Algebra: A Solution of the Fermion Doubling Problem

TL;DR

This paper develops a Dirac operator theory on the fuzzy sphere using the Ginsparg-Wilson algebra, effectively solving the fermion doubling problem and correctly reproducing anomaly and index theory.

Contribution

It systematically establishes a Ginsparg-Wilson algebra-based Dirac operator on the fuzzy sphere, eliminating fermion doubling and aligning with anomaly and index theory.

Findings

No fermion doubling in the fuzzy sphere Dirac operator

Reproduces the U(1)_A anomaly accurately

Establishes a Ginsparg-Wilson algebra framework for fuzzy sphere

Abstract

The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of the Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)_A anomaly and index theory correctly.