Ginsparg-Wilson Relation, Topological Invariants and Finite Noncommutative Geometry

TL;DR

This paper demonstrates how the Ginsparg-Wilson relation can be used to define chiral structures and topological invariants in finite noncommutative geometries, exemplified by a gauge theory on a fuzzy sphere.

Contribution

It introduces a noncommutative analog of the Ginsparg-Wilson relation, chirality operator, and index theorem for finite geometries, linking to classical topological invariants.

Findings

Established a finite noncommutative index theorem.

Constructed a noncommutative Ginsparg-Wilson relation and chirality operator.

Showed the topological invariant matches the first Chern class in the commutative limit.

Abstract

We show that the Ginsparg-Wilson (GW) relation can play an important role to define chiral structures in {\it finite} noncommutative geometries. Employing GW relation, we can prove the index theorem and construct topological invariants even if the system has only finite degrees of freedom. As an example, we consider a gauge theory on a fuzzy two-sphere and give an explicit construction of a noncommutative analog of the GW relation, chirality operator and the index theorem. The topological invariant is shown to coincide with the 1st Chern class in the commutative limit.