Construction of a topological charge on fuzzy S^2 x S^2 via Ginsparg-Wilson relation

TL;DR

This paper develops a method to define a topological charge on fuzzy S^2 x S^2 using a Ginsparg-Wilson Dirac operator, extending noncommutative geometry concepts to gauge theories.

Contribution

It introduces a novel topological charge construction on fuzzy S^2 x S^2 via a Ginsparg-Wilson relation, generalizing the 2nd Chern character in noncommutative geometry.

Findings

Topological charge defined on fuzzy S^2 x S^2 matches continuum limits.

Number of chiral zero modes calculated for nontrivial gauge configurations.

Discussion on extending the formulation to higher-dimensional fuzzy spheres.

Abstract

We construct a topological charge of gauge field configurations on a fuzzy S^2xS^2 by using a Dirac operator satisfying the Ginsparg-Wilson relation. The topological charge defined on the fuzzy S^2xS^2 can be interpreted as a noncommutative (or matrix) generalization of the 2nd Chern character on S^2xS^2. We further calculate the number of chiral zero modes of the Dirac operator in topologically nontrivial gauge configurations. Generalizations of our formulation to fuzzy (S^2)^k are also discussed.