Index theorem in spontaneously symmetry-broken gauge theories on a fuzzy 2-sphere

TL;DR

This paper explores the topological aspects of gauge configurations in a gauge-Higgs system on a fuzzy 2-sphere, linking noncommutative index theory with topological charge and symmetry breaking.

Contribution

It introduces a noncommutative index theorem for gauge theories on a fuzzy sphere, generalizing topological charge concepts to noncommutative geometry.

Findings

Topological charge is classified by the Dirac operator index satisfying Ginsparg-Wilson relation.

The noncommutative topological charge reduces to the winding number in the commutative limit.

Explicit calculation of topological charge for a family of configurations confirms the theoretical framework.

Abstract

We consider a gauge-Higgs system on a fuzzy 2-sphere and study the topological structure of gauge configurations, when the U(2) gauge symmetry is spontaneously broken to U(1) times U(1) by the vev of the Higgs field. The topology is classified by the index of the Dirac operator satisfying the Ginsparg-Wilson relation, which turns out to be a noncommutative analog of the topological charge introduced by 't Hooft. It can be rewritten as a form whose commutative limit becomes the winding number of the Higgs field. We also study conditions which assure the validity of the formulation, and give a generalization of the admissibility condition. Finally we explicitly calculate the topological charge of a one-parameter family of configurations.