Projectors for the fuzzy sphere

TL;DR

This paper constructs projector-valued fields for the fuzzy sphere, enabling the definition of non-commutative vector bundles and monopole configurations, and verifies their topological charges are integers in the classical limit.

Contribution

It introduces a method to derive projector-valued fields for the fuzzy sphere, facilitating non-commutative bundle constructions and topological charge calculations.

Findings

Defined non-commutative n-monopole configurations

Verified topological charges are integers in the classical limit

Established a framework for vector bundles on non-commutative spaces

Abstract

All fiber bundle with a given set of characteristic classes are viewable as particular projections of a more general bundle called a universal classifying space. This notion of projector valued field, a global definition of connections and gauge fields, can be useful to define vector bundles for non commutative base spaces. In this paper we derive the projector valued field for the fuzzy sphere, defining non-commutative n-monopole configurations, and check that in the classical limit, using the machinery of non-commutative geometry, the corresponding topological charges (Chern class) are integers.