On globally non-trivial almost-commutative manifolds

TL;DR

This paper explores the concept of globally non-trivial almost-commutative manifolds within noncommutative geometry, linking them to gauge theories and introducing principal modules and gauge modules as key constructs.

Contribution

It introduces the notion of principal modules and gauge modules, providing a framework for describing gauge theories on non-trivial almost-commutative manifolds.

Findings

Defined and studied globally non-trivial almost-commutative manifolds.

Established the concept of principal modules from fibre bundles and spectral triples.

Provided illustrative examples of the theory.

Abstract

Within the framework of Connes' noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almost-commutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a 'principal module', which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of 'gauge modules', and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.