Noncommutative generalization of SU(n)-principal fiber bundles: a review

TL;DR

This paper reviews noncommutative geometric constructions related to SU(n)-principal fiber bundles, emphasizing their applications in gauge theories, Higgs fields, and new mathematical insights like a novel approach to Chern classes.

Contribution

It introduces a noncommutative framework for fiber bundles that integrates gauge models, Higgs mechanisms, and new mathematical constructions such as a novel Chern class formulation.

Findings

Noncommutative algebra based on endomorphisms of SU(n)-vector bundles.

Inclusion of gauge theories, Higgs fields, and mass generation within the noncommutative setting.

New construction of Chern characteristic classes in noncommutative geometry.

Abstract

This is an extended version of a communication made at the international conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary fiber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)-vector bundle, and its differential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs fields and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes.