Noncommutative Geometry Approach to Principal and Associated Bundles

TL;DR

This paper reformulates key concepts of differential geometry within noncommutative geometry, introducing new notions of triviality and analyzing principal bundles through algebraic and geometric lenses, including explicit examples and connections.

Contribution

It introduces the concept of piecewise triviality for C*-algebra fiber products and explores non-proper free actions, expanding the understanding of principal bundles in noncommutative geometry.

Findings

Characterized principal actions via density conditions in C*-algebras

Defined piecewise triviality for fiber product bundles

Computed line bundle pairings using Dirac monopole connection

Abstract

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of local triviality to fibre products of C*-algebras. In the context of principal actions, we study in detail an example of a non-proper free action with continuous translation map, and examples of compact principal bundles which are piecewise trivial but not locally trivial, and neither piecewise trivial nor locally trivial, respectively. We show that the module of continuous sections of a vector bundle associated to a compact principal bundle is a cotensor product of the algebra of functions defined on the total space (that are continuous along the base and polynomial along the fibres) with the vector space of the representation. On the algebraic side, we review the formalism of connections for the universal differential algebras. In the differential geometry framework, we consider smooth connections on principal bundles as equivariant splittings of the cotangent bundle, as 1-form-valued derivations of the algebra of smooth functions on the structure group, and as axiomatically given covariant differentiations of functions defined on the total space. Finally, we use the Dirac monopole connection to compute the pairing of the line bundles associated to the Hopf fibration with the cyclic cocycle of integration over S^2.