Frames of group-sets and their application in bundle theory

Abstract

We study fiber bundles where the fibers are not a group $G$, but a free $G$-space with disjoint orbits. These bundles closely resemble principal bundles, hence we call them semi-principal bundles. The study of such bundles is facilitated by defining the notion of a basis of a $G$-set, in analogy with a basis of a vector space. The symmetry group of these bases is a wreath product. Similar to vector bundles, using the notion of a basis induces a frame bundle construction, which in this case results in a principal bundle with the wreath product as structure group. This construction can be formalized in the language of a functor, which retracts the semi-principal bundles to the principal bundles. In addition, semi-principal bundles support parallel transport just like principal bundles, and this carries over to the frame bundle.

28 pages, 3 figures

References in corpus (2)

• Genons, twist defects, and projective non-Abelian braiding statistics
• A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

Cited by in corpus (1)

• A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics