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Frames of group-sets and their application in bundle theory

Eric J. Pap, Holger Waalkens

TL;DR

The paper develops a framework for fiber bundles whose fibers are free $G$-spaces (semi-torsors), introducing bases for $G$-sets and a frame-bundle construction with structure group the wreath product $G\wr X$, where $X=F/G$. It proves that the frame construction defines a functor from semi-principal bundles to principal bundles, effectively retracting semi-principal bundles onto the classical theory, and shows that semi-principal bundles admit compatible connections and parallel transport that lift to their frame bundles. By analyzing the orbit structure via the quotient $F/G$ and using the wreath-product symmetry, the authors unify bundle-theoretic notions with group-set symmetry and provide equivalences of categories in a finite-index setting. The motivation from adiabatic quantum mechanics and eigenray bundles suggests practical relevance for gauge-like descriptions of parameter-dependent spectral data and broader applications in physics where fiber symmetries exceed those captured by ordinary principal bundles.

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.

Frames of group-sets and their application in bundle theory

TL;DR

The paper develops a framework for fiber bundles whose fibers are free -spaces (semi-torsors), introducing bases for -sets and a frame-bundle construction with structure group the wreath product , where . It proves that the frame construction defines a functor from semi-principal bundles to principal bundles, effectively retracting semi-principal bundles onto the classical theory, and shows that semi-principal bundles admit compatible connections and parallel transport that lift to their frame bundles. By analyzing the orbit structure via the quotient and using the wreath-product symmetry, the authors unify bundle-theoretic notions with group-set symmetry and provide equivalences of categories in a finite-index setting. The motivation from adiabatic quantum mechanics and eigenray bundles suggests practical relevance for gauge-like descriptions of parameter-dependent spectral data and broader applications in physics where fiber symmetries exceed those captured by ordinary principal bundles.

Abstract

We study fiber bundles where the fibers are not a group , but a free -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 -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.

Paper Structure

This paper contains 19 sections, 38 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Group bundles versus group-space bundles
  3. Group bundles
  4. Group-space bundles
  5. Bases of a group-set
  6. Basis of a group-set
  7. The space of bases of a group-set
  8. Wreath product action
  9. Functorial property
  10. Semi-principal bundles and their frame bundles
  11. The frame bundle of a semi-principal bundle
  12. Functoriality of taking the frame bundle
  13. Connections on a semi-principal bundle and its frame bundle
  14. Discussion
  15. Automorphism group of a group-set
  16. ...and 4 more sections

Key Result

Lemma 2.2

The transition functions of a group bundle with model $G$ take values in the automorphism group $\mathop{\mathrm{Aut}}\nolimits(G)$ of $G$.

Figures (3)

  • Figure 1: Representatives of the two non-isomorphic classes of $\mathbb{Z}_3$-bundles over the circle, see Ex. \ref{['ex:z3 bundles circle']}. The shaded cylinder and Möbius band are visual aids. In both cases, the unit section traces out the middle circle, whereas the other group elements follow the boundary.
  • Figure 2: Correspondence between classes of group-spaces and the classes of group-space bundles having these as fibers. This paper focuses on the gray parts, i.e. the semi-torsors and semi-principal bundles, which extend the torsors and principal bundles.
  • Figure 3: Illustration of the total space of $\Pi_2$ in Ex. \ref{['ex:winding torus 1']}. This space is topologically a torus, but the projection $\Pi_2$ indicates that it winds twice around its base circle. Points with the same color are projected to the same point in $S^1$.

Theorems & Definitions (89)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 2.4: $\mathbb{Z}_3$-bundles over the circle
  • Example 2.5: $\mathbb{Z}_2\times\mathbb{Z}_2$-bundles over the circle
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 79 more