Principal bundles and sections of fibre bundles: Reduction of the structure and the gauge group I

Abstract

In this chapter, we consider bundles p: E → B where a topological group G acts on the fibres through an action of G on the total space E. This is just an action E × G → E of G on E such that p(χs) = p(χ) for all χ ∈ E and s ∈ G. In particular, there is a restriction Eb × G → Eb of the globally defined action to each fibre Eb, b ∈ B, of the bundle. A principal G-bundle is a bundle p: P → B with an additional algebraic and continuity action property implying, for example, that all fibres are isomorphic to G by any map G → Pb of the form s → us for any u ∈ Pb and all s ∈ G. For a principal G-bundle p: P → B and a left G-space Y, we have the fibre bundle construction q: P[Y] = P×G Y → B. Vector bundles are examples of fibre bundles where G = GL(n), the general linear group, and Y is an n-dimensional vector space. The characterization of sections in Γ(B,P[Y]) by certain maps P →Y plays a fundamental role in applications of principal bundle theory. We outline two important aspects within principal bundle theory in this chapter. At first, the reduction of the structure group G of a principal bundle P. The second aspect is the study of AutG (P), the automorphism group of the principal G-bundle P. Chapter 4 of Fibre Bundles (Husemöller 1994) is a reference for this chapter. © Springer-Verlag Berlin Heidelberg 2008.