Principal Bundles and Sections of Fibre Bundles: Reduction of the Structure and the Gauge Group I

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(xs) = p(x) for all x ∈ E and s ∈ G. In particular, there is a restriction E _b × G → E _b of the globally defined action to each fibre E _b, 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 → P _b of the form s → us for any u ∈ P _b 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.