Adjoint bundle

Let $\pi:P\to M$ be a principal G-bundle for a Lie group $G$.

The adjoint bundle of $P$ is the associated bundle

where $G$ acts on itself on the left by conjugation:

Each fiber $\mathrm{Ad}(P)_x$ is (noncanonically) isomorphic to $G$, and the group multiplication on $G$ descends to give $\mathrm{Ad}(P)$ the structure of a smooth bundle of groups over $M$.

The group of smooth sections $\Gamma(\mathrm{Ad}(P))$ is naturally identified with the gauge group of $P$, and a section can be interpreted as a “gauge function” acting fiberwise (equivalently, as data defining a gauge transformation ).

Examples

1. Trivial bundle. If $P\cong M\times G$, then $\mathrm{Ad}(P)\cong M\times G$ as bundles of groups.
2. Abelian groups. If $G$ is abelian, the conjugation action is trivial, so $\mathrm{Ad}(P)\cong M\times G$ for every principal $G$-bundle $P$ (even if $P$ is nontrivial).
3. Frame bundle interpretation. If $P$ is the frame bundle of a rank-$n$ vector bundle $E$, then $\mathrm{Ad}(P)$ can be identified with the bundle of fiberwise linear automorphisms $\mathrm{Aut}(E)\to M$ (matrices transform by conjugation under change of frame).