Gauge transformation

Let $\pi:P\to M$ be a principal $G$ -bundle. A gauge transformation of $P$ is a principal bundle automorphism covering $\operatorname{id}_M$ : a smooth diffeomorphism $\Phi:P\to P$ satisfying

\pi\circ\Phi=\pi, \qquad \Phi(p\cdot g)=\Phi(p)\cdot g

for every $p\in P$ and $g\in G$ .

Equivariant maps and adjoint-bundle sections

Every gauge transformation can be written uniquely in the form

\Phi(p)=p\cdot u(p)

for a smooth map $u:P\to G$ satisfying the conjugation-equivariance condition

u(p\cdot g)=g^{-1}u(p)g.

Such maps $u$ are the same data as smooth sections of the adjoint bundle $\mathrm{Ad}(P)$ (often described explicitly in sections of Ad(P)).

Examples

Trivial bundle. For $P=M\times G$ , any smooth map $g:M\to G$ defines a gauge transformation by $\Phi(x,h)=(x,\,g(x)\,h).$
Abelian structure group. If $G$ is abelian (e.g. $U(1)$ ), the conjugation condition on $u$ becomes $u(p\cdot g)=u(p)$ , so gauge transformations correspond to smooth $G$ -valued functions on $M$ .
Frames of a vector bundle. For the frame bundle of a rank- $n$ vector bundle $E\to M$ , gauge transformations correspond to bundle automorphisms of $E$ covering $\mathrm{id}_M$ (locally given by $GL(n)$ -valued functions acting on frames).