Gauge transformation

Let $\pi:P\to M$ be a principal G-bundle . A gauge transformation is a special case of a principal bundle automorphism .

A gauge transformation of $P$ is a smooth map $\Phi:P\to P$ such that:

1. (Base-preserving) $\pi\circ\Phi=\pi$,
2. (Equivariance) $\Phi(p\cdot g)=\Phi(p)\cdot g$ for all $p\in P$ and $g\in G$,
3. (Invertibility) $\Phi$ is a diffeomorphism (equivalently, an automorphism in the principal-bundle sense).

Every gauge transformation can be written uniquely in the form

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

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

1. Trivial bundle. For $P=M\times G$, any smooth map $g:M\to G$ defines a gauge transformation by $\Phi(x,h)=(x,\,g(x)\,h).$
2. 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 $M$.
3. 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).