Gauge group action on connections by pullback

Let $G$ be a Lie group and let $\pi:P\to M$ be a principal G-bundle. Write $\mathrm{Conn}(P)$ for the set of principal connections on $P$ , and let $\mathrm{Gauge}(P)$ denote the gauge group, i.e. the group of $G$ -equivariant diffeomorphisms $\Phi:P\to P$ covering the identity on $M$ (so $\pi\circ\Phi=\pi$ and $\Phi(p\cdot g)=\Phi(p)\cdot g$ ).

For every $\Phi\in \mathrm{Gauge}(P)$ and every principal connection $\omega\in \mathrm{Conn}(P)$ , the pullback

\omega\cdot\Phi \;:=\; \Phi^*\omega

is again a principal connection on $P$ . This defines a right group action:

(\omega\cdot\Phi_1)\cdot\Phi_2 =\omega\cdot(\Phi_1\Phi_2), \qquad \omega\cdot\mathrm{id}=\omega.

Equivalently, if $\omega$ is viewed as a $G$ -equivariant horizontal distribution $H=\ker\omega\subset TP$ , then $\Phi$ sends horizontals to horizontals:

H^{\Phi^*\omega}_p \;=\; (d\Phi_p)^{-1}\bigl(H_{\Phi(p)}^{\omega}\bigr),

so the gauge group acts on the set of horizontal distributions defining connections.

Examples

Trivial bundle $P=M\times G$ . With the right principal action $(x,h)\cdot k=(x,hk)$ , a gauge transformation determined by $g:M\to G$ is $\Phi_g(x,h)=(x,g(x)h)$ . In the section $s(x)=(x,e)$ , pullback sends the local connection form $A$ to $A \longmapsto A^g := \mathrm{Ad}_{g^{-1}}A + g^{-1}dg.$
Abelian structure group. If $G$ is abelian (e.g. $U(1)$ ), then $\mathrm{Ad}_{g^{-1}}$ is trivial and the transformation law reduces to $A \longmapsto A + g^{-1}dg,$ i.e. gauge transformations act by translation by the closed Lie-algebra-valued form $g^{-1}dg$ , locally exact. For $G=U(1)$ , its periods lie in $2\pi i\mathbb Z$ .
Frame bundle viewpoint. If $P$ is a frame bundle of a vector bundle, a gauge transformation is a change of frame covering $\mathrm{id}_M$ . Pulling back the connection corresponds to the usual transformation rule for connection matrices under a change of frame.