Local gauge transformation law for a connection

Let $\pi:P\to M$ be a principal G-bundle with a principal connection $\omega$ . Fix an open set $U\subset M$ and a local section $s:U\to P$ . The associated local connection form is

A:=s^*\omega\in\Omega^1(U;\mathfrak g).

Let $g:U\to G$ be a smooth map and define a new local section by $s':=s\cdot g$ (right action of $G$ on $P$ ).

Lemma (Local gauge transformation law). The local connection form $A':=(s')^*\omega$ satisfies

A'=\mathrm{Ad}(g^{-1})A + g^{-1}dg.

In a matrix realization of $G$ , this is $A' = g^{-1}Ag + g^{-1}dg$ . The $1$ -form $g^{-1}dg$ is the pullback of the left Maurer–Cartan form and satisfies the Maurer–Cartan equation.

Examples

Abelian case (U(1)). Since Ad is trivial, the formula becomes $A'=A+g^{-1}dg$ , matching the usual shift of a $U(1)$ potential by an exact form locally.
Constant change of section. If $g$ is constant, then $dg=0$ and $A'=\mathrm{Ad}(g^{-1})A$ is just pointwise conjugation.
Producing a pure gauge potential. Starting from the trivial potential $A=0$ on a trivial bundle, the transformation gives $A'=g^{-1}dg$ , a pure gauge connection.