Gauge transform of a local connection form

Let $\pi:P\to M$ be a principal $G$-bundle with connection form $\omega\in \Omega^1(P;\mathfrak{g})$ as in a connection 1-form on a principal bundle .

Fix an open set $U\subset M$ and a local section $s:U\to P$. The local connection form on $U$ is

A gauge transformation on $U$ is a smooth map $g:U\to G$. It determines a new local section $s':U\to P$ by

The corresponding local connection form $A'=(s')^*\omega$ is related to $A$ by the gauge transformation rule

where $g^{-1}dg\in \Omega^1(U;\mathfrak{g})$ is the pullback of the left Maurer–Cartan form on $G$ along $g$, and $\mathrm{Ad}(g^{-1})$ acts pointwise on $\mathfrak{g}$.

This formula is the local manifestation of the global equivariance condition (R_h^*\omega=\mathrm{Ad}(h^{-1})\omega.

Examples

1. Abelian case $U(1)$. For $G=U(1)$, $\mathrm{Ad}$ is trivial, so $A'=A+g^{-1}dg$. Writing $g=e^{if}$ locally gives $g^{-1}dg=i\,df$, hence $A'=A+i\,df$.

2. Matrix group $GL(n)$. For $G=GL(n)\subset \mathrm{End}(\mathbb{R}^n)$, the adjoint action is conjugation, and the rule becomes

   $$A' = g^{-1}Ag + g^{-1}dg,$$

   where $A$ is a matrix-valued 1-form.

3. Constant gauge transformations. If $g$ is constant on $U$, then $dg=0$ and $A'=\mathrm{Ad}(g^{-1})A$; i.e. only the $\mathfrak{g}$-basis changes.