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 \coloneqq s^*\omega \in \Omega^1(U;\mathfrak{g}).

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

s'(x)\coloneqq s(x)\cdot g(x).

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

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

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

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 $A'=A+i\,df$ .

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.

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.