Local gauge transformation

Let $\pi:P\to M$ be a principal $G$-bundle, and let $U\subset M$ be an open set (typically from an open cover ) on which we choose an equivariant local trivialization $\psi:\pi^{-1}(U)\to U\times G$.

A local gauge transformation on $U$ is a smooth map

viewed as acting on the local product $U\times G$ by

Equivalently, it acts on the associated local section $s(x)=\psi^{-1}(x,e)$ by

If $\Phi$ is a global gauge transformation of $P$, then in any chosen trivialization over $U$ it is represented by a unique local gauge transformation $g:U\to G$ via

When a principal connection is present, local gauge transformations also act on the resulting local connection 1-forms by the usual formula

where $dg$ is computed using the exterior derivative .

Examples

1. Electromagnetism. For $G=U(1)$, a local gauge transformation is $g=e^{i\chi}$ with $\chi:U\to\mathbb{R}$, and the local potential transforms by $A\mapsto A+d\chi$.
2. Change of local frame. For a rank-$n$ vector bundle, a change of local frame on $U$ is given by $g:U\to GL(n)$; it is exactly a local gauge transformation for the associated frame bundle.
3. Transition functions as local gauge data. On an overlap $U_i\cap U_j$, the transition function $g_{ij}:U_i\cap U_j\to G$ describes how local sections differ, and can be read as the local gauge transformation relating two trivializations.