Local connection 1-form

Let $\pi:P\to M$ be a principal G-bundle with Lie algebra $\mathfrak g$ , and let $\omega\in\Omega^1(P;\mathfrak g)$ be a connection 1-form representing a principal connection.

Given a local section $s:U\to P$ over an open set $U\subseteq M$ , the local connection 1-form (also called the gauge potential in that trivialization) is the $\mathfrak g$ -valued 1-form

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

This is the local representative of the global connection in the trivialization determined by $s$ (compare local trivialization from a local section).

Change of section (local gauge transformation)

If $s'(x)=s(x)\cdot g(x)$ for a smooth map $g:U\to G$ (a local gauge transformation), then the pulled-back forms satisfy the standard transformation law

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

as recorded in the local gauge transformation law.

On overlaps

If $\{s_i:U_i\to P\}$ is a system of local sections and $g_{ij}:U_i\cap U_j\to G$ are the associated transition functions defined by $s_j=s_i\,g_{ij}$ (see transition functions from local sections), then on $U_i\cap U_j$ the local forms obey

A_j = g_{ij}^{-1} A_i g_{ij} + g_{ij}^{-1}dg_{ij}.

The corresponding local curvature 2-form is

F:=dA + A\wedge A,

as in the local curvature formula, and it transforms by conjugation (see local curvature transformation law).

Examples

Trivial principal bundle. If $P\cong M\times G$ is trivial, choosing the global section $s(x)=(x,e)$ identifies a principal connection with a single global $\mathfrak g$ -valued 1-form $A$ on $M$ . The flat connection corresponds to $A=0$ , while a pure gauge connection has the form $A=g^{-1}dg$ .

Dirac monopole on the Hopf bundle. For the Hopf fibration as a principal $U(1)$ -bundle, the Dirac monopole connection is described by two local connection 1-forms $A_N$ and $A_S$ on the northern and southern charts of $S^2$ , related on the overlap by a $U(1)$ -valued gauge function.

Levi-Civita connection in a local frame. For a Riemannian manifold, the Levi-Civita connection can be viewed as a principal connection on the orthonormal frame bundle. Choosing a local orthonormal frame gives a matrix-valued local connection 1-form whose entries are the usual connection coefficients; its curvature is the matrix of curvature 2-forms in that frame (see curvature 2-form in a frame).