Local curvature 2-form - Imported Knowlpedia Content

Let $\pi:P\to M$ be a principal $G$ -bundle with connection form $\omega$ and curvature $\Omega$ as in the curvature 2-form of a principal connection.

On an open set $U\subset M$ with local section $s:U\to P$ , define the local connection 1-form $A=s^*\omega\in \Omega^1(U;\mathfrak{g})$ . The local curvature 2-form on $U$ is

F \;\coloneqq\; s^*\Omega \in \Omega^2(U;\mathfrak{g}).

Then $F$ is given by the structure equation

F \;=\; dA \;+\; \tfrac12[A\wedge A],

where $d$ is the exterior derivative and the bracket uses the Lie bracket on $\mathfrak{g}$ .

If $s' = s\cdot g$ is another local section related by a gauge function $g:U\to G$ , then the associated local curvature forms satisfy the gauge transformation rule

F' = \mathrm{Ad}(g^{-1})F,

compatible with the gauge transformation of the local connection form.

Examples

Abelian case. If $G$ is abelian (e.g. $U(1)$ ), then $[A\wedge A]=0$ , hence $F=dA$ , and the gauge rule reduces to $F'=F$ .

Matrix-valued form convention. For $G\subset GL(n)$ , one often writes $F=dA + A\wedge A$ . This matches the formula above because for matrix-valued 1-forms, $[A\wedge A]=A\wedge A - (A\wedge A)^{\mathrm{op}} = 2A\wedge A$ .

Trivial (flat) local connection. If $A=0$ on $U$ , then $F=0$ on $U$ , so the connection is locally flat there.