Gauge group

Remarks

Equivalent forms: a gauge transformation $u$ is the same as a $G$-equivariant map $\sigma \colon P \to G$, with $\sigma(p g) = g^{-1} \sigma(p) g$, through $u(p) = p \, \sigma(p)$. So $\mathcal{G}(P)$ is the group of sections of the bundle $P \times_G G$ associated with the conjugation action. If $P = X \times G$ is trivial, then $\mathcal{G}(P)$ is the group $C^\infty(X, G)$ of smooth maps with pointwise product.

The gauge group acts on the space of connections on $P$ by pullback. Take a trivial bundle and a matrix group $G$, and write a connection as a $\mathfrak{g}$-valued $1$-form $A$. Then a map $g \colon X \to G$ sends $A$ to $g^{-1} A g + g^{-1} dg$, and the curvature changes to $g^{-1} F_A g$.