Section of Ad(P)

Let $\mathrm{Ad}(P)\to M$ be the adjoint bundle of a principal $G$-bundle $P$.

A section of $\mathrm{Ad}(P)$ is a smooth map

such that $\pi_{\mathrm{Ad}}\circ s=\mathrm{id}_M$, where $\pi_{\mathrm{Ad}}:\mathrm{Ad}(P)\to M$ is the bundle projection.

Equivalently, choose an open cover $\{U_i\}$ and local trivializations of $P$ with transition functions $g_{ij}:U_i\cap U_j\to G$. Then a section $s$ is represented by smooth maps $a_i:U_i\to G$ such that on overlaps

This is the “gauge function” gluing law: local representatives differ by conjugation with the bundle cocycle.

Under pointwise multiplication in the fibers, the set of sections $\Gamma(\mathrm{Ad}(P))$ is a group, canonically isomorphic to the gauge group of $P$.

Examples

1. Trivial or trivialized case. If $\mathrm{Ad}(P)\cong M\times G$, then sections are exactly smooth maps $a:M\to G$.
2. Abelian groups. If $G$ is abelian, conjugation is trivial, so every section is again just a smooth map $M\to G$, regardless of whether $P$ is trivial.
3. Central elements. If $z\in Z(G)$, then the constant choice “$z$ in every fiber” defines a global section of $\mathrm{Ad}(P)$; it corresponds to the gauge transformation $p\mapsto p\cdot z$.