Section of Ad(P) - Knowlpedia

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

s:M\to \mathrm{Ad}(P)

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

Gauge-group interpretation

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

Equivalent characterizations

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

a_j(x)=g_{ij}(x)^{-1}\,a_i(x)\,g_{ij}(x).

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

Examples

Trivial or trivialized case. If $\mathrm{Ad}(P)\cong M\times G$ , then sections are exactly smooth maps $a:M\to G$ .
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.
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 $p\mapsto p\cdot z$ .