Equivariant map associated to a section of an associated bundle

Let $\pi:P\to M$ be a principal G-bundle, written using the standard convention that $P$ carries a right $G$ -action (see the principal bundle action convention). Let $F$ be a smooth manifold with a left $G$ -action, as in the associated bundle fiber action convention.

E := P\times_G F \;\longrightarrow\; M,

Let $s:M\to E$ be a smooth section. Define a map $\Phi_s:P\to F$ by the rule:

given $p\in P$ with $\pi(p)=x$ , write the point $s(x)\in E_x$ as an equivalence class $[p,f]$ , and set $\Phi_s(p):=f$ .

This is well-defined and smooth, and it satisfies the equivariance condition

\Phi_s(p\cdot g) = g^{-1}\cdot \Phi_s(p) \qquad\text{for all }p\in P,\ g\in G,

i.e. $\Phi_s$ is an equivariant map with respect to the right action on $P$ and the given left action on $F$ .

Conversely, if $\Phi:P\to F$ is smooth and satisfies $\Phi(p\cdot g)=g^{-1}\cdot \Phi(p)$ , then

s_\Phi(x) := [p,\Phi(p)]\in E_x

is independent of the choice of $p\in P_x$ and defines a smooth section $s_\Phi:M\to E$ .

Together, these constructions give a natural bijection

\Gamma(E)\ \cong\ \{\,\Phi:P\to F\text{ smooth} \mid \Phi(p\cdot g)=g^{-1}\cdot \Phi(p)\,\}.

Examples

Trivial principal bundle reduces equivariance to an ordinary map. If $P=M\times G$ is trivial, then any section of $E=(M\times G)\times_G F \cong M\times F$ is the same as a smooth map $u:M\to F$ . The corresponding equivariant map is $\Phi_u(x,g) = g^{-1}\cdot u(x).$

Associated line bundle and equivariant complex-valued functions. Take $G=U(1)$ acting on $F=\mathbb C$ by scalar multiplication, and let $E=P\times_{U(1)}\mathbb C$ be the associated complex line bundle (see associated vector bundles for the general vector-bundle case). A section of $E$ corresponds to a smooth function $\Phi:P\to\mathbb C$ satisfying $\Phi(p\cdot e^{i\theta}) = e^{-i\theta}\,\Phi(p).$ This is the standard “equivariant function” description of sections of a line bundle.

Adjoint bundle: sections as conjugation-equivariant maps. Let $F=G$ with the conjugation action, so $E=P\times_G G$ is the adjoint bundle. A section then corresponds to a map $\Phi:P\to G$ satisfying $\Phi(p\cdot g)=g^{-1}\Phi(p)g.$ Interpreting such sections as bundle automorphisms recovers the usual identification of the gauge group with sections of the adjoint bundle (compare gauge transformations).