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 .

Form the associated bundle

as in the construction of associated bundles .

Construction: section gives an equivariant map

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

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

Converse: equivariant map gives a section

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

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

Examples

1. 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

   $$\Phi_u(x,g) = g^{-1}\cdot u(x).$$

2. 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.

3. 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 ).