Gauge equivalence classes of connections form an orbit space

Let $\pi:P\to M$ be a principal G-bundle . Write

for the set (indeed an affine space) of all principal connections on $P$.

Let $\mathcal G(P)$ be the gauge group of $P$, i.e. the group of principal bundle automorphisms covering the identity on $M$.

By the proposition that the gauge group acts on connections by pullback , there is a well-defined left action

Because this is a genuine group action, it determines an equivalence relation on $\mathrm{Conn}(P)$:

Corollary (orbit space of connections modulo gauge)

The quotient set

is therefore a well-defined orbit space: its elements are precisely the gauge equivalence classes of connections, i.e. the orbits of the action $u\cdot \omega := u^*\omega$.

In local data, this action is the familiar gauge transformation law for connection 1-forms: on a chart, a local gauge transformation $g:U\to G$ sends a local connection form $A$ to $A^g$, as in the local gauge transformation law .

Examples

1. Trivial bundle: gauge action on Lie algebra valued 1-forms.
   If $P=M\times G$ is the trivial principal bundle , then specifying a connection is equivalent to specifying a $\mathfrak g$-valued 1-form $A\in\Omega^1(M;\mathfrak g)$. The gauge group identifies with $C^\infty(M,G)$, and the action is

   $$A \longmapsto g^{-1}Ag + g^{-1}dg,$$

   exactly the transformation described by gauge transform of a local connection form .

2. Abelian case: $U(1)$ connections differ by exact 1-forms on a trivial bundle.
   For $G=U(1)$ on a trivial bundle, the adjoint term $g^{-1}Ag$ is just $A$, so the gauge action becomes

   $$A \longmapsto A + g^{-1}dg.$$

   Writing $g=e^{i\theta}$ locally, one has $g^{-1}dg = i\,d\theta$. Thus gauge-equivalent connections differ by an exact 1-form, and the orbit space records precisely the ambiguity coming from adding exact forms.

3. Flat connections and holonomy data.
   For a flat principal connection , gauge equivalence preserves the holonomy up to conjugation. On $M=S^1$, gauge classes of flat connections on the trivial bundle correspond to conjugacy classes in $G$ via the holonomy element around the loop (constructed as in holonomy from parallel transport around a loop ).