Space of connections modulo gauge

Remarks

The simplest case is $G = U(1)$, $X = S^1 = \R / 2\pi \Z$ and $P$ trivial. A connection is $A = i a(t) \, dt$ with $a$ a smooth real function, and a gauge transformation $g = e^{i \phi}$, with $\phi \colon \R \to \R$ smooth and $\phi(t + 2\pi) - \phi(t) \in 2\pi \Z$, sends $A$ to $A + i \, d\phi$. So the class of $A$ is determined by $\int_0^{2\pi} a(t) \, dt$ modulo $2\pi$, and $\mathcal{B}(P)$ is a circle; the map is the holonomy $A \mapsto \exp(-i \int_0^{2\pi} a \, dt) \in U(1)$.

In general $\mathcal{B}(P)$ is not a manifold, because the stabilisers of connections vary. The stabiliser of $A$ always contains the centre of $G$, acting by constant gauge transformations. The connection $A$ is called irreducible when its stabiliser is exactly the centre.

After completing in suitable Sobolev norms, the irreducible connections modulo gauge form a Hilbert manifold, with local charts given by the Coulomb slice $\set{A + a : d_A^* a = 0, \ a \text{ small}}$.