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  "body": "\n## Description\n\nLet $G$ be a compact Lie group with Lie algebra $\\mathfrak{g}$, let $X$ be a compact manifold, and let $P \\to X$ be a principal $G$-bundle.\nThe space $\\mathcal{A}(P)$ of [[connections|NJ21PN5ZGJ]] on $P$ is an affine space over the vector space $\\Omega^1(X; \\mathrm{ad} P)$ of $1$-forms with values in the adjoint bundle.\nThe [[gauge group|707DEWCCFX]] $\\mathcal{G}(P)$ acts on $\\mathcal{A}(P)$, and the *space of connections modulo gauge* is the [[quotient space|0BR3FY5Q8M]]\n$$\\mathcal{B}(P) = \\mathcal{A}(P) / \\mathcal{G}(P).$$\n\n## Remarks\n\nThe simplest case is $G = U(1)$, $X = S^1 = \\R / 2\\pi \\Z$ and $P$ trivial.\nA 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 $A + i \\, d\\phi$.\nSo 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 $A \\mapsto \\exp(-i \\int_0^{2\\pi} a \\, dt) \\in U(1)$.\n\nIn general $\\mathcal{B}(P)$ is not a manifold, because the [[stabilisers|0RGGF8ZS6D]] of connections vary.\nThe stabiliser of $A$ always contains the centre of $G$, acting by constant gauge transformations.\nThe connection $A$ is called irreducible when its stabiliser is exactly the centre.\n\nAfter completing in suitable Sobolev norms, the irreducible connections modulo gauge form a Hilbert manifold, with local charts given by the Coulomb slice $\\set{A + a : d_A^* a = 0, \\ a \\text{ small}}$.\n\n## Explanations\n\n### L1 human\n\nTwo connections that differ only by a change of gauge describe the same physical field, so one wants to count them once.\nThe space of connections modulo gauge is the result: one point for each field, not for each description of it.\nOn a circle, a $U(1)$ connection modulo gauge is described by a single angle, its holonomy around the circle.\nModuli spaces of instantons and of other special connections sit inside this quotient space.\n",
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