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  "body": "\n## Description\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] and $P \\to X$ a smooth [[principal $G$-bundle|8E37E74XVP]] over a smooth manifold $X$, with $G$ acting on $P$ on the right.\nThe *gauge group* $\\mathcal{G}(P)$ is the set of diffeomorphisms $u \\colon P \\to P$ such that $\\pi \\circ u = \\pi$ and $u(p g) = u(p) g$ for all $p \\in P$ and $g \\in G$, with composition as product.\n\nIt is a group: the identity map has both properties, and so does the composite of two such maps.\nThe inverse of such a map $u$ commutes with $\\pi$ and with the right action, because $u$ does.\n\n## Remarks\n\nEquivalent forms: a gauge transformation $u$ is the same as a $G$-equivariant map $\\sigma \\colon P \\to G$, with $\\sigma(p g) = g^{-1} \\sigma(p) g$, through $u(p) = p \\, \\sigma(p)$.\nSo $\\mathcal{G}(P)$ is the group of sections of the bundle $P \\times_G G$ associated with the conjugation action.\nIf $P = X \\times G$ is trivial, then $\\mathcal{G}(P)$ is the group $C^\\infty(X, G)$ of smooth maps with pointwise product.\n\nThe gauge group acts on the space of [[connections|NJ21PN5ZGJ]] on $P$ by pullback.\nTake a trivial bundle and a matrix group $G$, and write a connection as a $\\mathfrak{g}$-valued $1$-form $A$.\nThen a map $g \\colon X \\to G$ sends $A$ to $g^{-1} A g + g^{-1} dg$, and the [[curvature|ZAJFW1S078]] changes to $g^{-1} F_A g$.\n\n\n## Explanations\n\n### L1 human\n\nA principal bundle attaches a copy of the group $G$ to each point of $X$, with no preferred identity element in each copy.\nA gauge transformation changes the identification in each copy independently, point by point.\nPhysical quantities, such as the curvature of a connection up to conjugation, do not depend on these choices, so the gauge group is a symmetry group of gauge theory.\nFor a trivial bundle over $X$ it is simply the group of smooth maps from $X$ to $G$.\n",
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