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In physics the term local gauge transformation or gauge equivalence means essentially isomorphism or rather equivalence in an (infinity,1)-category: the configuration space of a physical theory is typically a groupoid (an orbifold) and a gauge transformation between configurations is a morphism in this groupoid.
More specifically, in physical theories called gauge theory – such as describing electromagnetism and Yang-Mills fields – the configuration space is a space of $G$-principal bundles (over spacetime $X$) with connection over a Lie group $G$ called the gauge group. A gauge transformation in the strict and original sense of the word is a morphism in the groupoid $G Bund_\nabla(X)$ of these bundles with connection.
In this context the connection $\nabla$ itself – usually thought of in terms of its local Lie-algebra valued 1-forms $(A_i \in \Omega^1(U_i \subset X, \mathfrak{g}))$ – is the gauge field . The equation
that characterizes morphisms $A_i \stackrel{g}{\to} A'_i$ in the groupoid of Lie-algebra valued forms is historically the hallmark of the “gauge principle” and is often what is meant specifically by gauge transformation .
But from there on the terminology generalizes to almost all physical theories. For one this is because the configuration space of many theories may be thought of as spaces of bundles with connections, notably also for gravity.
Moreover, many formal physical theories such as Chern-Simons theory, supergravity, etc. are described by higher categorical generalizations of bundles with connections: principal ∞-bundles with connection on a principal ∞-bundle such as the Chern-Simons circle 3-bundle. Their configuration spaces form not just groupoids but ∞-groupoids. The higher morphisms in these are called higher gauge transformations.
For instance the configuration space of the Kalb-Ramond field is the 2-groupoid of circle 2-bundles with connection over spacetime. An object in there is locally given by a 2-form $B_i \in \Omega^2(U_i \subset X)$. A 1-morphism in there is a first order gauge transformation $B_i \stackrel{\lambda}{\to} B'_i$ characterized by the equation $B_i' = B_i + d_{dR} \lambda_i$. A 2-morphism in there is a second order gauge transformation $\lambda_i \stackrel{g_i}{\Rightarrow} \lambda'_i$ characterized by $\lambda'_i = \lambda_i + d_{dR} g_i$.
The Lie algebroid of the groupoid of configurations and gauge transformations is known in physics in terms of its dual Chevalley-Eilenberg algebra called the BRST-complex. The degree 1 generators in this dg-algebra are hence the functions on infinitesimal gauge transformations. (A discussion of such infinitesimal transformations is here.) These graded functions on infintesimal gauge transformations are called ghost fields or ghosts for short, in the physics literature.
If the space of configurations is not just a groupoid in ordinary spaces but a groupoid in derived spaces such as derived smooth manifolds, then the CE-algebra of the corresponding derived ∞-Lie algebroid is called the BV-BRST complex.
See global gauge group.
The formulas for the local gauge transformations and higher gauge transformations for connections on bundles, connections on 2-bundles and connections on 3-bundles are discussed, respectively, at
The fully general description for connections on ∞-bundles is at
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Haag–Lopuszanski–Sohnius theorem on symmetries of an S-matrix
gauge field: models and components