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\begin{document}

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\section*{gauge transformation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{local_gauge_transformations}{Local gauge transformations}\dotfill \pageref*{local_gauge_transformations} \linebreak
\noindent\hyperlink{global_gauge_transformations}{Global gauge transformations}\dotfill \pageref*{global_gauge_transformations} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{for_local_connection_forms}{For local connection forms}\dotfill \pageref*{for_local_connection_forms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_electromagnetism}{In electromagnetism}\dotfill \pageref*{in_electromagnetism} \linebreak
\noindent\hyperlink{in_yangmills_theory}{In Yang-Mills theory}\dotfill \pageref*{in_yangmills_theory} \linebreak
\noindent\hyperlink{in_gravity_and_topological_field_theories}{In gravity and topological field theories}\dotfill \pageref*{in_gravity_and_topological_field_theories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{local_gauge_transformations}{}\subsubsection*{{Local gauge transformations}}\label{local_gauge_transformations}

In [[physics]] the term local \textbf{gauge transformation} or \textbf{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 \emph{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 field]]s -- the configuration space is a space of $G$-[[principal bundle]]s (over [[spacetime]] $X$) with [[connection on a bundle|connection]] over a [[Lie group]] $G$ called the \textbf{[[gauge group]]}. A \emph{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-form]]s $(A_i \in \Omega^1(U_i \subset X, \mathfrak{g}))$ -- is the \emph{[[gauge field]]} . The equation

\begin{displaymath}
A'_i = h_i^{-1} A_i h_i + h^{-1} d h_i \;\;\;\; ; \;\; h_i \in C^\infty(U_i,G)
\end{displaymath}
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 \emph{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 category theory|higher categorical]] generalizations of bundles with connections: [[principal ∞-bundle]]s 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 \textbf{[[higher gauge transformations]]}.

For instance the configuration space of the [[Kalb-Ramond field]] is the [[2-groupoid]] of [[circle n-bundle with connection|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 \textbf{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 \textbf{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 \emph{[[infinitesimal object|infinitesimal]]} gauge transformations. (A discussion of such infinitesimal transformations is .) These graded functions on infintesimal gauge transformations are called \textbf{[[ghost fields]]} or \textbf{ghosts} for short, in the physics literature.

If the space of configurations is not just a groupoid in ordinary spaces but a groupoid in \emph{derived} spaces such as [[derived smooth manifold]]s, then the CE-algebra of the corresponding [[derived ∞-Lie algebroid]] is called the [[BV-BRST complex]].

\hypertarget{global_gauge_transformations}{}\subsubsection*{{Global gauge transformations}}\label{global_gauge_transformations}

See [[global gauge group]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{for_local_connection_forms}{}\subsubsection*{{For local connection forms}}\label{for_local_connection_forms}

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

\begin{itemize}%
\item [[groupoid of Lie-algebra valued 1-forms]]


\item [[2-groupoid of Lie 2-algebra valued forms]]


\item [[3-groupoid of Lie 3-algebra valued forms]].



\end{itemize}
The fully general description for [[connections on ∞-bundles]] is at

\begin{itemize}%
\item [[∞-groupoid of ∞-Lie-algebra valued forms]].

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{in_electromagnetism}{}\subsubsection*{{In electromagnetism}}\label{in_electromagnetism}

(\ldots{})

\hypertarget{in_yangmills_theory}{}\subsubsection*{{In Yang-Mills theory}}\label{in_yangmills_theory}

(\ldots{})

\hypertarget{in_gravity_and_topological_field_theories}{}\subsubsection*{{In gravity and topological field theories}}\label{in_gravity_and_topological_field_theories}

\begin{itemize}%
\item [[general covariance]]

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[gauge]]


\item [[gauge parameter]]


\item [[gauge group]], [[gauge fixing]]


\item [[large gauge transformation]]


\item [[higher gauge transformation]]


\item [[gauge invariance]]


\item [[group averaging]]


\item [[Haag?Lopuszanski?Sohnius theorem]] on symmetries of an [[S-matrix]]


\item [[spontaneously broken symmetry]]


\item [[principle of equivalence]]


\item [[general covariance]]



\end{itemize}
[[!include gauge field - table]]

[[!redirects gauge transformations]]

[[!redirects gauge symmetry]] [[!redirects gauge symmetries]]

[[!redirects global gauge symmetry]] [[!redirects local gauge symmetry]] [[!redirects global gauge symmetries]] [[!redirects local gauge symmetries]]

[[!redirects global symmetry]] [[!redirects local symmetry]] [[!redirects global symmetries]] [[!redirects local symmetries]]

[[!redirects infinitesimal gauge transformation]] [[!redirects infinitesimal gauge transformations]]

[[!redirects infinitesimal gauge symmetry]] [[!redirects infinitesimal gauge symmetries]]

[[!redirects gauge equivalence]] [[!redirects gauge equivalences]]



\end{document}