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\section*{higher gauge field}

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

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

[[!include physicscontents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An ordinary [[gauge field]] (such as the [[electromagnetic field]] or the fields that induce the [[nuclear force]]) is a [[field (physics)|field (in the sense of physics)]] which is locally represented by a [[differential 1-form]] (the ``[[gauge potential]]'') and whose [[field strength]] is locally a [[differential 2-form]]. For instance, in the case of the [[electromagnetic field]] this [[differential 2-form]] is the \emph{[[Faraday tensor]]}.

Roughly speaking, a \emph{higher gauge field} is similarly a field which is locally represented by [[differential forms]] of higher degree.

An explanation as to why an ordinary [[gauge field]] has a [[gauge potential]] given locally by a [[differential 1-form]] $A$ is that the [[trajectory]] of a [[charged particle]] is a 1-dimensional [[curve]] in [[spacetime]] $X$, its \emph{[[worldline]]}, hence a [[smooth function]] $\gamma \colon \Sigma_1 \to X$, and the canonical way to produce an [[action functional]] on the [[mapping space]] of such curves is the [[integration of differential forms|integration of 1-forms]] over curves:

\begin{displaymath}
\exp(\tfrac{i}{\hbar} S_{gauge})
  \;\colon\;
  \gamma
  \mapsto
  P \exp\left(
    \int_{\Sigma_1} \gamma^\ast A
  \right)
  \,.
\end{displaymath}
This is the \emph{[[parallel transport]]} map or the \emph{[[holonomy]]} map, if $\Sigma_1$ is a [[closed manifold|closed manifold]]. The contribution to the [[Euler-Lagrange equation]] of the particle obtained from the [[variational calculus|variation]] of this [[action functional]] is the \emph{[[Lorentz force]]} which is exerted by the [[background field|background]] [[gauge field]] on the particle.

When one generalizes in this picture from 0-dimensional [[particles]] with 1-dimensional [[worldlines]] to $p$-dimensional particles (often called ``[[p-branes]]'') with $(p+1)$-dimensional [[worldvolumes]] $\gamma_{p+1} \colon \Sigma_{p+1} \to X$, then one needs, locally, a [[differential n-form|differential (p+1)-form]] $A_{p+1}$ on [[spacetime]] $X$

\begin{displaymath}
\exp(\tfrac{i}{\hbar} S_{higher\,gauge})
  \;\colon\;
  \gamma
  \mapsto
  P \exp\left(
    \int_{\Sigma_{p+1}} \gamma_{p+1}^\ast A_{p+1}
  \right)
  \,.
\end{displaymath}
The [[field strength]] or \emph{[[flux]]} of such a higher gauge field is, accordingly, locally the $(p+2)$-form $F_{p+2}$.

The archetypical example of such a higher gauge field is the (hypothetical) \emph{[[Kalb-Ramond field]]} or \emph{[[B-field]]} (a precursor of the [[axion]] field under [[KK-compactification]]) to which the charged 1-brane, the ``[[string]]'', couples. This is locally a [[differential 2-form]] $B_2$, and the gauge-coupling term in the [[action functional]] for the string is accordingly, locally, of the form

\begin{displaymath}
\exp(\tfrac{i}{\hbar} S_{stringy\,gauge})
  \;\colon\;
  \gamma
  \mapsto
  P \exp\left(
    \int_{\Sigma_{2}} \gamma_2^\ast B_2
  \right)
  \,.
\end{displaymath}
This continues: next one may consider ``2-branes'', i.e. \emph{[[membranes]]}, and these will couple to a 3-form gauge field. For instance, the membrane which gives the name to \emph{[[M-theory]]} (the [[M2-brane]]) couples to a 3-form field called the \emph{[[supergravity C-field]]}.

But there is an important further aspect to higher gauge fields which makes this simple picture of higher degree differential forms drastically more rich:

Where an ordinary [[gauge field]] has [[gauge transformations]] $A_1 \mapsto A'_1$ given locally by smooth functions (0-forms) $\lambda_0$ via the [[de Rham differential]] $d_{dR}$

\begin{displaymath}
A'_1 = A_1 + d_{dR} \lambda_0
\end{displaymath}
so a higher gauge field has [[higher gauge transformations]] given locally by $p$-forms $\lambda_p$:

\begin{displaymath}
A'_{p+1} = A_{p+1} + d_{dR} \lambda_{p}
  \,.
\end{displaymath}
But for $p \geq 0$ then a crucial new effect appears: these gauge transformations, being higher differential forms themselves, have ``[[higher gauge transformation|gauge-of-gauge transformations]]'' between them, given by lower degree forms.

This phenomenon implies that higher gauge fields have a rich \emph{global} (``topological'') structure, witnessed by the higher analog of their [[instanton sectors]]. Namely, while a higher gauge field to which a [[p-brane]] may couple is \emph{locally} given by a $(p+1)$-form $A_{p+1}$, as one moves across [[coordinate charts]] this form gauge transforms by a $p$-form, which then itself, as one passes along two charts, transforms by a $(p-1)$-form, and so on.

The global structure for higher gauge fields obtained by carrying out this globalization via [[higher gauge transformations]] is the higher analog of that of a [[fiber bundle]] with [[connection on a bundle]] in [[higher differential geometry]]. This is sometimes known as a \emph{[[gerbe]]} or, more generally, a \emph{[[principal infinity-bundle]]}.

In fact the situation that there is just one [[gauge potential]] of degree $(p+1)$ with [[field strength]] of degree $(p+2)$ is just the simplest case, the ``ordinary'' case. More abstractly one says that such higher gauge fields are [[cocycles]] in \emph{[[ordinary differential cohomology]]}.

More generally it may happen in [[higher gauge theory]] that the gauge potential is a [[formal linear combination]] of differential forms in various degrees.

The canonical example of this phenomenon is the [[RR-field]] in [[string theory]]. This has, locally, a gauge potential which is a differential form in every even degree, or every odd degree. If one is careful about the [[higher gauge transformations]] in this situation to find the correct global structure (the ``[[instanton sector]]'') of the higher gauge field, then one finds that this now is a [[cocycle]] in a [[differential cohomology|differential]] \emph{[[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]]}, namely, in what is called [[differential K-theory|differential]] [[topological K-theory]]. This may be understood as a higher and generalized form of the famous [[Dirac charge quantization]] condition for the [[electromagnetic field]], see \hyperlink{Freed00}{Freed 00}. A lot of the fine detail of the [[anomaly cancellation]] in [[type II string theory]] depends on being careful about the global nature of this [[K-theory|K-theoretic]] higher gauge [[RR-field]] (\hyperlink{DistlerFreedMoore09}{Distler-Freed-Moore 09})

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

\begin{itemize}%
\item [[Kalb-Ramond field]]


\item [[RR-field]]


\item [[supergravity C-field]]


\item [[string field theory|string field]]


\item [[AKSZ sigma-model]]



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Wilson surface]]


\item [[local prequantum field theory]]


\item [[higher category theory and physics]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Introduction and exposition includes

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{An invitation to higher gauge theory}, General Relativity and Gravitation 43 (2011), 2335-2392 (\href{https://arxiv.org/abs/1003.4485}{arXiv:1003.4485})


\item PhysicsForums-Insights \emph{\href{https://www.physicsforums.com/insights/higher-category-theory-physics/}{Why higher category theory in physics?}}


\item \emph{[[geometry of physics]]}


\item \emph{[[twisted smooth cohomology in string theory]]}


\item \emph{\href{https://www.physicsforums.com/insights/examples-prequantum-field-theories-ii-higher-gauge-fields/}{Examples of prequantum field theories II: Higher gauge fields}}



\end{itemize}
For technical introduction to the [[RR-field]] as a higher gauge field see

\begin{itemize}%
\item [[Daniel Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]}, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129--194 (\href{http://arxiv.org/abs/hep-th/0011220}{arxiv:hep-th/0011220})


\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides})



\end{itemize}
Discussion of higher gauge theory for [[Green-Schwarz mechanisms]]:

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{Twisted differential String and Fivebrane structures}, Commun. Math. Phys. 315 (2012), 169-213 (\href{https://arxiv.org/abs/0910.4001}{arXiv:0910.4001})


\item [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], \emph{Exploring 2-Group Global Symmetries} (\href{https://arxiv.org/abs/1802.04790}{arXiv:1802.04790})


\item [[Francesco Benini]], [[Clay Cordova]], [[Po-Shen Hsin]], \emph{On 2-Group Global Symmetries and Their Anomalies}, (\href{https://arxiv.org/abs/1803.09336}{arXiv:1803.09336})



\end{itemize}
For foundations of higher [[prequantum field theory]] see

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]}

\end{itemize}
For foundations of higher gauge theory formalized in [[homotopy type theory]] see

\begin{itemize}%
\item [[Urs Schreiber]], [[Mike Shulman]], \emph{[[schreiber:Quantum gauge field theory in Cohesive homotopy type theory]]}

\end{itemize}
For higher gauge theory in [[condensed matter physics]] see

\begin{itemize}%
\item Chenchang Zhu, Tian Lan, Xiao-Gang Wen, \emph{Topological non-linear σ-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems}, (\href{https://arxiv.org/abs/1808.09394}{arXiv:1808.09394})


\item J.P. Ang, Abhishodh Prakash, \emph{Higher categorical groups and the classification of topological defects and textures}, (\href{https://arxiv.org/abs/1810.12965}{arXiv:1810.12965})



\end{itemize}
[[!redirects higher gauge fields]]

[[!redirects higher gauge theory]] [[!redirects higher gauge theories]]

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



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