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\begin{document}

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\section*{Wilson loop}

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

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

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

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_1d_chernsimons_theory}{Relation to 1d Chern-Simons theory}\dotfill \pageref*{relation_to_1d_chernsimons_theory} \linebreak
\noindent\hyperlink{relation_to_defects}{Relation to defects}\dotfill \pageref*{relation_to_defects} \linebreak
\noindent\hyperlink{duality_with_t_hooft_operators_under_sduality_and_geometric_langlands}{Duality with `t Hooft operators under S-duality and geometric Langlands}\dotfill \pageref*{duality_with_t_hooft_operators_under_sduality_and_geometric_langlands} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_chernsimons_theory}{In Chern-Simons theory}\dotfill \pageref*{in_chernsimons_theory} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_chernsimons_theory_2}{In Chern-Simons theory}\dotfill \pageref*{in_chernsimons_theory_2} \linebreak
\noindent\hyperlink{in_qft_with_defects}{In QFT with defects}\dotfill \pageref*{in_qft_with_defects} \linebreak
\noindent\hyperlink{as_partition_functions_of_1d_chernsimons_theory}{As partition functions of 1d Chern-Simons theory}\dotfill \pageref*{as_partition_functions_of_1d_chernsimons_theory} \linebreak
\noindent\hyperlink{in_sduality_and_relation_to_t_hoofts_operators}{In S-duality and relation to t Hoofts operators}\dotfill \pageref*{in_sduality_and_relation_to_t_hoofts_operators} \linebreak


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

A \emph{Wilson loop} or \emph{Wilson line} is an [[observable]] in (both classical and quantum) [[gauge theory]] obtained from the [[holonomy]] of the [[gauge field|gauge]] [[connection on a bundle|connection]].

Hence if the gauge connection is given by a globally defined 1-form $A$, then the \textbf{Wilson loop} along a closed [[loop]] $C$ is the trace of the [[path-ordered exponential]]

\begin{displaymath}
W_C = Tr(\mathcal{P}exp(i\oint_C A_\mu d x^\mu))
\end{displaymath}
where $\mathcal{P}$ is the ``path-ordering operator'' and $A_\mu$ are the components of the connection.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_1d_chernsimons_theory}{}\subsubsection*{{Relation to 1d Chern-Simons theory}}\label{relation_to_1d_chernsimons_theory}

For $G$ a suitable [[Lie group]] (compact, semi-simple and simply connected) the Wilson loops of $G$-[[principal connections]] are equivalently the [[partition functions]] of a [[1-dimensional Chern-Simons theory]].

This appears famously in the formulation of [[Chern-Simons theory]] \href{Chern-Simons+theory#WithWilsonLineObservables}{with Wilson lines}. More detailes are at \emph{[[orbit method]]}.

\hypertarget{relation_to_defects}{}\subsubsection*{{Relation to defects}}\label{relation_to_defects}

Wilson loop insertions may be thought of or at least related to \emph{defects} in the sense of \emph{[[QFT with defects]]}.

\hypertarget{duality_with_t_hooft_operators_under_sduality_and_geometric_langlands}{}\subsubsection*{{Duality with `t Hooft operators under S-duality and geometric Langlands}}\label{duality_with_t_hooft_operators_under_sduality_and_geometric_langlands}

[[S-duality]] of 4d [[super Yang-Mills theory]] may exchange Wilson loop operators with [[`t Hooft operators]], in an incarnation of the [[geometric Langlands correspondence]] (\hyperlink{KapustinWitten06}{Kapustin-Witten 06})

[[!include geometric Langlands QFT -- table]]

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

\hypertarget{in_chernsimons_theory}{}\subsubsection*{{In Chern-Simons theory}}\label{in_chernsimons_theory}

In $SU(2)$-[[Chern-Simons theory]] the Wilson line observables compute the [[Jones polynomial]] of the given curve. See there for more details.

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

\begin{itemize}%
\item [[`t Hooft operator]]


\item [[Wilson surface]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item Kenneth Wilson, \emph{Confinement of quarks}, Physical Review \textbf{D 10} (8): 2445. \href{http://dx.doi.org/10.1103/PhysRevD.10.2445}{doi} (1974)


\item Yuri Makeenko, \emph{Methods of contemporary gauge theory}, Cambridge Monographs on Math. Physics, \href{http://books.google.com/books?id=9W-W2w75ulAC}{gBooks}


\item wikipedia \href{http://en.wikipedia.org/wiki/Wilson_loop}{Wilson loop}


\item R. Giles, \emph{Reconstruction of gauge potentials from Wilson loops}, Physical Review \textbf{D 24} (8): 2160, \href{http://dx.doi.org/10.1103/PhysRevD.24.2160}{doi}


\item A. Andrasi, J. C. Taylor, \emph{Renormalization of Wilson operators in Minkowski space}, Nucl. Phys. B516 (1998) 417, \href{http://arxiv.org/abs/hep-th/9601122}{hep-th/9601122}


\item Amit Sever, Pedro Vieira, Luis F. Alday, Juan Maldacena, [[Davide Gaiotto]], \emph{An Operator product expansion for polygonal null Wilson loops}, \href{http://arxiv.org/abs/1006.2788}{arxiv.org/abs/1006.2788}



\end{itemize}
\hypertarget{in_chernsimons_theory_2}{}\subsubsection*{{In Chern-Simons theory}}\label{in_chernsimons_theory_2}

The [[Poisson bracket]] of Wilson line observables in [[3d Chern-Simons theory]] was obtained in

\begin{itemize}%
\item W. Goldman, \emph{Invariant functions on Lie groups and Hamiltonian flow of surface group representations}, Inventiones Math., 85 (1986), 263--302.

\end{itemize}
For more see

\begin{itemize}%
\item [[Razvan Gelca]], [[Alejandro Uribe]], section 4 of \emph{Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$} (\href{http://arxiv.org/abs/1007.2010}{arXiv:1007.2010})

\end{itemize}
\hypertarget{in_qft_with_defects}{}\subsubsection*{{In QFT with defects}}\label{in_qft_with_defects}

Relation to [[QFT with defects]] is discussed in

slide 17 of

\begin{itemize}%
\item [[Constantin Bachas]], \emph{Conformal defects in/and String theory} (\href{http://www.ggi.fi.infn.it/talks/talk106.pdf}{pdf})

\end{itemize}
slide 5 of

\begin{itemize}%
\item [[Constantin Bachas]], \emph{Conformal defects in gauged WZW models} (\href{http://hep.physics.uoc.gr/conf09/TALKS/Bachas.pdf}{pdf})

\end{itemize}
\hypertarget{as_partition_functions_of_1d_chernsimons_theory}{}\subsubsection*{{As partition functions of 1d Chern-Simons theory}}\label{as_partition_functions_of_1d_chernsimons_theory}

Expression of Wilson loops as [[partition functions]] of [[1-dimensional Chern-Simons theories]] by the [[orbit method]] (as used notably in [[Chern-Simons theory]]) is in section 4 of

\begin{itemize}%
\item [[Chris Beasley]], \emph{Localization for Wilson Loops in Chern-Simons Theory}, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) \emph{Chern-Simons Gauge Theory: 20 Years After}, AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (\href{http://arxiv.org/abs/0911.2687}{arXiv:0911.2687})

\end{itemize}
referring to

\begin{itemize}%
\item S. Elitzur, [[Greg Moore]], A. Schwimmer, and [[Nathan Seiberg]], \emph{Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory}, Nucl. Phys. B 326 (1989) 108--134.

\end{itemize}
in the context of [[Chern-Simons theory]] and in more general gauge theory to

\begin{itemize}%
\item A. P. Balachandran, S. Borchardt, and A. Stern, \emph{Lagrangian And Hamiltonian Descriptions of Yang-Mills Particles}, Phys. Rev. D 17 (1978) 3247--3256

\end{itemize}
\hypertarget{in_sduality_and_relation_to_t_hoofts_operators}{}\subsubsection*{{In S-duality and relation to t Hoofts operators}}\label{in_sduality_and_relation_to_t_hoofts_operators}

\begin{itemize}%
\item [[Anton Kapustin]], [[Edward Witten]], \emph{Electric-Magnetic Duality And The Geometric Langlands Program}, Communications in Number Theory and Physics Volume 1 (2007) Number 1 (\href{http://arxiv.org/abs/hep-th/0604151}{arXiv:hep-th/0604151})

\end{itemize}
[[!redirects Wilson loop]] [[!redirects Wilson loops]] [[!redirects Wilson-loop]] [[!redirects Wilson-loops]]

[[!redirects Wilson line]] [[!redirects Wilson-line]] [[!redirects Wilson lines]] [[!redirects Wilson-lines]]

[[!redirects Wilson operator]] [[!redirects Wilson operators]]



\end{document}