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\section*{geometric Langlands correspondence}

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

\hypertarget{langlands_correspondence}{}\paragraph*{{Langlands correspondence}}\label{langlands_correspondence}

[[!include Langlands correspondence -- contents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - 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{AbelianCase}{Abelian case}\dotfill \pageref*{AbelianCase} \linebreak
\noindent\hyperlink{RelationToSDuality}{Relation to S-duality}\dotfill \pageref*{RelationToSDuality} \linebreak
\noindent\hyperlink{RelationToTDuality}{Relation to T-duality}\dotfill \pageref*{RelationToTDuality} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesOriginal}{Original}\dotfill \pageref*{ReferencesOriginal} \linebreak
\noindent\hyperlink{surveys_and_reviews}{Surveys and reviews}\dotfill \pageref*{surveys_and_reviews} \linebreak
\noindent\hyperlink{further_resources}{Further resources}\dotfill \pageref*{further_resources} \linebreak
\noindent\hyperlink{ReferencesInterpretationInStringTheory}{Interpretation in string theory}\dotfill \pageref*{ReferencesInterpretationInStringTheory} \linebreak
\noindent\hyperlink{global}{Global}\dotfill \pageref*{global} \linebreak
\noindent\hyperlink{local}{Local}\dotfill \pageref*{local} \linebreak


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

The [[conjecture|conjectural]] \emph{geometric Langlands correspondence} is meant to be an [[analogy|analog]] of the [[number theory|number theoretic]] [[Langlands correspondence]] under the [[function field analogy]], hence with [[number fields]] replaced by [[function fields]] and further replaced by [[rational functions]] on [[complex curves]]. The key to this analogy is the [[Weil uniformization theorem]] which expresses the [[moduli stack of G-principal bundles]] over an [[algebraic curve]] as a [[double coset space]] of various function rings (as discussed at \emph{\href{moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence}{Moduli of bundles over curves}}) of just the kind as they appear in the number-theoretic [[Langlands correspondence]] (for instance in the [[Artin reciprocity law]] and in the definition of [[automorphic representations]]).

[[!include Langlands analogies -- table]]

The original version of the conjectured statement of \emph{geometric Langlands duality} (going back to \hyperlink{BeilinsonDrinfeld9x}{Beilinson-Drinfeld 9x} and reviewed for instance in \hyperlink{Frenkel05}{Frenkel 05}) asserts that for $G$ a [[reductive group]] and for $\Sigma$ an [[algebraic curve]], then there is an [[equivalence of categories|equivalence]] of [[derived categories]] of, on the one hand, [[D-modules]] on the [[moduli stack of G-principal bundles]] on $\Sigma$, and, on the other hand, [[quasi-coherent sheaves]] on the ${}^L G$-[[moduli stack of local systems]] on $\Sigma$:

\begin{displaymath}
\mathcal{O}Mod(Loc_{{}^L G}(\Sigma))
    \stackrel{\simeq}{\longrightarrow}
  \mathcal{D} Mod( Bun_G(\Sigma))
\end{displaymath}
for ${}^L G$ the [[Langlands dual group]]. Moreover, the conjecture asserts that there is canonical such an equivalence which is a [[non-abelian cohomology|non-abelian]] analogue of the [[Fourier-Mukai transform|Fourier-Mukai]] [[integral transform]] and takes [[skyscraper sheaves]] on the left ([[categorification|categorified Dirac distributions]]) to what are called ``[[Hecke eigensheaves]]'' on the right. This equivalence is in turn supposed to be a certain limit of the more general [[quantum geometric Langlands correspondence]] that identifies [[twisted D-modules]] on both sides.

For the abelian case that $G$ is a [[torus]] the above equivalence has indeed been proven, given by a [[Fourier-Mukai transform]] (\hyperlink{Laumon85}{Laumon 85}, \hyperlink{Laumon96}{Laumon 96}, \hyperlink{Rothstein96}{Rothstein 96}), see also \hyperlink{AbelianCase}{below}.

However, in general the above version of the conjecture is false. For instance it fails in the case $G = SL_2$ and $\Sigma= \mathbb{P}^1$ (\hyperlink{Lafforgue09}{Lafforgue 09}).

A refined formulation of the conjecture due to (\hyperlink{ArinkinGaitsgory12}{Arinkin-Gaitsgory 12}), meant to fix this failure, replaces plain quasicoherent sheaves with certain ``nilpotent'' [[ind-objects]] of quasicoherent sheaves and refines [[derived categories]] to [[stable (∞,1)-categories]], to make the conjecture read

\begin{displaymath}
(Ind(\mathcal{O}Mod(Loc_{{}^L G}(\Sigma))))_{Nilp_{Glob}}
    \stackrel{\simeq}{\longrightarrow}
  \mathcal{D} Mod( Bun_G(\Sigma))
\end{displaymath}
(\hyperlink{ArinkinGaitsgory12}{Arinkin-Gaitsgory 12, conjecture 0.1.6}). This form is called the [[categorical geometric Langlands conjecture]].

Since [[D-modules]] on [[moduli stacks of G-principal bundles]] play a central role in [[gauge theory|gauge]] [[quantum field theory]] (in particular as [[Hitchin connections]] on bundles of [[conformal blocks]] of $G$-[[Chern-Simons theory]] [[holographic principle|holographically]] dual to the [[WZW model]] [[2d conformal field theory]]) and since the [[Langlands dual group]] also appears in [[electric-magnetic duality]], it has long been suggested (\href{S-duality#Atiyah77}{Atiyah 77}) that geometric Langlands duality has a distinguished meaning also in [[mathematical physics]] in general and in [[string theory]] in particular. One proposal for a realization of the correspondence as an incarnation of [[mirror symmetry]]/[[S-duality]] is due to (\hyperlink{KapustinWitten06}{Kapustin-Witten 06}), which however has not been turned into [[theorems]] yet. Another proposal for realizing the [[local Langlands correspondence|local]] correspondence via another incarnation of [[mirror symmetry]] is due to (\hyperlink{GerasimovLebedevOblezin09}{Gerasimov-Lebedev-Oblezin 09}).

The geometric Langlands conjecture has been motivated from the [[number theory|number theoretic]] [[Langlands correspondence]] via the [[function field analogy]] and some educated guessing, but there is to date no formalization of this analogy that would allow to regard number-theoretic and the geometric correspondence as two special cases of one ``global'' [[arithmetic geometry]]/[[global analytic geometry]] statement. Cautioning remarks on the accuracy of the analogy and on the rigour of the mirror-symmetric proposals may be found in (\hyperlink{Langlands14}{Langlands 14}). Some discussion of how to possibly start to go about making the analogy more systematic are at \emph{[[differential cohesion and idelic structure]]}.

[[!include function field analogy -- table]]

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

\hypertarget{AbelianCase}{}\subsubsection*{{Abelian case}}\label{AbelianCase}

In the case where $G$ is the [[multiplicative group]], hence where all bundles in question are [[line bundles]], geometric Langlands duality is well understood and is in fact a slight variant of a [[Fourier-Mukai transform]] (\hyperlink{Frenkel05}{Frenkel 05, section 4.4, 4.5}).

\hypertarget{RelationToSDuality}{}\subsubsection*{{Relation to S-duality}}\label{RelationToSDuality}

The [[Kapustin-Witten TQFT]] (\hyperlink{KapustinWitten06}{KapustinWitten 2007}) is supposed to exhibit geometric Langlands duality as a special case of [[S-duality]].

See also at \emph{\href{http://ncatlab.org/nlab/show/Kaluza-Klein+mechanism#Formalization}{KK-compactification -- Formalization}}

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

[[!include gauge theory from AdS-CFT -- table]]

\hypertarget{RelationToTDuality}{}\subsubsection*{{Relation to T-duality}}\label{RelationToTDuality}

In some cases the passage between a Lie group and its [[Langlands dual group]] can be understood as a special case of [[T-duality]]. (\hyperlink{DaenzerErp}{Daenzer-vanErp})

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

\begin{itemize}%
\item [[geometric class field theory]]


\item [[geometric Satake equivalence]]


\item [[Hitchin fibration]]


\item [[topologically twisted D=4 super Yang-Mills theory]]



\end{itemize}
[[duality in physics]], [[duality in string theory]]

\begin{itemize}%
\item [[S-duality]]

\begin{itemize}%
\item [[electro-magnetic duality]]

\begin{itemize}%
\item [[Montonen-Olive duality]]

\end{itemize}

\item [[Seiberg duality]]


\item \textbf{geometric Langlands correspondence}

\begin{itemize}%
\item [[S-duality]], [[Kapustin-Witten TQFT]]

\end{itemize}

\item [[quantum geometric Langlands correspondence]]



\end{itemize}


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

\hypertarget{ReferencesOriginal}{}\subsubsection*{{Original}}\label{ReferencesOriginal}

The conjecture goes back to

\begin{itemize}%
\item [[Alexander Beilinson]], [[Vladimir Drinfeld]], section 5.2.7 of \emph{Quantization of the Hitchin system and Hecke eigensheaves}, (\href{http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf}{pdf})

\end{itemize}
based on

\begin{itemize}%
\item [[Gérard Laumon]], sections 5.3 and 4.3.3. of \emph{Correspondance de Langlands g\'e{}om\'e{}trique pour les corps de fonctions}, Duke Math. Jour., vol. 54 (1987), 309-359

\end{itemize}
Proof in the abelian case is due to

\begin{itemize}%
\item [[Gérard Laumon]], \emph{Transformation de Fourier g\'e{}om\'e{}trique} Preprint IHES/85/M/52 (1985)


\item [[Gérard Laumon]], \emph{Transformation de Fourier g\'e{}n\'e{}ralis\'e{}e} (\href{http://arxiv.org/abs/alg-geom/9603004}{arXiv:alg-geom/9603004})


\item [[Gérard Laumon]], \emph{Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands}, \href{http://arxiv.org/abs/math.AG/0207078}{math.AG/0207078}


\item M. Rothstein. \emph{Sheaves with connection on abelian varieties}, Duke Math. J., 84(3):565--598, 1996

\emph{Correction to: ``Sheaves with connection on abelian varieties.'' Duke Math. J., 87(1):205--211, 1997.}



\end{itemize}
Proof that the original version of the conjecture is false in general is due to

\begin{itemize}%
\item [[Vincent Lafforgue]], \emph{Quelques calculs reli\'e{}s \`a{} la correspondance de Langlands g\'e{}om\'e{}trique pour $\mathbb{P}^1$ (version provisoire)} 2009 (\href{http://www.math.jussieu.fr/~vlafforg/}{web}, \href{http://vlafforg.perso.math.cnrs.fr/geom.pdf}{pdf})

\end{itemize}
The refined version of the conjecture stated in [[derived algebraic geometry]], called the [[categorical geometric Langlands conjecture]], is due to

\begin{itemize}%
\item [[Dennis Gaitsgory]], [[Nick Rozenblyum]], \emph{[[Notes on geometric Langlands]], (\href{http://www.math.harvard.edu/~gaitsgde/GL/}{web})}


\item [[Dima Arinkin]], [[Dennis Gaitsgory]], \emph{Singular support of coherent sheaves, and the geometric Langlands conjecture} (\href{http://arxiv.org/abs/1201.6343}{arXiv.1201.6343})



\end{itemize}
Other comments on the relation to [[TQFT]] include

\begin{itemize}%
\item [[Mikhail Kapranov]], \emph{Analogies between the Langlands Correspondence and Topological Quantum Field Theory}, in \emph{Functional Analysis on the Eve of the 21st Century} Progress in Mathematics Volume 131/132, 1995, pp 119-151

\end{itemize}
Comments on the development of the geometric duality by [[Robert Langlands]] himself:

\begin{itemize}%
\item [[Robert Langlands]], \emph{The Search for a Mathematically Satisfying Geometric Theory of Automorphic Forms}, Notes for a lecture at Mostow-Fest, Yale 2013 (\href{http://publications.ias.edu/rpl/paper/2578}{IAS page}, \href{http://www.youtube.com/watch?v=pfpzET8UkF4}{video}, \href{http://publications.ias.edu/sites/default/files/lecture_6.pdf}{pdf})


\item [[Robert Langlands]], \emph{[[Problems in the theory of automorphic forms -- 45 years later]]}, talk at \href{https://www.maths.nottingham.ac.uk/personal/ibf/files/sc3.html}{Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends}, Oxford, July 5 - July 8, 2014



\end{itemize}
Langlands's doubts about or dissatifaction with the ``geometric Langlands program'' expressed in these talks (where he suggests that his name not be associated with the ``geometric'' part of the program) eventually led to

\begin{itemize}%
\item [[Robert Langlands]], \emph{Об аналитическом виде геометрической теории автоморфных форм}, IAS 2018 (\href{http://publications.ias.edu/rpl/paper/2678}{ias:2678}, \href{http://publications.ias.edu/sites/default/files/iztvestiya_3.pdf}{pdf})

\end{itemize}
This in turn led to the reaction

\begin{itemize}%
\item [[Edward Frenkel]], \emph{Is there an analytic theory of automorphic functions for complex algebraic curves?} (\href{https://arxiv.org/abs/1812.08160}{arXiv:1812.08160})

\end{itemize}
See also the deformation to the [[quantum geometric Langlands correspondence]], such as

\begin{itemize}%
\item [[Mina Aganagic]], [[Edward Frenkel]], [[Andrei Okounkov]], \emph{Quantum $q$-Langlands Correspondence} (\href{https://arxiv.org/abs/1701.03146}{arXiv:1701.03146})

\end{itemize}
\hypertarget{surveys_and_reviews}{}\subsubsection*{{Surveys and reviews}}\label{surveys_and_reviews}

A classical survey is

\begin{itemize}%
\item [[Edward Frenkel]], \emph{Lectures on the Langlands Program and Conformal Field Theory}, in \emph{Frontiers in number theory, physics, and geometry II}, Springer Berlin Heidelberg, 2007. 387-533. (\href{http://arxiv.org/abs/hep-th/0512172}{arXiv:hep-th/0512172})

\end{itemize}
Another set of lecture notes on geometric Langlands and [[nonabelian Hodge theory]] is

\begin{itemize}%
\item [[Ron Donagi]], [[Tony Pantev]], \emph{Lectures on the geometric Langlands conjecture and non-abelian Hodge theory}, 2009 (\href{http://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf}{pdf})

\end{itemize}
More exposition of the relation to [[string theory]] and [[S-duality]] is in

\begin{itemize}%
\item [[Edward Frenkel]], \emph{Gauge Theory and Langlands Duality}, S\'e{}minaire Bourbaki no 1010, June 2009 (\href{http://arxiv.org/abs/0906.2747}{arXiv:0906.2747})

\end{itemize}
\hypertarget{further_resources}{}\subsubsection*{{Further resources}}\label{further_resources}

Collections of resources are here;

\begin{itemize}%
\item [[David Ben-Zvi]], Geometric Langlands -- Lectures and Resources (\href{http://www.math.utexas.edu/users/benzvi/Langlands.html}{web})


\item geometric Laglands \href{http://www.math.uchicago.edu/~mitya/langlands}{page}



\end{itemize}
Notes on two introductory lecture talks are here:

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/string/archives/000806.html}{Pantev on Langlands I}

\href{http://golem.ph.utexas.edu/string/archives/000807.html}{Pantev on Langlands II}



\end{itemize}
See also

\begin{itemize}%
\item Ng\^o{} Bo Ch\^a{}u, \emph{Le lemme fondamental pour les algebres de Lie}, \href{http://arxiv4.library.cornell.edu/abs/0806.4566}{arxiv/0806.4566}


\item James Arthur, \emph{The Work of Ng\^o{} Bo Ch\^a{}u}, Proc. ICM Hyderabad 2010, \href{http://www.icm2010.org.in/wp-content/icmfiles/laudaions/fields2.pdf}{pdf}


\item lecture notes on an introductory talk by [[Tony Pantev]]: \href{http://golem.ph.utexas.edu/string/archives/000806.html}{Pantev on Langlands I}, \href{http://golem.ph.utexas.edu/string/archives/000807.html}{Pantev on Langlands II}


\item [[Edward Frenkel]], \emph{Langlands correspondence for loop groups}, \href{http://math.berkeley.edu/~frenkel/NEWBOOK}{description}, \href{http://math.berkeley.edu/~frenkel/loop.pdf}{pdf}


\item [[Edward Frenkel]], a Bourbaki exposition, \href{http://math.berkeley.edu/~frenkel/BOOK/bourbaki.pdf}{pdf}


\item [[Edward Frenkel]], \emph{Langlands duality for representations of quantum groups}, \href{http://arxiv.org/abs/0809.4453}{arxiv/0809.4453}



\end{itemize}
\hypertarget{ReferencesInterpretationInStringTheory}{}\subsubsection*{{Interpretation in string theory}}\label{ReferencesInterpretationInStringTheory}

\hypertarget{global}{}\paragraph*{{Global}}\label{global}

An interpretation of the global geometric Langlands correspondence as describing [[S-duality]] of [[topological twist|topologically twisted]] [[super Yang-Mills theory]], incarnated in [[mirror symmetry]] on its [[KK-compactification]] to 2d [[sigma-models]] ([[A-model]]/[[B-model]]-type) was given in

\begin{itemize}%
\item [[Anton Kapustin]], [[Edward Witten]], \emph{Electric-Magnetic Duality And The Geometric Langlands Program} , Communications in number theory and physics, Volume 1, Number 1, 1--236 (2007) (\href{http://arxiv.org/abs/hep-th/0604151}{arXiv:hep-th/0604151})


\item [[Edward Frenkel]], [[Edward Witten]], \emph{Geometric Endoscopy and Mirror Symmetry} (\href{http://arxiv.org/abs/0710.5939}{arXiv:0710.5939})


\item [[Edward Witten]], \emph{Mirror Symmetry, Hitchin's Equations, And Langlands Duality} (\href{http://arxiv.org/abs/0802.0999}{arXiv:0802.0999})



\end{itemize}
and discussed in the bigger picture of [[S-duality]] arising as the conformal invariance of the [[6d (2,0)-superconformal QFT]] in

\begin{itemize}%
\item [[Edward Witten]], \emph{Geometric Langlands From Six Dimensions}, in Peter Kotiuga (ed.) \emph{A Celebration of the Mathematical Legacy of Raoul Bott}, AMS 2010 (\href{http://arxiv.org/abs/0905.2720}{arXiv:0905.2720})

\end{itemize}
Reflections on the history (and possible future) of this insight are in

\begin{itemize}%
\item interview with [[Edward Witten]] by [[Hirosi Ooguri]], 2014 (\href{http://www.ams.org/notices/201505/rnoti-p491.pdf}{pdf})

\end{itemize}
An exposition of the relation to [[S-duality]] and [[electro-magnetic duality]] is in (\hyperlink{Frenkel09}{Frenkel 09}) and in

\begin{itemize}%
\item [[Edward Frenkel]], \emph{What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?} KITP talk 2011 (\href{http://online.kitp.ucsb.edu/online/bblunch/frenkel/}{web})


\item [[Edward Frenkel]], \emph{Overview of the links between the Langlands program and 4D super Yang--Mills theory}, KITP talk 2010, \href{http://online.kitp.ucsb.edu/online/duallang_m10/frenkel}{video page}, notes \href{http://online.kitp.ucsb.edu/online/duallang_m10/frenkel/pdf/Frenkel_LanglandsQFT_KITP.pdf}{pdf}



\end{itemize}
Further developments are surveyed in

\begin{itemize}%
\item [[David Ben-Zvi]], \emph{[[Loop Groups, Characters and Elliptic Curves]]}, 2012

\end{itemize}
Further discussion is also in

\begin{itemize}%
\item Tamas Hausel, \emph{Global topology of the Hitchin system} (\href{http://arxiv.org/abs/1102.1717}{arXiv:1102.1717}, \href{http://geom.epfl.ch/files/content/sites/geom/files/Calgary0312.pdf}{pdf slides})


\item Kevin Setter, \emph{Topological quantum field theory and the geometric Langlands correspondence}. Dissertation (Ph.D.), California Institute of Technology 2013 (\href{http://resolver.caltech.edu/CaltechTHESIS:09192012-150137728}{web})



\end{itemize}
Discussion from the point of view of [[M-theory]] is in

\begin{itemize}%
\item [[Meng-Chwan Tan]], \emph{M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems} (\href{http://arxiv.org/abs/1301.1977}{arXiv:1301.1977})

\end{itemize}
A relation to [[T-duality]] (of the group manifolds!) is discussed in

\begin{itemize}%
\item [[Calder Daenzer]], [[Erik van Erp]], \emph{T-Duality for Langlands Dual Groups} (\href{http://arxiv.org/abs/1211.0763}{arXiv:1211.0763})

\end{itemize}
\hypertarget{local}{}\paragraph*{{Local}}\label{local}

Discussion of [[local Langlands correspondence|local]] [[archimedean field|Archimedean]] Langlands duality for [[Whittaker functions]] as [[mirror symmetry]] of a suitable [[A-model]] and [[B-model]] is discussed in

\begin{itemize}%
\item [[Anton Gerasimov]], Dimitri Lebedev, [[Sergey Oblezin]],

\emph{Archimedean L-factors and Topological Field Theories I} (\href{http://arxiv.org/abs/0906.1065}{arXiv:0906.1065})

\emph{Archimedean L-factors and Topological Field Theories II} (\href{http://arxiv.org/abs/0909.2016}{arXiv:0909.2016})

\emph{Parabolic Whittaker Functions and Topological Field Theories I} (\href{http://arxiv.org/abs/1002.2622}{arXiv:1002.2622})



\end{itemize}
[[!redirects geometric Langlands duality]]

[[!redirects geometric Langlands program]]

[[!redirects geometric Langlands theory]] [[!redirects geometric Langlands duality]] [[!redirects geometric Langlands]]

[[!redirects geometric Langlands conjecture]]



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