nLab geometric Langlands correspondence

Idea

The conjectural geometric Langlands correspondence is meant to be an analog of the 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 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).

analogies in the Langlands program:

arithmetic Langlands correspondence	geometric Langlands correspondence
ring of integers of global field	structure sheaf on complex curve $\Sigma$
Galois group	fundamental group of $\Sigma$
Galois representation	flat connection/local system on $\Sigma$
idele class group mod integral adeles	moduli stack of line bundles on $\Sigma$
nonabelian $\;$ “	moduli stack of vector bundles on $\Sigma$
automorphic representation	Hitchin connection D-module on bundle of conformal blocks over the moduli stack

The original version of the conjectured statement of geometric Langlands duality (going back to Beilinson-Drinfeld 9x and reviewed for instance in Frenkel 05) asserts that for $G$ a reductive group and for $\Sigma$ an algebraic curve, then there is an 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 $\Sigma$:

$$\mathcal{O}Mod(Loc_{{}^L G}(\Sigma)) \stackrel{\simeq}{\longrightarrow} \mathcal{D} Mod( Bun_G(\Sigma))$$

for ${}^L G$ the Langlands dual group. Moreover, the conjecture asserts that there is canonical such an equivalence which is a non-abelian analogue of the Fourier-Mukai integral transform and takes skyscraper sheaves on the left (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 (Laumon 85, Laumon 96, Rothstein 96), see also below.

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

A refined formulation of the conjecture due to (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

$$(Ind(\mathcal{O}Mod(Loc_{{}^L G}(\Sigma))))_{Nilp_{Glob}} \stackrel{\simeq}{\longrightarrow} \mathcal{D} Mod( Bun_G(\Sigma))$$

(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 quantum field theory (in particular as Hitchin connections on bundles of conformal blocks of $G$-Chern-Simons theory 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 (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 (Kapustin-Witten 06), which however has not been turned into theorems yet. Another proposal for realizing the local correspondence via another incarnation of mirror symmetry is due to (Gerasimov-Lebedev-Oblezin 09).

The geometric Langlands conjecture has been motivated from the 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 (Langlands 14). Some discussion of how to possibly start to go about making the analogy more systematic are at differential cohesion and idelic structure.

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Properties

Abelian case

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 (Frenkel 05, section 4.4, 4.5).

Relation to S-duality

The Kapustin-Witten TQFT (KapustinWitten 2007) is supposed to exhibit geometric Langlands duality as a special case of S-duality.

See also at KK-compactification – Formalization

geometric Langlands correspondence	S-duality in N=4 D=4 super Yang-Mills theory
Hecke transformation	't Hooft operator
local system/flat connection	electric eigenbrane (eigenbrane of Wilson operator)
Hecke eigensheaf	magnetic eigenbrane (eigenbrane of 't Hooft operator )

(Kapustin-Witten 06)

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6	F-theory perspective
11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$	compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance	M5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface	double dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence	D3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

gauge theory induced via AdS5-CFT4
type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

Relation to T-duality

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

Related concepts

• geometric class field theory

• geometric Satake equivalence

• Hitchin fibration

• topologically twisted D=4 super Yang-Mills theory

duality in physics, duality in string theory

• S-duality

  • electro-magnetic duality

    • Montonen-Olive duality

  • Seiberg duality

  • geometric Langlands correspondence

    • S-duality, Kapustin-Witten TQFT

  • quantum geometric Langlands correspondence

References

Original

The conjecture goes back to

• Alexander Beilinson, Vladimir Drinfeld, section 5.2.7 of Quantization of the Hitchin system and Hecke eigensheaves, (pdf)

based on

• Gérard Laumon, sections 5.3 and 4.3.3. of Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. Jour., vol. 54 (1987), 309-359

Proof in the abelian case is due to

• Gérard Laumon, Transformation de Fourier géométrique Preprint IHES/85/M/52 (1985)

• Gérard Laumon, Transformation de Fourier généralisée (arXiv:alg-geom/9603004)

• Gérard Laumon, Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands, math.AG/0207078

• M. Rothstein. Sheaves with connection on abelian varieties, Duke Math. J., 84(3):565–598, 1996

  Correction to: “Sheaves with connection on abelian varieties.” Duke Math. J., 87(1):205–211, 1997.

Proof that the original version of the conjecture is false in general is due to

• Vincent Lafforgue, Quelques calculs reliés à la correspondance de Langlands géométrique pour $\mathbb{P}^1$ (version provisoire) 2009 (web, pdf)

The refined version of the conjecture stated in derived algebraic geometry, called the categorical geometric Langlands conjecture, is due to

• Dennis Gaitsgory, Nick Rozenblyum, Notes on geometric Langlands, (web)

• Dima Arinkin, Dennis Gaitsgory, Singular support of coherent sheaves, and the geometric Langlands conjecture (arXiv.1201.6343)

Other comments on the relation to TQFT include

• Mikhail Kapranov, Analogies between the Langlands Correspondence and Topological Quantum Field Theory, in Functional Analysis on the Eve of the 21st Century Progress in Mathematics Volume 131/132, 1995, pp 119-151

Comments on the development of the geometric duality by Robert Langlands himself:

• Robert Langlands, The Search for a Mathematically Satisfying Geometric Theory of Automorphic Forms, Notes for a lecture at Mostow-Fest, Yale 2013 (IAS page, video, pdf)

• Robert Langlands, Problems in the theory of automorphic forms – 45 years later, talk at Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends, Oxford, July 5 - July 8, 2014

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

• Robert Langlands, Об аналитическом виде геометрической теории автоморфных форм, IAS 2018 (ias:2678, pdf)

This in turn led to the reaction

• Edward Frenkel, Is there an analytic theory of automorphic functions for complex algebraic curves? (arXiv:1812.08160)

See also the deformation to the quantum geometric Langlands correspondence, such as

• Mina Aganagic, Edward Frenkel, Andrei Okounkov, Quantum $q$-Langlands Correspondence (arXiv:1701.03146)

Proof of the general case:

• Dennis Gaitsgory, Proof of the geometric Langlands conjecture [web]

• Dennis Gaitsgory, Sam Raskin, Proof of the geometric Langlands conjecture I: construction of the functor [arXiv:2405.03599]

• Dima Arinkin, D. Beraldo, J. Campbell, L. Chen, Dennis Gaitsgory, J. Faergeman, K. Lin, Sam Raskin, Nick Rozenblyum, Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE [arXiv:2405.03648]

Surveys and reviews

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory, in Frontiers in number theory, physics, and geometry II, Springer Berlin Heidelberg, 2007. 387-533. (arXiv:hep-th/0512172)

• Ron Donagi, Tony Pantev, Lectures on the geometric Langlands

  conjecture and non-abelian Hodge theory_, 2009 (pdf)

  (with an eye towards nonabelian Hodge theory)

More on the relation to string theory and S-duality:

• Edward Frenkel, Gauge Theory and Langlands Duality, Séminaire Bourbaki no 1010, June 2009 (arXiv:0906.2747)

With emphasis on the role of magnetic monopoles and 't Hoof lines:

• Alexander B. Atanasov, Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence, 2018 (pdf, slides)

Further resources

Collections of resources are here;

• David Ben-Zvi, Geometric Langlands – Lectures and Resources (web)

• geometric Laglands page

Notes on two introductory lecture talks are here:

• Pantev on Langlands I

  Pantev on Langlands II

See also

• Ngô Bảo Châu, Le lemme fondamental pour les algebres de Lie, arxiv/0806.4566

• James Arthur, The Work of Ngô Bảo Châu, Proc. ICM Hyderabad 2010, pdf

• lecture notes on an introductory talk by Tony Pantev: Pantev on Langlands I, Pantev on Langlands II

• Edward Frenkel, Langlands correspondence for loop groups, description, pdf

• Edward Frenkel, a Bourbaki exposition, pdf

• Edward Frenkel, Langlands duality for representations of quantum groups, arxiv/0809.4453

Interpretation in string theory

Global

An interpretation of the global geometric Langlands correspondence as describing S-duality of 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

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) (arXiv:hep-th/0604151)

• Edward Frenkel, Edward Witten, Geometric Endoscopy and Mirror Symmetry (arXiv:0710.5939)

• Edward Witten, Mirror Symmetry, Hitchin’s Equations, And Langlands Duality (arXiv:0802.0999)

and discussed in the bigger picture of S-duality arising as the conformal invariance of the 6d (2,0)-superconformal QFT in

• Edward Witten, Geometric Langlands From Six Dimensions, in Peter Kotiuga (ed.) A Celebration of the Mathematical Legacy of Raoul Bott, AMS 2010 (arXiv:0905.2720)

Reflections on the history (and possible future) of this insight are in

• interview with Edward Witten by Hirosi Ooguri, 2014 (pdf)

An exposition of the relation to S-duality and electro-magnetic duality is in (Frenkel 09) and in

• Edward Frenkel, What Do Fermat’s Last Theorem and Electro-magnetic Duality Have in Common? KITP talk 2011 (web)

• Edward Frenkel, Overview of the links between the Langlands program and 4D super Yang–Mills theory, KITP talk 2010, video page, notes pdf

Further developments are surveyed in

• David Ben-Zvi, Loop Groups, Characters and Elliptic Curves, 2012

Further discussion is also in

• Tamas Hausel, Global topology of the Hitchin system (arXiv:1102.1717, pdf slides)

• Kevin Setter, Topological quantum field theory and the geometric Langlands correspondence. Dissertation (Ph.D.), California Institute of Technology 2013 (web)

Discussion from the point of view of M-theory is in

• Meng-Chwan Tan, M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems (arXiv:1301.1977)

• Meer Ashwinkumar, Meng-Chwan Tan, Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory (arXiv:1910.01134)

A relation to T-duality (of the group manifolds!) is discussed in

• Calder Daenzer, Erik van Erp, T-Duality for Langlands Dual Groups (arXiv:1211.0763)

Local

Discussion of local Archimedean Langlands duality for Whittaker functions as mirror symmetry of a suitable A-model and B-model is discussed in

• Anton Gerasimov, Dimitri Lebedev, Sergey Oblezin,

  Archimedean L-factors and Topological Field Theories I (arXiv:0906.1065)

  Archimedean L-factors and Topological Field Theories II (arXiv:0909.2016)

  Parabolic Whittaker Functions and Topological Field Theories I (arXiv:1002.2622)