Contents

duality

# Contents

## 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 correspondencegeometric Langlands correspondence
ring of integers of global fieldstructure sheaf on complex curve $\Sigma$
Galois groupfundamental group of $\Sigma$
Galois representationflat connection/local system on $\Sigma$
idele class group mod integral adelesmoduli stack of line bundles on $\Sigma$
nonabelian $\;$moduli stack of vector bundles on $\Sigma$
automorphic representationHitchin 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}$$\mathbb{P}_{\mathbb{F}_q}$ (projective line)Riemann sphere
$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)$\frac{\partial}{\partial z}$ (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
$\mathbb{Z}_p$ (p-adic integers)$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)formal disks in $X$
$\mathbb{Q}_p$ (p-adic numbers)$\mathbb{F}_q((z-x))$ (Laurent series around $x$)$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$$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 fieldgenus of an algebraic curvegenus 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}$$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
Galois group$\pi_1(\Sigma)$ fundamental group
Galois representationflat connection (“local system”) on $\Sigma$
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
number field Langlands correspondencefunction field Langlands correspondencegeometric 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 fieldsTamagawa-Weil for function fields
theta functions
Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functionsHasse-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.

gauge theory induced via AdS-CFT correspondence

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 invarianceM5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-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

$\,$

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)

## References

### Original

The conjecture goes back to

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

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)

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

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

### Surveys and reviews

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

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

### Further resources

Collections of resources are here;

Notes on two introductory lecture talks are here:

### 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

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

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

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

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

#### Local

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

Last revised on August 12, 2021 at 07:35:48. See the history of this page for a list of all contributions to it.