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geometric Langlands correspondence

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 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 Σ\Sigma
Galois groupfundamental group of Σ\Sigma
Galois representationflat connection/local system on Σ\Sigma
idele class group mod integral adelesmoduli stack of line bundles on Σ\Sigma
nonabelian \;moduli stack of vector bundles on Σ\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 GG a reductive group and for Σ\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 Σ\Sigma, and, on the other hand, quasi-coherent sheaves on the LG{}^L G-moduli stack of local systems on Σ\Sigma:

𝒪Mod(Loc LG(Σ))𝒟Mod(Bun G(Σ)) \mathcal{O}Mod(Loc_{{}^L G}(\Sigma)) \stackrel{\simeq}{\longrightarrow} \mathcal{D} Mod( Bun_G(\Sigma))

for LG{}^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 GG is a torus the abive 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 2G = SL_2 and Σ= 1\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(𝒪Mod(Loc LG(Σ)))) Nilp Glob𝒟Mod(Bun G(Σ)) (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).

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 GG-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 𝔽 q\mathbb{F}_q (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
\mathbb{Z} (integers)𝔽 q[z]\mathbb{F}_q[z] (polynomials, function algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational functions)meromorphic functions on complex plane
pp (prime number/non-archimedean place)x𝔽 px \in \mathbb{F}_pxx \in \mathbb{C}
\infty (place at infinity)\infty
Spec()Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
p() p()p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)z\frac{\partial}{\partial z} (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
p\mathbb{Z}_p (p-adic integers)𝔽 q[[tx]]\mathbb{F}_q[ [ t -x ] ] (power series around xx)[[zx]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX
p\mathbb{Q}_p (p-adic numbers)𝔽 q((zx))\mathbb{F}_q((z-x)) (Laurent series around xx)((zx))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((zx))\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)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field ) xGL 1(((zx)))\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
KK a number field (K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma)
𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)ΣP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
() pΦ()p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)z\frac{\partial}{\partial z}
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v (formal completion at vv)((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles) xΣ GL 1(((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
Galois theory
Galois groupπ 1(Σ)\pi_1(\Sigma) fundamental group
Galois representationflat connection (“local system”) on Σ\Sigma
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)
GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})Bun GL 1(Σ)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)\GL n(𝔸 K)//GL n(𝒪)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()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\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 Σ\Sigma
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on Σ\Sigma
higher dimensional spaces
zeta functionsHasse-Weil zeta function

Properties

Abelian case

In the case where GG 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 correspondenceS-duality in N=4 D=4 super Yang-Mills theory
Hecke transformation’t Hooft operator
local system/flat connectionelectric eigenbrane (eigenbrane of Wilson operator)
Hecke eigensheafmagnetic eigenbrane (eigenbrane of ’t Hooft operator )

(Kapustin-Witten 06)

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon 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 GBun_G, Donaldson theory

\,

gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^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 GBun_G and B-model on Loc GLoc_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)

duality in physics, duality in string theory

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 1\mathbb{P}^1 (version provisoire) 2009 (web, pdf)

The refined version of the conjecture stated in derived algebraic geometry 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 R. Langlands himself include

Surveys and reviews

A classical survey is

  • 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)

Another set of lecture notes on geometric Langlands and nonabelian Hodge theory is

More exposition of the relation to string theory and S-duality is in

Further resources

Collections of resources are here;

Notes on two introductory lecture talks are here:

See also

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)

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

  • 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)

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

Revised on December 13, 2014 21:27:26 by Urs Schreiber (127.0.0.1)