abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
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 $\Sigma$:
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
(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.
number fields (“function fields of curves over F1”) | function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves) | Riemann surfaces/complex curves | |
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affine and projective line | |||
$\mathbb{Z}$ (integers) | $\mathbb{F}_q[z]$ (polynomials, function 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 functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (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 | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (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 |
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).
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 |
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Hecke transformation | 't Hooft operator? |
local system/flat connection | electric eigenbrane (eigenbrane of Wilson operator) |
Hecke eigensheaf | magnetic eigenbrane (eigenbrane of 't Hooft operator? ) |
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
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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 |
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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 |
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
geometric Langlands correspondence
The conjecture goes back to
based on
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
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
Comments on the development of the geometric duality by R. Langlands himself include
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
See also the deformation to the quantum geometric Langlands correspondence, such as
A classical survey is
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
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:
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
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
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
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)