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 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).
The conjectured statement asserts roughly 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 βHecke eigensheavesβ 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.
Review of this idea includes (Frenkel 05). A refined formulation replaces plain quasicoherent sheaves with certain Ind-coherent sheaves and refines derived categories to stable (β,1)-categories, see (Arinkin-Gaitsgory 12). The geometric Langlands correspondence conjecture is a theorem in the abelian case, as discussed below.
Since D-modules on moduli stacks of G-principal bundles play a central role in gauge quantum field theory (in particular as prequantum line bundles on the phase space 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 βglobalβ 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).
number fields (βfunction fields of curves over F1β) | 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, 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) | β | |
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 $x$) | |
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (β$p$-arithmetic jet spaceβ 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) | $\frac{\partial}{\partial z}$ | β | |
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 | β | $\pi_1(\Sigma)$ fundamental group | |
Galois representation | β | flat connection (βlocal systemβ) on $\Sigma$ | |
class field theory | |||
class field theory | β | 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) | β | ||
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$ | β | $Bun_{GL_1}(\Sigma)$ (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) | β | $Bun_{GL_n(\mathbb{C})}(\Sigma)$ (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 | |||
zeta functions | |||
Dedekind zeta function | Weil zeta function | zeta function of a Riemann surface |
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 β Formalization
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 |
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
Discussion of what the geometric theory ultimately should be from the point of view of derived algebraic geometry are in
Dennis Gaitsgory, Nick Rozenblyum, Notes on Geometric Langlands β A study in derived algebraic geometry (web)
Dima Arinkin, Dennis Gaitsgory, Singular support of coherent sheaves, and the geometric Langlands conjecture (arXiv.1201.6343)
and in less category-theoretic terms in
See also
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 β Lectures and Resources (web)
geometric Laglands page
Notes on two introductory lecture talks are here:
See also
NgΓ΄ BαΊ£o ChΓ’u, Le lemme fondamental pour les algebres de Lie, arxiv/0806.4566
James Arthur, The Work of NgΓ΄ BαΊ£o ChΓ’u, 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, 1β236 (2007) (arXiv:hep-th/0604151)
Edward Frenkel, Edward Witten, Geometric Endoscopy and Mirror Symmetry (arXiv:0710.5939)
Edward Witten, Mirror Symmetry, Hitchinβs 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
An exposition of the relation to S-duality and electro-magnetic duality is in (Frenkel 09) and in
Edward Frenkel, What Do Fermatβs 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 YangβMills 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)