abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
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:
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 a reductive group and for 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 , and, on the other hand, quasi-coherent sheaves on the -moduli stack of local systems on :
for 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 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 and (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 -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.
In the case where 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
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
Kaluza-Klein compactification on | compactificationon elliptic fibration followed by T-duality |
7-dimensional supergravity | |
topological sector | |
7-dimensional Chern-Simons theory | |
AdS7-CFT6 holographic duality | |
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance | M5-brane worldvolume theory |
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 |
topological twist | |
topologically twisted N=2 D=4 super Yang-Mills theory | |
KK-compactification on Riemann surface | |
A-model on , Donaldson theory |
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 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
This in turn led to the reaction
See also the deformation to the quantum geometric Langlands correspondence, such as
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]
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:
With emphasis on the role of magnetic monopoles and 't Hoof lines:
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
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
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)
Last revised on May 10, 2024 at 06:02:32. See the history of this page for a list of all contributions to it.