Gauge theory and Langlands duality

Frenkel, Edward

Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1010, 35 p.

MR Zbl

@incollection{AST_2010__332__369_0,
     author = {Frenkel, Edward},
     title = {Gauge theory and {Langlands} duality},
     booktitle = {S\'eminaire Bourbaki : volume 2008/2009 expos\'es 997-1011  - Avec table par noms d'auteurs de 1848/49 \`a 2008/09},
     series = {Ast\'erisque},
     note = {talk:1010},
     pages = {369--403},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {332},
     year = {2010},
     mrnumber = {2648685},
     zbl = {1209.22009},
     language = {en},
     url = {http://www.numdam.org/item/AST_2010__332__369_0/}
}

TY  - CHAP
AU  - Frenkel, Edward
TI  - Gauge theory and Langlands duality
BT  - Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09
AU  - Collectif
T3  - Astérisque
N1  - talk:1010
PY  - 2010
SP  - 369
EP  - 403
IS  - 332
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2010__332__369_0/
LA  - en
ID  - AST_2010__332__369_0
ER  -

%0 Book Section
%A Frenkel, Edward
%T Gauge theory and Langlands duality
%B Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09
%A Collectif
%S Astérisque
%Z talk:1010
%D 2010
%P 369-403
%N 332
%I Société mathématique de France
%U http://www.numdam.org/item/AST_2010__332__369_0/
%G en
%F AST_2010__332__369_0

Frenkel, Edward. Gauge theory and Langlands duality, dans Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1010, 35 p. http://www.numdam.org/item/AST_2010__332__369_0/

Bibliographie
Cité par

[1] D. Arinkin - "Moduli of connections with a small parameter on a curve", preprint arXiv:mathAG/0409373.

[2] J. Arthur - "Unipotent automorphic representations: conjectures", Astérisque 171-172(1989), p. 13-71. | Numdam | MR | Zbl

[3] A. Beilinson & V. Drinfeld - "Quantization of Hitchin's integrable system and Hecke eigensheaves", preprint http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf.

[4] M. Bershadsky, A. Johansen, V. Sadov & C. Vafa - "Topological reduction of $4 DSYM$ to $2 D σ$ -models", Nuclear Phys. B 448 (1995), p. 166-186. | DOI | MR | Zbl

[5] K. Corlette - "Flat $G$ -bundles with canonical metrics", J. Differential Geom. 28 (1988), p. 361-382. | DOI | MR | Zbl

[6] P. Deligne Et Al. - Quantum fields and strings : A course for mathematicians, Vol. I and II, Amer. Math. Soc, Institute for Advanced Study, 1999.

[7] R. Donagi & T. Pantev - "Langlands duality for Hitchin systems", preprint arXiv:math. AG/0604617. | DOI | MR | Zbl

[8] V. G. Drinfel'D - "Langlands' conjecture for $G L (2)$ over functional fields", in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, 1980, p. 565-574. | MR | Zbl

[9] V. G. Drinfel'D, "Two-dimensional $ℓ$ -adic representations of the fundamental group of a curve over a finite field and automorphic forms on $G L (2)$ ", Amer. J. Math. 105 (1983), p. 85-114. | DOI | MR | Zbl

[10] V. G. Drinfel'D, "Moduli varieties of $F$ -sheaves", Funct. Anal Appl. 21 (1987), p. 107-122. | DOI | MR | Zbl

[11] V. G. Drinfel'D, "The proof of Petersson's conjecture for $G L (2)$ over a global field of characteristic $p$ ", Funct. Anal. Appl. 22 (1988), p. 28-43. | DOI | MR | Zbl

[12] F. Englert & ; P. Windey - "Quantization condition for $^{'} t$ Hooft monopoles in compact simple Lie groups", Phys. Rev D 14 (1976), p. 2728-2731. | DOI | MR

[13] E. Frenkel - "Recent advances in the Langlands program", Bull. Amer. Math. Soc. (N.S.) 41 (2004), p. 151-184. | DOI | MR | Zbl

[14] E. Frenkel, Langlands correspondence for loop groups, Cambridge Studies in Advanced Math., vol. 103, Cambridge Univ. Press, 2007. | MR | Zbl

[15] E. Frenkel, "Lectures on the Langlands program and conformai field theory", in Frontiers in number theory, physics, and geometry. II, Springer, 2007, p. 387-533. | DOI | MR | Zbl

[16] E. Frenkel & D. Gaitsgory - "Local geometric Langlands correspondence and affine Kac-Moody algebras", in Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser, 2006, p. 69-260. | DOI | MR | Zbl

[17] E. Frenkel, D. Gaitsgory & K. Vilonen - "On the geometric Langlands conjecture", J. Amer. Math. Soc. 15 (2002), p. 367-417. | DOI | MR | Zbl

[18] E. Frenkel & S. Gukov - " $S$ -duality of branes and geometric Langlands correspondence", to appear.

[19] E. Frenkel & E. Witten - "Geometric endoscopy and mirror symmetry", Commun. Number Theory Phys. 2 (2008), p. 113-283. | DOI | MR | Zbl

[20] D. Gaiotto & E. Witten - "Supersymmetric boundary conditions in $𝒩 = 4$ super Yang-Mills theory", J. Stat. Phys. 135 (2009), p. 789-855. | DOI | MR | Zbl

[21] D. Gaiotto & E. Witten, " $S$ -duality of boundary conditions in $N = 4$ super Yang-Mills theory", preprint arXiv:0807.3720. | DOI | MR | Zbl

[22] D. Gaitsgory - "On a vanishing conjecture appearing in the geometric Langlands correspondence", Ann. of Math. 160 (2004), p. 617-682. | DOI | MR | Zbl

[23] S. Gelbart - "An elementary introduction to the Langlands program", Bull. Amer. Math. Soc. (N.S.) 10 (1984), p. 177-219. | DOI | MR | Zbl

[24] S. I. Gelfand & Y. I. Manin - Homological algebra, Encyclopaedia of Math. Sciences, vol. 38, Springer, 1994. | MR | Zbl

[25] P. Goddard, J. Nuyts & D. I. Olive - "Gauge theories and magnetic charge", Nuclear Phys. B 125 (1977), p. 1-28. | DOI | MR

[26] S. Gukov & E. Witten - "Gauge theory, ramification, and the geometric Langlands program", in Current developments in mathematics, 2006, Int. Press, Somerville, MA, 2008, p. 35-180. | MR | Zbl

[27] S. Gukov & E. Witten, "Rigid surface operators", preprint arXiv:0804.1561. | DOI | MR | Zbl

[28] J. A. Harvey, G. Moore & A. Strominger - "Reducing $S$ duality to $T$ duality", Phys. Rev. D 52 (1995), p. 7161-7167. | DOI | MR

[29] T. Hausel & M. Thaddeus - "Mirror symmetry, Langlands duality, and the Hitchin system", Invent. Math. 153 (2003), p. 197-229. | DOI | MR | Zbl

[30] N. J. Hitchin - "The self-duality equations on a Riemann surface", Proc. London Math. Soc. 55 (1987), p. 59-126. | DOI | MR | Zbl

[31] N. J. Hitchin, "Stable bundles and integrable systems", Duke Math. J. 54 (1987), p. 91-114. | DOI | MR | Zbl

[32] N. J. Hitchin, "Langlands duality and $G_{2}$ spectral curves", Q. J. Math. 58 (2007), p. 319-344. | DOI | MR | Zbl

[33] A. Kapustin - "A note on quantum geometric Langlands duality, gauge theory, and quantization of the moduli space of flat connections", preprint arXiv:0811.3264.

[34] A. Kapustin & E. Witten - "Electric-magnetic duality and the geometric Langlands program", Commun. Number Theory Phys. 1 (2007), p. 1-236. | DOI | MR | Zbl

[35] M. Kashiwara & P. Schapira - Sheaves on manifolds, Grund. Math. Wiss., vol. 292, Springer, 1990. | MR | Zbl

[36] M. H. Krieger - "A 1940 letter of André Weil on analogy in mathematics", Notices Amer. Math. Soc. 52 (2005), p. 334-341. | MR | Zbl

[37] L. Lafforgue - "Chtoucas de Drinfeld et correspondance de Langlands", Invent. Math. 147 (2002), p. 1-241. | DOI | MR | Zbl

[38] L. Lafforgue, "Quelques calculs reliés à la correspondance de Langlands géométrique pour $ℙ^{1}$ ", preprint http://people.math.jussieu.fr/~vlafforg/geom.pdf.

[39] R. P. Langlands - "Problems in the theory of automorphic forms", in Lectures in modern analysis and applications, III, Lecture Notes in Math., vol. 170, Springer, 1970, p. 18-61. | MR | Zbl

[40] G. Laumon - "Correspondance de Langlands géométrique pour les corps de fonctions", Duke Math. J. 54 (1987), p. 309-359. | MR | Zbl

[41] G. Laumon, "Transformation de Fourier généralisée", preprint arXiv:alggeom/9603004. | Numdam | MR | Zbl

[42] S. Lysenko - "Geometric Waldspurger periods", Compos. Math. 144 (2008), p. 377-438. | DOI | MR | Zbl

[43] S. Lysenko, "Geometric theta-lifting for the dual pair $G S p_{2 n}, G O_{2 m}$ ", preprint arXiv:0802.0457. | Numdam | MR | Zbl

[44], "Geometric theta-lifting for the dual pair $S O_{2 m}, S p_{2 n}$ ", preprint arXiv:math/0701170.

[45] I. Mlrkovlć & ; K. Vilonen - "Geometric Langlands duality and representations of algebraic groups over commutative rings", Ann. of Math. 166 (2007), p. 95-143. | DOI | MR | Zbl

[46] C. Montonen & D. I. Olive - "Magnetic monopoles as gauge particles ?", Phys. Lett. B 72 (1977), p. 117-120. | DOI

[47] D. Nadler - "Microlocal branes are constructible sheaves", preprint arXiv:math/0612399. | DOI | MR | Zbl

[48] D. Nadler & E. Zaslow - "Constructible sheaves and the Fukaya category", J. Amer. Math. Soc. 22 (2009), p. 233-286. | DOI | MR | Zbl

[49] B. C. Ngô - "Le lemme fondamental pour les algèbres de Lie", preprint arXiv:0801.0446. | Numdam | MR | Zbl

[50] H. Osborn - "Topological charges for $N = 4$ supersymmetric gauge theories and monopoles of Spin 1", Phys. Lett. B 83 (1979), 321-326. | DOI

[51] M. Rothstein - "Connections on the total Picard sheaf and the $K P$ hierarchy", Acta Appl. Math. 42 (1996), p. 297-308. | DOI | MR | Zbl

[52] C. T. Simpson - "Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization", J. Amer. Math. Soc. 1 (1988), p. 867-918. | DOI | MR | Zbl

[53] C. T. Simpson, "Harmonic bundles on noncompact curves", J. Amer. Math. Soc.. 3 (1990), p. 713-770. | DOI | MR | Zbl

[54] A. Strominger, S.-T. Yau & E. Zaslow - "Mirror symmetry is $T$ -duality", Nuclear Phys. B 479 (1996), p. 243-259. | DOI | MR | Zbl

[55] C. Vafa & E. Witten - "A strong coupling test of $S$ -duality", Nuclear Phys. B 431 (1994), p. 3-77. | DOI | Zbl

[56] E. Witten - "Topological quantum field theory", Comm. Math. Phys. 117 (1988), p. 353-386. | DOI | Zbl

[57] E. Witten, "Mirror manifolds and topological field theory", in Essays on mirror manifolds, Int. Press, Hong Kong, 1992, p. 120-158. | Zbl

[58] E. Witten, "Gauge theory and wild ramification", Anal. Appl. (Singap.) 6 (2008), p. 429-501. | DOI | Zbl

[59] E. Witten, "Geometric Langlands and the equations of Nahm and Bogomolny", to appear. | DOI | Zbl

[60] E. Witten, "Geometric Langlands from six dimensions", preprint arXiv:0905.2720. | DOI | Zbl

[61] E. Witten, "Mirror symmetry, Hitchin's equations, and Langlands duality", preprint arXiv:0802.0999. | DOI | Zbl