Abstract
This is an introduction to the work of Beilinson and Drinfeld [6]_on the Langlands program.
Supported by the Spanish Ministerio de Educación y Ciencia [MTM2007-63582].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Arkhipov and D. Gaitsgory, Differential operators on the loop group via chiral algebras. Int. Math. Res. Not. 2002, no. 4, 165–210.
A. Beauville and Y. Laszlo, Un lemme de descente. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340.
K. Behrend, Differential Graded Schemes II: The 2-category of Differential Graded Schemes. Preprint. arXiv:math/0212226.
A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisque 100 (1982).
A. Beilinson, and J. Bernstein, A proof of Jantzen conjectures. I.M. Gelfand Seminar, 1–50, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc. Providence, RI, 1993.
A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, Preprint. Available at http://www.math.uchicago.edu/∼mitya/langlands.html.
A. Beilinson and V. Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, 51. AMS, Providence, RI, 2004. vi+375 pp.
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241 (1984), no. 2, 333–380.
R.E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068-3071.
A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic D-modules. Perspectives inMathematics, 2. Academic Press, Inc., Boston, MA, 1987. xii+355 pp.
R. Donagi and T. Pantev, Langlands duality for Hitchin systems, arXiv:math/0604617
V. Drinfeld, Two-dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2). Amer. J. Math. 105 (1983), no. 1, 85–114.
V.G. Drinfeld and C. Simpson, B-structures on G-bundles and local triviality. Math. Res. Lett. 2 (1995), no. 6, 823–829.
E. Frenkel, Lectures on the Langlands program and conformal field theory. Frontiers in number theory, physics, and geometry. II, 387–533, Springer, Berlin, 2007. arXiv:hep-th/0512172
E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture. J. Amer. Math. Soc. 15 (2002), no. 2, 367–417.
E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves. Second edition. Mathematical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, 2004. xiv+400 pp.
D. Gaitsgory, Notes on 2D conformal field theory and string theory. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 1017–1089, Amer. Math. Soc. Providence, RI, 1999. arXiv:math/9811061v2
D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence. Ann. of Math. (2) 160 (2004), no. 2, 617–682.
V. Ginzburg, Perverse sheaves on a loop group and Langlands’ duality, arXiv: alg-geom/9511007
V. Ginzburg, The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519.
N. Hitchin, Langlands duality and G 2 spectral curves. Q. J. Math. 58 (2007), no. 3, 319–344. arXiv:math/0611524
L. Ilusie, Complexe cotangent et déformations. I. Lecture Notes in Mathematics, Vol. 239. Springer-Verlag, Berlin-New York, 1971. xv+355 pp.
L. Ilusie, Complexe cotangent et déformations. II. Lecture Notes in Mathematics, Vol. 283. Springer-Verlag, Berlin-New York, 1972. vii+304 pp.
A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1 (2007), no. 1, 1-236. arXiv:hep-th/0604151
L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147 (2002), no. 1, 1–241.
G. Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math J. 54 (1987) 309–359.
G. Laumon, L. Moret-Bailly, Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 39. Springer-Verlag, Berlin, 2000. xii+208 pp.
Y. Laszlo, M. Olsson, The six operations for sheaves on Artin stacks. I. Finite coefficients. Publ. Math. Inst. Hautes Études Sci. 107 (2008), 109–168.
Y. Laszlo, M. Olsson, The six operations for sheaves on Artin stacks. II. Adic coefficients. Publ. Math. Inst. Hautes Études Sci. 107 (2008), 169–210.
Y. Laszlo, M. Olsson, Perverse t-structure on Artin stacks. Math. Z. 261 (2009), no. 4, 737–748.
I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math (2) 166 (2007), 95–143.
L. Moret-Bailly, Un problème de descente. Bull. Soc. Math. France 124 (1996), no. 4, 559–585.
M. Olsson, Sheaves on Artin stacks. J. reine angew. Math. 603 (2007), 55–112.
C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves. School on Algebraic Geometry (Trieste, 1999), 1–57. ICTP Lect. Notes, 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000. (Available at http://publications.ictp.it).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Basel AG
About this paper
Cite this paper
Gómez, T.L. (2010). Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture. In: Schmitt, A. (eds) Affine Flag Manifolds and Principal Bundles. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0346-0288-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0288-4_2
Publisher Name: Springer, Basel
Print ISBN: 978-3-0346-0287-7
Online ISBN: 978-3-0346-0288-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)