Advertisement

Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture

  • Tomás L. Gómez
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

This is an introduction to the work of Beilinson and Drinfeld [6]_on the Langlands program.

Mathematics Subject Classification (2000)

Primary 11R39 Secondary 14D20 

Keywords

Langlands program Chiral algebras G-opers Hitchin integrable system D-modules 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Arkhipov and D. Gaitsgory, Differential operators on the loop group via chiral algebras. Int. Math. Res. Not. 2002, no. 4, 165–210.Google Scholar
  2. [2]
    A. Beauville and Y. Laszlo, Un lemme de descente. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340.zbMATHMathSciNetGoogle Scholar
  3. [3]
    K. Behrend, Differential Graded Schemes II: The 2-category of Differential Graded Schemes. Preprint. arXiv:math/0212226.Google Scholar
  4. [4]
    A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisque 100 (1982).Google Scholar
  5. [5]
    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.Google Scholar
  6. [6]
    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.Google Scholar
  7. [7]
    A. Beilinson and V. Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, 51. AMS, Providence, RI, 2004. vi+375 pp.Google Scholar
  8. [8]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R.E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068-3071.Google Scholar
  10. [10]
    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.Google Scholar
  11. [11]
    R. Donagi and T. Pantev, Langlands duality for Hitchin systems, arXiv:math/0604617Google Scholar
  12. [12]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    V.G. Drinfeld and C. Simpson, B-structures on G-bundles and local triviality. Math. Res. Lett. 2 (1995), no. 6, 823–829.zbMATHMathSciNetGoogle Scholar
  14. [14]
    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/0512172Google Scholar
  15. [15]
    E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture. J. Amer. Math. Soc. 15 (2002), no. 2, 367–417.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    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.zbMATHGoogle Scholar
  17. [17]
    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/9811061v2Google Scholar
  18. [18]
    D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence. Ann. of Math. (2) 160 (2004), no. 2, 617–682.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    V. Ginzburg, Perverse sheaves on a loop group and Langlands’ duality, arXiv: alg-geom/9511007Google Scholar
  20. [20]
    V. Ginzburg, The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    N. Hitchin, Langlands duality and G 2 spectral curves. Q. J. Math. 58 (2007), no. 3, 319–344. arXiv:math/0611524zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    L. Ilusie, Complexe cotangent et déformations. I. Lecture Notes in Mathematics, Vol. 239. Springer-Verlag, Berlin-New York, 1971. xv+355 pp.Google Scholar
  23. [23]
    L. Ilusie, Complexe cotangent et déformations. II. Lecture Notes in Mathematics, Vol. 283. Springer-Verlag, Berlin-New York, 1972. vii+304 pp.Google Scholar
  24. [24]
    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/0604151Google Scholar
  25. [25]
    L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147 (2002), no. 1, 1–241.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    G. Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math J. 54 (1987) 309–359.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    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.Google Scholar
  28. [28]
    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.zbMATHMathSciNetGoogle Scholar
  29. [29]
    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.zbMATHMathSciNetGoogle Scholar
  30. [30]
    Y. Laszlo, M. Olsson, Perverse t-structure on Artin stacks. Math. Z. 261 (2009), no. 4, 737–748.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math (2) 166 (2007), 95–143.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    L. Moret-Bailly, Un problème de descente. Bull. Soc. Math. France 124 (1996), no. 4, 559–585.zbMATHMathSciNetGoogle Scholar
  33. [33]
    M. Olsson, Sheaves on Artin stacks. J. reine angew. Math. 603 (2007), 55–112.zbMATHMathSciNetGoogle Scholar
  34. [34]
    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).Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Tomás L. Gómez
    • 1
    • 2
  1. 1.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain
  2. 2.Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

Personalised recommendations