Selecta Mathematica

, Volume 17, Issue 1, pp 1–46 | Cite as

Algebras of twisted chiral differential operators and affine localization of \({\mathfrak {g}}\) -modules

  • Tomoyuki Arakawa
  • Dmytro Chebotarov
  • Fyodor Malikov


We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest” such modules are irreducible \({{\hat{{\mathfrak{g}}}}}\) -modules, and all irreducible \({{\mathfrak{g}}}\) -integrable \({{\hat{{\mathfrak{g}}}}}\) -modules at the critical level arise in this way.


Rings of differential operators Chiral differential operators Representations 

Mathematics Subject Classification (2010)

Primary 17B69 Secondary 14F10 17B67 


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  1. 1.
    Beilinson A., Bernstein J.: Localisation de \({\mathfrak{g}}\)-modules (French). C. R. Acad. Sci. Paris Se’r. I Math. 292(1), 15–18 (1981)MATHMathSciNetGoogle Scholar
  2. 2.
    Beilinson, A., Bernstein, J.: A proof of Jantzen conjectures. I. M. Gelfand Seminar, 1–50, Adv. Soviet Math. vol. 16, Part 1, American Mathematical Society, Providence, RI (1993)Google Scholar
  3. 3.
    Beilinson, A., Drinfeld, V.: Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, pp. vi+375. American Mathematical Society, Providence, RI. ISBN: 0-8218-3528-9 (2004)Google Scholar
  4. 4.
    Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprintGoogle Scholar
  5. 5.
    Bressler P.: The first Pontryagin class. Compos. Math. 143, 1127–1163 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Frenkel E.: Wakimoto modules, opers and the center at the critical level. Adv. Math. 195(2), 297–404 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Frenkel, E.: Lectures on the Langlands Program and Conformal Field Theory. Frontiers in Number Theory, Physics, and Geometry, vol. II, pp. 387–533. Springer, Berlin (2007)Google Scholar
  8. 8.
    Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves, 2nd edn, Mathematical Surveys and Monographs, vol. 58, AMS (2004)Google Scholar
  9. 9.
    Frenkel E.: Langlands Correspondence for Loop Groups. Cambridge University Press, Cambridge (2007)MATHGoogle Scholar
  10. 10.
    Feigin B., Frenkel E.: Representations of affine Kac-Moody algebras and bosonization. In: Brink, L., Friedan, D., Polyakov, A.M. (eds) V. Knizhnik Memorial Volume, pp. 271–316. World Scientific, Singapore (1990)Google Scholar
  11. 11.
    Feigin B., Frenkel E.: Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras. In: Tsuchiya, A., Eguchi, T., Jimbo, M. (eds) Infinite Analysis, Adv. Series Math. Phys., vol. 16, pp. 197–215. World Scientific, Singapore (1992)Google Scholar
  12. 12.
    Frenkel E., Gaitsgory D.: D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras. Duke Math. J. 125, 279–327 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Frenkel, E., Gaitsgory, D.: Weyl modules and opers without monodromy, arXiv:0706.3725Google Scholar
  14. 14.
    Frenkel, E., Gaitsgory, D.: Local Geometric Langlands Correspondence: the Spherical Case, posted on arXiv:0711.1132Google Scholar
  15. 15.
    Frenkel I., Zhu Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168 (1992)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gorbounov V., Malikov F., Schechtman V.: Gerbes of chiral differential operators. II. Vertex algebroids. Invent. Math. 155(3), 605–680 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gorbounov V., Malikov F., Schechtman V.: On chiral differential operators over homogeneous spaces. Int. J. Math. Math. Sci. 26(2), 83–106 (2001)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kac, V.: Vertex algebras for beginners. University Lecture Series, 10. American Mathematical Society, Providence, RI, pp. viii+141, ISBN: 0-8218-0643-2 (1997)Google Scholar
  19. 19.
    Linshaw, A.: Invariant chiral differential operators and W 3 algebra, arXiv:0710.0194Google Scholar
  20. 20.
    Li H., Yamskulna G.: On certain vertex algebras and their modules associated with vertex algebroids. J. Algebra 283(1), 367–398 (2005)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Malikov F.: Verma modules over Kac-Moody algebras of rank 2. Leningrad Math. J. 2(2), 269–286 (1991)MATHMathSciNetGoogle Scholar
  22. 22.
    Malikov F., Schechtman V.: Chiral de Rham complex II, D.B.Fuchs’ 60-th anniversary volume. Amer. Math. Soc. Transl. 194, 149–188 (1999)MathSciNetGoogle Scholar
  23. 23.
    Malikov F., Schechtman V., Vaintrob A.: Chiral de Rham complex II. Comm. Math. Phys. 204, 439–473 (1999)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rosellen, M.: A course in vertex algebra, arXiv:math/0607270Google Scholar
  25. 25.
    Zhu, M.: Vertex operator algebras associated to modified regular representations of affine Lie algebras. arXiv:math/0611517Google Scholar
  26. 26.
    Wakimoto M.: Fock representations of the affine Lie algebra \({A\sp {(1)}\sb 1}\). Comm. Math. Phys. 104(4), 605–609 (1986)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Zhu Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9(1), 237–302 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Tomoyuki Arakawa
    • 1
  • Dmytro Chebotarov
    • 2
  • Fyodor Malikov
    • 2
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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