A Finite Analog of the AGT Relation I: Finite W-Algebras and Quasimaps’ Spaces

  • Alexander Braverman
  • Boris Feigin
  • Michael Finkelberg
  • Leonid Rybnikov


Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on \({\mathbb{P}^2}\) . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties.

We propose a “finite analog” of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from \({\mathbb{P}^1}\) to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra \({U(\mathfrak{g},e)}\) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of [5] when P is the Borel subgroup. We prove our conjecture for G = GL(N), using the works of Brundan and Kleshchev interpreting the algebra \({U(\mathfrak{g},e)}\) in terms of certain shifted Yangians.


Modulus Space Borel Subgroup Verma Module Equivariant Cohomology Nekrasov Partition Function 
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  1. 1.
    Alday L.F., Gaiotto D., Gukov S., Tachikawa Y., Verlinde H.: Loop and surface operators in N = 2 gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91(2), 167–197 (2010)MathSciNetADSzbMATHGoogle Scholar
  3. 3.
    Alday L.F., Tachikawa Y.: Affine SL(2) conformal blocks from 4d gauge theories. Lett. Math. Phys. 94(1), 87–114 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Awata H., Yamada Y.: Five-dimensional AGT Relation and the deformed β-ensemble. Prog. Theor. Phys. 124, 227–262 (2010)ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Braverman, A.: Instanton counting via affine Lie algebras. I. Equivariant J-functions of (affine) flag manifolds and Whittaker vectors. In: Algebraic structures and moduli spaces, CRM Proc. Lecture Notes 38, Providence, RI: Amer. Math. Soc., 2004, pp. 113–132Google Scholar
  6. 6.
    Braverman, A.: Spaces of quasi-maps and their applications. In: International Congress of Mathematicians. Vol. II, Zürich: Eur. Math. Soc., 2006, pp. 1145–1170Google Scholar
  7. 7.
    Braverman A., Etingof P.: Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential. In: Studies in Lie theory, Progr. Math., 243, pp. 61–78. Birkhäuser Boston, Boston (2006)Google Scholar
  8. 8.
    Braverman A., Finkelberg M., Gaitsgory D.: Uhlenbeck spaces via affine Lie algebras. In: The unity of mathematics (volume dedicated to I. M. Gelfand’s 90th birthday), Progr. Math. 244, pp. 17–135. Birkhäuser Boston, Boston, MA (2006)Google Scholar
  9. 9.
    Braverman A., Finkelberg M., Gaitsgory D., Mirković I.: Intersection cohomology of Drinfeld’s compactifications. Selecta Math. (N.S.) 8(3), 381–418 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brundan J., Goodwin S.M.: Good grading polytopes. Proc. London Math. Soc. 94, 155–180 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Brundan, J., Goodwin, S., Kleshchev, A.: Highest weight theory for finite W-algebras. Int. Math. Res. Not. 2008, Art. ID rnn051, 53 pp. (2008)Google Scholar
  12. 12.
    Brundan J., Kleshchev A.: Representations of shifted Yangians and finite W-algebras. Mem. Amer. Math. Soc. 196, no. 918. Amer. Math. Soc., Providence, RI (2008)Google Scholar
  13. 13.
    Etingof P.: Whittaker functions on quantum groups and q-deformed Toda operators. In: Differential topology, infinite-dimensional Lie algebras, and applications, pp. 9–25. Amer. Math. Soc., Providence, RI (1999)Google Scholar
  14. 14.
    Feigin B., Frenkel E.: Representations of affine Kac-Moody algebras, bosonization and resolutions. Lett. Math. Phys. 19, 307–317 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Finkelberg, M., Mirković, I.: Semi-infinite flags. I. Case of global curve \({\mathbb{P}^1}\) . In: Differential topology, infinite-dimensional Lie algebras, and applications , Amer. Math. Soc. Transl. Ser. 2, 194, Providence,RI: Amer. Math. Soc., 1999, pp. 81–112Google Scholar
  16. 16.
    Feigin B., Finkelberg M., Kuznetsov A., Mirković I.: Semi-infinite flags. II. Local and global intersection cohomology of quasimaps’ spaces. In: Differential topology, infinite-dimensional Lie algebras, and applications, pp. 113–148. Amer. Math. Soc., Providence, RI (1999)Google Scholar
  17. 17.
    Feigin, B., Finkelberg, M., Negut, A., Rybnikov, L.: Yangians and cohomology rings of Laumon spaces. Selecta Math. [math.AG], (2011, to appear)
  18. 18.
    Futorny, V., Molev, A., Ovsienko, S.: Gelfand-Tsetlin bases for representations of finite W-algebras and shifted Yangians. In: “Lie theory and its applications in physics VII”, H. D. Doebner, V. K. Dobrev, eds., Proceedings of the VII International Workshop, Varna, Bulgaria, June 2007, Sofia: Heron Press, 2008, pp. 352–363Google Scholar
  19. 19.
    Givental A., Kim B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168(3), 609–641 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Kim B.: cohomology of flag manifolds G/B and quantum Toda lattices. Ann. of Math. 149((2), 129–148 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Laumon G.: Un Analogue Global du Cône Nilpotent. Duke Math. J 57, 647–671 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Laumon G.: Faisceaux Automorphes Liés aux Séries d’Eisenstein. Perspect. Math. 10, 227–281 (1990)MathSciNetGoogle Scholar
  23. 23.
    Mironov A., Morozov A.: On AGT relation in the case of U(3). Nucl. Phys. B 825, 1–37 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Marshakov A., Mironov A., Morozov A.: On non-conformal limit of the AGT relations. Phys. Lett. B 682(1), 125–129 (2009)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Maulik, D., Okounkov, A.: In preparationGoogle Scholar
  26. 26.
    Taki M.: On AGT Conjecture for Pure Super Yang-Mills and W-algebra. JHEP 1105, 038 (2011)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Alexander Braverman
    • 1
  • Boris Feigin
    • 2
    • 3
  • Michael Finkelberg
    • 3
    • 4
    • 5
  • Leonid Rybnikov
    • 3
    • 5
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.L.D. Landau Institute for Theor. Phys.MoscowRussia
  3. 3.Department of MathematicsNational Research University Higher School of EconomicsMoscowRussia
  4. 4.Independent Moscow UniversityMoscowRussia
  5. 5.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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