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.


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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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