Abstract
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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Communicated by N.A. Nekrasov
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Braverman, A., Feigin, B., Finkelberg, M. et al. A Finite Analog of the AGT Relation I: Finite W-Algebras and Quasimaps’ Spaces. Commun. Math. Phys. 308, 457–478 (2011). https://doi.org/10.1007/s00220-011-1300-3
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DOI: https://doi.org/10.1007/s00220-011-1300-3