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

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.