A Finite Analog of the AGT Relation I: Finite WAlgebras and Quasimaps' Spaces
Abstract
Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4dimensional supersymmetric gauge theory for a gauge group G with certain 2dimensional 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 Gbundles on {mathbb{P}^2} . More precisely, it predicts the existence of an action of the corresponding Walgebra 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 quasimaps 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 Walgebra {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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 December 2011
 DOI:
 10.1007/s0022001113003
 arXiv:
 arXiv:1008.3655
 Bibcode:
 2011CMaPh.308..457B
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematics  Representation Theory
 EPrint:
 minor changes