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

作者：Braverman Alexander^*; Feigin Boris; Finkelberg Michael; Rybnikov Leonid

来源：Communications in Mathematical Physics, 2011, 308(2): 457-478.

DOI：10.1007/s00220-011-1300-3

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 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 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(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(G, e) in terms of certain shifted Yangians.

• 出版日期2011-12