A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces

TL;DR

This paper proposes a finite analog of the AGT relation, connecting the equivariant intersection cohomology of quasi-map spaces to finite W-algebras, extending previous results and providing a proof for G=GL(N).

Contribution

It introduces a conjecture linking finite W-algebras to the intersection cohomology of quasi-map spaces, generalizing prior work and proving the case for G=GL(N).

Findings

Conjecture established for G=GL(N)

Finite W-algebra acts on cohomology of quasi-map spaces

Extends AGT-related structures to finite setting

Abstract

Recently Alday, Gaiotto and Tachikawa 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 arXiv:math/0401409 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.