A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
Alexander Braverman, Boris Feigin, Leonid Rybnikov, Michael Finkelberg
TL;DR
This work introduces a finite analog of the AGT correspondence by proposing that the equivariant intersection cohomology of parabolic quasi-map spaces carries an action of finite W-algebras, with a Whittaker vector and a Shapovalov-compatible pairing. The authors formulate this conjecture for general G and prove it in type A by realizing the W-algebra action explicitly on parabolic Laumon spaces for G=GL_N, using a Gelfand–Tsetlin–type description via shifted Yangians. They establish a precise isomorphism between IH_{G,P} and the universal Verma module M(gl_N,e), identify the fixed-point basis with GT patterns, and show the Whittaker vector corresponds to the sum of unit classes, matching the Shapovalov form to the geometric pairing. The paper also situates these finite-W constructions in the broader AGT framework, connecting Nekrasov partition functions, Virasoro/W-algebras, and affine versus finite reductions, thereby linking geometric representation theory with four-dimensional gauge theories through a combinatorial and algebro-geometric toolkit.
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.