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

Abstract

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 [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.

M-theoretic derivations of 4d-2d dualities: from a geometric Langlands duality for surfaces, to the AGT correspondence, to integrable systems

“…Our derivation physically reproduces the mathematical conjecture by Braverman et al in [20], that the Nekrasov instanton partition function for pure G is given by the norm of a coherent state in the Verma module over the Langlands dual affine W-algebra. Furthermore, the underlying Seiberg-Witten curve -interpreted as an N -or 2N -fold cover of the two-punctured Gaiotto curve C [16,3] -also arises naturally in our picture.…”

“…, α (M ) n M times ), (2.19) where n i > n i+1 . 7 This means that the Levi subgroup is 20) where N = n 1 + n 2 + · · · + n M . It is in this sense that the underlying 4d worldvolume defect can be characterized by the partition [n 1 , n 2 , .…”

“…The basis for the AGT correspondence for SU (N ) -as first pointed out by Alday and Tachikawa in [19] -is a conjectured relation between the equivariant cohomology of the moduli space of SU (N )-instantons and the integrable representations of an affine W N -algebra. This conjectured relation was first proved mathematically for finite W N -algebras in [20], and later proved mathematically for affine W N -algebras in [21,22]. An original effort to furnish a fundamental physical derivation of the AGT correspondence from the viewpoint of 6d N = (2, 0) SCFT was also undertaken by Yagi in [23,24], although certain assumptions made in loc.cit.…”

“…The insertion of P 2 then means that in computing the trace over H 0 BPS , one must consider only O 0 's which are invariant under the Z 2 -transformations ϕ → −ϕ 20 The reason why one has to add a twisted sector whilst projecting onto Z 2 -invariant states -like in any consistent 2d CFT with a cyclic identification along its spatial direction -is because the spacetime BPS states come from the topological sector of the sigma-model which is therefore conformal. 21 Unlike the commuting bose fields, the anti-commutativity of the fermi fields means that whenever a fermi field passes another in a correlation function as it is being translated around S 1 n , the correlation function picks up a minus sign.…”

The stringy instanton partition function

“…This introduces surface operators in the gauge theory [62] and provides a set-up to compute the quantum cohomology of their moduli spaces, such as for example Laumon spaces [65] and partial flag varieties [66]. Our approach should be compared with the results of [67].…”