Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A 2

Abstract

We construct a representation of the affine W-algebra of ${\mathfrak{g}}{\mathfrak{l}}_{r}$ on the equivariant homology space of the moduli space of U r -instantons, and we identify the corresponding module. As a corollary, we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r). Our approach uses a deformation of the universal enveloping algebra of W 1+∞, which acts on the above homology space and which specializes to $W({\mathfrak{g}}{\mathfrak{l}}_{r})$ for all r. This deformation is constructed from a limit, as n tends to ∞, of the spherical degenerate double affine Hecke algebra of GL n .