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

Understand

We construct a representation of the affine W-algebra of gl_r on the equivariant homology space of the moduli space of U_r-instantons on A^2, and 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).
• Another proof has been announced by Maulik and Okounkov.
• Our approach uses a suitable deformation of the universal enveloping algebra of the Witt algebra W_{1+\infty}, which is shown to act on the above homology spaces (for any r) and which specializes to all W(gl_r).