Instanton moduli spaces and W-algebras

Abstract: I will explain my recent joint work with Braverman-Finkelberg. Cohomology groups of Hilbert schemes of points on the complex plane are known to be a Fock space, that is a representation of Heisenberg algebra. This result is generalized to the moduli spaces of torsion free framed sheaves on the projective plane by Maulik-Okounkov and Schiffmann-Vasserot last year. The cohomology groups are equivariant ones under the natural torus action on moduli spaces. Heisenberg algebra is generalized to the W-algebra of general linear groups. Now we further generalize this result to moduli spaces of framed holomorphic principal bundles whose structure groups are ADE groups. The cohomology groups are now the equivariant intersection ones, and we have the W-algebra representation on them. This result confirms a special case of the so-called AGT conjecture for N=2 SUSY gauge theories.

here is rough plan

First day. Hilbert schemes and Heisenberg algebras: I will cover results in my book and a few new results, like Virasoro algebra.

Second day. 1st half : Uhlenbeck spaces I will explain basic properties of Uhlenbeck spaces.

2nd half : W-algebras I will explain basics on W-algebras

Third day. I will explain our new results.

nakajima@kurims.kyoto-u.ac.jp