# Chern-Simons Research Lectures on December 2nd, 4th, and 6th

Monday, Dec. 2, Evans 939, from 1-3pm

Wednesday, Dec. 4, Evans 891, from 2-4pm

Friday, Dec. 6, Evans 939, from 3-5pm

Speaker: Hiraku Nakajima, RIMS, Kyoto.

Title: Instanton moduli spaces and W-algebras.

Abstract:

These lectures are an exposition of 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.

Dec. 2: Hilbert schemes and Heisenberg algebras.

I will cover some results from my book (http://www.amazon.com/Lectures-Hilbert-Schemes-Surfaces-University/dp/0821819569) and a few new results, like Virasoro algebra.

Dec. 4: I will start with a review of basic properties of Uhlenbeck spaces.

It will follow by the review of basics on W-algebras

Dec. 6: I will explain our new results.