Speaker: Francesco Sala (Kavli IPMU)
Title: K-HA/CoHA of the stack of Higgs sheaves on a curve
Date (JST): Thu, Jan 12, 2017, 15:30 - 17:00
Place: Seminar Room A
Abstract: K-theoretical and Cohomological Hall algebras of preprojective algebras play a preeminent role in algebraic geometry, representation theory and mathematical physics. For example, if the preprojective algebra is the one of the Jordan quiver, the corresponding CoHA is the Maulik-Okounkov Yangian associated with the Jordan quiver. It acts on the equivariant cohomology of Hilbert schemes of points on the complex affine plane ("extending" the previous results of Nakajima, Grojnowski, Vasserot for actions of Heisenberg algebras) and of moduli spaces of framed sheaves on the complex projective plane. The latter action yields an action of W-algebras and hence provides a proof of the Alday-Gaiotto-Tachikawa conjecture for pure supersymmetric gauge theories on the real four-dimensional space.

In the first part of the talk, I will give an introduction to the subject from a mathematical and a physical point of view. In the second part of the talk, I will discuss K-HA/CoHAs associated with the (derived) stack of (nilpotent) Higgs sheaves on a smooth projective complex curve. Their representations can be realized by using the equivariant K- theory/cohomology of moduli spaces of (framed) torsion free sheaves on A-type ALE spaces (i.e., minimal resolutions of (p, p-1)-toric singularities). If time permits, I will state a list of conjectures about the interpretation of these algebras in terms of quantum groups and Yangians respectively, and I will discuss possible applications to the AGT conjectures for gauge theories on A-type ALE spaces.