**Mini-workshop on Quiver Varieties and Related Topics**

Mathematical Institute, University of Oxford, 24-25 June 2022

Speakers: Chris Beem (Oxford), Dylan Butson (Oxford), Alastair Craw (Bath), Adam Gyenge (Renyi Institute, Budapest), Hiraku Nakajima (Kavli IPMU), Yukari Ito (Kavli IPMU), Michael McBreen (Chinese University of Hong Kong), Richard Rimanyi (North Carolina)

If you are interested in participating, please email Balazs Szendroi on [surname]ATmathsDOToxDOTacDOTuk

All talks are in room L3.

Programme

- Friday 24 June
- 9.15am Adam Gyenge: The Heisenberg category of a category
- 10.45am Dylan Butson: Perverse coherent extensions on Calabi-Yau threefolds and representations of cohomological Hall algebras
- 12pm Chris Beem: Chiral algebras of class S and outer automorphism twists
- 2.15pm Hiraku Nakajima: Involutions on quiver varieties and bow varieties
- 3.45pm Richard Rimanyi: On stable envelopes for quiver and bow varieties

- Saturday 25 June
- 9.15am Alastair Craw: The birational geometry of quiver varieties
- 10.45am Yukari Ito: Existence of crepant resolutions
- 12pm Michael McBreen: Mirror symmetry for affine Slodowy slices

Some abstracts

- Adam Gyenge: The Heisenberg category of a category
- The Heisenberg algebra associated with a lattice is a much investigated object originating in quantum theory. Khovanov introduced recently a categorification of the infinite Heisenberg algebra associated with the free boson or, equivalently, a rank 1 lattice, using a graphical construction involving planar diagrams. We extend Khovanov's graphical construction to derived categories of smooth and projective varieties or, more generally, to categories having a Serre functor. In our case the underlying lattice will be the (numerical) Grothendieck group of the category. We also obtain a 2-representation of our Heisenberg category on a categorical analogue of the Fock space. Joint work with Clemens Koppensteiner and Timothy Logvinenko.

- Hiraku Nakajima: Involutions on quiver varieties and bow varieties
- (Q)uiver varieties of type A are also intersections of (N)ilpotent cone of GL and slices, affine (G)rassmannian slices of GL, and (C)oulomb branches of quiver gauge theories of type A. Either description, except (C), gives natural involutions whose fixed point sets are the same type of varieties (Q),(N),(G) associated with classical groups. I will explain how to describe involutions of (N) and (G) in terms of quiver varieties. These description will have variants in bow varieties, which are conjecturally give new examples of Coulomb branches.

- Alastair Craw: The birational geometry of quiver varieties
- I'll describe work in progress, joint with Gwyn Bellamy and Travis Schedler, in which we show that every small birational model of certain GIT quotients can be obtained by variation of GIT quotient. As an application, every projective crepant resolution of a Nakajima quiver variety is itself a Nakajima quiver variety. The key tool is a new construction of relative Mori Dream Spaces.

- Michael McBreen: Mirror Symmetry for Affine Slodowy Slices
- I will describe certain moduli spaces of Higgs bundles on the line, which may be viewed as multiplicative analogues of Slodowy slices. Work in progress with Roman Bezrukavnikov, Pablo Boixeda and Zhiwei Yun relates the category of microlocal sheaves on such spaces to modular representation theory. I will show how various surprising features of the modular theory find natural interpretations on the microlocal side.

Supported by EPSRC