01/10/2023: Clusters and twistors

01/11/2021, 01/13/2021, and 01/14/2021: From Donaldson-Thomas invariants to complex hyperkahler structures (3 lecture series)

### January 10, 2023

TITLE: Clusters and twistors

ABSTRACT: Given an ADE quiver Q I will explain how to construct a complex manifold Z with a map to P^{1} whose fibre over 0 is the stability space of the CY3 Ginzburg algebra of Q, quotiented by spherical twists, and whose general fibre is an etale cover of the cluster Poisson variety of Q. This is joint work with Helge Ruddat.

### January 11, 2021, January 13, 2021, and January 14, 2021

TITLE: From Donaldson-Thomas invariants to complex hyperkahler structures

ABSTRACT: I will report on an ongoing project which aims to use the DT invariants of a CY3 triangulated category to encode a geometric structure on its stability space. The basic idea is to interpret DT invariants as defining non-linear Stokes factors, as in the work of Gaiotto, Moore and Neitzke. Lecture 1 will be mostly background material: I will discuss stability conditions, the wall-crossing formula for DT invariants, and Stokes data. Lecture 2 will be about the particular type of complex hyperkahler structure we expect to find on stability space: I will give a local description involving Plebanski’s second heavenly equation and discuss a (partly conjectural) class of examples relating to moduli spaces of holomorphic connections on rank 2 vector bundles over Riemann surfaces. Lecture 3 will be about attempting to construct the complex hyperkahler structure on stability space from the DT invariants: this involves a class of Riemann-Hilbert problems for maps from the complex plane into a group of symplectic automorphisms; I will discuss their solutions in some simple examples.