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Tom Bridgeland: Lectures

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

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.

Slides of first lecture

Slides of second lecture

Slides of third lecture