Clusters, twistors and stability conditions

Helge Ruddat (University of Stavanger)

Wednesday 25th November 16:00-17:00
Maths 311B

Abstract

We consider a quiver $Q$ of ADE type and use cluster combinatorics  to define two complex manifolds $\mathcal{S}$ and $\mathcal{L}$. The space $ \mathcal{S}$ can be identified with a quotient of the space of stability conditions on the CY_3 category associated to $Q$. The space $\mathcal{L}$ has a canonical map to the complex cluster Poisson space $\mathcal{X}_\mathbb{C}$ which we prove to be a local homeomorphism. When $Q$ is of type $A$,  we give a geometric description of the spaces $\mathcal{S}$ and $\mathcal{L}$ as moduli spaces of meromorphic quadratic differentials and projective structures respectively. In the sequel paper we will introduce a space $\pi\colon Z\to \mathbb{C}$ whose fibre over over a point $\epsilon\in \mathbb{C}$ is isomorphic to  $\mathcal{S}$  when $\epsilon=0$ and to  $\mathcal{L}$ otherwise. The problem of constructing sections of this map gives a  geometric approach  to the Donaldson--Thomas  Riemann--Hilbert problems of [T. Bridgeland,  Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. math. 216, (2019)]. 

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