Clusters, twistors and stability conditions I

TL;DR

This paper explores the geometric structures related to ADE quivers using cluster combinatorics, defining complex manifolds linked to stability conditions and Poisson spaces, with special cases described via quadratic differentials.

Contribution

It introduces new complex manifolds associated with ADE quivers, connecting stability conditions, cluster spaces, and geometric structures, and establishes local homeomorphisms and descriptions for type A.

Findings

Identifies a quotient space of stability conditions with a complex manifold.

Establishes a local homeomorphism to the complex cluster Poisson space.

Provides geometric descriptions for type A quivers as moduli spaces.

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 $Z\to \mathbb C$ 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 Riemann-Hilbert problems defined by the Donaldson-Thomas invariants.