Clusters, twistors and stability conditions I

TL;DR

The paper develops a cluster-geometric framework for ADE quivers by constructing three complex manifolds— the complex cluster space $\mathcal{X}_{\mathbb{C}}$, the cluster stability space $\mathcal{S}$, and the log cluster space $\mathcal{L}$—and a cluster twistor space that interpolates between them. It shows $\mathcal{S}$ canonically realizes the space of stability conditions on the CY$_3$ category associated to $Q$ modulo the cluster Br group, while $\mathcal{L}$ maps holomorphically to $\mathcal{X}_{\mathbb{C}}$ via a local homeomorphism, thus linking stability conditions and cluster coordinates. In type $A$, the spaces admit geometric identifications with moduli of framed meromorphic quadratic differentials and framed meromorphic projective structures, enriching the link between cluster theory, differential geometry, and surface theory; the DT element and tropical dualities provide a robust combinatorial handle on these structures. The sequel promises a cluster-twistor construction $\mathcal{Z}\to \mathbb{C}$ that encodes Donaldson–Thomas invariants through Riemann–Hilbert problems and hyperkähler twistor theory, unifying algebraic, geometric, and physical perspectives on stability and wall-crossing.}

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 $ε\in \mathbb C$ is isomorphic to $\mathcal S$ when $ε=0$ 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.

Theorems & Definitions (96)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1
• Definition 2
• Theorem 2.1
• Lemma 1
• proof