Clusters, twistors and stability conditions I

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 $\hat{I}\mu \in \mathbb{C}$ is isomorphic to $\mathcal{S}$ when $\hat{I}\mu =0$ 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.

51 pages