Clusters, twistors and stability conditions I

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.