  We describe the relationship between two spaces associated to a quiver with
potential. The first is a complex manifold parametrizing Bridgeland stability
conditions on a triangulated category, and the second is a cluster variety with
a natural Poisson structure. For quivers of type $A$, we construct a local
biholomorphism from the space of stability conditions to the cluster variety.
The existence of this map follows from results of Sibuya in the classical
theory of ordinary differential equations.
