  We consider two interesting spaces associated to a quiver with potential: a
space of stability conditions and a cluster variety. In the case where the
quiver with potential arises from an ideal triangulation of a marked bordered
surface, we construct a natural map from a dense subset of the space of
stability conditions to the cluster variety. Using this construction, we give
solutions to a family of Riemann-Hilbert problems arising in Donaldson-Thomas
theory.
