We will describe the space of Bridgeland stability conditions
of the derived category of some CY3 algebras of quivers drawn on the
Riemann sphere. We give a biholomorphic map from the upper-half plane to
the space of stability conditions lifting the period map of a meromorphic
differential on a 1-dimensional family of elliptic curves. The map is
equivariant with respect to the actions of a subgroup of $\mathrm{PSL}(2,\mathbb Z)$ on the
left by monodromy of the rational elliptic surface and on the right by
autoequivalences of the derived category.
The complement of a divisor in the rational elliptic surface can be
identified with Hitchin's moduli space of connections on the projective
line with prescribed poles of a certain order at marked points. This is
the space of initial conditions of one of the Painleve equations whose
solutions describe isomonodromic deformations of these connections.