  Recently T. Bridgeland defined a complex hyperk\"ahler metric on the tangent
bundle over the space of stability conditions of a triangulated category, based
on a Riemann-Hilbert problem determined by the Donaldson-Thomas invariants.
This metric is encoded in a function $W(z,\theta)$ satisfying a heavenly
equation, or a potential $F(z,\theta)$ satisfying an isomonodromy equation.
After recasting the RH problem into a system of TBA-type equations, we obtain
integral expressions for both $W$ and $F$ in terms of solutions of that system.
These expressions are recognized as conformal limits of the `instanton
generating potential' and `contact potential' appearing in studies of
D-instantons and BPS black holes. By solving the TBA equations iteratively, we
reproduce Joyce's original construction of $F$ as a formal series in the
rational DT invariants. Furthermore, we produce similar solutions to deformed
versions of the heavenly and isomonodromy equations involving a non-commutative
star-product. In the case of a finite uncoupled BPS structure, we rederive the
results previously obtained by Bridgeland and obtain the so-called $\tau$
function for arbitrary values of the fiber coordinates $\theta$, in terms of a
suitable two-variable generalization of Barnes' $G$ function.
