  We study the Borel summation of the Gromov-Witten potential for the resolved
conifold. The Stokes phenomena associated to this Borel summation are shown to
encode the Donaldson-Thomas invariants of the resolved conifold, having a
direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland.
There exist distinguished integration contours for which the Borel summation
reproduces previous proposals for the non-perturbative topological string
partition functions of the resolved conifold. These partition functions are
shown to have another asymptotic expansion at strong topological string
coupling. We demonstrate that the Stokes phenomena of the strong-coupling
expansion encode the DT invariants of the resolved conifold in a second way.
Mathematically, one finds a relation to Riemann-Hilbert problems associated to
DT invariants which is different from the one found at weak coupling. The
Stokes phenomena of the strong-coupling expansion turn out to be closely
related to the wall-crossing phenomena in the spectrum of BPS states on the
resolved conifold studied in the context of supergravity by D. Jafferis and G.
Moore.
