Riemann–Hilbert problems for the resolved conifold and non-perturbative partition functions

We study the Riemann-Hilbert problems of [6] (T. Bridgeland, “Riemann-Hilbert problems from Donaldson–Thomas theory”, arxiv:1611.03697) in the case of the Donaldson–Thomas theory of the resolved conifold. We give explicit solutions in terms of the Barnes double and triple sine functions. We show that the τ-function of [6] is a non-perturbative partition function, in the sense that its asymptotic expansion coincides with the topological closed string partition function.