We study the Riemann-Hilbert problems of  (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 $\tau$-function of  is a non-perturbative partition function, in the sense that its asymptotic expansion coincides with the topological closed string partition function.
"Riemann–Hilbert problems for the resolved conifold and non-perturbative partition functions." J. Differential Geom. 115 (3) 395 - 435, July 2020. https://doi.org/10.4310/jdg/1594260015