Quantum Riemann-Hilbert problems for the resolved conifold

We study the quantum Riemann-Hilbert problems determined by the refined
Donaldson-Thomas theory on the resolved conifold. Using the solutions to
classical Riemann-Hilbert problems by Beidgeland, we give explicit solutions in
terms of multiple sine functions with unequal parameters. The new feature of
the solutions is that the valid region of the quantum parameter
$q^{\frac{1}{2}}=\exp(\pi i \tau)$ varies on the space of stability conditions
and BPS $t$-plane. Comparing the solutions with the partition function of
refined Chern-Simons theory and invoking large $N$ string duality, we find that
the solution contains the non-perturbative completion of the refined
topological string on the resolved conifold. Therefore solving the quantum
Riemann-Hilbert problems provides a possible non-perturbative definition for
the Donaldson-Thomas theory.