Quantum Riemann-Hilbert problems for the resolved conifold

Abstract

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 in [6] 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 q12=exp⁡(πiτ) 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.