Integrable hierarchy for the resolved conifold

We provide a direct proof of a conjecture of Brini relating the Gromov-Witten theory of the resolved confiold to the Ablowitz-Ladik integrable hierarchy. We use a functional representation of the Ablowitz-Ladik hierarchy as well as a difference equation for the Gromov-Witten potential. We provide the solution of the difference equation as an analytic function which is a specialization of a Tau function put forward by Bridgeland in the study of wall-crossing phenomena of Donaldson-Thomas invariants.