On the integrable hierarchy for the resolved conifold

Abstract

We provide a direct proof of a conjecture of Brini relating the Gromov–Witten (GW) theory of the resolved conifold to the Ablowitz–Ladik (AL) integrable hierarchy at the level of primaries. In doing so, we use a functional representation of the AL hierarchy as well as a difference equation for the GW potential. In particular, we express certain distinguished solutions of the difference equation in terms of 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.