On the integrable hierarchy for the resolved conifold

TL;DR

This paper proves a conjecture linking the Gromov-Witten theory of the resolved conifold to the Ablowitz-Ladik integrable hierarchy, using functional and difference equations, and relates solutions to Bridgeland's Tau functions.

Contribution

It provides a direct proof of the conjecture connecting Gromov-Witten invariants to an integrable hierarchy at the level of primaries, utilizing new functional representations.

Findings

Established a functional representation of the Ablowitz-Ladik hierarchy.

Expressed solutions of the Gromov-Witten difference equation via Bridgeland's Tau functions.

Confirmed the conjectural relationship between Gromov-Witten theory and integrable hierarchies.

Abstract

We provide a direct proof of a conjecture of Brini relating the Gromov-Witten theory of the resolved conifold to the Ablowitz-Ladik integrable hierarchy at the level of primaries. In doing so, we use a functional representation of the Ablowitz-Ladik hierarchy as well as a difference equation for the Gromov-Witten 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.