On fermionic representation of the Gromov-Witten invariants of the resolved Conifold

TL;DR

This paper demonstrates that the generating function for Gromov-Witten invariants of the resolved conifold can be expressed as a fermionic Bogoliubov transform, linking it to integrable systems and topological vertex formalism.

Contribution

It proves the fermionic form of the generating function is a tau function of the KP hierarchy, confirming conjectures and extending the fermionic representation framework.

Findings

The generating function is a Bogoliubov transform of the fermionic vacuum.

It is a tau function of the KP hierarchy.

The proof utilizes the gluing rule and fermionic formulas for topological vertices.

Abstract

We prove that the fermionic form of the generating function of the Gromov-Witten invariants of the resolved conifold is a Bogoliubov transform of the fermionic vacuum; in particular, it is a tau function of the KP hierarchy. Our proof is based on the gluing rule of the topological vertex and the formulas of the fermionic representations of the framed one-legged and two-legged topological vertex which were conjectured by Aganagic et al and proved in our recent work.