Integrable hierarchies and the mirror model of local CP1

TL;DR

This paper explores the Ablowitz-Ladik hierarchy's structure, its relation to the Gromov-Witten theory of local CP1, and derives genus zero mirror symmetry via Frobenius manifolds and duality concepts.

Contribution

It establishes new connections between the Ablowitz-Ladik hierarchy, Frobenius manifolds, and Gromov-Witten theory, including a local bi-Hamiltonian structure and a derivation of mirror symmetry.

Findings

Ablowitz-Ladik hierarchy as a reduction of the 2D Toda hierarchy.

Dispersionless limit described by a conformal Frobenius manifold.

Derivation of genus zero mirror symmetry for local CP1.

Abstract

We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and establish new results on its connection to the Gromov-Witten theory of local CP1. We first of all elaborate on the relation to the Toeplitz lattice and obtain a neat description of the Lax formulation of the AL system. We then study the dispersionless limit and rephrase it in terms of a conformal semisimple Frobenius manifold with non-constant unit, whose properties we thoroughly analyze. We build on this connection along two main strands. First of all, we exhibit a manifestly local bi-Hamiltonian structure of the Ablowitz-Ladik system in the zero-dispersion limit. Secondarily, we make precise the relation between this canonical Frobenius structure and the one that underlies the Gromov-Witten theory of the resolved conifold in the equivariantly Calabi-Yau case; a key role is played by Dubrovin's notion of "almost duality" of Frobenius manifolds. As a consequence, we obtain a derivation of genus zero mirror symmetry for local CP1 in terms of a dual logarithmic Landau-Ginzburg model.