Integrable hierarchies and the mirror model of local CP¹

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. Second, 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.