  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.
