Integrable hierarchies and the mirror model of local CP^{1}
Abstract
We study structural aspects of the AblowitzLadik (AL) hierarchy in the light of its realization as a twocomponent reduction of the twodimensional Toda hierarchy, and establish new results on its connection to the GromovWitten theory of local CP^{1}. 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 nonconstant unit, whose properties we thoroughly analyze. We build on this connection along two main strands. First of all, we exhibit a manifestly local biHamiltonian structure of the AblowitzLadik system in the zerodispersion limit. Second, we make precise the relation between this canonical Frobenius structure and the one that underlies the GromovWitten theory of the resolved conifold in the equivariantly CalabiYau 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 CP^{1} in terms of a dual logarithmic LandauGinzburg model.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 December 2012
 DOI:
 10.1016/j.physd.2011.09.011
 arXiv:
 arXiv:1105.4508
 Bibcode:
 2012PhyD..241.2156B
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Symplectic Geometry;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 27 pages, 1 figure