24 research outputs found
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. 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.Comment: 27 pages, 1 figur
Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed
We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson
brackets of hydrodynamic type vanishes for almost all degrees. This implies the
existence of a full dispersive deformation of a semisimple bihamiltonian
structure of hydrodynamic type starting from any infinitesimal deformation.Comment: 22 pages. v2: corrected typos. v3: small improvements of the
presentation. v4: typos, small improvements in the introduction and the
presentatio
Reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures
We study finite-dimensional reductions of the dispersionless 2D Toda
hierarchy showing that the consistency conditions for such reductions are given
by a system of radial Loewner equations. We then construct their Hamiltonian
structures, following an approach proposed by Ferapontov.Comment: 15 page
Bihamiltonian cohomology of KdV brackets
Using spectral sequences techniques we compute the bihamiltonian cohomology
groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In
particular this proves a conjecture of Liu and Zhang about the vanishing of
such cohomology groups.Comment: 16 pages. v2: corrected typos, in particular formulas (28), (78
The bi-Hamiltonian cohomology of a scalar Poisson pencil
We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless
Poisson pencil in a single dependent variable using a spectral sequence method.
As in the KdV case, we obtain that is isomorphic to
for , to for ,
, , , and vanishes otherwise