Tri-Hamiltonian Structure of the Ablowitz-Ladik Hierarchy

TL;DR

This paper establishes a local tri-Hamiltonian structure for the Ablowitz-Ladik hierarchy, computes its central invariants, and links it to Gromov-Witten invariants of local CP1, supporting Brini's conjecture.

Contribution

It constructs a novel local tri-Hamiltonian structure for the Ablowitz-Ladik hierarchy and relates it to Frobenius manifolds and Gromov-Witten invariants.

Findings

Central invariants equal to 1/24

Dispersionless limit matches Frobenius manifold structure

Supports Brini's conjecture connecting Gromov-Witten invariants and Ablowitz-Ladik hierarchy

Abstract

We construct a local tri-Hamiltonian structure of the Ablowitz-Ladik hierarchy, and compute the central invariants of the associated bihamiltonian structures. We show that the central invariants of one of the bihamiltonian structures are equal to 1/24, and the dispersionless limit of this bihamiltonian structure coincides with the one that is defined on the jet space of the Frobenius manifold associated with the Gromov-Witten invariants of local CP1. This result provides support for the validity of Brini's conjecture on the relation of these Gromov-Witten invariants with the Ablowitz-Ladik hierarchy.