The local Gromov-Witten theory of CP^1 and integrable hierarchies

TL;DR

This paper explores the connection between local Gromov-Witten invariants of certain Calabi-Yau bundles over P^1 and integrable hierarchies, constructing new hierarchies and relating them to known lattice systems.

Contribution

It explicitly constructs new Hamiltonian dispersionless hierarchies governing genus zero Gromov-Witten theory and relates the resolved conifold hierarchy to the Ablowitz-Ladik lattice.

Findings

New hierarchies for genus zero Gromov-Witten invariants

Relation between resolved conifold hierarchy and Ablowitz-Ladik lattice

Evidence supporting the higher genus Ablowitz-Ladik correspondence

Abstract

In this paper we begin the study of the relationship between the local Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full descendent genus zero theory. Our main tool is the application of Dubrovin's formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of the resolved conifold O(-1) + O(-1) -> P1 in the equivariantly Calabi-Yau case. For this example the relevant dispersionless system turns out to be related to the long-wave limit of the Ablowitz-Ladik lattice. This identification provides us with a complete procedure to reconstruct the dispersive hierarchy which should conjecturally be related to the higher genus theory of the resolved conifold. We give a complete proof of this conjecture for genus g<=1; our methods are based on establishing, analogously to the case of KdV, a "quasi-triviality" property for the Ablowitz-Ladik hierarchy at the leading order of the dispersive expansion. We furthermore provide compelling evidence in favour of the resolved conifold/Ablowitz-Ladik correspondence at higher genus by testing it successfully in the primary sector for g=2.