English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper we begin the study of the relationship between the localGromov-Witten theory of Calabi-Yau rank two bundles over the projective lineand the theory of integrable hierarchies. We first of all construct explicitly,in a large number of cases, the Hamiltonian dispersionless hierarchies thatgovern the full descendent genus zero theory. Our main tool is the applicationof Dubrovin's formalism, based on associativity equations, to the known resultson the genus zero theory from local mirror symmetry and localization. Thehierarchies we find are apparently new, with the exception of the resolvedconifold O(-1) + O(-1) -> P1 in the equivariantly Calabi-Yau case. For thisexample the relevant dispersionless system turns out to be related to thelong-wave limit of the Ablowitz-Ladik lattice. This identification provides uswith a complete procedure to reconstruct the dispersive hierarchy which shouldconjecturally 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 arebased on establishing, analogously to the case of KdV, a "quasi-triviality"property for the Ablowitz-Ladik hierarchy at the leading order of thedispersive expansion. We furthermore provide compelling evidence in favour ofthe resolved conifold/Ablowitz-Ladik correspondence at higher genus by testingit successfully in the primary sector for g=2.

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