The Local GromovWitten Theory of {{C}{P}^1} and Integrable Hierarchies
Abstract
In this paper we begin the study of the relationship between the local GromovWitten theory of CalabiYau 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 fulldescendent 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}_{{P}^1}(1) bigoplus {O}_{{P}^1}(1)}} in the equivariantly CalabiYau case. For this example the relevant dispersionless system turns out to be related to the longwave limit of the AblowitzLadik 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 "quasitriviality" property for the AblowitzLadik hierarchy at the leading order of the dispersive expansion. We furthermore provide compelling evidence in favour of the resolved conifold/AblowitzLadik correspondence at higher genus by testing it successfully in the primary sector for g = 2.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 August 2012
 DOI:
 10.1007/s0022001215179
 arXiv:
 arXiv:1002.0582
 Bibcode:
 2012CMaPh.313..571B
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 30 pages