MathDBAblowitz–Ladik conjecture for the resolved conifold
Ablowitz–Ladik conjecture for the resolved conifold
Let be the resolved conifold equipped with the anti-diagonal action, and consider its equivariantly Calabi–Yau case. Let denote its all-genus, full-descendent Gromov–Witten potential. Ablowitz–Ladik conjecture for the resolved conifold. The potential is the logarithm of a -function of the Ablowitz–Ladik hierarchy. The principal hierarchy for the resolved conifold coincides with the long-wave limit of the Ablowitz–Ladik lattice, providing a candidate dispersive deformation, but the supplied text does not establish that this conjecture has been proved.
Sources & referencesView supporting material
Primary source
Andrea Brini, “The local Gromov-Witten theory of CP^1 and integrable hierarchies”, arXiv:1002.0582 (2012).
Progress summary
The full conjecture remains open, but the primary and stationary parts are proved in every genus and earlier work verified the first few genera.
Brini formulated the conjecture that the resolved-conifold Gromov–Witten potential is the logarithm of an Ablowitz–Ladik tau-function. The full statement includes every genus and all descendants.
Known results
- Brini proved the descendent correspondence for genus .
- Brini found primary-sector agreement at genus , up to the constant-map contribution.
- Takasaki proved the stationary-sector correspondence in all genera.
2022 all-genus primary proof
A later paper gave a direct proof of the conjecture on the small phase space: the primary Gromov–Witten potential agrees with an Ablowitz–Ladik tau-function in every genus. This does not prove the full descendant statement, and no later source retrieved here reports such a proof, counterexample, or withdrawn claim.
Current status (as of August 2026): The primary and stationary sectors are settled to all genera, while the full all-genus, full-descendent conjecture remains open.
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