|Jul 08, 2022
The SYZ conjecture suggests a folklore that ``Lagrangian multi-sections are mirror to holomorphic vector bundles". In this talk, I am going to give a proof to this folklore for Lagrangian multi-sections inside cotangent bundle of vector spaces, which are equivariantly mirror to compact toric manifolds by the work of Fang-Liu-Treumann-Zaslow. I will also introduce the converse of this folklore, called the Lagrangian realization conjecture, stating that mirror of toric vector bundles can be quasi-represented by Lagrangian multi-sections. I will prove this conjecture in the case of rank 2 toric vector bundles over projective plane. This is a joint work with Yong-Geun Oh.