# MathSciDoc: An Archive for Mathematician ∫

#### Geometric Analysis and Geometric Topologymathscidoc:1605.15003

Journal of Geometry and Physics
We construct SYZ mirrors of the local Calabi--Yau manifolds of type $\widetilde{A}$ by developing an equivariant SYZ theory for the toric Calabi--Yau manifolds of infinite-type. The equations for the SYZ mirrors involve the Riemann theta functions and generating functions of the open Gromov--Witten invariants. We obtain explicit formulae for the generating functions which are open analogs of the Yau--Zaslow formula in dimension $2$, and show that they have nice modular properties. We also relate the SYZ mirror pairs with mirror symmetry for the abelian varieties and hypersurfaces therein.
ADE, SYZ, mirror symmetry, local Calabi-Yau, Yau-Zaslow formula, open Gromov-Witten invariant
@inproceedings{atsushilocal,