Lau, SiuKanazawa, Atsushi2019-04-222019-04-222019-05Atsushi Kanazawa, Siu Lau. "Local Calabi-Yau manifolds of affine type A˜ via SYZ mirror symmetry." Journal of Geometry and Physics, Volume 139, May 2019, Pages 103-138. https://doi.org/10.1016/j.geomphys.2018.12.0150393-0440https://hdl.handle.net/2144/34881We carry out the SYZ program for the local Calabi–Yau manifolds of type A˜ by developing an equivariant SYZ theory for the toric Calabi–Yau manifolds of infinite-type. Mirror geometry is shown to be expressed in terms of the Riemann theta functions and generating functions of open Gromov–Witten invariants, whose modular properties are found and studied in this article. Our work also provides a mathematical justification for a mirror symmetry assertion of the physicists Hollowood–Iqbal–Vafa (Hollowood et al., 2008).Mathematical sciencesPhysical sciencesMathematical physicsCalabi–Yau manifoldsSYZ mirror symmetryRiemann theta functionsToric geometryOpen Gromov-Witten invariantsAbelian varietiesArticle10.1016/j.geomphys.2018.12.015