Quantum corrections and wall-crossing via Lagrangian intersections

This article introduces the past and ongoing works on quantum corrections in SYZ from the author’s perspective. It emphasizes on a method of gluing local pieces of mirrors using isomorphisms between immersed Lagrangians, which is an ongoing joint work with Cho and Hong. It gives a canonical construction of mirrors and generalizes the SYZ setting

Generalized SYZ mirror transformation

Strominger-Yau-Zaslow proposed that mirror symmetry can be understood by torus duality. In this article we explain how it fits into a bigger framework, where tori are replaced by general Lagrangian immersions. The generalized construction is applicable to a wider class of geometries. We also give a brief introduction to our ongoing work on gluing local mirrors into global geometries

Moduli of Lagrangian immersions with formal deformations

Partly presented in the Gokova Geometry/Topology Conference 2017.We introduce a joint project with Cheol-Hyun Cho on the construction of quantum-corrected moduli of Lagrangian immersions. The construction has important applications to mirror symmetry for pair-of-pants decompositions, SYZ and wall-crossing. The key ingredient is Floer-theoretical gluing between local moduli spaces of Lagrangians with different topologies

Geometric transitions and SYZ mirror symmetry

We prove that the punctured generalized conifolds and punctured orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. This mathematically confirms the gauge-theoretic prediction by Aganagic-Karch-L\"ust-Miemiec, and also provides a supportive evidence to Morrison's conjecture that geometric transitions are reversed under mirror symmetry.Comment: v3: one compact example added. 25 pages, 12 figure

Local Calabi-Yau manifolds of type \tilde{A} via SYZ mirror symmetry

We carry out the SYZ program for the local Calabi--Yau manifolds of type $\widetilde{A}$ 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.Comment: v3: 43 pages, 12 figures, improved expositio

Moduli of Lagrangian immersions with formal deformations

We introduce a joint project with Cheol-Hyun Cho on the construction of quantum-corrected moduli of Lagrangian immersions. The construction has important applications to mirror symmetry for pair-of-pants decompositions, SYZ and wall-crossing. The key ingredient is Floer-theoretical gluing between local moduli spaces of Lagrangians with different topologies.Comment: 23 pages, 12 figures, partly presented in the Gokova Geometry/Topology Conference 201