11 research outputs found
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
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
SYZ mirror symmetry for hypertoric varieties
We construct a Lagrangian torus fibration on a smooth hypertoric variety and
a corresponding SYZ mirror variety using -duality and generating functions
of open Gromov-Witten invariants. The variety is singular in general. We
construct a resolution using the wall and chamber structure of the SYZ base.Comment: v_2: 31 pages, 5 figures, minor revision. To appear in Communications
in Mathematical Physic
Mirror of Atiyah flop in symplectic geometry and stability conditions
We study the mirror operation of the Atiyah flop in symplectic geometry. We
formulate the operation for a symplectic manifold with a Lagrangian fibration.
Furthermore we construct geometric stability conditions on the derived Fukaya
category of the deformed conifold and study the action of the mirror Atiyah
flop on these stability conditions.Comment: v2: 45 pages, 18 figures; revised expositio
Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P 1 a,b,c
This paper gives a new way of constructing Landau–Ginzburg mirrors usingdeformation theory of Lagrangian immersions motivated by the works of Seidel,Strominger –Yau–Zaslow and Fukaya–Oh–Ohta–Ono. Moreover, we construct acanonical functor from the Fukaya category to the mirror category of matrixfactorizations. This functor derives homological mirror symmetry under someexplicit assumptions.As an application, the construction is applied to spheres with three orbifoldpoints to produce their quantum-corrected mirrors and derive homological mirrorsymmetry. Furthermore, we discover an enumerative meaning of the (inverse)mirror map for elliptic curve quotients