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

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

Maurer-Cartan deformation of Lagrangians

The Maurer-Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;\Lambda)$ as an $A_\infty$-algebra. We identify the Maurer-Cartan algebra with the $0$-th cohomology of the Koszul dual dga of $CF(L,L;\Lambda)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer-Cartan algebra of $L$ and a certain analytic completion of the wrapped Floer cohomology of another Lagrangian $G$ when $G$ is \emph{dual} to $L$ in the sense to be defined. In view of mirror symmetry, this can be understood as specifying a local chart associated with $L$ in the mirror rigid analytic space. We examine the idea by explicit calculation of the isomorphism for several interesting examples.Comment: 51 pages, 12 figures. Comments are welcom

Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period

We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical-holomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces which relates the log descendant Gromov-Witten invariants with the oscillatory integrals of the bulk-deformed potentials.Comment: 44 pages, 9 figures, comments are welcom

Examples of Matrix Factorizations from SYZ

We find matrix factorization corresponding to an anti-diagonal in ${\mathbb C}P^1 \times {\mathbb C}P^1$, and circle fibers in weighted projective lines using the idea of Chan and Leung of Strominger-Yau-Zaslow transformations. For the tear drop orbifolds, we apply this idea to find matrix factorizations for two types of potential, the usual Hori-Vafa potential or the bulk deformed (orbi)-potential. We also show that the direct sum of anti-diagonal with its shift, is equivalent to the direct sum of central torus fibers with holonomy $(1,-1)$ and $(-1,1)$ in the Fukaya category of ${\mathbb C}P^1 \times {\mathbb C}P^1$, which was predicted by Kapustin and Li from B-model calculations

Gluing Localized Mirror Functors

We develop a method of gluing the local mirrors and functors constructed from immersed Lagrangians in the same deformation class. As a result, we obtain a global mirror geometry and a canonical mirror functor. We apply the method to construct the mirrors of punctured Riemann surfaces and show that our functor derives homological mirror symmetry.Comment: 69 pages, 39 figures, comments are welcom

Localized mirror functor constructed from a Lagrangian torus

Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold , we define a holomorphic function known as the Floer potential. We construct a canonical ∞ -functor from the Fukaya category of to the category of matrix factorizations of . It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations and prove homological mirror symmetry. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space.Accepted manuscrip