This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients. Contents 1. Introduction 1 Acknowledgement 10 2. Algebraic construction of localized mirror functor 10 3. Immersed Lagrangian Floer theory and generalized SYZ construction 17 4. Geometric construction of localized mirror functor 23 5. Generalized SYZ for a finite-group quotient 27

Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A ∞ -functor from the Fukaya category of X to the category of matrix factorizations of W . 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. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space. arXiv:1406.4597v3 [math.SG] 30 Sep 2016 1≤j≤n z δ(a j ,−1) j and m pq = T k/2 1≤j≤n z δ(a j ,1) j if i = 0, m qp = T k/2 z i and m pq = T k/2 if i = 0. δ(a, b) = 1 when a = b and zero otherwise. (2) m qp = m pq = 0 otherwise.

Abstract. For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov-Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.

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We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular fiber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in Gr(2, C n ) and OG(1, C 5 ) and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an effective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagrangians.