Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $\mathbb{P}^1_{a,b,c}$

Abstract

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.