Examples of Matrix Factorizations from SYZ

TL;DR

This paper constructs matrix factorizations for specific Lagrangian submanifolds in various orbifolds and projective spaces, linking symplectic geometry with algebraic models via Strominger-Yau-Zaslow transformations.

Contribution

It provides explicit matrix factorizations for anti-diagonals and circle fibers in weighted projective lines, extending the SYZ approach to orbifolds and confirming predictions from B-model calculations.

Findings

Matrix factorizations for anti-diagonals in ${ m CP}^1 imes { m CP}^1$

Matrix factorizations for circle fibers in weighted projective lines

Equivalence of certain Lagrangian sums with torus fibers in Fukaya category

Abstract

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.