Examples of Matrix Factorizations from SYZ

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.

References in corpus (3)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle
• Floer cohomology of real Lagrangians in the Fermat quintic threefold