SIGMA 8 (2012), 053, 24 pages      arXiv:1205.4495      https://doi.org/10.3842/SIGMA.2012.053
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

Examples of Matrix Factorizations from SYZ

Cheol-Hyun Cho, Hansol Hong and Sangwook Lee
Department of Mathematics, Research Institute of Mathematics, Seoul National University, 1 Kwanak-ro, Kwanak-gu, Seoul, South Korea

Received May 15, 2012, in final form August 12, 2012; Published online August 16, 2012

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.

Key words: matrix factorization; Fukaya category; mirror symmetry; Lagrangian Floer theory.

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