Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 053, 24 pages      arXiv:1205.4495
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

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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  1. Alston G., Floer cohomology of real Lagrangians in the Fermat quintic threefold, arXiv:1010.4073.
  2. Ashok S.K., Dell'Aquila E., Diaconescu D.E., Fractional branes in Landau-Ginzburg orbifolds, Adv. Theor. Math. Phys. 8 (2004), 461-513, hep-th/0401135.
  3. Auroux D., Mirror symmetry and $T$-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51-91, arXiv:0706.3207.
  4. Chan K., Leung N.C., Matrix factorizations from SYZ transformations, in Advances in Geometric Analysis, Adv. Lect. Math., Vol. 21, International Press, Somerville, MA, 2011, 203-224, arXiv:1006.3832.
  5. Chan K., Leung N.C., Mirror symmetry for toric Fano manifolds via SYZ transformations, Adv. Math. 223 (2010), 797-839, arXiv:0801.2830.
  6. Cho C.-H., Constant triangles in Fukaya category, in preparation.
  7. Cho C.-H., Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not. 2004 (2004), no. 35, 1803-1843, math.SG/0308224.
  8. Cho C.-H., Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle, J. Geom. Phys. 58 (2008), 1465-1476, arXiv:0710.5454.
  9. Cho C.-H., Oh Y.-G., Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), 773-814, math.SG/0308225.
  10. Cho C.-H., Poddar M., Holomorphic orbi-discs and Lagrangian Floer cohomology for toric orbifolds, arXiv:1206.3994.
  11. Fukaya K., Oh Y.-G., Ohta H., Ono K., Lagrangian Floer theory on compact toric manifolds. I, Duke Math. J. 151 (2010), 23-174, arXiv:0802.1703.
  12. Fukaya K., Oh Y.-G., Ohta H., Ono K., Lagrangian Floer theory on compact toric manifolds. II. Bulk deformations, Selecta Math. (N.S.) 17 (2011), 609-711, arXiv:0810.5654.
  13. Fukaya K., Oh Y.-G., Ohta H., Ono K., Lagrangian intersection Floer theory: anomaly and obstruction, AMS/IP Studies in Advanced Mathematics, Vol. 46, Amer. Math. Soc., Providence, RI, 2009.
  14. Gross M., The Strominger-Yau-Zaslow conjecture: from torus fibrations to degenerations, in Algebraic Geometry - Seattle 2005, Proc. Sympos. Pure Math., Vol. 80, Amer. Math. Soc., Providence, RI, 2009, Part 1, 149-192.
  15. Kapustin A., Li Y., D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Phys. 2003 (2003), no. 12, 005, 44 pages, hep-th/0210296.
  16. Kwon D., Oh Y.-G., Structure of the image of (pseudo)-holomorphic discs with totally real boundary condition, Comm. Anal. Geom. 8 (2000), 31-82.
  17. Oh Y.-G., Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), 949-993.
  18. Oh Y.-G., Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Int. Math. Res. Not. 1996 (1996), no. 7, 305-346.
  19. Orlov D.O., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), 240-262.
  20. Seidel P., Fukaya categories and Picard-Lefschetz theory, Zürich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.
  21. Strominger A., Yau S.T., Zaslow E., Mirror symmetry is $T$-duality, in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, 1999), AMS/IP Studies in Advanced Mathematics, Vol. 23, Editors C. Vafa, S.T. Yau, Amer. Math. Soc., Providence, RI, 2001, 275-295.

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