Browse > Article
http://dx.doi.org/10.4134/JKMS.j190589

COMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE  

Lee, Sangwook (School of Mathematics Korea Institute for Advanced Study)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.5, 2020 , pp. 1135-1165 More about this Journal
Abstract
Polishchuk-Zaslow explained the homological mirror symmetry between Fukaya category of symplectic torus and the derived category of coherent sheaves of elliptic curves via Lagrangian torus fibration. Recently, Cho-Hong-Lau found another proof of homological mirror symmetry using localized mirror functor, whose target category is given by graded matrix factorizations. We find an explicit relation between these two approaches.
Keywords
LG/CY correspondence; homological mirror symmetry;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Abouzaid and I. Smith, Homological mirror symmetry for the 4-torus, Duke Math. J. 152 (2010), no. 3, 373-440. https://doi.org/10.1215/00127094-2010-015   DOI
2 P. S. Aspinwall, Landau-Ginzburg to Calabi-Yau dictionary for D-branes, J. Math. Phys. 48 (2007), no. 8, 082304, 18 pp. https://doi.org/10.1063/1.2768185   DOI
3 D. Auroux, A beginner's introduction to Fukaya categories, in Contact and symplectic topology, 85-136, Bolyai Soc. Math. Stud., 26, Janos Bolyai Math. Soc., Budapest, 2014. https://doi.org/10.1007/978-3-319-02036-5_3
4 A. Caldararu and J. Tu, Curved A1 algebras and Landau-Ginzburg models, New York J. Math. 19 (2013), 305-342.
5 C.-H. Cho, H. Hong, and S.-C. Lau, Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $P^1_-a,b,c}$, J. Differential Geom. 106 (2017), no. 1, 45-126. http://projecteuclid.org/euclid.jdg/1493172094
6 C.-H. Cho, H. Hong, and S. Lau, Noncommutative homological mirror functor, arXiv:1512.07128.
7 T. Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), no. 2, 223-274. https://doi.org/10.1215/00127094-1415869   DOI
8 D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. https://doi.org/10.2307/1999875   DOI
9 K. Fukaya, Floer homology and mirror symmetry. II, in Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), 31-127, Adv. Stud. Pure Math., 34, Math. Soc. Japan, Tokyo, 2002. https://doi.org/10.2969/aspm/03410031
10 K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, 46.2, American Mathematical Society, Providence, RI, 2009.
11 D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503-531, Progr. Math., 270, Birkhauser Boston, Inc., Boston, MA, 2009. https://doi.org/10.1007/978-0-8176-4747-6_16
12 A. Polishchuk and E. Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2 (1998), no. 2, 443-470. https://doi.org/10.4310/ATMP.1998.v2.n2.a9   DOI
13 P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zurich, 2008. https://doi.org/10.4171/063
14 P. Seidel, Homological mirror symmetry for the quartic surface, Mem. Amer. Math. Soc. 236 (2015), no. 1116, vi+129 pp. https://doi.org/10.1090/memo/1116
15 P. Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727-769. https://doi.org/10.1090/S1056-3911-10-00550-3   DOI