Browse > Article
http://dx.doi.org/10.4134/JKMS.2006.43.5.951

A TOPOLOGICAL MIRROR SYMMETRY ON NONCOMMUTATIVE COMPLEX TWO-TORI  

Kim, Eun-Sang (Department of Applied Mathematics Hanyang University)
Kim, Ho-Il (Department of Mathematics Kyungpook National University)
Publication Information
Journal of the Korean Mathematical Society / v.43, no.5, 2006 , pp. 951-965 More about this Journal
Abstract
In this paper, we study a topological mirror symmetry on noncommutative complex tori. We show that deformation quantization of an elliptic curve is mirror symmetric to an irrational rotation algebra. From this, we conclude that a mirror reflection of a noncommutative complex torus is an elliptic curve equipped with a Kronecker foliation.
Keywords
noncommutative complex torus; mirror symmetry; Kronecker foliation;
Citations & Related Records

Times Cited By Web Of Science : 1  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
연도 인용수 순위
1 B. Blackadar, K-Theory for Operator Algebras, Math. Sci. Res. Inst. Publ. 5, Springer-Verlag, New York, 1986
2 C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Birkhauser, Boston, Inc., Boston, MA, 1985
3 A. Connes, Noncommutative Geometry, Academic Press, New York, 1994
4 A. Connes, A short survey of noncommutative geometry, J. Math. Phys. 41 (2000), no. 6, 3832-3866   DOI   ScienceOn
5 A. Connes, M. R. Douglas, and A. Schwarz, Noncommutative geometry and matrix theory: compactification on tori, J. High Energy Phys. (1998), no. 2, Paper 3, 35 pp
6 A. Connes and M. Rieffel, Yang-Mills for noncommutative two-tori, Contemp. Math. 62 (1987), 237-266   DOI
7 M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of I. C. M., Vol. 1,2 (Zurich, 1994), 120-139, Birkhauser, Basel. 1995
8 A. Polishchuk, Classification of holomorphic vector bundles on noncommutative two-tori, Doc. Math. 9 (2004), 163-181
9 A. Polishchuk and A. Schwarz, Categories of holomorphic vector bundles on non- commutative two-tori, Comm. Math. Phys. 236 (2003), no. 1, 135-159   DOI
10 A. Strominger, S. T. Yau, and E. Zaslow, Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243-259   DOI   ScienceOn
11 M. A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Ca- nad. J. Math. 40 (1988), no. 2, 257-338   DOI
12 M. A. Rieffel, Noncommutative tori- A case study of noncommutative differentiable manifolds, Contemp. Math. 105 (1990), 191-211   DOI
13 N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys. 1999, no. 9, Paper 32, 93 pp
14 H. Kajiura, Kronecker foliation, D1-branes and Morita equivalence of noncom- mutative two-tori, J. High Energy Phys. (2002), no. 8, Paper 50, 26 pp
15 M. Dieng and A. Schwarz, Differential and complex geometry of two-dimensional noncommutative tori, Lett. Math. Phys. 61 (2002), no. 3, 263-270   DOI
16 K. Fukaya, Floer homology of Lagrangian foliation and noncommutative mirror symmetry, Preprint 98-08, Kyoto Univ., 1998
17 C. Hofman and E. Verlinde, Gauge bundles and Born-Infeld on the non- commutative torus, Nuclear Phys. B 547 (1999), no. 1-2, 157-178   DOI   ScienceOn
18 H. Kajiura, Homological mirror symmetry on noncommutative two-tori, preprint
19 G. 't Hooft, Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus, Comm. Math. Phys. 81 (1981), no. 2, 267-275   DOI
20 A. N. Tyurin, Special Lagrangian geometry as a slight deformation of algebraic geometry (geometric quantization and mirror symmetry), Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 2, 141-224
21 A. Schwarz, Theta functions on noncommutative tori, Lett. Math. Phys. 58 (2001), no. 1, 81-90   DOI
22 M. A. Rieffel, $C^+$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415-429   DOI