JOURNAL BROWSE
Search
Advanced SearchSearch Tips
A TOPOLOGICAL MIRROR SYMMETRY ON NONCOMMUTATIVE COMPLEX TWO-TORI
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
A TOPOLOGICAL MIRROR SYMMETRY ON NONCOMMUTATIVE COMPLEX TWO-TORI
Kim, Eun-Sang; Kim, Ho-Il;
  PDF(new window)
 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;
 Language
English
 Cited by
1.
Mirror duality and noncommutative tori, Journal of Physics A: Mathematical and Theoretical, 2009, 42, 1, 015206  crossref(new windwow)
 References
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 crossref(new window)

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 crossref(new window)

7.
M. Dieng and A. Schwarz, Differential and complex geometry of two-dimensional noncommutative tori, Lett. Math. Phys. 61 (2002), no. 3, 263-270 crossref(new window)

8.
K. Fukaya, Floer homology of Lagrangian foliation and noncommutative mirror symmetry, Preprint 98-08, Kyoto Univ., 1998

9.
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 crossref(new window)

10.
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 crossref(new window)

11.
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

12.
H. Kajiura, Homological mirror symmetry on noncommutative two-tori, preprint

13.
M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of I. C. M., Vol. 1,2 (Zurich, 1994), 120-139, Birkhauser, Basel. 1995

14.
A. Polishchuk, Classification of holomorphic vector bundles on noncommutative two-tori, Doc. Math. 9 (2004), 163-181

15.
A. Polishchuk and A. Schwarz, Categories of holomorphic vector bundles on non- commutative two-tori, Comm. Math. Phys. 236 (2003), no. 1, 135-159 crossref(new window)

16.
M. A. Rieffel, $C^+$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415-429 crossref(new window)

17.
M. A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Ca- nad. J. Math. 40 (1988), no. 2, 257-338 crossref(new window)

18.
M. A. Rieffel, Noncommutative tori- A case study of noncommutative differentiable manifolds, Contemp. Math. 105 (1990), 191-211 crossref(new window)

19.
A. Schwarz, Theta functions on noncommutative tori, Lett. Math. Phys. 58 (2001), no. 1, 81-90 crossref(new window)

20.
N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys. 1999, no. 9, Paper 32, 93 pp

21.
A. Strominger, S. T. Yau, and E. Zaslow, Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243-259 crossref(new window)

22.
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