Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

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)
  • Published : 2006.09.30
Download PDF
⟨ Previous Next ⟩

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

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 https://doi.org/10.1063/1.533329
  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 https://doi.org/10.1090/conm/062/878383
  7. M. Dieng and A. Schwarz, Differential and complex geometry of two-dimensional noncommutative tori, Lett. Math. Phys. 61 (2002), no. 3, 263-270 https://doi.org/10.1023/A:1021272314232
  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 https://doi.org/10.1016/S0550-3213(99)00062-0
  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 https://doi.org/10.1007/BF01208900
  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 https://doi.org/10.1007/s00220-003-0813-9
  16. M. A. Rieffel, $C^+$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415-429 https://doi.org/10.2140/pjm.1981.93.415
  17. M. A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Ca- nad. J. Math. 40 (1988), no. 2, 257-338 https://doi.org/10.1007/BF01092890
  18. M. A. Rieffel, Noncommutative tori- A case study of noncommutative differentiable manifolds, Contemp. Math. 105 (1990), 191-211 https://doi.org/10.1090/conm/105/1047281
  19. A. Schwarz, Theta functions on noncommutative tori, Lett. Math. Phys. 58 (2001), no. 1, 81-90 https://doi.org/10.1023/A:1012515417396
  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 https://doi.org/10.1016/0550-3213(96)00434-8
  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

Cited by

  1. Mirror duality and noncommutative tori vol.42, pp.1, 2009, https://doi.org/10.1088/1751-8113/42/1/015206