Letters in Mathematical Physics

, Volume 104, Issue 11, pp 1425–1443 | Cite as

Modules Over the Noncommutative Torus and Elliptic Curves

  • Francesco D’AndreaEmail author
  • Gaetano Fiore
  • Davide Franco


Using the Weil–Brezin–Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely generated projective modules over the algebra A θ of the noncommutative torus. We show that such A θ -modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve E τ , under the condition that the deformation parameter θ and the modular parameter τ satisfy a non-trivial relation.

Mathematics Subject Classification

Primary 58B34 Secondary 46L87 53D55 


noncommutative torus imprimitivity bimodules elliptic curves Moyal deformation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Atiyah M.F.: Vector bundles over an elliptic curve. Proc. Lond. Math. Soc. 7, 414–452 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Birkenhake, C., Lange, H.: Complex Abelian Varieties. 2nd ed. Springer, Berlin (2004)Google Scholar
  3. 3.
    Connes A.: C *-algèbres et géométrie différentielle. C. R. Acad. Sci. Paris 290A, 599–604 (1980)MathSciNetGoogle Scholar
  4. 4.
    Connes A.: Noncommutative Geometry. Academic Press, New York (1994)zbMATHGoogle Scholar
  5. 5.
    Connes A., Rieffel M.A.: Yang-Mills for noncommutative two-tori. Contemp. Math. 62, 237–266 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Drinfeld V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1989)MathSciNetGoogle Scholar
  7. 7.
    Fiore, G.: On twisted symmetries and quantum mechanics with a magnetic field on noncommutative tori, PoS(CNCFG2010)018.
  8. 8.
    Fiore G.: On quantum mechanics with a magnetic field on \({\mathbb{R}^n}\) and on a torus \({\mathbb{T}^n}\) , and their relation. Int. J. Theor. Phys. 52, 877–896 (2013)MathSciNetCrossRefzbMATHADSGoogle Scholar
  9. 9.
    Folland, G.B.: Harmonic analysis in phase space. Ann. Math. Stud. 122; Princeton University Press, Princeton (1989)Google Scholar
  10. 10.
    Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C.: Moyal planes are spectral triples. Commun. Math. Phys. 246, 569–623 (2004)CrossRefzbMATHADSGoogle Scholar
  11. 11.
    Lechner, G., Waldmann, S.: Strict deformation quantization of locally convex algebras and modules. preprint arXiv:1109.5950 [math.QA]
  12. 12.
    Mahanta S., Suijlekom W.D.: Noncommutative tori and the Riemann–Hilbert correspondence. J. Noncommut. Geom. 3, 261–287 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Manin, Y.: Real multiplication and noncommutative geometry. In: The Legacy of Niels Henrik Abel, pp. 685–727. Springer, Berlin (2004)Google Scholar
  14. 14.
    Mumford D.: Tata Lectures on Theta I. Birkäuser, Basel (1983)CrossRefzbMATHGoogle Scholar
  15. 15.
    Plazas, J.: Arithmetic structures on noncommutative tori with real multiplication. Int. Math. Res. Not. (2008), rnm147Google Scholar
  16. 16.
    Polishchuk A., Schwarz A.: Categories of holomorphic vector bundles on noncommutative two-tori. Commun. Math. Phys. 236, 135–159 (2003)MathSciNetCrossRefzbMATHADSGoogle Scholar
  17. 17.
    Polishchuk A.: Noncommutative two-tori with real multiplication as noncommutative projective varieties. J. Geom. Phys. 50, 162–187 (2004)MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. 18.
    Rieffel M.A.: The cancellation theorem for projective modules over irrational rotation C *-algebras. Proc. Lond. Math. Soc. 47, 285–302 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rieffel, M.A.: Deformation quantization and operator algebras. Proc. Symp. Pure Math. 51, 411–423 (1990)Google Scholar
  20. 20.
    Rieffel, M.A.: Deformation quantization for actions of \({\mathbb{R}^d}\) . Mem. Am. Math. Soc. 106 (1993)Google Scholar
  21. 21.
    Swan R.G.: Vector bundles and projective modules. Trans. Am. Math. Soc. 105, 264 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Várilly, J.C.: An introduction to noncommutative geometry. EMS Lect. Ser. in Math. (2006)Google Scholar
  23. 23.
    Vlasenko, M.: The graded ring of quantum theta functions for noncommutative torus with real multiplication. Int. Math. Res. Not. 15825 (2006)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Francesco D’Andrea
    • 1
    • 2
    Email author
  • Gaetano Fiore
    • 1
    • 2
  • Davide Franco
    • 1
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Napoli Federico IINaplesItaly
  2. 2.I.N.F.N., Sezione di Napoli, Complesso MSANaplesItaly

Personalised recommendations