Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 060, 23 pages      arXiv:1305.7479
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Generalized Fuzzy Torus and its Modular Properties

Paul Schreivogl and Harold Steinacker
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria

Received June 19, 2013, in final form October 11, 2013; Published online October 17, 2013

We consider a generalization of the basic fuzzy torus to a fuzzy torus with non-trivial modular parameter, based on a finite matrix algebra. We discuss the modular properties of this fuzzy torus, and compute the matrix Laplacian for a scalar field. In the semi-classical limit, the generalized fuzzy torus can be used to approximate a generic commutative torus represented by two generic vectors in the complex plane, with generic modular parameter τ. The effective classical geometry and the spectrum of the Laplacian are correctly reproduced in the limit. The spectrum of a matrix Dirac operator is also computed.

Key words: fuzzy spaces; noncommutative geometry; matrix models.

pdf (533 kb)   tex (135 kb)


  1. Alexanian G., Balachandran A.P., Immirzi G., Ydri B., Fuzzy CP2, J. Geom. Phys. 42 (2002), 28-53, hep-th/0103023.
  2. Arnlind J., Choe J., Hoppe J., Noncommutative minimal surfaces, arXiv:1301.0757.
  3. Arnlind J., Hoppe J., Huisken G., Discrete curvature and the Gauss-Bonnet theorem, arXiv:1001.2223.
  4. Aschieri P., Grammatikopoulos T., Steinacker H., Zoupanos G., Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking, J. High Energy Phys. 2006 (2006), no. 9, 026, 26 pages, hep-th/0606021.
  5. Banks T., Fischler W., Shenker S.H., Susskind L., M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997), 5112-5128, hep-th/9610043.
  6. Chaichian M., Demichev A., Presnajder P., Sheikh-Jabbari M.M., Tureanu A., Quantum theories on noncommutative spaces with nontrivial topology: Aharonov-Bohm and Casimir effects, Nuclear Phys. B 611 (2001), 383-402, hep-th/0101209.
  7. Connes A., Douglas M.R., Schwarz A., Noncommutative geometry and matrix theory: compactification on tori, J. High Energy Phys. 1998 (1998), no. 2, 003, 35 pages, hep-th/9711162.
  8. Grosse H., Klimcík C., Presnajder P., On finite 4D quantum field theory in non-commutative geometry, Comm. Math. Phys. 180 (1996), 429-438, hep-th/9602115.
  9. Grosse H., Presnajder P., The Dirac operator on the fuzzy sphere, Lett. Math. Phys. 33 (1995), 171-181.
  10. Hofman C., Verlinde E., U-duality of Born-Infeld on the noncommutative two-torus, J. High Energy Phys. 1998 (1998), no. 12, 010, 21 pages, hep-th/9810116.
  11. Hofman C., Verlinde E., Gauge bundles and Born-Infeld on the non-commutative torus, Nuclear Phys. B 547 (1999), 157-178, hep-th/9810219.
  12. Hoppe J., Some classical solutions of membrane matrix model equations, in Strings, Branes and Dualities (Cargèse, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 520, Kluwer Acad. Publ., Dordrecht, 1999, 423-427, hep-th/9702169.
  13. Ishibashi N., Kawai H., Kitazawa Y., Tsuchiya A., A large-N reduced model as superstring, Nuclear Phys. B 498 (1997), 467-491, hep-th/9612115.
  14. Kimura Y., Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrix model, Progr. Theoret. Phys. 106 (2001), 445-469, hep-th/0103192.
  15. Landi G., Lizzi F., Szabo R.J., From large N matrices to the noncommutative torus, Comm. Math. Phys. 217 (2001), 181-201, hep-th/9912130.
  16. Madore J., The fuzzy sphere, Classical Quantum Gravity 9 (1992), 69-87.
  17. Nakahara M., Geometry, topology and physics, 2nd ed., Graduate Student Series in Physics, Institute of Physics, Bristol, 2003.
  18. Steinacker H., Emergent gravity and noncommutative branes from Yang-Mills matrix models, Nuclear Phys. B 810 (2009), 1-39, arXiv:0806.2032.
  19. Steinacker H., Emergent geometry and gravity from matrix models: an introduction, Classical Quantum Gravity 27 (2010), 133001, 46 pages, arXiv:1003.4134.
  20. Steinacker H., Non-commutative geometry and matrix models, PoS Proc. Sci. (2011), PoS(QGQGS2011), 004, 27 pages, arXiv:1109.5521.

Previous article  Next article   Contents of Volume 9 (2013)