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


SIGMA 6 (2010), 026, 17 pages      arXiv:0912.4185      https://doi.org/10.3842/SIGMA.2010.026
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

Spectral Distances: Results for Moyal Plane and Noncommutative Torus

Eric Cagnache and Jean-Christophe Wallet
Laboratoire de Physique Théorique, Bât. 210, CNRS, Université Paris-Sud 11, F-91405 Orsay Cedex, France

Received October 31, 2009, in final form March 20, 2010; Published online March 24, 2010

Abstract
The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak * topology. The case of the noncommutative torus is also considered and a formula for the spectral distance between some states is also obtained.

Key words: noncommutative geometry; non-compact spectral triples; spectral distance; noncommutative torus; Moyal planes.

pdf (338 kb)   ps (220 kb)   tex (23 kb)

References

  1. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, available at http://www.alainconnes.org/downloads.html.
  2. Connes A., Marcolli M., A walk in the noncommutative garden, math.QA/0601054.
  3. Landi G., An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics, New Series m: Monographs, Vol. 51, Springer-Verlag, Berlin, 1997.
  4. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhaüser Advanced Texts: Basler Lehrbücher, Birkhaüser Boston, Inc., Boston, MA, 2001.
  5. Connes A., Compact metric spaces, Fredholm modules, and hyperfiniteness, Ergodic Theory Dynam. Systems 9 (1989), 207-220.
  6. Connes A., Gravity coupled with matter and the foundation of noncommutative geometry, Comm. Math. Phys. 182 (1996), 155-176, hep-th/9603053.
  7. d'Andrea F., Martinetti P., A view on transport theory from noncommutative geometry, arXiv:0906.1267.
  8. Rieffel M.A., Metrics on state spaces, Doc. Math. 4 (1999), 559-600, math.OA/9906151.
  9. Dimakis A., Müller-Hoissen F., Striker T., Non-commutative differential calculus and lattice gauge theories, J. Phys. A: Math. Gen. 26 (1993), 1927-1949.
  10. Dimakis A., Müller-Hoissen F., Conne's distance function on one dimensional lattices, Internat. J. Theoret. Phys. 37 (1998), 907-913, q-alg/9707016.
  11. Bimonte G., Lizzi F., Sparano G., Distances on a lattice from non-commutative geometry, Phys. Lett. B 341 (1994), 139-146, hep-lat/9404007.
  12. Iochum B., Krajewski T., Martinetti P., Distances in finite spaces from noncommutative geometry, J. Geom. Phys. 37 (2001), 100-125, hep-th/9912217.
  13. Martinetti P., Wulkenhaar R., Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry, J. Math. Phys. 43 (2002), 182-204, hep-th/0104108.
  14. Martinetti P., Carnot-Carathéodory metric and gauge fluctuation in noncommutative geometry, Comm. Math. Phys. 265 (2006), 585-616, hep-th/0506147.
  15. Martinetti P., Spectral distance on the circle, J. Funct. Anal. 255 (2008), 1575-1612, math.OA/0703586.
  16. Gracia-Bondía J.M., Várilly J.C., Algebras of distributions suitable for phase-space quantum mechanics. I, J. Math. Phys. 29 (1988), 869-879.
  17. Várilly J.C., Gracia-Bondía J.M., Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988), 880-887.
  18. Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, hep-th/0307241.
  19. Gayral V., Iochum B., The spectral action for Moyal planes, J. Math. Phys. 46 (2005), 043503, 17 pages, hep-th/0402147.
  20. Gayral V., The action functional for Moyal planes, Lett. Math. Phys. 65 (2003), 147-157, hep-th/0307220.
  21. Rieffel M.A., Compact quantum metric spaces, in Operator Algebras, Quantization, and Noncommutative Geometry, Contemp. Math., Vol. 365, Amer. Math. Soc., Providence, RI, 2004, 315-330, math.OA/0308207.
  22. Cagnache E., d'Andrea F., Martinetti P., Wallet J.-C., The spectral distance on the Moyal plane, arXiv:0912.0906.
  23. Cagnache E., Masson T., Wallet J.-C., Noncommutative Yang-Mills-Higgs actions from derivation based differential calculus, J. Noncommut. Geom., to appear, arXiv:0804.3061.
  24. de Goursac A., Masson T., Wallet J.-C., Noncommutative ε-graded connections and application to Moyal space, arXiv:0811.3567.
  25. Wallet J.-C., Derivations of the Moyal algebra and noncommutative gauge theories, SIGMA 5 (2009), 013, 25 pages, arXiv:0811.3850.
  26. Schwartz L., Théorie des distributions, Hermann, Paris, 1966.
  27. Connes A., C*-algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A599-A604.
  28. Rieffel M.A., C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429.

Previous article   Next article   Contents of Volume 6 (2010)