Advertisement

Letters in Mathematical Physics

, Volume 109, Issue 12, pp 2665–2679 | Cite as

Scalar curvature of a Levi-Civita connection on the Cuntz algebra with three generators

  • Soumalya JoardarEmail author
Article
  • 78 Downloads

Abstract

A differential calculus on the Cuntz algebra with three generators coming from the action of rotation group in three dimensions is introduced. The differential calculus is shown to satisfy Assumptions I–IV of Bhowmick et al. (A new look at Levi-Civita Connection in NCG. arXiv:1606.08142) so that Levi-Civita connection exists uniquely for any pseudo-Riemannian metric in the sense of Bhowmick et al. [2] Scalar curvature is computed for the Levi-Civita connection corresponding to the canonical bilinear metric.

Keywords

Cuntz algebra Levi-Civita connection Scalar curvature 

Mathematics Subject Classification

46L87 58B34 

Notes

Acknowledgements

The author acknowledges support from Department of Science and Technology, India (DST/INSPIRE/04/2016/002469). He would like to thank J. Bhowmick for some fruitful discussions and both K. B. Sinha as well as Debashish Goswami for their words of encouragement. He would also like to thank an anonymous referee for his/her comments which helped in improving a previous version of the paper.

References

  1. 1.
    Arnlind, J., Wilson, M.: Riemannian curvature of the noncommutative 3-sphere. J. Noncommutative Geom. 11(2), 507–536 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bhowmick J., Goswami D., Joardar S.: A new look at Levi-Civita Connection in NCG, arXiv:1606.08142
  3. 3.
    Bhowmick J., Goswami D., Mukhopadhay S.: Levi-Civita Connections for a class of spectral triples, arXiv: 1809.06721
  4. 4.
    Carey, A., Phillips, J., Rennie, A.: Semifinite spectral triples associated with graph \(C^{\ast }\)-algebras. In: Albeverio, S., Marcolli, M., Paycha, S., Plazas, J. (eds.) Traces in Number Theory, pp. 35–56. Geometry and Quantum Fields, Vieweg, Germany (2008)Google Scholar
  5. 5.
    Connes, A.: Noncommutative Geometry. Academic Press Inc, San Diego (1994)zbMATHGoogle Scholar
  6. 6.
    Connes, A., Moscovici, H.: Modular curvature for noncommutative two-tori. J. Am. Math. Soc. 27, 639–684 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dabrowski, L., Sitarz, A.: Curved noncommutative torus and Gauss-Bonnet. J. Math. Phys. 54, 013518 (2013)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Fathizadeh, F., Khalkhali, M.: Scalar curvature for noncommutative four-tori. J. Noncommutative Geom. 9(2), 473–503 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Frohlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and noncommutative geometry. Commun. Math. Phys 203, 119–184 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    Landi, G.: An introduction to noncommutative spaces and their geometry. Lecture notes in Physics monographs, Springer, Berlin, Heidelberg , vol 51 (1997)Google Scholar
  11. 11.
    Landi, G., Viet, N.A., Wali, K.C.: Gravity and electromagnetism in noncommutative geometry. Phys. Lett. B 326, 45 (1994)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Majid, S.: Noncommutative Riemannian and spin geometry of the standard q-sphere. Commun. Math. Phys. 256, 255–285 (2005)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Rosenberg, J.: Levi-Civita’s theorem for noncommutative tori. SIGMA 9, 071 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.IIT KanpurKalyanpur, KanpurIndia

Personalised recommendations