Skip to main content
SpringerLink
Log in

Spin geometry of Dirac and noncommutative geometry of Connes

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The review is devoted to the interpretation of the Dirac spin geometry in terms of noncommutative geometry. In particular, we give an idea of the proof of the theorem stating that the classical Dirac geometry is a noncommutative spin geometry in the sense of Connes, as well as an idea of the proof of the converse theorem stating that any noncommutative spin geometry over the algebra of smooth functions on a smooth manifold is the Dirac spin geometry.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Connes, Noncommutative Geometry (Academic, San Diego, CA, 1994).

    MATH  Google Scholar 

  2. J. M. Gracia-Bondía, J. C. Várilly, and H. Figueroa, Elements of Noncommutative Geometry (Birkhäuser, Boston, 2001).

    Book  MATH  Google Scholar 

  3. M. Khalkhali, Basic Noncommutative Geometry (Eur. Math. Soc., Zürich, 2013).

    Book  MATH  Google Scholar 

  4. G. Landi, An Introduction to Noncommutative Spaces and Their Geometries (Springer, Berlin, 1997).

    MATH  Google Scholar 

  5. H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry (Princeton Univ. Press, Princeton, NJ, 1989).

    MATH  Google Scholar 

  6. A. G. Sergeev, “Quantum calculus and quasiconformal mappings,” Mat. Zametki 100 (1), 144–154 (2016) [Math. Notes 100, 123–131 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  7. A. G. Sergeev, “Quantization of the Sobolev space of half-differentiable functions,” Mat. Sb. 207 (10), 96–104 (2016) [Sb. Math. 207, 1450–1457 (2016)].

    Article  MathSciNet  Google Scholar 

  8. A. G. Sergeev, “Introduction to noncommutative geometry,” E-print (Steklov Math. Inst., Moscow, 2016), http://www.mi.ras.ru/noc/14_15/encgeom_2016.pdf

    Google Scholar 

  9. A. G. Sergeev, “Noncommutative geometry and analysis,” in Differential Equations. Mathematical Physics (VINITI, Moscow, 2017), Itogi Nauki Tekhn., Sovrem. Mat. Prilozh., Temat. Obzory 137, pp. 61–81.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. G. Sergeev

Authors
  1. A. G. Sergeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Sergeev.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 298, pp. 276–314.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sergeev, A.G. Spin geometry of Dirac and noncommutative geometry of Connes. Proc. Steklov Inst. Math. 298, 256–293 (2017). https://doi.org/10.1134/S0081543817060177

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543817060177

Search

Navigation