Skip to main content

Almost-Commutative Manifolds and Gauge Theories

  • Chapter
  • First Online:
Noncommutative Geometry and Particle Physics

Part of the book series: Mathematical Physics Studies ((MPST))

  • 1999 Accesses

Abstract

In this chapter we analyze the gauge theories corresponding (in the sense of Chap. 6) to a special class of noncommutative manifolds, to wit almost-commutative, or AC manifolds. We will see that this class leads to the usual gauge theories in physics.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 64.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Kaluza, T.: Zum Unitätsproblem in der Physik. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math.Phys.) 1921, 966–972 (1921)

    Google Scholar 

  2. Klein, O.: Quantentheorie und fünfdimensionale relativitätstheorie. Z. Phys. 37, 895–906 (1926)

    Article  MATH  ADS  Google Scholar 

  3. Iochum, B., Schucker, T., Stephan, C.: On a classification of irreducible almost commutative geometries. J. Math. Phys. 45, 5003–5041 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Connes, A.: Essay on Physics and Noncommutative Geometry. In: The Interface of Mathematics and Particle Physics (Oxford, 1988). The Institute of Mathematics and its Applications Conference Series, vol. 24, pp. 9–48. Oxford University Press, New York (1990)

    Google Scholar 

  5. Connes, A., Lott, J.: Particle models and noncommutative geometry. Nucl. Phys. Proc. Suppl. 18B, 29–47 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  6. Dubois-Violette, M., Madore, J., Kerner, R.: Classical bosons in a noncommutative geometry. Class. Quant. Grav. 6, 1709 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Dubois-Violette, M., Madore, J., Kerner, R.: Gauge bosons in a noncommutative geometry. Phys. Lett. B217, 485–488 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  8. Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry and new models of gauge theory. J. Math. Phys. 31, 323 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry of matrix algebras. J. Math. Phys. 31, 316 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Chamseddine, A.H., Connes, A.: Universal formula for noncommutative geometry actions: unifications of gravity and the standard model. Phys. Rev. Lett. 77, 4868–4871 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. van den Dungen, K., van Suijlekom, W.D.: Particle physics from almost commutative spacetimes. Rev. Math. Phys. 24, 1230004 (2012)

    Article  MathSciNet  Google Scholar 

  14. Ćaćić, B.: A Reconstruction theorem for almost-commutative spectral triples. Lett. Math. Phys. 100, 181–202 (2012)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Ćaćić, B.: Real structures on almost-commutative spectral triples. Lett. Math. Phys. 103, 793–816 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Boeijink, J., van den Dungen, K.: Notes on topologically non-trivial almost-commutative geometries. Work in progress.

    Google Scholar 

  17. Boeijink, J., van Suijlekom, W.D.: The noncommutative geometry of Yang–Mills fields. J. Geom. Phys. 61, 1122–1134 (2011)

    Article  MATH  ADS  Google Scholar 

  18. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  19. Gilkey, P.B.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. In: Mathematics Lecture Series. Publish or Perish Inc., Wilmington (1984)

    Google Scholar 

  20. Vassilevich, D.V.: Heat kernel expansion: user’s manual. Phys. Rept. 388, 279–360 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Walter D. van Suijlekom .

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

van Suijlekom, W.D. (2015). Almost-Commutative Manifolds and Gauge Theories. In: Noncommutative Geometry and Particle Physics. Mathematical Physics Studies. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9162-5_8

Download citation

Publish with us

Policies and ethics