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Noncommutative Riemannian Spin Manifolds

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Noncommutative Geometry and Particle Physics

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Abstract

We now extend our treatment of noncommutative geometric spaces from the finite case to the continuum. This generalizes spin manifolds to the noncommutative world.

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References

  1. Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3, 3–38 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  2. Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, New Jersey (1989)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Várilly, J.C.: Dirac operators and spectral geometry. Notes taken by Paweł Witkowski (2006)

    Google Scholar 

  5. Landsman, N.P.: Notes on noncommutative geometry. http://www.math.ru.nl/~landsman/ncg2010.html

  6. Jordan, P., Von Neumann, J.: On inner products in linear, metric spaces. Ann. Math. 36(2), 719–723 (1935)

    Google Scholar 

  7. Atiyah, M. F.: \(K\)-theory and reality. Quart. J. Math. Oxford Ser. 17(2), 367–386 (1966)

    Google Scholar 

  8. Atiyah, M., Bott, R.: On the periodicity theorem for complex vector bundles. Acta Math. 112, 229–247 (1964)

    MATH  MathSciNet  Google Scholar 

  9. Jost, J.: Riemannian Geometry and Geometric Analysis. Universitext, 3rd edn. Springer, Berlin (2002)

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  11. Várilly, J.C.: An Introduction to Noncommutative Geometry. Publishing House (EMS Series of Lectures in Mathematics), European Math. Soc (2006)

    Google Scholar 

  12. Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  13. Blackadar, B.: Operator algebras, of encyclopaedia of mathematical sciences. Theory of \(C^{\ast }\)-algebras and von Neumann Algebras, Operator Algebras and Non-commutative, Geometry, 3, vol. 122. Springer, Berlin (2006)

    Google Scholar 

  14. Takesaki, M.: Theory of operator algebras I, of encyclopaedia of mathematical sciences. Reprint of the first (1979) edn. Operator Algebras and Non-commutative, Geometry, 5, vol. 124. Springer, Berlin (2002)

    Google Scholar 

  15. Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36(11), 6194–6231 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Rieffel, M.A.: Metrics on state spaces. Doc. Math. (electronic) 4, 559–600 (1999)

    Google Scholar 

  17. D’Andrea, F., Martinetti, P.: A view on optimal transport from noncommutative geometry. Symmetry Integrability Geom. Methods Appl. (SIGMA) 6, 057, 24 (2010)

    Google Scholar 

  18. D’Andrea, F., Martinetti, P.: On Pythagoras theorem for products of spectral triples. Lett. Math. Phys. 103, 469–492 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Martinetti, P.: Carnot-Carathéodory metric and gauge fluctuation in noncommutative geometry. Commun. Math. Phys. 265, 585–616 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Martinetti, P., Wulkenhaar, R.: Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry. J. Math. Phys. 43, 182–204 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Connes, A.: On the spectral characterization of manifolds. J. Noncommut. Geom. 7, 1–82 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Connes, A.: Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182, 155–176 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Lord, S., Rennie, A., Várilly, J.C.: Riemannian manifolds in noncommutative geometry. J. Geom. Phys. 62, 1611–1638 (2012)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Fröhlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and non-commutative geometry. Commun. Math. Phys. 193 (1998)

    Google Scholar 

  25. Vanhecke, F.J.: On the product of real spectral triples. Lett. Math. Phys. 50, 157–162 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Dabrowski, L., Dossena, G.: Product of real spectral triples. Int. J. Geom. Meth. Mod. Phys. 8, 1833–1848 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Landi, G.: An introduction to noncommutative spaces and their Geometry. Springer, Berlin (1997)

    Google Scholar 

  28. 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 

  29. Connes, A., Marcolli, M.: Noncommutative Geometry. Quantum Fields and Motives. AMS, Providence (2008)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands

    Walter D. van Suijlekom

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  1. Walter D. van Suijlekom
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Correspondence to Walter D. van Suijlekom .

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van Suijlekom, W.D. (2015). Noncommutative Riemannian Spin Manifolds. In: Noncommutative Geometry and Particle Physics. Mathematical Physics Studies. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9162-5_4

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