Advertisement

Moduli Spaces of Dirac Operators for Finite Spectral Triples

  • Branimir Ćaćić

Abstract

The structure theory of finite real spectral triples developed by Krajewski and by Paschke and Sitarz is generalised to allow for arbitrary KO-dimension and the failure of orientability and Poincaré duality, and moduli spaces of Dirac operators for such spectral triples are defined and studied. This theory is then applied to recent work by Chamseddine and Connes towards deriving the finite spectral triple of the noncommutative-geometric Standard Model.

Keywords

Modulus Space Dirac Operator Direct Summand Noncommutative Geometry Complex Hilbert Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    [1] John W. Barrett, A Lorentzian version of the non-commutative geometry of the standard model of particle physics, J. Math. Phys. 48 (2007), no. 012303.CrossRefMathSciNetGoogle Scholar
  2. [2]
    [2] Ali H. Chamseddine and Alain Connes, Conceptual explanation for the algebra in the noncommutative approach to the Standard Model, Phys. Rev. Lett. 99 (2007), no. 191601.CrossRefMathSciNetGoogle Scholar
  3. [3]
    [3] Ali H., Why the standard model, J. Geom. Phys. 58 (2008), 38–47.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    [4] Ali H. Chamseddine, Alain Connes, and Matilde Marcolli, Gravity and the Standard Model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007), 991–1089.zbMATHMathSciNetGoogle Scholar
  5. [5]
    [5] Alain Connes, Geometry from the spectral point of view, Lett. Math. Phys. 34 (1995), no. 3, 203–238.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    [6] Alain Connes, Noncommutative geometry and reality, J. Math. Phys. 6 (1995), 6194–6231.Google Scholar
  7. [7]
    [7] Alain Connes, Noncommutative geometry and the Standard Model with neutrino mixing, JHEP 11 (2006), no. 81.Google Scholar
  8. [8]
    [8] Alain Connes and Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives, Colloquium Publications, vol. 55, American Mathematical Society, Providence, RI, 2007.Google Scholar
  9. [9]
    [9] George A. Elliott, Towards a theory of classification, Adv. in Math. 223 (2011), no. 1, 30–48.Google Scholar
  10. [10]
    [10] George A. Elliott, private conversation, 2008.Google Scholar
  11. [11]
    [11] Douglas R. Farenick, Algebras of Linear Transformations, Springer, New York, 2000.Google Scholar
  12. [12]
    [12] Bruno Iochum, Thomas Schücker, and Christoph Stephan, On a classification of irreducible almost commutative geometries, J. Math. Phys. 45 (2004), 5003–5041.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    [13] Jan-H. Jureit and Christoph A. Stephan, On a classification of irreducible almost commutative geometries, a second helping, J. Math. Phys. 46 (2005), no. 043512.CrossRefMathSciNetGoogle Scholar
  14. [14]
    [14] Jan-Hendrik Jureit, Thomas Schücker, and Christoph Stephan, On a classification of irreducible almost commutative geometries III, J. Math. Phys. 46 (2005), no. 072303.CrossRefMathSciNetGoogle Scholar
  15. [15]
    [15] Jan-Hendrik Jureit and Christoph A. Stephan, On a classification of irreducible almost commutative geometries IV, J. Math. Phys. 49 (2008), 033502.CrossRefMathSciNetGoogle Scholar
  16. [16]
    [16] Jan-Hendrik Jureit and Christoph A. Stephan, On a classification of irreducible almost commutative geometries, V (2009).Google Scholar
  17. [17]
    [17] Jan-H. Jureit and Christoph A. Stephan, Finding the standard model of particle physics, a combinatorial problem, Comp. Phys. Comm. 178 (2008), 230–247.CrossRefGoogle Scholar
  18. [18]
    [18] Thomas Krajewski, Classification of finite spectral triples, J. Geom. Phys. 28 (1998), 1–30.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    [19] Bing-Ren Li, Introduction to Operator Algebras, World Scientific, Singapore, 1992.Google Scholar
  20. [20]
    [20] Mario Paschke and Andrzej Sitarz, Discrete spectral triples and their symmetries, J. Math. Phys. 39 (1998), 6191–6205.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    [21] Mario Paschke and Andrzej Sitarz, The geometry of noncommutative symmetries, Acta Physica Polonica B 31 (2000), 1897–1911.zbMATHMathSciNetGoogle Scholar
  22. [22]
    [22] Thomas Schücker, Krajewski diagrams and spin lifts (2005), available at arXiv:hep-th/0501181v2.Google Scholar
  23. [23]
    [23] Andrzej Sitarz, Equivariant spectral triples, Noncommutative Geometry and Quantum Groups (Piotr M. Hajac and Wiesław Pusz, eds.), Banach Center Publ., vol. 61, Polish Acad. Sci., Warsaw, 2003, pp. 231–268.Google Scholar
  24. [24]
    [24] Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    [25] Christoph A. Stephan, Almost-commutative geometry, massive neutrinos and the orientability axiom in KO-dimension 6 (2006), available at arXiv:hep-th/0610097v1.Google Scholar

Copyright information

© Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011

Authors and Affiliations

  • Branimir Ćaćić
    • 1
  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadena, CAUSA

Personalised recommendations