Quantum Isometries of the Finite Noncommutative Geometry of the Standard Model

  • Jyotishman Bhowmick
  • Francesco D’Andrea
  • Ludwik Dąbrowski


We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M × F, where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.


  1. 1.
    Ahmad Q.R., et al. (SNO Collaboration): Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory. Phys. Rev. Lett. 89, 011301 (2002)ADSCrossRefGoogle Scholar
  2. 2.
    Banica T.: Le groupe quantique compact libre U(n). Commun. Math. Phys. 190, 143–172 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Banica T.: Quantum automorphism groups of small metric spaces. Pacific J. Math. 219, 27–51 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Banica T.: Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224, 243–280 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Banica T., Vergnioux R.: Invariants of the half-liberated orthogonal group. Ann. Inst. Fourier 60, 2137–2164 (2010)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bhowmick J., Goswami D., Skalski A.: Quantum Isometry Groups of 0-Dimensional Manifolds. Trans. AMS 363, 901–921 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bhowmick J., Goswami D.: Quantum Group of Orientation preserving Riemannian Isometries. J. Funct. Anal. 257, 2530–2572 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bhowmick J., Goswami D.: Quantum isometry groups of the Podles spheres. J. Funct. Anal. 258, 2937–2960 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bhowmick J., Goswami D.: Some counterexamples in the theory of quantum isometry groups. Lett. Math. Phys. 93(3), 279–293 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Bhowmick J., Skalski A.: Quantum isometry groups of noncommutative manifolds associated to group C*-algebras. J. Geom. Phys. 60(10), 1474–1489 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Bichon J.: Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc. 131(3), 665–673 (2003)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Chamseddine A.H., Connes A.: The Spectral Action Principle. Commun. Math. Phys. 186, 731–750 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Chamseddine A.H., Connes A.: Why the Standard Model. J. Geom. Phys. 58, 38–47 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Chamseddine A.H., Connes A., Marcolli M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1090 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    Connes A.: Noncommutative Geometry. Academic Press, London (1994)MATHGoogle Scholar
  16. 16.
    Connes A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    structuredConnes, A.: Noncommutative differential geometry and the structure of space-time. In: Proceedings of the Symposium on Geometry, Huggett, S.A. (ed.) et al., Oxford: Oxford Univ. Press, 1998, pp. 49–80Google Scholar
  18. 18.
    Connes A.: Noncommutative geometry and the Standard Model with neutrino mixing. JHEP 11, 081 (2006)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Connes, A.: On the spectral characterization of manifolds. http://arxiv.org/abs/0810.2088v1 [math.OA], 2008
  20. 20.
    Connes, A., Marcolli, M.: Noncommutative geometry, quantum fields and motives. Colloquium Publications, Vol. 55, Providence, RI: Amer. Math. Soc., 2008Google Scholar
  21. 21.
    Coquereaux R.: On the finite dimensional quantum group \({M_3\oplus (M_{2|1}(\Lambda^2))_0}\) . Lett. Math. Phys. 42, 309–328 (1997)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    D’Andrea F., Dąbrowski L., Landi G., Wagner E.: Dirac operators on all Podleś spheres. J. Noncomm. Geom. 1, 213–239 (2007)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    D’Andrea F., Dąbrowski L., Landi G.: The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere. Commun. Math. Phys. 279, 77–116 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Dąbrowski L., Landi G., Paschke M., Sitarz A.: The spectral geometry of the equatorial Podleś sphere. Comptes Rendus Acad. Sci. Paris 340, 819–822 (2005)MathSciNetGoogle Scholar
  25. 25.
    Dąbrowski L., Landi G., Sitarz A., van Suijlekom W., Várilly J.C.: The Dirac operator on SU q(2). Commun. Math. Phys. 259, 729–759 (2005)ADSCrossRefGoogle Scholar
  26. 26.
    Dabrowski L., Nesti F., Siniscalco P.: A Finite Quantum Symmetry of \({M(3,{\mathbb C})}\) . Int. J. Mod. Phys. A 13, 4147–4162 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Fukuda (Super-Kamiokande Collaboration) Y. et al.: Evidence for Oscillation of Atmospheric Neutrinos. Phys. Rev. Lett. 81, 1562–1567 (1998)ADSCrossRefGoogle Scholar
  28. 28.
    Goswami D.: Quantum Group of Isometries in Classical and Noncommutative Geometry. Commun. Math. Phys. 285, 141–160 (2009)MathSciNetADSCrossRefMATHGoogle Scholar
  29. 29.
    Goswami D.: Quantum Isometry Group for Spectral Triples with Real Structure. SIGMA 6, 007 (2010)MathSciNetGoogle Scholar
  30. 30.
    Kastler, D.: Regular and adjoint representation of SL q(2) at third root of unit. CPT internal report, 1995Google Scholar
  31. 31.
    Lizzi F., Mangano G., Miele G., Sparano G.: Fermion Hilbert space and fermion doubling in the noncommutative geometry approach to gauge theories. Phys. Rev. D 55, 6357–6366 (1997)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Maes A., Van Daele A.: Notes on compact quantum groups. Nieuw Arch. Wisk. 16, 73–112 (1998)MathSciNetMATHGoogle Scholar
  33. 33.
    Podles P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO (3 groups. Commun. Math. Phys. 170, 1–20 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  34. 34.
    Sołtan P.M.: Quantum SO(3) groups and quantum group actions on M 2. J. Noncommut. Geom. 4, 1–28 (2010)MathSciNetADSMATHGoogle Scholar
  35. 35.
    Van Daele A., Wang S.: Universal quantum groups. Int. J. Math. 7, 255–264 (1996)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Wang S.: Free products of compact quantum groups. Commun. Math. Phys. 167(3), 671–692 (1995)ADSMATHCrossRefGoogle Scholar
  37. 37.
    Wang S.: Quantum Symmetry Groups of Finite Spaces. Commun. Math. Phys. 195, 195–211 (1998)ADSMATHCrossRefGoogle Scholar
  38. 38.
    Wang S.: Structure and Isomorphism Classification of Compact Quantum Groups A u(Q) and B u(Q). J. Operator Theory 48, 573–583 (2002)MathSciNetMATHGoogle Scholar
  39. 39.
    Wang S.: Ergodic actions of universal quantum groups on operator algebra. Commun. Math. Phys. 203(2), 481–498 (1999)ADSMATHCrossRefGoogle Scholar
  40. 40.
    Woronowicz S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111, 613–665 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  41. 41.
    Woronowicz, S.L.: Compact quantum groups. In: Symétries quantiques (Les Houches, 1995), edited by A. Connes et al., Amsterdam: Elsevier, 1998, pp. 845–884Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Jyotishman Bhowmick
    • 1
  • Francesco D’Andrea
    • 2
  • Ludwik Dąbrowski
    • 2
  1. 1.Abdus Salam International Center for Theoretical Physics (ICTP)TriesteItaly
  2. 2.Scuola Internazionale Superiore di Studi Avanzati (SISSA)TriesteItaly

Personalised recommendations