Fluctuations of the Spectral Action

  • Michał EcksteinEmail author
  • Bruno Iochum
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 27)


As we have learned in Sect.  1.6 a given spectral triple \((\mathscr {A,H,D})\) ought to be considered as a representative of the entire family of triples \((\mathscr {A},\mathscr {H},\mathscr {D}_{\mathbb {A}})\), which yield equivalent geometries. It is therefore of utmost importance to understand how the spectral action is affected by the fluctuations of geometry. We explore the meromorphic structure of the fluctuated zeta function and, for regular spectral triples with simple dimension spectra, we provide a few formulae for the noncommutative integrals. Finally, we sketch the method of operator perturbations.


  1. 1.
    Azamov, N., Carey, A., Dodds, P., Sukochev, F.: Operator integrals, spectral shift and spectral flow. Can. J. Math. 61, 241–263 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chamseddine, A.H., Connes, A., van Suijlekom, W.D.: Inner fluctuations in noncommutative geometry without the first order condition. J. Geom. Phys. 73, 222–234 (2013)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chattopadhyay, A., Sinha, K.: Trace formulae in operator theory. In: Bhattacharyya, T., Dritschel, M. (eds.) Operator Algebras and Mathematical Physics, pp. 1–36. Birkhäuser, Basel (2015)Google Scholar
  4. 4.
    Connes, A., Chamseddine, A.H.: Inner fluctuations of the spectral action. J. Geom. Phys. 57(1), 1–21 (2006)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. Colloquium Publications, vol. 55. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  6. 6.
    Davies, E.B.: One-Parameter Semigroups. Academic Press, London (1980)zbMATHGoogle Scholar
  7. 7.
    Essouabri, D., Iochum, B., Levy, C., Sitarz, A.: Spectral action on noncommutative torus. J. Noncommutative Geom. 2(1), 53–123 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Géré, A., Wallet, J.C.: Spectral theorem in noncomutative field theories: Jacobi dynamics. J. Phys. Conf. Ser. 634, 012006 (2015)CrossRefGoogle Scholar
  9. 9.
    Iochum, B., Levy, C.: Spectral triples and manifolds with boundary. J. Funct. Anal. 260, 117–134 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Iochum, B., Levy, C.: Tadpoles and commutative spectral triples. J. Noncommutative Geom. 5(3), 299–329 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Iochum, B., Levy, C., Sitarz, A.: Spectral action on \(SU_q(2)\). Commun. Math. Phys. 289(1), 107–155 (2009)ADSCrossRefGoogle Scholar
  12. 12.
    Iochum, B., Levy, C., Vassilevich, D.: Spectral action beyond the weak-field approximation. Commun. Math. Phys. 316(3), 595–613 (2012)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Iochum, B., Levy, C., Vassilevich, D.: Spectral action for torsion with and without boundaries. Commun. Math. Phys. 310(2), 367–382 (2012)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Simon, B.: Trace Ideals and Their Applications. Mathematical Surveys and Monographs, vol. 120. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  15. 15.
    Skripka, A.: Asymptotic expansions for trace functionals. J. Funct. Anal. 266(5), 2845–2866 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    van Suijlekom, W.D.: Perturbations and operator trace functions. J. Funct. Anal. 260(8), 2483–2496 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    van Suijlekom, W.D.: Noncommutative Geometry and Particle Physics. Springer, Dordrecht (2015)CrossRefGoogle Scholar
  18. 18.
    Zagrebnov, V.: Topics in the Theory of Gibbs Semigroups. Leuven University Press, Louvain (2003)zbMATHGoogle Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Copernicus Center for Interdisciplinary StudiesKrakówPoland
  2. 2.Faculty of Physics, Astronomy and Applied Computer ScienceJagiellonian UniversityKrakówPoland
  3. 3.National Quantum Information Centre, Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and InformaticsUniversity of GdańskGdańskPoland
  4. 4.Aix-Marseille Univ, Université de Toulon, CNRS, CPTMarseilleFrance

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