Spectral Invariants

  • Walter D. van SuijlekomEmail author
Part of the Mathematical Physics Studies book series (MPST)


In the previous chapter we have identified the gauge group canonically associated to any spectral triple and have derived the generalized gauge fields that carry an action of that gauge group.


Spectral Invariants Generalized Gauge Fields Gauge Group Spectral Activity Asymptotic Taylor Expansion 
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.


  1. 1.
    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)CrossRefzbMATHMathSciNetADSGoogle Scholar
  2. 2.
    Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997)CrossRefzbMATHMathSciNetADSGoogle Scholar
  3. 3.
    Lizzi, F., Mangano, G., Miele, G., Sparano, G.: Fermion Hilbert space and fermion doubling in the noncommutative geometry approach to gauge theories. Phys. Rev. D55, 6357–6366 (1997)MathSciNetADSGoogle Scholar
  4. 4.
    Connes, A., Marcolli, M.: Noncommutative Geometry. Quantum Fields and Motives. AMS, Providence (2008)Google Scholar
  5. 5.
    Widder, D.V.: The Laplace Transform. Princeton Mathematical Series, vol. 6. Princeton University Press, Princeton (1941)Google Scholar
  6. 6.
    Connes, A., Chamseddine, A.H.: Inner fluctuations of the spectral action. J. Geom. Phys. 57, 1–21 (2006)CrossRefzbMATHMathSciNetADSGoogle Scholar
  7. 7.
    van Suijlekom, W.D.: Perturbations and operator trace functions. J. Funct. Anal. 260, 2483–2496 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Higson, N.: The residue index theorem of Connes and Moscovici. In: Surveys in Noncommutative Geometry, Vol. 6 of Clay Mathematics Proceedings, American Mathematical Society, Providence, RI, pp. 71–126 (2006)Google Scholar
  9. 9.
    Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7, 65–222 (1982)Google Scholar
  10. 10.
    Hansen, F.: Trace functions as Laplace transforms. J. Math. Phys. 47(11), 043504 (2006)Google Scholar
  11. 11.
    Skripka, A.: Asymptotic expansions for trace functionals. J. Funct. Anal. 266, 2845–2866 (2014)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gilliam, D.S., Hohage, T., Ji, X., Ruymgaart, F.: The Fréchet derivative of an analytic function of a bounded operator with some applications. Int. J. Math. Math. Sci. 17, Art. ID 239025 (2009)Google Scholar
  13. 13.
    Lifshits, I.: On a problem in perturbation theory (russian). Uspehi Matem. Nauk. 7, 171–180 (1952)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Krein, M.: On a trace formula in perturbation theory. Matem. Sbornik 33, 597–626 (1953)MathSciNetGoogle Scholar
  15. 15.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Cambridge Philos. Soc. 77, 43–69 (1975)CrossRefzbMATHMathSciNetADSGoogle Scholar
  16. 16.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry II. Math. Proc. Cambridge Philos. Soc. 78, 405–432 (1975)CrossRefzbMATHMathSciNetADSGoogle Scholar
  17. 17.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry III. Math. Proc. Cambridge Philos. Soc. 79, 71–99 (1976)CrossRefzbMATHMathSciNetADSGoogle Scholar
  18. 18.
    Yafaev, D.R.: Mathematical Scattering Theory, vol. 105 of Translations of Mathematical Monographs. General Theory. American Mathematical Society, Providence (1992). (Translated from the Russian by J.R Schulenberger)Google Scholar
  19. 19.
    Birman, M.S., Pushnitski, A.B.: Spectral Shift Function, Amazing and Multifaceted. Integr. Eqn. Oper. Theory 30, 191–199 (1998). [Dedicated to the memory of Mark Grigorievich Krein (1907–1989)]Google Scholar
  20. 20.
    Koplienko, L.S.: The trace formula for perturbations of nonnuclear type. Sibirsk. Mat. Zh. 25, 62–71 (1984)MathSciNetGoogle Scholar
  21. 21.
    Potapov, D., Skripka, A., Sukochev, F.: Spectral shift function of higher order. Invent. Math. 193, 501–538 (2013)CrossRefzbMATHMathSciNetADSGoogle Scholar
  22. 22.
    Carey, A., Phillips, J.: Unbounded Fredholm modules and spectral flow. Canad. J. Math. 50, 673–718 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Phillips, J., Raeburn, I.: An index theorem for Toeplitz operators with noncommutative symbol space. J. Funct. Anal. 120, 239–263 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Müller, W.: Relative zeta functions, relative determinants and scattering theory. Commun. Math. Phys. 192, 309–347 (1998)CrossRefzbMATHADSGoogle Scholar
  25. 25.
    Azamov, N.A., Carey, A.L., Sukochev, F.A.: The spectral shift function and spectral flow. Commun. Math. Phys. 276, 51–91 (2007)CrossRefzbMATHMathSciNetADSGoogle Scholar
  26. 26.
    Azamov, N.A., Carey, A.L., Dodds, P.G., Sukochev, F.A.: Operator integrals, spectral shift, and spectral flow. Canad. J. Math. 61, 241–263 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.A.: The local index formula in semifinite von Neumann algebras I. Spectral flow. Adv. Math. 202, 451–516 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.A.: The local index formula in semifinite von Neumann algebras II. The even case. Adv. Math. 202, 517–554 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Benameur, M.-T., Fack, T.: Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras. Adv. Math. 199, 29–87 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Carey, A.L., Phillips, J., Sukochev, F.A.: On unbounded \(p\)-summable Fredholm modules. Adv. Math. 151, 140–163 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Carey, A., Phillips, J., Sukochev, F.: Spectral flow and Dixmier traces. Adv. Math. 173, 68–113 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Gesztesy, F., Latushkin, Y., Makarov, K.A., Sukochev, F., Tomilov, Y.: The index formula and the spectral shift function for relatively trace class perturbations. Adv. Math. 227, 319–420 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Hermite, C.: Sur la formule d’interpolation de lagrange. J. Reine Angew. Math. 84, 70–79 (1878)CrossRefGoogle Scholar
  34. 34.
    Floater, M.S., Lyche, T.: Two chain rules for divided differences and Faà di Bruno’s formula. Math. Comp. 76, 867–877 (2007)CrossRefzbMATHMathSciNetADSGoogle Scholar
  35. 35.
    Donoghue Jr, W.F.: Monotone Matrix functions and Analytic Continuation. Die Grundlehren der Mathematischen Wissenschaften, Band 207. Springer, New York (1974)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.IMAPPRadboud University NijmegenNijmegenThe Netherlands

Personalised recommendations