Letters in Mathematical Physics

, Volume 98, Issue 3, pp 333–348 | Cite as

Spectral Action for Scalar Perturbations of Dirac Operators

Open Access
Article

Abstract

We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey–de Witt coefficients and make explicit calculations for the case of n-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes (J Geom Phys 57:121, 2006) and study the case of SUq(2).

Mathematics Subject Classification (2000)

58B34 81T75 

Keywords

spectral geometry noncommutative geometry 

References

  1. 1.
    Barth N.H.: Heat kernel expansion coefficient. I. An extension. J. Phys. A Math. Gen. 20, 857–874 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Chamseddine A., Connes A.: Inner fluctuations of the spectral action. J. Geom. Phys. 57, 121 (2006)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Connes, A.: On the spectral characterization of manifolds. arXiv:0810.2088.Google Scholar
  4. 4.
    Connes A.: Noncommutative Geometry and Reality. J. Math. Phys. 36, 619 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2) (1995)Google Scholar
  6. 6.
    Connes A., Marcolli M.: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society, New York (2007)Google Scholar
  7. 7.
    Da̧abrowski L., Landi G., Sitarz A., van Suijlekom W., Várilly J.C.: The Dirac operator on SU q(2). Commun. Math. Phys. 259, 729–759 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    Donnelly, H.: Heat equation asymptotics with torsion. Indiana Univ. Math. J. 34(1) (1985)Google Scholar
  9. 9.
    Essouabri D., Iochum B., Levy C., Sitarz A.: Spectral action on noncommutative torus. J. Noncommut. Geom. 2, 53123 (2008)MathSciNetGoogle Scholar
  10. 10.
    Gracia-Bondía J.M., Várilly J.C., Figueroa H.: Elements of Noncommutative Geometry. Birkhäuser, Boston (2001)MATHGoogle Scholar
  11. 11.
    Gusynin V.P., Gorbar E.V., Romankov V.V.: Heat kernel expansion for nonminimal differential operations and manifolds with torsion. Nucl. Phys. B 362, 449–471 (1991)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Hanisch F., Pfffle F., Stephan C.: The Spectral Action for Dirac Operators with Skew-Symmetric Torsion. Commun. Math. Phys. 300(3), 877–888 (2010)ADSMATHCrossRefGoogle Scholar
  13. 13.
    Iochum B., Levy C., Sitarz A.: Spectral action on SU q(2). Commun. Math. Phys. 289, 107–155 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Iochum, B., Levy, C., Vassilevich, D.: Spectral action for torsion with and without boundaries. arXiv:1008.3630Google Scholar
  15. 15.
    Kalau W., Walze M.: Gravity, noncommutative geometry and the Wodzicki residue. J. Geom. Phys. 16, 327–344 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Kastler D.: The Dirac operator and gravitation. Commun. Math. Phys. 166(3), 633–643 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Marcolli, M., Pierpaoli, E., Teh, K.: The spectral action and cosmic topology. arXiv:1005.2256Google Scholar
  18. 18.
    Nieh H.T., Yan M.L.: Quantized Dirac field in curved Riemann-Cartan background. Ann. Phys. 138, 237 (1982)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Vassilevich D.V.: Heat kernel expansion: user’s manual. Phys. Rep. 388C, 279–360 (2003)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Yajima S.: Mixed anomalies in 4 and 6 dimensional space with torsion. Prog. Theor. Phys. 79(2), 535–554 (1988)ADSCrossRefGoogle Scholar
  21. 21.
    Yajima S.: Evaluation of the heat kernel in Riemann-Cartan space. Class. Quantum Gravity 13(9), 2423–2435 (1996)MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institute of PhysicsJagiellonian UniversityKrakówPoland

Personalised recommendations