Skip to main content
SpringerLink
Log in

Periods and Motives in the Spectral Action of Robertson–Walker Spacetimes

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson–Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bloch S., Esnault H., Kreimer D.: On motives associated to graph polynomials. Commun. Math. Phys. 267(1), 181–225 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Brown F., Schnetz O.: A K3 in \({\phi^4}\). Duke Math. J. 161(10), 1817–1862 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chamseddine, A.H., Connes, A.: Spectral action for Robertson–Walker metrics. J. High Energy Phys. 10, 101 (2012)

  4. Chamseddine A.H., Connes A.: The spectral action principle. Commun. Math. Phys. 186(3), 731–750 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Connes A.: Geometry from the spectral point of view. Lett. Math. Phys. 34(3), 203–238 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Connes, A., Marcolli, M.: Renormalization and motivic Galois theory. Int. Math. Res. Not. 76, 4073–4091 (2004)

  7. Fan W., Fathizadeh F., Marcolli M.: Spectral action for Bianchi type-IX cosmological models. J. High Energy Phys. 10, 085 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  8. Fan, W., Fathizadeh, F., Marcolli M.: Modular forms in the spectral action of Bianchi IX gravitational instantons. arXiv:1511.05321

  9. Fathizadeh, F., Ghorbanpour, A., Khalkhali, M.: Rationality of spectral action for Robertson–Walker metrics. J. High Energy Phys. 12, 064 (2014)

  10. Golden J., Goncharov A.B., Spradlin M., Vergu C., Volovich A.: Motivic amplitudes and cluster coordinates. J. High Energy Phys. 01, 091 (2014)

    Article  ADS  Google Scholar 

  11. Goncharov, A.B., Spradlin, M., Vergu, C., Volovich, A.: Classical polylogarithms for amplitudes and Wilson loops. Phys. Rev. Lett. 105(15), 151605 (2010)

  12. Gracia-Bondia J.M., Varilly J.C., Figueroa H.: Elements of Noncommutative Geometry. Birkhäuser, Cambridge (2001)

    Book  MATH  Google Scholar 

  13. Kontsevich, M., Zagier, D.: Periods. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited: 2001 and Beyond, pp. 771–808, Springer, Berlin (2001)

  14. Marcolli M.: Feynman Motives. World Scientific, (2010)

  15. Marcolli, M.: Noncommutative Cosmology. World Scientific (2017)

  16. Rost, M.: The motive of a Pfister form. Preprint (1998). http://www.physik.uni-regensburg.de/~rom03516/motive.html

  17. Suijlekom W.: Noncommutative Geometry and Particle Physics. Springer, Berlin (2014)

    MATH  Google Scholar 

  18. Vishik, A.: Integral motives of quadrics. Max-Planck-Institut für Mathematik Bonn, Preprint MPI-1998-13, pp. 1–82 (1998)

  19. Vishik, A.: Motives of quadrics with applications to the theory of quadratic forms. In: Geometric methods in the algebraic theory of quadratic forms, pp. 25–101, Lecture Notes in Math., vol. 1835, Springer, Berlin (2004)

  20. Voevodsly V., Suslin A., Friedlander E.M.: Cyles, Transfers, and Motivic Homology Theories. Princeton University Press, Princeton (2000)

    Google Scholar 

  21. Wodzicki M.: Local invariants of spectral asymmetry. Invent. Math. 75(1), 143–177 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Wodzicki, M.: Noncommutative residue. I. Fundamentals. In: K-theory, Arithmetic and Geometry (Moscow, 1984–1986), pp. 320–399, Lecture Notes in Math., vol. 1289, Springer, Berlin (1987)

Download references

Author information

Author notes

  1. Farzad Fathizadeh

    Present address: Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK

Authors and Affiliations

  1. Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Farzad Fathizadeh & Matilde Marcolli

Authors
  1. Farzad Fathizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matilde Marcolli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farzad Fathizadeh.

Additional information

Communicated by Y. Kawahigashi

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fathizadeh, F., Marcolli, M. Periods and Motives in the Spectral Action of Robertson–Walker Spacetimes. Commun. Math. Phys. 356, 641–671 (2017). https://doi.org/10.1007/s00220-017-2991-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2991-x

Search

Navigation