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The Ponzano–Regge Model and Parametric Representation

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Abstract

We give a parametric representation of the effective noncommutative field theory derived from a \({\kappa}\) -deformation of the Ponzano–Regge model and define a generalized Kirchhoff polynomial with \({\kappa}\) -correction terms, obtained in a \({\kappa}\) -linear approximation. We then consider the corresponding graph hypersurfaces and the question of how the presence of the correction term affects their motivic nature. We look in particular at the tetrahedron graph, which is the basic case of relevance to quantum gravity. With the help of computer calculations, we verify that the number of points over finite fields of the corresponding hypersurface does not fit polynomials with integer coefficients, hence the hypersurface of the tetrahedron is not polynomially countable. This shows that the correction term can change significantly the motivic properties of the hypersurfaces, with respect to the classical case.

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References

  1. Aluffi, P., Marcolli, M.: Feynman motives and deletion-contraction relations. In: Topology of Algebraic Varieties and Singularities, Vol. 538, Contemporary Mathematics, 2011, pp. 21–64

  2. Aluffi P., Marcolli M.: A motivic approach to phase transitions in Potts models. J. Geom. Phys. 63, 6–31 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Belkale P., Brosnan P.: Matroids, motives, and a conjecture of Kontsevich. Duke Math. J. 116, 147–188 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bittner F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(2), 1011–1032 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bjorken, J., Drell, S.: Relativistic Quantum Fields, Maidenheach: McGraw-Hill, 1965

  6. Bloch S.: Motives associated to graphs. Jpn. J. Math. 2, 165–196 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bloch, S.: Motives associated to sums of graphs. In: The Geometry of Algebraic Cycles (Clay Mathematics Proceedings), Vol. 9, AMS, 2010, pp. 137–145

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Broadhurst D., Kreimer D.: Association of multiple zeta values with positive knots via feynman diagrams up to 9 loops. Phys. Lett. B 393, 403–412 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Brown F.: Multiple zeta values and periods of moduli spaces \({\mathfrak{M}_{0,n}}\) . Annales scientifiques de l’ENS 42(fascicule 3), 371–489 (2009)

    MATH  Google Scholar 

  11. Brown, F., Doryn, D.:Framings for graph hypersurfaces, arXiv:1301.3056[math.AG] (2013)

  12. Brown F., Schnetz O.: A K3 in ϕ4. Duke Math. J. 161(10), 1817–1862 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Brown, F., Schnetz, O.:Modular forms in quantum field theory, arXiv:1304.5342[math.AG] (2013)

  14. Connes, A., Marcolli, M.: Noncommutative geometry, quantum fields and motives, Vol. 55, Colloquium Publications, AMS, 2007

  15. Freidel L., Livine E.: Effective 3d quantum gravity and non-commutative quantum field theory. Phys. Rev. Lett. 96, 221301 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  16. Freidel L., Livine E.: Ponzano–Regge model revisited III: Feynman diagrams and effective field theory. Class Quant. Grav. 23, 2021 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Freidel L., Louapre D.: Ponzano–Regge model revisited I: Gauge fixing, observables and interacting spinning particles. Class. Quant. Grav. 21, 5685–5726 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Gillet H., Soulé C.: Descent, motives and K-theory. J. Reine Angew. Math. 478, 127–176 (1996)

    MATH  MathSciNet  Google Scholar 

  19. Goncharov A.B., Manin Yu.I.: Multiple ζ-motives and moduli spaces \({\overline{\mathfrak{m}}_{0,n}}\) . Compos. Math. Soc. 12(2), 569–618 (1999)

    MATH  Google Scholar 

  20. Gurau R., Rivasseau V.: Parametric representation of noncommutative field theory. Commun. Math. Phys. 272, 811–835 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Itzykson, C., Zuber, J.B.:Quantum Field Theory, New York: Dover Publications, 2006

  22. Krajewski T., Rivasseau V., Tanasa A., Wang Z.: . J. Noncommut. Geom. 4, 29–82 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Marcolli, M.: Feynman motives, Singapore: World Scientific, 2009

  24. Reine M.: Counting rational points over quiver moduli. Int. Math. Res. Lett. 70456, 1–19 (2006)

    Google Scholar 

  25. Stembridge J.: Counting points on varieties over finite fields related to a conjecture of Kontsevich. Ann. Comb. 2(4), 365–385 (1998)

    Article  MathSciNet  Google Scholar 

  26. Tanasa, A.: Overview of the parametric representation of renormalizable non-commutative field theory. J. Phys. Conf. Ser. 103(012012) (2008)

  27. Terasoma T.: Mixed tate motives and multiple zeta values. Invent. Math. 149(2), 339–369 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Dan Li

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  1. Dan Li
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Correspondence to Dan Li.

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Communicated by A. Connes

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Li, D. The Ponzano–Regge Model and Parametric Representation. Commun. Math. Phys. 327, 243–260 (2014). https://doi.org/10.1007/s00220-014-1945-9

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  • DOI: https://doi.org/10.1007/s00220-014-1945-9

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