Annales Henri Poincaré

, 12:829 | Cite as

The 1/N Expansion of Colored Tensor Models

Article

Abstract

In this paper, we perform the 1/N expansion of the colored three-dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with increasingly complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S3 contribute to the leading order in the large N limit.

References

  1. 1.
    ’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Gross D.J., Miljkovic N.: A nonperturbative solution of D = 1 string theory. Phys. Lett. B 238, 217 (1990)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Gross D.J., Klebanov I.R.: One-dimensional string theory on a circle. Nucl. Phys. B 344, 475–498 (1990)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2-D gravity and random matrices. Phys. Rept. 254, 1–133 (1995) [hep-th/9306153]MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    David F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985)ADSCrossRefGoogle Scholar
  6. 6.
    Kazakov V.A., Migdal A.A., Kostov I.K.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157, 295–300 (1985)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Boulatov D.V., Kazakov V.A., Kostov I.K. et al.: Analytical and numerical study of the model of dynamically triangulated random surfaces. Nucl. Phys. B 275, 641 (1986)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Kazakov V., Kostov I.K., Kutasov D.: A Matrix model for the two-dimensional black hole. Nucl. Phys. B 622, 141–188 (2002) [hep-th/0101011]MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Brezin E., Itzykson C., Parisi G., Zuber J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Gross M.: Tensor models and simplicial quantum gravity in > 2-D. Nucl. Phys. Proc. Suppl. 25, 144 (1992)ADSCrossRefGoogle Scholar
  11. 11.
    Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133 (1991)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Freidel L.: Group field theory: an overview. Int. J. Theor. Phys. 44, 1769 (2005) [arXiv:hep-th/0505016]MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Oriti, D.: The Group Field Theory Approach to Quantum Gravity: Some Recent Results. [arXiv:0912.2441 [hep-th]]Google Scholar
  15. 15.
    Boulatov D.V.: A model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992) [arXiv:hep-th/9202074]MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Freidel L., Louapre D.: Ponzano-Regge model revisited. I: Gauge fixing, observables and interacting spinning particles. Class. Quant. Grav. 21, 5685 (2004) [arXiv:hep-th/0401076]MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Baratin, A., Oriti, D.: Group Field Theory with Non-Commutative Metric Variables. [arXiv:1002.4723 [hep-th]]Google Scholar
  18. 18.
    Engle J., Pereira R., Rovelli C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008) [arXiv:0708.1236 [gr-qc]]MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Livine E.R., Speziale S.: A new spinfoam vertex for quantum gravity. Phys. Rev. D 76, 084028 (2007) [arXiv:0705.0674 [gr-qc]]MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Freidel L., Krasnov K.: A new spin foam model for 4D gravity. Class. Quant. Grav. 25, 125018 (2008) [arXiv:0708.1595 [gr-qc]]MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Geloun J.B., Gurau R., Rivasseau V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010) [arXiv:1008.0354 [hep-th]]CrossRefGoogle Scholar
  22. 22.
    Alexandrov, S., Roche, P.: Critical Overview of Loops and Foams. arXiv:1009.4475 [gr-qc]Google Scholar
  23. 23.
    Gurau, R.: Colored Group Field Theory. [arXiv:0907.2582 [hep-th]]Google Scholar
  24. 24.
    Gurau R.: Topological graph polynomials in colored group field theory. Ann. Henri Poincaré 11, 565 (2010) [arXiv:0911.1945 [hep-th]]MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Gurau R.: Lost in translation: topological singularities in group field theory. Class. Quant. Grav. 27, 235023 (2010) arXiv:1006.0714 [hep-th]MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Freidel L., Gurau R., Oriti D.: Group field theory renormalization—the 3D case: power counting of divergences. Phys. Rev. D 80, 044007 (2009) [arXiv:0905. 3772 [hep-th]]ADSCrossRefGoogle Scholar
  27. 27.
    Magnen J., Noui K., Rivasseau V., Smerlak M.: Scaling behaviour of three-dimensional group field theory. Class. Quant. Grav. 26, 185012 (2009) [arXiv: 0906.5477 [hep-th]]MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Geloun, J.B., Magnen, J., Rivasseau, V.: Bosonic Colored Group Field Theory. [arXiv:0911.1719 [hep-th]]Google Scholar
  29. 29.
    Geloun J.B., Krajewski T., Magnen J., Rivasseau V.: Linearized group field theory and power counting theorems. Class. Quant. Grav. 27, 155012 (2010) [arXiv:1002.3592 [hep-th]]ADSCrossRefGoogle Scholar
  30. 30.
    Gurau R., Rivasseau V.: Parametric representation of noncommutative field theory. Commun. Math. Phys. 272, 811 (2007) [arXiv:math-ph/0606030]MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    Lins, S.: Gems, Computers and Attractors for 3-Manifolds. Series on Knots and Everything, vol. 5. ISBN: 9810219075/ISBN-13: 9789810219079Google Scholar
  32. 32.
    Ferri, M., Gagliardi, C.: Crystallisation moves. Pac. J. Math. 100(1) (1982)Google Scholar
  33. 33.
    Bonzom V., Smerlak M.: Bubble divergences from cellular cohomology. Lett. Math. Phys. 93, 295 (2010) [arXiv:1004.5196 [gr-qc]]MathSciNetADSMATHCrossRefGoogle Scholar
  34. 34.
    Grosse H., Wulkenhaar R.: Renormalisation of phi**4 theory on noncommutative R**4 in the matrix base. Commun. Math. Phys. 256, 305 (2005) [arXiv:hep-th/0401128]MathSciNetADSMATHCrossRefGoogle Scholar
  35. 35.
    Gurau R., Magnen J., Rivasseau V., Vignes-Tourneret F.: Renormalization of non-commutative phi**4(4) field theory in x space. Commun. Math. Phys. 267, 515 (2006) [arXiv:hep-th/0512271]MathSciNetADSMATHCrossRefGoogle Scholar
  36. 36.
    Rivasseau V., Vignes-Tourneret F., Wulkenhaar R.: Renormalization of noncommutative phi**4-theory by multi-scale analysis. Commun. Math. Phys. 262, 565–594 (2006) [hep-th/0501036]MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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