Advertisement

Annales Henri Poincaré

, Volume 13, Issue 3, pp 399–423 | Cite as

The Complete 1/N Expansion of Colored Tensor Models in Arbitrary Dimension

  • Razvan GurauEmail author
Article

Abstract

In this paper we generalize the results of Gurau (arXiv:1011. 2726 [gr-qc], 2011), Gurau and Rivasseau (arXiv:1101.4182 [gr-qc], 2011) and derive the full 1/N expansion of colored tensor models in arbitrary dimensions. We detail the expansion for the independent identically distributed model and the topological Boulatov Ooguri model.

Keywords

Matrix Model Arbitrary Dimension Colored Graph Ribbon Graph Loop Quantum Cosmology 
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.

References

  1. 1.
    Gurau R.: The 1/N expansion of colored tensor models. Annales Henri Poincare 12, 829 (2011) arXiv:1011.2726 [gr-qc]MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Gurau, R., Rivasseau, V.: The 1/N expansion of colored tensor models in arbitrary dimension. (2011) arXiv:1101.4182 [gr-qc]Google Scholar
  3. 3.
    David F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985)ADSCrossRefGoogle Scholar
  4. 4.
    Gross M.: Tensor models and simplicial quantum gravity in >  2-D. Nucl. Phys. Proc. Suppl. 25, 144 (1992)ADSCrossRefGoogle Scholar
  5. 5.
    Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix mod. Mod. Phys. Lett. A 6, 1133 (1991)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Boulatov D.V.: A model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992) arXiv:hep-th/9202074MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Ooguri H.: Topological lattice models in four-dimensions. Mod. Phys. Lett. A 7, 2799 (1992) arXiv:hep-th/9205090MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Freidel L.: Group field theory: an overview. Int. J. Theor. Phys. 44, 1769 (2005) arXiv:hep-th/0505016MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Oriti, D.: The group field theory approach to quantum gravity: some recent results. arXiv:0912.2441 [hep-th]Google Scholar
  11. 11.
    Brezin E., Itzykson C., Parisi G., Zuber J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    ’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Gross D.J., Miljkovic N.: A nonperturbative solution of D = 1 string theory. Phys. Lett. B 238, 217 (1990)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Gross D.J., Klebanov I.R.: One-dimensional string theory on a circle. Nucl. Phys. B 344, 475 (1990)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2-D Gravity and random matrices. Phys. Rept. 254, 1 (1995) arXiv:hep-th/9306153MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Kazakov V.A., Migdal A.A., Kostov I.K.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157, 295 (1985)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Boulatov D.V., Kazakov V.A., Kostov I.K., Migdal A.A.: Analytical and numerical study of the model of dynamically triangulated random surfaces. Nucl. Phys. B 275, 641 (1986)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Kazakov V., Kostov I.K., Kutasov D.: A matrix model for the two-dimensional black hole. Nucl. Phys. B 622, 141 (2002) arXiv:hep-th/0101011MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    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/0401076MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Baratin A., Oriti D.: Group field theory with non-commutative metric variables. Phys. Rev. Lett. 105, 221302 (2010) arXiv:1002.4723 [hep-th]MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Engle J., Pereira R., Rovelli C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008) arXiv:0708.1236[gr-qc]MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Freidel L., Krasnov K.: A new spinfoam model for 4d gravity. Class. Quant. Grav. 25, 125018 (2008) arXiv:0708.1595 [gr-qc]MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Geloun J.B., Gurau R., Rivasseau V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010) arXiv:1008.0354 [hep-th]CrossRefGoogle Scholar
  25. 25.
    Oriti D., Tlas T.: Encoding simplicial quantum geometry in group field theories. Class. Quant. Grav. 27, 135018 (2010) arXiv:0912.1546 [gr-qc]MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    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/0401128MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    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/0512271MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Disertori M., Gurau R., Magnen J., Rivasseau V.: Vanishing of beta function of non commutative phi(4)**4 theory to all orders. Phys. Lett. B 649, 95 (2007) arXiv:hep-th/0612251MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Geloun J.B., Gurau R., Rivasseau V.: Vanishing beta function for Grosse-Wulkenhaar model in a magnetic field. Phys. Lett. B 671, 284 (2009) arXiv:0805.4362 [hep-th]MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    Conrady F., Freidel L.: On the semiclassical limit of 4d spin foam models. Phys. Rev. D 78, 104023 (2008) arXiv:0809.2280 [gr-qc]MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Barrett J.W., Dowdall R.J., Fairbairn W.J., Hellmann F., Pereira R.: Lorentzian spin foam amplitudes: graphical calculus and asymptotics. Class. Quant. Grav 27, 165009 (2010) arXiv:0907.2440 [gr-qc]MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Freidel, L., Oriti, D., Ryan, J.: A group field theory for 3d quantum gravity coupled to a scalar field. arXiv:gr-qc/0506067Google Scholar
  33. 33.
    Oriti D., Ryan J.: Group field theory formulation of 3d quantum gravity coupled to matter fields. Class. Quant. Grav 23, 6543 (2006) arXiv:gr-qc/0602010MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. 34.
    Dowdall, R.J.: Wilson loops, geometric operators and fermions in 3d group field theory. arXiv:0911.2391 [gr-qc]Google Scholar
  35. 35.
    Fairbairn W.J., Livine E.R.: 3d spinfoam quantum gravity: matter as a phase of the group field theory. Class. Quant. Grav 24, 5277 (2007) arXiv:gr-qc/0702125MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Di Mare A., Oriti D.: Emergent matter from 3d generalised group field theories. Class. Quant. Grav 27, 145006 (2010) arXiv:1001.2702 [gr-qc]MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Baratin, A., Girelli, F., Oriti, D.: Diffeomorphisms in group field theories. arXiv:1101.0590 [hep-th]Google Scholar
  38. 38.
    Ashtekar A., Campiglia M., Henderson A.: Loop quantum cosmology and spin foams. Phys. Lett. B 681, 347 (2009) arXiv:0909.4221 [gr-qc]MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    Ashtekar A., Campiglia M., Henderson A.: Casting loop quantum cosmology in the spin foam paradigm. Class. Quant. Grav. 27, 135020 (2010) arXiv:1001.5147 [gr-qc]MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    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
  41. 41.
    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
  42. 42.
    Geloun J.B., Magnen J., Rivasseau V.: Bosonic colored group field theory. Eur. Phys. J. C 70, 1119 (2010) arXiv:0911.1719 [hep-th]ADSCrossRefGoogle Scholar
  43. 43.
    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
  44. 44.
    Geloun, J.B., Bonzom, V.: Radiative corrections in the Boulatov-Ooguri tensor model: the 2-point function. arXiv:1101.4294[hep-th]Google Scholar
  45. 45.
    Alexandrov, S., Roche, P.: Critical overview of loops and foams. arXiv:1009.4475 [gr-qc]Google Scholar
  46. 46.
    Gurau R.: Colored group field theory. Commun. Math. Phys 304, 69 (2011) arXiv:0907.2582 [hep-th]]MathSciNetADSzbMATHCrossRefGoogle Scholar
  47. 47.
    Gurau R.: Topological graph polynomials in colored group field theory. Annales Henri Poincaré 11, 565 (2010) arXiv:0911.1945 [hep-th]MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Gurau R.: Lost in translation: topological singularities in group field theory. Class. Quant. Grav 27, 235023 (2010) arXiv:1006.0714 [hep-th]MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    Ambjorn J., Kristjansen C.F., Makeenko Yu.M.: Higher genus correlators for the complex matrix model. Mod. Phys. Lett. A 7, 3187 (1992) arXiv:hep-th/9207020MathSciNetADSCrossRefGoogle Scholar
  50. 50.
    Ambjorn, J., Chekhov, L., Kristjansen, C.F., Makeenko, Yu.: Matrix model calculations beyond the spherical limit. Nucl. Phys. B 404, 127 (1993). Erratum-ibid. B 449, 681 (1995). arXiv:hep-th/9302014Google Scholar
  51. 51.
    Gurau R.: A diagrammatic equation for oriented planar graphs. Nucl. Phys. B 839, 580 (2010) arXiv:1003.2187 [hep-th]MathSciNetADSzbMATHCrossRefGoogle Scholar
  52. 52.
    Gurau R., Rivasseau V.: Parametric representation of noncommutative field theory. Commun. Math. Phys 272, 811 (2007) arXiv:math-ph/0606030MathSciNetADSzbMATHCrossRefGoogle Scholar
  53. 53.
    Stillwell J.: The word problem and the isomorphism problem for groups. Bull. Am. Math. Soc. 6(1), 33–56 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Caravelli, F.: A simple proof of orientability in the colored Boulatov model. arXiv:1012.4087 [math-ph]Google Scholar
  55. 55.
    Ferri, M., Gagliardi, C.: Crystallisation moves. Pac. J. Math. 100(1) (1982)Google Scholar
  56. 56.
    Lins, S.: Gems, computers and attractors for 3-manifolds, (Series on Knots and Everything, Vol 5). ISBN:9810219075/ISBN-13:9789810219079Google Scholar
  57. 57.
    Bonzom, V., Smerlak, M.: Bubble divergences from twisted cohomology. arXiv:1008.1476 [math-ph]Google Scholar
  58. 58.
    De Pietri R., Freidel L., Krasnov K., Rovelli C.: Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space. Nucl. Phys. B 574, 785 (2000) arXiv:hep-th/9907154MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations