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.
Article PDF
Similar content being viewed by others
References
Gurau R.: The 1/N expansion of colored tensor models. Annales Henri Poincare 12, 829 (2011) arXiv:1011.2726 [gr-qc]
Gurau, R., Rivasseau, V.: The 1/N expansion of colored tensor models in arbitrary dimension. (2011) arXiv:1101.4182 [gr-qc]
David F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985)
Gross M.: Tensor models and simplicial quantum gravity in > 2-D. Nucl. Phys. Proc. Suppl. 25, 144 (1992)
Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix mod. Mod. Phys. Lett. A 6, 1133 (1991)
Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)
Boulatov D.V.: A model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992) arXiv:hep-th/9202074
Ooguri H.: Topological lattice models in four-dimensions. Mod. Phys. Lett. A 7, 2799 (1992) arXiv:hep-th/9205090
Freidel L.: Group field theory: an overview. Int. J. Theor. Phys. 44, 1769 (2005) arXiv:hep-th/0505016
Oriti, D.: The group field theory approach to quantum gravity: some recent results. arXiv:0912.2441 [hep-th]
Brezin E., Itzykson C., Parisi G., Zuber J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978)
’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)
Gross D.J., Miljkovic N.: A nonperturbative solution of D = 1 string theory. Phys. Lett. B 238, 217 (1990)
Gross D.J., Klebanov I.R.: One-dimensional string theory on a circle. Nucl. Phys. B 344, 475 (1990)
Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2-D Gravity and random matrices. Phys. Rept. 254, 1 (1995) arXiv:hep-th/9306153
Kazakov V.A., Migdal A.A., Kostov I.K.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157, 295 (1985)
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)
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/0101011
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
Baratin A., Oriti D.: Group field theory with non-commutative metric variables. Phys. Rev. Lett. 105, 221302 (2010) arXiv:1002.4723 [hep-th]
Engle J., Pereira R., Rovelli C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008) arXiv:0708.1236[gr-qc]
Livine E.R., Speziale S.: A new spinfoam vertex for quantum gravity. Phys. Rev. D 76, 084028 (2007) arXiv:0705.0674[gr-qc]
Freidel L., Krasnov K.: A new spinfoam model for 4d gravity. Class. Quant. Grav. 25, 125018 (2008) arXiv:0708.1595 [gr-qc]
Geloun J.B., Gurau R., Rivasseau V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010) arXiv:1008.0354 [hep-th]
Oriti D., Tlas T.: Encoding simplicial quantum geometry in group field theories. Class. Quant. Grav. 27, 135018 (2010) arXiv:0912.1546 [gr-qc]
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
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
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/0612251
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]
Conrady F., Freidel L.: On the semiclassical limit of 4d spin foam models. Phys. Rev. D 78, 104023 (2008) arXiv:0809.2280 [gr-qc]
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]
Freidel, L., Oriti, D., Ryan, J.: A group field theory for 3d quantum gravity coupled to a scalar field. arXiv:gr-qc/0506067
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/0602010
Dowdall, R.J.: Wilson loops, geometric operators and fermions in 3d group field theory. arXiv:0911.2391 [gr-qc]
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/0702125
Di Mare A., Oriti D.: Emergent matter from 3d generalised group field theories. Class. Quant. Grav 27, 145006 (2010) arXiv:1001.2702 [gr-qc]
Baratin, A., Girelli, F., Oriti, D.: Diffeomorphisms in group field theories. arXiv:1101.0590 [hep-th]
Ashtekar A., Campiglia M., Henderson A.: Loop quantum cosmology and spin foams. Phys. Lett. B 681, 347 (2009) arXiv:0909.4221 [gr-qc]
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]
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]
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]
Geloun J.B., Magnen J., Rivasseau V.: Bosonic colored group field theory. Eur. Phys. J. C 70, 1119 (2010) arXiv:0911.1719 [hep-th]
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]
Geloun, J.B., Bonzom, V.: Radiative corrections in the Boulatov-Ooguri tensor model: the 2-point function. arXiv:1101.4294[hep-th]
Alexandrov, S., Roche, P.: Critical overview of loops and foams. arXiv:1009.4475 [gr-qc]
Gurau R.: Colored group field theory. Commun. Math. Phys 304, 69 (2011) arXiv:0907.2582 [hep-th]]
Gurau R.: Topological graph polynomials in colored group field theory. Annales Henri Poincaré 11, 565 (2010) arXiv:0911.1945 [hep-th]
Gurau R.: Lost in translation: topological singularities in group field theory. Class. Quant. Grav 27, 235023 (2010) arXiv:1006.0714 [hep-th]
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/9207020
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/9302014
Gurau R.: A diagrammatic equation for oriented planar graphs. Nucl. Phys. B 839, 580 (2010) arXiv:1003.2187 [hep-th]
Gurau R., Rivasseau V.: Parametric representation of noncommutative field theory. Commun. Math. Phys 272, 811 (2007) arXiv:math-ph/0606030
Stillwell J.: The word problem and the isomorphism problem for groups. Bull. Am. Math. Soc. 6(1), 33–56 (1982)
Caravelli, F.: A simple proof of orientability in the colored Boulatov model. arXiv:1012.4087 [math-ph]
Ferri, M., Gagliardi, C.: Crystallisation moves. Pac. J. Math. 100(1) (1982)
Lins, S.: Gems, computers and attractors for 3-manifolds, (Series on Knots and Everything, Vol 5). ISBN:9810219075/ISBN-13:9789810219079
Bonzom, V., Smerlak, M.: Bubble divergences from twisted cohomology. arXiv:1008.1476 [math-ph]
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/9907154
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Carlo Rovelli.
Rights and permissions
About this article
Cite this article
Gurau, R. The Complete 1/N Expansion of Colored Tensor Models in Arbitrary Dimension. Ann. Henri Poincaré 13, 399–423 (2012). https://doi.org/10.1007/s00023-011-0118-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-011-0118-z