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Topological graph polynomials in colored group field theory. (English) Zbl 1208.81153
Summary: In this paper, we analyze the open Feynman graphs of the Colored Group Field Theory introduced in R. Gurau [Colored group field theory, arXiv:0907.2582 [hep-th]]. We define the boundary graph \({\mathcal G}_\partial\) of an open graph \({\mathcal G}\) and prove it is a cellular complex. Using this structure we generalize the topological (Bollobás-Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension.

MSC:
81T18 Feynman diagrams
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[1] Gurau, R.: Colored group field theory. arXiv:0907.2582 [hep-th] · Zbl 1214.81170
[2] Nakanishi, N., Graph Theory and Feynman Integrals (1970), New York: Gordon and Breach, New York · Zbl 0212.29203
[3] Itzykson, C.; Zuber, J.-B., Quantum Field Theory (1980), New York: McGraw and Hill, New York · Zbl 0453.05035
[4] David, F., A model of random surfaces with nontrivial critical behavior, Nucl. Phys. B, 257, 543 (1985)
[5] Ginsparg, P.: Matrix models of 2-d gravity. arXiv:hep-th/9112013 · Zbl 0985.82500
[6] Gross, M., Tensor models and simplicial quantum gravity in > 2-D, Nucl. Phys. Proc. Suppl., 25A, 144-149 (1992) · Zbl 0957.83511
[7] Sasakura, N., Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A, 6, 2613 (1991) · Zbl 1020.83542
[8] Connes, A., Noncommutative Geometry (1994), San Diego: Academic Press Inc., San Diego
[9] Douglas, M. R.; Nekrasov, N. A., Noncommutative field theory, Rev. Mod. Phys., 73, 977 (2001) · Zbl 1205.81126
[10] Grosse, H.; Wulkenhaar, R., Renormalization of \({\phi^4}\)-theory on noncommutative \({\mathbb{R}^4}\) in the matrix base, Commun. Math. Phys., 256, 2, 305 (2005) · Zbl 1075.82005
[11] Grosse, H.; Wulkenhaar, R., Power-counting theorem for non-local matrix models and renormalization, Commun. Math. Phys., 254, 1, 91 (2005) · Zbl 1079.81049
[12] Rivasseau, V.; Vignes-Tourneret, F.; Wulkenhaar, R., Renormalization of noncommutative \({\phi^4}\)-theory by multi-scale analysis, Commun. Math. Phys., 262, 565 (2006) · Zbl 1109.81056
[13] 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, 2, 515 (2006) · Zbl 1113.81101
[14] Gurau, R.; Magnen, J.; Rivasseau, V.; Tanasa, A., A translation-invariant renormalizable non-commutative scalar model, Commun. Math. Phys., 287, 275 (2009) · Zbl 1170.81041
[15] ’T Hooft, G., A planar diagram theory for strong interactions, Nucl. Phys. B, 72, 461 (1974)
[16] Grosse, H.; Wulkenhaar, R., The beta-function in duality-covariant noncommutative \({\phi^4}\)-theory, Eur. Phys. J., C35, 277 (2004)
[17] Disertori, M.; Rivasseau, V., Two and three loops beta function of non commutative phi(4)**4 theory, Eur. Phys. J. C, 50, 661 (2007) · Zbl 1248.81254
[18] 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) · Zbl 1248.81253
[19] Gurau, R.; Rosten, O. J., Wilsonian renormalization of noncommutative scalar field theory, JHEP, 0907, 064 (2009)
[20] Geloun, J. B.; Gurau, R.; Rivasseau, V., Vanishing beta function for Grosse-Wulkenhaar model in a magnetic field, Phys. Lett. B, 671, 284 (2009)
[21] Boulatov, D., A model of three-dimensional lattice gravity, Mod. Phys. Lett. A, 7, 1629-1646 (1992) · Zbl 1020.83539
[22] Freidel, L., Group field theory: an overview, Int. J. Phys., 44, 1769-1783 (2005) · Zbl 1100.83010
[23] Oriti, D.: Quantum Gravity. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Birkhauser, Basel (2007). arXiv:gr-qc/0512103 · Zbl 1120.83024
[24] De Pietri, R.; Petronio, C., Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4, J. Math. Phys., 41, 6671-6688 (2000) · Zbl 0971.81101
[25] Barrett, J., Nash-Guzman, I.: arXiv:0803.3319 (gr-qc)
[26] Engle, J.; Pereira, R.; Rovelli, C., The loop-quantum-gravity vertex-amplitude, Phys. Rev. Lett., 99, 161301 (2007) · Zbl 1228.83045
[27] Engle, J.; Pereira, R.; Rovelli, C., Flipped spinfoam vertex and loop gravity, Nucl. Phys. B, 798, 251 (2008) · Zbl 1234.83009
[28] Livine, E. R.; Speziale, S., A new spinfoam vertex for quantum gravity, Phys. Rev. D, 76, 084028 (2007)
[29] Freidel, L.; Krasnov, K., A new spin foam model for 4d gravity, Class. Quant. Grav., 25, 125018 (2008) · Zbl 1144.83007
[30] Conrady, F.; Freidel, L., On the semiclassical limit of 4d spin foam models, Phys. Rev. D, 78, 104023 (2008)
[31] Bonzom, V.; Livine, E. R.; Smerlak, M.; Speziale, S., Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model, Nucl. Phys. B, 804, 507 (2008) · Zbl 1190.83038
[32] Freidel, L.; Gurau, R.; Oriti, D., Group field theory renormalization—the 3d case: power counting of divergences, Phys. Rev. D, 80, 044007 (2009)
[33] Magnen, J., Noui, K., Rivasseau, V., Smerlak, M.: arXiv:0906.5477 [hep-th]
[34] Adbesselam, A.: On the volume conjecture for classical spin networks. arXiv: 0904.1734[math.GT]
[35] Geloun, J.B., Magnen, J., Rivasseau, V.: Bosonic Colored Group Field Theory. arXiv:0911.1719 [hep-th] · Zbl 1195.81093
[36] Kirchhoff, G., Uber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme gefürht wird, Ann. Phys. Chem., 72, 497-508 (1847)
[37] Tutte, W. T., Graph Theory (1984), Reading: Addison-Wesley, Reading
[38] ‘T Hooft, G.; Veltman, M., Regularization and renormalization of gauge fields, Nucl. Phys., B44, 1, 189-213 (1972)
[39] Crapo, H. H., The Tutte polynomial, Aequationes Mathematicae, 3, 211-229 (1969) · Zbl 0197.50202
[40] Sokal, A.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, Surveys in combinatorics 2005, pp. 173-226. London Math. Soc. Lecture Note Ser., vol. 327. Cambridge University Press, Cambridge (2005). arXiv:math/0503607 · Zbl 1110.05020
[41] Jackson, B., Procacci, A., Sokal, A.D.: Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights. arXiv:0810.4703v1 [math.CO] · Zbl 1257.05068
[42] Bollobás, B.; Riordan, O., A polynomial invariant of graphs on orientable surfaces, Proc. Lond. Math. Soc., 83, 513-531 (2001) · Zbl 1015.05024
[43] Bollobás, B.; Riordan, O., A polynomial of graphs on surfaces, Math. Ann., 323, 81-96 (2002) · Zbl 1004.05021
[44] Ellis-Monaghan, J., Merino, C.: Graph polynomials and their applications. I. The Tutte polynomial. arXiv:0803.3079 · Zbl 1221.05002
[45] Ellis-Monaghan, J., Merino, C.: Graph polynomials and their applications. II. Interrelations and interpretations. arXiv:0806.4699 · Zbl 1221.05003
[46] Gurau, R.; Rivasseau, V., Parametric representation of noncommutative field theory, Commun. Math. Phys., 272, 811 (2007) · Zbl 1156.81465
[47] Rivasseau, V.; Tanasa, A., Parametric representation of ‘critical’ noncommutative QFT models, Commun. Math. Phys., 279, 355 (2008) · Zbl 1158.81026
[48] Tanasa, A.: Parametric representation of a translation-invariant renormalizable noncommutative model. arXiv:0807.2779 [math-ph] · Zbl 1178.81262
[49] Krajewski, T., Rivasseau, V., Tanasa, A., Wang, Z.: Topological Graph Polynomials and Quantum Field Theory. Part I. Heat Kernel Theories. arXiv:0811.0186 [math-ph] · Zbl 1186.81095
[50] Bollobás, B.; Riordan, O., A Tutte polynomial for coloured graphs, Combin. Probab. Comput., 8, 45-93 (1999) · Zbl 0926.05017
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