The \(1/N\) expansion of colored tensor models. (English) Zbl 1218.81088
Summary: 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 \(S^3\) contribute to the leading order in the large \(N\) limit.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81V05 | Strong interaction, including quantum chromodynamics |
| 83C45 | Quantization of the gravitational field |
| 15B52 | Random matrices (algebraic aspects) |
References:
| [1] | ’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974) · doi:10.1016/0550-3213(74)90154-0 |
| [2] | Gross D.J., Miljkovic N.: A nonperturbative solution of D = 1 string theory. Phys. Lett. B 238, 217 (1990) · Zbl 1332.81175 · doi:10.1016/0370-2693(90)91724-P |
| [3] | Gross D.J., Klebanov I.R.: One-dimensional string theory on a circle. Nucl. Phys. B 344, 475-498 (1990) · doi:10.1016/0550-3213(90)90667-3 |
| [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] · doi:10.1016/0370-1573(94)00084-G |
| [5] | David F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985) · doi:10.1016/0550-3213(85)90363-3 |
| [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) · doi:10.1016/0370-2693(85)90669-0 |
| [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) · Zbl 0968.81540 · doi:10.1016/0550-3213(86)90578-X |
| [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] · Zbl 0988.81099 · doi:10.1016/S0550-3213(01)00606-X |
| [9] | Brezin E., Itzykson C., Parisi G., Zuber J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978) · Zbl 0997.81548 · doi:10.1007/BF01614153 |
| [10] | Gross M.: Tensor models and simplicial quantum gravity in > 2-D. Nucl. Phys. Proc. Suppl. 25, 144 (1992) · Zbl 0957.83511 · doi:10.1016/S0920-5632(05)80015-5 |
| [11] | Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133 (1991) · Zbl 1020.83537 · doi:10.1142/S0217732391001184 |
| [12] | Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991) · Zbl 1020.83542 · doi:10.1142/S0217732391003055 |
| [13] | Freidel L.: Group field theory: an overview. Int. J. Theor. Phys. 44, 1769 (2005) [arXiv:hep-th/0505016] · Zbl 1100.83010 · doi:10.1007/s10773-005-8894-1 |
| [14] | Oriti, D.: The Group Field Theory Approach to Quantum Gravity: Some Recent Results. [arXiv:0912.2441 [hep-th]] · Zbl 1223.83025 |
| [15] | Boulatov D.V.: A model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992) [arXiv:hep-th/9202074] · Zbl 1020.83539 · doi:10.1142/S0217732392001324 |
| [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] · Zbl 1060.83013 · doi:10.1088/0264-9381/21/24/002 |
| [17] | Baratin, A., Oriti, D.: Group Field Theory with Non-Commutative Metric Variables. [arXiv:1002.4723 [hep-th]] · Zbl 1448.83034 |
| [18] | Engle J., Pereira R., Rovelli C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008) [arXiv:0708.1236 [gr-qc]] · Zbl 1234.83009 · doi:10.1016/j.nuclphysb.2008.02.002 |
| [19] | Livine E.R., Speziale S.: A new spinfoam vertex for quantum gravity. Phys. Rev. D 76, 084028 (2007) [arXiv:0705.0674 [gr-qc]] · doi:10.1103/PhysRevD.76.084028 |
| [20] | Freidel L., Krasnov K.: A new spin foam model for 4D gravity. Class. Quant. Grav. 25, 125018 (2008) [arXiv:0708.1595 [gr-qc]] · Zbl 1144.83007 · doi:10.1088/0264-9381/25/12/125018 |
| [21] | Geloun J.B., Gurau R., Rivasseau V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010) [arXiv:1008.0354 [hep-th]] · doi:10.1209/0295-5075/92/60008 |
| [22] | Alexandrov, S., Roche, P.: Critical Overview of Loops and Foams. arXiv:1009.4475 [gr-qc] |
| [23] | Gurau, R.: Colored Group Field Theory. [arXiv:0907.2582 [hep-th]] · Zbl 1214.81170 |
| [24] | Gurau R.: Topological graph polynomials in colored group field theory. Ann. Henri Poincaré 11, 565 (2010) [arXiv:0911.1945 [hep-th]] · Zbl 1208.81153 · doi:10.1007/s00023-010-0035-6 |
| [25] | Gurau R.: Lost in translation: topological singularities in group field theory. Class. Quant. Grav. 27, 235023 (2010) arXiv:1006.0714 [hep-th] · Zbl 1205.83022 · doi:10.1088/0264-9381/27/23/235023 |
| [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]] · doi:10.1103/PhysRevD.80.044007 |
| [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]] · Zbl 1176.83066 · doi:10.1088/0264-9381/26/18/185012 |
| [28] | Geloun, J.B., Magnen, J., Rivasseau, V.: Bosonic Colored Group Field Theory. [arXiv:0911.1719 [hep-th]] · Zbl 1195.81093 |
| [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]] · Zbl 1195.81093 · doi:10.1088/0264-9381/27/15/155012 |
| [30] | Gurau R., Rivasseau V.: Parametric representation of noncommutative field theory. Commun. Math. Phys. 272, 811 (2007) [arXiv:math-ph/0606030] · Zbl 1156.81465 · doi:10.1007/s00220-007-0215-5 |
| [31] | Lins, S.: Gems, Computers and Attractors for 3-Manifolds. Series on Knots and Everything, vol. 5. ISBN: 9810219075/ISBN-13: 9789810219079 · Zbl 0868.57002 |
| [32] | Ferri, M., Gagliardi, C.: Crystallisation moves. Pac. J. Math. 100(1) (1982) · Zbl 0517.57003 |
| [33] | Bonzom V., Smerlak M.: Bubble divergences from cellular cohomology. Lett. Math. Phys. 93, 295 (2010) [arXiv:1004.5196 [gr-qc]] · Zbl 1197.81180 · doi:10.1007/s11005-010-0414-4 |
| [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] · Zbl 1075.82005 · doi:10.1007/s00220-004-1285-2 |
| [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] · Zbl 1113.81101 · doi:10.1007/s00220-006-0055-8 |
| [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] · Zbl 1109.81056 · doi:10.1007/s00220-005-1440-4 |
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