Abstract
Classes of renormalizable models in the Tensorial Group Field Theory framework are investigated. The rank d tensor fields are defined over d copies of a group manifold \({G_D=U(1)^D}\) or \({G_D= SU(2)^D}\) with no symmetry and no gauge invariance assumed on the fields. In particular, we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form \({p^{2a}, a\in (0,1]}\). This study is tailored for models equipped with Laplacian dynamics on G D (case a = 1) but also for more exotic nonlocal models in quantum topology (case 0 < a < 1). A generic model can be written \({(_{\dim G_D}\Phi^{k}_{d}, a)}\), where k is the maximal valence of its interactions. Using a multi-scale analysis for the generic situation, we identify several classes of renormalizable actions, including matrix model actions. In this specific instance, we find a tower of renormalizable matrix models parametrized by \({k \geq 4}\) . In a second part of this work, we study the UV behavior of the models up to maximal valence of interaction k = 6. All rank \({d \geq 3}\) tensor models proved renormalizable are asymptotically free in the UV. All matrix models with k = 4 have a vanishing β-function at one-loop and, very likely, reproduce the same feature of the Grosse–Wulkenhaar model (Commun Math Phys 256:305, 2005).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Di Francesco, P., Ginsparg, P.H., Zinn-Justin, J.: 2-D Gravity and random matrices. Phys. Rep. 254, 1 (1995). arxiv:hep-th/9306153
Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133 (1991)
Gross M.: Tensor models and simplicial quantum gravity in > 2-D. Nucl. Phys. Proc. Suppl. 25, 144 (1992)
Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)
Konopka, T., Markopoulou, F., Smolin, L.: Quantum Graphity. hep-th/0611197
’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)
Kazakov V.A.: Bilocal regularization of models of random surfaces. Phys. Lett. B 150, 282 (1985)
David F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985)
Knizhnik V.G., Polyakov A.M., Zamolodchikov A.B.: Fractal structure of 2D quantum gravity. Mod. Phys. Lett. A 3, 819 (1988)
David F.: Conformal field theories coupled to 2D gravity in the conformal gauge. Mod. Phys. Lett. A 3, 1651 (1988)
Distler J., Kawai H.: Conformal field theory and 2D quantum gravity or who’s afraid of Joseph Liouville?. Nucl. Phys. B 321, 509 (1989)
Duplantier, B.: Conformal Random Geometry. In: Bovier, A., Dunlop, F., den Hollander, F., van Enter, A., Dalibard, J. (eds.) Les Houches, Session LXXXIII, 2005. Mathematical Statistical Physics, pp. 101–217. Elsevier B. V., Amsterdam (2006). arXiv:math-ph/0608053 ]
Duplantier, B.: The Hausdorff dimension of two-dimensional quantum gravity. arXiv:1108.3327[math-ph]
Duplantier, B., Sheffield, S.: Schramm Loewner evolution and Liouville quantum gravity. Phys. Rev. Lett. 107, 131305 (2011). arXiv:1012.4800 [math-ph
Miermont, G.: The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210, 319–401 (2013). arXiv:1104.1606 [math.PR
Curien, N., Le Gall, J.-F., Miermont, G.: The Brownian Cactus I. Scaling limits of discrete cactuses. Ann. Inst. H. Poincaré Probab. Stat. 49, 307–609 (2013). arXiv:1102.4177 [math.PR
Le Gall, J.-F., Miermont, G.: Scaling limits of random trees and planar maps. In: Probability and Statistical Physics in Two or More Dimensions (Búzios, 2010), Clay Mathematics Proceedings, vol. 15, pp. 155–211. American Mathematical Society, Providence (2012) arXiv:1101.4856 [math.PR]
Le Gall, J.-F., Miermont, G.: Scaling limits of random planar maps with large faces. Ann. Probab. 39, 1–69 (2011). arXiv:0907.3262 [math.PR]
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 ]
Oriti, D.: The group field theory approach to quantum gravity. In: Oriti, D. (ed.) Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, pp 310–331. Cambridge University Press, Cambridge (2009). gr-qc/0607032 ]
Oriti, D.: A quantum field theory of simplicial geometry and the emergence of spacetime. J. Phys. Conf. Ser. 67, 012052 (2007). hep-th/0612301 ]
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. Europhys. Lett. 95, 50004 (2011). arXiv:1101.4182 [gr-qc
Gurau, R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. Annales Henri Poincare 13, 399 (2012). arXiv:1102.5759 [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 Poincare 11, 565 (2010). arXiv:0911.1945 [hep-th
Gurau, R.: Lost in translation: topological singularities in group field theory. Class. Quantum Gravity 27, 235023 (2010). arXiv:1006.0714 [hep-th
Gurau, R., Ryan, J.P.: Colored tensor models—a review. SIGMA 8, 020 (2012). arXiv:1109.4812 [hep-th
Bonzom, V., Gurau, R., Riello, A., Rivasseau, V.: Critical behavior of colored tensor models in the large N limit. Nucl. Phys. B 853, 174 (2011). arXiv:1105.3122 [hep-th
Gurau, R., Ryan, J.P.: Melons are branched polymers. Annates Henri Poincaré (to appear, 2014).arXiv:1302.4386 [math-ph]
Gurau, R.: The 1/N expansion of tensor models beyond perturbation theory. Commun. Math. Phys., online first. doi:10.1007/s00220-014-1907-2, arXiv:1304.2666 [math-ph]
Bonzom, V., Gurau, R., Rivasseau, V.: The ising model on random lattices in arbitrary dimensions. Phys. Lett. B 711, 88 (2012). arXiv:1108.6269 [hep-th
Benedetti, D., Gurau, R.: Phase transition in dually weighted colored tensor models. Nucl. Phys. B 855, 420 (2012). arXiv:1108.5389 [hep-th
Gurau, R.: The double scaling limit in arbitrary dimensions: a toy model. Phys. Rev.D 84, 124051 (2011). arXiv:1110.2460 [hep-th
Gurau, R.: A generalization of the Virasoro algebra to arbitrary dimensions. Nucl. Phys. B 852, 592 (2011). arXiv:1105.6072 [hep-th
Gurau, R.: The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders. Nucl. Phys. B 865, 133 (2012). arXiv:1203.4965 [hep-th
Gurau, R.: Universality for random tensors. accepted for pubilcation by Annales de l’Institut Henri Poincaré (B) Probability and Statistics, 1111.0519 [math.PR]
Gurau, R.: A review of the large N limit of tensor models. arXiv:1209.4295 [math-ph]
Gurau, R.: A review of the 1/N expansion in random tensor models. In: Proceedings ot the 17th International Congress on Mathematical Physics (ICMP12). arXiv:1209.3252 [math-ph]
Bonzom, V., Gurau, R., Rivasseau, V.: Random tensor models in the large N limit: uncoloring the colored tensor models. Phys. Rev. D 85, 084037 (2012). arXiv:1202.3637 [hep-th
Gordan, P.: Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist. J. Reine Angew. Math. 69, 323–354 (1868). (Available at the Gottinger DigitalisierungsZentrum (GDZ), at http://gdz.sub.uni-goettingen.de/en/gdz..)
Abdesselam A.: On the volume conjecture for classical spin networks. J. Knot Theory Ramif. 21(3), 1250022 (2012)
Ben Geloun, J., Rivasseau, V. A renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. 318, 69 (2013). arXiv:1111.4997 [hep-th
Ben Geloun, J., Rivasseau, V.: Addendum to ‘A Renormalizable 4-Dimensional Tensor Field Theory’. Commun. Math. Phys. 322, 957 (2013). arXiv:1209.4606 [hep-th
Rivasseau, V.: Towards renormalizing group field theory. PoS C NCFG2010, 004 (2010). arXiv:1103.1900 [gr-qc
Rivasseau, V.: Quantum gravity and renormalization: the tensor track. AIP Conf. Proc. 1444, 18 (2011). arXiv:1112.5104 [hep-th
Rivasseau, V.: The tensor track: an update. arXiv:1209.5284 [hep-th]
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 ]
Rivasseau, V.: Non-commutative renormalization. In: Duplantier, B., Rivasseav, V. (eds.) Quantum Spaces (séininaire Poincaré X), pp. 19–107. Bikhäuser, Basel (2007) arXiv:0705.0705 [hep-th]
Rivasseau, V.: From perturbative to constructive renormalization. Princeton series in Physics. Princeton University Press, Princeton (1991)
Ben Geloun, J., Magnen, J., Rivasseau, V.: Bosonic colored group field theory. Eur. Phys. J. C 70, 1119 (2010). arXiv:0911.1719 [hep-th
Ben Geloun, J., Krajewski, T., Magnen, J., Rivasseau, V.: Linearized group field theory and power counting theorems. Class. Quantum Gravity 27, 155012 (2010). arXiv:1002.3592 [hep-th
Ben Geloun, J., Gurau, R., Rivasseau, V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010). arXiv:1008.0354 [hep-th
Ben Geloun, J., Bonzom, V.: Radiative corrections in the Boulatov–Ooguri tensor model: the 2-point function. Int. J. Theor. Phys. 50, 2819 (2011). arXiv:1101.4294 [hep-th
Ben Geloun, J., Samary, D.O.: 3D tensor field theory: renormalization and one-loop β-functions. Annales Henri Poincare 14, 1599 (2013). arXiv:1201.0176 [hep-th
Ben Geloun, J., Livine, E.R.: Some classes of renormalizable tensor models. J. Math. Phys. 54, 082303 (2013). arXiv:1207.0416 [hep-th
Ben Geloun, J.: Two and four-loop β-functions of rank 4 renormalizable tensor field theories. Class. Quantum Gravity 29, 235011 (2012). arXiv:1205.5513 [hep-th
Carrozza, S., Oriti, D., Rivasseau, V.: Renormalization of tensorial group field theories: Abelian U(1) models in four dimensions. Commun. Math. Phys. 327, 603 (2014). arXiv:1207.6734 [hep-th
Samary, D.O., Vignes-Tourneret, F.: Just renormalizable TGFT’s on U(1)d with gauge invariance. Commun. Math. Phys. 329(2), 545–578 (2014). arXiv:1211.2618 [hep-th
Carrozza, S., Oriti, D., Rivasseau, V.: Renormalization of an SU(2) tensorial group field theory in three dimensions. Commun. Math. Phys. (to appear, 2014). arXiv:1303.6772 [hep-th
Samary, D.O.: Beta functions of U(1)d gauge invariant just renormalizable tensor models. Phys. Rev. D 88, 105003 (2013). arXiv:1303.7256 [hep-th
Grosse, H., Wulkenhaar, R.: Self-dual noncommutative \({\phi^4}\)-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory. Commun. Math. Phys. 329(3), 1069–1130 (2014) arXiv:1205.0465 [math-ph]
Avohou, R.C., Ben Geloun, J., Hounkonnou, M.N.: A polynomial invariant for rank 3 weakly-colored stranded graphs. arXiv:1301.1987 [math.CO]
Ryan, J.P.: Tensor models and embedded Riemann surfaces. Phys. Rev. D 85, 024010 (2012). arXiv:1104.5471 [gr-qc
Bollobas B., Riordan O.: A polynomial of graphs on surfaces. Math. Ann. 323, 81–96 (2002)
Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)
Rivasseau, V.: The Tensor Track, III. Fortschr. Phys. 62(2), 81–107 (2014) arXiv:1311.1461 [hep-th]
Gallavotti G., Nicolo F.: Renormalization theory in four-dimensional scalar fields. I. Commun. Math. Phys. 100, 545 (1985)
Ben Geloun, J., Ramgoolam, S.: Counting tensor model observables and branched covers of the 2-sphere. Ann. Inst. Henri Poincaré, D 1(1), 77–138 (2014) arXiv:1307.6490 [hep-th]
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). hep-th/0612251 ]
Ben Geloun, J., 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]
Feldman J., Trubowitz E.: The flow of an electron–phonon system to the superconducting state. Helvetica Physica Acta 64, 214 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Connes
Rights and permissions
About this article
Cite this article
Geloun, J.B. Renormalizable Models in Rank \({d \geq 2}\) Tensorial Group Field Theory. Commun. Math. Phys. 332, 117–188 (2014). https://doi.org/10.1007/s00220-014-2142-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2142-6