Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 069, 17 pages      arXiv:1603.07278
Contribution to the Special Issue on Tensor Models, Formalism and Applications

Random Tensors and Quantum Gravity

Vincent Rivasseau
Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, F-91405 Orsay Cedex, France

Received March 23, 2016, in final form July 06, 2016; Published online July 15, 2016

We provide an informal introduction to tensor field theories and to their associated renormalization group. We focus more on the general motivations coming from quantum gravity than on the technical details. In particular we discuss how asymptotic freedom of such tensor field theories gives a concrete example of a natural ''quantum relativity'' postulate: physics in the deep ultraviolet regime becomes asymptotically more and more independent of any particular choice of Hilbert basis in the space of states of the universe.

Key words: renormalization; tensor models; quantum gravity.

pdf (470 kb)   tex (44 kb)


  1. Ambjørn J., Simplicial Euclidean and Lorentzian quantum gravity, in General Relativity & Gravitation (Durban, 2001), World Sci. Publ., River Edge, NJ, 2002, 3-27, gr-qc/0201028.
  2. Ambjørn J., Durhuus B., Jonsson T., Quantum geometry. A statistical field theory approach, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1997.
  3. Ambjørn J., Görlich A., Jurkiewicz J., Loll R., Causal dynamical triangulations and the search for a theory of quantum gravity, Internat. J. Modern Phys. D 22 (2013), 1330019, 18 pages.
  4. Ambjørn J., Jurkiewicz J., Loll R., The universe from scratch, Contemp. Phys. 47 (2006), 103-117, hep-th/0509010.
  5. Avohou R.C., Rivasseau V., Tanasa A., Renormalization and Hopf algebraic structure of the five-dimensional quartic tensor field theory, J. Phys. A: Math. Theor. 48 (2015), 485204, 20 pages, arXiv:1507.0354.
  6. Bandieri P., Casali M.R., Cristofori P., Grasselli L., Mulazzani M., Computational aspects of crystallization theory: complexity, catalogues and classification of 3-manifolds, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 58 (2011), 11-45.
  7. Baratin A., Oriti D., Group field theory and simplicial gravity path integrals: a model for Holst-Plebanski gravity, Phys. Rev. D 85 (2012), 044003, 15 pages, arXiv:1111.5842.
  8. Ben Geloun J., Ward-Takahashi identities for the colored Boulatov model, J. Phys. A: Math. Theor. 44 (2011), 415402, 30 pages, arXiv:1106.1847.
  9. Ben Geloun J., Two- and four-loop $\beta$-functions of rank-4 renormalizable tensor field theories, Classical Quantum Gravity 29 (2012), 235011, 40 pages, arXiv:1205.5513.
  10. Ben Geloun J., Asymptotic freedom of rank 4 tensor group field theory, in Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., Vol. 11, World Sci. Publ., Hackensack, NJ, 2013, 367-372, arXiv:1210.5490.
  11. Ben Geloun J., On the finite amplitudes for open graphs in Abelian dynamical colored Boulatov-Ooguri models, J. Phys. A: Math. Theor. 46 (2013), 402002, 12 pages, arXiv:1307.8299.
  12. Ben Geloun J., Renormalizable models in rank $d\geq 2$ tensorial group field theory, Comm. Math. Phys. 332 (2014), 117-188, arXiv:1306.1201.
  13. Ben Geloun J., Bonzom V., Radiative corrections in the Boulatov-Ooguri tensor model: the 2-point function, Internat. J. Theoret. Phys. 50 (2011), 2819-2841, arXiv:1101.4294.
  14. Ben Geloun J., Krajewski T., Magnen J., Rivasseau V., Linearized group field theory and power-counting theorems, Classical Quantum Gravity 27 (2010), 155012, 14 pages, arXiv:1002.3592.
  15. Ben Geloun J., Livine E.R., Some classes of renormalizable tensor models, J. Math. Phys. 54 (2013), 082303, 25 pages, arXiv:1207.0416.
  16. Ben Geloun J., Magnen J., Rivasseau V., Bosonic colored group field theory, Eur. Phys. J. C Part. Fields 70 (2010), 1119-1130, arXiv:0911.1719.
  17. Ben Geloun J., Martini R., Oriti D., Functional renormalization group analysis of a tensorial group field theory on $\mathbb{R}^3$, Europhys. Lett. 112 (2015), 31001, 6 pages, arXiv:1508.01855.
  18. Ben Geloun J., Ramgoolam S., Counting tensor model observables and branched covers of the 2-sphere, Ann. Inst. Henri Poincaré D 1 (2014), 77-138, arXiv:1307.6490.
  19. Ben Geloun J., Rivasseau V., A renormalizable 4-dimensional tensor field theory, Comm. Math. Phys. 318 (2013), 69-109, arXiv:1111.4997.
  20. Ben Geloun J., Rivasseau V., Addendum to: A renormalizable 4-dimensional tensor field theory, Comm. Math. Phys. 322 (2013), 957-965, arXiv:1209.4606.
  21. Ben Geloun J., Samary D.O., 3D tensor field theory: renormalization and one-loop $\beta$-functions, Ann. Henri Poincaré 14 (2013), 1599-1642, arXiv:1201.0176.
  22. Benedetti D., Ben Geloun J., Oriti D., Functional renormalisation group approach for tensorial group field theory: a rank-3 model, J. High Energy Phys. 2015 (2015), no. 3, 084, 40 pages, arXiv:1411.3180.
  23. Benedetti D., Lahoche V., Functional renormalization group approach for tensorial group field theory: a rank-6 model with closure constraint, Classical Quantum Gravity 33 (2016), 095003, 35 pages, arXiv:1508.06384.
  24. Bonzom V., New $1/N$ expansions in random tensor models, J. High Energy Phys. 2013 (2013), no. 6, 062, 25 pages, arXiv:1211.1657.
  25. Bonzom V., Delepouve T., Rivasseau V., Enhancing non-melonic triangulations: a tensor model mixing melonic and planar maps, Nuclear Phys. B 895 (2015), 161-191, arXiv:1502.0136.
  26. Bonzom V., Gurau R., Riello A., Rivasseau V., Critical behavior of colored tensor models in the large $N$ limit, Nuclear Phys. B 853 (2011), 174-195, arXiv:1105.3122.
  27. Bonzom V., Gurau R., Rivasseau V., Random tensor models in the large $N$ limit: uncoloring the colored tensor models, Phys. Rev. D 85 (2012), 084037, 12 pages, arXiv:1202.3637.
  28. Bonzom V., Lionni L., Rivasseau V., Colored triangulations of arbitrary dimensions are stuffed Walsh maps, arXiv:1508.03805.
  29. Boulatov D.V., A model of three-dimensional lattice gravity, Modern Phys. Lett. A 7 (1992), 1629-1646, hep-th/9202074.
  30. Carrozza S., Tensorial methods and renormalization in group field theories, Springer Theses, Springer, Cham, 2014, arXiv:1310.3736.
  31. Carrozza S., Discrete renormalization group for ${\rm SU}(2)$ tensorial group field theory, Ann. Inst. Henri Poincaré D 2 (2015), 49-112, arXiv:1407.4615.
  32. Carrozza S., Oriti D., Rivasseau V., Renormalization of a ${\rm SU}(2)$ tensorial group field theory in three dimensions, Comm. Math. Phys. 330 (2014), 581-637, arXiv:1303.6772.
  33. Carrozza S., Oriti D., Rivasseau V., Renormalization of tensorial group field theories: Abelian ${\rm U}(1)$ models in four dimensions, Comm. Math. Phys. 327 (2014), 603-641, arXiv:1207.6734.
  34. Casali M.R., Cristofori P., Coloured graphs representing PL 4-manifolds, Electron. Notes Discrete Math. 40 (2013), 83-87.
  35. Casali M.R., Cristofori P., Cataloguing PL 4-manifolds by gem-complexity, Electron. J. Combin. 22 (2015), Paper 4.25, 25 pages, arXiv:1408.0378.
  36. Casali M.R., Cristofori P., Gagliardi C., Classifying PL 4-manifolds via crystallizations, results and open problems, in A Mathematical Tribute to Professor José Marí a Montesinos Amilibia, Editors M. Castrillń, E. Martín-Peinador, J.M. Rodríguez-Sanjurjo, J.M. Ruiz, Ciudad Universitaria, Madrid, 2016, 199-226, available at
  37. Dartois S., Gurau R., Rivasseau V., Double scaling in tensor models with a quartic interaction, J. High Energy Phys. 2013 (2013), no. 9, 088, 33 pages, arXiv:1307.5281.
  38. Delepouve T., Gurau R., Rivasseau V., Universality and Borel summability of arbitrary quartic tensor models, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), 821-848, arXiv:1403.0170.
  39. Delepouve T., Rivasseau V., Constructive tensor field theory: the $T^4_3$ model, Comm. Math. Phys. 345 (2016), 477-506, arXiv:1412.5091.
  40. Di Francesco P., Ginsparg P., Zinn-Justin J., $2$D gravity and random matrices, Phys. Rep. 254 (1995), 1-133, hep-th/9306153.
  41. 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 (2007), 95-102, hep-th/0612251.
  42. Disertori M., Rivasseau V., Two and three loops beta function of non commutative $\Phi_4^4$ theory, Eur. Phys. J. C Part. Fields 50 (2007), 661-671, hep-th/0610224.
  43. Donaldson S.K., An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), 279-315.
  44. Doplicher S., Fredenhagen K., Roberts J.E., Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994), 39-44.
  45. Dyson F.J., Disturbing the universe, Basic Books, 1979.
  46. Eichhorn A., Koslowski T., Continuum limit in matrix models for quantum gravity from the functional renormalization group, Phys. Rev. D 88 (2013), 084016, 15 pages, arXiv:1309.1690.
  47. Ferri M., Gagliardi C., Grasselli L., A graph-theoretical representation of PL-manifolds - a survey on crystallizations, Aequationes Math. 31 (1986), 121-141.
  48. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, hep-th/0505016.
  49. Freidel L., Gurau R., Oriti D., Group field theory renormalization in the 3D case: power counting of divergences, Phys. Rev. D 80 (2009), 044007, 20 pages, arXiv:0905.3772.
  50. Grosse H., Wulkenhaar R., The $\beta$-function in duality-covariant non-commutative $\phi^4$-theory, Eur. Phys. J. C Part. Fields 35 (2004), 277-282, hep-th/0402093.
  51. Grosse H., Wulkenhaar R., Renormalisation of $\phi^4$-theory on noncommutative ${\mathbb R}^4$ in the matrix base, Comm. Math. Phys. 256 (2005), 305-374, hep-th/0401128.
  52. Grosse H., Wulkenhaar R., Progress in solving a noncommutative quantum field theory in four dimensions, arXiv:0909.1389.
  53. Grosse H., Wulkenhaar R., Self-dual noncommutative $\phi^4$-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory, Comm. Math. Phys. 329 (2014), 1069-1130, arXiv:1205.0465.
  54. Grosse H., Wulkenhaar R., Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity, arXiv:1406.7755.
  55. Grosse H., Wulkenhaar R., On the fixed point equation of a solvable 4D QFT model, Vietnam J. Math. 44 (2016), 153-180, arXiv:1505.0516.
  56. Gurau R., Colored group field theory, Comm. Math. Phys. 304 (2011), 69-93, arXiv:0907.2582.
  57. Gurau R., The $1/N$ expansion of colored tensor models, Ann. Henri Poincaré 12 (2011), 829-847, arXiv:1011.2726.
  58. Gurau R., The complete $1/N$ expansion of colored tensor models in arbitrary dimension, Ann. Henri Poincaré 13 (2012), 399-423, arXiv:1102.5759.
  59. Gurau R., The $1/N$ expansion of tensor models beyond perturbation theory, Comm. Math. Phys. 330 (2014), 973-1019, arXiv:1304.2666.
  60. Gurau R., Universality for random tensors, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), 1474-1525, arXiv:1111.0519.
  61. Gurau R., Rivasseau V., The $1/N$ expansion of colored tensor models in arbitrary dimension, Europhys. Lett. 95 (2011), 50004, 5 pages, arXiv:1101.4182.
  62. Gurau R., Rivasseau V., The multiscale loop vertex expansion, Ann. Henri Poincaré 16 (2015), 1869-1897, arXiv:1312.7226.
  63. Gurau R., Ryan J.P., Colored tensor models - a review, SIGMA 8 (2012), 020, 78 pages, arXiv:1109.4812.
  64. Gurau R., Ryan J.P., Melons are branched polymers, Ann. Henri Poincaré 15 (2014), 2085-2131, arXiv:1302.4386.
  65. Gurau R., Schaeffer G., Regular colored graphs of positive degree, arXiv:1307.5279.
  66. Gurau R., Tanasa A., Youmans D.R., The double scaling limit of the multi-orientable tensor model, Europhys. Lett. 111 (2015), 21002, 6 pages, arXiv:1505.00586.
  67. Haag R., Local quantum physics. Fields, particles, algebras, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
  68. Krajewski T., Group field theories, PoS Proc. Sci. (2011), PoS(QGQGS2011), 005, 58 pages, arXiv:1210.6257.
  69. Krajewski T., Rivasseau V., Tanasa A., Combinatorial Hopf algebraic description of the multi-scale renormalization in quantum field theory, Sém. Lothar. Combin. 70 (2013), Art. B70c, 23 pages, arXiv:1211.4429.
  70. Krajewski T., Toriumi R., Polchinski's exact renormalisation group for tensorial theories: Gaussian universality and power counting, arXiv:1511.09084.
  71. Lahoche V., Constructive tensorial group field theory I: the $U(1)-T^4_3$ model, arXiv:1510.05050.
  72. Lahoche V., Constructive tensorial group field theory II: the $U(1)-T^4_4$ model, arXiv:1510.05051.
  73. Lahoche V., Oriti D., Renormalization of a tensorial field theory on the homogeneous space ${\rm SU}(2)/{\rm U}(1)$, arXiv:1310.3736.
  74. Oriti D., Group field theory as the microscopic description of the quantum spacetime fluid: a new perspective on the continuum in quantum gravity, PoS Proc. Sci. (2007), PoS(QG-Ph), 030, 38 pages, arXiv:0710.3276.
  75. Oriti D., A quantum field theory of simplicial geometry and the emergence of space-time, J. Phys. Conf. Ser. 67 (2007), 012052, 10 pages, hep-th/0612301.
  76. Oriti D., Tlas T., A new class of group field theories for first order discrete quantum gravity, Classical Quantum Gravity 25 (2008), 085011, 44 pages, arXiv:0710.2679.
  77. Raasakka M., Tanasa A., Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory, Sém. Lothar. Combin. 70 (2013), Art. B70d, 29 pages, arXiv:1306.1022.
  78. Rivasseau V., Towards renormalizing group field theory, PoS Proc. Sci. (2010), PoS(CNCFG2010), 004, 21 pages, arXiv:1103.1900.
  79. Rivasseau V., Quantum gravity and renormalization: the tensor track, AIP Conf. Proc. 1444 (2012), 18-29, arXiv:1112.5104.
  80. Rivasseau V., The tensor track: an update, in Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., Vol. 11, World Sci. Publ., Hackensack, NJ, 2013, 63-74, arXiv:1209.5284.
  81. Rivasseau V., The tensor theory space, Fortschr. Phys. 62 (2014), 835-840, arXiv:1407.0284.
  82. Rivasseau V., The tensor track, III, Fortschr. Phys. 62 (2014), 81-107, arXiv:1311.1461.
  83. Rivasseau V., Why are tensor field theories asymptotically free?, Europhys. Lett. 111 (2015), 60011, 6 pages, arXiv:1507.04190.
  84. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  85. Samary D.O., Beta functions of ${\rm U}(1)^d$ gauge invariant just renormalizable tensor models, Phys. Rev. D 88 (2013), 105003, 15 pages, arXiv:1303.7256.
  86. Samary D.O., Pérez-Sánchez C.I., Vignes-Tourneret F., Wulkenhaar R., Correlation functions of a just renormalizable tensorial group field theory: the melonic approximation, Classical Quantum Gravity 32 (2015), 175012, 18 pages, arXiv:1411.7213.
  87. Samary D.O., Vignes-Tourneret F., Just renormalizable TGFT's on ${\rm U}(1)^d$ with gauge invariance, Comm. Math. Phys. 329 (2014), 545-578, arXiv:1211.2618.
  88. Scorpan A., The wild world of 4-manifolds, Amer. Math. Soc., Providence, RI, 2005.
  89. Seiberg N., Emergent spacetime, hep-th/0601234.
  90. Sindoni L., Emergent models for gravity: an overview of microscopic models, SIGMA 8 (2012), 027, 45 pages, arXiv:1110.0686.
  91. Streater R.F., Wightman A.S., PCT, spin and statistics, and all that, Princeton Landmarks in Physics, Princeton University Press, Princeton, NJ, 2000.
  92. Wishart J., Generalized product moment distribution in samples, Biometrika 20A (1928), 32-52.

Previous article  Next article   Contents of Volume 12 (2016)