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

SIGMA 12 (2016), 070, 30 pages      arXiv:1603.01902
Contribution to the Special Issue on Tensor Models, Formalism and Applications

Flowing in Group Field Theory Space: a Review

Sylvain Carrozza
Université Bordeaux, LaBRI, UMR 5800, 33400 Talence, France

Received March 08, 2016, in final form July 13, 2016; Published online July 16, 2016

We provide a non-technical overview of recent extensions of renormalization methods and techniques to Group Field Theories (GFTs), a class of combinatorially non-local quantum field theories which generalize matrix models to dimension $d \geq 3$. More precisely, we focus on GFTs with so-called closure constraint, which are closely related to lattice gauge theories and quantum gravity spin foam models. With the help of recent tensor model tools, a rich landscape of renormalizable theories has been unravelled. We review our current understanding of their renormalization group flows, at both perturbative and non-perturbative levels.

Key words: group field theory; quantum gravity; quantum field theory; renormalization.

pdf (697 kb)   tex (547 kb)


  1. Alexandrov S., Geiller M., Noui K., Spin foams and canonical quantization, SIGMA 8 (2012), 055, 79 pages, arXiv:1112.1961.
  2. Ashtekar A., Lewandowski J., Differential geometry on the space of connections via graphs and projective limits, J. Geom. Phys. 17 (1995), 191-230, hep-th/9412073.
  3. Bagnuls C., Bervillier C., Exact renormalization group equations: an introductory review, Phys. Rep. 348 (2001), 91-157, hep-th/0002034.
  4. Bahr B., On background-independent renormalization of spin foam models, arXiv:1407.7746.
  5. Baratin A., Carrozza S., Oriti D., Ryan J., Smerlak M., Melonic phase transition in group field theory, Lett. Math. Phys. 104 (2014), 1003-1017, arXiv:1307.5026.
  6. Baratin A., Girelli F., Oriti D., Diffeomorphisms in group field theories, Phys. Rev. D 83 (2011), 104051, 22 pages, arXiv:1101.0590.
  7. Baratin A., Oriti D., Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model, New J. Phys. 13 (2011), 125011, 28 pages, arXiv:1108.1178.
  8. 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.
  9. Baratin A., Oriti D., Ten questions on group field theory (and their tentative answers), J. Phys. Conf. Ser. 360 (2012), 012002, 10 pages, arXiv:1112.3270.
  10. 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.
  11. Ben Geloun J., Renormalizable models in rank $d\geq 2$ tensorial group field theory, Comm. Math. Phys. 332 (2014), 117-188, arXiv:1306.1201.
  12. Ben Geloun J., A power counting theorem for a $p^{2a}\phi^4$ tensorial group field theory, arXiv:1507.00590.
  13. Ben Geloun J., Gurau R., Rivasseau V., EPRL/FK group field theory, Europhys. Lett. 92 (2010), 60008, 6 pages, arXiv:1008.0354.
  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., 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.
  17. Ben Geloun J., Martini R., Oriti D., Functional renormalisation group analysis of tensorial group field theories on $\mathbb{R}^d$, Phys. Rev. D 94 (2016), 024017, 45 pages, arXiv:1601.08211.
  18. Ben Geloun J., Rivasseau V., A renormalizable 4-dimensional tensor field theory, Comm. Math. Phys. 318 (2013), 69-109, arXiv:1111.4997.
  19. Ben Geloun J., Rivasseau V., Addendum to: A renormalizable 4-dimensional tensor field theory, Comm. Math. Phys. 322 (2013), 957-965, arXiv:1209.4606.
  20. 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.
  21. Ben Geloun J., Toriumi R., Parametric representation of rank $d$ tensorial group field theory: Abelian models with kinetic term $\sum_s\vert p_s\vert +\mu$, J. Math. Phys. 56 (2015), 093503, 53 pages, arXiv:1409.0398.
  22. 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.
  23. 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:1502.01365.
  24. 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.
  25. 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.
  26. Bonzom V., Gurau R., Ryan J.P., Tanasa A., The double scaling limit of random tensor models, J. High Energy Phys. 2014 (2014), no. 9, 051, 49 pages, arXiv:1404.7517.
  27. Bonzom V., Smerlak M., Bubble divergences from cellular cohomology, Lett. Math. Phys. 93 (2010), 295-305, arXiv:1004.5196.
  28. Bonzom V., Smerlak M., Bubble divergences: sorting out topology from cell structure, Ann. Henri Poincaré 13 (2012), 185-208, arXiv:1103.3961.
  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., Group field theory in dimension $4-\varepsilon$, Phys. Rev. D 91 (2015), 065023, 10 pages, arXiv:1411.5385.
  33. Carrozza S., Oriti D., Bounding bubbles: the vertex representation of $3d$ group field theory and the suppression of pseudomanifolds, Phys. Rev. D 85 (2012), 044004, 22 pages, arXiv:1104.5158.
  34. Carrozza S., Oriti D., Bubbles and jackets: new scaling bounds in topological group field theories, J. High Energy Phys. 2012 (2012), no. 6, 092, 42 pages, arXiv:1203.5082.
  35. 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.
  36. 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.
  37. De Pietri R., Freidel L., Krasnov K., Rovelli C., Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, Nuclear Phys. B 574 (2000), 785-806, hep-th/9907154.
  38. Dittrich B., The continuum limit of loop quantum gravity - a framework for solving the theory, arXiv:1409.1450.
  39. Dittrich B., Geiller M., A new vacuum for loop quantum gravity, Classical Quantum Gravity 32 (2015), 112001, 13 pages, arXiv:1401.6441.
  40. Dupuis M., Livine E.R., Holomorphic simplicity constraints for 4D spinfoam models, Classical Quantum Gravity 28 (2011), 215022, 32 pages, arXiv:1104.3683.
  41. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  42. Ferri M., Gagliardi C., Grasselli L., A graph-theoretical representation of PL-manifolds - a survey on crystallizations, Aequationes Math. 31 (1986), 121-141.
  43. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, hep-th/0505016.
  44. 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.
  45. Freidel L., Krasnov K., A new spin foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.
  46. Freidel L., Louapre D., Ponzano-Regge model revisited. I. Gauge fixing, observables and interacting spinning particles, Classical Quantum Gravity 21 (2004), 5685-5726, hep-th/0401076.
  47. Gielen S., Oriti D., Sindoni L., Cosmology from group field theory formalism for quantum gravity, Phys. Rev. Lett. 111 (2013), 031301, 4 pages, arXiv:1303.3576.
  48. Gielen S., Sindoni L., Quantum cosmology from group field theory condensates: a review, arXiv:1602.08104.
  49. Gurau R., Colored group field theory, Comm. Math. Phys. 304 (2011), 69-93, arXiv:0907.2582.
  50. Gurau R., A generalization of the Virasoro algebra to arbitrary dimensions, Nuclear Phys. B 852 (2011), 592-614, arXiv:1105.6072.
  51. Gurau R., The $1/N$ expansion of colored tensor models, Ann. Henri Poincaré 12 (2011), 829-847, arXiv:1011.2726.
  52. Gurau R., The complete $1/N$ expansion of colored tensor models in arbitrary dimension, Ann. Henri Poincaré 13 (2012), 399-423, arXiv:1102.5759.
  53. Gurau R., Universality for random tensors, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), 1474-1525, arXiv:1111.0519.
  54. 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.
  55. Gurau R., Ryan J.P., Colored tensor models - a review, SIGMA 8 (2012), 020, 78 pages, arXiv:1109.4812.
  56. Gurau R., Ryan J.P., Melons are branched polymers, Ann. Henri Poincaré 15 (2014), 2085-2131, arXiv:1302.4386.
  57. Krajewski T., Group field theories, PoS Proc. Sci. (2011), PoS(QGQGS2011), 005, 58 pages, arXiv:1210.6257.
  58. Krajewski T., Magnen J., Rivasseau V., Tanasa A., Vitale P., Quantum corrections in the group field theory formulation of the Engle-Pereira-Rovelli-Livine and Freidel-Krasnov models, Phys. Rev. D 82 (2010), 124069, 20 pages, arXiv:1007.3150.
  59. Krajewski T., Toriumi R., Polchinski's equation for group field theory, Fortschr. Phys. 62 (2014), 855-862.
  60. Krajewski T., Toriumi R., Polchinski's exact renormalisation group for tensorial theories: Gaussian universality and power counting, arXiv:1511.09084.
  61. Krajewski T., Toriumi R., Exact renormalisation group equations and loop equations for tensor models, SIGMA 12 (2016), 068, 36 pages, arXiv:1603.00172.
  62. Lahoche V., Constructive tensorial group field theory I: the $U(1)-T^4_3$ model, arXiv:1510.05050.
  63. Lahoche V., Constructive tensorial group field theory II: the $U(1)-T^4_4$ model, arXiv:1510.05051.
  64. Lahoche V., Oriti D., Rivasseau V., Renormalization of an Abelian tensor group field theory: solution at leading order, J. High Energy Phys. 2015 (2015), no. 4, 095, 41 pages, arXiv:1501.02086.
  65. Litim D.F., Optimization of the exact renormalization group, Phys. Lett. B 486 (2000), 92-99, hep-th/0005245.
  66. Magnen J., Noui K., Rivasseau V., Smerlak M., Scaling behavior of three-dimensional group field theory, Classical Quantum Gravity 26 (2009), 185012, 25 pages, arXiv:0906.5477.
  67. Morris T.R., The exact renormalization group and approximate solutions, Internat. J. Modern Phys. A 9 (1994), 2411-2450, hep-ph/9308265.
  68. Ooguri H., Topological lattice models in four dimensions, Modern Phys. Lett. A 7 (1992), 2799-2810, hep-th/9205090.
  69. 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.
  70. Oriti D., The group field theory approach to quantum gravity, in Approaches to Quantum Gravity - toward a New Understanding of Space, Time, and Matter, Cambridge University Press, Cambridge, 2009, 310-331, gr-qc/0607032.
  71. Oriti D., The microscopic dynamics of quantum space as a group field theory, in Foundations of Space and Time, Cambridge University Press, Cambridge, 2012, 257-320, arXiv:1110.5606.
  72. Oriti D., Pranzetti D., Sindoni L., Horizon entropy from quantum gravity condensates, Phys. Rev. Lett. 116 (2016), 211301, 6 pages, arXiv:1510.06991.
  73. Oriti D., Sindoni L., Wilson-Ewing E., Bouncing cosmologies from quantum gravity condensates, arXiv:1602.08271.
  74. Perez A., The spin foam approach to quantum gravity, Living Rev. Relativ. 16 (2013), 3, 128 pages, arXiv:1205.2019.
  75. Reisenberger M.P., Rovelli C., Spacetime as a Feynman diagram: the connection formulation, Classical Quantum Gravity 18 (2001), 121-140, gr-qc/0002095.
  76. Riello A., Self-energy of the Lorentzian Engle-Pereira-Rovelli-Livine and Freidel-Krasnov model of quantum gravity, Phys. Rev. D 88 (2013), 024011, 30 pages, arXiv:1302.1781.
  77. Rivasseau V., From perturbative to constructive renormalization, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1991.
  78. Rivasseau V., Quantum gravity and renormalization: the tensor track, AIP Conf. Proc. 1444 (2012), 18-29, arXiv:1112.5104.
  79. Rivasseau V., Why are tensor field theories asymptotically free?, Europhys. Lett. 111 (2015), 60011, 6 pages, arXiv:1507.04190.
  80. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  81. 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.
  82. Samary D.O., Closed equations of the two-point functions for tensorial group field theory, Classical Quantum Gravity 31 (2014), 185005, 29 pages, arXiv:1401.2096.
  83. 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.
  84. 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.
  85. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007, gr-qc/0110034.
  86. Wetterich C., Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993), 90-94.
  87. Wilson K.G., Kogut J., The renormalization group and the $\epsilon$ expansion, Phys. Rep. 12 (1974), 75-199.

Previous article  Next article   Contents of Volume 12 (2016)