Communications in Mathematical Physics

, Volume 330, Issue 2, pp 581–637 | Cite as

Renormalization of a SU(2) Tensorial Group Field Theory in Three Dimensions

  • Sylvain Carrozza
  • Daniele Oriti
  • Vincent Rivasseau


We address in this paper the issue of renormalizability for SU(2) Tensorial Group Field Theories (TGFT) with geometric Boulatov-type conditions in three dimensions. We prove that interactions up to ϕ 6-tensorial type are just renormalizable without any anomaly. Our new models define the renormalizable TGFT version of the Boulatov model and provide therefore a new approach to quantum gravity in three dimensions. Among the many new technical results established in this paper are a general classification of just renormalizable models with gauge invariance condition, and in particular concerning properties of melonic graphs, the second order expansion of melonic two point subgraphs needed for wave-function renormalization.


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  1. 1.
    Oriti, D.: The Group field theory approach to quantum gravity. In: Oriti, D. (ed.) Approaches to Quantum Gravity, pp. 310–331. University Press, Cambridge (2009). arXiv:gr-qc/0607032
  2. 2.
    Oriti, D.: Quantum gravity as a quantum field theory of simplicial geometry. In: Fauser, B. (ed.) Quantum Gravity, pp. 101–126. Birkhäuser, Basel (2007). [gr-qc/0512103]Google Scholar
  3. 3.
    Oriti, D.: The microscopic dynamics of quantum space as a group field theory. In: Ellis, G., Murugan, J., Weltman, A. (eds.) Foundations of Space and Time. Cambridge University Press, Cambridge (2012). arXiv:1110.5606 [hep-th]
  4. 4.
    Rivasseau, V.: Quantum gravity and renormalization: the tensor track. In: AIP Conference Proceedings, vol. 1444, p. 18 (2011). arXiv:1112.5104; The tensor track: an update. arXiv:1209.5284 [hep-th]
  5. 5.
    Thiemann T.: Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  6. 6.
    Ashtekar A., Lewandowski J.: Background independent quantum gravity: a status report. Class Quant. Grav. R53–R152 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Rovelli C.: Quantum Gravity. Cambridge University Press, Cambridge (2006)Google Scholar
  8. 8.
    David, F.: Planar diagrams, two-dimensional lattice gravity and surface models. Nucl. Phys. 45, B257 (1985)Google Scholar
  9. 9.
    Ginsparg, P.H.: Matrix models of 2-d gravity (1991). arXiv: hep-th/9112013
  10. 10.
    Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2-D Gravity and random matrices. Phys. Rept. 254, 1–133 (1995). hep-th/9306153Google Scholar
  11. 11.
    Ginsparg, P., Moore, G.: Lectures on 2D Gravity and 2D string theory (1993). hep-th/9304011Google Scholar
  12. 12.
    Perez, A.: The spin foam approach to quantum gravity. Liv. Rev. Relat. 16 (2013). arXiv:1205.2019
  13. 13.
    Rovelli, C.: Zakopane lectures on loop gravity. PoS QGQGS 2011, 003 (2011). arXiv:1102.3660 [gr-qc]
  14. 14.
    Freidel, L., Krasnov, K.: A new spin foam model for 4d gravity. Class. Quant. Grav. 25, 125018 (2008). arXiv:0708.1595
  15. 15.
    Engle, J., Pereira, R., Rovelli, C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008). arXiv:0708.1236
  16. 16.
    Engle, J., Livine, E., Pereira, R., Rovelli, C.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B 799, 136 (2008). arXiv:0711.0146
  17. 17.
    Ben Geloun, J., Gurau, R., Rivasseau, V.: EPRL/FK group field theory. Europhys. Lett. 92 60008 (2010). arXiv:1008.0354 [hep-th]
  18. 18.
    Baratin, A., Oriti, D.:Group field theory and simplicial gravity path integrals: a model for Holst–Plebanski gravity. Phys. Rev. D 85, 044003 (2012). arXiv:1111.5842 [hep-th]
  19. 19.
    Gross M.: Tensor models and simplicial quantum gravity in > 2-D. Nucl. Phys. Proc. Suppl. 25, 144–149 (1992)CrossRefGoogle Scholar
  20. 20.
    Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133–1146 (1991)MathSciNetGoogle Scholar
  21. 21.
    Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613–2624 (1991)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Gurau, R., Ryan, J. P.: Colored tensor models—a review. SIGMA 8, 020 (2012). arXiv:1109.4812 [hep-th]
  23. 23.
    Gurau, R.: Universality for random tensors. arXiv:1111.0519 [math.PR]
  24. 24.
    Gurau, R.: The 1/N expansion of colored tensor models. Annales Henri Poincare 12, 829 (2011). arXiv:1011.2726 [gr-qc]
  25. 25.
    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]Google Scholar
  26. 26.
    Gurau, R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. Ann. Henri Poincaré 13(3), 399–423 (2012). arXiv:1102.5759 [gr-qc]Google Scholar
  27. 27.
    Bonzom, V., Erbin, H.: Coupling of hard dimers to dynamical lattices via random tensors. J. Stat. Mech. 1209, P09009 (2012). arXiv:1204.3798 [cond-mat.stat-mech]
  28. 28.
    Bonzom, V., Gurau, R.,Smerlak, M.: Universality in p-spin glasses with correlated disorder. J. State. Mech. L02003 (2013). arXiv:1206.5539 [hep-th]
  29. 29.
    Oriti, D.: Group fieldtheory as the microscopic description of the quantum spacetime fluid: a new perspective on the continuum in quantum gravity. PoS QG -PH, 030 (2007). arXiv:0710.3276 [gr-qc]
  30. 30.
    Sindoni, L.: Gravity as an emergent phenomenon: a GFT perspective. arXiv:1105.5687 [gr-qc]
  31. 31.
    Gielen, S., Oriti, D., Sindoni, L.: Cosmology from group field theory formalism for quantum grauity. Phys. Rev. Lett. 111 031301 (2013). arXiv:1303.3576 [gr-qc]
  32. 32.
    Bonzom, V., Gurau, R., Riello, A., Rivasseau, V.: Critical behavior of colored tensor models in the large N limit. Nucl. Phys. B853, 174–195 (2011). arXiv:1105.3122 [hep-th]
  33. 33.
    Bonzom, V., Gurau, R., Rivasseau, V.: The Ising model on random lattices in arbitrary dimensions. Phys. Lett. B 711, 88–96 (2012). arXiv:1108.6269 [hep-th]Google Scholar
  34. 34.
    Benedetti, D., Gurau, R.: Phase transition in dually weighted colored tensor models. Nucl. Phys. B 855, 420–437 (2012). arXiv:1108.5389 [hep-th]
  35. 35.
    Ben Geloun, J.: Classical group field theory. J. Math. Phys. 53, 022901 (2012). arXiv:1107.3122 [hep-th]Google Scholar
  36. 36.
    Baratin, A., Girelli, F., Oriti, D.: Diffeomorphisms in group field theories. Phys. Rev. D 83, 104051 (2011). arXiv:1101.0590 [hep-th]
  37. 37.
    Gurau, R.: A generalization of the Virasoro algebra to arbitrary dimensions. Nucl. Phys. B 852, 592 (2011). arXiv:1105.6072 [hep-th]
  38. 38.
    Oriti, D., Sindoni, L.: Towards classical geometrodynamics from group field theory hydrodynamics. New J. Phys. 13, 025006 (2011). arXiv:1010.5149 [gr-qc]
  39. 39.
    Girelli, F., Livine, E.R., Oriti, D.: 4d deformed special relativity from group field theories. Phys. Rev. D 81, 024015 (2010). arXiv:0903.3475 [gr-qc]
  40. 40.
    Livine, E.R., Oriti, D., Ryan, J.P.: Effective hamiltonian constraint from group field theory. Class. Quant. Grav. 28, 245010 (2011). arXiv:1104.5509 [gr-qc]
  41. 41.
    Calcagni, G., Gielen, S., Oriti, D.: Group field cosmology: a cosmological field theory of quantum geometry. Class. Quant. Grav. 29, 105005 (2012). arXiv:1201.4151 [gr-qc]
  42. 42.
    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]Google Scholar
  43. 43.
    Riello, A.: Self-energy of the Lorentzian EPRL-FK spin foam model of quantum gravity. Phys. Rev. D 88, 024011 (2013). arXiv:1302.1781 [gr-qc]
  44. 44.
    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]
  45. 45.
    Ben Geloun, J., 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]Google Scholar
  46. 46.
    Bonzom, V., Smerlak, M.: Bubble divergences from cellular cohomology. Lett. Math. Phys. 93, 295–305 (2010). arXiv:1004.5196 [gr-qc]Google Scholar
  47. 47.
    Bonzom, V., Smerlak, M.: Bubble divergences from twisted cohomology. Commun. Math. Phys. 312(2), 399–426 (2012). arXiv:1008.1476 [math-ph]Google Scholar
  48. 48.
    Bonzom, V., Smerlak, M.: Bubble divergences: sorting out topology from cell structure. Ann. Henri Poincaré 13, 185–208 (2012). arXiv:1103.3961 [gr-qc]Google Scholar
  49. 49.
    Ben Geloun, J., Rivasseau, V.: A renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. (2012). doi: 10.1007/s00220-012-1549-1 arXiv:1111.4997 [hep-th]
  50. 50.
    Ben Geloun, J., Rivasseau, V.: Addendum to A renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. 322(3), 957–965 (2013). arXiv:1209.4606 [pdf, other]Google Scholar
  51. 51.
    Ben Geloun, J., Samary, D.O.: 3D tensor field theory: renormalization and one-loop β-functions. Ann. Henri poicaré 14(6), 1599–1642 (2013). arXiv:1201.0176 [hep-th]
  52. 52.
    Ben Geloun, J.: Two and four-loop β-functions of rank 4 renormalizable tensor field theories. Class. Quant. Grav. 29, 235011 (2012). arXiv:1205.5513 [hep-th]
  53. 53.
    Carrozza, S., Oriti, D.: Bounding bubbles: the vertex representation of 3d group field theory and the suppression of pseudo-manifolds. Phys. Rev. D 85, 044004 (2012). [arXiv:1104.5158 [hep-th]]
  54. 54.
    Carrozza, S., Oriti, D.: Bubbles and jackets: new scaling bounds in topological group field theories. JHEP 1206, 092 (2012). arXiv:1203.5082 [hep-th]
  55. 55.
    Carrozza, S., Oriti, D., Rivasseau, V.: Renormalization of tensorial group field theories: Abelian U(1) models in four dimensions. arXiv:1207.6734 [hep-th]
  56. 56.
    Geloun, J.B., Livine, E.R.: Some classes of renormalizable tensor models. J. Math. Phys. 54, 082303 (2013). arXiv:1207.0416 [hep-th]Google Scholar
  57. 57.
    Samary, D.O., Vignes-Tourneret, F.: Just Renormalizable TGFT’s on U(1)d with Gauge invariance. arXiv:1211.2618 [hep-th]
  58. 58.
    Rivasseau, V.: Constructive matrix theory. JHEP 0709, 008 (2007). arXiv:0706.1224 [hep-th]
  59. 59.
    Magnen, J., Rivasseau, V.: Constructive phi**4 field theory without tears. Annales Henri Poincare 9, 403 (2008). arXiv:0706.2457 [math-ph]Google Scholar
  60. 60.
    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]Google Scholar
  61. 61.
    Rivasseau, V., Wang, Z.: Loop vertex expansion for Phi**2K theory in zero dimension. J. Math. Phys. 51, 092304 (2010). arXiv:1003.1037 [math-ph]Google Scholar
  62. 62.
    Oriti, D.: Generalized group field theories and quantum gravity transition amplitudes. Phys. Rev. D73, 061502. gr-qc/0512069 (2006)Google Scholar
  63. 63.
    Oriti, D.: Group field theory and simplicial quantum gravity. Class. Quant. Grav. 27, 145017 (2010). arXiv:0902.3903 [gr-qc]
  64. 64.
    Oriti, D., Tlas, T.: Encoding simplicial quantum geometry in group field theories. Class. Quant. Grav. 27, 135018 (2010). arXiv:0912.1546 [gr-qc]
  65. 65.
    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]
  66. 66.
    Ferri M., Gagliardi C.: Crystallisation moves. Pac. J. Math. 100(1), 85–103 (1982)CrossRefMATHMathSciNetGoogle Scholar
  67. 67.
    Vince A.: n-Graphs. Disc. Math. 72(13), 367–380 (1988)Google Scholar
  68. 68.
    Vince A.: The classification of closed surfaces using colored graphs. Graphs Combin. 975–84 (1993)MATHMathSciNetGoogle Scholar
  69. 69.
    Boulatov, D.V: A Model of three-dimensional lattice gravity. Mod. Phys. Lett. A7, 1629–1646 (1992). arXiv:hep-th/9202074
  70. 70.
    Rivasseau, V.: From Perturbative to Constructive Renormalization, Princeton Series in Physics. Princeton University Press, Princeton (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Sylvain Carrozza
    • 1
    • 2
  • Daniele Oriti
    • 2
  • Vincent Rivasseau
    • 1
    • 3
  1. 1.Laboratoire de Physique Théorique, CNRS UMR 8627Université Paris SudOrsay CedexFrance
  2. 2.Max Planck Institute for Gravitational Physics, Albert Einstein InstituteGolmGermany
  3. 3.Perimeter InstituteWaterlooCanada

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