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Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions

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Abstract

We tackle the issue of renormalizability for Tensorial Group Field Theories (TGFT) including gauge invariance conditions, with the rigorous tool of multi-scale analysis, to prepare the ground for applications to quantum gravity models. In the process, we define the appropriate generalization of some key QFT notions, including connectedness, locality and contraction of (high) subgraphs. We also define a new notion of Wick ordering, corresponding to the subtraction of (maximal) melonic tadpoles. We then consider the simplest examples of dynamical 4-dimensional TGFT with gauge invariance conditions for the Abelian U(1) case. We prove that they are super-renormalizable for any polynomial interaction.

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References

  1. Thiemann T.: Modern canonical quantum general relativity. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  2. Ashtekar A., Lewandowski J.: Background independent quantum gravity: a status report. Class. Quantum Gravity 21, R53–R152 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Rovelli C.: Quantum gravity. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  4. Ambjorn, J., Goerlich, A., Jurkiewicz, J., Loll, R.: Nonperturbative quantum gravity. Phys. Rep. 519, 127–210 (2012). arXiv:1203.3591 [hep-th]

    Google Scholar 

  5. Hamber, H.: Quantum gravity on the lattice. Gen. Relativ. Gravit. 41, 817–876 (2009). arXiv:0901.0964 [gr-qc]

  6. Williams, R.M.: Recent progress in Regge calculus. Nucl. Phys. Proc. Suppl. 57, 73–81 (1997). gr-qc/9702006

    Google Scholar 

  7. Seiberg, N.: Emergent spacetime. hep-th/0601234

  8. Witten, E.: Quantum background independence in string theory. hep-th/9306122 [hep-th]

  9. Horowitz, G., Polchinski, J.: Gauge/gravity duality. In: Oriti, D. (ed.) Approaches to quantum gravity, pp. 169–186, Cambridge University Press, Cambridge (2009). gr-qc/0602037 [gr-qc]

  10. Dowker, F., Sorkin, R.: A spin-statistics theorem for certain topological geons. Class. Quantum Gravity 15, 1153–1167 (1998). gr-qc/9609064

    Google Scholar 

  11. Anspinwall, P., Greene, B., Morrison, D.: Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory. Nucl. Phys. B 416, 414–480 (1994). hep-th/9309097

    Google Scholar 

  12. Banks T.: Prolegomena to a theory of bifurcating universes: a nonlocal solution to the cosmological constant problem or little lambda goes back to the future. Nucl. Phys. B 309, 493 (1988)

    Article  ADS  Google Scholar 

  13. Coleman S.: Why there is nothing rather than something: a theory of the cosmological constant. Nucl. Phys. B 310, 643 (1988)

    Article  ADS  MATH  Google Scholar 

  14. Giddings S., Strominger A.: Baby universe, third quantization and the cosmological constant. Nucl. Phys. B 321, 481 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  15. David, F.: Planar diagrams, two-dimensional lattice gravity and surface models. Nucl. Phys. B 257, 45 (1985)

  16. Ginsparg, P.: Matrix models of 2d gravity. [arXiv: hep-th/9112013]

  17. Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Rep. 254, 1–133 (1995). hep-th/9306153

  18. Ginsparg, P., Moore, G.: Lectures on 2D gravity and 2D string theory (TASI 1992). hep-th/9304011

  19. Sorkin, R.: Ten theses on black hole entropy. Stud. Hist. Philos. Mod. Phys. 36, 291–301 (2005). hep-th/0504037 [hep-th]

    Google Scholar 

  20. Perez, A.: The spin foam approach to quantum gravity. Living Rev. Relativ. 16, 3 (2013). arXiv:1205.2019

  21. Rovelli, C.: Zakopane lectures on loop gravity. PoS(QG QGS 2011) 003. arXiv:1102.3660

  22. Oriti, D.: The group field theory approach to quantum gravity. In: Oriti, D. (ed.) Approaches to quantum gravity. Cambridge University Press, Cambridge (2009). arXiv: gr-qc/0607032

  23. Oriti, D.: Quantum gravity as a quantum field theory of simplicial geometry. In: Fauser, B. (ed.) et al. Quantum gravity, pp. 101–126. Birkhauser, Basel (2006). gr-qc/0512103 [gr-qc]

  24. Gurau, R., Ryan, J.: Colored tensor models - a review. SIGMA 8, 020 (2012). arXiv:1109.4812 [hep-th]

  25. 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]

  26. Rivasseau, V.: Quantum gravity and renormalization: the tensor track. arXiv:1112.5104 [hep-th]

  27. Gross M.: Tensor models and simplicial quantum gravity in >2-D. Nucl. Phys. Proc. Suppl. 25A, 144–149 (1992)

    Article  ADS  MATH  Google Scholar 

  28. Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A6, 1133–1146 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  29. Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A6, 2613–2624 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  30. Reisenberger, M., Rovelli, C.: Spacetime as a Feynman diagram: the connection formulation. Class. Quantum Gravity 18, 121–140 (2001). gr-qc/0002095

    Google Scholar 

  31. Baratin, A., Oriti, D.: Group field theory with non-commutative metric variables. Phys. Rev. Lett. 105, 221302 (2010). arXiv:1002.4723 [hep-th]

    Google Scholar 

  32. Baratin, A., Dittrich, B., Oriti, D., Tambornino, J.: Non-commutative flux representation for loop quantum gravity. Class. Quantum Gravity 28, 175011 (2011). arXiv:1004.3450 [hep-th]

  33. Boulatov, D.V.: A model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629–1646 (1992). arXiv:hep-th/9202074

    Google Scholar 

  34. Ooguri, H.: Topological lattice models in four dimensions. Mod. Phys. Lett. A7, 2799 (1992). hep-th/9205090

  35. De Pietri, R., Freidel, L., Krasnov, K., Rovelli, C.: Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space. Nucl. Phys. B 574, 785 (2000). arXiv: hep-th/9907154

  36. Perez, A., Rovelli, C.: A spin foam model without bubble divergences. Nucl. Phys. B 599, 255 (2001). arXiv: gr-qc/0006107

  37. Freidel, L., Krasnov, K.: A new spin foam model for 4d gravity. Class. Quantum Gravity 25, 125018 (2008). arXiv: 0708.1595

  38. Engle, J., Pereira, R., Rovelli, C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008). arXiv: 0708.1236

  39. Engle, J., Livine, E., Pereira, R., Rovelli, C.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B 799, 136 (2008). arXiv:0711.0146

  40. Ben Geloun, J., Gurau, R., Rivasseau, V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010). arXiv:1008.0354 [hep-th]

    Google Scholar 

  41. Baratin, A., Oriti, D.: Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model. New J. Phys. 13, 125011 (2011). arXiv:1108.1178 [gr-qc]

  42. Baratin, A., Oriti, D.: Group field theory and simplicial gravity path integrals: a model for Holst-Plebanski gravity. Phys. Rev. D85, 044003 (2012). arXiv:1111.5842 [hep-th]

  43. Oriti, D.: Group field theory as the microscopic description of the quantum spacetime fluid: a new perspective on the continuum in quantum gravity. Proceedings of Science PoS(QG-Ph)030. arXiv:0710.3276

  44. Sindoni, L.: Gravity as an emergent phenomenon: a GFT perspective. arXiv:1105.5687 [gr-qc]

  45. Konopka, T., Markopoulou, F., Smolin, L.: Quantum graphity. hep-th/0611197

  46. Konopka, T., Markopoulou, F., Severini, S.: Quantum graphity: a model of emergent locality. Phys. Rev. D77, 104029 (2008). arXiv:0801.0861 [hep-th]

  47. Ben Geloun, J.: Classical group field theory. J. Math. Phys. 53, 022901 (2012). arXiv:1107.3122 [hep-th]

    Google Scholar 

  48. Baratin, A., Girelli, F., Oriti, D.: Diffeomorphisms in group field theories. Phys. Rev. D 83, 104051 (2011). arXiv:1101.0590 [hep-th]

  49. Gurau, R.: A generalization of the Virasoro algebra to arbitrary dimensions. Nucl. Phys. B852, 592–614 (2011). arXiv:1105.6072 [hep-th]

  50. Oriti, D., Sindoni, L.: Towards classical geometrodynamics from group field theory hydrodynamics. New J. Phys. 13, 025006 (2011). arXiv:1010.5149 [gr-qc]

  51. Girelli, F., Livine, E., Oriti, D.: 4d Deformed special relativity from group field theories. Phys. Rev. D81, 024015 (2010). arXiv: 0903.3475 [gr-qc]

  52. Livine, E., Oriti, D., Ryan, J.: Effective Hamiltonian constraint from group field theory. Class. Quantum Gravity 28, 245010 (2011). arXiv:1104.5509 [gr-qc]

  53. Calcagni, G., Gielen, S., Oriti, D.: Group field cosmology: a cosmological field theory of quantum geometry. Class. Quantum Gravity 29, 105005 (2012). arXiv:1201.4151 [gr-qc]

  54. Bahr, B., Dittrich, B.: Improved and perfect actions in discrete gravity. Phys. Rev. D80, 124030 (2009). arXiv:0907.4323 [gr-qc]

  55. Dittrich, B., Eckert, F., Martin-Benito, M.: Coarse graining methods for spin net and spin foam models. arXiv:1109.4927 [gr-qc]

  56. 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

  57. Rivasseau, V.: Towards renormalizing group field theory. PoS CNCFG2010 (2010) 004. arXiv:1103.1900 [gr-qc]

  58. 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]

  59. Bonzom, V., Smerlak, M.: Bubble divergences from cellular cohomology. Lett. Math. Phys. 93, 295–305 (2010). arXiv:1004.5196 [gr-qc]

    Google Scholar 

  60. Bonzom, V., Smerlak, M.: Bubble divergences from twisted cohomology. Commun. Math. Phys. 312(2), 399–426 (2012). arXiv:1008.1476 [math-ph]

    Google Scholar 

  61. 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 

  62. 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]

  63. Carrozza, S., Oriti, D.: Bubbles and jackets: new scaling bounds in topological group field theories. JHEP 1206, 092 (2012). arXiv:1203.5082 [hep-th]

  64. Crane, L., Perez, A., Rovelli, C.: A finiteness proof for the Lorentzian state sum spinfoam model for quantum general relativity. Phys. Rev. Lett. 87, 181301 (2001). gr-qc/0104057 [gr-qc]

    Google Scholar 

  65. Perez, A.: Finiteness of a spinfoam model for euclidean quantum general relativity. Nucl. Phys. B599, 427–434 (2001). gr-qc/0011058 [gr-qc]

  66. Perini, C., Rovelli, C., Speziale, S.: Self-energy and vertex radiative corrections in LQG. Phys. Lett. B682, 78–84 (2009). arXiv:0810.1714 [gr-qc]

  67. Gurau, R.: Colored group field theory. Commun. Math. Phys. 304, 69 (2011). arXiv:0907.2582 [hep-th]

    Google Scholar 

  68. Ferri M., Gagliardi C.: Crystallisation moves. Pac. J. Math. 100(1), 85–103 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  69. Vince A.: n-Graphs. Discret. Math. 72(13), 367–380 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  70. Vince, A.: The classification of closed surfaces using colored graphs. Graphs Comb. 9, 75–84 (1993)

    Google Scholar 

  71. 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]

    Google Scholar 

  72. Gurau, R.: Universality for random tensors. arXiv:1111.0519 [math.PR]

  73. Gurau, R.: The 1/N expansion of colored tensor models. Ann. Henri Poincare 12, 829 (2011). arXiv:1011.2726 [gr-qc]

  74. 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 

  75. 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]

  76. 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]

  77. 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 

  78. Benedetti, D., Gurau, R.: Phase transition in dually weighted colored tensor models. Nucl. Phys. B855, 420–437 (2012). arXiv:1108.5389 [hep-th]

  79. Oriti, D.: Generalised group field theories and quantum gravity transition amplitudes. Phys. Rev. D73, 061502 (2006). gr-qc/0512069

  80. Oriti, D.: Group field theory and simplicial quantum gravity. Class. Quantum Gravity 27, 145017 (2010). arXiv:0902.3903 [gr-qc]

  81. Oriti, D., Tlas, T.: Encoding simplicial quantum geometry in group field theories. Class. Quantum Gravity 27, 135018 (2010). arXiv:0912.1546 [gr-qc]

  82. 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 

  83. Ben Geloun, J., Rivasseau, V.: A renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. 318(1), 69–109 (2013). arXiv:1111.4997 [hep-th]

    Google Scholar 

  84. Ben Geloun, J., Ousmane Samary, D.: 3D tensor field theory: renormalization and one-loop β-functions. Ann. Henri Poincaré 14(6), 1599–1642 (2013). arXiv:1201.0176 [hep-th]

  85. Ben Geloun, J., Livine, E.: Some classes of renormalizable tensor models. J. Math. Phys. 54, 082303 (2013). arXiv:1207.0416 [hep-th]

    Google Scholar 

  86. 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]

  87. Rivasseau, V.: Constructive matrix theory. JHEP 0709, 008 (2007). arXiv:0706.1224 [hep-th]

  88. Magnen, J., Rivasseau, V.: Constructive φ4 field theory without tears. Ann. Henri Poincare 9, 403 (2008). arXiv:0706.2457 [math-ph]

  89. Magnen, J., Noui, K., Rivasseau, V., Smerlak, M.: Scaling behaviour of three-dimensional group field theory. Class. Quantum Gravity 26, 185012 (2009). arXiv:0906.5477 [hep-th]

  90. Rivasseau, V., Wang, Z.: Loop vertex expansion for Phi2k theory in zero dimension. J. Math. Phys. 51, 092304 (2010). arXiv:1003.1037 [math-ph]

    Google Scholar 

  91. Bonzom, V., Erbin, H.: Coupling of hard dimers to dynamical lattices via random tensors. J. Stat. Mech (2012) PO9009. arXiv:1204.3798 hep-th]

  92. Bonzom, V., Gurau, R., Smerlak, M.: Universality in p-spin glasses with correlated disorder. J. Stat. Mech. (2013) LO2003. arXiv:1206.5539 [hep-th]

  93. Simon B.: P(Φ)2 Euclidean quantum field theory. Princeton University Press, Princeton (1974)

    Google Scholar 

  94. Rivasseau V.: From perturbative to constructive renormalization. Princeton University Press, Princeton (1991)

    Google Scholar 

  95. Freidel, L., Louapre, David: Ponzano-Regge model revisited I: gauge fixing, observables and interacting spinning particles. Class. Quantum Gravity 21, 5685–5726 (2004). arXiv:0401076 [hep-th]

  96. Ryan, J.: Tensor models and embedded Riemann surfaces. Phys. Rev. D85, 024010 (2012). arXiv:1104.5471 [gr-qc]

  97. Caravelli, F.: A simple proof of orientability in colored group field theory. SpringerPlus 20 1 (1:6). arXiv:1012.4087 [math-ph]

  98. Grosse, H., Wulkenhaar, R.: Renormalisation of φ 4-theory on noncommutative R4 in the matrix base. Commun. Math. Phys. 256, 305–374 (2005). hep-th/0401128

    Google Scholar 

  99. Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalization of noncommutative phi 4-theory by multi-scale analysis. Commun. Math. Phys. 262, 565–594 (2006). arXiv:0501036 [hep-th]

    Google Scholar 

  100. Rivasseau, V., Wang, Z.: Constructive renormalization for Φ42 theory with loop vertex expansion. J. Math. Phys. 53, 042302 (2012). arXiv:1104.3443 [math-ph]

    Google Scholar 

  101. Wang, Z.: Constructive renormalization of 2-dimensional Grosse-Wulkenhaar model. arXiv:1205.0196 [hep-th]

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Authors and Affiliations

  1. Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405, Orsay Cedex, France

    Sylvain Carrozza & Vincent Rivasseau

  2. Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, 14476, Golm, Germany

    Sylvain Carrozza & Daniele Oriti

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  1. Sylvain Carrozza
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  2. Daniele Oriti
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  3. Vincent Rivasseau
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Carrozza, S., Oriti, D. & Rivasseau, V. Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions. Commun. Math. Phys. 327, 603–641 (2014). https://doi.org/10.1007/s00220-014-1954-8

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