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

SIGMA 8 (2012), 020, 78 pages      arXiv:1109.4812
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Colored Tensor Models - a Review

Razvan Gurau a and James P. Ryan b
a) Perimeter Institute for Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
b) MPI für Gravitationsphysik, Albert Einstein Institute, Am Mühlenberg 1, D-14476 Potsdam, Germany

Received October 05, 2011, in final form March 13, 2012; Published online April 10, 2012

Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two-dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger-Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.

Key words: colored tensor models; 1/N expansion.

pdf (1282 kb)   tex (824 kb)


  1. Alspach B., The wonderful Walecki construction, Bull. Inst. Combin. Appl. 52 (2008), 7-20.
  2. Ambjørn J., Quantization of geometry, in Fluctuating Geometries in Statistical Mechanics and Field Theory (Les Houches, 1994), Editors F. David, P. Ginsparg, J. Zinn-Justin, North-Holland, Amsterdam, 1996, 77-193, hep-th/9411179.
  3. Ambjørn J., Simplicial Euclidean and Lorentzian quantum gravity, gr-qc/0201028.
  4. Ambjørn J., Durhuus B., Fröhlich J., Orland P., The appearance of critical dimensions in regulated string theories, Nuclear Phys. B 270 (1986), 457-482.
  5. Ambjørn J., Durhuus B., Jónsson T., Quantum geometry. A statistical field theory approach, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1997.
  6. Ambjørn J., Durhuus B., Jónsson T., Summing over all genera for d>1: a toy model, Phys. Lett. B 244 (1990), 403-412.
  7. Ambjørn J., Durhuus B., Jónsson T., Three-dimensional simplicial quantum gravity and generalized matrix models, Modern Phys. Lett. A 6 (1991), 1133-1146.
  8. Ambjørn J., Görlich A., Jurkiewicz J., Loll R., CDT - an entropic theory of quantum gravity, arXiv:1007.2560.
  9. Ambjørn J., Görlich A., Jurkiewicz J., Loll R., Gizbert-Studnicki J., Trzesniewski T., The semiclassical limit of causal dynamical triangulations, Nuclear Phys. B 849 (2011), 144-165, arXiv:1102.3929.
  10. Ambjørn J., Jordan S., Jurkiewicz J., Loll R., Second-order phase transition in causal dynamical triangulations, Phys. Rev. Lett. 107 (2011), 211303, 5 pages, arXiv:1108.3932.
  11. Ambjørn J., Jurkiewicz J., Scaling in four-dimensional quantum gravity, Nuclear Phys. B 451 (1995), 643-676, hep-th/9503006.
  12. Ambjørn J., Jurkiewicz J., Loll R., Reconstructing the universe, Phys. Rev. D 72 (2005), 064014, 24 pages, hep-th/0505154.
  13. Ambjørn J., Jurkiewicz J., Makeenko Y.M., Multiloop correlators for two-dimensional quantum gravity, Phys. Lett. B 251 (1990), 517-524.
  14. Bahr B., Dittrich B., Ryan J.P., Spin foam models with finite groups, arXiv:1103.6264.
  15. Baratin A., Girelli F., Oriti D., Diffeomorphisms in group field theories, Phys. Rev. D 83 (2011), 104051, 22 pages, arXiv:1101.0590.
  16. Baratin A., Oriti D., Group field theory with noncommutative metric variables, Phys. Rev. Lett. 105 (2010), 221302, 4 pages, arXiv:1002.4723.
  17. 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.
  18. Ben Geloun J., Classical group field theory, arXiv:1107.3122.
  19. Ben Geloun J., Ward-Takahashi identities for the colored Boulatov model, J. Phys. A: Math. Theor. 44 (2011), 415402, 30 pages, arXiv:1106.1847.
  20. 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.
  21. Ben Geloun J., Gurau R., Rivasseau V., EPRL/FK group field theory, Europhys. Lett. 92 (2010), 60008, 6 pages, arXiv:1008.0354.
  22. Ben Geloun J., Gurau R., Rivasseau V., Vanishing β-function for Grosse-Wulkenhaar model in a magnetic field, Phys. Lett. B 671 (2009), 284-290, arXiv:0805.4362.
  23. 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.
  24. 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.
  25. Benedetti D., Gurau R., Phase transition in dually weighted colored tensor models, Nuclear Phys. B 855 (2012), 420-437, arXiv:1108.5389.
  26. Benedetti D., Henson J., Imposing causality on a matrix model, Phys. Lett. B 678 (2009), 222-226, arXiv:0812.4261.
  27. Bernoulli J., Ars conjectandi, 1713.
  28. Bessis D., Itzykson C., Zuber J.B., Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980), 109-157.
  29. Bialas P., Burda Z., Phase transition in fluctuating branched geometry, Phys. Lett. B 384 (1996), 75-80, hep-lat/9605020.
  30. Bilke S., Burda Z., Krzywicki A., Petersson B., Tabaczek J., Thorleifsson G., 4d simplicial quantum gravity: matter fields and the corresponding effective action, Phys. Lett. B 432 (1998), 279-286, hep-lat/9804011.
  31. Blau M., Thompson G., Topological gauge theories from supersymmetric quantum mechanics on spaces of connections, Internat. J. Modern Phys. A 8 (1993), 573-585, hep-th/9112064.
  32. Bonzom V., Multicritical tensor models and hard dimers on spherical random lattices, arXiv:1201.1931.
  33. 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.
  34. Bonzom V., Gurau R., Rivasseau V., The Ising model on random lattices in arbitrary dimensions, arXiv:1108.6269.
  35. Bonzom V., Smerlak M., Bubble divergences from cellular cohomology, Lett. Math. Phys. 93 (2010), 295-305, arXiv:1004.5196.
  36. Bonzom V., Smerlak M., Bubble divergences from twisted cohomology, arXiv:1008.1476.
  37. Bonzom V., Smerlak M., Bubble divergences: sorting out topology from cell structure, Ann. Henri Poincaré 13 (2012), 185-208, arXiv:1103.3961.
  38. Boulatov D.V., A model of three-dimensional lattice gravity, Modern Phys. Lett. A 7 (1992), 1629-1646, hep-th/9202074.
  39. Boulatov D.V., Kazakov V.A., The ising model on a randomplanarlattice: the structure of the phase transition and the exact critical exponents, Phys. Lett. B 186 (1987), 379-384.
  40. Bouttier J., Di Francesco P., Guitter E., Counting colored random triangulations, Nuclear Phys. B 641 (2002), 519-532, cond-mat/0206452.
  41. Brézin É., Douglas M.R., Kazakov V., Shenker S.H., The Ising model coupled to 2D gravity. A nonperturbative analysis, Phys. Lett. B 237 (1990), 43-46.
  42. Brézin É., Itzykson C., Parisi G., Zuber J.B., Planar diagrams, Comm. Math. Phys. 59 (1978), 35-51.
  43. Brézin É., Kazakov V.A., Exactly solvable field theories of closed strings, Phys. Lett. B 236 (1990), 144-150.
  44. Caravelli F., A simple proof of orientability in colored group field theory, arXiv:1012.4087.
  45. 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.
  46. David F., A model of random surfaces with nontrivial critical behaviour, Nuclear Phys. B 257 (1985), 543-576.
  47. David F., Conformal field theories coupled to 2-D gravity in the conformal gauge, Modern Phys. Lett. A 3 (1988), 1651-1656.
  48. David F., Simplicial quantum gravity and random lattices, in Gravitation and Quantizations (Les Houches, 1992), Editors J. Zinn-Justin, B. Julia, North-Holland, Amsterdam, 1995, 679-749, hep-th/9303127.
  49. David F., Hagendorf C., Wiese K.J., A growth model for RNA secondary structures, J. Stat. Mech. Theory Exp. 2008 (2008), P04008, 43 pages, arXiv:0711.3421.
  50. De Pietri R., Petronio C., Feynman diagrams of generalized matrix models and the associated manifolds in dimension four, J. Math. Phys. 41 (2000), 6671-6688, gr-qc/0004045.
  51. Di Francesco P., Rectangular matrix models and combinatorics of colored graphs, Nuclear Phys. B 648 (2003), 461-496, cond-mat/0208037.
  52. Di Francesco P., Eynard B., Guitter E., Coloring random triangulations, Nuclear Phys. B 516 (1998), 543-587, cond-mat/9711050.
  53. Di Francesco P., Ginsparg P.H., Zinn-Justin J., 2D gravity and random matrices, Phys. Rep. 254 (1995), 1-133, hep-th/9306153.
  54. Di Mare A., Oriti D., Emergent matter from 3D generalized group field theories, Classical Quantum Gravity 27 (2010), 145006, 17 pages, arXiv:1001.2702.
  55. Dijkgraaf R., Verlinde H., Verlinde E., Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity, Nuclear Phys. B 348 (1991), 435-456.
  56. Disertori M., Gurau R., Magnen J., Rivasseau V., Vanishing of beta function of non-commutative Φ44 theory to all orders, Phys. Lett. B 649 (2007), 95-102, hep-th/0612251.
  57. Distler J., Kawai H., Conformal field theory and 2D quantum gravity, Nuclear Phys. B 321 (1989), 509-527.
  58. Dittrich B., Eckert F., Martin-Benito M., Coarse-graining spin nets and spin foams, in preparation.
  59. Douglas M.R., Shenker S.H., Strings in less than one dimension, Nuclear Phys. B 335 (1990), 635-654.
  60. Duplantier B., Conformal random geometry, in Mathematical Statistical Physics, Editors A. Bovier, F. Dunlop, F. den Hollander, A. van Enter, J. Dalibard, Elsevier, Amsterdam, 2006, 101-217, math-ph/0608053.
  61. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  62. Engle J., Pereira R., Rovelli C., Loop-quantum-gravity vertex amplitude, Phys. Rev. Lett. 99 (2007), 161301, 4 pages, arXiv:0705.2388.
  63. Fairbairn W.J., Livine E.R., 3D spinfoam quantum gravity: matter as a phase of the group field theory, Classical Quantum Gravity 24 (2007), 5277-5297, gr-qc/0702125.
  64. Ferri M., Gagliardi C., Crystallisation moves, Pacific J. Math. 100 (1982), 85-103.
  65. Fisher M.E., Barber M.N., Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett. 28 (1972), 1516-1519.
  66. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, hep-th/0505016.
  67. Freidel L., Gurau R., Group field theory renormalization in the 3D case: power counting of divergences, Phys. Rev. D 80 (2009), 044007, 20 pages, arXiv:0905.3772.
  68. Freidel L., Krasnov K., A new spin foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.
  69. Freidel L., Livine E.R., Ponzano-Regge model revisited. III. Feynman diagrams and effective field theory, Classical Quantum Gravity 23 (2006), 2021-2061, hep-th/0502106.
  70. Freidel L., Louapre D., Nonperturbative summation over 3D discrete topologies, Phys. Rev. D 68 (2003), 104004, 16 pages, hep-th/0211026.
  71. Fukuma M., Kawai H., Nakayama R., Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity, Internat. J. Modern Phys. A 6 (1991), 1385-1406.
  72. Gallavotti G., Nicolò F., Renormalization theory in four-dimensional scalar fields. I, Comm. Math. Phys. 100 (1985), 545-590.
  73. Girelli F., Livine E.R., A deformed Poincaré invariance for group field theories, Classical Quantum Gravity 27 (2010), 245018, 15 pages, arXiv:1001.2919.
  74. Goulden I.P., Jackson D.M., Combinatorial enumeration, Dover Publications Inc., Mineola, NY, 2004.
  75. Graham R.L., Knuth D.E., Patashnik O., Concrete mathematics. A foundation for computer science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994.
  76. Grimmett G.R., Stirzaker D.R., Probability and random processes, 3rd ed., Oxford University Press, New York, 2001.
  77. Gross D.J., Migdal A.A., Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 (1990), 127-130.
  78. Grosse H., Wulkenhaar R., Power-counting theorem for non-local matrix models and renormalisation, Comm. Math. Phys. 254 (2005), 91-127, hep-th/0305066.
  79. Grosse H., Wulkenhaar R., Renormalisation of φ4-theory on noncommutative R4 in the matrix base, Comm. Math. Phys. 256 (2005), 305-374, hep-th/0401128.
  80. Grosse H., Wulkenhaar R., The β-function in duality-covariant non-commutative φ4-theory, Eur. Phys. J. C Part. Fields 35 (2004), 277-282, hep-th/0402093.
  81. Gurau R., A generalization of the Virasoro algebra to arbitrary dimensions, Nuclear Phys. B 852 (2011), 592-614, arXiv:1105.6072.
  82. Gurau R., Colored group field theory, Comm. Math. Phys. 304 (2011), 69-93, arXiv:0907.2582.
  83. Gurau R., Double scaling limit in arbitrary dimensions: a toy model, Phys. Rev. D 84 (2011), 124051, 11 pages, arXiv:1110.2460.
  84. Gurau R., Lost in translation: topological singularities in group field theory, Classical Quantum Gravity 27 (2010), 235023, 20 pages, arXiv:1006.0714.
  85. Gurau R., The 1/N expansion of colored tensor models, Ann. Henri Poincaré 12 (2011), 829-847, arXiv:1011.2726.
  86. Gurau R., The complete 1/N expansion of colored tensor models in arbitrary dimension, Ann. Henri Poincaré 13 (2012), 399-423, arXiv:1102.5759.
  87. Gurau R., Topological graph polynomials in colored group field theory, Ann. Henri Poincaré 11 (2010), 565-584, arXiv:0911.1945.
  88. Gurau R., Magnen J., Rivasseau V., Vignes-Tourneret F., Renormalization of non-commutative Φ44 field theory in x space, Comm. Math. Phys. 267 (2006), 515-542, hep-th/0512271.
  89. Gurau R., Rivasseau V., Parametric representation of noncommutative field theory, Comm. Math. Phys. 272 (2007), 811-835, math-ph/0606030.
  90. 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.
  91. Hempel J., 3-manifolds, AMS Chelsea Publishing, Providence, RI, 2004.
  92. Heubach S., Li N.Y., Mansour T., Staircase tilings and k-Catalan structures, Discrete Math. 308 (2008), 5954-5964.
  93. Kazakov V.A., Bilocal regularization of models of random surfaces, Phys. Lett. B 150 (1985), 282284.
  94. Kazakov V.A., Ising model on a dynamical planar random lattice: exact solution, Phys. Lett. A 119 (1986), 140-144.
  95. Kazakov V.A., Staudacher M., Wynter T., Almost flat planar diagrams, Comm. Math. Phys. 179 (1996), 235-256, hep-th/9506174.
  96. Kazakov V.A., Staudacher M., Wynter T., Character expansion methods for matrix models of dually weighted graphs, Comm. Math. Phys. 177 (1996), 451-468, hep-th/9502132.
  97. Kazakov V.A., Staudacher M., Wynter T., Exact solution of discrete two-dimensional R2 gravity, Nuclear Phys. B 471 (1996), 309-333, hep-th/9601069.
  98. Knizhnik V.G., Polyakov A.M., Zamolodchikov A.B., Fractal structure of 2D-quantum gravity, Modern Phys. Lett. A 3 (1988), 819-826.
  99. Kolmogorov A.N., Foundations of the theory of probability, Chelsea Publishing Co., New York, 1956.
  100. Kozlov D., Combinatorial algebraic topology, Algorithms and Computation in Mathematics, Vol. 21, Springer, Berlin, 2008.
  101. 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.
  102. Laiho J., Coumbe D., Evidence for asymptotic safety from lattice quantum gravity, Phys. Rev. Lett. 107 (2011), 161301, 4 pages, arXiv:1104.5505.
  103. Le Gall J.F., Miermont G., Scaling limits of random planar maps with large faces, Ann. Probab. 39 (2011), 1-69.
  104. Lins S., Gems, computers and attractors for 3-manifolds, Series on Knots and Everything, Vol. 5, World Scientific Publishing Co. Inc., River Edge, NJ, 1995.
  105. Livine E.R., Oriti D., Ryan J.P., Effective Hamiltonian constraint from group field theory, Classical Quantum Gravity 28 (2011), 245010, 17 pages, arXiv:1104.5509.
  106. Livine E.R., Speziale S., New spinfoam vertex for quantum gravity, Phys. Rev. D 76 (2007), 084028, 14 pages, arXiv:0705.0674.
  107. 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.
  108. Makeenko Y., Loop equations and Virasoro constraints in matrix models, hep-th/9112058.
  109. Manes K., Sapounakis A., Tasoulas I., Tsikouras P., Recursive generation of k-ary trees, J. Integer Seq. 12 (2009), Article 09.7.7, 18 pages.
  110. Ooguri H., Topological lattice models in four dimensions, Modern Phys. Lett. A 7 (1992), 2799-2810, hep-th/9205090.
  111. Oriti D., Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity, Rep. Progr. Phys. 64 (2001), 1703-1757, gr-qc/0106091.
  112. Oriti D., The group field theory approach to quantum gravity, in Approaches to Quantum Gravity, Editor D. Oriti, Cambridge University Press, Cambridge, 2009, 310-331, gr-qc/0607032.
  113. Oriti D., Sindoni L., Toward classical geometrodynamics from the group field theory hydrodynamics, New J. Phys. 13 (2011), 025006, 44 pages, arXiv:1010.5149.
  114. Orland H., Zee A., RNA folding and large N matrix theory, Nuclear Phys. B 620 (2002), 456-476, cond-mat/0106359.
  115. Perini C., Rovelli C., Speziale S., Self-energy and vertexradiativecorrections in LQG, Phys. Lett. B 682 (2009), 78-84, arXiv:0810.1714.
  116. Rivasseau V., Towards renormalizing group field theory, PoS Proc. Sci. (2010), PoS(CNCFG2010), 004, 21 pages, arXiv:1103.1900.
  117. Rivasseau V., Vignes-Tourneret F., Wulkenhaar R., Renormalisation of noncommutative φ4-theory by multi-scale analysis, Comm. Math. Phys. 262 (2006), 565-594, hep-th/0501036.
  118. Rovelli C., Smerlak M., In quantum gravity, summing is refining, Classical Quantum Gravity 29 (2012), 055004, 7 pages, arXiv:1010.5437.
  119. Ryan J.P., Tensor models and embedded Riemann surfaces, arXiv:1104.5471.
  120. Ryan J.P., Tensor models and embedded Riemann surfaces in arbitrary dimension, in preparation.
  121. Salmhofer M., Renormalization. An introduction, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1999.
  122. Sasakura N., Super tensor models, super fuzzy spaces and super n-ary transformations, Internat. J. Modern Phys. A 26 (2011), 4203-4216, arXiv:1106.0379.
  123. Sasakura N., Tensor models and 3-ary algebras, arXiv:1104.1463.
  124. Sasakura N., Tensor model for gravity and orientability of manifold, Modern Phys. Lett. A 6 (1991), 2613-2623.
  125. Sasakura N., Tensor models and hierarchy of n-ary algebras, Internat. J. Modern Phys. A 26 (2011), 3249-3258, arXiv:1104.5312.
  126. Stanley R.P., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 2001.
  127. Szabo R.J., Wheater J.F., Curvature matrix models for dynamical triangulations and the Itzykson-Di Francesco formula, Nuclear Phys. B 491 (1997), 689-723, hep-th/9609237.
  128. 't Hooft G., A planardiagramtheoryforstronginteractions, Nuclear Phys. B 72 (1974), 461-473.
  129. Tanasa A., Generalization of the Bollobás-Riordan polynomial for tensor graphs, J. Math. Phys. 52 (2011), 073514, 17 pages, arXiv:1012.1798.
  130. Tanasa A., Multi-orientable group field theory, arXiv:1109.0694.

Previous article  Next article   Contents of Volume 8 (2012)