Communications in Mathematical Physics

, Volume 358, Issue 2, pp 589–632 | Cite as

The Full Ward-Takahashi Identity for Colored Tensor Models



Colored tensor models (CTM) is a random geometrical approach to quantum gravity. We scrutinize the structure of the connected correlation functions of general CTM-interactions and organize them by boundaries of Feynman graphs. For rank-D interactions including, but not restricted to, all melonic \({\varphi^4}\) -vertices—to wit, solely those quartic vertices that can lead to dominant spherical contributions in the large-N expansion—the aforementioned boundary graphs are shown to be precisely all (possibly disconnected) vertex-bipartite regularly edge-D-colored graphs. The concept of CTM-compatible boundary-graph automorphism is introduced and an auxiliary graph calculus is developed. With the aid of these constructs, certain U (∞)-invariance of the path integral measure is fully exploited in order to derive a strong Ward-Takahashi Identity for CTMs with a symmetry-breaking kinetic term. For the rank-3 \({\varphi^4}\) -theory, we get the exact integral-like equation for the 2-point function. Similarly, exact equations for higher multipoint functions can be readily obtained departing from this full Ward-Takahashi identity. Our results hold for some Group Field Theories as well. Altogether, our non-perturbative approach trades some graph theoretical methods for analytical ones. We believe that these tools can be extended to tensorial SYK-models.


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  1. 1.
    Ambjørn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A6, 1133–1146 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Geloun, J.B., Ramgoolam, S.: Counting tensor model observables and branched covers of the 2-sphere. Ann. Inst. Henri Poincar Comb. Phys. Interact., 1, 77–138 (2014). arXiv:1307.6490
  3. 3.
    Geloun, J.B., Rivasseau, V.: A renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. 318, 69–109 (2013). arXiv:1111.4997
  4. 4.
    Geloun, J.B., Rivasseau, V.: A renormalizable SYK-type tensor field theory (2017). arXiv:1711.05967
  5. 5.
    Bonzom, V., Gurău, R., Rivasseau, V.: Random tensor models in the large N limit: Uncoloring the colored tensor models. Phys. Rev. D85, 084037 (2012). arXiv:1202.3637
  6. 6.
    Bonzom, V., Lionni, L., Tanasă, A.: Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders. J. Math. Phys. 58(5), 052301 (2017). arXiv:1702.06944
  7. 7.
    Carrozza, S., Oriti, D., Rivasseau, V.: Renormalization of a SU(2) tensorial group field theory in three dimensions. Commun. Math. Phys. 330, 581–637 (2014). arXiv:1303.6772
  8. 8.
    Carrozza S., Tanasă A.: O(N) random tensor models. Lett. Math. Phys. 106(11), 1531–1559 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Casali, M.R., Cristofori, P., Gagliardi, C.: PL 4-manifolds admitting simple crystallizations: framed links and regular genus. J. Knot Theory Ramif. 25(01), 1650005 (2016). arXiv:1410.3321
  10. 10.
    Delepouve T., Rivasseau V.: Constructive tensor field theory: the \({T^{4}_3}\) model. Commun. Math. Phys. 345(2), 477–506 (2016)ADSCrossRefMATHGoogle Scholar
  11. 11.
    Francesco, P.D., Ginsparg, P.H., Zinn-Justin, J.: 2-D gravity and random matrices. Phys. Rep. 254, 1–133 (1995). arXiv:hep-th/9306153
  12. 12.
    Disertori, M., Gurău, R., Magnen, J., Rivasseau, V.: Vanishing of beta function of non commutative \({\Phi_4^4}\) theory to all orders. Phys. Lett. B649, 95–102 (2007). arXiv:hep-th/0612251
  13. 13.
    Disertori, M., Rivasseau, V.: Two and three loops beta function of non commutative \({\Phi_4^4}\) theory. Eur. Phys. J. C50, 661–671 (2007). arXiv:hep-th/0610224
  14. 14.
    Ferri M., Gagliardi C., Grasselli L.: A graph-theoretical representation of pl-manifolds—a survey on crystallizations. Aequ. Math. 31(1), 121–141 (1986)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Freedman M.H.: The topology of four-dimensional manifolds. J. Differ. Geom. 17(3), 357–453 (1982)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Freidel, L.: Group field theory: an Overview. Int. J. Theor. Phys. 44, 1769–1783 (2005). arXiv:hep-th/0505016
  17. 17.
    Gagliardi C.: How to deduce the fundamental group of a closed n-manifold from a contracted triangulation. J. Comb. Inf. Syst. Sci. 4(3), 237–252 (1979)MathSciNetMATHGoogle Scholar
  18. 18.
    Gross, D.J., Rosenhaus, V.: (2017) All point correlation functions in SYKGoogle Scholar
  19. 19.
    Grosse, H., Wulkenhaar, R.: The beta function in duality covariant noncommutative \({\phi^4}\) theory. Eur. Phys. J. C35, 277–282 (2004). arXiv:hep-th/0402093
  20. 20.
    Grosse, H., Wulkenhaar, R.: Renormalization of \({\phi^4}\) theory on noncommutative R 4 in the matrix base. Commun. Math. Phys. 256, 305–374 (2005). arXiv:hep-th/0401128
  21. 21.
    Grosse, H., Wulkenhaar, R.: Self-dual noncommutative \({\phi^4}\) -theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory. Commun. Math. Phys. 329, 1069–1130 (2014). arXiv:1205.0465
  22. 22.
    Gurău, R.: A generalization of the Virasoro algebra to arbitrary dimensions. Nucl. Phys. B852, 592–614 (2011). arXiv:1105.6072 [hep-th]
  23. 23.
    Gurău, R.: The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders. Nucl. Phys. B865, 133–147 (2012). arXiv:1203.4965 [hep-th]
  24. 24.
    Gurău R., Rivasseau V.: The multiscale loop vertex expansion. Ann. Henri Poincare 16(8), 1869–1897 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gurău, R.: Colored group field theory. Commun. Math. Phys. 304, 69–93 (2011). arXiv:0907.2582
  26. 26.
    Gurău, R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. Ann. Henri Poincare 13, 399–423 (2012). arXiv:1102.5759
  27. 27.
    Gurău, R., Ryan, J.P.: Colored tensor models: a review. SIGMA 8, 020 (2012). arXiv:1109.4812
  28. 28.
    Kirby, R.C., Siebenmann, L.: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Number 88. Princeton University Press, Princeton (1977)Google Scholar
  29. 29.
    Kitaev, A.: A simple model of quantum holography (lecture). (2015). Accessed 10 May 2017
  30. 30.
    Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B533, 168–177 (2002). arXiv:hep-th/0202039
  31. 31.
    Moise E.E.: Affine structures in 3-manifolds: V. The triangulation theorem and Hauptvermutung. Ann. Math. 56(1), 96–114 (1952)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    The on-line encyclopedia of integer sequences. Accessed 29 June 2016
  33. 33.
    Oriti, D.: Group field theory and loop quantum gravity. Gen. Relativ. 4, 125–151 (2017)Google Scholar
  34. 34.
    Ousmane Samary, D., Pérez-Sánchez, C.I., Vignes-Tourneret, F., Wulkenhaar, R.: Correlation functions of a just renormalizable tensorial group field theory: the melonic approximation. Class. Quantum Gravity 32(17), 175012 (2015). arXiv:1411.7213
  35. 35.
    Pérez-Sánchez, C.I.: Surgery in colored tensor models. J. Geom. Phys. 120, 262–289 (2017). arXiv:1608.00246
  36. 36.
    Pérez-Sánchez, C.I., Wulkenhaar, R.: Correlation functions of coloured tensor models and their Schwinger-Dyson equations (2017). arXiv:1706.07358
  37. 37.
    Pezzana M.: Sulla struttura topologica delle varietà compatte. Ati Sem. Mat. Fis. Univ. Modena 23(1), 269–277 (1975)MathSciNetMATHGoogle Scholar
  38. 38.
    Reisenberger, M.P., Rovelli, C.: Space-time as a Feynman diagram: The Connection formulation. Class. Quantum Gravity 18, 121–140 (2001). arXiv:gr-qc/0002095
  39. 39.
    Rivasseau, V.: The tensor track: an update. In: 29th International Colloquium on Group-Theoretical Methods in Physics (GROUP 29) Tianjin, China, August 20–26, 2012 (2012). arXiv:1209.5284 [hep-th]
  40. 40.
    Rivasseau V.: The tensor track, III. Fortsch. Phys. 62, 81–107 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Rivasseau, V.: The tensor track, IV. In: Proceedings, 15th Hellenic School and Workshops on Elementary Particle Physics and Gravity (CORFU2015): Corfu, Greece, September 1–25, 2015 (2016)Google Scholar
  42. 42.
    Sachdev S., Ye J.: Gapless spin fluid ground state in a random, quantum Heisenberg magnet. Phys. Rev. Lett. 70, 3339 (1993)ADSCrossRefGoogle Scholar
  43. 43.
    Samary, D.O.: Closed equations of the two-point functions for tensorial group field theory. Class. Quantum Gravity 31, 185005 (2014). arXiv:1401.2096
  44. 44.
    Smerlak, M.: Comment on ‘Lost in Translation: Topological Singularities in Group Field Theory’. Class. Quantum Gravity 28, 178001 (2011). arXiv:1102.1844
  45. 45.
    Tanasă, A.: Multi-orientable group field theory. J. Phys. A45, 165401 (2012). arXiv:1109.0694
  46. 46.
    Tanasă, A.: The multi-orientable random tensor model, a review. SIGMA 12, 056 (2016). arXiv:1512.02087
  47. 47.
    Witten, E.: An SYK-Like model without disorder (2016). arXiv:1610.09758

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Authors and Affiliations

  1. 1.Mathematisches InstitutWestfälische Wilhelms-UniversitätMünsterGermany

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