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Constructive Tensor Field Theory: the \({T^4_3}\) Model

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Abstract

We build constructively the simplest tensor field theory which requires some renormalization, namely the rank three tensor theory with quartic interactions and propagator inverse of the Laplacian on \({U(1)^3}\). This superrenormalizable tensor field theory has a power counting almost similar to ordinary \({\phi^4_2}\). Our construction uses the multiscale loop vertex expansion (MLVE) recently introduced in the context of an analogous vector model. However, to prove analyticity and Borel summability of this model requires new estimates on the intermediate field integration, which is now of matrix rather than of scalar type.

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References

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Oriti, D.: The microscopic dynamics of quantum space as a group field theory arXiv:1110.5606 [hep-th]

  5. Ambjorn J., Durhuus B., Jonsson T.: Three-Dimensional Simplicial Quantum Gravity And Generalized Matrix Models. Mod. Phys. Lett. A 6, 1133 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. 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]

    Article  ADS  Google Scholar 

  10. Gurau R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. Annales Henri Poincare 13, 399 (2012) arXiv:1102.5759 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Gurau R.: Universality for random tensors. Ann. Inst. H. Poincaré Probab. Statist. 50, 1474–1525 (2014) arXiv:1111.0519 [math.PR]

    Article  MathSciNet  MATH  Google Scholar 

  12. Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2-D gravity and random matrices. Phys. Rept. 254, 1 (1995) arXiv:hep-th/9306153

    Article  ADS  MathSciNet  Google Scholar 

  13. Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Grosse H., Wulkenhaar R.: Renormalisation of phi**4 theory on noncommutative R**4 in the matrix base. Commun. Math. Phys. 256, 305 (2005) arXiv:hep-th/0401128

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Disertori M., Gurau R., Magnen J., Rivasseau V.: Vanishing of beta function of non commutative Phi**4(4) theory to all orders. Phys. Lett. B 649, 95 (2007) hep-th/0612251

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Rivasseau, V.: Non-commutative Renormalization. arXiv:0705.0705 [hep-th]

  17. Grosse, H., Wulkenhaar, R.: Progress in solving a noncommutative quantum field theory in four dimensions. arXiv:0909.1389

  18. Grosse H., Wulkenhaar R.: Self-dual noncommutative \({\phi^4}\)-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory. Comm. Math. Phys. 329, 1069–1130 (2014) arXiv:1205.0465

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. 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]

    Article  ADS  Google Scholar 

  20. Rivasseau V.: Quantum Gravity and renormalization: the tensor track. AIP Conf. Proc. 1444, 18 (2011) arXiv:1112.5104 [hep-th]

    ADS  Google Scholar 

  21. Rivasseau, V.: The Tensor track: an update. arXiv:1209.5284 [hep-th]

  22. Rivasseau V.: The tensor track, III. Fortsch. Phys. 62, 81–107 (2014) arXiv:1311.1461 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Rivasseau V.: The tensor theory space. Fortsch. Phys. 62, 835 (2014) arXiv:1407.0284 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Benedetti D., Ben Geloun J., Oriti D.: Functional renormalisation group approach for tensorial group field theory: a rank-3 model. JHEP 1503, 084 (2015) arXiv:1411.3180 [hep-th]

    Article  MathSciNet  Google Scholar 

  25. Ambjorn, J.: Simplicial Euclidean and Lorentzian quantum gravity. arXiv:gr-qc/0201028

  26. Oriti, D.: Group field theory as the 2nd quantization of Loop. Quantum Grav. arXiv:1310.7786 [gr-qc]

  27. 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]

    Article  MATH  Google Scholar 

  28. Ben Geloun J., Rivasseau V.: A renormalizable 4-dimensional tensor field theory. Comm. Math. Phys. 318, 69–109 (2013) arXiv:1111.4997

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Ben Geloun J., Samary D.O.: 3D Tensor Field theory: renormalization and one-loop \({\beta}\)-functions. Ann. Henri Poincaré 14, 1599 (2013) arXiv:1201.0176

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Ben Geloun J.: Renormalizable models in rank \({d\geq 2}\) tensorial group field theory. Commun. Math. Phys. 332(1), 117–188 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Carrozza S., Oriti D., Rivasseau V.: Renormalization of tensorial group field theories: Abelian U(1) Models in Four Dimensions. Comm. Math. Phys. 327, 603–641 (2014) arXiv:1207.6734

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Carrozza S., Oriti D., Rivasseau V.: Renormalization of an SU(2) tensorial group field theory in three dimensions. Comm. Math. Phys. 330, 581–637 (2014) arXiv:1303.6772

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Carrozza, S.: Tensorial methods and renormalization in group field theories. Series: Springer Theses. arXiv:1310.3736 [hep-th]

  34. Reuter M., Saueressig F.: Quantum Einstein gravity. New J. Phys. 14, 055022 (2012) arXiv:1202.2274

    Article  ADS  MathSciNet  Google Scholar 

  35. Ben Geloun J.: Two and four-loop \({\beta}\)-functions of rank 4 renormalizable tensor field theories. Class. Quant. Grav. 29, 235011 (2012) arXiv:1205.5513

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Carrozza S.: Discrete Renormalization group for SU(2) tensorial group field theory. Ann. Inst. Henri. Poincaré Comb. Phys. Interact 2, 49–112 (2015) arXiv:1407.4615 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  37. Lahoche, V., Oriti, D., Rivasseau, V.: Renormalization of an Abelian tensor group field theory: solution at leading order. JHEP 1504, 095 (2015). doi:10.1007/JHEP04(2015)095 [arXiv:1501.02086 [hep-th]]

  38. Rivasseau, V.: Why are tensor field theories asymptotically free?. Europhys. Lett. 111(6), 60011 (2015). doi:10.1209/0295-5075/111/60011 [arXiv:1507.04190 [hep-th]]

  39. Glimm J., Jaffe A.M.: Quantum Physics. A Functional Integral Point Of View. Springer, New York (1987)

    MATH  Google Scholar 

  40. Rivasseau, V.: From Perturbative To Constructive Renormalization. Princeton University Press, (1991)

  41. Rivasseau V.: Constructive matrix theory. JHEP 0709, 008 (2007) arXiv:0706.1224

    Article  ADS  MathSciNet  Google Scholar 

  42. Magnen J., Rivasseau V.: Constructive phi**4 field theory without tears. Ann. Henri Poincaré 9, 403 (2008) arXiv:0706.2457 [math-ph]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Rivasseau V., Wang Z.: How to resum Feynman graphs. Ann. Henri Poincaré 15, 2069–2083 (2014) arXiv:1304.5913 [math-ph]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Magnen J., Noui K., Rivasseau V., Smerlak M.: Scaling behavior of three-dimensional group field theory. Class. Quant. Grav. 26, 185012 (2009) arXiv:0906.5477

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Gurau R.: The \({1/N}\) expansion of tensor models beyond perturbation theory. Comm. Math. Phys. 330, 973–1019 (2014) arXiv:1304.2666

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Delepouve, T., Gurau, R., Rivasseau, V.: Universality and borel Summability of arbitrary quartic tensor models. arXiv:1403.0170 [hep-th]

  47. Gurau R., Rivasseau V.: The multiscale loop vertex expansion. Ann. Henri Poincaré 16, 1869–1897 (2015) arXiv:1312.7226 [math-ph]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Rivasseau, V., Wang, Z.: Corrected loop vertex expansion for Phi42 Theory. arXiv:1406.7428 [math-ph]

  49. Lahoche, V.: Constructive tensorial group field theory I: the \({U(1)-T^4_3}\) model. arXiv:1510.05050 [hep-th]

  50. Lahoche, V.: Constructive Tensorial Group Field Theory II: The \({U(1)-T^4_4}\) Model. arXiv:1510.05051 [hep-th]

  51. Mayer J.E.: Montroll, Elliott molecular distributions. J. Chem. Phys. 9, 216 (1941)

    Article  Google Scholar 

  52. Brydges D., Federbush P.: A new form of the mayer expansion in classical statistical mechanics. J. Math. Phys. 19, 2064 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  53. Brydges D.: A short course on cluster expansions. In: Osterwalder, K., Stora, R. Session XLIII, 1984, Elsevier, Les Houches (1986)

    Google Scholar 

  54. Nelson, E.: A quartic interaction in two dimensions. Mathematical Theory of Elementary Particles, pp. 69-73. M.I.T. Press, Cambridge (1965)

  55. Simon, B.: The \({P(\Phi)_2}\) Euclidean (Quantum) Field Theory, pp. 392. Princeton University Press, princeton (1974) (Princeton Series in Physics)

  56. Brydges D., Kennedy T.: Mayer expansions and the Hamilton–Jacobi equation. J. Stat. Phys. 48, 19 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  57. Abdesselam, A., Rivasseau, V.: Trees, forests and jungles: a botanical garden for cluster expansions. In: Constructive physics results in field theory, statistical mechanics and condensed matter physics, pp. 7-36. Springer (1995). arXiv:hep-th/9409094

  58. Abdesselam A., Rivasseau V.: Explicit fermionic tree expansions. Lett. Math. Phys. 44, 77–88 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  59. Sokal A.D.: An improvement of Watson’s theorem on Borel summability. J. Math. Phys. 21, 261–263 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  60. Rivasseau V., de Calan C.: Local existence of the Borel transform in Euclidean \({\Phi^4_4}\). Comm. Math. Phys. Vol. 82(1), 69–100 (1981)

    Article  ADS  Google Scholar 

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Authors and Affiliations

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

    Thibault Delepouve & Vincent Rivasseau

  2. Centre de Physique Théorique, CNRS UMR 7644, École Polytechnique, 91128, Palaiseau Cedex, France

    Thibault Delepouve

  3. Perimeter Institute for Theoretical Physics, 31 Caroline St. N, N2L 2Y5, Waterloo, ON, Canada

    Vincent Rivasseau

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  1. Thibault Delepouve
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  2. Vincent Rivasseau
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Correspondence to Vincent Rivasseau.

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Communicated by A. Connes

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Delepouve, T., Rivasseau, V. Constructive Tensor Field Theory: the \({T^4_3}\) Model. Commun. Math. Phys. 345, 477–506 (2016). https://doi.org/10.1007/s00220-016-2680-1

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