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


SIGMA 12 (2016), 085, 31 pages      arXiv:1603.07312      https://doi.org/10.3842/SIGMA.2016.085
Contribution to the Special Issue on Tensor Models, Formalism and Applications

Constructive Tensor Field Theory

Vincent Rivasseau
Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, F-91405 Orsay Cedex, France

Received March 23, 2016, in final form August 18, 2016; Published online August 22, 2016

Abstract
We provide an up-to-date review of the recent constructive program for field theories of the vector, matrix and tensor type, focusing not on the models themselves but on the mathematical tools used.

Key words: constructive field theory; renormalization; tensor models.

pdf (637 kb)   tex (86 kb)

References

  1. Abdesselam A., Field theoretic cluster expansions and the Brydges-Kennedy forest sum formula, Slides of the talk for the conference ''Combinatorial Identities and Their Applications in Statistical Mechanics'' (April 2008, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK), available at http://people.virginia.edu/~aa4cr/Cambridge08Apr2008.pdf.
  2. Abdesselam A., Rivasseau V., Trees, forests and jungles: a botanical garden for cluster expansions, in Constructive Physics (Palaiseau, 1994), Lecture Notes in Phys., Vol. 446, Springer, Berlin, 1995, 7-36, hep-th/9409094.
  3. Ambjørn J., Simplicial Euclidean and Lorentzian quantum gravity, gr-qc/0201028.
  4. Ambjørn J., Durhuus B., Jónsson T., Three-dimensional simplicial quantum gravity and generalized matrix models, Modern Phys. Lett. A 6 (1991), 1133-1146.
  5. Ben Geloun J., Renormalizable models in rank $d\geq 2$ tensorial group field theory, Comm. Math. Phys. 332 (2014), 117-188, arXiv:1306.1201.
  6. Ben Geloun J., Ramgoolam S., Counting tensor model observables and branched covers of the 2-sphere, Ann. Inst. Henri Poincaré D 1 (2014), 77-138, arXiv:1307.6490.
  7. Ben Geloun J., Rivasseau V., A renormalizable 4-dimensional tensor field theory, Comm. Math. Phys. 318 (2013), 69-109, arXiv:1111.4997.
  8. Bergère M.C., David F., Ambiguities of renormalized $\varphi^{4}_{4}$ field theory and the singularities of its Borel transform, Phys. Lett. B 135 (1984), 412-416.
  9. Billionnet C., Renouard P., Analytic interpolation and Borel summability of the $({\lambda \over N}\Phi_{N}^{:4})_{2}$ models. I. Finite volume approximation, Comm. Math. Phys. 84 (1982), 257-295.
  10. Bonzom V., New $1/N$ expansions in random tensor models, J. High Energy Phys. 2013 (2013), no. 6, 062, 25 pages, arXiv:1211.1657.
  11. Bonzom V., Delepouve T., Rivasseau V., Enhancing non-melonic triangulations: a tensor model mixing melonic and planar maps, Nuclear Phys. B 895 (2015), 161-191, arXiv:1502.01365.
  12. Bonzom V., Gurau R., Rivasseau V., Random tensor models in the large $N$ limit: uncoloring the colored tensor models, Phys. Rev. D 85 (2012), 084037, 12 pages, arXiv:1202.3637.
  13. Brydges D.C., Kennedy T., Mayer expansions and the Hamilton-Jacobi equation, J. Statist. Phys. 48 (1987), 19-49.
  14. Carrozza S., Oriti D., Rivasseau V., Renormalization of a ${\rm SU}(2)$ tensorial group field theory in three dimensions, Comm. Math. Phys. 330 (2014), 581-637, arXiv:1303.6772.
  15. Carrozza S., Oriti D., Rivasseau V., Renormalization of tensorial group field theories: Abelian ${\rm U}(1)$ models in four dimensions, Comm. Math. Phys. 327 (2014), 603-641, arXiv:1207.6734.
  16. David F., Feldman J., Rivasseau V., On the large order behavior of $\phi^4_4$, Comm. Math. Phys. 116 (1988), 215-233.
  17. de Calan C., Rivasseau V., The perturbation series for $\Phi^{4}_{3}$ field theory is divergent, Comm. Math. Phys. 83 (1982), 77-82.
  18. Delepouve T., Gurau R., Rivasseau V., Universality and Borel summability of arbitrary quartic tensor models, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), 821-848, arXiv:1403.0170.
  19. Delepouve T., Rivasseau V., Constructive tensor field theory: the $T^4_3$ model, Comm. Math. Phys. 345 (2016), 477-506, arXiv:1412.5091.
  20. Di Francesco P., Ginsparg P., Zinn-Justin J., $2$D gravity and random matrices, Phys. Rep. 254 (1995), 1-133, hep-th/9306153.
  21. Eckmann J.P., Magnen J., Sénéor R., Decay properties and Borel summability for the Schwinger functions in $P(\phi)_{2}$ theories, Comm. Math. Phys. 39 (1975), 251-271.
  22. Feldman J., Magnen J., Rivasseau V., Sénéor R., Construction and Borel summability of infrared $\Phi^4_4$ by a phase space expansion, Comm. Math. Phys. 109 (1987), 437-480.
  23. Fröhlich J., Mardin A., Rivasseau V., Borel summability of the $1/N$ expansion for the $N$-vector [${\rm O}(N)$ nonlinear $\sigma $] models, Comm. Math. Phys. 86 (1982), 87-110.
  24. Gross M., Tensor models and simplicial quantum gravity in $>2$-D, Nuclear Phys. B Proc. Suppl. 25A (1992), 144-149.
  25. Grosse H., Wulkenhaar R., Renormalisation of $\phi^4$-theory on noncommutative ${\mathbb R}^4$ in the matrix base, Comm. Math. Phys. 256 (2005), 305-374, hep-th/0401128.
  26. Gurau R., The $1/N$ expansion of colored tensor models, Ann. Henri Poincaré 12 (2011), 829-847, arXiv:1011.2726.
  27. Gurau R., Colored group field theory, Comm. Math. Phys. 304 (2011), 69-93, arXiv:0907.2582.
  28. Gurau R., The complete $1/N$ expansion of colored tensor models in arbitrary dimension, Ann. Henri Poincaré 13 (2012), 399-423, arXiv:1102.5759.
  29. Gurau R., The $1/N$ expansion of tensor models beyond perturbation theory, Comm. Math. Phys. 330 (2014), 973-1019, arXiv:1304.2666.
  30. Gurau R., Universality for random tensors, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), 1474-1525, arXiv:1111.0519.
  31. 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.
  32. Gurau R., Rivasseau V., The multiscale loop vertex expansion, Ann. Henri Poincaré 16 (2015), 1869-1897, arXiv:1312.7226.
  33. Gurau R., Rivasseau V., Sfondrini A., Renormalization: an advanced overview, arXiv:1401.5003.
  34. Gurau R., Ryan J.P., Colored tensor models - a review, SIGMA 8 (2012), 020, 78 pages, arXiv:1109.4812.
  35. Gurau R.G., Krajewski T., Analyticity results for the cumulants in a random matrix model, Ann. Inst. Henri Poincaré D 2 (2015), 169-228, arXiv:1409.1705.
  36. Jaffe A., Divergence of perturbation theory for bosons, Comm. Math. Phys. 1 (1965), 127-149.
  37. Kruskal J.B., On the shortest spanning subtree of a graph and the traveling salesman problem, Proc. Amer. Math. Soc. 7 (1956), 48-50.
  38. Lahoche V., Constructive tensorial group field theory I: the $U(1)-T^4_3$ model, arXiv:1510.05050.
  39. Lahoche V., Constructive tensorial group field theory II: the $U(1)-T^4_4$ model, arXiv:1510.05051.
  40. Lahoche V., Oriti D., Renormalization of a tensorial field theory on the homogeneous space ${\rm SU}(2)/{\rm U}(1)$, arXiv:1506.08393.
  41. Lahoche V., Oriti D., Rivasseau V., Renormalization of an Abelian tensor group field theory: solution at leading order, J. High Energy Phys. 2015 (2015), no. 4, 095, 41 pages, arXiv:1501.02086.
  42. Lionni L., Rivasseau V., Note on the intermediate field representation of $\Phi^{2k}$ theory in zero dimension, arXiv:1601.02805.
  43. 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.
  44. Magnen J., Sénéor R., Phase space cell expansion and Borel summability for the Euclidean $\phi_{3}^{4}$ theory, Comm. Math. Phys. 56 (1977), 237-276.
  45. Nelson E., A quartic interaction in two dimensions, in Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965), M.I.T. Press, Cambridge, Mass., 1966, 69-73.
  46. Rivasseau V., Constructive matrix theory, J. High Energy Phys. 2007 (2007), no. 9, 008, 13 pages, arXiv:0706.1224.
  47. Rivasseau V., The tensor track, III, Fortschr. Phys. 62 (2014), 81-107, arXiv:1311.1461.
  48. Rivasseau V., Why are tensor field theories asymptotically free?, Europhys. Lett. 111 (2015), 60011, 6 pages, arXiv:1507.04190.
  49. Rivasseau V., Random tensors and quantum gravity, SIGMA 12 (2016), 069, 17 pages, arXiv:1603.07278.
  50. Rivasseau V., Tanasa A., Generalized constructive tree weights, J. Math. Phys. 55 (2014), 043509, 13 pages, arXiv:1310.2424.
  51. Rivasseau V., Vignes-Tourneret F., Constructive tensor field theory: the $T^4_4$ model, in preparation.
  52. Rivasseau V., Wang Z., Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension, J. Math. Phys. 51 (2010), 092304, 17 pages, arXiv:1003.1037.
  53. Rivasseau V., Wang Z., How to resum Feynman graphs, Ann. Henri Poincaré 15 (2014), 2069-2083, arXiv:1304.5913.
  54. Samary D.O., Beta functions of ${\rm U}(1)^d$ gauge invariant just renormalizable tensor models, Phys. Rev. D 88 (2013), 105003, 15 pages, arXiv:1303.7256.
  55. Samary D.O., Pérez-Sánchez C.I., Vignes-Tourneret F., Wulkenhaar R., Correlation functions of a just renormalizable tensorial group field theory: the melonic approximation, Classical Quantum Gravity 32 (2015), 175012, 18 pages, arXiv:1411.7213.
  56. Samary D.O., Vignes-Tourneret F., Just renormalizable TGFT's on ${\rm U}(1)^d$ with gauge invariance, Comm. Math. Phys. 329 (2014), 545-578, arXiv:1211.2618.
  57. Sasakura N., Tensor model for gravity and orientability of manifold, Modern Phys. Lett. A 6 (1991), 2613-2623.
  58. Sokal A.D., An improvement of Watson's theorem on Borel summability, J. Math. Phys. 21 (1980), 261-263.

Previous article  Next article   Contents of Volume 12 (2016)