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

SIGMA 12 (2016), 068, 36 pages      arXiv:1603.00172      https://doi.org/10.3842/SIGMA.2016.068
Contribution to the Special Issue on Tensor Models, Formalism and Applications

Exact Renormalisation Group Equations and Loop Equations for Tensor Models

Thomas Krajewski a and Reiko Toriumi b
a) Aix Marseille Université, Université de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille, France
b) Department of Physics, University of California Berkeley, USA

Received February 29, 2016, in final form July 05, 2016; Published online July 14, 2016

Abstract
In this paper, we review some general formulations of exact renormalisation group equations and loop equations for tensor models and tensorial group field theories. We illustrate the use of these equations in the derivation of the leading order expectation values of observables in tensor models. Furthermore, we use the exact renormalisation group equations to establish a suitable scaling dimension for interactions in Abelian tensorial group field theories with a closure constraint. We also present analogues of the loop equations for tensor models.

Key words: tensor models; group field theory; large $N$ limit; exact renormalisation equation.

pdf (730 kb)   tex (194 kb)

References

  1. Ambjørn J., Durhuus B., Jónsson T., Three-dimensional simplicial quantum gravity and generalized matrix models, Modern Phys. Lett. A 6 (1991), 1133-1146.
  2. Ambjørn J., Jurkiewicz J., Makeenko Y.M., Multiloop correlators for two-dimensional quantum gravity, Phys. Lett. B 251 (1990), 517-524.
  3. Bagnuls C., Bervillier C., Exact renormalization group equations: an introductory review, Phys. Rep. 348 (2001), 91-157, hep-th/0002034.
  4. Ben Geloun J., Martini R., Oriti D., Functional renormalisation group analysis of tensorial group field theories on $\mathbb{R}^d$, arXiv:1601.08211.
  5. Ben Geloun J., Rivasseau V., A renormalizable 4-dimensional tensor field theory, Comm. Math. Phys. 318 (2013), 69-109, arXiv:1111.4997.
  6. Benedetti D., Ben Geloun J., Oriti D., Functional renormalisation group approach for tensorial group field theory: a rank-3 model, J. High Energy Phys. 2015 (2015), no. 3, 084, 40 pages, arXiv:1411.3180.
  7. Benedetti D., Lahoche V., Functional renormalization group approach for tensorial group field theory: a rank-6 model with closure constraint, Classical Quantum Gravity 33 (2016), 095003, 35 pages, arXiv:1508.06384.
  8. Bonzom V., Revisiting random tensor models at large $N$ via the Schwinger-Dyson equations, J. High Energy Phys. 2013 (2013), no. 3, 160, 25 pages, arXiv:1208.6216.
  9. 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.
  10. Bonzom V., Gurau R., Ryan J.P., Tanasa A., The double scaling limit of random tensor models, J. High Energy Phys. 2014 (2014), no. 9, 051, 49 pages, arXiv:1404.7517.
  11. Braun J., Gies H., Scherer D.D., Asymptotic safety: a simple example, Phys. Rev. D 83 (2011), 085012, 15 pages, arXiv:1011.1456.
  12. Carrozza S., Tensorial methods and renormalization in group field theories, Springer Theses, Springer, Cham, 2014, arXiv:1310.3736.
  13. Carrozza S., Discrete renormalization group for ${\rm SU}(2)$ tensorial group field theory, Ann. Inst. Henri Poincaré D 2 (2015), 49-112, arXiv:1407.4615.
  14. Carrozza S., Group field theory in dimension $4-\varepsilon$, Phys. Rev. D 91 (2015), 065023, 10 pages, arXiv:1411.5385.
  15. Carrozza S., Flowing in group field theory space: a review, arXiv:1603.01902.
  16. 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.
  17. 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.
  18. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), 249-273, hep-th/9912092.
  19. Dartois S., Gurau R., Rivasseau V., Double scaling in tensor models with a quartic interaction, J. High Energy Phys. 2013 (2013), no. 9, 088, 33 pages, arXiv:1307.5281.
  20. Eichhorn A., Koslowski T., Continuum limit in matrix models for quantum gravity from the functional renormalization group, Phys. Rev. D 88 (2013), 084016, 15 pages, arXiv:1309.1690.
  21. Eichhorn A., Koslowski T., Towards phase transitions between discrete and continuum quantum spacetime from the renormalization group, Phys. Rev. D 90 (2014), 104039, 14 pages, arXiv:1408.4127.
  22. Gurau R., A diagrammatic equation for oriented planar graphs, Nuclear Phys. B 839 (2010), 580-603, arXiv:1003.2187.
  23. Gurau R., Colored group field theory, Comm. Math. Phys. 304 (2011), 69-93, arXiv:0907.2582.
  24. Gurau R., A generalization of the Virasoro algebra to arbitrary dimensions, Nuclear Phys. B 852 (2011), 592-614, arXiv:1105.6072.
  25. Gurau R., The complete $1/N$ expansion of colored tensor models in arbitrary dimension, Ann. Henri Poincaré 13 (2012), 399-423, arXiv:1102.5759.
  26. Gurau R., The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders, Nuclear Phys. B 865 (2012), 133-147, arXiv:1203.4965.
  27. Gurau R., The $1/N$ expansion of tensor models beyond perturbation theory, Comm. Math. Phys. 330 (2014), 973-1019, arXiv:1304.2666.
  28. Gurau R., Universality for random tensors, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), 1474-1525, arXiv:1111.0519.
  29. Gurau R., Rivasseau V., Sfondrini A., Renormalization: an advanced overview, arXiv:1401.5003.
  30. Krajewski T., Schwinger-Dyson equations in group field theories of quantum gravity, in Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., Vol. 11, World Sci. Publ., Hackensack, NJ, 2013, 373-378, arXiv:1211.1244.
  31. Krajewski T., Toriumi R., Polchinski's equation for group field theory, Fortschr. Phys. 62 (2014), 855-862.
  32. Krajewski T., Toriumi R., Polchinski's exact renormalisation group for tensorial theories: Gaussian universality and power counting, arXiv:1511.09084.
  33. 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.
  34. Oriti D., Group field theory and loop quantum gravity, arXiv:1408.7112.
  35. Polchinski J., Renormalization and effective Lagrangians, Nuclear Phys. B 231 (1984), 269-295.
  36. Polonyi J., Lectures on the functional renormalization group method, Cent. Eur. J. Phys. 1 (2003), 1-71, hep-th/0110026.
  37. Rivasseau V., From perturbative to constructive renormalization, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1991.
  38. Rivasseau V., The tensor track, III, Fortschr. Phys. 62 (2014), 81-107, arXiv:1311.1461.
  39. Rosten O.J., Fundamentals of the exact renormalization group, Phys. Rep. 511 (2012), 177-272, arXiv:1003.1366.
  40. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  41. 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.
  42. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
  43. Zinn-Justin J., Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys., Vol. 113, Oxford University Press, 2002.

Previous article  Next article   Contents of Volume 12 (2016)