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The multi-orientable random tensor model, a review. (English) Zbl 1385.60011

Summary: After its introduction (initially within a group field theory framework) in [the author, J. Phys. A, Math. Theor. 45, No. 16, Article ID 165401, 19 p. (2012; Zbl 1246.81172)], the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the \(1/N\) expansion and of the large \(N\) limit (\(N\) being the size of the tensor), the combinatorial analysis of the various terms of this expansion and finally, the recent implementation of a double scaling limit.

MSC:

60B20 Random matrices (probabilistic aspects)
05C90 Applications of graph theory
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81T99 Quantum field theory; related classical field theories

Citations:

Zbl 1246.81172
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References:

[1] Ambj{\o}rn, Jan and Durhuus, Bergfinnur and J{\'o}nsson, Th{\'o}rdur, Three-dimensional simplicial quantum gravity and generalized matrix models, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 6, 12, 1133-1146, (1991) · Zbl 1020.83537
[2] Avohou, Remi C. and Rivasseau, Vincent and Tanasa, Adrian, Renormalization and {H}opf algebraic structure of the five-dimensional quartic tensor field theory, Journal of Physics. A. Mathematical and Theoretical, 48, 48, 485204, 20 pages, (2015) · Zbl 1331.83021
[3] Baratin, Aristide and Oriti, Daniele, Ten questions on Group Field Theory (and their tentative answers, Journal of Physics: Conference Series, 360, 012002, 10 pages, (2012)
[4] Ben Geloun, Joseph and Ramgoolam, Sanjaye, Counting tensor model observables and branched covers of the 2-sphere, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 1, 1, 77-138, (2014) · Zbl 1288.15031
[5] Ben Geloun, Joseph and Rivasseau, Vincent, A renormalizable 4-dimensional tensor field theory, Communications in Mathematical Physics, 318, 1, 69-109, (2013) · Zbl 1261.83016
[6] Bonzom, Valentin and Combes, Fr{\'e}d{\'e}ric, The calculation of expectation values in {G}aussian random tensor theory via meanders, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 1, 4, 443-485, (2014) · Zbl 1309.05099
[7] Bonzom, Valentin and Combes, Fr{\'e}d{\'e}ric, Tensor models from the viewpoint of matrix models: the cases of loop models on random surfaces and of the {G}aussian distribution, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 2, 1, 1-47, (2015) · Zbl 1310.05084
[8] Bouttier, J. and Di Francesco, P. and Guitter, E., Geodesic distance in planar graphs, Nuclear Physics. B, 663, 3, 535-567, (2003) · Zbl 1022.05022
[9] Br{\'e}zin, {\'E}. and Kazakov, V. A., Exactly solvable field theories of closed strings, Physics Letters. B, 236, 2, 144-150, (1990)
[10] Carrozza, Sylvain, Tensorial methods and renormalization in group field theories, Springer Theses, xvi+226, (2014), Springer, Cham · Zbl 1338.81004
[11] Carrozza, Sylvain, Discrete renormalization group for {\( \rm SU(2)\)} tensorial group field theory, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 2, 1, 49-112, (2015) · Zbl 1319.81068
[12] Dartois, St{\'e}phane and Gurau, Razvan and Rivasseau, Vincent, Double scaling in tensor models with a quartic interaction, Journal of High Energy Physics, 2013, 9, no. 9, 088, 33 pages, (2013) · Zbl 1342.83079
[13] Dartois, St{\'e}phane and Rivasseau, Vincent and Tanasa, Adrian, The {\(1/N\)} expansion of multi-orientable random tensor models, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 15, 5, 965-984, (2014) · Zbl 1288.81070
[14] Di Francesco, P. and Ginsparg, P. and Zinn-Justin, J., {\(2\)}{D} gravity and random matrices, Physics Reports. A Review Section of Physics Letters, 254, 1-2, 1-133, (1995)
[15] Douglas, Michael R. and Shenker, Stephen H., Strings in less than one dimension, Nuclear Physics. B, 335, 3, 635-654, (1990)
[16] Flajolet, Philippe and Sedgewick, Robert, Analytic combinatorics, xiv+810, (2009), Cambridge University Press, Cambridge · Zbl 1165.05001
[17] Freidel, L., Group field theory: an overview, International Journal of Theoretical Physics, 44, 10, 1769-1783, (2005) · Zbl 1100.83010
[18] Freidel, Laurent and Gurau, Razvan, Group field theory renormalization in the 3D case: power counting of divergences, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 80, 4, 044007, 20 pages, (2009)
[19] Gross, David J. and Migdal, Alexander A., Nonperturbative two-dimensional quantum gravity, Physical Review Letters, 64, 2, 127-130, (1990) · Zbl 1050.81610
[20] Gurau, Razvan, Colored group field theory, Communications in Mathematical Physics, 304, 1, 69-93, (2011) · Zbl 1214.81170
[21] Gurau, Razvan, The {\(1/N\)} expansion of colored tensor models, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 12, 5, 829-847, (2011) · Zbl 1218.81088
[22] Gurau, Razvan and Rivasseau, V., The \(1/N\) expansion of colored tensor models in arbitrary dimension, Europhysics Letters, 95, 5, 50004, 5 pages, (2011)
[23] Gurau, Razvan and Ryan, James P., Colored tensor models – a review, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 8, 020, 78 pages, (2012) · Zbl 1242.05094
[24] Gurau, Razvan and Ryan, James P., Melons are branched polymers, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 15, 11, 2085-2131, (2014) · Zbl 1303.83012
[25] Gurau, Razvan and Schaeffer, Gilles, Regular colored graphs of positive degree, (None) · Zbl 1352.05090
[26] Gurau, Razvan and Tanasa, Adrian and Youmans, Donald R., The double scaling limit of the multi-orientable tensor model, Europhysics Letters, 111, 2, 21002, 6 pages, (2015)
[27] Fusy, Eric and Tanasa, Adrian, Asymptotic expansion of the multi-orientable random tensor model, Electronic Journal of Combinatorics, 22, 1, 1.52, 30 pages, (2015) · Zbl 1310.81117
[28] Approaches to quantum gravity: toward a new understanding of space, time and matter, (2009), Cambridge University Press, Cambridge · Zbl 1168.83002
[29] Oriti, Daniele, The quantum geometry of tensorial group field theories, Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., 11, 379-384, (2013), World Sci. Publ., Hackensack, NJ · Zbl 1298.83058
[30] Raasakka, Matti and Tanasa, Adrian, Combinatorial {H}opf algebra for the {B}en {G}eloun–{R}ivasseau tensor field theory, S\'eminaire Lotharingien de Combinatoire, 70, B70d, 29 pages, (2013) · Zbl 1297.05258
[31] Raasakka, Matti and Tanasa, Adrian, Next-to-leading order in the large {\(N\)} expansion of the multi-orientable random tensor model, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 16, 5, 1267-1281, (2015) · Zbl 1314.83022
[32] Rivasseau, Vincent, Non-commutative renormalization, Quantum Spaces, Prog. Math. Phys., 53, 19-107, (2007), Birkh\"auser, Basel · Zbl 1139.81047
[33] Rivasseau, Vincent, The tensor track, {III}, Fortschritte der Physik. Progress of Physics, 62, 2, 81-107, (2014) · Zbl 1338.83085
[34] Sasakura, Naoki, Tensor model for gravity and orientability of manifold, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 6, 28, 2613-2623, (1991) · Zbl 1020.83542
[35] Schaeffer, Gilles, Bijective census and random generation of {E}ulerian planar maps with prescribed vertex degrees, Electronic Journal of Combinatorics, 4, 1, 20, 14 pages, (1997) · Zbl 0885.05076
[36] Tanasa, Adrian, Multi-orientable group field theory, Journal of Physics. A. Mathematical and Theoretical, 45, 16, 165401, 19 pages, (2012) · Zbl 1246.81172
[37] Tanasa, Adrian, Combinatorics of random tensor models, Proceedings of the Romanian Academy. Series A. Mathematics, Physics, Technical Sciences, Information Science, 13, 1, 27-31, (2012)
[38] Tanasa, Adrian, Tensor models, a quantum field theoretical particularization, Proceedings of the Romanian Academy. Series A. Mathematics, Physics, Technical Sciences, Information Science, 13, 3, 225-234, (2012)
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