Abstract
In this paper we express some simple random tensor models in a Givental-like fashion i.e. as differential operators acting on a product of generic 1-Hermitian matrix models. Finally we derive Hirota’s equations for these tensor models. Our decomposition is a first step towards integrability of such models.
Article PDF
Similar content being viewed by others
References
J. Ambjørn, B. Durhuus and T. Jonsson, Three-dimensional simplicial quantum gravity and generalized matrix models, Mod. Phys. Lett. A 6 (1991) 1133 [INSPIRE].
N. Sasakura, Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A 6 (1991) 2613 [INSPIRE].
M.L. Mehta, Random Matrices, Pure and Applied Mathematics 142 (2004), Elsevier/Academic Press, Amsterdam, Netherlands.
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D Gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
A.S. Alexandrov, A. Mironov and A. Morozov, M-theory of matrix models, Theor. Math. Phys. 150 (2007) 153 [hep-th/0605171] [INSPIRE].
G. Borot, Formal multidimensional integrals, stuffed maps and topological recursion, arXiv:1307.4957 [INSPIRE].
S. Garoufalidis and M. Mariño, On Chern-Simons matrix models, math/0601390 [INSPIRE].
P. Di Francesco, 2D Quantum Gravity, Matrix Models and Graph Combinatorics, Nato Sci. Ser. II 221 (2006) 33 [math-ph/0406013] [INSPIRE].
S. Dartois, R. Gurau and V. Rivasseau, Double Scaling in Tensor Models with a Quartic Interaction, JHEP 09 (2013) 088 [arXiv:1307.5281] [INSPIRE].
R. Gurau and G. Schaeffer, Regular colored graphs of positive degree, arXiv:1307.5279.
V. Bonzom, R. Gurau, J.P. Ryan and A. Tanasa, The double scaling limit of random tensor models, JHEP 09 (2014) 051 [arXiv:1404.7517] [INSPIRE].
C. Itzykson and J.B. Zuber, The Planar Approximation. 2, J. Math. Phys. 21 (1980) 411 [INSPIRE].
J. Alfaro and I.K. Kostov, Generalized Hirota equations in models of 2D quantum gravity, hep-th/9604011 [INSPIRE].
A. Givental, Semisimple Frobenius structures at higher genus, Int. Math. Res. Not. 2001 (2001) 1265 [math/0008067] [INSPIRE].
A.B. Givental, Gromov-Witten invariants and quantization of quadratic hamiltonians, Moscow Math. J. 1 (2001) 551 [math/0108100] [INSPIRE].
B. Eynard, Topological expansion for the 1-Hermitian matrix model correlation functions, JHEP 11 (2004) 031 [hep-th/0407261] [INSPIRE].
B. Eynard and N. Orantin, Algebraic methods in random matrices and enumerative geometry, arXiv:0811.3531 [INSPIRE].
N. Orantin, Symplectic invariants, Virasoro constraints and Givental decomposition, arXiv:0808.0635 [INSPIRE].
P. Dunin-Barkowski, N. Orantin, S. Shadrin and L. Spitz, Identification of the Givental formula with the spectral curve topological recursion procedure, Commun. Math. Phys. 328 (2014) 669 [arXiv:1211.4021] [INSPIRE].
T. Delepouve, R. Gurau and V. Rivasseau, Universality and Borel Summability of Arbitrary Quartic Tensor Models, arXiv:1403.0170 [INSPIRE].
D. Oriti, Group field theory as the 2nd quantization of Loop Quantum Gravity, arXiv:1310.7786 [INSPIRE].
C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge, U.K. (2006).
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
W.T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962) 21.
V. Rivasseau, The Tensor Track, III, Fortsch. Phys. 62 (2014) 81 [arXiv:1311.1461] [INSPIRE].
R. Gurau, Colored Group Field Theory, Commun. Math. Phys. 304 (2011) 69 [arXiv:0907.2582] [INSPIRE].
R. Gurau and J.P. Ryan, Colored Tensor Models — a review, SIGMA 8 (2012) 020 [arXiv:1109.4812] [INSPIRE].
R. Gurau, Lost in Translation: Topological Singularities in Group Field Theory, Class. Quant. Grav. 27 (2010) 235023 [arXiv:1006.0714] [INSPIRE].
M. Pezzana, Sulla struttura topologica delle varietà compatte, Atti Sem. Mat. Fis. Univ. Modena 23 (1974) 269.
M. Ferri and C. Gagliardi, Crystallisation moves, Pacific J. Math. 100 (1982) 85.
S. Lins, Gems, Computers, and Attractors for 3-manifolds, World Scientific, (1995).
R. Gurau, The complete 1/N expansion of colored tensor models in arbitrary dimension, Annales Henri Poincaré 13 (2012) 399 [arXiv:1102.5759] [INSPIRE].
J.P. Ryan, Tensor models and embedded Riemann surfaces, Phys. Rev. D 85 (2012) 024010 [arXiv:1104.5471] [INSPIRE].
V. Bonzom, R. Gurau, A. Riello and V. Rivasseau, Critical behavior of colored tensor models in the large-N limit, Nucl. Phys. B 853 (2011) 174 [arXiv:1105.3122] [INSPIRE].
V. Bonzom, R. Gurau and V. Rivasseau, Random tensor models in the large-N limit: Uncoloring the colored tensor models, Phys. Rev. D 85 (2012) 084037 [arXiv:1202.3637] [INSPIRE].
J. Ben Geloun and V. Rivasseau, A Renormalizable 4-Dimensional Tensor Field Theory, Commun. Math. Phys. 318 (2013) 69 [arXiv:1111.4997] [INSPIRE].
S. Carrozza, Discrete Renormalization Group for SU(2) Tensorial Group Field Theory, arXiv:1407.4615 [INSPIRE].
V. Bonzom, R. Gurau and V. Rivasseau, The Ising Model on Random Lattices in Arbitrary Dimensions, Phys. Lett. B 711 (2012) 88 [arXiv:1108.6269] [INSPIRE].
V. Bonzom, R. Gurau and M. Smerlak, Universality in p-spin glasses with correlated disorder, J. Stat. Mech. (2013) L02003 [arXiv:1206.5539].
R. Gurau, The 1/N Expansion of Tensor Models Beyond Perturbation Theory, Commun. Math. Phys. 330 (2014) 973 [arXiv:1304.2666] [INSPIRE].
T. Krajewski, Group field theories, PoS(QGQGS 2011)005 [arXiv:1210.6257] [INSPIRE].
S. Carrozza, Tensorial methods and renormalization in Group Field Theories, arXiv:1310.3736.
D. Oriti, Disappearance and emergence of space and time in quantum gravity, Stud. Hist. Philos. Mod. Phys. 46 (2014) 186 [arXiv:1302.2849] [INSPIRE].
W. Kaminski, D. Oriti and J.P. Ryan, Towards a double-scaling limit for tensor models: probing sub-dominant orders, New J. Phys. 16 (2014) 063048 [arXiv:1304.6934] [INSPIRE].
V.A. Nguyen, S. Dartois and B. Eynard, An analysis of the intermediate field theory of T 4 tensor model, JHEP 01 (2015) 013 [arXiv:1409.5751] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1409.5621
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Dartois, S. A Givental-like formula and bilinear identities for tensor models. J. High Energ. Phys. 2015, 129 (2015). https://doi.org/10.1007/JHEP08(2015)129
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2015)129