TY - JOUR

T1 - TASI lectures on large N tensor models

AU - Klebanov, Igor R.

AU - Popov, Fedor

AU - Tarnopolsky, Grigory

N1 - Funding Information:
These notes are an expanded version of the lectures presented by IRK at the TASI 2017 summer school in June 2017 in Boulder, Colorado, and at the Abdus Salam ICTP 2018 Spring School in March 2018 in Trieste, Italy. IRK is grateful to the TASI 2017 co-organizers Mirjam Cvetic, Tom DeGrand and Oliver DeWolfe for creating a wonderful environment at the school. He is also grateful to the organizers of the 2018 ICTP Spring School, especially Atish Dabholkar, for the invitation and hospitality. Many thanks to the students at both school for the many good questions and useful discussions. We thank our collaborators K. Bulycheva, S. Giombi, A. Milekhin, K. Pakrouski, and S. Prakash, with whom some of the results reviewed here were obtained. We are grateful to D. Gross, R. Gurau, J. Maldacena, V. Rosenhaus, D. Stanford, and E. Witten, for illuminating discussions. The work of IRK and FP was supported in part by the US NSF under Grant No. PHY-1620059. The work of GT was supported by the MURI grant W911NF-14-1-0003 from ARO and by DOE grant de-sc0007870.
Publisher Copyright:
© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

PY - 2017

Y1 - 2017

N2 - The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large N limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large N limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N), O(N)2, or O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large N limits. In the latter example we review how the dominance of melonic diagrams follows when a special “tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large N limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2 ×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large N tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

AB - The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large N limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large N limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N), O(N)2, or O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large N limits. In the latter example we review how the dominance of melonic diagrams follows when a special “tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large N limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2 ×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large N tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

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M3 - Conference article

AN - SCOPUS:85078479216

SN - 1824-8039

VL - 305

JO - Proceedings of Science

JF - Proceedings of Science

T2 - 2017 Theoretical Advanced Study Institute Summer School "Physics at the Fundamental Frontier", TASI 2017

Y2 - 4 June 2017 through 1 July 2017

ER -