### Abstract

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.

Original language | English (US) |
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Journal | Proceedings of Science |

Volume | 305 |

State | Published - Jan 1 2017 |

Event | 2017 Theoretical Advanced Study Institute Summer School "Physics at the Fundamental Frontier", TASI 2017 - Boulder, United States Duration: Jun 4 2017 → Jul 1 2017 |

### All Science Journal Classification (ASJC) codes

- General

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## Cite this

*Proceedings of Science*,

*305*.