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)|
|Journal||Proceedings of Science|
|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