TASI Lectures on Large $N$ Tensor Models

TL;DR

The notes survey three canonical large $N$ limits—vector, matrix, and tensor—and expose how diagrammatics and solvability evolve across these classes, culminating in melonic dominance for tensor theories with a tetrahedral interaction. They connect tensor models to SYK-like dynamics via Schwinger–Dyson equations, and present both fermionic and complex/bipartite tensor variants that exhibit near-conformal infrared behavior and a rich spectrum of multi-particle operators. The work also discusses the parallels and distinctions with AdS/CFT, higher-spin dualities, and potential gravity interpretations, including the branched-polymer phases and vanishing Hagedorn temperature in the tensor setups. Finally, it surveys bosonic tensor theories, highlighting stability challenges and potential fixed points in certain regimes, thus outlining a broad landscape of solvable large $N$ theories with potential holographic connections.

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 $SU(N)^2\times O(N)\times U(1)$ 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.

Figures (31)

• Figure 1: All the melonic vacuum diagrams up to order $g^8$.
• Figure 2: A snail vs. a (water)melon.
• Figure 3: A configuration of spins in an $O(2)$ model on a two-dimensional lattice.
• Figure 4: A stack of $N$ D3-branes and the curved background it creates.
• Figure 5: The three vacuum diagrams up to order $g^2$: "figure eight," "melon," and "triple bubble."