A New Large N Expansion for General Matrix-Tensor Models

TL;DR

This work introduces a novel large-$N$ and large-$D$ expansion for general $ ext{O}(N)^{R}$ or $ ext{U}(N)^{R}$ invariant tensor models by enhancing the scaling of coupling constants. The expansion is governed by a graph index $ ext{ind}_{0}$ rather than Gurau’s degree, with leading graphs termed generalized melons and, in MST-prime cases, the prime-complete generalized melons (PCGMs) that capture planar-like physics within a tensor framework. A key result is that when the interaction is the complete vertex of order $R+1$ and $R$ is prime, the leading sector can be fully classified, showing a precise correspondence with mirror-melon constructions and enabling explicit SYK-like quantum-model realizations (Majorana and Dirac). The analysis reveals a noncommuting large-$N$ and large-$D$ limit, an $1/D^{1/(r+1)}$ expansion governed by the index, and a natural embedding of matrix-model diagrams within tensor models, offering a robust route to studying planar physics and holographic-inspired dynamics in higher dimensions. The paper also outlines open problems, including boundary-graph extensions, subleading corrections, and broader MST-interaction classifications for further exploration of matrix-tensor universality and SYK-like behavior.

Abstract

We define a new large $N$ limit for general $\text{O}(N)^{R}$ or $\text{U}(N)^{R}$ invariant tensor models, based on an enhanced large $N$ scaling of the coupling constants. The resulting large $N$ expansion is organized in terms of a half-integer associated with Feynman graphs that we call the index. This index has a natural interpretation in terms of the many matrix models embedded in the tensor model. Our new scaling can be shown to be optimal for a wide class of non-melonic interactions, which includes all the maximally single-trace terms. Our construction allows to define a new large $D$ expansion of the sum over diagrams of fixed genus in matrix models with an additional $\text{O}(D)^{r}$ global symmetry. When the interaction is the complete vertex of order $R+1$, we identify in detail the leading order graphs for $R$ a prime number. This slightly surprising condition is equivalent to the complete interaction being maximally single-trace.

Figures (20)

• Figure 1: Construction of a new bubble $\mathcal{B}$ from bubbles $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, such that $\deg \mathcal{B} = \deg \mathcal{B}_{1}+\deg\mathcal{B}_{2}$ and $\mathop{\mathrm{ind}}\nolimits_{0}\mathcal{B} = \mathop{\mathrm{ind}}\nolimits_{0}\mathcal{B}_{1}+\mathop{\mathrm{ind}}\nolimits_{0}\mathcal{B}_{2}$ (Proposition \ref{['addprop']}). Dashed lines represent edges of color 0.
• Figure 2: Bubble $\mathcal{B}'$ insertion (from left to right) and contraction (from right to left) on an edge of color 0 in a bubble $\mathcal{B}$. According to Proposition \ref{['addprop']}, the insertion (contraction) increases (decreases) the degree and the index of $\mathcal{B}$ by $\deg\mathcal{B}'$ and $\mathop{\mathrm{ind}}\nolimits_{0}\mathcal{B}'$ respectively.
• Figure 3: A mirror melon built from two identical bubbles.
• Figure 4: Edge-coloring for the complete graph $K_6$. Left: rule for the coloring of the edges of a particular color, here green. Center: full edge-coloring and standard vertex labeling, here $1=\text{green}$, $2=\text{red}$, $3=\text{blue}$, $4=\text{orange}$ and $5=\text{purple}$. Right: the equivalent $(\text{green},\text{red})$-polygonal representation in the shape of a six-sided polygon whose boundary is the face of colors green and red. This polygonal representation is natural when $R$ is prime.
• Figure 5: MST coloring of $K_{10}$.

Theorems & Definitions (26)

• Lemma 2.1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.2
• Proposition 2.4
• Lemma 2.3
• Proposition 2.5
• Proposition 3.1
• Lemma 3.1