Uncolored Random Tensors, Melon Diagrams, and the SYK Models

TL;DR

This work analyzes uncolored rank-3 tensor theories with O(N)^3 symmetry and demonstrates a melonic large-N limit analogous to the SYK model without disorder. It explores real and complex fermionic, and bosonic variants, derives Schwinger-Dyson equations and ladder kernels, and shows equivalence to SYK in the large-N limit while revealing a rich spectrum of gauge-invariant operators that would shape any gravity dual. The authors compute two-point functions, ladder resummations, and scaling dimensions of two-particle operators, including a gravity-like h=2 mode in several cases, and they extend the analysis to a 4−ε bosonic theory with a controlled fixed point. The results suggest a highly nontrivial holographic dual and motivate future work on higher-rank generalizations, supersymmetric extensions, and numerical investigations of chaos in tensor models.

Abstract

Certain models with rank-$3$ tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large $N$ limit, where $g^2 N^3$ is held fixed. In this limit the perturbative expansion in the quartic coupling constant, $g$, is dominated by a special class of "melon" diagrams. We study "uncolored" models of this type, which contain a single copy of real rank-$3$ tensor. Its three indexes are distinguishable; therefore, the models possess $O(N)^3$ symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large $N$ limit dominated by the melon diagrams. The quantum mechanics of a real anti-commuting tensor therefore has a similar large $N$ limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the $O(N)^3$ symmetry in our quantum mechanical model removes the non-singlet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anti-commuting tensor, which has $U(N)^2\times O(N)$ symmetry and argue that it is equivalent in the large $N$ limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension $d$. While the quartic interaction is not positive definite, we construct the large $N$ Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4-ε$ expansion.

Figures (16)

• Figure 1: Some melonic contributions to the free energy.
• Figure 2: The resolved propagator $\langle \phi^{abc} \phi^{a'b'c'} \rangle = \delta^{aa'}\delta^{bb'}\delta^{cc'}$.
• Figure 3: Three equivalent ways to represent the resolved vertex.
• Figure 4: The standard matrix model vertex obtained after "forgetting" the middle lines.
• Figure 5: A melonic second-order diagram and all its fat subgraphs.