Uncolored Random Tensors, Melon Diagrams, and the SYK Models

TL;DR

This paper explores uncolored rank-3 tensor models with large N limits dominated by melon diagrams, connecting them to SYK models and discussing their potential gravity duals and conformal properties.

Contribution

It introduces uncolored tensor models with $O(N)^3$ symmetry, analyzes their large N limit, and relates them to SYK models without disorder, also discussing their gauge invariance and potential gravity duals.

Findings

Uncolored tensor models exhibit a large N limit dominated by melon diagrams.

The models are related to SYK models with similar large N behavior.

The conformal invariance of the two-point function is supported by Schwinger-Dyson equations and $4- ext{epsilon}$ expansion.

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 $4-\epsilon$ expansion.