Contrasting SYK-like Models

TL;DR

This paper compares various SYK-like models, analyzing their symmetry properties, spectrum, and random matrix behavior, revealing insights into their structure, gauge invariance, and connections to quantum gravity and number theory.

Contribution

It provides a detailed comparison of SYK-like models, including gauge effects, singlet spectra, and random matrix classifications, advancing understanding of their large-N melonic behavior.

Findings

Ungauged tensor models can exhibit symmetry breaking.

Gauged models may have no singlets and be anomalous.

Certain tensor models share spectral properties with SYK and Riemann zeros.

Abstract

We contrast some aspects of various SYK-like models with large-$N$ melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even $N$ is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv:1706.05364, and reproduce the singlet states expected in $N=2$. In the second part of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of $N$. In an appendix, we contrast the (expected) spectra of AdS$_2$ quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.