Correlation functions of $\mathrm{U}(N)$-tensor models and their Schwinger-Dyson equations

TL;DR

This paper derives exact Schwinger-Dyson equations for correlation functions in $ ext{U}(N)$-tensor models, providing a non-perturbative analysis and extending to holographic tensor models related to the SYK model.

Contribution

It develops a complete set of analytic Schwinger-Dyson equations for $ ext{U}(N)$-tensor models with boundary graph classification, including explicit forms for ranks 3 and 4.

Findings

Derived exact Schwinger-Dyson equations for correlation functions.

Classified correlation functions by boundary graphs.

Extended analysis to Gurau-Witten holographic tensor models.

Abstract

We analyse the correlation functions of $\mathrm{U}(N)$-tensor models (or complex tensor models), which turn out to be classified by boundary graphs, and use the Ward-Takahashi identity and the graph calculus developed in [Commun. Math. Phys. (2018) 358: 589] in order to derive the complete tower of exact, analytic Schwinger-Dyson equations for correlation functions with connected boundary graphs. We write them explicitly for ranks $D=3$ and $D=4$. Throughout, we follow a non-perturbative approach to Tensor (Group) Field Theories. We propose the extension of this program to the Gurau-Witten model, a holographic tensor model based on the Sachdev-Ye-Kitaev model (SYK model).