# Ward identities and combinatorics of rainbow tensor models

## TL;DR

This paper explores the renormalization group completion of non-Gaussian tensor models, introducing methods to relate tensor and matrix models, and deriving new sum rules for Gaussian correlators to simplify their analysis.

## Contribution

It presents novel methods connecting tensor and matrix models, along with new sum rules for Gaussian correlators, aiding the understanding of tensor model integrability.

## Key findings

- Derived new factorization formulas for Gaussian correlators.
- Established sum rules as solutions to finite linear systems.
- Connected tensor model calculations to matrix model techniques.

## Abstract

We discuss the notion of renormalization group (RG) completion of non-Gaussian Lagrangians and its treatment within the framework of Bogoliubov-Zimmermann theory in application to the matrix and tensor models. With the example of the simplest non-trivial RGB tensor theory (Aristotelian rainbow), we introduce a few methods, which allow one to connect calculations in the tensor models to those in the matrix models. As a byproduct, we obtain some new factorization formulas and sum rules for the Gaussian correlators in the Hermitian and complex matrix theories, square and rectangular. These sum rules describe correlators as solutions to finite linear systems, which are much simpler than the bilinear Hirota equations and the infinite Virasoro recursion. Search for such relations can be a way to solving the tensor models, where an explicit integrability is still obscure.