Ward identities and combinatorics of rainbow tensor models

TL;DR

This work develops a Bogoliubov-Zimmermann-inspired framework for renormalization-group completeness in matrix and tensor models, focusing on the universal structure of Ward identities generated by keystone operators and their RG-descendants. It demonstrates how tree-descendant operators form a manageable RG-complete set in rainbow (RGB) tensor models, enabling finite linear relations among Gaussian averages and a potential path to solvability beyond the traditional Virasoro/topological-recursion toolkit. The paper lays out detailed analyses for Hermitian, complex, and rectangular matrix models, and extends these ideas to RG-closed tensor generalizations, culminating in the Aristotelian RGB model with explicit recursive relations and a roadmap for constructing Ward identities and RG-complete operator towers. The results suggest a promising direction toward tensor model solvability by leveraging RG structure, spectral-curve ideas, and W-representations, potentially paralleling matrix-model integrability in a higher-coloured quantum-mechanical setting.

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.