Blobbed topological recursion for the quartic melonic tensor model

TL;DR

This paper demonstrates that blobbed topological recursion applies to the quartic melonic tensor model, extending its use beyond matrix models to certain tensor models with disconnected spectral curves, and provides a method to evaluate tensor observables.

Contribution

It shows that blobbed topological recursion applies to a class of tensor models, specifically the quartic melonic case, via a multi-matrix model transformation.

Findings

Blobbed topological recursion applies to the quartic melonic tensor model.

The spectral curve of the model is disconnected, comprising several Gaussian Unitary Ensemble curves.

A method to evaluate tensor observables using the recursion-derived correlation functions is proposed.

Abstract

Random tensor models are generalizations of random matrix models which admit $1/N$ expansions. In this article we show that the topological recursion, a modern approach to matrix models which solves the loop equations at all orders, is also satisfied in some tensor models. While it is obvious in some tensor models which are matrix models in disguise, it is far from clear that it can be applied to others. Here we focus on melonic interactions for which the models are best understood, and further restrict to the quartic case. Then Hubbard-Stratonovich transformation maps the tensor model to a multi-matrix model with multi-trace interactions. We study this matrix model and show that after substracting the leading order, it satisfies the blobbed topological recursion. It is a new extension of the topological recursion, recently introduced by Borot and further studied by Borot and Shadrin. Here it applies straightforwardly, yet with a novelty as our model displays a disconnected spectral curve, which is the union of several spectral curves of the Gaussian Unitary Ensemble. Finally, we propose a way to evaluate expectations of tensorial observables using the correlation functions computed from the blobbed topological recursion.