Melonic phase transition in group field theory

TL;DR

This paper analyzes the melonic sector in group field theory, deriving a combinatorial formula for amplitudes, establishing bounds, and demonstrating a phase transition relevant to quantum gravity models.

Contribution

It introduces a new combinatorial formula for melonic amplitudes and proves the existence of a phase transition in group field theories.

Findings

Derived a graph polynomial for melonic amplitudes

Established bounds indicating a phase transition

Focused on models relevant to quantum gravity

Abstract

Group field theories have recently been shown to admit a 1/N expansion dominated by so-called `melonic graphs', dual to triangulated spheres. In this note, we deepen the analysis of this melonic sector. We obtain a combinatorial formula for the melonic amplitudes in terms of a graph polynomial related to a higher dimensional generalization of the Kirchhoff tree-matrix theorem. Simple bounds on these amplitudes show the existence of a phase transition driven by melonic interaction processes. We restrict our study to the Boulatov-Ooguri models, which describe topological BF theories and are the basis for the construction of four dimensional models of quantum gravity.