Scaling behaviour of three-dimensional group field theory

TL;DR

This paper rigorously analyzes the scaling behavior of a three-dimensional group field theory model, demonstrating its divergence properties and constructing its Borel sum to facilitate the renormalization process in quantum gravity.

Contribution

It provides the first rigorous bounds on Feynman amplitudes for the BFL model and constructs its Borel sum using cactus expansions, advancing the understanding of renormalization in group field theory.

Findings

BFL model is more divergent than Boulatov's original model.

Established optimal bounds on large order Feynman amplitudes.

Constructed the Borel sum of the perturbative series for the model.

Abstract

Group field theory is a generalization of matrix models, with triangulated pseudomanifolds as Feynman diagrams and state sum invariants as Feynman amplitudes. In this paper, we consider Boulatov's three-dimensional model and its Freidel-Louapre positive regularization (hereafter the BFL model) with a ?ultraviolet' cutoff, and study rigorously their scaling behavior in the large cutoff limit. We prove an optimal bound on large order Feynman amplitudes, which shows that the BFL model is perturbatively more divergent than the former. We then upgrade this result to the constructive level, using, in a self-contained way, the modern tools of constructive field theory: we construct the Borel sum of the BFL perturbative series via a convergent ?cactus' expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our method shows how the ?sum over trian- gulations' in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory.