Scaling behavior of three-dimensional group field theory

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 an ‘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 triangulations’ in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory.