Scaling behaviour of three-dimensional group field theory

Jacques Magnen (1) , Karim Noui (2) , Vincent Rivasseau (3) , Matteo Smerlak (4, 5)

1 CPHT - Centre de Physique Théorique [Palaiseau]
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 LPT - Laboratoire de Physique Théorique d'Orsay [Orsay]
4 CPT - Centre de Physique Théorique - UMR 6207
5 CPT - Centre de Physique Théorique - UMR 7332

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.

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Jacques Magnen, Karim Noui, Vincent Rivasseau, Matteo Smerlak. Scaling behaviour of three-dimensional group field theory. 2009. ⟨hal-00400149v1⟩