Scaling behaviour of three-dimensional group field theory

TL;DR

<p>We study the scaling behavior of three-dimensional group field theory by comparing Boulatov's original model with its positive regularization (the BFL model) under an ultraviolet cutoff. Using both perturbative bounds and a constructive cactus expansion, we establish that Boulatov scales as $\Lambda^{3/2}$ per vertex perturbatively, while the BFL regularization scales as $\Lambda^{3}$ per vertex, with the latter being perturbatively more divergent. The cactus expansion yields a convergent Borel sum for the BFL free energy with a Borel radius that scales as $\Lambda^{-3}$, thereby taming the sum over triangulations and providing a constructive route toward renormalization in group field theory. The results illuminate how constructive methods can control nonlocal, tensorial QFTs and set the stage for extending these techniques to higher-dimensional GFT models and EPR-FK spin-foam theories.</p>

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.

Figures (8)

• Figure 1: The covariance $C_{\phi}$, the Boulatov tetrahedral vertex $T[\phi]$, and the Freidel-Louapre pillow $P[\phi]$. The labels on the vertices match the ordering of the group elements in the integrand of $T[\phi]$ and $P[\phi]$.
• Figure 2: $(A_{p},B_{p})$-cut in a two-point graph: the dotted lines are the frontier lines, and the solid lines in $B_{p}$ represent the spanning tree $T_{p}$, the remaining lines (loop lines) being omitted. When the thinner line $l_{p+1}$ is deleted, $T_{p}$ splits into $R_{p}$ (bottom) and $S_{p}$ (top).
• Figure 3: The vertex operators. In the pillow case, the upper index labels the three different channels.
• Figure 4: The chains of graphs $G_{n}$ and $G'_{n}$, each with $n$ vertices.
• Figure 6: Slicing the BFL vertices with an intermediate field $\sigma$ over $\mathrm{SU}(2)^4$: the dashed lines are combined in the covariance $C_{\sigma}$.

Theorems & Definitions (12)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.1
• Lemma 3.2
• Theorem 5.1