A Renormalizable 4-Dimensional Tensor Field Theory

Abstract

41 pages, 9 figuresInternational audienceWe prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on $U(1)^4$ is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the $\phi^6$ rather than of the $\phi^4$ type, since two different $\phi^6$-type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent $(\int \phi^2)^2$ term, which can be interpreted as the generation of a scalar matter field out of pure gravity

Spin Foam Models with Finite Groups

“…Moreover, many techniques may be applied to these field theories that are idiosyncratic to quantum field theories (as opposed to quantum mechanics). Perhaps the most striking is the study of their renormalisation group properties [208][209][210][211][212][213][214][215][216][217][218], but also work has commenced on the study of mean field theory properties [219], matter coupling [220][221][222], various symmetry analyses [223,224], and instantonic field theory solutions [225][226][227][228] (including cosmological applications [229]). …”

Quantum gravity and renormalization: The tensor track

“…Renormalizable tensor models have been defined [20]. They provide random tensors with natural quantum field extensions equipped with their genuine renormalization group.…”

“…TFT can in particular include the study of renormalizable GFT models, which are similar to combinatorial models but with an additional gauge invariance. It should be relatively easy to define such renormalizable GFT models by slight modifications of [20], since combinatorial and GFT models share the same 1/N expansion [16]. However exclusive attention to GFT models and their richer geometric content could be premature until we have better understood the simpler combinatorial models and their phase transitions.…”

Functional Renormalization Group analysis of a Tensorial Group Field Theory on $\mathbb{R}^3$

“…Furthermore, GFT renormalization is also one of the two main strategies to define and study the renormalization of spin foam models, the other being through a generalised lattice gauge theory approach [22]. Most work in this direction has concerned a particular class of GFTs, called Tensorial Group Field Theories (TGFT's) [23][24][25][26][27][28][29][30][31][32][33], which incorporate recent advances in the statistical analysis of colored tensor models [34][35][36][37]. In particular, in TGFT framework, fields are endowed with tensorial transformation properties under the action of the group itself.…”

“…-We consider a rank-3 TGFT defined over R 3 endowed with a specific φ 4 interaction called "melonic" [34], shown in Fig.1. In general, rank-3 melonic interactions correspond to peculiar triangulations of the 3-sphere and are the most dominant objects in the large cut-off N limit [23,36,37] (in both simple tensor models and topological GFTs, but this result is expected to extend to a wider class of models). Written in momentum space 1 , the classical action of the model reads:…”