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

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We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hep-th]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop γ-and β-functions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave function renormalization is asymptotically free in the UV. Pacs numbers: 11.10.Gh, 04.60.-m We recall some definitions (see [41][54][45]):

Abstract:We set up the Functional Renormalisation Group formalism for Tensorial Group Field Theory in full generality. We then apply it to a rank-3 model over U(1) 3 , endowed with a kinetic term linear in the momenta and with nonlocal interactions. The system of FRG equations turns out to be non-autonomous in the RG flow parameter. This feature is explained by the existence of a hidden scale, the radius of the group manifold. We investigate in detail the opposite regimes of large cut-off (UV) and small cut-off (IR) of the FRG equations, where the system becomes autonomous, and we find, in both case, Gaussian and non-Gaussian fixed points. We derive and interpret the critical exponents and flow diagrams associated with these fixed points, and discuss how the UV and IR regimes are matched. Finally, we discuss the evidence for a phase transition from a symmetric phase to a broken or condensed phase, from an RG perspective, finding that this seems to exist only in the approximate regime of very large radius of the group manifold, as to be expected for systems on compact manifolds.

Classes of renormalizable models in the Tensorial Group Field Theory framework are investigated. The rank d tensor fields are defined over d copies of a group manifold GD = U (1) D or GD = SU (2) D with no symmetry and no gauge invariance assumed on the fields. In particular, we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form p 2a , a ∈]0, 1]. This study is tailored for models equipped with Laplacian dynamics on GD (case a = 1) but also for more exotic nonlocal models in quantum topology (case 0 < a < 1). A generic model can be written ( dim G D Φ k d , a), where k is the maximal valence of its interactions. Using a multi-scale analysis for the generic situation, we identify several classes of renormalizable actions including matrix model actions. In this specific instance, we find a tower of renormalizable matrix models parametrized by k ≥ 4. In a second part of this work, we focus on the UV behavior of the models up to maximal valence of interaction k = 6. All rank d ≥ 3 tensor models proved renormalizable are asymptotically free in the UV. All matrix models with k = 4 have a vanishing β-function at one-loop and, very likely, reproduce the same feature of the Grosse-Wulkenhaar model [Commun. Math. Phys. 256, 305 (2004)].

The Boulatov-Ooguri tensor model generates a sum over spacetime topologies for the D-dimensional BF theory. We study here the quantum corrections to the propagator of the theory. In particular, we find that the radiative corrections at the second order in the coupling constant yield a mass renormalization. They also exhibit a divergence which cannot be balanced with a counter-term in the initial action, and which usually corresponds to the wave-function renormalization.

Lattice gauge theories of permutation groups with a simple topological action (henceforth permutation-TFTs) have recently found several applications in the combinatorics of quantum field theories (QFTs). They have been used to solve counting problems of Feynman graphs in QFTs and ribbon graphs of large N , often revealing inter-relations between different counting problems. In another recent development, tensor theories generalizing matrix theories have been actively developed as models of random geometry in three or more dimensions. Here, we apply permutation-TFT methods to count gauge invariants for tensor models (colored as well as non-colored), exhibiting a relationship with counting problems of branched covers of the 2-sphere, where the rank d of the tensor gets related to a number of branch points. We give explicit generating functions for the relevant counting and describe algorithms for the enumeration of the invariants. As well as the classic count of Hurwitz equivalence classes of branched covers with fixed branch points, collecting these under an equivalence of permuting the branch points is relevant to the color-symmetrized tensor invariant counting. We also apply the permutation-TFT methods to obtain some formulae for correlators of the tensor model invariants.

A recent rank 4 tensor field model generating 4D simplicial manifolds has been proved to be renormalizable at all orders of perturbation theory [arXiv:1111.4997 [hep-th]]. The model is built out of φ 6 (φ(1) ) interactions and an anomalous term (φ 4 (2) ). The β-functions of this model are evaluated at two and four loops. We find that the model is asymptotically free in the UV for both the main φ 6(1/2) interactions whereas it is safe in the φ 4 (1) sector. The remaining anomalous term turns out to possess a Landau ghost.