Renormalizable Models in Rank {d \geq 2} Tensorial Group Field Theory

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 \({G_D=U(1)^D}\) or \({G_D= SU(2)^D}\) 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 \({p^{2a}, a\in (0,1]}\) . This study is tailored for models equipped with Laplacian dynamics on G _D (case a = 1) but also for more exotic nonlocal models in quantum topology (case 0 < a < 1). A generic model can be written \({(_{\dim G_D}\Phi^{k}_{d}, a)}\) , 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 \({k \geq 4}\) . In a second part of this work, we study the UV behavior of the models up to maximal valence of interaction k = 6. All rank \({d \geq 3}\) 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, 2005).