Renormalizable Models in Rank 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 or 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 . 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 , 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 . 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 tensor models proved renormalizable are asymptotically free in the UV. All matrix models with k = 4 have a vanishing beta-function at one-loop and, very likely, reproduce the same feature of the Grosse-Wulkenhaar model (Commun Math Phys 256:305, 2005).