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

Abstract

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)].

Parametric representation of rank d tensorial group field theory: Abelian models with kinetic term ∑sps+μ

“…In this section, we provide a lightning review of the basic definitions of objects in the above references. Most of our illustrations focus on the rank 3 situation which is already a nontrivial case mostly discussed in our following sections; we invite the reader to more illustrations in [27].…”

“…(i) With the exact power of momentum in the propagator, there exist rank d models that are renormalizable among which we have a rank 3 tensor model and several matrix models [27]. They will be the prototype models on which our following dimensional regularization procedure will be applied.…”

“…-In rank d ≥ 3: the interactions of the models considered are effective interactions obtained after integrating d colors in the rank d + 1 colored tensor model [54] as discussed in Section 2.1 (for a complete discussion, we refer to [27]). The above field ϕ is nothing but the remaining field ϕ 0 = ϕ.…”

“…It is only recent that the notion of large N expansion was extended to tensor models [43]- [45] (for however the class of colored models [46]- [48]). From this point, important progresses have been unlocked [49]- [63] and a renormalization program for tensor models uncovered (for a review, see [8,27]). …”

“…Section 2 covers definitions and terminologies associated with important graph concepts used throughout the text, for the rank d ≥ 3 colored tensor graphs and the rank d = 2 ribbon graphs. For notations closer to our discussion, we refer to the survey given in Section II in [27]. Section 3 presents the models including rank d = 2 matrix and d ≥ 3 tensor models that we shall study.…”