Renormalizable Models in Rank $d\ge 2$ 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 $G_D=U(1{)}^D$ 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 ${(}_{\dim G_D}{\Phi }_d^k,a)$, 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 $k\ge 4$. 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 $d\ge 3$ 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. {\bf 256}, 305 (2004)].

52 pages, 21 figures, 9 tables

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• Why are tensor field theories asymptotically free?
• Symmetry breaking in tensor models
• Renormalization of a tensorial field theory on the homogeneous space SU(2)/U(1)
• Renormalizable Tensor Field Theories
• GEMs and amplitude bounds in the colored Boulatov model
• Intermediate Field Representation for Positive Matrix and Tensor Interactions
• The Multiscale Loop Vertex Expansion
• New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications