Tensor models and group field theories: combinatorics, large $N$ and renormalization

Abstract

We provide a brief overview of tensor models and group field theories, focusing on their main common features. Both frameworks arose in the context of quantum gravity research, and can be understood as higher-dimensional generalizations of matrix models. We describe the common combinatorial structure underlying such models and review some of the key mathematical results that have been obtained in this area. Of central importance is the discovery of a new family of large $N$ expansions for random tensors. It has driven applications of tensor models to random geometry and non-perturbative local quantum field theory, and has also enabled the development of rigorous renormalization methods for group field theories and other non-local quantum field theories.