A complete theory of quantum gravity remains elusive despite many years of research and the pursuit of many different approaches. The group field theory (GFT) approach to quantum gravity is a generalization of matrix models for two-dimensional quantum gravity, incorporating ideas from canonical loop quantum gravity and spin foams. This review introduces the basic concepts in the GFT approach to quantum gravity and its relationship to other quantum gravity programs. The essential ingredients of the theory (namely a quantum field theory framework, the use of group structures, and combinatorial non-locality) are motivated, together with the central idea of “third” quantization to allow space-time topology to be dynamical. In this paper the space-time geometry is Euclidean, although the theory can also be developed for Lorentzian geometry. The GFT model for three-dimensional Riemannian geometry is discussed in some detail, firstly outlining the kinematic structures, then the classical dynamics and finally the quantum dynamics by a perturbative expansion of the partition function in Feynman diagrams. The development of a GFT formulation of four-dimensional quantum gravity is covered more briefly, as this theory is less well-developed than the three-dimensional case. The article closes with a selection of current research directions in the field and some important open problems. For example, in common with other discrete approaches to quantum gravity, it is currently not known how to derive effective matter field theories from the GFT.
For the entire collection see [Zbl 1247.83006].