Introduction and motivation

Group field theories (GFTs) were developed at first as a generalization of matrix models for 2d Quantum Gravity to 3 and 4 spacetime dimensions to produce a lattice formulation of topological theories. More recently, they have been developed further in the context of spin foam models for Quantum Gravity, as a tool to overcome the limitations of working with a fixed lattice in the non-topological case. In our opinion, however, GFTs should be seen as a fundamental formulation of Quantum Gravity and not just as an auxiliary tool. The bottom line of this perspective, here only tentatively outlined and still to be fully realized, hopefully, after much more work, can be summarized as follows: GFTs are quantum field theories of spacetime (as opposed to QFTs on spacetime), that describe the dynamics of both its topology and geometry in local, simplicial, covariant, algebraic terms, and that encompass ideas and insights from most of the other approaches to non-perturbative Quantum Gravity. We have just began to explore the structure of these models, but there is already some evidence, in our opinion, that in the GFT framework lies the potential for important developments.

The idea of defining a quantum field theory of geometry, i.e. a QFT on superspace (the space of 3-geometries) for given spatial topology, say S3, was already explored in the past. The context was then a global or “quantum cosmology” one.