3D spinfoam quantum gravity: matter as a phase of the group field theory

An effective field theory for matter coupled to three-dimensional quantum gravity was recently derived in the context of spinfoam models (Freidel and Livine 2006 Phys. Rev. Lett. 96 221301 (Preprint hep-th/0512113)). In this paper, we show how this relates to group field theories and generalized matrix models. In the first part, we realize that the effective field theory can be recast as a matrix model where couplings between matrices of different sizes can occur. In a second part, we provide a family of classical solutions to the three-dimensional group field theory. By studying perturbations around these solutions, we generate the dynamics of the effective field theory. We identify a particular case which leads to the action of Freidel and Livine (2006 Phys. Rev. Lett. 96 221301 ( Preprint hep-th/0512113)) for a massive field living in a flat non-commutative spacetime. The most general solutions lead to field theories with nonlinear redefinitions of the momentum which we propose to interpret as living on curved spacetimes. We conclude by discussing the possible extension to four-dimensional spinfoam models.