Reconstructing the metric in group field theory

We study a group field theory (GFT) for quantum gravity coupled to four massless scalar fields, using these matter fields to define a (relational) coordinate system. We exploit symmetries of the GFT action, in particular under shifts in the values of the scalar fields, to derive a set of classically conserved currents, and show that the same conservation laws hold exactly at the quantum level regardless of the choice of state. We propose a natural interpretation of the conserved currents which implies that the matter fields always satisfy the Klein--Gordon equation in GFT. We then observe that in our matter reference frame, the same conserved currents can be used to extract all components of an effective GFT spacetime metric. Finally, we apply this construction to the simple example of a spatially flat homogeneous and isotropic universe, where we derive an effective Friedmann equation directly from this metric. The Friedmann equation displays a bounce and a late-time limit equivalent to general relativity with a single scalar field. Our proposal goes substantially beyond the GFT literature in which only specific geometric quantities such as the total volume or volume perturbations could be defined, opening up the possibility to study more general geometries as emerging from GFT.