A group field theory for 3d quantum gravity coupled to a scalar field

TL;DR

This work defines a six-argument group field theory that extends the Boulatov construction to 3D Riemannian quantum gravity coupled to a scalar field. The model introduces a mass-insertion operator $P_\theta$ and a propagator $\mathcal{P}=(I+aK_{\theta})/(1-a^2)$ with $(K_{\theta})^2=I$, producing Feynman amplitudes identical to the Ponzano-Regge spin foam with massive spinless particles, and connects these to a non-commutative effective scalar field theory. The authors detail vertex and propagator structures, face amplitudes, and normalisation, and discuss regularisation via $U_q(\mathrm{SU}(2))$ and gauge-fixing issues; they also explore generalisations and alternatives, including multi-argument mass insertions and kinetic vs. interaction-term insertions. The work provides a concrete framework to couple matter to quantum gravity in the group field theory setting, offering a path to study observables, spinning matter, gauge fields, and the associated symmetries at the GFT level.

Abstract

We present a new group field theory model, generalising the Boulatov model, which incorporates both 3-dimensional gravity and matter coupled to gravity. We show that the Feynman diagram amplitudes of this model are given by Riemannian quantum gravity spin foam amplitudes coupled to a scalar matter field. We briefly discuss the features of this model and its possible generalisations.