3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory
Winston Fairbairn, Etera R. Livine
TL;DR
The paper addresses how to embed matter within three-dimensional quantum gravity by showing that an effective field theory (EFT) for matter coupled to 3d quantum gravity can be recast as a generalized matrix model, with matrix sizes tied to length quanta of the Ponzano–Regge geometry. It demonstrates that the EFT arises as perturbations around nontrivial classical solutions (instantons) of Boulatov's 3d GFT, yielding a nonlinear momentum redefinition $Q_f$ that encodes a curved background and higher-derivative corrections. This establishes a picture in which matter is an excited geometry phase of the group field theory, not an add-on field. The authors sketch a four-dimensional extension, anticipating string-like excitations of geometry and highlighting the role of nonperturbative GFT dynamics in connecting quantum gravity to emergent matter behavior.
Abstract
An effective field theory for matter coupled to three-dimensional quantum gravity was recently derived in the context of spinfoam models in 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 recasted 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 hep-th/0512113 for a massive field living in a flat non-commutative space-time. The most general solutions lead to field theories with non-linear redefinitions of the momentum which we propose to interpret as living on curved space-times. We conclude by discussing the possible extension to four-dimensional spinfoam models.