Group field theory and simplicial quantum gravity

TL;DR

This work constructs a 4d quantum gravity model within a generalized group field theory (GFT) framework that treats both group elements and bivector (Lie algebra) variables. By introducing projection maps that implement the discrete Plebanski (simplicity) constraints at the level of the path integral, the model yields Feynman amplitudes that take the explicit form of a simplicial gravity action, bridging BF theory and gravity in first order form. The resulting amplitudes factorize over dual faces with a Regge‑type action S_R(B_f, H_f) and include measure and quantum correction terms S_c, establishing a direct link to Regge calculus and dynamical triangulations (DT) as subsectors under certain restrictions. The Lorentzian extension, compatibility constraints between simplicity and parallel transport, and potential non‑commutative reformulations suggest a versatile framework that could unify spin foam, LQG, and simplicial gravity approaches, while leaving important questions about constraints, Immirzi parameter, and continuum limits to be explored.

Abstract

We present a new Group Field Theory for 4d quantum gravity. It incorporates the constraints that give gravity from BF theory, and has quantum amplitudes with the explicit form of simplicial path integrals for 1st order gravity. The geometric interpretation of the variables and of the contributions to the quantum amplitudes is manifest. This allows a direct link with other simplicial gravity approaches, like quantum Regge calculus, in the form of the amplitudes of the model, and dynamical triangulations, which we show to correspond to a simple restriction of the same.