Group field theories for all loop quantum gravity

TL;DR

This paper extends group field theories to encompass all loop quantum gravity states, providing a comprehensive combinatorial framework and linking spin foam models with tensor model techniques.

Contribution

It generalizes the combinatorics of group field theories to include all LQG states and demonstrates the use of tensor tools for effective construction.

Findings

Unified GFT formulation for all LQG states

Explicit GFT models for KKL spin foam and modifications

Detailed combinatorial analysis of spin foam and GFT structures

Abstract

Group field theories represent a 2nd quantized reformulation of the loop quantum gravity state space and a completion of the spin foam formalism. States of the canonical theory, in the traditional continuum setting, have support on graphs of arbitrary valence. On the other hand, group field theories have usually been defined in a simplicial context, thus dealing with a restricted set of graphs. In this paper, we generalize the combinatorics of group field theories to cover all the loop quantum gravity state space. As an explicit example, we describe the GFT formulation of the KKL spin foam model, as well as a particular modified version. We show that the use of tensor model tools allows for the most effective construction. In order to clarify the mathematical basis of our construction and of the formalisms with which we deal, we also give an exhaustive description of the combinatorial structures entering spin foam models and group field theories, both at the level of the boundary states and of the quantum amplitudes.