Wilson loops, geometric operators and fermions in 3d group field theory

TL;DR

This paper develops 3d group field theories incorporating Wilson loops, volume observables, and fermions, enabling a unified framework for quantum gravity with matter and summing over fermionic loops.

Contribution

It introduces a novel group field theory framework that includes Wilson loops, volume observables, and fermions, extending previous spin foam models with a natural sum over fermionic loops.

Findings

Defines group field theories with Wilson loops and volume observables.

Constructs a spin foam model for fermions coupled to gravity.

Includes sum over fermionic loops in perturbation theory.

Abstract

Group field theories whose Feynman diagrams describe 3d gravity with a varying configuration of Wilson loop observables and 3d gravity with volume observables at each vertex are defined. The volume observables are created by the usual spin network grasping operators which require the introduction of vector fields on the group. We then use this to define group field theories that give a previously defined spin foam model for fermion fields coupled to gravity, and the simpler quenched approximation, by using tensor fields on the group. The group field theory naturally includes the sum over fermionic loops at each order of the perturbation theory.