Quantum Gravity with Matter via Group Field Theory

TL;DR

This paper extends group field theory to include matter in 3D quantum gravity by using the Drinfeld double, introduces a new topological model, and explores models related to point particle Feynman diagrams.

Contribution

It develops a generalized GFT framework incorporating the Drinfeld double, and introduces a new topological model related to quantum gravity with matter.

Findings

Constructed GFT models using the Drinfeld double.

Identified a new topological model related to the Ponzano-Regge model.

Proposed models interpreted through point particle Feynman diagrams.

Abstract

A generalization of the matrix model idea to quantum gravity in three and higher dimensions is known as group field theory (GFT). In this paper we study generalized GFT models that can be used to describe 3D quantum gravity coupled to point particles. The generalization considered is that of replacing the group leading to pure quantum gravity by the twisted product of the group with its dual --the so-called Drinfeld double of the group. The Drinfeld double is a quantum group in that it is an algebra that is both non-commutative and non-cocommutative, and special care is needed to define group field theory for it. We show how this is done, and study the resulting GFT models. Of special interest is a new topological model that is the ``Ponzano-Regge'' model for the Drinfeld double. However, as we show, this model does not describe point particles. Motivated by the GFT considerations, we consider a more general class of models that are defined using not GFT, but the so-called chain mail techniques. A general model of this class does not produce 3-manifold invariants, but has an interpretation in terms of point particle Feynman diagrams.