Group field theory with non-commutative metric variables

TL;DR

This paper presents a dual formulation of group field theories as non-commutative field theories with Lie algebra variables, enabling new approaches to impose simplicity constraints and defining a new Barrett-Crane model.

Contribution

It introduces a non-commutative formulation of GFT with Lie algebra variables, facilitating the implementation of gravity constraints directly at the action level.

Findings

Feynman amplitudes as simplicial path integrals for BF theories

New GFT definition of the Barrett-Crane model

Potential for imposing gravity constraints at the GFT level

Abstract

We introduce a dual formulation of group field theories, making them a type of non-commutative field theories. In this formulation, the variables of the field are Lie algebra variables with a clear interpretation in terms of simplicial geometry. For Ooguri-type models, the Feynman amplitudes are simplicial path integrals for BF theories. This formulation suggests ways to impose the simplicity constraints involved in BF formulations of 4d gravity directly at the level of the group field theory action. We illustrate this by giving a new GFT definition of the Barrett-Crane model.