Barbero-Immirzi connections and how to build them

TL;DR

This paper presents a covariant formulation of Barbero-Immirzi connections in Loop Quantum Gravity, revealing their unique features across different dimensions and clarifying the role of the Barbero-Immirzi parameter.

Contribution

It introduces a covariant approach to define Barbero-Immirzi connections for any spin Lorentzian manifold, highlighting the special nature of the 4-dimensional case.

Findings

Barbero-Immirzi connections can be uniquely constructed from a spin connection in any dimension.

The real Barbero-Immirzi parameter is specific to 4-dimensional spacetime.

The formulation clarifies the geometric and algebraic properties of these connections.

Abstract

We introduce a covariant formulation of Barbero-Immirzi connections, which are used in Loop Quantum Gravity to describe gravity. We show that Barbero-Immirzi connections can be uniquely defined out of a given spin connection for any $(n+1)$-dimensional lorentzian manifold which is spin. A remarkable result is that the presence of a real Barbero-Immirzi parameter is a feature unique to the $4$-dimensional case.