Barbero–Immirzi connections and how to build them

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.