  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.
