Do Barbero-Immirzi connections exist in different dimensions and signatures?

TL;DR

This paper investigates the existence of Barbero-Immirzi connections across different dimensions and signatures, revealing they only exist in four dimensions with specific properties, impacting Loop Quantum Gravity formulations.

Contribution

It demonstrates that reductive splittings and BI connections are unique to four dimensions, providing insights into their globality and covariance in various signatures.

Findings

Reductive splittings exist only in 4D for certain signatures.

In 4D, BI connections can be globally defined and covariant.

Other dimensions require alternative mechanisms for BI-like connections.

Abstract

We shall show that no reductive splitting of the spin group exists in dimension 3 \leq m \leq 20 other than in dimension m = 4. In dimension 4 there are reductive splittings in any signature. Euclidean and Lorentzian signatures are reviewed in particular and signature (2, 2) is investigated explicitly in detail. Reductive splittings allow to define a global SU(2)-connection over spacetime which encodes in an weird way the holonomy of the standard spin connection. The standard Barbero-Immirzi (BI) connection used in LQG is then obtained by restriction to a spacelike slice. This mechanism provides a good control on globality and covariance of BI connection showing that in dimension other than 4 one needs to provide some other mechanism to define the analogous of BI connection and control its globality.