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

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.

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