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Keywords:
Barbero-Immirzi connection; global connections; Loop Quantum Gravity
Barbero-Immirzi connection; global connections; Loop Quantum Gravity
Summary:
We shall show that no reductive splitting of the spin group exists in dimension $3\le m\le 20$ 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 $\mbox{SU} (2)$-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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