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Do Barbero-Immirzi connections exist in different dimensions and signatures?
Communications in Mathematics, Tome 20 (2012) no. 1, pp. 3-11 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 53C07
Keywords: Barbero-Immirzi connection; global connections; Loop Quantum Gravity 
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Fatibene, L.; Francaviglia, M.; Garruto, S. Do Barbero-Immirzi connections exist in different dimensions and signatures?. Communications in Mathematics, Tome 20 (2012) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/COMIM_2012_20_1_a1/

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