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

L. Fatibene, Mauro Francaviglia, S. Garruto (2012)

Communications in Mathematics

We shall show that no reductive splitting of the spin group exists in dimension 3 m 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...

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