On a covariant formulation of the Barbero–Immirzi connection

The Barbero–Immirzi (BI) connection, as usually introduced out of a spin connection, is a global object though it does not transform properly as a genuine connection with respect to generic spin transformations, unless quite specific and suitable gauges are imposed. Here we shall investigate whether, and under which global conditions, a (properly transforming and hence global) SU(2)-connection can be canonically defined in a gauge covariant way. Such an SU(2)-connection locally agrees with the usual BI connection and it can be defined on pretty general bundles; in particular, triviality is not assumed. As a by-product we shall also introduce a global covariant SU(2)-connection over the whole spacetime (while for technical reasons the BI connection in the standard formulation is just introduced on a space slice) which restricts to the usual BI connection on a space slice.