Global Barbero-Immirzi Connections

TL;DR

This paper investigates the conditions under which a globally well-defined SU(2) connection, consistent with the Barbero-Immirzi connection, can be constructed in a gauge covariant manner without restrictive bundle assumptions.

Contribution

It provides a framework for defining a proper, globally consistent SU(2) connection that aligns with the BI connection and extends it to the entire spacetime.

Findings

Established conditions for global SU(2) connection construction

Introduced a covariant SU(2) connection over spacetime

Connected the global connection with the standard BI connection on space slices

Abstract

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. We shall here 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 in such a way that SU(2)-connection locally agrees with the usual BI connection and 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.