On a Covariant Formulation of the Barbero-Immirzi Connection

TL;DR

This paper investigates conditions under which a globally well-defined, covariant SU(2) connection can be constructed in the context of the Barbero-Immirzi formulation, extending the usual local BI connection to a spacetime-wide object.

Contribution

It introduces a gauge covariant, globally defined SU(2) connection compatible with the BI connection, without requiring bundle triviality, and extends it over the entire spacetime.

Findings

A covariant SU(2) connection can be globally defined under general conditions.

The global connection locally agrees with the standard BI connection on space slices.

A spacetime-wide covariant SU(2) connection is constructed, extending the usual BI connection.

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. Such 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.