Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

Abstract

Let \({\cal A}\) be the affine space of all connections in an SU(2) principal fibre bundle over ℝ3. The set of homogeneous isotropic connections forms a line l in \({\cal A}\). We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding \(l \hookrightarrow {\cal A}\) does not continuously extend to an embedding \(\overline{l} \hookrightarrow \overline{\cal A}\) of the respective compactifications. Here, the Bohr compactification \(\overline{l}\) corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and \(\overline{\cal A}\) to that of loop quantum gravity. Analogous results are given for the anisotropic case.