Differential geometric invariants for time-reversal symmetric Bloch-bundles: The "Real" case

Topological quantum systems subjected to an even (resp. odd) time-reversal symmetry can be classified by looking at the related "Real" (resp. "Quaternionic") Bloch-bundles. If from one side the topological classification of these time-reversal vector bundle theories has been completely described in De Nittis and Gomi [J. Geom. Phys. 86, 303–338 (2014)] for the "Real" case and in De Nittis and Gomi [Commun. Math. Phys. 339, 1–55 (2015)] for the "Quaternionic" case, from the other side it seems that a classification in terms of differential geometric invariants is still missing in the literature. With this article and its companion [G. De Nittis and K. Gomi (unpublished)] we want to cover this gap. More precisely, we extend in an equivariant way the theory of connections on principal bundles and vector bundles endowed with a time-reversal symmetry. In the "Real" case we generalize the Chern-Weil theory and we show that the assignment of a "Real" connection, along with the related differential Chern class and its holonomy, suffices for the classification of "Real" vector bundles in low dimensions.