Differential geometric invariants for time-reversal symmetric Bloch-bundles: the "Real" case
Giuseppe De Nittis, Kiyonori Gomi
备注：
50 pages. key words: Topological quantum systems, Bloch-bundle, "Real and "Quaternionic" vector bundles , equivariant connections, "Real" Chern-Weil theory. (v2) Version accepted for publication on J. Math. Pays. Introduction partially rewritten. minor corrections in the main body of the text
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 [DG1] for the "Real" case and in [DG2] 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 [DG3]) 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 showed 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.
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