Differential geometric invariants for timereversal symmetric Blochbundles: The "Real" case
Abstract
Topological quantum systems subjected to an even (resp. odd) timereversal symmetry can be classified by looking at the related "Real" (resp. "Quaternionic") Blochbundles. If from one side the topological classification of these timereversal vector bundle theories has been completely described in De Nittis and Gomi [J. Geom. Phys. 86, 303338 (2014)] for the "Real" case and in De Nittis and Gomi [Commun. Math. Phys. 339, 155 (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 With this article and its companion [G. De Nittis and K. Gomi (unpublished)] we want to cover this gap.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 timereversal symmetry. In the "Real" case we generalize the ChernWeil 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 May 2016
 DOI:
 10.1063/1.4948742
 arXiv:
 arXiv:1502.01232
 Bibcode:
 2016JMP....57e3506D
 Keywords:

 Mathematical Physics;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 Primary: 57R22;
 Secondary: 53A55;
 55N25;
 53C80
 EPrint:
 50 pages. key words: Topological quantum systems, Blochbundle, "Real and "Quaternionic" vector bundles , equivariant connections, "Real" ChernWeil theory. (v2) Version accepted for publication on J. Math. Pays. Introduction partially rewritten. minor corrections in the main body of the text