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

Abstract

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.

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

References in corpus (5)

• Classification of topological insulators and superconductors in three spatial dimensions
• On the K-theoretic classification of topological phases of matter
• Bott periodicity for $Z_2$ symmetric ground states of gapped free-fermion systems
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Cited by in corpus (7)

• Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII
• The FKMM-invariant in low dimension
• On the K-theoretic classification of dynamically stable systems
• The Geometry of (non-Abelian) Landau levels
• Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional "Quaternionic" case
• Topological Insulators and the Kane-Mele Invariant: Obstruction and Localisation Theory
• Supersymmetry in Hadron Spectroscopy and Solitons in 2 Dimensional Fermionic Media