
TL;DR
This paper introduces chiral vector bundles, a new mathematical framework for classifying topological quantum systems with chiral symmetry, extending existing classifications by incorporating additional topological invariants.
Contribution
It develops a classification scheme for chiral vector bundles using classifying spaces and cohomology, revealing richer topological invariants than traditional methods.
Findings
Classification over spheres and tori up to dimension 4
Reproduction of known AIII topological insulator classification
Identification of new odd-dimensional characteristic classes
Abstract
This paper focuses on the study of a new category of vector bundles. The objects of this category, called chiral vector bundles, are pairs given by a complex vector bundle along with one of its automorphisms. We provide a classification for the homotopy equivalence classes of these objects based on the construction of a suitable classifying space. The computation of the cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes: some of those just reproduce the ordinary Chern classes but there are also new odd-dimensional classes which take care of the extra topological information introduced by the chiral structure. Chiral vector bundles provide a geometric model for topological quantum systems in class AIII, namely for systems endowed with a (pseudo-)symmetry of chiral type. The classification of the chiral vector bundles over sphere and tori…
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