Chiral vector bundles

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 (explicitly computable up to dimension 4), recovers the commonly accepted classification for topological insulator of class AIII which is usually based on the K-group $K_1$. However, our classification turns out to be even richer since it takes care also for possible non-trivial Chern classes.