Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional "Quaternionic" case

TL;DR

This paper develops differential geometric invariants, specifically the Wess-Zumino and Chern-Simons invariants, to classify Quaternionic vector bundles over low-dimensional manifolds with involution, aiding in distinguishing inequivalent structures.

Contribution

It introduces a method to use Wess-Zumino and Chern-Simons invariants for classifying Quaternionic vector bundles on low-dimensional manifolds with involution.

Findings

Wess-Zumino term distinguishes inequivalent Quaternionic structures.

Chern-Simons invariant provides topological classification.

Applicable to manifolds of dimension two or three with finite fixed points.

Abstract

This paper is devoted to the construction of differential geometric invariants for the classification of "Quaternionic" vector bundles. Provided that the base space is a smooth manifold of dimension two or three endowed with an involution that leaves fixed only a finite number of points, it is possible to prove that the Wess-Zumino term and the Chern-Simons invariant yield topological quantities able to distinguish between inequivalent realization of "Quaternionic" structures.