# Spectral flow of monopole insertion in topological insulators

## TL;DR

This paper generalizes the concept of spectral flow induced by magnetic flux insertion from 2D topological insulators to higher dimensions using non-abelian monopoles, linking spectral flow to higher Chern and winding numbers.

## Contribution

It introduces a higher-dimensional framework for spectral flow via non-abelian monopoles and establishes a new index theorem connecting spectral flow and topological invariants.

## Key findings

- Spectral flow equals higher Chern numbers in even dimensions.
- Introduction of 'chirality flow' for odd dimensions.
- New index theorem relating spectral flow to unitary conjugation.

## Abstract

Inserting a magnetic flux into a two-dimensional one-particle Hamiltonian leads to a spectral flow through a given gap which is equal to the Chern number of the associated Fermi projection. This paper establishes a generalization to higher even dimension by inserting non-abelian monopoles of the Wu-Yang type. The associated spectral flow is then equal to a higher Chern number. For the study of odd spacial dimensions, a new so-called `chirality flow' is introduced which, for the insertion of a monopole, is then linked to higher winding numbers. This latter fact follows from a new index theorem for the spectral flow between two unitaries which are conjugates of each other by a self-adjoint unitary.