# Analysis of zero modes for Dirac operators with magnetic links

## TL;DR

This paper develops methods to approximate Dirac operators with magnetic links by smooth fields, proving spectral flow equivalence and identifying conditions for zero modes, advancing understanding of magnetic link effects on Dirac spectra.

## Contribution

It introduces an approximation technique for Dirac operators with magnetic links and establishes spectral flow equivalence between smooth and singular cases, revealing new zero modes.

## Key findings

- Spectral flow is preserved between smooth and singular magnetic link cases.
- Existence of smooth, compactly supported magnetic fields with non-trivial Dirac kernels.
- Criteria for the non-existence of zero modes based on Clifford analysis.

## Abstract

In this paper we provide a means to approximate Dirac operators with magnetic fields supported on links in $\mathbb{S}^3$ (and $\mathbb{R}^3$) by Dirac operators with smooth magnetic fields. We then proceed to prove that under certain assumptions, the spectral flow of paths along these operators is the same in both the smooth and the singular case. We recently characterized the spectral flow of such paths in the singular case. This allows us to show the existence of new smooth, compactly supported magnetic fields in $\mathbb{R}^3$ for which the associated Dirac operator has a non-trivial kernel. Using Clifford analysis, we also obtain criteria on the magnetic link for the non-existence of zero modes.