# Magnetic Zero-Modes, Vortices and Cartan Geometry

## TL;DR

This paper explores the relationship between magnetic zero-modes of the Dirac operator, vortex configurations on the sphere, and Cartan geometry, providing a new geometric perspective and explicit formulas for these zero-modes.

## Contribution

It introduces a novel geometric framework connecting magnetic zero-modes, vortices, and Cartan geometry, and derives explicit smooth formulas for zero-modes using homogeneous polynomials.

## Key findings

- Magnetic zero-modes relate to vortex configurations on the 2-sphere.
- A geometric interpretation via the pull-back of the round geometry on the 3-sphere.
- Explicit smooth formulas for zero-modes in terms of homogeneous polynomials.

## Abstract

We show that magnetic zero-modes of the Dirac operator on $\mathbb{R}^3$ which obey an additional non-linear equation are closely related to vortex configurations on the 2-sphere, and that both are best understood in terms of the geometry induced on the 3-sphere via pull-back of the round geometry with bundle maps of the Hopf fibration. We use this viewpoint to deduce a manifestly smooth formula for square-integrable magnetic zero-modes in terms of two homogeneous polynomials in two complex variables.