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Related Experiment Video

Updated: Feb 12, 2026

Preparation of Free-Surface Hyperbolic Water Vortices

Preparation of Free-Surface Hyperbolic Water Vortices

Published on: July 28, 2023

Magnetic zero-modes, vortices and Cartan geometry.

Calum Ross^1,2, Bernd J Schroers^1,2

¹1Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS UK.

Letters in Mathematical Physics

|April 3, 2018

Summary

Vortex configurations on a 2-sphere and magnetic zero-modes are closely related. This connection is revealed through the geometry of the 3-sphere and the Hopf fibration, leading to a smooth formula for these modes.

Keywords:

Cartan geometry Dirac operator Magnetic zero-modes Vortex equations

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Area of Science:

Mathematical Physics
Differential Geometry
Quantum Field Theory

Background:

Vortex configurations on spheres and magnetic zero-modes of Dirac operators are key concepts in theoretical physics.
Understanding their relationship requires advanced geometric techniques.

Purpose of the Study:

To establish a precise connection between vortex configurations and magnetic zero-modes.
To develop a geometric framework for understanding these phenomena.
To derive an explicit formula for magnetic zero-modes.

Main Methods:

Utilizing the geometry of the 3-sphere induced by the Hopf fibration.
Applying pull-back techniques to the round geometry.
Employing bundle maps within the Hopf fibration framework.

Main Results:

A close relationship is demonstrated between vortex configurations on the 2-sphere and magnetic zero-modes of the Dirac operator on a specific manifold.
Both are effectively understood via the geometry of the 3-sphere induced by the Hopf fibration.
A manifestly smooth formula for square-integrable magnetic zero-modes is deduced.

Conclusions:

The geometric perspective via the Hopf fibration provides a unified understanding of vortex configurations and magnetic zero-modes.
The derived formula offers a new analytical tool for studying these modes.