Magnetic Zero-Modes, Vortices and Cartan Geometry

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.

33 pages, 3 figures. LMP accepted version

References in corpus (3)

• Five Vortex Equations
• Dirac operators on the Taub-NUT space, monopoles and SU(2) representations
• Spectral Properties of Schwarzschild Instantons

Cited by in corpus (2)

• Linked and knotted synthetic magnetic fields
• Vortex Harmonic Spinors on the Nappi-Witten Space