### Abstract

We exhibit a close relation between vortex configurations on the 2-sphere and magnetic zero-modes of the Dirac operator on (Formula presented.) which obey an additional nonlinear equation. We show 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.

Original language | English |
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Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Letters in Mathematical Physics |

Early online date | 7 Nov 2017 |

DOIs | |

Publication status | E-pub ahead of print - 7 Nov 2017 |

### Keywords

- Cartan geometry
- Dirac operator
- Magnetic zero-modes
- Vortex equations

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Profiles

## Bernd Johannes Schroers

- School of Mathematical & Computer Sciences - Professor
- School of Mathematical & Computer Sciences, Mathematics - Professor

Person: Academic (Research & Teaching)