Magnetic zero-modes,vortices and Cartan geometry |
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Authors: | Calum Ross Bernd J. Schroers |
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Affiliation: | 1.Department of Mathematics,Heriot-Watt University,Edinburgh,UK;2.Maxwell Institute for Mathematical Sciences,Edinburgh,UK |
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Abstract: | We exhibit a close relation between vortex configurations on the 2-sphere and magnetic zero-modes of the Dirac operator on (mathbb {R}^3) 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. |
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