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Magnetic Zero-Modes, Vortices and Cartan Geometry

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arxiv 1705.09632 v2 pith:VXBVJVCP submitted 2017-05-26 hep-th gr-qcmath-phmath.MP

Magnetic Zero-Modes, Vortices and Cartan Geometry

classification hep-th gr-qcmath-phmath.MP
keywords geometrymagneticzero-modesspheretermsadditionalbestbundle
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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.

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