Spectral flow for Dirac operators with magnetic links

This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov-Bohm solenoids in Euclidean three-space, the flux carried by an oriented knot features a \(2\pi\)-periodicity of the...

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Bibliographic Details
Published in:arXiv.org 2019-02
Main Authors: Portmann, Fabian, Sok, Jérémy, Solovej, Jan Philip
Format: Article
Language:English
Subjects:
Euclidean geometry
Fluxes
Knot theory
Magnetic properties
Operators
Periodic variations
Solenoids
Spectra
Toruses
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Summary:This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov-Bohm solenoids in Euclidean three-space, the flux carried by an oriented knot features a \(2\pi\)-periodicity of the associated operator. For a given link one thus obtains a family of Dirac operators indexed by a torus of fluxes. We study the spectral flow of paths of such operators corresponding to loops in this torus. The spectral flow is in general non-trivial. In the special case of a link of unknots we derive an explicit formula for the spectral flow of any loop on the torus of fluxes. It is given in terms of the linking numbers of the knots and their writhes.
ISSN:2331-8422