Spectral Flow for Dirac Operators with Magnetic Links

  • Fabian Portmann
  • Jérémy Sok
  • Jan Philip SolovejEmail author


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 the 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 nontrivial. 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.


Dirac operators Knots Links Seifert surface Spectral flow Zero modes 

Mathematics Subject Classification

81Q10 58C40 58J30 57M25 



We would like to thank M. Goffeng, G. Grubb and R. Nest for fruitful discussions and helpful comments. We also thank the referee for various remarks and comments that helped us improving the paper. The authors acknowledge support from the ERC Grant No. 321029 “The mathematics of the structure of matter” and VILLUM FONDEN through the QMATH Centre of Excellence Grant No. 10059. This work has been done when all the authors were working at the University of Copenhagen.


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Authors and Affiliations

  1. 1.QMATH, Department of Mathematical SciencesUniversity of Copenhagen Universitetsparken 5CopenhagenDenmark
  2. 2.IBM SwitzerlandZurichSwitzerland
  3. 3.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland

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