Spectral Flow for Dirac Operators with Magnetic Links

Fabian Portmann; Jérémy Sok; Jan Philip Solovej

doi:10.1007/s12220-018-00128-5

Spectral Flow for Dirac Operators with Magnetic Links

Fabian Portmann, Jérémy Sok, Jan Philip Solovej^*

^*Corresponding author for this work

Department of Mathematical Sciences

Abstract

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 π-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.

Original language	English
Journal	Journal of Geometric Analysis
Number of pages	68
ISSN	1050-6926
DOIs	https://doi.org/10.1007/s12220-018-00128-5
Publication status	Published - 1 Jan 2020

Keywords

Dirac operators
Knots
Links
Seifert surface
Spectral flow
Zero modes

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10.1007/s12220-018-00128-5

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title = "Spectral Flow for Dirac Operators with Magnetic Links",

abstract = "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 π-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.",

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TY - JOUR

T1 - Spectral Flow for Dirac Operators with Magnetic Links

AU - Portmann, Fabian

AU - Sok, Jérémy

AU - Solovej, Jan Philip

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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 π-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.

AB - 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 π-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.

KW - Dirac operators

KW - Knots

KW - Links

KW - Seifert surface

KW - Spectral flow

KW - Zero modes

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DO - 10.1007/s12220-018-00128-5

M3 - Journal article

AN - SCOPUS:85074532755

SN - 1050-6926

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

ER -

Spectral Flow for Dirac Operators with Magnetic Links

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Keywords

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