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 af dette arbejde

Institut for Matematiske Fag

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.

Originalsprog	Engelsk
Tidsskrift	Journal of Geometric Analysis
Antal sider	68
ISSN	1050-6926
DOI	https://doi.org/10.1007/s12220-018-00128-5
Status	Udgivet - 1 jan. 2020

Adgang til dokumentet

10.1007/s12220-018-00128-5

Andre filer og links

Link to publication in Scopus

Citationsformater

@article{bb49e6f77d8140ca9574072e4b4fcae0,

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.",

keywords = "Dirac operators, Knots, Links, Seifert surface, Spectral flow, Zero modes",

author = "Fabian Portmann and J{\'e}r{\'e}my Sok and Solovej, {Jan Philip}",

year = "2020",

month = jan,

day = "1",

doi = "10.1007/s12220-018-00128-5",

language = "English",

journal = "Journal of Geometric Analysis",

issn = "1050-6926",

publisher = "Springer",

}

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

UR - http://www.scopus.com/inward/record.url?scp=85074532755&partnerID=8YFLogxK

U2 - 10.1007/s12220-018-00128-5

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

Abstract

Adgang til dokumentet

Andre filer og links

Fingeraftryk

Citationsformater