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 Euclidean three-space, the flux carried by an oriented knot features a 2π-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.

Self-adjointness and spectral properties of Dirac operators with magnetic links

“…In Theorem 29 we show that the Dirac operator associated to a magnetic circle always has a trivial kernel. This fact will be heavily used in two forthcoming publications [39,40].…”

“…The aforementioned 2π-periodicity in the flux 2πα suggests the study of the spectral flow of loops of Dirac operators, as 2πα runs from 0 to 2π. In [39] we will study the spectral flow for such paths of operators and obtain as a byproduct information about zero modes for the associated Dirac operators. The study of these newly discovered zero modes is then continued in [40], where we provide a regularization procedure to show that these zero modes persist when the singular magnetic fields are approximated by smooth versions.…”

Multi-valued vortex solutions to the Schrödinger equation and radiation

“…The second is the problem of calculating the vector potential A which is generated by a magnetic flux filament pointing along a knot curve. The magnetic scalar potential is discussed in many sources, for example [36,22,54,55,28,40]. A classical result for a vortex loop which is unknotted, ie.…”