Seminar Analysis: Jérémy Sok (University of Basel)

Dirac operators with magnetic links

We consider Dirac operators on the 3-sphere with singular magnetic fields which are supported on links, that is on one-dimensional manifolds which are diffeomorphic to finitely many copies of S¹. Each connected component carries a flux 2πα which exhibits a 2π-periodicity, just like Aharonov-Bohmsolenoids in the complex plane. We study the kernel of such operators through the spectral flow of loops corresponding to tuning some flux from 0 to 2π, that is the number of eigenvalues crossing 0 along the loop (counted algebraically). It turns out that the spectral flow is generically non-zero and depends on the shape of the curves and their linking number. Through the stereographic projection the result extends to R³. And then by smearing out the magnetic fields we obtain new solutions (ψ,A) to the zero-mode equation on R³:

σ·(-i∇+A)=0,
(ψ,A) ∈ H¹(R³)² × \dot{H}¹(R³)³ ∩ L⁶(R³)³,

where σ=(σ)_j=1...3 denotes the family of the Pauli matrices, A is the magnetic potential associated to the magnetic field ∇×A, and σ⋅(-i∇+A) is the corresponding Dirac operator in R³.

(Joint work with Fabian Portmann and Jan Philip Solovej)

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