Noninteger flux - why it does not work

TL;DR

This paper investigates the spectral properties of the Dirac operator on a punctured 2-sphere with non-integer magnetic flux, revealing that supersymmetry is broken due to the inability to extend the Hilbert space to include singular states.

Contribution

It demonstrates that for non-integer flux, the Dirac operator's spectral problem is well-defined only on a restricted Hilbert space, which breaks supersymmetry, unlike the integer flux case.

Findings

Spectral problem well-defined on nonsingular wave functions

Hilbert space not invariant under Dirac operator for non-integer flux

Supersymmetry broken in non-integer flux case

Abstract

We consider the Dirac operator on a 2-sphere without one point in the case of non-integer magnetic flux. We show that the spectral problem for the Hamiltonian (the square of Dirac operator) can always be well defined, if including in the Hilbert space only nonsingular on 2-sphere wave functions. However, this Hilbert space is not invariant under the action of the Dirac operator; the action of the latter on some nonsingular states produces singular functions. This breaks explicitly the supersymmetry of the spectrum. In the integer flux case, the supersymmetry can be restored if extending the Hilbert space to include locally regular sections of the corresponding fiber bundle. For non-integer fluxes, such an extention is not possible.