Noninteger flux - why it does not work

Details Noninteger flux - why it does not work Noninteger flux - why it does not work

2011-04-20

arxiv.org/abs/1104.3986v2

Physics

Mathematical Physics

10 pages. The final version to be published in Journ. Math. Phys

Scientific paper

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.

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