There is an issue that I have with the argument given in “Topological Degeneracy of non-Abelian States for Dummies” http://arxiv.org/abs/cond-mat/0607743 , regarding the ground state degeneracy of the Pfaffian state on a torus: This is They argue that adiabatic pairwise annihilation of quasiparticles around the $x$-direction (implemented by $T_x$) should be equivalent to inserting half a flux quantum into the y-direction “hole” of the torus, i.e.
Explicitly, this is described in the following paragraph, on page 9, Section 4: Consistency between non-Abelian statistics and charge fractionalization:
The issue I have with this is that the magnetic field in a vortex is localized at a point, and is not like the uniform field described by "insertion of half a flux quantum into a hole", and so the authors still have to prove that a quasiparticle encircling the appropriate fundamental cycle of the torus still picks up the same Aharonov Bohm phase, despite the fact that the magnetic fields are different in each case.
Moreover, I do not believe that it is possible for for half a flux quantum to exist: then the Chern number of the electromagnetic $U(1)$-bundle would then be 1/2, not an integer, which violates mathematics!!!
What's going on here?