  The Kitaev honeycomb model hosts a fascinating fractionalized state of matter
featuring emergent Majorana fermions and a vison particle that carries the flux
of an emergent gauge field. In the exactly solvable model these visons are
static but certain perturbations can induce their motion. We show that the
nature of the vison motion induced by a Zeeman field is sharply distinct in the
ferromagnetic vs the antiferromagnetic Kitaev models. Namely, in the
ferromagnetic model the vison has a trivial non-projective translational
symmetry, whereas in the antiferromagnetic Kitaev model it has a projective
translational symmetry with $\pi$-flux per unit cell. The vison band of the
ferromagnetic case has zero Berry curvature, and no associated intrinsic
contribution to the thermal Hall effect. In contrast, in the antiferromagnetic
case there are two gapped vison bands with opposite Chern numbers and an
associated intrinsic vison contribution to the thermal Hall effect. We discuss
these findings in the light of the physics of the spin liquid candidate
$\alpha$-RuCl$_3$.
