A short answer: The soliton in bosonic field theory can be fermionic because the model secretly contains massive fermions at high energies.
This is because in order to define an bosonic field theory, we need to non-perturbatively regulate the field theory. So let us put the bosonic field theory on a lattice to non-perturbatively regulate the theory (otherwise, the field theory is not even well defined at non-perturbative level). The claim is that to produce the needed topological term that makes the soliton an fermion, the lattice model must contain fermions with finite energy gap.
A more precise claim: Any gapped bosonic model on lattice that have unique ground state on closed space of any topology do not contain fermionic quasiparticles.
Basically, "gapped bosonic model on lattice that have unique ground state on closed space" implies that the model has no topological order. The only way to have emergent fermion from a bosonic lattice model is to have a non-trivial topological order. Also see a related discussion, where I claim that, in lattice bosonic model, the emergent fermion must appear together with emergent gauge theory at low energies. Skyrme-model contain no low energy gauge theory. This is why I claim that the Skyrme-model secretly contains massive fermions at high energies.
Note: Here we have assumed that far away from the soliton, the field is constant. There is another type of defect, such as vortex, whose field is not constant far away. Also we have assumed that there is no gauge field.
This post imported from StackExchange Physics at 2014-04-05 04:19 (UCT), posted by SE-user Xiao-Gang Wen