Fermions coupled to solitons on low-dimensional spheres

We examine models of Dirac fermions coupled to topological solitons on the circle S1 and the sphere S2. The fermion is coupled to a pseudoscalar kink in the (1+1)-dimensional model, and to an isovector modelling a baby Skyrmion in the (2+1)-dimensional model. In each case, we solve the spectrum of the fermionic Hamiltonian exactly when the soliton field is kept fixed as a background field, and the fermion dynamics do not cause any back-reaction on the soliton field. In the (1+1)-dimensional model, we then bring the kink field out of the background, fully coupling it to the fermion field. After a change of coordinates to a set of bosonic coordinates constructed out of bispinors, we demonstrate that solutions to the bosonic dynamical system can be understood analytically via the framework of elliptic functions. We show that for a particular class of solutions with no axial charge, we can recover the underlying fermion field from the bispinor solution. In the (2+1)-dimensional model we specialise to the case of the background soliton of topological degree 1 and exploit an SU(2) symmetry to describe the fermion spectrum.