Massless monopole-string-domain wall fermions and polyhedral vacuum fermions

Fermion zero modes of Bogomol’nyi-Prasad-Sommerfield monopole-string-domain wall composites in three spatial dimensions are studied. We analytically solve the Dirac equation and prove the existence of one fermion zero mode. Depending on mass parameters of bosons/fermions in the model, the zero modes are localized either on the monopoles, strings or domain walls, which we call monopole-string-domain wall fermions. We also show that in special cases, the zero modes can be confined within a finite vacuum region in the shape of an arbitrary convex polyhedron, which we call the polyhedral vacuum fermions. Furthermore, we show that fermionic superconducting currents do not generally flow on the host solitons except for the cases that the soliton network consists only of strings and domain walls and has translational symmetry about a spatial axis.