Zero modes in vortex-fermion system with compact extra space

The existence of fermionic zero modes is shown in the presence of vortex configuration of pure $SU(2)$ gauge field on the manifold $M_4 \times S^2$. From the perspective of four-dimensional effective theory, these zero modes are almost the same as the Jackiw-Rossi type zero modes of the vortex-fermion system.