Fermionic zero modes in the vortex field in arbitrary dimensions and index of the Dirac operator with Majorana-like interaction

In this work we consider fermionic zero modes in the external scalar and electromagnetic field forming the vortex on a sphere. We find the correspondence between the equations for the fermions in different dimensions, find their explicit expressions through the vortex fields in the case of massless fermions, asymptotics near the poles in the case of massive fermions, and check the number of the solutions by proving the index theorem for the fermions on a sphere. As a part of deriving the index, we write a detailed calculation of the Green function of the Heat equation.