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

TL;DR

This paper investigates fermionic zero modes in vortex fields on a sphere across various dimensions, deriving explicit solutions, analyzing asymptotics, and confirming the index theorem through detailed heat kernel calculations.

Contribution

It provides a comprehensive analysis of fermionic zero modes in vortex backgrounds on a sphere, including explicit solutions and index calculations for different dimensions.

Findings

Explicit solutions for massless fermions in vortex fields

Asymptotic behavior near poles for massive fermions

Verification of the index theorem on a spherical geometry

Abstract

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 case of massless fermions, asymptotics near the poles in case of massive fermions and check the number of the solutions by proving 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.