Index-like Theorem for Massless Fermions in Spherically Symmetric Monopole Backgrounds

TL;DR

This paper derives a formula for the dimension of the kernel of the Dirac operator for massless fermions in spherically symmetric monopole backgrounds, extending understanding of fermion-monopole interactions without Yukawa couplings.

Contribution

It provides a novel index-like theorem for non-Fredholm Dirac operators in monopole backgrounds, applicable to various representations and gauge symmetries.

Findings

Derived a formula for the kernel dimension of the Dirac operator.

Applicable to fermions in any representation of SU(N).

Includes monopoles that preserve non-abelian gauge symmetry.

Abstract

In this paper we study massless fermions coupled to spherically symmetric $SU(N)$ monopoles without Yukawa couplings between the Higgs and fermion fields. The corresponding Dirac operator is not Fredholm and the associated eigenfunctions are not $L^2$-normalizable. Here we derive a formula for the dimension of the plane-wave normalizable kernel of such a Dirac operator for fermions of any representation of $SU(N)$ in the presence of any spherically symmetric monopole background. Notably, our results also apply to fermions coupled to monopoles that preserve non-abelian gauge symmetry.