On the charge density and asymptotic tail of a monopole

TL;DR

This paper introduces a smooth, new definition of magnetic charge density for non-abelian monopoles using Dirac zero-modes, aligning with traditional asymptotic fields and simplifying calculations from spectral data.

Contribution

It presents a novel, smooth charge density definition based on Dirac zero-modes that matches the asymptotic magnetic field to all orders and simplifies spectral curve computations.

Findings

New smooth charge density aligns with asymptotic field expansions

Charge density derived from Dirac zero-modes simplifies calculations

Explicit examples demonstrate the method's effectiveness

Abstract

We propose a new definition for the abelian magnetic charge density of a non-abelian monopole, based on zero-modes of an associated Dirac operator. Unlike the standard definition of the charge density, this density is smooth in the core of the monopole. We show that this charge density induces a magnetic field whose expansion in powers of 1/r agrees with that of the conventional asymptotic magnetic field to all orders. We also show that the asymptotic field can be easily calculated from the spectral curve. Explicit examples are given for known monopole solutions.