Application of Harmonic Maps $CP^{(N-1)}$ on SU(N) Bogomolny Equation for BPS Magnetic Monopoles

TL;DR

This thesis explores simplifying SU(N) Bogomolny equations for BPS magnetic monopoles using harmonic maps to transform complex matrix equations into scalar equations, with a detailed example for SU(2).

Contribution

It introduces a method employing harmonic maps $CP^{(N-1)}$ to reduce the complexity of SU(N) Bogomolny equations for magnetic monopoles.

Findings

Harmonic maps effectively simplify SU(N) Bogomolny equations.

The method relates to $Gr(n,N)$ sigma-models.

Explicit example provided for SU(2) case.

Abstract

In this thesis we study dynamic of magnetic monopoles from Lagrangian density in Yang-Mills-Higgs field theory. In particular, we discuss BPS (Bogomolny Prasad Sommerfield) magnetic monopoles, described by SU(N) Bogomolny equations, which has field equations in form of non-linear coupled matrix field equations. One of the methods to simplify SU(N) Bogomolny equations is by using harmonic maps $CP^{(N-1)}$. This method has relation with $Gr(n,N)} \sigma$-model and can transform SU(N) Bogomolny equation into more simple scalar field equations that depends only on one variable. As an example, we consider the case of SU(2) Bogomolny equation.