A Generalised Self-Duality for the Yang-Mills-Higgs System

TL;DR

This paper introduces a generalized Yang-Mills-Higgs system with scalar fields forming an invertible matrix, leading to new self-duality equations and solutions like monopoles and toroidal magnetic fields, enhancing understanding of topological solitons.

Contribution

It presents a novel extension of the Yang-Mills-Higgs system by incorporating a symmetric invertible matrix of scalar fields, resulting in modified self-duality equations and explicit solutions.

Findings

The 't Hooft-Polyakov monopole becomes a self-dual solution in the new framework.

Construction of vacuum solutions with non-trivial toroidal magnetic fields.

The model maintains conformal invariance in three dimensions.

Abstract

Self-duality is a very important concept in the study and applications of topological solitons in many areas of Physics. The rich mathematical structures underlying it lead, in many cases, to the development of exact and non-perturbative methods. We present a generalization of the Yang-Mills-Higgs system by the introduction of scalar fields assembled in a symmetric and invertible matrix h of the same dimension as the gauge group. The coupling of such new fields to the gauge and Higgs fields is made by replacing the Killing form, in the contraction of the group indices, by the matrix h in the kinetic term for the gauge fields, and by its inverse in the Higgs field kinetic term. The theory is conformally invariant in the three dimensional space R^3. An important aspect of the model is that for practically all configurations of the gauge and Higgs fields the new scalar fields adjust themselves to solve the modified self-duality equations. We construct solutions using a spherically symmetric ans\"atz and show that the 't Hooft-Polyakov monopole becomes a self-dual solution of such modified Yang-Mills-Higgs system. We use an ans\"atz based on the conformal symmetry to construct vacuum solutions presenting non-trivial toroidal magnetic fields.