# Monopoles, Strings, and Necklaces in $SO(10)$ and $E_6$

## TL;DR

This paper explores various topological defects such as monopoles, strings, and necklaces arising from symmetry breaking in $SO(10)$ and $E_6$ Grand Unified Theories, including their stability, formation, and potential gravitational wave signatures.

## Contribution

It demonstrates the universal presence of certain monopoles, introduces novel necklace configurations, and analyzes their evolution and gravitational wave emissions within GUT models.

## Key findings

- Stable superheavy monopoles with Dirac and color magnetic charge are always present.
- Lighter monopoles with multiple Dirac charges can exist and survive inflation.
- A monopole-string system can decay via gravitational waves detectable by LISA.

## Abstract

We employ a variety of symmetry breaking patterns in $SO(10)$ and $E_6$ Grand Unified Theories to demonstrate the appearance of topological defects including magnetic monopoles, strings, and necklaces. We show that independent of the symmetry breaking pattern, a topologically stable superheavy monopole carrying a single unit of Dirac charge as well as color magnetic charge is always present. Lighter intermediate mass topologically stable monopoles carrying two or three quanta of Dirac charge can appear in $SO(10)$ and $E_6$ models respectively. These lighter monopoles as well as topologically stable intermediate scale strings can survive an inflationary epoch. We also show the appearance of a novel necklace configuration in $SO(10)$ broken to the Standard Model via $SU(4)_c\times SU(2)_L\times SU(2)_R$. It consists of $SU(4)_c$ and $SU(2)_R$ monopoles connected by flux tubes. Necklaces consisting of monopoles and antimonopoles joined together by flux tubes are also identified. Even in the absence of topologically stable strings, a monopole-string system can temporarily appear. This system decays by emitting gravity waves and we provide an example in which the spectrum of these waves is strongly peaked around $10^{-4}~{\rm Hz}$ with $\Omega_{\rm gw}h^2\simeq 10^{-12}$. This spectrum should be within the detection capability of LISA.