# Analytic crystals of solitons in the four dimensional gauged non-linear sigma model

## TL;DR

This paper presents the first analytic solutions of multi-solitons with topological charge in a 3+1 dimensional gauged non-linear sigma model, revealing crystalline structures and electromagnetic field configurations.

## Contribution

It introduces explicit analytic multi-soliton solutions with topological charge in a gauged non-linear sigma model, reducing complex equations to a solvable linear form.

## Key findings

- Solitons form crystalline, tube-shaped energy and charge densities.
- Electromagnetic fields are confined to the surface of the tubes.
- Solutions preserve topological charge and exhibit periodic structures.

## Abstract

The first analytic topologically non-trivial solutions in the (3+1)-dimensional gauged non-linear sigma model representing multi-solitons at finite volume with manifest ordered structures generating their own electromagnetic field are presented. The complete set of seven coupled non-linear field equations of the gauged non-linear sigma model together with the corresponding Maxwell equations are reduced in a self-consistent way to just one linear Schrodinger-like equation in two dimensions. The corresponding two dimensional periodic potential can be computed explicitly in terms of the solitons profile. The present construction keeps alive the topological charge of the gauged solitons. Both the energy density and the topological charge density are periodic and the positions of their peaks show a crystalline order. These solitons describe configurations in which (most of) the topological charge and total energy are concentrated within three-dimensional tube-shaped regions. The electric and magnetic fields vanish in the center of the tubes and take their maximum values on their surface while the electromagnetic current is contained within these tube-shaped regions. Electromagnetic perturbations of these families of gauged solitons are shortly discussed.