# Exact Self-Dual Skyrmions

## TL;DR

This paper introduces a novel Skyrme-type model with coordinate-dependent couplings on S^3, featuring a self-dual sector with exact solutions, and explores their geometric and energetic properties.

## Contribution

It presents a new self-dual Skyrme model with variable couplings and derives exact toroidal solutions with specific symmetry properties.

## Key findings

- Self-dual solutions saturate the Bogomolny bound.
- Solutions exhibit spherical, oblate, or prolate symmetry depending on parameters.
- The model's energy depends linearly on topological charge.

## Abstract

We introduce a Skyrme type model with the target space being the 3-sphere S^3 and with an action possessing, as usual, quadratic and quartic terms in field derivatives. The novel character of the model is that the strength of the couplings of those two terms are allowed to depend upon the space-time coordinates. The model should therefore be interpreted as an effective theory, such that those couplings correspond in fact to low energy expectation values of fields belonging to a more fundamental theory at high energies. The theory possesses a self-dual sector that saturates the Bogomolny bound leading to an energy depending linearly on the topological charge. The self-duality equations are conformally invariant in three space dimensions leading to a toroidal ansatz and exact self-dual Skyrmion solutions. Those solutions are labelled by two integers and, despite their toroidal character, the energy density is spherically symmetric when those integers are equal and oblate or prolate otherwise.