Self-Dual Skyrmions on the Spheres $S^{2N+1}$

TL;DR

This paper develops a framework for constructing self-dual solutions in higher-dimensional scalar field theories with target space on spheres, generalizing known three-dimensional Skyrmion solutions to higher dimensions.

Contribution

It introduces a new method to create self-dual sectors in higher-dimensional theories, enabling exact solutions with arbitrary topological charge.

Findings

Constructed self-dual sectors on $S^{2N+1}$ spheres.

Derived infinite exact self-dual solutions with arbitrary topological charge.

Extended known 3D Skyrmion results to higher dimensions.

Abstract

We construct self-dual sectors for scalar field theories on a $(2N+2)$-dimensional Minkowski space-time with target space being the $2N+1$-dimensional sphere $S^{2N+1}$. The construction of such self-dual sectors is made possible by the introduction of an extra functional on the action that renders the static energy and the self-duality equations conformally invariant on the $(2N+1)$-dimensional spatial submanifold. The conformal and target space symmetries are used to build an ansatz that leads to an infinite number of exact self-dual solutions with arbitrary values of the topological charge. The five dimensional case is discussed in detail where it is shown that two types of theories admit self dual sectors. Our work generalizes the known results in the three-dimensional case that leads to an infinite set of self-dual Skyrmion solutions.