Self-dual Skyrmions on the spheres S2N+1

Yuki Amari ,L A Ferreira

doi:10.1103/physrevd.97.085006

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.

Highlights

The beauty of self-duality is that it is characterized by first-order differential equations, such that their solutions solve the second-order Euler-Lagrange equations of the full theory
We have introduced Skyrme-type models in (2N þ 2)-dimensional Minkowski space-time with the target space being the spheres S2Nþ1
The models do not have a gauge symmetry, and in order to have finite-energy static solutions the fields must go to a constant at spatial infinity

Summary

INTRODUCTION

The beauty of self-duality is that it is characterized by first-order differential equations, such that their solutions solve the second-order Euler-Lagrange equations of the full theory. We consider the generic case of theories in (2N þ 2)-dimensional Minkowski space-time with target space S2Nþ1 In such cases the number of possibilities of splitting the density of topological charge is very large, leading to theories which are conformally invariant in IR2Nþ1. We use the conformal and target-space symmetries of the self-duality equations to construct infinite sets of exact self-dual solutions for these two types of theories. In Appendix A we give the proof of the conformal symmetry of the self-duality equations, and in Appendix B we solve some integrals relevant for the calculation of the topological charges of the solutions

SOLUTIONS ON S3

SOLUTIONS ON S5

Type I theory on S5

Type II theory on S5

H Þ μ2N μ2Nþ1 2

CONCLUSIONS