Self-dual Skyrmions on the spheres $S^{2N+1}$

We construct self-dual sectors for scalar field theories on a ($2N+2$)-dimensional Minkowski space-time with the 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 in 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 lead to an infinite set of self-dual Skyrmion solutions.

Figure 1

The isosurfaces of the topological charge density (2.1) for the three-dimensional solutions (2.9), for $a=1$. The densities correspond, from left to right, to $(m,n)=(2,2),(m,n)=(4,1)$, and $(m,n)=(1,4)$. The layers from the core of the figure to its outside are colored in the order yellow (1), blue (2), green (3), red (4), violet (5), and brown (6), where the $n$th layer denotes the isosurface with the density ${\mathcal{Q}}_3=4^{2-n}/{\pi }^2$, i.e., the yellow surface corresponds to ${\mathcal{Q}}_3=4/{\pi }^2$, the green surface to ${\mathcal{Q}}_3=(4\pi {)}^{-2}$, etc.

Figure 2

The isosurfaces of the topological charge density (3.20) for the five-dimensional solutions (3.17), with ${\tilde{\rho }}_a$ and ${\tilde{x}}_5$ defined in Eq. (3.29). The densities correspond, from left to right, to $(m,n_1,n_2)=(1,1,1)$, $(m,n_1,n_2)=(4,1,1)$, $(m,n_1,n_2)=(1,2,2)$, and $(m,n_1,n_2)=(1,4,1)$. The layers from the core of the figure to its outside are colored in the order yellow (1), blue(2), green (3), red (4), violet (5), and brown (6), where the $n$th layer denotes the isosurface with the density ${\mathcal{Q}}_5=5^{3-n}/(a^5{\pi }^3)$, i.e., the yellow surface corresponds to ${\mathcal{Q}}_5=25/(a^5{\pi }^3)$, the green surface to ${\mathcal{Q}}_5=1/(a^5{\pi }^3)$, etc.