$SO(4)$ gauged $O(5)$ Skyrmion on $\mathbb{R}^4$

TL;DR

This work constructs an $SO(4)$ gauged $O(5)$ Skyrme system on ${ m I space}\space R^4$ using Skyrme-Chern-Pontryagin densities to obtain finite-energy, topologically stable solitons. By imposing radial symmetry and employing a two-parameter scaling $(\alpha,\beta)$, the authors derive and solve the resulting ODEs for the gauge function $w(r)$ and the Skyrme profile $f(r)$, finding two solution branches distinguished by $w_\infty=\pm1$. The solutions resemble instantons in four dimensions with curvature decaying as $r^{-3}$ but differ in asymptotics due to a Pontryagin-induced potential; no Bogomol'nyi saturation occurs, and only one branch connects to the gauge-decoupled Skyrmion. The study highlights both the similarities and differences with lower-dimensional cases and points to future work on higher dimensions and instanton-gas configurations, potentially via bi-azimuthal symmetry. These results advance understanding of stabilized solitons in higher-dimensional, gauge-coupled Skyrme theories and their instanton-like structure.

Abstract

We have studied an $SO(4)$ gauged $O(5)$ Skyrmion on $\mathbb{R}^4$ which can be seen as a static soliton in $4+1$ dimensions. This is a sequel of the known $SO(D)$ gauged $O(D+1)$ Skyrmions on $\mathbb{R}^D$ in $D=2$ and in $D=3$, like which they are localised to an absolute scale and are topologically stable, their energies being bounded below by the winding number. In the absence of an analytic proof of existence some such solutions are constructed numerically. Two families of solutions are found, of which only one possesses a gauge decoupling limit. The curvatures of both of these solutions decay as $r^{-3}$, a property they share with the Yang-Mills instantons on $\mathbb{R}^4$.

Figures (9)

• Figure 1: The profile of the function $f(r)$ for a typical solution is shown for $\alpha=\beta=1$, both for the gauged and the ungauged cases.
• Figure 2: The profile of the function $w(r)$ for a typical solution is shown for $\alpha=\beta=1$, both for the gauged and the ungauged cases.
• Figure 3: Energy density ${\cal\tilde{E}}(r)$ for solutions with $\alpha=\beta=1$, both for the gauged and the ungauged cases.
• Figure 4: Energy $E$ versus the parameter $\alpha$ for fixed $\beta=0.1$, in logarithmic scale.
• Figure 5: Energy $E$ versus the parameter $\beta$ for fixed $\alpha=0.1$, in logarithmic scale.