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In the Rational map approximation for Skyrmions, stereographic coordinate $z$ describes the direction $\hat{\mathbf{x}}\in S^2$, and a degree-$B$ rational map $R(z)=p(z)/q(z)$ defines a unit vector ${\mathbf{n}}_R$. The Skyrme model field is approximated byIts baryon number is the degree $B$ of $R$. Angular integration reduces the energy to a radial variational problem,up to the conventional overall normalization, where the angular functional $\mathcal{I}$ depends only on $R$. One first minimizes $\mathcal{I}$ among degree-$B$ maps and then minimizes over the profile $f$. This efficiently captures the topology, energy, and polyhedral symmetries of many Skyrmions.

Let $\omega =e^{2\pi i/5}$. Since ${\omega }^5=1$,which is a fivefold spatial rotation accompanied by a target-space rotation. The real coefficients also give $R(\bar{z})=\overline{R(z)}$, while direct substitution givesTogether these transformations extend the cyclic symmetry to the stated $D_{5d}$ symmetry.

For $p=z^7-7z^2$ and $q=7z^5+1$, the Wronskian isBesides $z=0$, put $y=z^5$. ThenThus five zeros lie on the circleat arguments $2\pi k/5$, and five lie on the reciprocal circle at arguments $(2k+1)\pi /5$. The polynomial has degree eleven, so the twelfth zero lies at $z=\infty$. On the Riemann sphere, the zeros therefore form two opposite poles and two staggered pentagonal rings: the twelve vertices of an icosahedron.

The angular baryon-density factor is proportional to $|dR/dz{|}^2$ and vanishes at these critical directions. The Wronskian zeros therefore point toward twelve holes in the baryon-density surface. They are the face centers of the dodecahedral $B=7$ Skyrmion, equivalently the vertices of its dual icosahedron, and make its icosahedral symmetry visible directly in the rational map.

Under SU(2) as the three-sphere, a Skyrme model field with $U\to I$ at infinity defines $S^3\to S^3$. Choose $T_i=-i{\tau }_i$ and positive left volume form ${\theta }^1\wedge {\theta }^2\wedge {\theta }^3$. The normalized form is $-(24{\pi }^2{)}^{-1}\mathrm{tr}(U^{-1}dU{)}^3$, so the topological baryon number in the Skyrme model equals $\mathrm{deg}U$. Generator and orientation conventions fix the sign.

The particular quartic-derivative interaction in the Skyrme model, with $L_{\mu }=U^{†}{\partial }_{\mu }U$. Its static energy is positive and scales inversely with soliton size, opposing collapse under Derrick scaling. It is one choice of four-derivative mesonic interaction, not the whole general fourth-order chiral perturbation theory Lagrangian.

A Skyrmion is a finite-energy topological soliton of the Skyrme model. Its baryon number is the degree of the compactified spatial map $S^3\to SU(2)\simeq S^3$.

The unit Skyrmion has a spherically symmetric ansatz with Pauli matrices $\mathbf{\sigma }$, $f(0)=\pi$ and $f(\infty )=0$. The baryon number isThe radial profile is obtained from the Skyrme model variational equation. Spatial rotations and isospin rotations act on the same hedgehog orientation, leaving only three independent rotational collective coordinates.

For a smooth Skyrme model field $U:{\mathbb{R}}^3\to SU(2)$ with $U(\infty )=1$, compactification gives a map $S^3\to S^3$. With $L_i=U^{†}{\partial }_{i}U$, its degree of a map between oriented manifolds is $B=-(24{\pi }^2{)}^{-1}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$, with sign chosen so the standard decreasing hedgehog profile has $B=1$. The integrand is the pullback of the normalized volume form on $SU(2)$. Hence it is integer-valued and unchanged by smooth finite-energy homotopies. In the rational map approximation for Skyrmions, a degree-$N$ angular map and profile $f(0)=\pi ,f(\infty )=0$ give $B=-(2N/\pi )\int f^{\prime }{\sin }^{2}fdr=N$. This is a topological conservation law, not the Noether charge of isospin.

The standard low-energy $B=2$ Skyrmion in the ordinary Skyrme model has a toroidal baryon-density shape, with axial density symmetry $D_{\infty h}$. Its proper combined field symmetries have an $O(2)$-type stabilizer. The angular ansatz $R(z)=z^2$ in the rational map approximation for Skyrmions explains the axial relation $R(e^{i\theta }z)=e^{2i\theta }R(z)$: spatial rotation through $\theta$ is accompanied by isorotation through $2\theta$. The angular density vanishes at the two axial directions, yielding the toroidal hole. This describes the familiar minimal branch and a useful approximate field; it is not an assertion that every degree-two field is toroidal.

The massless derivative Skyrme model has chiral symmetry $U\mapsto LU{R}^{†}$. For the fixed finite-energy boundary condition $U(\infty )=1$, a transformation preserves the same vacuum exactly when $L{R}^{†}=1$, or $L=R$. This leaves conjugation $U\mapsto AU{A}^{†}$, the isospin action. Since $A$ and $-A$ act identically, its effective group is $SO(3)$. Together with spatial translations and rotations it generates the localized zero-mode orbit of a Skyrmion. Independent axial rotations change the massless vacuum and are not additional normalizable rigid orientation modes in that fixed-vacuum sector. A usual positive pion-mass term preserves only conjugation even before the boundary condition is imposed.