= Solution
The <Skyrme model> represents the three <pions> by a field in the <special unitary group> $SU(2)$,
$$
U(x,t)=\sigma(x,t)\mathbf1+i\boldsymbol\pi(x,t)\cdot\boldsymbol\tau,\qquad \sigma^2+|\boldsymbol\pi|^2=1,
$$
with <Pauli matrices> $\tau_a$. Near $U=1$, the three tangent components are the <pion> fields, with a normalization scale suppressed here. Define $L_\mu=U^\dagger\partial_\mu U$. In one useful sign convention with metric $(+---)$, the action consists of
$$
\mathcal L=-c_2\operatorname{tr}(L_\mu L^\mu)+c_4\operatorname{tr}([L_\mu,L_\nu][L^\mu,L^\nu])-c_0\operatorname{tr}(2\mathbf1-U-U^\dagger),
$$
where $c_2,c_4>0$ and the last term is optional, with $c_0\geq0$ proportional to a common <pion> mass squared. The first term is a <nonlinear sigma model> kinetic term, and the second is the four-derivative term of the <Skyrme model>. For a static configuration of size $s$, the two derivative contributions scale as $E_2\propto s$ and $E_4\propto1/s$, whereas a mass term scales as $s^3$. This <Derrick scaling> explains why the four-derivative term can stabilize a finite size instead of allowing collapse.
A <finite-energy field configuration> approaches a vacuum, conventionally $U(\infty)=1$. Compactifying space makes $U$ a map $S^3\to SU(2)\simeq S^3$. The <topological baryon number in the Skyrme model> is its <degree of a map between oriented manifolds>,
$$
\boxed{B=-\frac1{24\pi^2}\int\epsilon_{ijk}\operatorname{tr}(L_iL_jL_k)\,d^3x\in\mathbb Z.}
$$
The sign convention makes the standard decreasing hedgehog have $B=1$. Smooth evolution with the vacuum boundary condition preserves this integer. A <Skyrmion> is a localized soliton in such a sector; the unit soliton, after quantization, models a <nucleon>, and higher positive charges model multi-baryon systems. The topological conservation law is distinct from an ordinary <Noether charge> of <isospin>. Classical <pion> fields are bosonic, so obtaining fermionic <nucleons> also needs the quantum-statistics choice discussed below.
The derivative theory has global <chiral symmetry> $SU(2)_L\times SU(2)_R$, acting by $U\mapsto LUR^\dagger$. The simultaneous pair $(-1,-1)$ acts trivially, so the faithful connected action can also be viewed as $SO(4)$ on $(\sigma,\boldsymbol\pi)$. In the massless theory the choice of vacuum breaks this to the vector subgroup. For fixed $U(\infty)=1$, the <vacuum-preserving symmetry of the Skyrme model> requires $L=R$ and acts by $U\mapsto AUA^\dagger$. This is <isospin>, effectively $SO(3)$ because $A$ and $-A$ act the same way. A usual common pion-mass term explicitly preserves only this vector subgroup; full <chiral symmetry> is then an approximate massless-limit symmetry, not an exact symmetry of that term. Independent axial rotations change the vacuum and are not extra localized rigid-rotor coordinates in a sector with fixed boundary vacuum.
The space-time symmetry is the <Poincaré group>, comprising translations and <Lorentz transformations>, including spatial rotations. <Parity> acts as $U(t,\mathbf x)\mapsto U^\dagger(t,-\mathbf x)$, since <pions> are <pseudoscalars>. For a static finite-energy solution, translations change its position; spatial rotations and <isorotations> change its orientation. These transformations generate <collective coordinates>, but a particular field can be unchanged by certain combined transformations.
For the usual low-charge minimum branches of the standard model, the relevant shapes and density symmetries are the following. These are not claims about every field of a given degree, all excited solutions or arbitrary modified <pion> potentials.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-308-skyrmion-sketches.png]
{title=Schematic Skyrmion configurations}
{description=Schematic shapes of the <Skyrme baryon density> at low charge: sphere, torus, tetrahedral shell and cubic shell. They show shape and symmetry, rather than numerically computed density isosurfaces.}
For $B=1$, the <Skyrmion hedgehog ansatz> is
$$
U_1(\mathbf x)=\cos f(r)+i\sin f(r)\widehat{\mathbf x}\cdot\boldsymbol\tau,\qquad f(0)=\pi,\quad f(\infty)=0.
$$
Its <Skyrme baryon density> is spherical. Indeed $B=-(2/\pi)\int_0^\infty f'\sin^2 f\,dr=1$. A spatial rotation rotates the <pion> direction, so the field itself is invariant under a compensating <isorotation>, not under every spatial rotation alone. Its proper combined stabilizer is a diagonal $SO(3)$.
For $B=2$, the <toroidal two-Skyrmion> has a ring-shaped density with an axial hole and full density symmetry $D_{\infty h}$. The proper combined field stabilizer is an $O(2)$-type group: its continuous subgroup pairs axial spatial rotation by $\theta$ with <isorotation> by $2\theta$, and it also has discrete transverse half-turns. The angular approximation $R(z)=z^2$ makes the axial factor two explicit. A static two-baryon minimum is consequently not simply two separate round unit lumps.
For $B=3$, the <tetrahedral three-Skyrmion> has a tetrahedral shell with four face-hole directions and full density group $T_d$. Its proper rotational group is $T\cong A_4$, the <tetrahedral symmetry> group of order 12. The actual field symmetries again pair these rotations with <isorotations>.
For $B=4$, the <cubic four-Skyrmion> has a cubic shell with six face-hole directions. Its full density group is $O_h$, the <symmetry group of a cube>, and its proper group is $O\cong S_4$, the <rotational symmetry group of a cube> of order 24. Cubic and octahedral symmetry name the same point group; this particular density shape is cubic. Reflections in these full density groups should not be confused with the proper rotation-isorotation group used for angular-momentum quantization.
As a concrete independent check on these shape symmetries, the <rational map approximation for Skyrmions> uses
$$
U(r,z)=\cos f(r)+i\sin f(r)\widehat{\mathbf n}_{R(z)}\cdot\boldsymbol\tau,
$$
with $z=\tan(\theta/2)e^{i\varphi}$ and $\widehat{\mathbf n}_w=(2\operatorname{Re}w,2\operatorname{Im}w,1-|w|^2)/(1+|w|^2)$. Representative maps are
$$
R_1=z,\qquad R_2=z^2,\qquad R_3=\frac{i\sqrt3z^2-1}{z^3-i\sqrt3z},\qquad R_4=\frac{z^4+2i\sqrt3z^2+1}{z^4-2i\sqrt3z^2+1}.
$$
Their degrees are $1,2,3,4$, and with the stated radial boundary conditions their baryon numbers equal these degrees. The <angular Jacobian of a rational map> is $J_R=[(1+|z|^2)|R'|/(1+|R|^2)]^2$. It vanishes at branch directions. $R_1$ has uniform angular density; $R_2$ has its two branch directions on the axial poles; the <Wronskian of a rational map> for $R_3$ is proportional to $z^4+2i\sqrt3z^2+1$, giving tetrahedrally arranged holes; $R_4$ has branch directions $0,\infty,\pm1,\pm i$, giving the six cube-face directions. Also $R_4(iz)=1/R_4(z)$ pairs a spatial quarter-turn with an <isospin> half-turn. These give illustrative approximate fields, not exact analytic solutions or a proof of global <energy> minimality.
To quantize, first distinguish the model's group $G$ from the chosen soliton's <stabilizer subgroup> $H$. A family of the same static <energy> is
$$
U_{A,R,\mathbf X}(\mathbf x)=A U_0\!\left(R^{-1}(\mathbf x-\mathbf X)\right)A^\dagger.
$$
Ignoring translations for the moment, $G=SO(3)_{\rm space}\times SO(3)_{\rm iso}$, and different rigid orientations form the <Skyrmion collective-coordinate orbit> $G/H$. The connected stabilizer dimensions are $3,1,0,0$ for the four shapes, so their orientation-orbit dimensions are $3,5,6,6$; adding translations gives $6,8,9,9$. For the unit hedgehog, treating rotation and <isorotation> as six independent modes would double-count its locked orientation. For higher charges, arbitrary separation of constituent lumps is not an exact flat moduli space in this non-Bogomolny theory.
Let the orbit coordinates depend slowly on time. Integrating the kinetic terms gives a <collective-coordinate effective Lagrangian> $L_{\rm eff}=-M+g_{ab}(q)\dot q^a\dot q^b/2$, with positive kinetic metric after removing redundant stabilizer directions. Its angular blocks are the spatial, <isospin> and mixed <inertia tensors>. <Collective-coordinate quantization> produces a rotor <Hamiltonian> on this orbit, with translations giving center-of-mass <momentum>. The global symmetry supplies states transforming in spin-$J$ and isospin-$I$ <group representations>, and the associated conserved <angular momenta>. <Inertia tensors> determine <energy> splittings; group symmetry alone does not specify those tensors or their numerical energies.
The topology and the static stabilizer supply further <collective-rotation constraints for a Skyrmion>. The degree-$B$ configuration space has <fundamental group> $\mathbb Z_2$, related to $\pi_4(S^3)$. Choosing the nontrivial <Finkelstein-Rubinstein constraints> character gives fermionic unit baryons. Wavefunctions live on the appropriate cover, and a lifted stabilizer operation obeys
$$
\boxed{\widehat D^J(R)\widehat D^I(A)\Psi=\chi_{\rm FR}(A,R)\Psi,\qquad \chi_{\rm FR}\in\{1,-1\}.}
$$
The sign depends on whether the actual field-configuration loop is contractible, not just on whether the density looks symmetric. A $2\pi$ spatial rotation or <isorotation> has sign $(-1)^B$ in this fermionic choice, so
$$
\boxed{(-1)^{2J}=(-1)^{2I}=(-1)^B.}
$$
Thus odd <baryon number> requires half-integer spin and <isospin>; even <baryon number> requires integers. Discrete or continuous combined stabilizer symmetries impose additional restrictions on the allowed pairs and body-fixed rotor states. They must not simply be omitted, or replaced by trivial invariance under every density symmetry. The trivial character is a possible bosonic quantization but would not produce a fermionic <nucleon>. Improper field symmetries, implemented using the model <parity> transformation, can additionally constrain <parity> labels; scalar-density reflection symmetry alone is insufficient to infer those labels.
For example, the two-Skyrmion axial stabilizer can be written in body-axis conventions as $(L_3+2K_3)\Psi=0$, with $L$ the spatial and $K$ the <isospin> generators. A transverse spatial half-turn accompanied by an <isospin> half-turn has the nontrivial sign in the <Finkelstein-Rubinstein constraints>. The integer-spin scalar state $J=I=0$ is therefore excluded despite even $B$. A spin-one, isospin-zero state with zero axial body projection has the required minus sign under that half-turn; the spin-zero, isospin-one channel can also satisfy the constraints. Their relative energies require the <inertia tensors>.
For the unit hedgehog, these requirements reduce to <rotational quantization of a unit Skyrmion>: one orientation $A\in SU(2)$ with $\Psi(-A)=-\Psi(A)$ has $J=I=j=1/2,3/2,\ldots$. With $\hbar=1$ the energies are $M+j(j+1)/(2\Lambda)$, where $\Lambda$ is its <moment of inertia>. Since $U_1^\dagger(-\mathbf x)=U_1(\mathbf x)$, the purely rotational hedgehog band has positive <parity>; vibrational excitations need not share it. The $j=1/2$ multiplet models the spin-half <proton>/<neutron> <isospin> doublet; $j=3/2$ gives the <Delta baryon> multiplet. The familiar lowest rotor assignments for $B=2,3,4$ are respectively $(J,I)=(1,0),(1/2,1/2),(0,0)$. These assignments incorporate the corresponding field-stabilizer constraints and standard inertia ordering, not a prediction from density shape alone. Deformations, vibrational modes, radiation and binding dynamics lie beyond the rigid approximation. The two symmetry roles are therefore
$$
\boxed{\text{model symmetry labels multiplets; soliton symmetry restricts allowed states}.}
$$
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