### What is a Skyrmion?

In 1961 before the advent of quantum chromodynamics (QCD) the nuclear physicist T.H.R. Skyrme conjectured that the interior of a nucleus is dominated by a medium formed from three pion fields [1]. He introduced the Skyrme model, a non-linear sigma model, with the intention to describe baryons as the quantised soliton solutions of a field theory which involves only bosonic degrees of freedom. The model is understood as an intermediate between the traditional models which represent the nucleons as point particles interacting through a potential, and a complete description based on quarks and gluons [2]. The pion fields $\mathbf{\pi} = (\pi_1,\pi_2,\pi_3)$ are combined into a $SU(2)$-valued field \begin{equation} U(\mathbf x) = \sqrt{1 - \mathbf{\pi}(\mathbf x)\cdot\mathbf{\pi}(\mathbf x)}~\mathbb{1} + i \mathbf{\pi}(\mathbf x) \cdot \mathbf{\sigma}~, \end{equation} where $\mathbf{\sigma}$ is the vector of Pauli matrices and we have suppressed a possible time dependence of the fields. For static fields the energy in the Skyrme model is given by \begin{equation} E = \int d^3\mathbf r \left( -\frac{1}{2} \text{Tr}(R_i R_i) - \frac{1}{16} \text{Tr}([R_i,R_j][R_i,R_j]) \right)~, \end{equation} where we have introduced an associated current $R_i=(\partial_i U)U^\dagger$ and $[\cdot,\cdot]$ denotes the commutator. The vacuum is represented by $U(\mathbf x)=\mathbb{1}$. For the energy to be finite, $U$ must approach a constant at infinity [3]. The energy is invariant under translations and rotations in $\mathbb{R}^3$ and also under the transformation $U \to A U A^\dagger$ with $A \in SU(2)$, one may thus choose $U(\mathbf r \to \infty)=\mathbb{1}$. Effectively, due to this boundary condition space is then topologically (but not metrically) compactified to $\mathbb{S}^3$, and since the group manifold of $SU(2)$ is also $\mathbb{S}^3$, $U$ defines a map from $\mathbb{S}^3$ to $\mathbb{S}^3$. The structure of the homotopically distinct maps $U$ is given by the third homotopy group $\pi_3(\mathbb{S}^3)$ which happens to be isomorphic to $\mathbb{Z}$. The space of all maps $U: \mathbb{S}^3 \to \mathbb{S}^3$ decomposes into distinct subsets characterised by an integer-valued topological charge $B=\int d^3\mathbf r~\mathcal{B}$ with the topological charge density $\mathcal{B}$ given by \begin{equation} \mathcal{B} = -\frac{1}{24 \pi^2} \epsilon_{ijk} \text{Tr}(R_i R_j R_k)~. \end{equation} The minimal energy solutions for each $B$ are called Skyrmions and their energy is identified with their mass and $B$ with the Baryon number of the nucleus. The $B=1$ Skyrmion has the spherically symmetric hedgehog form [3] \begin{equation} U(\mathbf x) = \exp(i f(r) \hat{\mathbf{r}} \cdot \mathbf{\sigma})~, \end{equation} where $f(r)$ is a radial profile function obeying an ordinary differential equation with the boundary conditions $f(0)=\pi$ and $f(r\to\infty)=0$.Skyrmions in their original sense are therefore smooth, topologically stable extremal field configurations which are trivial at spatial infinity and have a finite energy. They are defined by surjective mappings into the order parameter space $\mathbb{S}^3$ and characterised by a non-trivial topological charge $B$. Since the $n$th homotopy group of $\mathbb{S}^n$ is isomorphic to $\mathbb{Z}$ for any $n\ge1$, a more permissive definition of Skyrmion is given by

A Skyrmion is a smooth field configuration defined by a topologically non-trivial, surjective mapping from a base manifold $\mathcal{M}$ into the order parameter space $\mathcal{T}\simeq\mathbb{S}^n$, trivial on the surface of $\mathcal{M}$ and characterised by a finite integer-valued topological charge.

Fig. 1 shows the construction recipe for $\mathcal{M} = \mathbb{R}^2$ and $\mathcal{T}=\mathbb{S}^2$. We start with the identity map, $\hat{\Omega}(\hat{\mathbf x})=\hat{\mathbf x}$, with $\hat{\mathbf x} \in \mathbb{S}^2$ which can be visualised as a hedgehog configuration (c.f. Fig. 1, top left). The stereographic projection $\mathcal{P}$ maps the sphere onto the two-dimensional plane, $\mathcal{P}(\hat{\mathbf x})\in \mathbb{R}^2$ and thus a Skyrmion configuration is given by the mapping $\hat{M}: \mathbb{R}^2 \to \mathbb{S}^2$, $\hat{M}: \mathbf r \mapsto \hat{\Omega} \circ \mathcal{P}^{-1}(\mathbf r)$. The corresponding topological charge $W$ is given by $\int d^2\mathbf r~\mathcal{W}$ where we have defined the topological charge density $\mathcal{W}$ as \begin{equation} \mathcal{W} = \frac{1}{8\pi} \hat{M} \cdot (\partial_x \hat{M} \times \partial_y \hat{M})~. \end{equation} $W$ counts the number of times the mapping $\hat{M}$ sweeps out the target manifold $\mathbb{S}^2$. From the construction recipe above it is obvious that for the Skyrmion (Fig. 1, lower left) $W=-1$.