Topological baryon number in the Skyrme model ($B=-\frac{1}{24{\pi }^2}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$)

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.

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 local Skyrme baryon density is $\mathcal{B}(x)=-{ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)/(24{\pi }^2)$, with $L_i=U^{†}{\partial }_{i}U$ and $B=\int \mathcal{B}{d}^{3}x$. It is invariant under global isorotations and transforms as a scalar under proper spatial rotations, so its contours reveal the geometric shape of a Skyrmion. For the rational map approximation for Skyrmions, $\mathcal{B}=-f^{\prime }{\sin }^{2}f{J}_R/(2{\pi }^2{r}^2)$, where $J_R$ is the angular Jacobian of a rational map. Integrating $J_R$ over the sphere gives $4\pi \mathrm{deg}R$, recovering the integer total charge. A decreasing profile and holomorphic angular map give a nonnegative density, but a general field may have regions of negative density; positivity is not a general topological theorem. The density is different from the energy density.