= Topological baryon number in the Skyrme model
{title2=$B=-\frac1{24\pi^2}\int\epsilon_{ijk}\operatorname{tr}(L_iL_jL_k)d^3x$}
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 $L_i=U^\dagger\partial_iU$, its <degree of a map between oriented manifolds> is $B=-(24\pi^2)^{-1}\int\epsilon_{ijk}\operatorname{tr}(L_iL_jL_k)d^3x$, 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 $B=-(2N/\pi)\int f'\sin^2f\,dr=N$. This is a topological conservation law, not the <Noether charge> of <isospin>.
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