Degree charge of an O3 sigma-model lump
= Degree charge of an O3 sigma-model lump
{title2=$Q=\frac1{4\pi}\int\mathbf n\cdot(\partial_1\mathbf n\times\partial_2\mathbf n)\,d^2x$}
A unit-vector field $\mathbf n:\mathbb R^2\to S^2$ approaching a fixed value at infinity extends to $S^2\to S^2$. Its <topological charge> is the <topological degree> $Q=(4\pi)^{-1}\int\mathbf n\cdot(\partial_1\mathbf n\times\partial_2\mathbf n)\,d^2x$. This normalized target area pullback counts the number of oriented coverings.