Degree charge of an O3 sigma-model lump ($Q=\frac{1}{4\pi }\int \mathbf{n}\cdot ({\partial }_1\mathbf{n}\times {\partial }_2\mathbf{n})d^{2}x$)

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 $Q=(4\pi {)}^{-1}\int \mathbf{n}\cdot ({\partial }_1\mathbf{n}\times {\partial }_2\mathbf{n})d^{2}x$. This normalized target area pullback counts the number of oriented coverings.

For $E=\frac{1}{2}\int (|{\partial }_1\mathbf{n}{|}^2+|{\partial }_2\mathbf{n}{|}^2)$, completing the square gives $E=\frac{1}{2}\int |{\partial }_1\mathbf{n}\pm \mathbf{n}\times {\partial }_2\mathbf{n}{|}^2\pm 4\pi Q$, hence $E\ge 4\pi |Q|$. Saturation yields the Bogomolny equations of the sigma-model lump. With the oriented stereographic projection $\mathbf{n}=(2\mathrm{Re}w,2\mathrm{Im}w,1-|w{|}^2)/(1+|w{|}^2)$, holomorphic rational maps have positive degree and saturate the bound; antiholomorphic maps have negative degree.