Why is $ \frac{1}{4\pi} \int \int_S dx \, dy\, \vec{n} \cdot ( \partial_x \vec{n} \times \partial_y \vec{n} ) = 1 $?

Question

I am trying to understand this notion of "topological charge" from a paper in Physics Review Letters. They talk about skyrmions and give this integral (here $\vec{n}$ is a vector field on the sphere, from unit vectors in 3-space, which is also a sphere.) : $$ \frac{1}{4\pi} \int \int_S dx \, dy\, \Big[ \vec{n} \cdot ( \partial_x \vec{n} \times \partial_y \vec{n} ) \Big] = 1 \in \mathbb{Z}$$ In this case, this is reduced to a vector calculus problem, there's lots of questions that this result was a whole number. The winding number discusses this notion for a curve in the plane winding around a point or since this an integral over $S^2$ we can talk about "degree". One other possible statement is: $$ \pi_3(S^3) = \mathbb{Z} $$ which is relatd to the Hopf fibration. In this case, there's no more integral. Discussions of K-theory are very technical, there's is also article on vector bundles.

Answer 1

Your wording is a bit vague and the physicists are being sloppy, as usual. $\vec n$ is simultaneously the position vector on the unit sphere and the unit outward normal field. You cannot parametrize the entire sphere by $(x,y)$, but what the integrand is supposed to represent (if you had a surface that could be parametrized by a region in the $xy$-plane) is the surface area element of the sphere, pulled back by the $(x,y)$-parametrization. Taking all these grains of salt into account, the integral is supposed to represent the surface area of the entire sphere, which, of course, is $4\pi$.

In general, if $\mathbf x(u,v)$ is parametrizing a surface $S\subset\Bbb R^3$ as $(u,v)$ varies in a region $D$ in the $uv$-plane, then for a vector field $\vec F$ on $S$, the integral $$\int_D \vec F(\mathbf x(u,v))\cdot \big(\frac{\partial\mathbf x}{\partial u}\times \frac{\partial\mathbf x}{\partial v}\big)du\,dv$$ gives the flux of $\vec F$ across $S$. The flux of the position vector on the unit sphere gives the surface area of the sphere.