= Bogomolny square completion for an Abelian Higgs vortex
{c}
{title2=$E\geq\pi|N|$}
With $D_i=\partial_i-iA_i$ and $B=\partial_1A_2-\partial_2A_1$, use the critical-coupling energy $E=\int[B^2/2+|D_i\phi|^2/2+(1-|\phi|^2)^2/8]d^2x$. For positive <vortex number>, integration by parts gives
$$
E=\int\left[\frac12|(D_1+iD_2)\phi|^2+\frac12\left(B-\frac{1-|\phi|^2}{2}\right)^2\right]d^2x+\frac12\int B\,d^2x.
$$
The boundary divergence vanishes for the usual decaying vortex fields, and <magnetic flux> is $2\pi N$. The two squares vanish exactly at the <Bogomolny vortex equations>. Reversing both signs treats negative $N$. Energy coefficients and the numerical bound change together under other normalizations.
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