= Critically coupled Abelian Higgs field equations
{title2=$D_\mu D^\mu\phi+\tfrac12(|\phi|^2-1)\phi=0$}
Use metric $(+--)$, $D_\mu=\partial_\mu-ia_\mu$ and Lagrangian $-f_{\mu\nu}f^{\mu\nu}/4+\overline{D_\mu\phi}D^\mu\phi/2-(1-|\phi|^2)^2/8$. The <Euler-Lagrange field equations> are $D_\mu D^\mu\phi+(|\phi|^2-1)\phi/2=0$ and $\partial_\mu f^{\mu\nu}=\operatorname{Im}(\bar\phi D^\nu\phi)$. Varying $\bar\phi$ gives a kinetic contribution $-D_\mu D^\mu\phi/2$ and potential contribution $(1-|\phi|^2)\phi/4$; variation of $a_\nu$ gives $\partial_\mu f^{\mu\nu}-\operatorname{Im}(\bar\phi D^\nu\phi)$. For static fields in <temporal gauge> $a_0=0$, the positive <Bogomolny vortex equations> imply the scalar equation because $(D_1-iD_2)(D_1+iD_2)=D_1^2+D_2^2+B$. They imply the spatial gauge equations because $\operatorname{Im}(\bar\phi D_1\phi)=-\partial_2|\phi|^2/2$ and $\operatorname{Im}(\bar\phi D_2\phi)=\partial_1|\phi|^2/2$, while $B=(1-|\phi|^2)/2$. The Gauss equation then vanishes identically.
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