Conformal-surface vortex square completion ($V\ge \pi |N|,B={\Omega }^{-1}{F}_{12}$)

For metric $g=\Omega (d{x}^2+d{y}^2)$, set $F_{12}={\partial }_x{A}_y-{\partial }_y{A}_x$ and use the physical magnetic field $B=F_{12}/\Omega$. At critical coupling, define $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$. The identity $|D_x\Phi {|}^2+|D_y\Phi {|}^2=|(D_x+i{D}_y)\Phi {|}^2+F_{12}|\Phi {|}^2+{\partial }_x{j}_y-{\partial }_y{j}_x$ completes the energy into nonnegative squares plus half the magnetic flux and half the current boundary integral. For unit-magnitude vacuum boundary data, the combined boundary contribution is $\frac{1}{2}\oint (A+j)=\pi N$, since $j=d\arg \Phi -A$ there. In the more restrictive covariantly decaying sector the current integral separately vanishes and the flux is $2\pi N$. Equality for positive $N$ requires $(D_x+i{D}_y)\Phi =0$ and $B=(1-|\Phi {|}^2)/2$. The factor $\Omega$ appears instead if the symbol $B$ denotes the coordinate component $F_{12}$.