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Write $F_{12}={\partial }_1{A}_2-{\partial }_2{A}_1$. Up to a total derivative, the covariant-gradient identity and completion of the magnetic and potential terms giveFor positive flux, equality in the Bogomolny bound therefore gives the Bogomolny vortex equationsThe opposite simultaneous signs describe negative winding.

For the orientation $d{x}^1\wedge d{x}^2>0$, the Hodge star operator isThus

A radial vortex of winding $N>0$ haswith $f(0)=a(0)=0$, $f(1)=1$, $a(1)=N$, and equations

For $e^{2\rho }=8/(1-|z{|}^2{)}^2$, put $u=\log |\Phi |$. Away from zeros, the first vortex equation givesSincethe field $\psi =u-\log (1-z\bar{z})+\log 2$ satisfies the Liouville equationFor holomorphic $g:D\to D$, direct use of the Cauchy-Riemann equations verifiesChoosing $g(z)=z^{N+1}$ gives the radial Witten hyperbolic vortexFor $N=1$ these reduce toSince $Bd{\mu }_g=dA=a_1^{\prime }(r)dr\wedge d\theta$,This directly verifies the Abelian Higgs vortex flux relation for unit winding.

Away from zeros of $\varphi$, writeThe first Bogomolny vortex equation, $(D_x+i{D}_y)\varphi =0$, givesIf the zeros $z_r$ have multiplicities $N_r$, the phase curl and the logarithmic singularities give, distributionally,Equating this with $\Omega (1-e^u)/2$ yields the Taubes equation

Choose positions $z_a$ and positive integer multiplicities $m_a$. Solve the Taubes equation with $h\to 0$ at infinity and $h=2m_a\log |z-z_a|+O(1)$ near each position. The planar existence theorem supplies such a solution for every finite multiset of positions; see the original existence theorem. Set $\chi ={\sum }_a{m}_a\arg (z-z_a)$, $\Phi =e^{h/2+i\chi }$ and $A=\nabla \chi -J\nabla h/2$, with $J(v_1,v_2)=(-v_2,v_1)$. The singular terms cancel in $A$, while $\Phi$ has precisely the selected zeros. These fields satisfy the Bogomolny vortex equations, with $B=(1-|\Phi {|}^2)/2$. Their vortex number is ${\sum }_a{m}_a$. Complex conjugation of $\Phi$ together with $A\mapsto -A$ gives antivortices.