= Prescribed-zero construction of planar Abelian Higgs vortices
{title2=$\Delta h+1-e^h=4\pi\sum_am_a\delta_{z_a}$}
Choose positions $z_a$ and positive integer multiplicities $m_a$. Solve the <Taubes equation> with $h\to0$ 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 https://doi.org/10.1007/BF01197552[the original existence theorem]. Set $\chi=\sum_am_a\arg(z-z_a)$, $\Phi=e^{h/2+i\chi}$ 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 $\sum_am_a$. Complex conjugation of $\Phi$ together with $A\mapsto-A$ gives antivortices.
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