Prescribed-zero construction of planar Abelian Higgs vortices ($\Delta h+1-e^h=4\pi {\sum }_a{m}_a{\delta }_{z_a}$)

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.