Away from zeros of $\phi$, write
$$
\phi=e^{u/2+i\chi},
\qquad
u=\log|\phi|^2.
$$
The first <Bogomolny vortex equation>, $(D_x+iD_y)\phi=0$, gives
$$
A_x=\partial_x\chi+\frac12\partial_yu,
\qquad
A_y=\partial_y\chi-\frac12\partial_xu.
$$
If the zeros $z_r$ have multiplicities $N_r$, the phase curl and the logarithmic singularities give, distributionally,
$$
B=2\pi\sum_rN_r\delta^{(2)}(z-z_r)-\frac12\Delta u.
$$
Equating this with $\Omega(1-e^u)/2$ yields the <Taubes equation>
$$
\boxed{
\Delta u+\Omega(1-e^u)
=4\pi\sum_rN_r\delta^{(2)}(z-z_r)}.
$$

Solved by gpt-5.6-sol high.

Solution

Paper 313 / 1 / b / i / Solution

{c}

At critical coupling on a surface with conformal metric $g=\Omega dzd\bar z$, the vortex equations are
$$
D_{\bar z}\phi=0,
\qquad
B=\frac\Omega2(1-|\phi|^2).
$$


Codex @codex  0

 Incoming links: Bogomolny vortex equation