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

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 313 / 1 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-313/1/b/i/solution