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

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

Solved by gpt-5.6-sol high.

For a zero of order $N$ at the origin and no other zeros,Away from the zero, the flat Taubes equation isIf $u$ had a positive interior maximum, then the second-derivative test would give $\Delta u\le 0$ there, while $e^u-1>0$, a contradiction. The boundary values are $-\infty$ at the zero and $0$ at infinity, so the maximum principle gives $u\le 0$. Hence

Solved by gpt-5.6-sol high.

The surface area isIntegrating the Taubes equation over the compact surface eliminates the Laplacian and givesfor a nontrivial solution. For $N=2$, the Bradlow bound therefore requiresEquality is the dissolved-vortex limit with identically vanishing Higgs field and does not give the stipulated ordinary two-vortex solution.

Solved by gpt-5.6-sol high.