Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 2 / a / Solution

Away from a zero of the Higgs field, writeSeparating real and imaginary parts of $(D_x+i{D}_y)\varphi =0$ givesThus $B=-\Delta u/2$ away from zeros. Combining this with $B=(1-e^u)/2$ givesIf $\varphi$ has a zero of multiplicity $N$ at the origin, its vortex number is the winding number $N$ and $u=2N\log r+O(1)$. Since $\Delta \log r=2\pi {\delta }^{(2)}(\mathbf{x})$ as a distributional identity, the complete Taubes equation isThe boundary conditions for a single $N$-vortex areThe latter is the finite-energy condition $|\varphi |\to 1$.