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

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 $|\varphi |\to 1$.

Suppose $u$ were positive somewhere. Since it tends to zero at infinity and to $-\infty$ at its vortex zero, it would attain a positive interior maximum away from the origin. At such a maximum the second-derivative test gives $\Delta u\le 0$, whereas the smooth Taubes equation givesThis contradiction with the maximum principle for subharmonic functions provesConsequently the Higgs magnitude of a vortex satisfies $|\varphi |=e^{u/2}\le 1$ everywhere.

For a rotationally symmetric vortex, the equation away from the origin isInsertThe prescribed zero fixes $\alpha =2N$. Since $N\ge 1$, $e^u=e^{\beta }{r}^{2N}[1+o(1)]$, whileMatching the singular and constant terms with $e^u-1=-1+o(1)$ givesThe undetermined $\beta$ is fixed by matching this local expansion to $u\to 0$ at infinity.

For a conformal rescaling of a Riemannian metric $h=\Omega (d{x}^2+d{y}^2)$, the Taubes equation in flat coordinates isThe first solution $u$ has $\Omega =1$ and vortex number $N$. For $\tilde{g}=e^{u}g$, the second solution therefore obeyswhere we used ${\Delta }_{\tilde{g}}=e^{-u}{\Delta }_g$ and converted the covariant delta distribution to the flat coordinate measure. Adding the equation for $u$ givesThus $u+\tilde{u}$ is again a flat-metric Taubes equation solution. Its vortex divisor is the union of the two divisors, with multiplicities added at coincident zeros, and its total vortex number isThis is vortex composition by conformal rescaling.