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

With $D_i\varphi =({\partial }_i-i{A}_i)\varphi$ and $B=F_{12}$, the critically coupled Abelian Higgs model energy in the conventions of the question isFinite energy requires $|\varphi |\to 1$, $D_i\varphi \to 0$, and $B\to 0$ at spatial infinity.

For the Derrick theorem test, preserve gauge covariance by definingThe magnetic, covariant-gradient, and potential energies scale asStationarity at $\lambda =1$ requires $E_B=E_V$, which is possible for a nonconstant finite-energy configuration. The oppositely scaling magnetic and potential terms therefore evade the usual Derrick obstruction and allow vortex solitons.

At infinity write $\varphi \to e^{i\chi }$. The condition $D\varphi \to 0$ gives $A\to d\chi$, and the phase winding defines the Abelian Higgs vortex numberBy Stokes theorem,

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