With $D_i\phi=(\partial_i-iA_i)\phi$ and $B=F_{12}$, the critically coupled <Abelian Higgs model> energy in the conventions of the question is
$$
E=\frac12\int_{\mathbb R^2}d^2x\left[
B^2+|D_1\phi|^2+|D_2\phi|^2
+\frac14(1-|\phi|^2)^2\right].
$$
Finite energy requires $|\phi|\to1$, $D_i\phi\to0$, and $B\to0$ at spatial infinity.

For the <Derrick theorem> test, preserve gauge covariance by defining
$$
\phi_\lambda(x)=\phi(\lambda x),
\qquad
A_i^{(\lambda)}(x)=\lambda A_i(\lambda x).
$$
The magnetic, covariant-gradient, and potential energies scale as
$$
E_B(\lambda)=\lambda^2E_B,
\qquad
E_D(\lambda)=E_D,
\qquad
E_V(\lambda)=\lambda^{-2}E_V.
$$
Stationarity 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 $\phi\to e^{i\chi}$. The condition $D\phi\to0$ gives $A\to d\chi$, and the phase winding defines the <Abelian Higgs vortex> number
$$
N=\frac1{2\pi}\oint_{S^1_\infty}d\chi\in\mathbb Z.
$$
By <Stokes theorem>,
$$
\boxed{\int_{\mathbb R^2}F
=\oint_{S^1_\infty}A=2\pi N}.
$$

Solved by gpt-5.6-sol high.

Solution

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

 Solution

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