Write $F_{12}=\partial_1A_2-\partial_2A_1$. Up to a total derivative, the covariant-gradient identity and completion of the magnetic and potential terms give
$$
\begin{aligned}
V={}&\frac12\int_D\left[
e^{-2\rho}\left(F_{12}-\frac{e^{2\rho}}2(1-|\Phi|^2)\right)^2
+|D_1\Phi+iD_2\Phi|^2\right]d^2x\\
&+\frac12\int_DF_{12},d^2x.
\end{aligned}
$$
For positive flux, equality in the <Bogomolny bound> therefore gives the <Bogomolny vortex equations>
$$
\boxed{D_1\Phi+iD_2\Phi=0,
\qquad
F_{12}=\frac{e^{2\rho}}2(1-|\Phi|^2)}.
$$
The opposite simultaneous signs describe negative winding.

For the orientation $dx^1\wedge dx^2>0$, the <Hodge star operator> is
$$
\boxed{*1=e^{2\rho}dx^1\wedge dx^2},
$$
$$
\boxed{*dx^1=dx^2,
\qquad *dx^2=-dx^1},
$$
$$
\boxed{*(dx^1\wedge dx^2)=e^{-2\rho}}.
$$
Thus
$$
\boxed{B=*dA=e^{-2\rho}F_{12}
=\frac12(1-|\Phi|^2)}.
$$

A radial vortex of winding $N>0$ has
$$
\boxed{\Phi=f(r)e^{iN\theta},
\qquad A=a(r)d\theta},
$$
with $f(0)=a(0)=0$, $f(1)=1$, $a(1)=N$, and equations
$$
\boxed{f'=\frac{N-a}{r}f,
\qquad
\frac{a'}r=\frac{e^{2\rho}}2(1-f^2)}.
$$

For $e^{2\rho}=8/(1-|z|^2)^2$, put $u=\log|\Phi|$. Away from zeros, the first vortex equation gives
$$
\Delta u=-\frac{e^{2\rho}}2(1-e^{2u}).
$$
Since
$$
\Delta[-\log(1-z\bar z)]=\frac4{(1-|z|^2)^2},
$$
the field $\psi=u-\log(1-z\bar z)+\log2$ satisfies the <Liouville equation>
$$
\boxed{\Delta\psi=e^{2\psi}}.
$$
For holomorphic $g:D\to D$, direct use of the <Cauchy-Riemann equations> verifies
$$
\boxed{e^{2\psi}=\frac{4|g'(z)|^2}{(1-|g(z)|^2)^2}}.
$$
Choosing $g(z)=z^{N+1}$ gives the radial <Witten hyperbolic vortex>
$$
\boxed{\Phi(z)=
\frac{(N+1)z^N(1-|z|^2)}{1-|z|^{2N+2}}},
$$
$$
\boxed{A=a_N(r)d\theta,
\qquad
a_N(r)=\frac{2r^2}{1-r^2}
-\frac{(2N+2)r^{2N+2}}{1-r^{2N+2}}}.
$$
For $N=1$ these reduce to
$$
f(r)=\frac{2r}{1+r^2},
\qquad
a_1(r)=\frac{2r^2}{1+r^2}.
$$
Since $Bd\mu_g=dA=a_1'(r)dr\wedge d\theta$,
$$
\int_DBd\mu_g
=2\pi\int_0^1a_1'(r)dr
=2\pi[a_1(1)-a_1(0)]
=\boxed{2\pi}.
$$
This directly verifies the <Abelian Higgs vortex> flux relation for unit winding.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 3 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-313/3/solution