Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 49 / 2 / iv / Solution

Since $|\Phi |\to 1$, the Higgs field is nonzero outside a sufficiently large disk. On a large positively oriented circle write $\Phi =\rho e^{i\chi }$ locally, where $\rho >0$. Its normalized phase is a map from the circle to the unit circle. The phase need not be a globally single-valued real function, but its total change is $2\pi N$ for an integer winding number $N$. This integer is the same on all sufficiently large circles because the intervening annulus contains no zeros.

The gauge covariant derivative givesIntegrate around the circle. The assumed rapid decay makes the integral of the second term tend to zero. For example, uniform $|D\Phi |=o(R^{-1})$ and $\rho \to 1$ on radius-$R$ circles suffice. HenceBy Stokes theorem, ${\oint }_{|x|=R}A={\int }_{|x|\le R}B{d}^{2}x$. Integrability of $B$ allows the limit, giving the magnetic flux quantization of an Abelian Higgs vortex:The topological charge $N$ is the degree of $\Phi /|\Phi |$ at infinity, counting net phase winding around the cores. For isolated zeros it is the sum of their signed local winding numbers. In the positive Abelian Higgs vortex construction, each core of multiplicity $m_a$ has local phase $m_a\arg (z-z_a)$, sothe number of vortices counted with multiplicity. Antivortices have the opposite sign. The integer is invariant under smooth gauge transformations defined throughout the disk, whose boundary winding is zero.