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

At this critical coupling, construct solutions of the Bogomolny vortex equations rather than superpose isolated vortices, which would not respect the nonlinear field equations. Define $D_j={\partial }_j-i{A}_j$ and $j_k=\mathrm{Im}(\overline{\Phi }{D}_k\Phi )$. Direct differentiation using $[D_1,D_2]\Phi =-iB\Phi$ givesIntegrating the last divergence away under the usual finite-energy vortex boundary conditions and completing the magnetic square yieldsThus positive vortices saturate this Bogomolny bound whenSuch fields solve the static second-order equations: under any compactly supported variation the flux is unchanged, and the first variation of both zero squares vanishes. The bound here is $2\pi N$, twice the bound in the frequently used half-normalized Abelian Higgs model energy.

For the prescribed-zero construction of planar Abelian Higgs vortices, choose arbitrary points $z_a\in \mathbb{C}$ and positive integer multiplicities $m_a$, permitting coincident vortices. Away from the zeros, write $\Phi =e^{h/2+i\chi }$, where $h=\log |\Phi {|}^2$. The first Bogomolny vortex equation givesChoose the phase winding $\chi ={\sum }_a{m}_a\arg (z-z_a)$ locally. Distributionally its curl contributes $2\pi {\sum }_a{m}_a{\delta }_{z_a}$, so the second Bogomolny vortex equation becomes the Taubes equationNear each zero, require $h=2m_a\log |z-z_a|+$ a smooth remainder. The planar existence theorem for this equation gives a solution for every finite multiset of prescribed points, unique in this decaying class; see the original theorem. This elliptic existence result is the nontrivial analytic ingredient of the construction, not a linear superposition assumption.

Reconstruct $\Phi =e^{h/2+i\chi }$ and $A$ from the formulas above. The phase factors have integer winding, so $\Phi$ is single-valued. Near a zero, the modulus and phase combine into $(z-z_a{)}^{m_a}$ times a smooth nonvanishing factor. The singular part of $\nabla \chi$ cancels the rotated logarithmic-gradient part of $A$, so the physical fields extend smoothly through the cores. This produces the desired Abelian Higgs vortices at the selected positions. Their vortex number is $N={\sum }_a{m}_a$. Replacing $(A,\Phi )$ by $(-A,\overline{\Phi })$ constructs the negative-sign antivortex family; the positive construction does not assert arbitrary mixtures of positive and negative cores solve the same first-order equations.