= Solution
Use metric $(+,-,-)$ and the paper's normalization. A static multi-vortex solution is a smooth <finite-energy field configuration> of <Abelian Higgs vortices> whose complex <Higgs field> has winding at spatial infinity and zeros in the plane. In a positive $N$-vortex sector, the zeros have total multiplicity $N$ and the <magnetic flux> is $2\pi N$; they may be separated or coincident. Finite <energy> requires $|\phi|\to1$, $D_i\phi\to0$ and $B=f_{12}\to0$ at infinity. Then $a_i$ asymptotically compensates the phase <gradient>, so Stokes' theorem relates flux to the phase winding. The positive <Bogomolny vortex equations> describe this same-sign sector; negative flux uses the opposite signs and is not covered by simply mixing positive and negative zero multiplicities.
For the <critically coupled Abelian Higgs field equations>, vary $\phi$ and $\bar\phi$ independently. After covariant integration by parts,
$$
\delta_{\bar\phi}S=\int d^3x\left[-\frac12D_\mu D^\mu\phi+\frac14(1-|\phi|^2)\phi\right]\delta\bar\phi.
$$
Thus the scalar equation and its <complex conjugate> are
$$
\boxed{D_\mu D^\mu\phi+\frac12(|\phi|^2-1)\phi=0.}
$$
Varying the <gauge field> gives $\partial_\mu f^{\mu\nu}\delta a_\nu$ from the Maxwell term. The matter variation is
$$
\delta_a\mathcal L_{\rm matter}=\frac i2\left(\bar\phi D^\nu\phi-\phi\overline{D^\nu\phi}\right)\delta a_\nu=-\operatorname{Im}(\bar\phi D^\nu\phi)\delta a_\nu.
$$
Consequently
$$
\boxed{\partial_\mu f^{\mu\nu}=j^\nu,\qquad j^\nu=\operatorname{Im}(\bar\phi D^\nu\phi).}
$$
These factors use the scalar kinetic coefficient $1/2$; importing the current from a convention with coefficient one would give an incorrect factor two. The current is invariant under <gauge transformations>, and its conservation follows from the field equations and antisymmetry of $f^{\mu\nu}$.
For the static purely magnetic solutions take <temporal gauge> $a_0=0$. The scalar equation reduces to
$$
(D_1^2+D_2^2)\phi+\frac12(1-|\phi|^2)\phi=0.
$$
Since $[D_1,D_2]=-iB$, applying $D_1-iD_2$ to the first <Bogomolny vortex equation> gives
$$
0=(D_1-iD_2)(D_1+iD_2)\phi=(D_1^2+D_2^2+B)\phi.
$$
The second equation sets $B=(1-|\phi|^2)/2$, proving the scalar equation, including at smooth vortex cores without dividing by $\phi$.
For the spatial gauge equations, put $k_i=\operatorname{Im}(\bar\phi D_i\phi)$. The first equation implies $D_1\phi=-iD_2\phi$, and $\partial_i|\phi|^2=2\operatorname{Re}(\bar\phi D_i\phi)$. Therefore
$$
k_1=-\frac12\partial_2|\phi|^2,\qquad k_2=\frac12\partial_1|\phi|^2.
$$
With the stated metric, $j^i=-k_i$ and the two static Maxwell equations are $\partial_2B=k_1$ and $\partial_1B=-k_2$. Both follow by differentiating $B=(1-|\phi|^2)/2$. The time component is the <Gauss law constraint in gauge theory>; it is identically satisfied because both the <electric field> and $j^0$ vanish in this static <temporal gauge>. Hence
$$
\boxed{\text{both Bogomolny vortex equations imply every static field equation}.}
$$
As an <energy> check, the <Bogomolny square completion for an Abelian Higgs vortex> gives $E\geq\pi N$ in this normalization, saturated by these equations. On the plane, the <Abelian Higgs vortex moduli space> admits arbitrary vortex positions, including coincidence; its static <energy> is $\pi N$, so there is no position-dependent static potential. Their slow dynamical interactions are a different question.
The <logarithmic form of a Bogomolny vortex> is only local away from zeros. Write $\phi=e^{u+i\chi}$ there, with $u=\log|\phi|$. Then
$$
D_i\phi=\phi\left[\partial_i u+i(\partial_i\chi-a_i)\right].
$$
Separating the real and imaginary parts of $(D_1+iD_2)\phi=0$ yields $u_1-\chi_2+a_2=0$ and $\chi_1-a_1+u_2=0$. Hence
$$
\boxed{a_1=\chi_1+u_2,\qquad a_2=\chi_2-u_1,\qquad \mathbf a=\nabla\chi-J\nabla u,}
$$
where $J(v_1,v_2)=(-v_2,v_1)$ is the positive quarter-turn. Under a <gauge transformation>, $\chi\mapsto\chi+\omega$ and $\mathbf a\mapsto\mathbf a+\nabla\omega$, as this relation requires.
At regular contour points, the stipulated orthogonality means $\nabla\chi\cdot\nabla u=0$. Since $J\nabla u$ is also perpendicular to $\nabla u$,
$$
\boxed{\mathbf a\cdot\nabla u=0.}
$$
Thus, in that gauge, the <gauge potential> is tangent to contours of $u$, equivalently contours of $|\phi|$, and is parallel to the phase <gradient> wherever that <gradient> is nonzero. This is a gauge-dependent statement about the vector potential, not an assertion that the <magnetic field> is tangent to these curves, nor that the potential is a pure <gradient> or divergence-free. The two <gradient> magnitudes are not fixed by contour orthogonality.
On a phase patch $B=\partial_1a_2-\partial_2a_1=-\Delta u$. Combining with the second equation gives $\Delta u+(1-e^{2u})/2=0$ away from the cores. At a zero of multiplicity $n_j$, $u\sim n_j\log|x-X_j|$, and the phase has winding $2\pi n_j$; $u$ is not a finite real smooth field there. Distributionally, the full equality is
$$
\Delta u+\frac12(1-e^{2u})=2\pi\sum_jn_j\delta^{(2)}(x-X_j).
$$
This is the <Taubes equation> in the paper's amplitude logarithm, rather than the often-used $\log|\phi|^2=2u$ convention. No single smooth globally defined phase or regular contour description at every zero has been assumed.
Back to article page
