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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 $|\varphi |\to 1$, $D_i\varphi \to 0$ and $B=f_{12}\to 0$ 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 $\varphi$ and $\bar{\varphi }$ independently. After covariant integration by parts,Thus the scalar equation and its complex conjugate areVarying the gauge field gives ${\partial }_{\mu }{f}^{\mu \nu }\delta a_{\nu }$ from the Maxwell term. The matter variation isConsequentlyThese 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 toSince $[D_1,D_2]=-iB$, applying $D_1-i{D}_2$ to the first Bogomolny vortex equation givesThe second equation sets $B=(1-|\varphi {|}^2)/2$, proving the scalar equation, including at smooth vortex cores without dividing by $\varphi$.

For the spatial gauge equations, put $k_i=\mathrm{Im}(\bar{\varphi }{D}_i\varphi )$. The first equation implies $D_1\varphi =-i{D}_2\varphi$, and ${\partial }_i|\varphi {|}^2=2\mathrm{Re}(\bar{\varphi }{D}_i\varphi )$. ThereforeWith the stated metric, $j^i=-k_i$ and the two static Maxwell equations are ${\partial }_{2}B=k_1$ and ${\partial }_{1}B=-k_2$. Both follow by differentiating $B=(1-|\varphi {|}^2)/2$. 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. HenceAs an energy check, the Bogomolny square completion for an Abelian Higgs vortex gives $E\ge \pi N$ 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 $\varphi =e^{u+i\chi }$ there, with $u=\log |\varphi |$. ThenSeparating the real and imaginary parts of $(D_1+i{D}_2)\varphi =0$ yields $u_1-{\chi }_2+a_2=0$ and ${\chi }_1-a_1+u_2=0$. Hencewhere $J(v_1,v_2)=(-v_2,v_1)$ 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 $\nabla u$,Thus, in that gauge, the gauge potential is tangent to contours of $u$, equivalently contours of $|\varphi |$, 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 $B={\partial }_1{a}_2-{\partial }_2{a}_1=-\Delta u$. 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 isThis is the Taubes equation in the paper's amplitude logarithm, rather than the often-used $\log |\varphi {|}^2=2u$ convention. No single smooth globally defined phase or regular contour description at every zero has been assumed.