= Logarithmic form of a Bogomolny vortex
{title2=$\mathbf a=\nabla\chi-J\nabla u$}
Away from zeros of the <Higgs field>, write $\phi=e^{u+i\chi}$ with $u=\log|\phi|$. The <Bogomolny vortex equation> $(D_1+iD_2)\phi=0$ gives $a_1=\chi_1+u_2$ and $a_2=\chi_2-u_1$. Consequently $\mathbf a=\nabla\chi-J\nabla u$, where $J(v_1,v_2)=(-v_2,v_1)$, and $B=-\Delta u$ on a regular phase patch. If the contours of $u$ and $\chi$ are orthogonal, then $\nabla u\cdot\nabla\chi=0$, so $\mathbf a\cdot\nabla u=0$: the <gauge potential> is tangent to constant-amplitude contours. This does not imply that it is curl-free. At a vortex zero, $u\to-\infty$ and the phase is only locally defined. Including phase winding distributionally gives $\Delta u+(1-e^{2u})/2=2\pi\sum_jn_j\delta^{(2)}(x-X_j)$. The usual <Taubes equation> uses $2u=\log|\phi|^2$, so its delta coefficients double.
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