 Logarithmic form of a Bogomolny vortex

Away from zeros of the Higgs field, write $\varphi =e^{u+i\chi }$ with $u=\log |\varphi |$. The Bogomolny vortex equation $(D_1+i{D}_2)\varphi =0$ 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 $\Delta u+(1-e^{2u})/2=2\pi {\sum }_j{n}_j{\delta }^{(2)}(x-X_j)$. The usual Taubes equation uses $2u=\log |\varphi {|}^2$, so its delta coefficients double.