Flux quantization and positive tension for an admitted dual vortex

In polar coordinates $(r,\theta )$ write the smooth one-form $\tilde{A}={\tilde{A}}_{r}dr+{\tilde{A}}_{\theta }d\theta$. For a disk of radius $R$, integration of ${\tilde{F}}_{r\theta }={\mathrm{\partial }}_r{\tilde{A}}_{\theta }-{\mathrm{\partial }}_{\theta }{\tilde{A}}_r$ first in $r$ and then in $\theta$ gives ${\int }_{\mathrm{\mid }x\mathrm{\mid }\le R}{\tilde{F}}_{12}{d}^{2}x={\int }_0^{2\pi }{\tilde{A}}_{\theta }(R,\theta )d\theta$: the $\theta$-derivative term cancels by single-valuedness, and the inner boundary term tends to zero because a smooth one-form has ${\tilde{A}}_{\theta }=O(r)$ at the origin.

On a sufficiently large circle choose a continuous phase $M\mathrm{/}\mathrm{\mid }M\mathrm{\mid }=e^{i\vartheta (\theta )}$ with $\vartheta (2\pi )-\vartheta (0)=2\pi n$. The tangential condition in [F1] implies ${\tilde{A}}_{\theta }(R,\theta )-{\mathrm{\partial }}_{\theta }\vartheta (\theta )\to 0$ uniformly in $\theta$ as $R\to \mathrm{\infty }$. Taking the limit in step 1.1 therefore gives ${\mathrm{\Phi }}_D={\int }_0^{2\pi }{\mathrm{\partial }}_{\theta }\vartheta d\theta =2\pi n$.

Since $n\mathrm{\ne }0$, step 2.1 gives nonzero flux, so the smooth function ${\tilde{F}}_{12}$ is nonzero at some point and hence on a neighborhood of that point. Its squared integral is strictly positive. By [F2] and $g_D>0$, the Maxwell term contributes strictly positive energy, while all remaining energy terms are nonnegative; thus $\sigma >0$. Finiteness of $\sigma$ is part of [F1]'s admitted-profile hypothesis.