Abelian Higgs model

The Abelian Higgs model couples a complex scalar field to a $U(1)$ gauge field and permits magnetic vortex solitons in two spatial dimensions.

An Abelian Higgs vortex has quantized magnetic flux and a Higgs field whose phase winds at spatial infinity. Its vortex number is $N=(2\pi {)}^{-1}\int F$ under the convention $D=d-iA$.

At critical coupling on a surface with conformal metric $g=\Omega dzd\bar{z}$, the vortex equations are

For $u=\log |\varphi {|}^2$ and zeros of multiplicities $N_r$ at $z_r$, the Taubes equation is

Integrating the Taubes equation on a compact surface of area $A$ gives $A\ge 4\pi N$. A nontrivial vortex with nonzero Higgs field requires the strict inequality $A>4\pi N$.