Hyperbolic vortex

At the integrable curvature scale on the Poincare disc, the vortex Taubes equation reduces to the Liouville equation. Holomorphic self-maps of the disc then generate explicit vortex solutions.

For metric $8|dz{|}^2/(1-|z{|}^2{)}^2$, these smooth fields satisfy the Bogomolny vortex equations with one simple zero at the origin. Their curvature coefficient is $F_{12}=4/(1+|z{|}^2{)}^2$, and their physical magnetic field is $(1-|z{|}^2{)}^2/[2(1+|z{|}^2{)}^2]=(1-|\Phi {|}^2)/2$. The magnetic flux is $2\pi$, and the critically coupled energy in the conformal-surface vortex square completion normalization is $\pi$.

For the disc metric $8|dz{|}^2/(1-|z{|}^2{)}^2$ and a holomorphic map $g:D\to D$, the Higgs magnitudesolves the critically coupled vortex equations. Choosing $g(z)=z^{N+1}$ gives a radial vortex of winding $N$.