Bogomolny vortex equation

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

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.

With $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$, use the critical-coupling energy $E=\int [B^2/2+|D_i\varphi {|}^2/2+(1-|\varphi {|}^2{)}^2/8]d^{2}x$. For positive vortex number, integration by parts givesThe boundary divergence vanishes for the usual decaying vortex fields, and magnetic flux is $2\pi N$. The two squares vanish exactly at the Bogomolny vortex equations. Reversing both signs treats negative $N$. Energy coefficients and the numerical bound change together under other normalizations.

For metric $g=\Omega (d{x}^2+d{y}^2)$, set $F_{12}={\partial }_x{A}_y-{\partial }_y{A}_x$ and use the physical magnetic field $B=F_{12}/\Omega$. At critical coupling, define $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$. The identity $|D_x\Phi {|}^2+|D_y\Phi {|}^2=|(D_x+i{D}_y)\Phi {|}^2+F_{12}|\Phi {|}^2+{\partial }_x{j}_y-{\partial }_y{j}_x$ completes the energy into nonnegative squares plus half the magnetic flux and half the current boundary integral. For unit-magnitude vacuum boundary data, the combined boundary contribution is $\frac{1}{2}\oint (A+j)=\pi N$, since $j=d\arg \Phi -A$ there. In the more restrictive covariantly decaying sector the current integral separately vanishes and the flux is $2\pi N$. Equality for positive $N$ requires $(D_x+i{D}_y)\Phi =0$ and $B=(1-|\Phi {|}^2)/2$. The factor $\Omega$ appears instead if the symbol $B$ denotes the coordinate component $F_{12}$.

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

Choose positions $z_a$ and positive integer multiplicities $m_a$. Solve the Taubes equation with $h\to 0$ at infinity and $h=2m_a\log |z-z_a|+O(1)$ near each position. The planar existence theorem supplies such a solution for every finite multiset of positions; see the original existence theorem. Set $\chi ={\sum }_a{m}_a\arg (z-z_a)$, $\Phi =e^{h/2+i\chi }$ and $A=\nabla \chi -J\nabla h/2$, with $J(v_1,v_2)=(-v_2,v_1)$. The singular terms cancel in $A$, while $\Phi$ has precisely the selected zeros. These fields satisfy the Bogomolny vortex equations, with $B=(1-|\Phi {|}^2)/2$. Their vortex number is ${\sum }_a{m}_a$. Complex conjugation of $\Phi$ together with $A\mapsto -A$ gives antivortices.

If $u$ solves the Taubes equation for a metric $g$ and $\tilde{u}$ solves it for $e^{u}g$, then $u+\tilde{u}$ solves it for $g$. The vortex divisors add, so the total vortex number is $N+\tilde{N}$.

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 $A>4\pi N$.

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 $N$.