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Put $h=|\Phi {|}^2$, $D_i={\partial }_i-i{A}_i$, $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$, and $W(h)=h(1-h{)}^2/4$. Because the inverse-density Abelian Higgs model contains $B^2/h$, its finite-energy admissible class matters at zeros of the Higgs field. First derive the formal equations on the open set where $h>0$; on the first-order solutions below the potentially singular quotients extend across the zeros.

For variations $\psi =\delta \Phi$ and $a_i=\delta A_i$,Covariant integration by parts in the scalar term, and ordinary integration by parts in the magnetic term, givewhere ${ϵ}_{12}=1$. HereThus the Euler-Lagrange field equations areEquivalently the gauge equation is ${\partial }_j(F_{ji}/h)+j_i=0$. The current sign follows from the explicit choice $D=\partial -iA$.

For the Bogomolny bound for the inverse-density Abelian Higgs model, computeusing $[D_1,D_2]\Phi =-iB\Phi$. Expanding the first square givessoAlso,The current boundary integral vanishes by the assumed rapid decay. Combining the identities therefore givesFor fixed positive flux the minimum is $\boxed{V_{\min }=\pi N}$, attained when the Bogomolny equations areThe quotient at a zero is understood through the finite-energy limit, not by dividing a literal nonzero numerator by zero.

These equations imply the second-order equations directly. Applying $D_1-i{D}_2$ to the first gives $(D_i{D}_i+B)\Phi =0$. On the second equation,which is exactly the scalar equation. The first equation also implies $j_1=-\frac{1}{2}{\partial }_{2}h$ and $j_2=\frac{1}{2}{\partial }_{1}h$. Since $B/h=(1-h)/2$, the gauge equations follow too. All these coefficients are regular across zeros for smooth first-order fields.

For the scalar reduction, write $\Phi =e^{u/2+i\chi }$ away from its zeros. Separating the real and imaginary parts of the first Bogomolny equation givesThus, with the oriented planar Hodge star $\ast d{x}^1=d{x}^2$, $\ast d{x}^2=-d{x}^1$,away from the zeros. A zero $p_j$ of multiplicity $n_j$ has $u=2n_j\log |x-p_j|+\text{smooth}$ and phase winding $2\pi n_j$. The multiplicities are positive: locally the first-order equation is a covariant Cauchy–Riemann equation, and a nonvanishing integrating factor turns its solutions into holomorphic functions. Distributionally,Consequently the logarithmic equation for inverse-density Abelian Higgs vortices isThe delta terms are essential for multi-vortex reconstruction; omitting them would lose the specified zeros and flux.

Here is an explicit regular reconstruction, which also removes all artificial singularities. Given distinct locations $p_j$ and integer multiplicities, putSince $\Delta u_0=4\pi {\sum }_j{n}_j{\delta }_{p_j}-g$, solving the smooth nonlinear elliptic problem for $v=u-u_0$ amounts toLet $z=x^1+i{x}^2$ and let $z_j$ represent $p_j$. A smooth solution with the topological decay reconstructsThese formulas are smooth at every prescribed zero and agree with the phase formula off the zeros. Taking a curl and using the equation for $v$ verifies $B=e^u(1-e^u)/2$; substitution verifies the other first-order equation. At infinity the phase winds $N$ times, producing flux $2\pi N$. Thus multi-vortex construction reduces to solving one scalar elliptic boundary-value problem for specified positions and multiplicities, followed by these explicit formulas.

For the topological branch, $u\le 0$: a positive interior maximum would have $\Delta u\le 0$, whereas the equation there gives $\Delta u=e^u(e^u-1)>0$. At infinity, the linearized scalar equation is $\Delta u-u=0$. Comparison barriers and elliptic estimates give the decaying tail for regular topological solutions, and hence the required decay of $B,D\Phi$ and $1-h$. The nonlinear elliptic problem determines $v$; the reconstruction formulas do not constitute an arbitrary superposition of single-vortex profiles.