= Solution
The <Bogomolny equations> are first-order equations obtained by expressing a static <energy> as nonnegative squares plus a term fixed by the <topological charge>. Vanishing of the squares minimizes the <energy> in that <topological sector>. In particular, the solutions satisfy the full second-order field equations, since their first variation vanishes for compactly supported variations preserving the boundary data. Both monopole and vortex realizations illustrate this mechanism.
For a <Yang-Mills theory> with gauge group <SU(2)> and an adjoint <Higgs field> $\Phi^a$, take the static, purely magnetic <energy>
$$
E=\int_{\mathbb R^3}\left[\frac12 B_i^aB_i^a+
\frac12(D_i\Phi)^a(D_i\Phi)^a+
\frac{\lambda_H}{4}(\Phi^a\Phi^a-v_H^2)^2\right]d^3x.
$$
Here $B_i^a=\frac12\epsilon_{ijk}F_{jk}^a$, and $e$ is the gauge coupling. In the Bogomolny limit $\lambda_H=0$, retain the boundary condition $|\Phi|\to v_H>0$; it breaks <SU(2)> to <U(1)> at infinity even though the scalar potential has been removed. The normalized <Higgs field> on the sphere at infinity defines a map $S^2_\infty\to S^2$ of <topological degree> $n$. In a consistent magnetic orientation the asymptotic flux is $g_m=4\pi n/e$, and
$$
\int_{S^2_\infty}\Phi^a B_i^a\,dS_i=v_Hg_m.
$$
For example, this equality follows by projecting the asymptotic <gauge curvature> on the Higgs direction: its flux is the integral of the target-sphere area form, divided by $e$.
The <gauge-theory Bianchi identity> $D_iB_i=0$ makes the cross term a divergence:
$$
B_i^a(D_i\Phi)^a=\partial_i(\Phi^aB_i^a).
$$
For either sign $s=\pm1$, complete the square:
$$
E=\frac12\int_{\mathbb R^3}|B-sD\Phi|^2d^3x+
s\int_{S^2_\infty}\Phi\cdot B\,dS.
$$
Choosing $s=\operatorname{sgn}n$ gives
$$
\boxed{E\geq\frac{4\pi v_H}{e}|n|,
\qquad B_i=sD_i\Phi\ \text{at equality}.}
$$
These are the <Bogomolny-Prasad-Sommerfield monopole> equations. A nonzero scalar potential would add a positive term; a nontrivial smooth monopole with a core cannot generally saturate this same bound at $\lambda_H>0$. The <Bogomolny monopole equations imply Yang-Mills-Higgs equations>: the <Bianchi identity> already gives $D_iD_i\Phi=0$, and the square-completion argument gives stationarity with respect to the gauge field as well.
An explicit charge-one solution is furnished by the <hedgehog ansatz for a monopole>. To fix its signs, use component conventions $D_i\Phi=\partial_i\Phi-eA_i\times\Phi$ and $F_{ij}=\partial_iA_j-\partial_jA_i-eA_i\times A_j$. With $\rho=ev_Hr$ and $\hat x=x/r$, put
$$
\Phi^a=v_HH(\rho)\hat x^a,\qquad
A_i^a=\frac{1-K(\rho)}{er}\epsilon_{iaj}\hat x^j.
$$
The radial and transverse parts of $D_i\Phi$ are $ev_H^2H'\hat x_i\hat x^a$ and $v_HHK(\delta_{ia}-\hat x_i\hat x^a)/r$. The corresponding parts of $B_i^a$ are $(1-K^2)\hat x_i\hat x^a/(er^2)$ and $-v_HK'(\delta_{ia}-\hat x_i\hat x^a)/r$. Therefore $B=D\Phi$ reduces to
$$
\frac{dK}{d\rho}=-KH,\qquad
\frac{dH}{d\rho}=\frac{1-K^2}{\rho^2}.
$$
The <Prasad-Sommerfield radial monopole solution> is
$$
\boxed{K(\rho)=\frac{\rho}{\sinh\rho},\qquad
H(\rho)=\coth\rho-\frac1\rho.}
$$
Direct differentiation verifies both equations. Near $\rho=0$, $K=1-\rho^2/6+O(\rho^4)$ and $H=\rho/3+O(\rho^3)$, so the apparent angular singularities disappear and the fields are smooth. At infinity $K$ decays exponentially while $H=1-1/\rho+O(e^{-2\rho})$, leaving an Abelian $1/(er^2)$ magnetic field and mass $4\pi v_H/e$.
Translations of this <magnetic monopole> give three <collective coordinates>; a residual <U(1)> phase provides another in the framed description, where gauge transformations are fixed at infinity. Multimonopole solutions of a common sign also saturate the linear bound. Their widely separated constituents have position and phase parameters, with no static force: the magnetic repulsion and massless-Higgs attraction cancel in this limit. Relative motion is described at low speeds by the <moduli-space approximation>. An opposite-sign monopole pair does not solve one common sign of the <Bogomolny equations> and is not protected by that no-force argument.
For the <Abelian Higgs model>, a second realization occurs in the plane. Let $\psi$ be a complex <Higgs field>, $D_i=\partial_i-ieA_i$, $B=\partial_1A_2-\partial_2A_1$, and choose the critically coupled static <energy>
$$
E=\int_{\mathbb R^2}\left[\frac12B^2+|D_i\psi|^2+
\frac{e^2}{2}(|\psi|^2-v_H^2)^2\right]d^2x.
$$
Different conventions for the scalar kinetic coefficient change the numerical coefficients of both the bound and the equations. This normalization gives scalar and gauge masses $\sqrt2ev_H$, explaining the critical balance of their long-distance forces. In three spatial dimensions the planar solutions describe straight strings with this <energy> per unit length.
Finite <energy> requires $|\psi|\to v_H$ and $D_i\psi\to0$. On a large circle the Higgs phase winds by $2\pi n$, so the <magnetic flux quantization of an Abelian Higgs vortex> gives
$$
\Phi_B=\int B\,d^2x=\oint A_i\,dx^i=\frac{2\pi n}{e}.
$$
Writing $J_i=\operatorname{Im}(\bar\psi D_i\psi)$, expansion and integration by parts give
$$
|D_1\psi|^2+|D_2\psi|^2
=|(D_1+iD_2)\psi|^2+eB|\psi|^2+
\partial_1J_2-\partial_2J_1.
$$
The divergence integrates to zero for the usual localized vortex boundary conditions. Thus the <Bogomolny square completion for an Abelian Higgs vortex> is
$$
E=\int_{\mathbb R^2}\left[
|(D_1+iD_2)\psi|^2+
\frac12\{B-e(v_H^2-|\psi|^2)\}^2\right]d^2x
+ev_H^2\Phi_B.
$$
For $n\geq0$ the <Bogomolny vortex equations> and bound are
$$
\boxed{(D_1+iD_2)\psi=0,\quad
B=e(v_H^2-|\psi|^2),\quad E=2\pi v_H^2n.}
$$
For $n<0$, reverse the sign in both first-order equations and obtain $E=2\pi v_H^2|n|$.
For a rotationally symmetric positive <vortex number>, use $\psi=v_H f(r)e^{in\theta}$ and the physical angular component $A_\theta=na(r)/(er)$. The <Bogomolny vortex equations> become
$$
f'=\frac{n(1-a)}r f,\qquad
a'=\frac{e^2v_H^2r}{n}(1-f^2),
$$
with $f(0)=a(0)=0$ and $f(\infty)=a(\infty)=1$. The regular core behaves as $f\sim cr^n$ and $a\sim e^2v_H^2r^2/(2n)$. Linearization at infinity gives exponentially decaying massive tails. Unlike the radial monopole, these planar radial profiles have no elementary closed form in general.
The general multi-vortex solution can be encoded by the <Taubes equation>. Away from zeros, write $\psi=v_He^{h/2+i\chi}$. The first equation gives
$$
A_1=\frac1e\left(\partial_1\chi+\frac12\partial_2h\right),\qquad
A_2=\frac1e\left(\partial_2\chi-\frac12\partial_1h\right).
$$
At zeros $z_a$ of multiplicity $m_a$, the phase contributes circulation $2\pi m_a$. The second equation becomes, including these <Dirac delta> terms,
$$
\Delta h+2e^2v_H^2(1-e^h)
=4\pi\sum_a m_a\delta^{(2)}(x-z_a),\qquad h\to0\ \text{at infinity}.
$$
For prescribed zeros the planar existence result gives a solution; the <prescribed-zero construction of planar Abelian Higgs vortices> reconstructs smooth fields, with $h=2m_a\log|x-z_a|+O(1)$ near each zero. Uniqueness is visible directly: for two solutions with the same zeros, their nonsingular difference $w$ satisfies $\Delta w=2e^2v_H^2(e^{h_1}-e^{h_2})$. Multiplying by $w$ and integrating gives
$$
-\int|\nabla w|^2d^2x
=2e^2v_H^2\int w(e^{h_1}-e^{h_2})\,d^2x\geq0,
$$
hence $w=0$. The <Abelian Higgs vortex moduli space> has arbitrary unordered positions of the $n$ zeros, giving $2n$ real position parameters. The saturated <energy> is independent of their positions, so critically coupled vortices have no static interaction energy. Away from critical coupling this cancellation is lost.
Back to article page
