= Bogomolny monopole equations imply Yang-Mills-Higgs equations
{c}
For zero scalar potential, the static field equations are $D_iD_i\Phi=0$ and $D_iF_{ij}=[\Phi,D_j\Phi]$. If $B_i=-D_i\Phi$, the <gauge-theory Bianchi identity> gives the first equation. For the second,
$$
D_iF_{ij}=-\frac12\epsilon_{ijk}[F_{ik},\Phi]
=[B_j,\Phi]=[\Phi,D_j\Phi],
$$
using $F_{ik}=\epsilon_{ik\ell}B_\ell$ and $\epsilon_{ijk}\epsilon_{ik\ell}=-2\delta_{j\ell}$. Thus the first-order equations imply the full second-order equations locally, independently of the radial ansatz.
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