= Higgs norm identity for a Bogomolny monopole
{title2=$\Delta|\Phi|^2=|B|^2+|D\Phi|^2$}
For the trace <inner product> and <adjoint covariant derivative>, compatibility gives $\Delta|\Phi|^2=2|D\Phi|^2+2\langle\Phi,D_iD_i\Phi\rangle$. The <gauge-theory Bianchi identity> and $B_i=\pm D_i\Phi$ imply $D_iD_i\Phi=0$. Therefore $\Delta|\Phi|^2=2|D\Phi|^2$, equal to the energy density $|B|^2+|D\Phi|^2$ on a <Bogomolny-Prasad-Sommerfield monopole>. If the energy density is instead defined with an overall factor $1/2$, this Laplacian is twice that density.
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