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    $\begingroup$ Sorry for not posting a full answer, please observe that $\hat{\Phi} B \hat{\Phi} = \mathrm{Tr}( \hat{\Phi} B ) \hat{\Phi}$ for any adjoint quantity $B=\sum_{a=1}^3 B^a t_a$, (such as the gauge potential). As for your second question, please see the remark in Manton and Sutcliffe following equation (8.70), where they explain that this solution is asymptotically valid in the region outside the core of the monopole, where the gauge field Abelianizes. $\endgroup$
    – David Bar Moshe
    Commented May 12, 2019 at 13:51
  • $\begingroup$ Well, about the second part... it's true, but $D_{\mu}\hat{\Phi}=0 $ is a weaker condition. The results must be valid even inside the core, so that the interpretation of the magnetic field as $b_i=-\frac{1}{2}\epsilon_{ijk}f_{jk}$ still holds (in the sense that it remains a valid interpretation).Also, outside of the core we could just use the stronger condition $D_{\mu}\Phi=0$, which must be satisfied far from the origin. $\endgroup$
    – Othin
    Commented May 14, 2019 at 14:43
  • $\begingroup$ I don't know that book, but is this in the case of a BPS monopole? Do you also have the Bogomol'nyi equation, $\vec{D}\Phi = \vec{B}$? $\endgroup$
    – Subhaneil Lahiri
    Commented May 19, 2019 at 16:32
  • $\begingroup$ Well, the solution is not assumed to be BPS here, it's the general case. $\endgroup$
    – Othin
    Commented May 22, 2019 at 0:36