= 't Hooft electromagnetic tensor
{c}
{title2=$\mathcal F_{jk}=\langle F_{jk},n\rangle-\kappa^{-2}\langle n,[D_jn,D_kn]\rangle$}
= 't Hooft tensor
{c}
{synonym}
For a nonvanishing adjoint <Higgs field>, put $n=\Phi/|\Phi|$. If an oriented orthonormal internal basis satisfies $[e_a,e_b]=\kappa\varepsilon_{abc}e_c$, the displayed tensor is gauge invariant. In a gauge with $n=e_3$, write $A_j=a_j^ae_a$; the transverse quadratic terms cancel, leaving $\mathcal F_{jk}=\partial_ja_k^3-\partial_ka_j^3$. Hence it is a <closed differential form> wherever $\Phi\ne0$. Under <trace normalization of su(2)> with $-\operatorname{tr}(XY)$, $\kappa^2=2$ and the commutator term has coefficient $1/2$. With $-2\operatorname{tr}(XY)$, its coefficient is one. Omitting this normalization factor generally destroys closure.
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