 't Hooft electromagnetic tensor

For a nonvanishing adjoint Higgs field, put $n=\Phi /|\Phi |$. If an oriented orthonormal internal basis satisfies $[e_a,e_b]=\kappa {\epsilon }_{abc}{e}_c$, the displayed tensor is gauge invariant. In a gauge with $n=e_3$, write $A_j=a_j^a{e}_a$; the transverse quadratic terms cancel, leaving ${\mathcal{F}}_{jk}={\partial }_j{a}_k^3-{\partial }_k{a}_j^3$. Hence it is a closed differential form wherever $\Phi \ne 0$. Under trace normalization of su(2) with $-\mathrm{tr}(XY)$, ${\kappa }^2=2$ and the commutator term has coefficient $1/2$. With $-2\mathrm{tr}(XY)$, its coefficient is one. Omitting this normalization factor generally destroys closure.