= BPST instanton
{c}
{title2=$A_\mu^a=\frac{2\eta^a_{\mu\nu}(x-a)^\nu}{|x-a|^2+\rho^2}$}
{wiki}
With <Skew-Hermitian> $SU(2)$ generators obeying $[T_a,T_b]=\epsilon_{abc}T_c$ and $\operatorname{tr}(T_aT_b)=-\delta_{ab}/2$, put
$$
A_\mu^a=\frac{2\eta^a_{\mu\nu}(x-a)^\nu}{|x-a|^2+\rho^2},\qquad
F_{\mu\nu}^a=-\frac{4\rho^2\eta^a_{\mu\nu}}{(|x-a|^2+\rho^2)^2}.
$$
The <gauge curvature> is self-dual. The identity $\sum_{a,\mu,\nu}(\eta^a_{\mu\nu})^2=12$ gives $F^2=192\rho^4/(|x-a|^2+\rho^2)^4$. Radial integration yields action $8\pi^2/g^2$ and charge one with $k=-(8\pi^2)^{-1}\int\operatorname{tr}(F\wedge F)$. The parameters are center $a\in\mathbb R^4$, size $\rho>0$, and global gauge orientation in a framed description. Replacing $\eta$ by $\bar\eta$ reverses duality and charge.
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