BPST instanton ($A_{\mu }^a=\frac{2{\eta }_{\mu \nu }^a(x-a{)}^{\nu }}{|x-a{|}^2+{\rho }^2}$)

With Skew-Hermitian $SU(2)$ generators obeying $[T_a,T_b]={ϵ}_{abc}{T}_c$ and $\mathrm{tr}(T_a{T}_b)=-{\delta }_{ab}/2$, putThe gauge curvature is self-dual. The identity ${\sum }_{a,\mu ,\nu }({\eta }_{\mu \nu }^a{)}^2=12$ 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 $k=-(8{\pi }^2{)}^{-1}\int \mathrm{tr}(F\wedge F)$. 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.