It is known that the SO(3) gauge bundle on the 4-manifold $M=\mathbb{CP}^2$, we can have the minimal instanton number 1.
Here we denote $SO(3)$ connection $A$ on a bundle $E$ over a $4$-manifold $M$, let $F(A)$ denote its curvature $2$-form.
$$p_1(A)=-\frac{1}{8\pi^2}\int_M\text{Tr}(F(A)\wedge F(A)).$$
Question, do we know the explicit global structure of gauge bundle of $SO(3)$ on this $M=\mathbb{CP}^2$ given the minimal instanton number 1? Say $$ A_\mu=? $$ for $\mu$ parametrizes the 4-coordinates of $M=\mathbb{CP}^2$? In principle, we should have the $SO(3)$ gauge flux through the 2-submanifold $\mathbb{CP}^1$ with $$ \int_{\mathbb{CP}^1} F(A)/(2\pi)= 1. $$ But how about the full functions of $A_\mu$ on $\mathbb{CP}^2$?