Let $(X,g)$ be a closed oriented smooth Riemannian manifold of dimension $4$, and let $E\to X$ be a smooth rank-$2$ Hermitian complex vector bundle whose structure group is reduced to $SU(2)$. Let $A$ be a smooth $SU(2)$-connection on $E$, and let
in the defining representation, and let $|\cdot|$ denote the pointwise norm on $\Lambda^2T^*X\otimes \mathfrak{su}(E)$ induced by $g$ and this inner product. Define the Yang-Mills energy by
\begin{align*}
k:=\langle c_2(E),[X]\rangle\in \mathbb Z.
\end{align*}
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Let
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\begin{align*}
F_A=F_A^+ + F_A^-
\end{align*}
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be the decomposition of $F_A$ into its self-dual and anti-self-dual parts with respect to the Hodge star operator determined by $g$ and the given orientation. Then
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\begin{align*}
YM(A)\ge 8\pi^2|k|.
\end{align*}
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Moreover, equality holds exactly when $F_A^-=0$ if $k>0$, exactly when $F_A^+=0$ if $k<0$, and exactly when $F_A=0$ if $k=0$.