Let $X$ be a closed oriented smooth four-manifold, and let $E\to X$ be a smooth complex vector bundle of rank $2$ whose structure group is reduced to $SU(2)$. Let $A$ be an $SU(2)$-connection on $E$, and let $F_A\in \Omega^2(X;\mathfrak{su}(E))$ denote its curvature. Assume the gauge-theory Chern-Weil convention for skew-Hermitian curvature in which the second Chern-Weil form is the degree-four component of

\begin{align*} \det\left(I+\frac{1}{2\pi}F_A\right), \end{align*}

with $\operatorname{tr}$ taken in the defining rank-$2$ complex representation. Then $\operatorname{tr}(F_A\wedge F_A)\in\Omega^4(X;\mathbb C)$ is integrated over $X$ by the oriented integration map on top-degree forms, denoted $\int_X^{\mathrm{or}}$. The second Chern number satisfies

\begin{align*} c_2(E)[X]=-\frac{1}{8\pi^2}\int_X^{\mathrm{or}} \operatorname{tr}(F_A\wedge F_A). \end{align*}

The integer

\begin{align*} k:=c_2(E)[X] \end{align*}

is called the instanton number of $E$.