On page 4 of https://arxiv.org/pdf/1009.5365.pdf, the authors write:
Given any $SO(3)$ connection $A$ on a bundle $E$ over a (not necessarily compact) $4$-manifold, $Z$, let $F(A)$ denote its curvature $2$-form. Define the Pontryagin charge of $A$ to be the real number $$p_1(A)=-\frac{1}{8\pi^2}\int_Z\text{Tr}(F(A)\wedge F(A)),$$ provided this integral converges. When $Z$ is closed, $p_1(A)=\langle p_1(E),[Z]\rangle\in\mathbb{Z}$.
Can you give a few examples of how to compute the Pontryagin charge? Thank you!
If $E$ is the direct sum of an $SO(2) = U(1)$ bundle $L$ and a trivial line bundle $\varepsilon^1$, then
$$p_1(E) = p_1(L\oplus\varepsilon^1) = p_1(L) = -c_2(L\otimes_{\mathbb{R}}\mathbb{C}) = -c_2(L\oplus\overline{L}) = -c_1(L)c_1(\overline{L}) = c_1(L)^2.$$
If you know enough about $L$ and $Z$, then this allows you to compute the Pontryagin charge.
Example: On $\mathbb{CP}^2$, if $E = \mathcal{O}(k)\oplus\varepsilon^1$, then
$$p_1(A) = \langle p_1(E), [\mathbb{CP}^2]\rangle = \langle c_1(\mathcal{O}(k))^2, [\mathbb{CP}^2]\rangle = \langle(k\alpha)^2, [\mathbb{CP}^2]\rangle = \langle k^2\alpha^2, [\mathbb{CP}^2]\rangle = k^2$$
where $\alpha = c_1(\mathcal{O}(1))$ is a generator for $H^2(\mathbb{CP}^2; \mathbb{Z}) \cong \mathbb{Z}$.