Let $X$ be a closed, orientable, smooth 4-manifold. Let us give an orientation to $X$ by setting $e^1, e^2, e^3, e^4$ to be an oriented local orthonormal basis for its cotangent bundle.
Does this induce an orientation on the space of self-dual forms $\Omega^2_+X$? How about $\mathcal{H}^2_+(X)$, the harmonic self-dual forms?
The question is motivated by understanding the Seiberg-Witten invariants. In particular, to orient the moduli space of solutions we need an orientation of the spaces $H^0(X;i\mathbb{R}), H^1(X;i\mathbb{R})$, and $H^2_+(X;i\mathbb{R})$. I am trying to understand which orientations we get for free from an orientation of $X$ and which we don't. For example, $H^0(X;i\mathbb{R})$ is oriented by the cohomology class $1$ after multiplication by $i$. I think $H^1(X;i\mathbb{R})\cong\mathcal{H}^1(X)$ is oriented for free because we have an orientation of the 1-forms.