Consider a Yang-Mills type theory, with algebra $\mathfrak{g}$, defined over a manifold $M$. The action functional$\newcommand{\tr}{\operatorname{tr}}$ is $$S[a] = \frac{1}{2g^2}\int_M\tr_\mathfrak{g}(f\wedge\star f),$$ where $f$ is the curvature of the connection $a$. You can add a $\theta$ term but it won't play a role. My question is what are the instantons over $M$ of arbitrary topology?
In the abelian case, $\mathfrak{g}=\mathbb{R}$, i.e. in the case of Maxwell theory, instantons exist only when $M$ has non-trivial topology (cf. this phys.SE answer). This is easy to see as the only solutions to Maxwell equations with finite, non-zero, action are harmonic 2-forms. This is all. Cool.
In the non-abelian case, I have only seen instantons on $M=\mathbb{R}^4$ (or at most, topologically trivial manifolds). They are given by the ADHM construction. But this isn't all there is to it, right?
More explicitly, let's see what happens over a fixed background flat connection, $A\in\Omega^1(M;\mathfrak{g})$. In this case $f_A$ is an element of $\mathrm{ker}(\mathrm{d}_A)$, where $\mathrm{d}_A = \mathrm{d}+A\wedge\phantom{a}$. $\mathrm{ker}(\mathrm{d}_A)$ has a Hodge decomposition similar to $\mathrm{ker}(\mathrm{d})$, as $$ \mathrm{ker}(\mathrm{d}_A)\cong \mathrm{H}^2(M;\mathfrak{g}|\mathrm{d}_A)\oplus \mathrm{d}_A \Omega^1(M;\mathfrak{g}),$$ where $\mathrm{H}^2(M;\mathfrak{g}|\mathrm{d}_A)$, is the cohomology with respect to $\mathrm{d}_A$. So one would write $$f_A = f_A^0 + \mathrm{d}_A a = f_A^0 + \mathrm{d}a+\frac{1}{2}[A\wedge a],$$ with $f_A^0 \in \mathrm{H}^2(M;\mathfrak{g}|\mathrm{d}_A)$.
Promoting the background connection to a dynamical one, the ADHM construction is concerned with $f = \mathrm{d}_a a\in\mathrm{d}_a\Omega^1(M;\mathfrak{g})$. $f$ can globally take this form if $M$ is such that $\mathrm{H}^2(M;\mathfrak{g}|\mathrm{d}_a)$ is trivial for all connections $a$, selecting thus, flat space. In other words, the ADHM construction, deals with topologically trivial instantons. What is known about topologically non-trivial instantons? I.e. those coming from $\mathrm{H}^2(M;\mathfrak{g}|\mathrm{d}_a)$?
As a concrete question, where the algebra and the manifold are tame enough and we know quite a lot about both, do we know, say, all the $\mathfrak{su}(2)$ instantons on the four-torus?
