In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with (i.e. have the same parameter space as) the instantons in 2D for the theory in which the complex projective $n$-space $CP_n$, is replaced by the infinite-dimensional manifold ($\Omega G$ of loops on the structure group $G$."

Question (1): What is the precise restriction of $G$ in Atiyah's on Instantons in 2D and in 4D? Must $G$ be simple compact Lie groups? Or must $G$ be classical groups? Or how general the $G$ can be?

In Atiyah's work [Ref. 1], he also mentions that Donaldson's work [Ref. 2] gives the proof only for the classical groups but it seems likely that his result holds for all $G$.

Question (2): What is the precise restriction of $G$ in Donaldson's work here? Must $G$ be simple compact Lie groups? Or must $G$ be classical groups? Or how general the $G$ can be?

Question (3): The instanton study here in 2D for Atiyah's theory in which the complex projective $n$-space $CP_n$, is replaced by the infinite-dimensional manifold ($\Omega G$ of loops on the structure group $G$.) How is this story of $CP_n$ v.s. $\Omega G=\Omega SU(n)$ here related to the $G=SU(n)$-Yang Mills theory? Here $CP_n$ is finite $n$-dimensional complex manifold, while $\Omega G$ is said to be infinite-dimensional.

Refs:

Instantons In Two-dimensions And Four-dimensions, 1984 - 15 pages Commun.Math.Phys. 93 (1984) 437-451, M.F. Atiyah

Instantons and geometric invariant theory, Comm. Math. Phys. Volume 93, Number 4 (1984), 453-460. S. K. Donaldson

Note: Classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces