# The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons

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:

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

2. 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

• Donaldson's proof uses the ADHM construction in an essential way. I do not know a version of the ADHM construction for groups other than the classical groups, but perhaps that is my ignorance speaking. I would be glad to learn that is the case. Perhaps there is a proof that works more generally following the analytical ideas he mentions but does not pursue. – Mike Miller Oct 17 '18 at 4:01
• @ArunDebray I see you have suggested the synonym (qft) $\to$ (quantum-field-theory). Perhaps mentioning this synonym also in the designated thread on meta might attract more users who can vote on (I am aware that this is not related to the topic of the question - but since this was the question where the (qft) tag was created, it seemed like a reasonable place to mention this. If needed, we can continue this discussion in chat.) – Martin Sleziak Oct 30 '18 at 5:18