It seems that holomorphic (or rational) maps play a crucial role to relate the following data:

1. Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$ in a 2 dimensional (2d) spacetime.

2. Instanton in a $n$-dimensionalcomplex Projective Space $$\mathbb{P}^n$$ in a 2 dimensional (2d) spacetime.

3. Instanton in an infinite-dimensional Kahler manifold, the loop group, $$ΩG$$ which is the loops on the structure group $G$, in a 2 dimensional (2d) spacetime.

4. Instanton in a Yang Mills (YM) theory with the group $G$, in a 4 dimensional (4d) spacetime.

Question 1: What are the constructions of holomorphic (or rational) maps $F$ from 1. to 2.:

(this shall be more straightforward), $$ F_{1 \to 2}: \mathbb{P}^j \to \mathbb{P}^n, $$ from 2. to 3.: $$ F_{2 \to 3}: \mathbb{P}^n \to ΩG, $$ from 2. to 4. $$ F_{2 \to 4}: \mathbb{P}^n \to G \text{ on YM instantons}? $$ Do we need to fix a particular structure group $G$? (Like special unitary groups?)

Question 2: For the above maps, $F_{I \to J}$, is that the instanton number 1 of the theory-$I$ always maps to the instanton number 1 of the theory-$J$? (Namely what is the ratio of the instanton numbers of the maps? Is that always 1:1?)

Ref: Instantons and geometric invariant theory_Comm. Math. Phys_(1984), 453-460_S. K. Donaldson