# Complex projective algebraic variety, moduli space of flat connections, and instantons

In Looijenga's work below, if I understand correctly, it shows that

Statement 1: At an algebraic variety, the moduli space of SU($$N$$) flat connections on a 2-torus $$T^2$$ is given by the space of $$CP^{N-1}$$.

In Friedman-Morgan-Witten's work below, roughly speaking, it shows that

Statement 2: Yang-Mills instanton on $$T^2 \times \Sigma$$ (where $$\Sigma$$ is a Riemann surface) is given by $$CP^{N-1}$$ instanton on $$\Sigma$$.

Questions:

1. How are these two works and two statements related to each other? Can we use one to derive the other?

2. Do we have an analogous statement (of 2) for Yang-Mills instanton on $$V^{4-d} \times M^d$$ such that this instanton is also given by some other theory X's instantons living on $$M^d$$? If so, what can the theory X be?

E. Looijenga, Root Systems And Elliptic Curves, Invent. Math. 38 (1977).

E. Looijenga, Invariant Theory For Generalized Root Sytems, Invent. Math. 61 (1980).

R. Friedman, J. Morgan, and E. Witten, Vector bundles and F theory, Commun. Math. Phys. 187 (1997) 679–743.