I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space: $$\Bbb E/S_n \cong \Bbb P^{n-1}. $$ This is related to a 4d SU(N) Yang-Mills theory compactified on a 2d 2-torus.

question 1. Do we have some examples where the moduli space of flat connection are nontrivial **Calabi-Yau** manifolds?

question 2. I also encounter the word moduli space of instantons.

What are the definitions of "moduli space of flat connections" v.s. moduli space of instantons"? What are the differences and their relations?

anti-self-dual 2-form ($*F_A = -F_A$). and $\Gamma$ in the last sentence is a finite subgroup of $SU(2)$; the instantons there considered need decay conditions on the curvature $F_A$ to form well-behaved moduli spaces (eg the curvature is $L^2$). $\endgroup$ – Mike Miller Oct 26 '18 at 0:36