# Moduli space is a Calabi-Yau manifold?

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?

• Instantons are connections on (bundles over) oriented Riemannian 4-manifolds satisfying $F_A^+ = 0$; that is, the curvature is a self-dual 2-form. Flat connections make sense on any space and any bundle; in differential geometry language they are connections satisfying $F_A = 0$. They are very simple examples of instantons. In the majority examples neither of these are smooth manifolds at all, though it is a beautiful theorem of Kronheimer (and I believe independently Nakajima - I am less familiar with his work) that the space of instantons on $(\Bbb C^2\setminus 0)/\Gamma$ is hyperkahler. – Mike Miller Oct 26 '18 at 0:30
• typo: curvature is an 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 $L^2$). – Mike Miller Oct 26 '18 at 0:36
• thanks Mike Miller, why dont you compose a deliberate answer then? ;) – annie heart Oct 26 '18 at 1:54