Questions: I would like to have a precise description of the meanings of the Moduli Space of Flat Connections, such that it is understandable by mathematical physicists and physicists.

For 3d Chern-Simons (CS) theory, I suppose that the following is an interpretation.

the Moduli Space of Flat Connections of CS theory = the phase space of the classical Chern-Simons field theory $\equiv$ the classical phase space

the quantization of the classical phase space = the Hilbert space of ground states and zero modes of quantum Chern-Simons theory.

By quantization, we mean that replacing the Poisson bracket in the classical phase space $\{x, p\}$; by the commutator of matrix operators $[X, P]$.

For 4d Yang-Mills (YM) theory, what would it be the Moduli Space of Flat Connections?

YM flat connections are in the classical phase space?

YM non-flat connections are also in the classical phase space?

the Moduli Space of YM theory = the phase space of the classical YM field theory $\equiv$ the classical phase space of both flat connections and non-flat connections?

the Moduli Space of Flat Connections of YM theory = the classical phase space of only the flat connections part?

What will be the quantization of the Moduli Space of YM theory?

What will be the quantization of the Moduli Space of Flat Connections of YM theory?

Flat Connectionsare toYM Theoryare likeConstant Functionsto theMassless Wave Equation, or likeClosed 1-Formsare toMaxwell Electromagnetism(in the vector potential form). Applying these analogies, I think some of your questions answer themselves. $\endgroup$ – Igor Khavkine Oct 29 '18 at 23:36