The Precise Meaning of the Moduli Space of Flat Connections? 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.


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*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?


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*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?
 A: Let $P \to M$ be a principal $G$-bundle. The moduli space of flat connections on $P$ is, by definition, the space $\mathcal{M} = \mathcal{C}_0 / \mathcal{G}$, where $\mathcal{C}_0$ denotes the subspace of flat connections on $P$ and $\mathcal{G}$ is the group of (local) gauge transformations. Whether $M$ is $3$ or $4$-dimensional does not make a difference (for the definition, the properties of $\mathcal{M}$ of course depend on the topology of $M$).
If you want to speak of a configuration or phase space, you need to split your equations into space and time direction (at least in the naive interpretation you need an evolution to have a meaningful notion of a phase space). So, for example, for 4-dimensional Yang-Mills you choose a splitting $M = \mathbb{R} \times \Sigma$ and decompose the YM-equations according to get the non-abelian analog of the Maxwell equations. The configuration space of the theory is then the space of $G$-connections over $\Sigma$ and phase space is the cotangent bundle.
