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.

• 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 Connections are to YM Theory are like Constant Functions to the Massless Wave Equation, or like Closed 1-Forms are to Maxwell Electromagnetism (in the vector potential form). Applying these analogies, I think some of your questions answer themselves. – Igor Khavkine Oct 29 '18 at 23:36

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.