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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?

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  • $\begingroup$ 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. $\endgroup$
    – Igor Khavkine
    Commented Oct 29, 2018 at 23:36

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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.

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    $\begingroup$ It is not necessarily the case that to speak of a phase space one needs to break the spacetime symmetry. I agree that this is what is often done, but as Crnkovic and Witten showed some 32 years ago, this is not actually required. (See @article{Crnkovic:1986ex, author = "Crnkovic, Cedomir and Witten, Edward", title = "{COVARIANT DESCRIPTION OF CANONICAL FORMALISM IN GEOMETRICAL THEORIES}", year = "1986", reportNumber = "Print-86-1309 (PRINCETON)"} $\endgroup$
    – José Figueroa-O'Farrill
    Commented Oct 29, 2018 at 23:17
  • $\begingroup$ thanks very much for both of your inputs. +1. $\endgroup$
    – wonderich
    Commented Oct 31, 2018 at 22:36

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