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There is a key result in 2+1 dimensional Chern-Simons theory, which was first discussed in ref.1.: the Hilbert space of the theory, when quantised on $T^2\times\mathbb R$, is isomorphic to $$ \frac{\Lambda_w}{W\ltimes k\Lambda^\vee_r}\tag{2.14} $$ where $\Lambda_w$ is the weight lattice of the gauge group $G$ (assumed simple and simply-connected), $W$ the Weyl group of $\mathfrak g$, $k\in\mathbb Z$ the coefficient of the Chern-Simons form, and $\Lambda_r^\vee$ the co-root lattice of $G$.

Most papers nowadays simply refer to the original reference, or reproduce the argument almost verbatim. The problem is that I cannot quite follow the original reference, and the review articles follow the exact same reasoning, so I cannot follow them either. I have several questions, and I will split them into several posts.

Let $A$ be a $\mathfrak g$-valued one-form that lives in $T^2\times\mathbb R$, and decompose it into its time and spatial components, $A=A_0\mathrm dt+\tilde A$. Due to the equations of motion, $\tilde A$ is $\tilde{\mathrm d}$-flat, $\tilde{\mathrm d}\tilde A+\tilde A^2\equiv 0$, with $\mathrm d=\mathrm dt\partial_t+\tilde{\mathrm d}$. The authors claim that, being $\tilde{\mathrm d}$-flat, it can be decomposed as $$ \tilde A=-\mathrm dUU^{-1}+U\theta(t) U^{-1}\tag{2.9} $$ where $U$ is a single-valued map from $T^2\times\mathbb R$ to $G$, and where $\theta(t)$ is a $\mathfrak g$-valued one form that depends on $t$ only.

Where does this come from?

The authors offer no reference nor explanation. Is this a general result? Does it work only for the torus? What about other surfaces? What about flat connections on a higher-dimensional manifold?

I guess the problem is purely two-dimensional, and the dependence on $t$ is parametrical. In other words, we should think of this as trying to solve $\tilde D\tilde A\equiv0$ on $T^2$, where everything depends implicitly on $t$, as a parameter. Thus, the claim is that $\theta$ is constant over $T^2$.

References.

  1. Elitzur, Moore, Schwimmer, Seiberg - Remarks on the canonical quantization of the Chern-Simons-Witten theory.
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  1. Flat $G$-principal connections on $X$ modulo gauge transformations are in bijection with group homomorphisms $\pi_1(X)\to G$ modulo conjugation. The proof of this is "well-known" and can be found e.g. in Kobayashi's and Nomizu's book on differential geometry. A nice exposition is in chapter 5 of "Moduli Spaces of Flat Connections" by Daan Michiels.

    The basic idea is that a flat connection is already determined by its holonomy around non-contractible loops, since it is trivial around contractible loops due to $\oint_\gamma A = \int_S F = 0$ where $\partial S = \gamma$ for some surface $S$ and through an application of (non-Abelian) Stokes. Due to the same argument the holonomies of homotopic loops are equal, and the homotopy classes of non-contractible loops are precisely the non-trivial elements of $\pi_1(X)$. Therefore a homomorphism $\pi_1(X)\to G$ is a neat way to enumerate all non-trivial holonomies.

    Since $\pi_1(T^2) = \mathbb{Z}^2$, such a homomorphism is given by two commuting elements $\Theta_1, \Theta_2\in G$, and these correspond to the holonomies of the connection along the two basic loops in $T^2$.

  2. If $G$ is compact and simply-connected, then there are no non-trivial $G$-principal bundles over compact orientable surfaces, i.e. the connection form is globally defined, or "single valued".

Define $\theta_1 = \mathrm{e}^{\Theta_1}$ and $\theta_2 = \mathrm{e}^{\Theta_2}$ and do so for every time-slice, obtaining functions $\theta_i : \mathbb{R}\to \mathfrak{g}$. Then $$ \theta(t) = \theta_1(t) \mathrm{d}\phi_1 + \theta_2(t) \mathrm{d}\phi_2,$$ with $\mathrm{d}\phi_i$ the two basic cohomologically non-trivial 1-forms on $T^2$ is a connection on $T^2$ with holonomies $\exp(\theta_1(t)),\exp(\theta_2(t))$ at each point in time.

All that is left is to observe that the claimed general solution is then simply a general gauge transformation of this connection.

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  • $\begingroup$ Nice, this actually (partially) answers what was going to be part 2. I feel the first sentence is the key, but I'm not sure where it comes from. Should it be obvious? Is it a well-known theorem? Why are we looking at homotopy instead of homology? Is it due to $\pi_1$ being abelian? $\endgroup$ – AccidentalFourierTransform Jul 5 '18 at 19:28
  • $\begingroup$ @AccidentalFourierTransform See my edit $\endgroup$ – ACuriousMind Jul 5 '18 at 19:49
  • $\begingroup$ @AccidentalFourierTransform Hurewicz' theorem states that if $X$ is path-connected then $H_1(X,\mathbb{Z})\simeq\pi_1(X)/[\pi_1(X),\pi_1(X)]$, i.e. the first holomogy group is isomorphic to the Abelianisation of the fundamental group (and since $X$ is path-connected we can omit the base point). $\endgroup$ – green.onion Jul 5 '18 at 21:00
  • $\begingroup$ @green.onion Indeed, that's why I was asking whether $\pi_1$ being abelian was the reason we were considering $\pi_1$ instead of $H$. Thanks anyway. Cheers! $\endgroup$ – AccidentalFourierTransform Jul 5 '18 at 21:14
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    $\begingroup$ @green.onion I asked why we were using $\pi_1$ instead of $H_1$ before ACM edited the answer :-) $\endgroup$ – AccidentalFourierTransform Jul 5 '18 at 23:24

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