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.
- Elitzur, Moore, Schwimmer, Seiberg - Remarks on the canonical quantization of the Chern-Simons-Witten theory.