Consider a U(1) Chern Simons theory on a torus $\mathbb{T}$: \begin{align} L &= \frac{k}{4\pi} \int_{\mathbb{T}} a \partial a \end{align} where a is some U(1) gauge field, $k\in\mathbb{Z}$ and we used the short hand notation $a \partial a \equiv \epsilon^{\mu \nu \lambda} a_\mu \partial_\nu a_\lambda$.
Consider Wilson Loops of the form \begin{equation} W(C) = \mathcal{P} e^{i \oint_C a \cdot dl}. \end{equation} Here $\mathcal{P}$ denotes path ordering and $C$ denotes some closed loop on the torus $\mathbb{T}$.
Consider the two non-contractible loops on $\mathbb{T}$ denoted by $\mathcal{a}$ and $\mathcal{b}$. (For a picture see: http://share.pdfonline.com/7e91df64f6e84f43bff166c6911972d6/torus_a_b.htm)
THe ground state manifold of of the theory is $|k|$-fold degenerate.
Consider a basis that consists of wrapping "quasiparticles" around the b loop of the torus: $\left| n \right\rangle$ with n=0...$|k|-1$. Then the Wilson loop operators act as \begin{align} W(b)|n \rangle &= |n + 1 \text{ mod } |k| \rangle, \nonumber \\ W(a) |n \rangle &= e^{2\pi i n /k} |n \rangle. \end{align}.
What is the reason for this? I assume it must somehow be related to the explicit construction of the ground state manifold?
Since I am working myself into Chern Simons theory at the moment I would also be happy for some advice on readable literature with a focus on condensed matter problems.
Best regards.
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