Allowed anyons for Chern-Simons at level $k.$

These representations are called integrable representations. In the case of a general compact semisimple Lie group, a highest weight representation descends from a highest weight: $$lambda = sum_i n_i w_i, quad i = 1, ...,r$$ Where $r$ is the rank, $w_i$ are the fundamental weights and $n_i in mathbb{Z}^+$. The above representation is integrable for a level $k$ if for all $i$ $$0le n_i le k$$ The reasons for this condition can be understood qualitatively as follows: The Gauss law constraint of the Chern-Simons theory on the disc in the presence of an infinitesimal Wilson loop at $x_0$ corresponding to the representation $lambda$ is given by: $$frac{k}{2pi} F^a_{12} = i T^a_{(lambda)} delta^2(x-x_0)$$ Witten equation 3.4. (Witten explains this subject in words in the next few paragraphs)

Where $ T^a_{(lambda)} $ is a generator of the Lie algebra in the representation $lambda$, which can always be diagonalized as: $$T^a_{(lambda)} = g H_{(lambda)} g^{-1}$$ Where $ H_{(lambda)} $ is in the Cartan subalgebra.

The holonomy of the connection solving the Gauss law has the form:

$$U = e^ {frac{2 pi i}{k} g H_{(lambda)} g^{-1} phi}$$ Where $phi$ is the rotation angle around the insertion point.

Since the (diagonal) matrix elements of $H_{(lambda)} $ are less or equal to the highest weight components, thus due to the pre-factor $frac{2 pi}{k}$, a change by integer multiples of $k$ does not change the holonomy. These representations are named integrable, because the level $k$ Kac-Moody algebras that are based on them generate representations the corresponding Kac-Moody groups, Please see Goddard and Olive.