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

Ref.1. proves that the allowed representations of Chern-Simons $\mathrm{SU}(2)_k$ are those with dimension $$\dim(R)\le k+1\tag{7.53}$$

Question: Is the generalisation of $(7.53)$ to arbitrary $N$ known? What about arbitrary (semisimple) Lie groups $G$?

Furthermore, the author also proves that the fusion rules for $\mathrm{SU}(2)_k$ are $$R_{j_1}\times R_{j_2}=\sum_{j_3=|j_1-j_2|}^{b(j_1,j_2)} R_{j_3}\tag{7.54}$$ with $b(j_1,j_2)=\min(j_1+j_2,k-j_1-j_2)$.

Question: Is the generalisation of $(7.54)$ to arbitrary $N$ known? i.e., where do we truncate the Littlewood-Richardson decomposition of $R_1\times R_2\in\mathrm{Rep}(\mathrm{SU}(N))^2$? As before, what about other Lie groups $G$?

References.

1. Pachos, J.K. - Introduction To Topological Quantum Computation.

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 $$i$$ $$0\le n_i \le k$$ 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: $$\frac{k}{2\pi} F^a_{12} = i T^a_{(\lambda)} \delta^2(x-x_0)$$ 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.
$$U = e^ {\frac{2 \pi i}{k} g H_{(\lambda)} g^{-1} \phi}$$ 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.
• Integrable representations are those representations which can be promoted to group representations. For example the representation $j = \frac{3}{4}$ of $SU(2)$ is not integrable. These representations have (at least) two characterizations: (1) the lowering and raising operators are nilpotent, thus after a finite number of applications the result becomes zero, (2) they have a group character. In the case of quantum groups, I think that it is meant an integrable representation possesses the two mentioned characteristics. May 7, 2018 at 15:23