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 $G$?


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

1 Answer 1


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.

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.

  • $\begingroup$ This is nice, thank you. Is there any relation between integrable representations and quantum groups? $\endgroup$ May 7, 2018 at 14:38
  • $\begingroup$ 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. $\endgroup$ May 7, 2018 at 15:23

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