How much of the Capelli-Itzykson-Zuber ADE-classification of su(2)-conformal field theories can one see perturbatively? In their celebrated work, Capelli Itzykson and Zuber established an 
ADE-classification of modular invariant CFTs with chiral algebra $\mathfrak{su}(2)_k$.

How much of that classification can one see using the tools of perturbative quantum field theory?

Presumably, one can't see the exceptional $E_6$, $E_7$, $E_8$ family... what about the $A_n$ versus $D_n$ families?
 A: The following facts might be useful to OP. Firstly, the simple simply-laced $ADE$ Lie groups are in one-to-one correspondence with finite subgroups $\Gamma$ of $SU(2)$. Secondly, for $c=1$ CFTs, the $ADE$ classification is realized by orbifold models based upon modding out string propagation on a $SU(2)$ group manifold by its finite subgroups $\Gamma$, see Ref. 1 and  Ref. 2, Chap.8. Thirdly, models of the $\hat{sl}(2)$ affine Kac-Moody algebra are discussed in Ref. 2, Chap.9.
Some References: 


*

*P. Ginsparg, Curiosities at $c = 1$, Nucl. Phys. B 295 (1988) 153.

*P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028.
A: The $A_n$ family is obtained by considering the usual WZW action for $G=SU(2)$ and $k=n-1$. A well-known example is the free boson at $R=\sqrt{2}$, it corresponds to an $A_2$ model: its partition function is a sum of characters for spin-0 and spin-1/2 representations.
The $D_{2\rho+2}$ and $D_{2\rho+1}$ families correspond to twisting the above theories by the non-trivial outer automorphism of $\hat{A}_1$, it turns out you can do that only for even levels.
Besides Ginsparg's lectures, you can take a look at chapter 17 in Di Francesco et al. book.
