mathematical physics - 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?