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.

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