The An 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=√2, it corresponds to an A2 model: its partition function is a sum of characters for spin-0 and spin-1/2 representations.
The D2ρ+2 and D2ρ+1 families correspond to twisting the above theories by the non-trivial outer automorphism of ˆA1, 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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