The classification of chiral WZW models by H4+ (BG,Z)

We axiomatize the defining properties of chiral WZW models. We show that such models are in almost bijective correspondence with pairs (G, k), where G is a connected Lie group and k ∈ H4+ (BG,Z) is a degree four cohomology class subject to a certain positivity condition. We find a couple extra models which satisfy all the defining properties of chiral WZW models, but which don’t come from pairs (G, k) as above. The simplest such model is the simple current extension of the affine VOA E8 ×E8 at level (2, 2) by the group Z2.