The classification of chiral WZW models by $H^4_+(BG,\mathbb Z)$

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 $k \in H^4_+(BG,\mathbb Z)$ 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 $E_8 \times E_8$ at level $(2,2)$ by the group $\mathbb Z_2$.