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

TL;DR

This paper classifies chiral WZW models using degree four cohomology classes of classifying spaces, establishing a near-bijective correspondence with pairs of Lie groups and cohomology classes, and identifies additional models beyond this classification.

Contribution

It axiomatizes chiral WZW models and links them to pairs of Lie groups and cohomology classes, also discovering models outside this framework.

Findings

Chiral WZW models correspond to pairs (G,k) with G a Lie group and k in H^4_+(BG,Z).

Identifies additional models not arising from the (G,k) classification.

Provides a near-bijective classification framework for chiral WZW models.

Abstract

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 $\mathbb Z_2$.