Four-dimensional Wess-zumino-witten Model and Abelian Extensions of Ω

We construct two abelian extensions of Ω3G, one by the abelian group Map(D4 0G,U(1)) and the other by Map((D )0G,U(1)), where D4 0G is the group of mappings from the unit four-disc D 4 to G = SU(N), N ≥ 3, that are equal to 1 on the boundary S3, and the prime indicates the opposite orientation of D4. They are homotopically equivalent to a U(1) principal bundle over Ω3G and the associated line bundles are dual to each other. These line bundles become principal ingredients to construct our four-dimensional WZW model as a functor from the category of conformally flat spin manifolds to the category of line bundles.