Four-dimensional Wess-Zumino-Witten actions

We shall give an axiomatic construction of Wess-Zumino-Witten actions valued in (G=SU(N)), (Ngeq 3). It is realized as a functor ({WZ}) from the category of conformally flat four-dimensional manifolds to the category of line bundles with connection that satisfies, besides the axioms of a topological field theory, the axioms which abstract Wess-Zumino-Witten actions. To each conformally flat four-dimensional manifold (Sigma) with boundary (Gamma=partialSigma), a line bundle (L=WZ(Gamma)) with connection over the space (Gamma G) of mappings from (Gamma) to (G) is associated. The Wess-Zumino-Witten action is a non-vanishing horizontal section (WZ(Sigma)) of the pull back bundle (r^{ast}L) over (Sigma G) by the boundary restriction (r). (WZ(Sigma)) is required to satisfy a generalized Polyakov-Wiegmann formula with respect to the pointwise multiplication of the fields (Sigma G). Associated to the WZW-action there is a geometric descrption of extensions of the Lie group (Omega^3G) due to J. Mickelsson. In fact we shall construct two abelian extensions of (Omega^3G) that are in duality.