Four-dimensional Wess-zumino-witten Actions and Central Extensions of Ω

We shall give an axiomatic construction of Wess-Zumino-Witten actions valued in G = SU(N), N ≥ 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 WessZumino-Witten actions. To each conformally flat four-dimensional manifold Σ with boundary Γ = ∂Σ, a line bundle L = WZ(Γ) with connection over the space ΓG of mappings from Γ to G is associated. The Wess-Zumino-Witten action is a non-vanishing horizontal section WZ(Σ) of the pull back bundle r∗L over ΣG by the boundary restriction r. WZ(Σ) is imposed to satisfy a generalized PolyakovWiegmann formula with respect to the pointwise multiplication of the fields ΣG. Associated to the WZW-action there is a geometric descrption of extensions of the group Ω3G . In fact we shall construct two central extensions of Ω3G by U(1) that are in duality. 1 1 This is the fully revised version of author’s previous article ”Four-dimensional WessZumino-Witten model and abelian extensions of ΩG”( math.DG/0105090) which contained some mistakes