Four-dimensional Wess-Zumino-Witten actions

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 the characteristics of Wess-Zumino-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 pullback bundle r∗L over ΣG by the boundary restriction r : ΣG −→ ΓG. WZ(Σ) is required to satisfy a generalized Polyakov-Wiegmann formula with respect to the pointwise multiplication of the fields ΣG. Associated to the WZW-action there is a geometric description of the extension of the Lie group Ω3G due to J. Mickelsson. In fact we have two abelian extensions of Ω3G that are in duality. MSC: 57R; 58E; 81E Subj. Class.: Global analysis, Quantum field theory