Four-dimensional Wess–Zumino–Witten actions

Abstract

We shall give an axiomatic construction of Wess–Zumino–Witten (WZW) 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 WZW 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 WZW 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 Ω 3G due to Mickelsson. In fact, we have two Abelian extensions of Ω 3G that are in duality.