Four-dimensional Wess-Zumino-Witten actions

Abstract

We shall give an axiomatic construction of Wess-Zumino-Witten actions valued in (G=SU(N)), (N\geq 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 (Σ) with boundary (Γ=\partialΣ), 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^{\ast}L) over (ΣG) by the boundary restriction (r). (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 descrption of extensions of the Lie group (Ω^3G) due to J. Mickelsson. In fact we shall construct two abelian extensions of (Ω^3G) that are in duality.