  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 (\Sigma) with boundary
(\Gamma=\partial\Sigma), 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.
