  We present and study a 4d Chern-Simons (CS) model whose gauge symmetry is
encoded in a balanced Lie group crossed module. Using the derived formal set-up
recently found, the model can be formulated in a way that in many respects
closely parallels that of the familiar 3d CS one. In spite of these formal
resemblance, the gauge invariance properties of the 4d CS model differ
considerably. The 4d CS action is fully gauge invariant if the underlying base
4fold has no boundary. When it does, the action is gauge variant, the gauge
variation being a boundary term. If certain boundary conditions are imposed on
the gauge fields and gauge transformations, level quantization can then occur.
In the canonical formulation of the theory, it is found that, depending again
on boundary conditions, the 4d CS model is characterized by surface charges
obeying a non trivial Poisson bracket algebra. This is a higher counterpart of
the familiar WZNW current algebra arising in the 3d model. 4d CS theory thus
exhibits rich holographic properties. The covariant Schroedinger quantization
of the 4d CS model is performed. A preliminary analysis of 4d CS edge field
theory is also provided. The toric and Abelian projected models are described
in some detail.
