4-d Chern-Simons Theory: Higher Gauge Symmetry and Holographic Aspects

We present and study a 4-d 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 3-d CS one. In spite of these formal resemblance, the gauge invariance properties of the 4-d CS model differ considerably. The 4-d CS action is fully gauge invariant if the underlying base 4-fold 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 4-d 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 3-d model. 4-d CS theory thus exhibits rich holographic properties. The covariant Schroedinger quantization of the 4-d CS model is performed. A preliminary analysis of 4-d CS edge field theory is also provided. The toric and Abelian projected models are described in some detail.