  We consider the volume of the gauge orbit space for gauge fields on
four-dimensional complex projective space. The analysis uses a parametrization
of gauge fields where gauge transformations act homogeneously on the fields,
facilitating a manifestly gauge-invariant analysis. The volume element contains
a four-dimensional Wess-Zumino-Witten (WZW) action for a hermitian
matrix-valued field. There is also a mass-like term for certain components of
the gauge field. We discuss how the mass term could be related to results from
lattice simulations as well as Schwinger-Dyson equations. We argue for a
kinematic regime where the Yang-Mills theory can be approximated by the 4d-WZW
theory. The result is suggestive of the instanton liquid picture of QCD.
Further it is also indicative of the mechanism for confinement being similar to
what happens in two dimensions.
