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

Gauge and Scalar Fields on $\mathbb{CP}^2$: A Gauge-invariant Analysis II. The measure for gauge fields and a 4d WZW theory