A gauge-invariant measure for gauge fields on $\mathbb{CP}^2$

We consider four-dimensional non-Abelian gauge theory living on a complex projective space $\mathbb{CP}^2$ as a way of gaining insights into (3+1)-dimensional QCD. In particular, we use a complex parametrization of gauge fields on which gauge transformations act homogeneously. This allows us to factor out the gauge degrees of freedom from the volume element leading to a manifestly gauge-invariant measure for the gauge-orbit space (the space of all gauge potentials modulo gauge transformations). The terms appearing in the measure that are of particular interest are mass-like terms for the gauge-invariant modes of the gauge fields. Since these mass terms come with dimensional parameters they are significant in the context of dimensional transmutation. Moreover, the existence of local gauge-invariant mass terms on $\mathbb{CP}^2$ could be related to Schwinger-Dyson calculations of the soft gluon mass. Finally, we argue that there is a kinematic regime in which the theory can be approximated by a 4d Wess-Zumino-Witten (WZW) theory. This result can be used to draw similarities between the mechanism of confinement in four and (2+1) dimensions.