Five-dimensional non-Abelian supersymmetric Chern-Simons action in Projective Superspace

In this paper, we derive the five-dimensional non-Abelian $\mathcal{N}=1$ Chern-Simons action from its established definition of variation with respect to an infinitesimal deformation of the vector prepotential in a projective superspace setting. It has been a long-standing open problem how to integrate this variation. Now, thanks to the simplicity of a projective superspace technique, we have been finally able to obtain this term. As a bonus, we also give the supersymmetric Yang-Mills action with a half measure as well.