$N=4$ Multiplets in $N=3$ Harmonic Superspace

We show that the $N=3$ harmonic superfield equations of motion are invariant with respect to the fourth supersymmetry. We also use the $SU(3)$ harmonics to analyze a more flexible form of superfield constraints for the Abelian $N=4$ vector multiplet and its $N=3$ decomposition. An unusual alternative representation of the $N=4$ supersymmetry is realized on infinite multiplets of analytic superfields in the $N=3$ harmonic superspace. An integer-valued parameter playing the role of a discrete coordinate parameterizes $U(1)$ charges of superfields in these multiplets. Each superfield term of the $N=3$ Yang–Mills action has an infinite-dimensional $N=4$ generalization. The gauge group of this model contains an infinite number of superfield parameters.