Three-dimensional $\mathcal{N}{=}4$ supersymmetry in harmonic $\mathcal{N}{=}3$ superspace

We consider the map of three-dimensional $\mathcal N=4$ superfields to the $\mathcal N=3$ harmonic superspace. The left and right representations of the $\mathcal N=4$ superconformal group are constructed on $\mathcal N=3$ analytic superfields. These representations are convenient for describing $\mathcal N=4$ superconformal couplings of Abelian gauge superfields to hypermultiplets. We investigate the $\mathcal N=4$ invariance in the non-Abelian $\mathcal N=3$ Yang–Mills theory.