Three-dimensional $\mathcal N=4$ superconformal superfield theories

The mirror map in the $D{=}3$, ${\mathcal N=4}$ supersymmetry connects the left and right $SU(2)$ automorphism groups and also the superfield representations of the corresponding ${\mathcal N=4}$ supermultiplets. The mirror ${\mathcal N=4}$ harmonic superspaces use the harmonics of two $SU(2)$ groups and two types of Grassmann analyticity. The irreducible left and right ${\mathcal N=4}$ supermultiplets are defined in these harmonic superspaces. We analyze the ${\mathcal N=4}$ superconformal interactions of the gauge and matter superfields and the spontaneous superconformal symmetry breaking. The most interesting superconformal action has the mirror symmetry and contains two nonlinear terms of the Abelian left and right gauge superfields and also the mixing ${\mathcal N=4}$ $BF$ interaction, which yields the topological masses of the gauge fields and the nontrivial interaction of the scalar and pseudoscalar fields. The minimal coupling of the left and right ${\mathcal N=4}$ hypermultiplets can be included in this Abelian gauge theory. We also consider the nonlinear ${\mathcal N=4}$ gauge superfield interactions.