Constrained superpotentials in harmonic gauge theories with eight supercharges

We consider $D$-dimensional supersymmetric gauge theories with eight supercharges $(D\leq6,\mathcal N=8)$ in the framework of harmonic superspaces. The effective Abelian low-energy action for $D=5$ contains the free and Chern–Simons terms. Effective $\mathcal N=8$ superfield actions for $D\leq4$ can be written in terms of superpotentials satisfying the superfield constraints and $(6-D)$-dimensional Laplace equations. The role of alternative harmonic structures is discussed.