Harmonic superpotentials and symmetries in gauge theories with eight supercharges

Models of interactions of D-dimensional hypermultiplets and supersymmetric gauge multiplets with $\mathcal{N}=8$ supercharges $(D{\leq} 6)$ can be formulated in the framework of harmonic superspaces. The effective Coulomb low-energy action for D=5 includes the free and Chern-Simons terms. We consider also the non-Abelian superfield D=5 Chern-Simons action. The biharmonic $D=3,\mathcal{N}=8$ superspace is introduced for a description of l and r supermultiplets and the mirror symmetry. The D=2,(4,4) gauge theory and hypermultiplet interactions are considered in the triharmonic superspace. Constraints for $D{=}1,\mathcal{N}{=}8$ supermultiplets are solved with the help of the $SU(2){\times}Spin(5)$ harmonics. Effective gauge actions in the full $D{\leq}3,\mathcal{N}{=}8$ superspaces contain constrained (harmonic) superpotentials satisfying the (6-D) Laplace equations for the gauge group U(1) or corresponding (6-D)p-dimensional equations for the gauge groups $[U(1)]^p$. Generalized harmonic representations of superpotentials connect equivalent superfield structures of these theories in the full and analytic superspaces. The harmonic approach simplifies the proofs of non-renormalization theorems.