The low-energy N = 4 SYM effective action in diverse harmonic superspaces

We review various superspace approaches to the description of the low-energy effective action in N = 4 super Yang–Mills (SYM) theory. We consider the four-derivative part of the low-energy effective action in the Coulomb branch. The typical components of this effective action are the gauge field F⁴/X⁴ and the scalar field Wess–Zumino terms. We construct N = 4 supersymmetric completions of these terms in the framework of different harmonic superspaces supporting N = 2,3,4 supersymmetries. These approaches are complementary to each other in the sense that they make manifest different subgroups of the total SU(4) R-symmetry group. We show that the effective action acquires an extremely simple form in those superspaces which manifest the non-anomalous maximal subgroups of SU(4). The common characteristic feature of our construction is that we restore the superfield effective actions exclusively by employing the N = 4 supersymmetry and/or superconformal PSU(2, 2| 4) symmetry.