2D (4,4)1 hypermultiplets (I): Diversity for N = 4 models

The structure of on-shell and off-shell 2D, (4,4) supersymmetric scalar multiplets is investigated, in components and in superspace. We reach the surprising result that there exist eight distinct on-shell versions related to each other by outer automorphisms of the N = 4 supersymmetry algebra. These provide a non-trivial generalization of the concept of twisted superfields from N = 2 to N = 4 systems. Due to the close relations between twisted multiplets and mirror symmetry in the case of the N = 2 theories, we conjecture on the possibility of non-trivial generalizations of mirror symmetry to the case of N = 4. The off-shell generalized tensor and relaxed N = 4 multiplets are introduced in superspace, and their universal invariant self-interaction is constructed.