Superconformal Chern-Simons-matter theories in N=4 superspace

© 2015 American Physical Society. In three dimensions, every known N=4 supermultiplet has an off-shell completion. However, there is no off-shell N=4 formulation for the known extended superconformal Chern-Simons (CS) theories with eight and more supercharges. To achieve a better understanding of this issue, we provide N=4 superfield realizations for the equations of motion which correspond to various N=4 and N=6 superconformal CS theories, including the Gaiotto-Witten theory and the Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. These superfield realizations demonstrate that the superconformal CS theories with N≥4 (except for the Gaiotto-Witten theory) require a reducible long N=4 vector multiplet, from which the standard left and right N=4 vector multiplets are obtained by constraining the field strength to be either self-dual or antiself-dual. Such a long multiplet naturally originates upon reduction of any off-shell N>4 vector multiplet to N=4 superspace. For the long N=4 vector multiplet we develop a prepotential formulation. It makes use of two prepotentials being subject to the constraint which defines the so-called hybrid projective multiplets introduced in the framework of N=4 supergravity-matter systems in arXiv:1101.4013. We also couple N=4 superconformal CS theories to N=4 conformal supergravity.