On superconformal Chern-Simons-matter theories in N=4 superspace

TL;DR

This paper develops N=4 superfield formulations for superconformal Chern-Simons theories in three dimensions, revealing the necessity of a long N=4 vector multiplet for theories with more than three supercharges.

Contribution

It introduces a prepotential formulation for the long N=4 vector multiplet and demonstrates how known theories like ABJM and Gaiotto-Witten fit into this framework.

Findings

Long N=4 vector multiplet is essential for N>3 superconformal CS theories.

Superfield realisations connect N=4 and N=6 theories, including ABJM.

Coupling to N=4 conformal supergravity is achieved.

Abstract

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 realisations for the equations of motion which correspond to various N=4 and N=6 superconformal CS theories, including the Gaiotto-Witten theory and the ABJM theory. These superfield realisations demonstrate that the superconformal CS theories with N>3 (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 anti self-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.