We classify canonical metric 3-algebra structures on matrix algebras and find novel three-dimensional conformally invariant actions in $\mathcal{N}=4$ projective superspace based on them. These can be viewed as Chern-Simons theories with special matter content and couplings. We explore the relations of these actions with the $\mathcal{N}=2$ actions based on generalized 3-Lie algebras found earlier and relate them to the original Bagger-Lambert-Gustavsson action.

• Received 4 February 2009

DOI:https://doi.org/10.1103/PhysRevD.79.086002