Reviving 3D $N$ = 8 superconformal field theories

We present a Lagrangian formulation for $N$ = 8 superconformal field theories in three spacetime dimensions that is general enough to encompass infinite-dimensional gauge algebras that generally go beyond Lie algebras. To this end we employ Chern-Simons theories based on Leibniz algebras, which give rise to L_∞ algebras and are defined on the dual space $g$^* of a Lie algebra $g$ by means of an embedding tensor map ϑ: $g$^* → $g$. We show that for the Lie algebra $sdif{f}_3$ of volume-preserving diffeomorphisms on a 3-manifold there is a natural embedding tensor defining a Leibniz algebra on the space of one-forms. Specifically, we show that the cotangent bundle to any 3-manifold with a volume-form carries the structure of a (generalized) Courant algebroid. The resulting $N$ = 8 superconformal field theories are shown to be equivalent to Bandos-Townsend theories. We show that the theory based on S³ is an infinite-dimensional generalization of the Bagger-Lambert-Gustavsson model that in turn is a consistent truncation of the full theory. We also review a Scherk-Schwarz reduction on S² × S¹, which gives the super-Yang-Mills theory with gauge algebra $sdif{f}_2$, and we construct massive deformations.