Reviving 3D $$ \mathcal{N} $$ = 8 superconformal field theories

Abstract

Abstract We present a Lagrangian formulation for $$ \mathcal{N} $$ 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 $$ \mathfrak{g} $$ g * of a Lie algebra $$ \mathfrak{g} $$ g by means of an embedding tensor map ϑ: $$ \mathfrak{g} $$ g * → $$ \mathfrak{g} $$ g . We show that for the Lie algebra $$ \mathfrak{sdif}{\mathfrak{f}}_3 $$ 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 $$ \mathcal{N} $$ N = 8 superconformal field theories are shown to be equivalent to Bandos-Townsend theories. We show that the theory based on S 3 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 S 2 × S 1, which gives the super-Yang-Mills theory with gauge algebra $$ \mathfrak{sdif}{\mathfrak{f}}_2 $$ sdif f 2 , and we construct massive deformations.