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

Abstract

We present a Lagrangian formulation for mathcal{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} * of a Lie algebra mathfrak{g} by means of an embedding tensor map ϑ: mathfrak{g} * → mathfrak{g} . We show that for the Lie algebra mathfrak{sdif}{mathfrak{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} = 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 mathfrak{sdif}{mathfrak{f}}_2 , and we construct massive deformations.