N = 8 Superconformal Chern--Simons Theories

TL;DR

The paper verifies that the Bagger-Lambert $SO(4)$ Chern-Simons theory realizes maximal ${\cal N}=8$ superconformal symmetry, $OSp(8|4)$, and preserves parity when the two $SU(2)$ factors are exchanged under reflection. It then systematically searches for generalizations to other gauge groups and representations, including $SO(n)\times SO(n)$, $USp(2n)\times USp(2n)$, and octonionic constructions, finding the fundamental identity is satisfied only in the known $SO(4)$ case and no new consistent theories arise. The authors argue for the uniqueness of the BL theory under reasonable irreducibility and finiteness assumptions and discuss potential connections to AdS/CFT and 3D gravity, clarifying that a gravity interpretation is unlikely due to the matter content. Overall, the work solidifies the symmetry structure of the BL theory and frames its role as a highly constrained, uniquely arising 3D CFT with strong implications for M-theory.

Abstract

A Lagrangian description of a maximally supersymmetric conformal field theory in three dimensions was constructed recently by Bagger and Lambert (BL). The BL theory has SO(4) gauge symmetry and contains scalar and spinor fields that transform as 4-vectors. We verify that this theory has OSp(8|4) superconformal symmetry and that it is parity conserving despite the fact that it contains a Chern--Simons term. We describe several unsuccessful attempts to construct theories of this type for other gauge groups and representations. This experience leads us to conjecture the uniqueness of the BL theory. Given its large symmetry, we expect this theory to play a significant role in the future development of string theory and M-theory.