N = 8 Superconformal Chern--Simons Theories

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.