Three-Algebras and N=6 Chern-Simons Gauge Theories

TL;DR

This work extends the Bagger–Lambert–Gustavsson framework to ${N}=6$ supersymmetry by relaxing antisymmetry and reality constraints on the 3-algebra, yielding a general scale-invariant Chern-Simons–matter theory with $SU(4)$ R-symmetry and a $U(1)$ global symmetry. The authors derive the ${N}=6$ SUSY transformations, a manifestly $SU(4)$ covariant Lagrangian with a twisted Chern-Simons term, and a potential dictated by a generalized fundamental identity for the 3-algebra, including a reality/antisymmetry condition on the structure constants $f^{ab\bar{c}\bar{d}}$. They show how a natural class of complex 3-algebras leads to ABJM–type theories with gauge groups such as $U(N)\times U(N)$ (and $U(N_1)\times U(N_2)$ in the general bi-fundamental construction), thereby providing a solid algebraic underpinning for M2-brane dynamics in $\mathbb{R}^8/\mathbb{Z}_k$ backgrounds. The paper also delineates an explicit infinite family of examples built from maps $X:V_1\to V_2$ yielding a concrete realization of the ${N}=6$ theories and clarifying the relationship to known ABJM models. This work thus positions 3-algebras as the natural language for interacting ${N}=6$ M2-brane theories and suggests avenues for broader gauge-group realizations and connections to related embedding-tensor classifications.

Abstract

We derive the general form for a three-dimensional scale-invariant field theory with N=6 supersymmetry, SU(4) R-symmetry and a U(1) global symmetry. The results can be written in terms of a 3-algebra in which the triple product is not antisymmetric. For a specific choice of 3-algebra we obtain the N=6 theories that have been recently proposed as models for M2-branes in an R^8/Z_k orbifold background.